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+{"text":"\\section{Introduction}\nOne important problem in machine learning is to find the minimum of the expected loss, \n\\begin{align}\\label{origin}\n\\min_{\\theta} \\mathbb{E}_{\\Xb,Y\\sim \\cD}\\left[l(Y,\\langle \\Xb,\\theta \\rangle) \\right]. \n\\end{align}\nHere $l(\\cdot,\\cdot)$ is a loss function and $(\\Xb,Y) \\in \\cX\\times \\cY \\subseteq \\mathbb{R}^d \\times \\cY$ has a distribution $\\cD$. In practice, the minimizer $\\theta^*$ needs to be estimated by observing $N$ samples $\\{\\xb_i,y_i\\}$ drawn from distribution $\\cD$. In many applications $N$ or $d$ are very large, so distributed algorithms are necessary in such case. Without loss of generality, assume that $N = nm$ and that the observations of $j$-th machine are $\\{\\xb_{ji},y_{ji}\\}_{i =1}^n$. We consider the high-dimensional learning problem where the dimension $d$ can be very large, and the effective variables are supported on $S:= \\text{support}\\{\\theta^*\\} = \\{i\\in [d]:\\theta^*_i \\ne 0\\}$ and $s:=|S|\\ll d$. Extensive efforts have been made to develop batch algorithms \\cite{friedman2007pathwise,Beck:09,xiao2013proximal}, which provide good convergence guarantees in optimization. However, when $N$ is large, batch algorithms are inefficiency, which takes at least $\\cO(N)$ time per iteration. Therefore, an emerging recent interest is observed to address this problem using the distributed optimization frameworks \\cite{jordan2016communication,lee2015distributed,wang2016efficient}, which is more efficient than the stochastic algorithms. One important issue of existing distributed optimization for sparse learning is that they did not take advantage of the sparse structure, thus they have the same communication costs with general dense problems. In this paper, we propose a novel communication-efficient distributed algorithm to explicitly leverage the sparse structure for solving large scale sparse learning problems. This allows us to reduce the communication cost from $\\cO(d)$ in existing works to $\\cO(s)$, while we still maintaining nearly the same performance under mild assumptions.\n\n\\noindent \\textbf{Notations}\nFor a sequence of numbers $a_n$, we use $\\cO(a_n)$ to denote a sequence of numbers $b_n$ such that $b_n\\leq C \\cdot a_n$ for some positive constant $C$. Given two sequences of numbers $a_n$ and $b_n$, we say $a_n\\lesssim b_n$ if $a_n =\\cO(b_n) $ and $a_n\\gtrsim b_n$ if $b_n =\\cO(a_n) $. The notation $a_n \\asymp b_n$ denotes that $a_n =\\cO(b_n) $ and $b_n =\\cO(a_n) $. For a vector $\\vb\\in \\mathbb{R}^d$, the $l_p$-norm of $\\vb$ is defined as $\\|\\vb\\|_p = ( \\sum_{i =1}^d|\\vb_i|^p)^{1\/p}$, where $p>0$; the $l_0$-norm of $\\vb$ is defined as the number of its nonzero entries; the support of $\\vb$ is defined as $\\text{supp}(\\vb) = \\{i:\\vb_i\\ne 0 \\}$. For simplicity, we use $[d]$ to denote the set $\\{1,\\cdots,d\\}$. For a matrix $A = (a_{ij})\\in \\mathbb{R}^{n_1\\times n_2}$, we define the $l_\\infty$-norm of $A$ as $\\|A\\|_\\infty = \\max_{i\\in[n_1],j\\in[n_2]}|a_{ij}|$. Given a number $k\\leq d$, the hard thresholding $\\cH_k(\\vb)$ of a vector $\\vb\\in\\RR^d $ is defined by keeping the largest $k$ entries of $\\vb$ (in magnitude) and setting the rest to be zero. Given a subset $S$ of index set $\\{1,\\cdots,d\\}$, the projection $\\cP_S(\\vb)$ of a vector $\\vb$ on $S$ is defined by \n\\begin{align*} \n\\cP_S(\\vb)_j =0,\\hspace{0.1in} \\text{if} \\hspace{0.05in} j\\notin S \\hspace{0.1in} \\text{and}\\hspace{0.1in} \\cP_S(\\vb)_j =\\vb_j, \\hspace{0.1in}\\text{if}\\hspace{0.05in} j\\in S.\n\\end{align*} \n$\\cP_S(\\vb)$ is also denoted as $(\\vb)_S$ for short.\n\n\\vspace{-0.1in}\n\\subsection{{Related work}}\nThere is much previous work on distributed optimizations such as (Zinkevich et al. \\cite{zinkevich2010parallelized}; Dekel et al. \\cite{dekel2012optimal}; Zhang et al. \\cite{zhang2012communication}; Shamir and Srebro \\cite{shamir2014distributed}; Arjevani and Shamir \\cite{arjevani2015communication}; Lee et al. \\cite{lee2015distributed}; Zhang and Xiao \\cite{zhang2015disco}). Initially, most distributed algorithms used averaging estimators formed by local machines (Zinkevich et al. \\cite{zinkevich2010parallelized}; Zhang et al. \\cite{zhang2012communication}). Then Zhang and Xiao \\cite{zhang2015disco}, Shamir et al. \\cite{shamir2014communication} and Lee et al. \\cite{lee2015communication} proposed more communication-efficient distributed optimization algorithms. More recently, using ideas of the approximate Newton-type method, Jordan et al. \\cite{jordan2016communication} and Wang et al. \\cite{wang2016efficient} further improved the computational efficiency of this type of method.\n\nMany gradient hard thresholding approaches are proposed in recent years such as (Yuan et al. \\cite{yuan2014gradient}; Li et al. \\cite{li2016stochastic}; Jain et al. \\cite{jain2014iterative}). They showed that under suitable conditions, the hard thresholding type first-order algorithms attain linear convergence to a solution which has optimal estimation accuracy with high probability. However, to the best of our knowledge, hard thresholding techniques applied to approximate Newton-type distributed algorithms has not been considered yet.  So in this paper, we present some initial theoretical and experimental results on this topic. \n\\iffalse\nInitially, averaging estimators formed by locally machines is a intuitive approach to distributed estimation (Zinkevich et al. \\cite{zinkevich2010parallelized}; Zhang et al. \\cite{zhang2012communication}).\n\\fi\n\\vspace{-0.15in}\n\n\\section{Algorithm} \n\\vspace{-0.05in}\nIn this section, we explain our approach to estimating the $\\theta^*$ that minimizes the expected loss. The detailed steps are summarized in Algorithm \\ref{algor}. \n\nFirst the empirical loss at each machine is defined as\n\\begin{align*}\\textstyle\n\\cL_j(\\theta) = \\frac{1}{n} \\sum_{i =1}^n l(y_{ji},\\langle \\xb_{ji},\\theta \\rangle), ~~\\text{where}~~ j\\in [m]. \n\\end{align*}\nAt the beginning of algorithm, we solve a local Lasso subproblem to get an initial point. Specifically, at iteration $h =0$, the master machine solves the minimization problem\n\\begin{align} \\label{initial}\n\\textstyle \\gamma^0 = \\argmin \\cL_1(\\theta) + \\mu_0\\|\\theta\\|_1. \n\\end{align}\nThe initial point $\\theta^0$ is formed by keeping the largest $k$ elements of the resulting minimizer $\\gamma^0$ and setting the other elements to be zero, i.e., $\\theta^0 = \\cH_k(\\gamma^0)$. Then, $\\theta^0$ is broadcasted to the local machines, where it is used to compute a gradient of local empirical loss at $\\theta^0$, that is, $\\nabla\\cL_j(\\theta^0)$. The local machines project $\\nabla\\cL_j(\\theta^0)$ on the support $S^0$ of $\\theta^0$ and transmit the projection $\\cP_{S^0}\\left[\\nabla\\cL_j(\\theta^0)\\right]$ back to the master machine. Later at $(h+1)$-th iteration ($h\\geq 0$), the master solves a shifted $l_1$ regularized minimization subproblem:\n\\begin{align}\\label{subproblem}\n\\textstyle \\nonumber \\gamma^{h+1}& = \\argmin_{ \\theta} \\hspace{0.1in}\\cL_1(\\theta) + \\mu_{h+1} \\|\\theta\\|_1\\\\\n&+\\Big\\langle \\cP_{S^h}\\left[{\\textstyle\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^h)}\\right] -\\nabla \\cL_1(\\theta^h),\\theta \\Big \\rangle.  \n\\end{align} \nAgain the minimizer $\\gamma^{h+1}$ is truncated to form $\\theta^{h+1}$, and this quantity is communicated to the local machines, where it is used to compute the local gradient as before. \n\nSolving subproblem $\\eqref{subproblem}$ is inspired by the approach of Wang et al. \\cite{wang2016efficient} and Jordan et al.\\cite{jordan2016communication}. Note that the formulation takes advantage of both global first-order information and local higher-order information. Specially, assuming the $\\mu_{h+1} = 0$ and $\\cL_j$ has an invertible Hessian, the solution of $\\eqref{subproblem}$ has the following closed form\n\\begin{align*}\n\\gamma^{h+1} = \\theta^{h} - \\nabla^2\\cL_1(\\theta^h) ^{-1} \\left(\\cP_{S^h}\\left[{\\textstyle\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^h)}\\right]\\right),\n\\end{align*}\nwhich is similar to a Newton updating step. Note that here we add a projection procedure $\\cP_{S^h}\\left[\\frac{1}{m}\\sum_{j=1}^m\\nabla\\cL_j(\\theta^h)\\right]$ to reduce the number of nonzeros that need to be communicated to the master machine. This procedure is reasonable intuitively. First, when $\\theta^h$ is close to $\\theta^*$, the elements of $\\frac{1}{m}\\sum_{j=1}^m\\nabla \\cL_j(\\theta^h)$ outside the support $S^{h}$ should be very small, so nominally little error is incurred in the truncation step. Second, when $\\theta^{h+1}$ is also close to $\\theta^*$, the lost part has even more minimal effects on the inner product in subproblem $\\eqref{subproblem}$. Third, we leave $-\\nabla \\cL_1(\\theta^h)$ in $\\eqref{subproblem}$ out of the truncation to maintain the formulation as unbiased.\n\n\\begin{algorithm}[!htb]\n\t\\caption{Two-way Truncation Distributed Sparse Learning}\n\t\\label{algor}\n\n\t\\begin{algorithmic}\n\t\t\\REQUIRE Loss function $l(\\cdot,\\cdot)$, data $\\{\\xb_{ji}, y_{ji}\\}_{i\\in[n],j\\in[m]}$. \n\t\t\\STATE \\hspace{-1.25em} \\textbf{\\underline{Local machines:}}\n\t\t\\STATE \\hspace{-1.25em} \\textbf{Initializaiton:} The master solves the local $l_1$ regularized loss minimization problem \\eqref{initial} to get a solution $\\gamma^0$. Set $\\theta^{0} = \\cH_k(\\gamma^{0})$. \n\t\t\\FOR{$h=0,1, \\dots$}\n\t\t\\FOR{$j=2,3, \\dots, m$}\n\t\t\\STATE   \\textbf{if} Receive $\\theta^h$ from the master \\textbf{then}\n\t\t\\STATE Calculate gradient $\\nabla \\cL_j(\\theta^h)$ and get the projection $\\cP_{S^{h}}\\left[ \\nabla\\cL_j(\\theta^h)\\right]$ of the gradient on support $S^h$ and transmit it to the master.\n\t\t\\STATE \\textbf{end}\n\t\t\n\t\t\\ENDFOR\n\t\t\\STATE  \\textbf{\\underline{Master:}}\n\t\t\\STATE   \\textbf{if} Receive $\\{\\nabla\\cL_j(\\theta^h)\\}_{j=2}^m$ from local machines \\textbf{then}\n\t\t\\STATE \\hspace{1.25em}Solve the shifted $l_1$ regularized problem\n\t\t\\STATE \\hspace{1.25em}\\eqref{subproblem} to obtain $\\gamma^{h+1}$.\n\t\t\\STATE \\hspace{1.25em}Do hard thresholding $\\theta^{h+1} =         \\cH_k(\\gamma^{h+1})$. \n\t\t\\STATE \\hspace{1.25em}Let $S^{h+1} = \\text{supp}(\\theta^{h+1})$. \n\t\t\\STATE \\hspace{1.25em}Broadcast $\\theta^{h+1}$ to every local machine. \n\t\t\\STATE \\textbf{end}\t\n\t\t\\ENDFOR\t\n\t\\end{algorithmic}\n\\end{algorithm}\n\\vspace{-0.1in}\n\\section{Theoretical Analysis}\n\\vspace{-0.05in}\n\\subsection{Main Theorem}\n\nWe present some theoretical analysis of the proposed algorithm in this section.\n\\vspace{-0.05in}\n\\begin{assumption}\\label{assum_smooth}\n\tThe loss $l(\\cdot,\\cdot)$ is a $L$-smooth function of the second argument, i.e.,\n\t\\begin{align*} {\\textstyle\n\tl^\\prime(x,y) - l^\\prime(x,z) \\leq L|y - z|, \\hspace{0.2in} \\forall x,y,z\\in \\RR}\n\t\\end{align*}\n\tMoreover, the third derivative with respect to its second argument, $\\partial^3 l(x,y)\/\\partial y^3$, is bounded by a constant $M$, i.e.,\n\t{\\begin{align*} \\textstyle\n\t|\\partial^3 l(x,y)\/\\partial y^3| \\leq M, \\hspace{0.2in} \\forall x,y\\in \\RR\n\t\\end{align*}}\n\\end{assumption}\n\n\\begin{assumption}\\label{assum_restrict}\n\tThe empirical loss function computed on the first machine satisfies that: $\\forall \\Delta \\in \\cC(S,3)$, we have\n\t\\begin{align*}\n\t\\cL_1(\\theta^* + \\Delta) - \\cL_1(\\theta^*) -\\langle \\nabla \\cL_1(\\theta^*),\\Delta \\rangle \\geq \\kappa \\|\\Delta\\|_2^2,\n\t\\end{align*}\n\twhere $\\cC(S,3)$ is defined as\n\t\\begin{align*}\n\t\\cC(S,3) = \\{\\Delta \\in \\RR^d |~ \\|\\Delta_{S^c}\\|_1 \\leq 3\\|\\Delta_S\\|_1\\}. \n\t\\end{align*}\t\n\\end{assumption}\n\n\\begin{assumption}\\label{assum_supp}\n\tThe $\\gamma^{h+1}$, $S^{h+1}$ and $S^h$ defined in Algorithm \\ref{algor} satisfy the following condition:\n\tthere exists some positive constants $H$ and $\\tau_1$ and $\\tau_2$ such that for $h\\geq H$, \n\t\\begin{align*}\n\t\\left\\|\\left(\\gamma^{h} - \\theta^*\\right)_{(S^h)^c}\\right\\|_1 &\\leq \\tau_1 \\left\\|\\gamma^{h} - \\theta^*\\right\\|_1\\\\\n\t\\left\\|\\left(\\gamma^{h+1} - \\theta^*\\right)_{S^{h+1}\\backslash S^{h}}\\right\\|_1 &\\leq \\tau_2\\left\\|\\gamma^{h+1}-\\theta^*\\right\\|_1.   \n\t\\end{align*}\n\\end{assumption}\n\\begin{remark}\n\tIn practice, both $\\tau_1$ and $\\tau_2$ are very small even after only one round of communication and will decrease to $0$ fast in the later steps. \n\\end{remark}\n\\vspace{-0.1in}\nFor simplicity, we define the following notation:\n\\begin{align*}\n\\overbar{\\cL_1}(\\theta^*,\\theta^h) &:= \\cL_1(\\theta^*) +\\left\\langle {\\textstyle \\frac{1}{m}\\sum_{j = 1}^m  \\nabla\\cL_j(\\theta^h) - \\nabla\\cL_1(\\theta^h)},\\theta \\right \\rangle,\\\\\n\\tilde{\\cL_1}(\\theta^*,\\theta^h) &:= \\cL_1(\\theta^*)\\\\\n&\\hspace{0.2in}+\\left\\langle \\cP_{S^h}\\left[{\\textstyle \\frac{1}{m}\\sum_{j = 1}^m  \\nabla\\cL_j(\\theta^h) - \\nabla\\cL_1(\\theta^h)}\\right],\\theta \\right \\rangle.\n\\end{align*}\n\nNow we state our main theorem. \n\\begin{thmi} \\label{main_theorem}\n\tSuppose that Assumption~\\ref{assum_smooth}, \\ref{assum_restrict}, and \\ref{assum_supp} hold. Let $k = C_1\\cdot s$ with $C_1>1$ and \n\t\\begin{align}\n\t\\nonumber& \\textstyle\\mu_{h+1}  = \\hspace{0.1in} 4\\left\\|{ \\frac{1}{m}\\sum_{j = 1}^m  \\nabla \\cL_j(\\theta^*)}\\right\\|_\\infty \\\\\n\t\\nonumber&\\textstyle +  2L\\left(\\max_{j,i}\\|x_{j,i}\\|_\\infty^2\\right)\\cdot \\Big[{ 2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho }\\Big]\\|\\theta^h - \\theta^*\\|_1\\\\\n\t&\\textstyle + 2M\\left(\\max_{j,i}\\|x_{j,i}\\|_\\infty^3 \\right)\\|\\theta^h - \\theta^*\\|_1^2, \\label{mu_def}\n\t\\end{align} \n\twhere $\\rho := \\tau_1+\\tau_2$.  \n\t\n\tThen with probability at least $1-\\delta$, we have that\n\t\n\t\\begin{align*} \n    &\\|\\theta^{h+1} - \\theta^*\\|_1  {\\textstyle \\leq \\frac{C_2 s}{\\kappa}\\left \\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right \\|_\\infty}\\\\\n    &\\hspace{0.2in}{\\textstyle +\\frac{C_2 s}{2\\kappa} L\\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^2\\cdot \\Big[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\Big]\\|\\theta^h - \\theta^*\\|_1}\\\\\n    &\\hspace{0.2in}{\\textstyle +\\frac{C_2 s}{2\\kappa}  M\\cdot  \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^3\\cdot \\|\\theta^h - \\theta^*\\|_1^2}, ~\\text{and}\\\\\n    &\\|\\theta^{h+1} - \\theta^*\\|_2 {\\textstyle \\leq \\frac{C_3\\sqrt{s}}{\\kappa}\\left \\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right \\|_\\infty }\\\\\n    &\\hspace{0.2in} {\\textstyle + \\frac{C_3\\sqrt{s}}{2\\kappa} L \\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^2\\cdot\\Big[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\Big]\\cdot \\|\\theta^h - \\theta^*\\|_1} \\\\\n    &\\hspace{0.2in} \\textstyle+\\frac{C_3\\sqrt{s}}{2\\kappa}M\\cdot \\max_{j,i}\\|\\xb_{ji}\\|_\\infty^3\\cdot \\|\\theta^h - \\theta^*\\|_1^2, \n    \\end{align*}\n\twhere $\\textstyle C_2 = 24\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\cdot \\sqrt{C_1+1}$ and $\\textstyle C_3= 24\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}$ are positive constants independent of $m,n,s,d$.\n\\end{thmi}\nThe theorem immediately implies the following convergence result.\n\\begin{cori} \\label{first_cor}\n\tSuppose that for all $h$\n\t\\begin{align}\\label{cor_ass}\n\t\\textstyle \\nonumber&M\\cdot  \\Big(\\max_{j,i}\\|x_{ji}\\|_\\infty^3\\Big) \\|\\theta^h - \\theta^*\\|_1\\leq\\\\ &  L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} \n\t+ \\rho\\right],\n\t\\end{align}\n\twhere $\\rho := \\tau_1+\\tau_2$.\n\t\n\tThen under the assumption of Theorem~\\ref{main_theorem} we have\n\t\\begin{align*} \n\t\\|\\theta^{h+1} - \\theta^*\\|_1 &\\leq \\textstyle \\frac{1-a_n^{h+1}}{1 - a_n}\\cdot \\frac{C_2 s}{\\kappa}\\cdot \\textstyle \\left\\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right\\|_\\infty \\\\\n\t& \\hspace{0.1in}+ a_n^{h +1} \\|\\theta^0 - \\theta^*\\|_1,\\\\\n\t|\\theta^{h+1} - \\theta^*\\|_2 &\\leq \\textstyle \\frac{1-a_n^{h+1}}{1 - a_n}\\cdot \\frac{C_3\\sqrt{s}}{\\kappa}\\cdot \\textstyle \\left\\|\\frac{1}{m}\\sum_{j=1}^m \\nabla \\cL_j(\\theta^*)\\right\\|_\\infty \\\\\n\t&\\hspace{0.1in}+ a_n^h b_n  \\|\\theta^0 - \\theta^*\\|_1,\n\t\\end{align*}\n\twhere \n   \\begin{align*}\n\t a_n =&\\textstyle \\frac{C_2 s}{\\kappa} L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2 \\cdot \\Bigg[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}+ \\rho\\Bigg]\n\t\\end{align*}\n\tand \n\t\\begin{align*}\n\t b_n =&\\textstyle \\frac{C_3\\sqrt{s}}{\\kappa} L\\cdot \\max_{j,i}\\|x_{ji}\\|_\\infty^2\\cdot\\Bigg[ 2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}+ \\rho \\Bigg],\n\t\\end{align*}\n\twhere $C_2 $ and $C_3$ are defined in Theorem~\\ref{main_theorem} and independent of $m,n,s,d$. \n\\end{cori}\n\\begin{remark}\n\tFrom the conclusion, we know that the hard thresholding parameter $k$ can be chosen as $C_1\\cdot s$, where $C_1$ can be a moderate constant larger than $1$. By contrast, previous work such as \\cite{li2016stochastic} solving a nonconvex minimization problem subject to $l_0$ constraint $\\|\\theta\\|_0 \\leq k$ requires that $k\\geq \\cO(\\kappa_s^2 s)$, where $\\kappa_s$ is the condition number of the object function.\n\tMoreover, instead of only hard thresholding on the solution of Lasso subproblems, we also do projection on the gradients in \\eqref{subproblem}. These help us reduce the communication cost from $\\cO(d)$ to $\\cO(s)$. \n\\end{remark}\n\n\\subsection{Sparse Linear Regression}\nIn the sparse linear regression, data $\\{\\xb_{ji}, y_{ji}\\}_{i\\in [n], j\\in[m]}$ are generated\naccording to the model\n\\begin{align} \\label{lin_model} \n\\textstyle y_{ji} = \\langle\\xb_{ji},\\theta^*\\rangle + \\epsilon_{ji},\n\\end{align}\nwhere the noise $\\epsilon_{ji}$ are i.i.d subgaussian random variables with zero mean. Usually the the loss function for this problem is the squared loss function $l(y_{ji},\\langle \\theta,\\xb_{ji}\\rangle) = \\frac{1}{2}(y_{ji}-\\langle \\theta,\\xb_{ji}\\rangle)^2$, which is $1$-smooth.\n\nCombining Corollary \\ref{first_cor} with some intermediate results obtained from \\cite{rudelson2011reconstruction, vershynin2010introduction} and \\cite{wainwright2009sharp}, we have the following bound for the estimation error. \n\\begin{cori}\n\tSuppose the design matrix and noise are subgaussian, Assumption \\ref{assum_supp} holds and $\\mu_{h+1}$ is defined as \\eqref{mu_def}. Then under the sparse linear model, we have the following estimation error bounds with probability at least $1-2\\delta$:\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1} - \\theta^*\\|_1 \\lesssim \\frac{1-a_n^{h+1}}{1-a_n} \\cdot\\frac{C_2s\\sigma\\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}}\\\\\n\t\\textstyle + a_n^{h+1} \\frac{s \\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(nd\/\\delta)}{n}}\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1} - \\theta^*\\|_2 \\lesssim\\frac{1-a_n^{h+1}}{1-a_n} \\cdot\\frac{C_3\\sqrt{s}\\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}}\\\\\n\t\\textstyle + a_n^{h}b_n \\frac{s \\sigma \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(nd\/\\delta)}{n}},\n\t\\end{align*}\n\twhere $C_2 $ and $ C_3$ are defined in Theorem~\\ref{main_theorem}, and where\n\t\\begin{align*} \n\t\\textstyle a_n =\\frac{C_2s}{\\kappa}\\sigma_X^2\\log\\left(\\frac{mnd}{\\delta}\\right)\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\right] \n\t\\end{align*}\n\tand\n\t\\begin{align*} \n\t\\textstyle b_n = \\frac{C_3\\sqrt{s}}{\\kappa}\\sigma_X^2\\log\\left(\\frac{mnd}{\\delta}\\right)\\left[2\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho \\right]. \n\t\\end{align*}\t\n\\end{cori}\n\n\\begin{remark}\nUnder certain conditions we can further simplify the bound and have an insight of the relation between $n,m,s,d$. When $n\\geq s^2\\log d$, it is easy to see by choosing\n\\begin{align*}\n\\mu_{h+1} \\asymp \\sqrt{\\frac{\\log d}{mn}} + \\sqrt{\\frac{\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1} \n\\end{align*}\nand $k = \\cO(s)$ there holds the following error bounds with high probabiltiy:\n\\begin{align*}\n&\\|\\theta^{h+1} - \\theta^*\\|_1 \\\\\n&\\hspace{0.2in} \\lesssim s\\sqrt{\\frac{\\log d}{mn}} + s\\sqrt{\\frac{\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1},\\\\\n&\\|\\theta^{h+1} - \\theta^*\\|_2 \\\\\n&\\hspace{0.2in}\\lesssim \\sqrt{\\frac{s\\log d}{mn}} +\\sqrt{\\frac{s\\log d}{n}} \\left[s\\left( \\sqrt{\\frac{\\log d}{n}} + \\rho \\right)\\right]^{h+1}.\n\\end{align*}\n\\end{remark}\n\n\\subsection{Sparse Logistic Regression}\nCombining Corollary \\ref{first_cor} with some intermediate results obtained from \\cite{wang2016efficient} and \\cite{regularizers2012m}, we now can give a similar result about the estimation error bound for sparse logistic regression. The explicit form is omitted due to the limitation of spaces.\n\\iffalse\n\\begin{cori}\n\tSuppose Assumption \\ref{assum_supp} holds and $\\mu_{h+1}$ is defined as \\eqref{mu_def}.\n\tIf the following condition holds for some $T\\geq 0$:\n\t\\begin{align*}\n\t\\|\\theta^T - \\theta^*\\|_1 \\leq 4\\sqrt{\\frac{\\log (2d\/\\delta)}{n}}. \n\t\\end{align*}\n\tThen under the sparse logistic model with random design, we have the following estimation error bound for all $h\\geq T$ with probability at least $1-2\\delta$:\t\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1}- \\theta^*\\|_1 \\lesssim\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\frac{1-a_n^{h-T+1}}{1-a_n}\\hspace{0.27in}\\\\\n\t\\cdot\\frac{24C_2s \\sigma_X}{\\kappa} \\sqrt{\\frac{\\log(d\/\\delta)}{mn}}+4a_n^{h-T+1}\\sqrt{\\frac{\\log(2d\/\\delta)}{n}}\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\t\\|\\theta^{h+1}- \\theta^*\\|_2 \\lesssim \\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\frac{1-a_n^{h-T+1}}{1-a_n}\\hspace{0.27in}\\\\\n\t\\cdot \\frac{24\\sqrt{s} \\sigma_X}{\\kappa}\\sqrt{\\frac{\\log(d\/\\delta)}{mn}} +4a_n^{h-T}\\sqrt{\\frac{\\log(2d\/\\delta)}{n}},\n\t\\end{align*}\n\twhere\n\t\\begin{align*} \\textstyle\n\ta_n = 6\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\hspace{1.5in}\\\\\n\t\\cdot\\frac{ C_2 s}{\\kappa}\\sigma_X^2\\log(\\frac{mnd}{\\delta})\\left[\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho\\right]\n\t\\end{align*}\n\tand\n\t\\begin{align*} \\textstyle\n\tb_n = 6\\sqrt{1+2(C_1-1)^{-\\frac{1}{2}}}\\hspace{1.5in}\\\\\n\t\\cdot\\frac{\\sqrt{s}}{\\kappa}\\sigma_X^2\\log(\\frac{mnd}{\\delta})\\left[\\sqrt{\\frac{\\log(2d\/\\delta)}{n}} + \\rho\\right]. \n\t\\end{align*}\n\\end{cori}\n\\fi\n\n\\section{Experiments}\n\\vspace{-0.02in}\nNow we test our algorithm on both simulated data and real data. In both settings, we compare our algorithm with various advanced algorithms. These algorithms are:\n\n\\begin{itemize}\n\t\\item[1.] EDSL: the state-of-the-art approach proposed by Jialei Wang et al. \\cite{wang2016efficient}.\n\t\\vspace{-0.05in}\n\t\\item[2.] Centralize: using all data, one machine solves the centralized loss minimization problem with $l_1$ regularization. This procedure is communication expensive or requires much larger storage.\n\t\\vspace{-0.05in} \n\t\\item[3.] Local: the first machine solves the local $l_1$ regularized loss minimization problem with only the data stored on this machine, ignoring all the other data.\n\t\\vspace{-0.05in}\n\t\\item[4.] Two-way Truncation: the proposed sparse learning approach which further improves the communication efficiency. \t\n\t\\vspace{-0.1in}\n\\end{itemize}\n\\subsection{Simulated data}\n\\begin{figure}[!htb]\n\t\\centering \n\t\\vspace{-0.1in}\n\t\\begin{center}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\hspace{-0.096in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure1a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|\/5}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure1b}.pdf}\n\t\t}\\\\\n\t\t\\vspace{0.1in} $m= 20, n =600, d = 20000, s=10, \\Xb \\sim \\cN(0,\\Sigma)$ \\vspace{0.1in}\\\\\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\t\n\t \\caption{Comparison among four algorithms in sparse linear regression setting}\\label{figure1}\n\n\\end{figure}\n\n\\begin{figure}[!htb] \n\n\t\\centering\n\t\\begin{center}\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\n\t\t\\hspace{-0.096in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure2a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[$\\Sigma_{ij} = 0.5^{|i -j|\/5}$]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure2b}.pdf}\n\t\t}\\\\\n\t\t\\vspace{0.1in} $m= 10, n =1000, d = 2000, s=20, \\Xb \\sim \\cN(0,\\Sigma)$ \\vspace{0.1in}\\\\\n\t\n\t\n\t\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\n\t\\caption{\\hspace{-0.1in} Comparison among four algorithms in sparse logistic regression setting}\\label{figure2}\n\n\\end{figure}\nThe simulated data $\\{\\xb_{ji}\\}_{i\\in [n], j\\in[m]}$ is sampled from multivariate Gaussian distribution with zero mean and covariance matrix $\\Sigma$. We choose two different covariance matrices: $|\\Sigma_{ij}| = 0.5^{|i-j|}$ for a well-conditioned situation and $|\\Sigma_{ij}| = 0.5^{|i-j|\/5}$ for an ill-conditioned situation. The noise $\\epsilon_{ji}$ in sparse linear model ($y_{ji} = \\langle\\xb_{ji},\\theta^*\\rangle + \\epsilon_{ji}$) is set to be a standard Gaussian random variable. We set the true parameter $\\theta^*$ to be $s$-sparse where all the entries are zero except that the first $s$ entries are i.i.d random variables from a uniform distribution in [0,1]. Under both two models, we set the hard thresholding parameter $k$ greater than s but less than $3 s$. \n\\vspace{-0.02in}\n\nHere we compare the algorithms in different settings of $(n,d,m,s)$ and plot the estimation error $\\|\\theta^h - \\theta^*\\|_2$ over rounds of communications. The results of sparse linear regression and sparse logistic regression are showed in Figure~\\ref{figure1} and Figure~\\ref{figure2}. We can observe from these plots that: \n\n\\begin{itemize}\n\t\\item First, there is indeed a large gap between the local estimation error and the centralized estimation error. The estimation errors of EDSL and the Two-way Truncation decrease to the centralized one in the first several rounds of communications.\n\n\t\\item Second, the Two-way Truncation algorithm is competitive with EDSL in both statistical accuracy and convergence rate as the theory indicated. Since it can converge in at least the same speed as EDSL's and requires less communication and computation cost in each iteration, overall it's more communicationally and computationally efficient.   \n\n\\end{itemize}\nThe above results support the theory that the Two-way Truncation approach is indeed more efficient and competitive to the centralized approach and EDSL.\n\\vspace{-0.05in}\n\n\\vspace{-0.05in}\n\n\\subsection{Real data}\n\\begin{figure}[!htb]\n\t\\vspace{-0.1in}\n\t\\begin{center}\n\t\t\\hspace{-0.08in}\\subfloat[dna (linear regression)]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure3a}.pdf}\n\t\t}\n\t\t\\hspace{-0.08in}\\subfloat[a9a (classification)]{\n\t\t\t\\includegraphics[width=0.46\\linewidth]{{figure3b}.pdf}\n\t\t}\n\t\n\t\n\t\n\t\\end{center}\n\t\\vspace{-0.1in}\n\t\\caption{Comparison among four algorithms on real datasets}\\label{fig:real_data1}   \n\t\\vspace{-0.5in}\n\\end{figure}\n\\vspace{0.4in}\nIn this section, we examine the above sparse learning algorithms on real-world datasets. The data comes from UCI Machine Learning Repository \\footnote{http:\/\/archive.ics.uci.edu\/ml\/} and the LIBSVM website \\footnote{https:\/\/www.csie.ntu.edu.tw\/$\\sim$cjlin\/libsvmtools\/datasets\/}. The high-dimensional data 'dna' and 'a9a' are used in the regression model and classification model respectively. We randomly partition the data in $[60\\%, 20\\%,20\\%]$ for training, validation and testing respectively. Here the data is divided randomly on $m = 10$ machines and processed by algorithms mentioned above. The results are summarized in Figure \\ref{fig:real_data1}. These results in real-world data experiments again validate the theoretical analysis that the proposed Two-way Truncation approach is a quite effective sparse learning method with very small communication and computation costs. \n\n\\section{Conclusions}\n\nIn this paper we propose a novel distributed sparse learning algorithm with Two-way Truncation.  Theoretically, we prove that the algorithm gives an estimation that converges to the minimizer of the expected loss exponentially and attain nearly the same statistical accuracy as EDSL and the centralized method. Due to the truncation procedure, this algorithm is more efficient in both communication and computation. Extensive experiments on both simulated data and real data verify this statement.\n\n\n\\section*{Acknowledgment}\n\nThe authors graciously acknowledge support from NSF Award CCF-1217751 and DARPA Young Faculty Award N66001-14-1-4047 and thank Jialei Wang for very useful suggestion. \n\n\\bibliographystyle{CADSLTT}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nIn the local Universe, there is a noticeable dearth of baryons within massive ($M_{DM}>10^{12.5}$ M$_{\\odot}$) galactic halos \\citep{Benson03,Croton06}. Feedback from supermassive black holes (SMBH) is commonly invoked to explain the missing baryons within the massive halos. Similarly, the tight correlation between the mass of a galaxy's SMBH and the total mass of the bulge and galaxy \\citep{Magorrian98,Gebhardt00} suggests that galaxies and SMBHs have evolved together. How SMBHs and galaxies co-evolve and regulate their mutual growth is an outstanding problem in modern astrophysics. Theoretical work suggests that the strong correlation between the mass of the SMBH and the velocity dispersion of the bulge ($M_{\\bullet}-\\sigma~$) may be achieved through quenching of star formation by powerful outflows driven once the galaxies reside on the $M_{\\bullet}-\\sigma~$ relationship \\citep{Hopkins06,Zubovas12,Zubovas14}.\n\nThe feedback that regulates the overall growth of a galaxy is expected to be most important when both the galaxies and the SMBHs were experiencing the majority of their growth \\citep{Zubovas12,Choi12,Barai17,Costa18}. Given that both quasar activity and star formation peak in peak in normal galaxies at cosmic noon  ($1.5<z<3$), studying in detail the nature of outflows from quasars and starbursts at these redshifts will provide important clues about the nature of feedback during this important epoch.\n\nFeedback can manifest itself in the form of galaxy-scale winds that expel gas from the galaxy. Moreover, winds can drive shocks and turbulence and prolong the time necessary for gas to cool, collapse and form stars. To date, most of the outflows studied in the distant ($z>1$) Universe have focused on optical emission lines that trace the warm ionized phase of outflows with typical densities $n_e\\sim 100-1000$ cm$^{-3}$ and temperatures $T=10^4$ K. At the epoch of peak quasar activity $z\\sim 2-3$, the bright emission lines such as [O{\\sc III}]\\xspace\\ $\\lambda$5007\\AA, H$\\alpha$ and H$\\beta$ are redshifted into near-infrared bands where they can be resolved spatially and spectrally. Using near-infrared spectroscopy, several authors have studied quasar-driven ionized winds on galaxy-wide scales both in radio-loud quasars (with powerful jets) and in radio-quiet quasars \\citep{Nesvadba08,Cano-Diaz12,Carniani15,Vayner17,Vayner21,Kakkad20}.\n\nRecent theoretical and observational works have suggested that the energy and momentum in galaxy outflows are shared between multiple phases of the gas. These gas phases span a wide range of densities and temperatures, from the dense cold molecular and neutral gas ($T\\sim10-300$ K) to diffuse hot post-shock medium ($T>10^7$ K; \\citealt{Crichton16,Hall19}). There have been several studies of multi-phase outflows in distant quasars, focusing on individual systems \\citep{Vayner17,Brusa18,Herrera-Camus19} and at studying the cold molecular and ionized gas phase in galactic outflows. It is unknown which phase of the outflow is responsible for the bulk of the momentum and mass of these multi-phase outflows, and it may well be a function of the conditions in the host galaxies. Theoretical work suggests that the molecular gas clouds may be disrupted and entrained by the outflow \\citep{Scannapieco15} or, instead, molecules may form within the outflowing gas \\citep{Richings18}. Observations of nearby galaxies suggest that the cold molecular gas phase (50-100 K) may be the dynamically dominant phase of quasar winds \\citep{Sun14,Cicone14,Alatalo11,Aalto12,Feruglio13,Morganti13,Veilleux17,Fiore17}. The warm (T$\\sim$500 K) molecular gas phase may also be important and carry a significant fraction of the momentum in a galactic outflow \\citep{Richings18}. In nearby galaxies, Spitzer observations of the warm molecular H$_{2}$\\ gas (T$\\sim$500 K) have found a significant amount of gas mass in galactic outflows \\citep{Beirao15,Dasyra14,Rogemar20}. \n\nBecause cold molecular gas is the fuel for star formation, the fate of the molecular gas phase is the key link in understanding the impact of quasar feedback on star formation. Since outflows are invariably multi-phase, understanding the relationship between molecular gas and atomic gas in outflows is crucial for accurately estimating the energetics of feedback and the fate of the interstellar medium (ISM). Statistical studies are lacking, as the study of multi-gas phase outflows in the distant Universe is still in its infancy. A recent attempt at comparing molecular gas properties of galaxies with powerful (L$_{bol}=10^{45-47}$ erg s$^{-1}$ ) active galactic nuclei (AGN) to star forming galaxies with similar stellar mass have found the AGN to reside in galaxies with lower molecular gas masses hinting at potential effects of feedback from outflows or radiation \\citep{Circosta21,Bischetti21}.\n\nTo address both the molecular gas reservoir and the measurement of cold molecular gas in galactic outflows requires kpc-scale spatial resolution observations that have the right sensitivity to detect the molecular gas or place stringent limits. The lowest energy transition of the H$_{2}$\\ molecule is the rotational quadrupole transitions that require gas temperatures $>100$ K to excite; hence H$_{2}$\\ is invisible in the cold molecular gas phase. The next most abundant molecule in molecular clouds is carbon monoxide (CO) that has a weak permanent dipole moment, with the near-ground rotational transitions having small excitation energies, enabling to trace colder molecular gas (5.5-55 K). In this paper, we present Atacama Large Millimeter Array (ALMA) observations of the cold molecular gas traced through rotational transitions of CO in six radio-loud quasars at $z=1.439-2.323$ with known powerful ionized gas outflows. We present observations, data reduction, and emission line analysis in Section \\ref{sec:obs}. We describe how we search for molecular outflows and calculate their energetics in Section \\ref{sec:dynamics}. We discuss individual objects in Section \\ref{sec:indiv_obj}. We compare the entire sample, discuss potential sources that drive the multi-phase gas outflows and the dominant source of molecular gas depletion in Section \\ref{sec:discussion}. We summarize our conclusions in Section \\ref{sec:conc}. We use an $\\rm H_{0}=67.8$ \\kms\\ Mpc$^{-1}$, $\\Omega_{\\rm m}=0.308$, $\\Omega_{\\Lambda}=0.692$ cosmology throughout this paper. \n\n\\section{Observations, Data Reduction \\& Line Fitting} \\label{sec:obs}\n\n\\begin{deluxetable*}{ccccccccc}\n\\centering\n\\tablecaption{Summary of ALMA observations \\label{tab:VLA-archive}}\n\\tablehead{\n\\colhead{Object} & \n\\colhead{Date} & \n\\colhead{Central frequency} &\n\\colhead{Continuum beam \\tablenotemark{a}}&\n\\colhead{Continuum Sensitivity}&\n\\colhead{Line beam}&\n\\colhead{Line sensitivity}&\n\\colhead{Channel width} \n\\\\\n\\colhead{} & \n\\colhead{} & \n\\colhead{(GHz)}& \n\\colhead{}&\n\\colhead{mJy}&\n\\colhead{}&\n\\colhead{mJy}&\n\\colhead{\\kms}}\n\\startdata\n7C 1354 & 2018 Jan 8-24  & 153.364 & 0.48\\arcsec$\\times$0.36\\arcsec &0.0069&0.59\\arcsec$\\times$0.47\\arcsec&0.08 & 34.0\\\\\n4C 22.44 & 2017 Dec 17-30 & 135.55 & 0.39\\arcsec$\\times$0.35\\arcsec&0.0068&0.408\\arcsec$\\times$0.366\\arcsec&0.083 & 66.0\\\\\n4C 05.84 & 2018 Jan 5-16  & 138.87 & 0.41\\arcsec$\\times$0.30\\arcsec&0.0075&0.438\\arcsec$\\times$0.54\\arcsec&0.134 & 33.0\\\\\n4C 09.17 & 2017 Dec 25 & \t148.24 & 0.32\\arcsec$\\times$0.23\\arcsec&0.0139&0.39\\arcsec$\\times$0.34\\arcsec&0.130 & 15.8\\\\\n & 2018 Jan 8 & \t & &&&\\\\\n3C 318 & 2014 Jul 6  & 134.34 & 0.25\\arcsec$\\times$0.18\\arcsec&0.0095&0.454\\arcsec$\\times$0.320\\arcsec&0.107 & 66.66\\\\\n3C 298 & 2016 Sep 9  & 141.85 & 0.25\\arcsec$\\times$0.18\\arcsec&0.046&0.39\\arcsec$\\times$0.30\\arcsec&0.22 & 34.0\n\\enddata\n\\tablenotetext{a}{From continuum images integrated over all frequencies and cleaned with robust = 0.5 parameter.}\n\\end{deluxetable*}\n\nA leading goal of the ALMA observational program was to detect molecular gas outflows in quasars with powerful ionized outflows that were previously detected via optical emission line (e.g., [O{\\sc III}]\\xspace, H$\\alpha$\\xspace) kinematics using integral field spectroscopy observations with adaptive optics \\citep{Vayner19b,Vayner19a}. All sources selected within this study display ionized gas outflows on kpc scales, with outflow rates in the range of 50-1000 \\myr, velocities $>$ 500 \\kms\\ with momentum fluxes $>$ 10$^{35} $ dyne and coupling efficiencies between the kinetic luminosity of the outflow and the bolometric luminosity of the quasar $>$0.05$\\%$. Given the similarities between the field-of-view and angular resolution between the Keck\/OSIRIS and ALMA observations we are able to study molecular gas outflows on similar spatial and dynamical time scales.\n\nALMA band 4 observations were conducted in Cycle 5 in the C43-5 configuration with a typical angular resolution of 0.4\\arcsec\\ and a maximum recoverable scale of 4-5.5\\arcsec, corresponding to rough physical scales of 3 kpc and 34-43 kpc, respectively. One 1.875 GHz spectral window was tuned to the redshifted frequency of CO (3-2) or CO (4-3) emission line with an effective velocity bandwidth of 4,000 \\kms, while additional three bands were tuned to the nearby continuum.\\\\  \n\nData reduction was performed using CASA (Common Astronomy Software Applications; \\citealt{McMullin07}) version 5.1.2-4. The ALMA automated pipeline was used to create the measurement sets (MS) for each observing block, which were then combined into a single measurement set for each source. We performed phase self-calibration for 4C 05.84 and 7C 1354+2552, while for 3C 318 and 4C 09.17, we performed both phase and amplitude self-calibration. For each source, the band 4 quasar continuum, was used as a self-calibration model using the CASA task \\textit{clean}. Given the similarity between the archival VLA images of the jets in these systems, we believe that the majority of the continuum in our sources comes from the synchrotron emission of the quasar core\/jets. While there is extended continuum emission for each source, the majority of the flux is associated with the unresolved core emission from the quasar making the modeling of the continuum relatively simple for self-calibration. We image the continuum with Briggs weighting using a robust value of 0.5 and a pixel size set to 1\/4th of the beam's full-width half max (FWHM). In this work, we also include our pilot observations of 3C 298 conducted in cycles 2 and 3 \\citep{Vayner17} with ALMA band 4. We achieve an SNR of 600-25,000 for the peak continuum flux. The continuum SNR improved by a factor of 1.2-5 from phase-only self-calibration and in the case of 4C09.17 after amplitude self-calibration the rms further improved by a factor of 1.5, while for 3C318 we did not see a significant improvement in the rms after amplitude calibration.\\\\\n\nWe performed continuum subtraction using the CASA task \\textit{uvcontsub} by fitting a first-order polynomial to line-free emission channels of the spectral window with the CO emission. We then subtracted the best fit continuum model from the full spectral window.\\\\\n\nWe imaged the cube using \\textit{clean} with a robust value of 1.5 to help improve detection of fainter and more diffuse emission, resulting in a larger beam than the continuum imaging. We used a spectral pixel size of either 16 \\kms, 34 \\kms, or 66 \\kms, depending on the signal-to-noise ratio (SNR) of the CO emission. We used a value of 0.05\\arcsec\\ for the spatial pixel size. For all sources except 4C 09.17, we used a wide circular aperture centered on the quasar for the cleaning mask with a radius of 1\/4 the primary beam size. For 4C 09.17, the CO emission was detected in individual 16 \\kms\\ channels in the first cleaning cycle, and a tight mask was designed for each channel encompassing the CO (4-3) emission.\n\n\nTo search for extended emission, we construct an SNR map for each channel in the data cube. An SNR map is made by first computing a standard deviation in a large aperture away from the phase center, followed by dividing the flux per spaxel by the standard deviation. SNR maps are constructed from data cubes that were not corrected for the primary beam's response. For any spaxel containing emission with a peak SNR$\\geq$4 per beam, we fit the emission line in neighboring spaxels that lie in the beam with a Gaussian plus constant continuum model using the least-squares fitting routine \\textit{curvefit} within \\textit{SciPy}. We create a 0$^{th}$ moment (flux) map by integrating the emission line in each spaxel with a successful emission line fit over the fitted Gaussian model and a 1$^{st}$ moment map by computing the line centroid's Doppler shift relative to the redshift of the quasar host galaxy. The integrated intensity maps are all optimally extracted, with a varying window for integrating the CO emission line based on the velocity offset and dispersion. We construct a 2$^{nd}$ moment map using the fit's velocity dispersion from the Gaussian model. All moment maps are constructed from data cubes that were corrected for the primary beam's response. Figure \\ref{fig:all_sources_CO} showcases the integrated CO (3-2) or CO (4-3) maps for sources with detected extended emission.\\\\\n\nHerein we use the quasar redshifts that are reported from Keck\/OSIRIS observations \\citep{Vayner19b} where they were derived from the centroid of the spatially unresolved ($<1.5$ kpc) [O{\\sc III}]\\xspace\\ or H$\\alpha$\\xspace emission lines, effectively originating from the narrow-line-region (NLR) of the quasar. The [O{\\sc III}]\\xspace\\ and H$\\alpha$\\xspace\\ redshifts agree within the centroiding uncertainty. For 3C 298 the redshift derived from the molecular gas disk traced through CO (3-2) is within 50 \\kms\\ of the redshift derived from the quasar emission \\citep{Vayner17}.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0in]{all_sources_ALMA.eps}\n    \\caption{ALMA band 4 observations of the sources within our sample. We present ALMA band 4 integrated intensity maps of CO emission. In contours we present the ALMA band 4 continuum emission that is dominated by synchrotron emission from the quasar jets. The contours stretch from a peak flux of 0.013, 0.0037, 0.2, 0.003, 0.045, and 0.002 Jy\/beam down to 2$\\sigma$ sensitivity in linear steps, from top left to bottom right, respectively. The ellipse in the lower-left corner of each map represents the beam of the emission line data. The star in the center represents the quasar's location, and the bar represents 1 arcsecond. The maps are at at a position angle of 0$^{\\circ}$, where north is oriented up, the only exception is 3C 298 that is at a position angle of 103$^{\\circ}$ East of North.}\n    \\label{fig:all_sources_CO}\n\\end{figure*}\n\n\n\\begin{deluxetable*}{lcclll@{\\extracolsep{-10pt}}c}[!th]\n\\tiny\n\\tablecaption{Sample properties \\label{tab:sample}}\n\\tablehead{\n\\colhead{Name} & \n\\colhead{RA} & \n\\colhead{DEC} &\n\\colhead{z\\tablenotemark{a}} &\n\\colhead{L$_{\\rm bol}$} &\n\\colhead{L$_{\\rm 178 MHz}$} &\n\\colhead{M$_{\\rm BH}$}\\\\\n\\colhead{} & \n\\colhead{J2000} &\n\\colhead{J2000} & \n\\colhead{} & \n\\colhead{($10^{46}$ erg s$^{-1}$ )} & \n\\colhead{($10^{44}$ erg s$^{-1}$ )} & \n\\colhead{M$_{\\odot}$}}\n\\startdata\n4C 09.17 & 04:48:21.74  & +09:50:51.46 & 2.1170 & 2.88$\\pm$0.14 &2.6 & 9.11  \\\\\n7C 1354+2552 & \t13:57:06.54 & +25:37:24.49 & 2.0068 & 2.75$\\pm$0.11 & 1.4& 9.86 \\\\\n3C 298 & 14:19:08.18 & +06:28:34.76  & 1.439\\tablenotemark{b} & 7.80$\\pm$0.30 & 12 &9.51 \\\\\n3C 318 & 15:20:05.48  & +20:16:05.49 & 1.5723 & 0.79$\\pm$0.04 & 4.0 &9.30 \\\\\n4C 22.44 & 17:09:55.01 & +22:36:55.66 & 1.5492 & 0.491$\\pm$0.019 & 0.6 &9.64 \\\\\n4C 05.84 & 22:25:14.70  & +05:27:09.06 & 2.320 & 20.3$\\pm$1.00& 4.5 &9.75  \\\\\n\\enddata\n\\tablenotetext{a}{Redshift relative to narrow-line region emission of the quasar, derived from 5007 \\AA\\ [O{\\sc III}]\\xspace\\ emission.}\n\\tablenotetext{b}{Redshift derived from host galaxy CO (3-2) emission \\citep{Vayner17}}\n\\end{deluxetable*}\n\n\n\\section{Dynamics of the molecular gas}\\label{sec:dynamics}\n\n\\subsection{Systemic molecular gas}\n\nFor each source, we search for emission at the systemic redshift of the quasar host galaxy. Narrow, CO line emission is found in the host galaxies of 4C 09.17A, 4C 09.17B, 3C 298, and 7C 1354+2552. The narrow CO emission in 3C 298 resembles a rotating disk that we modeled in \\citet{Vayner17}. The narrow emitting gas in the other systems does not show a smooth velocity gradient that would be indicative of a rotating galactic disk. \n\n\\noindent We compute the emission line luminosity using the following equation:\n\n\\begin{equation}\n   L^{'}_{CO}=3.25 \\times 10^{7}S_{CO}\\Delta v \\frac{D^{2}_{L}}{(1+z)^{3}\\nu_{obs}^{2}} \\rm K~km~s^{-1}~pc^{2},  \n\\end{equation} \n\n\\noindent \nwhere $\\nu_{obs}$ is the observed CO transition frequency, $D_{L}$ is the luminosity distance, and $S_{CO}\\Delta v$ is the line integrated flux in units of Jy \\kms. We convert the observed CO transition luminosity into CO (1-0) luminosity (L$^{'}_{CO(1-0)}$) by assuming that the low-J CO transitions are thermalized and are optically thick, so ${L}_{\\mathrm{CO}\\ 4\\mbox{--}3}^{{\\prime} }={L}_{\\mathrm{CO}\\ 3\\mbox{--}2}^{{\\prime} } ={L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime} }$. Using the ratios (${r}_{{\\rm{J}}1}={L}_{\\mathrm{CO}J\\to J-1}^{{\\prime} }\/{L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime} }$) from \\cite{CarillinWalter13} with $r_{31}$=0.97 and $r_{41}$=0.87 did not significantly change our results. Furthermore in 3C 298 we found that the molecular gas is consistent with being thermalized and optically thick \\citep{Vayner17}. These physical conditions are consistent with what is found for Mrk 231 \\citep{Feruglio15}. Finally, we convert the ${L}_{\\mathrm{CO}\\ 1\\mbox{--}0}^{{\\prime}}$ line luminosity into molecular gas mass using the CO-to-H$_{2}$ conversion factor: $\\alpha_{\\rm CO}$ with units of (K \\kms$\\rm pc^{2}$)$^{-1}$. For sources where we do not detect any narrow CO emission at the systemic redshift, we place a limit on the molecular gas mass over an aperture equal to the beam size for an emission line with a velocity FWHM of 250 \\kms. The molecular gas mass limits can be linearly scaled with a different $\\alpha_{CO}$\\ value. For sources with detected molecular gas at the quasar's systemic redshift, we compute the radius of the molecular gas region, which allows us to measure the gas surface density. All radii are computed using a curve of growth method. Effective radii refer to a region which encloses 50\\% of the flux, while a ``maximum\" extent refers to a size scale which encloses 90\\% of the flux. In all cases, narrow emission at the systemic redshift of the quasar is spatially resolved by our observations. We deconvolve the size of the beam from all radius measurements. For sources with no detected CO emission, we use the molecular gas mass limit and the beam of the observations as a proxy for the radius. Values associated with the molecular gas at the systemic redshift are summarised in Table \\ref{tab:narrow-prop}.\n\n\\begin{deluxetable*}{lcccccr}[!th]\n\\tablecaption{Properties of molecular gas at the systemic redshift of each quasar. $S_{\\nu}\\Delta V$ is the spatially and line-integrated CO flux. $M_{H_{2}}$ is the mass of gas at the systemic redshift of each galaxy, assuming an $\\alpha_{CO}$ of 0.8. $R$ is the radius of the region. V$\\rm_{\\sigma}$ is the velocity dispersion. SFR is the expected star formation rate based on the currently available molecular gas reservoir based on the KS law.  \\label{tab:narrow-prop}}\n\n\n\n\\tablehead{\n\\colhead{Source}&\n\\colhead{$S_{\\nu}\\Delta V$}&\n\\colhead{$M_{H_{2}}$}&\n\\colhead{R}&\n\\colhead{V$\\rm_{\\sigma}$}&\n\\colhead{$\\Sigma_{molecular}$}&\n\\colhead{SFR}\\\\\n\\colhead{}&\n\\colhead{Jy \\kms}&\n\\colhead{$\\times10^{9}$M$_{\\odot}$}&\n\\colhead{kpc}&\n\\colhead{\\kms}&\n\\colhead{M$_{\\odot}$ pc$^{-2}$}&\n\\colhead{\\myr}}\n\\startdata\n4C 09.17 A RL & 0.29$\\pm$0.03 & 3$\\pm$0.3 & 2.8  & 158.0$\\pm$14 & 137$\\pm$13 & 6 \\\\\n4C 09.17 B RQ & 0.88$\\pm$0.09 & 10$\\pm$1 & 1.1 & 143.5$\\pm$12 & 2357$\\pm$235 & 56 \\\\\n7C 1354 & 0.028 $\\pm$0.003 & 0.3$\\pm$0.1 & 2.0 & 44.10$\\pm$11 & 23$\\pm$2 & 1\\\\\n3C 298 & 0.63$\\pm$0.07 & 6.6$\\pm$1 & 1.6 & 42.35$\\pm$12.8 & 820$\\pm$10 & 24\\\\\n3C 318 & $<0.05$ & $<0.6$ & -- & -- & -- & --\\\\\n4C 22.44 & $<$0.05 & $<$ 1 & -- & -- & -- & --\\\\\n4C 05.84 & $<0.07$ & $<0.8$ & -- & -- & -- & --\\\\\n\\enddata\n\\end{deluxetable*}\n\n\\subsection{Observed Molecular Outflows}\\label{sec:search_outflow}\n\nIonized gas outflows typically show a peak emission line offset  $>$ 300 \\kms\\ from the quasar redshift measured from the narrow emission line region, with a velocity dispersion $>$ 250-300 \\kms\\ over the outflow region for the sources within our sample \\citep{Vayner19b}. Based on these observed velocities of the ionized gas outflows, we define the criteria for a molecular outflow to be any molecular gas that has a peak emission line offset relative to the redshift of the quasar host galaxy $|v| >300$ \\kms\\ or a spaxel with a velocity dispersion greater than 250 \\kms. The selected outflow criteria are typical of molecular outflows found in nearby galaxies \\citep{Fluetsch19}. Using these criteria, we detect molecular gas outflows in four quasars in the ALMA sample. For 3C 318 and 4C 05.84, we select outflows based on both broad ($\\sigma>250$ \\kms) and offset ($|v| > 500$ \\kms) emission and for 4C 09.17, based on broad ($\\sigma>300$ \\kms) emission. We include the molecular gas outflow in 3C 298 that was previously detected in \\citet{Vayner17} based on broad CO (3-2) and CO (5-4) emission. For each source we extract a spectrum by integrating over all spaxels satisfying this outflow criteria. We present the extent of the molecular outflows along with the spectra in Figures \\ref{fig:3C318_spec}, \\ref{fig:4C0917_spec}, \\ref{fig:4C0584_spec}, and \\ref{fig:3C298_spec}. \n\nWe potentially may be missing dense outflowing gas moving at slower speeds since our high velocity criteria is based on atomic gas observations, which may lead us to underestimate the total molecular gas in the outflow for the 4C 09.17 and 3C 298 systems where we detect CO emission moving at speeds $<$ 300 \\kms. Observations of denser molecular gas tracers, such as CS or HCN, could provide a more complete picture of the outflow. \\\\\n\nWe calculate the molecular gas mass in the outflows based on flux associated with the broad or highly offset emission line component. We use an $\\alpha_{CO}$\\ value of 0.8 \\msun$\\rm(K~kms^{-1}pc^{2})^{-1}$ for consistency with other works at low and high redshift \\citep{Herrera-Camus19,Fluetsch19}, this value is commonly adopted for the molecular gas in the ISM of nearby Ultra Luminous Infrared Galaxies \\citep{Bolatto13}. However, in some well-studied molecular outflows in nearby galaxies, the conversion factor can be much higher, $\\alpha_{\\rm CO}\\sim 2$ \\citep{Cicone18}, so we may be conservative with our mass estimates.\n\nWe compute the molecular gas outflow rate using\n\n\\begin{equation}\\label{equation:outflow-thin-shell}\n    \\dot{M}_{H_{2}}=\\frac{M_{H_{2}}v_{out}}{R_{out}}.\n\\end{equation}\n\nWe select this equation since the molecular gas outflows in 3C 318, 4C 05.84, and 4C 09.17-A RL are seen as a single high-velocity offset component that spans either blue- or red-shifted velocities. The molecular gas outflows in 4C 09.17B and 3C 298 may be closer in geometry to a filled wide-angle cone since they span a broader velocity range. In these two sources, the estimates obtained from equation \\ref{equation:outflow-thin-shell} should be multiplied by a factor of 3 if the outflows are closer to a filled cone. Here $R_{out}$ is the extent of the outflow where 90\\% of the flux associated with the outflow emission accumulates. The velocity is computed as $v_{out} = \\left | v_{r} \\right |  + FWHM\/2$, where $\\left | v_{r} \\right |$ and $FWHM$  are the radial velocity and full-width-half-maximum in units of \\kms\\ of the emission relative to the systemic redshift of the emission associated with the outflow component.\n\nIn addition to the outflow rates we also compute the momentum flux of the outflow using: \n\n\\begin{equation}\n    \\dot{P}_{H_{2}} = \\dot{M_{H_{2}}} \\times v_{out}\n\\end{equation}\n\n\\noindent and the kinetic luminosity:\n\n\\begin{equation}\n    \\dot{E}_{H_{2}} = \\frac{1}{2}\\times\\dot{M_{H_{2}}}\\times v_{out}^{2}.\n\\end{equation}\n\nThe CO line luminosity used to compute the molecular gas mass along with the spatial extent, velocity, outflow rate, and energetics are summarized for each source in Table \\ref{tab:outflow-prop}.\n\n\n\\begin{deluxetable*}{lllccccccccl}\n\\tablecaption{Multi-phase outflow properties. $S_{\\nu}\\Delta V$ is the spatially and line-integrated CO flux associated with the molecular outflow. R$\\rm_{out}$ is the radial extent of the outflow. V$\\rm_{out}$ is the velocity of the outflow. dM\/dt$\\rm_{H_{2}}$ and dM\/dt$\\rm_{Ionized}$ are the molecular and ionized outflow rates. $\\dot{P}_{H_{2}}$ is the momentum flux of the molecular outflow (assuming $\\alpha_{CO}$$=0.8$), while $\\dot{P}_{Ionized}$ is the momentum flux of the ionized outflow, using the dM\/dt$\\rm_{H\\alpha}$ outflow rate from \\citep{Vayner19a}. $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ is the ratio of the momentum flux of outflow to the momentum flux of the quasar accretion disk ($L_{bol}\/c$) using a sum of the ionized and molecular outflow momenta flux.\\label{tab:outflow-prop}}\n\n\\tablehead{\\colhead{Source}&\n\\colhead{$S_{\\nu}\\Delta V$}&\n\\colhead{M$\\rm_{H_{2}}$}&\n\\colhead{R$\\rm_{out}$}&\n\\colhead{V$\\rm_{out}$}&\n\\colhead{V$\\rm_{FWHM}$}&\n\\colhead{dM\/dt$\\rm_{H_{2}}$}&\n\\colhead{M$\\rm_{Ionized}$}&\n\\colhead{dM\/dt$\\rm_{Ionized}$}&\n\\colhead{$\\dot{P}_{H_{2}}$}&\n\\colhead{$\\dot{P}_{Ionized}$}&\n\\colhead{$\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$}\\\\\n\\colhead{}&\n\\colhead{Jy \\kms}&\n\\colhead{$\\times10^{9}$M$_{\\odot}$}&\n\\colhead{kpc}&\n\\colhead{\\kms}&\n\\colhead{\\kms}&\n\\colhead{\\myr} &\n\\colhead{$\\times10^{9}$M$_{\\odot}$} &\n\\colhead{\\myr}&\n\\colhead{$10^{35}$dyne}&\n\\colhead{$10^{35}$dyne}&\n\\colhead{}\n}\n\\startdata\n4C 05.84 & 0.1$\\pm$0.03 & 1.4$\\pm$0.2 & 8.6 & 653$\\pm$30 & 382.2$\\pm$47.4 & 110$\\pm$12 & 0.4$\\pm$0.3 & 870$\\pm$600 & 4.4$\\pm$0.6  & 40$\\pm$30& 0.7$\\pm$0.4 \\\\\n3C 318 & 0.25$\\pm$0.03 & 3$\\pm$0.3 & 20.2 & 1132$\\pm$44 & 528.7$\\pm$67.4 & 168$\\pm$18 & 0.32$\\pm$0.2 & 220$\\pm$150 & 12$\\pm$2 & 10$\\pm$7  & 8$\\pm$3 \\\\\n3C 298 & 0.3$\\pm$0.03 & 3$\\pm$0.3 & 1.6 & 394$\\pm$64 & 624.0$\\pm$49.0 & 780$\\pm$150 & 0.6$\\pm$0.3 & 750$\\pm$400 & 20$\\pm$7 & 77$\\pm$40  & 4$\\pm$2 \\\\\n4C 09.17 A & 0.11$\\pm$0.01 & 1.3$\\pm$0.1 & 2.8 &852$\\pm$77& 439.1$\\pm$122.6 & 400$\\pm$50 & 0.05$\\pm$0.02 & 50$\\pm$20 & 21$\\pm$4 & 2.1$\\pm$1  & 2.4$\\pm$0.5 \\\\\n4C 09.17 B & 2.3$\\pm$0.2 & 27$\\pm$3 & 4.9 & 456$\\pm$26  & 870.6$\\pm$47.2 & 2500$\\pm$300 & -- & -- & 73$\\pm$11 & --  & -- \\\\\n\\enddata\n\\end{deluxetable*}\n\n\\section{Individual Objects}\\label{sec:indiv_obj}\n\nIn this section, we outline the known properties of each quasar within our sample, focusing on their radio jet morphology, far-infrared properties, morphologies, the extent of the ionized gas outflows, and a description of what we detect with our ALMA band 4 observations.\n\n\\subsection{3C 318 (z=1.5734)}\\label{sec:3c318}\n\n3C 318 is a luminous radio-loud quasar at $z=1.5734$. The radio jet is double-sided, stretching in the southwest and northeast, with a bright core emission associated with the quasar's optical emission location. Within our ALMA continuum observations, we only detect the jet's southwest component. The northeast component of the jet blends with the bright unresolved core. 3C 318 has been detected with the \\textit{Herschel Space Telescope} and is known to be a bright far-infrared emitting source \\citep{Podigachoski15}. Resolved band 7 ALMA observations of the dust emission reveal a ring-like structure on kpc scale centered on the quasar \\citep{Barthel19}.\\\\\n\nWith the Keck\/OSIRIS, we detected ionized gas emission in nebular emission lines [O{\\sc III}]\\xspace 5007 \\AA, H$\\alpha$\\xspace, [N{\\sc II}]\\xspace 6585 \\AA, and [S{\\sc II}]\\xspace 6717, 6731 \\AA\\ with an extent of 4 kpc \\citep{Vayner19b}. We detected an ionized outflow extending in the SW and NE direction with a maximum extent of 3.2 kpc. \\\\\n\n3C 318 was known to have molecular emission detected at the CO (2-1) transition with PdBI \\citep{willott07}. The CO (2-1) emission was known to be blueshifted by 400 \\kms\\ and spatially offset from the quasar continuum by 2.4\\arcsec\\ to the west and 0.5\\arcsec\\ to the north with considerable uncertainty due to a coarse beam of 8.05\\arcsec$\\times$4.32\\arcsec. \\\\\n\nOur ALMA band 4 observations reveal an extended CO (3-2) emission with one component offset 1.7 kpc to the west and a second component offset 17 kpc towards the south. The emission is divided between two regions that have widths of 4 and 8 kpc, that show similar highly blueshifted emission relative to the quasar. The spatially integrated emission is blueshifted (-936.0 \\kms) and relatively broad with an FWHM of 534 \\kms. The spectrum and integrated intensity map of the detected CO emission are shown in Figure \\ref{fig:3C318_spec}. Likely the ALMA and PdBI observations trace the same molecular gas components. However, the differences in the beams, maximum recoverable scales, and SNR play a role in the observed line shift and integrated line flux. 3C 318 has also been observed with VLA targeting the CO (1-0) transition \\citep{Heywood13}. Emission associated with the CO (1-0) emission line at the velocity offset of the PdBI observations was found 0.33\\arcsec\\ north from the quasar continuum. We do not detect any CO (3-2) emission at that location. Using typical ratios between the CO (1-0) and CO (3-2) line luminosity, we would have expected to detect this component at an SNR of 100, integrated over an emission line equivalent with a velocity dispersion of 250 \\kms. \\\\\n\nThe separation between the two high velocity clumps roughly matches the maximum recoverable scale of the interferometric observations. We have attempted to recover the more diffuse emission between the two clumps by smoothing the data in the UV plane with Gaussian kernel using the \\textit{uvtaper} option within \\textit{tclean} in CASA. Using a uv-taper parameter of 0.5\\arcsec\\ and 1\\arcsec\\ on the sky, the fainter emission between the two clumps seen in Figure \\ref{fig:3C318_spec} is below the noise in the uv-tapered data. The total integrated flux from the CO (3-2) emission is within 10\\% of the original data, within the statistical noise of the observations, hence no additional ``diffuse\" emission was recovered. A loss of baselines resulted in an increase in noise with larger uv-taper parameters. Observations in a more compact ALMA configuration are necessary to detect the more diffuse emission. \\\\\n\nALMA's higher angular resolution and sensitivity lead us to speculate that the molecular gas emission in 3C 318 is associated with a molecular outflow rather than a merging system. This is supported by more accurate redshift measurement from the narrow-line region, allowing us to do better kinematics and dynamics measurement of the molecular gas emission. The Dark Energy Survey \\citep{DES2021} shows no apparent optical detection of a companion galaxy down to an $r$-band magnitude of 24 at a significance of 10$\\sigma$. Infrared observations with Spitzer and archival ALMA band 7 observations do not show any evidence for a galaxy at the spatial locations of the highly blueshifted CO (3-2) emission \\citep{Barthel19}. There may be a possibility that high-velocity emission is associated with an obscured galaxy near the 3C 318 quasar host galaxy. In recent years there have been several galaxies detected with ALMA that have faint or no counterparts in very deep optical imaging and are referred to as ``optically dark ALMA galaxies\" \\citep{Williams19,Zhou20}. These galaxies show high far-infrared luminosities and are characterized as dusty star-forming galaxies at high redshift ($z>2$). Such galaxies are typically relatively massive, with stellar masses of 10$^{10.2-11.5}$ M$_{\\odot}$\\ and contain a substantial amount of molecular gas. If the emission in 3C 318 is associated with such a galaxy, then the associated dust continuum emission would have been easily detectable in the ALMA band 7 observations.\\\\ \n\nWe divide the molecular gas mass by the area of the emitting region using the effective radius as the scale size. Using the Milky Way's hydrogen column density - $V$-band extinction relationship \\citep{Guver09} we convert the column density into a $V$-band extinction value for the molecular gas traced by CO. We measure an $\\rm A_{v}$ value in the CO gas of 1-4 magnitudes, which is a factor of 100 lower than those found in the dusty star-forming galaxies. Furthermore, we use the molecular gas mass of individual clumps and convert them into expected dust continuum emission in band 7 based on the relationship between ISM mass and dust emission of \\citet{Scoville17}. The expected continuum flux density is 0.017 mJy\/beam, which would be undetected in the band 7 observations with a sensitivity of 0.0297 mJy\/beam. We assume the clumps have the same surface brightness profile in both bands and in the dust and molecular gas emissions. Based on these calculations, the clumps are unlikely to be associated with typical dust-obscured galaxies at this redshift. Band 4 continuum is not optimal for detecting the dust continuum at this redshift since the dust emission is expected to be fainter at the longer wavelength of the Rayleigh-Jeans tail. Another possibility is that the emission may be associated with a tidal tail feature. However the FWHM of 534 \\kms\\ and an offset of -936.0 \\kms\\ relative to the redshift of the quasar are both larger than what would be expected for a tidal feature. Based on morphological identification of tidal tail from HST imaging and Keck\/OSIRIS observations, in two other systems, we found that the velocity dispersion in both the ionized and molecular gas mass is $\\sim$ 150 \\kms\\ with velocity offsets of -250 \\kms\\ \\citep{Vayner19b}. \\\\\n\nCombining rest-frame optical and sub-mm observations, we find that the ionized and molecular gas outflows in 3C 318 show different morphologies, spatial extent, and kinematics. The molecular gas outflow is far more extended, with a maximum distance of 21 kpc from the quasar, while the ionized outflow shows a maximum extent of only 3.2 kpc. The molecular outflow is also faster moving with blueshifted velocities up to -1200 \\kms. In contrast, the ionized gas outflow has a velocity of 703 \\kms\\ with a bi-conal morphology that is both blue- and red-shifted relative to the quasar in the SW and NE directions. We find no evidence for CO (3-2) emission at the quasar's systemic redshift associated with narrow emission. We place a limit on the molecular gas reservoir at the quasar's location over an aperture matching the size of the beam of $<0.7\\times10^{9}$ M$_{\\odot}$\\ at 2$\\sigma$ confidence.\\\\\n\nThe extent of the molecular outflow is similar to the cold gas outflow detected in $z=6.4$ quasar SDSS J1148+5251, through the 158 \\micron\\ [C II] emission line \\citep{Cicone15}. The morphology and kinematics of the molecular outflow in 3C 318 are also similar to the outflow recently detected in zC400528 \\citep{Herrera-Camus19} through CO (3-2) observations, where they see an extended emission entirely redshifted from the galaxy with a relatively collimated morphology similar to the case of 3C 318. The clumpy morphology and high velocity of the outflowing molecular gas are also similar to recent molecular outflows detected in PDS 456 \\citep{Bischetti19} and in the lensed quasar HS 0810+2554 \\citep{Chartas20}.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0in]{3C318_spec.eps}\n    \\caption{ALMA band 4 observations of 3C 318. On the left we show optimally extracted intensity map of the molecular outflow in the 3C 318 system, detected in the CO (3-2) line. The white contours outline the molecular outflow region. On the right we show a spectrum integrated over the entire molecular outflow region shown in the white contours, along with fit to the CO (3-2) emission line. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations. We detect no molecular CO (3-2) emission at the systemic redshift.}\n    \\label{fig:3C318_spec}\n\\end{figure*}\n\n\n\\subsection{4C 09.17 (z=2.117)}\\label{sec:4c0917}\n\n4C 09.17 is a radio-loud quasar at $z=2.117$. A one-sided jet extends towards the southwest, with a bright core emission associated with the quasar's optical emission location. The system is also bright at far-infrared wavelength \\citep{Podigachoski15}. \\\\\n\nWith Keck\/OSIRIS we detected ionized gas emission in nebular emission lines [O{\\sc III}]\\xspace\\ 5007 \\AA, H$\\alpha$\\xspace, and [N{\\sc II}]\\xspace\\ 6585 \\AA\\ \\citep{Vayner19b}. We found an ionized gas outflow extending towards the east with a maximum extent of 6 kpc.\\\\ \n\nWe detect broad, blueshifted emission resembling a molecular gas outflow in the host galaxy of the radio-loud (RL) quasar 4C 09.17. From here on, we refer to this object as 4C 09.17 A - RL. We also detect a very broad component in the merging radio-quiet (RQ) galaxy towards the northeast; from here on, we refer to this object as 4C 09.17 B - RQ. This galaxy is also detected in \\textit{K}-band imaging of \\citet{Armus97}, with a red optical to near-IR continuum color. In 4C 09.17 B - RQ, we detect very faint narrow [O{\\sc III}]\\xspace\\ emission in Keck\/OSIRIS, the ionized outflow is undetected. 4C 09.17 B - RQ also contains a narrower emission line component in CO (4-3) at a similar redshift to the narrow [O{\\sc III}]\\xspace\\ emission, which we use to calculate its redshift. The velocity offset between the 4C 09.17 A - RL and 4C 09.17 B - RQ is -593 \\kms. The majority of the narrow CO emission-line flux is found concentrated in 4C 09.17 B - RQ, within a 1 kpc radius region. The majority of the dust continuum detected at far-infrared wavelengths with the \\textit{Herschel Space Telescope} is likely associated with this galaxy. 4C 09.17B - RQ is highly obscured; the narrow CO emission component yields a line integrated gas column density of 3.4$\\rm \\times10^{24}~cm^{-2}$ computed by dividing the molecular gas mass by area of the emitting region. Using the Milky Way's hydrogen column density - $V$-band extinction relationship \\citep{Guver09}, we find a $V$-band extinction of 150 mag. The narrow CO emission is likely at the center of the merging galaxy because it roughly corresponds to the $K$-band continuum's peak location.\\\\\n\nFor the galaxies detected to the southwest and northwest of the quasar in \\citet{Armus97} and \\citet{Lehnert99} we detect narrow CO (4-3) emission near their optical locations. We do not detect any high velocity or broad molecular gas associated with these two systems. The detection of 3 galaxies found within 20 kpc of the quasar host galaxy from both ALMA and Keck\/OSIRIS observations makes the 4C 09.17 system likely a proto-group environment at z $\\sim 2.11$. \\\\\n\nWe find that the molecular gas outflow in 4C 09.17 A -RL is more compact than the ionized gas outflow. The ionized and molecular gas outflow show similar blueshifted velocities and velocity dispersion. The maximum extent of the molecular gas outflow is 2.8 kpc, while the ionized outflow extends to 6 kpc. We find that both the ionized and molecular outflow in this system are not along the path of the radio jet, but extend in the same eastern direction. Similar results have been found for a subset of nearby galaxies recently, where outflows appear to expand perpendicular to the path of the jet \\citep{Venturi21}. \\\\\n\nIn 4C 09.17 B - RQ, the molecular outflow extends 4.9 kpc from the narrow CO emission line component. The extent of the molecular outflow in 4C 09.17 B - RQ roughly matches the maximum extent of the $K$-band stellar continuum; hence the molecular outflow is occurring on galactic scales in this galaxy. Extinction within the outflowing gas can potentially prevent ionization by quasar photons and prevent the observer from detecting recombination photons. Spectra of the distinct regions detected in this system are shown in Figure \\ref{fig:4C0917_spec}.  \\\\\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=6.5 in]{4C0917_spec.eps}\n    \\caption{ALMA band 4 observations of 4C 09.17. On the left we show optimally extracted intensity map of CO emission in the 4C 09.17 system. The 4C 09.17 is a merger system with molecular outflows detected in both galaxies. The teal contours outline the molecular outflow in the radio-quiet galaxies 4C 09.17 B, while the purple contours outline the molecular outflow detected in the host galaxy of the radio loud quasar 4C 09.17. On the right we show the spectra extracted over the respective outflow regions along with the Gaussian fit model. Dashed lines represent the individual Gaussian components of the emission line fit, while the solid black line represents the sum of all components and a 0th order polynomial fit to any residual continuum. Component 1 (C1) in 4C 09.17 B is the fit to the narrow emission at the systemic redshift of the merging galaxy, that has a velocity offset of about 593 \\kms\\ relative to 4C 09.17A, while C2 is gas in the outflow. For 4C 09.17 A, C1 corresponds to the outflow gas while C2 is the narrow gas at the systemic redshift of the quasar host galaxy. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations.}\n    \\label{fig:4C0917_spec}\n\\end{figure*}\n\\subsection{4C 22.44 (z=1.5492)}\\label{sec:4c2244}\n\n\n4C 22.44 is a radio-loud quasar at z=1.5492. The radio jet in this system extends along the east-west direction with a length of 7\\arcsec\\ and a bright core emission associated with the quasar's optical emission location. With Keck\/OSIRIS we detected extended ionized gas in the [O{\\sc III}]\\xspace, H$\\alpha$\\xspace, and the [N{\\sc II}]\\xspace emission lines with an extent of 2 kpc. An ionized gas outflow is detected on a spatial scale of $<1$ kpc. We detect no emission in the CO (3-2) line with ALMA. We place a limit on the CO (3-2) line luminosity of 0.08 Jy \\kms\\ for an aperture matching the beam with a line velocity dispersion of 250 \\kms, which converts to a molecular gas mass limit of $<$ 1$\\times10^{9}$ M$_{\\odot}$.\n\n\\subsection{4C 05.84 (z=2.323)}\\label{sec:4c0584}\n\n4C 05.84 is a radio-loud quasar at z=2.323. The one-sided radio jet in this system extends towards the southwest with a maximum extent of 12 kpc. There is a bright radio core component associated with the quasar's optical emission location.\n\nWith Keck\/OSIRIS observations, we detected extended ionized gas on an 8 kpc scale in the [O{\\sc III}]\\xspace, H$\\alpha$\\xspace, and [N{\\sc II}]\\xspace emission lines \\citep{Vayner19a}. We detected a bi-conical ionized outflow extending along the northeast and southwest direction.\n\nIn our ALMA band 4 observations, we detect extended CO (4-3) emission that is offset in the western direction consisting of multiple clumps. The individual clump components and the spatially integrated emission is highly blueshifted relative to the quasar's systemic redshift. This emission is not associated with any known galactic component. The spectrum and integrated intensity map of the detected CO emission are shown in Figure \\ref{fig:4C0584_spec}. \n\nThe ionized and molecular gas outflows in this system are on similar scales. The ionized outflow extending towards the southwest direction shows a similar blueshifted velocity offset as the molecular outflow. The ionized outflow detected on spatial a scale $<1$ kpc is similarly blueshifted to the more extended ionized and molecular outflow. The ionized outflow appears to be more turbulent with a larger velocity dispersion than the molecular outflow. We find no evidence for CO (4-3) emission at the quasar's systemic redshift associated with narrow emission. We place a limit on the molecular gas reservoir mass of $<0.8\\times10^{9}$ M$_{\\odot}$\\ at the quasar's location with an aperture the size of the beam and using a velocity dispersion of 250 \\kms.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.5 in]{4C0584_spec.eps}\n    \\caption{ALMA band 4 observations of 4C 05.84. On the left we show optimally extracted intensity map of the molecular outflow in the 4C 05.84 system, detected in the CO (4-3) line. The white contours outline the molecular outflow region. On the right we show a spectrum integrated over the entire molecular outflow region, along with fit to the CO (4-3) emission line. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel shows the beam of our ALMA band 4 observations. We detect no molecular CO (4-3) emission at the systemic redshift.}\n    \\label{fig:4C0584_spec}\n\\end{figure*}\n\n\\subsection{3C 298 (z=1.439)}\\label{sec:3C298}\n\n3C 298 is a radio-loud quasar at z=1.439. The jets in the system extend in the east-west direction, with a length of about 16 kpc and a bright core emission associated with the quasar's optical emission location. The system is bright at far-infrared wavelengths. ALMA band 7 observations have revealed that the dust emission mostly comes from a kpc scale region centered on the quasar \\citep{Barthel18}.\n\nWith Keck\/OSIRIS, we detected extended ionized gas emission \\citep{Vayner19b} on scales up to 20 kpc from the quasar. A bi-conical ionized outflow is detected with a maximum extent of 3 kpc from the quasar along the jet's path. The redshifted cone is associated with the western jet component, while the blueshifted cone is associated with the eastern jet component. We also detected an ionized gas outflow in the merging system 8 kpc from the quasar.\n\nIn the ALMA band 4 observations, we detect a molecular gas emission in two distinct components, one centered on the quasar with a radius of 2.1 kpc and a second component 21 kpc from the quasar, offset by -250 \\kms. The component near the quasar shows both broad and narrow emission. The narrow emission is associated with a galactic disk \\citep{Vayner17}, while the broad component is associated with the outflow. The outflow emanates from the molecular disk centered on the quasar with a maximum extent of 1.6 kpc. The majority of the molecular gas in the outflow is on the blueshifted side of the disk and extends in the direction of the jet's western component. The ionized outflow is more extended than the molecular outflow, has a faster velocity, and appears to be more turbulent with a larger velocity dispersion. \n\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0 in]{3C298_spec.eps}\n    \\caption{ALMA band 4 observations of 3C 298. On the left we show optimally extracted intensity map of CO emission in the 3C 298 system, detected in our 2017 study of this object \\citep{Vayner17}. The purple contours outline the molecular outflow region, white contours outline total emission from the molecular disk, while the teal contours outline a star forming\/tidal tail feature. On the right we show a spectrum integrated over each distinct region along with fit to the CO (3-2) emission line. Dashed lines represent the individual Gaussian components of the emission line fit, while the solid black line represents the sum of all components and a 0th order polynomial fit to any residual continuum. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel represents the beam of the ALMA band 4 observations.}\n    \\label{fig:3C298_spec}\n\\end{figure*}\n\n\n\\subsection{7C 1354+2552 (z=2.0064)}\\label{sec:7C1354}\n\n7C 1354+2552 is a radio-loud quasar at z=2.0064. The system contains two jets that are perpendicular to each other. The east-west jet has a length of about 24 kpc, while the north-south jet has a length of 86 kpc. Only the east-west jet is detected in the continuum in our ALMA observations due to limited sensitivity to low-surface brightness emission on scales $>$ 6\\arcsec.\n\nUsing the Keck\/OSIRIS observations, we detected extended emission in the nebular emission lines [O{\\sc III}]\\xspace and H$\\alpha$\\xspace on scales of 4-6 kpc. An ionized outflow is detected on a spatial scale of $<1$ kpc. \n\nIn the ALMA band 4 observations, we detect molecular gas emission towards the northeast. The narrow emission line resides near the quasar's systemic redshift. We do not detect any broad emission from a turbulent molecular gas outflow. We detect no highly offset emission consistent with our outflow criteria; hence there is no evidence for a cold molecular gas outflow in this system at our observations' sensitivity. Spectra of the distinct region detected in this system is shown in Figure \\ref{fig:7C1354_spec}. The detected emission may be associated with a merging galaxy. With the large (8 kpc) separation from the quasar and the fact that the motion of the gas does not appear to follow the kinematics of the galactic disk on the eastern side \\citep{Vayner19b}, there is a possibility that this emission is associated with a merging galaxy.\n\n\n\\begin{figure*}[!th]\n    \\centering\n    \\includegraphics[width=7.0 in]{7C1354_spec.eps}\n    \\caption{ALMA band 4 observations of 7C 1354+2552. On the left we show optimally extracted intensity map of the molecular gas in the 7C 1354+2552 system, detected in the CO (4-3) line. The gas is narrow and is consistent with gravitational motion. Yellow contours outline the CO emitting region towards the north-west that is slightly redshifted, the spectrum associated with it is shown on right. The systemic redshift of the quasar host galaxy is at 0 \\kms. The ellipse in the lower left corner on the right panel represents the beam of the ALMA band 4 observations.}\n    \\label{fig:7C1354_spec}\n\\end{figure*}\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{What is driving the outflows?}\\label{sec:driving}\nSeveral powerful mechanisms can drive galactic-scale outflows. Quasars can reside in galaxies with powerful star formation activity \\citep{Duras17,Aird19,Circosta21,Bischetti21}, especially at redshifts near the peak of the star formation activity. Star formation can result in powerful galactic winds \\citep{Rupke18}. Radio-loud quasars that are optically luminous can drive galactic outflows by both jets \\citep{Wagner12,Mukherjee16} and radiation pressure \\citep{Murray95,Faucher12a,Zubovas12,Costa18} from the accretion disk. In this section, we explore primary mechanisms that are capable of driving the multi-phase galactic outflows. We combine the momentum flux and kinetic luminosities from the cold molecular and ionized gas phases to look at both the total impact of galactic winds on the quasar host galaxies and what mechanism may be responsible for driving the entirety of the outflowing gas in each system.\n\nTo understand the main driving mechanism of galactic outflows, we compare the momentum flux of the outflow ($\\dot{P}_{outflow}$ ) to the momentum input from the quasar accretion disk ($\\dot{P}_{Quasar}$ ) and to the momentum deposition from stellar feedback ($\\dot{P}_{SNe}$).\n\n\\subsection{Stellar feedback as potential driver of galactic outflows}\nTo explore whether star formation activity can drive the galactic scale outflows, we compare the momentum deposition from supernovae explosions, based on the star formation rate, to the energetics of the multi-phase gas outflow. To compute the energy deposition from supernovae, we use the results of recent simulations by \\citet{2015MNRAS.450..504M} and \\citet{2015ApJ...802...99K}, that predict the momentum deposition per unit of solar mass formed. We assume that one supernova explodes for every 100 M$_{\\odot}$\\ of stars. We assume an electron density of 100 cm$^{-3}$  in the interstellar medium with solar metallicity.\n\n\n\\begin{equation}\n    \\dot{P}_{SNe} = 7.5\\times 10^{33} \\frac{\\dot{M}_{SFR}}{1M_{\\odot}{\\rm yr}^{-1}} \\left(\\frac{n}{100\\,{\\rm cm}^{3}}\\right)^{-0.18}\n    \\left(\\frac{Z}{Z_\\odot}\\right)\n    \\rm dyne \\label{eq:SNe_mom}.\n\\end{equation} \n\nFor sources with resolved rest-frame far-infrared observations (3C 318 and 3C 298), we use the far-infrared luminosity and convert that into a star formation rate based on work by \\citet{Barthel18,Barthel19}. The total-infrared (8-1000 \\micron) derived star formation rates of 3C 298 and 3C 318 are 930\\myr\\ and 580 \\myr\\ derived using the \\citet{Kennicutt98} calibration \\citep{Podigachoski15}. Infrared-derived star formation rates have high uncertainties since it is not clear what fraction of the far-infrared emission comes from dust heating from massive stars and quasar heating on kpc scales \\citep{Symeonidis17,Symeonidis21}. In radio-loud objects synchrotron emission from the jets and the core of the quasar can also contribute to the far-infrared and mm-emission. In \\citet{Podigachoski15} correction for synchrotron emission was only applied to the 850 \\micron\\ flux of a handful of objects, including 3C 298 that is part of our study. We therefore use the infrared-derived star-formation rates as an upper limit on the actual star formation rates. The total infrared star formation rates are presented in Table \\ref{tab:SFR-rates}. \\\\\n\nAnother tracer of recent star formation comes from recombination lines of hydrogen. In Table \\ref{tab:SFR-rates} we present the star formation rates derived from the H$\\alpha$\\xspace line luminosity using the \\citep{Kennicutt98} calibration. The H$\\alpha$\\xspace derived star formation rates are derived from integrated H$\\alpha$\\xspace emission at the systemic redshift of the quasar host galaxy. The H$\\alpha$\\xspace and total infrared derived star formation rates show a significant discrepancy, with the far-infrared star formation rates being almost an order of magnitude higher. The discrepancy between these inferred star formation rates may be due to several reasons; dust extinction, different tracers of the star formation history, contamination of the far-infrared emission from quasar processes, or in addition a large difference between the resolution of the \\textit{Herschel Space Telescope} and Keck\/OSIRIS observations. Indeed, the far-infrared derived star formation rate in 4C 09.17 can be attributed to several galaxies falling within \\textit{Herschel Space Telescope} PSF leading to contamination of the far-infrared emission. The merging galaxy 4C 09.17 B-RQ contains higher molecular gas surface density, hence the larger fraction of the star formation rate among the galaxies. Contribution from nearby galaxies can also affect the far-infrared derived flux for 3C 298 and 3C 318, but we do not see such a strong over-density compared to 4C 09.17. Recent observations with ALMA of a high redshift ($z=4.4$) and luminous quasar have revealed multiple sources falling within 17 kpc from the quasar \\citep{Bischetti18}, which all contribute to the far-infrared emission detected with \\textit{Herschel Space Telescope} for this system. \n\nWe find that stellar processes can deposit a momentum flux of 7.5 - 10,000 $\\times10^{33}$ dyne, comparable to momentum fluxes of the multi-phase outflows. However, since these estimates rely on the maximum momentum flux from SNe and the maximum star formation rates, it is unlikely that star formation alone drives the outflows in these systems.\n\n\\begin{deluxetable*}{cccc}\n\\tiny\n\\tablecaption{Star formation rates based on H$\\alpha$\\xspace emission line \\citep{Vayner19b},infrared observations \\citep{Podigachoski15} and expected star formation rate using the molecular gas surface density based on the KS law. \\label{tab:SFR-rates}}\n\\tablehead{\n\\colhead{Name} & \n\\colhead{SFR [H$\\alpha$\\xspace]}&\n\\colhead{SFR [Total IR]}&\n\\colhead{SFR [KS]}\\\\\n\\colhead{} & \n\\colhead{\\myr}&\n\\colhead{\\myr}&\n\\colhead{\\myr}}\n\\startdata\n4C 09.17 A RL &  9$\\pm$1 & 1330 \\tablenotemark{a}  & 6  \\\\\n7C 1354+2552 & \t 29$\\pm$3 & -- & 1 \\\\\n3C 298 &  22$\\pm$2 & 930 & 24 \\\\\n3C 318 &  88$\\pm$9 & 580 & - \\\\\n4C 22.44 &  32$\\pm$3 & -- & - \\\\\n4C 05.84 &  11$\\pm$1 & $<$540 & - \\\\\n\\enddata\n\\tablenotetext{a}{Star formation rate likely contaminated by a merging galaxy}\n\\end{deluxetable*}\n\n\\subsection{Quasar as potential driver of galactic outflows}\nThe photon momentum fluxes of $10^{35-37}$ dyne and bolometric luminosities of $10^{46-47}$ erg s$^{-1}$ \\ for these quasars indicate that they have sufficient energy and momentum to drive the observed multi-phase gas outflows. To test which quasar mechanism is the primary driver of the observed galactic-scale outflows, we compare $\\dot{P}_{outflow}$  to $\\dot{P}_{Quasar}$  and the location and extent of the quasar jets. High ($>2$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$\\ on scales $>$ 1 kpc are generally attributed to energy-conserving outflows, where a radiatively-driven nuclear wind \\citep{Faucher12a} or a jet \\citep{Mukherjee16,Wagner12} drives a hot shock (T$>10^{7}$ K) in the interstellar medium that does not cool efficiently, maintaining nearly all of the kinetic energy provided to it. The swept-up material is shocked and is able to cool to produce a multi-phase outflow medium \\citep{Faucher12,Faucher12a,Zubovas12}, and may explain the presence of molecular gas moving at fast outflow velocities. To decipher whether the galactic-scale wind is ultimately powered by a jet or by radiation, we need to compare the location and extent of the quasar jet to the outflow and to search for evidence of a fast, radiatively driven wind in X-ray and UV spectrum of the quasar.\n\nHigh ($>2$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ on scales $<$ 1 kpc can be attributed to outflows driven by radiation pressure in a high column density environment, where the ``momentum boost\" is provided by photons scattering multiple times off dust grains as they are driving the outflow \\citep{Thompson15}. Detecting and resolving the hot shocked gas produced by the jet or the radiatively-driven nuclear wind would be helpful in understanding what is driving the outflow. The shocked hot gas can be detected through the Sunyaev\u2013Zeldovich effect \\citep{Chatterjee07} and has recently been found in the host galaxy of a luminous quasar at z=1.71 \\citep{Lacy19}. Additionally it was detected by stacking analysis of luminous quasars from the Atacama Cosmology Telescope, \\textit{Herschel Space Telescope} and VLA observations \\citep{Hall19}.\n\nLow ($\\ll1$) $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$\\ on scales $>$ 1 kpc can be attributed to outflows driven by a radiative shock produced where the shocked gas can cool efficiently, and kinetic energy is radiated away. Outflows driven through radiation pressure in a low column density environment can also produce an observed $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ $\\ll$1. \n\nTo compare the energetics of winds of our sample with other massive galaxies with AGN, we collated data on molecular and ionized outflows in galaxies at $1<z<3$. We recomputed the ionized outflow rates, luminosities, and momentum fluxes using a consistent prescription; details can be found in \\citet{Vayner19a}. To differentiate AGN outflow mechanisms we compare the momentum flux of the outflow against the photon momentum flux from the accretion disk, as well as and the kinetic luminosity of the outflow against the bolometric luminosity of the quasar (Figure \\ref{fig:energetics}). On Figure \\ref{fig:energetics} we overlay theoretical predictions of the expected $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ value based on energy-conserving outflows, radiative shock driving, and radiation pressure \\citep{Faucher12,Zubovas12,Zubovas14}. Minimum coupling efficiency between the quasar bolometric luminosity and outflow kinetic luminosity are prescribed in theoretical simulations when outflows directly impact the star-forming properties of the host galaxy. Based on theoretical models, minimum coupling efficiencies are presented in Figure \\ref{fig:energetics} \\citep{Hopkins10,Choi12,Costa18}. We present the molecular outflows and the total energetics from the multi-phase outflow in Figure \\ref{fig:energetics}. After combining the ionized and molecular outflows' energetics, the outflows exhibit a $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$ $>1$ on kpc scales. An energy-conserving shock is responsible for driving the outflow for 3C 318, 4C 09.17 A, and 3C 298. For 4C 05.84, it is still possible for an energy-conserving shock to drive the outflow; however, radiation pressure or a radiative shock model is still able to explain the observed $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$. We overlay the location of the ionized and molecular outflows in Figure \\ref{fig:CO_ionized_outflows}, to highlight the extent and morphologies of the multi-phase gas outflows.\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0 in]{energetics_comparison.eps}\n    \\caption{A comparison of the momentum flux (left) and kinetic luminosity (right) of the outflow to the momentum flux of the accretion disk and the quasar's bolometric luminosity. On the left, we plot lines of constant $\\frac{\\dot{P}_{outflow}}{\\dot{P}_{Quasar}}$, points above the 2:1 line represent outflows that are likely driven by an energy-conserving shock on kpc scale or radiation pressure on small ($<1$ kpc) scales. On the right, we plot lines of constant coupling efficiency between the outflow's kinetic luminosity and the quasar's bolometric luminosity. Blue circles and squares represent molecular outflows detected through CO emission and OH absorption, respectively. Red circles represent ionized outflows at cosmic noon, recomputed in the exact same manner in \\citep{Vayner19a}. Black stars are the ionized outflows of the parent sample for this study. Blue stars are molecular outflows derived in this study through CO emission, and orange stars are the energetics from the total (molecular + ionized) momentum flux and kinetic luminosity. Combining the energetics of the multi-phase outflows indicates that they are likely driven by an energy-conserving shock and have coupling efficiencies between 0.1-1\\%.}\n    \\label{fig:energetics}\n\\end{figure*}\n\n\\subsection{The nature of the multi-phase outflows}\n\nFor 3C 318, 4C 05.84, and 4C 09.17 A, compared to the ionized outflowing gas the molecular gas shows a smaller velocity range single component that spans only blueshifted emission from -200 to -1200 \\kms. The morphology of the outflowing molecular gas is much clumpier and more confined compared to the ionized outflows. The ionized outflows span a broader range of velocities and fill a larger volume likely residing in a wide-angle cone. The high velocity and clumpy molecular outflows are unexpected, appearing to be out of pressure equilibrium with their surroundings at the larger separations observed in 3C 318 and 4C 05.84. Recently high velocity and clumpy outflows have also been detected in one low redshift radio-quiet quasar and a higher redshift lensed quasar, appearing to have similar velocity offset, dispersion, and morphology to 3C 318, 4C 05.84 and 4C09.17A \\citep{Bischetti19,Chartas20}. There is no consensus on the observed spatial morphology and velocity structure of high redshift molecular outflows in luminous quasars due to the small number of detections. The velocity dispersions and offsets for 3C 318, 4C 05.84, and 4C 09.17 A are consistent with molecular outflows detected through OH absorption \\citep{Veilleux13}, which have been hypothesized to have a thin shell geometry. For 3C 298 and 4C 09.17 B, the molecular outflows show a broader velocity range, which likely indicates that these molecular outflows are more volume filling, similar to the ionized outflows. In case these outflows are indeed filled cones, then their outflow rates and energetics would scale by a factor of 3. We are unable to fully rule out a shell geometry for these outflows. It would require higher angular resolution observations to distinguish if they reside in a shell or in a filled cone\/sphere. The velocities of the molecular and ionized outflows are consistent with recent theoretical predictions by \\citet{Richings20} for multi-phase gas outflows driven by a quasar. \\\\\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=7.0in]{all_sources_ALMA_OSIRIS_overlay.eps}\n    \\caption{We present the comparison between the ionized outflow and the molecular gas distribution, extent, and morphology in sources where we detect a multi-phase gas outflow. In the background, we show the line integrated CO intensity tracing the cold molecular gas. Violet contours outline the location of the molecular outflows in the quasar host galaxies, the teal contour in the 4C09.17 system represent the molecular outflow in the merging galaxy. The white contours show the ionized outflow traced through the [O{\\sc III}]\\xspace\\ 5007 \\AA\\ emission line for all systems except 4C 05.84 where it is traced with H$\\alpha$\\xspace. The white star represents the quasar's location, while the bar to the right of each source represents 1 arcsecond.}\n    \\label{fig:CO_ionized_outflows}\n\\end{figure*}\n\nThe energy-conserving shocks responsible for the galactic scale outflows are either powered by radiatively driven outflow or through shock heating of the gas by the radio jet. In 3C 298 and 3C 318, the jet's path is consistent with both the ionized and molecular outflows, while in 4C 05.84, the jet is only consistent with the path of the ionized outflow; however, the size of the jet is still consistent with both the extent of the ionized and molecular gas outflows. In 4C 09.17, the jet is not consistent with either the molecular or ionized outflow path. The lack of correlation between the jet and the galactic outflows does not indicate that the jet could not have driven the outflow. At later stages in their evolution, the hot bubble cocoons shocked by the jet could be relatively spherical and volume filling relative to the thin jet that we observe at radio wavelength \\citep{Wagner12,Mukherjee16}. The lack of path correlation between the multiple phases of the outflow, the general asymmetry, and spatial differences in ionized and molecular outflow can be caused by the non-uniform cooling properties of the galactic outflow, subject to gas and dust extinction and surface brightness sensitivity of our observations. Some nearby galaxies show fast-moving outflows that are not correlated with the axis of the jet \\citep{Venturi21} but are rather found to expand perpendicular to the jet's path. None of the quasars appear to show evidence for a broad absorption line wind, based on their rest-frame UV spectra from SDSS \\citep{Paris18}. The X-ray observations are either missing or are too shallow to search for radiatively-driven nuclear winds in Fe absorption lines. Future space-based X-ray telescopes with larger effective apertures and higher spectral resolving power will be crucial to search for ultra-fast outflows and compare their energetics with the galactic scale winds to decipher whether the jet or quasar disk winds are the primary driver.\n\nAt the current evolutionary stage of the quasar host galaxies in our study, the dominant outflow component, in terms of mass, is the molecular gas. In fact, for sources with both a molecular outflow and a molecular gas reservoir at the systemic redshift, between 20-60\\% of the cold molecular gas reservoir is in the galactic outflow. For sources with no detected molecular reservoir at the systemic redshift, we are catching these systems when the majority of the cold molecular gas is in an outflow. This suggests that we are observing these systems at a phase where the quasar is responsible for removing a significant fraction of the gas in these galaxies, and therefore may be directly responsible for removing the fuel for subsequent star formation, in turn impacting the stellar growth of these galaxies.\n\nFor the galactic outflows in 3C 298 and 4C 05.84, we find that most of the momentum flux is in the ionized gas phase. For 3C 318, the ionized outflow can have higher energetics within the errors that are dominated by the measurement of the electron density. For 4C 09.17 A, the molecular outflow has the higher energetics. In 4C 09.17 B, we do not detect an ionized outflow, likely due to obscuration effects since we measure a high concentration of molecular gas with a high $\\rm A_{V}$ value near the center of galaxy and 4C 09.17 B galaxy also has a red $V-K$ color of $>$5.35. Hence it is unclear how the energetics are distributed between the ionized and molecular gas phase of the outflow. For most of our objects, the distribution in energetics is consistent with recent theoretical predictions by \\citet{Richings20}. The coupling efficiency between the multi-phase outflow and the bolometric luminosity of the quasar is consistent with theoretical predictions by \\citet{Hopkins10,Choi12,Costa18}, where a minimum of 0.1-0.5$\\%$ of the quasar's bolometric luminosity is expected to transfer into the kinetic luminosity of the outflow for there to impact star formation processes.\n\n\\subsection{Missing mass in galactic outflows}\n\nWe do not have any measurement of the neutral atomic gas mass in the outflow. Yet the neutral atomic phase likely exists within each outflow since we observe ionized emission through the optical [O{\\sc III}]\\xspace and H$\\alpha$\\xspace lines. Theoretical work by \\citet{Dempsey18} suggests that in the absence of a substantial amount of neutral gas, the [O{\\sc III}]\\xspace transition would not be present, and most of the gas would be over-ionized (e.g., into [O IV]). At the same time, theoretical work by \\citet{Richings18} has shown that a substantial amount of the molecular gas in an outflow is expected to be in the warmer molecular gas phase, with temperatures on the order of 400 K. The CO transition that trace the molecular gas reservoir within our study are only capable of tracing the cold molecular gas phase at temperatures of 40-70 K. The actual total momentum flux ratios are still likely lower limits, so energy-conserving outflows driven through a quasar mechanism are the most likely scenario for explaining the driving mechanism behind these galactic-scale outflows. Future observations with the MIRI instrument aboard \\textit{JWST} will trace the warm molecular gas phase through the rest-frame mid-infrared rotational transitions of hydrogen. Observations with future 30-meter class telescopes will be able to probe the neutral gas phase through the Na D, [OI] 6300 \\AA\\ lines. Furthermore, the Square Kilometer Array (SKA) will enable us to probe the neutral gas phase directly through the 21 cm hydrogen line. \n\n\\subsection{Molecular gas depletion time scales}\n\nThis section we explore the molecular gas depletion time scale in the galaxies within our sample. The molecular gas depletion time scale is defined as $t_{depletion, SFR} = M_{molecular}\/SFR$. We will also compare the star formation depletion scale to the outflow depletion scale defined in a similar manner; $t_{depletion, outflow} = M_{molecular}\/\\dot{M}_{outflow}$. We compare the depletion time scale of the molecular gas due to star formation and from both the ionized and the molecular gas outflows. In all cases, we find a molecular gas depletion time scale of $<$ 31 Myr, with $t_{depletion, outflow}$ having the shorter timescale. The infrared-derived star formation rates are two orders of magnitude too high to be supported by the current molecular surface gas density, suggesting that the IR emission is likely contaminated by the quasar. While the maximum derived star formation rate from the total-infrared emission can be comparable to the multi-phase gas outflow rates, at present time, star formation rates expected from the Kennicutt-Schmidt (KS) law are two orders of magnitude smaller. The gas depletion timescale due to the KS-derived star formation rates is about two orders of magnitude higher than the depletion time scale due to outflows. Due to the expected low star formation rate from the low molecular gas surface density, in the next few Myr, the depletion time scale will be dominated by the galactic scale outflows. The depletion time scales from each source for the systems in our survey are presented in Table \\ref{tab:depletion}. Not only are we catching these systems when a substantial fraction of the gas is in an outflow state, the rate of molecular gas depletion is dominated by the quasar driven outflows.\n\n\\begin{deluxetable*}{ccccc}\n\\tablecaption{Measured depletion time scales based on the star formation activity and multi-phase outflows in our sample. t$_{depletion,SFR}$ is the depletion time scale of the current molecular reservoir using the highest star formation rate observed. t$_{depletion,outflow}$ is the depletion time scale due to the outflows, and t$_{depletion,KS}$ is the depletion time scale based on the KS law for the observed molecular gas surface density. t$_{depletion,MS}$ is the expected depletion time scale for a galaxy at the measured stellar mass on the galaxy main sequence. \\label{tab:depletion}}\n\n\n\n\\tablehead{\n\\colhead{Source}&\n\\colhead{t$_{depletion,SFR}$}&\n\\colhead{t$_{depletion,outflow}$}&\n\\colhead{t$_{depletion,KS}$} &\n\\colhead{t$_{depletion,MS}$}\\\\\n\\colhead{}&\n\\colhead{Myr}&\n\\colhead{Myr}&\n\\colhead{Myr}&\n\\colhead{Myr}}\n\\startdata\n4C 09.17 A RL &  & 6$\\pm$1 & 500$\\pm$50 & 700 \\\\\n4C 09.17 B RQ &  & 4$\\pm$1 & 178$\\pm$18 & 700 \\\\\n7C 1354 & 10$\\pm$4 & 6$\\pm$3 & 300$\\pm$100& 700 \\\\\n3C 298 & 7$\\pm$1 & 4$\\pm$1 & 275$\\pm$40 & 700 \\\\\n3C 318 & $<$ 1 & $<$ 1.7$\\pm$0.6 & & 800 \\\\\n4C 22.44 & $<$ 31$\\pm$3 & $<$3$\\pm$2 & & 800 \\\\\n4C 05.84 & $<$1.5 & $<$0.8$\\pm$0.5 & & 700 \\\\\n\\enddata\n\\end{deluxetable*}\n\nIn the last several years, there have been many molecular gas observations in massive galaxies near cosmic noon. We can compare the depletion time scales that we observe within our systems to those that do not have powerful quasars. We compare the expected depletion time scale due to star formation for an average galaxy in our stellar mass range to the observed depletion time scale due to the outflows. \\citet{Tacconi18} measured the depletion time scale for galaxies at an extensive range of redshifts and stellar masses. The dynamical mass \\citep{Vayner19b} of the galaxies within our sample range from $10^{10.5-11.5}$ M$_{\\odot}$, with star formation rates of 30-1330 \\myr. This places them on the massive end of the galaxy luminosity function and the galaxy main sequence (MS) at $z=2$ that relates the star formation rate of a galaxy to its stellar mass. For galaxies with a mass in the range of $10^{10.5-11.5}$ M$_{\\odot}$, the expected depletion time scale is around 700-800 Myr for galaxies on the star-formation main sequence at $z\\sim2$. The range in the molecular gas depletion time scale for galaxies in the mass range of $10^{10.5-11.5}$ M$_{\\odot}$\\ is $0.3-3.5$ Gyr. Our systems appear to show a depletion time scale of the molecular reservoir that is least 100 times faster, indicating that powerful quasars can play a role in having shorter depletion times compared to star formation in a typical massive galaxy. While the H$\\alpha$\\xspace derived star formation rates place the sample quasar hosts on or below the star forming main sequence at z$\\sim$2, the much larger FIR-derived star formation rates place them well above the MS.  We therefore estimate a range of depletion times in Table \\ref{tab:depletion} covering the range of estimated star formation rates for each source.\n\nThe rapid depletion time scales of the cold molecular gas reservoir and the present amount of molecular gas has a profound implication for the evolution of these galaxies from cosmic noon to the present day. From the Keck\/OSIRIS observations, we learned that these galaxies are offset from the local $M_{\\bullet}-\\sigma~$ and $M_{\\bullet}-M_{*}~$ relationships \\citep{Vayner19b}, indicating that the galaxies are under-massive for the mass of their SMBH. We estimated that they require a continuous stellar mass growth of approximately 100 \\myr\\ between z=2 and z=0 to land onto the local scaling relations. For each system, we have computed the molecular gas at the systemic redshift of the quasar within the radius where the dynamical masses are calculated in \\cite{Vayner19b}. We find molecular gas mass in the range of $0.3-10\\times10^{9}$ M$_{\\odot}$, while in systems without detection of CO emission at the systemic redshift, we have placed limits of $<1\\times10^{9}$ M$_{\\odot}$. None of the systems have the molecular gas mass necessary to increase their stellar mass to bring them close to the local scaling relations between the SMBH mass and the stellar mass of the galaxy.\n\nFurthermore, the majority of the molecular gas is in the galactic outflows rather than in the host galaxy; hence the overall stellar mass is still not going to increase by a substantial amount through star-formation. This study further indicates that a large new fresh reservoir of gas is necessary to accrete from the circumgalactic medium to replenish the fuel necessary for future star formation, and\/or a large number of mergers is expected between z=2 and z=0 \\citep{Burke13,Cooke19}. Our results further indicate that strong quasar feedback occurs before galaxies assemble onto the local scaling relations.\n\n\n\\section{Conclusions}\\label{sec:conc}\n\nWe have conducted a study of the molecular gas properties in 6 radio-loud quasar host galaxies at $1.4<z<2.3$ through ALMA band 4 observation of the CO (3-2) and CO (4-3) rotational transitions at sub-arcsecond resolution. The survey aimed to study the molecular gas morphologies and kinematics to address the energetics and evolution of these massive galaxies. We resolve the molecular gas reservoirs in 5 galaxies and detect molecular outflows in 4 systems. Coupling these observations with high spatial resolution observations with Keck\/OSIRIS we are able to study the ionized gas and molecular gas to gauge the impact of quasar activity on the star formation properties of the host galaxies.\n\n\\begin{itemize}\n    \\item We detect CO emission on kpc scale in 5\/6 quasar host galaxies.\n    \\item We detect high-velocity clumps in 3C 318, 4C 09.17A, and 4C 05.84, offset up to -1200 km\/s while in 3C 298 and 4C 09.17B we detect broad CO emission lines. We interpret these spectral features as molecular outflows. We derive outflow rates of 168-2500 \\myr, combining the energetics of the ionized to the molecular gas outflows, we find that the dominant outflow mechanism is through an energy-conserving shock from the quasar jets.\n    \\item We find that the outflowing gas to be predominantly in the molecular gas phase while the momentum flux and kinetic luminosity resides in the ionized gas phase for 3\/4 multi gas-phase outflows.\n    \\item While the maximum star-formation rate from total infrared luminosity show a similar depletion time scales to the outflows, due to a much lower molecular gas surface densities than expected, the outflow depletion time scales are at least 50 times faster. The depletion time scale compared to the expected value for an average galaxy on the galaxy star formation main sequence at the stellar mass of our survey is about 100 times faster.\n    \\item A substantial accretion of fresh gas from the circumgalactic medium, and\/or stellar growth through dry mergers is necessary for these galaxies to achieve the expected mass based on their central SMBH mass using local scaling relations. The current molecular gas reservoir is insufficient to provide a significant amount of fuel for star formation.\n    \\item In the 4C 09.17 system, we find a molecular gas outflow in its quasar host galaxy, as well as a nearby dust-obscured galaxy, indicating evidence for feedback occurring well before the coalescence phase of this merger system.\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nThe formation process of low- to intermediate-mass stars is divided into several \nstages, ranging from the deeply embedded protostellar stage to the period when \na young star is dispersing its protoplanetary disk in which planets may have formed.\nDuring the protostellar phase, which is estimated to last $\\sim$ 0.5 Myr \n\\citep{evans09,dunham14}, the growing central source accretes dust and gas \nfrom a collapsing envelope. The material from the envelope is most likely accreted \nthrough a disk, feeding the growing star. A fraction of the mass is ejected in outflows, \nwhich carve openings into the envelope along the outflow axis. \nDespite our understanding of the basic processes operating in low-mass protostars, \nfundamental questions remain \\citep[e.g.,][]{dunham14}. In particular, it is not \nunderstood how the processes of infall, feedback from outflows, disk accretion, \nas well as the surrounding birth environment, affect mass accretion and determine the \nultimate stellar mass. The luminosity of protostars, which can be dominated by \naccretion, is observed to span more than three orders of magnitude, yet the underlying \nphysics of this luminosity range is also not understood \\citep{dunham10,offner11}. \nIt is in this protostellar phase that disks are formed, setting the stage for planet \nformation, yet how infall, feedback, accretion, and environment influence the \nproperties of disks and of planets that eventually form from them is unknown. \nThe large samples of well-characterized protostars identified from surveys with \n{\\it Spitzer} and {\\it Herschel} now provide the means to systematically study the \nprocesses controlling the formation of stars and disks; the goal of this work is to\nprovide such a characterization for the protostars found in the Orion A and B \nclouds, the largest population of protostars for any of the molecular clouds within \n500 pc of the Sun \\citep{kryukova12,dunham13,dunham15}.\n  \nIn protostars, dust in the disk and envelope reprocesses the shorter-wavelength \nradiation emitted by the central protostar and the accretion shock on the stellar surface \nand reemits it prominently at mid- to far-infrared wavelengths. As a result, the combined \nemission of most protostellar systems (consisting of protostar, disk, and envelope) peaks \nin the far-IR. \nYoung, deeply embedded protostars have spectral energy distributions (SEDs) \nwith steeply rising slopes in the infrared, peaking around 100 \\micron, and large \nfractional submillimeter luminosities \\citep[e.g.,][]{enoch09,stutz13}. Near 10 \n\\micron\\ and 18 \\micron, absorption by sub-micron-sized silicate grains causes broad \nabsorption features; in addition, there are several ice absorption features across \nthe infrared spectral range \\citep{boogert08, pontoppidan08}. These absorption \nfeatures are indicative of the amount of material along the line of sight, with the \ndeepest features found for the most embedded objects. In addition, due to the\nasymmetric radiation field, the orientation of a protostellar system to the line of \nsight, whether through a dense disk or a low-density cavity, plays a role in the\nappearance of the SED. It influences the near- to far-IR slope, the depth of the \nsilicate feature, the emission peak, and the fraction of light emitted at the longest \nwavelengths \\citep[see, e.g.,][]{whitney03a}. \n \nTo classify young stellar objects (YSOs) into observational classes, the near- to \nmid-infrared spectral index $n$ ($\\lambda F_{\\lambda} \\propto \\lambda^n$) \nfrom about 2 to 20 $\\mu$m has traditionally been used \\citep{lada87,adams87,\nandre94,evans09,dunham14}. This index is positive for a Class 0\/I protostar, \nbetween $-0.3$ and 0.3 for a flat-spectrum source, and between $-1.6$ and $\n-0.3$ for a Class II pre-main-sequence star. Class 0 protostars are distinguished \nfrom Class I protostars as having $L_{submm}\/L_{bol}$ ratios larger than 0.5\\%, \naccording to the original definition by \\citet{andre93}. Other values for this threshold \nthat have recently been used are 1\\% \\citep{sadavoy14} and even 3\\% \\citep{maury11}.\nAnother measure for the evolution of a young star is the bolometric temperature \n($T_{bol}$), which is the temperature of a blackbody with the same flux-weighted \nmean frequency as the observed SED \\citep{myers93}. A Class 0 protostar has \n$T_{bol}<70$ K, a Class I protostar 70 K $<T_{bol}<$ 650 K, and a Class II \npre-main-sequence star 650 K $<T_{bol}<$ 2800 K \\citep{chen95}. \nThese observational classes are inferred to reflect evolutionary stages, with the \ninclination angle to the line of sight being the major source of uncertainty in \ntranslating classes to ``stages'' \\citep{robitaille06,evans09}. Also the accretion\nhistory, which likely includes episodic accretion events and thus temporary \nincreases in luminosity, adds to this uncertainty \\citep{dunham10,dunham12}.\nProtostars with infalling envelopes of gas and dust correspond to Stages 0 and I, \nwith the transition from Stage 0 to I occurring when the stellar mass becomes \nlarger than the envelope mass \\citep{dunham14}. Young stars that have dispersed \ntheir envelopes and are surrounded by circumstellar disks correspond to Stage II. \n\nBy modeling the SEDs of protostars, properties of their envelopes, and to some\nextent of their disks, can be constrained. The near-IR is particularly sensitive to \nextinction and thus constrains the inclination angle and cavity opening angle, \nas well as the envelope density.  Mid-IR spectroscopy reveals the detailed emission \naround the silicate absorption feature and thus provides additional constraints for \nboth disk and envelope properties \\citep[see, e.g.,][]{furlan08}. At longer wavelengths, \nenvelope emission starts to dominate. Thus, photometry in the far-IR is necessary \nto determine the peak of the SED and constrain the total luminosity and envelope \nproperties.\n\nHere we present 1.2--870 $\\mu$m SEDs and radiative transfer model fits\nof 330 YSOs, most of them protostars, in the Orion star formation complex. \nThis is the largest sample of protostars studied in a single, nearby star-forming \nregion (distance of 420 pc; \\citealt{menten07,kim08}) and therefore significant \nfor advancing our understanding of protostellar structure and evolution.\nThese protostars were identified in {\\it Spitzer Space Telescope} \\citep{werner04} \ndata by \\citet{megeath12} and were observed at 70 and 160 \\micron\\ with \nthe Photodetector Array Camera and Spectrometer \\citep[PACS;][]{poglitsch10} \non the {\\it Herschel Space Observatory}\\footnote{{\\it Herschel} is an ESA \nspace observatory with science instruments provided by European-led \nPrincipal Investigator consortia and with important participation from NASA.}\n\\citep{pilbratt10} as part of the Herschel Orion Protostar Survey (HOPS), a \n{\\it Herschel} open-time key program \\citep[e.g.,][W. J. Fischer et al. 2016, \nin preparation; B. Ali et al. 2016, in preparation]{fischer10, stanke10, \nmanoj13, stutz13}. To extend the SEDs into the sub-mm, most of the\nYSOs were also observed in the continuum at 350 and 870~$\\mu$m with \nthe Atacama Pathfinder Experiment (APEX) telescope \\citep{stutz13}. \nOur sample also includes 16 new protostars identified in PACS data obtained \nby the HOPS program (\\citealt{stutz13,tobin15}; see section \\ref{sample}).\nWe use a grid of 30,400 protostellar model SEDs to find the best fit to the SED for each\nobject and constrain its protostellar properties. As mentioned above, the far-infrared \ndata add crucial constraints for the model fits, given that for most protostars the \nSED peaks in this wavelength region, and therefore, within the framework of the \nmodel grid, our SED fits yield the most reliable protostellar parameters to date \nfor these sources. \n\n\n\\section{Sample Description}\n\\label{sample}\n\nThe 488 protostars identified in {\\it Spitzer} data by \\citet{megeath12}\nrepresent the basis for the HOPS sample\\footnote{The selection of HOPS\ntargets is based on an earlier version of the {\\it Spitzer} Orion Survey,\nand in addition some objects likely in transition between Stages I and II\nwere included; thus, not all protostars in the HOPS sample are classified\nas protostars with envelopes in \\citet{megeath12}.} \\citep[see][]{fischer13,\nmanoj13,stutz13}. They have 3.6-24 $\\mu$m spectral indices $\\geq$ \n$-0.3$ and thus encompass flat-spectrum sources.  \nTo be included in the target list for the PACS observations, the predicted flux \nof a protostar in the 70 $\\mu$m PACS band had to be at least 42 mJy as\nextrapolated from the {\\it Spitzer} SED. Since targets were required to have \na 24 $\\mu$m detection, protostars in the Orion Nebula -- where the {\\it Spitzer} \n24 $\\mu$m data are saturated -- are excluded. In addition, after the PACS data \nwere obtained, several new point sources that were very faint or undetected \nin the {\\it Spitzer} bands were discovered in the {\\it Herschel} data \n\\citep{stutz13}. Fifteen of them were found to be reliable new protostars.\nOne more protostar, which was not included in the sample of \\citet{stutz13} due \nto its more spatially extended appearance at 70 $\\mu$m, was recently confirmed \nby \\citet{tobin15}. We have added these 16 protostars to the HOPS sample for \nthis work (see Table D\\ref{New_proto} in the Appendix). Most of these new \nprotostars have very red colors and are thus potentially the youngest protostars \nidentified in Orion \\citep[see][]{stutz13}. \n\nEach object in the target list was assigned a ``HOPS'' identification number, resulting\nin 410 objects with such numbers; HOPS 394 to 408 are the new protostars\nidentified by \\citet{stutz13}, and HOPS 409 is the new protostar from \\citet{tobin15}.\nFour of the 410 HOPS targets turned out to be duplicates, and 31 are likely \nextragalactic contaminants (see Appendix \\ref{exgal_not_modeled} for details). \nSome objects in the HOPS target list were not observed by PACS; of these \n33 objects, 16 are likely contaminants, while the remaining objects were \noriginally proposed but were not observed since they were too faint to have \nbeen detected with PACS in the awarded observing time. In addition, \n35 HOPS targets were not detected at 70 \\micron\\ (see Appendices\n\\ref{YSOs_not_modeled} and \\ref{exgal_not_modeled}); eight of these are \nconsidered extragalactic contaminants, while two of them (HOPS 349 and 381) \nhave only two measured flux values each, making their nature more uncertain. \nOne more target, HOPS 350, also has just two measured flux values (at 24 and \n70 $\\mu$m) and is therefore also excluded from the analysis of this paper.\nSimilarly, we excluded HOPS 352, since it was only tentatively detected at 24 \n\\micron\\ (it lies on the Airy ring of HOPS 84) and in none of the other data sets.\n\nTo summarize, starting from the sample of 410 HOPS targets, but excluding likely \ncontaminants and objects not observed or detected by PACS, there are 330 \nremaining objects that have {\\it Spitzer} and {\\it Herschel} data and are considered \nprotostars (based on their {\\it Spitzer} classification from \\citealt{megeath12}). \nThey form the sample studied in this work. Their SEDs are presented in the \nnext section, and in later sections we show and discuss the results of SED fits \nfor these targets. Their coordinates, SED properties, and classification, as well \nas their best-fit model parameter values, are listed in Table A\\ref{bestfit}. \nThe 41 likely protostars that lack PACS data (either not observed or not detected) \nare presented in Appendix \\ref{YSOs_not_modeled}.\n    \n\n\\section{Spectral Energy Distributions}\n\\label{SEDs}\n\n\\subsection{Data}\n\n\\begin{splitdeluxetable*}{lcccccccccccBlcccccc}\n\\tablewidth{0.9\\linewidth}\n\\tabletypesize{\\scriptsize}  \n\\tablecaption{SED Data for the HOPS targets\n\\label{SED_data}}\n\\tablehead{\n\\colhead{Object} & \\colhead{$J$ Flux} & \\colhead{$J$ Unc.} & \\colhead{$J$ Flag} &\n\\colhead{\\nodata} &\n\\colhead{[70] Flux} & \\colhead{[70] Unc.} & \\colhead{[70] Flag} &\n\\colhead{\\nodata} &\n\\colhead{[870] Flux} & \\colhead{[870] Unc.} & \\colhead{[870] Flag} &\n\\colhead{Object} & \\colhead{[5.4] Flux} & \\colhead{[5.4] Unc.} & \\colhead{\\nodata} & \n\\colhead{[35] Flux} & \\colhead{[35] Unc.} & \\colhead{IRS scaling}}\n\\startdata\n HOPS 1 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    3.697E+00 &    1.850E-01 &   1 & \\nodata &    6.354E-01 &    1.271E-01 &   2 &   HOPS 1 &  8.631E-03 &    6.069E-04 & \\nodata &    1.185E+00 &    6.460E-02 &  1.17 \\\\\n HOPS 2 &    2.770E-04 &    5.000E-05 &   1 & \\nodata &    5.188E-01 &    2.617E-02 &   1 & \\nodata &    3.865E-01 &    7.730E-02 &   2 &    HOPS 2 &  4.360E-02 &    3.757E-03 & \\nodata &    3.704E-01 &    3.035E-02 &  1.00 \\\\\n HOPS 3 &    2.198E-03 &    8.900E-05 &   1 & \\nodata &    3.187E-01 &    1.622E-02 &   1 & \\nodata &    1.201E-01 &    2.402E-02 &   1 &    HOPS 3 &  4.460E-02 &    7.925E-03 & \\nodata &    4.050E-01 &    2.443E-02 &  1.66 \\\\\n HOPS 4 &    3.820E-04 &    5.300E-05 &   1 & \\nodata &    6.116E-01 &    3.083E-02 &   1 & \\nodata &    1.840E-01 &    3.680E-02 &   2 &    HOPS 4 &  2.055E-02 &    1.240E-03 & \\nodata &    4.943E-01 &    3.484E-02 &  1.00 \\\\\n HOPS 5 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    7.103E-01 &    3.573E-02 &   1 & \\nodata &    6.973E-02 &    1.395E-02 &   2 &    HOPS 5 &  1.475E-02 &    2.695E-03 & \\nodata &    3.077E-01 &    4.484E-03 &  1.00 \\\\\n HOPS 6 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    9.110E-02 &    5.523E-03 &   1 & \\nodata &    2.311E-01 &    4.622E-02 &   2 &    HOPS 6 &  1.271E-03 &    3.935E-04 & \\nodata &    5.350E-02 &    5.244E-03 &  1.00 \\\\\n HOPS 7 &    0.000E+00 &    0.000E+00 &   3 & \\nodata &    1.342E+00 &    6.728E-02 &   1 & \\nodata &    3.577E-01 &    7.154E-02 &   2 &    HOPS 7 &  6.459E-04 &    3.090E-04 & \\nodata &    3.258E-01 &    2.114E-02 &  1.00 \\\\\n \\enddata\n\\tablecomments{\nEach object has up to 13 photometric data points and 16 IRS data points\n(see Section \\ref{method}). Here we only show some of the data points for\na few HOPS targets. For each measurement, we provide the measured flux\nin Jy, its uncertainty (also in Jy) and, for the photometry only, a flag value \n(0---not observed, 1---measured, 2---upper limit, 3---not detected). For those \nHOPS targets with IRS spectra, we also provide the scaling factor that was \napplied to all IRS fluxes in each spectrum to bring them in agreement \nwith the IRAC and MIPS fluxes (see Section \\ref{method} for details). \\\\\nTo convert the 2MASS magnitudes and the {\\it Spizter} magnitudes from\n\\citet{megeath12} to fluxes, we used the following zero points: 1594 Jy for\n$J$, 1024 Jy for $H$, 666.7 Jy for $K_s$, 280.9 Jy for [3.6], 179.7 Jy for\n[4.5], 115.0 Jy for [5.8], 64.1 Jy for [8], and 7.17 Jy for [24] (2MASS: \n\\citealt{cohen03}; IRAC: \\citealt{reach05}; MIPS: \\citealt{engelbracht07}). \\\\\n{\\it This table is published in its entirety in the machine-readable format on \nthe journal website. A portion is shown here for guidance regarding its \nform and content.}}\n\\end{splitdeluxetable*}\n\nIn order to construct SEDs for our sample of 330 YSOs, we combined \nour own observations with data from the literature and existing catalogs. \nFor the near-infrared photometry, we used $J$, $H$, and $K_s$ data from \nthe Two Micron All Sky Survey \\citep[2MASS;][]{skrutskie06}. For the \nmid-infrared spectral region, we used {\\it Spitzer} data from \\citet{kryukova12}\nand \\citet{megeath12}: the Infrared Array Camera \\citep[IRAC;][]{fazio04} \nprovided 3.6, 4.5, 5.8, and 8.0 \\micron\\ photometry, while the Multiband \nImaging Photometer for Spitzer \\citep[MIPS;][]{rieke04} provided 24 \\micron\\ \nphotometry. In addition, most of the YSOs in the HOPS sample \nwere also observed with the Infrared Spectrograph \\citep[IRS;][]{houck04} \non {\\it Spitzer} using the Short-Low (SL; 5.2-14 \\micron) and Long-Low \n(LL; 14-38 \\micron) modules, both with a spectral resolution of about 90\n(see, e.g., \\citet{kim16} for a description of IRS data reduction). \n{\\it Herschel} PACS data at 70, 100, and 160 \\micron\\ yielded far-infrared \nphotometric data points (B. Ali et al. 2016, in preparation; the 100 $\\mu$m \ndata are from the Gould Belt Survey; e.g., \\citealt{andre10}). Most YSOs \nwere also observed at 350 and 870 \\micron\\ (see \\citealt{stutz13}) by the \nAPEX telescope using the SABOCA and LABOCA instruments \n\\citep[][respectively]{siringo10, siringo09}. Thus, our SEDs have well-sampled\nwavelength coverage from 1.2 to 870 $\\mu$m; we did not include additional \ndata from the literature in order to preserve a homogeneous data set for \nall the objects in our sample.\n\nThe aperture radius used for the photometry varies depending on the\ninstrument and wave band. The photometry in the 2MASS catalog was\nderived from point-spread function (PSF) fits using data from 4\\arcsec\\ \napertures around each object (see the Explanatory Supplement to the 2MASS \nAll Sky Data Release and Extended Mission Products). \\citet{megeath12} used \nan aperture radius of 2{\\farcs}4 for IRAC and PSF photometry for MIPS\n24 $\\mu$m data. We used aperture radii of 9{\\farcs}6 and sky annuli \nof 9{\\farcs}6-19{\\farcs}2 for PACS 70 and 100 \\micron\\ images; we then applied \naperture correction factors of 0.7331 and 0.6944 to the 70 and 100 \\micron\\ fluxes,\nrespectively. For  PACS 160 \\micron, we used an aperture radius of \n12{\\farcs}8, a sky annulus of 12{\\farcs}8-25{\\farcs}6, and an aperture correction\nfactor of 0.6602. In some cases (background contamination, close companions)\nwe used PSF photometry at 70 and 160 $\\mu$m instead (see B. Ali et al. 2016, \nin preparation, for details). Finally, we adopted beam fluxes at 350 and 870 \n$\\mu$m (with FWHMs of 7{\\farcs}34 and 19\\arcsec, respectively).\nThe IRS SL module has a slit width of 3{\\farcs}6, while the LL module\nis wider, with a slit width of 10{\\farcs}5. Sometimes the flux level of the\ntwo segments did not match at 14 $\\mu$m (due to slight mispointings or \nmore extended emission from surrounding material measured in LL), \nand in these cases usually the SL spectrum was scaled by at most a \nfactor of $\\sim$ 1.4 (typically 1.1-1.2). In a few cases, especially when \nthe LL spectrum included substantial amounts of extended emission or \nflux from a nearby object, the LL spectrum was scaled down to match the \nflux level of the SL spectrum at 14 $\\mu$m, typically by a factor of 0.8-0.9.\nWe discuss how the different aperture sizes are accounted for in the model\nfluxes in section \\ref{model_ap}.\n\nThe SEDs of our HOPS sample are shown in Figure A\\ref{bestSEDs} together\nwith their best-fit models from our model grid (see sections below); the data\nare listed in Table \\ref{SED_data}.\nMany objects display a deep silicate absorption feature at 10 $\\mu$m and\nice features in the 5-8 $\\mu$m region, as expected for protostars. Those \nobjects with very deep 10 $\\mu$m features and steeply rising SEDs are likely \ndeeply embedded protostars, often seen at high inclination angles. \n\n\n\\subsection{Multiplicity and Variability}\n\nA large fraction (203 out of 330) of the young stars in our sample have at least one\n{\\it Spitzer}-detected source within a radius of 15\\arcsec; in most cases, this\n``companion'' is faint in the infrared and likely a background star or galaxy.\nThus, the emission at far-IR and sub-mm wavelengths is expected to be \ndominated by the protostar or pre-main-sequence star, and we can assume that \nthe SEDs are representative of the YSOs even if the nearby sources cannot be \nseparated at these wavelengths. There are a few YSOs that have objects \nseparated by just 1\\arcsec-3\\arcsec\\ and are only resolved in one or two IRAC \nbands (HOPS 22, 78, 108, 184, 203, 247, 293, 364); in these cases we used \nthe flux at the IRAC position that most closely matched those at longer wavelengths. \nWe note that some of these very close ``companions'' are likely outflow knots.\nThere are also unresolved binaries, which appear as single sources even in the \nIRAC observations \\citep{kounkel16}; in these cases our SEDs \nshow the combined flux in all wave bands. If two point sources are not fully resolved \nand the resulting blended source is elongated, no IRAC photometry was extracted.  \nIn such cases, a protostar may not have IRAC fluxes even though it was detected \nin the {\\it Spitzer} images.\n\nThere are also several protostars that lie close to other protostars:\nHOPS 66 and 370 ($d$=14.9\\arcsec), HOPS 76 and 78 ($d$=14.1\\arcsec),\nHOPS 86 and 87 ($d$=12.1\\arcsec), HOPS 117 and 118 ($d$=13.7\\arcsec),\nHOPS 121 and 123 ($d$=7.6\\arcsec), HOPS 124 and 125 ($d$=9.8\\arcsec),\nHOPS 165 and 203 ($d$=13.3\\arcsec), HOPS 175 and 176 ($d$=8.0\\arcsec), \nHOPS 181 and 182 ($d$=10.2\\arcsec), HOPS 225 and 226 ($d$=9.2\\arcsec), \nHOPS 239 and 241 ($d$=12.4\\arcsec), HOPS 262 and 263 ($d$=6.3\\arcsec), \nHOPS 316 and 358 ($d$=6.9\\arcsec), HOPS 332 and 390 ($d$=11.2\\arcsec), \nHOPS 340 and 341 ($d$=4.7\\arcsec), and HOPS 386 and 387 ($d$=9.9\\arcsec).\nHOPS 105 lies 8.7\\arcsec\\ to the north of an infrared-bright source, identified\nby \\citet{megeath12} as a young star with a protoplanetary disk. This source\nis brighter than HOPS 105 in all {\\it Spitzer} bands and at 70 $\\mu$m, but it is\nwell separated at all wavelengths. A similar situation applies to HOPS 128, which\nhas a disk-dominated source 6.3\\arcsec\\ to the southeast. \nHOPS 108 is 6.6\\arcsec\\ from HOPS 64, which is brighter than HOPS 108 out\nto 8 $\\mu$m, but not detected in the far-IR and sub-mm. HOPS 108 also lies\n16.6\\arcsec\\ from HOPS 369 and 28.2\\arcsec\\ from HOPS 370.\nHOPS 140 has two neighboring sources, at 9.6\\arcsec\\ and 13.9\\arcsec, that\nare likely surrounded by protoplanetary disks; they are both brighter than \nHOPS 140 out to 8 $\\mu$m, but at 70 $\\mu$m and beyond HOPS 140\ndominates.\nHOPS 144 lies 7.9\\arcsec\\ from HOPS 377; there is also a somewhat fainter,\nred source 11.7\\arcsec\\ to the northeast, which is not detected beyond 24 \n$\\mu$m. This source also lies 9.7\\arcsec\\ to the southwest of HOPS 143.\nHOPS 173 forms a small cluster with HOPS 174 (at 7.1\\arcsec) and HOPS 380\n(at 11.4\\arcsec); HOPS 174 is the brightest source out to 24 $\\mu$m, but at\n70 $\\mu$m HOPS 173 takes on this role. \nAlso HOPS 322, 323, and 389 form a group of protostars; HOPS 322 lies\n13.4\\arcsec\\ from HOPS 389 and 20.1\\arcsec\\ from HOPS 323, while\nHOPS 323 and 389 are 10.2\\arcsec\\ apart. HOPS 323 is the brightest source.\n\nThus, there are 45 targets in our sample that have an object within 15\\arcsec\\\nthat is bright in the mid- or far-IR and that is resolved with IRAC and MIPS.\nGiven that \\citet{megeath12} used PSF photometry for the MIPS 24 $\\mu$m \nobservations, they obtained reliable fluxes even for companions separated by \nless than 6\\arcsec, the typical PSF FWHM. For fluxes at 70 and 160 $\\mu$m, \nwe also used PSF photometry for objects that were point sources, but too close \nfor aperture photometry. In cases where the fluxes could not be determined \neven with PSF photometry, we had to adopt upper limits instead. Similarly, \nwe performed PSF photometry on protostars without companions, but \ncontaminated by extended or filamentary emission; if the PSF photometry\ndid not return a good fit, we used the flux value from aperture photometry \nas an upper limit.\n\nSince most of our targets have an IRS spectrum, in addition to data points from\nIRAC at 5.8 and 8 \\micron\\ and from MIPS at 24 \\micron, we can detect\ndiscrepancies if flux values at similar wavelengths, but from different instruments,\ndo not agree. They might be due to calibration or extraction problems in the \nIRS spectrum (for example, some extended emission around the target or a \nclose companion), but also to variability. We assumed the former scenario if \nthe flux deviations between IRS and IRAC and between IRS and MIPS were \nsimilar (and more than 10\\%, a conservative estimate for the typical calibration \nuncertainty), and in such cases scaled the IRS spectrum to the MIPS 24 \n\\micron\\ flux. Even though this scaling could mask actual variability, it \ncreates a representative SED for the YSO and yields an estimate of \nthe protostellar parameters from model fits of the SED.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[angle=90, scale=0.68]{HOPS_n_Tbol_Lbol_histogram.eps}\n\\caption{Histograms of the 4.5-24 $\\mu$m spectral indices ({\\it left}), bolometric \ntemperatures ({\\it middle}), and bolometric luminosities ({\\it right}) for the 330 \nYSOs in our sample.  \n\\label{HOPS_n_Tbol_Lbol_histo}}\n\\end{figure*}\n\nIn Appendix \\ref{variability} we identify potentially variable HOPS targets \nbased on their mid-IR fluxes and find that about 5\\% of the protostars \nwith IRS, IRAC, and MIPS data could be variable. The Young Stellar Object \nVariability (YSOVAR) program, which monitored large samples of protostars \nand pre-main-sequence stars in nearby star-forming regions with {\\it Spitzer} \nat 3.6 and 4.5 $\\mu$m \\citep{morales11, cody14, rebull14, guenther14, \npoppenhaeger15, wolk15, rebull15}, found that up to $\\sim$ 90\\% of flat-spectrum \nand Class I YSOs are variable on a timescale of days, with typical changes in \nbrightness of 10\\%-20\\%. On longer timescales (years as opposed to days), \n20\\%-40\\% of members of young clusters show long-term variability, with the \nhighest fraction for those clusters with more Class I protostars \\citep{rebull14}. \nIn Orion, the fraction of variable Class I protostars is $\\sim$ 85\\% \n\\citep{morales11}. Using a larger sample of protostars in Orion and IRAC \ndata at 3.6, 4.5, 5.8, and 8.0 $\\mu$m, \\citet{megeath12} found that, on a \ntimescale of about 6 months, 60\\%-70\\% of Orion protostars show brightness \nvariabilities of $\\sim$ 20\\%, with some as high as a factor of four.\nThus, given that our SEDs consist of noncontemporaneous data sets, small \nflux discrepancies should be common, but we also expect some protostars \nwith large mismatches.\n\nOne protostar with a large discrepancy between various data sets is HOPS 223.\nIt is an outbursting protostar (also known as V2775 Ori; \\citealt{caratti11}), \nand for its SED we had 2MASS, IRAC, and MIPS data from the pre-outburst \nphase available, while the IRS spectrum, PACS, and APEX data are from the \npost-outburst period. Thus, its SED does not represent an actual state of the \nobject, and the derived $T_{bol}$ and $L_{bol}$ values are unreliable. \nPre- and post-outburst SEDs and model fits for this protostar can be found\nin \\citet{fischer12}.\nHOPS 223 is the only protostar with an SED affected by extreme variability.\nA few more protostars, HOPS 71, 132, 143, 228, 274, and 299, show notable \ndiscrepancy between the IRAC and IRS fluxes, and to a minor extent between \nMIPS 24 $\\mu$m and IRS, and thus have somewhat unreliable SEDs and \nSED-derived parameters. \nHOPS 383, which was identified as an outbursting Class 0 protostar by\n\\citet{safron15}, does not appear variable in the SED presented here,\nsince we adopted post-outburst IRAC 3.6 and 4.5 $\\mu$m fluxes obtained \nby the YSOVAR program \\citep{morales11,rebull14} to construct a \nrepresentative post-outburst SED for this object.\n\n\\vspace{6ex}\n\n\\subsection{Determination of $L_{bol}$, $T_{bol}$, Spectral Index, and\nSED Classification}\n\nThe SEDs provide the means to determine $L_{bol}$, $T_{bol}$, and\nthe 4.5-24 $\\mu$m spectral indices for our sample of protostars. For measuring \nthe near- to mid-IR SED slope ($n= d \\log(\\lambda F_{\\lambda})\/d \\log(\\lambda)$), \nwe chose a spectral index between 4.5 and 24 $\\mu$m to minimize the effect of \nextinction on the short-wavelength data point; also, the IRAC 4.5 $\\mu$m fluxes \nfor our HOPS targets are more complete than the IRAC 3.6 $\\mu$m fluxes due \nto the lower extinction at this wavelength. \nFor calculating $L_{bol}$ and $T_{bol}$, we used all available fluxes for each \nobject, including the IRS spectrum, assumed a distance of 420 pc, and used trapezoidal \nsummation; for $T_{bol}$, we applied the equation from \\citet{myers93}.\nFigure \\ref{HOPS_n_Tbol_Lbol_histo} shows the distribution of $n_{4.5-24}$, \n$T_{bol}$, and $L_{bol}$ values for our targets. There is a peak in the distribution \nof spectral indices around 0, while the distribution of $T_{bol}$ values is relatively \nuniform from about 30 K to 800 K. The bolometric luminosities cover a wide range, \nwith a broad peak around 1 $L_{\\odot}$. The median $L_{bol}$, $T_{bol}$, and\n$n_{4.5-24}$ values are 1.1 $L_{\\odot}$, 146 K, and 0.68, respectively.\n\nOur distribution of $L_{bol}$ values is very similar to the observed luminosity function \nof Orion protostars presented in \\citet{kryukova12}; both distributions peak around\n1 $L_{\\odot}$\\ and include values from $\\sim$ 0.02 $L_{\\odot}$\\ up to several hundred $L_{\\odot}$.\nSome differences between the two distributions are expected, given that\n\\citet{kryukova12} only had {\\it Spitzer} 3.6-24 $\\mu$m data available and thus\nhad to extrapolate $L_{bol}$ from the measured near- to mid-infrared luminosity.\nThe main difference is a somewhat larger number of protostars with $L_{bol} \n\\lesssim$ 0.5 $L_{\\odot}$\\ for the \\citet{kryukova12} Orion sample; our median $L_{bol}$ \nvalue amounts to 1.1 $L_{\\odot}$, while their value is 0.8 $L_{\\odot}$. However, with the \ncontaminating sources removed from their sample (which tend to have lower \nluminosities; see \\citealt{kryukova12} for details), their median bolometric luminosity \nand our value match. \nOverall, given that Orion is considered a region of high-mass star formation, its \nluminosity function is similar to that of other regions where massive star forms \n\\citep{kryukova12,kryukova14}, and it is different from that of low-mass star-forming \nregions such as Taurus and Ophiuchus \\citep{kryukova12, dunham13}. Compared \nto the sample of 230 protostars in 18 different molecular clouds studied by \n\\citet{dunham13}, the observed (i.e., not extinction-corrected) $L_{bol}$ values \nof those protostars span from 0.01 to 69 $L_{\\odot}$, with a median value of 0.9 $L_{\\odot}$. \nHowever, given that almost half the protostars in the \\citet{dunham13} sample lack \nfar-IR and sub-mm data, the true luminosities are likely higher, which would bring the \nmedian closer to the Orion value. Finally, we note that the distribution of observed \n$T_{bol}$ values from \\citet{dunham13} is similar to our distribution for Orion protostars;  \nthe median $T_{bol}$ of their and our sample is 160 K and 146 K, respectively, and the \nbulk of their protostars also has $T_{bol}$ values between 30 K and 1000 K, with\na tail down to temperatures of $\\sim$ 10 K and another tail up to $T_{bol}$=2700~K.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[angle=90, scale=0.41]{HOPS_n_vs_Tbol.eps}\n\\caption{The 4.5-24 $\\mu$m spectral index versus the bolometric temperature\nfor the 330 YSOs in our sample. The dashed lines delineate \nthe regions that define the various SED classes (see text for details).\n\\label{HOPS_n_Tbol}}\n\\end{figure}\n\nTo separate our targets into Class 0, Class I, Class II, and flat-spectrum sources, \nwe used the 4.5 to 24 \\micron\\ spectral index ($n_{4.5-24}$) and\/or bolometric \ntemperature ($T_{bol}$): Class 0 protostars have $n_{4.5-24}>0.3$ and \n$T_{bol}<70$~K, Class I protostars have $n_{4.5-24}>0.3$ and $T_{bol}>70$~K, \nflat-spectrum sources have $-0.3 < n_{4.5-24} < 0.3$, and Class II pre-main-sequence \nstars have $n_{4.5-24}<-0.3$.\nBased on this, we identify 92 targets as Class 0 protostars, 125 as Class I protostars, \n102 as flat-spectrum sources, and 11 as Class II pre-main-sequence stars (see Table\nA\\ref{bestfit} and Figure \\ref{HOPS_n_Tbol}). \nThere are nine protostars with $T_{bol}$ values between 66.5 and 73.5 K (which \ncorresponds to a $\\pm$ 5\\% range around the Class 0--I boundary of 70 K); \nsix of them have $T_{bol}$ $>$ 70 K (HOPS 1, 18, 186, 256, 322, 370), and \nthe other three have $T_{bol}$ values just below 70 K (HOPS 75, 250, 361). \nThese protostars' classification is less firm than for the other HOPS targets. \nThere are also a few flat-spectrum sources whose classification is more \nuncertain: HOPS 45, 183, 192, 194, 210, 264, and 281 should be Class I \nprotostars based on their 4.5-24 $\\mu$m spectral index, but when considering \nthe IRS spectrum (specifically, the 5-25 $\\mu$m spectral index), they fall \ninto the flat-spectrum regime ($n_{5-25} < 0.3$). Also, for HOPS 45 and 194 \nthe $T_{bol}$ values are relatively high ($>$ 500 K).\nSimilarly, HOPS 33, 134, 242, 255, and 284 should be Class II pre-main-sequence\nstars based on their 4.5-24 $\\mu$m spectral index, but the spectral slope over \nthe IRS wavelength range suggests that they are flat-spectrum sources.\nIn these cases where the $n_{4.5-24}$ and $n_{5-25}$ spectral indices were \nsomewhat discrepant, we adopted the latter, and thus these objects were \nclassified as flat-spectrum sources. \n\nThere are five objects with $T_{bol}$ $<$ 70 K and $n_{4.5-24}$ $<$ 0\n(HOPS 164, 340, 341, 373, 405); despite their negative 4.5-24 $\\mu$m SED \nslopes, their SEDs either show or imply a deep silicate absorption feature at \n10 $\\mu$m, rise steeply in the mid- to far-IR, and their long-wavelength emission \nis strong. Thus, their $T_{bol}$ values are low, and we identify them as Class 0 \nprotostars, even though they have 4.5-24 $\\mu$m spectral indices not typical \nof embedded protostars. In particular, HOPS 341, 373, and 405 are likely young \nprotostars with dense envelopes (\\citealt{stutz13}; see also section \\ref{Class0}). \nIn the case of HOPS 373, the 4.5 $\\mu$m flux may be contaminated by bright\nH$_2$ emission from an outflow shock, rendering the $n_{4.5-24}$ value more \nunreliable. This might also explain the negative 4.5-24 $\\mu$m spectral index \nfor the other four protostars.\n\nFinally, the few Class II objects in our sample were thought to be potential\nprotostars prior to their observations with {\\it Herschel}. Their 4.5-24 \\micron\\ \nSED slopes are usually just slightly more negative than the cutoff for a \nflat-spectrum source ($-0.3$); three Class II pre-main-sequence stars \n(HOPS 22, 184, 201) have SEDs that are typical of disks with inner holes, \ndisplaying a 10 $\\mu$m silicate emission feature and a rising SED from \n12 to about 20 \\micron\\ \\citep[e.g.,][]{kim13}. The SEDs of the other\nClass II objects are similar to those of flat-spectrum sources; thus, they could\nhave (remnant) envelopes that contribute to their long-wavelength emission.\n\nOur HOPS sample is mostly complete in the number of Class 0, Class I, and \nflat-spectrum sources in the areas of Orion surveyed by {\\it Spitzer} excluding\nthe Orion Nebula \\citep[see][]{megeath12,stutz13}. Of the 357 unique YSOs \noriginally identified in {\\it Spitzer} data that were included in the HOPS sample and \nobserved with PACS, 322 were detected at least at 70 $\\mu$m, which amounts \nto a fraction of 90\\%. We removed likely contaminants and added 16 new\nprotostars discovered in PACS data to get to our sample of 330 YSOs,\nmost of which are protostars.\nOur lowest $L_{bol}$ source is HOPS 208, with $L_{bol}$= 0.017 $L_{\\odot}$. This \nprotostar also has the lowest PACS 70 $\\mu$m flux in our sample (8.2 mJy). \nOverall, our sample has 27 protostars with $L_{bol}<$ 0.1 $L_{\\odot}$, which places \nthem in the luminosity range of very low luminosity objects \n\\citep[VeLLOs;][]{diFrancesco07,dunham08}. The number of VeLLOs in our \nsample is likely larger, given that VeLLOs are defined as having internal \nluminosities less than 0.1 $L_{\\odot}$, and the bolometric luminosity has contributions \nfrom both the internal luminosity and that due to external heating \n\\citep[see][]{dunham08}. In addition, our sample could miss fainter flat-spectrum \nsources and Class 0 and Class I protostars. In fact, there are several faint YSOs \nwithout PACS data that were excluded from our sample, but do have {\\it Spitzer} \ndetections (see Appendix section \\ref{YSOs_not_modeled}).\n\n\n\\vspace{3ex}\n\n\\section{Model Grid}\n\\label{grid}\n\nTo characterize the SEDs of our HOPS sample in a uniform manner,\nwe fit the data to simple but physically plausible models. In this way we\ncan assess how well such simple models can fit the data, and how the\nquality of the fits changes with evolutionary class. We can also determine\nthe full range of physical parameters implied by the fits and the range of\nparameters for each protostellar class. There are degeneracies and biases \nin the fits, and the uncertainties in model parameters will vary from object \nto object, but our results represent a first step in estimating physical \nparameters that describe the protostars in our sample.\n\nWe use a large model grid calculated using the 2008 version of the\n\\citet{whitney03a,whitney03b} Monte Carlo radiative transfer code \n\\citep[see][]{stutz13}; an early version of the grid was presented in \n\\citet{ali10}. \nEach model consists of a central protostar, a circumstellar disk, and an envelope;\nthe radiation released by the star and the accretion is reprocessed by the\ndisk and envelope. The density in the disk is described by power laws in the \nradial and vertical directions, while the density distribution in the envelope \ncorresponds to that of a rotating, collapsing cloud core with constant infall\nrate (the so-called TSC model, after \\citealt{terebey84}; see also \n\\citealt{ulrich76,cassen81}). The envelope also contains an outflow cavity, \nwhose walls are assumed to follow a polynomial shape. At favorable inclination \nangles, this evacuated cavity allows radiation from the inner envelope and \ndisk regions to reach the observer directly. Also, radiation is scattered off the \ncavity walls and can increase the near-IR emission from a protostellar system.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.49]{Opacities_plot.eps}\n\\caption{Extinction opacities of the \\citet{ormel11} dust model ``icsgra3''\n({\\it black}) compared to other dust opacities from the literature: grains with\nthin ice mantles after 10$^5$ years of coagulation with a gas density of\n10$^6$ cm$^{-3}$ from \\citet{ossenkopf94} ({\\it orange});\ncase A model of carbon and silicate dust for R$_V$=5.5 from \\citet{draine03}\n({\\it green}); two extinction curves derived for star-forming regions by\n\\citet{mcclure09}, one for $0.76 < A_J < 2.53$ ({\\it blue}), and one\nfor $2.53 < A_J < 17.71$ ({\\it purple}).\n\\label{opacities}}\n\\end{figure}\n\nWe used dust opacities from \\citet{ormel11} to account for larger, icy grains\n(as opposed to the small grains made of amorphous silicates typically found in\nthe interstellar medium). We adopted their dust model that includes graphite \ngrains without ice coating and ice-coated silicates, with a size distribution that \nassumes growth of aggregates for $3 \\times 10^5$ years, when grains have \ngrown up to 3 $\\mu$m in size (``icsgra3''). Particle sizes range from 0.1 to \n3 $\\mu$m, with a number density that is roughly proportional to $a^{-2.3}$ \n(where $a$ is the particle radius).\nFigure \\ref{opacities} shows our adopted opacities compared to different \nones found in the literature. The opacities from \\citet{draine03} assume a \nmixture of small carbonaceous and amorphous silicate grains. Including larger \nand icy grains broadens the 10 $\\mu$m silicate feature (which is mostly due to \nthe libration mode of water ice) and causes additional absorption at 3 $\\mu$m \nand in the 40-60 $\\mu$m range (all mostly due to the presence of water ice). \nThe mid-IR opacities of the ``icsgra3'' dust model are similar to the ones \ndetermined by \\citet{mcclure09} for star-forming regions and also to those \nused by \\citet{tobin08} to model an edge-on Class 0 protostar; in the mid- to \nfar-IR, they resemble the opacities of \\citet{ossenkopf94}, which are often \nused to model embedded sources. In Figure \\ref{opacities}, we show model \n`OH5' from \\citet{ossenkopf94}, which is listed as the fifth model in their \nTable 1 and corresponds to grains with thin ice mantles after 10$^5$ years \nof coagulation and a gas density of 10$^6$ cm$^{-3}$. We could not use the \n`OH5' opacities for our model grid, since that opacity law does not include \nscattering properties (which are required by the Whitney Monte Carlo \nradiative transfer code). Other authors have modified the `OH5' dust  \nto include the scattering cross section and extend the opacities to shorter\nand longer wavelengths \\citep{young05,dunham10}.\n\n\\subsection{Model Parameters}\n\\label{model_parameters}\n\nThere are 3040 models in the grid; they cover 8 values for the total (i.e.,\nintrinsic) luminosity, 4 disk radii, 19 envelope infall rates (which correspond \nto envelope densities), and 5 cavity opening angles. Each model is calculated \nfor 10 different inclination angles, from 18.2\\degr\\ to 87.2\\degr, in equal steps \nin $\\cos(i)$ (starting at 0.95 and ending at 0.05), resulting in 30,400 different \nmodel SEDs. The values for the various model parameters are listed in Table \n\\ref{modelpars}. Since there are a large number of parameters that can be set \nin the Whitney radiative transfer models, we focused on varying those \nparameters that affect the SED the most, leaving the other parameters \nat some typical values. For example, we assumed a stellar mass of 0.5 $M_{\\odot}$, \na disk mass of 0.05 $M_{\\odot}$, and an envelope outer radius of 10,000 AU.\nThe stellar mass enters the code in two ways. First, it is needed to relate the \ndensity of the envelope to the infall rate (see Equation 1 below). Since we fit the \ndensity of the envelope, the infall rate plays no role in the best-fit envelope \nparameters; any stellar mass can be chosen to determine the infall rate for a \ngiven best-fit envelope density. Second, the stellar mass is combined with the \nstellar radius and disk accretion rate to set the disk accretion luminosity. Given \nthat the accretion luminosity is the actual parameter that influences the SED, \nit does not matter which of the three factors is varied. For simplicity and reasons \ndescribed below, we varied the disk accretion rate and the stellar radius, but\nleft the stellar mass constant, to achieve different values for this component \nof the luminosity.\n \nThe total luminosity for each system consists of the stellar luminosity \n(derived from a 4000 K stellar atmosphere model), the accretion luminosity \nresulting from material accreting through the disk down to the disk truncation \nradius, and the accretion luminosity from the hot spots on the stellar surface, \nwhere the accretion columns, which start at the magnetospheric truncation \nradius, land (these columns are not included in the modeled density distribution, \nsince they do not contain dust and do not have a source of opacity in the radiative \ntransfer models). Typically, the accretion luminosity from the hot spots is much \nlarger than the disk accretion luminosity; in our models, the former is about a factor \nof 9 larger than the latter.\nWe chose three different stellar radii, 0.67, 2.09, and 6.61 $R_{\\odot}$\\ (with the same \nstellar temperature), resulting in three different stellar luminosities. Since both \ncomponents of the accretion luminosity depend on the disk accretion rate, \nchoosing a total of eight different disk accretion rates (three for the 0.67 $R_{\\odot}$\\ \nstar, two for the 2.09 $R_{\\odot}$\\ star, and three for the 6.61 $R_{\\odot}$\\ star) results in \neight values for the total luminosity used in the grid (see Table \\ref{modelpars}). \nThe input spectrum produced by the central protostar depends on the relative \ncontributions from the intrinsic stellar luminosity (which peaks at 0.7~$\\mu$m) \nand the accretion luminosity (which is radiated primarily in the UV). In the \nmodels, it can be altered to some degree by choosing different combinations of \nthe disk accretion rate and stellar radius (the former affects only the accretion \nluminosity, while the latter affects both the stellar and accretion luminosity). \nHowever, the effect of the input spectrum on the output SED is negligible. \nConsequently, we cannot reliably measure the relative contributions of stellar \nand accretion luminosity through our SED fits. Instead, we adjusted the particular \nvalues for the stellar radius and disk accretion rate to set the values of the\ntotal luminosity.\n\nFor our model grid, we chose four values for the disk outer radius, which we\nset equal to the centrifugal radius ($R_c$). In a TSC model, the centrifugal \nradius is the position in the disk where material falling in from the envelope \naccumulates; due to envelope rotation, material from the envelope's\nequatorial plane lands at $R_c$, while material from higher latitudes falls\ncloser to the star. The disk could extend beyond $R_c$, but in our models \nit ends at $R_c$. In this work, we use the terms ``disk (outer) radius'' and \n``centrifugal radius'' interchangeably. The primary effect of $R_c$ is to set \nthe rotation rate of the infalling gas and thereby determine the density \nstructure of the envelope \\citep{kenyon93}.  \n \n\\begin{deluxetable*}{clcc}\n\\tablecaption{Model Parameters\n\\label{modelpars}}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Description} & \\colhead{Values} & \n\\colhead{Units}}\n\\startdata\n\\multicolumn{4}{c}{\\it{\\bf Stellar Properties}} \\\\ \n$M_{\\ast}$ &  Stellar mass & 0.5 & $M_{\\odot}$ \\\\\n$T_{\\ast}$ &  Stellar effective temperature & 4000 & K \\\\\n$R_{\\ast}$ &  Stellar radius & 0.67, 2.09, 6.61 & $R_{\\odot}$ \\\\ \n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Disk Properties}} \\\\ \n$M_{disk}$ &  Disk mass & 0.05 & $M_{\\odot}$ \\\\\n$R_{disk}$ &  Disk outer radius & 5, 50, 100, 500 & AU \\\\\nA & Radial exponent in disk density law & 2.25 & \\nodata \\\\\nB & Vertical exponent in disk density law  & 1.25 & \\nodata \\\\\n$\\dot{M}_{disk,1}$ & Disk-to-star accretion rate for $R_{star}$=0.67 $R_{\\odot}$\\ & \n0, $1.14 \\times 10^{-8}$, $5.17 \\times 10^{-8}$  & $M_{\\odot}$\\ yr$^{-1}$ \\\\\n$\\dot{M}_{disk,2}$ & Disk-to-star accretion rate for $R_{star}$=2.09 $R_{\\odot}$\\ & \n$3.67 \\times 10^{-7}$, $1.63 \\times 10^{-6}$ & $M_{\\odot}$\\ yr$^{-1}$ \\\\\n$\\dot{M}_{disk,3}$ & Disk-to-star accretion rate for $R_{star}$=6.61 $R_{\\odot}$\\ & \n$1.14 \\times 10^{-5}$, $5.15 \\times 10^{-5}$,  $1.66 \\times 10^{-4}$ & \n$M_{\\odot}$\\ yr$^{-1}$ \\\\\n$R_{trunc}$ & Magnetospheric truncation radius$^a$ & 3 & $R_{\\ast}$ \\\\\n$f_{spot}$ & Fractional area of the hot spots on the star$^b$ & 0.01 & \\nodata \\\\\n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Envelope Properties}} \\\\ \n$R_{env}$ &  Envelope outer radius$^c$ & 10,000 & AU \\\\\n$\\rho_{1000}$ & Envelope density at 1000 AU$^d$ & 0.0, 1.19 $\\times 10^{-20}$, \n1.78 $\\times 10^{-20}$, 2.38 $\\times 10^{-20}$, &  g cm$^{-3}$ \\\\\n& & 5.95 $\\times 10^{-20}$, 1.19 $\\times 10^{-19}$, 1.78 $\\times 10^{-19}$, \n&  g cm$^{-3}$ \\\\\n& & 2.38 $\\times 10^{-19}$, 5.95 $\\times 10^{-19}$, 1.19 $\\times 10^{-18}$,\n&  g cm$^{-3}$ \\\\\n& & 1.78 $\\times 10^{-18}$, 2.38 $\\times 10^{-18}$, 5.95 $\\times 10^{-18}$,\n&  g cm$^{-3}$ \\\\\n& & 1.19 $\\times 10^{-17}$, 1.78 $\\times 10^{-17}$, 2.38 $\\times 10^{-17}$,\n&  g cm$^{-3}$ \\\\\n& & 5.95 $\\times 10^{-17}$, 1.19 $\\times 10^{-16}$, 1.78 $\\times 10^{-16}$\n&  g cm$^{-3}$ \\\\\n$R_c$ & Centrifugal radius of TSC envelope & $= R_{disk}$ & AU \\\\ \n$\\theta$ & Cavity opening angle & 5, 15, 25, 35, 45 & degrees \\\\\n$b_{cav}$ & Exponent for cavity shape$^e$ (polynomial) & 1.5 & \\nodata \\\\\n$z_{cav}$ & Vertical offset of cavity wall & 0 & AU \\\\\n \\hline\n\\multicolumn{4}{c}{\\it{\\bf Derived Parameters}} \\\\ \n$L_{\\ast}$ &  Stellar luminosity$^f$ &  0.1, 1, 10 & $L_{\\odot}$ \\\\\n$L_{tot}$ & Total luminosity (star + accretion)$^g$ & 0.1, 0.3, 1.0, 3.1, \n         10.1, 30.2, 101, 303 & $L_{\\odot}$ \\\\\n\\hline\n\\multicolumn{4}{c}{\\it{\\bf Parameters for Model SEDs}} \\\\ \n$i$ & Inclination angle & 18.2, 31.8, 41.4, 49.5, 56.7, & degrees \\\\\n  &   &  63.3, 69.5, 75.6, 81.4, 87.2 & degrees \\\\\n  &  Aperture radii for model fluxes$^h$ &  420, 840, 1260, 1680, ..., 10080 & AU \\\\\n\\enddata\n\\tablecomments{\nThe dust opacities used for these models are those called ``icsgra3'' from\n\\citet{ormel11}.\\\\\n$^a$ This radius applies to the gas. The inner disk radius for the dust is equal to the \ndust destruction radius. The scale height of the disk at the dust sublimation radius is set \nto the hydrostatic equilibrium solution. \\\\\n$^b$ The hot spots are caused by the accretion columns that reach from the \nmagnetospheric truncation radius to the star. \\\\\n$^c$ The inner envelope radius is set to the dust destruction radius. \\\\\n$^d$ The actual input parameter for the Whitney code is the envelope infall\nrate, which can be derived from $\\rho_{1000}$ using Equation (2). The first six \n$\\rho_{1000}$ values correspond to envelope infall rates of 0, $5.0 \\times 10^{-8}$, \n$7.5 \\times 10^{-8}$, $1.0 \\times 10^{-7}$, $2.5 \\times 10^{-7}$, and $5.0 \\times 10^{-7}$  \n$M_{\\odot}$\\ yr$^{-1}$; the other values can be similarly deduced. \\\\\n$^e$ The cavity walls are assumed to have a polynomial shape; no material is assumed\nto lie inside the cavity. Also, the ambient density (outside the envelope) is 0. \\\\\n$^f$ The three values of $L_{\\ast}$ correspond to the three different stellar radii.\\\\\n$^g$ The total luminosities combine the stellar luminosities and the accretion luminosities\n(which depend on  $\\dot{M}_{disk}$). \\\\\n$^h$ For each model, the emitted fluxes are calculated for 24 apertures ranging from \n420 to 10080 AU, in steps of 420 AU. }\n\\end{deluxetable*}\n\nThe largest number of parameter values in our grid is for the envelope \ninfall rate. The envelope infall rate used as an input in the Whitney code \nsets the density of the envelope for a given mass of the protostar.  \nSince the SED depends on the density of the envelope (and not directly \non the infall rate, which is only inferred from the density and the acceleration \ndue to gravity from the central protostar), in this work we report a  \nreference envelope density instead of the envelope infall rate as one of our \nmodel parameters. For the TSC model, the envelope infall rate $\\dot{M}_{env}$ \nand the reference density at 1 AU in the limit of no rotation ($R_c$=0) are \nrelated as follows \\citep[see][]{kenyon93}:\n\\begin{equation}\n\\rho_1 = 5.318 \\times 10^{-14} \\left( \\frac{\\dot{M}_{env}}{10^{-5} \nM_{\\odot} \\, \\mathrm{yr}^{-1}} \\right) \\left(\\frac{M_{\\ast}}\n{1\\, M_{\\odot}}\\right)^{-1\/2} \\mathrm{g}\\, \\mathrm{cm}^{-3},\n\\end{equation} \nwhere $M_{\\ast}$ is the mass of the central protostar, which is assumed \nto be 0.5 $M_{\\odot}$\\ in our model grid. The density distribution in the\nenvelope follows a power law, $\\rho \\propto r^{-3\/2}$, at radii\nlarger than the centrifugal radius, $R_c$, but then flattens as a result\nof the rotation of the envelope. The density reported by $\\rho_1$ \nassumes a spherically symmetric envelope with a $-3\/2$ power-law \nexponent valid down to the smallest radii, and it is higher than the \nangle-averaged density of a rotating envelope at 1 AU. To quote \ndensities that are closer to actual values found in the modeled rotating \nenvelopes (which have $R_c$ values ranging from 5 to 500 AU), we \nreport $\\rho_{1000}$, the density at 1000 AU for a $\\rho \\propto r^{-3\/2}$ \nenvelope with a 0.5 $M_{\\odot}$\\ protostar:\n\\begin{eqnarray}\n\\rho_{1000} & = & \\rho_1 \\left(\\frac{1}{1000}\\right)^{3\/2} \\nonumber \\\\\n& = & 2.378 \\times 10^{-18} \\left( \n\\frac{\\dot{M}_{env}}{10^{-5} M_{\\odot} \\, yr^{-1}} \\right) \n\\mathrm{g}\\, \\mathrm{cm}^{-3}.\n\\end{eqnarray} \nThus, the range of reference densities probed in our model grid,\nfrom $1.2 \\times 10^{-20}$ to $1.8 \\times 10^{-16}$ g cm$^{-3}$ \n(see Table \\ref{modelpars}), would correspond to envelope\ninfall rates from $5.0 \\times 10^{-8}$ to $7.5 \\times 10^{-4}$ $M_{\\odot}$\\ yr$^{-1}$,\nassuming $M_{\\ast}$=0.5 $M_{\\odot}$\\ (this does not account for a reduction \nof the infalling mass due to clearing by outflow cavities).\nIn Figure \\ref{Rho_env_profiles}, we show the radial density profiles\nfor two TSC models with 5 AU and 500 AU centrifugal radii. The density \nprofiles are azimuthally symmetric and show the flattening of the density \ndistribution inside $R_c$ due to envelope rotation. These plots demonstrate\nthat the density $\\rho_1$ is much higher than the angle-averaged \ndensity at 1 AU; $\\rho_{1000}$ seems to yield more physical values for \nthe density in the envelope at 1000 AU, even for $R_c$ values of 500 AU.\n \n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.37,angle=90]{Envelope_rho_vs_radius_Rc5.eps}\n\\includegraphics[scale=0.37,angle=90]{Envelope_rho_vs_radius_Rc500.eps}\n\\caption{Envelope density versus radius for a model protostar with\n$\\dot{M}_{env}=1.0 \\times 10^{-6}$ $M_{\\odot}$\\ yr$^{-1}$,\n$M_{\\ast}$=0.5 $M_{\\odot}$, and $R_c$=5 AU ({\\it left}) and 500\nAU ({\\it right}) to show the difference between the reference densities\n$\\rho_1$ and $\\rho_{1000}$. The lines with different colors represent \nradial density profiles for different polar angles $\\theta$; the black line represents\nthe angle-averaged density profile (for equations see \\citealt{whitney03a,\nadams86}). The dashed line represents an $r^{-3\/2}$ power law. The vertical\ndotted line marks the location of the centrifugal radius.\n\\label{Rho_env_profiles}}\n\\end{figure*}\n \nAs can be seen from the values of the envelope density in Table \\ref{modelpars}, \nthere is one set of models with an envelope density of 0. These are models \nthat do not contain an envelope component; the entire excess emission \nis caused by the circumstellar disk. If an object is best fit by such a model, \nit would indicate that it is more evolved, having already dispersed its envelope.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Whitney_cavity_shapes.eps}\n\\caption{Schematic showing the shape of the cavity assumed in our models\nfor three cavity opening angles $\\theta$: 5\\degr, 25\\degr, and 45\\degr\\ \n(from left to right). The cavity walls are defined as a polynomial with exponent \n1.5 ($z \\propto \\tilde r^{1.5}$), with $\\tilde r_{max} = z_{max}\\, \\tan\\theta$, and \nare shown as solid lines. The outer envelope radius ($R_{env}$) at 10,000 AU \nis shown with a short-dashed line. The dotted lines show a different definition \nof the cavity size, where $\\tilde r_{max} = R_{env} \\sin\\theta$ \nand $z_{max} = R_{env} \\cos\\theta$.\n\\label{cavity_shape}}\n\\end{figure}\n\nThe cavities in our models range from 5\\degr\\ to 45\\degr\\ and are defined\nsuch that $z \\propto \\tilde r^{1.5}$, where $\\tilde r$ and $z$ are the cylindrical\ncoordinates for the radial and vertical direction, respectively, and $\\tilde r_{max} \n= z_{max}\\, \\tan\\theta$, with $\\theta$ defined as the cavity opening angle that \nis specified in the parameter file of the Whitney radiative transfer code. In this \ncode, $z_{max}$ is set to the envelope outer radius. Thus, a polynomial-shaped \ncavity, which is wider at smaller $\\tilde r$ values and then converges toward the \nspecified opening angle, is somewhat larger than this opening angle at the outer \nenvelope radius (see Figure \\ref{cavity_shape}). This effect is most noticeable at \nlarger cavity opening angles, but negligible for small cavities. A different definition \nof the cavity size, where $\\tilde r_{max} = R_{env} \\sin\\theta$ and \n$z_{max} = R_{env} \\cos\\theta$ (with $R_{env}$ as the envelope outer radius), \nresults in $z$ values that are a factor of $1\/\\cos\\theta$ larger, and thus the cavity \nreaches the specified opening angle at the outer envelope radius. For this \nwork, the adopted definition of the cavity opening angle is inconsequential, \nbut it becomes relevant when comparing the results of SED modeling to \nscattered light images that reveal the actual cavity shape and size. \nWe also note that in our models the cavities are evacuated of material,\nso there is no dust and gas inside the cavity; in reality, there might be\nsome low-density material left that would add to the scattered light\n\\citep[see][]{fischer14}.\n\nFigures \\ref{Models_inc} to \\ref{Models_Ltot} display a few examples\nof model SEDs from our grid to show the effect of changing those model \nparameters that influence the resulting SED the most. \nThe inclination angle has a strong effect on the near- and mid-infrared SED\n(Figure \\ref{Models_inc}). While a low inclination angle results in an overall \nflat SED in this wavelength region, increasing the inclination angle causes \na deeper silicate absorption feature at 10 $\\mu$m and a steep slope \nbeyond it. The far-infrared to millimeter SED is not affected by the \ninclination angle, since emission at these wavelengths does not suffer \nfrom extinction through the envelope.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_inc_example.eps}\n\\caption{A model from the grid seen at 10 different inclination\nangles to illustrate the effect of viewing angle on the SED. \nThe model has $L_{tot}$=10.1 $L_{\\odot}$, $R_c$=50 AU, \n$\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, \nand is seen at inclination angles 18\\degr, 32\\degr, 41\\degr, 49\\degr, \n57\\degr, 63\\degr, 69\\degr, 76\\degr, 81\\degr, and 87\\degr\\ (from top to bottom). \n\\label{Models_inc}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_cavity_example.eps}\n\\caption{Models from the grid to illustrate the effect of cavity opening\nangle on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$R_c$=50 AU, $\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$, i=63\\degr, \nbut each has a different cavity opening angle: 5\\degr\\ ({\\it red}),\n15\\degr\\ ({\\it yellow}), 25\\degr\\ ({\\it green}), 35\\degr\\ \n({\\it blue}), 45\\degr\\ ({\\it purple}).\n\\label{Models_cavity}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_Rdisk_example.eps}\n\\caption{Models from the grid to illustrate the effect of the centrifugal radius\n($=R_{disk}$) on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$\\rho_{1000}$=$1.2 \\times 10^{-18}$ g cm$^{-3}$,  $\\theta$=5\\degr, i=63\\degr, \nbut different disk radii: 5 AU ({\\it red}), 50 AU ({\\it yellow}), 100 AU ({\\it green}), \n500 AU ({\\it purple}).\n\\label{Models_Rdisk}}\n\\end{figure}\n\nThe cavity opening angle affects the SED shape at all wavelengths (Figure\n\\ref{Models_cavity}). A small cavity only minimally alters the SED compared\nto a case without a cavity; there is still a deep silicate absorption at 10 $\\mu$m \nand steep SED slope, but the cavity allows some scattered light to escape in\nthe near-IR. A larger cavity results in higher emission at near- and mid-infrared \nwavelengths and reduced emission in the far-infrared. \nThe effect of the cavity on the SED would change if a different shape\nfor the cavity walls were adopted. For example, cavities where the outer\nwall follows the streamlines of the infalling gas and dust evacuate less inner \nenvelope material than our polynomial-shaped cavities, resulting in deeper \nsilicate absorption features and steeper mid-infrared SED slopes for the \nsame cavity opening angle \\citep[see][]{furlan14}. Thus, our cavity \nopening angles are tied to our assumed cavity shape.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_density_example.eps}\n\\caption{Models from the grid to illustrate the effect of envelope\ndensity on the SED. The models have $L_{tot}$=10.1 $L_{\\odot}$, \n$R_c$=50 AU, $\\theta$=15\\degr, i=63\\degr, but different reference \ndensities $\\rho_{1000}$:\n0, $2.4 \\times 10^{-20}$, $1.2 \\times 10^{-19}$, $2.4 \\times 10^{-19}$, \n$1.2 \\times 10^{-18}$, $2.4 \\times 10^{-18}$, $1.2 \\times 10^{-17}$, \n$2.4 \\times 10^{-17}$,  and $1.2 \\times 10^{-16}$ g cm$^{-3}$ (the \npeak of the SED moves to longer wavelengths as $\\rho_{1000}$ \nincreases).\n\\label{Models_density}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.52]{Models_Ltot_example.eps}\n\\caption{Models from the grid to illustrate the effect of the total\nluminosity on the SED. The models have $R_c$=50 AU, $\\rho_{1000}$=\n$1.2 \\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, i=63\\degr, but \ndifferent values for the total luminosity: 0.1, 0.3, 1.0, 3.1, 10.1, 30.2, 101, \nand 303 $L_{\\odot}$\\ (from bottom to top).\n\\label{Models_Ltot}}\n\\end{figure}\n\nThe effect of the centrifugal radius is somewhat similar to those of the cavity\nopening angle and inclination angle, but less pronounced (Figure \n\\ref{Models_Rdisk}). Small disk radii imply more slowly rotating, less flattened \nenvelopes and depress the near- and mid-infrared fluxes more than larger \ndisk radii, but even with large disk radii (and more flattened envelopes) there \nis still sufficient envelope material along the line of sight to cause a pronounced \n10 $\\mu$m absorption feature. Overall, our models do not directly constrain the \nsize of the disk; the opacity is dominated by the envelope. Furthermore, the \nflattening of the envelope that is determined by $R_c$ has a similar effect on \nthe SED as changing the outflow cavity opening angle.  \n\nChanging the envelope density causes shifts in the SED in terms of both\nwavelength and flux level: the higher the envelope density, the less \nflux is emitted at shorter wavelengths, and the more the peak of the \nSED shifts to longer wavelengths (Figure \\ref{Models_density}). Deeply \nembedded protostars have SEDs that peak at $\\lambda >$ 100 $\\mu$m, \nsteep mid-IR SED slopes, and deep silicate absorption features.\nThe effect of the envelope density on the SED is different from that of the\ninclination angle, especially in the far-IR: while the SED is not very\nsensitive to the inclination angle in this wavelength region, the ratio of, \ne.g., 70 and 160 $\\mu$m fluxes changes considerably depending on \nthe envelope density. \n\nThe total luminosity of the source has an effect on the overall emission\nlevel of the protostar, but does not strongly affect the SED shape. The\nmain effect is that the peak of the SED shifts to longer wavelengths as \nthe luminosity decreases ($\\lambda_{peak} \\propto L^{-1\/12}$; \n\\citealt{kenyon93}). Especially when comparing models with $L_{tot}$ \nvalues that differ by a factor of a few, the SED shapes are similar \n(Figure \\ref{Models_Ltot}). Thus, one could scale a particular model by \na factor between $\\sim$ 0.5 and 2 and get a good representation of a \nprotostar that is somewhat fainter or brighter, without having to rerun \nthe model calculation with the different input luminosity.\n\n\n\\subsection{Model Apertures}\n\\label{model_ap}\n\nThe model fluxes are computed for 24 different apertures, ranging from \n420 to 10,080 AU in steps of 420 AU (which corresponds to 1\\arcsec\\ at\nthe assumed distance of 420 pc to the Orion star-forming complex). \nFor these SED fluxes, no convolution with a PSF is done, and therefore\nthe spatial distribution of the flux is solely due to the extended nature \nof protostars. Since the envelope outer radius is chosen to be 10,000 AU, \nthe largest aperture encompasses the entire flux emitted by each protostellar\nsystem. However, most of the near- and mid-infrared emission comes \nfrom smaller spatial scales, so an aperture of about 5000 AU will already\ncapture most of the flux emitted at these wavelengths.  \n\nFor a more accurate comparison of observed and model fluxes, in each \ninfrared photometric band where we have data available, we interpolate \nmodel fluxes from the two apertures that bracket the aperture used in \nmeasuring the observed fluxes (4\\arcsec\\ for 2MASS, 2{\\farcs}4\nfor IRAC, PSF photometry for MIPS 24 \\micron, with a typical FWHM of\n6\\arcsec, 9{\\farcs}6 for PACS 70 and 100 \\micron, 12{\\farcs}8 \nfor PACS 160 \\micron). For the IRS data points, we use fluxes interpolated \nfor a 5{\\farcs}3 aperture, since the spectra are composed of two segments,\nSL (5.2-14 \\micron; slit width of 3{\\farcs}6) and LL (14-38 \\micron, slit \nwidth of 10{\\farcs}5), and, if any flux mismatches were present, the SL \nsegment was typically scaled to match the LL flux level at 14 \\micron\\ \n\\citep[see, e.g.,][]{furlan08}. So, fluxes measured in an aperture with \na radius of 5{\\farcs}3 roughly correspond to fluxes from a 10{\\farcs}6-wide \nslit.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_PACS160.eps}\n\\caption{PACS 160 $\\mu$m fluxes versus aperture radius derived for \na model ($L_{tot}=1.0$ $L_{\\odot}$, $R_c=100$ AU, $\\rho_{1000}=2.378 \n\\times 10^{-18}$ g cm$^{-3}$, $\\theta$=15\\degr, $i= 63$\\degr) using \ndifferent methods. \nThe black symbols represent fluxes from the model SED, the blue symbols \nfluxes derived using aperture photometry on the model image convolved with \nthe PACS 160 $\\mu$m PSF, and the red symbols fluxes derived from the \nconvolved model image and then corrected for PSF losses (see text for details). \nThe maximum flux from the model SED was used to normalize all other fluxes.\nThe dotted line indicates an aperture radius of 12{\\farcs}8.\n\\label{Model_fluxes_PACS160}}\n\\end{figure}\n\nGiven that our targets are typically extended and that the near- to mid-infrared \ndata have relatively high spatial resolution, measuring fluxes in small apertures \n(a few arcseconds in radius) will truncate some of the object's flux, so it is\nimportant to choose similar apertures for the model fluxes. \nFrom about 30 to 100 $\\mu$m, the model fluxes calculated for smaller apertures \nare not very different from the total flux (i.e., the flux from the largest aperture),\nwhich is a result of the emission profile in the envelope and the lower spatial \nresolution at longer wavelengths. \nTo check whether extended source emission in the far-infrared might affect \nthe flux we measure in our models, we calculated a small set of model\nimages at 160 $\\mu$m, convolved them with the PACS 160 $\\mu$m\nPSF, and compared the fluxes from the model images to those written\nout for the model SEDs (which we refer to as ``SED fluxes''; these are\nthe fluxes from the models in the grid).\nModel images would be the most observationally consistent way to measure\nthe flux densities, but they are too computationally expensive and would not\nrepresent a significant gain.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_SABOCA.eps}\n\\caption{SABOCA (350 $\\mu$m) fluxes versus aperture radius derived \nfor the same model as in Figure \\ref{Model_fluxes_PACS160} using different\nmethods. The black symbols represent fluxes from the model SED, the \nblue symbols  fluxes derived using aperture photometry on the model \nimage convolved with a Gaussian PSF, and the red dot-dashed line the \nbeam flux (assuming a beam with a FWHM of 7{\\farcs}3). The maximum \nflux from the model SED was used to normalize all other fluxes.\nThe dotted line indicates an aperture radius of 3{\\farcs}65.\n\\label{Model_fluxes_SABOCA}}\n\\end{figure}\n\nIn Figure \\ref{Model_fluxes_PACS160} we show the fluxes derived for\na particular model at 160 $\\mu$m using different methods. The fluxes \nmeasured in the convolved model image are lower than the SED fluxes; this \nis caused by the wide PACS 160 $\\mu$m PSF, which spreads flux to very\nlarge radii. Since the shape of the PSF is known, we can correct for these \nPSF losses (assuming a point source and using standard aperture\ncorrections). The fluxes corrected for these PSF losses are very similar to the \nSED fluxes, typically within $\\sim$ 5-10\\% at apertures larger than 5\\arcsec.\nSince our observed fluxes correspond to these PSF-corrected fluxes (we apply \naperture corrections to our fluxes measured in a 12{\\farcs}8 aperture to \naccount for PSF losses), adopting the SED fluxes from the largest aperture \nwould yield model fluxes that are somewhat too high. Thus, we chose to adopt \nthe SED flux measured in a 12{\\farcs}8 aperture as a good approximation for \nthe model flux we would get if we had model images available for all models in \nthe grid and measured aperture-corrected fluxes in these images. We note that \nin our PACS data, the 160 $\\mu$m sky annulus, which extends from 12{\\farcs}8 \nto 25{\\farcs}6 (see B. Ali et al. 2016, in preparation), can include extended emission\nfrom surrounding material and also some envelope emission. In these cases, we \noften used PSF photometry to minimize contamination from nearby sources and \nnebulosity; however, PSF fitting was not used for more isolated sources since \nthe envelopes can be marginally resolved at 160 $\\mu$m and thus deviate \nslightly from the adopted PSF shape.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39,angle=90]{Model_fluxes_LABOCA.eps}\n\\caption{Similar to Figure \\ref{Model_fluxes_SABOCA}, but for the\nLABOCA (870 $\\mu$m) fluxes. The dotted line indicates an aperture \nradius of 9{\\farcs}5.\n\\label{Model_fluxes_LABOCA}}\n\\end{figure}\n\nFor the SABOCA and LABOCA data, beam fluxes were adopted; the\nFWHM of the SABOCA beam is 7{\\farcs}3, while for the LABOCA\nbeam it is 19\\arcsec. In order to determine which aperture radius\ncorresponds best to beam fluxes, we created a similar set of model \nimages as above at 350 and 870 $\\mu$m, convolved them with \nGaussian PSFs, and measured fluxes in the model images using \ndifferent apertures (see Figures \\ref{Model_fluxes_SABOCA} and\n\\ref{Model_fluxes_LABOCA}, where we show the results for one\nmodel). Fluxes measured in the convolved model image are smaller \nthan the SED fluxes, especially at aperture radii smaller than the \nFWHM of the beam. We find that the beam fluxes for SABOCA and \nLABOCA are best matched by SED fluxes from apertures with radii \nhalf the size of the FWHM of the beam, i.e., 3{\\farcs}65 for \nSABOCA and 9{\\farcs}5 for LABOCA (thus, the aperture sizes are\nthe same as the beam FWHM). \nThis is again an idealized situation, since the measured SABOCA\nand LABOCA beam fluxes also include extended emission (if the\nsource lies on top of background emission), and thus they could be \nhigher than those from the model.\n\n\\subsection{Effect of External Heating}\n\\label{model_ext_heat}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Ext_heat_models.eps}\n\\caption{{\\it Left:} Comparison of models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, \n$\\theta$=15\\degr, $\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$ ({\\it top}) or\n2.4 $\\times 10^{-20}$ g cm$^{-3}$ ({\\it bottom}), $i$=63\\degr, without external heating \n({\\it black}), with external heating by an ISRF equal to that in the solar neighborhood\n({\\it green, dashed line}), and with heating by an ISRF 10 times stronger ({\\it orange, \ndashed line}). {\\it Right:} Similar to the models in the left panels, but these models have \n$L_{tot}$=1.0 $L_{\\odot}$.\n\\label{models_ext_heat}}\n\\end{figure*}\n\nIn our models, the luminosity is determined by the central protostar and\nthe accretion; no external heating is included. The interstellar radiation\nfield (ISRF) could increase the temperature in the outer envelope regions,\nthus causing an increase in the longer-wavelength fluxes \\citep[e.g.,][]\n{evans01,shirley02,young03}.\nIt is expected that external heating has a noticeable effect only on \nlow-luminosity sources ($\\lesssim$ 1 $L_{\\odot}$), while objects with strong \ninternal heating are not affected by the ISRF. Moreover, the strength of \nthe ISRF varies spatially \\citep{mathis83}, and thus its effect on each \nindividual protostar is uncertain. Nonetheless, in the following we estimate\nthe effect of external heating on model fluxes by using a different set of\nmodels.\n \n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.7,angle=90]{Ext_heat_models_flux_ratios.eps}\n\\caption{Ratio of the excess emission due to external heating and the emission of \nthe protostar with external heating in different bands, for heating by an ISRF\nequal to that in the solar neighborhood ({\\it green diamonds}) and by an ISRF \n10 times stronger ({\\it orange squares}). The vertical lines show the range of\nflux excess ratios resulting from different viewing angles (inclination angles range \nfrom 18\\degr\\ to 87\\degr), while the symbols represent mean values. The top (bottom) \npanels are for models with $L_{tot}$=0.1 (1.0) $L_{\\odot}$. The four columns correspond \nto the four reference densities probed.\n\\label{models_ext_heat_ratios}}\n\\end{figure*}\n\nFor this model calculation, we used the 2012 version of the Whitney radiative \ntransfer code \\citep{whitney13}, which allows for the inclusion of external illumination \nby using the ISRF value in the solar neighborhood from \\citet{mathis83}; to vary \nthe ISRF strength, the adopted value can be scaled by a multiplicative factor \nand extinguished by a certain amount of foreground extinction. \nWe calculated a small number of models with and without external heating\nand then compared their far-infrared and submillimeter fluxes. One set of models \nhas $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, and four different \nreference densities $\\rho_{1000}$, ranging from 2.4 $\\times 10^{-17}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-20}$ g cm$^{-3}$. The other set has the same parameters\nexcept for $L_{tot}$, which is 1.0 $L_{\\odot}$. We calculated models without external \nheating, with heating from an ISRF equal to that in the solar neighborhood, and \nwith ISRF heating 10 times the solar neighborhood value. For these models, we \ndid not include any foreground extinction for the ISRF; thus, the ISRF heating in \nthese models can be considered an upper limit -- especially the 10-fold increase \nover the ISRF in the solar neighborhood represents an extreme value.\nFigure \\ref{models_ext_heat} shows a few examples of model SEDs with and\nwithout external heating. External heating results in flux increases in the far-IR\nand sub-mm; as expected, it affects low-luminosity sources more, and its effects \nare also more noticeable for higher-density envelopes.\n\nFor a more quantitative comparison of model fluxes in the far-IR and sub-mm,\nwe computed the fluxes for each model in six different bands, those of MIPS \n24 $\\mu$m, PACS 70, 100, and 160 $\\mu$m, and SABOCA (350 $\\mu$m) \nand LABOCA (870 $\\mu$m), using apertures as described in section \n\\ref{model_ap}. The model fluxes are affected by poorer signal-to-noise \nratios at the longest wavelengths, so the 870 $\\mu$m fluxes are less reliable.\nWe subtracted the fluxes of the models without external heating ($F_{\\rm no.ext.heating}$)\nfrom those with external heating ($F_{\\rm ext.heating}$) to determine the flux \nexcess due to external heating. The ratios of these excess fluxes and the model fluxes \nwith external heating ($(F_{\\rm ext.heating}-F_{\\rm no.ext.heating})\/F_{\\rm ext.heating}$)\nare shown in Figure \\ref{models_ext_heat_ratios}. Given that these ratios depend \non the inclination angle to the line of sight, we show them as average values for all\n10 inclination angles as well as the range subtended by all inclination angles.\nWe note overall smaller flux ratios at 350 $\\mu$m due to the smaller aperture \nsize chosen in this wave band (see section \\ref{model_ap}).\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Models_ext_heat_SED1.eps}  \n\\caption{{\\it Black and orange lines:} SEDs for models with $L_{tot}$=0.1 $L_{\\odot}$, \n$R_c$=100 AU,  $\\theta$=15\\degr, $i$=75\\degr, reference densities \n$\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$ ({\\it left}) and 2.4 $\\times 10^{-19}$ \ng cm$^{-3}$ ({\\it right}), without external heating ({\\it black}) and with heating by an ISRF \nscaled by a factor of 10 ({\\it orange}). The purple dashed lines show SEDs from our \nmodel grid (which does not include external heating) with model parameters\nchanged as indicated in the figure label; these models were chosen to closely \nmatch the model SEDs with external heating.\n\\label{ext_heat_SED1}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Models_ext_heat_SED2.eps}  \n\\caption{Similar to Figure \\ref{ext_heat_SED1}, but for model SEDs with\n$L_{tot}$=1.0 $L_{\\odot}$\\ ({\\it black and orange lines}). The light blue and purple \ndashed lines show SEDs from our model grid (no external heating) with the same \nmodel parameters as shown except for a reference density 2.5 times higher \n({\\it light blue}) and $\\theta$=25\\degr, $i=$81\\degr, and a higher luminosity \n({\\it purple}).\n\\label{ext_heat_SED2}}\n\\end{figure*}\n\nOur analysis shows that heating by the ISRF results in flux increases in the far-IR \nand sub-mm that are about a factor of 2-3 higher for envelopes of low-luminosity \nsources ($L_{tot}$=0.1 $L_{\\odot}$) than for those with higher luminosity. Also, the\neffect of external heating is more noticeable at longer wavelengths (where \napertures\/beams are also larger) than at shorter ones; given our chosen apertures, \nthe largest effect occurs at 160 and 870 $\\mu$m. We also note that the flux increases \ndue to heating by the ISRF are smallest for the lowest $\\rho_{1000}$ value probed, \n2.4 $\\times 10^{-20}$ g cm$^{-3}$; at 160 $\\mu$m, the flux increase is largest for \nintermediate envelope densities. Finally, the flux increases in the far-IR and sub-mm \nare far larger for a solar-neighborhood ISRF scaled by factor of 10 than for an \nunscaled ISRF; for the $L_{tot}$=0.1 $L_{\\odot}$\\ models, an unscaled ISRF increases \nthe fluxes from a few percent (at $\\lesssim$ 100 $\\mu$m) to 50\\% (at 870 $\\mu$m), \nwhile an ISRF scaled by a factor of 10 increases these fluxes by 30\\%-75\\%. Thus, \nfor low-luminosity protostars, up to $\\sim$ 75\\% of a protostar's 870 $\\mu$m flux \ncould be due to external heating, if the environment is dominated by an extremely \nstrong ISRF.\n\nTo estimate how the contribution of external heating would modify derived\nmodel parameters, in Figures \\ref{ext_heat_SED1} and \\ref{ext_heat_SED2} \nwe compare model SEDs that include external heating by an ISRF 10 times \nstronger than in the solar neighborhood and model SEDs without this additional \nheating. For the latter, we used models from our model grid and tried to reproduce\nthe SEDs with external heating. For the models with $L_{tot}$=0.1 $L_{\\odot}$, the\neffect of external heating can be reproduced by increasing the luminosity by\nfactors of a few, increasing $\\rho_{1000}$ by up to an order of magnitude,\nand increasing the cavity opening angle and inclination angle by a small\namount. For the $L_{tot}$=1.0 $L_{\\odot}$\\ models, just increasing the reference \ndensity by a factor of 2.5 results in a good match to the long-wavelength \nemission of our externally heated models; however, the shorter-wavelength \nflux is either under- or overestimated. A better match is achieved with models \nhaving the same reference density as the externally heated models, but with \nslightly larger cavity opening angles and inclination angles, and luminosities \nabout a factor of 2 larger.\nThus, if the far-IR and sub-mm fluxes were contaminated by emission resulting \nfrom extremely strong external heating, a model fit using models from our grid \n(which does not include external heating) could overestimate the envelope density \nby up to an order of magnitude and the luminosity by a factor of 2-5. The cavity \nopening and inclination angles would also be more uncertain, but not by much.  \nFor a more realistic scenario with more modest external heating (which would \nalso include the effect of local extinction), the effect on model parameters would\nbe smaller.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.4,angle=90]{Ext_heat_models_with_Av.eps}\n\\caption{Models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, \n$\\rho_{1000}$=2.4 $\\times 10^{-18}$ g cm$^{-3}$, $i$=63\\degr, without external \nheating ({\\it black}), with external heating by an ISRF 10 times stronger than in the \nsolar neighborhood ({\\it orange} to {\\it brown}, {\\it dashed lines}) and different \namounts of extinction applied to the ISRF (from $A_V = 2.5$ to $A_V = 50$, \n{\\it top} to {\\it bottom}).\n\\label{models_ext_heat_Av}}\n\\end{figure}\n\nFor the latter point, we explored the effect of extinction on the ISRF by calculating \na few more models with $L_{tot}$=0.1 $L_{\\odot}$, $R_c$=100 AU, $\\theta$=15\\degr, \n$\\rho_{1000} = 2.4 \\times 10^{-18}$ g cm$^{-3}$, an ISRF 10 times stronger than \nthat in the solar neighborhood, and $A_V$ values for the ISRF of 2.5, 10, 20, and 50. \nThe model SEDs are shown in Figure \\ref{models_ext_heat_Av}. Compared to ISRF\nheating without any foreground extinction, already $A_V=2.5$ causes a decrease \nby a factor of 1.5-2 in the overall emission at far-IR wavelengths. With $A_V$ of\n10 and 20, the far-IR emission decreases by factors of up to $\\sim$ 3.5 and 4, \nrespectively, compared to a strong ISRF that is not extinguished. The fraction of\nexcess emission due to external heating at 160 $\\mu$m decreases from an\naverage of 0.8 for $A_V$=0 (see Figure \\ref{models_ext_heat_ratios}) to 0.6,\n0.3, and 0.2 for $A_V$=2.5, 10, and 20, respectively. Therefore, considering that \ntypical $A_V$ values in Orion are $\\sim$ 10-20 mag \\citep{stutz15}, it is likely\nthat the effect of external heating on model parameters of low-luminosity sources\ndoes not exceed a factor of $\\sim$ 2 in luminosity and $\\sim$ 5 in envelope\ndensity.\n\n\n\\section{Fitting Method}\n\\label{method}\n\nA customized fitting routine determines the best-fit model from the grid \nfor each object in our sample of 330 YSOs (see Sections \\ref{sample} \nand \\ref{SEDs}) using both photometry and, where available, IRS spectroscopy.\nIdeally, an object has 2MASS, IRAC, IRS, MIPS, PACS, and SABOCA and LABOCA \ndata; in many cases, no submillimeter data are available, and in a few cases the \nobject is too faint to be detected by 2MASS. Of the 330 modeled objects, 40 do \nnot have IRS spectra. As a minimum, objects have some {\\it Spitzer} photometry \nand a measured flux value in the PACS 70 $\\mu$m band. No additional data \nfrom the literature were included in the fits to keep them homogeneous.\n\nIn order to reduce the number of data points contained in the IRS spectral\nwavelength range (such that the spectrum does not dominate over the photometry) \nand to exclude ice absorption features in the 5-8 \\micron\\ region and at 15.2 \\micron\\\nthat are usually observed, but not included in the model opacities, we rebin each \nIRS spectrum to fluxes at 16 wavelengths. These data points trace the continuum \nemission and the 10 and 20 \\micron\\ silicate features. Also, when rebinning the \nspectrum, we smooth over its noisy regions, and we scale the whole spectrum \nto match the MIPS 24 \\micron\\ flux if a similar deviation is also seen at the \nIRAC 5.8 and 8 \\micron\\ bands and is larger than 10\\%. Figure \\ref{IRS_rebin} \nshows three examples of our IRS spectra with the rebinned fluxes overplotted.\nOur selection of 16 IRS data points in addition to at most 13 photometric points \nspread from 1.1 to 870 \\micron\\ puts more emphasis on the mid-IR spectral \nregion in the fits. This wavelength region is better sampled by observations, \nmost of the emission is thermal radiation from the protostellar envelope and\ndisk (as opposed to some possible inclusion of  scattered light or thermal \nemission from surrounding material at shorter and longer wavelengths, \nrespectively), and it contains the 10 \\micron\\ silicate feature, which crucially \nconstrains the SED fits. As a result, most models are expected to reproduce \nthe mid-IR fluxes well and might fit more poorly in the near-IR and sub-mm.\n\n\\begin{figure}[!]\n\\centering\n\\includegraphics[scale=0.63]{HOPS_IRS_spectra_rebin.eps}\n\\caption{Three IRS spectra, one for HOPS 32 (Class 0 protostar; {\\it top}),\none for HOPS 84 (Class I protostar; {\\it middle}), and one for HOPS 105\n(flat-spectrum source; {\\it bottom}), overlaid with the rebinned data points\n({\\it filled circles}) used by the fitting routine. Note the different flux ranges\non the y axis in the three panels and thus the big differences in slopes\namong the three spectra.\n\\label{IRS_rebin}}\n\\end{figure}\n\nTo directly compare observed and model fluxes, we create model SEDs \nwith data points that correspond to those obtained from observations,\nfrom both photometry and IRS spectroscopy. For the former, the model \nfluxes are not only derived from the same apertures as the data (see \nsection \\ref{model_ap}), but also integrated over the various filter \nbandpasses, thus yielding model photometry. For the latter, the model \nfluxes are interpolated at the same 16 wavelength values as the IRS spectra.\n\nSince the model grid contains a limited number of values for the total luminosity\n(eight), but the objects we intend to fit have luminosities that likely do not\ncorrespond precisely to these values, we include scaling factors for the luminosity \nwhen determining the best-fit model. As long as these scaling factors are not far\nfrom unity, they are expected to yield SEDs that are very similar to those obtained \nfrom models using the scaled luminosity value as one of the input parameters.\nThe scaling factor can also be related to the distance of the source; for all model \nfluxes, a distance of 420 pc is assumed, but in reality the protostars in our sample\nspan a certain (presumably small) range of distances along the line of sight.\nFor example, a 10\\% change in distance would result in a $\\sim$ 20\\% change \nin flux values (scaling factors of 0.83 or 1.23). Here we report luminosities \nassuming a distance of 420 pc.\n\nIn addition to scaling factors, each model SED can be extinguished to account\nfor interstellar extinction along the line of sight. We use two foreground extinction \nlaws from \\citet{mcclure09} that were derived for star-forming regions: one applies \nto $0.76 \\leq A_J < 2.53$ (or $0.3 \\leq A_K < 1$), and the other one to $A_J \\geq 2.53$\n(or $A_K \\geq 1$). For $A_J < 0.76$, we use a spline fit to the Mathis $R_V=5$ \ncurve \\citep{mathis90}. Since the three laws apply to different extinction environments, \nwe use a linear combination of them to achieve a smooth change in the extinction law \nfrom the diffuse interstellar medium to the dense regions within molecular clouds. \nThus, to find a best-fit model for a certain observed SED, the model fluxes \n$F_{mod}(\\lambda)$ are scaled and extinguished  as follows:\n\\begin{equation}\nF_{obs}(\\lambda) = s F_{mod}(\\lambda) 10^{-0.4 A_{\\lambda}},\n\\label{F_scaled_ext}\n\\end{equation}\nwhere  $F_{obs}(\\lambda)$ and $F_{mod}(\\lambda)$ are the observed and model \nfluxes, respectively, $s$ is the luminosity scaling factor, and $A_{\\lambda}$\nis the extinction at wavelength $\\lambda$. We use three reddening laws,\n$k_{\\lambda}=A_{\\lambda}\/A_J$; by denoting them with the subscripts 1, 2,\nand 3, $A_{\\lambda}$ in the above equation becomes\n\\begin{eqnarray}\nA_{\\lambda} = A_J k_{1,\\lambda} \\quad {\\rm for}\\; A_J < 0.76 \\nonumber \\\\\nA_{\\lambda} = 0.76 k_{1,\\lambda} + (A_J - 0.76) k_{2,\\lambda} \n\\nonumber \\\\ {\\rm for}\\; 0.76 < A_J < 2.53 \\nonumber \\\\\nA_{\\lambda} = 0.76 k_{1,\\lambda} + 2.53 k_{2,\\lambda} + \n(A_J - 2.53) k_{3,\\lambda} \\nonumber \\\\ {\\rm for}\\; A_J > 2.53\n\\end{eqnarray}\nThus, equation \\ref{F_scaled_ext} can be written as\n\\begin{eqnarray}\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) \n= A_J k_{1,\\lambda} -2.5 \\log(s) \\nonumber \\\\ \n{\\rm for}\\; A_J < 0.76 \\nonumber \\\\\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) \n- 0.76 (k_{1,\\lambda}-k_{2,\\lambda}) = \\nonumber \\\\ \nA_J k_{2,\\lambda} -2.5 \\log(s) \\quad {\\rm for}\\; 0.76 < A_J < 2.53 \n\\nonumber \\\\\n2.5 \\log(F_{mod}(\\lambda)\/F_{obs}(\\lambda)) - 0.76 k_{1,\\lambda} \n- 2.53 (k_{2,\\lambda}-k_{3,\\lambda}) \\nonumber \\\\\n=  A_J k_{3,\\lambda} -2.5 \\log(s) \\quad {\\rm for}\\; A_J > 2.53 \\qquad\n\\end{eqnarray}\nThese are linear equations in $A_J$, with the left-hand side of the \nequations as the dependent variables and $k_{\\lambda}$ as the \nindependent variable. For each regime of $A_J$ values, a best-fit\nline can be determined that yields $A_J$ and $-2.5 \\log(s)$ from the \nslope and intercept, respectively, for each model that is compared to \nthe observations.\n\nFor each set of model fluxes and observed fluxes, we calculate three linear fits \n(using linear combinations of the three different extinction laws, as explained above),\nthus yielding three values for scaling factors and three for the extinction value. \nIf each extinction value is within the bounds of the extinction law that was \nused and smaller than a certain maximum $A_J$ value (which will be discussed\nbelow), and the scaling factor is in the range from 0.5 to 2.0, then the result \nwith the best linear fit will be used. \nHowever, if some of the values are not within their boundaries, then combinations\nof their limiting values are explored, and the set of scaling factor and extinction\nwith the best fit is adopted. For example, if a model has fluxes that are\nmuch higher than all observed fluxes, the linear fit described above will likely\nyield very large extinction values and small scaling factors. In this case the fitter\nwould only accept the smallest possible scaling factor (0.5) and the maximum \nallowed $A_J$ value as a solution (which will still result in a poor fit). \n\nFor each object, we allowed the model fluxes to be extinguished up to a maximum\n$A_J$ value derived from column density maps of Orion (\\citealt{stutz15}; see also\n\\citealt{stutz10,stutz13,launhardt13} for the methodology of deriving N$_H$ from \n160-500 \\micron\\ maps). We converted the total hydrogen column \ndensity from these maps to $A_V$ values ($A_V$=3.55 $A_J$)\nby using a conversion factor of \n$1.0 \\times 10^{21}$ cm$^{-2}$ mag$^{-1}$ \\citep{winston10, pillitteri13}. \nFor objects for which no column density could be derived, we set the maximum \n$A_J$ value to 8.45 (which corresponds to $A_V=30$). \n\nAfter returning a best-fit scaling factor and extinction value for each model, \neach data point is assigned a weight, and the goodness of the fit is \nestimated with \n\\begin{equation}\n R = \\frac{\\sum_{i=1}^{N} w_i |\\ln \\left(\\frac{F_{obs}(\\lambda_i)}\n{F_{mod}(\\lambda_i)}\\right)|}{N},\n\\end{equation}\nwhere $w_i$ are the weights, $F_{obs}(\\lambda_i)$ and $F_{mod}(\\lambda_i)$\nare the observed and the scaled and extinguished model fluxes, respectively, \nand N is the number of data points \\citep[see][]{fischer12}. Thus, $R$ is a measure \nof the average, weighted, logarithmic deviation between the observed and model SED. \nIt was introduced by \\citet{fischer12} since the uncertainty of the fit is dominated by \nthe availability of models in the grid (i.e, the spacing of the models in SED space) \nand not by the measurement uncertainty of the data, making the standard $\\chi^2$ \nanalysis less useful. Also, a statistic that measures deviations between models and \ndata in log space more closely resembles the assessment done by eye when \ncomparing models and observed SEDs in log($\\lambda F_{\\lambda}$) vs.\\ $\\lambda$ \nplots.\nWe set the weights $w_i$ to the inverse of the estimated fractional uncertainty \nof each data point; so, for photometry at wavelengths below 3 \\micron\\ they \nare equal to 1\/0.1, between 3 and 60 \\micron\\ they are 1\/0.05, at 70 and \n100 \\micron\\ they are 1\/0.04, at 160 \\micron\\ the weight is 1\/0.07, and \nfor photometry at 350 and 870 \\micron\\ they are 1\/0.4 and 1\/0.2, respectively. \nFor fluxes from IRS spectra the weights are 1\/0.075 for wavelength ranges \n8-12 \\micron\\ and 18-38 \\micron, while they are 1\/0.1 for the 5-8 \\micron\\ \nand 12-18 \\micron\\ regions. These IRS weights are also multiplied by 1.5 for\n high signal-to-noise spectra and by 0.5 for noisy spectra. In this way those \nparts of the IRS spectrum that most constrain the SED, the 10 \\micron\\ silicate \nabsorption feature and slope beyond 18 $\\mu$m, are given more weight; for \nhigh-quality spectra, the weights in these wavelength regions are the same as \nfor the 3-60 \\micron\\ photometry.\n\nFor small values, $R$ measures the average distance between model and data\nin units of the fractional uncertainty.\nIn general, the smaller the $R$ value, the better the model fit, but protostars with fewer \ndata points can have small $R$ values, while protostars with some noisy data can have \nlarger $R$ values (but still an overall good fit). We find a best-fit model for each object, \nbut we also record all those models that lie within a certain range of $R$ values from the \nbest-fit $R$. These models give us an estimate on how well the various model parameters\nare constrained (see Section \\ref{deltaR}).\n\nOur model grid is used to characterize the parameters that best describe the \nobserved SED of each object; the $R$ values rank the models for each \nobject and thus can be used to derive best-fit parameters, as well as estimates of \nparameter ranges. In several instances, better fits could be achieved if the model \nparameters were further adjusted, for example by testing more values of cavity \nopening angle or shape, or even changing the opacities (see, e.g., HOPS 68 \n\\citep{poteet11}, HOPS 223 \\citep{fischer12}, HOPS 59, 60, 66, 108, 368, 369, 370\n\\citep{adams12}, HOPS 136 \\citep{fischer14}, and HOPS 108 \\citep{furlan14}).\nHowever, for protostars that are well fit with one of the models from the grid or for \nwhich the grid yields a narrow range of parameter values, it is unlikely that a more \nextended model grid would yield much different best-fit parameters. Overall, our \nmodel fits yield good estimates of envelope parameters for a majority of the sample, \nand thus we can analyze the protostellar properties of our HOPS targets in a statistical \nmanner. \n\n\n\\section{Results of the Model Fits}\n\nThe best-fit parameters resulting from our models can be found in Table \nA\\ref{bestfit}, and Figure A\\ref{bestSEDs} shows the SEDs and best fits for our sample.\nIn this section we give an overview of the quality of the fits, the distributions \nof the best-fit model parameters, both for the sample as a whole and separated \nby SED class, the parameter uncertainties, and the various degeneracies between \nmodel parameters.\n\n\\subsection{Quality of the Fits}\n\\label{fit_quality}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{R_histogram.eps}\n\\caption{Histogram of the $R$ values of the best fits of the 330 \nYSOs in the HOPS sample that have {\\it Spitzer} and {\\it Herschel}\ndetections. \\label{R_histo}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.62,angle=90]{R_histogram_by_class.eps}\n\\caption{Histograms of the $R$ values of the best fits  shown separately\nfor the three classes of objects (Class 0, I, and flat-spectrum). The three \nfits with $R>8$ (two Class 0 protostars, one Class I protostar) are not shown.\n\\label{R_histo_by_class}}\n\\end{figure*}\n\nFigure \\ref{R_histo} displays the histogram of $R$ values of the best model\nfits for the 330 objects in our HOPS sample that have {\\it Spitzer} and \n{\\it Herschel} data (more than two data points at different wavelengths)\nand are not contaminants (see Section \\ref{sample}). The median $R$ \nvalue is 3.10, while the mean value is 3.29. Fitting a Gaussian to the \nhistogram at $R$ $\\leq$ 7 yields 3.00 and 2.24 as the center and FWHM \nof the Gaussian, respectively. \nThe distribution of $R$ values implies that, on average, the model deviates\nby about three times the average fractional uncertainty from the data.\nThis is not unexpected, given that we fit models from a grid to observed \nSEDs that span almost three orders of magnitude in wavelength range, \nwith up to 29 data points. The fewer the data points, the easier it is to \nachieve a good fit; in fact, the eight protostars with $R < 1$, HOPS 371, 391, \n398, 401, 402, 404, 406, and 409, have SEDs with measured flux values at \nonly 4-5 points.\nStarting at $R$ values of about 1, $R$ can be used as an indicator of the \ngoodness of fit. However, in some cases a noisy IRS spectrum can \nincrease the $R$ value of a fit that, judged by the photometry alone, does \nnot deviate much from the observed data points. In other cases, mismatches \nbetween different data sets, like offsets between the IRAC fluxes and the IRS\nspectrum, can result in larger $R$ values. These might be interesting protostars \naffected by variability and are thus ideal candidates for follow-up observations. \n\nWhen looking at the SED fits in Figure A\\ref{bestSEDs} (and the corresponding \n$R$ values in Table A\\ref{bestfit}), we estimate that an $R$ value of up to \n$\\sim$ 4 can identify a reliable fit (with some possible discrepancies between \ndata and model in certain wavelength regions). When $R$ gets larger than \nabout 5, the discrepancy between the fit and the observed data points usually\nbecomes noticeable; the fit might still reproduce the overall SED shape\nbut deviate substantially from most measured flux values. \n\nIn Figure \\ref{R_histo_by_class}, we show the histogram of $R$ values \nseparately for the three main protostellar classes in our sample. The \nmedian $R$ value decreases from 3.27 for the Class 0 protostars to 3.18 for \nthe Class I protostars to 2.58 for the flat-spectrum sources. There are \n4 Class 0 protostars and 4 Class I protostars with $R$ values between 1.0 \nand 2.0, but 17 flat-spectrum sources in this $R$ range. These numbers \ntranslate to 17\\% of the flat-spectrum sources in our sample, 4\\% of the \nClass 0 protostars, and 3\\% of the Class I protostars. When examining objects' \n$R$ values between 2.0 and 4.0, there are 51 Class 0 protostars (55\\% of \nClass 0 protostars in the sample), 91 Class I protostars (73\\% of the Class I \nsample), and 74 flat-spectrum sources (73\\% of the flat-spectrum sample).\n\nThus, close to 90\\% of flat-spectrum sources are fit reasonably well ($R$ \nvalues $<$ 4), representing the largest fraction among the different classes \nof objects in our sample. This could be a result of their source properties \nbeing well represented in our model grid, but also lack of substantial \nwavelength-dependent variability (see, e.g., \\citealt{guenther14}), which, \nif present, would make their SEDs more difficult to fit.  \nAbout three-quarters of Class I protostars also have best-fit models with $R < 4$; \nthis fraction drops to about two-thirds for the Class 0 protostars. The latter \ngroup of objects often suffers from more uncertain SEDs due to weak\nemission at shorter wavelengths (which, e.g., results in a noisy IRS\nspectrum); they might also be more embedded in extended emission, \nsuch as filaments, which can contaminate the far-IR to submillimeter fluxes. \nAnother factor that could contribute to poor fits is their presumably\nhigh envelope density, which places them closer to the limit in parameter\nspace probed by the model grid.\nOverall, 75\\% of the best-fit models of the protostars in our sample have\n$R < 4$.\n\nWhen examining the SED fits of objects with $R$ values larger than 5.0,\nseveral have very noisy IRS spectra (HOPS 19, 38, 40, 95, 164, 278, 316,\n322, 335, 359). In a few cases the measured PACS 100 and 160 $\\mu$m \nfluxes seem too high compared to the best-fit model (e.g., HOPS 189), \nwhich could be an indication of contamination by extended emission \nsurrounding the protostar. \n\nOf particular interest are objects where variability likely plays a role in \na poor fit. As mentioned in Section \\ref{SEDs}, variability among protostars\nis common; we found in Appendix \\ref{variability} that about 5\\% of our \ntargets display noticeable ($\\gtrsim$ 50\\%) mismatches between \nthe IRS, IRAC, and MIPS fluxes that could be due to intrinsic variability.\nThe SED fits of objects for which the flux mismatches between IRS and \nIRAC and between IRS and MIPS are different are particularly affected, \nsince in that case we did not scale the IRS spectrum to match the \nMIPS 24 $\\mu$m flux.\nHOPS 228 exemplifies such a case: there is a clear discrepancy\nbetween the IRAC and IRS fluxes (a factor of 2.1-2.7) and also\nbetween MIPS 24 $\\mu$m and IRS (a factor of 0.8); even though \nthe fit gives more weight to the IRS data, they are not fit well, especially \nthe silicate absorption feature. The $R$ value of 5.74 for the fit of\nHOPS 228 reflects the discrepant data sets and poor fit. \nHOPS 223 is another case where the IRS  fluxes do not match the \nshorter-wavelength data (they are more than an order of magnitude \nlarger); however, it is a known FU Ori source \\citep[see][]{fischer12},\nand the SED presented here contains both pre- and post-outburst data. \nThe model fit is very poor, which can also be gauged by the $R$ value \nof 8.41. \n\nThere are also objects with overall good fits whose SEDs show discrepancies\nthat may be signs of variability or contamination. \nFor example, for the Class I protostar HOPS 71 the IRAC fluxes are a factor \nof 1.8-2.4 lower than the IRS fluxes in the 5-8 $\\mu$m region, and also the \nMIPS flux is about 20\\% lower. The best-fit model ($R=3.63$) fits the SED \nextremely well beyond about 6 $\\mu$m, with some discrepancy at shorter \nwavelengths. There is a source just 11\\arcsec\\ from HOPS 71 that is detected\nin 2MASS and {\\it Spitzer} data, but not by PACS; this object, HOPS 72, is\nlikely an extragalactic object (see Appendix \\ref{exgal_not_modeled}) that\ncould contaminate the IRS fluxes. Thus, in this case, wavelength-dependent\ncontamination by a companion could explain the discrepancies observed in\nthe SED.\n\nAnother example is HOPS 124, which is a deeply embedded Class 0 protostar. \nFor this object, the mismatch between IRS and IRAC and MIPS fluxes \ndecreases with increasing wavelength (from a factor of 2.5 to a factor of 1.4); \nfor the SED fit, the IRS spectrum was scaled by 0.7 to match the MIPS 24 \n$\\mu$m flux. As with HOPS 71, there is a nearby source that could contaminate \nsome of the fluxes, especially at shorter wavelengths: HOPS 125, a flat-spectrum \nsource, lies 9.8\\arcsec\\ from HOPS 124 and is brighter than HOPS 124 out to \n$\\sim$ 20 $\\mu$m, but then much fainter at longer wavelengths. The best-fit \nmodel of HOPS 124 ($R=2.43$) matches the mid- to far-IR photometry and \nalso most of the IRS spectrum well.\n\nAs an example of a probably variable flat-spectrum source, HOPS 132 \nhas IRAC fluxes that lie a factor of 1.3-1.7 above those of IRS and a \nMIPS 24 $\\mu$m flux that is a factor of 0.6 lower. It does not have a close\ncompanion; the nearest HOPS source, HOPS 133, is 27\\arcsec\\ away.\nThe IRS spectrum was not scaled, and since the SED fitter gave more \nweight to the spectrum, it is fit well, but the IRAC photometry is underestimated\nand the MIPS photometry overestimated. Nonetheless, the $R$ value\nof the best fit is 2.87.\n\nOverall, the SED fits of objects that are likely variable or suffer from some\ncontamination are less reliable, but it is not always clear from the $R$ \nvalue of the best fit. The SED fitting procedures assume that the protostars \nare not variable, so when large mismatches between different data sets are \npresent, the fit will appear discrepant with at least some of the observed\ndata points, but the $R$ value would not end up particularly high \nif, e.g., the IRS spectrum was fit exceptionally well. However, given the\ndata sets we have for these protostars, our SED fits will still yield the best \npossible estimate for the protostellar parameters describing these systems.\n\n \n\\subsection{Overview of Derived Parameters}\n\\label{results_overview}\n\nThe histogram of best-fit $\\rho_{1000}$ values (which is the density of the \nenvelope at 1000 AU; see Section \\ref{model_parameters}) is shown in \nFigure \\ref{rho1000_histo}.\nThe median value of the distribution amounts to $5.9 \\times 10^{-19}$ \ng cm$^{-3}$; this corresponds to a $\\rho_1$ value of $1.9 \\times \n10^{-14}$ g cm$^{-3}$. There is a spread in values: 69 objects have \ndensities $\\rho_{1000}$ smaller than $5.0 \\times 10^{-20}$ g cm$^{-3}$\n(6 of them have actually no envelope), 89 fall in the $5.0 \\times\n10^{-20}$ to $5.0 \\times 10^{-19}$ g cm$^{-3}$ range, 96\nare between $5.0 \\times 10^{-19}$ and $5.0 \\times 10^{-18}$ g \ncm$^{-3}$, 60 between  $5.0 \\times 10^{-18}$ and $5.0 \\times \n10^{-17}$ g cm$^{-3}$, and 16 have $\\rho_{1000}$ values\nlarger than $5.0 \\times 10^{-17}$ g cm$^{-3}$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Rho1000_histogram.eps}\n\\caption{Histogram of the envelope reference density $\\rho_{1000}$ \nof the best fits for the 330 targets in our sample. \n\\label{rho1000_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Menv_histogram.eps}\n\\caption{Histogram of the envelope mass within 2500 AU derived for\nthe best fits for the 330 targets in our sample.\n\\label{Menv_histo}}\n\\end{figure}\n\nWe also calculated the envelope mass ($M_{env}$) within 2500 AU for \nthe best-fit models (see Figure \\ref{Menv_histo} for their distribution). \nThe 2500 AU radius is close to half the FWHM of the PACS 160 $\\mu$m \nbeam at the distance of Orion (i.e., $\\sim$ 6\\arcsec), and thus roughly \nrepresents the spatial extent over which we measure the SEDs.\nThis envelope mass is determined from the integrated envelope density \nof our best-fit models, with allowances made for outflow cavities, and thus \nonly valid in the context of our models. The median envelope mass within \n2500 AU amounts to 0.029 $M_{\\odot}$. The majority of protostars have \nmodel-derived masses in the inner 2500 AU of their envelopes around \n0.1~$M_{\\odot}$; just 22 objects have $M_{env}$ ($<$ 2500 AU) larger than \n1.0~$M_{\\odot}$. Of the 330 modeled objects, 291 have $M_{env}$ ($<$ 2500 AU)\nsmaller than 0.5 $M_{\\odot}$\\ (6 of these 291 objects have no envelope).  \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Ltot_histogram.eps}\n\\caption{Histogram of the total luminosities of the best fits for the 330 targets \nin our sample. \n\\label{Ltot_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Rdisk_histogram.eps}\n\\caption{Histogram of the disk radii of the best fits for the 330 targets \nin our sample. \n\\label{Rdisk_histo}}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Cavity_histogram.eps}\n\\caption{Histogram of the cavity opening angles of the best fits for the \n330 targets in our sample. \n\\label{Cav_histo}}\n\\end{figure}\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.7]{Rho1000_and_cavity_histogram.eps}\n\\includegraphics[scale=0.7]{Ltot_and_cavity_histogram.eps}\n\\includegraphics[scale=0.7]{Rdisk_and_cavity_histogram.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$ \n({\\it left}), the total luminosity ({\\it middle}), and the disk radius ({\\it right})\nof the best fits grouped by cavity opening angles. \n\\label{Pars_cav_histo}}\n\\end{figure*}\n\nFigure \\ref{Ltot_histo} contains the histogram of the total luminosities\nderived from the best-fit models. These luminosities consist of the stellar,\ndisk accretion, and accretion shock components. The median total luminosity \namounts to 3.02 $L_{\\odot}$, while the values cover four orders of magnitude, \nfrom 0.06 $L_{\\odot}$\\ (for HOPS 336) to 607 $L_{\\odot}$\\ (for HOPS 288 and 361). \nSince the minimum and maximum values for the total luminosity \nin our grid amount to 0.1 and 303.5 $L_{\\odot}$, respectively, and our scaling \nfactors range from 0.5 to 2.0, our fitting procedure can return best-fit\nluminosities that range from 0.05 to 607 $L_{\\odot}$. Thus, two protostars are\nreaching the upper limit allowed for total luminosities in our grid; it is possible \nthat even better fits could be achieved by increasing the luminosity further. \n\nFrom the distribution of best-fit outer disk radii in Figure \\ref{Rdisk_histo}, it is \napparent that most protostars are fit by small disks whose radius is only 5 AU. \nSince the outer disk outer radius is the centrifugal radius in our models, \ninfalling material from the envelope tends to accumulate close to the star \nfor most sources. Thus, the disk radius is tied to the envelope structure; \na small centrifugal radius implies higher envelope densities at smaller radii\nand a less flattened envelope structure. The median disk radius is 50 AU, \nbut the number of objects with disk radii $\\geq$ 50 AU is roughly evenly \nsplit among the values of 50, 100, and 500 AU.\n\nThe distribution of best-fit cavity opening angles is displayed in Figure \n\\ref{Cav_histo}. Most protostars seem to have either very small (5\\degr) \nor very large (45\\degr) cavities; the median value is 25\\degr.\nWhen dividing the envelope densities by cavity opening angle (see Figure\n\\ref{Pars_cav_histo}, left column), differences emerge: the distributions of \n$\\rho_{1000}$ values are significantly different when comparing objects \nwith $\\theta$=5\\degr\\ and $\\theta \\geq$35\\degr, objects with \n$\\theta$=15\\degr\\ and $\\theta$ $\\geq$ 25\\degr, and objects with \n$\\theta$=25\\degr\\ and $\\theta$=45\\degr. The Kolmogorov-Smirnov (K-S)\ntests yield significance levels that these subsamples are drawn from the\nsame parent population of $\\lesssim$ 0.015. Thus, there seems to be a \ndifference in the distribution of envelope densities among the best-fit \nmodels with smaller cavity opening angles and those with larger cavities. \nProtostars with larger cavities ($\\geq$ 35\\degr) tend to have higher envelope \ndensities (their median $\\rho_{1000}$ values are about an order of magnitude \nlarger compared to objects with cavities $\\leq$ 15\\degr). \n\nFigure \\ref{Pars_cav_histo} (middle column) also shows the distribution \nof total luminosities for the different cavity opening angles. The only \nsignificant difference can be found for the $\\theta$=5\\degr\\ histogram \nas compared to the histograms for larger $\\theta$ values (K-S test\nsignificance level $\\lesssim$ 0.03); the luminosities of models with \n$\\theta$=5\\degr\\ have a different distribution, and also their median \nvalue is 1.45 $L_{\\odot}$, as compared to $\\sim$ 3-5 $L_{\\odot}$\\ for the models\nwith larger cavities. So, protostars with small cavities seem to have lower\ntotal luminosities.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.55]{Inc_histogram.eps}\n\\caption{Histogram of the inclination angles of the best fits for the 330 \ntargets in our sample. The green dashed histogram represents the \ndistribution of uniformly (randomly) distributed inclination angles.\n\\label{Inc_histo}}\n\\end{figure}\n\nThe distribution of centrifugal radii for different cavity opening angles \n(right column in Figure \\ref{Pars_cav_histo}) shows that, independent of \ncavity size, most objects have $R_{disk}$ = 5 AU. However, the\ndistribution among the four different disk radii becomes flatter for \nthe largest cavity opening angles; the histograms for $\\theta$=35\\degr\\ \nand $\\theta$=45\\degr\\ are very similar (K-S test significance level of\n0.98). There is also no significant difference (K-S test values $>$ 0.075)\nbetween the $\\theta$=15\\degr\\ and $\\theta$=25\\degr\\ histograms and \nbetween the $\\theta$=5\\degr\\ and $\\theta \\geq $ 35\\degr\\ histograms. \nThe distributions of disk radii for the other cavity opening angles are all \ndifferent from one another (K-S test significance levels $<$ 0.015). \nOverall, Figure \\ref{Pars_cav_histo} shows that protostars best fit by \nmodels with large cavity opening angles are also fit by models with \nhigher envelope densities and larger centrifugal radii.  \n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.54]{Cumulative_inc.eps}\n\\caption{Cumulative distribution of the inclination angles of the best fits,\nnormalized by the total number of fits ({\\it solid line}), compared to the \ncumulative probability of finding an inclination angle below a given value\nfor randomly distributed inclinations ({\\it green dashed line}). \n\\label{Inc_CDF}}\n\\end{figure}\n\nIn Figure \\ref{Inc_histo}, we show the distribution of the inclination angles\nfor the best-fit models. There is a clear concentration of models in the \n60\\degr$-$70\\degr\\ range; the median inclination angle is 63\\degr.\nThis median value is close to 60\\degr, which is where the probability for \nisotropically distributed inclination angles reaches 50\\% (i.e., the probability \nof observing an inclination angle less than 60\\degr\\ is the same as the \nprobability of observing i$>$60\\degr). However, the details of the\ndistributions differ.\nThe cumulative probability of finding an inclination angle less than a certain value, \n$i_c$, is $1-\\cos(i_c)$, assuming a random distribution of inclination angles. For\ninclination angles $i_1$ and $i_2$, the probability for $i_1 < i < i_2$ is\n$\\cos(i_1)-\\cos(i_2)$. Thus, since the inclination angles in our model grid were \nchosen to be equally spaced in $\\cos(i)$ (there are five values $<$60\\degr\\ and \nfive values $>$60\\degr), one would expect a flat distribution in Figure \\ref{Inc_histo} \nif the best-fit inclination angles were randomly distributed (see the green dashed\nhistogram). However, we find a distribution peaked at 63\\degr\\ and 70\\degr. \nThis can also be seen in Figure \\ref{Inc_CDF}, where we compare our observed \ncumulative distribution of inclination angles to that of randomly distributed ones. \nOur distribution shows a deficit at inclination angles below 60\\degr\\ and \nis just slightly higher at large inclination angles. A K-S test of the two \ndistributions yields a 5.6\\% chance that they are drawn from the same parent\ndistribution.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.59]{Rho1000_and_inc_histogram.eps} \n\\includegraphics[scale=0.59]{Ltot_and_inc_histogram.eps}\n\\includegraphics[scale=0.59]{Cavity_and_inc_histogram.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$\n({\\it left}), total luminosity ({\\it middle}), and cavity opening angles \n({\\it right}) of the best fits divided by bins of inclination angles. \n\\label{Pars_inc_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.37,angle=90]{Ltot_vs_Lbol_and_inc.eps}\n\\includegraphics[scale=0.37,angle=90]{Ltot_vs_Lbol_and_Av.eps}\n\\caption{Ratio of the total luminosity from the best fits and the bolometric \nluminosity derived from the observed SEDs versus the inclination angle ({\\it \nleft}) and foreground extinction ({\\it right}) of the best fits. In the left\npanel, the open stars represent the median ratios at each inclination angle.\nIn the right panel, the open circles represent the median ratios for eight\nbins in $A_V$ values, represented by the horizontal lines bisecting\neach circle.\n\\label{Ltot_Lbol}}\n\\end{figure*}\n\nTo examine whether the distribution of envelope parameters changes with\ninclination angle (which could imply a degeneracy), Figure \\ref{Pars_inc_histo}\nshows the reference envelope density $\\rho_{1000}$, the total luminosity, \nand the cavity opening angle binned by three ranges of inclination angles.\nNone of the three model parameters show a significantly different distribution \nfor any of the inclination bins (K-S test significance levels are $\\gtrsim$ 0.1,\nexcept for the cavity opening angles for the lowest and middle inclination\nrange, for which the K-S test significance value is 0.02). \nThe median $\\rho_{1000}$ values for the $i=$18\\degr--41\\degr, 49\\degr--63\\degr, \nand 69\\degr--87\\degr\\ inclination bins are all $5.9 \\times 10^{-19}$ g cm$^{-3}$. \nEven though not shown in Figure \\ref{Pars_inc_histo}, the objects whose best-fit\nmodel does not include an envelope are only found at $i \\geq$ 49\\degr. It is\nnoteworthy that protostars with the highest envelope densities do not have \ninclination angles in the 69\\degr--87\\degr\\ range; it is not clear whether this \nis an observational bias, whether our observed sample does not contain high-density, \nedge-on protostars, or whether this is due to biases in the fitting procedure and\/or \nmodel grid.\nThe median values for the total luminosity do not differ by much for the \ndifferent bins of inclination angle, increasing from 2.9 to 4.1 $L_{\\odot}$\\ from\nthe lowest to the middle inclination range and then decreasing to 2.0 $L_{\\odot}$\\\nfor the highest inclination angles. The few protostars with very high $L_{tot}$\nvalues have large inclination angles ($i \\geq$ 49\\degr).\nFinally, the distribution of cavity opening angles is quite similar for different\nranges in inclination, except for a somewhat larger number of $\\theta =$ 45\\degr\\\nvalues at intermediate inclination angles. Half the objects in the $i=$18\\degr--41\\degr\\ \nand 69\\degr--87\\degr\\ inclination bins have $\\theta \\leq $ 15\\degr\\ (with the most\ncommon value 5\\degr), while almost half the objects at intermediate inclination \nangles have $\\theta \\geq $ 35\\degr\\ (the most common value is 45\\degr). \n\nIn Figure \\ref{Ltot_Lbol}, we show ratios of the total and bolometric luminosities\nas a function of inclination angle and foreground extinction ($i$ and $A_V$ are\nadopted from the best model fits). The total luminosity is the intrinsic luminosity \nfrom the best-fit model of each object, while the bolometric luminosity is derived \nby integrating the fluxes of the observed SED. It is expected that $L_{tot}$ is \nhigher than $L_{bol}$ for objects seen at higher inclination angles, since for \nthese objects a large fraction of the emitted flux is not directed toward the \nobserver (and thus deriving bolometric luminosities from observed fluxes will \nunderestimate the intrinsic source luminosity). Conversely, objects seen more \nface-on should have lower $L_{tot}$ values compared to $L_{bol}$. Our data \nand model fits yield $L_{tot}$ values that are usually higher than the $L_{bol}$\nvalues measured from the SED; the discrepancy is larger for the more \nhighly inclined sources. The median $L_{tot}\/L_{bol}$ ratio is 1.5 for \nprotostars with inclination angles in the 18\\degr--41\\degr\\ range, 2.5\nfor the i=49\\degr--63\\degr\\ range, and 3.5 for inclination angles \n$\\geq$ 69\\degr. \nThe fact that $L_{tot}>L_{bol}$ even for $i=$18\\degr--41\\degr\\ could\nbe related to the typically smaller cavity opening angles for this range of\ninclination angles (see Figure \\ref{Pars_inc_histo}); less flux, especially at\nshorter wavelengths, is detected since the opacity along the line of sight \nis still high due to the small cavities. \n\nForeground extinction also plays a role in increasing the $L_{tot}$\/$L_{bol}$\nratio. The median ratio of these luminosities increases from 1.8 for the $A_V$=\n0-5 mag range to 5.0 for $A_V$=25-30; it decreases somewhat for the next\n$A_V$ bin, but reaches 5.9 at $A_V$=40-50 (the 23 objects with $A_V >$ 50, \nnot shown in Figure \\ref{Ltot_Lbol}, have a median $L_{tot}$\/$L_{bol}$ ratio of 8.2). \nAmong the 22 objects with best-fit $A_V$ values of 0-5 mag and inclination angles \n$\\leq$ 50\\degr, only four have $L_{tot}\/L_{bol}$ ratios that are larger than \n1.5 (they are HOPS 57, 147, 199, and 201; in most cases the model overestimates \nthe near-IR emission).   \n\n\n\\subsection{Envelope Parameters for Different SED Classes}\n\\label{par_classes}\n\nFigures \\ref{Rho_class_histo}--\\ref{AV_class_histo} divide the histograms of the \nbest-fit reference density $\\rho_{1000}$, inclination angle, cavity opening angle, \ntotal luminosity, disk radius, and foreground extinction, respectively, by protostar \nclass. As explained in Section \\ref{SEDs}, we divided our targets into Class 0, \nClass I, flat-spectrum, and Class II objects based on their mid-infrared (4.5-24 \\micron) \nspectral index and bolometric temperature (see also Table A\\ref{bestfit}). Thus, \nClass 0 and I protostars have a spectral index $>$ 0.3, and Class 0 protostars have \n$T_{bol}$ values $<$ 70 K, but, as mentioned in Section \\ref{SEDs}, there are \na few protostars whose spectral index or $T_{bol}$ value places them very close \nto the transition region between Class 0 and I or between Class I and flat spectrum. \nGiven that our sample contains just eleven Class II pre-main-sequence stars, we did \nnot include them in the following histograms; they will be discussed in section \n\\ref{disk_sources}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Rho1000_histogram_by_class.eps}\n\\caption{Histograms of the envelope reference density $\\rho_{1000}$ \nof the best fits for the different SED classes. \n\\label{Rho_class_histo}}\n\\end{figure*}\n\nThe distributions of reference densities (Figure \\ref{Rho_class_histo}) are \ndifferent for all SED classes; none are consistent with being drawn from the \nsame parent population (K-S test significance level $<$ 0.01). Overall, Class 0 \nprotostars have higher envelope densities than Class I and flat-spectrum sources; \nthe median $\\rho_{1000}$ values decrease from 5.9 $\\times 10^{-18}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-19}$ g cm$^{-3}$ to 1.2 $\\times 10^{-19}$ g cm$^{-3}$ for \nthese three groups. The lower and upper quartiles for $\\rho_{1000}$ are \n1.8 $\\times 10^{-18}$ and 1.8 $\\times 10^{-17}$ g cm$^{-3}$ for the Class 0\nprotostars, and 2.4 $\\times 10^{-20}$ and 1.2 $\\times 10^{-18}$ g cm$^{-3}$ for the \nClass I and flat-spectrum objects.\nWe will discuss some implications of these differences in derived envelope\ndensities in section \\ref{SED_class_properties}.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65,angle=90]{Inc_histogram_by_class.eps}\n\\caption{Histograms of the inclination angles of the best fits for the different \nSED classes. \n\\label{Inc_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Cavity_histogram_by_class.eps}\n\\caption{Histograms of the cavity opening angles of the best fits for the different \nSED classes. \n\\label{Cav_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Ltot_histogram_by_class.eps}\n\\caption{Histograms of the total luminosity of the best fits for the different \nSED classes. \n\\label{Ltot_class_histo}}\n\\end{figure*}\n\nFor the inclination angles (Figure \\ref{Inc_class_histo}), the distributions \nare significantly different for all protostellar classes, too (K-S test significance \nlevel $\\ll$ 0.01). As was shown in Figure \\ref{Inc_histo}, a random distribution of \ninclination angles would result in equal numbers of protostars at each value; \nthere is a deficit of Class 0 and Class I protostars at $i \\lesssim$ 60\\degr, and \nthere are also few Class I protostars and hardly any flat-spectrum sources \nat the highest inclination angles. \nThe median inclination angle is highest for Class 0 protostars (70\\degr), then \ndecreases somewhat for Class I protostars (63\\degr) and even more for \nflat-spectrum sources (57\\degr). Similar to the envelope density, \nthe median inclination angle decreases as one progresses from Class 0 to \nflat-spectrum sources. \n\nIn the distributions of cavity opening angles (Figure \\ref{Cav_class_histo}), \nsignificant differences can be found between Class 0 and Class I protostars \nand between Class I protostars and flat-spectrum sources (K-S test significance \nlevel $\\ll$ 0.01). The median cavity opening angle is 15\\degr\\ for the Class I \nprotostars, but 25\\degr\\ for the other two classes. About 40\\% of Class I protostars \nhave $\\theta$=5\\degr, while the distribution among the different cavity opening \nangles is flatter for the other two object classes. The large fraction of Class I\nprotostars with small cavities could be the result of degeneracy in model parameters \n(see section \\ref{SED_class_properties}) or our assumptions on envelope geometry \n(see section \\ref{Model_problems}). There are notably few flat-spectrum sources \nwith a 5\\degr\\ cavity opening angle; most of them have cavity opening angles of \n15\\degr\\ or 45\\degr.\n\nWhen comparing the total luminosities for the different SED classes\n(Figure \\ref{Ltot_class_histo}), the distribution of $L_{tot}$ values\nis different for the Class 0 protostars when compared to the other two\nclasses (K-S test significance level $<$ 0.015), but similar for Class I \nprotostars and flat-spectrum sources. The median total luminosity for \nClass 0 protostars is 5.5 $L_{\\odot}$, compared to 2.0 $L_{\\odot}$\\ for Class I \nprotostars and 3.0 $L_{\\odot}$\\ for flat-spectrum sources. Both Class 0 and I \nprotostars cover close to the whole range of $L_{tot}$ values in the model grid \n($\\sim$ 0.06-600 $L_{\\odot}$), while flat-spectrum sources span a more limited \nrange, from 0.1 to 316 $L_{\\odot}$.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rdisk_histogram_by_class.eps}\n\\caption{Histograms of the disk radii of the best fits for the different \nSED classes. \n\\label{Rdisk_class_histo}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{AV_histogram_by_class.eps}\n\\caption{Histograms of the foreground extinction of the best fits for the different \nSED classes. \n\\label{AV_class_histo}}\n\\end{figure*}\n\nThe distribution of centrifugal radii for the whole sample showed a preference\nfor 5 AU (see Figure \\ref{Rdisk_histo}). When separating the best-fit disk\nradii by protostellar class (Figure \\ref{Rdisk_class_histo}), it is clear that the\ntrend for small centrifugal radii is driven by the flat-spectrum sources \nand also Class I protostars. The fraction of Class 0 protostars with $R_{disk}$=\n5 AU is 17\\%; it increases to 46\\% and 73\\% for Class I protostars and \nflat-spectrum sources, respectively. The median disk radius decreases \nfrom 100 AU for Class 0 protostars to 50 AU for Class I protostars to 5 AU for \nflat-spectrum sources. All three histograms are significantly different from \none another (K-S test significance level $\\ll$ 0.001). The unexpectedly small\ncentrifugal radii for Class I protostars and flat-spectrum sources could point\nto parameter degeneracies (see section \\ref{SED_class_properties}) or the\nneed to revise certain model assumptions (see section \\ref{Model_problems}).\n\nFinally, the distribution of best-fit foreground extinction values (Figure \n\\ref{AV_class_histo}) is similar for all three object classes (K-S test significance \nlevel $>$ 0.03). Even the median values are close: $A_V$=9.2 for Class 0 \nprotostars, $A_V$=8.9 for Class I protostars, and $A_V$=10.1 for flat-spectrum\nsources. Most objects are fit with relatively low foreground extinction values.\nAs can be seen from Figure \\ref{AV_models_maps}, the majority of protostars\nhave best-fit $A_V$ values well below the maximum $A_V$ values determined\nfrom column density maps, which were used as the largest allowed $A_V$\nvalues for the SED fitter. The ratio of model-derived $A_V$ to observationally\nconstrained maximum $A_V$ is lower than 0.5 for about 60\\% of the sample.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.53]{AV_model_AV_maps.eps}\n\\caption{Foreground extinction values $A_V$ from the best-fit models versus\nthe maximum $A_V$ value determined from column density maps of Orion.\nThe dashed line indicates where the two $A_V$ values are equal.\n\\label{AV_models_maps}}\n\\end{figure}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rho1000_vs_AV_by_class.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus the foreground extinction for the different \nSED classes. Note that there are a few objects at $A_V > 75$, but they are not shown\nfor overall clarity of the figure.\n\\label{Rho_AV_class}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.65, angle=90]{Rho1000_vs_inc_by_class.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus inclination angle for the different \nSED classes. The size of the plotting symbol increases with the number of objects\nhaving the same ($i$, $\\rho_{1000}$) combination; the legend in the leftmost panel\nshows which symbol size corresponds to which number of objects. \n\\label{Rho_inc_class}}\n\\end{figure*}\n\nIn Figure \\ref{Rho_AV_class}, we plot the reference densities $\\rho_{1000}$\nversus the foreground extinction for Class 0, Class I, and flat-spectrum sources.\nAs was already seen in Figure \\ref{AV_class_histo}, the extinction along the line\nof sight is similar for all three classes, with most objects in the $A_V \\sim$ 0-30\nregime. Class 0 protostars, which have higher envelope densities, tend to \nhave lower $A_V$ values from foreground extinction; the highest-density\nenvelopes are spread among a wide range of $A_V$ values. The result is\nsimilar for Class I protostars. Flat-spectrum sources display a range in \nenvelope densities at various foreground extinction values; the lowest-density\nenvelopes typically have $A_V < 20$.\nThus, foreground extinction does not seem to affect the classification of\nprotostars. This result is also supported by the statistical analysis of \\citet{stutz15},\nwho found that, for $A_V$ values up to 35, the misclassification of a Class I\nprotostar as a Class 0 protostar due to foreground extinction (which results in a\nlower $T_{bol}$) is low.\n\n\\begin{deluxetable*}{l|ccc}\n\\tablewidth{0.9\\linewidth}\n\\tablecaption{Median Best-Fit Parameter Values for the Three Protostellar Classes\n\\label{Median_par}}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Class 0} & \\colhead{Class I} & \n\\colhead{Flat-spectrum}}\n\\startdata\n$L_{tot}$ & 5.5 $L_{\\odot}$ & 2.0 $L_{\\odot}$ & 3.0 $L_{\\odot}$ \\\\\n$\\rho_{1000}$ & 5.9 $\\times 10^{-18}$ g cm$^{-3}$ & \n2.4 $\\times 10^{-19}$ g cm$^{-3}$ & 1.2 $\\times 10^{-19}$ g cm$^{-3}$ \\\\\n$\\theta$ & 25\\degr & 15\\degr & 25\\degr \\\\\n$R_{disk}$ & 100 AU & 50 AU & 5 AU \\\\\n$i$ & 70\\degr & 63\\degr & 57\\degr \\\\\n$A_V$ & 9.2 & 8.9 & 10.1 \\\\\n\\enddata\n\\end{deluxetable*}\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.5, angle=90]{Median_model_SEDs.eps}\n\\caption{Model SEDs for Class 0 protostars ({\\it red}), Class I protostars\n({\\it green}), and flat-spectrum sources ({\\it blue}) with parameter values \nequal to the median values for each SED class (see Table \\ref{Median_par}).\n\\label{median_SEDs}}\n\\end{figure*}\n\nWe found differences in the best-fit envelope densities and inclination angles\nfor the various protostellar classes.\nThe result that Class 0 protostars tend to have larger inclination angles and \nenvelope densities compared to Class I and flat-spectrum objects can also\nbe seen in Figure \\ref{Rho_inc_class}. There are very few Class 0 protostars\nwith low inclination angles; most have relatively high density and $i>$60\\degr. \nClass I protostars are best fit by somewhat lower inclination angles than Class 0 \nprotostars and also lower $\\rho_{1000}$ values. The best-fit reference density\nfor Class I protostars decreases as the inclination angle increases; thus, higher-density\nprotostars are typically classified as Class I protostars only if they are not seen at\nclose to edge-on orientations.\nFlat-spectrum sources are spread out in density--inclination space, but intermediate \ninclination angles and low envelope densities are common. There is a relatively\nlarge number of objects at $i=$18\\degr\\ and a deficit of objects at high inclination\nangles. The highest-density flat-spectrum sources are seen at inclination angles \n$<$ 50\\degr, while the lower-density objects cover almost the full range of \ninclination angles.\n\nThe median parameter values we determined from the best fits for the \nClass 0, Class I, and flat-spectrum sources (see Table \\ref{Median_par}) \ncan be used to show representative median SEDs for each protostellar class. \nIn Figure \\ref{median_SEDs}, we show model SEDs whose parameter values \nare equal to the median values found for each of the three protostellar classes. \nIt is apparent that the large envelope density and higher inclination angle for \nClass 0 protostars cause a deep absorption feature at 10 $\\mu$m and a steeply \nrising SED in the mid- and far-IR, with a peak close to 100 $\\mu$m. In Class I \nprotostars, the SED is less steep and peaks at a shorter wavelength than \nthe median SED of Class 0 protostars. Flat-spectrum sources show the \nstrongest near-IR emission of the three protostellar classes; their median\nSED is very flat out to 70 $\\mu$m, but at longer wavelengths it is very \nsimilar in shape and flux level to that of Class I protostars.\n\n\n\\subsection{Estimating Parameter Uncertainties}\n\\label{deltaR}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Inc_modes.eps}\n\\caption{Mode of the inclination angle of all models that lie within \n0.5, 1.0, 1.5, and 2 of the best-fit $R$ value (from left to right) versus \nthe best-fit inclination angle for all 330 objects in our sample. \nNote that for each data point, small random offsets in the x and y direction \nhave been applied to avoid overlap. Also, when two or more parameter \nvalues had the same frequency within a $\\Delta R$ bin (i.e., not a unique \nmode value), we computed the average of these values and used it for \nthe mode. The dashed line indicates where the mode and best-fit value \nare equal.\n\\label{inc_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Ltot_modes.eps}\n\\caption{Mode of the total luminosity of all models that lie within \n0.5, 1.0, 1.5, and 2 of the best-fit $R$ value (from left to right) versus \nthe best-fit total luminosity for all 330 objects in our sample. \n\\label{lum_modes}}\n\\end{figure*}\n\nGiven that the $R$ values are a measure of the goodness of fit in units of\nthe fractional uncertainty, we can use models that lie within a certain range\nof the best-fit $R$ value to estimate uncertainties for the various model \nparameters. For each modeled HOPS target, we tabulated the model\nparameters for all those models that lie within a difference of 0.5, 1.0, \n1.5, and 2.0 of the best-fit $R$. We then computed the mode (i.e., the \nvalue with the highest frequency) for the inclination angle, total luminosity, \n$\\rho_{1000}$, cavity opening angle, outer disk radius, and $A_V$ in \neach of the $\\Delta R$ bins for each object. \nFor any given protostar, the models in each $\\Delta R$ bin span certain \nranges in parameter values; while the modes do not capture the full extent of \nthese ranges, they convey the most common value within each parameter \nrange. The farther away a mode is from the best-fit value, the more poorly \nconstrained the model parameter. Conversely, if a mode of a certain\nparameter is close to or matches the best-fit value, especially for $\\Delta R=$ \n1.5 or 2, that particular model parameter is well constrained.\nIn Figures \\ref{inc_modes} to \\ref{AV_modes} we plot the mode versus the best-fit \nvalue for six model parameters and four $\\Delta R$ bins for all 330 targets \nin our sample. The larger $\\Delta R$, the larger the spread in modes is expected \nto be for each parameter value. \n\nFor example, Figure \\ref{inc_modes} shows that even when considering \nall models with an $R$ value of up to 2 larger than the best-fit $R$ \n($\\Delta R=$ 2), the inclination angle for objects with a best-fit $i$ of \n18\\degr\\ is well constrained; most modes lie at $i=$ 18\\degr, too, and \nonly a few modes can be found at larger inclination angles. However, \nobjects with best-fit $i$ values of 32\\degr\\ or 41\\degr\\ typically can also \nbe fit by models with lower inclination angles (the majority of modes lies \nbelow the line where mode and best-fit value are equal). Inclination \nangles $\\gtrsim$ 63\\degr\\ are better constrained, since their modes \nmostly lie at high inclination angle values, but there are protostars \nwith modes of $i=$18\\degr, too.\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Rho1000_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the reference density\n$\\rho_{1000}$.\n\\label{rho1000_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Cavity_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the cavity opening\nangle.\n\\label{cavity_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{Rdisk_modes.eps}\n\\caption{Similar to Figure \\ref{inc_modes}, but for the outer disk radius ($=R_c$).\n\\label{Rdisk_modes}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.69, angle=90]{AV_modes.eps}\n\\caption{Similar to Figure \\ref{lum_modes}, but for the foreground\nextinction.\n\\label{AV_modes}}\n\\end{figure*}\n\nThe modes for the total luminosity (Figure \\ref{lum_modes}) show a small \nspread for models within $\\Delta R$=0.5, but the spread increases as $R$ \nincreases, with some objects displaying up to an order of magnitude in \nvariation of $L_{tot}$.\nAs illustrated in Figure \\ref{rho1000_modes}, the reference density \n$\\rho_{1000}$ is usually well constrained; however, as $R$ increases, \nthe modes of the $\\rho_{1000}$ values are often lower than the best-fit \nvalues.\nFor the cavity opening angle (Figure \\ref{cavity_modes}), many models up \nto $\\Delta R$=2 have modes of $\\theta$=45\\degr, independent of the \nbest-fit value. Similarly for the centrifugal radius (Figure \\ref{Rdisk_modes}), \n$R_{disk}$=500 AU is a common mode. For all four disk radii, the modes \ntend to be larger than the best-fit values; in particular, objects with a best-fit \n$R_{disk}$ of 5 AU have a very uncertain disk radius. In general, it looks like \nour models do not constrain the disk radius and cavity opening angle well.\nThe foreground extinction (Figure \\ref{AV_modes}) displays a certain range \nof modes for each best-fit value, but objects with $A_V \\lesssim$ 20 typically\nhave more reliable $A_V$ values from their model fits. \n\nFigures \\ref{inc_modes} to \\ref{AV_modes} allow us to gauge general trends \nbetween best-fit values and modes for different model parameters. For results \non individual objects, we refer to Appendix \\ref{models_unc}, where we show \nplots of the difference between the modes and the best-fit values of the major \nmodel parameters for all modeled HOPS targets. In this way it is possible to \nestimate which models are better constrained and thus which objects have \nmore reliable SED fits. \nIn addition, in Appendix  \\ref{models_unc} we also include contour plots of \n$R$ values for different pairs of model parameters for a few targets to \nillustrate typical parameter degeneracies, which also contribute to parameter\nuncertainties.\n\n\n\\section{Discussion} \n  \n\\subsection{Deriving Envelope Parameters from a Model Grid}\n\nWe compared a grid of TSC models to each target by ranking the models \nusing a statistic, $R$, which measures the deviation between observed and  \nmodel fluxes in logarithmic space. We did not model each source by further \nadjusting the model parameters, but instead identified the best-fit SED from \nour model grid. Thus, we are bound by the range and sampling of parameters \nchosen for the grid, and while we constructed the grid with the aim of covering \nthe typical parameter space for protostars, it is limited to discrete values. It is \nlikely that many protostars have best-fit parameters that would fall between those \nsampled by the grid, and a few objects could have parameter values that lie \nbeyond the limits set by the grid. In addition, TSC models are axisymmetric \nand have mostly smooth density and temperature profiles, and they do not\ninclude external heating. They assume a rotating, infalling envelope with \nconstant infall rate and with the gravitational force dominated by the central\nprotostar, but the true envelope structure is likely more complex. \nThe models would not apply to the collapse of a cloud in an initial filamentary \nor sheet-like geometry or to multiple systems with, e.g., more than one \noutflow cavity \\citep[e.g.,][]{hartmann96,tobin12}.\n\nDespite the relatively simple models that we use, many of the observed SEDs are \nfit remarkably well: 75\\% of the fits have $R<4$. In those cases, the continuum \ntraced by the IRS spectrum, the silicate absorption feature at 10 $\\mu$m, and \nthe PACS fluxes are all accurately reproduced by the model. Even many \nflat-spectrum sources, which often do not display any spectral features in \nthe mid-infrared and have an overall flat SED out to 30 or 70 $\\mu$m, often \nhave models that fit them very well. About 75\\% of Class I protostars and  $\\sim$ \n70\\% of Class 0 protostars have $R < 4$, while close to 90\\% of flat-spectrum \nsources have $R$ values in this range. This validates the choice of parameter \nvalues for our model grid.  \nAdditional constraints, like limits on foreground extinction or information on the \ninclination and cavity opening angles from scattered light images or mapping of \noutflows, would allow us to further test and refine the models.  We have used limits \non the extinction in our analysis.  Although {\\it Hubble Space Telescope (HST)} \nscattered light images have been used to constrain models for one HOPS \nprotostar \\citep{fischer14}, scattered light images are not available for many \nof our targets. We therefore chose to focus on fitting the SEDs of all of our \ntargets in a uniform way to a well-defined set of models. Future studies will \nincorporate scattered light images and compare the results to those from \nthe SED fits (J. Booker et al. 2016, in preparation).\n\nThe best-fit models from our grid for the HOPS targets both reproduce the SEDs \nand yield estimates for their protostellar parameters, mostly envelope properties.  \nHowever, these are not necessarily unique fits to the data for three reasons. \nFirst, there are degeneracies in the model parameters; increasing the envelope \ndensity or inclination angle, or decreasing the cavity opening angle or disk radius, \nresults in a steeper mid-IR SED slope and deeper silicate features. Each of these \nparameters affects the SED differently (just the general trends are the same), and \nthe best fit for each object tries optimizing them. The next best fit, however, could \nbe a different combination of these parameters, especially if the SED is not \nwell constrained by observations (see Section \\ref{deltaR}).  \nSecond, although the TSC models reproduce the observed SEDs, other models \nwith different envelope geometries may also be able to fit the same SEDs. The \nmodeling presented here is only valid in the context of the TSC models of single \nstars, and the resulting derived properties are only valid within that framework.   \nLast, neglecting external heating could result in overestimated envelope densities\nand luminosities, with the most noticeable effects ($\\rho_{1000}$ and $L_{tot}$ too \nlarge by factors of a few) on low-luminosity sources exposed to strong radiation \nfields (see Section \\ref{model_ext_heat}). From the distribution of best-fit $L_{tot}$ \nvalues, we estimate that $\\sim$ 20\\% of HOPS targets in our sample could be \naffected by external heating. Even though we do not know the strength of \nexternal heating for each protostar, it is likely that external heating would only \nresult in relatively small changes in the derived envelope parameters for these \nprotostars.\n\n\n\\subsection{Envelope Properties and SED Classes}\n\\label{SED_class_properties}\n\nWhen comparing envelope parameters sorted by SED classes, we found that \nenvelope densities and inclination angles decrease from the sample of\nClass 0 protostars through that of Class I protostars to that of flat-spectrum objects. \nThe former is likely an evolutionary effect, while the latter confirms the results of \nprevious work \\citep[e.g.,][]{evans09} that  the inclination angle has an important \neffect on the SED and that the evolutionary state of an object cannot be derived \nfrom SED slopes alone. Thus, there is a difference between the ``stage'' and \n``class'' of an object \\citep{robitaille06}; Stage 0 and I objects are characterized \nby substantial envelopes, Stage II objects are surrounded by optically thick disks, \nwith possibly some remnant infalling envelopes, and Stage III objects have \noptically thin disks. \n\nIn general, the trends we see among model parameters are a consequence \nof the definition of a protostar based on its SED: in order to be classified \nas a Class 0 or I object, a protostar is required to have a near- to mid-infrared \nSED slope larger than 0.3. A protostellar  model with a small cavity opening \nangle, small centrifugal radius, and\/or high inclination angle will generate \nsuch an SED, since it increases the  optical depth along the line of sight. \nModels with a large cavity will only yield a rising SED in the 2$-$40 $\\mu$m \nspectral range if their  envelope density is large or the inclination angle is \nrelatively high. \n\nWe find that Class 0 protostars can be best fit not only by very high envelope \ndensities but also moderately high envelope densities and large inclination \nangles. The bolometric temperature, which is used to separate \nClass~0 from Class~I protostars, is inclination dependent; some Class~I \nprotostars are shifted to the Class~0 regime if they are viewed more edge-on. \nThe higher-density Class~I protostars tend to have lower inclination angles (but \nstill $>$ 50\\degr); thus, their evolutionary stage could be similar to more \nembedded protostars that are seen edge-on and classified as Class 0 protostars. \nConversely, some Class~0 objects with large inclination angles, but lower \nenvelope densities, could be in a later evolutionary stage than typical Class~0 \nprotostars. Similarly, Class I protostars with large $i$ and low $\\rho_{1000}$ \nvalues could be edge-on Stage II objects (whose infrared emission is dominated \nby a disk). Finally, low-inclination Stage 0 and I protostars can appear \nas a flat-spectrum sources \\citep{calvet94}.\n\n\\begin{figure*}[!]\n\\centering\n\\includegraphics[scale=0.58, angle=90]{Rho1000_vs_inc_cavity.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus inclination angle with the cavity\nsize indicated by the different symbol sizes and gray shades: symbols become\nlarger and lighter colored with increasing cavity size (5\\degr, 15\\degr, 25\\degr,\n35\\degr, 45\\degr). A small random offset in the x direction has been applied to \neach data point to prevent too much overlap.\n\\label{Rho1000_inc_cavity}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.58, angle=90]{Rho1000_vs_inc_Rdisk.eps}\n\\caption{Similar to Figure \\ref{Rho1000_inc_cavity}, but with the outer disk \nradius indicated by the different symbol sizes and gray shades: symbols \nbecome larger and lighter colored with increasing disk radius (5, 50, 100,\n500 AU).\n\\label{Rho1000_inc_Rdisk}}\n\\end{figure*}\n\nNevertheless, the observed trend in envelope densities suggests that the variations \nin the observed SEDs track, in great part, an evolution toward lower envelope \ndensities and lower infall rates. Assuming a certain mass for the central \nstar, the reference density in our models can be used to infer an envelope infall rate\n($\\dot{M}_{env} \\propto \\rho_{1000} \\sqrt{M_{\\ast}}$). \nAs mentioned in section \\ref{model_parameters}, this infall rate is model dependent \nand therefore tied to the assumptions of the models. With this in mind, the median \n$\\rho_{1000}$ values for the Class 0, Class I, and \nflat-spectrum protostars in our sample correspond to envelope infall rates of \n$2.5 \\times 10^{-5}$, $1.0 \\times 10^{-6}$, and $5.0 \\times 10^{-7}$ $M_{\\odot}$\\ yr$^{-1}$, \nrespectively, for a 0.5 $M_{\\odot}$\\ star. \nUsing a more realistic assumption of larger stellar mass for more evolved protostars, \nthe infall rates for Class I and flat-spectrum protostars would be larger than these \nvalues by a factor of a few. However, just larger stellar masses cannot explain the\nlarge decrease of a factor of 50 in the median envelope density from Class 0 to \nflat-spectrum protostars; to achieve such a decrease with a constant infall rate\nof $2.5 \\times 10^{-5}$ $M_{\\odot}$\\ yr$^{-1}$, the stellar mass would have to increase\nby a factor of 2500. Thus, within the context of our model fits, we can conclude that,\nas envelopes become more tenuous, the infall rates also decrease.\n\nOther trends are also apparent. Class 0 protostars and flat-spectrum sources show \na relatively flat distribution of cavity opening angles. On the other hand, the best \nfit for a large fraction of Class I protostars (40\\%) results in $\\theta$=5\\degr. \nThis could point to a  degeneracy in the models, since protostars with small cavity \nopening angles tend to have lower envelope densities (and also lower total \nluminosities); thus, the smaller cavity partly compensates for the lower opacity \nresulting from the lower envelope density (see also Figure \\ref{Rho1000_inc_cavity}). \n\nEven though our models do not yield reliable disk properties, we can make a\nstatement about the difference in the best-fit centrifugal radii (or $R_{disk}$),\nwhich are tied to the structure of the rotating envelope given by the model fits.  \nIt should be noted that the centrifugal radii set a lower limit to the disk radii, \nsince disks may spread outward due to viscous accretion.  Most Class I protostars \nand flat-spectrum sources are fit with a centrifugal radius of just 5 AU. Since \nthe smallest centrifugal radius in our model grid is 5 AU and the next value is 50 AU, \nwe can state that, except for Class 0 protostars, most protostars in our sample \nhave $R_{disk} < 50$~AU, and some might even have $R_{disk} <$ 5~AU.\n\nSmall disks of those sizes have been observed; radio interferometry  of the \nmultiple protostellar system L1551 IRS 5 shows disks whose semi-major axes \nare $\\lesssim$~20~AU \\citep{rodriguez98,lim06}. However, there is a degeneracy \nbetween the centrifugal radius and the envelope density; for protostars with low \nenvelope densities, the small centrifugal radius can compensate the decrease in \nopacity by concentrating more material closer to the star. As can be seen in Figure \n\\ref{Rho1000_inc_Rdisk}, most protostars with $R_{disk}=$ 5 AU also have lower \nenvelope densities. Inclination angle also plays a role; protostars seen at \n$i>$ 80\\degr\\ typically have larger centrifugal radii. In addition, our envelope\nmodels include outflow cavities, which allow some of the mid-IR radiation to\nescape. In order to generate model SEDs with silicate absorption features \nand steep mid-IR slopes at low to intermediate inclination angles, a lower \n$R_{disk}$ value is needed.  \nWe will discuss the potential implications of the small cavity sizes and $R_c$ \nvalues for Class I protostars and flat-spectrum sources in Section \n\\ref{Model_problems}.\n\n\n\\subsubsection{The Most Embedded Protostars}\n\\label{Class0}\n\nAmong the Class 0 protostars, there are protostars in the earliest evolutionary \nstages, when the envelope is massive and the protostar still has to accrete \nmost of its mass. \n\\citet{stutz13} identified 18 protostars with very red mid- to far-infrared\ncolors ($\\log(\\lambda F_{\\lambda}(70)\/\\lambda F_{\\lambda}(24))$\n$>$ 1.65), of which 11 were newly identified (see Table D\\ref{New_proto}). \n\\citet{tobin15} added an additional object.\nThese protostars were named PACS Bright Red sources (PBRs) by \\citet{stutz13}; \nthey are HOPS 169, 341, 354, 358, 359, 372, 373, 394, 397-405, 407, and 409. \nBased on their steep 24-70 $\\mu$m SEDs and large submillimeter luminosities, \nthey were interpreted as the youngest protostars in Orion with very dense \nenvelopes.\n\nFrom our best-fit models to the SEDs of the PBRs, we derive a median \n$\\rho_{1000}$ value of 1.2 $\\times$ 10$^{-17}$ g cm$^{-3}$, \nwhich is twice as high as the median value of all the Class 0 protostars in \nour sample. These fits also result in a median envelope mass within 2500 AU \nof 0.66 $M_{\\odot}$\\ for the PBRs, but the individual objects cover a large \nrange, from 0.07 to 1.83 $M_{\\odot}$. The median total luminosity amounts to \n5.6 $L_{\\odot}$\\ (with a range from 0.6 to 71.0 $L_{\\odot}$), which is very similar\nto the median $L_{tot}$ value for the Class 0 protostars in our sample. \nMost PBRs (14 out of 19 protostars) are fit by models with large inclination \nangles ($i \\geq$ 70\\degr), but, as shown in \\citet{stutz13}, high inclination\nalone cannot explain the redness of the PBRs. \nThus, our models confirm the results of \\citet{stutz13} that the PBRs\nare deeply embedded and thus likely among the youngest protostars\nin Orion.\n\n\n\\subsubsection{Flat-spectrum Sources}\n\\label{flat-spectrum}\n\nA particularly interesting group of protostars that are not easy to categorize\nare the flat-spectrum sources. They are thought to include objects in transition \nbetween Stages I and II, when the envelope is being dispersed \\citep{greene94}. \nIn particular, those with low envelope densities could be more evolved protostars, \nor they could be protostars that started out with more tenuous envelopes. \nOn the other hand, flat-spectrum sources could also be highly inclined \ndisk sources \\citep[see][]{crapsi08}, or protostars surrounded by dense \nenvelopes, but seen close to face-on \\citep{calvet94}. This type of \nmisclassification could have a large effect on the lifetimes of the earlier \nprotostellar stages and thus on the timeline of envelope dispersal. \nAmong the 330 HOPS targets in our sample, we identified 102 flat-spectrum \nsources based on their flat ($-0.3$ to $+0.3$) spectral index from 4.5 to 24 \n$\\mu$m (or 5-25 $\\mu$m in a few cases). Thus, they compose a fairly large \nfraction of our protostellar sample. Of these 102 objects, 47 have a negative \nspectral index and 55 have one between 0 and $+0.3$; 41 have a spectral \nindex between $-0.1$ and 0.1, which results in a very flat mid-infrared SED. \n\nDespite a flat SED slope between 4.5 and 24 $\\mu$m, many flat-spectrum\nsources display a weak silicate emission or absorption feature at 10 $\\mu$m,\nwhich may indicate the presence of a very tenuous infalling envelope or\nmay be the result of the viewing geometry. Some SEDs are very flat out to \n100 $\\mu$m, others have negative spectral slopes beyond 40 $\\mu$m, \nand again others a rising SED from the mid- to the far-IR.\nThere are also objects with more pronounced absorption features due to \nnot only silicates but also ices, as are typically found in Class 0 and I protostars,\nbut also edge-on disks (see HOPS 82, 85, 89, 90, 92, 129, 150, 200, 210, 211, \n281, 304, 331, and 363). Only two flat-spectrum sources have prominent silicate \nemission features, and their SEDs are reminiscent of protoplanetary disks \n(see HOPS 187 and 199). Thus, flat-spectrum sources likely include objects of a \nvariety of evolutionary stages.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[scale=0.39, angle=90]{Median_Rho1000_vs_inc.eps}\n\\caption{Median best-fit $\\rho_{1000}$ values at each inclination angle \nfor the Class 0 and I protostars ({\\it squares}) and the flat-spectrum sources\n({\\it circles}) in our sample. \n\\label{Rho_inc_flat-spectrum}}\n\\end{figure}\n\nThe latter conclusion can also be drawn when analyzing the distribution\nof envelope reference densities and inclination angles for flat-spectrum\nsources. In Figure \\ref{Rho_inc_class}, we showed that flat-spectrum\nsources typically have intermediate inclination angles and lower envelope\ndensities. To compare their properties more directly to Class 0 and I protostars,\nin Figure \\ref{Rho_inc_flat-spectrum} we show the median best-fit \n$\\rho_{1000}$ value at each best-fit inclination angle; it is larger for Class 0\nand I protostars than for flat-spectrum sources at all inclination angles.\nFor Class 0 and I protostars, the median $\\rho_{1000}$ value is highest\nat intermediate inclination angles, decreases at larger inclination angles,\nand then increases again for $i>$ 80\\degr. For flat-spectrum sources, the \nmedian $\\rho_{1000}$ value is relatively flat over the 18\\degr--63\\degr\\\nregion but has its peak value at $i=$41\\degr; it decreases for larger \ninclination angles. The only flat-spectrum source with a best-fit inclination \nangle of 81\\degr, HOPS 357, has a very low envelope density (the lowest \nvalue for this parameter in the model grid), and its spectrum displays a \ndeep silicate absorption feature.\n\nOverall, this shows that, while a range of envelope densities and inclination \nangles can explain flat-spectrum sources, their envelope densities are typically\nlower than for Class 0 and I protostars. The higher-density objects are seen at low\nto intermediate inclination angles, while only the lowest-density objects\nare seen closer to edge-on. Some of the high-density flat-spectrum sources\ncould actually be more embedded protostars (Stage 0 objects) seen\nface-on (which would be classified as Class 0 objects if seen at larger\ninclination angles). Thus, in terms of envelope evolution, they include a\ndiverse group of objects. \n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.71, angle=90]{Rho1000_vs_F350.eps}\n\\caption{Best-fit $\\rho_{1000}$ values versus the 350 $\\mu$m fluxes\nfor the Class 0 and I protostars ({\\it left}) and the flat-spectrum sources\n({\\it right}) in our sample. Detections at 350 $\\mu$m are shown with \ndiamonds, while upper limits are shown with arrows. The histograms \nshow the distribution of best-fit $\\rho_{1000}$ values for sources with \na 350 $\\mu$m flux measurement ({\\it thick solid line}) and with 350 \n$\\mu$m upper limits ({\\it shaded area}). \n\\label{Rho_F350}}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[scale=0.71, angle=90]{Rho1000_vs_F870.eps}\n\\caption{Similar to Figure \\ref{Rho_F350}, but for the 870 $\\mu$m fluxes.\n\\label{Rho_F870}}\n\\end{figure*}\n\nWe note that even though we find that flat-spectrum sources have in general\nlower envelope densities than Class 0 and Class I objects, their best fit does\ninclude an envelope in almost all cases; just 3 of the 102 flat-spectrum \nsources are best fit without an envelope. This seems to contradict recent\nfindings by \\citet{heiderman15}, who found that only about 50\\% of flat-spectrum\nsources were actually protostars surrounded by envelopes. This could be partly\nexplained by different criteria used to select flat-spectrum sources; in the \n\\citet{heiderman15} sample, flat-spectrum sources are selected by their\nextinction-corrected 2-24 $\\mu$m spectral index (see also \\citealt{evans09,\ndunham13}), while our sample uses a flat 4.5-24 $\\mu$m spectral index.\nMoreover, in their study \\citet{heiderman15} detected the presence of an \nenvelope via HCO$^+$ emission, and they found that almost all sources \ndetected in the sub-mm are also detected in HCO$^+$ (but the opposite \ndoes not always hold). For our sample of Orion protostars, we find that \n75\\% of Class 0+I protostars observed with SABOCA (350 $\\mu$m) are \ndetected, while only 47\\% of flat-spectrum sources have detections. For \nLABOCA observations (870 $\\mu$m), these two fractions amount to \n41\\% and 21\\%, respectively. Thus, we find that flat-spectrum sources \nhave a $\\sim$ 50\\% lower sub-mm detection rate than Class 0+I protostars. \nFlat-spectrum sources without sub-mm detections would likely also not \ndisplay HCO$^+$ emission and thus would be considered as protostars \nwithout envelopes by \\citet{heiderman15}.\n\nTo compare how our submillimeter detections correlate with the presence\nof an envelope, in Figures \\ref{Rho_F350} and \\ref{Rho_F870} we show the\nderived best-fit reference envelope densities as a function of 350 or 870 $\\mu$m\nfluxes for the combined Class 0+I sample and the flat-spectrum sources. We\nalso differentiate the distribution of envelope densities between measured\nflux values and upper limits; at 870 $\\mu$m, the upper limits are often cases\nwhere the sources are not detected due to confusion with bright, spatially\nvarying emission. We find that even protostars with upper limits at 350 and \n870 $\\mu$m are best fit with an envelope; however, the envelope density \nis lower for objects with upper limits in the sub-mm. This is especially evident \nfor Class 0+I protostars; for flat-spectrum sources, the distributions of envelope \ndensities for sub-mm detections and upper limits show  significant overlap. Four \ntimes as many flat-spectrum sources have upper limits instead of detections \nat 870 $\\mu$m, but their derived $\\rho_{1000}$ values span almost the full \nrange of values. Furthermore, the median $\\rho_{1000}$ value of 1.19 $\\times \n10^{-19}$ g cm$^{-3}$ for sources without detections is relatively close to the \nmedian value of 5.95 $\\times 10^{-19}$ g cm$^{-3}$ for the sources with \n870 $\\mu$m detections. Thus, our model fits do not rely on sub-mm detections \nto yield a best fit with an envelope; in most cases the near- to far-IR SED is \nsufficient to constrain the properties of the envelope.\n\n\n\\subsubsection{Sources without an Envelope and Class II Objects}\n\\label{disk_sources}\n\nAmong the six objects whose best-fit SED required no envelope ($\\rho_{1000}$\nvalue of 0), three are flat-spectrum sources (HOPS 47, 187, 265), two are Class \nII pre-main-sequence stars (HOPS 113, 293), and one is a Class I protostar (HOPS 232).  \nThe low 70 $\\mu$m fluxes of HOPS 47 and 265 constrained the best model to \none without an envelope. The SED of HOPS 187 looks like that of a transitional \ndisk, which are disks with gaps or holes in their inner regions (see \\citealt{espaillat14} \nand references therein). If HOPS 187 were a transitional disk, it would not have \nan envelope. HOPS 232 has a rising SED over the mid-IR spectral range; its best \nfit requires no envelope, but an edge-on disk with a high accretion luminosity.\n\nIt would be expected that the SEDs of Class II objects can be best fit by a \nmodel that does not include an envelope. This is the case for HOPS 113 and \n293. Of the nine remaining Class II objects in our sample, four have very \nlow envelope densities ($\\rho_{1000} \\sim (1-2.5) \\times 10^{-20}$ g cm$^{-3}$; \nHOPS 22, 26, 98, 283), while five have $\\rho_{1000}$ between $6.0 \\times \n10^{-20}$ and $1.8 \\times 10^{-19}$ g cm$^{-3}$ (HOPS 184, 201, 222, 272, 277).\nThe SEDs of HOPS 22, 184, and 201 are similar to those of transitional disks,\nwith some silicate emission at 10 $\\mu$m and a rising SED between about\n13 and 20 $\\mu$m. The best-fit models require some envelope emission\nto fit the long-wavelength data. \nHOPS 222, 272, and 277 lie close to the border between a Class II \npre-main-sequence star and a flat-spectrum source based on their \n4.5-24 $\\mu$m spectral index, and therefore they could have some \nenvelope material left, despite being classified as Class II objects.\n\nOverall, of the 330 YSOs in our sample, 319 were classified as\neither Class 0, Class I, or flat-spectrum protostars based on their SEDs. \nHowever, four of them are best fit without an envelope. Conversely, of the\n11 Class II objects in our sample, nine are best fit with an envelope; however,\nthree of these might be transitional disks. Thus, based on our model fits and\nSEDs, 321 of our 330 YSOs are protostars with envelopes, and nine are likely\npre-main-sequence stars with disks.\n\n\\clearpage\n\n\\subsection{The Total Luminosities of Protostars}\n\nThe luminosity distribution of protostars is a significant constraint on \nprotostellar evolution, and it is important to understand the effect of the \nenvelope on the observed luminosity \\citep[e.g.,][]{offner11}. The bolometric \nluminosity distribution of the HOPS protostars is very similar to that determined \nfor the {\\it Spitzer}-identified protostars by \\citet{kryukova12} with a \npeak near 1~$L_{\\odot}$\\ (Fig.\\ \\ref{HOPS_n_Tbol_Lbol_histo}). In contrast, the \ndistribution of the total luminosities from the models shows a peak near \n2.5~$L_{\\odot}$\\ (Fig.\\ \\ref{Ltot_class_histo}), indicating that the luminosities of \nprotostars may be systematically underestimated by the bolometric \nluminosities, which do not take into account the inclination angle (and thus\nbeaming of the radiation along the outflow cavities) as well as foreground\nextinction (see Fig.\\ \\ref{Ltot_Lbol} in section \\ref{results_overview}).\n\nHigher intrinsic luminosities for protostars could help address the \n``luminosity problem'' first pointed out by \\citet{kenyon90}, who found \nthat the luminosities of protostars are lower by about an order of magnitude \nthan a simple estimate of the expected accretion luminosity. However, \nan increase in the luminosity by a factor of 2.5-3 would not solve the \nproblem; solutions proposed by other authors, such as mass-dependent\naccretion rates \\citep{offner11} or episodic accretion events \\citep{dunham12}, \nare still needed.\n\nOur best-fit models also suggest that Class 0 protostars have a \ndifferent distribution of $L_{tot}$ values compared to Class I protostars or \nflat-spectrum sources. Their median total luminosity is higher, which could \nbe an indication of larger accretion luminosities for younger protostars. \nWe must bear in mind the caveats and degeneracies mentioned above; \nin particular, in some cases the higher luminosity could be related to the \nadoption of an overly large inclination angle, which results in most \nof the emitted radiation not reaching the observer. Nevertheless, these \ndifferences have potentially important implications for protostellar evolution, \nwhich will be discussed in a future publication (W. Fischer et al. 2016, in \npreparation).\n\n\n\\subsection{Potential Problems with TSC Models}\n\\label{Model_problems}\n\nAlthough the TSC models provide impressive fits to the SEDs, some of the \nobserved trends suggest problems with the models.  First, the distribution \nof inclination angles (Fig.\\ \\ref{Inc_histo}) deviates from what we expect from \na randomly oriented sample of protostars.  Although this could result from \nunintentional selection biases in our sample of protostars, it may also be the \neffect of applying the wrong envelope model to the data. \n\nFurthermore, our data show flat distributions in cavity opening angles \nfor Class~0 and flat-spectrum sources, but an excess of small cavities for the \nClass~I protostars (Figure \\ref{Cav_class_histo}). We also find that protostars \nwith large cavities often have high envelope densities (Figure \\ref{Rho1000_inc_cavity}).\nFor example, models with high envelope densities viewed more edge-on require \nlarge cavity opening angles and high $L_{tot}$ values to \ngenerate sufficient mid-IR flux; this is the case for a few of our highest-luminosity \nobjects (HOPS 87, 108, and 178). These trends do not support the notion of \nincreasing cavity size with later evolutionary stage, which would be expected if \noutflows play a major role in dispersing envelopes \\citep{arce06}. This may \nsuggest that cavity sizes are not growing with time; however, this may also imply  \na deviation from spherical symmetry for the initial configuration of the collapsing \nenvelopes.  Such a deviation may result if the envelope collapses from the \nfragmentation of a flattened sheet or elongated filament.  \n\nFinally, we find an excess of small values of $R_{disk}$, and therefore small \ncentrifugal radii, for Class I and flat-spectrum protostars (Figure \n\\ref{Rdisk_class_histo}). This is contrary to the expectation from the TSC \nmodel, in which the late stages of protostellar evolution are characterized \nby the infall of high angular momentum material from large radii and hence \nlarger values of $R_c$.  This may imply that disks sizes are small, but it \nmay also be the result of incorrect assumptions about the distribution of \nangular momentum in the TSC model.  \n\nIn total, these ``conundrums'' that arise from our model fits hint that the \ncurrent models do not realistically reproduce the structure of collapsing \nenvelopes. Future high-resolution observations at submillimeter and longer\nwavelengths that resolve the structure and motions of envelopes may \nprovide the means to develop more refined models that can fit the SEDs \nwith more realistic envelope configurations. \n\n\n\\vspace{2ex}\n\n\\section{Conclusions}\n\nWe have presented SEDs and model fits for 330 young stellar objects\nin the Orion A and B molecular  clouds. The SEDs include data from \n1.2 to 870 $\\mu$m, with near-infrared photometry from 2MASS, mid-infrared \nphotometry and spectra from the {\\it Spitzer Space Telescope}, far-infrared \nphotometry at 70, 100, and 160 $\\mu$m from the {\\it Herschel Space Observatory}, \nand submillimeter photometry from the APEX telescope. \nWe calculated bolometric luminosities ($L_{bol}$), bolometric temperatures\n($T_{bol}$), and 4.5-24 $\\mu$m spectral indices ($n_{4.5-24}$) for all 330 sources \nin our sample. From the distributions of these three parameters, we find that $L_{bol}$\nhas a broad peak near 1 $L_{\\odot}$\\ and extends from 0.02 to several hundred $L_{\\odot}$,\nwhile the distribution of $T_{bol}$ values is broad and flat from about 30 K to 800 K,\nwith a median value of 146 K. The 4.5-24 $\\mu$m spectral indices range from \n-0.75 to 2.6, with a peak near 0.\n\nBased on traditional classification schemes involving $n_{4.5-24}$\nand $T_{bol}$, we have identified 92 sources as Class 0 protostars \n($n_{4.5-24} > 0.3$ and $T_{bol} < 70$~K), 125 as Class I protostars \n($n_{4.5-24} > 0.3$ and $T_{bol} > 70$~K), and 102 as flat-spectrum sources \n($-0.3 < n_{4.5-24} < 0.3$). The remaining 11 sources are Class II pre-main-sequence \nstars with $n_{4.5-24} < -0.3$; most of them just missed the flat-spectrum cutoff,\nand three have SEDs typical of disks with inner holes. Considering these\ntransitional disks and YSOs whose best fit does not require an envelope, we\nfind that 321 of the 330 HOPS targets in our sample are protostars with\nenvelopes. Class 0 and I protostars often display a deep silicate absorption \nfeature at 10 $\\mu$m due to the presence of the envelope, while many \nflat-spectrum sources have a weak silicate emission or absorption feature \nat that wavelength. \n\nWe have used a grid of 30,400 protostellar model SEDs, calculated using the\n2008 version of the \\citet{whitney03a,whitney03b} Monte Carlo radiative \ntransfer code, to find the best-fit models for each observed SED. The grid \nis limited to discrete values for protostellar parameters, and their ranges \nwere chosen to represent typical protostars. Within the framework of these \nmodels, we find the following:\n\n\\begin{itemize}\n\\item{About 70\\% of Class 0 protostars, 75\\% of Class I protostars, and close to \n90\\% of flat-spectrum sources have reliable SED fits ($R < 4$, where $R$\nis a measure of the average distance between model and data in units\nof the fractional uncertainty). Thus, our model grid can reproduce most of the \nobserved SEDs of Orion protostars.}\n\\item{Our results show a clear trend of decreasing envelope densities as we \nprogress from Class 0 to Class I and then to flat-spectrum sources: we find that \nthe median $\\rho_{1000}$ values decrease from 5.9 $\\times 10^{-18}$ g cm$^{-3}$ \nto 2.4 $\\times 10^{-19}$ g cm$^{-3}$ to 1.2 $\\times 10^{-19}$ g cm$^{-3}$.\nThe decrease in densities implies a decrease in the infall rates of the protostars \nas they evolve.\nWe find that the PACS Bright Red sources (PBRs) have median $\\rho_{1000}$ \nvalues twice as high as the median value of the Class 0 protostars in our sample, \nsupporting the interpretation that they are likely the youngest protostars in Orion.}\n\\item{There are degeneracies in the parameters for models that reproduce the \nobserved SEDs. For example, increasing the mid-IR SED slope and \ndeepening the silicate absorption feature at 10 $\\mu$m of a model protostar \ncan be done by increasing the envelope density or inclination angle, \ndecreasing the cavity opening angle or centrifugal radius, or even increasing \nthe foreground extinction. Hence, the properties of a specific source may be fit \nby a wide range of parameters. The best-fit model parameters are particularly \nuncertain for objects whose SED is not well constrained by observations. \nBecause of these degeneracies, the observed classes contain a mixture \nof evolutionary stages.}\n\\item{We find that flat-spectrum sources are particularly well fit by our models.\nThey have, on average, lower envelope densities and intermediate inclination \nangles, so many flat-spectrum sources are likely more evolved protostars, \nbut this group also includes protostars with higher envelope densities (and\nsometimes larger cavity opening angles) seen at lower inclination angles. \nFlat-spectrum sources seen at $i>$ 65\\degr\\ have very tenuous envelopes. \nThus, the sample of flat-spectrum sources includes protostars \nat different stages in their envelope evolution. All but three of the flat-spectrum \nsources in our sample have envelopes in their best-fit models, indicating that, \nwith a small number of exceptions, these objects are protostars with infalling \ngas.}\n\\item{The luminosity function for the model luminosities peaks at a higher \nluminosity than that for the observed bolometric luminosities as a result of \nbeaming along the outflow cavities.  Furthermore, the total luminosity \ndetermined by the models is higher for Class 0 protostars: the median \ntotal luminosities are 5.5, 2.0, and 3.0 $L_{\\odot}$\\ for Class 0, Class I, and \nflat-spectrum sources, respectively.}\n\\item{Since heating by external radiation fields is not included \nin our model grid, we assessed its influence by adding an interstellar radiation \nfield to a set of models. We find that an ISRF ten times that typical of the solar \nneighborhood can substantially change the SEDs of  sources with internal \nluminosities of 0.1 $L_{\\odot}$. However, when we incorporate the effect of \nextinction on the external radiation field, the effect on the protostellar\nSEDs is smaller; the best-fit luminosities and envelope densities would be \noverestimated by factors of a few for $\\sim$ 0.1 $L_{\\odot}$\\ prototars and much\nless for higher-luminosity protostars. We estimate that the best-fit parameters\n(in particular, $L_{tot}$, $\\rho_{1000}$) of $\\sim$ 20\\% of the HOPS sources \ncould be affected by external heating.}\n\\item{Although the adopted TSC models reproduce the observed SEDs well, \nthere are trends that suggest inadequacies with these models.  First, the \ndistribution of best-fit inclination angles does not reproduce that expected \nfor randomly oriented protostars.  Second, although the distribution of outflow \ncavity sizes for flat-spectrum and Class~0 sources is flat, there is an excess \nof small cavities for Class I sources. This is in contradiction to the typical \npicture that outflow cavities grow as protostars evolve. Finally, the distribution \nof outer disk radii set by the rotation of the envelope is concentrated at small \nvalues ($<$ 50 AU) for the Class I and flat-spectrum sources but is slightly \ntilted toward large values ($>$ 50 AU) for Class 0 protostars. Again, this trend \ncontradicts the expected growth of disks as the infall region in protostellar \nenvelopes expands. These findings suggest that either the envelope structure \nof the adopted models is incorrect, or our understanding of the evolution of \nprotostars needs to be revised substantially.}\n\\end{itemize}\n\nOur work provides a large sample of protostars in one molecular cloud\ncomplex for future, more detailed studies of protostellar evolution. For example,\nusing additional constraints, such as from scattered light imaging, the \nstructure of envelope cavities and thus the role of outflows can be better \nunderstood. In addition, the detailed structure of the envelope and the disk \nembedded within, as well as multiplicity of the central source, can be studied \nwith high spatial resolution imaging such as ALMA can provide. With the \nanalysis of their SEDs presented in this work, the HOPS protostars constitute \nan ideal sample to derive a better understanding of the early evolution of \nyoung stars, when the assembly of the stellar mass and the initial \nstages of planet formation likely take place.\n\n\\vspace{1ex}\n\n\\acknowledgments\nSupport for this work was provided by NASA through awards issued by\nJPL\/Caltech.\nThe work of W.J.F. was supported in part by an appointment to the NASA \nPostdoctoral Program at Goddard Space Flight Center, administered by \nOak Ridge Associated Universities through a contract with NASA.\nJ.J.T. acknowledges support provided by NASA through Hubble Fellowship grant \n\\#HST-HF-51300.01-A awarded by the Space Telescope Science Institute, which is \noperated by the Association of Universities for Research in Astronomy, Inc., \nfor NASA, under contract NAS 5-26555. J.J.T acknowledges further support from \ngrant 639.041.439 from the Netherlands Organisation for Scientific Research (NWO).\nThe work of A.M.S. was supported by the Deutsche Forschungsgemeinschaft\npriority program 1573 (``Physics of the Interstellar Medium'').\nM.O. acknowledges support from MINECO (Spain) AYA2011-3O228-CO3-01 \nand AYA2014-57369-C3-3-P grants (co-funded with FEDER funds). \nWe thank Thomas Robitaille for helpful discussions regarding the model grid\nand model parameters.\nThis work is based on observations made with the {\\it Spitzer Space Telescope}, \nwhich is operated by the Jet Propulsion Laboratory (JPL), California Institute of \nTechnology (Caltech), under a contract with NASA; it is also based on\nobservations made with the {\\it Herschel Space Observatory}, a European Space\nAgency Cornerstone Mission with significant participation by NASA. \nThe {\\it Herschel} spacecraft was designed, built, tested, and launched under \na contract to ESA managed by the {\\it Herschel\/Planck} Project team by an industrial \nconsortium under the overall responsibility of the prime contractor Thales Alenia Space \n(Cannes), and including Astrium (Friedrichshafen) responsible for the payload module \nand for system testing at spacecraft level, Thales Alenia Space (Turin) responsible \nfor the service module, and Astrium (Toulouse) responsible for the telescope, with \nin excess of a hundred subcontractors.\nWe also include data from the Atacama Pathfinder Experiment, a collaboration \nbetween the Max-Planck Institut f\\\"ur Radioastronomie, the European Southern \nObservatory, and the Onsala Space Observatory. \nThis publication makes use of data products from the Two Micron All Sky Survey, \nwhich is a joint project of the University of Massachusetts and the Infrared Processing \nand Analysis Center\/Caltech, funded by NASA and the NSF. \n\n\\vspace{2ex}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Observations and reductions}\n\\subsection{Spectroscopy}\nNew electronic spectra were obtained at four observatories: with\nthe Ond\\v{r}ejov 2~m reflector (OND) and a~coud\\'e\nspectrograph, with the Dominion Astrophysical Observatory 1.22~m reflector (DAO)\nand a coud\\'e spectrograph, with the Cerro Armazones 1.5~m Hexapod Telescope (HPT),\nthe Bochum Echelle Spectroscopic Observer (BESO)\nspectrograph, which is similar to FEROS \\citep{beso}, and with the Cerro Tololo Inter-American\nObservatory (CTIO) 1.5~m reflector with the CHIRON echelle spectrograph\n\\citep{toko2013}.\nWe also used one archival ESO FEROS echelle spectrum \\citep{feros,feros2},\nthe medium-resolution CCD  spectra obtained and published by Christian Buil,\n\\footnote{For the description of his instrumentation and data reduction, see\n\\url{http:\/\/www.astrosurf.com\/buil\/us\/bestar.htm}\\,.} and a selection of\namateur CCD spectra with resolutions better than 10000 from the BeSS\nspectroscopic database \\citep{neiner2011}.\nTable~\\ref{jourv} lists a journal of all spectral observations used.\n\nThe initial reduction of all Ond\\v{r}ejov and DAO spectra (bias subtraction,\nflat-fielding, creation of 1D spectra, and wavelength calibration) was carried\nout in {\\tt IRAF}. Initial reduction of the HTP and CTIO spectra\nwas carried out at the respective observatories. Rectification, removal\nof residual cosmics and flaws, and RV measurements of all spectra were carried\nout with the Pascal program {\\tt SPEFO}  \\citep{sef0,spefo}, namely the latest\nversion 2.63 developed by J.~Krpata \\citep{spefo3}.\n{\\tt SPEFO}  displays direct and flipped traces of the line\nprofiles superimposed on the computer screen that the user can slide\nto achieve a precise overlapping of the parts of the profile for which the RV\nis to be measured.\nAll RVs measurements were carried out independently by PH and also by AH, who\nstudied the spectra as a~part of his student's research project. In addition to the wings of the H$\\alpha$  and \\ion{He}{i}~6678  emission, we also measured the bottom of the\n(sometimes asymmetric) cores of the H$\\alpha$, \\ion{He}{i}~6678, \\ion{Si}{ii}, and \\ion{Fe}{ii}\nshell absorptions. Both sets of measurements were\nintercompared, the larger deviations were carefully\nchecked, and the mean of the measurements for each line was then used here.\nOnly after these RV measurements were completed was a new program\nfor spectral reduction {\\tt reSPEFO}, a~modern replacement of {\\tt SPEFO} , written\nin JAVA and running on different platforms (Linux, Windows) developed\nby A.~Harmanec.\\footnote{\\url{https:\/\/astro.troja.mff.cuni.cz\/projects\/respefo}}\nIt can, among other things, import the spectra that were originally reduced in {\\tt SPEFO} \nand treat spectra stored as FITS files.\nWe used this new program to measure line intensities, equivalent widths, and\nthe $V\/R$ ratio of the double H$\\alpha$  emission.\\\\\n\n\\subsection{Photometry}\nWe attempted to collect all available observations with known\ndates of observations. Basic information about all data sets\ncan be found in Table~\\ref{jouphot}, and the details of the photometric\nreductions and standardisation are described in Appendix~\\ref{apb}.\n\nFor the convenience of other investigators, we also publish all our individual\nobservations together with their HJDs. All measured RVs are listed in Table~3,\nspectrophotometric quantities are provided in Table~4, and photometric observations\nare collected in Table~5.  These three tables are available in electronic form only.\n\n\\section{Long-term spectral, light, and colour changes of V1294~Aql}\nAs mentioned above, we attempted to collect and homogenise all available\nphotometric observations and measurements of RVs and line strengths from\nthe records with known dates of observations. Long-term behaviour of these\nquantities and their mutual correlations are discussed in the following\nsub-sections.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{vis-v.pdf}}\n\\caption{Time plot of the Otero visual brightness estimates\n(empty red circles) along with Johnson $V$ photometry from Hvar (blue dots),\n        SPM (green), \\c{C}anakkale (cyan), and TUG (magenta).}\n\\label{vis-v}\n\\end{figure}\n\n\\subsection{Value of visual estimates of brightness}\nThe visual estimates of brightness by skilful amateur observers are commonly\nused to determine the times of minima (or maxima) of periodic\nvariables and to monitor light changes of variables with a large\namplitude (over several magnitudes). The scatter band of\nvisual estimates is typically about 0\\m1 to 0\\m15, but it can be pushed down by\ntalented individuals who, moreover, follow a few principles:\n(a) to perform only one visual estimate per night (without recalling the previous\none), and (b) to reduce the estimates to Johnson $V$ magnitudes of the\ncomparison stars (known to one thousandth of a magnitude), not to\nthe Harvard scale of magnitudes, which is only accurate to 0\\m1.\nOne of us (SO) contacted the first author of this study back in 2003\nto inform him that he had observed another light decrease of V1294~Aql  for\nabout 0\\m3. We then agreed to test his ability to obtain accurate visual\nestimates via parallel photoelectric observations at Hvar,\nSan Pedro M\\'artir (SPM), Tubitak National Observatory\n(TUG), and \\c{C}anakkale. Figure~\\ref{vis-v}\nshows the comparison of visual estimates and Johnson $V$ photoelectric\nphotometry from several stations over the time interval covered by visual\nestimates. The visual estimates agree very well with\nthe general trend of variations that were recorded via photoelectric photometry, but\nthe deep light minimum appears broader than that recorded by photoelectric\nphotometry. One possible reason is that the minimum was observed\nclose to the end of visibility of the star in the sky and the visual\nestimates were not corrected for the differential extinction.\n\n\\subsection{Correlation between the long-term light and spectral changes\nin time}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{all.pdf}}\n\\caption{Time plot of available observations over the whole time interval\nof about 25000~d covered by the data. Top panel: Yellow brightness observations,\nwhich could be transformed into Johnson $V$ magnitude. Second and third panel:\nAvailable \\hbox{$B\\!-\\!V$}\\  and \\hbox{$U\\!-\\!B$}\\ colour index observations. Bottom panel: $V\/R$ changes in the peaks of the double H$\\alpha$  emission.\nIn the three panels with photometry, the differential observations\nare shown as blue dots, and all-sky observations are shown as black dots. In the bottom panel,\nblue circles denote the DAO spectra, red circles show the OND spectra, green circles represent BESO spectra,\nmagenta circles show BeSS spectra, black circles show Castanet Tolosan spectra, and the black crosses plot the data\nfrom the literature.}\n\\label{all}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=-90]{all-n.pdf}}\n\\caption{Time plot showing the correlation between the secular brightness\nand colour variations in the $V$ -band magnitudes and \\hbox{$B\\!-\\!V$}\\ and \\hbox{$U\\!-\\!B$}\\ colour\nindices (black shows all-sky and blue shows differential observations),\nand the $V\/R$ changes, EW, and strength of the\nH$\\alpha$  emission for the more recent electronic spectra. The colour symbols for \nspectra from different sources are the same as in Fig. 2.}\n\\label{all-n}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics{si1vr.pdf}}\n\\caption{Time plot of the $V\/R$ changes recorded for the electronic spectra\nin the \\ion{Si}{ii}~6347~\\AA\\ line. The colour symbols for spectra from different\nsources are the same as in Fig.~\\ref{all}.}\n\\label{si1vr}\n\\end{figure}\n\n\\begin{figure}[t]\n\\resizebox{\\hsize}{!}{\\includegraphics{rvhe.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{rvh3.pdf}}\n\\caption{Time plot of radial velocities. Top panel: RV of the \\ion{He}{i}\nshell lines. Bottom panel: RV measured on the steep wings of the H$\\alpha$  emission.\nData from individual instruments are shown by different symbols:\nthe circles are the same as in Fig. 2, and\nblack crosses plot data from \\citet{balle89}. The ranges on the two\naxes in the two plots are different.}\n\\label{rvtime}\n\\end{figure}\n\nFigure~\\ref{all} is a time plot of all available observations secured in\nor transformed into the Johnson \\hbox{$U\\!B{}V$}\\ magnitudes. The top panel\nshows that the usual brightness level in $V$ is occasionally\ndisturbed by rapid light decreases of different durations.\nMoreover, we note that there is also a secular, steady slow light decrease\nof the undisturbed brightness of the star outside the more rapid light\ndecreases until about HJD~2455000, when it suddenly changed to\na~steeper secular light increase. We return to this new phenomenon\nin a separate section below. The second panel of Fig.~\\ref{all} shows\nthat the \\hbox{$B\\!-\\!V$}\\ index followed the brightness changes, but with a small amplitude,\nwhile the \\hbox{$U\\!-\\!B$}\\ index showed a similar pattern of changes, but with a~larger\namplitude and with a~more or less steady secular reddening.\n\nAn enlarged Fig.~\\ref{all-n} covers only the more recent time interval,\nwhen electronic spectra became available. It shows that in the time intervals\nthat are sufficiently densely covered by the data, the sharp light decreases are\naccompanied by similarly sharp strengthening in the H$\\alpha$  emission.\n\nFigure~\\ref{si1vr} shows the $V\/R$ changes of \\ion{Si}{ii}~6347~\\AA\\ line.\nIt reveals a~pattern similar to that observed for H$\\alpha$, but the time coverage\nis less dense because these variations could not be measured in time intervals\nwhen the emission was faint.\n\nFigure~\\ref{rvtime} shows the variation of the shell absorption RVs\n\\citep[characterised by \\ion{He}{i} RVs, for which data are also\npublished by][]{balle89} and emission-line RVs measured on the wings of\nthe H$\\alpha$  line in electronic spectra.  We note that while the shell RVs\nshows large cyclic changes that have also been observed for a number of other Be stars,\nthe RVs measured on the wings of the H$\\alpha$  emission are secularly stable and\nshow only mild changes on a shorter timescale.\n\n\\subsection{Unusual colour variations}\n\n\\begin{figure*} [t]\n\\includegraphics[angle=0,scale=0.95]{ubbvc.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbvd.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbva.pdf}\n\\includegraphics[angle=0,scale=0.95]{ubbvb.pdf}\n\\caption{ \\hbox{$U\\!-\\!B$}\\ vs. \\hbox{$B\\!-\\!V$}\\ diagram for several distinct data subsets.\nTop panels: Older data until JD 2450000 (left). All-sky observations\nare shown as black circles, and data from stations defined\nin Table~\\ref{jouphot} are denoted as follows:\n01 (blue), 04 (green), 12 (red), and 26 (magenta).\nMore recent data from the interval of secular\nlight brightening, all from station 01 (blue) (right).\nThe bottom panels show \\hbox{$U\\!B{}V$}\\ observations from\nthe two sharp increases in the emission-line strength accompanied\nby light decreases. Time interval JD~2452741 -- 2453309, which\ncovers the first sharp light decrease (left, cf. Figs.~\\ref{vis-v} and \\ref{all-n}.)\nData from the time interval JD~2455357 -- 2456094 corresponding\nto the second sharp increase in the emission-line strength (right).\nData from stations 1, 30, 66, and 89 of Table~\\ref{jouphot} are shown\nby blue, red, green, and magenta dots, respectively. The main sequence\nand the supergiant sequence based on data from \\citet{golay74} (pp. 79-80)\nare shown, as is the reddening line.}\n\\label{ubbv}\n\\end{figure*}\n\n\n Many systematically studied Be stars are known to exhibit always the same\nand a rather clear type of either a positive or an inverse correlation\nbetween the long-term brightness variations, a~characteristic type of\nbehaviour in the colour-colour diagram, and the Balmer emission-line strength\nas defined by \\citet{hvar83,hec2000}. He identified these two types\nof correlation as an aspect effect.\nFor Be stars with an~inverse type of correlation, light decreases are followed\nby the rise of the Balmer emission-line strength and by a~shift along\nthe main sequence towards later spectral subclasses in the \\hbox{$U\\!-\\!B$}\\ versus \\hbox{$B\\!-\\!V$}\\\ndiagram. For a~positive type of correlation, the brightenings are followed\nby the rise of the emission strength and a~shift from the main sequence\ntowards the supergiant sequence in the \\hbox{$U\\!-\\!B$}\\ versus \\hbox{$B\\!-\\!V$}\\ diagram. The inverse\ncorrelation is observed for stars that are seen more or less equator-on (a growing\ngaseous envelope is attenuating the light of the central object), while\nthe positive correlation is observed for stars that are seen more pole-on (inner\noptically thick parts of the growing envelope mimic an~apparent increase in\nthe stellar radius). Several examples of both types of correlation in\nthe colour-colour diagram can be found, for instance, in Fig.~2 of\n\\citet{bozic2013}.\n\n  The situation is dramatically different for V1294~Aql. The \\hbox{$U\\!B{}V$}\\ observations\naccumulated over several decades cover a large part of the whole\ncolour-colour diagram, with a single clear pattern.\n\n  To understand better what is going on, we investigated\nthe colour-colour diagrams for different segments of the long-term\nchanges. Figure~\\ref{ubbv} shows the colour changes separately for\nthe old data secured before JD~2450000, for more recent data from the secular\nbrightness increase (observations after JD~2457000), and for\nobservations covering two episodes of a rapid increase and decrease in the H$\\alpha$  emission\nassociated with sharp light decreases. The\npattern is remarkably similar for both these episodes. Formally, it looks like\na~positive correlation. However, the phases of minimum brightness and\nmaximum strength of the emission correspond to data that are close\nto the main sequence, even below it when the reddening is considered.\nThe older data are all clustered above the supergiant sequence, while\nthe recent data lie along the supergiant sequence for late-B and early-A\nspectral classes.\nAll this indicates that we observe a combination of several different types\nof long-term changes.\n\n\\section{Duplicity of V1294~Aql}\n\n\\setcounter{table}{5}\n\\begin{table}\n\\begin{flushleft}\n\\caption{Orbital solutions based on the H$\\alpha$  emission RVs.}\n\\label{sol}\n\\begin{tabular}{lccrrccc}\n\\hline\\hline\\noalign{\\smallskip}\n Element                   &All RVs        &Hi-res. spectra only\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n$P$ (d)                    &$192.91\\pm0.18$& 192.91 fixed       \\\\\n$T_{\\rm super.conj.} ^*$   &$56318.5\\pm2.2$&$56316.2\\pm2.9$     \\\\\n $e$                       &0.0 fixed      & 0.0 fixed          \\\\\n$\\gamma$ (km~s$^{-1}$)             & $-6.27\\pm0.31$& $-5.52\\pm0.44$     \\\\\n$K_1$    (km~s$^{-1}$)             & $6.33\\pm0.41$ & $6.26\\pm0.61$      \\\\\nNo. of RVs                 & 172           & 38                 \\\\\nrms (km~s$^{-1}$)                  & 3.92          &  2.63              \\\\\n\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{flushleft}\n\\tablefoot{$^*$) All epochs are in HJD-2400000.}\n\\end{table}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{orbit.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{orbithe.pdf}}\n\\caption{Radial-velocity curve corresponding to the orbital solution, based on\nRVs measured on the steep wings of the H$\\alpha$  emission, plotted for\nphases from ephemeris (top; \\ref{efem}).\nOrbital curve based on RV of the \\ion{He}{i}~6678~\\AA\\ shell line\nprewhitened for the long-term changes, plotted for the same ephemeris (bottom).\nData from individual instruments are shown by different symbols. The circles \nare the same as in Fig. 2, and black triangle shows CTIO.}\n\\label{orbit}\n\\end{figure}\n\nThe idea that duplicity can be an important factor for the very existence\nof the Be phenomenon is not new. \\citet{plahor69,kriz69} and\n\\citet{plavec70} have argued that at least some Be stars could be binaries\nthat are observed in the later phases of mass exchange between the binary components.\n\\citet{krizhec75} and \\citet{hk76} formulated the general hypothesis that\nBe stars are mass-accreting components of binaries and showed that this idea \ncan also explain several types of time variations observed for Be\nstars.  Additional arguments were provided by \\citet{plavec76a} and\n\\citet{peters76}. However, as pointed out already by \\citet{plavec76b},\nif all Be stars have Roche-lobe filling secondaries,\nmore eclipsing binaries should be observed among them. Later investigations also led to the\nfinding that the presence of Roche-lobe filling secondaries can be excluded\nfor some Be stars that were found to be spectroscopic binaries, such as V744~Her = 88~Her\n\\citep{zarf7a,zarf7b} or V439~Her = 4~Her \\citep{zarf5,zarf6}.\nThis led \\citet{pols91} to suggest that many\nBe stars might be objects created by large-scale mass transfer that were\nobserved in phases after the mass transfer ceased. The expected\nsecondaries of such objects would be hot compact stars, white dwarfs\nin some cases. These are the most easily detectable sources in far-UV spectra. Evidence\nfor a hot secondary to the well-known Be binary $\\varphi$~Per was found\nfirst from the antiphase variation in the \\ion{He}{ii}~4686~\\AA\\ emission\nseen in the photographic spectra \\citep{poeckert81} and later from\n\\ion{He}{i}~6678~\\AA\\ emission in the electronic spectra obtained\nby \\citet{gies93}. Its ultimate direct detection as an O~VI subdwarf\ncame from the study of the far-UV spectra from the Hubble Space Telescope by\n\\citet{gies98}. The secondary was then resolved with optical spectro-interferometry\nby \\citet{mourard2015}. Detections for several other systems followed.\n\\citet{wang2018} carried out a systematic search for the presence of hot\nsecondaries and summarised our knowledge of already known cases. \\citet{wang2021}\ndetected nine new Be+sdO binaries from analyses of the Hubble Space Observatory\nspectra, and \\citet{klement2022} reported the first interferometric detection and\nsignatures of the orbital motion for three known Be+sdO systems. On the other\nhand, \\citet{boden2020} carried out a systematic search for Be stars with\nmain-sequence secondaries, with a completely null result. This constitutes\nindirect evidence that the mass exchange is or was behind the formation\nof binaries with Be primaries. \\citet{hast2021} carried out evolutionary\ncalculations of mass exchange in binaries in an effort to set some limits\non the fraction of Be stars produced by binary interaction. They found\nthat under certain conditions, this fraction can be quite high.\n\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{haprofs.pdf}}\n\\caption{Three Ond\\v{r}ejov H$\\alpha$  profiles. They are mutually shifted in\nordinate by 1.0 of the continuum level for better clarity.\nProfiles from HJD~2457128.5976 and 2457137.5762 have anomalously positive\nRVs of the emission wings, which stem from the episode of a large strengthening\nof the emission. The next profile, from HJD~2457154.5435, has a~normal\norbital RV.}\n\\label{haprofs}\n\\end{figure}\n\n\\begin{table}\n\\begin{flushleft}\n\\caption{Possible properties of the binary system:\nMass of the secondary $M_2$, mass ratio $M_2\/M_1$, semi-amplitude of\nthe RV curve of the secondary $K_2$, and the semi-major axis $a$ for\nseveral possible orbital inclinations $i$. The mass of the primary was\nassumed to be $M_1=16.9$~\\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$}\\ after \\citet{zorec2016}.}\n\\label{secmass}\n\\begin{tabular}{lccrrccc}\n\\hline\\hline\\noalign{\\smallskip}\n  $i$     & $ M_2$ &$M_2\/M_1$& $K_2$  & $a$\\\\\n($^\\circ$)& (\\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$})   &      & (km~s$^{-1}$)  & (\\hbox{$\\mathcal{R}^{\\mathrm N}_\\odot$}) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n 90.0  & 1.171 &  0.0693 & 90.43  &    368.69   \\\\\n 85.0  & 1.175 &  0.0695 & 90.07  &    368.72   \\\\\n 80.0  & 1.189 &  0.0704 & 88.99  &    368.82   \\\\\n 70.0  & 1.249 &  0.0739 & 84.73  &    369.23   \\\\\n 60.0  & 1.361 &  0.0805 & 77.77  &    369.98   \\\\\n\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{flushleft}\n\\tablefoot{We express the values of masses and radii\nin the nominal values \\hbox{$\\mathcal{M}^{\\mathrm N}_\\odot$}, and \\hbox{$\\mathcal{R}^{\\mathrm N}_\\odot$}\\ as defined by \\citet{prsa2016}.}\n\\end{table}\n\n\nOne always has to be cautious when analysing binaries with clear signatures\nof the presence of circumstellar matter in the system. The experience from\nour previous studies of individual Be stars \\citep{bozic95,zarf18,zarf20,\nzarf21,zarf24,zarf26} shows that the binary nature of particular Be stars\nis most easily detected via periodic RV variations of the steep\nemission wings of the H$\\alpha$  line and often also via the periodic changes\nin the $V\/R$ ratio of the double Balmer emission lines.\n\nPeriod analyses of all H$\\alpha$  emission-line RVs of V1294~Aql, using\nboth the \\citet{deeming75} and \\citet{stelling78} methods, revealed that\nthe RV of the H$\\alpha$ \\ emission wings varies with a period of 193~d\nand a~semi-amplitude of $\\sim 5$ km~s$^{-1}$.\nThe same periodicity is also detected in the RV of the H$\\alpha$  absorption core\nand in the absorption RVs of \\ion{Si}{ii} doublet at 6347 and 6371~\\AA\\,\nand \\ion{Fe}{ii}~6456~\\AA\\  after long-term changes are removed.\n\nUsing the program {\\tt FOTEL}  \\citep{fotel1,fotel2}, we derived the circular-orbit\norbital elements for all H$\\alpha$  emission-wing RVs and for those from the\nhigh-resolution spectra alone. The mutual agreement of the two solutions is\nvery satisfactory. They are presented in Table~\\ref{sol}, and\nthe corresponding RV curve is plotted in Fig.~\\ref{orbit}.\nIn the rest of this study, we adopt the following linear ephemeris:\n\n\\begin{equation}\nT_{\\rm super.conj.}={\\rm HJD}\\,2456318.5+192\\fd91\\times E \\label{efem}\n\\end{equation}\n\n\\noindent based on the solution for all spectra.\n\n To be fair, we note that some deviations from the mean RV curve in the upper\npanel of Fig.~\\ref{orbit} are rather large. This is, for instance,\nthe case of two Ond\\v{r}ejov spectra taken on HJD~2457128.6 and 2457137.6,\nwhen a very steep rise of the emission strength\nhad occurred. We remeasured these spectra several times, but the result was the\nsame. Their RVs are almost in anti-phase to the orbital RV curve near phase 0.3.\nWe show the corresponding line profiles in Fig.~\\ref{haprofs} together with\nanother profile, taken about two weeks later, which already gives a~RV in accord\nwith the orbital motion. The two peculiar RVs were given zero weight in the\norbital solution.\n\nWe tentatively adopted the mass of the Be primary after \\citet{zorec2016} and\nestimated the basic properties of the system for several\npossible orbital inclinations. Because no lines of the secondary were detected\nin the optical spectra and because no companions to Be stars were ever found\namong main-sequence objects \\citep{boden2020}, we conclude that the secondary\nis not a Roche-lobe filling object, but most probably a hot subdwarf star or\nwhite dwarf. It should be looked for in the far-UV spectral region.\nFor the Gaia DR2 parallax of 0\\farcs0007059, the projected angular separation\nof the binary components is 0\\farcs0048, which might be resolved with \npresent-day optical interferometers such as the currently tested interferometer\nSPICA \\citep{spica2020}.\n\n\n\\section{Correlations between orbital and long-term changes}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{v57500.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v57900.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v58300.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v58600.pdf}}\n\\caption{Phase plots of $V$ magnitude for several seasons of Hvar observations\nfor binary ephemeris (\\ref{efem}). From top to bottom: Data from\nJD~$2457568-57657$,\nJD~$2457931-58079$,\nJD~$2458318-58392$, and\nJD~$2458664-58740$.}\n\\label{vorb}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{haemisph.pdf}}\n\\caption{Phase plots of the strength of the H$\\alpha$  emission\nfor binary ephemeris (\\ref{efem}). Data from individual sources are\ndenoted as follows: Circles in blue show DAO spectra, red circles show OND spectra, green circles represent BESO spectra,\nmagenta circles show BeSS, and black circles show Castanet spectra. The black crosses show data from the literature.}\n\\label{haemisph}\n\\end{figure}\n\nInspection of the light, colour, and emission-line strength variations seems to\nindicate that the rapid episodes of large changes such as those near JD~2452900\nor JD~2457300 occurred during one binary orbital period. To investigate the problem,\nwe plot phase diagrams of the variability of the $V$ magnitude for several\nmore recent shorter time intervals in Fig.~\\ref{vorb}. The two\nlarge light decreases that are accompanied by strong increases in the H$\\alpha$  emission-line\nstrength apparently occurred around phases of elongations, with the Be primary receding from us.\nAt the same time, the plot shows that in another observing season, brightenings\nwere observed around similar orbital phases. The same is also confirmed by\na phase plot of the H$\\alpha$  emission-line strength; see Fig.~\\ref{haemisph}.\n\n  We also investigated the time behaviour of the $V\/R$ ratio of the double\nH$\\alpha$  emission. In this case as well, V1294~Aql  appears to be quite unusual.\nAs Figs.~\\ref{all} and \\ref{all-n} show, the $V\/R$ variations are different\nin different time intervals and do not recall either the long-term cyclic\nchanges known for Be stars with one-armed global oscillations or phase-locked\nchanges. In several panels of Fig.~\\ref{vrsets} we show enlarged plots of\nthe $V\/R$ changes. Instants of expected phase-locked $V\/R$ maxima predicted\nby the orbital ephemeris (\\ref{efem}) are shown by vertical lines.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics{vrsets.pdf}}\n\\caption{Enlarged subsets of time variability of the H$\\alpha$  $V\/R$ ratio\nwith the instants of the expected phase-locked maxima predicted by the\norbital ephemeris (\\ref{efem}).}\n\\label{vrsets}\n\\end{figure}\n\n\\section{Rapid changes}\nAlthough we collected a large number of photometric observations of V1294~Aql,\ntheir time distribution is not suitable for a search for rapid periodic changes. Perhaps the only observations suitable for\nsuch a search are the early $V$ observations by \\citet{lynds59}.\nHe himself concluded that his observations definitively indicate\nvariations in brightness, which appear to be somewhat erratic, however, and\nno period could be found. The observations were secured within one month\nduring a~time interval that was  not affected by secular variations. Our period analysis\nrevealed sinusoidal variations with a semi-amplitude of 0\\m0139(29).\nA~least-squares fit led to ephemeris (\\ref{efelynds}), the rms of one\nobservation being 0\\m0067. The corresponding phase plot is shown in Fig.~\\ref{rapid}.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{lynds.pdf}}\n\\caption{Possibly periodic rapid light changes based on \\citet{lynds59}\n$V$ magnitude photometry and plotted for ephemeris (\\ref{efelynds}).}\n\\label{rapid}\n\\end{figure}\n\n\\begin{equation}\nT_{\\rm light\\, min.}={\\rm HJD}\\,2436450.8053(92)+0\\fd64827(50)\\times E. \\label{efelynds}\n\\end{equation}\n\nThis indicates that a scatter of at least 0\\m03 is to be expected in individual\nobservations on longer timescales.\n\n\\citet{lefe2009} carried out an~automatic period search in the Hipparcos \\hbox{$H_{\\rm p}$}\\\nphotometry to find new periodic variables among OB stars. They identified\nV1294~Aql  as a~possible slowly pulsating B star (SPB) with a period of 7\\fd752.\nWe cannot confirm their result. They apparently did not take the secular\nlight change in \\hbox{$H_{\\rm p}$}\\ photometry into account; see the upper panel of Fig.~\\ref{all} here.\n\n\\citet{zorec2016} estimated the following physical properties of the Be component:\\\\\n$T_{\\rm eff}$  = ($30120\\pm2540$)~K, {\\rm log}~$g$  = ($4.08\\pm0.40$) [cgs],\nmass M = ($16.9\\pm2.7$)~M$_{\\odot}$, $v$~sin~$i$  = ($207\\pm18$)~km~s$^{-1}$,\ncritical rotational velocity $v=(517\\pm64)$~km~s$^{-1}$, and the\ninclination of the rotational axis $i=(37^\\circ\\pm9^\\circ)$.\nFor these values, the period 0\\fd648 appears as a reasonable rotation period of the Be component.\nIn passing we note that at the time of writing, the star has not been observed\nby the TESS satellite.\n\n\\section{Fourth timescale}\n\\citet{hec98} has called attention to the fact that the brightness of the\nBe star $\\omega$~CMa outside of the episodes of brightenings accompanied\nby the growth of emission-line strength (typical of the positive correlation\ndiscussed above) has been decreasing secularly. His observation was later\nconfirmed with more recent photometry \\citep{ghore2018, ghore2021}. These\nauthors and also \\citet{marr2021}, who studied another Be star, V2048~Oph = 66~Oph,\nmodelled the secular variability and the episodes of brightening and\nincreases in the Balmer emission-line strength with some success, estimating\nthe required viscosity values for individual episodes, and also discussing\nsome limitations of their effort. The yellow light curve of V2048~Oph\nis shown in the upper panel of Fig.~1 of \\citet{marr2021}. It shows a~secular\nlight decrease between 1980 and 2000, occasionally interrupted by brightenings\nreminiscent of a positive correlation. However, the strength of\nthe H$\\alpha$  emission is near its maximum over the same time interval of about\n20 years, and only then does it gradually decrease. No emission has been\nobserved since about 2010. However, as \\citet{marr2021} pointed out,\nthe outer parts of the disk are still seen in the radio wavelengths.\n\nWe collected and homogenised the $V$ photometry of V2048~Oph\nfrom the archive of \\hbox{$U\\!B{}V$}\\ photometry provided by J.R.~Percy, from Hvar,\nSPM, \\citet{john66}, \\citet{haupt74} and \\citet{kozok85} and transformed the Hipparcos\n\\hbox{$H_{\\rm p}$}\\ \\citep{esa97} photometry into Johnson $V$ and the observations of\n\\cite{hill76} secured in the DAO photometric system into \\hbox{$U\\!B{}V$} \\ using the transformation\nformul\\ae\\ provided by \\citet{hecboz01}.  The $V$ light curve of V2048~Oph\nbased on the above-mentioned data sets is shown in the upper panel\nof Fig.~\\ref{fourth}.\n\nA similar secular light decrease has also been reported for V744~Her = 88~Her,\na Be star with an inverse type of correlation \\citep{hecboz2013}. We show\nits light curve complemented by more recent observations, adapted from\nBo\\v{z}i\\'c et al. (in prep.) in Fig.~\\ref{fourth} as well. In the same figure,\nwe also show the $V$ and $B$ light curves of the Be star EW~Lac = HD~217050\nfrom another study in preparation. In this case, a secular increase\nin brightness is observed. We note that the large scatter around the mean trend\nis related to the known rapid light variability of EW~Lac on a timescale shorter than\none day. The fourth panel of Fig.~\\ref{fourth} shows the plot of Hvar $V$ photometry\nof $\\varphi$~And, another Be star with a positive type of correlation.\nA mild light decrease over several decades of observations is visible.\n\nSearching the literature, we found a few more examples.\nA secular light increase has also been observed for the Be star with a~positive\ncorrelation $\\gamma$~Cas \\citep[][Fig.~5]{gcas2002} over nearly 30000~days.\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{66ophv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{ewlacv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{ewlacb.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{phiandv.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics{v744herv.pdf}}\n\\caption{Secular photometric changes of several well-observed Be stars.}\n\\label{fourth}\n\\end{figure}\n\nFinally, as we have shown here, the secular light decrease of V1294~Aql \nhas recently changed to a secular light increase.\nAll this shows the large variety of different possible evolutions\nof the circumstellar disk, its replenishment, and a~gradual dissipation.\n\n\n\n\\section{Discussion}\nIn spite of the effort of several generations of stellar astronomers,\nthe engine leading to the occasional formation of circumstellar disks\naround Be stars has not been firmly identified so far.\nOne  possible explanation is based on the idea that Be stars are\nrapidly rotating non-radial pulsators (NRP) and that the additional\nforce needed to facilitate the outflow of gas and angular momentum transfer\nfrom the stellar equator arises from a constructive interference of two\nor more NRP modes\n\\citep{rivi98a,rivi98b,baade2017a,baade2017b,baade2020,borre2020,bartz2021}.\nEspecially the systematic photometries from space observatories were\nused and analysed to support this conjecture. Confirmation of this\nscenario would require the creation of new, self-consistent models,\nhowever, which would show that Be stars are indeed pulsationally unstable over the whole\narea that they occupy in the Hertzsprung-Russell (HR) diagram.\nIt should also be mentioned that \\citet{baade2017b} warned that evidence\nfor constructive interference of pulsational modes for a larger number of Be stars\nis lacking and pointed out additional problems such as the rotational splitting of modes\nand the presence of rapid changes in the circumstellar envelopes during active phases.\nAn alternative view was suggested by \\citet{hec98}, who argued that the dominant\nperiod of rapid changes undergoes small cyclic changes. Modelling such\na situation, he found that a standard period analysis of a~corresponding series\nof observations returns a multiperiodicity with several close periods.\nYet another possibility was suggested by \\citet{bebin2002}, who argued that\nthe presence of a secondary can facilitate outflow from the equatorial parts\nof the gaseous disk of the Be primary even in systems that are not filling\ntheir Roche lobes. This idea is problematic, however, in that the effect is\nrather small in the majority of cases.\n\n For a long time, various other suggestions have been made that\nthe Be phenomenon and observed variability patterns of Be stars might be causally\nrelated to their binary nature\n\\citep[e.g. ][among others]{plahor69,krizhec75,hk76,bebin87,pols91,\n  pano2018,boden2018,boden2020,langer2020,boden2021}. \nThe approach of \\citet{klement2019} is worth mentioning. These authors studied \nthe spectral energy distribution of several Be stars and provided arguments that \ntheir disks had to be truncated by the Roche lobes. This constitutes \nan~indirect argument for the presence of companions to these objects.\n\n  The phase-locked $V\/R$ changes observed for several Be binaries represent another\ninteresting phenomenon. As already noted above, they are usually\nobserved roughly in phase with the orbital RV changes\nof the Be stars in question \\citep{zarf6,zarf7b,zarf16,zarf21,stefl2007}. A phase-locked\nemission-line variation with a single maximum and minimum per orbital period\nwas also found by \\citet{borre2020} for $\\gamma$~Cas. These authors used\nan interesting detection technique. They analysed local pixel-per-pixel line\nfluxes across the H$\\alpha$  profile in a~series of higher-resolution BeSS spectra.\n\\citet{pano2018} modelled the phase-locked $V\/R$ changes as global oscillations\nin the circumstellar disks with two spiral patterns and concluded that the\nphase-locked $V\/R$ changes should exhibit two maxima and minima during one\norbital period. This clearly disagrees with the available observations\nmentioned above. The only case for which a double-wave $V\/R$ curve was detected is\nV696~Mon = HR~2142 \\citep{peters72,peters76}. There is a natural explanation\nof roughly sinusoidal phase-locked $V\/R$ changes in our view, as discussed\nin Appendix~C of \\citet{wolf2021}. The circumstellar disk probably\noccupies almost the whole volume of the Roche lobe near\nthe orbital plane. This causes there to be more gaseous material in the part\nof the disk facing the secondary than on the opposite side. Because the disk\nrotates, more emission power is available on the side facing the secondary,\nand this naturally leads to phase-locked $V\/R$ changes that are in phase\nwith the RV changes of the Be component.\n\n  To show how confusing the interpretation of $V\/R$ changes can be,\nwe compare the $V\/R$ changes of H$\\alpha$  and \\ion{He}{i}~6678~\\AA\\ for\ntwo shorter time intervals in Fig.~\\ref{fales}. In the first interval,\nthe variations for both lines are in phase, while in the second interval,\nantiphase variation is observed. We note that this is a consequence\nof the fact that the He shell RV becomes very negative (a~temporarily\nelongated envelope?) and apparently weakens the $V$ peak of a~relatively\nfaint He emission. This is best illustrated in Fig.~\\ref{vrdva}, where\nthe H$\\alpha$  and \\ion{He}{i}~6678  profiles for two dates are shown.\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{heh3vra.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{heh3vrb.pdf}}\n\\caption{Apparent $V\/R$ changes observed for the H$\\alpha$  and \\ion{He}{i}~6678 \nemission lines intercompared for two time segments. The variation\nin the shell He RV is also shown. An~apparently anti-phase behavior\nis observed in the second time interval, when the shell RV becomes\nquite negative and the He shell line blends with the $V$ peak of the\nfaint He emission. The same colours as in the previous time plots\nare used to distinguish spectra from individual observatories.}\n\\label{fales}\n\\end{figure}\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{havrline.pdf}}\n\\resizebox{\\hsize}{!}{\\includegraphics[angle=0]{hevrline.pdf}}\n\\caption{Apparent $V\/R$ changes observed for the H$\\alpha$  and \\ion{He}{i}~6678 \nemission lines intercompared for two dates. An~apparently anti-phase\nbehaviour is observed for the later date, when the \\ion{He}{i}~6678  shell RV becomes\nquite negative and the He shell line blends with the $V$ peak of the\nfaint He emission.}\n\\label{vrdva}\n\\end{figure}\n\nThis study of V1294~Aql  demonstrates very clearly how hard it is to identify\nand understand mutually coexisting and overlapping variability patterns\ngoverning the observed spectral, light, and colour changes. Attempts at modelling them\nquantitatively, planned for a~continuation of this study, are expected to shed\nmore light on the mysterious Be phenomenon. We also suggest that further\nmonitoring of the object with systematic photometry, high-resolution\nspectroscopy, and especially with the optical interferometry could help to\nreveal the secrets of this intriguing Be binary, or possibly a multiple system,\nas indicated by the analysis of the astrometric data \\citep{brandt2021}.\n\n\\begin{acknowledgements}\nWe gratefully acknowledge the use of the latest publicly available version\nof the program {\\tt FOTEL}  written by P.~Hadrava.\nWe thank A.~Aret, A.~Budovi\\v{c}ov\\'a, P.~Chadima, M.~Dov\\v{c}iak,\nJ.~Fuchs, P.~Hadrava, J.~Jury\\v{s}ek, E.~Kiran, L.~Kotkov\\'a,\nR.~K\\v{r}i\\v{c}ek, J.~Libich, J.~Nemravov\\'a, P.~Rutsch, S.~Saad, P.~\\v{S}koda,\nS.~\\v{S}tefl, and V.~Votruba, who obtained some of the Ond\\v{r}ejov spectra\nused in this study. J.R.~Percy kindly put the archive of his systematic\n\\hbox{$U\\!B{}V$}\\ observations of bright Be stars in three observatories at our disposal.\nWe also acknowledge the constructive suggestions of an anonymous referee\nto the first version of this study.\nOver the years, this long-term project was supported by the grants 205\/06\/0304,\n205\/08\/H005, P209\/10\/0715, and GA15-02112S  of the Czech Science Foundation,\nby the grants 678212 and 250015 of the Grant Agency of the Charles University\nin Prague, from the research project AV0Z10030501 of the Academy of Sciences\nof the Czech Republic, and from the Research Program MSM0021620860\n{\\sl Physical study of objects and processes in the solar system and\nin astrophysics} of the Ministry of Education of the Czech Republic.\nThe research of PK was supported by the ESA PECS grant 98058.\nHB, DR, and DS acknowledge financial support\nfrom the Croatian Science Foundation\nunder the project 6212 ``Solar and Stellar Variability\".\nThis work has made use of data from\nthe European Space Agency (ESA) mission Gaia\n(\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the Gaia\nData Processing and Analysis Consortium (DPAC;\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}).\nFunding for the DPAC has been provided by national institutions,\nin particular the institutions participating in\nthe Gaia Multilateral Agreement. We also used some spectra\nof the BeSS database, operated at LESIA,\nObservatoire de Meudon, France: \\url{http:\/\/basebe.obspm.fr}.\nFinally, we acknowledge the use of the electronic database from\nthe CDS, Strasbourg, and the electronic bibliography maintained by\nthe NASA\/ADS system.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nAlthough the fifth-generation (5G) wireless network is still under deployment, researchers have moved forward to define the next-generation or sixth-generation (6G) wireless network, with the aim for achieving more stringent performance, such as unprecedentedly high throughput, super-high reliability, ultra-low latency, extremely low power consumption, etc \\cite{9040264,8766143}.\nHowever, these targets may not be fully achieved by only relying on the existing technologies, such as massive multi-input multi-output (MIMO) and millimeter wave (mmWave) communications, which can attain enhanced performance but generally incur more substantial energy consumption and hardware cost.\nOn the other hand, wireless communication performance is fundamentally constrained by the wireless channel impairments such as path-loss, shadowing, and small-scale fading, which can be partially mitigated by conventional wireless communication techniques such as power control, adaptive modulation, diversity, dynamic beamforming, etc., but still remain random and uncontrolled at large.\nRecently, \\emph{intelligent reflecting surface} (IRS) has emerged as a promising technology to address the above issues by leveraging massive low-cost reflecting elements to flexibly and dynamically control the radio signal propagation environment in favor of wireless communications\/sensing, thus achieving substantially improved communication spectral\/energy efficiency and sensing accuracy cost-effectively \\cite{9326394,9724202}.\n\nThe existing works on IRS have mainly considered the wireless systems aided by \\emph{passive IRS}.\nSpecifically, as illustrated in Fig. \\ref{p-IRS}, the passive IRS is composed of a large number of passive reflecting elements with positive resistances (e.g., positive-intrinsic-negative (PIN) diodes, field-effect transistors (FETs), micro-electromechanical system (MEMS) switches) \\cite{9326394}.\nAs such, each passive element can reflect the incident signal with a desired phase shift, while it has no signal processing\/amplification capability due to the lack of transmit\/receive radio frequency (RF) chains. Moreover, compared with the conventional half-duplex active relay, the passive IRS operates in a full-duplex mode and hence is free of amplification\/processing noise as well as self-interference \\cite{9119122}.\nBy properly adjusting the individual phase shifts of all passive reflecting elements with the reflection amplitude no larger than one, the reflected signal by IRS can be added constructively with that from the other propagation paths for enhancing the signal power at the intended receiver \\cite{9362274,9241706} or destructively for suppressing the undesired interference \\cite{9171881}.\nRemarkably, it has been shown in \\cite{8811733} that the passive IRS beamforming can achieve a \\emph{squared power scaling order}, i.e., $\\mathcal{O}(N^2)$ with $N$ denoting the number of reflecting elements, which is even higher than that of the massive MIMO with active arrays.\nExtensive research has been conducted recently to efficiently incorporate passive IRS into wireless systems for various purposes, e.g., enhancing the communication throughput \\cite{9714463}, reducing the outage probability \\cite{9205879}, saving the transmit power \\cite{8741198}, and extending the range of active relays \\cite{9464248,9586067}, among others.\nHowever, the performance gain of passive IRS is fundamentally constrained by the severe \\emph{product-distance} path-loss of the reflected channel by IRS \\cite{8888223}.\nTwo practical approaches to dealing with this problem are, respectively, deploying more passive elements at each IRS to enhance its aperture\/beamforming gain and placing the passive IRSs closer to the transmitter and\/or receiver for reducing the reflected channel product-distance \\cite{8982186}. However, these solutions may not be suitable for practical scenarios when the space of IRS site is limited and\/or its location cannot be freely selected.\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Passive IRS.}] {\n\\label{p-IRS}\n\\includegraphics[width=2in]{p-IRS.pdf}  \n}}     \n{\\subfigure[{Active IRS.}] {\\label{a-IRS}\n\\includegraphics[width=2in]{a-IRS.pdf}     \n}}\n{\\subfigure[{Hybrid active-passive IRS.}] {\\label{h-IRS}\n\\includegraphics[width=2in]{h-IRS.pdf}     \n}}\n{\\caption{Different types of IRS architecture.}}\n\\end{figure}\n\nTo tackle the above issues, a new type of IRS, called \\emph{active IRS}, has been recently proposed (see, e.g., \\cite{9377648,9734027,9530750,9568854}). \nSpecifically, the active IRS comprises a number of active reflecting elements, each equipped with an active load (or called negative resistance) such as the tunnel diode and negative impedance converter.\nAs illustrated in Fig. \\ref{a-IRS}, each active load is connected to an additional power supply for signal amplification \\cite{8403249,9219017}.\nTherefore, the active IRS not only enables adjustable phase shifts as the passive IRS, but also allows the amplitude amplification (i.e., larger than one) of incident signals in a full-duplex mode, albeit at modestly higher hardware and energy cost than the passive IRS \\cite{9377648}.\nOn the other hand, compared to the active relay that attaches RF chains to the antennas, the active IRS does not entail costly and power-hungry RF chain components \\cite{9758764}.\nThe performance comparison between the active and passive IRSs has been recently studied in the literature.\nFor example, given the active-IRS location and power budget, it has been shown that the active IRS can achieve higher spectral efficiency \\cite{9377648}, energy efficiency \\cite{9568854}, and reliability \\cite{9530403} than the passive IRS.\nBesides, the authors in \\cite{9530750} further optimized the IRS placement for both the passive- and active-IRS aided systems for rate maximization. It was shown that the passive IRS achieves higher rate performance than the active IRS with their respectively optimized placement when the number of reflecting elements is large and\/or the active-element amplification power is small. \nMoreover, the authors in \\cite{9734027} considered the same power budget constraint for both the passive- and active-IRS aided systems, where both the base station's (BS's) transmit power and active IRS's amplification power are considered in the case of active IRS. It was revealed that the active IRS outperforms the passive IRS only when the number of reflecting elements is small and\/or the amplification power of the active IRS is sufficiently large.\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=5in]{sysmod.pdf}}\n\\caption{A hybrid active-passive IRS aided wireless communication system.}\\label{sysmod}\n\\end{figure}\nTo summarize, the existing works on IRS have shown that passive and active IRSs have complementary advantages.\nSpecifically, the passive IRS has a higher asymptotic beamforming gain than the active IRS (i.e., $\\mathcal{O}(N^2)$ versus $\\mathcal{O}(N)$), thus is more appealing when the number of reflecting elements $N$ is large \\cite{9530750,9377648}. In contrast, the active IRS provides additional power amplification gain, which leads to a much higher signal-to-noise ratio (SNR) than the passive IRS when $N$ is relatively small \\cite{9723093,9716895,8403249}.\nBesides, the active IRS generally incurs higher cost and power consumption than the passive IRS. These thus indicate that given a total budget on the IRS deployment cost (or equivalently the number of active\/passive reflecting elements to be deployed), the conventional IRS architectures with either passive or active elements only in general may not achieve the optimum communication performance.\n\nMotivated by the above, we propose in this paper a new \\emph{hybrid active-passive IRS}\\footnote{We use the term hybrid IRS to denote this new architecture hereafter for brevity.} architecture as shown in Fig. \\ref{h-IRS} to achieve the advantages of both passive and active IRSs for further improving the performance over that with active or passive IRS alone.\nSpecifically, the hybrid IRS is composed of two co-located sub-surfaces, each consisting of a certain number of passive and active reflecting elements, respectively.  \nIn particular, we design the active versus passive reflecting elements allocation at the hybrid IRS under a given deployment budget, for optimally balancing the trade-off between the unique power amplification gain of active IRS and the higher beamforming gain of passive IRS than active IRS.\nTo this end, we consider a hybrid active-passive IRS aided multi-user communication system as shown in Fig. \\ref{sysmod}, where a BS transmits independent data to a cluster of users.\nA hybrid IRS is properly deployed at the edge of this user cluster to serve the users in its half-space reflection region over different time slots.\nWe consider a given deployment budget for the hybrid IRS with different costs of each active and passive reflecting element based on practical models, where an active element generally incurs higher cost than its passive counterpart.\nTo reduce the real-time channel estimation overhead and avoid frequent IRS reflection adjustment, we consider a practical approach that designs the hybrid IRS beamforming based on the \\emph{statistical} channel state information (CSI) only, assuming the practical Rician fading channel model (i.e., only the channel path-loss parameters and Rician fading factors are assumed to be known), instead of requiring the knowledge of the instantaneous CSI of all links involved.\nIn the following, we summarize the main contributions of this paper.\n\n\\begin{itemize}\n\\item First, to guarantee the achievable rate performance of all users, we formulate an optimization problem to maximize the ergodic capacity of the worst-case user located at the boundary of the IRS reflection region.\nSpecifically, we assume the statistical CSI available only and jointly optimize the active\/passive reflecting elements allocation, their phase shifts, and the amplification factors of active elements, subject to various practical constraints on the active-element amplification factor and amplification power consumption, as well as the total active and passive elements deployment budget.\nThis problem, however, is shown to be non-convex, which is difficult to be optimally solved in general.\nTo address this difficulty, we approximate the ergodic capacity of the worst-case user with high accuracy and thereby reformulate the original problem in a simpler form.\n\n\\item Next, we propose an efficient algorithm to solve the reformulated problem. First, we jointly optimize all elements' phase shifts and the amplification factors of active elements based on the statistical CSI, and obtain a closed-form expression for the achievable ergodic capacity with a given elements allocation.\nThen, we apply the one-dimensional search to find the optimal active\/passive elements allocation to maximize the ergodic capacity. \nTo obtain useful insight into the optimal elements allocation, we further consider two special cases where the involved channels are line-of-sight (LoS) and follow Rayleigh fading, respectively. \nIt is shown that in the former case with LoS paths, only active elements need to be deployed when the total deployment budget is sufficiently small, while both active and passive elements should be deployed with a decreasing number ratio when the budget increases and exceeds a certain threshold.\n\n\\item Last, we present extensive numerical results to evaluate the effectiveness of our proposed hybrid IRS architecture and its optimized design for rate maximization.\nWe show that the hybrid IRS with optimized elements allocation outperforms the conventional active\/passive-only IRS architecture as well as other benchmarks. \nMoreover, the optimal active\/passive elements allocation under the general Rician fading channel and that under the LoS channel are presented, which are in accordance with our theoretical analysis.\nThe effects of several key parameters such as the Rician fading factor, active-element amplification power, and active\/passive-element deployment cost on the capacity performance and optimal active\/passive elements allocation are also investigated.\n\\end{itemize}\n\nIt is worth noting that the authors in \\cite{48550,9733238} considered a hybrid relay-reflecting intelligent surface architecture with a few relaying elements connected to power amplifiers and RF chains, which, however, significantly differs from our proposed hybrid IRS architecture with both active and passive reflecting elements. \nFor example, in \\cite{48550,9733238}, the relaying elements forward the signals with power-hungry RF chains, while the active reflecting elements in our proposed hybrid IRS architecture adopt the negative resistance components of much lower power consumption for signal amplification.\nThe remainder of this paper is organized as follows. The system model is first introduced in Section \\ref{sec_sysmod}, based on which we formulate an optimization problem and approximate the worst-case user's ergodic capacity in Section \\ref{sec_prob_approx}. \nIn Section \\ref{sec_design}, we elaborate the algorithm to solve the reformulated optimization problem, and present theoretical results that provide useful insight into the optimal IRS active\/passive elements allocation.\nSimulation results and pertinent discussions are presented in Section \\ref{sec_sim}. Finally, the conclusions are drawn in Section \\ref{sec_conclu}.\n\n\\emph{Notations}: \nSuperscript $(\\cdot)^H$ stands for the Hermitian transpose operation. $\\mathbb{C}^{a \\times b}$ denotes the space of $a \\times b$ complex-valued matrices, $\\mathbb{N}$ denotes the set of natural numbers, and $\\mathbb{R}^+$ denotes the set of positive real numbers. \nThe operation $\\mathbb{E}\\left\\{\\cdot\\right\\}$ returns the expected value of a random variable, $\\arg(\\cdot)$ returns the angle of a complex number, and ${\\operatorname{diag}}{(\\boldsymbol{x})}$ returns a diagonal matrix with the elements in $\\boldsymbol{x}$ on its main diagonal. The notation $\\jmath$ represents the imaginary unit, $\\otimes$ denotes the Kronecker product, $\\mathcal{C} \\mathcal{N}(\\mu,\\sigma^2)$ denotes the circularly symmetric complex Gaussian (CSCG) random variable with mean of $\\mu$ and variance of $\\sigma^2$,\n$[\\cdot]_{m,n}$ denotes the $(m,n)$-th entry of a matrix, $[\\cdot]_{m}$ denotes the $m$-th entry of a vector, and $\\lfloor\nx\\rfloor$ denotes the largest integer that does not exceed the real number $x$.\n\n\\section{System Model}\\label{sec_sysmod}\nAs illustrated in Fig.~\\ref{sysmod}, we consider a hybrid active-passive IRS aided wireless communication system, where a single-antenna BS transmits independent data to a cluster of single-antenna users\\footnote{The proposed hybrid IRS and its results obtained in this paper can be extended to the case with multi-antenna BS and multiple IRSs\/user clusters, which will be investigated in our future work.}.\nWe assume that the BS and its served users are separated by long distance and the direct links between them are negligible due to the large channel path-loss and\/or severe blockage. \nAs such, a hybrid IRS comprising both active and passive reflecting elements is properly placed to serve the users in its half-space reflection region.\nMoreover, we consider the time division multiple access (TDMA) scheme, where the users are served by the BS over orthogonal time slots\\footnote{Under this setup, the TDMA scheme has been shown to outperform the non-orthogonal multiple access (NOMA) and orthogonal frequency division multiple access (OFDMA) schemes in terms of energy and spectral efficiency \\cite{8970580}.}. \nTo guarantee the system performance among all users, we consider the worst-case user performance at the boundary of the IRS reflection region with the IRS-user distance $d_{\\mathrm{IU}}$ and BS-IRS distance $D_{\\mathrm{BI}}$ (see Fig. \\ref{sysmod}).\n\nFor ease of implementation, we assume that the hybrid IRS comprises two co-located sub-surfaces with $N_{\\mathrm{pas}}$ passive and $N_{\\mathrm{act}}$ active reflecting elements, respectively (see Fig. \\ref{sysmod}).\nSpecifically, we denote by \n$\\mathbf{\\Psi}^{\\mathrm{pas}}\\triangleq {\\operatorname{diag}}{(e^{\\jmath\\varphi^{\\mathrm{pas}}_1},\\cdots,e^{\\jmath\\varphi^{\\mathrm{pas}}_{N_{\\mathrm{pas}}}})}$\nthe reflection matrix of the passive sub-surface, where $\\varphi^{\\mathrm{pas}}_n$ denotes the phase shift of the $n$-th passive element with $n\\in\\mathcal{N}_{\\mathrm{pas}}\\triangleq \\{1,\\cdots,N_{\\mathrm{pas}}\\}$, and the reflection amplitude of each passive element is set as one (i.e., its maximum value). The cost of each passive element is denoted by $W_{\\mathrm{pas}}$.\nOn the other hand, as the active sub-surface can simultaneously amplify the signal and tune its phase shift, we denote by $\\mathbf{\\Psi}^{\\mathrm{act}}\\triangleq\\mathbf{A}^{\\mathrm{act}}\\mathbf{\\Phi}^{\\mathrm{act}}$ the reflection matrix of the active sub-surface, where $\\mathbf{A}^{\\mathrm{act}}\\triangleq{\\operatorname{diag}}{(\\alpha_1,\\cdots,\\alpha_{N_{\\mathrm{act}}})}$ and $\\mathbf{\\Phi}^{\\mathrm{act}}\\triangleq\n{\\operatorname{diag}}{(e^{\\jmath\\varphi^{\\mathrm{act}}_1},\\cdots,e^{\\jmath\\varphi^{\\mathrm{act}}_{N_{\\mathrm{act}}}})}$\ndenote respectively its reflection amplification matrix and phase-shift matrix with $\\alpha_n$ and $\\varphi^{\\mathrm{act}}_n$ representing the amplification factor and phase shift of each active element $n\\in\\mathcal{N}_{\\mathrm{act}}\\triangleq\\{1,\\cdots,N_{\\mathrm{act}}\\}$. \nTo ensure that each active reflecting element amplifies the signal, we impose a constraint on the amplification factor for each active element as $\\alpha_n\\geq \\alpha_{\\min},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha_{\\min}\\geq 1$ \\cite{9377648}. Moreover, the limited load of each active element \\cite{8403249} leads to the following constraint on its maximum amplification factor: $\\alpha_n\\leq\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha_{\\max}>\\alpha_{\\min}$.\nLet $W_{\\mathrm{act}}$ represent the deployment cost of each active element, which in general is larger than that of each passive element (i.e., $W_{\\mathrm{act}}>W_{\\mathrm{pas}}$) due to its more sophisticated hardware (i.e., additional power amplifier and amplification control circuit \\cite{9377648}) and higher static operation power (e.g., 6-20 mW for active element \\cite{7920385} versus 5 mW for passive element \\cite{8888223}). \nMoreover, we denote $W_0$ as the total deployment budget for the hybrid IRS such that $N_{\\mathrm{act}}W_{\\mathrm{act}}+N_{\\mathrm{pas}}W_{\\mathrm{pas}}\\leq W_0$.\n\nWe assume the practical Rician fading channel model for all involved links. As such, the baseband equivalent channel from the BS to the active IRS sub-surface, denoted by $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}}\\times 1}$, can be modeled as \n\\begin{equation}\n    \\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}=\\sqrt{\\frac{K_{1}}{K_{1}+1}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_{1}+1}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}},\n\\end{equation}\nwhere $K_{1}$ is the Rician fading factor of the BS$\\to$IRS link, $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the LoS component, and $\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the non-LoS (NLoS) component.\nSpecifically, the LoS component can be modeled as $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}={h}_{\\mathrm{BI}}^{\\mathrm{act}}\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}},\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)$, where ${h}_{\\mathrm{BI}}^{\\mathrm{act}}\\triangleq\\sqrt{\\beta}\/D_{\\mathrm{BI}}e^{-\\jmath\\frac{2\\pi}{\\lambda}D_{\\mathrm{BI}}}$ denotes the complex channel gain with $\\lambda$ and $\\beta$ representing respectively the carrier wavelength and the reference channel gain at a distance of 1 meter (m); and $\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\in[0, \\pi]$ represents the azimuth (elevation) angle-of-arrival (AoA) at the IRS.\nMoreover, $\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, \\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)$ denotes the receive response vector, which is given by $\\boldsymbol{a}_{\\mathrm{r}}\\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, \\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}, N_{\\mathrm{act}}\\right)=\\boldsymbol{u}\\left(\\frac{2 d_{\\mathrm{I}}}{\\lambda} \\cos \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right), N_{\\mathrm{x}}\\right) \\otimes$ $\\boldsymbol{u}\\left(\\frac{2 d_{\\mathrm{I}}}{\\lambda} \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right) \\sin \\left(\\vartheta_{\\mathrm{BI}}^{\\mathrm{r}}\\right), N_{\\mathrm{y}}\\right)$, with $\\boldsymbol{u}(\\varsigma, M)=[1, e^{-\\jmath \\pi \\varsigma},$ $\\ldots, $ $e^{-(M-1) \\jmath \\pi \\varsigma}]^{T}$ representing the steering vector function, $d_{\\mathrm{I}}$, $N_{\\mathrm{x}}$, and $N_{\\mathrm{y}}$ denoting the distance between adjacent reflecting elements and the number of active reflecting elements along the $x$- and $y$-axis on IRS surface, respectively.\nBesides, the NLoS component, $\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}$, follows the complex Gaussian distribution with each entry $[\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}]_{n}\\sim\\frac{\\sqrt{\\beta}}{D_{\\mathrm{BI}}}\\mathcal{C} \\mathcal{N}(0,1), \\forall n\\in\\mathcal{N}_{\\mathrm{act}}$.\nSimilarly, the baseband equivalent channel from the active IRS sub-surface to the worst-case user, $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}}\\times 1}$, can be modeled as\n\\begin{equation}\n    \\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}=\\sqrt{\\frac{K_{2}}{K_{2}+1}}\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_{2}+1}}\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}},\n\\end{equation}\nwhere $K_{2}$ denotes the Rician fading factor of the IRS$\\to$user link, $\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ denotes the LoS component, and $\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ denotes the NLoS component.\nThe baseband equivalent channel from the BS to IRS passive sub-surface, $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}$, and that from the IRS passive sub-surface to the worst-case user, $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}}$, can be defined similarly as $\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}$ and $\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}}$, with the details omitted for brevity. \n\nBased on the above, the received signal at the worst-case user aided by the hybrid IRS is given by\n\\begin{equation}\n    y = (\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}s+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}s+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{z}_{\\mathrm{I}}+z_0,\n   \n\\end{equation}\nwhere $s\\in\\mathbb{C}$ denotes the information symbol with $\\mathbb{E}\\left\\{\\left|s\\right|^{2}\\right\\}=P_{\\mathrm{B}}$, and $P_{\\mathrm{B}}$ denotes the transmit power of the BS. \nMoreover, $\\mathbf{z}_{\\rm{I}}\\in\\mathbb{C}^{N_{\\mathrm{act}} \\times 1}$ is the thermal noise introduced by the active elements due to signal amplification, which is assumed to follow the independent CSCG distribution, i.e., $\\mathbf{z}_{\\rm{I}} \\sim \\mathcal{C} \\mathcal{N}\\left(\\mathbf{0}_{N_{\\mathrm{act}}}, \\sigma_{\\mathrm{I}}^{2} \\mathbf{I}_{N_{\\mathrm{act}}}\\right)$ with $\\sigma_{\\mathrm{I}}^{2}$ denoting the amplification noise power, and $z_0\\sim\\mathcal{C} \\mathcal{N}\\left(0, \\sigma_0^{2}\\right)$ is the additive white Gaussian noise (AWGN) at the user. \nNote that at the user's receiver, the desired signal is superposed by the reflected signals over both active and passive elements, while the noise is due to the amplification noise at active elements and the thermal noise at the receiver.\n\nAs such, the receiver SNR at the worst-case user is given by\n\\begin{equation}\n    \\gamma = \\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2}.\\label{snr_hybrid}\n\\end{equation}\nThus, the ergodic capacity achieved by the worst-case user in the hybrid IRS aided wireless communication system is given by\n\\begin{equation}\n    C=\\mathbb{E}\\left\\{\\log _{2}(1+\\gamma)\\right\\},\\label{ergo_capa}\n   \n\\end{equation}\nwhere the expectation is taken over the random NLoS components in all channels involved.\n\\section{Problem Formulation and Ergodic Capacity Analysis}\\label{sec_prob_approx}\nWe aim to maximize the ergodic capacity of the worst-case user subject to a total deployment budget of $W_0$ by optimizing the numbers of active and passive elements,\n$N_{\\mathrm{act}}$ and $N_{\\mathrm{pas}}$, the IRS phase shifts, \\{$\\mathbf{\\Phi}^{\\mathrm{act}}$, $\\mathbf{\\Psi}^{\\mathrm{pas}}$\\}, and the active-element amplification matrix, $\\mathbf{A}^{\\mathrm{act}}$. This problem can be formulated as follows.\n\\begin{align}\n    &\\hspace{-0.2cm}\\mathrm{(P1)}~~~~~\\max_{\\mathbf{\\Phi}^{\\mathrm{act}},\\mathbf{\\Psi}^{\\mathrm{pas}},\\mathbf{A}^{\\mathrm{act}},N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\mathbb{E}\\left\\{\\log _{2}\\left(1\\!+\\!\\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\!+\\!(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\!+\\!\\sigma_0^2}\\right)\\right\\} \\label{obj_func_orig}\\\\\n    &\\qquad\\qquad~~~~\\!~\\quad~\\text{s.t.} \\qquad\\qquad\\qquad~~0<\\varphi_n^{\\mathrm{act}}\\leq 2\\pi,\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{cons_phase_act}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~0<\\varphi_n^{\\mathrm{pas}}\\leq 2\\pi,\\forall n\\in\\mathcal{N}_{\\mathrm{pas}},\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~\\alpha_{\\min}\\leq \\alpha_n\\leq \\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{cons_alpha}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~ \\mathbb{E}\\left\\{P_{\\mathrm{B}}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2+\\sigma_{\\mathrm{I}}^2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\right\\}\\leq P_{\\mathrm{I}},\\label{cstr_power_HI}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~N_{\\mathrm{act}}W_{\\mathrm{act}}+N_{\\mathrm{pas}}W_{\\mathrm{pas}}\\leq W_0,\\label{cons_C}\\\\\n    &\\qquad\\qquad~~~~~~~~\\quad~\\qquad\\qquad\\qquad~~N_{\\mathrm{act}}\\in\\mathbb{N},N_{\\mathrm{pas}}\\in\\mathbb{N}\\label{cons_C_AnP},\n\\end{align}\nwhere the constraint \\eqref{cstr_power_HI} indicates that the average amplification power of all active elements over the Rician fading channels is constrained by the average amplification power budget, $P_{\\mathrm{I}}$.\nNote that different from the existing works on IRS-aided wireless communications that generally require the instantaneous CSI knowledge (e.g., \\cite{9133142}), we consider a practical scenario where only the statistical CSI, i.e., $\\{\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}},\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}},$ $\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}},$ $\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}},$ $K_1,$ $K_2\\}$ is known \\textit{a priori}, which suffices for the design of active\/passive elements allocation for maximizing the ergodic capacity (in the worst case).\nThis approach also reduces the real-time channel estimation overhead and avoids frequent IRS reflection adjustment for each user. \n\nFor problem (P1), note that it includes the conventional IRS architectures with all-passive and all-active elements as\ntwo special cases. Specifically, when $N_{\\mathrm{act}}=0$, the hybrid IRS reduces to the conventional passive IRS and we have $\\mathbf{\\Psi}^{\\mathrm{act}}=\\mathbf{0}_{N_{\\mathrm{act}}\\times N_{\\mathrm{act}}}$. On the other hand, when $N_{\\mathrm{pas}}=0$, it reduces to the conventional active IRS and thus $\\mathbf{\\Psi}^{\\mathrm{pas}}=\\mathbf{0}_{N_{\\mathrm{pas}}\\times N_{\\mathrm{pas}}}$.\nHowever, problem (P1) is challenging to solve even in the above two special cases, since the phase shifts are coupled with the amplification factors in the function of the ergodic capacity (see \\eqref{snr_hybrid} and \\eqref{ergo_capa}). Moreover, the numbers of active and passive elements are discrete, rendering the design objective a complicated function and the constraints in \\eqref{cstr_power_HI}--\\eqref{cons_C_AnP} non-convex.\n\nTo address the above issues, we first analyze the ergodic capacity to approximate the objective function of problem (P1) in a simpler form.\n\n{\\color{black}\\begin{lemma}\\label{lem_C_approx}\n\\textbf{\\emph{(Ergodic Capacity Approximation)}} \\emph{The ergodic capacity in \\eqref{obj_func_orig} can be approximated by}\n\\begin{align\n    C\\approx \\tilde{C}\\triangleq\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right),\\label{sig_approx}\n\\end{align}\n\\emph{where}\n\\begin{align}\n    &x_{\\mathrm{L}} \\triangleq \\frac{K_1K_2P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\left|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\!+\\!(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}\\right|^2,\\label{x_1}\\\\\n    &x_{\\mathrm{NL,act}}\\triangleq\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\frac{K_1\\beta}{d_{\\mathrm{IU}}^2}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\frac{K_2\\beta}{D_{\\mathrm{BI}}^2}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\!+\\!\\frac{\\beta^2}{D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}\\Big),\\label{x_2}\\\\\n    &x_{\\mathrm{NL,pas}}\\triangleq\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\frac{K_1\\beta}{d_{\\mathrm{IU}}^2}\\|\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}\\|^2\\!+\\!\\frac{K_2\\beta}{D_{\\mathrm{BI}}^2}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\|^2\\!+\\!\\frac{\\beta^2N_{\\mathrm{pas}}}{D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}\\Big),\\label{x_3}\\\\\n    &{z_{\\mathrm{L,act}}}\\triangleq{\\frac{K_2\\sigma_{\\rm I}^2}{K_2\\!+\\!1}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2},\\quad\n    {z_{\\mathrm{NL,act}}}\\triangleq{\\frac{\\sigma_{\\rm I}^2}{K_2\\!+\\!1}\\|(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}.\n \\end{align}\n\\end{lemma}}\n\\begin{proof}\nSee Appendix \\ref{proof_lem1}.\n\\end{proof}\nThe accuracy of the approximation in Lemma \\ref{lem_C_approx} will be numerically verified in Section \\ref{sec_sim}.\nSimilarly, the average amplification power consumption can be expressed as\n\\begin{align}\n    \\!\\!\\!\\!\\!\\mathbb{E}\\left\\{P_{\\mathrm{B}}\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\sigma_{\\mathrm{I}}^2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2\\right\\}\\!=\n    \\!\\frac{P_{\\mathrm{B}}}{K_2\\!+\\!1}\\left(K_2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\mathbb{E}\\left\\{\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\right\\}\\right)\\!+\\!\\sum_{n=1}^{N_{\\mathrm{act}}}\\sigma_{\\mathrm{I}}^2\\alpha^2_{n}\\label{power_decomp}.\n\\end{align}\nAs such, problem (P1) is reformatted as follows by using Lemma \\ref{lem_C_approx} and substituting \\eqref{power_decomp} into \\eqref{cstr_power_HI},\n\\begin{align}\n    &\\mathrm{(P2)}\\max_{\\mathbf{\\Phi}^{\\mathrm{act}},\\mathbf{\\Psi}^{\\mathrm{pas}},\\mathbf{A}^{\\mathrm{act}},N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right) \\\\\n    &\\qquad\\qquad~~~~\\!~\\text{s.t.} \\qquad\\qquad~~\\eqref{cons_phase_act}-\\eqref{cons_alpha}, \\eqref{cons_C},\\eqref{cons_C_AnP}\\nonumber,\\\\\n    &\\qquad~~~~~~~\\qquad\\qquad\\qquad~~ \\frac{P_{\\mathrm{B}}}{K_2\\!+\\!1}\\left(K_2\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\!+\\!\\mathbb{E}\\left\\{\\|\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}\\|^2\\right\\}\\right)\\!+\\!\\sum_{n=1}^{N_{\\mathrm{act}}}\\sigma_{\\mathrm{I}}^2\\alpha^2_{n}\\leq\\! P_{\\mathrm{I}}.\\label{cstr_power_HI_decomp}\n\\end{align}\n\n\\section{Optimal Solution to Problem (P2)}\\label{sec_design}\nIn this section, we aim to optimally solve problem (P2) and gain useful insight into the optimal active\/passive elements allocation at the hybrid IRS.\nSpecifically, we first obtain the optimal hybrid IRS beamforming and active\/passive elements allocation based on the statistical CSI under the general Rician fading channel model. Then, we consider two special channel setups, i.e., the LoS and Rayleigh fading channel models, to gain useful insight into the optimal active\/passive elements allocation.\n\n\\subsection{IRS Phase Shift Optimization}\nGiven any feasible active\/passive elements allocation and active-element amplification factors, it can be easily shown that the approximated ergodic capacity in \\eqref{sig_approx} is maximized when the LoS components of the IRS-associated channels, i.e., $(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}$ and $(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}$, are phase-aligned. The optimal IRS phase shifts are thus given by\n\\begin{align}\n    &\\varphi_{n}^{\\mathrm{act}} = \\arg([\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}}]_n)-\\arg([\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}]_n),\\forall n\\in\\mathcal{N}_{\\mathrm{act}},\\label{opt_phase_1}\\\\\n    &\\varphi_{n}^{\\mathrm{pas}} = \\arg([\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}}]_n)-\\arg([\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}]_n),\\forall n\\in \\mathcal{N}_{\\mathrm{pas}},\\label{opt_phase_2}\n\\end{align}\nwhere the optimal phase shifts of the active and passive sub-surfaces have the similar form.\nThen, by substituting the optimal phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} into \\eqref{sig_approx}, we have\n\\begin{align}\n     &x_{\\mathrm{L}}=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n+N_{\\mathrm{pas}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\\\\\n     &x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}=\\gamma_2\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n^2+N_{\\mathrm{pas}}\\right)P_{\\mathrm{B}}\\beta^2(K_1+K_2+1)\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\\label{x_NLoS}\\\\\n     &z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}=\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n^2\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^{2},\\label{n_act}\n\\end{align}\nwhere \n\\begin{equation}\n    \\gamma_1 \\triangleq \\frac{K_1K_2}{(K_1+1)(K_2+1)}, \\gamma_2 \\triangleq \\frac{1}{(K_1+1)(K_2+1)}.\\label{gamma_approx}\n\\end{equation}\nNote that the considered hybrid IRS beamforming with the statistical CSI only is designed based on the LoS components while ignoring the NLoS components of the involved channels.\n\n\\subsection{Amplification Factor Optimization for Active Elements}\n\nIn this subsection, we optimize the amplification factor given any feasible active\/passive elements allocation and optimal IRS phase shifts (see \\eqref{opt_phase_1} and \\eqref{opt_phase_2}). To this end, we first present an important lemma as follows.\n\\begin{lemma}\\label{lem_min}\n\\textbf{\\emph{(Minimum amplification power for active elements)}}\n\\emph{Given the total deployment budget $W_0$, the hybrid IRS should employ passive elements only (i.e., $N_{\\mathrm{act}}=0$) when}\n\\begin{equation\n    P_{\\mathrm{I}} < \\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right),\\label{lem_pas\n\\end{equation}\n\\emph{and employ active elements (i.e., $N_{\\mathrm{act}}>0$) otherwise.}\n\\end{lemma}\n\\begin{proof}\n{It can be shown that when \\eqref{lem_pas} holds, the maximum amplification factor that $P_{\\mathrm{I}}$ can support is smaller than its allowable minimum value, i.e., $\\alpha_{n}<\\alpha_{\\min},\\forall n\\in \\mathcal{N}_{\\mathrm{act}}$, thus making it infeasible to satisfy $\\alpha_{n}\\geq\\alpha_{\\min}$.}\n\\end{proof}\n\nNext, we consider the non-trivial case with $P_{\\mathrm{I}} \\geq \\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)$ and hence $N_{\\mathrm{act}}>0$. Given the optimal phase shift design in \\eqref{opt_phase_1} and \\eqref{opt_phase_2}, problem (P2) reduces to the following problem for optimizing the amplification factors of the $N_{\\mathrm{act}}$ active elements.\n\\begin{align}\n    &\\mathrm{(P3)}~~~~~\\max_{\\mathbf{A}^{\\mathrm{act}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}}{z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}+\\sigma_0^2}\\right) \\\\\n    &\\qquad\\!~~~~\\quad~\\text{s.t.} \\qquad~~\\eqref{cons_alpha}\\nonumber,\\\\\n    &\\qquad~\\qquad\\qquad\\qquad\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\\leq P_{\\mathrm{I}}.\\label{cstr_power_HI_decomp_2}\n\\end{align}\nThe constraints in \\eqref{cons_alpha} and \\eqref{cstr_power_HI_decomp_2} indicate that the amplification factors are bounded by both the hardware limitation and the power budget for all active elements. \nTo simplify the analysis, we have the following lemma.\n\\begin{lemma}\\label{lem_P_I_range}\n\\textbf{\\emph{(Favorable amplification power condition)}}\n\\emph{For problem (P3), the constraint in \\eqref{cons_alpha} is always satisfied if}\n\\begin{equation}\n\\alpha^2_{\\min}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\\leq P_{\\mathrm{I}}\\leq W_0\\alpha^2_{\\max}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\/W_{\\mathrm{act}}.\\label{cons_pb\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst, it can be shown that when \n\\begin{equation}\n    P_{\\mathrm{I}}\\geq W_0\\alpha^2_{\\max}\\left(P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}\\right)\/W_{\\mathrm{act}},\\label{lem_max\n\\end{equation}\nthe amplification factors of $N_{\\mathrm{act}}$ active elements are equal to their maximum value, i.e., $\\alpha_{n}=\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$, where the power amplification is constrained by the maximum load.\nThen, combining \\eqref{lem_max} and Lemma \\ref{lem_min}, there always exists a feasible $\\alpha_{n}\\in[\\alpha_{\\min},\\alpha_{\\max}]$ such that the constraint in \\eqref{cons_alpha} of problem (P3) is satisfied if \\eqref{cons_pb} holds by choosing an appropriate $N_{\\mathrm{act}}\\in[0,W_0\/W_{\\mathrm{act}}]$.\n\\end{proof}\n\nLemma \\ref{lem_P_I_range} gives the favorable amplification power condition for active elements that are able to operate in the amplification mode (i.e., $\\alpha_{n}\\geq\\alpha_{\\min}$) and make full use of the amplification power (i.e., $\\alpha_{n}\\leq\\alpha_{\\max}$).\nIn contrast, if the amplification power is too large, i.e., satisfying \\eqref{lem_max}, the amplification factor of each active element is constrained by its hardware limitation, i.e., $\\alpha_n=\\alpha_{\\max},\\forall n\\in\\mathcal{N}_{\\mathrm{act}}$. On the other hand, if the amplification power is insufficient, i.e., when \\eqref{lem_pas} holds, the active elements cannot operate in the amplification mode, i.e., $\\alpha_n< \\alpha_{\\min},\\exists n\\in \\mathcal{N}_{\\mathrm{act}}$, thus making problem (P3) infeasible. \nTherefore, in the sequel, we only consider the case where $P_{\\mathrm{I}}$ satisfies the favorable amplification power condition given in \\eqref{cons_pb} in order to draw useful insight into the optimal IRS active\/passive elements allocation.\n\\begin{lemma}\\label{lem1}\n\\textbf{\\emph{(Optimal amplification factors)}}\n\\emph{Given \\eqref{cons_pb} and $N_{\\mathrm{act}}>0$, the optimal amplification factor for each active element in problem (P3) is given by}\n\\begin{align}\n    \\alpha_{n}=\\alpha^* \\triangleq \\sqrt{\\frac{P_{\\mathrm{I}}\/N_{\\mathrm{act}}}{P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}}}, \\forall n \\in\\mathcal{N}_{\\mathrm{act}}.\\label{opt_a_n}\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nSee Appendix \\ref{proof_lem2}.\n\\end{proof}\nLemma \\ref{lem1} shows that all active reflecting elements should adopt a common amplification factor due to the same path-loss over the LoS channels associated with the active elements.\nMoreover, it is observed from \\eqref{opt_a_n} that given the amplification power $P_{\\mathrm{I}}$, the optimal amplification factor, $\\alpha^*$, decreases with the number of active elements or signal power.\n\\begin{remark}\n\\textbf{\\emph{(Amplification noise power)}}\n\\emph{By substituting $\\alpha^*$ in \\eqref{opt_a_n} into \\eqref{n_act}, the amplification noise power is given by}\n\\begin{equation}\n    z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}=\\frac{\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})P_{\\mathrm{I}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^{2}}{P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}},\\label{noise_term}\n\\end{equation}\n\\emph{where the indicator function $\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})=1$ when $N_{\\mathrm{act}}>0$ and $\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})=0$ otherwise, i.e., the noise term exists only when the hybrid IRS consists of active elements.\nNote that the amplification noise power in \\eqref{noise_term} is a constant that depends on the total amplification power, $P_{\\mathrm{I}}$, and the path-loss of the BS-IRS and IRS-user channels.\n\n\\end{remark}\nBased on Lemma \\ref{lem1}, the ergodic capacity of the worst-case user with the optimal IRS phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} and amplification factors of active elements in \\eqref{opt_a_n} is given by\n\\begin{equation}\n\\tilde C^*\\!=\\!\\log _{2}\\left(1+\\frac{\\frac{P_{\\mathrm{B}} \\beta^{2}}{D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}}\\left(\\gamma_{1} \\left(\\sqrt{A_{\\mathrm{sum}}N_{\\mathrm{act}}}+N_{\\mathrm{pas}}\\right)^{2}+\\gamma_{2}\\left(A_{\\mathrm{sum}}\\!+\\!N_{\\mathrm{pas}}\\right)\\right)}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2}\\beta \/ d_{\\mathrm{IU}}^{2}\\!+\\!\\sigma_{0}^{2}}\\right),\\label{capa_approx2}\n\\end{equation}\nwhere $A_{\\mathrm{sum}}\\triangleq\\frac{\\mathcal{I}_{\\mathbb{R}^+}(N_{\\mathrm{act}})P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}$.\n\n\\subsection{Active\/Passive Elements Allocation Optimization}\\label{ele_alloc}\nGiven the optimal IRS phase shifts and amplification factors of active elements, in the next, we focus on optimizing the active\/passive elements allocation for maximizing the ergodic capacity subject to the total deployment budget constraint, which is formulated as follows.\n\\begin{align}\n    &\\mathrm{(P4)}~~~~~\\max_{N_{\\mathrm{act}},N_{\\mathrm{pas}}}\n    \\quad~~\\log _{2}\\left(1+\\frac{\\frac{P_{\\mathrm{B}} \\beta^{2}}{D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}}\\left(\\gamma_{1} \\left(\\sqrt{A_{\\mathrm{sum}}N_{\\mathrm{act}}}+N_{\\mathrm{pas}}\\right)^{2}+\\gamma_{2}\\left(A_{\\mathrm{sum}}\\!+\\!N_{\\mathrm{pas}}\\right)\\right)}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2}\\beta \/ d_{\\mathrm{IU}}^{2}\\!+\\!\\sigma_{0}^{2}}\\right) \\\\\n    &\\qquad~~~\\!~\\quad~\\text{s.t.}\\qquad\\quad \\eqref{cons_C},\\eqref{cons_C_AnP}\\nonumber.\n\\end{align}\n\nProblem (P4) is a non-convex optimization problem due to the discrete active\/passive-element number and the non-concave objective function, making it difficult to obtain a closed-form expression for the optimal solution in general.\nAlthough the optimal number of active (passive) elements can be numerically obtained by applying the one-dimensional search over $N_{\\mathrm{act}}=\\{0,1,\\cdots,\\lfloor\\frac{W_0}{W_{\\mathrm{act}}}\\rfloor\\}$ and hence $N_{\\mathrm{pas}} = \\lfloor\\frac{W_0-N_{\\mathrm{act}}W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor$, it yields little useful insight into the optimal IRS active\/passive elements allocation. \nThus, we consider two special channel setups in the following to obtain closed-form expressions for their corresponding optimal active\/passive elements allocation.\n\n\\subsubsection{LoS channel model}\nFor the LoS channels with $K_1\\to\\infty$ and $K_2\\to\\infty$, we obtain from \\eqref{gamma_approx} that $\\gamma_1\\to 1$ and $\\gamma_2\\to 0$. Then, the approximated ergodic capacity, $\\tilde C$ in \\eqref{sig_approx}, is equal to the exact capacity, $C$ in \\eqref{ergo_capa}.\nNote that the term $A_{\\mathrm{sum}}$ is a positive constant, i.e., $A_{\\mathrm{sum}}=\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta \/ D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}$ when $N_{\\mathrm{act}}>0$, and $A_{\\mathrm{sum}}=0$ when $N_{\\mathrm{act}}=0$.\nTherefore, in the following, we solve problem (P4) in two cases, corresponding to the case of passive IRS with $N_{\\mathrm{act}}=0$ and the case of hybrid IRS with $N_{\\mathrm{act}}>0$ and $N_{\\mathrm{pas}}>0$, respectively. \nTo address the discrete active\/passive elements deployment cost in constraints \\eqref{cons_C} and \\eqref{cons_C_AnP}, we first relax the integer values $N_{\\mathrm{act}}$ and $N_{\\mathrm{pas}}$ into continuous values, $\\tilde N_{\\mathrm{act}}$ and $\\tilde N_{\\mathrm{pas}}$, and then apply the integer rounding technique to reconstruct them based on their optimal continuous values.\nMoreover, it can be shown that the equality in \\eqref{cons_C} holds in the optimal solution to problem (P4), i.e., $\\tilde N_{\\mathrm{pas}}=\\frac{W_0-W_{\\mathrm{act}}\\tilde N_{\\mathrm{act}}}{W_{\\mathrm{pas}}}$.\n\nFirst, consider the case of $\\tilde N_{\\mathrm{act}}=0$. By substituting $\\tilde N_{\\mathrm{pas}}=\\frac{W_0}{W_{\\mathrm{pas}}}$, $\\gamma_1\\to 1$, and $\\gamma_2\\to 0$ into \\eqref{capa_approx2}, the hybrid IRS reduces to the passive IRS for which the achievable capacity under the LoS channels is given by\n\\begin{equation}\n   C = C^*_{\\text {L,pas}} \\triangleq  \\log _{2}\\left(1+\\frac{W_0^{2} P_{\\mathrm{B}} \\beta^{2}}{W_{\\mathrm{pas}}^{2} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2} \\sigma_{0}^{2}}\\right).\\label{C_p}\n\\end{equation} \nNext, when $\\tilde N_{\\mathrm{act}}>0$ under the LoS channel model, the achievable capacity in \\eqref{capa_approx2} reduces to\n\\begin{equation}\n    C = C_{\\text {L,hyb}} \\triangleq \\log _{2}\\left(1+\\frac{P_{\\mathrm{B}}\\beta^2(\\sqrt{A_{\\mathrm{sum}}\\tilde N_{\\mathrm{act}}}+\\tilde N_{\\mathrm{pas}})^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2}\\right).\\label{C_h}\n\\end{equation}\nBy substituting $\\tilde N_{\\mathrm{pas}}=\\frac{W_0-W_{\\mathrm{act}}\\tilde N_{\\mathrm{act}}}{W_{\\mathrm{pas}}}$ into \\eqref{C_h}, the optimal solution to problem (P4) given $\\tilde N_{\\mathrm{act}}>0$ can be obtained by solving the following equivalent maximization problem.\n\\begin{align}\n    &\\mathrm{(P5)}~~~~~\\max_{\\tilde N_{\\mathrm{act}}}\n    \\quad~~\\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2\\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)^2\\label{obj_p3}\\\\\n    &\\qquad~~~\\!~\\quad~\\text{s.t.} ~~~~~0< \\tilde N_{\\mathrm{act}}\\leq\\frac{W_0}{W_{\\mathrm{act}}},\n\\end{align}\nwhere\n\\begin{align}\n    \\xi_1=\\frac{P_{\\mathrm{B}}\\beta^2W_{\\mathrm{act}}^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2W_{\\mathrm{pas}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2},\n    \\qquad \\xi_2 = \\frac{\\sqrt{A_{\\mathrm{sum}}}W_{\\mathrm{pas}}}{W_{\\mathrm{act}}},\n    \\qquad \\text{and } \\xi_3 = \\frac{W_0}{W_{\\mathrm{act}}}.\n\\end{align}\n\\begin{theorem}\\label{the_opt_N}\n\\textbf{\\emph{(Optimal active\/passive elements allocation)}} \\emph{Under the LoS channel model and given $\\tilde N_{\\mathrm{act}}>0$, the optimal solution to problem (P4) is}\n\\begin{equation}\\label{opt_na}\n\\begin{cases}\n&\\tilde N^*_{\\mathrm{act}}=\\frac{W_0}{W_{\\mathrm{act}}},\\tilde N^*_{\\mathrm{pas}}=0, \\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~~~\\emph{if } W_0< \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}},\\\\\n&\\tilde N^*_{\\mathrm{act}}=\\frac{P_{\\mathrm{I}}W_{\\mathrm{pas}}^2\/W_{\\mathrm{act}}^2}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\tilde N^*_{\\mathrm{pas}}=\\frac{W_0}{W_{\\mathrm{pas}}}-\\frac{P_{\\mathrm{I}}W_{\\mathrm{pas}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\qquad \\emph{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{proof_lem3}.\n\\end{proof}\n\\begin{remark}\n\\emph{Theorem \\ref{the_opt_N} shows that when the total deployment budget is small, i.e., $W_0< \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, only active elements should be employed since they can provide a signal amplification gain and thus a higher rate than passive elements. In contrast, if the total deployment budget is sufficiently large, i.e., $W_0\\geq \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, the optimal number of active elements should balance the amplification gain of active elements and the beamforming gain of passive elements, which is independent of $W_0$ but determined by other parameters, such as $P_{\\mathrm{I}}$, $W_{\\mathrm{act}}$ and $W_{\\mathrm{pas}}$.\nThis is because the performance bottleneck of active elements becomes the limited amplification power when $W_0$ is large, where the limited power can only support partial active elements to operate with the optimal amplification factor.}\n\\end{remark\n\nBased on Theorem \\ref{the_opt_N} and substituting the optimal number of active elements, $\\tilde N_{\\mathrm{act}}=\\frac{P_{\\mathrm{I}} W_{\\mathrm{pas}}^{2} \/ W_{\\mathrm{act}}^{2}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, into \\eqref{C_h}, the achievable capacity of the worst-case user is obtained as\n\\begin{equation}\nC^*_{\\text {L,hyb}} = \\log _{2}\\left(1+\\frac{P_{\\mathrm{B}} \\beta^{2}\\left(\\frac{A_{\\text {sum }} W_{\\mathrm{pas}}}{4 W_{\\mathrm{act}}}+\\frac{W_0}{W_{\\mathrm{pas}}}\\right)^{2} \/ D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\text {sum }} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\\label{C_h_2}\n\\end{equation}\n\n\\begin{corollary}\n\\emph{For the case of LoS channels, given the optimal IRS phase shifts in \\eqref{opt_phase_1} and \\eqref{opt_phase_2} and active\/passive elements allocation in \\eqref{opt_na}, the optimal amplification factor of each active element is given by}\n\\begin{equation\n    \\alpha_{n}=\\frac{2 W_{\\mathrm{act}}}{W_{\\mathrm{pas}}} ,\\forall  n \\in\\mathcal{N}_{\\mathrm{act}}\n\\end{equation}\n\\emph{for achieving the capacity in \\eqref{C_h_2}.}\n\\end{corollary}\n\nMoreover, when $W_0<\\frac{W_{\\mathrm{pas}}^2P_{\\mathrm{I}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2}$ and hence $N^*_{\\mathrm{act}}=W_0\/W_{\\mathrm{act}}$, the achievable capacity of the worst-case user aided by the active IRS under the LoS channel model is given by\n\\begin{equation}\n    C^*_{\\text {L,act}} \\triangleq \\log _{2}\\left(1+\\frac{W_0A_{\\mathrm{sum}} P_{\\mathrm{B}} \\beta^{2} \/W_{\\mathrm{act}} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\\label{C_a}\n\\end{equation}\n\nNext, we compare the achievable capacity of the hybrid IRS with that of the fully-active and fully-passive IRSs with respect to (w.r.t.) different values of the deployment budget under the LoS channels. To this end, we define the following active\/passive elements allocation thresholds for the deployment budget.\n\\begin{align}\n    W_{\\mathrm{A-H}}&\\triangleq\\frac{W_{\\mathrm{pas}}^2P_{\\mathrm{I}}\/W_{\\mathrm{act}}}{4P_{\\mathrm{B}}\\beta\/D_{\\mathrm{BI}}^2+4\\sigma_{\\mathrm{I}}^2},\\label{B_H-A}\\\\\n    W_{\\mathrm{H-P}}&\\triangleq\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2d_{\\mathrm{IU}}^2}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}^2\\beta}+\\frac{W_{\\mathrm{pas}}^2\\sigma_0d_{\\mathrm{IU}}}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}}\\sqrt{\\frac{\\sigma_0^2d_{\\mathrm{IU}}^2}{\\sigma_{\\mathrm{I}}^2\\beta^2}+\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta^2\/D_{\\mathrm{BI}}^2+\\sigma_{\\mathrm{I}}^2\\beta}}.\\label{condition_H-P}\n\\end{align}\nThen, we have the following result.\n\\begin{theorem}\\label{lem_thres}\n\\textbf{\\emph{(Capacity comparison among different IRS architectures)}}\n\\emph{Under the LoS channel model, the capacity comparison among different IRS architectures is given as follows.}\n\n\\emph{1) When $0\\leq W_0< W_{\\mathrm{A-H}}$, $C^*_{\\text {L,act}}=C^*_{\\text {L,hyb}}>C^*_{\\text {L,pas}}$;}\n\n\\emph{2) When $W_{\\mathrm{A-H}}\\leq W_0\\leq W_{\\mathrm{H-P}}$, $C^*_{\\text {L,hyb}}>C^*_{\\text {L,act}}$ and $C^*_{\\text {L,hyb}}>C^*_{\\text {L,pas}}$;}\n\n\\emph{3) When $W_0>W_{\\mathrm{H-P}}$, $C^*_{\\text {L,pas}}=C^*_{\\text {L,hyb}}>C^*_{\\text {L,act}}$.}\n\\end{theorem}\n\\begin{proof}\nSee Appendix \\ref{proof_lem4}.\n\\end{proof}\n\\begin{remark}\n\\textbf{\\emph{(IRS architecture selection)}}\n\\emph{Theorem \\ref{lem_thres} shows that the optimal architecture selection for the IRS (i.e., passive, active, or hybrid) is determined by the total deployment budget. Specifically, when $W_0$ is small, the hybrid IRS reduces to the fully-active IRS, which is the optimal architecture to achieve the maximum capacity. This is because when $N_{\\mathrm{pas}}$ is small, the active IRS can provide a high power amplification gain while passive elements only have limited beamforming gain. In contrast, when $W_0$ is sufficiently large, the hybrid IRS reduces to the fully-passive IRS. This is expected since in this case, the amplification factor for the active elements is limited, which may not be able to mitigate the amplification noise. However, when $W_{\\mathrm{A-H}}\\leq W_0\\leq W_{\\mathrm{H-P}}$, there in general exists a trade-off between increasing the power amplification gain (i.e., assigning more active elements) and beamforming gain (i.e., assigning more passive elements); hence, it is necessary to design the optimal active\/passive elements allocation for the hybrid IRS to maximize the capacity.}\n\\end{remark}\n\\begin{remark}\n\\textbf{\\emph{(Deployment budget thresholds)}}\n\\emph{It is observed from \\eqref{B_H-A} and \\eqref{condition_H-P} that $W_{\\mathrm{A-H}}$ and $W_{\\mathrm{H-P}}$ both increase with the amplification power, $P_{\\mathrm{I}}$, and the passive-element cost, $W_{\\mathrm{pas}}$, while they decrease with the active-element cost, $W_{\\mathrm{act}}$. \nThis can be explained as follows. \nFirst, a higher amplification power, $P_{\\mathrm{I}}$, allows more active elements to operate with sufficiently large amplification factors. \nSecond, higher passive-element cost, $W_{\\mathrm{pas}}$, and\/or lower active-element cost, $W_{\\mathrm{act}}$, make it more desirable to deploy active elements.}\n\\end{remark}\n\n\\subsubsection{Rayleigh fading channel model}\nFor the case of Rayleigh fading channels with $K_1=K_2=0$, we obtain from \\eqref{gamma_approx} that $\\gamma_1=0$ and $\\gamma_2=1$. As such, the ergodic capacity under the Rayleigh fading channels is given by\n\\begin{equation}\n    \\tilde C=C_{\\mathrm{NL}}\\triangleq\\log _{2}\\left(1+\\frac{\\left(A_{\\mathrm{sum}}+{N}_{\\mathrm{pas}}\\right) \\frac{P_{\\mathrm{B}} \\beta^{2}}{D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right).\n\\end{equation}\n\n\\begin{lemma}\\label{lem_opt_ea_NLoS}\n\\textbf{\\emph{(Optimal active\/passive elements allocation for Rayleigh fading channels)}}\n\\emph{Under the Rayleigh fading channel model and favorable amplification power condition in \\eqref{cons_pb}, the optimal number of active and passive elements are given by}\n\\begin{equation}\\label{opt_na_NLoS}\n\\begin{cases}\n& N^*_{\\mathrm{act}}=1, N^*_{\\mathrm{pas}}=\\lfloor\\frac{W_0-W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor, \\qquad~~~\\emph{if } W_0> \\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}\\sigma_0^2-W_{\\mathrm{act}}\\sigma_0^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2},\\\\\n& N^*_{\\mathrm{act}}=0, N^*_{\\mathrm{pas}}=\\lfloor\\frac{W_0}{W_{\\mathrm{pas}}}\\rfloor,\\qquad~~~~~~~\\emph{otherwise}.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFirst, when $N_{\\mathrm{act}}>0$, it can be shown that\n\\begin{equation}\n    C_{\\mathrm{NL}}\\leq C^*_{\\mathrm{NL,hyb}}\\triangleq\\log _{2}\\left(1+\\frac{\\left(A_{\\mathrm{sum}}+{N}_{\\mathrm{pas,max}}\\right) \\frac{P_{\\mathrm{B}} \\beta^{2}}{D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}\\right),\\label{C_NLoS_hyb}\n\\end{equation}\nwhere ${N}_{\\mathrm{pas,max}}=\\lfloor\\frac{W_0-W_{\\mathrm{act}}}{W_{\\mathrm{pas}}}\\rfloor$ and $N_{\\mathrm{act}}=1$.\nSecond, when $N_{\\mathrm{act}}=0$, we obtain that $A_{\\mathrm{sum}}=0$ and the ergodic capacity achieved by passive IRS is given by\n\\begin{equation}\n    C_{\\mathrm{NL}}=C^*_{\\mathrm{NL,pas}}\\triangleq\\log _{2}\\left(1+\\frac{W_0P_{\\mathrm{B}} \\beta^{2}}{W_{\\mathrm{pas}} \\sigma_{0}^{2} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}\\right). \\label{C_NLoS_pas}\n\\end{equation}\nThen, by comparing $C^*_{\\mathrm{NL,hyb}}$ in \\eqref{C_NLoS_hyb} and $C^*_{\\mathrm{NL,pas}}$ in \\eqref{C_NLoS_pas}, it can be shown that when $W_0> \\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}\\sigma_0^2-W_{\\mathrm{act}}\\sigma_0^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2}$, we have $C^*_{\\mathrm{NL,hyb}}>C^*_{\\mathrm{NL,pas}}$. Based on the above, the optimal active\/passive elements allocation under the Rayleigh fading channel model is given in \\eqref{opt_na_NLoS}, thus completing the proof.\n\\end{proof}\n\\begin{remark}\n\\textbf{\\emph{(Active\/passive elements allocation under the Rayleigh fading channel model)}}\n\\emph{Lemma \\ref{lem_opt_ea_NLoS} shows that under the Rayleigh fading channel model (or rich scattering environment), most of the deployment budget should be assigned to passive elements, while at most one active element needs to be deployed.\nSpecifically, when the amplification power is sufficiently small and\/or the deployment budget is sufficiently large, all the deployment budget should be assigned to passive elements. This is because the power amplification gain provided by active elements cannot mitigate their amplification noise and\/or the beamforming gain of passive elements (which prevails active-element power amplification).\nOn the other hand, when the amplification power is large and\/or  the deployment budget is small, one active element is enough to achieve the amplification power gain because it has no active-element beamforming gain under the Rayleigh fading channels, thus making it desirable to assign most of the deployment budget to passive elements.\n}\n\\end{remark}\n\n\\section{Simulation Results}\\label{sec_sim}\nSimulation results are presented in this section to evaluate the effectiveness of the proposed hybrid IRS architecture and active\/passive elements allocation design. \nIf not specified otherwise, the system setups are as follows.\nWe consider a two-dimensional (2D) Cartesian coordinate system, where the reference points of the BS, the hybrid IRS, and the worst-case user are located at $(0,0)$ m, $(60,0)$ m, and $(60,20)$ m, respectively. \n\nThe deployment costs of each active\/passive element are set as $W_{\\mathrm{act}} = 5$ and $W_{\\mathrm{pas}} = 1$, respectively, by taking into account the fact that the active element in general incurs higher static operation power and hardware cost.\nFor each active element, the feasible region of the amplification factor is in the range of $[\\alpha_{\\min},\\alpha_{\\max}]=[0,14]$ dB \\cite{8403249}.\nThe system operates at a carrier frequency of 6 GHz with the wavelength of $\\lambda=0.05 \\mathrm{~m}$. \nWe consider the same Rician fading factor for the BS-IRS and IRS-user channels, i.e., $K_1=K_2=K$.\nOther parameters are set as $d_{\\mathrm{I}}=\\frac{\\lambda}{4}$, $\\beta=-30$ dB, $\\sigma_{\\mathrm{I}}=\\sigma_0=-80$ dBm, $P_{\\mathrm{B}}=15$ dBm, and $P_{\\mathrm{I}} = 5$ dBm \\cite{9377648}.\n\\subsection{Accuracy of Ergodic Capacity Approximation}\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=2.7in]{fig1.pdf}}\n\\caption{Accuracy of the ergodic capacity approximation given in \\eqref{sig_approx}.}\\label{eval_approx}\n\\end{figure}\n\nFirst, we evaluate in Fig. \\ref{eval_approx} the accuracy of the ergodic capacity approximation presented in Lemma \\ref{lem_C_approx} (see \\eqref{sig_approx}). \nBy setting equal active\/passive elements allocation, i.e., $N_{\\mathrm{act}}=N_{\\mathrm{pas}}$, the actual ergodic capacity in \\eqref{ergo_capa} is obtained based on 1000 independent channel realizations under different Rician factors of $K\\in\\{0$ dB, $10$ dB, $+\\infty\\}$ ($K\\to +\\infty$ corresponds to the LoS channels).\nIt is observed that the approximated ergodic capacity in \\eqref{sig_approx} is close to the exact capacity in \\eqref{ergo_capa} under different Rician factors. This is because when the number of reflecting elements is large, the variance of random channels is averaged due to the law of large numbers.\n\\subsection{Effect of Active\/passive Elements Allocation on Ergodic Capacity}\n\\begin{figure}[ht]\n\\centerline{\\includegraphics[width=2.7in]{fig2.pdf}}\n\\caption{Capacity performance versus budget percentage of active elements.}\\label{rate_ratio}\n\\end{figure}\n\nNext, we investigate the effect of the active\/passive elements allocation at the hybrid IRS on the capacity performance.\nWe set the total deployment budget $W_0=3000$, the amplification power $P_{\\mathrm{I}}=15$ dBm, and denote $\\rho$ as the percentage of the budget assigned to active elements ($0\\leq \\rho\\leq 1$).\nIn Fig. \\ref{rate_ratio}, we plot the ergodic capacity versus $\\rho$ with different Rician factors $K\\in\\{0$, $15$ dB, $+\\infty\\}$.\nIt is observed that for the cases of $K\\to\\infty$ and $K=15$ dB (with LoS channel components), the ergodic capacity first increases with $\\rho$, then decreases after it exceeds a threshold (i.e., $\\rho^*=0.35$ for $K\\to\\infty$, and $\\rho^*=0.18$ for $K=15$ dB). This shows that for the general Rician fading channels, the active\/passive elements allocation has a significant effect on the capacity maximization.\nThis is expected because if the budget assigned to active elements is too small, the active-element power amplification gain is not fully exploited due to the small number of active elements. However, if the budget assigned to active elements is too large, the performance bottleneck of active elements becomes the limited amplification power that cannot support all active elements to operate in the amplification mode, thus making it desirable to assign partial budget to passive elements for achieving higher beamforming gain.\nMoreover, the optimal budget assigned to active elements increases with the Rician factor because the active elements can achieve both the power amplification gain and beamforming gain over the LoS paths but have no beamforming gain over NLoS paths.\nLast, when $K=0$ (corresponding to the Rayleigh fading channels), the total deployment budget should be assigned to passive elements to maximize the ergodic capacity since the beamforming gain of passive elements is larger than the amplification gain of active elements.\n\\subsection{Effect of Total Deployment Budget}\n\\begin{figure}[t] \\centering    \n{\\subfigure[{LoS channels with $K\\to +\\infty$.}] {\n\\label{fig_3_LoS}\n\\includegraphics[width=2.7in]{fig3_LoS.pdf}  \n}}     \n{\\subfigure[{Rician fading channels with $K=10$ dB.}] {\\label{fig_3_Rician}\n\\includegraphics[width=2.7in]{fig3_Rician.pdf}  \n}}\n{\\caption{The ergodic capacity achieved by hybrid IRS, fully-active IRS, and fully-passive IRS versus deployment budget.}}\n\\end{figure}\n\nNext, we compare the proposed optimal active\/passive elements allocation (EA) design against three benchmarks: 1) Fully-active IRS with all the deployment budget assigned to active elements; 2) Fully-passive IRS with all the deployment budget assigned to active elements; 3) Hybrid IRS under equal EA for which the deployment budget is equally assigned to active and passive elements. We apply the optimal IRS beamforming design in \\eqref{opt_phase_1}, \\eqref{opt_phase_2}, and \\eqref{opt_a_n} for the proposed hybrid IRS architecture and the benchmarks.\n\nFigs. \\ref{fig_3_LoS} and \\ref{fig_3_Rician} show the ergodic capacity of the worst-case user versus the total deployment budget under the LoS and Rician fading channel models, respectively.\nFirst, it is observed that the ergodic capacity achieved by the hybrid IRS architecture with optimal active\/passive elements allocation is always larger than or equal to that achieved by the fully-active or fully-passive IRSs. This is expected because the hybrid IRS provides an extra degree of freedom for elements allocation to balance the trade-off between the power amplification and beamforming gains.\nSecond, the hybrid IRS with the optimal active\/passive elements allocation reduces to the fully-active IRS when the deployment budget is sufficiently small (i.e., $W_0<1250$ for the LoS channels, and $W_0<600$ for the Rician fading channels with $K=10$ dB). \nThis is because the active elements has a higher power amplification gain as compared to the passive-element beamforming gain when the budget is small.\nThird, one can observe that the hybrid IRS with the optimal active\/passive elements allocation outperforms that with equal elements allocation, which shows the effectiveness of elements allocation optimization for the hybrid IRS.\nLast, it is also observed that given the same deployment budget, the achievable capacity under the LoS channels is higher than that under Rician fading channels since the IRS phase shifts are designed based on the LoS channel components or statistical CSI only.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{LoS channels with $K\\to +\\infty$.}] {\n\\label{fig_4_LoS}\n\\includegraphics[width=2.7in]{fig4_LoS.pdf}  \n}}     \n{\\subfigure[{Rician fading channels with $K=10$ dB.}] {\\label{fig_4_Rician}\n\\includegraphics[width=2.7in]{fig4_Rician.pdf}     \n}}\n{\\caption{The optimal numbers of active and passive elements versus total deployment budget.}}\n\\end{figure}\n\nFigs. \\ref{fig_4_LoS} and \\ref{fig_4_Rician} present the optimal numbers of active and passive elements for maximizing the ergodic capacity of the worst-case user under different total deployment budgets with $W_{\\mathrm{act}}=5$ and $W_{\\mathrm{pas}}=1$.\nIt is observed that as $K$ increases, the deployment budget is first assigned to active elements only and then assigned to more passive elements after it exceeds a threshold.\nIn addition, we observe that given the same deployment budget, \nthe optimal number of active elements for the LoS channels, i.e., $N^*_{\\mathrm{act}}=250$ is higher than that for the Rician fading channels with $K=10$ dB, i.e., $N^*_{\\mathrm{act}}=120$.\n\\subsection{Effect of Rician Factor}\nIn Figs. \\ref{fig5_capa_vs_K} and \\ref{fig5_ea_vs_K}, we investigate the effect of the Rician factor of the BS-IRS and IRS-user channels on the achievable capacity and active\/passive elements allocation. The IRS beamforming design and the active-element amplification factors are given by \\eqref{opt_phase_1}, \\eqref{opt_phase_2}, and \\eqref{opt_a_n}, respectively.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus Rician factor.}] {\n\\label{fig5_capa_vs_K}\n\\includegraphics[width=2.63in]{fig5_capa_vs_K.pdf}  \n}}     \n{\\subfigure[Optimal number of active and passive elements versus Rician factor.] {\\label{fig5_ea_vs_K}\n\\includegraphics[width=2.93in]{fig5_ea_vs_K.pdf}     \n}}\n{\\caption{Effect of the Rician factor on ergodic capacity and active\/passive elements allocation.}}\n\\end{figure}\n\nSpecifically, Fig. \\ref{fig5_capa_vs_K} presents the ergodic capacity of the worst-case user under different Rician factors. It is observed that the proposed hybrid IRS architecture with the optimal active\/passive elements allocation outperforms other benchmarks under different channel conditions, while the active IRS achieves a higher capacity than the passive IRS only when the channels are LoS-dominant (i.e., Rician factor $K$ is large). \nFig. \\ref{fig5_ea_vs_K} plots the optimal numbers of active and passive elements of the hybrid IRS under different Rician factors. \nIt is observed that with an increasing Rician factor, the optimal number of active elements first increases and then keeps unchanged after it exceeds a threshold, where the NLoS channel components are negligible as compared to the LoS channel components. \nThe reason is that for the NLoS paths, the active elements have no beamforming gain but can achieve power amplification gain due to the high amplification power, while for the LoS paths, active elements can achieve both the power amplification and beamforming gains with multiple elements. \nTherefore, as $K$ increases, more deployment budget should be assigned to the active elements so as to achieve a higher beamforming gain of the active elements.\n\n\\subsection{Effect of Amplification Power}\nMoreover, we evaluate the effect of the amplification power of active elements on the ergodic capacity and active\/passive elements allocation of the proposed hybrid IRS architecture under the general Rician fading channel model with $K=15$ dB.\n\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus amplification power.}] {\n\\label{fig6_capa_vs_P_I}\n\\includegraphics[width=2.63in]{fig6_capa_vs_P_I.pdf}  \n}}     \n{\\subfigure[Optimal numbers of active and passive elements versus amplification power.] {\\label{fig6_ea_vs_P_I.pdf}\n\\includegraphics[width=2.93in]{fig6_ea_vs_P_I.pdf}     \n}}\n{\\caption{Effect of the amplification power on ergodic capacity and elements allocation.}}\n\\end{figure}\n\nIn Fig. \\ref{fig6_capa_vs_P_I}, we plot the ergodic capacity versus the amplification power $P_{\\mathrm{I}}$ for different schemes.\nFirst, it is observed that the ergodic capacity of the worst-case user aided by the fully-active or hybrid IRSs monotonically increases with the amplification power. Second, we observe that the hybrid IRS with the optimal active\/passive elements allocation achieves a much higher capacity than the fully-active IRS when $P_{\\mathrm{I}}$ is small and significantly outperforms the fully-passive IRS when $P_{\\mathrm{I}}$ is large. This shows the flexibility and advantages of the hybrid IRS under different amplification powers.\n\nIn Fig. \\ref{fig6_ea_vs_P_I.pdf}, we plot the optimal numbers of active and passive elements for the hybrid IRS architecture versus the amplification power. \nOne can observe that the optimal number of active elements increases with the amplification power, which is expected because a higher amplification power can support more active elements to operate in the amplification mode with optimal amplification factors. \n\\subsection{Effect of Active\/Passive-element Deployment Cost Ratio}\n\\begin{figure}[t] \\centering    \n{\\subfigure[{Ergodic capacity versus active\/passive-element cost ratio.}] {\n\\label{fig7_capa_vs_c_ratio}\n\\includegraphics[width=2.63in]{fig7_capa_vs_c_ratio.pdf}  \n}}     \n{\\subfigure[Optimal numbers of active and passive elements versus deployment cost ratio.] {\\label{fig7_ea_vs_c_ratio}\n\\includegraphics[width=2.93in]{fig7_ea_vs_c_ratio.pdf}     \n}}\n{\\caption{Effect of the deployment cost ratio on ergodic capacity and elements allocation.}}\n\\end{figure}\nIn Fig. \\ref{fig7_capa_vs_c_ratio}, we compare the ergodic capacity of the worst-case user versus different ratios of active-over-passive deployment cost, i.e., $W_{\\mathrm{act}}\/W_{\\mathrm{pas}}$, where we set $W_{\\mathrm{pas}}=1$, $K=15$ dB, $W_0=1500$, and $P_{\\mathrm{B}}=10$ dBm.\nFirst, the hybrid IRS with the optimal active\/passive elements allocation has the highest capacity as compared to other benchmarks. Besides, one can observe that as $W_{\\mathrm{act}}$ increases, the ergodic capacity decreases when $W_{\\mathrm{act}}$ is small, then remains unchanged when $W_{\\mathrm{act}}$ is sufficiently large, which can be explained as follows.\nWhen the deployment budget is small, a larger $W_{\\mathrm{act}}$ means that a smaller number of active elements can be deployed, thus resulting in reduced ergodic capacity.\nWhen $W_{\\mathrm{act}}$ exceeds a threshold, all the deployment budget is assigned to passive elements for maximizing the ergodic capacity. As such, increasing $W_{\\mathrm{act}}$ will not change the ergodic capacity and elements allocation.\n\\section{Conclusions}\\label{sec_conclu}\nIn this paper, we proposed a new hybrid active-passive IRS architecture and studied its optimal active\/passive elements allocation in a hybrid IRS aided wireless system based on the statistical CSI only.\nSpecifically, under the general Rician fading channel model, we formulated an optimization problem to maximize the ergodic capacity of the worst-case user, by jointly optimizing the active\/passive elements allocation, their phase shifts, and the amplification factors of active elements, subject to various practical constraints on the active-element amplification factor and amplification power consumption, as well as the total active and passive elements deployment budget.\nTo solve this problem, we first approximated the ergodic capacity in a simpler form and then proposed an efficient algorithm to obtain the optimal hybrid IRS beamforming and active\/passive elements allocation.\nMoreover, it was shown that when all channels are of LoS, only active elements need to be deployed when the total deployment budget is sufficiently small, while both active and passive elements should be deployed with a decreasing number ratio when the budget increases and exceeds a certain threshold.\nLast, numerical results demonstrated the performance gains achieved by the proposed hybrid IRS architecture with the optimal active\/passive elements allocation against the benchmarks of the fully-active and fully-passive IRSs, as well as the hybrid IRS with equal active\/passive elements allocation. This validated that the hybrid IRS can flexibly balance the trade-off between the peculiar power amplification gain of active IRS and superior beamforming gain of passive IRS.\n\n\\appendices\n\\section{Proof of Lemma \\ref{lem_C_approx}}\\label{proof_lem1}\nFirst, based on the Lemma 1 of \\cite{6816003}, we obtain the following approximation for the ergodic capacity. \n\\begin{align}\n    C&=\\mathbb{E}\\left\\{\\log _{2}\\left(1+\\frac{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2}{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2}\\right)\\right\\} \\\\\n    &\\approx\\log_2\\left(1+\\frac{\\mathbb{E}\\left\\{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2\\right\\}}{\\mathbb{E}\\left\\{\\sigma_{\\mathrm{I}}^2\\|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2+\\sigma_0^2\\right\\}}\\right).\\label{C_approx}\n\\end{align}\nThen, we focus on the derivation of the desired signals at the worst-case user, which can be expressed as\n\\begin{align}\n    &\\mathbb{E}\\left\\{P_{\\mathrm{B}}|(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{act}}+(\\mathbf{h}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\mathbf{h}_{\\mathrm{BI}}^{\\mathrm{pas}}|^2\\right\\}\\\\\n    &=\\frac{P_{\\mathrm{B}}}{(K_1+1)(K_2+1)}\\mathbb{E}\\Big\\{\\Big|\\underbrace{\\sqrt{K_1K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{1}}+\\underbrace{\\sqrt{K_1}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{2}}\\nonumber\\\\\n    &+\\underbrace{\\sqrt{K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{3}}+\\underbrace{(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{act}}}_{x_{4}}+\\underbrace{\\sqrt{K_1K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{5}}\\\\\n    &+\\underbrace{\\sqrt{K_1}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\bar{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{6}}+\\underbrace{\\sqrt{K_2}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{7}}+\\underbrace{(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{pas}})^H\\mathbf{\\Psi}^{\\mathrm{pas}}\\tilde{\\mathbf{h}}_{\\mathrm{BI}}^{\\mathrm{pas}}}_{x_{8}}\\Big|^2\\Big\\}\\nonumber\\\\\n    &=\\frac{P_{\\mathrm{B}}}{(K_1\\!+\\!1)(K_2\\!+\\!1)}\\Big(\\mathbb{E}\\left\\{\\left|x_{1}\\!+\\!x_{5}\\right|^2\\right\\}\\!+\\!\\mathbb{E}\\Big\\{\\left|x_{2}\\right|^2\\!+\\!\\left|x_{3}\\right|^2\\!+\\!\\left|x_{4}\\right|^2\\!+\\!\\left|x_{6}\\right|^2\\!+\\!\\left|x_{7}\\right|^2\\!+\\!\\left|x_{8}\\right|^2\\Big\\} \\Big)\\\\\n    &=x_{\\mathrm{L}}+x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}},\\label{lem_C_approx_1}\n\\end{align}\nwhere $x_{\\mathrm{L}}$, $x_{\\mathrm{NL,act}}$, and $x_{\\mathrm{NL,pas}}$ are defined in \\eqref{x_1}, \\eqref{x_2}, and \\eqref{x_3}, respectively.\nNext, the noise introduced by active reflecting elements and that at the receiver can be expressed as\n\\begin{align}\n    &\\mathbb{E}\\left\\{\\sigma_{\\rm I}^2\\left\\|({\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\right\\|^2+\\sigma_0^2\\right\\}\\\\\n    &=\\mathbb{E}\\left\\{\\sigma_{\\rm I}^2\\left\\|\\sqrt{\\frac{K_2}{K_2+1}}(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}+\\sqrt{\\frac{1}{K_2+1}}(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\right\\|^2\\right\\}+\\sigma_0^2\\\\\n    &=\\underbrace{\\frac{K_2\\sigma_{\\rm I}^2}{K_2+1}\\|(\\bar{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}_{z_{\\mathrm{L,act}}}+\\underbrace{\\frac{\\sigma_{\\rm I}^2}{K_2+1}\\|(\\tilde{\\mathbf{h}}_{\\mathrm{IU}}^{\\mathrm{act}})^H\\mathbf{\\Psi}^{\\mathrm{act}}\\|^2}_{z_{\\mathrm{NL,act}}}+\\sigma_0^2.\\label{noise_approx}\n\\end{align}\nLast, the proof is completed by substituting \\eqref{lem_C_approx_1} and \\eqref{noise_approx} into \\eqref{C_approx}.\n\\section{Proof of Lemma \\ref{lem1}}\\label{proof_lem2}\nFirst, given favorable amplification power condition in \\eqref{cons_pb}, it can be verified that at the optimal solution to problem (P3), the constraint \\eqref{cstr_power_HI_decomp_2} is always active, i.e., $\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha^2_{n}=A_{\\mathrm{sum}}$. Then, by substituting $A_{\\mathrm{sum}}$ into \\eqref{x_NLoS} and \\eqref{n_act}, we can show that the NLoS part of the desired signal, $x_{\\mathrm{NL,act}}+x_{\\mathrm{NL,pas}}$, and the amplification noise , $z_{\\mathrm{L,act}}+z_{\\mathrm{NL,act}}$, are constants. Thus, problem (P3) can be solved by maximizing the ergodic capacity due to the LoS channel component, $x_{\\mathrm{L}}$. \nBy using the Cauchy-Schwarz inequality, we have\n\\begin{align}\n    x_{\\mathrm{L}}&=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\alpha_n+N_{\\mathrm{pas}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}\\\\\n    &=\\gamma_1\\left(\\sum_{n=1}^{N_{\\mathrm{act}}}\\left(\\alpha_n+N_{\\mathrm{pas}}\/N_{\\mathrm{act}}\\right)\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}}\\\\\n    &\\leq \\gamma_1N_{\\mathrm{act}}^2\\left(\\alpha^*+N_{\\mathrm{pas}}\/N_{\\mathrm{act}}\\right)^2P_{\\mathrm{B}}\\beta^2\/D^2_{\\mathrm{BI}}d^2_{\\mathrm{IU}},\n\\end{align}\nwhere the equality holds if and only if $\\alpha_{n}=\\alpha^*,\\forall n \\in\\mathcal{N}_{\\mathrm{act}}$ with $\\alpha^*=\\sqrt{\\frac{P_{\\mathrm{I}}\/N_{\\mathrm{act}}}{P_{\\mathrm{B}}\\beta\/D^2_{\\mathrm{BI}}+\\sigma^2_{\\mathrm{I}}}}$ in \\eqref{opt_a_n}, which thus completes the proof.\n\\section{Proof of Theorem \\ref{the_opt_N}}\\label{proof_lem3}\n\\begin{table}[t]\\centering\n\\caption{Variations of the Receiver SNR under Different Conditions.}\\label{opt_N_A}\n\\begin{center}\n\\begin{tabular}{|c|c|}\n\\hline \nCondition & Variations of $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$ \\\\\n\\hline\n$0<\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}<n_{\\mathrm{A},2}$ & Increase for $\\tilde N_{\\mathrm{act}}\\in(0,\\frac{W_0}{W_{\\mathrm{act}}}]$\\\\\n\\hline \n$n_{\\mathrm{A},2}\\leq \\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}\\leq n_{\\mathrm{A},3}$ & Increase for  $\\tilde N_{\\mathrm{act}}\\in(0,n^2_{\\mathrm{A},2}]$, decrease for $\\tilde N_{\\mathrm{act}}\\in(n^2_{\\mathrm{A},2},\\frac{W_0}{W_{\\mathrm{act}}}]$\\\\\n\\hline \n$\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}>n_{\\mathrm{A},3}$ &\\!\\! Increase for $\\tilde N_{\\mathrm{act}}\\in(0,n^2_{\\mathrm{A},2}]\\cup(n^2_{\\mathrm{A},3},\\frac{W_0}{W_{\\mathrm{act}}}]$, decrease for $\\tilde N_{\\mathrm{act}}\\in(n^2_{\\mathrm{A},2},n^2_{\\mathrm{A},3}]$ \\\\\n\\hline \n\\end{tabular}\n\\end{center\n\\end{table}\nFor the receiver SNR, $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}}) = \\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2\\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)^2$, its first-order derivative over $\\sqrt{\\tilde N_{\\mathrm{act}}}$ can be expressed as\n\\begin{equation}\n    \\frac{\\partial \\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})}{\\partial \\sqrt{\\tilde N_{\\mathrm{act}}}}=2\\xi_1\\left(-\\tilde N_{\\mathrm{act}}+\\xi_2 \\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_3\\right)(-2  \\sqrt{\\tilde N_{\\mathrm{act}}}+\\xi_2).\\label{deri_snr}\n\\end{equation}\n\nFirst, it can be shown that \\eqref{deri_snr} equals to 0 for $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},1}\\triangleq\\frac{\\xi_2-\\sqrt{\\xi_2^2+4\\xi_3}}{2}$, $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},2}=\\frac{\\xi_2}{2}$, and $\\sqrt{\\tilde N_{\\mathrm{act}}}=n_{\\mathrm{A},3}\\triangleq\\frac{\\xi_2+\\sqrt{\\xi_2^2+4\\xi_3}}{2}$ with $n_{\\mathrm{A},1}<0<n_{\\mathrm{A},2}<n_{\\mathrm{A},3}$.\nThen, we summarize in Table \\ref{opt_N_A} the optimal number of active elements under different conditions of $\\tilde N_{\\mathrm{act}}$.\n\nBased on Table \\ref{opt_N_A}, when $n_{\\mathrm{A},2}\\leq \\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}\\leq n_{\\mathrm{A},3}$, it is obtained that $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$ increases for $0<\\tilde N_{\\mathrm{act}}\\leq n_{\\mathrm{A},2}^2$ and decreases for $n_{\\mathrm{A},2}^2<\\tilde N_{\\mathrm{act}}\\leq\\frac{W_0}{W_{\\mathrm{act}}}$, thus leading to the optimal number of active elements, $\\tilde N^*_{\\mathrm{act}}=n_{\\mathrm{A},2}^2$, for maximizing $\\gamma_{\\mathrm{hyb}}(\\tilde N_{\\mathrm{act}})$. Following the similar procedure, the optimal number of active elements under different conditions can be obtained.\nNote that when $\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}>n_{\\mathrm{A},3}$, we have\n\\begin{equation}\n    \\gamma_{\\mathrm{hyb}}(n_{\\mathrm{A},2}^2)-\\gamma_{\\mathrm{hyb}}(\\frac{W_0}{W_{\\mathrm{act}}})=\\frac{\\xi_2^2}{4}+\\xi_3-\\xi_2\\sqrt{\\xi_3}=(\\frac{\\xi_2}{2}-\\xi_3)^2\\geq 0,\n\\end{equation}\nsuch that $\\gamma_{\\mathrm{hyb}}(n_{\\mathrm{A},2}^2)\\geq \\gamma_{\\mathrm{hyb}}(\\frac{W_0}{W_{\\mathrm{act}}})$.\nMoreover, when $0<\\sqrt{\\frac{W_0}{W_{\\mathrm{act}}}}<n_{\\mathrm{A},2}$, i.e., the deployment cost $W_0 \\leq \\frac{W_{\\mathrm{pas}}^{2} P_{\\mathrm{I}} \/ W_{\\mathrm{act}}}{4 P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+4 \\sigma_{\\mathrm{I}}^{2}}$, all the deployment budget should be assigned to active elements, i.e., $\\tilde N^*_{\\mathrm{act}}=\\frac{W_0}{W_{\\mathrm{act}}}$.\nBased on the above, the optimal active\/passive elements allocation for the hybrid IRS is given by \\eqref{opt_na}, thus completing the proof.\n\\section{Proof of Theorem \\ref{lem_thres}}\\label{proof_lem4}\nFirst, we compare the achievable capacity by the fully-passive and fully-active IRSs in \\eqref{C_p} and \\eqref{C_a}, respectively. By solving\n\\begin{equation}\n    \\frac{W_0A_{\\mathrm{sum}} P_{\\mathrm{B}} \\beta^{2} \/W_{\\mathrm{act}} D_{\\mathrm{BI}}^{2} d_{\\mathrm{IU}}^{2}}{A_{\\mathrm{sum}} \\sigma_{\\mathrm{I}}^{2} \\beta \/ d_{\\mathrm{IU}}^{2}+\\sigma_{0}^{2}}>\\frac{W_0^2P_{\\mathrm{B}}\\beta^2}{W_{\\mathrm{pas}}^2D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2\\sigma_0^2},\n\\end{equation}\nit is obtained that when\n\\begin{equation}\n    W_0 > W_{\\mathrm{A-P}}\\triangleq\\frac{W_{\\mathrm{pas}}^2\/W_{\\mathrm{act}}}{\\frac{\\beta\\sigma_{\\mathrm{I}}^2}{d_{\\mathrm{IU}}^2\\sigma_0^2}+\\frac{P_{\\mathrm{B}} \\beta \/ D_{\\mathrm{BI}}^{2}+\\sigma_{\\mathrm{I}}^{2}}{P_{\\mathrm{I}}}},\n\\end{equation}\nthe fully-passive IRS outperforms the fully-active IRS in term of the capacity.\nSecond, it is obtained from \\eqref{deri_snr} that when $W_0>W_{\\mathrm{A-H}}$, the hybrid IRS outperforms the fully-active IRS, and achieves its maximum.\nThird, we compare the achievable capacity by the hybrid IRS and passive IRS. By comparing \\eqref{C_p} and \\eqref{C_h_2}, it can be shown that when\n\\begin{equation}\n    \\frac{W_0^2P_{\\mathrm{B}}\\beta^2}{W_{\\mathrm{pas}}^2D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2\\sigma_0^2}>\\frac{P_{\\mathrm{B}}\\beta^2\\left(\\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}}{4W_{\\mathrm{act}}}+\\frac{W_0}{W_{\\mathrm{pas}}}\\right)^2\/D_{\\mathrm{BI}}^2d_{\\mathrm{IU}}^2}{A_{\\mathrm{sum}}\\sigma_{\\mathrm{I}}^2\\beta\/d_{\\mathrm{IU}}^2+\\sigma_0^2},\\label{con_hp}\n\\end{equation}\nthe fully-passive IRS achieves the maximum capacity.\nThe condition in \\eqref{con_hp} can be re-expressed as\n\\begin{equation}\\label{ineq_H-P}\n    \\frac{\\sigma_{\\mathrm{I}}^2\\beta}{d_{\\mathrm{IU}}^2}W_0^2-\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2}{2W_{\\mathrm{act}}}W_0-\\frac{A_{\\mathrm{sum}}W_{\\mathrm{pas}}^4\\sigma_0^2}{16W_{\\mathrm{act}}^2}>0,\n   \n\\end{equation}\nwhich leads to\n\\begin{equation}\n    W_0<\\frac{W_{\\mathrm{pas}}^2\\sigma_0^2d_{\\mathrm{IU}}^2}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}^2\\beta}-\\frac{W_{\\mathrm{pas}}^2\\sigma_0d_{\\mathrm{IU}}}{4W_{\\mathrm{act}}\\sigma_{\\mathrm{I}}}\\sqrt{\\frac{\\sigma_0^2d_{\\mathrm{IU}}^2}{\\sigma_{\\mathrm{I}}^2\\beta^2}+\\frac{P_{\\mathrm{I}}}{P_{\\mathrm{B}}\\beta^2\/D_{\\mathrm{BI}}^2+\\sigma_{\\mathrm{I}}^2\\beta}}<0,\n   \n\\end{equation}\nand $W_0>W_{\\mathrm{H-P}}$. \nLast, we discuss the relations of $W_{\\mathrm{A-P}},W_{\\mathrm{A-H}}$ and $W_{\\mathrm{H-P}}$\nto facilitate our analysis, we list all possible permutations of the above thresholds in Table \\ref{per_thres}.\n\\begin{table}[t]\\centering\n\\caption{Possible relations for budget thresholds.}\\label{per_thres}\n\\begin{tabular}{|c|c|}\n\\hline\n\\begin{tabular}[c]{@{}c@{}}Permutations\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}$W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}},W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}},W_{\\mathrm{A-P}}<W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}},$\\\\ $W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}}<W_{\\mathrm{A-H}},W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-H}},W_{\\mathrm{H-P}}<W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}.$\\end{tabular} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\noindent It can be shown that only one permutation of the above thresholds is feasible, i.e., $W_{\\mathrm{A-H}}<W_{\\mathrm{A-P}}<W_{\\mathrm{H-P}}$. \nSpecifically, the hybrid IRS reduces to the fully-active IRS when $W_0<W_{\\mathrm{A-H}}$; it reduces to the fully-passive IRS when $W_0>W_{\\mathrm{H-P}}$; and it outperforms the fully-active and fully-passive IRSs with the optimal number of active elements of $\\tilde N_{\\mathrm{act}}=\\tilde N^*_{\\mathrm{act}}$ otherwise. \nMoreover, all other permutations can be verified to be infeasible by contradiction. For example, for the permutation $W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_{\\mathrm{A-P}}$, when $W_{\\mathrm{A-H}}<W_{\\mathrm{H-P}}<W_0<W_{\\mathrm{A-P}}$, we obtain that the hybrid IRS outperforms the fully-active IRS when $W_0>W_{\\mathrm{A-H}}$, the fully-passive IRS outperforms hybrid IRS when $W_0>W_{\\mathrm{H-P}}$, and the fully-active IRS outperforms the fully-passive IRS when $W_0<W_{\\mathrm{A-P}}$, which contradicts each other and thus is infeasible.\nCombining the above leads to the desired result.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFollowing the canonical approach to a quantum theory of gravity, a\n(Dirac) observable is by definition a self-adjoint operator on the full \nHilbert space (not only on the Hilbert space induced from the full Hilbert\nspace by restricting to the space of solutions of the quantum constraints)\nwhich weakly commutes with the constraint operators. Equivalently, an\nobservable leaves the physical Hilbert space of solutions to the quantum\nconstraints invariant.\n\nThere are no Dirac observables known in neither classical nor quantum\ngravity, except for the asymptotically flat case where it is well-known that\nthe Poincar\\'e generators at spatial infinity form a closed Dirac observable\nalgebra.\n\nIn the present paper we address the question of how to quantize these\ngenerators. We work with the real version \\cite{Barbero} of the\noriginally complex connection formulation of general relativity \\cite{0}.\nThe associated real connection representation has been made a solid\nfoundation of the quantum theory in the series of papers \n\\cite{1,2,3,4,5,6,7}. This rigorous mathematical framework which is based \non the earlier pioneering work on loop variables, for instance \n\\cite{Gambini,RS} for gauge theories and gravity respectively (the latter\nof which was designed for the complex variables and thus was lacking an \nappropriate inner product), equips us with the tools necessary to ask the \nquestion of \nwhether the various operators constructed are densely defined, symmetric,\nself-adjoint, diffeomorphism invariant and so forth.\n\nBut even though this kinematical framework is available, it is still quite\nsurprising that something like an ADM energy operator can actually be\nconstructed. The reason for this is that in the representation under\nconsideration the ADM energy function turns out ot be a rather\nnon-polynomial function of the canonical momenta and thus it is far from\nclear how to define it. It turns out that the same technique that \nenabled one to define the Wheeler-DeWitt operator \nfor 3+1 Lorentzian gravity \\cite{8,9,10}, the Wheeler-DeWitt operator for \n2+1 Euclidean gravity \\cite{10a}, length operators \\cite{11}, and matter \nHamiltonians when coupled to \ngravity \\cite{12} can be employed to define Poincar\\'e quantum operators.\\\\\n\\\\\nThe plan of the present paper is as follows :\n\nIn section 2 we review the necessary mathematical background from\n\\cite{1,2,3,4,5,6,7,15}.\n\nIn section 3 we regularize the ADM energy operator. There are at least\ntwo natural orderings, one in which the operator becomes densely defined \non the full physical Hilbert space as defined in \\cite{15} and one in which \nit is not. Nevertheless the latter\noperator should be the physically relevant one because when restricting the \nHilbert space to a subspace which is suggested to us given the \nasymptotically flat structure available, then this operator turns out to be \npositive-semidefinite by \ninspection and essentially self-adjoint on the physical Hilbert space.\nThis result reveals that several of the speculations spelled out in \n\\cite{Smolin} and which were based on several unproved assumptions \nwere pre-mature : after taking the quantum dynamics of the theory and the \nquantum asymptotic and regularity conditions on the Hilbert space \nappropriately into account, the quantum positivity of energy theorem is \n{\\em not} violated.\\\\ \nIt should be said from the outset, however, that \nthe ``quantum positivity of energy theorem\" that we provide rests, \nbesides on a quite particular regularization procedure of the ADM energy \noperator \nwhich exploits the fall-off behaviour of the fields at spatial infinity \nquite crucially, on one additional technical assumption (the {\\em tangle \nassumption}, see below) whose physical significance is unclear.\nAlthough it can be motivated also given the structure available at \nspatial infinity, it should be stressed that without this assumption \nthe positivity theorem would not hold.\n\nIn section 4 we naturally extend the unitary representation of the \ndiffeomorphism group to include asymptotic translations and rotations \nand compute the symmetry algebra between time translations and the \nspatial Euclidean group, that is, we verify the algebra of the little \ngroup of the Poincar\\'e group. We find no anomaly. As is well-known, the\nlittle group suffices \nto induce the unitary irreducible representations of the Poincar\\'e\ngroup. In this paper we do not, however, address the more difficult problem\nof how to define a boost quantum operator.\n\n\n\n\\section{Preliminaries}\n\nWe begin with a compact review of the relevant notions from \\cite{7,15}.\nThe interested reader is urged to consult these papers and references \ntherein.\\\\ \n\\\\ \nWe assume that spacetime is of the form $M=\\Rl\\times \\Sigma$ where \n$\\Sigma$ has an asymptotically flat topology, that is, there is \na compact set $B\\subset\\Sigma$ such that $\\Sigma-B$ is homeomorphic with\nof $\\Rl^3$ with a compact ball cut out. We also assume that $\\partial\\Sigma$\nis homeomorphic with the 2-sphere. The case of more than one component of\n$\\partial\\Sigma$ (e.g. several asymptotic ends or horizons etc.) can be \ntreated in a similar way.\\\\\nDenote by $a,b,c,..$ spatial tensor indices and by $i,j,k,..$ $su(2)$ \nindices. The gravitational phase space is described by a canonical pair \n$(A_a^i,E^a_i\/\\kappa)$ where $A_a^i$ is an $SU(2)$ connection on the \nhypersurface\n$\\Sigma$, $E^a_i$ is an $\\mbox{ad}(SU(2))$ transforming vector density\nand $\\kappa$ is the gravitational coupling constant.\nThis means that the symplectic structure is given by\n$\\{A_a^i(x),E^b_j(y)\\}=\\kappa\\delta(x,y)\\delta^b_a\\delta^i_j$. The \nrelation with the co-triad $e_a^i$, the extrinsic curvature $K_{ab}$\nand the three-metric $q_{ab}=e^i_a e^i_b$ is \n$E^a_i=\\frac{1}{2}\\epsilon^{abc}\\epsilon_{ijk}e^j_b e^k_c$ and \n$A_a^i=\\Gamma_a^i+\\mbox{sgn}(\\det((e_c^j)))K_{ab} e^b_j$ where\n$\\epsilon^{abc}$ is the metric independent, completely skew tensor density\nof weight one, $\\Gamma_a^i$ is the spin-connection of $e_a^i$ and $e^a_i$\nis the inverse of the matrix $e_a^i$. \n\nBy $\\gamma$ we will denote in the sequel a closed, piecewise analytic graph\nembedded into a d-dimensional smooth manifold $\\Sigma$ (the case of interest\nin general relativity is $d=3$).\nThe set of its edges will be denoted $E(\\gamma)$ and the set of its \nvertices $V(\\gamma)$. By suitably subdividing edges into two halves we \ncan assume that all of them are outgoing from a vertex (the remaining \nendpoint of the so divided edges is not a vertex of the graph because \nit is a point of analyticity). Let $A$ be a $G$-connection for a compact \ngauge group\n$G$ (the case of interest in general relativity is $G=SU(2)$). We will denote\nby $h_e(A)$ the holonomy of $A$ along the edge $e$. Let $\\pi_j$ be the \n(once and for all fixed representant of the equivalence class of the \nset of) $j$-th irreducible representations of $G$ (in general relativity\n$j$ is just a spin quantum number) and label each edge $e$ of $\\gamma$ \nwith a label $j_e$. Let $v$ be an \n$n$-valent vertex of $\\gamma$ and let $e_1,..,e_n$ be the edges incident \nat $v$. Consider the decomposition of the tensor product \n$\\otimes_{k=1}^n\\pi_{j_{e_k}}$ into irreducibles and denote by \n$\\pi_{c_v}(1)$ the\nlinearly independent projectors onto the irreducible representations $c_v$ \nthat appear. \n\\begin{Definition} \\label{def0}\nAn extended spin-network state is defined by\n\\begin{equation} \\label{1}\nT_{\\gamma,\\vec{j},\\vec{c}}(A):=\n\\mbox{tr}(\\otimes_{v\\in V(\\gamma)}\n[\\pi_{c_v}(1)\\cdot\\otimes_{e\\in E(\\gamma),v\\in e}\\pi_{j_e}(h_e(A))])\n\\end{equation}\nwhere $\\vec{j}=\\{j_e\\}_{e\\in E(\\gamma)},\\;\\vec{c}=\\{c_v\\}_{v\\in \nV(\\gamma)}$. In what follows we will use a compound label\n$I\\equiv(\\gamma(I),\\vec{j}(I),\\vec{c}(I))$. An ordinary spin-network\nstate is an extended one with all vertex projectors corresponding\nto singlets.\n\\end{Definition}\nThus, a spin-network state is a particular function of \nsmooth connections restricted to a graph with definite transformation \nproperties under gauge transformations at the vertices. Their importance is \nthat they form\nan orthonormal basis for a Hilbert space ${\\cal H}\\equiv{\\cal H}_{aux}$, \ncalled the auxiliary Hilbert space. Orthonormality means that\n\\begin{equation} \\label{2}\n<T_{\\gamma,\\vec{j},\\vec{c}},T_{\\gamma',\\vec{j}',\\vec{c}'}>\n\\equiv\n<T_{\\gamma,\\vec{j},\\vec{c}},T_{\\gamma',\\vec{j}',\\vec{c}'}>_{aux}\n=\\delta_{\\gamma\\gamma'}\\delta_{\\vec{j},\\vec{j}'}\n\\delta_{\\vec{c},\\vec{c}'}\\;.\n\\end{equation}\nAnother way to describe $\\cal H$ is by displaying it as a space of\nsquare integrable functions $L_2({\\overline {{\\cal A}\/{\\cal G}}},d\\mu_0)$. Here ${\\overline {{\\cal A}\/{\\cal G}}}$ is a space \nof distributional connections modulo gauge transformations, typically \nnon-smooth and $\\mu_0$ is a \nrigorously defined, $\\sigma$-additive, diffeomorphism invariant probability\nmeasure on ${\\overline {{\\cal A}\/{\\cal G}}}$. The space ${\\overline {{\\cal A}\/{\\cal G}}}$ is the maximal extension of \nthe space ${{\\cal A}\/{\\cal G}}$ of smooth connections such that (the Gel'fand \ntransform of) spin-network functions \nare still continuous. The inner product can be extended, with the same\northonormality relations, to any smooth (rather than analytic) graph with a \nfinite number of edges and to non-gauge-invariant functions. It is only  \nthe latter description of $\\cal H$ which enables one to verify that \nthe inner product $<.,.>$ is the unique one that incorporates the \ncorrect reality conditions that $A,E$ are in fact real valued. The\ninner product (\\ref{2}) was postulated earlier (see remarks in \\cite{RDP})\nfor the {\\em complex} connection formulation. But\nit was not until the construction of the Ashtekar-Lewandowski measure $\\mu_0$\nthat one could show that this inner product is actually the correct one \nfor the real connection formulation only.\n\nWe will denote by $\\Phi$ the finite linear combinations of spin-network \nfunctions and call it the space of cylindrical functions. A function \n$f_\\gamma$ is said to be cylindrical with respect to a graph \n$\\gamma$ whenever it is a linear combination of spin-network functions on \nthat graph such that $\\pi_{j_e}$ is not the trivial representation for no\n$e\\in E(\\gamma)$. \nThe space $\\Phi$ can be equipped with one of the standard nuclear topologies\ninduced from $G^n$ because on each graph $\\gamma$ every cylindrical \nfunction $f_\\gamma$ becomes a function $f_n$ on $G^n$ where $n$ is the \nnumber of \nedges $e$ of $\\gamma$ through the simple relation $f_\\gamma(A)=\nf_n(h_{e_1}(A),..,h_{e_n}(A))$. This \nturns it into a topological vector space. By $\\Phi'$ we mean the topological\ndual of $\\Phi$, that is, the bounded linear functionals on $\\Phi$. General\ntheorems on nuclear spaces show that the inclusion $\\Phi\\subset{\\cal \nH}\\subset\\Phi'$ (Gel'fand triple) holds.\n\nSo far we have dealt with solutions to the Gauss constraint only, that is,\nwe have explicitly solved it by dealing with gauge invariant functions only.\nWe now turn to the solutions to the diffeomorphism constraint (we follow \n\\cite{15}).\\\\ \nRoughly speaking one constructs a certain subspace $\\Phi_{Diff}$ of \n$\\Phi'$ by ``averaging spin-network states\nover the diffeomorphism group\" by following the subsequent recipe :\\\\\nTake a spin-network state $T_I$ and consider its orbit $\\{T_I\\}$ under the \ndiffeomorphism group. Here we mean orbit under asymptotically identity \ndiffeomorphisms only ! Then construct the distribution\n\\begin{equation} \\label{3}\n[T_I]:=\\sum_{T\\in\\{T_I\\}} T\n\\end{equation}\nwhich can be explicitly shown to be an \nelement of $\\Phi'$. Any other vector is averaged by \nfirst decomposing it into spin-network states and then averaging those \nspin-network states separately. Certain technical difficulties having to \ndo with superselection rules and graph symmetries \\cite{7} were removed in \n\\cite{15}.\n\nAn inner product on the space of the so constructed states is given by\\\\\n\\begin{equation} \\label{4}\n<[f],[g]>_{Diff}:=[f](g)\n\\end{equation}\nwhere the brackets stand for the averaging \nprocess and the right hand side means evaluation of a distribution on a \ntest function. The completion of $\\Phi_{Diff}$ with respect to \n$<.,.>_{Diff}$ is denoted ${\\cal H}_{Diff}$.\n\nFinally, the Hamiltonian constraint is solved as follows \\cite{10} :\nOne can explicitly write down an algorithm of how to construct the most\ngeneral solution. It turns out that \none can construct ``basic\" solutions $s_\\mu\\in\\Phi'$ which are\nmutually orthonormal with respect to $<.,.>_{Diff}$ (in a generalized sense)\nand diffeomorphism invariant.\nThe span of these solutions is equipped \nwith the natural orthonormal basis $s_\\mu$ (in the generalized sense).\nOne now defines a ``projector\" \n\\begin{equation} \\label{5}\n\\hat{\\eta}f:=[[f]]:=\\sum_\\mu s_\\mu <s_\\mu,[f]>_{Diff}\n\\end{equation}\nfor each $f\\in\\Phi$ and so obtains a \nsubspace $\\Phi_{Ham}\\subset\\Phi'$. \nThe physical inner product \\cite{15} is defined by \n\\begin{equation} \\label{6}\n<[[f]],[[g]]>_{phys}:=[[f]]([g])\\;.\n\\end{equation}\nFinally, the physical Hilbert space is just the completion of \n$\\Phi_{Ham}$ with respect to $<.,.>_{Ham}$.\n\n\n\n\\section{Regularization of the ADM Hamiltonian}\n\nThere are many ways to write the ADM-Hamiltonian \nwhich are all classically weakly identical. We are going to choose a form \nwhich is a pure surface integral and which depends only on $E^a_i$ \nbecause in this case the associated operator will be almost diagonal in a \nspin-network basis so that we can claim that spin-network states really\ndo provide a non-linear Fock representation for quantum general relativity\nas announced in \\cite{8,9}.\n\nThe appropriate form of the classical symmetry generators was derived in \n\\cite{13,13a}.\nAlthough that paper was written for the {\\em complex} Ashtekar variables,\nall results can be taken over by carefully removing factors of $i$ at \nvarious places. We find for the surface part of the Hamiltonian (expression \n(4.31) in \\cite{13}, we \nuse that $\\tiN=N\/\\sqrt{\\det(q)},\\;D_a\\tiN=(D_a N)\/\\sqrt{\\det(q)}$ where\n$N$ is the scalar lapse function)\n\\begin{equation} \\label{20}\nE(N)=-\\frac{2}{\\kappa}\\int_{\\partial\\Sigma} dS_a\\frac{N}{\\sqrt{\\det(q)}}\nE^a_i\\partial_b E^b_i\\;.\n\\end{equation}\nIt is easy, instructive and for the sign of the ADM energy crucial to see \nthat (\\ref{20}) really equals \nthe classical expression $+\\frac{1}{\\kappa}\\int_{\\partial\\Sigma}dS_a \n(q_{ab,b}-q_{bb,a})$ due to ADM :\nUsing that $E^a_i=\\frac{1}{2}\\epsilon^{abc}\\epsilon_{ijk} e_a^j e_b^k$\nwe have the chain of identities\n\\begin{eqnarray} \\label{7}\n&&-\\frac{2}{\\sqrt{\\det(q)}}E^a_i\\partial_b E^b_i\n=-\\mbox{sgn}(\\det(e))e^a_i\\epsilon^{bcd}\\epsilon_{ijk}[e_c^j e_d^k]_{,b}\n\\nonumber\\\\\n&=&-2\\mbox{sgn}(\\det(e))e^a_i\\epsilon^{bcd}\\epsilon_{ijk}e_c^j e_{d,b}^k\n\\nonumber\\\\\n&=& -2\\mbox{sgn}(\\det(e))q^{af}\\epsilon^{bcd}\\epsilon_{ijk}e_f^i e_c^j \ne_{d,b}^k\n=-2q^{af}\\epsilon^{bcd}\\sqrt{\\det(q)} \\epsilon_{fce}e^e_k e_{d,b}^k\n\\nonumber\\\\\n&=&-4q^{ac}\\delta^b_{[c}\\delta^d_{e]}\\sqrt{\\det(q)} e^e_i e_{d,b}^i\n\\nonumber\\\\\n&=& 4\\sqrt{\\det(q)} q^{ac}  q^{ed} e_d^i e_{[c,e]}^i\n=2\\sqrt{\\det(q)} q^{ac}  q^{bd} e_d^i[e_{c,b}^i-e_{b,c}^i]\n\\nonumber\\\\\n&=& \\sqrt{\\det(q)} q^{ac}  q^{bd} \n[2e_{(d}^i e_{c),b}^i+2e_{[d}^i e_{c],b}^i\n-2e_{(d}^i e_{b),c}^i-2e_{[d}^i e_{b],c}^i]\n\\nonumber\\\\\n&=& \\sqrt{\\det(q)} q^{ac}  q^{bd} \n[(q_{cd,b}-q_{bd,c})\n+2e_{[d}^i e_{c],b}^i]\\;\n\\end{eqnarray}\nNow we expand $e_a^i(x)=\\delta_a^i+\\frac{f_a^i(x\/r)}{r}+o(1\/r^2)$ where\n$r^2=\\delta_{ab}x^a x^b$ defines the asymptotic Cartesian frame. The function\n$f_a^i(x\/r)$ only depends on the angular coordinates of the asymptotic \nsphere and is related to the analogous expansion \n$q_{ab}(x)=\\delta_{ab}+\\frac{f_{ab}(x\/r)}{r}+o(1\/r^2)$ by \n$f_{ab}\\delta^b_i=f_a^i$. Consider now the \nremainder in the last line of (\\ref{7}). Its $o(1\/r^2)$ part vanishes \nbecause $f_{[ab]}=0$ and this concludes the proof.\n\nIn the sequel we focus on the energy functional $E_{ADM}=E(N=1)$. We will \nquantize it in two different ways correspoinding to two quite different \nfactor orderings. Each of the orderings has certain advantages and \ncertain disadvantages which we will point out. \n\n\n\\subsection{Ordering I : No state space restriction}\n\nIn this subsection we will derive a form of the operator which is densely \ndefined on the whole Hilbert space $\\cal H$ (and extends to the spaces \n${\\cal H}_{Diff},\\;{\\cal H}_{phys}$ defined above) without imposing any further\nrestriction that corresponds to asymptotic flatness.\n\nUsing again that\n$E^a_i=\\frac{1}{2}\\epsilon_{ijk}\\epsilon^{abc}e^j_b e^k_c$ we can write \nit as\n\\begin{equation} \\label{21}\nE_{ADM}=\\lim_{S\\to \\partial\\Sigma} E_{ADM}(S) \n\\mbox{ where } E_{ADM}(S)=-\\frac{2}{\\kappa}\\int_S\n\\frac{1}{\\sqrt{\\det(q)}} \\epsilon^{ijk}e^j\\wedge e^k \\partial_b E^b_i\n\\end{equation}\nand $S$ is a closed 2-surface which is topologically a sphere.\nThe idea is to point-split expression (\\ref{21}) and to use that\n$$\n[\\mbox{sgn}(\\det(e))e_a^i](x)=\\frac{1}{2\\kappa}\\{A_a^i(x),V(x,\\epsilon)\\}\n$$\nwhere $V(x,\\epsilon)=\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)\\sqrt{\\det(q)}(y)$\nand $\\chi_\\epsilon$ is the (smoothed out) characteristic function of a \nbox of coordinate volume $\\epsilon^3$. Since \n$$\n\\lim_{\\epsilon\\to 0}\\frac{\\chi_\\epsilon(x,y)}{\\epsilon^3}=\\delta(x,y)\n\\mbox{ so that } \\lim_{\\epsilon\\to 0}\\frac{V(x,\\epsilon)}{\\epsilon^3}\n=\\sqrt{\\det(q)}(x)\n$$\nwe have\n\\begin{eqnarray} \\label{22}\n&&E_{ADM}(S)\\nonumber\\\\\n&=&\\lim_{\\epsilon\\to 0}\n-\\frac{2}{\\kappa}\\int_S \n\\frac{1}{\\epsilon^3\\sqrt{\\det(q)}(x)}\n\\epsilon^{ijk}e^j(x)\\wedge e^k(x)\\int_\\Sigma d^3y\n\\chi_\\epsilon(x,y)(\\partial_b E^b_i)(y)\n\\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}-\\frac{2}{\\kappa}\\int_S \n\\frac{\\epsilon^{ijk}}{V(x,\\epsilon)}\ne^j(x)\\wedge e^k(x)\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)(\\partial_b E^b_i)(y)\n\\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}\n-\\frac{1}{2\\kappa^3}\\int_S \n\\frac{\\epsilon^{ijk}}{V(x,\\epsilon)}\n\\{A^j(x),V(x,\\epsilon)\\}\\wedge  \\{A^k(x),V(x,\\epsilon)\\}\n\\int_\\Sigma d^3y \\chi_\\epsilon(x,y) (\\partial_b E^b_i)(y) \\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}\n-\\frac{2}{\\kappa^3}\\int_S \\epsilon^{ijk} \n\\{A^j(x),\\sqrt{V(x,\\epsilon)}\\}\\wedge \\{A^k(x),\\sqrt{V(x,\\epsilon)}\\}\n\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)\n(\\partial_b E^b_i)(y) \\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}\n\\frac{4}{\\kappa^3}\\int\n\\mbox{tr}(\\{A(x),\\sqrt{V(x,\\epsilon)}\\}\\wedge \n\\{A(x),\\sqrt{V(x,\\epsilon)}\\} \\int_\\Sigma d^3y \\chi_\\epsilon(x,y)\n(\\partial_b E^b)(y)) \\nonumber\\\\\n&=& -\\lim_{\\epsilon\\to 0}\n\\frac{4}{\\kappa^3}\\int_S \n\\mbox{tr}(\\{A(x),\\sqrt{V(x,\\epsilon)}\\}\\wedge \n\\{A(x),\\sqrt{V(x,\\epsilon)}\\} \\int_\\Sigma d^3y \n[\\partial_{y^b}\\chi_\\epsilon(x,y)] E^b(y)) \\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0} E^\\epsilon_{ADM}(S)\n\\end{eqnarray}\nwhere in the second before the last step we have taken a trace with \nrespect to generators $\\tau_i$ of $su(2)$ obeying \n$[\\tau_i,\\tau_j]=\\epsilon_{ijk}\\tau_k$ and in \nthe last step we have performed an integration by parts\n(the boundary term at $\\partial\\Sigma$ does not contribute for finite $S$\nand $\\epsilon$ sufficiently small).\nThus, we absorbed the $1\/\\sqrt{\\det(q)}$ into a square-root within a \nPoisson-bracket and simultanously the singular $1\/\\epsilon^3$ into a \nvolume functional. Classically we could have dropped the $1\/\\sqrt{\\det(q)}$\n(although the integrand would then no longer be a density of weight one \nand is strictly speaking not the boundary integral of a variation of the \nHamiltonian constraint) due to the classical boundary conditions which \ntell us that $\\det(q)$ tends to $1$. \\\\\nWe now quantize $E^\\epsilon_{ADM}(S)$. This consists of two parts : In the \nfirst we focus on the volume integral in (\\ref{22}) and replace $E^a_i$ by\n$\\hat{E}^a_i=-i\\hbar\\kappa\\delta\/\\delta A_a^i$. In the second step we \ntriangulate $S$ exactly as the hypersurface of 2+1 gravity in \\cite{10a}, \nreplace the volume \nfunctional by the volume operator and Poisson brackets by commutators times\n$1\/(i\\hbar)$. \n\nSo let $f_\\gamma$ be a function cylindrical with \nrespect to a graph $\\gamma$. Since we are only interested in the \nlimit $S\\to\\partial\\Sigma$ we may assume that \\\\\n1) $\\gamma$ lies entirely within the closed ball whose boundary is $S$ but\\\\\n2) $\\gamma$ may intersect $S$ at an endpoint of one of its edges and may \neven have edges that lie entirely inside $S$.\\\\\nFurthermore we can label the edges of $\\gamma$ in \nsuch a way that an edge either intersects $S$ transversally (with an \norientation outgoing from the intersection point with $S$) or lies entirely \nwithin $S$.\\\\ \nComing to the first step we have for the $y$ integral involved in\n$E^\\epsilon_{ADM}(S)$ :\n\\begin{eqnarray} \\label{23}\n&& \\int_\\Sigma d^3y [\\partial_a\\chi_\\epsilon(x,y)]\n\\hat{E}^a_(y)_i f_\\gamma \\nonumber\\\\\n&=& -i\\hbar\\kappa \\sum_{e\\in E(\\gamma)}\\int_\\Sigma d^3y \n[\\partial_a\\chi_\\epsilon(x,y)]\n\\int_0^1 dt \\dot{e}^a(t) \\delta(y,e(t))X^i_e(t) f_\\gamma \n\\nonumber\\\\\n&=& -i\\hbar\\kappa \\sum_{e\\in E(\\gamma)}\\int_0^1 dt \\dot{e}^a(t)\n[\\partial_{y^a}\\chi_\\epsilon(x,y)_{y=e(t)}]\nX^i_e(t) f_\\gamma \n\\nonumber\\\\\n&=& -i\\hbar\\kappa \\sum_{e\\in E(\\gamma)}\\int_0^1 dt \n[\\frac{d}{dt}\\chi_\\epsilon(x,e(t))]\nX^i_e(t) f_\\gamma \n\\nonumber\\\\\n&=& -i\\hbar\\kappa \\sum_{e\\in E(\\gamma)} \\lim_{n\\to\\infty}\n\\sum_{k=1}^n [\\chi_\\epsilon(x,e(t_k))-\\chi_\\epsilon(x,e(t_{k-1}))]\nX^i_e(t_{k-1}) f_\\gamma \n\\end{eqnarray}\nwhere $E(\\gamma)$ is the set of edges of $\\gamma$, $X^i_e(t):=\n[h_e(0,t)\\tau_i h_e(t,1)]_{AB} \\partial\/\\partial[h_e(0,1)]_{AB}$\nand $0=t_0<t_1<..<t_n=1$ is an arbitrary partition of the interval \n$[0,1]$.\\\\\nIt is important for what follows that for each $t,e,i$ $X^i_e(t) f_\\gamma$\nis still a function cylindrical with respect to $\\gamma$.\n\nWe now come to the second step. This involves, first of all, a \ntriangulation of the {\\em two-dimensional} surface $S$ in adaption to the\ngraph $\\gamma$. Besides the prescription explained in detail in \\cite{10a}\nwhich deals with the triangulation of $S$ in the neighbourhood of a \nvertex formed by edges of $\\gamma$ that lie entirely within $S$ we \njust need to deal with the case that a vertex $v$ of $\\gamma$ has also edges\nincident at it which lie entirely inside the open ball with boundary $S$ \nexcept for the one point $v$. In case that there are at least {\\em two}\nedges $e_1,e_2$ of $\\gamma$ incident at $v$ such that $e_1,e_2\\subset S$ we can \nstill take over the triangulation from \\cite{10a}. However, if there is\nonly {\\em one or no} such edge (indeed, since we do not allow for gauge \ntransformations at spatial infinity we can allow for open edges that lie \nentirely within or end at $S$ without ruining gauge invariance) we need \nan additional prescription : In case there is only one edge $e\\subset S$ \nincident\nat $v$, choose an arbitrary edge $e'$ not intersecting $\\gamma$ except \nat $v$ such that the tangents of the edges $e,e',e^{\\prime\\prime}$ are \npositively \noriented at $v$ where $e^{\\prime\\prime}$ is any of the edges of $\\gamma$ \nincident at $v$ but transversal to $S$.\\\\ \nIn case there is no edge $e\\subset S$ incident\nat $v$, choose two arbitrary edges $e,e'$ not intersecting $\\gamma$ \nexcept at $v$ such that  \nthe tangents of the edges $s,e',e^{\\prime\\prime}$ are positively \noriented at $v$ where \n$e^{\\prime\\prime}$ is any of the edges of $\\gamma$ incident at $v$ but \ntransversal to $S$.\nThese arbitrary edges will disappear from the stage again at the end of \nthe calculation.\\\\\nGiven this set-up, at each vertex $v$ of $\\gamma$ that lies inside $S$ we \nhave now at least to edges $e_1,e_2$ incident at it that are inside $S$ \nand we can \ndefine the triangles $\\Delta$ associated with pairs of edges incident at \n$v$ and inside $S$ exactly as in \\cite{10a}. As in $\\cite{10a}$ we then \nhave two segments $s_1(\\Delta),s_2(\\Delta)$ for each triangle $\\Delta$\nwhich are actually segments of edges of $\\gamma$ incident at $v$ and that \nlie inside $S$. Now observe that\n\\begin{eqnarray} \\label{24}\n&& \\epsilon^{ij} h_{s_i(\\Delta)}\\{h_{s_i(\\Delta)}^{-1},V(v,\\epsilon)\\}\nh_{s_j(\\Delta)}\\{h_{s_j(\\Delta)}^{-1},V(v,\\epsilon)\\}\n\\nonumber\\\\\n&=&\\delta^2\\epsilon^{ij}\\dot{s}^a_i(0)\\dot{s}^b_j(0)\n\\{A_a(v),\\sqrt{V(v,\\epsilon)}\\}\\{A_b(v),\\sqrt{V(v,\\epsilon)}\\}\n+o(\\delta^2)\\nonumber\\\\\n&=& 2\\mbox{vol}(\\Delta)\\epsilon^{ab}\n\\{A_a(v),\\sqrt{V(v,\\epsilon)}\\}\\{A_b(v),\\sqrt{V(v,\\epsilon)}\\}\n\\end{eqnarray}\nwhere $\\delta$ is a small parameter corresponding to the parameter length\nof the $s_i(\\Delta)$ and $\\epsilon^{ab}$ the metric independent totally \nskew tensor density of weight one on $S$. Altogether we therefore conclude \nthat the surface integral \n$\\int_S \\{A(x),\\sqrt{V(x,\\epsilon)}\\}\\wedge \\{A(x),\\sqrt{V(x,\\epsilon)}\\}$\ninvolved in $E_{ADM}(S)$ can be quantized by\n\\begin{equation} \\label{25}\n-\\frac{1}{2\\hbar^2} \\sum_{v\\in V(\\gamma)} \\frac{4}{E(v)}\\sum_{v(\\Delta)=v}\n\\epsilon^{ij} \nh_{s_i(\\Delta)}[h_{s_i(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}]\nh_{s_j(\\Delta)}[h_{s_j(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}] f_\\gamma\\;.\n\\end{equation}\nWe do not wish to give a full derivation of (\\ref{25}) for which the \nreader should consult \\cite{10a}, however, a few remarks are in order\nwhich intuitively explain (\\ref{25}) :\n\\begin{itemize}\n\\item[1)] First of all as already mentioned, we do not work with a fixed \ntriangulation but with a whole finite family of triangulations that \ndepend on the \ngraph $\\gamma$ of the state that we act on. More precisely, a particular \nmember of the family of triangulations is defined \nsuch that for each vertex $v$ of $\\gamma$ which lies in $S$ we a) pick one \npair $e_1,e_2$ of edges of $\\gamma$ incident at it which are such that \n$e_2$ is next to $e_1$ to its right while the tangents of $e_1,e_2$ \nenclose an angle less than or equal to $\\pi$ (with respect to \n$\\delta_{ab}$), b) take proper \nsubsegments $s_i(\\Delta)\\subset e_i$ incident at $v$, c) construct two more\nsegments $s_{\\bar{i}}(t)=2v-s_i(t)$, d) construct four obvious \ntriangles from $s_i,s_{\\bar{i}}$ which saturate $v$ and e) \nchoose a triangulation which embeds those four triangles for each $v$\nas basic ones and is otherwise arbitrary only subject to the restriction \nthat none of the remaining $\\Delta$ has its basepoint on $\\gamma$. \n\\item[2)] Next, each of the four triangles \n$\\Delta_{12},\\Delta_{2\\bar{1}},\\Delta_{\\bar{1}\\bar{2}},\\Delta_{\\bar{2}1}$\ncontributes classically the same to the surface \nintegral $\\int_S=\\sum_\\Delta \\int_\\Delta$ as the triangle $\\Delta_{12}$\nso that we can classically replace these four terms by\n$4\\int_{\\Delta_{12}}$. This explains the factor of $4$ in (\\ref{25}).\n\\item[3)] Now, since we do not want to distinguish one particular pair of \nedges as \ncompared to any other, we average over the choice of pairs which explains \nthe factor of $1\/E(v)$ where $E(v)=n(v)$ or $E(v)=n(v)-1$ is the \npossible number of pairs depending on whether the angles less than or \nequal to $\\pi$ add up to $2\\pi$ (with respect to $\\delta_{ab}$) or not and \n$n(v)$ is the valence of $v$.\n\\item[4)] Finally, the fact that we sum over vertices of $\\gamma$ only comes \nfrom the presence of the volume operator which has non-trivial action at \nvertices only. Therefore the contribution of all other triangles which \ndefine the triangulation drop out (all triangles such that there is no \nedge of $\\gamma$ transversal to $S$ at its basepoint since the \nthree-dimensional volume operator annihilates co-planar vertices).\n\\end{itemize}\nIt should be mentioned that any of the so-defined ``averaged family of \ntriangulations of $S$ adapted to $\\gamma$\" has as classical continuum limit \nthe original integral $\\int_\\Sigma$ !\\\\\nPutting (\\ref{23}) and (\\ref{25}) together we obtain as the final result\n\\begin{eqnarray} \\label{26}\n\\hat{E}^{\\epsilon,n}_{ADM}(S)f_\\gamma&=&\n\\frac{4}{\\kappa^3}(-\\frac{1}{2\\hbar^2}) (-i\\hbar\\kappa)\n\\sum_{v\\in V(\\gamma)} \\frac{4}{E(v)}\\sum_{v(\\Delta)=v}\\times\\nonumber\\\\\n&\\times&\\epsilon^{ij} \n\\mbox{tr}(h_{s_i(\\Delta)}[h_{s_i(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}]\nh_{s_j(\\Delta)}[h_{s_j(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}]\n\\times\\nonumber\\\\\n&\\times&\\sum_{e\\in E(\\gamma)} \n\\sum_{k=1}^n [\\chi_\\epsilon(v,e(t_k))-\\chi_\\epsilon(v,e(t_{k-1}))]\nX^i_e(t_{k-1}) f_\\gamma \\;.\n\\end{eqnarray}\nNow we perform the limit $n\\to\\infty$ and $\\epsilon\\to 0$ in reversed \norder\\footnote{One can also make $\\epsilon$ $y$-dependent \nin (\\ref{22}) in a state-dependent way which then leads to a dependence \n$\\epsilon=o(1\/n)$. If one then takes $n\\to\\infty $ one gets  \nclassically back to the original expression for \nthe ADM energy. If one takes $n$ large but finite then one arrives \nat the same result as below on the quantum level \nwithout that an interchange of limits is necessary. Therefore \nthe calculations that follow are justified. The state-dependence of the \nregularization drops out in the final expression because before actually\ntaking the limit $n\\to\\infty$ the operator will be $n$-independent. The\nfamily of operators so obtained for each state are consistently defined \nas we will see. See \\cite{10a,12} for a more detailed explanation.} : \nKeeping $n$ fixed, for small enough $\\epsilon$ only the term with\n$k=1$ in the sum survives provided that $e(0)=v$. Therefore also the sum\nover $k$ and with it the $n$ dependence drops out. Also the operator\n$\\hat{V}(v,\\epsilon)$ actually has a limit as $\\epsilon\\to 0$ which we call\n$\\hat{V}_v$ and which is defined by $\\lim_{\\epsilon\\to 0}\n\\hat{V}(v,\\epsilon)f_\\gamma=:(\\hat{V}_v)_\\gamma f_\\gamma$ for all $\\gamma$. \nThat \nthe limit exists relies on the fact that either $\\gamma$ has $v$ as a vertex\nor it does not. In the latter case the limit just vanishes, in the former \ncase for sufficiently small $\\epsilon$ the $\\epsilon$ box around $v$ does \nnot include any other vertex of $\\gamma$ other than $v$ and so there is only\none contribution $(\\hat{V}_v)_\\gamma f_\\gamma$ which is  constant as \n$\\epsilon\\to 0$. The family of operators $(\\hat{V}_v)_\\gamma$ is \nconsistently defined because $\\hat{V}$ is. For a more explicit formula \nin terms of analytic germs of edges see \\cite{10a}.\\\\\nNow, the limit $n\\to\\infty$ is trivial and the resulting operator derived \nfor arbitrary but finite $S$ can be extended to $\\partial\\Sigma$. The \nresult is\n\\begin{eqnarray} \\label{27}\n\\hat{E}_{ADM} f_\\gamma&=&\n\\frac{2i}{\\hbar\\kappa^2}\n\\sum_{v\\in V(\\gamma)} \\frac{4}{E(v)}\\sum_{v(\\Delta)=v}\\times\\nonumber\\\\\n&\\times&\\epsilon^{ij} \n\\mbox{tr}(h_{s_i(\\Delta)}[h_{s_i(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}]\nh_{s_j(\\Delta)}[h_{s_j(\\Delta)}^{-1},\\sqrt{\\hat{V}(v,\\epsilon)}]\n\\times\\nonumber\\\\\n&\\times&\\sum_{e\\in E(\\gamma),e(0)=v} \nX^i_e f_\\gamma \n\\end{eqnarray}\nwhere $X^i_e(0)=X^i_e=X^i(h_e))$ and $X^i(g)$ is the right invariant \nvector field at $g\\in SU(2)$.\\\\\nThe virtue of (\\ref{27}) is that it displays the ADM energy operator as a\ndensely defined operator on all of the Hilbert space. Also, the \ndependence on the ``arbitrarily short edges $s_i(\\Delta)$ drops out at \nthe end of the calculation because of gauge invariance as explained \nin \\cite{11}. The disadvantage is that the operator (\\ref{27}) is not \na manifestly positive semi-definite operator. This is, however, not \nsurprising because even the classical ADM energy is not a positive \nsemi-definite functional on the full phase space of general relativity.\nIt is only when evaluating it on a) asymptotically flat b) solutions of the \nEinstein equations which c) satisfy an energy condition for allowed matter \nand d) allow for a regular initial data set, that the positive energy \ntheorem has been proved \\cite{Yau,Witten} and in fact one can easily\nproduce negative ADM energy when one of these conditions is violated.\nAs we did not impose any (quantum analogue of) such restrictions we \ncannot expect to find a manifestly non-negative operator. \\\\\nIn the next subsection we will derive another quantization of the \nADM energy which is only densely defined on a subspace of the Hilbert space,\nhowever, the definition of that subspace a quite natural quantum \ntranslation of the classical condition that there be an asymptotically flat \nregular initial data set. The virtue will be that the ADM operator \nacquires non-negative discrete spectrum on that subspace of the Hilbert \nspace thus proving a ``Quantum Positivity of Energy Theorem\".\n\n \n\\subsection{Ordering II : Restrictions on the State Space}\n\nIn order to make sense of the operator to be defined in this section\nwe need to give some definition of ``asymptotically flat state\". The \nfollowing definition is a first attempt towards a precise notion of \n``quantum asymptotic flatness\" to be considerably refined in future \npublications. \n\\begin{Definition} \\label{def1}\nAn asymptotically flat state $\\Psi$ on the Hilbert space $\\cal H$ is \na distribution in $\\Phi'$ satisfying the following conditions :\\\\\ni) $\\Psi$ is a normalized solution $\\hat{\\eta}\\xi$ to all quantum \nconstraints \n$\\hat{U}(\\varphi)-1=0,\\;\\hat{H}(N)=0$ (the diffeomorphism and Hamiltonian \nconstraint) of general \nrelativity where $\\varphi,N$ are arbitrary \nthree-diffeomorphisms and lapse functions subject to the \ncondition that they be pure gauge, that is, $\\varphi(x)\\to \n\\mbox{id}$ and $N(x)\\to 0$ as $x\\to \\partial \\Sigma$ and \nwhere $\\hat{\\eta}$ is \nthe operator \\cite{15} \nthat maps \nelements of $\\Phi$ to solutions to all constraints.\\\\\nii) $\\Psi=\\hat{\\eta}\\xi$ is asymptotically flat. That is, consider any \ncompact region $R$, surface $S$ and loop $\\alpha$ \n(which is embedded in the graph underlying the definition of $\\xi$)\nof $\\Sigma-\\partial\\Sigma$ \nand which are large as compared to Planck volume, area and length \nrespectively as measured by the \nEuclidean metric $q^{(0)}_{ab}:=\\delta_{ab}$.\nThen, as $R,S,\\alpha$ tend to $\\partial\\Sigma$, the quantities\n$\\Psi([\\hat{V}(R)-V_0(R)]\\xi),\\Psi([\\hat{A}(S)-A_0(S)]\\xi),\n\\Psi([\\mbox{tr}(h_\\alpha)-2]\\xi)$ respectively \nare of order $\\ell_p^3,\\ell_p^2,\\ell_p\/L(c)$ respectively \nwhere $V_0(R),A_0(S),L_0(\\alpha)$ are the volume, area and length of \n$R,S,\\alpha$ as measured by $q^{(0)}_{ab}$.\\\\\niii) $\\Psi$ transfroms according to an unitary, irreducible \nrepresentation of the Poincar\\'e group at spatial infinity.\\\\\niv) $\\Psi$ satisfies the dominant energy condition :\nLet $\\hat{H}_{matter}'(N)$ be the dual of the matter part of the \nHamiltonian constraint (not Hamiltonian !) and \n$\\hat{V}_{matter}'(\\vec{N})$ be the dual of the matter part of the  \nDiffeomorphism constraint.\\\\ \nThe four-vector $N^\\mu=(N,\\vec{N})$ is said to \nbe a future directed timelike vector in a state $\\Psi$ if a) $N>0$ and   \nb) there exists $\\epsilon>0$ such that $-t^2 \nN^2(x)+\\Psi[\\hat{L}(c(\\vec{N},x,t))^2\\xi]<0$ for each $0<t<\\epsilon$\nwhere $\\hat{L}(c)$ is the length operator \\cite{11} and $c(\\vec{N},x,t)$\nis the segment of the integral curve of $\\vec{N}$ beginning in $x$ and \nending after a parameter distance $t$.\\\\\n$\\Psi$ is said to satisfy the dominant energy condition provided that\nfor every four-vector $N^\\mu$ which is future directed and timelike for \n$\\Psi$ then there is an $\\epsilon>0$ such that the four vector \n$\\hat{P}^\\mu(x,t):=(\\hat{H}_{matter}'(N_{x,t})\n+\\hat{V}_{matter}'(\\vec{N}_{x,t}),0)$ is \neither zero or \nfuture directed and timelike for every $x\\in\\Sigma$ and $0<t<\\epsilon$ in \nthe state $\\Psi$ where \n$N_{x,t}(y)=\\chi_t(x,y)N(y)$ and likewise for $\\vec{N}_{x,t}(y)$\n($\\chi_t(x,y)$ is the characteristic function of a box of coordinate volume\n$t^3$ and centre $x$).\nIn other words,\n$\\Psi([\\hat{H}_{matter}(N_{x,t})+\\hat{V}_{matter}(\\vec{N}_{x,t})]\\xi)>0$ for \neach $x,0<t<\\epsilon$. Here we have adopted the convention that the \nsignature of the Lorentz metric be $-,+,+,+$.\\\\\nThe subspace of $\\xi$'s \nin $\\cal H$ satisfying these conditions will be called ${\\cal H}_{af}$\nwhere ``af\" stands for asymptotically flat.\n\\end{Definition}\nCondition i) makes sure that $\\Psi$ is a solution of the ``Quantum \nEinstein Equations\". Condition ii) is a possible way of defining \nasymptotic flatness $q_{ab}\\to\\delta_{ab},K_{ab}\\to 0$ (although not very \ncarefully, no fall-off and parity conditions were imposed \\cite{13,13a} and\ncertainly this condition needs to be refined in future publications. For \ninstance, one might imagine that the error of $\\Psi(\\hat{V}(R)\\xi)\/V_0(R)\n-1$ is even smaller than $\\ell_p^3\/V_0(R)$ in the sense that it \ncould depend on some negative power of the value\nof the radius $r$ (with respect to $\\delta_{ab}$) at the center of $R$).\nCondition iii) makes contact with physics and allows us to identify certain \nstates with elementary particles. In particular, in the present context of \npure gravity we should be able to isolate the graviton (spin 2,\nmassless) states. Notice that in order to allow for non-trivial \nrepresentations of the little subgroup of the Lorentz group we must\nspecify the appropriate Diffeomorphism group in the group averaging \nprocess in order to arrive at the diffeomorphism invariant states \\cite{7}\nwhich means, roughly speaking, that we include only those diffeomorphisms \nin the averaging process that approach identity at $\\partial\\Sigma$.\nFinally, Condition iv) is the most imprecise one and attemts at defining\na possible quantum analogue for the dominant energy condition : recall \nthat the classical energy momentum tensor $T_{\\mu\\nu}$ is said to satisfy \nthe dominant energy condition if for every future directed timelike \nvector field $v^\\mu$, the vector field $T^\\mu_\\nu v^\\nu$ is zero or future \ndirected and timelike. Now if $t^\\mu=N n^\\mu+N^\\mu$ is the future \ndirected timelike foliation vector field underlying the split \n$M=\\Rl\\times\\Sigma$ and $n^\\mu$ the normal vector field of $\\Sigma$ \nthen we may pick a frame such that $n^\\mu T_{\\mu\\nu}t^\\nu=N H_{matt}+N^a \n(V_{matt})_a$ is the only non-vanishing component of the vector \n$T_{\\mu\\nu}t^\\nu$ and so we need to ask that it be non-negative, in \nparticular, for $\\vec{N}=0$, we ask that the matter Hamitonian densities \nbe non-negative. This condition is, of course, incomplete since it is frame \ndependent and needs to be improved in the future. Notice that $(t^2 \ng_{\\mu\\nu}t^\\mu t^\\nu)(x)=-t^2 N^2(x)+t^2 (q_{ab}N^a \nN^b)(x) \\approx -N^2(x)t^2+ L(c(\\vec{N},x,t))^2$ which motivates our \ndefinition of future directedness of $t^\\mu$. \nWe conclude with the remark that Condition iv) is certainly satisfied in the \nvaccum case that we are interested in in the present paper.\\\\\n\\\\\nOne might think that a state satisfying all those conditions is rather \nhard to construct. Let us pause for a moment to argue that it is \nrather simple :\n\nConsider for simplicity a $\\Sigma$ with \ntopology of $\\Rl^3$ and distribute a countable number of vertices $v_n$ \nrandomly into $\\Sigma$ \nwith an average next neighbour distance of a Planck length as measured by\n$q^{(0)}_{ab}$. Make $v_n$ the only vertex of a \ngraph $\\gamma_n$ which is four-valent and non-co-planar (for instance,\n$\\gamma_n=\\alpha_n\\cap\\beta_n$ where $\\alpha_n,\\beta_n$ are two kinks \nwith vertex $v_n$). The graphs $\\gamma_n$ are supposed to be  \ncontained in a box $B_n$ of $q^{(0)}$ volume $k\\ell_p^3$ for some \npositive number $k$ and the $B_n$ are mutually \nnon-inersecting. Consider normalized vectors $f_n$ which are finite linear \ncombination of spin-network states\ndefined on $\\gamma_n$ and which are eigenstates of the volume \noperator $\\hat{V}(R)$ for any region $R$, all with the same eigenvalue \n$\\lambda_n\\ell_p^3=\\lambda\\ell_p^3>0$ if $v_n\\in R$. Thus we have \n$\\hat{V}(B_m)f_n=\\lambda\\ell_p^3\\delta_{m,n}f_n$. Consider the \ninfinite product state $\\xi:=\\prod_{n=1}^\\infty f_n$ which is a regular \n(non-cylindrical) spin-network state on the infinite graph \n$\\gamma=\\cup_n \\gamma_n$ and which is in fact\nnormalized, $||\\xi||=1$ thanks to the disjointness of the graphs $\\gamma_n$\nbecause of which $||\\xi||=\\prod_n||f_n||$ due to the properties of the \nAshtekar-Lewandowski measure. We now choose $k:=\\lambda$\nand find that for any macroscopic $R$, that is, any $R$ that contains \nmany of the boxes $B_n$ it holds that \n$\\hat{V}(R)\\xi=V_0(R)[1+o(\\ell_p^3\/V_0(R)]\\xi$. Now, since no state\nwhich is cylindrical with respect to any of the \ngraphs $\\gamma_n$ can be in the image of the Hamiltonian constraint\n\\cite{8,9,10} it follows from its definition \\cite{15} that the $\\hat{\\eta}$\noperator reduces to group averaging with respect to the diffeomorphism group\nbecause of which the group averaged diffeomorphism invariant state \n$\\Psi=[\\xi]=\\hat{\\eta}\\xi$ is normalized as well with respect to the \nphysical inner product \\cite{15} $||\\Psi||_{phys}^2=\\Psi(\\xi)=||\\xi||^2=1$.\nThus indeed $\\Psi([\\hat{V}(R)-V_0(R)]\\xi)=o(\\ell_p^3\/V_0(R))$ is satisfied.\nIt is clear that the construction can be repeated for the surface operator\nas well because most of the intersections of the macroscopic surface \n$S$ with the $\\gamma_n$ will not be in vertices of the $\\gamma_n$ so \nthat $\\hat{V}(R),\\hat{A}(S)$ can be simultanously diagonalized up to \nerrors of order of $\\ell_p^2\/A_0(S)$. Thus, almost every $f_n$ can be \nchosen as a simultanous eigenvector of $\\hat{V}(R),\\hat{V}(S)$. Finally,\nany macroscopic, for simplicity non-self-intersecting (any loop is \nproduct of these), loop $\\alpha$ on our particular $\\gamma$ is of the \nproduct form $\\alpha=\\circ^n \n\\alpha_n^{k_n},\\;\\alpha_n\\subset\\gamma_n,\\;k_n\\in\\{0,1\\}$ where $\\;k_n=0$ \nexcept for finitely many. The $SU(2)$ Mandelstam algebra is too \ncomplicated as to exhibit an explicit solution for $SU(2)$ so let us \nargue with an $U(1)$ substitute that the condition stated in definition\n(\\ref{def1}) is reasonable. For $U(1)$ we have  \n$h_\\alpha=\\prod_{k_n=1}h_{\\alpha_n}$. Now, if we \nchoose for simplicity $\\alpha_n=\\gamma_n$ then \n$f_n=\\sum_{k=-N}^N a_k h_{\\alpha_n}^k$ where $\\chi_k(g)=g^k$ is the \ncharacter of the irreducible representation of $U(1)$ with weight $k$.\nSince $T=\\chi_k\\chi_l=\\chi_{k+l}$ the condition stated in the definition \namounts to asking that (for $U(1)$) $1=\\prod_{k_n=1}\\sum_k \n|a_k|^2=\\prod_{k_n=1}\\sum_{k=-N+1}^N\\bar{a}_k a_{k+1}$ up to some \ncorrections. Indeed, if we could choose all $a_k$ to be equal \n($=1\/\\sqrt{2N+1}$) then the error would be  $1-\\prod_{k_n=1} [1-1\/(2N+1)]$ \nwhich is small provided that $\\sum_{k_n=1}1=o(L(\\alpha)\/\\ell_p)<<N$.\nFor instance we may choose $N=[L\/\\ell_p]^2$ where $L$ is the bound on the  \nlength of a macrosccopic loop that we wish to consider. \nThis will suffice to motivate definition (\\ref{def1}). Obviously one has \nto refine it but this seems impossible without the notion of coherent states\nand will be left for future publications \\cite{16}.\\\\\n\\\\\nWe will now show that on an asymptotically flat state the ADM energy\noperator as defined below is non-negative. We will not show that vanishing\nenergy corresponds to Minkowski space. Our definition of asymptotically\nflat states as of yet is not restrictive or precise enough for that. \nAs we will see, in order for the ADM operator to be densely defined we need \nthat the following stronger condition :\\\\ \n\\\\\n{\\em ii)' A state $\\Psi=\\hat{\\eta}\\xi$ is said to be asymptotically flat\nprovided ii) of Definition \\ref{def1} holds and in addition :\\\\\nLet $\\gamma$ be the (infinite) graph on which $\\xi$ depends, $p$ a \npoint in $\\partial\\Sigma\\cap V(\\gamma)$ and $B_t,\\;t\\in [0,1]$ any\nhomotopy of regions in $\\Sigma$ such that $p\\in B_t$ for each $t$ and\n$B_0=\\{p\\}$. Then we require that for each such $p$ there exists \n$\\epsilon>0$ such that $\\xi$ is,\nfor each $0<t<\\epsilon$, a finite linear combination of eigenstates \nwith non-vanishing and $t-$independent eigenvalues of the volume \noperator $\\hat{V}(B_t)$.}\\\\\n\\\\\nIt is not clear that condition ii) implies ii)'\nand if that should not be the case then we must add the requirement ii)'\nstated as an additional restriction on ${\\cal H}_{af}$ ! Notice, however, \nthat ii)' is not unreasonable in the asymptotically flat context.\\\\\nIt will turn out in the course of the derivation that \nthe positivity of energy theorem then holds if we impose one additional \ncondition on the so already restricted space of states.\n\nWe write expression (\\ref{22}) this time in the form (setting \n$S=\\partial\\Sigma$ right from the beginning)  \n\\begin{eqnarray} \\label{28}\nE_{ADM}&=&\\lim_{\\epsilon\\to 0}\n-\\frac{2}{\\kappa}\\int_{\\partial\\Sigma} dS_a E^a_i(x)\n\\frac{1}{\\epsilon^3\\sqrt{\\det(q)}(x)}\n\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)(\\partial_b E^b_i)(y)\n\\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}-\\frac{2}{\\kappa}\\int_{\\partial\\Sigma} dS_a E^a_i(x)\n\\frac{1}{V(x,\\epsilon)}\n\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)(\\partial_b E^b_i)(y)\n\\nonumber\\\\\n&=& \\lim_{\\epsilon\\to 0}\n-\\frac{2}{\\kappa}\\int_{\\partial\\Sigma} dS_a(x) E^a_i(x)\n\\frac{1}{V(x,\\epsilon)} \\int_\\Sigma d^3y \n\\chi_\\epsilon(x,y)[G_i(y)-\\epsilon_{ijk}A_b^j(y) E^b_k(y)] \\nonumber\\\\\n&=:& \\lim_{\\epsilon\\to 0} E^\\epsilon_{ADM}\n\\end{eqnarray}\nwhere $G_i=\\partial_a E^a_i+[A_a,E^a]_i$ is the Gauss law constraint.\nRecall that $G_i=0$ only needs to hold in the interior of $\\Sigma$ \nbecause the Lagrange multiplier $\\Lambda^i$ of the Gauss constraint falls \noff like \n$1\/r^2$ so that at $\\partial\\Sigma$ every function of $E^a_i,A_a^i$ is \ngauge invariant. More precisely we have the following : It is of \ninterest by itself to derive the quantum Gauss law operator on a \nfunction of {\\em smooth} connections cylindrical with respect to a graph\n$\\gamma$ :\n\\begin{eqnarray} \\label{28a}\n&&\\int_\\Sigma d^3x \\Lambda^i(x)\\hat{G}_i(x)f_\\gamma\\nonumber\\\\\n&=&-i\\hbar\\kappa\\sum_{e\\in E(\\gamma)}\\int d^3x\\Lambda^i(x)\\int_0^1 dt\n\\dot{e}^a(t)([\\partial_{x^a}\\delta_{ik}+\\epsilon_{ijk}\nA_a^j(x)]\\delta(x,e(t))) (X^k_e(t)f_\\gamma)\\nonumber\\\\\n&=&-i\\hbar\\kappa\\sum_{e\\in E(\\gamma)}\\int d^3x\\Lambda^i(x) \\int_0^1 dt\n([-\\frac{d}{dt}\\delta_{ik}+\\epsilon_{ijk}\n\\dot{e}^a(t)A_a^j(x)]\\delta(x,e(t))) (X^k_e(t)f_\\gamma)\\nonumber\\\\\n&=&-i\\hbar\\kappa\\{\\sum_{e\\in E(\\gamma)}\\int d^3x\\Lambda^i(x)\n[-\\delta(x,e(1))X_e^i(1)+\\delta(x,e(0))X^i_e(0)\\nonumber\\\\\n&& + \\int_0^1 dt\\delta(x,e(t))(\\dot{X}^i_e(t)+\\epsilon_{ijk}\n\\dot{e}^a(t)A_a^j(x)X^k_e(t)]f_\\gamma\\}\\nonumber\\\\\n&=&-i\\hbar\\kappa\\{\\sum_{e\\in E(\\gamma)}\n[-\\Lambda^i(e(1))X_e^i(1)+\\Lambda^i(e(0))X^i_e(0)\\nonumber\\\\\n&& + \\int_0^1 dt\\Lambda^i(e(t))(\\dot{X}^i_e(t)+\\epsilon_{ijk}\n\\dot{e}^a(t)A_a^j(e(t))X^k_e(t)]f_\\gamma\\} \\;.\n\\end{eqnarray}\nHere we have made use of $\\partial_x\\delta(x,y)=-\\partial_y\\delta(x,y)$ \nin the third step which holds on spaces of test functions of rapid decrease. \nBut since for smooth connections $h_e(t,t+\\delta t)^{\\pm 1}=1\\pm\\dot{e}^a(t)\n\\delta t A_a(e(t))+o(\\delta t^2)$ we have\n\\begin{eqnarray} \\label{28b}\n&&X^i_e(t+\\delta t)-X^i_e(t)\n=\\mbox{tr}(h_e(0,t+\\delta t)\\tau_i h_e(t+\\delta t,1)\n\\partial_{h_e(0,1)})-X^i(t)\\nonumber\\\\\n&=&\\mbox{tr}(h_e(0,t)h_e(t,t+\\delta t)\\tau_i h_e^{-1}(t,t+\\delta t)\nh_e(t,1)\\partial_{h_e(0,1)})-X^i(t)\\nonumber\\\\\n&=&\\delta \nt\\dot{e}^a(t)\\mbox{tr}(h_e(0,t)[A_a(e(t)),\\tau_i]h_e(t,1)\\partial_{h_e(0,1)})\n+o(\\delta t^2)\\nonumber\\\\\n&=&-\\epsilon_{ijk}\\delta \nt\\dot{e}^a(t)A^j_a(e(t))\\mbox{tr}(h_e(0,t)\\tau_k h_e(t,1)\\partial_{h_e(0,1)})\n+o(\\delta t^2)\\nonumber\\\\\n&=&-\\epsilon_{ijk}\\delta t\\dot{e}^a(t)A^j_a(e(t)) X^k_e(t)\n+o(\\delta t^2)\\;.\n\\end{eqnarray}\nThis shows that the $t-$ integral in (\\ref{28a}) vanishes identically for \nsmooth connections. Now $X^i_e(0)=X^i(h_e)=:X^i_e$ where $X^i(g)$ is the \nright invariant vector field at $g\\in SU(2)$ and \n$X^i_e(1)=-X^i(h_e^{-1})$. Thus, when splitting each edge into \ntwo halves $e=e_1\\circ e_2^{-1}$ \nwhere both $e_1,e_2$ are outgoing at the vertex different from their \nintersection point then $e(1)=e_2(0)$ and the right invariance of $X$\nnow implies $X(h_e)=X(h_{e_1}),\\;-X(h_e^{-1})=X(h_{e_2})$. Summarizing we \nfind that the Quantum Gauss Constraint is given by \n\\begin{equation} \\label{28c}\n\\hat{G}(\\Lambda)f_\\gamma=-i\\hbar\\kappa\\sum_{e\\in E(\\gamma)} \n\\Lambda^i(e(0))X^i_e \nf_\\gamma =-i\\hbar\\kappa\\sum_{v\\in V(\\gamma)} \\Lambda^i(v) X^i_v f_\\gamma\n\\end{equation}\nwhere $-iX_v=-i\\sum_{e(0)=v}X_e$ is the total ``internal\" angular momentum \noperator.\nNotice that (\\ref{28c}) can be extended from smooth to distributional \nconnections and that no assumption on asymptotic behaviour or smoothness \nof $\\Lambda$ had to be made. \nThe Quantum Gauss Constraint is obviously a self-adjoint \noperator on ${\\cal H}$ and anomaly free : it is trivial to check that\n$[\\hat{G}(\\Lambda),\\hat{G}(\\Lambda')]=\\hat{G}([\\Lambda,\\Lambda'])$\nprecisely mirroring the classical constraint algebra. Moreover, the \nQuantum Gauss law constraint is \nidentically satisfied as $\\gamma$ tends to $\\partial\\Sigma$ because \n$\\Lambda_{|\\partial\\Sigma}=0$, that is, there are no internal charges in \ngeneral relativity \\cite{0}. This allows quantum states of \ndistributional connections to be non-gauge invariant at spatial infinity,\na fact that we are going to exploit in the sequel.\n\nLet now the state $\\xi\\in {\\cal H}_{af}$ be considered as a \nfunction $f_\\gamma$ cylindrical with respect to a graph $\\gamma$ which\nis a finite subgraph of the graph on which $\\xi$ depends and which \ninersects $\\partial\\Sigma$. Because $\\xi\\in {\\cal H}_{af}$ we know that \n$f_\\gamma$ is a {\\em finite linear combination of eigenstates of the volume \noperator $\\hat{V}(R)$ with non-zero eigenvalue for sufficiently small \nregions $R$ and such that $R\\cap \nV(\\gamma)_{|\\partial\\Sigma}\\not=\\emptyset$}. \nConsider first the volume integral in (\\ref{28}). \nSetting $\\Lambda(y)=\\chi_\\epsilon(x,y)$\nthere is an obvious quantization for the term proportional to $G_i$\nin view of (\\ref{28c}). However, for the remainder we have, again \non functions of smooth connections only to begin with\n\\begin{eqnarray} \\label{29}\n&& \\epsilon_{ijk}\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)\nA_a^j(y)\\hat{E}^a_k(y) f_\\gamma \\nonumber\\\\\n&=& -i\\hbar\\kappa\\epsilon_{ijk} \\sum_{e\\in E(\\gamma)}\\int_\\Sigma d^3y \n\\chi_\\epsilon(x,y)A_a^j(y)\n\\int_0^1 dt \\dot{e}^a(t) \\delta(y,e(t))X^k_e(t) f_\\gamma \n\\nonumber\\\\\n&=& -i\\hbar\\kappa \\epsilon_{ijk}\\sum_{e\\in E(\\gamma)}\n\\int_0^1 dt \\dot{e}^a(t)A_a^j(e(t))\\chi_\\epsilon(x,e(t))\nX^k_e(t) f_\\gamma\\nonumber\\\\ \n&=& 2i\\hbar\\kappa \\sum_{e\\in E(\\gamma)}\\lim_{n\\to\\infty}\n\\sum_{k=1}^n \\chi_\\epsilon(x,e(t_{k-1}))\n\\mbox{tr}(\\tau_i[h_e(t_{k-1},t_k)-1])X^i_e(t_{k-1}) f_\\gamma \\;.\n\\end{eqnarray}\nNow recall that $x\\in\\partial\\Sigma$ and that\n$\\gamma\\subset B(\\partial\\Sigma)$. Pick a particular edge $e$ in (\\ref{29}).\nThen for sufficiently small $\\epsilon$ the corresponding term either \nvanishes or $e(0)=x$ and the term becomes \n$\\mbox{tr}(\\tau_i[h_e(0,t(\\epsilon))-1])X^i_e f_\\gamma+o(\\epsilon^2)$\nwhere $t(\\epsilon)$ is the largest value of $t$ such that\n$\\chi_\\epsilon(x,e(t))=1$. Now for a classical, smooth connection which \napproaches \ninfinity as $1\/r^2$ the term $h_e(0,t(\\epsilon))-1$ is at most of order \n$\\epsilon\/r$ (the ``length\" of $e(\\epsilon)$ is at most of order $r$) and so \nvanishes even at finite $\\epsilon$ because $x\\to \\partial\\Sigma$. Therefore   \nexpression (\\ref{29}) vanishes and the volume integral contribution of \n$\\hat{E}_{ADM}$ becomes \n\\begin{equation} \\label{30a}\n\\int_\\Sigma d^3y \\chi_\\epsilon(x,y)\n\\partial_a\\hat{E}^a_i(y) f_\\gamma \\nonumber\\\\\n=-i\\hbar\\kappa \\sum_{v\\in V(\\gamma)}\n\\chi_\\epsilon(x,v)X^i_v f_\\gamma\n\\end{equation}\nwhich one can extend to non-smooth connections.\n\nWe turn to the surface integral of (\\ref{28}) and write \n$f_{\\gamma,v}=X^i_v f_\\gamma$ \nWe have, ordering the $1\/\\hat{V}(x,\\epsilon)$ to the left\n\\begin{eqnarray} \\label{30}\n&&\\int_{\\partial\\Sigma} \\chi_\\epsilon(x,v) \ndS_a\\frac{1}{\\hat{V}(x,\\epsilon)}\\hat{E}^a_i f_{\\gamma,v}\\nonumber\\\\\n&=&-i\\hbar\\kappa\\sum_{e\\in E(\\gamma)}\\int_0^1 dt \\dot{e}^a(t)\n\\int_{\\partial\\Sigma}dS_a(x)\\delta(x,e(t))\n\\frac{\\chi_\\epsilon(x,v)}{\\hat{V}(x,\\epsilon)}X^i_e(t)f_{\\gamma,v}\n\\nonumber\\\\\n&=&-i\\hbar\\kappa\\sum_{e\\in E(\\gamma),e(0)\\in\\partial\\Sigma} \n\\mbox{sgn}(\\partial\\Sigma,e) \n\\frac{\\chi_\\epsilon(e(0),v)}{\\hat{V}(e(0),\\epsilon)}X^i_e f_{\\gamma,v}\n\\nonumber\\\\\n&=&i\\hbar\\kappa\n\\sum_{e\\in E(\\gamma),e(0)\\in\\partial\\Sigma,e\\not\\subset\\partial\\Sigma}\n\\frac{\\chi_\\epsilon(e(0),v)}{\\hat{V}(e(0),\\epsilon)}X^i_e f_{\\gamma,v}\n\\end{eqnarray}\nwhere $\\mbox{sgn}(S,e)$ is the sign of the intersection of $e$ with the \nsurface $S$ (which is outward oriented) at\n$e(0)$ which is thus $-1$ because all edges $e$ are outgoing from a vertex \n$e(0)$ and, because of $\\gamma\\subset B(\\partial\\Sigma)$, they are thus \nrunning away from \n$\\partial\\Sigma$ (there is no contribution from edges that run inside \n$\\partial\\Sigma$ because $\\mbox{sgn}(\\partial\\Sigma,e)_{|e(t)}=0$ for all \n$t$ as was shown in \\cite{ALVol}). Thus, only edges which run transversally\ninto $\\partial\\Sigma$ contribute to the sum in (\\ref{30}).\\\\\nWe can take now the limit $\\epsilon\\to 0$. Notice that for small enough\n$\\epsilon$ we have $\\chi_\\epsilon(e(0),v)=1$ as $\\epsilon\\to 0$ provided\nthat $e(0)=v$ is a vertex of $\\gamma$. Moreover, for small enough\n$\\epsilon$, $v$ is the only vertex of $\\gamma$ in the $\\epsilon-$box \naround $v$. Thus, we may replace $\\hat{V}(v,\\epsilon)$ by $\\hat{V}_v$\ninterpreting the operator $1\/\\hat{V}_v$ by its spectral resolution. Now \nthe only critical point is whether the two operators $X_e^i,X_v^i$\nthat are to the right of $1\/\\hat{V}_v$ will leave the crucial property of\n$f_\\gamma$ intact, namely, that \n$f_\\gamma=\\sum_{i=1}^N f_i$ with $\\hat{V}_v f_i\n=\\lambda_i f_i,\\;\\lambda_i\\not=0$. But this is \neasily seen to be the case provided we impose the following additional \nrestriction on ${\\cal H}_{af}$ :\\\\\n\\\\\n{\\bf Tangle Property}\\\\\n{\\it An asymptotically flat state is a linear combination of diffeomorphism \ngroup averaged elements of the kinematical Hilbert space ${\\cal H}_{af}$ \neach of which depends on an (infinite) graph $\\gamma$ all of \nwhose intersections with $\\partial\\Sigma$ are transversal, that is, there are\nno edges of $\\gamma$ which lie inside $\\partial\\Sigma$. Thus, any path\nalong $\\gamma$ between 2 distinct points of $\\gamma\\cap\\partial\\Sigma$\nis a generalized tangle \\cite{BaTan}, that is, a piecewise analytic path \nintersecting $\\partial\\Sigma$ transversally and points of non-analyticity\nare vertices of $\\gamma$, equivalently, intersections of tangles.\nThe subspace of ${\\cal H}_{af}$ having the tangle property will be called\n${\\cal H}_{tangle}$}.\\\\\n\\\\\nThe tangle property is a rather natural assumption about states $\\xi$ \nbecause curves of non-zero parameter length running in $\\partial\\Sigma$ \nthemselves between vertices automatically have infinite length with \nrespect to $\\delta_{ab}$. This is not the case for curves between \nvertices which just approach $\\partial\\Sigma$ but otherwise lie \nin the interiour.\\\\\nUnder this additional assumption, we have in the limit $\\epsilon\\to 0$, \nputting (\\ref{28c}), (\\ref{30})\nand the remaining pre-factor of $-2\/\\kappa$ from (\\ref{28}) together\n\\begin{equation} \\label{31}\n\\hat{E}_{ADM}(S)f_\\gamma=-2\\hbar^2\\kappa\\sum_{v\\in \\partial\\Sigma\\cup \nV(\\gamma)} \\frac{1}{\\hat{V}_v}X^i_v X^i_v f_\\gamma\\;.\n\\end{equation}\nBut recognizing $-iX_v=\\hat{J}_v$ as the total angular momentum of \n$f_\\gamma$ at $v$ we finally find\n\\begin{equation} \\label{32}\n\\hat{E}_{ADM} f_\\gamma=2\\hbar^2\\kappa\\sum_{v\\in\\partial\\Sigma\\cup V(\\gamma)}\n\\frac{1}{\\hat{V}_v}\\hat{J}^i_v \\hat{J}^i_v f_\\gamma\\;.\n\\end{equation}\nExpression (\\ref{32}) defines a self-consistent family of operators \n$\\{\\hat{E}_{ADM,\\gamma}\\}$ of operators which can be extended to infinite\ngraphs and thus to ${\\cal H}_{tangle}$ provided that the number of \npunctures of $\\gamma$ with $\\partial\\Sigma$ is finite. This defines \nthe domain of $\\hat{E}_{ADM}$.\\\\ \nA number of remarks are in order : \n\\begin{itemize}\n\\item\nThe operator $\\hat{J}_v^i$ is the infinitesimal generator of gauge \ntransformations at $v$ and therefore commutes with $\\hat{V}_v$ because the \nvolume \noperator is gauge invariant proving that (\\ref{32}) is densely defined on \nthe restricted state space. If we would not require the tangle property\nthen while the volume integral in (\\ref{28}) gives essentially $X_v$,\nthe surface integral does not, rather it gives \n$X'_v=X_v-\\sum_{e(0)=v,e\\subset\\partial\\Sigma}X_v$ which does not commute \nwith\n$\\hat{V}_v$ and so may map $f_\\gamma$ into a linear combination of states \nsome of which may acquire zero volume and so (\\ref{32}) would blow up.\nThus, the tangle property is {\\em sufficient} for $\\hat{E}_{ADM}$ to be \na densely defined operator. Although we do not have a proof, it is almost\ngranted that $X'_v f_\\gamma$ will contain zero volume eigenstates in its \nexpansion and so the tangle property would also be {\\em necessary} !\n\\item\nAs a striking bonus of the tangle property we easily see that (\\ref{32})\nindeed defines a positive-semidefinite self-adjoint operator on the \nHilbert space ${\\cal H}_{tangle}$ : First of all, since the \nvolume operator and the $X^i_v$ are defined in terms of the operators \n$X^i_e$ it follows that the family of operators (\\ref{32}) is \nconsistently defined because the $X^i_e$ are. Next, since \n$[X^i_v,\\hat{V}_v]=0$ we see that we may order (\\ref{32}) symmetrically \ninvolving only terms of the form \n$\\hat{J}^i_v\\frac{1}{\\hat{V}_v}\\hat{J}_v^i$ and so (\\ref{32}) defines a \nsymmetric operator because it leaves the graph $\\gamma$ invariant\n\\cite{2,3,4}. Finally, \nthe Laplacian $-\\Delta_v:=J_v^i J_v^i$ has non-negative\ndiscrete spectrum $j_v(j_v+1)$ where $j_v$ is the spin of the contractor \nof the generalized (non-gauge-invariant) spin-network states into\nwhich $f_\\gamma$ can be decomposed.\nMoreover, since $-\\Delta_v$ and $\\hat{V}_v$ commute,\nthey can be simultanously diagonalized. Thus, if $\\lambda_v$ is \nthe eigenvalue of the volume operator then the simultanous eigenstate\n$f_\\gamma$ is an eigenstate of the ADM energy with eigenvalue\n$\\sum_v\\frac{j_v(j_v+1)}{\\lambda_v}$. This provides an explicit \ndiagonalization of $\\hat{E}_{ADM}$ on ${\\cal H}_{tangle}$ and demonstrates \nthat it is a self-adjoint operator on ${\\cal H}_{tangle}$.\nNow, since states in ${\\cal H}_{tangle}$ are not diffeomorphism averaged at \nspatial infinity the part of the graph which is responsible for the spectrum\nof the ADM energy is untouched by the diffeomorphism group averaging. Next, \nthe map $\\hat{\\eta}$ that \nmakes a diffeomorphism invariant state a solution of the Hamiltonian \nconstraint is a generalized projector at each vertex of the graph \nseparately \\cite{8,9,10,15}. Therefore, it is the identity map at those \nvertices of \n$\\gamma$ that lie in $\\partial\\Sigma$ because for a gauge transformation \n$N\\to\\partial\\Sigma$ the action of the constraint at those vertices is \ntrivially zero. \nThus the part of the state responsible for the spectrum \nand the adjointness relations of $\\hat{E}_{ADM}$ \nis unchanged by the map $\\hat{\\eta}$ and are thus preserved when we go to \nthe physical Hilbert space ${\\cal H}_{phys}=\\hat{\\eta}{\\cal H}_{tangle}$.\nMoreover, the spectrum is entirely discrete and \nnon-negative, thus we have proved {\\em A Quantum Positivity of Energy \nTheorem} ! \n\\item\nAstonishingly, the proof of this theorem turned out to be surprisingly \nsimple : the proof of the classical positivity theorem is much more \ncomplicated and uses the boundary conditions and the Einstein equations \nat various stages. Why did we not need (a quantum analogue of) these\nassumptions ? The answer is that we actually did use them : We used them\nin the definition of an asymptotically flat state, in particular, that\nthe volume of the state be non-vanishing and that it be a solution of the \nquantum Einstein equations. Since the energy functional is really given\nby $E(N)=H(N)+E_{ADM}(N)$ where the Hamiltonian constraint only vanishes \non a solution, likewise the ADM energy operator $\\hat{E}_{ADM}$ only \nrepresents energy if we apply it to a solution $\\Psi$ of $\\hat{H}'(N)\\Psi=0$.\nHere we have written the dual $\\hat{H}'(N)$ on $\\Phi'$ of $\\hat{H}(N)$ \nbecause, as stated above, solutions $\\Psi$ lie actually in $\\Phi'$\n\\cite{7,10}. Finally, we used a regularization of the operator consisting\n1) in the restriction of the space of states to functions of classical \nconnections (this means here that they are smooth and decay at \ninfinity as $1\/r^2$) and then 2) in the extension of the expression \nfor the operator obtained to all of \n${\\overline \\A}$. In the derivation of the operator we made crucial use of the fact \nthat a classical connection decays at infinity.\\\\\nNow a subtle issue is the following : by definition a solution $\\Psi$ \nsatisfies $\\hat{H}'(N)\\Psi=0$ for any $N$ which vanishes at \n$\\partial\\Sigma$. But how about lapse functions $N$ that approach a \nconstant value at $\\partial\\Sigma$ ? It is now not a consequence of \nthe formalism any longer that $\\hat{H}'(N)\\Psi=0$ should hold, very much\nlike in the case of the Gauss constraint. The only guideline of what\nto do is the classical theory and there it is indeed true that on \nclassical solutions $(A_0,E_0)$ to the Einstein equations $E(N)$ just\nequals $E_{ADM}(N)_{A=A_0,E=E_0}$ so that we will require that \n$\\hat{H}'(N)\\Psi=0$ even for asymptotically constant lapse.\\\\\nNow, by definition the operator $\\hat{\\eta}$ acts like the identity \noperator at $\\partial\\Sigma$.\nTherefore we conclude that any physical state has the property that it is \nannihilated by $\\hat{H}'(N)$\neven for $N=const.$ and moreover it is a linear combination of \neigenstates with non-zero eigenvalue of the volume operator $\\hat{V}_v$\nfor each vertex of the graph $\\gamma$ on which $\\xi$ depends \n(with $\\hat{\\eta}\\xi=\\Psi$) such that $v\\in\\partial\\Sigma$. One might \nsuspect that the number of states that satisfy this condition is rather \ntiny but the contrary is the case : the volume operator has the \nparticular property that it does not change the graph or the labelling of \nthat graph with spin quantum numbers. Now there exist an infinite number\nof states which are annihilated by $\\hat{H}'(N)$ just because the graph\nor its labellings is of a particular type (see \\cite{10} where such \nstates were labelled by ``spin-webs\", more precisely, ``sources\" \nof spin-webs) and so one can construct eigenstates of the volume \noperator of such states while they are still annihilated by $\\hat{H}'(N)$.\nAs an aside, this might shed some light on the issue of how to interprete \nthose special solutions of the Hamiltonian constraint.\\\\\nThese remarks are sufficient to show that then \n$\\hat{E}_{ADM}'(N)=\\hat{E}_{ADM}(N)$. \n\\item\nThe fact that the ADM energy operator is essentially diagonal on spin-network\nstates can be interpreted as saying that the spin-network representation \nis the {\\em non-linear Fock representation} of quantum general relativity.\nNamely, if we compare the spin that the edges of a spin-network carry \nwith the occupation number of momentum modes of, say, the free Maxwell \nfield then we may interprete this spin essentially as the occupation number\nof a gravitational mode which is not labelled by a momentum but by an edge.\n\\item\nNotice that each vertex of $\\gamma$\nwhich lies in $\\partial\\Sigma$ must be at least three-valent (not \nfour-valent \nbecause the function does not need to be gauge invariant at infinity) in \norder that it can \nhave non-vanishing volume. Also notice that the gravitational energy is\nquantized in quanta of the Planck mass : the eigenvalues of the \nvolume operator are multipla of $\\ell_p^3$ so the eigenvalues of \nenergy, according to (\\ref{32}) are multipla of $\\hbar^2\\kappa\/\\ell_p^3=\n\\sqrt{\\hbar\/\\kappa}=m_p$.\n\\item \nFor a classical connection every function at $\\partial\\Sigma$ is gauge\ninvariant because it decays like $1\/r^2$ which results in a holonomy\nof order $exp(i\/r)\\to 1$, i.e., a trivial holonomy. In quantum theory \nthis is lifted by the distributional nature of a connection, smooth \nconnections are assigned zero volume by the Ashtekar Lewandowski measure \nin the space of distributional connections and are unimportant.\n\\item\nGauge invariant states\nat $\\partial\\Sigma$ correspond to vanishing energy eigenvalue. Thus,\nenergy seems to sit at non-gauge invariant vertices.\nWe may interprete this observation as follows : The gravitational energy \nin a state labelled by an (infinite) graph $\\gamma$ is concentrated \nat the vertices of $\\gamma$, and energy flows from vertex to vertex along \nthe edges of $\\gamma$ in quantized packages labelled by the spin of those \nedges. Non-zero energy at a vertex corresponds to lack of gauge \ninvariance at this vertex meaning that the spins that flow out or into\na vertex do not add up to zero. Now in the interiour of $\\Sigma$ the \nQuantum Gauss Constraint requires that all spins add up to zero. We \ninterprete this as the connection dynamics version of the \ngeometrodynamics result that there \nis no energy location in general relativity in the interior of $\\Sigma$,\ngravitational energy can be gauged away locally, it is pure gauge. However,\nwhile it can be pushed around at one's will, one cannot entirely delete\nit, one can push it all the way to spatial infinity where it eventually\nshows up in the form of a non-zero net spin flow at the vertices of $\\gamma$\nat $\\partial\\Sigma$. \n\\item\nThe fact that (representations of) the $SU(2)$ gauge group of general \nrelativity should play an essential role for the energy is very unexpected\nfrom a geometrodynamics point of view where one never even talks about \nthe $SU(2)$ gauge freedom. Even the classical ADM expression \n$\\int_{\\partial\\Sigma} dS_a (q_{ab,b}-q_{bb,a})$ is manifestly gauge \ninvariant, so how\ndid the $SU(2)$ gauge group enter the stage\\footnote{This question actually \narises already in connection with the spectrum of the geometrical\noperators volume, area and length \\cite{RSVol,JVol,RDP,TTVol,11}, however,\nsince irreducible representations also carry gauge invariant information\nthe fact that these operators have a spectrum which is determined by \nspin-quantum numbers of {\\em gauge-invariant states} is maybe not that \nsurprising. What {\\em is} surprising for the energy operator is that\nit is the spins of non-gauge invariant states which determines the \nspectrum.} ? The answer is the following :\nNotice that while the classical ADM expression is manifestly gauge invariant,\nit is not at all covariant : The derivatives that appear in\n$\\int_{\\partial\\Sigma} dS_a (q_{ab,b}-q_{bb,a})$ are not covariant \nderivatives (in fact if they were covariant derivatives then the energy would\nvanish identically). This does not need to concern us because at \n$\\partial\\Sigma$ the diffeomorphisms that underly the diffeomorphism \nconstraint must die off. Now as we showed in (\\ref{7}), when writing \n$\\int_{\\partial\\Sigma} dS_a (q_{ab,b}-q_{bb,a})$ in terms of $E^a_i$\nthis lack of three-diffeomorphism covariance, by means of the boundary \nconditions, gets translated into non-gauge-invariance while installing\ncovariance because in (\\ref{7}) a correction term drops out \ndue to the boundary conditions which explains why one and same \nfunction can be written in a manifestly gauge invariant or diffeomorphism \ncovariant way but not both. Indeed, the expression\n$-2\\int_{\\partial\\Sigma} dS_a E^a_i \\partial_b E^b_i\/\\sqrt{\\det(q)}$ is \nmanifestly \nthree-diffeomorphism covariant but it fails to be gauge-invariant. \nThis is not unexpected : after all the triad formulation reduces the local \ndiffeomorphism gauge freedom to local rotation freedom.\nIn conclusion, the fact that states with non-zero energy\nare not gauge invariant is in fact very natural.\n\\item\nAnother function for which $SU(2)$ gauge transformations and \ndiffeomorphisms get\nmixed up is the classical vector constraint $V_a=\\mbox{tr}(F_{ab} E^b)$.\nStrictly speaking this constraint function does not generate \ndiffeomorphisms but only on gauge invariant functions.\n\\item\nNotice that although\n$-2\\int_{\\partial\\Sigma} dS_a E^a_i \\partial_b E^b_i\/\\sqrt{\\det(q)}$ is not \ngauge invariant the quantum expression (\\ref{32}) in fact is. The reason \nwhy that is \npossible lies in the structure of quantum theory : in the classical theory\nwe only have functions on phase space. In quantum theory those functions \nget translated into operators on a Hilbert space but values of those \nfunctions really correspond to expectation values. Thus non-gauge invariant\nfunctions correspond to expectation values of either a gauge-invariant \noperator in a non-gauge-invariant state or vice versa. The quantization\n(\\ref{32}) picks the latter possibility.\n\\item \nIn principle we have now solved the ``problem of time\" : Since we \nhave a true Hamiltonian we can introduce the {\\em Schroedinger time parameter}\n$t$ and our state vectors $\\Psi\\in\\Phi'$, being distributions which are\ninvariant under asymptotically identity diffeomorphisms are \nsupposed to satisfy the {\\em non-stationary Schroedinger equation}\n\\begin{equation} \\label{32a}\n\\hat{E}_{ADM}(N=1)\\Psi=-i\\hbar\\partial_t\\Psi\\;.\n\\end{equation}\nNotice that, as we showed above, the ADM energy operator is its own dual \nso that (\\ref{32a}) makes sense. \n\\item\nIn the next section the operator (\\ref{32}) will be shown to commute with \nall quantum constraints of general relativity and it is therefore a\n{\\em Strong Quantum Dirac Observable} for Quantum Gravity in the strict \nsense of the word. This is not unexpected because it is built purely from\nmomentum operators.\n\\end{itemize}\n\n\n\\section{The Poincar\\'e Algebra}\n\n\nWe now wish to define the rest of the quantum generators of the little \ngroup of the\nasymptotic Poincar\\'e group and check whether their algebra is anomalous \nor not \\footnote{By little group of the Poincar\\'e group we \nmean the group generated by the four-translations and the little subgroup of \nthe connected component of the Lorentz Group. The latter, as is \nwell-known, is the stabilator subgroup of the Lorentz group associated \nwith a standard four vector $\\bar{s}$. In the massive case $\\bar{s}^2>0$\nthe standard vector is associated with the spin of the particle in the rest\nframe and the covering group of the stabilator group is given by by $SU(2)$.\nIn the massless case the standard vector is associated with the helicity\nof the particle (spin in momentum direction) and \nthe covering group of the stabilator group is given by by $U(1)$, \nphysically important representations being two-valued. Thus, the \nrotations at spatial infinity determine the unitary irreducible \nrepresentation of the particle state in question.}. \nThis is enough to construct particle states since the \nirreducible unitary representations of the little group induce a unique\nunitary irreducible representation of the full Poincar\\'e group. So far \nwe did not construct an operator corresponding to a boost generator which\nis more difficult to obtain than the ADM energy operator.\n\nFirst of all we must clarify on which space to represent the Poincar\\'e \ngroup, respectively its generators. To that end it is helpful to remember\nhow the classical Poincar\\'e generators are realized as a subalgebra of \nthe Poisson algebra \\cite{13,13a}.\\\\ \nLet $H(N),V(\\vec{N})$ be the Hamiltonian and diffeomorphism constraint \nfunctional respectively. Both functionals are integrals over $\\Sigma$ of \nlocal \ndensities and both converge and are functionally differentiable only\nif the lapse and shift functions $N,\\vec{N}$ vanish at $\\partial\\Sigma$.\nIn order to be able to describe the Poincar\\'e group corresponding to\nthe asymptotically constant or even diverging functions ($x^a$ is a \ncartesian frame at spatial infinity)\n$N=a+\\chi_a x^a,N^a=a^a+\\epsilon^{abc}\\phi_b x^c$ where $(a,a^a)$ is \na four translation, $\\phi^a$ are rotation angles and $\\chi^a$ are boost\nparameters, one proceeds as follows : let $S$ be a bounded two-surface\nthat is topologically a sphere and let $B(S)$ be the (intersection of \n$\\Sigma$ with the) closed ball such that $\\partial B(S)=S$. For \neach $S$ one defines \n$E(N,S):=H(N,S)+E_{ADM}(N,S)+B(N,S),\\;P(N,S):=V(\\vec{N},S)+P_{ADM}(N,S)$\nwhere the parameter $S$ means that volume integrals are restricted to\n$B(S)$ only (a classical regularization of the divergent integrals) and the \n``counter-terms\" $E_{ADM}(N,S),B(N,S),P_{ADM}(\\vec{N},S)$ are the surface \nintegrals defined in \\cite{13} and correspond to ADM energy, boost and \nmomentum. One can show that $\\lim_{S\\to\\partial\\Sigma} E(N,S), \n\\lim_{S\\to\\partial\\Sigma} P(\\vec{N},S)$ exist. Moreover, for each finite $S$,\n$E(N,S),P(\\vec{N},S)$ are functionally differentiable so that it is \nmeaningful to compute the Possion brackets\n\\begin{eqnarray} \\label{33}\n\\{E(M,S),E(M,S)\\}=P(q^{ab}(MN_{,b}-M_{,b}N),S)\\nonumber\\\\\n\\{E(M,S),P(\\vec{N},S)\\}=E({\\cal L}_{\\vec{N}}M,S)\\nonumber\\\\\n\\{P(\\vec{M},S),P(\\vec{N},S)\\}=P({\\cal L}_{\\vec{M}}\\vec{N},S)\\;.\n\\end{eqnarray}\nThe crucial point is that one computes the Poisson brackets a) at finite \n$S$ and b) on the full phase space and then takes the limit \n$S\\to\\partial\\Sigma$ or restricts to the constraint surface of the phase \nspace (where $H(N,S)=V(\\vec{N},S)=0$). Notice that the numerical value \nof, say, $E(N,S)$ equals $H(N,S)$ for a gauge transformation for which $N\\to \n0$ as $S\\to\\partial\\Sigma$. On the other hand, on the constraint surface \nfor a symmetry for which $N\\not\\to 0$ as $S\\to \\partial\\Sigma$ it equals a \ntime translation or a boost respectively. A similar remark holds for \n$P(\\vec{N},S)$. One therefore interpretes (\\ref{33}) as follows :\nif $M,N$ are both pure gauge then the constraint algebra closes. If $M$\nis a symmetry and $N$ pure gauge then energy (or boost generator) are gauge \ninvariant.\nIf $M,N$ are both symmetry then time translations commute with each other,\ntime translations and boosts give a spatial translation and a boost with a \nboost gives a rotation, in other words the symmetry algebra closes.\n\nIn quantum theory we will therefore proceed as follows : \\\\\nRecall \\cite{8,9,10,15} that the Hamiltonian constraint $\\hat{H}(N)$\n(for asymptotically vanishing $N$) is only well-defined on the subspace\nof $\\Phi'$ corresponding to distributions on $\\Phi$\nwhich are invariant under diffeomorphisms that approach identity at \n$\\partial\\Sigma$. Thus we can expect the symmetry algebra to hold only on\nsuch distributions as well. In fact, we will just choose $\\Psi$ to be a \nsolution to all constraints.\\\\ \nNext, in view of the fact that even the classical symmtry algebra \nonly holds provided one first computes Poisson brackets at finite $S$ and \nthen takes the limit, we will check the quantum algebra first \nby evaluating $\\Psi$ on $\\hat{E}(N_S) f_S$ \nfor functions $f_S\\in\\Phi$ cylindrical with \nrespect to a graph which lies in the interior of $B(S)$ (it may \nintersect $S$ in such a way that the volume operator does not vanish\nat the intersection point for none of the eigenvectors into which\n$f_S$ maybe decomposed) and lapse functions $N_S$ which grow at infinity like\nsymmetries but which are supported in $B(S)\\cup S$ {\\em including $S$}, and \nthen to take the limit $S\\to\\partial\\Sigma$ \n(the support fills all of $\\Sigma$ as $S\\to \\partial\\Sigma$ in this \nprocess). \\\\ \n\nWe come to the definition of $\\hat{E}(N),\\hat{P}(\\vec{N})$. \nFirst we treat the spatial Euclidean group.\\\\\nThe unitary representation of the diffeomorphism group defined by\n$\\hat{U}(\\varphi) f_\\gamma=f_{\\varphi(\\gamma)}$ which was for matters of \nsolving the diffeomorphism constraint so far only defined for \ndiffeomorphisms that approach asymptotically the identity, can easily \nbe extended to three-diffeomorphisms which correpond to asymptotic spatial\ntranslations or rotations. Instead of defining the generator \n$\\hat{P}(\\vec{N})$ though \n(which does not exist on $\\cal H$ \\cite{7}) we content ourselves with the \nexponentiated version $\\hat{U}(\\varphi(\\vec{N}))$ where $\\varphi(\\vec{N})$\nis the diffeomorphism generated by the six parameter shift vector field\n$N^a=a^a+\\epsilon_{abc} \\phi^b x^c$ for some cartesian frame $x^a$ possibly \ncorrected by an asymptotically vanishing vector field corresponding to a \ngauge transformation. It is trivial to check that \n\\begin{equation} \\label{34}\n\\hat{U}(\\varphi(\\vec{N}))\n\\hat{U}(\\varphi(\\vec{N}'))\n\\hat{U}(\\varphi(\\vec{N}))^{-1}\n\\hat{U}(\\varphi(\\vec{N}'))^{-1}\n=\\hat{U}(\\varphi({\\cal L}_{\\vec{N}}\\vec{N}'))\n\\end{equation}\nwhere $\\cal L$ denotes the Lie derivative \nso that there are no anomalies coming from the spatial Euclidean group.\nThis expression was derived by applying it to any function $f_S$ cylindrical \nwith respect to a graph with support in $B(S)$. \n\nWe now turn to the time translations. As already mentioned we will not \nconsider boosts in this paper so that $\\chi_a\\equiv 0$ in \nthe four parameter family of lapse functions $N=a+\\chi_a x^a$\n(modulo a correction which vanishes at $\\partial\\Sigma$).\nDefine the operator on ${\\cal H}$\n\\begin{equation} \\label{35}\n\\hat{E}(N):=\\hat{H}(N)\n+\\hat{E}_{ADM}(N)\n\\end{equation}\nwhere $\\hat{H}(N)$ is the Lorentzian Hamiltonian constraint.\nNotice that $\\hat{E}(N)$ just as the Hamiltonian constraint in \n\\cite{8,9,10,15} carries a certain prescription dependence \nwhich is removed by evaluating its dual on $\\Phi_{Diff}$. We will not\nrepeat these details here and refrain from indicating this prescription\ndependence in \\ref{35}, however, the prescription dependence has \nconsequences for the commutator algebra that we will dicuss below in \ngreat detail.\n\nLet us verify the commutators between the time \ntranslations among themselves and between time taranslations and spatial \ntranslations and rotations. We have \n\\begin{eqnarray} \\label{36}\n&&\\Psi([\\hat{E}(M),\\hat{E}(N)]f_\\gamma)=\n\\Psi([\\hat{H}(M),\\hat{H}(N)]f_\\gamma)\n+\\Psi([\\hat{E}_{ADM}(M),\\hat{E}_{ADM}(N)]f_\\gamma)\n\\nonumber\\\\\n&+&\\Psi(\\{[\\hat{E}_{ADM}(M),\\hat{H}(N)]+[\\hat{H}(M),\\hat{E}_{ADM}(N)]\\}\nf_\\gamma)\\;.\n\\end{eqnarray}\nThe first term vanishes for the same reason as in \\cite{8,9,10,15} \nalthough one needs one additional argument : the \nHamiltonian constraint does not act at vertices that it creates. Therefore,\nit can be written as a double sum over vertices $v,v'$ of $\\gamma$ alone and \neach of these terms is of the form \n$$\n(M(v)N(v')-M(v')N(v))\n\\Psi([\\hat{H}_{v',\\gamma(v)}\\hat{H}_{v,\\gamma}-\n\\hat{H}_{v,\\gamma(v')}\\hat{H}_{v',\\gamma}]f_\\gamma)\n$$\nwhere the notation means that $\\hat{H}_{v,\\gamma}$ is a family of\nconsistently defined operators each of which  \nacting on cylindrical functions which depend on the graph $\\gamma$ and \n$\\gamma(v)$ is a graph on which $\\hat{H}_{v,\\gamma}f_\\gamma$ depends. \nThis expression clearly is non-vanishing only if $v\\not=v'$ but then \nit can be shown that the operators $\\hat{H}_{v,..}$ and \n$\\hat{H}_{v',..}$ actually commute. Now still this does not show that\nthe term above vanishes however, it can be shown that \n$\\hat{H}_{v',\\gamma(v)}\\hat{H}_{v,\\gamma}f_\\gamma$ and \n$\\hat{H}_{v,\\gamma(v')}\\hat{H}_{v',\\gamma}f_\\gamma$ are related by a \ndiffeomorphism \\cite{9}. Now in \\cite{9} that was enough to show that the \ncommutator vanishes because we were dealing there only with vertices \nwhich do not intersect $S$ as otherwise both lapse functions identically \nvanish for a pure gauge transformation. Thus the diffeomorphism that \nrelates the two terms above could be chosen to have support inside \n$B(S)$ and $\\Psi$ is invariant under such diffeomorphisms. In the present \ncontext that \ndoes not need to be true. However, the crucial point is now that by the \ntangle property all edges of $\\gamma$ that intersect $S$ must intersect\n$S$ transversally. Therefore the arcs that the Hamiltonian constraint \nattaches to $\\gamma$ and whose position is the only thing by which the \ntwo above vectors differ {\\em lie inside $B(S)$ and do not intersect $S$}. \nTherefore, again the two vectors are related by a diffeomorphism which \nhas support inside $B(S)$, that is, they are related by a gauge \ntransformation and therefore the commutator vanishes.\n\nWe turn to the second term in (\\ref{36}). Now we obtain a double sum\nover vertices of $\\gamma$ which lie in $S$ and each term is \nof the form\n$$\n(M(v)N(v')-M(v')N(v))\n\\Psi([\\hat{E}_{v',ADM},\\hat{E}_{v,ADM}]f_\\gamma)\n$$\nwhich is significantly simpler than before because $\\hat{E}_{v,ADM}$\ndoes not alter the graph. Notice that the commutator makes sense because\n$\\hat{E}_{ADM,v}$ leaves the span of non-zero volume eigenvectors invariant.\nNow for $v\\not=v'$ the commutator trivially vanishes, this time without \nemploying diffeomorphism invariance of $\\Psi$.\n\nFinally the last term in (\\ref{36}) is a double sum over vertices \n$v,v'$ of $\\gamma$, where $v$ must lie in $S$, of the form\n\\begin{equation} \\label{37}\n(M(v)N(v')-M(v')N(v))\n\\Psi([\\hat{H}_{v',\\gamma},\\hat{E}_{v,ADM}]f_\\gamma)\\;.\n\\end{equation}\nThe fact that $\\hat{E}_{ADM}$ does not alter the graph was used to write\n(\\ref{37}) as a commutator without employing diffeomorphism \ninvariance of $\\Psi$. Now it may happen that, although $f_\\gamma$ is in \nthe domain of $\\hat{E}_{v,ADM}$, that $\\hat{H}_{v,\\gamma}f_\\gamma$ is not \nany longer in the domain and so (\\ref{37}), for $v=v'$, is in danger of \nbeing \na meaningless product of something that blows up times zero while that\ncannot happen for $v\\not=v'$. However, \nsince $\\Psi$ is a solution we conclude first of all that \n(\\ref{37}) equals\n\\begin{equation} \\label{38}\n-(M(v)N(v')-M(v')N(v))\n[\\hat{E}_{v,ADM}\\Psi](\\hat{H}_{v',\\gamma} f_\\gamma)\n\\end{equation}\nand since $\\Psi$ is also in the domain of $\\hat{E}_{ADM}$ both \n$\\hat{E}_{v,ADM}\\Psi$ and $\\hat{H}_{v',\\gamma} f_\\gamma$\nare well-defined elements of $\\Phi'$ and $\\Phi$ respectively we conclude \nthat in case $v=v'$ (\\ref{37}) indeed vanishes. On the other hand,\nthe same argument as before shows that the commutator trivially vanishes for \n$v\\not=v'$.\n\nLet us now check the commutator between time translations and spatial\ntranslations and rotations $\\varphi$. We have\n\\begin{eqnarray} \\label{39}\n&&\\Psi([\\hat{U}(\\varphi)^{-1}\\hat{E}(N)\\hat{U}(\\varphi)-\\hat{E}(N)]f_\\gamma)\n\\nonumber\\\\\n&=& \\sum_{v\\in V(\\gamma)}[N(\\varphi(v))\n\\Psi(\\hat{U}(\\varphi^{-1})\\hat{H}_{\\varphi(v),\\varphi(\\gamma)}\nf_{\\varphi(\\gamma)})\n-N(v)\\Psi(\\hat{H}_{v,\\gamma}f_\\gamma)]\n\\nonumber\\\\\n&+& \n\\sum_{v\\in \nV(\\gamma)\\cap \nS}[N(\\varphi(v))\\Psi(\\hat{U}(\\varphi^{-1})\\hat{E}_{ADM,\\varphi(v)} \nf_{\\varphi(\\gamma)})-N(v)\\Psi(\\hat{H}_{ADM,v}f_\\gamma)]\\;.\n\\end{eqnarray}\nSince $\\hat{E}_{ADM}$ does not change the graph on which a function depends\nwe have identically \n$\\hat{U}(\\varphi^{-1})\\hat{E}_{ADM,\\varphi(v)} f_{\\varphi(\\gamma)}\n=\\hat{E}_{ADM,v} f_\\gamma$.\\\\  \nNow, as explained in more detail in \\cite{9}, the operator $\\hat{H}(N)$\ndepends on a certain prescription of how to attach loops to graphs. Since \nin the interiour of $B(S)$ there is no background metric\navailable, this prescription can only be topological in nature and therefore\ngraphs differing\nby a diffeormorphism $\\varphi$ are assigned graphs by $\\hat{H}(N)$ which\nare diffeomorphic by a diffeomorphism $\\varphi'$ which may not coincide with\n$\\varphi$. That is, in the interiour of $B(S)$, $\\hat{H}(N)$ is \nonly covariant up to a diffeomorphism. On the other hand,\nsince one has the fixed background metric $\\delta_{ab}$ at $S$ one can \nmake $\\hat{H}(N)$ precisely covariant at $S$, that is, the prescription \nsatisfies $\\varphi_{|S}=\\varphi'_{|S}$.\nTherefore, with this sense of covariance of $\\hat{H}(N)$ \nit is true that \n$\\hat{U}(\\varphi^{-1})\\hat{H}_{\\varphi(v),\\varphi(\\gamma)}f_{\\varphi(\\gamma)}$\nand $\\hat{H}_{v,\\gamma}f_\\gamma$ differ at most by a diffeomorphism \nwith support in the interiour of $B(S)$.\\\\ \nIn conclusion we obtain\n$$\n\\Psi([\\hat{E}(N),\\hat{U}(\\varphi)]f_\\gamma)=\n\\Psi(\\hat{E}(\\varphi^\\star N-N)f_\\gamma)\n$$ \nwhich is what we were looking for.\n\nWe conclude that the little algebra of the Poincar\\'e algebra is faithfully \nimplemented.\\\\ \\\\\n\\\\\n\\\\\n{\\large Acknowledgements}\\\\\n\\\\\nThis research project was supported in part by DOE-Grant\nDE-FG02-94ER25228 to Harvard University.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMost of the accretion onto the supermassive black hole (SMBH) found in the center of most massive galaxies is heavily \nobscured by the surrounding dust and gas (e.g., \\citealp{fabian99a}). In the local Universe, $\\sim$75\\% of the Seyfert 2 galaxies\nare heavily-obscured ($N_H$$>$10$^{23}$~cm$^{-2}$; \\citealp{risaliti99}). Many of these, if at $z$$\\gtrsim$1, where most of the black hole growth\noccurs, would not be identified in X-rays even in very deep ($>$1 Msec) Chandra or XMM\/Newton exposures \\citep{treister04}. Locating and \nquantifying this heavily obscured SMBH growth, in particular at high redshifts, is currently one of the fundamental problems \nin astrophysics. \n\nBecause the energy absorbed at optical to X-ray wavelengths is later re-emitted in the mid-IR, it is expected that all Active Galactic Nuclei (AGN), even\nthe most obscured ones, should be very bright mid-IR sources (e.g., \\citealp{martinez06}). Hence, it is not surprising\nthat a large number of heavily obscured --- even Compton-thick ($N_H$$>$10$^{24}$cm$^{-2}$) --- AGN have been found amongst the Luminous and\nUltra-luminous Infrared Galaxies ((U)LIRGs; L$_{IR}$$>$10$^{11}$ and $>$10$^{12}$L$_\\odot$ respectively), both locally \\citep{iwasawa09} and at \nhigh redshift \\citep{bauer10}. Deep X-ray observations performed using the XMM-Newton (e.g., \\citealp{braito03,braito04}), Chandra \\citep{teng05} and \nSuzaku \\citep{teng09} observatories have shown that most ULIRGs are intrinsically faint X-ray sources, most likely due to the effects of obscuration, while \ntheir X-ray spectra show combined signatures of starburst and AGN activity. The key features observed in the X-ray spectra of ULIRGs are \na soft thermal component, typically associated with star formation, a heavily-obscured (N$_H$$\\sim$10$^{24}$ cm$^{-2}$) power-law associated \nwith the AGN direct emission, and a prominent emission line at $\\sim$6.4~keV, identified with fluorescence emission from iron in the \nK$_\\alpha$ ionization level, originating either in the accretion disk or in the surrounding material \\citep{matt91}. \n\nThe presence of heavily-obscured AGN among the most extreme ULIRGs at $z$$\\simeq$1-2 has recently been established from deep\nSpitzer observations \\citep{daddi07,fiore08,treister09c}. Most of these sources have very high, quasar-like, intrinsic luminosities, and hence most likely\ndo not constitute the bulk of the heavily-obscured AGN population \\citep{treister10}. Establishing the fraction of (U)LIRGs that host a lower luminosity AGN is \na more challenging task. Recent works based on X-ray stacking \\citep{fiore09} and using 70-$\\mu$m selected sources \\citep{kartaltepe10} report \na steep decrease in the fraction of AGN with decreasing IR luminosity, going from $\\sim$100\\% at L$_\\textnormal{IR}$=10$^{13}$~L$_\\odot$ \nto $<$10\\% at L$_\\textnormal{IR}$=10$^{10}$~L$_\\odot$. In the local Universe, \\citet{schawinski10} found that the incidence of low-luminosity, Seyfert-like, \nAGN as a function of stellar mass is more complicated, and is influenced by other parameters. For example, the dependence of AGN fraction on stellar mass\ncan be opposite if galaxy morphology is considered (increases with decreasing mass in the early-type galaxy population).\n\nIn this work, we estimate the fraction of heavily-obscured AGN in mid-IR-luminous and massive galaxies at high redshift, few of which are individually \ndetected in  X-rays. The main goal is to constrain the amount of obscured SMBH accretion happening in distant galaxies. This can be done thanks\nto the very deep X-ray observations available in the Chandra Deep Fields and the very low and stable Chandra background, which allows for the efficient\nstacking of individually undetected sources. Throughout this letter, we assume a $\\Lambda$CDM cosmology with $h_0$=0.7, $\\Omega_m$=0.27\nand $\\Omega_\\Lambda$=0.73, in agreement with the most recent cosmological observations \\citep{hinshaw09}.\n\n\\section{Analysis and Results}\n\nBy stacking individually-undetected sources selected at longer wavelengths, it is possible to detect very faint X-ray emitters using Chandra observations. For \nexample, this technique was used successfully by \\citet{brandt01} in the Chandra Deep Field North (CDF-N) to measure\nthe average X-ray emission from a sample of Lyman break galaxies at $z$$\\simeq$2--4 and by \\citet{rubin04} to detect\nX-rays from red galaxies at $z$$\\sim$2. More recently, samples of heavily-obscured AGN candidates selected based\non their mid-IR properties have been studied in X-rays via Chandra stacking (e.g., \\citealp{daddi07,fiore08,treister09c}).\n\nThe 4 Msec Chandra observations of the Chandra Deep Field South (CDF-S), are currently\nthe deepest view of the X-ray sky. In addition, the CDF-S has been observed extensively at many wavelengths. The multiwavelength data available on the (E)CDF-S were \npresented by \\citet{treister09c}. Very relevant for this work are the deep Spitzer observations available in this field, using both the Infrared Array Camera (IRAC) and the \nMultiband Imaging Photometer for Spitzer (MIPS), from 3.6 to 24~$\\mu$m. Also critical is the availability of good quality photometric \nredshifts ($\\Delta$$z$\/(1+$z$)=0.008 for $R$$<$25.2) obtained thanks to deep observations in 18 medium-band optical filters performed using \nSubaru\/Suprime-Cam \\citep{cardamone10}. \n\nWe generated our sample starting with the 4959 Spitzer\/MIPS 24~$\\mu$m sources in the region covered by the Chandra observations that have\nphotometric redshift $z$$>$0.5, and hence rest-frame E$>$4~keV emission falling in the high-sensitivity Chandra range. In addition, sources \nindividually detected in X-rays and reported in the catalogs of \\citet{luo08}, \\citet{lehmer05} or \\citet{virani06} were removed from our \nsample, as these sources will otherwise dominate the stacked spectrum. We then inspected the remaining sources to eliminate\nindividual detections in the 4 Msec data not present in the 2 Msec catalog of \\citet{luo08}. We further excluded 28 sources that meet the \nselection criteria of \\citet{fiore08} for heavily obscured AGN, $f_{24}$\/$f_R$$>$1000 and $R$-$K$$>$4.5 (Vega), because we expect these sources to \ncontain an intrinsically luminous AGN (quasar), while the aim of this work is to find additional hidden accretion in less luminous objects. The median \nredshift of the sources in our final sample is 1.32 (average $z$=1.5) with a standard deviation of 0.77.\n\nIn order to perform X-ray stacking in the rest-frame, we started with the regenerated level 2 merged event files created by the Chandra X-ray \nCenter\\footnote{Data publicly available at http:\/\/cxc.harvard.edu\/cda\/whatsnew.html}. For each source, we extracted all events in a circle of \n30$''$ radius centered in the optical position. The energy of each event was then converted to the rest frame using the photometric redshift of the \nsource. Using standard CIAO \\citep{fruscione06} tools we then generated seven X-ray images for each source covering the energy range from 1-8 keV \nin the rest-frame with a fixed width of 1 keV. Images for individual sources were then co-added to measure the stacked signal. Total counts were measured\nin a fixed 5$''$ aperture, while the background was estimated by randomly placing apertures with same the area, 5$''$ to 30$''$ away from the center.\n\nSeveral groups have found (e.g, \\citealp{kartaltepe10} and references therein) that the fraction of galaxies containing an AGN is a strong function of\ntheir IR luminosity. Hence, it is a natural choice to divide our sample in terms of total IR luminosity. The infrared luminosity was estimated from the \nobserved 24~$\\mu$m luminosity assuming the relation found by \\citet{takeuchi05}: $\\log$(L$_{IR}$)=1.02+0.972 $\\log$(L$_{12~\\mu m}$). We further \nassumed that the $k$ correction between observed-frame 24~$\\mu$m and rest-frame 12~$\\mu$m luminosity for these sources  \nis negligible, as shown by \\citet{treister09c}. We then separated our sample in 4 overlapping bins: $L_{IR}$$>$10$^{11}$$L_\\odot$, \n$L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$, 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ and $L_{IR}$$>$10$^{10}$$L_\\odot$ \nand stacked them independently. The number of sources in each sample is 670, 1545, 2342 and 3887 respectively.\n\nIn Fig.~\\ref{obs_spec} we present the stacked spectra as a function of rest-frame energy, both in total counts and normalized at 1 keV to highlight the \ndifference in spectral shape among the different samples. At E$\\gtrsim$5 keV, the spectra begin to diverge, where we expect the AGN emission to\ndominate even for heavily-obscured sources. There is a clear trend, with more high energy X-ray emission with increasing IR luminosity.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.22\\textwidth]{f1a.eps}\n\\includegraphics[width=0.22\\textwidth]{f1b.eps}\n\\end{center}\n\\caption{{\\it Left panel:} Stacked background-subtracted Chandra counts as a function of rest-frame energy from 1 to 8 keV. Samples were selected\nbased on their IR luminosity in the following overlapping bins: $L_{IR}$$>$10$^{11}$$L_\\odot$ ({\\it filled circles}), $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ \n({\\it squares}), 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ ({\\it triangles}) and $L_{IR}$$>$10$^{10}$$L_\\odot$ ({\\it open circles}).\n{\\it Right panel}: same as left panel but normalized at 1 keV in order to highlight the differences in spectral shape among the different samples.\nThe largest differences are at E$\\gtrsim$5 keV, where there is a clear trend in the relative intensity as a function of IR luminosity,\nsuggesting a larger fraction of AGN in the most luminous IR sources.}\n\\label{obs_spec}\n\\end{figure}\n\n\n\\section{Discussion}\n\nThe spectra shown in Fig.~\\ref{obs_spec} cannot be directly interpreted, as the detector-plus-telescope response information is lost after the conversion to \nrest-frame energy and stacking. Hence, we perform simulations assuming different intrinsic X-ray spectra in order to constrain the nature of the sources \ndominating the co-added signal. We use the XSPEC code \\citep{arnaud96} to convolve several intrinsic input spectra with the latest response \nfunctions\\footnote{Obtained from http:\/\/cxc.harvard.edu\/caldb\/calibration\/acis.html} for the Chandra ACIS-I camera used in the CDF-S observations. We\nthen compare these simulated spectra with the observations in our sample of IR-selected sources.\n\nThe low energy spectrum of (U)LIRGs is dominated by a combination of a thermal plasma component with temperatures $kT$$\\simeq$0.7~keV, particularly\nimportant at E$<$3 keV, and the emission from high-mass X-ray binaries (HMXBs) at 1$<$E (keV)$<$10 (e.g., \\citealp{persic02}). For each source, we\ngenerated a simulated spectrum using a combination of these two components, taking into account the luminosity and redshift of the source. For\nthe HMXB population we assumed a power-law given by $\\Gamma$=1.2 and cutoff energy $E_c$=20 keV, consistent with recent observations \n(e.g., \\citealp{lutovinov05}). This component was normalized assuming the relation between IR and X-ray luminosity in starburst galaxies found \nby \\citet{ranalli03}. For the thermal component, we assumed a black body with temperature $kT$=0.7 keV. The normalization of this component\nwas then adjusted to match the observations at $E$$<$3 keV.\n\nIn order to compute the possible contribution from heavily-obscured AGN to the stacked spectrum we assumed the observed X-ray spectrum of the nearby\nULIRG IRAS19254-7245, as observed by Suzaku \\citep{braito09}. In addition to the starburst emission described above, the X-ray spectrum is described by\nan absorbed, Compton-thick, power-law with $\\Gamma$=1.9, $N_H$=10$^{24}$~cm$^{-2}$, and a possible scattered component, characterized by a \npower-law with $\\Gamma$=1.9, no absorption, and 1\\% of the intensity of the direct emission. The resulting simulated spectral components and the comparison\nwith the observed stacked spectrum for sources with the four samples defined above are shown in Fig.~\\ref{simul_spec}. \n\n\\begin{figure}\n\\begin{center}\n\\plotone{f2.eps}\n\\end{center}\n\\caption{Stacked background-subtracted Chandra counts as a function of rest-frame energy, as in Fig.~\\ref{obs_spec}. {\\it Black data points (filled circles)} show\nthe stacked X-ray signal for sources binned by IR luminosity. The {\\it cyan dashed lines (stars)} shows the simulated spectra for the HMXB population\nnormalized using the \\citet{ranalli03} relation between star-formation rate and X-ray luminosity. The {\\it blue dashed lines (open squares)} show simulated thermal spectra\ncorresponding to a black body with $kT$=0.7 keV, required to explain the E$<$3 keV emission. An absorbed AGN spectrum, given by a power-law with $\\Gamma$=1.9 and a \nfixed $N_H$=10$^{24}$~cm$^{-2}$, is shown by the {\\it red dashed lines (open circles)}. In addition, a scattered AGN component, characterized by a 1\\% reflection of the underlying \nunobscured power-law, is shown by the {\\it green dashed lines (open triangles)}. The resulting summed spectrum ({\\it black solid lines}) is in very good \nagreement with the observed counts. The strong detection in the stacked spectrum at E$>$5 keV, in particular at the higher IR luminosities, confirms the presence of a significant \nnumber of heavily-obscured AGN in these samples.}\n\\label{simul_spec}\n\\end{figure}\n\nIt is not possible to explain the observed stacked spectral shape using only a plausible starburst spectrum without invoking an AGN component, which dominates \nat E$>$5~keV. The average intrinsic rest-frame 2-10 keV AGN luminosity needed to explain the observed spectrum, assuming that every source in the sample contains an AGN of the \nsame luminosity, is 6$\\times$10$^{42}$~erg~s$^{-1}$ for sources with $L_{IR}$$>$10$^{11}$$L_\\odot$, 3$\\times$10$^{42}$~erg~s$^{-1}$ \nfor sources with $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$, 5$\\times$10$^{41}$~erg~s$^{-1}$ in the sample with 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$\nand 7$\\times$10$^{41}$~erg~s$^{-1}$ for sources with $L_{IR}$$>$10$^{10}$$L_\\odot$. All of these are (intrinsically) very low-luminosity AGN; even if there is a range, it is extremely\nunlikely to include high-luminosity quasars like those discussed in previous stacking papers. An alternative possibility is that the extra emission at E$>$5~keV is due entirely to \nthe Fe~K$\\alpha$ line, provided the errors in the photometric redshifts in these samples are significantly larger than the values reported by \\citet{cardamone10}. \nRegardless of the template assumed for the AGN emission, we obtain similar values for the average AGN luminosity in each sample.\n\nThe median hard X-ray luminosity for the Chandra sources with measured photometric redshifts in the catalog of \\citet{luo08} is 4.1$\\times$10$^{43}$ erg~s$^{-1}$ for \nthe sources in the $L_{IR}$$>$10$^{11}$$L_\\odot$ sample, 3.5$\\times$10$^{43}$~erg~s$^{-1}$ in the $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ group, 5.7$\\times$10$^{42}$~erg~s$^{-1}$ for \nsources with 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ and 1.6$\\times$10$^{43}$~erg~s$^{-1}$ in the $L_{IR}$$>$10$^{10}$$L_\\odot$ sample. Hence, \nif the heavily-obscured AGN in our stacked samples have the same median intrinsic luminosity this would indicate that 15\\% (98 sources) of the 670 galaxies \nwith $L_{IR}$$>$10$^{11}$$L_\\odot$ contain a heavily-obscured AGN. This fraction is $\\sim$9\\% (132 and 205 sources respectively) in the $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ and\n5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ samples. For sources with $L_{IR}$$>$10$^{10}$$L_\\odot$ this fraction is $<$5\\%. The integrated intrinsic \nX-ray emission in the rest-frame 2-10 keV band due to the heavily-obscured AGN in this sample, obtained by multiplying the intrinsic X-ray luminosity by the number of sources \nand dividing by the studied area, is $\\sim$4.6$\\times$10$^{46}$~erg~cm$^{-2}$~s$^{-1}$~deg$^{-2}$. For comparison, the total emission from all the X-ray \ndetected AGN in the CDF-S is 1.63$\\times$10$^{47}$~erg~cm$^{-2}$~s$^{-1}$~deg$^{-2}$. Hence, this extra AGN activity can account for $\\sim$22\\% of the total SMBH accretion.\nAdding this to the obscured SMBH growth in X-ray detected AGN \\citep{luo08}, we confirm that most SMBH growth, $\\sim$70\\%, is significantly obscured and missed by even the \ndeepest X-ray surveys \\citep{treister04,treister10}.\n\nPerforming a similar study on the 28 sources with $f_{24}$\/$f_R$$>$1000 and $R$-$K$$>$4.5 that we previously excluded, we find a very hard X-ray\nspectrum, harder than that of the $L_{IR}$$>$10$^{11}$$L_\\odot$ sources. This spectrum is consistent with a population of luminous AGN with intrinsic rest-frame 2-10 keV \nluminosity $\\sim$2$\\times$10$^{43}$~erg~s$^{-1}$ and negligible contribution from the host galaxy, except at E$<$2 keV where the thermal component is $\\sim$30\\% of the\ntotal emission. This result justifies our choice of removing these sources from our study (otherwise they would dominate the stacked signal), while at the same time it confirms the\nAGN nature of the vast majority of these sources, in contrast to the suggestion that the extra IR emission could be due to star-formation processes \\citep{donley08, pope08,georgakakis10}. \nA similar result for these high-luminosity sources was found by \\citet{fiore10}: In a sample of 99 mid-IR excess sources in the COSMOS field he found a strong stacked signal \nat E$\\sim$6~keV, which he interpreted as due to the Fe~K$\\alpha$ line, a clear signature of AGN emission and high obscuration (see discussion below).\n\n\\subsection{Multiwavelength Properties}\n\nBy design, none of the sources in our sample are individually detected in X-rays, nor do they satisfy the selection criteria of \\citet{fiore08}. However, it is interesting to investigate\nif they present other AGN signatures. For example, 237 out of the 1545 sources with $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ in our sample (15\\%) are found inside the\nAGN IRAC color-color region defined by \\citet{stern05}. For comparison, in the sample of 2342 sources with 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ --- \nin which from the stacked hard X-ray signal we determined a negligible AGN fraction --- there are 327 galaxies (14\\%) in the \\citet{stern05} region. This suggests that the \nIRAC color-color diagram cannot be used to identify heavily-obscured low-luminosity AGN, because the near-IR emission in these sources is dominated by the host \ngalaxy \\citep{cardamone08}. At longer wavelengths, 83 of the 1545 sources with $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ were detected in the deep VLA observations of the \nCDF-S \\citep{kellermann08}. In contrast, only 33 sources in the 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ sample were detected in these \nobservations. Using the $q_{24}$ ratio between 1.4 GHz and 24~$\\mu$m flux densities (e.g., \\citealp{appleton04}), we find that in the $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$ \nsample, only 14 sources have $q_{24}$$<$-0.23 and can be considered ``radio-loud'' \\citep{ibar08}, and in the 5$\\times$10$^{10}$$L_\\odot$$>$$L_{IR}$$>$10$^{10}$$L_\\odot$ \nsample only 10 sources have $q_{24}$$<$-0.23. Hence, we conclude that the fraction of bona fide radio-loud sources is negligible and that in most cases the radio emission \nis produced by star-formation processes.\n\n\\subsection{AGN Fraction Versus Stellar Mass}\n\nIn order to investigate the fraction of heavily-obscured AGN as a function of other galaxy parameters, we performed X-ray stacking of samples sorted by stellar mass.\nStellar masses were taken from \\citet{cardamone10}, who performed spectral fitting to the extensive optical and near-IR spectro-photometry using FAST \\citep{kriek09} and the stellar \ntemplates of \\citet{maraston05} assuming the \\citet{kroupa01} initial mass function and solar metallicity. We further restricted our sample to sources with $z$$<$1.2, for \nwhich photometric redshifts and stellar masses are very well determined ($\\Delta$z\/(1+$z$)=0.007). We then divided the sample into three mass bins: M$>$10$^{11}$M$_\\odot$, \n10$^{11}$$>$M (M$_\\odot$)$>$10$^{10}$ and 10$^{10}$$>$M (M$_\\odot$)$>$10$^{9}$. The resulting stacked X-ray spectra are shown in Fig.~\\ref{obs_spec_mass}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.2]{f3a.eps}\n\\includegraphics[scale=0.2]{f3b.eps}\n\\end{center}\n\\caption{Stacked Chandra counts for galaxies binned as a function of their stellar mass. The {\\it left panel} shows the spectra for the following\nbins: M$>$10$^{11}$M$_\\odot$ ({\\it red squares}), 10$^{10}$$<$M$<$10$^{11}$M$_\\odot$ ({\\it blue triangles}) and 10$^{9}$$<$M$<$10$^{10}$M$_\\odot$ ({\\it black circles}).\n{\\it Right panel:} same but normalized at 1 keV. In the M$>$10$^{11}$M$_\\odot$ sample, the strong excess at E=6-7~keV, which we associate with the Fe~K$\\alpha$ line, is an \nindicator of AGN activity. Similarly, for the sources with 10$^{10}$$<$M$<$10$^{11}$M$_\\odot$ there is a hard X-ray spectrum, also suggesting a significant AGN fraction.\nThese preliminary results indicate that these heavily-obscured moderate-luminosity AGN are predominantly present in the most massive galaxies.}\n\\label{obs_spec_mass}\n\\end{figure}\n\n\nFor sources with M$>$10$^{11}$M$_\\odot$, there is a significant excess at 6-7~keV, above a spectrum that otherwise declines with increasing energy. This might \nbe due to the presence of the Fe~K$\\alpha$ line, a clear indicator of AGN activity.  Contrary to the case of stacking as a function of IR luminosity (Fig.~\\ref{simul_spec}), \nhere we do not find evidence for an absorbed power-law ---the 6-7 keV feature is simply too sharply peaked. Possibly the restriction to $z$$<$1.2 for the mass-binned stacking, where \nphotometric redshifts are most accurate, reveals an emission line that is broadened by less accurate photometric redshifts in the full sample. That is, the feature in the $L_{IR}$-binned stack \nthat we interpreted as a heavily absorbed power law may instead be an Fe~K$\\alpha$ line broadened artificially by bad photometric redshifts. In the 10$^{11}$$>$M (M$_\\odot$)$>$10$^{10}$ \nsample we found a significant hardening of the X-ray spectrum (Fig.~\\ref{obs_spec_mass}), suggesting the presence of a significant fraction of AGN. In contrast, only a soft spectrum, \nconsistent with star-formation emission, can be seen for sources with 10$^{10}$$>$M (M$_\\odot$)$>$10$^{9}$. Taken together, these results indicate that AGN are predominantly present \nin the most massive galaxies, in agreement with the conclusions of \\citet{cardamone10b} and others. This will be elaborated in a paper currently in preparation.\n\n\\subsection{Space Density of Heavily-Obscured AGN}\n\nThe fraction of Compton-thick AGN in the local Universe is still heavily debated. \\citet{treister09b} reported a fraction of $\\sim$8\\% in a flux-limited sample of sources\ndetected in the Swift\/BAT all-sky \\citep{tueller08} and  International Gamma-Ray Astrophysics Laboratory (INTEGRAL; \\citealp{krivonos07}) surveys. From an INTEGRAL volume-limited \nsurvey at $z$$<$0.015, \\citet{malizia09} found a higher fraction of 24\\%, suggesting that even surveys at E$>$10~keV are potentially \nbiased against the detection of Compton-thick AGN. The fraction of moderate-luminosity Compton-thick sources in our sample of sources \nwith $L_{IR}$$>$5$\\times$10$^{10}$$L_\\odot$, relative to all AGN in the CDF-S, is $\\sim$25\\% (132\/525), assuming that Compton-thick and Compton-thin AGN have similar \nmedian  intrinsic luminosities. This indicates that there is no major evolution in the number of moderate-luminosity heavily-obscured AGN from $z$=0 to 2. In contrast, at higher \nluminosities, \\citet{treister10} reported that the ratio of obscured to unobscured quasars increased from $\\sim$1 at $z$=0 to $\\sim$2-3 at $z$$\\simeq$2. Hence, although all these \nestimates are still uncertain, it appears that the evolution of Compton-thick AGN depends strongly on their luminosity. We further speculate that this is indication that the \ntriggering of low-luminosity AGN is not related to the major merger of gas-rich galaxies as found by \\citet{treister10} for high-luminosity quasars or that the time delay\nbetween galaxy interactions and black hole growth is long \\citep{schawinski10b}.\n\n\\acknowledgements\n\nWe thank the referee, Fabrizio Fiore, for very useful and constructive comments. Support for the work of ET and KS was provided by the National\nAeronautics and Space Administration through Chandra\/Einstein\nPost-doctoral Fellowship Award Numbers PF8-90055 and PF9-00069\nrespectively issued by the Chandra X-ray Observatory Center, which is\noperated by the Smithsonian Astrophysical Observatory for and on\nbehalf of the National Aeronautics Space Administration under contract\nNAS8-03060. CMU and CC acknowledge support from NSF grants \nAST-0407295, AST-0449678, AST-0807570 and Yale University.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the rapid development of Internet of Things, the smart in-vehicle applications (i.e., autonomous driving, image assisted navigation and multimedia entertainment) have been widely applied to smart vehicles \\cite{Feng}\\cite{Zhang}, which can provide more comfortable and safer environment for drivers and passengers. Since these in-vehicle applications will consume huge computation resource and require low execution latency, the cloud server is employed to afford these complex computing tasks, which results in serious burden on backhaul network \\cite{Huang0}. Luckily, vehicular edge computing (VEC) with powerful computing capacity is deployed in the roadside densely and attached in the road side unit (RSU). Therefore, it is worthy to study how to efficiently utilize the computing capacity of VEC server to support low latency and energy-efficiency in-vehicle service. \n\nTo improve the offloading efficiency of vehicle terminals, the researchers have proposed many offloading and resource allocation methods in the VEC-based network. Zhang et al. \\cite{Mao} proposed a hierarchical cloud-based VEC offloading framework to reduce the execution delay, where a back server was utilized to offer extra computation resource for the VEC server. However, the priority of tasks were not considered in the offloading process, which may cause that delay sensitive tasks (i.e., assisted imaged navigation) cannot be processed timely for high-speed vehicles. To further reduce the task execution delay, Liu et al. \\cite{Liu} studied the task offloading problem by treating the vehicles and the RSUs as the two sides of a matching problem to minimize the task execution delay. But the computation capacity of VEC server should be fully exploited to decrease the energy consumption of in-vehicle applications. In addition, Li et al. \\cite{Li} considered the influence of time-varying channel on the task offloading strategies and formulated the problem of joint radio and computation resource allocation. However, the variety of vehicle speeds was not considered, which may cause that the high-speed vehicles cannot obtain the sufficient computation and wireless resource. To accelerate the radio and computation allocation process, Dai et al. \\cite{Dai} proposed a low-complexity algorithm to jointly optimize server selection, offloading ratio and computation resource, but the task handover between VEC servers may result in the increase in delay and occurrence of lost packets. To make a more accurate offloading decision, Sun et al. \\cite{Sun} proposed the task offloading algorithm by determining where the tasks were performed and the execution order of the tasks on the VEC server, but the proposed heuristic algorithm may not obtain a satisfying result when the vehicle's channel state fluctuated frequently. Considering the fluctuation of channel state, Zhan et al. \\cite{Zhan} proposed a deep reinforcement learning-based offloading algorithm to make the task offloading decision, which can be applied in the dynamic environment caused by vehicle's mobility. \n\nIn general, there are still some problems to be solved for the computing task offloading and resource allocation for the in-vehicle applications: (1) The impact of vehicle speed on task delay constraints have been neglected, which results in mismatching between vehicle speed and its allocated computation and wireless resource and cannot guarantee the delay requirements. (2) The vast fluctuation of vehicle's channel state caused by fast mobility has not been considered, which may result in offloading failure. (3) The computation capacity of VEC server has not been fully exploited because of the inaccurate resource allocation strategy. Inspired by the above work, we propose a vehicle speed aware computing task offloading and resource allocation strategy based on multi-agent reinforcement learning. Our work is novel in the following aspects.\n\n\\textbf{(1)Vehicle speed aware delay constraint model:} Different types of tasks and vehicle speeds demand various delay requirements for in-vehicle applications. Therefore, we fully analyze the internal relationship among vehicle speed, task type and delay requirements to propose a vehicle speed aware delay constraint model, which can make the task offloading and resource allocation process more accurately. \n\n\\textbf{(2)The calculation of energy consumption and delay for different types of tasks:} Based on the bandwidth and delay requirements, in-vehicle computing tasks are classified into three types: critic application, high-priority application and low-priority application. For different types of tasks, we calculate the energy consumption and delay based on task transmission and execution process for different offloading positions, respectively.\n\n\\textbf{(3)Multi-agent reinforcement learning based solution to our formulated problem:} A joint optimization of offloading and resource allocation problem is formulated by a Markov decision process (MDP) process, with the objective to minimize the energy consumption subject to delay constraint. In addition, multi-agent reinforcement learning is applied to solve the high-dimension action decision-making.\n\n\n\\section{System Model}\n\\subsection{System Framework}\nThe scenario considered in this paper is the computing task offloading of vehicles on urban roads in VEC network, as shown in Figure \\ref{system}. RSU is located along the roadside, and the coverage areas of adjacent RSUs do not overlap. Therefore, according to the coverage areas of RSUs, the road can be divided into several adjacent segments, where the vehicle can only establish the wireless link with the RSU of the current road segment. Each RSU is equipped with a VEC server whose powerful computation capacity can help the vehicle quickly handle computing tasks. Since the delay constraint of the computing task is extremely short, it can be assumed that the vehicle can still receive the task processing result from the previous VEC server when the vehicles travels to the road segment horizon. In addition, the computing task can also be executed locally to alleviate the traffic and computing burden of VEC server.\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=8.5cm]{figure-2.eps}\n\t\\centering\n\t\\caption{The architecture of computing task offloading in VEC network}\n\t\\label{system}\n\\end{figure}\n\nBecause the VEC server can real-time perceive the state information of vehicles and owns powerful computing processing capacity, so it can generate the optimal computing task offloading and resource allocation strategy (including computing resource allocation and wireless resource allocation) for each vehicle. First, the state information of the vehicle, including task queue, speed, location, remaining computing capacity and wireless resources, is reported to the RSU in real time. The RSU will forward the received state information to the VEC server, which utilizes the extracted state information to calculates the task execution delay and energy consumption to formulate the optimized problem. The goal is to reduce the energy consumption of all vehicles by optimizing task offloading and resource allocation decisions without exceeding delay constraint. Then, according to the results of the joint task offloading and resource allocation, the vehicle's computing tasks can be executed locally or offloaded to the VEC server.\n\n\\subsection{Vehicle Speed Aware Delay Constraint Model}\n\nVehicle' computing tasks can be divided into three types: critical application (CA), high-priority application (HPA) and low-priority application (LPA), which needs different bandwidth and delay requirements \\cite{Dzi}. We denote these three task types by $\\phi_1$, $\\phi_2$ and $\\phi_3$, respectively. CA task generally refers to the autonomous driving and road safety tasks, which needs ultra delay to ensure the safety of vehicle driving. Therefore, this type of task needs to be executed locally and delay threshold is set to $Th{{r}_{1}}$. For the HPA task, it mainly involves image assisted navigation, parking navigation and some optional security applications. The delay tolerance for HPA task is related to the current vehicle speed. For vehicles with low speed, it doesn't matter even if the computing task takes a little longer time. More wireless and computation resource can be allocated to vehicles with high speed, whose tasks can be processed preferentially. The delay threshold of HPA task is set to $Th{{r}_{2}}$ when the vehicle speed reaches at the maximum road speed limit ${{v}_{\\max }}$. LPA tasks generally includes multimedia and passenger entertainment activities, so the requirement for delay threshold is relatively slack. The delay threshold is set to $Th{{r}_{3}}$.\n\nIn this paper, it is assumed that the generated computing tasks are independent task sequence. The computing task of the vehicle $k$ at time $t$ to be processed is defined as ${{\\mathcal{I}}_{t}}(k)$. For HPA task, when the vehicle's speed is low, the delay threshold of the computing task can be relatively longer. With the increase of speed, the information of the vehicle received from surrounding environment will increase rapidly at the same delay because of the longer distance traveled. Therefore, the delay of the computing task that the vehicle needs to be processed should be reduced rapidly. When the speed reaches a higher level, with the increase of speed, the increase amplitude of the vehicle's information received from the surrounding environment will gradually decrease. Therefore, the delay of the computing task that the vehicle needs to be processed will be reduced slowly.\n\nTherefore, we select one-tailed normal function to describe the relationship between delay constraint, $\\mathcal{T}({{v}_{\\mathcal{I}_t(k)}})$, and speed for task $\\mathcal{I}_t(k)$ of $\\phi _2$, as follows:\n\\begin{equation}\\small\n\\begin{aligned}\n\\mathcal{T}({{v}_{\\mathcal{I}_t(k)}})&=Th{{r}_{2}}\\frac{1}{\\sqrt{2\\pi }\\alpha }\\exp (-\\frac{v_{{{\\mathcal{I}}_{t}}(k)}^{2}}{2{{\\alpha }^{2}}})\/(\\frac{1}{\\sqrt{2\\pi }\\alpha }\\exp (-\\frac{v_{\\max }^{2}}{2{{\\alpha }^{2}}})) \\\\ \n& =\\exp (-\\frac{v_{{{\\mathcal{I}}_{t}}(k)}^{2}-v_{\\max }^{2}}{2{{\\alpha }^{2}}})Th{{r}_{2}},\\text{ }if\\text{  }{{\\mathcal{I}}_{t}}\\text{(}k\\text{)}\\in \\phi _2 \\\\ \n\\end{aligned}\n\\end{equation}\nwhere ${{v}_{\\mathcal{I}_t(k)}}$ is the current vehicle speed and ${{\\alpha }^{2}}$ is the variance of the normal function. $v_{max}$ is the road maximum speed. To ensure that the probability that vehicle speed is within the maximum speed exceeds 95\\%, we denote $\\alpha ={{v}_{\\max }}\/1.96$. Therefore, We employ $\\Upsilon({{v}_{\\mathcal{I}_t(k)}})$ to represent the delay threshold of computing task ${{\\mathcal{I}}_{t}}(k)$ of all task types, as follows:\n\\begin{equation}\n\\begin{aligned}\n& \\Upsilon ({{\\mathcal{I}}_{t}}\\text{(}k\\text{)})=\\left\\{ \\begin{aligned}\n& Th{{r}_{1}},\\text{ }if\\text{  }{{\\mathcal{I}}_{t}}\\text{(}k\\text{)}\\in {{\\phi }_{1}} \\\\ \n& \\mathcal{T}\\text{(}{{v}_{{{\\mathcal{I}}_{t}}\\text{(}k\\text{)}}}\\text{), }if\\text{  }{{\\mathcal{I}}_{t}}\\text{(}k\\text{)}\\in {{\\phi }_{2}} \\\\ \n& Th{{r}_{3}},\\text{ }if\\text{  }{{\\mathcal{I}}_{t}}\\text{(}k\\text{)}\\in {{\\phi }_{3}} \\\\ \n\\end{aligned} \\right. \\\\ \n& \\text{           } \\\\ \n\\end{aligned}\n\\end{equation}\n\n\\section{Delay and Energy Consumption of Different Offloading Positions}\nFor the generated computing task, the task handover between VEC server is generally not considered \\cite{Zhang} to ensure its successful transmission. For HPA and LPA computing tasks generated by the in-vehicle applications at time $t$, there are usually three ways to handle them, which are hold on, offloading to VEC server and local execution, as shown in Figure \\ref{delay}.\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=8.5cm]{figure-1.eps}\n\t\\centering\n\t\\caption{Delay and energy consumption of different offloading pcositions}\n\t\\label{delay}\n\\end{figure}\nWhen the vehicle's current remaining computing resource and wireless resource and VEC server's remaining computing resource are insufficient to process the new computing task, the vehicle's computing task can choose to wait for a certain time until computing and wireless resources are released. \n\\subsection{Delay of task execution}\n\\subsubsection{Offloading to the VEC server}\nFor the local VEC server, we denote the set of vehicles in the service area by $\\mathbb{Q}$ and the number of these vehicles is $K$. When the computing task ${{\\mathcal{I}}_{t}}(k)$ belonging to task type $\\phi_2$ or $\\phi_3$ is offloaded to VEC server, the task completion time contains upload time, execution time and download time. For the task type $\\phi_2$, since the size of output file is much smaller than that of input file, the download time can be ignored. Considering that task upload, execution and download cannot be executed simultaneously within one transport time interval (TTI), the consumed time, $TR_{{{\\mathcal{I}}_{t}}(k)}$, needs to round up, which can be described as: \n\\begin{equation}\\small\nTR_{{{\\mathcal{I}}_{t}}(k)}^{{}}=\\left\\{ \\begin{aligned}\n& \\left\\lceil \\frac{{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,up}} \\right\\rceil +\\left\\lceil \\frac{{{\\kappa }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{b_{{{\\mathcal{I}}_{t}}(k)}^{VEC}{{f}^{VEC}}} \\right\\rceil ,\\text{  }if\\text{ }{{\\mathcal{I}}_{t}}(k)\\in \\phi_2 \\\\ \n& \\left\\lceil \\frac{{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,up}} \\right\\rceil +\\left\\lceil \\frac{{{\\kappa }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{b_{{{\\mathcal{I}}_{t}}(k)}^{VEC}{{f}^{VEC}}} \\right\\rceil +\\left\\lceil \\frac{{{\\omega }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,down}} \\right\\rceil ,\\\\&\\text{ }if\\text{ }{{\\mathcal{I}}_{t}}(k)\\in \\phi_3 \\\\ \n\\end{aligned} \\right.\n\\end{equation}\nwhere ${{c}_{{{\\mathcal{I}}_{t}}(k)}}$ is the file size of task ${{\\mathcal{I}}_{t}}(k)$ and ${{\\kappa }_{{{\\mathcal{I}}_{t}}(k)}}$ is the calculation density of processing the task ${{\\mathcal{I}}_{t}}(k)$. ${{\\omega }_{{{\\mathcal{I}}_{t}}(k)}}$ is the scaling ratio of the downloaded task size relative to the uploaded task size. $b_{{{\\mathcal{I}}_{t}}(k)}^{VEC}$ indicates the proportion of computing resource allocated by VEC server to the task ${{\\mathcal{I}}_{t}}(k)$. ${{f}^{VEC}}$ denotes the CPU frequency of VEC server. The transmission capacity between vehicle and server can be obtained from the number of allocated channels, channel bandwidth, transmission power and noise power \\cite{Huang}. For uplink channel $n$ of VEC server allocated to vehicle $k$, the available uplink transmission capacity, $r_{k,n}^{VEC,up}$, can be expressed as\n\\begin{equation}\\small\nr_{k,n}^{VEC,up}=\\omega _{VEC}^{up}{{\\log }_{2}}\\left( 1+\\frac{P\\cdot h_{k,n}^{VEC,up}}{{{\\sigma }^{2}}+I_{k,n}^{VEC,up}} \\right),\\text{  }for\\text{  }n\\in \\mathbb{N}_{VEC}^{up}\n\\end{equation}\nwhere $\\omega _{VEC}^{up}=B_{VEC}^{up}\/N_{VEC}^{up}$. $B_{VEC}^{up}$ is the uplink bandwidth of VEC server and $N_{VEC}^{up}$ is the number of total channels of VEC server. $\\sigma^2$ denotes noise power and $P$ is transmission power. $I_{k,n}^{VEC,up}$ denotes the interference on channel $n$. $\\mathbb{N}_{VEC}^{up}$ indicates the uplink channel set of VEC server. Let $z_{k,n}^{VEC,up}$ indicate whether the uplink channel $n$ is allocated to the vehicle $k$. If it is allocated, $z_{k,n}^{VEC,up} = 1$, otherwise, $z_{k,n}^{VEC,up} = 0$. Then the uplink transmission capacity between vehicle $k$ and VEC server, $r_{k}^{VEC,up}$, can be depicted as\n\\begin{equation}\nr_{k}^{VEC,up}=\\sum\\limits_{n\\in \\mathbb{N}_{VEC}^{up}}{z_{k,n}^{VEC,up}r_{k,n}^{VEC,up}}\n\\end{equation}\n\nSimilarly, the transmission capacity of downlink channel $n$ of VEC server allocated to vehicle $k$, $r_{k,n}^{VEC,down}$, can be expressed as\n\\begin{equation}\n\\begin{aligned}\nr_{k,n}^{VEC,down}&=\\omega _{VEC}^{down}{{\\log }_{2}}\\left( 1+\\frac{P\\cdot h_{k,n}^{VEC,down}}{{{\\sigma }^{2}}+I_{k,n}^{VEC,down}} \\right),\\\\& \\text{  }for\\text{  }n\\in \\mathbb{N}_{VEC}^{down}\n\\end{aligned}\n\\end{equation}\nwhere $\\omega _{VEC}^{down}=B_{VEC}^{down}\/N_{VEC}^{down}$. Then the downlink transmission capacity between vehicle $k$ and VEC server, $r_{k}^{VEC,down}$, can be depicted as\n\\begin{equation}\nr_{k}^{VEC,down}=\\sum\\limits_{n\\in {{\\mathbb{N}_{VEC}^{down}}}}{z_{k,n}^{VEC,down}r_{k,n}^{VEC,down}}\n\\end{equation}\n\n\\subsubsection{Local Execution}\nFor computing task ${{\\mathcal{I}}_{t}}(k)$ executed locally, the consumed time, $TL_{{{\\mathcal{I}}_{t}}(k)}$, can be expressed as:\n\\begin{equation}\nTL_{{{\\mathcal{I}}_{t}}(k)}^{{}}=\\left\\lceil \\frac{{{\\kappa }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{{{f}^{k}}} \\right\\rceil \n\\end{equation}\nwhere ${{f}^{k}}$ is the CPU frequency of vehicle $k$. At time $t$, computing task ${{\\mathcal{I}}_{t}}(k)$ can select to hold on, be offloaded to the local VEC server and be executed locally. Therefore, for computing task ${{\\mathcal{I}}_{t}}(k)$, the total delay from generation to completion, $D({{\\mathcal{I}}_{t}}(k))$, is derived by\n\\begin{equation}\\small\n\\begin{aligned}\nD({{\\mathcal{I}}_{t}}(k))&=t-t_{{{\\mathcal{I}}_{t}}(k)}^{g}+\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold}{{T}_{h}}\\\\&+(1-\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold})[\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC}TR_{{{\\mathcal{I}}_{t}}(k)}^{{}}+(1-\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC})T{{L}_{{{\\mathcal{I}}_{t}}(k)}}]\n\\end{aligned}\n\\end{equation}\nwhere $t_{k,i}^{g}$ is the generated time of task ${{\\mathcal{I}}_{t}}(k)$.  $\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold}$ indicates whether task ${{\\mathcal{I}}_{t}}(k)$ holds on. If it holds on, ${{\\mathcal{I}}_{t}}(k) = 1$, otherwise, ${{\\mathcal{I}}_{t}}(k) = 0$. ${{T}_{h}}$ denotes the waiting time. $\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC}$ indicates whether task ${{\\mathcal{I}}_{t}}(k)$ is offloaded to VEC server. If it is allocated, $\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC} = 1$, otherwise, $\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC} = 0$.\n\n\\subsection{Energy Consumption of Task Execution}\n\\subsubsection{Offloading to VEC server}\nWhen the computing task is offloaded to the VEC server, the energy consumption origins from uploading and downloading computing task. For the task type $\\phi_2$, since the size of output file is much smaller than that of input file, the energy consumed by downloading computing task can be ignored. Therefore, the energy consumption of task $\\mathcal{I}_{t}(k)$ belonging to $\\phi_2$ or $\\phi_3$ offloaded to VEC server, $ER_{{{\\mathcal{I}}_{t}}(k)}$, can be depicted as\n\\begin{equation}\nER_{{{\\mathcal{I}}_{t}}(k)}^{{}}=\\left\\{ \\begin{aligned}\n& P\\frac{{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,up}},\\text{  }if\\text{  }{{\\mathcal{I}}_{t}}(k)\\in \\phi_2 \\\\ \n& P(\\frac{{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,up}}\\text{+}\\frac{{{\\omega }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}}{r_{k}^{VEC,down}}),\\text{  }if\\text{  }{{\\mathcal{I}}_{t}}(k)\\in \\phi_3 \\\\ \n\\end{aligned} \\right.\n\\end{equation}\n\n\\subsubsection{Local Execution}\nWhen computing task ${{\\mathcal{I}}_{t}}(k)$ is executed locally, the consumed energy can be calculated according to the assigned computation resource, $EL_{{{\\mathcal{I}}_{t}}(k)}$, which can be expressed as\n\\begin{equation}\nEL_{{{\\mathcal{I}}_{t}}(k)}^{{}}={{\\xi }_{{{\\mathcal{I}}_{t}}(k)}}{{\\kappa }_{{{\\mathcal{I}}_{t}}(k)}}{{c}_{{{\\mathcal{I}}_{t}}(k)}}{{({{f}^{k}})}^{2}}\n\\end{equation}\nwhere ${\\xi }_{\\mathcal{I}_{t}(k)}$ is the energy density of processing task ${{\\mathcal{I}}_{t}}(k)$ \\cite{Din}.\nAccording to the different offloading positions of computing task ${{\\mathcal{I}}_{t}}(k)$, including VEC server and local device, the consumed energy of all vehicles served by local VEC server, $E(t)$, can be derived by\n\\begin{equation}\nE(t)=\\sum\\limits_{k\\in \\mathbb{Q}}{(1-\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold})[\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC}E{{R}_{{{\\mathcal{I}}_{t}}(k)}}}+(1-\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC})E{{L}_{{{\\mathcal{I}}_{t}}(k)}}]\n\\end{equation}\n\n\\section{Delay and Energy-Efficiency Driven Computing Task Offloading and Resource Allocation Algorithm Based on Multi-Agent Reinforcement Learning}\n\\subsection{Problem Formulation}\nWe formulate the optimized problem of reducing the energy consumption of each vehicle without exceeding delay constraint by carrying out optimal computing task offloading and resource allocation strategy, which can be described as follows:\n\\begin{equation}\\label{problem}\n\\begin{aligned}\n& \\underset{{{X}_{t}}(k,n),\\forall k,n}{\\mathop{\\min }}\\,\\sum\\limits_{t=1}^{T}{E(t)} \\\\ \n& s.t. \\\\ \n& (c1)\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC}+\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold}\\le 1,\\forall k\\in \\mathbb{Q} \\\\ \n& (c2)\\sum\\limits_{k}{b_{{{\\mathcal{I}}_{t}}(k)}^{VEC}}\\le 1,\\forall k\\in \\mathbb{Q} \\\\ \n& (c3)\\sum\\limits_{k}{z_{k,n}^{VEC,up}}\\le 1,\\forall k\\in \\mathbb{Q},n\\in \\mathbb{N}_{VEC}^{up} \\\\ \n& (c4)\\sum\\limits_{k}{z_{k,n}^{VEC,down}}\\le 1,\\forall k\\in \\mathbb{Q},n\\in \\mathbb{N}_{VEC}^{down} \\\\ \n& (c5)D({{\\mathcal{I}}_{t(k)}})\\le \\Upsilon({{v}_{\\mathcal{I}_t(k)}}),\\forall k\\in \\mathbb{Q} \\\\ \n\\end{aligned}\n\\end{equation}\nwhere ${{X}_{t}}(k,n)=(\\tau _{{{\\mathcal{I}}_{t}}(k)}^{VEC},\\tau _{{{\\mathcal{I}}_{t}}(k)}^{Hold},z_{k,n}^{VEC,up},z_{k,n}^{VEC,down})$. Constraint (c1) implies that computing task ${{\\mathcal{I}}_{t(k)}}$ cannot be offloaded to the local VEC server, executed locally and hold on simultaneously. Constraint(c2) indicates that computation capacity allocated to computing task ${{\\mathcal{I}}_{t(k)}}$ by VEC server cannot exceed its own computing capacity. Constraint (c3) and constraint (c4) indicate that each channel must be assigned to one and only one vehicle at each scheduling period. Constraint (c5) indicates that computing task ${{\\mathcal{I}}_{t(k)}}$ should be completed within the delay constraint.\n\n\\subsection{Deep Reinforcement Learning-Based Solution Method}\nEquation (\\ref{problem}) is a multi-vehicle cooperation and competition problem, which is obviously a NP-hard problem. Therefore, we employ the deep reinforcement learning method to solve the proposed computing task offloading and resource allocation problem. First, we formulate our problem as a MDP to accurately describe the offloading and resource allocation decision process and utilize the multi-agent deep deterministic policy gradient (MADDPG) \\cite{Lowe} to find the optimal policy for the MDP. In what follows, we will present the elements of MDP, including state space, action space and reward function.\n\\subsubsection{State Space}\nWe defined the state space of vehicle $k$ as ${{s}_{k}}(t)$, including the state information of vehicle $k$, other vehicles and VEC server, which is depicted as\n\\begin{equation}\\small\n\\begin{aligned}\n{{s}_{k}}(t)&=[{{v}_{1}}(t),...,{{v}_{K}}(t),{{d}_{1}}(t),...,{{d}_{K}}(t),{{c}_{1}}(t),...,{{c}_{K}}(t), \\\\&r{{b}_{VEC}}(t),\ns\\tau _{k}^{Hold}(t),s\\tau _{k}^{VEC}(t),sb_{k}^{VEC}(t), sz_{1}^{VEC,up}(t),\\\\&...,sz_{N_{VEC}^{up}}^{VEC,up}(t),sz_{1}^{VEC,down}(t),...,sz_{N_{VEC}^{down}}^{VEC,down}(t)] \\\\ \n\\end{aligned}\n\\end{equation}\nwhere ${{v}_{k}}(t),{{d}_{k}}(t),{{c}_{k}}(t)$ are vehicle speed, position and file size to be processed of vehicle $k$ at time $t$, respectively. $r{{b}_{VEC}}(t)$ is the current remaining computation capacity of VEC server at time $t$. $s\\tau _{k}^{(\\centerdot )}(t)$ indicates whether the vehicle $k$ selects the offloading position $(\\centerdot)$ at time $t$. If it is selected, $s\\tau _{k}^{(\\centerdot )}(t) = 1$, otherwise,  $s\\tau _{k}^{(\\centerdot )}(t)$ = 0. $sb_{k}^{VEC}(t)$is the ratio of computation resource allocated by VEC server to vehicle $k$ at time $t$. $sz_{1}^{VEC,up}(t),...,sz_{N_{VEC}^{up}}^{VEC,up}(t)$ indicates whether the uplink channel resource of VEC is available at time $t$. If it is available, the value is 1, otherwise, the value is 0. $sz_{1}^{VEC,down},...,sz_{N_{VEC}^{down}}^{VEC,down}(t)$ indicates whether the downlink channel resource of VEC is available at time $t$. If it is available, the value is 1, otherwise, the value is 0. Therefore, the state space of the system can be defined as: ${{S}_{t}}=({{s}_{1}}(t),...{{s}_{k}}(t)...,{{s}_{K}}(t))$.\n\n\\subsubsection{Action Space}\nSince vehicle $k$ cannot offload multiple tasks simultaneously at time $t$, which means that task ${{\\mathcal{I}}_{t(k)}}$ and vehicle $k$ have a one-to-one correspondence. Therefore, We can represent the action space of task ${{\\mathcal{I}}_{t(k)}}$ with the one of vehicle $k$. For vehicle $k$, the action space, $a_k(t)$, contains whether to hold on and offload task to VEC server, computation resource allocated by VEC server and the uplink and downlink channels allocated by VEC server, which can be expressed as\n\\begin{equation}\\small\n\\begin{aligned}\na_{k}^{{}}(t)&=[\\tau _{k}^{Hold}(t),\\tau _{k}^{VEC}(t),b_{k}^{VEC}(t),z_{k,1}^{VEC,up}(t),...,z_{k,N_{VEC}^{up}}^{VEC,up}(t),\\\\&z_{k,1}^{VEC,down}(t),...,z_{k,N_{VEC}^{down}}^{VEC,down}(t)]\n\\end{aligned}\n\\end{equation}\nTherefore, the action space of the system can be defined as: ${{A}_{t}}=\\{{{a}_{1}}(t),...{{a}_{k}}(t)...,{{a}_{K}}(t)\\}$.\n\n\\subsubsection{Reward}\nThe goal of this paper is to reduce the energy consumption of each vehicle terminal without exceeding task delay constraint, which can be realized by allocating the computation resource and wireless resource of the system. Therefore, we set rewards based on constraint conditions and objective function to accelerate the training speed. After taking action ${{a}_{k}}(t)$, if the state of vehicle $k$ does not satisfy the constraints (c1)-(c4), the reward function can be defined as \n\\begin{equation}\\small\n\\begin{aligned}\n& {{r}_{k}}(t)={{\\ell }_{1}}+{{\\Gamma }_{1}}\\cdot (s\\tau _{k}^{VEC}+s\\tau _{k}^{Hold}-1)\\cdot {{\\Lambda }_{(s\\tau _{k}^{VEC}+s\\tau _{k}^{Hold}\\le 1)}}\\\\&+{{\\Gamma }_{2}}\\cdot (\\sum\\limits_{k}{sb_{k}^{VEC}}-1)\\cdot {{\\Lambda }_{(\\sum\\limits_{k}{sb_{k}^{VEC}}\\le 1)}}\\\\& \n+{{\\Gamma }_{3}}\\cdot (\\sum\\limits_{k}{sz_{k,n}^{VEC,up}}-1)\\cdot {{\\Lambda }_{(\\sum\\limits_{k}{sz_{k,n}^{VEC,up}}\\le 1)}}\\\\&+{{\\Gamma }_{4}}\\cdot (\\sum\\limits_{k}{sz_{k,n}^{VEC,down}}-1)\\cdot {{\\Lambda }_{(\\sum\\limits_{k}{sz_{k,n}^{VEC,down}}\\le 1)}} \\\\ \n\\end{aligned}\n\\end{equation}\nwhere ${{\\Lambda }_{(\\centerdot )}}$ indicates that if the condition $(\\centerdot)$ is not satisfied, the value is -1, otherwise, the value is 0. ${{\\ell }_{1}},{{\\Gamma }_{1}},{{\\Gamma }_{2}},{{\\Gamma }_{3}},{{\\Gamma }_{4}}$ is experimental parameters. After taking action ${{a}_{k}}(t)$, if the state of vehicle $k$ satisfy all constraints (c1)-(c4), the reward function can be defined as \n\\begin{equation}\n{{r}_{k}}(t)={{\\ell }_{2}}+\\exp (Th{{r}_{k}}(t)-\\Upsilon({{v}_{\\mathcal{I}_t(k)}}))\n\\end{equation}\nwhere ${{\\ell }_{2}}$ is experimental parameters. After taking action ${{a}_{k}}(t)$, if the state of vehicle $k$ satisfy all constraints (c1)-(c5), the reward function can be defined as \n\\begin{equation}\nr(t)={{\\ell }_{3}}\\text{+}{{\\Gamma }_{5}}\\cdot \\exp ({{E}_{k}}(t))\n\\end{equation}\nwhere ${{\\ell }_{3}},{{\\Gamma }_{5}}$ denote experimental parameters.\n\\subsubsection{Joint Delay and Energy-Efficiency Algorithm Based on MADDPG}\nThe centralized training process is composed of $K$ agents, whose network parameter are $\\theta =\\{{{\\theta }_{1}},...,{{\\theta }_{K}}\\}$. We denote $\\mu =\\{{{\\mu }_{{{\\theta }_{1}}}},...,{{\\mu }_{{{\\theta }_{K}}}}\\}$ (abbreviated as $\\mu_i$) as the set of all agent deterministic policies. So for the deterministic policy ${{\\mu }_{k}}$ of agent $k$, the gradient can be depicted as\n\\begin{equation}\n\\begin{aligned}\n&{{\\nabla }_{{{\\theta }_{k}}}}J({{\\mu }_{k}})=\\\\&{{\\mathbb{E}}_{S,A\\sim \\mathcal{D}}}[{{\\nabla }_{{{\\theta }_{k}}}}{{\\mu }_{k}}({{a}_{k}}|{{s}_{k}}){{\\nabla }_{{{a}_{k}}}}Q_{k}^{\\mu }(S,{{a}_{1}},...,{{a}_{K}}){{|}_{{{a}_{k}}={{\\mu }_{k}}({{s}_{k}})}}]\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{D}$ is experience replay buffer, which contains a series of $(S,A,{{S}^{'}},R)$. $Q_{k}^{\\mu }(s,{{a}_{1}},...,{{a}_{K}})$ is the Q-value function. For the critic network, it can be updated according to the loss function as follows\n\\begin{equation}\n\\begin{aligned}\n& \\mathcal{L}({{\\theta }_{k}})={{\\mathbb{E}}_{S,A,R,{{S}^{'}}}}{{[Q_{k}^{\\mu }(S,a_{1}^{{}},...,a_{K}^{{}})-y)}^{2}}] \\\\ \n& \\text{where} \\;\\; y=r_{k}^{{}}+\\gamma Q_{k}^{{{\\mu }^{'}}}({{S}^{'}},a_{1}^{'},...,a_{K}^{'}){{|}_{a_{j}^{'}=\\mu _{j}^{'}({{s}_{j}})}} \\\\ \n\\end{aligned}\n\\end{equation}\nwhere $\\gamma $ is the discount factor. The action network is updated by minimizing the policy gradient of the agent, which can be expressed as\n\\begin{equation}\n{{\\nabla }_{{{\\theta }_{k}}}}J\\approx \\frac{1}{X}\\sum\\limits_{j}{{{\\nabla }_{{{\\theta }_{k}}}}}{{\\mu }_{k}}(s_{k}^{j}){{\\nabla }_{{{a}_{k}}}}Q_{k}^{\\mu }({{S}^{j}},a_{1}^{j},...,a_{K}^{j}){{|}_{{{a}_{k}}={{\\mu }_{k}}(s_{k}^{j})}}\n\\end{equation}\nwhere $X$ is the size of mini-batch, $j$ is the index of samples. The specific joint delay and energy-efficiency algorithm based on MADDPG (JDEE-MADDPG) is shown in Algorithm 1.\n\\begin{algorithm}[!h]\n\t\\caption{JDEE-MADDPG}\n\t\\label{alg1}\n\n\t\\textbf{Initialize:} the positions, speed, task queue, computing resources and wireless resources of all vehicles. Initialize the computing and wireless resources of the VEC server. Initialize the weights of actor and critic networks.\n\t\n\t\\For{{episode= 1:M}}\n\t{\n\t\tInitialize a random process $\\mathcal{N}$ for action exploration;\n\t\n\t\tReceive initial state $S$;\n\t\t\n\t\t\\For{each vehicle $k=1,...,K$}\n\t\t{\n\t\t\tExecute actions ${{a}_{k}}$ and obtain new state $s_{k}^{'}$;\n\t\t\t\n\t\t\t\\uIf{ the $s_{k}^{'}$ does not satisfy constraints (c1)-(c4) in Eq.(12):}\n\t\t\t\t{\n\t\t\t\t \tObtain the reward of vehicle $k$ based on Eq.(15);\n\t\t\t    }\n\t\t\t\\uElseIf{the $s_{k}^{'}$ satisfy all constraints (c1)-(c4) in Eq.(12):}\t \n\t\t\t\t{\n\t\t\t\t\tObtain the reward of vehicle $k$ based on Eq.(16);\n\t\t\t\t}\n\t\t\t\\uElseIf{the $s_{k}^{'}$ satisfy all constraints (c1)-(c5) in Eq.(12):}\n\t\t\t\t{\n\t\t\t\t\tObtain the reward of vehicle $k$ based on Eq.(17);\n\t\t\t\t}\n\n\t\t\t \\textbf{end}\n\t\t\t \n             Obtain\tthe action $A$, new state ${{S}^{'}}$ and reward $R$;\n             \n             Store $(S,A,{{S}^{'}},R)$ in replay buffer $\\mathcal{D}$;\n              \n\t\t}\n\t\t\\For{each vehicle $k=1,...,K$}\n\t\t{\n\t\t\tSample a random mini-batch of $X$ samples $({{S}^{j}},{{A}^{j}},{{R}^{j}},{{S}^{'}}^{j})$ from $\\mathcal{D}$;\n\t\t\t\n\t\t\tUpdate the critic network by minimizing the loss function, Eq.(19);\n\t\t\t\n\t\t\tUpdate actor network using the sampled policy gradient, Eq.(20);\n\t\t} \n\t\tUpdate the target network parameters of each vehicle $k$: $\\theta _{k}^{'}\\leftarrow \\delta {{\\theta }_{k}}+(1-\\delta )\\theta _{k}^{'}$\n\t}\n\\end{algorithm}\n\n\n\\section{Simulation Results }\n\\subsection{Parameter Setting}\nThe specific simulation parameters are presented in Table I and Table II. The algorithms compared in this section are as follows: \n\n\\textbf{All Local Execution (AL):} All computation tasks are executed locally.\n\n\\textbf{All VEC Execution (AV):} The CA tasks are executed locally, while HPA and LPA tasks are executed in VEC server. The resource allocation strategy is based on the size of task.\n\n\\textbf{Random Offloading (RD):} The HPA and LPA tasks are executed locally and in VEC server based on the uniform distribution. The resource allocation strategy is based on the size of task.\n\n\\textbf{Energy and Delay Greedy (EDG):} The offloading strategy is based on vehicle's channel state and resource allocation strategy is based on the size of task, in order to decrease the energy cost and execution delay in each step. \n\n\\begin{table}[!h]\n\t\\caption{Simulation Parameter Configuration}\n\t\\begin{tabular}{|c|c|ccc}\n\t\t\\cline{1-2}\n\t\t\\bf{Parameter}                                                                  & \\bf{Value}                                &  &  &  \\\\ \\cline{1-2}\n\t\tNumber of vehicles                                                                    & 5, 7 ,9, 11, 13                               &  &  &  \\\\ \\cline{1-2}\n\t\tSize of   task queue                                                                  & 10                                            &  &  &  \\\\ \\cline{1-2}\n\t\tSize of task   input                                                                  & {[}0.2, 1{]} Mb                               &  &  &  \\\\ \\cline{1-2}\n\t\tSpeed of vehicle                                                                      & {[}30, 50{]}, {[}50, 80{]}, {[}30, 80{]} Km\/h &  &  &  \\\\ \\cline{1-2}\n\t\tRSU's   coverage range                                                                & 500 m                                         &  &  &  \\\\ \\cline{1-2}\n\t\tRSU's   bandwidth                                                                     & 100 MHz                                       &  &  &  \\\\ \\cline{1-2}\n\t\tChannel model                                                                         & Typical Urban                                 &  &  &  \\\\ \\cline{1-2}\n\t\t\\begin{tabular}[c]{@{}c@{}}Transmission power between\\\\  vehicle and RSU\\end{tabular} & 0.5 W                                         &  &  &  \\\\ \\cline{1-2}\n\t\t\\begin{tabular}[c]{@{}c@{}}Computation   capacity \\\\ of VEC server\\end{tabular}       & 10 G Cycles\/s                                   &  &  &  \\\\ \\cline{1-2}\n\t\t\\begin{tabular}[c]{@{}c@{}}Computation   capacity \\\\ of vehicle\\end{tabular}          & 1, 1.2, 1.4, 1.6, 1.8 G Cycles\/s                &  &  &  \\\\ \\cline{1-2}\n\t\tComputation   density                                                                 & {[}20, 50{]} Cycles\/bit                       &  &  &  \\\\ \\cline{1-2}\n\t\tWaiting   time of hold on                                                             & 20, 50 ms                                     &  &  &  \\\\ \\cline{1-2}\n\t\tDelay Threshold                                 & 10, 40, 100ms                                     &  &  &  \\\\ \\cline{1-2}\t\t\n\t\t\\begin{tabular}[c]{@{}c@{}}Output data   size\/ input \\\\ data size ratio\\end{tabular}  & 0.1                                           &  &  &  \\\\ \\cline{1-2}\n\t\tEnergy density {\\cite{Dai}}                                             & $1.25\\times 10^{-26}$ J\/Cycle                            &  &  &  \\\\ \\cline{1-2}\n\t\tParameters of reward                                                                  &  \\begin{tabular}[c]{@{}c@{}}${{\\Gamma }_{1}}=0.8$, ${{\\Gamma }_{2}},{{\\Gamma }_{3}},{{\\Gamma }_{4}},{{\\Gamma }_{5}}=0.5$ \\\\  ${{\\ell }_{1}}=-0.4,{{\\ell }_{2}}=-0.2,{{\\ell }_{3}}=0.5$ \\end{tabular}                                       &  &  &  \\\\ \\cline{1-2}\n\t\\end{tabular}\n\\end{table}\n\n\\begin{table}[!h]\n\t\\caption{The Neural Network and Training Parameters}\n\t\\begin{tabular}{|c|c|c|c|}\n\t\t\\hline\n\t\t\\textbf{Parameter}                 & \\textbf{Value} & \\textbf{Parameter}        & \\textbf{Value}  \\\\ \\hline\n\t\tLayers                             & 3              & Layer Type                & Fully Connected \\\\ \\hline\n\t\t\\multicolumn{1}{|c|}{Hidden Units} & 512            & Learning Rate of   Critic & 0.001           \\\\ \\hline\n\t\tOptimizer                          & Adam           & Learning Rate of Actor    & 0.0001          \\\\ \\hline\n\t\tEpisode                            & 140000         & Activation Function       & Relu            \\\\ \\hline\n\t\tMini-batch                          & 128            & Buffer Size               & 20000           \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Performance Evaluation}\nWe validate the algorithm performance in term of convergence property, task completion delay and energy consumption  under different simulation configurations. \n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[width=8.6cm]{figure0.eps}\n\t\\centering\n\t\\caption{Convergence property of different numbers of vehicles.}\n\t\\label{convergence}\n\\end{figure}\n\\begin{figure*}[!h]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figure1.eps}\n\t\\centering\n\t\\caption{Average task completion delay of different algorithms: (a) the number of vehicles is 5, (b) the number of vehicles is 7, (c) the number of vehicles is 9, (d) the number of vehicles is 11, (e) the number of vehicles is 13.}\n\t\\label{figure1}\n\\end{figure*}\n\\begin{figure*}[!h]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figure2.eps}\n\t\\centering\n\t\\caption{Average task energy consumption of different algorithms: (a) the number of vehicles is 5, (b) the number of vehicles is 7, (c) the number of vehicles is 9, (d) the number of vehicles is 11, (e) the number of vehicles is 13.}\n\t\\label{figure2}\n\\end{figure*}\n\\begin{figure*}[!h]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figure3.eps}\n\t\\centering\n\t\\caption{Average task completion delay and energy consumption of different algorithms: (a)(b) the vehicle speed range is [30, 50], (c)(d) the vehicle speed range is [50, 80].}\n\t\\label{figure3}\n\\end{figure*}\n\nIn Figure \\ref{convergence}, we present the convergence performance of our proposed JDEE-MADDPG algorithm with the different numbers of vehicles. It can be seen that with the increase of training episodes, the average reward of vehicles rises gradually and preserves a stable positive reward eventually. In the initial stage, the average reward of our proposed JDEE-MADDPG algorithm with less vehicles is higher than that with more vehicles, because the increase of vehicles means higher dimension of state space and action space and our proposed JDEE-MADDPG algorithm needs to take more explorations. Therefore, it will result in that the average reward of large number of vehicles is lower than that of small number of vehicles. As the training episode increases, our proposed JDEE-MADDPG algorithm gradually achieves the convergence state, where the average reward of 5 vehicles is the highest, while the average reward of 13 vehicles is the lowest. The reason is that more vehicles implies that they will compete the limited computation resource and wireless resource more fiercely, which causes that the reward of energy consumption and delay decreases. \n\nIn Figure \\ref{figure1}, we present the comparison of average task completion delay with different numbers of vehicles when the vehicle speed range is from 30 to 80 Km\/h. It can be seen that compared with AL, AV and RD algorithms, our proposed JDEE-MADDPG algorithms can always preserve a lower level of task completion delay for each client. This is because that our proposed JDEE-MADDPG algorithm can allocate the computation resource and wireless resource to vehicles more accurately based on the task priority, task size, vehicle speed and vehicle's channel state. In addition, the task completion delay of some vehicles of EDG algorithm is less than that of our proposed JDEE-MADDPG algorithm, because our proposed algorithm sacrifices a little task completion delay to decrease the energy consumption of vehicle terminals without exceeding the task delay constraint.  \n\n\nIn Figure \\ref{figure2}, we show the comparison of average task energy consumption with different numbers of vehicles when the vehicle speed range is from 30 to 80 Km\/h. It can be observed that compared with other algorithms, our proposed JDEE-MADDPG algorithm can always preserve a lower level of energy consumption. This is because that our proposed JDEE-MADDPG algorithm can always make the optimal offloading and resource allocation strategy based on the task priority, task size, vehicle speed and vehicle's channel state and reduce the energy consumption of all vehicles as soon as possible.\n\n\nIn Figure \\ref{figure3}, we compare the average task completion delay and energy consumption with different vehicle speed range when the number of vehicles is 9. Compared with AL, AV and RD algorithm, our proposed JDEE-MADDPG algorithm performs better in terms of delay and energy consumption. This is because that our proposed JDEE-MADDPG algorithm can utilize more information of vehicle terminals and VEC server, i.e., vehicle position, vehicle speed, task queue, channel state and remaining computation resource to make the optimal offloading and resource allocation strategy. Besides, the reason that some vehicles' task completion delay of our proposed JDEE-MADDPG algorithm remains higher than that of EDG algorithm is that our JDEE-MADDPG algorithm usually allocate more wireless and computation resource of VEC server to the vehicles with high speed without exceeding task delay constraint, which may cause that task completion delay of some vehicles is higher than that of EDG algorithm. \n\n\n\n\\section{Conclusion}\nIn this paper, we propose a vehicle speed aware computing task offloading and resource allocation algorithm to achieve the goal of energy-efficiency for all vehicles within task delay constraint. First, we establish the vehicle speed-based delay constraint model based on task types and vehicle speed. And then we calculate the task completion delay and energy consumption for different offloading positions based on the allocated computation and wireless resource. Finally, we formulate the mathematical model with the objective to minimize energy consumption of all vehicles subject to the delay constraint. The MADDPG method is utilized to obtain the offloading and resource allocation strategy. Simulation results show that the proposed JDEE-MADDPG algorithm can decrease energy consumption and task completion delay compare with other algorithms under different numbers of vehicles and vehicle speed ranges.\n\n\\section*{Acknowledgement}\nThis research work was supported in part by the National Science Foundation of China (61701389, U1903213), the Natural Science Basic Research Plan in Shaanxi Province of China (2018JQ6022) and the Shaanxi Key R\\&D Program (2018ZDCXL-GY-04-03-02).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite{LSY1, LSY2}, the authors introduced the notion of \\textit{holomorphic homogeneous regular}. Then in \\cite{Y:squeezing}, the equivalent notion of \\textit{uniformly squeezing} was introduced. Motivated by these studies, in \\cite{DGZ1}, the authors introduced the \\textit{squeezing function} as follows.\n\nDenote by $\\ball(r)$ the ball of radius $r>0$ centered at the origin 0. Let $\\Omega$ be a bounded domain in $\\cv^n$, and $p\\in \\Omega$. For any holomorphic embedding $f:\\Omega\\rightarrow \\ball(1)$, with $f(p)=0$, set\n$$s_{\\Omega,f}(p):=\\sup\\{r>0:\\ \\ball(r)\\subset f(\\Omega)\\}.$$\nThen, the squeezing function of $\\Omega$ at $p$ is defined as\n$$s_\\Omega(p):=\\sup_f\\{s_{\\Omega,f}(p)\\}.$$\n\nMany properties and applications of the squeezing function have been explored by various authors, see e.g. \\cite{DGZ1,DGZ2,DF:Bergman,DFW:squeezing,FS:squeezing,FW:squeezing,KZ:squeezing}.\n\nIt is clear that squeezing functions are invariant under biholomorphisms, and they are positive and bounded above by 1. It is a natural and interesting problem to study the uniform lower and upper bounds of the squeezing function.\n\nIt was shown recently in \\cite{KZ:squeezing} that the squeezing function is uniformly bounded below for bounded convex domains (cf. \\cite[Theorem 1.1]{Frankel}). On the other hand, in \\cite{DGZ1}, the authors showed that the squeezing function is not uniformly bounded below on certain domains with non-smooth boundaries, such as punctured balls. In \\cite{DF}, the authors constructed a smooth pseudoconvex domain in $\\cv^3$ on which the quotient of the Bergman metric and the Kobayashi metric is not bounded above near an infinite type point. By \\cite[Theorem 3.3]{DGZ2}, the squeezing function is not uniformly bounded below on this domain.\n\nThese studies raise the question: Is the squeezing function always uniformly bounded below near a smooth finite type point? In this paper, we answer the question negatively. More precisely, we have the following\n\n\\begin{thm}\\label{T:main}\nLet $\\Omega$ be a bounded domain in $\\cv^3$, and $q\\in \\partial\\Omega$. Assume that $\\Omega$ is smooth and pseudoconvex in a neighborhood of $q$ and the Bloom-Graham type of $\\Omega$ at $q$ is $d<\\infty$. Moreover, assume that the regular order of contact at $q$ is greater than $2d$ along two smooth complex curves not tangent to each other. Then the squeezing function $s_\\Omega(p)$ has no uniform lower bound near $q$.\n\\end{thm}\n\n\\begin{rmk}\nThe proof gives the estimate $s_\\Omega(p)\\leq C \\delta^{\\frac{1}{2d(2d+1)}}$ for some points approaching the boundary.\n\\end{rmk}\n\nIn section \\ref{S:prelim}, we recall some preliminary notions and results. In section \\ref{S:proof}, we prove Theorem \\ref{T:main}.\n\n\\section{Preliminaries}\\label{S:prelim}\n\nLet $\\Omega$ be a bounded domain in $\\cv^n$, $n\\ge 2$, and $q\\in \\partial\\Omega$. Assume that $\\Omega$ is smooth and pseudoconvex in a neighborhood of $q$. The \\textit{Bloom-Graham type} of $\\Omega$ at $q$ is the maximal order of contact of complex manifolds of dimension $n-1$ tangent to $\\partial \\Omega$ at $q$ (see e.g. \\cite{BG}). Choose local coordinates $(z,t)\\in \\cv^{n-1}\\times \\cv$ such that the complex manifold of dimension $n-1$ with the maximal order of contact is given by $\\{t=0\\}$. Then $\\Omega$ is locally given by $\\rho(z,t)<0$, where $\\rho(z,t)=\\Re t+P(z)+Q(z,t)$ with $Q(z,0)\\equiv 0$ and $\\deg P(z)=d$. (We say that the degree of $P$ is $d$ if the Taylor expansion of $P$ has no nonzero term of degree less than $d$.) Since $\\Omega$ is pseudoconvex, we actually have $d=2k$ (see e.g. \\cite{D'Angelo}).\n\nFor $1\\le k\\le n-1$, let $\\varphi:\\cv^k\\rightarrow \\cv^n$ be analytic with $\\varphi(0)=q$ and $\\textup{rank} d\\varphi(0)=k$. Then the \\textit{regular order of contact} at $q$ along the $k$-dimensional complex manifold defined by $\\varphi$ is defined as $\\deg \\rho\\circ\\varphi$ (see e.g. \\cite{D'Angelo}).\n\nDenote by $\\Delta$ the unit disc in $\\cv$. Let $p\\in \\Omega$ and $\\zeta\\in \\cv^n$. The \\textit{Kobayashi metric} is defined as\n$$K_\\Omega(p,\\zeta):=\\inf\\{\\alpha:\\ \\alpha>0,\\ \\exists\\ \\phi:\\Delta\\rightarrow \\Omega,\\ \\phi(0)=p,\\ \\alpha \\phi'(0)=\\zeta\\}.$$\nThen the \\textit{Kobayashi indicatrix} is defined as (see e.g. \\cite{K:indicatrix})\n$$D_\\Omega(p):=\\{\\zeta\\in \\cv^n:\\ K_\\Omega(p,\\zeta)<1\\}.$$\nFor each unit vector $e\\in \\cv^n$, set $D_\\Omega(p,e):=\\max\\{|\\eta|:\\ \\eta\\in \\cv,\\ \\eta e\\in D_\\Omega(p)\\}$. By the definition of Kobayashi indicatrix, the following three lemmas are clear.\n\n\\begin{lem}\\label{L:ball}\n$D_{\\ball(r)}(0)=\\ball(r)$.\n\\end{lem}\n\n\\begin{lem}\\label{L:indicatrix}\nLet $\\Omega_1$ and $\\Omega_2$ be two domains in $\\cv^n$ with $\\Omega_1\\subset \\Omega_2$. Then for each $p\\in \\Omega_1$, $D_{\\Omega_1}(p)\\subset D_{\\Omega_2}(p)$.\n\\end{lem}\n\n\\begin{lem}\\label{L:indicatrix1}\nLet $\\Omega$ be a domain in $\\cv^n$ and $f:\\Omega\\rightarrow \\cv^n$ a biholomorphic map. Then for each $p\\in \\Omega$, $D_{f(\\Omega)}(f(p))=f'(p)D_\\Omega(p)$.\n\\end{lem}\n\nWe also need the following localization lemma (see e.g. \\cite[Lemma 3]{FL:metrics}). We will use $\\gtrsim$ (resp. $\\lesssim$, $\\simeq$) to mean $\\ge$ (resp. $\\le$, $=$) up to a positive constant.\n\n\\begin{lem}\\label{L:localization}\nLet $\\Omega$ be a bounded domain in $\\cv^n$, $q\\in \\partial \\Omega$ and $U$ a neighborhood of $q$. If $V\\subset\\subset U$ and $q\\in V$, then\n$$K_\\Omega(p,\\zeta)\\simeq K_{\\Omega\\cap U}(p,\\zeta),\\ \\ \\ \\forall\\ p\\in V,\\ \\zeta\\in \\cv^n.$$\n\\end{lem}\n\nBy the above lemma, when we consider the size of the Kobayashi indicatrix in the next section, we will work in $\\Omega\\cap U$.\n\n\\section{Estimate of the squeezing function}\\label{S:proof}\n\nWe first choose local coordinates adapted to our purpose.\n\n\\begin{lem}\\label{L:normal}\nLet $\\Omega$ be a bounded domain in $\\cv^{n+1}$, $n\\ge 1$, and $q\\in \\partial\\Omega$. Assume that $\\Omega$ is smooth and pseudoconvex in a neighborhood of $q$ and the Bloom-Graham type of $\\Omega$ at $q$ is $2k$, $k\\ge 1$. Then there exist local coordinates $(z,t)=(z_1,\\cdots,z_n,u+iv)$ such that $q=(0,0)$ and $\\Omega$ is locally given by $\\rho(z,t)<0$ with\n\\begin{equation}\\label{E:normal}\n\\rho(z,t)=u+P(z)+Q(z)+vR(z)+u^2+v^2+o(u^2,uv,v^2,u|z|^{2k}),\n\\end{equation}\nwhere $P(z)$ is plurisubharmonic, homogeneous of degree $2k,$ but not pluriharmonic, $\\deg Q(z)\\ge 2k+1$ and $\\deg R(z)\\ge k+1$.\n\\end{lem}\n\\begin{proof}\nBy assumption, we have a local defining function of the form\n$$\\rho(z,t)=u+P(z)+Q(z)+au^2+buv+cv^2+uA(z)+vB(z)+o(|t|^2),$$\nwhere $P(z)$ is plurisubharmonic, homogeneous of degree $2k,$ but not pluriharmonic,  $\\deg Q(z)\\ge 2k+1$,\nand $\\deg A(z),B(z)\\ge 1$. By changing $t$ to $t+dt^2$ and multiplying with $1+eu$ or $1+ev$, we can freely change the quadratic terms in $u,v$. Thus, we can assume that\n$$\\rho(z,t)=u+P(z)+Q(z)+u^2+v^2+uA(z)+vB(z)+o(|t|^2).$$\nMultiplying with $1-A(z)$, we get a new $A$, say $A'$ of degree at least $2$. Multiplying with $1-A'(z)$ we get\na new $A$ of degree at least $4$. Continuing, we can further assume that\n$$\\rho(z,t)=u+P(z)+Q(z)+u^2+v^2+vB(z)+o(|t|^2, u|z|^{2k}).$$\n\nWrite $B(z)=B_s(z)+B'(z)$, where $B_s(z)$ is the lowest order homogeneous term of degree $s\\ge 1$. Assume that $B_s(z)$ is pluriharmonic. Then there exists a holomorphic function $F(z)=A(z)-iB_s(z)$. Change again $\\rho$ to add the term $uA(z)$ with this new $A(z)$. Then $\\rho$ takes the form\n$$\\begin{aligned}\n\\rho(z,t)&=u+P(z)+Q(z)+u^2+v^2+uA(z)+vB_s(z)+vB'(z)+o(|t|^2,u|z|^{2k})\\\\\n&=u+P(z)+Q(z)+u^2+v^2+\\Re(tF(z))+vB'(z)+o(|t|^2,u|z|^{2k}).\n\\end{aligned}$$\nBy absorbing $\\Re(tF(z))$ into $u$, we get\n$$\\rho(z,t)=u+P(z)+Q(z)+u^2+v^2+vB'(z)+o(|t|^2,u|z|^{2k}).$$\nContinuing this process, we can assume that $\\rho$ takes the form\n$$\\rho(z,t)=u+P(z)+Q(z)+u^2+v^2+vB_l(z)+vB'(z)+o(|t|^2,u|z|^{2k}),$$\nwhere $B_l(z)$ is not pluriharmonic and $l\\le k$ or $l\\ge k+1$. We assume the first alternative, otherwise the proof is done. We will arrive at a contradiction to pseudoconvexity.\n\nNote that $P(z)$ is plurisubharmonic but not pluriharmonic. This implies that there exists a complex line through the origin on which the restriction of $P$ is subharmonic, but not harmonic (cf. \\cite{Forelli}). Pick a tangent vector $\\xi=(\\xi_1,\\dots \\xi_n)$ so that the Levi form of $P$ calculated at a point $\\eta\\xi$, $\\eta=|\\eta|e^{i\\theta}$, in the direction of $\\xi$ is $|\\eta|^{2k-2}G(\\theta)\\|\\xi\\|^2$. Here $G$ is a smooth nonnegative function which at most vanishes at finitely many angles. Choose $\\lambda$ such that $\\sigma=(\\xi,\\lambda)$ is a complex tangent vector to $\\partial \\Omega$, i.e.\n$$\\sum_{j=1}^n \\frac{\\partial \\rho}{\\partial z_j}\\xi_j+ \\frac{\\partial \\rho}{\\partial t}\\lambda=0.$$\nThen we have $|\\lambda|=O(|\\eta|^{2k-1}+|v||\\eta|^{l-1}+|t|^2+|u||\\eta|^{2k-1})\\|\\xi\\|$.\n\nThe Levi form of $\\rho$ at a boundary point $(\\eta\\xi,t)$, along the tangent vector $\\sigma$ is\n$$\\begin{aligned}\n\\mathcal L(r,\\sigma)=&|z|^{2k-2}G(\\theta)\\|\\xi\\|^2+|\\lambda|^2+\\mathcal L(vB_l(z),\\sigma)+\\cdots\\\\\n=&|z|^{2k-2}G(\\theta)\\|\\xi\\|^2+|\\lambda|^2+\\Re(\\sum_{j=1}^n\\frac{\\partial B_l}{\\partial z_j}i\\xi_j\\overline{\\lambda})+v\\sum_{k,m}\\frac{\\partial^2 B_l}{\\partial z_k\\overline z_m}\\xi_k\\overline{\\xi}_m+\\cdots.\n\\end{aligned}$$\n\nSince $B_l$ is not pluriharmonic, we can assume after changing $\\xi$ slightly that $\\frac{\\partial^2 B_l}{\\partial z_k\\partial \\overline{z}_m}\\xi_k\\overline{\\xi}_m\\neq 0$. Next choose $v=\\pm C|\\eta|^k$ with $C>\\max_\\theta\\{G(\\theta)\\}$. The second term is $o(|\\eta|^{2k-2}\\|\\xi\\|^2)$ and the third terms is $o(|\\eta|^{k+l-2}\\|\\xi\\|^2)$. The last term is $\\simeq (|\\eta|^{k+l-2}\\|\\xi\\|^2)$ and, since $l\\le k$, at least $O(|\\eta|^{2k-2}\\|\\xi\\|^2)$. Thus we have $\\mathcal L(r,\\sigma)<0$. This is a contraction.\n\\end{proof}\n\nBy Lemma \\ref{L:normal}, we can choose local coordinates $(z,w,t)=(z,w,u+iv)$ near $q$ such that $q=(0,0,0)$ and $\\Omega$ is locally given by $\\rho(z,w,t)<0$, where\n\\begin{equation}\\label{E:rho}\n\\rho(z,w,t)=u+P(z,w)+Q(z,w)+vR(z,w)+u^2+v^2+o(u^2,uv,v^2,u|(z,w)|^{2k}).\n\\end{equation}\nHere $P(z,w)$ is homogeneous of degree $2k$ with $P(z,0)=P(0,w)=0$, $\\deg Q(z,w)\\ge 2k+1$ with $\\deg Q(z,0)\\ge 4k+1$ and $\\deg Q(0,w)\\ge 4k+1$, and $\\deg R(z,w)\\ge k+1$. Set $p=(0,0,-\\delta)$ with $0<\\delta\\ll 1$.\n\n\\begin{lem}\\label{L:10}\nLet $\\zeta_1=(1,0,0)$ and $\\zeta_2=(0,1,0)$. Then $K_\\Omega(p,\\zeta_1),K_\\Omega(p,\\zeta_2)\\lesssim \\delta^{-\\frac{1}{4k+1}}$.\n\\end{lem}\n\\begin{proof}\nConsider the linear map $\\phi:\\Delta\\rightarrow \\cv^3$ with $\\phi(\\tau)=(\\beta\\tau,0,-\\delta)$ for $\\tau\\in \\Delta$, and $|\\beta|=\\epsilon \\delta^{\\frac{1}{4k+1}}$ for $0<\\epsilon\\ll 1$. Then\n$$\\rho\\circ\\phi(\\tau)\\le -\\delta+C|\\beta\\tau|^{4k+1}+o(\\delta)<-\\delta+\\epsilon\\delta+o(\\delta)<0.$$\nTherefore, $K_\\Omega(p,\\zeta_1)\\lesssim \\delta^{-\\frac{1}{4k+1}}$. The argument in the direction $\\zeta_2$ is similar.\n\\end{proof}\n\nLet $(a,b,0)$ be a point so that $P(a \\tau,b \\tau)$ is a subharmonic homogeneous polynomial of degree $2k$ which is not harmonic. Then both $a$ and $b$ must be nonzero. By scaling in each variable, we can assume that $a=b=1\/{\\sqrt {2}}$. \n\n\\begin{lem}\\label{L:11}\nLet $\\zeta=\\frac{1}{\\sqrt{2}}(1,1,0).$  Then $K_\\Omega(p,\\zeta)\\gtrsim \\delta^{-\\frac{1}{4k}}$.\n\\end{lem}\n\\begin{proof}\nFor $z,w$ small, we have\n$$\\begin{aligned}\nv^2\/2+vR(z,w)&\\ge v^2\/2-C|v|\\|z,w\\|^{k+1}+C^2\\|z,w\\|^{2k+2}-C^2\\|z,w\\|^{2k+2}\\\\\n&\\ge -C^2\\|z,w\\|^{2k+2}.\n\\end{aligned}$$\nTherefore,\n$$\\begin{aligned}\n\\rho&\\ge u+P(z,w)+Q(z,w)-C^2\\|z,w\\|^{2k+2}+u^2+v^2\/2+o(u^2,uv,v^2,u|(z,w)|^{2k})\\\\\n&\\ge u+P(z,w)+\\tilde{Q}(z,w)+u^2\/2+v^2\/4+o(u|(z,w)|^{2k})=:\\tilde{\\rho}.\n\\end{aligned}$$\n\nConsider an analytic map $\\phi:\\Delta\\rightarrow \\Omega$ with\n$$\\phi(\\tau)=(\\beta\\tau+f(\\tau),\\beta\\tau+g(\\tau),-\\delta+h(\\tau)),\\ \\ \\ |f(\\tau)|,|g(\\tau)|,|h(\\tau)|\\le |\\tau|^2.$$\nThen $\\tilde{\\rho}\\circ\\phi(\\tau)\\leq\\rho\\circ\\phi(\\tau)<0$. And we have\n$$\\begin{aligned}\n\\tilde{\\rho}(\\phi(\\tau))=&-\\delta+\\Re h(\\tau)+P(\\beta\\tau+f(\\tau),\\beta\\tau+g(\\tau))+\\tilde{Q}(\\beta\\tau+f(\\tau),\\beta\\tau+g(\\tau))\\\\\n&+u^2\/2+v^2\/4+o(u|(z,w)|^{2k})<0,\n\\end{aligned}$$\nand\n\\begin{equation}\\label{E:average}\n\\frac{1}{2\\pi}\\int_0^{2\\pi} \\tilde{\\rho}(\\phi(|\\tau|e^{i\\theta}))d\\theta<0.\n\\end{equation}\n\nNote that, by the homogeneous expansion of $P(z,w)$, we have for $|\\tau|<1$ small\n\\begin{equation}\\label{E:xitau}\n|\\beta\\tau|^{2k}-\\sum_{i=0}^{2k-1} |\\beta|^i|\\tau|^{4k-i}\\lesssim \\delta.\n\\end{equation}\nChoose $|\\tau|=c|\\beta|$ for some small constant $c>0.$  Then \\eqref{E:xitau} gives\n$$\\left|\\frac{\\beta}{2}\\right|^{4k}\\lesssim \\delta.$$\nHence, $K_\\Omega(p,u)\\gtrsim \\delta^{-\\frac{1}{4k}}$. \n\\end{proof}\n\n\\begin{lem}\\label{L:squeezing}\nLet $D$ be a bounded domain in $\\cv^n$, $n\\ge 2$, containing the origin. Assume that there exist two linearly independent nonzero vectors $\\zeta_1,\\zeta_2\\in D$ and $\\epsilon>0$ such that $\\epsilon(\\zeta_1+\\zeta_2)\\not\\in D$. Then there does not exist a linear map $L:D\\rightarrow \\cv^n$, with $L(0)=0$, such that $\\ball(3\\epsilon)\\subset L(D)\\subset \\ball(1)$.\n\\end{lem}\n\\begin{proof}\nLet $L:D\\rightarrow \\cv^n$ be a linear map with $L(0)=0$ and suppose $\\ball(3\\epsilon)\\subset L(D)$. Since $\\epsilon(\\zeta_1+\\zeta_2)\\not\\in D$ and $L$ is linear, we have $\\epsilon(L(\\zeta_1)+L(\\zeta_2))\\not\\in L(D)$. This implies that $\\epsilon(L(\\zeta_1)+L(\\zeta_2))\\not\\in \\ball(3\\epsilon)$ and thus $\\|L(\\zeta_1)+L(\\zeta_2)\\|\\ge 3$. However,\n$\\|L(\\zeta_1)+L(\\zeta_2)\\|\\leq \\|L(\\zeta_1)\\|+\\|L(\\zeta_2)\\|\\leq 1+1=2.$\nThis completes the proof.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{T:main}]\nChoose local coordinates $(z,w,t)$ such that $q=(0,0,0)$ and let $p=(-\\delta,0,0)$ for $\\delta>0$ small. Let $\\zeta_1=(1,0,0)$ and $\\zeta_2=(0,1,0)$, the two directions along which the regular order of contact at $q$ is greater than $2d=4k$. By Lemma \\ref{L:10}, $K_\\Omega(p,\\zeta_1),K_\\Omega(p,\\zeta_2)\\lesssim \\delta^{-\\frac{1}{4k+1}}$. By Lemma \\ref{L:11}, $K_\\Omega(p,\\frac{1}{\\sqrt{2}}(\\zeta_1+\\zeta_2))\\gtrsim \\delta^{-\\frac{1}{4k}}$.\n\nChoose $\\lambda>0$ with $\\lambda\\gtrsim \\delta^{\\frac{1}{4k+1}}$ such that $\\lambda \\zeta_1,\\lambda \\zeta_2\\in D_\\Omega(p)$. Then for $\\epsilon \\simeq \\delta^{\\frac{1}{4k(4k+1)}}$, we have $\\epsilon (\\lambda \\zeta_1+ \\lambda \\zeta_2)\\not\\in D_\\Omega(p)$. Thus, by Lemma \\ref{L:squeezing}, there does not exist a linear map $L:D_\\Omega(p)\\rightarrow \\cv^3$ such that $\\ball(3\\epsilon)\\subset L(D_\\Omega(p))\\subset \\ball(1)$.\n\nLet $f$ be a biholomorphism of $\\Omega$ into $\\ball(1)$ such that $f(p)=0$ and $\\ball(c)\\subset f(\\Omega)$ for some $c>0$. Set $L=f'(p)$. Then, by Lemmas \\ref{L:ball}, \\ref{L:indicatrix} and \\ref{L:indicatrix1}, $\\ball(c)\\subset L(D_\\Omega(p))\\subset \\ball(1)$. Therefore, we have $c\\lesssim \\delta^{\\frac{1}{4k(4k+1)}}$. Since $f$ is arbitrary, we get $s_\\Omega(p) \\lesssim \\delta^{\\frac{1}{4k(4k+1)}}$. Since $\\delta$ can be arbitrarily small, this completes the proof.\n\\end{proof}\n\n\\begin{rmk}\nTheorem \\ref{T:main} does not hold if only assuming that the regular order of contact at $q$ is greater than $2d$ along one smooth complex curve. For instance, consider $\\Omega$ given by\n$$\\{(z,w,t)\\in \\cv^3:\\ |t|^2+|z|^2+|w|^6<1\\}.$$\nThen at $q=(0,0,1)$, the Bloom-Graham type is $2$ and the regular order of contact along $(0,1,0)$ is $6>4$. But $\\Omega$ is a bounded convex domain and thus the squeezing function has a uniform lower bound by \\cite{KZ:squeezing}.\n\\end{rmk}\n\n\\begin{rmk}\nUsing similar arguments, one can extend Theorem \\ref{T:main} to higher dimensions as follows.\n\n\\begin{thm}\nLet $\\Omega$ be a bounded domain in $\\cv^n$, $n\\ge 4$, and $q\\in \\partial\\Omega$. Assume that $\\Omega$ is smooth and pseudoconvex in a neighborhood of $q$ and the Bloom-Graham type of $\\Omega$ at $q$ is $d$. Moreover, assume that the regular order of contact at $q$ is $d$ along a two-dimensional complex surface $\\Sigma$ and the regular order of contact at $q$ is greater than $2d$ along two smooth complex curves not tangent to each other contained in $\\Sigma$. Then the squeezing function $s_\\Omega(p)$ has no uniform lower bound near $q$.\n\\end{thm}\n\\end{rmk}\n\n\\begin{rmk}\nAfter the completion of this work, it was brought to our attention by Gregor Herbort that a similar comparison result to \\cite{DF} was obtained for the following domain in \\cite{DFH:Bergman}:\n$$\\Omega:=\\{(z,w,t)\\in \\cv^3:\\ \\Re t+|z|^{12}+|w|^{12}+|z|^2|w|^4+|z|^6|w|^2<0\\}.$$\nTherefore, by our remark in the introduction, the squeezing function does not have a uniform lower bound on this domain. More generally, we have the following\n\\begin{thm}\nLet $\\Omega$ be a bounded domain in $\\cv^3$, and $q\\in \\partial\\Omega$. Assume that $\\Omega$ is smooth and pseudoconvex in a neighborhood of $q$ and the Bloom-Graham type of $\\Omega$ at $q$ is $d<\\infty$. Let $\\rho$ be a defining function of $\\Omega$ near $q$ in the normal form \\eqref{E:normal} and assume that the leading homogeneous term $P(z)$ only contains positive terms. Moreover, assume that the regular order of contact at $q$ is greater than $d$ along two smooth complex curves not tangent to each other. Then the squeezing function $s_\\Omega(p)$ has no uniform lower bound near $q$.\n\\end{thm}\n\\begin{proof}[Sketch of proof]\nIn Lemma \\ref{L:10}, we get $K_\\Omega(p,\\zeta_1),K_\\Omega(p,\\zeta_2)\\lesssim \\delta^{-\\frac{1}{2k+1}}$, by the same argument. In Lemma \\ref{L:11}, we get $K_\\Omega(p,\\zeta)\\gtrsim \\delta^{-\\frac{1}{2k}}$, by noticing that instead of \\eqref{E:xitau} we have $|\\xi\\tau|^{2k}\\lesssim \\delta$ since all terms of $P(z)$ are positive. Then arguing exactly as in the proof of Theorem \\ref{T:main}, we get $s_\\Omega(p) \\lesssim \\delta^{\\frac{1}{2k(2k+1)}}$.\n\\end{proof}\n\\end{rmk}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nRecent advances in computer vision and machine learning have dramatically increased the accuracy of face recognition technologies \\cite{schroff2015facenet, masi2018deep, deng2019arcface}. Face recognition is already commonly used in commercial products such as Apple's FaceID \\cite{faceid} or at border checkpoints in airports where the portrait on a digitized biometric passport is compared with the holder's face. Most people have little to no concerns about these specific applications as they are limited in scope and have a single well defined goal. As the technologies mature however it becomes possible to deploy them on a much larger scale. The most well known example of this is the large scale use of CCTV cameras equipped with intelligent analysis software. In this way, facial recognition checkpoints are deployed at areas like gas stations, shopping centers, and mosque entrances \\cite{larson2018china, NOS}.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/overview_.png}\n    \\setlength{\\abovecaptionskip}{-2pt}\n    \\setlength{\\belowcaptionskip}{-2pt}\n    \\caption{Overview of our proposed approach. For each subject, we automatically extract one high quality frame (or face crop) and encrypt it for storage. Access to the images is only possible after legal authorization in case of an investigation.}\n    \\label{fig:overview}\n\\end{figure}\nFrom a domestic security point of view, these technologies are extremely useful. They can be used to find missing children, identify and track criminals or locate potential witnesses. There are many examples where CCTV footage in combination with facial recognition software has supported and accelerated criminal investigations. In 2016 for example, the ``man in the hat'' responsible for the Brussels terror attacks was identified thanks to FBI facial recognition software \\cite{maninhat}. \n\\\\\n\\newline\nThe proliferation of facial recognition technology however also raises many valid privacy concerns. A fundamental human rights principle is that surveillance should be necessary and proportionate. This principle was adopted by the UN Human Rights Council (HRC) in the ``the right to privacy in the digital age'' resolution which states that ``States should ensure that any interference with the right to privacy is consistent with the principles of legality, necessity and proportionality''\\cite{unitednations}.\n\\\\\n\\newline\nGovernments have to balance both aspects of this technology before they implement a certain solution. On the one hand, there are the obvious advantages of a large scale blanket surveillance system but this clearly violates the proportionality principle. On the other hand, a recent study showed that a majority of the public considers it to be acceptable for law enforcement to use facial recognition tools to assess security threats in public spaces as long as it is within a clearly defined regulatory framework \\cite{smith2019more}.\n\\\\\nAn interesting research direction is the development of more privacy-friendly alternatives that can still support law enforcement in criminal investigations. In this work we present an intelligent frame selection approach that can be used as a building block in a more privacy-friendly alternative to large scale face recognition. The problem of key frame selection is defined as selecting a single frame out of a video stream that represents the content of the scene \\cite{wolf1996key}. In the context of facial recognition, our goal is to record a single clear crop of each person visible in the stream without revealing his or her identity. In such a way, the minimization strategy of privacy preserving technologies \\cite{DomingoFerrer2020} is implemented by collecting the minimal necessary amount of personal data.\nAccording to \\cite{duncan2007engineering}, data minimization is an unavoidable first step to engineer systems in line with the principles of privacy by design \\cite{cavoukian2009privacy}. Next, to ensure the confidentiality of the collected data, all images are encrypted (i.e. ``hide'' strategy) and access can only be provided to law enforcement after legal authorization as shown in figure \\ref{fig:overview}.\n\\\\\n\\newline\nExtracting face crops which are suitable for recognition is a difficult problem as surveillance footage is typically blurry because of the motion of the subjects. In addition, it is hard to quantify the quality of a single frame as it depends on multiple aspects such as the angle, illumination and head position. Frame selection techniques have been used before in the context of face recognition  to reduce the computational cost or increase the recognition rate \\cite{wong_cvprw_2011, anantharajah2012quality, qi2018boosting, vignesh2015face, best2017automatic, hernandez2020biometric, terhorst2020ser}. In this work we introduce a novel deep learning based technique to perform intelligent face recognition. The key contribution is a new face image quality assessment (FQA) approach inspired by anomaly detection. Unlike previous solutions, our system is trained completely unsupervised without the need for labeled face images and without access to a face recognition system. Our approach can be used for the same tasks as previous frame selection techniques but in this work we propose to use it as a building block for a more privacy-friendly alternative to large scale face recognition.\n\\\\\n\\newline\nThis paper is organized as follows. An overview of the related work is presented in section \\ref{section:related}. Next, we propose a privacy preserving alternative to large scale facial recognition in section \\ref{section:privacy_preserving_appr}. Our novel FQA method is introduced in section \\ref{section:approach} and experimentally validated in  section \\ref{section:experimental_setup} and \\ref{section:results}. We conclude in section \\ref{section:conclusion} and give some pointers for future research.\n\n\\section{Related work}\n\\label{section:related}\nDeep learning has dominated the field of facial recognition in the past years. Recent techniques can reach accuracies of 99.63\\% \\cite{schroff2015facenet} on the Labeled Faces in The Wild dataset \\cite{huang2008labeled}, the most commonly used benchmark dataset. These data-driven methods outperform techniques based on engineered features by a large margin \\cite{masi2018deep}. Face recognition is typically subdivided into face verification and face identification. Face verification is the task of deciding whether two pictures show the same person where face identification tries to determine the identity of the person on an image.\n\\\\\n\\newline\nA large amount of research is dedicated to extracting high quality face image representations. These can then be used to calculate the similarity between two face crops or as input to a classification model. Different loss functions have been designed to get meaningful face image representations. First attempts used softmax loss \\cite{sun2014deep} while recent approaches focus on euclidean distance-based loss \\cite{schroff2015facenet}, cosine-margin-based loss \\cite{wang2017normface} and variations of the softmax loss \\cite{deng2019arcface}. Given these high quality feature embeddings, we can directly perform face verification by calculating the distance between two images (i.e. one-to-one matching). Face identification requires a database which contains a reference image of every identity. To identify a person on an image, the probe image's embedding should be compared with all images in the reference database (i.e. one-to-many matching).\n\\\\\n\\newline\nFace images are a popular biometric since they can be collected in unconstrained environments and without the user's active participation. These properties in combination with the widespread deployment of surveillance cameras give rise to some severe privacy concerns.  As a result, researchers explore techniques to secure the data collected by surveillance cameras and are developing privacy preserving facial recognition systems. The techniques differ in exactly what privacy-sensitive aspects they protect. Some techniques avoid deducing soft biometrics (e.g. age, gender, race) from data collected for verification or recognition purposes \\cite{mirjalili2018gender, mirjalili2020privacynet}. Other techniques focus on face image de-identification \\cite{newton2005preserving}, which eliminates the possibility of a facial recognition system to identify the subject while still preserving some facial characteristics. A very powerful technique hides the input image and the facial recognition result from the server that performs the recognition using a cryptographic enhanced facial recogniton system \\cite{erkin2009privacy}.\n\\\\\n\\newline\nA lot of effort has gone into the development of face image quality assessment (FQA) metrics. Face image quality is defined as the suitability of an image for consistent, accurate and reliable recognition \\cite{hernandez2020biometric}. FQA methods aim to predict a value which describes the quality of a probe image. Most previous FQA techniques focus on improving the performance of a face recognition system \\cite{wong_cvprw_2011, anantharajah2012quality, qi2018boosting, vignesh2015face, best2017automatic, hernandez2020biometric, terhorst2020ser}. An FQA system is then used as a first step to make sure that we only feed high quality images to the face recognition model, therefore increasing the accuracy and reducing the computational cost. A first family of FQA techniques (Full-reference and reduced-reference FQA) assumes the presence of a high quality sample of the probe image's subject. These methods do not work for unknown subjects, which is necessary for our purpose as we will explain in the next section. A second family of FQA techniques develops hand-crafted features to assess the quality of a face image \\cite{anantharajah2012quality, REFICAO} while more recent studies apply data driven methods and report a considerable increase in performance. Different studies \\cite{qi2018boosting, vignesh2015face} propose a supervised approach where a model is trained to predict the distance between the feature embeddings of two images. Since two samples are necessary to perform a comparison, one low quality sample can affect the quality score of a high quality sample. This is commonly solved by assuming that an image compliant with the ICAO guidelines \\cite{REFICAO} represents perfect quality \\cite{hernandez2020biometric}. In the work of Hernandez-Ortega et al. a pretrained resnet-50 network \\cite{he2016deep} is modified by replacing the classification layer with two regression layers which output the quality score. Alternatively, it is also possible to use human labeled data \\cite{best2017automatic}. The most similar to our work is \\cite{terhorst2020ser} which also proposes an unsupervised approach. Here the quality of an image is measured as its robustness in the embedding space, which is calculated by generating embeddings using random subnetworks of a selected face recognition model.\n\\\\\n\\newline\nCompared to previous work, we introduce a novel completely unsupervised FQA method based on a variational autoencoder. We assume no access to the identities of the people and show that it works well for unseen people. Unlike \\cite{terhorst2020ser}, no facial recognition model is used to calculate a quality score. In contrast to previous work our main goal is not necessarily to improve the facial recognition accuracy but instead we use it as a building block to enable a more privacy-friendly alternative to large scale face recognition, as explained in the next section.\n\n\\section{Frame selection as an alternative to face recognition}\n\\label{section:privacy_preserving_appr}\nIn this section we introduce a framework based on \\cite{simoens2013scalable} that uses intelligent frame selection in the context of face recognition to build a more privacy-friendly alternative to large scale face recognition. Instead of proactively trying to recognize individuals, we follow the presumption of innocence principle and do not indiscriminately perform recognition of people as they go about their daily business. Instead, our system uses key frame extraction to capture a high quality snapshot of every person passing by the camera. These snapshots are encrypted and stored locally on the camera or securely transmitted to a cloud back-end for storage. In a normal situation this data is then automatically deleted after a well defined time period as defined in article 17 of the GDPR (``Right to be forgotten''). In this case, the identity of the people will never be known and no human operator will have access to the unencrypted pictures. The images can only be decrypted after legal authorization in case of a criminal investigation or another situation where access to the identities present in a certain location at a certain time is warranted. The high quality images can then be used as input for a face recognition system or to aid the investigation process. Since only high quality crops are stored, the storage overhead is much lower than in a system where the full video is stored. This system is summarized in Figure \\ref{fig:overview} using a video from the ChokePoint dataset \\cite{wong_cvprw_2011}.\n\\\\\n\\newline\nIt is important to note that we need to store at least one crop for each person visible in the video. It is not enough to use a fixed threshold to decide whether a frame should be stored or not, instead we have to store the best frame for each individual even if this best frame still has a relatively low quality compared to images of other individuals (for example because the person never perfectly faces the camera). As a generalization we could also decide to store a short video clip of a few seconds before and after the best frame has been captured.\n\\\\\n\\newline\nAn obvious disadvantage of our approach is that it is not possible to proactively recognize people for example to detect wanted individuals. On the other hand it does support the criminal investigation after the fact. Our system is therefore complementary and more suited to low risk areas where a full blown face recognition system would violate the proportionality principle. \n\n\n\\section{Face image quality assessment}\n\\label{section:approach}\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/FQA_overview.png}\n    \\caption{Overview of our face quality-aware frame selection system.}\n    \\label{fig:fqa_overview}\n\\end{figure}\nThe system proposed in the previous section relies on a Face Quality Assessment block to decide which crops to store. Any FQA method can be used but in this section we introduce a novel technique based on a variational autoencoder. Compared to other FQA methods, this has the benefit of being completely unsupervised. We do not assume access to a face recognition system or to the identities of the people in the dataset. Our method also generalizes well to other people outside of the original training dataset.\n\\\\\n\\newline\n An overview of the proposed system is depicted in figure \\ref{fig:fqa_overview}. The first step is to detect all faces in each frame and track them across subsequent frames. We use the MTCNN model \\cite{zhang2016joint} to detect faces in a still video frame. The output is a list of bounding boxes around the detected faces. To track a subject across a video, we simply calculate the euclidean distance between the bounding boxes of subsequent frames. Bounding boxes that are close to each other are considered to correspond to the same subject. To evaluate the quality of a face crop, we calculate the reconstruction probability of a variational autoencoder (VAE) trained on a dataset of high quality images. The VAE reconstruction probability is commonly used as an anomaly detection metric \\cite{an2015variational} to determine how different an input is to the data seen during training. By training the VAE on a publicly available dataset of high quality face images, we reformulate the FQA task as an anomaly detection task (i.e. how different is this face crop from the high quality face crops seen during training?). The next paragraph explains the VAE and reconstruction probability in more details.\n\\\\\n\\newline\nA variational autoencoder (VAE) \\cite{kingma2013auto} is a probabilistic variant of the standard autoencoder (AE). The encoder and decoder are modeled by probabilistic distributions rather than deterministic functions. The encoder $f_{\\phi}(x)$ models the posterior $q_{\\phi}(z | x)$ of the latent variable $z$ and a decoder $f_{\\theta}(z)$ models the likelihood $p_{\\theta}(x | z)$ of the data $x$ given the latent variable $z$. The prior distribution of the latent variable $p_{\\theta}(z)$ is chosen as a Gaussian distribution $\\mathcal{N}(0, I)$. The posterior $q_{\\phi}(z | x)$ and likelihood $p_{\\theta}(x | z)$ are isotropic multivariate normal distributions $\\mathcal{N}(\\mu_{z}, \\sigma_{z})$ and $\\mathcal{N}(\\mu_{x}, \\sigma_{x})$ respectively. Figure \\ref{fig:vae} shows the process of a forward pass of an image $x$ through the VAE, the arrows represent a sampling process. To train a VAE using backpropagation, every operation should be differentiable which is not the case for the sampling operations: $z \\sim \\mathcal{N}(\\mu_{z}, \\sigma_{z})$ and $\\hat{x} \\sim \\mathcal{N}(\\mu_{x}, \\sigma_{x})$. Applying the re-parameterization trick fixes this problem. A dedicated random variable $\\epsilon \\sim \\mathcal{N}(0, 1)$ is sampled such that the sampling operations can be rewritten as: $z \\sim \\mu_{z} + \\epsilon \\cdot \\sigma_z$ and $\\hat{x} \\sim \\mu_{x} + \\epsilon \\cdot \\sigma_x$. The VAE training objective is written as the expected log likelihood minus the KL divergence between the posterior and the prior as described in equation \\ref{eq:train_obj}. \n\\begin{equation}\n    \\mathcal{L}(x) = E_{q_{\\phi}(z|x)}(p_{\\theta}(x|z)) - KL(q_{\\phi}(z|x)|p_{\\theta}(z))\n    \\label{eq:train_obj}\n\\end{equation}\nThe first term is the reconstruction term and forces a good reconstruction $\\hat{x}$ of the input data $x$. The KL term regularizes the distribution of the latent space by forcing it to be Gaussian. By training a generative model, like a VAE, the model learns to approximate the training data distribution. When a large reconstruction error is observed, this is an indication that the data is not sampled from the data distribution the VAE was trained on.\n\\\\\n\\newline\nThe reconstruction probability is a generalization of the reconstruction error by taking the variability of the latent space and the reconstruction into account \\cite{an2015variational}. First, an image $x$ is fed to the encoder which generates the mean vector $\\mu_z$ and standard deviation vector $\\sigma_z$. Next, $L$ samples $\\{z^0, z^1,...,z^l\\}$ are drawn from the latent distribution $\\mathcal{N}(\\mu_z,\\sigma_z)$. All samples $z^l$ are fed into the decoder to get the distribution of the reconstruction of $x$ which is described by the mean $\\mu^{l}_{\\hat{x}}$ and the standard deviation $\\sigma^{l}_{\\hat{x}}$. The reconstruction probability is the probability of $x$ averaged over L samples as described by equation \\ref{eq:r_prop}.\n\\begin{equation}\n    \\mathrm{RP}(x) = \\frac{1}{L} \\sum^{L}_{l=1} \\mathcal{N}(x | \\mu^{l}_{\\hat{x}}, \\sigma^{l}_{\\hat{x}})\n    \\label{eq:r_prop}\n\\end{equation}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=225pt]{figures\/VAE.png}\n    \\caption{Variational autoencoder with encoder $f_{\\phi}(x)$ and decoder $f_{\\theta}(z)$, each arrow represents a sampling processes.}\n    \\label{fig:vae}\n\\end{figure}\n\\newline\nThe reconstruction probability was originally developed as an anomaly score by \\cite{an2015variational}. When a VAE is trained solely on samples of ``normal'' data, the latent distribution learns to represent these samples in a low dimensional space. Accordingly samples from ``normal'' data result in a high reconstruction probability while anomalies result in low reconstructing probabilities. We define the biometric quality of a face image as the reconstruction probability calculated by a VAE trained on high quality face images. Correspondingly, a high reconstruction probability is expected for high quality face images. Note that there is no explicit definition of face image quality and the quality score is independent of any face recognition model. The definition of a high quality face image is derived directly from the training data. \n\\\\\n\\newline\nThe encoder $f_{\\phi}(x)$ consists of 5 consecutive blocks of a convolution layer, batch normalization and a leaky ReLU activation function with at the end two fully connected layers. The outputs of the encoder are the parameters defining $q_{\\phi}(z|x)$. The decoder $f_{\\theta}(z)$ consists of 5 blocks of a transposed convolution layer, batch normalization and a leaky ReLU activation function with again two fully connected layers at the end. The outputs of the decoder are the parameters defining $p_{\\theta}(x|z)$. To calculate the reconstruction probability, $L$ is set to 10. The CelebA dataset \\cite{liu2015faceattributes} consisting of 202,599 face images serves as training data. The Adam  optimization algorithm \\cite{kingma2014adam} was applied with a learning rate of 0.005 and a batch size of 144. The VAE was trained for 10 epochs. \nEach image is cropped by MTCNN \\cite{zhang2016joint}, resized to 64x64 pixels and converted to grayscale.\n\n\\section{Experimental setup}\n\\label{section:experimental_setup}\nIn this section, we isolate the FQA block for evaluation. According to the national institute of standards and technology (NIST), the default way to quantitatively evaluate a FQA system is analyzing the error vs. reject curve (ERC) \\cite{grother2007performance, grother2020ongoing}. As defined in section \\ref{section:related}, FQA indicates the suitability of an image for recognition. The ERC measures to what extent the rejection of low quality samples increases the verification performance as measured by the false non-match rate (FNMR). The FNMR is the rate at which a biometric matcher miscategorizes two signals from the same individual as being from different individuals \\cite{Schuckers2010}. Face verification consists of calculating a comparison score of two images and comparing this score with some threshold. The comparison score is defined as the euclidean distance between the FaceNet \\cite{schroff2015facenet} embeddings of the two images. To avoid a low quality sample affecting the verification performance of a high quality sample, one high quality reference image for every subject is required. For this, an ICAO compliant image is typically used \\cite{hernandez2020biometric}. We used the BioLab framework \\cite{ferrara2012face} to calculate an average ICAO compliance score for all images. For every subject,  the image with the highest ICAO score is selected as a reference image. Note that it is not possible to use the BioLab framework as face quality assessment method directly because it cannot assess all images accurately and it is unable to operate in real-time. Now, assume a set of genuine image pairs $(x_i^{(1)}, x_i^{(2)})$ (i.e. two images of the same person) of length N. Every image pair $(x_i^{(1)}, x_i^{(2)})$ constitutes a distance $d_i$ (i.e. comparison score). To determine if two images match, the distance between the two images is compared with a threshold $d_t$. Using quality predictor function $F$ (i.e. a FQA method), a quality value $q_i$ is calculated for each image pair. Since $x_i^{(1)}$ is always a high quality image from the reference database, the quality $q_i$ of image pair $(x_i^{(1)}, x_i^{(2)})$ can be written as:\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=225pt]{figures\/example.png}\n    \\caption{Sample images from the VggFace2 dataset ranked by different quality metrics.}\n    \\label{fig:example_images}\n\\end{figure}\n\\begin{equation}\n    q_i = q_i^{(2)} = F(x_i^{(2)}).\n\\end{equation}\nNow consider $R$ as a set of low quality entities composed by the samples that correspond with a predicted quality value below some threshold:\n\\begin{equation}\n    R(r) = \\{i : q_i < F^{-1}(r),  \\forall i < N \\}.\n\\end{equation}\n$F^{-1}$ is the inverse of the empirical cumulative distribution function of the predicted quality scores. The parameter $r$ is the fraction of images to discard, such that $F^{-1}(r)$ equals the quality threshold that corresponds with rejecting a fraction $r$ of all images. Then the FNMR can be written as:\n\\begin{equation}\n    \\mathrm{FNMR} = \\frac{|d_i : d_i \\geq d_t, i \\notin R(r)|}{|d_i : d_i \\geq -\\infty, i \\notin R(r)|}\n\\end{equation}\nThe value of $r$ is manipulated to calculate the FMNR for different fractions of rejected images. The value of $d_t$ is fixed and computed using the inverse of the empirical cumulative distribution function of the distances between reference and probe images $M^{-1}$:\n\\begin{equation}\n    d_t = M^{-1}(1 - f).\n\\end{equation}\nPractically, $f$ is chosen to give some reasonable FNMR. As suggested by \\cite{frontex} a maximum FNMR of 0.05 is maintained.\n\\section{Results}\n\\label{section:results}\n\\subsection{VggFace2}\nThe VggFace2 dataset \\cite{Cao18} is designed for training face recognition models and consists of more than 3.3 million face images of more than 9000 identities. In the following experiments, our approach (VAE) is compared to FaceQNet (FQN) \\cite{hernandez2020biometric}, a method based on the stochastic embedding robustness (SER) \\cite{terhorst2020ser} and a general image quality assessment (IQA) system \\cite{talebi2018nima}. FaceQNet is a fine-tuned resnet-50 network \\cite{he2016deep} which is trained on large amounts of face images for a face recognition task. The classification layer is then replaced by two layers designed for quality regression. The ground truth quality score is defined as the euclidean distance between the feature embeddings of the probe image and an ICAO compliant image of the same subject. The face image quality assessment method based on stochastic embedding robustness (SER) calculates a quality score by measuring the variations of the embeddings generated by random subnetworks of a resnet-101 \\cite{he2016deep} model trained for facial recognition. The general image quality assessment (IQA) system \\cite{talebi2018nima} does not take face features into account and predicts the general image quality. For all conducted experiments, images were cropped by the MTCNN face detector \\cite{zhang2016joint}.\n\\\\\n\\newline\nFigure \\ref{fig:example_images} shows five images from the VggFace2 dataset \\cite{Cao18} ranked by different quality metrics. This allows a first qualitative evaluation of the five metrics. As presented on the figure, all metrics agree on assigning the highest quality to the first image. All FQA metrics assign a low quality value to the third image because the person looks down, the general IQA method does not take this aspect into account and assigns a higher quality value. For the same reason, the IQA method assigns a low quality value to the fifth image in contrast to all FQA algorithms. On the fourth image, our method agrees with the IQA method and selects it as worst quality image as opposed to the other FQA metrics. We could assume that our method attaches more importance to blurriness compared to the other FQA algorithms. It is remarkable that the second image is ranked as second worst by FQN since the person looks straight into the camera.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=225pt]{figures\/FNMR_01.png}\n    \\caption{ERC with an initial FNMR of 0.01.}\n    \\label{fig:vggface2_fnmr_01}\n\\end{figure}\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=225pt]{figures\/FNMR_001.png}\n    \\caption{ERC with an initial FNMR of 0.001.}\n    \\label{fig:vggface2_fnmr_001}\n\\end{figure}\n\\\\\n\\newline\nIn a second experiment, we evaluate our proposed FQA algorithm on still face images by analyzing the ERC plots as explained in section \\ref{section:experimental_setup}. For 103.008 images of 300 subjects from the VggFace2 dataset \\cite{Cao18}, a high quality reference image is selected using the ICAO compliance scores. The ERC plots in figures \\ref{fig:vggface2_fnmr_01} and \\ref{fig:vggface2_fnmr_001} display the FNMR for different fractions of rejected images. The red line with the label ``PERFECT'' represents a quality measure which correlates perfectly with the distance scores. When an initial FNMR of 0.01 is set, in an ideal scenario, the FNMR will be zero after rejecting 1\\% of all images. The closer an ERC is to the red line, the better the performance of the used FQA algorithm. For an initial FNMR of 0.01 our approach clearly outperforms FaceQNet, SER and the general image quality assessment program. We hypothesize that SER would perform better when the same type of embeddings were used for verification as quality estimation. In the conducted experiments SER uses ArcFace \\cite{deng2018arcface} embeddings to estimate face image quality while the FNMR is calculated using FaceNet embeddings. For an initial FNMR of 0.001, the difference with the other approaches is smaller. It is important to note that our model is considerably smaller than FaceQNet and the resnet-101 model \\cite{he2016deep} used by SER.\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=225pt]{figures\/cp_map_0.jpg}\n    \\caption{Consecutive face crops from one tracked identity in the ChokePoint dataset. The crop with the green border corresponds with the highest quality calculated by our FQA algorithm.}\n    \\label{fig:cp_map_0}\n\\end{figure}\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=225pt]{figures\/cp_plot_0.png}\n    \\caption{Logarithm of the reconstruction probability (i.e. face quality) for consecutive face crops.}\n    \\label{fig:cp_plot_0}\n\\end{figure}\nFaceQNet comprises 7 times more parameters than our VAE and resnet-101 even 14 times more. Additionally, our method is trained completely unsupervised without the need for ground truth quality values while FaceQNet relies on distance scores as ground truth quality values. The ground truth generation process used by FaceQNet also indicates the dependency on one or more face recognition models. This dependency is even more prominent for SER since a specific face recognition network is used for generating the quality predictions.\n\\subsection{ChokePoint}\nIn a third experiment, we focus on the ChokePoint dataset \\cite{wong_cvprw_2011}. This dataset is designed for conducting experiments on face identification and verification under real-world surveillance conditions. The dataset consists of 48 videos or 64,204 face images. Both crowded and single-identity videos are included. We now evaluate our system qualitatively on selecting one high quality face crop of each identity in a video stream. Figure \\ref{fig:cp_map_0} shows consecutive face crops of an example video. The crop outlined in green is the frame that corresponds with the highest quality value calculated by our FQA algorithm. Figure \\ref{fig:cp_plot_0} shows how the quality score changes over time as the subject moves through the scene. We define the quality score as the logarithm of the reconstruction probability. We can see that initialy the quality score decreases as the person is moving through a darker area looking down. The shades and the angle of the face makes these crops less useful for face recognition. As the person moves closer to the camera, the brightness increases and the subject becomes clearly visible. This is also reflected in an increasing quality score. The highest score is obtained when the person is close to the camera and is looking in the general direction of the camera. As the person turns away, the score again decreases. This qualitative example shows that our model is indeed able to assign understandable and meaningful scores to each frame. We made videos of this and other examples publicly available \\footnote{\\url{https:\/\/drive.google.com\/drive\/folders\/1GRlFRSxHRfBnfTpI5DG2v2rN3nTCg5Y0?usp=sharing}}.   \n\\subsection{Bicycle parking}\nFinally, we also validated our approach on video data from security cameras in a bicycle parking lot. This to investigate how well the model generalizes to data collected in the real world. Figure \\ref{fig:bicycle_park} shows three  example frames with its corresponding frame crop and quality score. Even though these images are very different from the images the VAE was originaly trained on, we can see that the model generalizes well and is able to assign useful scores to each crop.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=225pt]{figures\/bicycle_park.png}\n    \\caption{Three frames from the footage of a security camera in a bicycle parking lot. The corresponding frame crops and quality scores are depicted below each frame.}\n    \\label{fig:bicycle_park}\n\\end{figure}\n\n\\section{Conclusion and future work}\n\\label{section:conclusion}\nIn this study, a novel face image quality assessment method is proposed based on a variational autoencoder's reconstruction probability. This is, by our knowledge, the first time a generative model like a VAE is used to tackle the problem of face image quality assessment. We demonstrate, by quantitative and qualitative results, that our method can be used as a biometric quality predictor. Unlike other data driven approaches, no facial recognition model is used for training and no explicit definition of face quality is given. Our FQA algorithm is used as a building block in a privacy-friendly alternative to large scale facial recognition. Instead of identifying all detected faces in a video stream, our system saves one high quality face crop without revealing the person's identity. This face crop is encrypted and access is only granted after legal authorization. In such a way, the system still supports criminal investigations while not violating the proportionality principle.\n\\\\\n\\newline\nIn future work, we will further optimize the VAE architecture keeping the constraints on model size and computational complexity in mind as the final goal would be to deploy the model on a stand-alone edge device. It would be interesting to investigate different hardware platforms such as FPGAs that allow the model to process data in real-time with a small energy consumption, making it possible to embed our system in low cost surveillance camera's. Moreover, our method should be evaluated on other datasets and in combination with alternative feature extractors.\n\n\\section{ Acknowledgments}\nWe thank Klaas Bombeke (MICT UGent) for providing us with the dataset used in the experiment of Figure~9, and Stijn Rammeloo from Barco for the feedback on the initial manuscript. This research was funded by the imec.ICON SenseCity project and the Flemish Government (AI Research Program).\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\subsection{Gauge fields}\n\nWe will consider $SU(N)$\ngauge fields \nin a four dimensional torus of size $L^4$. The twisted boundary\nconditions are implemented by requiring the field to gauge-transform\nunder the displacement of a period\n\\begin{equation}\n  A_\\mu(x+L\\hat \\nu) = \\Omega_\\nu(x)A_\\mu(x)\\Omega_\\nu^+(x) + \n  \\Omega_\\nu(x)\\partial_\\mu\\Omega_\\nu^+(x) \\,,\n\\end{equation}\nwhere $\\Omega_\\mu(x)$ are known as the twist matrices. The uniqueness\nof $A_\\mu(x+L\\hat\\mu+L\\hat\\nu)$ requires that the twist matrices have\nto obey the relation \n\\begin{equation}\n  \\label{eq:consistency}\n  \\Omega_\\mu(x+L\\hat \\nu)\\Omega_\\nu(x) = e^{2\\pi\\imath n_{\\mu\\nu}\/N}\n  \\Omega_\\nu(x+L\\hat \\mu)\\Omega_\\mu(x)\\,,\n\\end{equation}\nwhere $n_{\\mu\\nu}$ is an anti-symmetric tensor of integers modulo $N$\ncalled the twist tensor. It is easy to check that under a gauge\ntransformation, $\\Lambda(x)$, the twist matrices change according to\n\\begin{equation}\n  \\Omega_\\nu(x) \\longrightarrow \\Omega'_\\nu(x) = \\Lambda(x+L\\hat \\nu)\n  \\Omega_\\nu(x)\\Lambda^+(x)\\,,\n\\end{equation}\nbut the twist tensor $n_{\\mu\\nu}$ remains unchanged. Therefore all the\nphysics of the twisted boundary conditions is contained in the twist\ntensor, and the particular choice of twist matrices is \nirrelevant. One can restrict the gauge transformations to those that\nleave the twist matrices unchanged. It is easy to check that the\nnecessary and sufficient condition for the gauge transformations is to\nobey the periodicity condition\n\\begin{equation}\n  \\label{eq:gauge}\n  \\Lambda(x+L\\hat\\nu) = \\Omega_\\nu(x)\\Lambda(x)\\Omega_\\nu^+(x)\\,.\n\\end{equation}\n\nThe reader interested in knowing more about the twisted boundary\nconditions is invited to consult the review~\\cite{ga:torus}. Here we\nwill use a particular setup: we choose to twist only one plane\n(the $x_1-x_2$ plane) by choosing $n_{12} = -n_{21} = 1$, while the\nrest of the components of the twist tensor will be zero. This means\nthat our gauge connections will still be periodic in the $x_3$ and\n$x_4$ directions. As we will\nsee, this choice guarantees that the action has a unique minimum\n(modulo gauge transformations), and therefore it turns out to be a\nconvenient choice for perturbative studies. This is the reason why the very\nsame choice has been made before to define the Twisted Polyakov Loop\nrunning coupling scheme~\\cite{deDivitiis:1994yp}, or for other\nperturbative studies~\\cite{Luscher:1985wf}. We will closely follow the\nnotation and steps \npresented in~\\cite{Perez:2013dra}, a reference that the reader\ninterested in more details should consult.\n\nA convenient implementation of twisted boundary conditions consists in\nusing space-time independent twist matrices. In particular for the\nperiodic directions we set the twist matrices to one\n\\begin{subequations}\n\\begin{eqnarray}\n  \\Omega_{1,2}(x) &=& \\Omega_{1,2} \\\\\n  \\Omega_{3,4}(x) &=& 1\\,.\n\\end{eqnarray}\n\\end{subequations}\n\nWe will use latin indexes ($i,j,\\dots=1,2$) to run over the directions in the\ntwisted plane, while greek indexes ($\\mu,\\nu,\\dots=0,\\dots,3$) will\nrun over the four space time directions. The consistency relation\nEq.~(\\ref{eq:consistency}) implies the \nfollowing condition for the twist matrices\n\\begin{equation}\n  \\Omega_1\\Omega_2 = e^{2\\pi\\imath \/N}\n  \\Omega_2\\Omega_1.\n\\end{equation}\nNotice that the boundary conditions for the gauge\nfield with this choice of the twist matrices are\n\\begin{equation}\n  \\label{eq:twalg}\n  A_\\mu(x+L\\hat k) = \\Omega_kA_\\mu(x)\\Omega_k^+\\,,\n\\end{equation}\nand $A_\\mu=0$ is a valid connection. In fact we\nwill show that it is the only connection compatible with the boundary\nconditions that does not depend on $x$, and therefore it is the\nunique minimum of the action modulo gauge transformations. \n\nEq.~(\\ref{eq:twalg}) defines a generalization of the Dirac algebra. It\ncan be shown~\\cite{ga:torus} that there is a\nunique solution for the matrices $\\Omega_i$ modulo similarity\ntransformations. Introducing the \\emph{color momentum}, $\\tilde p_i =\n\\frac{2\\pi\\tilde n_i}{NL}$ with $n_i=0,\\dots,N-1$\nit is easy to check that the $N^2$ matrices\n\\begin{equation}\n  \\label{eq:defG}\n  \\Gamma(\\tilde p) = \\frac{\\imath}{\\sqrt{2N}}e^{\\imath \\alpha(\\tilde\n    p)} \\Omega_1^{-\\tilde \n    n_2}\\Omega_2^{\\tilde n_1}\\,,\n\\end{equation}\nwhere $\\alpha(\\tilde p)$ are arbitrary phases, are linearly\nindependent and obey the relation \n\\begin{equation}\n  \\Omega_i \\Gamma(\\tilde p) \\Omega_i^+ = e^{\\imath L\\tilde p_i}\n  \\Gamma(\\tilde p)\\,.\n\\end{equation}\nMoreover all but\n$\\Gamma(\\tilde p=0)$ are traceless, and therefore they can be used as\na basis of the Lie algebra of the gauge group. This means that any\ngauge connection can be expanded as\n\\begin{equation}\n  A_\\mu^a(x)T^a = \\sum_{\\tilde p}'\\hat A_\\mu(x,\\tilde p)e^{\\imath\n    \\tilde px}\\Gamma(\\tilde p).\n\\end{equation}\nThe prime over the sum means that the term $\\tilde p_i=0$ is\nabsent in the sum, as required for a $SU(N)$ gauge group. Notice that\nthe coefficients $\\hat A_\\mu(x,\\tilde p)$ are functions (not \nmatrices) periodic in $x$. Therefore one can do an usual Fourier\nexpansion and obtain\n\\begin{equation}\n  A_\\mu^a(x)T^a = \\frac{1}{L^4} \\sum_{p,\\tilde p}'\\tilde A_\\mu(p,\\tilde\n  p)e^{\\imath \n    (p+\\tilde p)x}\\Gamma(\\tilde p)\\,,\n\\end{equation}\nwith the usual spatial momentum \n\\begin{equation}\n  p_\\mu = \\frac{2\\pi n_\\mu}{L}\\quad (n_\\mu\\in \\mathbb Z)\\,.\n\\end{equation}\nFinally we define the \\emph{total} momentum as the sum of the color\nand space momentum $P_i = p_i+\\tilde p_i$, $P_{3,4} = p_{3,4}$. Noting\nthat any $P_\\mu$ can be \nuniquely decomposed in the space momentum and color momentum degrees\nof freedom we can safely write $\\Gamma(P)$ instead of $\\Gamma(\\tilde\np)$. Our main conclusion is that any gauge connection compatible\nwith our choice of boundary conditions can be written as \n\\begin{equation}\n  \\label{eq:gaugetw}\n  A_\\mu^a(x)T^a = \\frac{1}{L^4} \\sum_{P}'\\tilde A_\\mu(P)\n  e^{\\imath Px}\\Gamma(P)\\,.\n\\end{equation}\nIn particular the only connection that does not depend on $x$ is given\nby $\\tilde A_\\mu(P) = 0$. In general the matrices $\\Gamma(P)$ are not\nanti-hermitian, but one can choose the phases \n$\\alpha(P)$ of equation~(\\ref{eq:defG}) so that this condition is\nenforced\n\\begin{equation}\n  \\alpha(P) = \\frac{\\theta}{2}P_1P_2\\qquad \n  \\left(\n    \\theta = \\frac{NL^2}{2\\pi}\n  \\right)\\,.\n\\end{equation}\nIn this case, the Fourier coefficients $\\tilde A_\\mu(P)$ satisfy the\nusual relation\n\\begin{equation}\n  \\tilde A_\\mu(P)^* = \\tilde A_\\mu(-P)\\,,\n\\end{equation}\nand the $\\Gamma$ matrices are normalized according to\n\\begin{equation}\n  {\\rm Tr}\\left\\{ \\Gamma(P)\\Gamma(-P)\\right\\} = -\\frac{1}{2}\\,.\n\\end{equation}\n\nWe finally note that a simlar expansion is possible on the lattice,\nwith the only difference that the space momentum will be restricted\nto the Brillouin zone. \n\n\\subsection{Matter fields}\n\\label{sc:fermions}\n\nThe inclusion of matter fields interacting with gauge fields with\ntwisted boundary conditions is not completely straightforward. To\nunderstand why it is better first to consider how to include \nfermion fields in the fundamental representation. Since the twist\nmatrices tell us how fields change under translations, one naively\nexpects \n\\begin{equation}\n  \\psi(x+L\\hat i) = \\Omega_i\\psi(x)\\,,\n\\end{equation}\nbut one can easily see that this choice is not consistent, \nsince the value of the field $\\psi(x+L\\hat i+L\\hat j)$ depends on the\norder in which we perform the translations due to the\nnon-commutativity of the twist matrices. This difficulty can be\navoided by introducing more fermions, or what usually is called a\n``smell'' degree of freedom~\\cite{Parisi:1984cy}. If\n$\\alpha,\\beta=1,\\dots,N_s$ are indices that run over the $N_s$ smells of\nfermions, and $a,b=1,\\dots,N$ run over the color degrees of freedom,\nthe boundary conditions of the fermions read\n\\begin{equation}\n  \\psi^a_\\alpha(x+L\\hat i) =\n  e^{\\imath\\theta_i}(\\Omega_i)_{ab}(\\Omega^*_i)_{\\alpha\\beta} \n  \\psi^b_\\beta(x) \\,.\n\\end{equation}\nThis means that a fermion smell becomes a linear combination of the\ngauge transformed fermion smells under a translation. $\\theta_i$\nare in principle arbitrary, but introduced for \nconvenience to remove the zero-momentum modes of the Dirac\noperator. These phases have to be chosen\nsuch that they are not elements of the gauge group\n(i.e. $e^{\\imath\\theta} \\not\\in SU(N)$).\nThis choice of boundary conditions for the  \nfermion fields is consistent, but they require the number of\nsmells to be equal to the number of colors. One can easily extend the\nconstruction to the case when the ratio $N_s\/N$ is an integer, but in\ngeneral one can not have an arbitrary number of fermions in the\nfundamental representation.\n\nOn the other hand fermions in the adjoint representation transform in\nthe same way as the gauge fields, and therefore any number of fermions\nwould be compatible with the twisted boundary conditions. \n\nRegardless of the representation but assuming that the matter fields\nare compatible with the twisted boundary conditions, $\\mathcal O(a)$\nimprovement for massless Wilson quarks is automatically\nsatisfied since fields live on a torus, and the boundary conditions do\nnot break chiral symmetry (see~\\cite{Sint:2010eh,Frezzotti:2003ni}). \n\n\n\n\n\\subsection{Cutoff effects in the twisted running coupling}\n\nThe comparison of the lattice and the continuum computations of\n$\\mathcal E(t)$ can give us an idea of the size of cutoff effects (to\nleading order in $g_0^2$) of the twisted gradient flow coupling. We\nare going to study in detail the case of lattice simulations using the\nWilson action, the Wilson flow, and the clover definition for the\nobservable. If we\ndefine \n\\begin{equation}\n  \\hat{\\mathcal N}_T(c,a\/L) = \n  \\frac{c^4}{128}\\sum_P' e^{-\\frac{c^2L^2}{4}\\hat P^2}\n  \\frac{\\mathring P^2 C^2 - (\\mathring P_\\mu C_\\mu)^2}{\\hat P^2}\\,,\n\\end{equation}\nthe quantity\n\\begin{equation}\n  Q(c, a\/L) = \\left|\\frac{\\hat{\\mathcal N}_T(c,a\/L) - \\mathcal N_T(c)}\n{{\\mathcal N}_T(c)}\\right|\\,,\n\\end{equation}\nquantifies to leading order the size of cutoff effects as a function of the\nlattice size and the scheme parameter $c$. A global picture of cutoff\neffects for the groups $SU(2)$ and $SU(3)$ \ncan be seen in the figure~\\ref{fig:cut1}. \n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig\/cutoff}\n  \\caption{Cutoff effects to leading order of perturbation theory in\n    the twisted gradient flow coupling. As we \n  see, for $c\\in[0.3-0.5]$ cutoff effects are below the 7\\% for an\n  $L\/a=8$ lattice.}\n  \\label{fig:cut1}\n\\end{figure}\n\nThese figures may lead to the conclusion that a large value of $c$ is\noptimal. But from the point of view of lattice simulations, it is\nknown~\\cite{Fritzsch:2013je} that larger values of $c$ lead to larger\nstatistical errors when computing the coupling via lattice\nsimulations. For the typical lattice sizes ($L\/a\\sim 10-20$) that one\nuses in step scaling studies the values $c\\in[0.3,0.5]$ seem reasonable. \n\n\\subsection{Improved coupling definition}\n\nIf one is computing $t^2\\langle E(t)\\rangle$ non-perturbatively via\nlattice simulations, and one is using the Wilson action, the Wilson\nflow and the clover observable for the evaluation of the energy\ndensity observable, one can alternatively define the coupling via\n\\begin{equation}\n\\label{eq:latcou}\n  g_T^2(L) = \\hat{\\mathcal N}_T^{-1}(c,a\/L)t^2\\langle E(t) \\rangle\n  \\Big|_{t=c^2L^2\/8} \\,.\n\\end{equation}\nThis last coupling definition has the same properties, but one expects\nan improved scaling towards the continuum limit, since the leading\norder cutoff effects $\\propto g_0^2$ have been removed thanks to\nthe lattice factor $\\hat{\\mathcal N}_T(c,a\/L)$.\n\nIn a similar way, any choice of discretizations that define a coupling\ncan be normalized with a factor computed on the lattice\n(cf. section~\\ref{sc:disc}), leading to an improved scaling towards\nthe continuum. \n\n\n\n\n\\subsection{Perturbative behavior of the gradient flow in a twisted box: continuum}\n\n\\subsubsection{Generalities and gauge fixing}\n\nWe are interested in the perturbative expression for  $\\langle E(t)\n\\rangle$, and in order to avoid some difficulties in the definition of\npropagators, it turns out to be convenient to fix the gauge of the\nflow field $B_\\mu(x,t)$. This can be achieved by studying the modified\nflow equation\n\\begin{equation}\n    \\frac{{\\rm d} B_\\mu^{(\\alpha)}(x,t)}{{\\rm d}t} = D_\\nu^{(\\alpha)}\n    G_{\\nu\\mu}^{(\\alpha)}(x,t) +  \n  \\alpha D_\\mu^{(\\alpha)}\\partial_\\nu B_\\nu^{(\\alpha)}(x,t) \\,.\n\\end{equation}\nThe superscript ${(\\alpha)}$ recalls that covariant derivatives and field\nstrength are made of the modified flow field $B_\\mu^{(\\alpha)}(x,t)$,\nsolution of the previous equation. A solution of this modified flow\nequation $B_\\mu^{(\\alpha)}(x,t)$ can be transformed in a solution of the\noriginal flow equation~(\\ref{eq:flow}) by a time dependent gauge\ntransformation~\\cite{Luscher:2011bx}\n\\begin{equation}\n  B_\\mu = \\Lambda B_\\mu^{(\\alpha)}\\Lambda^{-1} + \n  \\Lambda \\partial_\\mu\n  \\Lambda^{-1} \\,,\n\\end{equation}\nwhere \n\\begin{equation}\n  \\frac{{\\rm d} \\Lambda}{{\\rm d}t} =\n  \\alpha \\Lambda \\partial_\\mu B_\\mu \\,;\\quad\n  \\Lambda\\big|_{t=0} = 1\\,.\n\\end{equation}\n\nTherefore gauge invariant quantities are independent of $\\alpha$. Note\nthat the previously defined gauge transformation \n$\\Lambda(x)$ belongs to the restricted set of gauge transformations\nthat leave the twist matrices invariant (see\nequation~(\\ref{eq:gauge})), and the boundary conditions of\n$B_\\mu^{(\\alpha)}$ are also independent of $\\alpha$. \n\n\\subsubsection{Flow field and energy density to leading order}\n\nThe particular choice $\\alpha=1$ simplify the computations, and we\nwill use it for the rest of this section. The modified flow equation\nreads in this case\n\\begin{equation}\n  \\label{eq:flowmd}\n  \\frac{{\\rm d} B_\\mu}{{\\rm d}t} = D_\\nu G_{\\nu\\mu} +\n  D_\\mu\\partial_\\nu B_\\nu \\,.\n\\end{equation}\nIn perturbation theory one re-scales the gauge potential with the bare\ncoupling $A_\\mu \\rightarrow g_0A_\\mu$, and the flow field has an\nasymptotic expansion in the bare coupling\n\\begin{equation}\n  \\label{eq:flowfg0}\n  B_\\mu(x, t) = \\sum_{n=1}^{\\infty} B_{\\mu,n}(x, t) g_0^n \\,.\n\\end{equation}\nTo leading order our flow equation~(\\ref{eq:flowmd}) is just the heat\nequation\n\\begin{eqnarray}\n  \\label{eq:flowlo}\n   \\frac{{\\rm d} B_{\\mu,1}(x,t)}{{\\rm d}t} &=& \\partial_\\nu^2\n   B_{\\mu,1}(x,t) \\\\\n   B_{\\mu,1}(x,0) &=& A_{\\mu}(x)\\, ,\n\\end{eqnarray}\nexpanding $B_{\\mu,1}(x,t)$ in our preferred basis~(\\ref{eq:gaugetw}) one\ncan easily solve~(\\ref{eq:flowlo}) and obtain\n\\begin{equation}\n  B_{\\mu,1}(x,t) = \\frac{1}{L^4}\n  \\sum_P' e^{-P^2t} \\tilde A_\\mu(P) e^{\\imath Px}\n  \\Gamma(P)\\,.\n\\end{equation}\n\nFinally our observable of interest also has an expansion in powers of\n$g_0$\n\\begin{equation}\n  \\langle E(t)\\rangle = -\\frac{1}{2}\\langle\n  {\\rm Tr}\\{G_{\\mu\\nu}(x, t)G_{\\mu\\nu}(x,t)\\}\\rangle = \\mathcal E(t) + \\mathcal\n  O(g_0^4)\\,.\n\\end{equation}\nOne can easily obtain \n\\begin{eqnarray}\n  \\mathcal E(t) &=& \\frac{g_0^2}{2}\\langle \n  \\partial_\\mu B_{\\nu,1}\\partial_\\mu B_{\\nu,1} - \\partial_\\mu\n  B_{\\nu,1}\\partial_\\nu B_{\\mu,1} \n  \\rangle  \\\\\n  \\nonumber\n  &=& \\frac{-g_0}{2L^8}\\sum_{P,Q}'e^{-(P^2+Q^2)t}e^{\\imath (P+Q)x} \n  \\left( P_\\alpha Q_\\alpha\\delta_{\\mu\\nu} -\n    P_\\mu Q_\\nu\\right) \\langle \\tilde A_\\mu(P)\\tilde A_\\nu(Q)\\rangle \n  {\\rm Tr}(\\hat\\Gamma(P)\\hat\\Gamma(Q))\\,.\n\\end{eqnarray}\nFinally using the expression for the gluon propagator\n\\begin{equation}\n  \\langle \\tilde A_\\mu(P)\\tilde A_\\nu(Q) \\rangle =\n  L^4 \\delta_{P_\\alpha, -Q_\\alpha} \\frac{1}{P^2}\n  \\left[ \\delta_{\\mu\\nu} - (1-\\lambda^{-1})\\frac{P_\\mu P_\\nu}{P^2}\\right]\n  \\frac{1}{{\\rm Tr}({\\Gamma(-P)\\Gamma(P)})} + \\mathcal O(g_0^2)\n\\end{equation}\none gets\n\\begin{equation}\n  \\label{eq:et}\n  \\mathcal E(t) = \n  \\frac{3g_0^2}{2L^4}\\sum_P' e^{-2P^2t}\\,.\n\\end{equation}\n\n\\subsection{Perturbative behavior of the gradient flow in a twisted box: lattice}\n\nWhen defining the gradient flow in the lattice one has to make several\nchoices. These basically correspond to the particular discretizations\nof the action whose gradient is used to define the flow, as well as\nthe discretization of the energy density and the choice of action that\none simulates (i.e. Wilson\/improved actions). \n\nFirst we will analyze the popular case where the Wilson action is\nsimulated, and one uses the same action to define the flow (in this\ncase is called the Wilson flow). The clover definition of the\nobservable has been a typical choice~\\cite{Luscher:2010iy} for a \ndiscretization of\nthe energy density. Later we will comment on the general case. \n\n\\subsubsection{Generalities and gauge fixing}\n\nOn the lattice the gradient flow is substituted by a discretized\nversion. There are several possibilities: one can use the Wilson\naction  \n\\begin{equation}\n  S_w(V) = \\frac{1}{g_0^2} \\sum_{\\rm p} {\\rm Re}\\{{\\rm Tr}(1-U_{\\rm p})\\}\n\\end{equation}\nwhere the sum runs over the oriented plaquettes, and define the flow\nequation by equating the time derivative of the links with\nthe gradient of the Wilson action\n\\begin{equation}\n  \\label{eq:flowlat}\n  a^2\\partial_t V_\\mu(x,t) = -g_0^2 \\{T^a\\partial_{x,\\mu}^a S_w(V)\\}\n  V_\\mu(x,t) \\,,  \\qquad V_\\mu(x,0) = U_\\mu(x)  \\,.\n\\end{equation}\nIn this case the gradient flow is usually referred as the Wilson\nflow. Some explanations of our notation are in order. \nIf\n$f(U_\\mu(x))$ is an arbitrary function of the link variable\n$U_\\mu(x)$, the components of its Lie-algebra valued derivative\n$\\partial_{x,\\mu}^a $  \nare defined as \n\\begin{equation}\n   \\partial_{x,\\mu}^a f(U_\\mu(x)) = \\left.\\frac{ {\\rm d} f(e^{\\epsilon\n            T^a}U_\\mu(x))}{ {\\rm d}\\epsilon} \\right|_{\\epsilon=0}\\,. \n\\end{equation}\nIn perturbation theory one is interested in a neighborhood of the\nclassical vacuum configuration. In this neighborhood the lattice  \nfields $U_\\mu(x)$ and $V_\\mu(x,t)$ are parametrized as follows:\n\\begin{align}\n  U_\\mu(x)   &= \\exp\\{ag_0 A_\\mu(x)\\}   \\;, &\n  V_\\mu(x,t) &= \\exp\\{ag_0 B_\\mu(x,t)\\} \\;.\n\\end{align}\n\nAgain it is convenient to study a modified flow equation where the\ngauge degrees of freedom are damped. We will consider\n\\begin{equation}\n  \\label{eq:flowlatmd}\n  a^2\\partial_t V_\\mu^\\Lambda(x,t) = g_0^2 \\left\\{ \n        -\\big[ T^a\\partial_{x,\\mu}^a S_w(V^\\Lambda) \\big] \n        + a^2\\hat D_\\mu^{\\Lambda}\\big[\\Lambda^{-1}(x,t)\\dot \\Lambda(x,t)\\big] \n                                           \\right\\} V_\\mu^\\Lambda(x,t) \\,,\n\\end{equation}\nwith $V_\\mu^\\Lambda(x,0) = U_\\mu(x)$ and the forward lattice covariant\nderivative \n$\\hat D_\\mu^{\\Lambda}$ acting on Lie-algebra valued functions according to\n\\begin{equation}\n  \\hat D_\\mu f(x) = \\frac{1}{a}\\left[\n    V_\\mu(x,t)f(x+\\hat\\mu)V_\\mu^{-1}(x,t) - f(x)\n  \\right] \\,.\n\\end{equation}\n\nSolutions of the modified and original flow equations are related by a\ngauge transformation\n\\begin{equation}\n  V_\\mu(x,t) = \\Lambda(x,t)V_\\mu^\\Lambda(x,t)\\Lambda^{-1} (x+\\hat\\mu,t)\\,.\n\\end{equation}\nThe most natural choice for $ \\Lambda(x,t)$ is the same functional\nused for gauge fixing \n\\begin{equation}\n  \\label{eq:lam}\n  \\Lambda^{-1}\\frac{{\\rm d} \\Lambda}{{\\rm d}t} = \\alpha\n  \\hat\\partial^*_\\mu B_\\mu(x,t)\\,,\\qquad \n  \\Lambda\\big|_{t=0} = 1\\,.\n\\end{equation}\nwhere $\\hat \\partial, \\hat \\partial^*$ denote the forward\/backward \nfinite differences. We again note that the boundary conditions of\n$V_\\mu^\\Lambda(x,t)$ are independent of $\\alpha$, since $\\Lambda(x,t)$\nbelongs to the restricted class of gauge transformations that leave\nthe twist matrices unchanged.\n\n\n\\subsubsection{Flow field and energy density to leading order}\n\nAgain the choice $\\alpha=1$ turns out to be convenient and we\nwill stick to it from now on.\n\nThe modified flow equation reads\n\\begin{equation}\n  a^2\\partial_t V_\\mu(x,t) = g_0^2 \\left\\{ -[T^a\\partial_{x,\\mu}^a\n      S_w(V)] + a^2\\hat D_\\mu(\\hat\\partial_\\nu^* B_\\nu ) \n      \\right\\}\n  V_\\mu(x,t) \\,,  \\qquad V_\\mu(x,0) = U_\\mu(x) \\,.\n\\end{equation}\nThe flow field can be expanded in powers of $g_0$\n(equation~(\\ref{eq:flowfg0})) and to first order in $g_0$ we have\n\\begin{equation}\n  \\label{eq:wflowlato1}\n  \\partial_t B_{\\mu,1}(x,t) = \\hat \\partial_\\nu\\hat\\partial_\\nu^*\n  B_{\\mu,1}(x,t) \\,.\n\\end{equation}\nExpanding the flow field in our favorite Lie-algebra basis\n(equation~(\\ref{eq:gaugetw})) one can write the solution to the\nprevious equation\n\\begin{equation}\n  B_{\\mu,1}(x,t) = \\frac{1}{L^4}\\sum_P' e^{-\\hat P^2t} \\tilde A_\\mu(P)\n  e^{\\imath Px} \\Gamma(P)\\,,\n\\end{equation}\nwhere \n\\begin{equation}\n  \\hat P_\\mu = \\frac{2}{a}\\sin\\left(a\\frac{P_\\mu}{2}\\right)\n\\end{equation}\nis the usual lattice momentum.\n\nWe can choose among different discretizations for the energy\ndensity. The most popular one consists in using the clover definition\nfor $G_{\\mu\\nu}(x,t)$~\\cite{Luscher:2010iy}. To leading order we have\n\\begin{eqnarray}\n  \\nonumber\n  \\hat G_{\\mu\\nu}(x,t) &=& \\frac{g_0}{2}\\,\\mathring\\partial_\\mu\\left[B_{\\nu,1}(x,t) + \n    B_{\\nu,1}(x-\\hat \\nu,t)\\right] \\\\\n  &-&\n  \\frac{g_0}{2}\\,\\mathring\\partial_\\nu\\left[B_{\\mu,1}(x,t) + \n    B_{\\mu,1}(x-\\hat \\mu,t)\\right] + \\mathcal O(g_0^2) \\,,\n\\end{eqnarray}\nwhere $\\mathring\\partial_{\\mu} = \\tfrac{1}{2}(\\hat \\partial_\\mu +\n\\hat \\partial^*_\\mu)$ is the symmetric finite difference. The energy\ndensity computed with the clover definition for the field strength\ntensor reads \n\\begin{equation}\n  \\langle E^{\\rm cl}(t)\\rangle = -\\frac{1}{2}\\langle {\\rm Tr}\\{ \\hat\n  G_{\\mu\\nu} \\hat G_{\\mu\\nu}\\} \\rangle = \\mathcal E^{\\rm cl}(t, a\/L) +\n  \\mathcal O(g_0^2) \n\\end{equation}\nUsing the definitions\n\\begin{subequations}\n\\begin{eqnarray}\n  \\mathring P_\\mu &=& \\frac{1}{a}\\sin\\left(aP_\\mu\\right)\\,,\\\\\n  C_\\mu &=& \\cos\\left(a\\frac{P_\\mu}{2}\\right)\\,,\n\\end{eqnarray}\n\\end{subequations}\nand the lattice gluon propagator, one can easily obtain\n\\begin{equation}\n  \\label{eq:etcl}\n  \\hat{\\mathcal E}^{\\rm cl}(t, a\/L) = \n  \\frac{g_0^2}{2L^4}\\sum_{P}' e^{-2\\hat P^2t}\n  \\frac{\\mathring P^2 C^2 - (\\mathring P_\\mu C_\\mu)^2}{\\hat P^2}\\,.\n\\end{equation}\n\n\n\\subsubsection[Some comments on different\n  discretizations]{Some comments on different\n  discretizations\\protect\\footnote{The author wants to thank S. Sint for his\n    help in understanding the points discussed in this section.}}\n\\label{sc:disc}\n\nIn general the lattice computation of the leading order behavior of\nthe energy density involves several choices of discretization: the\naction that one simulates (labelled (a)), the action whose gradient\ndefines the flow evolution (labelled (f)), and finally the\ndiscretization used to compute the observable (labelled (O)). To\nleading order, these three choices can be expressed as \nchoice of ``actions'' \n\\begin{subequations}\n  \\begin{eqnarray}\n    S_a[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(a)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2)\\,,\\\\\n    S_f[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(f)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2)\\,, \\\\\n    S_O[\\tilde A_{\\mu}] &=& \\frac{1}{4L^4}\\sum_{P}' \\tilde A_\\mu(-P)\n    K_{\\mu\\nu}^{(O)}(P) \\tilde \n    A_\\nu(P) + \\mathcal O(g_0^2) \\,.\n  \\end{eqnarray}\n\\end{subequations}\n\nThe matrices $K^{(a)}$ and $K^{(f)}$ may (and should) contain a gauge\nfixing part, but not the one corresponding to the observable\n$K^{(O)}$. In this way final results will be independent of the\nchoices of gauge.\nThe inverse of the $K_{\\mu\\nu}^{(a)}$ defines the lattice gluon propagator\n\\begin{eqnarray}\n  \\langle A_\\mu(-P)A_\\nu(P)\\rangle &=& D_{\\mu\\nu}(P)\\,, \\\\\n  K_{\\mu\\alpha}^{(a)}(P)D_{\\alpha\\nu}(P) &=& \\delta_{\\mu\\nu}\\,.\n\\end{eqnarray}\n\nUsing this notation it is trivial to obtain the form of the flow field\nto leading order\n\\begin{equation}\n  \\tilde B_{\\mu,1}(P) = \\left( \\exp\\{-t K^{(f)}(P)\\}\\right)_{\\mu\\nu}\n  \\tilde A_\\nu(P) = H_{\\mu\\nu}(t,P) \\tilde A_\\nu(P) \\,, \n\\end{equation}\nand noting that the reality of the action requires that $H^+(t,P) =\nH(t,-P)$,  we can write the expression of the energy density to\nleading order as \n\\begin{eqnarray}\n  \\mathcal E(t,a\/L) &=& g_0^2 \\langle S_O[\\tilde\n  B_{\\mu,1}]\\rangle \\\\ \n  &=& \\frac{g_0^2}{2L^4} \\sum_P' {\\rm Tr}\\{ H^+(t,P)K^{(O)}(P)H(t,P) \n  D(P)\\}\\,.\n\\end{eqnarray}\n\nThis formula allows an easy evaluation of the energy density, to\nleading order in perturbation theory, for any choice of\ndiscretizations. One general point that one can make is that if one\nuses the Wilson flow the matrix $H(t,P)$ can be chosen to be\nproportional to the identity (by an appropriate gauge choice), and\ntherefore commutes with any other matrix. Moreover if the \naction that one simulates is the same as the one that we use to\ncompute the observable, the product of matrices $DK^{O}$ together with\nthe trace simply result in a factor 3, and therefore one obtains\n\\begin{equation}\n  \\mathcal E(t,a\/L) = \\frac{3g_0^2}{2L^4} \\sum_P' e^{-2t\\hat P^2}\\,.\n\\end{equation}\n\nThis means that without changing the flow, improving\nthe action and the observable leads to exactly the same cutoff effects\nthan if one does not improve anything (to leading order). \n\n\\subsection{Tests}\n\nIn order to check the previous computations one can perform several\nconsistency checks. First it is obvious that the continuum result\n(equation~(\\ref{eq:et})) is recovered from the lattice one \n(equation~(\\ref{eq:etcl})) if one takes the limit $a\/L \\rightarrow\n0$. In the infinite volume limit boundary conditions are irrelevant,\nand therefore for $L\\rightarrow \\infty$ one should recover the result\nof~\\cite{Luscher:2010iy} that reads\n\\begin{equation}\n  \\mathcal E^{(L=\\infty)}(t) = \\frac{3g_0^2(N^2-1)}{128\\pi^2 t^2}\\,.\n\\end{equation}\n\nThis result is reproduced from our expression\nequation~(\\ref{eq:et}) by simply noting that \n\\begin{equation}\n  P_\\mu = \\frac{2\\pi}{L}\\left( n_\\mu + \\frac{\\tilde n_\\mu}{N}\\right)\\,,\n\\end{equation}\nand therefore \n\\begin{equation}\n  \\frac{1}{L^4} \\sum_P' \\xrightarrow[L\\rightarrow \\infty]{}\n  \\frac{1}{(2\\pi)^4}\\sum_{\\tilde p_i}' \\int_{-\\infty}^\\infty {\\rm d}^4P\\,.\n\\end{equation}\nFinally recalling that there are $N^2-1$ terms in the sum over $\\tilde\np_i$ (the term $\\tilde p_i=0$ is explicitly excluded) one obtains\n\\begin{equation}\n  \\mathcal E(t)\\xrightarrow[L\\rightarrow \\infty]{}\n  \\frac{3g_0^2}{32\\pi^4}\\sum_{\\tilde p_i}' \\int_{-\\infty}^\\infty {\\rm d}^4P\n  e^{-2P^2 t} = \\frac{3g_0^2(N^2-1)}{128\\pi^2 t^2}\\,.\n\\end{equation}\n\nTo check the lattice computations we have performed some dedicated\npure gauge lattice \nsimulations. We use the plaquette action of an $SU(2)$ gauge theory in\ntwo different volumes $L\/a=4^4$ and $L\/a= \n6^4$. We collect $10,000$ measurements of $\\langle E^{\\rm\n  cl}(t)\\rangle$ for different values of $t$ and $\\beta = 2\/g_0^2 =\n40, 80,  120,  200,  400,  560,  800,  960,  1120, 1280$. In these\nlarge-$\\beta$ simulations the \nmeasured $\\langle E^{\\rm cl}(t)\\rangle$ should\nreproduce the perturbative expression\n(equation~(\\ref{eq:etcl})). Being more precise, we will study\nnumerically the\nquantity\n\\begin{equation}\n  R(g_0, t) = \\frac{\\langle E^{\\rm cl}(t)\\rangle - \\mathcal E^{\\rm\n      cl}(t)}{\\mathcal E^{\\rm cl}(t)}\\,.\n\\end{equation}\nWe expect that $R(g_0,t) = \\mathcal O(g_0^2)$, and therefore by fitting\nthe data from the simulations to a linear behavior \n\\begin{equation}\n  R(g_0, t)  = m(t)g_0^2 + n(t)\n\\end{equation}\none should obtain an intercept $n(t)$ compatible with zero within\nerrors. Indeed this is the case, for different values of $t$ and $L$,\nas the reader can check in table~(\\ref{tab:fit}). A couple of\nrepresentative fits are shown in the figure~\\ref{fig:fit}.\n\n\\input{fit-table.tex}\n\n\\begin{figure}\n  \\centering\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/L4c030}\n    \\caption{Fit for $L=4^4$ and $c=0.3$.}\n  \\end{subfigure}\n  \\begin{subfigure}[b]{0.45\\textwidth}\n    \\includegraphics[width=\\textwidth]{fig\/L6c050}\n    \\caption{Fit for $L=6^4$ and $c=0.5$}\n  \\end{subfigure}\n\n  \\caption{Some representative fits to the large-$\\beta$\n    simulations. The plots show the function $R(g_0, t)$ at fixed\n    $t=c^2L^2\/8$ versus $g_0^2$.} \n  \\label{fig:fit}\n\\end{figure}\n\n\n\n\n\n\n\\section{Introduction}\n\n\\input{intro.tex}\n\n\\section{Twisted boundary conditions}\n\\label{sc:tw}\n\\input{bc.tex}\n\n\\section{The gradient flow in a twisted box}\n\\label{sc:flow}\n\\input{flow.tex}\n\n\\section{Running coupling definition}\n\\label{sc:coupling}\n\\input{coupling.tex}\n\n\\section{$SU(2)$ running coupling}\n\\label{sc:run}\n\\input{running.tex}\n\n\n\\section{Conclusions}\n\\input{conclusions.tex}\n\n\n\\section*{Acknowledgments}\n\nThis work has a large debt with\nM. Garc\\'ia Perez and A. Gonz\\'alez-Arroyo for sharing some of their results\nand notes before publication and for the many illuminating\ndiscussions. The help and advice of R. Sommer and U. Wolff was\ninvaluable in many of the steps of this work. \n\nI also want to thank my colleagues at DESY\/HU, specially\nP. Korcyl, P. Fritzsch, S. Sint and R. Sommer for the very many interesting\ndiscussions and their careful reading of the manuscript. D. Lin was very\nkind reading and helping to improve a manuscript of this work. \n\n\n\n\\subsection{Numerical computation of the step scaling function and\n  running coupling}\n\n\\subsubsection{Simulation details}\n\nWe will simulate $SU(2)$ YM theory using the Wilson action\n\\begin{equation}\n  S = \\frac{\\beta}{4}\\sum_{\\rm p} {\\rm Tr}\\left\\{ 1-U_{\\rm p}\\right\\}\n\\end{equation}\nwhere the sum runs over all oriented plaquettes. We simulate lattices\nof size $L\/a=20, 24, 30, 36$, and in order \nto compute the step scaling function also lattices of half this\nsize ($L\/a=10, 12, 15, 18$). The range of values of $\\beta$ (between 2.75\nand 12.0) translate to renormalized couplings\n$g_{\\rm TGF}^2(L)$ between 7.5 and 0.6 (for $c=0.3$), enough to cover both the\nnon-perturbative and perturbative regions of the \ntheory. Appendix~\\ref{ap:values} collects the values of the $g^2_{\\rm\n  TGF}(L)$ of our simulations. \n\nWe will use a combination of heatbath~\\cite{Creutz:1980zw,Fabricius:1984wp,Kennedy:1985nu} and\noverrelaxation~\\cite{Creutz:1987xi} as suggested\nin~\\cite{Wolff:1992nq}. In particular we \nchoose to do one heatbath sweep followed by $L\/a$ overrelaxation\nsweeps. Since measuring the coupling (i.e. integrating the flow\nequations) is numerically more expensive \nthan the Monte Carlo updates, we repeat this process 50 times between\nmeasurements. \n\nIn total we collect 2048 measurements of the coupling for each lattice\nsize, each value of $\\beta$, and several values of\n$c\\in[0.3,0.5]$. These measurements are collected in $N_r$ \nparallel runs (replica) of length $N_{\\rm MC}$ each so that $N_r\\times\nN_{\\rm MC} = 2048$. \nWe check that there are no autocorrelation between\nmeasurements (i.e. $\\tau_{\\rm int}=0.5$ within errors), even for our\nlarger lattices and larger values of $c$. We conclude that we can\nsafely consider the measurements independent. \n\nThe Wilson flow equations are integrated using the adaptive step size\nintegrator described in appendix D of~\\cite{Fritzsch:2013je}. With\nthis scheme we \nmake sure that the integration error in each step is not larger than\n$10^{-6}$. \n\n\\subsubsection{Data analysis}\n\nFor each $L\/a$ we have computed the value of the twisted gradient flow\ncoupling at different values of\n$\\beta$ (we call it $g^2_{\\rm TGF}(\\beta;L\/a)$). These data are fitted to a\nPad\\'e-like ansatz\n\\begin{equation}\n  \\label{eq:pade}\n  g^2_{\\rm TGF}(\\beta;L\/a) = \\frac{4}{\\beta} \\quad\n  \\frac{\\sum_{n=0}^{M-1} a_n\\beta^n + \\beta^M}\n  {\\sum_{n=0}^{M-1} b_n\\beta^n + \\beta^M}\\,.\n\\end{equation}\nThis fit imposes the one-loop constraint to the data (i.e. $g^2_{\\rm\n  TGF}(\\beta;L\/a) \\rightarrow 4\/\\beta$ at large $\\beta$), and has\na total of $2M$ free fit parameters. \n\nAlternatively, and to estimate the dependence of our results on the \nchoice of functional form used to fit the data, we use a different Pad\\'e\ninspired functional form\n\\begin{equation}\n  \\label{eq:taylor}\n  g^2_{\\rm TGF}(\\beta;L\/a) = \\frac{4}{\\beta} \\quad\n  \\frac{1}\n  {1 + \\sum_{n=1}^{M} c_n\/\\beta^n}\\,,\n\\end{equation}\nthat also ensures the correct one-loop behavior at large $\\beta$.\n\nWe obtain good fits ($\\chi^2\/{\\rm\nndof}\\sim 0.6-1.9$) with $M=2$ when using the functional form of\nEq.~\\ref{eq:pade} to fit the lattice data (i.e. 4 fitting\nparameters). When using the functional form of Eq.\\ref{eq:taylor} we\nneed $M=4$ to accurately describe the data on the small lattices\n($L\/a=10,12$) and $M=6$ for the larger ones\n($L\/a=15,18,20,24,30,36$). It is important to \nstress that the data are statistically uncorrelated, since they\ncorrespond to different simulations. \n\nIn the figures~\\ref{fig:fit_l24} we show a couple of these fits. Our\nworst fit corresponds to the $L\/a=24$ lattice and \nthe Pade fit gives a $\\chi^2\/{\\rm ndof}=1.69$, while the Taylor fit\nresults in a fit quality of $\\chi^2\/{\\rm ndof}=1.9$. We see how\nin this case the two different functional forms interpolate\ndifferently between the data, giving us confidence that if one\nestimates the error of the interpolation using both functional forms,\none will be on the safe side\\footnote{We point that probably a more\nsophisticated analysis technique (or simply, simulating an additional\nlattice to avoid having large gaps in the data), might result in a\nmore precise result.}. \n\\begin{figure}\n  \\centering\n        \\begin{subfigure}[t]{0.45\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/fit_l24}\n          \\caption{}\n        \\end{subfigure}\n        \\begin{subfigure}[t]{0.45\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/fit_l36}\n          \\caption{}\n        \\end{subfigure}\n  \\caption{Some examples of our fits to interpolate the values of the\n    renormalized coupling for different values of $\\beta$. \n    (a): Our worst fits corresponds to the $L\/a=24$. As we\n            can see there is a difference between the different\n            interpolating functions between the data points. We\n            stress that this systematic effects is taken into account\n            in our analysis by using both functional forms to estimate\n          the error of the interpolations (see the text for more details).\n    (b): Fits to the data of the $L\/a=36$ lattice. As we can\n            see, in this case both interpolating functions agree\n            within errors, although the polynomial fits tends to have\n            larger errors.\n}\n  \\label{fig:fit_l24}\n\\end{figure}\n\nWe use resampling methods to propagate errors by using 4000 bootstrap\nsamples. All fitting parameters derived from our original data are\ncomputed for each bootstrap sample. Interpolation points are computed\nfor each bootstrap sample and each functional form. The final error of\nthe interpolated point is computed using \\emph{both} functional forms\nand \\emph{all} bootstrap samples,\nand therefore takes into account not only the statistical uncertainty,\nbut also the systematic effect due to the dependence of the\ninterpolating functional form. \n\n\\subsubsection{Step scaling function}\n\nWe will first show the continuum extrapolations of the step scaling\nfunction $\\Sigma(u,a\/L)$ at some representative values of\n$u=7.5, 3.75, 1.5$. Figure~\\ref{fig:ss} shows that these\nextrapolations are mild. We have used the value $c=0.3$ that gives a\nprecision in the data for the renormalized coupling between $0.15\\%$\nand $0.25\\%$.\n\nOne of the advantages of the use of the\ntwisted boundary conditions is the absence of $\\mathcal O(a)$ cutoff\neffects, that are present for example in the Schr\\\"odinger functional\ndue to boundary effects. Here the invariance under\ntranslations guarantee that the continuum limit can be safely taken by\na linear extrapolation in $(a\/L)^2$.\n\n\n\\begin{figure}\n  \\centering\n        \\begin{subfigure}{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_75}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_375}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{\\textwidth}\n          \\centering\n          \\includegraphics[width=\\textwidth]{fig\/extra_15}\n        \\end{subfigure}\n  \\caption{Examples of the continuum extrapolation of the step scaling\n  function. The three figures corresponds (from top to bottom) to the\n  values $u=7.5, 3.75, 1.5$. We recall that we use a scale factor\n  $s=1\/2$, and the scheme is defined by the parameter $c=0.3$.} \n  \\label{fig:ss}\n\\end{figure}\n\n\n\\subsubsection{Running coupling}\n\nAs a final application, we will compute the running coupling.\nWe will fix the scheme by setting \n$c=0.3$. We start our recursion in a volume $L_{\\rm max}$ defined by the\ncondition\n\\begin{equation}\n  g^2_{\\rm TGF}(L_{\\rm max})\\Big|_{c=0.3} = 7.5\\,.\n\\end{equation}\nThe lattice step scaling function and its continuum limit is computed\nas described in the previous sections. As figure~\\ref{fig:ss}\nshows, the extrapolations towards the \ncontinuum are rather flat. The continuum limit values are used to\nfurther compute the values of the step scaling function at larger\nrenormalization scales (smaller volumes), up to $L_{\\rm min} = L_{\\rm\n  max}\/2^{26}$, \nwhere $g^2_{\\rm TGF}(L_{\\rm min})|_{c=0.3}=0.5324(84)$.\n\nSince the same functional form (fitting parameters) are used\nrecursively to compute the values of the coupling at different scales,\none has to propagate errors taking into account the correlations\ncorrectly. This is done in the spirit of the resampling methods in the\nmost naive way: one uses as input for the coupling at a scale $L$ all\nthe bootstrap samples of the coupling from the scale $2L$. We recall\nhere that these bootstrap samples carry the information not only of\nthe statistical uncertainties, but also of the dependence of our\nresults on the functional form chosen to fit the data.\nOur results have carefully taken into account the two sources of\nsystematic uncertainty: the continuum extrapolation and the choice of\nfitting function for our lattice data.\n\nFigure~\\ref{fig:gvsL} shows the running of the coupling from the low\nenergies to the high energies, over a factor $2^{26}$ change in scale,\nwhile table~\\ref{tab:gvsL} contains the numerical values of the\ncoupling at different renormalization scales. The fact that the\nabsolute error in the renormalized coupling tends to be constant a\nlarge energies (small volumes), is a consequence of the error\npropagation, that dominates for large energies the error budget. \n\\begin{figure}\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig\/gvsL}\n  \\caption{$g^2_{\\rm TGF}(L)$ as a function of the renormalization\n    scale $\\log(L\/L_{\\rm min})$, and a comparison with the two loop\n    perturbative prediction. Errors are plotted, but compatible with\n    the size of the points.}\n  \\label{fig:gvsL}\n\\end{figure}\n\nAs a further consistency test, we have repeated the full running of\nthe coupling using as scale factor to define the step scaling function\n$s=2$ (i.e. we run from high to low energies), obtaining\nconsistent results. \n\n\\begin{table}\n  \\centering\n  \\begin{tabular}{l|llllll}\n    \\toprule\n    $L=L_{\\rm max}\/2^k$ & $k=0$ & $k=1$ & $k=2$ & $k=3$ & $k=4$ & $k=5$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 7.5 & 4.824(17) & 3.581(15) & 2.858(12) &\n    2.383(10) & 2.0464(95) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$ & $k=11$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 1.7949(94) & 1.5995(94) & 1.4432(93) &\n    1.3153(92) & 1.2085(90) & 1.1181(89) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=12$ & $k=13$ & $k=14$ & $k=15$ & $k=16$ & $k=17$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 1.0405(87) & 0.9732(86) & 0.9143(84) &\n    0.8621(83) & 0.8158(83) & 0.7742(82) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=18$ & $k=19$ & $k=20$ & $k=21$ & $k=22$ & $k=23$ \\\\\n    $g^2_{\\rm TGF}(L)$ & 0.7368(82) & 0.7028(82) & 0.6720(82) &\n    0.6437(82) & 0.6178(82) & 0.5939(83) \\\\\n    \\midrule\n    $L=L_{\\rm max}\/2^k$ & $k=24$ & $k=25$ & $k=26$ &  &  &  \\\\\n    $g^2_{\\rm TGF}(L)$ & 0.5718(83) & 0.5514(84) & 0.5324(84) &&& \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Values of the renormalized twisted gradient flow coupling\n    as a funtion of the renormalization scale $\\mu = 1\/cL$ for\n    $c=0.3$. The final error at large scales\n    (small volumes) is dominated by the error propagation.}\n  \\label{tab:gvsL}\n\\end{table}\n\nThe $\\Lambda$ parameter can be extracted, in units of $L_{\\rm max}$\nvia\n\\begin{equation}\n  \\Lambda = \\mu(\\beta_0g^2(\\mu))^{-\\beta_1\/2\\beta_0^2} e^{-1\/2\\beta_0g^2(\\mu)}\n  e^{-\\int_0^{g^2(\\mu)}\\left\\{\\frac{1}{\\beta(x)}+\\frac{1}{\\beta_0x^3}-\\frac{\\beta_1}{\\beta_0^2x}\\right\\}}\\,,\n\\end{equation}\nusing that $\\mu = 1\/cL$. The previous formula is exact, but the last\nexponential is essentially unknown analytically. Nevertheless if one\nuses a value of $g^2_{\\rm TGF}(L)$ where the difference between the two loop\nand the non-perturbative results are negligible, the effect of the\nlast exponential is also negligible. Of course this is\nmore certain the smaller the coupling, but since the relative error of\nthe coupling grows as the coupling decreases, this would translate in\na larger error for the $\\Lambda$ parameter. Below we quote a couple of\nvalues as example. \n\\begin{eqnarray*}\n  \\Lambda L_{\\rm max} = 1.509(44)\\quad (@ g_{TGF}^2(L) = 1.7949(94))\\,,\\\\\n  \\Lambda L_{\\rm max} = 1.57(13)\\quad  (@ g_{TGF}^2(L) = 1.0405(87))\\,.\n\\end{eqnarray*}\n\nWe want to end this section with a small comment on the use of\ndifferent values of $c$. The main point has already been\nraised in~\\cite{Fritzsch:2013je}: the larger the value of $c$, the larger\nthe (relative) statistical error of the coupling, but the scaling\ntowards the continuum seems better. This general behavior is\nconsistent with the leading order in perturbation theory as we have\nseen. We will simply say that the relative error in the raw data\nincreases with $c$, and roughly one can say that for $c=0.4$ the\nrelative error is two times larger than for $c=0.3$, while for\n$c=0.5$ the error is three times larger. This statement seem to hold\ntrue independently of the volume (i.e. of the value of $g^2_{\\rm TGF}$). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The criterion illustrated by the cuprate example}\n\\label{app:example}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=.7]{fig4a.eps}\n\\includegraphics[scale=.7]{fig4b.eps}\n\\caption{(a)~The 2D nodal band structure and its projections on $(1\\bar{1})$ and $(01)$\nsurfaces for a model $d$-wave superconductor. The first BZ and one extended zone are drawn. The solid black curves denote the Fermi surface and the shaded region is filled in the normal state.\nRed arrows are the unit vectors in \\Eq{n}. The blue dots with $\\pm$ signs represent nodes with\nvorticity $\\pm 2$ respectively. Top left: the slanted thin black line segment is the surface\nBZ of the $(1\\bar{1})$ edge. The black arrow with letter ``P'' indicates the direction of projection.\nThick red line segments mark the surface momenta with zero energy bound states. Top right:\nthe thin black horizontal line segment is the BZ of the $(01)$ edge.\nThe $\\pm$ vorticity nodes overlap after projection.\nThe  vorticity of the node enclosed by the parallelogram is equal to the winding number difference along the two side blue segments because the winding along the top and bottom segments cancel due to the periodicity in momentum space. This can be explicitly seen by following the turning of the red arrows (the actual winding number is twice of the winding shown by the arrows, due to the spin degeneracy). (b) The (11) edge bandstructure. The surface flat bands are marked red.  In constructing this figure we have used $\\epsilon(\\v k)=-\\cos k_x-\\cos k_y-\\mu$ and $\\Delta(\\v k)=\\Delta_0 (\\cos k_x-\\cos k_y)$ in \\Eq{dwave}. Here  $\\mu=0.45,\\Delta_0=0.1$.}\n\\label{fig:d-wave}\n\\end{center}\n\\end{figure}\n\nThe idea behind the criterion presented in the main text is \nbest illustrated by using the cuprate superconductor as an example.\nThe Bogoliubov-de Gennes (BdG) Hamiltonian of the cuprate superconductor read\n\\be\nH_{\\rm cuprates}(\\v k)=\\epsilon(\\v k)\\s_0\\otimes\\tau_3+\\Delta(\\v k)\\s_0\\otimes\\tau_1,\\label{dwave}\n\\ee\nwhere $\\s_0$ is the identity matrix acting in the spin space and $\\tau_{1,3}$ are $2\\times 2$ Pauli matrices in the Nambu space, $\\epsilon(\\v k)$ is the normal state dispersion satisfying $\\e(-\\v k)=\\e(\\v k)$ and $\\Delta(\\v k)$ is the d-wave gap function. (Since the cuprates are quasi two dimensional materials, we shall use two dimensional notations in the following discussions.) The Fermi surface and the gap nodes are shown in \\Fig{fig:d-wave}a, therefore $d=2,q=0$. In the same figure, the normalized vector  \\be \\hat{n}(\\v k)=(\\epsilon(\\v k),\\Delta(\\v k))\/\\sqrt{\\epsilon(\\v k)^2+\\Delta(\\v k)^2}\\label{n}\\ee is plotted as a function of $\\v k$ over two BZs (see red arrows). Inspecting these arrows one notices each node is a ``vortex'' in $\\hat{n}(\\v k)$. Around each vortex the arrows exhibit non-zero winding. The total winding number associated with each node is given by \n\\be\nw={2\\over 2\\pi}\\oint d\\v p\\cdot [n_1(\\v k)\\gr_{\\v k} n_2(\\v k)-n_2(\\v k)\\gr_{\\v k} n_1(\\v k)].\n\\label{w}\n\\ee (The extra factor of 2 is due to spin degeneracy). Clearly each node is characterized by an even integer winding number. The BdG Hamiltonian defined on all one ($=d-q-1$) dimensional loop enclosing the node are topologically nontrivial.\n\nNow consider the bandstructure projected along the $(1\\bar{1})$ direction. For each transverse momentum $k$ along $(11)$ we have a 1D chain running in $(1\\bar{1})$.\nSo long as $k$ does not coincide with the projection of the nodes the spectrum is fully gapped and characterized by the winding number defined in \\Eq{w}. Any two chains whose $k$ straddle the projection of a node their winding numbers must differ by $\\pm 2$ (see \\Fig{fig:d-wave}a captions), hence at least one of them is topologically non-trivial and possess $E=0$ end states\nwhen the boundary condition along $(1\\bar{1})$ changes from closed to open.\n~This implies  E=0 bound states exists for {\\em intervals} of $k$. Therefore $d_{E=0}$ is indeed $q+1=1$. An example of the (11) boundary bandstructure is shown in \\Fig{fig:d-wave}b. The $k$ intervals showing the flat bands are represented by the thick red line segments in the top left corner of \\Fig{fig:d-wave}a.  The only edges which do not possess ZBABS are the $\\{10\\}$ (Miller's notation is used) edges where the projection of positive and negative nodes overlap (see top right corner of \\Fig{fig:d-wave}a).\nFor the real material $d=3, q=1$ and the only modification is $d_{E=0}$ changes from 1 to 2.\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{intro}Introduction}\nHawks and Doves, also known as Chicken, or the Snowdrift game, is a two-person, symmetric\ngame with the following payoff bi-matrix:\n\n\\begin{table}[!ht]\n\\begin{center}\n{\\normalsize\n\\begin{tabular}{c|cc}\n & H & D\\\\\n\\hline\n{\\rule[-3mm]{0mm}{8mm}}\nH & ($\\frac{G-C}{2},\\frac{G-C}{2}$) & ($G,0$)\\\\\n{\\rule[-3mm]{0mm}{4mm}}\nD & ($0,G$) & ($\\frac{G}{2},\\frac{G}{2}$)\\\\\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\\vspace{-0.4cm}\n\n\n\\noindent In this matrix, H stands for ``hawk'', and D stands for ``dove''.\nMetaphorically, a hawkish behavior means a strategy of fighting, while a dove, when facing a confrontation, will always yield.\nAs in the \n\\textit{Prisoner's Dilemma} \\cite{axe84}, this game, for all its simplicity, appears to\ncapture some important features of  social interactions. In this sense, it applies\nin many situations in which ``parading'', ``retreating'', and ``escalading'' are common.\nOne striking example of a situation\nthat has been thought to lead to a Hawk-Dove dilemma is the Cuban missile crisis in 1962\n \\cite{poundstone92}.\nIn the payoff matrix above, $G > 0$ is the gain that a hawk obtains when it meets a dove; the dove retreats and looses nothing.\nIf a dove meets another dove, one of them, or both, will retreat and they will gain half of\nthe price each ($G\/2$) in the average. Finally, when a hawk meets another hawk, they \nboth fight and each obtains an average payoff of $(G-C)\/2$, where $C$ is the cost of any\ninjury that might occur in the fight. It is assumed that $C > G$, i.e. the cost of injury\nalways exceeds the prize of the fight.\nThe game has the same structure as the Prisoner's Dilemma in that if both players\ncooperate (i.e. they play dove), they both gain something, although there is a strong motivation\nto act aggressively (i.e. to play the hawk strategy). However, in this game one makes the\nassumption that one player is willing to cooperate, even if the other does not, and that mutual defection, i.e. result (H,H), is detrimental to both players. \n\nIn contrast to the Prisoner's Dilemma which has a unique Nash equilibrium that corresponds to\nboth players defecting, the Hawk-Dove game has two Nash equilibria in pure strategies\n(H,D) and (D,H), and a third equilibrium in mixed strategies where strategy H is played\nwith probability $G\/C$, and strategy D with probability $1 - G\/C$. Note that we only consider\none-shot games in this work; repeated games are not taken into account.\n\nConsidering now not just two players but rather a large, mixing population of identical players,\n\\textit{evolutionary\ngame theory} \\cite{hofb-sigm-book-98} prescribes that the only evolutionarily\nstable strategy (ESS) of the population is the mixed strategy, giving rise, at equilibrium,\nto a frequency of hawks in the population equal to $G\/C$.\nIn the case of the Prisoner's Dilemma, one finds a unique ESS with all the individuals defecting.\nHowever,\nin 1992, Nowak and May \\cite{nowakmay92} showed\nthat cooperation in the population is sustainable in the Prisoner's Dilemma under certain conditions,\nprovided that the network of the interactions between players has a lattice spatial structure. Killingback and Doebeli \\cite{KD-96} extended the\nspatial approach to the Hawk-Dove game and found that a planar lattice structure\nwith only nearest-neighbor interactions may favor cooperation, i.e. the fraction of doves in\nthe population is often higher than what is predicted by evolutionary game theory. In addition,\ncomplex dynamics resembling phase transitions were observed, which is not the case in the\nmixing population.\nIn a more recent work however, Hauert and Doebeli  \\cite{hauer-doeb-2004} were led to a different conclusion, namely that\nspatial structure does not seem to favor cooperation in the Hawk-Dove game. \nAdditional\nresults on the Hawk-Dove game on a two-dimensional lattice have been recently obtained by Sysi-Aho et al.\n\\cite{myopic-hd-05} using a simple local decision rule for the players that does not reduce to the customary\n\\textit{replicator} or \\textit{imitation} dynamics \\cite{hofb-sigm-book-98}. They concluded that,\nwith their player's decision rule, cooperation persists, giving results different from those obtained\nwith the replicator dynamics. \nThese apparently\ncontradictory results aroused our curiosity, and motivated us to study the situation in a more general\nsetting, in which the mixing population and the lattice are special cases. \n\nFollowing pioneering work by sociologists in the sixties such as that of Milgram\n\\cite{milgram67}, in the last few years it has become apparent that the topological structures of social\ninteractions networks have particular, and partly unexpected, properties that are a consequence\nof their \\textit{small-world} characteristics. Roughly speaking, small-world networks are\ngraphs that have a short \\textit{average path length}, i.e. any node is relatively close to any other\nnode, like random graphs and unlike regular lattices.\nHowever, in contrast with random graphs, they also have a certain amount of local structure,\nas measured, for instance, by a quantity called the \\textit{clustering coefficient} (an excellent\nreview of the subject is \\cite{newman-03}).\nIn the same vein, many real conflicting situations in economy and sociology are not well \ndescribed neither by a fixed\ngeographical position of the players in a regular lattice, nor by a mixing population or a\nrandom graph. Starting from the two limiting cases of a random-graph and the two-dimensional lattice, our objective\nhere is to study the Hawk-Dove game on small-world networks in order to cover the ``middle ground''\nbetween them. Although the Watts--Strogatz networks \\cite{watts-strogatz-98} used here are not faithful representations\nof the structure of real social networks, they are a useful first step toward a better understanding\nof evolutionary games on networks. While we study here the Hawk-Dove game, this class of networks has been previously used for the Prisoner's \nDilemma in \\cite{social-pd-kup-01,pd-dyn-sw-02,watts99}. The work of \\cite{social-pd-kup-01} is especially relevant\nfor our present study, while the two others deal either with special features such as\n``influential individuals''\\cite{pd-dyn-sw-02}, or refer to iterated versions of the game \\cite{watts99}.\n\nRecently, Santos and Pacheco \\cite{santos-pach-05} have investigated both the Prisoner's\nDilemma and Hawk-Dove games on\nfixed scale-free networks. The main observation from their simulations is that, at least on preferential attachment\nnetworks, the amount of cooperative behavior is much higher than in either mixing or \nlattice-structured populations. In the abstract, and in some particular social situation\nin which some individuals have an unusually high number of contacts than the rest, this is an interesting result.\nHowever, scale-free graphs, which characterize the web and Internet among others, are not a realistic \nmodel of most observed social networks for various reasons (see \\cite{jin-gir-newman-01,ebel-dav-born-03}),\nwhich is why we do not comment further on the issue.\n\n\\section{\\label{sect:model}The Model}\nIn this section we present our network models and their dynamical properties.\n\n\\subsection{\\label{pop-topo}Population Topologies}\nWe consider a population $P$ of $N$ players where\neach individual $i$ in $P$ is represented as a vertex $v_i$  of a graph $G(V,E)$,\nwith $v_i \\in V, \\; \\forall i \\in P$. An interaction between two players $i$ and\n$j$ is represented by the undirected edge $e_{ij} \\in E, \\; e_{ij} \\equiv e_{ij}$.\nThe number of neighbors of player $i$ is the degree $k_i$ of vertex $v_i$. The average\ndegree of the network will be called $\\bar k$.\n\nWe shall use three main graph population structures: regular lattices, random graphs, and small-worlds.\nIn fact, our goal is to explore significant population network structures that somehow fall\nbetween the regular lattice and random graph limits, including the bounding cases.\n\nOur regular lattices are two-dimensional with $k_i=8, \\; \\forall v_i \\in V$  and periodic boundary conditions. \nThis neighborhood is usually called the Moore neighborhood and comprises nine individuals, including the central node.\nWe would like to stress that we believe regular lattice structures are not realistic representations\nof most actual population structures, especially human, except when mobility and dispersal ability\nof the individuals are limited as, for example, in plant ecology and territorial animals. \nThe main reasons why lattices have been so heavily used is that they are more amenable to mathematical analysis and are easier to simulate.\nWe include them here for two reasons: as an interesting bounding case, and to allow comparison with previous work.\n \nThe small-world networks used here are similar to the graphs proposed by Watts and Strogatz\n\\cite{watts-strogatz-98}. However, there are two main differences (see \\cite{boccara-04}). First, we start from a \ntwo-dimensional regular lattice substrate, instead of a one-dimensional lattice. This does not\nmodify the main features of the resulting graphs, as observed in \\cite{watts99}, and as measured by us.\nThe reason for starting from a two-dimensional lattice is to keep with the customary ordered population\ntopology that is used in structured evolutionary games.  Although\nthey have been used as a starting point for the Prisoner's Dilemma by Abramson and Kuperman \\cite{social-pd-kup-01}, one-dimensional lattices do not\nmake much sense in a social or biological setting, although after some rewiring the effect of\nthe substrate becomes almost negligible. \n\nThe second difference is in the rewiring process. The\nalgorithm used here comes from \\cite{boccara-04} and works as follows: starting from a regular two-dimensional lattice with\nperiodic boundary conditions, visit each edge and, with probability $p$, replace it  by\nan edge between two randomly selected vertices, with the constraint that two vertices are\nnot allowed to be connected by more than one edge. As in the original Watts--Strogatz model,\nthe average vertex degree $\\bar k$ does not change, and the process may produce disconnected graphs, \nwhich have been avoided in our simulations.\nThe advantage of this construction is that, for $p \\rightarrow 1$,\nthe graph approaches a classical Erd\\\"os--R\\'enyi random graph, while this is not the case for\nthe original Watts--Strogatz construction, since in the latter, the degree of any vertex is \nalways larger than or\nequal to $k\/2$, $k$ being the degree of a vertex in the original lattice. \n\nWe would like to point out that it is known that Watts--Strogatz small worlds \nare not adequate representations of social networks. Although they share some common statistical\nproperties with the latter, i.e. high clustering and short average path length, they lack other features that characterize\nreal social networks such as clusters, and dynamical self-organization \\cite{ebel-dav-born-03}. In \nspite of these shortcomings, they are\na convenient first approximation for studying the behavior of agents in situations where the interaction\nnetwork is neither regular, nor random. Note also that once fixed, the interaction network\ndoes not change during the system evolution in our study, only the strategies may evolve. Evolutionary games\non dynamic networks have been studied, for instance in \\cite{zimm-et-al-04,luthi-giac-tom-05,games-ecal-05}.\n\n\\subsection{\\label{dyn}Population Dynamics}\n\\subsubsection{Local Dynamics}\nThe local dynamics of a player $i$ only depends on its own strategy $s_i \\in \\{H,D\\}$, and on\nthe strategies of the $k_i$ players in its neighborhood $N(i)$. Let us call $M$ the payoff matrix\nof the game (see section \\ref{intro}). The quantity\n$$G_i(t) = \\frac{1}{k_i} \\sum _{j \\in N(i)} s_i(t)\\; M\\; s_{j}^T(t)$$\n\\noindent is the average payoff collected by player $i$ at time step $t$.\nNote that in our study, $i \\notin N(i)$ meaning that self-interaction is not considered when\ncalculating the average payoff of an individual.\nSelf-interaction has  traditionally been taken into account in some previous work on the\nPrisoner's Dilemma\ngame on grids \\cite{nowakmay92,nowaketal94} on the grounds that, in biological applications,\nseveral entities may occupy a single patch in the network. Nowak et al. find that\nself-interaction does not qualitatively change the results in the Prisoner's Dilemma game. In the Hawk-Dove game\nself-interaction is usually not considered; moreover, in this work we wish to compare results\nwith those of \\cite{KD-96,hauer-doeb-2004}, where self-interaction is not included.\n\n\nWe use three types of rules to update a player's strategy. The rules are among those employed by\nHauert et al. \\cite{hauer-doeb-2004} to allow for comparison of the results in regular lattices\nand in small-world networks. Decision rules based on the player's satisfaction degree, \nsuch as those used in \\cite{zimm-et-al-04,games-ecal-05,luthi-giac-tom-05,myopic-hd-05} are not examined here.\nThe rules are the following:\n\\begin{enumerate}\n\\item replicator dynamics;\n\\item proportional updating;\n\\item best-takes-over.\n\\end{enumerate}\n\n\nThe \\textit{replicator dynamics} rule aims at maximal consistency with the original\nevolutionary game theory equations. Player $i$ is updated by drawing\nanother player $j$ at random from the neighborhood $N(i)$\nand replacing $s_i$ by $s_j$ with probability $p_j = \\phi(G_j - G_i)$ \\cite{hofb-sigm-book-98}.\n\n\nThe \\textit{proportional updating} rule is also a stochastic rule. All the players in the neighborhood $N(i)$,\nplus the player $i$ itself compete for the strategy $i$ will take at the next time step, each with a probability $p_j$ given by\n$$ p_j = \\frac{G_j}{\\sum_l G_l}, \\;\\; l,j \\in \\{N(i) \\cup i \\}.$$\n\\noindent Negative payoffs cannot be used with this rule, since the probabilities of replication\nmust be $p_j \\ge 0$. In order to avoid negative, or zero, values, the payoffs have been\nshifted by an amount equal to the cost $C$ which, of course, leaves the game's Nash equilibria invariant.\n\n\n\nIn \\textit{best-takes-over}, the strategy $s_i(t)$ of individual $i$ at time step $t$ will\nbe\n$$s_i(t) = s_j(t-1),$$\nwhere\n$$j \\in \\{N(i) \\cup i\\} \\;s.t.\\; G_j = \\max \\{G_k(t-1)\\}, \\; \\forall k \\in \\{N(i) \\cup i\\}.$$\n\\noindent That is, individual $i$ will adopt the strategy of the player with the highest\npayoff among its neighbors.\nIf there is a tie, the winner individual is chosen uniformly at random between the best, and its strategy\nreplaces the current strategy of player $i$, otherwise the rule is deterministic. It should be\nnoted that this rule does not fit to the usual continuous evolutionary game theory which\nleads to replicator dynamics, since the update\ndecision is a step function.\n\n\\subsubsection{Global Dynamics}\nCalling $C(t) = (s_1(t), s_2(t), \\ldots , s_N(t))$ a \\textit{configuration} of the population\nstrategies at time\nstep $t$, the global \\textit{synchronous} system dynamics is implicitly given by:\n$$C(t) = F(C(t-1)), \\;\\; t =1,2, \\ldots $$\n\\noindent where $F$ is the evolution operator.\n\nSynchronous update, with its idealization of a global clock, is customary \nin spatial evolutionary games, and most results have been obtained using this model \n\\cite{nowakmay92,KD-96}.\nHowever, perfect synchronicity is only an abstraction. Indeed, in some biological\nand, particularly, sociological environments, agents normally act at different and possibly uncorrelated\ntimes, which seems to preclude a faithful globally synchronous simulation in most\ncases of interest \\cite{hubglance93}. In spite of this, it has been shown that the\nupdate mode does not fundamentally alter the results, as far\nas evolutionary games are concerned \\cite{nowaketal94,hauer-doeb-2004}. In this paper we\npresent results for both synchronous and asynchronous dynamics.\n\nAsynchronous dynamics must nevertheless be further qualified, since there \nare many ways for serially updating the strategies\nof the agents. Here we use the discrete update dynamics that makes the least assumption\nabout the update sequence: the next cell to be updated is chosen\nat random with uniform probability and with replacement.\nThis corresponds to a binomial distribution of the updating probability and is a good approximation of a continuous-time Poisson\nprocess. This asynchronous update is analogous to the one used by Hauert et al. \n\\cite{hauer-doeb-2004}, which will allow us to make meaningful comparisons.\n\n\\section{\\label{sim}Simulation Results}\nIn order to analyze the influence of the structure of the network\non the proportion of cooperation (i.e. dove behavior),\n2500 players were organized into 5 different networks:\na 50 by 50 toroidal lattice where every cell is connected to its 8 nearest neighbors,\nthree different small-world networks, and the random graph.\nThe three categories of small worlds are obtained by rewiring each edge\nwith a certain probability $p$ using the technique described under \\ref{pop-topo}.\nThe values used are $p \\in \\{0.01,0.05,0.1\\}$.\nThe random graph is generated by first creating the lattice\nand then rewiring each link, in the same manner used to construct\nsmall worlds, but with probability $p=1$. Although our population size is smaller\nthan that used in \\cite{hauer-doeb-2004}, which is $10000$, results turn out to be\n qualitatively similar and comparable.\nFor each of the 5 networks mentioned above and for all update policies,\n50 runs of 5000 time steps each were executed.\nIn the following figures, the curves indicating the proportion of doves in the population\nwere obtained by averaging over the last 10 time steps of each run, well after all transients\nhave decayed.\nAt the beginning of each run, we generate a new network of the type being studied\nand randomly initialize it with 50\\% doves and 50\\% hawks. For completeness, we mention\nthat experiments with 10\\% and 90\\% initial cooperators respectively, give results that\nare qualitatively indistinguishable from the 50\\% case in the long run. Therefore, we do not include the \ncorresponding graphs for reasons of space.\n\nIn the following figures, the dashed diagonal line going from a fraction of cooperators\nof $1$ for $r=0$, to a fraction of $0$ for $r=1$, represents the equation $1-G\/C = 1-r$,\nwhich is the equilibrium fraction of cooperators as a function of $r$ given by the standard replicator-dynamics equations \\cite{hofb-sigm-book-98},\nand it is reported here for the sake of comparison. \nIt should be noted, however, that the simulations are not expected to fit this line. The reason is\nthat the analytic solution is obtained under two main hypotheses: the population size is very\nlarge, and individuals are matched pairwise randomly. These conditions are not satisfied by\nthe finite-size, discrete systems used for the simulations, and thus one should not expect strict\nadherence to the mean-field equations. On the other hand, the type of \\textit{mesoscopic} system\nsimulated here is probably closer to reality, where finiteness and discreteness are the rule.\nAnother reason why we do not expect the results of the simulations to closely fit the\ntheoretical solution is that two of the local update rules\n(best-takes-over and proportional updating) do not reduce to the standard replicator dynamics.\n\nThis section is subdivided into three separate parts, one per decision rule previously\nmentioned under \\ref{dyn}.\n\n\\subsection{Replicator Dynamics}\nTo determine the probability $p_j$ for replacing an individual $i$, having a gain $G_i$, \nby one of its randomly chosen neighbors $j$, whose gain is $G_j$,\nwe use the the previously introduced function $\\phi(G_j - G_i)$ as follows:\n\\begin{eqnarray}\np_j = \\phi(G_j - G_x) =\n\\begin{cases} \\textrm{\\large{$\\frac{G_j - G_i}{{d}_{max}}$}} & \\textrm{if $G_j - G_i > 0$}\\\\\\\\\n0 & \\textrm{otherwise}\n\\end{cases}\n\\label{repl_dyn_eq}\n\\end{eqnarray}\nwhere ${d}_{max} = \\frac{G+C}{2}$ is the largest difference in gain\nthere can be between two players.\n\nWith this definition of $\\phi$, individual $i$ imitates neighbor $j$'s strategy\nwith a certain probability proportional to the difference of their average payoffs\nand only if $j$ has a higher gain than $i$.\nNotice that if $i$ and $j$ have the same average payoffs, $i$'s strategy is left untouched, while\nif $G_j -G_i = {d}_{max}$, $i$ necessarily adopts $j$'s strategy.\n\nNow taking a look at figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync},\nwe clearly observe that for both synchronous and asynchronous dynamics,\ncooperation is globally inhibited by spatial structure, confirming the results of \\cite{hauer-doeb-2004}.\nEven the case of the random graph generates higher rates of hawks.\nFurther details as to why this may occur can be found in section \\ref{disc}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_dPercentage}}\\\\\n(a) & &(b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{repl_dyn_async}(Color online) asynchronous replicator dynamics updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{repl_dyn_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{repl_dyn_sync}(Color online) synchronous replicator dynamics updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\nWe note in passing that the experimental curve corresponding to the random graph limit appears\nto be close to the curve corresponding to the pair approximation calculation in Hauert and Doebeli's\nwork \\cite{hauer-doeb-2004}. This is not surprising, given that pair approximation works\nbetter in random graphs than in regular lattices, unless higher-order effects are taken into\naccount \\cite{van-baalen-00}. Since the curves for the random graphs in \nfigures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync} are averages over many graph realizations,\neach pair has some probability to contribute in the simulation,\nwhich explains the resemblance between our experimental curves and the calculations of \\cite{hauer-doeb-2004}.\n\n\\subsection{Proportional Updating}\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{prop_async}(Color online) asynchronous proportional udpating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{prop_sync_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{prop_sync}(Color online) synchronous proportional updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\nFigures \\ref{prop_async} and \\ref{prop_sync} show that,\nwhen using the proportional updating rule,\nspatial structure neither favors nor inhibits dove-like behavior\ncontrary to what \\cite{KD-96} and \\cite{hauer-doeb-2004} seem to suggest.\nIndeed, for low values of $r$, the more the network is structured,\nthe higher the proportion of doves.\nHowever as $r$ increases, the tendency is reversed,\nthus giving a lower percentage of doves in the lattice and small-world networks\nthan present in the random graph topology.\nThis phenomenon is even more marked when using the asynchronous update.\n\nThus when using the proportional updating rule, if spatial structure should favor one strategy over the other for a given value of $r$,\nit would be the one that is already present in greater numbers\nwhen the topology is a random graph.\n \n\nAnother interesting aspect observed is the higher percentage of doves\nwhen updating asynchronously compared to the synchronous equivalent.\nThis will be discussed in more detail in section \\ref{disc}.\n\n\n\\subsection{Best-takes-over}\nAs pointed out by Hauert and Doebeli \\cite{hauer-doeb-2004}, the best-takes-over rule lacks stochasticity,\nwhich in figures \\ref{bto_async} and \\ref{bto_sync}, translates into discontinuous jumps.\n\nNote that when updating synchronously, best-takes-over is the only rule,\nout of the three studied here, where spatial structure actually favors cooperation, as remarked\nin \\cite{KD-96}, where this was the local update rule used. In fact, the same qualitative results\nwere found in \\cite{hauer-doeb-2004}; however, they appear in the \"supplementary material\" section,\nnot in the main text.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_async_uc_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_async_uc_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{bto_async}(Color online) asynchronous best-takes-over updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_3d}}\\protect & \\hspace*{1cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_dPercentage}}\\\\\n(a) & & (b)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{bto_sync}(Color online) synchronous best-takes-over updating;\n(a) frequency of doves as a function of the gain-to-cost ratio $r$ for differents topologies:\nlattice ($p=0$), small worlds ($p=0.01$, $p=0.05$, $p=0.1$), random graph ($p=1$);\n(b) small world with $p = 0.05$ compared to the grid ($p=0$) and random graph ($p=1$) cases.\nBars indicate standard deviations and the diagonal dashed line is $1-r$ (see text).}\n\\end{figure}\n\n\\subsection{Time Evolution}\n\\label{tev}\n\nWhile the figures in the previous subsections summarize the results at system stability,\nhere we describe the dynamical behavior of populations through the first $100$ time\nsteps, where fluctuations might influence the system dynamics.\n\nWe have studied both asynchronous and synchronous dynamics for\nthe three update rules in three topologies each: lattice, random graph, and a small\nworld with $p=0.05$. This was done for $r=0.7$, where defection predominates. The results are \nrelatively uninteresting for the replicator and proportional updates in all topologies. One\nobserves in the average a monotone decrease of cooperation starting with $50\\%$ at time $0$ until\nthe curve flattens out at the values reported in figures \\ref{repl_dyn_async} to \\ref{prop_sync}.\nThe only difference is that\nthe variance is more pronounced in the proportional case, as one would expect looking\nat standard deviations in figures \\ref{repl_dyn_async} to \\ref{prop_sync}.\n\nThe situation is different, and more interesting, in the case of best-takes-over\nupdate whose determinism causes stronger variations. The most striking feature is a \nsudden drop of cooperation at the beginning\nof the simulation, followed by an increase and by fluctuations whose amplitude diminishes\nover time. The effect is much more pronounced with synchronous dynamics, shown in Fig. \\ref{time-ev}, than\nwith the asynchronous one. The behavior appears in all three topologies but the drop is stronger\nin lattices and small worlds with respect to the random graph at earlier times. As time\ngoes by, fluctuations remain larger in the random graph case. Nevertheless, no experiment\nled to total extinction of cooperators at $r=0.7$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccccc}\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_0_dPercentage_time_ev}}\\protect &\n\\hspace*{0.3cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_005_dPercentage_time_ev}}\\protect &\n\\hspace*{0.3cm} &\n\\mbox{\\includegraphics[width=5.5cm, height=5.5cm]{bto_sync_07_1_dPercentage_time_ev}}\\\\\n(a) & & (b) & & (c)\n\\end{tabular}\n\\end{center}\n\\caption{\\label{time-ev} time evolution (first $100$ steps) of the proportion of doves\nfor best-takes-over update; synchronous evolution with $r=0.7$. (a) lattice structure;\n(b) small world with $p=0.05$; (c) random graph. Ten randomly chosen evolutions are shown in\neach case.}\n\\end{figure}\n\n\n\n\\section{\\label{disc}Analysis and Discussion}\nIf we take a closer look when comparing Fig. \\ref{prop_async} and Fig. \\ref{prop_sync},\nwe notice that, for proportional dynamics, asynchronous updating allows for better cooperation than its\nsynchronous counterpart. The reason for this difference can be intuitively understood\nin the following manner:\nwhen updating asynchronously, let us suppose a player $y$ has just imitated the strategy\nof one of its neighbors $x$. Another way of viewing this change, is to say that player $x$ has ``infected''\nindividual $y$ with its strategy. If $x$ is a dove player, making $y$ a dove as well,\nnot only does the percentage of doves increase in the population, but\nthe next time either $x$ or $y$ is evaluated for an update, it will be able to take advantage of the other one's\npresence to help increase its payoff. Hence, the two players mutually reinforce each other.\nMeanwhile, if $y$ is infected by $x$ and turns into a hawk, on the one hand $x$ has successfully propagated\nhis strategy thus increasing the overall amount of hawks in the population,\nbut on the other hand this propagation will cause him to have a lower payoff than he previously had.\nNot only is $x$'s payoff negatively affected, but $x$'s prescence also harms $y$'s payoff.\n\nThe same reasoning cannot be held when updating synchronously.\nIndeed, a player $x$ may change strategies\nat the same time it infects its neighbor $y$.\nSo if $x$'s initial strategy was $D$, it might switch\nto $H$ as it infects its neighbor $y$, in which case\n$x$ will no longer have a positive effect on $y$'s payoff\ncontrary to what happens in asychronous updating.\n\nWhen applying the replicators dynamics rule, the small drop of the percentage of doves  seen\non the very left of figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync} is due to the fact that for\n$r=0$ the game is somewhat degenerated. Indeed, any cluster of more than one hawk will either reduce to\na single hawk or totally disappear, since a dove, no matter what its neighborhood comprises,\nwill always have a gain of zero while a hawk that interacts with at least one other hawk will have a negative payoff.\nThe remaining lone hawks will however survive but will not be able to propagate (having a gain \nexactly equal to that of their neighboring doves). The system is thus found locked in a configuration\nof a very high proportion of doves with a significant number of isolated hawks.\n\nIf $r > 0$, lone hawks always have a higher payoff than the doves in their surroundings and will thus infect one\nof its neighbors with its strategy. However for $0 < r \\leq 0.1 $, once the pair of hawks is established, their payoff\nis lower than the one of any of the doves connected to either one them. Even a dove that interacts with both\nhawks has an average payoff still greater than what a hawk composing the\npair receives.\nConsequently, when $0 < r \\leq 0.1$, clusters of hawks first start by either disappearing\nor reducing to single hawks like previously explained for the $r=0$ case, but then these lone hawks\nwill become pairs of hawks.\nIf the updates are done synchronously, a pair of hawks will either vanish\nor reduce back to a single hawk. One can clearly see that in the long run, hawks will become extinct.\nNow if the updates are done asynchronously, a pair cannot totally disappear since only one\nplayer is updated at a time. However, this mechanism of  a pair reducing to a single hawk and\nturning back to a pair again\nwill cause the small groups of two hawks to move across the network and ``collide'' with each other\nforming larger groups that reduce back to a single-pair hawk formation. Therefore, after a large\nnumber of time steps, only a very few hawks will survive.  \n\nIf we take another look at figures \\ref{repl_dyn_async} and \\ref{repl_dyn_sync},\nwe notice that when the population of players is constrained to a lattice-like structure,\nthe proportion of doves is reduced to zero for values of\nthe gain-to-cost ratio greater or equal to approximately $0.8$,\nwhile this not the case when the topology is a random graph.\nLet us try to give a qualitative explanation of the two different behaviors:\nthe first thing to be pointed out is that in the case of the replicators dynamics,\nif a dove is surrounded by 8 hawk-neighbors,\nit is condemned to die for values of $r$ greater than $\\frac{7}{9}$ whatever the topology may be.\nHowever, this does not explain why for these same values, doves no longer exist on\nsquare lattices or small worlds but are able to survive on random graphs.\nIf the population were mixing, $r=0.8$ would induce a proportion of doves equal to $20\\%$.\nTherefore, let us suppose that at a certain time step,\nthere is approximately $20\\%$ of doves in our population.\nFurthermore, as pointed out by Hauert and Doebeli \\cite{hauer-doeb-2004}, in the Hawk-Dove\ngame on lattices, the doves are usually spread out and form many small-isolated patches.\nThus, we will also suppose $20\\%$ of doves in the  population\nimplies that in a set of players comprising an individual and its immediate eight neighbors,\nthere are about two doves.\nHence, a D-player has on average one dove and seven hawks in its neighborhood.\nIn the lattice network, this pair of doves can be linked in two different manners\n(see Fig. \\ref{2D_lattice_configs}), having either two or four common neighbors,\nthus an average of three.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\mbox{\\includegraphics[width=3.5cm, height=3.5cm]{lattice_2D_a}}\\protect\n&  \\hspace{1cm} & \\mbox{\\includegraphics[width=3.5cm, height=3.5cm]{lattice_2D_b}}\\\\\n(a) & \\hspace{1cm} & (b)\\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{2D_lattice_configs}(Color online) lattice: two possible configurations.}\n\\end{figure}\n\nMore generally, if we denote $\\Gamma$ the clustering coefficient of the graph and $\\overline{k}$\nthe average degree, a pair of doves will have on average $\\Gamma(\\overline{k} - 1)$\ncommon neighbors.\nLet us denote $x$ one of the two doves composing the pair,\n$H_x$ a hawk linked to $x$ but not to the other dove of the pair and $H_{x,y}$ one that is connected to both doves.\nIf $\\frac{2}{3} < r < \\frac{7}{8}$ and assuming that the hawks surrounding the pair of doves are not interacting with any other doves\n(this gives the pair of doves a maximum chance of survival), we have\n$$G_{H_x} < G_x < G_{H_{x,y}},$$\nwhere $G_\\alpha$ is the average payoff of player $\\alpha$\n\nConsequently, according to Eq. (\\ref{repl_dyn_eq}),\n$x$ can infect $H_x$, and $H_{x,y}$ can infect $x$.\n\nLet us now calculate for what values of $r$ the probability that $x$ invades the site\nof at least one $H_x$ is less than an $H_{x,y}$ infecting $x$.\nTo do so, let us distinguish the case of the asynchronous udating policy from the synchronous one.\n\n\\subsubsection*{Asynchronous Dynamics}\nThe probability that an $H_x$ neighbor is chosen to be updated and adopts strategy $D$ is given by\n\\begin{equation}\n\\underbrace{\\frac{(1 - \\Gamma)(\\overline{k} - 1)}{N}}_{(\\ast)} \\underbrace{\\frac{1}{\\;\\overline{k}\\;}}_{(\\ast\\ast)} \\;\\phi(G_x - G_{H_x}),\n\\label{H2D_async}\n\\end{equation}\nwhere $N$ is the size of the population, $(\\ast)$ the probability an $H_x$ hawk is chosen to be updated (among the $N$ players),\n$(\\ast\\ast)$ the probability the chosen $H_x$ hawk compares its payoff with player $x$,\nand finally $\\phi$ is the function defined in Eq. (\\ref{repl_dyn_eq}).\n\nThe probability that $x$ is chosen to be updated and is infected by one of the $H_{x,y}$ hawks is given by\n\\begin{equation}\n\\underbrace{\\frac{1}{N}}_{(\\ast)} \\underbrace{\\frac{\\Gamma(\\overline{k}-1)}{\\overline{k}}}_{(\\ast\\ast)} \\;\\phi(G_{H_{x,y}} - G_x),\n\\label{D2H_async} \n\\end{equation}\nwhere $(\\ast)$ is the probability $x$ is chosen to be updated,\n$(\\ast\\ast)$ the probability it measures itself against an $H_{x,y}$ neighbor,\nand $\\phi$ the function defined by Eq. (\\ref{repl_dyn_eq}).\n\nFor a square lattice with a Moore neighborhood ($\\Gamma = \\frac{3}{7}$ and $\\overline{k} = 8$),\nexpressions \\ref{H2D_async} and \\ref{D2H_async} give us $r > \\frac{46}{59} \\approx 0.78$,\nwhereas for a random graph, $\\Gamma  = \\frac{\\overline{k}}{N-1} = \\frac{8}{2499} \\simeq 0.003 \\approx 0$ implies that\na pair of doves does not have any common hawk neighbors enabling them to survive\nif $r < \\frac{7}{8}$.\nAs for the small-world cases, the clustering coefficient is very close to that of the lattice, generating a behavior\npratically identical to the latter.\nThis gives a qualitative explanation for the difference observed in Fig. \\ref{repl_dyn_async}.\n\n\\subsubsection*{Synchronous Dynamics}\nThe probability that at least one $H_x$ adopts strategy $D$ is given by\n\\begin{equation}\n1 - \\overbrace{[1 - \\underbrace{\\frac{1}{\\;\\overline{k}\\;}\\;\\phi(G_x - G_{H_x})}_{(\\ast)}]^{(1-\\Gamma)(\\overline{k}-1)}}^{(\\ast\\ast)},\n\\label{H2D_sync}\n\\end{equation}\nwhere $(\\ast)$ is the probability a specific $H_{x}$ turns into a dove and $(\\ast\\ast)$ the probability none of the $H_x$\nadopt strategy $D$.\n\nThe probability that $x$ adopts the hawk strategy is given by\n\\begin{equation}\n\\underbrace{\\frac{\\Gamma(\\overline{k} - 1)}{\\overline{k}}}_{(\\ast)} \\;\\phi(G_{H_{x,y}} - G_x),\n\\label{D2H_sync}\n\\end{equation}\nwhere $(\\ast)$ is the probability player $x$ compares its payoff with one of its $H_{x,y}$ neighbors.\n\nFor a square lattice with a Moore neighborhood ($\\Gamma = \\frac{3}{7}$ and $\\overline{k} = 8$),\nexpressions \\ref{H2D_sync} and \\ref{D2H_sync} yield\n$$\n1 - \\left[1 - \\frac{1}{8}\\left(\\frac{-8G+7C}{G+C}\\right)\\right]^4 < \\frac{3}{8}\\left(\\frac{9G - 6C}{G+C}\\right),\n$$\nand given that $\\frac{G}{C} = r$, we obtain\n$$\n1 - \\left[1 - \\frac{1}{8}\\left(\\frac{-8r+7}{r+1}\\right)\\right]^4 < \\frac{3}{8}\\left(\\frac{9r - 6}{r+1}\\right),\n$$\nwhich is true for about $r > 0.775$. This also holds for the small-world cases, since, once again,\nthey have a $\\Gamma$ close to the one of the lattice. \n\nFor a random graph of $N=2500$ nodes and $\\overline{k}=8$, we have $\\Gamma \\approx 0$.\nTherefore, a pair of doves has a negligible probability of having a hawk neighbor in common and thus cannot be infected by the H strategy if $r < \\frac{7}{8}$.\nThis enables a small percentage of doves to survive on the random graph topology contrary to the lattice and small-world\nnetworks (see Fig. \\ref{repl_dyn_sync}). \n\nIn a few words, whether the update policy is asynchronous or synchronous, as soon as $r > \\frac{7}{9}$,\nisolated doves, as well as pairs of doves surrounded by hawks, will end up disappearing in the\nlattice and small-world cases due to the high clustering coefficient.\nHowever, in the random graph scenario, although isolated doves are also bound to die if $r > \\frac{7}{9}$,\npairs of doves have a more than even chance of surviving (at least as long as $r < \\frac{7}{8}$).\n\n\\section{\\label{concl}Conclusions}\nIn this work we clarify previous partially contradictory results on cooperation in populations playing the\nHawk-Dove game on regular grids. Furthermore, we notably extend the study to Watts--Strogatz small-world graphs, as\nthese population structures lie between the two extreme cases of regular lattices and random graphs,\nand are a first simple step towards real social interaction networks. This allows us to\nunravel the role of network clustering on cooperation in the Hawk-Dove game.\nWe find that, in general, spatial structure on the network of interactions in the  game\neither favors or inhibits cooperation with respect to the perfectly mixed case.\nThe influence it has depends not only on the rule that determines a player's future strategy,\nbut also on the value of the gain-to-cost ratio $G\/C$ and to a lesser degree,\non the synchronous and asynchronous timing of events.\n\nIn the case of the best-takes-over rule, dove-like behavior is advantaged if synchronous\nupdate is used but the rule is noisy due to its discrete nature.\nIn the case of the proportional update\nrule, giving the network a regular structure tends to increase\nthe percentage of the strategy that would already be in majority on a random graph\nconfiguration of the population. The more important the structure, in terms of clustering\ncoefficient, the higher the percentage of the dominant strategy. In fact, cooperation predominates\nfor low to medium $r$ values, while for higher $r$ values cooperation falls below the \nlarge population, mixing case.\nFinally, the replicators dynamics rule tends to favor hawks over doves on\nspatially structured topologies such as small worlds and square lattices, thus\nconfirming previous results for regular lattices and extending them to small-world networks.\nIn the end, although small-world topologies show behaviors that are somewhat in between\nthose of the random graph and the two-dimensional lattice, they usually tend more\ntowards the latter, at least in terms of cooperation level.\n\nIn this work, we have used static\nnetwork structures, which is a useful step but it is not realistic enough as the interactions\nthemselves help shape the network. In future work we shall\nextend the study using more faithful social network structures, including their dynamical aspects.\n\n\n\\begin{small}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Noncommutative spacetime and nonabelian gauge fields}\n\\hspace*{12pt}Between 1921-1948, Kaluza, Klein and Thiry \\cite{KK} had shown that the Hilbert-Einstein action in the spacetime extended with a circle ${\\cal M}^4 \\times S^1$ consisted of gravity, electromagnetism and a Brans-Dicke scalar. In 1968, R.Kerner \\cite{Kerner} had generalized the Kaluza-Klein theory to include nonabelian gauge fields. Today, the multi-dimensional theories are studied widely as candidates of unified theories of interactions. However, these theories have a weakness of containing an infinite tower of massive fields leading to theoretical and observational obstacles.\n\nIn the 1980's, Connes had put forward the new concept of spacetime based on noncommutative geometry(NCG) \\cite{Co}.In 1986, Connes and Lott \\cite{CoLo} applied the idea to the spacetime extended by two discrete points ${\\cal M}^4 \\times Z_2$ and shown that Higgs fields emerged naturally in a gauge theory with a quartic potential. The most attractive feature of NCG with discrete extra dimensions is that it does not contains an infinte tower of massive fields.\n \nIn 1993, Chamsedine, Felder and Fr\\\"ohlich \\cite{CFF} made the first attempt to generalize the Hilbert-Einstein action to NCG, leading to no new physical content. In 1994, G.Landi, N.A.Viet and K.C.Wali \\cite{LVW} had overcome this no-go result and derived the zero mode sector of the Kaluza-Klein theory from the generalized Hilbert-Einstein's action. Viet and Wali \\cite{VW1} have generalized this model further and obtained a full spectum consisting of bigravity, bivector and biscalar. In each pair, one field is massless and the other one is masive.\n\nThe incorporation of the nonabelian gauge fields in Viet-Wali's model is a not trivial task. Recently, Viet and Du \\cite{VietDu} have successfully derived the nonabelian gauge interaction from the Hilbert-Einstein's action. However, it is possible to do so only in two following cases:\n\ni. The gauge vector fields must be abelian on one sheet of spacetime and nonabelian on the other one. This is exactly the case of the electroweak gauge fields on the two copies of Connes-Lott's spacetime of chiral spinors.\n\nii. The gauge vector field must be the same on both copies of spacetime of chiral spinors. This is also the case of QCD of strong interaction.\n\nSo, NCG can \"explain\" the specific gauge symmetry structure of the Standard Model.\n \nIn this article, we propose a new noncommutative spacetime structure ${\\cal M}^4 \\times Z_2 \\times Z_2$, which is the ordinary spacetime extended by two discrete extra dimension, each consists of two discrete points. In other words, this noncommutative spacetime consists of two copies of Connes-Lott's spacetime. The generalized  Hilbert-Einstein action in this new spacetime contains all the known interactions of Nature and the observed Higgs field. In a more general case, this theory can also lead to multigravity, which might be necessary to explain the dark matter and inflationary cosmology related observations. \n\\section{A new model of noncommutative spacetime and gravity}\n\\hspace*{12pt}The noncommutative spacetime ${\\cal M}^4 \\times Z_2 \\times Z_2$ is the usual four dimensional spinor manifold extended by two extra discrete dimensions given by two differential elements $DX^5$ and $DX^6$ in addition to the usual four dimensional ones $dx^\\mu$. Each extra dimension consists of only two points. This structure can also be viewed as four sheeted space-time having a noncommutative differential structure with the following spectral triplet:\n\ni) The Hilbert space ${\\mathcal H}= {\\mathcal H}^v \\oplus {\\cal H}^w $ which is a direct sum of two Hilbert spaces ${\\mathcal H}^u = {\\cal H}^u_L \\oplus {\\cal H}^u_R, u = v,w$, which are direct sums of the Hilbert spaces of left-handed and right-handed spinors. Thus the wave functions $ \\Psi \\in {\\mathcal H}$  can be represented as follows \n\\begin{equation}\n    \\Psi(x) =  \\pmatrix{\n    \\Psi^v(x) \\cr\n    \\Psi^w(x) \\cr\n    } ~~,~~ \n    \\Psi^u(x) = \\pmatrix{\n        \\psi^v_L(x) \\cr\n        \\psi^w_R(x) \\cr\n        }  \\in {\\mathcal H}^u ~;~ u=v,w,\n\\end{equation}\nwhere the functions $\\psi^u_{L,R}(x) \\in {\\cal H}^u_{L,R}$ are defined on the 4-dimensional spin manifold ${\\cal M}^4$.\n\nii) The algebra ${\\cal A}={\\cal A}^v \\oplus {\\cal A}^w ; {\\cal}^u = {\\cal A}^u_L \\oplus {\\cal A}^u_R$ contains the 0-form ${\\cal F}$ \n\\begin{equation}\\label{0form}\n   {\\mathcal  F}(x) =  \\pmatrix{\n    F^v(x) & 0 \\cr\n    0 & F^w(x) \\cr\n    } ~,~ F^u(x) =  \\pmatrix{\n        f^u_L(x) & 0 \\cr\n        0 & f^u_R(x) \\cr\n        } \\in {\\cal A}^u,\n\\end{equation}\nwhere $f^u_{L,R}(x)$ are real valued function operators defined on the ordinary spacetime ${\\cal M}^4$ and acting on the Hilbert spaces ${\\cal H}^u_{L,R}$.\n\niii) The Dirac operator ${\\cal D} = \\Gamma^P D_P = \\Gamma^\\mu \\partial_\\mu + \\Gamma^5 D_5+ \\Gamma^6 D_6, P=0,1,2,3,5,6 $ is defined as follows\n\\begin{eqnarray}\\label{Dirac1}\n{\\cal D} &=& \\pmatrix{\nD & m_1 \\theta_1 \\cr\nm_1 \\theta_1 & D \\cr\n}~,~ D = \\pmatrix{\nd & m_2 \\theta_2 \\cr\nm_2 \\theta_2 & d \\cr} ~,~ d = \\gamma^\\mu \\partial_\\mu \\\\ \nD_\\mu {\\cal F} &=& \\pmatrix{\n\\partial_\\mu F^v(x) & 0 \\cr\n0 & \\partial_\\mu F^w(x) \\cr \n} ~,~\n\\partial_\\mu F^u = \\pmatrix{\n\\partial_\\mu f^u_L(x) & 0 \\cr\n0 & \\partial_\\mu f^u_R(x) \\cr \n} \\\\\nD_{6} {\\cal F} &=& m_1(F^v - F^w){\\bf r} ~,~ D_5 F^u = m_2(f^u_L-f^u_R) {\\bf r} ~,~ {\\bf r} = \\pmatrix{\n1 & ~0\\cr\n0 & -1\\cr\n} \n\\end{eqnarray}\nwhere $\\theta_1, \\theta_2$ are Clifford elements $ \\theta^2_1 = \\theta^2_1=1$, $m_1, m_2$ are parameters with dimension of mass. \n\nThe construction of noncommutative Riemannian geometry in the Cartan formulation is given in \\cite{VW1} in a perfect parallelism with the ordinary one. Here we will use the following flat and curved indices to extend the 4 dimensions with 5-th and 6-th dimensions.\n\\begin{eqnarray}\nE,F,G = A, \\dot{6} ~,~ & A,B,C = a, \\dot{5}&~,~ a,b,c =0,1,2,3 \\\\\nP,Q,R = M,6 ~,~ &M,N,L = \\mu, 5 &~,~\\mu,\\nu, \\rho=0,1,2,3.\n\\end{eqnarray}\n\nThe construction of noncommutative Riemannian geometry \\cite{VW1} is in a perfect parallelism with the ordinary one. The starting point is the locally flat reference frame, which is a linear transformation of the curvilinear one with the vielbein coefficients. For transparency, let us write down the vielbein in 4,5 and 6 dimensions as follows \n\\begin{equation}\ne^a = dx^\\mu e^a_\\mu(x) ~,~ E^A = DX^M E^A_M(x) ~,~{\\cal E}^E = DX^P {\\cal E}^E_P(x),\n\\end{equation}\nwhere $e^a_\\mu(x), E^A_M(x), {\\cal E}^E_P$ are 4,5 and 6-dimensional vielbeins.\nThe Levi-Civita connection 1-forms $\\Omega^\\dagger_{EF} = - \\Omega_{FE}$ are introduced as a direct generalization of the ordinary case. With a condition \\cite{VW1}, which is a generalization of the torsion free condition one can determine the Levi-Civita connection 1-forms and hence the Ricci curvature tensor from the generalized Cartan structure equations    \n\\begin{eqnarray}\n{\\cal T}^E &= &  DE^E + E^F \\Omega^E_F \\label{Torsion}\\\\\n{\\cal R}^{EF} &=& D\\Omega^{EF}+ \\Omega^E_G \\wedge \\Omega^{GF} \\label{Curv}\n\\end{eqnarray}\nThen we can calculate the Ricci scalar curvature $R=\\eta^{EG} \\eta^{FH} R_{EFGH}$.\n\nThe construction of our model is carried out in two subsequent steps. First, we construct the 6-dimensional Ricci curvature with an ansatz containing one 5-dimensional gravity and two 5-dimensional vectors fields, where one is abelian and the other is nonabelian to use Viet-Du's results. Then\n\\begin{equation}\nR_6 = R_5 - {1 \\over 4}  G^{MN} G_{MN} = R_5 + {\\cal L}_g(5)\n\\end{equation}\nwhere $G_{MN}; M,N=0,1,2,3,5$ is the 5-dimensional covariant field streng tensor of the nonabelian $SU(2)\\times U(1)$ gauge fields.\n\nIn the second step, the gravity sector is reduced further to 4-dimensional gravity  nonabelian gauge $SU(3)$ vector of strong interaction \n\\begin{equation}\nR_5 = R_4 - {1 \\over 4} Tr H^{\\mu \\nu} H_{\\mu \\nu} ~~,~~ H_{\\mu \\nu} = \\partial_\\mu B_\\nu - \\partial_\\nu B_\\mu + i g_S [B_\\mu, B_\\nu],\n\\end{equation}\nwhere $B_\\mu = B^i_\\mu(x) \\lambda^i$ are the gluon field and $\\lambda^i, i=1,..,8$ are the GellMann matrices.\n\nConnes-Lott's procedure can be applied now to reduce the 5-dimensional gauge Lagrangian ${\\cal L}_g(5)$ to the 4-dimensional electroweak gauge-Higgs sector of the Standard Model as follows\n\\begin{equation}\n{\\cal L}_g(5) = -{1 \\over 4}( F^{\\mu\\nu} F_{\\mu\\nu}+ G^{\\mu \\nu} G_{\\mu \\nu}) + {1 \\over 2} \\nabla^\\mu \\bar{H} \\nabla_\\mu H + V(\\bar{H}, H),  \n\\end{equation} \nwhere $H$ is a Higgs doublet, $\\nabla_\\mu$ is the gauge covariant derivative and $V(\\bar{H}, H)$ is the usual quartic potential of the Higgs field.\n\\section{Multigravity in noncommutative spacetime}\n\n\\hspace*{12pt}In Section 2, we have presented the minimal ansatz to include the all known interactions and the Higgs fields. New cosmological observations might shed light to more detailed structure of the new commutative spacetime. In principles, in a more general case, our model can adopt up to 4 gravitational fields, one of those is massless while the other ones are massive. \n\nFrom theoretical points of view, this model can provide a geometric construction approach to the massive gravity, which has\nrecently attracted a lot of attention as a candidate theory of modified gravity \\cite{DeRham}. From the viewpoints of modern cosmology, multigravity might give new explanations the existence of dark matter and inflationary  cosmology.\n  \n\\section{Summary and discusions}\nWe have presented a new noncommutative spacetime ${\\cal M}^4 \\times Z_2 \\times Z_2$, which can unify all the known interactions and Higgs field on a geometric foundation. This is very similar to the foundation of Einstein's general relativity.  This model unifies all forces in nature without resorting to infinite tower of massive fields.\n\nIn the most general case, this theory can contain more (but still a finite number) degrees of freedom, including four different massless and massive gravity fields, Brans-Dicke scalars and more gauge fields. The model can provide a geometric foundation to the theories of massive and modified gravity. The reality might be just a special case of the most general theory. The cosmological observations might help us to see more details of this theory.\n\nThere are some issues, at the moment we are not able to answer such as the physical meaning of the sixth dimension and the energy scale of this theory. It is worth to quote the following relation from the work by Viet and Du \\cite{VietDu}\n\\begin{equation}\ng = 8 m \\sqrt{\\pi G_N},\n\\end{equation}\nwhere $g$ the weak coupling constant and $G_N$ is the Newton constant. This relation must hold when the theory becomes valid. One can speculate this might happen at an energy scale, which is million times lower that the Planck scale. That might be the case in some evolution stage of our universe after the Big Bang. All the above perspectives would merit more research.\n\\section*{Acknowledgments}\nThe discussions with Pham Tien Du, Do Van Thanh (College of Natural Sciences, VNU) and Nguyen Van Dat (ITI-VNU) are greatly appreciated. The author would also like to thank Jean Tran Thanh Van for the hospitality at Quy Nhon and supports. The work is partially supported by ITI-VNU and Department of Physics, College of Natural Sciences, VNU.\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe past decades have seen a proliferation of research using evolutionary theory to study social traits, in the fields of biology, animal behavior, and even social science \\cite{Nowak1998,Nowak2004,Hilbe2018a,Ohtsuki2006,Weitz2016,Tilman2020,Allen2017a}.\nMost of this theoretical development has been based on mathematical models that assume either infinite populations \\cite{Taylor1978,Schuster1983,Nowak2006a,Weitz2016,Tilman2020}  or finite populations of constant size \\cite{Hilbe2018a,Ohtsuki2006,Nowak2004}.\nDespite these simplifying assumptions, mathematical models provide rich insights into how exogenous and intrinsic factors drive evolutionary dynamics of social behavior. \nFor example, \nthe literature has produced a rich set of explanations for cooperation based on repeated interactions, the establishment of reputations, and various forms of population structure \\cite{Nowak1992,Ohtsuki2006,Tarnita2009,Allen2017a,McAvoy2021,Su2022,Su2022nhb,Cooney2019,Hilbe2018a,Santos2018,Nowak2005,Nowak1993,Nowak2006fiverule,Stewart2013}. Several key theoretical insights have been validated by controlled experiments on human subjects \\cite{Gachter2009,Yamauchi2011,Yoelia2013,Greiner2005}. This field of research has been so successful that the question of how cooperation can be favored by natural selection, famously posed by Darwin, is now not only resolved, but resolved in several distinct ways applicable in different contexts.\n\nHere we reveal an qualitatively different and pervasive mechanism that can promote cooperation by natural selection or payoff-biased imitation. \nMost mechanisms known to support cooperation boil down to some form of population structure \\cite{Kay2020} -- either physical limitations on social interactions, reproduction, or imitation, or structure imposed by tags or reputations. By contrast, here we describe a much more simple scenario that can favor cooperation in a population that lacks any form of exogenous or endogenous structure. We show that demographic stochasticity, which is \\textit{a priori} a realistic feature of any natural population, can by itself promote social behaviors that would otherwise be suppressed in idealized populations of constant (or infinite) size.\n\nThere is precedent for the idea that demographic stochasticity alters evolutionary dynamics. The fact that mortality, reproduction, and migration are subject to demographic fluctuations in populations -- as well as processes of imitation and innovation -- is known influence the dynamics of competing types under frequency-independent selection \\cite{Parsons2007,Parsons2007a,Parsons2010,McKane2005,Butler2009,Hallatschek2007,Stollmeier2018,Wienand2017,Taitelbaum2020,Chotibut2017} and also frequency-dependent selection \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015,Huang2015}.\nFor example, when a population contains two types with the same expected number of offspring, one type can be favored when the population size is small, and the other type favored when the population size is near to its carrying capacity \\cite{Parsons2007,Parsons2007a,Parsons2010}.\nAnd a few studies have shown that demographic stochasticity can even reverse the direction of natural selection, promoting a type that would otherwise be disfavored without stochasticity  \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}.\n\nNonetheless, prior work on selection with demographic stochasticity has either assumed constant fitness, in which one's fitness is independent of the composition of the population, or assumed different carrying capacities for different phenotypes, e.g.,~producers enjoy a larger carrying capacity than non-producers \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}.\nMost models of demographic stochasticity \nalso assume that offspring numbers follow a Poisson distribution \\cite{Constable2016,Huang2015,Parsons2010,Parsons2007,Parsons2007a,Houchmandzadeh2012,Houchmandzadeh2015}, so that the mean and variance in offspring number are identical. But empirical field studies have found that over-dispersion in offspring number (variance exceeding mean) is commonplace across diverse taxa \\cite{Zuur2009,Linden2011,VerHoef2007,Richards2008}.\n\n\n\nIn this paper, we develop a general framework to study evolutionary dynamics with demographic stochasticity, which can capture both frequency-dependent fitness, arising from social interactions, as well as over-dispersion in the number of offspring. We provide a simple analytical condition that governs the long-term outcome of competition between multiple types. Applied to pairwise social interactions involving cooperation or defection, we find that demographic stochasticity can favor cooperators provided the offspring variance is sufficiently large, even without any other mechanisms. For more general pairwise payoff structures, we show that demographic stochasticity can reverse the stability of equilibria, from coexistence to bi-stability and vice versa, or from dominance of one type to dominance of another. Our analysis highlights the profound effects of demographic stochasticity on the evolution of interacting types in a population.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figure_main\/main_figure1.pdf}\n    \\caption{\\textbf{Evolutionary dynamics with demographic stochasticity.} \n    (A) Competition between cooperators (blue circle) and defectors (red circle) in a stochastic population of non-constant size.  Each individual $i$ derives payoff $\\pi_i$ from pairwise game-play with each other individual in the population. The number of offspring produced by an individual within time $\\Delta t$ has mean $(B+s\\pi_i)\\Delta t$ and variance $(\\delta_1B+\\delta_2 s\\pi_i)\\Delta t$, which are both higher for defectors than for cooperators. When selection is weak ($s \\ll \\alpha$), the population quickly reaches carrying capacity (during time period I) while the frequency of cooperators and defectors remains unchanged from its initial value ($p_0=1\/2$ shown here). Thereafter (time period II) the population remains near carrying capacity ($M \\approx 1000$ shown here), while the frequency of cooperators and defectors slowly vary until either cooperators go extinct (example in panel B) or defectors go extinct (panel C).\n    Parameters: $b=3$, $c=1$, $s=0.01$, $\\delta_1=\\delta_2=1$, $x_0=y_0=10$, $\\lambda=1\\times10^{-3}$, $B=2$, $D=1$.}\n    \\label{fig1}\n\\end{figure}\n\n\\section{Model}\n\nWe first consider an evolving population of two types: cooperators (C) and defectors (D).\nEach individual interacts pairwise with each other, in which the cooperator pays a cost $c$ to bring his opponent a benefit $b$ ($b>c$), and the defector pays no cost and provides no benefit. In other words, pairwise interactions follow a simple ``donation game\", which provides a minimal model for studying the evolution of cooperation \\cite{rapoport1965prisoner}.\nFollowing all pairwise interactions, \neach individual obtains an average payoff that will determine their reproductive output (or, equivalently, the number of individuals who copy their type by social contagion). In a population with $x$ cooperators and $y$ defectors, the cooperator's payoff (denoted by $\\pi_C$) and the defector's payoff (denoted by $\\pi_D$) are \n\n\n\\begin{subequations}\n\\begin{align}\n    \\pi_C=&\\frac{x}{x+y}b-c, \\\\ \\pi_D=&\\frac{x}{x+y}b.\n\\end{align}\n\\end{subequations}\n\n\nIn a classic Moran model, each birth event is followed by a death event, and so the population size remains constant. Here we remove this constraint by decoupling the birth and death events. Births are assumed to follow a continuous-time Markov process with independent and stationary increments (see Section S1 in Supplementary Information), such that the expected number of offspring individual $i$ produces per unit time is \\begin{equation}\n     \\mathbb{E}(\\xi_i)=B+s\\pi_i,\n     \\label{eq:mean}\n\\end{equation}\nwhere $B$ is a baseline number of offspring, $\\pi_i$ is individual $i$'s payoff, and the parameter $s>0$ is the intensity of selection. Note that the baseline birth rate is the same for all individuals, regardless of type, and it does not depend upon payoffs from social interactions. The selection intensity $s$ measures to what degree the payoff derived from social interactions affects the offspring number. In this paper we focus on the case of weak selection ($s\\ll 1$), a regime widely adopted in the literature \\cite{Allen2017a,McAvoy2021,Nowak2004,Ohtsuki2006}. Since the defector's payoff $\\pi_D$ is larger than the cooperator's payoff $\\pi_C$ in any population state, defectors always have a greater expected fecundity (Fig.~\\ref{fig1}). \n\nTo fully describe the birth process, we also specify the variance in the number of offspring. We are particularly interested in cases of over-dispersion, which can be modelled in many alternative ways \\cite{Linden2011,VerHoef2007}, such as a quasi-Poisson model (variance proportional to mean), mixed-effects Poisson model, and negative binomial model (variance a quadratic function of mean). Here we study a general class of Markov birth models by stipulating\n\\begin{equation}\n    {\\rm Var}(\\xi_i)=\\delta_1 B+\\delta_2 s\\pi_i,\n    \\label{eq:variance}\n\\end{equation}\nwhere parameters $\\delta_1$ and $\\delta_2$ measure the magnitude of offspring variance ${\\rm Var}(\\xi_i)$ relative to the mean $\\mathbb{E}(\\xi_i)$. The parameter $\\delta_1$ controls how  offspring variance is influenced by the baseline birth rate; and $\\delta_2$ controls how  offspring variance is influenced by payoffs from social interactions. Specific choices of $\\delta_1$ and $\\delta_2$ reduce to well-known classical models, such as a deterministic system ($\\delta_1=\\delta_2=0$) or a Poison birth process ($\\delta_1=\\delta_2=1$). In the regime of weak selection, the number of offspring produced per unit time is over-dispersed whenever  $\\delta_1>1$.\n\n\n\nDeath events are modelled as a Poisson process, arising from two rates that are summed. First, an individual dies at constant baseline rate, $D$. Second, in order to model competition for limited resources, additional deaths occur at rate $\\lambda$ times the current total population size.\n\n\\section{Results}\n\n\n\n\n\n\\subsection{Evolution of cooperation with demographic stochasticity}\n\\label{section3.1}\nLet $x$ and $y$ denote the number of cooperators and defectors respectively, which will change over time. \nGiven the class of models described above for the payoff-dependent birth-process and the population-size dependent death process,  the evolutionary dynamics of $x$ and $y$ can be approximated by a two-dimensional It\\^o stochastic differential equation (see Section S1 in Supplementary Information):\n\\begin{subequations}\n    \\begin{align}\n    {\\rm d}x&=x\\left[\\alpha+s\\pi_{\\rm C}-\\lambda (x+y)\\right] {\\rm d}t+ \\sqrt{x\\left[\\delta_1 B+\\delta_2 s\\pi_{\\rm C}+D+\\lambda(x+y)\\right]}{\\rm d}W^{(1)}_t, \\label{eq:ito_diffusion_a} \\\\\n    {\\rm d}y&=y\\left[\\alpha+s\\pi_{\\rm D}-\\lambda (x+y)\\right]{\\rm d}t+ \\sqrt{y\\left[\\delta_1 B+\\delta_2 s\\pi_{\\rm D}+D+\\lambda(x+y) \\right]}{\\rm d}W^{(2)}_t, \\label{eq:ito_diffusion_b} \n    \\end{align} \\label{eq:ito_diffusion} \n\\end{subequations}\nwhere $\\alpha =B-D>0$ indicates the net growth rate from baseline birth and death events, and $W_t^{(1)}$ and $W_t^{(2)}$ are independent standard Wiener processes. Although the birth process can be over-dispersed in our model (when $\\delta_1>1$), deaths follow a simple Poisson process with variance equal to mean. \n\nTo study how the relative abundance of cooperators and the total population size evolve over time, we make the co-ordinate transformation $(p,n)=(x\/(x+y),x+y)$. Applying It\\^o's lemma in Eq.~\\ref{eq:ito_diffusion}, the system can then be described by the equations\n\\begin{subequations}\n\\begin{align}\n          {\\rm d}p=&scp(1-p)\\left(-1 +\\frac{\\delta_2}{n}\\right){\\rm d}t+\\frac{y}{n^2} \\sqrt{x(\\delta_1 B+D+\\lambda n) }{\\rm d}W^{(1)}_t  \\notag\\\\\n          &-\\frac{x}{n^2}\\sqrt{y(\\delta_1 B + D+\\lambda n)}{\\rm d}W^{(2)}_t, \\label{eq:transformed_a}\\\\\n          {\\rm d}n=&[n\\alpha+s(b-c)pn-\\lambda n^2]{\\rm d}t+ \\sqrt{x(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(1)}_t   \\notag \\\\\n          & +\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t. \\label{eq:transformed_b}\n\\end{align}\n\\label{eq:transformed}\n\\end{subequations}\n\nThe simple case in which stochasticity is absent (i.e., $\\delta_1=\\delta_2=0$ for births, and no variance for deaths) provides a deterministic reference point for comparison to any stochastic system. \nIn the deterministic system, ${\\rm d}p$ is always negative and the abundance of cooperators continuously decreases until cooperators reach extinction. Thus, cooperation is never favored by natural selection in the deterministic limit. Moreover, in this deterministic limit, changes in the total population size $n$ depend on both $p$ and $n$. But for sufficiently weak selection intensity ($s\\ll \\alpha$), changes in the total population size $n$ are much more rapid than changes in the cooperator frequency, $p$.\nIn the regime of weak selection, before $p$ changes its value at all, $n$ has grown logistically to its equilibrium value $(\\alpha+s(b-c)p)\/\\lambda$, which we denote by $M$. $M$ is called carrying capacity, and it describes the maximum number of individuals that the environment can sustain.  When the net growth rate is much larger than selection intensity, $\\alpha \\gg s$, the carrying capacity is well approximated by $M\\approx \\alpha\/\\lambda$. \n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.75]{figure_main\/heat_simu.pdf}\n    \\caption{\\textbf{Demographic stochasticity can favor the evolution of cooperation.} Colors represent the fixation probability of cooperation relative to neutral drift, $\\rho - p_0$, as a function of parameters $\\delta_2$ and $\\delta_1$. We say that selection favors cooperation when cooperators are more likely to fix than under neutrality (blue regions). Panel (A) shows exact solutions sampled from the stochastic differential equation (Eq.~\\ref{eq:ito_diffusion}), whereas panel (B) shows the analytical approximation in the regime of weak selection (Eq.~\\ref{eq:fixation prob}). The dashed line indicates the separation between regimes that favor cooperation (blue) or favor defection (red).\n       \n    Parameters: $B=2$, $D=1$, $s=0.005$, $b=1.1$, $c=1$, $\\lambda=5\\times 10 ^{-3}$, $x_0=y_0=50$.}\n    \\label{fig2}\n\\end{figure}\n\nFor a stochastic system ($\\delta_1 \\ne 0$ and $\\delta_2 \\ne 0$) the trajectories of $p$ and $n$ are not determined by the initial conditions alone, but depend upon chance events. We quantify the evolutionary advantage of cooperators by studying the fixation probability -- namely, the chance of absorption into the full-cooperation state ($p=1$). \nStarting from $x_0$ cooperators and $y_0$ defectors initially (thus $p_0=x_0\/(x_0+y_0)$ and $n_0=x_0+y_0$), the fixation probability, denoted by $\\rho(x_0,y_0)$ or $\\rho(p_0,n_0)$, is the probability that at some time $t$ defectors become extinct while cooperators still exist, that is $y(t)=0$ but $x(t)>0$ \\cite{Czuppon2018}. \nIn the regime $s \\ll \\alpha$ the fixation probability can be calculated by separating the time-scale of changes in $p$ versus changes in $n$ \\cite{Parsons2017}. This analysis is tantamount to assuming that the total population size $n$ rapidly reaches its carrying capacity, while $p$ remains unchanged from $p_0$, and that subsequently $p$ evolves in one dimension while the population size remains near the slow manifold $n=M$ (see Fig.~\\ref{fig1} and Supplementary Fig.~S3). Under this analysis, we can approximate the fixation probability by a simple expression (Section S2.1 in Supplementary Information) \n\\begin{equation}\n    \\rho(p_0,n_0) \\approx p_0+\\frac{sc}{(\\delta_1+1) B}\\left(\\delta_2-M\\right)p_0(1-p_0).\n    \\label{eq:fixation prob}\n\\end{equation}\nWe performed numerical simulations, drawing sample paths from the full SDE system given by  Eq.~\\ref{eq:ito_diffusion}, to verify the accuracy of this analytic approximation for the fixation probability  (Fig.~\\ref{fig2}).\n\nNote that fixation probability does not depend on the initial population size, but rather on the initial frequency of cooperators. In the absence of selection ($s=0$), the fixation probability equals the initial frequency of cooperators, $p_0$. And so we say that cooperation is favored by selection if the fixation probability exceeds $p_0$, which will occur whenever\n\\begin{equation}\n    \\delta_2>M. \\label{eq:condition}\n\\end{equation}\nThis simple condition tells us when demographic stochasticity causes selection to favor cooperators, even though selection disfavors cooperation in a deterministic setting. In particular, demographic stochasticity can favor the fixation of cooperators when the offspring variance is sufficiently large -- in particular, when $\\delta_2$ exceeds the carrying capacity $M$. What matters for the direction of selection, then, is the size of the offspring variance arising from payoffs in social interactions, relative to its mean.\n\n\n\nWe can gain some useful intuition for the forces that govern the fate of cooperators by considering the deterministic part of Eq.~\\ref{eq:transformed_a}. The first term in this equation, $-scp(1-p)$, represents the deterministic contribution to the evolution of cooperator frequency, which always opposes cooperators. Whereas the second term in this equation, $\\delta_2scp(1-p)\/n$, arises from demographic stochasticity and it always favors cooperators. Whether or not cooperation is favored overall depends upon the balance between these two forces -- the deterministic force suppressing cooperation and demographic stochasticity that favors cooperation. For $\\delta_2<M$, the deterministic disadvantage is the stronger force and cooperators are net disfavored (recall that $n$ rapidly reaches carrying capacity $n=M$ before cooperators change frequency). However, if $\\delta_2>M$, the stochastic advantage matters more, so that cooperators are favored, which constitutes an evolutionary reversal compared to a classical model without demographic stochasticity.\n\n\nOther model parameters, $s$, $c$, $p_0$, $\\delta_1$ and $B$, do not produce a reversal in the direction of selection for cooperation, but they nonetheless influence the fixation probability. For example, increasing $\\delta_1$ or increasing the baseline birth rate $B$ moves the fixation probability towards the neutral value, $p_0$. Moreover, in the regime where demographic stochasticity favors cooperation, $\\delta_2>M$, the fixation probability is increased yet further when the selection intensity $s$ is large or when the cost of cooperation $c$ is large (Eq.~\\ref{eq:fixation prob}). Both of these results contravene the classical intuition that selection and the cost of cooperation should disfavor cooperators. \nWe have performed simulations to verify the effects of all these parameters, in comparison to the analytical approximation (Supplementary Fig.~S2).\n\n\n\\subsection{An explicit birth-death process}\n\nOur model of demographic stochasticity is quite general, stipulating only several properties of the Markov birth and death processes for competing types. We have analyzed this class of models by approximation, using a stochastic differential equation. In this section we construct an explicit example of birth and death processes that satisfy our model stipulations, and we compare the predictions of our SDE analysis to individual-based simulations of the discrete stochastic process.\n\nMost prior studies of demographic stochasticity are based on a reproduction process with a single offspring per birth event, which naturally leads to a Poisson birth process \\cite{Feller1950,Huang2015,Constable2016,Parsons2010,Parsons2007,Parsons2007a}. The Poisson process occurs as a special case within our family of models, when $\\delta_1=\\delta_2=1$. In this case, our analysis shows that demographic stochasticity alone cannot favor cooperation, because $\\delta_2<M$. \nWe will therefore consider non-Poisson birth process, in which the offspring produced per unit time is over-dispersed. This is a realistic scenario for many species, especially pelagic organisms, that have heavy-tailed offspring distributions \\cite{Davis1956,Eldon2006}; as well as for social contagion \\cite{brady2017emotion,schroder2021social}.\n\nWe will define a birth process by two factors: the times of birth events and the litter size (offspring number) in each such birth event. A natural way to describe this is through a compound Poisson process \\cite{Last2017}. Specifically, for individual $i$ with payoff $\\pi_i$, the times of birth events obey a Poisson process with intensity $\\theta_i$. In each such birth event, the number of offspring produced (litter size) is also stochastic. We consider two cases: the litter size itself follows a Poisson distribution with mean $\\mu_i$, or the litter size follows a negative binomial distribution with parameters $q_i$ and $m$ ($q_i\\in [0,1]$ and $m\\in \\mathbb{N}^*$). Both of these distributions have been used to model litter sizes in empirical studies \\cite{Sheldon2003,Richards2008,Linden2011}. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.75]{figure_main\/compound_poisson_2.pdf}\n    \\caption{\\textbf{Selection for cooperation in a compound Poisson birth process.}\n    We simulated a compound Poisson birth process with either a Poisson-distributed litter size (A) or a negative binomial litter size (B). The parameters of the birth process ($\\theta_i$ and $\\mu_i$ in panel A; $\\theta_i$ and $q_i$ in panel B) can be chosen to satisfy our general conditions for the mean and variance in total offspring produced per unit time, for any choice of $\\delta_1>1$, $\\delta_2$, and $B$.  \n    Two examples with the parameters that correspond to $(\\delta_1=6,\\delta_2=60)$ and $(\\delta_1=6,\\delta_2=140)$ are shown in each panel. Blue squares indicate the fixation probability of cooperators, starting from an initial population with $x_0=y_0=50$, observed in $5\\times 10^7$ replicate Monte Carlo simulations, with carrying capacity either $M=100$ or $M=200$. Selection favors cooperation if the fixation probability $\\rho$ exceeds the initial fraction of cooperators,  0.5 (horizontal dashed line).\n    The solid lines plot our analytical approximation for the fixation probability (Eq.~\\ref{eq:fixation prob}). As predicted by our analysis, cooperation is favored when $\\delta_2>M$.\n    Parameters: $B=2$, $D=1$, $\\delta_1=6$, $s=0.001$, $b=1.1$, $c=1$, $m=5$ (negative binomial), $x_0=y_0=50$, $\\lambda=1\/100$ ($M=100$) or $\\lambda=1\/200$ ($M=200$).}\n    \\label{fig3}\n\\end{figure}\n\nThe parameters of the compound Poisson process depend upon an individual's payoff and the selection intensity. For the Poisson-Poisson case (the litter size follows a Poisson distribution) the reproductive process of individual $i$ is characterized by parameters $\\theta_i$ and $\\mu_i$, and we assume that the payoff $\\pi_i$ affects both $\\theta_i$ and $\\mu_i$ linearly\n\\begin{subequations}\n\\begin{align}\n    \\theta_i&=\\theta_0+k_\\theta s\\pi_i, \\\\\n    \\mu_i&=\\mu_0+k_\\mu s\\pi_i.\n\\end{align}\n\\end{subequations}\nFor the Poisson-negative binomial case (the litter size follows a negative binomial distribution), we assume that all individuals share the same $m$ and that payoffs affect $q_i$ and $\\theta_i$ as follows:\n\\begin{subequations}\n\\begin{align}\n    \\theta_i&=\\theta_0+k_\\theta s\\pi_i, \\\\\n    q_i&=q_0+k_q s\\pi_i.\n\\end{align}\n\\end{subequations}\n\nGiven these equations, we can always choose parameters of the compound Poisson process\nthat satisfy our general stipulations on the mean and variance in the total offspring produced per unit time (Eq.~\\ref{eq:mean} and Eq.~\\ref{eq:variance}), provided $\\delta_1>1$ and $\\delta_2>0$ (see Section S3 in Supplementary Information). Note that for both of these compound Poisson birth processes (Poisson-Poisson and Poisson-Negative-Binomial) the total number of offspring produced per unit time must be over-dispersed ($\\delta_1>1$).\n\nWe can compare Monte-Carlo simulations of these explicit population processes (discrete state, continuous time) to the analytical prediction for the fixation probability that we derived from a stochastic differential equation (Eq.~\\ref{eq:fixation prob}). We find good agreement between the individual-based simulations and analytic approximations, for carrying capacities as small as $M=100$ or $M=200$  (Fig.~\\ref{fig3}). Note that in both cases shown in Fig.~\\ref{fig3}, for sufficiently large $\\delta_2$ we have $k_\\theta<0$ and $k_m>0$ or $k_q>0$. In other words, higher payoffs reduce the rate of birth events but increase the mean litter size per birth event; and when these effects are strong enough, then selection favors cooperation.\n\n\n\n\n\n\\subsection{Intuition for the effects of demographic stochasticity}\n\nThere is a simple intuition for how demographic stochasticity can favor cooperation in our class of models, even though cooperation is always disfavored in models with constant (or infinite) population size. The key insight has to do with the rapid growth of the total population size to carrying capacity, followed by slow dynamics in the frequency of cooperators near the manifold $n=M$. Importantly, during the slow dynamics there are still small fluctuations that move the population off the manifold $n=M$, followed by a rapid return back to carrying capacity. These small fluctuations have the effect of inducing an advective force pushing the frequency of cooperators $p$ in one direction or another. \n\nTo be more precise, we have already noted that the total population size $n$ equilibrates much more quickly than the frequency of cooperators $p$ (Eq.~\\ref{eq:transformed}), in the regime we study $\\alpha \\gg s$. And so, given an arbitrary initial state $p_0$ and $n_0$, $n$ will quickly converge to the slow manifold\n\\begin{equation}\n    n=\\frac{\\alpha+s(b-c)p_0}{\\lambda}\\approx \\frac{\\alpha}{\\lambda}=M,\n    \\label{eq:slow manifold}\n\\end{equation}\nwhile $p$ does not change from $p_0$ (see example in Supplementary Fig.~S3B). After the population size reaches carrying capacity, trajectories then move along the slow manifold until one type or the other fixes ($p=0$ or $p=1$). We focus on the dynamics on the slow manifold, which simplifies the analysis to a one-dimensional system \\cite{Parsons2017}. \n\nIn the co-ordinate system $(x,y)$, the slow manifold is defined $x+y=M$, and the fast manifolds are lines connecting the origin to points on the slow manifold (see Fig.~\\ref{fig4}A). \nGiven any initial conditions, the trajectory will rapidly approach the slow manifold along one of these lines, and then subsequently move within the slow manifold. However, unlike the case of a strictly constant population size, the system with demographic stochasticity does not lie precisely on the slow manifold at all times. Small fluctuations take the system off the slow manifold briefly, and then the system rapidly  returns to the slow manifold. Critically, the position where the system returns to the slow manifold, after a fluctuation, is not necessarily the same as where it started. In fact, there can be a systematic deviation in the position on the slow manifold that arises from stochastic fluctuations and rapid returns -- which produces an advective force on the frequency $p$ along the slow manifold (see Fig.~\\ref{fig4}B, C, D). It is this systematic deviation, caused by demographic stochasticity, that introduces a force favoring cooperation.\n\nIn particular, fluctuations from $x+y=M$ follow a two-dimensional Gaussian distribution with variance $x(\\delta_1B+\\delta_2s\\pi_C+D+\\lambda n)$ in the $x$-direction and variance $y(\\delta_1B+\\delta_2 s\\pi_D+D+\\lambda n)$ in the $y$-direction. In Fig.~\\ref{fig4}, we illustrate the fluctuation starting from state $x=y=M\/2$ (see Supplementary Information Section S2.2 for the analysis of any other states).\nWhen $\\pi_C=\\pi_D$, the Gaussian fluctuation is isotropic, and so a fluctuation followed by return along a fast-manifold line produces no expected change in the resulting position on the slow manifold (see Fig.~\\ref{fig4}B). However, whenever $\\pi_C \\ne \\pi_D$, the two-dimensional Gaussian fluctuation has an ellipsoid shape, and fluctuation followed by rapid return produces an expected change in the frequency of cooperators, $p$, along the slow manifold. In particular, when $\\pi_C < \\pi_D$, the expected change due to demographic stochastic favors cooperators, whereas if $\\pi_C > \\pi_D$ the expected change favors defectors (Fig.~\\ref{fig4}C,D). In general, we can analytically calculate the adjective force along the slow manifold that arises from these stochastic fluctuations and rapid returns (Section S2.2 in Supplementary Information).  \n\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.65]{figure_main\/fig4.pdf}\n    \\caption{\\textbf{How demographic stochasticity can favor cooperation or defection.}  \n    (A) The system features a separation of timescales, where the total number of individuals $n=x+y$ changes much faster than the fraction of cooperators $p=x\/(x+y)$.\n    Starting from $x_0$ and $y_0$ cooperators and defectors, trajectories rapidly converge to the slow manifold ($x+y=M$) along the fast manifold $x\/y=x_0\/y_0$.\n    (B, C, D) Stochastic fluctuations away from the slow manifold, followed by rapid return, can induce an advective force on the frequency of cooperators.\n    For simplicity we consider constant payoffs, where $\\pi_C$ and $\\pi_D$ are independent of the number of cooperators and defectors. The ellipses illustrate the variance-covariance structure  of two-dimensional Gaussian fluctuations around the slow manifold from a given point $x=M\/2$ and $y=M\/2$ (red point $O$). \n    (B) When $\\pi_C=\\pi_D$, fluctuations from point $O$ are isotropic, shown as a circle.\n    We consider four representative fluctuations from point $O$, $X_{-},X_{+},Y_{-},Y_{+}$, and the following points of return $X_{-}',X_{+}',Y_{-}',Y_{+}'$ to the slow manifold.\n    For isotropic fluctuations there is no expected change in $p$ after return to the slow manifold.\n    (C) For $\\pi_C<\\pi_D$, the Gaussian fluctuations are an-isotropic, shown as an ellipse, with larger fluctuations in the number of defectors. This asymmetry leads to an expected increase in cooperator frequency $p$ after return to the slow manifold, as indicated by the blue arrow.\n    (D) For $\\pi_C>\\pi_D$, the larger fluctuation occurs in the number of cooperators, which leads to an expected decrease in cooperator frequency after return to the slow manifold. These effects of an-isotropic noise are similar to those discussed by \\cite{Constable2016}, but they arise here even when both types have the same baseline birth rate and the same carrying capacity, under weak selection.}\n    \\label{fig4}\n\\end{figure}\n\n\nFor the donation game we have studied so far, cooperators always have a lower payoff than defectors regardless of the population state. And so the advective force arising from demographic stochasticity always favors cooperation, regardless of $p$. If this force is large enough relative to the deterministic force favoring defectors, then it can produce a net advantage for cooperators. For other types of pairwise games, however, the direction of deterministic selection  ($\\pi_C$ vs $\\pi_D$) may depends on the current frequency $p$ in the population, and so the noise-induced advection may change sign along the slow manifold, producing complicated effects on long-term dynamics. We investigate these effects of demographic noise on evolutionary dynamics for general two-player games in the next section.  \n\n\n\n\\subsection{General evolutionary game dynamics with demographic stochasticity}\n\nFor an arbitrary two-player game that gives rise to payoffs, the two-dimensional system can be simplified to a one-dimensional system by separation of timescales, provided selection is weak enough, $s \\ll \\alpha$.  Suppose the game has the following payoff structure:\n\\begin{equation}\n    \\begin{array}{cc}\n       & \\begin{array}{cc}\n          {\\rm C} & {\\rm D} \n      \\end{array} \\\\ \\begin{array}{c}\n           {\\rm C}\\\\{\\rm D} \n      \\end{array}\n     & \\left(\\begin{array}{cc}\n          a&b \\\\\n          c&d \n     \\end{array}\\right).\n\\end{array}\n\\end{equation}\nPlayers have two strategies, which we still generically call cooperation (C) or defection (D). When two cooperators interact, both of them receive payoff $a$. When a cooperator interacts with a defector, the cooperator receives $b$ and the defector $c$. Mutual defection brings payoff $d$ to both players. The average payoff for a cooperator or defector in a population are respectively\n\\begin{equation}\n\\begin{split}\n    \\pi_C&=\\frac{xa+yb}{x+y}, \\\\\n    \\pi_D&=\\frac{xc+yd}{x+y}.\n\\end{split}\n\\end{equation}\n\nSimilar to Section \\ref{section3.1}, we can describe the system by a stochastic differential equation:\n\\begin{subequations}\n\\begin{align}\n          {\\rm d}p=&sp(1-p)\\left(1-\\frac{\\delta_2}{n}\\right)(\\pi_{\\rm C}-\\pi_{\\rm D}){\\rm d}t+\\frac{1-p}{n} \\sqrt{x(\\delta_1 B+D+\\lambda n)}{\\rm d}W^{(1)}_t \\notag \\\\\n          &-\\frac{p}{n}\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t, \\label{eq:transformed_general_a}\\\\\n          {\\rm d}n=&[n\\alpha+s(p\\pi_{\\rm C}+(1-p)\\pi_{\\rm D})pn-\\lambda n^2]{\\rm d}t+ \\sqrt{x(\\delta_1 B+D+\\lambda n)}{\\rm d}W^{(1)}_t\\notag \\\\\n          &+\\sqrt{y(\\delta_1 B +D+\\lambda n)}{\\rm d}W^{(2)}_t. \\label{eq:transformed_general_b}\n\\end{align}\n\\label{eq:transformed_general}\n\\end{subequations}\nSince the population size quickly equilibrates to the carrying capacity  $M\\approx\\alpha\/\\lambda$, we substitute $n=M$ into Eq.~\\ref{eq:transformed_general_a} which yields a one-dimensional equation for the evolution of $p$ along the slow manifold:\n\\begin{subequations}\n\\begin{align}\n    {\\rm d}p=&sp(1-p)\\left[\\left(1-\\frac{\\delta_2}{M}\\right)\\left(b-d+(a-b-c+d)p\\right) \\right]{\\rm d}t \\notag \\\\\n    &+\\sqrt{\\frac{(\\delta_1+1) B p(1-p)}{M}}\\left(\\sqrt{1-p}{\\rm d}W^{(1)}_t-\\sqrt{p}{\\rm d}W^{(2)}_t\\right).\n\\end{align}\n\\label{eq:general one dimensional}\n\\end{subequations}\n\nIn the case of deterministic births and deaths ($\\delta_1=\\delta_2=0$ and neglecting variance in the death process), this equation simplifies to the classic replicator equation \\cite{Schuster1983,Nowak2006a}. For general games there may be interior equilibrium points, and so the fixation probability is no longer a good measure to describe long-term evolutionary outcomes. Instead, we analyze the dynamics from two perspectives. One is from the perspective of the deterministic behavior on the slow manifold,\nwhich neglects stochasticity altogether in Eq.~\\ref{eq:general one dimensional} and studies the equilibria of the resulting ordinary differential equation. The other, more nuanced perspective accounts for stochasticity. Since $p=0$ and $p=1$ are the only absorbing states, any trajectory will finally reach one of these states and then become invariant. However, we can impose a reflecting condition on the boundary, which is equivalent to assuming that, when the number of one phenotype reaches zero, a new mutant of this phenotype arises instantly. The resulting evolutionary process of $p$ becomes an ergodic Markov process which has a unique stationary distribution $v^*(p)$. A frequency $p$ with greater probability density means that trajectories spend more time there. Derivation of the stationary distribution $v^*(p)$ under reflecting boundaries is given in Section S4.1 of Supplementary Information. \n\nWhen we ignore the stochastic terms, then Eq.~\\ref{eq:general one dimensional} is an ODE with the same equilibrium points and stabilities as the classic replicator equation, provided $\\delta_2<M$. Whereas if $\\delta_2>M$, then the equilibrium points are the same as the classic replicator equation, but the stabilities are reversed: equilibrium points that are classically unstable become stable, and conversely. And so the value of $\\delta_2$, which determines the payoff-component of offspring variance, can reverse the evolutionary outcome, even from a deterministic perspective.\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width=\\textwidth]{figure_main\/main_figure4.pdf}\n    \\caption{\\textbf{General evolutionary game dynamics with demographic stochasticity.} \n    We consider three types of representative games, such as prisoner's dilemma (A, D), snowdrift game (B, E), and stag-hunt games (C, F).\n    In the prisoner's dilemma games, when demographic stochasticity is absent or does not meet $\\delta_2>M$, defectors dominate the population (see trajectories sampled in A, left part).\n    While the evolutionary direction can be reversed for $\\delta_2>M$, where cooperation becomes the dominant strategy (see trajectories sampled in A, right part).\n    Shown in (D) is the stationary distribution of cooperators for $\\delta_2=0$, $\\delta_2=25000$, and  $\\delta_2=50000$.\n    Analogously, in the snowdrift game, the demographic stochasticity with $\\delta_2>M$ changes the equilibrium from the coexistence of two strategies (B, left part) to the bi-stability (B, right part), which suggests the transformation of a snowdrift game to a stag-hunt game.\n    We also find that with demographic stochasticity, the evolution in the stag-hunt games proceed ``as if\" the population are playing snowdrift games. \n    Parameters: $B=2$, $D=1$, $s=10^{-3}$, $\\delta_1=2.5$, $\\lambda=10^{-4}$, $x_0=y_0=100$. }\n    \\label{fig5}\n\\end{figure}\n\nMore generally, we can classify three different deterministic scenarios based on the payoff matrix of the two-player, two-action game. For dominance games (Fig.~\\ref{fig5}A), one strategy is always dominant. Here, without loss of generality, we assume defection dominates cooperation ($a<c$ and $b<d$, e.g.,~a prisoner's dilemma). If $\\delta_2<M$, then all trajectories will converge to the full-defector state ($p=0$ stable and $p=1$ unstable). However, if $\\delta_2>M$, cooperation becomes the dominant strategy and all trajectories converge to full-cooperator state ($p=1$ stable and $p=0$ unstable). For coexistence games ($a<c$ and $b>d$, e.g.,~a snowdrift game), the best response is to choose the opposite strategy of the opponent (Fig.~\\ref{fig5}B). If $\\delta_2<M$, there is only one stable equilibrium, $p^*=(d-b)\/(a-b-c+d)$. All trajectories will converge to $p^*$ and therefore cooperators and defectors stably coexist. If $\\delta_2>M$, $p^*$ becomes unstable and $p=0$ and $p=1$ are each stable. Thus, all trajectories converge to either the full-cooperator or the full-defector state, similar to the outcome of a classic coordination game. For coordination games (Fig.~\\ref{fig5}C), the best response is to choose the same strategy as the opponent ($a>c$ and $d>b$, e.g., a stag-hunt game). In this case, $\\delta_2<M$ leads to an unstable internal equilibrium $p^*$ with stable boundaries ($p=0$ and $p=1$). But for $\\delta_2>M$, $p^*$ becomes stable while $p=0$ and $p=1$ are unstable. Most trajectories fluctuate around $p^*$ for a long time, showing similar behavior as a classic coexistence game. In summary, in a population with sufficiently large offspring variance ($\\delta_2>M$), the outcome of each type of game has the dynamical properties classically associated with the opposite type of game in a deterministic setting. In other words, demographic stochasticity effectively transforms the payoff structure of a game in the following way\n\n\\begin{equation}\n    \\begin{pmatrix}\n    a&b\\\\c&d\n    \\end{pmatrix} \\Rightarrow \\begin{pmatrix}\n    -a&-b\\\\-c&-d\n    \\end{pmatrix}.\n\\end{equation}\n\nWe can also characterize general two-player games in term of the stationary frequency distribution of strategies, with reflecting boundaries. This description accounts for more details in the stochastic dynamics, and it reveals a similar, transformative effect of large offspring variance. \nIf $\\delta_2$ is sufficiently large, namely $\\delta_2>M$, then modes of the stationary distribution can be moved from one boundary to the other boundary (dominance games, Fig.~\\ref{fig5}D), from the interior to the two boundaries (coexistence games, Fig.~\\ref{fig5}E), or from the two boundaries to the interior (coordination games, Fig.~\\ref{fig5}F). \nThese results reflect our ODE-based analysis above, and they show that sufficient offspring variance can reverse the evolutionary dynamics in an interacting population. These dramatic effects extend to games with more than two actions, such as rock-paper-scissors (Supplementary Fig.~S4).\n\n\n\n\nThese two analytical perspectives  underscore that large offspring variance can reshape the payoff structure of a game, producing dynamics classically seen in an entirely different game type.  So far, we have focused on the scaling factor $\\delta_2$, which governs how offspring variance grows with payoff, as opposed to $\\delta_1$, which governs the baseline offspring variance. The value of $\\delta_1$ can also profoundly influence evolutionary outcomes, although this cannot be seen from a deterministic perspective alone because $\\delta_1$ has no effect on stabilities of equilibria. Analysis of the stationary distribution shows that a large baseline variance ($\\delta_1B$) can transform any game into a coordination game (see Section S4.1 in Supplementary Information). An example of this result is shown in Fig.~\\ref{fig5}F, where even though $\\delta_2=25,000$ exceeds the carrying capacity, the stationary distribution is not unimodal around intermediate frequency. This is because the effect of  $\\delta_2$ here is offset by the effect of $\\delta_1$.\nThese results show that demographic noise, especially when offspring variance is high, can qualitatively change the evolutionary outcomes compared to  predictions of traditional analysis by replicator equations for fixed or infinite population size \\cite{Hofbauer1998}.\n\n\n\n\n\n\n\n\n\\section{Discussion}\n\nThe question of how cooperation can be maintained is a longstanding and active area of research, spanning multiple disciplines. A large literature has produced compelling explanations for cooperation, but these typically rely on some form of population structure or repeated interactions. Here, we find that even in a well-mixed population with one-shot interactions, natural stochasticity in the total population size alone can favor cooperation that would otherwise be suppressed. For other types of social interactions, as well, demographic stochasticity can reverse the direction of evolutionary trajectories and produce behavioral outcomes that contravene classical expectations.\n\n\nIt is intuitively easier to invade a noisy population than a stable population. And so natural selection near carrying capacity prefers types not only with higher fecundity (greater mean offspring number), but also with lower reproductive noise (smaller offspring variance) \\cite{Parsons2007,Parsons2007a}. The reversal in the direction of selection in a stochastic population reflects this basic trade-off between offspring mean and offspring variance. A larger payoff produces higher fecundity but also greater noise in the reproduction process. Whether it is the mean or the variance in offspring number that dominates the course of evolution is determined by their relative importance, which is governed by $\\delta_2$ in our model.\nClassical models of populations with constant (or infinite) size neglect the effects of offspring variance altogether; but more realistic models, we have seen, permit regimes where offering variance is more important than fecundity.\n\n\n\nAlthough demographic noise has been studied extensively in population models, the underlying mechanism for our results is qualitatively different from those explored in prior studies. Most research on demographic noise has been restricted constant fitness for competing types \\cite{Parsons2007,Parsons2007a,Parsons2010,McKane2005,Butler2009,Hallatschek2007,Stollmeier2018,Wienand2017,Taitelbaum2020,Chotibut2017}, which does not provide a model of social interactions. However, Constable et al.~ analyzed a frequency-dependent fitness model, and they also found that demographic noise can reverse the direction of selection \\cite{Constable2016}. Their model is based on the production and consumption of a public good. One phenotype produces the public good, at a cost that reduces its baseline birth rate, while the other phenotype does not produce the public good. \nThey analyze the case when ``cooperators\" (who produce the public good) have a larger intrinsic carrying capacity than non-producers, and the larger carrying capacity then yields an evolutionary advantage by making producers more robust against invasion. \nThis mechanism is thus a stochastic form of $r$ versus $K$ selection \\cite{pianka1970r}, and it occurs when births and deaths follow Poisson processes. By contrast, in our model, the evolutionary advantage of cooperators arises even though both types have the same baseline birth rate and the same carrying capacity; \n\nand it arises only when the birth process related to payoff is sufficiently over-dispersed. \nThis mechanism is thus fundamentally different from a trade-off between baseline birth rate and carrying capacity of competing types in a Poisson model \\cite{Constable2016,Houchmandzadeh2012,Houchmandzadeh2015}, and it is more closely related to phenomena in population models with heavy-tailed offspring distributions \\cite{schweinsberg2000necessary,Eldon2006,Sargsyan2008,der2012dynamics}.\n\nAside from promoting cooperation in the prisoner's dilemma, demographic stochasticity also transforms outcomes in other forms of social interaction. Stochasticity can convert a snowdrift game into a stage-hunt game, for example, so that the stable co-existence expected in a deterministic or Poisson setting is transformed into bi-stability. Here, again, the underlying mechanism that reverses the evolutionary outcome is over-dispersion in the offspring contribution related to payoff, even when both types have the same baseline birth rate and carrying capacity.\n\nAll of our analyses have assumed a fast-growing population ($\\alpha \\gg s$), which rapidly reaches carrying capacity before any change in the relative frequencies of competing types. The dynamics of competition may be more complicated in a stochastic, slow-growing population, because their analysis cannot be reduced to a one-dimensional slow manifold. In this regime, fixation will take place before reaching carrying capacity. We can nonetheless derive approximations for the fixation probability in this regime as well (Section S4.2 in Supplementary Information), and, in the case of the donation game, we find that cooperation will be favored by selection provided $\\delta_2$ exceeds the initial population size, $\\delta_2>n_0$. This condition is typically easier to satisfy than Eq.~\\ref{eq:condition}, and it is confirmed by both numerical simulations and Monte Carlo simulations of the compound Poisson process (Supplementary Fig.~S5 and Fig.~S6). After cooperators or defectors fix, in this regime of a slow-growing population, the population will tend to grow logistically to its carrying capacity; but in this case the carrying capacity is larger for cooperators (Supplementary Fig.~S7), which provides an additional evolutionary advantage and greater chance of long-term persistence (Supplementary Fig.~S8).\n\n\n\nOur results highlight the strong impact of stochasticity on evolutionary outcomes in populations. The demographic stochasticity we have studied arises from intrinsic properties of birth and death processes, which have size of order $O(\\sqrt{n})$. As the population size grows towards infinity this form of stochasticity has little influence on evolutionary dynamics, which is consistent with the recent finding that migration in finite, group-structured populations can favor cooperators provided the population size is not too large \\cite{Braga2022}. Aside from intrinsic stochasticity during reproduction, real populations may also be subject to external noise, arising from exogenous variation in environmental conditions. Unlike demographic noise, exogenous noise  can be substantial even in population of arbitrary large size. Prior studies on environmental fluctuations, including fluctuations in selection intensity \\cite{Assaf2013}, carrying capacity \\cite{Wienand2017,Taitelbaum2020}, and payoff structure \\cite{Stollmeier2018}, have analyzed their effects by imposing an external noise term onto an otherwise classical, deterministic and continuous system of equations.  The effects of exogenous noise on discrete stochastic systems remain less explored, and they are likely to differ qualitatively from stochastic perturbations of continuous systems  \\cite{Durrett1994}. Coupling intrinsic demographic noise with external environmental noise may produce even more complicated effects, which remains a topic for future research.\n\nThe impact of stochasticity on strategic outcomes likely extends beyond the two-player\/two-action games we focused on, to include many aspects of non-human and human social behavior. Even if behavioral spread is caused by biased imitation, there is nonetheless variance in number of individuals who imitate a type, as well as physical variation in population sizes of interacting social groups as individuals move between social settings. Empirical data has documented burstiness, a form of over-dispersion, in  social interactions \\cite{Stehle2010,Goh2008}. Likewise, in the context of behavior during an epidemic, there is evidence of super-spreading individuals that cause over-dispersion in infectiousness \\cite{Tkachenko2021,Kirkegaard2021}, which may influence frequency-dependent competition among co-circulating variants. Extending our model and analysis to these settings remains an open topic for future research.\n\n\n\n\n\\clearpage\n\\bibliographystyle{unsrtnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\nQuivers with relations play a fundamental role in the representation theory of finite dimensional algebras.\n\nEvery basic finite dimensional algebra $\\Lambda$ over an algebraically closed field $\\Bbbk$ is isomorphic to a path algebra $\\Bbbk Q\/R$ of a finite quiver $Q$ with admissible relations $R$. Moreover, the modules of $\\Lambda$ are in bijection with the representations of $(Q,I)$ over $\\Bbbk$. Without relations, we get a correspondence only to hereditary algebras \\cite{Gabriel}; see also \\cite[Ch.~III.1]{ARS}. \n\nNon-hereditary algebras are central in most fields of modern representation theory of algebras. For one, higher homological algebra requires algebras of global dimension at least 2 \\cite{Jasso, Kvamme}. There is a rich tradition of studying classes of non-hereditary algebras, such as gentle \\cite{AH, AS}, clannish \\cite{C-B}, Schur \\cite{Erdmann}, preprojective \\cite{Ringel}, and self--injective \\cite{SY} algebras.\n\nContinuous quivers and their representations were first explicitly studied in \\cite{IRT}. They are a natural generalisation of quivers, replacing finite sets of vertices with uncountably infinite sets. In the process, one gains intuition about what characteristics of representation theory come from innate properties of algebraic structures, and what comes from the discrete examples that are usually studied.\n\nOne parameter persistence modules are often defined over the real line so that persistence modules coincide with pointwise finite-dimensional representations of a continuous quiver of type $\\mathbb{A}$ (see, for example, \\cite{CdSGO}).\nIn \\cite{BBOS} the authors consider $m\\times n$ rectangular grid quivers which have the commutativity relation on each square.\nThe authors of \\cite{BBH} study homological approximations in order to obtain new invariants of these representations (persistence modules).\n\n\nGiven the important role of quiver relations in the representation theory of finite-dimensional algebra, it is natural to ask if relations can be extended to the continuous setting. This has already been done in a restricted sense by the second author and Zhu in \\cite{RZ}. We give a more general definition that works with any underlying quiver. To capture the full generality we actually go beyond quivers and consider categories instead.\n\nIn starting this work, we were motivated by two areas of study that we intend to lift to the continuous setting: gentle algebras and $d$-cluster-tilting subcategories. In gentle algebras, the relations appear in the definition, and are always generated by compositions of two arrows. This type of relations are generalized as \\emph{point relations} in \\Cref{subsec:point relations}. An important class of $d$-cluster-tilting subcategories appear in the module category of type A algebras, with relations consisting of all paths above a certain length \\cite{Vaso}. This type of relations is generalized as \\emph{length relations} in \\Cref{subsec:length}.\n\n\n\\subsection{Contributions}\nIn \\Cref{sec:definition and general}, we give essential background, before stating our main definition.\n\n\\begin{definition}[\\Cref{def:admissible}]\nLet $\\mathcal{C}$ be a category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    We say $\\mathcal{I}$ is \\emph{admissible} if the following are satisfied.\n    \\begin{enumerate}\n        \\item For each $f$ in $\\mathcal{I}$, there exists a finite collection of morphisms $g_1,\\ldots,g_n$ not in $\\mathcal{I}$ such that $f = g_n\\circ\\cdots\\circ g_1$.\n        \\item For each nonzero endomorphism $f$, if $f$ is not an isomorphism then there exists $n\\geq 2\\in\\mathbb{N}$ such that $f^n\\in\\mathcal{I}(X,X)$.\n        \\end{enumerate}\n\\end{definition}\n\n\\Cref{sec:general results} gives some general results on the quotient category $\\mathcal{C} \/ \\mathcal{I}$, summarized here.\n\n\n\\begin{theorem}[\\Cref{prop:connected,prop:basic,prop:radical commutes with admissible}]\n    Let $\\mathcal{C}$ be a  category. Let $\\mathcal{I}$ be an admissible ideal of morphisms.\n    \n    \\begin{enumerate}\n        \\item If $\\mathcal{C}$ is connected, then $\\mathcal{C} \/ \\mathcal{I}$ is also connected.\n        \\item If $\\mathcal{C}$ is Krull--Remak--Schmidt--Azumaya, then $\\mathcal{C} \/ \\mathcal I$ is also Krull--Remak--Schmidt--Azumaya.\n        \\item If all endomorphism rings of $\\mathcal{C}$ are artinian, then \\[\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I})=\\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}.\\]\n    \\end{enumerate}\n\\end{theorem}\n\nIn \\Cref{sec:relations} we give two important classes of relations that can generate admissible ideals. The first is \\emph{point relations}, which generalizes relations of length two. The idea is that certain paths through a vertex in the quiver are excluded, but others not. For an illustration see \\Cref{fig:point intro}.\n\n\\begin{figure}[h!]\n    \\centering\n    \\begin{subfigure}[b]{0.3\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n        \\node (center) at (0,0) {$\\bullet$};\n        \\node at (0,.3) {$P$};\n        \\node (left) at (-1,0) {$\\circ$};\n        \\node (right) at (1,0) {$\\circ$};\n        \\draw[->] (left)--(center);\n        \\draw[->] (center) -- (right);\n    \\end{tikzpicture}\n    \\caption{Discrete case}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.3\\textwidth}\n    \\centering\n    \\begin{tikzpicture}[decoration={markings, mark=at position 0.5 with {\\arrow{>}}}]\n        \\node (center) at (0,0) {$\\bullet$};\n        \\node (left) at (-1,0) {};\n        \\node (right) at (1,0) {};\n        \\node at (0,.3) {$P$};\n        \\draw[very thick, postaction=decorate] (left.center) to[bend left] (center.center);\n        \\draw[very thick, postaction=decorate] (center.center) to[bend right] (right.center);\n    \\end{tikzpicture}\n    \\caption{Continuous case}\n    \\end{subfigure}\n    \\caption{An illustration of point relations in the discrete and continuous case. In both cases, the relations contain all paths passing through the point $P$.\n    See \\Cref{fig:arrow and line} on page~\\pageref{fig:arrow and line} for an explanation of our drawing conventions.}\n    \\label{fig:point intro}\n\\end{figure}\n\n\n\\begin{theorem}[\\Cref{thm:point relations are admissible}]\n    Let $\\{\\mathcal P_\\alpha\\}$ be an admissible collection of point relations in $\\mathcal{C}$, such that any cycles are either isomorphisms (and hence trivial) or contained in  at least one $\\mathcal P_\\alpha$.\n    Then $\\mathcal{I}=\\langle \\bigcup_\\alpha \\mathcal P_\\alpha \\rangle$ is an admissible ideal in $\\mathcal{C}$.\n\\end{theorem}\n\nThe other class of relations we define are \\emph{length relations}. This is a generalisation of relations generated by paths containing at least $n$ arrows, where $n$ is a natural number.\n\n\\begin{theorem}[\\Cref{thm:length relations are admissible}]\n    A length relation generates an admissible ideal.\n\\end{theorem}\n\\Cref{sec:examples} contains multiple examples of how relations work, including a sketch of their Auslander--Reiten theory.\n\n\\subsection{Future Work}\nThe present paper is a precursor to future work on generalizations of non-hereditary structures.\nOf note, the authors will consider point relations, such as \\Cref{ex:crossing real lines}, that generalize gentle algebras. They will also study the modding out by length relations, such as \\Cref{ex:length relations}(\\ref{ex:length relations:continuous A}), to generate higher cluster tilting subcategories.\n\n\\subsection{Acknowledgements}\nThe idea for this project was conceived at the Hausdorff Research Institute of Mathematics, KMJ visited JDR at Ghent University during this project, and JDR visited KMJ at Aarhus University during this project.\nThe authors thank each of these institutions for their hospitality.\nKMJ is supported by the Norwegian Research Council via the project Higher Homological Algebra and Tilting Theory (301046).\nJDR is supported at Ghent University by BOF grant 01P12621.\nThe authors would like to thank Jenny August, Raphael Bennett-Tennenhaus, Charles Paquette, Amit Shah, Emine Y{\\i}ld{\\i}r{\\i}m, and Shijie Zhu for helpful discussions.\n\n\\subsection{Conventions}\nWe work over $\\Bbbk=\\overline{\\Bbbk}$ be a field of characteristic 0.\nBy $\\mathsf{Vec}(\\Bbbk)$ and $\\mathsf{vec}(\\Bbbk)$ we denote the categories of $\\Bbbk$-vector spaces and finite-dimensional $\\Bbbk$-vector spaces, respectively.\nFor a $\\Bbbk$-algebra $\\Lambda$, denote by $\\mathsf{J}(\\Lambda)$ the Jacobson radical of $\\Lambda$.\nWe assume $\\mathcal{C}$ is a $\\Bbbk$-linear category.\n\nRecall that a category $\\mathcal{C}$ is called \\emph{Krull--Remak--Schmidt--Azumaya} if any object $X$ is isomorphic to a arbitrary sum $\\bigoplus X_i$, where each $\\End_\\mathcal{C} X_i$ is a local ring, which itself is unique up to isomorphism.\nIf every object is instead isomorphic to a finite sum $\\bigoplus_{i=1}^n X_i$ as above, we say $\\mathcal{C}$ is \\emph{Krull--Remak--Schmidt}.\n\nFinally, we consider (discrete) quivers, continuous generalizations of such quivers, and combinations of the two.\nWhen we draw an arrow, we  use a thin line with an arrow head at the end to indicate the direction.\nWhen we draw a continuous line segment, we use a bold line with the arrow head in the middle to indicate the direction;\nsee \\Cref{fig:arrow and line}.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{tikzpicture}[decoration={markings, mark=at position 0.5 with {\\arrow{>}}}]\n        \\node (left-start) at (0,0) {$\\bullet$};\n        \\node (left-end) at (2,0) {$\\bullet$};\n        \\node (right-start) at (3,0) {$\\bullet$};\n        \\node (right-end) at (5,0) {$\\bullet$};\n        \\draw[->] (left-start)--(left-end);\n        \n        \\draw[very thick, postaction=decorate] (right-start.center)--(right-end.center);\n        \n    \\end{tikzpicture}\n    \\caption{On the left, how we draw arrows. On the right, how we draw line segments.}\n    \\label{fig:arrow and line}\n\\end{figure}\n\n\n\\section{Definition and General Results}\\label{sec:definition and general}\n\n\\subsection{$\\Bbbk$-linear categorization}\n\n\\begin{definition}\\label{def:categorification}\n    Let $Q$ be a (finite) quiver and $\\Bbbk Q$ its path algebra.\n    Let $\\mathcal{Q}$ be the category whose indecomposable objects are the vertices of $Q$ and morphisms between indecomposables $i$ and $j$ are given by\n    \\begin{displaymath}\n        \\Hom_{\\mathcal{Q}}(i,j) = e_j \\Bbbk Q e_i.\n    \\end{displaymath}\n    The objects in $\\mathcal{Q}$ are finite direct sums of the indecomposables (and 0).\n    The morphisms in $\\mathcal{Q}$ are given by extending bilinearly.\n    We call $\\mathcal{Q}$ the \\emph{$\\Bbbk$-linear categorification} of $Q$.\n\\end{definition}\n\n\\begin{example}\\label{ex:discrete categorification}\nLet $Q$ be the following quiver:\n\n\\begin{center}\n\\begin{tikzpicture}[xscale=1.5, yscale=.5]\n    \\node (1) at (0,1) {1};\n    \\node (2) at (1,2) {2};\n    \\node (3) at (1,0) {3};\n    \\node (4) at (2,1) {4};\n    \n    \\draw[->] (1) -- node[pos=0.6, above]{$\\alpha_1$} (2);\n    \\draw[->] (1) -- node[pos=0.6, below]{$\\beta_1$} (3);\n    \\draw[->] (2) -- node[pos=0.4, above]{$\\alpha_2$} (4);\n    \\draw[->] (3) -- node[pos=0.4, below]{$\\beta_2$} (4);\n\\end{tikzpicture}\n\\end{center}\nThen the $\\Bbbk$-linear categorification $\\mathcal{Q}$ is a category with indecomposable objects $1, 2, 3$ and $4$. The morphisms in $\\mathcal{Q}$ are given by paths in $Q$, so for example we have $\\Hom(1,4) \\cong\\Bbbk^2$, while  $\\Hom(4,1) = 0$\n\\end{example}\n\n\\begin{proposition}\\label{lem:correspondence between paths and morphisms}\n    There is a bijection between nonzero elements in $\\Bbbk Q$ and nonzero morphisms in $\\mathcal{Q}$.\n\\end{proposition}\n\\begin{proof}\n    A non-zero element in $\\Bbbk Q$ is a finite sum of paths in $Q$. We can therefore define a map $F$ from the elements in $\\Bbbk Q$ to morphisms in $\\mathcal{Q}$ by specifying the action of $F$ on paths in $Q$. We let this mapping be determined by $\\Hom_{\\mathcal{Q}}(i,j) = e_j \\Bbbk Q e_i$. This map is a bijection by bilinearity of $\\mathcal{Q}$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{prop:same representations}\n    Let $\\mathsf{Mod}(\\mathcal{Q})$ be the category of functors $\\mathcal{Q}\\to \\mathsf{Vec}(\\Bbbk)$.\n    Then there exists an isomorphism of categories $\\Phi:\\mathsf{Mod}(\\mathcal{Q}) \\to \\mathsf{Rep}(Q)$.\n\\end{lemma}\n\n\\begin{proof}\n    Let $F$ be a functor in $\\mathsf{Mod}(\\mathcal{Q})$.\n    We now define the corresponding representation $V=\\Phi(F)$.\n    Let $M$ be the representation of $Q$ over $\\Bbbk$ where $V(i)=F(i)$ for each $i\\in Q_0$.\n    For a path $\\rho$ in $Q$, let $V(\\rho)$ be the $\\Bbbk$-linear map $F(\\rho)$.\n    \n    Let $f:F\\to G$ be a morphism in $\\mathsf{Mod}(\\mathcal{Q})$.\n    Then $\\Phi(f):\\Phi(F)\\to \\Phi(G)$ is defined by the $f_i:F(i)\\to G(i)$ for each $i\\in Q_0$.\n    Straightforward computations show that $\\Phi$ respects composition and so it is a functor.\n    \n    Define $\\Phi^{-1}:\\mathsf{Rep}(Q)\\to \\mathsf{Mod}(\\mathcal{Q})$ in the following way.\n    For a representation $V$ of $Q$, let $F=\\Phi^{-1}(V)$ be determined by $F(i)=V(i)$, for each $i\\in Q_0$, and $F(\\rho)=V(\\rho)$ for each path in $Q$.\n    Morphisms are defined similarly.\n    One may check $\\Phi^{-1}\\Phi$ and $\\Phi\\Phi^{-1}$ are the identity functors on $\\mathsf{Mod}(\\mathcal{Q})$ and $\\mathsf{Rep}(Q)$, respectively.\n\\end{proof}\n\n\\subsection{The Jacobson Radical}\\label{sec:jacobson radical}\n\n\\begin{definition}\\label{def:radical of C}\n    Let $\\mathcal{C}$ be a category.\n    The \\emph{radical} $\\mathsf{Rad}(\\mathcal{C})$ of $\\mathcal{C}$ is the ideal consisting of\n    \\begin{displaymath}\n        \\mathsf{Rad}_{\\mathcal{C}}(X,Y) := \\left\\{ f\\in\\Hom_{\\mathcal{C}}(X,Y) \\mid \\forall g\\in\\Hom_{\\mathcal{C}}(Y,X),\\, f\\circ g\\in\\mathsf{J}(\\End_{\\mathcal{C}}(Y))  \\right\\},\n    \\end{displaymath}\n    for each pair of objects $X,Y$ in $\\mathcal{C}$.\n\\end{definition}\n\n\\begin{proposition}[\\cite{Krause}]\\label{prop:Krause}\n    Let $X$ and $Y$ be objects in $\\mathcal{C}$.\n    Then $\\mathsf{Rad}_{\\mathcal{C}}(X,Y)=\\mathsf{J}(\\Hom_{\\mathcal{C}}(X,Y))$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:easy radical}\n    Let $f:X\\to Y$ be an morphism for indecomposable objects $X,Y$ in $\\mathcal{C}$.\n    Then $f\\in\\mathsf{Rad}(\\mathcal{C})$ if and only if $f$ is not an isomorphism.\n\\end{proposition}\n\\begin{proof}\n    Suppose $f$ is not an isomorphism.\n    If $\\mathcal{C}$ does not have cycles we are done.\n    If $\\mathcal{C}$ has cycles, let $g:Y\\to X$ be a nonzero morphism.\n    Then $f\\circ g\\in\\mathsf{J}(\\End_{\\mathcal{C}}(Y))$ and so $f\\in \\mathsf{Rad}_{\\mathcal{C}}(X,Y)$.\n    Reversing the argument shows that if $f\\in\\mathsf{Rad}_{\\mathcal{C}}(X,Y)$ then $f$ is not an isomorphism.\n\\end{proof}\n\nRecall that $\\mathcal{C}$ is semi-simple if every object in $\\mathcal{C}$ is a finite direct sum of simple objects and all such direct sums exist.\n\n\\begin{proposition}\\label{prop:C is basic}\n    If $\\mathcal{C}$ is Krull--Remak--Schmidt, then $\\mathcal{C}\/ \\mathsf{Rad}(\\mathcal{C})$ is semi-simple.\n\\end{proposition}\n\\begin{proof}\n    Let $\\mathcal{C}$ be a Krull--Remak--Schmidt category.\n    Let $X$ and $Y$ be indecomposables in $\\mathcal{C}$ such that $X\\not\\cong Y$.\n    Then $\\Hom_{\\mathcal{C}}(X,Y)=\\mathsf{Rad}_{\\mathcal{C}}(X,Y)$ and so $\\Hom_{\\mathcal{C} \/ \\mathsf{Rad}(\\mathcal{C})}(X,Y)=0$.\n    Extending bilinearly we see $\\mathcal{C} \/ \\mathsf{Rad}(\\mathcal{C})$ is semi-simple.\n\\end{proof}\n\n\\begin{remark}\\label{rmk:categorification is basic}\n    It follows immediately from \\Cref{prop:C is basic} that if $Q$ is a finite acyclic quiver then $\\mathcal{Q} \/\\mathsf{Rad}(\\mathcal{Q})$ is semi-simple.\n\\end{remark}\n\n\\subsection{Admissible Ideals}\\label{sec:admissible ideals}\n\n\\begin{definition}[\\cite{Krause}]\n    Let $\\{\\mathcal{C}_i\\}_{i\\in I}$ be a family of full additive subcategories of $\\mathcal{C}$. We have an \\emph{orthogonal decomposition} $\\coprod_{i\\in I} \\mathcal{C}_i$ of $\\mathcal{C}$ if every object $X$ in $\\mathcal{C}$ is isomorphic to a direct sum $\\bigoplus_{i\\in I} X_i$, where $X_i$ is an object of $\\mathcal{C}_i$, and for $X_i\\in \\mathcal{C}_i,X_j\\in C_j$ we have $\\Hom_{\\mathcal{C}}(X_i,X_j)=0$ when $i\\neq j$.\n    \n    We say $\\mathcal{C}$ is \\emph{connected} if the only orthogonal decomposition of $\\mathcal{C}$ is the trivial one. \n\\end{definition}\n\nAn \\emph{ideal} $\\mathcal{I}$ of a category $C$ is a collection of morphisms is $\\mathcal{C}$ such that for any $f\\in \\mathcal{I}$ and for any $g$ and $h$ such that the composition $gfh $ is defined, the composition $gfh\\in \\mathcal{I}$.\nFor an ideal $\\mathcal{I}$ of $\\mathcal{C}$, we denote by $\\mathcal{I}(X,Y)$ the morphisms in $\\Hom_{\\mathcal{C}}(X,Y)\\cap \\mathcal{I}$.\n\n\\begin{remark}\n    For an ideal $\\mathcal{I}$ of $\\mathcal{C}$, the category $\\mathcal{C} \/\\mathcal{I}$ has the same objects as $\\mathcal{C}$.\n    The morphisms of $\\mathcal{C} \/ \\mathcal{I}$ are given by $\\Hom_{\\mathcal{C}}(X,Y) \/ \\mathcal{I}(X,Y)$.\n    A representation $V: \\mathcal{C} \/ \\mathcal{I} \\to \\Bbbk\\text{vec}$ is also a representation of $\\mathcal{C}$ by precomposition with the quotient functor $\\pi$. Thus we obtain a representation $\\widetilde{V}:\\mathcal{C} \\stackrel{\\pi}{\\to} \\mathcal{C} \/ \\mathcal{I} \\stackrel{V}{\\to} \\Bbbk\\text{vec}$.\n    Hence\n    we may consider the representations of $\\mathcal{C}\/\\mathcal{I}$ as a subcategory of the representations of $\\mathcal{C}$.\n    In particular, representations of $\\mathcal{C}\/\\mathcal{I}$ are those representations $V$ of $\\mathcal{C}$ such that if $f\\in\\mathcal{I}$ then $V(f)=0$.\n\\end{remark}\n\n\\begin{definition}\\label{def:admissible}\n    Let $\\mathcal{C}$ be a $\\Bbbk$-linear, Krull--Remak--Schmidt--Azumaya category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    We say $\\mathcal{I}$ is \\emph{admissible} if the following are satisfied.\n    \\begin{enumerate}\n        \\item For each $f$ in $\\mathcal{I}$, there exists a finite collection of morphisms $g_1,\\ldots,g_n$ not in $\\mathcal{I}$ such that $f = g_n\\circ\\cdots\\circ g_1$.\n        \\item For each indecomposable $X$ in $\\mathcal{C}$, the endomorphism ring $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite-dimensional.\n    \\end{enumerate}\n\\end{definition}\n\nWe remark that, in \\Cref{def:admissible}(2), we do not want to require that there is some $n$ that works for all nonisomorphism endomorphisms $f$.\nSee \\Cref{ex: cycles length} for an explicit example why.\n\n\n\\begin{lemma}\\label{lem: admissible in radical}\n\tLet $\\mathcal{C}$ be a Krull--Remak--Schmidt category and $\\mathcal{I}$ an ideal in $\\mathcal{C}$.\n    If $\\mathcal{I}$ is admissible, then $\\mathcal{I}$ is contained in the radical of $\\mathcal{C}$.\n\\end{lemma}\n\n\\begin{proof}\n\tLet $f:X\\rightarrow Y$ be a morphism in $\\mathcal{I}$ between indecomposable objects. Then we know by \\Cref{prop:easy radical} that if $f$ is not contained in the radical, it is an isomorphism. However, if $f$ is an isomorphism, we have $1_X\\in\\mathcal{I}$.\n\tThen \\emph{every} morphism to \/ from $X$ is in $\\mathcal{I}$.\n\tThus, if $1_x=g_n\\circ\\cdots\\circ g_1$ for any composition, both $g_n,g_1\\in \\mathcal{I}$, which contradicts condition \\Cref{def:admissible}(1).\n\tHence $\\mathcal{I}(X,Y)\\subseteq \\mathsf{Rad}(X,Y)$ for indecomposable $X,Y$. \n\t\n\tNow let $X=\\bigoplus_{i=0}^m X_i$ and $Y=\\bigoplus_{j=0}^n Y_j$, where each $X_i$ and $Y_j$ is indecomposable. Consider $f\\in \\mathcal{I}(X,Y)$. We can rewrite $f$ as $f=(f_{ij})$, where $f_{ij}:X_i\\rightarrow Y_j$. By composition with the canonical injections and projections, we see that $f_{ij}\\in \\mathcal{I}(X_i,Y_j)$, so by the above, $f_{ij}\\in \\mathsf{Rad}(X_i,Y_j)$.\n\tThen by linearity, $f\\in \\mathsf{Rad}(X,Y)$.\n\\end{proof}\n\n\nLet $Q$ be a finite quiver and let $\\mathcal{Q}$ be its $\\Bbbk$-linear categorification. Suppose $I$ is an ideal of the path algebra $\\Bbbk Q$. We show how to build an ideal $\\mathcal{I}$ in $\\mathcal{Q}$ from $I$. \n\nFrom the definition of the $\\Bbbk$-linear categorification, we know that each path in $I$ corresponds to a non-zero morphism in $\\mathcal{Q}$, see \\Cref{lem:correspondence between paths and morphisms}. We (na\\\"ively) let $\\mathcal{I}$ be the set of morphisms obtained by mapping $I$ to $\\mathcal{Q}$. We now show that $\\mathcal{I}$ is an ideal of the category $\\mathcal{Q}$. \n\nBy $\\Bbbk$-linearity of $Q$, it is enough to consider morphisms between indecomposable objects.\nSuppose $f\\in\\mathcal{I}(i,j)$ for some $i,j\\in Q_0=\\operatorname{Ind}\\mathcal{Q}$, and let $g:j\\rightarrow k$ and $h:l\\rightarrow i$ be two nonzero morphisms in $\\mathcal{Q}$.\nBy \\Cref{lem:correspondence between paths and morphisms} we know that $f$ corresponds to an element $\\rho$ in $e_j\\Bbbk Qe_i$.\nFurther, $g$ corresponds to an element $\\psi$ in $\\Bbbk Qe_j$ and $h$ corresponds to an element $\\phi$ in $e_i\\Bbbk Q$.\nEach of $\\rho$, $\\phi$, and $\\psi$ are are sums of paths in $Q$ from the respective source and to the respective target.\nWithout loss of generality, due to $\\Bbbk$-linearity, suppose each of $\\rho$, $\\phi$, and $\\psi$ is a path in $Q$.\nWe see $\\psi\\rho\\phi$ is an element of $I$ since $I$ is a two-sided ideal containing $\\rho$.\nThe image of the composition $\\psi\\rho\\phi$ is the composition $gfh$, which must therefore be in $\\mathcal{I}$.\n\n\\begin{proposition}\\label{prop:admissible is correct}\n    Let $Q$ be a finite quiver and $I$ an ideal of $\\Bbbk Q$ as a path algebra.\n    Let $\\mathcal{Q}$ be the $\\Bbbk$-linear category induced by $Q$ and let $\\mathcal{I}$ be the ideal induced by $I$ in $\\mathcal{Q}$.\n    Then $\\mathcal{I}$ is an admissible ideal of $\\mathcal{Q}$ as in \\Cref{def:admissible} if and only if $I$ is an admissible ideal of $\\Bbbk Q$.\n\\end{proposition}\n\n\\begin{proof}\n    Let $I$ be an admissible ideal of $\\Bbbk Q$ and $\\rho\\in I$.\n    We first prove $\\mathcal I$ satisfies property (1) of \\Cref{def:admissible}.\n    Without loss of generality, assume $\\rho$ is a path in $Q$.\n    Then $\\rho = \\alpha_n \\alpha_{n-1}\\cdots \\alpha_2\\alpha_1$, where each $\\alpha_i$ is an arrow in $Q$.\n    Now, let $f$ be the morphism in $\\mathcal{Q}$ corresponding to $\\rho$ and $g_i$ the morphism in $\\mathcal{Q}$ corresponding to $\\alpha_i$, for each $i$.\n    Then we know each $g_i\\notin \\mathcal I$ and have satisfied property (1) of \\Cref{def:admissible}.\n    Reversing the argument proves the converse.\n    \n    If $Q$ has no cycles then the proposition immediately holds for property (2) of \\Cref{def:admissible}.\n    So, suppose $Q$ has at least one oriented cycle.\n    Since $I \\subset \\mathsf{Rad}^n(\\Bbbk Q)$, for some $n\\geq 2$, we see that $\\mathcal I$ must immediately satisfy property (2) of \\Cref{def:admissible}.\n    \n    Now suppose $\\mathcal{I}$ satisfies \\Cref{def:admissible}(2).\n    Since $Q$ is finite, there are finitely many cycles.\n    For each cycle $\\rho$ at each vertex $i$, let \n    \\[ m_\\rho= \\min_m \\{ m \\mid \\rho^m \\in \\mathcal{I}(i,i)\\}.\\]\n    We know such an $m_\\rho$ exists since $\\End_{\\mathcal{Q}}(i) \/ \\mathcal{I}(i,i)$ is finite-dimensional.\n    Let $n_\\rho$ be $m_\\rho$ time the length of $\\rho$.\n    Then let \\[ N = \\max\\left(\\max_\\rho \\{n_\\rho\\}\\cup \\{\\text{length of longest path without cycles in }Q\\}\\right).\\]\n    Thus, $\\mathsf{Rad}^N(\\Bbbk Q)\\supset I$.\n    This concludes the proof.\n\\end{proof}\n\n\\begin{remark}\nThe second half of the proof above can be extended to more general quivers.\nSuppose $\\mathcal{Q}$ is the $\\Bbbk$-linear categorification of a (not necessarily finite) quiver $Q$ with finitely many cycles. Suppose that for each cycle $\\rho$ in the quiver with corresponding morphism $f_\\rho:X\\rightarrow X$, there is some $n\\geq 2$ such that $f_\\rho\\in \\mathcal{I}(X,X)$.\nThen $\\mathcal{I}$ satisfies criterion (2) in \\Cref{def:admissible}. \n\nFor the majority of our examples, this will be the criterion we actually use.\n\\end{remark}\n\n\\begin{example}\\label{ex:discrete ideal}\n    Consider the quiver from \\Cref{ex:discrete categorification}. Let $I$ be the commutative ideal generated by $\\{\\alpha_2\\alpha_1-\\beta_2\\beta_1\\}$.\n    \n    In the $\\Bbbk$-linear categorification, the relation $\\alpha_2\\alpha_1-\\beta_2\\beta_1$ can be written as $\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2\\end{smallmatrix}\\right]\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -\\beta_1\\end{smallmatrix}\\right]$. The ideal generated by this morphism fulfills the criteria for being an admissible ideal. \n\\end{example}\n\n\\subsection{General Results}\\label{sec:general results}\n\n\\begin{proposition}\\label{prop:connected}\n    Let $\\mathcal{C}$ be connected, and let $\\mathcal{I}$ be an admissible ideal of morphisms. Then $\\mathcal{C} \/ \\mathcal{I}$ is connected.\n\\end{proposition}\n\n\\begin{proof}\n Assume towards a contradiction that $\\mathcal{C} \/ \\mathcal{I}$ is not connected; then there exists a decomposition of $\\mathcal{C} \/ \\mathcal{I}$ into mutually orthogonal subcategories $\\mathcal{C}'_1, \\cdots, \\mathcal{C}'_n$. We can lift these subcategories to subcategories $\\mathcal{C}_1, \\cdots, \\mathcal{C}_n$ of $\\mathcal{C}$. As $\\mathcal{C}$ is connected, these subcategories cannot all be mutually orthogonal, so assume that there exists some morphism $f:X_i\\rightarrow X_j$, with $X_i\\in \\mathcal{C}_i, X_j\\in \\mathcal{C}_j$ and $i\\neq j$. To preserve mutual orthogonality in $\\mathcal{C}\/\\mathcal{I}$, we must have $f\\in \\mathcal{I}$. Then since $I$ is an admissible ideal, we can write $f=g_m\\circ \\cdots \\circ g_1$, where each of the $g_1, \\cdots, g_m$ are not in $\\mathcal{I}$.\n \n Consider $g_1: X_i\\rightarrow Y$ and denote its image in $\\mathcal{C} \/ \\mathcal{I}$ by $\\overline{g_1}$. As $\\mathcal{C} \/ \\mathcal{I}$ is not connected, we can write $Y=\\bigoplus_{i=1}^n Y_i$, with $Y_i\\in \\mathcal{C}'_i$ and $\\overline{g_1}=(g^1_1,\\cdots g_1^n)$. Now, as $\\overline{g_1}$ is nonzero, $g_1^k$ is nonzero for some $k$. If $k\\neq i$, we have reached a contradiction. If $k=i$, we can repeat the argument with $\\overline{g_2}|_{Y_i}$, eventually reaching a contradiction. \n\\end{proof}\n\n\\begin{proposition}\\label{prop:basic}\n    Let $\\mathcal{C}$ be Krull--Remak--Schmidt--Azumaya and $\\mathcal{I}$ an admissible ideal.\n    Then $\\mathcal{C} \/ \\mathcal I$ is Krull--Remak--Schmidt--Azumaya.\n\\end{proposition}\n\\begin{proof}\nLet $X$ be an object in $\\mathcal{C} \/ \\mathcal I$. Then $X$ is an object in $\\mathcal{C}$, which we assume to be Krull--Remak--Schmidt--Azumaya. It follows that $X=\\bigoplus_{\\alpha} X_\\alpha$, where $\\End_\\mathcal{C}(X_\\alpha)$ is local. If we can show that $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_\\alpha)$ is local, we are done.\n\nWe know that $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_{\\alpha})=\\End_\\mathcal{C}(X_{\\alpha})\/\\mathcal{I}(X_{\\alpha},X_{\\alpha})$. By property 1 of \\Cref{def:admissible}, the identity on $X_{\\alpha}$ cannot be an element of $\\mathcal{I}(X_{\\alpha},X_{\\alpha})$, so $\\End_{\\mathcal{C}\/\\mathcal{I}}(X_{\\alpha})$ is a nonzero quotient ring of a local ring, which is local by the ideal correspondence theorem for quotient rings.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:radical commutes with admissible}\n    Let $\\mathcal{C}$ be a category such that all endomorphism rings are artinian and $\\mathcal{I}$ an admissible ideal.\n    Then \\[\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I})=\\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}.\\] \n\\end{proposition}\n\\begin{proof}\nThe equation $\\mathsf{Rad}(\\mathcal{C} \/ \\mathcal{I}) = \\mathsf{Rad}(\\mathcal{C}) \/ \\mathcal{I}$ holds if and only if $\\mathsf{Rad}_{\\mathcal{C} \/ \\mathcal{I}}(X,Y) = \\mathsf{Rad}_\\mathcal{C}(X,Y) \/ \\mathcal{I}$ holds for all pairs of objects $X,Y\\in \\mathcal{C}$.\n\nFirst note that if $\\End_{\\mathcal{C} }(Y)$ is artinian, then\n\\begin{align*}\n\t \\mathsf{Rad}_\\mathcal{C}(Y,Y) \/ \\mathcal{I} &= \\mathsf{J}(\\End_{\\mathcal{C} }(Y))\/ \\mathcal{I}=  \\mathsf{J}(\\End_{\\mathcal{C} }(Y)+\\mathcal{I})\/\\mathcal{I} \\\\ \n\t &= \\mathsf{J}(\\End_{\\mathcal{C} \/ \\mathcal{I}}(Y)) =\\mathsf{Rad}_{\\mathcal{C} \/ \\mathcal{I}}(Y,Y).\n\\end{align*}\nNow we can see that \n\\begin{align*}\n\tf \\in \\mathsf{Rad}_\\mathcal{C}(X,Y) \/ \\mathcal{I} &\\Leftrightarrow f = f'+\\mathcal{I}(X,Y)\\text{, with } f'\\in \\mathsf{Rad}_\\mathcal{C}(X,Y) \\\\\n\t& \\Leftrightarrow  f = f'+\\mathcal{I}(X,Y)\\text{, s.t.\\ } f'\\circ g'\\in \\mathsf{J}(\\End_{\\mathcal{C} }(Y))\\, \\forall\\, g'\\in \\Hom_\\mathcal{C}  (Y,X)\\\\\n\t& \\Leftrightarrow f\\circ (g'+\\mathcal{I}(Y,X))\\in \\mathsf{J}(\\End_{\\mathcal{C} }(Y)  )\/\\mathcal{I}\\, \\forall \\, g'\\in \\Hom_\\mathcal{C}  (Y,X)\\\\\n\t& \\Leftrightarrow f\\circ g \\in \\mathsf{J}(\\End_{\\mathcal{C} \/\\mathcal{I}}(Y)  )\\, \\forall\\, g\\in \\Hom_{\\mathcal{C}\/\\mathcal{I}}  (Y,X)\\\\\n\t& \\Leftrightarrow f \\in \\mathsf{Rad}_{\\mathcal{C}\/\\mathcal{I}}(X,Y). \n\\end{align*}\n\\end{proof}\nIn particular, if $\\mathcal{C}$ is $\\Bbbk$-linear and Krull--Remak--Schmidt, then all endomorphism rings are artinian and the above Proposition holds.\n\nThe following lemma is useful for our examples in \\Cref{sec:examples}.\n\\begin{lemma}\\label{lem:stack admissible}\n    Let $\\mathcal{C}$ be a small, $\\Bbbk$-linear, Krull--Remak--Schmidt--Azumaya category and $\\mathcal{I}$ an admissible ideal in $\\mathcal{C}$.\n    Let $\\mathcal{J}$ be an admissible ideal in $\\mathcal{C}\/\\mathcal{I}$ and $\\widetilde{\\Jay}$ the set of morphisms $f$ in $\\mathcal{C}$ such that $f+\\mathcal{I}\\in\\mathcal{J}$ in $\\mathcal{C}\/\\mathcal{I}$. Then the following hold:\n    \\begin{enumerate}\n        \\item $\\widetilde{\\Jay}$ contains $\\mathcal{I}$.\n        \\item $\\widetilde{\\Jay}$ is an ideal.\n        \\item $\\widetilde{\\Jay}$ is admissible in $\\mathcal{C}$.\n        \\item $(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J} \\simeq \\mathcal{C}\/\\widetilde{\\Jay}$.\n    \\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n    \\textbf{1.} Let $f\\in\\mathcal{I}$.\n    Then $f\\mapsto 0$ in $\\mathcal{C}\/\\mathcal{I}$.\n    Since all zero morphisms in $\\mathcal{C}\/\\mathcal{I}$ are in $\\mathcal{J}$, we see $f\\in \\widetilde{\\Jay}$.\n    \n    \\textbf{2.} Let $f:X\\to Y$ be nonzero in $\\widetilde{\\Jay}$ and let $g:Y\\to Z$ be nonzero in $\\mathcal{C}$.\n    Then $f+\\mathcal{I}$ and $g+\\mathcal{I}$ are in $\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$.\n    So, $(g+\\mathcal{I})\\circ(f+\\mathcal{I})$ is in $\\mathcal{J}$ and is equal to $gf + \\mathcal{I}$.\n    Then $gf \\in \\widetilde{\\Jay}$.\n    \n    \\textbf{3.} Let $f\\in\\widetilde{\\Jay}$.\n    Then $f+\\mathcal{I}\\in\\mathcal{J}$ and, by assumption, there exists $(g_1+\\mathcal{I}),\\ldots,(g_n+\\mathcal{I})$ in $\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})\\setminus\\mathcal{J}$ such that \\[ f + \\mathcal{I} = (g_n+\\mathcal{I})\\circ (g_{n-1}+\\mathcal{I})\\circ\\cdots\\circ (g_2+\\mathcal{I})\\circ(g_1+\\mathcal{I}).\\]\n    Then for each $g_i$ there is $h_i\\in\\mathcal{I}$ such that $g_i+h_i\\mapsto g_i+\\mathcal{I}$ and \\[f = (g_n+h_n)\\circ(g_{n-1}+h_{n-1})\\circ\\cdots\\circ(g_2+h_2)\\circ (g_1+h_1). \\]\n    Since each $g_i\\notin \\mathcal{J}$, we know each $g_i+h_i\\notin\\widetilde{\\Jay}$ and so $f$ is a finite composition of morphisms not in $\\widetilde{\\Jay}$.\n    Additionally, for any nonzero, nonisomorphism endomorphism $f$, we have $f^n\\in\\mathcal{I}$ for some $n\\in\\mathbb{N}$.\n    Then $f^n\\in\\widetilde{\\Jay}$ by statement 1.\n    Therefore, $\\widetilde{\\Jay}$ is admissible.\n    \n    \\textbf{4.} Recall $\\mathsf{Ob}((\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}) = \\mathsf{Ob}(\\mathcal{C}\/\\widetilde{\\Jay})$.\n    We now produce a bijection between $\\mathsf{Mor}((\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J})$ and  $\\mathsf{Mor}(\\mathcal{C}\/\\widetilde{\\Jay})$ by producing bijections \\[ \\phi_{X,Y}:\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y) \\to \\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}}(X,Y) \\] for each ordered pair $X,Y$ of objects.\n    \n    Let $f+\\widetilde{\\Jay}\\in\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y)$.\n    Then there exists $g\\in\\widetilde{\\Jay}\\subset \\mathsf{Mor}(\\mathcal{C})$ such that $f+g\\mapsto f+\\widetilde{\\Jay}\\in\\mathsf{Mor}(\\mathcal{C}\/\\widetilde{\\Jay})$.\n    If $g\\in\\mathcal{I}$ then $f+g\\mapsto f+\\mathcal{I}\\in\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$; otherewise $f+g\\mapsto f+g+\\mathcal{I}\\in\\mathsf{Mor}(\\mathcal{C}\/\\mathcal{I})$.\n    In either case, $f+g\\mapsto f+\\mathcal{J}$ in $\\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathsf{J}}(X,Y)$.\n    We define $\\phi_{X,Y}(f+\\widetilde{\\Jay}):= f+\\mathcal{J}$.\n    \n    It is immediate that $\\phi_{X,Y}$ is injective.\n    Suppose $f+\\mathcal{J}\\in\\Hom_{(\\mathcal{C}\/\\mathcal{I})\/\\mathcal{J}}$.\n    Then there exists $g+\\mathcal{I}$ in $\\Hom_{\\mathcal{C}\/\\mathcal{I}}(X,Y)$ such that $f+g+\\mathcal{I}\\mapsto f+\\mathcal{J}$.\n    Then there exists $h\\in\\Hom_{\\mathcal{C}}(X,Y)$ such that $f+g+h\\mapsto f+g+\\mathcal{I}$.\n    But this means $g+h\\in\\widetilde{\\Jay}$ and so $f+(g+h)\\mapsto f+\\widetilde{\\Jay}$ in $\\Hom_{\\mathcal{C}\/\\widetilde{\\Jay}}(X,Y)$.\n    Thus, $\\phi_{X,Y}$ is surjective.\n\\end{proof}\n\n\\section{Relations}\\label{sec:relations}\n\nIn this section we look at two types of admissible ideals: those generated by point relations (\\Cref{subsec:point relations}) and those generated by length relations (\\Cref{subsec:length}).\nThese generalize a relation generated by a single path of length two and relations generated by all paths of a particular length, respectively.\n\n\\subsection{Point Relations}\\label{subsec:point relations}\nHere we generalize relations generated by a single path of length 2 to a point relation (\\Cref{def:point relation}) and prove this generates an admissible ideal (\\Cref{thm:point relations are admissible}).\nWe then give examples that point to a continuous version of a gentle algebra (\\Cref{ex:monomial points,ex:crossing real lines}).\n\n\\begin{definition}\\label{def:decomposition point}\n    Let $f:X\\to Y$ be a nonisomorphism between indecomposables in $\\mathcal{C}$.\n    A \\emph{decomposition point} $Z$ in $\\mathcal{C}$ is an indecomposable object such that there exists nonisomorphisms $g:X\\to Z$ and $h:Z\\to Y$ where $f=h\\circ g$.\n\\end{definition}\n\n\\begin{definition}\\label{def:acyclic morphism}\n    Let $f:X\\to Y$ be a nonisomorphism of indecomposables in $\\mathcal{C}$, $Z$ a decomposition point of $f$, and $f=g\\circ h$ such a decomposition.\n    We call $f$ an \\emph{acyclic morphism} if for all pairs $g':X\\to Z$ and $h':Z\\to Y$ such that $f=h'\\circ g'$ then $h'$ and $g'$ are scalar multiples of $h$ and $g$, respectively.\n\\end{definition}\n\nNote that an acyclic morphism cannot be irreducible. (I.e., it must be a path of length at least 2 in a quiver.)\n\n\\begin{definition}\\label{def:point relation}\n    Let $f$ be an acyclic morphism and $Z$ a decomposition point of $f$.\n    Let $P$ be the set of all nonisomorphisms $g$ between indecomposables satisfying the following.\n    \\begin{itemize}\n        \\item There exists $h_1$ and $h_2$ morphisms of indecomposables such that $f=h_2\\circ g\\circ h_1$.\n        \\item We have $Z$ as a decomposition point of $g$.\n    \\end{itemize}\n    Let $\\mathcal P_{f,Z}$ be the ideal in $\\mathcal{C}$ generated by $P$.\n    We call $\\mathcal P_{f,Z}$ the \\emph{point relation through $Z$ by $f$}.\n\\end{definition}\n\n\\begin{definition}\\label{def:admissible point relations}\n    Let $\\{\\mathcal P_\\alpha\\}$ be a collection of point relations in $\\mathcal{C}$.\n    We say $\\{\\mathcal P_\\alpha\\}$ is \\emph{admissible} if each morphism of indecomposables appears in at most finitely-many $\\mathcal P_\\alpha$.\n\\end{definition}\n\n\\begin{theorem}\\label{thm:point relations are admissible}\n    Let $\\{\\mathcal P_\\alpha\\}$ be an admissible collection of point relations in $\\mathcal{C}$ and let $\\mathcal{I}=\\langle \\bigcup_\\alpha P_\\alpha\\rangle$.\n    Suppose also that for each indecomposable $X$ in $\\mathcal{C}$, we have $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite dimensional.\n    Then $\\mathcal{I}$ is an admissible ideal.\n\\end{theorem}\n\\begin{proof}\nWe satisfy \\Cref{def:admissible}(2) by assumption.\n\nNow suppose $f\\in \\mathcal{I}$; we show that $f$ can be written as a finite composition of morphisms not in $\\mathcal{I}$.\nSince $f$ is a finite sum of morphisms of indecomposables, we assume without loss of generality $f$ is a morphism of indecomposables.\nThen, by assumption there are at most finitely-many $\\mathcal P_\\alpha$ such that $f\\in \\mathcal P_\\alpha$.\n\nWe proceed by induction beginning with $f$ is only in $\\mathcal P_1$.\nLet $f_1$, $P_1$, and $Z_1$ be as in \\Cref{def:point relation}.\nThen there exists morphisms $h_1$ and $h_2$ in $Mor(\\mathcal{C})$ and $g\\in P_1$ such that $f = h_2\\circ g \\circ h_1$.\nFurther, $g=h'_2\\circ h'_1$ where the target of $h'_1$ is $Z_\\alpha$ and the source of $h'_2$ is $Z_\\alpha$.\nSo, let $g_1 = h'_1\\circ h_1$ and let $g_2=h_2\\circ h'_2$.\nNote that neither $g_1$ nor $g_2$ is in $\\mathcal P_1$.\nFurther, neither $g_1$ nor $g_2$ is in $\\mathcal{I}$ or else $f$ would be in another $\\mathcal P_\\alpha$ as well.\nThus, we have our desired decomposition.\n\nNow assume that if $f$ is in $n$ of the $\\mathcal P_\\alpha$, then $f$ is a finite composition of morphisms not in $\\mathcal{I}$.\nSuppose $f$ is in $n+1$ of the $\\mathcal P_\\alpha$ and denote one of them by $\\mathcal P_1$.\nLet $f_1$, $P_1$, and $Z_1$ be as before for $\\mathcal P_1$.\nWe find $g_1$ and $g_2$ as before, but they may be in $\\mathcal{I}$.\nHowever, each $g_1$ and $g_2$ may only be in $n$ or fewer $\\mathcal P_\\alpha$ and so are a finite composition of morphisms not in $\\mathcal{I}$.\nTherefore, $f$ is a finite composition of morphisms not in $\\mathcal{I}$.\n\\end{proof}\n\n\\begin{example}[Discrete quiver]\\label{ex:monomial points}\nLet $Q$ be a discrete quiver. Then any quadratic monomial relation in $Q$ corresponds to a point relation in the $\\Bbbk$-linear categorification of $Q$.\n\nIn particular, any gentle algebra can be obtained by considering a quiver with point relations. \n\\end{example}\n\n\\begin{example}[Continuous ``gentle'', crossing real lines]\\label{ex:crossing real lines}\nConsider two copies of the real line, labeled $\\mathbb{R}$ and $\\mathbb{R}'$. We label the numbers in $\\mathbb{R}$ by $x$ and the numbers in $\\mathbb{R}'$ by $x'$. Identify $0$ and $0'$ and label the category of $\\Bbbk$-representations  of the resulting partially ordered set by $\\mathcal{C}$. \n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[inner sep = 0cm, outer sep = 0cm, xscale = 1, decoration={\n    markings,\n    mark=at position 0.5 with {\\arrow{>}}}, very thick\n    ]\n    \\coordinate (topleft) at (0,1);\n    \\coordinate (bottomleft) at (0,0);\n    \\coordinate (topright) at (10,1);\n    \\coordinate (bottomright) at (10,0);\n    \\coordinate (mid) at (5,0.5);\n    \\draw[blue, postaction=decorate] (topleft) ..controls +(4,0).. (mid); \n    \\draw[red, postaction=decorate] (mid) ..controls +(1,-.5).. (bottomright);\n    \\draw[red, postaction=decorate] (bottomleft)  ..controls +(4,0).. (mid); \n    \\draw[blue, postaction=decorate] (mid) ..controls +(1,.5).. (topright);\n    \\draw[fill=white] (mid) circle[radius=1mm];\n    \\end{tikzpicture}\n    \\caption{The category considered in \\Cref{ex:crossing real lines}. The two copies of the real line have been drawn in different colours}\n    \\label{fig:my_label}\n\\end{figure}\n\nLet $\\mathcal P$ be the point relation at $0$ generated morphisms starting in $\\mathbb{R}_{<0}$ and ending in  $\\mathbb{R}'_{>0}$. Dually, let $\\mathcal P'$ be the point relation at $0$ generated morphisms starting in $\\mathbb{R}'_{<0}$ and ending in  $\\mathbb{R}_{>0}$. The collection $\\{\\mathcal P, \\mathcal P'\\}$ generates an admissible ideal.\nIn later work, we will argue that this $\\mathcal{C}$ with this ideal yields a continuous generalization of a gentle algebra. \n\n\\begin{remark}\nIf we do not assume that $\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)$ is finite dimensional in our hypothesis of \\Cref{thm:point relations are admissible}, it is possible that we do not have an admissible ideal.\nSee \\Cref{ex:big wedge}.\n\\end{remark}\n\n\\end{example}\n\\subsection{Length Relations}\\label{subsec:length}\nWe now generalize relations generated by all paths of a certain length to length relations.\nTo do this we define a way of measuring length in our category (\\Cref{def:weakly archimedean monoid,def:category with length in Lambda}) and provide examples (\\Cref{ex:weakly Archimedian monoids,ex:categories with length in Lambda}).\nThen we define the length relations (\\Cref{def:length relation}) and provide examples (\\Cref{ex:length relations}) and prove that length relations generate admissible ideals (\\Cref{thm:length relations are admissible}).\nIn \\Cref{apx:length} we discuss the proof of \\Cref{thm:length relations are admissible} (\\Cref{subsec:need weakly archimedean}), why we require the specific setup that we have (\\Cref{sec:more on length}), and compare our notion of length to the notion of a metric on a category, introduced by Lawvere \\cite{L73} (\\Cref{subsec:length vs metric}).\n\nRecall a commutative monoid $\\Lambda$ is a set with an associative, commutative, binary operation $+_{\\Lambda}:\\Lambda\\times\\Lambda \\to \\Lambda$ and an identity 0.\n\\begin{definition}\\label{def:weakly archimedean monoid}\n    Let $\\Lambda$ be a commutative monoid.\n    We say $\\Lambda$ is \\emph{weakly Archimedian} if it satisfies the following.\n    \\begin{itemize}\n        \\item There is a total order $\\leq$ on $\\Lambda$.\n        \\item If $\\lambda\\neq 0$ then $\\lambda>0$.\n        \\item If $\\lambda_1>\\lambda_2$ then, for any $\\lambda_3$, we have $\\lambda_1+\\lambda_3> \\lambda_2+\\lambda_3$ or $\\lambda_1+\\lambda_3 = \\lambda_2+\\lambda_3=\\max \\Lambda$.\n        \\item For all $0<\\lambda_1<\\lambda_2$ in $\\Lambda$, there exists $n\\in\\mathbb{N}$ such that\n        \\begin{displaymath}\n            n\\lambda_1 := \\underbrace{\\lambda_1 +_{\\Lambda} \\lambda_1 +_{\\Lambda} \\cdots +_{\\Lambda} \\lambda_1}_{n} \\geq \\lambda_2.\n        \\end{displaymath}\n    \\end{itemize}\n\\end{definition}\n\n\\begin{example}\\label{ex:weakly Archimedian monoids}\n    We give three examples, two of which the reader might expect.\n    \\begin{enumerate}\n        \\item The set $\\mathbb{N}$ with the usual total order and $+_{\\mathbb{N}}$ given in the usual way is weakly Archimedian.\n        \\item The set $\\mathbb{R}_{\\geq 0}$ with the usual total order and $+_{\\mathbb{R}}$ given in the usual way is weakly Archimedian.\n        \\item\\label{ex:weakly Archimedian monoids:with max} Let $\\Lambda = \\{0,1,2,\\ldots,n-1,n,\\infty\\}$. Let $+_{\\Lambda}$ be given by\n        \\begin{displaymath}\n            \\lambda_1+_{\\Lambda}\\lambda_2 = \\begin{cases}\n                \\lambda_1+_{\\mathbb{N}}\\lambda_2 & (\\lambda_1+_{\\mathbb{N}}\\lambda_2)\\leq n \\\\\n                \\infty & \\text{otherwise}.\n            \\end{cases}\n        \\end{displaymath}\n        For the total order, we say $0<1<2<\\cdots<n-1<n<\\infty$.\n        Then $\\Lambda$ is weakly Archimedian.\n    \\end{enumerate}\n\\end{example}\nWhen the weakly Archimedian monoid \\textcolor{karin}{structure} is clear, we write $+$ instead of $+_{\\Lambda}$.\n\n\\begin{definition}\\label{def:stem category}\n    Let $\\widehat{\\mathcal{C}}$ be a small category and ignore any additional structure.\n    The \\emph{additive $\\Bbbk$-linearization} $\\mathcal{C}$ of $\\widehat{\\mathcal{C}}$ is the category such that\n    \\begin{itemize}\n        \\item $\\mathsf{Ob}(\\mathcal{C})$ contains finite direct sums of objects in $\\widehat{\\mathcal{C}}$ and\n        \\item $\\Hom_{\\mathcal{C}}(X,Y) = \\langle \\Hom_{\\widehat{\\mathcal{C}}}(X,Y) \\rangle_\\Bbbk$ for indecomposables $X,Y\\in\\mathsf{Ob}(\\mathcal{C})$.\n    \\end{itemize}\n    The other Hom spaces and composition are given by extending bilinearly and using composition in $\\widehat{\\mathcal{C}}$.\n    \n    Let $\\mathcal{C}$ be a small $\\Bbbk$-linear category.\n    We say subcategory $\\widehat{\\mathcal{C}}$ of $\\mathcal{C}$ is a \\emph{stem category} of $\\mathcal{C}$ if $\\mathcal{C}$ is isomorphic to the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{C}}$.\n\\end{definition}\nThis is similar to the use of ``stem category'' in \\cite{Bongartz}.\nWe note that Bongartz considered stem categories of locally bounded categories.\nWe say $\\mathcal{C}$ is \\emph{locally bounded} if, for each object $X\\in\\mathsf{Ob}(\\mathcal{C})$, we have \\[\\dim_{\\Bbbk}\\left( \\bigoplus_{Y\\in\\mathsf{Ob}(\\mathcal{C})} \\Hom_{\\mathcal{C}}(X,Y)\\oplus\\Hom_{\\mathcal{C}}(Y,X) \\right) <\\infty. \\]\nImmediately, we note that when we consider the $\\Bbbk$-linearization of $\\mathbb{R}$ as a category, this condition fails.\nIn fact, many categories in the present paper are not locally bounded, especially those considered in \\Cref{sec:examples}.\n\n\\begin{remark}\\label{rmk:quiver additive linearization}\n    If we consider a quiver $Q$ as a category $\\widehat{\\mathcal{Q}}$, then the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$ is precisely the $\\Bbbk$-linear categorification $\\mathcal{Q}$ of $Q$.\n\\end{remark}\n\n\\begin{definition}\\label{def:category with length in Lambda}\n    Let $\\mathcal{C}$ be a $\\Bbbk$-linear category, $\\widehat{\\mathcal{C}}$ be a stem category of $\\mathcal{C}$, and $\\Lambda$ a weakly archimedian monoid.\n    \n    We say \\emph{$\\mathcal{C}$ has length in $\\Lambda$} if there is a function $\\ell:\\mathsf{Mor}(\\widehat{\\mathcal{C}})\\to\\Lambda$ satisfying the following.\n    \\begin{enumerate}\n        \\item If $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ is an isomorphism then $\\ell(f)=0$.\n        \\item For each $f,g\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ such that $g\\circ f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$, we have $\\ell(g\\circ f)=\\ell(g)+\\ell(f)$.\n        \\item For each $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ with $\\ell(f)>0$ and for each $\\lambda<\\ell(f)$ there are $g,h\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$ such that $f=h\\circ g$ and either $\\ell(g)=\\lambda$ or $\\ell(h)=\\lambda$.\n    \\end{enumerate}\n\\end{definition}\n\n\\begin{example}\\label{ex:categories with length in Lambda}\n    We give some existing examples of \\Cref{def:category with length in Lambda}.\n    \\begin{enumerate}\n        \\item\\label{ex:categories with length in Lambda:classic} Let $Q$ be a quiver, $\\mathcal{Q}$ the $\\Bbbk$-linear categorification of $Q$, and $\\widehat{\\mathcal{Q}}$ a stem category of $\\mathcal{Q}$ whose morphisms are generated by arrows.\n        Let $\\Lambda=\\mathbb{N}$ and set $\\ell(\\alpha)=1$ for each morphism in $\\widehat{\\mathcal{Q}}$ from an arrow $\\alpha$ in $Q$.\n        Then $\\mathcal{Q}$ has length in $\\mathbb{N}$.\n        \\item\\label{ex:categories with length in Lambda:AR} Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of a continuous quiver $\\widehat{\\mathcal{Q}}$ of type $A$ as in \\cite{IRT}.\n        Then $\\widehat{\\mathcal{Q}}$ is a stem category of $\\mathcal{Q}$.\n        \n        Define $\\ell: \\mathsf{Mor}(\\widehat{\\mathcal{Q}})\\to\\mathbb{R}_{\\geq 0}$ by $\\ell(g_{x,y}) = |x-y|$ where $g_{x,y}$ is the unique nonzero morphism in $\\widehat{\\mathcal{Q}}$ from $x$ to $y$.\n        Then $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n        Note $\\mathcal{Q}$ is also a category with a metric.\n        See \\Cref{subsec:length vs metric}.\n    \\end{enumerate}\n\\end{example}\n\nWe now define a length relation.\n\\begin{definition}\\label{def:length relation}\n    Let $\\Lambda$ be a weakly Archimedian monoid and $\\mathcal{C}$ a category with length in $\\Lambda$ with stem category $\\widehat{\\mathcal{C}}$.\n    Consider $\\Lambda_1,\\Lambda_2$ subsets of $\\Lambda$ such that $\\Lambda_1\\amalg\\Lambda_2=\\Lambda$, $|\\Lambda_1|\\geq 2$, and for all $\\lambda_1\\in\\Lambda_1, \\lambda_2\\in\\Lambda_2$ we have $\\lambda_1<\\lambda_2$.\n    Then the set $\\ell^{-1}(\\Lambda_2)$ in $\\widehat{\\mathcal{C}}$ generates an ideal $\\mathcal{I}$ in $\\mathcal{C}$.\n    We call $\\mathcal{I}$ a \\emph{length relation}.\n\\end{definition}\n\n\\begin{remark}\\label{rmk:length is not a number}\n    It is possible that $\\Lambda_1$ has no maximum element and $\\Lambda_2$ has no minimum element.\n    (Consider, for example, $\\Lambda=\\mathbb{Q}_{\\geq 0}$.)\n    Thus, we may not always be able to say that we are taking ``paths longer than $\\lambda$'' for some $\\lambda\\in\\Lambda$.\n\\end{remark}\n\n\\begin{example}\\label{ex:length relations}\n    We give three examples of length relations.\n    \\begin{enumerate}\n        \\item Let $Q$ be a quiver and $\\mathcal{Q}$ its $\\Bbbk$-linear categorification, which has length in $\\mathbb{N}$ (\\Cref{ex:categories with length in Lambda}(\\ref{ex:categories with length in Lambda:classic})).\n        Let $\\widehat{\\mathcal{Q}}$ be the stem category of $\\mathcal{Q}$ seen, effectively, as $Q$ embedded in $\\mathcal{Q}$.\n        Let $\\Lambda_1 = \\{0,1,2\\}$ and $\\Lambda_2=\\{3,4,5,\\cdots\\}$.\n        Then $\\mathcal{I}=\\langle \\ell^{-1}(\\Lambda_2)\\rangle $ is the set of morphisms in $\\mathcal{Q}$ generated by paths with length $\\geq 3$ in $Q$.\n        \\item Any Nakayama algebra where the relations have constant length $l$ can be realized as the $\\Bbbk$-linear categorification of its underlying quiver with length relations of length $l$ in $\\mathbb{N}$.\n        \\item\\label{ex:length relations:continuous A} Let $\\mathcal{Q}$ and $\\widehat{\\mathcal{Q}}$ be as in \\Cref{ex:categories with length in Lambda}(\\ref{ex:categories with length in Lambda:AR}).\n        Recall $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n        Let $\\Lambda_1 =[0,4]$ and $\\Lambda_2=(4,+\\infty)$.\n        Then $\\langle\\ell^{-1}(\\Lambda_2)\\rangle$ is the set of morphisms in $\\mathcal{Q}$ of length \\emph{strictly greater than $4$}.\n    \\end{enumerate}\n\\end{example}\n\n\\begin{theorem}\\label{thm:length relations are admissible}\n    Let $\\Lambda$ be a weakly Archimedian monoid, $\\mathcal{C}$ a category with length in $\\Lambda$ with stem category $\\widehat{\\mathcal{C}}$, and $\\mathcal{I}$ a length relation.\n    If $\\End_{\\widehat{\\mathcal{C}}}(X)$ is a finitely-generated monoid, for each $X\\in\\mathsf{Ob}(\\widehat{\\mathcal{C}})$, then $\\mathcal{I}$ is an admissible ideal.\n\\end{theorem}\n\\begin{proof}\n    If $\\mathcal{I}=\\emptyset$ then condition (1) is vacously satisfied.\n    Assume $\\mathcal{I}\\neq\\emptyset$, let $f\\in \\mathcal{I}$ such that $f\\in\\mathsf{Mor}(\\widehat{\\mathcal{C}})$, and let $\\lambda\\in\\Lambda_1$ such that $\\lambda>0$.\n    Then there is $n\\in\\mathbb{N}$ such that $n\\lambda \\geq \\ell(f)$.\n    Thus, there is some decomposition $f=g_n\\circ\\cdots\\circ g_1$ where $\\ell(g_i)\\in\\Lambda_1$ for each $g_i$.\n    Thus, each $g_i$ is not in $\\mathcal{I}$.\n    \n    Since $\\End_{\\widehat{\\mathcal{C}}}(X)$ is a finitely-generated monoid, let $m$ be the number of generators and let $\\{f_i\\}_{i=1}^m$ be the set of generators.\n    Let\n    \\[ N = \\max_i \\{ \\min_n \\{ n\\ell(f_i) \\mid n\\ell(f_i)\\in\\Lambda_2\\}\\}.\\]\n    Then\n    \\[ \\dim_{\\Bbbk}(\\End_{\\mathcal{C}}(X) \/ \\mathcal{I}(X,X)) \\leq m\\cdot N + 1,\\]\n    where we need the ``$+1$'' to account for the identity in $\\End_{\\widehat{\\mathcal{C}}}(X)$.\n    Therefore, $\\mathcal{I}$ is an admissible ideal.\n\\end{proof}\n\n\n\n\\section{Examples}\\label{sec:examples}\n\\begin{figure}[b]\n    \\centering\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \n\\begin{tikzpicture}[xscale=2]\n\\node (1) at (0,0) {1};\n\\node (2) at (1,1) {2};\n\\node (3) at (1,0) {3};\n\\node (4) at (1,-1) {4};\n\\node (5) at (2,0) {5};\n\n\\draw[->] (1) -- node[pos=0.6, above]{$\\alpha_1$} (2);\n\\draw[->] (1) -- node[pos=0.7, above]{$\\beta_1$} (3);\n\\draw[->] (1) -- node[pos=0.6, above]{$\\gamma_1$} (4);\n\\draw[->] (2) -- node[pos=0.4, above]{$\\alpha_2$} (5);\n\\draw[->] (3) -- node[pos=0.3, above]{$\\beta_2$} (5);\n\\draw[->] (4) -- node[pos=0.4, above]{$\\gamma_2$} (5);\n\\end{tikzpicture}\n        \\caption{}\n        \\label{fig:three-path-discrete}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.4\\textwidth}\n        \\centering\n        \\begin{tikzpicture}[xscale=2, inner sep = 0cm, outer sep = 0cm, very thick,decoration={\n    markings,\n    mark=at position 0.5 with {\\arrow{>}}}\n    ]\n\\node (start) at (0,0) {$\\bullet$};\n\\node (end) at (2,0) {$\\bullet$};\n\\node at (-.1,0){$0$};\n\\node at (2.1,0){$1$};\n\n\\draw[postaction={decorate}] (start) .. controls (.5,1.3) and (1.5,1.3).. node[above=2pt, pos=0.6]{$\\alpha$} (end);\n\\draw[postaction={decorate}] (start) -- node[pos=0.6, above=2pt]{$\\beta$} (end);\n\\draw[postaction={decorate}] (start) .. controls (.5,-1.3) and (1.5,-1.3).. node[pos=0.6, above=2pt]{$\\gamma$} (end);\n\\end{tikzpicture}\n        \\caption{}\n        \\label{fig:three-path-continuous}\n    \\end{subfigure}\n    \\caption{The quivers considered in \\Cref{ex: discrete 3 paths,ex: cts 3 paths}, respectively.}\n    \\label{fig:three-path-quivers}\n\\end{figure}\n\n\\begin{example}\\label{ex: discrete 3 paths}\nConsider the (discrete) quiver $Q$ shown in \\Cref{fig:three-path-discrete}.\nThe relation $\\alpha_2\\alpha_1-2\\beta_2\\beta_1+3\\gamma_2\\gamma_1$ generates an admissible ideal in the classical sense. In the $\\Bbbk$-linear categorification of $Q$, we rewrite the relation as the composition of morphisms\n\\[1\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -2\\beta_1\\\\3\\gamma_1\\end{smallmatrix}\\right]} 2\\oplus 3\\oplus 4\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2 &\\gamma_2\\end{smallmatrix}\\right]}  5,\n    \\]\n    which generates an admissible ideal\n    \n\\end{example}\n\n\n\\begin{example} \\label{ex: cts 3 paths}\n\nConsider the continuous analogue of \\Cref{ex: discrete 3 paths}, displayed in \\Cref{fig:three-path-continuous}.\nWe consider a similar relation $\\alpha-2\\beta+3\\gamma$.\nLet $X$, $Y$, and $Z$ be points on the interior of the paths $\\alpha$, $\\beta$, and $\\gamma$, respectively.\nThen $\\alpha=\\alpha_2\\alpha_1$ where $\\alpha_1:0\\to X$ and $\\alpha_2:X\\to 1$.\nWe similarly write $\\beta=\\beta_2\\beta_1$ and $\\gamma=\\gamma_2\\gamma_1$.\nThen the relation $\\alpha-2\\beta+3\\gamma$ can be written as the composition\n\\[0\n\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_1 \\\\ -2\\beta_1\\\\3\\gamma_1\\end{smallmatrix}\\right]}\nX\\oplus Y\\oplus Z\n\\xrightarrow{\\left [ \\begin{smallmatrix}\\alpha_2 & \\beta_2 &\\gamma_2\\end{smallmatrix}\\right]}\n1,\n    \\]\nand it generates an admissible ideal.\n\\end{example}\n\n\\begin{example}[Real line with point relations on integers]\\label{ex:real line integers}\nLet $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}$ as a category where paths move upwards.\nFor a point $c\\in \\mathbb{R}$, let the (unique) point relation at $c$ be $\\mathcal P_c$. The collection of point relations on the integers, $\\{\\mathcal P_z\\}_{z\\in \\mathbb{Z}}$ generates an admissible ideal by \\Cref{thm:point relations are admissible}.\n\nThe ``Auslander--Reiten space'' of the representations of this quiver is shaped like a mountain range; it is a set of triangles joined at their bottom vertices, see \\Cref{fig:AR mountain range}\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}{\\textwidth}\n    \\centering\n    \\begin{tikzpicture}\n        \\draw[dotted, thick] (-1.5,0) -- (0,0)--(-1.5,1);\n        \\draw[dotted,thick] (10.5,1) -- (9,0) -- (10.5,0);\n        \\draw[] (0,0) -- (1.5,1)--(3,0)--(4.5,1)--(6,0)--(7.5,1)--(9,0)--(0,0);\n    \\end{tikzpicture}\n    \\caption{The ``Auslander--Reiten space'' of representations of the quiver from \\Cref{ex:real line integers}}\n    \\label{fig:AR mountain range}\n    \\end{subfigure}\n    \n    \\begin{subfigure}{\\textwidth}\n    \\begin{tikzpicture}\n    \\draw[color=white] (0,0)--(0,1.3);\n        \\draw[dotted, thick] (-1.5,0) -- (0,0)--(-1,1)--(-1.5,1);\n        \\draw[dotted,thick] (10.5,1) -- (10,1) -- (9,0) -- (10.5,0);\n        \\draw[] (0,0) -- (1,1)--(2,1)--(3,0)--(4,1)--(5,1)--(6,0)--(7,1)--(8,1)--(9,0)--(0,0);\n    \\end{tikzpicture}\n    \n    \\caption{The ``Auslander--Reiten space'' for representations of $\\mathbb{R}$ modulo paths longer than $s$ and modulo paths that go through any $x\\in r\\mathbb{Z}$, where $r>s\\in\\mathbb{R}_{>0}$.}\\label{fig:AR chopped mountains}\n    \\end{subfigure}\n    \\caption{The ``Auslander--Reiten spaces'' of representations of the quivers in \\Cref{ex:real line integers,ex:real line points}}\n    \\label{fig:AR spaces}\n\\end{figure}\n\\end{example}\n\n\\begin{example}[Circle with length\/Kupisch relations]\\label{ex:circle length}\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of a continuous quiver $\\widehat{\\mathcal{Q}}$ of type $\\widetilde{A}$ as in \\cite{HR}.\n    Define $\\ell: \\mathsf{Mor}(\\widehat{\\mathcal{Q}})\\to\\mathbb{R}_{\\geq 0}$ by $\\ell(f)=\\phi-\\theta + 2n\\pi$ where $f:e^{i\\theta}\\to e^{i\\phi}$, and $0\\leq \\phi-\\theta<2\\pi$, and $n$ is the number of full rotations around the circle at $e^{i\\theta}$ before moving to $e^{i\\phi}$.\n    Then $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    If $\\mathcal{Q}$ is acyclic, we may replace $\\mathbb{R}_{\\geq 0}$ with $\\Lambda=[0,2\\pi)\\cup\\{\\infty\\}$ and define $+_\\Lambda$ similarly to \\Cref{ex:weakly Archimedian monoids}(\\ref{ex:weakly Archimedian monoids:with max}).\n    \n    \\begin{figure}\n        \\centering\n        \\begin{tikzpicture}[very thick, decoration={markings,mark=at position 0.5 with {\\arrow{>}}}]\n            \\draw[radius=3cm] (0,0) circle;\n            \\draw[radius=2.8cm, color=red,postaction=decorate] (180:2.8cm) arc[start angle =180, delta angle= 180] node[above left, pos=0.666, red]{\\XSolidBrush};\n            \\draw[radius=3.2cm, color=blue, postaction=decorate] (0:3.2cm) arc[start angle = 0, delta angle= 90] node[above right, pos=0.666, blue]{\\CheckmarkBold};\n            \\foreach \\x in {0, 30,...,360}\n            \t\\draw[thin] (\\x:2.9cm) -- (\\x:3.1cm);  \n        \\end{tikzpicture}\n        \\caption{The circle with length relations as described in \\Cref{ex:circle length}. In this figure, the relations have length $\\frac{2\\pi}{3}$.}\n        \n        \\label{fig:circle length}\n    \\end{figure}\n    \n    Now assume $\\widehat{\\mathcal{Q}}$ has cyclic orientation.\n    Let $\\kappa$ be a Kupisch function as in \\cite[Definition~3.9]{RZ}.\n    That is, $\\kappa:\\mathbb{R}\\to\\mathbb{R}_{> 0}$ is a function such that $\\kappa(t)+t>t$ and $\\kappa(t+1)=\\kappa(t)$, for all $t\\in\\mathbb{R}$.\n    This yields a map $\\mathbb{S}^1\\to \\mathbb{R}_{>0}$ where $\\mathbb{S}^1=[0,1]\/\\{0\\sim 1\\}$.\n    If $\\kappa$ is constant with value $a$, then this yields a length relation where $\\Lambda_1=[0,a]$ and $\\Lambda_2=(a,+\\infty)$.\n    If $\\kappa$ is not constant, then we do not have a length relation.\n    However, if $\\kappa$ does not have any separation points \\cite[Definition~4.2]{RZ}, then $\\kappa$ still induces an admissible ideal.\n\\end{example}\n\n\\begin{example}[Real line with length and point relations]\\label{ex:real line points}\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}$ as a category where paths move upwards.\n    Let $r,s$ be positive real numbers and for each $x\\in r\\mathbb{Z}\\subset \\mathbb{R}$, let $\\mathcal P_x$ be the (unique) point relation in $\\mathcal{Q}$ through $x$ and $\\mathcal{I}$ the admissible ideal generated by $\\bigcup_{x\\in r\\mathbb{Z}} \\mathcal{P}_x$.\n    Let $\\mathcal{J}$ be the the length relation in $\\mathcal{Q}\/\\mathcal{I}$ obtained by modding out by paths of length greater than $s$.\n    By \\Cref{thm:length relations are admissible,thm:point relations are admissible} with \\Cref{lem:stack admissible} we obtain an admissible ideal $\\widetilde{\\Jay}$ given by the point relations at each $x\\in r\\mathbb{Z}$ and paths of length greater than $s$.\n    \n    If $r\\leq s$ then $\\mathcal{C}\/\\mathcal{I} = \\mathcal{C}\/\\widetilde{\\Jay}$ since we cannot have a morphism of length greater than $r$ in $\\mathcal{C}\/\\mathcal{I}$ anyway.\n    If $r>s$ then we obtain paths of of length less than or equal to $s$ that do not pass through any $x\\in r\\mathbb{Z}$.\n    The ``Auslander--Reiten space'' for the case $r>s$ is in \\Cref{fig:AR chopped mountains}; notice the similarity with \\Cref{fig:AR mountain range}.\n\\end{example}\n\n\\begin{example}[Complications with Cycles]\\label{ex: cycles length}\n    For each $n\\in\\mathbb{N}$, let $\\mathcal{C}_n$ be circle whose radius is $\\frac{1}{2}e^{-n}$.\n    Let $\\mathcal{C}$ be the additive $\\Bbbk$-linearization of $\\mathbb{R}_{\\leq 0}\\amalg(\\coprod_{n\\in\\mathbb{N}} \\mathcal{C}_n)\/ \\sim$, where $\\mathbb{R}\\ni -n\\sim 0\\in\\mathcal{C}_n$.\n    See \\Cref{fig:cycles with decreasing length} for a visual depiction.\n    \n    We see that $\\mathcal{C}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    Let $\\mathcal{I}$ be a length relation.\n    Since our length is in $\\mathbb{R}_{\\geq 0}$ we can say we are modding out by length $>L$ or $\\geq L$ for some $L>0\\in\\mathbb{R}$.\n    \n    Notice that for each $N\\in\\mathbb{N}$, there exists some $\\mathcal{C}_n$ with radius $r$ such that $Nr < L$.\n    Therefore, there is no natural number $n$ such that for all nonisomorphism endomorphisms $f$ we have $f^n\\in\\mathcal{I}$.\n    \n    \\begin{figure}\n        \\centering\n        \\begin{tikzpicture}[scale=2, decoration={markings, mark=at position 0.6 with {\\arrow{>}}}]\n    \\foreach \\x in {0, -1,...,-3}{\n    \t\\node[outer sep=4+\\x] (\\x) at (\\x,0){$\\bullet$};\n    \t\\node at (\\x.south){\\x};}\n    \\node (-4) at (-4,0){$\\cdots$};\n    \\foreach \\x [remember=\\x as \\y (initially 0)]in {-1, -2,...,-4}\n    \t\\draw[very thick, postaction=decorate] (\\x.center)--(\\y.center);\n    \\foreach \\x [evaluate=\\x as \\r using .5*e^(\\x\/2)] in {0, -1,...,-3}\n         \\draw[radius=-\\r, very thick, postaction=decorate](\\x)+(0,\\r ) circle;\n    \\end{tikzpicture}\n        \\caption{Illustration of $\\mathcal{C}$ in \\Cref{ex: cycles length}, where we have glued smaller and smaller circles to each non-positive integer $n$ in $\\mathbb{R}_{\\leq 0}$.}\n        \\label{fig:cycles with decreasing length}\n    \\end{figure}\n\\end{example}\n\n\\begin{example}[Big wedge]\\label{ex:big wedge}\n    Let $\\mathcal{C}$ be a cyclic continuous quiver of type $\\widetilde{A}$ as in \\cite{HR}.\n    Let $\\widehat{\\mathcal{Q}} = (\\coprod_{\\mathbb{N}} \\mathcal{C} )\/\\sim$ where we join all the copies of the $\\mathcal{C}$ together at one point.\n    Denote the wedge point by $X$.\n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$.\n    Let us discuss the contruction of an admissible ideal out of point relations and a length relation.\n\n    Notice $\\mathcal{Q}$ has length in $\\mathbb{R}_{\\geq 0}$.\n    However, since the endomorphism ring of $X$ is an infinitely-generated monoid in $\\widehat{\\mathcal{Q}}$, we see that $\\End_{\\mathcal{C}}(X)\/ \\mathcal{I}(X,X)$ is not finite-dimensional.\n    Instead, we must add a point relation on all but finitely-many different copies of $\\mathcal{C}$.\n    If we do not have a point relation on all the cycles, we also need a length relation.\n    Without such a combination, it is not possible to build an admissible ideal out of point relations and a length relation.\n\\end{example}\n\n\\subsection{The Real Plane}\\label{subsec:real plane}\n    We now consider a continuous version of the grid quiver with commutativity relations as examined in \\cite{BBOS}.\n    Let $\\widehat{\\mathcal{Q}}$ be the category whose objects are points in $\\mathbb{R}^2$.\n    \n    We now define the $\\Hom$ set between an arbitrary pair of points $(x,y)$ and $(z,w)$.\n    Hom sets are given by considering paths made of up of horizontal and vertical line segments.\n    For a pair $(x,y)$ and $(z,w)$, consider the set $P_{x,y}^{z,w}$ of all finite sequences $\\{(x_i,y_i)\\}_{i=i}^n$ such that\n    \\begin{itemize}\n        \\item $(x_1,y_1)=(x,y)$ and $(x_n,y_n)=(z,w)$,\n        \\item $x_1\\leq x_2\\leq\\cdots \\leq x_n$ and $y_1\\leq y_2\\leq\\cdots \\leq y_n$,\n        \\item $(x_i,y_i)\\neq (x_{i+1},y_{i+1})$ for all $1\\leq i<n$,\n        \\item $x_1=x_2$ or $y_1=y_2$,\n        \\item for all $1\\leq i<n-1$, if $x_i=x_{i+1}$ then $y_{i+1}=y_{i+2}$, and\n        \\item for all $1\\leq i<n-1$, if $y_i=y_{i+1}$ then $x_{i+1}=x_{i+2}$\n    \\end{itemize}\n    We define \\[\\Hom_{\\widehat{\\mathcal{Q}}}((x,y),(z,w))=P_{x,y}^{z,w}.\\]\n    \n    Note that $P_{x,y}^{z,w}$ may be empty.\n    If either $x>z$ or $y>w$, then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (z,w)) = \\emptyset.\\]\n    If (i) $x=z$ and $y< w$ or (ii) $x< z$ and $y=w$, then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (z,w)) = \\left\\{ \\{ (x,y), (z,w) \\}\\right\\}. \\]\n    If $(x,y)=(z,w)$ then\n    \\[ \\Hom_{\\widehat{\\mathcal{Q}}}((x,y), (x,y)) = \\left\\{ \\{ (x,y) \\}\\right\\}. \\]\n    Composition is given by concatenating sequences and, if necessary, deleting a repeated term.\n    \n    Let $\\mathcal{Q}$ be the additive $\\Bbbk$-linearization of $\\widehat{\\mathcal{Q}}$ and let $(x,y),(z,w)\\in \\mathbb{R}^2$ such that $x<z$ and $y<w$.\n    Then, we define\n    \\[\n    \\mathcal{I}((x,y),(z,w)) = \\left\\langle \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n \\mid \\{(x_i,y_i)\\}_{i=1}^m \\neq \\{(x'_j,y'_j)\\}_{j=1}^n \\right\\rangle.\n    \\]\n    If $x=z$ or $y=w$, then $\\mathcal{I}((x,y),(z,w))=0$.\n    \n    Consider\n    \\begin{align*}\n        \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n &\\in \\mathcal{I}((x,y),(z,w)) \\\\\n        \\{(z_k,y_k)\\}_{k=1}^p - \\{(z'_l,z'_l)\\}_{l=1}^q &\\in \\mathcal{I}((z,w),(u,v)).\n    \\end{align*}\n    We show the composition is in $\\mathcal{I}((x,y),(u,v))$.\n    \\begin{align*}\n       & (\\{(z_k,y_k)\\}_{k=1}^p - \\{(z'_l,z'_l)\\}_{l=1}^q)\\circ (\\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n) \\\\\n       =& \\{(z_k,y_k)\\}\\circ \\{(x_i,y_i)\\} - \\{(z_k,y_k)\\}\\circ \\{(x'_j,y'_j)\\} \\\\& - \\{(z'_l,z'_l)\\}\\circ \\{(x_i,y_i)\\} + \\{(z'_l,z'_l)\\}\\circ \\{(x'_j,y'_j)\\}.\n    \\end{align*}\n    We know \n    \\begin{align*}\n        \\{(z_k,y_k)\\}\\circ \\{(x_i,y_i)\\} & \\neq \\{(z_k,y_k)\\}\\circ \\{(x'_j,y'_j)\\} \\\\\n        &\\text{and} \\\\\n        \\{(z'_l,z'_l)\\}\\circ \\{(x_i,y_i)\\} & \\neq \\{(z'_l,z'_l)\\}\\circ \\{(x'_j,y'_j)\\}.\n    \\end{align*}\n    Thus, the composition is in $\\mathcal{I}((x,y),(u,v))$.\n    \n    Consider $\\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n\\in\\mathcal{I}((x,y),(z,w))$.\n    By assumption there is $1\\leq k \\leq m$ and $1\\leq l \\leq n$ such that $(x_k,y_k)\\neq (x'_l,y'_l)$.\n    Let\n    \\begin{align*}\n        f:(x,y)\\to (x_k,y_k)\\oplus(x'_l,y'_l) &= \\left[\\begin{matrix} \\{(x_i,y_i)\\}_{i=1}^k \\\\ - \\{(x'_j,y'_j)\\}_{j=1}^l \\end{matrix} \\right] \\\\\n        g:(x_k,y_k)\\oplus(x'_l,y'_l) \\to (z,w) &= \\left[\\begin{matrix} \\{(x_i,y_i)\\}_{i=k}^m & \\{(x'_j,y'_j)\\}_{j=l}^n \\end{matrix} \\right].\n    \\end{align*}\n    Notice $f\\notin \\mathcal{I}((x,y),(x_k,y_k)\\oplus(x'_l,y'_l))$, $g\\notin \\mathcal{I}((x_k,y_k)\\oplus(x'_l,y'_l), (z,w))$, and\n    \\[ \\{(x_i,y_i)\\}_{i=1}^m - \\{(x'_j,y'_j)\\}_{j=1}^n = g\\circ f.\\]\n    Thus, every morphism in $\\mathcal{I}$ is given by a finite composition of morphisms not in $\\mathcal{I}$.\n    Furthermore, there are no cycles in $\\mathcal{Q}$.\n    Thus, $\\mathcal{I}$ is an admissible ideal.\n    \n    The resulting $\\mathcal{Q}\/\\mathcal{I}$ is the category where the objects are fnite direct sums of points in $\\mathbb{R}^2$ and Hom spaces between points is given by\n    \\[\n        \\Hom_{\\mathcal{Q}\/\\mathcal{I}}((x,y),(z,w)) = \\begin{cases}\n            \\Bbbk & x\\leq z \\text{ and }y\\leq w \\\\\n            0 & \\text{otherwise}.\n        \\end{cases}\n    \\]\n    This means that $\\mathcal{Q}\/\\mathcal{I}$ is the continuous generalization of (finite) discrete commutative grid quivers as examined in \\cite{BBOS}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nTwo structures $R$ and $S$ are called {\\em equimorphic}, denoted by $R\\approx S$, when each embeds in the other; we may also say that one is a {\\em sibling} of the other. Equimorphic finite structures are necessarily isomorphic, but this is no longer the case for infinite structures. For instance, the rational numbers, considered as a linear order, has continuum many siblings, up to isomorphism. The number of isomorphism classes of siblings of a structure $S$ is called the {\\em sibling number} of $S$ and we denote it by $Sib(S)$. \nA {\\em relation} $R$ is a pair $(V,E)$ where $V$ is a non-empty set, called the {\\em domain} of $R$, and $E\\subseteq V^n$ for some positive integer $n$; the integer $n$ is called the arity of $E$ and $E$ is called an $n$-ary relational symbol. A {\\em binary relation} is a relation $(V,E)$ where $E$ is of arity 2. The cardinality of a relation is the cardinality of its domain. \nWhen studying infinite relations, Thomass\\'e \\cite{TH2} made the following conjecture:\n\n\\begin{conjecture} [Thomass\\'e's Conjecture, \\cite{TH2}]\nLet R be a countable relation. Then $Sib(R)=1$, $\\aleph_0$ or $2^{\\aleph_0}$. \n\\end{conjecture}\n\nThere is also an alternate version of the conjecture as follows:\n\n\\begin{conjecture}[The Alternate Thomass\\'e Conjecture]\nLet R be a relation of any cardinality. Then $Sib(R)=1$ or $\\infty$.  \n\\end{conjecture}\n\nThe alternate Thomass\\'{e} conjecture is connected to a conjecture of Bonato-Tardif \\cite{BT} ({\\em The Tree Alternative Conjecture}) stating that the sibling number of a tree of any cardinality is one or infinite in the category of trees. The connection between these two conjectures is through this fact that each sibling of a tree $T$ is a tree if and only if $T\\oplus 1$, the graph obtained by adding an isolated vertex to $T$, does not embed in $T$. Therefore, for a tree $T$ not equimorphic to $T\\oplus 1$, the tree alternative conjecture and the alternate Thomass\\'e conjecture are equivalent. The tree alternative conjecture has been verified for rayless trees by Bonato and Tardif \\cite{BT}, for rooted trees by Tyomkyn \\cite{TY} and for scattered trees by Laflamme, Pouzet and Sauer \\cite{LPS}. Besides, the alternate Thomass\\'e conjecture was proved for rayless graphs by Bonato, Bruhn, Diestel and Spr\\\"ussel \\cite{BBDS}, for countable $\\aleph_0$-categorical structures by Laflamme, Pouzet, Sauer and Woodrow \\cite{LPSW}, for countable cographs by Hahn, Pouzet and Woodrow \\cite{HPW} and for countable universal theories by Braunfeld and Laskowski \\cite{BL}. Furthermore, Thomass\\'e's conjecture for countable chains and the alternate Thomass\\'e conjecture for all chains have been proved by Laflamme, Pouzet and Woodrow \\cite{LPW}. Tateno \\cite{TAT} claimed a counterexample to the tree alternative conjecture in his thesis.\nAbdi, Laflamme, Tateno and Woodrow \\cite{ALTW} revisited and verified Tateno's claim and provided locally finite trees having an arbitrary finite number of siblings. Moreover, using an adaptation, Abdi, Laflamme, Tateno and Woodrow \\cite{ALTW} provided countable partial orders with a similar conclusion which disproves Thomass\\'e's conjecture. It turns out that both conjectures of Bonato-Tardif and Thomass\\'e  are false. \nThis is a major development in the program of understanding siblings of a given mathematical structure. We now know that the simple dichotomy one vs infinite for the number of siblings is no longer valid, and the situation is much more intricate. While counting the number of siblings provides a good first insight into the siblings of a structure, in particular understanding which structures exactly do satisfy the dichotomy, the equimorphy programme is now ready to move on and focus on the actual structure of those siblings. In this paper, we show that a countable $NE$-free poset fits in the programme.  \n \n\\section{Overview}\n\nIn order-theoretic mathematics, the class of series-parallel posets is the smallest class of finite posets obtained from the one element poset by iteration of two operations: direct sum and linear sum. This class of posets was introduced by Lawler and studied by Valdes and Tarjan.  \nGrillet and Heuchenne introduced $NE$-free posets and later Habib and Jegou \\cite{HJ} showed that $NE$-free posets are generalisations of series-parallel posets by giving a recursive construction of $NE$-free posets.  \nFinite and some infinite $NE$-free posets can be obtained by direct sum and linear sum. However, there are infinite $NE$-free posets which need another operation, namely sum over a labelled chain in which the chain has no least element. In order to count the siblings of countable $NE$-free posets, we first determine their structure. For this purpose, we use modular decomposition of $NE$-free posets in Section \\ref{DT}. Section \\ref{StructureNE} determines the structure of $NE$-free posets by classifying them. \nThen Section \\ref{SibNE} will provide a proof of the main result of this article: a countable $NE$-free poset has one or infinitely many siblings. In Section \\ref{Finiteposetsub}, we will show that a countable $NE$-free poset $P$ whose comparability graph has one sibling is a finite poset substitution of chains or antichains and that Thomass\\'{e}'s conjecture holds for $P$. This is a generalisation of a result of \\cite{LPW} for countable chains whose comparability graph is a clique. More, the result of Section \\ref{Finiteposetsub} leaves Thomass\\'e's conjecture open for a countable $NE$-free poset whose comparability graph has infinitely many siblings. \n\n\n\n\\section{Partial Orders and $NE$-Free Posets} \n\n\nA \\textit{poset (partially ordered set)} is a binary relation $P=(V,\\leq)$ where $\\leq$ is reflexive, antisymmetric and transitive. If a poset $P$ is given, by $\\leq_P$, we mean the order of $P$. Let $P=(V,\\leq)$ be a poset. By an element $x$ of $P$ we mean $x\\in V$. By $\\leq^*$ we mean the partial order obtained by reversing the order $\\leq$, that is $x\\leq^* y$ if and only if $y\\leq x$ where $x, y\\in P$. We denote the poset $(V,\\leq^*)$ by $P^*$. Any substructure, resp induced substructure, of $P$ is called a \\textit{subposet}, resp an {\\em induced subposet}, of $P$. For two subposets $P_1$ and $P_2$ of $P$ we write $P_1 \\leq P_2$ if for every $x\\in P_1$ and every $y\\in P_2$ we have $x \\leq y$. Note that $P_1$ or $P_2$ might be empty. The empty set is both greater than and less than any set. Therefore, $\\leq$ on set is not a partial order.  Two elements $x, y \\in P$ are called \\textit{comparable} if $x \\leq y$ or $y \\leq x$, otherwise they are \\textit{incomparable}, denoted by $x \\perp y$. A {\\em chain}, resp an {\\em antichain}, is a poset whose elements are pairwise comparable, resp incomparable. Two  subposets $P_1, P_2$ of $P$ are {\\em comparable} if $P_1 \\leq P_2$ or $P_2 \\leq P_1$, and {\\em incomparable}, denoted by $P_1\\perp P_2$, if $x\\perp y$ for every $x\\in P_1$ and every $y\\in P_2$. Two elements $x, y \\in P$ are \\textit{compatible} if the pair $\\{x, y\\}$ has a lower bound, that is, there is $z\\in P$ such that $z\\leq x$ and $z\\leq y$.  The graph $CG(P)=(V,E)$ such that for two distinct $x, y \\in P$, $xy \\in E$ if $x < y$ or $y < x$, is called the \\textit{comparability graph} of $P$. Note that a partial order is reflexive meaning that each element of a poset is comparable to itself, however, in order to get simple graphs, our definition of a comparability graph prevents having loops in the graphs. A \\textit{connected}, resp \\textit{disconnected}, poset is a poset whose comparability graph is connected, resp disconnected. By a \\textit{component} of a poset $P=(V,\\leq)$, we mean an induced subposet $Q=(W,\\leq)$ of $P$ where $W$ is the vertex set of a component of $CG(P)$. We say two elements $x, y$ of $P$ are connected if they belong to the same component of $P$; equivalently, they are connected in $CG(P)$ by a path and we say $x, y$ are disconnected when they belong to different components of $P$. \n\nLet us denote by an `$N$' the following poset on four elements $\\{a, b, c, d\\}$ such that $a < b$, $c < b$, $c < d$, $a\\perp c$, $b \\perp d$ and $a \\perp d$. An $NE$-\\textit{free poset} is a poset which does not embed an $N$. For instance, a chain is an $NE$-free poset. \n\n\n\n\n\n\\subsection{Graph Substitution and Poset Substitution} \\label{GPSubstitution}\n\nLet $G$ be a graph. By $V(G)$, resp $E(G)$, we mean the set of vertices, resp edges, of $G$. For simplicity, by $x\\in G$ we mean that $x$ is a vertex of $G$. The following definition is from \\cite{CD}. \n\n\\begin{definition} [Graph Substitution] \\label{Graphsub}\nLet $K$ be a graph and let $(H_v)_{v\\in K}$ be a pairwise disjoint family of graphs. We denote by $G:=K[H_v\/v : v\\in K]$ the graph resulting from the substitution of $H_v$ for $v\\in K$ as follows: \n\\begin{enumerate}\n    \\item $V(G)=\\bigcup_{v\\in K} V(H_v)$;\n    \\item $xy \\in E(G)$ if and only if either\n    \\begin{enumerate}\n        \\item $xy\\in E(H_v)$ for some $v\\in K$; or\n        \\item for two distinct $u, v \\in K$ we have $x \\in H_u$, $y \\in H_v$ and $uv\\in E(K)$. \n    \\end{enumerate}\n\\end{enumerate} \nThen we say $G$ is a {\\em graph substitution} of the $H_v$ for $K$. We call $K$ the {\\em context graph} and the $H_v$ the {\\em graph blocks} of $G$. If $K$ is a clique (complete graph), resp an independent set, then $G$ is called a {\\em complete}, resp {\\em direct}, {\\em sum} of graphs.\n\\end{definition} \n\nA poset substitution applies a {\\em composition operation} which is the substitution of a poset $P$ for an element $x$ in a poset $Q$. In the resulting poset, denoted by $Q[P\/x]$, the element $x$ is replaced by $P$, and all elements in $Q$ comparable to $x$ are comparable to all elements of $P$. \nA \\textit{chain} ({\\em linearly ordered set}), resp an \\textit{antichain}, is a poset whose elements are pairwise comparable, resp incomparable. Any substructure of a chain is called a {\\em subchain}. \n\n\n\\begin{definition} [Poset Substitution] \\label{PosetSub} \nLet $Q$ be a poset and $\\{P_v\\}_{v\\in Q}$ be a family of pairwise disjoint posets. For each $v\\in Q$ let $\\leq_v$ be the order of $P_v$. The {\\em poset substitution} of the $P_v$ for $Q$ is a poset, denoted by $P:=Q[P_v\/v : v \\in Q]$, defined as follows:  \n\\begin{enumerate}\n    \\item $P=\\bigcup_{v \\in Q} P_v$\n    \\item for $x, y \\in P$, $x \\leq_P y$ if and only if \n     \\begin{enumerate} \n        \\item $x, y \\in P_v$ for some $v \\in Q$ and $x \\leq_v y$; or\n        \\item for two distinct $u, v\\in Q$, we have  $x \\in P_u$, $y \\in P_v$ and $u \\leq_Q v$. \n    \\end{enumerate} \n\\end{enumerate}\nThe poset $Q$ is called the {\\em context poset} of $P$ and each $P_v$ is called a {\\em poset block} of $P$.   If $Q$ is a chain, resp an antichain, $P$ is called a {\\em linear}, resp {\\em direct, sum} of posets, denoted by $P=+_{v\\in Q} P_v$, resp $P=\\bigoplus_{v\\in Q} P_v$ and each $P_v$ is called a {\\em summand}, resp {\\em component}, of the linear, resp direct, sum.\n\\end{definition}\n\n\n\n\n\n\\section{Modular Decomposition} \\label{DT}\n\n\nCourcelle and Delhomm\\'{e} \\cite{CD} investigated the modular decomposition of infinite (mainly countable) graphs. The notion of modular decomposition is essential for establishing structural properties of graphs and related objects, in particular of partial orders and their comparability graphs. This fact is a special case of a general result about modular decomposition of binary relations. In particular,\neach $NE$-free poset can be represented by a special type of ordered and labelled tree which is called \\textit{decomposition tree}.  Due to the importance of such trees,  in this section, we give a detailed presentation of them. The materials  of this section are taken from \\cite{CD} and \\cite{HPW}. Indeed, we construct the decomposition tree of an $NE$-free poset in a way similar to the method employed by \\cite{HPW} for cographs. However, one can find more details about the decomposition tree of a countable graph in \\cite{CD}.\n\n\n\n\n\\subsection{Modules} \\label{Modules}\n\n\nLet $S$ be a set. A \\textit{binary structure over} $S$ is a pair $\\mathbb{B}:=(V,d)$ where $V$, the {\\em domain} of $\\mathbb{B}$, is a non-empty set and $d$ is a map from $V\\times V$ into $S$. A subset $M$ of $V$ is a \\textit{module} of $\\mathbb{B}$ if $d(x,y)=d(x,y')$ and $d(y,x)=d(y',x)$ for every $x \\in V\\setminus M$ and every $y, y' \\in M$. Modules are also called {\\em intervals}. The sets $\\emptyset$, $V$ and the singletons are called \\textit{trivial} modules. A binary structure is called {\\em indecomposable} if all its modules are trivial. If moreover it has more than two vertices it is called {\\em prime}. A non-empty module is called \\textit{strong} if it is comparable w.r.t inclusion to every module which it meets.\n\n\n\n\\begin{lemma} [\\cite{HPW}] \\label{Intersectionunionstrong}\nThe intersection of any set of strong modules is either empty or a strong module.\n\\end{lemma}\n\n\n\nLet $A$ be a non-empty subset of $V$. Let $S_{\\mathbb{B}}(A)$ be the intersection of strong modules of $\\mathbb{B}$  containing $A$. We write $S_{\\mathbb{B}}(x,y)$ instead of $S_{\\mathbb{B}}(\\{x,y\\})$. According to  Lemma \\ref{Intersectionunionstrong}, $S_{\\mathbb{B}}(A)$ is a strong module. A module is called \\textit{robust} if it is either a singleton or the smallest strong module containing two distinct elements. Hence, if $A$ is a non-trivial robust module, then there exist $x\\neq y \\in A$ such that $A$ is strong and every strong module containing $x$ and $y$ contains $A$, that is, $A=S_{\\mathbb{B}}(x,y)$.  A binary structure $\\mathbb{B}$ is called {\\em robust} if its domain is a robust module of $\\mathbb{B}$.  \n\nLet $A$ be a strong module and $x, y \\in A$. We set $x \\equiv_A y$ if either $x=y$ or there is a strong module containing $x$ and $y$ and properly contained in $A$. It is proven (Lemma 6.5 in \\cite{HPW}) that the relation $\\equiv_A$ is an equivalence relation on $A$ whose equivalence classes are strong modules. Also, if $A$ has more than one element, then there are at least two classes if and only if $A$ is robust. Moreover, these classes are the maximal strong modules properly contained in $A$. \nThe equivalence classes of a strong module $A$ are called \\textit{components} of $A$. We say that a module is {\\em limit} if it is a strong module such that none of its non-empty proper strong submodules is maximal (see \\cite{HPW}).\n\n\\begin{proposition} [\\cite{HPW}] \\label{Nonlimitrobust}\nLet $\\mathbb{B}=(V,d)$ be a binary structure and A a subset of V. Then A is a non-trivial robust module if and only if $A$ is not a limit module.\n\\end{proposition}\n\n\n\nBy Proposition \\ref{Nonlimitrobust}, a module is limit if and only if it is not robust. \nIf $A$ is a robust module of a binary structure with at least two elements, then there are two distinct components $I, J$ of $A$. Let $x, x' \\in I$ and $y, y' \\in J$. Since $J$ is a module and $x \\in A\\setminus J$, we have $d(x,y)=d(x,y')$. Since $I$ is a module and $y' \\in A\\setminus I$, we have $d(x,y')=d(x',y')$. So, $d(x,y)=d(x',y')$ meaning that the values $d(x,y)$ for $x \\in I$ and $y \\in J$ depend only upon $I$ and $J$. Therefore, the binary structure on $A$ induces a binary structure on the set $A\/\\equiv_A$ of components of $A$, called the \\textit{Gallai quotient} of $A$.  The following proposition is given in \\cite{HPW} as an observation without proof, however, we give a proof for completeness.  \n\n\\begin{proposition} \\label{Gallai}\nLet $A$ be a robust module of a binary structure $\\mathbb{B}$. The strong modules of the Gallai quotient of A are trivial: the empty set, the whole set $A\/\\equiv_A$ and the singletons.  \n\\end{proposition}\n\n\\begin{proof}\nLet $\\mathcal{B}$ be a strong module of $A\/\\equiv_A$ containing more than one component of $A$. Notice that $A$ has at least two components. Assume that $\\mathcal{B}$ contains two distinct components $I$ and $J$ of $A$. Let $K \\in A\/\\equiv_A \\setminus \\mathcal{B}$ and $I_1, I_2\\in \\mathcal{B}$. We have $d(K,I_1)=d(K,I_2)$ and $d(I_1,K)=d(I_2,K)$. It means that for every $x \\in K$ and every $y, y' \\in \\bigcup \\mathcal{B}$, we have $d(x,y)=d(x,y')$ and $d(y,x)=d(y',x)$. Thus, $\\bigcup\\mathcal{B}$ is a module of $A$. We prove that $\\bigcup\\mathcal{B}$ is a strong module of $A$. Let $N$ be a module of $A$ such that $N \\cap (\\bigcup\\mathcal{B}) \\neq \\emptyset$. Without loss of generality assume that $N$ meets $I$. Since $I$ is strong, $N \\subseteq I$ or $I \\subseteq N$. In the first case $N \\subseteq \\bigcup\\mathcal{B}$. In the second case if $I \\subset N$, then $N=A$ because $I$ is maximal and hence $\\bigcup\\mathcal{B}\\subseteq N$, so, $\\bigcup \\mathcal{B}$ is strong. Note that the components of $A$ are the maximal strong modules properly contained in $A$. Since $I$ is a proper subset of the strong module $\\bigcup\\mathcal{B}$, it follows that $\\bigcup\\mathcal{B}=A$. Equivalently, $\\mathcal{B}=A\/\\equiv_A$.\n\\end{proof} \n\nThe central result of the decomposition theory of binary structures describes the structure of the Gallai quotient. Let $\\mathbb{B}=(V,d)$ be a binary structure. We say that $\\mathbb{B}$ is {\\em constant} with value $\\alpha$ if $d(x,y)=\\alpha$ for all $x\\neq y\\in V$, and {\\em linear} with values $\\{\\alpha, \\beta\\}$, $\\alpha\\neq \\beta$, if $\\{(x,y)\\in V\\times V : x\\neq y, d(x,y)=\\alpha\\}$ is a linear order and $\\{(x,y)\\in V\\times V : x\\neq y, d(x,y)=\\beta\\}$ is the opposite linear order.  \n\n\n\\begin{theorem} [\\cite{HPW}] \\label{Primeconstantlinear}\nThe strong modules of a binary structure $\\mathbb{B}$ are trivial if and only if $\\mathbb{B}$ is prime,  constant or linear. \n\\end{theorem} \n\nLet $A$ be a non-trivial robust module of a binary structure $\\mathbb{B}$. The {\\em type} of $A$, denoted by $t(A)$, is {\\em prime} if $A\/\\equiv_A$ is prime, otherwise its type is $\\alpha$ if $A\/\\equiv_A$ is  constant with value $\\alpha$, and $\\{\\alpha, \\beta\\}$ if $A\/\\equiv_A$ is linear with values $\\{\\alpha, \\beta\\}$, $\\alpha\\neq \\beta$. By Proposition \\ref{Gallai},  the strong modules of the Gallai quotient $A\/\\equiv_A$ are trivial. Hence, the Gallai quotient $A\/\\equiv_A$ is either prime, constant or linear. \n\nLet $A$ be a non-trivial robust module of an $NE$-free poset $P$. Note that $P$ can be regarded as a binary structure $\\mathbb{B}=(P,d)$ where for $x, y\\in P$, $d(x,y)=0$ if $x\\perp y$, $d(x,y)=+1$ if $x <_P y$ and $d(x,y)=-1$ if $y <_P x$. By Proposition \\ref{Gallai}, the strong modules of $A\/\\equiv_A$ are trivial and by Theorem \\ref{Primeconstantlinear}, $A\/\\equiv_A$ is either prime, constant or linear. Further, a theorem due to Kelly \\cite{KE} asserts that any indecomposable partial order with at least three elements embeds `$N$'. By definition, a partial order with only two elements is not prime. Therefore, the Gallai quotient $A\/\\equiv_A$ is either constant or linear. In the first case, the quotient $A\/\\equiv_A$ is an antichain i.e. $d(I,J)=0$ for each $I\\neq J\\in A\/\\equiv_A$. So, the type of $A$ is 0 in this case. In the second case, the quotient $A\/\\equiv_A$ is a linearly ordered set, that is, for two $I\\neq J\\in A\/\\equiv_A$, we have $I <_P J$ or $J <_P I$. Thus, $\\{(I,J)\\in A\/\\equiv_A\\times A\/\\equiv_A : I\\neq J, d(I,J)=+1\\}$ is a linear order and $\\{(I,J)\\in A\/\\equiv_A\\times A\/\\equiv_A: I\\neq J, d(I,J)=-1\\}$ is the opposite linear order. Hence, the type of $A$ is $\\{-1,+1\\}$ in this case.  \n\n\\subsection{Decomposition Tree of $NE$-Free Posets} \\label{DTNE}\n\nOur terminology in this section borrows from \\cite{TH2} and \\cite{HPW}. We follow the way of construction employed by \\cite{HPW} in which a decomposition tree is a meet-tree ordered by reverse inclusion of robust modules. Let $P=(V,\\leq)$ be a poset.  We use the notations $P^x=\\{y \\in V : x \\leq y\\}$ and $P_x=\\{y \\in V : y \\leq x\\}$. A \\textit{forest} is a poset $F$ such that for every $x\\in F$, the set $F_x$ is a chain; the elements of $F$ are called \\textit{nodes}.  A \\textit{tree} is a forest whose elements are pairwise compatible. Let $(T,\\leq)$ be a tree. \nA {\\em branch} of a node $x\\in T$ is an induced and maximal (under inclusion) subtree of $T^x\\setminus \\{x\\}$. A {\\em limit branch} is a branch of a node $x\\in T$ with no minimal element. For two nodes $x, y$ of $T$, we say $x$ is a \\textit{child} of $y$ if $y < x$ and there is no $z\\in T$ such that $y < z < x$. A tree is a \\textit{meet-tree} if any two nodes $x, y$ have a greatest lower bound, called their \\textit{meet} and denoted by $x \\wedge y$. \nLet $T$ be a meet-tree. A \\textit{leaf} in $T$ is a maximal node, and a \\textit{root} is a minimum one. An \\textit{internal} node is one that is not a leaf. A tree has at most one root. We observe that if a node $x$ of $T$ is the meet of a finite set $X$ of the maximal nodes of $T$, denoted by $Max(T)$, then $x$ is the meet of a subset $X'$ of $X$ with at most two nodes. \nWe say that a meet-tree $T$ is \\textit{ramified} if every node of $T$ is the meet of a finite set of maximal nodes of $T$.  \n\nLet $T$ be a ramified meet-tree and $T':=T\\setminus Max(T)$. A $\\{0,\\{-1,+1\\}\\}$-\\textit{valuation} is a map $v :T'\\to \\{0,\\{-1,+1\\}\\}$. The valuation is \\textit{dense} if for every $x<y$ in $T'$, there is some $z$ with $x< z \\leq y$ such that $v(z)\\neq v(x)$. A ramified meet-tree densely valued by $\\{0,\\{-1,+1\\}\\}$  is called a \\textit{decomposition tree}. \n\nLet $P$ be an $NE$-free poset. We construct the decomposition tree $T=(R(P), \\leq,v)$ of $P$ as follows:\nThe nodes of the tree $T$ are the robust modules of $P$ denoted by $R(P)$. The order $\\leq$ on the nodes of the tree is reverse inclusion meaning that for two robust modules $A, B$ of $P$, $A \\leq B$ if and only if $B \\subseteq A$. To see that this order gives a tree, first let $A$ be a non-empty robust module of $P$. For two $C_1, C_2 \\in R(P)$ with $C_1, C_2 \\leq A$ we have $A \\subseteq C_1$ and $A \\subseteq C_2$. Since $A$ is non-empty, it follows that there exists some $x\\in A$. So, $x \\in C_1 \\cap C_2$. This means that $C_1 \\subseteq C_2$ or $C_2 \\subseteq C_1$ because both $C_1$ and $C_2$ are strong. Hence, the down-set of a node of the tree $T$ is a chain. Now, suppose that $A, B$ are two robust modules which are not comparable. Let $x \\in A\\setminus B$ and $y \\in B\\setminus A$. Then $x$ and $y$ are distinct. Set $C$ to be the robust module determined by $\\{x, y\\}$. We must have that $C$ is comparable to $A$ and $B$. Moreover, since $x\\in A$, $y \\in C$ and $y \\notin A$, we obtain $A \\subset C$. Interchanging the roles of $x$ and $y$, we get $B \\subset C$. Under reverse inclusion, $C$ is the meet of $A$ and $B$, that is $C=A \\wedge B$. The robust modules of $P$ consisting of singletons are exactly the leaves of $T$. Robust modules of $P$ which are not singletons are valued with one of two values 0 and $\\{-1, +1\\}$. If $A$ is a non-trivial robust module of $P$, then the {\\em value} $v(A)$ of $A$ is the type of $A$ that is $v(A):=\\{-1,+1\\}$, resp $v(A):=0$, if $t(A)$ is $\\{-1,+1\\}$, resp 0. It implies that if $A$ is the least strong module containing two distinct elements $x, y$, then $v(A)=\\{-1,+1\\}$, resp $v(A)=0$, if $x$ and $y$ are comparable, resp incomparable. Note that the map $v$ is defined on the set of nodes of $T$ which are not maximal, that is $v : T\\setminus Max(T) \\to \\{0,\\{-1,+1\\}\\}$.   Given a node $A\\in T$ which is a robust module of $P$, the union of the robust modules properly contained in $A$ is a maximal proper strong submodule of $A$ which might be limit itself.  \n \n\nThe following proposition implies that the valuation of $T$ is dense. \n\n\\begin{proposition} [\\cite{HPW}] \\label{Types}\nLet A, C be two non-trivial robust modules of a binary structure such that $A\\subset C$ and A and C have the same non-prime type $\\{\\alpha, \\beta\\}$. Then, there is a robust module B with $A\\subset B\\subset C$ whose type is distinct from the type of $\\{\\alpha, \\beta\\}$.  \n\\end{proposition}\n\n\\begin{corollary} \\label{Dense}\nLet P be an NE-free poset and T its decomposition tree. The valuation v of T is dense. \n\\end{corollary}\n\n\\begin{proof}\nRecall that for each non-trivial robust module $A$ of $P$, we have either $t(A):=0$, or $t(A):=\\{-1,+1\\}$. Now, let $A\\subset C$ be a non-trivial robust modules of $P$. If $v(A)\\neq v(C)$, then by setting $B:=C$, the result follows. Now assume that $v(A)=v(C)$. Then, both $\\mathbb{B}_{\\upharpoonright A\/\\equiv_A}$ and $\\mathbb{B}_{\\upharpoonright C\/\\equiv_C}$ are either constant or linear. Therefore, $A, C$ have the same non-prime type $\\{\\alpha,\\beta\\}$ ($\\alpha=\\beta=0$ or $\\alpha=-1, \\beta=+1$). By Proposition \\ref{Types}, there exists a robust module $B$ with $A\\subset B\\subset C$ whose type is distinct from the type $\\{\\alpha,\\beta\\}$, that is $t(B)\\neq t(A), t(C)$. This implies that $v(B)\\neq v(A)$. Therefore, the valuation $v$ is dense.\n\\end{proof}\n \n \n\n\n \n\n\n\n\n \n  \n    \n    \n\n \n  \n   \n\n \n  \n\n \n\n\n \n  \n   \n\n \n  \n   \n\n \n  \n\n\n\n\n\n\n\\section{Structure of $NE$-Free Posets} \\label{StructureNE}\n\nIn this section we classify $NE$-free posets and show that each $NE$-free poset is either a direct sum or a linear sum of $NE$-free posets or a sum over a labelled chain with no least element. \nIndeed, all the three operations are based on substituting smaller posets for three sorts of posets: chains, antichains and posets obtained from labelled chains. A \\textit{cograph} is a graph with no induced subgraph isomorphic to a path $P_4$ with four vertices  \\cite{J}.\n\n\n\n\n\\begin{proposition} \\label{Subcograph}\nLet $G=K[H_v\/v : v\\in K]$ be a graph substitution. $G$ is a cograph if and only if its context graph and graph blocks are cographs. \n\\end{proposition}\n\n\\begin{proof}\nLet $P$ be a copy  of $P_4$. \n\n($\\Rightarrow$) If $G$ is a cograph, then it is clear that all its graph blocks are cographs. If $K$ embeds $P$, then let $v_1, v_2, v_3, v_4$ be the vertices of $P$ in $K$. For every $1\\leq i \\leq 4$, pick an element $x_i\\in H_{v_i}$. Then, the $x_i$ form a copy of $P_4$ in $G$, a contradiction.  \n\n($\\Leftarrow$) For the sake of a contradiction, assume that $P$ is embedded in $G$. We show that no graph block $H_v$ contains more than one element of $P$. By assumption, no $H_v$ embeds $P$. Suppose $|P\\cap H_v|=3$ and let $a, b, c \\in H_v \\cap P$. Then $d \\in P$ belongs to some $H_u$ where $u \\neq v$. Without loss of generality assume that the vertex $d$ is connected to $c$. This means that $uv\\in E(K)$. Consequently, $d$ is connected to $a$ and $b$. But, then $a, b, c$ and $d$ do not form a path as $d$ has degree 3. Now suppose $|P\\cap H_v|=2$ and let $a,b \\in P\\cap H_v$. Suppose $b$ and $c$ are adjacent where $c \\in H_u$ with $u \\neq v$. It follows that $a$ and $c$ are adjacent as well. The vertex $d$ cannot be adjacent to $c$ as then $deg(c)=3$. Therefore, $d$ must be adjacent to  both $a$ and $b$. In this case the vertices $a, b, c$ and $d$ form a cycle. Thus, every vertex of $P$ belongs to a unique $H_v$.  It follows that $K$ itself embeds $P$, a contradiction.   \n\\end{proof} \n\nThe only partial order $P$ on $P_4$ with $CG(P)=P_4$ results in an `$N$'. On the other hand, the comparability graph of an `$N$' is $P_4$. So, we get the following result. \n\n\\begin{corollary} \\label{SubNEposet}\nLet $P=Q[P_v\/v : v\\in Q]$ be a poset substitution. $P$ is $NE$-free if and only if its context poset and poset blocks are $NE$-free. \n\\end{corollary}\n\n\n$NE$-free posets which are not direct or linear sums require poset labelled sums. \nLet $(I,\\leq)$ be a chain and $r$ a map from $I$ to $\\{-1, 0, +1\\}$. Let $Q^I_r=(I,\\leq')$ be defined as follows: for $i<j$, $i\\perp j$ if $r(i)=0$, $i<'j$ if $r(i)=-1$ and $j<'i$ if $r(i)=+1$. We prove in the following lemma that $Q^I_r$ is an $NE$-free poset. Let $\\{(P_i,\\leq_i)\\}_{i\\in I}$ be a disjoint family of non-empty posets. We call the poset substitution of the $P_i$ for $Q^I_r$ i.e. $P=Q^I_r[P_i\/i:i\\in I]$, a {\\em poset labelled sum} of the $P_i$ and denote it by $P=\\sum_{i\\in I} P_i$. The $P_i$ are called the {\\em summands} of $P$. We may view such a sum as the poset associated to the {\\em  labelled chain} $C:=(I,\\leq,\\ell)$ where $\\ell(i)=(P_i,r(i))$ and denote it by $P=\\sum C$. \n\n\\begin{lemma} \n$Q^I_r$ is an NE-free poset. Hence, a poset labelled sum $\\sum_{i\\in I}P_i$ is NE-free if and only if its summands are NE-free. \n\\end{lemma}\n\n\\begin{proof}\nIn order to prove that $Q^I_r=(I,\\leq')$ is a poset, it suffices to show that $\\leq'$ is transitive. Let $i, j, k\\in I$ and assume that $i <' j$ and $j <' k$. If $i<j<k$, then $r(i)=-1$ and if $k<j<i$, then $r(k)=+1$. In both cases we get $i <' k$. Now, assume that $i<j$ and $k<j$. Since $i <' j$, we have $r(i)=-1$. Further, since $(I,\\leq)$ is a chain, we have $i<k$ or $k<i$. In the first case, we get $i <' k$. In the second case, we have $k<i<j$ and regarding $j <' k$ we get $r(k)=+1$. Thus, $i <' k$. Note that the case $j<i$, $j<k$ is not possible because then we get $r(j)=+1$ and $r(j)=-1$, a contradiction. \n\nNow, suppose that a copy of $N$ embeds in $Q^I_r$. Let $i\\in N$ be such that $i\\leq j$ for each $j\\in N$. Such an element exists because $(I,\\leq)$ is a chain. Since $i$ is comparable w.r.t $\\leq'$ to at least one $j\\neq i$ in $N$, we have $r(i)=-1$ or $r(i)=+1$. But then $i$ is comparable w.r.t $\\leq'$ to each element of $N$, a contradiction. Thus, $Q^I_r$ is $NE$-free. By Corollary \\ref{SubNEposet}, it implies that $P=\\sum_{i\\in I} P_i$ is $NE$-free if and only if each $P_i$ is $NE$-free.   \n\\end{proof}\n\n\n\n\\subsection{Classification} \\label{Classification} \n\nThe classification of $NE$-free posets is based on the structure of their decomposition tree. Let $I$ be a chain and $r:I\\to\\{-1,0,+1\\}$ a map. We set $I_0:=\\{i\\in I : r(i)=0\\}$ and $I_1:=\\{i\\in I : |r(i)|= 1\\}$. In order to classify $NE$-free posets, we will need poset labelled sums indexed by infinite chains $C=(I,\\leq,r)$ with no least element where $r$ is a mapping from $I$ to $\\{-1, 0 , +1\\}$ such that both $I_0$ and $I_1$ are coinitial in $I$. We aim to show that if $T$ has no least element, then $P$ is a poset labelled sum. The crucial ingredient that we will use is the following (see \\cite{BD}, page 1747 Lemma 2.2). \n\n\n\n\\begin{lemma} [\\cite{BD}] \\label{BD}\nLet $\\mathbb{B}=(V,d)$ be a binary structure. The collection of robust modules of $\\mathbb{B}$ containing a given element is a chain covering the elements of $V$ (i.e. every element of V belongs to some member of the collection). \n\\end{lemma}\n\n\nLet $P$ be an $NE$-free poset. If $T$ is the decomposition tree of $P$, then the collection $\\mathfrak{C}_x$ of robust modules of $P$ containing a given element $x\\in P$ corresponds to a maximal chain in $T$ whose greatest element is $\\{x\\}$. Assume that $\\mathfrak{C}_x$ has no least element and take an arbitrary $y\\in P$. Let $\\mathfrak{C}_y$ be the collection of robust modules of $P$ containing $y$. Since $\\mathfrak{C}_y$ and $\\mathfrak{C}_x$ meet at $\\{x\\}\\wedge \\{y\\}$ in $T$, it follows that $\\mathfrak{C}_y$ does not have a least element as well. \n\n\n\n\\begin{lemma} \\label{Posetreduced}\nLet P be an NE-free poset such that for some (every) $x\\in P$, the chain of robust modules of $P$ containing $x$ has no least element. Then P is the sum $\\sum C$ of a labelled chain $C=(I,\\leq,\\ell)$ where $(I,\\leq)$ is an infinite chain with no least element and for each $i\\in I$, $\\ell(i)=(P_i,r(i))$ where $P_i$ is an NE-free poset and r is a mapping from I to $\\{-1,0,+1\\}$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in $I$. \n\\end{lemma} \n\n\\begin{proof} \nLet $P$ satisfy the condition of the lemma and $\\mathfrak{C}$ be the chain of robust modules of $P$ containing $x$ under reverse inclusion. Let $v$ be the valuation of the decomposition tree $T$ of $P$. Let $A\\in\\mathfrak{C}$ be given. If $v(A)=0$, then let $P_A$ be the poset obtained by restricting $P$ to $A \\setminus [x]_{\\equiv_A}$ and if $v(A)=\\{-1, +1\\}$, then let $P_{A^-}$, resp $P_{A^+}$, be the poset obtained by restricting $P$ to $\\{y \\in A : y\\not\\equiv_A x\\ \\text{and}\\ y <_P x\\}$, resp $\\{y\\in A : y\\not\\equiv_A x, \\ \\text{and}\\ x <_P y\\}$. For $A\\in \\mathfrak{C}$ with $v(A)=\\{-1, +1\\}$, replace $A$ by $A^-$ if $P_{A^+}=\\emptyset$, by $A^+$ if $P_{A^-}=\\emptyset$ and by the two-element chain $A^-<A^+$ if both $P_{A^-}$ and $P_{A^+}$ are non-empty. Let $I$ consist of $\\{x\\}$, the $A\\in\\mathfrak{C}$ with $v(A)=0$ and the replacements $A^-$ and $A^+$ for $A\\in\\mathfrak{C}$ with $v(A)=\\{-1, +1\\}$. For $A_1, A_2\\in I$, if $A_1$, resp $A_2$, is determined by some robust module $A$, resp $B$, where $A\\neq B$, then define $A_1 < A_2$ if and only if $B \\subset A$ ($A <_T B$). It follows that $(I,\\leq)$ is a chain with no least element because $\\mathfrak{C}$ has no least element. \n\nLet $I$ be the chain constructed above. Define $r:I\\to\\{-1,0,+1\\}$ by $r(A)=0$ if $v(A)=0$, $r(A^-)=-1$ and $r(A^+)=+1$. Since $\\mathfrak{C}$ has no least element and $v$ is dense on the elements of $\\mathfrak{C}$, it follows that $I_0$ and $I_1$ are coinitial in $I$. \n\nLet $I$ be the chain constructed above and for each $A\\in I$ let $P_A$ be the poset obtained above. Consider $\\{P_A\\}_{A\\in I}$ which is a family of pairwise disjoint $NE$-free posets. We have $P=\\bigcup_{A\\in I} P_A$. Let $A, B\\in I$ be given such that $A<B$. If $r(A)=0$ then $v(A)=0$ and we have $A <_T B$. So, $P_A\\perp P_B$ in this case. If $r(A)=-1$, resp $r(A)=+1$, then $A$ is determined by a robust module with value $\\{-1,+1\\}$ and we have $P_A <_P P_B$, resp $P_B <_P P_A$. This means that $P=Q^I_r[P_A\/A : A\\in I]$, that is $P$ is a poset labelled sum. In other words $P=\\sum C$ where $C:=(I,\\leq,\\ell)$ is a labelled chain with $\\ell(A)=(P_A,r(A))$. Note that $(I,\\leq)$ is an infinite chain with no least element and $r$ is a mapping from $I$ to $\\{-1,0,+1\\}$ such that $I_0$ and $I_1$ are coinitial in $I$. \n\\end{proof}\n\n\\begin{definition} \\label{CCGC}\nWe say that a poset $P$ has {\\em property CCGC} if the complement of the comparability graph of $P$ is connected, that is, $(CG(P))^c$ is connected. \n\\end{definition}\n\nIn Subsection \\ref{Linearsum}, when determining the sibling number of a linear sum $P$ of $NE$-free posets, we will see that the property CCGC of the summands of $P$ plays a crucial role. We also prove that an $NE$-free poset satisfying Lemma \\ref{Posetreduced} has property CCGC. \n\n\\begin{proposition} \\label{Plsccgc}\nLet P be an NE-free poset whose decomposition tree has no least element. Then P is connected and it has property CCGC. \n\\end{proposition}\n\n\\begin{proof}\nLet $T$ be the decomposition tree of $P$. By assumption $T$ has no least element. Set $G:=CG(P)$. Take $x, y\\in G$ and consider the robust module $A$ determined by $x$ and $y$. If $v(A)=\\{-1,+1\\}$, then $x$ and $y$ are comparable meaning that $xy\\in E(G)$. If $v(A)=0$, then since $T$ has no least element and $v$ is dense, we can find a robust module $B\\supset A$, equivalently $B <_T A$, with $v(B)=\\{-1,+1\\}$. Take some $z\\in B$. Then $x$ and $z$ are comparable as well as $y$ and $z$. Therefore, there is a path in $G$ connecting $x$ and $y$. Hence, $P$ is connected. \n\n\nNow we prove that $G^c$ is connected. Let $x, y\\in G$ be given. Let $A=S_\\mathbb{B}(x,y)$ as computed for $G$. If $v(A)=0$, then $x$ and $y$ are incomparable w.r.t $\\leq_P$. Thus, $xy\\in E(G^c)$ meaning that in this case $x$ and $y$ are connected in $G^c$. If $v(A)=\\{-1,+1\\}$, then by density of $v$ and the fact that $T$ has no least element we can find a robust module $B$ of $P$ with $B<_T A$ and $v(B)=0$. Take some $z\\in B$. We have $z\\perp x$ and $z\\perp y$. Therefore, $xz, zy\\in E(G^c)$ meaning that $x$ and $y$ are connected in $G^c$. This completes the proof. \n\\end{proof}\n\nWe have the following classification of $NE$-free posets which will be used in determining the sibling number of an $NE$-free poset.  \n\n\\begin{theorem} \\label{Pstructure}\nLet P be an NE-free poset with more than one element. Then either \n\\begin{enumerate}\n    \\item P is a direct sum of at least two non-empty connected NE-free posets, or\n    \\item P is a linear sum of at least two non-empty NE-free posets with property CCGC, or\n    \\item P is the sum $\\sum C$ of a labelled chain $C=(I,\\leq,\\ell)$ such that $(I,\\leq)$ is an infinite chain with no first element, and each label $\\ell(i)$ is the pair $(P_i,r(i))$ made of a non-empty NE-free poset $P_i$ and an element $r(i)\\in\\{-1,0 , +1\\}$ in such a way that r is a mapping on the chain $(I,\\leq)$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in I. \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nLet $T$ be the decomposition tree of $P$. If $P$ is the least element of $T$, then $P$ is a robust module with value  0 or $\\{-1, +1\\}$. In the first, resp second, case, each maximal proper robust submodule of $P$ is connected, resp has property CCGC, the quotient $P\/\\equiv_P$ is an antichain, resp a chain, and $P$ is a direct, resp linear, sum of its components, resp summands. Finally, if $T$ has no least element, then by Lemma \\ref{Posetreduced}, $P$ satisfies (3). \n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Siblings of Countable $NE$-Free Posets} \\label{SibNE}\n\nIn this section our aim is to prove the following theorem which shows that the alternate Thomass\\'{e} conjecture holds for countable $NE$-free posets.\n\n\\begin{theorem} \\label{AltThomasseNEP}\nA countable $NE$-free poset has one or infinitely many siblings.\n\\end{theorem}\n\nA {\\em quasi-order} is a reflexive and transitive binary relation. A {\\em well-quasi-order (wqo)} is a quasi-order $\\mathcal{Q}$ such that any infinite sequence of elements of $\\mathcal{Q}$ contains an infinite increasing subsequence w.r.t $\\leq_\\mathcal{Q}$. \nWe denote the class of countable $NE$-free posets by $\\mathcal{N}_{\\leq\\omega}$. \nThomass\\'{e} \\cite{TH2} proved that  $\\mathcal{N}_{\\leq\\omega}$  is well-quasi-ordered under embeddability. Hence, embeddability is a well-founded relation on $\\mathcal{N}_{\\leq\\omega}$. Therefore, in order to prove that a countable $NE$-free poset $P$ has one or infinitely many siblings, we suppose that this property holds for all $NE$-free posets which strictly embed in $P$ and then show that the property holds for $P$. In order to tackle the problem we use the classification of $NE$-free posets. Recall by Theorem \\ref{Pstructure} that an $NE$-free poset with more that one element is either a direct sum of at least two non-empty connected $NE$-free posets or a linear sum of at least two non-empty $NE$-free posets with property CCGC, or it is the sum of a labelled chain such that the chain has no first element. We argue by cases.  We start with the last case and prove that such a countable poset has continuum many siblings. Next, direct and linear sums will be considered.  \n\n\n\n\n\n\\subsection{Sum Over a Labelled Chain with no Least Element}  \n\n\nLet $P$ be a countable $NE$-free poset such that its decomposition tree has no least element. In this case by Lemma \\ref{Posetreduced}, $P$ is the sum $\\sum C$ of a  labelled chain $C=(I,\\ell)$ such that $I$ has no first element and $\\ell(i)=(P_i,r(i))$ where $r:I\\to\\{-1,0,+1\\}$ is a map such that $I_0$ and $I_1$ are coinitial in $I$. We show that $P$ has continuum many siblings in this case. The proof is based on constructing continuum many pairwise non-isomorphic labelled chains $C_\\alpha$. We quote some results in \\cite{HPW} and also translate some results there into the context of $NE$-free posets to show that the sums $\\sum C_\\alpha$ are pairwise non-isomorphic siblings of $\\sum C$. We also provide the notions needed. \n\nA {\\em labelled chain} is a pair $C=(I,\\ell)$ where $I$ is a chain and $\\ell$ is a map from $I$ to a quasi-order. Let $\\mathcal{Q}$ be a quasi-order. A {\\em $\\mathcal{Q}$-embedding}, (or an {\\em embedding} when the quasi-order $\\mathcal{Q}$ is definite), of a labelled chain  $C=(I,\\ell)$ into another labelled chain $C'=(I',\\ell')$, denoted by $C\\leq C'$, is an order embedding $f:I\\to I'$ such that $\\ell(i)\\leq_\\mathcal{Q} \\ell'(f(i))$ for all $i\\in I$. Two labelled chains $C, C'$ are {\\em equimorphic}, denoted by $C\\approx C'$, when they are mutually embeddable. They are {\\em isomorphic} when there is an isomorphism $\\phi$ from $I$ to $I'$ such that $\\ell'\\circ \\phi=\\ell$. \nA labelled chain  $C=(I,\\ell)$ is {\\em additively indecomposable}, or briefly {\\em indecomposable}, if for every partition of $I$ into an initial segment $J$ and a final segment $F$, $C\\leq C_{\\upharpoonright J}$ or $C\\leq C_{\\upharpoonright F}$. We say that $C$ is {\\em left-indecomposable}, resp {\\em strictly left-indecomposable}, if $C$ embeds in every non-empty initial segment, resp if $C$ is left-indecomposable and embeds in no proper final segment. The {\\em right-indecomposability} and the {\\em strict right-indecomposability} are defined similarly. \n\nThe {\\em sum} $\\sum_{i\\in I} C_i$ of labelled chains over a chain (not over a labelled chain) is defined as in the case of chains. If $I$ is the $n$-element chain $\\underline{n}:=\\{0,1,\\ldots, n-1\\}$ with $0<1<\\cdots<n-1$, then the sum is rather denoted by $C_0+C_1+\\cdots+C_{n-1}$. \nLet $C=(I,\\ell)$ be a labelled chain and $J, F$ be a partition of $I$ into an initial and a final segment, respectively. Let $C_0, C_1$ be obtained by restricting $C$ to $J, F$, respectively. With the notion of sum, $C$ is indecomposable if  $C=C_0+C_1$ implies that $C\\leq C_0$ or $C\\leq C_1$. \n\nLet $C=(I,\\ell)$ be a labelled chain. Recall that $I^*$ is obtained from $I$ by reversing the order of $I$. Define $C^*$ to be the labelled chain $(I^*,\\ell)$. \nFor two labelled chains $C_0$ and $C_1$, we define $C_0+^* C_1$ to be the sum $C_1^*+C_0^*$. For a chain $J$ define $\\sum^*_{j\\in J} C_j=\\left(\\sum_{j\\in J} C_j\\right)^*=\\sum_{j\\in J^*} C_j^*$. \nA sequence $(C_n)_{n<\\omega}$ of labelled chains is {\\em quasi-monotonic} if $\\{m : C_n\\leq C_m\\}$ is infinite for each $n<\\omega$, its sum $\\sum_{n<\\omega} C_n$  is right indecomposable or equivalently $\\sum_{n<\\omega}^* C_n$ is left indecomposable (\\cite{HPW}).  \n\nLet $C=(I,\\ell)$ be a labelled chain. Two elements $x, y\\in I$, $x\\leq y$, are {\\em equivalent}, denoted by $x\\equiv_C y$, if $C$ does not embed in the restriction of $C$ to the interval determined by $x$ and $y$, that is, $C \\not\\leq C_{\\upharpoonright[x,y]}$. If $C$ is indecomposable, then this relation is an equivalence relation. Equivalence classes are intervals of $I$. Further, either all the elements of $I$ are equivalent, that is, there is only one equivalence class, or the quotient $I\/\\equiv_C$ is dense and if $J$ is any equivalence class, $C$ does not embed into $C_{\\upharpoonright J}$. \n\nAccording to Laver \\cite{LAV1}, the class $\\mathcal{C}_{\\leq\\omega}$ of countable chains quasi-ordered by embeddability is wqo. Moreover, the class $\\mathcal{Q}^{\\mathcal{C}_{\\leq\\omega}}$ of countable chains labelled by a wqo $\\mathcal{Q}$ is wqo. Hence, the class of countable chains labelled by $\\mathcal{N}_{\\leq\\omega}$ is wqo.  \n\n\n\nWe list the properties we need. These properties are due to Laver \\cite{LAV1} and they are given in \\cite{HPW}. We quote the proof given in \\cite{HPW}. \n\n\\begin{lemma} [\\cite{HPW}]  \\label{Labelledchainsproperties}\nLet $C=(I,\\ell)$ be a countable chain labelled over a wqo $\\mathcal{Q}$.   \n\\begin{enumerate}\n   \\item If I has no least element, then there is some initial segment J of I such that $C_{\\upharpoonright J}$ is left indecomposable;\n    \\item If C is left indecomposable, then C is an $\\omega^*$ sum $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and the set of m such that $C_n$ embeds into $C_m$ is infinite. \n   \n    \\end{enumerate}\n\\end{lemma}\n\nWe translate a result in \\cite{HPW} to the context of $NE$-free posets. \n\n\\begin{lemma}  \\label{Finalsegmentsiso} \nLet $P, P'$ be two isomorphic NE-free posets such that their decomposition tree has no least element. If $P=\\sum C$ and $P=\\sum C'$ where $C=(I,\\ell)$ and $C'=(I',\\ell')$ are two labelled chains, then there are two infinite initial segments J of I and $J'$ of $I'$ and an isomorphism h of the induced poset labelled chains $C_{\\upharpoonright J}$ and $C'_{\\upharpoonright J'}$. \n\\end{lemma}\n\n\\begin{proof}\nSince $P$ and $P'$ are isomorphic, their decomposition trees are isomorphic. We may assume that they are identical. Let $T$ be such a tree. Then, $(I,\\leq)$ and $(I'\\leq')$ correspond to maximal chains in $T$ and by Lemma \\ref{Posetreduced} they do not have a first element. If these chains are identical, then there is nothing to prove ($J=J'=I$). If not, then since $T$ has no least element, these chains meet in some element $x\\in I\\cap I'$ corresponding to a node in $T$. Set $J=J'=I_x$ and let $h$ be the identity on the initial segment $I_x$. \n\\end{proof}\n\nThe following lemma appears as Lemma 4 in page 39 of \\cite{LPSZ}. \n\n\\begin{lemma} [\\cite{LPSZ}] \\label{Continuummaps}\nThere is a set of $2^{\\aleph_0}$ maps  $f$ in $\\{0,1\\}^\\mathbb{N}$ such that for every pair f, $f'$ of distinct maps, and every isomorphism h from a final segment F of $\\mathbb{N}$ onto another $F'$, there is some $n\\in F$ such that $f(n)\\neq f'(h(n))$. \n\\end{lemma} \n\n\nThe following lemma gives a construction of siblings of a countable $NE$-free poset which are  sums of labelled chains with no least element. The proof is adapted with some passages quoted from \\cite{HPW}. \n\n\\begin{lemma} \\label{Nonisoplc}\nLet $C=(I,\\ell)$ be a countable labelled chain with no least element where $\\ell(i)$ is the pair $(P_i,r(i))$ made of a non-empty NE-free poset and $r(i)\\in \\{-1,0,+1\\}$ such that $I_0=\\{i\\in I : r(i)=0\\}$ and $I_1=\\{i\\in I : |r(i)|=1\\}$ are coinitial in I. Then, there are $2^{\\aleph_0}$  labelled chains $C_f$ where $f:\\mathbb{N}\\to\\{0,1\\}$, such that $\\sum C_f\\ncong \\sum C_g$ for $f\\neq g$. \n\\end{lemma}\n\n\\begin{proof}\nLet $C=(I,\\ell)$ be a countable labelled chain satisfying the conditions of the lemma. We select an initial segment $J$ of $I$ such that the restriction $C_J:=(J,\\ell_{\\upharpoonright J})$ is left indecomposable (this is due to Lemma \\ref{Labelledchainsproperties} (1)). Let $Ev(J)$ be the set of $i\\in J$ such that $P_i$ is a chain or an antichain of even size. Since $I$ has no least element and $I_0$ and $I_1$ are coinitial in $I$, there is an infinite descending sequence $(a_n)_{n<\\omega}$ of elements of $J$ coinitial in $I$ such that $|r(a_n)|=1$. We insert infinitely many elements $b_n, c_n$ such that $b_n$ covers $a_n$ and $c_n$ covers $b_n$. Set $\\Bar{J}:=J\\cup \\{b_n, c_n : n<\\omega\\}$, $F:=I\\setminus J$ and $\\Bar{I}:=\\Bar{J}\\cup F$. To each map $f:\\mathbb{N}\\to \\{0,1\\}$ we associate a labelled chain $C_f=(\\Bar{I},\\ell_f)$ with  $\\ell_f(i):=(\\Bar{P_i},r_f(i))$ as follows: If $i\\in I\\setminus Ev(J)$, then we set $\\ell_f(i):=\\ell(i)$. If $i\\in Ev(J)$, then set $r_f(i):=r(i)$ and let $\\Bar{P_i}$ be an extension of $P_i$ to an extra element in such a way that the new poset is a chain if $P_i$ is a chain and an antichain, otherwise. If $i\\in \\{b_n, c_n\\}$, recalling that $|r(a_n)|=1$, we set $r_f(b_n)=0$ and $r_f(c_n)=r(a_n)$, $\\Bar{P_i}$ being an antichain, resp a chain, of size $2f(n)+2$ if $i=b_n$, resp $i=c_n$. It is clear that $\\bar{I}_0=\\{i\\in \\bar{I} : r_f(i)=0\\}$ and $\\bar{I}_1=\\{i\\in \\bar{I} : |r_f(i)|=1\\}$ are coinitial in $\\bar{I}$. \n\n\n \n\n\n\nIf $f$ and $f'$ are two maps from $\\mathbb{N}$ to $\\{0,1\\}$ such that the sums $\\sum C_f$ of $C_f=(\\Bar{I},\\ell_f)$ and $\\sum C_{f'}$ of $C_{f'}=(\\Bar{I},\\ell_{f'})$ are isomorphic, then according to Lemma \\ref{Finalsegmentsiso}, there are two initial segments $\\Bar{J}$ and $\\Bar{J'}$ of $\\Bar{I}$ and an isomorphism $h$ from $\\Bar{J}$ onto $\\Bar{J'}$ preserving the labels, that is, the labels $\\ell_f(i)=(\\Bar{P_i},r_f(i))$ and $\\ell_{f'}(h(i))=(\\Bar{P}_{{h(i)}},r_{f'}(h(i)))$, $i\\in \\Bar{J}$, are isomorphic, meaning that $\\Bar{P_i}$ and $\\Bar{P}_{h(i)}$ are isomorphic and $r_f(i)=r_{f'}(h(i))$ for each $i\\in \\Bar{J}$. If $i=b_n$, resp $i=c_n$, then necessarily $h(b_n)=b_m$, resp $h(c_n)=c_m$, for some $m$. Indeed, $\\Bar{P}_{h(i)}$  must be an antichain, resp chain, of even size. So, $h(b_n), h(c_n)\\in Ev(\\Bar{J})$. Further, recall that for $i\\in Ev(J)$, $\\Bar{P_i}$ is a chain or antichain of odd size. Since $r_{f'}(h(b_n))=r_f(b_n)=0$, resp $r_{f'}(h(c_n))=r_f(c_n)=\\pm 1$ where $\\pm 1$ means $-1$ or $+1$, hence $h(b_n)$, resp $h(c_n)$, must be some $b_m$, resp some $c_m$. Consequently, there are two final segments $F$ and $F'$ of $\\mathbb{N}$, and an isomorphism $h$ from $F$ to $F'$ such that $f'(h(n))=f(n)$ for every $n\\in F$. By applying Lemma \\ref{Continuummaps}, we get $2^{\\aleph_0}$ pairwise non-isomorphic labelled chains $C_f$ having the same properties as does $C$.\n\\end{proof} \n\nIn Lemma \\ref{Nonisoplc}, we observe that the order on $P_f:=\\sum C_f$ extends both $\\sum C_{f\\upharpoonright\\Bar{J}}$ and $\\sum C_{f\\upharpoonright F}$ where the relation between an element $x$ of $\\sum C_{f\\upharpoonright \\Bar{J}}$ and an element $y$ of $\\sum C_{f\\upharpoonright F}$ is determined as follows: $x <_{P_f} y$ if and only if $r_f(i)=-1$ where $i\\in\\Bar{J}$ and $x\\in \\Bar{P_i}$; and $y <_{P_f} x$ if and only if $r_f(i)=+1$ where $i\\in \\Bar{J}$ and $x\\in \\Bar{P_i}$. We denote this by $\\mp$ and obtain $\\sum C_f=\\sum C_{f\\upharpoonright\\Bar{J}}\\mp\\sum C_{f\\upharpoonright F}$. \n\nLet $C=(I,\\ell)$ be a labelled chain and $n$ be a positive integer. The {\\em ordinal product} $\\underline{n}.C$ is the labelled chain $(\\underline{n}.I,\\ell_n)$ where $\\underline{n}.I$ is the ordinal product of the $n$-element chain $\\underline{n}:=\\{0, \\ldots, n-1\\}$, $0<1<\\cdots<n-1$, and the chain $I$, that is, the ordinal sum of $C$ copies of $\\underline{n}$, and $\\ell_n(m,i)=\\ell(i)$ for every $m\\in\\underline{n}$, $i\\in I$. \n\n\n\\begin{lemma} [\\cite{HPW}]  \\label{ncopies}\nLet $C=(I,\\ell)$ be a countably infinite labelled chain. If the labels belong to a b.q.o and C is indecomposable, then for every positive integer n, the ordinal product $\\underline{n}.C$ embeds in C. \n\\end{lemma} \n\n\nNow we show that the sum of labelled chains $C_f$ in Lemma \\ref{Nonisoplc} are equimorphic to the sum of the labelled chain $C$.  \n\n\\begin{lemma} \\label{ReducedequimorphP}\nLet $C=(I,\\ell)$ be a countable labelled chain such that I has no first element and the labels belong to $\\mathcal{N}_{\\leq\\omega}\\times\\{-1, 0,+1\\}$. Then, there is an initial segment J of I such that the restriction $C_J=(J,\\ell_{\\upharpoonright J})$ is left indecomposable. If J is such an initial segment, then for every map $f:\\mathbb{N}\\to \\{0,1\\}$,  $\\sum C_f\\approx \\sum C$ where $C_f$ is constructed as in Lemma \\ref{Nonisoplc}.\n\\end{lemma}\n\n\\begin{proof}\nThe existence of $J$ follows from the fact that the class of countable chains labelled by $\\mathcal{N}_{\\leq\\omega}\\times \\{-1, 0, +1\\}$ is b.q.o and Lemma \\ref{Labelledchainsproperties} (1). Let $f:\\mathbb{N}\\to\\{0,1\\}$ and $C_f=(\\Bar{I},\\ell_f)$ be the labelled  chain defined in Lemma \\ref{Nonisoplc} and set $P_f:=\\sum C_f$. We prove that $\\sum C_f\\leq \\sum C$.  Let $C_{f\\upharpoonright\\Bar{J}}$, resp $C_{f\\upharpoonright F}$, be the restriction of $C_f$ to $\\Bar{J}$, resp $F$. We have $C_f=C_{f\\upharpoonright\\Bar{J}}+C_{f\\upharpoonright F}$. By the observation given after Lemma \\ref{Nonisoplc}, we get $\\sum C_f=\\sum C_{f\\upharpoonright\\Bar{J}}\\mp\\sum C_{f\\upharpoonright F}$. \n\nTo conclude, it suffices to prove that $\\sum C_{f\\upharpoonright\\Bar{J}}$ embeds into $\\sum C_{\\upharpoonright J}$ and $\\sum C_{f\\upharpoonright F}$ embeds into $\\sum C_{\\upharpoonright F}$. Only the first statement needs a proof because $f$ does not affect $F$. In order to prove it, we define an auxiliary labelled chain $D=(M,d)$ as follows: the domain $M$ is $\\underline{2}.J\\cup X$ where $X=\\{b_n, c_n : n<\\omega\\}$, $b_n$ covers $(1,a_n)$ and $c_n$ covers $b_n$. The labelling $d$ is defined by $d(i)=\\ell(j)$ if $i=(m,j)\\in \\underline{2}.J$ and $d(i)=\\ell_f(i)$ if $i\\in X$. We prove that the following inequalities hold.\n$$\\sum C_{f\\upharpoonright\\Bar{J}}\\leq\\sum D\\leq \\sum\\underline{2}.C_{\\upharpoonright J}\\leq \\sum C_{\\upharpoonright J}.$$\n\nFor the first inequality, define $h:\\Bar{J}\\to \\underline{2}.J\\cup X$ by $h(j)=(0,j)$ for $j\\in J$ and $h(i)=i$ for $i\\in X$. Then $h$ is an embedding and due to the definition of $d$, it follows that $C_{f\\upharpoonright \\Bar{J}}\\leq D$ and consequently $\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum D$.\n\nFor the second inequality, recall that $C_{\\upharpoonright J}$ is left indecomposable. It can be verified that $\\underline{2}.C_{\\upharpoonright J}$ is left indecomposable.  Thus, by Lemma \\ref{Labelledchainsproperties} (2), we can write $\\underline{2}.C_{\\upharpoonright J}$ as $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and for each $n<\\omega$, the set of $m$ such that $C_n$ embeds into $C_m$ is infinite. For each $n<\\omega$, let $J_n$ be the domain of $C_n$ and let $L_n:=J_n\\cup \\{b_m, c_m : a_m\\in J_n\\}$. Set $D_n:=(L_n, \\ell_{f\\upharpoonright L_n})$. Then, $D=\\sum^*_{n<\\omega} D_n$. We define an embedding from $\\sum D$ to $\\sum \\underline{2}.C_{\\upharpoonright J}$ by induction. Let $n<\\omega$. First we show that the poset labelled sum $\\sum D_n$ embeds in the poset labelled sum obtained by a finite sum of $C_m$, that is $\\sum D_n\\leq \\sum(C_{m+k}\\cdots+C_m)$ for some $m, k<\\omega$. The  labelled chain $D_n$ consists of $C_n$ plus finitely many elements, each labelled by a 2 or 4-element chain or antichain. Thus, $D_n$ can be written \n\\begin{multline*} \nD_{n,0}+\\alpha_{(0,0)}+\\alpha_{(1,0)}+D_{n,1}+\\alpha_{(0,1)}+\\alpha_{(1,1)}+\\cdots+D_{n,k_n-1}+ \\\\ \\alpha_{(0,k_n-1)}+  \\alpha_{(1,k_n-1)}+D_{n,k_n}\n\\end{multline*}\n\nwith $D_{n,0}+\\cdots+D_{n,k_n}=C_n$ and for each $0\\leq i\\leq k_n-1$, $\\alpha_{(0,i)}$, resp $\\alpha_{(1,i)}$, is a singleton belonging to $X:=\\{b_n, c_n : n\\in\\mathbb{N}\\}$ labelled by a 2 or 4-element antichain, resp chain.  Suppose that the poset labelled sum\n$$\\sum(D_{n,t} + \\cdots+D_{n,k_n-1}+\\alpha_{(0,k_n-1)}+\\alpha_{(1,k_n-1)}+D_{n,k_n})$$\nwhere $0< t \\leq k_n$, has been embedded in $\\sum(C_{\\varphi(t)}+\\cdots+C_m)$ for some $m$. \nWe know that $\\alpha_{(1,t-1)}$ is an $l$-element chain where $l=2$ or 4. Since $C_n$ embeds in infinitely many $C_m$, the coinitial sequence $(a_n)_{n<\\omega}$ has infinitely many elements with the same label of $c_{n,t-1}$ where $c_{n,t-1}\\in X$ is the singleton corresponding to $\\alpha_{(1,t-1)}$. Select $l$ elements $j_1, \\ldots, j_l$ of $J$ such that $j_1,\\ldots,j_l<L_{n,t}$ where $L_{n,t}$ is the domain of $D_{n,t}$ and $r(j_s)=r_f(c_{n,t-1})$. Without loss of generality assume that $j_l < \\cdots < j_1$. If $r(j_s)=-1$, resp $+1$, then, $P_{l1}:=P_{j_l}+\\cdots+P_{j_1}$, resp $P_{1l}:=P_{j_1} + \\cdots + P_{j_l}$, contains a chain of size $l$. Thus, $\\alpha_{(1,t-1)}$ embeds in $P_{l1}$, resp $P_{1l}$. Similarly, for $\\alpha_{(0,t-1)}$ which is an $l$-element antichain, choose $s_l < \\cdots < s_1\\in J$ with $s_1 < j_l$ and $r(s_k)=0$. Such elements exist since $I_0=\\{i\\in I : r(i)=0\\}$ is coinitial in $I$. It is clear that $\\alpha_{(0,t-1)}$ embeds in $P_{s_1}\\oplus \\cdots \\oplus P_{s_l}$. Now, consider $D_{n,t-1}$. We have $D_{n,t-1}\\leq C_n$. Since the set of $m$ such that $C_n$ embeds in $C_m$ is infinite, we can find $p$ such that $\\sum (D_{n,t-1}+\\cdots+D_{n,k_n})$ embeds in $\\sum(C_{m+p}+\\cdots+C_m)$. Then set $\\varphi(t-1)=m+p$. This proves the induction step. Thus, $\\sum D_n$ embeds in $\\sum(C_{m+k}+\\cdots+C_m)$ for some $k$.  \n\nNow, let $n<\\omega$ and suppose that the poset labelled sum $\\sum(D_{n-1}+\\cdots+D_0)$ has been embedded in $\\sum(C_{\\varphi(n-1)}+\\cdots+C_0)$. By the argument above, $\\sum D_n\\leq \\sum (C_{m+k}+\\cdots+C_m)$ for some $m>\\varphi(n-1)$ and $k$. Hence, $\\sum(D_n+\\cdots+D_0)\\leq \\sum(C_{m+k}+\\cdots+C_0)$. Set $\\varphi(n)=m+k$. This proves the second inequality.\n\nFor the last inequality, since $C_{\\upharpoonright J}$ is left indecomposable, by Lemma \\ref{ncopies} we have $\\underline{2}.C_{\\upharpoonright J}\\leq C_{\\upharpoonright J}$ which implies that $\\sum \\underline{2}.C_{\\upharpoonright J}\\leq \\sum C_{\\upharpoonright J}$. \n\\end{proof} \n\n\nLemmas \\ref{Nonisoplc} and \\ref{ReducedequimorphP} imply the following. \n\n\\begin{theorem}  \\label{Reduced}\nLet P be the sum of a countable labelled chain $C:=(I,\\ell)$ where $I$ has no least element and $\\ell(i)=(P_i,r(i))\\in \\mathcal{N}_{\\leq\\omega}\\times\\{-1, 0, +1\\}$ such that r takes 0 and $\\pm 1$ densely. Then  $Sib(P)=2^{\\aleph_0}$.    \n\\end{theorem}\n\n\n\n\n\n\\subsection{Direct Sum}\n \nThroughout this subsection $P$ is a countable direct sum of at least two non-empty connected $NE$-free posets. Thus, it is disconnected.  \nIn this section we prove that $P$ has one or infinitely many siblings on condition that this property holds for each component of $P$. \n\n\\begin{lemma} \\label{Connecteddisconnected}\nIf some sibling of $P$ is  connected, then $Sib(P)=\\infty$.\n\\end{lemma}\n\n\\begin{proof}\nLet $P'\\approx P$ where $P'$ is connected. So, $P'$ embeds into some component $Q$ of $P$. Since $P$ has at least two non-empty components, $P'\\oplus 1\\hookrightarrow Q\\oplus 1\\hookrightarrow P\\hookrightarrow P'$. So, for every $n$, $P'\\oplus \\Bar{K}_n\\hookrightarrow P'$ where $\\Bar{K}_n$ is an antichain of size $n$. Since $P'$ is connected and $P'\\approx P'\\oplus \\Bar{K}_n$, $P'$ and consequently $P$ has infinitely many pairwise non-isomorphic siblings. \n\\end{proof}\n\n\n\\begin{lemma} \\label{Infinitecomponent} \nIf some component of P has infinitely many siblings, then \\\\ $Sib(P)=\\infty$. \n\\end{lemma}\n\n\\begin{proof}\nLet $P$ satisfy the conditions of the lemma. Let $Q$ be a  component of $P$ with infinitely many pairwise non-isomorphic siblings $Q_1, Q_2, Q_3, \\ldots$. For each $n$, let $P_n$ be the poset resulting from $P$ by replacing every component of $P$ which is equimorphic to $Q$ by $Q_n$. Now suppose that $P_m\\cong P_n$ for $m\\neq n$. Consider a component $Q'$ of $P_m$ equimorphic to $Q_m$. We know that $Q'$ is isomorphic to some component $Q''$ of $P_n$. We have $Q''\\cong Q'\\approx Q_m\\approx Q$ which implies that $Q''=Q_n$ because $Q''$ is a component of $P_n$. But then $Q_m\\cong Q_n$, a contradiction. \nTherefore, the resulting posets $P_n$ are pairwise non-isomorphic siblings of $P$. \n\\end{proof}\n\n\\begin{lemma} \\label{increasing}\nSuppose that P has infinitely many non-trivial components. Then $Sib(P)=\\infty$. \n\\end{lemma}\n\n\\begin{proof}\nSince $P$ has infinitely many non-trivial components and $\\mathcal{N}_{\\leq\\omega}$ is w.q.o, there is an increasing sequence $(P_n)_{n<\\omega}$  of non-trivial components of $P$. Let $\\mathcal{Q}$ be the direct sum of the non-trivial components of $P$ other than the $P_n$. We have $P=\\bigoplus_n P_n\\oplus \\mathcal{Q}\\oplus A$ where $A$ is the direct sum of the trivial components of $P$. Notice that since $P$ is countable, so is $A$. Therefore, $A$ embeds in $\\bigoplus_n P_{2n+1}$. Also $\\bigoplus_n P_n$ embeds into $\\bigoplus_n P_{2n}$. Hence, $P$ embeds into $P':=\\bigoplus_n P_n\\oplus \\mathcal{Q}$. That is, $P'\\approx P$. Similarly, $P'\\oplus \\Bar{K}_n \\approx P'\\approx P$ where $\\Bar{K}_n$ is an antichain of size $n$. Since $P'\\oplus\\Bar{K}_n$ has exactly $n$ trivial components, it follows that $P$ has infinitely many pairwise non-isomorphic siblings. \n\\end{proof}\n\nHence, by Lemmas  \\ref{Connecteddisconnected}, \\ref{Infinitecomponent} and \\ref{increasing} we have the following. \n\n\\begin{proposition} \\label{InfinitesiblingnumberofP}\nLet P be a countable direct sum of NE-free posets with at least two non-empty components. If P is a sibling of some connected NE-free poset, or some component of P has infinitely many siblings, or P has infinitely many non-trivial components, then $Sib(P)=\\infty$. \n\\end{proposition}\n\n\n\n\n\\begin{lemma} \\label{Finitenontrivial} \nIf  P has only finitely many non-trivial components and each component has only one  sibling, then $Sib(P)=1$.\n\\end{lemma}\n\n\\begin{proof}\nSet $P:=\\bigoplus_{i<l}P_i\\oplus A$ where each $P_i$ is non-trivial and $A$ is the direct sum of the trivial components of $P$. If $l=0$, then $P$ is an antichain and $Sib(P) = 1$. Assume that $l>0$ and that $P' \\subseteq P$ induces a sibling\nof $P$ via an embedding $f$. We prove that $f$ induces a bijection on the set of indices of components of $P$. First note that since the components of $P$ are connected, for each $i$, there is $j$ such that $f(P_i)\\subseteq P_j$. For each $i$, define $\\hat{f}(i)=j$ where $j$ is such that $f(P_i)\\subseteq P_j$. Suppose that for $i\\neq j$, $\\hat{f}(i)=\\hat{f}(j)=k$. It follows that $f$ embeds $P_i\\oplus P_j$ in $P_k$. We first show that $k$ cannot be $i$ or $j$. Suppose, without loss of generality, that $k=i$. Then $P_i\\oplus 1\\hookrightarrow P_i\\oplus P_j\\hookrightarrow P_i$ and by Lemma  \\ref{Connecteddisconnected}, $P_i$ has infinitely many siblings, a contradiction. So, $k\\neq i, j$. It follows that $\\hat{f}(k)\\neq k$ because otherwise $f$ embeds $P_i\\oplus P_k$ in $P_k$ which implies that $P_k$ has infinitely many siblings, a contradiction. Further, $\\hat{f}(k)\\neq i, j$ because otherwise $P_i\\oplus P_j\\approx P_k$. By a similar argument, $\\hat{f}^2(k)\\neq i, j, k, \\hat{f}(k)$. Iterating, the $P_{\\hat{f}^n(k)}$ provide infinitely many non-trivial components  of $P$, a contradiction. Thus, $\\hat{f}$ is injective on the set of indices of non-trivial components of $P$ and since there are finitely many such indices, $\\hat{f}$ is bijective. Extending $\\hat{f}$ to the set $I$ of indices of components of $P$, it follows that $\\hat{f}$ is bijective on $I$. Pick $i<l$ and consider the orbit $\\hat{f}.i$ of $i$ under $\\hat{f}$. We have $P_i\\approx P_j$ where $j\\in \\hat{f}.i$ and by assumption, $P_i\\cong P_{\\hat{f}.i}$ for each $i<l$. Hence $P'$ is isomorphic to $P$. \n\\end{proof}\n\n\nNow we are ready to conclude the following. \n\n\\begin{proposition} \\label{Directsumsiblings}\nLet $P$ be a countable direct sum of at least two non-empty connected NE-free posets. If each component of P has one or infinitely many siblings, then $Sib(P)=1$ or $\\infty$. \n\\end{proposition}\n\n\\begin{proof}\nIf $P$ has some component with infinitely many siblings or $P$ has infinitely many non-trivial components, then $Sib(P)=\\infty$ by Proposition \\ref{InfinitesiblingnumberofP}. If $P$ has only finitely many non-trivial components each of which has only one sibling, then $Sib(P)=1$ by Lemma \\ref{Finitenontrivial}. This completes the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Linear Sum} \\label{Linearsum}\n\n\nBy Theorem \\ref{Pstructure}, we know that an $NE$-free poset with more than one element which is not a direct sum or the sum of a labelled chain, is a linear sum $P$ of at least two non-empty $NE$-free posets with property CCGC. In this section we prove that such a countable linear sum $P$ of $NE$-free posets has one or infinitely many siblings on condition that this property holds for each summand of $P$. In this section, by a linear sum, we always mean a linear sum of $NE$-free posets consisting of summands with property CCGC. \n\n\\begin{lemma} \\label{Uniquesummand}\nLet $P=+_{i\\in I} P_i$ and $Q=+_{j\\in J} Q_j$ be two linear sums. Every embedding f from P to Q induces an order preserving map $\\hat{f}:I\\to J$.  Moreover, If f maps distinct summands of P into distinct summands of Q, resp f is an isomorphism, then $\\hat{f}$ is a chain embedding, resp a chain isomorphism.  \n\\end{lemma}\n\n\\begin{proof}\nWe show that for every $i\\in I$ there is $j\\in J$ such that $f(P_i)\\subseteq Q_j$. Let $i\\in I$ be given. Pick $x, y\\in P_i$. Set $G_i=CG(P_i)$. Since $G_i^c$ is connected, $x, y$ are connected in $G_i^c$ by a path $x=x_0, x_1, \\ldots, x_{n-1}=y$. This means that $x_m\\perp x_{m+1}$ in $P_i$ for every $0\\leq m\\leq n-2$. We have $f(x_m)\\perp f(x_{m+1})$ for every $0\\leq m\\leq n-2$. Now suppose that $f(x)\\in Q_j$ for some $j\\in J$. Since the summands of $Q$ are comparable to each other, $f(x_m)\\in Q_j$ for every $0\\leq m\\leq n-1$ because otherwise for two $m, m+1$, $f(x_m)$ and $f(x_{m+1})$ are comparable. This implies that $f(P_i)\\subseteq Q_j$. Now define $\\hat{f}:I\\to J$ by $\\hat{f}(i)=j$ where $j\\in J$ is such that $f(P_i)\\subseteq Q_j$. The mapping $\\hat{f}$ is well-defined by above argument. If $i \\leq_I i'$, then $P_i \\leq_P P_{i'}$ which implies that $f(P_i)\\leq_Q f(P_{i'})$ because $f$ is order-preserving. Consequently, $Q_j \\leq_Q Q_{j'}$ where $f(P_i)\\subseteq Q_j$ and $f(P_{i'})\\subseteq Q_{j'}$. Hence, $\\hat{f}(i) \\leq_J \\hat{f}(i')$. Moreover, if $f$ does not embed two distinct summands of $P$ into the same summand of $Q$, then $\\hat{f}$ is injective meaning that it is a chain embedding. In particular, if $f$ is an isomorphism, then so is $f^{-1}$ and for every $i$ there is $j$ such that $f(P_i)\\subseteq Q_j$ and $f^{-1}(Q_j)\\subseteq P_i$ which imply that $f(P_i)=Q_j$. Thus, $\\hat{f}$ defined as above is a chain isomorphism. \n\\end{proof}\n\n\n\n\\begin{lemma} \\label{InfiniteCproperty}\nLet P be a linear sum. Suppose that some summand of P has infinitely many pairwise non-isomorphic siblings with property CCGC. Then P has infinitely many siblings.\n\\end{lemma} \n\n\\begin{proof}\nSet $P=+_{i\\in I} P_i$.  Let $P_j$ be a summand  of $P$ such that  $\\{P_{jn}\\}_{n<\\omega}$ is an infinite family of pairwise non-isomorphic siblings of $P_j$ with property CCGC. Let $J=\\{i\\in I : P_i\\approx P_j\\}$. For each $n<\\omega$, set $P^n:=+_{i\\in I} Q_i^n$ where $Q_i^n=P_i$ if $i\\notin J$ and $Q_i^n=P_{jn}$ if $i\\in J$. Notice that for each $i\\in I$, $Q_i^n$ has property CCGC. Taking each summand $P_i$ to the summand $Q_i^n$ and vice versa, it is clear that $P^n\\approx P$ for every $n<\\omega$. Now suppose that $P^m\\cong P^n$ for $m\\neq n$ and let $f:P^m\\to P^n$ be an isomorphism. Since $f$ is an isomorphism and both $P^m, P^n$ are linear sums whose summands have property CCGC, by Lemma \\ref{Uniquesummand}, $f$ induces a chain isomorphism $\\hat{f}:I\\to I$ and for each $i\\in I$, $f(Q_i^m)= Q^n_{\\hat{f}(i)}$. More, we have $Q_i^m\\cong Q_{\\hat{f}(i)}^n$. Let $i\\in J$. Then $P_{jm}= Q_i^m \\cong Q_{\\hat{f}(i)}^n = P_{jn}$, a contradiction. The last equality is due to the fact that $Q^n_{\\hat{f}(i)}\\approx P_{jm}\\approx P_j$. This implies that the $P^n$ are pairwise non-isomorphic. Hence, $Sib(P)=\\infty$. \n\\end{proof}\n\nThe following proposition provides sufficient conditions to obtain infinitely many siblings for a countable linear sum. \n\n\\begin{proposition} \\label{InfiniteDproperty}\nLet P be a countable linear sum. If some summand $P_j$ of P is the sum of a labelled chain with no least element; or $P_j$ is disconnected and has infinitely many siblings with property CCGC; or $P_j$ is disconnected and has at least one connected sibling, then P has infinitely many siblings. \n\\end{proposition}\n\n\\begin{proof}\nBy Proposition \\ref{Plsccgc} the sum of a countable labelled chain with no least element has property CCGC and by Theorem \\ref{Reduced} it has continuum many siblings with the same properties. If $P$ contains a summand $P_j$ which is the sum of a labelled chain with no least element or $P_j$ is disconnected having  infinitely many siblings with property CCGC, then the statement is true by Lemma \\ref{InfiniteCproperty}.\n\nAssume that there is some  \ndisconnected summand $P_j$ of $P$ which is equimorphic to some connected $NE$-free poset $Q$. Then $Q\\approx P_j'$ for some connected component $P_j'$ of $P_j$. Consequently, the $\\Bar{K}_n\\oplus Q$, $n<\\omega$, where $\\Bar{K}_n$ is an antichain of size $n$, are pairwise non-isomorphic siblings of $Q$ and thus of $P_j$ which have property CCGC. Hence, in this case $P$ has infinitely many siblings by Lemma \\ref{InfiniteCproperty}. \n\\end{proof}\n\nLet $P$ be a countable linear sum with more than one element such that all its summands have only one sibling. Then $P$ does not contain a summand which is the sum of a labelled chain with no least element. That is, each summand of $P$ is either a singleton or a disconnected $NE$-free poset. Obviously, $P$ does not embed in a singleton and if $P$ embeds in a disconnected summand $P_j$, then $P_j$ has infinitely many siblings by Lemma \\ref{Connecteddisconnected} which contradicts our assumption. In other words, each summand of $P$ strictly embeds into $P$. In case a linear sum has only finitely many non-trivial summands having only one sibling, we show that $P$ has one or infinitely many siblings. However, in light of the following result, we prove a stronger statement.\n\n\\begin{theorem} [\\cite{LPW},  Corollary 3.6] \\label{Dichchain}\nIf C is a countable chain, then $Sib(C)=1$, $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{theorem}\n\n\n\\begin{lemma} \\label{Finitesummand}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum with only finitely many non-trivial summands each of which has only one sibling. Then $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $f$ be an embedding of $P$ and $\\hat{f}$ be defined as in Lemma \\ref{Uniquesummand}. Let $K$ be the chain of indices $i\\in I$ such that $P_i$ is  non-trivial. If $K=\\emptyset$, then $P=I$  and there is nothing to prove (the statement holds for a countable chain by Theorem \\ref{Dichchain}). Assume that $K\\neq\\emptyset$. We know that $\\hat{f}$ is order-preserving. We show that $\\hat{f}$ is injective. For the sake of a contradiction, assume that $\\hat{f}(i)=\\hat{f}(j)=k$ for some $i < j\\in I$. It implies that $k\\in K$. Since $P_k$ is non-trivial, $\\hat{f}(k), \\hat{f}^2(k), \\ldots\\in K$. Since $K$ is a finite chain and $\\hat{f}$ is order-preserving, we conclude that for some $l\\in K$ and some integer $m$, $\\hat{f}^m(k)=\\hat{f}^{m+1}(k)=l$. Without loss of generality, assume that $i\\leq_I l$. Let $\\Bar{I}=\\{i'\\in I : i\\leq_I i' \\leq_I l\\}$. We have $\\hat{f}^{m+1}(\\Bar{I})=l$. Now consider $P'=+_{i' \\in \\Bar{I}} P_{i'}$. From $\\hat{f}^{m+1}(\\Bar{I})=l$, we conclude that $f^{m+1}$ embeds $P'$ into $P_l$. Clearly, $P_l$ embeds into $P'$. That is $P'\\approx P_l$. Since $l\\in K$, $P_l$ is disconnected. Also, $P_l$ is equimorphic to a connected $NE$-free poset $P'$ which implies that $Sib(P_l)=\\infty$ by Lemma \\ref{Connecteddisconnected}, a contradiction. Therefore, $\\hat{f}$ is a chain embedding of $I$. Further, since $K$ is finite, $\\hat{f}$ is the identity on $K$. Consequently, $f(P_k)\\subseteq P_k$ for each $k\\in K$.\n\nSet $K:=\\underline{n}$. We may represent $P$ as $P=I_0+P_0+\\cdots+P_{n-1}+I_n$ where the $P_k$ are the non-trivial summands of $P$, $I_0=\\{x\\in P : \\{x\\} <_P P_0\\}$,  $I_k=\\{x\\in P : P_{k-1} <_P \\{x\\} <_P P_k\\}$ for every $1\\leq k\\leq n-1$ and $I_n=\\{x\\in P : P_{n-1} <_P \\{x\\}\\}$. Notice that each $I_k$ is a countable chain. By the above argument, each sibling of $P$ is of the form $Q=J_0+Q_0+\\cdots+Q_{n-1}+J_n$ where $J_k\\approx I_k$, $k\\in\\underline{n+1}$, and $Q_k\\approx P_k$, $k\\in\\underline{n}$.  Since $Sib(P_k)=1$, we have $Q_k\\cong P_k$. Moreover, each countable chain has one or countably many or continuum many siblings by Theorem \\ref{Dichchain}.  Hence, $Sib(P)=\\max\\{Sib(I_k) : k\\in\\underline{n+1}\\}$. It follows that $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{proof}\n\n\n\n\nIt remains to verify the case in which a linear sum contains infinitely many non-trivial summands, each having only one sibling. However, we will show that when a linear sum has infinitely many non-trivial summands, it has continuum many siblings, regardless of the sibling number of its summands. \nWe may consider a linear sum $P=+_{i\\in I}P_i$ as the sum $\\sum C$ of a labelled chain $C=(I,\\ell)$ where $\\ell:I\\to \\mathcal{N}_{\\leq\\omega}\\times \\{-1\\}$ is defined by $\\ell(i)=(P_i,r(i))$ and $r(i)=-1$ for each $i\\in I$. Since the map $r$ is constant, we may abuse the notation and write $\\ell(i)=P_i$. Note that the labels belong to a b.q.o. We denote by $K$ the chain of $i\\in I$ such that $P_i$ is non-trivial. Our proof contains two main ingredients: Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}. \n\nLet $n\\geq 1$ and consider the chains $\\underline{n}$ and $\\underline{n+2}$. We denote by $A_n$, resp $B_n$, the poset obtained by replacing each $m\\in \\underline{n}$, resp $0 < m < n+1$, with a two-element antichain. Note that each $A_n$ and $B_n$ is a linear sum whose summands have property CCGC with context chains $\\underline{n}$ and $\\underline{n+2}$, respectively.  \nTwo elements $i$ and $j$ of $K$ are equivalent, denoted by $i\\equiv_K j$, just in case every element of the interval in $I$ with endpoints $i$ and $j$ is in $K$. It is clear that $\\equiv_K$ is an equivalence relation. We denote by $\\mathcal{K}$ the set of equivalence classes of $K$. The equivalence classes of $K$ are maximal intervals of $I$ that are contained in $K$. \n\n\\begin{lemma} \\label{ConstructionLS}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum such that K is coinitial in I and without least element. Then, there are $2^{\\aleph_0}$ pairwise non-isomorphic linear sums $P_f=\\sum C_f$ where $f\\in \\{0,1\\}^\\mathbb{N}$. \n\\end{lemma}\n\n\\begin{proof}\nIt follows that $I$ has no least element. Set $C:=(I,\\ell)$ where $\\ell(i)=P_i$. \nRecall that in this case, by Lemma \\ref{Labelledchainsproperties} (1), there is an initial segment $J$ of $I$ such that $C_{\\upharpoonright J}$ is left indecomposable. For each $K'\\subset J$ with $K'\\in\\mathcal{K}$, if $|K'|=2$ or $|K'|=4$, then insert a new element $d$ such that $d$ covers the greatest element of $K'$ and set $\\ell'(d):=A_1$. Thus, $C_{\\upharpoonright K'}$ is replaced with a  labelled chain which is isomorphic to $A_3$ or $A_5$. Let $Y$ be the collection of all these new elements $d$. Set $J':=J\\cup Y$.   \nWe select an infinite descending sequence $(a_n)_{n<\\omega}$ of elements of $K\\cap J$ coinitial in $J$. To each map $f:\\mathbb{N}\\to\\{0,1\\}$ we associate a  labelled chain $C_f$ as follows. Let $f:\\mathbb{N}\\to\\{0,1\\}$. For $i\\in I\\setminus Y$ set $\\ell_f(i):=\\ell(i)$ and for $i\\in Y$ set $\\ell_f(i):=\\ell'(i)$.  For each $n<\\omega$, set $k(n):=2f(n)+4$ and insert elements $b_{1,n} < \\cdots < b_{k(n),n}$ such that $b_{1,n}$ covers $a_n$ and $b_{i+1,n}$ covers $b_{i,n}$ for each $1\\leq i\\leq k(n)-1$. Set $\\ell_f(b_{1,n}):=b_{1,n}$, $\\ell_f(b_{k(n),n}):=b_{k(n),n}$ and $\\ell_f(b_{i,n}):=A_1$ where $1<i<k(n)$. Indeed, the chain $b_{1,n}<\\cdots<b_{k(n),n}$ is the context chain of $B_{2f(n)+2}$. Let $b_n$ be the chain $b_{1,n}<\\cdots<b_{k(n),n}$. Then, $(b_n)_{n<\\omega}$ is a sequence of chains such that $b_{n+1}<b_n$.  Set $X:=\\bigcup_{n<\\omega} b_n$ which is coinitial in $J$. Set $\\Bar{J}:=J'\\cup X$, $F:=I\\setminus \\Bar{J}$ and $\\Bar{I}:=F\\cup \\Bar{J}$. Then, $C_f:=(\\Bar{I},\\ell_f)$ is a labelled chain. Thus, we obtain a sum $\\sum C_f$ which is a linear sum whose summands have property CCGC. That is, $\\sum C_f=+_{i\\in\\Bar{I}}P_i$ such that $K$ has no least element and $K$ is coinitial in $\\Bar{I}$.\n\nSuppose that $f, g$ are two maps from $\\mathbb{N}$ to $\\{0,1\\}$ such that $\\sum C_f\\cong \\sum C_g$ by some isomorphism $\\phi$. Since the summands of these linear sums have property CCGC,  by Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $\\Bar{I}$ preserving the labels, that is $\\ell_f(i)=\\ell_g(h(i))$ for each $i\\in \\Bar{I}$. If for each $n<\\omega$, $h(b_n)\\subset F$, then $h(\\Bar{I})\\subseteq F$ because $X$ is coinitial in $\\Bar{I}$. But this means that $h$ is not surjective, a contradiction. Therefore, $h(b_n)\\cap \\Bar{J}\\neq\\emptyset$ for some $n$. Since each $b_n$ is the context chain of $B_2$ or $B_4$ and since there is no  labelled chain $C_{\\upharpoonright K'}$, $K'\\in\\mathcal{K}$, in $C_{g\\upharpoonright\\Bar{J}}$ which is isomorphic to $A_2$ or $A_4$, it follows that $h(b_n)=b_m$ for some $m$ and consequently, for each $k\\geq 1$, $h(b_{n+k})=b_{m+k}$. Thus, $h$ may be regarded as an isomorphism from a final segment $D$ of $\\mathbb{N}$ onto another $D'$ such that $g(h(n))=f(n)$ for each $n\\in D$. By Lemma \\ref{Continuummaps}, there are $2^{\\aleph_0}$ maps in $\\{0,1\\}^\\mathbb{N}$ such that for two distinct maps $f$ and $g$, we have $\\sum C_f\\ncong \\sum C_g$. For each $f\\in\\{0,1\\}^\\mathbb{N}$, set $P_f:=\\sum C_f$. Then, the set $\\{P_f\\}_{f\\in\\{0,1\\}^\\mathbb{N}}$ has the desired properties. \n\\end{proof}\n\n\nIn lemma \\ref{ConstructionLS}, we observe that the order on $P_f:=\\sum C_f$ extends both $\\sum C_{f\\upharpoonright\\Bar{J}}$ and $\\sum C_{\\upharpoonright F}$ as follows: if $x\\in\\sum C_{f\\upharpoonright\\Bar{J}}$ and $y\\in \\sum C_{\\upharpoonright F}$, then $x\\in P_i$ and $y\\in P_j$ for some $i\\in \\Bar{J}$ and $j\\in F$. We have $i<j$ and since $r(i)=-1$, it follows that $x <_{P_f} y$. Therefore,  $\\sum C_f =\\sum C_{f\\upharpoonright\\Bar{J}}+\\sum C_{\\upharpoonright F}$.\n\n\n\nIn order to show that the $\\sum C_f$ in Lemma \\ref{ConstructionLS} are embeddable into $\\sum C$, first we prove the following claim.\n\n\\begin{claim} \\label{J'embedsJ}\nWe have $C_{\\upharpoonright J'}\\leq C_{\\upharpoonright J}$. Moreover, $C_{\\upharpoonright J'}$ is left indecomposable.\n\\end{claim}\n\n\\begin{proof}\nWe show that the following inequalities hold.\n$$ C_{\\upharpoonright J'}\\leq  \\underline{2}.C_{\\upharpoonright J}\\leq  C_{\\upharpoonright J}.$$\n\nFor the first inequality, recall $Y$ in Lemma \\ref{ConstructionLS}  which is countable. Let $d_0, \\ldots, d_n, \\ldots$ be an enumeration of $Y$. For each $n$, let $c_n\\in J$ be such that $d_n$ covers $c_n$. Therefore, $c_n\\in K$. Define an embedding $\\varphi:C_{\\upharpoonright J'}\\to\\underline{2}.C_{\\upharpoonright J}$ as follows. Define $f:J'\\to\\underline{2}.J$ by $f(j)=(0,j)$ if $j\\in J$ and $f(d_n)=(1,c_n)$ for each $n$. Since $\\ell'(d_n)$ is a 2-element antichain and $\\ell((1,c_n))=\\ell(c_n)$ is a direct sum of at least two non-empty posets, it follows that $\\ell'(d_n)$ embeds in $\\ell(f(d_n))$. Therefore, $C_{\\upharpoonright J'}\\leq \\underline{2}.C_{\\upharpoonright J}$. \n\nThe second inequality is because $\\underline{2}.C_{\\upharpoonright J}\\leq C_{\\upharpoonright J}$, a fact which follows from Lemma \\ref{ncopies} since $C_{\\upharpoonright J}$ is left indecomposable.  \n\nWe know that $C_{\\upharpoonright J}$ is left indecomposable. Let $L'+R'$ be a partition of $J'$ into an initial and a final segment and set $L:=L'\\setminus Y$ and $R:=R'\\setminus Y$. Then $L+R$ is a partition of $J$ into an initial and a final segment. By the inequality above and the fact that $C_{\\upharpoonright J}$ is left indecomposable, we have $C_{\\upharpoonright J'}\\leq C_{\\upharpoonright J}\\leq C_{\\upharpoonright L}\\leq C_{\\upharpoonright L'}$. Thus, $C_{\\upharpoonright J'}$ is left indecomposable. \n\\end{proof}\n\n\n\n\\begin{lemma} \\label{Noleastlinearcontinuum}\nLet $P=+_{i\\in I}P_i$ be a countable linear sum such that K is coinitial in I and has no least element. Then $Sib(P)=2^{\\aleph_0}$. \n\\end{lemma}\n\n\\begin{proof}\nLet $f:\\mathbb{N}\\to\\{0,1\\}$ and $C_f$ be the labelled chain constructed in Lemma \\ref{ConstructionLS}. We prove that $\\sum C_f\\leq \\sum C$. We have $C_f= C_{f\\upharpoonright \\Bar{J}}+C_{\\upharpoonright F}$ because $f$ does not change $F$. By the observation given after Lemma \\ref{ConstructionLS} we have $\\sum C_f =\\sum C_{f\\upharpoonright\\Bar{J}}+\\sum C_{\\upharpoonright F}$. It suffices to prove that $\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum C_{\\upharpoonright J}$.  We prove that \n$\\sum C_{f\\upharpoonright \\Bar{J}}\\leq \\sum C_{\\upharpoonright J'}$ because then the result follows from Claim \\ref{J'embedsJ}. \n\nBy Claim \\ref{J'embedsJ}, $C_{\\upharpoonright J'}$ is left indecomposable, so, we can write it as $\\sum^*_{n<\\omega} C_n$ where each $C_n$ is indecomposable and the set of $m$ such that $C_n\\leq C_m$ is infinite. \nFor each $n$, let $J_n$ be the domain of $C_n$. Set $L_n:=J_n\\cup\\bigcup_{a_m\\in J_n} b_m$ and $D_n:=(L_n,\\ell_{f\\upharpoonright L_n})$. Then we have $C_{f\\upharpoonright \\Bar{J}}=\\sum^*_{n<\\omega} D_n$. We define an embedding from $\\sum C_{f\\upharpoonright \\Bar{J}}$ into $\\sum C_{\\upharpoonright J'}$ by induction. Let $n$ be given. Suppose that $\\sum(D_{n-1}+\\cdots+D_0)$ has been embedded into $\\sum(C_{\\varphi(n-1)}+\\cdots+C_0)$. The labelled chain $D_n$ consists of $C_n$ plus finitely many elements, each labelled by a singleton or a 2-element antichain. Thus, $D_n$ can be written\n$D_{n,0}+\\alpha_{n,0}+D_{n,1}+\\cdots+\\alpha_{n,k_n-1}+D_{n,k_n}$\nwith $D_{n,0}+D_{n,1}+\\cdots+D_{n,k_n}=C_n$ and each $\\alpha_{n,i}$ is a finite chain whose elements are labelled by either a singleton or an antichain of size 2. Suppose that $\\sum(D_{n,i}+\\alpha_{n,i}+\\cdots+\\alpha_{n,k_n-1}+D_{n,k_n})$ has been embedded into $\\sum(C_{m+k}+\\cdots+C_m)$ for some $m > \\varphi(n-1)$ and some $k$. We know that the sequence $(a_n)_{n<\\omega}$ is coinitial in $J'$ and the $P_{a_n}$ are non-trivial. Pick a chain $a_{n_i}<\\cdots<a_{n_0}< J_{m+k}$ of elements of $(a_n)_{n<\\omega}$ such that its length equals to the size of the context chain of $\\alpha_{n,i-1}$.  Then, $\\alpha_{n,i-1}$ embeds in $P_{a_{n_i}}+\\cdots+P_{a_{n_0}}$.  We have $D_{n,i-1}\\leq C_n$. Further, $C_n$ embeds in infinitely many $C_m$. Therefore, we can find sufficiently large $l$ such that $J_l < a_{n_i}$ and $D_{n,i-1}$ embeds in $C_l$. Hence, $\\sum D_n$ embeds in $\\sum(C_{m+p}+\\cdots+C_m)$ for some $p$. Then by setting $\\varphi(n)=m+p$ we have $\\sum(D_n+\\cdots+D_0)\\leq \\sum(C_{\\varphi(n)}+\\cdots+C_0)$. This completes the induction step as well as the proof.\n\\end{proof}\n\nLet $P$ be a poset. It is easy to verify that $P$ and $P^*$ have the same sibling number.  With this observation, if in a countable linear sum $P=+_{i\\in I} P_i$, the chain $K$ has no greatest element and it is cofinal in $I$, then we may consider $P^*$ in which $K^*$ has no least element and it is coinitial in $I^*$. Then, by applying Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, we get continuum many siblings for $P^*$ and thus for $P$. \n\nIn both Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, $K$ has no least (greatest) element and it is coinitial (cofinal) in $I$. This is not necessarily the case in a general linear sum. However, the following lemma implies that we can always find some interval $J$ of $I$ in which $K$ is coinitial (cofinal) in $J$ with no least (greatest) element.  \n\n\\begin{lemma} \\label{Interval}\nLet I be an infinite chain. Then there exists an interval J of I such that J has no least or greatest element. \n\\end{lemma}\n\n\\begin{proof}\nFor the sake of a contradiction, assume that for each interval $J$ of $I$, $J$ has least and greatest element. Set $I_0:=I$ and let $x_0$, resp $y_0$, be the least, greatest, element of $I_0$. For $n\\geq 1$, set $I_{n+1}:=I_n\\setminus\\{x_n,y_n\\}$ and let $x_{n+1}$, resp $y_{n+1}$, be the least, resp greatest, element of $I_{n+1}$. Since $I$ is infinite, $I_n\\neq\\emptyset$ for every $n<\\omega$. Then, the initial interval $J:=\\{x\\in I : \\exists n, x < x_n\\}$ has no greatest element, a contradiction. \n\\end{proof}\n\n\n\n\\begin{proposition} \\label{Infinitesummand}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum with infinitely many non-trivial summands. Then, P has continuum many siblings. \n\\end{proposition}\n\n\\begin{proof}\nLet $K$ be the chain of indices $i\\in I$ such that $P_i$ is non-trivial. By assumption, $K$ is countably infinite. Let $I_1$, resp $I_2$, be the maximal interval of $I$ such that $I_1<_I K$, resp $K<_I I_2$. Then $I=I_1+\\hat{I}+I_2$ where $\\hat{I}$ is the minimal interval of $I$ containing $K$. By Lemma \\ref{Interval}, without loss of generality, there exists a final interval $J$ of $\\hat{I}$ and a final interval $\\hat{K}$ of $K$ such that $\\hat{K}$ has no least element and it is coinitial in $J$. Thus, if we set \n$$J:=\\{i\\in\\hat{I} : \\exists (k_n)_{n<\\omega},\\ \\forall n,\\  k_n\\in K\\ \\text{and}\\ k_{n+1} < k_n < i\\},$$\nthen $J\\neq\\emptyset$ and it is maximal among all final intervals of $\\hat{I}$ in which there is a descending sequence of elements of $K$ with no least element. Set $L:=\\hat{I}\\setminus J$. Then, $L\\cap K$ is well-ordered because otherwise there is a descending sequence of elements of $K$ in $L$ which contradicts the maximality of $J$. We have $\\hat{I}=L+J$ and also $I=I_1+L+J+I_2$. By Lemmas \\ref{ConstructionLS} and \\ref{Noleastlinearcontinuum}, the linear sum $Q:=+_{j\\in J} P_j$ has $2^{\\aleph_0}$ pairwise non-isomorphic siblings $\\{Q_\\alpha\\}_{\\alpha < \\mathfrak{c}}$ where $\\mathfrak{c}$ is the continuum. For each $\\alpha<\\mathfrak{c}$ set $\\hat{Q}_\\alpha:=Q_\\alpha+(+_{i\\in I_2} P_i)$. Suppose that $\\hat{Q}_\\alpha\\cong \\hat{Q}_\\beta$ for two $\\alpha\\neq \\beta$ by some isomorphism $\\phi$. Since the $\\hat{Q}_\\alpha$ are linear sums whose summands have property CCGC, by Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $J+I_2$. If for some $x\\in J$, $h(x)\\in I_2$, then $x\\notin K$. Since by assumption there is some $y\\in J\\cap K$ with $x<_I y$, we have $h(x)<_I h(y)$. But then $h(y)$ must be the index of a non-trivial summand in $+_{i\\in I_2} P_i$ which is not possible. Therefore, $h(J)=J$ meaning that $Q_\\alpha\\cong Q_\\beta$, a contradiction. Hence, the $\\hat{Q}_\\alpha$ are pairwise non-isomorphic. Next, for each $\\alpha<\\mathfrak{c}$ set $\\hat{P}_\\alpha:=(+_{i\\in L}P_i)+\\hat{Q}_\\alpha$. Suppose that $\\hat{P}_\\alpha\\cong \\hat{P}_\\beta$ for two $\\alpha\\neq\\beta$ by an isomorphism $\\phi$. By Lemma \\ref{Uniquesummand}, $\\phi$ induces an isomorphism $h$ on $L+J+I_2$. Suppose that for some $x\\in J$, $h(x)\\in L$. By definition of $J$, select a descending sequence $(k_n)_{n<\\omega}$ of elements of $K$ which has no least element in $J$ and $k_n < x$ for each $n$. Then $(h(k_n))_{n<\\omega}$ is a coinitial sequence of elements of $K$ in $L$ with no least element. But this is impossible because $L\\cap K$ is well-ordered. Thus, $h(J+I_2)=J+I_2$ meaning that $\\hat{Q}_\\alpha\\cong \\hat{Q}_\\beta$, a contradiction. Finally, for each $\\alpha<\\mathfrak{c}$ set $P_\\alpha:=(+_{i\\in I_1} P_i)+\\hat{P}_\\alpha$. By an argument similar to what we gave for the $\\hat{Q}_\\alpha$, it is proven that the $P_\\alpha$ are pairwise non-isomorphic siblings of $P$. \n\\end{proof}\n\n\n\n\nNow we are ready to conclude the main result of this section. \n\n\\begin{proposition} \\label{Linearsumsiblings}\nLet P be a countable linear sum of at least two non-empty NE-free posets with property CCGC. If each summand has one or infinitely many siblings, then, so does P. \n\\end{proposition}\n\n\\begin{proof}\nLet $P=+_{i\\in I} P_i$ be a countable linear sum. \n\n\\noindent\\textbf{Case 1} Some summand $P_j$ has infinitely many siblings. Then either $P_j$ is the sum of a  labelled chain with no least element or $P_j$ is disconnected and has infinitely many siblings with property CCGC. In this case $Sib(P)=\\infty$ by Proposition \\ref{InfiniteDproperty}.  \n\n\\noindent\\textbf{Case 2} All summands of $P$ have only one sibling. \n\n\\noindent\\textbf{Subcase 2.1} $P$ has only finitely many non-trivial summands. In this case $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$ by Lemma \\ref{Finitesummand}. \n\n\\noindent\\textbf{Subcase 2.2} $P$ has infinitely many non-trivial summands. In this case $Sib(P)=2^{\\aleph_0}$ by Proposition \\ref{Infinitesummand}. \n\\end{proof}\n\n\n\\subsection{Proof of Theorem \\ref{AltThomasseNEP}}\n\nIn this short section, we prove that the alternate Thomass\\'{e} conjecture holds for a countable $NE$-free poset $P$, that is $P$ has one or infinitely many siblings.\n\n\\begin{proof} (of Theorem \\ref{AltThomasseNEP})\nLet $P$ be a countable $NE$-free poset. Note that the class of countable $NE$-free posets is w.q.o under embeddability and thus well-founded. Therefore, we can use induction. For each countable $NE$-free poset $Q$, let $\\mathcal{S}(Q)$ be the statement ``$Sib(Q)=1$ or $\\infty$\". The statement is true for a singleton. Let $P$ have more than one element and assume that $\\mathcal{S}(Q)$ holds for each $Q$ strictly embedding in $P$. By Theorem \\ref{Pstructure}, $P$ is either (1) a direct sum of at least two non-empty connected $NE$-free posets, or (2) a linear sum of at least two non-empty $NE$-free posets with property CCGC, or (3) the sum $\\sum C$ of a  labelled chain $C=(I,\\ell)$ with no least element such that each $\\ell(i)$ is the pair $(P_i,r(i))$ made of an $NE$-free poset $P_i$ and  $r(i)\\in\\{-1, 0, +1\\}$ where $r$ takes 0 and $\\pm 1$ densely. For case (3), by Theorem \\ref{Reduced} we have $Sib(P)=2^{\\aleph_0}$. Assume that $P$ satisfies (1), resp (2). Then, $P$ is disconnected, resp connected. If $P$ embeds in some of its components, resp disconnected summands, then $P$ is equimorphic to a connected, resp disconnected, $NE$-free poset and $Sib(P)=\\infty$ by Lemma \\ref{Connecteddisconnected}. Otherwise, each component, resp summand, of $P$ strictly embeds in $P$. By induction hypothesis and Proposition \\ref{Directsumsiblings}, resp Proposition \\ref{Linearsumsiblings}, $\\mathcal{S}(P)$ holds. \n\\end{proof} \n\n\n\n\\section{Finite Poset Substitution of Chains or Antichains}  \\label{Finiteposetsub}\n\n\nIn this section, we prove that a countable $NE$-free poset whose comparability graph has one sibling, is a finite poset substitution of chains or antichains and that it has one, countably many or else continuum many siblings. Moreover, the comparability graph of $NE$-free posets of this section is a finite graph substitution of cliques or independent sets. Thus, the result of this section is a generalisation of Thomass\\'e's conjecture for countable chains whose comparability graph is a clique. \n\n\nThe following theorem proven in \\cite{HPW} describes the structure of countable cographs $G$ with only one sibling which will be used in determining the structure of a poset $P$ with $CG(P)=G$.  \n\n\\begin{theorem} [\\cite{HPW}] \\label{Lexico}\nLet $G$ be a countable cograph. $G$ has only one sibling if and only if $G$ is a finite graph substitution of cliques or independent sets.\n\\end{theorem}  \n\nLet $P$ be a countable $NE$-free poset such that its comparability graph $G$ has one sibling. Then, by Theorem \\ref{Lexico}, $G=K[H_v\/v : v\\in K]$ where $K$ is a finite graph and each  $H_v$ is a clique or an independent set. We know that $G$ is a cograph, thus,\nby Proposition \\ref{Subcograph}, so is $K$. Throughout this section let $T:=T(K)$ be the decomposition tree  of $K$. For every $M\\in T$, denote by $P(M)$ the poset obtained by restricting $P$ to the graph blocks $H_v$ of $G$ where $v\\in M$. For every leaf $\\{u\\}$ of $T$ we write $P_u$ instead of $P(\\{u\\})$. We verify the following statement by induction on the vertices of $T$. For every $M\\in T$, let \n\\begin{center} \n     $\\mathcal{S}_{fps}(M)$ : $P(M)$ is a finite poset substitution of chains or antichains. \n\\end{center}\nThe statement $\\mathcal{S}_{fps}(M)$  also provides a structure of the poset $P(M)$ which will be used in the proof of the following statement: \n\\begin{center}\n    $\\mathcal{S}_{sib}(M)$ : $Sib(P(M))=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.\n\\end{center}\nSince $T$ is finite, this proves $\\mathcal{S}_{fps}(M)$ and $\\mathcal{S}_{sib}(M)$ for all $M\\in T$. In particular, $P$ has one, countably many or continuum many siblings. \n\n\n \nWe say that a node $M$ of $T$ is \\textit{edge minimal} if $M$ does not have a module $\\{u, v\\}$ of $K$ satisfying the following:   \n\\begin{enumerate}\n    \\item $u$ and $v$ are adjacent and both $H_u$ and $H_v$ are cliques;\n    \\item $u$ and $v$ are non-adjacent and both $H_u$ and $H_v$ are independent sets.  \n\\end{enumerate} \nWe call $T$ \\textit{leaf minimal} if every node of $T$ is edge minimal. It follows that if $M\\in T$ is edge minimal, then for every $u, v\\in M$ with property (1) or (2), $\\{u,v\\}$ is not a module of $K$, thus, there exists some $w\\in K$ adjacent to one and only one of $u$ or $v$. Since $M$ is a module of $K$, we conclude that $w\\in M$ meaning that $\\{u,v\\}$ is not a module of $M$. Therefore, $M$ does not have two children $\\{u\\}, \\{v\\}$ as leaves of $T$ such that $u, v$ satisfy (1) or (2). \nWe say that the context graph $K$ of $G=K[H_v\/v : v\\in K]$ where the graph blocks are cliques or independent sets, is the \\textit{smallest} when $G$ cannot be represented as $K'[H'_v\/v : v\\in K']$ where $K'$ is a graph,  $|K'|<|K|$ and each $H'_v$ is a clique or an independent set. By the following lemma, it turns out that the smallest possible context graph $K$ for $G$ implies that $T$ is leaf minimal. \n\n\n\\begin{lemma} \\label{Kminimal} \nLet $G=K[H_v\/v : v\\in K]$ where $K$ is a finite cograph and each $H_v$ is either a clique or an independent set and $T$ be the decomposition tree of $K$. If $K$ is the smallest, then $T$ is leaf minimal. \n\\end{lemma}\n\n\\begin{proof} \nSuppose $T$ is not leaf minimal. Then some vertex $M\\in T$ is not edge minimal. Let $u, v\\in M$ form a module such that either they are adjacent and both $H_u, H_v$ are cliques or they do not form an edge and both $H_u, H_v$ are independent sets. In either case let $H_a$ be the graph obtained by restricting $G$ to the graph blocks $H_u$ and $H_v$. We know that in the first, resp second, case, $H_a$ is a clique, resp an independent set. Also, since $N:=\\{u, v\\}$ is a module, for $w\\in K\\setminus N$,  we have $uw\\in E(K)$ if and only if $vw\\in E(K)$. Let $K'$ be a graph whose vertex set is $V':=(K\\setminus N)\\cup \\{a\\}$ and its edge set is obtained by replacing all $wu, wv\\in E(K)$ with $wa$ where $w\\in K\\setminus N$. Let $x\\in G$. Then $x\\in H_w$ for some $w\\in K$. If $w\\in K\\setminus N$, then $w\\in V'$ and $x\\in H_w$; if $w=u$ or $w=v$, then $x\\in H_a$ and $a\\in V'$. Now, let $xy\\in E(G)$. Then either $x, y\\in H_w$, $w\\in K$, and $xy\\in E(H_w)$; or $x\\in H_w$, $y\\in H_{w'}$, $w, w'\\in K$ and $ww'\\in E(K)$. In the first case, if $w\\in K\\setminus N$, then $w\\in V'$ and $xy\\in E(H_w)$; and if $w=u$ ($w=v$), then $H_u$ ($H_v$) is a clique and we have $xy\\in E(H_a)$. In the second case when $w, w'\\in K\\setminus N$, then $w, w'\\in V'$ and $ww'\\in E(K')$; if $w\\in K\\setminus N$ and $w'\\in N$, then $wa\\in E(K')$; and finally if $w,w'\\in N$, then $x, y \\in H_a$ and $uv\\in E(K)$ which means that $xy\\in E(H_a)$.  Hence, $G=K'[H'_v\/v : v\\in K']$. Since $K'$ is a graph, $|K'|<|K|$ and each $H_w$, $w\\in K'$, is a clique or independent set, we get a contradiction because $K$ is the smallest context graph of $G$.     \n\\end{proof}     \n\n\nWe say that a node $M\\in T$ is \\textit{gs-connected}, resp \\textit{gs-disconnected}, if  $G(M)=M[H_v\/v : v\\in M]$ is connected, resp disconnected. Note that if $v(M)=1$, then each pair $\\{u, v\\}$ of elements of $M$ forms an edge meaning that $M$ is gs-connected. Similarly, if $v(M)=0$, then $M$ contains two non-adjacent elements $u, v$ meaning that $M$ is gs-disconnected.  So, when $M$ is gs-disconnected, then $P(M)$ has at least two components. \nWe will use the edge minimality of the nodes of $T$ to argue by induction on those vertices. \n\n\n\n\\begin{lemma} \\label{Disconnectedmodule}\nLet $M\\in T$ be such that $v(M)=1$ and $N$ a gs-disconnected child of $M$.\n\\begin{enumerate}\n    \\item If for some $y\\in P(M\\setminus N)$ and some $x\\in P(N)$ we have $x <_P y$, resp $y <_P x$, then $P(N) <_P \\{y\\}$, resp $\\{y\\} <_P P(N)$. \n    \\item If $N'$ is another gs-disconnected child of $M$, then we have $P(N) <_P P(N')$ or $P(N') <_P P(N)$. \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1) \nSuppose some $y\\in P(M\\setminus N)$ and some $x\\in P(N)$ satisfy $x <_P y$. Since $N$ is gs-disconnected, there is some $z\\in P(N)$ such that $x$ and $z$ belong to different components of $P(N)$ that is $x \\perp z$. Also, $z\\in H_u$ and $y\\in H_v$ for some $u, v\\in M$. Since $v(M)=1$,  $u$ and $v$ are adjacent. Thus, $zy\\in E(G(M))$. In other words, $z$ and $y$ are comparable w.r.t $\\leq_P$. We have $z <_P y$ because otherwise we get $x <_P y <_P z$ which yields that $x$ and $z$ are comparable, a contradiction. It follows that for every element $z\\in P(N)$ in a component other than the component to which $x$ belongs, we have $z <_P y$. By a similar argument, for every element  $x\\in P(N)$ in a component other than the component to which $z$ belongs, we have $x <_P y$. Hence, $P(N) <_P \\{y\\}$ in this case. Similarly, if there is some $x\\in P(N)$ and $y\\in P(M\\setminus N)$ satisfying  $y <_P x$, then $\\{y\\} <_P P(N)$. \n\n(2) Now let $N'$ be another gs-disconnected child of $M$. Since $v(M)=1$, every element of $P(N)$ is comparable to every element of $P(N')$. Take some $x\\in P(N)$ and suppose that for some $y\\in P(N')$ we have $x <_P y$. Since $N'$ is gs-disconnected, (1) implies that $\\{x\\} <_P P(N')$. Since $N$ is gs-disconnected, take some $z\\in P(N)$ such that $x$ and $z$ belong to different components of $P(N)$. We have $z\\perp x$. Therefore, $\\{z\\} <_P P(N')$ because otherwise we get $x <_P y <_P z$, a contradiction. So, for every $z\\in P(N)$ in a component of $P(N)$ other than the component to which $x$ belongs, we have $\\{z\\} <_P P(N')$. Similarly, for every $x$ in a component of $P(N)$ other than the component $z$ belongs to, we have $\\{x\\} <_P P(N')$. It follows that $P(N) <_P P(N')$. The argument for the other case is similar.    \n\\end{proof}\n\nWith the above lemma at our disposal, we can prove $\\mathcal{S}_{fps}(K)$. The proof is by induction on the vertices of $T$.  \n \n\n\\begin{proposition} \\label{Finiteps} \nLet $P$ be a countable $NE$-free poset whose comparability graph is a finite graph substitution of cliques or independent sets. Then,  $P$ is a finite poset substitution of chains or antichains. \n\\end{proposition} \n\n\n\n\\begin{proof}\nSuppose $G:=CG(P)$ is of the form $K[H_v\/v : v\\in K]$ where $|K|<\\infty$ and each $H_v$ is a clique or independent set. We know that $K$ is a cograph. Assume that $K$ is the smallest. Let $T$ be the decomposition tree of $K$. By Lemma \\ref{Kminimal}, $T$ is leaf minimal. For every $M\\in T$ let $\\mathcal{S}_{fps}(M)$ be the statement ``$P(M)$ is a finite poset substitution of chains or antichains\". \n\nIt is easy to verify that a poset whose comparability graph is a clique, resp an independent set, is a chain, resp an antichain. \nTherefore, the statement is true for the leaves of $T$. Let $M\\in T$ be non-trivial and $M_1, \\ldots, M_k$ the children of $M$. Also, assume that $\\mathcal{S}_{fps}(M_i)$ holds for every $1\\leq i\\leq k$. We consider the following cases. \n\n\\noindent\\textbf{Case 1} $v(M)=0$. In this case $M$ is gs-disconnected. Let $x\\in P(M)$. Then there is some $u\\in M$ such that $x\\in H_u$.  Since the $M_i$ are the children of $M$, $x\\in P(M_i)$ for some $1\\leq i\\leq k$. Moreover, if $x\\in P(M_i)$ and $y\\in P(M_j)$, $i\\neq j$, then $x$ and $y$ are disconnected because $v(M)=0$. Thus, $P(M)=\\bigoplus_i P(M_i)$. That is $P(M)$ is the substitution of the $P(M_i)$ for a finite antichain. Since $\\mathcal{S}_{fps}(M_i)$ holds for every $i$, $\\mathcal{S}_{fps}(M)$ is true in this case.\n\n\\noindent\\textbf{Case 2} $v(M)=1$.   \nSince $M$ is edge minimal,  at most one of its children is a leaf $\\{u\\}$ such that $H_u$ is a clique, equivalently $P_u$ is a chain. If there is a leaf $u$ such that $P_u$ is a chain, we may suppose without loss of generality that $M_k=\\{u\\}$. Let $i, j\\neq k$ be given with $i\\neq j$. Then both $M_i$ and $M_j$ are gs-disconnected. By Lemma \\ref{Disconnectedmodule}, we have $P(M_i) <_P P(M_j)$ or $P(M_j) <_P P(M_i)$.  Without loss of generality assume that $P(M_i) <_P P(M_j)$ if $i < j < k$. So, $P(\\bigcup_{i<k}M_i)=+_{i<k}P(M_i)$. Again, by Lemma \\ref{Disconnectedmodule}, for every $i < k$,  $P(M_i)$ is in a gap of $P_u$. Set $P_u^1:=\\{x\\in P_u : \\{x\\} <_P P(M_1)\\}$, for every $1 < i < k$ set\n$$P_u^i:=\\{x\\in P_u : P(M_{i-1}) <_P \\{x\\} <_P P(M_i)\\}$$\nand also set $P_u^k:=\\{x\\in P_u : P(M_{k-1}) <_P \\{x\\} \\}$.  \nThen, we have\n$$P(M)=P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k,$$\nwhere each $P_u^i$ is a chain. That is $P(M)$ is an alternate substitution of the $P(M_i)$ and the $P_u^i$ for a finite chain. \nSince $\\mathcal{S}_{fps}(M_i)$ holds for every $i$ and since each $P_u^i$ is a chain, $\\mathcal{S}_{fps}(M)$ is true in this case.  \n\nSince the statement is true for every $M\\in T$, $\\mathcal{S}_{fps}(K)$ is true which means that $P$ is a finite poset substitution of chains or antichains. \n\\end{proof}\n\nWe observed in Proposition \\ref{Finiteps} that for every $M\\in T$, $P(M)$ is a linear or direct sum of posets. Determining the structure of the siblings of $P(M)$ will enable us to obtain its sibling number. We show that a sibling of $P(M)$ has (up to isomorphism) the same context poset of $P(M)$ (a chain or antichain) replaced with siblings of poset blocks of $P(M)$. The following lemma shows that every embedding of $P(M)$ induces an injective mapping on the poset blocks of $P(M)$. Note that for each $u\\in M$, $P_u$ is a chain or an antichain. When we say $P_u$ and $P_v$, $u, v\\in M$, have the same {\\em type}, it means that either both $P_u$ and $P_v$ are chains or both are antichains. \n\n\n\\begin{lemma} \\label{Incomparableelement} \nLet $M\\in T$ be edge minimal. The images of two distinct poset blocks of $P(M)$ under any embedding of $P(M)$ do not intersect the same poset block. \n\\end{lemma}\n\n\\begin{proof}\nLet $h$ be an embedding of $P(M)$. It suffices to show that for two distinct $u, v\\in M$ and $x\\in P_u, y\\in P_v$, there exists some $z\\in P(M)$ other than $x, y$ such that $z$ is comparable to one and only one of $x$ or $y$. Then $h(z)$ is comparable to one and only one of $h(x)$ or $h(y)$ meaning that $h(P_u)$ and $h(P_v)$ cannot intersect the same poset block because $P(M)$ is a poset substitution by Proposition \\ref{Finiteps}.  \n\nNote that a singleton as a poset block of $P(M)$ is both a chain and an antichain. Let two distinct $u, v\\in M$ and $x\\in P_u, y\\in P_v$ be given. We consider the following cases:\n\n\\noindent\\textbf{Case 1} $u, v$ are adjacent.  If both $P_u, P_v$ are chains, then the edge minimality of $M$ implies that $\\{u, v\\}$ is not a module of $M$. So, there is some $w\\in M$ such that $w$ is adjacent to one of $u$ or $v$ and not to the other. Then picking some $z\\in P_w$ proves the statement. If both $P_u, P_v$ are antichains, then one of them is non-trivial since otherwise they can be regarded as chains. Assume, without loss of generality, that $P_u$ is non-trivial. Then there is some $z\\in P_u$ such that $z\\perp x$ but $z$ and $y$ are comparable. Now suppose, without loss of generality, that $P_u$ is an antichain and $P_v$ is a chain. $P_u$ must be non-trivial since otherwise it can be regarded as a chain. Again, take some $z\\in P_u$ with $z\\perp x$. We know that $z$ and $y$ are comparable. \n\n\\noindent\\textbf{Case 2} $u, v$ are non-adjacent. If both $P_u, P_v$ are antichains, then the edge minimality of $M$ implies that $\\{u,v\\}$ is not a module of $M$. So, there is some $w\\in M$ such that $w$ is adjacent to one of $u$ or $v$ but not to the other. Then pick some $z\\in P_w$ which is a witness to the statement. Note that this also proves the case both $P_u, P_v$ are singletons. So, assume that both $P_u, P_v$ are chains and one of them, say without loss of generality $P_v$, is non-trivial. Take some $y\\neq z\\in P_v$ comparable to $y$. However, we have $z\\perp x$. Finally, if $P_u$ and $P_v$ are of different types, then both of them are non-trivial. Therefore, there exists an element $z$ comparable to one and only one of $x$ or $y$. \n\\end{proof}\n\nBy the lemma above one can define a graph isomorphism on $M$. \n\n\\begin{lemma} \\label{Distinctobjects}\nLet $M\\in T$ be edge minimal. Each embedding of $P(M)$ induces a graph isomorphism of $M$. Moreover, the embedding preserves the types of the poset blocks of $P(M)$. \n\\end{lemma}  \n\n\\begin{proof} \nLet $h$ be an embedding of $P(M)$. First note that $M$ is finite. Therefore, there is a linear order $\\leq_M$ on $M$. Define the mapping $\\hat{h}:M\\to M$ as follows: for each $u\\in M$, select the least $v\\in M$ w.r.t $\\leq_M$ such that $h(P_u)\\cap P_v\\neq \\emptyset$ and define $\\hat{h}(u)=v$. If $h(P_u)\\cap P_v\\neq\\emptyset$ and $h(P_u)\\cap P_w\\neq\\emptyset$, then $v \\leq_M w$ and $w \\leq_M v$ which implies $v=w$. That is $\\hat{h}$ is well-defined. For $u\\neq v\\in M$, by Lemma \\ref{Incomparableelement}, $h(P_u)$ and $h(P_v)$ do not intersect the same poset block. Therefore, the mapping $\\hat{h}$ is injective and since $M$ is finite $\\hat{h}$ is onto. So, $\\hat{h}$ is a bijection on $M$. Further, $u, v\\in M$ are adjacent if and only if $\\hat{h}(u)$ and $\\hat{h}(v)$ are adjacent because $h$ preserves the order $\\leq_P$. Thus, $\\hat{h}$ is a graph isomorphism of $M$. \n\nSince $\\hat{h}$ is well-defined, it is not the case that for some $u\\in M$, $h(P_u)\\cap P_v\\neq \\emptyset$ and $h(P_u)\\cap P_w\\neq\\emptyset$ where $v\\neq w$. It follows that $h(P_u)\\subseteq P_{\\hat{h}(u)}$ for every $u\\in M$. The inclusion also implies that each poset block of $P(M)$ of any type is mapped into a poset block of $P(M)$ of the same type because a non-trivial chain cannot embed in an antichain and a non-trivial antichain cannot embed in a chain. \n\\end{proof}\n\nOne more fact resulting from the lemma above is that any embedding of $P(M)$, $M\\in T$, sends a singleton poset block of $P(M)$ to a singleton poset block because otherwise we get infinitely many chains or antichains as poset blocks of $P(M)$. \n\n\n\\begin{lemma} \\label{Bijectiononconnected}\nLet $M\\in T$ be edge minimal such that $v(M)=0$. Every embedding of $P(M)$ permutes the gs-connected children of $M$ and fixes the possible gs-disconnected child of $M$.  Hence, a sibling of $P(M)$ is a direct sum of the siblings of the $P(M_i)$ where $M_i$ is a child of $M$.  \n\\end{lemma}\n\n\\begin{proof}\nLet $h$ be an embedding of $P(M)$. \nLet $X:=\\{M_1, \\ldots, M_n\\}$ be the set of gs-connected children of $M$ and $\\{u\\}$ the possible gs-disconnected child of $M$. Note that if the gs-disconnected child $\\{u\\}$ of $M$ exists, then $P_u$ is non-trivial and also all $P(M_i)$ are non-trivial because otherwise the edge minimality of $M$ fails. Let $M_i\\in X$ be given and suppose that for some $x\\in P(M_i)$, $h(x)\\in P(M_j)$. For every $y\\in P(M_i)$, $y$ is connected to $x$, so $h(y)$ is connected to $h(x)$ because $h$ preserves the path connecting $x$ and $y$. Therefore,  $h(P(M_i))\\subseteq P(M_j)$ and since $P(M_i)$ is non-trivial, $h(P(M_i))\\cap P_u=\\emptyset$. Since $X$ is finite, there is a linear order $\\leq_X$ on $X$. For every $M_i\\in X$, select the least $M_j\\in X$ w.r.t $\\leq_X$ such that $h(P(M_i))\\cap P(M_j)\\neq\\emptyset$ and define $\\hat{h}(M_i)=M_j$. \nSince $M$ is gs-disconnected, it is not the case that for some $i$ and some $j\\neq k$, $h(P(M_i))\\cap P(M_j)\\neq \\emptyset$ and $h(P(M_i))\\cap P(M_k)\\neq\\emptyset$. So, $\\hat{h}$ is well-defined. Since $T$ is leaf minimal, the children of $M$ are edge minimal. Hence, each $P(M_i)$ is a finite poset substitution of chains or antichains. Let $\\lambda_i$ be the cardinal of $M_i$ for each $i$. We have $\\lambda_i > 0$ for every $i$. Now, suppose for two $i\\neq j$, $h(P(M_i)), h(P(M_j))\\subseteq P(M_k)$. By Lemma \\ref{Distinctobjects}, we have $\\lambda_i + \\lambda_j\\leq \\lambda_k$.  This means that $\\lambda_k > \\lambda_i, \\lambda_j$. A contradiction is immediate when $k=i$ or $k=j$ because then $\\lambda_k > \\lambda_k$. Assume that $k\\neq i, j$. Since $\\lambda_k > \\lambda_i, \\lambda_j$, $P(M_k)$ cannot be embedded into  $P(M_i)$ or $P(M_j)$. If $P(M_k)$ embeds in $P(M_k)$, then by Lemma \\ref{Distinctobjects}, $\\lambda_i+\\lambda_j+\\lambda_k \\leq \\lambda_k$, a contradiction to $\\lambda_i, \\lambda_j\\neq 0$.  It follows that $P(M_k)$ embeds into some $P(M_l)$ where $l\\neq i, j, k$. Continuing this, we get infinitely many gs-connected children of $M$, a contradiction. Thus, $\\hat{h}$ is injective and since $X$ is finite, $\\hat{h}$ is a bijection on $X$. It also follows that $h(P_u)\\subseteq P_u$. Define $\\hat{h}(\\{u\\})=\\{u\\}$.  \n\n\nSince $X$ is finite, for every $i$, there is some integer $m_i > 0$ such that $\\hat{h}^{m_i}(M_i)=M_i$. This means that for every $i$, $P(M_i)\\approx P(\\hat{h}.i(M_i))$ where $\\hat{h}.i$ is the orbit of $M_i$ under $\\hat{h}$. Hence, a sibling of $P(M)$ is of the form $\\bigoplus_iQ(M_i)\\oplus P'_u$ where $Q(M_i)\\approx P(M_i)$ for every $i$ and $P'_u\\cong P_u$ because $P_u$ has only one sibling.  \n\\end{proof} \n\nFor case $v(M)=1$ where $M\\in T$ is edge minimal, we provide a result similar to Lemma \\ref{Bijectiononconnected}.  \n\n\\begin{lemma} \\label{Bijectionondisconnected}\nLet $M\\in T$ be edge minimal such that $v(M)=1$ and $h$ an embedding of $P(M)$. \n\\begin{enumerate}\n    \\item For every gs-disconnected child $M_i$ of $M$ we have $h(P(M_i))\\subseteq P(M_i)$.\n    \\item For any chain $P_u^i$ in the representation $P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k$ of $P(M)$ where $\\{u\\}$ is the unique possible gs-connected child of $M$, $h(P_u^i)\\subseteq P_u^i$. \n\\end{enumerate}\nHence, a sibling of $P(M)$ is of the form \n$$Q_u^1 + Q(M_1) + Q_u^2 + Q(M_2) + \\cdots + Q(M_{k-1}) + Q_u^k $$\nwhere $Q(M_i)\\approx P(M_i)$ and $Q_u^i\\approx P_u^i$ for every $i$. \n\\end{lemma}\n\n\\begin{proof}\nBy Proposition \\ref{Finiteps}, $P(M)$ is of the form \n$$P_u^1 + P(M_1) + P_u^2 + P(M_2) + \\cdots + P(M_{k-1}) + P_u^k\\ $$\nwhere each $M_i$ is a gs-disconnected child of $M$ and the $P_u^i$ are the intervals of the unique chain $P_u$ where $\\{u\\}$ is the possible gs-connected child of $M$.   \n\n(1) Let $i$ and $x\\in P(M_i)$ be given. Since $M_i$ is gs-disconnected, take some $y$ in a component of $P(M_i)$ other than the component to which $x$ belongs. So, $x\\perp y$. We have $h(x)\\perp h(y)$. This implies that $h(x)\\notin P_u^j$ where $1\\leq j \\leq k$. Suppose that $h(x)\\in P(M_j)$ where $i\\neq j$. Without loss of generality assume that $P(M_i) <_P P(M_j)$. Since $h(x)\\perp h(y)$, we have $h(y)\\in P(M_j)$. So, for every $y\\in P(M_i)$ in a component other than the component to which $x$ belongs, we have $h(y)\\in P(M_j)$. Exchanging the role of $x$ and $y$, for every $x\\in P(M_i)$ in a component other than the component to which $y$ belongs, we have $h(x)\\in P(M_j)$. It means that $h(P(M_i))\\subseteq P(M_j)$. Let $\\lambda_i, \\lambda_j$ be the number of elements of $M_i, M_j$, respectively. By Lemma \\ref{Distinctobjects} we have $\\lambda_i \\leq \\lambda_j$. Thus, it is not the case that $P(M_j)$ embeds in $P(M_j)$ by $h$ because then we get $\\lambda_i+\\lambda_j \\leq \\lambda_j$, a contradiction to $\\lambda_i > 0$. So, $h(P(M_j))\\subseteq P(M_l)$ where $j < l$. Continuing this, we get infinitely many gs-disconnected children of $M$, a contradiction.  \n\n(2) Let $i$ and some $x\\in P_u^i$ be given. Suppose $h(x)\\in P_u^j$ or $h(x)\\in P(M_j)$ where $j\\neq i$. Assume that $i < j$. The case $j<i$ is similar. Pick some gs-disconnected $M_l$ where $i \\leq l < j$. Since $M_l$ is gs-disconnected, $P(M_l)\\neq\\emptyset$. Pick some $y\\in P(M_l)$. We have $x <_P y$. By (1), $h(y)\\in P(M_l)$. It follows that $h(y) <_P h(x)$, equivalently, $y <_P x$, a contradiction. \nNow assume that $h(x)\\in P(M_i)$ (the argument for $P(M_{i-1})$ is similar). Since $M_i$ is gs-disconnected, take some $y\\in P(M_i)$ such that $y$ belongs to a component of $P(M_i)$ other than the component to which $h(x)$ belongs. We have $x <_P y$ implying that $h(x) <_P  h(y)$. Also, $h(x) <_P h^2(x)$. Moreover, by (1), $h_{\\upharpoonright P(M_i)}$ is an embedding from $P(M_i)$ into $P(M_i)$. This means that $h(x)$ and $y$ are mapped into the same component of $P(M_i)$. This contradicts Lemma \\ref{Bijectiononconnected} because $v(M_i)=0$. \n\\end{proof} \n\n\n\nNow we are ready to deduce Thomass\\'{e}'s Conjecture for a countable $NE$-free poset whose comparability graph has one sibling. Bear in mind that by Theorem \\ref{Dichchain}, a countable chain has either 1, $\\aleph_0$ or $2^{\\aleph_0}$ siblings. \n\n\n\\begin{theorem} \\label{GtoP} \nLet $P$ be a countable $NE$-free poset whose comparability graph has one sibling. Then $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$.  \n\\end{theorem}  \n\n\\begin{proof}\nSet $G:=CG(P)$. By Theorem \\ref{Lexico}, $G$ is of the form $K[H_v\/v : v\\in K]$ where $K$ is finite and each $H_v$ is a clique or an independent set. Indeed, $K$ is a cograph. Assume that $K$ is the smallest. Let $T$ be the decomposition tree of $K$. By Lemma \\ref{Kminimal},  $T$ is leaf minimal. For each $M\\in T$ let $\\mathcal{S}_{sib}(M)$ be the statement: ``$Sib(P(M))=1$  or $\\aleph_0$ or $2^{\\aleph_0}$\".   We prove that $\\mathcal{S}_{sib}(K)$ holds.\n\nFor every leaf $\\{v\\}$ of $T$, $P_v$ is a countable chain or antichain. Therefore, $\\mathcal{S}_{sib}(\\{v\\})$ holds since an antichain has only one sibling and the statement holds for a countable chain by Theorem \\ref{Dichchain}. Now, take $M\\in T$ such that $M$ is not a leaf of $T$ and suppose that for each child $N$ of $M$, $\\mathcal{S}(N)$ holds. If $v(M)=0$, then by Lemma \\ref{Bijectiononconnected},  each sibling of $P(M)$ is a finite direct sum of siblings of the $P(N)$ where $N$ is a child of $M$; and if $v(M)=1$, then by Lemma \\ref{Bijectionondisconnected}, each sibling of $P(M)$ is an alternate substitution of chains and the siblings of the $P(N)$ for a finite chain where $N$ is a non-trivial child of $M$. By the induction hypothesis and the fact that the statement holds for a countable chain,  $Sib(P(M))$ is the maximum of the sibling numbers of the components or the summands of $P(M)$. Thus, $\\mathcal{S}_{sib}(M)$ holds. Since the statement is true for every $M\\in T$,  $\\mathcal{S}_{sib}(K)$ holds.   \n\\end{proof}\n\n\n\n\\section{Open Cases} \n\nWe just proved the alternate Thomass\\'{e} conjecture for countable $NE$-free posets. We saw how the siblings of the sum of a labelled chain with no least element were constructed by its decomposition tree. The same technique was used for countable linear sums with infinitely many non-trivial summands. Also, there were other cases in both direct sums and linear sums in which we determined the exact sibling number. Moreover, Section \\ref{Finiteposetsub} asserts that Thomass\\'e's conjecture holds for countable $NE$-free posets whose comparability graph has one sibling. Posing all these restrictions, we may ask the following.\n\n\\begin{problem}\nLet P be a countable direct sum of connected $NE$-free posets or a countable linear sum of $NE$-free posets with property CCGC such that the comparability graph of $P$ has infinitely many siblings. Is it true that $Sib(P)=1$ or $\\aleph_0$ or $2^{\\aleph_0}$?\n\\end{problem}\n\n\\vspace{0.5cm}\n\\noindent{\\bf \\large Acknowledgements}\n\nI would like to thank my PhD supervisors Professor Robert Woodrow and Professor Claude Laflamme for suggesting this problem and for their help and advice. \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{section:introduction}\n\nThe decomposition theorem for one-parameter persistence modules states that any such module decomposes uniquely as a direct sum of interval modules, which are completely characterized by their supports \\cite{gabriel,webb,zomorodian-carlsson,crawley-boevey}.\nThis theorem is of fundamental importance in topological inference and topological data analysis, where it is used, in particular, to distinguish between prominent topological features, which correspond to features of the process that generated the data, and spurious topological features, due to noise \\cite{edelsbrunner-letscher-zomorodian,edelsbrunner-harer,chazal-guibas-oudot-skraba}.\nThe consistency of this method is guaranteed by the stability theorem \\cite{cohensteiner-edelsbrunner-harer,damico-frosini-landi,chazal-silva-glisse-oudot,chazal-cohen-steiner-glisse-guibas-oudot,bauer-lesnick}, which states that decomposing a one-parameter persistence module into its indecomposable summands is a stable operation.\n\nMulti-parameter persistence modules arise naturally when dealing with data where both metric and density structure is relevant, or with otherwise parametrized data \\cite{carlsson-zomorodian,botnan-lesnick-2,biasotti-cerry-frosini-giorgi-landi,lesnick-wright,mcinnes-healy,vipond}.\nWhen one moves from one parameter to multiple parameters, the approach to understanding a persistence module by completely characterizing its indecomposable summands is less promising: classifying all indecomposable multi-parameter persistence modules is thought to be an infeasible task as it involves classifying all indecomposable representations of certain posets of wild representation type \\cite{Nazarova}.\nNevertheless, one can still consider computable descriptors\nderived from the decomposition into indecomposables \\cite{dey-xin}.\nA natural example is the descriptor given by the supports of the indecomposable summands, and a natural question is whether this descriptor is stable.\n\nFrom the results in this paper we can conclude that, for general two-parameter persistence modules, this descriptor is highly unstable.\nAs such, our results provide further motivation for approaches to understanding the structure and stability of multi-parameter persistence modules that do not rely on decomposing by arbitrary indecomposables\n\\cite{carlsson-zomorodian,\nlesnick-wright-2,\nscolamiero-chacholski-lundman-ramanujan-oberg,\nkim-memoli,\nasashiba2019approximation,\nmiller,\nmccleary2021edit,\nbotnan-oppermann-oudot,\nblanchette-brustle-hanson,\noudot-scoccola,\nasashiba-escolar-nakashima-yoshiwaki,\nbotnan-oppermann-oudot-scoccola,\ncacholski-guidolin-ren-scolamiero-tombari,\ncerri-difabio-ferri-frosini-landi,\nbiasotti-cerri-frosini-giorgi-landi},\nand give a sense of how complicated arbitrary indecomposables can be, complementing results such as the ones in \\cite{carlsson-zomorodian,buchet-escolar,moore}.\n\n\n\\subparagraph{Contributions}\nFor definitions, see \\cref{section:background}.\nThe following is our main result.\n\n\\begin{theorem}\n    \\label{theorem:main-theorem}\n    Consider the set of isomorphism classes of finitely presentable two-parameter persistence modules, metrized with the interleaving distance.\n    For every $\\epsilon > 0$ there exists a dense and open subset $O^\\epsilon$ with the property that every $N \\in O^\\epsilon$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{theorem}\n\nRecall that, in a topological space, a \\define{generic} property is one that holds for all points of some dense and open set.\nOur main result then implies that, for every $\\epsilon > 0$, the property of decomposing as a direct sum of an indecomposable and an $\\epsilon$-trivial module is generic.\n\nTo put this result into context, we mention a similar instance of a generic property of persistence modules. \nThe set of all finitely presentable one-parameter persistence modules that are \\emph{staggered} (meaning that the finite endpoints of the intervals in their barcode are all distinct) is dense but not open.\nNevertheless, for every $\\epsilon > 0$ there exists a dense and open set of modules that decomposes as $A \\oplus B$ with $A$ staggered and $B$ $\\epsilon$-trivial, as a direct consequence of the isometry theorem \\cite{lesnick,chazal-silva-glisse-oudot,bubenik-scott,bauer-lesnick}.\n\n\n\\smallskip\nWe prove \\cref{theorem:main-theorem} by combining the following two results, which are of independent interest.\nThe first one states that, in the space of finitely presentable two-parameter persistence modules, indecomposable modules are dense.\n\n\\begin{restatable}{proposition}{indecomposabledense}\n    \\label{theorem:indecomposables-dense}\n    Let $N : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable.\n    For every $\\epsilon > 0$, there exists an indecomposable persistence module $M : \\Rbf^2 \\to \\mathbf{vec}$ such that $d_I(M,N) \\leq \\epsilon$.\n\\end{restatable}\n\nThe second result relates the property of being close to an indecomposable to the property of decomposing as a direct sum of an indecomposable and a nearly trivial persistence module.\n\n\\begin{restatable}{proposition}{stabilityindecomposability}\n    \\label{proposition:stability-indecomposability}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$ be finitely presentable and indecomposable.\n    For every $\\epsilon > 0$ there exists $\\delta > 0$ such that every persistence module $N : \\Rbf^n \\to \\mathbf{vec}$ with $d_I(M,N) < \\delta$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{restatable}\n\n\\subparagraph{Discussion}\n\\cref{proposition:stability-indecomposability} motivates the following definition.\n\n\\begin{definition}\n    Let $\\epsilon > 0$.\n    A persistence module $M : \\Rbf^n \\to \\mathbf{vec}$ is \\define{$\\epsilon$-indecomposable} if we have $M \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n\\end{definition}\nOur main theorem then implies that, for every $\\epsilon > 0$, two-parameter persistence modules are generically $\\epsilon$-indecomposable.\nSince there are no general strong relationships between meager and measure-zero sets \\cite{oxtoby}, our results do not imply that random two-parameter persistence modules, according to some suitable probability measure, are $\\epsilon$-indecomposable.\nOne could then ask the following general questions:\nWhat are interesting and useful probability measures on spaces of multi-parameter persistence modules?\nAccording to these probability measures, how do random multi-parameter persistence modules decompose?\n\nWe prove \\cref{proposition:stability-indecomposability} in the generality of multi-parameter persistence modules, while we have given \\cref{theorem:indecomposables-dense} and consequently \\cref{theorem:main-theorem} only in the case of two-parameter persistence.\nWe believe the results hold for any number of parameters greater than one.\nSimilarly, the results extend to classes of modules more general than finitely presentable ones.\nWe leave these extensions for future work.\n\n\\subparagraph{Structure of the paper}\nIn \\cref{section:background}, we recall necessary background and introduce notation.\nIn \\cref{section:almost-indecomposable-open}, we prove \\cref{proposition:stability-indecomposability}.\nIn \\cref{section:indecomposables-dense}, we prove \\cref{theorem:indecomposables-dense} and \\cref{theorem:main-theorem}.\nIn \\cref{section:proof-main-lemma} we prove a key lemma that allows us to ``tack'' indecomposables together.\n\n\n\n\\ifarxivversion\n\\subparagraph{Acknowledgements}\nWe thank H\\r{a}vard Bjerkevik for interesting conversations and useful observations.\nL.~S.~thanks\nNicolas Berkouk,\nMathieu Carri\u00e8re,\nRen\u00e9 Corbet,\nChristian Hirsch,\nClaudia Landi,\nVadim Lebovici,\nDavid Loiseaux,\nEzra Miller,\nSteve Oudot,\nFran\u00e7ois Petit,\nand Alexander Rolle\nfor discussions during the \\emph{Metrics in Multiparameter Persistence workshop} (Lorentz Center, 2021).\nThis research has been conducted while U.~B.~was participating in the program \\emph{Representation Theory: Combinatorial Aspects and Applications};\nU.~B.~thanks the Centre for Advanced Study (CAS) at the Norwegian Academy of Science and Letters for their hospitality and support.\nThis research has been supported by the DFG Collaborative Research Center SFB\/TRR 109 \\emph{Discretization in Geometry and Dynamics}.\n\\fi\n\n\\section{Background}\n\\label{section:background}\n\nAlthough we use some notions from category theory, we only assume familiarity with basic concepts; in particular: categories, isomorphisms, functors, functor categories, direct sums, and kernels and cokernels.\nSee, e.g., \\cite{maclane} for an introduction.\n\nThroughout the paper, we fix a field $\\kbb$ and let $\\mathbf{vec}$ denote the category of finite dimensional $\\kbb$-vector spaces.\n\nAn \\define{extended metric space} consists of a set $A$ together with a function $d : A \\times A \\to \\Rbb \\cup \\{\\infty\\}$ such that, for all $a,b \\in A$ we have $d(a,b) = d(b,a) \\geq 0$ and $d(a,b) = 0$ if and only if $a=b$; and for all $a,b,c \\in A$ we have $d(a,c) \\leq d(a,b) + d(b,c)$, with the convention that $r + \\infty = \\infty + r = \\infty$ for all $r \\in \\Rbb \\cup \\{\\infty\\}$.\n\n\\subparagraph{Posets}\nWe let $\\Rbf$ denote the poset of real numbers with its standard ordering and reserve the notation $\\Rbb$ for the metric space of real numbers.\nLet $n \\in \\Nbb$ and consider the product poset $\\Rbf^n$.\nBy an abuse of notation, if $r \\in \\Rbf$, we interpret it as an element $r \\in \\Rbf^n$ all of whose coordinates are equal to $r$.\nThus, if for instance $s \\in \\Rbf^n$ and $r \\in \\Rbf$, then $s + r \\in \\Rbf^n$ denotes the element $(s_1 + r, s_2 + r, \\dots, s_n + r)$. \n\nWe interpret any poset $\\Pscr$ as a category with objects the elements of the poset and exactly one morphism $i \\to j \\in \\Pscr$ whenever $i \\leq j$.\n\n\\subparagraph{Persistence modules}\nLet $\\Pscr$ be a poset.\nA pointwise finite dimensional \\define{$\\Pscr$-persistence module} is a functor $M : \\Pscr \\to \\mathbf{vec}$.\nNote that all persistence modules in this paper are assumed to be pointwise finite dimensional and that, for the sake of brevity, we may omit this qualifier.\nWhen the indexing poset $\\Pscr$ is clear from the context, we may refer to a $\\Pscr$-persistence module simply as a persistence module or as a module.\nThe collection of all $\\Pscr$-persistence modules forms a category, where the morphisms are given by natural transformations.\n\nIf $M : \\Pscr \\to \\mathbf{vec}$ is a $\\Pscr$-persistence module and $i \\leq j \\in \\Pscr$, we let $\\phi^M_{i,j} : M(i) \\to M(j)$ denote the structure morphism corresponding to the morphism $i \\to j$ in $\\Pscr$ seen as a category.\n\nLet $\\Pscr$ be a poset and let $i \\in \\Pscr$.\nDefine the persistence module $\\Psf_i : \\Pscr \\to \\mathbf{vec}$ by\n\\[\n    \\Psf_i(j) =\n    \\begin{cases}\n        \\kbb & i \\leq j\\\\\n        0 & \\text{else},\n    \\end{cases}\n\\]\nwith all structure morphisms that are not forced to be zero being the identity $\\kbb \\to \\kbb$.\n\nA $\\Pscr$-persistence module is \\define{finitely presentable} if it is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$, where $I$ and $J$ are finite multisets of elements of $\\Pscr$.\n\n\n\\subparagraph{Restrictions and extensions}\nIf $\\Qscr \\subseteq \\Pscr$ is an inclusion of posets and $M : \\Pscr \\to \\mathbf{vec}$ is a $\\Pscr$-persistence module, the \\define{restriction} of $M$ to $\\Qscr$, denoted $M|_\\Qscr : \\Qscr \\to \\mathbf{vec}$, is the $\\Qscr$-persistence module obtained by precomposing $M : \\Pscr \\to \\mathbf{vec}$ with the inclusion $\\Qscr \\to \\Pscr$.\n\nWe consider two main types of subposets of $\\Rbf^n$.\nIn one case, we let $\\{r_1 < r_2 < \\dots < r_k\\}$ be a finite set of real numbers and consider the product poset $\\{r_1 < r_2 < \\dots < r_k\\}^n \\subseteq \\Rbf^n$.\nWe refer to a subposet of $\\Rbf^n$ obtained in this way as a \\define{finite grid}.\nIn the other case, we let $\\{r_i\\}_{i \\in \\Zbb}$ be a countable set of real numbers without accumulation points and such that $r_i < r_{i+1}$ for all $i \\in \\Zbb$, and again consider the product poset $\\{r_i\\}^n \\subseteq \\Rbf^n$.\nWe refer to a subposet of $\\Rbf^n$ obtained in this way as a \\define{countable grid}.\nA \\define{regular grid} is any countable grid of the form $(s \\Zbf)^n = \\{s \\cdot m : m \\in \\Zbf\\}^n \\subseteq \\Rbf^n$ for $s > 0 \\in \\Rbb$, where $\\Zbf$ denotes the poset of integers.\nNote that, in the definitions of finite and countable grid, one could take a product of different subposets of $\\Rbf$, instead of an $n$-fold product of the same poset.\nSince we do not need this generality, we choose the more restrictive definition for simplicity.\n\n\nLet $\\Pscr \\subseteq \\Rbf^n$ be a finite or countable grid.\nGiven $M : \\Pscr \\to \\mathbf{vec}$ define $\\widehat{M} : \\Rbf^n \\to \\mathbf{vec}$\nby\n\\[\n    \\widehat{M}(r) =\n    \\begin{cases}\n        M\\left(\\sup\\{ p \\in \\Pscr : p \\leq r\\}\\right) & \\text{if there exists $p \\in \\Pscr$ such that $p \\leq r$}\\\\\n        0 & \\text{else};\n    \\end{cases}\n\\]\nfor its structure morphisms use the structure morphisms of $M$ and the fact that $\\sup\\{p \\in \\Pscr : p \\leq r\\} \\leq \\sup\\{p \\in \\Pscr : p \\leq s\\}$ whenever $r \\leq s$ and there exists $p \\in \\Pscr$ such that $p \\leq r$.\nWe refer to $\\widehat{M}$ as the \\define{extension} of $M$ along the inclusion $\\Pscr \\subseteq \\Rbf^n$.\nAs a side remark, we note that this notion of extension is an instance of the notion of left Kan extension from category theory, but we do not make use of this fact.\n\nIf $\\Pscr \\subseteq \\Rbf^n$ is a finite or countable grid and $M : \\Rbf^n \\to \\mathbf{vec}$, define the \\define{restriction-extension} of $M$ along $\\Pscr$, denoted $M_\\Pscr : \\Rbf^n \\to \\mathbf{vec}$, as the extension along $\\Pscr \\subseteq \\Rbf^n$ of the restriction $M|_\\Pscr : \\Pscr \\to \\mathbf{vec}$.\nGiven $M,N : \\Rbf^n \\to \\mathbf{vec}$ and a morphism $f : M \\to N$, there is a morphism $f_\\Pscr : M_\\Pscr \\to N_\\Pscr$ given by extending the restriction $f|_\\Pscr : M|_\\Pscr \\to N|_\\Pscr$.\nIt follows immediately that this construction is functorial in the sense that given modules $A,B,C : \\Rbf^n \\to \\mathbf{vec}$ and morphism $f : A \\to B$ and $g : B \\to C$, we have $g_\\Pscr \\circ f_\\Pscr = (g \\circ f)_\\Pscr : A_\\Pscr \\to C_{\\Pscr}$.\n\n\n\\begin{lemma}\n    \\label{lemma:fp-is-extension-finite-poset}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$.\n    Then $M$ is finitely presentable if and only if there exists a finite grid $\\Pscr \\subseteq \\Rbf^n$ such that $M \\cong M_\\Pscr$.\n\\end{lemma}\n\\begin{proof}\n    Assume that $M$ is finitely presentable, so that $M$ is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$ with $I$ and $J$ finite subsets of $\\Rbf$.\n    Consider the set $S = \\{x_k \\in \\Rbf : x \\in I \\cup J, \\, 1 \\leq k \\leq n\\}$ of all the coordinates of the points in $I \\cup J$, and the finite grid $\\Pscr = S^n \\subseteq \\Rbf^n$.\n    It is straightforward to see that $M \\cong M_\\Pscr$.\n\n    Now assume that $M \\cong \\widehat{N}$ with $\\Pscr \\subseteq \\Rbf^n$ a finite grid and $N : \\Pscr \\to \\mathbf{vec}$.\n    Since the poset $\\Pscr$ has finitely many elements and $N$ is pointwise finite dimensional, there exists an epimorphism $e : \\bigoplus_{i \\in I} \\Psf_i \\to N$, for some finite multiset $I$ of elements of $\\Pscr$.\n    Similarly, if $K$ is the kernel of the morphism $e$, there must exist an epimorphism $\\bigoplus_{j \\in J} \\Psf_j \\to K$, with $J$ finite.\n    Putting these two morphisms together we see that $N$ is isomorphic to the cokernel of a morphism $\\bigoplus_{j \\in J} \\Psf_j \\to \\bigoplus_{i \\in I} \\Psf_i$.\n    It is easy to see that $M$ is then isomorphic to the cokernel of the induced morphism $\\bigoplus_{j \\in J} \\widehat{\\Psf_j} \\to \\bigoplus_{i \\in I} \\widehat{\\Psf_i}$, and that $\\widehat{\\Psf_r} = \\Psf_r : \\Rbf^n \\to \\mathbf{vec}$ for all $r \\in \\Pscr$.\n\\end{proof}\n\nIt is straightforward to see that, in a grid extension persistence module, the structure maps that do not cross the grid are isomorphisms, as recorded in the following lemma.\n\n\\begin{lemma}\n    \\label{lemma:structure-map-iso}\n    Let $\\Pscr$ be a finite or countable grid and let $M : \\Rbf^n \\to \\mathbf{vec}$ be isomorphic to the extension of a $\\Pscr$-persistence module.\n    Let $r < s \\in \\Rbf^n$ such that every $p \\in \\Pscr$ with $p \\leq s$ also satisfies $p \\leq r$.\n    Then the structure morphism $\\phi^M_{r,s} : M(r) \\to M(s)$ is an isomorphism.\n    \\qed\n\\end{lemma}\n\n\\subparagraph{Decomposition of persistence modules}\nThe proofs of the results in this section are standard, we include them in \\cref{appendix}, for completeness.\nLet $\\Pscr$ be a poset and let $M : \\Pscr \\to \\mathbf{vec}$.\nWe say that $M$ is \\define{decomposable} if there exist $A,B : \\Pscr \\to \\mathbf{vec}$ non-zero such that $M \\cong A \\oplus B$.\nIf $M$ is non-zero and not decomposable, we say that $M$ is \\define{indecomposable}.\n\n\nThe next two results follow from \\cite{botnan-crawleybovey} and the Krull--Remak--Schmidt--Azumaya theorem~\\cite{azumaya}.\n\n\n\\begin{restatable}{theorem}{theoremdecomposition}\n    \\label{theorem:decomposition}\n    Let $\\Pscr$ be a poset and let $M : \\Pscr \\to \\mathbf{vec}$ be a pointwise finite dimensional $\\Pscr$-persistence module.\n    There exists a set $I$ and an indexed family of indecomposable $\\Pscr$-persistence modules $\\{M_i\\}_{i \\in I}$ such that $M \\cong \\bigoplus_{i \\in I} M_i$.\n    Moreover, if $M \\cong \\bigoplus_{j \\in J} M_j$ for another indexed family of indecomposable $\\Pscr$-persistence modules $\\{M_j\\}_{j \\in J}$, then there exists a bijection $f : I \\to J$ such that $M_i \\cong M_{f(j)}$ for all $i \\in I$.\n\\end{restatable}\n\n\\begin{restatable}{lemma}{indecomposablelocalring}\n    \\label{lemma:indecomposable-local-ring}\n    Let $\\Pscr$ be a poset.\n    A persistence module $M : \\Pscr \\to \\mathbf{vec}$ is indecomposable if and only if its endomorphism ring $\\End(M)$ is local.\n\\end{restatable}\n\n\\begin{proof}\n    This is a direct consequence of \\cite[Theorem~1.1]{botnan-crawleybovey}.\n\\end{proof}\n\nThe following result states that direct sum decompositions behave well with respect to restrictions and extensions; its proof is straightforward.\n\n\\begin{lemma}\n    \\label{lemma:decomposition-restriction-extension}\n    Let $\\Pscr \\subseteq \\Rbf^n$ be a subposet.\n    Let $M : \\Rbf^n \\to \\mathbf{vec}$ decompose as $M \\cong \\bigoplus_{i\\in I} M_i$ for some indexed family $\\{M_i\\}_{i \\in I}$ of persistence modules.\n    Then $M|_\\Pscr \\cong \\bigoplus_{i\\in I} (M_i)|_\\Pscr$.\n    Similarly, if $\\Pscr$ is a finite or countable grid and $M : \\Pscr \\to \\mathbf{vec}$ decomposes as $M \\cong \\bigoplus_{i\\in I} M_i$ for some indexed family $\\{M_i\\}_{i \\in I}$ of $\\Pscr$-persistence modules, then $\\widehat{M} \\cong \\bigoplus_{i\\in I} \\widehat{M_i}$.\n    \\qed\n\\end{lemma}\n\nThe next result asserts that finitely presentable persistence modules decompose as a finite direct sum of finitely presentable modules.\nThis holds in the generality of persistence modules indexed by an arbitrary poset, but we do not need this generality.\n\n\\begin{restatable}{lemma}{decompositionfp}\n    \\label{lemma:decomposition-fp}\n    Let $n \\geq 1$ and let $M : \\Rbf^n \\to \\mathbf{vec}$.\n    If $M$ is finitely presentable, then $M \\cong \\bigoplus_{i \\in I} M_i$ with $\\{M_i\\}_{i \\in I}$ a finite family of finitely presentable indecomposable $\\Rbf^n$-persistence modules.\n\\end{restatable}\n\n\\subparagraph{Interleavings, interleaving distance, and space of persistence modules}\nLet $n \\in \\Nbb$.\nIf $M : \\Rbf^n \\to \\mathbf{vec}$ is a $\\Rbf^n$-persistence module and $r \\in \\Rbf$, the \\define{$r$-shift} of $M$ is the persistence module $M[r] : \\Rbf^n \\to \\mathbf{vec}$ with $M[r](s) = M[s+r]$ for all $s \\in \\Rbf^n$ and $\\phi^{M[r]}_{s,t} = \\phi^M_{s+r,t+r}$ for all $s\\leq t \\in \\Rbf^n$.\nIf $r \\geq 0 \\in \\Rbf$, then there is a natural transformation $\\eta^M_r : M \\to M[r]$ with component $M(a) \\to M[r](a) = M(a+r)$ given by the structure morphism $\\phi^M_{a,a+r}$.\n\nLet $M,N : \\Rbf^n \\to \\mathbf{vec}$ and $\\epsilon \\geq 0 \\in \\Rbf$.\nAn \\define{$\\epsilon$-interleaving} between $M$ and $N$ consists of a pair of morphisms $f : M \\to N[\\epsilon]$ and $g : N \\to M[\\epsilon]$ such that $g[\\epsilon] \\circ f = \\eta^M_{2\\epsilon}$ and $f[\\epsilon] \\circ f = \\eta^N_{2\\epsilon}$.\nThe \\define{interleaving distance} between $M$ and $N$ is\n\\[\n    d_I(M,N) = \\inf\\left(\\{\\epsilon \\geq 0 : \\text{ exists an $\\epsilon$-interleaving between $M$ and $N$ }\\} \\cup \\{\\infty\\}\\right) \\in \\Rbb \\cup \\{\\infty\\}.\n\\]\n\nBy composing interleavings, one shows that $d_I$ satisfies the triangle inequality.\nUsing \\cref{lemma:structure-map-iso}, one sees that if $M, N : \\Rbf^n \\to \\mathbf{vec}$ are finitely presentable and $d_I(M,N) = 0$, then $M$ and $N$ are isomorphic \\cite[Corollary~6.2]{lesnick}.\nThis implies that $d_I$ defines an extended metric on the set of isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules.\nWe mention here that the collection of all isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules is indeed a set, as opposed to a proper class, but we shall not delve into the details as they are standard and not relevant to the results presented here.\n\nWe consider the set of isomorphism classes of finitely presentable $\\Rbf^n$-persistence modules which we topologize with the topology generated by the open balls with respect to the interleaving distance.\n\nLet $\\epsilon > 0$.\nA persistence module $M : \\Rbf^n \\to \\mathbf{vec}$ is \\define{$\\epsilon$-trivial} if, for every $r \\in \\Rbf^n$, the structure morphism $\\phi^M_{r,r+\\epsilon} : M(r) \\to M(r + \\epsilon)$ is the zero morphism.\nNote that this is equivalent to $M$ being $\\epsilon\/2$-interleaved with the zero module.\nThus, if $d_I(M,0) < \\epsilon\/2$, then $M$ is $\\epsilon$-trivial.\n\n\n\\subparagraph{Shifts and restriction-extensions}\nIf $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n \\subseteq \\Rbf^n$ is a countable grid and $r \\in \\Rbf$, define the \\define{shifted} countable grid $\\Pscr + r = (\\{s_i + r\\}_{i \\in \\Zbb})^n \\subseteq \\Rbf^n$.\nLet $M : \\Rbf^n \\to \\mathbf{vec}$ and $s \\in \\Rbf$.\nIf $a \\in \\Rbf^n$, then $\\sup\\{ p + r : p + r \\leq a + s\\} = \\sup\\{p + r - s : p + r - s \\leq a\\} + s$.\nThus,\n\\[\n    M_{\\Pscr + r}[s] = M[s]_{\\Pscr + (r-s)},\n\\]\na fact that we use repeatedly in \\cref{section:almost-indecomposable-open}.\nIn particular, given $N : \\Rbf^n \\to \\mathbf{vec}$ and a morphism $f : M \\to N[s]$, we get a morphism\n\\begin{equation}\n    \\label{equation:shift-and-restriction-extension}\n    f_\\Pscr : M_{\\Pscr + r} \\to N[s]_{\\Pscr + r} = N_{\\Pscr + r + s}[s].\n\\end{equation}\n\n\\section{Nearly indecomposables are open}\n\\label{section:almost-indecomposable-open}\n\nBefore we can prove \\cref{proposition:stability-indecomposability}, we need a definition and a lemma.\n\n\\begin{definition}\n    Let $\\alpha > 0$.\n    A countable grid $\\Pscr = \\left(\\{s_i\\}_{i \\in \\Zbb}\\right)^n \\subseteq \\Rbf^n$ has \\define{mesh width $\\alpha$} if for all $i \\in \\Zbb$ we have $s_{i+1} - s_i \\leq \\alpha$.\n\\end{definition}\n\nNote that, for any fixed $\\alpha > 0$, every countable grid $S^n = \\left(\\{s_i\\}_{i \\in \\Zbb}\\right)^n$ is included in some countable grid with mesh width $\\alpha$, for example, the countable grid $(S \\cup (\\alpha\\Zbf))^n$.\n\n\\begin{lemma}\n    \\label{lemma:factor-bounded-grid}\n    Let $\\alpha > 0$, let $\\Pscr$ be a countable grid with mesh width $\\alpha$, and let $L : \\Rbf^n \\to \\mathbf{vec}$.\n    For every $r < \\alpha$ there exists a morphism $m : L_{\\Pscr + r}[r] \\to L_\\Pscr[\\alpha]$ rendering the following square commutative:\n    \\[\n    \\begin{tikzpicture}\n        \\matrix (m) [matrix of math nodes,row sep=4em,column sep=7em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n        {  L_{\\Pscr}  &  L_{\\Pscr}[\\alpha] \\\\\n           L[r]_\\Pscr & L_{\\Pscr + r}[r],\\\\};\n        \\path[line width=0.75pt, -{>[width=8pt]}]\n        (m-1-1) edge node [above] {$\\eta^{L_\\Pscr}_\\alpha$} (m-1-2)\n        (m-1-1) edge node [left] {$(\\eta^L_r)_\\Pscr$} (m-2-1)\n        (m-2-2) edge node [right] {$m$} (m-1-2)\n        (m-2-1) edge [-,double equal sign distance] (m-2-2)\n        ;\n    \\end{tikzpicture}\n    \\]\n\\end{lemma}\n\\begin{proof}\n    We start by defining the morphism $m$, which is equivalently a morphism $m : L_{\\Pscr+r} \\to L_\\Pscr[\\alpha-r]$.\n    If $a \\in \\Rbf^n$, then\n    $L_{\\Pscr + r}(a) = L(\\sup\\{ s + r : s \\in \\Pscr, s + r \\leq a\\})$ and\n    \\[\n        L_{\\Pscr}[\\alpha-r](a) = L_{\\Pscr}(a + \\alpha - r) = L(\\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}).\n    \\]\n    Let $s_0$ such that $s_0 + r = \\sup\\{ s + r : s \\in \\Pscr, s + r \\leq a\\}$ so that $L_{\\Pscr + r}(a) = L(s_0 + r)$.\n    If $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n$, then $s_0 = (s_{i_1}, \\dots, s_{i_n})$.\n    Let $s_1 = (s_{i_1+1}, \\dots, s_{i_n+1})$.\n    Note that, since $\\Pscr$ has mesh width $\\alpha$, we have $s_1 - \\alpha \\leq s_0$.\n    Thus, $s_1 - \\alpha + r \\leq a$, which implies $s_1 \\leq a + \\alpha - r$.\n    This means that $s_1 \\leq \\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}$.\n    We can then consider the composite:\n    \\[\n        L_{\\Pscr + r}(a) = L(s_0) \\to L(s_1) \\to L(\\sup\\{s : s \\in \\Pscr, s \\leq a + \\alpha - r\\}) = L_{\\Pscr}[\\alpha-r](a),\n    \\]\n    given by the structure morphisms of $L$.\n    Let us denote the above composite by $m_a$.\n    Since the morphism was constructed only using the structure morphisms of $L$, it is straightforward to check that the morphisms $m_a : L_{\\Pscr + r}(a) \\to L_{\\Pscr}[\\alpha-r](a)$ assemble into a natural transformation $m : L_{\\Pscr+r} \\to L_\\Pscr[\\alpha-r]$.\n    By the same reason, it follows that $m$ renders commutative the square in the statement.\n\\end{proof}\n\n\n\\stabilityindecomposability*\n\\begin{proof}[Proof outline]\\renewcommand{\\qedsymbol}{}\n    Let $M$ be as in the statement.\n    Let us start by giving an outline of the proof strategy.\n    We start by arguing that the persistence module $M$ is isomorphic to the extension of a persistence module over a sufficiently fine countable grid.\n    We then choose a sufficiently small $\\delta > 0$ and consider an arbitrary module $N$ at interleaving distance at most $\\delta$ from $M$.\n    Next, we use \\cref{lemma:structure-map-iso}, which says that the extension of a module over a grid can only change when one of its coordinates crosses a point in the grid; this is used to show that, after restricting to appropriate grids, a restriction of $M$ is indecomposable and isomorphic to a direct summand of a restriction of $N$.\n    This allows us to find a decomposition of $N$ into two summands, one of which is indecomposable and is related to a restriction of $M$.\n    Finally, we show that the other summand is close to the zero module in the interleaving distance.\n\\end{proof}\n\n\n\\begin{proof}\n    By \\cref{lemma:fp-is-extension-finite-poset}, the module $M$ is isomorphic to the extension of its restriction to some finite grid $\\Qscr = \\{r_1 < r_2 < \\dots < r_k\\}^n \\subseteq \\Rbf^n$.\n    Let\n    \\[\n        \\tau = \\min_{1\\leq i \\leq k-1} r_{i+1} - r_i\\;\\;, \\;\\; \\alpha = \\min\\left(\\epsilon\/4\\; , \\tau\\right)\\;\\;, \\;\\; \\delta < \\alpha\/4,\n    \\]\n    and let $\\Pscr = (\\{s_i\\}_{i \\in \\Zbb})^n$ be a countable grid with mesh width $\\alpha$ and containing $\\Qscr$.\n    Note that $M$ is isomorphic to the extension of its restriction to $\\Pscr$, that is $M \\cong M_\\Pscr$.\n\n    Assume given $N : \\Rbf^n \\to \\mathbf{vec}$ and a $\\delta$-interleaving $f : M \\to N[\\delta]$ and $g : N \\to M[\\delta]$ between $M$ and $N$.\n    The interleaving equalities $g[\\delta] \\circ f = \\eta^M_{2\\delta}$ and $f[2\\delta] \\circ g[\\delta] = \\eta^{N[\\delta]}_{2\\delta}$ induce, by \\cref{equation:shift-and-restriction-extension}, the following commutative diagram of $\\Rbf^n$-persistence modules:\n    \\begin{equation}\n        \\label{equation:decomposition-N}\n        \\begin{tikzpicture}[baseline=(current  bounding  box.center)]\n            \\matrix (m) [matrix of math nodes,row sep=4em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {      & N_{\\Pscr + \\delta}[\\delta]  & & N_{\\Pscr + 3\\delta}[3\\delta] \\\\\n                 M_{\\Pscr} &  & M_{\\Pscr + 2\\delta}[2\\delta] &  \\\\};\n            \\path[line width=0.75pt, -{>[width=8pt]}]\n            (m-2-1) edge node [above] {$\\cong$} node [below] {$(\\eta^M_{2\\delta})_{\\Pscr}$} (m-2-3)\n            (m-2-1) edge [>->] node [above left] {$f_{\\Pscr}$} (m-1-2)\n            (m-1-2) edge [->>] node [above right] {$g[\\delta]_{\\Pscr}$} (m-2-3)\n            (m-1-2) edge node [above] {$(\\eta^{N[\\delta]}_{2\\delta})_{\\Pscr}$} (m-1-4)\n            (m-2-3) edge node [below right] {$f[2\\delta]_\\Pscr$} (m-1-4)\n            ;\n        \\end{tikzpicture}\n    \\end{equation}\n    The bottom horizontal morphism is an isomorphism by \\cref{lemma:structure-map-iso} and the fact that $2\\delta < \\alpha\/2 \\leq \\tau\/2$.\n    This implies that the left diagonal morphism is a split monomorphism and that the middle diagonal morphism is a split epimorphism.\n\n    In particular, it follows that $M_{\\Pscr}$ is a direct summand of $N_{\\Pscr+\\delta}[\\delta]$, so that $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ for some $X$.\n    At the same time, by \\cref{theorem:decomposition}, there exists a decomposition $N \\cong \\bigoplus_{i \\in I} N_i$ with $N_i$ indecomposable for all $i \\in I$.\n    Using \\cref{lemma:decomposition-restriction-extension}, we see that, by restricting to $\\Pscr$ and extending, we get a decomposition $N_{\\Pscr+\\delta} \\cong \\bigoplus_{i \\in I} (N_i)_{\\Pscr+\\delta}$, where now the summands $\\{(N_i)_{\\Pscr+\\delta}\\}_{i \\in I}$ may not be indecomposable anymore.\n    Nevertheless, since $M_\\Pscr \\cong M$ is indecomposable by assumption, there exists $i \\in I$ such that $(N_i)_{\\Pscr+\\delta}\\cong M_\\Pscr[-\\delta] \\oplus X'$ for some $X'$, using the fact that $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ and \\cref{theorem:decomposition}.\n    We consider the decomposition\n    \\[\n        N_{\\Pscr + \\delta}\\;\\; \\cong \\;\\;(N_i)_{\\Pscr + \\delta} \\oplus \\bigoplus_{j \\in I \\setminus i} (N_j)_{\\Pscr + \\delta} \\;\\;\\cong \\;\\;(M_\\Pscr[-\\delta] \\oplus X') \\oplus \\bigoplus_{j \\in I \\setminus i} (N_j)_{\\Pscr + \\delta},\n    \\]\n    and let $A = N_i$ and $B = \\bigoplus_{j \\in I \\setminus i} N_j$.\n    Since this will be of use later, we note here that, by construction, $B_\\Pscr$ is isomorphic to a summand of $X$.\n    \n    We have thus decomposed $N \\cong A \\oplus B$ with $A$ indecomposable.\n    To conclude, it remains to show that $B$ is $\\epsilon$-trivial, and for this it is enough to show that $d_I(B,0) < \\epsilon\/2$.\n    By definition of $\\Pscr$, we have $d_I(B,B_\\Pscr) < \\alpha \\leq \\epsilon\/4$, which reduces the problem to showing that $d_I(B_\\Pscr,0) < \\epsilon\/4$, by the triangle inequality.\n    Since $B_\\Pscr$ is a summand of $X$, it is enough to prove that $d_I(X,0) < \\epsilon\/4$.\n    We will show that $\\eta^X_{\\alpha} : X \\to X[\\alpha]$ is the zero morphism, which implies that $d_I(X,0) \\leq \\alpha\/2 \\leq \\epsilon\/8 < \\epsilon\/4$.\n\n    By the naturality of shifting, that is, by the commutativity of the following square\n    \\[\n    \\begin{tikzpicture}\n        \\matrix (m) [matrix of math nodes,row sep=4em,column sep=7em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n        {  M_\\Pscr \\oplus X  &  M_\\Pscr[\\alpha] \\oplus X[\\alpha] \\\\\n           N_{\\Pscr + \\delta} [\\delta] & N_{\\Pscr + \\delta}[\\delta + \\alpha],\\\\};\n        \\path[line width=0.75pt, -{>[width=8pt]}]\n        (m-1-1) edge node [above] {$\\left(\\eta^{M_\\Pscr}_\\alpha, \\eta^X_\\alpha\\right)$} (m-1-2)\n        (m-1-1) edge node [left] {$\\cong$} (m-2-1)\n        (m-1-2) edge node [left] {$\\cong$} (m-2-2)\n        (m-2-1) edge node [above] {$\\eta^{N_{\\Pscr+\\delta}[\\delta]}_\\alpha$} (m-2-2)\n        ;\n    \\end{tikzpicture}\n    \\]\n    it is sufficient to show that the following composite is the zero morphism:\n    \\begin{equation}\n        \\label{equation:sufficient-zero-morphism}\n        X \\rightarrowtail M_{\\Pscr} \\oplus X \\xrightarrow{\\cong} N_{\\Pscr+\\delta}[\\delta] \\xrightarrow{\\eta^{N_{\\Pscr+\\delta}[\\delta]}_\\alpha} N_{\\Pscr+\\delta}[\\delta + \\alpha].\n    \\end{equation}\n    Note that, by the construction of the splitting $N_{\\Pscr+\\delta}[\\delta] \\cong M_{\\Pscr} \\oplus X$ and the commutativity of the right triangle in \\cref{equation:decomposition-N}, the following composite is the zero morphism\n    \\begin{equation}\n        \\label{equation:zero-morphism}\n        X \\rightarrowtail M_{\\Pscr} \\oplus X \\xrightarrow{\\cong} N_{\\Pscr+\\delta}[\\delta]\n        \\xrightarrow{(\\eta^{N[\\delta]}_{2\\delta})_{\\Pscr}} N_{\\Pscr + 3\\delta}[3\\delta].\n    \\end{equation}\n    Thus, it suffices to see that the right-most morphism in \\cref{equation:sufficient-zero-morphism} factors through the right-most morphism in \\cref{equation:zero-morphism}.\n    Letting $L = N[\\delta]$ and using \\cref{equation:shift-and-restriction-extension},\n    this translates to the claim that the morphism \n    $\\eta^{L_{\\Pscr}}_\\alpha : L_{\\Pscr} \\to L_{\\Pscr+\\delta}[\\delta + \\alpha]$ factors through $(\\eta^{L}_{2\\delta})_{\\Pscr} : L_{\\Pscr} \\to L_{\\Pscr + 2\\delta}[2\\delta]$, which follows from \\cref{lemma:factor-bounded-grid} by letting $r = 2\\delta = \\alpha\/2 < \\alpha$.\n\\end{proof}\n\n\\begin{corollary}\n    \\label{corollary:open-with-indecomposables}\n    Let $n \\geq 1$.\n    For every $\\epsilon \\geq 0$, there exists an open set $O^\\epsilon$ of isomorphism classes of finitely presentable $n$-parameter persistence modules, containing the set of all indecomposable finitely presentable modules as a subset, and such that every $N \\in O^\\epsilon$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $d_I(B,0) < \\epsilon$.\n\\end{corollary}\n\\begin{proof}\n    Fix $\\epsilon > 0$.\n    By \\cref{proposition:stability-indecomposability}, for every indecomposable finitely presentable $M : \\Rbf^n \\to \\mathbf{vec}$ there exists $\\delta^M > 0$ such that, for all finitely presentable $N : \\Rbf^n \\to \\mathbf{vec}$ in the open $\\delta^M$-ball around $M$ in the interleaving distance, it holds that $N$ decomposes as $N \\cong A \\oplus B$ with $A$ indecomposable and $B$ $\\epsilon$-trivial.\n    It is then clear that the set\n    \\[\n        O^\\epsilon := \\bigcup_{\\substack{M \\text{ finitely presentable} \\\\ \\text{and indecomposable}}} \\{N : \\text{ $N$ is finitely presentable and } d_I(M,N) < \\delta^M \\}\n    \\]\n    satisfies the conditions in the statement.\n\\end{proof}\n\n\n\n\\section{Indecomposables are dense}\n\\label{section:indecomposables-dense}\n\n\nThe proof of \\cref{theorem:indecomposables-dense} depends on the following key lemma, which lets us ``tack together'' two indecomposable modules to obtain a third indecomposable module that is at small interleaving distance from the direct sum of the initial modules.\nThe proof is in \\cref{section:proof-main-lemma}.\n\n\\begin{restatable}[Tacking Lemma]{lemma}{mainlemma}\n    \\label{lemma:main-lemma-attaching}\n    Let $A,B : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and isomorphic to extensions of $\\Pscr$-persistence modules for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    For every $\\delta > 0$ there exists a regular grid $\\Qscr \\supseteq \\Pscr$ and $M : \\Rbf^2 \\to \\mathbf{vec}$ with $M$ indecomposable, finitely presentable, isomorphic to the extension of a $\\Qscr$-persistence module, and such that\n    \\[\n        \\mathsf{im}\\left(\\eta^M_\\delta\\right) \\cong \\mathsf{im}\\left(\\eta^A_\\delta\\right) \\oplus \\mathsf{im}\\left(\\eta^B_\\delta\\right).\n    \\]\n\\end{restatable}\n\nWe also need the following two simple lemmas.\n\n\\begin{lemma}\n    \\label{lemma:image-to-interleaving}\n    Let $A : \\Rbf^n \\to \\mathbf{vec}$ and let $\\delta \\geq 0$.\n    Consider $\\mathsf{im}(\\eta^A_\\delta ) : \\Rbf^n \\to \\mathbf{vec}$, the image of the morphism of persistence modules $\\eta^A_\\delta : A \\to A[\\delta]$ given by the structure morphisms.\n    Then $A$ and $\\mathsf{im}(\\eta^A_\\delta)$ are $\\delta$-interleaved.\n    As a consequence, if $B : \\Rbf^n \\to \\mathbf{vec}$ satisfies $\\mathsf{im}(\\eta^A_\\delta) \\cong \\mathsf{im}(\\eta^B_\\delta)$, then $A$ and $B$ are $2\\delta$-interleaved.\n\\end{lemma}\nIn fact, the $2\\delta$ can be improved to $\\delta$ using the notion of asymmetric interleaving \\cite[Section~2.6.1]{lesnick-thesis}, but this is not necessary for our purposes.\n\\begin{proof}\n    The second statement follows directly from the first one.\n    By construction, there is a canonical epimorphism $A \\to \\mathsf{im}(\\eta^A_\\delta)$, and thus we get a morphism $f : A \\to \\mathsf{im}(\\eta^A_\\delta)[\\delta]$, by composing with $\\eta^{\\mathsf{im}(\\eta^A_\\delta)}_\\delta: \\mathsf{im}(\\eta^A_\\delta) \\to \\mathsf{im}(\\eta^A_\\delta)[\\delta]$.\n    Similarly, there is a canonical monomorphism $g : \\mathsf{im}(\\eta^A_\\delta) \\to A[\\delta]$.\n    A straightforward check shows that $f$ and $g$ form a $\\delta$-interleaving.\n\\end{proof}\n\n\\begin{lemma}\n    \\label{lemma:eta-direct-sum}\n    Let $A,B : \\Rbf^n \\to \\mathbf{vec}$ and $\\epsilon \\geq 0$.\n    Then $\\mathsf{im}\\left(\\eta^A_\\epsilon\\right) \\oplus\\mathsf{im}\\left(\\eta^B_\\epsilon\\right) \\cong \\mathsf{im}\\left(\\eta^{A \\oplus B}_\\epsilon\\right)$.\n    \\qed\n\\end{lemma}\n\n\\indecomposabledense*\n\\begin{proof}\n    We start by reducing the problem to the case in which the given module is an extension of a module defined over a regular grid.\n    Consider the regular grid $\\Pscr = ((\\epsilon\/2)\\Zbf)^2$ and note that $d_I(N_\\Pscr,N) \\leq \\epsilon\/2$.\n    Let $L = N_\\Pscr$.\n    It is easy to see that $L$ is finitely presentable.\n\n    If $L = 0$, then take $M$ to be the persistence module such that $M(x,y) = \\kbb$ if $0 \\leq x < \\epsilon$ and $y \\geq 0$, and $0$ otherwise, with all the structure morphisms that are not forced to be the zero morphism being the identity $\\kbb \\to \\kbb$.\n    It is straightforward to see that $M$ is indecomposable and that $d_I(M,0) = \\epsilon\/2$.\n    This case now follows from $d_I(N,M) \\leq d_I(N,L) + d_I(L,M) \\leq \\epsilon\/2 + d_I(0,M) \\leq \\epsilon$.\n\n    We now assume that $L$ is non-zero.\n    By \\cref{lemma:decomposition-fp}, there exist finitely presentable indecomposables $X_1, \\dots, X_k : \\Rbf^2 \\to \\mathbf{vec}$ such that $L \\cong X_1 \\oplus \\dots \\oplus X_k$.\n    Note that, for all $1 \\leq i \\leq k$, the persistence module $X_i$ is isomorphic to the extension of a $\\Qscr$-persistence module for any grid $\\Qscr \\supseteq \\Pscr$.\n    Let us define $M_i : \\Rbb^2 \\to \\mathbf{vec}$ for $1 \\leq i \\leq k$ inductively.\n    Let $M_1 = X_1$ and, for $2 \\leq i \\leq k$, let $M_{i}$ be obtained by setting $A = X_i$, $B = M_{i-1}$, and $\\delta = \\epsilon\/4$ in \\cref{lemma:main-lemma-attaching}.\n    Thus, $M_i$ is finitely presentable, indecomposable, and satisfies\n    \\[\n        \\mathsf{im}\\left(\\eta^{M_i}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_i}_{\\epsilon\/4}\\right) \\oplus \\mathsf{im}\\left(\\eta^{M_{i-1}}_{\\epsilon\/4}\\right).\n    \\]\n    Let $M = M_k$.\n    It follows by induction and \\cref{lemma:eta-direct-sum} that \n    \\[\n        \\mathsf{im}\\left(\\eta^{M}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_1}_{\\epsilon\/4}\\right) \\oplus \\dots \\oplus \\mathsf{im}\\left(\\eta^{X_k}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^{X_1 \\oplus \\dots \\oplus X_k}_{\\epsilon\/4}\\right) \\cong \\mathsf{im}\\left(\\eta^L_{\\epsilon\/4}\\right).\n    \\]\n    Using \\cref{lemma:image-to-interleaving}, we get that $M$ and $L$ are $\\epsilon\/2$-interleaved.\n    By the triangle inequality, we get $d_I(M,N) \\leq d_I(M,L) + d_I(L,N) \\leq \\epsilon$, as required.\n\\end{proof}\n\nWe can now give the proof of our main result.\n\n\\begin{proof}[Proof of \\cref{theorem:main-theorem}]\n    Given $\\epsilon > 0$, the set $O^\\epsilon$ of \\cref{corollary:open-with-indecomposables} is open and contains all indecomposables finitely presentable modules.\n    Since, when $n=2$, the set of indecomposables is dense by \\cref{theorem:indecomposables-dense}, the result follows.\n\\end{proof}\n\n\n\\section{Tacking indecomposables together}\n\\label{section:proof-main-lemma}\n\nIn this section, we prove \\cref{lemma:main-lemma-attaching} (the Tacking Lemma).\nWe first summarize the strategy.\n\n\n\\begin{proof}[Proof outline]\\renewcommand{\\qedsymbol}{}\nWe start by defining a persistence module $\\Gsf$ on a finite grid which allows us to ``tack together'' indecomposable modules.\nNow, fix modules $A$ and $B$ as in the statement.\nThe tacking works in three steps.\nIn \\cref{lemma:add-1-dim-corner}, we replace $A$ and $B$ by modules which admit a ``thin corner'' (\\cref{definition:1-dim-corner}); the purpose of this thin corner is that it allows us to attach a copy of $\\Gsf$ to each of the modules, which we do in the next step.\nIn \\cref{lemma:add-antenna-attachment}, we replace the modules by modules that admit a ``horizontal antenna attachment'' (\\cref{definition:horizontal-antenna-attachment}); the purpose of this horizontal antenna attachment is that it allows us to tack together the two modules with a third copy of $\\Gsf$. \nFinally, in \\cref{lemma:actual-tacking}, we construct $M$ by tacking together the two modules.\n\\cref{figure:diagram-proof-main-lemma} shows a diagrammatic description of these steps.\n\\end{proof}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{pictures\/diagram-steps.eps}\n    \\caption{A schematic summary of the main steps in the proof of the tacking lemma.\n    The transition from (0.) to (1.) corresponds to \\cref{lemma:add-1-dim-corner}; the transition from (1.) to (2.) corresponds to \\cref{lemma:add-antenna-attachment}; the transition from (2.) to (3.) corresponds to \\cref{lemma:actual-tacking}.}\n    \\label{figure:diagram-proof-main-lemma}\n\\end{figure}\n\n\\begin{definition}\n\\label{def:G}\n    Let $\\Pscr = \\{0, 1, 2, 3, 4\\}^2$ and define $\\Gsf : \\Pscr \\to \\mathbf{vec}$ as follows:\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n               \\kbb & \\kbb^2 & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n               0    & \\kbb   & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n               0    & 0      & \\kbb   & \\kbb^2 & \\kbb \\\\\n               0    & 0      & 0      & \\kbb   & \\kbb \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-1) edge [-,double equal sign distance] (m-1-2)\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n\n            (m-2-2) edge [-,double equal sign distance] (m-2-3)\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n\n            (m-3-3) edge [-,double equal sign distance] (m-3-4)\n\n            (m-5-5) edge [-,double equal sign distance] (m-4-5)\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n            (m-2-5) edge [-,double equal sign distance] (m-1-5)\n\n            (m-4-4) edge [-,double equal sign distance] (m-3-4)\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n\n            (m-3-3) edge [-,double equal sign distance] (m-2-3)\n\n            (m-2-1) edge node [left] {\\footnotesize $0$} (m-1-1)\n            (m-2-1) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-2-2)\n            (m-2-2) edge node [right] {\\footnotesize $(1,-1)$} (m-1-2)\n            (m-2-3) edge node [right] {\\footnotesize $(1,-1)$} (m-1-3)\n            (m-2-4) edge node [right] {\\footnotesize $(1,-1)$} (m-1-4)\n\n            (m-5-4) edge node [above] {\\footnotesize $0$} (m-5-5)\n            (m-5-4) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-4-4)\n            (m-4-4) edge node [above] {\\footnotesize $(1,-1)$} (m-4-5)\n            (m-3-4) edge node [above] {\\footnotesize $(1,-1)$} (m-3-5)\n            (m-2-4) edge node [above] {\\footnotesize $(1,-1)$} (m-2-5)\n\n            (m-3-2) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-2-2)\n            (m-3-2) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-3-3)\n\n            (m-4-3) edge node [left] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-3-3)\n            (m-4-3) edge node [above] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-4-4)\n\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n\n            (m-4-2) edge (m-3-2)\n            (m-5-2) edge (m-4-2)\n\n            (m-5-3) edge (m-4-3)\n\n            (m-5-1) edge (m-5-2)\n            (m-5-2) edge (m-5-3)\n            (m-5-3) edge (m-5-4)\n\n            (m-4-1) edge (m-4-2)\n            (m-4-2) edge (m-4-3)\n        \n            (m-3-1) edge (m-3-2)\n            ;\n        \\end{tikzpicture}\n    \\]\n\\end{definition}\n\n\n\n\\begin{restatable}{lemma}{Gindecomposable}\n    \\label{lemma:G-is-indecomposable}\n    The persistence module $\\Gsf$ is indecomposable.\n    More specifically, the morphism of rings $\\End(\\Gsf) \\to \\End(\\Gsf(4,4)) \\cong \\kbb$ given by evaluation at $(4,4)$ is an isomorphism.\n\\end{restatable}\n\n\\begin{proof}\n\n\n    Let $f : \\Gsf \\to \\Gsf$ be an endomorphism.\n    If $f(4,4) : \\kbb \\to \\kbb$ is given by multiplication by $\\alpha$, then, for all $0 \\leq j \\leq 4$, we have that $f(4,j) : \\kbb \\to \\kbb$ and $f(j,4) : \\kbb \\to \\kbb$ must also be given by multiplication by $\\alpha$.\n\n    Assume that $f(1,2) : \\kbb \\to \\kbb$ is given by multiplication by $\\beta$ and that $f(2,1) : \\kbb \\to \\kbb$ is given by multiplication by $\\gamma$.\n    Since the structure morphisms $\\kbb = \\Gsf(1,2) \\to \\Gsf(4,4) = \\kbb$ is the identity, we must have $\\beta = \\alpha$.\n    By an analogous argument, we have $\\gamma = \\alpha$.\n    This also implies that for all $(j,k)$ with $\\Gsf(j,k) = \\kbb^2$ the endomorphism $f(j,k)$ is given by multiplication by $\\alpha$.\n   \n    \n    To conclude, note that, by the definition of the structure morphisms $\\Gsf(0,3) \\to \\Gsf(1,3)$ and $\\Gsf(3,0) \\to \\Gsf(3,1)$, we have that $f(0,3)$ and $f(3,0)$ are also given by multiplication by $\\alpha$.\n    Thus, $\\End(\\Gsf) \\cong \\End(\\Gsf(4,4)) \\cong \\kbb$, and thus $\\Gsf$ is indecomposable, by \\cref{lemma:indecomposable-local-ring}.\n\\end{proof}\n\nThe crucial property of $\\Gsf$ of importance to us, other than being indecomposable, is the fact that there are two pairs of adjacent copies of the ground field $\\kbb$ connected by a zero map, on the bottom right at $(0,3)$ and $(0,4)$, and on the top left at $(3,0)$ and $(4,0)$.\n\n\\begin{definition}\n    \\label{definition:1-dim-corner}\n    Let $X : \\Rbf^2 \\to \\mathbf{vec}$.\n    We say that $X$ \\define{admits a thin corner} if there exists $\\epsilon > 0$, $X' : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$, and $(x,y) \\in (\\epsilon \\Zbf)^2$ such that $X$ is isomorphic to the extension of $X'$, and we have $X'(x,y) = \\kbb$ and $X'(x-\\epsilon,y) = X'(x,y-\\epsilon) = 0$.\n\\end{definition}\n\nSee \\cref{figure:diagram-proof-main-lemma}(1.) for a schematic depiction of two modules admitting a thin corner.\n\n\\begin{lemma}\n    \\label{lemma:add-1-dim-corner}\n    Let $A : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and isomorphic to the extension of a $\\Pscr$-persistence module for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    For every $\\delta > 0$, there exists $A_1 : \\Rbf^2 \\to \\mathbf{vec}$ indecomposable, finitely presentable, admitting a thin corner, isomorphic to an extension of a $\\Qscr$-persistence module for some regular grid $\\Qscr \\supseteq \\Pscr$, and such that $\\mathsf{im}\\left(\\eta^A_\\delta\\right) \\cong \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n\\end{lemma}\n\\begin{proof}\n    Assume that $\\Pscr = (\\epsilon \\Zbf)^2$.\n    Let $m \\in \\Nbb$ be such that $\\epsilon\/m < \\delta$.\n    Let $\\tau = \\epsilon\/m$.\n    Since $A$ is finitely presentable, there exists $(x,y) \\in (\\tau\\Zbf)^2$ such that $A(x,y) \\neq 0$ and $A(x-\\tau,y) = 0 = A(x,y-\\tau)$.\n    Thus, on the regular grid $((\\tau\/2)\\Zbf)^2$, and around index $(x,y)$ (highlighted entry), the module $A$ restricts as follows:\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=1.5em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  A(x-\\tau,y+\\tau) & A(x-\\tau\/2,y+\\tau) & A(x,y+\\tau) \n                   & A(x+\\tau\/2,y+\\tau)   & A(x+\\tau,y+\\tau) \\\\\n               0 & 0 & A(x,y+\\tau\/2)\n                   & A(x+\\tau\/2,y+\\tau\/2) & A(x+\\tau,y+\\tau\/2) \\\\\n               0    & 0      & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| A(x,y) & A(x+\\tau\/2,y) & A(x+\\tau,y) \\\\\n               0    & 0      & 0      & 0 & A(x+\\tau,y-\\tau\/2) \\\\\n               0    & 0      & 0      & 0   & A(x+\\tau,y-\\tau) \\\\};\n            \\path[line width=0.5pt, -{>[width=4pt]}]\n            (m-1-1) edge node [above] {$\\cong$} (m-1-2)\n            (m-2-1) edge node [above] {$\\cong$} (m-2-2)\n            (m-3-1) edge node [above] {$\\cong$} (m-3-2)\n            (m-4-1) edge node [above] {$\\cong$} (m-4-2)\n            (m-5-1) edge node [above] {$\\cong$} (m-5-2)\n\n            (m-1-2) edge (m-1-3)\n            (m-2-2) edge (m-2-3)\n            (m-3-2) edge (m-3-3)\n            (m-4-2) edge (m-4-3)\n            (m-5-2) edge (m-5-3)\n\n            (m-1-3) edge node [above] {$\\cong$} (m-1-4)\n            (m-2-3) edge node [above] {$\\cong$} (m-2-4)\n            (m-3-3) edge node [above] {$\\cong$} (m-3-4)\n            (m-4-3) edge node [above] {$\\cong$} (m-4-4)\n            (m-5-3) edge node [above] {$\\cong$} (m-5-4)\n\n            (m-1-4) edge (m-1-5)\n            (m-2-4) edge (m-2-5)\n            (m-3-4) edge (m-3-5)\n            (m-4-4) edge (m-4-5)\n            (m-5-4) edge (m-5-5)\n\n            (m-5-1) edge node [left] {$\\cong$} (m-4-1)\n            (m-5-2) edge node [left] {$\\cong$} (m-4-2)\n            (m-5-3) edge node [left] {$\\cong$} (m-4-3)\n            (m-5-4) edge node [left] {$\\cong$} (m-4-4)\n            (m-5-5) edge node [left] {$\\cong$} (m-4-5)\n\n            (m-4-1) edge (m-3-1)\n            (m-4-2) edge (m-3-2)\n            (m-4-3) edge (m-3-3)\n            (m-4-4) edge (m-3-4)\n            (m-4-5) edge (m-3-5)\n\n            (m-3-1) edge node [left] {$\\cong$} (m-2-1)\n            (m-3-2) edge node [left] {$\\cong$} (m-2-2)\n            (m-3-3) edge node [left] {$\\cong$} (m-2-3)\n            (m-3-4) edge node [left] {$\\cong$} (m-2-4)\n            (m-3-5) edge node [left] {$\\cong$} (m-2-5)\n\n            (m-2-1) edge (m-1-1)\n            (m-2-2) edge (m-1-2)\n            (m-2-3) edge (m-1-3)\n            (m-2-4) edge (m-1-4)\n            (m-2-5) edge (m-1-5)\n            ;\n        \\end{tikzpicture}\n    \\]\n    where the isomorphisms are due to the fact that $A$ is an extension of a persistence module on the regular grid $(\\tau\\Zbf)^2$.\n\n    Let $\\kbb \\to A(x,y)$ be any non-zero morphism.\n    We now define a persistence module $A'$, and then define $A_1$ to be an extension of $A'$.\n    Let $A' : ((\\tau\/2)\\Zbf)^2 \\to \\mathbf{vec}$ coincide with the restriction of $A$ to $((\\tau\/2)\\Zbf)^2$ at all places, except at $(x,y)$ where it takes the value $\\kbb$.\n    The structure morphisms $A'(x,y) \\to A'(x,y+\\tau\/2) = A(x,y+\\tau\/2) $ and $A'(x,y) \\to A'(x+\\tau\/2, y) = A(x+\\tau\/2, y)$ are defined by composing the corresponding structure morphisms of $A$ with the map $\\kbb \\to A(x,y)$.\n    Thus, around index $(x,y)$ (again highlighted), the module $A'$ looks as follows:\n    \\[  \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=1.5em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            {  A(x-\\tau,y+\\tau) & A(x-\\tau\/2,y+\\tau) & A(x,y+\\tau) \n                   & A(x+\\tau\/2,y+\\tau)   & A(x+\\tau,y+\\tau) \\\\\n               0 & 0 & A(x,y+\\tau\/2)\n                   & A(x+\\tau\/2,y+\\tau\/2) & A(x+\\tau,y+\\tau\/2) \\\\\n               0    & 0      & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| \\kbb & A(x+\\tau\/2,y) & A(x+\\tau,y) \\\\\n               0    & 0      & 0      & 0 & A(x+\\tau,y-\\tau\/2) \\\\\n               0    & 0      & 0      & 0   & A(x+\\tau,y-\\tau) \\\\};\n            \\path[line width=0.5pt, -{>[width=4pt]}]\n            (m-1-1) edge (m-1-2)\n            (m-2-1) edge (m-2-2)\n            (m-3-1) edge (m-3-2)\n            (m-4-1) edge (m-4-2)\n            (m-5-1) edge (m-5-2)\n\n            (m-1-2) edge (m-1-3)\n            (m-2-2) edge (m-2-3)\n            (m-3-2) edge (m-3-3)\n            (m-4-2) edge (m-4-3)\n            (m-5-2) edge (m-5-3)\n\n            (m-1-3) edge (m-1-4)\n            (m-2-3) edge (m-2-4)\n            (m-3-3) edge (m-3-4)\n            (m-4-3) edge (m-4-4)\n            (m-5-3) edge (m-5-4)\n\n            (m-1-4) edge (m-1-5)\n            (m-2-4) edge (m-2-5)\n            (m-3-4) edge (m-3-5)\n            (m-4-4) edge (m-4-5)\n            (m-5-4) edge (m-5-5)\n\n            (m-5-1) edge (m-4-1)\n            (m-5-2) edge (m-4-2)\n            (m-5-3) edge (m-4-3)\n            (m-5-4) edge (m-4-4)\n            (m-5-5) edge (m-4-5)\n\n            (m-4-1) edge (m-3-1)\n            (m-4-2) edge (m-3-2)\n            (m-4-3) edge (m-3-3)\n            (m-4-4) edge (m-3-4)\n            (m-4-5) edge (m-3-5)\n\n            (m-3-1) edge (m-2-1)\n            (m-3-2) edge (m-2-2)\n            (m-3-3) edge (m-2-3)\n            (m-3-4) edge (m-2-4)\n            (m-3-5) edge (m-2-5)\n\n            (m-2-1) edge (m-1-1)\n            (m-2-2) edge (m-1-2)\n            (m-2-3) edge (m-1-3)\n            (m-2-4) edge (m-1-4)\n            (m-2-5) edge (m-1-5)\n            ;\n        \\end{tikzpicture}\n    \\]\n    We have the following isomorphisms of endomorphism rings:\n    \\[\n        \\End(A) \\cong \\End\\left(A|_{((\\tau\/2)(2\\Zbf+1))^2}\\right) = \\End\\left(A'|_{((\\tau\/2)(2\\Zbf+1))^2}\\right) \\cong \\End\\left(A'|_{((\\tau\/2)\\Zbf)^2}\\right).\n    \\]\n    The first isomorphism is due to the fact that $A$ is isomorphic to the extension of a $(\\tau\\Zbf)^2$-persistence module.\n    The equality is because\n    $A|_{((\\tau\/2)(2\\Zbf+1))^2} = A'|_{((\\tau\/2)(2\\Zbf+1))^2}$.\n    For the last isomorphism, note that the value of $A'$ at any element of the grid $((\\tau\/2)\\Zbf)^2$ other than $(x,y)$ is isomorphic, through a structure morphism of $A'$, to the value of $A'$ at an element of the grid $((\\tau\/2)(2\\Zbf+1))^2$, by construction.\n    Moreover, by construction, the structure morphism $A'(x,y) = \\kbb \\to A'(x+\\tau\/2, y+\\tau\/2)$ is non-zero, and thus a monomorphism; and $(x+\\tau\/2, y+\\tau\/2) \\in ((\\tau\/2)(2\\Zbf+1))^2$.\n    It follows that the action of an endomorphism of $A'$ is completely determined by the action of the endomorphism on the restriction $A'|_{((\\tau\/2)(2\\Zbf+1))^2}$, proving the last isomorphism.\n\n    Thus, $A'$ is indecomposable by \\cref{lemma:indecomposable-local-ring} and the fact that $A$ is indecomposable.\n    Define $A_1 = \\widehat{A'}$, the extension of $A'$ to $\\Rbf^2$.\n    Note that $A_1$ is indecomposable, finitely presentable, and satisfies $\\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big) = \\mathsf{im}\\left(\\eta^{A}_\\delta\\right)$.\n    Moreover, $A_1$ admits a thin corner, by construction.\n\\end{proof}\n\n\\begin{definition}\n    \\label{definition:horizontal-antenna-attachment}\n    Let $X : \\Rbf^2 \\to \\mathbf{vec}$ and $(x,y) \\in \\Rbf^2$.\n    We say that $X$ \\define{admits a horizontal antenna attachment} at $(x,y)$ if there exists $\\epsilon > 0$ and $X' : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$, such that $(x,y) \\in (\\epsilon \\Zbf)^2$, $X$ is isomorphic to the extension of $X'$, and we have $X'(x,y) = \\kbb$, $X'(x-k\\epsilon,y) = 0$ for all $k \\geq 1 \\in \\Nbb$, and the structure morphism $X'(x,y) \\to X'(x,y+\\epsilon)$ is zero.\n\\end{definition}\n\nFor example, the extension $\\widehat\\Gsf: \\Rbf^2 \\to \\mathbf{vec}$ admits a horizontal antenna attachment.\nSee \\cref{figure:diagram-step-3} for a schematic depiction of two modules admitting a horizontal antenna attachment.\n\n\\begin{lemma}\n    \\label{lemma:add-antenna-attachment}\n    Let $A : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable, indecomposable, and admitting a thin corner.\n    For every $\\delta > 0$, there exists $A_1 : \\Rbf^2 \\to \\mathbf{vec}$ indecomposable, finitely presentable, admitting a horizontal antenna attachment, and such that $\\mathsf{im}\\left(\\eta^A_\\delta\\right) \\cong \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n\\end{lemma}\n\n\\begin{proof}\n    Since $A$ admits a thin corner, we may assume the following:\n    there exists a regular grid $(\\epsilon \\Zbf)^2$ for some $\\epsilon > 0$ and $X : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$ such that $A \\cong \\widehat{X}$, and such that there exists $(x,y) \\in (\\epsilon \\Zbf)^2$ with $X(x,y) = \\kbb$ and $X(x-\\epsilon,y) = X(x,y-\\epsilon) = 0$.\n\n    Let $m \\in \\Nbb$ be such that $\\epsilon\/m < \\delta$ and let $\\tau = \\epsilon\/m$.\n    We make use of the regular grid $((\\tau\/5)\\Zbf)^2$.\n    By extending $X$ to $\\Rbf^2$ and restricting to $((\\tau\/5)\\Zbf)^2$, we get a module $X' : ((\\tau\/5)\\Zbf)^2 \\to \\mathbf{vec}$ that, when restricted to the finite grid\n    \\begin{align*}\n        \\Tscr = \\{(&x-\\tau\/5,x,x+\\tau\/5,x+2\\tau\/5,x+3\\tau\/5,x+4\\tau\/5)\\}\\\\\n        &\\times \\{(y-\\tau\/5,y,y+\\tau\/5,y+2\\tau\/5,y+3\\tau\/5,y+4\\tau\/5)\\},\n    \\end{align*}\n    looks as follows (index $(x,y)$ highlighted):\n    \\[\\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            { 0 & \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n              0 & \\kbb & \\kbb & \\kbb & \\kbb & \\kbb \\\\\n              0 &  \\kbb    & \\kbb   & \\kbb & \\kbb & \\kbb \\\\\n             0 &   \\kbb    & \\kbb      & \\kbb   & \\kbb & \\kbb \\\\\n             0 &  |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]| \\kbb    & \\kbb      & \\kbb      & \\kbb   & \\kbb \\\\\n             0 &      0    & 0         & 0         & 0      &  0 \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n            (m-1-5) edge [-,double equal sign distance] (m-1-6)\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n            (m-2-4) edge [-,double equal sign distance] (m-2-5)\n            (m-3-4) edge [-,double equal sign distance] (m-3-5)\n            (m-5-6) edge [-,double equal sign distance] (m-4-6)\n            (m-4-6) edge [-,double equal sign distance] (m-3-6)\n            (m-3-6) edge [-,double equal sign distance] (m-2-6)\n            (m-2-6) edge [-,double equal sign distance] (m-1-6)\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n            (m-2-2) edge [-,double equal sign distance] (m-1-2)\n            (m-2-2) edge [-,double equal sign distance] (m-2-3)\n            (m-2-3) edge [-,double equal sign distance] (m-1-3)\n            (m-2-4) edge [-,double equal sign distance] (m-1-4)\n            (m-2-5) edge [-,double equal sign distance] (m-1-5)\n            (m-5-5) edge [-,double equal sign distance] (m-5-6)\n            (m-5-5) edge [-,double equal sign distance] (m-4-5)\n            (m-4-5) edge [-,double equal sign distance] (m-4-6)\n            (m-3-5) edge [-,double equal sign distance] (m-3-6)\n            (m-2-5) edge [-,double equal sign distance] (m-2-6)\n            (m-3-3) edge [-,double equal sign distance] (m-2-3)\n            (m-3-3) edge [-,double equal sign distance] (m-3-4)\n            (m-4-4) edge [-,double equal sign distance] (m-3-4)\n            (m-4-4) edge [-,double equal sign distance] (m-4-5)\n            (m-3-2) edge [-,double equal sign distance] (m-2-2)\n            (m-4-2) edge [-,double equal sign distance] (m-3-2)\n            (m-5-2) edge [-,double equal sign distance] (m-4-2)\n            (m-4-3) edge [-,double equal sign distance] (m-3-3)\n            (m-5-3) edge [-,double equal sign distance] (m-4-3)\n            (m-5-4) edge [-,double equal sign distance] (m-4-4)\n            (m-5-2) edge [-,double equal sign distance] (m-5-3)\n            (m-5-3) edge [-,double equal sign distance] (m-5-4)\n            (m-5-4) edge [-,double equal sign distance] (m-5-5)\n            (m-4-2) edge [-,double equal sign distance] (m-4-3)\n            (m-4-3) edge [-,double equal sign distance] (m-4-4)\n            (m-3-2) edge [-,double equal sign distance] (m-3-3)\n\n            (m-1-1) edge (m-1-2)\n            (m-2-1) edge (m-2-2)\n            (m-3-1) edge (m-3-2)\n            (m-4-1) edge (m-4-2)\n            (m-5-1) edge (m-5-2)\n            (m-6-1) edge (m-6-2)\n\n            (m-6-2) edge (m-6-3)\n            (m-6-3) edge (m-6-4)\n            (m-6-4) edge (m-6-5)\n            (m-6-5) edge (m-6-6)\n\n            (m-2-1) edge (m-1-1)\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n            (m-6-1) edge (m-5-1)\n\n            (m-6-2) edge (m-5-2)\n            (m-6-3) edge (m-5-3)\n            (m-6-4) edge (m-5-4)\n            (m-6-5) edge (m-5-5)\n            (m-6-6) edge (m-5-6)\n            ;\n        \\end{tikzpicture}\n    \\]\n    Let $A' : ((\\tau\/5)\\Zbf)^2 \\to \\mathbf{vec}$ coincide with $X'$ at all places, except at the subgrid $\\Tscr$, where we use a copy of the module $\\Gsf$ as follows (index $(x,y)$ highlighted):\n    \\[ \\footnotesize\n        \\begin{tikzpicture}\n            \\matrix (m) [matrix of math nodes,row sep=3em,column sep=3em,minimum width=2em,nodes={text height=1.75ex,text depth=0.25ex}]\n            { 0 & \\kbb & \\kbb   & \\kbb   & \\kbb   & \\kbb \\\\\n              0 & \\kbb & \\kbb^2 & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n              0 & 0    & \\kbb   & \\kbb^2 & \\kbb^2 & \\kbb \\\\\n              0 & 0    & 0      & \\kbb   & \\kbb^2 & \\kbb \\\\\n              0 & |[fill=lightgray!25, rectangle, outer sep = 2pt, minimum size = 0]|0    & 0      & 0      & \\kbb   & \\kbb \\\\\n              0 & 0    & 0      & 0      & 0      & 0    \\\\};\n            \\path[line width=0.5pt, -{>[width=6pt]}]\n            (m-1-2) edge [-,double equal sign distance] (m-1-3)\n            (m-1-3) edge [-,double equal sign distance] (m-1-4)\n            (m-1-4) edge [-,double equal sign distance] (m-1-5)\n            (m-1-5) edge [-,double equal sign distance] (m-1-6)\n\n            (m-2-3) edge [-,double equal sign distance] (m-2-4)\n            (m-2-4) edge [-,double equal sign distance] (m-2-5)\n\n            (m-3-4) edge [-,double equal sign distance] (m-3-5)\n\n            (m-5-6) edge [-,double equal sign distance] (m-4-6)\n            (m-4-6) edge [-,double equal sign distance] (m-3-6)\n            (m-3-6) edge [-,double equal sign distance] (m-2-6)\n            (m-2-6) edge [-,double equal sign distance] (m-1-6)\n\n            (m-4-5) edge [-,double equal sign distance] (m-3-5)\n            (m-3-5) edge [-,double equal sign distance] (m-2-5)\n\n            (m-3-4) edge [-,double equal sign distance] (m-2-4)\n\n            (m-2-2) edge node [left] {\\footnotesize $0$} (m-1-2)\n            (m-2-2) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-2-3)\n            (m-2-3) edge node [right] {\\footnotesize $(1,-1)$} (m-1-3)\n            (m-2-4) edge node [right] {\\footnotesize $(1,-1)$} (m-1-4)\n            (m-2-5) edge node [right] {\\footnotesize $(1,-1)$} (m-1-5)\n\n            (m-5-5) edge node [above] {\\footnotesize $0$} (m-5-6)\n            (m-5-5) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 1\\end{pmatrix}$} (m-4-5)\n            (m-4-5) edge node [above] {\\footnotesize $(1,-1)$} (m-4-6)\n            (m-3-5) edge node [above] {\\footnotesize $(1,-1)$} (m-3-6)\n            (m-2-5) edge node [above] {\\footnotesize $(1,-1)$} (m-2-6)\n\n            (m-3-3) edge node [left] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-2-3)\n            (m-3-3) edge node [above] {\\footnotesize $\\begin{pmatrix} 1\\\\ 0\\end{pmatrix}$} (m-3-4)\n\n            (m-4-4) edge node [left] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-3-4)\n            (m-4-4) edge node [above] {\\footnotesize $\\begin{pmatrix} 0\\\\ 1\\end{pmatrix}$} (m-4-5)\n\n            (m-3-2) edge (m-2-2)\n            (m-4-2) edge (m-3-2)\n            (m-5-2) edge (m-4-2)\n\n            (m-4-3) edge (m-3-3)\n            (m-5-3) edge (m-4-3)\n\n            (m-5-4) edge (m-4-4)\n\n            (m-5-2) edge (m-5-3)\n            (m-5-3) edge (m-5-4)\n            (m-5-4) edge (m-5-5)\n\n            (m-4-2) edge (m-4-3)\n            (m-4-3) edge (m-4-4)\n        \n            (m-3-2) edge (m-3-3)\n\n            (m-1-1) edge (m-1-2)\n            (m-2-1) edge (m-2-2)\n            (m-3-1) edge (m-3-2)\n            (m-4-1) edge (m-4-2)\n            (m-5-1) edge (m-5-2)\n            (m-6-1) edge (m-6-2)\n\n            (m-6-2) edge (m-6-3)\n            (m-6-3) edge (m-6-4)\n            (m-6-4) edge (m-6-5)\n            (m-6-5) edge (m-6-6)\n\n            (m-2-1) edge (m-1-1)\n            (m-3-1) edge (m-2-1)\n            (m-4-1) edge (m-3-1)\n            (m-5-1) edge (m-4-1)\n            (m-6-1) edge (m-5-1)\n\n            (m-6-2) edge (m-5-2)\n            (m-6-3) edge (m-5-3)\n            (m-6-4) edge (m-5-4)\n            (m-6-5) edge (m-5-5)\n            (m-6-6) edge (m-5-6)\n            ;\n            ;\n        \\end{tikzpicture}\n    \\]\n    We have the following isomorphisms of endomorphism rings:\n    \\[\n        \\End\\left(X'\\right) \\cong \\End\\left(X'|_{((\\tau\/5)(5\\Zbf + 4))^2}\\right) = \\End\\left(A'|_{((\\tau\/5)(5\\Zbf + 4))^2}\\right) \\cong \\End\\left(A'|_{((\\tau\/5)\\Zbf)^2}\\right).\n    \\]\n    The first isomorphism is due to the fact that $X'$ is isomorphic to the restriction of an extension of an $(\\epsilon\\Zbf)^2$-persistence module, and $\\epsilon$ is an integer multiple of $\\tau$, by construction.\n    The equality is because \n    $X'|_{((\\tau\/5)(5\\Zbf + 4))^2} = A'|_{((\\tau\/5)(5\\Zbf + 4))^2}$, by construction.\n    For the last isomorphism, note that the value of $A'$ at any element of the grid $((\\tau\/5)\\Zbf)^2$ not belonging to $\\Tscr$ is isomorphic, through a structure morphism of $A'$, to the value of $A'$ at an element of the grid $((\\tau\/5)(5\\Zbf+4))^2$, by construction.\n    Moreover, the action of an endomorphism of $A'$ on $A'|_\\Tscr$ is determined by the action of the endomorphism on $A'(x+4\\tau\/5,y+4\\tau\/5)$, by \\cref{lemma:G-is-indecomposable}; and $(x+4\\tau\/5,y+4\\tau\/5) \\in ((\\tau\/5)(5\\Zbf+4))^2$.\n    It follows that the action of an endomorphism of $A'$ is completely determined by the action of the endomorphism on the restriction $A'|_{((\\tau\/5)(5\\Zbf+4))^2}$, proving the last isomorphism.\n\n    Thus, $A'$ is indecomposable by \\cref{lemma:indecomposable-local-ring} and the fact that $A$, and hence $X'$, is indecomposable.\n    Define $A_1$ to be the extension of $A'$ to the full poset $\\Rbf^2$.\n    Note that $A_1$ is indecomposable, finitely presentable, and such that $\\mathsf{im}\\left(\\eta^{A}_\\delta\\right) = \\mathsf{im}\\big(\\eta^{A_1}_\\delta\\big)$.\n    Moreover, $A_1$ admits a horizontal antenna attachment at $(x,y+3\\tau\/5)$, by construction.\n\\end{proof}\n\n\\begin{lemma}\n    \\label{lemma:sheaf-condition-persistence-module}\n    Let $\\epsilon > 0$ and let $a \\in \\epsilon \\Zbf$.\n    Let $s(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x \\leq a\\}$, $g(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x \\geq a\\}$, and $e(a) = \\{(x,y) \\in (\\epsilon \\Zbf)^2 : x = a\\}$, which are all subposets of $(\\epsilon \\Zbf)^2$.\n    Assume given persistence modules $X : s(a) \\to \\mathbf{vec}$ and $Y : g(a) \\to \\mathbf{vec}$ such that $X|_{e(a)} = Y|_{e(a)}$.\n    Then, there exists a unique persistence module $C : (\\epsilon\\Zbf)^2 \\to \\mathbf{vec}$ such that $C|_{s(a)} = X$ and $C|_{g(a)} = Y$.\n    If, moreover, $X$ is indecomposable and, for every indecomposable summand $I$ of $Y$, we have that $I|_{e(a)} \\neq 0$, then $C$ is indecomposable.\n\\end{lemma}\n\\begin{proof}\n    The statement about the existence and uniqueness of $C$ is clear.\n    Now, assume that $C \\cong S \\oplus T$.\n    On one hand, we have $X \\cong S|_{s(a)} \\oplus T|_{s(a)}$.\n    Since $X$ is indecomposable, without loss of generality, it must be the case that $T_{g(a)} = 0$.\n    In particular, $T_{e(a)} = 0$.\n    On the other hand, $Y \\cong S|_{g(a)} \\oplus T|_{g(a)}$, and, since all the indecomposable summands $I$ of $Y$ satisfy $I|_{e(a)} \\neq 0$, we must have $T|_{g(a)} = 0$.\n    Thus, $T = 0$ and $C$ does not admit any non-trivial decompositions.\n\\end{proof}\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=1\\linewidth]{pictures\/step-3.eps}\n    \\caption{A schematic summary of the main constructions in the proof of \\cref{lemma:actual-tacking}.}\n    \\label{figure:diagram-step-3}\n\\end{figure}\n\n\n\\begin{lemma}\n    \\label{lemma:actual-tacking}\n    Let $A,B : \\Rbf^2 \\to \\mathbf{vec}$ be finitely presentable indecomposable, and isomorphic to extensions of $\\Pscr$-persistence modules for some regular grid $\\Pscr \\subseteq \\Rbf^2$.\n    Assume that $A$ and $B$ both admit horizontal antenna attachments at points in the grid $\\Pscr$.\n    For every $\\delta > 0$, there exists a regular grid $\\Qscr \\supseteq \\Pscr$ and $M : \\Rbf^2 \\to \\mathbf{vec}$ with $M$ is indecomposable, finitely presentable, isomorphic to the extension of a $\\Qscr$-persistence module, and such that\n    \\[\n        \\mathsf{im}\\left(\\eta^M_\\delta\\right) \\cong \\mathsf{im}\\left(\\eta^A_\\delta\\right) \\oplus \\mathsf{im}\\left(\\eta^B_\\delta\\right).\n    \\]\n\\end{lemma}\n\\begin{proof}\n    Without loss of generality, we may assume that $A$ and $B$ are extensions of modules $A'$ and $B'$ defined over a regular grid $\\Qscr = (\\epsilon\\Zbf)^2$ with $\\epsilon < \\delta\/5$.\n    Let $(x,y) \\in \\Qscr$ and $(w,z) \\in \\Qscr$ be horizontal antenna attachments for $A$ and $B$, respectively.\n    Without loss of generality, we may assume that $y \\geq z$.\n    Choose $a,b \\in \\epsilon\\Zbf$ with $a < \\min(x,w)$ and $b > y$.\n    Let the subposets $s(a)$, $g(a)$, and $e(a)$ of $(\\epsilon\\Zbf)^2$ be as in \\cref{lemma:sheaf-condition-persistence-module}.\n\n    We now use \\cref{lemma:sheaf-condition-persistence-module} to construct a persistence module $M$ as in the statement.\n    In order to do this, we construct persistence modules $X$ and $Y$ over the grids $s(a)$ and $g(a)$, respectively.\n    \\cref{figure:diagram-step-3} is a schematic summary of these constructions.\n\n    Define a persistence module $Y_1 : g(a) \\to \\mathbf{vec}$ as follows:\n    \\[\n    Y_1(c,d) =\n    \\begin{cases}\n        \\kbb & \\text{if $d = y$ and $a \\leq c \\leq x$} \\\\\n        A'(c,d) & \\text{else}\n    \\end{cases}\n    \\]\n    The structure morphisms are taken to be those of $A'$ wherever that makes sense.\n    The structure morphisms $\\kbb = Y_1(c,y) \\to Y_1(c+\\epsilon,y) = \\kbb$ for $a \\leq c \\leq x-\\epsilon$ are taken to be the identity, and the structure morphisms $Y_1(c,y) \\to Y_1(c,y+\\epsilon)$ and $Y_1(c,y-\\epsilon) \\to Y_1(c,y)$ are taken to be zero.\n    It is straightforward to check that $Y_1$ is a well-defined persistence module and that $\\End(Y_1) \\cong \\End(A')$.\n    Thus, $Y_1$ is indecomposable.\n    Define $Y_2$ in an entirely analogous way using $B'$ and its horizontal antenna attachment.\n    Let $Y = Y_1 \\oplus Y_2$.\n\n    Consider the following subgrid of $\\epsilon \\Zbf$:\n    \\[\n        \\Tscr = \\{(a-5\\epsilon,a-4\\epsilon,a-3\\epsilon,a-2\\epsilon,a-\\epsilon)\\} \\times \\{(b, b+\\epsilon, b+2\\epsilon, b+3\\epsilon, b+4\\epsilon)\\}.\n    \\]\n    Let us assume that $y > z$; the case $y=z$ is similar.\n    Define a persistence module $X : s(a) \\to \\mathbf{vec}$ as follows:\n    \\[\n    X(c,d) =\n    \\begin{cases}\n        \\Gsf\\left(\\frac{(c,d)-(a-5\\epsilon,b)}{\\epsilon}\\right) & \\text{if $(c,d) \\in \\Tscr$} \\\\\n        \\kbb & \\text{if $c=a-\\epsilon$ and $y \\leq d \\leq b$} \\\\\n        \\kbb & \\text{if $c=a-2\\epsilon$ and $z \\leq d \\leq b$} \\\\\n        \\kbb & \\text{if $(c,d)=(a-\\epsilon,z)$} \\\\\n        0 & \\text{else}\n    \\end{cases}\n    \\]\n    For the structure morphisms corresponding to pairs of elements of $\\Tscr$ we use the structure morphisms of $\\Gsf$; for the remaining vertical structure morphisms we use the identity of $\\kbb$ when possible and the zero morphism otherwise; having done this, for the remaining horizontal morphisms we use the identity of $\\kbb$ when possible (taking commutativity into account) and the zero morphism otherwise (in particular, all horizontal maps with second coordinate different from $y$ and $z$).\n    \n    It is again straightforward to check that $X$ is a well-defined persistence module and that $\\End(X) = \\End(\\Gsf)$.\n    Thus, $X$ is indecomposable.\n\n    Finally, define $C : (\\epsilon \\Zbf)^2 \\to \\mathbf{vec}$ using \\cref{lemma:sheaf-condition-persistence-module}; it follows that $C$ is indecomposable.\n    Finally, extend $C$ to the poset $\\Rbf^2$ to get a module $M$, which satisfies the conditions in the statement by construction.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of \\cref{lemma:main-lemma-attaching}]\n    Let $A$ and $B$ be as in the statement.\n    Using \\cref{lemma:add-1-dim-corner}, we may assume that $A$ and $B$ admit a thin corner, and using \\cref{lemma:add-antenna-attachment}, we may assume that $A$ and $B$ admit horizontal antenna attachments.\n    \\cref{lemma:actual-tacking} now finishes the proof.\n\\end{proof}\n\n\\ifarxivversion\n\\bibliographystyle{alpha}\n\\fi\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\\label{sec_1}\nUltrafast pump-probe spectroscopy is a good tool to investigate the nonequilibrium properties of a given system since a pump pulse triggers ultrafast processes and a subsequent probe pulse monitors the pump-induced dynamical processes~\\cite{Mukamel1995, Diels1996, Krausz2009, Giannetti2016}.\nEspecially, by using femtosecond pulses, nonequilibrium dynamics of electrons can be detected since the timescale of the motion of electrons is of the order of a femtosecond.\nHowever, increasing the resolution of optical measurements in both the time and energy domains is difficult and limited by the uncertainty principle.\n\nRecently, ultrafast spectroscopic techniques have been advanced by using a transform-limited pulse, i.e., a pulse that has the minimum possible duration for a given spectral bandwidth, and have opened a new door to make both temporal and spectral resolutions as high as possible~\\cite{Diels1996}.\nThese techniques can disclose new ultrafast nonequilibrium phenomena.\nIn fact, by applying these techniques, interference in the energy domain has been observed in atomic systems and nanometric tips~\\cite{Wollenhaupt2002, Lindner2005, Kiffner2006, Milosevic2006, Kruger2018}.\nThis interference is applied to control the atomic storage medium for recording the information of optical pulses~\\cite{Leung1982, Carlson1983, Hemmer1994, Fleischauer2002, Ohmori2009}.\nHowever, as far as we know, there has been no such report on transient interference of pump-probe spectroscopy of band and Mott insulators both experimentally and theoretically.\n\nIn this paper, we investigate ultrafast pump-probe spectroscopy of band and Mott insulators and propose transient interference between temporary well-separated pulses in electron systems as in the case of atomic systems.\nWe formulate such transient interference in pump-probe spectroscopy of a two-band model.\nWe find that the existence of a continuum structure in the excitation spectrum is important for generating the transient interference since the continuum structure acts as a medium for storing the spectral information of the pump pulse and for creating interference between temporary well-separated pump and probe photons.\nThe information persists due to a memory effect, i.e., a relaxation process of electron systems.\nAs a result, the time-domain pump-probe spectrum depends on both probe energy $\\omega$ and the central frequency of the pump and probe pulses $\\Omega$ and thus oscillates with a frequency\n\\begin{align}\n\\label{probe_dep}\n\\omega_0 = \\omega - \\Omega.\n\\end{align}\nIn order to demonstrate the transient interference in the presence of electron correlation, we perform numerical calculations of the pump-probe spectrum in a one-dimensional (1D) half-filled Hubbard model.\nMoreover, we find that bosons coupled to electrons in the two-band model make an additional contribution to the interference.\nBased on the result, we speculate that the transient interference will be observed in Mott insulators strongly correlated to magnons.\nFor the observation of the proposed transient interference, high resolution of measurements of both time and energy is required in ultrafast pump-probe spectroscopy.\nRecently, oscillations of electronic states above the charge-transfer gap in a two-dimensional (2D) Mott insulator Nd$_2$CuO$_4$ were observed on the reflectivity changes detected by pump-probe measurement with ultrashort pulses~\\cite{Miyamoto2018}. \nThe time and energy resolutions of the measurement are as high as $10 \\text{fs}$ and $0.01 \\text{eV}$, respectively.\nBy extracting the oscillatory components from the pump-probe spectrum, the oscillation component with the frequency indicated by Eq.~(\\ref{probe_dep}) was found~\\cite{Miyamoto2018}.\nWe propose that the transient interference will be one of the possible origins of the observed oscillations.\n\nThis paper is organized as follows. We introduce a two-band model, which is a minimal model to describe the interference effect by two photon pulses through an electron system, and show the pump-probe absorption spectrum in Sec.~\\ref{sec_2}. In Sec.~\\ref{sec_3}, we calculate the time-dependent optical conductivity at half filling just after pumping. The effect of bosons coupled to electrons on the pump-probe spectrum is discussed in Sec.~\\ref{sec_4}. Finally, a summary is given in Sec.~\\ref{sec_5}.\n\n\\section{Two-band model}\\label{sec_2}\nWe first introduce a two-band model, which is the minimal model to describe the interference effect by two photon pulses through an electron system, and analytically calculate the pump-probe absorption spectrum.\nWith the assumption of dipole transitions, the Hamiltonian of the two-band model under the time-dependent electric field reads\n\\begin{align*}\n\\mathcal{H} =& \\sum_{\\bm{k}} \\varepsilon_{\\bm{k}} c_{\\text{c}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} + \\sum_{\\bm{k}} \\varrho_{\\bm{k}} c_{\\text{v}\\bm{k}}^\\dag c_{\\text{v}\\bm{k}} \\nonumber\\\\\n&-\\sum_{\\bm{k}} \\left( d_{\\text{cv}} \\mathcal{E}(t) c_{\\text{c}\\bm{k}}^\\dag c_{\\text{v}\\bm{k}} + d_{\\text{cv}}^* \\mathcal{E}(t) c_{\\text{v}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\right), \\nonumber\n\\end{align*}\nwhere $c_{\\text{c(v)}\\bm{k}}$ is an annihilation operator for fermions in the conduction (valence) band with momentum $\\bm{k}$. The energies of the conduction and valence bands are $\\varepsilon_{\\bm{k}} = \\varepsilon + \\frac{\\hbar ^2 \\bm{k}^2}{2m_{\\text{c}}}$ and $ \\varrho_{\\bm{k}} = \\varrho + \\frac{\\hbar ^2 \\bm{k}^2}{2m_{\\text{v}}}$,\nwhere $\\varepsilon$ and $\\varrho$ are the minimum and maximum of the conduction and valence bands, respectively, and $m_c$ and $m_v$ are the effective masses of electrons in the conduction and valence bands, respectively.\nWe introduce the interband dipole matrix element $d_{\\text{cv}}$ and external electric field $\\mathcal{E}(t)$. Hereafter, we set $\\hbar=1$.\n\nBy taking the long-wave-length limit of the electric field, the optical Bloch equation is written as~\\cite{HaugKoch}\n\\begin{align} \\label{Eq_bloch_eq_1}\n&\\left( \\frac{\\partial}{\\partial t} + i\\{ \\varepsilon _{\\bm{k}}-\\varrho_{\\bm{k}}-i \\gamma \\} \\right) p_{\\text{vc}}^0(\\bm{k},t) = d_{\\text{cv}} \\mathcal{E}(t) \\{ 1-2f_c(\\bm{k}) \\}\n\\end{align}\nand\n\\begin{align} \\label{Eq_bloch_eq_2}\n\\left(  \\frac{\\partial}{\\partial t} +  \\Gamma \\right) f_{\\text{c}}(\\bm{k},t) = -2\\text{Im}\\left[ d_{\\text{cv}} \\mathcal{E}(t) p_{\\text{vc}}^{0*}(\\bm{k},t) \\right],\n\\end{align}\nwhere $f_{\\text{c}}(\\bm{k}) = \\langle c_{\\text{c}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\rangle$ and $p_{\\text{vc}}^0(\\bm{k}) = \\langle c_{\\text{v}\\bm{k}}^\\dag c_{\\text{c}\\bm{k}} \\rangle$, with $\\langle \\cdots \\rangle$ representing the expectation value.\nWe introduce a phenomenological damping rate $\\Gamma$ for $f_{\\text{c}}$ and dephasing rate $\\gamma$ for $p_{\\text{vc}}^0$.\nWe consider an electric field $\\mathcal{E}(t) = \\frac{1}{2}\\left(\\mathcal{\\tilde{E}}(t)e^{-i\\Omega t} + \\mathcal{\\tilde{E}}^*(t)e^{i\\Omega t}\\right)$, where $\\mathcal{\\tilde{E}}(t)=2\\left\\{ \\mathcal{\\tilde{E}}_{\\text{p}}(t)e^{i\\bm{k}_p\\cdot \\bm{r}} + \\mathcal{\\tilde{E}}_{\\text{t}}(t)e^{i\\bm{k}_t\\cdot \\bm{r}} \\right\\}$, and the electric field and wave vector of the pump (probe) pulse are $\\mathcal{\\tilde{E}}_{\\text{p}}$ and $\\bm{k}_{\\text{p}}$ ($\\mathcal{\\tilde{E}}_{\\text{t}}$ and $\\bm{k}_{\\text{t}}$), respectively.\nIntroducing an expansion parameter $\\lambda$ through $\\mathcal{E}(t) \\rightarrow \\lambda \\mathcal{E}(t)$, we obtain $p_{\\text{vc}}^0 = \\lambda p_{\\text{vc}}^{0(1)} + \\lambda^2 p_{\\text{vc}}^{0(2)} + \\lambda^3 p_{\\text{vc}}^{0(3)} + \\cdots,\\; f_{\\text{c}} = \\lambda f_{\\text{c}}^{(1)} + \\lambda^2 f_{\\text{c}}^{(2)} + \\lambda^3 f_{\\text{c}}^{(3)} + \\cdots$.\nThe shape of the probe pulse is represented by the delta function $\\mathcal{\\tilde{E}}_{\\text{t}}(t) = \\mathcal{\\tilde{E}}_{\\text{t}} \\delta (t -\\tau)$ $(\\tau >0)$, where $\\tau$ is the delay time between the pump and probe pulses.\nThe pump-induced absorption change is given by $\\alpha = -\\textrm{Im}\\left[ d_{cv}^*\\chi(\\bm{k},\\omega) \\right].$\nTaking $\\mathcal{\\tilde{E}}_{\\text{p}}(t)=\\mathcal{\\tilde{E}}_{\\text{p}}e^{-\\sigma|t|}$ and with the rotating-wave approximation, the probe susceptibility is given by (see Appendix A)\n\\begin{align} \\label{chi0}\n&\\chi (\\bm{k},\\omega)\\simeq \\frac{p_{\\text{vc}}^{0(3)}(\\bm{k},\\omega)}{\\mathcal{E}_{\\text{t}}(\\omega)}\\nonumber \\\\\n&=\\frac{8d_{\\text{cv}}\\left|d_{\\text{cv}}\\right| {}^2 \\left| \\mathcal{\\tilde{E}}_p\\right| {}^2  e^{-\\left(\\sigma - \\gamma \\right)\\tau} e^{i \\tau  \\left( - \\Omega +\\varepsilon _k-\\varrho   _k\\right)}  \\Gamma \\sigma }{\\left(i \\gamma +\\omega -\\varepsilon _k+\\varrho _k\\right)(i\\Gamma +i\\sigma + \\omega -\\Omega ) v_{\\bm{k}}^+ u_{\\bm{k}}^+ u_{\\bm{k}}^-}+\\cdots,\n\\end{align}\nwhere $u_{\\bm{k}}^{\\pm}=i\\gamma \\pm i\\sigma + \\Omega - \\varepsilon _{\\bm{k}} + \\varrho _{\\bm{k}}$ and $v_{\\bm{k}}^+ = i\\gamma +i\\Gamma -i\\sigma + \\Omega - \\varepsilon _{\\bm{k}}+ \\varrho _{\\bm{k}}$. \nIn the limit $\\gamma\\rightarrow 0$, the pole of the energy denominator $\\omega = \\varepsilon_{\\bm{k}} - \\varrho _{\\bm{k}}$ in the third term of $\\chi (\\bm{k},\\omega)$ gives rise to an oscillatory behavior of $e^{i(\\omega -\\Omega)\\tau}$ with decay $e^{-(\\sigma-\\gamma)\\tau}$.\nThis is the oscillation component indicated by Eq.~(\\ref{probe_dep}).\nSince the timescale where the oscillation persists is on the order of $\\gamma^{-1}$, real-time ultrafast dynamics should be observed with high accuracy~\\cite{Rhodes2013}.\n\nIn order to maintain the oscillation in the two-band model, we have to select a proper set of parameters that leads to the coherence and memory effect in the energy domain.\nFirst of all, we examine the coherence in the energy domain.\nWhen $\\sigma \\gg 1\/\\tau$, i.e., the pulse duration is much shorter than the time delay $\\tau$, we obtain $\\Delta t \\sim 0$, where $\\Delta t$ is the uncertainty in the time domain.\nSimultaneously, the energy uncertainty $\\Delta E \\sim \\infty$, leading to low energy resolution.\nAs a result, the interference in the energy domain is invisible.\nThis corresponds to the fact that the interference pattern vanishes in Young's double-slit experiment if the path of light is measured~\\cite{Englert1996, Durr1998}.\nIn fact, if the electric field of the pump pulse is represented by the $\\delta$ function, $p_{vc}^{0(3)}(\\bm{k},\\omega)$ completely cancels out $\\mathcal{E}_t(\\omega)$, which means that $\\chi(\\bm{k},\\omega)$ does not have the interference term $e^{i(\\omega -\\Omega)\\tau}$ (see Appendix A).\nIn contrast, when $\\sigma \\lesssim 1\/\\tau$, the coherence in the energy domain is obtained, which leads to the interference in energy space.\n\nSecond, we examine the memory effect.\nWhen $\\sigma \\ll \\gamma$, i.e., the pulse duration is longer than the dephasing time, $\\Delta t \\sim \\infty$ and $\\Delta E\\sim 0$ are simultaneously obtained.\nThis leads to the relaxation that holds true as long as electrons have well-defined energies, and their energy changes are slow with the timescale of $1\/\\Delta \\epsilon$, where $\\Delta \\epsilon$ is the characteristic energy exchange in a scattering event~\\cite{Aihara1982, Kuznetsov1991, Kuznetsov1991_2, Rossi2002, SchaeferWegener}.\nWhen $\\sigma \\gtrsim \\gamma$, the relaxation involving electrons with ill-defined energies starts to contribute to the memory effect.\nTherefore, if $\\sigma$ and $1\/\\tau$ are carefully controlled to realize $1\/\\tau \\gtrsim \\sigma \\gtrsim \\gamma$, both coherence in the energy domain and the memory effect are relevant, and the interference in the energy domain is maintained for the time $\\gamma^{-1}$.\nUsually, $\\gamma$ of a given system cannot be changed. However, if we make use of the quantum Zeno effect~\\cite{Misra1977, Itano1990, Kaulakys1997, Streed2006}, we might be able to control $\\gamma$, which can help us to observe our finding.\n\n\\section{Hubbard model}\\label{sec_3}\nPump-probe spectroscopy has been performed in strongly correlated systems to investigate exotic phenomena~\\cite{Okamoto2010, Okamoto2011, Filippis2012, Matsueda2012, Zala2013, Golez2014, Eckstein2014, Novelli2014, Prelovsek2015, Giannetti2016, Bittner2017, Bittner2018, Miyamoto2018}.\nEven in correlated electron systems, there is a continuum structure in the excitation spectrum. This indicates that interference effects similar to those in the two-band model may be realized, which will be demonstrated by using a 1D half-filled Hubbard model, which is given by\n\\begin{align}\nH=-t_\\mathrm{h}\\sum_{i,\\sigma} \\left( c^\\dagger_{i,\\sigma} c_{i+1,\\sigma} + \\mathrm{H.c.}\\right) + U\\sum_i n_{i,\\uparrow}n_{i,\\downarrow},\n\\label{H}\n\\end{align}\nwhere $c^\\dagger_{i\\sigma}$ is the creation operator of an electron with spin $\\sigma$ at site $i$, $n_{i,\\sigma}=c^\\dagger_{i,\\sigma}c_{i,\\sigma}$, $n_i=\\sum_\\sigma n_{i,\\sigma}$, and $t_\\mathrm{h}$ and $U$ are the nearest-neighbor hopping and on-site Coulomb interaction, respectively. Taking $t_\\mathrm{h}$ to be the unit of energy ($t_\\mathrm{h}=1$), we use $U=10$.\n\nWe investigate the probe-energy dependence of the optical conductivity of a Hubbard open chain with $L=10$, where $L$ is the number of sites.\nWe assume that both the pulses have the same shape of the vector potential given by $A(t)=A_0 e^{-(t-t_0)^2\/(2t_\\mathrm{d}^2)} \\cos[\\Omega(t-t_0)]$.\nWe set $A_0=0.1$, $t_0=3.0$, $t_\\mathrm{d}=0.5$, and $\\Omega=E_g =7.1$ for the pump pulse and $A_0=0.001$, $t_0=\\tau+3.0$, $t_\\mathrm{d}=0.02$, and $\\Omega=E_g =7.1$ for the probe pulse unless otherwise specified, where $E_g$ is the energy of the Mott gap.\nAn external spatially homogeneous electric field applied along the chain in the Hamiltonian can be incorporated via the Peierls substitution in the hopping terms as $c_{i,\\sigma}^\\dag c_{i+1,\\sigma} \\rightarrow e^{iA(t)}c_{i,\\sigma}^\\dag c_{i+1,\\sigma}$.\nUsing the method discussed in Refs.~\\cite{Lu2015, Shao2016}, we obtain the optical conductivity in the nonequilibrium system, $\\sigma(\\omega,\\tau) = \\frac{j_\\text{probe}(\\omega,\\tau) }{i(\\omega +i\\eta)LA_\\text{probe}(\\omega)}$, where $j_\\text{probe}(\\omega,\\tau)$ is the Fourier transform of the current induced by the probe pulse and $A_{\\text{probe}}(\\omega)$ is the Fourier transform of the vector potential of the probe pulse (see Appendix B for details).\n\nTo trace the temporal evolution of the system, we employ the time-dependent Lanczos method to evaluate $|\\psi (t)\\rangle$. \nHere $|\\psi(t+\\delta{t})\\rangle\\simeq\\sum_{l=1}^{M}{e^{-i\\epsilon_l\\delta{t}}}|\\phi_l\\rangle\\langle\\phi_l|\\psi(t)\\rangle$, where $\\epsilon_l$ and $|\\phi_l\\rangle$ are eigenvalues and eigenvectors of the tridiagonal matrix generated in the Lanczos iteration, respectively, $M$ is the dimension of the Lanczos basis, and $\\delta t$ is the minimum time step. We set $M=50$ and $\\delta t=0.02$.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=20pc]{Fig_OC_L10p_wpump10_wprobe_t_high.eps}\n    \\caption{$\\text{Re} \\sigma (\\omega,\\tau)$ in the 1D half-filled Hubbard chain with $L=10$ and $U=10$, before pumping ($\\tau<0$) and after pumping ($\\tau=10$,\\;20,\\;30, and 40). Since the system is weakly excited, the dashed line for $\\tau<0$ is almost overlapped by the solid lines above $\\omega =7$.}\n    \\label{band}\n\\end{figure}\n\nFigure~\\ref{band} shows the real part of the time-dependent optical conductivity $\\text{Re} \\sigma(\\omega,\\tau)$ of the Hubbard model.\nPhotoinduced decreases in the spectral weights at absorption peaks above the Mott gap are small since the system is weakly excited. \nThe pump photon excites carriers into an optically allowed odd-parity state.\nThe probe pulse couples in part to the odd-parity state, resulting in an excitation from the optically allowed state to an optically forbidden even-parity state.\nIn 1D Mott insulators with open boundary conditions, the optically forbidden state is located slightly above the optically allowed state~\\cite{Mizuno2000}.\nLow-energy in-gap excitation comes from the excitation from the optically allowed to forbidden state~\\cite{Lu2015}.\nInside the Mott gap, we find photoinduced low-energy spectral weights at $\\omega \\simeq1.2$, 2.2, and 3.3.\nThese energies correspond to the energy differences between the optically allowed populated state at $\\omega=7.1$ and the optically forbidden states.\n\nFigures~\\ref{probe_energy_dep_w0}(a)-\\ref{probe_energy_dep_w0}(e) show the $\\tau$ dependence of $\\text{Re} \\sigma(\\omega,\\tau)$ above the Mott gap with probe energy $\\omega=7.10$, $7.92$, $8.98$, $10.08$, and $11.18$, respectively, whose energies agree with the peak energies of the absorption spectrum in Fig.~\\ref{band}.\nWe find that the frequencies of the oscillations depend on $\\omega$.\nThe larger $\\omega$ is, the larger the frequency is, which is consistent with our argument in the two-band model discussed above.\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[clip, width=20pc]{OC_power.eps}\n    \\caption{$\\text{Re} \\sigma(\\omega,\\tau)$ in the 1D half-filled Hubbard chain with $L=10$ and $U=10$ for (a) $\\omega=7.1$, (b) $\\omega=7.92$, (c) $\\omega=8.98$, (d) $\\omega=10.08$, and (e) $\\omega=11.18$. The power spectra of $\\text{Re} \\sigma (\\omega,\\tau)$ for (f) $\\omega=7.1$, (g) $\\omega=7.92$, (h) $\\omega=8.98$, (i) $\\omega=10.08$, and (j) $\\omega=11.18$.}\n    \\label{probe_energy_dep_w0}\n\\end{figure}\n\nIn order to further examine the probe-energy dependence, we show the power spectra of $\\text{Re} \\sigma(\\omega,\\tau)$ with respect to $\\tau$ in Figs.~\\ref{probe_energy_dep_w0}(f)-\\ref{probe_energy_dep_w0}(j) for $\\omega=7.1$, 7.92, 8.98, 10.08, and 11.18, respectively.\nWe discuss two possible contributions to the power spectra.\nThe first one is the contribution from the Rabi oscillation, whose frequencies are related to the low-energy in-gap states at $\\omega=$1.2, 2.2, and 3.3.\nIn fact, we find the Rabi-oscillation contributions to the spectral weights at $\\omega_0=$1.2, 2.2, and 3.3 in Figs.~\\ref{probe_energy_dep_w0}(f)-\\ref{probe_energy_dep_w0}(j).\nSince our system is of finite size, energy levels are discretized.\nTherefore, there are oscillations with resonant frequencies between the levels.\nIn the thermodynamic limit, the number of the levels is infinite, and thus we expect that the contributions from a huge number of such resonances with various frequencies cancel out, giving rise to an inifinite number of infinitesimal weights in the power spectra.\nThus, we consider that the Rabi-oscillation contribution to the power spectra is visible only in finite-size systems and negligible in the thermodynamic limit.\n\nThe second one is the contribution from the interference effect, which gives rise to the $\\omega$ dependence of the pump-probe spectra as discussed in the two-band model. \nThe oscillations with the frequencies $\\omega-\\Omega$ appear at $\\omega_0=7.92-7.10=0.82$, $8.98-7.10=1.88$, $10.08-7.10=2.98$, and $11.18-7.10=4.08$.\nThese energies correspond to the energy difference between the levels at $\\omega=\\Omega=7.1$ and the excited states above the Mott gap, all of which belong to the same electronic states with odd parity.\nWe consider that this origin makes the dominant contribution to the power spectra in the thermodynamic limit.\nIn order to induce the transient interference, we should use the pump pulse whose spectrum covers some energy levels. Then we can store the information of the pump pulse in electronic states with a wide range of energies above the Mott gap.\n\nAccording to the two possible contributions to the power spectra, in Fig.~\\ref{probe_energy_dep_w0}(g), for example, we find peak structures at $\\omega_0=0.82$, 1.2, and 2.2.\nThe peak structures at $\\omega_0=1.2$ and 2.2 come from the Rabi oscillation of the two odd- and even-parity states.\nOn the other hand, the origin of the structure at $\\omega_0=0.82$ is the interference because $\\omega_0=0.82$ corresponds to one of the energy differences between the odd-odd states mentioned above.\nSimilarly, Figs.~\\ref{probe_energy_dep_w0}(h)-\\ref{probe_energy_dep_w0}(j) are understood in the same way (see Appendix B for details).\n\n\\section{electron-boson coupling in the two-band model} \\label{sec_4}\n\nFinally, we discuss the effect of bosons coupled to electrons on the probe-energy-dependent oscillation.\nNonequilibrium electron dynamics coupled to a boson driven by a laser has been extensively studied.\nFurthermore, since non-Markovian relaxation is important in electron systems coupled to a bosonic environment, open quantum systems with non-Markovian properties have been studied for a long time~\\cite{Jaynes1963, Caldeira1985, Leggett1987, BreuerPetruccione, Zurek2003, Reiter2014, Seetharam2015, Nazir2016, deVega2017}.\nThe additional Hamiltonian due to boson degrees of freedom is \n\\begin{align}\\label{H_ph}\n\\mathcal{H}_{\\text{ph}} = \\sum_{\\bm{q}} \\omega_{\\bm{q}} a_{\\bm{q}}^\\dag a_{\\bm{q}} + \\sum_{\\bm{k},\\bm{q}} g_{\\bm{q}} (a_{-\\bm{q}}^\\dag + a_{\\bm{q}}) (c_{\\text{c}\\bm{k+q}}^\\dag c_{\\text{c}\\bm{k}} + c_{\\text{v}\\bm{k+q}}^\\dag c_{\\text{v}\\bm{q}}),\n\\end{align}\nwhere $a_{\\bm{q}}$ is an annihilation operator for bosons with momentum $\\bm{q}$, $\\omega_{\\bm{q}}$ is the boson frequency, and $g_{\\bm{q}}$ is an electron-boson coupling constant.\n\nWe examine the two-band model with electron-boson coupling under the application of the exponential pump pulse.\nTotal polarization is given by $p_{\\text{vc}}^{}(\\bm{k},t)= p_{\\text{vc}}^{0}(\\bm{k},t) + p_{\\text{vc}}^{b}(\\bm{k},t)$, where $p_{\\text{vc}}^{0}(\\bm{k},t)$ is from the one-particle contribution, as discussed above, and $p_{\\text{vc}}^{b}(\\bm{k},t)$ is from the electron-boson coupling.\nSolving the kinetic equation with $\\mathcal{H}_{\\text{ph}}$ (see Appendix A), the probe susceptibility is given by\n\\begin{align} \\label{Eq_chi_b_main}\n&\\chi ^{b}(\\bm{k},\\omega)\\simeq \\frac{p_{\\text{vc}}^{b(3)}(\\bm{k},\\omega)}{\\mathcal{E}_{\\text{t}}(\\omega)}\\nonumber =\\sum _{\\bm{q}} g_{\\bm{q}}^2\\mathcal{N}_{\\bm{q}} \\cdot 4i\\sigma d_{\\text{cv}}\\left| d_{\\text{cv}}\\right|{}^2 \\left| \\mathcal{\\tilde{E}}_p\\right| {}^2 \\nonumber \\\\\n&\\times \\Biggl[\n\\frac{e^{-\\tau \\left(\\sigma-\\gamma\\right)} e^{i \\tau  \\left( -\\Omega +\\varepsilon_{\\bm{k}} -\\varrho _{\\bm{k}}\\right)} \\left(-i\\gamma -2i \\Gamma - \\omega + \\varepsilon _{\\bm{k}}- \\varrho _{\\bm{k}}\\right) }{\\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k}}+\\varrho _{\\bm{k}}\\right){}^2 \\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k+q}}+\\varrho _{\\bm{k}}+\\omega _{\\bm{q}}\\right) v_{\\bm{k}}^+}\\nonumber \\\\\n&\\times \\frac{\\left(2 i \\gamma +2 \\omega -\\varepsilon _{\\bm{k}}-\\varepsilon _{\\bm{k+q}}+\\varrho _{\\bm{k}}+\\varrho _{\\bm{k-q}}+2 \\omega _{\\bm{q}}\\right)}{(i\\Gamma +i\\sigma + \\omega -\\Omega ) \\left(i \\gamma +\\omega -\\varepsilon _{\\bm{k}}+\\varrho _{\\bm{k-q}}+\\omega _{\\bm{q}}\\right) u_{\\bm{k}}^+u_{\\bm{k}}^-}\\Biggr]\\nonumber \\\\\n&+\\cdots,\n\\end{align}\nwhere $\\mathcal{N}_{\\bm{q}}=\\frac{1}{e^{\\omega_{\\bm{q}}\/k_BT}-1}$.\nIn the limit $\\gamma \\rightarrow 0$, the pole of the energy denominator $\\omega = \\varepsilon_{\\bm{k}} - \\varrho _{\\bm{k}}$ gives rise to an oscillatory behavior of $e^{i(\\omega -\\Omega)\\tau}$ with decay $e^{-(\\sigma -\\gamma)\\tau}$, which is the same behavior as the third term in Eq.~(\\ref{chi0}).\nTherefore, the information of pump and probe pulses is transmitted with the help of boson-assisted electron scattering, which gives one of the possible origins of the transient interference.\n\nIn Mott insulators, magnons are strongly coupled to photo-excited electrons in 2D Mott insulators, in contrast to the 1D Mott insulator where spin and charge degrees of freedom are separated.\nTherefore, the interference proposed in this work will be easily realized in the 2D Mott insulators.\nWe thus speculate that the oscillations observed by the pump-probe spectroscopy of the 2D Mott insulator Nd$_2$CuO$_4$~\\cite{Miyamoto2018} come from the interference effect. \nIn order to confirm this speculation, we need to investigate theoretically the pump-probe spectrum of the 2D half-filled Hubbard model, but it remains for a future work.\n\n\\section{summary}\\label{sec_5}\nIn summary, we suggested the transient interference in the energy domain between temporary well-separated light pulses using electronic states of band and Mott insulators as a medium, which manifests as the oscillation of the pump-probe spectrum whose frequency is indicated by Eq.~(\\ref{probe_dep}).\nThis interference could be observed only after recent developments of ultrafast spectroscopic techniques.\nThe transient interference reflects the universal property of interference between two photon pulses mediated by electron systems, which does not depend on the details of the electron systems. Therefore, the interference is also realized in the presence of electron correlation since there is a continuum structure. We examined this by calculating the pump-probe spectrum in the 1D half-filled Hubbard model.\nTo verify our prediction, we suggested an experiment for Nd$_2$CuO$_4$ with varying pump-pulse duration and delay. \nSince our theory predicts the transient oscillation even in the 1D Mott insulators, we proposed a pump-probe experiment in Sr$_2$CuO$_3$.\nFurthermore, we found that bosons coupled to electrons in the two-band model make the additional contribution to the transient interference.\nBased on the result, both magnons coupled to electrons and the continuum structure in electronic excitation spectrum would be possible origins of the oscillation observed in Nd$_2$CuO$_4$.\n\n\\begin{acknowledgments}\nWe would like to thank H. Okamoto, T. Miyamoto, I. Eremin, and P. Prelov\\v{s}ek for fruitful discussions. This work was supported by CREST (Grant No. JPMJCR1661), the Japan Science and Technology Agency, the creation of new functional devices and high-performance materials to support next-generation industries (CDMSI), and the challenge of basic science exploring extremes through multi-physics and multi-scale simulations to be tackled by using a post-K computer.\n\\end{acknowledgments}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion and Future Work}\n\\label{sec:conclusions}\n\nWe presented the first distributed optimal primal-dual resource allocation\nalgorithm for uplink OFDM systems. The key features of the proposed\nalgorithm include: (a) incorporating practical OFDM system\nconstraints such as self-noise and maximum SNR constraints, (b)\nreduced primal-dual algorithm which eliminates unnecessary variable\nupdates, (c) distributed implementation by the end users and base\nstation, (d) simple local updates with limited message passing, (e)\nglobal convergence despite of the existence of multiple global\noptimal solutions, and (f) fast convergence compared with the\nstate-of-art centralized optimal algorithm. Currently we are\nextending this work in several directions, including proving the\ntheoretical convergence speed of the algorithm.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLeft Handed Materials (LHM) are a new kind of materials which were theoretically envisionned by Veselago \\cite{veselago} as early as in 1967. Such materials have simultaneously negative relative permittivity ($\\varepsilon_{\\mathrm{r}}$) and negative relative permeability ($\\mu_{\\mathrm{r}}$). This theoretical curiosity became a real field of research in 2000 after Pendry showed the potential of LHM to overcome the diffraction limit \\cite{pendry_prl00} and Smith \\textit{et al.} \\cite{smith_nr} proposed a first realization for such extra-ordinary materials based on periodic lattices combining Split Ring Resonators (SRR: Concentric annular rings with splits) and wires. The latter work can be considered as the experimental foundation of LHM (as the first experimental evidence of negative refraction) and it is based on a theoretical study by Pendry \\textit{et al.} which shows that negative permittivity could be obtained by a periodic arrangement of parallel wires and that a periodic lattice of SRR had a negative magnetic response around its resonance frequency \\cite{pendry_ieee99}.\n\n\\noindent In the recent years, there has been a keen interest in wave propagation in periodically structured media \\cite{wavesPC}. Investigation of photonic crystals has paved the way to the theoretical prediction and experimental realization of photonic band gaps \\cite{yablono87,john,krauss,zengerle87,gralak2000,notomi2002,luo2002} i.e. ranges of frequencies for which light, or a light polarization, is disallowed to propagate. Soon after, the focus has been extended to the study of acoustic waves in periodic media, and the existence of phononic band gaps has been verified both theoretically and experimentally \\cite{2-ross1,2-ross2,acoustic,zhang2004,hu2004,feng2006}. Recently, the interest was even extended to the study of different types of waves, e.g. liquid surface waves \\cite{chou97,chou98,mciver2000,torres2000,hu2003} or biharmonic waves \\cite{ross-platonic,sasha-prsa,farhat-apl,farhat-epl,farhat-prl2009} in perforated thin plates. It has been shown that complete bandgaps also exist for these waves when propagating through a periodic lattice of vertically standing rods or over a periodically perforated thin plate \\cite{torres2000,farhat-apl}. In addition, many interesting phenomena have been reported, including negative refraction  \\cite{veselago,pendry2000,maystre2004,guenneau2005,smith2000,ramak2005}, the superlensing effect and cloaking \\cite{schurig,norris,farhatprl,farhatwm, farhatpre}. The essential condition for the AANR effect is that the constant frequency surfaces (EFS: equifrequency surfaces) should become convex everywhere about some point in the reciprocal space, and the size of these EFS should shrink with increasing frequency \\cite{zengerle87,gralak2000,notomi2002}.\\\\\n\n\\noindent In this paper, we focus on the application of split ring resonator (SRR) structures \\cite{pendry_ieee99,Seb_SRR,guenneau_physb06} to the domain of elastic waves. We first derive the homogenized governing equations of bending waves propagating within a thin-plate with a doubly periodic square array of freely vibrating holes shaped as SRR, from the generalized biharmonic equation, and an asymptotic analysis involving three scales (one for the thickness of the thin-cut of each SRR, one for the array-pitch, and one for the wave wavelength). We then present an analysis of dispersion curves. To do this, we set the spectral problem for the biharmonic operator within a doubly periodic square array of SRR, homogeneous stress-free boundary conditions are prescribed on the contour of each resonator and the standard Floquet-Bloch conditions are set on the boundary of an elementary cell of the periodic structure. Such a structure presents an elastic bandgap at low frequencies. It turns out that the asymptotic analysis of our structure allows us to get analytically the frequency of the first localized mode and then the frequency of the first band gap.\nThe aim of our work is actually to demonstrate the AANR effect at low frequencies for elastic thin perforated plates as well as their superlensing properties. Ultra-refraction is also considered and shows the versatility and power of using such structured media to realize new functionalities for surface elastic waves.\n\n\\section{Homogenization of a thin-plate with an array of stress-free SRR inclusions near resonance}\n\nThe equations for bending of plates can be found in many textbooks \\cite{timoshenko,graff}.\nThe wavelength $\\lambda$ is supposed to be large enough compared to the thickness\n of the plate $H$ and small compared to its in-plane dimension $L$, i.e. $H \\ll \\lambda \\ll L$.\nIn this case we can adopt the hypothesis of the theory of Von-Karman\n\\cite{timoshenko,graff}. In this way, the mathematical setup is essentially two-dimensional,\nthe thickness $H$ of the plate appearing simply as a parameter in the governing equation.\n\nWe would like to homogenize a periodically structured thin-plate involving resonant elements. The resonances are associated with fast-oscillating displacement fields in thin-bridges of perforations shaped as split ring resonators (SRR), and we filter these oscillations by introducing a third scale in the usual two-scale expansion. We start with the Kirchhoff-Love equation and we consider an open bounded region $\\Omega_f\\in\\mathbb{R}^2$. This region is e.g. a slab lens consisting of a square array of SRR shaped as the letter $C$.\n\n\\noindent When the bending wave penetrates the structured area $\\Omega_f$ of the plate\nwhose geometry is shown in Figs. \\ref{fig6}(c)-(d), it\nundergoes fast periodic oscillations. To filter these oscillations,\nwe consider an asymptotic expansion of the associated vertical\ndisplacement $U_\\eta$ solution of the biharmonic equation given in (\\ref{bihaeta}) in terms of a\nmacroscopic (or slow) variable ${\\bf x}=(x_1,x_2)$\nand a microscopic (or fast) variable ${\\bf\nx}_\\eta={\\bf x}\/\\eta$, where $\\eta$ is a small positive\nreal parameter.\n\n\\begin{figure}[]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics[angle=0]{Fig1.eps}}\n\\caption{(a) Geometry of a split ring resonator C consisting of a disc $\\Sigma$ connected to a thin ligament $\\Pi_\\eta$ of length $l$ and thickness $\\eta h$ where $0<\\eta\\ll 1$ in a unit cell Y; (b) Helmholtz resonator consisting of a mass connected to a wall via a spring which models resonances of SRR in (a); (c) Doubly periodic square lattice of SRR with the first Brillouin zone $\\Gamma$XM\nin reciprocal space; (d) Geometry of the thin plate of thickness $H$, with a source on the left side of a platonic crystal (PC) slab occupying\nthe region $\\Omega_f$.}\n\\label{fig6}\n\\end{center}\n\\end{figure}\n\n\n\n\\noindent With all the above assumptions, the out-of-plane\ndisplacement ${\\bf u}_\\eta=(0,0,U_\\eta(x_1,x_2))$ in the\n$x_3$-direction (along the vertical axis) is solution of\n(assuming a time-harmonic dependence $\\exp(-i\\omega t)$\nwith $\\omega$ the angular wave frequency):\n\\begin{equation}\n\\begin{array}{ll}\n\\displaystyle{\\frac{\\partial^2}{\\partial x_1\\partial x_1}\\left({{D}_\\eta}\\left(\\frac{\\partial^2}{\\partial x_1\\partial x_1}+\\nu_\\eta \\frac{\\partial^2}{\\partial x_2\\partial x_2}\\right)\\right)U_\\eta} \\nonumber \\\\\n+\\displaystyle{\\frac{\\partial^2}{\\partial x_2\\partial x_2}\\left({{D}_\\eta}\\left(\\frac{\\partial^2}{\\partial x_2\\partial x_2}+\\nu_\\eta \\frac{\\partial^2}{\\partial x_1\\partial x_1}\\right)\\right)U_\\eta} \\nonumber \\\\\n+\\displaystyle{2\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\left({D}_{\\eta}\\left(1-\\nu_\\eta\\right)\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\right)U_\\eta\n-\\,\\beta_\\eta^4\\,U_\\eta}\n=0 \\; ,\n\\end{array}\n\\label{bihaeta}\n\\end{equation}\ninside the heterogeneous isotropic region $\\Omega_f$ (the platonic crystal, PC, in Fig. \\ref{fig6}(d)), where\n$$D_\\eta=D(\\frac{{\\bf x}}{\\eta}) \\; , \\; \\nu_\\eta=\\nu(\\frac{{\\bf x}}{\\eta}) \\; \\hbox{ and }\n\\; \\beta^4_\\eta=\\beta_0^4\\rho(\\frac{{\\bf x}}{\\eta}) \\; , $$\nare nondimensionalized spatially varying parameters related to the flexural rigidity of the plate, its Poisson ratio and the\nwave frequency, respectively. In most cases, $D$ and $\\nu$ take piecewise constant values, with\n$D>0$ and $-1\/2<\\nu<1\/2$. Note that $\\beta_0^2=\\omega\\sqrt{\\,\\rho_0 H\/D_0}$, where $D_0$ is the\nflexural rigidity of the plate outside the platonic crystal, $\\rho_0$ its density\nand H its thickness.\n\nRemark that (\\ref{bihaeta}) is written in weak form and we notably retrieve the classical boundary conditions for a\nhomogeneous plate with stress-free inclusions (vanishing of bending moments and shearing stress for vanishing\n$D_\\eta$ and $\\nu_\\eta$ in the soft phase) \\cite{graff}.\nSince there is only one phase in the problem which we consider (homogeneous medium outside freely-vibrating inclusions),\nit is also possible to recast  (\\ref{bihaeta}) as\n\\begin{equation}\n(\\sqrt{D_\\eta}\\nabla^2 U_\\eta+\\beta_\\eta^2)(\\sqrt{D_\\eta}\\nabla^2 U_\\eta-\\beta_\\eta^2) U_\\eta=0 \\; , \\hbox{ in\n$\\Omega_f\\setminus\\overline{\\Theta_\\eta}$} \\; , \\;\n\\label{6-helmholtz}\n\\end{equation}\nsince $D_\\eta$ vanishes inside the inclusions $\\Theta_\\eta=\\bigcup_{i\\in\\mathbb{Z}^2}\\{ \\eta (i+C) \\}$ and it is a constant in the matrix.\nBear in mind that the number of SRR in $\\Omega_f$ is an integer which scales as $\\eta^{-2}$.\nNote also that the vanishing of bending moment and shearing stress deduced from (\\ref{bihaeta}) requires that\n\\begin{equation}\n\\begin{array}{ll}\n\\left((1+\\nu_\\eta)\\left(\\frac{\\partial^2}{\\partial x_1^2}+\\frac{\\partial^2}{\\partial x_2^2}\\right)\n+2(1-\\nu_\\eta)\\left(\\frac{\\partial^2}{\\partial x_1\\partial x_2}\\right)\\right)U_\\eta=0 \\; ,\n\\nonumber \\\\\n\\left((3-\\nu_\\eta)\\left(\\frac{\\partial^3}{\\partial x_1^3}+\\frac{\\partial^3}{\\partial x_2^3}\\right)\n+(1+\\nu_\\eta)\\left(\\frac{\\partial^3}{\\partial x_1^2\\partial x_2}\n+\\frac{\\partial^3}{\\partial x_1\\partial x_2^2}\\right)\\right)U_\\eta=0 \\; ,\n\\end{array}\n\\end{equation}\nAt the boundary $\\partial\\Theta_\\eta$ of $\\Theta_\\eta$, which is consistent with our former work on thin perforated plates \\cite{farhat-epl}.\n\n\\noindent In the present case, perforations are shaped as split ring resonators, and each SRR $C$ can be modeled as\n\\begin{equation}\nC=\\{a < \\sqrt{x_1^2+x_2^2} < b\\} \\setminus\\overline{\\Pi_\\eta} \\;\n\\end{equation}\nwhere $a$ and $b$ are functions of variables $x_1,x_2$,\nunless the ring is circular and\n\\begin{equation}\n\\Pi_\\eta = \\Bigl \\{ (x_1,x_2) \\, : \\, 0<x_1 < l \\, , \\, \\mid x_2\n\\mid < \\eta h\/2 \\Bigr \\} \\; , \\label{thindom}\n\\end{equation}\nis a thin ligament of length $l=b-a$ between the ends of the letter $C$,\nsee Fig. \\ref{fig6}(a).\n\nOur aim is to show that the homogenized multi-structured platonic structure within $\\Omega_f$ is characterized by an effective density which can take negative values near the fundamental resonance of the SRR. To do this, we need to perform a homogenization of a periodic structure involving resonant elements. The resonances are associated with fast-oscillating fields in thin-bridges of SRR perforations of the plate, and we filter these oscillations by introducing a third scale in the usual two-scale expansion:\n\n\\begin{equation}\n\\begin{array}{lll}\n\\forall {\\bf x} \\in \\Omega_f, \\;\\; U_\\eta ({\\bf x}) &=\n\\displaystyle{U_0 ({\\bf x},{{\\bf x}\\over\\eta},{x_2\\over\\eta^2}) +\n\\eta U_1 ({\\bf\nx}, {{\\bf x}\\over\\eta},{x_2\\over\\eta^2})} \\nonumber \\\\\n& + \\displaystyle{\\eta^2 U_2 ({\\bf x},{{\\bf\nx}\\over\\eta},{x_2\\over\\eta^2})} +...\n\\end{array}\n\\end{equation}\n\n\\noindent where $U_i : \\Omega_f \\times Y\\times[-h\/2,h\/2]\\longmapsto\\mathbb{C}$ is a smooth function of variables $({\\bf x},{\\bf y},\\xi)=(x_1,x_2,y_1,y_2,\\xi)$, independent of $\\eta$, such that $\\forall {\\bf x}\\in\\Omega_f,\\; \\phi_i({\\bf x},\\cdot,\\xi)$ is $Y$-periodic, and $h$ denotes the thickness of the thin-cut of the SRR.\n\n\\noindent The differential operator is rescaled accordingly as $\\nabla=\\nabla_{\\bf x}+\\frac{1}{\\eta}\\nabla_{\\bf y}+\\frac{1}{\\eta^2}\\nabla_{\\xi}$, so that Eq. (\\ref{6-helmholtz}) can be reexpressed as\n\\begin{equation}\n\\begin{array}{lll}\n&\n\\displaystyle{\\left\\{\\sqrt{D_\\eta}\\left(\\nabla_{\\bf\nx}+\\frac{1}{\\eta}\\nabla_{\\bf y} +\\frac{1}{\\eta^2}\\nabla_\\xi\\right)\n\\cdot \\left(\\nabla_{\\bf x}+\\frac{1}{\\eta}\\nabla_{\\bf y} +\n\\frac{1}{\\eta^2}\\nabla_{\\xi}\n\\right) +\\beta_\\eta^2 \\right\\}} \\nonumber \\\\\n&\\times\\displaystyle{\\left\\{ \\sqrt{D_\\eta}\\left( \\nabla_{\\bf\nx}+\\frac{1}{\\eta}\\nabla_{\\bf y} +\\frac{1}{\\eta^2}\\nabla_\\xi\\right)\n\\cdot \\left(\\nabla_{\\bf x}+\\frac{1}{\\eta}\\nabla_{\\bf y} +\n\\frac{1}{\\eta^2}\\nabla_{\\xi}\n\\right) -\\beta_\\eta^2 \\right\\}} \\nonumber \\\\\n&\\displaystyle{\\left\\{\\sum_{i=0}^\\infty \\eta^i U_i({\\bf\nx},\\frac{{\\bf x}}{\\eta},\\frac{x_2}{\\eta})\\right\\}} =0 \\; ,\n\\end{array}\n\\label{helmholtzrescaled}\n\\end{equation}\nwith the notations $\\nabla_{\\bf x}=(\\frac{\\partial}{\\partial x_1},\\frac{\\partial}{\\partial x_2})$, $\\nabla_{\\bf y}=(\\frac{\\partial}{\\partial y_1},\\frac{\\partial}{\\partial y_2})$ and $\\nabla_\\xi=(0,\\frac{\\partial}{\\partial\\xi})$.\n\n\\noindent Collecting terms of same power of $\\eta$ in Eq. (\\ref{helmholtzrescaled}), we obtain the\nhomogenized problem \\cite{hombook},\n\\begin{equation}\n(\\sqrt{D_{hom}}\\nabla^2 U_\\eta+\\beta_0^2{\\sqrt{\\rho_{hom}}})(\\sqrt{D_{hom}}\\nabla^2 U_\\eta-\\beta_0^2{\\sqrt{\\rho_{hom}}}) U_\\eta=0 \\; , \\hbox{ in\n$\\Omega_f\\setminus\\overline{\\Theta_\\eta}$} \\; , \\;\n\\label{bihahom}\n\\end{equation}\nin the limit when $\\eta$ tends to zero, in the platonic crystal $\\Omega_f$. \nThis is the homogenized biharmonic equation with the homogenized\nplate rigidity $D_{hom}={(\\int_{Y^*}D^{-1}(y_1,y_2)dy_1dy_2)}^{-1}$ \nwhere $Y^*=Y \\setminus \\bar{C}$. We point out that the circular\ngeometry of SRR is essential, if one would take elliptical SRR, then\n$D_{hom}$ would be a rank-2 anisotropic tensor.\n\n\\noindent The homogenized density $\\rho_{hom}$ is given in the form\n\\begin{equation}\n\\sqrt{\\rho_{hom}}(\\beta)=1-\\sum_{m=1}^{\\infty}\n\\frac{\\beta^2}{\\beta^2-{\\beta_m}^2} {\\Vert V_m\n\\Vert}_{L^2(0,l)}^2 \\; , \\label{negdens}\n\\end{equation}\n\n\\noindent where the eigen-solutions $V_m$ correspond to longitudinal vibrations of the thin cut $\\Pi_\\eta$ within the split ring resonator.\n\n\\noindent It is clear from Eq. (\\ref{negdens}) that $\\sqrt{\\rho_{hom}}(\\beta)$ takes negative values near resonances $\\beta^2=\\beta_m^2$. For our purpose, it is enough to look at the first few resonant frequencies $\\beta_m^2$ (the higher the frequency the worse the asymptotic approximation). These frequencies are associated with vibrations $V_m$ of the thin domain $\\Pi_\\eta$ \\cite{Seb_SRR}:\n\\begin{eqnarray}\nV_m''(x_1) + {\\beta_m}^2 V_m(x_1) & = & 0 \\, , \\, 0<x_1<l \\, , \\label{6-sys1}\\\\\nV_m(0) & = & 0 \\; , \\label{6-sys2}\\\\\n\\eta h V_m'(l) & = & \\mathrm{area}(\\Xi) {\\beta_m}^2 V_m(l) \\, ,\n\\label{6-sys3}\n\\end{eqnarray}\nwhere $\\eta h$ and $l$ are the thickness and the length of the thin ligament $\\Pi_\\eta$, and $\\Xi$ is the central disc within the SRR. The ligament $\\Pi_\\eta$ is connected to $\\Xi$, hence $V(l)=V$, where $V$ is the vibration of the stress-free air cavity $\\Xi$. Note that the derivation of Eq. (\\ref{6-sys3}) required a boundary layer analysis, and we refer to \\cite{kozlov99} for more details.\n\n\\noindent The solution of the problem (\\ref{6-sys1})-(\\ref{6-sys3}) has the form\n\\begin{equation}\nV_m(x_1)= A \\sin(\\beta_m x_1) \\; , \\label{asymfield}\n\\end{equation}\nwhere $\\beta_m$ (square root of frequency) is given as the solution of the following equation\n\\begin{equation}\n\\eta h\\cot(\\beta_m l) = \\mathrm{area}(\\Xi) \\beta_m \\; .\n\\label{tranres}\n\\end{equation}\n\nNote also that a similar problem to\nEqs. (\\ref{6-sys1})-(\\ref{6-sys3}) is deduced from Eq. (\\ref{helmholtzrescaled}) with a minus sign in (\\ref{6-sys1}), which leads to\na $\\sinh$ function in (\\ref{tranres}), but this does not give any additional resonant frequencies. The resonant\nfrequencies of $\\sqrt{\\rho_{hom}}$ also lead to negative values of $\\rho_{hom}$, in the same way that a\nproduct of two simultaneously negative square roots give a negative refractive index in the metamaterial's literature\n\\cite{veselago,ramak2005}.\n\n\nIn the sequel, we focus our analysis on the Floquet-Bloch bending wave problem and give some numerical results and comparisons with the homogenization theory.\n\n\\section{Numerical analysis of SRR platonic crystals}\n\n\\subsection{Dispersive properties}\n\n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.65}{\\includegraphics[angle=0]{Fig2.eps}}\n\\mbox{}\\vspace{-5.5cm}\\caption{(a) Band diagram for the doubly periodic square lattice representing the normalized frequency $\\beta^2$ versus the Bloch vector {\\bf k} in the first Brillouin zone $\\Gamma$XM. (b) Map of the eigen-mode corresponding to the eigen-frequency $\\beta^2=2.95$ in a unit cell for a SRR with an outer radius $0.48d$ and inner radius $0.1d$ and thickness of the thin ligament equal to $0.05d$.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\n\n\n\nTo investigate numerically the stop-band properties for out-of-plane bending waves propagating within the array of SRR, we use the Finite Element Method (by implementing the weak form of (\\ref{6-helmholtz}) in the commercial software COMSOL and its corresponding Floquet-Bloch boundary conditions as well as the PMLs: perfectly matched layers).\n\nIn Fig. \\ref{fig1}(a), we give the band diagram for normalized eigen-frequencies $\\beta^2$ as a function of the projection of the Bloch vector ${\\bf k}$ on the first Brillouin zone $\\Gamma$XM. We consider a square array of normalized pitch $d$=1 (take for example $d=1$cm for comparisons with feasible experiments) with embedded SRR of inner radius $0.1d$ and outer radius $0.48d$ with a thin cut (ligament) of thickness $0.05d$.\\\\\nThis figure displays two full platonic band gaps for the range of normalized frequencies $[2.39, 2.92]$ and $[4.23, 24.32]$. As depicted in a previous study \\cite{Seb_SRR}, the low frequency band gap (the first one) is associated with the resonant modes of a single SRR (microscopic structure), while the second one is due to a Bragg scattering phenomenon (macroscopic structure). We also plot in Fig. \\ref{fig1}(b) the corresponding localized eigen-function (the first eigen-mode) which corresponds in the context of continuum mechanics, to oscillations of the central region of the SRR as a rigid solid connected by the thin cut $\\Pi_{\\varepsilon}$ to the fixed rigid region around it \\cite{kozlov99}. In our context of bending waves, this can be interpreted as a collective vibration of the plate elements together, up and down in each SRR.\n\n\\begin{figure}[]\n\\begin{center}\n\\scalebox{0.65}{\\includegraphics[angle=0]{Fig3.eps}}\n\\caption{(a) Band diagram of surface mode along the $\\Gamma$X direction. (b) 1D profile along the $x_2$ direction of the mode. (c) Supercell showing the configuration where the plasmon mode was excited from the top layer with PMLs in the bottom layer to avoid any reflection and Floquet-Bloch boundary conditions on the left and right sides to account for the infinite array in $x_1$\ndirection.\n}\n\\label{fig5_2}\n\\end{center}\n\\end{figure}\n\n\n\nThe transcendental equation (\\ref{tranres}) can be further simplified if we look at the first low frequency, for which we deduce the explicit asymptotic approximation\n\\begin{equation}\n\\beta_1^2 \\sim \\frac{\\eta h}{\\mathrm{area}(\\Xi)l} \\; .\n\\label{asymfor}\n\\end{equation}\n\n\\noindent Our numerical estimate is\n\\begin{equation}\n\\beta_1^2 \\sim \\frac{0.05}{0.38} \\frac{1}{\\pi\n0.1^2}=4.19 \\; ,\\label{numestimate1}\n\\end{equation}\nwhich is in excellent agreement with the finite element value for the plasmon frequency $\\beta^2=4.23$ occurring at the $M$ point when ${\\bf k}=(\\pi,\\pi)$.\n\nIt is interesting to also analyze the dispersive properties of surface elastic modes (waves exponentially localized at the interface of a platonic crystal). These modes originate from interference effect in phononic or platonic crystals and are important, for example, for superlensing effect. The dispersion diagram of these modes is shown in Fig. \\ref{fig5_2}(a), where the frequency is plotted versus Bloch vector in the $\\Gamma$M direction. The red dotted-dashed line reprsents the surface mode's dispersion and shows that's standing in the elastic bandgap (between the first and the second propagating modes). This means, for instance, that this mode is of evanescent nature. Figure \\ref{fig5_2}(b) shows the flexural displacement at the interface between the platonic crystal and free-space [Fig. \\ref{fig5_2}(c)], validating the decaying nature of the mode.\n\n\n\\subsection{(Super-)Lensing effect via negative effective density of a platonic crystal}\n\n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.6}{\\includegraphics[angle=0]{Fig4.eps}}\n\\mbox{}\\vspace{-3cm}\\caption{(a) Pattern of the energy of the bending wave at the frequency $\\beta=1.98$ where AANR occurs. (b) and (c) give respectively the profile of the energy along the dotted white line at the source (red curve) and the image (green curve) axis, respectively.}\n\\label{fig2}\n\\end{center}\n\\end{figure}\n\n\nWe then consider a 2D finite platonic crystal perforating a thin elastic plate. The crystal consists of 230 holes with a C-shaped cross-section with identical parameters as those of Fig. \\ref{fig1}(b). The crystal has a four-fold (square) symmetry with pitch $d=1$. A flexural waves source of wavenumber $\\beta=1.98$ is located at a distance $0.1d$ above the top of the crystal. The field-maps in Fig. \\ref{fig2} (a) show the existence of an image of the source, located twice the width of the crystal away in the lower side. As we can see in Fig. \\ref{fig2} (a), the amplitude of the field is the highest near the thin ligaments $\\Pi_{\\varepsilon}$ of the SRR. Formula (\\ref{negdens}) shows that the resonance of the field with the microscopic structure of SRR is responsible for the appearance of negative refraction through negative effective density $\\rho_{hom}$. Finally, the resolution of the image $\\delta$ is enhanced compared to that obtained using an array of perforations with a circular or square cross-section, as demonstrated in Fig. \\ref{fig2}(b), (c), whereby the full width at half maximum of the image point $\\delta\\approx\\lambda\/3$.\n\n\\noindent We note that at the plasmon frequency, the vibration of the thin ligaments is enhanced, which is in accordance with the behavior of the field as observed on Fig. \\ref{fig2}. It is therefore important to incorporate the resonances within the thin-bridges as we did, to model the frequency at which negative refraction occurs.\n\n\n\\subsection{Gaussian beam at oblique incidence on a platonic crystal}\n\n\\begin{figure}[h]\n\\begin{center}\n\\scalebox{0.65}{\\includegraphics[angle=0]{Fig5.eps}}\n\\mbox{}\\vspace{-4.5cm}\\caption{Bending displacement when the platonic crystal is excited by a gaussian beam at an incidence angle $\\theta_{\\mathrm{inc}}=25$ degrees. (b) Snapshot of the bending wave in the presence of an effective medium with negative elastic index $n_{\\mathrm{e}}=-1.5+10^{-3}j$, with $n_{\\mathrm{e}}=\\sqrt{12\\rho(1-\\nu^2)\/(EH^2)}$.}\n\\label{fig3}\n\\end{center}\n\\end{figure}\n\n\nThe interaction of a Gaussian beam (GB) with the platonic crystal around the AANR is considered in this section. Refraction of GB (whose size is in the order of the wavelength) obliquely incident on the SRR platonic crystal is demonstrated to be negative.\n\n\\noindent As known, considering the interaction of a GB with any optical or mechanical media, reveals much of their propagating and scattering properties \\cite{gbeam,gbeam2}. These are paraxial solutions of the biharmonic scalar wave  equation $(\\Delta^2-\\beta^4)W=0$, and thus have the form\n\\begin{equation}\nU(x,y)=u(x,y)e^{-j\\beta x}+n.p.\n\\label{gaussian}\n\\end{equation}9\nwith the wavenumber $\\beta$, the coefficient $u(x=0,y)$ has the well know Gaussian distribution in the $x-y$ plane. $n.p.$ denotes a non propagating contribution due to the fourth order nature of the biharmonic equation. It could be neglected in the real implementation. The expression of the obtained elastic beam is thus\n\\begin{equation}\nu(x,y)=u_0\\frac{\\omega_0}{\\omega(x)}e^{\\left(-\\frac{y^2}{\\omega^2(x)}-ik\\frac{y^2}{2R(x)}+i\\zeta(x)\\right)}\\, ,\n\\label{gaussian}\n\\end{equation}\nwith $\\omega(x)=\\omega_0\\sqrt{1+\\left(\\frac{x}{x_R}\\right)^2}$,\n$x_R=\\frac{\\pi\\omega_0^2}{\\lambda}$ and $\\omega_0$ is called\nthe waist of the beam. The radius of the curvature is given by\n$R(x)=x\\left[1+\\left(\\frac{x_R}{x}\\right)^2\\right]$ and the\nGouy phase shift is $\\zeta(x)=\\mathrm{atan}\\frac{x}{x_R}$.\n\n\\noindent Figure \\ref{fig3} gives the snapshot of the\ndisplacement field $U$ in the presence of the platonic crystal\nat oblique incidence with angle\n$\\theta_{\\mathrm{inc}}=25$ degrees; a reflected beam could be\ndistinguished as well as a refracted one which turns out to be\nin the negative refraction regime. To further verify this\nclaim, another simulation using, now, an homogeneous effective\nslab of the same size, with negative elastic index $n_{\\mathrm{e}}=\\omega\/\\beta^2=\\sqrt{12\\rho(1-\\nu^2)\/(EH^2)}$,\nis shown in Fig. \\ref{fig3}(b). The patterns of the wave are identical in \\ref{fig3}(a) and\n\\ref{fig3}(b) and show that the elastic index of the SRR crystal could be\ndescribed as negative: $n_{\\mathrm{e}}=-1.5+10^{-3}j$. The\nrobustness of the AANR was further verified by changing the\nangle of incidence $\\theta_{\\mathrm{inc}}$ and the wavelength\nof operation around their main values, and the effect still\ncould be observed. \n\n\n\n\\subsection{Ultrarefraction effect via vanishing effective density of a platonic crystal}\n\n\\begin{figure}[h]\n\\begin{center}\n\\mbox{}\\vspace{-4.5cm}\\scalebox{0.65}{\\includegraphics[angle=0]{Fig6.eps}}\n\\caption{Pattern of the elastic displacement (a) and energy of the bending wave (b) at the frequency where ultrarefraction occurs when a point source is located at the center of the platonic crystal.}\n\\label{fig4}\n\\end{center}\n\\end{figure}\n\n\nTo conclude this section, let us consider another anomalous refractive effect, namely, the ultra-refraction of bending waves. This phenomenon generally occurs when the effective elastic index $n_{\\mathrm{e}}$ defined above, tends to zero or equally speaking the group velocity ${\\bf v}_{\\mathrm{g}}\\approx0$. In this case, independently of the angle of incidence of an incident ray, propagating towards the crystal, it will be refracted normally to it (parallel to the normal). And the operation point could be easily deduced from the dispersion diagram by picking a point where the mode's dispersion curve becomes flat ($\\nabla_{\\bf k}\\beta^2\\approx0$). This allows us to build an elastic antenna (for instance by placing a cylindrical source of flexural waves in the center of the crystal). The wave will be then refracted along the normal to the crystal, permitting thus highly directive beam as could be seen from Fig. \\ref{fig4}(a) and (b) where the displacement snapshot and its energy are respectively plotted around a wavelength $\\lambda=2\\pi\/\\beta\\approx3$. This corresponds as previously stated to the $M$ point in the diagram of Fig. \\ref{fig1}(a) at the maximum. Using formula (\\ref{negdens}), one can indeed achieve ultrarefraction when the effective density of the platonic crystal $\\rho_{hom}$ is close to zero, which happens for\n$\\beta_2^2\\sim 4.37$. To get this estimate, one need replace the boundary condition (\\ref{6-sys2}) by $V_m(0)=-\\rm{area}(\\Sigma)\/\\rm{area}(Y-C)$ which amounts to assuming that the field takes the same {\\it non zero} values on the opposite edges of the basic cell.\nThis leads to the new frequency estimate\n\\begin{equation}\n\\beta_2^2 \\sim \\frac{\\eta h}{\\mathrm{area}(\\Xi)l}\n\\left( 1 + \\frac{\\rm{area}(\\Sigma)}{\\rm{area}(Y-C)}\n\\right) \\; ,\n\\label{asymforbest}\n\\end{equation}\nwhich is a refinement of (\\ref{asymfor}).\nThe numerical result $\\beta_2^2\\sim 4.37$ is in good agreement with finite element computations (4.385).\n\n\n\n\n\\section{Concluding remarks}\n\n\\noindent In conclusion, we have proposed an original route towards elastic negative refraction (superlensing) and ultrarefraction based on excitation of flexural surface modes. To analyze these effects, we have derived the homogenized biharmonic equation using a multi-scale asymptotic approach. We found that the homogenized elastic parameters are described by a scalar flexural plate rigidity (since the perforations have a circular geometry owing to the fact that thin-bridges come unseen in the second scale asymptotics) and a scalar density, the latter taking negative values near thin-bridges resonances (unveiled by the third scale). We then performed numerical computations based on the finite element method which confirmed that the asymptotic model gives accurate predictions on dispersive properties of the elastic Lamb modes. \n\nWe believe that such a micro-structured plate could be manufactured easily, having in mind some potential applications in superlensing or directive elastic antennas. The range of industrial applications is vast, and our proof of concept should foster research efforts in this emerging area of acoustic and seismic metamaterials.\n\n\n\\acknowledgments\nS.G. would like to acknowledge a funding of the European Research Council through ERC grant ANAMORPHISM.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction: muscle fatigue and digital human modeling}\n\\subsection{Muscle fatigue}\nMuscle fatigue is defined as the decrease in maximum force \\cite{Bigland-Ritchie1995}. Work-related muscle fatigue contributes to occupational Musculoskeletal Disorders (MSDs) \\cite{Chaffin1999}, which makes up the vast proposition of the occupational diseases \\cite{Eurogip2015}. As illustrated by Armstrong \\cite{Armstrong1993}, improper physical work requirements lead to muscle fatigue. It is important to quantify fatigue and to determine the limits of acceptable work requirements \\cite{Chaffin1999}. Proper work design would reduce the risk of excessive physical workload.\n\nIn the effort to involve muscle fatigue analysis into work design, there are two key problems that bother us. First, in actual working scene, the motion adopted by workers to finish a task would be arbitrary rather than routine and repeated. This makes it difficult to evaluate the exact workload carried by a certain muscle. Fatigue analysis, without the exact information about muscle workload, would be inaccurate. Second, muscle fatigue process varies a lot among human groups. The utilization of fatigue analysis would be limited without proper consideration about demographical human attribute.    \n\\subsection{Digital human modeling}\nDigital human modeling (DHM) technique offers an efficient way to simulate ergonomics issues in the process of work design. The integration of biomechanical models with DHM systems allows us to evaluate musculoskeletal workload in manual work simulations. Related software such as Jack \\cite{Jack2017}, Delmia \\cite{Delmia2017}, 3DSSPP \\cite{3DSSPP1999}, Anybody \\cite{Damsgaard2006}, OpenSim \\cite{Delp2007} are available for work design. All these softwares render realistic mannequins to visualize work tasks. Backward or inverse dynamics methods are used to calculate the muscle-tendon reaction force \\cite{Chaffin2008}. \n\nAmong the mentioned software, the mannequin used by Jack, Delmia and 3DSSPP shall be settled by the gender and the percentile of body height and weight in a given anthropometric database (USA, Canadian, German, Korean, etc). This makes it easy to apply analysis for a specific group of people. Unfortunately, as a vital parameter of work design, muscle force capacity is not included in the database. It is unreasonable to assume the same muscle capacity among different anthropometric groups while the other measures diverse. Further muscle analysis on the basis of this muscle capability would be low-effective. \n \nIn Jack, Delmia and 3DSSPP, a motive task is simulated by the congregation of a set of static tasks with a certain posture. Each working posture is evaluated separately without considering the history of the motion. The external loads, the duration time and the frequency of the posture is identified. By applying strength models or inverse dynamic models, load of several major joints are calculated. In 3DSSPP and Jack, fatigue analysis is available based on the static joint load, task duration and frequency. This method goes well for simple and repetitive tasks. But when it comes to arbitrary task, there would be a great lack of accuracy.\n \nThe simulation and analysis by OpenSim and Anybody are more specified. The mannequin is constituted of concrete bones and muscles where musculoskeletal geometry is scaled and adaptive to subjects. The motions obtained by a motion capture system or computed along a simulated task permitted us to have the kinematic information, such as positions, velocities and accelerations of a motion. The inertial properties of body segments are estimated. By applying the Newtonian principles, the prediction of the resultant extrinsic forces and moments are then available.\n\nAlthough the joint reaction force and muscle load are accessible in OpenSim and Anybody, the accumulation effect has not been taken into account; and no accurate fatigue analysis is available.\n\\subsection{Objectives}\nIn this research, a plug-in to OpenSim is written to involve the muscle fatigue analysis to an arbitrary task. Concrete muscle force capability change is specified and the influence of demographic human attributes is considered. This work is promising to offer a virtual work design platform that helps to predict muscle fatigue.\n\\section{Methodology: OpenSim human modeling and muscle fatigue analysis}\n\\subsection{Human modeling and dynamic simulation in OpenSim}\\label{chap2.1}\nAs mentioned above, OpenSim is a digital human modeling platform. It allows users to build and analyze different musculoskeletal models. A model consists of a set of rigid segments connected by joints. Muscles and ligaments span the joints, develop force, and generate movements of the joints. After the build-up of a musculoskeletal model, OpenSim takes experimentally-measured kinematics, reaction forces and moments as input data. This is usually obtained by motion capture system from a subject. The experimental kinematics ($i.e.$, trajectories of markers, joint centers, and joint angles) are used to adjust and scale the musculoskeletal model to match the dimensions of the subject\\cite{Delp2007}. \n\nFor dynamic simulation, an inverse kinematics problem is solved to find the model joint angles that best reproduce the experimental kinematics. Then a residual reduction algorithm is used to adjust the kinematics so that they are more dynamically consistent with the experimental reaction forces and moments. Finally, a  computed muscle control (CMC) algorithm is used to find a set of muscle excitations and distribute forces across synergistic muscles to generate a forward dynamic simulation that closely tracks the motion \\cite{Delp2007}. In this way, the workload of each muscle is accessible along an arbitrary motion, which paves way for the fatigue analysis. \n\\subsection{Muscle fatigue analysis}\\label{chap2.2}\n Ma et al. \\cite{Ma2009} proposed a \"Force-load fatigue model\" based on mechanical parameters. It depicts how muscle force declines with time with consideration of relative workload and intrinsic human attribute. The model was described as a differential function (Eq. \\ref{eq}). According to this model, during a fatiguing process, muscle force capability ($F_{cem}(t)$) changes depending on a) Maximal (or initial) muscle force capability, $F_{max}$; b) External load on the muscle, $F_{Load}(t)$ and c) Intrinsic muscle fatigability, $k$. For detailed explanation of this model was introduced in Ma et al. \\cite{Ma2009,Ma2011} and \\cite{Sakka2015} for static and dynamic cases, respectively.\n \\begin{eqnarray}\\label{eq}\n\\frac{d F_{cem}(t)}{d t}=-k\\frac{F_{cem}(t)}{F_{max}}F_{Load}(t)\n\\end{eqnarray}\n\nThis model has been mathematical validated in Ma et al. \\cite{Ma2009} with static MET models and other existing dynamic theoretical models.\n \nIn this model, intrinsic human attribute concerning to fatigue rate is taken into consideration, which is referred to fatigability. The definition of fatigability is proposed by Ma \\& Chang \\cite{Ma2014} \n``Muscle fatigability describes a tendency of a muscle from a given subject to get tired or exhausted, and it should only be determined by the physical and psychological properties of the individual subject\". According to this model, the decrease rate of muscle capability is in proportion with both work load and current muscle capability. The proportion coefficient $k$ quantifies the tendency of muscle strength descending, and is noted as fatigability. \n\nFatigability varies significantly among human groups. For example, females are found to be more fatigue-resistant than males \\cite{Yoon2015}; and the older groups shows significantly much less force loss than the younger group after a certain exercises \\cite{Kent-Braun2002}. Fatigability $k$ has been determined by comparing the Force-Load muscle fatigue model with the empirical maximal endurance models \\cite{Ma2011}. The determined value of $k$ varies from 0.87 $min^{-1}$ to 2.15 $min^{-1}$ for general muscle groups. Ma et al \\cite{Ma2013} also conducted experiments to measure fatigability. In a static drilling task, the fatigability of 40 male workers was identified to be $1.02 \\pm 0.49$ $min^{-1}$ for the upper limbs.\n\\subsection{Muscle fatigue analysis in OpenSim}\nThe object of this research is a concrete analysis of muscle fatigue. In another word, how force capability of each muscle declines during an arbitrary motion. According to the Force-load muscle fatigue model, this objective can be achieved on condition of two values: workload on each muscle along the motion and the maximal muscle capability. \n\\subsubsection{Workload on each muscle}\nThe muscle force generation dynamics can be divided into activation dynamics and contraction dynamics \\cite{Zajac1989}. Activation dynamics corresponds to the transformation of neural excitation to activation of the muscle fibers. Muscle contraction dynamics corresponds to the transformation of activation to muscle force. \n\n \\begin{figure}[h!]\n \\label{Fig1}\n \\centering\n \\includegraphics[width=1\\textwidth]{Figure1.jpg}\n  \\caption{Muscle force generation dynamics. Adapted from Zajac (1989) \\cite{Zajac1989}. }\n \\end{figure}\n Activation dynamics is related to calcium release, diffusion and uptake from the sarcoplasmic reticulum \\cite{Winters1995}. It is modeled by a first-order differential equation \\cite{Thelen2003}:\n \\begin{eqnarray}\\label{eq2}\n \\frac{da}{dt}=\\frac{u-a}{\\tau(u,a)}\n \\end{eqnarray}\n where $u$ is excitation (from 0 to 1), $a$ is activation (from 0 to 1), and $\\tau$ is a variable time constant.\n\nMuscle contraction dynamics deals with the force-length-velocity relationships and the elastic properties of muscles and tendons. In OpenSim, it is modeled by a lumped-parameter model \\cite{Delp2007}:\n\n\\begin{eqnarray}\n\\frac{dl_{m}}{dt}=f_{v}^{-1}(l_{m}, l_{mt},a)\n\\end{eqnarray}\nwhere $l_{m}$ is the muscle length, $l_{mt}$ is the muscle-tendon actuator length, and $f_{v}$ is the force velocity relation for muscle.\n\nAs mentioned in chapter \\ref{chap2.1}, OpenSim develops a CMC algorithm to calculate muscle activation and therefore to distribute joint force among a series of muscles. \nBy applying the CMC algorithm, the workload on each muscle is accessible. \n\\subsubsection{The Maximal muscle force capability of each muscle}\n %\nMuscle activation depends on the neural excitation level. In the case of a certain muscle contraction speed and muscle length, muscle force increases with muscle activation. Full activation (i.e., $a(t) = 1$) happens when a muscle contractile component has been maximally excited (i.e., $u(t) = 1$) for a long time \\cite{Zajac1989}. During an arbitrary motion, muscle kinematics changes from time to time. We calculate the maximal muscle force $F_{max}$ based on the CMC algorithm, in addition that the activation level of each muscle is preset to full level ($a(t) = 1$).\n \nAs the final step, a Plug-in is written based on the Force-load muscle fatigue model, with the required inputs obtained by the above methods. \n\\section{Data and Simulation}\nThe plug-in is tested on a three-dimensional, 29 degree-of-freedom human model developed by Stanford \\cite{Hamner2010}. The model, as the other OpenSim musculoskeletal models, is made up of bodies, joints, and muscle-tendon actuators. Specifically, this model consists of 20 body segments, 19 joints and 92 muscle actuators, as shown in Figure~2. The inertial parameters for the body segments in the model are based on average anthropometric data obtained from five subjects (age 26 $\\pm$ 3 years, height 177 $\\pm$ 3 cm, and weight 70.1 $\\pm$ 7.8 kg).\n\n\n \\begin{figure}[h!]\n \\label{Fig2}\n \\centering\n \\includegraphics[width=0.35\\textwidth]{Figure2.jpg}\n  \\caption{Full body running digital human modeling. Muscles are represented by red lines.}\n \\end{figure}\n\n\nThis model is developed to study the muscles' contribution to the acceleration of the body during running. It covers the muscles that needed for arbitrary running motions. These muscles could be classified into three groups: torso-core muscle group, pelvis-femur muscle group, and lower knee extremities muscle group. \n\n\nThe simulation data is also from the project of Hamner et al. \\cite{Hamner2010}. It is recorded from a healthy male subject (height 1.83 m, mass 65.9 kg) running on a treadmill at 3.96 m\/s. A total of 41 reflective markers are placed on the subject's anatomical landmarks during the experiment to scale the model to the subject's anthropometry. Ground reaction forces and markers' trajectories are recorded. The recorded motion lasts for 10 seconds. \n \n\nIn our study, 10 muscles are selected from the three muscle groups to conduct muscle fatigue analysis. The basic characteristics of these muscles are listed in Table~\\ref{tab1}.\n\n\\begin{table}[h!]\n\\label{tab1}\n\\begin{center}\n\\caption{Basic characteristics of the analyzed muscles.}\n\\begin{tabular}{l@{\\qquad}l@{\\qquad}l@{\\qquad}l}\n\\hline\n\\rule{0pt}{20pt} \nMuscle name & \\tabincell{l}{Appertained \\\\ group} & \\tabincell{l}{Optimal fiber\\\\length ($m$)} & \\tabincell{l}{Maximal isometric\\\\force ($N$)} \\\\[2pt]\n\\hline\\rule{0pt}{12pt}\nErector spinae        & Torso-core    & 0.120 & 2500.0 \\\\\nExternal oblique      & Torso-core    & 0.120 & 900.0  \\\\\nInternal oblique      & Torso-core    & 0.100 & 900.0  \\\\\nAdductor magnus       & Pelvis-femur  & 0.131 & 488.0  \\\\\nGlute maximus         & Pelvis-femur  & 0.142 & 573.0  \\\\\nGlute medius          & Pelvis-femur  & 0.065 & 653.0  \\\\\nTibialis posterior    & Lower knee    & 0.031 & 1588.0 \\\\\nLateral gastrocnemius & Lower knee    & 0.064 & 683.0  \\\\\nExtensor digitorum    & Lower knee    & 0.102 & 512.0  \\\\\nSoleus                & Lower knee    & 0.050 & 3549.0  \\\\\n[2pt]\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\section{Results of simulation}\nDuring the arbitrary running, workloads on muscles vary from moment to moment. A typical muscle workload change is shown in Figure~\\ref{Fig3}.\n\n \\begin{figure}[h!]\n \\label{Fig3}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{Figure3.jpg}\n  \\caption{Workload on erector spinae muscle during 10s arbitrary running at 3.96 m\/s.}\n \\end{figure}\n\n\\subsection{General muscle force decline}\n\nIn order to investigate the fatigue process, the motion data is duplicated to 10~min. During the running process, muscle force capabilities decline with time. After input the fatigability ($k$) of the subject, the detail information about force capability changes is accessible. A general view of force capability changes of the selected muscles are shown in Figure~4. Here the fatigability is set to 1.0 $min^{-1}$. Generally, the muscles' capabilities reduce to 60\\% to 70\\% of their maximum after running  for 10 min.\n\n \\begin{figure}[h!]\n \\label{Fig4}\n\\begin{centering}\n \\includegraphics[width=0.7\\textwidth]{Figure4.jpg}\n  \\caption{Force capability lines of ten muscles during 10 min running. $k$ = 1.0 $min^{-1}$. \\qquad \\qquad Force of each muscle is normalized by its maximum.}\n  \\end{centering}\n \\end{figure}\n\\subsection{Comparison among different muscles}\nThe initial and ending force capabilities of the selected muscles are shown in Table~2.\n\n\\begin{table}[h!]\n\\label{tab2}\n\\begin{center}\n\\caption{General information of muscle force capabilities}\n\\begin{tabular}{l@{\\qquad}l@{\\qquad}l@{\\qquad}l@{\\qquad}l}\n\\hline\n\\rule{0pt}{20pt} \nMuscle name & \\tabincell{l}{Appertained \\\\ group} & \\tabincell{l}{Initial \\\\ capabilities\\\\($N$)} & \\tabincell{l}{Finial \\\\ capabilities\\\\($N$)} & \\tabincell{l}{Proportion\\\\of reduction}\\\\[2pt]\n\\hline\\rule{0pt}{12pt}\nErector spinae        & Torso-core    & 38703.5 & 23563.3 & 39.1\\% \\\\\nExternal oblique      & Torso-core    & 19876.2 & 12926.2 & 35.0\\%\\\\\nInternal oblique      & Torso-core    & 18374.8 & 11705.9 & 36.3\\%\\\\\nAdductor magnus       & Pelvis-femur  & 10923.7 & 7322.3  & 33.0\\%\\\\\nGlute maximus         & Pelvis-femur  & 12304.1 & 8096.3  & 34.2\\%\\\\\nGlute medius          & Pelvis-femur  & 19134.0 & 11817.8 & 38.2\\%\\\\\nTibialis posterior    & Lower knee    & 36901.9 & 22624.1 & 38.7\\%\\\\\nLateral gastrocnemius & Lower knee    & 18231.9 & 12481.3 & 31.5\\%\\\\\nExtensor digitorum    & Lower knee    & 8940.7  & 5558.1  & 37.8\\%\\\\\nSoleus                & Lower knee    & 95221.7 & 60708.3 & 36.3\\%\\\\\n[2pt]\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe proportion of force capability reduction is between 30\\% to 40\\% for each of the ten muscles. Erector spinae loses the maximal proportion of force. As far as the selected muscles, torso-core muscle group fatigues no less than the pelvis-femur or the lower knee group (average fatigue level: (36.8 $\\pm$ 2.1)\\%, (35.1 $\\pm$ 2.7)\\%, (36.1 $\\pm$ 3.2)\\%, respectively).  \n\\subsection{Influence of fatigability index $k$}\nFatigability is a subject-specific parameter that might also varies between muscle groups. As mentioned in Chapter~\\ref{chap2.2}, the determined value of $k$ varies from 0.87 $min^{-1}$ to 2.15 $min^{-1}$ for general muscle groups. In the current study, we examine the influence of fatigability by comparing the fatigue level when $k$ = 1.0 $min^{-1}$ with that when $k$ = 2.0 $min^{-1}$. A typical comparison is shown in Figure~5.\n\n \\begin{figure}[h!]\n \\label{Fig5}\n \\centering\n \\includegraphics[width=0.8\\textwidth]{Figure5.jpg}\n  \\caption{Force capability lines of erector spinae with different preset $k$ values. Force is normalized by the maximum.}\n \\end{figure}\n\nTable~3 manifests the fatigue level comparisons of all the ten muscles. Generally, the muscle reduces to 40\\% to 50\\% of its initial maximal capability. The relative sort of muscles' fatigue level remains unchanged.  \n\n \\begin{table}[h!]\n \\label{tab3}\n \\begin{center}\n \\caption{Comparison of muscle force capabilities between different fatigabilities}\n \\begin{tabular}{l@{\\qquad}l@{\\qquad}p{2.2cm}@{\\qquad}c}\n \\hline\n \\rule{0pt}{15pt} \n ~Muscle name & \n ~Appertained group~ &\n \\multicolumn{2}{c}{~Proportion of capability loss~}\\\\\n\\cline{3-4} \n \\rule{0pt}{10pt} \n & &  \\centering $k$ = 1.0 $min^{-1}$ & $k$ = 2.0 $min^{-1}$\\\\  \n \\hline\n \\rule{0pt}{12pt} \nErector spinae        & Torso-core    &  \\centering 39.1\\%  &  62.9\\% \\\\\n External oblique      & Torso-core    &  \\centering 35.0\\%  &  57.7\\% \\\\\n Internal oblique      & Torso-core    &  \\centering 36.3\\%  &  59.4\\% \\\\\n Adductor magnus       & Pelvis-femur  &  \\centering 33.0\\%  &  55.1\\% \\\\\n Glute maximus         & Pelvis-femur  &  \\centering 34.2\\%  &  56.7\\% \\\\\n Glute medius          & Pelvis-femur  &  \\centering 38.2\\%  &  61.9\\% \\\\\n Tibialis posterior    & Lower knee    &  \\centering 38.7\\%  &  62.4\\% \\\\\n Lateral gastrocnemius & Lower knee    &  \\centering 31.5\\%  &  53.1\\% \\\\\n Extensor digitorum    & Lower knee    &  \\centering 37.8\\%  &  61.4\\% \\\\\n Soleus                & Lower knee    &  \\centering 36.3\\%  &  59.4\\% \\\\\n \\hline\n \\end{tabular}\n \\end{center}\n \\end{table}\n\\section{Discussions}\nAccording to the simulation data, after 10 minutes' running at the speed of 3.96 $m\/s$, a healthy male subject is likely to lose 30\\% to 40\\% of his maximal muscle capability. The requirement of the running task at the current posture is about 5\\% of his maximal muscle capability. If the task continues, there would be a time in future when the subject's muscle capability reduces to near to the required workload. Muscles will enter into a risk zone \\cite{Ma.R2012,Ma2009a}. Damage will occur to muscles, which increases the risk of MSDs. The subject would change his posture unconsciously \\cite{fuller2009}. It is important to predict the exhausting time in the early process of work design.\n\nIn Hammer's research \\cite{Hamner2010}, the quadriceps (pelvis-femur muscle group) and plantar flexors (lower knee muscle group) are the major contributors to acceleration of the body mass center during running, compared with the torso-core muscle group (erector spinae and iliopsoas). While in the current study, erector spinae, who loses 39.2\\% of its maximal capability, is found to be more fatigue-exposed than the others. Also, torso-core muscle group fatigues no less than the other two muscle groups. This phenomenon indicates that torso-core muscles undertake other supportive functions than contributing to body accelerating, such as counterbalancing the vertical angular momentum of the legs.   \n\n\nA subject with a higher fatigability value ($k$ = 2.0 $min^{-1}$) losses about 20\\% of his maximal capability more than the subject with a lower fatigability value ($k$ = 1.0 $min^{-1}$). The influence of fatigability is evident. It is essential to study the fatigability among different human groups.\n\nIt should be noticed that the current study considers no effect of fatigue recovery. Future study should integrate the muscle recovery for more accurate calculation.\n\\section{Conclusions and implications}\nIn this study, a plug-in to OpenSim is written on the base of the Force-load muscle fatigue model and muscle force generation dynamics to obtain the concrete information about how force-output capability of each muscle declines along time. \n\nSimulation on a three-dimensional, 29 degree-of-freedom human model shows that the force-output capability of ten selected muscles reduced to 60\\% - 70\\% after 10 minutes' running at the speed of 3.96 $m\/s$, with a fatigability value of 1.0 $min^{-1}$. Torso-core muscle group, which has been found to contribute less to the body's acceleration in previous research, shows no less proportion of force loss than the other two groups. The difference in fatigue level caused by the change of fatigability is evident, which emphasizes the necessity of the study and determination of fatigability among different human groups.\n\nThis work offers a virtual work design platform that helps to predict muscle fatigue and thereby to control the MSDs risks at the early stage of work design. In future works, the Motion Capture System of the \\'Ecole Centrale de Nantes will be used to acquire data for industrial tasks. A force platform will be used to validate the muscle properties during experiments. The study of the fatigability $k$ as a function of the people will also be addressed in these new experiments.\n\\section{Acknowledgments}\nThis work was supported by the National Natural Science Foundation of China under Grant numbers 71471095, by Chinese State Scholarship Fund, and by INTERWEAVE Project (Erasmus Mundus Partnership Asia-Europe)  under Grants number  IW14AC0456 and IW14AC0148.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{section-1}\n\nMotivated by its interest in Differential Geometry and General Relativity, in the last years a great deal of research has been devoted to the study of boundary value problems associated with the Minkowski-curvature equation \n\\begin{equation}\\label{eq-inizio}\n\\mathrm{div}\\,\\Biggl{(} \\dfrac{\\nabla u}{\\sqrt{1- | \\nabla u |^{2}}}\\Biggr{)} + f(x,u) = 0, \\qquad x \\in \\Omega \\subseteq \\mathbb{R}^N,\n\\end{equation}\nboth in the ODE case ($N=1$) and in the PDE case ($N \\geq 2$); see, for instance, \\cite{Az-14,BeJeTo-13,BeMa-07,BoCoNo-pp,BoGa-19ccm,CoObOmRi-13,Ma-13} and the references therein.\nQuite frequently, it is assumed that \\eqref{eq-inizio} admits the trivial solution (i.e.~$f(x,u) \\equiv 0$) and existence\/multiplicity of positive solutions of \\eqref{eq-inizio} is investigated.\n\nOn this line of research, in the recent paper \\cite{BoFe-PP} we searched for positive periodic solutions (both harmonic and subharmonic) to the parameter-dependent equation\n\\begin{equation}\\label{eq-ode}\n\\Biggl{(} \\frac{u'}{\\sqrt{1-(u')^{2}}}\\Biggr{)}' + \\lambda  a(x) g(u) = 0,\n\\end{equation}\nwhere $\\lambda > 0$, $a(x)$ is a $T$-periodic sign-changing (i.e.~indefinite) weight function and  \n$g(u)$ is a nonlinear term satisfying $g(0) = 0$.\nAmong other results, we proved therein that a two-solution theorem holds for the $T$-periodic boundary value problem associated with equation \\eqref{eq-ode}: more precisely, for weight functions $a(x)$ satisfying the mean value condition $\\int_{0}^{T} a(x)\\,\\mathrm{d}x < 0$\nand for a large class of nonlinear terms $g(u)$ which are superlinear at zero (namely, $g(u)\/u \\to 0$ for $u \\to 0^{+}$), two positive $T$-periodic solutions of \\eqref{eq-ode} exist, whenever the parameter $\\lambda$ is large enough (see \\cite[Theorem~3.1]{BoFe-PP} for the precise statement of this result). We refer the reader to the introduction of \\cite{BoFe-PP} for several comments about this solvability pattern, arising as a result of a delicate interplay between the behaviors of the nonlinear differential operator driving equation \\eqref{eq-ode} and the nonlinear term $a(x)g(u)$ when $u \\to +\\infty$.\n\nThe technical tool used in \\cite{BoFe-PP} to prove the above mentioned result is topological degree theory in Banach spaces, along a line of research started in \\cite{FeZa-15jde} and later developed and applied in several different situations (cf.~\\cite{Fe-18}), always dealing with nonlinear BVPs for semilinear equations of the type\n\\begin{equation}\\label{eq-ode2}\nu'' + q(x)g(u) = 0,\n\\end{equation}\nwhere $q(x)$ is an indefinite weight function. As well known, the first step within this approach is the formulation of the differential equation as a nonlinear functional equation in a Banach space: this can be done in a standard way when considering equation \\eqref{eq-ode2}, since the differential operator $Lu = -u''$ is a (linear) Fredholm operator of index zero. Then, depending on the invertibility\/non-invertibility of $L$ (and, hence, on the boundary conditions), either classical Leray--Schauder degree theory or Mawhin's coincidence degree theory apply. \n\nAs far as the strongly nonlinear equation \\eqref{eq-ode} is concerned, different strategies can be followed to achieve this goal. In \\cite{BoFe-PP}, we chose to write \\eqref{eq-ode} as the equivalent planar system\n\\begin{equation*}\nu' = \\frac{v}{\\sqrt{1+v^2}}, \\qquad v' = -\\lambda a(x)g(u),\n\\end{equation*}\nin order to directly apply coincidence degree theory in the product space, as proposed in the recent paper \\cite{FeZa-17tmna}.\nThis approach, which looks very natural when dealing with the periodic problem, has the drawback of not being suited for other boundary conditions. In particular, in spite of the well-known strong analogies existing in this setting between the periodic and the Neumann boundary value problem (see, for instance, \\cite{BoZa-15}), the possibility of proving the Neumann counterpart of the result in \\cite{BoFe-PP} is not discussed therein.\n\nThe aim of this brief paper is to provide a positive answer to this question. \nMore generally, we deal with the Neumann boundary value problem for the PDE version of equation \\eqref{eq-ode}, namely\n\\begin{equation}\\label{eq-main-pde}\n\\begin{cases}\n\\, \\mathrm{div}\\,\\Biggl{(} \\dfrac{\\nabla u}{\\sqrt{1- | \\nabla u |^{2}}}\\Biggr{)} + \\lambda a(|x|) g(u) = 0, & \\text{in $B$,} \\\\\n\\, \\partial_{\\nu}u=0, & \\text{on $\\partial B$,}\n\\end{cases}\n\\end{equation}\nwhere $B$ is a ball of the $N$-dimensional Euclidean space and $a(\\vert x \\vert)$ is a (sign-changing) radial weight function. \n\nIn this framework, we prove the following two-solution theorem for positive radial solutions of \\eqref{eq-main-pde}.\n\n\\begin{theorem}\\label{th-main}\nLet $N \\geq 1$ be an integer and let $B \\subseteq \\mathbb{R}^N$ be an open ball of center the origin and radius $R>0$.\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^{1}$-function such that\n\\begin{itemize}[leftmargin=30pt,labelsep=12pt,itemsep=5pt]\n\\item [$(a_{*})$]\nthere exist $m\\geq 1$ closed and pairwise disjoint intervals $I^{+}_{1},\\ldots,I^{+}_{m}$ in $\\mathopen{[}0,R\\mathclose{]}$ such that\n\\begin{align*}\n&\\qquad\\qquad a(r)\\geq0, \\; \\text{ for a.e.~$r\\in I^{+}_{i}$,} \\quad a\\not\\equiv0 \\; \\text{ on $I^{+}_{i}$,} \\quad \\text{for $i=1,\\ldots,m$,} \\\\\n&\\qquad\\qquad a(r)\\leq0, \\; \\text{ for a.e.~$r\\in \\mathopen{[}0,R\\mathclose{]}\\setminus\\bigcup_{i=1}^{m}I^{+}_{i}$;}\n\\end{align*}\n\\item [$(a_{\\#})$] $\\displaystyle \\int_{B} a(|x|) \\,\\mathrm{d}x < 0$.\n\\end{itemize}\nLet $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying \n\\begin{itemize}[leftmargin=30pt,labelsep=12pt,itemsep=5pt]\n\\item [$(g_{*})$] $g(0)=0$ and $g(u)>0$, for all $u > 0$;\n\\item [$(g_{0})$] $\\displaystyle \\lim_{u\\to 0^{+}} \\dfrac{g(u)}{u} = 0$ and $\\displaystyle \\lim_{\\substack{u\\to0^{+} \\\\ \\omega\\to1}}\\dfrac{g(\\omega u)}{g(u)}=1$;\n\\item [$(g_{\\infty})$] $\\displaystyle\\lim_{\\substack{u\\to+\\infty \\\\ \\omega\\to1}}\\dfrac{g(\\omega u)}{g(u)}=1$.\n\\end{itemize}\nThen, there exists $\\lambda^{*}>0$ such that for every $\\lambda>\\lambda^{*}$ there exist at least two positive radial solutions of problem \\eqref{eq-main-pde}.\n\\end{theorem}\n\nNotice that no Sobolev subcriticality assumptions are required for the nonlinear term: for instance, the function $g(u) = u^{p}$ enters the setting of Theorem~\\ref{th-main} for every $p > 1$. The only restrictions on the growth at infinity come from assumption\n$(g_{\\infty})$: as an example, a function behaving, for $u$ large, like $e^u$ cannot be treated. Observe that a dual condition is required also at $u = 0$; for some comments about these assumptions (of regular-oscillation type) we refer to \\cite[p.~452--453]{BoFeZa-16}.\n\n\\begin{figure}[!htb]\n\\begin{tikzpicture}[scale=1]\n\\begin{axis}[\n  tick label style={font=\\scriptsize},\n  axis y line=left, \n  axis x line=middle,\n  xtick={0.359781, 1.39176, 2.60244, 4.3119 ,5},\n  ytick={-1,0,1},\n  xticklabels={,,,,$5$},\n  yticklabels={$-1$,$0$,$1$},\n  xlabel={\\small $|x|$},\n  ylabel={\\small $a(|x|)$},\nevery axis x label\/.style={\n    at={(ticklabel* cs:1.0)},\n    anchor=west,\n},\nevery axis y label\/.style={\n    at={(ticklabel* cs:1.0)},\n    anchor=south,\n},\n  width=5.5cm,\n  height=4.5cm,\n  xmin=0,\n  xmax=6,\n  ymin=-1.5,\n  ymax=1.5]\n\\addplot graphics[xmin=0,xmax=5,ymin=-1.1,ymax=1.1] {BoFe-fig01-a.pdf};\n\\end{axis}\n\\end{tikzpicture}\n\\qquad\n\\begin{tikzpicture}[scale=1]\n\\begin{axis}[\n  tick label style={font=\\scriptsize,major tick length=3pt},\n          scale only axis,\n  enlargelimits=false,\n  xtick={0, 0.359781, 1.39176, 2.60244, 4.3119 ,5},\n  xticklabels={$0$,,,,,$5$},\n  ytick={0,8},\n  max space between ticks=50,\n                minor y tick num=3,                \n  xlabel={\\small $|x|$},\n  ylabel={\\small $u(|x|)$},\nevery axis x label\/.style={\nbelow,\nat={(1.9cm,0.1cm)},\n  yshift=-8pt\n  },\nevery axis y label\/.style={\nbelow,\nat={(0cm,1.4cm)},\n  xshift=-3pt},\n  y label style={rotate=90,anchor=south},\n  width=3.8cm,\n  height=2.8cm,  \n  xmin=0,\n  xmax=5,\n  ymin=0,\n  ymax=8]\n\\addplot graphics[xmin=0,xmax=5,ymin=0,ymax=8] {BoFe-fig01-profile.pdf};\n\\end{axis}\n\\end{tikzpicture}\n\\vspace{10pt}\\\\\n\\includegraphics[width=0.46\\linewidth]{BoFe-fig01-graph1.pdf}\n\\includegraphics[width=0.46\\linewidth]{BoFe-fig01-graph2.pdf}\n\\caption{The picture shows the graph of the weight function $a(|x|)=(\\cos(||x|-5|^{3\/2}+1)$ in $\\mathopen{[}0,R\\mathclose{]}=\\mathopen{[}0,5\\mathclose{]}$ and the graphs of the two radial solutions of problem \\eqref{eq-main-pde} where $N=2$, $g(u)=u^{2}+u^{3}$ and $\\lambda=0.1$. We notice that the large solution appears quite ``sharp-cornered'', in agreement with the analysis in \\cite[Section~4]{BoFe-PP} and \\cite[Section~3.2]{BoGa-19ccm}.}    \n\\label{fig-01}\n\\end{figure}\n\nThe proof of our main result is based again on topological degree theory, but, of course, with some differences with respect to the approach in \\cite{BoFe-PP}. In particular, after having converted the radial Neumann problem \\eqref{eq-main-pde} into the (singular) one-dimensional problem\n\\begin{equation}\\label{eq-main}\n\\begin{cases}\n\\, \\Biggl{(} r^{N-1}\\dfrac{u'}{\\sqrt{1-(u')^{2}}}\\Biggr{)}' + \\lambda r^{N-1} a(r) g(u) = 0,\\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation}\nwe write it as a fixed point equation in the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$, by exploiting a strategy comparable to the one used, for instance, in \\cite{BeJeMa-10,GaMaZa-93} (see also Remark~\\ref{rem-2.1}). At this point, Leray--Schauder degree theory can be applied: the computation of the degree on three different balls of the space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ leads to the result, similarly as in \\cite{BoFe-PP,BoFeZa-16}.\n\nThe paper is organized as follows. In Section~\\ref{section-2}, we describe the abstract setting and we present two lemmas for the computation of the degree. We give here all the details of the arguments, since it seems that no appropriate reference exists for the auxiliary results that we need. In Section~\\ref{section-3}, we give the proof of Theorem~\\ref{th-main}, by showing the applicability of the above mentioned lemmas, under the present assumptions. Compared to the case $N=1$, some technical difficulties arise when $N \\geq 2$. First, the convexity (respectively, concavity) of the solutions is not a priori prescribed by the negativity (respectively, positivity) of the weight function $a(r)$. Second, \nwhen the weight function is positive near the center of the ball, as expected, some further care is needed in the estimates in order to exclude the possible appearance of a ``sudden loss of energy'' for the solutions (cf.~\\cite{GaMaZa-97}). We stress that, due to the peculiar features of the Minkowski curvature operator, this can be successfully done even with no subcriticality assumptions on the nonlinear term.\n\n\n\\section{The abstract setting}\\label{section-2}\n\nIn this section we present an abstract formulation of the problem and we prove some preliminary results based on the theory of the topological degree.\n\n\\subsection{The fixed point reduction}\\label{section-2.1}\n\nThroughout this section, we deal with the boundary value problem \n\\begin{equation}\\label{eq-phi}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + r^{N-1} f(r,u) = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $N \\geq 1$ is an integer,\n\\begin{equation*}\n\\varphi(s) = \\frac{s}{\\sqrt{1-s^2}}, \\quad s \\in \\mathopen{]}-1,1\\mathclose{[},\n\\end{equation*}\nand $f \\colon \\mathopen{[}0,R\\mathclose{]} \\times \\mathbb{R} \\to \\mathbb{R}$ is an $L^{1}$-Carath\\'{e}odory function (i.e.~$f(\\cdot,u)$ is measurable for every $u \\in \\mathbb{R}$, $f(r,\\cdot)$ is continuous for a.e. $r \\in \\mathopen{[}0,R\\mathclose{]}$, and for every $K > 0$\nthere exists $\\eta_k \\in L^1(0,R)$ such that $|f(r,u)| \\leq \\eta_k(r)$ for a.e. $r \\in \\mathopen{[}0,R\\mathclose{]}$ and for every \n$| u| \\leq K$). Let us recall that a solution of \\eqref{eq-phi} is a continuously differentiable function $u\\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$, with $u'(0) = u'(R) = 0$ and $|u'(r)| < 1$ for every $r \\in \\mathopen{]}0,R\\mathclose{[}$, such that the map $r \\mapsto r^{N-1}\\varphi(u')$ is absolutely continuous on $\\mathopen{[}0,R\\mathclose{]}$ and the differential equation in \\eqref{eq-phi} is satisfied almost everywhere (see also Remark \\ref{regolarita}).\n\nIt is convenient for the sequel to embed problem \\eqref{eq-phi} into the two-parameter family of problems\n\\begin{equation}\\label{eq-phi-theta}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta r^{N-1} \\bigl{[} f(r,u) + \\alpha v(r) \\bigr{]} = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$, $\\alpha\\geq0$ and $v(r)$ is a fixed Lebesgue integrable function.\nNotice that \\eqref{eq-phi} is exactly \\eqref{eq-phi-theta} with $\\vartheta=1$ and $\\alpha=0$.\n\nOur aim is to write \\eqref{eq-phi-theta} as a fixed point problem in the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ of continuous functions in $\\mathopen{[}0,R\\mathclose{]}$, endowed with the usual norm $\\|\\cdot\\|_{\\infty}$. Accordingly, for $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$ and $\\alpha\\geq0$, we introduce the operator\n\\begin{equation*}\nT_{\\vartheta,\\alpha} \\colon \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}) \\to \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}),\n\\end{equation*}\ndefined as\n\\begin{equation}\\label{operator}\n\\begin{aligned}\n(T_{\\vartheta,\\alpha} u) (r) &= u(0) - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} \\bigl{[} f(\\zeta,u(\\zeta)) + \\alpha v(\\zeta) \\bigr{]} \\,\\mathrm{d}\\zeta\n \\\\ &\\quad + \\int_{0}^{r} \\varphi^{-1} \\biggl{(} -\\dfrac{\\vartheta}{\\zeta^{N-1}} \\int_{0}^{\\zeta}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi \\biggr{)} \\,\\mathrm{d}\\zeta,\n\\end{aligned}\n\\end{equation}\nfor every $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$.\nNotice that the definition of $T_{\\vartheta,\\alpha}$ is well-posed, since the function\n\\begin{equation*}\n\\mathopen{]}0,R\\mathclose{]} \\ni \\zeta \\mapsto \\dfrac{1}{\\zeta^{N-1}} \\int_{0}^{\\zeta}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi\n\\end{equation*}\ncontinuously extends up to $\\zeta=0$. For future convenience, we also observe that $T_{\\vartheta,\\alpha} u\\in\\mathcal{C}^{1}(\\mathopen{[}0,R\\mathclose{]})$ with\n\\begin{equation}\\label{diff-operator}\n(T_{\\vartheta,\\alpha} u)' (r) = \n\\begin{cases}\n\\, \\displaystyle \\varphi^{-1} \\biggl{(} -\\dfrac{\\vartheta}{r^{N-1}} \\int_{0}^{r}\\xi^{N-1} \\bigl{[} f(\\xi,u(\\xi)) + \\alpha v(\\xi) \\bigr{]}\\,\\mathrm{d}\\xi \\biggr{)}, &\\text{if $r\\in\\mathopen{]}0,R\\mathclose{]}$,} \\\\\n\\, 0, &\\text{if $r=0$,}\n\\end{cases}\n\\end{equation}\nfor every $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$.\n\nThe following result holds true.\n\n\\begin{theorem}\\label{th-operator}\nLet $f \\colon \\mathopen{[}0,R\\mathclose{]} \\times \\mathbb{R} \\to \\mathbb{R}$ be an $L^{1}$-Carath\\'{e}odory function, $v \\in L^1(0,R)$, \n$\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$, and $\\alpha\\geq0$. Then, the operator $T_{\\vartheta,\\alpha}$ defined in \\eqref{operator} is completely continuous. Moreover, $u(r)$ is a solution of \\eqref{eq-phi-theta} if and only if $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is a fixed point of $T_{\\vartheta,\\alpha}$.\n\\end{theorem}\n\n\\begin{proof}\nThe continuity of the operator $T_{\\vartheta,\\alpha}$ follows straightforwardly by observing that $T_{\\vartheta,\\alpha}$ is a composition of continuous maps. We claim that $T_{\\vartheta,\\alpha}$ sends bounded sets into  relatively compact sets. As a well-known consequence of Ascoli--Arzel\\`{a} theorem, this is true if we prove that $\\{T_{\\vartheta,\\alpha}u_{n}\\}_{n}$ and $\\{(T_{\\vartheta,\\alpha}u_{n})'\\}_{n}$ are uniformly bounded for every bounded sequence $\\{u_{n}\\}_{n}$ in $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$. This easily follows from the regularity assumptions on $f(r,u)$ and $v(r)$.\n\nWe prove the equivalence between the Neumann boundary value problem \\eqref{eq-phi-theta} and the fixed point problem $u=T_{\\vartheta,\\alpha}u$.\nLet $u(r)$ be a solution of \\eqref{eq-phi-theta}, hence $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$. By integrating the equation in \\eqref{eq-phi-theta} in $\\mathopen{[}0,r\\mathclose{]}$, recalling that $u'(0)=0$, and dividing by $r^{N-1}$, we obtain\n\\begin{equation}\\label{eq-phi-th2.1}\n\\varphi(u'(r)) = -\\dfrac{\\vartheta}{r^{N-1}} \\int_{0}^{r} \\zeta^{N-1} \\bigl{[} f(\\zeta,u(\\zeta)) + \\alpha v(\\zeta) \\bigr{]} \\,\\mathrm{d}\\zeta, \\quad \\text{for all $r\\in\\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation}\nSince $u'(R)=0$ we deduce that\n\\begin{equation}\\label{eq-averR}\n\\dfrac{1}{R^{N-1}} \\int_{0}^{R}\\zeta^{N-1} f(\\zeta,u(\\zeta))\\,\\mathrm{d}\\zeta =0.\n\\end{equation}\nBy applying $\\varphi^{-1}$ to \\eqref{eq-phi-th2.1} and integrating in $\\mathopen{[}0,r\\mathclose{]}$, we thus deduce that $u=T_{\\vartheta,\\alpha}u$.\n\nOn the other hand, let $u\\in\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ be such that $u=T_{\\vartheta,\\alpha}u$. Therefore, we have $u\\in\\mathcal{C}^{1}(\\mathopen{[}0,R\\mathclose{]})$ and $u'$ is given by the right-hand side of formula \\eqref{diff-operator}. In particular, $u'(0)=0$ and $|u'(r)|<1$ for every $r\\in\\mathopen{]}0,R\\mathclose{[}$. \nNext, by computing $u(0)=(Tu)(0)$, we obtain that \\eqref{eq-averR} holds and so $u'(R)=0$.\nBy computing $\\varphi(u')$, multiplying by $r^{N-1}$, and differentiating, we finally obtain that $u(r)$ solves \\eqref{eq-phi-theta}. The proof is thus completed.\n\\end{proof}\n\n\\begin{remark}\\label{regolarita}\nLet us notice that, for every solution $u(r)$ of \\eqref{eq-phi-theta}, from \\eqref{eq-phi-th2.1} it follows that $\\varphi(u')$ is absolutely continuous on $\\mathopen{[}0,R\\mathclose{]}$. Therefore, since $\\varphi^{-1}$ is smooth, $u'$ is absolutely continuous as well.\nIn particular, if both $f(r,u)$ and $v(r)$ are continuous functions, then $u \\in \\mathcal{C}^2(\\mathopen{[}0,R\\mathclose{]})$ (cf.~\\cite[Remark~3.3]{CoCoRi-14}).\n$\\hfill\\lhd$\n\\end{remark}\n\nAs a consequence of Theorem~\\ref{th-operator}, if $\\Omega\\subseteq \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is an open and bounded set such that\n\\begin{equation*}\nu \\neq T_{\\vartheta,\\alpha} u, \\quad \\text{for all $u\\in\\partial\\Omega$,}\n\\end{equation*}\nthe Leray--Schauder degree $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{\\vartheta,\\alpha},\\Omega,0)$ is well-defined (we refer to \\cite{De-85} for a classical reference about topological degree theory).\n\n\\begin{remark}\\label{rem-2.1}\nDealing with the (one-dimensional) periodic boundary value problem\n\\begin{equation}\\label{eq-per}\n\\begin{cases}\n\\, \\bigl{(} \\varphi(u')\\bigr{)}' +  f(r,u) = 0, \\\\\n\\, u(0) = u(R), \\; u'(0) = u'(R),\n\\end{cases}\n\\end{equation} \na similar strategy can be followed in order to obtain a functional analytic formulation. Precisely, it can be seen (cf.~\\cite{BeMa-07,MaMa-98}) that $u(r)$ is a solution of \\eqref{eq-per} if and only if \n$u \\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ is a fixed point of the operator $\\widetilde T \\colon \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}) \\to \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ defined as \n\\begin{align*}\n (\\widetilde T u) (r) &= u(0) -\\int_{0}^{R} f(\\zeta,u(\\zeta))  \\,\\mathrm{d}\\zeta\n \\\\ &\\quad + \\int_{0}^{r} \\varphi^{-1} \\biggl{(} - \\int_{0}^{\\zeta} f(\\xi,u(\\xi))\\,\\mathrm{d}\\xi + Q_{\\varphi}\\left( \\int_0^{\\odot} \nf(\\eta,u(\\eta))\\,\\mathrm{d}\\eta\\right) \\biggr{)} \\,\\mathrm{d}\\zeta,\n\\end{align*}\nwhere, for each $h \\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$, $Q_\\varphi(h) = Q_\\varphi(h(\\odot)) \\in \\mathbb{R}$ is defined as the unique solution of the equation\n\\begin{equation*}\n\\int_0^R \\varphi^{-1}\\left( -h(r) + Q_\\varphi(h)\\right)\\,\\mathrm{d}r = 0.\n\\end{equation*}\nIt is worth noticing that, compared with the Neumann case, this formulation is slightly less transparent: indeed, due to the non-locality of the\nperiodic boundary conditions, the additional term $Q_\\varphi$ appears.  \nAn alternative fixed point formulation for \\eqref{eq-per}, relying on a direct use of coincidence degree theory for the equivalent planar system\n$u' = \\varphi^{-1}(v)$, $v' = - f(r,u)$, has been recently proposed in \\cite{FeZa-17tmna}.\n$\\hfill\\lhd$\n\\end{remark}\n\n\n\\subsection{Two degree lemmas}\\label{section-2.2}\n\nTaking advantage of the abstract setting just presented, we now prove two lemmas for the computation of the degree on open balls $B(0,d)$ (with center $0$ and radius $d>0$) of the Banach space $\\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ in the framework of the Neumann problem\n\\begin{equation}\\label{eq-phi-ag}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\lambda r^{N-1} a(r)g(u) = 0, \\\\\n\\, u'(0) = u'(R) = 0,\n\\end{cases}\n\\end{equation} \nwhere $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ is an $L^1$-function, $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ is a continuous function satisfying $g(0) = 0$, and $\\lambda>0$. \nNotice that, to enter the setting of the previous section, we need for the nonlinearity appearing in the equation to be defined for every $u \\in \\mathbb{R}$; accordingly, we set\n\\begin{equation*}\nf(r,u) = \\begin{cases}\n\\, \\lambda a(r) g(u), & \\text{if $u\\geq0$,} \\\\\n\\, -u,& \\text{if $u<0$.} \n\\end{cases}\n\\end{equation*}\nObserve that, due to the assumptions on $a(r)$ and $g(u)$, the function $f(r,u)$ is $L^{1}$-Carath\\'{e}odory.\n\nRecalling the definition \\eqref{operator} of $T_{\\vartheta,\\alpha}$, we thus compute the Leray--Schauder degree of the map $\\mathrm{Id}-T_{1,0}$. Observe that, by standard maximum principle arguments (based on the monotonicity of the map $r \\mapsto r^{N-1}\\varphi(u'(r))$ when $u(r)<0$), $u = T_{1,0} u$ implies that $u(r)$ is a \\emph{non-negative} solution of \\eqref{eq-phi} and thus solves \\eqref{eq-phi-ag}.\n\nThe first lemma gives conditions for zero degree.\n\n\\begin{lemma}\\label{lem-deg0}\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^1$-function, let $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying $g(0) = 0$, and $\\lambda>0$. \nLet $d>0$ and assume that there exists a non-negative function $v \\in L^1(0,R)$, with $v\\not\\equiv0$, such that the following properties hold:\n\\begin{itemize}\n\\item[$(H_{1})$]\nIf $\\alpha \\geq 0$ and $u(r)$ is a non-negative solution of\n\\begin{equation}\\label{eq-lem-deg0}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\lambda r^{N-1} a(r) g(u) + \\alpha r^{N-1} v(r) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nthen $\\|u\\|_{\\infty}\\neq d$.\n\\item[$(H_{2})$]\nThere exists $\\alpha_{0} \\geq 0$ such that problem \\eqref{eq-lem-deg0}, with $\\alpha=\\alpha_{0}$, has no non-negative solutions $u(r)$ with $\\|u\\|_{\\infty}\\leq d$.\n\\end{itemize}\nThen, it holds that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0) = 0$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, recalling Theorem~\\ref{th-operator}, we have that $u(r)$ is a solution of \n\\begin{equation}\\label{eq-lem-deg0f}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + r^{N-1} \\bigl{[} f(r,u) + \\alpha v(r) \\bigr{]}= 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nif and only if $u = T_{1,\\alpha} u$.\nBy maximum principle arguments, since $v \\geq 0$, every solution of \\eqref{eq-lem-deg0f} is non-negative and thus solves \\eqref{eq-lem-deg0}.\nTherefore, for every $\\alpha \\geq 0$, condition $(H_{1})$ ensures that \n\\begin{equation*}\nu \\neq T_{1,\\alpha} u, \\quad \\text{for all $u\\in \\partial B(0,d)$.}\n\\end{equation*}\nWe deduce that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,\\alpha},B(0,d),0)$ is well-defined for every $\\alpha \\geq 0$. As a final step, we apply the homotopy invariance property of the degree to conclude that\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0)=\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,\\alpha_{0}},B(0,d),0)=0,\n\\end{equation*}\nwhere the last equality follows from hypothesis $(H_{2})$.\n\\end{proof}\n\nThe second lemma ensures non-zero degree. Notice that here we need to assume further conditions on $a(r)$ and $g(u)$.\n\n\\begin{lemma}\\label{lem-deg1}\nLet $a \\colon \\mathopen{[}0,R\\mathclose{]} \\to \\mathbb{R}$ be an $L^1$-function satisfying $(a_{\\#})$, let $g \\colon \\mathopen{[}0,+\\infty\\mathclose{[} \\to \\mathopen{[}0,+\\infty\\mathclose{[}$ be a continuous function satisfying $(g_{*})$, and $\\lambda>0$. Let $d>0$ and assume that the following property holds:\n\\begin{itemize}\n\\item[$(H_{3})$]\nIf $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$ and $u(r)$ is a non-negative solution of\n\\begin{equation}\\label{eq-lem-deg1}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta \\lambda r^{N-1} a(r) g(u) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nthen $\\|u\\|_{\\infty} \\neq d$.\n\\end{itemize}\nThen, it holds that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0) = 1$.\n\\end{lemma}\n\n\\begin{proof}\nFirst, recalling Theorem~\\ref{th-operator}, we have that $u(r)$ is a solution of\n\\begin{equation}\\label{eq-lem-deg11f}\n\\begin{cases}\n\\, \\bigl{(} r^{N-1}\\varphi(u')\\bigr{)}' + \\vartheta r^{N-1} f(r,u) = 0, \\\\\n\\, u'(0)=u'(R)=0,\n\\end{cases}\n\\end{equation}\nif and only if $u = T_{\\vartheta,0} u$.\nBy maximum principle arguments, every solution of \\eqref{eq-lem-deg11f} is non-negative and thus solves \\eqref{eq-lem-deg1}.\n\nHypothesis $(H_{3})$ implies that $\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{\\vartheta,0},B(0,d),0)$ is well-defined for every $\\vartheta\\in\\mathopen{]}0,1\\mathclose{]}$.\nLet us consider the case $\\vartheta=0$. The fixed point problem $u = T_{0,0} u$ reduces to\n\\begin{equation*}\nu(r) = (T_{0,0} u)(r) = u(0) - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,u(\\zeta)) \\,\\mathrm{d}\\zeta, \\quad u\\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]}),\n\\end{equation*}\nwhose solutions are constant functions.\nObserving that\n\\begin{equation*}\n(\\mathrm{Id}-T_{0,0}) u \\equiv  - \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,s) \\,\\mathrm{d}\\zeta, \\quad \\text{for all $u\\in \\mathcal{C}(\\mathopen{[}0,R\\mathclose{]})$ with $u\\equiv s\\in\\mathbb{R}$,}\n\\end{equation*}\nwe consider the function\n\\begin{equation*}\nf^{\\#}(s) = \\dfrac{1}{R^{N-1}} \\int_{0}^{R} \\zeta^{N-1} f(\\zeta,s) \\,\\mathrm{d}\\zeta =\n\\begin{cases}\n\\, -\\dfrac{R}{N} s, & \\text{if $s \\leq 0$,} \\\\\n\\, \\dfrac{\\lambda}{R^{N-1}} \\biggl{(}\\displaystyle \\int_{0}^{R} \\zeta^{N-1}a(\\zeta) \\,\\mathrm{d}\\zeta \\biggr{)} g(s), & \\text{if $s \\geq 0$.}\n\\end{cases}\n\\end{equation*}\nBy hypothesis $(a_{\\#})$ we deduce that\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) \\,\\mathrm{d}r = \\dfrac{1}{\\omega_{N}} \\int_{B}a(|x|) \\,\\mathrm{d}x < 0\n\\end{equation*}\n(where $\\omega_{N}$ is the measure of the unit sphere in $\\mathbb{R}^{N}$) \nand, consequently, we obtain that $f^{\\#}(s)s < 0$ for $s \\neq 0$.\nAs a consequence of the excision property, we can reduce the study of the degree of $\\mathrm{Id}-T_{0,0}$ in the set $B(0,d)\\cap\\mathbb{R}=\\mathopen{]}-d,d\\mathclose{[}$.\nSince $f^{\\#}(s)$ has no zeros on $\\partial(B(0,d)\\cap\\mathbb{R})=\\{\\pm d\\}$ and, more precisely, $f^{\\#}(d)<0<f^{\\#}(-d)$, we deduce that\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{0,0},B(0,d),0) = \\mathrm{deg}_{\\mathrm{B}}(-f^{\\#},\\mathopen{]}-d,d\\mathclose{[},0) = 1.\n\\end{equation*}\nTherefore, by the homotopy invariance property of the degree we infer that\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d),0) = \\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{0,0},B(0,d),0) = 1.\n\\end{equation*}\nThis concludes the proof.\n\\end{proof}\n\n\n\\section{Proof of Theorem~\\ref{th-main}}\\label{section-3}\n\nWe look for radial solutions of problem \\eqref{eq-main-pde}, meant as solutions of the one-dimensional boundary value problem \\eqref{eq-main}. To investigate this, we rely on the abstract setting presented in Section~\\ref{section-2}.\n\nWe divide the arguments of the proofs into some steps. \n\n\\subsubsection*{Step~1. Fixing the constants $\\delta^{*}$ and $\\lambda^{*}$.}\nWe first set $I^{+}_{i}=\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$ for $i=1,\\ldots,m$, and we assume that these intervals are ordered in the natural way, that is\n\\begin{equation*}\n0 \\leq \\sigma_{1} < \\tau_{1} < \\ldots < \\sigma_m < \\tau_m \\leq R.\n\\end{equation*}\nNotice that, whenever $a(r)$ vanishes on one or more non-degenerate intervals, the choice of some of the values $\\sigma_{i}$ and $\\tau_{i}$ is not unique.\nLet $\\varepsilon>0$ be such that\n\\begin{equation*}\n\\varepsilon<\\dfrac{ |I^{+}_{i} |}{4} \\quad \\text{ and } \\quad \\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon}  r^{N-1} a(r)\\,\\mathrm{d}r > 0, \\quad \\text{for every $i = 1,\\ldots,m$,}\n\\end{equation*}\nand let us define\n\\begin{equation*}\n\\delta^{*} = \\dfrac{2^{N-1}\\varepsilon^{N}}{R^{N-1}}\n\\end{equation*}\nand\n\\begin{equation*}\n\\delta_{*} = \\min_{i=1,\\ldots,m} \n\\dfrac{\\delta^{*}}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}}},\n\\end{equation*}\nif $\\sigma_{1}\\neq0$, or \n\\begin{equation*}\n\\delta_{*} = \\min \\left\\{ \n\\dfrac{\\delta^{*}-\\gamma}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{1}| \\dfrac{\\tau_{1}^{2N-2}}{\\gamma^{N-1}(2\\varepsilon)^{N}}},\n\\min_{i=2,\\ldots,m} \\dfrac{\\delta^{*}}{1+2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}}} \\right\\}.\n\\end{equation*}\nif $\\sigma_{1}=0$, where $\\gamma=\\min\\{\\delta^{*},\\tau_{1}\\}\/2$. Notice that, in both cases, $\\delta_{*}\\in\\mathopen{]}0,\\delta^{*}\\mathclose{[}$.\nMoreover, let\n\\begin{equation*}\n\\lambda^{*} = \\max_{i=1,\\ldots,m}\\dfrac{2 R^{N-1}\\varphi(1\/2)}{ \\min \\bigl{\\{} g(u) \\colon u \\in \\mathopen{[} \\delta_{*},\\delta^{*} \\mathclose{]} \\bigr{\\}} \\displaystyle{\\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon} r^{N-1} a(r)\\,\\mathrm{d}r}}.\n\\end{equation*}\nFrom now on, let $\\lambda > \\lambda^{*}$ be fixed.\n\n\\subsubsection*{Step~2. Computation of the degree in $B(0,\\delta^{*})$.}\nWe are going to apply Lemma~\\ref{lem-deg0} to the open ball $B(0,\\delta^{*})$ taking as $v(r)$ the indicator function of the set $\\bigcup_{i} I^{+}_{i}$.\n\nFirst we verify condition $(H_{1})$. We suppose by contradiction that there exist $\\alpha \\geq 0$ and a non-negative solution $u(r)$ to \\eqref{eq-lem-deg0} such that $\\|u\\|_{\\infty} = \\delta^{*}$. \n\n\\smallskip\n\\noindent\n\\textit{Claim~1.} There exists $i=\\{1,\\ldots,m\\}$ such that\n\\begin{equation}\\label{eq-cl1}\n\\max_{r\\in I^{+}_{i}} u(r) = \\delta^{*}.\n\\end{equation}\nSince $v\\equiv0$ on $\\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$, by conditions $(a_{*})$ and $(g_{*})$ we deduce that the map $r\\to r^{N-1}\\varphi(u'(r))$ is non-increasing on each interval $I^{+}_{i}$ and non-decreasing on each interval $J \\subseteq \\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$. \nWe show that\n\\begin{equation}\\label{eq-convexity}\n\\max_{r\\in J}u(r) = \\max_{r\\in\\partial J} u(r),\n\\end{equation}\nfor every interval $J \\subseteq \\mathopen{[}0,R\\mathclose{]}\\setminus \\bigcup_{i} I^{+}_{i}$.\nIndeed, let $J=\\mathopen{[}\\tau,\\sigma\\mathclose{]}$ and $\\hat{r}\\in\\mathopen{]}\\tau,\\sigma\\mathclose{[}$. If $u'(\\hat{r}) \\geq 0$, then $u'(r) \\geq 0$ for all $r\\in\\mathopen{[}\\hat{r},\\sigma\\mathclose{]}$, and so $u(\\hat{r})\\leq u(\\sigma)$. Analogously, if $u'(\\hat{r}) \\leq 0$, then $u'(r) \\leq 0$ for all $r\\in\\mathopen{[}\\tau,\\hat{r}\\mathclose{]}$, and so $u(\\hat{r})\\leq u(\\tau)$. Therefore, \\eqref{eq-convexity} holds. \n\nWe further observe that if $\\tau=0$ then $u'(\\tau)=0$ and so $\\max_{r\\in J}u(r) = u(\\sigma)$; if $\\sigma=R$ then $u'(\\sigma)=0$ and so $\\max_{r\\in J}u(t) = u(\\tau)$. As a consequence of this and \\eqref{eq-convexity}, \\eqref{eq-cl1} follows.\n\n\nFrom now on we focus on the behavior of $u(r)$ on $I^{+}_{i}=\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$. \n\n\\smallskip\n\\noindent\n\\textit{Claim~2.} It holds that\n\\begin{equation}\\label{estim-u'}\n|u'(r)| \\leq \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\, u(r), \\quad \\text{for all $r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$.}\n\\end{equation}\nIndeed, let us fix $r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$. If $u'(r)=0$ then the estimate is obvious. If $u'(r)>0$, by using the monotonicity of the map $r\\to r^{N-1}\\varphi(u'(r))$, we have that \n\\begin{equation*}\nu'(\\xi) \\geq \\varphi^{-1} \\biggl{(} \\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1} \\varphi(u'(r)) \\biggr{)} \\geq \\varphi^{-1} \\bigl{(} \\varphi(u'(r)) \\bigr{)} = u'(r), \\quad \\text{for all $\\xi\\in\\mathopen{]}\\sigma_{i},r\\mathclose{]}$.}\n\\end{equation*}\nBy integrating the above inequality in $\\mathopen{[}\\sigma_{i},r\\mathclose{]}$ we obtain \n\\begin{equation*}\nu(r)\\geq u(r)-u(\\sigma_{i})= \\int_{\\sigma_{i}}^{r} u'(\\xi)\\,\\mathrm{d}\\xi\n\\geq (r-\\sigma_{i}) u'(r)\\geq 2\\varepsilon u'(r) \\geq \\dfrac{(2\\varepsilon)^{N}}{\\tau_{i}^{N-1}} u'(r),\n\\end{equation*}\nwhere the last inequality follows from $2\\varepsilon<\\tau_{i}$. This implies~\\eqref{estim-u'}. As last case, if $u'(r)<0$, by arguing as above, we have that \n\\begin{align*}\n-u'(\\xi) &\\geq \\varphi^{-1} \\biggl{(} \\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1} \\varphi(-u'(r)) \\biggr{)} \n\\geq -\\biggl{(}\\dfrac{r}{\\xi}\\biggr{)}^{\\!N-1}  u'(r)\n\\\\\n& \\geq -\\biggl{(}\\dfrac{r}{\\tau_{i}}\\biggr{)}^{\\!N-1}  u'(r), \\quad \\text{for all $\\xi\\in\\mathopen{[}r,\\tau_{i}\\mathclose{]}$,}\n\\end{align*}\nwhere we have used the oddness of $\\varphi$ and $\\varphi^{-1}$, and the elementary inequality\n\\begin{equation*}\n\\varphi^{-1} \\bigl{(} \\vartheta \\varphi(s) \\bigr{)} \\geq \\vartheta s, \\quad \\text{for all $\\vartheta\\in\\mathopen{[}0,1\\mathclose{]}$ and $s\\in\\mathopen{[}0,1\\mathclose{[}$,}\n\\end{equation*}\ncoming from the convexity of $\\varphi$ in $\\mathopen{[}0,1\\mathclose{[}$.\nThen, by integrating in $\\mathopen{[}r,\\tau_{i}\\mathclose{]}$ we obtain \n\\begin{equation*}\nu(r)\\geq u(r)-u(\\tau_{i}) = - \\int_{r}^{\\tau_{i}} u'(\\xi)\\,\\mathrm{d}\\xi \n\\geq -(\\tau_{i}-r) \\biggl{(}\\dfrac{r}{\\tau_{i}}\\biggr{)}^{\\!N-1}u'(r) \n\\geq - \\dfrac{(2\\varepsilon)^{N}}{\\tau_{i}^{N-1}} u'(r),\n\\end{equation*}\nfinally implying \\eqref{estim-u'}.\n\nFor further convenience, we observe that from \\eqref{estim-u'} we have in particular that\n\\begin{equation}\\label{u'_1_2}\n|u'(r)| \\leq \\dfrac{1}{2}, \\quad \\text{for all $r \\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$.}\n\\end{equation}\n\n\\smallskip\n\\noindent\n\\textit{Claim~3.} It holds that\n\\begin{equation}\\label{eq-delta}\n\\min_{r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}} u(r)\\geq \\delta_{*}.\n\\end{equation}\nTo show this, let $r^{*}\\in I^{+}_{i}$ and $\\check{r}\\in\\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$ be such that\n\\begin{equation*}\nu(r^{*}) = \\max_{r\\in \\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}} u(r) = \\delta^{*}, \\quad u(\\check{r}) = \\min_{r\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}} u(r).\n\\end{equation*}\nThe case $r^{*}=\\check{r}$ is trivial. So we consider the two cases: $r^{*}<\\check{r}$ and $r^{*}>\\check{r}$. \n\nIf $\\sigma_{i} \\leq r^{*} < \\check{r}\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$, then $u'(r^{*}) \\leq 0$. \nUsing the monotonicity of $r\\mapsto r^{N-1} \\varphi(u'(r))$ in $\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$, we deduce that\n\\begin{equation}\\label{eq-monot1}\nu'(\\xi) \\geq \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\zeta}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(u'(\\zeta)) \\biggr{)}, \\quad \\text{for all $\\xi,\\zeta\\in \\mathopen{]}\\sigma_{i},\\tau_{i}\\mathclose{]}$ with $\\xi\\leq\\zeta$,}\n\\end{equation}\nand so, from \\eqref{eq-monot1} with $\\xi=r^{*}$, $u'(\\zeta) \\leq 0$ for all $\\zeta\\in \\mathopen{[}r^{*},\\check{r}\\mathclose{]}$, in particular $u'(\\check{r}) \\leq 0$. \nAssume now $i\\geq 2$, or $i =1$ and $\\sigma_{1}\\neq 0$. \nAn integration of \\eqref{eq-monot1} with $\\zeta=\\check{r}$ on $\\mathopen{[}r^{*},\\check{r}\\mathclose{]}$ leads to\n\\begin{align*}\n\\delta^{*} - u(\\check{r}) &= - \\int_{r^{*}}^{\\check{r}} u'(\\xi) \\,\\mathrm{d}\\xi \n\\leq \\int_{r^{*}}^{\\check{r}} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\check{r}}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})) \\biggr{)} \\,\\mathrm{d}\\xi\n\\\\ &\\leq \\int_{r^{*}}^{\\check{r}} \\biggl{(} \\dfrac{\\check{r}}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})) \\,\\mathrm{d}\\xi\n\\leq (\\check{r}-r^{*}) \\biggl{(} \\dfrac{\\check{r}}{r^{*}}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r}))\n\\\\ &\\leq |I^{+}_{i}| \\biggl{(} \\dfrac{\\tau_{i}}{\\sigma_{i}}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\check{r})).\n\\end{align*}\nUsing \\eqref{u'_1_2} together with the fact that\n\\begin{equation*}\n\\varphi(s) \\leq 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)}s, \\quad \\text{for all $s\\in\\biggl{[}0,\\dfrac{1}{2}\\biggr{]}$,}\n\\end{equation*}\nand estimate \\eqref{estim-u'}, we obtain\n\\begin{equation*}\n\\delta^{*} - u(\\check{r}) \\leq 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{i}| \\dfrac{\\tau_{i}^{2N-2}}{\\sigma_{i}^{N-1}(2\\varepsilon)^{N}} \\, u(\\check{r}).\n\\end{equation*}\nThen, \\eqref{eq-delta} holds.\nConsider now the case $i=1$ and $\\sigma_{1}=0$. Then, recalling that $u'(\\xi)\\leq 0$ on $\\mathopen{[}0,\\tau_{1}\\mathclose{]}$, we have $r^{*}=0$ and $\\check{r}=\\tau_{1}-2 \\varepsilon$. Now, arguing as above and using the fact that $|u'(\\xi)|<1$ on $\\mathopen{[}0,\\tau_{1}\\mathclose{]}$, we deduce\n\\begin{align*}\n\\delta^{*} - u(\\tau_{1}-2\\varepsilon) &= - \\int_{0}^{\\tau_{1}-2 \\varepsilon} u'(\\xi) \\,\\mathrm{d}\\xi \n= - \\int_{0}^{\\gamma} u'(\\xi) \\,\\mathrm{d}\\xi - \\int_{\\gamma}^{\\tau_{1}-2\\varepsilon} u'(\\xi) \\,\\mathrm{d}\\xi\n\\\\&\\leq \\gamma + \\int_{\\gamma}^{\\tau_{1}-2\\varepsilon} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\tau_{1}-2\\varepsilon}{\\xi}\\biggr{)}^{\\! N-1} \\varphi(-u'(\\tau_{1}-2\\varepsilon)) \\biggr{)} \\,\\mathrm{d}\\xi\n\\\\ &\\leq \\gamma + 2\\varphi\\biggl{(}\\dfrac{1}{2} \\biggr{)} |I^{+}_{1}| \\dfrac{\\tau_{1}^{2N-2}}{\\gamma^{N-1}(2\\varepsilon)^{N}} \\, u(\\tau_{1}-2 \\varepsilon).\n\\end{align*}\nThen, \\eqref{eq-delta} holds.\n\nOn the other hand, if $\\tau_{i}\\geq r^{*} > \\check{r}\\in \\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$, then $u'(r^{*}) \\geq 0$. Notice that this can happen only when $i\\geq 2$, or $i=1$ and $\\sigma_{1}\\neq 0$.\nUsing the monotonicity of $r\\mapsto r^{N-1} \\varphi(u'(r))$ in $\\mathopen{[}\\sigma_{i},\\tau_{i}\\mathclose{]}$, we deduce that\n\\begin{equation}\\label{eq-monot2}\n\\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\xi}{\\zeta}\\biggr{)}^{\\! N-1} \\varphi(u'(\\xi)) \\biggr{)} \\geq u'(\\zeta), \\quad \\text{for all $\\xi,\\zeta\\in \\mathopen{]}\\sigma_{i},\\tau_{i}\\mathclose{]}$ with $\\xi\\leq\\zeta$,}\n\\end{equation}\nand so, from \\eqref{eq-monot2} with $\\zeta=r^{*}$, $u'(\\xi) \\geq 0$ for all $\\xi\\in \\mathopen{[}\\check{r},r^{*}\\mathclose{]}$, in particular $u'(\\check{r}) \\geq 0$. \nAn integration of \\eqref{eq-monot2} with $\\xi=\\check{r}$ on $\\mathopen{[}\\check{r},r^{*}\\mathclose{]}$ and an application of \\eqref{estim-u'} lead to\n\\begin{align*}\n\\delta^{*} - u(\\check{r}) &= \\int_{\\check{r}}^{r^{*}} u'(\\zeta) \\,\\mathrm{d}\\zeta\n\\leq \\int_{\\check{r}}^{r^{*}} \\varphi^{-1} \\biggl{(} \\biggl{(} \\dfrac{\\check{r}}{\\zeta}\\biggr{)}^{\\! N-1} \\varphi(u'(\\check{r})) \\biggr{)} \\,\\mathrm{d}\\zeta\n\\\\ &\\leq \\int_{\\check{r}}^{r^{*}}\\varphi^{-1} \\bigl{(} \\varphi(u'(\\check{r})) \\bigr{)} \\,\\mathrm{d}\\zeta\n\\leq |I^{+}_{i}| \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\, u(\\check{r})\n\\leq |I^{+}_{i}| \\dfrac{\\tau_{i}^{N-1}}{(2\\varepsilon)^{N}} \\dfrac{\\tau_{i}^{N-1}}{\\sigma_{i}^{N-1}} \\, u(\\check{r}).\n\\end{align*}\nThen, we have \\eqref{eq-delta}.\n\n\\smallskip\n\nWe are now ready to verify condition $(H_{1})$ of Lemma~\\ref{lem-deg0}. We integrate the equation in \\eqref{eq-lem-deg0} on $\\mathopen{[}\\sigma_{i}+2\\varepsilon,\\tau_{i}-2\\varepsilon\\mathclose{]}$ and, recalling \\eqref{u'_1_2}, \\eqref{eq-delta} and that $\\varphi$ is odd, we obtain\n\\begin{equation*}\n\\lambda \\min \\bigl{\\{} g(u) \\colon u \\in \\mathopen{[}\\delta_{*},\\delta^{*} \\mathclose{]}\\bigr{\\}} \\int_{\\sigma_{i}+2\\varepsilon}^{\\tau_{i}-2\\varepsilon} r^{N-1}a(r)\\,\\mathrm{d}r \\leq 2 R^{N-1}\\varphi \\biggl{(} \\dfrac{1}{2}\\biggr{)}, \n\\end{equation*}\na contradiction with respect to $\\lambda > \\lambda^{*}$.\n\nAs for assumption $(H_{2})$, we integrate the equation in \\eqref{eq-lem-deg0} in $\\mathopen{[}0,R\\mathclose{]}$ and, passing to the absolute value, we deduce\n\\begin{equation*}\n\\alpha \\| r^{N-1}v \\|_{L^{1}} \\leq \\lambda R^{N-1} \\|a\\|_{L^{1}} \\max_{u \\in \\mathopen{[}0,\\delta^{*}\\mathclose{]}} g(u).\n\\end{equation*}\nA contradiction follows for $\\alpha$ sufficiently large. From Lemma~\\ref{lem-deg0} we thus obtain\n\\begin{equation}\\label{deg-B0rho*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,\\delta^{*})) = 0.\n\\end{equation}\n\n\\subsubsection*{Step~3. Fixing the constant $d^{*}$ and computation of the degree in $B(0,d^{*})$.}\nFirst, we claim that there exists $d^{*}\\in\\mathopen{]}0,\\delta^{*}\\mathclose{[}$ such that, for every $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$, every non-negative  solution $u(r)$ of \\eqref{eq-lem-deg1} with $\\|u\\|_{\\infty} \\leq d^{*}$ is such that $u\\equiv0$.\n\nWe assume, by contradiction, that there exists a sequence $\\{u_{n}\\}_{n}$ of non-negative solutions of \\eqref{eq-lem-deg1} for $\\vartheta=\\vartheta_{n}$ satisfying $0<\\|u_{n}\\|_{\\infty}=d_{n}\\to0$.\nWe define\n\\begin{equation*}\nv_{n}(r) = \\dfrac{u_{n}(r)}{d_{n}}, \\quad r\\in\\mathopen{[}0,R\\mathclose{]},\n\\end{equation*}\nand observe that $v_{n}(r)$ is a non-negative solution of the Neumann problem associated with\n\\begin{equation}\\label{eq-v_n}\n\\Biggl{(} \\dfrac{r^{N-1} v_{n}'}{\\sqrt{1-(u_{n}')^{2}}}\\Biggr{)}' + \\vartheta_{n} \\lambda r^{N-1} a(r) q(u_{n}(r)) v_{n} = 0,\n\\end{equation}\nwhere we set $q(u) = g(u)\/u$ for $u > 0$ and $q(0) = 0$. \nIntegrating equation \\eqref{eq-v_n} between $0$ and $r$ and dividing by $r^{N-1}$ we obtain that\n\\begin{equation*}\n\\dfrac{v_{n}'(r)}{\\sqrt{1-u_{n}'(r)^{2}}} = -\\dfrac{1}{r^{N-1}} \\vartheta_{n} \\lambda \\int_{0}^{r} \\xi^{N-1}a(\\xi)q(u_{n}(\\xi))v_{n}(\\xi) \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation*}\nPassing to the absolute value we have\n\\begin{equation*}\n|v_{n}'(r)| \\leq \\dfrac{|v_{n}'(r)|}{\\sqrt{1-u_{n}'(r)^{2}}}\n\\leq \\lambda \\int_{0}^{R} |a(\\xi)| |q(u_{n}(\\xi))||v_{n}(\\xi)| \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{]}0,R\\mathclose{]}$.}\n\\end{equation*}\nTherefore, using the first condition in $(g_{0})$ and the fact that $\\| v_{n} \\|_{\\infty} \\leq 1$, we obtain that $v_{n}' \\to 0$ uniformly.\nAs a consequence, $v_{n} \\to 1$ uniformly in $\\mathopen{[}0,R\\mathclose{]}$, since\n\\begin{equation*}\n|v_{n}(r) - 1 | = |v_{n}(r) - v_{n}(\\hat{\\eta}_{n}) | \\leq \\int_{0}^{R} | v_{n}'(\\xi) | \\,\\mathrm{d}\\xi, \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nwhere $\\hat{\\eta}_{n}\\in \\mathopen{[}0,R\\mathclose{]}$ is such that $u_{n}(\\hat{\\eta}_{n}) = \\|u_{n}\\|_{\\infty} = d_{n}$.\nAn integration of equation \\eqref{eq-v_n} in $\\mathopen{[}0,R\\mathclose{]}$ gives\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) g(u_{n}(r))\\,\\mathrm{d}r = 0\n\\end{equation*}\nand hence\n\\begin{equation*}\n\\int_{0}^{R} r^{N-1} a(r) g(d_{n})\\,\\mathrm{d}r + \\int_{0}^{R} r^{N-1} a(r)\\bigl{[}g(d_{n} v_{n}(r)) - g(d_{n})\\bigr{]}\\,\\mathrm{d}r=0.\n\\end{equation*}\nDividing by $g(d_{n}) > 0$, we have\n\\begin{equation*}\n0 < - \\int_{0}^{R} r^{N-1} a(r)\\,\\mathrm{d}r \\leq R^{N-1}\\|a\\|_{L^{1}} \\sup_{r\\in \\mathopen{[}0,R\\mathclose{]}}\\biggl{|}\\dfrac{g(d_{n} v_{n}(r))}{g(d_{n})} - 1\\biggr{|}.\n\\end{equation*}\nUsing the second condition in $(g_{0})$ and recalling that $v_{n} \\to 1$ uniformly, we find a contradiction. The claim is thus proved and we can fix $d^{*}\\in\\mathopen{[}0,\\delta^{*}\\mathclose{[}$.\n\nFinally, condition $(H_{3})$ of Lemma~\\ref{lem-deg1} is trivially satisfied for $d=d^{*}$, and therefore\n\\begin{equation}\\label{deg-B0d*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,d^{*})) = 1.\n\\end{equation}\n\n\\subsubsection*{Step~4. Fixing the constant $D^{*}$ and computation of the degree in $B(0,D^{*})$.}\nFirst, we claim that there exists $D^{*}>\\delta^{*}$ such that, for every $\\vartheta\\in \\mathopen{]}0,1\\mathclose{]}$, every non-negative solution $u(r)$ of \\eqref{eq-lem-deg1} satisfies $\\|u\\|_{\\infty} < D^{*}$.\n\nWe assume, by contradiction, that there exists a sequence $\\{u_{n}\\}_{n}$ of non-negative solutions of \\eqref{eq-lem-deg1} for $\\vartheta=\\vartheta_{n}$ and $\\lambda=\\lambda_{n}$ satisfying $\\|u_{n}\\|_{\\infty}=D_{n}\\to+\\infty$. We proceed similarly to the previous step. We define\n\\begin{equation*}\nv_{n}(r) = \\dfrac{u_{n}(r)}{D_{n}}, \\quad r\\in\\mathopen{[}0,R\\mathclose{]},\n\\end{equation*}\nwhich solves the Neumann problem associated with equation \\eqref{eq-v_n}. Since $\\|u_{n}'\\|_{\\infty} \\leq 1$, we easily find $\\|v_{n}'\\|_{\\infty} \\to 0$ and, consequently, $v_{n} \\to 1$ uniformly in $\\mathopen{[}0,R\\mathclose{]}$ (proceeding as shown in Step~3).\nIntegrating equation \\eqref{eq-v_n} and dividing by $g(D_{n}) > 0$, we thus obtain\n\\begin{equation*}\n0 < - \\int_{0}^{R} r^{N-1}a(r)\\,\\mathrm{d}r \\leq R^{N-1}\\|a\\|_{L^{1}} \\sup_{r\\in \\mathopen{[}0,R\\mathclose{]}}\\biggl{|}\\dfrac{g(D_{n} v_{n}(r))}{g(D_{n})} - 1\\biggr{|}.\n\\end{equation*}\nUsing $(g_{\\infty})$, a contradiction easily follows. The claim is thus proved and we can fix $D^{*}>\\delta^{*}$.\n\nFinally, condition $(H_{3})$ of Lemma~\\ref{lem-deg1} is trivially satisfied for $d=D^{*}$, and therefore\n\\begin{equation}\\label{deg-B0D*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,D^{*})) = 1.\n\\end{equation}\n\n\\subsubsection*{Step~5. Concluding the proof.}\nStarting from the formulas \\eqref{deg-B0rho*}, \\eqref{deg-B0d*}, \\eqref{deg-B0D*} proved in the last three steps, we apply the additivity property of the coincidence degree to obtain\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,\\delta^{*}) \\setminus \\overline{B(0,d^{*})}) = -1\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathrm{deg}_{\\mathrm{LS}}(\\mathrm{Id}-T_{1,0},B(0,D^{*}) \\setminus \\overline{B(0,\\delta^{*})}) = 1.\n\\end{equation*}\nAs a consequence of the existence property of the degree, there exist two solutions $u_{s}\\in B(0,\\delta^{*}) \\setminus \\overline{B(0,d^{*})}$ and $u_{\\ell}\\in B(0,D^{*}) \\setminus \\overline{B(0,\\delta^{*})}$ of problem \\eqref{eq-phi}. Then, $u_{s}(r)$ and $u_{\\ell}(r)$ satisfy \n\\begin{equation*}\nd^{*} < \\| u_{s} \\|_{\\infty} < \\delta^{*} < \\| u_{\\ell} \\|_{\\infty} < D^{*}.\n\\end{equation*}\nBy maximum principle arguments, both these solutions are non-negative and hence they solve \\eqref{eq-main}.\nIt thus remains to show that they are positive, that is, $u(r) > 0$ for every $r \\in [0,R]$, for both $u = u_s$ and $u = u_{\\ell}$.\n\nBy contradiction, assume that $u(r_0) = 0$ for some $r_0 \\in [0,R]$. Then $u'(r_0) = 0$, coming from the boundary conditions if $r_0 = 0$ and from the fact that $u(r)$ is non-negative if $r_0 \\in (0,R]$.\n\nIf $r_0 = 0$, integrating the equation we find\n\\begin{equation*}\nu(r) = - \\int_{0}^r \\varphi^{-1}\\biggl{(}\\dfrac{1}{\\zeta^{N-1}} \\int_{0}^{\\zeta} \\xi^{N-1}a(\\xi)g(u(\\xi))\\,\\mathrm{d}\\xi \\biggr{)} \\,\\mathrm{d}\\zeta, \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nimplying, since $|\\varphi^{-1}(s)|\\leq |s|$ for all $s \\in \\mathbb{R}$, that\n\\begin{equation*}\n|u(r)| \\leq \\int_{0}^r \\int_{0}^{\\zeta} |a(\\xi)| |g(u(\\xi))|\\,\\mathrm{d}\\xi \\,\\mathrm{d}\\zeta \n\\leq R \\int_{0}^r |a(\\xi)| |g(u(\\xi))|\\,\\mathrm{d}\\xi, \n\\end{equation*}\nfor all $r \\in \\mathopen{[}0,R\\mathclose{]}$.\nRecalling that $g(u)\/u \\to 0$ for $u \\to 0^{+}$, we finally obtain \n\\begin{equation*}\n|u(r)| \\leq M R \\int_{0}^r |a(\\xi)| |u(\\xi)|\\,\\mathrm{d}\\xi , \\quad \\text{for all $r \\in \\mathopen{[}0,R\\mathclose{]}$,}\n\\end{equation*}\nwhere $M > 0$ is a suitable constant. By Gronwall's lemma, $u(r) = 0$ for every $r \\in \\mathopen{[}0,R\\mathclose{]}$, a contradiction.\n\nIn the case $r_0 \\in \\mathopen{]}0,R\\mathclose{]}$, the contradiction is reached again using the assumption $g(u)\/u \\to 0$ for $u \\to 0^{+}$, which as well known implies\n (together with the smoothness of $\\varphi^{-1}$)\nthat the only solution of the planar system\n\\begin{equation*}\nu' = \\varphi^{-1}\\biggl{(} \\dfrac{v}{r^{N-1}} \\biggr{)}, \\qquad v' = -r^{N-1} a(r) g(u),\n\\end{equation*}\nsatisfying the initial condition $(u(r_0),v(r_0)) = (0,0)$ is the trivial one.\n\\qed\n\n\\bibliographystyle{elsart-num-sort}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStable broadband sources are essential for a variety of applications, from spectroscopy \\cite{Liu2013} to imaging \\cite{Kano2005} and nonlinear optical parametric amplifiers \\cite{Reed1995}. Supercontinuum (also known as white light) generation offers a promising tool for these multidisciplinary uses. Sapphire is regarded as the crystal of choice for visible supercontinuum generation \\cite{Yakovlev1994} and is the focus of our particular study. Supercontinua are generated in sapphire through the process of single filamentation. This general process encompasses a wide variety of linear and nonlinear effects including self-steepening, self-phase modulation, dispersion, four-wave mixing, Raman excitation, second and third harmonic generation, and plasma generation, absorption, and refraction \\cite{Alfano2006}. Since in the most general case it is diffcult to separate the exact contribution of each of these, individual optimization of each effect is neither particularly feasible nor extremely desirable. However, each of these effects can be highly influenced by the spatial distribution of the beam. One of the most basic effects of spatial shaping is to focus the beam tighter, thereby generating more plasma and influencing the resultant spectrum. Although studies with Bessel \\cite{Majus2013} and Laguerre-Gauss beams \\cite{Sztul2006a} have failed to find significant spectral deviations or improvements in effciency from the Gaussian regime, systematic spatial optimization has not been performed. \n\nPrevious work that has shown promise in this direction includes a study performed with microlenses generated via spatial light modulators \\cite{Borrego-Varillas2014}. Moreover, there have been several successful studies of spectral pump-beam optimization in filamentation \\cite{Thompson2017, Ackermann2006} and other nonlinear effects, such as the control of second harmonic generation \\cite{Thompson2016} and the enhancement of spontaneous Raman signals through a turbid medium \\cite{Thompson2016a}. We extend these results and methods to the theoretically challenging regime of supercontinuum generation by using the wavefront shaping algorithm to influence the supercontinuum spectrum.\n\n\n\nThere are several aspects of interest here, including reshaping the physics of filamentation and drawing attention to the potentials of non-Gaussian beams. More commercial applications are also possible. We envision that this work may lead to universally stronger seeds for spectroscopic applications that depend on nonlinear effects for a large signal-to-noise. Our preliminary results are a proof-of-concept that the idea of spatial optimization has merit and may be further expanded with additional work.\n\n\n\n\\section{Experimental Setup}\n\nThe experimental setup is depicted in Fig. \\ref{fig:setup}. We used a Ti:Sapphire regenerative amplifier (Coherent, Legend) to produce infrared ($\\lambda = 802$ nm) $35$ fs pulses with a $1$ kHz repetition rate and $4$ W average power that we attenuated to produce supercontinuum. We then investigated two regimes of supercontinuum generation: chirped and unchirped. In the first case, we added positive chirp by changing the grating distance within the compressor unit of the amplifier. This produced pulses of 900 fs FWHM duration, measured using a commercial autocorrelator (Pulse Check; A.P.E.).\n\n\n\\begin{figure\n\t\\centering\n\t\n\n\n\t\\includegraphics[width=0.5\\textwidth]{setup}\n\t\\caption{Setup for generating supercontinua (SC) from shaped pulses; a photograph of the generated SC is shown in the inset. The angle of the spatial light modulator (SLM) is greatly exaggerated. OAP stands for off-axis parabola and was used to collimate the SC spectrum after the crystal.}\n\t\\label{fig:setup}\n\t\n\n\\end{figure}  \n\nDifferent powers\/pulse were needed to produce supercontinua in each regime: we used 5-6 $\\mu$J in the chirped and 1 $\\mu$J in the unchirped cases. In both regimes, we used a spatial phase-only light modulator (Hamamatsu; x10468; abbreviated SLM) to shape the originally Gaussian beam. The beam has a 15 mm diameter prior to being shaped by the SLM; this is the greatest we could expand the beam without clipping on the SLM screen. We then split the beam and focused part of it with a 5 cm focal length lens to generate a supercontinuum in a 3 mm thick single crystal c-cut sapphire plate (Newlight Photonics; SAP0030-C; Toronto, ON). The resultant diverging supercontinuum was collimated by an off-axis paraboloid to a 1-inch diameter. The pump beam was then filtered out via a 750 nm short pass filter (Semrock; FF01-745\/SP-25; Rochester, NY). The filtered beam was subsequently refocused with a 20 cm focal length lens into a multi-mode 600 $\\mu$m core fiber. Since the supercontinuum light should roughly focus to 5 $\\mu$m, the fiber core is of sufficient size to collect all the light and not be affected by any spatial phase changes the SLM adds. We checked this assumption by translating the fiber in x-y dimensions in the focal plane; this operation revealed no unmeasured light. Hence, we are confident that the VCSA algorithm is not optimizing the light-collection system.\n\nThe other part of the beam was sent through a long (2 m focal length) lens to be very loosely focused onto a CCD array (Spiricon; SP620U) and recorded by the computer. These images did not take part in the spatial optimization at all -- they are there to help visualize the effect of different phase maps on the beam's spatial profile in the focus. \n\n\\section{Optimization Details}\n\nFor all optimization regimes, we used a variation of the continuous sequential algorithm (abbreviated VCSA) \\cite{Vellekoop2008, Thompson2015}. The VCSA groups pixels on the SLM together and cycles through $2\\pi$ phase values in 8 steps. The algorithm then compares spectrometer output in a particular spectral range before and after adding different phase values. If the average of the spectrometer reading in that spectral range improves, then the algorithm keeps the phase value. This cycle is repeated three times and results averaged to minimize influence from shot-to-shot fluctuations and other noise. The algorithm then moves on to another pixel group and repeats the process. Each iteration takes 12 seconds, with the spectrometer integration time forming the largest limit on speed. \n\nFor all results given in this paper, we employed the ``spiral out\" method of this algorithm, which starts with large pixel groups (of 264 $x$ 300 pixels) in the center and spirals out to the edges, as in Figure \\ref{fig:algorithm}. It then starts a new stage at the center with smaller pixel groups (of 132 $x$ 150 pixels) and spirals out until it is forced to repeat itself with even smaller groups of pixels (of 72 $x$ 60). The final run consists of groups of 24 $x$ 24 pixels. In total, we let the algorithm optimize for roughly half an hour for the results given in this paper. We do not consider time to be a major limit in our experiment, as there are no discernible differences between spectra taken at the beginning of the day and those taken at the end. Further, for spectroscopic applications, it will not be necessary to quickly reoptimize the masks so long as the user takes care to produce a bank of working masks that they may easily switch between.\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=0.5\\textwidth]{algorithm_3}\n\t\\caption{The smaller the block the longer the program takes to finish; the process can be stopped at any time if the user is satisfied with the results. Our SLM is comprised of 792 $x$ 600 pixels total.}\n\t\\label{fig:algorithm}\n\t\n\t\n\t\n\\end{figure} \n\n\\section{Preliminary Results}\n\nUsing these methods, we were able to obtain a general 10\\% broadening of the spectral width of the supercontinuum generation for highly positively chirped pulses (900 fs), as shown in Fig. \\ref{fig:chirpedresults}. However, this regime is tricky to work with as the damage threshold for these focusing conditions in sapphire is near the critical power of self-focusing. This makes further optimization difficult but not impossible, potentially under different focusing conditions.\n\nFor 35 fs unchirped pulses, we discovered that it is possible to shift the supercontinuum spectral cutoff peak between 450 and 650 nm, as our preliminary results indicate in Figure  \\ref{fig:results}. The region from 450-500 nm is completely absent in the supercontinuum spectrum generated without any phase mask applied and so represents a significant broadening ($>\\approx 20\\%$). In this case, the effect of the added phase mask on the supercontinuum spectrum is easily noticeable by eye and hence can not be due to any limitations in our light-collection system.\n\n Further, the phase masks shown in Figure \\ref{fig:results} generate the same spectrum from day-to-day without any special additional environmental control, making our experiment repeatable in a variety of conditions. However, the spatial profile of the shaped beam is very sensitive to the alignment of the pump beam on the SLM screen. This is because any displacement in this region will result in different parts of the beam obtaining different phase values, and hence not reproducing the original phase-optimized beam. In this case, each phase mask will need to be re-optimized to obtain a tailored spectrum.\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=0.5\\textwidth]{chirped_results2}\n\t\\caption{SC spectrum before (blue line) and after (red line, dash-dot) spatially optimizing the pump pulse. The range of optimization was 500--550 nm.}\n\t\\label{fig:chirpedresults}\n\t\n\t\n\t\n\t\n\t\n\\end{figure} \n\n\n\n\\begin{figure\n\t\\centering\n\t\n\t\n\t\n\n\t\\includegraphics[width=1\\textwidth]{dotdash_results2}\n\t\\caption{(a) Measured supercontinuum spectrum for different optimization regimes -- the supercontinuum cutoff peak is spectrally shifted as the spatial shape changes. Each entry in the legend corresponds to the optimization range of that particular run of the algorithm (i.e. for the second entry, the algorithm attempts to optimize the average spectrometer-measured counts in the range of 450 -- 500 nm). All spectrums were taken with the phase masks and profiles in (b). (b) SLM phase masks (top), beam profiles in the focus magnified approximately 40 times and with the left three profiles integrated 5x longer than the right-most profile (middle), and true-color photographs of the resultant supercontinuum (bottom; taken with a Sony DSLR camera) for different optimization regimes.}\n\t\\label{fig:results}\n\t\n\t\n\t\n\\end{figure} \n\n\\section{Conclusions}\n\nOur preliminary results indicate that spatial beam shaping has a substantial untapped potential in optimizing supercontinuum generation by enhancing a particular spectral region. We envision that this technique can dramatically improve the ability to tailor the supercontinuum spectrum for any particular application. For example, we can provide significantly stronger seed pulses for optical parametric amplifiers and substantially enhance signals in broadband coherent anti-Stokes Raman spectroscopy\/microscopy. An SLM provides a much more flexible platform, as compared to a micro-structured fiber, to tailor the spectral properties of the supercontinuum\n\\cite{Zheltikov2006}. The user will simply load the SLM with the phase mask for the particular spectral range they desire. By pre-generating optimal phase masks, the frequency can be tuned at the 10 Hz refresh rate of the SLM.\n\nIn the future, we envision that this will lead to higher available powers for various nonlinear spectroscopy experiments and hence a greater signal-to-noise, paving the way for future precision measurements. Further work will include explorations of the theoretical foundations of spatial effects in high-order nonlinear optical interactions, which we initiated in \\cite{Thompson2016}. We also plan thorough investigations of algorithmic shaping in the IR.\n\nThis research was partially supported by the NSF (Grant \\# PHY-1307153, DBI-1532188, and ECCS-1509268), the US Department of Defense (award FA9550-15-1-0517), the Welch Foundation (Awards No. A-1547 and No. A-1261), the Cancer Prevention Research Institute of Texas (grant RP160834), and the Office of Navel Research (Award No. N00014-16-1-3054).\n\n\\bibliographystyle{jmo_test2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe $\\ast$-involution, sometimes referred to as Kashiwara's involution, is an involution on the crystal $B(\\infty)$ that is induced from a subtle involutive antiautomorphism of $U_q(\\mathfrak{g})$.  \nThe importance of $\\ast$ in the theory of crystal bases and their applications cannot be understated.  Here are just a few of its applications.\n\\begin{enumerate}\n\\item Saito~\\cite{Sai94} used the involution during the proof that Lusztig's PBW basis has a crystal structure isomorphic to $B(\\infty)$, provided that $\\mathfrak{g}$ is a finite-dimensional semisimple Lie algebra.\n\\item Kamnitzer and Tingley~\\cite{KT09} generalize the definition of the crystal commutor of Henriques and Kamnitzer \\cite{HK06} in terms of the $\\ast$-involution. This leads to a proof by Savage~\\cite{Sav09} that the category of crystals forms a coboundary category over any symmetrizable Kac-Moody algebra.\n\\item In affine type $A$, Jacon and Lecouvey~\\cite{JL09} prove that $\\ast$ coincides with the Zelevinsky involution~\\cite{MW86,Zel80} on the set of simple modules for the affine Hecke algebra.\n\\end{enumerate}\nSeveral combinatorial realizations of the $\\ast$-involution are known in the literature.  For example, Lusztig~\\cite{Lusztig93} gave a description of the behavior of $\\ast$ on Lusztig's PBW basis in the finite types, Kamnitzer~\\cite{Kamnitzer07} showed that $\\ast$ acts on an MV polytope by negation, Kashiwara and Saito \\cite{KS97} gave a description of $\\ast$ in terms of quiver varieties~\\cite{KS97}, and Jacon and Lecouvey~\\cite{JL09} give a description of the involution in terms of the multisegment model.  Such model-specific calculations of the $*$-crystal operators are important as, {\\it a priori}, the algorithm for computing the action of these operators is not efficient \\cite[Thm. 2.2.1]{K93} (see also \\cite[Prop. 8.1]{K95}).\n\n\nIn this paper, the authors continue their development of the rigged configuration model of $B(\\infty)$~\\cite{SalS15,SalS15II}.  Rigged configurations are sequences of partitions, one for every node of the underlying Dynkin diagram, where each part is paired with an integer, satisfying certain conditions.  These objects arose as an important tool in mathematical physics from the studies of the Bethe Ansatz by Kerov, Kirillov, and Reshetikhin~\\cite{KKR86,KR86}, and they have been shown to correspond to the action and angle variables of box-ball systems~\\cite{KOSTY06}. Additionally, rigged configurations have been used extensively in the theory of Kirillov-Reshetikhin crystals \\cite{HKOTT02,HKOTY99,OSS13,OSS03,OSS03II,Sakamoto14,S05,S06,SchillingS15,SW10,Scrimshaw15}. During the course of this study, a crystal structure was given to rigged configurations~\\cite{S06, SchillingS15}.\n\nOur description of $\\ast$ is as nice as one could hope: in contrast to the definition of $e_a$ and $f_a$ on rigged configurations, one interchanges ``label'' and ``colabel'' to obtain a definition of $e_a^*$ and $f_a^*$ (see Definition \\ref{def:RC_star_crystal_ops}).  In turn, applying $\\ast$ to a rigged configuration replaces all labels with its corresponding colabels and leaves the partitions fixed (see Corollary~\\ref{cor:RC_star_involution}).\n\nThe method of proof applied here is to use a classification theorem of $B(\\infty)$ asserted by Tingley and Webster~\\cite{TW} by translating the $\\ast$-involution directly into the classification theorem of Kashiwara and Saito~\\cite{KS97} without the use of Kashiwara's embedding.  This classification theorem requires several assertions to be satisfied, and proving these assertions hold in $\\RC(\\infty)$ with our new $\\ast$-crystal operators consumes most of Section \\ref{sec:*crystal}.\n\n\nThe (conjectural) bijection $\\Phi$ between $U_q'(\\mathfrak{g})$-rigged configurations and tensor products of Kirillov-Reshetikhin crystals~\\cite{OSS13, OS12, OSS03, OSS03II, OSS03III, S05, SchillingS15, SS2006, Scrimshaw15} is given roughly as follows. It removes the largest row with a colabel of 0, which is the minimal colabel, for each $e_a$ from $b$ to the highest weight element in $B(\\Lambda_1)$, where $b$ is the leftmost factor in the tensor product.  \nLet $\\theta$ be the involution on $U_q'(\\mathfrak{g})$-rigged configurations which interchanges labels with colabels on classically highest weight $U_q'(\\mathfrak{g})$-rigged configurations and let $\\widetilde{\\ast}^L$ denote the involution which is the composition of Lusztig's involution and the map sending the result to the classically highest weight element~\\cite{S05, SS2006} (where it is also denoted by $\\ast$). It is known that $\\Phi \\circ \\theta = \\widetilde{\\ast}^L \\circ \\Phi$ on classically highest weight elements. In particular, the latter map reverses the order of the tensor product. Thus, given the description of the crystal commutor, our work suggests there is a strong link between the $\\ast$-involution and the bijection $\\Phi$. We hope this could lead to a more direct description of the bijection $\\Phi$, its related properties, and a (combinatorial) proof of the $X = M$ conjecture of~\\cite{HKOTT02, HKOTY99}.\n\nAnother model for $B(\\infty)$ uses marginally large tableaux, as developed by Hong and Lee \\cite{HL08, HL12}.  It is known that the bijection $\\Phi$ mentioned above can be extended to a $U_q(\\mathfrak{g})$-crystal isomorphism between rigged configurations and marginally large tableaux~\\cite{SalS15III} when $\\mathfrak{g}$ is of finite classical type or type $G_2$.  An ambitious hope of this paper is that it may lead to a description of the $\\ast$-crystal structure on marginally large tableaux.  (In finite type $A$, this result is in~\\cite{CT15}.) However, this appears to be a hard problem as the bijection $\\Phi$ is highly recursive and depends on conditions on colabels, many of which can change under applying the $\\ast$-crystal operators.\n\nThere is also a model for $B(\\infty)$ using Littelmann paths constructed by Li and Zhang~\\cite{LZ11}.  From~\\cite{PS15}, natural virtualization maps arise to the embeddings on the underlying geometric information. The virtualization map on rigged configurations is also quite natural, giving evidence that rigged configurations encode more geometry than their combinatorial origins and description suggests. This is also evidence that there exists a straightforward and natural explicit combinatorial bijection between rigged configurations and the Littelmann path model. Thus this work could potentially lead to a description of the $\\ast$-crystal on the Littelmann path model.\n\nIn a similar vein, the virtualization map is known to act naturally on MV polytopes~\\cite{JS15,NS08}, also reflecting the geometric information of the root systems via the Weyl group. This is evidence that there should be a natural explicit combinatorial bijection between MV polytopes and rigged configurations (and the Littelmann path model). Moreover, considering the $\\ast$-involution, which acts by negation on MV polytopes~\\cite{Kamnitzer07, Kamnitzer10}, this work gives further evidence that such a bijection should exist. Furthermore, this bijection would suggest a natural generalization beyond finite type, which the authors expect to recover the KLR polytopes of~\\cite{TW}.\n\nThis paper is organized as follows.\nIn Section~\\ref{sec:background}, we give the necessary background on crystals and the $\\ast$-involution.\nIn Section~\\ref{sec:RC}, we give background information on the rigged configuration model for $B(\\infty)$.\nIn Section~\\ref{sec:*crystal}, we give the proof of our main theorem and some consequences.\nIn Section~\\ref{sec:hw_crystals}, we give a description of highest weight crystals using the $\\ast$-crystal structure and describe the natural projection from $B(\\infty)$ in terms of rigged configurations.\n\n\n\n\n\n\n\n\n\\section{Crystals and the $\\ast$-involution}\n\\label{sec:background}\n\nLet $\\mathfrak{g}$ be a symmetrizable Kac-Moody algebra with quantized universal enveloping algebra $U_q(\\mathfrak{g})$ over $\\mathbf{Q}(q)$, index set $I$, generalized Cartan matrix $A = (A_{ij})_{i,j\\in I}$, weight lattice $P$, root lattice $Q$, fundamental weights $\\{\\Lambda_i \\mid i \\in I\\}$, simple roots $\\{\\alpha_i \\mid i\\in I\\}$, and simple coroots $\\{h_i \\mid i\\in I\\}$.  There is a canonical pairing $\\langle\\ ,\\ \\rangle\\colon P^\\vee \\times P \\longrightarrow \\mathbf{Z}$ defined by $\\langle h_i, \\alpha_j \\rangle = A_{ij}$, where $P^{\\vee}$ is the dual weight lattice.\n\nAn \\defn{abstract $U_q(\\mathfrak{g})$-crystal} is a set $B$ together with maps\n\\[\n e_i, f_i \\colon B \\longrightarrow B\\sqcup\\{0\\},\\qquad\n\\varepsilon_i,\\varphi_i\\colon B \\longrightarrow \\mathbf{Z} \\sqcup \\{-\\infty\\},\\qquad\n\\mathrm{wt}\\colon B \\longrightarrow P\n\\]\nsatisfying certain conditions (see \\cite{HK02,K95}).  Any $U_q(\\mathfrak{g})$-crystal basis, defined in the classical sense (see \\cite{K91}), is an abstract $U_q(\\mathfrak{g})$-crystal.  In particular, the negative half $U_q^-(\\mathfrak{g})$ of the quantized universal enveloping algebra of $\\mathfrak{g}$ has a crystal basis which is an abstract $U_q(\\mathfrak{g})$-crystal.  We denote this crystal by $B(\\infty)$ (rather than the using the entire tuple $(B(\\infty),e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$), and denote its highest weight element by $u_\\infty$.  As a set, one has\n\\[\nB(\\infty) = \\{  f_{i_d} \\cdots  f_{i_2} f_{i_1} u_\\infty : i_1,\\dots,i_d \\in I, \\ d \\ge 0 \\}.\n\\]\nThe remaining crystal structure on $B(\\infty)$ is\n\\begin{align*}\n\\mathrm{wt}( f_{i_d} \\cdots  f_{i_2} f_{i_1} u_\\infty) &= -\\alpha_{i_1}-\\alpha_{i_2}-\\cdots-\\alpha_{i_d} ,\\\\\n\\varepsilon_i(b) &= \\max\\{ k \\in \\mathbf{Z} :  e_ib \\neq 0 \\}, \\\\\n\\varphi_i(b) &= \\varepsilon_i(b) + \\langle h_i,\\mathrm{wt}(b) \\rangle. \n\\end{align*}\nWe say that $b\\in B(\\infty)$ has \\defn{depth} $d$ if $b = f_{i_d} \\cdots  f_{i_2} f_{i_1} u_\\infty$ for some $i_1,\\dots,i_d \\in I$.\n\n\n\nThere is a $\\mathbf{Q}(q)$-antiautomorphism $*\\colon U_q(\\mathfrak{g}) \\longrightarrow U_q(\\mathfrak{g})$ defined by\n\\[\nE_i \\mapsto E_i, \\qquad\nF_i \\mapsto F_i, \\qquad\nq \\mapsto q, \\qquad\nq^h \\mapsto q^{-h}.\n\\]\nThis is an involution which leaves $U_q^-(\\mathfrak{g})$ stable.  Thus, the map $\\ast$ induces a map on $B(\\infty)$, which we also denote by $*$, and is called the \\defn{$\\ast$-involution} or \\defn{star involution} (and is sometimes known as Kashiwara's involution).  Denote by $B(\\infty)^*$ the image of $B(\\infty)$ under $*$. \n\n\\begin{thm}[\\cite{K93,Lusztig90}]\nWe have\n$\nB(\\infty)^* = B(\\infty).\n$\n\\end{thm}\n\nThis induces a new crystal structure on $B(\\infty)$ with Kashiwara operators\n\\[\n e_i^* = * \\circ  e_i \\circ *,\\ \\ \\ \\ \\ \\ \n f_i^* = * \\circ  f_i \\circ *,\n\\]\nand the remaining crystal structure is given by\n\\[\n\\varepsilon_i^* = \\varepsilon_i \\circ * , \\qquad \\qquad\n\\varphi_i^* = \\varphi_i \\circ *,\n\\]\nand weight function $\\mathrm{wt}$, the usual weight function on $B(\\infty)$.\nAdditionally, for $b\\in B(\\infty)$ and $i\\in I$, define\n\\begin{equation}\n\\label{eq:jump}\n\\kappa_i(b) := \\varepsilon_i(b) + \\varepsilon_i^*(b) + \\langle h_i, \\mathrm{wt}(b)\\rangle.\n\\end{equation}\nThis was called the {\\it $i$-jump} in \\cite{LV11}.\n\n\n\n\n\n\nWe will appeal to the following statement from \\cite{CT15}, which was proven in a dual form in \\cite{TW} based on Kashiwara and Saito's classification theorem for $B(\\infty)$ from \\cite{KS97}.  First, a \\defn{bicrystal} is a set $B$ with two abstract $U_q(\\mathfrak{g})$-crystal structures $(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$ and $(B,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt})$ with the same weight function.  In such a bicrystal $B$, we say $b\\in B$ is a \\defn{highest weight element} if $e_ib = e_i^\\star b = 0$ for all $i \\in I$.\n\n\\begin{prop}\n\\label{prop:star_properties}\nFix a bicrystal $B$ with highest weight $b_0$ such the crystal data is determined by setting $\\mathrm{wt}(b_0) = 0$.  Assume further that, for all $i \\neq j$ in $I$ and all $b\\in B$, \n\\begin{enumerate}\n\\item\\label{item:star1} $f_ib$, $f_i^\\star b \\neq 0$;\n\\item\\label{item:star2} $f_i^\\star f_jb = f_jf_i^\\star b$;\n\\item\\label{item:star3} $\\kappa_i(b) \\ge 0$;\n\\item\\label{item:star4} $\\kappa_i(b) = 0$ implies $f_ib = f_i^\\star b$;\n\\item\\label{item:star5} $\\kappa_i(b) \\ge 1$ implies $\\varepsilon_i^\\star(f_ib) = \\varepsilon_i^\\star(b)$ and $\\varepsilon_i(f_i^\\star b) = \\varepsilon_i(b)$;\n\\item\\label{item:star6} $\\kappa_i(b) \\ge 2$ implies $f_if_i^\\star b = f_i^\\star f_ib$.\n\\end{enumerate}\nThen \n\\[\n(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt}) \\cong (B,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt}) \\cong B(\\infty),\\]\nwith $e_i^\\star = e_i^*$ and $f_i^\\star = f_i^*$.\n\\end{prop}\n\nHowever, we will need to slightly weaken the assumptions of Proposition~\\ref{prop:star_properties}.\n\n\\begin{prop}\n\\label{prop:weaker_conditions}\nLet $(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt})$ and $(B^\\star,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt})$ be highest weight abstract $U_q(\\mathfrak{g})$-crystals with the same highest weight vector $b_0 \\in B \\cap B^\\star$, where the remaining crystal data is determined by setting $\\mathrm{wt}(b_0) = 0$. Suppose also that (\\ref{item:star1})--(\\ref{item:star6}) are satisfied. Then\n\\[\n(B,e_i,f_i,\\varepsilon_i,\\varphi_i,\\mathrm{wt}) \\cong (B^\\star,e_i^\\star,f_i^\\star,\\varepsilon_i^\\star,\\varphi_i^\\star,\\mathrm{wt}) \\cong B(\\infty),\n\\]\nwith $e_i^\\star = e_i^*$ and $f_i^\\star = f_i^*$.\n\\end{prop}\n\n\\begin{proof}\nWe prove that $B \\cap B^\\star$ is closed under $f_i$ and $f_i^\\star$ by using induction on the depth and making repeated use of conditions (\\ref{item:star1})--(\\ref{item:star6}) above. The base case is depth $0$, where we just have $b_0$. Suppose all elements of depth at most $d$ in $B$ and $B^\\star$ are in $B \\cap B^\\star$. Next, fix some $b \\in B \\cap B^\\star$ at depth $d$. If $\\kappa_i(b) = 0$, then $f_i b = f_i^\\star b$ for all $i\\in I$. If $i \\neq j$, then $f_i^\\star f_j b' = f_j f_i^\\star b'$. Hence $f_i^* b \\in B \\cap B^\\star$, and $f_j b \\in B \\cap B^\\star$ by our induction assumption and that $B$ (resp., $B^\\star$) is closed under $f_j$ (resp., $f_i^\\star$). A similar argument shows that $f_i b \\in B \\cap B^\\star$ if $\\kappa_i(b) \\geq 1$ since $\\kappa_i(b') \\geq 2$. Therefore $B = B \\cap B^\\star = B^\\star$ since $B \\cap B^\\star$ is closed under $f_i$ and $f_j^\\star$ and generated by $b_0$ (along with $B$ and $B^\\star$). Thus the claim follows by Proposition~\\ref{prop:star_properties}.\n\\end{proof}\n\n\n\n\n\\section{Rigged configurations}\n\\label{sec:RC}\n\nLet $\\mathcal{H} = I \\times \\mathbf{Z}_{>0}$. A rigged configuration is a sequence of partitions $\\nu = (\\nu^{(a)} \\mid a \\in I)$ such that each row $\\nu_i^{(a)}$ has an integer called a \\defn{rigging}, and we let $J = \\bigl(J_i^{(a)} \\mid (a, i) \\in \\mathcal{H} \\bigr)$, where $J_i^{(a)}$ is the multiset of riggings of rows of length $i$ in $\\nu^{(a)}$. We consider there to be an infinite number of rows of length $0$ with rigging $0$; i.e., $J_0^{(a)} = \\{0, 0, \\dotsc\\}$ for all $a \\in I$. The term rigging will be interchanged freely with the term \\defn{label}.  We identify two rigged configurations $(\\nu, J)$ and $(\\widetilde{\\nu}, \\widetilde{J})$ if \n\\[\nJ_i^{(a)} = \\widetilde{J}_i^{(a)}\n\\]\nfor any fixed $(a, i) \\in \\mathcal{H}$. Let $(\\nu, J)^{(a)}$ denote the rigged partition $(\\nu^{(a)}, J^{(a)})$.\n\nDefine the \\defn{vacancy numbers} of $\\nu$ to be \n\\begin{equation}\n\\label{eq:vacancy}\np_i^{(a)}(\\nu) = p_i^{(a)} = - \\sum_{(b,j) \\in \\mathcal{H}} A_{ab} \\min(i, j) m_j^{(b)},\n\\end{equation}\nwhere $m_i^{(a)}$ is the number of parts of length $i$ in $\\nu^{(a)}$.  The \\defn{corigging}, or \\defn{colabel}, of a row in $(\\nu,J)^{(a)}$ with rigging $x$ is $p_i^{(a)} - x$.  In addition, we can extend the vacancy numbers to\n\\[\np_{\\infty}^{(a)} = \\lim_{i\\to\\infty} p_i^{(a)} =  - \\sum_{b \\in I} A_{ab} \\lvert \\nu^{(b)} \\rvert\n\\]\nsince $\\sum_{j=1}^{\\infty} \\min(i,j) m_j^{(b)} = \\lvert \\nu^{(b)} \\rvert$ for $i \\gg 1$.  Note this is consistent with letting $i = \\infty$ in Equation~\\eqref{eq:vacancy}.\n\nLet $\\RC(\\infty)$ denote the set of rigged configurations generated by $(\\nu_{\\emptyset}, J_{\\emptyset})$, where $\\nu_{\\emptyset}^{(a)} = 0$ for all $a \\in I$, and closed under the crystal operators as follows.\n\n\\begin{dfn}\n\\label{def:RC_crystal_ops}\nFix some $a \\in I$, and let $x$ be the smallest rigging in $(\\nu,J)^{(a)}$.\n\\begin{itemize}\n\\item[\\defn{$e_a$}:] If $x =0$, then $e_a(\\nu, J) = 0$. Otherwise, let $r$ be a row in $(\\nu, J)^{(a)}$ of minimal length $\\ell$ with rigging $x$. Then $e_a(\\nu, J)$ is the rigged configuration which removes a box from row $r$, sets the new rigging of $r$ to be $x+1$, and changes all other riggings such that the coriggings remain fixed.\n\n\\item[\\defn{$f_a$}:] Let $r$ be a row in $(\\nu, J)^{(a)}$ of maximal length $\\ell$ with rigging $x$. Then $f_a(\\nu, J)$ is the rigged configuration which adds a box to row $r$, sets the new rigging of $r$ to be $x-1$, and changes all other riggings such that the coriggings remain fixed.\n\\end{itemize}\n\\end{dfn}\n\n\nWe define the remainder of the crystal structure on $\\RC(\\infty)$ by\n\\begin{gather*}\n\\varepsilon_a(\\nu, J) = \\max \\{ k \\in \\mathbf{Z} \\mid e_a^k(\\nu, J) \\neq 0 \\}, \\hspace{20pt} \\varphi_a(\\nu, J) = \\inner{h_a}{\\mathrm{wt}(\\nu,J)} + \\varepsilon_a(\\nu, J),\n\\\\ \\mathrm{wt}(\\nu, J) = -\\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a.\n\\end{gather*}\nFrom this structure, we have $p_\\infty^{(a)} = \\inner{h_a}{\\mathrm{wt}(\\nu,J)}$ for all $a\\in I$.\n\n\\begin{thm}[{\\cite{SalS15, SalS15II}}]\n\\label{thm:binf_isomorphism}\nLet $\\mathfrak{g}$ be of symmetrizable type. Then $\\RC(\\infty) \\cong B(\\infty)$ as $U_q(\\mathfrak{g})$-crystals.\n\\end{thm}\n\n\\begin{prop}[{\\cite{SalS15, S06}}]\n\\label{prop:ep_phi}\nLet $(\\nu, J) \\in \\RC(\\infty)$ and fix some $a \\in I$. Let $x$ denote the smallest label in $(\\nu,J)^{(a)}$. Then we have\n\\[\n\\varepsilon_a(\\nu, J) = -\\min(0, x) \\hspace{40pt} \\varphi_a(\\nu, J) = p_{\\infty}^{(a)} - \\min(0, x).\n\\]\n\\end{prop}\n\nIt is a straightforward computation from the vacancy numbers to show that\n\\begin{equation}\n\\label{eq:convexity_exact}\n\\inner{h_a}{\\lambda} - \\sum_{b \\in I} A_{ab} m_i^{(b)} = -p_{i-1}^{(a)} + 2 p_i^{(a)} - p_{i+1}^{(a)}.\n\\end{equation}\nFrom this, we obtain the well-known convexity properties of the vacancy numbers.\n\n\\begin{lemma}[Convexity]\n\\label{lemma:convexity}\nIf $m_i^{(a)} = 0$, then we have\n\\[\n2 p_i^{(a)} \\geq p_{i-1}^{(a)} + p_{i+1}^{(a)}.\n\\]\nMoreover, $p_{i-1}^{(a)} \\geq p_i^{(a)} \\leq p_{i+1}^{(a)}$ if and only if $p_{i-1}^{(a)} = p_i^{(a)} = p_{i+1}^{(a)}$.\n\\end{lemma}\n\nIn the sequel, we will refer to this lemma simply as convexity as we will frequently use it.\n\n\n\n\n\n\n\\section{Star-crystal structure}\n\\label{sec:*crystal}\n\n\\begin{dfn}\n\\label{def:RC_star_crystal_ops}\nFix some $a \\in I$, and let $x$ be the smallest \\emph{co}rigging in $(\\nu,J)^{(a)}$.\n\\begin{itemize}\n\\item[\\defn{$e_a^*$}:] If $x =0$, then $e_a(\\nu, J) = 0$. Otherwise let $r$ be a row in $(\\nu, J)^{(a)}$ of minimal length $\\ell$ with corigging $x$. Then $e_a(\\nu, J)$ is the rigged configuration which removes a box from row $r$ and sets the new corigging of $r$ to be $x+1$.\n\n\\item[\\defn{$f_a^*$}:] Let $r$ be a row in $(\\nu, J)^{(a)}$ of maximal length $\\ell$ with corigging $x$. Then $f_a(\\nu, J)$ is the rigged configuration which adds a box to row $r$ and sets the new colabel of $r$ to be $x-1$.\n\\end{itemize}\n\\end{dfn}\n\nIf $e_a^*$ removes a box from a row of length $\\ell$ in $(\\nu, J)$, then the the vacancy numbers change by the formula\n\\begin{equation}\n\\label{eq:change_vac_e}\n\\widetilde{p}_i^{(b)} = \\begin{cases}\np_i^{(b)} & \\text{if } i \\leq \\ell, \\\\\np_i^{(b)} + A_{ab} & \\text{if } i > \\ell.\n\\end{cases}\n\\end{equation}\nOn the other hand, if $f_a^*$ adds a box to a row of length $\\ell$, then the vacancy numbers change by\n\\begin{equation}\n\\label{eq:change_vac_f}\n\\widetilde{p}_i^{(b)} = \\begin{cases}\np_i^{(b)} & \\text{if } i < \\ell, \\\\\np_i^{(b)} - A_{ab} & \\text{if } i \\geq \\ell.\n\\end{cases}\n\\end{equation}\nSimilar equations hold for $e_a$ and $f_a$ respectively. So the riggings of unchanged rows are changed according to Equation~\\eqref{eq:change_vac_e} and Equation~\\eqref{eq:change_vac_f} under $e_a$ and $f_a$, respectively. \n\n\\begin{remark}\nBy Equation~\\eqref{eq:change_vac_e} and Equation~\\eqref{eq:change_vac_f}, the crystal operators $e_a$ and $f_a$ preserve all colabels of $(\\nu, J)$ other than the row changed in $(\\nu, J)^{(a)}$.\n\\end{remark}\n\n\\begin{ex}\n\\label{ex:running}\nConsider type $D_4$ with Dynkin diagram\n\\[\n\\begin{tikzpicture}[xscale=2,yscale=.75]\n\\node[circle,fill,scale=.45,label={below:$1$}] (1) at (0,0) {};\n\\node[circle,fill,scale=.45,label={below:$2$}] (2) at (1,0) {};\n\\node[circle,fill,scale=.45,label={right:$3$}] (3) at (2,1) {};\n\\node[circle,fill,scale=.45,label={right:$4.$}] (4) at (2,-1) {};\n\\path[-]\n (1) edge (2)\n (2) edge (3)\n (2) edge (4);\n\\end{tikzpicture}\n\\]\nLet $(\\nu,J)$ be the rigged configuration\n\\begin{align*}\n(\\nu, J) \n&= f_2^*f_3^* f_1^* f_2^* f_2^* f_4^* f_3^* f_1^* f_2^* (\\nu_{\\emptyset}, J_{\\emptyset}) \\\\\n&=\n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-2,-1}{-3,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\end{align*}\nThen\n\\[\nf_2^*(\\nu, J) = \\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{4,1}{-3,-1}{-5,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\n\\end{ex}\n\nLet $\\RC(\\infty)^*$ denote the closure of $(\\nu_{\\emptyset}, J_{\\emptyset})$ under $f_a^*$ and $e_a^*$. We define the remaining crystal structure by\n\\begin{gather*}\n\\varepsilon_a^*(\\nu, J) = \\max \\{ k \\in \\mathbf{Z} \\mid (e_a^*)^k(\\nu, J) \\neq 0 \\}, \n\\hspace{20pt} \n\\varphi_a^*(\\nu, J) = \\inner{h_a}{\\mathrm{wt}(\\nu,J)} + \\varepsilon_a^*(\\nu, J),\n\\\\ \\mathrm{wt}(\\nu, J) = -\\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a.\n\\end{gather*}\n\n\\begin{remark}\n\\label{remark:duality}\nWe will say an argument holds by duality when we can interchange:\n\\begin{itemize}\n\\item ``label'' and ``colabel'';\n\\item $e_a$ and $e_a^*$;\n\\item $f_a$ and $f_a^*$.\n\\end{itemize}\nFor an example, compare the proof of Proposition~\\ref{prop:ep_phi_star} with~\\cite[Thm.~3.8]{Sakamoto14}.\n\\end{remark}\n\n\\begin{lemma}\nThe tuple $(\\RC(\\infty)^*, e_a^*, f_a^*, \\varepsilon_a^*, \\varphi_a^*, \\mathrm{wt})$ is an abstract $U_q(\\mathfrak{g})$-crystal.\n\\end{lemma}\n\n\\begin{proof}\nThe proof that $(\\RC(\\infty)^*, e_a^*, f_a^*, \\varepsilon_a^*, \\varphi_a^*, \\mathrm{wt})$ is an abstract $U_q(\\mathfrak{g})$-crystal is dual to that $\\RC(\\infty)$ is an abstract $U_q(\\mathfrak{g})$-crystal under $e_a$ and $f_a$ in~\\cite[Lemma~3.3]{SalS15}.\n\\end{proof}\n\n\n\\begin{prop}\n\\label{prop:ep_phi_star}\nLet $(\\nu, J) \\in \\RC(\\infty)$ and fix some $a \\in I$. Let $x$ denote the smallest colabel in $(\\nu,J)^{(a)}$. Then we have\n\\[\n\\varepsilon_a^*(\\nu, J) = -\\min(0, x), \\hspace{40pt} \\varphi_a^*(\\nu, J) = p_{\\infty}^{(a)} - \\min(0, x).\n\\]\n\\end{prop}\n\n\\begin{proof}\nThe following argument for $\\varepsilon_a^*$ is essentially the dual to that given in~\\cite[Thm.~3.8]{Sakamoto14}. We include it here as an example of Remark~\\ref{remark:duality}.\n\nIt is sufficient to prove $\\varepsilon_a^*(\\nu, J) = -\\max(0, x)$ since $p_{\\infty}^{(a)} = \\inner{h_a}{\\mathrm{wt}(\\nu, J)}$. If $x \\geq 0 = \\varepsilon_a^*(\\nu, J)$, then $e_a^*(\\nu, J) = 0$ by definition. Thus we proceed by induction on $\\varepsilon_a^*(\\nu,J)$ and assume $x < 0$. Let $(\\nu',J') = e_a^*(\\nu, J)$ and $y'$ denote the resulting colabel from a colabel $y$. In particular, we have $x' = x + 1$ and all other colabels follow Equation~\\eqref{eq:change_vac_e}. Next, let $y$ denote the colabel of a row of length $j$. For $j  < \\ell$, we have $y > x$ (equivalently $y \\geq x - 1$) because we chose $\\ell$ as large as possible. Thus $y' = y$, and hence $y' = y \\geq x + 1 = x'$. For $j \\geq \\ell$, we have $y \\geq x$ by the minimality of $x$ and $y' = y + 2$. Hence, $y' = y + 2 \\geq x + 1 = x'$, and so $\\varepsilon_a^*(\\nu', J') = \\varepsilon_a^*(\\nu, J) - 1$ as desired.\n\\end{proof}\n\n\nThe rest of this section will amount to showing that Conditions~(\\ref{item:star1})--(\\ref{item:star6}) of Proposition~\\ref{prop:weaker_conditions} hold.\nNote that using Proposition \\ref{prop:ep_phi_star} and \\cite[Prop. 4.2]{SalS15}, we can rewrite Equation \\eqref{eq:jump} as \n\\begin{equation}\n\\label{eq:RCjump}\n\\begin{aligned}\n\\kappa_a(\\nu,J) &= -\\min(0,x_\\ell) - \\min(0,x_c) + \\langle h_a , \\mathrm{wt}(\\nu,J) \\rangle, \\\\\n&= -\\min(0,x_\\ell) - \\min(0,x_c) + p_\\infty^{(a)},\n\\end{aligned}\n\\end{equation}\nwhere $x_\\ell$ and $x_c$ are the smallest label and colabel, respectively, in $(\\nu,J)^{(a)}$.\n\n\n\\begin{lemma}\n\\label{lemma:kappa0}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) = 0$. Then $f_a(\\nu, J) = f_a^*(\\nu, J)$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $f_a$ adds a box to a row of length $i$ with rigging $x$. Recall that $x = -\\varepsilon_a(\\nu, J)$. Suppose the longest row of $\\nu^{(a)}$ has length $\\ell > i$ and let $x_\\ell$ denote any rigging of the longest row. Therefore, we have $x_\\ell > x$ by the definition of $f_a$, and we have $p_\\ell^{(a)} \\leq p_{\\infty}^{(a)}$ by convexity. Thus from the definition of $\\varepsilon_a^*$, we have\n\\begin{equation}\n\\label{eq:contradiction_kappa0}\n\\varepsilon_a^*(\\nu, J) \\geq x_\\ell - p_\\ell^{(a)} > x - p_{\\infty}^{(a)} = -\\varepsilon_a(\\nu, J) - p_{\\infty}^{(a)}.\n\\end{equation}\nThis implies $\\varepsilon_a^*(\\nu, J) + \\varepsilon_a(\\nu, J) + p_{\\infty}^{(a)} > 0$, which is a contradiction. Therefore $f_a$ must add a box to one of the longest rows of $\\nu^{(a)}$. Moreover, if $p_\\ell^{(a)} < p_{\\infty}^{(a)}$, then Equation~\\eqref{eq:contradiction_kappa0} would still hold and result in a contradiction. Similar statements holds for $f_a^*$ by duality.\n\nTherefore $f_a$ and $f_a^*$ act on the longest row of $\\nu^{(a)}$ and $p_i^{(a)} = p_{i+1}^{(a)} = p_{\\infty}^{(a)}$. Let $x$ and $x^*$ denote the label of the row on which $f_a$ and $f_a^*$ act, respectively. Both of these labels decrease by $1$ after applying $f_a$ and $f_a^*$, respectively, by Equation~\\eqref{eq:colabel_change} and Equation~\\eqref{eq:label_change}, respectively. So it is sufficient to show $x = x^*$. Note that $x \\leq x^*$ as the smallest colabel is the one with the largest rigging. Suppose $x < x^*$, then we have\n\\[\n\\varepsilon_a^*(\\nu, J) \\geq x^* - p_i^{(a)} > x - p_{\\infty}^{(a)} \\geq -\\varepsilon_a(\\nu, J) - p_{\\infty}^{(a)},\n\\]\nwhich is a contradiction. Therefore we have $f_a = f_a^*$.\n\\end{proof}\n\n\\begin{ex}\n\\label{ex:kappa0}\nLet $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}.  Then $\\kappa_2(\\nu,J) = 0$, and \nand \n\\[\nf_2(\\nu,J) = \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{4,1}{-3,-1}{-5,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\nOne can check that this agrees with $f_2^*(\\nu,J)$ from Example \\ref{ex:running}.\n\\end{ex}\n\n\n\n\\begin{lemma}\n\\label{lemma:kappa1}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) \\geq 1$. Then\n\\[\n\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J),\n\\hspace{40pt}\n\\varepsilon_a(f_a^*(\\nu,J)) = \\varepsilon_a(\\nu,J).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $x^c$ denote the smallest colabel of $(\\nu,J)^{(a)}$.\nLet $x$ and $i$ denote the rigging and length of the row on which $f_a$ acts.  By the minimality of $x^c$, we have $x_c := p_i^{(a)} - x \\geq x^c$. Note that the colabel of the row after the application of $f_a$ becomes\n\\begin{equation}\n\\label{eq:new_colabel}\n\\widetilde{x}_c := p_{i+1}^{(a)} - 2 - (x - 1) = p_{i+1}^{(a)} - x - 1,\n\\end{equation}\nwhich implies that\n\\begin{equation}\n\\label{eq:colabel_change}\n\\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1.\n\\end{equation}\n\nThe remainder of the proof will be split into two cases: $x^c = p_i^{(a)} - x$ and $x^c < p_i^{(a)} - x$.\n\n\\case{$x^c = p_i^{(a)} - x$}\n\nFirst, consider the case $p_{i+1}^{(a)} \\leq p_i^{(a)}$. We also assume there exists a row of length $\\ell > i$ of $\\nu^{(a)}$, and let $x_\\ell$ denote the rigging of that row. Thus $x < x_\\ell$ and $x^c \\leq p_\\ell^{(a)} - x_\\ell$ by the definition of $f_a$ and the minimality of $x^c$. Hence \n\\[\np_i^{(a)} - x = x^c \\leq p_\\ell^{(a)} - x_\\ell < p_\\ell^{(a)} - x,\n\\]\nwhich is equivalent to $p_i^{(a)} < p_\\ell^{(a)}$. It must be the case, then, that $p_{i+1}^{(a)} > p_i^{(a)}$ by convexity, which is impossible. Thus the longest row of $\\nu^{(a)}$ must be of length $i$. Convexity implies $p_i^{(a)} = p_{i+1}^{(a)} = p_\\infty^{(a)}$, which results in\n\\begin{align*}\n\\kappa_a(\\nu, J) & = p_{\\infty}^{(a)} - \\min(x, 0) - \\min(x^c, 0)\n\\\\ & = p_i^{(a)} - \\min(x, 0) - \\min\\bigl(p_i^{(a)} - x, 0\\bigr).\n\\end{align*}\nSince $x$ was the rigging chosen by $f_a$, we must have $x \\leq 0$. Additionally, if $p_i^{(a)} - x = x^c \\leq 0$, we have\n\\[\n1 \\leq \\kappa_a(\\nu, J) = p_i^{(a)} - x - (p_i^{(a)} - x) = 0,\n\\]\nwhich is a contradiction. Thus $x^c \\geq 1$, which implies $\\varepsilon_a^*(\\nu, J) = 0 = \\varepsilon_a^*\\bigl(f_a(\\nu, J))$ since $\\widetilde{x}_c \\geq 0$.\n\nNext if $p_{i+1}^{(a)} = p_i^{(a)} + 1$, then $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$ since $\\widetilde{x}_c = x^c$, all other coriggings are fixed, and $\\widetilde{x}_c = x_c$ by Equation~\\eqref{eq:colabel_change} .\n\nSo we now assume $p_{i+1}^{(a)} \\geq p_i^{(a)} + 2$, which implies $\\widetilde{x}_c > x^c$. If there is another row with a corigging of $x^c$ or $x^c \\geq 0$, then $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$. So assume $f_a$ acts on the only row with a corigging of $x^c < 0$. Note that $p_i^{(a)} - x = x^c < 0$ and $x \\leq 0$ implies $p_i^{(a)} < 0$.\n\nWe have $m_i^{(a)} = 1$ as, otherwise, we would either have a second corigging of $x^c$ or a smaller corigging from the minimality of $x$. Thus, by Equation~\\eqref{eq:convexity_exact}, \n\\begin{align*}\n-2 - \\sum_{b \\neq a} A_{ab} m_i^{(b)} &= -p_{i-1}^{(a)} + 2p_i^{(a)} - p_{i+1}^{(a)} \\\\\n&\\leq -p_{i-1}^{(a)} + 2p_i^{(a)} - p_i^{(a)} - 2 \\\\\n&= -p_{i-1}^{(a)} + p_i^{(a)} - 2.\n\\end{align*}\nSince $m_i^{(b)} \\geq 0$ and $-A_{ab} \\geq 0$ for all $a \\neq b$, we then have\n\\[\n0 \\leq -p_{i-1}^{(a)} + p_i^{(a)},\n\\]\nor, equivalently, $p_{i-1}^{(a)} \\leq p_i^{(a)}$. If $i$ is the length of the smallest row, then $0 \\leq p_{i-1}^{(a)} \\leq p_i^{(a)} < 0$ by convexity, which is a contradiction. Thus let $x_\\ell$ denote rigging of the longest row in $\\nu^{(a)}$ such that $\\ell < i$, and by convexity, we have $p_\\ell^{(a)} \\leq p_i^{(a)}$. By the definition of $f_a$, we have $x_\\ell \\geq x$. Thus, we have $p_\\ell^{(a)} - x_\\ell \\leq p_i^{(a)} - x$. However, by the unique minimality of $x^c$, we have $p_\\ell^{(a)} - x_\\ell > x^c = p_i^{(a)} - x$. This is a contradiction. Therefore $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$.\n\n\\case{$x^c < p_i^{(a)} - x$}\n\nAssume $\\varepsilon_a^*\\bigl(f_a(\\nu,J)\\bigr) \\neq \\varepsilon_a^*(\\nu, J)$.  Then\n\\begin{equation}\n\\label{eq:difference_epsilon_change}\np_{i+1}^{(a)} - p_i^{(a)} - 1 < x^c+ x - p_i^{(a)} < 0,\n\\end{equation}\nas, otherwise, the new corigging is not smaller than the minimal corigging (i.e., $\\widetilde{x}_c < x^c$), which occurs on a different row and does not change under $f_a$. We rewrite Equation~\\eqref{eq:difference_epsilon_change} as\n\\begin{equation}\n\\label{eq:rewrite_less_than}\np_{i+1}^{(a)} - 1 < x^c + x < p_i^{(a)}.\n\\end{equation}\nSuppose there exists a row of length $\\ell > i$ in $\\nu^{(a)}$.  Then $x_\\ell > x$ and $x^c \\leq p_\\ell^{(a)} - x_\\ell$. Therefore,\n\\begin{equation}\n\\label{eq:double_inequality}\np_{i+1}^{(a)} - \\ell < p_\\ell^{(a)} - x_\\ell + x < p_\\ell^{(a)},\n\\end{equation}\nwhich implies $p_{i+1}^{(a)} \\leq p_j^{(a)}$ for all $\\ell \\geq j > i$ by convexity. Note that Equation~\\eqref{eq:double_inequality} implies that $\\ell > i+1$ since, otherwise, we would have $p_{i+1}^{(a)} < p_{i+1}^{(a)}$. We also have $p_{i+1}^{(a)} \\leq p_j^{(a)}$ for all $j > i$ if there does not exist a row of length $\\ell > i$. Since there does not exist a row of length $i+1$, we must have $p_i^{(a)} \\leq p_{i+1}^{(a)} \\leq p_{i+2}^{(a)}$ by convexity. Yet, Equation~\\eqref{eq:rewrite_less_than} implies\n\\[\np_{i+1}^{(a)} < p_i^{(a)} \\leq p_{i+1}^{(a)},\n\\]\nbut this is a contradiction. Therefore $\\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr) = \\varepsilon_a^*(\\nu, J)$.\n\\end{proof}\n\n\n\\begin{ex}\n\\label{ex:kappa1}\nAgain, let $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}.  Then $\\varepsilon_3(\\nu,J) = 0$, $\\varepsilon_3^*(\\nu,J) = 1$, and $\\kappa_3(\\nu,J) = 1$.  We have\n\\begin{align*}\nf_3(\\nu,J) &=\n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-1,-1}{-1,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{3}{-1}{-2}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}\\\\\nf_3^*(\\nu,J) &= \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=6cm]\n \\rpp{3,1}{-2,-1}{-2,-1}\n \\end{scope}\n \\begin{scope}[xshift=14cm]\n \\rpp{3}{0}{-2}\n \\end{scope}\n \\begin{scope}[xshift=20cm]\n \\rpp{1}{0}{0}\n \\end{scope}\n\\end{tikzpicture}.\n\\end{align*}\nThen $\\varepsilon_3^*\\bigl(f_3(\\nu,J)\\bigr) = 1$ and $\\varepsilon_3\\bigl(f_3^*(\\nu,J)\\bigr) = 0$.\n\\end{ex}\n\n\n\\begin{lemma}\n\\label{lemma:kappa2}\nFix $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$. Assume $\\kappa_a(\\nu, J) \\geq 2$. Then\n\\[\nf_af_a^*(\\nu, J) = f_a^* f_a(\\nu, J).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSuppose $f_a$ (resp., $f_a^*$) acts on row $r$ of length $i$ (resp., row $r^*$ of length $i^*$) with rigging $x$ (resp., $x^*$). Without loss of generality, let $r$ (resp., $r^*$) be the northernmost such row in the diagram of $\\nu^{(a)}$. Let $x_c^* = p_{i^*}^{(a)} - x^*$ and $x_c = p_i^{(a)} - x$. Note that $x \\leq x^*$ and $x_c^* \\leq x_c$. Applying $f_a^*$, the new rigging (and the only changed rigging) is\n\\begin{equation}\n\\label{eq:label_change}\n\\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1.\n\\end{equation}\nRecall that Equation~\\eqref{eq:colabel_change} gives the new corigging (and only changed corigging)\n\\[\n\\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1\n\\]\nafter applying $f_a$. We split the proof into three cases: the first two are cases in which $r \\neq r^*$ and the last is when $r = r^*$.\n\n\\case{$f_a$ acts on row $r \\neq r^*$ in $f_a^*(\\nu, J)$}\n\nSuppose $f_a f_a^*(\\nu, J) \\neq f_a^* f_a(\\nu, J)$.  This is equivalent to $f_a^*$ acting on row $r' \\neq r^*$ in $f_a(\\nu, J)$. Note that we must have $r' = r$, since $f_a$ preserves all other colabels. From Equation~\\eqref{eq:colabel_change}, we must have $p_{i+1}^{(a)} < p_i^{(a)} + 2$, as otherwise $\\widetilde{x}_c > x_c \\geq x_c^*$, which would imply $r = r' = r^*$ and be a contradiction. Next, consider when $p_{i+1}^{(a)} = p_i^{(a)} + 1$. Thus we have $\\widetilde{x}_c = x_c$. Since $r' \\neq r^*$, we must have $i = i^*$ and $x_c^* = \\widetilde{x}_c = x_c$. However, this contradicts the assumption $r \\neq r^*$ as we have $x = x^*$.\n\nHence $p_{i+1}^{(a)} \\leq p_i^{(a)}$. Suppose $i$ is the length of the longest row of $\\nu^{(a)}$.  Then $p_i^{(a)} = p_{i+1}^{(a)} = p_{\\infty}^{(a)}$ by convexity. Moreover, we have $\\widetilde{x}_c = x_c - 1 = x_c^*$ since $r, r' \\neq r^*$. Note that since $f_a$ (resp., $f_a^*$) acts on $r$ (resp., $r^*$), we must have $x \\leq 0$ (resp., $x_c^* \\leq 0$). Therefore, we have\n\\begin{align*}\n2 \\leq \\kappa_a(\\nu, J) & = p_i^{(a)} - \\min(x, 0) - \\min(x_c^*, 0)\n\\\\ & = p_i^{(a)} - x - x_c^*\n\\\\ & = p_i^{(a)} - x - (p_i^{(a)} - x - 1) = 1,\n\\end{align*}\nwhich is a contradiction.\n\nSuppose there exists a row $r_\\ell$ of length $\\ell > i$ in $\\nu^{(a)}$. Let $x_\\ell$ denote the rigging of $r_\\ell$, and note $x < x_\\ell$ by our assumption. Therefore, we have\n\\[\np_{i+1}^{(a)} - x - 1  = \\widetilde{x}_c \\leq x_c^* \\leq p_\\ell^{(a)} - x_l < p_\\ell^{(a)} - x,\n\\]\nwhich implies $p_{i+1}^{(a)} \\leq p_\\ell^{(a)}$. Assume there exists a row of length $i+1$ in $\\nu^{(a)}$ with rigging $x_{i+1}$.  It follows that\n\\[\np_{i+1}^{(a)} - x > p_{i+1}^{(a)} - x_{i+1} \\geq x_c^* = p_{i^*}^{(a)} - x^*,\n\\]\nwhich is equivalent to\n\\[\np_{i+1}^{(a)} - p_{i^*}^{(a)} > x - x^*.\n\\]\nFurthermore,\n\\[\np_{i+1}^{(a)} - x - 1 = \\widetilde{x}_c \\leq x_c^* = p_{i^*}^{(a)} - x^*,\n\\]\nwhich results in\n\\begin{equation}\n\\label{eq:changing_f_star_inequality}\nx - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1.\n\\end{equation}\nAdditionally, Equation~\\eqref{eq:changing_f_star_inequality} is necessarily a strict inequality if $i^* > i$ because it must be the case that $\\widetilde{x}_c < x_c^*$.\nHence\n\\[\np_{i+1}^{(a)} - p_{i^*}^{(a)} > x - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nwhich is a contradiction for $i^* > i$ as the right inequality becomes a strict inequality.\n\nNext, note that $\\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x$ since $f_a$ acts on $r$. Hence \n\\begin{equation}\n\\label{eq:unchanged_f_inequality}\np_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x - x^*,\n\\end{equation}\nwhich is a strict inequality for $i \\leq i^*$. Thus \n\\[\np_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 \\geq x - x^* \\geq p_{i+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nor, equivalently, $p_{i^*+1}^{(a)} \\geq p_{i+1}^{(a)}$. Since $i \\leq i^*$, we have $p_{i^*+1}^{(a)} > p_{i+1}^{(a)}$, and hence $i = i^*$ cannot occur\n\nNow suppose $i^* < i$. Therefore $x_c < p_{i+1}^{(a)} - x_{i+1}$, which implies\n\\[\np_{i+1}^{(a)} - x - 1  = \\widetilde{x}_c \\leq x_c^* < p_{i+1}^{(a)} - x_{i+1} < p_{i+1}^{(a)} - x.\n\\]\nSo $p_{i+1}^{(a)} < p_{i+1}^{(a)}$, which is a contradiction.\n\nFinally, if there does not exist a row of length $i+1$, then $p_i^{(a)} = p_{i+1}^{(a)} = p_\\ell^{(a)}$ by convexity, so the argument given above will still yield a contradiction. Hence, $f_a f_a^*(\\nu,J) = f_a^* f_a(\\nu, J)$.\n\n\\case{$f_a$ acts on row $r' \\neq r$ in $f_a^*(\\nu, J)$}\n\nNote that $r' = r^*$, where $r^* \\neq r$, as $f_a^*$ fixes all other riggings. So from Lemma~\\ref{lemma:kappa1}, we have\n\\[\n\\widetilde{x}^* = -\\varepsilon_a\\bigl(f_a^*(\\nu, J)\\bigr) = -\\varepsilon_a(\\nu, J) = x,\n\\]\nand hence $i \\leq i^*$. Therefore $x < x^*$ as $x = x^*$ implies $r = r^*$. Thus,\n\\[\nx  = \\widetilde{x}^* = x^* + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1 > x + p_{i^*+1}^{(a)} - p_{i^*}^{(a)} - 1,\n\\]\nand hence $p_{i^*+1}^{(a)} \\leq p_{i^*}^{(a)}$. Dually, $f_a^*$ acts on row $r$ in $f_a(\\nu, J)$ since this would contradict $f_a$ acting on $r^* \\neq r$ from the previous case. Similarly, \n\\[\n\\widetilde{x}_c = -\\varepsilon_a^*\\bigl(f_a^*(\\nu, J)\\bigr) = -\\varepsilon_a^*(\\nu, J) = x_c^*\n\\]\nby the dual version of Lemma~\\ref{lemma:kappa1}, implying $i^* \\leq i$. Hence, $i = i^*$ and\n\\[\np_i^{(a)} - x^* = x_c^* = \\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = - x +  p_{i+1}^{(a)} - 1,\n\\]\nwhich yields $x - x^* = p_{i+1}^{(a)} - p_i^{(a)} - 1$. Thus\n\\begin{align*}\nx & = \\widetilde{x}^* = x^* + p_{i+1}^{(a)} - p_i^{(a)} - 1\n\\\\ p_i^{(a)} - x^* = x_c^* & = \\widetilde{x}_c = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_{i+1}^{(a)} - x - 1 \\leq p_i^{(a)} - x - 1,\n\\end{align*}\nwhich implies $x - x^* \\leq -1$. Hence,\n\\[\np_i^{(a)} - p_{i+1}^{(a)} = x^* - x - 1 \\leq -2,\n\\]\nand this contradicts $0 \\leq p_i^{(a)} - p_{i+1}^{(a)}$. \n\n\n\\case{$r = r^*$}\n\nFrom $x = x^*$ and $i = i^*$, we have \n\\begin{align*}\n\\widetilde{x}^* & = x^* + p_{i+1}^{(a)} - p_i^{(a)} - 1 = x + p_{i+1}^{(a)} - p_i^{(a)} - 1,\n\\\\ \\widetilde{x}_c & = x_c + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_i^{(a)} - x  + p_{i+1}^{(a)} - p_i^{(a)} - 1 = p_{i+1}^{(a)} - x - 1,\n\\\\ x_c^* & = x_c = p_i^{(a)} - x.\n\\end{align*}\n\nIf $p_{i+1}^{(a)} \\leq p_i^{(a)} + 1$, then $\\widetilde{x}^* \\leq x$ and $\\widetilde{x}_c \\leq x_c^*$. Hence, $f_a$ and $f_a^*$ select row $r$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively, and so we have $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$. Next, consider the case when $p_{i+1}^{(a)} \\geq p_i^{(a)} + 2$.  Then $\\widetilde{x}^* > x$ and $\\widetilde{x}_c > x_c^*$. If $m_i^{(a)} \\geq 2$, then there exists a row $r' \\neq r$ such that $x^* = x$ and $x_c^* = x_c$. So $f_a$ and $f_a^*$ select row $r'$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively, and thus we have $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$.\nIf $m_i^{(a)} = 1$, then, as in Lemma~\\ref{lemma:kappa1}, we have\n\\[\n-2 - \\sum_{b \\neq a} A_{ab} m_i^{(b)} = -p_{i-1}^{(a)} + 2p_i^{(a)} - p_{i+1}^{(a)} \\leq -p_{i-1}^{(a)} + p_i^{(a)} - 2.\n\\]\nfrom Equation~\\eqref{eq:convexity_exact}.  This implies $p_{i-1}^{(a)} \\leq p_i^{(a)}$.  We consider the case when $r$ is the smallest row of $\\nu^{(a)}$, which implies $r$ is the unique row with rigging $x$ and corigging $x_c^*$. Moreover, we have $0 \\leq p_{i-1}^{(a)} \\leq p_i^{(a)}$ by convexity. Because we are acting on $r$ by $f_a$ and $f_a^*$, we have $x \\leq 0$ and $x_c^* \\leq 0$. Hence,\n\\[\n0 \\leq p_i^{(a)} \\leq p_i^{(a)} - x = x_c^* \\leq 0 \\implies \n0 = p_i^{(a)} = x = x_c^*. \n\\]\nTherefore $f_a$ (resp., $f_a^*$) acts on a row of length $0$ in $f_a^*(\\nu, J)$ (resp., $f_a(\\nu, J)$) as all other riggings (resp., coriggings) are positive. Moreover, the resulting rigging is $-1$ in both cases, and so $f_a f_a^*(\\nu, J) = f_a^* f_a(\\nu, J)$.\n\nNow assume there exists a row $r_\\ell$ of length $\\ell < i$ in $\\nu^{(a)}$, and without loss of generality, suppose $\\ell$ is maximal. Let $x_\\ell$ denote the rigging of $r_\\ell$, and by the definition of $f_a$ and $f_a^*$, we have $x_\\ell \\geq x$ and $p_\\ell^{(a)} - x_\\ell \\geq x_c^* = p_i^{(a)} - x$. By convexity, we have $p_\\ell^{(a)} \\leq p_{i-1}^{(a)} \\leq p_i^{(a)}$. Therefore, we have\n\\[\np_i^{(a)} - x \\leq p_\\ell^{(a)} - x_\\ell \\leq p_i^{(a)} - x,\n\\]\nand so $p_i^{(a)} - x = p_\\ell^{(a)} - x_\\ell$. Moreover, if $p_\\ell^{(a)} < p_i^{(a)}$, then we have $x_\\ell < x$, which cannot occur, and hence we also have $x = x_\\ell$. Therefore $f_a$ and $f_a^*$ acts on $r_\\ell$ in $f_a^*(\\nu, J)$ and $f_a(\\nu, J)$, respectively. Thus we have $f_a f_a^*(\\nu, J) = f_a^*f_a(\\nu, J)$.\n\\end{proof}\n\n\n\\begin{ex}\n\\label{ex:kappa2}\nContinuing our running example, let $(\\nu,J)$ be the rigged configuration of type $D_4$ from Example \\ref{ex:running}.  Then $\\kappa_4(\\nu,J) = 2$ and \n\\[\nf_4^*f_4(\\nu,J) = f_4f_4^*(\\nu,J) = \n\\begin{tikzpicture}[scale=.35,anchor=top,baseline=-18]\n \\rpp{2}{0}{-1}\n \\begin{scope}[xshift=5cm]\n \\rpp{3,1}{-1,-1}{-1,-1}\n \\end{scope}\n \\begin{scope}[xshift=11.5cm]\n \\rpp{2}{0}{-1}\n \\end{scope}\n \\begin{scope}[xshift=16.5cm]\n \\rpp{3}{-1}{-2}\n \\end{scope}\n\\end{tikzpicture}.\n\\]\nWith $a=4$, we have the diagram\n\\[\n\\begin{tikzpicture}[xscale=2,yscale=1.5,font=\\normalsize]\n\\node (t) at (0,0) {$(\\nu,J)$};\n\\node (s) at (-1,-1) {$f_4^*(\\nu,J)$};\n\\node (a) at (1,-1) {$f_4(\\nu,J)$};\n\\node (ss) at (-2,-2) {$f_4^*f_4^*(\\nu,J)$};\n\\node (aa) at (2,-2) {$f_4f_4(\\nu,J)$};\n\\node (as) at (0,-2) {$f_4^*f_4(\\nu,J) = f_4f_4^*(\\nu,J)$};\n\\node (sss) at (-2,-3) {$\\substack{f_4^*f_4^*f_4^*(\\nu,J) \\\\ = f_4f_4^*f_4^*(\\nu,J)}$};\n\\node (asa) at (0,-3) {$\\substack{f_4^*f_4^*f_4(\\nu,J) \\\\ = f_4^*f_4f_4^*(\\nu,J) \\\\ = f_4f_4^*f_4(\\nu,J) \\\\ = f_4f_4f_4^*(\\nu,J)}$};\n\\node (aaa) at (2,-3) {$\\substack{f_4^*f_4f_4(\\nu,J) \\\\ = f_4f_4f_4(\\nu,J)}$};\n\\foreach \\x in {-2,0,2}\n {\\node at (\\x,-4) {$\\vdots$};}\n\\path[->]\n (t) edge (s)\n (t) edge (a)\n (s) edge (ss)\n (s) edge (as)\n (a) edge (aa)\n (a) edge (as)\n (ss) edge (sss)\n (as) edge (asa)\n (aa) edge (aaa)\n (sss) edge (-2,-3.85)\n (asa) edge (0,-3.85)\n (aaa) edge (2,-3.85);\n\\end{tikzpicture}\n\\]\nAs discussed in \\cite[Cor. 2.8]{CT15}, $\\kappa_4(\\nu,J)$ counts how many times one must apply either $f_4$ or $f_4^*$ to $(\\nu,J)$ to reach a point where $f_4$ and $f_4^*$ have the same affect.\n\\end{ex}\n\n\n\n\n\\begin{thm}\nLet $e_a$ and $f_a$ be the crystal operators given by Definition~\\ref{def:RC_crystal_ops}, and let $e_a^*$ and $f_a^*$ be given by Definition~\\ref{def:RC_star_crystal_ops}. Then we have\n\\[\ne_a^*  = * \\circ e_a \\circ *, \\ \\ \\ \\ \\ \\\nf_a^*  = * \\circ f_a \\circ *.\n\\]\n\\end{thm}\n\n\\begin{proof}\nWe show the conditions of Proposition~\\ref{prop:star_properties} hold for $\\RC(\\infty)$ with the given crystal operations. Fix some $(\\nu, J) \\in \\RC(\\infty)$ and $a \\in I$.\n\nWe first note the fact that $f_a(\\nu, J)$, $f_a^*(\\nu, J) \\neq 0$ follows immediately from the definitions. So we have Condition~(\\ref{item:star1}). Now let $b \\in I$.  As $f_b$ acts on labels and preserves colabels in $(\\nu,J)^{(k)}$, for $k \\neq b$ in $I$, and $f_a^*$ acts on colabels and preserves labels in $(\\nu,J)^{(k)}$, for $k \\neq a$ in $I$, it follows that $f_a^* f_b (\\nu, J) = f_b f_a^* (\\nu, J)$ for all $a \\neq b$. Hence Condition~(\\ref{item:star2}) is satisfied.\n\nLemma~\\ref{lemma:kappa0} implies Condition~(\\ref{item:star4}).\n\nLemma~\\ref{lemma:kappa1} implies Condition~(\\ref{item:star5}).\n\nLemma~\\ref{lemma:kappa2} implies Condition~(\\ref{item:star6}).\n\nThus it remains we prove Condition~(\\ref{item:star3}), that $\\kappa_a(\\nu,J) \\ge 0$. We prove this by induction on the depth of $(\\nu,J)$.  Observe that $\\kappa_a(\\nu_\\emptyset,J_\\emptyset) = 0$, which is our base case.  Now suppose $\\kappa_a(\\nu,J) \\ge 0$ for all $(\\nu,J) \\in \\RC(\\infty)$ at depth at most $d$.  It suffices to show that $\\kappa_a\\bigl(f_a(\\nu,J)\\bigr) \\ge 0$ and $\\kappa_a\\bigl(f_a^*(\\nu,J)\\bigr) \\ge 0$.\n\nNote that all labels, except for the row of $\\nu^{(a)}$ at which the box was added, possibly change by adding $-A_{ab}$ under $f_a$ by Equation~\\eqref{eq:change_vac_f}. Additionally, $p_{\\infty}^{(b)}$ changes by $-A_{ab}$. Thus for $b \\neq a$ a label, and hence possibly $-\\varepsilon_b(\\nu, J)$, increases by $-A_{ab}$ and the colabels, and hence $-\\varepsilon_b^*(\\nu, J)$, stay fixed. Therefore, by the above and its dual, for $a \\neq b$, we have\n\\[\n\\kappa_b\\bigl(f_a(\\nu, J)\\bigr), \\kappa_b\\bigl(f_a^*(\\nu, J)\\bigr) \\geq \\kappa_b(\\nu, J) \\geq 0,\n\\]\nsince $-A_{ab} \\geq 0$. Now it is sufficient to show that $\\varepsilon_a^*(\\nu, J)$ increases by at least $1 - \\kappa_a(\\nu, J)$ because $\\varphi_a(\\nu, J) = p_{\\infty}^{(a)} + \\varepsilon_a(\\nu, J)$ decreases by $1$ after the application of $f_a$. If $\\kappa_a(\\nu, J) = 0$, then Lemma~\\ref{lemma:kappa0} gives $f_a(\\nu, J) = f_a^*(\\nu, J)$, and so $\\varepsilon_a^*(\\nu, J)$ is increased by $1$. By Lemma~\\ref{lemma:kappa1}, we have $\\varepsilon_a^*(\\nu, J) = \\varepsilon_a^*\\bigl(f_a(\\nu, J)\\bigr)$ when $\\kappa_a(\\nu, J) \\geq 1$. Note that by our assumption, this is all possible values. Therefore, both $\\kappa_a\\bigl(f_a(\\nu, J)\\bigr), \\kappa_a\\bigl(f_a^*(\\nu, J)\\bigr) \\geq 0$, as required.\n\nThus Conditions~(\\ref{item:star1})--(\\ref{item:star6}) are satisfied. Moreover, $\\RC(\\infty)$ and $\\RC(\\infty)^*$ are both generated by the highest weight element $(\\nu_{\\emptyset}, J_{\\emptyset})$. Hence $\\RC(\\infty) = \\RC(\\infty)^*$ and the result follows from Proposition~\\ref{prop:weaker_conditions}.\n\\end{proof}\n\nNow from Definition~\\ref{def:RC_crystal_ops} and Definition~\\ref{def:RC_star_crystal_ops}, we have that the $*$-involution is given as follows.\n\n\\begin{cor}\n\\label{cor:RC_star_involution}\nThe $*$-involution on $\\RC(\\infty)$ is given by replacing every rigging $x$ of a row of length $i$ in $(\\nu, J)^{(a)}$ by the corresponding corigging $p_i^{(a)} - x$ for all $(a, i) \\in \\mathcal{H}$.\n\\end{cor}\n\nLet $\\mathfrak{g}$ and $\\widehat{\\mathfrak{g}}$ be symmetrizable Kac-Moody algebras such that there exists a folding of the Dynkin diagram of $\\mathfrak{g}$ to the Dynkin diagram of $\\widehat{\\mathfrak{g}}$ with corresponding index sets $I$ and $\\widehat{I}$, respectively.  Consider the map $\\phi \\colon \\widehat{I} \\searrow I$ induced by such a Dynkin diagram folding and consider a sequence $(\\gamma_a \\in \\mathbf{Z}_{>0})_{a \\in I}$ such that the map $\\Psi \\colon P \\longrightarrow \\widehat{P}$ given by\n\\[\n\\Lambda_a \\mapsto \\gamma_a \\sum_{b \\in \\phi^{-1}(a)} \\Lambda_b\n\\]\nalso satisfies\n\\[\n\\alpha_a \\mapsto \\gamma_a \\sum_{b \\in \\phi^{-1}(a)} \\alpha_b.\n\\]\nThis induces a \\defn{virtualization map} $v$ of $B(\\infty)$ of type $\\mathfrak{g}$ to that of type $\\widehat{\\mathfrak{g}}$. In particular, on $\\RC(\\infty)$, the image $(\\widehat{\\nu}, \\widehat{J})$ of a rigging configuration $(\\nu, J)$ is given by\n\\[\n\\widehat{m}_{\\gamma_a i}^{(b)} = m_i^{(a)},\n\\hspace{30pt}\n\\widehat{J}_{\\gamma_a i}^{(b)} = \\gamma_a J_i^{(a)},\n\\]\nfor all $b \\in \\phi^{-1}(a)$. We refer the reader to~\\cite{OSS03III, SalS15III, SchillingS15} for more details.\n\n\\begin{cor}\nLet $v$ be a virtualization map on $B(\\infty)$ of type $\\mathfrak{g}$ to $\\widehat{\\mathfrak{g}}$. Then\n\\[\n\\ast \\circ v = v \\circ \\ast.\n\\]\n\\end{cor}\n\n\\begin{proof}\nThis follows from the fact\n$\n\\widehat{p}_{\\gamma_a i}^{(b)} = \\gamma_a p_i^{(a)}\n$\nfor all $b \\in \\phi^{-1}(a)$ and Corollary~\\ref{cor:RC_star_involution}.\n\\end{proof}\n\n\n\n\n\n\\section{Highest weight crystals}\n\\label{sec:hw_crystals}\n\nWe wish to classify the subcrystal of $\\RC(\\infty)$ which is isomorphic to $B(\\lambda)$ with respect to the $\\ast$-crystal structure.  In particular, defining $B(\\lambda)$ requires the additional condition that $\\varphi_a^*(\\nu, J) = \\max\\{k \\in \\mathbf{Z} \\mid (f_a^*)^k(\\nu, J) \\neq 0\\}$.  For example, the condition $\\varphi_a(\\nu,J) = \\max\\{ k \\in \\mathbf{Z} \\mid f_a^k(\\nu,J) \\neq 0 \\}$ means, for all riggings $x$ corresponding to a row of length $i$ in $\\nu^{(a)}$, we have $x \\leq p_i^{(a)}$.  If we consider the natural dual to this, we have $p_i^{(a)} - x \\leq p_i^{(a)}$, or equivalently $x \\geq 0$. We show this is the correct condition by proving the dual version of~\\cite[Thm.~6.1]{SalS15}.\n\n\nFor any $\\lambda \\in P^+$, we define\n\\[\n\\RC(\\lambda) := \\{ (\\nu, J) \\in \\RC(\\infty) \\mid \\max J_i^{(a)} \\leq p_i^{(a)}(\\nu; \\lambda) \\text{ for all } (a, i) \\in \\mathcal{H} \\},\n\\]\nwhere\n\\begin{equation}\n\\label{eq:vacancy_numbers}\np_i^{(a)}(\\nu; \\lambda) := \\inner{h_a}{\\lambda} - \\sum_{b \\in I} A_{ab} \\sum_{j \\in \\mathbf{Z}_{>0}} \\min(i,j) m_j^{(b)}.\n\\end{equation}\nNote that Equation~\\eqref{eq:vacancy_numbers} differs from Equation~\\eqref{eq:vacancy} by\n\\[\np_i^{(a)}(\\nu) + \\inner{h_a}{\\lambda} = p_i^{(a)}(\\nu; \\lambda).\n\\]\nWhen there is no danger of confusion, we will simply write $p_i^{(a)} = p_i^{(a)}(\\nu; \\lambda)$.\n\nWe consider a crystal structure on $\\RC(\\lambda)$ as that inherited from $\\RC(\\infty)$ under the natural projection except with $\\mathrm{wt}(\\nu, J) = \\lambda - \\sum_{a \\in I} \\lvert \\nu^{(a)} \\rvert \\alpha_a$.\n\n\\begin{thm}[{\\cite{SalS15,SalS15II,S06}}]\nWe have\n$\n\\RC(\\lambda) \\cong B(\\lambda).\n$\n\\end{thm}\n\nUsing the $\\ast$-crystal structure, we easily obtain~\\cite[Prop.~8.2]{K95}.  (We refer the reader to \\cite{K95} or \\cite{HK02} for an exposition on the tensor product of crystals.  Note that we are using the opposite, anti-Kashiwara, convention.  The precise definition in this setting may be found, for example, in \\cite{SalS15}.)\n\n\\begin{prop}\nLet $\\lambda \\in P^+$. Then we have\n\\[\n\\RC(\\lambda) \\cong \\{ t_\\lambda \\otimes (\\nu,J) \\in T_\\lambda \\otimes \\RC(\\infty) \\mid \\varepsilon_a^*(\\nu,J) \\le \\langle h_a , \\lambda \\rangle \\text{ for all } a \\in I\\}.\n\\]\n\\end{prop}\n\n\\begin{proof}\nFix some $(\\nu, J) \\in \\RC(\\infty)$. Let $x$ be a rigging of a row of length $i$. We have\n\\[\n\\inner{h_a}{\\lambda} \\geq \\varepsilon_a^*(\\nu, J) = -\\min(0, p_i^{(a)} - x)\n\\]\nif and only if\n\\[\np_i^{(a)} + \\inner{h_a}{\\lambda} \\geq x.\n\\]\nRecall that the left-hand side is the vacancy numbers in $\\RC(\\lambda)$ by Equation~\\eqref{eq:vacancy_numbers}, and so we have the defining relation for $\\RC(\\lambda)$.\n\\end{proof}\n\nBy letting $\\pi_{\\lambda} \\colon B(\\infty) \\longrightarrow B(\\lambda)$ be the natural projection, we can rephrase the last proposition as\n\\[\n\\tau(b^*) = \\varepsilon(b), \\qquad \\qquad \\varepsilon(b^*) = \\tau(b),\n\\]\nwhere $\\varepsilon(b) = \\sum_{a \\in I} \\varepsilon_a(b) \\Lambda_a$ and $\\tau(b) = \\min \\{ \\lambda \\mid \\pi_{\\lambda}(b) \\in B(\\lambda) \\}$.   In~\\cite{SalS15II}, $\\tau$ was called the difference statistic and can be explicitly given on rigged configurations by\n\\[\n\\tau(\\nu, J) = \\sum_{a \\in I} \\min_{i \\in \\mathbf{Z}_{>0}} \\{p_i^{(a)} - \\max J_i^{(a)} \\} \\Lambda_a.\n\\]\n\n\n\n\n\nNow we formalize the dual version of $\\RC(\\lambda)$.\n\n\\begin{dfn}\nLet $\\RC(\\lambda)^*$ denote the closure of $(\\nu_{\\emptyset}, J_{\\emptyset})$ under $e_a^*$ and the following modified $f_a^*$, both using the vacancy numbers given by Equation~\\eqref{eq:vacancy_numbers} to determine the colabels. Consider $f_a^*$ as in Definition~\\ref{def:RC_star_crystal_ops} except define $f_a^*(\\nu, J) = 0$ if in the result, there exists a rigging $x < 0$.\n\\end{dfn}\n\nNote that the condition that $x \\geq 0$ is equivalent to $p_i^{(a)} - x \\leq p_i^{(a)}$. Hence, by duality, the proof of~\\cite[Lemma~3.6]{S06} holds, and we obtain the following.\n\n\\begin{lemma}\n\\label{lemma:lower_regular}\nLet $(\\nu, J) \\in \\RC(\\lambda)^*$. Then\n\\[\n\\varphi_a^*(\\nu, J) = \\max\\{ k \\in \\mathbf{Z} \\mid (f_a^*)^k (\\nu, J) \\neq 0 \\}\n\\]\nfor all $a \\in I$.\n\\end{lemma}\n\nLet $\\RC_{\\lambda}(\\infty)^* = T_{\\lambda} \\otimes \\RC(\\infty)$ with the $\\ast$-crystal structure. Let $C = \\{c\\}$ be the crystal given by\n\\[\n\\mathrm{wt}(c) = 0, \\qquad\n\\varphi_a^*(c) = \\varepsilon_a^*(c) = 0, \\qquad\nf_a^*c = e_a^*c = 0,\n\\]\nfor all $a \\in I$. Nakashima \\cite[Thm. 3.1]{N99} has shown that the connected component generated by $c \\otimes t_{\\lambda} \\otimes u_{\\infty}$ is isomorphic to $B(\\lambda)$.\n\nIn~\\cite{SalS15}, the map $\\psi_{\\lambda,\\mu} \\colon \\RC(\\lambda) \\longrightarrow \\RC(\\mu)$, for $\\lambda \\leq \\mu$ in $P^+ \\sqcup \\{\\infty\\}$, is the identity map on rigged configurations. This follows because $e_a$ and $f_a$ are determined by the riggings alone, not the vacancy numbers, and so preserving the labels is sufficient to show $\\psi_{\\lambda,\\mu}$ commutes with the crystal operators. However, for the $\\ast$-crystal structure, we need to preserve coriggings, and as such, we need to take into account the shift in vacancy numbers. Thus, define a map $\\psi_{\\lambda,\\mu}^* \\colon \\RC(\\lambda)^* \\longrightarrow \\RC(\\mu)^*$ as the identity on the partitions but with new riggings\n\\[\nx' = x + \\langle h_a , \\mu - \\lambda\\rangle,\n\\]\nwhere we make the convention that $\\inner{h_a}{\\infty} = 0$. Note that $\\psi_{\\lambda,\\mu}^*$ commutes with the crystal operators (however, it only becomes a crystal embedding after an appropriate tensor product is taken to shift weights).\n\nWith this modification, Proposition~\\ref{prop:ep_phi_star}, and Lemma~\\ref{lemma:lower_regular}, we have the dual argument of~\\cite[Thm.~6.1]{SalS15}. \n\n\\begin{thm}\nLet $C_{\\emptyset}^*$ denote the connected component of $C \\otimes \\RC_{\\lambda}(\\infty)^*$ generated by $c \\otimes (\\nu_{\\emptyset}, J_{\\emptyset})$. The map $\\Psi \\colon C_{\\emptyset}^* \\longrightarrow \\RC(\\lambda)^*$ given by \n\\[\nc \\otimes (\\nu_{\\lambda}, J_{\\lambda}) \\mapsto (\\psi_{\\lambda,\\infty}^*)^{-1}(\\nu_{\\lambda}, J_{\\lambda})\n\\]\nis a weight-preserving bijection which commutes with $e_a^*$ and $f_a^*$ for every $a\\in I$.\n\\end{thm}\n\n\\begin{cor}\nLet $\\mathfrak{g}$ be of symmetrizable type. Then $\\RC(\\lambda)^* \\cong B(\\lambda)$.\n\\end{cor}\n\nHence, we can now construct an explicit crystal isomorphism $\\RC(\\lambda)^* \\cong \\RC(\\lambda)$ by passing through $\\RC(\\infty)$.\n\n\\begin{cor}\nLet $\\mathfrak{g}$ be a symmetrizable Kac-Moody algebra and let $\\lambda \\in P^+$. Define $\\Xi \\colon \\RC(\\lambda) \\longrightarrow \\RC(\\lambda)^*$ by $\\Xi(\\nu, J) = (\\nu, J')$, where the resulting riggings are\n\\[\nx' = x + \\inner{h_a}{\\lambda}.\n\\]\nThen $\\Xi$ is a crystal isomorphism.\n\\end{cor}\n\n\\begin{proof}\nWe have $\\Xi = (\\psi_{\\lambda,\\infty}^*)^{-1} \\circ \\psi_{\\lambda,\\infty}$.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nBranching programs are one of the well known models of computation. These models have been shown useful in a variety of domains, such as hardware verification, model checking, and other CAD applications (see for example the book by Wegener \\cite{Weg00}). It is known that the class of Boolean  functions computed by polynomial size branching programs are coincide with the class of functions computed by non-uniform log-space machines. Moreover branching program is a convenient model for  considering different (natural)  restrictive  variants and different complexity measures such as size (number of inner nodes), length, and width.\n\nOne of important restrictive branching programs is oblivious read once branching programs, also known in applied computer science as Ordered Binary Decision Diagrams (OBDD) \\cite{Weg00}.  Since the length of an OBDD is at most linear (in the length of the input), the main complexity measure is ``width''.\n\nOBDDs also can be seen as nonuniform automata (see for example \\cite{ag05}).  During the last decades different variants of OBDDs were considered, i.e. deterministic, nondeterministic, probabilistic, and quantum, and\nmany results have been proved on comparative power of deterministic, nondeterministic, and randomized OBDDs \\cite{Weg00}. For example, Ablayev and Karpinski \\cite{ak96} presented the first function that is polynomially easy for  randomized and exponentially hard for deterministic and even nondeterministic OBDDs. More specifically, it was proven that OBDD variants of  $ \\mathsf{coRP} $ and $\\mathsf{NP}$ are different.\n\nIn the last decade quantum model of OBDD came into play \\cite{agk01},\\cite{nhk00},\\cite{SS05}.  It was proven that quantum OBDDs can be exponentially cheaper than classical ones and it was shown that this bound is tight \\cite{agkmp05}.\n\n\nIn this paper we present the first results on comparative complexity  for classical and quantum OBDDs computing partial functions. Then, we focus on the  width complexity of deterministic and nondeterministic OBDDs, which have been investigated in different papers (see for more information and citations \\cite{hs00}, \\cite{hs03}). Here we present very strict hierarchies for the classes of Boolean functions computed by deterministic and nondeterministic OBDDs.\n\nThe paper is organized as follows. The Section 2 contains the definitions and notations used in the paper. In Section 3, we compare classical and exact quantum OBDDs. We consider a partial function depending on parameter $k$ such that, for any $k>0$, this function is computed by an exact quantum OBDD of width $2$ but deterministic or bounded error probabilistic OBDDs need width $2^{k+1}$. Also it is easy to show that nondeterministic OBDD needs width $k+1$. In Section 4, we consider quantum and classical nondeterminism. We show that quantum nondeterministic OBDDs can be more efficient than their classical counterparts. We present  an explicit function which is computed by a quantum nondeterministic OBDD with constant width but any classical nondeterministic OBDD needs non-constant width. The Section 5 contains our results on hierarchies on the sublinear (5.1) and larger (5.2) widths of deterministic and nondeterministic OBDDs.\n\nThe proofs of lower bounds results (Theorem \\ref{th-DOBDD1} and Lemma \\ref{peq1s}) are based on Pigeonhole principle. The lower bound on Theorem \\ref{th-POBDD1} uses the technique of Markov chains. \n\n\\section{Preliminaries}\n\nWe refer  to \\cite{Weg00} for more information on branching programs. The main model investigated throughout the paper is OBDD (Ordered Binary Decision Diagram), a restricted version of branching program.\n\nIn this paper we use following notations for vectors. We use subscript for enumerating elements of vector and strings and superscript for enumerating vectors and strings.  For a binary string $\\nu$, $\\#_1(\\nu)$ and $\\#_0(\\nu)$ are the number of $1$'s and $0$'s in $\\nu$, respectively. We denote $\\#^k_{0}(\\nu)$ and $\\#^k_{1}(\\nu)$ to be the numbers of $1$'s and $0$'s in the first $k$ elements of string $\\nu$, respectively.\n \n\nFor a given $ n>0 $, a probabilistic OBDD $ P_n $ with width $d$, defined on $ \\{0,1\\}^n $, is a 4-tuple\n$\n\tP_n=(T,v^0,Accept{\\bf,\\pi}),\n$\nwhere\n\\begin{itemize}\n\t\\item $ T = \\{ T_j : 1 \\leq j \\leq n \\mbox{ and } T_j = (A_j(0),A_j(1))  \\} $ is an ordered pairs of (left) stochastic matrices representing the transitions, where, at the $j$-th step, $ A_j(0) $ or $ A_j(1) $, determined by the corresponding input bit, is applied.\n\t\\item $ v^0 $ is a zero-one initial stochastic vector (initial state of $P_n$).\n\t\\item $ Accept \\subseteq \\{1,\\ldots,d\\} $ is the set of accepting nodes.\n\\item $ \\pi $ is a permutation of $ \\{1,\\ldots,n\\} $ defining the order of testing the input bits.\n\\end{itemize}\n\n  For any given input $ \\nu \\in  \\{0,1\\}^n $, the computation of $P_n$ on $\\nu$ can be traced by a stochastic vector which is initially $ v^0 $. In each step $j$, $1 \\leq j \\leq n$, the input bit $ x_{\\pi(j)} $ is tested and then the corresponding stochastic operator is applied:\n\\[\n\tv^j = A_{j}(x_{\\pi(j)}) v^{j-1},\n\\]\nwhere  $ v^j $ represent the probability distribution vector of nodes after the $ j$-th steps, $ 1 \\leq j \\leq n $.\nThe accepting probability of $ P_n $ on $ \\nu $ is \n\\[\n\t\\sum_{i \\in Accept} v^n_i.\n\\]\n\nWe say that a function $f$ is computed by $ P_n $ with bounded error if there exists an $ \\varepsilon \\in (0,\\frac{1}{2}] $ such that $ P_n $ accepts all inputs from $ f^{-1}(1) $ with a probability at least $ \\frac{1}{2}+\\varepsilon $ and $ P_n $ accepts all inputs from $ f^{-1}(0) $ with a probability at most $ \\frac{1}{2}-\\varepsilon $. We say that $P_n$ computes $f$ \\textit{exactly} if $ \\varepsilon =1\/2 $. \n\nA deterministic OBDD is a probabilistic OBDD restricted to use only 0-1 transition matrices. In the other words, the system is always being in a single node and, from each node, there is exactly one outgoing transition for each tested input bit.\n\nA nondeterministic OBDD (NOBDD) can have the ability of making more than one outgoing transition for each tested input bit from each node and so the program can follow more than one computational path and if one of the path ends with an accepting node, then the input is accepted (rejected, otherwise). \n\n\n\\begin{itemize}\n\\item \n\nAn OBDD is called {\\em stable} if each transition set $T_j$ is identical for each level.\n\\item\n\nAn OBDD is called ID (ID-OBDD) if the input bits are tested in order $\\pi=(1,2,\\dots, n)$.\nIf a {\\em stable}  ID-OBDD has a fixed width and transition rules for each $ n $, then it can be considered as a realtime finite automaton.\n\n\\end{itemize}\n\nQuantum computation is a generalization of classical computation \\cite{Wat09A}. Therefore, each quantum model can simulate its probabilistic counterparts. In some cases, on the other hand, the quantum models are defined in a restricted way, e.g., using only unitary operators during the computation followed by a single measurement at the end, and so they may not simulate their probabilistic counterparts. Quantum automata literature has a lot of such results such as \\cite{KW97,AF98,BC01B}. A similar result was also given for OBDDs in \\cite{SS05}, in which a function with a small size of deterministic OBDD was given but the quantum OBDD defined in a restricted way needs exponential size to solve this function.\n\nQuantum OBDDs that defined with the general quantum operators, i.e. superoperator \\cite{Wat03,Wat09A,YS11A}, followed by a measurement on the computational basis at the end can simulate its classical counterpart with the same size and width. So we can always follow that any quantum class contains its classical counterpart. \n\nIn this paper, we follow our quantum results based stable ID-OBDDs, which are realtime quantum finite automata. But, we give the definition of quantum OBDDs for the completeness of the paper.\n\nA quantum OBDD is the same as a probabilistic OBDD with the following modifications:\n\\begin{itemize}\n\t\\item The state set is represented by a $ d $-dimensional Hilbert space over field of complex numbers. The initial one is $ \\ket{\\psi}_0=\\ket{q_0}$ where $ q_0 $ corresponds to the initial node.\n\t \n\t \\item Instead of a stochastic matrix, we apply a unitary matrix in each step. That is, $ T = \\{ T_j : 1 \\leq j \\leq n \\mbox{ and } T_j = (U_j^0,U_j^1)  \\} $, where, at the $j$-th step, $ U_j^0 $ or $ U_j^1 $, determined by the corresponding input bit, is applied,\n\t \n\t \\item At the end, we make a measurement on the computational basis.\n\\end{itemize}\n\n\n\nThe state of the system is updated as follows after the $ j$-th step:\n\\[\n\t\\ket{\\psi}_j = U_j^{x_{\\pi(j)}} (\\ket{\\psi}_{j-1}),\n\\]\nwhere $ \\ket{\\psi}_{j-1} $ and $ \\ket{\\psi}_j $ represent the state of the system after the $ (j-1)$-th  and $ j$-th steps, respectively, where $ 1 \\leq j \\leq n $.\n\n\n\nThe accepting probability of the quantum program on $ \\nu $ is calculated from $\\ket{\\psi}_n=(z_1, \\dots, z_d)$ as\n\\[\n\t\\sum_{i \\in Accept} |z_i|^2.\n\\]\n\n\n\\section{Exact Quantum OBDDs.}\n\nIn \\cite{AY12}, Ambainis and Yakary{\\i}lmaz defined a new family of unary promise problems: For any $ k>0 $, $ A^k = (A^k_{yes},A^k_{no}) $ such that $ A^k_{yes} = \\{ a^{(2i)2^k} : i \\geq 0 \\} $ and $ A^k_{no} = \\{ a^{(2i+1)2^k} : i \\geq 0 \\} $. They showed that each member of this family ($A^k$) can be solved exactly by a 2-state realtime quantum finite automaton (QFA) but any exact probabilistic finite automaton (PFA) needs at least $ 2^{k+1} $ states. Recently, Rashid and Yakary{\\i}lmaz \\cite{RY14A} showed that bounded-error realtime PFAs also needs at least $ 2^{k+1} $ states for solving $A^k$.\\footnote{The same result is also followed for two-way nondeterministic finite automata by Geffert and Yakary{\\i}lmaz \\cite{GY14A}.} Based on this promise problem, we define a partial function:\n\\[\n\t\\mathtt{PartialMOD^k_n} (\\nu)  = \\left\\lbrace \\begin{array}{lcll}\n\t\t1 & , & \\mbox{if } \\#_1(\\nu) = 0 & (mod\\textrm{ }2^{k+1}) \\\\\n\t\t0 & , & \\mbox{if } \\#_1(\\nu) = 2^{k} & (mod\\textrm{ }2^{k+1}) \\\\\n\t\t* & , & \\multicolumn{2}{l}{\\mbox{otherwise}} \\\\\t\n\\end{array}\t \\right. ,\n\\]\nwhere the function is not defined for the inputs mapping to ``*''. We call the inputs where the function takes the value of 1 (0) as yes-instances (no-instances).\n\n\\begin{theorem}\n\tFor any $k \\geq 0$, $ \\mathtt{PartialMOD^k_n} $ can be solved by a stable quantum ID-OBDD with width 2 exactly.\n\\end{theorem}\n\n\nThe OBDD can be construct by the same way as QFA, which solves promise problem $ A^k $ \\cite{AY12}.\n\n\n\nWe show that the width of deterministic, or bounded-error stable probabilistic OBDDs that solve $ \\mathtt{PartialMOD^k_n} $  cannot be less than $ 2^{k+1} $.\n\n\n\\begin{remark} \n Note that, a proof for deterministic OBDD is not similar to the proof for automata because potentially nonstability can give profit. Also this proof is different from proofs for total functions (for example, $ MOD_p$) due to the existence of incomparable inputs.\nNote that, classical one-way communication complexity techniques also fail for partial functions  (for example, it can be shown that communication complexity of $ \\mathtt{PartialMOD^k_n} $ is $1$), and we need to use more careful analysis in the proof. \n\\end{remark}\n \n \n Deterministic stable ID-OBDD with width $2^{k+1}$ for $ \\mathtt{PartialMOD^k_n} $ can be easy constructed. We left open the case for bounded-error non-stable probabilistic OBDDs.\n\n\\begin{theorem}\n\\label{th-DOBDD1}\n\tFor any $k\\geq 0$, there are infinitely many $n$ such that any deterministic  OBDD computing the partial function $ \\mathtt{PartialMOD^k_n} $ has width at least $2^{k+1}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\nu\\in\\{0,1\\}^n, \\nu=\\sigma\\gamma$. We call  $\\gamma$ valid for $\\sigma$ if $\\nu \\in (\\mathtt{PartialMOD^k_n})^{-1}(0) \\cup (\\mathtt{PartialMOD^k_n})^{-1}(1).$ We call two substrings $\\sigma'$ and $\\sigma''$ comparable  if for all $\\gamma$ it holds that $\\gamma$ is valid for $\\sigma'$  iff $\\gamma$ is valid for $\\sigma''$.\nWe call two substrings $\\sigma'$ and $\\sigma''$ nonequivalent if they are comparable and there exists a valid substring $\\gamma$ such that   $\\mathtt{PartialMOD^k_n}(\\sigma'\\gamma)\\neq\\mathtt{PartialMOD^k_n}(\\sigma''\\gamma)$ .  \n\nLet $P$ be a deterministic OBDD computing the partial function $ \\mathtt{PartialMOD^k_n} $. Note that paths associated with nonequivalent strings must lead to different nodes. Otherwise, if $\\sigma$ and $\\sigma'$ are nonequivalent, there exists a valid string $\\gamma$ such that $\\mathtt{PartialMOD^k_n}(\\sigma\\gamma)\\neq\\mathtt{PartialMOD^k_n}(\\sigma'\\gamma)$ and computations on these inputs lead to the same final node.\n\nLet $N=2^k$ and $\\Gamma=\\{\\gamma: \\gamma\\in \\{0,1\\}^{2N-1}, \\gamma=0\\dots  0 1\\dots 1\\}$.\nWe will naturally identify any string $\\nu$  with the element $a=\\#_1(\\nu) \\pmod{2N}$ of additive group $\\mathbb{Z}_{2N}$. We call two strings of the same length different if the numbers of ones by modulo $2N$ in them are different. We denote by $\\rho(\\gamma^1,\\gamma^2)=\\gamma^1-\\gamma^2$ the distance between numbers $\\gamma^1, \\gamma^2.$ Note that $\\rho(\\gamma^1,\\gamma^2)\\neq \\rho(\\gamma^2,\\gamma^1) $.\n\nLet the width of $P$ is $t<2N.$ On each step $i$ ($i=1,2,\\dots$) of the proof we will count the number of different strings,  which lead to the same node \n(denote this node $v_i$). On $i$-th step we consider $(2N-1)i$-th level of $P$.\n\nLet $i=1$.\nBy pigeonhole principle there exist two different strings $\\sigma^1$ and $ \\sigma^2$ from the set $\\Gamma$ such that corresponding  paths lead to the same node $v_1$ of the $(2N-1)$-th level of $P$.  Note that $\\rho(\\sigma^1, \\sigma^2)\\neq N,$ because  in this case $\\sigma^1$ and $\\sigma^2$ are nonequivalent and cannot lead to the same node.\n\nWe will show  by induction that on each step of the proof the number of different strings which lead to the same node increases.\n\nStep 2. By pigeonhole principle there exist two different strings $\\gamma^1$ and $ \\gamma^2$ from the set $\\Gamma$ such that corresponding  paths from the node $v_1$ lead to the same node $v_2$ of the $(2N-1)2$-th level of $P$.  In this case, the strings $\\sigma^1 \\gamma^1,\\sigma^2 \\gamma^1,\\sigma^1 \\gamma^2 $, and $\\sigma^2 \\gamma^2$ lead to the node $v_2$. Note that $\\rho(\\gamma^1, \\gamma^2)\\neq N,$ because  in this case $\\sigma^1\\gamma^1$ and $\\sigma^1\\gamma^2$ are nonequivalent and cannot lead to the same node. \n\nAdding  the same number does not change the distance between the numbers, so we have\n   \\[\\rho(\\sigma^1+\\gamma^1, \\sigma^2+\\gamma^1)=\\rho(\\sigma^1,\\sigma^2)\\]\n   and\n\\[\\rho(\\sigma^1+\\gamma^2, \\sigma^2+\\gamma^2)=\\rho(\\sigma^1,\\sigma^2).\\]\nLet  $\\gamma^2>\\gamma^1$. Denote $\\Delta=\\gamma^2-\\gamma^1 $. Let us count the number of different numbers among $\\sigma^1+\\gamma^1,$ $\\sigma^2+\\gamma^1  $, $\\sigma^1+\\gamma^1+\\Delta  $, and $\\sigma^2+\\gamma^1+\\Delta  $.\nBecause $\\sigma^1$ and $\\sigma^2$ are different and $\\rho(\\sigma^1, \\sigma^2)\\neq N$,  the numbers from the pair $\\sigma^1+\\gamma^1  $, and $\\sigma^2+\\gamma^1  $  coincide with corresponding numbers from the pair $\\sigma^1+\\gamma^1+\\Delta  $ and $\\sigma^2+\\gamma^1+\\Delta  $ iff $\\Delta=0 \\pmod {2N}$. But  $\\Delta \\neq 0 \\pmod {2N}$ since the  numbers $\\gamma^1 $ and $ \\gamma^2$ are different and  $\\gamma^1, \\gamma^2<2N.$ The numbers  $\\sigma^1+\\gamma^1+\\Delta  $ and $\\sigma^2+\\gamma^1+\\Delta   $  cannot be a permutation of numbers   $\\sigma^1+\\gamma^1  $ and $\\sigma^2+\\gamma^1  $  since $\\rho(\\gamma^1, \\gamma^2)\\neq N$ and    $\\rho(\\sigma^1, \\sigma^2)\\neq N$.\nIn this case, at least 3 numbers from $\\sigma^1+\\gamma^1  $, $\\sigma^2+\\gamma^1  ,$ $\\sigma^1+\\gamma^2  $, and $\\sigma^2+\\gamma^2  $ are different.\n\nStep of induction.  Let the numbers $\\sigma^1, \\dots, \\sigma^i$ be different on the step $i-1$ and the corresponding paths lead to the same node $v_{i-1}$ of the $(2N-1)(i-1)$-th level of $P$. \n\nBy pigeonhole principle there exist two different strings $\\gamma^1 $ and $ \\gamma^2$ from the set $\\Gamma$ such that  the corresponding  paths from the node $v_{i-1}$ lead to the same node $v_i$ of the $(2N-1)i$-th level of $P$.  So paths $\\sigma^1\\gamma^1,\\dots, \\sigma^i\\gamma^1,\\sigma^1\\gamma^2,\\dots, \\sigma^i\\gamma^2 $ lead to the same node $v_i$. Let us estimate a number of different strings among them. Note that  $\\rho(\\gamma^1, \\gamma^2)\\neq N$ since, in this case, the strings $\\sigma^1\\gamma^1$ and $ \\sigma^1\\gamma^2$ are nonequivalent but lead to the same node. \n\nThe numbers $\\sigma^1, \\dots,\\sigma^{i}$ are different and $\\rho(\\sigma^l, \\sigma^j)\\neq N$ for each pair $(l,j) $ such that $ l\\neq j$.\nLet $\\sigma^1 < \\dots <\\sigma^i$. We will show that among $\\sigma^1+\\gamma^1  ,\\dots, \\sigma^i+\\gamma^1   $ and $\\sigma^1+\\gamma^1+\\Delta  , \\dots, \\sigma^i+\\gamma^1+\\Delta  $ at least $i+1$ numbers are different.\n\n\nThe sequence of numbers $\\sigma^1+\\gamma^1  ,\\dots, \\sigma^i+\\gamma^1  $ coincide with the the sequence $\\sigma^1+\\gamma^1+\\Delta  , \\dots, \\sigma^i+\\gamma^1+\\Delta  $ iff $\\Delta=0 \\pmod {2N}$. But $\\Delta\\neq 0 \\pmod {2N}$ since $\\gamma^1$ and $ \\gamma^2$ are different and  $\\gamma^1, \\gamma^2<2N.$\n \nSuppose that the sequence $\\sigma^1+\\gamma^1+\\Delta  , \\dots, \\sigma^i+\\gamma^1+\\Delta  $  is a permutation of the sequence  $\\sigma^1+\\gamma^1  $,$\\dots,$ $\\sigma^i+\\gamma^1  $. In this case, \nwe have numbers $a_0, \\dots, a_r$ from $\\mathbb{Z}_{2N}$  such that all $a_j$ are from the sequence $\\sigma^1+\\gamma^1  $,$\\dots,$ $\\sigma^i+\\gamma^1  $, $a_0=a_r=\\sigma^1+\\gamma^1  $, and $a_j=a_{j-1}+\\Delta  $, where $j=1, \\dots, r$. \nIn this case, $r\\Delta=2Nm.$\nBecause $N=2^k$, $\\Delta<2N$, and $\\Delta\\neq N$  we have that $r$ is even. For $z=r\/2$ we have $z\\Delta=Nm$.\nSince all numbers from $\\sigma^1+\\gamma^1,\\dots, \\sigma^i+\\gamma^1$ are different, we have that $\\rho(a_0,a_z)=N$.\nSo we have that $a_0 $ and $ a_z$ are  nonequivalent but the corresponding strings lead to the same node $v_i$. \nSo after $i$-th step, we have that at least $i+1$ different strings lead to the same node $v_i.$\n\nOn the $N$-th step, we have that $N+1$ different strings lead to the same node $v_N$. Among these strings, there must be at least two nonequivalent strings. Thus we can follow that $P$ cannot compute the function $\\mathtt{PartialMOD^k_n}$ correctly.  \n\\qed\\end{proof}\n\n\\begin{theorem}\n\\label{th-NOBDD1}\n\tFor any $k\\geq 0$, there are infinitly many $n$ such that any nondeterministic  OBDD computing the partial function $ \\mathtt{PartialMOD^k_n} $ has width at least $2^{k+1}$.\n\\end{theorem}\n\nThe proof is similar to deterministic case with the following modifications. \nLet $P$ -- NOBDD, that computes $ \\mathtt{PartialMOD^k_n} $. \nWe consider only accepting pathes in $ P $. Note that if $P$ computes $ \\mathtt{PartialMOD^k_n} $ correctly then accepting paths associated with nonequivalent strings can not pass through the same nodes. Denote $N=2^k$.  Let $\\Gamma=\\{\\gamma: \\gamma\\in \\{0,1\\}^{2N-1}, \\gamma=\\underbrace{0\\dots  0}_{2N-1-j} \\underbrace{1\\dots 1}_{j}, j=0, \\dots,2N-1\\}$.\n\n\n Denote $V_{l}$ -- a set of nodes on the $l$-th level of $ P $. Assume that the width of $P$ is less than $2N$, that is, $|V_l|<2N$ for each $ l=0, \\dots,n $.\n\nDenote $V_l(\\gamma, V)$ -- a set of nodes of $l$-th level through which  accepting paths, leading from the nodes of set $ V $ and corresponding to string $\\gamma$, pass.\n\nOn step $i$ ($ i=1,2,\\dots $) of the proof we consider $(2N-1)i$-th level of $P$.\nBecause $|V_{2N-1}|<2N$, on the first step of the proof  we have that there exists two different strings $\\sigma^1,\\sigma^2$ from the set $\\Gamma$ such that $ V_{2N-1}(\\sigma^1, V_0) \\cap V_{2N-1}(\\sigma^2, V_0) \\neq \\emptyset$.  Denote this nonempty intersection $G_1$. Next we continue our proof considering only  accepting paths which  pass through the nodes of $G_1.$\n\nUsing the same ideas as for deterministic case we can show that the number of strings with different numbers of ones by modulo $2N$, such that corresponding accepting paths  lead to the same set of nodes, increases with each step of the proof. \n\nOn the $N$-th step of the proof we will have that there must be two different nonequivalent strings such that corresponding accepting paths  lead to the same set  $G_N$ of nodes.  That implies that $P$ does not compute the function $\\mathtt{PartialMOD^k_n}$ correctly. \n\n\n\n\n\\begin{theorem}\n\\label{th-POBDD1}\nFor any $k\\geq 0$, there are infinitely many $n$ such that any stable probabilistic OBDD computing $ \\mathtt{PartialMOD^k_n} $ with bounded error has width at least $2^{k+1}$. \n\\end{theorem}\n\nThe proof of the Theorem is based on the technique of Markov chains and the details are given in Appendix \\ref{app-POBDD1}.\n\n\\section{Nondeterministic Quantum and Classical OBDDs.}\n\nIn \\cite{YS10A}, Yakary{\\i}lmaz and Say showed that nondeterministic QFAs can define a super set of regular languages, called exclusive stochastic languages \\cite{Paz71}. This class contains the \\textit{complements} of some interesting languages:\n$ PAL = \\{ w \\in \\{0,1\\}^* : w = w^r \\} $, where $w^r$ is a reverse of $w$,   $ O = \\{ w \\in \\{0,1\\}^* : \\#_1(w)=\\#_0(w)\\} $, $ SQUARE = \\{ w \\in \\{0,1\\}^* : \\#_1(w)=(\\#_0(w))^2\\} $, and $ POWER = \\{ w \\in \\{0,1\\}^* : \\#_1(w)=2^{\\#_0(w)}\\} $.\n\nBased on these languages, we define three symmetric functions for any input $ \\nu \\in \\{0,1\\}^n $:\n\t \\[ \\mathtt{NotO_n}(\\nu) = \\left\\lbrace \\begin{array}{lll}\n\t\t0 & , & \\mbox{if } \\#_0(\\nu) = \\#_1(\\nu) \\\\\n\t\t1 & , & \\mbox{otherwise }\n\\end{array}\t \\right., \\]\n\t\\[ \\mathtt{NotSQUARE_n}(\\nu) = ~~~~ \\left\\lbrace \\begin{array}{lll}\n\t\t0 & , & \\mbox{if } (\\#_0(\\nu))^2 = \\#_1(\\nu) \\\\\n\t\t1 & , & \\mbox{otherwise }\n\\end{array}\t \\right.,  \\]\n\t\\[  \\mathtt{NotPOWER_n}(\\nu) = \\left\\lbrace \\begin{array}{lll}\n\t\t0 & , & \\mbox{if } 2^{\\#_0(\\nu)} = \\#_1(\\nu) \\\\\n\t\t1 & , & \\mbox{otherwise }\n\\end{array}\t \\right..  \\]\n\n\n\\begin{theorem}\n\tBoolean Functions $ \\mathtt{NotO_n} $, $ \\mathtt{NotSQUARE_n} $, and $ \\mathtt{NotPOWER_n} $ can be solved by a nondeterministic quantum OBDD with constant width.\n\\end{theorem}\n\nFor all these three functions, we can define nondeterministic quantum (stable ID-) OBDD with constant width based on nondeterministic QFAs for languages $ O $, $SQUARE$, and $POWER$, respectively \\cite{YS10A}. \n\n\nThe complements of $ PAL, O, SQUARE $ and $POWER$ cannot be recognized by classical nondeterministic finite automata. But, for example, the function version of the  complement of $PAL$, $ \\mathtt{NotPAL_n} $, which returns 1 only for the non-palindrome inputs, is quite easy since it can be solved by a deterministic OBDD with width $3$. Note that the order of such an OBDD is not natural $(1,\\dots,n)$. However, as will be shown soon, this is not the case for the function versions of the complements of the other three languages.\n\n\\begin{theorem}\\label{nondetermenisticBoundTheorem}\nThere are infinitely many $n$ such that any NOBDD $P_n$ computing $ \\mathtt{NotO_n} $ has width at least $\\lfloor\\log n\\rfloor -1 $.\n\\end{theorem}\n\nThe proof of Theorem is based on the complexity properties of Boolean function $\\mathtt{NotO_n}$. At first we will discuss complexity properties of this function in Lemma \\ref{leqN}. After that we will prove claim of Theorem.\n\t\n\\begin{lemma}\\label{leqN}\nThere are infinitely many $n$ such that any OBDD computing $ \\mathtt{NotO_n} $ has width at least $n\/2 + 1$. (For the proof see \\ref{app-leqN}).\n\\end{lemma}\n{\\em Proof of Theorem \\ref{nondetermenisticBoundTheorem}}\\quad\\quad\n Let function $\\mathtt{NotO_n}$ is computed  by  $NOBDD$ $P_n$ of  width $d$, then by the same way as in the proof of Theorem \\ref{th-NOBDD1} we have\n$d\\geq\\log (n\/2 + 1)>\\log n -1.$\n\\hfill$\\Box$\\\\\n\nBy the same way we can show that there are infinitely many $n$ such that any NOBDD $P_n$ computing function $\\mathtt{NotSQUARE_n}$ has width at least  $ \\Omega(\\log (n)) $ and any NOBDD $P'_n$ computing function $\\mathtt{NotPOWER_n}$ has width at least  $ \\Omega(\\log\\log (n)) $.\n\n\n\\section{Hierarchies for Deterministic and Nondeterministic OBDDs}\n\nWe denote $\\mathsf{OBDD^d}$ and $\\mathsf{NOBDD^d}$ to be the sets of Boolean functions that can be computed by $OBDD$ and $NOBDD$ of width $d=d(n)$, respectively, where $n$ is the number of variables.\nIn this section, we present some width hierarchies for $\\mathsf{OBDD^d}$ and $\\mathsf{NOBDD^d}$. Moreover, we discuss relations between these classes.\nWe consider $\\mathsf{OBDD^d}$ and $\\mathsf{NOBDD^d}$ with small (sublinear) widths and large widths.\n\n\\subsection{Hierarchies and relations for small width OBDDs}\n\n \nWe have the following width hierarchy for deterministic and nondeterministic models.\n\n\\begin{theorem}\\label{smallwh}\nFor any integer $n$, $d=d(n)$, and $1<d\\leq n\/2$, we have\n\\begin{eqnarray}\n\\mathsf{OBDD^{d-1}} & \\subsetneq  & \\mathsf{OBDD^{d}}\\label{st1s} \\mbox{ and}\n\\\\\n\\mathsf{NOBDD^{d-1}} & \\subsetneq &  \\mathsf{NOBDD^{d}}\\label{st2s} .\n\\end{eqnarray}\n\\end{theorem}\n\n\\paragraph{Proof of Theorem \\ref{smallwh}.}\n\nIt is obvious that $\\mathsf{OBDD^{d-1}}\\subseteq \\mathsf{OBDD^{d}}$ and $\\mathsf{NOBDD^{d-1}}\\subseteq \\mathsf{NOBDD^{d}}$. Let us show the inequalities of these classes. For this purpose we use the complexity properties of known Boolean function $\\mathtt{MOD^k_{n}}$. \n\nLet $k$ be a number such that $1<k\\leq n\/2$. For any given input $\\nu\\in\\{0,1\\}^n$,\n\n\\begin{displaymath}\n\\mathtt{MOD^k_{n}}(\\nu) = \\left\\{ \\begin{array}{ll}\n1, & \\textrm{if } \\#_{1}(\\nu)= 0\\textrm{ } (mod\\textrm{ } k), \\\\\n0, & \\textrm{otherwise}\n\\end{array} \\right. .\n\\end{displaymath}\n\n\\begin{lemma}\\label{peq2s}\n\nThere is an OBDD (and so a NOBDD) $P_n$ of width $d$ which computes Boolean function $\\mathtt{MOD^k_{n}}$ and $d=k$.\n\\end{lemma}\n\n{\\em Proof}\\quad\\quad \nIn each level, $P_n$ counts  number of $1$'s by modulo $k$. $P_n$ answers $1$ iff  the number in the last step is zero. It is clear that the width of $P_n$ is $k$. \n\\hfill$\\Box$\\\\\n\n\\begin{lemma}\\label{peq1s}\nAny OBDD and NOBDD computing $\\mathtt{MOD^k_{n}}$ has width at least $ k$.\n\\end{lemma}\n\n\n{\\em Proof}\\quad\\quad\nThe proof is based on Pigeonhole principle. Let $P$ be a deterministic OBDD computing the function $ \\mathtt{MOD^k_{n}} $. \nFor each input $\\nu$ from $ (\\mathtt{MOD^k_{n}})^{-1}(1) $ there must be exactly one path in $P$ leading from source node to accepting node.  Let us consider $k$ inputs $\\{\\nu^1, \\nu^2,\\dots, \\nu^{k} \\}$ from this set such that the last $k$ bits in $\\nu^j (j=1,\\dots, k)$ contains exactly $j$ 1's and $(k-j)$ 0's. Let us consider the $(n-k)$-th level of $P$.\nThe acceptance paths for different inputs from $\\{\\nu^1, \\nu^2,\\dots, \\nu^{k} \\}$ must pass trough  different nodes  of \n the $(n-k)$-th level of $P$. So the width of the $(n-k)$-th level of $P$ is at least $k$.\n\nThe proof for nondeterministic case is similar to deterministic one. For each input from  $ (\\mathtt{MOD^k_{n}})^{-1}(1) $  for the function $ \\mathtt{MOD^k_{n}} $ there must be  at least one path in $P$ leading from the source node to an accepting node labelling this input.  The accepting paths for different inputs from the set $\\{\\nu^1,\\nu^2,\\dots, \\nu^{k} \\}$  must go through different nodes of $(n-k)$-th level of $P$.\n\\hfill$\\Box$\\\\\n\nBoolean function $\\mathtt{MOD^d_{n}} \\in \\mathsf{OBDD^{d}}$ and  $\\mathtt{MOD^d_{n}} \\in \\mathsf{NOBDD^{d}}$ due to Lemma \\ref{peq2s} and Boolean function $\\mathtt{MOD^d_{n}} \\not\\in \\mathsf{OBDD^{d-1}}$ and $\\mathtt{MOD^d_{n}} \\not\\in \\mathsf{NOBDD^{d-1}}$ due to  Lemma \\ref{peq1s}.\nThis  completes the proof of the Theorem \\ref{smallwh}. \\qed\n\nWe have the following relationships between deterministic and nondeterministic models.\n\n\\begin{theorem}\\label{smallwh2}\nFor any integer $n$, $d=d(n)$, and $d'=d'(n)$ such that $d\\leq n\/2$ and $O(\\log^2 d \\log\\log d)<d'\\leq d-1$, we have\n\\begin{eqnarray}\n\\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}} \\subsetneq  \\mathsf{OBDD^{d}} \\mbox{ and} \\label{st4s}\n\\\\\n \\mathsf{OBDD^{d}} \\textrm{ and }   \\mathsf{NOBDD^{d'}} \\textrm{ are not comparable.} \\label{st3s}\n\\end{eqnarray}\n\n\\end{theorem}\n\n\\paragraph{Proof of Theorem \\ref{smallwh2}}\nWe start with (\\ref{st4s}). \nBy the same way as in the proof of Theorem \n\\ref{th-NOBDD1}, we can show that $\\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}\\subseteq\\mathsf{OBDD^{d}}$ and, from Lemma \\ref{peq1s},  we know that $\\mathtt{MOD^d_{n}} \\notin \\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}$. Then we have $\\mathsf{OBDD^{d}}\\neq \\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}$.\n\nWe continue with (\\ref{st3s}). Let $k$ be even and $1<k\\leq n$. For any given input $\\nu\\in\\{0,1\\}^n$,\n\\begin{displaymath}\n\\mathtt{NotO^k_{n}}(\\nu) = \\left\\{ \\begin{array}{ll}\n0, & \\textrm{if } \\#^k_{0}(\\nu)= \\#^k_{1}(\\nu)=k\/2 \\\\\n1, & \\textrm{otherwise}\n\\end{array} \\right. .\n\\end{displaymath}\nNote that function $\\mathtt{NotO^n_{n}}$ is identical to $\\mathtt{NotO_{n}}$.\n\\begin{lemma}\\label{pneq1s}\nAny OBDD computing $\\mathtt{NotO^k_{n}}$ has width at least $ k\/2+1$.\n\\end{lemma}\n{\\em Proof}\\quad\\quad \nThe proof can be followed by the same technique given in the proof of Lemma \\ref{leqN}.\n\\hfill$\\Box$\\\\\n\\begin{lemma}\\label{pneq4s}\nThere is NOBDD $P_n$ of width $d$ that computes Boolean function $\\mathtt{NotO^k_{n}}$ and $d\\leq O(\\log^2 k \\log\\log k)$.  (For the proof see \\ref{app-pneq4s}).\n\\end{lemma}\n\nRemember that $O(\\log^2 d \\log\\log d)\\leq d'\\leq d-1$, and, by Lemma \\ref{peq2s} and Lemma \\ref{peq1s}, we have $\\mathtt{MOD^d_{n}} \\in \\mathsf{OBDD^{d}}$ and $\\mathtt{MOD^d_{n}} \\notin\\mathsf{NOBDD^{d'}}$; by Lemma \\ref{pneq4s}, we have $\\mathtt{NotO_{n}^{2d-1}}\\in\\mathsf{NOBDD^{d'}}$; and, by Lemma \\ref{pneq1s}, we have $\\mathtt{NotO_{n}^{2d-1}} \\notin \\mathsf{OBDD^{d}}$. Therefore, we cannot compare these classes and so we can follow Theorem \\ref{smallwh2}. \\qed\n\n\n\\subsection{Hierarchies and relations for large width OBDDs}\n\nIn this section, we consider OBDDs of large width. We obtain some hierarchies which are different from the ones in the previous section (Theorem \\ref{smallwh}). \n\n\n\\begin{theorem}\\label{hi1}\nFor any integer $n$, $d=d(n)$, $16\\leq d \\leq 2^{n\/4}$, we have\n\\begin{eqnarray}\n\\mathsf{OBDD^{\\lfloor d\/8 \\rfloor-1}}  \\subsetneq  \\mathsf{OBDD^{d}} \\mbox{ and }\\label{st1}\\\\\n\\mathsf{NOBDD^{\\lfloor d\/8 \\rfloor-1}}  \\subsetneq  \\mathsf{NOBDD^{d}},\\label{st2}\n\\end{eqnarray}\n\n\\end{theorem}\n\n\\paragraph{Proof of Theorem \\ref{hi1}.}\n\nIt is obvious that $\\mathsf{OBDD^{\\lfloor d\/8 \\rfloor-1}}\\subseteq \\mathsf{OBDD^{d}}$ and $\\mathsf{NOBDD^{\\lfloor d\/8 \\rfloor-1}} $ $ \\subseteq  \\mathsf{NOBDD^{d}}$. \n\n We define Boolean function $\\mathtt{EQS^k_{n}}$ as a modification of  Boolean function  {\\em Shuffled Equality} which was defined in \\cite{ak96} and \\cite{A97}. The proofs of inequalities are based on the complexity properties of $\\mathtt{EQS^k_{n}}$.\n\nLet $k$ be multiple of $4$ such that $4\\leq k\\leq 2^{n\/4}$. The Boolean Function $\\mathtt{EQS_{n}}$ depends only on the first $k$ bits. \n\nFor any given input $\\nu\\in\\{0,1\\}^n$, we can define two binary strings $\\alpha(\\nu)$ and $\\beta(\\nu)$ in the following way. We call odd bits of the input {\\em marker bits} and even bits  {\\em value bits}. For any $i$ satisfying $1\\leq i \\leq k\/2$, the value bit $\\nu_{2i}$ belongs to $\\alpha(\\nu)$  if the corresponding marker bit $\\nu_{2i-1}=0$ and $\\nu_{2i}$ belongs to $\\beta(\\nu)$ otherwise. \n\n\\begin{displaymath}\n\\mathtt{EQS^k_{n}}(\\nu) = \\left\\{ \\begin{array}{ll}\n1, & \\textrm{if } \\alpha(\\nu)= \\beta(\\nu)\\\\\n0, & \\textrm{otherwise}\n\\end{array} \\right. .\n\\end{displaymath}\n\n\\begin{lemma}\\label{peq1}\n\nThere is OBDD $P_n$ of width $8\\cdot 2^{k\/4}-5$ which computes Boolean function $\\mathtt{EQS^k_{n}}$. (For the proof see \\ref{app-peq1}).\n\\end{lemma}\n\n\n\\begin{lemma}\\label{peq2}\nThere are infinitely many $n$ such that any OBDD and NOBDD $P_n$ computing $ \\mathtt{EQS^k_{n}} $ has width at least $2^{k\/4}$. (For the proof see \\ref{app-peq2}). (For the proof see \\ref{app-peq2})\n\\end{lemma}\n\nBoolean function $\\mathtt{EQS^{4\\lceil\\log (d+5)\\rceil-12}_n} \\in \\mathsf{OBDD^{d}}$ and  $\\mathtt{EQS^{4\\lceil\\log (d+5)\\rceil-12}_n} \\in \\mathsf{NOBDD^{d}}$ due to Lemma \\ref{peq1}.\n\nBoolean function $\\mathtt{EQS^{4\\lceil\\log (d+5)\\rceil-12}_n} \\not\\in \\mathsf{OBDD^{\\lfloor d\/8 \\rfloor-1}}$ and $\\mathtt{EQS^{4\\lceil\\log (d+5)\\rceil-12}_n} \\not\\in \\mathsf{NOBDD^{\\lfloor d\/8 \\rfloor-1}}$ due to Lemma \\ref{peq2}. So $\\mathsf{OBDD^{\\lfloor d\/8 \\rfloor-1}}\\neq \\mathsf{OBDD^{d}}$ and $\\mathsf{NOBDD^{\\lfloor d\/8 \\rfloor-1}} $  $ \\neq \\mathsf{NOBDD^{d}}$. These inequalities prove Statements (\\ref{st1}) and (\\ref{st2}) and complete the proof of the Theorem \\ref{hi1}. \\qed\n   \nIn the following theorem, we present a relationship between deterministic and nondeterministic models.\n\n\\begin{theorem}\\label{hi3}\nFor any integer $n$, $d=d(n)$, and $d'=d'(n)$ satisfying $ d\\leq 2^{n\/4}$ and $O(\\log^4 (d+1) \\log\\log (d+1))<d'<d\/8-1$,  we have\n\\begin{eqnarray}\n\\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}} \\subsetneq  \\mathsf{OBDD^{d}}\\mbox{ and } \\label{st4}\\\\\n\\mathsf{OBDD^{d}} \\textrm{ and }   \\mathsf{NOBDD^{d'}} \\textrm{ are not comparable.}\\label{st3}\n\\end{eqnarray}\n\n\\end{theorem}\n\n\\paragraph{Proof of Theorem \\ref{hi3}.}\nWe start with (\\ref{st4}). By the same way as in proof of Theorem \n\\ref{th-NOBDD1}, we can show that $\\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}\\subseteq\\mathsf{OBDD^{d}} $. By Lemma \\ref{peq2},  we have $\\mathtt{EQS_n^{4\\lceil\\log (d+5)\\rceil-12}} \\notin \\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}$ which means $\\mathsf{OBDD^{d}} $ $ \\neq \\mathsf{NOBDD^{\\lfloor\\log (d)\\rfloor}}$.\n\nNow we continue with (\\ref{st3}).\nWe use the complexity properties of Boolean function $\\mathtt{NotEQS^k_{n}}$, which is the inversion of $\\mathtt{EQS^k_{n}}$. \n\n\n\n\\begin{lemma}\\label{pneq2}\nThere are infinitely many $n$ such that any OBDD $P_n$ computing $ \\mathtt{NotEQS^k_{n}} $ has width at least $2^{k\/4}$.\n\\end{lemma}\n\n{\\em Proof}\\quad\\quad\nWe can prove it by the same way as in the proof of Lemma \\ref{peq2}.\n\\hfill$\\Box$\\\\\n\n\\begin{lemma}\\label{pneq1}\nThere is a NOBDD $P_n$ of width $d$ computing Boolean function $\\mathtt{NotEQS^k_{n}}$ where $d\\leq O(k^4 \\log k)$. (For the proof see \\ref{app-pneq1}).\n\\end{lemma}\n\nRemember that $O(\\log^4 (d+1) \\log\\log (d+1))\\leq d'\\leq \\lfloor d\/8\\rfloor-1$, and, by Lemma \\ref{peq2} and Lemma \\ref{peq1}, we have $\\mathtt{EQS_n^{4\\lceil\\log (d+5)\\rceil-12}} \\in \\mathsf{OBDD^{d}}$ and $\\mathtt{EQS_n^{4\\lceil\\log (d+5)\\rceil-12}} \\notin\\mathsf{NOBDD^{d'}}$; by Lemma \\ref{pneq1}, we have $\\mathtt{NotEQS_n^{4\\lceil\\log(d)\\rceil+4}}\\in\\mathsf{NOBDD^{d'}}$; and, by Lemma \\ref{pneq2}, we have $\\mathtt{NotEQS_n^{4\\lceil\\log(d)\\rceil+4}} \\notin \\mathsf{OBDD^{d}}$. Therefore we cannot compare this classes and so we can follow Theorem \\ref{hi3}. \\qed\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\n\\newmaterial{\nThe privacy and security of Bitcoin have been extensively\nstudied~\\cite{BKP14,KKM14,MPJLMVS13,AKRSC13,MGGR13,BSCG0MTV14,BNMCKF14} and\ndocumented~\\cite{BMCNKF15}. While the focus of previous work has been on the\narchitectural vulnerabilities of the cryptocurrency, no work has been done\nto analyze implementation vulnerabilities of the Stratum mining protocol, the\nmain Bitcoin mining option.}\n\n\nHowever, mining activities have important privacy\nimplications~\\cite{MiningSecret}.\n\\newmaterial{\nLearning the payouts of miners can make them targets of hacking and coin\ntheft~\\cite{BitcoinTheft,ListBitcoinTheft},\nkidnapping~\\cite{williams2011kidnappings}, and, in countries where Bitcoin is\nillegal~\\cite{WikiIllegal,BitcoinIllegal}, expose them to arrest and equipment\nconfiscation~\\cite{BitcoinArrests,BitcoinCrackdown}. For instance, in\ncountries like Venezuela, intelligence police hunt Bitcoin miners to extort,\nsteal mining equipment or prosecute~\\cite{VenezuelaBitcoin}.}\n\nIn this paper we study the vulnerabilities of the \\textit{Stratum}\nprotocol~\\cite{stratum_proto}, the de facto mining standard for pooled Bitcoin\nmining~\\cite{stratum_GBT} as well as alternative coins mining, e.g.,\nLitecoin~\\cite{litecoin_stratum}, Ethereum~\\cite{ethereum_stratum} and\nMonero~\\cite{monero_stratum}; currently the altcoins with the most market\ncapitalization \\cite{crypto_market_cap}. Stratum replaced the original\n\\textit{getwork} protocol of Bitcoin mining~\\cite{stratumDescription}, to\nenable miners to fetch jobs from mining pool servers more reliably and\nefficiently. In Stratum, the miners solve assigned {\\it jobs} and send their\nresults back in the form of \\textit{shares}.  The mining pool server then\ncompensates the miner according to the difficulty of the assigned jobs and the\nnumber of shares accepted.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{stratum}\n\\caption{Model of system that consists of a pool, miners, and adversary.  The\npool and the miners communicate over the Stratum protocol, to assign jobs and\nsubmit results (shares).  The adversary can eavesdrop, recover ISP logs, inject\nand modify the Stratum communications of victim miners.}\n\\label{fig:system}\n\\vspace{-15pt}\n\\end{figure}\n\n\nWe show that the lack of cryptographic protection of communications has made\nthe Stratum protocol vulnerable to several exploitation possibilities, see\nFigure~\\ref{fig:system}. An attacker able to observe the traffic between a\nminer and a pool server can accurately infer the earnings of the miner. We show\nthat this result holds even if the attacker has only limited access to the\ntransmitted packets, e.g., metadata stored in ISP logs.  In addition, we show\nthat active attackers, able to interfere with the Stratum traffic of miners,\nmay steal computational resources and bring forth financial loss to their\nvictims.\n\nThese attacks, especially given the wide adoption of the Stratum protocol, show\nthat Bitcoin and altcoin solutions fail to ensure the monetary privacy and\nsecurity of the vital miner community. Furthermore, the attacks reveal that\neven an exhaustive use of encryption will fail to ensure miner privacy,\nas access to only the timestamp of mining protocol traffic can enable an\nattacker to predict the payouts of a victim miner. In addition, the significant\noverhead of encryption makes such a solution unappealing to pools, that need to\nhandle mining traffic from thousands of miners simultaneously, e.g., more than\n16,000 for the F2Pool pool~\\cite{MinerCount,Corti,F2PoolShare}. In\n$\\S$~\\ref{sec:eval:bedrock} we show that complete encryption of all Stratum\ntraffic imposes a daily overhead of 1.36 hours on a pool server handling 16,000\nminers, while TLS imposes a daily overhead of 1.01 hours.\n\n\\newmaterial{\nFurthermore, Tor does not address the above vulnerabilities. In fact, sending\nStratum traffic over Tor would enable an adversary to launch the ISP Log attack\nnot only from the same network with the victim, but also from adversary\ncontrolled Tor exit nodes, that can inspect the cleartext Stratum traffic to\nthe destination.  Also, Bitcoin over Tor has been shown to be vulnerable to\nseveral attacks~\\cite{BP15}, and, even without an adversary, Tor may introduce\ndelays that can lead to miners losing blocks.}\n\n\\noindent\n{\\bf Our Contributions}.\nIn this paper we introduce the following contributions:\n\n\\begin{compactitem}\n\n\n\n\\item\n{\\bf Passive attacks}.\nWe show that F2Pool's Stratum implementation leaks sensitive miner information\nnot only through cleartext communications but also indirectly, through hashrate\ndependent inter-packet timing.  We introduce {\\it StraTap} and {\\it ISP Log},\npassive attacks where adversaries accurately infer the earnings of victim\nminers, given either captured transmissions of those miners, or only their log\nmetadata.\n\n\\item\n{\\bf Active attack}.\nWe propose \\textit{BiteCoin}, a payout hijack attack that enables an adversary\nable to access the communications of a victim miner, to steal its resources and\nmining payouts.  To implement BiteCoin, we have developed {\\it WireGhost}, a\nTCP hijacking tool that surreptitiously modifies TCP packets, without imposing\ndisconnections or session resets. \n\n\\item\n{\\bf Bedrock}.\nWe develop {\\it Bedrock}, a Stratum extension that addresses the proposed\nattacks.  Bedrock seeks to assuage the efficiency concerns of Bitcoin, by\nimposing minimal modifications and encryption overhead to the Stratum protocol.\nWe introduce the concept of {\\it mining cookies}, secret values that miners\nneed to include in the computed puzzles. Mining cookies prevent both passive\nand active attacks on share submission packets, without encrypting the vast\nmajority of the pool communications.\n\n\\item\n{\\bf Results}.\nWe have collected 138MB of Stratum traffic traces from mining equipment in the\nUS and Venezuela, and release it for public use~\\cite{StratumData}. We have\nimplemented the developed attacks and report results from their deployment on\nAntMiner mining equipment.  We show that StraTap and ISPLog achieve low payout\nprediction errors, and that BiteCoin can efficiently hijack job assignments and\nshare submissions from a victim miner. We show that Bedrock prevents these\nattacks, and is resilient to active attacks even when the adversary breaks its\ncrypto tools. Bedrock imposes a 12.03s daily overhead on a single pool server\nthat handles 16,000 miners simultaneously.\n\n\\end{compactitem}\n\n\\noindent\nThe attacks and defenses introduced in this paper apply to the Stratum\nprotocol, thus to most of the large mining\npools~\\cite{f2pool_help,AntPool,GHash.io,SlushPool,BTCC}. These attacks work\neven on miners that are behind a NAT, or that are firewalled.  Furthermore,\nwhile we focus our experiments on the popular AntMiner mining equipment, our\nattacks are general and apply to other manufacturers as well. \\newmaterial{We\nhave notified F2Pool about these vulnerabilities.}\n\n\\vspace{-5pt}\n\n\\section{Model and Background}\n\\label{sec:model}\n\n\nThe Bitcoin mining ecosystem consists of miners and pools, see\nFigure~\\ref{fig:system}. The communication between pools and miners takes place\nalmost exclusively over the Stratum, that we study in the next section.  The\nmain task of Bitcoin miners (or mining nodes) is to permanently insert new\nconsistent data into the network. Miners collect transaction data from other\nnodes, validate it and insert it in a structure called \\textit{block}. The\nminers need to solve a cryptographic puzzle based on the block, before the\nblock is permanently inserted into the storing structure of the network, called\n\\textit{the blockchain}.\n\n\\subsection{Mining Pools}\n\\label{sec:model:pool}\n\n\\newmaterial{\nAs specialized, more powerful miners were designed, and the difficulty of\nmining blocks increased, it became increasingly difficult for individual, {\\it\nsolo} miners to successfully mine and receive timely payouts for their work.\nThe concept of {\\it mining pools} has emerged in order to address this problem:\nenable miners to combine their resources, then split the reward according to\nthe amount of work they have performed. Pools and miners form a master\/slave\nparadigm, where pools parallelize the mining work among multiple miners. Thus,\ninstead of randomly receiving a large reward only once several years, pooled\nminers receive smaller rewards, on a regular basis (e.g., once per day).}\n\n\\newmaterial{\nPopular mining pools today interact with thousands of miners. For instance,\nF2Pool has more than 16,000 miners~\\cite{MinerCount,Corti,F2PoolShare}. As per\nthe Bitcoin specification, about 150 blocks are to be mined every day, i.e., 1\nblock every 10 minutes. Since each block is, at the time of writing, worth 12.5\nBTC , at an exchange rate of $\\approx$ \\$1,100 per BTC, about 2 million dollars\nare distributed each day.  The larger mining pool servers control about 20\\% of\nthese earnings~\\cite{PoolDistribution}.}\n\n\\newmaterial{\nThe mining pool rewards a {\\it share} to the miner who reports a proof of a\nunit of solved work.  Pools often offer a {\\it variable share difficulty}\nfeature: adaptively assign the share target to miners, according to their\ncomputation ability. Pools perform this in order to ensure that (i) the\nassigned work is not too difficult, thus enable miners to prove computation\nprogress and gain regular payouts, and that (ii) the work is not too easy, thus\nreduce the overhead on the pool, to process shares submitted by thousands of\nminers.} \n\n\\noindent\n{\\bf Hashrate to BTC conversion}.\nThe pool will reward the miner according to the number of shares submitted and\naccepted.  Each pool has a different payout policy. One of the most popular\npolicies is ``Pay Per Share'' (PPS) which usually has a pool fee attached to\nit. Since the final calculation for share payment is dependent on the actual\nmining activity of the pool, it is difficult to provide an exact estimate of\neach share payout.  However, most pools publish a hashrate to BTC (Bitcoin)\nrate of conversion for miner verification purposes. At the time of writing this\npaper, the conversion rate on the F2Pool was 0.00246248 BTC per\nTH\/s~\\cite{f2pool_help}.\n\n\\subsection{The Coinbase Transaction}\n\\label{sec:model:coinbase}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{coinbase}\n\\caption{Coinbase transaction format. The fields in the table are the\nunderlying Bitcoin transaction fields. Stratum overlays the coinbase fields\n($coinbase1$, $extranonce1$, $extranonce2$, $coinbase2$) on top of the Bitcoin\ncoinbase fields. Bedrock (see $\\S$~\\ref{sec:bedrock}) introduces the\n{\\it mining cookie} concept, whose value will overwrite the currently\nunused ``previous hash'' field within $coinbase1$.}\n\\label{fig:coinbase}\n\\vspace{-15pt}\n\\end{figure}\n\nAs mentioned above, each block collects a set of previous transactions in the\nBitcoin network. The first such transaction is special, called the {\\it\ncoinbase} transaction (see Figure~\\ref{fig:coinbase}): it specifies that the\npool will receive the value of this block (currently 12.5 Bitcoins) when this\nblock is mined. The Stratum coinbase consists of 4 fields, {\\it coinbase1},\n{\\it extranonce1}, {\\it extranonce2}, {\\it coinbase2}, overlayed on top of the\nBitcoin coinbase information. $coinbase1$ covers the first 5 fields  of the\ninput transaction (version, input count, previous transaction hash, previous\ntransaction index and input scriptlen) and part of the script, in the original\ncoinbase transaction specification. Except for the version number, these\nparameters are meaningless to all clients and pools since the coinbase\ntransaction does not have an input transaction.\n\nThe $extranonce1$ and $extranonce2$ fields are also overlayed on the unused\nscript.  $extranonce1$ needs to be unique (pseudo-random) per stratum\nconnection.  $extranonce2$ is used in the mining puzzle and needs to be\nincremented by the miner once the $nonce$ parameter is exhausted (see next\nparagraph). The rest of the coinbase transaction is packed in the $coinbase2$\nparameter.\n\nBedrock, our secure Stratum extension, leverages the unused ``previous\ntransaction hash'' field (Figure~\\ref{fig:coinbase}), to include the value of\nthe mining cookie, see $\\S$~\\ref{sec:bedrock}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{puzzle}\n\\vspace{-15pt}\n\\caption{Bitcoin block puzzle. The root of the Merkle tree built over\nthe coinbase and the mined transactions is the third field of the block\npuzzle. The miner iterates over the $nonce$ (last field) and over the\nextranonce2 value, part of the coinbase transaction.}\n\\label{fig:puzzle}\n\\vspace{-10pt}\n\\end{figure}\n\n\\subsection{The Bitcoin Puzzle}\n\\label{sec:model:puzzle}\n\nThe goal of the mining process is to make it difficult for a minority of\nmalicious nodes to insert bogus data inside invalid blocks. It achieves this by\ntransforming each block (collection of transactions in the Bitcoin network)\ninto a cryptographic ``puzzle''. The puzzle is designed such that the\nprobability of finding a solution by a mining node is proportional to its\ncomputational power.  A Bitcoin puzzle consists of a $target$ value and a\ntuple F = (block version number || hash of previous block || RMT ||\ntimestamp || Nbits ), || denotes concatenation, see\nFigure~\\ref{fig:puzzle} for an illustration.  \n\n\nSpecifically, F contains the block version number, the hash of the previous\nblock in the blockchain, the root of a Merkle tree (RMT) described next, a\ntimestamp, and the final target value in the form of the number of leading bits\nthat need to be 0 for the block to be considered ``mined''. The\nMerkle tree is built over the transactions that are being mined into the\ncurrent block, including the coinbase transaction, see Figure~\\ref{fig:puzzle}.\nGiven the F value, and the above mentioned $target$, the miner iterates\nover the $nonce$ and $extranonce2$ (see coinbase transaction) values until it\nidentifies a pair such that\n\\vspace{-5pt}\n\n\\begin{equation}\nH^2(nonce || F) < target\n\\label{eq:puzzle}\n\\end{equation}\n\n\\noindent\nwhere $H^2$ denotes the double (SHA-256) hash. The block is said to be\n``mined'' when $H^2(nonce || F)$ is less than the target corresponding to the\nabove Nbits value.\n\n\\noindent\n{\\bf The target and the difficulty}.\nWhile the $Nbits$ value specifies when the block is mined, pools set the above\n$target$ parameter to a larger value (fewer leading bits 0) to enable miners\nto prove and be rewarded for progress. The {\\it target of difficulty 1},\ndenoted $target\\_1$, is defined by pools as the number $2^{224} - 1$, i.e. a\n256 bit number with 32 bits of leading zeros followed by 224 bits of ones.  The\n\\textit{difficulty} value is a measure of how hard it is to solve a puzzle for\na given $target$ value.  Accordingly, the relationship between the $difficulty$\nand $target$ values is given by the formula:\n\\vspace{-5pt}\n\n\\begin{equation}\ndifficulty = \\frac {target\\_1}{target} = \\frac{2^{224} - 1}{target}\n\\label{eq:difficulty}\n\\end{equation}\n\n\\vspace{-5pt}\n\n\\subsection{\\newmaterial{Stratum}}\n\\label{sec:stratum}\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{timeline_650}\n\\caption{Stratum protocol timeline over 24 hours captured between an AntMiner\nS7 and the F2Pool pool. While we observe several difficulty change packets\n(orange bars) throughout the day, most are concentrated after the initial\nsubscription protocol (tall blue bar). The majority of share submissions\nare accepted (green bars); only a few are rejected (red bars) or ignored\n(black bars).}\n\\label{fig:timeline:general}\n\\vspace{-15pt}\n\\end{figure}\n\nStratum is a clear text communication protocol between the pool and the\nminers~\\cite{stratum_proto}, built over TCP\/IP and using the JSON-RPC format.\n\\newmaterial{\nThe official Stratum protocol documentation is not detailed and is often\noutdated~\\cite{stratum_proto}.  In this section we describe the\nF2Pool~\\cite{f2pool_help} implementation of the Stratum protocol, as observed\nfrom Stratum packets we captured over 13 days between an AntMiner S7 device and\nthe F2Pool mining pool (see $\\S$~\\ref{sec:implementation:passive}).}\nFigure~\\ref{fig:timeline:general} illustrates the timeline of captured Stratum\nprotocol packets over a 24 hour interval. The ability to capture, understand,\nmodify and inject these messages into a communication stream will be\ninstrumental for the passive and active attacks described in\n$\\S$~\\ref{sec:attacks:passive} and $\\S$~\\ref{sec:attacks:active}.\n\n\n\n\\noindent\n{\\bf Miner subscription}.\n\\newmaterial{\nTo register with the pool, the miner first subscribes through a\n\\textbf{connection subscription} message\n\\vspace{-5pt}\n\\[\n{\\tt mining.subscribe,\\ params},\n\\]\nthat describes the miner capabilities. The server responds with a\n\\textbf{subscription response} message,\n\\vspace{-5pt}\n\\[\n{\\tt result, \\{methods\\},\\ extranonce1,\\ extranonce2.size},\n\\]\nwhere the first field is a list method names used by the server pool, the\nsecond field (see $\\S$~\\ref{sec:model:puzzle}), should be random and unique per\nconnection, but F2Pool sets to constant ``$\\backslash x30\\backslash x30$'', and\nthe third is the size of the {\\tt extranonce2} (4B in F2Pool).}\n\n\\noindent\n\\textbf{Miner authorization}.\n\\newmaterial{\nFollowing the subscription exchange, Stratum authenticates the miner with the\npool, through a \\textbf{miner authorization request} message\n\\vspace{-5pt}\n\\[\n{\\tt mining.authorize,\\ account.minerID,\\ password},\n\\]\nwhose first field, the username, consists of two fields, that enable a user to\nregister multiple miners with the same account. While the password\nfield is transferred in cleartext, it is currently ignored by pools.\nThe pool responds with a \\textbf{status result} that notifies the miner of the\nresult of the authorization request.} In Figure~\\ref{fig:timeline:general}\nand~\\ref{fig:timeline:rate}, the miner subscription and authorization messages\nare shown as a single blue bar (the tallest), seen at the beginning of the\ninterval and each time the miner reconnects to the pool, e.g., after an\nInternet disconnection or power outage.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{timeline_1h8m_250MHz_indicator}\n\\caption{Timeline of miner operating at 250MHz. We emphasize the effect\nof successive share difficulty notification messages: the miner's share submission\nrate slows down. We also point the multiple miner subscribe and authorization\nprocedures (tall blue bars) due to repeated miner disconnections.}\n\\label{fig:timeline:rate}\n\\vspace{-15pt}\n\\end{figure}\n\n\\noindent\n{\\bf Share difficulty notification}.\n\\newmaterial{\nFollowing a successful authorization, the pool sends a \\textbf{difficulty\nnotification} message to the miner\n\\vspace{-5pt}\n\\[\n{\\tt set.difficulty,\\ difficulty},\n\\]\nthat specifies the minimum share difficulty that the server will be willing to\naccept.}\nFigure~\\ref{fig:timeline:general} and~\\ref{fig:timeline:rate} show\nthe difficulty notification messages as orange bars. They can occur\nthroughout the day as the pool seeks to adjust the miner's rate of share\nsubmissions.\n\n\\noindent\n{\\bf Job assignment}.\n\\newmaterial{\nThe pool assigns jobs (puzzles) to the miner through \\textbf{mining job\nnotification} messages\n\\vspace{-5pt}\n\\[\n{\\tt mining.notify,\\ job\\_id,\\ params,\\ clean\\_jobs}\n\\]\nthat specify the puzzle parameters, i.e., the fields of the $F$ value in\nEquation~\\ref{eq:puzzle} (see $\\S$~\\ref{sec:model:puzzle}),\nand a boolean that indicates if the miner should drop all previous jobs and\nwork exclusively on the one specified by this message.}\n\n\n\\noindent\n\\textbf{Share submission}.\n\\newmaterial{\nOnce the miner finds a solution that satisfies Equation~\\ref{eq:puzzle}, it\nsends a {\\bf share submission} message to the pool for verification and credit:\n\\[\n{\\tt mining.submit,\\ account.minerID,\\ job\\_id,\\ time,}\n\\]\n\\vspace{-20pt}\n\\[\n{\\tt nonce,\\ extranonce2}\n\\]\nthat specifies the miner's username, the job id received in the previous mining\njob notification, and the parameters of the puzzle solution: the $nonce$ and\n$extranonce2$ parameters, see $\\S$~\\ref{sec:model:puzzle}.  The pool uses these\nvalues to reconstruct the $F$ value (see $\\S$~\\ref{sec:model:puzzle}), and\nverifies that Equation~\\ref{eq:puzzle} is satisfied.}\n\nThe pool responds with a status result message, illustrated in\nFigure~\\ref{fig:timeline:general}: green bars denote accepted shares, red bars\ndenote rejected shares, and black bars denote ignored shares. Shares can be\nrejected due to stale work, i.e., being submitted too late.\nThe miner continues to mine current jobs until it receives a job message from\nthe pool that requires it to invalidate previous jobs (see the ``clean jobs''\nflag in the job assignment message).\n\n\\vspace{-5pt}\n\n\\section{Adversary Model}\n\\label{sec:model:adversary}\n\nWe consider adversaries that can launch both passive and active attacks against\nthe Bitcoin network, see Figure~\\ref{fig:system}. We assume that the pool and\nthe miner are honest. However, adversaries can target the communications of\nspecific, victim miners. Adversaries can own mining equipment, can eavesdrop\nand interfere with existing communications, and may even obtain data from ISPs.\nWe now detail each of these adversarial capabilities.\n\n\\noindent\n{\\bf Eavesdropping capabilities}.\nWe consider first an adversary who can access the entire communication of a\nvictim miner. Such adversaries include over-controlling governments, or\nattackers who gain control to equipment on the same LAN as the victim. We\nassume that such an adversary can capture and inspect all the packets sent and\nreceived by the victim miner.\n\n\\noindent\n{\\bf ISP log capabilities}.\nWe also consider adversaries with access to ISP logs, that include entries for\nthe communications of the miners in the ISP's subnet and the pool. This\ncapability is more restrictive than the eavesdropping capability, in terms of\nthe data that can be extracted from the miner-to-pool communications. This is\nbecause ISP logs usually contain only metadata~\\cite{dataRetention}, in order\nto comply with law enforcement requests~\\cite{desimmone2010pitting}. However,\nthese capabilities may enable the adversary to target more victims (i.e., all\nthe miners whose traffic was logged).\n\nPotential perpetrators include insiders (e.g., ISP employees) and government\norganizations that can subpoena the logs.  While law enforcement insiders have\nbeen shown to abuse collected data~\\cite{nsaSpy}, agencies have also been\nhacked in the past. Stolen data may then be sold, auctioned, or even made\npublic, thus becoming accessible to a broader range of adversaries.\n\nWe assume that this adversary has access to packet metadata that includes\ntimestamps, source and destination IPs and ports, and connection flags. As we\nshow in $\\S$~\\ref{sec:attacks:passive:isp}, these values enable the attacker to\nidentify mining traffic via well known pool IP\/port pairs, and identify the\nstart of mining sessions.\n\n\\noindent\n{\\bf Active attack capabilities}.\nWe further consider an adversary that can modify the communication stream\nbetween the server pool and a mining client. Potential such adversaries include\nattackers that are on the same network as the victim miner, rogue employees at\nan intermediate ISP, or a government backed agency. In\n$\\S$~\\ref{sec:attacks:active}, we show that such an adversary can add jobs and\nreplace submitted shares.  While this may allow for trivial denial of service\nattacks as well, in this paper we do not consider DoS attacks. We note that DoS\ntechniques exist with lesser technical requirements~\\cite{caooff,qian2012off}.\n\n\\vspace{-5pt}\n\n\\subsection{Relevance of Attacks}\n\n\\newmaterial{\nIn the following section we introduce passive and active attacks against miners\nthat use the Stratum protocol, whose goal ranges from inferring to stealing the\npayouts of the victim miners. Inferring the payouts of miners exposes their\nowners to a suite of attacks. Adversaries can hack the computers or accounts of\nminer owners in order to steal their\npayouts~\\cite{BitcoinTheft,ListBitcoinTheft}.  Passive attacks can also enable\nthe adversary to identify miners worthy of being targeted with resource\nhijacking attacks, e.g., Bitecoin, see $\\S$~\\ref{sec:attacks:active}.  In\naddition, this knowledge enables adversaries to target miner owners for\nequipment theft, arson~\\cite{Arson}, kidnapping~\\cite{williams2011kidnappings},\nand, in countries where Bitcoin is illegal~\\cite{WikiIllegal,BitcoinIllegal},\nfor extortion and prosecution~\\cite{VenezuelaBitcoin}.}\n\n\\vspace{-5pt}\n\n\\section{Passive Attacks}\n\\label{sec:attacks:passive}\n\n\\vspace{-5pt}\n\nIn this section we show that an attacker that observes even partial traffic\nof a victim miner, can infer the payouts received by the miner. We introduce\ntwo passive attacks, that make different assumptions on the adversary's\ncapabilities. First, the StraTap attack assumes an adversary able to capture\nand inspect entire packets transmitted between a pool and a victim miner.\nSecond, the ``ISP Log'' attack assumes an adversary that is able to inspect\nonly packet metadata, i.e., IP addresses, port numbers, and connection flags.\nIn the following we detail each attack.\n\n\\vspace{-5pt}\n\n\\subsection{The StraTap Attack}\n\\label{sec:attacks:passive:full}\n\n\\vspace{-5pt}\n\nWe consider first an eavesdropping adversary, see\n$\\S$~\\ref{sec:model:adversary}.  Given access to all the packets sent and\nreceived by the victim miner, the attacker counts the share submission messages\nalong with their associated difficulty (as described in\n$\\S$~\\ref{sec:stratum}). The attacker uses this data to estimate the hashrate\nof the victim miner.\n\nSpecifically, the probability of randomly finding a hash with the appropriate\ndifficulty is given by the ratio between the target (i.e.  the number of hashes\nwith the appropriate number of leading zeros according to the assigned job),\nand the total number of possible hashes. Hence, the probability of a miner\nfinding a share \\newmaterial{with a single hash} is $p = \\frac{target}{ 2^{256}\n- 1}$.  The expected number of hashes, $E$, that the miner needs to calculate\nbefore finding a valid share is then $1\/p$. Then, we derive the following for\n$E$:\n\n\\begin{equation}\n\\begin{split}\nE &= \\frac{ 2^{256} - 1}{target\\_1}\\times \\frac{target\\_1}{target} = \n\\\\\n&= \\frac{ 2^{256} - 1}{2^{224} - 1} \\times difficulty \\approx difficulty \\times 2^{32}\n\\end{split}\n\\label{eq:e}\n\\end{equation}\n\n\\noindent\nThe second equality follows from Equation~\\ref{eq:difficulty}, and the fact\nthat the target of difficulty 1, $target\\_1$ is $2^{224} -1$, see\n$\\S$~\\ref{sec:model:puzzle}.\nIf we divide Equation~\\ref{eq:e} by the hashrate of the miner, we obtain a\nformula that allows us to compute the expected time to find a share at a given\ndifficulty:\n\n\\begin{equation}\n\\begin{split}\\\ntime &= \\frac{E}{hashrate} = \\frac{difficulty \\times 2^{32}}{hashrate}\n\\end{split}\n\\label{eq:expected_time_to_share}\n\\end{equation}\n\n\\noindent\nThus, $hashrate = difficulty \\times \\frac{2^{32}}{time}$.\nThe attacker obtains the $difficulty$ value by inspecting the \\textit{share\ndifficulty notification} messages (see $\\S$~\\ref{sec:stratum}).  In\naddition, the attacker estimates the $time$ value as the ratio of the length of\ntime between consecutive share difficulty notification messages (orange bars in\nFigure~\\ref{fig:timeline:general}) and the number of shares submitted and\naccepted during that interval: \\[ \\overline{time} = \\frac{\\text{total time\nbetween difficulty changes}}{\\text{number of submitted (and approved) shares}}\n\\] The attacker obtains the accepted share count by inspecting the share\nsubmission messages and their corresponding status results, see\n$\\S$~\\ref{sec:stratum}.\n\nGiven the inferred $hashrate$ of the miner, the attacker uses the hashrate to\npayout conversion (see the corresponding paragraph in $\\S$~\\ref{sec:stratum})\nto predict the amount of Bitcoins received by the victim. In\n$\\S$~\\ref{sec:eval:passive:stratap} we experimentally evaluate the accuracy\nprediction of the StraTap attack.\n\n\\subsection{The ISP Log Attack}\n\\label{sec:attacks:passive:isp}\n\nWe now consider an attacker with ISP log data capabilities, see\n$\\S$~\\ref{sec:model:adversary}.\nThe ISP Log attack proceeds as follows. First, the attacker identifies the\nbeginning of the connection between the victim miner and the pool. This is the\ntime when a 3-way handshake connection is established, whose first step is a\n``connection subscription'' message as described in $\\S$~\\ref{sec:stratum}.\nThen, the attacker predicts the hashrate of the miner based on statistics over\nthe inter-packet times logged for the miner.\n\n\nIn early experiments we have observed that predictors that use statistics over\nlong time intervals are inaccurate.  To address this problem, we have\nidentified and exploited a vulnerability of the Stratum protocol. Specifically,\nwe observed that the first share difficulty notification message (see\n$\\S$~\\ref{sec:stratum}) following a successful miner subscription, sets the\ndifficulty to the minimum value acceptable by the pool (e.g., 1024 for F2Pool).\nIn addition, in $\\S$~\\ref{sec:eval:passive:isp} we show that the pool sends its\nsecond share difficulty notification after approximately 50 share submission\nmessages (for difficulty 1024) received from the miner.\n\n\\begin{figure}[t]\n\\begin{sequencediagram}\n  \\newinst[0.01]{a}{Miner}\n  \\newinst[1.5]{m}{MITM}\n  \\newinst[1.6]{b}{Pool Server}\n\n  \\mess{a}{SEQ:x, LEN:y}{m}\n  \\mess{m}{SEQ:x, LEN: y + z }{b}\n  \\mess{b}{ACK: x + y + z}{m}\n  \\mess{m}{ACK: x+ y + z - z }{a}\n  \\mess{a}{SEQ:x+y}{m}\n  \\mess{m}{SEQ:x+y+z}{b}\n\\end{sequencediagram}\n\\caption{WireGhost illustration: TCP hijacking with active re-synchronization,\nwhen the man-in-the-middle (MITM) adversary adds $z$ bytes to an existing\npacket originating from the client. WireGhost modifies the sequence numbers, to\nhide the difference in packet sizes.}\n\\label{fig:active:tcp_hijack}\n\\vspace{-5pt}\n\\end{figure}\n\nThen, the attacker estimates the time taken by share submissions for jobs of\ndifficulty 1024, i.e., over the first 50 packets sent by the miner following\nits subscription and authorization process.  It then uses the process outlined\nin the above StraTap attack to predict the miner's hashrate and payout. The\nattacker can repeat this process when observing subsequent 3-way handshake\nconnection protocols of the victim miner, e.g., when a disconnection occurs, in\norder to improve its estimates of the miner hashrate. In\n$\\S$~\\ref{sec:eval:passive:isp} we show that even when the ISP Log attack\nperforms a single hashrate inference attack per day, its daily miner payout\nprediction achieves a mean percentage error of -9.49\\%.\n\n\\vspace{-10pt}\n\n\\section{The BiteCoin Attack}\n\\label{sec:attacks:active}\n\n\\vspace{-5pt}\n\nWe consider now an active attacker with the ability to capture and modify the\ncommunication stream between the pool and the victim miner, see\n$\\S$~\\ref{sec:model:adversary}. In the following, we first focus on the\nchallenge to hijack and maintain the TCP connection between the miner and the\npool, then introduce BiteCoin, an attack that hijacks payments from victim\nminers.\n\n\\vspace{-5pt}\n\n\\subsection{WireGhost: TCP Hijack with Re-Sync}\n\n\\vspace{-5pt}\n\n\\noindent\n{\\bf Existing tools}.\nTraditional TCP hijacking attack tools seldom consider the need to\npreserve the status of the communication parties. For instance, in tools like\nShijack~\\cite{toolShijack} and Juggernaut~\\cite{toolJuggernaut}, once the TCP\nsequence mangling is performed, the generated ack storm is eliminated by\nresetting the connection with one of the peers. The tool Hunt~\\cite{toolHunt}\ndoes have a re-synchronization functionality but it is limited to the Telnet\nprotocol and requires victim interaction in the form of a social engineering\nattack to be successful. Stratum active attacks require that the original\nmining connection is maintained and that the re-synchronization needs to be\ndone completely unattended. For instance, the $extranonce1$ parameter will be\ndifferent for each connection, thus the attacker should not force\ndisconnections.\n\n\n\\noindent\n{\\bf WireGhost}.\nWe have developed WireGhost, a TCP hijack tool that maintains\nthe status of the hijacked connection, without having to reset the\ncommunication streams.  To address ack storms that would occur due to\ncommunication changes (e.g., packet modification, injection, removal), WireGhost\nmodifies the TCP sequence of packets according to the payload modification\nperformed by the attacker, see Figure~\\ref{fig:active:tcp_hijack}.\nSpecifically, if the attacker inserts data into the TCP payload (including\ninjecting new packets), WireGhost subtracts the appropriate number of bytes\nfrom the pool server's ack sequence for all the packets that follow the\nmodified (or inserted) one. It then adds the same amount of bytes \nto the sequence number for all the following packets originating from the\nclient. WireGhost performs the opposite mathematical operations when the\nattacker removes data from the TCP payload. \n\n\\begin{figure}[t]\n\\includegraphics[width=0.49\\textwidth]{bitecoin}\n\\caption{BiteCoin attack illustration. The attacker, a subscribed miner,\nforwards job assignments received from the pool to the victim miner. It\nthen hijacks the victim's share submissions and sends them as its own\nto the pool, to get the credit.}\n\\label{fig:bitecoin}\n\\vspace{-15pt}\n\\end{figure}\n\n\\vspace{-5pt}\n\n\\subsection{BiteCoin}\n\n\\vspace{-5pt}\n\n\nWe have used WireGhost to develop BiteCoin, an attack tool that enables an\nactive adversary to steal CPU cycles and payouts from victim miners.  We\nconsider an attacker who subscribes a device under his control as a miner to\nthe pool, see Figure~\\ref{fig:bitecoin} for an illustration.\n\nGiven access to the communication medium between the victim miner and the pool,\nBiteCoin first detects the miner subscription protocol (the 3-way handshake).\nIt then uses WireGhost to hijack the TCP connection between the miner and the\npool. Then, when the attacker device receives a job assignment message from the\npool, it directly injects it into the TCP connection of the victim miner and\nthe pool.  The victim miner receives this job assignment packet as if it was\ncoming from the pool.\n\nOnce the victim miner computes a share for this job, it packs it into a share\nsubmission message and sends it over its TCP connection to the pool. BiteCoin\nintercepts this share submission packet of the victim, and modifies it by\nchanging the victim's username to its own. It then sends this modified share\nsubmission over its own TCP connection to the pool.  BiteCoin also sends to the\npool a mangled copy of the victim's original share submission, to ensure that\nit is rejected.  In $\\S$~\\ref{sec:implementation:bitecoin} we detail our\nBiteCoin implementation, and in $\\S$~\\ref{sec:eval:bitecoin} we present results\nover its deployment.\n\n\n\\vspace{-10pt}\n\n\\section{Bedrock: Secure Stratum}\n\\label{sec:bedrock}\n\n\\vspace{-5pt}\n\nWe now study defenses against the proposed attacks. We first describe the\nrequirements of a private and secure mining protocol, then introduce Bedrock, a\nStratum extension, and discuss its defenses.\n\n\\vspace{-5pt}\n\n\\subsection{Solution Requirements}\n\\label{sec:bedrock:requirements}\n\n\\vspace{-5pt}\n\nA private and secure Stratum protocol should satisfy the following informal\nrequirements:\n\n\\begin{compactitem}\n\n\\item\n{\\bf Security}.\nThe solution needs to protect both against the Stratum attacks that we\nintroduced in $\\S$~\\ref{sec:attacks:passive} and $\\S$~\\ref{sec:attacks:active},\nand against attacks that target the solution itself.\n\n\\item\n{\\bf Efficiency}.\nEncryption of all the Stratum messages is not only inefficient, but also\ninsecure: in $\\S$~\\ref{sec:eval:passive:isp} we show that the ISP Log attack\ncan predict the miner's profits while knowing only the miner's transmission\ntimestamps.\n\n\\item\n{\\bf Adoptability}.\nThe solution should introduce minimal modifications to the Stratum protocol,\nin order to simplify its adoptability by pools and miners.\n\n\\end{compactitem}\n\n\\vspace{-5pt}\n\n\\subsection{The Solution}\n\\label{sec:bedrock:solution}\n\nWe introduce Bedrock \\footnote{In geology, the bedrock is a hard stratum.}, a\nsecure and efficient extension of the Stratum protocol. Bedrock seeks to\nprevent adversaries from inferring the hashrates of miners, and to efficiently\nauthenticate Stratum messages.\n\n\\begin{figure}[h]\n\\vspace{-15pt}\n\\renewcommand{\\baselinestretch}{0.5}\n\\begin{minipage}{0.49\\textwidth}\n\\begin{algorithm}[H]\n\\begin{tabbing}\nXXX\\=X\\=X\\=X\\=X\\=X\\= \\kill\n\n1.{Implementation\\ \\mbox{\\bf{PoolServer}}}\\\\\n2.\\>{\\mbox{\\bf{generateCookie}}(Miner $M$)\\{}\\\\\n3.\\>\\>{$R_M$ := getRandom(256);}\\\\\n4.\\>\\>{$C_M$ := $H^2(R_M, M.uname)$;}\\\\\n5.\\>\\>{$K_M$ := $M.key$;}\\\\\n6.\\>\\>{store($M.uname$, $K_M$, $R_M$, $target$);}\\\\\n7.\\>\\>{sendEncrypted($M$, $E_{K_M}(R_M)$);}\\\\\n\n8.\\>{\\mbox{\\bf{verifyJob}}(Miner $M$,\\ $nonce$,\\ $extranonce2$)\\{}\\\\\n9.\\>\\>{($K_M$,\\ $R_M$,\\ $target$) := getMParams($M.uname$);}\\\\\n10.\\>\\>{$C_M$ := $H^2(R_M, M.uname)$;}\\\\\n11.\\>\\>{$F$ := computeF($C_M$,\\ $extranonce2$);}\\\\\n12.\\>\\>{\\mbox{\\bf{if}}\\ ($H^2(nonce || F) < target$)}\\\\\n13.\\>\\>\\>{sendToMiner($M$, result, ``accept'');}\\\\\n14.\\>\\>{\\mbox{\\bf{else}}\\ sendToMiner($M$, result, ``reject'');}\\\\\\\\\n\n15.{Implementation\\ \\mbox{\\bf{Miner}}}\\\\\n16.\\>{$K_M$ : int[256] \\% key\\ shared\\ with\\ pool}\\\\\n17.\\>{$C_M$ : int[256] \\% mining\\ cookie;}\\\\\n18.\\>{\\mbox{\\bf{solvePuzzle}}($target$: int)\\{}\\\\\n19.\\>\\>{\\mbox{\\bf{do}}}\\\\\n20.\\>\\>\\>{$randPerm$ := newPseudoRandPerm(32);}\\\\\n21.\\>\\>\\>{$extranonce2$ := getRandom(32);}\\\\\n22.\\>\\>\\>{$F$ := computeF($C_M$,\\ $extranonce2$);}\\\\\n23.\\>\\>\\>{\\mbox{\\bf{while}}\\ ($randPerm$.isNext())\\{}\\\\\n24.\\>\\>\\>\\>{$nonce$ := $randPerm$.next();}\\\\\n25.\\>\\>\\>\\>{\\mbox{\\bf{if}}\\ ($H^2(nonce || F) < target$)}\\\\\n26.\\>\\>\\>\\>\\>{sendToPool($uname$, $nonce$, $extranonce2$);}\\\\\n27.\\>\\>{\\mbox{\\bf{while}} ($clean\\_jobs$ != 1)}\n\\vspace{-25pt}\n\\end{tabbing}\n\\caption{Bedrock pseudo-code for cookie generation and job verification (pool\nside), and job solving (miner side).}\n\\label{alg:bedrock}\n\\end{algorithm}\n\\end{minipage}\n\\normalsize\n\\vspace{-15pt}\n\\end{figure}\n\n\nBedrock has 3 components, each addressing different Stratum vulnerabilities.\nThe first component authenticates and obfuscates the job assignment and share\nsubmission messages. The second component secures the share difficulty\nnotifications, and the third component secures the pool's inference of the\nminer's capabilities.  In the following we detail each component. We assume\nthat the pool shares a unique secret symmetric key $K_M$ with each miner $M$.\nThe miner and the pool create the key during the first authorization protocol\n(see $\\S$~\\ref{sec:stratum}), e.g., using authenticated Diffie-Hellman).\n\n\\vspace{-5pt}\n\n\\subsubsection{Mining Cookies}\n\\label{sec:bedrock:solution:cookies}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47\\textwidth]{{bedrock.puzzle}.pdf}\n\\vspace{-15pt}\n\\caption{Bedrock puzzle illustration. The cookie $C_M$ is generated on the\npool, while the $nonce$ and $extranonce2$ are generated on the miner.  The\ncoinbase transaction contains both $C_M$ and $extranonce2$, see\nFigure~\\ref{fig:coinbase}.}\n\\label{fig:bedrock:puzzle}\n\\vspace{-15pt}\n\\end{figure}\n\nThe share submission packets are particularly vulnerable. First, they can\nreveal the target value, thus the difficulty of the jobs on which the miner\nworks and then the miner's hashrate (see $\\S$~\\ref{sec:bedrock:discussion}).\nSecond, share submissions can be hijacked by an active adversary, see\n$\\S$~\\ref{sec:attacks:active}. Encryption of share submissions\nwill prevent these attacks, but it will strain the pool's resources.\n\nTo efficiently address these vulnerabilities, we introduce the concept of {\\it\nmining cookie}, a secret that each miner shares with the pool, see\nFigure~\\ref{fig:bedrock:puzzle} and Algorithm~\\ref{alg:bedrock}. The miner uses\nits mining cookie as an additional, secret field in the Bitcoin puzzle. Without\nknowledge of the mining cookie, an adversary cannot infer the progress made by\nthe miner, thus its hashrate and payout, thus cannot hijack shares submitted by\nthe miner.\n\nSpecifically, let $R_M$ be a random cookie seed that the pool generates for a\nminer $M$ Algorithm~\\ref{alg:bedrock}, line 3). The pool associates $R_M$ with\n$M$, and stores it along with $M$'s symmetric key $K_M$, and its current\n$target$ value (line 6). The pool computes $M$'s cookie as $C_M = H^2(R_M,\nM.uname)$ (line 4),  where $M.uname$ is the username of the miner. It then\nsends $R_M$ to $M$, encrypted with the key $K_M$ (line 7), see\n$\\S$~\\ref{sec:bedrock:solution:set}. The miner similarly uses $R_M$ and its\n$username_M$ to compute $C_M$.\n\nTo minimally modify Bitcoin, Bedrock stores the cookie as part of the coinbase\ntransaction (see Figure~\\ref{fig:coinbase}), in the place of its unused {\\it\nprevious hash} field. This field is unused since the coinbase transaction does\nnot have a need for a meaningful input address hash (see\n$\\S$~\\ref{sec:model:coinbase}).  Thus, the puzzle remains the same: The miner\niterates over the $nonce$ and $extranonce2$ values, and reports the pairs that\nsolve the puzzle, along with its username, in share submission packets (lines\n23-26).\n\nTo verify the shares, the pool retrieves the miner's key $K_M$, random seed\n$R_M$ and $target$ values (line 9).  It uses $R_M$ to reconstruct the cookie\n(line 10) and uses $target$, and the reported $nonce$ and $extranonce2$ values,\nto reconstruct and verify the puzzle lines 11 ans 12).\n\n\n\\noindent\n{\\bf Random iterators}.\nIn the Bitcoin protocol and the Stratum implementation on F2Pool, the $nonce$\nand $extranonce2$ values are incremented sequentially: once the miner exhausts\n$nonce$, it increments $extranonce2$, then continues to iterate over a reset\n$nonce$ value.  In $\\S$~\\ref{sec:bedrock:discussion} we show that this further\nexposes the miner to hashrate inference attacks. We address this problem by\nrequiring the miner to choose random values for $nonce$ and $extranonce2$ at\neach puzzle iteration. To prevent the miner from recomputing an expensive\nMerkle tree root at each iteration, we iterate through the $nonce$ space\nusing a pseudo random permutation (lines 20, 24).\n\n\\noindent\n{\\bf Cookie refresh}.\nWhen a miner mines the current block, i.e., when $H^2(nonce || F || C_M)$ is\nless than the target corresponding to the $Nbits$ value, see\n$\\S$~\\ref{sec:model:puzzle}, the puzzle solution needs to be published in the\nblockchain.  The published block needs to include all the fields that defined\nthe puzzle (see $\\S$~\\ref{sec:model:puzzle}), including the miner's cookie, to\nbe publicly verified.\n\nTo prevent an adversary who monitors the blockchain to learn the\nmining cookie of a victim miner and then launch a successful BiteCoin attack\n(see $\\S$~\\ref{sec:bedrock:discussion}), Bedrock changes the mining cookie of\nthe miner once the miner mines the current block. This is an infrequent\nevent: for an AntMiner S7 mining equipment, with a hashrate of 4.73 TH\/s, and\nthe current Bitcoin network difficulty (2.58522748405e+11),\nEquation~\\ref{eq:expected_time_to_share} shows that the expected time to mine a\nblock is 7.44 years. This is a very low lower bound since it assumes a\nconstant difficulty. In reality, the difficulty has increased exponentially\nsince the creation of Bitcoin.\nTo change the cookie, the pool invokes generateCookie (line 2).\n\n\n\\vspace{-5pt}\n\n\\subsubsection{Protect Communicated Secrets}\n\\label{sec:bedrock:solution:set}\n\n\\vspace{-5pt}\n\nStratum's share difficulty notification messages reveal the difficulty assigned\nby the pool to the miner and that the miner uses in the subsequent jobs.\nKnowledge of the puzzle difficulty value coupled with the (regulated) share\nsubmission rate, will enable the adversary to infer the hashrate of the miner\n(see Equation~\\ref{eq:e}), thus its payout.  In addition, Bedrock also needs to\ncommunicate secret values (e.g., the random $R_M$, see\n$\\S$~\\ref{sec:bedrock:solution:cookies}). Bedrock addresses these problems by\nextending Stratum's set difficulty notifications to the following {\\bf mining\nencrypted} message:\n\n\\vspace{-10pt}\n\n\n\\[\n{\\tt mining.encrypted,\\ E_{K_M}(param\\_list)}\n\\]\n\n\\noindent\nwhere (\\textit{param\\_list}) is a list of values that need protection, i.e.,\ndifficulty values and the secret $R_M$. Specifically, \\textit{param\\_list} can\ncontain any number of sensitive values in the format {\\tt\n[[``difficulty'',1024],[``secret'',$R_M$]]}. \n\n\n\\vspace{-5pt}\n\n\\subsubsection{Secure Hashrate Computation}\n\n\\vspace{-5pt}\n\nThe hashrate inference protocol following a miner subscription and\nauthorization, as documented in $\\S$~\\ref{sec:attacks:passive:isp} and\n$\\S$~\\ref{sec:eval:passive:isp} can be exploited also by an adversary to infer\nthe miner's hashrate. To address this vulnerability, Bedrock requires the miner\nto directly report its hashrate during the initial subscription message, along\nwith other miner capabilities. The miner can locally estimate its hashrate,\ne.g., by creating and executing random jobs with a difficulty of 1024. The\nminer also needs to factor in its communication latency to the pool, which it\ncan infer during the subscription protocol.  The miner sends its hashrate\nencrypted, using the ``mining encrypted'' message defined above.\n\n\nIf subsequently, the pool receives share submissions from the miner, outside\nthe desired rate range, it can then adjust the difficulty (through the above\nencrypted share difficulty notifications) in order to reflect its more accurate\ninference of the miner's hashrate.\n\n\\vspace{-5pt}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Security Discussion}\n\\label{sec:bedrock:discussion}\n\n\\vspace{-5pt}\n\nWe now discuss attacks against Stratum and Bedrock, and detail the defenses\nprovided by Bedrock.\n\n\\noindent\n\\textbf{Target reconstruction attack}.\nAn attacker that can inspect cleartext subscription response, job assignment\nand share submission packets, can reconstruct the job (i.e., puzzle) solved by\nthe victim miner: Recover $extranonce1$ from an early miner subscription\nmessage, $coinbase1$, $coinbase2$ and the Merkle tree branches from a job\nassignment, and $nonce$ and $extranonce2$ from a subsequent share submission\npacket. The attacker then reconstructs the $F$ field of the puzzle (see\n$\\S$~\\ref{sec:model:puzzle}) and uses it to infer the miner's hashrate, even\nwithout knowing the puzzle's associated $target$ value. Specifically, the\nattacker computes the double hash of $F$ concatenated with $nonce$, then\ncounts the number of leading zeroes to obtain an upper bound on the job's\ntarget. The attacker then uses recorded inter-share submission time stats and\nEquation~\\ref{eq:e} to estimate the miner hashrate.\n\nBedrock thwarts this attack through its use of the cookie $C_M$, a secret known\nonly by the miner and the pool. The cookie is part of the puzzle. Without its\nknowledge, the attacker cannot reconstruct the entire puzzle, thus infer the\ntarget.\n\n\\noindent\n\\textbf{Brute force the cookie}.\nThe attacker can try to brute force the cookie value. To gain confidence, the\nattacker uses the fields from multiple jobs assigned to the same miner to try\neach candidate cookie value. A candidate is considered ``successful'' if it\nproduces a high target value for all the considered jobs. However, in\n$\\S$~\\ref{sec:implementation} we leverage the unused, 256-bit long ``previous\nhash'' field of the coinbase transaction, to store the mining cookie. Brute\nforcing this field is consider unfeasible.\n\n\\noindent\n{\\bf Resilience to cryptographic failure}.\nWe assume now an adversary that is able to break the encryption employed by the\npool and the miner, e.g., due to the use of weak random values.  Giechaskiel et\nal.~\\cite{GCR16} studied the effect of broken cryptographic primitives on\nBitcoin, see $\\S$~\\ref{sec:related}. While such an adversary can compromise the\nprivacy of the miner, by recovering the miner's cookie, he will be prevented\nfrom launching active attacks. This is because the miner's cookie is a function\nof both a random number and the miner's username.\n\nSpecifically, if the attacker hijacks a miner's share submission, the pool\nwould use the attacker's username instead of the victim's username to construct\nthe cookie, the coinbase transaction and eventually the header block. The share\nwill only validate if the attacker managed to find a username that produced a\ndouble hash that was still smaller than the target corresponding to the\ndifficulty set by the pool.  However, the attacker will need to find such\nusernames for each hijacked share.  If the attacker was able to quickly find\nsuch partial collisions, it would be much easier to simply compute the shares\nwithout doing any interception and hijacking.\n\nWe further consider an attacker able to break the hash function (invert and\nfind collisions). Such an attacker can recover a miner's $R_M$ value, then find\na username that produces a collision with the miner's cookie $C_M$.  We observe\nhowever that such an attacker could then be able to also mine blocks quickly,\ne.g., by inverting hash values that are smaller than the target corresponding\nto the $Nbits$ value.\n\n\\vspace{-5pt}\n\n\n\n\n\n\n\n\n\n\\subsection{Limitations}\n\n\\vspace{-5pt}\n\n\\noindent\n\\textbf{Opportunistic cookie discovery}.\nWhen the miner mines the current block, i.e., the double hash of the puzzle is\nsmaller than the target corresponding to $Nbits$, the miner's cookie is\npublished in the blockchain.  An adversary who has captured job assignments and\nshare submissions from the miner, just before this takes place, can use them,\nalong with the published cookie, to reconstruct the entire puzzle and infer the\nminer's hashrate.\n\nThis opportunistic attack may take years (e.g., 7.44 years\nfor an AntMiner S7, see $\\S$~\\ref{sec:bedrock:solution:cookies}), while, from\nour experience, mining equipment has a useful lifetime of around 2 years.\n\\newmaterial{However, this attack may be more effective against an entity\nthat owns many homogeneous miners: an adversary may only need days to infer\nthe rate of a single miner.}\n\nHowever, to address this limitation, each miner could, at random intervals,\nchange its operation frequency to a randomly chosen value within an\n``acceptable'' operation range. Assuming that the adversary only captures a\nlimited window of the victim miner's communications, he will only be able to\n(i) recover temporary, past hashrate values of the miner, and (ii) reconstruct\nthe miner's payouts over the monitored interval. Since the miner changes its\noperation frequency, once a new cookie is assigned, the adversary will not be\nable to predict the miner's future hashrates and payouts.\n\n\\noindent\n{\\bf Verification scope}.\nWe have only investigated the implementation of Stratum in the pool F2Pool.\nHowever, the identified privacy issues also likely affect other pools, as any\nobfuscation to the set difficulty messages would break the compatibility with\nthe Stratum protocol implemented in current mining equipment.\n\n\\vspace{-5pt}\n\n\\section{Implementation and Testbed}\n\\label{sec:implementation}\n\n\\vspace{-5pt}\n\nIn our experiments, we have used AntMiner S7, a specialized FPGA device for\nBitcoin mining that achieves a hashrate of 4.73 TH\/s at 700MHz~\\cite{s7_miner}.\nWe have configured the device for mining on the F2Pool pool, using the Stratum\nprotocol~\\cite{f2pool_help}.\n\n\n\\subsection{Passive Attacks}\n\\label{sec:implementation:passive}\n\nIn order to collect experimental traffic for the passive attacks, we have\nleveraged the ability of the AntMiner S7 device to operate at different chip\nclock frequencies in order to simulate miner devices with different\ncapabilities. Specifically, we carried out 24 hour long mining experiments with\nthe AntMiner S7 operating at frequencies ranging from 100 MHz to 700MHz, with\n50MHz granularity. We have used Tcpdump~\\cite{jacobson2003tcpdump} to capture\n138MB of Stratum traffic of AntMiner S7 devices in the US (May 27 - June\n8, 2016) and Venezuela (March 8 - April 2, 2016).  We have sliced the resulting\npcap files into 24 hour intervals using editcap, then processed the\nresults using python scripts with the scapy library~\\cite{biondi2011scapy}.\n\nIn addition to the mining traffic, for each of the 24 hour runs, we collected\nthe empirical payout as reported by the pool, as well as the device hashrate\nreported by its internal functionality. We used 24 hour runs because the pool\nuses 24 hour cycles for executing payouts. We have manually synchronized the\nruns and payout cycles so as to easily correlate the data collected with its\ncorresponding payout.\n\n\\noindent\n{\\bf StraTap attack}.\nTo implement the StraTap attack, we have created a script that selects packets\nfrom the captured traces with the ``set\\_difficulty'' pattern (invoked method\nof the share difficulty notification messages). This pattern signals our script\nto perform a share submission count reset, as well as a new recording of the\nnew difficulty.\n\n\\noindent\n{\\bf ISP Log attack}.\nFor the ISP Log attack, we used packets sent after the 3-way handshake\ninitiated by the pool. In addition, to compute more accurate inter-packet\ntimes, we only considered packets that had the PUSH flag set (captured by most\nfirewall logs, e.g., Snort IDS), thus with non-empty payloads (i.e., no ack\npackets that originated on the miner).  The PUSH flag is used to mitigate the\neffects of delays on the processing of share submissions, that may end up\ncausing share rejections. By setting the PUSH flag, miners try to increase the\nchance that their shares are quickly processed.\n\n\\vspace{-5pt}\n\n\\subsection{BiteCoin Attack Implementation}\n\\label{sec:implementation:bitecoin}\n\n\\vspace{-5pt}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{bitecoin_system}\n\\caption{Architecture of BiteCoin attack implementation.}\n\\label{fig:bitecoin:system}\n\\vspace{-15pt}\n\\end{figure}\n\nThe BiteCoin attack system is illustrated in Figure~\\ref{fig:bitecoin:system}.\nWe have built WireGhost using the iptables nfqueue target, in order to pass\npackets into user space. Once it receives network segments in the user space,\nit uses the scapy python library to parse and modify packets. Additionally, it\nuses the the python nfqueue bindings in order to pass a verdict to the packets.\n\nIn order to test BiteCoin and WireGhost, we set up the victim miner behind an\nattacker controlled server that performed ``source NAT'' and packet forwarding\nfor it. This architecture allowed us to emulate an active attacker\nintercepting the communication between the miner and the pool. We have\nimplemented the attacker as a python script that connects to the F2Pool using\nStratum, then intercepts and modifies job assignments and share submissions on\nthe victim's connection to the pool. While the attacker script does not perform\nany mining, in $\\S$~\\ref{sec:eval:bitecoin} we show that it is able to steal\nthe victim's hashing power.\n\n\\vspace{-5pt}\n\n\\subsection{Bedrock Implementation}\n\n\\vspace{-5pt}\n\nOne requirement of Bedrock is to minimally disrupt the Stratum protocol, see\n$\\S$~\\ref{sec:bedrock:requirements}. Thus, instead of designing the cookie to\nbe an external field, we seek to leverage unused fields of the coinbase\ntransaction.  An obvious candidate for the cookie placement is the input script\nwhere the $extranonce1$ and $extranonce2$ reside. However, most pools have\nalready started using this space for their own internal procedures, e.g., in\nF2Pool, to store the miner's name.\n\nInstead, Bedrock uses the yet unused, 32 byte (256 bit) long ``previous input\naddress'' field of the coinbase transaction, see Figure~\\ref{fig:coinbase}.\nSince the coinbase transaction rewards the pool with the value of the mined\nblock (if that event occurs), its input is not used. We have investigated the\nStratum implementation of several pools, including F2Pool~\\cite{f2pool_help},\nGHash.io~\\cite{GHash.io}, SlushPool~\\cite{SlushPool} and have confirmed that\nnone of them use this field. In addition, we note that the size of this field\nmakes it ideal to store the output of a double SHA-256 hash.\n\n\\begin{table}\n\\centering\n\\textsf{\n\\small\n\\begin{tabular}{l r r}\n\\textbf{Freq(MHz)} &  \\textbf{Hashrate(GHz)}  & \\textbf{StraTap Hashrate(GHz)}\\\\\n\\midrule\n700  &   4720.55 & 4571.48\\\\\n650   &  4371.85 & 4309.96\\\\\n600   &  4040.49 & 4151.27\\\\\n550   &  3693.90 & 3624.13\\\\\n500    & 3365.38 & 3524.57\\\\\n450   &  3030.01 & 3154.80\\\\\n400  &   2689.34 & 2696.72\\\\\n350    & 2364.61 & 2382.17\\\\\n300    & 2023.65 & 2039.55\\\\\n250    & 1687.17 & 1699.91\\\\\n200    & 1347.14 & 1274.29\\\\\n150    & 1010.19 & 1007.06\\\\\n100   &  672.55 & 703.28\\\\\n\\end{tabular}\n}\n\\caption{Operation frequency, actual hashrate and StraTap inferred hashrate.\nWe observe the correlation between the actual and the average hashrate, that\nallowed StraTap to achieve a good payout estimate.\n}\n\\label{table:freq_hashrate_payout}\n\\vspace{-15pt}\n\\end{table}\n\n\n\\vspace{-5pt}\n\n\\section{Evaluation}\n\\label{sec:eval}\n\n\\vspace{-5pt}\n\nIn this section we evaluate the StraTap, ISP Log and BiteCoin attacks, as well\nas the performance of Bedrock. We use the mean squared error (MSE) and the mean\npercentage error (MPE) to evaluate the accuracy of the predictions made by the\npassive attacks. Specifically, let $P = \\{ P_1, .., P_n\\}$ be a set of observed\ndaily payments over $n$ days, and let $\\bar{P} = \\{ \\bar{P_1}, .., \\bar{P_n}\n\\}$ be the corresponding predicted daily payments for the same days. Then,\nMSE($\\bar{P},P$) = $\\frac{1}{n} \\sum_i^n (\\bar{P_i} - P_i)^2$, and\nMPE($\\bar{P},P$) = $\\frac{100\\%}{n} \\sum_i^n \\frac{P_i - \\bar{P_i}}{P_i}$.\n\n\\vspace{-5pt}\n\n\\subsection{The StraTap Attack}\n\\label{sec:eval:passive:stratap}\n\n\\vspace{-5pt}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{isplog_stratap_payout}\n\\caption{Payout prediction by StraTap and ISP Log attacks, compared to\nempirical payout, in mili Bitcoin (mBTC), as a function of the miner's\nfrequency of operation (MHz). The \\textit{actual payout} series (red diamonds)\ncorresponds to daily payouts collected from the F2Pool account records. The\nStraTap payout series (blue disks) shows daily payout predictions based on\nentire Stratum messages intercepted. The ISP Log series (green triangles) shows\nthe daily payout prediction when using the average inter-packet times over 50\npackets. StraTap's prediction error ranges between 1.75-6.5\\% (MSE=0.062,\nMPE=3.46\\%). ISP Log has an error between 0.53 - 34.4\\% (MSE = 2.02, MPE =\n-9.49\\%).}\n\\label{fig:eval:stratap}\n\\vspace{-15pt}\n\\end{figure}\n\nWe have used the StraTap script described in\n$\\S$~\\ref{sec:implementation:passive} to calculate the average time of share\ncreation for each of the detected intervals of constant difficulty. For each of\nthe 24 hour runs, we also calculated the weighted average difficulty as well as\nthe weighted average hashrate for the entire run. In addition, we have also\nused Equation~\\ref{eq:e}, along with the computed average time and recorded\ndifficulty values, to compute a prediction of the weighted average hashrate of\nthe miner.\n\nTable~\\ref{table:freq_hashrate_payout} shows the AntMiner's frequency of\noperation, the output hashrate achieved at that frequency, and the predicted\nhashrate. As expected, there is a linear relationship between the frequency of\noperation and the device's hashrate achieved. As a consequence, this\nrelationship is preserved across the empirical payout reported by the pool\noperators.\n\nSpecifically, we have used the pool's hashrate to BTC conversion (see\n$\\S$~\\ref{sec:stratum}) to predict the miner's resulting daily payout.\nFigure~\\ref{fig:eval:stratap} shows the data series for the empirical and\npredicted payouts, versus the operation frequency of the miner.  The StraTap\nattack achieves a prediction error of between 1.75\\% and 6.5\\%, with a mean\nsquare error (MSE) of 0.062 and mean percentage error (MPE) of -3.46\\%. Thus,\nStraTap's predictions tend to be slightly larger than the actual payout values.\n\n\\subsection{The ISP Log Attack}\n\\label{sec:eval:passive:isp}\n\n\\vspace{-5pt}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]\n{\\label{fig:timeline:hashrate:200MHz}{\\includegraphics[width=0.49\\textwidth]{timeline_80packet_200MHz_indicator}}}\n\\vspace{-5pt}\n\\subfigure[]\n{\\label{fig:timeline:hashrate:600MHz}{\\includegraphics[width=0.49\\textwidth]{timeline_125packet_600MHz_indicator}}}\n\\vspace{-5pt}\n\\caption{Timelines that focus on the interval between the first two share\ndifficulty notifications, following a miner subscription and authorization\nprotocol, when\n(a) the miner operates at 200MHz and\n(b) the miner operates at 600MHz.\nWhile the intervals between the first two such notifications at both\nfrequencies contain approximately 50 share submission packets, this interval is\nsignificantly shorter at 600MHz. This is because at 600MHz the miner can solve\nthe 1024 difficulty puzzles much faster than at 200MHz.  The ``ISP Log'' attack\nexploits this observation to infer the hashrate of the miner, while only\ncounting packets (i.e., without being able to inspect them).}\n\\label{fig:timeline:hashrate}\n\\vspace{-15pt}\n\\end{figure}\n\nWe first present results of our analysis of F2Pool's hashrate inference\nprotocol. We then show the ability of the ISP Log attack to leverage these\nfindings to infer the miner's daily payouts, given only metadata of the miner's\npackets.\n\n\\begin{table}\n\\centering\n\\textsf{\n\\small\n\\begin{tabular}{l r r}\n\\textbf{Freq(MHz)} &  \\textbf{\\# of Packets}  & \\textbf{Time Interval}\\\\\n\\midrule\n100 & 57 & 288.872897148\\\\\n150 & 56 & 256.145660877\\\\\n200 & 51 & 153.622557878\\\\\n250 & 63 & 146.007184982\\\\\n300 & 55 & 131.089562893\\\\\n350 & 62 & 146.259056807\\\\\n400 & 54 & 101.954112053\\\\\n450 & 67 & 104.665092945\\\\\n500 & 50 & 58.2229411602\\\\\n550 & 62 & 76.0586118698\\\\\n600 & 54 & 50.7432210445\\\\\n650 & 56 & 45.6691811085\\\\\n\\end{tabular}\n}\n\\caption{Number of share submission packets for the initial $1024$ difficulty\nperiod, as well as the length of the time interval when the pool accepted those\nshares, for various miner frequencies of operation. At any miner operation\nfrequency, at least $50$ share submission packets are accepted, irrespective of\nwait time.  This process enables the pool and the ISP Log attack to infer the\nminer's hashrate.\n}\n\\label{table:timeline_1024}\n\\vspace{-15pt}\n\\end{table}\n\n\\noindent\n{\\bf Hashrate inference protocol}.\nAs mentioned in $\\S$~\\ref{sec:attacks:passive:isp}, immediately following the\nminer subscription and authorization, the pool sets the difficulty to 1024, and\nchanges it only after receiving a sufficient number of share submissions to\ninfer the miner's hashrate. For instance,\nFigure~\\ref{fig:timeline:hashrate:200MHz} shows that when the miner operates at\n200MHz, the number of share submissions between the first two share difficulty\nnotification messages is similar to the number of share submissions when the\nminer operates at 600MHz (Figure~\\ref{fig:timeline:hashrate:600MHz})\n(approximately 50).  However, the time interval between the first two share\ndifficulty notifications is much shorter at 600MHz: the miner can compute 50\nshares at the constant difficulty 1024 much faster than when operating at\n200MHz.\n\nMore general, Table~\\ref{table:timeline_1024} shows the number of share\nsubmission packets sent for this initial measurements period for each of the\nfrequencies analyzed.  We observe that the pool requires that this process\ngenerates at least $50$ share submissions, irrespective of the miner operation\nfrequency. The pool waits up to 288 seconds to receive the required number of\nshares, before sending the second share difficulty notification.\n\nWe conjecture that the pool uses this process in order to infer the hashrate of\nthe miner, which it needs in order to assign jobs (puzzles) that a miner can\nsolve at a ``desirable'' rate. Specifically, large pools handle thousands of miners\nsimultaneously~\\footnote{The Bitcoin network currently has around 100,000\nminers~\\cite{MinerCount,Corti}, of which at least 16\\% work with\nF2Pool~\\cite{F2PoolShare}.}. In order to minimize the time it takes to process\nshare submissions received from thousands of miners, the pool\nneeds to regulate the rate at which a miner submits shares, irrespective of the\nminer's computing capabilities. Figure~\\ref{fig:timeline:rate} illustrates\nthis share submission rate control. In our experiments we observed that for F2Pool,\nthis rate ranges to between 1 to 4 share submissions per minute. A second reason\nfor this process stems from the need of miners to prove computation progress\nand gain regular payouts.\n\n\\begin{figure}[t!]\n\\vspace{-15pt}\n\\centering\n\\includegraphics[width=0.43\\textwidth]{passive_blind_restricted_30_R}\n\\vspace{-5pt}\n\\caption{\n1st, 2nd and 3rd quartile for the inter-packet times of the first 50\npackets during the initial difficulty setting procedure, as a function of the\nminer's operating frequency. We observe a monotonically decreasing tendency\nof the inter-packet times, with an increase in the miner capabilities.\nThis suggests that inter-packet time stats over the first 50 packets can provide\na good hashrate estimator for the ISP Log attack.\n\\label{fig:isp:log:predictor}}\n\\vspace{-15pt}\n\\end{figure}\n\n\\noindent\n{\\bf ISP Log attack results}.\nWe have implemented the ISP Log attack using statistics of the inter-packet\narrival time of the first 50 packets sent by the miner to the pool, after a\ndetected 3-way miner subscription and authorization protocol.\nFigure~\\ref{fig:isp:log:predictor} shows the 1st, 2nd (median) and 3rd\nquartiles of the inter-packet times, for the first 50 packets, when the miner\noperates at frequencies ranging from 100 to 650 MHz. The linearly decreasing\nbehavior of the median, 1st and 3rd quartiles indicates that statistics over\nthe inter-packet times of the first 50 packets, may make a good predictor.\n\n\n\nTo confirm this, we have used the mean inter-packet time over the first 50\npackets to predict the miner's hashrate and then its payout.\nFigure~\\ref{fig:eval:stratap} compares the ISP Log attack daily payout\nprediction with that of StraTap and with the empirical payout. The ISP Log has\nan error that ranges between 0.53\\% and 34.4\\%, with an MSE of 2.02 and MPE of\n-9.49\\% . Thus, ISP Log over predicts the daily payouts, and, as expected, it\nexceeds the error of the StraTap attack. \n\n\\vspace{-5pt}\n\n\\subsection{BiteCoin: Proof of Concept}\n\\label{sec:eval:bitecoin}\n\n\\vspace{-5pt}\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]\n{\\label{fig:bitecoin:timeline:attacker}{\\includegraphics[width=0.49\\textwidth]{bitecoin_attacker_marked}}}\n\\vspace{-5pt}\n\\subfigure[]\n{\\label{fig:bitecoin:timeline:victim}{\\includegraphics[width=0.49\\textwidth]{bitecoin_victim}}}\n\\vspace{-5pt}\n\\caption{Greedy BiteCoin attack timelines for\n(a) adversary and (b) victim miner. In a 5h interval, the attacker hijacked 342\njob assignments and 72 corresponding share submissions of the victim miner.\n23 shares (the green clumps marked with red arrows) were accepted by the pool.}\n\\label{fig:bitecoin:timeline}\n\\vspace{-15pt}\n\\end{figure}\n\nWe have experimented with the BiteCoin implementation described in\n$\\S$~\\ref{sec:implementation:bitecoin}. Specifically, the attacker greedily\ninjected all the jobs assigned by the pool into the victim communication stream\nduring the attack time and without any modification. Our implementation\ninjected a total of 342 job assignments in a time interval of 5 hours, from\nhour 21:25 to 02:24. The attacker monitored the share submissions from the\nvictim, and hijacked shares corresponding to the injected jobs.\n\nFigure~\\ref{fig:bitecoin:timeline} shows the results of this attack. The\nadversary, whose timeline is shown in\nFigure~\\ref{fig:bitecoin:timeline:attacker}, hijacked 72 share submissions from\nthe victim miner. 23 shares (the green clumps marked with red arrows) were\naccepted by the pool, i.e., as if they were mined by the attacker and not by\nthe victim. 49 shares were rejected. Figure~\\ref{fig:bitecoin:timeline:victim}\nshows the timeline of the attack from the perspective of the victim miner.\n\nThe gaps are likely due to the script trying to get some constant work in.\nEvery disconnection and reconnection of the attacker will trigger a subscribe\nprotocol where the first job has the true flag set. This would explain why\nthere are no hijacked shares between around 22:00 and 1:00 in the attacker\ntimeline and also the gap of any activity in the victim timeline. These\nconstant reconnects may have constantly blanked the job pool of the victim\nuntil the attacker was able to maintain its connection to submit the shares.\n\n\n\\vspace{-5pt}\n\n\\subsection{The Bedrock Evaluation}\n\\label{sec:eval:bedrock}\n\n\\vspace{-5pt}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{overhead}\n\\caption{Overhead comparison of Bedrock and a complete encryption approach, for\nminer and pool. Bedrock imposes a small daily overhead on both the pool (12.03s\nto handle 16,000 miners) and miner (0.002s). However, a solution that\nencrypts all Stratum packets imposes a daily overhead of 1.36 hours on the\npool.}\n\\label{fig:eval:bedrock}\n\\vspace{-15pt}\n\\end{figure}\n\nWe measured Bedrock's encryption times when using AES-256 in CBC mode on the\nAntMiner S7 and on a server with 40 cores Intel(R) Xeon(R) CPU E5-2660 v2 @\n2.20GHz and 64 GB RAM. The AntMiner was able to encrypt 1024-blocks at\n32,231.09 Kb\/sec while the server was able to encrypt at 86,131.07 Kb\/sec for\nthe same block size.\n\n\nBased on the collected data, Stratum generates an average of 31.63 set\ndifficulty messages per day. Figure~\\ref{fig:eval:bedrock} shows that Bedrock\nimposes a 0.002s decryption overhead per day on an AntMiner S7, while on a pool\nusing the above server to handle 16,000 miners, it imposes an encryption\noverhead of 12.03 seconds per day.\n\nIn contrast, a solution that encrypts each Stratum packet imposes an overhead\nof 0.13 seconds per day on the AntMiner, and an unacceptable 1.36 hours per day\non the pool server, to handle 16,000 miners.\n\n\\vspace{-15pt}\n\n\\subsubsection{TLS Overheads}\n\n\\newmaterial{\nWe also compare Bedrock against Stratum protected with TLS. We have used a\nreplay of a 24 hour subset of our Stratum traffic dataset\n($\\S$~\\ref{sec:implementation:passive}), sent over TLS between a laptop used as\nminer (AntMiner does not support TLS) and the server above, used as the pool.}\n\n\\noindent\n\\newmaterial{\n{\\bf Computation overheads}.\nTo measure the TLS computation overheads, we have used\nTcpdump~\\cite{jacobson2003tcpdump} to capture the times when Stratum\/TLS\npackets leave from and arrive at the pool application, and also captured the\ntime when the packets are sent from\/received by the pool TLS socket.  We have\ncomputed the total daily pool side TLS overhead of sending and receiving\nStratum packets (job assignment, share submission, notifications, set\ndifficulty change, etc). Figure~\\ref{fig:eval:bedrock} shows the difference\nbetween this overhead and the same overhead but when using bare TCP. It shows\nthat the daily computation overhead imposed by TLS on the pool, through the\ntraffic of 16,000 miners, is 1.01 hours.} This amounts to a\ncomputational overhead percentage of at least 4.3\\%.\n\n\n\\noindent\n\\newmaterial{\n{\\bf Bandwidth overhead}.\nIn addition, we have measured the bandwidth overhead imposed by TLS. The total\nminer-to-pool payload (single miner) for cleartext Stratum\/TCP traffic is\n465,875 bytes and for Stratum\/TLS is 738,873 bytes. The total pool-to-miner\npayload of Stratum\/TCP is 3,852,795 bytes while for Stratum\/TLS is 4,062,956\nbytes. Thus, TLS imposes a 58\\% overhead on the miner-to-pool bandwidth, for a\ntotal of 4.05GB daily overhead on the pool from 16,000 miners. This uplink\noverhead is significant, especially for miners in countries with poor Internet\nconnectivity.}\n\n\\newmaterial{\nTLS imposes a 5\\% overhead on the pool-to-miner bandwidth, for a total of\n3.13GB daily overhead on the pool. The TLS overhead is much larger in\nminer-to-pool communications, even though there are more pool-to-miner packets.\nThis is because the miner-to-pool share submission packets are much smaller\nthan the pool-to-miner job assignments, thus the TLS overhead (125 to 160\nbytes) becomes a significant factor for them.} In contrast, the\npercentage bandwidth overhead for Bedrock is only 0.04\\%.\n\n\\noindent\n\\newmaterial{\n{\\bf Conclusions}.\nBedrock is more efficient than blanket encryption and TLS. While the pool could\nuse more equipment to handle encryption more efficiently, blanket encryption\nand TLS do not address the hashrate inference vulnerability. In addition, TLS\nimposes a significant uplink bandwidth overhead on miners.}\n\n\\vspace{-10pt}\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\vspace{-5pt}\n\n\\noindent\n{\\bf Bitcoin mining attacks}.\nDecker and Wattenhofer~\\cite{DW13} study Bitcoin's use of a multi-hop broadcast\nto propagate transactions and blocks through the network to update the ledger\nreplicas, then study how the network can delay or prevent block propagation.\nHeilman et al.~\\cite{HKZG15} propose eclipse attacks on the Bitcoin network,\nwhere an attacker leverages the reference client's policy for updating peers to\nmonopolize all the connections of a victim node, by forcing it to accept only\nfraudulent peers. The victim can then be exploited to attack the mining and\nconsensus systems of Bitcoin.  Bissas et al.~\\cite{BLOAH16} present and\nvalidate a novel mathematical model of the blockchain mining process and use it\nto conduct an economic evaluation of double-spend attacks, both with and\nwithout a concurrent eclipse attack.\n\nCourtois and Bahack~\\cite{CB14} propose a practical block withholding attack,\nin which dishonest miners seek to obtain a higher reward than their relative\ncontribution to the network. They also provide an excellent background\ndescription of the motivation and functionality of mining pools and the mining\nprocess.\n\n\\noindent\n{\\bf Bitcoin anonymity}.\nSignificant work has focused on breaking the anonymity of Bitcoin\nclients~\\cite{BKP14,KKM14,MPJLMVS13,AKRSC13}. For instance, Biryukov et\nal.~\\cite{BKP14} proposed a method to deanonymize Bitcoin users, which allows\nto link user pseudonyms to the IP ad- dresses where the transactions are\ngenerated. Koshy et al.~\\cite{KKM14} use statistical metrics for mappings of\nBitcoin to IP addresses, and identify pairs that may represent ownership\nrelations.\n\nSeveral solutions arose to address this problem. Miers et al.~\\cite{MGGR13}\nproposed ZeroCoin, that extends Bitcoin with a cryptographic accumulator and\nzero knowledge proofs to provide fully anonymous currency transactions.\nBen-Sasson et al.~\\cite{BSCG0MTV14} introduced Zerocash, a decentralized\nanonymous payment solution that hides all information linking the source and\ndestination of transactions. Bonneau et al.~\\cite{BNMCKF14} proposed\nMixcoin, a currency mix with accountability assurances and randomized fee\nbased incentives.\n\nOur work is orthogonal to previous work on Bitcoin anonymity, as it identifies\nvulnerabilities in Stratum, the communication protocol employed by\n\\newmaterial{cryptocurrency} mining solutions. As such, our concern is for the\nprivacy and security of the miners, as they generate coins. Our attacks are\nalso more general, as they apply not only to Bitcoin, but to a suite of other\npopular altcoin solutions,\ne.g.,~\\cite{litecoin_stratum,ethereum_stratum,monero_stratum} that build on\nStratum.\n\n\\noindent\n{\\bf Effects of broken crypto on Bitcoin}.\nGiechaskiel et al.~\\cite{GCR16} systematically analyze the effects of broken\ncryptographic primitives on Bitcoin. They reveal a wide range of possible\neffects that range from minor privacy violations to a complete breakdown of the\ncurrency. Our attacks do not need broken crypto to succeed.  However, we show\nthat Bedrock, our secure Stratum extension is resilient to broken crypto\nprimitives.\n\n\\vspace{-10pt}\n\n\\section{Conclusions}\n\n\\vspace{-5pt}\n\nIn this paper we have shown that the lack of security in Stratum, Bitcoin's\nmining communication protocol, makes miners vulnerable to a suite of passive\nand active attacks, that expose their owners to hacking, coin and equipment\ntheft, loss of revenue, and prosecution. We have implemented and shown that the\nattacks that we introduced are efficient. Our attacks reveal that encryption is\nnot only undesirable, due to its significant overheads, but also ineffective:\nan adversary can predict miner earnings even when given access to only the\ntimestamps of miner communications. We have developed Bedrock, a minimal and\nefficient Stratum extension that protects the privacy and security of mining\nprotocol participants. We have shown that Bedrock imposes an almost negligible\ncomputation overhead on the mining participants and is resilient to active\nattacks even if the used cryptographic tools are compromised.\n\n\\section{Acknowledgments}\n\n\\newmaterial{\nWe thank the shepherds and the anonymous reviewers for their excellent\nfeedback.  We thank Patrick O'Callaghan for suggesting this problem and for\ninsightful discussions. This research was supported by NSF grants 1526494 and\n1527153.}\n\n\\vspace{-10pt}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe problem of designing auctions that maximize the seller's revenue in settings with many heterogeneous goods has attracted a large amount of interest in the last years, both from the Computer Science as well as the Economics community (see e.g.~\\citep{Manelli:2006vn,Pavlov:2011fk,Hart:2012uq,Hart:2012zr,Daskalakis:2012fk,Daskalakis:2013vn,gk2014,Menicucci:2014jl,Daskalakis:2014fk}). Here the seller faces a buyer whose true values for the $m$ items come from a probability distribution over $\\mathbb R_+^m$ and, based only on this incomplete prior knowledge, he wishes to design a selling mechanism that will maximize his expected revenue. For the purposes of this paper, the prior distribution is a product one, meaning that the item values are independent. The buyer is additive, in the sense that her happiness from receiving any subset of items is the sum of her values of the individual items in that bundle. The buyer is also selfish and completely rational, thus willing to lie about her true values if this is to improve her own happiness. So, the seller should also make sure to give the right incentives to the buyer in order to avoid manipulation of the protocol by misreporting. \n\nThe special case of a single item has been very well understood since the seminal work of~\\citet{Myerson:1981aa}. However, when one moves to settings with multiple goods, the problem becomes notoriously difficult and novel approaches are necessary. Despite the significant effort of the researchers in the field, essentially only specialized, partial results are known: there are exact solutions for two items in the case of identical uniform distributions over unit-length intervals~\\citep{Pavlov:2011fk,Manelli:2006vn}, exponential over $[0,\\infty)$ \\citep{Daskalakis:2013vn} or identical Pareto distributions with tail index parameters $\\alpha\\geq 1\/2$ \\citep{Hart:2012uq}. For more than two items, optimal results are only known for uniform values over the unit interval~\\citep{gk2014}, and due to the difficulty of exact solutions most of the work focuses in showing approximation guarantees for simple selling mechanisms~\\citep{Hart:2012uq,Li:2013ty,Babaioff:2014ys,g2014,Bateni:2014ph,Rubinstein:2015kx}. This difficulty is further supported by the complexity ($\\#P$-hardness) results of~\\citet{Daskalakis:2012fk}. It is important to point out that even for two items \\emph{we know of no general and simple, closed-form conditions framework under which optimality can be extracted when given as input the item distributions, in the case when these are not necessarily identical.} This is our goal in the current paper. \n \n\\paragraph{Our contribution}\nWe introduce general but simple and clear, closed-form distributional conditions that can guarantee optimality and immediately give the form of the revenue-maximizing selling mechanism (its payment and allocation rules), for the setting of two goods with values distributed over bounded intervals (Theorem~\\ref{th:characterization_main}). For simplicity and a clearer exposition we study distributions supported over the real unit interval $[0,1]$. By scaling, the results generalize immediately to intervals that start at $0$, but more work would be needed to generalize them to arbitrary intervals. We use the closed forms to get optimal solutions for a wide class of distributions satisfying certain simple analytic assumptions (Theorem~\\ref{th:characterization_iid} and Sect.~\\ref{sec:non-iid}). As useful examples, we provide exact solutions for families of monomial ($\\propto x^c$) and exponential ($\\propto e^{-\\lambda x}$) distributions (Corollaries~\\ref{th:optimal_two_power} and \\ref{th:optimal_two_expo} and Sect.~\\ref{sec:non-iid}), and also near-optimal results for power-law ($\\propto (x+1)^{-\\alpha}$) distributions (Sect.~\\ref{sec:approximate_convex_fail}). This last approximation is an application of a more general result (Theorem~\\ref{th:two_iid_approx}) involving the relaxation of some of the conditions for optimality in the main Theorem~\\ref{th:characterization_main}; the ``solution'' one gets in this new setting might not always correspond to a feasible selling mechanism, however it still provides an upper bound on the optimal revenue as well as hints as to how to design a well-performing mechanism, by ``convexifying'' it into a feasible mechanism (Sect.~\\ref{sec:approximate_convex_fail}).\n\nParticularly for the family of monomial distributions it turns out that the optimal mechanism is a very simple deterministic mechanism that offers to the seller a menu of size just $4$ (using the menu-complexity notion of Hart and Nisan \\citep{Hart:2012ys,Wang:2013ab}): fixed prices for each one of the two items and for their bundle, as well as the option of not buying any of them. For other distributions studied in the current paper randomization is essential for optimality, as is generally expected in such problems of multidimensional revenue maximization (see e.g.~\\citep{Hart:2012zr,Pavlov:2011fk,Daskalakis:2013vn}). For example, this is the case for two i.i.d. exponential distributions over the unit interval $[0,1]$, which gives the first such example where determinism is suboptimal even for regularly\\footnote{A probability distribution $F$ is called \\emph{regular} if $t-\\frac{1-F(t)}{f(t)}$ is increasing. This quantity is known as the \\emph{virtual valuation}.} i.i.d.\\ items. \nA point worth noting here is the striking difference between this result and previous results~\\citep{Daskalakis:2013vn,g2014} about i.i.d.\\ exponential distributions which have as support the entire $\\mathbb R_+$: the optimal selling mechanism there is the deterministic one that just offers the full bundle of both items.\n\nAlthough the conditions that the probability distributions must satisfy are quite general, they leave out a large class of distributions. For example, they do not apply to power-law distributions with parameter $\\alpha>2$. In other words, this work goes some way towards the complete solution for arbitrary distributions for two items, but the general problem is still open. In this paper, we opted towards simple conditions rather than full generality, but we believe that extensions of our method can generalize significantly the range of distributions; we expect that a proper ``ironing'' procedure will enable our technique to resolve the general problem for two items.\n\n\\paragraph{Techniques}\nThe main result of the paper (Theorem~\\ref{th:characterization_main}) is proven by utilizing the \\emph{duality} framework of~\\citep{gk2014} for revenue maximization, and in particular using complementarity: the optimality of the proposed selling mechanism is shown by verifying the existence of a dual solution with which they satisfy together the required complementary slackness conditions of the duality formulation. Constructing these dual solutions explicitly seems to be a very challenging task and in fact there might not even be a concise way to do it, especially in closed-form. So instead we just prove the existence of such a dual solution, using a \\emph{max-flow min-cut} argument as main tool (Lemma~\\ref{lemma:coloring}, Fig.~\\ref{fig:flows_graph}). This is, in a way, an abstraction of a technique followed in~\\citep{gk2014} for the case of uniform distributions which was based on Hall's theorem for bipartite matchings. Since here we are dealing with general and non-identical distributions, this kind of refinement is essential and non-trivial, and in fact forms the most technical part of the paper. Our approach has a strong geometric flavor, enabled by introducing the notion of the \\emph{deficiency} of a two-dimensional body (Definition~\\ref{def:deficiency}, Lemma~\\ref{lemma:no_positive_def}), which is inspired by classic matching theory~\\citep{Ore:1955fk,Lovasz:1986qf}. \n\n\\subsection{Model and Notation}\nWe study a two-good monopoly setting in which a seller deals with a buyer who has values $x_1, x_2\\in I$ for the items, where $I=[0,1]$. The seller has only an incomplete knowledge of the buyer's preference, in the form of two independent distributions (with densities) $f_1$, $f_2$ over $I$ from which $x_1$ and $x_2$ are drawn, respectively. The cdf of $f_j$ will be denoted by $F_j$. As in the seminal work of~\\citet{Myerson:1981aa}, the density functions will be assumed to be absolutely continuous and positive.\nWe will also use vector notation $\\vecc x=(x_1,x_2)$. For any item $j\\in\\{1,2\\}$, index $-j$ will refer the complementary item, that is $3-j$, and as it's standard in game theory $\\vecc x_{-j}=x_{-j}$ will denote the remaining of vector $\\vecc x$ if the $j$-th coordinate is removed, so $\\vecc x=(x_j,x_{-j})$ for any $j=1,2$.\n\nThe seller's goal is to design a selling mechanism that will maximize his revenue. Without loss\\footnote{This is due to the celebrated Revelation Principle~\\citep{Myerson:1981aa}.} we can focus on direct-revelation mechanisms: the bidder will be asked to submit bids $b_1,b_2$ and the mechanism consists simply of an allocation rule $a_1,a_2:I^2\\to I$ and a payment function $p:I^2\\to\\mathbb R_+$ such that $a_j(b_1,b_2)$ is the probability of item $j$ being sold to the buyer (notice how we allow for randomized mechanisms, i.e.~lotteries) and $p(b_1,b_2)$ is the payment that the buyer expects to pay; it is easier to consider the expected payment for all allocations, rather than individual payments that depend on the allocation of items. The reason why the bids $b_j$ are denoted differently than the original values $x_j$ for the items is that, since the bidder is a rational and selfish agent, she might lie and misreport $b_j\\neq x_j$ if this is to increase her personal gain given by the quasi-linear \\emph{utility} function \n\\begin{equation}\n\\label{eq:utility}\nu(\\vecc b;\\vecc x)\\equiv a_1(\\vecc b) x_1+a_2(\\vecc b) x_2-p(\\vecc b),\n\\end{equation}\nthe expected happiness she'll receive by the mechanism minus her payment. \nThus, we will demand our selling mechanisms to satisfy the following standard properties: \n\\begin{itemize}\n\\item \\emph{Incentive Compatibility (IC)}, also known as truthfulness, saying that the player would have no incentive to misreport and manipulate the mechanism, i.e.\\ her utility is maximized by truth-telling: $u(\\vecc b;\\vecc x)\\leq u(\\vecc x;\\vecc x)$ \n\\item \\emph{Individual Rationality (IR)}, saying that the buyer cannot harm herself just by truthfully participating in the mechanism: $u(\\vecc x;\\vecc x)\\geq 0$.\n\\end{itemize}\nIt turns out the critical IC property comes without loss\\footnote{Also due to the Revelation Principle.} for our revenue-maximization objective, so for now on we will only consider truthful mechanisms, meaning we can also relax the notation $u(\\vecc b;\\vecc x)$ to just $u(\\vecc x)$.\n\nThere is a very elegant and helpful analytic characterization of truthfulness, going back to~\\citet{Rochet:1985aa} (for a proof see e.g.~\\citep{Hart:2012uq}), which states that the player's utility function must be \\emph{convex} and that the allocation probabilities are simply given by the utility's derivatives, i.e.\\ $\\partial u(\\vecc x)\/\\partial x_j=a_j(\\vecc x)$. Taking this into consideration and rearranging~\\eqref{eq:utility} with respect to the payment, we define\n$$\n\\mathcal R_{f_1,f_2}(u)\\equiv \\int_0^1\\int_0^1\\left(\\frac{\\partial u(\\vecc x)}{\\partial x_1}x_1+\\frac{\\partial u(\\vecc x)}{\\partial x_2}x_2-u(\\vecc x) \\right)f_1(x_1)f_2(x_2)\\,dx_1\\,dx_2\n$$\nfor every absolutely continuous function $u:I^2\\longrightarrow\\mathbb R_+$. If $u$ is convex with partial derivatives in $[0,1]$ then $u$ is a valid utility function and \\emph{$\\mathcal R_{f_1,f_2}(u)$ is the expected revenue of the seller under the mechanism induced by $u$}. Let $\\ensuremath{\\text{\\rm\\sc Rev}}(f_1,f_2)$ denote the best possible such revenue, i.e.\\ the supremum of $\\mathcal R_{f_1,f_2}(u)$ when $u$ ranges over the space of all feasible utility functions over $I^2$. So the problem we want to deal with in this paper is exactly that of $\\sup_u \\mathcal R_{f_1,f_2}(u)$.\n\nWe now present the condition on the probability distributions which will enable our technique to provide a closed-form of the optimal auction.\n\n\\begin{assumption}\n\\label{assume:upwards_def}\n\\label{assume:regularity}\nThe probability distributions $f_1,f_2$ are such that functions $h_{f_1,f_2}(\\vecc x)-f_2(1)f_1(x_1)$ and $h_{f_1,f_2}(\\vecc x)-f_1(1)f_2(x_2)$ are nonnegative, where\n\\begin{equation}\n\\label{eq:def_h}\nh_{f_1,f_2}(\\vecc x)\\equiv 3 f_1(x_1)f_2(x_2)+x_1f_1'(x_1)f_2(x_2)+x_2f'_2(x_2)f_1(x_1).\n\\end{equation}\nFunction $h_{f_1,f_2}$ will also be assumed to be absolutely continuous with respect to each of its coordinates.\n\\end{assumption}\nWe will drop the subscript $f_1,f_2$ in the above notations whenever it is clear which distributions we are referring to. Assumption~\\ref{assume:upwards_def} is a slightly stronger condition than $h(\\vecc x)\\geq 0$ which is a common regularity assumption in the economics literature for multidimensional auctions with $m$ items: $(m+1)f(\\vecc x)+\\nabla f(\\vecc x)\\cdot \\vecc x\\geq 0$, where $f$ is the joint distribution for the item values (see e.g.~\\citep{Manelli:2006vn,Pavlov:2011fk,McAfee:1988nx}). In fact, \\citet{Manelli:2006vn} make the even stronger assumption that for each item $j$, $x_j f_j(x_j)$ is an increasing function. Even more recently, that assumption has also been deployed by \\citet{Wang:2013ab} in a two-item setting as one of their sufficient conditions for the existence of optimal auctions with small-sized menus.  It has a strong connection with the standard single-dimensional regularity condition of~\\citet{Myerson:1981aa}, since for $m=1$ condition $h(\\vecc x)\\geq 0$ gives that $f(x)\\left( x-\\frac{1-F(x)}{f(x)} \\right)$ is increasing, thus ensures the single-crossing property of the virtual valuation function (see also the discussion in \\citep[Sect.~2]{Manelli:2006vn}). \n\nStrengthening the regularity condition $h(\\vecc x)\\geq 0$ to that of Assumption~\\ref{assume:upwards_def} is essentially only used \nas a technical tool within the proof of Lemma~\\ref{lemma:no_positive_def}, and\nas a matter of fact we don't really need it to hold in the entire unit box $I^2$ but just in a critical sub-region $D_{1,2}$ which corresponds to the valuation subspace where both items are sold with probability $1$ (see Fig.~\\ref{fig:Exp_Uniform} and Sect.~\\ref{sec:partition_optimal}). As mentioned earlier in the Introduction, we introduce this technical conditions in order to simplify our exposition and enforce the clarity of the techniques, but we believe that a proper ``ironing''~\\citep{Myerson:1981aa} process can probably bypass these restrictions and generalize our results.\nThe critical Assumption~\\ref{assume:upwards_def} is of course satisfied by all distributions considered in the results of this paper, namely monomial $\\propto x^c$ for any power $c\\geq 0$ (Corollary~\\ref{th:optimal_two_power}), exponential $\\propto e^{-\\lambda x}$ with rates $\\lambda\\leq 1$ (Corollary~\\ref{th:optimal_two_expo}), power-law $\\propto (t+1)^{-\\alpha}$ with parameters $\\alpha\\leq 2$ (Example~\\ref{example:power-law}), as well as combinations of these (see Example~\\ref{example:uniform-expo}). However, there is still a large class of distributions not captured by Assumption~\\ref{assume:upwards_def} as it is, e.g.\\ exponential with rates larger than $1$, power-law with parameters greater than $2$ and some beta-distributions (take, for example, $\\propto x^2(1-x)^2$). See Footnote~\\ref{foot:alter-assumption-monotone} for an alternative condition that can replace Assumption~\\ref{assume:upwards_def}.\n\\section{Sufficient Conditions for Optimality}\nThis section is dedicated to proving the main result of the paper:\n\\begin{theorem}\n\\label{th:characterization_main}\nIf there exist decreasing, concave functions $s_1,s_2:I\\to I$, with $s_1'(t),s_2'(t)> -1$ for all $t\\in I$, such that for almost every\\footnote{Everywhere except a subset of zero Lebesgue measure.} (a.e.) $x_1,x_2\\in I$\n\\begin{equation}\n\\label{eq:1slice_gen_functions}\n\\frac{s_1(x_2)f_1(s_1(x_2))}{1-F_1(s_1(x_2))} =2+\\frac{x_2f_2'(x_2)}{f_2(x_2)}\n\\quad\\text{and}\\quad\n\\frac{s_2(x_1)f_2(s_2(x_1))}{1-F_2(s_2(x_1))} =2+\\frac{x_1f_1'(x_1)}{f_1(x_1)}, \n\\end{equation}\nthen \nthere exists a constant $p\\in[0,2]$ such that \n\\begin{equation}\n\\label{eq:2slice_gen}\n\\int_{D}h(\\vecc x)\\,dx_1\\,dx_2 \n=f_1(1)+f_2(1)\n\\end{equation}\nwhere $D$ is the region of $I^2$ enclosed by curves\\footnote{See Fig.~\\ref{fig:Exp_Uniform}.} $x_1+x_2=p$, $x_1=s_1(x_2)$ and $x_2=s_2(x_1)$ and including point $(1,1)$, i.e.~$D=\\sset{\\vecc x\\in I\\fwh{x_1+x_2\\geq p\\lor x_1\\geq s_1(x_2) \\lor x_2\\geq s_2(x_1)}}$,   \nand the optimal selling mechanism is given by the utility function\n\\begin{equation}\n\\label{eq:optimal_auction_gen}\nu(\\vecc x)=\\max\\sset{0,x_1-s_1(x_2),x_2-s_2(x_1),x_1+x_2-p}.\n\\end{equation}\nIn particular, if $p\\leq\\min\\sset{s_1(0),s_2(0)}$, then the optimal mechanism is the deterministic full-bundling with price $p$. \n\\end{theorem}\n\nNotice that for any $s\\in I$ we have\n\\begin{align*}\n\\int_s^1h(\\vecc x)\\,dx_1 &=\\int_s^1 3f_1(x_1)f_2(x_2)+x_1f_1'(x_1)f_2(x_2)+x_2f_2'(x_2)f_1(x_1) \\,dx_1\\\\\n\t\t\t\t&=3f_2(x_2)(1-F_1(s))+f_2(x_2)\\int_s^1x_1f_1'(x_1)\\,dx_1+x_2f_2'(x_2)(1-F_1(s))\\\\\n\t\t\t\t&=3f_2(x_2)(1-F_1(s))+f_2(x_2)\\left(\\left[x_1f_1(x_1)\\right]_s^1-(1-F_1(s))\\right)+x_2f_2'(x_2)(1-F_1(s))\\\\\n\t\t\t\t&=2f_2(x_2)(1-F_1(s))+f_2(x_2)(f_1(1)-sf_1(s))+x_2f_2'(x_2)(1-F_1(s))\\\\\n\t\t\t\t&=(1-F_1(s))f_2(x_2)\\left[2+\\frac{x_2f_2'(x_2)}{f_2(x_2)}-\\frac{sf_1(s)}{1-F_1(s)} \\right] +f_1(1)f_2(x_2)\n\\end{align*}\nwhich means that an equivalent way of looking at~\\eqref{eq:1slice_gen_functions} is, more simply, by\n\\begin{equation}\n\\label{eq:1slice_gen_integrals}\n\\int_{s_1(x_2)}^1h(\\vecc x)\\,dx_1=f_1(1)f_2(x_2)\n\\quad\\text{and}\\quad\n\\int_{s_2(x_1)}^1h(\\vecc x)\\,dx_2=f_2(1)f_1(x_1).\n\\end{equation}\nThis also means that~\\eqref{eq:1slice_gen_integrals} can take the place of~\\eqref{eq:1slice_gen_functions} in the statement of Theorem~\\ref{th:characterization_main} whenever this gives an easier way to solve for functions $s_1$ and $s_2$.\n\\subsection{Partitioning of the Valuation Space}\n\\label{sec:partition_optimal}\n\\begin{figure}\n\\centering\n\\includegraphics[width=10cm]{Figures\/Exp_Uniform.pdf}\n\\caption{\\footnotesize The valuation space partitioning of the optimal selling mechanism for two independent items, one following a uniform distribution and the other an exponential with parameter $\\lambda=1$. Here $s_1(t)=(2-t)\/(3-t)$, $s_2(t)=2-W(2e)\\approx 0.625$  and $p\\approx 0.787$. In region $D_{1}$ (light grey) item $1$ is sold deterministically and item $2$ with a probability of $-s_1'(x_2)$, in $D_{2}$ (light grey) only item $2$ is sold and region $D_{1,2}$ (dark grey) is where the full bundle is sold deterministically, for a price of $p$.}\n\\label{fig:Exp_Uniform}\n\\end{figure}\nDue to the fact that the derivatives of functions $s_j$ in Theorem~\\ref{th:characterization_main} are above $-1$, each curve $x_1=s_1(x_2)$ and $x_2=s_2(x_1)$ can intersect the full-bundle line $x_1+x_2=p$ at most at a single point. So let $x_2^*=x_2^*(p), x_1^*=x_1^*(p)$ be the coordinates of these intersections, respectively, i.e.~$s_1(x_2^*)=p-x_2^*$ and $s_2(x_1^*)=p-x_1^*$. If such an intersection does not exist, just define $x_2^*=0$ or $x_1^*=0$.\n\nThe construction and the optimal mechanism given in Theorem~\\ref{th:characterization_main} then gives rise to the following partitioning of the valuation space $I^2$ (see Fig.~\\ref{fig:Exp_Uniform}):\n\\begin{itemize}\n\\item Region $\\bar D=I^2\\setminus D$ where no item is allocated\n\\item Region $D_1=\\sset{\\vecc x\\in I^2\\fwh{x_1\\geq s_1(x_2)\\land x_2\\leq x_2^*}}$ where item $1$ is sold with probability $1$ and item $2$ with probability $-s_1'(x_2)$ for a price of $s_1(x_2)-x_2s_1'(x_2)$\n\\item Region $D_2=\\sset{\\vecc x\\in I^2\\fwh{x_2\\geq s_2(x_1)\\land x_1\\leq x_1^*}}$ where item $2$ is sold with probability $1$ and item $1$ with probability $-s_2'(x_1)$ for a price of $s_2(x_1)-x_1s_2'(x_1)$\n\\item Region $D_{1,2}=D\\setminus{D_1\\union D_2}=\\sset{\\vecc x\\in I^2\\fwh{x_1+x_2\\geq p \\land x_1\\geq x_1^* \\land x_2\\geq x_2^*}}$ where both items are sold deterministically in a full bundle of price $p$.\n\\end{itemize}\n\nUnder this decomposition, by \\eqref{eq:1slice_gen_integrals}:\n$$\n\\int_{D_{1}}h(\\vecc x)\\,dx_1\\,dx_2=\\int_{0}^{x_2^*}\\int_{s_1(x_2)}^1h(\\vecc x)\\,dx_1\\,dx_2=f_1(1)F_2(x_2^*)\n$$\nso expression~\\eqref{eq:2slice_gen} can be written equivalently as\n\\begin{equation}\n\\label{eq:2slice_gen_bundle_region}\n\\int_{D_{1,2}}h(\\vecc x)\\,dx_1\\,dx_2 \n= f_1(1)(1-F_2(x_2^*))+f_2(1)(1-F_1(x_1^*)).\n\\end{equation}\n\\subsection{Duality}\n\\label{sec:duality}\nThe major underlying tool to prove Theorem~\\ref{th:characterization_main} will be the duality framework of~\\citep{gk2014}. For completeness we briefly present here the formulation and key aspects, and the interested reader is referred to the original text for further details. \n\nRemember that the revenue optimization problem we want to solve here is to maximize $\\mathcal R(u)$ over the space of all convex functions $u:I^2\\longrightarrow\\mathbb R_+$ with\n\\begin{equation}\n\\label{eq:allocs_probs_01}\n0\\leq \\frac{\\partial u(\\vecc x)}{\\partial x_j}\\leq 1,\\qquad j=1,2,\n\\end{equation}\nfor a.e. $\\vecc x\\in I^2$. First we relax this problem by dropping the convexity assumption and replacing it with (absolute) continuity. We also drop the lower bound in~\\eqref{eq:allocs_probs_01}. Then this new relaxed program is dual to the following: minimize $\\int_0^1\\int_0^1 z_1(\\vecc x)+z_2(\\vecc x)\\,d\\vecc x$ where the new dual variables $z_1,z_2:I^2\\longrightarrow\\mathbb R_+$ are such that $z_j$ is (absolutely) continuous with respect to its $j$-coordinate and the following conditions are satisfied for all $x_1,x_2\\in I$:\n\\begin{align}\nz_j(0,x_{-j}) &=0, &&j=1,2, \\label{eq:dual_const_1}\\\\\nz_j(1,x_{-j}) &\\geq f_j(1)f_{-j}(x_{-j}), &&j=1,2, \\label{eq:dual_const_2}\\\\\n\\frac{\\partial z_1(\\vecc x)}{\\partial x_2}+\\frac{\\partial z_2(\\vecc x)}{\\partial x_2} &\\leq 3 f_1(x_1)f_2(x_2)+ x_1f_1'(x_1)f_2(x_2)+x_2f_1(x_1)f_2'(x_2).\\label{eq:dual_const_3}\n\\end{align}\nWe will refer to the first optimization problem, where $u$ ranges over the relaxed space of continuous, nonnegative functions with derivatives at most $1$, as the \\emph{primal program} and to the second as the \\emph{dual}. Intuitively, every dual solution $z_j$ must start at zero and grow all the way up to $f_j(1)f_{-j}(x_{-j})$ while travelling in interval $I$, in a way that the sum of the rate of growth of both $z_1$ and $z_2$ is never faster than the right hand side of~\\eqref{eq:dual_const_3}.\nIn~\\citep{gk2014} is proven that indeed these two programs satisfy both weak duality, i.e.~for any feasible $u,z_1,z_2$ we have\n$$\n\\mathcal R(u)\\leq \\int_{0}^1\\int_0^1 z_1(\\vecc x)+z_2(\\vecc x)\\,d\\vecc x\n$$ \nas well as complementary slackness, in the form of the even stronger following form of $\\varepsilon$-complementarity: \n\n\\begin{lemma}[Complementarity]\\label{lemma:complementarity}\nIf $u,z_1,z_2$ are feasible primal and dual solutions, respectively, $\\varepsilon>0$ and the following complementarity constraints hold for a.e. $\\vecc x\\in I^2$,\n\\begin{align}\n  u(\\vecc x)   \\left( h(\\vecc x)\n    -\\frac{\\partial z_1(\\vecc x)}{\\partial x_1}-\\frac{\\partial z_2(\\vecc x)}{\\partial x_2}\n  \\right) &\\leq \\varepsilon f_1(x_1)f_2(x_2), \\label{eq:e_compl_2}\\\\\nu(1,x_{-j})  \\left( z_j(1, x_{-j})\n    -f_j(1)f_{-j}(x_{-j}) \\right) &\\leq \\varepsilon f_j(1)f_{-j}(x_{-j}) \\label{eq:e_compl_3}, &&j=1,2,\\\\\n  z_j(\\vecc x)  \\left( 1 - \\frac{\\partial\n        u(\\vecc x)}{\\partial x_j} \\right)   &\\leq \\varepsilon f_1(x_1)f_2(x_2),  &&j=1,2, \\label{eq:e_compl_4}\n\\end{align}\nwhere $h$ is defined in~\\eqref{eq:def_h}, then the values of the primal and dual programs differ by at most\n$7\\varepsilon$. In particular, if the conditions are satisfied\nwith $\\varepsilon=0$, both solutions are optimal.\n\\end{lemma}   \n\nOur approach into proving Theorem~\\ref{th:characterization_main} will be to show the existence of a pair of dual solutions $z_1,z_2$ with respect to which the utility function $u$ given by the theorem indeed satisfies complementarity. Notice here the existential character of our technique: our duality approach offers the advantage to use the proof of just the existence of such duals, without having to explicitly describe them and compute their objective value in order to prove optimality, i.e.~that the primal and dual objectives are indeed equal. Also notice that the utility function $u$ given by Theorem~\\ref{th:characterization_main} is convex by construction, so in case someone shows optimality for $u$ in the relaxed setting, then $u$ must also be optimal among all feasible mechanisms.\n\nDefine function $W:I^2\\to\\mathbb R_+$ by\n$$\nW(\\vecc x)=\n\\begin{cases}\nh(\\vecc x), &\\text{if}\\;\\; \\vecc x\\in D,\\\\\n0, &\\text{otherwise},\n\\end{cases} \n$$\nwhere $D$ is defined in Sect.~\\ref{sec:partition_optimal} (see Fig.~\\ref{fig:Exp_Uniform}).\nIf one could decompose $W$ into functions $w_1,w_2:I^2\\to\\mathbb R_+$ such that\n\\begin{align}\nw_1(\\vecc x)+w_2(\\vecc x) &=W(\\vecc x)\\label{eq:Wdecomp_sum} \\\\\n\\int_0^1w_j(\\vecc x)\\,d x_j &= f_j(1)f_{-j}(x_{-j}) \\label{eq:Wdecomp1}, \\qquad j=1,2,\n\\end{align}\nfor all $\\vecc x\\in I$, and $w_j$ is almost everywhere continuous with respect  to its $j$-th coordinate, then by defining \n$$\nz_j(\\vecc x)=\\int_0^{x_j} w_j(t,x_{-j})\\,dt\n$$\nwe'll have\n\\begin{align}\n\\frac{\\partial z_1(\\vecc x)}{\\partial x_1}+\\frac{\\partial z_2(\\vecc x_2)}{\\partial x_2} &=\n\\begin{cases}\n h(\\vecc x), & \\text{for}\\;\\; \\vecc x\\in D,\\\\\n 0, &\\text{otherwise},\n \\end{cases}\n \\label{eq:prop_dual_1}\n \\\\\nz_j(0,x_{-j}) &=0, && j=1,2, \\label{eq:prop_dual_2} \\\\\nz_j(1,x_{-j}) &= f_j(1)f_{-j}(x_{-j}), && j=1,2. \\label{eq:prop_dual_3} \n\\end{align}\nIf the requirements of Theorem~\\ref{th:characterization_main} hold, then it is fairly straightforward to get such a decomposition in certain regions. In particular, we can set $w_1=w_2=0$ in $I^2\\setminus D$, $w_1=W=h$ and $w_2=0$ in $D_1$ and $w_2=W=h$ and $w_1=0$ in $D_2$. Then, by~\\eqref{eq:1slice_gen_integrals}, it is not difficult to see that indeed conditions~\\eqref{eq:Wdecomp_sum}--\\eqref{eq:Wdecomp1} are satisfied. However, \\emph{it is highly non-trivial how to create such a decomposition in the remaining region $D_{1,2}$} and that is what the proof of Lemma~\\ref{lemma:coloring} achieves, with the assistance of the geometric Lemma~\\ref{lemma:no_positive_def}, in the remaining of this section. This is the most technical part of the paper.\n\nIn any case, if we are able to get such a decomposition, by the previous discussion that would mean that functions $z_1,z_2:I^2\\to\\mathbb R_+$ are \\emph{feasible dual} solutions: it is trivial to verify that properties~\\eqref{eq:prop_dual_1}--\\eqref{eq:prop_dual_3} satisfy the dual constraints \\eqref{eq:dual_const_1}--\\eqref{eq:dual_const_3}.  But most importantly, the \\emph{equalities} in properties~\\eqref{eq:prop_dual_1}--\\eqref{eq:prop_dual_3} and the way $w_1$ and $w_2$ are defined  in regions $D_1$ and $D_2$ tell us something more: that this pair of solutions would satisfy complementarity with respect to the primal given in~\\eqref{eq:optimal_auction_gen} and whose allocation is analyzed in detail in Sect.~\\ref{sec:partition_optimal}, thus proving that this mechanism is optimal and thus establishing Theorem~\\ref{th:characterization_main}.\n\n\\subsection{Deficiency}\n\\label{sec:nodef}\nThe following notion will be the tool that gives a very useful geometric interpretation to the rest of the proof of Theorem~\\ref{th:characterization_main} and it will be critical into proving Lemma~\\ref{lemma:coloring}.   \n\\begin{definition}\n\\label{def:deficiency}\nFor any body $S\\subseteq I^2$ define its \\emph{deficiency} (with respect to distributions $f_1,f_2$) to be\n$$\n\\delta(S)\\equiv \\int_S h(\\vecc x)\\,d\\vecc x - f_2(1)\\int_{S_1}f_1(x_1)\\,dx_1-f_1(1)\\int_{S_2}f_2(x_2)\\,dx_2,\n$$\nwhere $S_1$, $S_2$ denote $S$'s projections to the $x_1$ and $x_2$ axis, respectively.\n\\end{definition}\n\\begin{lemma}\n\\label{lemma:no_positive_def}\nIf the requirements of Theorem~\\ref{th:characterization_main} hold, then no body $S\\subseteq D_{1,2}$ has positive deficiency.\n\\end{lemma}\n\\begin{proof}\nTo get to a contradiction, assume that there is body $S\\subseteq D_{1,2}$ with $\\delta(S)>0$. \nFirst, we'll show that without loss $S$ can be assumed to be upwards closed. Intuitively, we'll show that one can push mass of $S$ to the right or upwards, without reducing its deficiency. By Assumption~\\ref{assume:upwards_def} function $h(\\vecc x)-f_2(1)f_1(x_1)$ is nonnegative. Then, if there exists a nonempty horizontal line segment $\\slice{S}{x_2}{t}$ of $S$ at some height $x_2=t$, then we can assume that this line segment fills the entire available horizontal space of $D_{1,2}$: if that was not the case, and there existed a small interval $[\\alpha,\\beta]\\times{t}$ that was not in $S$, then we could add it to it, not increasing the projection towards the $x_2$-axis (it is already covered by the other existing points at $x_2=t$) and the projection towards the $x_1$-axis is increased at most by $\\beta-\\alpha$, leading to a change to the overall deficiency by at most $\\int_{\\alpha}^{\\beta}h(\\vecc x)\\,dx_1-f_2(1)\\int_{\\alpha}^{\\beta}f_1(x_1)\\,dx_1$, which is nonnegative\\footnote{We must mention here that the assumption of the nonnegativity of $h(\\vecc x) -f_2(1)f_1(x_1)$ could be replaced by that of $h(\\vecc x)-f_2(1)f_1(x_1)$ being increasing with respect to $x_1$ and the argument would still carry through: we can move entire columns  of $S$ to the right, pushing elements horizontally; the projection towards axis $x_2$ again remains unchanged, and because of the monotonicity of $h(\\vecc x)-f_2(1)f_1(x_1)$, the overall deficiency will not decrease since we are integrating over higher values of $x_1$.\n\nThis means that the monotonicity of $h(\\vecc x)-f_j(x_j)j_{-j}(1)$ with respect to $x_j$ can replace its nonnegativity in the initial Assumption~\\ref{assume:upwards_def} (while still maintaining the regularity requirement of $h(\\vecc x)$ being nonnegative) without affecting the main results of this paper, namely Theorems \\ref{th:characterization_main}, \\ref{th:characterization_iid} and \\ref{th:two_iid_approx}.\n\\label{foot:alter-assumption-monotone}\n}.\n\nSo $S$ can be assumed to be the intersection of $D_{1,2}$ with a box, i.e.~$S=[t_1,1]\\times[t_2,1]\\inters D_{1,2}$, where $t_1\\geq x_1^*$ and $t_2\\geq x_2^*$. This also means that its projections are $S_1=[t_1,1]$ and $S_2=[t_2,1]$.\nNow consider the lowest horizontal slice $\\slice{S}{x_2}{t_2}$ of $S$. It obviously lies within $D_{1,2}$. But from condition~\\eqref{eq:1slice_gen_integrals} so do all horizontal line segments of the form $[s_1(x_2),1]$ for any $x_2\\in[x_2^*, t_2]$: $s_1(x_2)$ is decreasing and specifically less steeply than the line $-x_2+p$ which is the boundary of $D_{1,2}$. So, by adding all these segments to $S$ we won't increase the projections towards the $x_1$-axis (these are covered already by $\\slice{S}{x_2}{t_2}$, which has to be a superset of $[s_1(t_2),1]$, otherwise it would have a negative deficiency, see~\\eqref{eq:1slice_gen_integrals}) and the new projections towards the $x_2$-axis are dominated by the increase of the area of $S$ (this segments have nonnegative deficiency). So, $S$ can be assumed to project in the entire boundaries $[x_1^*,1]$ and $[x_2^*,1]$ of $D_{1,2}$ and thus, since $h$ is nonnegative, $S$ can be assumed to fill the entire $D_{1,2}$ region. But by the definition of price $p$ in Theorem~\\ref{th:characterization_main}, $\\delta(D_{1,2})=0$ which concludes the proof. \n\\end{proof}\n\\subsection{Dual Solution and Optimality}\n\\label{sec:optimality}\nNotice that Theorem~\\ref{th:characterization_main} ensures the existence of a full-bundling price in~\\eqref{eq:2slice_gen}. This needs to be proven. Indeed,\nquantity $\\int_Dh(\\vecc x)\\,d\\vecc x$ continuously (weakly) increases as $p$ decreases, and for $p=0$ %\n\\begin{align*}\n\\int_Dh(\\vecc x)\\,d\\vecc x &=\\int_0^1\\int_0^1 3f_1(x_1)f_2(x_2)+x_1f_1'(x_1)f_2(x_2)+yf_2'(x_2)f_1(x_1)\\,dx_1\\,dx_2\\\\\n\t\t&=3+(f_1(1)-1)+(f_2(1)-1)=1+f_1(1)+f_2(1)\n\t\t>f_1(1)+f_2(1)\n\\end{align*}\nwhile for $p=\\hat x_1+\\hat x_2$, where $(\\hat x_1,\\hat x_2)$ is the unique point of intersection of the curves $x_2=s_1(x_1)$ and $x_1=s_2(x_2)$ in $I^2$ (such a point certainly exists because $s_1$ and $s_2$ are defined over the entire $I$), \n\\begin{align*}\n\\int_{D_{1,2}}h(\\vecc x)\\,d\\vecc x &=\\int_{\\hat x_2}^1\\int_{\\hat x_1}^1 h(\\vecc x)\\,d\\vecc x\n\t\t\\leq \\int_{\\hat x_2}^1\\int_{s_1(x_2)}^1 h(\\vecc x)\\,d\\vecc x\n\t\t= \\int_{\\hat x_2}^1f_1(1)f_2(x_2)\\,dx_2\\\\\n\t\t&= f_1(1)(1-F_2(\\hat x_2))\n\t\t\\leq f_1(1)(1-F_2(\\hat x_2))+f_2(1)(1-F_1(\\hat x_1)),\n\\end{align*}\nthe first inequality holding because $h$ is nonnegative and $s_1(x_2)\\leq s_1(\\hat x_2)=\\hat x_1$ ($s_1$ is decreasing), and the second equality by substituting~\\eqref{eq:1slice_gen_integrals}, and from~\\eqref{eq:2slice_gen_bundle_region} this means that $\\int_{D}h\\,d\\vecc x\\leq f_1(1)+f_2(1)$.\n\nCombining the above, indeed there must be a $p\\in[0,\\hat x_1+\\hat x_2]$ such that $\\int_{D}h\\,d\\vecc x=f_1(1)+f_2(1)$. In fact, using this argument, if for $p=\\min\\sset{s_1(0),s_2(0)}$ it is $\\int_{D}h\\,d\\vecc x<f_1(1)+f_2(1)$ then $p$ must go below this value to get a solution, meaning that the full-bundling region will cover the rest of the regions $D_1$ and $D_2$, i.e.~$D=D_{1,2}$, and the mechanism defined by~\\eqref{eq:optimal_auction_gen} is a deterministic full-bundling.  \n\nThe following lemma will complete the proof of Theorem~\\ref{th:characterization_main}. It is the most technical part of this paper, and utilizes a max-flow min-cut argument in order to prove the existence of a feasible dual pair $z_1,z_2$ that satisfies the complementarity conditions with respect to the utility function given by Theorem~\\ref{th:characterization_main}, thus establishing optimality. It is inspired by the bipartite matching approach in~\\citep{gk2014} where Hall's theorem is used in order to prove existence, in the special case of uniformly distributed items. Here we need to abstract and generalize our approach in order to incorporate general distributions in the most smooth way possible. The proof has a strong geometric flavor, which is achieved by utilizing the notion of deficiency that was introduced in Sect.~\\ref{sec:nodef} and using Lemma~\\ref{lemma:no_positive_def}.  \n\\begin{lemma}\n\\label{lemma:coloring}\nAssume that the conditions of Theorem~\\ref{th:characterization_main} hold. Then for arbitrary small $\\varepsilon>0$, there exist feasible dual solutions $z_1,z_2$ which are $\\varepsilon$-complementary to the (primal) $u$ given by~\\eqref{eq:optimal_auction_gen}. Therefore, the mechanism induced by $u$ is optimal.\n\\end{lemma}\n\\begin{proof}\nFollowing the discussion in Sect.~\\ref{sec:duality}, we would like to decompose $W$ into the desired functions $w_1$ and $w_2$ within $D_{1,2}$, i.e.~such that they satisfy~\\eqref{eq:Wdecomp_sum}--\\eqref{eq:Wdecomp1}. In fact, we are aiming for $\\varepsilon$-complementarity, so we can relax conditions~\\eqref{eq:Wdecomp1} a bit: \n\\begin{equation}\n\\int_0^1w_j(\\vecc x)\\,dx_j \\leq f_j(1)f_{-j}(x_{-j})+\\varepsilon' \n\\label{eq:relax_dual_boundary}\n\\end{equation}\nTo be precise, the $\\varepsilon$-complementarity of Lemma~\\ref{lemma:complementarity} dictates that regarding these conditions we must show that for a.e.\\ $\\vecc x\\in D_{1,2}$ property~\\eqref{eq:e_compl_3} holds (conditions~\\eqref{eq:e_compl_2} and~\\eqref{eq:e_compl_4} are immediately satisfied with strong equality, by~\\eqref{eq:prop_dual_1} and the fact that within $D_{1,2}$ both items are sold deterministically with probability $1$.).\nBut since $u(\\vecc x)\\leq x_1+x_2 \\leq 2$ for all $x_1,x_2\\in I$ ($u$'s derivatives are at most $1$ with respect to any direction) and also exists $M>0$ such that $f_1(1)f_2(x_2),f_2(1)f_1(x_1)\\geq M$ for all $\\vecc x\\in D_{1,2}$ (the density functions are continuous over the closed interval $I$ and positive\\footnote{We would like to note here that this is the only point in the paper where the fact that the densities are \\emph{strictly} positive is used. As a matter of fact, a closer look will reveal that the proof just needs the property to hold in the closure of $D_{1,2}$ and not necessarily in the entire domain $I^2$. This allows the consideration of a wider family of feasible distributional priors, for example the monomial distributions of Corollary~\\ref{th:optimal_two_power}: their densities $f(t)=(c+1)t^c$ may vanish at $t=0$ but these ``problematic'' points happen to lie outside the area $D_{1,2}$ where both items are sold.}), indeed~\\eqref{eq:relax_dual_boundary} is enough to guarantee $\\varepsilon$ complementarity if one ensures $\\varepsilon'\\leq \\varepsilon M\/2$. So, the remaining of the proof is dedicated into constructing nonnegative, a.e.~continuous functions $w_1$ and $w_2$ over $D_{1,2}$, such that $w_1+w_2=h$ and~\\eqref{eq:relax_dual_boundary} are satisfied.\n\nWe will do that by constructing an appropriate graph and recovering $w_1$ and $w_2$ as ``flows'' through its nodes, deploying the min-cut max-flow theorem to prove existence. To start, we pick an arbitrary small $\\delta>0$ and discretize $I^2$ into a lattice of $\\delta$-size boxes $[(i-1)\\delta,i\\delta] \\times [(j-1)\\delta,j\\delta]$, where $i,j=1,2,\\dots,1\/\\delta$, selecting $\\delta$ such that $1\/\\delta$ is an integer. Denote the intersection of such a box with $D_{1,2}$ by $B_{i,j}$. Also, let $B^1_i$ denote the projection of all nonempty $B_{i,j}$'s, as $j$ ranges, towards the $x_1$-axis and $B^2_j$ towards the $x_2$-axis, as $i$ ranges. Note that these are well-defined in this way, since by the geometry of region $D_{1,2}$ two nonempty $B_{i,j}$, $B_{i',j'}$ will have the same vertical projection if $i=i'$ and the same horizontal if $j=j'$. Also, it is a simple fact to observe that all $B^1_i$ and $B^2_j$ are single-dimensional real intervals of length at most $\\delta$.\n\nNow let's construct a directed graph $G=(V,E)$, together with a capacity function $c(e)$ for all edges $e\\in E$. Initially, for any pair $(i,j)$ such that $B_{i,j}$ has positive (two-dimensional Lebesgue) measure we insert a node $v(i,j)$ in $V$. We'll call these nodes \\emph{internal} and we'll denote them by $V_o$. Also, for any internal node $v(i,j)$ we add nodes $v_1(i)$ and $v_2(j)$ corresponding to entire columns and rows, calling them \\emph{column} and \\emph{row} vertices and denoting them by $V_1$ and $V_2$, respectively. Finally there are two special nodes, a source $\\sigma$ and a destination $\\tau$. From the source to all internal nodes $v=v(i,j)$ we add an edge $(\\sigma,v)$ with capacity equal to the area of $B_{i,j}$ under $h$, i.e.~$c(\\sigma,v)=\\int_{B_{i,j}}h(\\vecc x)\\,d\\vecc x$. From any internal node $v=v(i,j)$ to its external column and row nodes $v_1=v_1(i)$ and $v_2=v_2(j)$ we add edges with capacities $c(v,v_1)=c(v,v_2)=c(\\sigma,v)$ equal to the internal node's incoming edge capacity from the source. Finally, for all external nodes $v_1(i)\\in V_1$ and $v_2(j)\\in V_2$ we add edges towards the destination $\\tau$ with capacities $c(v_1,\\tau)=f_2(1)\\int_{B^1_i}f_1(x_1)\\,dx_1$ and $c(v_2,\\tau)=f_1(1)\\int_{B^2_j}f_2(x_2)\\,dx_2$, respectively. The structure of graph $G$ is depicted in Fig.~\\ref{fig:flows_graph}.\n \\begin{figure}\n \\centering\n \\includegraphics[width=10cm]{Figures\/flows.pdf}\n \\caption{\\footnotesize The graph $G$ in the proof of Lemma~\\ref{lemma:coloring}. Every internal node $B_{i,j}$ of region $D_{1,2}$ can receive at most $\\int_{B_{i,j}}h(\\vecc x)\\,d\\vecc x$ flow from the source node $\\sigma$ and can send at most that amount to each one of its neighbouring external nodes $B_{i}^1$ and $B_{j}^2$. Every external node $B^1_i$ and $B^2_j$ is connected to the destination $\\tau$ with edges of capacity $f_2(1)\\int_{B_i^1}f_1(x_1)\\,dx_1$ and $f_1(1)\\int_{B_j^2}f_2(x_2)\\,dx_2$, respectively. Internal $B_{i,j}$'s are two-dimensional intersections of $\\delta$-boxes with $D_{1,2}$, while the external ones, $B^1_i$ and $B^2_j$ are single dimensional intervals of length $\\delta$.}\n \\label{fig:flows_graph}\n \\end{figure}\n\nAs a first observation, notice that the maximum flow that can be sent from $\\sigma$ within the graph is $\\int_{D_{1,2}}h(\\vecc x)\\,dx_1\\,dx_2$ and the maximum flow that $\\tau$ can receive is \n$$f_2(1)\\int_{x_1^*}^1f_1(x_1)\\,dx_1+f_1(1)\\int_{x_2^*}^1f_2(x_2)\\,dx_2$$ (remember that the projection of $D_{1,2}$ to the $x_1$-axis is $[x_1^*,1]$ and to the $x_2$-axis $[x_2^*,1]$). But, from the way the entire region $D$ is constructed, we know that the above two quantities are equal (see~\\eqref{eq:2slice_gen_bundle_region}). Let's denote this value by $\\psi$. Next, we will prove that indeed one can create a feasible flow through $G$ that achieves that maximum value $\\psi$. From the max-flow min-cut theorem, it is enough to show that the minimum $(\\sigma,\\tau)$-cut of $G$ has a value of at least $\\psi$. To do that, we'll show that $(\\sigma,V\\setminus\\{\\sigma\\})$ is a minimum cut of $G$.\n\nIndeed, let $(S,V\\setminus S)$ be a $(\\sigma,\\tau)$-cut of $G$. First, let there be an edge $(v,v_j)$ crossing the cut, i.e.~$v\\in S$ and $v_j\\notin S$, with $v$ internal node and $v_j$ external. Then, by moving $v$ at the other side of the cut, i.e.~removing it from $S$, we would create at most a new edge contributing to the cut, namely $(\\sigma,v)$ but also destroy at least one edge $(v,v_j)$. Since the capacities of these two edges are the same, the overall effect would be to get a new cut with weakly smaller value. So, from now on we can assume that for all edges $(v,v_j)$ of $G$, if $v\\in S$ then also $v_j\\in S$. Under this assumption, if $S_{o}=V_{o}\\inters S$ denotes the set of internal nodes belonging at the left side of the cut, for every $v\\in S_{o}$ all edges $(v,v_j)$ adjacent to $v$ will not cross the cut. However, this means that all edges $(v_j,\\tau)$, where $v_j\\in N(v)$\\footnote{$N(v)$ denotes the set of neighbours of $v$ in graph $G$.}, do contribute to the cut. But then, if we remove all nodes in $S_{o}$, together with their neighbouring external nodes $N(S_{o})$ at the other side of the cut, we increase the cut's value by at most $\\sum_{v\\in S_{o}}c(\\sigma,v)$ and at the same time reduce it by at least $\\sum_{v_j\\in N(S_{o})}c(v_j,\\tau)$. However, by the way graph $G$ is constructed, this corresponds to an overall increase in the cut of at least \n$$\\int_B h(\\vecc x)\\,d\\vecc x -f_2(1) \\int_{B_{1}}f_1(x_1)\\,dx_1 -f_1(1)\\int_{B_{2}}f_2(x_2)\\,dx_2,$$ \nwhere $B=\\union_{v(i,j)\\in S_{o}}B_{i,j}$ is the region of $D_{1,2}$ covered by the boxes of nodes in $S_{o}$ and $B_1$, $B_2$ are the projections of this body to the horizontal and vertical axis, respectively. From Lemma~\\ref{lemma:no_positive_def} this difference must be nonpositive, thus this change results in a cut of an even (weakly) smaller value. The above arguments show that indeed the cut that has only $\\sigma$ remaining at its left side is a minimum one.\n\nSo, there must be a flow $\\phi:E\\longrightarrow\\mathbb R_+$, achieving to transfer a total value of $\\psi$ through $G$. As we argued above though, by the construction of $G$, in order to achieve this value of $\\psi$ the full capacity of \\emph{all} edges $(\\sigma, v)$ as well as that of all $(v_j,\\tau)$ must be used. So, this flow $f$ manages to elegantly separate all incoming flow $\\phi(\\sigma,v(i,j))=\\int_{B_{i,j}}h(\\vecc x)\\,d\\vecc x$ towards an internal box of $D_{1,2}$, into a sum of flows $\\phi(v(i,j),v_1(i))+\\phi(v(i,j),v_2(j))$ towards its external neighbours. But this is exactly what we need in order to construct our feasible dual solution! For simplicity, denote this incoming flow $\\phi(i,j)$ and the outgoing ones $\\phi_1(i,j)$ and $\\phi_2(i,j)$, respectively. Then, define the functions $w_1$, $w_2$ throughout $D_{1,2}$ by\n$$\nw_1(\\vecc x)=\\frac{\\phi_1(i,j)}{\\phi(i,j)}h(\\vecc x)\\qquad\\text{and}\\qquad w_2(\\vecc x)=\\frac{\\phi_2(i,j)}{\\phi(i,j)}h(\\vecc x),\n$$\nwhere $B_{i,j}$ is the discretization box where point $\\vecc x$ of $D_{1,2}$ belongs to. In that way, first notice that we achieve $w_1+w_2=h$. Secondly, functions $w_1$ and $w_2$ are almost everywhere continuous, since the values of the flows are constant within the boxes, and our discretization is finite. The only remaining property to prove is~\\eqref{eq:relax_dual_boundary}. \n\nFix some height $x_2=\\tilde x_2$ such that this horizontal line intersects $D_{1,2}$. We'll prove that \n$$\\int_{0}^{1}w_1(x_1,\\tilde x_2)\\,dx_1-f_1(1)f_2(\\tilde x_2)\\leq \\varepsilon'.$$ \nValue $\\tilde x_2$ falls within some interval of the discretization, let $\\tilde x_2\\in[(\\tilde j-1)\\delta,\\tilde j\\delta]=B_{\\tilde j}^2$. The average value of function $f_1(1)f_2(x_2)$ (with respect to $x_2$) within this interval is $$\\frac{1}{\\delta}f_1(1)\\int_{B^2_{\\tilde j}}f_2(x_2)\\,dx_2=c(v_2(\\tilde j),\\tau)\/\\delta$$ and the average value of $\\int_0^1w_1(\\vecc x)\\,dx_1$ is \n$$\n\\frac{1}{\\delta}\\int_{B^2_{\\tilde j}}\\int_0^1w_1(\\vecc x)\\,dx_1 = \\frac{1}{\\delta}\\sum_i\\int_{B_{i,\\tilde j}}w_1(\\vecc x)\\,d\\vecc x  = \\frac{1}{\\delta}\\sum_i\\frac{\\phi_1(i,j)}{\\phi(i,j)}\\int_{B_{i,\\tilde j}}h(\\vecc x)\\,d\\vecc x \n= \\sum_i \\phi_1(i,\\tilde j)\/\\delta.\n$$\nBut since the sum of the outgoing flows over any horizontal line of internal nodes of the graph (here $j=\\tilde j$) must equal the outgoing flow of the corresponding external node (here $v_2(\\tilde j)$), the above quantities are equal. Thus, by selecting the discretization parameter $\\delta$ small enough, we can indeed make the values $\\int_{0}^{1}w_1(x_1,\\tilde x_2)\\,dx_1$ and $f_1(1)f_2(\\tilde x_2)$ to be $\\varepsilon'$ close to each other\n\\footnote{This should feel intuitively clear, and it relies on the uniform continuity of functions $f_2$ and $h$, but we also give a formal proof in Appendix~\\ref{append:technical_flow_rest}.}.\n\\end{proof}\n\n\n\\section{The Case of Identical Items}\nIn this section we focus on the case of identically distributed values, i.e.~$f_1(t)=f_2(t)\\equiv f(t)$ for all $t\\in I$, and we provide clear and simple conditions under which the critical property~\\eqref{eq:1slice_gen_functions} of Theorem~\\ref{th:characterization_main} holds.  \n\nFirst notice that in this case the regularity Assumption~\\ref{assume:regularity} gives $3+\\frac{x_1f'(x_1)}{f(x_1)}+\\frac{x_2f'(x_2)}{f(x_2)}\\geq 0$ a.e. in $I^2$ (since $f$ is positive) and thus $\\frac{tf'(t)}{f(t)}\\geq -\\frac{3}{2}$ for a.e. $t\\in I$. An equivalent way of writing this is that $t^{3\/2}f(t)$ is increasing, which interestingly is the complementary case of that studied by~\\citet{Hart:2012uq} for two i.i.d. items: they show that when $t^{3\/2}f(t)$ is decreasing, then deterministically selling in a full bundle is optimal.\n\n\\begin{theorem}\n\\label{th:characterization_iid}\nAssume that $G(t)=tf(t)\/(1-F(t))$ and $H(t)=tf'(t)\/f(t)$ give rise to well defined, differentiable functions over $I$, $G$ being strictly increasing and convex, $H$ decreasing and concave, with $G+H$ increasing and $G(1)\\geq 2+H(0)$. Then the requirements of Theorem~\\ref{th:characterization_main} are satisfied. In particular \n$$s(t)=G^{-1}(2+H(t))$$\nand, if  \n\\begin{equation}\n\\label{eq:two_iid_full_bundle_price}\n\\int_0^1\\int_0^1h(\\vecc x)\\,d\\vecc x-\\int_0^p\\int_0^{p-x_2}h(\\vecc x)\\,d\\vecc x-2f(1)\n\\end{equation}\nis nonpositive for $p=s(0)$ then the optimal selling mechanism is the one offering deterministically the full bundle for a price of $p$ being the root of \\eqref{eq:two_iid_full_bundle_price} in $[0,s(0)]$, otherwise the optimal mechanism is the one defined by the utility function\n$$\nu(\\vecc x)=\\max\\sset{0,x_1-s(x_2),x_2-s(x_1),x_1+x_2-p}\n$$\nwith $p=x^*+s(x^*)$, where $x^*\\in [0,s(0)]$ is the constant we get by solving \n\\begin{equation}\n\\label{eq:price_bundle_eq_iid}\n\\int_{x^*}^{s(x^*)}\\int_{s(x^*)+x^*-x_2}^1h(\\vecc x)\\,d\\vecc x+\\int_{s(x^*)}^1\\int_{x^*}^1h(\\vecc x)\\,d\\vecc x= 2f(1)(1-F(x^*)).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFunction $G$ is strictly monotone, thus invertible and has a range of $[G(0),G(1)]=[0,G(1)]\\supseteq [0,2+H(0)]$. By Assumption~\\ref{assume:regularity} and the previous discussion, it must be $tf'(t)\/f(t)\\geq -3\/2$, so $2+H(t)\\geq 1\/2>0$ for all $t\\in I$. Thus, $s(t)=G^{-1}(2+H(t))$ is well defined and furthermore it is decreasing, since $G$ is increasing and $H$ decreasing. Also, by the way $s$ is defined we get that for all $t$: $G(s(t))=2+H(t)$, which is exactly condition~\\eqref{eq:1slice_gen_functions} of Theorem~\\ref{th:characterization_main}. \n\nIt remains to be shown that $s$ is concave and that $s'(t)>-1$. From the definition of $s$, $s'(t)=H'(t)\/G'(s(t))$. Function $H$ is decreasing and concave, so $H'(t)$ is negative and decreasing, and function $G$ is increasing and convex and $s$ decreasing, so $G'(s(t))$ is positive and decreasing. Combining these we get that the ratio $H'(t)\/G'(s(t))$ is decreasing, proving that $s$ is concave. Finally, notice that since we are in a two item i.i.d. setting, the only part of curve $x_2=s(x_1)$ that matters and may appear in the utility of the resulting mechanism~\\eqref{eq:optimal_auction_gen} is the one where $x_1\\leq x_2$ (curves $x_2=s(x_1)$ and $x_1=s(x_2)$ will intersect on the line $x_1=x_2$), so we only have to show that $s'(t)>-1$ for $t\\leq s(t)$. Indeed, in that case $G'(t)\\leq G'(s(t))$, so $s'(t)=H'(t)\/G'(s(t))\\geq H'(t)\/G'(t)$  and thus it is enough to show that $H'(t)-G'(t)\\geq 0$ which we know holds since $H+G$ is assumed to be increasing.\n\n\\end{proof}\n\n\n\n\\begin{corollary}[Monomial Distributions]\n\\label{th:optimal_two_power}\nThe optimal selling mechanism for two items with i.i.d.\\ values from the family of distributions with densities $f(t)=(c+1)t^c$, $c\\geq 0$, is deterministic. In particular, it offers each item for a price of $s=\\sqrt[c+1]{\\frac{c+2}{2c+3}}$ and the full bundle for a price of $p=s+x^*$, where $x^*$ is the solution to~\\eqref{eq:price_bundle_eq_iid}.\n\\end{corollary}\n\\begin{proof}\nFor two monomial i.i.d.\\ items with $f_1(t)=f_2(t)=(c+1)t^c$ we have $h(\\vecc x)=(c+1)^2 (2 c+3) x_1^c x_2^c\\geq 0$, thus  $h(\\vecc x)-f_2(1)f_1(x_1)=(c+1)^2 x_1^c \\left((2 c+3) x_2^c-1\\right)$ which is nonnegative for all $x_2\\geq \\sqrt[c]{1\/(2c+3)}\\equiv \\omega$. So, in order to make sure that Assumption~\\ref{assume:upwards_def} is satisfied, it is enough to show that $x^*\\geq\\omega$ because then $D_{1,2}\\subseteq [\\omega,1]^2$. We'll soon show that this is indeed satisfied for all $c\\geq 0$.\n\nApplying Theorem~\\ref{th:characterization_iid} we compute: $G(t)=(c+1)t^{c+1}\/(1-t^{c+1})$ which is strictly increasing and convex in $I$ and $H(t)=c$ which is constant and thus decreasing and concave. Also, it is trivial to deduce that $G+H$ is increasing and $\\lim_{t\\to 1^{-}}G(t)=\\infty>2+c=2+H(0)$. Then, it is valid to compute $G^{-1}(t)=\\left(\\frac{3+2c}{2+c}\\right)^{-\\frac{1}{1+c}}$ and thus $s(t)=\\sqrt[c+1]{\\frac{c+2}{2c+3}}$ which is constant. \n\nRegarding the computation of the full-bundle price $p$, condition~\\eqref{eq:price_bundle_eq_iid} gives rise to quantity\n$$\n\\int_{x^*}^{s}\\int_{s+x^*-x_2}^1x_1^cx_2^c\\,d\\vecc x+\\int_{s}^1\\int_{x^*}^1x_1^cx_2^c\\,d\\vecc x- \\frac{2}{c+1}(1-{x^*}^{c+1}),\n$$\nwhich by plugging-in $x^*=\\omega$ and using the values of $s$ and $\\omega$ (as functions of $c$) one can see that  it is positive for all $c\\geq 0$. So, by the discussion in the beginning of Sect.~\\ref{sec:optimality} it can be deduced that the solution to~\\eqref{eq:price_bundle_eq_iid}  will be such that $x^*>\\omega$. \n\\end{proof}\nNotice that for $c=0$ the setting of Corollary~\\ref{th:optimal_two_power} reduces to a two uniformly distributed goods setting, and gives the well-known results of $s=2\/3$ and $p=(4-\\sqrt{2})\/3$ (see e.g.~\\citep{Manelli:2006vn}). For the linear distribution $f(t)=2t$, where $c=1$, we get $s=\\sqrt{3\/5}$ and $p\\approx1.091$.\n\n\\begin{corollary}[Exponential Distributions]\n\\label{th:optimal_two_expo}\nThe optimal selling mechanism for two items with exponentially i.i.d.\\ values over $[0,1]$, i.e.\\ having densities $f(t)=\\lambda e^{-\\lambda t}\/(1-e^{-\\lambda})$, with $0<\\lambda\\leq 1$, is the one having $s(t)=\\frac{1}{\\lambda}\\left[2-\\lambda t-W\\left(e^{2-\\lambda-\\lambda t}(2-\\lambda t)\\right)\\right]$ and a price of\n$p=x^*+s(x^*)$ for the full bundle, where $x^*$ is the solution to~\\eqref{eq:price_bundle_eq_iid}. Here $W$ is Lambert's product logarithm function\\footnote{Function $W$ can be defined as the solution to $W(t)e^{W(t)}=t$.}.\n\\end{corollary}\n\\begin{proof}\nFor two i.i.d. exponentially distributed items with $f_1(t)=f_2(t)=\\lambda e^{-\\lambda t}\/(1-e^{-\\lambda})$ we have \n$$\n\\hspace{-0.5cm}\nh(\\vecc x)-f_2(1)f_1(x_1)=\\frac{\\lambda^2}{\\left(e^\\lambda-1\\right)^2} e^{2-\\lambda (x_1+x_2)} (3-\\lambda (x_1+x_2)-e^{\\lambda x_2})\\geq \\frac{\\lambda^2}{\\left(e^\\lambda-1\\right)^2} e^{2-\\lambda (x_1+x_2)} (2-\\lambda (x_1+x_2))\\geq 0\n$$ \nfor all $x_1,x_2\\in I$, since $\\lambda\\leq 1$.\n\nApplying Theorem~\\ref{th:characterization_iid} we compute: $G(t)=\\lambda t\/(1-e^{-\\lambda(1-t)})$ which is strictly increasing and convex in $I$ and $H(t)=-\\lambda t$ which is decreasing and concave. Also, $G(t)+H(t)=\\lambda te^{-\\lambda(1-t)}\/(1-e^{-\\lambda(1-t)})$ is increasing and $\\lim_{t\\to 1^{-}}G(t)=\\infty>2=2+H(0)$. Then, it is valid to compute $G^{-1}(t)=t\/\\lambda-W\\left(t e^{t-\\lambda}\\right)\/\\lambda$ and thus $s(t)=\\frac{1}{\\lambda}\\left[2-\\lambda t-W\\left(e^{2-\\lambda-\\lambda t}(2-\\lambda t)\\right)\\right]$.\n\\end{proof}\n\nFor example, for $\\lambda=1$ we get $s(t)=2-t-W\\left(e^{1-t} (2-t)\\right)$ and $p\\approx 0.714$. \nInterestingly, to our knowledge this is the first example for an i.i.d.\\ setting with values coming from a regular, continuous distribution over an interval $[0,b]$, where an optimal selling mechanism is \\emph{not} deterministic. Also notice how this case of exponential i.i.d.\\ items on a bounded interval is different from the one on $[0,\\infty)$: by \\citep{Daskalakis:2013vn,g2014} we know that at the unbounded case the optimal selling mechanism for two exponential i.i.d.\\ items is simply the deterministic full-bundling, but in our case of the bounded $I$ this is not the case any more.\n\n\\section{Non-Identical Items}\n\\label{sec:non-iid}\n\nAn interesting aspect of the technique of Theorem~\\ref{th:characterization_iid} is that it can readily be used also for non identically distributed values. One just has to define $G_j(t)\\equiv tf_j(t)\/(1-F_j(t))$ and $H_j(t)=tf_j'(t)\/f_j(t)$ for both items $j=1,2$ and check again whether $G_1,G_2$ are strictly increasing and convex and $H_1,H_2$ nonnegative, decreasing and concave. Then, we can get $s_j(t)=G_j^{-1}(2+H_{-j}(t))$ and check if $s_j(1)> -1$ and the price $p$ of the full bundle can be given by~\\eqref{eq:2slice_gen}. Again, a quick check of whether full bundling is optimal is to see if for $p=\\min\\sset{s_1(0),s_2(0)}$ expression $\\int_0^1\\int_0^1h(\\vecc x)\\,d\\vecc x-\\int_0^p\\int_0^{p-x_2}h(\\vecc x)\\,d\\vecc x-f_1(1)-f_2(1)$ is nonpositive.\n\n\\begin{example}\n\\label{example:uniform-expo}\nConsider two independent items, one having uniform valuation $f_1(t)=1$ and one exponential $f_2(t)=e^{-t}\/(1-e^{-1})$. Then we get that $s_1(t)=(2-t)\/(3-t)$, $s_2(t)=2-W(2e)\\approx 0.625$ and $p \\approx0.787$. The optimal selling mechanism offers either only item $2$ for a price of $s_2\\approx 0.625$, or item $1$ deterministically and item $2$ with a probability $s_1'(x_2)$ for a price of $s_1(x_2)-x_2s_1'(x_2)$, or the full bundle for a price of $p\\approx 0.787$. You can see the allocation space of this mechanism in Fig.~\\ref{fig:Exp_Uniform}.\n\\end{example}\n\n\\section{Approximate Solutions}\n\\label{sec:approximate_convex_fail}\nIn the previous sections we developed tools that, under certain assumptions, can give a complete closed-form description of the optimal selling mechanism. However, remember that the initial primal-dual formulation upon which our analysis was based, assumes a relaxed optimization problem. Namely, we dropped the convexity assumption of the utility function $u$. In the results of the previous sections this comes for free: the optimal solution to the relaxed program turns out to be convex anyways, as a result of the requirements of Theorem~\\ref{th:characterization_main}. But what happens if that was not the case? The following tool shows that even in that case our results are still applicable and very useful for both finding good upper bounds on the optimal revenue (Theorem~\\ref{th:two_iid_approx}) as well as designing almost-optimal mechanisms that have provably very good performance guarantees (Sect.~\\ref{sec:convexification}).\n\\begin{theorem}\n\\label{th:two_iid_approx}\nAssume that all conditions of Theorem~\\ref{th:characterization_main} are satisfied, except from the concavity of functions $s_1,s_2$. Then, the function $u$ given by that theorem might not be convex any more and thus not a valid utility function, but it generates an \\emph{upper bound} to the optimal revenue, i.e.\\ $\\ensuremath{\\text{\\rm\\sc Rev}}(f_1,f_2)\\leq\\mathcal R_{f_1,f_2}(u)$. In particular, this is the case if all the requirements of Theorem~\\ref{th:characterization_iid} hold except the concavity of $H$.\n\\end{theorem}\n\\begin{proof}\nThe proof is a straightforward result of the duality framework (see Sect.~\\ref{sec:duality}): By dropping only the concavity requirement of functions $s_1$ and $s_2$ but satisfying all the remaining conditions of Theorem~\\ref{th:characterization_main}, we still construct an optimal solution to the pair of primal-dual programs, meaning that function $u$ produced in \\eqref{eq:optimal_auction_gen} maximizes $\\mathcal R_{f_1,f_2}(u)$ over the space of all functions $u: I^2\\longrightarrow \\mathbb R_+$ with partial derivatives in $[0,1]$ (see~\\eqref{eq:allocs_probs_01}); the only difference is that $u$ might not be convex since $s_1,s_2$ might not be concave any more. The actual optimal revenue objective $\\ensuremath{\\text{\\rm\\sc Rev}}(f_1,f_2)$ has the extra constraint of $u$ being convex, thus, given that it is a maximization problem, it has to be that $\\ensuremath{\\text{\\rm\\sc Rev}}(f_1,f_2)\\leq \\mathcal R_{f_1,f_2}(u)$. Finally, it is easy to verify in the proof of Theorem~\\ref{th:characterization_iid} that dropping just the concavity requirement for $H$ can only affect the concavity of functions $s_1,s_2$ and hence the convexity of $u$.\n\\end{proof}\n\n\\begin{example}[Power-Law Distributions]\n\\label{example:power-law}\n A class of important distributions that falls into the description of Theorem~\\ref{th:two_iid_approx} are the power-law distributions with parameters $0<\\alpha\\leq 2$. More specifically, these are the distributions having densities $f(t)=c\/(t+1)^\\alpha$, with the normalization factor $c$ selected so that $\\int_0^1f(t)\\,dt=1$, i.e.~$c=(a-1)\/(1-2^{1-\\alpha})$. It is not difficult to verify that these distributions satisfy Assumption~\\ref{assume:upwards_def}. For example, for $\\alpha=2$ one gets $f(x)=2\/(x+1)^2$, the \\emph{equal revenue} distribution shifted in the unit interval. For this we can compute via~\\eqref{th:two_iid_approx} that $s(t)=\\frac{1}{2} \\sqrt{5+2 t+t^2}-\\frac{1}{2} (1+t)$ and $p\\approx 0.665$, which gives an upper bound of $R_{f,f}(u)\\approx 0.383$ to the optimal revenue $\\ensuremath{\\text{\\rm\\sc Rev}}(f,f)$.\n\\end{example}\n\n\\subsection{Convexification}\n\\label{sec:convexification}\nThe approximation results described in Theorem~\\ref{th:two_iid_approx} can be used not only for giving upper bounds on the optimal revenue, but also as a \\emph{design} technique for good selling mechanisms. Since the only deviation from a feasible utility function is the fact that function $s$ is not concave (and thus $u$ is not convex), why don't we try to ``convexify'' $u$, by replacing $s$ by a concave function $\\tilde s$? If $\\tilde s$ is ``close enough'' to the original $s$, by the previous discussion this would also result in good approximation ratios for the new, feasible selling mechanism. \n\nLet's demonstrate this by an example, using the equal revenue distribution $f(t)=2\/(t+1)^2$ of the previous example. We need to replace $s$ with a concave $\\tilde s$ in the interval $[0,x^*]$. So let's choose $\\tilde s$ to be the concave hull of $s$, i.e.~the minimum concave function that dominates $s$. Since $s$ is convex, this is simply the line that connects the two ends of the graph of $s$ in $[0,x^*]$, that is, the line\n$$\n\\tilde s(t)=\\frac{s(0)-s(x^*)}{x^*}(x^*-t)+s(x^*).\n$$\nA calculation shows that this new \\emph{valid} mechanism has an expected revenue which is within a factor of just $1+3\\times 10^{-9}$ of the upper bound given by $s$ using Theorem~\\ref{th:two_iid_approx}, rendering it essentially optimal.\n\n\\paragraph{Acknowledgements:} We thank Anna Karlin, Amos Fiat, Costis Daskalakis and Ian Kash for insightful discussions. We also thank the anonymous reviewers for their useful comments on the conference version of this paper.\n\n\\bibliographystyle{abbrvnat} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Appendix: Experimental Details}\n\\label{sec:appendix_exp_details}\n\n\\subsection{Models}\nWe use the following families of architectures.\nThe PyTorch~\\cite{paszke2017automatic} specification of our ResNets and CNNs\nare available at \\url{https:\/\/gitlab.com\/harvard-machine-learning\/double-descent\/tree\/master}.\n\n\\paragraph{ResNets.}\nWe define a family of ResNet18s of increasing size as follows.\nWe follow the Preactivation ResNet18 architecture of \\cite{he2016identity},\nusing 4 ResNet blocks, each consisting of two BatchNorm-ReLU-Convolution layers. The layer widths for the 4 blocks are $[k, 2k, 4k, 8k]$ for varying $k \\in \\mathbb{N}$ and the strides are [1, 2, 2, 2].\nThe standard ResNet18 corresponds to $k=64$ convolutional channels in the first layer.\nThe scaling of model size with $k$ is shown in Figure~\\ref{fig:resnet_params}.\nOur implementation is adapted from \\url{https:\/\/github.com\/kuangliu\/pytorch-cifar}.\n\n\\paragraph{Standard CNNs.}\nWe consider a simple family of 5-layer CNNs,\nwith four Conv-BatchNorm-ReLU-MaxPool layers and a fully-connected output layer.\nWe scale the four convolutional layer widths as $[k, 2k, 4k, 8k]$. The MaxPool is [1, 2, 2, 8]. For all the convolution layers, the kernel size = 3, stride = 1 and padding=1. \nThis architecture is based on the ``backbone'' architecture from \n\\cite{mcnn}.\nFor $k=64$, this CNN has 1558026 parameters and can reach $>90\\%$ test accuracy on CIFAR-10 (\\cite{cifar}) with data-augmentation.\nThe scaling of model size with $k$ is shown in Figure~\\ref{fig:mcnn_params}.\n\n\n\\paragraph{Transformers.}\nWe consider the encoder-decoder Transformer model from \\cite{Vaswani}\nwith 6 layers and 8 attention heads per layer, as implemented by fairseq \\cite{ott2019fairseq}.\nWe scale the size of the network by modifying the embedding dimension ($d_{\\text{model}}$),\nand scale the width of the fully-connected layers proportionally ($d_{\\text{ff}} = 4 d_{\\text{model}}$).\nWe train with 10\\% label smoothing and no drop-out, for 80 gradient steps.\n\n\\begin{figure}[!h]\n    \\centering\n    \\begin{subfigure}{.3\\textwidth}\n      \\includegraphics[width=\\textwidth]{mcnn_parameters}\n      \\caption{5-layer CNNs}\n    \\label{fig:mcnn_params}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}{.3\\textwidth}\n      \\includegraphics[width=\\textwidth]{resnet_parameters}\n      \\caption{ResNet18s}\n    \\label{fig:resnet_params}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}{.3\\textwidth}\n      \\includegraphics[width=\\textwidth]{nlp_parameters}\n      \\caption{Transformers}\n    \\label{fig:resnet_params2}\n    \\end{subfigure}\n    \\label{fig:nparams}\n    \\caption{Scaling of model size with our\n    parameterization of width \\& embedding dimension.}\n\\end{figure}\n\n\\subsection{Image Classification: Experimental Setup}\nWe describe the details of training for CNNs and ResNets below.\n\n\\textbf{Loss function:} Unless stated otherwise, we use the cross-entropy loss for all the experiments.\n\n\\textbf{Data-augmentation:} In experiments where data-augmentation was used, we apply \\texttt{RandomCrop(32, padding=4)} and \\texttt{RandomHorizontalFlip}. In experiments with added label noise, the label for all augmentations of a given training sample are given the same label.\n\n\\textbf{Regularization:} No explicit regularization like weight decay or dropout was applied unless explicitly stated.\n\n\\textbf{Initialization:} We use the default initialization provided by PyTorch for all the layers.\n\n\\textbf{Optimization:}\n\\begin{itemize}\n    \\item \\textbf{Adam:} Unless specified otherwise, learning rate was set at constant to $1\\mathrm{e}{-4}$ and all other parameters were set to their default PyTorch values. \n    \\item \\textbf{SGD:} Unless specified otherwise, learning rate schedule inverse-square root (defined below) was used with initial learning rate\n    $\\gamma_{0} = 0.1$ and updates every $L = 512$ gradient steps. No momentum was used.\n\\end{itemize}\nWe found our results are robust to various other \nnatural choices of optimizers and learning rate schedule.\nWe used the above settings because\n(1) they optimize well,\nand (2) they do not require experiment-specific hyperparameter tuning, and allow us to use the same optimization across many experiments.\n\n\\textbf{Batch size}: All experiments use a batchsize of 128.\n\n\\textbf{Learning rate schedule descriptions:}\n\\begin{itemize}\n    \\item \\textbf{Inverse-square root $(\\gamma_0, L)$}:\n    At gradient step $t$, the learning rate is set to\n    $\\gamma(t) := \\frac{\\gamma_0}{\\sqrt{1+\\lfloor t \/ 512\\rfloor}}$.\n    We set learning-rate with respect to number of gradient steps, and not epochs, in order to allow comparison between experiments with varying train-set sizes.\n    \\item \\textbf{Dynamic drop ($\\gamma_0$, drop, patience)}: Starts with an initial learning rate of $\\gamma_0$ and drops by a factor of 'drop' if the training loss has remained constant or become worse for 'patience' number of gradient steps.\n\\end{itemize}\n\n\\subsection{Neural Machine Translation: Experimental Setup}\n\nHere we describe the experimental setup for the neural machine translation experiments.\n\n{\\bf Training procedure.}\n\nIn this setting, the distribution $\\mathcal D$ consists of triples \\[\n    (x, y, i)\\;:\\;x \\in V_{src}^*,\\;y \\in V_{tgt}^*,\\;i \\in \\{0, \\dots, |y|\\}\n\\] where $V_{src}$ and $V_{tgt}$ are the source and target vocabularies, the string $x$ is a sentence in the source language, $y$ is its translation in the target language, and $i$ is the index of the token to be predicted by the model. We assume that $i|x, y$ is distributed uniformly on $\\{0, \\dots, |y|\\}$.\n\nA standard probabilistic model defines an autoregressive factorization of the likelihood: \\[\n    p_M(y|x) = \\prod_{i = 1}^{|y|} p_M(y_i|y_{<i}, x).\n\\]\nGiven a set of training samples $S$, we define \\[\n    \\operatorname{Error}_S(M) = \\frac 1{|S|} \\sum_{(x, y, i) \\in S} -\\log p_M(y_i|y_{<i}, x).\n\\]\nIn practice, $S$ is \\textit{not} constructed from independent samples from $D$, but rather by first sampling $(x, y)$ and then including all $(x, y, 0), \\dots, (x, y, |y|)$ in $S$. \n\nFor training transformers, we replicate the optimization procedure specified in \\cite{Vaswani} section 5.3, where the learning rate schedule consists of a ``warmup'' phase with linearly increasing learning rate followed by a phase with inverse square-root decay. We preprocess the data using byte pair encoding (BPE) as described in \\cite{Sennrich2015NeuralMT}. We use the implementation provided by fairseq (\\url{https:\/\/github.com\/pytorch\/fairseq}).\n\n\n\\paragraph{Datasets.}\nThe IWSLT '14 German to English dataset contains TED Talks as described in \\cite{cettoloEtAl:EAMT2012}. The WMT '14 English to French dataset is taken from \\url{http:\/\/www.statmt.org\/wmt14\/translation-task.html}.\n\n\\iffalse\nWe train on the label-smoothed loss: \\[\n    -\\mathcal L(M, (x, y, i)) = (1 - \\lambda) \\log p_M(y_i|x, y_{<i}) + \\lambda \\frac{1}{|V_{\\text{tgt}}|} \\sum_{k = 1}^{|V_{\\text{tgt}}|} \\log p_M(y_k|x, y_{<k})\n\\]\nwhere we fix $\\lambda = 0.1$ in our experiments. \n\\fi\n\n\\iffalse\nWe define the per-token \\textit{label-smoothed} loss to be \\[\n    -\\mathcal L(M, (x, y, i)) = (1 - \\lambda) \\log p_M(y_i|x, y_{<i}) + \\lambda \\frac{1}{|V_{\\text{tgt}}|} \\sum_{k = 1}^{|V_{\\text{tgt}}|} \\log p_M(y_k|x, y_{<k})\n\\]\nwhere $\\lambda \\in [0, 1]$ is the factor of label-smoothing.\n\nIn practice, we do \\textit{not} take independent samples from $\\mathcal D$, but rather first sample sentence pairs $(x^{(1)}, y^{(1)}), \\dots, (x^{(N)}, y^{(N)}) \\sim p_{\\mathcal D}(x, y)$ and compute the empirical loss: \\[\n    \\frac 1 {\\sum_{n = 1}^N |y^{(n)}|}  \\sum_{n = 1}^N \\sum_{i = 1}^{|y^{(n)}|}\\mathcal L(M, (x, y, i))\n\\]\nHowever, assuming that sentence length is bounded by some $L$, the covariance between samples more than $L$ apart is 0, so the law of large numbers still holds.\n\nOne benefit of this setting is that we directly report test loss as a measure of generalization performance in our experiments. \nUnlike with images, log-likelihood is a preferred choice over accuracy in natural language tasks due to large vocabulary size (one can also view this as inherent uncertainty in the hypothetical ground truth distribution $p(y|x)$).\nLabel smoothing ensures that test loss is meaningful in the overfitting limit because it prevents diverging test loss on incorrectly labelled examples as final layer weights grow to infinity.\n\\fi\n\n\n\n\n\n\n\\subsection{Per-section Experimental Details}\n\\label{sec:secexp}\nHere we provide full details for experiments in the body, when not otherwise provided.\n\n\\textbf{Introduction: Experimental Details}\nFigure~\\ref{fig:errorvscomplexity}: All models were trained using Adam with learning-rate 0.0001 for 4K epochs.\nPlotting means and standard deviations for 5 trials, with random network initialization.\n    \n\\textbf{Model-wise Double Descent: Experimental Details}\nFigure~\\ref{fig:mcnn_cf100}:\nPlotting means and standard deviations for 5 trials, with random network initialization.\n\n\\textbf{Sample-wise Nonmonotonicity: Experimental Details}\nFigure~\\ref{fig:sample-modeld}: All models are trained with SGD for 500K epochs,\nand data-augmentation. Bottom: Means and standard deviations from 5 trials\n with random initialization, and random subsampling of the train set.\n\n\\newpage\n\\input{related.tex}\n\n\\newpage\n\\input{rffs}\n\n\n\n\n\\newpage\n\n\\section{Appendix: Additional Experiments}\n\\subsection{Epoch-wise Double Descent: Additional results}\n\\label{subsec:appendix_epoch}\nHere, we provide a rigorous evaluation of epoch-wise double descent for a variety of optimizers and learning rate schedules. We train ResNet18 on CIFAR-10 with data-augmentation and 20\\% label noise with three different optimizers---Adam, SGD, SGD + Momentum (momentum set to 0.9) and three different learning rate schedules---constant, inverse-square root, dynamic drop for differnet values of initial learning rate. We observe that double-descent occurs reliably for all optimizers and learning rate schedules and the peak of the double descent curve shifts with the interpolation point. \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{adam_const__train_test}\n        \\caption{Constant learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{adam_gammaT__train_test}\n        \\caption{Inverse-square root learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{adam_dynamic__train_test}\n        \\caption{Dynamic learning rate}\n    \\end{subfigure}\n    \\caption{{\\bf Epoch-wise double descent} for ResNet18 trained with Adam and multiple learning rate schedules\n    }\n    \\label{fig:epochDD-adam-lr}\n\\end{figure}\n\nA practical recommendation resulting from epoch-wise double descent is that stopping the training when the test error starts to increase may not always be the best strategy. In some cases, the test error may decrease again after reaching a maximum, and the final value may be lower than the minimum earlier in training. \n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{sgd_const__train_test}\n        \\caption{Constant learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{sgd_gammaT__train_test}\n        \\caption{Inverse-square root learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{sgd_dynamic__train_test}\n        \\caption{Dynamic learning rate}\n    \\end{subfigure}\n    \\caption{{\\bf Epoch-wise double descent} for ResNet18 trained with SGD and multiple learning rate schedules\n    }\n    \\label{fig:epochDD-sgd-lr}\n\\end{figure}\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{momentum_const__train_test}\n        \\caption{Constant learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{momentum_gammaT__train_test}\n        \\caption{Inverse-square root learning rate}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{momentum_dynamic__train_test}\n        \\caption{Dynamic learning rate}\n    \\end{subfigure}\n    \\caption{{\\bf Epoch-wise double descent} for ResNet18 trained with SGD+Momentum and multiple learning rate schedules\n    }\n    \\label{fig:epochDD-mom-lr}\n\\end{figure}\n\n\\subsection{Model-Wise Double Descent: Additional Results}\n\\label{subsec:appendix_model}\n\\subsubsection{Clean Settings With Model-wise Double Descent}\n\n\\textbf{CIFAR100, ResNet18}\n\n\\begin{figure}[!ht]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{app-clean-cifar100-ocean}\n\\end{minipage}%\n\\\\ \\vspace{0.25cm}\n\\includegraphics[width=0.7\\textwidth]{app-clean-cifar100-dynamics}\n\\caption{{\\bf Top:} Train and test  performance as a function of both model size and train epochs. {\\bf Bottom:} Test error dynamics of the same model (ResNet18, on CIFAR-100 with no label noise, data-augmentation and Adam optimizer trained for 4k epochs with learning rate 0.0001). Note that even with optimal early stopping this setting exhibits double descent.}\n\\label{fig:app-cifar100-clean}\n\\end{figure}\n\n\\textbf{CIFAR100, Standard CNN}\n\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{\\textwidth}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{pct-cifar100-mcnn-p0-sgd-noaug-ocean}\n\\end{minipage}%\n\\\\ \\vspace{0.25cm}\n\\includegraphics[width=0.7\\textwidth]{pct-cifar100-mcnn-p0-sgd-noaug}\n\\caption{{\\bf Top:} Train and test performance as a function of both model size and train epochs. {\\bf Bottom:} Test error dynamics of the same models.\n5-Layer CNNs, CIFAR-100 with no label noise, no data-augmentation\nTrained with SGD for 1e6 steps.\nSame experiment as Figure~\\ref{fig:mcnn_cf100}.\n}\n\\label{fig:app-cifar100-clean-sgd}\n\\end{figure}\n\\newpage\n\n\n\\subsubsection{Weight Decay}\n\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{0.55\\textwidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/weight_decay_dynamics1}\\\\\n  \\includegraphics[width=\\textwidth]{figures\/weight_decay_dynamics2}\\\\\n\\end{minipage}\\hfill\n\\begin{minipage}{0.4\\textwidth}\n\\includegraphics[width=\\textwidth]{figures\/weight_decay}\n\\end{minipage}\n\\caption{{\\bf Left:} Test error dynamics\nwith weight decay of 5e-4 (bottom left) and without weight decay (top left). \n{\\bf Right:} Test and train error and \\emph{test loss} for models with\nvarying amounts of weight decay.\nAll models are 5-Layer CNNs on CIFAR-10 with 10\\% label noise,\ntrained with data-augmentation and SGD for 500K steps.\n}\n\\label{fig:cnn-weight-decay}\n\\end{figure}\n\nHere, we now study the effect of varying the level of regularization on test error. We train CIFAR10 with data-augmentation and 20\\% label noise on ResNet18 for weight decay co-efficients $\\lambda$ ranging from 0 to 0.1. We train the networks using SGD + inverse-square root learning rate. Figure below shows a picture qualitatively very similar to that observed for model-wise double descent wherein \"model complexity\" is now controlled by the regularization parameter. This confirms our generalized double descent hypothesis along yet another axis of Effective Model Complexity.\n\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[t]{.45\\textwidth}\n        \\centering\n  \\includegraphics[width=\\textwidth]{weight_decay_epochDD_ocean_plot}\n        \\caption{Test Error}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[t]{.45\\textwidth}\n        \\centering\n  \\includegraphics[width=\\textwidth]{weight_decay_epochDD_ocean_plot_train}\n        \\caption{Train Loss}\n    \\end{subfigure}\n    \\caption{Generalized double descent for weight decay. We found that using the same initial learning rate for all weight decay values led to training instabilities. This resulted in some noise in the Test Error (Weight Decay $\\times$ Epochs) plot shown above.}\n    \\label{fig:regular_ocean}\n\\end{figure}\n\n\n\n\\newpage\n\n\\subsubsection{Early Stopping does not exhibit double descent}\n\n\\textbf{Language models}\n\n\\begin{figure}[!h]\n        \\centering\n  \\includegraphics[width=0.8\\textwidth]{figures\/nlp-sml-dynamics-appendix}\n    \\caption{Model-wise test error dynamics for a subsampled \\iwslt~dataset.\n    Left: 4k samples, Right: 18k samples. \n    Note that with optimal early-stopping, more samples is always better.}\n    \\label{fig:sml-nlp-dynamics}\n\\end{figure}\n\n\\begin{figure}[h]\n        \\centering\n  \\includegraphics[width=0.8\\textwidth]{figures\/nlp-dynamics-appendix}\n    \\caption{Model-wise test error dynamics for a \\iwslt\\  de-en and subsampled WMT`14 en-fr datasets.\n    {\\bf Left}: \\iwslt, {\\bf Right}: subsampled (200k samples) WMT`14. \n    Note that with optimal early-stopping, the test error is much lower for this task.}\n    \\label{fig:nlp-dynamics}\n\\end{figure}\n\n\\textbf{CIFAR10, 10\\% noise, SGD}\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}{0.7\\textwidth}\n  \\centering\n  \\includegraphics[width=0.8\\textwidth]{cnn-sgd-ocean}\n\\end{minipage}%\n\\\\ \\vspace{0.25cm}\n\\includegraphics[width=0.7\\textwidth]{figures\/cnn-sgd--dynamics}\n\\caption{{\\bf Top:} Train and test  performance as a function of both model size and train epochs. {\\bf Bottom:} Test error dynamics of the same model (CNN, on CIFAR-10 with 10\\% label noise, data-augmentation and SGD optimizer with learning rate $\\propto 1\/\\sqrt{T}$).}\n\\label{fig:app-cnn10-10}\n\\end{figure}\n\n\\newpage\n\n\\subsubsection{Training Procedure}\n\n\\begin{figure}[h]\n        \\centering\n  \\includegraphics[width=0.5\\textwidth]{adv_training_2}\n  \\caption[width=0.8\\linewidth]{\n{\\bf Model-wise double descent for adversarial training}\nResNet18s on CIFAR-10 (subsampled to 25k train samples)\nwith no label noise.\nWe train \nfor L2 robustness of radius ${\\epsilon}=0.5$ and ${\\epsilon}=1.0$,\nusing 10-step PGD (\\cite{goodfellow2014explaining, madry2017towards}).\nTrained using SGD (batch size 128) with learning rate $0.1$ for 400 epochs,\nthen $0.01$ for 400 epochs.\n  }\n    \\label{fig:adv_training}\n\\end{figure}\n\n\n\n\n\\begin{figure}[!h]\n        \\centering\n  \\includegraphics[width=0.6\\textwidth]{cifar10_mcnn_aug_pVar}\n    \\caption{}\n    \\label{fig:mcnn_aug_large}\n\\end{figure}\n\n\n\\newpage\n\\subsection{Ensembling}\n\n\n\\begin{figure}[!h]\n        \\centering\n  \\includegraphics[width=0.7\\textwidth]{ens_intro_resnet_p15_clean}\n    \\caption{{\\bf Effect of Ensembling (ResNets, 15\\% label noise). }\n    Test error of an ensemble of 5 models, compared to the base models.\n    The ensembled classifier is determined by plurality vote over the 5 base models.\n    Note that emsembling helps most around the critical regime.\n    All models are ResNet18s trained on CIFAR-10 with 15\\% label noise,\n    using Adam for 4K epochs\n    (same setting as Figure~\\ref{fig:errorvscomplexity}).\n    Test error is measured against the original (not noisy) test set,\n    and each model in the ensemble is trained using a train set with independently-sampled 15\\% label noise.\n   } \n    \\label{fig:ens_resnet}\n\\end{figure}\n\n\\begin{figure}[!h]\n        \\centering\n  \\includegraphics[width=0.7\\textwidth]{ens_mcnn_cf100}\n    \\caption{{\\bf Effect of Ensembling (CNNs, no label noise). }\n    Test error of an ensemble of 5 models, compared to the base models.\n    All models are 5-layer CNNs trained on CIFAR-10 with no label noise,\n    using SGD and no data augmentation.\n    (same setting as Figure~\\ref{fig:mcnn_cf100}).\n   } \n    \\label{fig:ens_mcnn}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\iffalse\n\\begin{figure}[!h]\n        \\centering\n  \\includegraphics[width=0.5\\textwidth]{cifar10_mcnn_aug_pVar}\n    \\caption{5-layer CNNS on CIFAR, with data-augmentation.\n    }\n    \\label{fig:mcnn_aug_large}\n\\end{figure}\n\\fi\n\n\n\n\n\n\n\n\\newpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Summary table of experimental results}\n\\label{sec:bigtable}\n\n\\footnotesize\n\\begin{tabular}{@{}llllllll@{}} \n    \\toprule\n    $ $ & $ $ & $ $ & $ $ & $ $ & \\multicolumn{2}{c}{Double-Descent} \\\\ \n    \\cmidrule(l){6-7}\n    Dataset     & Architecture & Opt. & Aug. & \\% Noise & Model    & Epoch     & Figure(s)\\\\ \n    \\midrule\n    CIFAR 10    & CNN           & SGD  & \\cmark    & 0     & \\xmark & \\xmark & \n                                                                        \\ref{fig:cnn-cifar}, \n                                                                        \\ref{fig:mcnn_aug_large}\\\\\n                &               &       & \\cmark    & 10    & \\cmark & \\cmark &\n                                                                        \\ref{fig:cnn-cifar}, \n                                                                        \\ref{fig:mcnn_aug_large},\n                                                                        \\ref{fig:more-mdd1appendix}\\\\\n                &               &       & \\cmark    & 20    & \\cmark & \\cmark &\n                                                                        \\ref{fig:cnn-cifar}, \n                                                                        \\ref{fig:mcnn_aug_large}\\\\\n                &               &       &           & 0     & \\xmark & \\xmark & \\ref{fig:cnn-cifar},\n                                                                            \\ref{fig:app-cnn10-10}\\\\\n                &               &       &           & 10    & \\cmark & \\cmark & \\ref{fig:cnn-cifar}\\\\\n                &               &       &           & 20    & \\cmark & \\cmark & \\ref{fig:cnn-cifar}\\\\\n                &               & SGD + w.d. & \\cmark & 20 & \\cmark & \\cmark      & \n                                                                        \\ref{fig:cnn-weight-decay}\\\\\n                &               & Adam  &           & 0     & \\cmark & -- & \\ref{fig:app-cnn10-10} \\\\\n                & ResNet        & Adam  & \\cmark    & 0     & \\xmark & \\xmark &                   \n                                                                \\ref{fig:resnet-cifar},\n                                                                \\ref{fig:epochDD-noise-dataset-arch}\\\\\n                &               &       & \\cmark    & 5     & \\cmark & -- &\n                                                                \\ref{fig:resnet-cifar}\\\\\n                &               &       & \\cmark    & 10    & \\cmark & \\cmark &\n                                                                \\ref{fig:resnet-cifar},\n                                                                \\ref{fig:epochDD-noise-dataset-arch}\\\\\n                &               &       & \\cmark    & 15    & \\cmark & \\cmark &\n                                                                \\ref{fig:resnet-cifar},\n                                                                \\ref{fig:unified}\\\\\n                &               &       & \\cmark    & 20    & \\cmark & \\cmark &\n                                                                \\ref{fig:resnet-cifar},\n                                                                \\ref{fig:epochDD-three-regimes},\n                                                                \\ref{fig:epochDD-noise-dataset-arch}\\\\\n               \n               \n                &               & Various & \\cmark  & 20    & --    & \\cmark    \n                                                                & \\ref{fig:epochDD-adam-lr},\n                                                                \\ref{fig:epochDD-sgd-lr},\n                                                                \\ref{fig:epochDD-mom-lr}\\\\\n                \n    (subsampled) & CNN          & SGD     & \\cmark         & 10    & \\cmark & -- & \\ref{fig:sample-modeld}\\\\\n                &               & SGD     & \\cmark         & 20    & \\cmark & -- & \n                                                                \\ref{fig:sample-modeld},\n                                                                \\ref{fig:grid}\\\\\n    (adversarial) & ResNet      & SGD     &           & 0     & Robust err.     & -- & \\ref{fig:adv_training}\\\\\n    \\addlinespace[0.5em]\n    CIFAR 100   \n                & ResNet        & Adam  & \\cmark    & 0     & \\cmark & \\xmark & \\ref{fig:resnet-cifar},\n                                                                    \\ref{fig:app-cifar100-clean},\n                                                                    \\ref{fig:epochDD-noise-dataset-arch}\\\\\n                &               &       & \\cmark    & 10    & \\cmark & \\cmark & \n                                                                    \\ref{fig:resnet-cifar},\n                                                                    \\ref{fig:epochDD-noise-dataset-arch}\\\\\n                &               &       & \\cmark    & 20    & \\cmark & \\cmark & \n                                                                    \\ref{fig:resnet-cifar},\n                                                                    \\ref{fig:epochDD-noise-dataset-arch}\\\\\n                & CNN        & SGD  &     & 0     & \\cmark & \\xmark & \\ref{fig:app-cifar100-clean-sgd}\\\\\n    \\addlinespace[0.5em]\n    IWSLT '14 de-en & Transformer & Adam &              & 0     & \\cmark & \\xmark & \n                                                                            \\ref{fig:more-mdd2},\n                                                                            \\ref{fig:nlp-dynamics}\\\\\n    (subsampled) &  Transformer & Adam &                & 0     & \\cmark & \\xmark & \\ref{fig:nlpdd}, \\ref{fig:sml-nlp-dynamics} \\\\\n    WMT '14 en-fr & Transformer & Adam  &               & 0     & \\cmark & \\xmark & \n                                                                            \\ref{fig:more-mdd2},\n                                                                            \\ref{fig:nlp-dynamics}\\\\\n    \\bottomrule\n\\end{tabular} \\\\\n\\normalsize\n\\section{Conclusion and Discussion}\n\\label{sec:discuss}\nWe introduce a generalized double descent hypothesis: models and training procedures exhibit atypical behavior\nwhen their Effective Model Complexity\nis comparable to the number of train samples.\nWe provide extensive evidence for our hypothesis\nin modern deep learning settings,\nand show that it is robust to choices of dataset,\narchitecture, and training procedures.\nIn particular, we demonstrate ``model-wise double descent''\nfor modern deep networks and\ncharacterize the regime where bigger models can perform worse.\nWe also demonstrate ``epoch-wise double descent,''\nwhich, to the best of our knowledge, has not been previously proposed.\nFinally, we show that the double descent phenomenon can\nlead to a regime where training on more data leads to worse test performance.\nPreliminary results suggest that double descent also holds\nas we vary the amount of regularization for a fixed model\n(see Figure~\\ref{fig:regular_ocean}).\n\nWe also believe our characterization of the critical regime provides a useful\nway of thinking for practitioners---if a model and training procedure are\njust barely able to fit the train set, then small changes\nto the model or training procedure may yield unexpected behavior\n(e.g. making the model slightly larger or smaller, changing regularization, etc. may hurt test performance).\n\n\\paragraph{Early stopping.} \nWe note that many of the phenomena that we highlight\noften do not occur with optimal early-stopping.\nHowever, this is consistent with our\ngeneralized double descent hypothesis: \nif early stopping prevents models from reaching $0$ train error\nthen we would not expect to see double-descent, since \nthe EMC does not reach the number of train samples.\nFurther, we show at least one setting where model-wise double descent can still occur\neven with optimal early stopping\n(ResNets on CIFAR-100 with no label noise, see Figure~\\ref{fig:app-cifar100-clean}).\nWe have not observed settings where more data hurts\nwhen optimal early-stopping is used.\nHowever, we are not aware of reasons which preclude this from occurring.\nWe leave fully understanding the optimal early stopping behavior of double descent\nas an important open question for future work.\n\n\n\n\n\\paragraph{Label Noise.}\nIn our experiments, we observe double descent most strongly in settings with label noise.\nHowever, we believe this effect is not fundamentally about label noise,\nbut rather about \\emph{model mis-specification}.\nFor example, consider a setting where the label noise is not truly random,\nbut rather pseudorandom (with respect to the family of classifiers being trained).\nIn this setting, the performance of the Bayes optimal classifier would not change\n(since the pseudorandom noise is deterministic, and invertible),\nbut we would observe an identical double descent as with truly random label noise.\nThus, we view adding label noise as merely a proxy for making distributions ``harder''---\ni.e. increasing the amount of model mis-specification.\n\n\\paragraph{Other Notions of Model Complexity.}\nOur notion of \\emph{Effective Model Complexity} is related to classical complexity notions\nsuch as Rademacher complexity, but differs in several crucial ways:\n(1) EMC depends on the \\emph{true labels} of the data distribution,\nand (2) EMC depends on the training procedure, not just the model architecture.\n\nOther notions of model complexity which do not incorporate features (1) and (2)\nwould not suffice to characterize the location of the double-descent peak.\nRademacher complexity, for example, is determined by the\nability of a model architecture to fit a randomly-labeled train set.\nBut Rademacher complexity and VC dimension are both insufficient to \ndetermine the model-wise double descent peak location, since they do\nnot depend on the distribution of labels--- and our experiments show that adding label noise\nshifts the location of the peak.\n\nMoreover, both Rademacher complexity and VC dimension\ndepend only on the model family and data distribution, and not\non the training procedure used to find models.\nThus, they are not capable of capturing train-time double-descent effects,\nsuch as ``epoch-wise'' double descent, and the effect of data-augmentation on\nthe peak location.\n\\section{Epoch-wise Double Descent}\n\\label{sec:epoch-dd}\n\n\n\n\n\n\nIn this section, we demonstrate a novel form of double-descent with respect to training epochs, which is consistent with our unified view of effective model complexity (EMC) and the generalized double descent hypothesis.\nIncreasing the train time increases the EMC---and thus a sufficiently large model transitions from under- to over-parameterized over the course of training.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[t]{.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{epochDD_regimes}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.48\\textwidth}\n        \\centering\n        \\includegraphics[width=\\textwidth]{epochDD_ocean_plot}\n    \\end{subfigure}\n    \\caption{\n    {\\bf Left: } \n    Training dynamics for models in three regimes.\n    Models are ResNet18s on CIFAR10 with 20\\% label noise,\n    trained using Adam with learning rate $0.0001$, and data augmentation.\n    {\\bf Right:} Test error over (Model size $\\x$ Epochs).\n    Three slices of this plot are shown on the left.\n    }\n    \\label{fig:epochDD-three-regimes}\n\\end{figure}\n\nAs illustrated in Figure~\\ref{fig:epochDD-three-regimes}, \nsufficiently large models can\nundergo\na ``double descent'' behavior where test error first decreases then increases near the interpolation\nthreshold, and then decreases again.\nIn contrast, for ``medium sized'' models, for which training to completion will only barely reach $\\approx 0$ error, the\ntest error as a function of training time will follow a classical U-like curve where it is better to stop early.\nModels that are too small to reach the approximation threshold will remain in the ``under parameterized'' regime where increasing\ntrain time monotonically decreases test error.\nOur experiments\n(Figure~\\ref{fig:epochDD-noise-dataset-arch}) show that many settings\nof dataset and architecture exhibit epoch-wise double descent, in the presence of label noise.\nFurther, this phenomenon\nis robust across optimizer variations and learning rate schedules (see additional experiments in Appendix \\ref{subsec:appendix_epoch}).\nAs in model-wise double descent, the test error peak is accentuated with label noise. \n\n\\iffalse\n\nIndeed, we observe three distinct train-time behaviors\ndepending on the model size and label noise (Figure~\\ref{fig:epochDD-three-regimes}):\n\\begin{enumerate}\n    \\item Small models (which are under-parameterized throughout training):\n    Test error decreases monotonically throughout training. \n    \\item Intermediate-size models\n    (which transition from under- to critically parameterized over training): Test error decreases but starts diverging close to interpolation (conventional understanding of when over-fitting sets in).\n    \\item Large models (which transition from under- to over-parameterized throughout training):\n    Test error decreases, increases or plateaus near the interpolation threshold, then decreases again.\\todo{added \"or plateaus\"--Boaz}\n\\end{enumerate}\n\n\\fi\n\n\n\n\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{epochDD_cf10_labelnoise_train_test}\n        \\caption{ResNet18 on CIFAR10.}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{epochDD_cf100_labelnoise_train_test}\n        \\caption{ResNet18 on CIFAR100.}\n    \\end{subfigure}\n    \\begin{subfigure}[t]{.32\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\linewidth]{epochDD_cf10_mcnn_labelnoise_train_test}\n        \\caption{5-layer CNN on CIFAR 10.}\n    \\end{subfigure}\n    \\caption{{\\bf Epoch-wise double descent} for ResNet18 and CNN (width=128).\n    ResNets trained using Adam with learning rate $0.0001$,\n    and CNNs trained with SGD with inverse-squareroot learning rate.\n    }\n    \\label{fig:epochDD-noise-dataset-arch}\n\\end{figure}\n\n\nConventional wisdom suggests that training is split into two phases: (1) In the first phase, the network learns a function with a small generalization gap (2) In the second phase, the network starts to over-fit the data leading to an increase in test error.\nOur experiments suggest that this is not the complete picture---in some regimes, the test error decreases again and may achieve a lower value at the end of training as compared to the first minimum (see Fig \\ref{fig:epochDD-noise-dataset-arch} for 10\\% label noise).\n\n\n\n\n\\section{Generalized Double Descent}\n\\label{sec:general}\n\nHere we present a generic (though informal) way of unifying the three types of\ndouble-descent phenomenon we observe. In particular, we introduce a notion of ``Effective Model Complexity'' along which double-descent occurs.\n\n\\pnote{Todo, rewrite this.}\n\n\\begin{definition}[Learning Algorithm]\nA \\emph{learning algorithm $\\cA$}\n$$\\cA: (\\cX \\x \\cY)^* \\to M$$\nis a function mapping $n$ samples $\\{(x_i, y_i)\\}$\nto a \\emph{Model} $M: \\cX \\to \\cY$.\n\\end{definition}\nWe will consider learning-algorithms which take as input\n$n$ i.i.d. samples $(x_i, y_i) \\sim \\cD$ from a distribution $\\cD$,\nand output a model $M$ (for example, by training a neural-network via SGD on the samples).\n\n\n\\begin{definition}[Model Complexity]\nFor a learning-algorithm $\\cA$, distribution $\\cD$, loss function $\\cL$, and $0 \\leq {\\epsilon} \\ll 1$,\ndefine the \n\\emph{ModelComplexity}$_{\\cD, \\cL, {\\epsilon}}(\\cA)$\nas the maximum number of samples from $\\cD$ for which $\\cA$ outputs a model with small train loss.\nFormally,\n$$\n\\mathrm{ModelComplexity}_{\\cD, \\cL, {\\epsilon}}(\\cA)\n:=\n\\argmax_n  \\Pr_{\\substack{\n\\{(x_i, y_i)\\} \\sim \\cD^n\\\\\nM \\gets \\cA(\\{(x_i, y_i)\\})\n}}\\left[ \\frac{1}{n} \\sum_i \\cL( M(x_i), y_i )  \\leq {\\epsilon} \\right] \\geq 1-o(1)\n$$\n\\end{definition}\n\nThroughout, we consider learning-algorithms which output models by \ntraining a neural network, optimizing the train loss via SGD or variants.\nIn this case, let $\\cA_{NN}[\\text{Model-Size, Train-Steps}]$ be the learning algorithm\nwhich trains a neural-network of a given size and architecture for the given number of train steps.\n\nFor $n$ training samples,\nwe consider a learning-algorithm $\\cA$ to be ``overparmeterized'' if\n$$\n\\mathrm{ModelComplexity}_{\\cD, \\cL, {\\epsilon}}(\\cA)\n\\gg n\n$$\nand ``underparameterized'' if\n$\\mathrm{ModelComplexity}_{\\cD, \\cL, {\\epsilon}}(\\cA) \\ll n$,\nand ``critically parameterized'' if\n$\\mathrm{ModelComplexity}_{\\cD, \\cL, {\\epsilon}}(\\cA) \\approx n$.\n\n\nWe can now state our main thesis in terms of the above.\n\\begin{claim}[Phase Transision, Informal]\nNeural-network learning algorithms $\\cA_{NN}$ undergo\na \\emph{phase transition} in their test performance \nin the ``critically parameterized'' regime, when\nthe \\emph{ModelComplexity} is equal to the size of the train set.\nThat is, when\n$$\n\\mathrm{ModelComplexity}_{\\cD, \\cL, {\\epsilon}}(\\cA_{NN})\n= n\n$$\nwhere $n$ is the number of training examples.\n\\end{claim}\n\n\n\n\\section{Introduction}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{errorvscomplexity}\n    \\caption{{\\bf Left:} Train and test error as a function of model size,\n    for ResNet18s of varying width \n    on CIFAR-10 with 15\\% label noise.\n   \n   \n    {\\bf Right:}\n    Test error, shown for varying train epochs.\n   \n   \n   \n    All models trained using Adam for 4K epochs.\n   \n    The largest model (width $64$) corresponds to standard ResNet18. \n   \n    }\n    \\label{fig:errorvscomplexity}\n\\end{figure}\n\nThe \\emph{bias-variance trade-off} is a fundamental concept in classical statistical learning theory (e.g., \\cite{hastie2005elements}).\nThe idea is that models of higher complexity have lower bias but higher variance.\nAccording to this theory, once model complexity passes a certain threshold, models ``overfit'' with the variance term dominating the test error, and hence from this point onward, increasing model complexity will only \\emph{decrease} performance  (i.e., increase test error).\nHence conventional wisdom in classical statistics is that, once we pass a certain threshold, \\emph{``larger models are worse.''}\n\n\n\nHowever, modern neural networks exhibit no such phenomenon.\nSuch networks have millions of parameters, more than enough to fit even random labels~(\\cite{ZhangBHRV16}), \nand yet they perform much better on many tasks than smaller models.\nIndeed, conventional wisdom among practitioners is that \\textit{``larger models are better'}' (\\cite{krizhevsky2012imagenet}, \\cite{gpipe}, \\cite{Inception}, \\cite{gpt}).\nThe effect of training time on test performance is also up for debate. \nIn some settings, ``early stopping'' improves test performance, while in other settings \ntraining neural networks to zero training error only improves performance.\nFinally, if there is one thing both classical statisticians and deep learning practitioners agree on is \\textit{``more data is always better''}.\n\n\nIn this paper, we present empirical evidence that both reconcile and challenge some of the above ``conventional wisdoms.'' \nWe show that many deep learning settings have two different regimes.\nIn the \\emph{under-parameterized} regime, where the model complexity is small compared to the number of samples, the test error as a function of model complexity follows the U-like behavior predicted by the classical bias\/variance tradeoff.\nHowever, once model complexity is sufficiently large to\n\\emph{interpolate} i.e., achieve (close to) zero training error, then increasing complexity only \\emph{decreases} test error, following the modern intuition of ``bigger models are better''.\nSimilar behavior was previously observed in\n\\cite{opper1995statistical, opper2001learning},\n\\cite{advani2017high},\n\\cite{spigler2018jamming}, and \\cite{geiger2019jamming}.\nThis phenomenon was first postulated in generality by \\cite{belkin2018reconciling}  who named it ``double descent'', and\ndemonstrated it\nfor \ndecision trees,\nrandom features,\nand\n2-layer neural networks\nwith $\\ell_2$ loss, on a variety of learning tasks including MNIST and CIFAR-10.\n\n\\iffalse\n\\begin{figure}[t]\n\\centering\n\\begin{minipage}{.4\\textwidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{figures\/bias-var-trade}\n\\end{minipage}\\hspace{1cm}\n\\begin{minipage}{.4\\textwidth}\n  \\centering\n  \\vspace{-.15cm}\n  \\includegraphics[width=\\textwidth]{effnet}\n\\end{minipage}\n    \\caption{Left: A classical ``bias-variance'' tradeoff where after a certain point, increasing model complexity hurts test performance; figure from \\cite{hastie2005elements}. Right: Illustration of modern over-parameterized neural networks, where increasing complexity empirically results in better performance; data from \\cite{tan2019efficientnet}.}\n    \\label{fig:classicalvsmodernfig}\n    \n\\end{figure}\n\\fi\n\n\n\\begin{figure}[t!]\n\\centering\n\\begin{minipage}{.512\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\textwidth]{Intro-ocean-test}\n\\end{minipage}%\n\\begin{minipage}{.488\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\textwidth]{Rn-cifar10-p15-adam-aug-train}\n\\end{minipage}\n\\caption{{\\bf Left:} Test error as a function of model size and train epochs. The horizontal line corresponds to model-wise double descent--varying model size while training for as long as possible.\nThe vertical line corresponds to epoch-wise double descent,\nwith test error undergoing double-descent as train time increases.\n{\\bf Right} Train error of the corresponding models.\nAll models are Resnet18s trained on CIFAR-10 with 15\\% label noise,\ndata-augmentation, and Adam for up to 4K epochs.}\n\\label{fig:unified}\n\\end{figure}\n\n\n\\paragraph{Main contributions.} We show that double descent is a robust phenomenon that occurs in a variety of tasks, architectures, and\noptimization methods (see Figure~\\ref{fig:errorvscomplexity} and Section~\\ref{sec:model-dd}; our experiments are summarized in Table~\\ref{sec:bigtable}).\nMoreover, we propose a much more general notion of ``double descent'' that goes beyond varying the number of parameters. We define the \\emph{effective model complexity (EMC)} of a training procedure as the maximum number of samples on which it can achieve close to zero training error. \nThe EMC depends not just on the data distribution and the architecture of the classifier but also on the training procedure---and in particular increasing training time will increase the EMC.\n\nWe hypothesize that for many natural models and learning algorithms, double descent occurs as a function of the EMC.\nIndeed we observe ``epoch-wise double descent''  when we keep the model fixed and increase the training time, with performance following a classical U-like curve in the underfitting stage (when the EMC is smaller than the number of samples) and then improving with training time once the EMC is sufficiently larger than the number of samples (see Figure~\\ref{fig:unified}). \nAs a corollary, early stopping only helps in the relatively narrow parameter regime of critically parameterized models.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{minipage}[c]{0.5\\textwidth}\n        \\includegraphics[width=\\textwidth]{NLP-small-data-vary-MDD}\n    \\end{minipage}\\hfill\n    \\begin{minipage}[c]{0.45\\textwidth}\n       \\caption{Test loss (per-token perplexity) as a function of Transformer model size (embedding dimension $d_{model}$)\n        on language translation (\\iwslt~German-to-English).\n        The curve for 18k samples is generally lower than the one for 4k samples, but also shifted to the right,\n        since fitting 18k samples requires a larger model.\n        Thus, for some models, the performance for 18k samples is \\emph{worse} than for 4k samples.}\n    \\label{fig:moresamplesareworse}\n    \\end{minipage}\n\\end{figure}\n\n\\paragraph{Sample non-monotonicity.}\nFinally, our results shed light on test performance\nas a function of the number of train samples.\nSince the test error peaks around the point where EMC matches the number of samples (the transition from the under- to over-parameterization), increasing the number of samples has the effect of shifting this peak to the right.\nWhile in most settings increasing the number of samples decreases error, this shifting effect\ncan sometimes result in a setting where \\emph{more data is worse!}\nFor example, Figure~\\ref{fig:moresamplesareworse} demonstrates cases in which increasing the number of samples by a factor of $4.5$ results in worse test performance.\n\n\n\n\\section{Our results}\n\nTo state our hypothesis more precisely, we define the notion of \\emph{effective model complexity}.\nWe define a \\emph{training procedure}  $\\mathcal{T}$ to be any procedure that takes as input \na set $S= \\{ (x_1,y_1),\\ldots,(x_n,y_n)\\}$ of labeled training samples and outputs a classifier $\\mathcal{T}(S)$\nmapping data to labels.\nWe define the \\emph{effective model complexity} of $\\mathcal{T}$ (w.r.t. distribution $\\mathcal D$) to be the maximum number of samples $n$ on which $\\mathcal{T}$ achieves on average $\\approx 0$ \\emph{training error}. \n\n\\newcommand{\\mathrm{EMC}}{\\mathrm{EMC}}\n\\begin{definition}[Effective Model Complexity]\nThe \\emph{Effective Model Complexity} (EMC) of a training procedure $\\cT$, with respect to distribution $\\cD$ and parameter $\\epsilon>0$,\nis defined as:\n\\begin{align*}\n    \\mathrm{EMC}_{\\cD,{\\epsilon}}(\\cT)\n    :=  \\max \\left\\{n ~|~ \\mathbb{E}_{S \\sim \\cD^n}[ \\mathrm{Error}_S( \\cT( S )  ) ] \\leq {\\epsilon} \\right\\}\n    \\end{align*}\n    where $\\mathrm{Error}_S(M)$ is the mean error of model $M$ on train samples $S$.\n\\end{definition}\n\nOur main hypothesis can be informally stated as follows:\n\n\\begin{hypothesis}[Generalized Double Descent hypothesis, informal] \\label{hyp:informaldd}\nFor any natural data distribution $\\cD$, neural-network-based training procedure $\\cT$, and small $\\epsilon>0$,\nif we consider the task of predicting labels based on  $n$ samples from $\\cD$ then:\n\\begin{description}\n    \\item[Under-paremeterized regime.]  If~$\\mathrm{EMC}_{\\cD,\\epsilon}(\\cT)$ is sufficiently smaller than $n$, any perturbation of $\\cT$ that increases its effective complexity will decrease the test error.\n    \\item[Over-parameterized regime.] If $\\mathrm{EMC}_{\\cD,\\epsilon}(\\cT)$ is sufficiently larger than $n$,\n    any perturbation of $\\cT$ that increases its effective complexity will decrease the test error.\n    \n    \\item[Critically parameterized regime.] If $\\mathrm{EMC}_{\\cD,\\epsilon}(\\cT) \\approx n$, then\n    a perturbation of $\\cT$ that increases its effective complexity\n    might decrease {\\bf or increase} the test error.\n\\end{description}\n\\end{hypothesis}\n\n\\iffalse\nHypothesis~\\ref{hyp:informaldd} is stated informally\nas we are yet to fully understand the behavior at the critically parameterized regime.\nFor example, it is an open question what determines the width of the \\emph{critical interval}---the interval around the point  in which $\\mathrm{EMC}_{\\cD,\\epsilon}(\\cT) = n$, where increasing complexity might hurt performance. Specifically, we lack a formal definition for ``sufficiently smaller'' and ``sufficiently larger''. Another parameter that lacks principled understanding is the choice of $\\epsilon$. In our experiments, we use $\\epsilon=0.1$ heuristically.\n\nWhile in both the under-parameterized and over-parameterized regimes increasing complexity helps performance, the dynamics of the learning process seem very different in these two regimes\nFor example, in the over-parameterized regime, the gain comes not from classifying more training samples correctly but rather from increasing the confidence for the training samples that have already been correctly classified. \\gal{iffalsed previous stuff}\n\\fi\n\nHypothesis~\\ref{hyp:informaldd} is informal in several ways.\nWe do not have a principled way to choose the parameter $\\epsilon$ (and currently heuristically use $\\epsilon=0.1$). \nWe also are yet to have a formal specification for ``sufficiently smaller'' and ``sufficiently larger''.\nOur experiments suggest that there is a \\emph{critical interval}\naround the \\emph{interpolation threshold} when $\\mathrm{EMC}_{\\cD,\\epsilon}(\\cT) = n$: below and above\nthis interval increasing complexity helps performance,\nwhile within this interval it may hurt performance.\nThe width of the critical interval depends on both the distribution and\nthe training procedure in ways we do not yet completely understand.\n\n\nWe believe Hypothesis~\\ref{hyp:informaldd} sheds light on the interaction between optimization algorithms, model size, and test performance and helps reconcile some of the competing intuitions about them.\nThe main result of this paper is an experimental validation of Hypothesis~\\ref{hyp:informaldd} under a variety of settings, where we considered several natural choices of datasets, architectures, and optimization algorithms,\nand we changed the ``interpolation threshold''\nby varying the number of model parameters,\nthe length of training, the amount of label noise in the distribution, and the number of train samples.\n\n\n{\\bf Model-wise Double Descent.}\nIn Section~\\ref{sec:model-dd},\nwe study the test error of models of increasing size,\nfor a fixed large number of optimization steps.\nWe show that ``model-wise double-descent'' occurs\nfor various modern datasets\n(CIFAR-10, CIFAR-100, \\iwslt~de-en, with varying amounts of label noise),\nmodel architectures (CNNs, ResNets, Transformers),\noptimizers (SGD, Adam),\nnumber of train samples,\nand training procedures (data-augmentation, and regularization).\nMoreover, the peak in test error systematically occurs at the interpolation threshold.\nIn particular, we demonstrate realistic settings in which\n\\emph{bigger models are worse}.\n\n{\\bf Epoch-wise Double Descent.}\nIn Section~\\ref{sec:epoch-dd},\nwe study the test error of a fixed, large architecture over the course of training.\nWe demonstrate, in similar settings as above,\na corresponding peak in test performance when\nmodels are trained just long enough to reach $~\\approx 0$ train error.\nThe test error of a large model\nfirst decreases (at the beginning of training), then increases (around the critical regime),\nthen decreases once more (at the end of training)---that is,\n\\emph{training longer can correct overfitting.}\n\n\n{\\bf Sample-wise Non-monotonicity.}\nIn Section~\\ref{sec:samples},\nwe study the test error of a fixed model and training procedure, for varying number of train samples.\nConsistent with our generalized double-descent hypothesis,\nwe observe distinct test behavior in the ``critical regime'',\nwhen the number of samples is near the maximum that the model can fit.\nThis often manifests as a long plateau region,\nin which taking significantly more data might not help when training to completion\n(as is the case for CNNs on CIFAR-10).\nMoreover, we show settings (Transformers on \\iwslt~en-de),\nwhere this manifests as a peak---and for a fixed architecture and training procedure, \\emph{more data actually hurts.}\n\n{\\bf Remarks on Label Noise.} \nWe observe all forms of double descent most strongly in settings\nwith label noise in the train set\n(as is often the case when collecting train data in the real-world).\nHowever, we also show several realistic settings with a test-error peak even without label noise:\nResNets (Figure~\\ref{fig:resnet-cifar-left}) and CNNs (Figure~\\ref{fig:app-cifar100-clean-sgd}) on CIFAR-100;\nTransformers on \\iwslt~ (Figure~\\ref{fig:more-mdd2}).\nMoreover,  all our experiments demonstrate distinctly different test behavior\nin the critical regime--- often manifesting as a ``plateau'' in the test error in the noiseless case which \ndevelops into a peak with added label noise.\nSee Section~\\ref{sec:discuss} for further discussion.\n\n\\section{Related work}\nModel-wise double descent was first proposed as a general phenomenon\nby \\cite{belkin2018reconciling}.\nSimilar behavior had been observed in\n\\cite{opper1995statistical, opper2001learning},\n\\cite{advani2017high},\n\\cite{spigler2018jamming}, and \\cite{geiger2019jamming}.\nSubsequently, there has been a large body of work studying the double descent phenomenon.\nA growing list of papers that theoretically analyze it in the tractable setting of linear least squares regression includes \\cite{belkin2019two, hastie2019surprises, bartlett2019benign, muthukumar2019harmless, bibas2019new, Mitra2019UnderstandingOP, mei2019generalization}. Moreover, \\cite{geiger2019scaling} provide preliminary results for model-wise double descent in convolutional networks trained on CIFAR-10. Our work differs from the above papers in two crucial aspects: First, we extend the idea of double-descent beyond the number of parameters to incorporate the training procedure under a unified notion of ``Effective Model Complexity\", leading to novel insights like epoch-wise double descent and sample non-monotonicity.\nThe notion that increasing train time corresponds to\nincreasing complexity was also presented in~\\cite{nakkiran2019sgd}.\nSecond, we provide an extensive\nand rigorous demonstration of double-descent for modern practices spanning a variety of architectures, datasets optimization procedures.\nAn extended discussion of the related work is provided in Appendix \\ref{sec:related_appendix}.\n\n\n\n\\subsubsection*{Acknowledgments}\nWe thank Mikhail Belkin for extremely useful discussions in the early stages of this work.\nWe thank Christopher Olah for suggesting the Model Size $\\x$ Epoch visualization, which led to the investigation of epoch-wise double descent,\nas well as for useful discussion and feedback.\nWe also thank Alec Radford, Jacob Steinhardt, and Vaishaal Shankar for helpful discussion and suggestions.\nP.N. thanks OpenAI, the Simons Institute, and the Harvard Theory Group for a research environment that enabled this kind of work.\n\nWe thank Dimitris Kalimeris, Benjamin L. Edelman, and Sharon Qian,\nand Aditya Ramesh for comments on an early draft of this work.\n\nThis work supported in part by NSF grant CAREER CCF 1452961, BSF grant 2014389, NSF USICCS proposal 1540428, a Google Research award, a Facebook research award, a Simons Investigator Award,\na Simons Investigator Fellowship,\nand NSF Awards\nCCF 1715187,\nCCF 1565264, CCF 1301976, IIS 1409097,\nand CNS 1618026. Y.B. would like to thank the MIT-IBM Watson AI Lab for contributing computational resources for experiments.\n\n\n\n\\section{Model-wise Double Descent}\n\\label{sec:model-dd}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}[t]{.47\\textwidth}\n        \\centering\n          \\includegraphics[width=\\textwidth]{cifar100_resnet_pVar}\n          \\caption[width=0.8\\linewidth]{{\\bf CIFAR-100.}\n          There is a peak in test error even with no label noise.\n          } \\label{fig:resnet-cifar-left}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}[t]{.47\\textwidth}\n        \\centering\n          \\includegraphics[width=\\textwidth]{cifar10_resnet_pVar}\n          \\caption[width=0.8\\linewidth]{{\\bf CIFAR-10.}\n        There is a ``plateau'' in test error around the interpolation point with no label noise,\n        which develops into a peak for added label noise.\n          }\n    \\end{subfigure}\n    \\caption{{\\bf Model-wise double descent for\n    ResNet18s.} Trained on CIFAR-100 and CIFAR-10, with varying label noise. Optimized using Adam with LR $0.0001$ for 4K epochs, and data-augmentation.\n    }\n    \\label{fig:resnet-cifar}\n\\end{figure}\n\nIn this section, we study the test error of models of increasing size,\nwhen training to completion (for a fixed large number of optimization steps). \nWe demonstrate model-wise double descent across different architectures,\ndatasets, optimizers, and training procedures.\nThe critical region exhibits distinctly different test behavior\naround the interpolation point and there is often a peak in test error that becomes more prominent in settings with label noise.\n\nFor the experiments in this section (Figures~\\ref{fig:resnet-cifar}, \\ref{fig:cnn-cifar},\n\\ref{fig:more-mdd1appendix},\n\\ref{fig:mcnn_cf100},\n\\ref{fig:more-mdd2}), notice that all modifications which increase the interpolation threshold\n(such as adding label noise, using data augmentation, and increasing the number of train samples)\nalso correspondingly shift the peak in test error towards larger models.\nAdditional plots showing the early-stopping behavior of these models,\nand additional experiments showing double descent\nin settings with no label noise (e.g. Figure~\\ref{fig:app-cifar100-clean})\nare in Appendix~\\ref{subsec:appendix_model}.\nWe also observed model-wise double descent for adversarial training,\nwith a prominent robust test error peak even in settings without label noise.\nSee Figure~\\ref{fig:adv_training} in Appendix~\\ref{subsec:appendix_model}.\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{subfigure}{.47\\textwidth}\n        \\centering\n  \\includegraphics[width=\\textwidth]{cifar10_mcnn_noaug_pVar}\n        \\caption{Without data augmentation.}\n    \\end{subfigure}\\hfill\n    \\begin{subfigure}{.47\\textwidth}\n        \\centering\n  \\includegraphics[width=\\textwidth]{cifar10_mcnn_aug_pVar_trunc}\n        \\caption{With data augmentation.}\n    \\end{subfigure}\n    \\caption{{\\bf Effect of Data Augmentation.}\n    5-layer CNNs on CIFAR10, with and without data-augmentation.\n    Data-augmentation shifts the interpolation threshold to the right,\n    shifting the test error peak accordingly.\n    Optimized using SGD for 500K steps.~\n     See Figure~\\ref{fig:mcnn_aug_large} for larger models.\n    }\\label{fig:cnn-cifar}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\captionsetup{calcwidth=.9\\linewidth}\n\\begin{minipage}[t]{0.47\\textwidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{cifar10_mcnn_sgd_adam}\n  \\caption[width=0.8\\linewidth]{{\\bf SGD vs. Adam.}\n  5-Layer CNNs on CIFAR-10 with no label noise, and no data augmentation.\n  Optimized using SGD for 500K gradient steps, and Adam for 4K epochs.}\\label{fig:more-mdd1appendix}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{.47\\textwidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{cifar100_mcnn_noaug}\n    \\caption{{\\bf Noiseless settings.} \n    5-layer CNNs on CIFAR-100 with no label noise;\n    note the peak in test error.\n    Trained with SGD and no data augmentation.\n    See Figure~\\ref{fig:app-cifar100-clean-sgd}\n    for the early-stopping behavior of these models.\n    }\n    \\label{fig:mcnn_cf100}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[h]\n    \\centering\n    \\begin{minipage}[c]{0.5\\textwidth}\n        \\includegraphics[width=\\textwidth]{nlp_de_en_fr}\n    \\end{minipage}\\hfill\n    \\begin{minipage}[c]{0.45\\textwidth}\n      \\caption[0.9\\textwidth]{{\\bf Transformers on language translation tasks:} Multi-head-attention encoder-decoder Transformer model trained for 80k gradient steps with labeled smoothed cross-entropy loss on \\iwslt\\ German-to-English (160K sentences) and WMT`14 English-to-French (subsampled to 200K sentences) dataset. Test loss is measured as per-token perplexity.}\n      \\label{fig:more-mdd2}\n    \\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\paragraph{Discussion.}\nFully understanding the mechanisms behind model-wise double descent\nin deep neural networks remains an important open question.\nHowever, an analog of model-wise double descent occurs even\nfor linear models. A recent stream of theoretical works\nanalyzes this setting\n(\\cite{bartlett2019benign,muthukumar2019harmless,belkin2019two,mei2019generalization,hastie2019surprises}).\nWe believe similar mechanisms may be at work in deep neural networks.\n\nInformally, our intuition is that for model-sizes at the interpolation threshold, there is effectively\nonly one model that fits the train data and this interpolating model is very sensitive\nto noise in the train set and\/or model mis-specification.\nThat is, since the model is just barely able to fit the train data,\nforcing it to fit even slightly-noisy or mis-specified\nlabels will destroy its global structure, and result in high test error.\n(See Figure~\\ref{fig:ens_resnet} in the Appendix for an experiment\ndemonstrating this noise sensitivity, by showing that ensembling helps significantly\nin the critically-parameterized regime).\nHowever for over-parameterized models,\nthere are many interpolating models that fit the train set,\nand SGD is able to find one that ``memorizes''\n(or ``absorbs'') the noise while still performing well on the distribution.\n\nThe above intuition is theoretically justified for linear models.\nIn general, this situation manifests even without label noise for linear models\n(\\cite{mei2019generalization}),\nand occurs whenever there is \\emph{model mis-specification}\nbetween the structure of the true distribution and the model family.\nWe believe this intuition extends to deep learning as well,\nand it is consistent with our experiments.\n\n\n\n\\iffalse\n\\textcolor{blue}{ {\\bf Omit this paragraph? --Preetum.}\nThe situation for linear models is most clear with added label noise.\nHere, the fundamental reason why there is high test error\nat the interpolation threshold is: when the number of samples $n$\nis equal to the dimension $d$ of the linear model,\nthen there is exactly one model which fits the train set--- and this \ninterpolating model is very sensitive\nto noise in the train set.\nThat is, since the linear model is just barely able to fit the train data,\nforcing it to fit even slightly-noisy\nlabels will destroy its global structure, and result in high test error.\nHowever for ``overparameterized'' linear models, when $d \\gg n$,\nthere are many interpolating models that fit the train set,\nand selecting the minimum-norm one will often\nachieve good test performance\\footnote{Note that for linear problems,\nthe minimum-norm solution is the one found by gradient descent from $0$ initialization.}.\nThat is, overparameterized linear models are able to\nfit the label noise, while maintaining good performance on the distribution.\nIn general, this situation manifests even without label noise, whenever there is\n\\emph{model mis-specification} between the structure of the true distribution, and the\nlinear model family.\n}\n\nWe believe similar mechanisms may be at work in deep neural networks.\nInformally, our intuition is:\nfor model-sizes at the interpolation threshold, there is effectively\nonly one model that fits the train data.\nThis model is very sensitive to [noise in] the specific train samples.\nOverparameterized models, however, have the capacity to ``memorize the noise'',\nwithout affecting their global structure as much.\nMoreover, SGD (through unexplained means) manages to find such a minima.\nFor overparameterized models, there are many minima which fit the train data,\nan\nand SGD (through unexplained means) is able to ``memorize the noise''\nwithout destroying the global structure of the model.\n\\fi\n\n\n\n\\section{On The Mechanisms and Characterizations of Double Descent}\n\\label{sec:morediscuss}\n\n\\paragraph{On Effective Model Complexity.}\nWe stress that Effective Model Complexity is \\emph{not} a single-letter characterization\nof test error.\nFor example, two training procedures with the same EMC can output models with\nvery different test errors (e.g. consider a CNN and a fully-connected net, which both are capable of interpolating the same number of samples from a natural image distribution.\nHere, we would expect the CNN to have much lower test error than the fully-connected net.\nOur generalized double desccent hypothesis applies as we vary EMC in a ``smooth'' way--- say, by only\nchanging the model by a small amount--- and does not apply if we say, jump discontinuously from a CNN to a fully-connected net.\n\\section{Extended discussion of related work}\n\\label{sec:related_appendix}\n\n\n\n\\paragraph{\\cite{belkin2018reconciling}:} \nThis paper proposed, in very general terms, that the apparent contradiction between traditional notions of the bias-variance trade-off and empirically successful practices in deep learning can be reconciled under a double-descent curve---as model complexity increases, the test error follows the traditional ``U-shaped curve'', but beyond the point of interpolation, the error starts to \\textit{decrease}. This work provides empirical evidence for the double-descent curve with fully connected networks trained on subsets of MNIST, CIFAR10, SVHN and TIMIT datasets. They use the $l_2$ loss for their experiments. They demonstrate that neural networks are not an aberration in this regard---double-descent is a general phenomenon observed also in linear regression with random features and random forests.\n\n\\paragraph{Theoretical works on linear least squares regression:}\nA variety of papers have attempted to theoretically analyze this behavior in restricted settings, particularly the case of least squares regression under various assumptions on the training data, feature spaces and regularization method.\n\n\\begin{enumerate}\n    \\item \\cite{advani2017high, hastie2019surprises} both consider the linear regression problem stated above and analyze the generalization behavior in the asymptotic limit $N, D \\rightarrow \\infty$ using random matrix theory. \\cite{hastie2019surprises} highlight that when the model is mis-specified, the minimum of training error can occur for over-parameterized models\n    \\item \\cite{belkin2019two} Linear least squares regression for two data models, where the input data is sampled from a Gaussian and a Fourier series model for functions on a circle. They provide a finite-sample analysis for these two cases\n    \\item \\cite{bartlett2019benign} provides generalization bounds for the minimum $l_2$-norm interpolant for Gaussian features\n    \\item \\cite{muthukumar2019harmless} characterize the fundamental limit of of any interpolating solution in the presence of noise and provide some interesting Fourier-theoretic interpretations.\n    \\item \\cite{mei2019generalization}: This work provides asymptotic analysis for ridge regression over random features\n\\end{enumerate}\n\nSimilar double descent behavior was investigated in \\cite{opper1995statistical, opper2001learning}\n\n\\cite{geiger2019jamming} showed that deep fully connected networks trained on the MNIST dataset with hinge loss exhibit a ``jamming transition'' when the number of parameters exceeds a threshold that allows training to near-zero train loss. \\cite{geiger2019scaling} provide further experiments on CIFAR-10 with a convolutional network. They also highlight interesting behavior with ensembling around the critical regime, which is consistent with our\ninformal intuitions in Section~\\ref{sec:model-dd} and our experiments in Figures \\ref{fig:ens_resnet}, \\ref{fig:ens_mcnn}.\n\n\\cite{advani2017high, geiger2019jamming, geiger2019scaling} also point out that double-descent is not observed when optimal early-stopping is used.\n\n\n\n\n\n\n\n\\section{Random Features: A Case Study}\n\\label{sec:rff}\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{rff_fmnist.png} \\\\\n   \n   \n    \\caption{\\textbf{Random Fourier Features} on the Fashion MNIST dataset. The setting is equivalent to two-layer neural network with $e^{-ix}$ activation, with randomly-initialized first layer that is fixed throughout training. The second layer is trained using gradient flow.}\n    \\label{fig:rff}\n\\end{figure}\n\n\nIn this section, for completeness sake, we show that both the model- and sample-wise double descent phenomena are not unique to deep neural networks---they exist even in the setting of Random Fourier Features of \\cite{rahimi2008random}. This setting is equivalent to a two-layer neural network with $e^{-ix}$ activation. The first layer is initialized with a $\\mathcal{N}(0, \\frac{1}{d})$ Gaussian distribution  and then fixed throughout training. The width (or embedding dimension) $d$ of the first layer parameterizes the model size. The second layer is initialized with $0$s and trained with MSE loss.\n\nFigure \\ref{fig:rff} shows the grid of Test Error as a function of\nboth number of samples $n$ and model size $d$.\nNote that in this setting $\\mathrm{EMC} = d$ (the embedding dimension). As a result, as demonstrated in the figure, the peak follows the path of $n=d$. Both model-wise and sample-wise (see figure~\\ref{fig:rff-samples})  double descent phenomena are captured, by horizontally and vertically crossing the grid, respectively. \n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{figures\/rff_fmnist_samples.png} \\\\\n   \n   \n    \\caption{Sample-wise double-descent slice for Random Fourier Features on the Fashion MNIST dataset. In this figure the embedding dimension (number of random features) is 1000.}\n    \\label{fig:rff-samples}\n\\end{figure}\n\\section{Sample-wise Non-monotonicity}\n\\label{sec:samples}\n\nIn this section, we investigate the effect of varying the number of train samples,\nfor a fixed model and training procedure.\nPreviously, in model-wise and epoch-wise double descent, we explored behavior in the critical regime, where $\\mathrm{EMC}_{\\cD, {\\epsilon}}(\\cT) \\approx n$, by varying the EMC.\nHere, we explore the critical regime by varying the number of train samples $n$.\nBy increasing $n$, the same training procedure $\\cT$ can switch from being effectively over-parameterized\nto effectively under-parameterized.\n\n\nWe show that increasing the number of samples has two different effects on the test error vs. model complexity graph.\nOn the one hand, (as expected) increasing the number of samples shrinks the area under the curve.\nOn the other hand, increasing the number of samples also has the effect of ``shifting the curve to the right'' and increasing the model complexity at which test error peaks.\n\\begin{figure}[H]\n    \\centering\n    \\begin{subfigure}[c]{.45\\textwidth}\n        \n         \\centering\n       \n         \\includegraphics[width=\\textwidth]{mcnn_p10_big}\n         \\includegraphics[width=\\textwidth]{mcnn_p20_big}\n         \\caption{Model-wise double descent for 5-layer CNNs on CIFAR-10, for varying dataset sizes.  \n        \n         {\\bf Top:}\n         There is a range of model sizes (shaded green)\n         where training on $2\\x$ more samples does not improve test error.\n         {\\bf Bottom:}\n         There is a range of model sizes (shaded red)\n         where training on $4\\x$ more samples does not improve test error.}\n         \\label{fig:sample-modeld}\n    \\end{subfigure}\\hspace{0.7cm}\n    \\begin{subfigure}[c]{.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{nlp_sampledd_test-annotated}\n         \\caption{\\textbf{Sample-wise non-monotonicity.}\n        Test  loss (per-word perplexity) as a function of number of train samples, for two transformer models trained to completion on IWSLT'14. For both model sizes, there is a regime where more samples hurt performance. Compare to Figure~\\ref{fig:moresamplesareworse}, of model-wise double-descent in the identical setting.}\n        \\label{fig:nlpdd}\n    \\end{subfigure}\n    \\caption{Sample-wise non-monotonicity.}\n\\end{figure}\n\nThese twin effects are shown in Figure~\\ref{fig:sample-modeld}.\nNote that there is a range of model sizes\nwhere the effects ``cancel out''---and\nhaving $4\\x$ more train samples does not help test\nperformance when training to completion.\nOutside the critically-parameterized regime, for sufficiently under- or over-parameterized models, having more samples helps.\nThis phenomenon is corroborated in Figure~\\ref{fig:grid},\nwhich shows test error as a function of both model and sample size,\nin the same setting as Figure~\\ref{fig:sample-modeld}.\n\n\n\n\\begin{figure}[h]\n    \\centering\n      \\includegraphics[width=\\textwidth]{joint_grid}\n    \\caption{{\\bf Left:} Test Error as a function of model size and\n    number of train samples, for 5-layer CNNs on CIFAR-10 + 20\\% noise.\n    Note the ridge of high test error again lies along the interpolation threshold.\n    {\\bf Right: } Three slices of the left plot,\n    showing the effect of more data for models of different sizes.\n    Note that, when training to completion, more data helps for\n    small and large models, but does not help\n    for near-critically-parameterized models (green).\n    }\n    \\label{fig:grid}\n\\end{figure}\n\nIn some settings, these two effects combine to yield a regime of model sizes\nwhere more data actually hurts test performance as in\nFigure~\\ref{fig:moresamplesareworse} (see also \nFigure~\\ref{fig:nlpdd}).\nNote that this phenomenon is not unique to DNNs:\nmore data can hurt even for linear models \n(see Appendix~\\ref{sec:rff}).\n\n\\section{Experimental Setup}\n\n\nWe briefly describe the experimental setup here; full details are in Appendix~\\ref{sec:appendix_exp_details}\n\\footnote{The raw data from our experiments\nare available at:\n\\url{https:\/\/gitlab.com\/harvard-machine-learning\/double-descent\/tree\/master}}.\nWe consider three families of architectures: ResNets, standard CNNs, and Transformers.\n\\textbf{ResNets:}\nWe parameterize a family of ResNet18s (\\cite{he2016identity})\nby scaling the width (number of filters) of convolutional layers.\nSpecifically, we use layer widths $[k, 2k, 4k, 8k]$ for varying $k$.\nThe standard ResNet18 corresponds to $k=64$.\n\\textbf{Standard CNNs:}\nWe consider a simple family of 5-layer CNNs, with 4 convolutional layers of widths $[k, 2k, 4k, 8k]$ for varying $k$, and a fully-connected layer. For context, the CNN with width $k=64$, can reach over $90\\%$ test accuracy on CIFAR-10 with data-augmentation.\n\\textbf{Transformers:}\nWe consider the 6 layer encoder-decoder from \\cite{Vaswani},\nas implemented by \\cite{ott2019fairseq}.\nWe scale the size of the network by\nmodifying the embedding dimension $d_{\\text{model}}$,\nand setting the width of the fully-connected layers proportionally\n($d_{\\text{ff}} = 4\\cdot d_{\\text{model}}$).\nFor ResNets and CNNs, we train with cross-entropy loss, and the following optimizers:\n(1) Adam with learning-rate $0.0001$ for 4K epochs; (2) SGD with learning rate $\\propto \\frac{1}{\\sqrt{T}}$ for 500K gradient steps.\nWe train Transformers for 80K gradient steps, with 10\\% label smoothing and no drop-out.\n\n\\paragraph{Label Noise.}\nIn our experiments, label noise of probability $p$ refers to\ntraining on a samples which have the correct label\nwith probability $(1-p)$, and a uniformly random incorrect label otherwise (label noise is sampled only once and not per epoch).\nFigure~\\ref{fig:errorvscomplexity} plots test error on the noisy distribution, while the remaining figures plot test error with respect to the clean distribution (the two curves are just linear rescaling of one another).","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Pseudo-code of proposed methods}\n\\label{appendix:ap_pseudocode}\nIn this section, we provide the pseudo-codes of methods proposed in the main paper. First, Algorithm~\\ref{alg:ap_white} shows the pseudo-code of ODI for white-box attacks in Section~\\ref{sec_ODS_white}. The line 4-6 in the algorithm describes the iterative update by ODI.\n\n\n\\begin{algorithm}[htbp]\n   \\caption{Initialization by ODS (ODI) for white-box attacks }\n   \\label{alg:ap_white}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} A targeted image $\\bm{\\mathrm{x}}_{org}$, a target classifier $\\bm{\\mathrm{f}}$, perturbation set $B(\\bm{\\mathrm{x}}_{org})$, number of ODI steps $N_{\\text{ODI}}$, step size $ \\eta_{\\text{ODI}}$, number of restarts $N_R$\n   \\STATE {\\bfseries Output:} Starting points $\\{x^{start}_i \\}$ for adversarial attacks\n   \\FOR{$i=1$ {\\bfseries to} $N_R$}\n   \\STATE Sample $\\bm{\\mathrm{x}}_{0}$ from $B(\\bm{\\mathrm{x}}_{org})$, and sample $\\bm{\\mathrm{w}}_{\\text{d}} \\sim U(-1,1)^C$\n   \\FOR{$k=0$ {\\bfseries to} $N_{\\text{ODI}}-1$}\n   \\STATE  $\\bm{\\mathrm{x}}_{k+1}  \\gets \\mathrm{Proj}_{B(\\bm{\\mathrm{x}}_{org})} \\left( x_{k} + \\eta_{\\text{ODI}} \\, \\mathrm{sign}(\\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}}_k,\\bm{\\mathrm{f}},\\bm{\\mathrm{w}}_{\\text{d}}) ) \\right)$\n   \\ENDFOR\n   \\STATE $\\bm{\\mathrm{x}}^{start}_i \\gets x_{N_{\\text{ODI}}}$\n   \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\nWe also describe the algorithm of Boundary-ODS, used in Section~\\ref{sec_black_decision} of the main paper.  Algorithm~\\ref{alg:ap_boundary} shows pseudo-code of Boundary-ODS. \nThe original Boundary Attack~\\citep{Brendel18} first sampled a random noise vector $\\bm{\\mathrm{q}}$ from a Gaussian distribution $\\mathcal{N}(\\mathbf{0}, \\mathbf{I})$ and then orthogonalized the vector to keep the distance from the original image (line 7 in Algorithm~\\ref{alg:ap_boundary}).\nAfter that, the attack refined the vector $\\bm{\\mathrm{q}}$ to reduce the distance from the original image such that the following equation holds:\n\\begin{equation}\n\\label{eq_decision_update}\nd(\\bm{\\mathrm{x}},\\bm{\\mathrm{x}}_{adv}) - d(\\bm{\\mathrm{x}},\\bm{\\mathrm{x}}_{adv}+\\bm{\\mathrm{q}}) = \\epsilon \\cdot d(\\bm{\\mathrm{x}},\\bm{\\mathrm{x}}_{adv})\n\\end{equation}\nwhere $d(a,b)$ is the distance between $a$ and $b$.\nWe replace the random Gaussian sampling to ODS as in the line 5 and 6 of Algorithm~\\ref{alg:ap_boundary}.\nSampled vectors by ODS yield large changes for outputs on the target model and increase the probability that the updated image is adversarial (i.e. the image satisfies the line 9 of Algorithm~\\ref{alg:ap_boundary}), so ODS makes the attack efficient.\n\n\\begin{algorithm}[htbp]\n   \\caption{Boundary Attack~\\citep{Brendel18} with sampling update direction by ODS }\n   \\label{alg:ap_boundary}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} A targeted image $\\bm{\\mathrm{x}}$, a label $y$, a target classifier $\\bm{\\mathrm{f}}$, a set of surrogate models $\\mathcal{G}$\n   \\STATE {\\bfseries Output:} attack result $\\bm{\\mathrm{x}}_{adv}$\n   \\STATE Set the starting point $\\bm{\\mathrm{x}}_{adv} = \\bm{\\mathrm{x}}$ which is adversary\n   \\WHILE {$k<$ number of steps}\n   \\STATE  Choose a surrogate model $\\bm{\\mathrm{g}}$ from $\\mathcal{G}$, sample $\\bm{\\mathrm{w}}_{\\text{d}} \\sim U(-1,1)^C$\n   \\STATE  Set $\\bm{\\mathrm{q}} = \\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}}_{adv},\\bm{\\mathrm{g}},\\bm{\\mathrm{w}}_{\\text{d}})$\n   \\STATE  Project $\\bm{\\mathrm{q}}$ onto a sphere around the original image $\\bm{\\mathrm{x}}$\n   \\STATE Update $\\bm{\\mathrm{q}}$ with a small movement toward the original image $\\bm{\\mathrm{x}}$ such that Equation~(\\ref{eq_decision_update}) holds\n   \\IF{$\\bm{\\mathrm{x}}_{adv}+\\bm{\\mathrm{q}}$ is adversarial}\n   \\STATE   Set $\\bm{\\mathrm{x}}_{adv}=\\bm{\\mathrm{x}}_{adv}+\\bm{\\mathrm{q}}$ \n   \\ENDIF\n   \\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\section{Details of experiment settings}\n\\label{appendix:ap_parameter}\n\n\n\\subsection{Hyperparameters and settings for attacks in Section~\\ref{sec_white_various}}\n\\label{appendix:ap_parameter_whiteall}\nWe describe hyperparameters and settings for PGD and C\\&W attacks in Section~\\ref{sec_white_various}. \n\nMultiple loss functions $L(\\cdot)$ can be used for PGD attacks, including the cross-entropy loss, and the margin loss defined as $\\max_{i \\neq y} f_{i}(\\bm{\\mathrm{x}}) - f_{y}(\\bm{\\mathrm{x}})$. %\nWe use the margin loss for PGD attacks to make considered attacking methods stronger.\n\n\n\nPGD attacks have three hyperparameters: pertubation size $\\epsilon$, step size $\\eta$ and number of steps $N$. \nWe chose $\\epsilon=0.3,8\/255,4\/255$, $\\eta= 0.02, 2\/255, 0.5\/255$ and $N=40,20,50$ for MadryLab (MNIST), MadryLab (CIFAR-10), ResNet152 Denoise (ImageNet), respectively. We use the whole test set except for ImageNet, where the first 1000 test images are used. \n\nFor C\\&W attacks, we define na\\\"{i}ve random initialization to make sure the starting points are within an $\\ell_2$ $\\epsilon$-radius ball: we first sample Gaussian noise $\\bm{\\mathrm{w}} \\sim  \\mathcal{N}(\\mathbf{0}, \\mathbf{I})$ and then add the clipped noise $\\epsilon \\cdot \\bm{\\mathrm{w}} \/ \\|\\bm{\\mathrm{w}}\\|_2$ to an original image. We set the perturbation radius of initialization $\\epsilon$ by reference to attack bounds in other studies: $\\epsilon= 2.0, 1.0, 5.0$ for MadryLab (MNIST), MadryLab (CIFAR-10), ResNet152 Denoise (ImageNet), respectively.\nwe also set hyperparameters of C\\&W attacks as follows: max iterations are 1000 (MNIST) and 100 (CIFAR-10 and ImageNet), search step is 10, learning rate is 0.1, and initial constant is 0.01. The attack is performed for the first 1000 images (MNIST and CIFAR-10) and the first 500 images (ImageNet).\n\n\n\\subsection{Hyperparameter tuning for tuned ODI-PGD in Section~\\ref{sec_sota} }\n\\label{appendix:ap_parameter_ODI}\nWe describe hyperparameter tuning for our tuned ODI-PGD in Section~\\ref{sec_sota}. \nWe summarize the setting in Table~\\ref{tab_para}. \n\n\\begin{table*}[ht]\n\\caption{Hyperparameter setting for tuned ODI-PGD in Section~\\ref{sec_sota}.}\n\\label{tab_para}\n\\begin{center}\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{tabular}{c|cc|ccc}\n\\toprule\n  & \\multicolumn{2}{c|}{\\text{ODI}}  & \\multicolumn{3}{c}{PGD} \\\\ \nmodel & \\begin{tabular}{c} total step \\\\ $N_{\\text{ODI}}$ \\end{tabular}  & \\begin{tabular}{c} step size \\\\ $\\eta_{\\text{ODI}}$ \\end{tabular} & \n  optimizer & \n \\begin{tabular}{c} total step \\\\ $N$ \\end{tabular}   & \n \\begin{tabular}{c} step size (learning rate) \\\\ $\\eta_k$  \\end{tabular}                    \\\\ \\midrule\n\\begin{tabular}{c}MNIST\n\\end{tabular} & 50 & \n    0.05   & Adam & 950  & \n  $\\begin{array}{l@{~}l}\n    0.1 & (k < 475) \\\\\n    0.01 & (475 \\leq k < 712) \\\\\n    0.001 & (712 \\leq k)\n  \\end{array}$ \\\\ \\hline\n\\begin{tabular}{c}CIFAR-10\n\\end{tabular}  & 10  & \n    8\/255 & \\begin{tabular}{c}sign \\\\ function \\end{tabular} & 140 & \n  $\\begin{array}{l@{~}l}\n    8\/255 & (k < 46) \\\\\n    0.8\/255  & (46 \\leq k <  92) \\\\\n    0.08\/255 & (92 \\leq k)\n  \\end{array}$ \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{table*}\nFor ODI, we increase the number of ODI step $N_{\\text{ODI}}$ to obtain more diversified inputs than ODI with $N_{\\text{ODI}}=2$. In addition, we make step size $\\eta_{\\text{ODI}}$ smaller than $\\epsilon$ on MNIST, because $\\epsilon$-ball with $\\epsilon=0.3$ is large and $\\eta_{\\text{ODI}}=0.3$ is not suitable for seeking the diversity within the large  $\\epsilon$-ball. \nIn summary, we set $(N_{\\text{ODI}},\\eta_{\\text{ODI}})=(50,0.05), (10, 8\/255)$ for the MNIST model and the CIFAR-10 model, respectively.\n\n\nWe tune hyperparameters of PGD based on Gowal et al.~\\citep{MT19}.  \nWhile several studies used the sign function to update images for the PGD attack, some studies \\citep{SPSA18,MT19} reported that updates by Adam optimizer~\\citep{Adam15} brought better results than the sign function. Following the previous studies~\\citep{SPSA18,MT19}, we consider the sign function as an optimizer and the choice of an optimizer as a hyperparameter. \nWe use Adam for the PGD attack on the MNIST model and the sign function on the CIFAR-10 model. \n\nWe adopt scheduled step size instead of fixed one. \nBecause we empirically found that starting from large step size brings better results, we set the initial step size $\\eta_0$ as $\\eta_0=\\epsilon$ on CIFAR-10. We update step size at $k=0.5N, 0.75N$ on MNIST and $k=N\/3, 2N\/3$ for on CIFAR-10. \nWhen we use Adam, step size is considered as learning rate.\nFinally, we set PGD step $N$ as $N_{\\text{ODI}}+N=1000$ on MNIST and $150$ on CIFAR-10.\n\n\\subsection{Setting for training on ImageNet in Section~\\ref{sec_black_limited}}\n\\label{appendix:ap_parameter_blackOOD}\nWe describe the setting of training of surrogate models on ImageNet in the experiment of Section~\\ref{sec_black_limited}.\nWe use the implementation of training provided in PyTorch with default hyperparameters.\nNamely, training epochs are 90 and learning rates are changed depending on epoch: 0.1 until 30 epochs, 0.01 until 60 epochs, 0.001 until 90 epochs. Batch size is 256 and weight decay 0.0001.\n\n\n\n\n\\section{Additional results and experiments for ODI with white-box attacks}\n\\label{appendix:ap_white}\n\n\\subsection{Diversity offered by ODI}\n\\label{appendix:ap_white_diversity}\n\\label{sec_append_white_diversity}\nWe empirically demonstrate that ODI can find a more diverse set of starting points than random uniform initialization, as pictorially shown in the left figures of Figure~\\ref{figure1} of the main paper. \n\nAs an example of target models, we train a robust classification model using adversarial training~\\citep{madry17} on CIFAR-10. We adopted popular hyperparameters for adversarial training under the $\\ell_\\infty$ PGD attack on CIFAR-10: perturbation size $\\epsilon = 8\/255$, step size $\\eta=2\/255$, and number of steps $N=10$. Training epochs are 100 and learning rates are changed depending on epoch: 0.1 until 75 epochs, 0.01 until 90 epochs, and 0.001 until 100 epochs. Batch size is 128 and weight decay 0.0002.\n\nOn the target model, we quantitatively evaluate the diversity of starting points by each initialization in terms of pairwise distances of output values $\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})$. Each initialization is bounded within $\\ell_\\infty$ $\\epsilon$-ball with $\\epsilon=8\/255$.\nWe pick 100 images on CIFAR-10 and run each initialization 10 times to calculate the mean pairwise distances among outputs for different starting points. As a result, the mean pairwise distance obtained from ODI is 6.41, which is about 15 times larger than that from uniform initialization (0.38). This corroborates our intuition that starting points obtained by ODI are more diverse than uniform initialization. We note that PGD does not generate diverse samples. When we use PGD with 2 steps as an initialization, the mean pairwise distance is only 0.43. \n\nWe also visualize the diversity offered by ODI. \nFirst, we focus on loss histogram of starting points by ODI and na\\\"{i}ve uniform initialization. We pick an image from the CIFAR-10 test dataset and run each initialization 100 times. Then, we calculate loss values for starting points to visualize their diversity in the output space. The left panel of Figure~\\ref{fig_div_visualize} is the histogram of loss values for each initialization. We can easily observe that images from na\\\"{i}ve initialization concentrate in terms of loss values (around $-1.0$), whereas images from ODI-2 (ODI with 2 steps) are much more diverse in terms of the loss values. We also observe that images from PGD-2 also take similar loss values. \nBy starting attacks from these initial inputs, we obtain the histogram of loss values in the center panel of Figure~\\ref{fig_div_visualize}. We can observe that ODI-PGD generates more diverse results than PGD with na\\\"{i}ve initialization (PGD-20).\n\nIn addition, we apply \nt-SNE~\\citep{tSNE} to the output logits for starting points by each initialization.\nWe visualize the embedding produced by t-SNE in the right panel of Figure~\\ref{fig_div_visualize}.\nAs expected, starting points produced by ODI are more diversified than those by na\\\"{i}ve initialization.\n\n\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{ccc}\n\n      \\begin{minipage}{0.35\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/neurips_losshist1_camera.png}\n        \\end{center}\n      \\end{minipage}\n      \\begin{minipage}{0.35\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/neurips_losshist2_camera.png}\n        \\end{center}\n      \\end{minipage}\n      \\hspace{0.3cm}\n      \\begin{minipage}{0.23\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.9\\textwidth]{image\/ODI_loss_tsne.png}\n        \\end{center}\n      \\end{minipage}\n    \\end{tabular}\n    \\caption{(Left): Histogram of loss values evaluated at starting points by ODI, na\\\"{i}ve uniform initialization and PGD. PGD-2 means 2-step PGD with na\\\"{i}ve initialization. The loss function is the margin loss. (Right) Histogram of loss values after attacks with 20 total steps. ODI-PGD-18 means 18-step PGD with 2-step ODI. \n    (Right): Embedding for starting points sampled on each initialization produced by t-SNE.\n    }\n    \\label{fig_div_visualize}\n\\end{figure*}\n\n\n\n\\subsection{Analysis of the sensitivity to hyperparameters of ODI}\n\\label{appendix:ap_white_sensitivity}\nFor ODI, we mainly set the number of ODI steps $N_{\\text{ODI}}=2$ and step size $\\eta_{\\text{ODI}}=\\epsilon$. \nTo validate the setting, we confirm that ODI-PGD is not sensitive to these hyperparameters. \nWe attack adversarially trained models on CIFAR-10 introduced in Section~\\ref{sec_append_white_diversity}, and adopt the same attack setup for ODI-PGD on CIFAR-10 as Section~\\ref{sec_white_various}.\nWe test $N_{\\text{ODI}}=2,4,8,16$ and $\\eta_{\\text{ODI}} = \\epsilon, \\epsilon\/2, \\epsilon\/4, \\epsilon\/8$, but \n exclude patterns with  $N_{\\text{ODI}} \\cdot \\eta_{\\text{ODI}} < 2 \\epsilon$ to make $N_{\\text{ODI}} \\cdot \\eta_{\\text{ODI}}$ larger than or equal to the diameter of the $\\epsilon$-ball. \nWe calculate the mean accuracy for five repetitions of the attack, each with 20 restarts. \n\n\n\\begin{table}[htb]\n\\caption{The sensitivity to the number of ODI steps $N_{\\text{ODI}}$ and step size $\\eta_{\\text{ODI}}$.  \nWe repeat each experiment 5 times to calculate statistics.\n}\n\\begin{center}\n\\begin{tabular}{cc|ccc}\n\\toprule\n$N_{\\text{ODI}}$ & $\\eta_{\\text{ODI}}$ & mean & max & min  \\\\ \\midrule\n2 &$\\epsilon$ & 44.46\\% &44.50\\% & 44.45\\% \\\\\n4 &$\\epsilon \/2$ & 44.47\\% &44.50\\% & 44.42\\% \\\\\n4 &$\\epsilon $ & 44.42\\% &44.48\\% & 44.40\\% \\\\\n8 &$\\epsilon \/4$ & 44.47\\% &44.52\\% & 44.44\\% \\\\\n8 &$\\epsilon \/2$ & 44.42\\% &44.48\\% & 44.36\\% \\\\\n8 &$\\epsilon $ &  44.46\\% &44.49\\% & 44.42\\% \\\\\n16 &$\\epsilon \/8$ & 44.46\\% &44.50\\% & 44.43\\% \\\\\n16 &$\\epsilon \/4$ & 44.46\\% &44.50\\% & 44.40\\% \\\\\n16 &$\\epsilon \/2$ &  44.45\\% &44.48\\% & 44.43\\% \\\\\n16 &$\\epsilon$ &  44.44\\% &44.47\\% & 44.41\\% \\\\\n \\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_sensi}\n\\end{table}\n\nTable~\\ref{tab_sensi} shows the mean accuracy under ODI-PGD for different hyperparameters. \nThe maximum difference in the mean accuracy among different hyperparameters of ODI is only 0.05\\%.\nAlthough large $N_{\\text{ODI}}$ and $\\eta_{\\text{ODI}}$ will be useful to find more diversified starting points, the performance of ODI is not very sensitive to hyperparameters.\nThus, we restrict $N_{\\text{ODI}}$ to a small value to give fair comparison in terms of computation time as much as possible. \nTable~\\ref{tab_sensi} also shows that the difference between the maximum and minimum accuracy is about 0.1\\% for all hyperparameter pairs. This result supports the stability of ODI.\n\n\\subsection{Accuracy curve for adversarial attacks with ODI}\n\\label{appendix:ap_white_accuracycurve}\nIn Section~\\ref{sec_white}, we experimentally represented that the diversity offered by ODI improved white-box $\\ell_\\infty$ and $\\ell_2$ attacks. \nwe describe the accuracy curve with the number of restarts for attacks with ODI and na\\\"{i}ve initialization.\n\nFigure~\\ref{fig_Linf} shows how the attack performance improves as the number of restarts increases in the experiment of Section~\\ref{sec_white_various}. \nAttacks with ODI outperforms those with na\\\"{i}ve initialization with the increase of restarts in all settings. These curves further corroborate that restarts facilitate the running of attack algorithms, and ODI restarts are more effective than na\\\"{i}ve ones. We note that the first restart of ODI is sometimes worse than na\\\"{i}ve initialization. It is because diversity can cause local optima, i.e. random directions of ODI are not always useful. With the increase of restarts, at least one direction is useful and the accuracy drops.\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{cccc}\n      \\rotatebox[origin=c]{90}{PGD}&\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_PGD_mnist.png}\n        \\end{center}\n      \\end{minipage}\n\n      &\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_PGD_cifar.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_PGD_image.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\\n      \\rotatebox[origin=c]{90}{C\\&W}&\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_cw_mnist.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_cw_cifar.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.28\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/white_cw_image.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\ \n        &   \\begin{tabular}{c} MNIST \n           \\end{tabular}&\n           \\begin{tabular}{c} CIFAR-10 \n           \\end{tabular}&\n           \\begin{tabular}{c} ImageNet \n           \\end{tabular}\n    \\end{tabular}\n    \\caption{The attack performance against number of restarts for attacks with ODI. (Top): the model accuracy for PGD, (Bottom):  the average of minimum $\\ell_2$ perturbations for C\\&W. %\n    }%\n    \\label{fig_Linf}\n\\end{figure*}\n\nNext, we describe the accuracy curve for the comparison between state-of-the-are attacks and ODI-PGD in Section~\\ref{sec_sota}. \nTo emphasize the stability of the improvement, \nwe evaluate the confidence intervals of our results against MadryLab's MNIST and CIFAR-10 models. We run tuned ODI-PGD attack with 3000 restarts on MNIST and \n100 restarts on CIFAR-10. \nThen, we sample 1000 runs on MNIST and 20 runs on CIFAR-10 from the results to evaluate the model accuracy, \nand re-sample 100 times to calculate statistics. \nFigure~\\ref{fig_SOTAmadry} shows the accuracy curve under tuned ODI-PGD. \nWe observe that confidence intervals become tighter as the number of restarts grows, and tuned ODI-PGD consistently outperforms the state-of-the-art attack after 1000 restarts on MNIST and 20 restarts on CIFAR-10.\n\n\n\\begin{figure}[ht]\n\\centering\n\n    \\begin{tabular}{cc}\n      \\begin{minipage}{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=.9\\textwidth]{image\/white_tuned_mnist.png}\n        \\end{center}\n      \\end{minipage}\n        &\n      \\begin{minipage}{0.45\\hsize}\n        \\begin{center}\n          \\includegraphics[width=.9\\textwidth]{image\/white_tuned_cifar.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\\n      MadryLab (MNIST) & MadryLab (CIFAR-10) \n    \\end{tabular}\n    \\caption{Model accuracy under tuned ODI-PGD and the current state-of-the-art attacks~\\citep{MT19}. \n    The solid lines represent values from Table~\\ref{tab_sota} and \n    the error bars show 95\\% confidence intervals. }\n    \\label{fig_SOTAmadry}\n\\end{figure}\n\n\n\\subsection{Tighter estimation of robustness for various models}\n\\label{appendix:ap_white_recent}\nOne important application of powerful adversarial attacks is to evaluate and compare different defense methods. In many previous works on defending against adversarial examples, PGD attack with na\\\"{i}ve uniform initialization (called na\\\"{i}ve-PGD) is a prevailing benchmark and its attack success rate is commonly regarded as a tight estimation on (worst-case) model robustness. In this section, we conduct a case study on six recently published defense methods~\\citep{UAT19,carmon19,scatter19,metric19,free19,YOPO19} to show that ODI-PGD outperforms na\\\"{i}ve-PGD in terms of upper bounding the worst model accuracy under all possible attacks.\n\n\\paragraph{Setup}\nWe use pre-trained models from four of those studies,\nand train the other two models~\\citep{free19,YOPO19}\n using the settings and architectures described in their original papers. We run attacks with $\\epsilon = 8\/255$ on all test images. Other attack settings are the same as the experiment for CIFAR-10 in Section~\\ref{sec_white_various}. Apart from comparing ODI-PGD and na\\\"{i}ve-PGD, we also evaluate PGD attack without restarts (denoted as $\\text{PGD}_{1}$) as it is adopted in several existing studies \\citep{UAT19,carmon19,scatter19,YOPO19}.\n \n\n\\begin{table*}[ht]\n\\caption{Accuracy of models after performing ODI-PGD and na\\\"{i}ve-PGD attacks against recently proposed defense models.}\n\\begin{center}\n\\begin{tabular}{c|ccc|cc}\n\\toprule\nmodel& (1) $\\text{PGD}_{1}$ & (2)  $\\text{na\\\"{i}ve-PGD}$ \n& (3) $\\text{ODI-PGD}$ \n&(1)$-$(2) &(2)$-$(3)\n\\\\ \\midrule\nUAT~\\citep{UAT19} & 62.63\\% & 61.93\\% & {\\bf 57.43\\%} & 0.70\\% & 4.50\\% \\\\\nRST~\\citep{carmon19} & 61.17\\% & 60.77\\% &  {\\bf 59.93\\%} & 0.40\\% &  0.84\\%  \\\\\nFeature-scatter~\\citep{scatter19} & 59.69\\% & 56.49\\% &  {\\bf 39.52\\%} & 3.20\\% &  16.97\\%  \\\\\nMetric learning~\\citep{metric19} & 50.57\\% & 49.91\\% &  {\\bf 47.64\\%} & 0.56\\% &  2.27\\%  \\\\\nFree~\\citep{free19} & 47.19\\% & 46.39\\% &{\\bf 44.20\\%} & 0.80\\% & 2.19\\%  \\\\\nYOPO~\\citep{YOPO19} & 47.70\\% & 47.07\\% & {\\bf 45.09\\%} & 0.63\\% & 1.98\\% \\\\\n \\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_recent}\n\\end{table*}\n\n\n\n\\paragraph{Results}\nAs shown in Table~\\ref{tab_recent}, ODI-PGD uniformly outperforms na\\\"{i}ve-PGD against all six recently-proposed defense methods, lowering the estimated model accuracy by 1--17\\%. In other words, ODI-PGD provides uniformly tighter upper bounds on the worst case model accuracy than na\\\"{i}ve-PGD. \nAdditionally, The accuracy ranking of the defence methods for ODI-PGD is different from na\\\"{i}ve-PGD and $\\text{PGD}_{1}$.\nThese results indicate that ODI-PGD might be {a better benchmark for comparing and evaluating different defense methods}, rather than na\\\"{i}ve-PGD and $\\text{PGD}_{1}$.\n\n\n\\section{Additional results and experiments for ODS with black-box attacks}\n\\label{appendix:ap_black}\n\n\n\n\\subsection{Diversified samples by ODS}\n\\label{appendix:ap_black_diversity}\nWe empirically show that ODS can yield diversified changes in the output space of the target model, as shown in the right figures of Figure~\\ref{figure1} of the main paper. \nSpecifically, we evaluate the mean pairwise distance among outputs for different perturbations by ODS and \ncompare it with the distance among outputs for random Gaussian sampling. \n\nWe use pre-trained Resnet50~\\citep{resnet16} and VGG19~\\citep{VGG19} model \nas the target and surrogate models, respectively. \nWe pick 100 images on ImageNet validation set and sample perturbations 10 times by each sampling method. For comparison, we normalize the perturbation to the same size in the input space. \nThen, the obtained pairwise distance on the target model by ODS is 0.79, which is 10 times larger than the pairwise distance by random Gaussian sampling (0.07). This indicates that the diversity by ODS is transferable. \n\n\\subsection{Success rate curve in Section~\\ref{sec_black_score} and Section~\\ref{sec_black_decision} }\n\\label{appendix:ap_black_successcurve}\nIn Section~\\ref{sec_black_score}, we demonstrated that SimBA-ODS outperformed state-of-the-art attacks in terms of the query-efficiency. As an additional result, we give the success rate curve of score-based attacks with respect to the number of queries in the experiments. Figure~\\ref{fig_append_simba} shows how the success rate changes with the number of queries for SimBA-ODS and SimBA-DCT for the experiment of  Table~\\ref{tab_black_score}. SimBA-ODS especially brings query-efficiency at small query levels. In Figure~\\ref{fig_append_square}, we also describe the success rate curve for the experiment of Table~\\ref{tab_black_square}. ODS-RGF outperforms other methods in the $\\ell_2$ norm.\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{cc}\n      \\begin{minipage}{0.4\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.9\\textwidth]{image\/simba_untarget.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.4\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.9\\textwidth]{image\/simba_target.png}\n        \\end{center}\n      \\end{minipage}\\\\\n      untargeted & targeted\n    \\end{tabular}\n    \\caption{Relationship between success rate and number of queries for score-based SimBA-ODS and SimBA-DCT.}\n    \\label{fig_append_simba}\n\\end{figure*}\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{cccc}\n      \\begin{minipage}{0.22\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1\\textwidth]{image\/curve_score_untargeted.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.22\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1\\textwidth]{image\/curve_score_targeted.png}\n        \\end{center}\n      \\end{minipage} &\n      \\begin{minipage}{0.22\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1\\textwidth]{image\/curve_score_linf_untargeted.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.22\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1\\textwidth]{image\/curve_score_linf_targeted.png}\n        \\end{center}\n      \\end{minipage}\\\\\n      untargeted ($\\ell_2$) & targeted  ($\\ell_2$) & untargeted ($\\ell_\\infty$) & targeted  ($\\ell_\\infty$)\n    \\end{tabular}\n    \\caption{Relationship between success rate and number of queries for SimBA-ODS, ODS-RGF, and Square Attack. Each attack is evaluated with norm bound $\\epsilon=5 (\\ell_2), 0.05 (\\ell_\\infty)$.}\n    \\label{fig_append_square}\n\\end{figure*}\n\nIn Section~\\ref{sec_black_decision}, we demonstrated that Boundary-ODS outperformed state-of-the-art attacks in terms of median $\\ell_2$ perturbation. Here, we depict the relationship between the success rate and perturbation size (i.e. the frequency distribution of the perturbations) to show the consistency of the improvement. \nFigure~\\ref{fig_append_decision_curve} describes the cumulative frequency distribution of $\\ell_2$ perturbations for each attack at 10000 queries.\nBoundary-ODS consistently decreases $\\ell_2$ perturbations compared to other attacks in both untargeted and targeted settings.  \n\n\n\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{cc}\n      \\begin{minipage}{0.4\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.9\\textwidth]{image\/decision_l2hist_untargeted.png}\n        \\end{center}\n      \\end{minipage}\n      &\n      \\begin{minipage}{0.4\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.9\\textwidth]{image\/decision_l2hist_targeted.png}\n        \\end{center}\n      \\end{minipage}\\\\\n      untargeted & targeted\n    \\end{tabular}\n    \\caption{Cumulative frequency distribution of $\\ell_2$ perturbations at 10000 queries for decision-based attacks.}\n    \\label{fig_append_decision_curve}\n\\end{figure*}\n\n\n\n\\subsection{Comparison of ODS with TREMBA}\n\\label{appendix:ap_black_TREMBA}\nWe run experiments to compare ODS with TREMBA, which is a state-of-the-art attack with surrogate models, as we mentioned in Section~\\ref{sec_black_score_RGF}. \nTREMBA leverages surrogate models to learn a\nlow-dimensional embedding \nso as to obtain initial adversarial examples using a transfer-based attack and then update them using a score-based attack. \nAlthough TREMBA uses random sampling, ODS does not work well with TREMBA because random sampling of TREMBA is performed in the embedding space. In addition, it is difficult to directly compare attacks with ODS (e.g., ODS-RGF) and TREMBA because we do not discuss the combination of ODS with transfer-based attacks in this paper. \n\nHowever, we can start attacks with ODS (e.g., ODS-RGF) from images generated by any transfer-based attacks and compare the attack with TREMBA. \nWe generate starting points by SI-NI-DIM~\\citep{NesterovTransfer2020} (Scale-Invariant Nesterov Iterative FGSM integrated with diverse input method), which is a state-of-the-art transfer-based attack, and run ODS-RGF from these starting points. \n\nWe adopt the same experiment setup as TREMBA~\\citep{Huang20TREMBA}: we evaluate attacks against four target models (VGG19, ResNet34, DenseNet121, MobileNetV2) for 1000 images on ImageNet and use four surrogate models (VGG16, Resnet18, Squeezenet~\\citep{squeezenet} and Googlenet~\\citep{googlenet} ).\nWe set the same hyperparameters as in the original paper~\\citep{Huang20TREMBA} for TREMBA. For fair comparisons, we set the same sample size (20) and use the same surrogate models as TREMBA for ODS-RGF. We also set step size of ODS-RGF as 0.001. \nAs for SI-NI-DIM, we set hyperparameters referring to the paper~\\citep{NesterovTransfer2020}: maximum iterations as 20, decay factor as 1, and number of scale copies as 5.\nWe describe results in Table~\\ref{fig_black_TREMBA}. We can observe that \nODS-RGF with SI-NI-DIM is comparable to TREMBA. \n\nWe note that ODS is more flexible than TREMBA in some aspects.  First, TREMBA is specific in the $\\ell_\\infty$-norm, whereas ODS can be combined with attacks at least in $\\ell_\\infty$ and $\\ell_2$-norms. \nIn addition, TREMBA needs to train a generator per target class in targeted settings, whereas ODS does not need additional training. \n\n\n\\begin{table}[htb]\n\\caption{Comparison of ODS-RGF with TREMBA against four target models. The first two rows and bottom two rows describe results for untargeted (U) attacks and targeted (T) attacks, respectively. Targeted class for targeted attacks is class 0. }\n\\begin{center}\n\\setlength{\\tabcolsep}{2pt}\n\\begin{tabular}{cc|cc|cc|cc|cc}\n\\toprule\n&& \\multicolumn{2}{c|}{VGG19}& \\multicolumn{2}{c|}{ResNet34}& \\multicolumn{2}{c|}{DenseNet121}& \\multicolumn{2}{c}{MobilenetV2}\\\\\n&attack& success & query& success & query& success & query& success & query\\\\ \\hline\n\\multirow{2}{*}{U}&TREMBA~\\citep{Huang20TREMBA}  & {\\bf 100.0\\%} & 34\n&  {\\bf 100.0\\%} & 161 &  {\\bf 100.0\\%} & 157 &  {\\bf 100.0\\%}& 63\\\\\n&SI-NI-DIM~\\citep{NesterovTransfer2020} + ODS-RGF& {\\bf 100.0\\%} & {\\bf 18}\n&  99.9\\% & {\\bf 47} &   99.9\\%  & {\\bf 50} &  {\\bf 100.0\\%}& {\\bf 29} \\\\ \\hline\n\\multirow{2}{*}{T}&TREMBA~\\citep{Huang20TREMBA} & {98.6\\%} & 975\n&  {96.7\\%} & {\\bf 1421} &  {\\bf 98.5\\%} & {\\bf 1151} &  {\\bf 99.0\\%}& {\\bf 1163}\\\\\n&SI-NI-DIM~\\citep{NesterovTransfer2020} + ODS-RGF& {\\bf 99.4\\%} & {\\bf 634}\n& {\\bf 98.7\\%} & {1578}   & {98.2\\%} & 1550 & {98.3\\%}& {2006} \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{fig_black_TREMBA}\n\\end{table}\n\n\n\\subsection{Performance of ODS against different target models}\n\\label{appendix:ap_black_targetchange}\nIn this paper, we used pre-trained ResNet50 model as the target model for all experiments in Section~\\ref{sec_black_experiment}. Here we set pre-trained VGG19 model as the target model and run experiments to show that the efficiency of ODS is independent with target models.\nAs surrogate models, we replace VGG19 with ResNet50, i.e. we use four pre-trained models (ResNet50, ResNet34, DenseNet121, MobileNetV2).\n\nWe run experiments for SimBA-ODS in Section~\\ref{sec_black_score} and Boundary-ODS in Section~\\ref{sec_black_decision}. \nAll settings except the target model and surrogate models are the same as the previous experiments.\nIn Table~\\ref{tab_black_score_VGG} and \\ref{tab_black_decision_VGG}, ODS significantly improves attacks against VGG19 model for both SimBA and Boundary Attack. This indicates that the efficiency of ODS does not depend on target models. \n\n\\begin{table}[htb]\n\\caption{Query counts and $\\ell_2$ perturbations for score-based Simple Black-box Attacks (SimBA) against pre-trained VGG19 model on ImageNet.}\n\\begin{center}\n\\setlength{\\tabcolsep}{4pt}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n&&  \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{3-8}\n& num. of &success  & average & median $\\ell_2$& success  & average & median $\\ell_2$  \\\\ \\\nattack& surrogates& rate & query &  distance &  rate & query & distance  \\\\ \\midrule\nSimBA-DCT~\\citep{Guo19} &0& {\\bf 100.0\\%} & 619 & 2.85 & {\\bf100.0\\%} & 4091 &6.81 \\\\\nSimBA-ODS &4& {\\bf 100.0\\%} & {\\bf 176} & {\\bf 1.35} & 99.7\\% & {\\bf 1779 } & {\\bf 3.31} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_score_VGG}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for decision-based Boundary Attacks against pre-trained VGG19 model on ImageNet.}\n\\begin{center}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n&&  \\multicolumn{6}{c}{number of queries} \\\\\n& num. of & \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ %\nattack&surrogates & 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\nBoundary\\citep{Brendel18} &0& 45.62&11.79&4.19 & 75.10&41.63&27.34 \\\\\nBoundary-ODS&4 &  {\\bf 6.03} & {\\bf 0.69} & {\\bf 0.43} & {\\bf 24.11} & {\\bf 5.44} & {\\bf 2.97}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_decision_VGG}\n\\end{table}\n\n\\subsection{Effect of the choice of surrogate models}\n\\label{appendix:ap_black_surrogatechange}\nIn Section~\\ref{sec_black_score} and \\ref{sec_black_decision}, we mainly used four pre-trained models as surrogate models. To investigate the effect of the choice of surrogate models, we run attacks with seven different sets of surrogate models.\nAll settings except surrogate models are the same as the previous experiments.\n\nTable~\\ref{tab_black_score_surrogate} and \\ref{tab_black_decision_surrogate} shows results for SimBA-ODS and Boundary-ODS, respectively. First, the first four rows in both tables are results for a single surrogate model. The degree of improvements depends on the model. ResNet34 gives the largest improvement and VGG19 gives the smallest improvement. Next, the fifth and sixth rows show results for sets of two surrogate models. By combining surrogate models, the query efficiency improves, especially for targeted attacks. This means that the diversity from multiple surrogate models is basically useful to make attacks strong. Finally, the performances in the seventh row are results for four surrogate models, which are not always better than results for the combination of two models (ResNet34 and DenseNet121). When the performances for each surrogate model are widely different, the combination of those surrogate models could be harmful.\n\n\\begin{table}[htb]\n\\caption{Query counts and $\\ell_2$ perturbations for SimBA-ODS attacks with various sets of surrogate models. In the column of surrogate models, R:ResNet34, D:DenseNet121, V:VGG19, M:MobileNetV2.}\n\\begin{center}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n&& \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{3-8}\nsurrogate& & success  & average & median $\\ell_2$& success  & average & median $\\ell_2$  \\\\ \\\nmodels&num.&  rate & query &  distance &  rate & query & distance  \\\\\n\\midrule\nR&1& {\\bf 100.0\\%}  & {274} & 1.35 & 95.3\\%& 5115 & 3.50  \\\\\nD&1& {\\bf 100.0\\%}  &342&1.38 &96.7\\% & 5282 & 3.51\\\\\nV&1& {\\bf 100.0\\%}  &660 &1.78 &88.0\\% & 9769 & 4.80 \\\\\nM&1& {\\bf 100.0\\%}  &475&1.70 &95.3\\%& 6539 & 4.53\\\\\nR,D&2& {\\bf 100.0\\%} &{\\bf 223}&{\\bf 1.31} & 98.0\\%& {\\bf 3381} & {\\bf 3.39}\\\\\nV,M&2& {\\bf 100.0\\%} &374&1.60 & 96.3\\%& 4696 & 4.27\\\\\nR,V,D,M &4&  {\\bf 100.0\\%}& {241} &  1.40 & {\\bf 98.3\\%} & {3502} & {3.55}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_score_surrogate}\n\\end{table}\n\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for Boundary-ODS attacks with various sets of surrogate models. In the column of surrogate models, R:ResNet34, D:DenseNet121, V:VGG19, M:MobileNetV2.}\n\\begin{center}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n& & \\multicolumn{6}{c}{number of queries} \\\\ \nsurrogate& & \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ \nmodels & num. & 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\nR&1 & 9.90&1.41&0.79 & 31.32&11.49&7.89\\\\\nD&1 & 10.12&1.39&0.76 &32.63&11.30&7.44\\\\\nV&1 & 22.68&3.47&1.52 & 49.18&24.26&17.75 \\\\\nM&1 & 20.67&2.34&1.10 & 44.90&18.62&12.01\\\\\nR,D&2 & {\\bf 7.53}&1.07&0.61 & {\\bf 26.00} & 8.08& 6.22\\\\\nV,M&2 & 17.60& 1.70& 0.92&39.63&14.97&9.21 \\\\\nR,V,D,M &4 &   7.57 & {\\bf 0.98} & {\\bf 0.57} &  27.24 & {\\bf 6.84} & {\\bf 3.76}\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_decision_surrogate}\n\\end{table}\n\nIn Section~\\ref{sec_black_score_RGF}, we compared ODS-RGF with P-RGF only using the ResNet34 surrogate model. To show the effectiveness of ODS-RGF is robust to the choice of surrogate models, we evaluate ODS-RGF with different surrogate models. Table~\\ref{fig_black_RGF_VGG} shows the query-efficiency of ODS-RGF and P-RGF with the VGG19 surrogate model. We can observe that ODS-RGF outperforms P-RGF for all settings \nand the results are consistent with the experiment in Section~\\ref{sec_black_score_RGF}.\n\n\\begin{table}[htb]\n\\caption{Comparison between ODS-RGF and P-RGF with the VGG19 surrogate model. Settings in the comparison are the same as Figure~\\ref{fig_black_RGF}. }\n\\begin{center}\n\\setlength{\\tabcolsep}{3.7pt}\n\\begin{tabular}{ccc|ccc|ccc}\n\\toprule\n& & &\\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{4-9}\n  &  &num. of & & average & median $\\ell_2$ & success  &  average & median $\\ell_2$   \\\\ \nnorm& attack &surrogates & success  & queries & perturbation & success  &  queries & perturbation  \\\\ \n\\midrule\n\\multirow{3}{*}{$\\ell_2$}&RGF & 0& \\textbf{100.0\\%} & 633 & 3.07 &\\textbf{99.3\\%} &3141 & 8.23\\\\\n&P-RGF~[25] &1& \\textbf{100.0\\%} &467 & 3.03\n&{97.0\\%} &3130 & 8.18\\\\\n&ODS-RGF& 1& \\textbf{100.0\\%} &\\textbf{294}& \\textbf{2.24}\n&{98.0\\%} &\\textbf{2274} & \\textbf{6.60}\\\\ \\hline\n\\multirow{3}{*}{$\\ell_\\infty$} &RGF  &0& {97.0\\%} & 520 & - \n&{25.0\\%} & 2971   & -  \\\\\n&P-RGF~[25]  &1& {98.7\\%} &337& - & {29.0\\%} & 2990 & -\\\\\n&ODS-RGF& 1& \\textbf{99.7\\%} &\\textbf{256}& - & \\textbf{45.7\\%} & \\textbf{2116} & -  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{fig_black_RGF_VGG}\n\\end{table}\n\n\\subsection{Effect of the number of surrogate models for the experiment in Section~\\ref{sec_black_limited}}\n\\label{appendix:ap_black_OOD_surrogatenum}\nWe described that surrogate models with limited out-of-distribution training dataset are still useful for ODS in Section~\\ref{sec_black_limited}. \nIn the experiment, we used five surrogate models with the same ResNet18 architecture.  \nHere, we reveal the importance of the number of surrogate models through experiments with the different number of models. \nTable~\\ref{tab_black_limited_append} shows the result for Boundary-ODS with the different number of surrogate models. With the increase of the number of models, the query efficiency consistently improves.\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for Boundary-ODS attacks with different number of surrogate models against out-of-distribution images on ImageNet.}\n\\begin{center}\n\\begin{tabular}{c|ccc|ccc}\n\\toprule\n& \\multicolumn{6}{c}{number of queries} \\\\ \nnum. of&  \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ \nsurrogates & 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\n1  & 19.45 & 2.90 & 1.66 & 47.86& 25.30 & 20.46\\\\ \n2  & 15.45 & 2.42 & 1.35 & 43.45 & 19.30 & 13.78\\\\ \n3  & 13.75& 1.96&1.14& {\\bf 41.63}& 16.91 & 11.14\\\\ \n4  & 14.23& 1.86 & 1.21 & 41.65& 14.86 & 9.64\\\\ \n5  & {\\bf 11.27}& {\\bf 1.63}& {\\bf 0.98}& 41.67& {\\bf 13.72} & {\\bf 8.39}\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_limited_append}\n\\end{table}\n\n\n\\subsection{Score-based attacks with ODS against out-of-distribution images}\n\\label{appendix:ap_black_OOD_scorebased}\nIn Section~\\ref{sec_black_limited}, we demonstrated that the decision-based Boundary-ODS attack works well even if we only have surrogate models trained with limited out-of-distribution dataset.\nHere, we evaluate score-based SimBA-ODS with these surrogate models. Except surrogate models, we adopt the same setting as Section~\\ref{sec_black_score}.\n\nIn Table~\\ref{tab_black_OOD_SimBA}, SimBA-ODS with out-of-distribution dataset outperforms SimBA-DCT in untargeted settings. \nIn targeted settings, while SimBA-ODS improves the $\\ell_2$ perturbation, the average queries for SimBA-ODS are comparable with SimBA-DCT. \nWe hypothesize that it is because ODS only explores the subspace of the input space. The restriction to the subspace may lead to bad local optima. \nWe can mitigate this local optima problem by applying random sampling temporally when SimBA-ODS fails to update a target image in many steps in a low.\n\nWe note that decision-based Boundary-ODS with OOD dataset is effective, as shown in Section~\\ref{sec_black_limited}. \nWe hypothesize that the difference in effectiveness is because Boundary-ODS does not use scores of the target model and thus does not trap in local optima.\n\n\n\n\\begin{table}[htb]\n\\caption{Query counts and $\\ell_2$ perturbations for SimBA-ODS attacks with surrogate models trained with OOD images on ImageNet.}\n\\begin{center}\n\\setlength{\\tabcolsep}{4pt}\n\\begin{tabular}{c|ccc|ccc}\n\\toprule\n& \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{2-7}\n& success  & average & median $\\ell_2$ & success  & average & median $\\ell_2$  \\\\ \\\nattack&  rate & queries & perturbation &  rate & queries &  perturbation  \\\\ \\midrule\nSimBA-DCT~\\citep{Guo19} & {\\bf 100.0\\%} & 909 & 2.95 & {\\bf 97.0\\%} & 7114 &7.00 \\\\\nSimBA-ODS \n(OOD dataset) & {\\bf 100.0\\%} & {\\bf 491} & {\\bf 1.94} & 94.7\\% & {\\bf 6925} & {\\bf 4.92} \\\\ \\hline\n\\begin{tabular}{c}\nSimBA-ODS (full dataset) \n\\end{tabular} & {100.0\\%} & {242} & {1.40} & {98.3\\%} & {3503} & {3.55} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_OOD_SimBA}\n\\end{table}\n\n\n\\subsection{ODS against robust defense models}\n\\label{appendix:ap_black_defensemodel}\nIn this paper, we mainly discuss transferability when surrogate models and the target model are trained with similar training schemes. On the other hand, it is known that transferability decreases when models are trained with different training schemes, \ne.g. the target model uses adversarial training and surrogate models use natural training. \nIf all surrogate models are trained with natural training scheme, \nODS will also not work against adversarially trained target models. \nHowever, we can mitigate the problem by simultaneously using surrogates obtained with various training schemes (which are mostly publicly available). In order to confirm this, we run an experiment to attack a robust target model using SimBA-ODS with both natural and robust surrogate models (a natural model and a robust model). In Table~\\ref{tab_black_advmodel}, the first row shows the attack performance of SimBA-DCT (without surrogate models) and the others show the performance of SimBA-ODS. \nIn the fourth row of Table~\\ref{tab_black_advmodel}, SimBA-ODS with natural and robust surrogate models significantly outperforms SimBA-DCT without surrogate models. This suggests that if the set of surrogates includes one that is similar to the target, ODS still works (even when some other surrogates are \"wrong\").\nWhile the performance with natural and robust surrogate models slightly underperforms single adversarial surrogate model in the third row, dynamic selection of surrogate models during the attack will improve the performance, as we mentioned in the conclusion of the paper.\n\n\\begin{table*}[htbp]\n\\caption{Transferability of ODS when training schemes of surrogate models are different from the target model. \nR50 shows pretrained ResNet50 model, and R101(adv) and R152(adv) are adversarially trained ResNeXt101 and ResNet152 denoise models from~\\citep{featureDenoise19}, respectively. All attacks are held in the same setting as Section~\\ref{sec_black_score}. \n}\n\\begin{center}\n\\begin{tabular}{cc|ccc}\n\\toprule\ntarget & surrogate & success rate  & average queries & median $\\ell_2$ perturbation \\\\ \n \\midrule\nR101(adv) &- &  89.0\\% & 2824& 6.38\\\\\nR101(adv) &R50 & 80.0\\% & 4337& 10.15\\\\\nR101(adv) &R152(adv) & \\textbf{98.0\\%} &\\textbf{1066} & \\textbf{4.93}\\\\ \nR101(adv) &R50, R152(adv) & \\textbf{98.0\\%} &1304 & {5.62}\\\\\n\\bottomrule         \n\\end{tabular}\n\\end{center}\n\\label{tab_black_advmodel}\n\\end{table*}\n\n\\section{Relationship and Comparison between ODS and MultiTargeted}\n\\label{appendix:ap_multitargeted}\nIn this section, we describe that ODS gives better diversity than the MultiTargeted attack~\\citep{MT19} for initialization and sampling. \n\nMultiTargeted is a variant of white-box PGD attacks, which maximizes $f_t(\\bm{\\mathrm{x}})-f_y(\\bm{\\mathrm{x}})$ where $\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})$ is logits, $y$ is the original label and $t$ is a target label. The target label is changed per restarts. In other words, MultiTargeted moves a target image to a particular direction in the output space, which is represented as like $\\bm{\\mathrm{w}}_{\\text{d}}=(1,0,-1,0)$ where 1 and -1 correspond to the target and original label, respectively. Namely, the procedure of MultiTargeted is technically similar to ODS. \n\n\nHowever, there are some key differences between MultiTargeted and ODS. One of the difference is the motivation. MultiTargeted was proposed as a white-box attack and the study only focused on $\\ell_p$-bounded white-box attacks. On the other hand, our study gives broader application for white- and black-box attacks. As far as we know, ODS is the first method which exploits the output diversity for initialization and sampling.\n\nAnother difference is the necessity of the original label of target images. \nODS does not require the original class of the target image,\nand thus ODS is applicable for black-box attacks \neven if surrogate models are trained with out-of-distribution training dataset, as shown in Section~\\ref{sec_black_limited}.\nOn the other hand, \nsince MultiTargeted exploits the original label of target images to calculate the direction of the attack, \nwe cannot apply MultiTargeted to sampling for black-box attacks against out-of-distribution images. \n\n\nFinally, the level of diversity is also different. \nAs we mentioned in Section~\\ref{sec_related}, the direction of MultiTargeted is restricted to away from the original class.\nThis restriction could be harmful for diversity because the subspace to explore directions is limited. \nTo show this statement, we apply MultiTargeted to initialization for white-box attacks and sampling for black-box attacks, and demonstrate that ODI provides better diversity than MultiTargeted for initialization and sampling (especially for sampling).\n\n\\paragraph{Initialization in white-box settings}\nWe apply MultiTargeted to initalization for white-box attacks in Section~\\ref{sec_white_various}. \nTable~\\ref{tab_appendix_MTwhite} represents the comparison of the attack performance with initialization by MultiTargeted and ODI.\nFor PGD attacks, MultiTargeted is slightly better than ODI. We hypotheses that it is because MultiTargeted was developed as a variant of PGD attacks and the initialization by MultiTargeted also works as an attack method. \nOn the other hand, ODI outperforms MultiTargeted for C\\&W attacks. In this setting, MultiTargeted does not work as an attack method, and thus the difference in the diversity makes the difference in the performance.\n\\begin{table*}[htb]\n\\caption{Comparison of model performance under attacks with MultiTargeted (MT) and ODI. The values are model accuracy (lower is better) for PGD and the average of the minimum $\\ell_2$ perturbations (lower is better) for C\\&W. All results are the average of three trials. Results for ODI are from Table~\\ref{tab_Linf}.\n}\n\\begin{center}\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{tabular}{c|cc|cc}\n\\toprule\n & \\multicolumn{2}{|c}{PGD } &\n\\multicolumn{2}{|c}{C\\&W} \\\\ \nmodel &  MT & ODI & MT & ODI \n\\\\ \\midrule\nMNIST & $\\textbf{89.95}\\pm 0.05\\%$ & ${90.21}\\pm 0.05\\%$ &  ${2.26}\\pm0.01$ &  $\\textbf{2.25}\\pm0.01$  \\\\ \nCIFAR-10 &  $\\textbf{44.33}\\pm 0.01\\%$ &${44.45}\\pm 0.02\\%$ & $0.69\\pm0.01$  &$\\textbf{0.67}\\pm0.00$ \\\\ \nImageNet  &  $\\textbf{42.2}\\pm 0.0\\%$ &  ${42.3}\\pm 0.0\\%$& $2.30\\pm0.01$ & $\\textbf{1.32}\\pm0.01$ \\\\ \n\\bottomrule         \n\\end{tabular}\n\\end{center}\n\\label{tab_appendix_MTwhite}\n\\end{table*}\n\n\n\\paragraph{Sampling in black-box settings} \nWe use MultiTargeted for sampling on the Boundary Attack in Section~\\ref{sec_black_decision} (called Boundary-MT), and compare it with Boundary-ODS. Table~\\ref{tab_append_MTblack} and Figure~\\ref{fig_append_MTblack} show the results of the comparison. While Boundary-MT outperforms the original Boundary Attack, Boundary-ODS finds much smaller adversarial perturbation than Boundary-MT. \n\nIn Figure~\\ref{fig_append_MTblack}, Boundary-MT slightly outperforms Boundary-ODS at small queries. We hypotheses that it is because MultiTargeted not works for providing diversity, but works for the approximation of gradients of the loss function. However, with the number of queries, the curve of Boundary-MT is saturated, and Boundary-MT underperforms Boundary-ODS. This is an evidence that the restriction of directions is harmful for sampling. %\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for Boundary Attack with ODS and MultiTargeted (MT).}\n\\begin{center}\n\\begin{tabular}{c|ccc|ccc}\n\\toprule\n&  \\multicolumn{6}{c}{number of queries} \\\\\n&  \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ \nattack & 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\nBoundary~\\citep{Brendel18} & 45.07& 11.46 & 4.30 & 73.94 &41.88 &27.05  \\\\\nBoundary-ODS &  {\\bf 7.57} & {\\bf 0.98} & {\\bf 0.57} & {\\bf 27.24} & {\\bf 6.84} & {\\bf 3.76}\\\\\nBoundary-MT & 7.65&2.20&2.01&28.16&18.48&16.59  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_append_MTblack}\n\\end{table}\n\\begin{figure}[ht]\n\\centering\n\n    \\begin{tabular}{cc}\n      \\begin{minipage}{0.35\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/boundary_untarget_MT.png}\n        \\end{center}\n      \\end{minipage}\n        &\n      \\begin{minipage}{0.35\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/boundary_target_MT.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\\n      Untargeted & Targeted \n    \\end{tabular}\n    \\caption{Relationship between median $\\ell_2$ perturbations and the number of queries for Boundary Attack with ODS and MultiTargeted. Error bars show 25\\%ile and 75\\%ile of $\\ell_2$ perturbations. }\n    \\label{fig_append_MTblack}\n\\end{figure}\n\n\\section{Introduction}\nDeep neural networks have achieved great success in image classification. However, it is known that they are vulnerable to adversarial examples~\\citep{Szegedy13} --- small perturbations imperceptible to humans that cause classifiers to output wrong predictions. \nSeveral studies have focused on improving model robustness against these malicious perturbations. Examples include adversarial training~\\citep{madry17,Ian15}, %\ninput purification using generative models~\\citep{song2017pixeldefend,samangouei2018defense}, \nregularization of the training loss~\\citep{regGrad18,regStGrad18,regCurv19,regLinear19}, \nand certified defenses~\\citep{Certify17,Certify18_2,Certify19}. \n\n\nStrong attacking methods are crucial for evaluating the robustness of classifiers and defense mechanisms. \nMany existing adversarial attacks rely on random sampling, i.e., adding small random noise to the input.\nIn white-box settings, random sampling is widely used for random restarts~\\citep{PGD17,Dist19,fab19,MT19} to find a diverse set of starting points for the attacks.\nSome black-box attack methods also use random sampling to explore update directions~\\citep{Brendel18,Guo19} or to estimate gradients of the target models~\\citep{ilyas2018blackbox,IEM2018PriorCB,autozoom2019}. \nIn these attacks, random perturbations are typically sampled from a na\\\"{i}ve uniform or Gaussian distribution in the input pixel space.\n\n\\begin{figure*}[htbp]\n\\centering\n    \\begin{tabular}{ccc|ccc}\n\n      \\begin{minipage}{0.2\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.88\\textwidth]{image\/input_target.png}\n        \\end{center}\n      \\end{minipage}\n        &\n      \\begin{minipage}{0.2\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.908\\textwidth]{image\/output_target.png}\n        \\end{center}\n      \\end{minipage} &&&\n      \\begin{minipage}{0.2\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.908\\textwidth]{image\/output_target.png}\n        \\end{center}\n      \\end{minipage} &\n      \\begin{minipage}{0.2\\hsize}\n        \\begin{center}\n          \\includegraphics[width=0.908\\textwidth]{image\/output_surrogate.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\ \n     Input space& Output space&&& Output space &Output space \n      \\\\\n     &&&& (surrogate model) &(target model)\n    \\end{tabular}\n    \\caption{Illustration of the differences between random sampling (blue dashed arrows) and ODS (red solid arrows). In each figure, the black `o' corresponds to an original image, and white `o's represent sampled perturbations. (Left): white-box setting. Perturbations by ODS in the input space are crafted by maximizing the distance in the output space. (Right): black-box setting. Perturbations crafted on the surrogate model transfer well to perturbations on the target model. \n    }\n    \\label{figure1}\n    \\vskip -0.0in\n\\end{figure*}\n\nRandom sampling in the input space, however, may not sufficiently explore the output (logits) space of a neural network --- diversity in the input space does not directly translate to diversity in the output space of a deep nonlinear model. We illustrate this phenomenon in the left panel of Figure~\\ref{figure1}. When we add random perturbations to an image in the input space (see dashed blue arrows in the first plot of Figure~\\ref{figure1}), the corresponding output logits could be very similar to the output for the original image (as illustrated by the second plot of Figure~\\ref{figure1}). \nEmpirically, we observe that this phenomenon can negatively impact the performance of attack methods.\n\nTo overcome this issue, we propose a sampling strategy designed to obtain samples that are diverse in the output space. Our idea is to perturb an input away from the original one as measured directly by distances in the output space (see solid red arrows in the second plot in Figure~\\ref{figure1}). \nFirst, we randomly specify a direction in the output space. Next, we perform gradient-based optimization to generate a perturbation in the input space that \nyields a large change in the specified direction. \nWe call this new sampling technique \\uline{Output Diversified Sampling} (ODS).\n\n\nODS can improve adversarial attacks under both white-box and black-box settings. For white-box attacks, we exploit ODS to initialize the optimization procedure of finding adversarial examples (called ODI). ODI typically provides much more diverse (and effective) starting points for adversarial attacks. \nMoreover, this initialization strategy is agnostic to the underlying attack method, and can be incorporated into most optimization-based white-box attack methods.\nEmpirically, we demonstrate that ODI improves the performance of $\\ell_\\infty$ and $\\ell_2$ attacks compared to na\\\"{i}ve initialization methods.\nIn particular, the PGD attack with ODI outperforms the state-of-the-art MultiTargeted attack~\\citep{MT19} against pre-trained defense models, with 50 times smaller computational complexity on CIFAR-10.\n\n\nIn black-box settings, we cannot directly apply ODS because we do not have access to gradients of the target model. As an alternative, we apply ODS to surrogate models and observe that the resulting samples are diverse with respect to the target model: diversity in the output space transfers (see the rightmost two plots in Figure~\\ref{figure1}). \nEmpirically, \nwe demonstrate that ODS can reduce the number of queries needed for a score-based attack (SimBA~\\citep{Guo19}) by a factor of two on ImageNet. \n{ODS also shows better query-efficiency than P-RGF~\\citep{Cheng19prior}, which is another method exploiting surrogate models to improve a black-box attack.}\nThese attacks with ODS achieve better query-efficiency than the state-of-the-art Square Attack~\\citep{ACFH2019square}.\nIn addition, ODS with a decision-based attack (Boundary Attack~\\citep{Brendel18}) reduces the median perturbation distances of adversarial examples by a factor of three compared to the state-of-the-art HopSkipJump~\\citep{chen2019hop} and Sign-OPT~\\citep{cheng20sign} attacks. \n\n\n\n\n\\section{Preliminaries}\nWe denote an image classifier as $\\bm{\\mathrm{f}}: \\bm{\\mathrm{x}}\\in[0,1]^D \\mapsto \\bm{\\mathrm{z}}\\in \\mathbb{R}^{C}$, where $\\bm{\\mathrm{x}}$ is an input image, $\\bm{\\mathrm{z}}$ represents the logits, and $C$ is the number of classes.\nWe use\n$ h(\\bm{\\mathrm{x}}) = \\argmax_{c=1,\\ldots,C} f_{c}(\\bm{\\mathrm{x}})$ to denote the model prediction, where $f_{c}(\\bm{\\mathrm{x}})$ is the $c$-th element of $\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})$. %\n\nAdversarial attacks can be classified into targeted and untargeted attacks.\nGiven an image $\\bm{\\mathrm{x}}$, a label $y$ and a classifier $\\bm{\\mathrm{f}}$, \nthe purpose of untargeted attacks  \nis to find an adversarial example $\\bm{\\mathrm{x}}^{\\text{adv}}$ that is similar to $\\bm{\\mathrm{x}}$ but causes misclassification $ h(\\bm{\\mathrm{x}}^{\\text{adv}}) \\neq y $. \nIn targeted settings, attackers aim to change the model prediction $h(\\bm{\\mathrm{x}}^{\\text{adv}})$ to a particular target label $t \\neq y$. \nThe typical goal of adversarial attacks is to find an adversarial example $\\bm{\\mathrm{x}}^{\\text{adv}}$ within $B_{\\epsilon}(\\bm{\\mathrm{x}}) = \\{ \\bm{\\mathrm{x}} + \\bm{\\delta} : \\|\\bm{\\delta}\\|_{p} \\leq \\epsilon \\}$, i.e., the $\\epsilon$-radius ball around an original image $\\bm{\\mathrm{x}}$. Another common setting is to find a valid adversarial example with the smallest $\\ell_p$ distance from the original image. \n\n\\paragraph{White-box attacks}\nIn white-box settings, attackers can access full information of the target model. One strong and popular example is the Projected Gradient Descent (PGD) attack~\\citep{madry17}, which iteratively applies the following update rule:\n\\begin{equation}\n\\bm{\\mathrm{x}}^{\\text{adv}}_{k+1}  \n=  \\mathrm{Proj}_{B_{\\epsilon}(\\bm{\\mathrm{x}})} \\left( \\bm{\\mathrm{x}}^{\\text{adv}}_k + \\eta \\, \\mathrm{sign} \\left(\\nabla_{\\bm{\\mathrm{x}}^{\\text{adv}}_k} L(\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}}^{\\text{adv}}_k),y) \n \\right) \\right)\n\\label{eq_pgd}\n\\end{equation}\nwhere $\\mathrm{Proj}_{B_{\\epsilon}(\\bm{\\mathrm{x}})}(\\bm{\\mathrm{x}}^{\\text{adv}}) \\triangleq \\argmin_{\\bm{\\mathrm{x}}' \\in B_{\\epsilon}(\\bm{\\mathrm{x}})} \\|\\bm{\\mathrm{x}}^{\\text{adv}}-\\bm{\\mathrm{x}}'\\|_{p}$, $\\eta$ is the step size, and $L(\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}}),y)$ is a loss function, e.g. the margin loss defined as $\\max_{i \\neq y} f_{i}(\\bm{\\mathrm{x}}) - f_{y}(\\bm{\\mathrm{x}})$. \nTo increase the odds of success, the procedure is restarted multiple times with uniformly sampled initial inputs from $B_{\\epsilon}(\\bm{\\mathrm{x}})$.\n\n\n\\paragraph{Black-box attacks}\nIn black-box settings, the attacker only has access to outputs of the target model without knowing its architecture and weights. Black-box attacks can be largely classified into transfer-based, score-based, and decision-based methods respectively. Transfer-based attacks craft white-box adversarial examples with respect to surrogate models, and transfer them to the target model. The surrogate models are typically trained with the same dataset as the target model so that they are close to each other.\nIn score-based settings, attackers can know the output scores (logits) of the classifier; while for decision-based settings, attackers can only access the output labels of the classifier. \nFor these two approaches, attacks are typically evaluated in terms of query efficiency, i.e. the number of queries needed to generate an adversarial example and its perturbation size.\n\nRecently, several studies~\\citep{Cheng19prior,subspaceattack,Cai19transferSMBdirect} employed surrogate models to estimate the gradients of the loss function of the target model.\nSome attack methods used random sampling in the input space, \nsuch as the decision-based Boundary Attack~\\citep{Brendel18} and the score-based Simple Black-Box Attack~\\citep{Guo19}.\n\n\\section{Output Diversified Sampling}\n\nAs intuitively presented in Figure~\\ref{figure1}, random sampling in the input space does not necessarily produce samples with high diversity as measured in the output space. To address this problem, we propose Output Diversified Sampling (ODS). Given an image $\\bm{\\mathrm{x}}$, a classifier $\\bm{\\mathrm{f}}$ and the direction of diversification $\\bm{\\mathrm{w}}_{\\text{d}}  \\in \\mathbb{R}^{C}$, \nwe define the normalized perturbation vector of ODS as follows:\n\\begin{equation}\n\\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}},\\bm{\\mathrm{f}},\\bm{\\mathrm{w}}_{\\text{d}}) = \\frac{\\nabla_{\\bm{\\mathrm{x}}} (\\bm{\\mathrm{w}}_{\\text{d}}^\\intercal \\bm{\\mathrm{f}}(\\bm{\\mathrm{x}}))}{\\| \\nabla_{\\bm{\\mathrm{x}}} (\\bm{\\mathrm{w}}_{\\text{d}}^\\intercal \\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})) \\|_2},\n\\end{equation}\nwhere $\\bm{\\mathrm{w}}_{\\text{d}}$ is sampled from the uniform distribution over $[-1,1]^C$. \nBelow we show how to enhance white- and black-box attacks with ODS.\n\n\n\n\\subsection{Initialization with ODS for white-box attacks}\n\\label{sec_ODS_white}\nIn white-box settings, we utilize ODS for initialization (ODI) to generate output-diversified starting points. Given an original input $\\bm{\\mathrm{x}}_{\\text{org}}$ and the direction for ODS $\\bm{\\mathrm{w}}_{\\text{d}}$, we try to find a restart point $\\bm{\\mathrm{x}}$ that is as far away from $\\bm{\\mathrm{x}}_{\\text{org}}$ as possible by maximizing $\\bm{\\mathrm{w}}_{\\text{d}}^\\intercal (\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})-\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}}_{\\text{org}}))$ via the following iterative update:\n\\begin{equation}\n\\label{eq_linf}\n\\bm{\\mathrm{x}}_{k+1}  = \\mathrm{Proj}_{B(\\bm{\\mathrm{x}}_{\\text{org}})} \\left( \\bm{\\mathrm{x}}_{k} + \\eta_{\\text{ODI}} \\,  \\mathrm{sign}(\\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}}_k,\\bm{\\mathrm{f}},\\bm{\\mathrm{w}}_{\\text{d}})) \\right)\n\\end{equation}\nwhere $B(\\bm{\\mathrm{x}}_{\\text{org}})$ is the set of allowed perturbations, which is typically an $\\epsilon$-ball in $\\ell_p$ norm, and $\\eta_{\\text{ODI}}$ is a step size. When applying ODI to $\\ell_2$ attacks, we omit the sign function. \nAfter some steps of ODI, we start an attack from the restart point obtained by ODI. \nWe sample a new direction $\\bm{\\mathrm{w}}_{\\text{d}}$ for each restart in order to obtain diversified starting points for the attacks.\nWe provide the pseudo-code for ODI in Algorithm~\\ref{alg:ap_white} of the Appendix. \n\n\n\nOne sampling step of ODI costs roughly the same time as one iteration of most gradient-based attacks (e.g., PGD). \nEmpirically, we observe that the number of ODI steps $N_{\\text{ODI}}=2$ is already sufficient to obtain diversified starting points (details of the sensitivity analysis are in Appendix~\\ref{appendix:ap_white_sensitivity}), and fix $N_{\\text{ODI}}=2$ in all our experiments unless otherwise specified. We emphasize that ODS is not limited to PGD, and can be applied to a wide family of optimization-based adversarial attacks.\n\n\\textbf{Experimental verification of increased diversity:} \nWe quantitatively evaluate the diversity of starting points in terms of pairwise distances of output values $\\bm{\\mathrm{f}}(\\bm{\\mathrm{x}})$, confirming the intuition presented in the left two plots of Figure~\\ref{figure1}. We take a robust model on CIFAR-10 as an example of target models, and generate starting points with both ODI and standard uniform initialization to calculate the mean pairwise distance. The pairwise distance (i.e. diversity) obtained by ODI is 6.41, which is about 15 times larger than that from uniform initialization (0.38). In addition, PGD with the same steps as ODI does not generate diverse samples (pairwise distance is 0.43). \nDetails are in Appendix~\\ref{appendix:ap_white_diversity}.\n\n\\subsection{Sampling update directions with ODS for black-box attacks}\n\\label{sec_ODS_black}\nIn black-box settings, we employ ODS to sample update directions instead of random sampling. \nGiven a target classifier $\\bm{\\mathrm{f}}$, we cannot directly calculate the ODS perturbation  $\\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}},\\bm{\\mathrm{f}},\\bm{\\mathrm{w}}_{\\text{d}})$ because gradients of the target model $\\bm{\\mathrm{f}}$ are unknown. Instead, we introduce a surrogate model $\\bm{\\mathrm{g}}$ and calculate the ODS vector $\\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}},\\bm{\\mathrm{g}},\\bm{\\mathrm{w}}_{\\text{d}})$. \n\nODS can be applied to attack methods that rely on random sampling in the input space. Since many black-box attacks use random sampling to explore update directions~\\citep{Brendel18,Guo19} or estimate gradients of the target models~\\citep{ilyas2018blackbox,autozoom2019,chen2019hop}, ODS has broad applications. \nIn this paper, we apply ODS to two popular black-box attacks that use random sampling: decision-based Boundary Attack~\\citep{Brendel18} and score-based Simple Black-Box Attack (SimBA~\\citep{Guo19}). In addition, we compare ODS with P-RGF~\\citep{Cheng19prior}, which is an another attack method using surrogate models.\n\nTo illustrate how we apply ODS to existing black-box attack methods, we provide the pseudo-code of SimBA~\\citep{Guo19} with ODS in Algorithm~\\ref{alg:black}.\nThe original SimBA algorithm picks an update direction $\\bm{\\mathrm{q}}$ randomly from a group of candidates $Q$ that are orthonormal to each other. We replace it with ODS, as shown in the line 5 and 6 of Algorithm~\\ref{alg:black}. For other attacks, we replace random sampling with ODS in a similar way.\nNote that in Algorithm~\\ref{alg:black}, we make use of multiple surrogate models and uniformly sample one each time, since we empirically found that using multiple surrogate models can make attacks stronger.\n\n\\textbf{Experimental verification of increased diversity:} \nWe quantitatively evaluate that ODS can lead to high diversity in the output space of the target model, as shown in the right two plots of Figure~\\ref{figure1}. We use pre-trained Resnet50~\\citep{resnet16} and VGG19~\\citep{VGG19} models on ImageNet as the target and surrogate models respectively. We calculate and compare the mean pairwise distances of samples with ODS and random Gaussian sampling. The pairwise distance (i.e. diversity) for ODS is 0.79, which is 10 times larger than Gaussian sampling (0.07). Details are in Appendix~\\ref{appendix:ap_black_diversity}. We additionally observe that ODS does not produce diversified samples when we use random networks as surrogate models. This indicates that good surrogate models are crucial for transferring diversity. \n\n\n\\begin{algorithm}[tb]\n   \\caption{Simple Black-box Attack~\\citep{Guo19} with ODS }\n   \\label{alg:black}\n\\begin{algorithmic}[1]\n   \\STATE {\\bfseries Input:} A targeted image $\\bm{\\mathrm{x}}$, loss function $L$, a target classifier $\\bm{\\mathrm{f}}$, a set of surrogate models $\\mathcal{G}$\n   \\STATE {\\bfseries Output:} attack result $\\bm{\\mathrm{x}}_{\\text{adv}}$\n   \\STATE Set the starting point $\\bm{\\mathrm{x}}_{\\text{adv}} = \\bm{\\mathrm{x}}$ \n   \\WHILE {$\\bm{\\mathrm{x}}_{\\text{adv}}$ is not adversary}\n   \\STATE  Choose a surrogate model $\\bm{\\mathrm{g}}$ from $\\mathcal{G}$, and sample $\\bm{\\mathrm{w}}_{\\text{d}} \\sim U(-1,1)^C$\n   \\STATE  Set $\\bm{\\mathrm{q}} = \\bm{\\mathrm{v}}_{\\text{ODS}}(\\bm{\\mathrm{x}}_{\\text{adv}},\\bm{\\mathrm{g}},\\bm{\\mathrm{w}}_{\\text{d}})$\n   \\FOR{$\\alpha \\in \\{\\epsilon, -\\epsilon\\}$}\n   \\IF{$L(\\bm{\\mathrm{x}}_{\\text{adv}}+\\alpha \\cdot \\bm{\\mathrm{q}}) > L(\\bm{\\mathrm{x}}_{\\text{adv}})$}\n   \\STATE   Set $\\bm{\\mathrm{x}}_{\\text{adv}}=\\bm{\\mathrm{x}}_{\\text{adv}}+\\alpha \\cdot \\bm{\\mathrm{q}}$ and {\\bf break}\n   \\ENDIF\n   \\ENDFOR\n   \\ENDWHILE\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\\section{Experiments in white-box settings}\n\\label{sec_white}\nIn this section, we show that the diversity offered by ODI can improve white-box attacks for both $\\ell_\\infty$ and $\\ell_2$ distances. \nMoreover, we demonstrate that a simple combination of PGD and ODI achieves new state-of-the-art attack success rates. All experiments are for untargeted attacks.\n\n\n\n\\subsection{Efficacy of ODI for white-box attacks}\n\\label{sec_white_various}\nWe combine ODI with two popular attacks: PGD attack~\\citep{madry17} with the $\\ell_\\infty$ norm and C\\&W attack~\\citep{cw17} with the $\\ell_2$ norm. \nWe run these attacks on MNIST, CIFAR-10 and ImageNet. \n\n\\paragraph{Setup}  \nWe perform attacks against three adversarially trained models from MadryLab\\footnote{\\url{https:\/\/github.com\/MadryLab\/mnist_challenge} and \\url{https:\/\/github.com\/MadryLab\/cifar10_challenge}. We use their secret model.}~\\citep{madry17} for MNIST and CIFAR-10 and the Feature Denoising ResNet152 network\\footnote{\\url{https:\/\/github.com\/facebookresearch\/ImageNet-Adversarial-Training}.}~\\citep{featureDenoise19} for ImageNet. For PGD attacks, we evaluate the model accuracy with 20 restarts, where starting points are uniformly sampled over an $\\epsilon$-ball for the na\\\"{i}ve resampling. For C\\&W attacks, we calculate the minimum $\\ell_2$ perturbation that yields a valid adversarial example among 10 restarts for each image, and measure the average of the minimum perturbations. Note that the original paper of C\\&W~\\citep{cw17} attacks did not apply random restarts. Here for the na\\\"{i}ve initialization of C\\&W attacks we sample starting points from a Gaussian distribution and clip it into an $\\epsilon$-ball (details in Appendix~\\ref{appendix:ap_parameter_whiteall}). \n\nFor fair comparison, we test different attack methods with the same amount of computation. Specifically, we compare $k$-step PGD with na\\\"{i}ve initialization (denoted as PGD-$k$) against ($k$-2)-step PGD with 2-step ODI (denoted as ODI-PGD-($k$-2)). We do not adjust the number of steps for C\\&W attacks because the computation time of 2-step ODI are negligible for C\\&W attacks. %\n\n\n\\begin{table*}[htb]\n\\caption{Comparing different white-box attacks. We report model accuracy (lower is better) for PGD and average of the minimum $\\ell_2$ perturbations (lower is better) for C\\&W. All results are the average of three trials.\n}\n\\begin{center}\n\\begin{tabular}{c|cc|cc}\n\\toprule\n & \\multicolumn{2}{|c}{PGD } &\n\\multicolumn{2}{|c}{C\\&W} \\\\ \nmodel &  na\\\"{i}ve (PGD-$k$) & ODI (ODI-PGD-($k$-2)) & na\\\"{i}ve & ODI \n\\\\ \\midrule\nMNIST & $90.31\\pm 0.02\\%$ & $\\textbf{90.21}\\pm 0.05\\%$ &  ${2.27}\\pm0.00$ &  $\\textbf{2.25}\\pm0.01$  \\\\ \nCIFAR-10 &  $46.06\\pm 0.02\\%$  &$\\textbf{44.45}\\pm 0.02\\%$ & $0.71\\pm0.00$ & $\\textbf{0.67}\\pm0.00$ \\\\ \nImageNet  &  $43.5\\pm 0.0\\%$  & $\\textbf{42.3}\\pm 0.0\\%$& $1.58\\pm0.00$ & $\\textbf{1.32}\\pm0.01$ \\\\ \n\\bottomrule         \n\\end{tabular}\n\\end{center}\n\\label{tab_Linf}\n\\end{table*}\n\n\n\n\n\n\\paragraph{Results} \nWe summarize all quantitative results in Table~\\ref{tab_Linf}. %\nAttack performances with ODI are better than na\\\"{i}ve initialization for all models and attacks. \nThe improvement by ODI on the CIFAR-10 and ImageNet models is more significant than on the MNIST model. We hypothesize that this is due to the difference in model non-linearity.  When a target model includes more non-linear transformations,\nthe difference in diversity between the input and output space could be larger, in which case ODI will be more effective in providing a diverse set of restarts. \n\n\n\n\\subsection{Comparison between PGD attack with ODI and state-of-the-art attacks}\n\\label{sec_sota}\nTo further demonstrate the power of ODI, we perform ODI-PGD against MadryLab's robust models~\\citep{madry17} on MNIST and CIFAR-10  \nand compare ODI-PGD with state-of-the-art attacks.\n\n\\paragraph{Setup}\nOne state-of-the-art attack we compare with is the well-tuned PGD attack~\\citep{MT19}, which achieved 88.21\\% accuracy for the robust MNIST model. The other attack we focus on is the MultiTargeted attack~\\citep{MT19}, which obtained 44.03\\% accuracy against the robust CIFAR-10 model. \nWe use all test images on each dataset and perform ODI-PGD under two different settings. One is the same as Section~\\ref{sec_white_various}. \nThe other is ODI-PGD with tuned hyperparameters, e.g. increasing the number of steps and restarts. Please see Appendix~\\ref{appendix:ap_parameter_ODI} for more details of tuning.\n\n\n\\begin{table*}[htbp]\n\\caption{Comparison of ODI-PGD with state-of-the-art attacks against pre-trained defense models. The complexity rows display products of the number of steps and restarts. Results for ODI-PGD are the average of three trials.\nFor ODI-PGD, the number of steps is the sum of ODS and PGD steps. \n}\n\\begin{center}\n\\setlength{\\tabcolsep}{4.5pt}\n\\begin{tabular}{cc|cccc}\n\\toprule\nmodel&  & \\begin{tabular}{c}ODI-PGD \\\\ \n(in Sec.~\\ref{sec_white_various}) \\end{tabular} & \\begin{tabular}{c}tuned  \\\\ODI-PGD  \\end{tabular}\n  & \\begin{tabular}{c}tuned PGD \\\\~\\citep{MT19} \\end{tabular}\n  & \\begin{tabular}{c}MultiTargeted \\\\~\\citep{MT19} \\end{tabular} \\\\ \\midrule\n \\multirow{2}{*}{MNIST} &accuracy & $90.21\\pm 0.05\\%$  & $\\textbf{88.13}\\pm 0.01\\%$  & 88.21\\% & 88.36\\%  \\\\\n &complexity & \n$40 \\times 20$ &  $1000\\times 1000 $ & $1000 \\times 1800$  &\n$1000 \\times 1800$    \\\\ \\hline \n\\multirow{2}{*}{CIFAR-10}&accuracy&  $44.45\\pm 0.02\\%$ & $\\textbf{44.00} \\pm 0.01\\%$  & 44.51\\%  & 44.03\\% \\\\\n  &complexity & $20 \\times 20$ & $150 \\times 20$ & $1000 \\times 180$ & $1000 \\times 180$    \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_sota}\n\\end{table*}\n\n\n\n\\paragraph{Results}\nWe summarize the comparison between ODI-PGD and state-of-the-art attacks in Table~\\ref{tab_sota}. \nOur tuned ODI-PGD reduces the accuracy \nto 88.13\\% for the MNIST model, and to 44.00\\% for the CIFAR-10 model. These results outperform existing state-of-the-art attacks. \n\n\nTo compare their running time, we report the total number of steps (the number of steps multiplied by the number of restarts) as a metric of complexity, because the total number of steps is equal to the number of gradient computations (the computation time per gradient evaluation is comparable for all gradient-based attacks).\nIn Table~\\ref{tab_sota}, the computational cost of tuned ODI-PGD is smaller than that of state-of-the-art attacks, and especially 50 times smaller on  CIFAR-10. \nSurprisingly, even without tuning ODI-PGD (in the first column) can still outperform  tuned PGD~\\citep{MT19}\nwhile also being \ndrastically more efficient computationally.\n\n\n\\section{Experiments in black-box settings}\n\\label{sec_black_experiment}\nIn this section, we demonstrate that \nblack-box attacks combined with ODS significantly reduce the number of queries needed to generate adversarial examples. \nIn experiments below, we randomly sample 300 correctly classified images from the ImageNet validation set. We evaluate both untargeted and targeted attacks. For targeted attacks, we uniformly sample target labels.\n\n\\subsection{Query-efficiency of score-based attacks with ODS}\n\\label{sec_black_score}\n\\subsubsection{Applying ODS to score-based attacks}\n\\label{sec_black_score_simba}\nTo show the efficiency of ODS, we combine ODS with the score-based Simple Black-Box Attack (SimBA)~\\citep{Guo19}. \nSimBA randomly samples a vector and \neither adds or subtracts the vector to the target image to explore update directions. \nThe vector is sampled from a pre-defined set of  orthonormal vectors in the input space. \nThese are the discrete cosine transform (DCT) basis vectors in the original paper~\\citep{Guo19}. We replace the DCT basis vectors with ODS sampling  (called SimBA-ODS), %\n\n\n\\paragraph{Setup}\nWe use pre-trained ResNet50 model as the target model \nand select four pre-trained models \n(VGG19, ResNet34, DenseNet121~\\citep{densenet}, MobileNetV2~\\citep{mobilenetv2}) as surrogate models.\nWe set the same hyperparameters for SimBA as~\\citep{Guo19}: the step size is $0.2$ and the number of iterations (max queries) is 10000 (20000) for untargeted attacks and 30000 (60000) for targeted attacks. As the loss function in SimBA, we employ the margin loss for untargeted attacks and the cross-entropy loss for targeted attacks.\n\n\\paragraph{Results}\nFirst, we compare  SimBA-DCT~\\citep{Guo19} and SimBA-ODS. Table~\\ref{tab_black_score} reports the number of queries and the median $\\ell_2$ perturbations. Remarkably, SimBA-ODS reduces the average number of queries by a factor between 2 and 3 compared to SimBA-DCT for both untargeted and targeted settings. This confirms that ODS not only helps white-box attacks, but also leads to significant improvements of query-efficiency in black-box settings. In addition, SimBA-ODS decreases the average perturbation sizes by around a factor of two, which means that ODS helps find better adversarial examples that are closer to the original image. \n\n\\begin{table}[htb]\n\\caption{Number of queries and size of $\\ell_2$ perturbations for score-based attacks.}\n\\begin{center}\n\\setlength{\\tabcolsep}{3.9pt}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n&& \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{3-8}\n& num. of & success  & average & median $\\ell_2$ & success  & average & median $\\ell_2$  \\\\ \\\nattack& surrogates &  rate & queries &  perturbation &  rate & queries &  perturbation  \\\\\n\\midrule\nSimBA-DCT~\\citep{Guo19} &0& {\\bf 100.0\\%} & 909 & 2.95 & 97.0\\% & 7114 &7.00 \\\\\nSimBA-ODS &4& {\\bf 100.0\\%} & {\\bf 242} & {\\bf 1.40} & {\\bf 98.3\\%} & {\\bf 3503} & {\\bf 3.55} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_score}\n\\end{table}\n\n\n\\subsubsection{Comparing ODS with other methods using surrogate models}\n\\label{sec_black_score_RGF}\nWe consider another black-box attack that relies on surrogate models: P-RGF~\\citep{Cheng19prior}, which improves over the original RGF (random gradient-free) method for gradient estimation. \nP-RGF exploits prior knowledge from surrogate models to estimate the gradient more efficiently than RGF. Since RGF uses random sampling to estimate the gradient, we propose to apply ODS to RGF (new attack named ODS-RGF) and compare it with P-RGF under $\\ell_2$ and $\\ell_\\infty$ norms.\n\n\n\\begin{table}[htbp]\n\\caption{ Comparison of ODS-RGF and P-RGF on ImageNet. Hyperparameters for RGF are same as~\\citep{Cheng19prior} :max queries are 10000, sample size is 10, step size is 0.5 ($\\ell_2$) and 0.005 ($\\ell_\\infty$), and epsilon is $\\sqrt{0.001 \\cdot 224^2 \\cdot 3}$ ($\\ell_2$) and 0.05 ($\\ell_\\infty$). }\n\\begin{center}\n\\setlength{\\tabcolsep}{3.5pt}\n\\begin{tabular}{ccc|ccc|ccc}\n\\toprule\n& & &\\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted}  \\\\ \\cline{4-9}\n  &  &num. of & success& average & median $\\ell_2$ & success  &  average & median $\\ell_2$   \\\\ \nnorm& attack &surrogates & rate  & queries & perturbation & rate  &  queries & perturbation  \\\\ \n\\midrule\n\\multirow{3}{*}{$\\ell_2$}&RGF & 0& \\textbf{100.0\\%} & 633 & 3.07 &\\textbf{99.3\\%} &3141 & 8.23\\\\\n&P-RGF~[25] &1& \\textbf{100.0\\%} &211 & 2.08 \n&{97.0\\%} &2296 & 7.03\\\\\n&ODS-RGF& 1& \\textbf{100.0\\%} &\\textbf{133}& \\textbf{1.50}\n&\\textbf{99.3\\%} &\\textbf{1043} & \\textbf{4.47}\\\\ \\hline\n\\multirow{3}{*}{$\\ell_\\infty$} &RGF  &0& {97.0\\%} & 520 & - \n&{25.0\\%} & 2971   & -  \\\\\n&P-RGF~[25]  &1& {99.7\\%} &88& - & {65.3\\%} & 2123 & -\\\\\n&ODS-RGF& 1& \\textbf{100.0\\%} &\\textbf{74}& - & \\textbf{92.0\\%} & \\textbf{985} & -  \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{fig_black_RGF}\n\\end{table}\n\nFor fair comparison, we use a single surrogate model as in~\\citep{Cheng19prior}.\nWe choose pre-trained ResNet50 model as the target model and ResNet34 model as the surrogate model. \nWe give query-efficiency results of both methods in Table~\\ref{fig_black_RGF}. The average number of queries required by ODS-RGF is less than that of P-RGF in all settings. This suggests ODS-RGF can estimate the gradient more precisely than P-RGF by exploiting diversity obtained via ODS and  surrogate models. \nThe differences between ODS-RGF and P-RGF are significant in targeted settings, since ODS-RGF achieves smaller perturbations than P-RGF (see median perturbation column). \nTo verify the robustness of our results, we also ran experiments using VGG19 as a surrogate model and obtained similar results. %\n\n\nWe additionally consider TREMBA~\\citep{Huang20TREMBA}, a black-box attack (restricted to the $\\ell_\\infty$-norm) that is state-of-the-art among those using surrogate models. In TREMBA, a\nlow-dimensional embedding is learned via surrogate models so as to obtain initial adversarial examples\nwhich\nare then updated using a score-based attack. %\nOur results show that ODS-RGF combined with SI-NI-DIM~\\citep{NesterovTransfer2020}, which is a state-of-the-art transfer-based attack, is comparable to TREMBA even though ODS-RGF is not restricted to the $\\ell_\\infty$-norm. \nResults and more details are provided in Appendix~\\ref{appendix:ap_black_TREMBA}.\n\n\n\\subsubsection{Comparison of ODS with state-of-the-art score-based attacks}\nTo show the advantage of ODS and surrogate models, \n we compare SimBA-ODS and ODS-RGF with the Square Attack~\\citep{ACFH2019square}, which is a state-of-the-art attack for both $\\ell_{\\infty}$ and $\\ell_2$ norms when surrogate models are not allowed.\nFor comparison, we regard SimBA as $\\ell_2$ bounded attacks: the attack is successful when adversarial $\\ell_2$ perturbation is less than a given bound $\\epsilon$.\nWe set $\\epsilon=5 \\ (\\ell_2)$ and $0.05 \\ (\\ell_\\infty)$ as well as other hyperparameters according to the original paper~\\citep{ACFH2019square}, except that we set the max number of queries to $20000$ for untargeted attacks and $60000$ for targeted attacks. For ODS-RGF, we use four surrogate models as discussed in Section~\\ref{sec_black_score_simba} for SimBA-ODS. %\n\n\n\n\\begin{table}[htbp]\n    \\caption{Number of queries for attacks with ODS versus the Square Attack. }\n    \\begin{center}\n    \\label{tab_black_square}\n    \\begin{tabular}{ccc|cc|cc}\n    \\toprule\n    &&& \\multicolumn{2}{c}{untargeted} &\\multicolumn{2}{|c}{targeted} \\\\ \\cline{4-7}\n    && num. of & success   & average &  success   & average \\\\ \n    norm &attack& surrogates&  rate  & queries &  rate  & queries  \\\\ \\midrule\n    \\multirow{3}{*}{$\\ell_2$}&Square~\\citep{ACFH2019square} &0&  {99.7\\%} & 647\n    & {96.7\\%}  & {11647} \\\\\n    &SimBA-ODS &4& {99.7\\%} &  {237}  & {90.3\\%}  & {2843} \\\\\n    &ODS-RGF & 4& {\\bf 100.0\\%} & {\\bf 144}  & {\\bf 99.0\\%} & {\\bf 1285}\\\\\n     \\hline\n    \\multirow{2}{*}{$\\ell_\\infty$} &Square~\\citep{ACFH2019square} &0&  {\\bf 100.0 \\%} & {\\bf  60} & {\\bf 100.0\\%}  & {2317} \\\\\n    &ODS-RGF &4&  {\\bf 100.0 \\%} & 78 & {97.7\\%}& {\\bf 1242} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\end{center}\n\\end{table}\n\nAs shown in Table~\\ref{tab_black_square}, \nthe number of queries required for ODS-RGF and SimBA-ODS are lower than that of the Square Attack under the $\\ell_2$ norm. The improvement is especially large for ODS-RGF. The difference between ODS-RGF and SimBA-ODS mainly comes from different base attacks (i.e., RGF and SimBA).\nFor the $\\ell_\\infty$ norm setting, ODS-RGF is comparable to the Square Attack. We hypothesize that the benefit of estimated gradients by RGF decreases under the $\\ell_\\infty$ norm due to the sign function. However, because ODS can be freely combined with many base attacks, a stronger base attack is likely to  further improve query-efficiency.\n\n\\subsection{Query-efficiency of decision-based attacks with ODS}\n\\label{sec_black_decision}\nWe demonstrate that ODS also improves query-efficiency for decision-based attacks. We combine ODS with the decision-based Boundary Attack~\\citep{Brendel18}. \nThe Boundary Attack starts from an image which is adversarial, and iteratively updates the image to find smaller perturbations. \nTo generate the update direction, the authors of~\\citep{Brendel18} sampled a random noise vector from a Gaussian distribution $\\mathcal{N}(\\mathbf{0}, \\mathbf{I})$ each step.\nWe replace this random sampling procedure with sampling by ODS (we call the new method Boundary-ODS). We give the pseudo-code of Boundary-ODS in Algorithm~\\ref{alg:ap_boundary} (in the Appendix).\n\n\\paragraph{Setup}\nWe use the same settings as the previous section for score-based attacks: 300 validation images on ImageNet, pre-trained ResNet50 target model, and four pre-trained surrogate models.\nWe test on both untargeted and targeted attacks. In targeted settings, we give randomly sampled images with target labels as initial images.\nWe use the implementation in Foolbox~\\citep{foolbox2017} for Boundary Attack with default parameters, which is more efficient than the original implementation.\n\nWe also compare Boundary-ODS with two state-of-the-art decision-based attacks: \nthe HopSkipJump attack~\\citep{chen2019hop} and the Sign-OPT attack~\\citep{cheng20sign}. We use the implementation in ART~\\citep{art2018} for HopSkipJump and the author's implementation for Sign-OPT. We set default hyperparameters for both attacks.\n\n\\paragraph{Results}\nTable~\\ref{tab_black_decision} summarizes the median sizes of $\\ell_2$ adversarial perturbations obtained with a fixed number of queries. Clearly, Boundary-ODS significantly improves query-efficiency compared to the original Boundary Attack. In fact, Boundary-ODS outperforms state-of-the-art attacks: it decreases the median $\\ell_2$ perturbation at 10000 queries to less than one-third of previous best untargeted attacks and less than one-fourth of previous best targeted attacks. \nWe additionally describe the relationship between median $\\ell_2$ perturbations and the number of queries in Figure~\\ref{fig_black_decision}. Note that Boundary-ODS outperforms other attacks, especially in targeted settings. Moreover, Boundary-ODS only needs fewer than 3500 queries to achieve the adversarial perturbation obtained by other attacks with 10000 queries.\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for Boundary-ODS and decision-based state-of-the-art attacks.}\n\\begin{center}\n\\begin{tabular}{cc|ccc|ccc}\n\\toprule\n& & \\multicolumn{6}{c}{number of queries} \\\\\n& num. of & \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ \nattack &surrogates& 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\nBoundary~\\citep{Brendel18} & 0& 45.07& 11.46 & 4.30 & 73.94 &41.88 &27.05  \\\\\nBoundary-ODS & 4&  {\\bf 7.57} & {\\bf 0.98} & {\\bf 0.57} & {\\bf 27.24} & {\\bf 6.84} & {\\bf 3.76}\\\\\nHopSkipJump~\\citep{chen2019hop} & 0& 14.86 &3.50 & 1.79 & 65.88 & 33.98 & 18.25 \\\\\nSign-OPT~\\citep{cheng20sign} & 0& 21.73 & 3.98 & 2.01 & 68.75 & 36.93&22.43\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_decision}\n\\end{table}\n\n\\begin{figure}[htbp]\n\\centering\n\n    \\begin{tabular}{cc}\n      \\begin{minipage}{0.35\\hsize} %\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/decision_untargeted.png}\n        \\end{center}\n      \\end{minipage}\n        &\n      \\begin{minipage}{0.35\\hsize}\n        \\begin{center}\n          \\includegraphics[width=1.0\\textwidth]{image\/decision_targeted.png}\n        \\end{center}\n      \\end{minipage}\n      \\\\\n      Untargeted & Targeted \n    \\end{tabular}\n    \\caption{Relationship between median $\\ell_2$ perturbations and the number of queries for decision-based attacks. Error bars show 25th and 75th percentile of $\\ell_2$ perturbations. } %\n    \\label{fig_black_decision}\n\\end{figure}\n\n\n\n\n\\subsection{Effectiveness of ODS with out-of-distribution images}\n\\label{sec_black_limited}\nAlthough several studies use prior knowledge from surrogate models to improve performance of black-box attacks, there is a drawback---those approaches require a dataset to train surrogate models. \nIn reality, it is typically impossible to obtain the same dataset used for training the target model. We show that ODS is applicable even when we only have a limited dataset that is out-of-distribution (OOD) and may contain only images with irrelevant labels.\n\nWe select 100 ImageNet classes which do not overlap with the classes used in the experiments of Section~\\ref{sec_black_decision}.  \nWe train surrogate models using an OOD training dataset with these 100 classes. \nWe train five surrogate models with the same ResNet18 architecture because multiple surrogate models provide diversified directions. \nThen, we run Boundary-ODS with the trained surrogate models under the same setting as Section~\\ref{sec_black_decision}. As shown in Table~\\ref{tab_black_limited1}, although Boundary-ODS with the OOD training dataset underperforms Boundary-ODS with the full dataset, it is still significantly better than the original Boundary Attack with random sampling. This demonstrates that the improved diversity achieved by ODS improves black-box attacks even if we only have OOD images to train a surrogate. \n\n\n\\begin{table}[htb]\n\\caption{Median $\\ell_2$ perturbations for Boundary-ODS with surrogate models trained on OOD images.}\n\\begin{center}\n\\begin{tabular}{c|ccc|ccc}\n\\toprule\n&  \\multicolumn{6}{c}{number of queries} \\\\\n&  \\multicolumn{3}{c}{untargeted} &\\multicolumn{3}{|c}{targeted} \\\\ \nattack & 1000 & 5000 & 10000 & 1000 & 5000 & 10000 \\\\ \\midrule\nBoundary~\\citep{Brendel18} & 45.07& 11.46 & 4.30 & 73.94 &41.88 &27.05  \\\\\nBoundary-ODS (OOD dataset) &  {\\bf 11.27}& {\\bf 1.63}& {\\bf 0.98}&{\\bf 41.67}&{\\bf 13.72} & {\\bf 8.39}\\\\ \\hline\nBoundary-ODS (full dataset in Sec.~\\ref{sec_black_decision}) &  7.57& 0.98& 0.57& 27.24& 6.84& 3.76\\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\label{tab_black_limited1}\n\\end{table}\n\n\n\n\\section{Related works}\n\\label{sec_related}\n\nThe closest approach to ours is the white-box MultiTargeted attack~\\citep{MT19}. This attack changes the target class of attacks per restart, and it can be regarded as a method which aims to obtain more diversified starting points. However, MultiTargeted attack is limited to the setting of $\\ell_p$-bounded white-box attacks. In contrast, ODS can be applied to more general white- and black-box attacks. In addition, ODS does not require the original class of the target image, therefore it is more broadly applicable. Further discussion are in Appendix~\\ref{appendix:ap_multitargeted}.\nAnother related work is Interval Attack~\\citep{wang19symbolic} which generates diverse starting points by leveraging symbolic interval propagation. While Interval Attack shows good performances against MNIST models, the attack is not scalable to large models. \n\nODS utilizes surrogate models, which are commonly used for black-box attacks. Most previous methods exploit surrogate models to estimate gradients of the loss function on the target model~\\citep{trans_papernot17,trans_liu17,Brunner19GuessingSmart,Cheng19prior,subspaceattack,Cai19transferSMBdirect}.\nSome recent works exploit surrogate models to train other models~\\citep{Du2020Query-efficient,Huang20TREMBA}  \nor update surrogate models during attacks~\\citep{Suya20hybridBlack}. Integrating ODS with these training-based methods is an interesting direction for future work.\n\n\n\\section{Conclusion}\nWe propose ODS, a new sampling strategy for white- and black-box attacks. \nBy generating more diverse perturbations as measured in the output space, ODS can create more effective starting points for white-box attacks. Leveraging surrogate models, ODS also improves the exploration of the output space for black-box attacks. Moreover, ODS for black-box attacks is applicable even if the surrogate models are trained with out-of-distribution datasets. Therefore, black-box attacks with ODS are more practical than other black-box attacks using ordinary surrogate models. Our empirical results demonstrate that ODS with existing attack methods outperforms state-of-the-art attacks in various white-box and black-box settings. \n\nWhile we only focus on ODS with surrogate models trained with labeled datasets, ODS may also work well using unlabeled datasets, which we leave as future work. One additional direction is to improve the efficiency of ODS by selecting suitable surrogate models with reinforcement learning. \n\n\\section*{Broader Impact}\nThe existence of adversarial examples is a major source of concern for machine learning applications in the real world. For example, imperceptible perturbations crafted by malicious attackers could deceive safety critical systems such as autonomous driving and facial recognition systems. Since adversarial examples exist not only for images, but also for other domains such as text and audio, the \npotential impact is large. \nOur research provides new state-of-the-art black-box adversarial attacks in terms of query-efficiency and makes adversarial attacks more practical and strong. \nWhile all experiments in this paper are for images, the proposed method is also applicable to other modalities. Because of this, our research could be used in harmful ways by malicious users. \n\nOn the positive side, strong attacks are necessary to develop robust machine learning models. \nFor the last few years, several researchers have proposed adversarial attacks which break previous defense models.\nIn response to these strong attacks, new and better defense mechanisms have been developed. \nIt is this feedback loop between attacks and defenses that advances the field. %\nOur research not only provides a state-of-the-art attack, \nbut also sheds light on a new perspective, namely the importance of diversity, for improving adversarial attacks. This may have a long term impact on inspiring more effective defense methods.\n\n\\section*{Acknowledgements and Disclosure of Funding}\nThis research was supported in part by AFOSR (FA9550-19-1-0024), NSF (\\#1651565, \\#1522054, \\#1733686), ONR, and FLI. \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIt has been over 12 years since blockchain was invented to serve as the public transaction ledger of Bitcoin~\\cite{bitcoin}. Since, blockchain technologies have evolved significantly. This includes hundreds of public blockchain platforms, thousands of cryptocurrencies~\\cite{cryptocurrency}, tens of thousands of decentralized applications (DApps) and blockchain-based services~\\cite{dapp}, millions of smart contracts~\\cite{he2020characterizing}, hundreds of millions of user accounts~\\cite{eos-account}, and billions of transactions~\\cite{eth-tx}.\nThese technologies and entities have formed a symbiotic relationship, which we refer to as the \\textit{\\textbf{blockchain ecosystem}}.\n\nThe global blockchain economy is estimated to be worth 1.8 trillion USD at the end of May 2021~\\cite{cryptocurrency}.\nAs a result, its continuing evolution has increased its complexity, which \\textit{itself needs considerable effort to understand: particularly its characteristics, operations, trends and security issues}.\n\nDue to this, the blockchain ecosystem has attracted a lot of attention from the research community. A large number of studies have focused on exploring the characteristics of blockchain networks~\\cite{huang2020understanding, chen2020traveling, wu2019t}, detecting attacks and vulnerabilities~\\cite{chen2018understanding, suevil, ji2020deposafe}, deanonymizing users and tracking money-flow~\\cite{ober2013structure, awan2017blockchain, moser2018empirical, wang2020identifying}, and so on.\nHowever, little is known (at a comprehensive level) on the evolution of the overall blockchain ecosystem. Although several prior studies have performed measurement studies on transactions and smart contracts, they characterize activities based on either a static snapshot~\\cite{chen2020traveling}, or based on a single platform~\\cite{chen2018understanding, huang2020understanding}. \nFor example, Chen et al.~\\cite{chen2018understanding} and Huang et al.~\\cite{huang2020understanding} characterized Ethereum and EOSIO, respectively, based on large-scale transaction analysis.\nTo the best of our knowledge, \\textit{no existing studies have characterized the evolution of blockchain ecosystems comprehensively at scale, longitudinally and across multiple representative platforms.}\n\n\\textbf{This Work.}\nTo fill this gap, we perform a multi-dimensional and large-scale study covering all of the transactions on three of the most representative blockchains: Bitcoin, Ethereum and EOSIO. \nBitcoin is the first distributed blockchain implementation.\nEthereum is the second largest blockchain system after Bitcoin, and the first second-generation blockchain platform that supports smart contract functionality~\\cite{eth-blockchain-2}.\nEOSIO claims to be the third-generation blockchain, and adopts Delegate Proof of Stake (DPoS) consensus, which significantly improves throughput and enables new applications~\\cite{eos-dapp-prosperity-1, eos-dapp-prosperity-2}.\n\nWe first collect the largest ever dataset across these three platforms (\\textbf{see \\S\\ref{sec:study-design}}), with over 35 billion traces in total. \nWe use this dataset to provide a high-level characterization and temporal analysis of the three platforms (\\textbf{\\S\\ref{sec:blockchain-evolution}}), including assessing the behavior of money transfers (\\textbf{\\S\\ref{sec:blockchain-evolution:mtg}}), account creation (\\textbf{\\S\\ref{sec:blockchain-evolution:acg}}) and contract invocation (\\textbf{\\S\\ref{sec:blockchain-evolution:cig}}).\nFollowing this, we explore ``outliers'' across the evolution of the three blockchains. We identify these as either misbehavior by attackers or highly popular ``killer'' DApps that trigger major fluctuations in activity (\\textbf{\\S\\ref{sec:abnormal-behaviors}}).\nAmong many interesting results, the following are the most prominent:\n\\begin{itemize}\n    \\item \\textit{The overall blockchain ecosystem has shown significant growth over the last decade. Ethereum and EOSIO are on their way to becoming key decentralized systems, featuring powerful and popular DApps.} \n    Bitcoin has shown a steady upward trend, with the number of transactions rising 15K times over 12 years. The number of transactions in Ethereum and EOSIO rose 7.6K times and 2.2K times, respectively.\n    However, along with this growth in the number of transactions, the proportion of money transferred has dropped to 16.5\\% and 15.1\\% for Ethereum and EOSIO till Nov. 2019, while the proportion of smart contract invocation has reached over 80\\%.\n    Smart contracts and DApps are therefore becoming increasingly popular.\n    \n    \\item \\textit{Many activities in the blockchain ecosystem follow the Pareto principle across their evolution, and we are witnessing increasing centralization.}\n    For all three types of activities, i.e., money transfer, account creation, and contract invocation, the Pareto principle is followed (across all blockchains). This applies to every month we analyze, indicating that a small group of accounts are increasingly controlling the entire network. \n    \n    \\item \\textit{Interactions among smart contracts are increasingly popular.}\n    During the development of Ethereum and EOSIO, the emergence of ``killer'' DApps has led to geater interactions between smart contracts. \n    In other words, contract invocation transactions are more likely to be initiated by another smart contract than by a human.\n    However, Ethereum and EOSIO show quite different behaviors regarding to the DApp evolution. For example, in Ethereum, the explosion of Decentralized Finance (DeFi) DApps accounts for over 1\/3 of contract invocations since Mar. 2020. In contrast, nearly half of (49.04\\%) EOSIO contract invocation transactions were related to gambling DApps.\n    \n    \\item \\textit{87 ``outliers'' are identified from our time-series analysis. These have a significant impact that either facilitates or impedes the progress of the blockchain ecosystem}. 48 of these are due to emerging DApps, while 39 of them are introduced by attacks or scams.\n\\end{itemize}\n\nTo the best of our knowledge, this is the first longitudinal and large-scale study across these three-generations of blockchain. Our results motivate the need for further research efforts. Particularly, we argue it is necessary to measure and mitigate the widely unexplored trends and misbehaviors of these blockchain technologies. \n\n\\section{Background}\n\\label{sec:background}\n\n\\subsection{Blockchain and its Evolution}\n\\label{sec:background:blockchain-evolution}\n\nBlockchain has a history spanning 12 years. \n\\textit{Bitcoin}, as the first platform that utilized blockchain, is widely called \\textit{Blockchain 1.0}. \nFollowing the growth of cryptocurrencies, users \nstarted to explore the possibility of decentralized applications (DApps). This led to the development of Ethereum. As a representative system of \\textit{Blockchain 2.0}, \\textit{Ethereum} is widely adopted by developers. In fact, millions of smart contracts have already been deployed on Ethereum. However, the performance of Ethereum is limited by its consensus protocol (Proof-of-Work, PoW). As a result, in 2018, EOSIO came online, which adopts Delegated Proof-of-Stake (DPoS) consensus. This has been termed \\textit{Blockchain 3.0}, improving transaction throughput significantly. To boost performance, Ethereum 2.0 (Phase 0 test) has also adopted Proof-of-Stake (PoS) consensus in Dec. 2020.\nAlthough many kinds of other blockchain systems have emerged, \\textit{we believe these three blockchains are the most representative and popular ones, making them ideal for studying the evolution of the overall blockchain ecosystem}.\n\n\\begin{table*}[tbp]\n\\caption{A Comparison of Bitcoin, Ethereum and EOSIO.}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{lccccccc}\t\n\\toprule\n                  &\\textbf{\\begin{tabular}[c]{@{}c@{}}Native\\\\ Cryptocurrency\\end{tabular}}       & \\textbf{Model} & \\textbf{Consensus} & \\textbf{\\begin{tabular}[c]{@{}c@{}}Support for\\\\ Smart Contract\\end{tabular}} & \\textbf{TPS}~\\cite{garriga2020blockchain} & \\textbf{Resource} &              \\textbf{Role}        \\\\\n\\midrule                \n\\textbf{Bitcoin}   &  Bitcoin  & UTXO           & PoW                & No                                                                            & 7            & Transaction Fee   & Input \/ Output       \\\\\n\\textbf{Ethereum}  &  Ether    & Account\/Balance        & PoW                & Yes                                                                           & 15           & Gas               & EOA \/ Smart Contract \\\\\n\\textbf{EOSIO}     &  EOS      & Account\/Balance        & DPoS               & Yes                                                                           & 1000+         & CPU \/ NET         & Account             \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{table:basic-info}\n\\end{table*}\n\n\n\\subsection{Bitcoin, Ethereum and EOSIO}\n\\label{sec:background:general}\n\nWe next briefly compare the working mechanisms of Bitcoin, Ethereum and EOSIO, as summarized in Table~\\ref{table:basic-info}.\n\n\n\\subsubsection{Record-Keeping Models}\n\\label{sec:background:general:model}\n\nTwo types of record-keeping models are adopted in existing blockchain networks. The first method is called the UTXO (Unspent Transaction Output) Model~\\cite{utxo}, and the second one is the Account\/Balance Model~\\cite{account-based}. Bitcoin is the only one that adopts the UTXO model, which is illustrated in Fig.~\\ref{fig:utxo} in the Appendix \\S\\ref{sec:appendix:money-transfer}.\nSpecifically, each transaction is composed of a set of \\textit{inputs} and \\textit{outputs}, which all correspond to a certain amount of Bitcoin. The UTXO model ensures that the sum of amount represented by inputs is equal to the outputs.\nMoreover, each output is locked by a \\textit{scriptPubKey}. Only the one with the corresponding public key can unlock the output and spend the money by an input. Therefore, in this paper, we refer to each of the inputs and outputs as $pubkey$.\n\nHowever, Bitcoin suffers from several drawbacks: \n1) the script in Bitcoin is not Turing-complete~\\cite{turing}, which means it cannot perform more sophisticated functions (e.g., smart contracts); \n2) due to its consensus algorithm, the transactions per second (TPS) is low; \nand \n3) UTXOs are stateless and can be executed in parallel, which is not well suited for many applications.\nTherefore, Ethereum and EOSIO adopt the Account\/Balance model. It works similarly to a debit card transaction: the bank, equivalent to a consensus algorithm, tracks the balance of each debit card to make sure enough money exists before approving the transaction.\n\n\\subsubsection{Accounts and Smart Contracts}\n\\label{sec:background:general:account}\nIn Ethereum, there are two types of accounts: \\textit{External Owned Accounts} (EOA) and \\textit{smart contracts}.\nSmart contracts contain immutable and executable bytecode, while an EOA does not and is controlled directly by a private key.\nA smart contract can be created by either EOA or another smart contract. \nIn contrast, the smart contract in EOSIO is \\textit{updatable}. That is because an executable smart contract's bytecode in EOSIO is equivalent to an attribute of an account, like one's balance. Therefore, in EOSIO, one can easily update the smart contract in accounts he\/she owns after paying the required cost. \n\n\\subsubsection{Transactions}\n\\label{sec:background:general:tx}\nAccounts in Ethereum and EOSIO can interact with each other by invoking \\textit{transactions}, including money transfers, invoking contracts, or creating an account. \nIn Ethereum, the transactions initiated by EOA and smart contracts are termed \\textit{external transaction} and \\textit{internal transaction}, respectively. \nIn EOSIO, a transaction consists of one or more \\textit{action(s)} and an action can be invoked by an account.\n\nTo better illustrate the working mechanisms, we depict the process of transferring tokens between accounts in Ethereum and EOSIO in Fig.~\\ref{fig:transfer-token} (see Appendix \\S\\ref{sec:appendix:money-transfer}).\nSpecifically, when $account_a$ transfers 1 Ether to $account_b$, it initiates a transaction with an empty \\textit{input} field and 1 Ether filled in the \\textit{value} field. Specifically, the data in the \\textit{input} field can be parsed by the recipient as a function signature with the corresponding parameters, and the \\textit{value} indicates the attached Ether.\nThe situation in EOSIO is more complicated. \nAn official smart contract, named \\texttt{eosio.token}, issues EOS tokens and maintains a balance table for all holders. If $account_a$ intends to transfer EOS to $account_b$, it has to request the \\texttt{eosio.token} to update the corresponding rows. After updating, the \\texttt{eosio.token} would immediately \\textit{notify} both of them that the transfer request is handled properly. Once $account_b$ received the notification, it would perform the following movements, e.g., initiating another action to others.\nNote that, when someone intends to create an account, they need to request another official account (called \\texttt{eosio}) to perform the action, instead of transferring an arbitrary amount of EOS to a non-existent address like in Ethereum.\n\n\n\\subsubsection{Consensus Protocol \\& Resource Model}\n\\label{sec:background:general:consensus}\nThe main difference between EOSIO and the other two platforms is the consensus protocol and the resource model. PoW requires miners to calculate a string of meaningless hash values which must be preceded by a certain number of zero. Therefore, the longer the prefix of zeros, the more calculation is required, which leads to higher delays and resource-consumption.\nIn contrast, the DPoS is more resource\u2013conservative and efficient. Compared to PoW, the theoretic TPS in EOSIO is hundreds-fold, reaching up to 1000+. \n\nBitcoin and Ethereum adopt a similar resource model, i.e., \\textit{transaction fees} and \\textit{gas}. Both are used to incentivize miners to pack the transactions with the highest transaction fee or gas. \nIn EOSIO, the resources are named as \\textit{CPU} and \\textit{NET}, which represent the allowed time to execute an action, and the space can be occupied within a single transaction. However, these two types of resource do not charge anything from initiators. They can mortgage a certain amount of EOS in exchange for \\textit{CPU} and \\textit{NET}, which can be reused in each transaction.\nNote that, the price of CPU fluctuates according to the total value of mortgaged EOS for CPU, and NET can be freely used if some is available. \nIn other words, the resource price in EOSIO is \\textit{nearly free in normal conditions}.\n\n\n\n\n\\section{Data Collection}\n\\label{sec:study-design}\n\n\\subsection{Methodology}\n\\label{sec:study-design:data-collection}\n\nTo obtain all transactions across Bitcoin, Ethereum, and EOSIO, we deploy their official clients to synchronize data, i.e., \\textit{Bitcoin Core}~\\cite{bitcoin-core}, \\textit{OpenEthereum}~\\cite{openethereum}, and \\textit{nodeos}~\\cite{nodeos}. \nTo conduct a fine-grained analysis, we also collect all \\textit{traces}, i.e., internal transactions in Ethereum and actions in EOSIO (see \\S\\ref{sec:background:general:tx}). \nAs the traces are too large to crawl directly from mainstream browsers (like \\textit{etherscan}~\\cite{etherscan}), we collect them by replaying each transactions. Specifically, \\textit{trace\\_transaction} in the trace module of OpenEthereum takes a transaction as an input, and replays it to collect all internal transactions that result from the given transaction. \nFor obtaining actions in EOSIO, we take advantage of the approach proposed by \\cite{huang2020understanding} to customize the core service daemon (nodeos) to accelerate the data collection process.\n\nTo facilitate our analysis of major events across the blockchains (see \\S\\ref{sec:abnormal-behaviors}), we further crawl the available information (i.e, name, category, and corresponding addresses or accounts) of all DApps from three well-known platforms: DAppTotal~\\cite{dapptotal}, DappRadar~\\cite{dappradar}, and DappReview~\\cite{dappreview}. \n\n\\begin{table}[t]\n\\caption{Dataset overview (as of March 31st, 2020).}\n\\centering\n\\resizebox{\\columnwidth}{!}{%\n\\begin{tabular}{lccc}\n\\toprule\n\\textbf{Category} & \\textbf{Bitcoin} & \\textbf{Ethereum} & \\textbf{EOSIO} \\\\\n\\midrule\nLaunch Date       & 2009.01.09     & 2015.07.30     & 2018.06.09 \\\\\nBlocks          &  623,837     & 9,782,602 & 113,124,658 \\\\\n\\midrule\nTraces            & 2,371,617,384     & 1,655,111,086     & 31,588,572,466 \\\\\nAccount           & 630,562,205      & 68,790,386        & 1,816,578      \\\\\nSmart Contract    & -                & 24,201,516        & 6,139          \\\\\n\\midrule\nDApp              & -                & 3,292             & 652            \\\\\nDApp Addresses    & -                & 13,218            & 1,556          \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\label{table:dataset}\n\\end{table}\n\n\n\\subsection{Dataset Overview}\n\\label{sec:study-design:overview}\n\n\\subsubsection{Dataset.} In total, we have collected over \\textit{35 Billion traces}. Table~\\ref{table:dataset} summarizes the dataset. To the best of our knowledge, this is the \\textit{largest ever} dataset studied in the research community. The \\textit{trace} is the basic unit of transaction in each blockchain. For Bitcoin, a trace is an input or an output; for Ethereum, traces are composed of all the external transactions, as well as the internal ones; for EOSIO, a trace corresponds to an action.\n\nObviously, \\textit{although EOSIO has the latest launch time, the numbers of traces and blocks are much higher than the other two blockchains.}\nAs Bitcoin adopted UTXO model, which does not have the concept of account, the number of Bitcoin accounts indicates the distinct \\textit{pubkeys} of inputs and outputs. \nIn Bitcoin, an entity may have multiple pubkeys to avoid reuse (to protect anonymity). Thus, the number of accounts in Bitcoin is tens of Ethereum's and hundreds of EOSIO's.\nMoreover, we can observe a significant distinction in the number of accounts and smart contracts between Ethereum and EOSIO. As explained in \\S\\ref{sec:background:general:account}, this is because: 1) creating an account in EOSIO charges not only transaction fee; and 2) EOSIO smart contract is updatable. Both of them impede the explosion of the number of accounts in EOSIO.\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/traces.pdf}}\n\\caption{The evolution of traces on a monthly basis.}\n\\label{fig:traces}\n\\end{figure}\n\n\\subsubsection{General Trends.}\n\\label{sec:dataset:overview:general-trend}\nFig.~\\ref{fig:traces} shows the number of traces collected on a monthly basis, and the corresponding Transaction per Second (TPS). \nThe numbers of traces in all three platforms have an upward trend. Interestingly, it only took EOSIO three months to catch up and overtake the other two platforms. However, it took Ethereum almost three years to surpass Bitcoin.\nCompared to the relatively steady growth of Bitcoin, Ethereum and EOSIO experience sudden increases. For example, in Oct. 2016, the TPS of Ethereum reached up to 73.0, around 15.9 times higher than the previous month. \nInterestingly, this explosive growth was actually due to a DoS~\\cite{eth-dos} attack, which far exceeded the theoretical TPS it could handle.\nAs for EOSIO, the TPS reached over 1,240 in Nov. 2019, and finally reached over 3,080. This resulted from the launch of a spam DApp, which will be detailed in \\S\\ref{sec:abnormal-behaviors:case-study:EIDOS}.\n\n\n\n\n\n\\section{Graph-based Evolution Analysis}\n\\label{sec:blockchain-evolution}\n\n\\subsection{Method}\nWe seek to characterize the evolution of the blockchain ecosystems by investigating three representative behaviors: \\textit{money transfers}, \\textit{account creation}, and \\textit{contract invocation}, corresponding to the types of interactions between accounts (see \\S\\ref{sec:background:general:tx}).\nTo this end, we represent these activities in three graph structures: the \\textit{Money Transfer Graph (MTG)}, \\textit{Account Creation Graph (ACG)}, and \\textit{Contract Invocation Graph (CIG)}. \nFor each blockchain platform, we will denote each graph using its first three lowercase letters in subscript, e.g., $CIG_{btc}$ for Bitcoin.\nWe build these three types of graphs on a monthly basis and specify the month (formatted as $yyyy-MM$) in superscript, e.g., $CIG_{btc}^{2019-10}$.\nOverall, we have created 372 graphs (135 for Bitcoin, 171 for Ethereum and 66 for EOSIO).\n\n\\begin{table*}[t]\n\\caption{Notations with their corresponding explanations.}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{llll}\n\\toprule\n\\textbf{Notations} & \\textbf{Explanations}            & \\textbf{Notations} & \\textbf{Explanations}                                                                                                                                                       \\\\ \\midrule\n$MTG$              & Money Transfer Graph             & $\\alpha$           & The exponent of fitting function in degree\/indegree\/outdegree distribution                                                                                                  \\\\\n$ACG$       & Account Creation Graph           & $R$                & The Pearson's Correlation Coefficient between indegree and outdegree                                                                                                        \\\\\n$CIG$       & Contract Invocation Graph        & $WCC$, $SCC$       & Weakly and strongly connected component                                                                                                                                     \\\\\n$T_m$              & Money transfer transactions      & $(v_i, v_j, w, t)$ & \\begin{tabular}[c]{@{}l@{}}Formal definition of edges, indicating at timestamp $t$, $v_i$ initiates\\\\ $w$ times of corresponding type of transactions to $v_j$\\end{tabular} \\\\\n$T_a$              & Account creation transactions    & $d$                & Date, converting from timestamp $t$ and formatting as ($yyyy-MM$)                                                                                                           \\\\\n$T_c$              & Contract invocation transactions & $V$, $E$           & The node\/edge set of the graph                                          \\\\ \\bottomrule                                                                                                   \n\\end{tabular}\n}\n\\label{table:notations}\n\\end{table*}\n\n\n\\subsection{Metrics and Notations}\n\\label{sec:blockchain-evolution:metrics}\n\nFollowing previous studies~\\cite{chen2018understanding, huang2020understanding}, we measure the graphs using the following four metrics. For each graph, we later inspect each of these metrics in-turn. Table~\\ref{table:notations} summarizes the key notations used in this paper.\n\n\\subsubsection{The number of traces.}\nWe used simplified notations $T_m$, $T_a$ and $T_c$ to refer to the \\textit{money transfer traces}, \\textit{account creation traces} and \\textit{contract invocation traces}, respectively. \\textit{This metric reflects the popularity of a certain type of activity along the timeline}. For example, a popular gambling DApp will introduce large number of $T_c$ (as the player invokes the contracts of the gambling DApp) and $T_m$ (as the player sends and receives money from the gambling DApp).\nMoreover,  for each kind of trace, we tag the \\textit{role of invokers}, i.e., from a smart contract or a user account. The trends can reflect whether the corresponding blockchain is mainly used as a \\textit{payment network} or a \\textit{decentralized system with emerging and powerful applications (i.e., smart contracts)}.\n\n\\subsubsection{Distribution of node degree.}\n\\label{sec:blockchain-evolution:metrics:degree}\nThe \\textit{indegree} and \\textit{outdegree} of a node represent how many edges are directed to, or pointed from that node. The \\textit{degree} of a node is the sum of its indegree and outdegree. \n\\textit{The node degrees distribution reflects the centralization of the blockchain in terms of certain activities.}\nTo measure this in our later analysis, we plot the degree\/indegree\/outdegree distribution for each graph, in which the y-axis is the proportion of nodes and x-axis is the size of degree\/indegree\/outdegree. On the basis of these plots, we take the logarithm of both axes, and perform a fit with function $y \\sim x^{\\alpha}$.\nThe \\textit{the lower the $\\alpha$, the shorter the tail, and the less centralized the corresponding traces}.\nWe illustrate a concrete example of degree distribution in \\S\\ref{sec:blockchain-evolution:mtg:degree}.\n\n\n\\subsubsection{Pearson Correlation Coefficient ($R$)}\nOn top of the indegree and outdegree of different nodes, we further measure the \\textit{Pearson Correlation Coefficient}~\\cite{pearson} (denoted as $R$ in the following) between them.\n$R$ reflects whether the distribution of outdegree and indegree have a linear relationship. It is between $1$ and $-1$ to indicate strong positive or negative correlation.\nFor example, a popular DApp interacts frequently with users, leading to both high indegree and outdegree for all participants. To this end, the $R$ would be close to 1, indicating strong positive correlation. \nThus, \\textit{$R$ indicates the consistency of indegree and outdegree of nodes within a graph}.\n\n\\subsubsection{Weakly connected component (WCC) \\& Strongly connected component (SCC)}\n\\label{sec:metric:wcc}\nIn graph theory, if there exists a \\textit{path} between $v_i$ to $v_j$ in a directed graph, there exists a set of edges end-to-end whose starting and ending points are $v_i$ and $v_j$, respectively.\nIn WCC, there exists a path between two arbitrary nodes but without differentiating the starting point and the end point. \nIn SCC, however, it requires that there exists a bi-directional path between any two nodes.\nTo this end, if the number of WCCs drops, it means there exists a node with extensive coverage; and if the number increases, it means such a node disappears.\nFurthermore, the number of SCCs examines the possibility of two-way interaction of that node.\nConsequently, \\textit{the number of WCCs and SCCs reflects the connectivity between nodes in terms of one-way and two-way interaction, respectively}. For example, a spammer would initiate a large amount of $T_m$ or $T_c$ to deliver its message, but few accounts would reply. Thus, the number of WCCs and SCCs would respectively drop and rise due to many one-way edges emerging.\n\n\n\n\\section{Evolution of Money Transfers}\n\\label{sec:blockchain-evolution:mtg}\n\n\n\\subsection{Graph Construction}\n\\label{sec:blockchain-evolution:mtg:construction}\n\nTo build $MTG$, we first define the money transfer trace, i.e., $T_m$ in Bitcoin, Ethereum and EOSIO.\nSpecifically, in Ethereum, a transaction with a certain amount of Ether and a blank input field is regarded as a $T_m$. The payer and payee can be identified easily by their addresses.\nIn EOSIO, we further parse the following notifications initiated from \\texttt{eosio.token} after transfer requests. Hence, we can extract the real participants and construct a $T_m$ directly between them.\nFor Bitcoin, a transaction may have multiple inputs and outputs, which are regarded as two sets: payers and payees. Therefore, \nwe follow previous work~\\cite{akcora2018forecasting} and insert an additional node and label it by the corresponding transaction id, $txid$.\n\nAs defined in Table~\\ref{table:notations}, each $T_m$ can be formally represented as $(v_i, v_j, w, t)$, indicating at time $t$, $v_i$ transferred $w$ tokens to $v_j$.\nThen, by converting the timestamp $t$ to date $d$ (formatted as $yyyy-MM$) and adding $w$ up as $w^*$, we can group $T_m$s with the same payer and payee on a monthly basis for each platform. This can be formalized as $\\{(v_i, v_j, w^*, d) | v_i, v_j \\in V, w^* \\in \\mathbb{R}^+\\}$.\nConsequently, $MTG$s can be defined as a \\textit{weighted directed graph}: $MTG = (V, E)$, where $E$ is the above mentioned set and $V$ is a set of nodes represented as: $(id, label)$. Specifically, $id$ is the identifier to distinguish nodes within a platform; and the $label$ is the category of DApps to which the $id$ belongs, according to the off-chain information we collected.\nConsequently, we have created 214 MFGs, 135 for Bitcoin, 57 for Ethereum and 22 for EOSIO.\nWe spend the rest of this section inspecting $MTG$ using the four metrics defined in \\S\\ref{sec:blockchain-evolution}.\n\n\n\\subsection{Number of Traces}\n\\label{sec:blockchain-evolution:mtg:traces}\n\n\n\\subsubsection{Overall evolution.}\nThe number of $T_m$ over time across three platforms is shown in Fig.~\\ref{fig:traces-mtg}.\nThe number of $T_m$ in Bitcoin has always been greater than Ethereum's.\nSince Oct. 2018, the number of $T_m$ in EOSIO has surpassed the other two platforms. Though there was a decline from late 2018, its number has reached 2.1 billion in $MTG_{eos}^{2019-11}$, which is more than an order of magnitude higher than Bitcoin and Ethereum at the same time.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=1\\columnwidth]{images\/traces-mtg.pdf}}\n\\caption{The evolution of monthly money transfer traces across three blockchains.}\n\\label{fig:traces-mtg}\n\\end{figure}\n\nWe further show the proportion of $T_m$ out of all traces by a dashed line in Fig.~\\ref{fig:traces-mtg}. Note that we omit the percentage of Bitcoin, as it is always 100\\% (the Bitcoin network is solely composed of $T_m$).\nFor Ethereum, we observe three troughs in late-2015, Oct. 2016 and Nov. 2017. These were caused by Dwarfpool~\\cite{dwarf}, a DoS attack~\\cite{eth-dos}, and a DApp called CryptoKitties~\\cite{cryptokitties}, respectively. \nTo be specific, the mining pool and the DoS attack both invoked a huge number of $T_c$ (see \\S\\ref{sec:blockchain-evolution:cig}). The DoS attack took advantage of the flawed design of the price of gas (see \\S\\ref{sec:background:general:consensus}).\nCryptoKitties is a collectible game where players can trade and exchange \\textit{kitties}. However, \\textit{kitties} are represented by alternative tokens, whose transfer is regarded as $T_c$ in our paper.\nExcept for these three troughs, the overall share of $T_m$ in Ethereum has been gradually decreasing, which we believe is due to the emergence of DeFi since May 2017. According to our data, in just six months, the number of DeFi related $T_c$ had risen by a remarkable 486.3\\% and another 7,420.4\\% for the next half year.\n\nIn contrast, the amount of $T_m$ in EOSIO surged in late-2018, which was due to the popularity of gambling DApps. This type of DApps requires frequent and abundant money transferred between players and contracts. Our collected data indicates there were 237 million $T_m$ related to gambling DApps in EOSIO in Nov. 2018 (equivalent to 23.4 times of all $T_m$ in Ethereum in the same month).\nAfter this, the share of $T_m$ also started to go down until Nov. 2019. The emergence of EIDOS\\footnote{This accepts a user's transfer of EOS and returns it back immediately. In the meanwhile, it initiates another transaction with a certain amount of EIDOS tokens, issued by themselves but can be traded in exchanges. Therefore, the ratio of $T_m$ to $T_c$ related to EIDOS is around 2 to 1.} \\cite{eidos} forced the percentage of $T_m$ to around 66\\% (i.e. two-thirds) of all traces in each month (see \\S\\ref{sec:abnormal-behaviors:case-study:EIDOS}).\n\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-mtg-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:percentage-mtg-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-mtg-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:percentage-mtg-eos}\n     \\end{subfigure}\n        \\caption{Percentage of initiators' role who invokes money transfer traces.}\n        \\label{fig:percentage-mtg}\n\\end{figure}\n\n\n\n\\subsubsection{The role of initiators.}\nFig.~\\ref{fig:percentage-mtg} shows the percentage of the role of initiators, from whom the $T_m$ is initiated, in Ethereum and EOSIO.\nAn obvious trough existed in Oct. 2016 from Fig.~\\ref{fig:percentage-mtg-eth}. \nThis is because the DoS attack requires frequent invocations of $T_m$, which can be easily achieved by smart contracts instead of regular accounts (i.e., human).\nExcept for that, we observe that users in Ethereum prefer transferring money on their own, though the ratio was still gradually going down.\nAccording to our data, along with the increase of the absolute number of $T_m$ (see Fig.~\\ref{fig:traces-mtg}), since 2018, the number of $T_m$ initiated by EOA has dropped by 67.6\\%.\nMoreover, in EOSIO, we see a surge of $T_m$ initiated from smart contracts in late-2018, as the contracts of gambling DApps needed to invoke $T_m$ to return money to their winners. Our data illustrates that among these 237M pieces of gambling related $T_m$, 78.38\\% of them were initiated by gambling DApps.\n\n\\textbf{\\textit{Insight: }}\n\\textit{\nThe activity of the blockchain ecosystems has shown continuous growth in terms of transferring native tokens, in-part incentivized by killer DApps.\nThe proportion of transfer transactions reveals that both Ethereum and EOSIO has been experiencing a decline, due to them departing from pure value-transfer networks.\n}\n\n\\subsection{Degree}\n\\label{sec:blockchain-evolution:mtg:degree}\nAs explained in \\S\\ref{sec:blockchain-evolution:metrics}, we use $\\alpha$ to measure the degree distribution. \nWe first illustrate its meaning using a concrete example. \nFig.~\\ref{fig:alpha-eg} illustrates the $\\alpha$ of indegree and outdegree distribution in $MTG_{eos}^{2019-10}$ and $MTG_{eth}^{2019-10}$.\nEach dot in Fig.~\\ref{fig:alpha-eg} shows that there is $y$ percent of nodes with the degree\/indegree\/outdegree valued as $x$ (in the corresponding month and platform). \nThe distribution of node degree is quite different across the blockchains.\nNote, the larger the $\\alpha$, the more obvious the long-tail distribution, indicating less variabilty in the nodes' degree. \nMore importantly, a high $\\alpha$ means greater centralization.\nFor example, we identify a family of accounts in EOSIO that invoke spam advertisements in Oct. 2019 (more details in \\S\\ref{sec:abnormal-behaviors:case-study:spam}).\nDue to this, Fig.~\\ref{fig:alpha-eg-eos-in} has points in the top left corner. These are nodes with a small indegree, representing victims of this spam attack. In contrast, we see a set of points located in the bottom right corner of Fig.~\\ref{fig:alpha-eg-eos-out}, i.e., the spammers.\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.4\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-eg-eos-in.pdf}\n         \\caption{EOSIO indegree}\n         \\label{fig:alpha-eg-eos-in}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.4\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-eg-eos-out.pdf}\n         \\caption{EOSIO outdegree}\n         \\label{fig:alpha-eg-eos-out}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.4\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-eg-eth-in.pdf}\n         \\caption{Ethereum indegree}\n         \\label{fig:alpha-eg-eth-in}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.4\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-eg-eth-out.pdf}\n         \\caption{Ethereum outdegree}\n         \\label{fig:alpha-eg-eth-out}\n     \\end{subfigure}\n        \\caption{Indegree and outdegree distribution for $MTG_{eos}^{2019-10}$ and $MTG_{eth}^{2019-10}$.}\n        \\label{fig:alpha-eg}\n\\end{figure}\n\n\n\\subsubsection{Overall evolution.}\nTo study this centralization, we further calculate the $\\alpha$ of the degree\/indegree\/outdegree distribution for each platform on a monthly basis. The trend of the $\\alpha$s are shown in Fig.~\\ref{fig:alpha-mtg}.\nThe general trend shows that, in Bitcoin, except for the infant stage (prior to 2012), the $\\alpha$ of indegree and outdegree distributions were quite stable and changed almost simultaneously.\nMoreover, the $\\alpha$ of Ethereum's indegree distribution shows an overall gradual upward trend, while that of outdegree was gradually stabilizing. This signals growing centralization of indegree distribution.\nIn EOSIO, the $\\alpha$ metrics fluctuate significantly though. Compared to Ethereum, the indegree was less centralized but the outdegree was exactly in the opposite position.\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-mtg-btc.pdf}\n         \\caption{Bitcoin}\n         \\label{fig:alpha-mtg-btc}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.63\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-mtg-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:alpha-mtg-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.36\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-mtg-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:alpha-mtg-eos}\n     \\end{subfigure}\n        \\caption{$\\alpha$ of degree\/indegree\/outdegree distribution of blockchain platforms over time in terms of $MTG$.}\n        \\label{fig:alpha-mtg}\n\\end{figure}\n\n\\subsubsection{Per-chain analysis.}\nWhen we look in detail on the per-chain basis, subtle differences arise.\nAs we see from Fig.~\\ref{fig:alpha-mtg-btc}, prior to 2012, \nBitcoin was still in its infancy and the number of $T_m$ per month was so small that a few transactions could cause a large shift in the $\\alpha$.\nFrom Fig.~\\ref{fig:alpha-mtg-eth}, we see that prior to mid-2018, the $\\alpha$ of outdegree distribution went higher than that of the indegree distribution. Combining with the historical prices of Ether~\\cite{eth-cap}, we see that the price was going up and reached a local maximum from mid-2017. This would motivate frequent buy and sell transactions through exchanges, who typically have multiple entries used to collect deposit requests from users. Hence, this has resulted in more centralized indegree distribution than outdegree. According to our data, the exchange related $T_m$ during late 2017 accounted for 15.7\\% --- 48.18\\% of the total DApp-related $T_m$.\nFor EOSIO,  except for Oct. 2019, we see the trend of $\\alpha$ was similar with the Ethereum's (see Fig.~\\ref{fig:alpha-mtg-eos},). Also, if we combine the historical price of EOS~\\cite{eos-cap}, we see that there were some price peaks, especially before Dec. 2018 and Jun. 2019.\nHowever, the existence of exchanges in EOSIO cannot drive the $\\alpha$ so dramatically. \nOther driving forces include the growth of gambling DApps in Sep. 2018,\ni.e., around 6.9 million transactions were initiated from gambling DApps. The top 1\\% accounted for 98.1\\% of them, which explains the  greater centralization of outdegree.\n\n\\textbf{\\textit{Insight: }}\n\\textit{\nCompared to the relative stable degree distribution in Bitcoin, the degree distribution in Ethereum and EOSIO has been significantly affected by the price of tokens. This has resulted in greater centralization (with some dominant accounts) that are reflected in the fluctuation of $\\alpha$.\n}\n\n\\subsection{Pearson Correlation Coefficient ($R$)}\n\\label{sec:blockchain-evolution:mtg:pearson}\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/pearson-mtg.pdf}}\n\\caption{Pearson's correlation between indegree and outdegree of blockchain platforms over time.}\n\\label{fig:pearson-mtg}\n\\end{figure}\n\nThe correlation ($R$) between indegree and outdegree of nodes for these platforms are depicted in Fig.~\\ref{fig:pearson-mtg}.\nAfter the infant stage of Bitcoin,\nits $R$ was relatively high, reflecting a strong positive correlation of indegree and outdegree. However, this may be due to the topology of $MTG_{btc}$, i.e., the node\nmust have one indegree (the referring output is directed from its $txid$) and zero\/one outdegree (depending on if it is (un)spent).\nIn Ethereum, after the DoS attack, there were two peaks that indicate a strong correlation: Mar. 2017 (0.521) and Dec. 2017 (0.607). These are caused by ShapeShift~\\cite{shapeshift} and Cryptokitties, respectively. The former is an exchange which allows users to exchange cryptocurrencies. Though Cryptokitties led to frequent token exchanges, it still required a huge amount of Ether transfer to perform game logic, like trading and breeding \\textit{kitties}. In terms of $T_m$, the core contract of Cryptokitties accounted for 469K and 458K outdegree and indegree, respectively.\nHowever, this has been gradually decreasing since 2018, due to the emergence of DeFi, which is more likely to initiate $T_c$ instead of $T_m$ (see \\S\\ref{sec:blockchain-evolution:cig}).\n\nAs for EOSIO, the overall tendency of $R$ shows there was nearly no correlation between indegree and outdegree, except for two months: Apr. 2019 and Nov. 2019. One previous work~\\cite{huang2020understanding} analyzed the EOS Global event that happened in Apr. 2019. This was a bot-like account that sent and received a huge number of EOS to fake an exaggerated trading volume. This resulted in huge outdegree and indegree.\nThe peak in Nov. 2019 resulted from EIDOS, a spam airdrop event, whose contract received a lot of EOS from investors and returned them back immediately (detailed in \\S\\ref{sec:abnormal-behaviors:case-study:EIDOS}).\n\n\\textbf{\\textit{Insight: }}\n\\textit{\nBitcoin has a relatively high $R$, but Ethereum's $R$ has been gradually decreasing. This may be related to the emergence of popoular DeFi DApps.\nIn contrast, EOSIO has maintained a relatively low level of $R$ in transferring money, except for the peaks imported by exchanges and spams.\n}\n\n\n\\subsection{Connected Components}\n\\label{sec:blockchain-evolution:mtg:cc}\nFig.~\\ref{fig:cc-mtg} shows the evolution of the numbers of WCC and SCC.\nGenerally, these numbers reflect the (dis)apperance of nodes who could import huge amounts of (bi-)directional edges (see \\S\\ref{sec:metric:wcc}).\nNote that, due to the impossibility of constructing bi-directional edges in $MTG_{btc}$ for the Bitcoin network, we only measure its number of WCC.\nUntil Apr. 2018, the number of WCCs for Ethereum and Bitcoin have been almost equal (48,529 for Ethereum and 55,387 for Bitcoin). In contrast, the number of WCC for EOSIO was only 230 at most (in Apr. 2019). There are two possible reasons: \n1) a relatively lower number of accounts and smart contracts in EOSIO compared to Ethereum and Bitcoin (see Table~\\ref{table:dataset}); \nand 2) $T_m$ is more likely to be invoked in EOSIO (see Fig.~\\ref{fig:traces-mtg}).\nIn Oct. 2019, the number of WCC in EOSIO dropped to 3. This was caused by an enormous amount of spam advertisements (detailed in \\S\\ref{sec:abnormal-behaviors:case-study:spam}) that carry tiny EOS but cover many accounts. Astonishingly, this pushes the size of WCC as 579,964, covering more than 99.9\\% of nodes in $MTG_{eos}^{2019-10}$.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/cc-mtg.pdf}}\n\\caption{The number of WCC and SCC in different blockchain platforms over time.}\n\\label{fig:cc-mtg}\n\\end{figure}\n\nAs to SCC, the difference between EOSIO and Ethereum has reduced, which suggests that one-way directed edges in $T_m$ still dominate. In EOSIO, we also see a sudden drop in Nov. 2019 (the number of SCC is 55,452), as the EIDOS enforces the bi-directional $T_m$.\n\n\\textbf{\\textit{Insight: }}\n\\textit{The number of WCC was relatively low in EOSIO compared to other two platforms, suggesting that accounts in EOSIO have stronger connections than those in Bitcoin and Ethereum. That said, violent fluctuations were introduced by certain misbehaviors.\nMoreover, the numbers of WCCs and SCCs have an upward trend, meaning that the platforms are constantly flooded with new users. However, our results show that the existing contracts do not perform many money transfers with them. \n}\n\n\\section{Evolution of Account Creation}\n\\label{sec:blockchain-evolution:acg}\n\n\\subsection{Graph Construction}\nThough we uniformly defined accounts in all three platforms (see \\S\\ref{sec:study-design:overview}), the concept of an `accounts' in Bitcoin is still vague. Therefore, we only discuss $T_a$, \\textit{account creation transactions}, in Ethereum and EOSIO.\nIn Ethereum, we focus on the smart contract, which can be created by an internal transaction whose type is \\textit{create} (see \\S\\ref{sec:background:general:tx}).\nAs for EOSIO, we parse the notifications followed by function \\texttt{newaccount} in theofficial account \\texttt{eosio} to parse the actual creator and the created account (explained in \\S\\ref{sec:background:general:tx}).\n\nSimilar to the definition of $T_m$ (see \\S\\ref{sec:blockchain-evolution:mtg:construction} and Table~\\ref{table:notations}), $T_a$ is also defined as $(v_i, v_j, w, t)$. However, as each $v_j$ can only be created once, the $w$ is always $1$ here.\nTherefore, we can group $T_a$s by only converting $t$ as $d$: $\\{(v_i, v_j, d) | v_i, v_j \\in V\\}$.\nEventually, we obtain an \\textit{unweighted directed graph} $ACG$ for each month: $ACG = (V, E)$, where $E$ is the above set and $V$ is the set composed of nodes as introduced in \\S\\ref{sec:blockchain-evolution:mtg:construction}.\nIn total, we have created 79 ACGs: 57 for Ethereum and 22 for EOSIO.\nWe spend the rest of this section inspecting $ACG$ using three of the metrics defined in \\S\\ref{sec:blockchain-evolution}.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/traces-acg.pdf}}\n\\caption{The evolution of account creation traces.}\n\\label{fig:traces-acg}\n\\end{figure}\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-acg-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:percentage-acg-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-acg-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:percentage-acg-eos}\n     \\end{subfigure}\n        \\caption{The distribution of initiators of $T_a$.}\n        \\label{fig:percentage-acg}\n\\end{figure}\n\n\n\\subsection{Number of Traces}\n\\label{sec:blockchain-evolution:acg:traces}\nThe number of $T_a$ and its corresponding proportion to the number of all traces in the corresponding month are shown in Fig.~\\ref{fig:traces-acg}.\nTaking the number of $T_a$ in Ethereum as a whole, we can observe a conspicuous upwards trend. Contrary to Ethereum, the number of newly created accounts in EOSIO has remained at a relatively low but stable level. The exception is during Apr. 2019, where the EOS Global contract created more than 241K bot-accounts.\nMoreover, in EOSIO, except for the first month, the ratio of $T_a$ was almost negligible (0.17\\% in Aug. 2018 was the maximum ratio).\nContrary, there was always a certain percentage of $T_a$ in Ethereum for each month (around 1\\% -- 2.5\\%), though the percentage was much lower than $T_m$'s (see Fig.~\\ref{fig:traces-mtg}).\nThe main reason is that creating an account in Ethereum is almost free compared to EOSIO\\footnote{\nCreating an EOSIO account will cost 5.39 USD~\\cite{eos-create-account}, while it takes less than 0.01 USD in Ethereum on that day.\n}, which may make users more reluctant to reuse existing accounts for different purposes.\n\n\nFig.~\\ref{fig:percentage-acg} further shows the role of the initiator who invokes $T_a$s.\nEspecially in Mar. 2020, the ratio of Ethereum accounts created by smart contracts is 33.7 times higher than that of EOAs. Such an astonishing ratio shows a dramatic contrast to the ratio in $MTG$ (see Fig.~\\ref{fig:percentage-mtg-eth}).\nAfter manual investigation, we find that this is due to a contract named GasToken~\\cite{gastoken}, which helps users to spend less gas to complete the transaction --- this created a huge amount of accounts since 2019. \nThe $T_a$ it initiated accounted for 62\\% of all $T_a$ in Mar. 2020.\nMoreover, we find that the ratio of $T_a$ initiated from contracts has little correlation with the prosperity of DApps.\nOverall, the DApp related account creation transactions has never accounted for higher than 15\\% in each month.\n\n\\textbf{\\textit{Insight: }}\n\\textit{In Ethereum and EOSIO, the absolute number and ratio of account creation transactions are relative low compared to transfer transactions. Creating accounts is less necessary in Ethereum, and EOSIO charges expensive fees.\nMoreover, accounts are more likely to be created by contracts, indicating the tendency for more relationships between smart contract.\n}\n\n\n\\subsection{Degree}\n\\label{sec:blockchain-evolution:acg:degree}\nAs each node can only be created once, this leads to a constant indegree value for each node. Hence, we only measure the distribution of outdegree and degree, and the evolution of the corresponding metrics $\\alpha$, as shown in Fig.~\\ref{fig:alpha-acg}.\n\nFor Ethereum, prior to mid-2017, the values of $\\alpha$ show volatility, which we believe is caused by the relatively small number of $T_a$ during this period (see Fig.~\\ref{fig:traces-acg}). From then on, both platforms exhibit an upwards trend for the $\\alpha$ of the outdegree distribution, especially in EOSIO. In other words, the account creation behavior has become increasingly centralized. For example, account \\texttt{genialwombat} which belongs to Wombat Wallet~\\cite{wombat}, offers a free account creation service for its users. This wallet has created more than 32.2K accounts. Furthermore, the top 1\\% of accounts in EOSIO have created 97.3\\% of all accounts in Mar. 2020.\nInterestingly, we can observe two sudden decreases in both sub-graphs: Aug. 2018 in Ethereum and Apr. 2019 in EOSIO. Combining the off-chain information we collected, we conclude that these two outliers were caused by an attack against a Fomo3D-like game called Last Winner~\\cite{last-winner} and the EOS Global event, respectively. The former took advantage of lots of newly created accounts to attack the DApp. This account creation process followed a multi-level structure~\\cite{last-winner} leading to invoking enormous $T_a$.\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.63\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-acg-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:alpha-acg-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.36\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-acg-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:alpha-acg-eos}\n     \\end{subfigure}\n        \\caption{$\\alpha$ of degree\/outdegree distribution of blockchain platforms over time in terms of $ACG$.}\n        \\label{fig:alpha-acg}\n\\end{figure}\n\n\\textbf{\\textit{Insight: }}\n\\textit{Except for the early stage of Ethereum, the $\\alpha$ of the outdegree distributions show an upward trend for both platforms. The exceptions are two troughs, which resulted from attacks against Last Winner and EOS Global, respectively.\nEspecially in EOSIO, the upwards $\\alpha$ indicates the emergence of large account creation providers which made the invocations of account creation transactions more centralized.}\n\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/cc-acg.pdf}}\n\\caption{The number of WCC in different blockchain platforms over time in terms of $ACG$.}\n\\label{fig:cc-acg}\n\\end{figure}\n\n\\subsection{Connected Components}\n\\label{sec:blockchain-evolution:acg:cc}\nFig.~\\ref{fig:cc-acg} shows the number of WCCs in Ethereum and EOSIO.\nEthereum shows an upward trend followed by a decline in terms of the number of WCC. \nIn EOSIO, it was also gradually decreasing (due to the emergence of account creation services).\nMoreover, we observe an obvious trough during Oct. 2016 in Ethereum, which had 294 WCCs.\nWe believe this is related to the DoS attack, which created a large number of accounts and prevented others from creating accounts properly.\nBesides, the strange 1 WCC in EOSIO in Jun. 2018 was because the account creation tree composed of invoked $T_a$ has a common root node: official account \\texttt{eosio}.\n\n\n\n\\textbf{\\textit{Insight: }}\n\\textit{The account creation process has become more centralized, causing the number of WCC to go down continuously. This further reflects the tighter relationship between smart contracts in terms of account creation behavior.\n}\n\n\n\\section{Evolution of Contract Invocation}\n\\label{sec:blockchain-evolution:cig}\n\n\n\\subsection{Graph Construction}\nSimilar to $ACG$, we do not discuss $T_c$ in Bitcoin.\nWe treat all the traces in Ethereum that are targeted to any smart contract as $T_c$.\nAs for EOSIO, we excluded the transfer actions to \\texttt{eosio.token} and the account creation action to \\texttt{eosio}. All the other actions are treated as $T_c$.\nConsequently, we collect all the $T_c$ as $(v_i, v_j, w, t)$.\nIdentical to the process introduced in \\S\\ref{sec:blockchain-evolution:mtg:construction}, the $CIG$ is defined as a \\textit{weighted directed graph}: $CIG = (V, E)$, where $E = \\{(v_i, v_j, w^*, d) | v_i, v_j \\in V, w^* \\in \\mathbb{R}^+\\}$ and $V$ is the set of nodes.\nIn total, we have created 79 CIGs, 57 for Ethereum and 22 for EOSIO.\n\n\\subsection{Number of Traces}\n\\label{sec:blockchain-evolution:cig:traces}\nFig.~\\ref{fig:traces-cig} depicts the number of $T_c$ and its corresponding ratio across time.\nAll these four lines fluctuate wildly.\nFor Ethereum, except for the spike in Oct. 2016 caused by the DoS attack which led to the number of $T_c$ exceeding 189 million, its absolute number shows a gradual upward trend.\nMoreover, the growth of the ratio is dramatic: from the launch of Ethereum to Mar. 2020, the ratio grew from 9.60\\% to 84.69\\%, thereby starting to dominate Ethereum's network. \n\nAs for EOSIO, its absolute number was also rising and was about an order of magnitude more than that of Ethereum, reaching almost two orders of magnitude after Nov. 2019. However, the trends between its ratio and absolute numbers are inconsistent. To be specific, we can see an obvious spike on Jul. 2018 that resulted from two accounts: \\texttt{blocktwitter}\nand \\texttt{chaintwitter}, which imported more than 64.9M and 21.0M $T_c$. \nThen, during late-2018, a gambling DApp led to frequent invocations between accounts, thereby resulting in a trough during this period. Since then, the ratio was continuously rising till Nov. 2019, since when the EIDOS's mechanism almost fixes the ratio to 33.3\\%.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/traces-cig.pdf}}\n\\caption{The evolution of contract invocation traces.}\n\\label{fig:traces-cig}\n\\end{figure}\n\n\nThe role of invokers who initiated $T_c$ is also interesting, as shown in Fig.~\\ref{fig:percentage-cig}.\nThe DoS attack in Ethereum (Oct. 2016) is mainly initiated by smart contracts. \nSince then, smart contracts increasingly invoke other smart contracts.\nBut in EOSIO, a different tendency is observed: users in EOSIO preferred to invoke smart contracts from their own accounts. In May 2019, 41.42\\% of $T_c$ was initiated by regular accounts. \nThe reason is unclear, however, from Nov. 2019, EIDOS started to dominate the ecosystem, resulting in the percentage of smart-contract-invoking invocations rising to 98.27\\%.\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-cig-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:percentage-cig-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.8\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/percentage-cig-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:percentage-cig-eos}\n     \\end{subfigure}\n        \\caption{The distribution of initiators of $T_c$.}\n        \\label{fig:percentage-cig}\n\\end{figure}\n\n\\textbf{\\textit{Insight: }}\n\\textit{The number of smart contract invocations has been increasing across both platforms, confirming their growing use.\nHowever, the number and the corresponding ratio were inconsistent in EOSIO, as they were significantly affected by gambling DApps and EIDOS.}\n\n\n\\subsection{Degree}\n\\label{sec:blockchain-evolution:cig:degree}\nFig.~\\ref{fig:alpha-cig} illustrates the degree distribution.\nInterestingly, the $\\alpha$ of outdegree distribution of EOSIO is overall higher than that of Ethereum. The $\\alpha$ of the indegree distribution, however, is on the opposite position. It indicates that, \\textit{most accounts in Ethereum invoke fewer contracts than accounts in EOSIO, but there exist some \\textit{super nodes} that accept most of the $T_c$ in Ethereum.} \nMoreover, in EOSIO, the $\\alpha$ of the outdegree distribution is almost higher than the indegree's all the time. This illustrates the existence of some killer DApps that invokes many $T_c$.\nThe significant upward trend in 2020 in Fig.~\\ref{fig:alpha-cig-eos} reflects that the initiation of $T_c$ became more centralized. Recall the mechanism of EIDOS we briefly introduced in \\S\\ref{sec:blockchain-evolution:mtg:traces}, which initiates lots of EIDOS token transfers to benefit users. Alone, it accounted for 92.2\\% of $T_c$ in Nov. 2019, which indicates the tendency towards centralization in EOSIO.\nAdditionally, in Ethereum, for the $\\alpha$ of the outdegree distribution, the trough during the whole 2018 in Fig.~\\ref{fig:alpha-cig-eth} is highly conspicuous. The reason is the emergence of some popular DeFi DApps who attract users to invoke them; this finally makes the distribution of outdegree less centralized.\n\n\n\\begin{figure}[t]\n     \\centering\n     \\begin{subfigure}[t]{0.63\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-cig-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:alpha-cig-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.36\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/alpha-cig-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:alpha-cig-eos}\n     \\end{subfigure}\n        \\caption{$\\alpha$ of degree\/indegree\/outdegree distribution of blockchain platforms over time in terms of $CIG$.}\n        \\label{fig:alpha-cig}\n\\end{figure}\n\n\\textbf{\\textit{Insight: }}\n\\textit{The $\\alpha$ of the outdegree\/indegree distributions in EOSIO and Ethereum follow opposite trends. This reflects the centralization of the outdegree in EOSIO, caused by popular gambling DApps and mis-behaviors; and the centralization of the indegree in Ethereum due to the emergence of DeFi DApps.\n}\n\n\n\\subsection{Pearson Coefficient ($R$)}\n\\label{sec:blockchain-evolution:cig:pearson}\nFig.~\\ref{fig:pearson-cig} gives the $R$ between indegree and outdegree in terms of contract invocation.\nWe see there was little correlation between the indegree and outdegree in the $CIG_{eth}$ and $CIG_{eos}$, and there never existed a negative correlation.\nThat said, in some months, the value of $R$ rises sharply. This is caused by popular DApps or malicious behaviors; these have such a large indegree and outdegree that they are able to affect the entire network.\nFor example, in Aug. 2018, a gambling game called \\textit{Last Winner} launched in Ethereum; and in Aug. 2019, a popular DApp \\textit{pornhashbaby} appeared in EOSIO. The former resulted in 876K indegree and 311K outdegree, accounting for more than 45\\% gambling related $T_c$ in $CIG_{eth}^{2018-08}$.\n\n\\begin{figure}[t]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/pearson-cig.pdf}}\n\\caption{Pearson's correlation between indegree and outdegree of blockchain platforms over time in $CIG$.}\n\\label{fig:pearson-cig}\n\\end{figure}\n\n\\textbf{\\textit{Insight: }}\n\\textit{The indegree and outdegree of nodes in both platforms had little correlation in terms of contract invocation. However, this metric can be severely impacted by misbehaviors or popular DApps.\n}\n\n\\subsection{Connected Components}\n\\label{sec:blockchain-evolution:cig:cc}\nFig.~\\ref{fig:cc-cig} shows the number of WCCs and SCCs in Ethereum and EOSIO in $CIG$s.\nThese have been gradually increasing. \nThe peak on Jan. 2018 in Ethereum's WCC is obvious, caused by the sudden decrease of users in Bittrex due to a hack event~\\cite{bittrex-hack}.\nMoreover, the number of WCCs in EOSIO is nearly two orders of magnitude smaller than Ethereum's.\nCombining Fig.~\\ref{fig:traces-cig} and Fig.~\\ref{fig:alpha-cig-eos}, we conclude that \\textit{users on EOSIO were calling contracts more frequently}.\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/cc-cig.pdf}}\n\\caption{The number of WCC and SCC in Ethereum and EOSIO over time in terms of $CIG$}\n\\label{fig:cc-cig}\n\\end{figure}\n\nBesides, some accounts initiated a large number of $T_c$, \nwhich covered numerous contracts and kept the number of WCC at a low level. The maximum WCC has covered at least 99.24\\% of nodes in the corresponding months. However, for SCC, we can conclude that the callback from the smart contract is unusual. This may be due to two reasons: 1) the initiating account is a regular account, thus it is unable to callback by a $T_c$; and 2) the callback trace just sends some native tokens, which is classified as a $T_m$. \nNote that there is a missing point of \\#SCC in Nov. 2016. This is because Ethereum officially deployed a sweeper contract\\footnote{0xa43ebd8939d8328f5858119a3fb65f65c864c6dd} that initiated more than 16 million one-way $T_c$ to remove all the redundant and meaningless contracts created by the DoS attack in October. This resulted in a large number of SCCs ($>$16M).\n\n\\textbf{\\textit{Insight: }}\n\\textit{\nThe growing number of accounts, indicate smart contracts are gaining increasing popularity.\n}\n\n\\section{General Overview of the Evolution}\n\\label{sec:overview}\n\n\n\n\\haoyu{may provide the general overview here. e.g., the number of overall TX, account, smart contract, etc.} \\haoyu{we can also calculate the TPS of the three chains and provide an evolution graph}\n\n\\haoyu{then, we say we will further provide detailed analysis of the TX from three different perspectives in the following section}\n\n\\haoyu{can we merge the graph construction section and the graph analysis section?}\n\\fi\n\n\n\n\n\n\n\n\n\n\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{DApp Ecosystem}\n\\label{sec:dapp}\nSmart contract has been first used in blockchain in Ethereum, in which the ability of interacting among smart contracts makes the appearance of DApp possible. Moreover, the resource model of EOSIO encourages the emergence of gambling and DeFi DApp that requires frequent money transfer or contract invocation.\nTherefore, based on the $MTG$ and $CIG$ for both above platforms and the DApp information we collected (see \\S\\ref{sec:study-design:data-collection}), we extracted all the DApp-related traces for each month.\nSpecifically, if the $label$ attribute of either one of the two nodes of a trace contains a DApp's information, we regard this trace as a DApp-related trace and classify it into the corresponding category by $label$.\nThen, we utilized these DApp-related traces to construct a series of $MTG$ and $CIG$ like the methodology we adopted in \\S\\ref{sec:blockchain-evolution}.\nIn this section, we would first give an overview of the DApp information we collected. Then, we would picture the overall ecosystem of DApps in Ethereum and EOSIO. Finally, we will pick out the DApp-related traces belonging to the two most representative categories: gambling and DeFi, to explore some interesting findings.\n\n\\subsection{DApp Overview}\n\\label{sec:dapp:overview}\nAs we mentioned in \\S\\ref{sec:study-design:data-collection}, we have collected the largest-ever dataset on DApp information on both Ethereum and EOSIO. Moreover, we made a great effort on verifying the correctness of the DApps belonging categories.\nTable~\\ref{table:dapp} shows brief descriptions of the categories which we have adopted, as well as the number of the corresponding DApps.\n\n\\begin{table}[tbp]\n\\caption{The description of categories, as well as the number of corresponding DApps.}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{llcc}\n\\toprule\n\\textbf{Category}  & \\textbf{Desc.} & \\textbf{\\#Ethereum (\\#Addr)} & \\textbf{\\#EOSIO (\\#Addr)} \\\\\n\\midrule\n\\textbf{DeFi}      & Financial products that directly build on blockchain technology     & 73 (4,386)                   & 8 (23)                    \\\\\n\\textbf{Exchange}  & A palace allows user to exchange fiat currency and cryptocurrency   & 162 (328)                    & 40 (96)                   \\\\\n\\textbf{Finance}   & DApps that are related to financial product                         & 95 (316)                     & 10 (19)                   \\\\\n\\textbf{Gambling}  & It requires players to invest tokens and may return more            & 377 (1,107)                  & 372 (1,052)               \\\\\n\\textbf{Game}      & All non-gambling games, e.g., collectible card games                & 1,109 (5,126)                & 97 (163)                  \\\\\n\\textbf{High-Risk} & Applications that are often characterized by Ponzi schemes          & 335 (453)                    & 47 (68)                   \\\\\n\\textbf{Platform}  & On which users can develop or deploy other applications or services & 72 (158)                     & 16 (34)                   \\\\\n\\textbf{Social}    & Social applications (like dating) that adopt blockchain technology  & 76 (134)                     & 20 (31)                   \\\\\n\\textbf{Token}     & Unofficial token issued on the blockchain platform                  & 854 (909)                    & 3 (4)                     \\\\\n\\textbf{Tool}      & Provide users with convenient and efficient tools                   & 139 (301)                    & 38 (65)                   \\\\\n\\textbf{EIDOS}     & A malicious but phenomenal DApp in EOSIO                            & -                            & 1 (1)                    \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{table:dapp}\n\\end{table}\n\nAs we can see from Table~\\ref{table:dapp}, in EOSIO, the number of gambling DApp has absolutely dominated the EOSIO's DApp, and it was also popular in Ethereum. However, game DApps in Ethereum has occupied larger share, around 38.78\\% addresses are related to game DApps. Moreover, 73 DeFi DApps in Ethereum has accounted for about one third of DApp addresses. We speculate that the reason for this phenomenon is that decentralized exchange needs different smart contracts to act as different trading pairs. For example, Uniswap~\\cite{}, a well-known decentralized exchange, has 410 contract address according to our dataset.\nLast but not least, we can not omit the EIDOS DApp in EOSIO, which receives EOS tokens transferred from any accounts, and then immediately returns all EOS tokens and attaches a certain amount (independent of the number of the transferred EOS tokens) of EIDOS tokens. Since EIDOS tokens can be circulated on the secondary market and have a certain value, users can transfer EOS to this type of DApps at a very high frequency and get profit with no cost. To this end, EIDOS has caused huge amount of spam transactions, which will be detailed in \\S\\ref{sec:dapp:evolution} and \\S\\ref{sec:abnormal-behaviors:case-study:EIDOS}.\n\n\n\\subsection{Overall Evolution}\n\\label{sec:dapp:evolution}\nAfter extracting all the DApp-related traces, we analyzed the percentage of $T_m$ and $T_c$ belonging to different DApp categories, which are shown in Fig.~\\ref{fig:dapp-mtg} and Fig.~\\ref{fig:dapp-cig}, respectively.\n\n\\subsubsection{DApp's Money Transfer}\n\\label{sec:dapp:evolution:mtg}\nFrom the perspective of money transfer, we can easily observe from Fig.~\\ref{fig:dapp-mtg-eos} that gambling DApp dominates EOSIO's DApp ecosystem overwhelmingly. To be specific, from the third month of its very beginning, i.e., August, 2018, the gambling DApps were continuously accounting for more than 90\\% of $T_m$ in the corresponding months. Astonishingly, the number has reached up to the highest point in 2018-11: 99.01\\%.\nHowever, from 2019-11, the EIDOS began to dominate the whole ecosystem by an overwhelming margin, around 99.39\\% in November 2019.\nExcept for these two categories, we can see exchanges also play an important role in EOSIO's ecosystem. Prior to 2019-11, exchange-related transfers began to become more active, which indicates users began to buy and sell EOS tokens more frequently. According to our statistics, the most active exchanges are: DEXEOS~\\cite{}, WhaleEx~\\cite{}, Findex~\\cite{} and Newdex~\\cite{}.\nAlso, we can see 79.51\\% of $T_m$ was related to game DApps only in 2018-07. After a comprehensive analysis, we found it was caused by a game DApp called PumpDumpWars~\\cite{}, \\he{which is a xxx}.\n\nFrom Fig.~\\ref{fig:dapp-mtg-eth}, we can see that platform DApps (a mining pool called Dawrfpool~\\cite{}) took up all the initial market share till mid-2017. However, its absolute number was actually relatively small. According to the statistics, it had took up at most 20.22\\% of all $T_m$, which happened in 2016-02 (see Fig.~\\ref{fig:no-dapp}).\nIn addition, we can see a spike of tool DApps in mid-2017. It is a DApp called Ethereum Name Service (ENS)~\\cite{}, which provides services that exchange between the abstract Ethereum addresses to human-readable strings.\nMoreover, two obvious spikes related to game and gambling DApps in December 2017 and August 2018 can be easily noticed, which are resulted from the prosperity of CryptoKitties~\\cite{} and Last Winner~\\cite{}.\nDeFi has started to flourish from the beginning of 2018, and became more and more important. At meanwhile, gambling DApps also started to capture the market of game DApps. We think the reason may be the popularity of Fomo3D~\\cite{}, a phenomenal gambling DApp in Ethereum at mid-2018, and prosperity of gambling DApps on EOSIO platform, which both contributed to the prevalence of gambling DApps.\n\nMoreover, Fig.~\\ref{fig:no-dapp} illustrates the percentage of non-DApp-related traces in corresponding months, which shows a significant difference between DApp ecosystems of Ethereum and EOSIO.\nWe can easily observe that more than three quarters of $T_m$ in EOSIO was related to DApps at most time. But instead in Ethereum, less than one quarter of $T_m$ was related to DApps.\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/dapp-mtg-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:dapp-mtg-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/dapp-mtg-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:dapp-mtg-eos}\n     \\end{subfigure}\n        \\caption{Percentage of different categories of DApps in terms of money transferring.}\n        \\label{fig:dapp-mtg}\n\\end{figure}\n\n\\textbf{\\textit{Insight:}} In terms of DApp-related $T_m$, gambling DApps nearly dominated the EOSIO's DApp ecosystem until the appearance of EIDOS. Along with the prosperity of gambling DApps, exchanges also play a vital role in EOSIO. Moreover, Except for the beginning four months, no less than 80.64\\% of $T_m$ was related to kinds of DApps.\nCompared to EOSIO, the ecosystem of Ethereum's DApps was less active. However, there did not exist a certain category of DApps that dominated a period within Ethereum. In other words, The rise and fall of percentage of various categories of DApps reflected the diversity of this ecosystem.\n\n\n\\subsubsection{DApp's Contract Invocation}\n\\label{sec:dapp:evolution:cig}\nFig.~\\ref{fig:dapp-cig} reveals the contract invocation behavior related to DApps on both platforms. \nSpecifically, Fig.~\\ref{fig:dapp-cig-eos} reveals the DApp-related $T_c$ in EOSIO. Compared with the money transfer in EOSIO (see Fig.~\\ref{fig:dapp-mtg-eos}), the biggest distinction is the appearance of traces related to social and platform DApps in 2018-07 and almost whole year of 2019, respectively. According to our statistics, we figured out the social DApp are ChallengeDapp~\\cite{} and KARMA~\\cite{}, and the platform DApp is pornhashbaby~\\cite{}, which provides the pornography storing and sharing service but was shut down in November 2019.\n\nSimilarly, the DApp-related $T_c$ in Ethereum (see Fig.~\\ref{fig:dapp-cig-eth}) are more complicated than the money transfer's.\nWe can easily observe that the number of $T_c$ categorized as token DApps increased suddenly. That is because we regard all the unofficial token transfers (see category \\textbf{Token} in \\S\\ref{sec:dapp:overview}) in Ethereum as contract invocation.\nExcept the emergence of token DApps, the DeFi has been capturing more and more market. As DeFi requires frequently interactions between DeFi DApps, which pushed the number higher than ever.\nMoreover, we can observe that the gambling-related $T_c$ has occupied lots of shares which original belonged to platform DApps (mainly Dawrfpool) before mid-2017. We think the reason is these gambling DApps accepted unofficial tokens as betting from players, which would only increase the number of $T_c$.\nAdditionally, we can obtain some abnormal spikes: `tool' in 2017-05, `gambling' in 2019-09, and `game' in 2019-11, which are caused by ENS, Fairwin~\\cite{} (a well-known but transitory gambling DApp), and Gods Unchained TCG~\\cite{} (a collectable trading cards game, like the CryptoKitties), respectively.\n\nTalking about the percentage of non-DApp $T_c$, it has completely different characteristics from that of money transfer. \nSpecifically, in Ethereum, the percent of non-DApp $T_c$ has dropped to nearly 50\\% compared with 80\\% of non-DApp $T_m$ at the same time period. For EOSIO, however, the opposite is true.\nThe main reason for this phenomenon we believe is that traces related to unofficial tokens transfer is more active in Ethereum compared to Ether, while the opposite is true in EOSIO.\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/dapp-cig-eth.pdf}\n         \\caption{Ethereum}\n         \\label{fig:dapp-cig-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.49\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/dapp-cig-eos.pdf}\n         \\caption{EOSIO}\n         \\label{fig:dapp-cig-eos}\n     \\end{subfigure}\n        \\caption{Percentage of different categories of DApps in terms of contract invocation.}\n        \\label{fig:dapp-cig}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centerline{\\includegraphics[width=0.7\\columnwidth]{images\/no-dapp.pdf}}\n\\caption{Percentage of non-DApp money transferring and contract invocation traces in Ethereum and EOSIO.}\n\\label{fig:no-dapp}\n\\end{figure}\n\n\n\\textbf{\\textit{Insight:}} In EOSIO's DApp ecosystem, the characteristics of $T_c$ is similar to that of $T_m$. However, the pornography platform occupied a certain share till its close in November. Comparing with $T_m$, the percentage of DApp-related $T_c$ is a little bit lower, but still higher than that of Ethereum.\nEthereum still illustrated the diversity mentioned in \\S\\ref{sec:dapp:evolution:mtg}. However, the tokens' related transfer has accounted for a large proportion, and DeFi has been dominating the ecosystem, reaching 66.38\\% in March 2020. Moreover, the game DApps, which initiated and received lots of $T_c$ to perform the game logic, can not be neglected from the late-2017 till the time of writing.\n\n\n\\subsection{Gambling \\& Defi DApps}\n\\label{sec:dapp:defi-gambling}\nWe decided to extract the sub-figures of $MTG$ and $CIG$ that are composed of all the traces related to gambling and DeFi DApps respectively to delve deeper for these two representative categories.\n\n\\subsubsection{Gambling Ecosystem}\n\\label{sec:dapp:defi-gambling:gambling}\nGambling DApps has emerged since the very beginning of these two platforms and always holds a certain of shares of DApp ecosystems, especially in EOSIO (see \\S\\ref{sec:dapp:evolution}). Therefore, we firstly characterized sub-graphs related to gambling DApps.\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-basic-mfg-node.pdf}\n         \\caption{$MTG$}\n         \\label{fig:gambling-basic-mfg-node}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-basic-cig-node.pdf}\n         \\caption{$CIG$}\n         \\label{fig:gambling-basic-cig-node}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-basic-mfg-trace.pdf}\n         \\caption{$MTG$}\n         \\label{fig:gambling-basic-mfg-trace}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-basic-cig-trace.pdf}\n         \\caption{$CIG$}\n         \\label{fig:gambling-basic-cig-trace}\n     \\end{subfigure}\n        \\caption{Number of nodes and traces within sub-$MTG$ and sub-$CIG$ that are related to gambling DApps.}\n        \\label{fig:gambling-basic}\n\\end{figure}\n\nFig.~\\ref{fig:gambling-basic} shows the number of nodes included in each months' gambling-related $MTG$ and $CIG$, as well as the number of traces. \nWe can see that gambling DApps were much more popular in EOSIO in terms of both absolute number and relative percentage at least prior to November 2019.\nTo be specific, though gambling DApps has emerged earlier in Ethereum and held a certain of market (see Fig.~\\ref{fig:dapp-cig-eth}), there was only 10.84\\% of $T_c$ in April 2017 related to the gambling DApps as shown in Fig.~\\ref{fig:gambling-basic-cig-trace}, which is the maximum point in the earlier stage of Ethereum.\nIn late-2018, gambling DApps related $T_c$ has occupied up to 88.53\\% in EOSIO in October 2018. The percentage for $T_m$ went even higher, reached up to 94.67\\% at the same month shown in Fig.~\\ref{fig:gambling-basic-mfg-trace}.\nInterestingly, we can see three strange spikes in Fig.~\\ref{fig:gambling-basic-mfg-node}: August 2018 and May 2019 in Ethereum, April and June in 2019 in EOSIO. For Ethereum, the reason lies behind were the Last Winner~\\cite{} and CoinGathernator~\\cite{}.\nHowever, in EOSIO, it is because the \\textit{spam advertisements} initiated from some gambling DApps. To be specific, the content of advertisements was embedded in the transaction in which it carries a tiny amount of EOS to attract the players' attention. We can see from Fig.~\\ref{fig:gambling-basic-mfg-trace}, the few pieces of advertisement did not have ability to change the whole ecosystem but covered lots of users in that month, i.e., 61.53\\% and 73.95\\% of users who included in $MTG_{eos}^{2019-04}$ and $MTG_{eos}^{2019-06}$, respectively.\n\n \\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-alpha-mfg.pdf}\n         \\caption{$MTG$}\n         \\label{fig:gambling-alpha-mfg}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/gambling-alpha-cig.pdf}\n         \\caption{$CIG$}\n         \\label{fig:gambling-alpha-cig}\n     \\end{subfigure}\n        \\caption{$\\alpha$ of indegree\/outdegree distribution within sub-$MTG$ and sub-$CIG$ that are related to gambling DApps. Blue lines are for Ethereum, and red lines are for EOSIO.}\n        \\label{fig:gambling-alpha}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\centerline{\\includegraphics[width=0.5\\columnwidth]{images\/gambling-pearson-eth.pdf}}\n\\caption{R of indegree and outdegree distribution in Ethereum related to gambling DApps.}\n\\label{fig:gambling-pearson-eth}\n\\end{figure}\n\nWhen we delve deeper to investigate the change of $\\alpha$ of indegree\/outdegree distributions, we can easily observe from Fig.~\\ref{fig:gambling-alpha-cig} that the $\\alpha$ of outdegree distribution in Ethereum is much more higher than that of indegree distribution. Combining with the relative small number of $T_c$ in gambling DApps (see Fig.~\\ref{fig:gambling-basic-cig-trace}), it is easy to draw conclusion: a few small number of gambling DApps in Ethereum has received almost all the gambling related $T_c$, and the players are dispersed among the Ethereum which leads to such a small $\\alpha$ of outdegree distribution.\nMoreover, Fig.~\\ref{fig:gambling-pearson-eth} depicts the R between indegree and outdegree distribution in Ethereum in terms of gambling DApps. During the most active period of gambling DApps in Ethereum, i.e., mid-2017, the R of $CIG$ even shows a negative correlation, which indicates that those gambling DApps who received lots of $T_c$ hardly initiated another $T_c$.\n\nRecall the appearance of Fairwin in September 2019 which leads to conspicuous spikes in Fig.~\\ref{fig:dapp-cig-eth} and Fig.~\\ref{fig:gambling-alpha-cig}. Within our analysis, the percentage of $T_c$ has reached up to 25.81\\%, while the value of $\\alpha$ of indegree distribution was -4.988. Moreover, the R of $CIG$ and $MTG$ were opposite, which are -0.078 and 0.997, respectively.\nThese values jointly indicates this gambling DApp in Ethereum centralized more than one quarter of $T_c$ in that month. However, the number of scattered players are so high (see the spike in Fig.~\\ref{fig:gambling-basic-mfg-node} and Fig.~\\ref{fig:gambling-basic-cig-node}) to pull the $\\alpha$ down. In addition, the two values of R indicate that Fairwin did not initiated much $T_c$ but a high number of $T_m$.\n\n\\textbf{\\textit{Insight:}} Gambling DApps dominated the EOSIO ecosystem, though some of the high coverage are due to the spam advertisements behavior. Contrary to the long-term prosperity of gambling DApps in EOSIO, Ethereum's popular ones are more in a transitory situation. In other words, some killer gambling DApps emerged and attracted lots of attention from players in Ethereum, but they disappeared suddenly.\n\n\\subsubsection{DeFi Ecosystem}\n\\label{sec:dapp:defi-gambling:defi}\nDeFi, as the most popular category of DApps now, has attracted lots of investors, as well as the malicious users. For example, Lendf.me, a famous lending platform based on decentralized finance technology, has been attack in April 2020, which leads to more than 25 million USD financial loss~\\cite{}.\nMoreover, from Fig.~\\ref{fig:dapp-mtg-eth} and Fig.~\\ref{fig:dapp-cig-eth}, we can easily observe the liveness of DeFi DApps. Thus, in this section, we would explore the characteristics of DeFi DApps in Ethereum and EOSIO.\n\n\\begin{figure}\n     \\centering\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/defi-basic-mfg-node-scale.pdf}\n         \\caption{$MTG$}\n         \\label{fig:defi-basic-mfg-node}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/defi-basic-cig-node.pdf}\n         \\caption{$CIG$}\n         \\label{fig:defi-basic-cig-node}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/defi-basic-mfg-trace-scale.pdf}\n         \\caption{$MTG$}\n         \\label{fig:defi-basic-mfg-trace}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.45\\textwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/defi-basic-cig-trace.pdf}\n         \\caption{$CIG$}\n         \\label{fig:defi-basic-cig-trace}\n     \\end{subfigure}\n        \\caption{Number of nodes and traces within sub-$MTG$ and sub-$CIG$ that are related to DeFi DApps.}\n        \\label{fig:defi-basic}\n\\end{figure}\n\nFig.~\\ref{fig:defi-basic} shows the number of related nodes, $T_m$, and $T_c$ in $MTG$ and $CIG$.\nWe can easily observe that, DeFi related traces are more classified into the $CIG$.\nAs we have observed in Fig.~\\ref{fig:dapp-mtg-eos} and Fig.~\\ref{fig:dapp-cig-eos}, DeFi was account for a tiny fraction of $T_m$ and $T_c$ in EOSIO: less than 1\\% for both types of graphs. However, we can still see three conspicuous spikes in Fig.~\\ref{fig:defi-basic-mfg-node} in 2018-11, 2019-05 and 2019-11, respectively. After the manual analysis, we found that the first two spikes were caused by Chintai~\\cite{} and EOS REX~\\cite{} (resource exchange platform in EOSIO), respectively. However, for November 2019, the number of nodes 7,366 did not increase too many compared to the preceding month (7,324 nodes in October 2019). Therefore, we think this spike is due to the congestion resulted by EIDOS.\nAs for Ethereum, steep growth trends in both $MTG$ and $CIG$ can be observed. Except the trough in late-2019, which is caused by the decline of coverage of USDT (a well-known stable coin in Ethereum), DeFi has accounted for 34.62\\% of $T_c$ in March 2020 and still kept an upward tendency. Compared to contract invocation, DeFi led to less money (Ether) transfer, however, it still account for about 7\\% of $T_m$ at the same time.\n\n\\begin{figure}[tbp]\n\\centerline{\\includegraphics[width=0.5\\columnwidth]{images\/defi-alpha-cig.pdf}}\n\\caption{$\\alpha$ of indegree\/outdegree distribution within sub-$CIG$ that are related to DeFi DApps.}\n\\label{fig:defi-alpha-cig}\n\\end{figure}\n\nAs the DeFi DApps are mainly interacted with each other in $CIG$, we only measured the $\\alpha$ of indegree\/outdegree distribution in $CIG$ for both platforms as shown in Fig.~\\ref{fig:defi-alpha-cig}.\nThe point in 2019-01 is obvious, the $\\alpha$ of outdegree and indegree distribution moved to opposite directions simultaneously. We think the reason was the appearance of a DeFi called Chintai~\\cite{}, which is a platform where users can lend their EOS on the market to earn interest.\nMoreover, for both platforms, we can observe an interesting phenomenon: the $\\alpha$ of indegree distribution nearly always higher than that of outdegree distribution. This indicates there were several head DeFi DApps that absorbed users' contract invocation requests.\nAccording to our statistics, the most 10 popular DeFi DApps in Ethereum's history and the number of its received $T_c$, as well as the covered users, are shown in Table~\\ref{table:top10-defi}.\nWe can see that, 4 out of top 5 are the stable coins, which provide a relative stable price by some financial measures. The most widely used stable coin is USDT~\\cite{}, which has more than 3 million users who have invoked more than 4.3 million pieces of $T_c$.\nMore than stable coins, exchanges and money markets also appeared in the top 10. Specifically, exchanges here refer to decentralized exchanges (DEX), which utilized smart contracts to automatically perform matchmaking. However, several blogs~\\cite{} and previous works~\\cite{} have reported and studied the vulnerability existed in DEXes. As for money market, it provides a platform to allow user lends and loans cryptocurrency with it.\n\n\\begin{table}[tbp]\n\\caption{Top 10 DeFi DApps in Ethereum and its corresponding number of $T_c$ and nodes.}\n\\centering\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{ccccccccccc}\n\\toprule\n                   & \\textbf{USDT} & \\textbf{USDC} & \\textbf{IDEX} & \\textbf{PAX} & \\textbf{DAI} & \\textbf{Oasis} & \\textbf{CDP Portal} & \\textbf{Uniswap} & \\textbf{TUSD} & \\textbf{Bancor} \\\\\n\\midrule\n\\textbf{Category}  & \\begin{tabular}[c]{@{}c@{}}Stable\\\\ Coin\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Stable\\\\ Coin\\end{tabular} & Exchanges     & \\begin{tabular}[c]{@{}c@{}}Stable\\\\ Coin\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Stable\\\\ Coin\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Money\\\\ Market\\end{tabular} & \\begin{tabular}[c]{@{}c@{}}Money\\\\ Market\\end{tabular} & Exchanges        & \\begin{tabular}[c]{@{}c@{}}Stable\\\\ Coin\\end{tabular} & Exchanges       \\\\\n\\textbf{\\# $T_c$} & 4,335,452     & 624,793       & 603,680       & 451,442      & 406,847      & 261,888        & 261,423             & 148,544          & 146,433       & 121,159         \\\\\n\\textbf{\\# Nodes}  & 3,074,449     & 508,592       & 273,815       & 407,656      & 301,911      & 106,761        & 122,859             & 41,131           & 105,198       & 34,501         \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{table:top10-defi}\n\\end{table}\n\n\n\\begin{figure}[tbp]\n\\centerline{\\includegraphics[width=0.5\\columnwidth]{images\/defi-cc-cig.pdf}}\n\\caption{The number of WCC and SCC within $CIG$ that are related to DeFi DApps for both platforms.}\n\\label{fig:defi-cc-cig}\n\\end{figure}\n\nFig.~\\ref{fig:defi-cc-cig} depicts the number of WCC and SCC related to the DeFi DApps in $CIG$. We can observe that the numbers of SCC in Ethereum were overall higher than EOSIO's. We think this must be due to the frequent interaction between users and DeFi DApps, as the corresponding numbers of WCC were only fluctuating around a relative low level.\nThe three spikes in the number of SCC in EOSIO exactly corresponds with the events mentioned in Fig.~\\ref{fig:defi-basic-mfg-node}.\n\n\\textbf{\\textit{Insight:}} DeFi in Ethereum has been being under a flourishing development, some of head DeFi DApps has attracted lots of users who initiate $T_c$ to interact with them. At meanwhile, it did not be developed well in EOSIO, in which only the unabiding DeFi DApps emerged.\n\n\n\\subsection{Summary}\n\\he{high level summary for this whole section, TBD}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\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{Characterizing Outliers}\n\\label{sec:abnormal-behaviors}\n\n\n\\subsection{Definition of Outlier}\n\nOur previous exploration suggests that some events (e.g., DoS attacks) and influential accounts have a large impact on ecosystems. \nIn this section, we explore such events and addresses. \\textit{Note our outlier analysis is not intended to identify attacks, as many attack events may not cause noticeable changes to the overall ecosystems.} Instead, our purpose is to pinpoint what kinds of events\/DApps can facilitate or impede the development of the blockchain ecosystems.\nFor all metrics considered, some data points deviate significantly from the average~\\cite{outlier}. We refer to these points as \\textit{outliers}. \n\n\n\\subsection{Detecting and Labelling Outliers}\n\\label{sec:abnormal-behaviors:method}\n\n\n\\subsubsection{Detecting Outliers.} \nWe use a simple but efficient algorithm, \\textit{z-score}~\\cite{z-score}, to flag outliers in each time series. Z-scores quantify the unusualness of an observation. Z-scores are the number of standard deviations above and below the mean that each value falls. Specifically, if the value of a metric for a month is $x$, the z-score would be: $z=\\frac{x-\\mu}{\\sigma}$, where $\\mu$ is the average of $x$ and corresponding values before and after three months (7 values in total), and $\\sigma$ is the standard deviation for these 7 values.\nTherefore, the absolute value of the z-score represents the distance between the $x$ and its adjacent elements mean in units standard deviation.\n\n\\subsubsection{Labelling Outlier}\n\\label{sec:abnormal-behaviors:detecting:labelling}\nWe propose a semi-automated method to label outliers.\nOur idea is that \\textit{outliers result from suddenly popular DApps or misbehaviors}. Thus, removing the responsible node(s) can make the absolute value of z-score smaller than a threshold.\nFor each metric, we have different strategies to identify the responsible node(s).\n\nSpecifically, (1) for the number of trace metric, the node with the highest degree (named the \\textit{supernode}) in the corresponding graph will be identified; \n(2) for $\\alpha$ of the degree distribution, there are two cases: trough or peak, corresponding to dispersion and centralization, respectively. Therefore, for the former, the strategy tries to identify the node(s) who import many nodes with a small degree; for the latter, the supernode with the highest degree\/indegree\/outdegree will be labeled;\n(3) based on our extensive manual investigation, a node with high indegree and outdegree simultaneously is likely to push the $R$ higher. Therefore, the strategy focuses on the node with the highest product of indegree and outdegree;\n(4) the significant change in the number of connected components is always related to the (dis)appearance of nodes that are directly connected to lots of nodes in the network. To this end, the nodes with the most one-way or two-way edges with the others will be identified.\n\nAfter the node(s) are identified according to the above strategy, it and its connected edges in the corresponding graph will be removed. We then recalculate the corresponding metric to see if the absolute value of z-score drops lower than a threshold. If not, the above process is repeated until the outlier disappears.\nTo guarantee accuracy, we manually recheck all removed nodes to confirm they are the actual responsible ones.\nIn other words, for each outlier, \\textit{our method can extract a set of responsible nodes that are related to attack\/spam events or killer DApps.}\n\nWe classify the outliers into two major groups: \\textit{Killer DApps} and \\textit{Misbehaviors}, as shown in Table~\\ref{table:category-outliers}.\nThe Killer DApp category indicates the outlier is due to the activity of a popular DApp. \nThe misbehavior category is divided into three sub-categories: \\textit{Attack}, \\textit{Resource Manipulation}, and \\textit{Spam}. An attack means the outlier has been caused by a well-known attack, like the DAO reentrancy attack~\\cite{the-dao}. Resource manipulation indicates the outlier is caused by an exploitation on the resource model, like EIDOS (see \\S\\ref{sec:background:general:consensus}). The spam sub-category covers spam advertisements and bot accounts. \nNote that the z-score algorithm also flags some outliers in the infant stage of the blockchains. We omit these, as a few transactions can lead to a volatile swing on the value of a metric.\n\n\n\\subsection{Results}\n\\label{sec:abnormal-behaviors:overall}\n\n\n\\begin{table*}[t]\n\\caption{The overall result of outlier labelling.}\n\\centering\n\\resizebox{0.7\\textwidth}{!}{%\n\\begin{tabular}{l|ccccccc|ccc|c}\n\\toprule\t\n                  & \\multicolumn{7}{c|}{\\textbf{Killer DApp}}                                         & \\multicolumn{3}{c|}{\\textbf{Misbehavior}}                                                                 & \\multirow{2}{*}{\\textbf{Total}} \\\\ \\cline{2-11}\n                  & DeFi       & Exchange   & Gambling   & Game       & Platform   & Token      & Tool       & Attack     & \\begin{tabular}[c]{@{}c@{}}Resource\\\\Manipulation\\end{tabular} & Spam        &                                 \\\\ \\midrule\n\\textbf{Bitcoin}  & 0          & 0          & 2          & 2          & 5          & 0          & 0          & 3          & 0                                                                              & 1           & \\textbf{13}                      \\\\ \\midrule\n\\textbf{Ethereum} & 2          & 10          & 3          & 6          & 0          & 8          & 2          & 2          & 13                                                                             & 3           & \\textbf{49}                     \\\\ \\midrule\n\\textbf{EOSIO}    & 0          & 0          & 5          & 0          & 1          & 1          & 1          & 0          & 2                                                                              & 15          & \\textbf{25}                     \\\\ \\midrule\n\\textbf{Total}    & \\textbf{2} & \\textbf{10} & \\textbf{10} & \\textbf{8} & \\textbf{6} & \\textbf{9} & \\textbf{3} & \\textbf{5} & \\textbf{15}                                                                    & \\textbf{19} & \\textbf{87}                    \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\label{table:category-outliers}\n\\end{table*}\n\n\nTable~\\ref{table:category-outliers} shows the overall results (see Table~\\ref{table:anomalies-1} to Table~\\ref{table:anomalies-3} in Appendix for details), 87 outliers are identified in total: 13 for Bitcoin, 49 for Ethereum, and 25 for EOSIO. \nAs concluded before, Bitcoin is the most stable platform. Though EOSIO fluctuates in several plots, due to its relative short history, its number of outliers is below Ethereum's.\n\nAmong the 87 outliers, 45\\% (39) belongs to the misbehavior category. Moreover, most of them are in the resource manipulation and spam sub-categories. Ethereum suffers more from malicious exploitation on resources. The main reason is the DoS attack, which was so powerful that it significantly altered many metrics in Oct. 2016. Due to the resource model in EOSIO (see \\S\\ref{sec:background:general:consensus}), it is more likely to suffer from spam.\nThis is partly due to the EOS Global event, which created many useless accounts and transferred EOS between them. The spam advertisements in EOSIO also played an important role (detailed in \\S\\ref{sec:abnormal-behaviors:case-study:spam}).\nMoreover, we observe that, compared to EOSIO, Ethereum is impacted more by killer DApps. These apps are evenly distributed across categories. This likely indicates that Ethereum is still preferred by DApp developers. The only killer app in EOSIO is the pornhashbaby, which brought significant traffic until it was shutdown.\n\n\n\n\\subsection{Case Studies}\n\\label{sec:abnormal-behaviors:case-study}\nWe next briefly introduce two case studies, while the details can be found in Appendix \\S\\ref{sec:appendix:case-study}.\n\n\\subsubsection{EIDOS Event}\n\\label{sec:abnormal-behaviors:case-study:EIDOS}\nFrom Fig.~\\ref{fig:traces} we observe that, in Nov. 2019, the number of traces in EOSIO jumped by 1,009.82\\% compared to the preceding month (reaching 3.21 billion pieces). Surprisingly, over 99\\% of the transactions are related to the EIDOS contract. However, only a few hundred accounts interacted with the contract. This suggests that billions of transactions were introduced by a small number of addresses.\nWe further calculate the total volume of money transferred to the EIDOS contract, and the average EOS per $T_m$. After a slightly increase in Dec. 2019, both of these two metrics dropped significantly. In Mar. 2020, only 1.43 million EOS were transferred to EIDOS (around 6.53\\% of the peak). Meanwhile, the average EOS per $T_m$ decreased to $6 \\times 10^{-4}$, while $1 \\times 10^{-4}$ is the minimum allowed transfer amount in EOSIO. We conclude that players were becoming rational and centralized, and a number of services have emerged~\\cite{eidos-miner-1, eidos-miner-2} that allow participants to maximize benefits at minimal cost.\nEIDOS also overtook other DApps, e.g., previously popular gambling DApps, and has had a significant negative impact on the entire ecosystem.\n\n\\subsubsection{Spam Advertisement}\n\\label{sec:abnormal-behaviors:case-study:spam}\nDuring the analysis of Fig.~\\ref{fig:alpha-mtg-eos}, we have observed two strange spikes in the $\\alpha$ of the outdegree distribution (Mar. and Oct. 2019).\nWe discover that these two outliers are due to \\textit{spam advertisement}.\nSpammers have taken advantage of the resource model in EOSIO, which allows users to initialize spam advertisements to cover huge amounts of accounts; specifically, in the memo field~\\cite{eos-memo} of transactions, users can write use this to free-text. We see that spammer write things like bait-and-switch advertisements.\nUsing $MTG$ and $CIG$, we further propose an automated method to identify spammers (see Appendix \\S\\ref{sec:appendix:case-study:spam} for details). The key idea is that within a given time frame, the spammer would initiate spam advertisements carrying identical spam content by the lowest cost to cover as many accounts as possible. Thus, we can group together small transactions with identical content to flag spam candidates.\nThrough this, we identify \\textbf{206} distinct spam accounts in EOSIO, and we identify two account family: \\texttt{peostoken} and \\texttt{defind.io}, who are responsible for the spikes located in Mar. 2019 and Oct. 2019.\n\n\n\\section{Discussion}\n\n\n\\subsection{Cross-chain Comparison}\n\n\\noindent\n\\textbf{Bitcoin.} Bitcoin is a value-transfer network, and does not support features like smart contracts. Moreover, its simple but effective transaction fee mechanism guarantees that it is hard to be abused. To this end, Bitcoin has been in a steady growth and has been hard to be manipulate according to its number of transfer transactions and degree distribution.\n\n\n\\noindent\n\\textbf{Ethereum.} Ethereum is still evolving, especially in terms of contract invocation. Prior to 2018, the most dominant transaction in Ethereum is the Ether transfer. The price of Ether would always incentivize a more frequent and centralized Ether transfer between users and exchanges. Since then, the emergence of DeFi has increased the number of interactions between smart contracts. Along with the growing number of transactions, the relatively stable ratio of newly created smart contracts also implies the liveness in Ethereum.\n\n\\noindent\n\\textbf{EOSIO.}\nVia DPoS, EOSIO is able to carry out hundreds of times the traces than Bitcoin and Ethereum (who adopt PoW). \nPrior to Nov. 2019, though there was a slight decline in the number of transfers and contract invocations, the popularity of gambling DApps has attracted many users. However, the appearance of EIDOS almost collapsed the entire network: The TPS was pushed to its limit, and all other normal transactions suffered high congestion. \n\n\n\n\\subsection{Threats to Validity and Future Work}\nThis study carries several limitations.\nFirst, we primarily study the evolution of the ecosystems based on graph analysis and, by definition, the metrics used are limited. Although we have considered the most widely used metrics, more metrics (e.g., global clustering coefficient~\\cite{clustering}, page rank~\\cite{page-rank}, reciprocity~\\cite{reciprocity}) could be adopted to gain further understanding of ecosystems.\nSecond, in several cases, we have relied on heuristics and manual validation due to the paucity of ground-truth data. For example, during the outlier analysis, we have manually confirmed the underlying reasons for several cases. We agree that automated approaches (e.g., via machine learning) could be used to facilitate the process.\nThird, our study relies solely on transaction analysis. We believe additional code analysis (of the smart contracts) could help us to gain a better understanding of ecosystems.\nFinally, although some highly influential attacks and unreported scams (e.g., the scam advertisements in EOSIO) have been discovered by us, we do not consider a number of known attacks in the community (as they did not have a noticeable impact on the overall ecosystem). \nFor future work, we plan to address the above limitations.\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\\textbf{Blockchain Transaction Analysis.}\n\\label{sec:related:tx-analysis}\nSome previous works have focused on blockchain transaction analysis for a single blockchain platform, e.g., Bitcoin~\\cite{awan2017blockchain,ober2013structure, ron2013quantitative, houstudy}, Ethereum~\\cite{chen2018understanding, wu2019t, chen2020traveling, kiffer2018analyzing, lee2020measurements, zhao2021temporal}, EOSIO~\\cite{huang2020understanding, perez2020revisiting} and Monero~\\cite{moser2018empirical}.\nOber et al.~\\cite{ober2013structure} and Awan et al.~\\cite{awan2017blockchain} both analyzed the topology and dynamics of the Bitcoin transaction graph.\nChen et al.~\\cite{chen2018understanding} used graph analysis to explore the characteristics of transactions in Ethereum.\nSimilarly, Chen et al.~\\cite{chen2020traveling} and Huang et al.~\\cite{huang2020understanding} also adopted graph analysis, but they focused on the behaviors of ERC20 tokens and fraudulent activities in EOSIO, respectively.\n\n\\noindent\n\\textbf{Smart Contract \\& DApp Analysis.}\n\\label{sec:related:sc-analysis}\nThere are some works \\cite{wu2019empirical, wang2020identifying} dedicated to exploring the ecosystem of DApps.\nFor example, Wu et al.~\\cite{wu2019empirical} defined the popularity of DApps by some metrics across different categories. They measured the reuse of DApps, and the transformation of a traditional application to a DApp. Wang et al.~\\cite{wang2020identifying} classify DApps and users behaviors according to the patterns extracted from transactions.\nMoreover, some works~\\cite{suevil, chen2020survey, he2020characterizing, ji2020deposafe, he2021eosafe} have focused on the security issue of DApps across platforms.\nFor example, Chen et al.~\\cite{chen2020survey} conducted a detailed survey about vulnerabilities and defenses related to smart contracts in DApps. Su et al.~\\cite{suevil} and He et al.~\\cite{he2020characterizing} measured the attack instances and clone behaviors against DApps, respectively.\n\n\n\\section{Conclusion}\n\nThis paper has presented a longitudinal comparative study of three representative blockchains: Bitcoin, Ethereum and EOSIO. \nWe utilized billions of transaction records to offer a unique insight into the overall ecosystems.\nThese have revealed complementary trends amongst the three platforms, highlighting their differing evolutions. \nAlthough our findings have revealed some promising trends for these technologies, a number of ``outliers'' have also been identified and analyzed, e.g., abuse by spammers.\nWe believe that our research efforts contribute positively to inform best operational practices and will benefit the wider research community and regulators. \n\n\\section{Money Transfer in Bitcoin, Ethereum and EOSIO}\n\\label{sec:appendix:money-transfer}\nFig.~\\ref{fig:utxo} shows the UTXO model adopted by Bitcoin, and Fig.~\\ref{fig:transfer-btc} depicts the corresponding topology of $MTG_{btc}$.\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.9\\columnwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/utxo.pdf}\n\t\t\\caption{UTXO model of Bitcoin.}\n\t\t\\label{fig:utxo}\n\t\\end{subfigure}\n     \\begin{subfigure}[t]{0.9\\columnwidth}\n\t\t\\includegraphics[width=\\textwidth]{images\/transfer-btc.pdf}\n\t\t\\caption{Its Structure of Money Transfer Graph.}\n\t\t\\label{fig:transfer-btc}\n\t\\end{subfigure}\n        \\caption{UTXO model and its corresponding structure of Money Transfer Graph.}\n        \\label{fig:bitcoin}\n\\end{figure}\n\nFig.~\\ref{fig:transfer-eth} and Fig.~\\ref{fig:transfer-eos} respectively illustrate the transferring Ether and EOS in Ethereum and EOSIO.\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.9\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/transfer-eth.pdf}\n         \\caption{Transferring Ether in Ethereum}\n         \\label{fig:transfer-eth}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.9\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/transfer-eos.pdf}\n         \\caption{Transferring EOS in EOSIO}\n         \\label{fig:transfer-eos}\n     \\end{subfigure}\n        \\caption{Transferring official tokens.}\n        \\label{fig:transfer-token}\n\\end{figure}\n\n\n\\newpage\n\\section{Outliers}\n\\label{sec:appendix:anomalies}\nThe detected outliers and their corresponding sub-category (category is omitted, details in Table~\\ref{table:category-outliers}), as well as the reason, are shown in Table~\\ref{table:anomalies-1} to Table~\\ref{table:anomalies-3}.\n\n\\begin{table*}[h]\n\\caption{The detected outliers and its corresponding sub-category, as well as the specific reason.}\n\\centering\n\\resizebox{0.68\\textwidth}{!}{%\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{Chain}                 & \\textbf{Date} & \\textbf{Metric}     & \\textbf{Figure}              & \\textbf{Sub-category}                                                          & \\textbf{DApp \/ Event}                      \\\\\n\\midrule\n\\multirow{8}{*}{\\textbf{BTC}}  & 2012-02       & indegree $\\alpha$    & Fig.~\\ref{fig:alpha-mtg-btc}            & Platform                                                                       & Deepbit                                    \\\\\n                               & 2015-07       & outdegree $\\alpha$   & Fig.~\\ref{fig:alpha-mtg-btc}            & Game                                                                           & 1Luc*        \\\\\n                               & 2015-12       & \\# WCC              & Fig.~\\ref{fig:cc-mtg}                   & Game                                                                           & 1Luc*         \\\\\n                               & 2012-02      & degree $\\alpha$     & Fig.~\\ref{fig:alpha-mtg-btc}                & Platform                                                                      & Deepbit        \\\\\n                               & 2012-02      & outdegree $\\alpha$  & Fig.~\\ref{fig:alpha-mtg-btc}                & Platform                                                                      & Deepbit        \\\\\n                               & 2010-11      & \\# traces          & Fig.~\\ref{fig:traces-mtg}                 & Spam                                                                          & 14mU** \\\\\n                               & 2013-07       & \\# traces          & Fig.~\\ref{fig:traces}                     & Gambling                                                                      & Satoshi Dice\n                               \\\\\n                               & 2015-07       & \\# traces         & Fig.~\\ref{fig:traces}                     & Gambling                                                                      & LuckyBit hot wallet\n                               \\\\\n                               & 2012-01       & \\# traces         & Fig.~\\ref{fig:traces}                     & Platform                                                                      & Deepbit\n                               \\\\\n                               & 2012-02        & \\# traces        & Fig.~\\ref{fig:traces}              & Platform                                                                              & Deepbit   \\\\\n                               & 2015-07        & \\# traces         & Fig.~\\ref{fig:traces}             & Attack                                                                            & Spam Attack  \\\\\n                               & 2015-07        & \\# traces         & Fig.~\\ref{fig:traces-mtg}         & Attack                                                                            & Spam Attack \\\\\n                               & 2015-07        & \\# traces         & Fig.~\\ref{fig:traces}             & Attack                                                                            & Spam Attack \\\\\n\\midrule\n\\multirow{30}{*}{\\textbf{EOS}} & 2018-10       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Gambling                                                                       & EOSTiger                                   \\\\\n                               & 2018-10       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Gambling                                                                       & Dice                                       \\\\\n                               & 2018-10       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Gambling                                                                       & eos sicbo                                  \\\\\n                               & 2018-10       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Gambling                                                                       & EOS.Win                                    \\\\\n                               & 2019-04       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-08       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Platform                                                                       & pornhashbaby                               \\\\\n                               & 2019-04       & indegree $\\alpha$    & Fig.~\\ref{fig:alpha-cig-eos}            & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       &$R$                  & Fig.~\\ref{fig:pearson-mtg}              & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-11       &$R$                  & Fig.~\\ref{fig:pearson-mtg}              & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & EIDOS                                      \\\\\n                               & 2019-04       & \\# WCC              & Fig.~\\ref{fig:cc-mtg}                   & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-10       & outdegree $\\alpha$   & Fig.~\\ref{fig:alpha-mtg-eos}            & Spam                                                                           & defind.io                                  \\\\\n                               & 2019-04       & \\# traces           & Fig.~\\ref{fig:traces-acg}               & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       & trace ratio         & Fig.~\\ref{fig:traces-acg}               & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       & From Account        & Fig.~\\ref{fig:percentage-acg-eos}       & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       & degree $\\alpha$      & Fig.~\\ref{fig:alpha-acg-eos}            & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       & outdegree $\\alpha$   & Fig.~\\ref{fig:alpha-acg-eos}            & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-04       & \\# SCC              & Fig.~\\ref{fig:cc-acg}                   & Spam                                                                           & EOS Global                                 \\\\\n                               & 2019-11       & degree $\\alpha$     & Fig.~\\ref{fig:alpha-cig-eos}                & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & EIDOS                                    \\\\\n                               & 2019-10      &  degree $\\alpha$    & Fig.~\\ref{fig:alpha-mtg-eos}                & Spam                                                                          & defind.io                             \\\\\n                               & 2019-10       & From Account        & Fig.~\\ref{fig:percentage-acg-eos}          & Spam                                                                           & defind.io                                         \\\\\n                               & 2019-11       & \\# WCC             & Fig.~\\ref{fig:cc-cig}                         & Token                                                                             & krownairdrop                                  \\\\\n                               & 2019-05       & \\# WCC            & Fig.~\\ref{fig:cc-mtg}                         & Spam                                                                          & EOS Global                            \\\\\n                               & 2019-05       & \\# SCC            & Fig.~\\ref{fig:cc-mtg}                         & Spam                                                                          & EOS Global                                    \\\\\n                               & 2019-05       & \\# SCC            & Fig.~\\ref{fig:cc-cig}                         & Tool                                                                          & AirDropsDAC                           \\\\\n                               & 2018-11       & $R$               & Fig.~\\ref{fig:pearson-mtg}                    & Gambling                                                                      & BET24                         \\\\\n\\midrule\n\\multirow{7}{*}{\\textbf{ETH}} & 2016-06       & \\# WCC              & Fig.~\\ref{fig:cc-mtg}                   & Attack                                                                         & The DAO                                    \\\\\n                               & 2016-10       & \\# traces           & Fig.~\\ref{fig:traces-mtg}               & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10       & From Smart Contract & Fig.~\\ref{fig:percentage-mtg-eth}       & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n\\bottomrule\n\\multicolumn{6}{l}{\\begin{tabular}[c]{@{}l@{}}* 1LuckyR1fFHEsXYyx5QK4UFzv3PEAepPMK\\\\ ** 14mUbjiofYY2F6h3ZGUSoTo3kxdqtajVTp\\end{tabular}}                                                                                                                                                  \n\\end{tabular}%\n}\n\\label{table:anomalies-1}\n\\end{table*}\n\n\\newpage\n\\begin{table*}[h]\n\\caption{The detected outliers and its corresponding sub-category, as well as the specific reason.}\n\\centering\n\\resizebox{0.68\\textwidth}{!}{%\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{Chain}                 & \\textbf{Date} & \\textbf{Metric}     & \\textbf{Figure}              & \\textbf{Sub-category}                                                          & \\textbf{DApp \/ Event}                      \\\\\n\\midrule\n\\multirow{50}{*}{\\textbf{ETH}} & 2018-08       & From Smart Contract & Fig.~\\ref{fig:percentage-mtg-eth}       & Gambling                                                                       & Last Winner                                \\\\\n                               & 2016-08       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Exchange                                                                       & ReplaySafeSplit                            \\\\\n                               \t& 2017-05       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Tool                                                                           & ENS                                        \\\\\n                               & 2018-08       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & Gambling                                                                       & Last Winner                                \\\\\n                               & 2019-09       &$R$                  & Fig.~\\ref{fig:pearson-cig}              & DeFi                                                                           & NEST Protocol                              \\\\\n                               & 2016-10       & From Smart Contract & Fig.~\\ref{fig:percentage-cig-eth}       & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10   & \\# traces           \t& Fig.~\\ref{fig:traces} & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular}\t& DoS Attack\\\\\n                               & 2016-10       & \\# traces           & Fig.~\\ref{fig:traces-cig}               & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2019-11       & trace ratio         & Fig.~\\ref{fig:traces-cig}               & Game                                                                           & Gods Unchained                             \\\\\n                               & 2018-01       & \\# SCC              & Fig.~\\ref{fig:cc-cig}                   & Attack                                                                         & Bittrex Hacked                             \\\\\n                               & 2019-02       & \\# SCC              & Fig.~\\ref{fig:cc-cig}                   & Game                                                                           & CryptoKitties                              \\\\\n                               & 2019-02       & \\# SCC              & Fig.~\\ref{fig:cc-cig}                   & Token                                                                          & ethairdrop                                 \\\\\n                               & 2019-09       & \\# SCC              & Fig.~\\ref{fig:cc-cig}                   & DeFi                                                                           & USDT                                       \\\\\n                               & 2018-05       & indegree $\\alpha$    & Fig.~\\ref{fig:alpha-cig-eth}            & Token                                                                          & EOS                                        \\\\\n                               & 2018-05       & indegree $\\alpha$    & Fig.~\\ref{fig:alpha-cig-eth}            & Token                                                                          & NePay                                      \\\\\n                               & 2019-06       & degree $\\alpha$      & Fig.~\\ref{fig:alpha-cig-eth}            & Token                                                                          & MGC TOKEN                                  \\\\\n                               & 2019-06       & degree $\\alpha$      & Fig.~\\ref{fig:alpha-cig-eth}            & Token                                                                          & VOKEN                                      \\\\\n                               & 2016-10       & From Smart Contract & Fig.~\\ref{fig:percentage-acg-eth}       & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10       & From EOA            & Fig.~\\ref{fig:percentage-acg-eth}       & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2018-07       & From EOA            & Fig.~\\ref{fig:percentage-acg-eth}       & Spam                                                                           & 0x004b * \\\\\n                               & 2019-05       & From EOA            & Fig.~\\ref{fig:percentage-acg-eth}       & Spam                                                                           & 0x8c4b ** \\\\\n                               & 2019-09       & From EOA            & Fig.~\\ref{fig:percentage-acg-eth}       & Spam                                                                           & 0x8c4b ** \\\\\n                               & 2016-10       & \\# traces           & Fig.~\\ref{fig:traces-acg}               & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10       & degree $\\alpha$      & Fig.~\\ref{fig:alpha-acg-eth}            & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10       & outdegree $\\alpha$   & Fig.~\\ref{fig:alpha-acg-eth}            & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2016-10       & \\# SCC              & Fig.~\\ref{fig:cc-acg}                   & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack                                 \\\\\n                               & 2019-10\t   & \\# WCC\t                & Fig.~\\ref{fig:cc-mtg}      \t& Gambling\t& FairWin \\\\\n\\bottomrule\n\\multicolumn{6}{l}{\\begin{tabular}[c]{@{}l@{}}* 0x004bd3562a42c8a7394794849b8ff5ad71c527b2\\\\ ** 0x8c4b7870fc7dff2cb1e854858533ceddaf3eebf4\\end{tabular}}  \n\\end{tabular}%\n}\n\\label{table:anomalies-2}\n\\end{table*}\n\n\\newpage\n\\begin{table*}[h]\n\\caption{The detected outliers and its corresponding sub-category, as well as the specific reason.}\n\\centering\n\\resizebox{0.68\\textwidth}{!}{%\n\\begin{tabular}{lccccc}\n\\toprule\n\\textbf{Chain}                 & \\textbf{Date} & \\textbf{Metric}     & \\textbf{Figure}              & \\textbf{Sub-category}                                                          & \\textbf{DApp \/ Event}                      \\\\\n\\midrule\n\\multirow{20}{*}{\\textbf{ETH}}  & 2016-11\t& \\# SCC\t& Fig.~\\ref{fig:cc-mtg}    \t& Exchange\t& ReplaySafeSplit\\\\\n                               & 2017-01\t& \\# WCC\t& Fig.~\\ref{fig:cc-acg} \t& Exchange\t& Yunbi \\\\\n                               & 2017-08  &\t\\# WCC \/ \\# nodes\t & Fig.~\\ref{fig:cc-acg}  &Exchange\t& Bittrex \\\\\n                               & 2018-05\t& degree $\\alpha$   \t& Fig.~\\ref{fig:alpha-cig-eth} \t& Token\t& NePay \\\\\n                               & 2016-05\t& degree $\\alpha$   \t& Fig.~\\ref{fig:alpha-mtg-eth}\t \t& Exchange & ShapeShift \\\\\n                               & 2016-05\t& outdegree $\\alpha$\t\t& Fig.~\\ref{fig:alpha-mtg-eth} \t\t & Token\t & TheDAO\\\\\n                               & 2018-01\t& \\# traces           \t& Fig.~\\ref{fig:traces-mtg} \t & Exchange & Binance \\\\\n                               \t& 2018-01\t&  trace ratio           \t& Fig.~\\ref{fig:traces-mtg} \t & Exchange & Binance \\\\\n                               \t& 2018-01\t&  trace ratio           \t& Fig.~\\ref{fig:traces-mtg} \t & Game &\tCryptoKitties \\\\\n                               \t& 2018-05  & \\# SCC & Fig.~\\ref{fig:cc-mtg} \t & Exchange & Binance \\\\\n                               \t& 2018-11\t& \\# WCC\t\t& Fig.~\\ref{fig:cc-mtg}      \t \t& Game\t& MegaCryptoPolis \\\\\n                               \t& 2019-02  &\t\\# traces           \t& Fig.~\\ref{fig:traces} \t & Game &\tDragonereum \\\\\n                               \t& 2019-02  &\t\\# traces           \t& Fig.~\\ref{fig:traces-cig} \t & Game &\tDragonereum \\\\\n                               \t& 2016-10  & \\# SCC           & Fig.~\\ref{fig:cc-cig}               & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack \\\\\n                               \t& 2016-10  & \\# WCC \/ \\# nodes          & Fig.~\\ref{fig:cc-acg}               & \\begin{tabular}[c]{@{}c@{}}Malicious\\\\ Exploitation\\\\ on Resource\\end{tabular} & DoS Attack \\\\\n                               \t& 2020-03 & \t\\# SCC              \t & Fig.~\\ref{fig:cc-acg}\t\t& Token &\tGasToken.io \\\\\n                               \t& 2016-02 & $R$        & Fig.~\\ref{fig:pearson-cig}        & Tool  & EthereumAlarmClock \\\\\n                               \t& 2016-03 & $R$        & Fig.~\\ref{fig:pearson-mtg}        & Exchange  & ShapeShift \\\\\n                               \t& 2016-03 & $R$        & Fig.~\\ref{fig:pearson-mtg}        & Exchange  & Poloniex \\\\\n\\bottomrule\n\\end{tabular}%\n}\n\\label{table:anomalies-3}\n\\end{table*}\n\n\\newpage\n\\section{EIDOS \\& Spam Advertisements}\n\\label{sec:appendix:case-study}\nIn \\S\\ref{sec:abnormal-behaviors:case-study}, we have briefly introduced the conclusion after we studied the EIDOS project and spam advertisements behavior in EOSIO.\nHere, we will provide the full details for them.\n\n\\subsection{EIDOS Event}\n\\label{sec:appendix:case-study:EIDOS}\nAs we briefly explained in \\S\\ref{sec:background:blockchain-evolution}, EIDOS attracts users to transfer EOS to it inexhaustibly. The mechanism of EIDOS is illustrated in Fig.~\\ref{fig:eidos-mechanism}.\n\n\\begin{figure*}[h]\n\\centerline{\\includegraphics[width=0.7\\textwidth]{images\/eidos-mechanism.pdf}}\n\\caption{The mechanism of EIDOS.}\n\\label{fig:eidos-mechanism}\n\\end{figure*}\n\nWe can see from Fig.~\\ref{fig:eidos-mechanism}, once the user transfers arbitrary amount of EOS by $T_m$ to its main contract: \\texttt{eidosonecoin} (see steps 1 to 3), the EOS would be immediately returned by another $T_m$ (see steps 4 to 6). Then, EIDOS would initiate a $T_c$ that invokes the \\texttt{transfer} function in its owned smart contract (see steps 9 and 10). In the \\texttt{transfer} function, EIDOS would increase the user's balance of EIDOS token whose value is not related to the amount of EOS in the previous $T_m$. Users can spend these EIDOS tokens in exchanges or other contracts that accept EIDOS token.\nNote that the EIDOS token is issued by the \\texttt{issue} function and part of them will be transferred to its official team: \\texttt{eidosoneteam} (see steps 7 and 8). Therefore, there also exists an another two $T_c$s which invoke \\texttt{issue} and \\texttt{transfer}. \nHowever, the issuance of EIDOS is not executed each time, thus the ratio of $T_m$ to $T_c$ is roughly two to one, which were illustrated by the `trace ratio' in Fig.~\\ref{fig:traces-mtg} and Fig.~\\ref{fig:traces-cig}.\n\nAccording to our statistics, in November 2019, the number of traces in EOSIO has jumped by 1009.82\\% over the preceding month, reaching up to 3.21 billion pieces. Therefore, we further extract all the traces related to EIDOS, and calculate several related metrics as shown in Fig.~\\ref{fig:eidos-statistics}.\n\n\\begin{figure}[h]\n     \\centering\n     \\begin{subfigure}[t]{0.49\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/eidos-user.pdf}\n         \\caption{Amount of users.}\n         \\label{fig:eidos-user}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.49\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/eidos-trace.pdf}\n         \\caption{Amount of $T_m$.}\n         \\label{fig:eidos-trace}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.49\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/eidos-volume.pdf}\n         \\caption{Transferred volume in EOS.}\n         \\label{fig:eidos-volume}\n     \\end{subfigure}\n     \\begin{subfigure}[t]{0.49\\columnwidth}\n         \\centering\n         \\includegraphics[width=\\textwidth]{images\/eidos-eos-per-trace.pdf}\n         \\caption{Average transferred EOS per $T_m$.}\n         \\label{fig:eidos-eos-per-trace}\n     \\end{subfigure}\n        \\caption{The statistics of EIDOS project.}\n        \\label{fig:eidos-statistics}\n\\end{figure}\n  \nWe see that the number of users decreased dramatically in Mar. 2020: the number has dropped to 643 (around 95.13\\% compared to Nov. 2019). In the meanwhile, the number of related $T_m$ has increased by 150.43\\%. These two opposite changes indicate that the players of EIDOS tend to be specialized. In other words, some accounts started to a provide service, which allows users to invest their owned resources and get dividends back. Actually, there indeed existed some service providers~\\cite{eidos-miner-1, eidos-miner-2}. These can also be observed from the $\\alpha$ of outdegree distribution during the corresponding time period (see Fig.~\\ref{fig:alpha-mtg-eos}).\n\nMoreover, these players were becoming \\textit{rational}. As we have explained in \\S\\ref{sec:abnormal-behaviors:case-study:EIDOS}, the amount of the returned EIDOS token has nothing to do with the amount of invested EOS. Thus, as we can see from Fig.~\\ref{fig:eidos-volume} and Fig.~\\ref{fig:eidos-eos-per-trace}, after an ephemeral increase in the volume of invested EOS and the amount of EOS per invested traces, the two curves start to go down sharply. In Mar. 2020, the average EOS per trace was as low as $6 \\times 10^{-4}$, while $1 \\times 10^{-4}$ is the minimum allowed transferred amount in EOSIO and the most rational choice.\nActually, according to our data, in Nov. 2019, 99.80\\% of $T_m$ were valued below 1 EOS. But in Oct. 2019, this percentage was up to 74.81\\%.\nTherefore, we conclude that EIDOS has captured the whole EOSIO ecosystem via meaningless $T_m$ and $T_c$ transactions, thereby slowing the TPS of other users.\n\n\n\\subsection{Spam Advertisement}\n\\label{sec:appendix:case-study:spam}\nDuring the analysis of Fig.~\\ref{fig:alpha-mtg-eos}, we have observed two strange spikes, located in Mar. 2019 and Oct. 2019, in the $\\alpha$ of outdegree distribution. After the analysis of our method mentioned in \\S\\ref{sec:abnormal-behaviors:method} and manual verification, we figured out these two outliers resulted from \\textit{spam advertisements}.\n\nTo be specific, taking the advantage of memo field in EOSIO transactions~\\cite{eos-memo} where the initiator can freely write anything, an account or a smart contract is able to initiate lots of $T_m$ carrying with $1 \\times 10^{-4}$ EOS and bait-and-switch advertisement to cover as many users as possible.\n\nBased on our investigation, we proposed a detection methodology. Specifically, our methodology traverses the $MTG$ and $CIG$ for each month, and detects if node $N$ obeys to the following rules:\n\\begin{enumerate}\n\t\\item The average amount of transferred EOS to each user initiated from $N$ is no more than $x$;\n\t\\item $N$ invokes up to $y$ pieces of $T_m$ to each user within a single month;\n\t\\item More than $z$ users who received $T_m$ from $N$ did not in turn initiates $T_m$ or $T_c$ to $N$;\n\t\\item The memo should carry the leading content, typically is an exaggerated statement and a URL.\n\\end{enumerate}\n\nThe $x$, $y$ and $z$ are heuristically set as 0.001, 30, and 500, which are relative conservative and may lead to false positives.\nThus we further conduct a manual verification focusing on the memo content and the patterns of behavior.\nWe discover \\textbf{206} distinct accounts in total that have initiated spam advertisements at least for one month. The distribution of the first appearance time for these 206 accounts is shown in Fig.~\\ref{fig:ad-distribution}.\nWe can see the obvious two spikes correspond to the outliers located in Mar. 2019 and Oct. 2019, respectively.\nHowever, except for these two spikes and the peak in the late-2018 (caused by gambling DApps), the advertisement accounts are not so active, especially after EIDOS.\n\nAfter investigation, we figure out these two spikes are caused by two clusters of advertisement accounts: \\texttt{peostoken} family and \\texttt{defind.io} family.\nTherefore, we have built a family tree to explore the creation relationship, taking advantage of our $ACG$, which is shown in Fig.~\\ref{fig:ad-family-tree}.\nWe observe that there are clusterings (framed by red boxes in the figure) in which the proportion of advertisements nodes are relatively high. We have marked the \\texttt{peostoken} family and \\texttt{defind.io} family in Fig.~\\ref{fig:ad-family-tree}, which were the reasons behind the two outliers. These two clusters are composed of accounts with multi-level creation relationships.\n\n\n\\begin{figure*}[h]\n\\centerline{\\includegraphics[width=0.7\\textwidth]{images\/ad-distribution.pdf}}\n\\caption{The distribution of these 206 detected advertisement accounts according their first appearance time.}\n\\label{fig:ad-distribution}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\centerline{\\includegraphics[width=\\columnwidth]{images\/ad-family-tree.pdf}}\n\\caption{The family tree of spam advertisement related accounts, where the red nodes are advertisement nodes.}\n\\label{fig:ad-family-tree}\n\\end{figure}\n\n\nIn addition, we have observed some interesting characteristics for spam advertisement accounts:\n\\begin{itemize}\n\t\\item The language in memo content is mainly in English and Chinese, as well as a small percentage of Korean and Japanese.\n\t\\item It is possible for the same account to send out memos with different content under different time periods, and even change the language used in the memo.\n\t\\item Most of the advertisements are promoting their own gambling games, and some are promoting their own tokens or services (e.g., airdrop services). Moreover, advertisements promoting DeFi-related DApps have been appearing since Sep. 2019.\n\t\\item \\texttt{eostribalert} and \\texttt{eostribealrt} are a pair of accounts that launch advertisements. The memo in the latter one contains a URL and convinces users to apply for TLOS token. Immediately after the former one also starts sending out a large number of $T_m$ containing a memo stating that the URL is fraudulent information. However, the difference of number of $T_m$ they sent is very large, 5.3K and 101K times, respectively.\n\\end{itemize}\n\nWe conclude that spam advertisement behavior was popular in EOSIO. They mainly target users who speak English and Chinese. Moreover, those advertisement accounts usually have relationships in terms of account creation behavior.\n\n\n\\end{document}\n\\endinput\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\\label{Sec:Intro}\n\nThe search for life on other worlds looms large in NASA's 30-year Strategic Plan.\\cite{2014arXiv1401.3741K,2015arXiv150704779D} To enable this, NASA is studying a Large UV-Optical-IR Surveyor (LUVOIR) that would use advanced coronagraphs for starlight suppression.\\cite{2014arXiv1401.3741K} ATLAST is one concept for LUVOIR. Alternatively, a Habitable Exoplanet Imaging Mission\\cite{NASATownhall:2015tm} has been proposed. This might pair a smaller aperture space telescope with a starshade flying in formation. Alternatively, an off-axis non-segmented telescope with a coronagraph has been suggested. In any case, better detectors than exist today are highly desirable.\n\nOur emphasis in this paper is on ATLAST\\cite{Bolcar:2015td}, and specifically on ATLAST's detectors for spectroscopic biosignature characterization in the VISIR (hereafter just ``biosignature characterization'') . Although not discussed here, other ATLAST technology needs include precision large-scale optics, ultra-stable structures, starlight suppression, and mirror coatings (See Ref.~\\citenum{Bolcar:2015td}). This emphasis sets aside the importance of the UV to ATLAST's overall mission. Within the ATLAST study, detector and other technology development is envisioned across ATLAST's $90~\\textrm{nm}-2.5~\\mu\\textrm{m}$ ``stretch goal'' wavelength range.\\cite{Bolcar:2015td}\n\nThis paper closely follows the SPIE presentation. We begin with an introduction to biosignature characterization, and show that biosignature characterization is ultra-low background astronomy. The extremely low background count rates motivate the need for further work on VISIR detectors. In Sec.~\\ref{ATLASTNeeds}, we briefly summarize ATLAST's VISIR detector needs in the context of existing technologies.\n\nSec.~\\ref{DetStatus} discusses what are arguably the two most mature detector technologies for biosignature characterization in greater detail. These are electron multiplying CCDs (EMCCDs) for the visible and HgCdTe avalanche photodiode (APD) arrays for the VISIR. We also include a more speculative discussion of what might be achieved in conventional HgCdTe arrays with appropriately optimized readout integrated circuits (ROIC). \n\n\\section{BIOSIGNATURE CHARACTERIZATION}\\label{BioSig}\n\nBiosignature characterization uses low resolution spectroscopy, $R=\\lambda\/\\Delta\\lambda>70$ (required) or $R>150$ (goal), to characterize atmospheric features that are either required for life, or caused by it.\\footnote{For purposes of this discussion, the word ``life'' refers to life as we know it on earth today.} Fig.~\\ref{EarthAsExoplanet} shows several important biosignatures overlaid on a spectrum of earth as it would appear if seen as an exoplanet.\n\nTo make this spectrum, Turnbull~\\etal\\cite{2006ApJ...644..551T} observed the night side of the moon and used modeling to solve for the earth's contribution as it would appear to a distant observer. We define a life detection as consisting of; (1) a rocky planet, (2) with water vapor, (3) a primary biosignature, and (4) a confirming biosignature to rule out false positives. Lacking a confirming biosignature, one could attempt to increase the statistical significance of a result by resolving the temporal dependence of a feature. Arguments for a biological source could also be strengthened by placing the detection in a more comprehensive geological and astrophysical context by measuring other atmospheric gases including ${\\rm CO_2}$, and characterization of the host star's energy distribution.\n\nWith regard to false positives, methane is thought to be particularly important because it is difficult to simultaneously maintain significant concentrations of oxygen, ozone, and methane. Non-equilibrium concentrations are most straightforwardly explained by biological processes. The methane feature at 2.32~$\\mu$m is unfortunately blended with water vapor. However, there is another methane feature between 3~$\\mu$m and 3.5~$\\mu$m that might be better if the observatory could observe it. The spectrum shows a few other features, notably ${\\rm CO_2}$ and ${\\rm O_4}$. Although these features do not provide as much information as the primary and secondary biosignatures, they can still be useful, especially when no confirming biosignature is available.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth]{f10.eps}\n\\caption{This figure overlays a number of important biosignatures on Turnbull~\\etal's\\cite{2006ApJ...644..551T} spectrum of earth seen as an exoplanet. In addition to the features shown here, there is a strong $\\textrm{O}_3$ bandhead at about 260~nm that is considered a primary biosignature. The vegetation red edge (VRE) is caused by chlorophyl from plants. The individual spectral features are discussed in the text.\\label{EarthAsExoplanet}}\n\\end{center}\n\\end{figure}\n\n\\section{ULTRA-LOW BACKGROUND}\\label{ULB}\n\nEven using a $\\geq 8~\\textrm{m}$ space telescope, biosignature characterization is extreme ultra-low background astronomy, potentially requiring days to observe each exoEarth candidate. Consider a simple toy model with these assumptions; a perfect coronagraph, 25\\% efficient integral field unit (IFU) spectrograph, $\\lambda=550$~nm, pixel size $=0.7\\times 1.22\\lambda\/D$, $R=150$, and the background is $3\\times$ the earth's zodiacal light. With these assumptions, the background count rate is $<0.001~\\textrm{cts}~s^{-1}~\\textrm{pix}^{-1}$. More sophisticated models that include the effects of imperfect coronagraphs and simulated exoEarths reach the same conclusion: biosignature characterization is extremely photon starved.\\cite{Stark:2015er} \n\nFor such extremely low count rates, a single photon detector (SPD) is clearly preferred. Better than photon counting, an SPD counts individual photons without adding appreciable noise from any source. The needed SPD combines high QE, zero read noise, ultra-low dark current, and ultra-low spurious count rate. In an EMCCD, clock induced charge (CIC) is one example of spurious counts that it would be beneficial to reduce. In IR APD arrays, glow from non-optimized ROICs is another example of spurious counts that it would be beneficial to eliminate. We discuss both EMCCDs and IR APD arrays in more detail later.\n\n\\section{ATLAST Detector Needs}\\label{ATLASTNeeds}\n\nThe ATLAST technology development plan has been discussed elsewhere at this conference.\\cite{Bolcar:2015td} Tab.~\\ref{ReqTab} summarizes the ATLAST detector needs and ``technology gaps''. Because detectors for the UV through near-IR are equally important to the ATLAST mission, we show them all here. However, this presentation is focused specifically on the VISIR, which we highlight with a red box. The grayed out technologies are no less important to the mission.\n\\begin{table}[t]\n\\begin{center}\n\\caption{Detector Technology Components and Identified ``Gaps''$\\rm ^a$}\\label{ReqTab}\n\\includegraphics[width=\\columnwidth]{f21.eps}\n\\end{center}\n\\end{table}\n\nBroadly speaking, the need is for $\\textrm{QE}>80\\%$ SPDs from 400~nm through 1.8~$\\mu$m (2.5~$\\mu$m goal). The $\\rm 2K\\times 2K$ pixel format is needed if used with an IFU. For space flight, the detectors need to be radiation hard in the L2 radiation environment and able to survive launch loads and vibration. \n\nThe ATLAST study team has indicated a strong preference for non-cryogenic detectors if they can enable the science. This is driven by goals that include: (1) simplifying the system engineering, (2) simplifying the integration and test flow, and (3) completely retiring the risks associated with cooling the detectors to $\\textrm{T}\\sim 100~\\textrm{mK}$.\n\nCoronagraphs capable of achieving contrasts of $10^{-10}$ require wavefront error stability at the level of tens of picometers. Exported vibrations from a conventional cryocooler would present an obvious threat to achieving this. If cryogenic detectors are to be used, then cooling technology development is needed to provide essentially zero vibration cooling. If the cooling challenges could be creatively overcome, then cryogenic detectors including microwave kinetic inductance devices (MKID) and transition edge sensor (TES) arrays might become attractive. \n\nOnce cooled, Both TESs and MKIDs already function as true SPDs with built in energy resolution. Both MKIDs\\cite{2013PASP..125.1348M} and TESs\\cite{1999ApJ...521L.153R,Romani:2001wl} have been used for refereed astrophysics publications. For biosignature characterization, both would require further development to improve parameters including their VISIR energy resolution and the efficiency of coupling light to detector elements. However, since this publication is about ATLAST, we defer further discussion of cryogenic detectors to a later publication.\n\nConsistent with ATLAST's preference for non-cryogenic detectors if possible, we take it as a requirement that the detectors will be operated at a temperature that can be achieved using only passive cooling, $\\textrm{T}\\gtrsim 30~\\textrm{K}$. In early JWST studies, this emerged as a practical detector temperature that could be achieved with margin at L2.\n\n\\section{Candidate VISIR Detector Technologies for ATLAST}\\label{DetTech}\n\nAlthough no completely satisfactory VISIR detector candidate exists for ATLAST today, Tab.~\\ref{DetCandidates} summarizes a number of promising technologies. In making this list, we limited consideration to detectors that we believe to be at least NASA TRL-3. This unavoidably leaves some lower TRL, but nevertheless promising technologies off the list. We encourage all efforts that aim to meet the needs outlined in Sec.~\\ref{ATLASTNeeds}, even if the specific technology does not appear in this table. Tab.~\\ref{DetCandidates} includes a few detectors that we will not be discussing further here because they would operate at $\\textrm{T}\\leq 30~\\textrm{K}$. These are MKIDs, TESs, superconducting nanowire single photon detectors (SNSPD), and Si:As hybrids.\n\\begin{table}[t]\n\\begin{center}\n\\caption{ATLAST VISIR Detector Candidates}\\label{DetCandidates}\n\\includegraphics[width=0.65\\columnwidth]{f31.eps}\n\\end{center}\n\\end{table}\n\nTab.~\\ref{DetCandidates} attempts to condense a wide trade space into a simple graphic for presentation purposes. Both dark and light blue represent existing technologies that we believe hold significant promise. Dark blue is arguably higher TRL than light blue for biosignature characterization. Yellow indicates a technology that we did not discuss to comply with presentation time limits, but that nevertheless may hold promise for further investigation.\n\nTwo of the technologies that we discuss in depth, EMCCDs and IR APDs, are both shaded dark blue. Of these, EMCCDs are currently closer to meeting performance requirements in the visible. IR APDs are the most mature non-cryogenic candidate spanning the VISIR. Although HgCdTe hybrids are shaded yellow in Tab.~\\ref{DetCandidates}, we also discuss these in Sec.~\\ref{DetStatus} because we plan to investigate them further in our labs at Goddard.\n\n\\section{Status of a Few Detector Candidates}\\label{DetStatus}\n\n\\subsection{EMCCD Status}\\label{EMCCDStatus}\n\nFor over a decade, EMCCDs have been leading candidates for low background photon counting in the visible. Starting in the early 2000s, several groups have explored individual photon counting with EMCCDs. In 2004, Daigle~\\textit{et al.}\\cite{Daigle:2004jka} studied how an e2v CCD97 camera, ``operating in pure photon counting mode would behave based on experimental data.'' In 2006, Wen~\\textit{et al.}\\cite{Wen:2006gv} evaluated an e2v CCD201 for space astronomy and published images of a test pattern showing that the EMCCD operated as a photon counter. Over the ensuing decade, steady progress has been made, and today it is possible to buy a commercial EMCCD camera from NuVu Cameras that uses shaped clocks and high readout rates to achieve $\\rm CIC<0.002~cts~pix^{-1}~frame^{-1}$. EMCCDs have been baselined for WFIRST's coronagraph and several presentations at this conference discuss WFIRST's EMCCD efforts. \\cite{Harding:2015tz,Bush:2015uf}.\n\nWhen new and un-degraded by the space radiation environment, EMCCDs are arguably able to meet even ATLAST's challenging performance needs. However, like any conventional n-channel CCD, they will degrade when irradiated. This is a consequence of the phosphorus that is used to dope the n-type channels. Radiation damage, including charge transfer efficiency degradation and pixel operability degradation, has been one of the major reasons for replacing the Hubble Space Telescope's (HST) CCDs. Understanding how radiation effects EMCCDs is important to both WFIRST and ATLAST. Although the ATLAST detector requirements will ultimately be more challenging than those for WFIRST, WFIRST nevertheless provides a valuable early opportunity to understand the issues and address them. For WFIRST, JPL has begun radiation testing and mitigation studies.\\cite{Bush:2015uf}\n\nAlthough EMCCDs are promising detectors for ATLAST, more work in selected areas would be very beneficial. These include efforts aimed at: (1) improving radiation tolerance, (2) further reducing clock induced charge, and (3) improving the red QE from about 850~nm to the bandgap wavelength. Any improvements in radiation tolerance will lead to longer usable life at L2. Further reduction in CIC is important for ATLAST because it is currently a major component of the noise budget. Improving the red QE is important because the strongest water line that is in band for a silicon detector is found at about 950~nm, where the QE of conventional CCDs tends to be falling rapidly.\n\n\\subsection{HgCdTe Photodiode Status}\\label{MCTStatus}\n\nCompared to many other detector materials, HgCdTe has shown very good QE from 400~nm through 2.5~$\\mu$m and beyond (Fig.~\\ref{QE}). For example, the James Webb Space Telescope's (JWST) near-IR detectors achieve $\\rm QE>70\\%$ from $0.6-1~\\mu\\rm m$, and $\\rm QE>80\\%$ from $1-2.5~\\mu$m. For non-astronomical applications, the major vendors have delivered HgCdTe detectors that function at wavelengths at least as short as 400~nm. This impressive QE and high overall maturity begs the question, could conventional HgCdTe photodiode arrays achieve much lower noise than they do today if paired with the right readout approach?\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.8\\columnwidth]{f42.eps}\n\\caption{AR coated HgCdTe has demonstrated $\\rm QE > 70\\%$ from $0.6-1~\\mu\\rm m$ and $\\rm QE > 80\\%$ from $1-2.5~\\mu\\rm m$ for JWST. The major vendors claim that using optimized designs, they can extend this performance to about 400~nm. We show the full potential range here, although the QE performance from $400-600$~nm needs to be confirmed in an astronomical detector.\\label{QE}}\n\\end{center}\n\\end{figure}\n\nSince the mid-1980s, most low background astronomy arrays have used a source-follower per detector architecture (SFD; Fig.~\\ref{SFD}). The SFD architecture has the advantages that it is simple, low power, low glow (when properly designed), and has met performance requirements up to and including those for WFIRST. A major factor driving ROIC design up to the present day has been the need to multiplex a large number of pixels out through a small number of video outputs. This necessitates very wide measurement bandwidth in the video electronics to reproduce the complicated waveform as the output changes from pixel to pixel.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.5\\columnwidth]{f50.eps}\n\\caption{Since the mid 1980s, most HgCdTe arrays for low background astronomy have used an SFD architecture to multiplex many pixels onto a few video outputs. However, the overall SFD system may not be optimal for achieving the lowest possible noise. To understand the full potential of HgCdTe photodiode arrays as ultra-low noise detectors, it would be helpful to better understand the noise that originates in: (a) the photodiode itself, (b) the resistive contact, (c) the pixel source-follower, and (d) the output source follower if one is used. Armed with comprehensive understanding of the noise components, it might be possible to design other ROIC architectures to achieve the noise floor that is set by the photodiode alone. This figure is based closely on a corresponding figure from Ref.~\\citenum{Loose:2003vh}.\\label{SFD}}\n\\end{center}\n\\end{figure}\n\nA good start toward understanding the full potential of HgCdTe photodiode arrays as ultra-low noise detectors would be detailed characterization of existing JWST and WFIRST arrays aimed at separating the noise contributions from elements $a-e$ of Fig.~\\ref{SFD}. The aim would be an itemized noise budget rather than the lumped ``read noise'' that is conventionally reported.\n\nAlthough SFD arrays are well adapted to many kinds of astronomy, the current SFD design may not be optimal for achieving the lowest possible noise. The fundamental noise floor of an \\eg JWST HgCdTe photodiode (a) is potentially of order $\\sqrt{i_d t}$, where $i_d$ is the dark current and $t$ is integration time. The JWST NIRCam arrays have $i_d\\sim 0.001~e^-~s^{-1}~\\textrm{pixel}^{-1}$. Although conventional HgCdTe photodiode arrays will never function as ideal SPDs on account of leakage current at temperatures $\\rm T>30~K$, it is possible that today's H2RG and H4RG detectors are not yet approaching the fundamental noise limits of the photodiodes themselves. It would be interesting to see what could be achieved if noise from the resistive interconnect (b) could be reduced, and\/or different ROICs and readout strategies could substantially reduce or eliminate $1\/f$ noise from (c) the pixel source follower. The output source follower (d) can already be bypassed in many cases.\n\n\\subsection{IR APD Status}\\label{IRAPDStatus}\n\nHgCdTe APD arrays are a promising technology that initially entered astronomy for comparatively high background applications including adaptive optics and interferometry\\cite{Finger:2012dv} and wavefront sensing and fringe tracking\\cite{Finger:2014we}. More recently, they have been used at the telescope to provide diffraction-limited imaging via the ``lucky imaging'' technique.\\cite{2014SPIE.9154E..19A} Although HgCdTe APD arrays have been made by DRS, Raytheon, and Teledyne; those made by Selex are the focus of most attention in astronomy now.\n\nA group at the University of Hawaii has been evaluating Selex SAPHIRA arrays for applications that include low background astronomy.\\cite{2014SPIE.9154E..19A} With appropriately optimized process, the HgCdTe itself is probably capable of the same QE performance as the JWST arrays. Moreover, because gain is built into the pixels before the first amplifier, they promise photon counting and potentially even single photon detection if ``dark current'' can be reduced to acceptable levels.\n\n``Dark current'' is the most significant obstacle to using Selex APD arrays for ultra-low background astronomy today. The $\\sim 10-20~e^-~s^{-1}~\\textrm{pixel}^{-1}$ gain corrected ``dark current'' that has been reported\\cite{2014SPIE.9154E..19A} is almost certainly dominated by glow from the ROIC. The ROIC in current devices was not optimized for ultra-low background, or even low background astronomy. Work continues at the University of Hawaii to try to disentangle ROIC glow from more fundamental leakage currents in current generation APD arrays. On the longer term, work is also underway aimed at optimizing the ROIC design.\n\nAlthough HgCdTe APD arrays hold out the promise of read noise below that which can be achieved using conventional photodiode; like conventional photodiodes there will ultimately be a leakage current noise floor that is determined by thermally activated defect states in the HgCdTe. However, it is likely that today's performance is still far from that floor, and more work is needed to better understand the full potential of HgCdTe APD arrays for ultra-low background astronomy in the context of missions like ATLAST.\n\n\\section{Summary}\\label{Summary}\n\nThe search for life on other worlds promises to be a major focus for astrophysics in coming decades. In space, the tools will include new observatories like ATLAST equipped with high performance coronagraphs and\/or starshades. These will enable biosignature characterization of exoEarths.\n\nFortunately, good detector prototypes exist, although more work is needed to mature them for ATLAST. In the VISIR, these include EMCCDs and IR APD arrays. More speculatively, HgCdTe photodiode arrays may still have room for improvement, even beyond the impressive performance that has been shown for JWST and that is expected for WFIRST. Although the challenges are real, they are solvable. To get from where we are today to what is needed to fully enable ATLAST, focused strategic investment in VISIR detectors is needed.\n\n\\acknowledgments\nThis work was supported by NASA as part of the Goddard Space Flight Center Science and Exploration Directorate Life Finder Detectors Science Task Group (STG). \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\footnotetext[12]{Based on observations obtained with MegaPrime\/MegaCam, a joint project of CFHT and CEA\/DAPNIA, at\nthe Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the\nInstitut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the\nUniversity of Hawaii.}\nSystematic deep photometric observations of the surroundings of the Andromeda galaxy have revolutionized our knowledge of its satellite system. Before 2004, 12 dwarf galaxies were known to be M31 companions, including only 6 dwarf spheroidal galaxies, while 12 new dwarf galaxies have been discovered to inhabit this region of the Local Group sky over the last five years. A dedicated scan of the Sloan Digital Sky Survey (SDSS), targeting a region within $2^\\circ$ of M31's major axis, unveiled the presence of Andromeda~IX and~X \\citep{zucker04,zucker07}. In parallel, a contiguous mapping of the region within $\\sim50{\\rm\\,kpc}$ of Andromeda, performed with the Wide Field Camera on the Isaac Newton Telescope, revealed the picture of a strikingly substructured inner stellar halo \\citep{ferguson02}, and led to the discovery of Andromeda~XVII \\citep{irwin08}.\n\nMapping the outer regions of M31's halo was the goal of an extension to this survey. Performed with the MegaPrime\/MegaCam $1\\,\\textrm{deg}^2$ camera on the Canada-France-Hawaii Telescope (CFHT), it mapped a quarter of the M31 stellar halo, out to $\\sim150{\\rm\\,kpc}$ \\citep{ibata07}. Confirming the clumpy nature of Andromeda's stellar halo, the survey has also revealed the presence of numerous other dwarf galaxies: Andromeda~XI, XII, XIII, XV, XVI, XVIII, XIX and~XX \\citep{martin06b,ibata07,mcconnachie08}. At about the same time, Andromeda~XIV was discovered serendipitously by \\citet{majewski07}, just outside of the edge of the MegaCam survey.\n\nOur understanding of these systems remains sparse, mainly due to the distance at which they reside, translating into difficult photometric observations that cannot easily reach much deeper than the horizontal branch at the distance of M31. Spectroscopic observations are also limited to the handful of bright red giant branch (RGB) stars that can be targeted in each system. Although progress is expected along these lines in the coming years, much can already be said of the generic properties of these new objects from the current survey data alone. Their absolute magnitude ranges from a very faint $M_V=-6.4$ for And~XII and And~XX \\citep{martin06b,mcconnachie08} to a surprisingly bright $M_V\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}-9.7$ for And~XVIII \\citep{mcconnachie08}, patently showing the incompleteness of the M31 satellite luminosity function at the faint end, in regions that have so far only been surveyed with photographic plates.\n\nGiven their relative faintness and significant sizes, these new systems are usually assumed to be dwarf spheroidal galaxies, in other words dwarf galaxies that are devoid of any significant amount of gas. This is currently consistent with the analysis of \\textsc{Hi} surveys but the upper limits on their \\textsc{Hi} content remains relatively high ($2-3\\times10^5{\\rm\\,M_\\odot}$; \\citealt{grcevich09}). Therefore, it cannot be entirely ruled out that some of these galaxies still contain non-negligeable amounts of gas.\n\nBuilding upon the previous MegaCam survey, we have initiated the Pan-Andromeda Archaeological Survey (PAndAS), a Large Program using the CFHT MegaCam imager to map the entire stellar halo of M31 and M33 out to distances of $\\sim 150$ and $\\sim50{\\rm\\,kpc}$ respectively. \nHere, we report on the discovery of two new dwarf galaxies in the vicinity of the Andromeda and Triangulum galaxies based on the first year of PAndAS data. These systems, dubbed Andromeda~XXI and~XXII are sparse but unmistakable overdensities of stars that are also aligned along a RGB at the distance of M31. Section~2 of this paper briefly summarizes the PAndAS data while section~3 presents the new systems and details their properties. Section~4 briefly discusses the new discoveries and section~5 concludes the paper.\n\nThroughout this paper, the distance moduli of M31 and M33 are assumed to be $(m-M)_0=24.47\\pm0.07$ and $(m-M)_0=24.54\\pm0.06$, or $783\\pm25$ and $809\\pm24{\\rm\\,kpc}$ respectively \\citep{mcconnachie05}.\n\n\\section{The PAndAS survey}\nPAndAS builds upon our previous CFHT\/MegaCam surveys of M31 whose results are presented in \\citet{martin06b}, \\citet{ibata07} and \\citet{mcconnachie08}. We use a similar observational set-up and refer the reader to these papers for a description of the observing strategy, data reduction and data quality. Maps showing early results and the complete survey area to date can be found in \\citet{mcconnachie09} and led to the discovery of And~XXI and And~XXII. The survey consists of contiguous exposures, performed with the $1\\times1$ deg$^2$ MegaPrime\/MegaCam camera mounted on the CFHT. The camera is a mosaic of 36 $2048\\times4612$ CCDs with a pixel size of 0.187\\,arsec. Small gaps within the survey lead to a scientifically useable field-of-view of $0.96\\times0.94\\,\\textrm{deg}^2$ for each of the current 250\\,pointings of the survey.\n\nEach field has been observed for 1350\\,s in each of the MegaCam $g$ and $i$ filters, split into $3\\times450$\\,s dithered subexposures. Good seeing ($<0.8''$) ensures that the photometry reaches $g\\sim25.5$ and $i\\sim24.5$ with a signal-to-noise ratio of 10 and guarantees that the star\/galaxy separation only degrades at magnitudes fainter than $g\\sim25.0$ and $i\\sim24.0$. Data were preprocessed (de-biased, flat-fielded and fringe corrected) by the Elixir system at CFHT that also determines the photometric zero point of the observations, and then processed using a version of the Cambridge Astronomical Survey Unit (CASU) photometry pipeline \\citep{irwin01}, adapted to CFHT\/MegaCam observations. The pipeline registers and stacks the images, and also generates catalogues with object morphological classification before creating band-merged $g$, $i$ products.\n\nIn the following, dereddened magnitudes ($g_0$ and $i_0$) have been determined from the \\citet{schlegel98} $E(B-V)$ extinction maps, using the following correction coefficients: $g_0=g-3.793E(B-V)$ and $i_0=i-2.086E(B-V)$, listed in their Table~6.\n\n\\section{Andromeda~XXI and Andromeda~XXII}\n\\subsection{Discovery}\n\nTwo new dwarf galaxies were directly identified on matched-filter surface density maps of RGB candidate stars selected to have colors consistent with the average (generally) metal-poor locus of M31 dwarf spheroidals. Satellites in the very sparsely populated outer halo regions correspond to stellar overdensities that are striking enough for them to be easily spotted, even down to magnitudes of $M_V\\sim-6.5$. This was the case for And~XII \\citep{martin06b} and And~XX \\citep{mcconnachie08} and also here with Andromeda~XXII. An automated search is certainly warranted to better quantify the detection limits of the survey in uncovering new dwarf galaxies. However, it is beyond the scope of this initial discovery paper and will be presented elsewhere.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\hsize,angle=270]{f1.ps}\n\\caption{\\label{map} Map of dwarf galaxies located in the surroundings of M31 and M33. M31 is located at the center of the coordinate system and known satellites are represented by empty circles. The two new discoveries, And~XXI and And~XXII, are represented by filled circles. The current extent of the PAndAS survey is shown as dotted squares, with each square representing a single $1\\times1\\,\\textrm{deg}^2$ MegaCam field. The ellipse represents the extent of M31's \\textsc{Hi} disk. Constellation boundaries, taken from \\citet{davenhall97}, are shown as thin lines.}\n\\end{center}\n\\end{figure}\n\nThe locations of the two new satellites are shown on the map of the PAndAS survey (Figure~\\ref{map}). As with many previous detections, these two systems are noticeably close to the edge of the survey, suggesting that the survey limit of $150{\\rm\\,kpc}$ in projected distance from M31 does not represent the true extent of Andromeda's satellite system (see also Figure~2 of \\citealt{mcconnachie09}). In the spirit of previous conventions, we dub these two satellites Andromeda~XXI and Andromeda~XXII (And~XXI and And~XXII; see the discussion below, in Appendix~\\ref{naming_conventions}).\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.44\\hsize,angle=270]{f2a.ps}\n\\includegraphics[width=0.44\\hsize,angle=270]{f2b.ps}\n\\caption{\\label{map_AndXXI}\\emph{Left panel:} Spatial distribution of stellar sources around And~XXI. Small dots represent all stars in the PAndAS survey whereas large dots correspond to likely RGB stars of the dwarf galaxy, selected within the dashed box shown on the CMD of the middle panel. These stars are clearly clumped into an overdensity of stars. MegaCam CCDs are shown as dashed rectangles and white regions correspond to holes in-between CCDs or holes in the survey. Open circles correspond to regions that are lost to the survey due to the presence of saturated bright stars. The central dashed ellipse corresponds to the region within two half-light radii of the dwarf galaxy, assuming the structural parameters listed in Table~\\ref{parameters}. \\emph{Right panels:} Color-magnitude diagrams within two half-light radii of And~XXI (middle panel) and, for comparison, of a field region at a distance of $\\sim20'$ covering the same area after correcting from gaps in the survey coverage (right-most panel). The galaxy's RGB is clearly visible as an overdensity of stars with $0.8\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} g-i\\lta1.5$ and $i\\gta21.2$ that does not appear in the reference CMD.}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.44\\hsize,angle=270]{f3a.ps}\n\\includegraphics[width=0.44\\hsize,angle=270]{f3b.ps}\n\\caption{\\label{map_AndXXII} Same as Figure~\\ref{map_AndXXI} but for And~XXII. Although this system is much fainter, it still appears as a spatial overdensity of stars (left panel) that are aligned along a RGB in the CMD (middle panel), a feature that does not appear in the reference CMD (right panel).}\n\\end{center}\n\\end{figure*}\n\nBoth systems appear as overdensities of stars on the sky, as is visible in the left panels of Figures~\\ref{map_AndXXI} and~\\ref{map_AndXXII}. These stars are also aligned along a RGB that would be at, or close to, the distance of M31 or M33. The CMDs within 2 half-light radii of the dwarfs (determined in \\S~\\ref{strParam} below) are shown in the middle panels of these figures and, when compared to the CMD of reference fields chosen in an annulus covering the same area at a distance of $\\sim20'$ from the dwarfs' centers (right panels), indeed reveal an alignment of stars that follow the typical shape of a RGB. Isolating these stars enhances the contrast of the overdensity of stars on the sky (large symbols in the left panels).\n\nAnd~XXI is typical of the relatively bright dwarf galaxies that we have found before (such as And~XV or And~XVI). The overdensity of stars in the CMD region $24.5<i<25.0$ and $g-i\\sim1.0$ corresponds to its horizontal branch but the degrading star\/galaxy separation at these magnitudes, as well as the rapidly dropping completeness, makes it difficult to extract any accurate information from this region of the CMD. With only $\\sim20$\\,stars visible along its RGB, And~XXII is one of the faintest M31 satellites found so far.\n\n\\begin{table}\n\\caption{Derived properties of the satellites}\n\\label{parameters}\n\\begin{tabular}{l|cc}\n\\hline\\hline\n & And~XXI & And~XXII \\\\\n$\\alpha$ (J2000) & $23^{\\rm h}54^{\\rm m}47.7^{\\rm s}\\pm 1.0^{\\rm s}$ & $01^{\\rm h}27^{\\rm m}40.0^{\\rm s} \\pm 0.6^{\\rm s}$\\\\\n$\\delta$ (J2000) & $42\\deg28'15''\\pm15''$ & $28\\deg05'25'' \\pm6''$ \\\\\n$(l,b)$ ($^\\circ$) & $(111.9,-19.2)$ & $(132.6,-34.1)$ \\\\\n$E(B-V)^\\mathrm{(a)}$ & 0.094 & 0.076 \\\\\n$(m-M)_0$ & $24.67\\pm0.13$ & [24.50] $\\lta25.07^\\mathrm{(b)}$ \\\\\n$D$ (kpc) & $859\\pm51$ & [794] $\\lta1033^\\mathrm{(b)}$ \\\\\n$r_\\mathrm{M31}$ (kpc) & $152\\pm31$ & [224] $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 315$\\\\\n$r_\\mathrm{M33}$ (kpc) & --- & [42] $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 227$\\\\\n$\\langle{\\rm[Fe\/H]}\\rangle$ & $-1.8$ & $[-1.8]$ $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}-2.5$ \\\\\n$r_h$ (arcmin) & $3.5\\pm0.3$ & $0.94\\pm0.10$ \\\\\n$r_h$ (pc) & $875\\pm127$ & [217] $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 282$ \\\\\nPA (North to East) ($^\\circ$) & $110\\pm15$ & $114\\pm15$ \\\\\n$\\epsilon=1-b\/a$ & $0.20\\pm0.07$ & $0.56\\pm0.11$ \\\\\n$M_V$ & $-9.9\\pm0.6$ & $[-6.5\\pm0.8]$ $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} -7.0\\pm0.8$\\\\\n$L_V$ (${\\rm\\,L_\\odot}$) & $7.8\\pm1.9\\times10^5$ & $[0.3\\pm0.1\\times10^5]$ $\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 0.6\\times10^5$\\\\\n$\\mu_{V,0}$ (mag\/arsec$^2$) & $27.0\\pm0.4$ & $26.7\\pm0.6$\\\\\n\\end{tabular}\\\\\n(a) \\citet{schlegel98}\\\\\n(b) Given the highly unreliable distance estimate of And~XXII from its TRGB, parameters between square brackets are determined assuming  the average distance of M31\/M33. The limit on all parameters, determined from the distance limit measured from the brightest star along And~XXII's RGB, are also given in the same column.\\\\\n\\end{table}\n\n\nAll the parameters of the two dwarf galaxies determined in this section of the paper are summarized in Table~\\ref{parameters}.\n\n\\subsection{Distances and metallicities}\n\\label{distances}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=0.30\\hsize,angle=270]{f4a.ps}\n\\includegraphics[width=0.30\\hsize,angle=270]{f4b.ps}\n\\caption{\\label{LF} \\emph{Left panels:} The CMD (left), luminosity function (LF, top right) and metallicity distribution (MD, bottom right) of And~XXI's stars, located within two half-light radii and within the two dashed lines overlaid on the CMD. An additional cut to remove stars fainter than $i_0=23.5$ has also been applied to determine the MD. Both the LF and MD panels also show the scaled contribution of fore\/background contamination as a dotted histogram, which is clearly negligible. The LF shows an increase of counts as one reaches the TRGB, which is estimated to be at $I_0=20.63$ and indicated by the arrow. This corresponds to a distance modulus of $(m-M)_0=24.67\\pm0.13$. The MD is used to determine the average metallicity of the dwarf $\\langle{\\rm[Fe\/H]}\\rangle=-1.8$, also indicated by the arrow. The \\citet{dotter08} 13\\,Gyr isochrone with $[\\alpha\/\\textrm{Fe}]=+0.0$ overlaid on the CMD corresponds to this metallicity and distance modulus and is in very good agreement with the satellite's RGB. \\emph{Right panels:} Same as the left panels, but for And~XXII. In this case, the LF is so underpopulated that we indicate both the magnitude of the brightest star that is likely to be a RGB star ($I_0\\simeq21.0$, filled arrow) and the average of the M31\/M33 distance that we conservatively assume for And~XXII throughout the paper ($I_0=20.46$, hollow arrow).}\n\\end{center}\n\\end{figure*}\n\nIt is difficult to accurately estimate distances to dwarf galaxies without photometric observations that reach down to their horizontal branches. Indeed, the tip of the RGB (TRGB), which is often used as a distance indicator since its magnitude in the I-band only slightly varies with the properties of the dwarfs' stellar populations (e.g. \\citealt{bellazzini01}), is not sampled well enough in faint systems that suffer from `CMD shot-noise': these systems do not contain enough stars for their CMD to be reliably sampled by their few stars, especially along their RGB \\citep{martin08b}. In the case of And~XXI, this is not so much of an issue as it contains a reasonable number of RGB stars and the TRGB is therefore well defined. From the luminosity function (LF) of stars within two half-light radii of the dwarf's center (dashed ellipse on the map of Figure~\\ref{map_AndXXI}) and within the two dashed lines of the dwarf's de-reddened CMD presented in Figure~\\ref{LF}, we estimate $I_{0,\\mathrm{TRGB}}=20.63\\pm0.04$ by applying the algorithm of \\citet{mcconnachie04}\\footnote{The $gi$ MegaCam magnitudes were first converted to the INT $Vi'$ system following the color equations derived by \\citet{ibata07} and then to the Landolt $VI$ system following the transformations of \\citet{mcconnachie04}.}. The complete LF is shown on the top-right panels of the same figure. Along with the absolute TRGB magnitude of $M_I=-4.04\\pm0.12$ \\citep{bellazzini01}, we determine a distance modulus of $(m-M)_0=24.67\\pm0.13$, or a distance of $D=859\\pm51{\\rm\\,kpc}$.\\footnote{The $g$-band LF of the dwarf, going deeper than the combined $g$- and $i$-band catalogue used to create the CMD of Figure~\\ref{map_AndXXI}, also shows a peak consistent with that of And~XXI's horizontal branch at the measured distance modulus.}\n\nIn And~XXII, on the other hand, the brightest possible member RGB star that can be seen in the CMD of stars within two half-light radii is much less likely to be close to the TRGB and thus to yield a reliable distance. It can only provide an upper limit to the distance modulus of this dwarf. This star, with $I_0=21.03$ (see the top-right panel of Figure~\\ref{LF}) converts to a maximum distance modulus of $(m-M)_{0,\\mathrm{max}}=25.07$ or $D_\\mathrm{max}=1033{\\rm\\,kpc}$. This limit to And~XXII's distance clearly makes this dwarf galaxy a likely member of the M31\/M33 satellite system. The $g$-band LF of And~XXII also shows a tentative detection of its horizontal branch, at a distance of $\\sim940{\\rm\\,kpc}$, very near the limit of the survey. Although this measurement is very uncertain, the absence of a clear horizontal branch at brighter magnitude would imply that And~XXII is at least as distant as And~XXI. For the moment, we will assume a distance to And~XXII that is the same as that to M31\/M33 ($(m-M)_0\\simeq24.50$ or $794{\\rm\\,kpc}$) until deeper photometric data provide a better constraint, keeping in mind that the dwarf is likely to be more distant than this.\n\nFigure~\\ref{LF} also presents the metallicity distribution (MD) of the stars that were used to construct the LF of the two satellites. The metallicity estimates of individual stars were obtained by comparison with the \\citet{dotter08} set of $13{\\rm\\,Gyr}$ isochrones\\footnote{These isochrones are available in the SDSS photometric system that is different, but very close, to the MegaCam one. The isochrones are therefore  recomputed for the MegaCam system from the color equations provided in the headers of the MegaCam fits images. See \\url{http:\/\/www.cfht.hawaii.edu\/Science\/CFHTLS-DATA\/megaprimecalibration.html} for more details.} with $[\\alpha\/\\textrm{Fe}]=+0.0$\\footnote{Using the larger alpha-enhancement of $[\\alpha\/\\textrm{Fe}]=+0.4$ usually found in metal-poor dwarf galaxies (e.g. \\citealt{tolstoy09}) yields {\\rm[Fe\/H]}\\ values that are ~0.2\\,dex more metal-poor but are a noticeably worse fits to the RGB of And~XXI.}, placed at the distance moduli determined above. Each star is given a metallicity determined by the ratio of the distances to its two closest isochrones. These have a metallicity ranging from ${\\rm[Fe\/H]}=-2.5$ to ${\\rm[Fe\/H]}=+0.5$, with steps of 0.1\\,dex. There is no significant contamination of these MD from the field population (dotted histograms), except for the very few, most metal-rich stars to appear in And~XXI's MD. When this contamination is removed, both galaxies are determined to be quite metal-poor, with an average metallicity of $\\langle{\\rm[Fe\/H]}\\rangle=-1.8$. The corresponding isochrones, overlaid on the CMDs of the same figure, show good agreement with the location of the observed RGB stars that revealed the satellites. Of course, if the distance modulus of And~XXII is larger than the $(m-M)_0=24.50$ value adopted here, that would translate into a lower metallicity for this system, with a minimum $\\langle{\\rm[Fe\/H]}\\rangle=-2.5$ for the maximum distance obtained above. It should also be noted that, given the photometric uncertainties and the small color differences between metal-poor isochrones, the determination of photometric metallicities does not give a good handle on the metallicity spread in these systems and the width of the MDs should not be interpreted at face value.\n\n\\subsection{Structural parameters}\n\\label{strParam}\nTo determine the structural properties of the two dwarf galaxies, we apply the maximum likelihood technique developed by \\citet{martin08b} that determines the best exponential profile (and background level) to fit the spatial distribution of a galaxy's stars. Compared to the original paper, the algorithm has been modified to account for incompleteness in the surroundings of the satellites, produced by gaps between the MegaCam CCDs, the survey edges, and the presence of saturated bright stars. The first effect is patently important in the case of And~XXI, which has a CCD gap running through its center. The best exponential profile is determined iteratively, after filling holes from the parameters of the best previous model. This step is performed by first binning the survey region around the dwarfs with small $0.1\\times0.1\\textrm{arcmin}^2$ pixels. Each pixel that falls over the edge of the survey, in a CCD gap, or behind a bright star is filled with a number of stars randomly generated from a Poisson distribution of mean the model density at this pixel's central position, computed from the best model from the previous iteration. These stars are then randomly distributed within the pixel. Convergence on the structural parameters model is usually achieved within five iterations but we proceed with eight iterations to secure convergence. Given the stochasticity introduced by filling the holes with artificial stars, we determine the contribution of the gap to the uncertainties on the model parameters via a Monte Carlo approach, performing 100 fits for each galaxy.\n\nSo as to limit the impact of background\/foreground contamination, we apply the maximum likelihood fit to these stars that fall within the selection box overlaid on the CMDs of Figures~\\ref{map_AndXXI} and~\\ref{map_AndXXII}. These have a faint limit fixed at $i=24.0$ so that fainter regions where star\/galaxy separation becomes an issue are avoided. The centroid ($\\alpha_0$, $\\delta_0$), half-light radius ($r_h$), ellipticity (defined as $\\epsilon=1-b\/a$ with $a$ and $b$ being, respectively, the major and minor axes scale lengths of the system) and position angle (PA) for which convergence was achieved are listed in Table~\\ref{parameters}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\hsize,angle=270]{f5.ps}\n\\caption{\\label{profiles} Stellar profiles of And~XXI and And~XXII. In both panels, the stellar density measured in fixed elliptical annuli from the PAndAS data is shown as dots, using the best ellipticity, position angle and centroid found by the fitting algorithm (the uncertainties are derived assuming Poisson statistics). These data points are corrected for incompleteness produced by CCD gaps, the edges of the survey and bright stars. The full lines represent the best models constructed from the best fit half-light radius and background level (the background level is represented by the thin dashed line). Although they are not fits to the data points, the profiles are in good agreement with them. Regions where some deviations are observed (at the center of And~XXI and in the outer part of And~XXII) are regions significantly affected by incompleteness.}\n\\end{center}\n\\end{figure}\n\nIn Figure~\\ref{profiles}, the fitted exponential profiles are compared to the data, which are binned following the best centroid, ellipticity and position angle determined by the fit. The binned data points have been corrected for spatial incompleteness produced by CCD gaps, survey edges and bright stars. For both dwarfs, the agreement between profile and data is quite good. There are regions --- in particular in the inner parts of the And~XXI profile and the outer parts of the And~XXII one --- where deviations from the model are significant but these correspond to regions were a large completeness correction was applied.\n\n\\subsection{Magnitudes}\nTo measure the total magnitude of each dwarf galaxy, we first determine the flux contribution of all stars brighter than $i_0=23.5$ (where we know the survey to be complete\\footnote{We have nevertheless checked the validity of our parameters with the fainter $i_0=24.0$ limit and find that, with this deeper limit, none of the derived parameters deviate by more than $\\pm0.1$\\,mag. from those listed in Table~1.}; \\citealt{ibata07}) and within its half-light radius, following the best structural parameters determined above. We then correct this number for background\/foreground contamination by counting the flux within the annulus between 16 and $20'$, scaled to the area covered within the half-light radius. Both these fluxes are also corrected for incompleteness coming from bright stars, CCD gaps and survey edges. For And~XXI and And~XXII, we measure partial magnitudes of $M_V=-7.7$ and $-4.3$ respectively. These numbers then need to be corrected to include the flux contribution of all stars fainter than $i_0=23.5$. In order to do this, we use the PandAS observations of another Andromeda satellite, And~III, that has also been observed in the survey under the same conditions as And~XXI and And~XXII. In the case of this satellite, we measure a difference of 2.2 magnitudes between the flux of stars brighter than the $i_0=23.5$ limit and the total magnitude of the dwarf $M_V=-10.2\\pm0.3$ measured by \\citet{mcconnachie06b}. This leads to total magnitudes of $M_V=-9.9$ and $-6.5$ for the two new dwarfs when we account for this difference. We estimate the uncertainties on these values to be $\\pm0.6$ for And~XXI and $\\pm0.8$ for And~XXII, stemming from the uncertainties on the structural parameters ($\\pm0.2$), the distance estimates ($\\pm0.1$ and $\\pm0.3$), the CMD shot-noise of these sparsely populated dwarf galaxie ($\\pm0.2$ and $\\pm0.4$), and the total magnitude of And~III used for the calibration ($\\pm0.3$) all added in quadrature, to which we add a systematic uncertainty of $\\sim0.2$\\,magnitudes to account for possible differences between the And~XXI and And~XXII luminosity functions and that of And~III.\n\nFrom the total magnitude, we determine the surface brightness of the dwarfs, following equation (6) of \\citet{martin08b} with the number of stars $N_*$ being replaced by the total luminosity of the dwarf, $L_V$, listed in Table~\\ref{parameters}. This equation accounts for the structure of the dwarf and in particular its ellipticity, and yields central surface brightnesses of $\\mu_{V,0}=27.0\\pm0.4$ and $26.7\\pm0.4\\,\\textrm{mag}\/\\textrm{arcsec}^2$ respectively for And~XXI and And~XXII. Their low surface brightnesses make them invisible on photographic plate and, contrary to the brighter And~XVIII \\citep{mcconnachie08}, they do not appear on the Digitized Sky Survey.\n\n\\section{Discussion}\n\n\\subsection{Properties of the new dwarf galaxies}\nThe measured magnitude and surface brightness of both new dwarf galaxies are within the range of values measured for dwarf galaxies previously found and, as such, And~XXI and And~XXII are very typical Andromeda satellites. The structures of these two dwarf galaxies are also fairly typical of that of M31's satellites. Although the ellipticity of And~XXII is quite high, it is not the only elliptical dwarf spheroidal galaxy known to orbit Andromeda. And~III, for instance, has a similarly large ellipticity ($\\epsilon=0.52\\pm0.02$, \\citealt{mcconnachie06b}).\n\nWith a half-light radius of almost $1{\\rm\\,kpc}$, And~XXI supports the size difference that has previously been noted to distinguish Andromeda from Milky Way satellites: an M31 dwarf galaxy is typically 2--3 times larger than its Galactic counterpart \\citep{mcconnachie06b}. This distinction still holds for fainter dwarf galaxies like the ones recently discovered. And~XXI is the fourth largest dwarf spheroidal galaxy known in the Local Group after And~XIX ($r_h=1.7\\pm0.1{\\rm\\,kpc}$; \\citealt{mcconnachie08}), And~II ($r_h=1.12\\pm0.02{\\rm\\,kpc}$; based on the exponential profile fit of \\citet{mcconnachie06b} for consistency) and Sagittarius, which is clearly in the final throes of its tidal disruption by the Milky Way. It is also interesting to notice that two of these galaxies (And~XIX and And~XXI) are recent discoveries, hinting that some other large satellites could still elude us, especially given that we are biased against finding large faint galaxies, as they are more diffuse and therefore have a lower surface brightness. Galaxies as large and diffuse as And~XIX would also be very hard to find around the Milky Way but, on the contrary, it is unlikely that Milky Way satellites that share the properties of And~II and And~XXI ($r_h\\sim1{\\rm\\,kpc}$, $M_V\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$}-9.0$) could have been missed in the quarter of the Galactic sky covered by the SDSS \\citep{koposov08,walsh09}. The largest Galactic satellite known to date was indeed discovered in the SDSS, but its half-light radius is only a mere $564\\pm36{\\rm\\,pc}$. Such a distinction is puzzling and can be used, in conjunction with spectroscopic data, to provide strong constraints on the type of dark matter haloes that these satellites should inhabit \\citep{penarrubia08a}.\n\n\\subsection{And~XXII: an M33 satellite?}\nFor And~XXII, the most intriguing question is whether it is a satellite of M33 or M31. M33 is a small spiral galaxy with a mass approximately one tenth the mass of M31. It resides at a distance of $204{\\rm\\,kpc}$ from the center of M31, presumably well-inside the M31 dark halo whereas measurements of its line-of-sight velocity and proper motion suggest that it is on a highly elongated orbit around M31.  The implication is that M33 is a satellite of M31 and likely passed much closer to M31 in the past than its present distance.  This association is supported by the existence of a strong warp in the M33 HI disk \\citep{rogstad76} as well as the presence of a stellar stream extending from its stellar disk \\citep{mcconnachie09}.\n\nIf And~XXII is shown to be a satellite of M33, then it would expand the known size of the M33 system by 50\\%, from its recently discovered outermost globular cluster \\citep{huxor09}. The existence of a satellite at this radius would also place a constraint on the strength of an M33--M31 interaction since such an interaction naturally strips material from the outer parts of M33. Finally, we point out that And XXII may be one of the few examples of a satellite's satellite, a phenomenon that naturally arises in the hierarchical clustering scenario of structure formation.\n\nAnd~XXII is, in projection, much closer to M33 than to M31. However, since the distance to And~XXII has not been determined, the true And~XXII\/M31 and And~XXII\/M33 distances are not known.  This is illustrated in the top panel of Figure~\\ref{forces} that shows these distances as a function of the heliocentric distance to And~XXII, $D_{\\rm AndXXII}$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\hsize]{f6.ps}\n\\caption{\\label{forces} \\emph{Top panel:} The And~XXII\/M31 (full line) and And~XXII\/M33 distance (dash-dotted line) as a function of the And~XXII heliocentric distance. The maximum distance we estimate for And~XXII, $D_\\mathrm{AndXXII}=1033{\\rm\\,kpc}$ is represented by the dotted line. \\emph{Bottom panel:} The ratio of the M33's dark matter halo density to that of M31 as a function of the heliocentric distance to And~XXII. There is a reasonable distance range ($720\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} D_\\mathrm{AndXXII}\\lta930{\\rm\\,kpc}$) over which the density of the M33 dark matter halo dominates, hinting at a more likely association of And~XXII to Triangulum in this region.}\n\\end{center}\n\\end{figure}\n\nCosmological simulations suggest that the density of satellites traces the dark matter density \\citep{sales07a}. If true, the ratio of the M33 halo density to the M31 halo density at a given location provides a rough measure of the probability for a dwarf at this location to be a satellite of M33 relative to the probability that it is a satellite of M31. This ratio is shown in the bottom panel of Figure~\\ref{forces}, as a function of the heliocentric distance to And~XXII. Here, we have assumed that the halos of M31 and M33 have NFW density profiles \\citep{navarro97} defined by a virial mass, $M_{\\rm vir}$, and a virial radius, $R_{\\rm vir}$.  By definition, the mean density within the virial radius is a factor $\\Delta$ times the background density where we take $\\Delta\\simeq 95$ \\citep{maccio08}, as appropriate for WMAP5 values of the cosmological parameters \\citep{komatsu09}. We assume $M_{\\rm vir}=2\\times10^{12}{\\rm\\,M_\\odot}$ for M31 and a mass that is ten times lower for M33, consistent with current observational constraints \\citep{mcconnachie06c}. Concentration parameters are chosen as $c=10$ and $c=9$ respectively, following the relations of \\citet{neto07} and \\citet{maccio08}, which leads to virial radii of $R_{\\rm vir}= 327{\\rm\\,kpc}$ for M31 and $R_{\\rm vir} = 152{\\rm\\,kpc}$ for M33. Figure~\\ref{forces} suggests that if the distance to And~XXII is found to be between $720{\\rm\\,kpc}$ and $930{\\rm\\,kpc}$, then it is more likely to be a satellite of M33 than of M31\\footnote{It should be noted that recent modeling of the Andromeda gravitational potential suggests that it could be less massive and with a larger concentration than assumed here \\citep{geehan06,seigar08}. This might expand, or otherwise narrow the range over which And~XXII is more likely to be an M33 satellite. However, given the lack of a similarly accurate model for M33, we currently favor cosmologically-motivated parameters for both galaxies, while we note that better models are in the process of being developed, partially constrained by PAndAS data.}.\n\nAn alternative argument can be made based on the notion of a Hill sphere which roughly defines the region where M33 rather than M31 dominates the dynamics of small systems.  In the limit of circular orbits (admittedly, a bad assumption for this problem) the radius of the Hill sphere for M33 is given by $r_{\\rm Hill} = D_{\\rm M33-M31} (M_{\\rm M33}\/3M_{\\rm M31})^{1\/3}$. The range of distances where And~XXII is likely to be an M33 satellite is then narrower ($757<D_{\\rm AndXXII}<856{\\rm\\,kpc}$).\n\nWhile our arguments are admittedly simplistic, they illustrate the importance an accurate distance measurement for And XXII will have in settling this issue.  A measurement of the line-of-sight velocity for And~XXII will also provide valuable information as will detailed modelling of the M31--M33 interaction.\n\n\\section{Conclusion}\nIn this paper, we present the discovery of two Local Group dwarf galaxies from the first year data of the Pan-Andromeda Archaeological Survey (PAndAS). Combined with previous installments of the MegaCam survey \\citep{ibata07}, the M31 and M33 surroundings have been mapped over $\\sim220$ deg$^2$ and unveiled 10 new dwarf galaxies. The two new satellites, Andromeda~XXI and Andromeda~XXII, have properties that are typical of other Andromeda satellites:\n\n\\begin{itemize}\n\\item Andromeda~XXI is a reasonably bright galaxy, with $M_V=-9.9\\pm0.6$, located slightly farther than M31 at $859\\pm51{\\rm\\,kpc}$, or at $152\\pm31{\\rm\\,kpc}$ from Andromeda itself and is metal-poor ($\\langle{\\rm[Fe\/H]}\\rangle=-1.8$, assuming $[\\alpha\/\\textrm{Fe}]=+0.0$). Its size is its most extraordinary property and, with a half-light radius of $875\\pm127{\\rm\\,pc}$, it is the third largest dwarf spheroidal galaxy of the Local Group, emphasizing the greater size of M31 satellites with respect to Milky Way satellites.\n\\item Andromeda~XXII is a much fainter system, with a total magnitude of only $M_V=-6.5\\pm0.8$; it is also metal-poor with $\\langle{\\rm[Fe\/H]}\\rangle=-1.8$ (also under the assumption of $[\\alpha\/\\textrm{Fe}]=+0.0$. Its physical properties are made difficult to constrain by the absence of a good distance estimate originating from the low number of stars that populate its red giant branch. An accurate distance estimate will have to rely on the magnitude of And~XXII's horizontal branch from deeper photometry. We nevertheless raise the interesting possibility that, given its proximity to M33, at least in projection, And~XXII could be the first ever discovered Triangulum satellite. We show that the dwarf galaxy has a reasonably large heliocentric distance range ($720\\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} D_\\mathrm{AndXXII}\\lta930{\\rm\\,kpc}$) over which it would more probably be a satellite of M33 than of M31.\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\n\\label{sec:YM:Introduction\nInfrared QCD is a strongly correlated system that is governed by confinement and spontaneous chiral symmetry breaking, whose understanding and resolution require numerical non-perturbative first principle approaches. Functional diagrammatic approaches such as the functional renormalisation group (fRG) and Dyson-Schwinger equations (DSE), or $n$-particle irreducible approaches (nPI) potentially offer both, analytic access to the mechanisms behind the infrared dynamics of QCD as well as its quantitative numerical resolution. For recent reviews on fRG and DSE approaches to QCD see~\\cite{Fischer:2018sdj, Dupuis:2020fhh}.\n\nThe diagrammatic nature of functional approaches is best implemented within gauge-fixed QCD, for a recent discussion of gauge invariant alternatives see~\\cite{Dupuis:2020fhh} and references therein. A specifically well-suited gauge fixing is the Landau gauge, in particular for numerical applications. The latter require truncations to the infinite hierarchy of coupled loop equations for correlation functions in functional approaches. Then, the strongly correlated nature of infrared QCD begs the question of whether the truncations commonly used for explicit numerical solutions transport the underlying gauge symmetry of QCD: are the correlation functions computed \\textit{gauge consistent}. Naturally, the gauge consistency of the correlation functions is essential for confinement, both its manifestation in gauge-fixed approaches as well as its understanding.\n\nIn the present work, we put forward a systematic approach towards the evaluation of the above question: First, one computes transverse and longitudinal correlation functions in QCD with functional approaches. Then, the gauge consistency of the results is tested by inserting them into the Slavnov-Taylor identities (STIs). The STIs encode the changes of gauge-fixed correlation functions under gauge or BRST-transformations. \n\nThe evaluation of the latter test comes with an intricacy. The STIs also constitute a set of functional relations for correlation functions. These relations can be derived from the BRST identity of the effective action, the BRST master equation, similarly to deriving functional relations from the fRG-flow of the effective action or the functional DSE for the latter. In the case of the STIs, longitudinal correlation functions are related to combinations of transverse and longitudinal correlation functions. In summary, fRG equations, DSEs, and STIs represent different resummation schemes for correlation functions. While their exact solutions agree, any approximations thereof will not coincide in general. To exemplify this situation, let us emphasise that one can easily guarantee the STIs for a finite set of correlation functions in a given order of the truncation if simply using the STIs for the computation of the longitudinal correlation functions. Evidently, this resolution of the STIs does not guarantee the gauge consistency of the transverse correlation functions. This is best seen in the Landau gauge, where the set of functional relations for transverse correlation functions is decoupled from the system of longitudinal ones. Instead, gauge consistency should rather be evaluated by the \\textit{smallness} of the deviations of correlation functions within \\textit{all} functional relations at hand: fRG flows, DSEs, STIs and $n$PI-relations. For a detailed discussion of these subtleties, see \\cite{Dupuis:2020fhh, Fischer:2008uz}.\n\n\nIn the present work, we study Landau gauge Yang-Mills theory within a systematic vertex expansion with the functional renormalisation group. Longitudinal correlation functions are also computed from their respective Slavnov-Taylor identities, which follow from the BRST master equation for the effective action. This task is facilitated with the code framework \\textit{QMeS}-Derivation \\cite{Pawlowski:2021tkk, github:QMeS}, that allows for the derivation of fRG and DSE equations as well as (m)STIs. We use an advanced truncation for these computations, even though fully quantitative computations are still more advanced, see in particular \\cite{Cyrol:2016tym, Huber:2020keu}. The respective results are also compared with the corresponding lattice results. In \\Cref{sec:YM:NumericalResults} we discuss the consequences for the gauge consistency of the present results as well as extensions of the current computations. \\Cref{sec:YM:Conclusion} contains a brief conclusion to the present work. The Appendices \\Cref{app:YM:Implementation}, \\Cref{app:YM:FRGYM} and \\Cref{app:YM:mSTI} contain some technical details of the computation and the diagrammatic functional equations within the truncation used throughout this work are shown in \\Cref{app:YM:Diags}. \n\n\n\\section{Slavnov-Taylor Identities in Functional Approaches\n\\label{sec:YM:Flows\n\nWe consider $SU(N_c)$ Yang-Mills theory in Euclidean space-time within the Landau gauge. The explicit results are obtained in the physical (QCD) case $N_c=3$, but they readily extend to general $N_c$. The approximation used here carries the large $N_c$ scaling which is present in the dominant tensor structure, for a respective discussion see \\cite{Corell:2018yil}. The gauge-fixed classical action, including the ghost term and BRST source terms, is given by \n\\begin{align}\nS= &\\,\n\\int_x \\Biggl[\\frac{1}{4}F^a_{\\mu\\nu}F^a_{\\mu\\nu}+\\partial_\\mu \\bar{c}^a D^{ab}_\\mu c^b+B^a\\partial_\\mu A_\\mu^a-\\frac{\\xi}{2}B^aB^a\\nonumber\\\\[4pt]\n&\\hspace{-.2cm}-Q_{A,\\mu}^a D^{ab}_\\mu c^b -\\frac{1}{2}g f^{abc}Q_c^ac^bc^c+\\frac{1}{\\xi}Q_{\\bar{c}}\\partial_\\mu A^a_\\mu\\Biggr]\\,, \n\\label{eq:Sclass}\\end{align}\nwhere we have introduced the shorthand notation $\\int_x = \\int d^4 x$. The covariant derivative $D$ in the adjoint representation and the field strength tensor are defined as,\n\\begin{align}\\nonumber \nD^{ab}_\\mu c^b  = &\\, \\partial_\\mu c^a + g f^{acb}A^c_\\mu c^b\\,,\\\\[1ex]\nF_{\\mu\\nu}^a = &\\, \\partial_\\mu A_\\nu^a-\\partial_\\nu A_\\mu^a + g f^{abc}A_\\mu^bA_\\nu^c\\,, \n\\end{align}\nand the Landau gauge is implemented with $\\xi\\rightarrow 0$. The action \\labelcref{eq:Sclass} depends on the superfield $\\Phi$, \n\\begin{align}\\label{eq:Phi}\n\t\\Phi=(A_\\mu, c, \\bar c, B)\\,,\\qquad  Q =(Q_A, Q_c,Q_{\\bar c})\\,, \n\\end{align}\nand we have already introduced source terms the BRST transformations (Becchi, Rouet, Stora, Tyutin)~\\cite{Becchi:1975nq,  Tyutin:1975qk} of the fields  $\\delta_{\\text{BRST}}\\Phi_a = \\mathfrak{s}\\Phi_a\\delta \\lambda$ in a general covariant gauge, coupled to the currents $Q_{\\Phi_a}$. The respective transformations are given by \n\\begin{align}\n\\mathfrak{s}A_\\mu^a &= D^{ab}_\\mu c^b\\,,\\nonumber\\\\[10pt]\n\\mathfrak{s}c^a &= \\frac{1}{2} g f^{abc}c^bc^c\\,,\\nonumber\\\\[10pt]\n\\mathfrak{s}\\bar{c}^a &= B^a\\,,\\nonumber\\\\[10pt]\n\\mathfrak{s} B^a &= 0\\,,\n\\label{eq:YM:YMBRSTtrafo}\n\\end{align}\nwith the Nakanishi-Lautrup field $B^a$. Evidently, we do not need $Q_B$, as the respective BRST transformation vanishes identically. Moreover, with $\\mathfrak{s}^2=0$ these additional terms are also BRST invariant.  Our general conventions and notation follows~\\cite{Pawlowski:2021tkk}.\n\n\n\\subsection{Functional Renormalisation Group}\n \nIn the functional renormalisation group approach, an infrared cutoff is introduced by adding momentum-dependent mass terms $\\Delta S_k$ to the classical action $S[\\Phi,Q]$ defined in \\labelcref{eq:Sclass}, \n\\begin{align}\n\\Delta S_k = \\int \\left(\\frac{1}{2}(R_A)^{ab}_{\\mu\\nu}\\, A^a_\\mu A^b_\\nu + (R_c)^{ab}\\,\\bar{c}^ac^b\\right)\\,,\n\\label{eq:YM:cutoff}\n\\end{align}\nwith the infrared cutoff scale $k$. The regulators $R=(R_A,R_c)$ suppress quantum fluctuations for momenta $p^2 \\lesssim k^2$ for the respective fields, and vanish for $p^2 \\gtrsim k^2$. The specific form of the regulators used in the present work can be found in \\Cref{app:YM:reg}. The infrared regularised classical action $S+\\Delta S_k$ is then used in the path integral representation of the generating functional $Z[J,Q]$. Here, the super-current $J$ includes currents for all component fields $J=(J_A, J_c, J_{\\bar c}, J_B)$. \n\nThe functional flow equation is derived for the scale-dependent effective action  \n\\begin{align}\\label{eq:G}\n\\Gamma_k[\\Phi,Q]= \\int J\\cdot \\Phi -\\log Z[J,Q] - \\Delta S_k[\\Phi]\\,, \n\\end{align}\nthe modified Legendre transform of the Schwinger functional $\\log Z$. Its logarithmic scale derivative provides us with the one-loop exact  fRG equation,\n\\begin{align}\n\\partial_t \\Gamma_k = \\int_x\\left[\\frac{1}{2}(\\dot{R}_A)^{ab}_{\\mu\\nu}\\,(G_{AA})_{\\mu\\nu}^{ab}-(\\dot{R}_c)^{ab} (G_{c\\bar{c}})^{ab}\\right](x,x)\\,,\n\\label{eq:YM:fRG}\n\\end{align}\nwith $\\dot R=\\partial_t R$,  the (negative) RG time $t = \\ln(k\/\\Lambda)$, and the full ghost and gluon propagators \n\\begin{align}\n\tG_{AA} = \\left[\\frac{1}{\\Gamma^{(2)}+R}\\right]_{AA} \\,,\\qquad \n\tG_{c\\bar c}= \\left[ \\frac{1}{\\Gamma^{(2)}+R} \\right]_{c\\bar c}\\,. \n\t\\label{eq:YM:propeq}\n\\end{align}\nHere, $\\Lambda$ is a reference scale, typically chosen to be the initial cutoff scale deep in the ultraviolet (UV). If the initial scale is chosen sufficiently large, the effective action tends towards the local UV effective action $\\Gamma_\\Lambda$ that consists out of all UV-relevant terms. For Yang-Mills theory this UV effective action includes all terms in the classical action as well as a mass term for the gluon. The latter term originates in the breaking of gauge invariance via the regulator term for $k\\neq 0$. In turn, for $k\\to 0$, this breaking disappears and we are left with the full BRST invariant effective action. For a full derivation and further discussions, see e.g.~\\cite{Pawlowski:2021tkk, LectureNotesprep} and the QCD-related reviews \\cite{Litim:1998nf, Berges:2000ew, Pawlowski:2005xe, Gies:2006wv, Rosten:2010vm, Braun:2011pp,  Dupuis:2020fhh}.\n\nAt this point we would like to remark on some implicit assumptions within the derivation of the flow equation \\labelcref{eq:YM:fRG}: it implies that the only $k$-dependence of the \\textit{renormalised} generating functional originates in the cutoff term. Then, in particular, the renormalisation procedure is assumed to be $k$-independent, for a detailed discussion see e.g.~\\cite{Pawlowski:2005xe, Rosten:2010vm}. While the self-consistency of this assumptions is readily shown for the wave function renormalisations and vertices, the absence of a mass renormalisation for the gluon is less trivial and potentially affects the underlying BRST invariance. This question is of utmost importance for the interpretation of Yang-Mills theory in the presence of the infrared cutoff term as a deformation of Yang-Mills theory, rather than a massive extension of Yang-Mills theory, for the Curci-Ferrari (CF) model see~e.g.~\\cite{Tissier:2010ts, Tissier:2011ey, Serreau:2012cg, Reinosa:2017qtf, Pelaez:2021tpq}. In the CF model, masses for ghost and gluons are part of the gauge fixing. In contrast, one may also simply add a gluonic mass term after the gauge fixing, and the two procedures agree in the Landau gauge. Note also, that the latter procedure is similar to adding the regulator term to the gauge-fixed action in the fRG approach, a subtle difference being the mass renormalisation and the related classical BRST symmetry. \n\nBoth procedures have to be contrasted to massive extensions of Yang-Mills theory, where the mass is added to the classical action without a gauge fixing, bound to introduce non-localities.\n\nWe close with the remark that the massless limit of massive extensions of Yang-Mills theory is an intricate one as it defines a flow in theory space. Instead, the removal of the momentum-local infrared regulator in Yang-Mills theory is smooth, as the cutoff term can be interpreted as the local deformation of Yang-Mills theory. Still, even for local deformations the correct description of the massless limit is intricate as we shall see later, see in particular \\Cref{sec:YM:FunRel}. \n\n\n\n\\subsection{STI \\& mSTI}\\label{sec:YM:STI+mSTI}\n\nClassical BRST symmetry with the infinitesimal transformations \\labelcref{eq:YM:YMBRSTtrafo} leads to symmetry identities on the quantum level, that can be formulated in terms of a master equation, see e.g.~\\cite{Zinn-Justin:1974ggz, Zinn-Justin:1999euj},\n\\begin{align}\n\\int_x \\left( \\frac{\\delta \\Gamma}{\\delta Q_{A,\\mu}^a} \\frac{\\delta \\Gamma}{\\delta A_\\mu^a} +\\frac{\\delta \\Gamma}{\\delta Q_c^a} \\frac{\\delta \\Gamma}{\\delta c^a}+\\frac{\\delta \\Gamma}{\\delta Q_{\\bar{c}}^a} \\frac{\\delta \\Gamma}{\\delta \\bar{c}^a}\\right)= 0\\,, \n\\label{eq:YM:STI}\n\\end{align}\nand integrating out the (Gau{\\ss}ian) Nakanishi-Lautrup field $B$ converts \\Cref{eq:YM:STI} into \n\\begin{align}\n\\int_x \\left( \\frac{\\delta \\Gamma}{\\delta Q_{A,\\mu}^a} \\frac{\\delta \\Gamma}{\\delta A_\\mu^a} +\\frac{\\delta \\Gamma}{\\delta Q_c^a} \\frac{\\delta \\Gamma}{\\delta c^a}-\\frac{1}{\\xi} (\\partial_\\nu A^a_\\nu)\\partial_\\mu \\frac{\\delta \\Gamma}{\\delta Q_{A,\\mu}^a}\\right)= 0\\,.\n\\label{eq:YM:STIwo}\n\\end{align}\nIn the derivation of \\labelcref{eq:YM:STIwo} one also uses the fact that the anti-ghost field only appears linearly in the action. \n\nFinally, the introduction of the cutoff term \\labelcref{eq:YM:cutoff} leads to a modification of the symmetry identities \\labelcref{eq:YM:STI} or \\labelcref{eq:YM:STIwo}, the \\textit{modified Slavnov-Taylor identity} (mSTI). It is derived similarly to the flow equation \\labelcref{eq:YM:fRG}: there the right hand side can be interpreted as the equation which governs the violation of scale invariance. For the mSTI, the additional term in comparison to \\labelcref{eq:YM:STI} or \\labelcref{eq:YM:STIwo} originates from the lack of BRST invariance of the regulator term. As for the flow equation, it is a  one-loop exact equation. In the presence of the Nakanishi-Lautrup field $B$, the mSTI takes the concise form, \n\\begin{align}\n&\\int_x  \\Gamma_{Q_i}  \\Gamma_{\\Phi_i} =\\int_{x,y}\\, \\left( R\\, G\\right)_{\\Phi_i \\Phi_j}\\, \\Gamma_{Q_{j} \\Phi_i}\n\\label{eq:YM:mSTI}\n\\end{align}\nwith the notation  \n\\begin{align}\n\t\\Gamma_{Q_1\\cdots Q_n\\Phi_{n+1}\\cdots \\Phi_{n+m}} \n\t= \\frac{ \\delta\\Gamma[\\Phi,Q]}{ \\delta {Q_1}\\cdots \\delta {Q_n}  \\delta\\Phi_{n+1} \\cdots \\delta{\\Phi_{n+m}}}\\,,\n\t\\end{align}\nfor mixed derivatives of the effective action with respect to BRST currents and fields. The integral $\\int_{x,y}$ on the right-hand side of \\labelcref{eq:YM:mSTI} constitutes a space-time trace of the product of operators $(R\\,G)(x,y)$ and $\\Gamma_{Q \\Phi}(y,x)$. On both sides of \\labelcref{eq:YM:mSTI}, a sum over species of fields as well as internal and Lorentz indices is implied. For example, the left-hand side of \\labelcref{eq:YM:mSTI} is simply that of \\labelcref{eq:YM:STI}, if re-instating all indices. \n\nAs already mentioned above, the right-hand side of \\labelcref{eq:YM:mSTI} originates in the breaking of BRST invariance induced by the regulator term, similarly to the right-hand side of the flow equation, which manifests the breaking of scale invariance induced by  the regulator term. We emphasise, that \\labelcref{eq:YM:mSTI} is derived with the same implicit assumption of the independence of the UV renormalisation procedure of the cutoff term. We will discuss the self-consistency of this assumption in \\Cref{sec:YM:FunRel}. \n\nFor $k\\rightarrow 0$ the mSTI reduces to the STI. Thus, satisfying the mSTI at all scales $k$, guarantees gauge invariance of observables at $k=0$. Modified Slavov-Taylor identities, Ward identities and Nielsen identities have been studied intensively in the fRG approach as they are key to quantitatively reliable approximations in gauge theories. For a partial use for momentum-dependent correlation functions in QCD see \\cite{Cyrol:2016tym, Cyrol:2017ewj}, for a full derivation and further discussions in the present context, see e.g.~\\cite{Pawlowski:2021tkk, LectureNotesprep}. For generic fRG literature on modified symmetry identities see e.g.\\ \\cite{Bonini:1993sj,  Ellwanger:1994iz, Becchi:1996an, DAttanasio:1996tzp, Reuter:1997gx, Freire:2000bq, Igarashi:1999rm, Igarashi:2001ey, Pawlowski:2003sk, Pawlowski:2005xe, Igarashi:2009tj, Donkin:2012ud, Lavrov:2012xz, Sonoda:2013dwa, Safari:2015dva, Cyrol:2016tym, Safari:2016gtj, Igarashi:2016gcf, Cyrol:2017ewj, Asnafi:2018pre, Igarashi:2019gkm, Barra:2019rhz, Lavrov:2020exa, Pawlowski:2020qer}, a complete list of references can be found in the recent review see \\cite{Dupuis:2020fhh}.  \n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonDressingPlot}\n\t\t\\caption{Gluon dressing $1\/Z_A(p)$.}\n\t\t\\label{fig:gluon_dressing}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GhostDressingPlot}\n\t\t\\caption{Ghost dressing $1\/Z_c(p)$.}\n\t\t\\label{fig:ghost_dressing}\n\t\\end{subfigure}\n\t\\caption{Propagator dressings form the fRG for scaling (solid) and decoupling (dashed) solutions as a function of momentum $p$ in comparison to the lattice results (squares) of~\\cite{Sternbeck:2006cg}. The global normalisation of the lattice results was fixed by our scaling solution, c.f.~\\Cref{app:YM:scalesetting}.\\hspace*{\\fill}}\n\t\\label{fig:YM:propdressings}\n\\end{figure*}\n\\subsection{Vertex Expansion and Truncations}\n\\label{sec:YM:VertexExpansion}\nWe expand the effective action in terms of $\\Phi$ and $Q$ vertices. At vanishing BRST current, $Q=0$, this entails, \n\\begin{align}\n\\Gamma_k[\\Phi,0] = \\sum_{n=1}^{\\infty}\\int \\frac{1}{n!}\\Gamma_{\\Phi_1\\cdots\\Phi_n}\\Phi_n\\dots\\Phi_1\\,,\n\\label{eq:YM:vertexexpansion}\n\\end{align}\nwith the superfield $\\Phi$ introduced \\labelcref{eq:Phi}, the 1PI correlation functions $\\Gamma_{\\Phi_1\\cdots\\Phi_n}$, and the normalisation $\\Gamma_k[0,0]=0$. The prefactor $1\/n!$ is a vector-factorial, where each component corresponds to the factorial of the number of fields of the same species in the summand. Including BRST and mixed vertices is done analogously. Inserting the vertex expansion \\labelcref{eq:YM:vertexexpansion} into the fRG equation \\labelcref{eq:YM:fRG}, one sees readily, that a full solution of the theory requires the complete set of 1PI correlation functions. Specifically, the flow of an $n$-point correlation function $\\Gamma_{\\Phi_1\\cdots\\Phi_n}$ depends on correlation functions $\\Gamma_{\\Phi_1\\cdots\\Phi_m}$ with $2\\leq m\\leq n+2$. This leads to an infinite tower of coupled integral-differential equations. For most practical purposes this tower has to be truncated. Similar considerations apply to all closed functional master equations and in particular to the set of DSEs, whose towers of coupled integral equations also satisfy $2\\leq m\\leq n+2$. For a complete survey on results for Landau gauge correlation functions we refer to the reviews \\cite{Dupuis:2020fhh, Alkofer:2000wg, Fischer:2006ub, Binosi:2009qm, Maas:2011se, Boucaud:2011ug, Huber:2018ned, \nPelaez:2021tpq}. \n\nFor this work, we restrict ourselves to the truncation shown in \\Cref{fig:YM:flowequations} in the appendix. The gluon and ghost propagators are obtained from their respective two-point functions via \\labelcref{eq:YM:propeq}. The two-point functions read \n\\begin{align}\n\\Gamma_{AA,\\mu\\nu}^{ab}(p) =&\\, \\Pi^\\perp_{\\mu\\nu}(p)Z_A(p) p^2\\delta^{ab}+\\Pi^\\parallel_{\\mu\\nu}(p)\\Gamma_{AA}^\\parallel(p)\\delta^{ab}\\,,\\nonumber\\\\[10pt]\n\\Gamma_{\\bar{c}c}^{ab}(p) = & \\,Z_c(p)p^2\\delta^{ab}\\,,\n\\label{eq:propsplit}\\end{align}\nwith the transverse and longitudinal projection operators,\n\\begin{align}\n\t\\Pi^{\\perp}_{\\mu \\nu}(p) = \\,\\left(\\delta_{\\mu\\nu}-\\frac{p_\\mu p_\\nu}{p^2}\\right)\\,,\\qquad \\Pi^\\parallel_{\\mu \\nu}(p) =\\, \\frac{p_\\mu p_\\nu}{p^2}\\,.\n\t\\label{eq:YM:ProjOps}\n\\end{align}  \nThe ghost and gluon dressing functions obtained from our computation are depicted in \\Cref{fig:YM:propdressings}. The respective gluon propagator is shown in \\Cref{fig:gluon_prop}. There, the present results are compared to lattice data. For recent results see in particular~\\cite{Cyrol:2016tym} (fRG) and~\\cite{Huber:2020keu, Aguilar:2021lke, Horak:2021pfr, Aguilar:2021uwa, Horak:2022myj} (DSE). \n\n\nThe longitudinal scalar part $\\Gamma_{AA}^\\parallel$ in \\labelcref{eq:propsplit} contains the gauge fixing part that diverges in the Landau gauge limit $\\xi\\to 0$ and a regular contribution, that originates from the breaking of BRST invariance for finite $k$ and vanishes in the limit $k\\to 0$. For $k\\neq 0$ we have  \n\\begin{align}\\label{eq:Gammaparal}\n\t\\Gamma_{AA}^\\parallel(p) = \\frac{1}{\\xi} p^2 + \\Gamma_{AA,\\textrm{reg}}^\\parallel \\,.\n\\end{align}\nWe remark that in the Landau gauge limit the gluon propagator is transverse and does not depend on $\\Gamma_{AA,\\textrm{reg}}^\\parallel$, even though the latter contribution does not vanish. This is one of the properties that singles out the Landau gauge not only as the technically least difficult Lorenz gauge, but also suggests the best convergence of the results for correlation functions in a systematic vertex expansions: for any other choice of $\\xi$ the regular part $\\Gamma_{AA,\\textrm{reg}}^\\parallel$ feeds back into the dynamics of the system, even though its integrated impact has to vanish at vanishing cutoff scale. A more detailed discussion including the consistency of different functional relations (fRG, DSE, mSTI) is deferred to \\Cref{sec:YM:FunRel}. \n\n\n\\begin{figure*}\n\t\\centering\n\t\\begin{minipage}[t]{0.45\\textwidth}\n\t\t\\begin{figure}[H]\n\t\t\t\\includegraphics[width=\\linewidth]{figures\/GluonPropPlotPepe}\n\t\t\t\\caption{Scaling (solid) and decoupling (dashed) gluon propagator as a function of momentum $p$ in comparison to the lattice data (squares) from~\\cite{Sternbeck:2006cg} and (triangles) from~\\cite{Aguilar:2021okw}. The global normalisation of the lattice data was fixed by our scaling solution, c.f.~\\Cref{app:YM:scalesetting}.\\hspace*{\\fill}}\n\t\t\t\\label{fig:gluon_prop}\n\t\t\\end{figure}\n\t\\end{minipage}\n\t\\hspace{0.05\\textwidth}\n\t\\begin{minipage}[t]{0.45\\textwidth}\n\t\t\\begin{figure}[H]\n\t\t\t\\includegraphics[width=\\linewidth]{figures\/CouplingsPlot}\n\t\t\t\\caption{Running couplings as defined in \\labelcref{eq:YM:couplingsrelation} obtained from different Yang-Mills vertex dressings as a function of momentum $p$.\\hspace*{\\fill}}\n\t\t\t\\label{fig:YM:couplings}\n\t\t\\end{figure}\n\t\\end{minipage}\n\\end{figure*}\nUtilising the projection operators \\labelcref{eq:YM:ProjOps}, we can split correlation functions into their transverse and longitudinal parts,\n\\begin{align}\n\\Gamma_{\\Phi_1\\cdots\\Phi_n} =&\\, \\Gamma_{\\Phi_1\\cdots\\Phi_n}^\\perp +\\Gamma_{\\Phi_1\\cdots\\Phi_n}^\\parallel\\,, \n\\end{align}\nwhere $\\Gamma_{\\Phi_1\\cdots\\Phi_n}^\\perp$ is the completely transverse part of the correlation function, structurally given by $\\Gamma_{\\Phi_1\\cdots\\Phi_n}^\\perp = (\\Pi^\\perp)^n\\Gamma_{\\Phi_1\\cdots\\Phi_n}$.  The longitudinal part simply is the complement, and hence is built up from correlation functions with at least one longitudinal leg. \n\nWe will explain this splitting at the example of the ghost-gluon vertex. Its complete basis is spanned by only two tensor structures:  the classical one, proportional to the anti-ghost momentum, and a longitudinal non-classical one, which is proportional to the gluon momentum, \n\\begin{align}\\label{eq:ghostgluon}\n\\Gamma_{A\\bar{c}c,\\mu}^{abc}(p,q) &= i f^{abc}\\Bigl(\\lambda_{A \\bar{c} c,\\text{cl}}(p,q) q_\\mu +\\lambda_{A \\bar{c} c,\\text{ncl}}(p,q)p_\\mu\\Bigr)\\,.\n\\end{align}\nHowever, we can rewrite this in terms of the projection operator defined in \\labelcref{eq:YM:ProjOps}, which is transverse in the gluon momentum $p$,\n\\begin{align}\n\\Gamma_{A\\bar{c}c,\\mu}^{abc}(p,q) =&\\, i f^{abc}\\Bigl(\\lambda_{A \\bar{c} c}(p,q)\\, \\Pi^{\\perp}_{\\mu \\nu}(p) q_\\nu \\nonumber\\\\[4pt]\n&\\hspace{2cm}+ \\bar \\lambda_{A \\bar{c} c,1}(p,q)\\,p_\\mu\\Bigr)\\,, \n\\label{eq:YM:ghostgluonsplit}\n\\end{align}\nwith \n\\begin{align}\n\\lambda_{A \\bar{c} c}(p,q) &= \\lambda_{A \\bar{c} c,\\text{cl}}(p,q)\\,,  \\nonumber\\\\[1ex]\n\\bar \\lambda_{A \\bar{c} c,1}(p,q) &=  \\frac{p\\cdot q}{p^2} \\lambda_{A \\bar{c} c,\\text{cl}}(p,q) + \\lambda_{A \\bar{c} c,\\text{ncl}}(p,q)\\,.\n\\label{eq:YM:nonclDressing}\n\\end{align}\n\nThe singularity of such a split with projection operators at $p=0$ is reflected in the prefactor $p\\cdot q\/p^2$ of the classical dressing in $\\bar \\lambda_{A \\bar{c} c,1}(p,q)$. It is matched by the respective one in the transverse projection operator multiplying $\\lambda_{A \\bar{c} c}(p,q)$. Therefore, for non-singular dressings in \\labelcref{eq:ghostgluon} we only have a parameterisation singularity.\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GhostGluonPlot}\n\t\t\\caption{Ghost-gluon vertex dressings.}\n\t\t\\label{fig:ghost_gluon}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/ThreeGluonPlot}\n\t\t\\caption{Three-gluon vertex dressings.}\n\t\t\\label{fig:three_gluon}\n\t\\end{subfigure}\n\t\\caption{Dressings of the three-point functions as a function of momentum $p$. Results from the fRG are shown as solid lines, while STI results are shown as dashed lines. The transverse component is black and the longitudinal ones are depicted in green and blue. Please note that the transverse components are only visible in the insets.\\hspace*{\\fill} }\n\t\\label{fig:YM:longcouplingsfrg}\n\\end{figure*}\nIn the present work, we consider relations between the dressings at the symmetric point $p^2=q^2 = -2 p\\cdot q$. In the ultraviolet limit, the non-classical dressing vanishes and we are left with the classical one. Hence, for the present purpose it is convenient to simply use the longitudinal projection of the classical tensor structure, $\\Pi_{\\mu\\nu}^\\parallel(p) q_\\nu$, for the longitudinal part. This leads us to,   \n\\begin{align}\n\t\\Gamma_{A\\bar{c}c,\\mu}^{abc}(p,q) =&\\Bigl(\\lambda_{A \\bar{c} c}(p,q)\\, \\Pi^{\\perp}_{\\mu \\nu}(p)  \\nonumber\\\\[4pt]\n\t&\\hspace{.5cm}+  \\lambda_{A \\bar{c} c,1}(p,q)\\,\\Pi^{\\parallel}_{\\mu \\nu}(p) \\Bigr)\\,\\, i f^{abc} q_\\nu\\,, \n\t\\label{eq:YM:ghostgluonsplitP}\n\\end{align}\nsee also \\cite{Cyrol:2016tym}. The number index in $\\lambda_{A \\bar{c} c,1}$ indicates the number of longitudinally projected gluons of the classical tensor structure of the vertex, and no index corresponds to the dressing of the fully transverse tensor structure. The dressing $\\lambda_{A \\bar{c} c,1}$ is given by  \n\\begin{align}\\label{eq:lambda1}\n\t\\lambda_{A \\bar{c} c,1}(p,q) = \\lambda_{A \\bar{c} c,\\text{cl}}(p,q) + \\frac{p^2}{p\\cdot q} \\lambda_{A \\bar{c} c,\\text{ncl}}(p,q)\\,.\n\t\\end{align}\nWe have in particular $\\bar \\lambda_{A \\bar{c} c,1}(p,q) = -2 \\lambda_{A \\bar{c} c,1}(p,q)$ at the symmetric point. Moreover, for all momenta except those with $p\\cdot q =0$, the dressing in \\labelcref{eq:lambda1} has the ultraviolet limit $\\lambda_{A \\bar{c} c,1}\\to \\lambda_{A \\bar{c} c,\\textrm{cl}}$ which facilitates the following discussions. In contradistinction, we find $\\bar \\lambda_{A \\bar{c} c,1}\\to (p\\cdot q)\/p^2 \\lambda_{A \\bar{c} c,\\textrm{cl}}$. In conclusion, the transverse ghost-gluon dressing is equivalent to the classical one, whereas the longitudinal dressing is a combination of the classical and non-classical dressing. For a comparison of the ghost-gluon dressings in \\labelcref{eq:YM:ghostgluonsplitP}, see \\Cref{fig:ghost_gluon} and \\Cref{fig:YM:accnonclcouplingsfrg} in the \\Cref{app:YM:FRGYM}. For recent results see in particular~\\cite{Cyrol:2016tym} (fRG) and~\\cite{Huber:2020keu, Aguilar:2021okw, Eichmann:2021zuv} (DSE). \n\nWe now proceed by parametrising the BRST and gluonic vertices in terms of transverse and longitudinal projections analogously to the parameterisation of the ghost-gluon vertex discussed above. With the anti-ghost shift symmetry of the effective action, we can relate the BRST vertices $\\Gamma_{cQ_A}$ and $\\Gamma_{AcQ_A}$ to the ghost two-point and the ghost-gluon vertex. A derivation thereof can be found in \\Cref{app:YM:shiftsym} and we arrive at \n\\begin{align}\n\\Gamma^{ab}_{c \\,Q_A,\\mu}(p) =& -i Z_c(p) p_\\mu \\delta^{ab}\\,,\\nonumber\\\\[10pt]\n\\Gamma^{abc}_{Ac\\,Q_A,\\mu\\nu}(p,q) =&\\, -f^{abc}\\Bigr(\\Pi^{\\perp}_{\\mu \\nu}(p)\\lambda_{A \\bar{c} c}(p,-q-p)\\nonumber\\\\[4pt]\n&\\hspace{.5cm}+ \\Pi^{\\parallel}_{\\mu \\nu}(p)\\lambda_{A \\bar{c} c,1}(p,-q-p)\\Bigl)\\,.\n\\label{eq:YM:BRSTVertices1}\n\\end{align}\nNotably, these identities even hold diagramatically true for the flows of the dressings. The other BRST two-point function and vertex are parameterised as,\n\\begin{align}\n\\Gamma^{ab}_{A Q_{\\bar{c},}\\mu}(p) &= -i p_\\mu \\frac{1}{\\xi} \\delta^{ab}\\,,\\nonumber\\\\[4pt]\n\\Gamma_{cc\\,Q_c}^{abc}(p,q) &= -i f^{abc}\\lambda_{Q_ccc}(p,q)\\,.\n\\label{eq:YM:BRSTVertices2}\n\\end{align}\n\n\nIn the present work we approximate the three- and four-gluon vertices with their fully dressed classical tensor structures, noted as  $\\tau_{A^3,\\text{cl}}$ and $\\tau_{A^4,\\text{cl}}$. The purely transverse parts are obtained by contracting the classical tensor structure with three and four transverse projection operators respectively. The complements now contain mixed vertices with longitudinal and transverse legs as well as a purely longitudinal part. The parts with at least three longitudinal legs do not contribute since they do not feed back into the diagrams and are therefore not included in our truncation. Thus we parameterise the gluonic vertices as,\n\\begin{align}\\label{eq:3-4}\n\\Gamma_{A^3}(p,q) = &\\left(\\lambda_{A^3}\\left(\\Pi^\\perp\\right)^3+\\lambda_{A^3,1}\\left(\\Pi^\\perp\\right)^2\\Pi^\\parallel\\right.\\nonumber\\\\[4pt]\n&\\left.+\\lambda_{A^3,2}\\Pi^\\perp\\left(\\Pi^\\parallel\\right)^2\\right) \\tau_{A^3,\\text{cl}}+\\cdots \\,\\nonumber\\\\[10pt]\n\\Gamma_{A^4}(p,q,r)   =&\\left(\\lambda_{A^4}\\left(\\Pi^\\perp\\right)^4+\\lambda_{A^4,1}\\left(\\Pi^\\perp\\right)^3\\Pi^\\parallel\\right.\\nonumber\\\\[4pt]\n&\\left.+\\lambda_{A^4,2}\\left(\\Pi^\\perp\\right)^2\\left(\\Pi^\\parallel\\right)^2\\right) \\tau_{A^4,\\text{cl}}+\\cdots \\,\n\\end{align}\nwhere indices and momenta are dropped and the permutations are implicit for the sake of simplicity. The dots indicate terms proportional to $(\\Pi^\\parallel)^n\\tau_{A^3,\\text{cl}}$ and $(\\Pi^\\parallel)^n\\tau_{A^4,\\text{cl}}$ with $n\\geq 3$, i.e. with at least $3$ longitudinally projected gluon legs, as well as further, non-classical,  tensor structures. We emphasise that for $p\\cdot q= 0$ or $p\\cdot r=0$ the parameterisation in \\labelcref{eq:3-4} may lead to parameterisation singularities for the dressings, that are then seen in the respective projections of the diagrams in the functional relations. For further details on the parameterisation, see \\Cref{app:YM:tensors} and e.g.\\ \\cite{Eichmann:2021zuv} for a similar parametrisation. The longitudinal four-gluon vertices are approximated by their STI-values, see \\labelcref{eq:YM:4glSTI}. For a comparison of the dressings in \\labelcref{eq:3-4} see \\Cref{fig:three_gluon} and \\Cref{fig:fourgluon}. For recent results see in particular~\\cite{Cyrol:2016tym} (fRG) and~\\cite{Eichmann:2014xya, Huber:2020keu, Aguilar:2021lke, Eichmann:2021zuv} (DSE). \n\nThe remaining dressings are obtained from their respective fRG equations, that are depicted in \\Cref{app:YM:Diags} in \\Cref{fig:YM:flowequations}. They are evaluated at a symmetric point for three(four)-point functions,\n\\begin{align}\np_i\\cdot p_j = \\begin{cases}\n\\;\\;\\;\\;p^2  &\\text{for $i=j$,}\\\\[10pt]\n-\\frac{1}{n-1} p^2& \\text{otherwise, where $n=3(4)$.}\n\\end{cases}\n\\label{eq:YM:symmPoint}\n\\end{align} \nFor simplicity, the vertex dressings feeding back into the fRG equations are evaluated at the average momentum configuration $\\lambda_i(\\bar{p})$ with,\n\\begin{align}\n\\bar{p}^2 =  \\frac{1}{n} \\sum_{i=1}^{n}p_i^2\\,,\n\\label{eq:YM:avmomconfig}\n\\end{align}\nwhich has been shown to be a good approximation, see e.g.~\\cite{Huber:2015fna}.\nFrom the transverse dressings we can derive momentum dependent running couplings,\n\\begin{align}\n\\alpha_{A\\bar{c}c}(p) &= \\frac{1}{4\\pi}\\frac{\\lambda_{A \\bar{c} c}(p)^2}{Z_A(p)Z_c(p)^2}\\,,\\nonumber\\\\[10pt]\n\\alpha_{A^3}(p) &= \\frac{1}{4\\pi}\\frac{\\lambda_{A^3}(p)^2}{Z_A(p)^3}\\,,\\nonumber\\\\[10pt]\n\\alpha_{A^4}(p) &= \\frac{1}{4\\pi}\\frac{\\lambda_{A^4}(p)}{Z_A(p)^2}\\,.\n\\label{eq:YM:couplingsrelation}\n\\end{align}\nThese vertex couplings are perturbatively two-loop degenerate (two-loop universality), leading to \n\\begin{align}\n\\alpha_{A\\bar{c}c}(p)\\approx \\alpha_{A^3}(p)\\approx \\alpha_{A^4}(p)\\,,\\quad\\text{for} \\quad p\\gtrsim  5\\,\\text{GeV}.\n\\label{eq:YM:couplingsdegeneracy}\n\\end{align}\nThis property is reflected well in our truncation, see \\Cref{fig:YM:couplings}. We emphasise that  \\labelcref{eq:YM:couplingsdegeneracy} is non-trivial, as our initial condition only feature constant vertices. These initial conditions are chosen or tuned such that \\labelcref{eq:YM:couplingsdegeneracy} is fulfilled. Satisfying \\labelcref{eq:YM:couplingsdegeneracy} over several orders of magnitude is already a significant indication for gauge consistency. For a detailed discussion of this aspect see~\\cite{Cyrol:2016tym} and more information on the tuning procedure is given in \\Cref{sec:YM:transverse}. \n\n\\subsection{Functional Relations and Consistency Constraints}\n\\label{sec:YM:FunRel}\n\nThe computation of functional relations is facilitated in Landau gauge Yang-Mills theory, if contrasted with general covariant gauges: in the Landau gauge the set of all transverse functional relations is closed, i.e.\n\\begin{subequations}\n\t\\label{eq:FunRelP}\n\\begin{align}\n\\Gamma_{(n)}^\\perp = \\text{funRel}^\\perp_{(n)}[\\{\\Gamma_{(2\\leq m\\leq n+2)}^\\perp\\}]\\,,\n\\label{eq:YM:closedtransversal}\n\\end{align}\nwhere $\\{\\Gamma_{(2\\leq m\\leq n+2)}^\\perp\\}$ indicates the set of all transverse $m$-point correlation functions with $2 \\leq m\\leq n+2$.  In contradistinction, the longitudinal correlation functions satisfy \n\\begin{align}\\label{eq:YM:long}\n\\Gamma_{(n)}^\\parallel = \\text{funRel}^\\parallel_{(n)}[\\{\\Gamma_{(2 < m\\leq n+2)}^\\parallel\\},\\{\\Gamma_{(2\\leq m\\leq n+1)}^\\perp\\}]\\, .\n\\end{align}\n\\end{subequations}\nNote that the longitudinal two-point function $\\Gamma_{(2)}^\\parallel$ does not feed back into the longitudinal relations \\labelcref{eq:YM:long} and \\labelcref{eq:YM:mSTISpace}. \\Cref{eq:YM:closedtransversal} follows from the fact that the gluon propagator $G_A$ as well as $(G\\,\\dot R\\, G)_{AA}$ are transverse in the Landau gauge and so are all internal legs. Also, the propagator only depends on $\\Gamma_{(2)}^\\perp$. The structure \\labelcref{eq:FunRelP} applies to fRG equations and  DSEs, for more details see \\cite{Fischer:2008uz, Cyrol:2016tym, Dupuis:2020fhh, LectureNotesprep}. \n\nThe  mSTIs in \\labelcref{eq:YM:mSTI} involve longitudinal and transverse correlation functions. They are similar to \\labelcref{eq:YM:long}, reading \n\\begin{align}\n\\Gamma_{(n)}^\\parallel = \\text{mSTI}_{(n)}[\\{\\Gamma_{(2 < m\\leq n+2)}^\\parallel\\},\\{\\Gamma_{(2\\leq m\\leq n+1)}^\\perp\\} ]\\,. \n\\label{eq:YM:mSTISpace}\n\\end{align}\nNote that \\labelcref{eq:YM:mSTISpace} constitutes yet another tower of functional relations for longitudinal correlation functions. Naturally, as for truncated towers of fRG equations and DSEs, its solution for a given truncation will in general differ from both, the solution of the longitudinal fRG equations and the longitudinal DSEs (or any other tower of longitudinal functional relations). Moreover, for a finite set of longitudinal and transverse correlation functions, one may simply use the (m)STI for computing the longitudinal correlation functions. Such a scheme seemingly builds in gauge consistency by construction. Naturally, the longitudinal correlation functions will then in general not satisfy the respective fRG equations or DSEs. In conclusion, it is not the violation of the mSTI alone, which comprises the information about gauge consistency, but the combination of all functional relations.\n\\begin{figure}[t]\n \t\\includegraphics[width=0.98\\linewidth]{figures\/FourGluonPlot}\n\t\\caption{Dressings of the four-point function as a function of momentum $p$. The result from the fRG is shown as a solid line, while STI results are shown as dashed lines. The transverse component is black and the longitudinal ones are depicted in green and blue.\\hspace*{\\fill}}\n\t\\label{fig:fourgluon}\n\\end{figure}\n\nHowever, even though \\labelcref{eq:YM:mSTISpace} is very similar to the fRG equations and DSEs for the longitudinal dressings, there is an important structural difference which is most apparent in the Landau gauge limit: while \\labelcref{eq:YM:closedtransversal} and \\labelcref{eq:YM:long} only depend on the transverse part of the regulator, the mSTI \\labelcref{eq:YM:mSTISpace} depends on the longitudinal part for regulators whose longitudinal part diverges with $1\/\\xi$. A natural class of regulators with this property is that of \\textit{RG-adapted} or spectrally adapted regulators, see \\cite{Pawlowski:2005xe, Gies:2002af}. In this case, the regulators are proportional to the full dispersion of the respective field, which is detailed in \\Cref{app:YM:reg}. This choice ensures, that the effective action in the presence of the regulator has the underlying RG-invariance of the full theory at vanishing regulator, hence the name \\textit{RG-adapted}. Specifically, the longitudinal part of the gluon regulator reads \n\\begin{align}\n\t(R_A)^{\\parallel, ab}_{\\mu\\nu}(p) =&\\,\\delta^{ab} \\Pi^\\parallel_{\\mu \\nu}(p)\\Gamma^{\\parallel}_{AA}(p) \\, r\\left(\\frac{p^2}{k^2}\\right) \\,.\n\t\\label{eq:YM:longregs}\n\\end{align}\nThe class of regulators with \\labelcref{eq:YM:longregs} leads to \n\\begin{align}\\label{eq:RGTL}\n\t\\lim_{\\xi\\to 0} \tp_\\mu (G_{AA} R_A)_{\\mu\\nu} = p_\\nu \\frac{r_A}{1+r_A}(p^2\/k^2)\\,,\n\\end{align}\nand hence one of the internal lines in the mSTI diagrams has a longitudinal part. In turn, for gluon regulators with a regular longitudinal part we have \n\\begin{align}\\label{eq:RGT}\n\tp_\\mu (G_{AA} R_A)_{\\mu\\nu} \\propto \\xi p_\\nu\\,,\n\\end{align}\nand hence all internal lines are transverse in the Landau gauge. Then, the mSTI has the same structure as \\labelcref{eq:YM:long} in terms of contributing correlation functions. \n\nInterestingly, the fRG equations and DSEs (in the presence of a regulator), \\labelcref{eq:FunRelP}, are the same in the Landau gauge limit for regulators with either  \\labelcref{eq:RGTL} or \\labelcref{eq:RGT}. However, it can be shown, that the contributions of the longitudinal part of the propagator \\labelcref{eq:RGTL}  do not cancel in the mSTI. In particular, the effective gluon mass $m_k^2$ from the STI only receives contributions from the longitudinal part of $(R_k G)_{AA}$. In conclusion, only for gluon regulators whose longitudinal part contains the gauge fixing term $1\/\\xi p^2 r(p^2\/k^2)$, the effective mSTI gluon mass and that from the fRG flow equation or DSE agree, and the implicit assumption of $k$-independence of the renormalisation procedure is self-consistent. In short, only for RG-adapted regulators we do not implicitly introduce a gluon mass (counter) term on the level of the classical action. \n\nThis intricacy emphasises the importance of gauge consistency of the regularisation procedure. The comparison of functional mSTI and fRG\/DSE always allows us to choose a longitudinal regulator $R_A^\\parallel$ such that all functional relations are compatible. However, this also implies that gauge consistency of truncated fRG flows or DSEs should rather be checked with a set of correlation functions and not a single one.   \n\nThroughout this work we will use \\textit{RG-adapted} regulators \\labelcref{eq:YM:longregs}, which are also \\textit{gauge consistent}, as discussed above. A more detailed study of this intricacy and the general regulator (shape) dependence is still under investigation and will be published elsewhere. \n\nWe conclude this section with two remarks. Firstly, for regular correlation functions, one can use the STIs for also extracting a part of the respective transverse correlation functions. Loosely speaking, regularity implies  \n\\begin{align}\\label{eq:regular}\n\t|\\partial_{p_i} \\Gamma_{(n)}| < \\infty\\,, \n\\end{align}\nin which case the longitudinal and transverse parts are linked. However, in the present case of Yang-Mills theories, it can be shown that regular correlation functions are at odds with confinement. In turn, in perturbation theory, one can show that the regularity assumption holds true. Indeed, it is also related to the identification of running couplings \\labelcref{eq:YM:couplingsrelation}. This adds yet another layer of complexity to the current situation: for perturbative and semi-perturbative momenta regularity holds true and we should see the respective relations between longitudinal and transverse correlation functions. \n\nIn summary, this leaves us with the necessity as well as a large variety of non-trivial consistency checks. We also emphasise that great care is needed in the interpretation of violations of the mSTI or alternatively of other functional relations for longitudinal correlation functions within truncations. \n\n\n\\subsection{Confinement}\nConfinement describes the effect that no coloured particles can be detected directly by experiments, i.e.\\ the absence of coloured states from the physical asymptotic spectrum of our theory, and is in one-to-one correspondence to a \\textit{dynamical} mass gap $m_\\textrm{gap}^\\textrm{\\tiny YM}$ in the physical spectrum of Yang-Mills theory. It is fundamentally linked to the non-Abelian gauge group of QCD, and hence it is most cleanly studied in Yang-Mills theory in the absence of dynamical quarks that complicate the matter. \n\nIn Landau gauge Yang-Mills theory confinement is in one-to-one correspondence to a mass gap (finite correlation length $\\xi_\\textrm{conf}$) in the gluon propagator, as has been shown in \\cite{Braun:2007bx, Fister:2013bh}. There, it has been also shown that the confinement-deconfinement temperature $T_\\textrm{conf}$ is proportional to the (inverse) screening length, $T_\\textrm{conf}\\propto 1\/\\xi_\\textrm{conf}$. Moreover, it also sets the scale of the glueball spectrum, $m_\\textrm{glueball}\\propto 1\/\\xi_\\textrm{conf}$, and in conclusion $m^\\textrm{\\tiny YM}_\\textrm{gap}\\propto 1\/\\xi_\\textrm{conf}$. Finally, it also sets the saturation scale of the effective charge, see \\cite{Aguilar:2009nf}. \n\n\\begin{figure}[t]\n\t\\includegraphics[width=0.98\\linewidth]{figures\/GluonMassAllPlotnew}\n\t\\caption{Effective gluon mass $m_\\perp^2$, defined in \\labelcref{eq:mPerp}, at $k=k_{\\mathrm{min}}$ from different fRG solutions as a function of the difference $m_{\\bot,\\Lambda}^2-m_{\\mathrm{ref}}^2$ of the mass parameters at the initial cutoff scale $\\Lambda$. More details on the effective gluon mass at a finite cutoff scale are deferred to \\Cref{app:YM:reg}. The reference mass $m_{\\mathrm{ref}}^2$ indicates the initial mass of the extrapolated UV value for the mass parameter of the true scaling solution, see also~\\Cref{app:YM:reg}; and the light blue triangle indicates our approximate scaling solution. The black markers correspond to Higgs-type and the blue markers to confining solutions. The dark blue square is used as a representative decoupling solution. The black pentagon is used as a representative Higgs-type solution. \\hspace*{\\fill}}\n\t\\label{fig:gluon_mass_tuning}\n\\end{figure}\n\nIn conclusion, while the mass gap $1\/\\xi_\\textrm{conf}$ in the Landau gauge gluon propagator is a property of a gauge-fixed correlation function, it is directly linked to the \\textit{dynamical} mass gap $m_\\textrm{gap}^\\textrm{\\tiny YM}$ in Yang-Mills theory. At small momenta, the mass gap manifests itself as an effective gluon mass parameter $m_\\perp^2$ in the gluon propagator or transverse gluon two-point function, \n\\begin{align}\\label{eq:mPerp}\n\tm_\\perp^2 = \\Gamma_{AA}^\\perp(p=0)\\,, \n\\end{align}\nsee also \\labelcref{eq:mPerpPar}. The effective gluon mass $m_\\perp^2 $ is bounded from below by the screening length $\\xi_\\textrm{conf}$ (subject to an appropriate normalisation of the gluon propagator). \n\nEvidently, the dynamics of Yang-Mills theory for momenta below the confining mass gap is dominated by the infrared aspects of the gauge fixing at hand. Even if restricting ourselves to the Landau gauge, we may encounter different IR solutions, see e.g.~\\cite{Fischer:2008uz}. An investigation of the different types of solutions within our set-up can be found in \\Cref{sec:YM:Discussiondec}. In the present fRG set-up the mass parameter \\labelcref{eq:mPerp} is approximated by its counterpart at the lowest infrared cutoff scale considered, $ k_{\\min}= 135$\\,MeV. The full range of solutions in terms of the effective mass parameter $m^2_{\\perp,k_{\\min}}$ is depicted in \\Cref{fig:gluon_mass_tuning}: right of the minimum the theory approaches massive Yang-Mills theory, while left of the minimum confining Yang-Mills theory is found. The Landau gauge solution found in lattice simulation is best approximated for effective gluon masses about the minimum. In the remainder of this section we discuss the different branches of the effective gluon mass curve depicted in \\Cref{fig:gluon_mass_tuning}. \n\nOne possible mechanism for confinement has been proposed by Kugo and Ojima \\cite{Kugo:1979gm}. The Kugo-Ojima confinement scenario can generally be broken down into three criteria. First, if we have a global, non-perturbative BRST charge, i.e.\\ unbroken BRST symmetry, it can be used to construct the physical state space $\\mathcal{H}_{phys}$. Second, if the global colour charge is unbroken, $\\mathcal{H}_{phys}$ only contains colourless states. And third, the cluster decomposition principle has to be violated in the total state space $\\mathcal{H}$, but not in $\\mathcal{H}_{phys}$. \n\nFor the latter, it has already been argued in \\cite{Nakanishi:1990qm} that this is indeed fulfilled in the covariant operator formulation of QCD. From the second criterion, we can deduce direct implications for our infrared (IR) Green's functions. It can be shown that this requires an IR enhancement of the ghost propagator. This yields a unique IR renormalisation condition. We shall refer to this as the \\textit{scaling} solution. This enhancement is accompanied by a dynamical creation of an effective gluon mass,\n\\begin{align}\n&\\lim\\limits_{p\\rightarrow 0} Z_c(p^2) \\propto (p^2)^\\kappa\\,,\\nonumber\\\\[10pt]\n&\\lim\\limits_{p\\rightarrow 0} Z_A(p^2) \\propto (p^2)^{-2\\kappa}\\,,\n\\label{eq:YM:scalingexponentsscaling}\n\\end{align}\nwith $1\/2 < \\kappa < 1$, see \\cite{vonSmekal:1997ohs, vonSmekal:1997ern} and the reviews \\cite{Alkofer:2000wg, Fischer:2006ub}. This scaling behaviour can be tuned in the present fRG set-up, and we approximate it by the leftmost gluon mass parameter in  \\Cref{fig:gluon_mass_tuning}, represented by the blue triangle. While only an approximation to full scaling even for $p\\to 0$, this solution shows scaling for the physical momenta considered, $p\\gtrsim k_\\textrm{min}$.  The scaling exponents obtained from our results can be found in \\labelcref{eq:YM:scalingexponents}. \n\nThe Kugo-Ojima confinement scenario rests upon at least one non-trivial assumption: the existence of a global BRST charge for the standard Landau gauge BRST symmetry with the \\textit{local} or infinitesimal BRST transformations \\labelcref{eq:YM:YMBRSTtrafo}, for a detailed discussion see also~\\cite{Fischer:2008uz}. First of all, the BRST charge may not exist or may be broken. Second of all, specific realisations of the Landau gauge may include infrared modifications such as in the modified Gribov-Zwanziger scenario. For a review on the latter see~\\cite{Vandersickel:2012tz}, for recent progress within a BRST-invariant formulation see e.g.~\\cite{Capri:2015ixa, Pereira:2016fpn, Capri:2016aif, Capri:2016jgn, Dudal:2017jfw, Capri:2021pye}. \n\nAll these scenarios are signalled by the absence of the Kugo-Ojima scaling: while the gluon propagator still exhibits a mass gap, it settles at a finite value in the infrared, which is well-described by an effective gluon mass in the infrared. We rush to add that there is no trace of this mass in the ultraviolet which is the qualitative difference to massive Yang-Mills theory. The respective ghost exhibits a finite infrared enhancement but shows no scaling. Still, for these infrared solutions of Yang-Mills theories in the Landau gauge the infrared dynamics is decoupled, and they have been called \\textit{decoupling} or \\textit{massive} solutions for the above reasons. In particular, they are realised within lattice simulations in the Landau gauge as well as in many functional implementations. An appealing scenario discussed in functional approaches is the dynamical emergence of the gluonic mass gap via the Schwinger mechanism \\cite{Schwinger:1962tn, Schwinger:1962tp, Jackiw:1973tr, Eichten:1974et, Poggio:1974qs}, for more recent works see e.g.~\\cite{, Cornwall:1981zr, Cornwall:1988ad, Aguilar:2011xe, Aguilar:2021uwa}, or gluon condensation, see~\\cite{Horak:2022aqx}. In the present work we consider the decoupling solution with the smallest dynamically created effective gluon mass $m^2_{\\perp,k_{\\mathrm{min}}}$ indicated by the blue square data point in \\Cref{fig:gluon_mass_tuning}. In the present approximation this solution resembles best that seen on the lattice. \n\nLastly, we consider solutions with an \\textit{explicit} effective gluon mass. This explicit mass is present in massive extensions of Yang-Mills theory as studied in e.g.~\\cite{Tissier:2010ts, Tissier:2011ey, Serreau:2012cg, Reinosa:2017qtf} in the Curci-Ferrari gauge (CF model), for a recent review see \\cite{Pelaez:2021tpq}. These works aim at modelling Yang-Mills theory as the CF model with a specific mass parameter, which allows to use perturbative expansion techniques.  Alternatively, explicit mass solutions can be generated by a Higgs-type mechanism. In the approximation of the present paper these two physically distinct cases are not distinguished and we call them \\textit{Higgs-type} solutions, see also \\cite{Cyrol:2016tym}. While this type of solution is realised for asymptotically large UV masses $m^2_{\\bot,\\Lambda}$ in massive extensions of  Yang-Mills theory, the transition from the \\textit{Higgs-type} branch to the confining decoupling branch  happens around the minimum in \\Cref{fig:gluon_mass_tuning}, but is not determined yet. We remark, that the mass parameter used in the CF model for modelling Yang-Mills theory is also within this regime. It trivially equals that seen on the lattice as the CF mass is adjusted such that the lattice data are reproduced best. \n\nThe IR behaviour of the decoupling and Higgs-type solution is characterized by\n\\begin{align}\n&\\lim\\limits_{p\\rightarrow 0} Z_c(p^2) \\propto 1\\,,\\nonumber\\\\[10pt]\n&\\lim\\limits_{p\\rightarrow 0} Z_A(p^2) \\propto (p^2)^{-1}\\,.\n\\end{align}\n\n\nMore details in general and on the investigation of the different solutions can be found in \\cite{Fischer:2006ub, Fischer:2008uz, Cyrol:2016tym, Reinosa:2017qtf}. \n\nWe close this section with the discussion of a rather interesting aspect of the results depicted in  \\Cref{fig:gluon_mass_tuning}. They demonstrate a smooth deformation or tuning from the scaling solution towards decoupling solutions in a vertex expansion. The present approximation assumes regular vertices, i.e. the longitudinal ones are directly related to the transverse ones. With such longitudinal vertices, the STIs cannot hold true for small momenta as regular vertices entail a vanishing transverse gluon mass gap, which is at odds with smooth deformations. This suggests that also the vertices in the scaling solution exhibit not only the scaling irregularities but further ones, see~e.g.~\\cite{Eichmann:2021zuv}. \n\n\n\\section{Numerical Results\n\\label{sec:YM:NumericalResults\n\n\nIn this work, we solve the transverse sector of Yang-Mills correlation functions self-consistently in the Landau gauge. Here, self-consistency refers to the fact that all correlation functions computed are fed back to the fRG loops. We also compute the longitudinal dressings and the BRST dressings both from the fRG and the mSTI. We proceed by comparing the results obtained with both approaches. All momentum-dependent correlation functions are evaluated at the symmetric point. As before, for a complete survey on results for Landau gauge correlation functions we refer to the reviews \\cite{Dupuis:2020fhh, Alkofer:2000wg, Fischer:2006ub, Binosi:2009qm, Maas:2011se, Boucaud:2011ug, Huber:2018ned, \n\tPelaez:2021tpq}. \n\n\n\n\n\\subsection{Transverse correlation functions}\n\\label{sec:YM:transverse}\nAs stated in \\Cref{sec:YM:FunRel}, the transverse sector of Yang-Mills theory is closed in Landau gauge. Thus, in a given approximation to the transverse sector, we deal with a closed and finite system of momentum-dependent self-consistent coupled differential equations. The mSTI for the gluon propagator, together with regularity, leads to a non-zero transverse gluon mass $m^2_{\\perp,\\Lambda}$ at the cutoff. Its exact value is uniquely fixed by demanding a scaling, decoupling or Higgs-type solution. \n\nThe constant initial values of the vertex dressings are chosen such that the respective momentum-dependent couplings are perturbatively degenerate at $k=k_{\\mathrm{min}}$: they satisfy the two-loop exact relation from \\labelcref{eq:YM:couplingsdegeneracy} for perturbative momenta, thus rendering only the initial value of the ghost-gluon dressing a free parameter. It is chosen to be $\\lambda_{A \\bar{c} c,\\Lambda}(p) = 1$ at the cutoff. \n\nAlthough the initial gluon mass parameter as well as the initial vertex dressings are uniquely fixed by the two aforementioned conditions, their exact values are determined via a fine-tuning procedure: one chooses a set of initial conditions for the vertex dressings and varies the initial gluon mass parameter thereby obtaining the different types of solutions and finally, the scaling solution, see e.g. \\Cref{fig:gluon_mass_tuning}. For the respective  scaling solution the difference from the coupling relations \\labelcref{eq:YM:couplingsdegeneracy} is used for retuning the initial vertex dressings.  This process is iterated until the relations \\labelcref{eq:YM:couplingsdegeneracy} are satisfied. The main difficulty in the selection or adaptation of the initial parameters after each step in the process is the fact, that the flow equations are a set of coupled differential equations. This means varying only one initial condition influences the outcome of the computation non-linearly, especially the scaling mass parameter is very sensitive to such a retuning. This also implies that the existence of a set of constant, i.e. momentum-independent, initial conditions leading to the momentum-dependent relations \\labelcref{eq:YM:couplingsdegeneracy} within a truncation over several orders of magnitude is a highly non-trivial result. This has first been achieved in \\cite{Cyrol:2016tym} within the fRG, for a respective DSE work see \\cite{Huber:2020keu}, for fRG results in full QCD see \\cite{Cyrol:2017ewj}. \nThe initial conditions used for our approximation of the scaling solution can be found in \\Cref{app:YM:tuning}. The vertex dressings of the three(four)-point functions generally depend on two(three) momentum variables. To reduce the numerical effort we approximate the momentum dependencies of the dressings to one variable, which we chose to be the average momentum $\\bar{p}$ at the symmetric point, see \\labelcref{eq:YM:avmomconfig}. The symbolic equations for these quantities can be found in figure \\Cref{fig:YM:flowequations} in the appendix and were derived with \\textit{QMeS}-Derivation \\cite{Pawlowski:2021tkk, github:QMeS}. More details on the projection procedure of the equations are given in \\Cref{app:YM:tensors}.\n\n\nThe results for the transverse correlation functions obtained from this setup are shown in \\Cref{fig:YM:propdressings}, \\Cref{fig:gluon_prop} and \\Cref{fig:YM:couplings}. The vertex dressings are depicted in \\Cref{fig:YM:longcouplingsfrg} and \\Cref{fig:fourgluon}. We adapted our scales to that of the lattice data \\cite{Sternbeck:2006cg}, and more details regarding the scale setting and global normalisation can be found in \\Cref{app:YM:scalesetting}.\n\n\nThe couplings are degenerate for scales $p \\gtrsim 2$\\,GeV in agreement with the perturbative STIs. In this region, the correlation functions from the scaling and decoupling solution also agree very well with the lattice result. At smaller momenta, the results are not comparable due to the different gauge fixing procedures on the lattice and in functional computations, see e.g.~\\cite{Maas:2019ggf, Li:2021wol}, as well as the increasing systematic error of the functional computations for momenta $p\\ll 1$\\,GeV in the deep infrared. Beyond the deep infrared regime, we find that the scaling solution matches the lattice results at intermediate scales $p\\approx1\\,$GeV much better than the decoupling solution obtained from the fRG, see \\Cref{fig:gluon_dressing} (dressing) and \\Cref{fig:gluon_prop} (propagator). This is to be expected, as the approximation used here is more amiable to the scaling solution. In turn, the decoupling solutions in the present fRG approximation lack  the mechanism of dynamical mass generation leading to the necessary irregularities in the vertices, see \\cite{Cyrol:2016tym}. \n\nA comparison of the transverse dressings from the fRG obtained from our setup to vertex dressings from lattice computations \\cite{Cucchieri:2006tf, Cucchieri:2008qm, Maas:2016pr} and to DSE dressings from \\cite{Huber:2020keu} are depicted in \\Cref{app:YM:FRGYM} in \\Cref{fig:YM:latticedressings}, \\Cref{fig:YM:dsepropdressings},  \\Cref{fig:YM:dseprop} and \\Cref{fig:YM:dsedressings}.\n\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonTwoPointPlot}\n\t\t\\caption{Gluon two-point function.}\n\t\t\\label{fig:gluon_two-point}\n\t\\end{subfigure}\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonMassPlot}\n\t\t\\caption{Absolute value of effective gluon mass squared.}\n\t\t\\label{fig:gluon_mass_flow}\n\t\\end{subfigure}%\n\t\\caption{The gluon two-point function (\\labelcref{fig:gluon_two-point}) as a function of momentum $p$ and the corresponding effective mass (\\labelcref{fig:gluon_mass_flow}) as a function of RG-scale $k$. Results from the fRG are shown as solid lines, while mSTI results are shown as dashed lines. The transverse component is black and the longitudinal ones are depicted in blue. Please note that the transverse two-point function is only displayed in the inset. \\hspace*{\\fill}}\n\t\\label{fig:YM:gluonlongmassplot}\n\\end{figure*}\n\\subsection{Longitudinal \\& BRST correlation functions}\nThe BRST symmetric Yang-Mills action additionally contains source terms $Q_i$ coupled to the respective BRST transformations of the fields. These vertices are fully dressed, see \\Cref{sec:YM:VertexExpansion}, and this dressing carries the non-trivial deformation of the classical BRST transformations in the quantised theory. Then (quantum) BRST symmetry, and thus the mSTIs, relate the longitudinal parts of vertex functions to themselves and the transverse parts, see \\labelcref{eq:YM:mSTISpace}.\n\nEvidently, for a self-consistent check of the (m)STIs we also need to compute the longitudinal and BRST dressings from the fRG. The symbolic equations were also derived using \\textit{QMeS-Derivation} and are depicted in \\Cref{fig:YM:flowequations} in the \\Cref{app:YM:Diags}. Again, the average momentum approximation at the symmetric point \\labelcref{eq:YM:avmomconfig} was used to simplify the momentum-dependence of the vertex dressings.\n\nDiagrammatically, the fRG equations of the transverse and longitudinal quantities are identical, the vertices contributing in the diagrams are however different. An illustrative and important example for these differences is given by the transverse and longitudinal gluon mass. In \\Cref{app:YM:twopointmSTI} we derive the transverse and longitudinal one-loop effective gluon mass from the fRG and mSTI.\n\nAs a first application we discuss the three-gluon dressing with one longitudinal leg. It vanishes at the symmetric point,\n\\begin{align}\n\\lambda_{A^3,1} \\propto \\mathcal{P}^{abc}_{A^3,1,\\mu\\nu\\rho}(p,q) \\Gamma_{A^3,\\mu\\nu\\rho}^{abc}(p,q) &= 0\\,, \n\\label{eq:YM:AAA1nonrunning}\n\\end{align}\nrendering this dressing RG-invariant, $\\dot{\\lambda}_{A^3,1}  = 0$. We are led to \n\\begin{align}\n\\lambda_{A^3,1,k}(p)\\equiv \\lambda_{A^3,1,\\Lambda} \\equiv\\lambda_{A^3,\\Lambda}\\,.\n\\end{align}\nThe same holds true for the three-gluon dressing with three longitudinal legs.\n\nWe use the same initial values for the longitudinal dressings as for the transverse ones. As the longitudinal gluon two-point function does not feed back into the flows, its initial value is chosen such that $\\Gamma^{\\parallel}_{AA,k_{\\mathrm{min}}}(p) = 0$. \n\nThe fine-tuning procedure described in \\Cref{sec:YM:transverse} also applies to the longitudinal sector: the initial values of the longitudinal dressings and of the longitudinal gluon mass at the cutoff have to be tuned such that they satisfy the respective STIs perturbatively at $k_{\\text{min}}$. In this work, we refrain from tuning the longitudinal sector and rather choose regular vertices, where the initial values of the respective longitudinal and transverse vertex dressings agree. We thus expect a small deviation of the STIs for perturbative momenta. This deviation is directly related to the functional nature of the the mSTIs and the fRG equations that are solved within a truncation. \n\nLongitudinal fRG equations are coupled to the transverse sector and to themselves, see \\Cref{sec:YM:FunRel}. In contrast to those in the transverse sector, the equations decouple hierarchically due to the non-running of the three-gluon vertex dressing with one longitudinal leg. Hence we can solve them successively. The projection procedure for the derivation of longitudinal fRG equations can be found in \\Cref{app:YM:tensors}. \n\nThe longitudinal system is solved self-consistently, the input being the transverse dressings. Moreover, we compute the longitudinal four-gluon dressings from the STI \\labelcref{eq:YM:4glSTI} as these relations are computationally far simpler accessible. They are depicted in \\Cref{fig:fourgluon}. The results for the longitudinal fRG dressings are shown in \\Cref{fig:YM:longcouplingsfrg} and \\Cref{fig:YM:gluonlongmassplot}. A discussion is postponed to \\Cref{sec:YM:Discussion}.\n\nWe also compute the BRST dressings from the fRG. Here, shift symmetry allows for the identification of the $Z_{cQ_A}$, $\\lambda_{AcQ_A}$  and $\\lambda_{AcQ_A,1}$ with the ghost and transverse and longitudinal ghost-gluon dressing, see \\Cref{app:YM:shiftsym}, thus one only has to compute the remaining dressing, $\\lambda_{Q_ccc}$. Furthermore, we show in \\Cref{sec:YM:AccmSTIsection} that this dressing is also related to the longitudinal ghost-gluon dressing. \n\n\n\n\\subsection{Modified Slavnov-Taylor Identitites}\n\nIn this section we insert the fRG results into the (m)STI equations and recompute longitudinal correlation functions. Hence, we solve the STIs within our truncation, which offers a non-trivial check of gauge consistency of our solution. A discussion and comparison of the correlation functions obtained from the fRG and from the mSTI can be found in \\Cref{sec:YM:Discussion} and a comparison of thereof for different types of solutions is given in \\Cref{sec:YM:Discussiondec}.\n\nFor the derivation procedure and the diagrammatic contributions, see \\Cref{app:YM:mSTI}. All (m)STIs are evaluated at the symmetric point \\labelcref{eq:YM:symmPoint} and the dressings are computed on the average momentum configuration \\labelcref{eq:YM:avmomconfig}.\n\nNumerically, it is not feasible to integrate the equations down to $k=0$, and we keep a small, yet finite, cutoff $k_{\\text{min}}=0.135\\,$GeV. Consequently, we do not check the gauge consistency of our results with the STIs but rather with the mSTIs at $k=k_{\\text{min}}$. In the following we will refer to expressions of the general form \n\\begin{align}\n\t\\label{eq:shortmSTI}\n\\textrm{STI $-$ Diagrams} \\overset{!}{=} 0\\,, \n\\end{align}\nas mSTIs. In \\labelcref{eq:shortmSTI}, Diagrams stands for the one-loop diagrammatic corrections of the STI due to the BRST symmetry breaking cutoff terms \\labelcref{eq:YM:mSTI}. The symbolic equations are depicted in \\Cref{app:YM:Diags} in \\Cref{fig:YM:mSTIequations}. \n\nAs briefly discussed before, the mSTIs carry the underlying gauge invariance of the theory. Hence, the offer non-trivial gauge consistency checks of the correlation functions of a specific solution (e.g. scaling, decoupling or Higgs-type) of the theory obtained numerically within a truncation. Deviations of the mSTIs can be traced back to three sources: numerical precision, the truncation, and the chosen solution. \n\nThe first can be further split into infrared cutoff effects, that might break the mSTIs for momenta around the IR cutoff scale, $p \\approx k_{\\text{min}}$ and overall numerical precision. Accordingly, a quantitative check of the mSTIs is only possible at momentum scales, that are at least one order of magnitude larger than the cutoff $k_{\\text{min}}$. \n\nTruncation artefacts manifest themselves in a twofold way: effects related to dropped vertices and sub-leading tensor structures should only result in a deviation in the infrared and the semi-perturbative regime, $p\\ll 5\\,$GeV. \n\nNonetheless, even the most sophisticated truncation in the fRG does not lead to correlation functions that fully satisfy the mSTIs: both constitute different resummations and can only agree within approximations that lead to the same full resummations in both classes of functional equations. Thus, one expects an overall deviation of the mSTIs on all momentum scales. Lastly, deviations in the mSTIs might occur due to the chosen solution of Yang-Mills theory, i.e. the scaling, the decoupling or a Higgs-type solution. Generally, we expect the mSTIs to be best fulfilled by the true scaling solution. Moving away from that (along the x-axis in \\Cref{fig:gluon_mass_tuning}) deviations of the mSTIs should occur at larger and larger momentum scales.\n\nThis entails, that an interpretation of the mSTI results is a difficult task as a clear disentanglement of all of the mentioned sources necessitates further computations of the different types of solutions within different truncations, and with different IR cutoffs and numerical precision. \n\n\n\\subsubsection{Gluon Two-Point and Gluon Mass STI}\nThe STI for the longitudinal gluon two-point function is simply given by vanishing quantum corrections of the longitudinal two-point function,\n\\begin{align}\\label{eq:STI2gluon}\n\\Gamma^{\\parallel}_{AA,\\text{reg}} (p) = 0\\,, \n\\end{align}\nwhere the full longitudinal gluon two-point function consists out of the loop contribution $\\Gamma^{\\parallel}_{AA,\\text{reg}}$ and the gauge fixing, see \\labelcref{eq:Gammaparal}. This also implies a vanishing longitudinal gluon mass,\n\\begin{align}\nm_\\parallel^2 = \\Gamma^{\\parallel}_{AA} (p =0) = 0\\,.\n\\label{eq:YM:massSTI}\n\\end{align}\nBoth, the full longitudinal gluon two-point function $ \\Gamma^{\\parallel}_{AA} (p)$ and the longitudinal gluon mass $m_\\parallel^2 $ are depicted in \\Cref{fig:YM:gluonlongmassplot}. In \\Cref{fig:gluon_two_point_mSTI} we depict the the deviation of the two-point function mSTI as defined in \\labelcref{eq:shortmSTI}, the cutoff dependence of transverse (fRG) and longitudinal (mSTI) masses is depicted in \\Cref{fig:YM:gluonlongmassplot}. Technical details can be found in \\Cref{app:YM:reg}. A discussion of these findings is deferred to \\Cref{sec:YM:Discussion}. \n\\begin{figure}[t]\n\t\\includegraphics[width=0.98\\linewidth]{figures\/GluonTwoPointmSTIPlot}\n\t\\caption{Absolute value of the gluon two-point mSTI as a function of momentum $p$ for different values of $k$.\\hspace*{\\fill}}\n\t\\label{fig:gluon_two_point_mSTI}\n\\end{figure}\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonMassdecPlot}\n\t\t\\caption{Absolute value of individual contributions. fRG results are shown with solid lines and mSTI results with dashed lines.\\hspace*{\\fill}}\n\t\t\\label{fig:gluon_mass_flow_abs}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonMassMeasuredecPlot}\n\t\t\\caption{Normalized difference.}\n\t\t\\label{fig:gluon_mass_flow_relative}\n\t\\end{subfigure}\n\t\\caption{Comparison of the flow of the effective transverse (fRG) and longitudinal gluon mass (mSTI) as a function of the cutoff scale $k$. The transverse part is taken from the fRG. The colors differentiate the different solutions (scaling, decoupling, Higgs-type).\\hspace*{\\fill}}\n\t\\label{fig:YM:massMeasuredec}\n\\end{figure*}\n\n\\subsubsection{Ghost-Gluon STI}\n\\label{sec:YM:AccmSTIsection}\nThe STI for the ghost-gluon vertex relates the longitudinal part of ghost-gluon vertex to the BRST projected vertex $Q_ccc$. In our truncation we are led to  \n\\begin{align}\\label{eq:STIGhostGluon}\n2 p_\\mu \\Gamma_{A\\bar{c}c,\\mu} (p,q) Z_c(p)+  q^2 \\Gamma_{ccQ_c}(p,-p-q) Z_c(q)=0\\,.\n\\end{align}\nThus we can derive the STI for the respective dressings at the symmetric point, \n\\begin{align}\n\\lambda_{A \\bar{c} c,1}(p) -\\lambda_{Q_ccc}(p) =0\\,.\n\\label{eq:YM:AccSTI}\n\\end{align}\nA formulation of the ghost-gluon STI in terms of the classical and non-classical tensor basis is presented in \\Cref{app:YM:AccmSTI}. The respective dressings are depicted in \\Cref{fig:YM:longcouplingsfrg} and \\Cref{fig:YM:accnonclcouplingsfrg}.\n\n\n\\subsubsection{Three-Gluon STI}\n\nThe STI for the three-gluon vertex with two longitudinal legs at the symmetric point is given by,\n\\begin{align}\n&\\lambda_{A^3,2}(p) - \\frac{Z_A(p)}{Z_c(p)} \\lambda_{Q_ccc}(p) = 0\\,.\n\\end{align}\nAfter applying the symmetric point ghost-gluon STI in our truncation, \\labelcref{eq:YM:AccSTI}, and dividing by propagator dressings, one obtains the relation for the couplings,\n\\begin{align}\n\\frac{\\lambda^2_{A^3,2}(p)}{4\\pi Z^3_A(p)} = \\frac{\\lambda^2_{A\\bar{c}c,1}(p)}{4 \\pi Z_A(p)Z^2_c(p)} \\,.\n\\end{align}\nUsing the ghost-gluon STI to simplifly the equation is a valid approximation as is shown in \\Cref{sec:YM:DiscussionAcc} and \\Cref{fig:YM:ThreePointmSTIPlots}. Assuming regularity, i.e. agreement of transverse and longitudinal projections of correlation functions (at least for large momenta), the above relation is similar to the well-known perturbative relation presented in \\labelcref{eq:YM:couplingsdegeneracy}.\n\n\nThe dressings of the different longitudinal and transverse projection of the classical tensor structure of the three gluon vertex are shown in \\Cref{fig:YM:longcouplingsfrg}. We remark, that no STI is obtained for the projection with two transverse and one longitudinal leg at the symmetric point, since this projection renders zero.\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GhostGluonmSTIPlotnewMeasure}\n\t\t\\caption{Ghost-gluon vertex dressing. The result is numerically compatible with zero, which becomes evident in the inset.\\hspace*{\\fill}}\n\t\t\\label{fig:ghost_gluon_mSTI}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/ThreeGluonmSTIPlotnewMeasure}\n\t\t\\caption{Three-gluon vertex dressing. The difference gets maximal in the IR, a logarithmic version is shown in the inset.\\hspace*{\\fill}}\n\t\t\\label{fig:three_gluon_mSTI}\n\t\\end{subfigure}\n\t\\caption{Normalized difference of the longitudinal three-point dressings between the fRG and the mSTI at different RG-scales $k$ as a function of momentum $p$.\\hspace*{\\fill}}\n\t\\label{fig:YM:ThreePointmSTIPlots}\n\\end{figure*}\n\n\n\\subsubsection{Four-Gluon STI}\nThe four-gluon STI relates different longitudinal four-gluon dressings to longitudinal and transverse three-gluon and ghost-gluon dressings. We have again used the symmetric point ghost-gluon STI \\labelcref{eq:YM:AccSTI} and arrive at the four-gluon STIs at the symmetric point. The one for the four-gluon vertex with one longitudinal leg, $\\lambda_{A^4,1}(p)$, is given by, \n\\begin{align}\n&\\lambda_{A^4,1}(p) - \\frac{1}{Z_c(p)}\\lambda_{A \\bar{c} c}(\\frac{\\sqrt{10}}{3}p)\\lambda_{A^3}(\\frac{\\sqrt{10}}{3}p) = 0\\,,\n\\end{align}\nThe STI for the four-gluon vertex with two longitudinal legs, $\\lambda_{A^4,2}(p)$, is given by, \n\\begin{align}\\nonumber \n&\\lambda_{A^4,2}(p) - \\frac{32}{33  Z_c(p)}\\lambda_{A \\bar{c} c,1}\\left(\\frac{\\sqrt{10}}{3}p\\right)\\lambda_{A^3}\\left(\\frac{\\sqrt{10}}{3}p\\right)\\\\[1ex] \n&-\\frac{1}{33  Z_c(p)}\\lambda_{A \\bar{c} c}\\left(\\frac{\\sqrt{10}}{3}p\\right)\\lambda_{A^3,1}\\left(\\frac{\\sqrt{10}}{3}p\\right) = 0\\,, \n\\label{eq:YM:4glSTI}\n\\end{align}\nIt is worth mentioning that the momentum argument of the dressings in the STIs at the symmetric point are quite different from the ones usually present in the literature, as e.g.~in~\\labelcref{eq:YM:couplingsdegeneracy}. Using the STI for the three-gluon dressing, the STIs for the four-gluon dressing with one and two longitudinal legs are equivalent to the form in \\labelcref{eq:YM:couplingsdegeneracy} assuming regularity of vertices for large momenta.\n\n\\subsection{Discussion}\\label{sec:YM:Discussion}\n\nWe now compare the longitudinal results of different correlation functions obtained from the fRG and mSTI approach by studying the normalised difference of the respective quantities.\n\n\n\n\\subsubsection{Longitudinal Gluon Mass\n\\label{sec:YM:DiscussionMass\n\nThe longitudinal mSTI mass, $m^2_{\\parallel}$, and the effective transverse fRG mass, $m^2_\\perp$ exhibit the same running for large $k$ which is shown in \\Cref{fig:YM:gluonlongmassplot}. This is in good agreement with the one-loop running, which is the same for both masses, for further details, see \\Cref{app:YM:twopointmSTI}. At about cutoff scales $k\\approx5\\,$GeV, the two mass start deviating from each other. This is triggered by the difference in the IR of the longitudinal and transverse vertices, cf. \\Cref{fig:YM:longcouplingsfrg}. \n\nWe remark, that the values of the longitudinal mSTI and transverse fRG masses are finite and approximately agree at $k_{\\mathrm{min}}$, while the longitudinal fRG mass is zero. This is due to the regularity of the vertices in the present approximation at finite cutoff. In the scaling solution, irregular vertices emerge at $k\\to 0$, while the decoupling solution does not incorporate the irregular vertices required for confinement in the current approximation. The STI, \\labelcref{eq:YM:massSTI}, however, renders the longitudinal gluon mass zero at $k=0$. Evidently, this deviation at least partially stems from our truncation in the vertex sector in the fRG equations, see also the discussion in \\Cref{sec:YM:DiscussionAA}. To circumvent this discrepancy, we have extracted the longitudinal mSTI mass at $p=4\\,k_\\textrm{min}$, in order to avoid the influence of any IR cutoff effects that are present for $p\\lesssim k_\\textrm{min}$. For more details, see \\Cref{app:YM:reg}.\n\n\\subsubsection{Longitudinal Gluon Two-Point Function\n\\label{sec:YM:DiscussionAA\n\nThe solution of the longitudinal two-point fRG equation constitutes the simplest case possible since it does not at all feed back into fRG diagrams in the Landau gauge. We also remark that the longitudinal propagator does feed back to the mSTI, but only its classical gauge fixing part. In summary this entails, that the longitudinal gluon two-point function is obtained by simply integrating the flow equation, and choosing the initial condition for $\\Gamma^{\\parallel}_{AA,\\mu\\nu,\\text{reg},\\Lambda}(p)$ such, that one obtains zero for the evaluated integral. This procedure not only solves the fRG equation but also satisfies the STI at $k_{\\mathrm{min}}$, hence disregarding the small breaking of the STI for small momentum scales. \n\nBy comparing the longitudinal gluon two-point function from the fRG and from the mSTI, one can see that they do not agree well on all scales $k$, see \\Cref{fig:YM:gluonlongmassplot} and \\Cref{fig:gluon_two_point_mSTI}. They are only identical at $k=k_{\\mathrm{min}}$ for $p \\gtrsim 0.5\\,$GeV. For small momenta one can see a deviation of order $\\mathcal{O}(10^{-2}\\,\\text{GeV}^2)$. Albeit small, the deviation is qualitatively similar to the deviation in the three-gluon mSTI, \\Cref{fig:YM:ThreePointmSTIPlots}. Since the longitudinal three-gluon dressing enters the mSTI equation for the gluonic two-point function, but not vice versa, one can therefore conclude, that both deviations are due to the truncation of the fRG equation for the longitudinal three-gluon and four-gluon dressing.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GluonTwoPointmSTIdecPlot}\n\t\t\\caption{Absolute value of gluon two-point STI. The inset shows a logarithmic scale.\\hspace*{\\fill}}\n\t\t\\label{fig:gluon_two-point_mSTI}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/GhostGluonmSTIdecPlotnewMeasure}\n\t\t\\caption{Normalized difference of the ghost-gluon STI. The result is numerically compatible with zero, which is evident from the inset, where the full data range is displayed.\\hspace*{\\fill}}\n\t\t\\label{fig:ghost_gluon_mSTI_relative}\n\t\\end{subfigure}\n\t\\caption{Results for the STI of the gluon two-point and the ghost-gluon vertex dressing as a function of momentum $p$. The colors differentiate IR scenarios.\\hspace*{\\fill}}\n\t\\label{fig:YM:MassPlot}\n\\end{figure*}\n\n\\subsubsection{Longitudinal Ghost-Gluon Dressing\n\\label{sec:YM:DiscussionAcc\n\nIn a general covariant gauge, only $\\lambda_{A \\bar{c} c}$ requires renormalisation. The non-classical dressing $\\lambda_{A \\bar{c} c, ncl}$ vanishes for large momenta. The non-classical dressing obtained from the fRG and mSTI is depicted in \\Cref{fig:YM:accnonclcouplingsfrg}.\n\nWe can see that our results fulfil this property perturbatively since the non-classical dressing is approximately zero for large momenta, $p\\gg 5\\,$GeV. In the infrared, it is however non-trivial. It is worth mentioning that the longitudinal and transverse ghost-gluon dressings do not agree well at any momentum scale $p$.\n\nFrom  \\Cref{fig:YM:longcouplingsfrg},  \\Cref{fig:YM:ThreePointmSTIPlots}, and \\Cref{fig:YM:accnonclcouplingsfrg} in the appendix, we can see that the mSTI and fRG dressing agree perfectly for all momenta $p$ implying that there is a non-trivial cancellation of diagrams within the two functional approaches. Indeed, \\Cref{fig:YM:ThreePointmSTIPlots} merely depicts the numerical accuracy of our computation.\n\n\nGenerally, this non-trivial result implies BRST symmetry being conserved by the respective fRG computation and thus strongly hints at gauge consistency of our setup.\n\n\n\\subsubsection{Longitudinal Three-Gluon Dressing\n\\label{sec:YM:DiscussionAAA\nThe results for the different three-gluon dressings are shown in \\Cref{fig:YM:longcouplingsfrg}. As demonstrated in \\labelcref{eq:YM:AAA1nonrunning} the three-gluon dressing with one longitudinal leg, $\\lambda_{A^3,1}$, does not run. One can see that the dressing with two longitudinal legs from the fRG agrees with the transverse fRG dressing, $\\lambda_{A^3}$, and the longitudinal STI dressing for large momenta $p\\gg 5\\,$GeV. The longitudinal STI dressing and the transverse fRG dressing however agree up to even smaller scales $p\\gg 1.5\\,$GeV. \n\nIn the infrared one can observe that the general behaviour of the longitudinal fRG and STI dressings agree. The STI dressing diverges for $p\\rightarrow 0$ whereas the fRG dressing forms a maximum at $p \\approx 0.2\\,$GeV. The minimum of both dressings is given at approximately the same momentum, $p\\approx 1\\,$GeV.\n\nIn contrast to that, the transverse dressing gets smaller and even becomes negative for $p\\rightarrow 0$. Thus we can observe a clear splitting between the transverse and longitudinal sector in both approaches which is due to the different contributions of longitudinal and transverse vertices in transverse and longitudinal functional equations. \n\nThe slight deviation in the UV can be explained by the logarithmic finetuning procedure: we have chosen the same initial values for the longitudinal and transverse dressings. The contributions on the level of correlation functions to the flows are however different. By choosing different (constant) initial values for the longitudinal dressings at the cutoff it is nevertheless possible to generate a better agreement of the dressings in the UV at $k=k_{\\mathrm{min}}$.\n\nAs stated in the previous section, the three-gluon mSTI shows a deviation at $p\\approx5\\,$GeV. This might be due to the fact that we did not include a full tensor basis of the gluonic sector and even approximated the longitudinal four-gluon dressings with their STI values in the fRG equation of the three-gluon vertex.\n\nFor momenta $p>5\\,$GeV, the three-gluon mSTI is however fulfilled, where the small deviation for very large momenta stems from the tuning procedure, as described before.\n\\begin{figure*}[t]\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/ThreeGluondecPlotnewMeasure}\n\t\t\\caption{Absolute value of individual contributions: fRG (solid lines), STI (dashed lines). \\hspace*{\\fill}}\n\t\t\\label{fig:three_gluon_msti_abs}\n\t\\end{subfigure}%\n\t\\hspace{0.05\\textwidth}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{figures\/ThreeGluonmSTIdecPlotnewMeasure}\n\t\t\\caption{Normalized difference.}\n\t\t\\label{fig:three_gluon_mSTI_relative}\n\t\\end{subfigure}\n\t\\caption{Results for the longitudinal three-gluon dressing $\\lambda_{A^3,2}(p)$ as a function of the momentum $p$. The colors differentiate IR scenarios.\\hspace*{\\fill}}\n\t\\label{fig:YM:theree_gluon_msti_rel}\n\\end{figure*}\n\n\\subsection{Discussion of Different Solutions\n\\label{sec:YM:Discussiondec\n\nIn this section, we present a comparison of the mSTIs for different types of solutions, i.e. the scaling, decoupling, and a Higgs-type solution, that are highlighted amongst the range of solutions in \\Cref{fig:gluon_mass_tuning}.\n\n \nStudying the normalized difference of the transverse andf longitudinal gluon mass, see \\Cref{fig:YM:massMeasuredec}, where the longitudinal mass is obtained from the mSTI and the effective transverse mass from the fRG, one can see that all three solutions agree equally well up until $p \\approx 20\\,$GeV. There, the Higgs-type solution starts to deviate. It is expected that for a more massive Higgs-type solution, the deviation starts at an even larger momentum scale $p$. Vice versa, the scaling solution starts to deviate at a smaller scale than the decoupling solution. \n\nThe ghost-gluon mSTI is fulfilled for all three types of solutions, the small deviations shown in \\Cref{fig:YM:MassPlot} simply depict the numerical precision in the computation.\n\nGenerally, we do not expect the mSTIs to be fulfilled for scales $p \\ll 1\\,$GeV, since non-classical vertices and tensor structures that were not taken into account in our truncation, contribute significantly in this regime. We furthermore expect IR cutoff effects contributing below this scale, being within one order of magnitude of $k_{\\mathrm{min}}$.\n\nHowever, we can clearly see the expected divergence in the gluon two-point, \\Cref{fig:YM:MassPlot}, and three-point mSTI, \\Cref{fig:YM:theree_gluon_msti_rel}, for momenta $p \\ll 1\\,$GeV.\n\nComparing the longitudinal three-gluon dressing  $\\lambda_{A^3,2}$ from the fRG and from the STI for different types of solutions, one can see that the scaling solution yields the best qualitative agreement of the dressings.\n\n\n\n\\section{Conclusion\n\\label{sec:YM:Conclusion\nWe studied the gauge consistency of functional approaches to Yang-Mills within the Landau gauge. For this purpose, we resolved the transverse and longitudinal sector by solving the corresponding fRG flow equations for the dressings of transverse and longitudinal propagators and vertices in a sufficiently advanced truncation. To quantify the violation of gauge consistency in such a set-up, we also computed longitudinal dressings with the associated modified Slavnov-Taylor identities. The results agree within numerical accuracy for momentum scale $p\\gtrsim  1\\,\\mathrm{GeV}$. We found that the agreement of STI dressings and fRG dressings differ for the solutions branches, i.e. for scaling, decoupling and Higgs type solutions: in general, the scaling solution shows the best agreement. In turn,  the longitudinal fRG dressings for decoupling and Higgs-type solutions start to deviate from the ST dressings for successively larger momenta, the farer away they are from scaling. Interestingly, the longitudinal three-gluon vertex dressing is an exception, as there the deviation of fRG and STI dressings happens at roughly the same scale. This structure deserves further investigation, one of the reasons being that the STIs used in the present work are the standard Landau gauge STIs and the decoupling and Higgs-type solutions triggered here are effectively induced by an explicit mass parameter as in the Curci-Ferrari (CF) model. This suggests to repeat the present investigation on the basis of the CF BRST transformations.\n\nIn summary, these comparisons provide us with a tool to control the gauge consistency in truncations of the quantum effective action. In our opinion, the present level of gauge consistency supports the reliability of such truncations in applications of non-perturbative functional methods to Yang-Mills theory.\n\nMonitoring the mSTIs in such a manner provides additional, much needed, guidance to improve on truncations. Alternatively, they can be used as constraints to determine correlation functions. \n\nThis work provides the foundation for an advanced study thereof in QCD, where the gauge consistency of the coupled self-consistent quark-gluon system poses an intricate problem. We hope to report on this in the near future.\n\n\\section*{Acknowledgments}\n We thank G.~Eichmann, J.~Horak, J.~Papavassiliou and U.~Reinosa for discussions. This work is done within the fQCD-collaboration~\\cite{fQCD} and we thank the members for discussion and collaborations on related projects. This work is supported by EMMI, and is part of\nand supported by the DFG Collaborative Research Centre \"SFB 1225 (ISOQUANT)\". It is also supported by Germany's Excellence Strategy EXC 2181\/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster).\nN.~W.~is additionally supported by the Hessian collaborative research cluster ELEMENTS and by the DFG Collaborative Research Centre \"CRC-TR 211 (Strong-interaction matter under extreme conditions)\".\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe micro-lens array (MLA) based plenoptic cameras, including the conventional plenoptic camera \\cite{ng2006digital} and the focused plenoptic camera \\cite{lumsdaine2009focused}, capture the radiance information of rays in both spatial and angular dimensions, e.g. the 4D light field data \\cite{levoy1996light,gortler1996lumigraph}. The data from the plenoptic camera is equivalent to narrow baseline images of traditional cameras with projection centers on the lens aperture plane. The measurement of the same position in multiple directions allows or strengths applications on computer photography, such as digital refocusing \\cite{ng2005fourier}, depth estimation \\cite{Jeon2015Accurate}, saliency detection \\cite{li2014saliency} and so on. However, the angular and spatial resolution of a light field data is limited by the physical parameters of the plenoptic camera. Recent work proposed the methods on light field registration and stitching to expand the field of view \\cite{johannsen2015linear,birklbauer2014panorama,guo2015enhancing}. To support these applications, calibrating the plenoptic camera and decoding accurate 4D light field in metric distance from the 2D image sensor are crucial.\n\nTo calibrate the plenoptic camera, it is essential to build a model to relate the measurement on the 2D raw image and the rays in the 3D space. Prior work dealt with the intrinsic calibration on the plenoptic cameras in different optical designs \\cite{dansereau2013decoding,bok2014geometric,heinze2015automated,vaish2004using}. However, the parameters of the proposed models are redundant or incomplete, and the models are still improvable on the description of plenoptic cameras. Some of the calibration methods have issues on the initialization estimation or the optimization procedure.\n\nIn this paper, we propose a novel unconstrained TPP model with 7 parameters to describe the light field structure inside and outside the plenoptic camera concisely. The 7 parameters are sufficient to constrain the rays in a 4D light field. Based on the 7-parameter TPP model, the pixels on the raw image can be related to the rays by a virtual MLA directly. We deduce the projective transformation \\cite{hartley2003multiple} on the reconstructed 3D points with different TPP parameters, which is the theoretical foundation of the closed-form solution of our calibration method. Then we employ a nonlinear optimization to refine the parameters via minimizing the re-projection error on the raw images. We conduct experiments on both simulated data and a physical plenoptic camera.\n\nIn summary, our main contributions are listed as follows:\n\n(1) We simplify the plenoptic camera system as a 7-parameter unconstrained TPP coordinate and deduce a projective transformation matrix to relate the measurement on the image sensor to the scene points in 3D space.\n\n(2) We solve the parameters of the TPP using a robust and efficient method, which consists of a linear initialization and an optimization via re-projection error.\n\nThe remainder of this paper is organized as follows: Section~\\ref{sec:RelatedWork} summarizes the related work on the plenoptic camera models and calibration methods. Section~\\ref{sec:TPPModel} describes the 7-parameter unconstrained TPP model and its relationship with the physical plenoptic camera, and derives the projective transformation involved the TPP's parameters. Section~\\ref{sec:Solve} provides the details of our proposed calibration method. Section~\\ref{sec:ExpResults} shows the calibration results on simulated data and real scene data.\n\n\n\n\\section{Related Work}\n\\label{sec:RelatedWork}\n\nTo acquire light field, there are various imaging systems developed from the traditional camera. Wilburn et al. \\cite{wilburn2005high} presented a camera array to obtain light field with high spatial and angular resolution. Prior work dealt with the calibration of the camera arrays \\cite{vaish2004using}. Unfortunately, applications on camera arrays are limited by its high cost and complex control. In contrast, a MLA enables a single camera to record 4D light field more conveniently and efficiently, though the baseline and spatial resolution is smaller than the camera array. Recent work devoted to calibrate the intrinsic parameters of the plenoptic cameras in two designs \\cite{ng2006digital,lumsdaine2009focused}, which are quite different according to the image structure of the micro lenses. Moreover, in traditional multi-view geometry, multiple cameras in different poses are defined as a set of unconstrained rays, which is known as as Generalized Camera Model (GCM) \\cite{pless2003using}. The ambiguity of the reconstructed scene was discussed in traditional topics. For a plenoptic camera, the different views of the same scene point are obtained, and the calibration of a plenoptic camera can use the theory on traditional multi-view for reference.\n\nSome work explored the calibration on the focused plenoptic camera, where the multiple projections of the same scene point are convenient to be recognized. Johannsen et al. \\cite{johannsen2013calibration} proposed the method on the intrinsic parameters calibration of a focused plenoptic camera. By reconstructing 3D points from the parallax in adjacent micro-lens images, the parameters including the depth distortion were estimated by nonlinear optimization directly without a linear initialization. However, the geometry center of the micro image was on its micro-lens's optical axis in their method. This assumption caused inaccuracy on the reconstructed points and was compensated by the depth distortion coefficients. Hahne et al. \\cite{hahne2015refocusing} discussed the influence of the deviation of the micro image's centers and the optical center of its micro lens. Heinze et al. \\cite{heinze2015automated} applied a similar method with \\cite{johannsen2015linear} and proposed a linear initialization for the intrinsic parameters.\n\nSome work explored the calibration on the conventional plenoptic camera, where the sub-aperture images are easy to be synthesized. Dansereau et al. \\cite{dansereau2013decoding} presented a model to decode the pixels into rays for a conventional plenoptic camera, where the 12-free-parameter transformation matrix was connected with the reference plane outside the camera. However, the calibration method was initialized using traditional camera calibration techniques and there were redundant parameters in the decoding matrix. Bok et al. \\cite{bok2014geometric} formulated a geometric projection model consisting of a main lens and a MLA to estimate the intrinsic and extrinsic parameters by utilizing raw images directly, including an analytical solution and a nonlinear optimization. Moreover, Thomason et al. \\cite{thomason2014calibration} concentrated on the misalignment of the MLA and estimated its position and orientation.\n\nDifferent from the previous models, we represent the image sensor, the MLA and the main lens as a simple TPP model with 7 parameters. The 7 parameters are connected with the physical parameters of the plenoptic camera and sufficient to relate the pixels on the raw image to the rays without redundancy. To reveal the relationship between the light field data and the scene structure, we explore the ray-ray association on the TPP coordinate. Then the 3D projective transformation of the reconstructed structure with different TPP parameters is deduced. Based on the projective transformation, we solve a linear initialization for the intrinsic and extrinsic parameters. In our method, the prior scene points are support by a planar calibration board in different poses. The solved initialization is refined by minimizing the re-projection error using Levenberg-Marquardt algorithm. Theoretical derivation and experimental results demonstrate the validity of our calibration method.\n\n\n\n\\section{Unconstrained TPP model}\n\\label{sec:TPPModel}\n\nThe distance of the traditional TPP model is normalized as 1 unit to describe a set of rays \\cite{levoy1996light,gortler1996lumigraph}. To describe the decoded rays of a plenoptic camera, a TPP model with free parameters are needed, e.g. the unconstrained TPP model. As shown in Fig.\\ref{fig:TPP_coordinate}, we define a TPP coordinate, where $\\vec{r}\\!=\\! \\left( x, y, u, v, f \\right)\\!\\! ^\\mathrm{T}$ defines a ray passing $\\left(x, y, 0 \\right)\\!\\! ^\\mathrm{T}$ and $\\left(u, v, f \\right)\\!\\! ^\\mathrm{T}$. In this section, we discuss the 3D projective transformation of reconstructed points in light field based on the TPP model. Then we establish the relationship between the TPP parameters and physical parameters of a focused plenoptic camera.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60mm]{fig1_TPP_coordinate}\n\\vspace{-3mm}\n\\caption{\nAn illustration of TPP coordinate system. The origins of two coordinates $x$-$y$ and $u$-$v$ lie on the axis $Z$. Axis $x$ and axis $y$ are parallel to axis $u$ and axis $v$ respectively. The distance between $x$-$y$ plane and $u$-$v$ plane is $f$.}\n\\label{fig:TPP_coordinate}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\\subsection{Projective Transformation on TPP}\n\\label{subsec:tpp}\n\nLet $\\vec{r}$ pass the point $\\left(X, Y, Z\\right)^\\mathrm{T}$, we have:\n\n\\vspace{-2mm}\n\\begin{equation}\n   \\underbrace{ \\left[  \\begin{array}{cccc}\n  f & 0 & x\\!-\\!u & -\\!fx \\\\\n  0 & f & y\\!-\\!v & -\\!fy \\end{array} \\right] }_{\\bm{M}}\n  \\left[ \\begin{array} {c}\n  \tX \\\\ Y \\\\ Z \\\\ 1 \\end{array} \\right] = \\bm{0},\n  \\label{eq_Mx=0}\n\\end{equation}\n\n\\noindent where $\\left(X, Y, Z\\right)^\\mathrm{T}$ can be solved iff there are at least two rays and any two of the rays $\\vec{r}_i$ and $\\vec{r}_j$ satisfy $\\frac{u_i-u_j}{x_i-x_j}=\\frac{v_i-v_j}{y_i-y_j}$.\n\nTransforming $\\vec{r}$ into $\\vec{r}^\\prime \\!\\!=\\!\\! \\left( k_{\\!x}x, k_{\\!y}y, k_{\\!u}u\\!+\\!u_0, k_{\\!v}v\\!+\\!v_0, f\\prime \\right)\\!\\! ^\\mathrm{T}$, the intersection point $\\left(X, Y, Z \\right)\\!\\!^\\mathrm{T}$ is changed to be $\\left(X^\\prime, Y^\\prime, Z^\\prime \\right)\\!\\! ^\\mathrm{T}$, which satisfies:\n\n\\vspace{-2mm}\n\\begin{equation}\n  \\setlength\\arraycolsep{4.0pt}\n  \\underbrace{ \\left[  \\begin{array}{cccc}\n  fk_uk_x & 0 & k_x u_0 & 0 \\\\\n  0 & fk_vk_x & k_x v_0 & 0 \\\\\n  0 & 0 & f^\\prime k_x & 0 \\\\\n  0 & 0 & k_x\\!-\\!k_u & fk_u \\end{array} \\right] }_{=:P\\left( \\bm{\\mathscr{X}}, f \\right)}\n  \\left[\\!\\! \\begin{array} {c}\n  \tX \\\\ Y \\\\ Z \\\\ 1 \\end{array} \\!\\!\\right] = s\n  \t\\left[\\!\\! \\begin{array} {c}\n  \tX^\\prime \\\\ Y^\\prime \\\\ Z^\\prime \\\\ 1 \\end{array} \\!\\!\\right],\n  \\label{eq_PX=sX_}\n\\end{equation}\n\\vspace{-1mm}\n\n\\noindent where $\\mathscr{X}\\!\\!\\!=\\!\\! \\left( k_x, k_y, k_u, k_v, u_0, v_0, f^\\prime \\right)$ is the transformation parameters of the TTP coordinate and $s$ is a scalar factor. To make the set of transformed rays $\\vec{r}^\\prime$ all intersect at the point $\\left(X^\\prime, Y^\\prime, Z^\\prime \\right)^\\mathrm{T}$, $\\mathscr{X}$ must satisfy $k_u\/k_x\\!=\\! k_v\/k_y$. Equation \\ref{eq_PX=sX_} indicates that $\\mathscr{X}$ affects the geometric structure of the recovered scene, {\\it i.e.} the intersections of rays. In addition, the non-zero elements in the last row of $\\bm{P}$ is equivalent to the refraction of a lens in a traditional camera. Therefore, by transforming the coordinate of the TPP via $\\mathscr{X}$, the light field inside a camera can be transformed into the real world scene, where the scale of $u$-$v$ handles the projective transformation and the scale of $x$-$y$ handles the zoom of the recovered scene. The transformations on the scene structure with single parameter in $\\mathscr{X}$ are shown in Fig.\\ref{fig:projective_distortion} separately.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=90mm]{fig2_projective_distortion}\n\\vspace{-4mm}\n\\caption{\nTPP light field recording a Lambertian cube. The left one is the original cube and the others are distorted cubes with the change of $f$, $k_{u\\!v}\\left(k_{uv}\\!\\!=\\!\\! k_u\\!\\!=\\!\\! k_v\\right)$, $\\left(u_0, v_0 \\right)^\\mathrm{T}$ respectively.}\n\\label{fig:projective_distortion}\n\\vspace{-3mm}\n\\end{figure}\n\nIn Section~\\ref{sec:Solve}, we will discuss the calibration method using the projective transformation in Eq.\\ref{fig:projective_distortion}.\n\n\n\n\\subsection{TPP Coordinate Inside and Outside the Camera}\n\\label{subsec:tpp2}\n\nWe model the main lens as a thin lens and the micro-lens as a pinhole, thus every pixel on the image sensor can be regarded as a ray passing through the coordinate on the image sensor and the optical center of its corresponding micro-lens \\cite{dansereau2013decoding,johannsen2013calibration,bok2014geometric}.\nThe TPP coordinate system consists of a image sensor and a MLA, {\\it i.e.} the $x^\\prime y^\\prime u^\\prime v^\\prime$ coordinate shown in Fig.\\ref{fig:TPP2System}. Moreover, there is another TPP coordinate $xyuv$ outside the camera, where the $x$-$y$ is related to the image sensor and the $u$-$v$ is related to the MLA. The $x$-$y$ and $u$-$v$ planes can be regarded as a zoomed raw image and a virtual MLA with larger diameter respectively.\n\nDue to the refraction of the main lens, the rays and the scene structure in the two TPP coordinates are different. Obviously, there is a projective transformation on the reconstructed 3D points with different TPP parameters (Eq.\\ref{eq_PX=sX_}). By transforming the rays $\\vec{r}^\\prime$ in $x^\\prime y^\\prime u^\\prime v^\\prime$ coordinate to $\\vec{r}$ in $xyuv$ coordinate via $\\mathscr{X}$, {\\it i.e.} reparameterizing the coordinate of TTP, the projective distortion of the reconstructed points can be rectified. Therefore, the complex compositions of the plenoptic camera can be equivalently replaced by two parallel planes. In other words, there is a similar two-plane light field structure in the real world scene with the one inside the camera. Then we discuss the relationship between the physical structure inside a focused plenoptic camera and the virtual structure outside the camera.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=80mm]{fig3_TPP2System}\n\\vspace{-3mm}\n\\caption{\nA focused plenoptic camera with a MLA. There are two TPP coordinates, {\\it i.e.} $x^\\prime y^\\prime u^\\prime v^\\prime$ inside the camera and $xyuv$ in the real world scene.}\n\\label{fig:TPP2System}\n\\end{figure}\n\n\nIn the plenoptic camera, a ray which passes the pixel $\\left(x, y \\right)$ on the image coordinate and the micro-lens with label $\\left(i, j \\right)$ can be represented as a virtual ray $\\left(x, y, i, j, 1 \\right)^{\\mathrm{T}}$ where $i \\! \\in \\! \\mathbb{Z}$, $j \\! \\in \\! \\mathbb{Z}$. There is a geometric relationship between the two coordinates $xyuv$ and $x^\\prime y^\\prime u^\\prime v^\\prime$. Let $\\mathscr{X}_{in}\\!\\!=\\!\\!\\left( k_{x\\!,in},k_{y\\!,in},k_{u\\!,in},k_{v\\!,in}, u_{in}, v_{in}, f_{\\!in} \\right)\\!\\! ^\\mathrm{T}$ and $\\mathscr{X}_{\\!out}\\!\\!=\\!\\!( k_{x\\!,out},k_{y\\!,out},k_{u\\!,out},k_{v\\!,out}, u_{out}$, $v_{out}, f_{\\!out} )\\!^\\mathrm{T}$ be the parameters of TPP inside and outside the camera respectively, thus the virtual ray $\\left(x, y, i, j, 1 \\right)\\!\\! ^\\mathrm{T}$ is related to two physical rays $\\left(k_{x,in}x, k_{y,in}y, k_{u,in}i+\\!u_{in}, k_{v,in}j+\\!v_{in}, f_{in} \\right)\\! ^\\mathrm{T}$ and $(k_{x,out}x,\\!k_{y,out}y, k_{u,out}i+u_{out}$, $k_{v,out}j+v_{out}, f_{out} )^\\mathrm{T}$respectively. The parameters $k_{x\\!,in}$ (or $k_{y\\!,in}$), $f_{in}$ can be regarded as the diameter of micro lens and the distance between the image sensor and the MLA. With the image sensor origin $\\left( X_{os}, Y_{os}, Z_{os} \\right)\\!\\!^{\\mathrm{T}}$  and the reference micro-lens $\\left( X_{oa}, Y_{oa}, Z_{oa} \\right)\\!\\! ^{\\mathrm{T}}$ whose label is $\\left(0,0\\right)$ (in the main lens coordinate), we can get the relationship between $\\mathscr{X}_{in}$ and $\\mathscr{X}_{out}$:\n\n\\vspace{-1mm}\n\\begin{equation}\n\\vspace{-3.0pt}\n\\frac{k_{x,in}}{k_{x,out}} = \\frac{k_{y,in}}{k_{y,out}} = \\frac{F}{Z_{os}-F},\t\n\\vspace{-3.0pt}\n\\end{equation}\n\n\\begin{equation}\n\\vspace{-3.0pt}\n\\frac{k_{u,in}}{k_{u,out}} = \\frac{k_{v,in}}{k_{v,out}} = \\frac{F}{Z_{oa}-F},\n\\vspace{-3.0pt}\n\\end{equation}\n\n\\begin{equation}\n\\vspace{-3.0pt}\n\\left[\\!\\! \\begin{array} {c}\nu_{out}-u_{in} \\\\v_{out}-v_{in}\n\\end{array} \\!\\!\\right] = \\frac{Z_{oa}}{F-Z_{oa}}\n\\left[\\!\\! \\begin{array} {c}\nX_{oa} \\\\ Y_{oa}\n\\end{array} \\!\\!\\right] - \\frac{Z_{os}}{F-Z_{os}}\n\\left[\\!\\! \\begin{array} {c}\nX_{os} \\\\ Y_{os}\n\\end{array} \\!\\!\\right]\n\\vspace{-3.0pt}\n\\end{equation}\n\n\\begin{equation}\n\\vspace{-1.0pt}\nf_{in} = Z_{oa}\\!\\!-\\!Z_{os},\\quad\nf_{out} = \\frac{F}{F\\!\\!-\\!Z_{oa}} Z_{oa}- \\frac{F}{F\\!\\!-\\!Z_{os}}Z_{os},\\\\\n\\vspace{-0.0pt}\n\\end{equation}\n\nIn addition, to simplify the discussion, we assume that the layout of the micro-lens array is square-like. For hexagon-like configuration, the label of micro-lens is different due to the layout.\n\n\n\n\\section{Calibration Method}\n\\label{sec:Solve}\n\nTo decode the virtual ray $\\left(x, y, i, j, 1 \\right)^{\\!\\mathrm{T}}$ into the ray in the real world scene, we need to calibrate the parameters of TPP coordinate system, {\\it i.e.} the intrinsic parameters of the plenoptic camera. The theorem in Section~\\ref{sec:TPPModel} indicates that given an arbitrary setting of TPP parameters, a set of 3D points can be recovered, and there is a projective transformation between the real scene points and the recovered points. Moreover, the projective transformation is determined by the TPP parameters.\n\nThis section provides the details of how to solve the parameters effectively, including a linear closed-form solution and a nonlinear optimization to minimize the re-projection error.\n\n\n\\subsection{Linear Initialization}\n\\label{subsec:linearInit}\n\nGiven an arbitrary setting $\\mathscr{X}^\\prime \\!\\!=\\!\\! \\left( k_x^\\prime, k_y^\\prime, k_u^\\prime, k_v^\\prime, u_0^\\prime, v_0^\\prime, f^\\prime \\right)^\\mathrm{T}$\\!\\!,\nwe decode the virtual ray $\\left(x, y, i, j, 1 \\right)^{\\!\\mathrm{T}}$\\!\\! into the ray passing\n$\\left( k_x^\\prime x,k_y^\\prime y, 0 \\right)^{\\mathrm{T}}$\\!\\! and $(k_u^\\prime i + u_0^\\prime,k_v^\\prime j +v_0^\\prime$,$f^\\prime)^{\\mathrm{T}}$. Then the distorted 3D point $\\bm{X}_d$ can be reconstructed.\nObviously, there is a projective transformation between $\\bm{X}_d$ and the real scene point $\\bm{X}_c$ (Eq.\\ref{eq_PX=sX_}). Therefore, using a transformation parameter setting $\\mathscr{X}_d \\!\\!=\\!\\! \\left(k_x, k_y, k_u, k_v, u_0, v_0, f^\\prime \\right)^\\mathrm{T}$, $\\bm{X}_c$ can be transformed to $\\bm{X}_d$. Then we assume that the points in the world coordinate $\\bm{X}_w$ is related to the TPP coordinate by a rigid motion, $\\bm{X}_c \\!=\\! \\bm{R}\\bm{X}_w+\\bm{t}$, with rotation $\\bm{R} \\in SO(3)$ and translation $\\bm{t}\\in \\mathbb{R}^3 $. Let's denote the $i^\\mathrm{th}$ column of $\\bm{R}$ by $\\bm{r}_i$. Here we assume that $k_x^\\prime \\!=\\! k_y^\\prime \\!=\\! k_{xy}^\\prime$, $k_u^\\prime \\!=\\! k_v^\\prime \\!=\\! k_{uv}^\\prime$, and the same as $\\mathscr{X}$. The relationship between the $\\bm{R}$, $\\bm{t}$, $\\bm{X}_w$, $\\bm{X}_d$ and $\\mathscr{X}_d$ is:\n\n\\vspace{-2mm}\n\\begin{equation}\ns\\bm{X}_d \\!= \\bm{P} \\left( \\mathscr{X}_d, f \\right) \\left[\\!\\!  \\begin{array} {cccc}\n\\bm{r_1} & \\bm{r_2} & \\bm{r_3} & \\bm{t} \\\\\n0 & 0 & 0 & 1 \\end{array} \\!\\!\\right]\n  \\left[ \\!\\!\\! \\begin{array}{c}\nX_w \\\\ Y_w \\\\ Z_w \\\\ 1 \\end{array} \\!\\!\\! \\right],\n\\label{eq_sXd=PRtX}\n\\end{equation}\n\n\\noindent where $f$ is the distance of the calibrated two parallel planes. Obviously, there is a $4\\times3$ homography matrix:\n\n\\vspace{-3mm}\n\\begin{equation}\n\\bm{H} \\!=\\! \\bm{P} \\left[ \\begin{array} {ccc}\n\\bm{r}_1 & \\bm{r}_2 & \\bm{t} \\\\\n0 & 0 & 1\n\\end{array}  \\right].\n\\label{eq_H=Prt}\n\\end{equation}\n\nWe assume that the calibration board plane is $Z=0$ on the world coordinate, thus $Z_w=0$. Let's denote the $i^\\mathrm{th}$ prior point by $\\bm{X}_{\\!w\\!,i}\\!=\\!( X_{\\!w\\!,i}, Y_{\\!w\\!,i}, 1 )^{\\!\\mathrm{T}}$. Combining Eq.\\ref{eq_Mx=0} and Eq.\\ref{eq_sXd=PRtX}, we have:\n\n\\vspace{-1mm}\n\\begin{equation}\n\\bm{M}_i \\, \\bm{H} \\left[ \\!\\! \\begin{array}{c}\nX_w \\\\ Y_w \\\\ 1\t\n\\end{array}\\!\\! \\right] = \\bm{0},\n\\label{eq_MiHXw=0}\n\\end{equation}\n\n\\begin{equation}\n\\left[\\! \\begin{array}{c}\n\\bm{M}_1 \\otimes \\left( X_{w\\!,1} \\,\\,\\, Y_{w\\!,1} \\,\\,\\, 1 \\right) \\\\\n\\vdots  \\\\\n\\bm{M}_n \\otimes \\left( X_{w\\!,n} \\,\\,\\, Y_{w\\!,n} \\,\\,\\, 1 \\right)\n\\end{array} \\!\\right] \\overrightarrow{\\bm{H}}  = \\bm{0},\n\\label{eq_MH=0}\n\\end{equation}\n\n\\noindent where $\\bm{M}_i$ is a $2m_i\\times4$ matrix which contains $\\bm{X}_{w,i}$'s $m_i$ decoded rays from the raw image, $\\overrightarrow{\\bm{H}}$ is a $12\\times1$ matrix stretched on row from $\\bm{H}$, and $\\otimes$ is a direct product operator. The homography $\\bm{H}$ multiplied by an unknown factor can be estimated by Eq.\\ref{eq_MH=0}. Let's denote the $i^\\mathrm{th}$ column vector of $\\bm{H}$ be $\\bm{h}_i= \\left(h_{1i}, h_{2i}, h_{3i}, h_{4i} \\right)\\!\\! ^\\mathrm{T}$. Utilizing the orthogonality and identity of $\\bm{R}$, we have:\n\n\\vspace{-1mm}\n\\begin{equation}\n\\begin{aligned}\n&\\bm{h}_1^{\\mathrm{T}} \\bm{P}^{-\\mathrm{T}} \\bm{P}^{-1} \\bm{h}_2 = 0, \\\\\n&\\bm{h}_1^{\\mathrm{T}} \\bm{P}^{-\\mathrm{T}} \\bm{P}^{-1} \\bm{h}_1 = \\bm{h}_2^{\\mathrm{T}} \\bm{P}^{-\\mathrm{T}} \\bm{P}^{-1} \\bm{h}_2,\n\\end{aligned}\n\\label{eq_hPPh}\n\\end{equation}\n\n\\noindent where $\\bm{P}\\!=\\! \\bm{P}\\left(\\mathscr{X}_d, f \\right)$. Let's denote $\\bm{P}^{-\\!\\mathrm{T}} \\!\\bm{P}^{-\\!1}$ by a symmetric matrix $\\bm{Q}$, thus:\n\n\\vspace{-3mm}\n\\begin{equation}\n\\setlength{\\arraycolsep}{4.0pt}\n\\bm{Q} = \\left[\\!\\! \\begin{array} {cccc}\n\\frac{1}{ f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2} } & 0 & - \\frac{u_0}{f^{\\!\\prime} \\! f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2}} & 0 \\\\\n0 & \\frac{1}{ f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2} } & - \\frac{v_0}{f^{\\!\\prime} \\! f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2}} & 0 \\\\\n- \\frac{u_0}{f^{\\!\\prime} \\!   f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2}} &\\! -\\frac{v_0}{\\!f^{\\!\\prime} \\! f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2}} &  \\frac{1}{f^{\\!\\prime 2}k_{\\!x\\!y}^2} \\!+\\! \\frac{u_0^2+v_0^2}{f^{\\!\\prime2} \\! f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2}} \\!+\\! \\frac{ \\left( k_{\\!u\\!v}\\!-\\!k_{\\!x\\!y} \\right) ^2 }{ f^{\\!\\prime 2}\\! f^{2} k_{\\!x\\!y}^{2}k_{\\!u\\!v}^{2} }  & \\frac{k_{\\!u\\!v}\\!-k_{\\!x\\!y}}{ f^{\\!\\prime} \\! f^{2} k_{\\!x\\!y} k_{\\!u\\!v}^{2} } \\\\\n0 & 0 & \\frac{k_{\\!u\\!v}-k_{\\!x\\!y}}{ f^{\\!\\prime} \\! f^{2} k_{\\!x\\!y}k_{\\!u\\!v}^{2} } & \\frac{1}{ f^{2}k_{\\!u\\!v}^{2} }\n\\end{array}\t\\!\\!\\right].\n\\label{eq_Q=PP}\n\\end{equation}\n\nNote that there are only six distinct non-zero elements in $\\bm{Q}$, denoted by $\\bm{q}= \\left(q_{11}, q_{13}, q_{23}, q_{33}, q_{34}, q_{44} \\right)^\\mathrm{T}$. To solve $\\bm{Q}$, we have:\n\n\\vspace{-2mm}\n\\begin{equation}\n\\setlength{\\arraycolsep}{8.0pt}\n\\left[ \\begin{array}{cc}\nh_{11}h_{12}+h_{21}h_{22} &  h_{11}^2\\!\\!-\\!h_{12}^2\\!+\\!h_{21}^2\\!\\!-\\!h_{22}^2\\\\\nh_{11}h_{32}+h_{12}h_{31} &  2\\left( h_{11}h_{31}\\!\\!-\\!h_{12}h_{32} \\right)\\\\\nh_{21}h_{32}+h_{22}h_{31} &  2\\left( h_{21}h_{31}\\!\\!-\\!h_{22}h_{32} \\right)\\\\\nh_{31}h_{32}              &  h_{31}^2\\!\\!-\\!h_{32}^2\\\\\nh_{31}h_{42}+h_{32}h_{41} &  2\\left( h_{31}h_{41}\\!\\!-\\!h_{32}h_{42} \\right)\\\\\nh_{41}h_{42}              &  h_{41}^2\\!\\!-\\!h_{42}^2\\\\\n\\end{array} \\right]^{\\!\\!\\mathrm{T}} \\left[ \\!\\!\n\\begin{array}{c}\nq_{11} \\\\ q_{13} \\\\ q_{23} \\\\ q_{33} \\\\ q_{34} \\\\ q_{44}\n\\end{array} \\!\\! \\right] = \\bm{0}.\n\\label{eq_hq=0}\n\\end{equation}\n\n\nBy stacking at least three such equations as Eq.\\ref{eq_hq=0}, we will have in general a unique non-zeros solution for $\\bm{q}$ denoted as $\\hat{\\bm{q}}= \\left(\\hat{q}_{11}, \\hat{q}_{13}, \\hat{q}_{23}, \\hat{q}_{33}, \\hat{q}_{34}, \\hat{q}_{44}\\right)\\!\\! ^\\mathrm{T}$, which is defined up to an unknown scale factor $\\lambda \\left( \\lambda \\bm{q}=\\hat{\\bm{q}}\\right)$. Once $\\bm{Q}$ is estimated, we can solve all the parameter $\\mathscr{X}_d$:\n\n\\begin{equation}\n\\begin{aligned}\n& \\lambda = f^{\\prime 2} \/ \\hat{q}_{11} \\left[ \\left(\\hat{q}_{33}\\hat{q}_{44}\\!-\\!\\hat{q}_{34}^2 \\right)-\\hat{q}_{44}\/\\hat{q}_{11} \\left( \\hat{q}_{13}^2+\\hat{q}_{23}^2 \\right) \\right], \\\\\n& k_{xy} = \\sqrt{ \\hat{q}_{44} \/ \\hat{q}_{11}} ,\\\\\n& k_{uv} = \\sqrt{ \\hat{q}_{44} \/ \\hat{q}_{11} } \\left( 1+f^\\prime \\hat{q}_{34} \/ \\hat{q}_{44} \\right),\\\\\n& u_0 = -f^\\prime \\hat{q}_{13} \/ \\hat{q}_{11},\\\\\n& v_0 = -f^\\prime \\hat{q}_{23} \/ \\hat{q}_{11},\\\\\n& f = \\left( \\sqrt{ \\lambda \/ \\hat{q}_{44} } \\right) \/ k_{uv}.\n\\end{aligned}\n\\label{eq_linearSln}\n\\end{equation}\n\nAfter solving $\\mathscr{X}_d$, the virtual ray $\\left(x,y,i,j,1\\right)^\\mathrm{T}$ is related to a ray passing the\n$\\left( k_{xy}^\\prime \/ k_{xy} x, k_{xy}^\\prime \/ k_{xy} y, 0 \\right)^\\mathrm{T}$ and \n$\\left(k_{u\\!v}^\\prime \/ k_{u\\!v} i \\!+\\! u_0^\\prime \\!-\\! u_0, k_{u\\!v}^\\prime \/ k_{u\\!v} j \\!+\\! v_0^\\prime \\!-\\! v_0, f \\right)^\\mathrm{T}$\nin the real world scene in metric distance, and the 3D points $\\bm{X}_c$ are recovered. Let's denote the parameters of the TPP in the real scene world by $\\mathscr{X}\\!\\!=\\!\\! \\left( k_{x\\!y}^\\prime \/ k_{x\\!y}, k_{x\\!y}^\\prime \/ k_{x\\!y}, k_{u\\!v}^\\prime \/ k_{u\\!v}, k_{u\\!v}^\\prime \/ k_{u\\!v}, u_0^\\prime\\!\\!-\\! u_0,  v_0^\\prime\\!\\!-\\! v_0, f \\right)\\!^\\mathrm{T}$. Then the extrinsic parameters $\\bm{R}_i$, $\\bm{t}_i$\nfor the $i^\\mathrm{th}$ raw image is computed:\n\n\\vspace{-2mm}\n\\begin{equation}\n\\begin{aligned}\n& \\bm{r}_1 = \\bm{P}^{-1} \\bm{h}_1 \/ \\left\\| \\bm{P}^{-1} \\bm{h}_1 \\right\\|,\\\\\n& \\bm{r}_2 = \\bm{P}^{-2} \\bm{h}_2 \/ \\left\\| \\bm{P}^{-2} \\bm{h}_2 \\right\\|,\\\\\n& \\bm{r}_3 = \\bm{r}_1 \\times \\bm{r}_2,\\\\\n& \\bm{t}\\,\\,\\, = \\bm{P}^{-1} \\bm{h}_3 \/ \\left\\| \\bm{P}^{-1} \\bm{h}_1 \\right\\|.\n\\end{aligned}\n\\label{eq_r1r2r3}\n\\end{equation}\n\n\n\\subsection{Nonlinear Optimization}\n\\label{subsec:nonlinearOpt}\n\nThe optical property of the lenses and the physical machining error of MLA lead to the distortion of the rays. Distortion is in primarily radially symmetric due to the symmetric design of the plenoptic camera. Moreover, the $x$-$y$ plane and  $u$-$v$ plane are related to the image sensor and the MLA respectively. Therefore, we employ the radial distortion on the two coordinates of TPP \\cite{Zhang2000A}:\n\n\\vspace{-2mm}\n\\begin{equation}\n\\begin{aligned}\n& \\hat{x}^d = (\\hat{x}-x_c)(1+s_1r_{\\!x\\!y}^2+s_2r_{\\!x\\!y}^4)+x_c,& \\\\\t\n& \\hat{y}^d = (\\hat{y}-y_c)(1+s_1r_{\\!x\\!y}^2+s_2r_{\\!x\\!y}^4)+y_c,& \\\\\n& \\hat{u}^d = (\\hat{u}-u_c)(1+t_1r_{\\!u\\!v}^2+t_2r_{\\!u\\!v}^4)+u_c,& \\\\\n& \\hat{v}^d = (\\hat{v}-v_c)(1+t_1r_{\\!u\\!v}^2+t_2r_{\\!u\\!v}^4)+v_c,&\n\\end{aligned}\n\\label{eq_dirtortion}\n\\end{equation}\n\n\\noindent where $\\left(x_c, y_c\\right)\\!\\! ^\\mathrm{T}$ and $\\left(u_c, v_c\\right)\\!\\! ^\\mathrm{T}$ are the offsets as the origin of the distortion on two planes, \n$\\left(\\hat{x}^d, \\hat{y}^d\\right)\\!\\! ^\\mathrm{T}$ and $\\left(\\hat{u}^d, \\hat{v}^d\\right)\\!\\! ^\\mathrm{T}$ are the distorted points, $r_{\\!xy}\\!=\\!\\sqrt{\\left(\\hat{x}\\!-\\!x_c\\! \\right)^2 \\!+\\! \\left(\\hat{y}\\!-\\!y_c\\! \\right)^2}$ and $r_{\\!uv}\\!=\\!\\sqrt{\\left(\\hat{u}\\!-\\!u_c\\! \\right)^2 \\!+\\! \\left(\\hat{v}\\!-\\!v_c\\! \\right)^2}$. The parameters $s_i$ and $t_i$ are the distortion coefficients.\n\nWe minimize the following cost function with initialization solved in Section~\\ref{subsec:linearInit} to refine the intrinsic and extrinsic parameters, including the distortion coefficients:\n\n\\vspace{-2mm}\n\\begin{equation}\n\\sum_{i=1}^{n}\\!\\sum_{j=1}^{p_i} { \\left\\| \\bm{x}_{i,j} - \\hat{\\bm{x}}_{i,j}^d \\left( \\mathscr{X}, s_1, s_2, t_1, t_2, \\bm{R}_i, \\bm{t}_i, \\bm{X}_{w,i} \\right) \\right\\| },\n\\label{eq_re-projectionError}\n\\end{equation}\n\n\\noindent where $\\bm{x}_{i,j}$ is the $j^\\mathrm{th}$ projections of the prior scene point $\\bm{X}_{w,i}$ on the image coordinate, and $p_i$ is the number of the projections of $\\bm{X}_{w,i}$. In Eq.\\ref{eq_re-projectionError}, $\\bm{R}$ is parameterized by Rodrigues formula \\cite{faugeras1993three}. Equation \\ref{eq_re-projectionError} is the re-projection error in traditional computer vision. In addition, the Jacobian matrix of the cost function is simple and sparse. It can be solved with the LM Algorithm based on the trust region method \\cite{madsen2004methods}. We use MATLAB's lsqnonlin function to complete the optimization.\n\n\\subsection{Summary}\n\\label{subsec:summary}\n\nOur proposed calibration procedure is listed as follows:\n\n\\begin{enumerate}\n\\item Take at least 3 raw images with different poses of the calibration board by moving either the calibration board or the camera.\n\\item Detect the multiple projections corresponding to the scene points.\n\\item Calculate the $4\\times3$ homography $\\bm{H}_i$ for the $i^\\mathrm{th}$ raw image via Eq.\\ref{eq_MH=0}.\n\\item Estimate the intrinsic and extrinsic parameters via Eqs.\\ref{eq_hq=0}, \\ref{eq_linearSln} and \\ref{eq_r1r2r3}.\n\\item Refine all the parameters via Eq.\\ref{eq_re-projectionError} using LM Algorithm.\n\\end{enumerate}\n\n\n\\section{Experimental Results}\n\\label{sec:ExpResults}\n\nIn experiments, we apply our calibration method on the simulated data and the real world scene data. The prior scene points $\\bm{X}_w$ are obtained by a planar calibration board with a circular grid pattern (Fig.\\ref{fig:physicalCamera}). Due to the inevitable misalignment of the MLA and the image sensor, a preprocess on the raw image is needed (Section~\\ref{subsec:rect}).\n\n\n\\subsection{Rectification}\n\\label{subsec:rect}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=50mm]{fig4_misalignment_MLA}\n\\vspace{-3mm}\n\\caption{\nThe misalignment of the MLA in a plenoptic camera.}\n\\label{fig:misalignment_MLA}\n\\end{figure}\n\n\nIn a physical plenoptic camera, there is a slight rotation between the MLA and the image sensor \\cite{thomason2014calibration}, as shown in Fig.\\ref{fig:misalignment_MLA}. Let's denote the optical center and diameter of the micro-lens by $(x_g, y_g, z_g )^\\mathrm{T}$ and $d_m$ respectively, we have:\n\n\\begin{equation}\n\\left[ \\begin{array}{c;{2pt\/2pt}c}\n             & x_m \\\\\t\n\\bm{R}_{m\\!l\\!a} & y_m \\\\\t\n             & L\n\\end{array} \\right]\t\\left[\\!\\! \\begin{array}{c}\nid_{m} \\\\ jd_{m} \\\\ 0 \\\\ 1\n\\end{array} \\!\\!\\right] = \\left[\\! \\begin{array}{c}\nx_g \\\\ y_g \\\\ z_g \\\\ 1\t\n\\end{array} \\!\\right],\n\\label{eq_RtXg}\n\\end{equation}\n\n\\noindent where $\\bm{R}_{mla} \\!\\!\\in\\!\\! SO(3)$, and $\\left( x_m,y_m,L \\right)\\!\\! ^\\mathrm{T}$ is the offset between the reference micro-lens and the main lens (Fig.\\ref{fig:misalignment_MLA}). Therefore, the geometric center of the micro-lens image $(x_{g}^\\prime, y_{g}^\\prime)\\! ^\\mathrm{T}$ is:\n\n\\vspace{-1mm}\n\\begin{equation}\n\\left[\\! \\begin{array}{c}\nx_g^{\\,\\prime}\t\\\\ y_g^{\\,\\prime}\n\\end{array} \\!\\right] = \\frac{L+l}{z_g} \\left[\\! \\begin{array}{c}\nx_g\t\\\\ y_g\n\\end{array} \\!\\right].\n\\label{eq_xg_}\n\\end{equation}\n\nThe centers of the micro images $\\left(x_g^\\prime, y_g^\\prime \\right)\\!\\!^\\mathrm{T} $ are recognized by a raw image shooting a white scene or other pure color \\cite{cho2013modeling,Chunping2016Decoding}. The misalignment of the MLA makes the diameter of the micro-lens image non-uniform. As shown in Fig.\\ref{fig:slopes_centers}, the slopes fitted by a line of the centers of micro-lens images are descending linearly. The rate of descending is related to the rotation of the MLA. The range of the slopes of the rectified images is smaller than the slopes of the original images. It indicates that the diameters of the rectified micro images are more uniform. Taking the misalignment of the MLA into account \\cite{thomason2014calibration}, we rectify the raw image by a homography to make the micro-lens images uniform. More importantly, by Eq.\\ref{eq_RtXg} and Eq.\\ref{eq_xg_}, a homography with 8 degree of freedom is sufficient to preprocess the raw image, thus the two non-parallel light field planes are reparameterized to parallel planes. The experiments in Section~\\ref{subsec:simu} and \\ref{subsec:physical} are based on the rectification.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=60mm]{fig5_slopes_centers}\n\\caption{\nThe slopes fitted by 88 line of micro-lens images' centers from the original and rectified white image of a physical focused plenoptic camera. Every slope is fitted by 115 centers from the same row of micro-images.}\n\\label{fig:slopes_centers}\n\\end{figure}\n\n\n\\subsection{Simulated data}\n\\label{subsec:simu}\n\nWe first verify the calibration method on the simulated images rendered in MATLAB. The image sensor resolution is $4008\\times2672$ with 9 $\\mu m$ pixel width. The focused plenoptic camera consists of a main lens with 50 $mm$ focal length and a MLA with 300 $\\mu m$ diameter and 2.726 $mm$ focal length in hexagon layout. The calibration board is a pattern with $5\\times5$ points of $54.0\\times54.0 \\, mm$ cells. We render 12 raw images with different poses of the board.\n\nFor a focused plenoptic camera, to recognize the multiple projections of the same scene point, we preprocess the raw image using a white image and then use template matching by normalized cross-correlation (NCC).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=105mm]{fig67_simuResult}\n\\caption{\nThe calibration errors on simulated data with different numbers of poses (the left-two) and different noise levels (the right-two).}\n\\label{fig:SimuResults}\n\\end{figure}\n\nWe test the performance with respect to the numbers of poses of the calibration board. We vary the number of images from 3 to 12. The calibration results with increasing number of poses are shown in Fig.\\ref{fig:SimuResults}. The errors decrease when more images are used. From to 3 to 4, the errors of most parameters decrease significantly. When the number of poses is more than 6, all parameters tend to be stable. In addition, the ground truth are calculated by the input paramters of the simulation and the equations in Section~\\ref{subsec:tpp2}. \n\nIn addition, we add the projections of the total 12 raw images with Gaussian noise with 0 mean and standard deviation varied from 0.1 pixels to 0.8 pixels. The results are shown in Fig.\\ref{fig:SimuResults}. Due to that there are at least 12 projections of the same scene point in single raw image, the calibration results are still reasonable with different noise levels. It verifies the robustness of the calibration method.\n\n\n\n\\subsection{Physical camera}\n\\label{subsec:physical}\n\n\\vspace{-3mm}\n\\begin{figure}\n\\centering\n\\includegraphics[width=90mm]{Fig8_physicalCamera}\n\\vspace{-2mm}\n\\caption{\nThe self-assembly focused plenoptic camera and the MLA inside the camera.}\n\\label{fig:physicalCamera}\n\\end{figure}\n\n\nWe capture raw images of calibration board and real scene using a self-assembly focused plenoptic camera. The camera and the MLA are shown in Fig.\\ref{fig:physicalCamera}. The camera consists of a GigE camera with a CCD image sensor whose resolution is $4008\\times2672$ with 9 $mm$ pixel width, a Nikon AF Nikkor f\/1.4D F-mount lens with 50 $mm$ focal length, and a MLA with 300 $\\mu m$ diameter and 2.726 $\\mu m$ focal length in hexagon layout. The calibration board is $10\\times10$ points with 2$cm$ $\\times$ 2 $cm$ cells (Fig.\\ref{fig:physicalCamera}). We shoot 9 raw images with different poses of the board, and 9 raw images with the same real scene and a calibration board. All the raw images are preprocessed by the method mentioned in Section~\\ref{subsec:rect}.\n\n\\begin{table}[tbp]\n\\caption{Calibration results of a physical camera with different numbers of poses.}\n\\centering\n\\tiny\n\\label{cp_tab4}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c}\n\\hline\nParame & \\multicolumn{2}{|c|}{3 poses} & \\multicolumn{2}{|c|}{5 poses} & \\multicolumn{2}{|c|}{7 poses} & \\multicolumn{2}{|c}{9 poses} \\\\\n\\cline{2-9}\n-ter & $4\\times 5$  & $8\\times10$ & $4\\!\\times\\! 5$ & $8\\!\\times\\! 10$ & $4\\!\\times\\! 5$  & $8\\!\\times\\! 10$ & $4\\!\\times\\! 5$  & $8\\!\\times\\! 10$ \\\\\n\\hline\n$k_x$  & 31.1525 & 32.2787 & 29.0403 & 28.7948 & 27.5295 & 27.7046 & 27.1743 & 27.0686\\\\\n$k_y$  & 30.9687 & 32.1397 & 29.0474 & 28.8172 & 27.5140 & 27.6846 & 27.1752 & 27.0632\\\\\n$k_u$  & 1230.23 & 1271.44 & 1152.98 & 1143.66 & 1097.09 & 1103.30 & 1084.01 & 1079.72\\\\\n$k_v$  & 1224.51 & 1267.02 & 1153.47 & 1144.48 & 1096.8 & 1102.77 & 1084.28 & 1079.66\\\\\n$u_0$\/pixel  & -3364.99 & -3598.28 & -3440.39 & -3604.41 & -3475.55 & -3685.79 & -3508.18 & -3691.41\\\\\n$v_0$\/pixel  & -7162.50 & -7417.96 & -6960.49 & -7075.08 & -6891.50 & -7043.57 & -6867.73 & -6987.77\\\\\n$f$\/pixel    & 31098.7 & 31605.3 & 30309.7 & 30162.6 & 29878.2 & 30012.6 & 29742.5 & 29701.1\\\\\n\\hline\n$s_1$  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n$s_2$  & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n$t_1$  & -2.44e-13 & -2.54e-13 & -3.81e-13 & -3.84e-13 & -3.85e-13 & -3.85e-13 & -3.72e-13 & -3.80e-13\\\\\n$t_2$  & 1.79e-22 & 1.60e-22 & 3.059e-22 & 2.89e-22 & 3.65e-22 & 3.32e-22 & 3.69e-22 & 3.47e-22\\\\\n\\hline\nRMS    & 0.30394 & 0.31505 &  0.27223 & 0.29952 & 0.26344 & 0.29151 & 0.24877 & 0.27846\\\\\n\\hline\n\\end{tabular}\n\\label{tab:RealResult}\n\\vspace{-2mm}\n\\end{table}\n\n\\begin{figure}[tbp]\n\n\\centering\n\\subfigure[\\,]{\n\\label{fig:RealResultPoses}\n\\includegraphics[width=0.55\\textwidth]{fig12_RealResultPoses}}\n\\vspace{-1mm}\n\\subfigure[\\,]{\n\\label{fig:RealResultErrors}\n\\includegraphics[width=0.42\\textwidth]{fig12_RealResultErrors}}\n\\caption{\n(a) shows the estimated poses of the 9 raw images with a calibration board (the black parallelogram on the top). (b) shows the histograms of the distribution of the re-projection error. The histograms are calculated without or with the distortion coefficients, and the mean errors are 0.31623 and 0.27846 pixels respectively.}\n\\label{fig:RealResult}\n\\end{figure}\n\nThe results of the estimated intrinsic parameters are listed in Tab.1. We apply our calibration method to the first 3, 5, 7, and all 9 raw images in different poses. As shown in Tab.\\ref{tab:RealResult}, the re-projections RMS error is less than 0.3 pixels when at least 5 poses are used. The RMS errors are quite consistent with different numbers of poses. With the same number of poses, the parameters are close when the number of the calibration points is changed from $4\\times 5$ points to $8\\times 10$ points. The poses estimated using the 9 raw images with a calibration board and the histogram of the re-projection errors are shown in Fig.\\ref{fig:RealResult}. Fig.\\ref{fig:RealResultErrors} shows that most of the errors are less than 0.5 pixels, and the re-projection errors decrease with the optimization on distortion.\n\n\\begin{figure}\n\\vspace{-2mm}\n\\centering\n\\includegraphics[width=120mm]{fig11_RenderPipeline}\n\\caption{\nThe refocus rendering pipeline of a focused plenoptic camera.}\n\\label{fig:RenderPipeline}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\subfigure[\\,]{\n\\label{fig:RenderImgsSub1}\n\\includegraphics[width=0.52\\textwidth]{fig10_RenderImgs}}\n\\subfigure[\\,]{\n\\label{fig:RenderImgsSub2}\n\\includegraphics[width=0.45\\textwidth]{fig10_pose2fig}}\n\\vspace{-4mm}\n\\caption{\n(a) shows the rendered images from the physical focused plenoptic camera. The top are the most left view and the most right view and the bottom is the stitched image of the different 9 poses. (b) shows two of the views of the estimated posed from the 9 raw images.}\n\\label{fig:RenderImgs}\n\\end{figure}\n\nAfter the estimation of $\\mathscr{X}$, the virtual optical centers of the MLA are calculated, thus the rays' directions are obtained \\cite{perwass2012single}. The refocus rendering pipeline of a focused plenoptic camera is shown in Fig.\\ref{fig:RenderPipeline}. The rendered images by ray tracing using 9 raw images in different poses are shown in Fig.\\ref{fig:RenderImgs}.\n\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nIn this paper, we present a novel unconstrained TPP model to describe the relationship between the 4D rays and the 3D scene structure by a projective transformation. To calibrate the focused plenoptic camera, we simplify the imaging system as a 7-parameter TPP model. Compared with the previous calibration method on the focused plenoptic cameras, we substitute the refraction of the main lens and simplify the image system as a 7-intrinsic-parameter unconstrained TPP model, which is closer to the optical path and the imaging principles. We derive the closed-form solution for the intrinsic and extrinsic parameters and then refine the parameters by minimizing the re-projections error via LM algorithm. Both simulated data and real data verify the robustness and validity of our proposed method. Due to the image features, the multiple projections are more convenient to be recognized in a focused plenoptic camera. The recognition of projections in a conventional plenoptic camera is mentioned in the work \\cite{bok2014geometric,bergamasco2015adopting}. Moreover, the TPP coordinate system for the conventional plenoptic camera can be regarded as the main lens plane and the focal plane of MLA outside the camera. Therefore, the calibration for a conventional plenoptic camera can be completed by estimating the view coordinates, the focal length and the principle points of the sub-aperture images.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION}\nIt is known that several deeply-embedded young stellar objects are driving highly-collimated molecular outflows with an extremely-high velocity (EHV) and jet-like component running along the axes of the lobes \\citep[e.g.,][]{Bac96}.  \nSince the EHV molecular jets having the terminal velocities of 50--150 km s$^{-1}$ are concentrated within the narrow angles along the axes of the lobes and have large momenta comparable to those of the slowly moving (20--30 km s$^{-1}$) ^^ ^^ classical'' outflows, the EHV jets are considered to be closely connected to the ^^ ^^ primary jet'' which is responsible for driving molecular outflows.\nTherefore, studying the physical and kinematical properties of the EHV jet will allow us to understand the properties of the ^^ ^^ primary jet'' which provide clues to constrain the launching mechanism of the outflows.\n\nThe low-luminosity (7.5 $L_{\\odot}$; \\citet{Tob07}) class 0 source L1448C (also known as L1448-mm) in the Perseus molecular cloud complex (D$\\sim$250 pc: e.g., \\citet{Eno06}) is a spectacular example of an outflow with an EHV jet.\nThe EHV component of this source was identified as the secondary peaks of the CO $J$=2--1 spectra at ${\\sim}{\\pm}$60 km s$^{-1}$ from the cloud systemic velocity \\citep{Bac90}.\nThe CO emission in the EHV range was found to be confined to a series of discrete clumps, called ^^ ^^ bullets\", which are aligned along the axes of the outflow lobes and are symmetrically placed with respect to the central source.\nThe EHV bullets are also observed in several transitions of the SiO \\citep[e.g.][]{Bac91, Dut97}.\nSince the SiO emission has been barely detected in quiescent dark clouds because of its very low abundance (of the order of 10$^{-12}$; \\citet{Ziu89,Mar92}), the detection of SiO in EHV bullets suggests the presence of shocks that enhanced the SiO abundance in bullet gas by a factor of $\\gtrsim$10$^4$.\nAlthough the lower transition of SiO, i.e. $J$=2--1,  was observed not only in the EHV bullets but also in the lower velocity component that delineates the tips and walls of the outflow cavities, $J$=5--4 emission was confined to a pair of EHV bullets located at the closest positions to the star \\citep{Bac91}. \nThis suggests that the excitation condition of the EHV jet varies along the axes, and that the jet gas is highly excited in the close vicinity of the driving source.\nRecently, higher excitation SiO up to $J$=11-10 has been observed by \\citet{Nis07}.\nTheir results have revealed that the innermost pair of bullets, labeled B1 and R1 by \\citet{Bac90}, have a density of $n_{\\rm H_2}\\sim10^6$ cm$^{-3}$ and a kinetic temperature of $T_{\\rm kin}\\gtrsim$ 500 K, which is denser and warmer than the bullets in the downstream.\nIt is also known that the innermost pair of bullets, B1 and R1 are resolved into two clumps, BI-BII and RI-RII, respectively, in the higher resolution ($\\sim$2\\arcsec) interferometric SiO $J$=2--1 observations \\citep{Gui92, Gir01}.\nThe high resolution SiO data exhibit the kinematic structure of the EHV jet near the source, which shows an apparent acceleration of the jet up to 70 km s$^{-1}$ within a region of 6$''$ ($\\sim$2000 AU).\nThe proper motion measurement of the SiO clumps carried out by \\citet{Gir01} suggests that the outflow axis is inclined by 21$^{\\circ}$ with respect to the plane of the sky, and therefore, the SiO clumps in the EHV jet are likely to be moving with absolute velocities of 180 km s$^{-1}$.\n\n\nIn this paper, we present the SiO $J$=8--7, CO $J$=3--2, and 350 GHz continuum images obtained with the Submillimeter Array\\footnote{The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.} (SMA) \\citep{Ho04} at $\\sim$1 arcsecond resolution, which is a factor of three higher than the previous SiO $J$=2--1 images \\citep{Gui92, Gir01}.\nA high angular resolution is crucial for studying the structure and kinematics of the jet near the base, at which the jet velocity increases up to the terminal velocity.\nIn addition, higher transitions of SiO and CO in the submillimeter waverange enable us to segregate the dense and warm gas in the EHV jet from the lower excitation gas in the cavity wall.\n\n\\section{OBSERVATIONS}\n\nThe observations of the SiO $J$=8--7 and CO $J$=3--2 lines and 350 GHz continuum emission were carried out with the SMA on 2006 December 5 in the extended configuration and on 2006 December 25 in the compact configuration.\nThe two array configurations provided projected baselines ranging from 12\\,m to 222\\,m.\nSince the primary beam of the SMA antenna has a size of $\\sim$35\\arcsec, two pointings separated by 17\\arcsec  were observed in order to cover the EHV bullet pair closest to the central source.\nThe receivers have two sidebands; the lower and upper sidebands covered the frequency ranges from 345.5 to 347.5 GHz, and from 355.5 to 357.5 GHz, respectively.\nThe SiO $J$=8--7 and CO $J$=3--2 lines were observed simultaneously in the lower sideband.\nThe SMA correlator divides each sideband of 2 GHz bandwidth into 24 ^^ ^^ chunks\" of 104 MHz width.\nWe used the configuration that gave 256 channels to all chunks, which provided a uniform frequency resolution of 406.25\\,kHz across a 2\\,GHz-wide band.\nThe corresponding velocity resolution was 0.35 km s$^{-1}$.\nWe used Titan for flux calibration, and a pair of quasars 3C84 and 3C111 for amplitude and phase calibrations.\nThe flux calibration was estimated to be accurate to 25\\%.\nThe band pass was calibrated by observing 3C273.\n\nThe calibrated visibility data were Fourier transformed and CLEANed using the MIRIAD package.\nThe velocity-channel maps of the SiO and CO were made with a velocity interval of 1 km s$^{-1}$.\nThe synthesized beam size of the SiO map was 0\\farcs96$\\times$0\\farcs84 with a position angle of $-$84$^{\\circ}$ and that of the CO map was 0\\farcs96$\\times$0\\farcs86 with a position angle of $-$76$^{\\circ}$ with uniform weighting.\nThe rms noise level of the velocity-channel map at 1.0 km s$^{-1}$ width was 0.15 Jy beam$^{-1}$.\nA non-linear joint deconvolution, MOSSDI, which is based on the CLEAN-based algorithm, and is part of the MIRIAD package \\citep{Sau96}, was used for deconvolving the images. \n\nThe 350 GHz continuum map was obtained by averaging the line-free chunks of both sidebands.\nTo improve the signal to noise ratio, the upper and lower sidebands data were combined.\nThe synthesized beam size of the map made with uniform weighting was 0\\farcs93$\\times$0\\farcs83 with a position angle of $-$86$^{\\circ}$.\nThe rms noise levels of the 345 GHz continuum maps was 6.4 mJy beam$^{-1}$.\n\n\\section{RESULTS}\n\\subsection{Continuum emission}\n\nThe 350 GHz continuum map reveals a bright compact source at the center.\nThis source has a peak intensity of 352 mJy beam$^{-1}$, and is surrounded by spatially extended emission.\nIn addition to this bright source, there is a faint emission peak of $\\sim$5$\\sigma$ level at $\\sim$8\\farcs3 southeast of the center.\nRecent {\\it Spitzer Space Telescope} observations at mid-infrared wavelengths have resolved L1448C into two components, L1448C(N) and L1448C(S) \\citep{Jor06}.\nThe bright submillimeter source at the center corresponds to L1448C(N), which is considered to be the driving source of the highly-collimated molecular outflow and is also referred to as L1448-mm, and the southern faint source corresponds to L1448C(S).\nDust continuum emission from L1448C(S) was also detected by \\citet{Jor07} at 230 GHz and 350 GHz with the SMA, and by \\citet{Mau10} at 107 GHz with the Plateau de Bure Interferometer (PdBI).\n\nThe visibility amplitudes plot for L1448C(N) as a function of {\\it uv} distance (Fig.~\\ref{fig2}) suggests that this source consists of two components; one is from a spatially extended envelope that dominates the flux at a {\\it uv} distance of $<$50 k$\\lambda$, and the other is from a compact source that is prominent at $>$50 k$\\lambda$.\nThe visibility amplitude profile was fit by two circular gaussian components; one is an extended component with a deconvolved size of $\\sim$3.6\\arcsec and a flux of $\\sim$440 mJy, and the other is a compact component with a deconvolved size of $\\sim$0.3\\arcsec and a flux of $\\sim$330 mJy.\nThe peak flux values of the extended and compact components correspond to $\\sim$27 mJy and $\\sim$330 mJy beam$^{-1}$, respectively, per 0\\farcs93$\\times$0\\farcs83 beam.\nAs shown in Fig. 1a, the extended component appears as a bump in the northwest and a tail in the southeast (both of them are in the 3$\\sigma$ level) of the central source, suggesting that this emission extends along the outflow axis.\nSimilar emission feature along the outflow axis is also seen in the maps of 3 mm \\citep{Gui92}, 2.6 mm \\citep{Bac95}, 1.4 mm \\citep{Sch04}, and 1.3 mm \\citep{Mau10} observed with 1.5--3\\arcsec resolution.\nSuch a faint elongated feature was clearly seen in L1157-mm, and was interpreted as the edges of the cavity excavated by the outflow \\citep{Gue97}.\nIt is, therefore, possible that the extended emission in L1448C(N) also delineates the inner part of the envelope which is disturbed by the outflow, although the cavity-like structure is not clearly seen.\nAn alternative interpretation is that the faint emission comes from an embedded companion.\nA faint secondary peak seen in the 230 GHz map of \\citet{Jor07} implies this possibility.\nHowever, the position of the secondary peak at 230 GHz in \\citet{Jor07} is 1\\arcsec offset toward the northwest from that of our 350 GHz map.\nAs discussed in \\citet{Jor07} such a small difference in position between the 350 GHz peak and 230 GHz peak could be introduced by the extended emission from the envelope component sampled by different {\\it uv} coverages.\nSince the total flux at 350 GHz observed with the SMA (extended + compact components) corresponds to 43 \\% of the flux observed by \\citet{Hat05} at 850 $\\mu$m with SCUBA at the JCMT (1.737 Jy per 14\\arcsec  beam), it is possible that the extended component that is not sampled well with the SMA affects the morphology of the faint component.\nAlthough the northwestern bump may harbor an embedded companion, it is likely that most of the extended emission arises from the inner part of the envelope.\nTherefore, the former scenario, which attributes the extended component to the envelope--outflow interacting region, is more preferable in this case.\n\nThe visibility amplitude profile at {\\it uv} distance longer than $\\sim$50 k$\\lambda$ implies that the compact component is not point-like but has a spatially resolved structure.\nIn order to exclude the contamination from the extended envelope component, we made a map using the visibility data with the {\\it uv} distances greater than 70 k$\\lambda$ (see Fig.1b).\nThe synthesized beam of this map is 0\\farcs70$\\times$0\\farcs50 with a position angle of $-$87.2$^{\\circ}$.\nIt is shown that the compact source has an elongation along the axis perpendicular to the outflow axis.\nA two dimensional Gaussian fit to the visibility data with {\\it uv} distances larger than 70 k$\\lambda$ yields the deconvolved major and minor axes of 0.37\\arcsec (90 AU) and 0.26\\arcsec (65 AU), respectively, with a position angle of $\\sim$70$^{\\circ}$.\nThe source position derived from the fit is $\\alpha$(J2000) = 3$^h$25$^m$38.873$^s$, $\\delta$(J2000)=30$^{\\circ}$44$'$05\\farcs35.\nThis position agrees well (less than 0\\farcs1) with the 3.6 cm continuum position observed with a smaller beam of 0\\farcs31$\\times$0\\farcs27 \\citep{Rei02}.\n\nThe 350 GHz flux of L1448C(S) was measured to be $\\sim$60 mJy, which is consistent to that reported by \\citet{Jor07}.\nL1448C(S) was detected in 230 GHz by \\citet{Jor07} but not by \\citet{Mau10} at the same frequency.\nThe non-detection of this source by \\citet{Mau10} is probably because this source is located near the edge of the primary beam of the PdBI ($\\sim$22\\arcsec).\nIf the response of their primary beam is taken into account, the 230 GHz continuum source with a flux of 12.8$\\pm$3 mJy detected by \\citet{Jor07} could be below the detection limit of their observations ($\\sim$8.4 mJy beam$^{-1}$).\nAlthough L1448C(S) is bright in mid-infrared, its sub-mm flux is more than ten times weaker than L1448C(N).\nIt is unlikely that such a small sub-mm flux is due to the effect of missing flux, because single-dish measurements at 450 and 350 $\\mu$m \\citep{Cha00} showed no hint of the secondary component to the south of L1448C(N) even though the angular resolution ($\\sim$8\\arcsec) was comparable to the separation of two sources.\n\n\n\\subsection{The SiO Jet}\n\nThe SiO $J$=8--7 emission was detected in two velocity ranges from $-$70 to $-$12 km s$^{-1}$ (blueshifted) and from 20 to 71 km s$^{-1}$ (redshifted) with respect to the systemic velocity of $V_{\\rm LSR}{\\sim}$5.0 km s$^{-1}$.\nFig.~\\ref{fig3} shows velocity channel maps of the SiO $J$=8--7 emission at 10 km s$^{-1}$ intervals.\nIt is shown that the SiO emission comes from the jetlike narrow region with its blueshifted part to the northwest and the redshifted part to the southeast of L1448C(N).\nThe SiO $J$=8--7 jet is partially resolved along its minor axis; after deconvolution from the SMA beam, the width of the SiO jet is $\\sim$0.8\\arcsec ($\\sim$200 AU) FWHM on average.\nIn order to estimate the missing flux, we have smoothed the SMA map to a resolution of 14\\arcsec and compared it with the SiO $J$=8--7 spectra observed with the James Clerk Maxwell Telescope \\citep[JCMT;][]{Nis07}.\nIt is found that 85--100 \\% of the single-dish flux is recovered by the SMA, implying that almost all the SiO $J$=8--7 emission arises from the narrow jet.\nSiO $J$=8--7 emission mainly comes from the B1 and R1 ^^ ^^ bullets'' identified in the single-dish CO $J$=2--1 map by \\citet{Bac90}.\nThe SiO jet consists of a chain of knots with a typical size scale of $\\sim$1--1.5\\arcsec.\nA comparison with the previous SiO $J$=2--1 maps with $\\sim$3\\arcsec resolution \\citep{Gui92, Gir01} reveals that the three pairs of knots close to L1448C(N) seen in the SiO $J$=8--7 map corresponds to the inner pair BI and RI in the SiO $J$=2--1 map, and the two pairs in the downstream correspond to the outer pair BII and RII.\nThe innermost knot pair BIa and RIa are located within 1\\arcsec (250 AU) from L1448C(N).\nThe high-resolution image also shows that the SiO jet is not straight.\nA close-up view of the high velocity component (Fig.~\\ref{fig5}) shows that the jet changes its position angle from +15$^{\\circ}$ at BII, $-$25$^{\\circ}$ at BI, $-$20$^{\\circ}$at RI, to $-$5$^{\\circ}$ at RII.\nThe kinks between BI and BII, and RI and RII are also seen in the previous SiO $J$=2--1 maps of $\\sim$3\\arcsec resolution \\citep{Gui92, Gir01}.\nHowever, it is more obvious in the higher resolution image.\nIn addition, it is clear that the jet axes in BI and RI are also misaligned by 5$^{\\circ}$.\n\\subsection{CO Jet and outflow}\n\nThe CO $J$=3--2 emission was detected in the wide velocity range from $-$77 km s$^{-1}$ to $+$79 km s$^{-1}$ with respect to the systemic velocity.\nIn the velocity ranges of ${\\Delta}V < {\\pm}$40 km s$^{-1}$ (Fig. 4a--4d), the CO $J$=3--2 emission delineates two V-shaped structures  open to the northwest and southeast with a common apex at the position of L1448C(N).\nThe opening angles of the V-shape features become narrower as the velocity offset increases, suggesting that the CO emission in the V-shaped features comes from the limb-brightened shells.\nThe largest opening angle of the shell is $\\sim$60$^{\\circ}$ in the blueshifted lobe, while it is $\\sim$40$^{\\circ}$ in the redshifted lobe.\nIn the higher velocity ranges of ${\\Delta}V = {\\pm}$41--70 km s$^{-1}$ (Fig. 4e, 4f, and 4g), the CO emission comes from a narrow jet-like region.\nThe CO flux recovered by the SMA in each velocity range was estimate by comparing the CO spectra observed by the SMA with those observed by the JCMT.\nThe SMA map was smoothed to be a 14\\arcsec resolution so as to match with the beam of the JCMT.\nIt is found that the recovered flux is only $\\sim$20~\\% in the lowest velocity ranges (${\\Delta}V = {\\pm}$1--10 km s$^{-1}$; Fig. 4a), and is $\\sim$50\\% in the next velocity range of ${\\Delta}V = {\\pm}$11--20 km s$^{-1}$ (Fig. 4b). \nThe edges of the shells in Fig. 4a and 4b look very steep probably because significant amount of the CO emission from spatially extended component was filtered out by the interferometer.\nIn fact, previous examples of the L1157 outflow and IRAS 04166+2706 outflow revealed that the edges of the bipolar cavities were emphasized in the maps made with the interferometer data alone, while the cavities were filled by the diffuse CO emission when the single-dish data were added \\citep{Gue96,San09}.\nOn the other hand, in the higher velocity ranges with ${\\Delta}V > {\\pm}$20 km s$^{-1}$, 80--100\\% of the CO flux was recovered by the SMA.\nThis suggests that almost all the CO $J$=3--2 flux in the extremely high velocity ranges (Fig. 4e, 4f, and 4g) comes from the narrow jet.\n\nAs in the case of the SiO jet, the CO $J$=3--2 jet also shows clumpy structure.\nIn addition, most of the knots seen in the SiO $J$=8--7 map have their counterparts in the CO map.\nHowever, the innermost knots BIa and RIa, which are significant in the SiO map at ${\\Delta}V = {\\pm}$21--60 km s$^{-1}$ are barely seen in the CO map.\nThis is probably because most of the CO molecules in these knots are excited to the levels higher than $J$=3 because of high density and high temperature.\nSimilar feature with strong SiO and weak CO in the close vicinity of the protostar was also observed in the highly collimated jet in the HH211 outflow \\citep{Pal06, Lee07b}.\nAnother difference between the CO jet and SiO jet is seen between BIc and BIIa, and RIc and RIIa. \nThe CO map reveals the knot pair labeled BI-II and RI-II, while the SiO map shows only faint emission (Fig.~\\ref{fig5}).\nIn the CO $J$=3--2, the overall distribution of the EHV jet is more continuous as compared to the SiO distribution. \n\nAs in the case of the SiO jet, the blue and red axes of the CO jet are also misaligned by $\\sim$5$^{\\circ}$.\nThe kinks between BI and BII, and RI and RII are also seen in the CO jet.\nIn addition, the ridge of the CO emission wiggles along the jet axis.\nThis wiggling feature is clearly seen in the maps of the highest velocity ranges (Fig.~\\ref{fig6}).\nIt is likely that each knot has been ejected in slightly different direction.\nSince the typical knot separation, 2--3\\arcsec corresponds to a time interval of 15--20 yr, the observed jet wiggling suggests that the direction of jet ejection also varies in a similar time scale.\nOn the other hand, the CO emission with lower intensity that is surrounding the emission ridge tends to extend linearly along the axes.\nThe transverse width of the lower level CO emission component increases with a distance from the source.\nThis is probably because significant part of the CO emission comes from the outflow shell even in the extremely high velocity ranges (see the next section).\nThere is faint CO emission with the highest velocity seen ahead of the SiO jet.\nSince this highest velocity emission is spatially extended, it is likely that it arises from the highest velocity part of the shells.\n\n\\subsection{Kinematics of the jet along its axis}\n\nPosition-velocity (P-V) diagrams of the SiO and CO emission along the jet axes (the position angle is $-$25$^{\\circ}$ for the blueshifted part and is $-$20$^{\\circ}$ for the redshifted part) are shown in Fig.~\\ref{fig7}.\nDue to the change of the position angle, the outer knots BIIb and RIIb do not appear in these P-V diagrams.\n\nThe velocity structure of the jet is well traced in the SiO.\nThe jet velocity rapidly increases within $\\backsimeq$1\\arcsec from the star, and reaches close to its highest velocity (${\\pm}$65 km s$^{-1}$ from $V_{\\rm sys}$) at $\\sim$5\\arcsec from the star, which corresponds to the positions of the BIc and RIc knots.\nThe velocity dispersion is extremely large (${\\Delta}V{\\sim}$50 km s$^{-1}$ at 1$\\sigma$ level) at the base, while it narrows to ${\\Delta}V{\\sim}$20 km s$^{-1}$ at the positions of the BIc and RIc knots. \nIn the positions further downsteam, from BIc to BIIa and from RIc to RIIa, the observed radial velocity slightly decreases in the blueshifted side, while it increases in the redshifted side.\nThese radial velocity changes are probably due to the change of inclination angle, because the position angle of the jet also changes at these positions.\nThe velocity pattern shown in the SiO is similar to those of the atomic jets from class I source and T Tauri stars, in which the low-velocity components with broad line widths are located near the base and the high-velocity components with narrower line widths are located further from the source \\citep[e.g.][]{Pyo02}.\nIn addition to the global velocity structure, each knot shows its internal velocity gradient with its higher velocity in the upstream side and lower velocity in the downstream side.\n\n\\subsection{Kinematics of the CO outflow shells}\n\nIn the CO $J$=3--2, the jet shows a similar velocity pattern with a similar velocity centroid as the SiO.\nIn addition to the jet, the P-V map of the CO shows extended low velocity features (slower than $\\pm$15 km s$^{-1}$) and linear velocity features with the velocity magnitude increasing with the distance from the source (^^ ^^ Hubble-law''). \nThese Hubble-law features are obviously different from the jets, and are not seen in the SiO, suggesting that they are from the outflow shells.\nIt should be noted that the radial velocity offsets of the linear velocity components become larger than 50 km s$^{-1}$ at around $\\backsimeq$10\\arcsec from the star, and contaminate the CO map in the EHV range.\n\nThe observed velocity pattern of the outflow shell is different from that predicted by the jet-driven bow shock model, in which the P-V structure shows a convex spur with high-velocity components at the jet head \\citep[e.g.][]{Mas93, Che94}; it is rather similar to the velocity pattern produced by the wide-angle wind model \\citep{Shu91, Shu00, Li96}.\nIn the wide-angle wind model, an outflow shell consists of ambient material swept-up by a radial wind from a young star.\nA ^^ ^^ Hubble-law'' in the shell velocity is expected if the shell is expanding into an ambient medium with a radial density profile of ${\\propto}r^{-2}$ \\citep{Shu91}.\nThe shape of the shell is determined by the combination of the poloidal density profiles of the wind and ambient gas.\nIt is approximately parabolic for an $X$-wind type of wide opening angle wind with an angle-dependent density profile of ${\\propto}1\/{\\rm sin}^2{\\theta}$ (where $\\theta$ is an angle measured from the axis of the flow) expanding into an ambient medium with a ${\\propto}{\\rm sin}^2{\\theta}\/r^2$ density profile, which is appropriate for magnetized cores \\citep{Li96}.\nTherefore, we adopted the simplified analytical model of a wind-driven model proposed by \\citet{Lee00} to examine whether the observed morphology and kinematics of the CO outflow shells can be explained by means of wide-opening angle wind model.\nIn the cylindrical coordinate system, the structure and velocity of the shell can be written as follows:\n \\begin{equation} z=CR^2, {\\it v}_R={\\it\nv}_0R, {\\it v}_z={\\it v}_0z, \\end{equation} \nwhere z is the distance along the outflow axis; R is the radial size of the outflow perpendicular to z ;\n$C$ and {\\it v}$_0$ are free parameters which describe the spatial and velocity distributions of the outflow shell, respectively.\nThe observed outflow shell features and the velocity patterns of the redshifted and blueshifted lobes were successfully reproduced by the model curves with $C$=0.8 arcsec$^{-1}$and {\\it v}$_0$=5.0 km s$^{-1}$ arcsec$^{-1}$, and $C$=0.6 arcsec$^{-1}$ and {\\it v}$_0$=5.0 km s$^{-1}$ arcsec$^{-1}$, respectively.\nHere the inclination angle of the outflow axis with respect to the plane of the sky was assumed to be 21$^{\\circ}$, which is derived from the proper motion measurement for the SiO $J$=2--1 jet by \\citet{Gir01}.\nThe dynamical age of the outflow shell is given by 1\/$v_0$ and is estimated to be $\\sim$240 yr.\nThis number is roughly consistent with the dynamical age of $\\sim$500 yr derived from the extent of the shell ($\\sim$25\\arcsec = 6300 AU), the mass-weighted-mean radial velocity of the shell ($\\sim$22 km s$^{-1}$), and the inclination angle of the outflow axis ($\\sim$21$^{\\circ}$).\nThe model curves that delineate the outer boundaries of the lobes projected onto the plane of the sky are shown in Fig.~\\ref{fig8} on top of the contours of the outflow shells.\nThe model curves of the P-V maps are shown in Fig.~\\ref{fig7}.\nHowever, this simplified wind-driven model has difficulty in reproducing several observed features.\nFirst, the observed CO intensity drops sharply at the systemic velocity and does not extend to the opposite velocity ranges, while the model curves on the P-V maps predict the emission at $V_{\\rm LSR}{\\sim}$10 km s$^{-1}$ in the blueshifted lobe and at $V_{\\rm LSR}{\\sim}$0 km s$^{-1}$ in the redshifted lobe.\nSecond, the shapes of the shells in the different velocities cannot be reproduced (Fig.~\\ref{fig9}).\nThe observed CO emission shows V-shaped distributions in the velocity ranges close to the systemic velocity.\nOn the other hand, the model curves predict elliptical shapes (Fig. 9a and 9b).\nIn addition, the transverse width of the CO shell becomes narrower as the velocity offset increases, while the model predict the opposite trend.\nIt should be noted that the CO emission near the protostar in Fig. 9c and 9d arises from the jet.\n\n\n\\subsection{Kinematics across the jet}\n\nIn order to search for the signs of jet rotation, the velocity gradient along the minor axes of the jet was examined.\nThe P-V diagrams of the SiO and CO at the positions of innermost two pairs of knots (Fig.~\\ref{fig11}) show no clear velocity gradient at the positions of BIa, RIa, and RIb.\nOn the other hand, there is some hint of velocity gradient in the SiO at the position of BIb;  the southwestern side of the jet tends to be more blueshifted than the northeastern part.\nHowever, this velocity gradient is in the opposite sense to the rotation pattern of the NH$_3$ core with its blueshifted part in the northeast and the redshifted part in the southwest \\citep{Cur99}.\nThis suggests that the observed velocity gradient is not due to the rotation.\n\n\n\\subsection{Physical properties of the EHV jet}\n\nThe physical parameters of the EHV jet were estimated using the CO flux measured in the velocity ranges of ${\\Delta}V$=50--70 km s$^{-1}$, in which most of the CO flux was recovered by the SMA.\nWe assumed that the CO emission in these velocity ranges is optically thin, and that the excitation condition of the CO follows the LTE.\nA fractional abundance of the CO and a mean atomic weight were adopted to be 10$^{-4}$ and 1.41, respectively.\nThe kinetic temperature of molecular gas in the EHV jetwas derived to be 500--1200 K from the far infrared lines of CO, H$_2$O, and H$_2$ \\citep{Nis99,Nis00}, and $>$500 K from the millimeter and submillimeter SiO lines \\citep{Nis07}.\nHowever, it is uncertain whether the CO (3-2) emission arises from the same gas component that contributes to the higher transition lines in the far-infrared.\nIn fact, \\citet{Gus08a} have modeled the multi-transition SiO data observed by \\citet{Nis07} using their face-on C-type shock models, and obtained much lower temperature of 70--90 K.\nThis implies that the most of the gas in the EHV jet is not so warm as $>$500 K. \nTherefore, an excitation temperature of $\\sim$100 K was assumed for the calculation.\n\nThe mass for each of the blueshifted and the redshifted jets is estimated to be $\\sim$10$^{-3}$ $M_{\\odot}$, which is a few times higher than the bullet mass estimated by \\citet{Bac90} using an assumption of $T_{\\rm ex}{\\sim}$20 K.\nThe momentum and kinetic energy of the bipolar jet are estimated to be 0.3 $M_{\\odot}$ km s$^{-1}$ and 5.0$\\times$10$^{44}$ erg, respectively.\nHere, the calculation was done using the de-projected jet velocity under the assumption that the jet axis is inclined by 21$^{\\circ}$ from the plane of the sky \\citep{Gir01}.\nThe dynamical timescale of the jet derived from the length ($\\sim$ 20\\arcsec = 5000 AU) and mass weighted mean (de-projected) velocity is estimated to be only $\\sim$150 yr.\nBecause of the rather large mass and short timescale, the obtained mass loss rate is also large, $\\sim$10$^{-5}$ $M_{\\odot}$ yr$^{-1}$.\nFurthermore, the high velocity of $\\sim$160 km s$^{-1}$ (corrected for the inclination of 21$^{\\circ}$) brings extremely large momentum supply rate of $\\sim$2${\\times}$10$^{-3}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$ and mechanical luminosity of $\\sim$26$L_{\\odot}$ for the jet.\n\nIt should be noted that the derived total mechanical luminosity of $\\sim$26 $L_{\\odot}$ is a factor of 3.5 larger than the bolometric luminosity of the central source (7.5 $L_{\\odot}$).\nOne possible reason for such a discrepancy is that the mass of the jet was overestimated.\nSince the CO $J$=3--2 emission comes from both the jet and shell, the high velocity part of the shell may contaminate the CO flux that was used to calculate the mass of the jet.\nHowever, this effect should not be significant, because the CO flux from the shell does not dominate  the CO flux at ${\\Delta}V >$50 km s$^{-1}$ velocity range.\nThe second possible reason is the excitation temperature.\nThe assumed excitation temperature of $T_{\\rm ex}{\\sim}$100 K is much lower then the kinetic temperature of $>$500 K derived from the far infrared measurements \\citep{Nis99,Nis00}.\nHowever, the assumption with higher excitation temperature increases the mass and dynamical parameters (i.e. momentum, kinetic energy, mass loss rate, momentum rate, and mechanical luminosity).\nFor example, the assumption of $T_{\\rm ex}$= 500 K increases the mass and dynamical parameters by a factor of 3.8, and brings an extremely large mechanical luminosity of $\\sim$90 $L_{\\odot}$.\nThis implies that the bulk of the molecular gas in the jet is not so warm as 500 K, and that the warm gas which contributes to the far infrared emission is not the major component.\nOn the other hand, the lower excitation temperature of 40--50 K reduces the mass.\nHowever this effect is limited to a factor of 1.5 at the most.\nAnother possible reason is the CO abundance, which was assumed to be 10$^{-4}$.\nThis value can be as high as 4$\\times$10$^{-4}$ in the chemical model of protostellar winds proposed by \\citet{Gla91}, in which molecules such as CO and SiO are formed via gas-phase reactions in an initially atomic protostellar wind.\nIf the higher CO abundance of 4$\\times$10$^{-4}$is adopted, the mass and all dynamical parameters are reduced by a factor of 4, and the mechanical luminosity of the jet becomes $\\sim$7 $L_{\\odot}$, which is comparable to the bolometric luminosity of the central source.\nHowever, this value for the mechanical luminosity of $\\sim$7 $L_{\\odot}$ should be the lower limit, because it is derived under the assumption of the optically thin CO emission.\nThe physical parameters derived using $T_{\\rm ex}$ = 100 K and CO\/H$_2$ abundance ratio of 4$\\times$10$^{-4}$ are given Table 1.\n\n\\subsection{Physical parameters of the outflow shell}\n\nThe mass, momentum, and kinetic energy of the outflow shell were estimated using the CO flux measured in the velocity ranges of $\\pm$1--40 km s$^{-1}$ from the systemic velocity.\nThe CO emission is assumed to be optically thin and in the LTE condition with an excitation temperature of 40 K, which is derived from the observed peak brightness temperature of the CO $J$=3--2 line.\nSince most of the gas in the shell component is considered to be the swept-up ambient material, the canonical CO abundance of 10$^{-4}$ was used for the calculations.\nThe dynamical parameters of the outflow shell are summarized in Table 2. \nSince significant fraction of the CO flux is missed in the low velocity ranges, Table 2 show the parameters with and without correction for the effect of missing flux. \nThe inclination angle of the outflow axis is assumed to be 21$^{\\circ}$ as in the case of the EHV jet.\nThe dynamical timescale of the outflow shell is adopted to be $\\sim$240 yr, which is derived from the modeling descried in the previous section.\nTable 2 shows that the dynamical parameters of the outflow shell are comparable to those of the EHV jet listed in Table 1.\nIf the effect of the missing flux is corrected, the outflow shell has a momentum supply rate of  $\\sim$1.8$\\times$10$^{-3}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, and a mechanical luminosity of $\\sim$8.6 $L_{\\odot}$.\nThe mechanical lumiosity derived here is comparable to the bolometric luminosity of the central source and the mechanical luminosity of the EHV jet.\nIt should be noted that the dynamical parameters for the redshifted outflow are affected by the contamination from the L1448C(S) outflow (see the next section).\nHowever, the effects of L1448C(S) outflow to the dynamical parameters are not so significant, because this outflow is seen only in the low velocity ranges.\n\n\n\\subsection{CO outflow from L1448C(S)}\n\nIn the CO maps in the low velocity ranges such as Fig. 4a and 4b, there is a  blueshifted component in the redshifted lobe.\nThis blueshifted component shows a triangle shape with its apex at the position of L1448C(S), and extends to the northeast direction with a position angle of $\\sim$40$^{\\circ}$.\nIt is likely that this blueshifted component is related to the activity of L1448C(S).\nFig.~\\ref{fig11}, which provides a close-up view of the L1448C(S) region, shows a redshifted counterpart to the southwest of L1448C(S), a compact component at $\\sim$1\\arcsec southwest of L1448C(S) and another extended component to the southwest of the V-shaped shell of the L1448C(N) outflow.\nThis NE-SW outflow from L1448C(S) is also highly collimated.\nThe opening angle of the lobes is $\\sim$40$^{\\circ}$, which is similar to the redshifted lobe of the L1448C(N) outflow.\nHowever, this outflow is seen only in the velocity ranges slower than 15 km s$^{-1}$, and has no high-velocity jet-like component.\nThere is no SiO $J$=8--7 emission either.\n\nThe redshifted part of the L1448C(S) outflow overlaps with the western wall of the L1448C(N) outflow.\nAt the place where the two outflows are superposed, the CO emission is significantly enhanced and the wall of the L1448C(N) outflow lobe is bending.\nTherefore, it is possible that the two outflows are intersecting, although three dimensional geometries of two outflows are uncertain.\n\nThe physical parameters of this outflow were estimated assuming that the CO emission is optically thin and that the excitation condition of the CO is in LTE.\nThe fractional abundance of the CO was adopted to be the canonical value, 10$^{-4}$, because the V-shaped morphology and low velocity suggest that the bulk of the L1448C(S) outflow is likely to be the swept-up ambient gas.\nA mean atomic weight of the gas was assumed to be 1.41.\nThe excitation temperature was assumed to be 20 K, which is same as the rotational temperature of the dense gas surrounding L1448C derived from the NH$_3$ observations \\citep{Cur99}.\nThe mass of the blueshifted component is estimated to be 5.3$\\times$10$^{-4}$ $M_{\\odot}$ and that of the redshifted component is 2.7$\\times$10$^{-4}$ $M_{\\odot}$.\nSince the CO $J$=3--2 emission is assumed to be optically thin, the values derived here are the lower limits.\nIn addition, the mass of the redshifted component is likely to be underestimated, because we exclude the region where the two outflows are superposed.\nIt is also possible that part of the flux from the spatially extended component is missing.\nThe dynamical parameters of the flow are summarized in Table~\\ref{table3}.\nTable 3 gives two kinds of values; one is corrected for the inclination effect  and the other is without this correction (uncorrected).\nHere, the inclination angle of the L1448C(S) outflow from the plane of the sky was assumed to be  32.7$^{\\circ}$, which corresponds to the mean inclination angle from assuming randomly oriented outflows.\nWe also assumed that the outflow lobes have a size of $\\sim$20\\arcsec (5000 AU).\nThe actual size of this outflow is not clear because of confusion with L1448C(N) outflow and its sidelobes.\nHowever, there is no counterpart of this outflow in the single-dish CO $J$=3--2 map \\citep{Hat07}.\nThis suggests that the spatial extent of this outflow is not large, or the spatially extended component is fainter than the sensitivity limit of the single-dish measurement. \nIn the later case, the faint component would not make significant contribution to the dynamical parameters even though its spatial extension is large.\n\n\\section{DISCUSSION}\n\n\\subsection{Compact disk around L1448C(N)}\n\nThe compact component of dust continuum emission is partially resolved by the 0\\farcs7$\\times$0\\farcs5 beam.\nThe observed structure is elongated perpendicular to the outflow axis, suggesting that this component traces the disk surrounding the protostar. \nIf the beam-deconvolved major axis, $\\sim$90 AU, represents the diameter of the disk, the disk size of L1448C(N) is comparable to those around the youngest protostars such as HH211 \\citep{Lee07b} and HH212 \\citep{Cod07, Lee07a, Lee08}.\nIf the measured flux (330 mJy) comes from the region with 0\\farcs37$\\times$0\\farcs26 size, the brightness temperature corresponds to $\\sim$35 K.\nThe spectral energy distribution (SED) of the compact source from 8.3 GHz to 350 GHz is shown in Fig.~\\ref{fig12}.\nThe measured flux densities are fit by a single power law with a spectral index  $\\alpha$ of 1.98 (solid line in Fig~\\ref{fig12}), which agrees with the previous result ($\\alpha$ = 1.84) of \\citet{Sch04}.\nThis spectral index is smaller than $\\alpha$ = 3.4 derived from the photometric broadband measurement including the contribution of the larger scale envelope component \\citep{Fro05}.\nSince the emission at cm wavelengths is likely to be the free-free emission from shock-ionized gas,\nthe contribution of the free-free component at mm and sub-mm wave ranges was estimated by using the equations given by  \\citet{Cur90}.\nUsing the parameters of the stellar wind in \\citet{Cur90}, the flux densities of the free-free emission at mm and sub-mm wave ranges were estimated to be less than 1 mJy (dotted line in Fig~\\ref{fig12}).\nTherefore, the small index number is unlikely to be due to the contribution of the free-free component.\nThe power law fit assumes that the emission is optically thin and that the Rayleigh-Jeans approximation is valid.\nHowever, the observed index $\\alpha\\sim$2 is close to the value for the blackbody radiation, suggesting that the emission is optically thick.\nIn addition, a brightness temperature of 35 K suggests that the Rayleigh-Jeans approximation is not applicable in the mm and sub-mm wave ranges.\nTherefore, we applied an optically thick fit without Rayleigh-Jeans approximation using the formula,\n\\begin{equation}\nS_{\\nu}={\\Omega}_s B_{\\nu}[1-{\\rm exp}(-{\\tau}_{\\nu})],\n\\end{equation}\nWhere $S_{\\nu}$ is the flux density, ${\\Omega}_s$ is the source size, B$_{\\nu}$ is the Planck function, and ${\\tau}_{\\nu}$ is the dust optical depth, which is assumed to follow the power low, ${\\tau}_{\\nu} {\\propto} {\\nu}^{\\beta}$.\nAssuming that the source size is 0\\farcs37$\\times$0\\farcs26 and a dust temperature of 40 K, the SED fit provides $\\beta$ = 1.3 and ${\\tau}_{350 {\\rm GHz}}$ = 7.5 (dash-dotted line in Fig~\\ref{fig12}).\nThe average optical depth at 350 GHz for a disk of mass (gas+dust) $M_D$ and radius $R_D$ is given by \n\\begin{equation}\n<\\tau_{350 {\\rm GHz}}> = \\left( \\frac{0.5}{{\\rm cos}  \\theta} \\right) \\left( \\frac{M_D}{0.1 M_{\\odot}} \\right) \\left( \\frac{R_D}{100 {\\rm AU}} \\right) ^{-2},\n\\end{equation}\nwhere $\\theta$ is the disk inclination angle to the line of sight \\citep{Jor07}.\nUsing this relation, the mass of the disk with ${\\tau}_{350 {\\rm GHz}}$ = 7.5, $R_D$ = 45 AU, and $\\theta$ = 69$^{\\circ}$ (assuming that the disk is perpendicular to the jet axis) is estimated to be 0.11 $M_{\\odot}$.\nThe mass derived here is approximately twice as large as the lower limit of 0.047 $M_{\\odot}$, which is derived under the assumption of optically thin emission.\n\n\\subsection{Stellar mass loss rate and its implication to the protostellar evolution}\n\nSince protostellar jet is considered to be closely linked to the mass accretion,  the stellar mass-loss rate gives us a rough estimate of the mass accretion rate onto the star.\nTheoretical estimate for the ratio of mass outflow to mass accretion rate ($\\dot{M}_{\\rm out}\/\\dot{M}_{\\rm acc}$) is ${\\sim}$1\/3 for an X-wind type magneto-centrifugal wind \\citep[e.g.][]{Shu94}.\nIf the $\\dot{M}_{\\rm out}\/\\dot{M}_{\\rm acc}$ ratio is assumed to be $\\sim$0.3, the total mass-loss rate (blue + red) derived from the  CO flux, 2.4$\\times$10$^{-6}$ $M_{\\odot}$ yr$^{-1}$, gives us the mass accretion rate of  8$\\times$ 10$^{-6}$ $M_{\\odot}$ yr$^{-1}$.\nIn spite of rather high accretion rate, the observed bolometric luminosity is only 7.5 $L_{\\odot}$, suggesting that the mass of the central star is still very small.\nIf most of the observed bolometric luminosity is released by means of accretion, the mass of the central star is calculated by using the relation, $M_*$ = $L_{\\rm acc}R_{*}\/G\\dot{M}_{\\rm acc}$, where $M_*$ is the mass of the central star, $L_{\\rm acc}$ is the accretion luminosity, and $\\dot{M}_{\\rm acc}$ is the mass accretion rate onto the protostar.\nThe radius of the protostar, $R_*$, is considered to be $\\sim$1 $R_{\\odot}$ in the earliest evolutionary stage with very low mass and $\\sim$3 $R_{\\odot}$ in the later stage \\citep[e.g.][]{Sta88}.\nThe mass of the central star is estimated to be 0.03--0.09 $M_{\\odot}$ for an accretion luminosity of 7.5 $L_{\\odot}$.\nWith a stellar mass of 0.03--0.09 $M_{\\odot}$, the Keplerian velocity at the surface of the protostar becomes $\\sim$80 km s$^{-1}$.\nIn this case, the jet velocity to Keplerian velocity ratio becomes $\\sim$2, which is reasonable if the jet is launched by magneto-centrifugal force \\citep{Shu94,Pud07}.\nIf a constant mass accretion rate of 8$\\times$ 10$^{-6}$ $M_{\\odot}$ yr$^{-1}$ is assumed, the age of the central star is estimated to be (4--12)$\\times$10$^3$ yr.\nThe timescale derived here is consistent to the kinematic age of the larger scale outflow of $\\sim$0.3 pc scale \\citep{Bac90}.\nHowever, morphology of the EHV jets implies that the mass accretion was variable.\nThe highly-colimated EHV jets terminate at $\\sim$20\\arcsec from the source, suggesting that L1448C(N) experienced lower activity phase in the past and enhanced its activity significantly in the last $\\sim$150 yr.\n\n\n\\subsection{Clumpy structure in the L1448C(N) jet}\n\nHigh resolution SiO and CO maps show that the EHV bullets B1 and R1 identified by \\citet{Bac90} consist of chains of knots.\nIf the BI-II and RI-II knots are included, the knots are aligned with almost equal intervals of $\\sim$2\\arcsec (500 AU).\nSimilar knotty structure with semi-regular intervals is also seen in the SiO and CO jets in HH211 \\citep{Hir06, Pal06, Lee07b} and HH212 \\citep{Cod07, Lee07a}.\nIn the case of HH211 and HH212, the knots seen in the SiO and CO have their counterparts in the near infrared H$_2$ emission except the innermost knots pairs that were highly obscured.\nIn the case of the L1448C(N) outflow, H$_2$ emission knots are seen only in the northern blueshifted side and not in the southern redshifted side \\citep{Dav94, Eis00}.\nThis is probably because the axis of the L1448C(N) jet is inclined from the plane of the sky and the near infrared emission in the southern part is obscured by the dense gas envelope traced by the NH$_3$ emission \\citep{Cur90}.\nIn the northern side, the morphology of the SiO jet coincides well with that of the H$_2$ jet \\citep{Eis00}, which also shows a kink between BI and BII components.\nTherefore it is likely that the knots in the L1448 jet are the internal bow shocks in the jet beam as in the cases of HH211\\citep{Hir06, Pal06, Lee07b} and HH212 \\citep{Cod07, Lee07a}.\nIn fact, some of the SiO knots are partially resolved in the transverse direction, and the RII-a knot shows an arc-shaped structure typical of a bow shock (Fig. 3f).\nThe SiO emission is weak at the positions of the BI-II and RI-II knots, suggesting that the shocks at these positions are rather weak as compared to the other knot positions.\nSince the jet is deflected at the positions of BI-II and RI-II knots, it is likely that the jet material there is impacting less dense material surrounding the jet beam.\n\n\n\\subsection{Jet bending}\n\nAs shown in Fig. 5,  the blue part and the red part of the jet are misaligned by  $\\sim$5$^{\\circ}$ and forming a C-shaped structure bending toward the west.\nSuch a C-shaped bending of the jet could be due to the Lorenz forces between the jet and interstellar magnetic field \\citep{Fen98}, the orbital motion of the jet source in a binary system \\citep{Fen98, Mas02}, or dynamical pressure from external medium \\citep{Fen98}. \nIn the case of the Lorenz forces, a C-shaped bending is expected if the poloidal current in the jet and counter jet flows in the same direction \\citep{Fen98}.\nHowever, this mechanism is difficult to account for the observed bending of the L1448C(N) jet, because typical interstellar magnetic field with several tens of microgauss is not strong enough to bend the jet beam with a density of $>$10$^6$ cm$^{-3}$.\n\nIf the C-shaped bending is produced by the orbital motion of a binary system, the orbital radius and orbital velocity can be estimated by using the analytical model of \\citet{Mas02}.\nHere the jet is assumed to be ejected at a velocity of $v_j$ from one of the binary protostars in a circular orbit of radius $r_0$ and orbital velocity $v_0$.\nThe $z$-axis is parallel to the orbital rotation axis, and $z$=0 is the orbital plane.\nAs shown in Fig. 3 of \\citet{Mas02}, the deflection angle $\\alpha$ of the jet beam near the source is approximated by the $x$=${\\kappa}z\/{\\rm cos}i$ line, where $\\kappa$=$v_0\/v_j$ and $i$ is the inclination angle of the orbital axis with respect to the plane of the sky.\nIn the case of the L1448C(N) jet, the deflection angle $\\alpha$ is estimated to be 2.5$^{\\circ}$, which corresponds to the half of the misalignment angle.\nUsing the jet velocity of $\\sim$160 km s$^{-1}$, the orbital velocity is calculated to be 6.5 km s$^{-1}$.\nSince the mass of the protostar with jet is only 0.03--0.09 $M_{\\odot}$,  the total mass of the binary system is considered to be less than 0.18 $M_{\\odot}$.\nTherefore, the radius and period of the orbital morion are estimated to be smaller than 4.2 AU and 20 yr, respectively.\nHowever, such a short period orbital motion cannot account for the observed C-shaped pattern, because\nthe C-shaped bending is seen in the BI and RI parts of the jet with a length of $\\sim$2000 AU with a dynamical time scale of  47 yr.\nIn order to produce the C-shaped pattern with the orbital motion, the orbital period needs to be longer than twice of the dynamical time scale.\n \nIn the case of dynamical pressure of external medium, ambient gas with $n$(H$_2$)$<$10$^4$ cm$^{-3}$ cannot account for bending the jet with a density of  $>$10$^6$ cm$^{-3}$,  unless the protostar is moving with a velocity that is comparable to the jet velocity.\nOn the other hand, the dynamical pressure caused by the outflow from the nearby protostar, L1448N, cannot be ruled out.\nAs shown in the CO map of \\citet{Bac90}, the redshifted lobe of the L1448N outflow overlaps with the blue lobe of the L1448C(N) outflow.\nThe interaction between the two outflows from L1448C(N) and L1448N has been suggested by \\citet{Bac95}, because the large scale outflow from L1448C(N) shows a considerable bending at the place where the two outflows are overlapping.\nSince the redshifted emission from L1448N outflow reaches close to the position of L1448C(N) \\citep{Bac95}, it is possible that the jet from L1448C(N) is propagating under the influence of the L1448N outflow.\nIn this case, the dynamical pressure from L1448N outflow acts from north to south, and deflects the jet beams to the west if they were ejected to the the northwest and southeast directions. \n\n\\subsection{Deflection and wiggling of the jet}\n\nIn addition to the C-shaped bending, the jet is also deflected toward the east by $\\sim$40$^{\\circ}$ at the position of BI-II and toward the south by $\\sim$15$^{\\circ}$ at the RI-II position.\nSince both sides of the jet are deflected at almost same distance from the central star, the jet deflection is likely to be caused by some variability intrinsic to the driving source rather than by external perturbation.\nThe observed morphology is similar to the S-shaped point-reflection symmetric pattern that is expected if the disk is  precessing or wobbling.\nAlthough the jet is not exactly the S-shape but asymmetric in deflection angle, this is probably because of the projection effect.\nIn a binary protostellar system with a disk misaligned with the orbital plane of the binary, the disk wobbles with a period approximately half of the binary orbital period and precesses with a period of $\\sim$20 orbital period \\citep{Bat00}.\nS-shaped point symmetry will be observed if the precession or wobbling time scale is longer than four times of the dynamical timescale.\nSince the jet deflection occurs at BI-II and RI-II the time scale of which is $\\sim$50 yr, the time scale of the precession or wobbling should be longer than $\\sim$200 yr.\nTherefore, if the deflection is due to the precession, the lower limit of the binary orbital period is $\\sim$10 yr.\nSince this orbital period of $\\sim$10 yr is comparable to the period of small scale wiggling  shown in Fig. 6, 15--20 yr, this orbital motion can also explain the wiggling feature.\nIf the binary system consists of equal mass protostars with 0.03--0.09 $M_{\\odot}$, the  orbital radius is estimated to be 2.4--4.2 AU.\nOn the other hand, if the jet deflection is due to the wobbling of the disk, the orbital period and the separation of the binary are estimated to be $\\sim$400 yr and 30 AU, respectively.\nSince the estimated separation of the binary is smaller than the size of the disk observed with the 350 GHz continuum emission,  it is possible that the observed 90 AU scale disk harbors two sources separated by 60 AU.\nHowever, the binary with a separation of $>$60 AU cannot account for the small scale wiggling feature.\n\n\\subsection{Velocity variation of the jet}\n\nThe P-V diagram of the SiO shows that the velocity of the jet varies semi-periodically.\nThe velocity variation is more obvious in Fig.~\\ref{fig13}, which plots the velocity centroid of the SiO emission in the redshifted part of the jet as a function of the distance from L1448C(N).\nIt is shown that the typical amplitude of the variation in velocity centroid is $\\sim$7 km s$^{-1}$.\nSuch a velocity variation is expected if the jet is precessing, the jet is launched from an orbiting object,  or the ejection velocity itself varies as a function of time \\citep[e.g.][]{Smi97}.\nThe period of the velocity variation estimated from the de-projected jet velocity and knot separation is $\\sim$15--20 yr.\nSince this time scale is much shorter than the precession time scale that is estimated to be 200 yr, it is unlikely that the velocity variation is caused by the precession of the jet.\nOn the other hand, the orbital motion of the driving source can account for the velocity variation; the binary system with an orbital period of $\\sim$15--20 yr, an orbital radius of 2.4--4.2 AU, a total mass of 0.06--0.18 $M_{\\odot}$ has an orbital velocity of $\\sim$4.7--6.2 km s$^{-1}$, which is comparable to the amplitude of the radial velocity variation.\nHowever, the orbital motion cannot explain the relation between the SiO intensity and velocity gradient.\nAs shown in the P-V map (Fig.~\\ref{fig7}), each SiO knot has its higher velocity in the upstream side and lower velocity in the downstream side.\nThe opposite velocity gradient is always seen in the faint emission between the knots.\nSuch a structure is more likely to be formed by means of periodic variation of the ejection velocity. \nIn such a case, the SiO knots are considered to be formed as the fast moving material plunge into the slow moving material in the downstream \\citep[e.g.][]{Sto93, Sut97}.\nThe periodic variation in the ejection velocity is probably due to the modulation of mass accretion by means of companion.\nIn such a case, the variation amplitude of the jet velocity corrected for the inclination is calculated to be $\\sim$20 km s$^{-1}$. \nThis velocity amplitude corresponds to the shock velocity, which is consistent to the velocity of C-type shocks that can account for the excitation conditions of the far infrared molecular lines \\citep{Nis99, Nis00}.\nSimilar velocity gradients in the knots with the faster part in the upstream side and the slower part in the downstream was also observed in the optical jet of HH111 \\citep{Rag02} and in the CO and SiO jets from IRAS 04166+2706 \\citep{San09}.\n\n\n\\subsection{Driving mechanism of the CO outflow}\n\nThe P-V diagram of the CO $J$=3--2 along the axis (Fig.~\\ref{fig7}) exhibits two kinematic components, i.e. the EHV jet with an almost constant velocity and the outflow shell with a parabolic velocity pattern.\nAlthough highly-collimated jet is clearly seen in both SiO and CO, the parabolic velocity pattern seen in the outflow shells is reproduced by the wind-driven model.\nTherefore, the observational results require a wide-opening angle wind and a collimated jet at the same time.\n\nOne possible mechanism to explain the observed jet+shell structure is the ^^ ^^ unified model'' proposed by \\citet{Sha06}, in which highly-collimated jet component is explained as an on-axis density enhancement of the {\\it X}-wind type of wide opening angle wind launched magnetocentrifugally.\nIn this model, the jet along the axis corresponds to the densest part of the primary wind, and the shell is mostly consisted of the swept-up ambient material.\nTheir numerical model successfully reproduced the structure of a dense and narrow jet surrounded by a conical shell. \nThe other models that can explain the two components structure are proposed by \\citet{Mac08} and \\citet{Ban06}.\nThe model proposed by \\citet{Mac08} predicts two distinct flows from the adiabatic core and the protostar.\nThe flow from the adiabatic core driven by the magnetocentrifugal mechanism has a low velocity and a wide opening angle, while the flow from the protostar, which is mainly driven by the magnetic pressure gradient force, has a high velocity and is well collimated.\nOn the other hand, a model proposed by \\citet{Ban06} predicts the structure with the jet powered by magnetocentrifugal force enclosed by the large-scale outflow driven by the magnetic pressure.\nAlthough these two models reproduce the jet+shell structure similar to the observational results, the velocities of the shells predicted in these model are rather small ($\\sim$5 km s$^{-1}$ for the model of \\citet{Mac08}) because of the shallow gravitational potential at the launching point.\nIn the case of the L1448C(N) outflow, the terminal velocity of the outflow shellreaches to ${\\Delta}V {\\sim} {\\pm}$70 km s$^{-1}$ without inclination correction, which is comparable to the velocity of the EHV jet.\nIn order to launch such a high velocity wind, the launching point of this wind should be close to the launching point of the EHV jet.\nTherefore, the models with two components launched from two different regions do not explain the jet+shell structure in the L1448C(N) outflow.\nIn the case of the {\\it X}-wind with density stratification \\citep{Sha06}, the rather high velocity in the shell component is naturally explained because the shell is driven by the high-velocity primary wind launched at the same region as the EHV jet.\n\n\n\\subsection{Origin of the SiO in the jet}\n\nIt is considered that the SiO molecules observed in jets and outflows are formed as a consequence of grain sputtering in a C-shock releasing Si-bearing material into the gas phase, followed by the reaction with O and OH \\citep{Sch97, Cas97, Gus08a}.\nThe multi-transition SiO lines from the L1448C(N) jet observed by \\citet{Nis07} have been successfully modeled by the steady-state C-shock model of  \\citet{Gus08a}  with a pre-shock density of $\\sim$10$^5$ cm$^{-3}$ and a shock velocity of 30--45 km s$^{-1}$.\nHowever, the conversion of Si into SiO is initially rather slow in their models, and the predicted  SiO line emission predominantly arises from postshock gas  $>$100 yr after the passage of shocks.\nSince most of the SiO knots in the L1448C(N) jet have dynamical timescale shorter than the SiO formation timescale, the steady-state C-type shock models of \\citet{Gus08a} does not account for the SiO in the knots close to the central star, especially in the innermost knot pair with extremely short time scale less than 10 yr.\nOne possible explanation is that the SiO molecules existed on the grain mantles and are released into the gas-phase by means of shocks as suggested by \\citet{Gus08b}.\n\nAnother possibility is the formation of SiO in high density primary jet \\citep{Sha06}.\n\\citet{Gla89, Gla91} studied the formation of molecules in protostellar winds, which are originally neutral atomic, and found that significant quantities of SiO can be formed quickly in the close vicinity ($<$0.1 AU) of the central star if the mass-loss rate is high ($>$10$^{-6}$ $M_{\\odot}$).\nSince the mass-loss rate of the L1448C(N) jet is high enough, this scenario of in situ formation also can be the origin of the SiO in the jet.\nThe chemical models of \\citet{Gla89, Gla91} also predict significant amount of CO synthesized in the winds; the CO abundance reaches an equilibrium value of 4$\\times$10$^{-4}$ under the conditions in which observable amount of SiO is formed.\nThe morphological and kinematical similarity of the CO and SiO jets supports the idea that both CO and SiO are formed in the protostellar wind.\n\n\n\\subsection{Properties of L1448C(S)}\n\nThe secondary source L1448C(S) is located at $\\sim$8\\farcs3 (2000 AU) southeast of L1448C(N).\nThe 350 GHz continuum flux of this source is $\\sim$60 mJy, which is five times smaller than the flux from L1448C(N).\nIf an optically thin condition is assumed, the mass of the circumstellar material surrounding L1448C(S) is estimated from the observed flux using the dust mass opacity of 1.75 cm$^2$ g$^{-1}$ \\citep{Oss94} and the formula given by \\citet{Jor07}.\nFor a dust temperature of $\\sim$40 K, the estimated mass in the optically thin limit is 8.6$\\times$10$^{-3}$ $M_{\\odot}$.\nSince L1448C(S) is associated with a molecular outflow, it is highly probable that this source is a protostar rather than a mere dust clump at the cavity wall as claimed by \\citet{Mau10}.\nThe NH$_3$ data of \\citet{Cur99} suggest that L1448C(S) is formed in the same dense core as L1448C(N).\nHowever, the SED of L1448C(S) is significantly different from that of L1448C(N).\nIn table 4, the broad-band spectra of L1448C(N) and L1448C(S) measured at different wavelengths are listed.\nIn the near-infrared, L1448C(S) is much dimmer than L1448C(N); only L1448C(N) appeared in the $K_s$ band image of \\citet{Tob07}.\nOn the contrary, in the mid-inrfrared at three IRAC bands, band 2 (4.5 $\\mu$m), band 3 (5.8 $\\mu$m), and band 4 (8.0 $\\mu$m), L1448C(S) becomes brighter than L1448C(N); especially in the bands 3 and 4, L1448C(S) is more than six times brighter than L1448C(N).\nIn the MIPS 24 $\\mu$m image, L1448C(S) is also seen clearly  \\citep{Tob07, Reb07}.\nThe flux from L1448C(S) at 24 $\\mu$m looks similar to that from L1448C(N), although the accurate flux value of each source is not easy to measure because of the confusion.\nIn the sub-mm and mm wavebands, L1448C(S) is much weaker than L1448C(N).\nThe masses of the circumstellar material surrounding L1448C(S), $\\sim$0.01 $M_{\\odot}$, is approximately 10 times smaller than that of L1448C(N), $\\sim$0.1 $M_{\\odot}$.\nDue to the small amount of circumstellar material, the central star of L1448C(S) is likely to be less obscured in the mid-infrared as compared to that of L1448C(N) enshrouded by the thick cocoon.\nThe outflow activities in two sources are also different significantly.\nThe CO outflow from L1448C(S) is compact and substantially weaker than the L1448C(N) outflow.\nThe momentum flux of the L1448C(S) outflow is only $\\sim$10$^{-6}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, which is two or three orders of magnitude smaller than that of the L1448C(N) outfow and is comparable to those of class I outflows studied by \\citet{Bon96}.\nIn addition, there is no EHV component nor SiO emission associated with the outflow from L1448C(S).\n\nThe small amount of circumstellar material of less than 0.01 $M_{\\odot}$ suggests that L1448C(S) may have accumulated most of its circumstellar mass.\nThe less energetic outflow also support the idea that L1448C(S) has a nature close to class I rather than class 0.\nThese results imply that two sources, L1448C(N) and L1448C(S) are formed in different epochs in the same dense core.\nAnother possibility is the effect of the L1448C(N) outflow.\nSince L1448C(S) is located at the same line of sight as the L1448C(N) outflow, it is possible that the outflowing gas has stripped away the dense gas surrounding L1448C(S).\nAlthough the three-dimensional geometries of the sources and outflows are not clear, the high rotational temperature of NH$_3$ observed at the position of L1448C(S) \\citep{Cur99} suggests the possibility that the gas around L1448C(S) is heated by the interaction with the jet from L1448C(N).\nIn such a case, the {\\it apparent} age of L1448C(S) is older than that of L1448C(N) simply because the amount of material left around L1448C(S) is smaller than that around L1448C(N).\nThe effect of outflow is also proposed to explain the difference of the apaprent evolutionary stage of protostellar pair L1448N(A) and L1448N(B), which is located at $\\sim$75\\arcsec northwest of L1448C \\citep{Oli06}.\n\nThe third scenario is the disintegration of an unstable multiple system as proposed by \\citet{Rei00}.\nSince nonhierarchical triple systems are unstable, they break up, ejecting the lightest member, while the remaining two members form a close binary system with a highly eccentric orbit.\nIn this scenario, the disks around escaping stars will be highly truncated; the typical disk size is expected to be around half of the distance between the stars in the close triple encounter.\nIf L1448C(S) is the escaping member, the small amount of its circumstellar material can be explained by means of disk truncation.\nThis scenario also implies that L1448C(N) is a close binary system.\nThe observed deflection, wiggling, and periodic velocity variation of the jet suggest the possibility that L1448C(N) is a close binary system with an orbital radius of $\\sim$2--4 AU.\nTherefore, it is possible that such a close binary system was formed by means of disintegration of a triple system.\nIn order to assess this scenario, kinematic information of L1448C(S) becomes important.\nAlthough previous NH$_3$ results of \\citet{Cur99} did not show peculiar motions in the dense core containing L1448C(N) and L1448C(S), a detailed study with higher angular resolution would be helpful.\n\n\n\n\n\\section{CONCLUSIONS}\nThe central region of the highly-colimated molecular outflow driven by L1448C was mapped in the SiO $J$=8--7, CO $J$=3--2, and 350 GHz continuum emission with the SMA at $\\sim$1 arcsecond resolution.\nOur main conclusions are the following:\n\\begin{enumerate}\n\\item The 350 GHz continuum emission was detected from two {\\it Spitzer} sources L1448C(N) and L1448C(S). \nThe continuum emission from L1448C(N) consists of an extended component  and a compact component. \nThe compact component is elongated perpendicular to the outflow axis, and is likely to be a circumstellar disk with a size of $\\sim$90 AU. \nThe spectral index of this compact component derived from the data from 86 GHz to 350 GHz is ${\\alpha}{\\sim}$2, suggesting the possibility that the continuum emission is optically thick at 350 GHz.\nThe mass of the disk is estimated to be $\\sim$0.11 $M_{\\odot}$.\n\\item The continuum flux from L1448C(S) is $\\sim$60 mJy, which is $\\sim$10 times lower than the flux from L1448C(N), although L1448C(S) is brighter than L1448C(N) in the mid infrared wavebands.\nThe mass of the circumstellar material surrounding L1448C(S) is estimated to be 8.6$\\times$10$^{-3}$ $M_{\\odot}$.\n\\item A narrow jet from L1448C(N) along the outflow axis was observed in the SiO and the high-velocity CO. The width of the jet measured in the SiO images is $\\sim$200 AU FWHM on average. \nThe jet consists of a chain of emission knots with an inter-knot spacing of $\\sim$500 AU.\nIt is likely that the knots in the L1448 jet are the internal bow shocks in the jet beam.\n\\item The dynamical timescale of the innermost pair of knots, which are significant in the SiO but barely seen in the CO, is only $\\sim$10 yr. \nIt is likely that the SiO may have been formed quickly in the protostellar wind through the gas-phase reaction, or been formed on the dust grain and directly released into the gas phase by means of shocks.\n\\item The low velocity CO emission delineates two V-shaped shells with a common apex at  L1448C(N). The kinematics of this shell component is reproduced by the model of wide opening angle wind. \nTherefore, the outflow from L1448C(N) consists of both highly-collimated jets and shells produced by wide-opening angle wind.\nThe observed jet+shell structure can be explained by the ^^ ^^ unified model'' proposed by \\citet{Sha06}, in which highly-collimated jet components are explained as an on-axis density enhancement of the {\\it X}-wind type of wide opening angle wind.\n\\item The Jet from L1448C(N) is extremely active with a momentum supply rate of $\\sim$5$\\times$10$^{-4}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$ and a mechanical luminosity of $\\sim$7 $L_{\\odot}$. \nThe mass accretion rate derived from the mass loss rate  is $\\sim$10$^{-5}$ $M_{\\odot}$ yr$^{-1}$.\nSuch a high mass-accretion rate and a rather low bolometric luminosity of the central source, 7.5 $L_{\\odot}$, imply that the central protostar is still in the very early phase of its evolution with a mass of 0.03--0.09 $M_{\\odot}$ and a dynamical age of (4--12)$\\times$10$^3$ yr.\n\\item The blue part and the red part of the jet are misaligned by $\\sim$5$^{\\circ}$ and forming a C-shaped bending toward the west.\nThe possible origin of this bending is the dynamical pressure caused by the outflow from the nearby protostar, L1448N.\n\\item The jet is deflected toward the east in the blueshifted part and toward the south in the redshifted part. \nIn addition, the jet is wiggling with a period of $\\sim$15--20 yr.\nThe deflection and wiggling of the jet can be explained if the driving source is a member of the binary system with an orbital radius of 2--4 AU.\n\\item The jet shows a semi-periodic variation in radial velocity with an amplitude of $\\sim$7 km s$^{-1}$.\nEach SiO knot has its higher velocity in the upstream side and lower velocity in the downstream side. The opposite velocity gradient is seen in the faint emission between the knots.\nIt is likely that the ejection velocity varies periodically by means of modulation of mass accretion.   \n\\item The bipolar outflow in the NE-SW direction centered at L1448C(S) was discovered in the CO $J$=3--2. \nThis provides a strong evidence that L1448C(S) is a protostar.\nThe momentum flux of this outflow is only $\\sim$10$^{-6}$ $M_{\\odot}$ km s$^{-1}$ yr$^{-1}$, which is two to three orders of magnitude smaller than that of L1448C(N) outflow, and is comparable to those of class I outflow.\n\\item L1448C(S) is surrounded by a rather small amount of circumstellar material of less than 0.01 $M_{\\odot}$ and is powering a less energetic outflow, suggesting that this source has a nature close to class I rather than class 0, even though this source is formed in the same dense core as L1448C(N).\nOne possible scenario to explain this dichotomy is the effect of the outflow from L1448C(N); significant amount of material in the envelope surrounding L1448C(S) might have been stripped off by the powerful outflow from L1448C(N).\nAnother possibility is the disintegration of an unstable multiple system, in which L1448C(S) is an escaping member with a truncated disk.\n\n\\end{enumerate}\n\n\\acknowledgments\n\nWe wish to thank all the SMA staff in Hawaii, Cambridge, and Taipei for their \nenthusiastic help during these observations. \nN. Hirano thanks M. Machida and S. Inutsuka for fruitful discussion.\nN. Hirano is supported by NSC grant 96-2112-M-001-023.\n\n \\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Intro}\nGiven a (commutative) integral domain $D$ with fraction field $K$, we define $\\textnormal{Int}(D) := \\{f \\in K[X] \\mid f(D) \\subseteq D\\}$, which is the ring of integer-valued polynomials on $D$. Integer-valued polynomials and the properties of $\\textnormal{Int}(D)$ have been well studied; the book \\cite{CaCh} covers the major theory in this area and provides an extensive bibliography. In recent years, researchers have begun to study a generalization of $\\textnormal{Int}(D)$ to polynomials that act on a $D$-algebra rather than on $D$ itself \\cite{EFJ}, \\cite{EvrJoh}, \\cite{Fri1}, \\cite{Fri1Corr}, \\cite{Fri2}, \\cite{LopWer}, \\cite{Per1}, \\cite{PerDivDiff}, \\cite{PerFinite}, \\cite{PerWer}, \\cite{PerWer1}, \\cite{Wer}. For this generalization, we let $A$ be a torsion-free $D$-algebra such that $A \\cap K = D$, and let $B = K \\otimes_D A$, which is the extension of $A$ to a $K$-algebra. By identifying $K$ and $A$ with their images under the injections $k \\mapsto k \\otimes 1$ and $a \\mapsto 1 \\otimes a$, we can evaluate polynomials in $K[X]$ at elements of $A$. This allows us to define $\\textnormal{Int}_K(A) := \\{f \\in K[X] \\mid f(A) \\subseteq A\\}$, which is the ring of integer-valued polynomials on $A$ with coefficients in $K$. With notation as above, the condition $A \\cap K = D$ ensures that $D[X] \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)$.\n\n\\begin{Def}\\label{Nontrivial}\nWe say that $\\textnormal{Int}_K(A)$ is \\textit{nontrivial} if $\\textnormal{Int}_K(A) \\ne D[X]$.\n\\end{Def}\n\nThe goal of this paper is to determine when $\\textnormal{Int}_K(A)$ is nontrivial. Some results in this direction were proved by Frisch in \\cite[Lem. 4.1]{Fri2} and \\cite[Thm. 4.3]{Fri2}; these are restated below in Proposition \\ref{Fin gen nontrivial}. In the traditional case, necessary and sufficient conditions for $\\textnormal{Int}(D)$ to be nontrivial were given by Rush in \\cite{Rush}. Using Rush's criteria, we prove (Theorem \\ref{Nontriv fin gen criterion}) that when $D$ is any integral domain and $A$ is finitely generated as a $D$-module, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. Part of this work involves conditions under which we have $D[X] \\subseteq \\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$ for some $n$, where $M_n(D)$ is the algebra of $n \\times n$ matrices with entries in $D$. This led us to investigate whether having $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$ implies that $A \\cong M_n(D)$. While this is not true in general, the result does hold if $D$ is a Dedekind domain and $A$ can be embedded in $M_n(D)$ (Theorem \\ref{Uniqueness of M_n(D)}).\n\nIf we drop the assumption that $A$ is finitely generated as a $D$-module, determining whether $\\textnormal{Int}_K(A)$ is nontrivial becomes more complicated. However, when $D$ is Dedekind, we are able to give necessary and sufficient conditions for $\\textnormal{Int}_K(A)$ to be nontrivial (Theorem \\ref{criterion}). Our work on this topic also allows us to prove that if $D$ is Dedekind, then $\\textnormal{Int}_K(A)$ has Krull dimension 2 (Corollary \\ref{Krull dimension}). This generalizes another theorem of Frisch \\cite[Thm. 5.4]{Fri1} where it was assumed that $A$ was finitely generated as a $D$-module.\n\n\\section{Integral Algebras of Bounded Degree}\\label{Non-trivial}\nThroughout, $D$ denotes an integral domain with field of fractions $K$, and $A$ denotes a $D$-algebra. We will always assume that $A$ satisfies certain conditions, which we call our \\textit{standard assumptions}.\n\n\\begin{Def}\\label{Standard assumptions}\nWhen $A$ is a torsion-free $D$-algebra such that $A \\cap K = D$, we say that $A$ is a $D$-algebra with \\textit{standard assumptions}. When $A$ is finitely generated as a $D$-module, we say that $A$ is of \\textit{finite type}.\n\\end{Def}\n\nAs mentioned in the introduction, the condition that $A \\cap K = D$ implies that\n\\begin{equation*}\nD[X] \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)\n\\end{equation*}\nand it is natural to consider when $D[X] = \\textnormal{Int}_K(A)$ or $\\textnormal{Int}_K(A) = \\textnormal{Int}(D)$. This latter equality is investigated in \\cite{IntdecompII}, where the following theorem is proved. Unless stated otherwise, all isomorphisms are ring isomorphisms.\n\n\\begin{Thm}\\label{Int_K(A)=Int(D)}\n\\cite[Thms. 2.10, 3.10]{IntdecompII} Let $D$ be a Dedekind domain with finite residue rings. Let $A$ be a $D$-algebra of finite type with standard assumptions. For each maximal ideal $P$ of $D$, let $\\widehat{A}_P$ and $\\widehat{D}_P$ be the $P$-adic completions of $A$ and $D$, respectively. Then, the following are equivalent.\n\\begin{enumerate}[(1)]\n\\item $\\textnormal{Int}_K(A) = \\textnormal{Int}(D)$.\n\\item For each nonzero prime $P$ of $D$, there exists $t \\in \\mathbb{N}$ such that $A\/PA \\cong \\bigoplus_{i=1}^t D\/P$.\n\\item For each nonzero prime $P$ of $D$, there exists $t \\in \\mathbb{N}$ such that $\\widehat{A}_P \\cong \\bigoplus_{i=1}^t \\widehat{D}_P$.\n\\end{enumerate}\n\\end{Thm}\n\nIn this paper, we examine the containment $D[X] \\subseteq \\textnormal{Int}_K(A)$. In the traditional setting of integer-valued polynomials, the ring $\\textnormal{Int}(D)$ is said to be \\textit{trivial} if $\\textnormal{Int}(D) = D[X]$, and we adopt the same terminology for $\\textnormal{Int}_K(A)$. Clearly, for $\\textnormal{Int}_K(A)$ to be nontrivial it is necessary that $\\textnormal{Int}(D)$ be nontrivial, so we begin by reviewing the situation for $\\textnormal{Int}(D)$. Section I.3 of \\cite{CaCh} and a paper by Rush \\cite{Rush} give several results regarding the triviality or non-triviality of $\\textnormal{Int}(D)$. We will summarize these theorems after recalling several definitions.\n\n\\begin{Def}\nAn ideal $\\mathfrak{a}$ of $D$ is said to be the colon ideal or conductor ideal of $q \\in K$ if \n$$\\mathfrak{a} = (D :_D q) = \\{d \\in D \\mid dq \\in D\\}.$$\nFor a commutative ring $R$, we denote by $\\textnormal{nil}(R)$ the nilradical of $R$, which is the set of all nilpotent elements of $R$, or, equivalently, the intersection of all nonzero prime ideals of $R$. For $x \\in \\textnormal{nil}(R)$, we let $\\nu(x)$ equal the nilpotency of $x$, i.e., the smallest positive integer $n$ such that $x^n = 0$. If $I\\subseteq R$ is an ideal, let $V(I) = \\{P \\in \\text{Spec}(R) \\mid P \\supseteq I\\}$.\n\\end{Def}\n\nThe following proposition summarizes several sufficient and necessary conditions on $D$ in order for $\\textnormal{Int}(D)$ to be nontrivial.\n\\begin{Prop}\\label{IntD nontrivial}\\mbox{}\n\\begin{enumerate}[(1)]\n\\item \\cite[Cor. I.3.7]{CaCh} If $D$ is a domain with all residue fields infinite, then $\\textnormal{Int}(D)$ is trivial.\n\\item \\cite[Prop. I.3.10]{CaCh} Let $D$ be a domain. If there is a proper conductor ideal $\\mathfrak{a}$ of $D$ such that $D\/\\mathfrak{a}$ is finite, then $\\textnormal{Int}(D)$ is nontrivial.\n\\item \\cite[Thm. I.3.14]{CaCh} Let $D$ be a Noetherian domain. Then, $\\textnormal{Int}(D)$ is nontrivial if and only if there is a prime conductor ideal of $D$ with finite residue field.\n\\item \\cite[Cor. 1.7]{Rush} Let $D$ be an integral domain. Then, the following are equivalent:\n\\begin{enumerate}[(i)]\n\\item $\\textnormal{Int}(D)$ is nontrivial.\n\\item There exist $a,b\\in D$ with $b\\notin aD$ such that the two sets $\\{\\;|D\/P|\\; \\mid P \\in V((aD:b))\\}$ and $\\{\\nu(x) \\mid x\\in {\\rm nil}(D\/(aD:b))\\}$ are bounded.\n\\end{enumerate}\n\\end{enumerate}\n\\end{Prop}\n\nIf $A$ is finitely generated as a $D$-module, Frisch has shown that the analogs of the above conditions in Proposition \\ref{IntD nontrivial} hold for $\\textnormal{Int}_K(A)$:\n\n\\begin{Prop}\\label{Fin gen nontrivial}\nLet $D$ be a domain. Let $A$ be a $D$-algebra of finite type with standard assumptions.\n\\begin{enumerate}[(1)]\n\\item \\cite[Lem. 4.1]{Fri2} Assume there is a proper conductor ideal $\\mathfrak{a}$ of $D$ such that $D\/\\mathfrak{a}$ is finite. Then, $\\textnormal{Int}_K(A)$ is nontrivial.\n\\item \\cite[Thm. 4.3]{Fri2} Assume that $D$ is Noetherian. Then, $\\textnormal{Int}_K(A)$ is nontrivial if and only if there is a prime conductor ideal of $D$ with finite residue field.\n\\end{enumerate}\n\\end{Prop}\n\nIn particular, \\cite[Thm. 4.3]{Fri2} shows that for a Noetherian domain $D$ and a finitely generated algebra $A$, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. In Theorem \\ref{Nontriv fin gen criterion}, we will show that this holds even if $D$ is not Noetherian. Additionally, we can weaken our assumptions on $A$. Recall the following definition, which can be found in \\cite{Jac} or \\cite{Lam}, among other sources.\n\n\\begin{Def}\nLet $R$ be a commutative ring and $A$ an $R$-algebra. We say that $A$ is an \\emph{algebraic algebra} (over $R$) if every element of $A$ satisfies a polynomial equation with coefficients in $R$. We say that $A$ is an \\emph{algebraic algebra of bounded degree} if there exists $n\\in\\mathbb{N}$ such that the degree of the minimal polynomial equation of each of its elements is bounded by $n$. If we insist that each element of $A$ satisfy a monic polynomial with coefficients in $R$, then we say that $A$ is an \\emph{integral algebra} over $R$.\n\\end{Def}\n\nAlgebraic algebras are usually discussed over fields, in which case an algebraic algebra is also an integral algebra. Over a domain however, the two structures are not equivalent. For example, $A = \\mathbb{Z}[\\frac{1}{2}]$ is an algebraic algebra over $\\mathbb{Z}$ that is not an integral algebra. In this case, $A$ does not satisfy our standard assumption that $A \\cap \\mathbb{Q}$ should equal $\\mathbb{Z}$. However, if we instead take $A = \\mathbb{Z} \\oplus \\mathbb{Z}[\\frac{1}{2}]$ (so that $B = \\mathbb{Q} \\otimes_\\mathbb{Z} A \\cong \\mathbb{Q} \\oplus \\mathbb{Q}$, $D$ is the diagonal copy of $\\mathbb{Z}$ in $B$, and $K$ is the diagonal copy of $\\mathbb{Q}$ in $B$), then $A$ is an algebraic algebra over $D$, $A$ is not an integral algebra over $D$, and $A \\cap K = D$.\n\nNote also that if $A$ is finitely generated as a $D$-module, then $A$ is an integral algebra of bounded degree, with the bound given by the number of generators (see \\cite[Thm. 1, Chap. V]{BourbakiAlg} or \\cite[Prop. 2.4]{AtMc}). However, the converse does not hold. For instance, $A = D[X_1, X_2, \\ldots]\/(\\{X_i X_j \\mid i, j \\geq 1 \\})$ is not finitely generated, but if $f \\in A$ with constant term $d \\in D$, then $f$ satisfies the polynomial $(X-d)^2$. Thus, this $A$ is an integral algebra of bounded degree. \n\nFor our purposes, the importance of having a bounding degree $n$, is that it guarantees that $\\textnormal{Int}_K(A)$ contains $\\textnormal{Int}_K(M_n(D))$, where $M_n(D)$ denotes the algebra of $n \\times n$ matrices with entries in $D$.\n\n\\begin{Lem}\\label{Matrix containment}\nLet $D$ be a domain and let $A$ be a $D$-algebra with standard assumptions. Assume that $A$ is an integral $D$-algebra of bounded degree $n$. Then, $\\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$.\n\\end{Lem}\n\\begin{proof}\nLet $a \\in A$ and let $\\mu_a \\in D[X]$ be monic of degree $n$ such that $\\mu_a(a) = 0$. Let $f(x) = g(X)\/d \\in \\textnormal{Int}_K(M_n(D))$, where $g \\in D[X]$ and $d \\in D \\setminus \\{0\\}$. By \\cite[Lem. 3.4]{FriSep}, $g$ is divisible modulo $dD[X]$ by every monic polynomial in $D[X]$ of degree $n$; hence, $\\mu_a$ divides $g$ modulo $d$. It follows that $g(a) \\in dA$ and $f(a) \\in A$. Since $a$ was arbitrary, $f \\in \\textnormal{Int}_K(A)$.\n\\end{proof}\n\n\\begin{Rem}\nThe converse of Lemma \\ref{Matrix containment} does not hold, even in the case when $\\textnormal{Int}_K(M_n(D))$ is nontrivial, as Example \\ref{integralalgebraboundeddegree not necessary} below will show.\n\\end{Rem}\n\nThus, in the case of an integral algebra of bounded degree $n$, to prove that $\\textnormal{Int}_K(A)$ is nontrivial it suffices to show that $\\textnormal{Int}_K(M_n(D))$ is nontrivial. This task is more tractable, because the polynomials given in the next definition can be used to map $M_n(D)$ into $M_n(P)$, where $P$ is a maximal ideal of $D$ with a finite residue field.\n\n\\begin{Def}\\label{BCL polynomials}\nFor each prime power $q$ and each $n > 0$, let \n\\begin{equation*}\n\\phi_{q,n}(X) = (X^{q^n} - X)(X^{q^{n-1}}-X) \\cdots (X^q - X).\n\\end{equation*}\n\\end{Def}\n\n\\begin{Lem}\\label{BCL lemma}\n\\cite[Thm. 3]{BrawCarLev} Let $\\mathbb{F}_q$ be the finite field with $q$ elements. Then, $\\phi_{q,n}$ sends each matrix in $M_n(\\mathbb{F}_q)$ to the zero matrix. Consequently, if $P \\subset D$ is a maximal ideal of $D$ with residue field $D\/P \\cong \\mathbb{F}_q$, then $\\phi_{q, n}$ maps $M_n(D)$ into $M_n(P)$.\n\\end{Lem}\n\n\\begin{Prop}\\label{Nontriv matrix criterion}\nLet $D$ be a domain. If $\\textnormal{Int}(D)$ is nontrivial, then $\\textnormal{Int}_K(M_n(D))$ is nontrivial, for all $n \\geq 1$.\n\\end{Prop}\n\\begin{proof}\nLet $n \\geq 1$ be fixed. Since $\\textnormal{Int}(D)$ is nontrivial, by \\cite[Cor. 1.7]{Rush} there exist $a,b\\in D$ with $b\\notin aD$ such that $\\{\\;|D\/P|\\; \\mid P \\in V((aD:b))\\}$ and $\\{\\nu(x) \\mid x\\in \\textnormal{nil}(D\/(aD:b))\\}$ are bounded. Let $I = (aD:b)$. Note that, because the former condition holds, each prime ideal containing $I$ is maximal, so the nilradical of $D\/I$ is equal to the Jacobson radical of $D\/I$.\n\nLet $\\{q_1,\\ldots,q_s\\}=\\{\\;|D\/P|\\; \\mid P\\in V(I)\\}$. By Lemma \\ref{BCL lemma}, we have $\\phi_{q,n}(M_n(D))\\subseteq M_n(P)$ for each maximal ideal $P\\subset D$  whose residue field has cardinality $q$. Then\n\\begin{equation*}\ng(X)=\\prod_{i=1,\\ldots,s}\\phi_{q_i,n}(X)\n\\end{equation*}\nis a monic polynomial such that $g(M_n(D))\\subseteq \\prod_{i}M_n(P_i)\\subseteq M_n(J)$, where $J=\\sqrt{I}$. Considering everything modulo $I$, we have $\\overline{g}(M_n(D\/I))\\subseteq M_n(J\/I)$.  \n\nNow, since $\\{\\nu(x) \\mid x\\in \\textnormal{nil}(D\/I)\\}$ is bounded, the nilpotency of every element in $J\/I$ is bounded by some positive integer $t$. It is a standard exercise that a matrix over a commutative ring with nilpotent entries is a nilpotent matrix. Moreover, it easily follows that the nilpotency of every matrix in $M_n(J\/I)$ is bounded by some $m\\in\\mathbb{N}$, depending only on $t$ and $n$. Hence, $\\overline{g}(X)^{m}$ maps every matrix $M_n(D\/I)$ to $0$, so that $g(X)^{m}$ maps $M_n(D)$ into $M_n(I)$. Finally, it is now easy to see that the polynomial $\\frac{b}{a}\\cdot g(X)^m$ is in $\\textnormal{Int}_K(M_n(D))$ but not in $D[X]$.\n\\end{proof}\n\nCombining Lemma \\ref{Matrix containment} with Proposition \\ref{Nontriv matrix criterion}, we obtain our desired theorem.\n\n\\begin{Thm}\\label{Nontriv fin gen criterion}\nLet $D$ be a domain and let $A$ be $D$-algebra with standard assumptions. Assume that $A$ is an integral $D$-algebra of bounded degree. Then, $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial. In particular, if $A$ is finitely generated as a $D$-module, then $\\textnormal{Int}_K(A)$ is nontrivial if and only if $\\textnormal{Int}(D)$ is nontrivial.\n\\end{Thm}\n\nLemma \\ref{Matrix containment} shows that, for an integral algebra $A$ of bounded degree $n$, the following containments hold:\n\\begin{equation*}\nD[X] \\subseteq \\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D).\n\\end{equation*}\nWhile our focus has been on whether $\\textnormal{Int}_K(A)$ equals $D[X]$, for the remainder of this section we will consider the containment $\\textnormal{Int}_K(M_n(D)) \\subseteq \\textnormal{Int}_K(A)$. In particular, we will examine to what extent $\\textnormal{Int}_K(M_n(D))$ is unique among rings of integer-valued polynomials. That is, if $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$, then can we conclude that $A \\cong M_n(D)$? In general, the answer is no, as we show below in Example \\ref{Quaternion example}. However, in Theorem \\ref{Uniqueness of M_n(D)} we will prove that for $D$ Dedekind, if $A$ can be embedded in $M_n(D)$, then having $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$ implies that $A \\cong M_n(D)$.\n\nWe first recall the definition of a null ideal of an algebra.\n\n\\begin{Def}\\label{Null ideal}\nLet $R$ be a commutative ring and $A$ an $R$-algebra. The \\textit{null ideal} of $A$ with respect to $R$, denoted $N_R(A)$, is the set of polynomials in $R[X]$ that kill $A$. That is, $N_R(A) = \\{ f \\in R[X] \\mid f(A) = 0\\}$. In particular, $N_{D\/P}(A\/PA) = \\{f \\in (D\/P)[X] \\mid f(A\/PA) = 0\\}$ denotes the null ideal of $A\/PA$ with respect to $D\/P$.\n\\end{Def}\n\nThere is a close relationship between polynomials in rings of integer-valued polynomials and polynomials in null ideals.\n\n\\begin{Lem}\\label{Null ideal lemma}\nLet $D$ be a domain and let $A$ and $A'$ be $D$-algebras with standard assumptions.\n\\begin{enumerate}[(1)]\n\\item Let $g(X)\/d \\in K[X]$, where $g \\in D[X]$ and $d \\ne 0$. Then, $g(X)\/d \\in \\textnormal{Int}_K(A)$ if and only if the residue of $g$ (mod $d$) is in $N_{D\/dD}(A\/dA)$. \n\\item $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(A')$ if and only if $N_{D\/dD}(A\/dA) = N_{D\/dD}(A'\/dA')$ for all $d \\in D$.\n\\end{enumerate}\n\\end{Lem}\n\\begin{proof}\nNotice that $g \\in \\textnormal{Int}_K(A)$ if and only if $g(A) \\subseteq dA$ if and only if $g(A\/dA) = 0$ mod $d$. This proves (1), and (2) follows easily.\n\\end{proof}\n\n\\begin{Ex}\\label{Quaternion example}\nLet $D = \\mathbb{Z}_{(p)}$ be the localization of $\\mathbb{Z}$ at an odd prime $p$. Take $A$ to be the quaternion algebra $A = D \\oplus D\\mathbf{i} \\oplus D\\mathbf{j} \\oplus D\\mathbf{k}$, where $\\mathbf{i}$, $\\mathbf{j}$, and $\\mathbf{k}$ are the imaginary  quaternion units satisfying $\\mathbf{i}^2 = \\mathbf{j}^2 = -1$ and $\\mathbf{i}\\mathbf{j} = \\mathbf{k} = -\\mathbf{j}\\mathbf{i}$. It is well known (cf.\\ \\cite[Exercise 3A]{Goodearl} or \\cite[Sec. 2.5]{DavSarVal}) that $A\/p^k A \\cong M_2(\\mathbb{Z}\/p^k \\mathbb{Z}) \\cong M_2(D\/p^k D)$ for all $k > 0$. By Lemma \\ref{Null ideal lemma}, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}_\\mathbb{Q}(M_2(D))$. However, $A$ contains no nonzero nilpotent elements (and is in fact contained in the division ring $\\mathbb{Q} \\oplus \\mathbb{Q}\\mathbf{i} \\oplus \\mathbb{Q}\\mathbf{j} \\oplus \\mathbb{Q}\\mathbf{k}$) and so cannot be isomorphic to $M_2(D)$.\n\\end{Ex}\n\nThus, in general, $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$ does not imply that $A \\cong M_n(D)$. However, as mentioned above, we do have such an isomorphism if $A$ can be embedded in $M_n(D)$. Proving this theorem involves some results of Racine \\cite{Racine}, \\cite{Racine2} about maximal subalgebras of matrix rings, which we now summarize.\n\n\\begin{Prop}\\label{Racine classification}\\mbox{}\n\\begin{enumerate}[(1)]\n\\item (\\cite[Thm. 1]{Racine}) Let $\\overline{A}$ be a maximal $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$. Let $V$ be an $\\mathbb{F}_q$-vector space of dimension $n$, so that $M_n(\\mathbb{F}_q)\\cong{\\rm End}_{\\mathbb{F}_q}(V)$. Then, $\\overline{A}$ is one of the following two types.\n\\begin{itemize}\n\\item[(I)] The stabilizer of a proper nonzero subspace of $V$. That is, $\\overline{A} = S(W) = \\{\\varphi\\in {\\rm End}_{\\mathbb{F}_q}(V) \\mid \\varphi(W)\\subseteq W\\}$, where $W$ is a proper nonzero $\\mathbb{F}_q$-subspace of $V$.\n\n\\item[(II)] The centralizer of a minimal field extension of $\\mathbb{F}_q$. That is, $\\overline{A} = C_{{\\rm End}_{\\mathbb{F}_q}(V)}(\\mathbb{F}_{q^l})=\\{\\varphi\\in{\\rm End}_{\\mathbb{F}_q}(V) \\mid \\varphi x=x\\varphi, \\forall x\\in \\mathbb{F}_{q^l} \\}$, where $l\\in\\mathbb{Z}$ is a prime dividing $n$.\n\\end{itemize}\n\n\\item (\\cite[Theorem p.\\ 12]{Racine2}) Let $D$ be a Dedekind domain and let $A$ be a maximal $D$-subalgebra of $M_n(D)$. Then, there exists a maximal ideal $P$ of $D$ such that $A\/P A$ is a maximal subalgebra of $M_n(D\/P)$. \n\\end{enumerate}\n\\end{Prop}\n\nRacine's classification allows us to establish a partial uniqueness result for the null ideal of $M_n(\\mathbb{F}_q)$, and hence for $\\textnormal{Int}_K(M_n(D))$.\n\n\\begin{Lem}\\label{FqsubalgebrasMnFq}\nLet $\\overline{A}$ be an $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$ such that $N_{\\mathbb{F}_q}(\\overline{A})=N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Then $\\overline{A} = M_n(\\mathbb{F}_q)$.\n\\end{Lem}\n\\begin{proof}\nSuppose the claim is not true, so that $\\overline A$ is contained in a maximal $\\mathbb{F}_q$-subalgebra of $M_n(\\mathbb{F}_q)$; hence, without loss of generality, we may assume that $\\overline A\\subsetneq M_n(\\mathbb{F}_q)$ is a maximal $\\mathbb{F}_q$-subalgebra. We will show that $N_{\\mathbb{F}_q}(\\overline{A})$ properly contains $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Note that $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q)) = (\\phi_{q,n}(X))$ by \\cite[Thm. 3]{BrawCarLev}, where $\\phi_{q,n}$ is the polynomial from Definition \\ref{BCL polynomials}.\n\nLet $V$ be an $\\mathbb{F}_q$-vector space of dimension $n$, so that $M_n(\\mathbb{F}_q)\\cong{\\rm End}_{\\mathbb{F}_q}(V)$. Assume first that $\\overline{A} = S(W)$ is of Type I as in Proposition \\ref{Racine classification}, and let $m = \\dim_{\\mathbb{F}_q}(W)$. Note that conjugating $\\overline{A}$ by an element of $GL(n, q)$ will change the matrices in $\\overline{A}$, but not the polynomials in the null ideal $N_{\\mathbb{F}_q}(\\overline{A})$. Moreover, up to conjugacy by an element in $GL(n, q)$, we may assume that $W$ has basis $e_1, e_2, \\ldots, e_m$, where $e_i$ is the standard basis vector with $1$ in the $i^\\text{th}$ component and 0 elsewhere. Under this basis, the matrices in $\\overline{A}$ are block matrices of the form \n$\\big(\\begin{smallmatrix}\nA_1 & A_2 \\\\ 0 & A_3\n\\end{smallmatrix}\\big)$ \nwhere $A_1$ is $m \\times m$ and $A_3$ is $(n-m) \\times (n-m)$. \n\nOne consequence of this representation is that every matrix in $S(W)$ has a reducible characteristic polynomial. As shown in the proof of \\cite[Thm. 3]{BrawCarLev}, $\\phi_{q, n}$ is the least common multiple of all monic polynomials in $\\mathbb{F}_q[X]$ of degree $n$. Hence, $\\phi_{q, n} \\in N_{\\mathbb{F}_q}(\\overline{A})$, because the characteristic polynomial of each matrix in $\\overline{A}$ divides $\\phi_{q,n}$. However, if $\\phi$ is the quotient of $\\phi_{q,n}$ by an irreducible polynomial in $\\mathbb{F}_q[X]$ of degree $n$, then $\\phi \\in N_{\\mathbb{F}_q}(\\overline{A})$, but $\\phi \\notin N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$. Thus, $N_{\\mathbb{F}_q}(\\overline{A})$ properly contains $N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q))$.\n\nNow, assume that $\\overline{A}$ is of Type II of Proposition \\ref{Racine classification}, so that $\\overline{A} = C_{{\\rm End}_{\\mathbb{F}_q}(V)}(\\mathbb{F}_{q^l})$ for some prime $l$ dividing $n$. Then, by \\cite[Thm. VIII.10]{McD}, we have $\\overline{A} \\cong M_{n\/l}(\\mathbb{F}_{q^l})$, and so \n\\begin{equation*}\nN_{\\mathbb{F}_q}(\\overline{A})=(\\phi_{q^l,n\/l}(X))\\supsetneq (\\phi_{q,n}(X)) = N_{\\mathbb{F}_q}(M_n(\\mathbb{F}_q)).\n\\end{equation*}\nAs before, the null ideal of $\\overline{A}$ strictly contains the null ideal of $M_n(\\mathbb{F}_q)$.\n\\end{proof}\n\n\\begin{Thm}\\label{Uniqueness of M_n(D)}\nLet $D$ be a Dedekind domain with finite residue fields. Let $A$ be a $D$-algebra of finite type with standard assumptions. Assume that $n \\geq 1$ is such that $A$ can be embedded in $M_n(D)$. Then, $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$ if and only if $A \\cong M_n(D)$.\n\\end{Thm}\n\\begin{proof}\nClearly, $A \\cong M_n(D)$ implies that $\\textnormal{Int}_K(A) = \\textnormal{Int}_K(M_n(D))$. So, assume that $\\textnormal{Int}_K(M_n(D)) = \\textnormal{Int}_K(A)$. As we will prove shortly in Lemma \\ref{wellbehaviourlocalization}, $\\textnormal{Int}_K(A)$ (and likewise $\\textnormal{Int}_K(M_n(D))$) is well-behaved with respect to localization at primes of $D$: for each prime $P$ of $D$, we have $\\textnormal{Int}_K(A)_P = \\textnormal{Int}_K(A_P)$. Hence, $\\textnormal{Int}_K(M_n(D_P)) = \\textnormal{Int}_K(A_P)$ for each $P$. Since $D$ is Dedekind, $D_P$ is a discrete valuation ring, so there exists $\\pi \\in D_P$ such that $PD_P = \\pi D_P$. Moreover, we have $D_P\/\\pi D_P \\cong D\/P$ and $A_P \/ \\pi A_P \\cong A\/PA$, so that $N_{D_P\/\\pi D_P}(A_P \/ \\pi A_P) = N_{D\/P}(A\/PA)$ (and likewise for $M_n(D)$). By Lemma \\ref{Null ideal lemma} (2), we conclude that the null ideals $N_{D\/P}(M_n(D\/P))$ and $N_{D\/P}(A\/P A)$ are equal for all maximal ideals $P$ of $D$.\n\nNow, suppose by way of contradiction that the image of $A$ in $M_n(D)$ does not equal $M_n(D)$. As in Lemma \\ref{FqsubalgebrasMnFq}, we may assume that the image of $A$ in $M_n(D)$ is a maximal $D$-subalgebra of $M_n(D)$. By Proposition \\ref{Racine classification}, there exists a maximal ideal $P$ of $D$ such that $A\/P A$ is isomorphic to a maximal subalgebra of $M_n(D\/P)$. By Lemma \\ref{FqsubalgebrasMnFq}, the null ideals $N_{D\/P}(A\/P A)$ and $N_{D\/P}(M_n(D\/P))$ are not equal. This is a contradiction. Therefore, $A \\cong M_n(D)$.\n\\end{proof}\n\n\\section{General Case}\\label{General case section}\n\nWe return now to the study of when $\\textnormal{Int}_K(A)$ is nontrivial. Because of Theorem \\ref{Nontriv fin gen criterion}, $A$ being an integral $D$-algebra of bounded degree can be sufficient for $\\textnormal{Int}_K(A)$ to be nontrivial, but it is not necessary. There exist $D$-algebras $A$ that are neither finitely generated, nor algebraic over $D$ (let alone integral or of bounded degree), but for which $\\textnormal{Int}_K(A)$ is nontrivial, as the next example shows.\n\n\\begin{Ex}\\label{integralalgebraboundeddegree not necessary}\nLet $D = \\mathbb{Z}$ and let $A = \\prod_{i \\in \\mathbb{N}} \\mathbb{Z}$ be an infinite direct product of copies of $\\mathbb{Z}$. Then, the element $(1, 2, 3, \\ldots)$ cannot be killed by any polynomial in $\\mathbb{Z}[X]$, so $A$ is not algebraic over $\\mathbb{Z}$. However, since operations in $A$ are done component-wise, any polynomial in $\\textnormal{Int}(\\mathbb{Z})$ is also in $\\textnormal{Int}_\\mathbb{Q}(A)$. Hence, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z})$, so in particular $\\textnormal{Int}_\\mathbb{Q}(A)$ is nontrivial. \n\\end{Ex}\n\nUltimately, the previous example works because for each prime $p$ there exists a polynomial that sends each element of $A\/pA$ to 0. More explicitly, each element of $\\prod_{i \\in \\mathbb{N}} \\mathbb{F}_p$ is killed by the polynomial $X^p-X$. This suggests that for $\\textnormal{Int}_K(A)$ to be nontrivial, it may be enough that there exists a finite index prime $P$ of $D$ with $A\/PA$ algebraic of bounded degree over $D\/P$ (since $D\/P$ is a field in this case, this is equivalent to having $A\/PA$ be integral of bounded degree over $D\/P$). We will prove below in Theorem \\ref{criterion} that if $D$ is a Dedekind domain, then this condition is necessary and sufficient for $\\textnormal{Int}_K(A)$ to be nontrivial.\n\nOur work will involve localizing $D$, $A$, and $\\textnormal{Int}_K(A)$ at $P$, and exploiting properties of $D_P$. In \\cite[Prop. 3.2]{Wer}, it is shown that if $D$ is Noetherian and $A$ is a free $D$-module of finite rank, then $\\textnormal{Int}_K(A)_P = \\textnormal{Int}_K(A_P)$ (in fact, \\cite[Prop. 3.2]{Wer} will hold if $A$ is merely finitely generated as a $D$-module). The next lemma shows that we can drop this finiteness assumption if $D$ is Dedekind.\n\n\\begin{Lem}\\label{wellbehaviourlocalization}\nLet $D$ be a Dedekind domain and $A$ a $D$-algebra with standard assumptions. Let $P$ be a prime ideal of $D$. Then $\\textnormal{Int}_K(A_P) = \\textnormal{Int}_K(A)_P$.\n\\end{Lem}\n\\begin{proof}\nThe containment $\\textnormal{Int}_K(A)_P \\subseteq \\textnormal{Int}_K(A_P)$ follows from the proof of \\cite[Prop. 3.2]{Wer}, which itself is an adaptation of a technique of Rush involving induction on the degrees of the relevant polynomials (see \\cite[Thm. I.2.1]{CaCh} or \\cite[Prop. 1.4]{Rush}). \n\nFor the other inclusion, let $f \\in \\textnormal{Int}_K(A_P)$ and write $f(X)=\\frac{g(X)}{d}$ for some $g \\in D[X]$ and $d\\in D \\setminus \\{0\\}$. Since $D$ is Dedekind, we may write $dD=P^aI$, where $a \\geq 0$ and $I$ is an ideal of $D$ coprime with $P$ (possibly equal to $D$ itself). If $a=0$ then $f \\in D_P[X] \\subseteq \\textnormal{Int}_K(A)_P$. If $a \\geq 1$, let $c\\in I \\setminus P$. We claim that $cf \\in\\textnormal{Int}_K(A)$, from which the statement follows since $c \\in D \\setminus P$. \n\nIf $Q \\subset D$ is a prime ideal different from $P$, then $cf \\in D_Q[X] \\subseteq \\textnormal{Int}_K(A_Q)$; that is, $cf(A_Q) \\subset A_Q$. Now, $f(A) \\subseteq f(A_P) \\subseteq A_P$ by assumption, so $cf(A) \\subset cA_P = A_P$, since $c \\notin P$. Since $A=\\bigcap_{Q\\in{\\rm Spec}(D)} A_Q$, it follows that $cf(A)\\subset A$, and we are done.\n\\end{proof}\n\n\nRecall (Definition \\ref{Null ideal}) that the null ideal of $A$ in $R$ is $N_R(A) = \\{ f \\in R[X] \\mid f(A) = 0\\}$.\n\n\\begin{Prop}\\label{equivalent conditions}\nLet $D$ be a Dedekind domain and $A$ a $D$-algebra with standard assumptions. Let $P$ be a prime ideal of $D$. Then, the following are equivalent.\n\\begin{enumerate}[(1)]\n\\item $N_{D\/P}(A\/PA) \\supsetneq (0)$.\n\\item $D_P[X] \\subsetneq \\textnormal{Int}_K(A_P)$.\n\\item $D\/P$ is finite and $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree.\n\\end{enumerate}\n\\end{Prop}\n\\begin{proof}\n$(1) \\Rightarrow (2)$ Let $g\\in D[X]$ be a monic pullback of a nontrivial element $\\overline{g}\\in N_{D\/P}(A\/PA)$ and let $\\pi\\in P\\setminus P^2$. Then, $g(A_P)\\subseteq PA_P=\\pi A_P$, so $\\frac{g(X)}{\\pi}\\in \\textnormal{Int}_K(A_P)\\setminus D_P[X]$. \n\n$(2) \\Rightarrow (1)$ Let $f(X)=\\frac{g(X)}{d} \\in \\textnormal{Int}_K(A_P) \\setminus D_P[X]$, with $g \\in D[X] \\setminus P[X]$ and  $d\\in P$. Let $v_P$ denote the canonical valuation on $D_P$. If $v_P(d)=e>1$ and $\\pi \\in P\\setminus P^2$, then $\\pi^{e-1} f(X)$ is still an element of $\\textnormal{Int}_K(A_P)$ which is not in $D_P[X]$. So, $g(A_P) \\subseteq \\frac{d}{\\pi^{e-1}} A_P \\subseteq \\pi A_P$. Hence, $\\overline{g}\\in(D_P\/PD_P)[X]\\cong(D\/P)[X]$ is a nontrivial element of $N_{D\/P}(A\/PA)$.\n\n$(1) \\Leftrightarrow (3)$ Note that\n\\begin{equation*}\nN_{D\/P}(A\/PA)=\\bigcap_{\\overline{a}\\in A\/PA}N_{D\/P}(\\overline{a})=\\bigcap_{\\overline{a}\\in A\/PA}(\\mu_{\\overline{a}}(X))\n\\end{equation*}\nwhere, for each $\\overline{a}\\in A\/PA$, $\\mu_{\\overline{a}}\\in (D\/P)[X]$ is the minimal polynomial of $\\overline{a}$ over the field $D\/P$.\n\nIf $N_{D\/P}(A\/PA)$ is nonzero, then it is equal to a principal ideal generated by a monic non-constant  polynomial $\\overline{g}\\in (D\/P)[X]$. Since $N_{D\/P}(A\/PA)\\subseteq N_{D\/P}(D\/P)$, it follows that $D\/P$ is finite (if not, then $N_{D\/P}(D\/P) = (0)$, because the only polynomial which is identically zero on an infinite field is the zero polynomial). Moreover, each element $\\overline{a}\\in A\/PA$ is algebraic over $D\/P$ (otherwise the corresponding $N_{D\/P}(\\overline{a})$ is zero) and its degree over $D\/P$ is bounded by $\\deg(\\overline{g})$. \n\nConversely, assume $D\/P$ is finite and $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree $n$. Then,  there are finitely many polynomials over $D\/P$ of degree at most $n$, and the product of all such polynomials is a nontrivial element of $N_{D\/P}(A\/PA)$.\n\\end{proof}\n\nWe can now establish the promised criterion for $\\textnormal{Int}_K(A)$ to be nontrivial.\n\n\\begin{Thm}\\label{criterion}\nLet $D$ be a Dedekind domain and let $A$ be a $D$-algebra with standard assumptions. Then $\\textnormal{Int}_K(A)$ is nontrivial if and only if there exists a prime ideal $P$ of $D$ of finite index such that $A\/PA$ is a $D\/P$-algebraic algebra of bounded degree.\n\\end{Thm}\n\\begin{proof}\nClearly, $D[X]\\subsetneq \\textnormal{Int}_K(A)$ if and only if there exists a prime ideal $P\\subset D$ such that the two $D$-modules $D[X]$ and $\\textnormal{Int}_K(A)$ are not equal locally at $P$, that is, $D_P[X]\\subsetneq \\textnormal{Int}_K(A)_P$. Since $\\textnormal{Int}_K(A)_P=\\textnormal{Int}_K(A_P)$ by Lemma \\ref{wellbehaviourlocalization}, we can apply Proposition \\ref{equivalent conditions} and we are done.\n\\end{proof}\n\n\\begin{Ex}\\label{Nontriv examples}\nTheorem \\ref{criterion} applies to the following examples.\n\\begin{itemize}\n\\item[(1)] Let $D=\\mathbb{Z}$ and $A=\\overline{\\mathbb{Z}}$, the absolute integral closure of $\\mathbb{Z}$. Then, for each $n\\in\\mathbb{N}$, there exists $\\alpha\\in\\overline{\\mathbb{Z}}$ of degree $d>n$ such that $O_{\\mathbb{Q}(\\alpha)}=\\mathbb{Z}[\\alpha]$. It follows that for each prime $p\\in\\mathbb{Z}$, $\\overline{\\mathbb{Z}}\/p\\overline{\\mathbb{Z}}$ is an algebraic $\\mathbb{Z}\/p\\mathbb{Z}$-algebra of unbounded degree. Thus, $\\textnormal{Int}_{\\mathbb{Q}}(\\overline{\\mathbb{Z}})=\\mathbb{Z}[X]$.\n\n\\item[(2)] Let $D=\\mathbb{Z}_{(p)}$ and $A=\\mathbb{Z}_p$. Then, $\\mathbb{Z}_p\/p\\mathbb{Z}_p\\cong\\mathbb{Z}\/p\\mathbb{Z}$, so $\\mathbb{Z}_{(p)}[X]\\subsetneq \\textnormal{Int}_{\\mathbb{Q}}(\\mathbb{Z}_p)$.\n\n\\item[(3)] Let $D=\\mathbb{Z}$ and $A=\\widehat{\\mathbb{Z}}=\\prod_{p\\in\\mathbb{P}}\\mathbb{Z}_p$, the profinite completion of $\\mathbb{Z}$, where $\\mathbb{P}$ denotes the set of all prime numbers. For each prime $p\\in\\mathbb{Z}$, we have $p\\widehat{\\mathbb{Z}}=\\prod_{p'\\not=p}\\mathbb{Z}_{p'}\\times p\\mathbb{Z}_p$, so $\\widehat{\\mathbb{Z}}\/p\\widehat{\\mathbb{Z}}\\cong \\mathbb{Z}_p\/p\\mathbb{Z}_p\\cong\\mathbb{Z}\/p\\mathbb{Z}$. Thus, $\\mathbb{Z}[X]\\subsetneq\\textnormal{Int}_{\\mathbb{Q}}(\\widehat{\\mathbb{Z}})$.\n\\end{itemize}\n\\end{Ex}\n\nIf $\\widehat{A}$ is the $P$-adic completion of a $D$-algebra $A$, then we can say more about $\\textnormal{Int}_K(\\widehat{A})$. The following lemma also appears in \\cite{IntdecompII}. We include it in its entirety since the proof is quite short.\n\n\\begin{Lem}\\label{DVR lemma}\nLet $D$ be a discrete valuation ring (DVR) with maximal ideal $P = \\pi D$. Let $A$ be a $D$-algebra with standard assumptions, and let $\\widehat{A}$ be the $P$-adic completion of $A$. Then, $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$.\n\\end{Lem}\n\\begin{proof}\nThe containment $\\textnormal{Int}_K(\\widehat{A}) \\subseteq \\textnormal{Int}_K(A)$ is clear, since $A$ embeds in $\\widehat{A}$. Conversely, let $f \\in \\textnormal{Int}_K(A)$ and $\\alpha \\in \\widehat{A}$. Suppose $f(X) = g(X)\/\\pi^k$, where $g \\in D[X]$ and $k \\in \\mathbb{N}$. If $k = 0$, then the conclusion is clear, so assume that $k > 1$. \n\nVia the canonical projection $\\widehat{A} \\to A\/\\pi^k A$, we see that there exists $a\\in A$ such that $\\alpha \\equiv a \\pmod{\\pi^k \\widehat{A}}$. Since the coefficients of $g$ are central in $A$, we get $g(\\alpha)\\equiv g(a) \\pmod{\\pi^k \\widehat{A}}$. Thus, $f(\\alpha)=f(a)+\\lambda\/\\pi^k$, where $\\lambda\\in\\pi^k \\widehat{A}$, so that $f(\\alpha)\\in \\widehat{A}$. Hence, $f \\in \\textnormal{Int}_K(\\widehat{A})$ and $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$.\n\\end{proof} \n\nThus, in Example \\ref{Nontriv examples} (2), we have $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z}_{(p)})$. Moreover, in Example \\ref{Nontriv examples} (3) we have $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}(\\mathbb{Z})$ (see also \\cite{ChabPer} where the profinite completion of $\\mathbb{Z}$ was considered in order to study the polynomial overrings of $\\textnormal{Int}(\\mathbb{Z})$). A more general example, which results in proper containments among all of $D[X]$, $\\textnormal{Int}_K(A)$, and $\\textnormal{Int}(D)$, is the following.\n\n\\begin{Ex}\\label{DVR example}\nLet $D$ be a DVR with maximal ideal $P = \\pi D$ and finite residue field. Let $A$ be a $D$-algebra of finite type with standard assumptions and such that $\\textnormal{Int}_K(A) \\subsetneq \\textnormal{Int}(D)$. Let $\\widehat{A}$ be the $P$-adic completion of $A$. Then, $P$ satisfies the conditions of Theorem \\ref{criterion} with respect to $A$, so $D[X] \\subsetneq \\textnormal{Int}_K(A)$; and $\\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A)$ by Lemma \\ref{DVR lemma}. Thus,\n\\begin{equation*}\nD[X] \\subsetneq \\textnormal{Int}_K(\\widehat{A}) = \\textnormal{Int}_K(A) \\subsetneq \\textnormal{Int}(D).\n\\end{equation*}\nIn general, $\\widehat{A}$ is not finitely generated as a $D$-module (this is the case, for instance, when $A$ is countable but $\\widehat{A}$ is uncountable). So, $\\widehat{A}$ can provide an example of a $D$-algebra that is not finitely generated and for which the integer-valued polynomial ring is properly contained between $D[X]$ and $\\textnormal{Int}(D)$.\n\\end{Ex}\n\n\\begin{Rem}\\label{Quaternion example again}\nLemma \\ref{DVR lemma} also gives us another approach to Example \\ref{Quaternion example}. With notation as in that example, we have $\\widehat{A} \\cong M_2(\\mathbb{Z}_p)$ (indeed, this follows from the fact that $A\/p^k A \\cong M_2(\\mathbb{Z}\/p^k \\mathbb{Z})$ for all $k > 0$). Thus, $\\textnormal{Int}_\\mathbb{Q}(A) = \\textnormal{Int}_\\mathbb{Q}(M_2(\\mathbb{Z}_p)) = \\textnormal{Int}_\\mathbb{Q}(M_2(\\mathbb{Z}_{(p)}))$ even though $A \\not\\cong M_2(\\mathbb{Z}_{(p)})$.\n\\end{Rem}\n\nWe close this paper by using the conditions of Proposition \\ref{equivalent conditions} to prove that when $D$ is Dedekind, $\\textnormal{Int}_K(A)$ has Krull dimension 2. This result was shown by Frisch \\cite[Thm. 5.4]{Fri1} in the case where $A$ is of finite type. Our work does not require $A$ to be finitely generated, and somewhat surprisingly does not require a full classification of the prime ideals of $\\textnormal{Int}_K(A)$.\n\nRecall that a nonzero prime ideal $\\mathfrak{P}$ of $\\textnormal{Int}_K(A)$ is called unitary if $\\mathfrak{P} \\cap D \\ne (0)$, and is called non-unitary if $\\mathfrak{P} \\cap D = (0)$.\n\n\\begin{Thm}\\label{Height thm}\nLet $D$ be a Dedekind domain and let $A$ be a $D$-algebra with standard assumptions. Let $\\mathfrak{P}$ be a nonzero prime ideal of $\\textnormal{Int}_K(A)$.\n\\begin{enumerate}[(1)]\n\\item If $\\mathfrak{P}$ is non-unitary, then $\\mathfrak{P}$ has height 1.\n\\item If $\\mathfrak{P}$ is unitary, then let $P = \\mathfrak{P} \\cap D$.\n\\begin{itemize}\n\\item[(i)] If $P$ does not satisfy any of the conditions of Proposition \\ref{equivalent conditions}, then $\\mathfrak{P}$ has height 2.\n\n\\item[(ii)] If $P$ satisfies one of the conditions of Proposition \\ref{equivalent conditions}, then $\\mathfrak{P}$ is maximal \nand has height at most 2.\n\\end{itemize}\n\\end{enumerate}\n\\end{Thm}\n\\begin{proof}\n(1) Following \\cite[Lem. 5.3]{Fri1}, the non-unitary prime ideals of $\\textnormal{Int}_K(A)$ are in one-to-one correspondence with the prime ideals of $K[X]$. Since $K[X]$ has dimension 1, the non-unitary primes of $\\textnormal{Int}_K(A)$ are all of height 1.\n\n(2) Let $P$ be a nonzero prime of $D$. Assume first that $P$ does not satisfy any of the conditions of Proposition \\ref{equivalent conditions}. Then, $D_P[X] = \\textnormal{Int}_K(A_P) = \\textnormal{Int}_K(A)_P$. It follows that the unitary primes of $\\textnormal{Int}_K(A)$ are in one-to-one correspondence with the primes of $D_P[X]$. Since $D$ is Dedekind, we know that $D_P[X]$ has dimension 2; hence, all the primes of $\\textnormal{Int}_K(A)$ under consideration have height 2.\n\nFor the remainder of the proof, assume that $P = \\mathfrak{P} \\cap D$ does satisfy the conditions of Proposition \\ref{equivalent conditions}. Since $\\mathfrak{P} \\cap D = P$, the prime ideal $\\mathfrak{P}$ survives in $\\textnormal{Int}_K(A)_P=\\textnormal{Int}_K(A_P)$ and clearly its extension $\\mathfrak{P}^e$ is still a prime unitary ideal (so, $\\mathfrak{P}^e\\cap D_P=PD_P$). It is sufficient to show that $\\mathfrak{P}^e$ is a maximal ideal, so we may work over the localizations. Thus, without loss of generality we will assume that $D$ is a DVR. In particular, this means that $P =\\pi D$, for some $\\pi\\in D$.\n\nLet $\\overline{g}\\in N_{D\/P}(A\/PA)$, $\\overline{g}\\not=0$, and let $g\\in D[X]$ be a pullback of $g(X)$. Then $g(A) \\subseteq PA=\\pi A$. Consequently, for each $f\\in\\textnormal{Int}_K(A)$ we have $(g\\circ f)(A)\\subset \\pi A$. Consider the ideal $\\mathfrak{A} = \\{F \\in \\textnormal{Int}_K(A) \\mid F(A) \\subseteq PA\\}$ of $\\textnormal{Int}_K(A)$. Because $P = \\pi  D$ is principal, we have $\\mathfrak{A} = \\pi \\textnormal{Int}_K(A)$, which is contained in $\\mathfrak{P}$. Hence, for each $f \\in \\textnormal{Int}_K(A)$, $g \\circ f \\in \\mathfrak{P}$. \n\nNow, if we consider the $D\/P$-algebra $\\textnormal{Int}_K(A)\/\\mathfrak{P}$, we see that each element of $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is annihilated by $\\overline{g}(X)$. But $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is a domain, and for it to be annihilated by a nonzero polynomial, it must be finite. Thus, in fact $\\textnormal{Int}_K(A)\/\\mathfrak{P}$ is a finite field, and so $\\mathfrak{P}$ is maximal.\n\nFinally, to show that $\\mathfrak{P}$ has height at most 2, let $\\mathfrak{Q}$ be a prime of $\\textnormal{Int}_K(A)$ such that $(0) \\subsetneq \\mathfrak{Q} \\subseteq \\mathfrak{P}$. If $\\mathfrak{Q}$ is unitary, then we have $\\mathfrak{Q} \\cap D = P$, and by our work above $\\mathfrak{Q}$ is maximal, hence equal to $\\mathfrak{P}$. If $\\mathfrak{Q}$ is non-unitary, then it has height 1 by part (1) of the theorem. It follows that $\\mathfrak{P}$ has height at most 2.\n\\end{proof}\n\n\\begin{Cor}\\label{Krull dimension}\nLet $D$ be a Dedekind domain with quotient field $K$. Let $A$ be a $D$-algebra with standard assumptions. Then, $\\textnormal{Int}_K(A)$ has Krull dimension 2.\n\\end{Cor}\n\\begin{proof}\nIf $\\textnormal{Int}_K(A) = D[X]$, then its dimension equals that of $D[X]$, which is 2. So, assume that $\\textnormal{Int}_K(A)$ is nontrivial. By Theorem \\ref{criterion}, there exists a prime $P$ of $D$ that satisfies the conditions of Proposition \\ref{equivalent conditions}.\n\nLet $\\mathfrak{P} = \\{f \\in \\textnormal{Int}_K(A) \\mid f(0) \\in P \\}$. Since $\\textnormal{Int}_K(A) \\subseteq \\textnormal{Int}(D)$, $\\mathfrak{P}$ is an ideal of $\\textnormal{Int}_K(A)$, and it is easily seen to be prime and unitary, with $\\mathfrak{P} \\cap D = P$. Moreover, it contains the non-unitary ideal $XK[X] \\cap \\textnormal{Int}_K(A)$. Hence, $\\mathfrak{P}$ has height at least 2, and so $\\dim(\\textnormal{Int}_K(A)) \\geq 2$. However, $\\dim(\\textnormal{Int}_K(A)) \\leq 2$ by Theorem \\ref{Height thm}, so we conclude that $\\dim(\\textnormal{Int}_K(A)) = 2$.\n\\end{proof}\n\n\\subsection*{Acknowledgments}\n\\noindent This research has been supported by  the grant ``Assegni Senior'' of the University of Padova. The authors wish to thank the referee for several suggestions which improved the quality of the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{S}{cene} text recognition is an essential process in computer vision tasks. Many practical applications such as traffic sign reading, product recognition, intelligent inspection, and image searching, benefit from the rich semantic information of scene text. With the development of scene text detection methods \\cite{gomez2017textproposals,khare2016blind,sun2015robust,zhu2016could}, scene character recognition has emerged at the forefront of this research topic and is regarded as an open and very challenging research problem \\cite{su2017accurate}.\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-a.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-b.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/regular-c.jpg}\n\\end{minipage}%\n}%\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-d.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-e.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-f.jpg}\n\\end{minipage}%\n}%\n\\subfigure[]{\n\\begin{minipage}[c]{0.15\\textwidth}\n\\centering\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-g.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-h.jpg}\n  \\includegraphics[width=2.5cm,height=1.5cm]{picture\/irregular-i.jpg}\n\\end{minipage}%\n}%\n\n\\caption{Examples of regular and irregular scene text. (a) Regular text. (b) Slanted and perspective text. (c) Curved text.}\n\\label{fig:1-scene-text}\n\\end{figure}\n\nNowadays, regular text recognition methods \\cite{bissacco2013photoocr,neumann2012real,shi2017end,su2017accurate,wang2012end} have achieved notable success. Moreover, methods based on convolutional neural networks \\cite{bissacco2013photoocr,jaderberg2016reading,wang2012end} have been broadly applied. Integrating recognition models with recurrent neural networks \\cite{he2016reading,shi2017end,shi2016robust} and attention mechanisms \\cite{cheng2017focusing,cheng2017arbitrarily,lee2016recursive,yang2017learning} yields better performance for these models.\n\nNevertheless, most current recognition models remain too unstable to handle multiple disturbances from the environment. Furthermore, the various shapes and distorted patterns of irregular text cause additional challenges in recognition. As illustrated in Fig.  \\ref{fig:1-scene-text}, scene text with irregular shapes, such as perspective and curved text, is still very challenging to recognize.\n\nReading text is naturally regarded as a multi-classification task involving sequence-like objects \\cite{shi2017end}. Usually, the characters in one text are of the same size. However, characters in different scene texts can vary in size. Therefore, we propose the multi-object rectified attention network (MORAN), which can read rotated, scaled and stretched characters in different scene texts. The MORAN consists of a multi-object rectification network \\textbf{(MORN)} to rectify images and an attention-based sequence recognition network \\textbf{(ASRN)} to read the text. We separate the difficult recognition task into two parts. First, as one kind of spatial transformer, the MORN rectifies images that contain irregular text. As Fig. \\ref{fig:2-MORAN-system} shows, after the rectification by the MORN, the slanted text becomes more horizontal, tightly-bounded, and easier to read. Second, ASRN takes the rectified image as input and outputs the predicted word.\n\nThe training of the MORN is guided by the ASRN, which requires only text labels. Without any geometric-level or pixel-level supervision, the MORN is trained in a weak supervision way. To facilitate this manner of network training, we initialize a basic coordinate grid. Every pixel of an image has its own position coordinates. The MORN learns and generates an offset grid based on these coordinates and samples the pixel value accordingly to rectify the image. The rectified image is then obtained for the ASRN.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=8.5cm,height=3.2cm]{picture\/Moran-system.jpg}\n\n\\put(11,33){Input}\n\\put(10,28){Image}\n\\put(45,33){Rectified}\n\\put(48,28){Image}\n\\put(82,26){Result}\n\n\\put(79,17){JOHNNY}\n\\put(28,26){MORN}\n\\put(66,26){ASRN}\n\n\\put(29,7){Weak}\n\\put(25,3){Supervision}\n\n\\put(65,7){Text Label}\n\\put(64,3){Supervision}\n\n\\end{overpic}\n\n\\caption{Overview of the MORAN. The MORAN contains a MORN and an ASRN. The image is rectified by the MORN and given to the ASRN. The dashed lines show the direction of gradient propagation, indicating that the two sub-networks are jointly trained.}\n\\label{fig:2-MORAN-system}\n\\end{figure}\n\nWith respect to the ASRN, a decoder with an attention mechanism is more likely to predict the correct words because of the rectified images. However, Cheng et al. \\cite{cheng2017focusing} found that existing attention-based methods cannot obtain accurate alignments between feature areas and targets. Therefore, we propose a fractional pickup method to train the ASRN. By adopting several scales of stretch on different parts of the feature maps, the feature areas are changed randomly at every iteration in the training phase. Owing to training with fractional pickup, the ASRN is more robust to the variation of context. Experiments show that the ASRN can accurately focus on objects.\n\nIn addition, we designed a curriculum learning strategy for the training of the MORAN. Because the MORN and ASRN are mutually beneficial in terms of performance, we first fix one of them to more efficiently optimize the other. Finally, the MORN and ASRN are optimized in an end-to-end fashion to improve performance. In short, the contributions of our research are as follows:\n\n\\begin{itemize}\n\n\\item We propose the MORAN framework to recognize irregular scene text. The framework contains a multi-object rectification network (MORN) and an attention-based sequence recognition network (ASRN). The image rectified by the MORN is more readable for the ASRN.\n\n\\item Trained in a weak supervision way, the sub-network MORN is flexible. It is free of geometric constraints and can rectify images with complicated distortion.\n\n\\item We propose a fractional pickup method for the training of the attention-based decoder in the ASRN. To address noise perturbations, we expand the visual field of the MORAN, which further improves the sensitivity of the attention-based decoder.\n\n\\item We propose a curriculum learning strategy that enables the MORAN to learn efficiently. Owing to the training with this strategy, the MORAN outperforms state-of-the-art methods on several standard text recognition benchmarks, including the IIIT5K, SVT, ICDAR2003, ICDAR2013, ICDAR2015, SVT-Perspective, and CUTE80 datasets.\n\n\\end{itemize}\n\nThe rest of the paper is organized as follow. Section 2 reviews related work. Section 3 details the proposed method. Experimental results are given in Section 4, and the conclusions are presented in Section 5.\n\n\\section{Related Work}\n\\label{section:Related work}\nIn recent years, the recognition of scene text has greatly advanced because of the rapid development of neural networks \\cite{gu2017recent}. Zhu et al. \\cite{zhu2016scene} and Ye et al. \\cite{ye2015text} have provided an overview of the major advances in the field of scene text detection and recognition. Based on the sliding window method \\cite{wang2011end,wang2010word}, pattern features extracted by a neural network become dominant with respect to the hand crafted features, such as the connected components \\cite{neumann2012real}, strokelet generation \\cite{yao2014strokelets}, histogram of oriented gradients descriptors \\cite{dalal2005histograms,su2014accurate}, tree-structured models \\cite{shi2014end}, semi-markov conditional random fields \\cite{seok2015scene} and generative shape models \\cite{lou2016generative}. For instance, Bissacco \\cite{bissacco2013photoocr} applied a network with five hidden layers for character classification. Using convolutional neural networks (CNNs), Jaderberg et al. \\cite{Jaderberg2015Deep} and Yin et al. \\cite{yin2017scene} proposed respective methods for unconstrained recognition.\n\nWith the widespread application of recurrent neural networks (RNNs)  \\cite{cho2014learning,hochreiter1997long}, CNN-based methods are combined with RNNs for better learning of context information. As a feature extractor, the CNN obtains the spatial features of images. Then, the RNN learns the context of features. Shi et al. \\cite{shi2017end} proposed an end-to-end trainable network with both CNNs and RNNs, named CRNN. Guided by the CTC loss \\cite{graves2006connectionist}, the CRNN-based network learns the conditional probability between predictions and sequential labels.\n\nFurthermore, attention mechanisms \\cite{bahdanau2014neural} focus on informative regions to achieve better performance. Lee et al. \\cite{lee2016recursive} proposed a recursive recurrent network with attention modeling for scene text recognition. Yang et al. \\cite{yang2017learning} addressed a two-dimensional attention mechanism. Cheng et al. \\cite{cheng2017focusing} used the focusing attention network (FAN) to correct shifts in attentional mechanisms and achieved more accurate position predictions.\n\nCompared with regular text recognition work, irregular text recognition is more difficult. One kind of irregular text recognition method is the bottom-up approach \\cite{cheng2017arbitrarily,yang2017learning}, which searches for the position of each character and then connects them. Another is the top-down approach \\cite{liu2016star,shi2016robust}. This type of approach matches the shape of the text, attempts to rectify it, and reduces the degree of recognition difficulty.\n\nIn the bottom-up manner, a two-dimensional attention mechanism for irregular text was proposed by Yang et al. \\cite{yang2017learning}. Based on the sliced Wasserstein distance \\cite{rabin2011wasserstein}, the attention alignment loss is adopted in the training phase, which enables the attention model to accurately extract the character features while ignoring the redundant background information. Cheng et al. \\cite{cheng2017arbitrarily} proposed an arbitrary-orientation text recognition network, which uses more direct information of the position to instruct the network to identify characters in special locations.\n\nIn the top-down manner, STAR-Net \\cite{liu2016star} used an affine transformation network that transforms the rotated and differently scaled text into more regular text. Meanwhile, a ResNet \\cite{he2016deep} is used to extract features and handle more complex background noise. RARE \\cite{shi2016robust} regresses the fiducial transformation points on sloped text and even curved text, thereby mapping the corresponding points onto standard positions of the new image. Using thin-plate-spline \\cite{bookstein1989principal} to back propagate the gradients, RARE is end-to-end optimized.\n\nOur proposed MORAN model uses the top-down approach. The fractional pickup training method is thus designed to improve the sensitivity of the MORAN to focus on characters. For the training of the MORAN, we propose a curriculum learning strategy for better convergence.\n\n\\section{Methodology}\n\nThe MORAN contains two parts. One is the MORN, which is trained in a weak supervision way to learn the offset of each part of the image. According to the predicted offsets, we apply sampling and obtain a rectified text image. The other one is ASRN, a CNN-LSTM framework followed by an attention decoder. The proposed fractional pickup further improves attention sensitivity. The curriculum learning strategy guides the MORAN to achieve state-of-the-art performance.\n\n\\subsection{Multi-Object Rectification Network}\n\\label{section:Multi-Object Rectification Network}\nCommon methods to rectify patterns such as the affine transformation network, are limited by certain geometric constraints. With respect to the affine transformation, it is limited to rotation, scaling, and translation. However, one image may have several kinds of deformations, and the distortion of scene text will thus be complicated. As shown in Fig. \\ref{fig:compare-stn}, the characters in the image become slanted after rectification by the affine transformation. The black edges introduce additional noise. Therefore, transformations with geometric constraints can not cover all complicated deformations.\n\nAnother method that is free of geometric constraints, is the deformable convolutional network \\cite{deformable2017}. Using deformable convolutional kernels, the feature extractor automatically selects informative features. We attempted to combine the recognition network with a deformable convolutional network. However, as a sequence-to-sequence problem, irregular text recognition is more challenging. The network sometimes failed to converge. The best accuracy rate on IIIT5K we achieved was only 78.1\\%, which is far behind the state-of-the-art result (91.2\\%).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5cm,height=5cm]{picture\/compare-stn.jpg}\n\\caption{Comparison of the MORN and affine transformation. The MORN is free of geometric constraints. The main direction of rectification predicted by the MORN for each character is indicated by a yellow arrow. The offset maps generated by the MORN are visualized as a heat map. The offset values on the boundary between red and blue are zero. The directions of rectification on both sides of the boundary are opposite and outward. The depth of the color represents the magnitude of the offset value. The gradual-change in color indicates the smoothness of the rectification.}\n\\label{fig:compare-stn}\n\\end{figure}\n\nBecause the recognition models remain inadequately strong to handle multiple disturbances from various shapes, we consider rectifying images to reduce the difficulty of the recognition. As demonstrated in Fig. \\ref{fig:3-MORAN-overview}, the MORN architecture rectifies the distorted image. The MORN predicts the offset of each part of the image without any geometric constraint. Based on the predicted offsets, the image is rectified and becomes easier to recognize.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=17cm]{picture\/Moran-overview.jpg}\n\\caption{Overall structure of MORAN. }\n\\label{fig:3-MORAN-overview}\n\\end{figure*}\n\nFurthermore, the MORN predicts the position offsets but not the categories of characters. The character details for classification are not necessary. We hence place a pooling layer before the convolutional layer to avoid noise and reduce the amount of calculation.\n\n\\begin{table}[h]\n\\centering\n\\caption{Architecture of the MORN}\n\\label{table:The architecture of MORN}\n\\begin{tabular*}{8.5cm}{c|p{3.5cm}<{\\centering}|c}\n\\hline\nType & Configurations  & Size \\\\\n\\hline\nInput & - & 1$\\times$32$\\times$100 \\\\\n\\hline\nMaxPooling & k2, s2 & 1$\\times$16$\\times$50 \\\\\n\\hline\nConvolution  & maps:64, k3, s1, p1 & 64$\\times$16$\\times$50 \\\\\n\\hline\nMaxPooling & k2, s2 & 64$\\times$8$\\times$25 \\\\\n\\hline\nConvolution  & maps:128, k3, s1, p1 & 128$\\times$8$\\times$25 \\\\\n\\hline\nMaxPooling & k2, s2 & 128$\\times$4$\\times$12 \\\\\n\\hline\nConvolution  & maps:64, k3, s1, p1 & 64$\\times$4$\\times$12 \\\\\n\\hline\nConvolution  & maps:16, k3, s1, p1 & 16$\\times$4$\\times$12 \\\\\n\\hline\nConvolution  & maps:2, k3, s1, p1 & 2$\\times$4$\\times$12 \\\\\n\\hline\nMaxPooling & k2, s1 & 2$\\times$3$\\times$11 \\\\\n\\hline\nTanh & - & 2$\\times$3$\\times$11 \\\\\n\\hline\nResize & - & 2$\\times$32$\\times$100 \\\\\n\\hline\n\\end{tabular*}\n\\begin{tablenotes}\n\\item Here k, s, p are kernel, stride and\npadding sizes, respectively. For example, $k3$ represents a $3\\times3$ kernel size.\n\\end{tablenotes}\n\\end{table}\n\nThe architecture of the MORN is given in Table\\ref{table:The architecture of MORN}. Each convolutional layer is followed by a batch normalization layer and a ReLU layer except for the last one. The MORN first divides the image into several parts and then predicts the offset of each part. With an input size of $32\\times100$, the MORN divides the image into $3\\times 11 = 33$ parts. All the offset values are activated by $Tanh(\\cdot)$, resulting in values within the range of $(-1, 1)$. The offset maps contain two channels, which denote the x-coordinate and y-coordinate respectively. Then, we apply bilinear interpolation to smoothly resize the offset maps to a size of $32\\times100$, which is the same size of the input image. After allocating the specific offset to each pixel, the transformation of the image is smooth. As demonstrated in Fig.\\ref{fig:compare-stn}, the color depth gradually changes on both sides of the boundary between the red and blue colors in the heat map, which evidences the smoothness of the rectification. There are no indented edges in the rectified image.\n\nMoreover, because every value in the offset maps represents the offset from the original position, we generate a basic grid from the input image to represent the original positions of the pixels. The basic grid is generated by normalizing the coordinate of each pixel to $[-1, 1]$. The coordinates of the top-left pixel are $(-1, -1)$, and those of the bottom-right one are $(1, 1)$. Pixels at the same positions on different channels have the same coordinates. Similar to the offset maps, the grid contains two channels, which represent the x-coordinate and y-coordinate, respectively. Then, the basic grid and the resized offset maps are summed as follows,\n\\begin{equation}\noffset_{(c,i,j)}^{'} = offset_{(c,i,j)}+basic_{(c,i,j)} ,  c = 1,2\n\\end{equation}\nwhere $(i,j)$ is the position of the $i$-th row and $j$-th column.\n\nBefore sampling, the x-coordinate and y-coordinate on the offset maps are normalized to $[0, W]$ and $[0, H]$, respectively. Here, $H\\times W$ is the size of the input image. The pixel value of $i$-th row and $j$-th column in rectified image $I'$ is,\n\\begin{equation}\nI'_{(i, j)} = I_{(i^{'}, j^{'})} \\label{sampling}\n\\end{equation}\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\ni^{'} = offset_{(1, i, j)}^{'} \\\\\nj^{'} = offset_{(2, i, j)}^{'}\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere $I$ is the input image. Further, $i^{'}$ is obtained from the first channel of the offset maps, whereas $j^{'}$ is from the second channel. Both $i^{'}$ and $j^{'}$ are real values as opposed to integers so rectified image $I'$ is sampled from $I$ using bilinear interpolation.\n\nBecause Equation (\\ref{sampling}) is differentiable, the MORN can back-propagate the gradients. The MORN can be trained in a weak supervision way with images and associated text labels only, which means that it does not need pixel-level labeling information about the deformation of the text.\n\nAs Fig. \\ref{fig:4-ori-rectified-img} shows, the text in the input images is irregular. However, the text in the rectified images is more readable. Slanted or perspective texts become tightly bound after rectification. Furthermore, redundant noise is eliminated by the MORN for the curved texts. The background textures are removed in the rectified images of Fig. \\ref{fig:4-ori-rectified-img} (b).\n\n\\begin{figure}[h]\n\\centering\n\\subfigure[Perspective texts]{\n\\begin{minipage}[c]{0.45\\textwidth}\n\\centering\n  \\includegraphics[width=7cm,height=4cm]{picture\/ori_img.jpg}\n\\end{minipage}%\n}%\n\n\\subfigure[curved texts]{\n\\begin{minipage}[c]{0.45\\textwidth}\n\\centering\n  \\includegraphics[width=7cm,height=4cm]{picture\/ori_img_curve.jpg}\n\\end{minipage}%\n}%\n\n\n\\caption{Results of the MORN on challenging image text. The input images are shown on the left and the rectified images are shown on the right. The heat maps visualize offset maps as well as Fig. \\ref{fig:compare-stn}. (a) Slanted and perspective text. (b) Curved text, which is more challenging for recognition. Removed background textures are indicated by red circles.}\n\\label{fig:4-ori-rectified-img}\n\\end{figure}\n\nThe advantages of the MORN are manifold. 1) The rectified images are more readable owing to the regular shape of the text and the reduced noise. 2) The MORN is more flexible than the affine transformation. It is free of geometric constraints, which enables it to rectify images using complicated transformations. 3) The MORN is more flexible than methods using a specific number of regressing points. Existing method \\cite{shi2016robust} cannot capture the text shape in details if the width of the image is large. Thus the MORN has no limit with respect to the image size, especially the width of the input image. 4) The MORN does not require extra labelling information of character positions. Therefore, it can be trained in a weak supervision way by using existing training datasets.\n\n\\subsection{Attention-based Sequence Recognition Network}\n\\label{section:Attention-based Sequence Recognition Network}\nAs Fig. \\ref{fig:3-MORAN-overview} shows, the major structure of the ASRN is a CNN-BLSTM framework. We adopt a one-dimensional attention mechanism at the top of CRNN. The attention-based decoder, proposed by Bahdanau et al. \\cite{bahdanau2014neural}, is used to accurately align the target and label. It is based on an RNN and directly generates the target sequence $(y_{1},y_{2} ...,y_{N})$. The largest number of steps that the decoder generates is $T$. The decoder stops processing when it predicts an end-of-sequence token $``EOS\"$ \\cite{sutskever2014sequence}. At time step $t$, output $y_t$ is,\n\\begin{equation}\ny_{t} = Softmax(W_{out}s_{t}+b_{out})\n\\end{equation}\nwhere $s_{t}$ is the hidden state at time step $t$. We update $s_{t}$ using GRU \\cite{cho2014learning}. State $s_{t}$ is computed as:\n\\begin{equation}\ns_t = GRU(y_{prev}, g_{t}, s_{t-1})\n\\end{equation}\nwhere $y_{prev}$ denotes the embedding vectors of the previous output $y_{t-1}$ and $g_{t}$ represents the glimpse vectors, respectively calculated as,\n\\begin{equation}\ny_{prev} = Embedding(y_{t-1})\n\\end{equation}\n\\begin{equation}\ng_{t} = \\sum_{i=1}^L(\\alpha_{t,i} h_{i}) \\label{equ-g}\n\\end{equation}\nwhere $h_{i}$ denotes the sequential feature vectors and $L$ is the length of the feature maps. In addition, $\\alpha_{t,i}$ is the vector of attention weights as follows,\n\\begin{equation}\n\\alpha_{t,i} = {exp(e_{t,i})}  \/  {\\sum_{j=1}^L(exp(e_{t,j}))} \\label{equ-alpha}\n\\end{equation}\n\\begin{equation}\ne_{t,i} = Tanh(W_{s}s_{t-1}+W_{h}h_{i}+b) \\label{equ-e}\n\\end{equation}\n\nHere, $W_{out}$, $b_{out}$, $W_{s}$, $W_{h}$ and $b$ are trainable parameters. Note that $y_{prev}$ is embedded from the ground truth of the last step in the training phase, whereas the ASRN only uses the predicted output of the last step as $y_{t-1}$ in the testing phase.\n\nThe decoder outputs the predicted word in an unconstrained manner in lexicon-free mode. If lexicons are available, we evaluate the probability distributions for all words and choose the word with the highest probability as the final result.\n\nThe architecture of the ASRN is given in Table\\ref{table:The architecture of ASRN}. Each convolutional layer is followed by a batch normalization layer and a ReLU layer.\n\\begin{table}[t]\n\\centering\n\\caption{Architecture of the ASRN}\n\\label{table:The architecture of ASRN}\n\\begin{tabular*}{8.5cm}{c|p{3.5cm}<{\\centering}|c}\n\\hline\nType & Configurations  & Size \\\\\n\\hline\nInput & - & 1$\\times$32$\\times$100 \\\\\n\\hline\nConvolution  & maps:64, k3, s1, p1 & 64$\\times$32$\\times$100 \\\\\n\\hline\nMaxPooling & k2, s2 & 64$\\times$16$\\times$50 \\\\\n\\hline\nConvolution  & maps:128, k3, s1, p1 & 128$\\times$16$\\times$50 \\\\\n\\hline\nMaxPooling & k2, s2 & 128$\\times$8$\\times$25 \\\\\n\\hline\nConvolution  & maps:256, k3, s1, p1 & 256$\\times$8$\\times$25 \\\\\n\\hline\nConvolution  & maps:256, k3, s1, p1 & 256$\\times$8$\\times$25 \\\\\n\\hline\nMaxPooling & k2, s2x1, p0x1 & 256$\\times$4$\\times$26 \\\\\n\\hline\nConvolution  & maps:512, k3, s1, p1 & 512$\\times$4$\\times$26 \\\\\n\\hline\nConvolution  & maps:512, k3, s1, p1 & 512$\\times$4$\\times$26 \\\\\n\\hline\nMaxPooling & k2, s2x1, p0x1 & 512$\\times$2$\\times$27 \\\\\n\\hline\nConvolution  & maps:512, k2, s1 & 512$\\times$1$\\times$26 \\\\\n\\hline\nBLSTM & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\nBLSTM & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\nGRU & hidden unit:256 & 256$\\times$1$\\times$26 \\\\\n\\hline\n\\end{tabular*}\n\\begin{tablenotes}\n\\item Here, k, s, p are kernel, stride and\npadding sizes, respectively. For example, $s2\\times1$ represents a $2\\times1$ stride size. ``BLSTM\" stands for bidirectional-LSTM. ``GRU\" is in attention-based decoder.\n\\end{tablenotes}\n\\end{table}\n\n\\subsection{Fractional Pickup}\n\nThe decoder in the ASRN learns the matching relationship between labels and target characters in images. It is a data-driven process. The ability to choose regions that are focus-worthy is enhanced by the feedback of correct alignment.\n\nHowever, scene text is surrounded by various types of noise. Often, the decoder is likely to be deceived into focusing on ambiguous background regions in practical applications. If the decoder generates an incorrect region of focus, the non-corresponding features are chosen, which can cause a failed prediction.\n\nSome challenging samples for recognition are presented in Fig. \\ref{fig:5-fractional-pickup}. In this figure, the images contain text with shadows and unclear boundaries between characters or complicated backgrounds. Moreover, the focus regions generated by the decoder are narrow, which increases the probability of drifting from the correct regions.\n\n\\begin{figure}[t]\n\\centering\n\\rule{8cm}{0.05em}\n\\\\\n-------------------------------------------------------------\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_1.jpg}\n\n\\put(13,39){Without FP}\n\\put(67,39){With FP}\n\n\\put(20,-3){\\color{blue}{hot}\\color{red}{l}}\n\\put(72,-3){\\color{blue}{hotel}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_2.jpg}\n\\put(20,-3){\\color{blue}{a}\\color{red}{r}\\color{blue}{ge}\\color{red}{h}}\n\\put(72,-3){\\color{blue}{angels}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\begin{overpic}[width=8cm,height=3cm]{picture\/fp_3.jpg}\n\\put(20,-3){\\color{red}{e}\\color{blue}{aser}}\n\\put(72,-3){\\color{blue}{laser}}\n\\end{overpic}\n\n-------------------------------------------------------------\n\n\\caption{Difference in $\\alpha_{t}$ for training with and without fractional pickup. Here $\\alpha_{t}$ is visualized as a heat map. We delete the $\\alpha_{t}$ corresponding to ``EOS\".}\n\\label{fig:5-fractional-pickup}\n\\end{figure}\n\nWe propose a training method called fractional pickup that fractionally picks up the neighboring features in the training phase. An attention-based decoder trained by fractional pickup method can perceive adjacent characters. The wider field of attention contributes to the robustness of the MORAN.\n\nWe hence adopt fractional pickup at each time step of the decoder. In other words, a pair of attention weights are selected and modified at every time step. At time step $t$, $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ are updated as,\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n\\alpha_{t,k}^{,} = \\beta \\alpha_{t,k}+(1-\\beta)\\alpha_{t,k+1} \\\\\n\\alpha_{t,k+1}^{,} = (1-\\beta) \\alpha_{t,k}+\\beta\\alpha_{t,k+1}\n\\end{aligned}\n\\right.\n\\end{equation}\nwhere decimal $\\beta$ and integer $k$ are randomly generated as,\n\\begin{equation}\n\\beta = rand(0,1)\n\\end{equation}\n\\begin{equation}\nk = rand[1,T-1]\n\\end{equation}\nHere, T is the maximum number of steps of the decoder.\n\n\\textbf{Variation of Distribution}\nFractional pickup adds randomness to $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ in the decoder. This means that, even for the same image, the distribution of $\\alpha_{t}$ changes every time step in the training phase. As noted in Equation (\\ref{equ-g}), the glimpse vectors $g_{t}$ grabs the sequential feature vectors $h_{i}$ according to the various distributions of $\\alpha_{t}$, which is equivalent to the changes in feature areas. The randomness of $\\beta$ and $k$ avoids over-fitting and contributes to the robustness of the decoder.\n\n\\textbf{Shortcut of Forward Propagation}\nSequential feature vector $h_{i}$ is the output of the last bidirectional-LSTM in the ASRN. As shown in Fig. \\ref{fig:short-cut}, for step $k+1$ in the bidirectional-LSTM, a shortcut connecting to step $k$ is created by fractional pickup. The shortcut retains some features of the previous step in the training phase, which is the interference to the forget gate in bidirectional-LSTM. Therefore, fractional pickup provides more information about the previous step and increases the robustness for the bidirectional-LSTM in the ASRN.\n\n\\begin{figure}\n\\centering\n\\begin{overpic}[width=4cm]{picture\/short-cut.jpg}\n\\put(-5,65){$h_{i}$}\n\\end{overpic}\n\\caption{Fractional pickup creates a shortcut of forward propagation. The shortcut is drawn as a red arrow.}\n\\label{fig:short-cut}\n\\end{figure}\n\n\\textbf{Broader Visual Field}\nTraining with fractional pickup disturbs the decoder through the local variation of $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$. Note that $\\alpha_{t,k}$ and $\\alpha_{t,k+1}$ are neighbors. Without fractional pickup, the error term of sequence feature vector $h_{k}$ is,\n\\begin{equation}\n\\delta_{h_{k}} = \\delta_{g_{t}}\\alpha_{t,k}\n\\end{equation}\nwhere $\\delta_{g_{t}}$ is the error term of glimpse vector $g_{t}$. $\\delta_{h_{k}}$ is only relevant to $\\alpha_{t,k}$. However, with fractional pickup, the error item becomes,\n\\begin{equation}\n\\delta_{h_{k}} = \\delta_{g_{t}}(\\beta \\alpha_{t,k}+(1-\\beta)\\alpha_{t,k+1})\n\\end{equation}\nwhere $\\alpha_{t,k+1}$ is relevant to $h_{k+1}$, as noted in Equations (\\ref{equ-alpha}) and (\\ref{equ-e}), which means $\\delta_{h_{k}}$ is influenced by the neighbouring features. Owing to the disturbance, back-propagated gradients are able to dynamically optimize the decoder over a broader range of neighbouring regions.\n\nThe MORAN trained with fractional pickup method generates a smoother $\\alpha_{t}$ at each time step. Accordingly, it extracts features not only of the target characters, but also of the foreground and background context. As demonstrated in Fig. \\ref{fig:5-fractional-pickup}, the expanded visual field enables the MORAN to correctly predict target characters. To the best of our knowledge, this is the first attempt to adopt a shortcut in the training of the attention mechanism.\n\n\\subsection{Curriculum Training}\n\\label{section:curriculum-training}\n\nThe MORAN is end-to-end trainable with random initialization. However, end-to-end training consumes considerable time. We found that the MORN and ASRN can hinder each other during training. A MORN cannot be guided to rectify images when the input images have been correctly recognized by the high-performance ASRN. For the same reason, the ASRN will not gain robustness because the training samples have already been rectified by the MORN. The reasons above lead to inefficient training.\n\nTherefore, we propose a curriculum learning strategy to guide each sub-network in MORAN. The strategy is a three-step process. We first optimize the MORN and ASRN respectively and then join them together for further end-to-end training. The difficulty of training samples is gradually increased. The training set is denoted as $D = \\left \\{I_{i}, Y_{i} \\right \\}, i=1...N $. We minimize the negative log-likelihood of conditional probability of $D$ as follows:\n\n\\begin{equation}\nLoss = -\\sum_{i=1}^N{ \\sum_{t=1}^{\\left| Y_{i} \\right|}{\\log p(Y_{i,t} \\left| \\right. I_{i}; \\theta)} }\n\\end{equation}\nwhere $Y_{i,t}$ is the ground truth of the $t$-th character in $I_{i}$. $\\theta$ denotes the parameters of MORAN.\n\n\\textbf{First Stage for ASRN}\nWe first optimize the ASRN by using regular training samples. The dataset released by Gupta et al. \\cite{gupta2016synthetic} has tightly bounded annotations, which makes it possible to crop a text region with a tightly bounded box. The ASRN is first trained with these regular samples. Then, we simply crop every text using a minimum circumscribed horizontal rectangle to obtain irregular training samples. The commonly used datasets released by Jaderberg et al. \\cite{jaderberg2014synthetic} and Gupta et al. \\cite{gupta2016synthetic} offer abundant irregular training samples. We use them for the following training. Taking advantage of them, we optimize ASRN, which thus achieves higher accuracy.\n\n\\textbf{Second Stage for MORN}\nThe ASRN trained using regular training samples is chosen to guide the MORN training. This ASRN is not adequately robust for irregular text recognition so it is able to provide informative gradients for the MORN. We fix the parameters of this ASRN, and stack it after the MORN. If the transformation of the MORN does not reduce the difficulty of recognition, few meaningful gradients will be provided by the ASRN. The optimization of MORN would not progress. Only the correct transformation that decreases difficulty for recognition will give positive feedback to the MORN.\n\n\\textbf{Third Stage for End-to-end Optimization}\nAfter the MORN and ASRN are optimized individually, we connect them for joint training in an end-to-end fashion. Joint training enables MORAN to complete end-to-end optimization and outperform state-of-the-art methods.\n\n\\section{Experiments}\nIn this section we describe extensive experiments conducted on various benchmarks, including regular and irregular datasets. The performances of all the methods are measured by word accuracy.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Comparison of pooling layers in lexicon-free mode. ``No\", ``AP\" and ``MP\" respectively indicate no pooling layer, an average-pooling layer and a max-pooling layer at the top of the MORN. The kernel size is 2. ``s\" represents the stride. }\n\\label{table:comparison-of-pooling}\n\\begin{tabular}{|c| c | c | c | c | c | c | c | c | c}\n\\hline\n\\multirow{2}{*}{} & s & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 & IC15 \\\\\n\\cline{2-9}\n\\hline\nNo & - & 85.7 & 87.9 & 92.9 & 91.5 & 75.8 & 65.9 & 59.4  \\\\\nAP & 2 & 89.2 & 87.4 & 94.8 & 91.1 & 75.9 & 71.1 & 64.6 \\\\\nAP & 1 & 89.3 & 87.9 & 94.7 & 91.6 & 75.9 & 72.9 & 64.9 \\\\\nMP & 2 & 90.4 & 88.2 & 94.5 & 91.8 & \\textbf{76.1} & 76.4 & 68.4 \\\\\nMP & 1 & \\textbf{91.2} & \\textbf{88.3} & \\textbf{95.0} & \\textbf{92.4} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[t]\n\\centering\n\\caption{Performance of the MORAN. }\n\\label{table:Performance-of-MORAN}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 & IC15 \\\\\n\\cline{2-8}\n\\hline\nEnd-to-end training & 89.9 & 84.1 & 92.5 & 90.0 & 76.1 & 77.1 & 68.8 \\\\\n\\hline\nOnly ASRN & 84.2 & 82.2 & 91.0 & 90.1 & 71.0 & 64.6 & 65.6 \\\\\nMORAN without FP & 89.7 & 87.3 & 94.5 & 91.5 & 75.5 & 77.1 & 68.6 \\\\\nMORAN with FP & \\textbf{91.2} & \\textbf{88.3} & \\textbf{95.0} & \\textbf{92.4} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Datasets}\n\n\\textbf{IIIT5K-Words (IIIT5K)} \\cite{mishra2012scene} contains 3000 cropped word images for testing. Every image has a 50-word lexicon and a 1000-word lexicon. The lexicon consists of a ground truth and some randomly picked words.\n\n\\textbf{Street View Text (SVT)} \\cite{wang2011end} was collected from the Google Street View, consisting of 647 word images. Many images are severely corrupted by noise and blur, or have very low resolutions. Each image is associated with a 50-word lexicon.\n\n\\textbf{ICDAR 2003 (IC03)} \\cite{lucas2003icdar} contains 251 scene images that are labeled with text bounding boxes. For fair comparison, we discarded images that contain non-alphanumeric characters or those have less than three characters, following Wang, Babenko, and Belongie \\cite{wang2011end}. The filtered dataset contains 867 cropped images. Lexicons comprise of a 50-word lexicon defined by Wang et al. \\cite{wang2011end} and a ``full lexicon\". The latter lexicon combines all lexicon words.\n\n\\textbf{ICDAR 2013 (IC13)} \\cite{karatzas2013icdar} inherits most of its samples from IC03. It contains 1015 cropped text images. No lexicon is associated with this dataset.\n\n\\textbf{SVT-Perspective (SVT-P)} \\cite{quy2013recognizing} contains 645 cropped images for testing. Images are selected from side-view angle snapshots in Google Street View. Therefore, most images are perspective distorted. Each image is associated with a 50-word lexicon and a full lexicon.\n\n\\textbf{CUTE80} \\cite{risnumawan2014robust} contains 80 high-resolution images taken in natural scenes. It was specifically collected for evaluating the performance of curved text recognition. It contains 288 cropped natural images for testing. No lexicon is associated with this dataset.\n\n\\textbf{ICDAR 2015 (IC15)} \\cite{karatzas2015icdar} contains 2077 cropped images including more than 200 irregular text. No lexicon is associated with this dataset.\n\n\\subsection{Implementation Details}\n\\textbf{Network: }Details about the MORN and the ASRN of MORAN are given in Table\\ref{table:The architecture of MORN} and Table\\ref{table:The architecture of ASRN} respectively. The number of hidden units of GRU in the decoder is $256$. The ASRN outputs 37 classes, including 26 letters, 10 digits and a symbol standing for $``EOS\"$.\n\n\\textbf{Training Model: }As stated in Section  \\ref{section:curriculum-training}, the training of the MORAN is guided by a curriculum learning strategy. The training data consists of 8-million synthetic images released by Jaderberg et al. \\cite{jaderberg2014synthetic} and 6-million synthetic images released by Gupta et al. \\cite{gupta2016synthetic}. No extra data is used. We do not use any geometric-level or pixel-level labels in our experiments. Without any fine-tuning for each specific dataset, the model is trained using only synthetic text. With ADADELTA \\cite{zeiler2012adadelta} optimization method, we set learning rate to $1.0$ at the beginning and decreased it to $0.01$ in the third stage of the curriculum learning strategy. Following the similar settings in \\cite{liu2016star}, we found that a decreased learning rate contributes to better convergence. The batch size was set to 64. We trained the model for 600,000, 20,000 and 300,000 iterations respectively in three stages of the curriculum learning strategy. The training totally consumed 30 hours.\n\n\\textbf{Implementation: }We implemented our method under the framework of PyTorch \\cite{pytorch}. CUDA 8.0 and CuDNN v7 backends are used in our experiments so our model is GPU-accelerated. All the images are resized to $32\\times 100$. With an NVIDIA GTX-1080Ti GPU, the MORAN takes 10.4ms to recognize an image containing five characters in lexicon-free mode.\n\n\\subsection{Performance of the MORAN}\n\nWe used a max-pooling layer at the top of the MORN. To evaluate the effectiveness of this technique, a comparison of pooling layers with different configurations is shown in Table \\ref{table:comparison-of-pooling}. The accuracy is the highest when we use a max-pooling layer with a kernel size of 2 and stride of 1.\n\nBefore conducting a comparison with other methods, we list three results with a progressive combination of methods in Table \\ref{table:Performance-of-MORAN}. The MORAN trained in an end-to-end manner already achieves very promising performance. In  curriculum learning, the first experiment is carried out using only an ASRN. Then, a MORN is added to the bottom of the above network to rectify the images. The last result is from the entire MORAN, including the MORN and ASRN trained with the fractional pickup method. The contribution of each part of our method is hence clearly demonstrated. For ICDAR OCR tasks, we report the total edit distance in Table \\ref{table:Performance-of-MORAN(TED)}.\n\n\\begin{table}[h]\n\\centering\n\\caption{Performance of the MORAN (total edit distance). }\n\\label{table:Performance-of-MORAN(TED)}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IC03 & IC13 & IC15 \\\\\n\\cline{2-4}\n\\hline\nEnd-to-end training & 29.1 & 57.7 & 368.8 \\\\\n\\hline\nOnly ASRN & 33.8 & 69.1 & 376.8 \\\\\nMORAN without FP & 22.7 & 45.3 & 345.2 \\\\\nMORAN with FP & \\textbf{19.8} & \\textbf{42.0} & \\textbf{334.0} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{Comparisons with Rectification Methods}\n\\textbf{Affine Transformation}: The results using the affine transformation are provided by Liu et al. \\cite{liu2016star}. For fair comparison, we replace the ASRN by the R-Net proposed by Liu et al. \\cite{liu2016star}. A direct comparison of the results is shown in Table \\ref{table:comparison-stn}. As demonstrated in Fig.\\ref{fig:compare-stn} and described in Section  \\ref{section:Multi-Object Rectification Network}, affine transformation is limited by the geometric constraints of rotation, scaling and translation. However, the distortion of scene text is complicated. The MORAN is more flexible than affine transformation. It is able to predict smooth rectification for images free of geometric constraints.\n\n\\begin{table}[h]\n\\centering\n\\caption{Comparison with STAR-Net. }\n\\label{table:comparison-stn}\n\\begin{tabular}{|p{2.2cm}<{\\centering}|p{0.9cm}<{\\centering}|p{0.6cm}<{\\centering}|p{0.6cm}<{\\centering}| p{0.6cm}<{\\centering}|p{1.1cm}<{\\centering}|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P \\\\\n\\cline{2-6}\n\\hline\nLiu et al. \\cite{liu2016star} & 83.3 & 83.6 & 89.9 & \\textbf{89.1} & 73.5 \\\\\n\\hline\nOurs & \\textbf{87.5} & \\textbf{83.9} & \\textbf{92.5} & \\textbf{89.1} & \\textbf{74.6} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\textbf{RARE} \\cite{shi2016robust}: The results of RARE given by Shi et al. \\cite{shi2016robust} are in the Table \\ref{table:Results on general benchmarks} and Table \\ref{table:Results on irregular text}. We directly compare the network using exactly the same recognition network as that proposed in RARE. The results are shown in Table \\ref{table:comparison-rare}.\n\nThe MORAN has some benefits and drawbacks comparing with RARE. RARE using fiducial points can only capture the overall text shape of an input image, whereas the MORAN can rectify every character in an image. As shown in Fig. \\ref{fig:comparison-other}, all the characters in the image rectified by the MORAN are more normal in appearance than those of RARE. Furthermore, the MORAN without any fiducial points is theoretically able to rectify text of infinite length.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Comparison with RARE. }\n\\label{table:comparison-rare}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\multirow{1}{*}{Method} & IIIT5K & SVT & IC03 & IC13 & SVT-P & CUTE80 \\\\\n\\cline{2-7}\n\\hline\nShi et al. \\cite{shi2016robust} & 81.9 & 81.9 & 90.1 & 88.6 & 71.8 & 59.2 \\\\\n\\hline\nOurs & \\textbf{87.9} & \\textbf{83.9} & \\textbf{92.7} & \\textbf{90.0} & \\textbf{73.2} & \\textbf{72.6} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{figure}[h]\n\\centering\n\\begin{overpic}[width=8.5cm,height=5cm]{picture\/rare-a-wide.jpg}\n\n\\put(70,50){Predict:\\color{red}{stink}\\color{green}{er}}\n\\put(75,30){GT:denver}\n\\put(70,10){Predict:\\color{green}{denver}}\n\n\\end{overpic}\n\n\\caption{Comparison of the MORAN and RARE. All characters are cropped for further comparison. The recognition results are on the right. ``GT\" denotes the ground truth.}\n\n\\label{fig:comparison-other}\n\\end{figure}\n\nThe training of MORAN is more difficult than that of RARE. We thus designed a curriculum learning strategy to enable the stable convergence of the MORAN. In terms of RARE, although it is end-to-end optimized with special initialization, randomly initialized network may result in failure of convergence.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Results on general benchmarks. ``50\" and ``1k\" are lexicon sizes. ``Full\" indicates the combined lexicon of all images in the benchmarks. ``None\" means lexicon-free.}\n\\label{table:Results on general benchmarks}\n\\begin{tabular}{| c | c c c | c c | c c c | c |}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{3}{c|}{IIIT5K} & \\multicolumn{2}{c|}{SVT} &\\multicolumn{3}{c|}{IC03} & IC13 \\\\\n\\cline{2-10}\n & 50 & 1k & None & 50 & None & 50 & Full & None & None \\\\\n\\hline\nAlmaz\\'{a}n et al \\cite{almazan2014word} & 91.2 & 82.1 & - & 89.2 & - & - & - & - & - \\\\\nYao et al. \\cite{yao2014strokelets} & 80.2 & 69.3 & - & 75.9 & - & 88.5 & 80.3 & - & - \\\\\nR.-Serrano et al. \\cite{rodriguez2015label} & 76.1 & 57.4 & - & 70.0 & - & - & - & - & - \\\\\nJaderberg et al. \\cite{jaderberg2014deep} & - & - & - & 86.1 & - & 96.2 & 91.5 & - & - \\\\\nSu and Lu \\cite{su2014accurate} & - & - & - & 83.0 & - & 92.0 & 82.0 & - & - \\\\\nGordo \\cite{gordo2015supervised} & 93.3 & 86.6 & - & 91.8 & - & - & - & - & -  \\\\\nJaderberg et al. \\cite{Jaderberg2015Deep} & 95.5 & 89.6 & - & 93.2 & 71.7 & 97.8 & 97.0 & 89.6 & 81.8 \\\\\nJaderberg et al. \\cite{jaderberg2016reading} & 97.1 & 92.7 & - & 95.4 & 80.7* & \\textbf{98.7} & \\textbf{98.6} & 93.1* & 90.8* \\\\\nShi, Bai, and Yao \\cite{shi2017end} & 97.8 & 95.0 & 81.2 & \\textbf{97.5} & 82.7 & \\textbf{98.7} & 98.0 & 91.9 & 89.6 \\\\\nShi et al. \\cite{shi2016robust} & 96.2 & 93.8 & 81.9 & 95.5 & 81.9 & 98.3 & 96.2 & 90.1 & 88.6 \\\\\nLee and Osindero \\cite{lee2016recursive} & 96.8 & 94.4 & 78.4 & 96.3 & 80.7 & 97.9 & 97.0 & 88.7 & 90.0 \\\\\nLiu et al. \\cite{liu2016star} & 97.7 & 94.5 & 83.3 & 95.5 & 83.6 & 96.9 & 95.3 & 89.9 & 89.1 \\\\\nYang et al. \\cite{yang2017learning} & 97.8 & 96.1 & - & 95.2 & - & 97.7 & - & - & -\\\\\nYin et al. \\cite{yin2017scene} & 98.7 & 96.1 & 78.2 & 95.1 & 72.5 & 97.6 & 96.5 & 81.1 & 81.4 \\\\\nCheng et al. \\cite{cheng2017focusing} & 98.9 & 96.8 & 83.7 & 95.7 & 82.2 & 98.5 & 96.7 & 91.5 & 89.4 \\\\\nCheng et al. \\cite{cheng2017arbitrarily} & \\textbf{99.6} & \\textbf{98.1} & 87.0 & 96.0 & 82.8 & 98.5 & 97.1 & 91.5 & - \\\\\n\\hline\nOurs & 97.9 & 96.2 & \\textbf{91.2} & 96.6 & \\textbf{88.3} & \\textbf{98.7} & 97.8 & \\textbf{95.0} & \\textbf{92.4} \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\\subsection{Results on General Benchmarks}\nThe MORAN was evaluated on general benchmarks in which most of the testing samples are regular text and a small part of them are irregular text. The MORAN was compared with 16 methods and the results are shown in Table \\ref{table:Results on general benchmarks}.\n\nIn Table \\ref{table:Results on general benchmarks}, the MORAN outperforms all current state-of-the-art methods in lexicon-free mode. As Jaderberg \\cite{jaderberg2016reading} treated each word as a category and the model cannot predict out-of-vocabulary words, we highlight these results by adding an asterisk. FAN \\cite{cheng2017focusing} trained with pixel-level supervision is also beyond the scope of consideration. We hence compare the MORAN with the baseline of FAN.\n\n\\begin{table*}[t]\n\\centering\n\\caption{Results on irregular datasets. ``50\" is lexicon sizes. ``Full\" indicates the combined lexicon of all images in the benchmarks. ``None\" means lexicon-free.}\n\\label{table:Results on irregular text}\n\\begin{tabular}{|c|p{1.cm}<{\\centering} p{1.cm}<{\\centering} p{1.cm}<{\\centering} |p{1.5cm}<{\\centering}|p{1.5cm}<{\\centering}|}\n\\hline\n\\multirow{2}{*}{Method} & \\multicolumn{3}{c|}{SVT-Perspective} & CUTE80 & IC15 \\\\\n\\cline{2-6}\n & 50 & Full & None & None & None \\\\\n\\hline\nABBYY et al. \\cite{wang2011end} & 40.5 & 26.1 & - & - & - \\\\\nMishra et al. \\cite{mishra2012scene} & 45.7 & 24.7 & - & - & - \\\\\nWang et al. \\cite{wang2012end} & 40.2 & 32.4 & - & - & - \\\\\nPhan et al. \\cite{quy2013recognizing} & 75.6 & 67.0 & - & - & - \\\\\nShi et al. \\cite{shi2016robust} & 91.2 & 77.4 & 71.8 & 59.2 & - \\\\\nYang et al. \\cite{yang2017learning} & 93.0 & 80.2 & 75.8 & 69.3 & - \\\\\nLiu et al. \\cite{liu2016star} & \\textbf{94.3} & 83.6 & 73.5 & - & - \\\\\nCheng et al. \\cite{cheng2017focusing} & 92.6 & 81.6 & 71.5 & 63.9 & 66.2 \\\\\nCheng et al. \\cite{cheng2017arbitrarily} & 94.0 & 83.7 & 73.0 & 76.8 & 68.2 \\\\\n\\hline\nOurs & \\textbf{94.3} & \\textbf{86.7} & \\textbf{76.1} & \\textbf{77.4} & \\textbf{68.8}\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\\subsection{Results on Irregular Text}\n\n\\begin{figure}[t]\n\\rule{8.25cm}{0.05em}\n\\\\\n\\\\\n\\\\\n\\centering\n\\begin{overpic}[width=8.5cm,height=8cm]{picture\/irregular-wide.jpg}\n\n\\put(5,100){Input Image}\n\\put(35,100){Rectified Images}\n\\put(67,101){Ground Truth}\n\\put(70,97){Prediction}\n\\put(0,94){-----------------------------------------------------------------}\n\n\\put(75,90){west}\n\\put(75,84){\\color{blue}{west}}\n\\put(70,77){---------------}\n\n\\put(75,71){united}\n\\put(75,64){\\color{blue}{united}}\n\\put(70,60){---------------}\n\n\n\\put(75,55){arsenal}\n\\put(75,50){\\color{blue}{arsenal}}\n\\put(70,45){---------------}\n\n\n\\put(75,40){football}\n\\put(75,35){\\color{blue}{football}}\n\\put(70,30){---------------}\n\n\n\\put(72,26){manchester}\n\\put(72,20){\\color{blue}{m}\\color{red}{essageid}}\n\\put(70,15){---------------}\n\n\n\\put(71,10){briogestone}\n\\put(71,4){\\color{red}{contracers}}\n\n\\put(0,-2){-----------------------------------------------------------------}\n\\end{overpic}\n\n\\caption{Effects of different curve angles of scene text. The first four rows are text with small curve angles and the last two rows are text with large curve angles. The MORAN can rectify irregular text with small curve angles.}\n\n\\label{fig:6-irregular-samples}\n\\end{figure}\n\nThe MORAN was also evaluated on irregular text datasets to reveal the contribution of the MORN. The results on SVT-Perspective, CUTE80 and IC15 are shown in Table \\ref{table:Results on irregular text}. The MORAN is still the best of all methods.\n\nFor the SVT-Perspective dataset, many samples are low-resolution and perspective. The result of the MORAN with 50-word lexicon is the same as that of the method of Liu et al. \\cite{liu2016star}. However, the MORAN outperforms all methods in the setting without any lexicon.\n\nIn addition to perspective text, the MORAN is able to recognize curved text. Some examples are demonstrated in Fig. \\ref{fig:6-irregular-samples}. The MORAN is able to rectify most curved text in CUTE80 and correctly recognize them. It is hence adequately robust to rectify text with small curve angle.\n\n\\subsection{Limitation of the MORAN}\n\nFor fair comparisons and good repeatability, we chose the widely used training datasets, which contain only horizontal synthetic text. Therefore, because of complicated background, the MORAN will fail when the curve angle is too large. Such cases are given in the last two rows of Fig. \\ref{fig:6-irregular-samples}. MORAN mistakenly regards the complicated background as foreground. However, such samples are rare in training datasets.\n\nFurthermore, with the existing training datasets and without any data augmentation, the MORAN focuses more on horizontal irregular text. Note that there are many vertical text in IC15. However, the MORAN is not designed for vertical text. Our method was proposed for the complicated deformation of text within a cropped horizontal rectangle.\n\nThe experiments above are all based on cropped text recognition. A MORAN without a text detector is not an end-to-end scene text recognition system. Actually, in more application scenarios, irregular and multi-oriented text are challenging both for detection and recognition, which have attracted great interest. For instance, Liu et al. \\cite{yuliang2017detecting} and Ch'ng et al. \\cite{CK2017} released complicated datasets. Sain et al. \\cite{sain2018multi} and He et al. \\cite{he2018multi} proposed methods to improve the performance of multi-oriented text detection. Therefore, scene text recognition still remains a challenging problem waiting for solutions.\n\n\\section{Conclusion}\nIn this paper, we presented a multi-object rectified attention network (MORAN) for scene text recognition. The proposed framework involves two stages: rectification and recognition. First, a multi-object rectification network, which is free of geometric constraints and flexible enough to handle complicated deformations, was proposed to transform an image containing irregular text into a more readable one. The rectified patterns decrease the difficulty of recognition. Then, an attention-based sequence recognition network was designed to recognize the rectified image and outputs the characters in sequence. Moreover, a fractional pickup method was proposed to expand the visual field of the attention-based decoder. The attention-based decoder thus obtains more context information and gains robustness. To efficiently train the network, we designed a curriculum learning strategy to respectively strengthen each sub-network. The proposed MORAN is trained in a weak-supervised way, which requires only images and the corresponding text labels. Experiments on both regular and irregular datasets, including IIIT5K, SVT, ICDAR2003, ICDAR2013, ICDAR2015, SVT-Perspective and CUTE80, demonstrate the outstanding performance of the MORAN.\n\nIn future, it is worth extending this method to deal with arbitrary-oriented text recognition, which is more challenging due to the wide variety of text and background. Moreover, the improvements in end-to-end text recognition performance come not just from the recognition model, but also from detection model. Therefore, finding a proper and effective way to combine the MORAN with a scene text detector is also a direction worth of study.\n\n\\section*{Acknowledgement}\nThis research was supported by the National Key R\\&D Program of China (Grant No.: 2016YFB1001405), GD-NSF (Grant No.: 2017A030312006), NSFC (Grant No.: 61472144, 61673182), GDSTP (Grant No.: 2015B010101004, 2015B010130003, 2017A030312006), GZSTP (Grant No.: 201607010227).\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Sample structures and low field transport data}\n\\begin{figure*}[h!]\n\\includegraphics[width=16cm] {FigS1.png}\n\\caption{(color online) (a) A schematic drawing of the sample structure. The samples studied here are from the same growth in Ref.\\cite{Oh_AM}, and their thicknesses are also characterized therein. A 20-QL In$_2$Se$_3$ \/15-QL (Sb$_0.65$In$_0.35$)$_2$Te$_3$ layer was first grown on a sapphire (Al$_2$O$_3$) substrate as a buffer layer to match the lattice constant of Sb$_2$Te$_3$. This buffer layer helps to reduce the defect density in the Sb$_2$Te$_3$ layer and allows tuning the carriers from p-type to n-type through titanium doping. Then, the Sb$_2$Te$_3$ layer is deposited and capped \\textit{in situ} with another 15-QL (Sb$_0.65$In$_0.35$)$_2$Te$_3$ layer to protect it from aging and maintain a symmetric environment between its top and bottom surfaces. (b) Carrier concentrations in the studied samples. The carrier densities in the samples are characterized with magneto-transport at low temperature with the Van der Pauw geometry. The carrier types are different in these two samples due to different amounts of Titanium dopants. From the sign of the slope in the $R_{Hall}$ vs $B$ plot, the 8-QL sample is determined to be p-doped, and the 10-QL sample is n-doped. The extracted carrier densities from the slope of the data give $n_e= 2.2 \\times 10^{11}$ cm$^{-2}$ and $n_h=3.1 \\times 10^{11}$ cm$^{-2}$ for the 10-QL and 8-QL samples, respectively. }\n\\end{figure*}\n\n\\newpage\n\\section{Additional data on the 8-QL sample}\n\\begin{figure*}[h!]\n\\includegraphics[width=16cm] {FigS2.png}\n\\caption{(color online) Normalized magneto-infrared spectra of the 8-QL sample for (a)(c) transmission and (b) reflection measurements outside and inside the substrate Reststrahlen band, respectively. All the modes are labeled with a Latin or Greek letter. The blue dash lines are guides to the eye showing the mode evolution in magnetic field. Compared to the 10-QL sample, the A mode is not observable here, probably due to the broad linewidth. All spectra are shifted for clarity.}\n\\end{figure*}\n\n\\section{Symmetrization and anti-symmetrization of the surface state dispersion}\nThe conduction surface state (CSS) and valence surface state (VSS) can be written generally as \\cite{eh_Theory1}:\n\\begin{align*}\n    E_{CSS}=E_0+Ck^2+\\sqrt{A^2k^2+(M-Bk^2)^2},\\\\\n    E_{VSS}=E_0+Ck^2-\\sqrt{A^2k^2+(M-Bk^2)^2}.\n\\end{align*}\nThe first term is the overall offset $E_0$, and the second term is the band asymmetry term described by $C$. The last term is the dominant component of the SS, which is described by the linear band parameter $A$, the Dirac mass $M$, and band inversion parameters $B$ and exhibits a electron-hole (e-h) symmetry. We can arrive at the e-h symmetry part $E_{sym}$ and e-h asymmetry part $E_{asy}$ by\n\\begin{align*}\n    &E_{sym}=(E_{CSS}-E_{VSS})\/2=\\sqrt{A^2k^2+(M-Bk^2)^2},\\\\\n    &E_{asy}=(E_{CSS}+E_{VSS})\/2=Ck^2+E_0.\n\\end{align*}\nFrom here, it is evident to see that the anti-symmetrized part leads to the band asymmetry term $Ck^2$ and an offset $E_0$, and the symmetrized part to the dominant dispersion also with an offset from the Dirac mass $M$. For clear comparisons between the dispersions of different thicknesses, we shift all the $E_{Sym}$ and $E_{Asy}$ to the same origin in Fig. 4(c-d) in the main text. \n\nFinally, we note that when the gap $M$ is sufficiently large, we can Taylor expand $E_{sym}$:\n\\begin{align*}\n    E_{sym}=\\sqrt{A^2k^2+(M-Bk^2)^2}=M+\\frac{(A^2-2M B)k^2}{2M}+\\dots,\n\\end{align*}\n which explains the quadratic behavior in the ultrathin film limit.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRandom matrix theory (RMT)~\\cite{RMT} accurately describes eigenvalue\ncorrelations in complex systems.  More precisely, if we consider a\nquantum system governed by a Hamiltonian $H$ whose classical\ncounterpart is chaotic, then the statistical properties of the\neigenvalue spectrum of $H$ can be modeled by an ensemble of matrices\nwith random entries (distributed according to some statistical weight)\nand with the same global symmetries as $H$. This description is\ninsensitive to the details of the interaction and predicts universal\nfeatures that are unveiled when different spectra are rescaled\n(unfolded) to the same mean density.\n\nAmong the many applications of RMT in mathematics and in physics, a\nparticularly interesting one is relevant for the description of the\nspectrum of the Dirac operator in quantum chromodynamics.  QCD\nin the $\\varepsilon$-regime can be described by a chiral RMT with the\nsame chiral and flavor symmetries as QCD~\\cite{chRMT}.  This approach\ncan also be extended to non-vanishing temperature and\/or chemical\npotential and has been confirmed in many numerical studies.  In this\nformulation, the anti-Hermitian massless Dirac operator $D=\\gamma_\\mu\n\\left(\\partial_\\mu + i A_\\mu \\right)$ is described in terms of a\nmatrix with an off-diagonal block structure,\n\\begin{equation}\n  \\label{dmatrix}\n  D\\to \\left( \n    \\begin{tabular}{cc}\n      0 & $i W$ \\\\\n      $i W^\\dagger$ & 0 \\\\\n    \\end{tabular}\n  \\right) \\: , \n\\end{equation}\nwhere $W$ is a complex $(n+\\nu)\\times n$ matrix and $\\nu$ plays the\nrole of the topological charge.\n\nDepending on the color gauge group $G$ and on the fermion field\nrepresentation, the Dirac operator may also be invariant under some\ndiscrete antiunitary symmetries, leading to the following symmetry\nclasses~\\cite{Verbaarschot:1994qf}:\n\\begin{enumerate}\n\\item For $G=\\text{SU}(2)$ and fermions in the fundamental\n  representation, the pseudo-real nature of the group generators\n  allows us to recast the Dirac operator in a form with real matrix\n  entries. The corresponding matrix ensemble is the chiral orthogonal\n  ensemble (chOE) with Dyson index $\\beta_D=1$.\n\\item For $G=\\text{SU}(N_C)$ with $N_C \\ge 3$ and fermions in the\n  fundamental representation, the Dirac operator generically has\n  complex entries.  The appropriate matrix ensemble is the chiral\n  unitary ensemble (chUE) with Dyson index $\\beta_D=2$.\n\\item For gauge group $G=\\text{SU}(N_C)$ and fermions in the adjoint\n  representation, the generators are antisymmetric matrices with\n  imaginary entries, and the Dirac operator can be written as a matrix\n  of real quaternions. The associated matrix ensemble is the chiral\n  symplectic ensemble (chSE) with Dyson index $\\beta_D=4$.\n\\end{enumerate}\nThe behavior of the universal quantities depends on the symmetry\nclasses listed above.  In particular, the probability density $P(s)$\nfor the spacing $s$ of adjacent unfolded levels can be computed\nexactly and is well approximated by the Wigner surmise,\n\\begin{align}\n  \\label{eq:Wigner}\n  P(s) = a\\,s^{\\beta_D}e^{-bs^2}\\quad\\text{with}\\quad\n  a=2\\,\\frac{\\Gamma^{\\beta_D+1} \\left( \\beta_D\/2 +1 \\right)}\n  {\\Gamma^{\\beta_D+2} \\left( (\\beta_D+1)\/2 \\right)} \\:,\\quad\n  b= \\frac{\\Gamma^2\\left( \\beta_D\/2+1 \\right)}\n    {\\Gamma^2 \\left( (\\beta_D +1)\/2 \\right)}\\:.\n\\end{align}\nFor quantum systems whose classical analog is integrable, $P(s)$ is\ngiven by the result for a Poisson process, $P(s)=e^{-s}$.\n\nThe massless staggered Dirac operator on a lattice in $d$ dimensions\nwith lattice spacing $a$,\n\\begin{equation}\n  \\label{eq:DKS}\n  (D_\\text{KS})_{x,y} = \\frac{1}{2a} \\sum_{\\mu=1}^{d}\n  (-1)^{\\sum\\limits_{\\nu<\\mu}\\!\\! x_\\nu} \\left[ \\delta_{x+\\hat\\mu,y} \n    U_\\mu^\\dagger(x) - \\delta_{x-\\hat\\mu,y} U_\\mu(x-\\hat\\mu) \\right] \\: , \n\\end{equation}\nwhich is widely used in numerical simulations, exhibits a peculiar\nfeature: For gauge group SU(2) and fundamental fermions, its\nantiunitary symmetry is that of the chSE \\cite{Halasz:1995vd} instead\nof the chOE symmetry of the continuum operator.\\footnote{A similar\n  situation occurs for adjoint fermions: The staggered Dirac operator\n  has chOE symmetry in this case, as opposed to the chSE symmetry of\n  the continuum operator.}  This discrepancy is due to the replacement\nof the $\\gamma$-matrices by the staggered phases in $D_\\text{KS}$.\nFig.~\\ref{fundsmallbetafig} confirms the expectation using $P(s)$ as\nan example.\n\n\\begin{figure}[-t]\n  \\includegraphics[width=.33\\textwidth]{fundL10beta3}\n  \\includegraphics[width=.33\\textwidth]{fundL14beta3}\n  \\includegraphics[width=.33\\textwidth]{fundL16beta3}\n  \\caption{The distribution of the unfolded level spacing $s$ obtained\n    for different lattice sizes from the SU(2) staggered Dirac\n    operator with fundamental fermions and gauge action parameter\n    $\\beta=4\/g^2=3.0$ (histograms) is consistent with the chSE, which\n    is not the symmetry of the continuum Dirac operator.}\n  \\label{fundsmallbetafig}\n\\end{figure}\n\nA transition of the symmetry properties of the staggered Dirac\noperator from chSE to chOE is expected in the continuum limit.  A\nfirst indication of such a transition has been reported in\nRef.~\\cite{Follana:2006zz}.  We plan to study this transition in more\ndetail, but here we first consider a numerically cheaper case, namely\nthe free limit.  This limit is approached by increasing $\\beta$ at\nfixed (or mildly varying) lattice size, i.e., the physical volume is\nshrinking to zero.  From the RMT point of view, this limit is\ninteresting since it might result in a transition to Poisson behavior\nin, e.g., $P(s)$.  We shall see that the situation is actually a bit\nmore complicated.\n\n\n\\section{The Dirac spectrum of vacuum configurations and the influence\n  of Polyakov loops}\n\\label{sec:vac}\n\nIn this section we present a short theoretical interlude in\npreparation for our numerical results.  Let us consider a particular\ngauge configuration.  For reasons that will become clear below, we now\nconstruct a corresponding vacuum configuration (i.e., a configuration\nwith all plaquettes equal to unity) that is built from uniform links\nin an Abelian subgroup of SU(2) which reproduce the average traced\nPolyakov loops $P_\\mu$ (for all directions $\\mu$) of the configuration\nunder consideration.  The eigenvalue spectrum of the staggered Dirac\noperator \\eqref{eq:DKS} can be computed analytically for such a vacuum\nconfiguration.  If the lattice extent in the $\\mu$-direction is\n$L_\\mu$, we obtain\n\\begin{equation}\n  \\label{clustersandpolyakovloops}\n  \\lambda=\\pm i \\sqrt{\\sum_{\\mu=1}^d \\sin^2 \\left[ \\frac{2 \\pi}{L_\\mu}\n      \\left( k_\\mu + c_\\mu + \\frac{\\arccos P_\\mu}{2 \\pi} \\right)\\right] }\n  \\quad \\text{with }\\: k_\\mu\\in\\mathbb N\\:,\\; 0 \\le k_\\mu < \\frac{L_\\mu}2\\:,\n\\end{equation}\nwhere $c_\\mu = 0$ $\\left(c_\\mu= \\frac{1}{2}\\right)$ for (anti-)\nperiodic boundary conditions (b.c.s) of the Dirac operator in\ndirection $\\mu$.  In the free limit, i.e., for $\\beta \\to \\infty$, the\nPolyakov loops take values in the center $\\mathbb Z_2$ of SU(2), i.e.,\n$P_\\mu=\\pm1$.  In this case it is clear from\nEq.~\\eqref{clustersandpolyakovloops} that changing $P_\\mu$ from $+1$\nto $-1$, or vice versa, is equivalent to switching between periodic\nand antiperiodic b.c.s in that direction.  In the following we always\nuse (anti-) periodic b.c.s for $\\mu=1$, 2, 3 ($\\mu=4$).  Close to the\nfree limit, i.e., for large values of $\\beta$, the distribution of\n$P_\\mu$ is peaked at $\\pm1$.  The eigenvalues predicted by\nEq.~\\eqref{clustersandpolyakovloops} are degenerate, see below.\n\n\n\\section{Numerical results for the eigenvalue spectrum close to the\n  free limit}\n\n\\begin{figure}[-t]\n  \\includegraphics[width=.33\\textwidth]{polyakov1}\n  \\includegraphics[width=.33\\textwidth]{polyakov2}\n  \\includegraphics[width=.33\\textwidth]{polyakov6}\n  \\includegraphics[width=.33\\textwidth]{polyakov8}\n  \\includegraphics[width=.33\\textwidth]{polyakov18}\n  \\includegraphics[width=.33\\textwidth]{L6_polyakov5}\n  \\caption{Separation of scales in the level spacings close to the\n    free limit.  The eigenvalues obtained for each configuration\n    (black dots) arrange themselves in clusters of eight.  These\n    clusters are spread about the well-separated plateaux\n    corresponding to the free case (dashed blue lines).\n    Eq.~\\protect\\eqref{clustersandpolyakovloops} yields an accurate\n    prediction for the location of each cluster (solid red\n    lines). The different plateau structures are due to the different\n    signs of $P_\\mu\\approx\\pm1$ in each configuration.  The last plot\n    confirms that the agreement between the data and\n    Eq.~\\protect\\eqref{clustersandpolyakovloops} persists on larger\n    lattices, for which the theoretical formula predicts more\n    clusters.}\n  \\label{threescalesfig}\n\\end{figure}\n\nWhen $\\beta$ is increased to very large values, we observe that the\neigenvalue spectrum of $D_\\text{KS}$ arranges itself as shown in\nFig.~\\ref{threescalesfig}.  Only the eigenvalues with positive\nimaginary part are plotted, and an overall double (Kramers) degeneracy\n\\cite{Hands:1990wc} has been divided out from all of our results.  The\ndashed blue lines, which will be called \\emph{plateaux} in the\nfollowing, correspond to the highly degenerate eigenvalues predicted\nby Eq.~\\eqref{clustersandpolyakovloops} in the free limit, i.e., for\n$P_\\mu=\\pm1$.  The numerically obtained eigenvalues form\n\\emph{clusters} consisting of eight eigenvalues each, and these\nclusters are located close to the plateaux of the free limit.  We\nobserve a clear separation of three energy scales (from largest to\nsmallest),\n\\begin{enumerate}\\itemsep-1mm\n\\item the spacings between the plateaux of the free limit,\n\\item the spacings between adjacent clusters (which, by definition, do\n  not overlap), and\n\\item the spacings between adjacent eigenvalues within a cluster.\n\\end{enumerate}\n\nThe question now arises to what extent the locations of the clusters\nfor a particular configuration can be described by the levels obtained\nfrom Eq.~\\eqref{clustersandpolyakovloops} for the vacuum configuration\nconstructed as described in Sec.~\\ref{sec:vac}.  The answer is given\nby the solid red lines, which correspond to the predictions of\nEq.~\\eqref{clustersandpolyakovloops} using the $P_\\mu$ computed from\nthe configuration under consideration.  These lines are essentially\nhidden by the data points and thus give very good approximations to\nthe cluster locations.  This statement holds for all configurations,\nsome of which are shown in Fig.~\\ref{threescalesfig}.\n\nWe can understand the observation that the clusters contain eight\nnearly degenerate eigenvalues.  A careful analysis shows that the\neigenvalues predicted by Eq.~\\eqref{clustersandpolyakovloops} have a\nmultiplicity of $2^d$. For $d=4$, this predicts an eightfold\ndegeneracy in addition to Kramers' degeneracy.  A small perturbation\nof the vacuum configuration lifts this eightfold degeneracy but not\nthe Kramers degeneracy, which is exact for $D_\\text{KS}$.\n\nWe also remark that in the continuum limit, in which the lattice\nspacing goes to zero at fixed physical volume, the eigenvalues of\n$D_\\text{KS}$ should arrange themselves in multiplets corresponding to\nthe taste degeneracy of staggered fermions.  This effect was observed\nfor SU(3)~\\cite{quadruplets} (quadruplets) and\nSU(2)~\\cite{Follana:2006zz} (doublets) for improved versions of\n$D_\\text{KS}$, but it should not be confused with the effect we are\nstudying here.\n\nOur observations may be related to other recent work~\\cite{chisbconf}\ndiscussing the connection between the spectrum of the Dirac operator\n(which is relevant for chiral symmetry breaking) and the Polya\\-kov\nloop (which is an order parameter for confinement in the quenched\ntheory).\n\n\n\\section{Spectral fluctuations on different scales}\n\nWe now turn to a study of spectral correlations close to the free\nlimit, using again the nearest-neighbor spacing distribution $P(s)$ as\nan example.  To construct $P(s)$, the average spectral density must be\nseparated from the spectral fluctuations by an unfolding procedure.\nBecause of the separation of scales observed above, a uniform\nunfolding of the entire spectral density is not sensible close to the\nfree limit.  Rather, we should consider the spectral fluctuations\nseparately on the three scales we identified.\n\nFirst, we construct $P(s)$ for the level spacings within the clusters\nby unfolding the spectral density only within a given cluster and then\naveraging $P(s)$ over all clusters.  Fig.~\\ref{still_chse} (left)\nshows that $P(s)$ within the clusters continues to agree with the chSE\neven for very large values of $\\beta$.  This is consistent with the\ntheoretical expectation, since the perturbation that lifts the\ndegeneracy of the eigenvalues in each cluster has the same symmetries\nas the full $D_\\text{KS}$ operator, which are those of the chSE.\n\nSecond, to construct $P(s)$ for the spacings between clusters, we\ndefine a cluster by the average of its eight members and unfold the\ndensity of the clusters.  Fig.~\\ref{still_chse} (right) shows that the\nresulting $P(s)$ differs from the chSE.  It also differs from the\nPoisson distribution, but we believe that this is due to the small\nlattice size and to the fact that on a $10^4$ lattice the free\nstaggered operator has many ``accidental'' degeneracies.  These\ndegeneracies can be removed by choosing a lattice with\n$L_\\mu=2\\ell_\\mu$, where the $\\ell_\\mu$ are four different prime\nnumbers.  As an example, we generated quenched configurations close to\nthe free limit ($\\beta=10000$) for a $34\\times38\\times46\\times58$\nlattice, computed the averaged traced Polyakov loops $P_\\mu$ and used\nthese to calculate the ``cluster spectrum'' according to\nEq.~\\eqref{clustersandpolyakovloops}.  The resulting $P(s)$ is shown\nin Fig.~\\ref{poissonlargelatticefig} (left) and now agrees with the\nPoisson distribution.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=.33\\textwidth]{fundL10beta10000}\n  \\hspace*{20mm}\n  \\includegraphics[width=.33\\textwidth]{clusters_L10_beta_10000}\n  \\caption{$P(s)$ for the eigenvalue spacings within the clusters\n    (left) and for the spacings between clusters (right), both for\n    $L^4=10^4$ and $\\beta=10000$.}\n  \\label{still_chse}\n\\end{figure}\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=.33\\textwidth]{incommensurable_sizes_clusters}\n  \\hspace*{20mm}\n  \\includegraphics[width=.33\\textwidth]{incommensurable_sizes_plateaux}\n  \\caption{$P(s)$ for the ``cluster spectrum'' predicted by\n    Eq.~\\protect\\eqref{clustersandpolyakovloops} for a single\n    configuration on a $ 34 \\times 38 \\times 46 \\times 58 $ lattice at\n    $\\beta=10000$ (left) and $P(s)$ for the free Dirac eigenvalues (or\n    plateaux) on the same lattice (right).  Both agree with the\n    Poisson distribution.}\n  \\label{poissonlargelatticefig}\n\\end{figure}\n\nThird, we consider $P(s)$ for the spacings between the free\neigenvalues (or plateaux), which are known analytically\n(Eq.~\\eqref{clustersandpolyakovloops} with $P_\\mu=\\pm1$).  Again it is\nsensible to remove accidental degeneracies by choosing a ``prime\nlattice''.  The result for a $34\\times38\\times46\\times58$ lattice,\nobtained after unfolding the free eigenvalues, is shown in\nFig.~\\ref{poissonlargelatticefig} (right) and agrees with Poisson as\nexpected \\cite{GoesToPoisson}.\n\nAlthough the two plots in Fig.~\\ref{poissonlargelatticefig} look very\nsimilar, it should be noted that they come from data at very different\nscales.  The average spacing between the levels of the free spectrum\nis more than ten times larger than the average spacing between the\nlevels predicted by Eq.~(\\ref{clustersandpolyakovloops}).\n\n\n\\section{Summary and outlook}\n\nWe have investigated the spectrum of the staggered Dirac operator with\nSU(2) gauge fields close to the free limit.  Three different energy\nscales emerge:\n\\begin{enumerate}\n\\item Overall plateau structure: The spectrum arranges itself in\n  clusters of eight eigenvalues each, lying close to the plateaux\n  predicted for the free Dirac operator. The plateau structure only\n  depends on the lattice geometry (i.e., on the $L_\\mu$ and on the\n  b.c.s) and on the signs of the average traced Polyakov loops in the\n  different directions.  (Note that the distribution of the traced\n  Polyakov loops is peaked at $\\pm1$, corresponding to the center\n  elements of SU(2).)\n\\item Plateau-breaking and cluster separation at an intermediate\n  scale: At a finer scale, the spread of the clusters about the\n  plateaux of the free limit is due to the deviations of the $P_\\mu$\n  from $\\pm1$ and can be accurately modeled by\n  Eq.~\\eqref{clustersandpolyakovloops}.\n\\item Eigenvalue splitting within the clusters: The system dynamics\n  removes the degeneracy of the eight eigenvalues belonging to the\n  same cluster.\n\\end{enumerate}\nIn the regime we have studied, these three scales are well separated\nand can be unambiguously disentangled from each other.\n\nThe nearest-neighbor spacing distribution $P(s)$ computed within the\nclusters shows a behavior compatible with the chSE, consistent with\nthe symmetries of the staggered Dirac operator.  For large enough\n``prime lattices'', the spacing distributions between the clusters and\nbetween the plateaux tends to the Poisson distribution.  In the near\nfuture, we will also present a study of the spectrum of the Dirac\noperator for adjoint fermions close to the free limit.  Ultimately, of\ncourse, we would like to obtain a more detailed understanding of the\ncontinuum limit, in which a chSE to chOE (for SU(2) with fundamental\nfermions) or chOE to chSE (for adjoint fermions) transition is\nexpected.\n\n\\acknowledgments\n\nWe thank J.J.M.~Verbaarschot for helpful discussions and acknowledge\nsupport from DFG (FB, SK, TW) and from the Alexander von Humboldt\nFoundation (MP).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Field-aligned downward current sheets}\nThe observations concerning auroral electron fluxes and the related currents are the following:\n\n\\subsection{Upward current region properties}\nDownward electrons\/upward currents occupy an extended spatial interval of  low density and barely structured fluxes. The variation of the perpendicular (to the main field) magnetic field is smooth; it changes about linearly from $-\\delta B_\\perp$ to $+\\delta B_\\perp$ signalling that the spacecraft has crossed a homogeneous broad structureless upward sheet current carried by the as well structureless medium energy electron flux which, in the energy-time spectrum occupies a narrow band of constant energy and small energy spread. \n\n\nAbsence of an ionospheric electron background at (FAST) spacecraft altitude either suggests that the ionosphere does not reach up to those altitudes ($\\sim2000- 3000$ km) which sometimes, in a diffusive model of the ionosphere, is interpreted as presence of a field aligned electric potential which holds the ionospheric electrons down while attracting magnetospheric electrons. The validity of such an assumption can be questioned in terms of tail reconnection as the inflow of reconnection accelerated electrons from the tail does not require such an electric potential field, the origin of which is  difficult to justify over a region of upward current extension while being natural when considering tail reconnection where it simply maps the large reconnection affected interval of the cross tail current down into the ionosphere. The small number of downward electrons does not require any presence of electric fields. The flux consists of nearly mono-energetic auroral electrons. These form a field-aligned beam and are accompanied by observed low frequency Langmuir-wave excitation which allows for the determination of the beam density being roughly $N_\\downarrow\\approx 10^6$ m$^{-3}$ (one electron per cubic centimetre). \n\n\\subsection{Upward topside electrons}\nFigure \\ref{fig3} shows simultaneous upward\/downward FAST measurements of electron fluxes when crossing a very active substorm topside auroral ionosphere. The upward-electron downward-current region behaves  differently. Its spatial extension is narrow. In view of the electron flux it consists of a large number of very closely spaced spikes. The flux in each spike (generally) maximizes at the lowest energies $\\epsilon_e\\lesssim 0.1$ keV.  Electron number densities are high estimated to be around $N_\\uparrow \\sim 10^7$ m$^{-3}$ (ten per cubic centimetre) or higher. The total integrated up and down currents must be similar for perfect closure. This is however not guaranteed for the divergence of currents in the ionosphere perpendicular to the magnetic field, current dissipation, and the high spatial structuring of the downward currents such that it cannot be checked whether the indication of the different downward current sheets all belong to closure of the single upward current. Some uncertainty in comparison remains, which however for our purposes does not matter.\n\nThe important observation is the high local structure of the downward currents, their obvious spatial closeness, and their differences in energies and flux level which is reflected in both the flux fluctuations across the narrow downward current region, and in the high spatial fluctuation of the main-field-perpendicular magnetic component $\\mathbf{b}_\\perp$ {(from here on denoting the magnetic variation $\\delta B_\\perp= b_\\perp$)} which indicates the crossing of many downward current sheets or filaments.  All these downward currents flow parallel while being closely spaced in the direction perpendicular to the main field $\\mathbf{B}_0$.  Electrodynamics requires that they should attract each other and merge. Why is this not happening in the auroral downward current region?\n\nOne might argue that the acceleration of electrons in the ionosphere below observation altitude is probably highly localized, depending on processes in the resistive ionospheric plasma.  Therefore there would be no need for upward escaping electrons to merge laterally. This argument is invalid because they transport current. Lorentz attraction forces the currents to approach each other to form a broad unstructured downward current sheet. This is, however, inhibited by the strong main auroral geomagnetic field $B_0$. The argument that this should also happen in the upward current region fails because the current sheet there is broad by its origin from the tail reconnection site.\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.75\\textwidth,clip=]{fig2.jpg}}\n\\caption{Full sequence of FAST measurements across dow-up-auroral current system on 02-01-1997  \\citep[after][]{treumann2011}. {$(a)$ Magnetic field component $b_\\perp$ transverse to main field $B_0$, $(b)$ electric field fluctuation wave form $\\delta E$, $(c)$ low frequency electric fluctuation spectrum, $(d)$ high frequency electric spectrum, showing emission of auroral kilometric radiation bands $(e)$ electron energy spectrum, $(f)$ electron flux versus pitch-angle, $(g)$ ion energy flux, $(h)$ ion flux versus pitch-angle. The most intriguing part here is the smoothness of the magnetic signature of the upward current in its linear course showing that the upward current is a broad homogeneous current sheet. The downward current region (DCR) flanks the upward current region to both its sides, is comparably narrow in its spatial extent, and exhibits strong current and flux variations. This is seen in the electron flux panels $(e-f)$. Downward fluxes around few keV are relatively smooth indicating a relatively stable tail reconnection over observation time, upward fluxes have maximum at low energy and are highly variable in time and space.}The magnetic field being the integrated response to the spatial flux fluctuations exhibits a much smoother course which is inverse with respect to that of the upward current thus indicating the reversed current direction. Note the low energies of the upward electron fluxes in panel five as well as the clear separation of upward and downward fluxes as seen in the left part.} \\label{fig2}\n\\end{figure*}\n\nSince anti-parallel currents reject each other the transition region between upward and downward currents is quiet. {This is seen in panels $(b - f)$ of Fig. \\ref{fig2} and} is in contrast to our previous investigation where we assumed that reconnection would happen there between parallel kinetic Alfv\\'en waves. The Lorentz force between two equally strong sheet currents $\\mathbf{J}_\\|$ is \n\\begin{equation}\n\\mathbf{J}_\\| \\times \\mathbf{b}_\\perp= -\\nabla_\\perp b_\\perp^{2}\/\\mu_0  \n\\end{equation}\nwhere $\\mathbf{b}_\\perp$ is the magnetic field between the two currents {(in the following we suppress the index $\\perp$ on the magnetic field component $\\mathbf{b}$)}, and  $\\nabla_\\perp$ refers to the gradient in the direction from current sheet to current sheet. The current consists, however, of gyrating electrons whose Lorentz force is the cross product of the azimuthal gyration speed times the very strong stationary field with gradient $\\nabla_c$ taken only over the gyro-radius $r_{ce}=v_{e\\perp}\/\\omega_{ce}$ of the electrons. For a separation of the sheet current exceeding the electron gyroradius and low current density the sheet currents will approach each other only on very long diffusive time scales of no interest. For a thin current sheet only a few gyroradii thick the condition for this time to be long is simply that the electron inertial length exceeds the gyroradius or\n\\begin{equation}\nv_{e\\perp}\/c\\ll\\omega_{ce}\/\\omega_e\n\\end{equation}\nwhich holds under very weak conditions in the topside auroral ionosphere. This implies that downward current sheets separated by say an electron inertial length $\\lambda_e=c\/\\omega_e$ will not merge under no circumstance. They remain separated over the observational spacecraft crossing time scales. It is their secondary magnetic field $\\mathbf{b}$ which will undergo reconnection without affecting the ambient magnetic field which just serves as guide field directed along the current flow. This distinguishes topside reconnection from other guide field mediated reconnection. {One may note, however, that sometimes in simulations when plasmoids form \\citep[cf., e.g.,][and others]{malara1991} parallel currents apparently do not attract each other. This happens, when the Lorentz force between the parallel currents does not overcome the mechanical forces exerted by the massive plasmoids, i.e. forces induces by their inertia and impulse. The Lorentz force is then too weak to push the parallel currents toward each other, an effect which can also be observed in highly turbulent plasmas. Such cases, when the currents remain close enough will, by the mechanism proposed below, be subject to reconnection between the opposing fields of the parallel current, leading to a cascade in the current structure towards smaller scales and to local reconnection as a main dissipation process of magnetic and turbulent energy.}\n\n\n\\subsection{Kinetic (shear) Alfv\\'en waves}\nIn the complementary wave picture of field-aligned currents in the auroral region, the current is carried by (kinetic) Alfv\\'en waves in the frequency range well below the local ion-cyclotron frequency. In addition, a large number of low-frequency electromagnetic waves are known to be present there \\citep{labelle2002}. We are in the downward current region with highly sheared upward particle flow along the magnetic field consisting of moderately fast electrons and much slower ions. Such flows are capable of generating Alfv\\'en waves \\citep{hasegawa1982} on perpendicular scales of the ion inertial length $\\lambda_i=c\/\\omega_i=\\lambda_e\\sqrt{m_i\/m_e}$ and below and long wavelength parallel to the ambient field. For the current-carrying electrons such waves are about stationary magnetic structures. \n\nThese Alfv\\'en waves cannot be body waves like in the solar wind \\citep{goldstein2005,narita2020} because they are strictly limited to the narrow field-aligned current sheets. Since they propagate in the strong auroral geomagnetic field, they are rather different from the usual kind of kinetic Alfv\\'en waves which one refers to in solar wind turbulence \\citep{goldstein2005}, where the magnetic field ist very weak and the turbulence is dominated by the mechanics of the flow \\citep{maiorano2020}. There the ion-temperature plays an important role imposing kinetic effects on the wave. \n\nUnder auroral conditions, in particular close to the ionosphere, the magnetic field is so strong that thermal ion effects on the wave are barely important. Their mass effect enters the Alfv\\'en speed. Instead, however, under those conditions electron inertia on scales $\\lambda_i\\sim\\Delta\\gtrsim\\lambda_e= c\/\\omega_e$ below the ion scale comes into play. For sufficiently narrow field-aligned current sheets of width the order of inertial scales, the field does not allow the electrons to leave their flux tube unless they have large perpendicular moment. Field-aligned electrons remain inside their gyration flux tube, and the currents cannot react to merge with neighbouring parallel current sheets. The Lorentz force on the field-aligned current in the magnetic field of its neighbour is not strong enough to move the currents. In this case the kinetic Alfv\\'en waves transporting the currents in pulses become inertially dominated with dispersion relation\n\\begin{equation}\n\\omega^2= \\frac{k_\\|^2V_A^2\\big(1+k_\\perp^2\\rho_i^2\\big)}{1+k_\\perp^2\\lambda_e^2}\n\\end{equation}\nwhere $\\rho_i$ is some modified ion gyro-radius \\citep[cf., e.g.][]{baumjohann1996} containing kinetic temperature contributions. For the cold ions in the topside auroral ionosphere the term containing $\\rho_i$ in the numerator vanishes. The kinetic Alfv\\'en wave under those conditions becomes an inertial or shear wave. It propagates at a reduced though still fast speed along the magnetic field, being of very long parallel wavelength. It also propagates slowly perpendicular to the magnetic field at short wavelength $\\lambda_\\perp\\sim\\lambda_e\/2\\pi$. It is, in principle, this wave which carries the current. Thus the current is not stationary on time scales long compared to the inverse frequency $\\Delta t>\\omega^{-1}$ but can be considered stationary for shorter time scales  $\\omega\\Delta t < 1$. \n\nThe above dispersion relation, neglecting the ion contribution in the numerator, gives the well known relations for the parallel and perpendicular energy transport in the shear wave\n\\begin{equation}\n\\frac{\\partial\\omega}{\\partial k_\\|}=\\frac{V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}},\\qquad \\frac{\\partial\\omega}{\\partial k_\\perp}=-\\frac{\\partial\\omega}{\\partial k_\\|}\\frac{k_\\|}{k_\\perp} \\frac{1}{\\big[1+1\/(k_\\perp\\lambda_e)^{2}\\big]}\n\\end{equation}\nEnergy transport in the perpendicular direction is smaller than parallel by the ratio of wave numbers.\n\nRepeating that we are in the downward current upward electron flux region causality requires that the upward electrons carry information from the ionosphere to the magnetosphere. Hence the kinetic Alfv\\'en waves in this region also propagate upward being produced in the topside by the transverse shear on ion-inertial scales below $\\lambda_i$. \n\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.8\\textwidth,clip=]{fig3.jpg}}\n\\caption{Sequence of downward (top)  and upward (bottom)  auroral electron fluxes observed by FAST on 02-07-1997 in the topside auroral region when crossing a substorm aurora \\citep[after][]{treumann2011a}. The sequence distinguishes nicely between the intense  downward electron fluxes at energies $\\epsilon_e\\sim 10$ keV and downward fluxes at energies $\\epsilon_e<0.1$ keV. Upward fluxes are confined to narrow spatial regions, downward fluxes are distributed over a much wider domain. In the transition regions between both domains one observes flux mixing which indicates that the current systems are not simply two dimension and also that there are many overlapping flux and current sources which the one-domensional path of the spacecraft does not resolve spatially. } \\label{fig3}\n\\end{figure*}\n\n\\section{Reconnection under auroral conditions}\nAssuming stationarity of the field aligned current $\\mathbf{J}=J_\\|\\mathbf{B}_0\/B_0$ in two adjacent but separated parallel sheets and assuming, for simplicity, that the two currents are of equal strength, reconnection will  occur in the central region of separation of the current sheets. (Figure \\ref{fig4} shows a two-cross section schematic of the downward current-field configuration for two closely spaced current sheets.)  Here the two magnetic fields of the field-aligned currents are antiparallel. According to the above discussion, we are in the downward current region \\citep[as in the case of our previous radiation model][]{treumann2011}. In fact an analogue model would apply to the upward current region. The current flows in direction $z$; the direction of $y$ is longitudinal (eastward), $x$ is latitudinal (northward). If the sheet ist extended mostly in $y$ the antiparallel magnetic fields are in $y$ along the sheets. They will touch each other and reconnect between the two sheets thereby forming reconnection X-points with field component $\\pm b_x$ and extended magnetic field free electron exhausts in $x$ and in $y$, which contain the local main-field parallel reconnection electric field, and accelerate electrons along the ambient field. Tjhese exhausts will propagate along the main field together with the wave. Plasmoids might also form in the separation between the sheets perpendicular to the ambient field,  and the presence of the strong ambient field will impose electron gyration and scatter of electrons  causing secondary effects like bursts of field aligned energetic electrons. Moreover, the exhausts will serve as source of radiation and various kinds of electrostatic instabilities (for instance Bernstein modes). \n\n{There are two essential differences between this type of reconnection and ordinary reconnection models. The first is that the ambient field serves as a strong guide field which, as noted, inhibits the adjacent field-aligned current to merge. The second is that initially the set-up lacks the presence of any central current sheet which in conventional models of reconnection is crucial and imposed from the beginning. In topside reconnection such a current flowing along the magnetic field inside the separation region would imply a return current which, however, is absent. Return currents flow through the bottomside ionosphere and close in the upward current region. Nevertheless formally a fictitious return current forms locally and temporarily in the centre of the separator, which can be assumed as distributed over the separating region and belonging to the antiparallel fields $\\pm b$.}\n\n{This current builds up dynamically and locally during the reconnection process itself when the two kinetic Alfv\\'en waves slowly move perpendicular. This is a difficult dynamical problem in that reconnection will set on when the encountering magnetic fields exceed some threshold. Since electrons in this region are magnetized by the strong ambient field, they gyrate but do not take notice of the weak field $b$ of the kinetic Alfv\\'en waves which is transported across the separating region by the perpendicular phase and group speeds of the waves to get into contact and merge.} \n\n{The reconnection process is thus solely between the two waves, primarily not affecting the ambient field and not based on any real central primary current sheet. Observations so far do not resolve the magnetic nor the particle effects of such fictitious return currents though some of the structure seen in the low energy electron fluxes in Fig. \\ref{fig3} could be interpreted as such without proof. In fact, in order to avoid formation of the fictitious return current, which would imply that this current would be equally strong in the gap between the current sheets, reconnection is required over the full length of half a wavelength along $z$. Thus it necessarily generates elongated field-aligned vertical X-lines and electron exhausts in $z$.}\n\n \\subsection{First step}\nAll these effects are of vital interest. However, one particularly interesting question concerns the dissipation produced by this kind of reconnection. It is frequently argued that it leads to sliding of main-field field lines. In order to understand such a mechanism one needs to know the anomalous resistance caused by reconnection. In electrodynamic formulation, reconnection is conventionally dealing solely with the merging and energy transfer of fields. The microscopic mechanism of energy transfer is accounted for in the transport coefficients. Hence the appropriate way of inferring their value is referring to the electromagnetic energy exchange. This leads to the application of Poynting's theorem\n\\begin{equation}\n\\frac{\\partial b^2}{\\partial t}=-\\mu_0\\,\\eta_{an}J_\\|^2-\\nabla_\\perp\\cdot\\big(\\mathbf{E}_\\|\\times\\mathbf{b}\\big)-\\nabla_\\perp\\cdot\\big(\\mathbf{E}_{rec}\\times\\mathbf{B}_0\\big)\n\\end{equation}\nwhere the contribution of the electric field to the left-hand side is neglected as it is relativistically small, and $\\mathbf{b}$ is the magnetic field of the field-aligned current. It allows for a convenient estimate of the anomalous resistivity $\\eta_{an}$ in reconnection without going into any microscopic detail of the mechanism of its generation. The electric field in this expression is along the ambient magnetic field, essentially being the electric field of the kinetic Alfv\\'en wave. Estimates of this parallel field have been provided by \\citet{lysak1996} and were taken as the important agent for accelerating auroral electrons. \n\nThe above expression shows that reconnection in this case is a two-step process. In the first step the parallel field $E_\\|$ along the ambient magnetic guide field sets up reconnection. In the second step the reconnection electric field $E_{rec}$ and exhaust have evolved. The cross-product with the main magnetic field then modifies the dynamics of the exhaust. \n\n\\subsection{Anomalous collision frequency}\nIn this subsection we are not interested in this effect here as it is overwritten once reconnection really sets on but enters in the determination of the perpendicular inflow speed.  It causes it to be different from tailward reconnection. Instead we proceed to an estimate of the anomalous collision frequency.  \n\nThe parallel electric field $E_\\|$ of the kinetic Alfv\\'en wave plays an important role in the first step of the topside reconnection process. Since this field is parallel to the ambient geomagnetic field $\\mathbf{B}_0$, the cross product with the wave magnetic field is responsible for the two current-sheet magnetic field components $\\pm\\mathbf{b}$ to approach each other in the region between the sheets. Hence, referring to this fact, the second term on the right can be expressed through the perpendicular velocity $\\mathbf{V}_\\perp=\\mathbf{E}_\\|\\times\\mathbf{b}\/b^2$, and we have\n\\begin{equation}\n\\nabla_\\perp\\cdot\\big(\\mathbf{E}_\\|\\times\\mathbf{b}\\big)=\\nabla_\\perp\\cdot\\big(\\mathbf{V}_\\perp b^2\\big)\n\\end{equation}\n\nIn order to get some information about the perpendicular velocity $\\mathbf{V}_\\perp$ which according to our coordinate system points to the centre of the region which separates the two current sheets, i.e. along $y$, we refer to the wave picture, noting that these pictures are equivalent: the field-aligned current $J_\\|$ is carried by (upward topside ionospheric) electrons, on the other hand these electrons are transported (or pushed) by the kinetic Alfv\\'en wave. {In fact, of course, $V_\\perp$ is counted from each of the two parallel currents as pointing to the center of the separating sheet. It thus in our water-bag model changes abruptly sign in the center where due to the two antiparallel magnetic fields which collide there a fictitious weak return current of strength $j_\\|\\approx 2b\/\\mu_0\\delta$ arises, with $\\delta$ the fictitious width of this narrow current layer which we do not explicitly consider. The simplest is in our water-bag model to assume that for closely separated parallel current sheets we have essentially \n$\\delta\\to\\alpha\\Delta$, with $\\alpha\\lesssim1$, and a return current distributed over almost the entire separation width. One should also keep in mind that any field-aligned current carried by the Alfv\\'en wave is a current pulse with both $E_\\|$ and $b$ changing direction (oscillating) over half the wavelength. Thus $V_\\perp$ for each current pulse on one ambient field line has same sign over the full wave length while maximizing twice. On using this equivalence the perpendicular velocities $\\pm V_\\perp$ are just the perpendicular phase speeds of the kinetic Alfv\\'en waves on the two adjacent current sheets}\n\\begin{equation}\nV_\\perp\\sim\\frac{\\omega}{k_\\perp}\\approx \\frac{V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|}{k_\\perp}\\ll V_A\n\\end{equation}\n{This velocity apparently diverges for $k_\\perp\\to0$ which, however, is not the case because the kAW is a surface wave being defined only for $k_\\perp\\neq0$ while becoming a body wave for $k_\\perp\\to0$ carrying no current anymore. Its most probable wavenumber is about $k_\\perp\\lambda_e\\sim1$ attributing to the parallel phase and group velocity $\\sim V_A\/\\sqrt{2}$ and a perpendicular group velocity $\\sim-V_A(k_\\|\\lambda_e)\/2^{3\/2}$. However, since $V_\\perp$ indeed transports not only the field but also energy, one may argue that the use of the latter expression would be more appropriate than the phase speed. Since this does not make any big difference for our purposes, we in the following for reasons of simplicity understand $V_\\perp$ as phase speed. For more precise expressions one may replace it in the following with the perpendicular group speed}. \n\nThe velocity $V_\\perp$ is small because $k_\\|\\ll k_\\perp$, i.e. the kinetic Alfv\\'en wave is long-wavelength parallel to the ambient field but of short perpendicular wavelength, a very well-known property. Moreover, $V_\\perp(z)$ may vary along the ambient field but, in the frame of the wave, which corresponds to a water-bag model, is constant in the perpendicular direction. Hence, of the above vector product just remains the variation of the magnetic field $b(x)$ over the distance between the two current sheets. This insight enables us to rewrite Poynting's equation as\n\\begin{equation}\n\\frac{\\partial b^2}{\\partial t}\\approx -\\mu_0\\,\\eta_{an}J_\\|^2-V_\\perp\\frac{\\partial b^2}{\\partial x}\n\\end{equation}\nwhich, assuming a stationary state, enables us to estimate the anomalous resistivity of stationary reconnection (in the wave frame) where the inflow of magnetic energy attributed by the current, i.e. the field-aligned electron flux whose origin is found in reconnection in the magnetotail, is balanced by anomalous energy transfer to the plasma in the region separating the two current sheets. Putting the left-hand side to zero we thus find that in this kind of topside reconnection the anomalous resistivity is bound from above as\n\\begin{equation}\n\\eta_{an}\\lesssim \\frac{4V_A}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|}{k_\\perp}\\frac{b^2}{\\mu_0 \\Delta J_\\|^2}{\\approx \\frac{2V_Ab^2}{\\mu_0\\Delta J_\\|^2} \\frac{k_\\|^2\\lambda_e^2}{(1+k_\\perp^2\\lambda_e^2)^{3\/2}}}\n\\end{equation}\nwhere $\\Delta$ is the spatial separation of the two field-aligned current sheets, and we have taken into account that each of the two identical current layers contributes a field $b$. {The second part of this expression makes use of the perpendicular group speed. This resistivity is small as $k_\\|^2\\lambda_e^2\/\\Delta$ but finite. It gives rise to a finite diffusion coefficient that can be interpreted as an anomalous diffusivity for the ambient magnetic field in the auroral topside ionosphere, caused by topside reconnection between anti-parallel current sheets in the downward current region. We might note at this occasion that the restriction to the downward current region is motivated by the observation of narrow current sheets in the downward current region. Observations do not suggest that similarly narrow current sheets evolve in the upward current region. If this would be the case, the same arguments would apply there, causing reconnection and a similar anomalous resistivity.}  \n\n\\begin{figure*}[t!]\n\\centerline{\\includegraphics[width=0.75\\textwidth,clip=]{fig4.jpg}}\n\\caption{Schematic of the field configuration between two parallel field aligned flux tubes in the downward current region. \\emph{Left}: Geometry along the ambient field. {Currents are in red. Included would be the (red dashed) fictitious return current which locally would correspond to the antiparallel wave magnetic fields $\\pm b$. This current would be local over the wavelength of the inertial Alfv\\'en wave. In any stationary reconnecting current picture it would be this current whose magnetic field reconnects. However, here this current does not exist in the exhaust. It is completely reconnected and gives rise to the reconnection electric field $E_z$ instead (dashed red) in the exhaust along the main magnetic field.  Electrons are directly accelerated by it along $\\mathbf{B}_0$.} \\emph{Right}: Reconnection geometry with perpendicular velocity $\\mathbf{V}_\\perp$, field free exhaust, reconnection fields $\\pm b_x$ indicated, and $E_z$. {The anomalous collisions caused in the exhaust volume also permit for weak diffusion of the ambient field. This may cause what is believed to be magnetic field diffusion, a  very slow process compared to the wave\/current induced spontaneous reconnection.}} \\label{fig4}\n\\end{figure*}\n\nWhat concerns the spatial separation of the current sheets (see Fig. \\ref{fig3}), the best available observations (FAST) do not resolve any single sheets; it can however be assumed that their scales are the order of or below the ion-inertial length, such that $\\Delta\\lesssim \\lambda_i\\sim$ several to many $\\lambda_e$. This may overestimate the real value but has been accounted for in writing the expression as an upper limit. Determination of the anomalous resistivity thus requires knowledge of the field aligned current density, current sheet separation, and the transverse magnetic field component of the sheet current. We then can estimate the anomalous collision frequency $\\nu_{an}= \\eta_{an}\\epsilon_0\\omega_e^2$ in this kind of reconnection\n\\begin{equation}\n\\nu_{an}\\lesssim \\frac{V_A\/\\alpha^2\\lambda_e}{\\sqrt{1+k_\\perp^2\\lambda_e^2}}\\frac{k_\\|\\Delta}{k_\\perp\\lambda_e }\n\\end{equation}\nwhere we used that $J_\\|\\approx 2b\/\\alpha\\Delta$. Note that $V_A=B_0\/\\sqrt{\\mu_0 m_i N}$ is based on the ambient magnetic field and plasma density. This simple estimate shows that reconnection in this case can, under stationary condition be described as being equivalent to a diffusive process based on the anomalous collision frequency which is provided by the merging of the transverse magnetic fields of the two neighbouring field-aligned current sheets. Since the related diffusivity is felt in the entire region it is remarkable that it could effect also the main ambient guide field. In other words, topside reconnection could become responsible for diffusion of the main magnetic field lines in a locally restricted domain possibly causing effects on a larger scale in the auroral region.   \n\nReal reconnection will not occur between field-aligned current sheets of same strength. Thus the above resistivity respectively the collision frequency must be reduced by another factor proportional to the involved current and field fractions. \n\n\\subsection{Second step: Reconnection electric field}\n{So far we just investigated the energy balance in order to obtain an anomalous collision frequency in this kind of reconnection. Reconnection however manifests itself in X points generating transverse magnetic fields and in addition electric fields. Since there is no primary return current flowing, it cannot be used as input into the two-dimensional reconnection equation for the vector potential $A_z$\n\\begin{equation}\n\\nabla^2 A_z=-\\mu_0j_z(x), \\qquad \\nabla= (\\partial_x,\\partial_y,0), \\qquad j_z(x)= -2\\epsilon b(x)\/\\mu_0\\Delta\n\\end{equation}\nwithout prescribing the built-up of the central current profile $j_z(x)$, which is possible only when assuming that the $b$ is independent of $x$, in which case it provides the usual stationary tearing mode solution \\citep[see, e.g.,][]{schindler1974} rewritten for electrons alone. Under these simplifying restrictions the two components of the reconnected magnetic field including the X point are given by $\\mathbf{b}=(\\partial_yA_z,-\\partial_xA_z,0)$, which to refer to suffices for our qualitative considerations. The a priori assumption of a return current is, however, incorrect. On the topside there may weak local return currents exist filling the separations between the narrow downward current sheets, but the main return current flows in the upward current region and is distributed over a wide domain. Hence just a fraction $\\epsilon$ of return current can flow in the gap, as included in the last expression. The electric field in this case primarily has only one component, which is along the main field and is given by $E_z=-\\partial_t A_z-\\nabla U$ where $U$ is the scalar electrostatic gauge potential which may occur if an inhomogeneity exists or the system is not ideally symmetric. This field adds to the field aligned kinetic Alfv\\'en wave electric field and contributes to electron acceleration. It is the wanted reconnection electric field and can be much larger than the small linear wave electric field. Unfortunately its precise knowledge  requires solution of the equation for the vector potential $A_z$ and some interpretation of the time derivative operator. The latter can be transformed into a spatial derivative $\\partial_t= \\pm\\mathbf{V}_\\perp\\cdot\\nabla$, still requiring the solution $A_z(x,y)$. }\n\n{The important conclusion in the case of topside reconnection is rather different from usual reconnection. It tells that the exhaust is, over half the wavelength of the inertial Alfv\\'en wave free of wave magnetic fields $b$, while being bounded by the reconnected wave fields $\\pm b_x$. The exhaust instead contains the reconnection electric field, by being along the main field, does directly contribute to acceleration respectively deceleration of electrons  (and also ions) along the main magnetic field, one of the most important and still unresolved problems in auroral physics. There acceleration is attributed to a variety of waves, reaching from kinetic Alfv\\'en through whistlers and several electrostatic waves to electron and ion holes. Except for the latter nonlinear structures, all wave electric fields are quite weak, and in addition fluctuate. Acceleration thus becomes a second order process.} \n\n{In case of the topside reconnection, a mesoscale first-order electric field $E_z$ is produced which directly accelerates particles, depending on its direction along the main field. Moreover, the source of the accelerated particles is the gap region between the two current sheets, the so-called exhaust, such that the kinetic Alfv\\'en wave electric field and the reconnection electric field do barely interfere. Hence the full strength of the reconnection exhaust field acts accelerating. One may thus conclude that topside reconnection, if it takes place, will substantially contribute to auroral particle acceleration. \n }\n\n{In order to circumvent the above named difficulty of calculating $A_z$ and to obtain an estimate of the reconnection electric field, we may return to the induction equation in its integral form where the electric field is given by the integral over the surface of the reconnection site\n\\begin{equation}\n\\oint \\mathbf{E}\\cdot d\\mathbf{s}=-\\frac{d\\Phi}{dt}= -\\frac{d}{dt}\\int\\mathbf{b}\\cdot d\\mathbf{F}\n\\end{equation}\nand the right-hand side is the exchange of magnetic flux in the reconnection process within the typical time $dt=\\tau_{rec}$. This time is not necessarily the same as the anomalous collision time. The magnetic flux is given by \n$\\Delta\\Phi\\approx 4\\pi b\\Delta\/k_\\|$. The line integral over the boundary of the reconnection site becomes $\\approx4\\pi E_z\/k_\\|+2\\Delta \\delta E_x$. Under ideally symmetric conditions the second term would vanish because the two contributions of the $x$ integration would cancel out. If some asymmetry is retained then a finite component $\\delta E_x$ arises. Taken these together yields dimensionally (not caring for the signs)\n\\begin{equation}\n4\\pi E_z\/k_\\|+2\\Delta \\delta E_x \\approx 4\\pi b\\Delta\/k_\\|\\tau_{rec}\n\\end{equation}\nNeglecting the small second term on the left then gives a simple order of magnitude estimate of the reconnection electric field\n\\begin{equation}\nE_z\\approx \\frac{b\\Delta}{\\tau_{rec}}\n\\end{equation}\nwhich could have been guessed from the beginning. This contains the reconnection time $\\tau_{rec}$ which  so far is undetermined. It can be taken for instance as the above derived anomalous collision time $\\tau_{an}=\\nu_{an}^{-1}$. Below we derive another characteristic time. Which one has to be chosen, cannot decided from these theoretical order of magnitude  estimates. It is either due to observation or numerical simulations.} \n\nThe small additional term $2\\Delta\\delta E_x =-U$ is a potential field produced by a possibly present asymmetry between the original current sheets or some gradient in the particle density. Such a gradient can be produced, if a substantial part of the electron component in the gap is accelerated away along the main field, causing a dilution of plasma in the exhaust. Being perpendicular to the magnetic fields $\\mathbf{B}_0$ and $\\mathbf{b}$ it leads to weak shear motions and circulation of the electrons inside the gap-exhaust region, which should observationally be detectable. \n\n\n\\subsection{Reconnection time}\nIn the above we have made use of the notion of reconnection time $\\tau_{rec}$. Here we attempt a clarification of this time. Topside reconnection will not be stationary. It should vary on the time scale of the kinetic Alfv\\'en frequency respectively moving together with the latter along the magnetic field. This motion should mainly be upward since causality requires that the wave transports information back upward with the upward moving electrons in the downward current region. It will thus be modulated and lead to quasi-periodic acceleration and generate medium energy electron bursts ejected from the local electron exhaust reconnection region along the sheet current magnetic field. These bursts flow perpendicular to the ambient field, start gyrating and immediately become scattered along the ambient field spiralling mainly upward into the weak ambient field region. Their pitch-angle distribution should obey a well defined downward loss-cone. \n\nWith the above estimate of the anomalous resistivity in this kind of reconnection, we can proceed asking for the typical reconnection time scale. For this purpose we return to Poynting's full theorem and take its variation with respect to the stationary state, indexing the latter with 0 while keeping the slow perpendicular velocity $V_\\perp$ fixed but varying the resistivity. We need to express the parallel current through the resistivity. This can be done via the electric field $E_\\|$ to obtain\n\\begin{equation}\nJ_\\|^2=\\eta^{-2}E_\\|^2=\\eta^{-2}b^2V_\\perp^2\n\\end{equation}\nThis procedure, after some straightforward and simple algebra and rearranging, leads to the following expressions\n\\begin{eqnarray}\n\\frac{d(\\delta b)^2}{dt}\\equiv\\Big(\\frac{\\partial}{\\partial t}+V_\\perp\\nabla_\\perp\\Big)(\\delta b)^2&=&- 2\\mu_0J_{\\|0}^2\\delta\\eta\\\\\n\\delta\\eta&=&-\\frac{V_\\perp}{\\mu_0J^2_{\\|0}\\Delta} (\\delta b)^2\n\\end{eqnarray}\nand we obtain dimensionally for the typical time of reconnection\n\\begin{equation}\n\\tau_{rec}\\sim\\frac{2\\Delta}{V_\\perp}\n\\end{equation}\nThis seems a trivial result, but it tells that reconnection is a process which annihilates the excess magnetic field which is provided by the perpendicular inflow under the condition that we are close to a stationary state. This time can be compared with the times of energy flow in the shear Alfv\\'en wave. Since clearly $V_\\perp\\approx\\partial\\omega\/\\partial k_\\perp$, one obtains\n\\begin{equation}\n\\tau_{rec}\\approx\\frac{2\\Delta}{V_A}\\frac{k_\\perp}{k_\\|}\\frac{\\big(1+k_\\perp^2\\lambda_e^2\\big)^\\frac{3}{2}}{k_\\perp^2\\lambda_e^2}\n\\end{equation}\na time the length of which depends essentially on the spacing of the current sheets. Since $V_A$ is large, there will be a balance between the spacing and the domain of the kinetic Alfv\\'en wave spectrum which allows reconnection to occur in the topside. Let the vertical topside width be $L_z$ and the Alfv\\'en time $\\tau_A=L_z\/V_A$ then we have the condition \n\\begin{equation}\n\\frac{\\tau_{rec}}{\\tau_A}\\approx \\frac{2\\Delta}{L_z}\\frac{k_\\perp}{k_\\|}\\frac{\\big(1+k_\\perp^2\\lambda_e^2\\big)^\\frac{3}{2}}{k_\\perp^2\\lambda_e^2}< 1\n\\end{equation}\nfor reconnection to occur in topside parallel field-aligned current sheets. This essentially is a condition on the spacing $\\Delta$ of the sheets, meaning that \n\\begin{equation}\n\\frac{\\sqrt{32}\\,\\Delta}{L_z}< \\frac{k_\\|}{k_\\perp}=\\frac{\\lambda_\\perp}{\\lambda_\\|}\\ll 1\n\\end{equation}\nAny current sheet separation is strictly limited. Since it must be larger than the upward electron gyro-radius we have $\\Delta>r_{ce}$. Both conditions are easily satisfied.\n\n\n\n\\subsection{Conclusions}\n{In the present letter we propose that reconnection might occur not only in given current sheets but also in the topside ionosphere-magnetosphere auroral transition region  where the main magnetic field is very strong, almost vertical, and directly connects to the tail reconnection region. It serves  as a guide for any particle flow exchange between the topside ionosphere and the tail plasma sheet, exchange between low frequency electromagnetic waves (in our case kinetic Alfv\\'en waves) trapped in flux tubes and the accompanying field-aligned current sheets, and ultimately as an inhibitor for the field-aligned parallel current sheets to merge. This  enables reconnection in the gap between the current sheets between the oppositely directed magnetic field of the sheets respectively the kinetic Alfv\\'en wave magnetic fields.} \n\n{Dealing with reconnection, one is not primarily interested in the change of magnetic topology but in energy transformation from magnetic into kinetic, diffusion of plasma and magnetic field across the reconnection region, generation of electric fields, and ultimately selective particle acceleration as these are the observed effects. }\nThe generality of reconnection is not the best argument. The decades old claim that reconnection converts magnetic energy into mechanical energy is no fundamental insight; in all processes involving reconnection, the main energy is stored in the basic mechanical motion and by no means in the magnetic field. This motion, convection in inhomogeneous media with boundaries, like the magnetotail or the magnetopause, or turbulence necessarily produces currents and transports magnetic fields to let them get into contact. The amount of energy released by reconnection is in all cases just the minor electromagnetic part, a fraction of the mechanical energy. \n\n{Topside reconnection is expected predominantly in the downward current region, which observationally seems to be highly structured, consisting of several adjacent parallel current sheets. Similar  conditions may also occur in the upward current region though no such structuring is obvious from observations. If it exist, then the physics will be similar. We have shown that topside reconnection is possible, generates a elongated field-aligned regions (exhausts) where the fields of parallel current sheets merge, anomalous collisions are generated, energy is exchanged and dissipated, and most important a first order reconnection electric field $E_{rec}$ is produced in the exhaust along the ambient magnetic field but  restricted to the gap region between the current sheets. This field is capable of accelerating electrons along the main field, as is most desired by all auroral physics. Here it comes out as a natural result of topside reconnection.} Topside reconnection generates parallel electron beams, it lifts the escaping electrons in the exhaust into an elevated parallel energy level. These beams then cause a wealth of auroral effects in the environment and when impinging onto the upper ionosphere. Acceleration of electrons by the reconnection electric field leaves behind an electron depleted exhaust mainly containing only an anisotropic electron component whose pitch angle distribution peaks at perpendicular energies.  \n\n{It is instructive to briefly inspect Fig. \\ref{fig3}. It shows the downward (upper panel) and upward (lower panel) electron fluxes. In addition to the temporally\/spatially highly structured fluxes, still obeying the spatial differences between the downward and upward current regions imposed by the tail-source of the downward fluxes, resulting  from  variations in tail-reconnection, or several tail-reconnection sites, one occasionally observes the simultaneous presence of upward and downward fluxes in the downward current region. One particular case it at $t\\approx 60$ s. The upward electron fluxes maximize below $\\sim0.1$ keV. Simultaneously a banded flux of downward electrons with central energy $\\sim 0.3$ keV appears in the upper panel. This event  is indicated as flux mixing. It could also be understood as acceleration of electrons resulting from the local reconnection in the gap between current sheets. }\n\nAside of acceleration, radiation generation may be taken as signature of topside reconnection. Radiation is preferrably generated by the electron cyclotron maser mechanism. It requires low electron densities, strong magnetic fields, and a rather particular particle distribution with excess energy in its component perpendicular to the ambient magnetic field \\citep{sprangle1977,melrose1985}. Such a state in dilute plasmas lacks sufficiently many electrons for re-absorbing the spontaneously emitted radiation while the excited state causes inversion of the absorption coefficient. These conditions allow for the plasma to become an emitter \\citep{twiss1958,schneider1959,gaponov1959} by the electron cyclotron maser mechanism \\citep{wu1979} based on a loss-cone distribution \\citep{louarn1996}. It requires weakly relativistic electrons \\citep[see][for reviews]{melrose1985,treumann2006} and a low density electron background embedded into a strong field. It nicely comes up for the weak auroral kilometric background radiation but fail explaining the intense narrow band observed and drifting emission seen in panel $d$ of Fig. \\ref{fig2}. \n\nTo explain the latter, in earlier work we referred to electron hole formation \\citep{pottelette2005,treumann2011}. Hole models favourably apply to electron depleted exhausts in topside reconnection where densities become low \\citep[see, e.g.][]{treumann2013} and the remaining trapped electron component  maximizes at perpendicularly speeds having large anisotropy. Intense narrow band drifting emissions in the frequency range 300-600 kHz may be a signature of topside reconnection in the strong main auroral field. They were originally attributed to Debye scale electrostatic electron holes \\citep{ergun1998b,pottelette1999} observed by Viking \\citep{deferaudy1987} and  FAST \\citep{carlson1998,ergun1998a,pottelette2005} but are to small-scale for radiation sources. Topside reconnection exhausts instead have dimensions along the magnetic field of half a kinetic Alfv\\'en length and transverse scales of few ion inertial lengths $\\lambda_i$ or $\\sim 100\\lambda_e$. Such scales can host and amplify one or more radiation wave lengths.\n\nOf course, details of this process should be developed both analytically as far as possible, and by numerical simulations. If confirmed, this mechanism would also map to any astrophysical moderately or strongly magnetized object with appropriate modification. \n\nThe present qualitative considerations which we spiced with a few simple estimates based on energy conservation arguments just propose that reconnection in the topside auroral ionosphere is a process which has so far been missed and probably is that mechanism which releases the largest amount of so-called magnetically stored energy available and from the smallest spatial regions. Reconnection in much weaker fields like in turbulence and broad current sheets will be substantially less efficient because of the weakness of the reconnecting magnetic fields. Nevertheless in very large extended systems with reconnection proceeding on the microscales  \\citep{treumann2015} with the total number of reconnection regions very large, the emission measure is large as well, and radiation from reconnection may become a non-negligible signature even in weak fields. However, in very strong fields like those in magnetized planets and magnetized stars (predominantly neutron stars, white dwarfs but also including outer atmospheres of magnetized stars like the sun) reconnection following our argumentation may be more important than so far assumed. \n\n\n\\section*{Acknowledgments}\n This work was part of a brief Visiting Scientist Programme at the International Space Science Institute Bern. RT acknowledges the interest of the ISSI directorate as well as the generous hospitality of the ISSI staff, in particular the assistance of the librarians Andrea Fischer and Irmela Schweitzer, and the Systems Administrator Saliba F. Saliba. We acknowledge discussions with R. Nakamura, and Y. Narita. RT acknowledges the cooperation with R. Pottelette two decades ago  on the data reduction and the radiation and electron hole problems.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCollections of interacting self-motile objects\nfall into the class of systems known as active\nmatter \\cite{1,2},\nwhich can be biological in nature\nsuch as swimming bacteria \\cite{3} or animal herds \\cite{4},\na social system such as pedestrian or traffic flow \\cite{5},\nor a robotic swarm \\cite{6,7}.\nThere are also a wide range of artificial active matter\nsystems such as self-propelled colloidal particles \\cite{8,9,10}.\nStudies of these systems have  generally focused on the case\nwhere the motile objects interact with either a  smooth or\na static substrate; however, the field is now advancing to a point where\nit is possible to ask\nhow such systems behave\nin more complex static or dynamic environments.\n\nOne subclass of active systems\nis a collection of interacting disks that undergo either run-and-tumble \\cite{11,12} or\ndriven diffusive \\cite{13,14,15} motion.\nSuch systems\nhave been shown to exhibit a transition\nfrom a uniform density liquid state\nto a motility-induced phase separated state\nin which the disks form dense clusters surrounded\nby a low density gas phase \\cite{9,10,11,12,13,14,15,16,17,18}.\nRecently it was shown that when phase-separated run-and-tumble disks are\ncoupled to a random pinning substrate, a transition to a uniform density liquid state\noccurs as a function of the maximum force exerted by the substrate \\cite{19}.\nIn other studies of run-and-tumble disks driven over an obstacle array by a dc driving force,\nthe onset of clustering coincides with a drop in the net disk transport since a large cluster\nacts like a rigid object that can only move through the obstacle array with difficulty; in\naddition, it was shown that the disk transport was maximized at an optimal\nactivity level or disk running time \\cite{20}.\nStudies of flocking or swarming disks that obey modified Vicsek models of\nself-propulsion  \\cite{21}\ninteracting with obstacle arrays\nindicate \nthat there is an optimal\nintrinsic noise level at which collective swarming occurs \\cite{22,23},\nand that transitions between swarming and non-swarming states can occur as\na function of increasing substrate disorder \\cite{24}.\nThe dynamics in such swarming models differ from those of\nthe active disk systems, so it is not clear whether the  same behaviors will occur across\nthe two different systems.\n\nA number of studies have already considered\nactive matter such as bacteria or run-and-tumble disks\ninteracting with periodic obstacle arrays \\cite{25} or\nasymmetric arrays  \\cite{26,27,28,29}.\nSelf-ratcheting behavior occurs for the asymmetric arrays\nwhen the combination of broken detailed balance and the substrate asymmetry\nproduces\ndirected or ratcheting motion of the active matter particles\n\\cite{30,31},\nand it is even possible to couple passive particles to the active matter particles\nin such arrays in order to shuttle cargo across the sample \\cite{29}.\nIn the studies described above, the substrate is static, and external driving\nis introduced via fluid flow or chemotactic effects; however, it is also possible\nfor the substrate itself to be dynamic, such as in the case of\ntime dependent optical traps  \\cite{32,33} or a traveling wave substrate.\nTheoretical and experimental studies of colloids in traveling wave potentials\nreveal a rich variety of dynamical phases,\nself-assembly behaviors, and directed transport \\cite{34,35,36,37,38,39,40}.\n\nHere we examine a two-dimensional system of run and tumble active\nmatter disks that can exhibit motility induced phase separation\ninteracting with a periodic quasi-one dimensional (q1D) traveling wave substrate.\nIn the low activity limit, the substrate-free system forms a\nuniform liquid state, while in the presence of a substrate,\nthe disks are readily trapped by the substrate minima\nand swept through the system by the traveling wave.\nAs the activity increases, a partial decoupling transition of the disks and the substrate\noccurs, producing a drop in the net effective transport.  This transition is correlated\nwith the onset of the phase separated state,\nin which the clusters act as large scale composite objects that cannot be transported\nas easily as individual disks by the traveling wave.\nWe also find that the net disk transport is optimized at particular\ntraveling wave speeds, disk run length, and substrate strength.\nIn the phase separated state we observe an interesting effect where\nthe center of mass of each cluster moves in the direction opposite to that in which\nthe traveling wave is moving, and we also find reversals\nto states in which the clusters and the traveling wave move in the same direction.\nThe reversed motion of the clusters arises due to asymmetric growth and shrinking rates\non different sides of the cluster.\nThe appearance of backward motion of the cluster center of mass\nsuggests that certain biological or social active systems can move against biasing\ndrifts by forming large collective objects or swarms.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig1.png}\n\\caption{Schematic of the system.\n  Red spheres represent the active run and tumble disks in a two-dimensional system\n  interacting with a periodic q1D traveling wave potential\n  which is moving in the positive $x$-direction (arrow) with a wave speed of $v_{w}$.\n}\n\\label{fig:1}\n\\end{figure}\n\n\n\\section{Simulation}\nWe model a two-dimensional system of $N$ run and tumble disks\ninteracting with a q1D traveling wave periodic substrate,\nas shown in the schematic in Fig.~\\ref{fig:1} where the substrate\nmoves to the right at a constant velocity $v_{w}$.\nThe dynamics of each disk is governed by the following overdamped equation of motion:\n\\begin{equation}\n\\eta \\frac{d{\\bf r}_i}{dt} = {\\bf F}^{\\rm inter}_i + {\\bf F}^m_i + {\\bf F}^{s}_i ,\n\\end{equation}\nwhere the damping constant is $\\eta = 1.0$.\nThe disk-disk repulsive interaction force ${\\bf F}^{\\rm inter}_i$\nis modeled as a harmonic spring,\n${\\bf F}^{\\rm inter}_i=\\sum_{j\\neq i}^N\\Theta(d_{ij}-2R)k(d_{ij}-2R){\\bf \\hat d}_{ij}$,\nwhere $R=1.0$ is the disk radius, $d_{ij}=|{\\bf r}_i-{\\bf r}_j|$ is the distance between\ndisk centers, ${\\bf \\hat d}_{ij}=({\\bf r}_i-{\\bf r}_j)\/d_{ij}$, and the spring constant\n$k=20.0$\nis large enough to prevent significant disk-disk overlap under the conditions we\nstudy\nyet small enough to permit a computationally efficient time step\nof $\\delta t=0.001$ to be used.\nWe consider a sample of size $L \\times L$ with $L=300$, and describe the disk density in\nterms of the area coverage\n$\\phi = N\\pi R^2\/L^2$.\nThe run and tumble self-propulsion is modeled with a motor force  ${\\bf F}^{m}_i$\nof fixed magnitude $F^m=1.0$ that acts in a randomly chosen direction during\na run time of $\\tilde{t}_{r}$.\nAfter this run time, the motor force instantly reorients into a new\nrandomly chosen direction for the next run time.\nWe take $\\tilde{t}_r$ to be uniformly distributed over the range\n[$t_r,2t_r$], using run times ranging from $t_r=1 \\times 10^3$ to $t_r=3 \\times 10^5$.\nFor convenience we describe \nthe activity in terms of the run length $r_l=F^mt_r\\delta t$,\nwhich is the distance a disk would move during a single run time\nin the absence of a substrate or other disks.\nThe substrate is modeled as a time-dependent\nsinusoidal force $F^s_i(t)=A_{s}\\sin(2\\pi x_i-v_w t)$\nwhere $A_s$ is the substrate strength and $x_i$ is the $x$ position of disk $i$.  We take\na substrate periodicity of $a = 15$ so that the system contains\n$20$ minima.\nThe substrate travels at a constant velocity of $v_{w}$ in the positive $x$-direction.\nWe measure the average drift velocity of the disks in the direction of the traveling wave,\n$\\langle V\\rangle = N^{-1}\\sum^{N}_{i}{\\bf v}_{i}\\cdot {\\hat {\\bf x}}$.\nWe  vary the run length, substrate strength, disk density, and wave speed.\nIn each case   \nwe wait for a fixed time\nof $5 \\times 10^6$ simulation time steps\nbefore taking measurements to avoid any transient effects.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig2.png}\n\\caption{(a) The average velocity per disk $\\langle V\\rangle$\n  vs wave speed $v_{w}$ for a system with\n  $N=13000$ disks, $\\phi  = 0.45376$, and\n  $r_l = 300$ for varied substrate strengths of $A_s=0.5$ to 3.0.\n  The dashed line indicates the limit in which all the disks move at the wave\n  speed, $\\langle V\\rangle = v_{w}$.\n  (b) The corresponding number $\\tilde{C}_L$ of disks that are in a cluster\nvs wave speed. \nThe inset shows the regions of cluster and non-cluster (NC) states\nas a function of $v_{w}$ vs $A_{s}$. \n(c) The number $\\tilde{P}_6$  of sixfold-coordinated\ndisks vs wave speed $v_w$.\n}\n\\label{fig:2}\n\\end{figure}\n\n\\section{Results}\nIn Fig.~\\ref{fig:2}(a) we plot the average velocity per disk\n$\\langle V\\rangle$ versus wave speed $v_{w}$ at different substrate strengths\n$A_{s}$ for a system containing $N=13000$ active disks,\ncorresponding to $\\phi = 0.45376$,\nat $r_l = 300$, a running length at which the substrate-free system\nforms a phase separated state.\nThe number of disks that are in\nthe largest cluster, $\\tilde{C}_L$, serves as an effective measure\nof whether the system is in a phase separated state or not.  We measure $\\tilde{C}_L$\nusing the cluster identification algorithm described in Ref.~\\cite{Hermann}, and call\nthe system phase separated when $\\tilde{C}_L\/N>0.55$.  In Fig.~\\ref{fig:2}(b) we plot\n$\\tilde{C}_L$ versus $v_w$ at varied $A_s$,\nand in Fig.~\\ref{fig:2}(c) we show the corresponding number of sixfold-coordinated\ndisks, $\\tilde{P}_6=\\sum_i^N\\delta(z_i-6)$, where $z_i$ is the coordination number of\ndisk $i$ determined from a Voronoi construction \\cite{cgal}.  In phase separated states,\nmost of the disks within a cluster have $z_i=6$ due to the triangular ordering of the\ndensely packed state.\nIn Fig.~\\ref{fig:2}(a), the linearly increasing dashed line denotes\nthe limit in which all the disks\nmove with the substrate so that $\\langle V\\rangle = v_{w}$ .\nAt $A_{s} = 3.0$, $\\langle V\\rangle$ initially increases linearly, following the dashed\nline, up to $v_{w} = 1.25$, \nindicating that there is a complete locking of the\ndisks to the substrate.\nFor $v_w > 1.25$, there is a slipping process in which the disks\ncannot keep up with the traveling wave and jump to the next well.\nA maximum in $\\langle V\\rangle$ appears near $v_{w} = 2.0$,\nand there is a sharp drop in $\\langle V\\rangle$ near $v_{w} = 5.0$,\nwhich also coincides with a sharp increase in  $\\tilde{C}_L$ and $\\tilde{P}_{6}$.\nThe  $\\langle V\\rangle$ versus $v_{w}$ curves for $A_{s} > 1.0$\nall show similar trends, with a sharp drop\nin $\\langle V\\rangle$ accompanied by an increase in\n$\\tilde{C}_{L}$ and $\\tilde{P}_{6}$, showing that the onset of clustering\nresults in a sharp decrease in $\\langle V\\rangle$.\nFor $A_{s} \\leq 1.0$, the substrate is weak enough that the system\nremains in a cluster state even at $v_{w} = 0$,\nindicating that a transition from a cluster to a non-cluster state\ncan also occur as a function of substrate strength.\nIn the inset of Fig.~\\ref{fig:2}(b) we show the regions in which\nclustering and non-clustering states appear as a function of $v_w$ versus $A_s$.\nAt $v_{w} = 0$, there is a substrate-induced transition from a\ncluster to a non-cluster state near $A_{s} = 1.0$,\nwhile for higher $A_s$, the location of the transition shifts linearly to higher $v_w$\nwith increasing $A_s$.\nSince the motor force is $F^m=1.0$,\nwhen $A_{s} < 1.0$ individual disks\ncan escape from the substrate minima,\nso provided that $r_{l}$ is large enough, the disks can freely move\nthroughout the entire system and form a cluster state.\nFor $A_{s} > 1.0$, the disks are confined by the substrate minima,\nbut when $v_w$ becomes large enough, the disks can readily escape\nthe minima and again form a cluster state.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig3.png}\n\\caption{(a) $\\langle V\\rangle\/v_{w}$\n  vs $A_{s}$ for the system in Fig.~\\ref{fig:2}\n  at $v_{w} = 0.6$.\n  (b) The corresponding normalized $C_{L}$ showing\n  that the  transition from a cluster to a non-cluster state\n  coincides with an increase in $\\langle V\\rangle\/v_w$.\n  (c) $\\langle V\\rangle\/v_{w}$ vs $A_s$ for the same system with $v_w=2.0$\n  where the cluster to non-cluster transition occurs at a higher value of $A_{s}$.\n  (d) The corresponding normalized $C_{L}$ vs $A_s$.\n}\n\\label{fig:3}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig4.png}\n\\caption{ The real space positions of the active disks for the system in Fig.~\\ref{fig:3}(a,b)\n  with $v_{w} = 0.6$.\n  (a) At $A_{s} = 0.75$, a phase separated state appears.\n(b) At $A_{s} = 2.5$, the disks are strongly localized in the substrate minima and move\nwith the substrate.\n}\n\\label{fig:4}\n\\end{figure}\n\nTo highlight the correlation between the changes in\nthe transport and the onset of clustering, in\nFig.~\\ref{fig:3}(a,b)\nwe plot $\\langle V\\rangle\/v_{w}$ and the normalized\n$C_{L}=\\tilde{C}_L\/N$ versus $A_{s}$ at a fixed value of  $v_{w} = 0.6$\nfrom the system in Fig.~\\ref{fig:2}.\nHere\nthe cluster to non-cluster transition occurs at $A_{s} = 1.25$,\nas indicated by the drop in $C_{L}$\nwhich also coincides with a jump in $\\langle V\\rangle\/v_{w}$.\nFor this value of $v_{w}$, a complete locking between the disks and the traveling\nwave occurs for $A_{s} \\geq 3.0$, where\n$\\langle V\\rangle\/v_{w} = 1.0$.\nIn Fig.~\\ref{fig:3}(c,d) we plot $\\langle V\\rangle\/v_w$ and $C_L$ versus $A_s$ for the\nsame system at\n$v_{w} = 2.0$, where the cluster to non-cluster transition\noccurs at a higher value of $A_{s} = 2.0$.\nThis transition again coincides with a sharp\nincrease in $\\langle V\\rangle\/v_{w}$.\nIn Fig.~\\ref{fig:4}(a) we show images of the disk configurations for\nthe system in Fig.~\\ref{fig:3}(a,b) with $v_w=0.6$ at\n$A_{s} = 0.75$,\nwhere the disks form a cluster state,\nwhile in Fig.~\\ref{fig:4}(b), at $A_s=2.5$ in the same system,\nthe clustering is lost and the disks are\nstrongly trapped in the substrate minima, forming\nchain like states that move with the substrate.\nThese results indicate that the clusters act as composite objects that\nonly weakly couple to the substrate.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig5.png}\n\\caption{(a) $\\langle V\\rangle$ vs $r_{l}$\n  in samples with $A_{s} = 2.0$ and $\\phi = 0.453$ for varied $v_{w}$ from\n  $v_w=0.25$ to $v_w=6.0$.\n(b) The corresponding $\\tilde{C}_{L}$ vs $r_{l}$.}\n\\label{fig:5}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig6.png}\n\\caption{A sample with $A_s=2.0$ and $\\phi=0.453$\n  for $r_{l} = 5$ (red squares) and $r_l=200$ (blue circles).\n  (a) $\\langle V\\rangle$ vs $v_{w}$.\n(b) $C_{L}$ vs $v_{w}$.\n}\n\\label{fig:6}\n\\end{figure}\n\nWe next examine the case with a fixed\nsubstrate strength of $A_{s} = 2.0$ and varied $r_{l}$.\nFigure~\\ref{fig:5}(a) shows\n$\\langle V\\rangle$ versus $r_{l}$\nfor $v_{w}$ values ranging from $v_w=0.25$\nto $v_w=6.0$, and Fig.~\\ref{fig:5}(b) shows the corresponding $\\tilde{C}_{L}$\nversus $r_{l}$.\nFor $v_{w} < 3.0$ the system remains in a non-cluster state for all values of $r_{l}$, while\nfor $v_{w} \\geq 3.0$ there is a transition\nfrom a non-cluster to a cluster state\nwith increasing $r_l$ as indicated by the simultaneous drop in\n$\\langle V\\rangle$ and increase in $\\tilde{C}_{L}$.\nIn Fig.~\\ref{fig:6}(a,b) we plot $\\langle V\\rangle$ and\n$C_{L}$ versus $v_{w}$ at $A_{s} = 2.0$\nfor $r_{l} = 200$ and $r_l=5.0$.\nThe system is in a non-cluster state for all $v_w$ when\n$r_{l} = 5.0$,\nand there is a peak in $\\langle V\\rangle$ near $v_{w} = 1.0$,\nwhile for $r_{l} = 200$ there is a transition to a cluster state\nclose to $v_{l} = 3.0$\nwhich coincides with a drop in $\\langle V\\rangle$ that is much sharper than the decrease\nin $\\langle V\\rangle$ with increasing $v_w$ for the $r_{l} = 5$ system.\nIn general, when $r_{l}$ is small, the net transport of disks through the sample is\ngreater than in samples with larger $r_l$.\nThe fact that the net disk transport varies with varying $r_l$ suggests\nthat traveling wave substrates could be used as a method for separating\ndifferent types of active matter, such as clustering and non-clustering species.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig7.png}\n\\caption{\n  (a) $\\langle V\\rangle$ vs $v_{w}$ at $\\phi = 0.56$ and\n  $r_{s} = 300$ for varied $A_{s} = 0.5$ to $A_s=4.0$.\n(b) The corresponding $\\tilde{C}_{L}$ vs $v_{w}$.\n(c) $\\langle V\\rangle$ vs $\\phi$ for $A_{s} = 2.5$, $r_{l} = 300$ and $v_{w} = 2.0$.\n  (d) The corresponding\n  $\\tilde{C}_{l}$ (blue circles) and $\\tilde{P}_{6}$ (red squares) vs $\\phi$\n  where the onset of clustering occurs\n  near $\\phi = 0.6$ at the same point for which there is a drop in\n $\\langle V\\rangle$ in panel (c).\n}\n\\label{fig:7}\n\\end{figure}\n\nWhen we vary the disk density $\\phi$ while holding $r_l$ fixed, we find\nresults similar to those described above.\nIn Fig.~\\ref{fig:7}(a) we plot\n$\\langle V\\rangle$ versus $v_{w}$\nat $\\phi = 0.56$ for varied $A_{s}$ from $A_s=0.5$ to $A_s=4.0$, where we find\na similar trend in which\n$\\langle V\\rangle$ increases with increasing wave speed\nwhen the disks are strongly coupled to the substrate.\nA transition to a cluster state occurs at higher $v_w$ as shown in\nFig.~\\ref{fig:7}(b) where we plot $\\tilde{C}_{L}$ versus $v_{w}$ for the same\nsamples.  The increase in $\\tilde{C}_{L}$ at the cluster state onset\ncoincides\nwith a drop in $\\langle V\\rangle$.\nIn Fig.~\\ref{fig:7}(c) we plot $\\langle V\\rangle$ versus $\\phi$\nfor a system with fixed $v_{w} = 2.0$, $A_{s} = 2.5$, and $r_{l} = 300$, while\nin Fig.~\\ref{fig:7}(d) we show the corresponding\n$\\tilde{C}_{L}$ and $\\tilde{P}_{6}$ versus $\\phi$.\nA transition from the\nnon-cluster to the cluster state occurs\nnear $\\phi=0.6$, which correlates with a sharp drop in $\\langle V\\rangle$ and\na corresponding increase\nin $\\tilde{C}_{L}$ and $\\tilde{P}_{6}$.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig8.png}\n\\caption{\n  The disk positions on the traveling wave substrate for the system in\n  Fig.~\\ref{fig:7}(a) at $\\phi = 0.56$.  Colors indicate disks belonging to\n  the five largest clusters.\n  (a) Complete locking\n  at $A_{s} = 4.0$ and $v_{w} = 1.0$,\n  where the transport efficiency is $\\langle V\\rangle\/v_{w} = 0.998$.\n  (b) Partial locking\n  at $A_{s} = 2.0$ and $v_{w} = 1.5$, \n  where $\\langle V\\rangle\/v_{w} =  0.41$.\n  (c) Weak locking\n  at $A_{s} = 1.0$ and $v_{w} = 0.6$  \n  with $\\langle V\\rangle\/v_{w} = 0.078$.\n}\n\\label{fig:8}\n\\end{figure}\n\nIn Fig.~\\ref{fig:8}(a) we show the disk configurations from the system in\nFig.~\\ref{fig:7}(a) at $A_{s} = 4.0$ and $v_{w} = 1.0$.\nHere $\\langle V\\rangle\/v_{w} = 0.998$,\nindicating that the disks are almost completely locked with\nthe traveling wave motion\nand there is little to no slipping of the disks out of the substrate minima.  \nIn Fig.~\\ref{fig:8}(b), the same system at\n$A_{s} = 2.0$ and $v_{w} = 1.5$\nhas a transport efficiency of $\\langle V\\rangle\/v_{w} = 0.41$.\nNo clustering occurs but there are numerous disks that \nslip as the traveling wave moves.\nAt $A_{s} = 1.0$ and $v_{w} = 0.6$ in\nFig.~\\ref{fig:8}(c)\nthere is a low transport efficiency of\n$\\langle V\\rangle\/v_w=0.078$.\nThe system forms a cluster state and smaller numbers of individual disks\noutside of the cluster are transported by the traveling wave.\n\n\\section{Forward and Backward Cluster Motion}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig9.png}\n\\caption{\n  The center of mass $X_{\\rm COM}$ location of a cluster vs\n  time in simulation time steps for a system with $\\phi = 0.454$ and $r_{l} = 300$.\n(a) At $A_{s} = 1.25$ and $v_{w} = 0.6$, the cluster moves in the negative $x$-direction,\n  against the direction of the traveling wave.\n  (b) At $A_{s} = 0.5$ and $v_{w} = 4.0$, the cluster\n  is stationary.\n  (c) At $A_{s} = 3.0$ and $v_{w} = 7.0$, the cluster moves in the positive\n  $x$-direction, with the traveling wave.\n  The dip indicates the point at which the center of mass\n  passes through the periodic boundary conditions.\n}\n\\label{fig:9}\n\\end{figure}\n\nIn general, we find that when the traveling wave is moving in the positive $x$-direction,\n$\\langle V\\rangle > 0$; however, within the\ncluster phase,\nthe center of mass motion of a cluster can be\nin the positive or negative $x$ direction or the cluster can be almost stationary.\nBy using the cluster algorithm we can track the $x$-direction motion of the\ncluster center of mass $X_{\\rm COM}$ over fixed time periods,\nas shown in Fig.~\\ref{fig:9}(a) for a system with\n$\\phi = 0.454$, $r_{l} = 300$, $A_{s} = 1.25$, and $v_{w} = 0.6$.\nDuring the course of $6\\times 10^6$ simulation time steps\nthe cluster moves in the negative $x$-direction a distance of\n$235$ units, corresponding to a space containing 16 potential minima.\nEven though the net disk flow is in the positive\n$x$ direction, the cluster itself drifts in the negative $x$ direction.\nIn Fig.~\\ref{fig:9}(b) at $A_{s} = 0.5$ and $v_{w} = 4.0$,\nthe disks are weakly coupled to the substrate\nand the cluster is almost completely stationary.\nFigure~\\ref{fig:9}(c) shows that at $A_{s} = 3.0$ and $v_{w} = 7.0$,\nthe cluster center of mass motion is now\nin the positive $x$ direction, and the cluster\ntranslates a distance equal to almost $20$ substrate minima\nduring the time period shown.\nThe apparent dip in the center of mass motion\nis due to the periodic boundary conditions.\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig10.png}\n\\caption{\n  Height field of the direction and magnitude of the center of mass motion $V_{\\rm COM}$\n  as a function of $A_s$ vs $v_w$ for the cluster\n  obtained after $4\\times 10^6$ simulation steps.\n  The gray area indicates a regime in which there is no cluster formation.\n}\n\\label{fig:10}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3.5in]{Fig11.png}\n\\caption{ The disk positions for the system in Figs.~\\ref{fig:9} and\n  ~\\ref{fig:10}.\n  (a)\n  At  $A_{s} = 1.0$ and $v_{w} = 0.4$,  the cluster drifts in the negative $x$-direction.\n  (b)\n  At $A_{s} = 3.0$ and $v_{w} = 5.0$, \n  the cluster drifts in the positive $x$-direction.\n}\n\\label{fig:11}\n\\end{figure}\n\n\nWe have conduced a series of simulations and measured\nthe direction and amplitude $V_{\\rm COM}$ of the center\nof mass motion, as plotted\nin Fig.~\\ref{fig:10} as a function of $A_{s}$  versus wave speed\nfor the system in Fig.~\\ref{fig:7}.\nThe gray area indicates a region in which clusters do not occur,\nand in general we find that the negative cluster motion occurs\nat lower wave speeds while the positive motion occurs\nfor stronger substrates and higher wave speeds.\nThere are two mechanisms that control the cluster center of mass motion.\nThe first is the motion of the substrate itself,\nwhich drags the cluster in the positive $x$ direction, and\nthe second is the manner in which the cluster grows or shrinks\non its positive $x$ and negative $x$ sides.\nAt lower substrate strengths and low wave speeds,\nthe disks in the cluster are weakly coupled to the substrate\nso the cluster does not move with the substrate.\nIn this case the disks can leave or\njoin the cluster anywhere around its edge; \nhowever, disks tend to join the cluster at a higher rate on its\nnegative $x$ side since individual disks, driven by the moving\nsubstrate, collide with the negative $x$ side of the cluster and can become\ntrapped in this higher density area.  The positive $x$ side of the cluster\ntends to shed disks at a higher rate since the disks can be carried\naway by the moving substrate into the low density gas region.\nThe resulting asymmetric growth rate causes the cluster to drift in\nthe negative $x$ direction.\nThere is a net overall transport of disks in the positive $x$ direction due to the\nlarge number of gas phase disks outside of the cluster region which follow\nthe motion of the substrate.\nFigure~\\ref{fig:11}(a) shows the\ndisk positions at $A_{s} = 1.0$ and $v_{w} = 0.4$, where the cluster\nis drifting in the negative $x$ direction.     \nFor strong substrate  strengths, all the disks that are outside of the cluster become\nstrongly confined in the q1D substrate minima,\nand the disk density inside the cluster itself starts to become modulated by the substrate.\nUnder these conditions, the cluster is dragged along with the traveling substrate\nin the positive $x$ direction, as illustrated\nin Fig.~\\ref{fig:11}(b) for $A_{s} = 3.0$ and $v_{w} = 5.0$. \nThese results\nsuggest that it may be possible for\ncertain active matter systems to \ncollectively form a cluster state in order to \nmove against an external bias\neven when isolated individual particles on average move with the\nbias.\n\n\n\\section{Summary}\n\nWe have examined run and tumble active matter disks interacting with\ntraveling wave periodic substrates.\nWe find that\nin the non-phase separated state,\nthe disks couple to the traveling waves,\nand that at the transition\nto the cluster state,\nthere is a partial decoupling from the substrate and the net transport of disks by\nthe traveling wave  is strongly reduced.\nWe also find a transition from a cluster\nstate to a periodic quasi-1D liquid\nstate for increasing substrate strength,\nas well as a transition back to a cluster state\nfor increasing traveling wave speed.\nWe show that there is a transition\nfrom a non-cluster to a cluster state as a function of increasing\ndisk density which is correlated with a drop in the net disk transport.\nSince disks with different run times drift with different velocities,\nour results indicate that traveling wave substrates\ncould be an effective method for separating active matter particles with different mobilities.\nWithin the regime in which the system forms a cluster state,\nwe find that as a function of wave speed and substrate strength,\nthere are weak substrate regimes where the center of mass of the\ncluster moves in the opposite direction from that of the traveling wave,\nwhile for stronger substrates,\nthe cluster center of mass moves in the same direction as the traveling wave.\nThe reversed cluster motion occurs due to the\nspatial asymmetry of the rate at which disks leave or join the cluster.\nThis suggests that collective clustering could be an effective method\nfor forming an emergent object that\ncan move  against gradients or drifts even\nwhen individual disks on average move with the drift.\n\n\\acknowledgments\nThis work was carried out under the auspices of the \nNNSA of the \nU.S. DoE\nat \nLANL\nunder Contract No.\nDE-AC52-06NA25396.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the past few years, the field of text generation witnesses many significant advances, including but not limited to neural machine translation \\cite{Transformer:17,gehring2017convolutional}, dialogue systems \\cite{liu-etal-2018-knowledge,zhang2019memory} and text generation \\cite{clark-etal-2018-neural,guo2018long}.\nBy utilizing the power of the sequence-to-sequence (S2S) framework \\cite{Sutskever:14}, generation models can predict the next token based on the previous generated outputs and contexts.\nHowever, S2S models are not perfect.\nOne of the obvious drawbacks is that S2S models tend to be short-sighted on long context and are unaware of global knowledge.\nTherefore, how to incorporate global or local knowledge into S2S models has been a long-standing research problem.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/figure0.png}\n  \\caption{\\label{example-table} An example from a novel called ``Fights Break Sphere''. The relations between \\textcolor{y}{Yanxiao}, \\textcolor{x}{Xuner} and \\textcolor{n}{Nalanyanran} are evolutionary. And the characteristic of \\textcolor{y}{Yanxiao} changes over time.}\n\\end{figure}\n\nThere lie two different directions to include knowledge into S2S models.\nOn the one hand, many efforts \\cite{zhang-etal-2018-improving,guan2019story,li-etal-2019-incremental} have been taken to address the short-sighted problem in text generation by explicitly modeling unstructured context. Nevertheless, these approaches rely heavily on the quality and scale of unstructured context, and become intractable when applying to scenarios where the context length increases drastically (e.g., commenting a full-length novel).\nOn the other hand, researchers~\\citep{beck-etal-2018-graph,marcheggiani-perez-beltrachini-2018-deep,li-etal-2019-coherent} try to combine knowledge with S2S by employing pre-processed structured data~(e.g., knowledge graph) which naturally avoid the difficulty for context length. However, those models are oriented to static knowledge, and hence hardly model events where temporal knowledge evolution occurs.\n\n\n\n\nThe dynamic knowledge evolution is very common in full-length novels. In a novel, a knowledge graph can be constructed by using entities (characters, organizations, locations etc.) as vertices together with the entity relations as edges. Obviously, a single static knowledge graph is hardly to represent the dynamic story line full of dramatic changes. For example, a naughty boy can grow up into a hero, friends may become lovers, etc. In this paper, we proposes \\textbf{E}volutionary \\textbf{K}nowledge \\textbf{G}raph~(\\textbf{EKG}) which contains a series of sub-graph for each time step. Figure 1 illustrates EKG for the novel ``Fights Break Sphere''. At three different scenes, ``Yanxiao'', the leading role of the novel, is characterized as ``proud boy'', ``weak fighter'', and ``magic master'' separately. At the same time, the relation between ``Yanxiao'' and ``Xuner'' is evolved from friend into lovers, and finally get married, and the relation between ``Yanxiao'' and ``Nalanyanran'' is changed over time as ``engagement$\\rightarrow$divorce$\\rightarrow$friend''.\n\nEKG is important for commenting passage of novels since it is the dramatic evolution and comparison in the storyline but not the static facts resonate the readers most. As illustrated in Figure~\\ref{example-table}. When commenting passage sampled from the $T$-th chapter of a novel, the user-A refers to a historical fact that \"Nalanyanran\" has abandoned ``Yanxiao'', while the user-B refers to the future relation between ``Yanxian'' and a related entity ``Xuner'' that they will go through difficulties together. In this paper, EKG is trained within a multi-task framework to represent the latent dynamic context, and then a novel graph-to-sequence model is designed to select the relevant context from EKG for comment generation.\n\n\\subsection{Related Work}\ngraph-to-sequence model has been proposed for text generation.~\\citet{song-etal-2018-graph}, ~\\citet{beck-etal-2018-graph}, and~\\citet{guo-etal-2019-densely} used the graph neural networks to solve the AMR-to-text problem.~\\citet{bastings-etal-2017-graph} and ~\\citet{GraphSeq2Seq:18} utilized graph convolutional networks to incorporate syntactic structure into neural attention-based encoder-decoder models for machine translation. In comment generation, Graph2Seq~\\citep{li-etal-2019-coherent} is proposed to generate comments by modeling the input news as a topic graph. These methods are using static graph, and did not involve the dynamic knowledge evolution.\n\nRecently, more research attention has been focused on dynamic knowledge modeling.~\\citet{taheri2019www} used gated graph neural networks to learn the temporal dynamics of an evolving graph for dynamic graph classification.~\\citet{KnowEvolve:17, trivedi2018dyrep, kumar2019kdd} learned evolving entity representations over time for dynamic link prediction. Unlike the EKG in this paper, they did not model the embeddings of the relations between dynamic entities.\n~\\citet{iyyer-etal-2016-feuding} proposed an unsupervised deep learning algorithm to model the dynamic relationship between characters in a novel without considering the entity embedding. Unlike these methods, our EKG-based model represents the temporal evolution of entities and relations simultaneously by learning their temporal embeddings, and hence has an advantage in supporting text generation tasks.\n\nTo our knowledge, few studies make use of evolutionary knowledge graph for text generation. This may due to the lack of datasets involving dynamic temporal evolution. We observed that novel commenting need to understand long context full of dramatic changes, and hence build such a dataset by collecting full-length novels and real user comments. The dataset with its EKG will be made publicly available, and more details can be found in Section~\\ref{sect:Dataset}.\n\nThe main contributions of our work are three-fold:\n\n\\begin{itemize}\n  \\item We build a new dataset to facilitate the research of evolutionary knowledge based text generation.\n  \\item We propose a multi-task framework for the learning of evolutionary knowledge graph to model the long and dynamic context.\n  \\item We propose a novel graph-to-sequence model to incorporate evolutionary knowledge graph for text generation.\n\n\\end{itemize}\n\n\\section{Dataset Development and Evolutionary Knowledge Graph Building}\\label{sect:Dataset}\nTo facilitate the research of modeling knowledge evolution for text generation, we build a dataset called \\textit{GraphNovel} by collecting full-length novels and real user comments. Together with the corresponding EKG embeddings, we will make the dataset public available soon.\nWe detail the collection of the dataset below.\n\n\\subsection{Data collection}\n\\label{sub:collect}\nThe data is collected from well-known Chinese novel websites. To increase the diversity of data, top-1000 hottest novels are crawled with different types including science fiction, fantasy, action, romance, historical, and so on. Then we filter out novels due to the following 3 considerations: 1) the number of chapters is less than 10, 2) few entities are mentioned and 3) lack of user comments. Each remained novel includes chapters in chronological order, a set of user-underlined passages, and user comments for the passages.\n\nThen, we use the lexical analysis tool~\\citep{jiao2018LAC} to recognize three types of entities (persons, organizations, locations) from each novel. Due to many of the nickname in novels, the identified entities from the tool contains much noise. To improve the knowledge quality, human annotators are asked to verify the entities, and add missing ones. Then all the paragraphs containing mentions of two entities are identified, and will later serve as a representation of the entity relations at that specific time step.\n\nAs for the highlighted novel passage and user comments, three criteria are used to select high quality and informative data: 1) The selected passage must contain at least one entity; 2) the selected passage must be commented by at least three users; 3) comments related to the same passage are ranked according to the upvotes, and the bottom 20\\% are dropped. Notably, those highlighted passages have a degree of redundancy because users tend to highlight and comment passages at very similar positions. Thus we merge passages which have more than 50\\% overlapping rate. This operation can effectively reduce the quantity of passages by 30\\%.\n\n\n\n\\subsection{Core Statistics}\nThe dataset contains 203 novels, 349,695 highlighted passages and 3,136,210 comments totally. Due to diverse genre of novels in our dataset, the number of entities and relations per novel varies widely with range [$10^2$, $10^5$]. And the number of comments per passage changes a lot with a range of [$3$, $2\\times{10^3}$], because it depends on how interesting the corresponding passage is.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=\\textwidth]{images\/figure2.png}\n  \\caption{Architecture of our model. First, the EKG is trained under a multi-task learning framework. Then a graph-to-sequence model is traind to utilize the EKG for text generation.}\\label{fig:model}\n\\end{figure*}\n\nWe partitioned the dataset into non-overlapping training, validation, and test portions, along novels (See Table~\\ref{dataset-table} for detailed statistics). Five most relevant comments for each passage in validation and test set are selected by human annotators. While the comments in train set are all preserved in order to ensure its flexible use.\n\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{l|r|r|r}\n\\hline\n & \\textbf{train} & \\textbf{valid} & \\textbf{test} \\\\\n\\hline\n{\\small{\\# novels}} & 173 &  10 & 20 \\\\\n\\hline\n{\\small{\\# passages}} & 324,803 & 7,976 & 16,916 \\\\\n\\hline\n{\\small{\\# comments}} & 3,011,750 & 39,880 & 84,580 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. length}\\\\\\small{~~of context}} & 520,571 & 305,492 & 277,847 \\\\\n\\hline\n{\\small{Avg. \\# entities}} & 383.4 & 720.6 & 281.7 \\\\\n\\hline\n{\\small{Avg. \\# relations}} & 9,013 & 1,9919 & 7,084 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# comments}\\\\\\small{~~per passage}} & 9.3 & 5.0 & 5.0 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# entities}\\\\\\small{~~per passage}} & 2.6 & 3.1 & 3.4 \\\\\n\\hline\n\\tabincell{l}{\\small{Avg. \\# relations}\\\\\\small{~~per passage}} & 4.4 & 9.2 & 11.0 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{dataset-table} Statistics of Dataset. }\n\\end{table}\n\n\\subsection{Build up Evolutionary Knowledge Graph}\n\\label{sub:collect}\n\n\nThen for each novel, we build up a knowledge graph which consists of a sequence of sub-graphs. Obviously, it is sensible to build up a sub-graph for each import scene of the novel, and build up more sub-graphs around the critical transitions in the stroryline. In this paper, we assume each chapter usually represents an integral scene, and hence build a sub-graph for each chapter of the novel.\n\nThen for each chapter, the entities being mentioned in the chapter are the vertices of the corresponding sub-graph. And if a paragraph consists two of the entities, an edge is created between the two entities. In such a way, a sequence of sub-graphs are constructed, and form our EKG. In the next section, we will formulate the embedding computation of the EKG, and its application for comment generation.\n\n\\section{Model Formalism}\\label{sec:formalism}\nIn this section, we formulate our approach in details. First, the training of EKG embedding is presented. Then a graph-to-sequence model is shown to utilize the EKG for comment generation. The architecture of the model is shown in Figure~\\ref{fig:model}.\n\n\\subsection{Definition}\nFor a novel with $n_e$ entities and $n_r$ relations, define a global evolutionary knowledge graph: $G_{ekg}^{global} = \\{G(t)\\}|_{t:1\\rightarrow{T}}$,\nwhere $T$ is the number of chapters\\footnote{We will cluster successive chapters into a longer one if they are too short.} (or time periods); $G(t)=\\langle{V(t), E(t)}\\rangle$, is a temporal knowledge graph of the chapter $t$; $V(t)=\\{v_1(t), v_2(t),...,v_{n_e}(t)\\}$ is the set of vertices and $E(t)=\\{e_1(t), e_2(t),...,e_{n_r}(t)\\}$ is the set of edges between two vertices.\n\nGiven a passage $C$ from the chapter $t$ with $c_e$ entities and $c_r$ relations, a local EKG related to it can is a sub-graph of the global EKG: $G_{ekg}^{local}=\\{G_c(t)\\}|_{t:1\\rightarrow{T}}$, where $G_c(t)=\\langle{V_c(t), E_c(t)}\\rangle$; $V_c(t)$ is a subset of $V(t)$ with size of $c_e$ and $E_c(t)$ is a subset of $E(t)$ with size of $c_r$. Then the local EKG of the passage is a sequence of local temporal knowledge sub-graphs with $T \\times c_e$ vertex embeddings and $T \\times c_r$ edge embeddings.\n\\subsection{EKG Embedding Training}\nInspired by the consistent state-of-the-art performance in language understanding tasks, we use off-the-shelf Chinese BERT model~\\citep{devlin-etal-2019-bert} to calculate the initial semantic representation of sentences. Considering the fact that entities are either out-of-vocabulary or associated with special semantics within the novel context, we propose the following algorithm to jointly learn entity and relation embeddings in EKG:\n\n\\paragraph{Vertex embedding learning.} The passages containing mentions of entity $v$ will contribute to learn the embedding of $v$. Specifically, the $i$-th passage is tokenized while the entity mention $v_i$ is masked with token ``[MASK]''. Then resulted tokens are fed into the pre-trained Chinese BERT model, and $f_{v_i}$ is obtained as the output feature corresponding to the mask token.\nWithin the chapter $t$, there exists $N_{v}^{t}$ sentences containing vertices $v$. The embedding of $v$ is learned by optimizing the following softmax loss summation which models the probabilities to predict the masked entities as $v$.\n\\begin{equation}\\label{node_learn}\n  L_{v}^{t} = -\\sum_{i=1}^{N_{v}^{t}}{\\log{p_{v_i}^{t}}}\n\\end{equation}\n\\begin{equation}\\label{node_softmax}\n  p_{v_i}^{t} = softmax(W_{v}^{t}\\cdot{f_{v_i}})\n\\end{equation}\nwhere $W_v^t$ is learnable parameter and denotes the embedding of $v$ from the chapter $t$. Usually the semantic representations of entities change smoothly over time, so we propose a temporally smoothed softmax loss to retain the similarity of entity embeddings from successive time periods:\n\\begin{equation}\\label{node_improve}\n   \\tilde{L}_{v}^{t} = -\\sum_{i=1}^{N_{v}^{t}}{(\\lambda{_0}\\log{p_{v_i}^{t-1}}+\\lambda{_1}\\log{p_{v_i}^{t}}+\\lambda{_2}\\log{p_{v_i}^{t+1}})}\n \n\\end{equation}\nwhere $\\lambda_0$, $\\lambda_1$ and $\\lambda_2$ are smooth factors; and only valid probabilities are included when $t=1$ or $T$. Then the overall loss for all time periods and all vertices is:\n\\begin{equation}\\label{node_overall}\n  L_{vertex} = \\sum_{t=1}^{T}\\sum_{v=1}^{n_e}{\\tilde{L}_{v}^{t}}\n\\end{equation}\n\n\n\\paragraph{Edge embedding learning.} Since the number of relations equal the number of co-occurrence for any two entities, it is infeasible to employ an embedding matrix to model the relation. Therefore, a \\textit{Relation Network} (RN) is proposed to learn the edge embeddings in the TKGs as shown in Figure~\\ref{fig:rn}. Specifically, the RN takes two vertex embeddings as input, and feed them into the first hidden layer to obtain the embedding of the edge $r$. Then the embeddings of two vertices and one edge are concatenated and fed into second hidden layer to reconstruct the sentence. We also use the pre-trained BERT~\\citep{devlin-etal-2019-bert} to obtain the representation $f_c$ for the whole sentence. The $f_c$ is taken from the final hidden state corresponding to ``[CLS]'' token because it aggregates sequence representation.\n\nA reconstruction loss applied to the network is optimized to jointly learn RN and edge embeddings:\n\\begin{equation}\\label{edge_l0}\n  L_r = \\max(d(f_{p^+}, f_c)- d(f_{p^-}, f_c) + \\alpha, 0)\n\\end{equation}\nwhere $p$ stands for a pair of vertices; $p^+$ represents a positive pair with two vertex related to the edge, and $p^-$ represents a negative pair with one vertex related to the edge and the other unrelated. The overall loss for learning edge embedding is:\n\\begin{equation}\\label{edge_loss}\n  L_{edge} = \\sum_{r=1}^{N_r}L_r\n\\end{equation}\n\nCombining the vertex and edge loss above, our final multi-task loss is:\n\\begin{equation}\\label{tkg_loss}\n  L_{EKG} = L_{vertex} + \\lambda_{r}L_{edge}\n\\end{equation}\nwhere $\\lambda_{r}$ is a hyperparameter to be tuned.\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/rn.png}\n  \\caption{Relation Network with a reconstruction loss.The edge embedding is shown in \\textcolor{red}{red color}.}\\label{fig:rn}\n\\end{figure}\n\n\\subsection{Graph-to-sequence modeling}\nAfter EKG embedding learning, we propose a graph encoder to utilize the embeddings of the EKG for comment generation. From the learned $G_{ekg}^{local}$, we can obtain vector sequences $V_i=\\{v_i(1),..,,v_i(2), v_i(T)\\}$ for each vertex, and $E_i=\\{e_i(1),e_i(2),...e_i(T)\\}$ for each edge. All these sequences are fed into Bi-LSTM to integrate information from all time periods. Then final representation of the vertices and edges are taken from the final hidden state of Bi-LSTM corresponding to the time step $t$.\n\nFurther, our graph encoder employs graph convolutional networks~\\citep{GNN:18} to aggregate the structured knowledge from the EKG and then is combined into a widely used encoder-decoder framework~\\citep{Transformer:17} for generation.\n\nour graph encoder is based on the implementation of GAT~\\citep{v2018graph}. The input to a single GAT layer is a set of vertices and edge features, denoted as $\\mathbf{F}_v=\\{\\vec{v}_1, \\vec{v}_2,..., \\vec{v}_{c_e}\\}$, and $\\mathbf{F}_r=\\{\\vec{r}_1, \\vec{r}_2,..., \\vec{r}_{c_r}\\}$, where ${c_e}$ is the number of vertices, and ${c_r}$ is the number of edges from the passage. The layer produces a new set of vertex features, $\\mathbf{F'}_v=\\{\\vec{v'}_1, \\vec{v'}_2,..., \\vec{v'}_{c_e}\\}$ as its output.\n\nIn order to aggregate the structured knowledge from input features and transform them into higher-level features, we then perform self-attention on both the vertices and the edges to compute attention coefficients\n\\begin{equation}\\label{gat_atten1}\n  \\alpha_{ij}^e = g(\\mathbf{W}\\vec{v}_i, \\mathbf{W}\\vec{v}_j)\n\\end{equation}\n\n\\begin{equation}\\label{gat_atten2}\n  \\alpha_{ij}^r = h(\\mathbf{W}\\vec{v}_i, \\mathbf{W}\\vec{r}_{ij})\n\\end{equation}\nwhere $\\mathbf{W}$ is learnable parameter; $g$ and $h$ are mapping functions; $\\alpha_{ij}^e$ and $\\alpha_{ij}^r$ indicate the importance of neighbor features to vertex $i$; they are normalized using softmax.\n\nOnce obtained, the normalized attention coefficients are used to compute a linear combination of the features corresponding to them, to serve as the final output features for every vertex:\n\\begin{equation}\\label{gat_output}\n  \\vec{v'}_i=\\sum_{j\\in{\\mathcal{N}_i}}\\alpha_{ij}^e\\mathbf{W}\\vec{v}_j + \\alpha_{ij}^r\\mathbf{W}\\vec{r}_{ij}\n\\end{equation}\n\nThen the graph encoder is combined into the encoder-decoder framework~\\citep{Transformer:17}, in which a self-attention based encoder is used to encode the passage. To aggregate structured knowledge, the encoding of all vertices from graph encoder are concatenated with the output of the passage encoder, and fed into a Transformer decoder for text generation. The whole graph-to-sequence model is trained to minimize the negative log-likelihood of the related comment.\n\\section{Experiment}\nIn this section, we first introduce the experimental details, and then present results from automatic and human evaluations.\n\\subsection{Details}\nWe train all the models using training set and tune hyperparamters based on validation set. The automatic and human evaluations are carried out based on test set.\nDuring the training of the EKG, we first learn the vertex embeddings and fix them during the subsequent training of edge embeddings. Our GAT-based graph encoder is based on the entities from the passage. we keep $K$ entities for each passage: if the number of entities\\footnote{Note that the number will not be zero because the passage contains as least one entity in our dataset.} is smaller than $K$, we use breadth-first searching on the global graph to fill the gap, otherwise we filter the low-order entities according to entity frequency. We set $K$ to 5 which is selected by validation. Label smoothing is used in the smoothed softmax loss. We denote our full model as ``\\textbf{EKG+GAT(V+E)}'', its variant which only use the first term in Equ.~\\ref{gat_output} as ``\\textbf{EKG+GAT(V)}'' and name the other variant ``\\textbf{EKG}'', which does not use GAT-based graph encoder and feeds the encoding of vertices into the Transformer decoder directly.\n\\subsection{Hyperparameters}\nIn our model, we set $\\lambda_1=0.5$, $\\lambda_2=1.0$, $\\lambda_3=0.3$ for the smoothed softmax loss; set $\\alpha$ to 0.0 for the reconstruction loss and $\\lambda_r$ to 1.0 for the multi-task loss; the number of self-attention layers in our passage encoder is 6; the number of Bi-LSTMs is 2 and the length of its hidden state is 768; and the GAT-based graph encoder has two layers. To stabilize the training, we use Adam optimizer~\\citep{adam:14} and follow the learning rate strategy in~\\citet{klein-etal-2017-opennmt} by increasing the learning rate linearly during the first $5000$ steps for warming-up and then decaying it exponentially. For inference, the maximum length of decoding is 50; beam searching is used with beam size 4 for all the models.\n\\subsection{Evaluation metrics}\nwe use both automatic metrics and human evaluations to evaluate the quality of generated novel comments.\n\n\\noindent{\\textbf{Automatic metrics}}:~1) \\textbf{BLEU} is commonly employed in evaluating translation systems. It is also introduced into comment generation task~\\citep{qin-etal-2018-automatic, yang-etal-2019-read}. we use $multi{-}bleu.perl$\\footnote{https:\/\/github.com\/moses-smt\/mosesdecoder\/blob\/master\/scripts\/generic\/multi-bleu.perl} to calculate the BLEU score. 2) \\textbf{ROUGE-L}(\\citep{lin-2004-rouge}) uses longest common subsequence to calculate the similar score between candidates and references. For calculation, we use a python package called $pyrouge$\\footnote{https:\/\/pypi.org\/project\/pyrouge\/}. These metrics also support the multi-reference evaluations on our dataset.\n\n\\noindent{\\textbf{Human evaluations}}:~1) \\textbf{Relevance}: This metric evaluates how relevant is the comment to the passage. It measures the degree that the comment is about the main storyline of the novel.2) \\textbf{Fluency}: This metric evaluates whether the sentence is fluent and judges whether the sentence follows the grammar and whether the sentence has clear logic. 3) \\textbf{Informativeness}: This metric evaluates how much structured knowledge the comment contains. It measures whether the comment reflects the evolution of entities and relations, or is just a general description that can be used for many passages. All these metrics have three gears, the final scores are projected to 0$\\sim$3.\n\n\\subsection{Baseline Models}\nWe describe three kinds of models used as baselines. All the baselines are implemented according to the related works and tuned on the validation set.\n\\begin{table*}[!t]\n\\centering\n\\begin{tabular}{lccccc}\n\\hline \\textbf{Model} & \\textbf{BLEU} &\\textbf{ROUGE-L}\n& \\textbf{Relevance} & \\textbf{Fluency} & \\textbf{Informativeness}\\\\ \\hline\n\\textbf{Seq2Seq}\\citep{qin-etal-2018-automatic} & 2.59 & 14.71 & 0.12 & 1.71 & 0.09\\\\\n\\textbf{Attn}\\citep{qin-etal-2018-automatic} & 3.71 & 16.44 & 0.34 & 1.70 & 0.33 \\\\\n\\textbf{Trans}\\citep{Transformer:17} & 6.11 & 19.21 & 0.57 & 1.62 & 0.58 \\\\\n\\textbf{Trans.+CTX}\\citep{zhang-etal-2018-improving} & 6.52 & 19.11 & 0.68 & 1.68 & 0.67\\\\\n\\textbf{Graph2Seq}\\citep{li-etal-2019-coherent} & 4.93 & 16.91 & 0.35 & 1.69 & 0.31 \\\\\n\\textbf{Graph2Seq++}\\citep{li-etal-2019-coherent} & 5.56 & 17.51 & 0.85 & 1.67 & 0.60\\\\\n\\hline\n\\textbf{EKG} & 6.59 & 20.00 & 0.81 & \\textbf{1.83} & 0.64 \\\\\n\\textbf{EKG+GAT(V)} & 6.72 & 20.09 & 0.88 & 1.77 & 0.70 \\\\\n\\textbf{EKG+GAT(V+E)} & \\textbf{7.01} & \\textbf{20.10} & \\textbf{0.89}  & 1.74 & \\textbf{0.75}  \\\\\n\\hline\n\\textbf{Human Performance} & 100 & 100 & 1.09 & 1.85 & 1.04\\\\\n\\hline\n\n\\end{tabular}\n\\caption{\\label{metric-table} Automatic metrics and human evaluations. }\n\\end{table*}\n\\begin{itemize}\n  \\item \\textbf{Seq2Seq models}~\\citep{qin-etal-2018-automatic}:~those models generate comments for news either from the title or the entire article. Considering there are no titles in our dataset, We compare two kinds of models from their work. 1) \\textbf{Seq2Seq:} it is a basic sequence-to-sequence model~\\citep{Sutskever:14} that generate comments from the passage; 2) \\textbf{Attn:} sequence-to-sequence model with an attention mechanism\\citep{Bahdanau:14}. For the input of the attention model, we append the related entities to the back of the passage.\n\n  \\item \\textbf{Self-attention models}:~our model includes a graph encoder to encode knowledge from graph, and a passage encoder use multiple self-attention layers. To show the power of graph encoder, we use the encoder-decoder framework (\\textbf{Trans.})~\\citep{Transformer:17} for passage-based comparison. Also we introduce an improved Transformer~\\citep{zhang-etal-2018-improving} with a context encoder to represent document-level context and denote it \\textbf{Trans.+CTX}. For the context input, we use up to 512 tokens before the passage as context.\n\n  \\item \\textbf{Graph2Seq}~\\citep{li-etal-2019-coherent}:~this is a graph-to-sequence model that builds a static topic graph for the input and generates comments based on representations of entities only. A two-layer transformer encoder is used in their work. For fair comparison, we use 6-layer transformer encoder to replace the original and denote the new model as \\textbf{Graph2Seq++}.\n\\end{itemize}\n\n\n\\subsection{Evaluation results}\nTable~\\ref{metric-table} shows the results of both automatic metrics and human evaluations.\n\nIn automatic metrics, our proposed model has best BLEU and ROUGE-L scores.\nFor BLEU, our full model EKG+GAT(V+E) achieves 7.01 score, which is 0.59 higher than that of the best baseline Trans.+CTX.\nThe Graph2Seq++ has a BLEU score 5.56 which is obviously lower than the EKG+GAT(V+E). The main reason is that the Graph2Seq++ is based on static graph and cannot make use of the dynamic knowledge.\nFor Rouge-L, our models all have ROUGE-L scores higher than 20\\%, and is 0.79\\% slightly better than that of Trans., which is the best among all baselines; the ROUGE-L score of Trans. is higher than of the Trans.+CTX, which is opposite of that in BLEU.\n\nIn human evaluations, we randomly select 100 passages from the test set and run all the models in Table~\\ref{metric-table} to generate respective comments. We also provide one user comment for each passage to get evaluations of human performance. All these passage-comment pairs are labeled by human annotators.\nIn relevance metric, our full model EKG+GAT(V+E) has better relevance score than all the baselines. It means our model can generate more relevant comments and better reflect the main storyline of the novel. For all this, there still exists significant gaps when compared to the human performance.\nIn fluency and informativeness metrics, our EKG+GAT(V+E) model has achieves higher score compared to all baselines. It illustrates that the generated comments by our proposed model are more fluent and contains more attractive information.\n\n\\begin{table*}[!t]\\small\n\\centering\n\\begin{tabular}{|l|}\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P1:}~\\textbf{\u8fd9\u4eba\u5bb6\u59d3\u66fe,\u4f4f\u5728\u53bf\u57ce\u4ee5\u5357\u4e00\u767e\u4e09\u5341\u91cc\u5916\u7684\u8377\u53f6\u5858\u90fd\u3002}\\\\\n(This family, surnamed Zeng, lives in Heyetangdu, 130 miles \\textcolor{orange}{south of the county}.)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T1:}~\u539f\u6765\u662f\u8fd9\u4e48\u6765\u7684\u3002~(That's how it turned out.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G1:}~\u8fd9\u4e2a\u5730\u65b9\u5440!~(This is the place !)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E1:}~\u4ed6\u4e00\u76f4\u601d\u5ff5\u7740\u5bb6\u91cc\u4eba\u3002~(He has been \\textcolor{cyan}{missing his family}.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P2]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P2:}~\\textbf{\u56fd\u85e9\u4eca\u65e5\u4e43\u6234\u5b5d\u4e4b\u8eab,\u8001\u6bcd\u5e76\u672a\u5b89\u846c\u59a5\u5e16,\u600e\u5fcd\u79bb\u5bb6\u51fa\u5c71?}\\\\\n(Today Guofan is wearing mourning. \\textcolor{cyan}{My mother hasn't been buried yet}. How can I leave home ?)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T2:}~\u771f\u662f\u4e00\u4e2a\u806a\u660e\u4eba~(He is so clever.)\\\\}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G2:}~\u8001\u592a\u592a\u4e5f\u662f\u4e2a\u597d\u4eba~(The old lady is also a good person.)\\\\}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E2:}~\u8fd9\u4e2a\u65f6\u5019\u7684\u56fd\u5bb6\u5df2\u7ecf\u6709\u4e86\u53d8\u5316~(\\textcolor{green}{The country is changing} at this time.)}\\end{CJK*} ~\\textcolor{blue}{\\textbf{[P3]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P3:}~\\textbf{\u9762\u4e34\u5927\u654c,\u66fe\u6697\u81ea\u4e0b\u5b9a\u51b3\u5fc3,\u4e00\u65e6\u57ce\u7834,\u7acb\u5373\u81ea\u520e,\u8ffd\u968f\u5854\u9f50\u5e03\u3001\u7f57\u6cfd\u5357\u4e8e\u5730\u4e0b\u3002}}\\end{CJK*} \\\\\n(\\textcolor{green}{Facing the enemy}, Zeng made up his mind to \\textcolor{red}{commit suicide as soon as the city broke}, following Ta Qibu and Luo Zenan.)\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T3:}~\u8fd9\u5c31\u662f\u539f\u6765\u6218\u4e89\u7684\u6837\u5b50}\\end{CJK*} (This is what the war looks like.)\\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G3:}~\u4e00\u4e2a\u4eba\u7684\u547d\u8fd0\u603b\u662f\u5982\u6b64}\\end{CJK*}(One's destiny is always like this.)\\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E3:}~\u81ea\u7acb\u4e8e\u5357\u57ce,\u81ea\u7834\u800c\u7acb~(He established himself in \\textcolor{orange}{the south of the country} throught constant breakthroughs.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P1]}}\\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{P4:}~\\textbf{\u66fe\u56fd\u85e9\u7684\u8138\u4e0a\u9732\u51fa\u4e00\u4e1d\u6d45\u6d45\u7684\u7b11\u610f,\u5934\u4e00\u6b6a,\u5012\u5728\u592a\u5e08\u6905\u4e0a.}\\\\\n(Zeng Guofan smiled slightly. His head tilted and fell on the chair.)\n}\\end{CJK*} \\\\\n\\hline\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{T4:}~\u8fd9\u4e00\u6bb5\u63cf\u5199\u771f\u7684\u5f88\u6709\u753b\u9762\u611f~(This description is really picturesque.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{G4:}~\u8fd9\u4e2a\u4eba\u7684\u5fc3\u601d\u7f1c\u5bc6~(This man is very thoughtful.)}\\end{CJK*} \\\\\n\\begin{CJK*}{UTF8}{gkai}\\tabincell{l}{\\textbf{E4:}~\u4ed6\u4e00\u751f\u5fe0\u541b\u4e3a\u56fd,\u5c31\u8fd9\u6837\u8d70\u4e86~(He was \\textcolor{red}{loyal to his country} all his life; he is gone.)}\\end{CJK*}~\\textcolor{blue}{\\textbf{[P3]}}\\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{case-table}Comments generated by Trans.+CTX~(\\textbf{T}), Graph2Seq++~(\\textbf{G}) and our EKG+GAT(V+E)~(\\textbf{E}). The passages (i.e., P1, P2, P3, P4) are extracted from the same novel called \\emph{Zeng Guofan}. We highlight the passage corresponding to the generated comment from our model~\\textbf{E} with \\textcolor{blue}{blue color}. Moreover, the relevant fragments are marked with a same color.}\n\\end{table*}\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=0.5\\textwidth]{images\/ablation.png}\n  \\caption{Ablation results about number of entities (a) and number of time periods (b).}\\label{fig:ablation}\n\\end{figure}\n\\subsection{Analysis and Discussion}\n\\paragraph{Ablation study:} we compare the results of EKG, EKG+GAT(V), and EKG+GAT(V+E). The EKG, which does not use graph encoder, can achieve 6.59 BLEU score, which is 1.03 higher than that of Graph2Seq++. Then the BLEU score can be further improved to 6.72 by introducing a vertex-only variant EKG+GAT(V). Comparing EKG+GAT(V) and EKG+GAT(V+E) to the EKG, the BLEU scores increase 0.13 and 0.42 respectively; it indicates the usefulness of the graph encoder and that the evolutionary knowledge from edges can be treated as a good supplement to that of vertices. In human evaluations, EKG+GAT(V+E) and EKG+GAT(V) have higher relevance and informativeness scores than that of the EKG. It also indicates that the graph encoder can effectively utilize the evolutionary knowledge from vertices and edges, and make the generated comments more relevant and informative.\n\\paragraph{Analysis of the number of entities:} the corresponding local EKG is constructed based on the entities from the passage. To explore the influence of the number ($N$) of entities, we report BLEU scores of our full model based on different number\\footnote{We do not report the BLEU score of the full model when $N=1$ because there are no edges included.} in Figure~\\ref{fig:ablation}(a). The best BLEU score is achieved at $N=5$. The BLEU score at $N=0$ belongs to the Transformer(Trans.). And our full model is robust to the number of entities because the BLEU scores are stable when N is in the range $[2, 7]$.\n\\paragraph{Analysis of the number of time periods:} we also report the BLEU scores under the different number of time periods in Figure~\\ref{fig:ablation}(b). Our full model achieves the best BLEU score of 7.01 at $N=4$, which is 0.41 higher than that of the static graph at $N=1$. It illustrates that the dynamic knowledge is useful for improving the performance.\n\n\\paragraph{Case study:} we provide a case study here. Four passages that need to be commented are extracted from a novel chronologically and shown in Table~\\ref{case-table}. For comparison, we use \\textbf{T}rans.+CTX and \\textbf{G}raph2Seq++, which have the best relevance and informativeness scores among baselines respectively. To start with, within each case, we find that the generated comments from our model are more informative, while the generated outputs from other models tend to be general or common replies, which proves the effectiveness of our knowledge usage.\n\nFrom another perspective, we observe that our model can make use of the dynamics of knowledge. Let us take a look at \\textbf{P3}, our generated comment describes that \\textit{Zeng Guofan} was born in the south of the country, which is in accordance with the passage described in \\textbf{P1}. Similar interactions can be found in all four cases, which support our claims above.\n\\section{Conclusion}\nIn this paper, we propose to encode evolutionary knowledge for automatic commenting long novels. We learn an \\textit{Evolutionary Knowledge Graph} under a multi-task framework and then design a graph-to-sequence model to utilize the EKG for generating comments. In addition, we collect a new generation dataset called \\textit{GraphNovel} to advance the corresponding research. Experimental results show that our EKG-based model is superior to several strong baselines on both automatic metrics and human evaluations. In the future, we plan to develop new graph-based encoders to generate personalized comments with this dataset.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nAnisotropy of molecular interactions plays an important role in many\nphysical, chemical and biological processes. Attractive forces are\nresponsible for the tendency toward particle association, while the\ndirectionality of the resulting bonds determines the geometry of the\nresulting clusters. Aggregation may thus lead to very different structures:\nin particular, chains, globular forms, and bi- or three-dimensional\nnetworks. Understanding the microscopic mechanisms underlying such phenomena\nis clearly very important both from a theoretical and a technological point\nof view. Polymerization of inorganic molecules, phase behaviour of\nnon-spherical colloidal particles, building up of micelles, gelation,\nformation of $\\alpha $-helices from biomolecules, DNA-strands, and other\nordered structures in living organisms, protein folding and crystallization,\nself-assembly of nanoparticles into composite objects designed for new\nmaterials, are all subjects of considerable interest, belonging to the same\nclass of systems with anisotropic interactions.\n\nModern studies on these complex systems strongly rely upon computer\nsimulations, which have provided a number of useful information about many\nproperties of molecular fluids.\n\nNevertheless, analytic models with explicit expressions for structural and\nthermodynamic properties still represent an irreplaceable tool, in view of\ntheir ability of capturing the essential features of the investigated\nphysical systems.\n\nAt the lowest level in this hierarchy of minimal models on assembling\nparticles lies the problem of the formation of linear aggregates, from\ndimers \\cite{Spinozzi02,Giacometti05} up to polymer chains. This topic has\nbeen extensively investigated, through both computer simulations and\nanalytical methods. In the latter case a remarkable example is Wertheim's\nanalytic solution of the \\textit{mean spherical approximation} (MSA)\nintegral equation for dipolar hard spheres (DHS), i.e. hard spheres (HS)\nwith a point dipole at their centre \\cite{Wertheim71} (hereafter referred to\nas I). For the DHS model, several studies predict chain formation, whereas\nlittle can be said about the existence of a fluid-fluid coexistence line,\nsince computer simulations and mean field theories provide contradictory\nresults \\cite{Weis93,Leeuwen93,Sear96,Camp00,Tlusty00}. On the other hand,\nfor mesoscopic fluids the importance of combining \\textit{short-ranged}\nanisotropic attractions and repulsions has been well established \\cite%\n{Gazzillo06,Fantoni07}, and hence the long-range of the dipolar interaction\nis less suited for the mesoscopic systems considered here, at variance with\ntheir atomistic counterpart.\n\nThe aim of the present paper is to address both the above points, by\nstudying a model with anisotropic surface adhesion that is amenable to an\nanalytical solution, within an approximation which is expected to be valid\nat significant experimental regimes.\n\nIn the isotropic case, the first model with `surface adhesion' was\nintroduced long time ago by Baxter \\cite{Baxter68,Baxter71}. The interaction\npotential of these `sticky hard spheres' (SHS) includes a HS repulsion plus\na spherically symmetric attraction, described by a square-well (SW) which\nbecomes infinitely deep and narrow, according to a limiting procedure\n(Baxter's sticky limit) that keeps the second virial coefficient finite.\n\nPossible anisotropic variations include `sticky points'\\ \\cite%\n{Sciortino05,Bianchi06,Michele06,Lomakin99,Starr03,Zhang04,Glotzer04a,Glotzer04b,Sciortino07}%\n, `sticky patches'\\ \\cite%\n{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07} and, more\nrecently, `Gaussian patches'\\ \\cite{Wilber06,Doye07}. The most common\nversion of patchy sticky models refers to HS with one or more `uniform\ncircular patches', all of the same species. This kind of patch has a\nwell-defined circular boundary on the particle surface, and is always\nattractive, with an `uniform' strength of adhesion, which does not depend on\nthe contact point within the patch \\cite{Jackson88}.\n\nIn the present paper we consider a `dipolar-like' SHS model, where the sum\nof a uniform surface adhesion (isotropic background) plus an appropriate\ndipolar sticky correction -- which can be both positive or negative,\ndepending on the orientations of the particles -- yields a nonuniform\nadhesion. Although the adhesion varies continuously and no discontinuous\nboundary exists, the surface of each molecule may be regarded as formed by\ntwo hemispherical `patches' (colored red and blue, respectively, in the\nonline Figure 1). One of these hemispheres is `stickier' than the other, and\nthe entire molecular surface is adhesive, but its stickiness is nonuniform\nand varies in a dipolar fashion. By varying the dipolar contribution, the\ndegree of anisotropy can be changed, in such a way that the total sticky\npotential can be continuously tuned from very strong attractive strength\n(twice the isotropic one) to vanishing adhesion (HS limit). The physical\norigin of this model may be manifold (non-uniform distribution of surface\ncharges, or hydrophobic attraction, or other physical mechanisms), one\nsimple realization being as due to an `extremely screened' attraction. The\npresence of a solvent together with a dense ionic atmosphere could induce\nany electrostatic interaction to vanish close to the molecular surface, and\n-- in the idealized sticky limit -- to become \\textit{truncated }exactly at\ncontact.\n\nFor this model, we solve analytically the molecular Ornstein-Zernike (OZ)\nintegral equation, by using a truncated \\textit{Percus-Yevick} (PY)\napproximation, \\textit{with orientational linearization }(PY-OL), since it\nretains only the lowest order terms in the expansions of the correlation\nfunctions in angular basis functions. This already provides a clear\nindication of the effects of anisotropy on the adhesive adhesion.\n\nThe idea of an anisotropic surface adhesion is not new. In a series of\npapers on hydrogen-bonded fluids such a water, Blum and co-workers \\cite%\n{Cummings86,Wei88,Blum90} already studied models of spherical molecules with\nanisotropic pair potentials, including both electrostatic multipolar\ninteractions and sticky adhesive terms of multipolar symmetry. Within\nappropriate closures, these authors outlined the general features of the\nanalytic solutions of the OZ equation by employing a very powerful formalism\nbased upon expansions in rotational invariants. In particular, Blum,\nCummings and Bratko \\cite{Blum90} obtained an analytic solution within a\nmixed MSA\/PY closure (extended to mixtures by Protsykevich \\cite%\n{Protsykevich03}) for molecules which have surface adhesion of dipolar\nsymmetry and at most dipole-dipole interactions. From the physical point of\nview, our model -- with `dipolar-like' adhesion resulting from the sum of an\nisotropic plus a dipolar term -- is different and more specifically\ncharacterized with respect to the one of Ref. [32], whose adhesion has a\nsimpler, strictly `dipolar', symmetry. From the mathematical point of view,\nhowever, the same formalism employed by Blum \\textit{et al.} \\cite{Blum90}\ncould also be applied to our model. Unfortunately, the solution given in\nRef. [32] is not immediately usable for the actual computation of\ncorrelation functions, since the explicit determination of the parameters\ninvolved in their analytical expressions is lacking.\n\nIn the present paper we adopt a simpler solution method, by extending the\nelegant approach devised by Wertheim for DHS within the MSA closure \\cite%\n{Wertheim71}, and, most importantly, we aim at providing a \\textit{complete}\nanalytic solution -- including the determination of all required parameters\n-- within our PY-OL approximation.\n\nThe paper is organized as follows. Section II defines the model. In Section\nIII we recall the molecular OZ integral equation and the basic formalism. In\nSection IV we present the analytic solution. Numerical exact results for\nsome necessary parameters, as well as very accurate analytic approximations\nfor them, will be shown in Section V. Some preliminary plots illustrating\nthe effects of the anysotropic adhesion on the local structure are reported\nin Section VI. Phase stability is breafly discussed in Section VII, while\nfinal remarks and conclusions are offered in Section VIII.\\bigskip\n\n\\bigskip\n\n\\section{HARD SPHERES WITH\\ ADHESION OF\\ DIPOLAR-LIKE\\ SYMMETRY}\n\nLet the symbol $i\\equiv \\left( \\mathbf{r}_{i},\\Omega _{i}\\right) $ (with $%\ni=1,2,3,\\ldots $) denote both the position $\\mathbf{r}_{i}$ of the molecular\ncentre and the orientation $\\Omega _{i}$ of molecule $i$; for linear\nmolecules, $\\Omega _{i}\\equiv \\left( \\theta _{i},\\varphi _{i}\\right) $\nincludes the usual polar and azimuthal angles. Translational invariance for\nuniform fluids allows to write the dependence of the pair correlation\nfunction $g\\left(1,2\\right)$ as \n\\begin{equation*}\n(1,2)=(\\mathbf{r}_{12},\\Omega _{1},\\Omega _{2})=(r,\\Omega _{1},\\Omega _{2},%\n\\widehat{\\mathbf{r}}_{12})=(r,\\Omega _{1},\\Omega _{2},\\Omega _{r}),\n\\end{equation*}%\nwith $\\mathbf{r}_{12}=\\mathbf{r}_{2}-\\mathbf{r}_{1}$, $r=|\\mathbf{r}_{12}|$,\nand $\\Omega _{r}$ being the solid angle associated with $\\widehat{\\mathbf{r}}%\n_{12}=\\mathbf{r}_{12}\/r.$\n\nIn the spirit of Baxter's isotropic counterpart \\cite{Baxter68,Gazzillo04},\nour model is defined by the Mayer function given by \n\\begin{equation}\nf^{\\mathrm{SHS}}(1,2)=f^{\\mathrm{HS}}(r)+t\\ \\epsilon (1,2)\\ \\sigma \\delta\n\\left( r-\\sigma \\right) ,  \\label{eq2}\n\\end{equation}%\nwhere $f^{\\mathrm{HS}}(r)=\\Theta \\left( r-\\sigma \\right) -1$ is its HS\ncounterpart, $\\Theta $ is the Heaviside step function ($\\Theta (x<0)=0$, $%\n\\Theta (x>0)=1$) and $\\delta \\left( r-\\sigma \\right) $ the Dirac delta\nfunction, which ensures that the adhesive interaction occurs only at contact\n($\\sigma $ being the hard sphere diameter). An appropriate limit of the\nfollowing particular square well potential of width $R-\\sigma $ \n\\begin{equation*}\n\\Phi ^{\\mathrm{SW}}\\left( 1,2\\right) =\\left\\{ \n\\begin{array}{ccc}\n+\\infty \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ } &  & 0<r<\\sigma \\\\ \n-k_{B}T\\ \\ln \\left[ 1+t\\ \\epsilon (1,2)\\ \\frac{\\sigma }{R-\\sigma }\\ \\ \\right]\n&  & \\sigma <r<R \\\\ \n0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ } &  & \\text{ \\ \\ \\ \\ \\ \\ \\ }r>R\\text{ ,}%\n\\end{array}%\n\\right.\n\\end{equation*}%\ncan be shown to lead to Eq. (\\ref{eq2}).\n\nThe angular dependence is buried in the angular factor%\n\\begin{equation}\n\\epsilon (1,2)=1+\\alpha D(1,2),\n\\end{equation}%\nincluding the dipolar function%\n\\begin{equation*}\nD(1,2)=D(\\Omega _{1},\\Omega _{2},\\Omega _{r})=3(\\mathbf{u}_{1}\\cdot \\hat{%\n\\mathbf{r}})(\\mathbf{u}_{2}\\cdot \\hat{\\mathbf{r}})-\\mathbf{u}_{1}\\cdot \n\\mathbf{u}_{2}\n\\end{equation*}%\nwhich stems from the dipole-dipole potential $\\phi ^{\\mathrm{dip-dip}%\n}(1,2)=-\\mu ^{2}D(1,2)\/r^{3}$ ($\\mu $ is the magnitude of the dipole moment)\nand is multiplied by the tunable \\textit{anisotropy parameter} $\\alpha $. In\nthe isotropic case, $\\alpha =0$, one has $\\epsilon (1,2)=1$. Here and in the\nfollowing, $\\hat{\\mathbf{r}}$ coincides with $\\hat{\\mathbf{r}}_{12}=-\\hat{%\n\\mathbf{r}}_{21}$ , while $\\mathbf{u}_{i\\text{ }}$is the versor attached to\nmolecule $i$ (drawn as yellow arrow in Figure 1) which completely determines\nits orientation $\\Omega _{i}$. Note the symmetry $D(2,1)=D(1,2)$.\n\nThe condition $\\epsilon (1,2)\\geq 0$ must be enforced in order to preserve a\ncorrect definition of the sticky limit, ensuring that the total sticky\ninteraction remains attractive for all orientations, and the range of\nvariability $-2\\leq D(1,2)\\leq 2$ yields the limitation $0\\leq \\alpha \\leq \n\\frac{1}{2}$ on the anisotropy degree. The stickiness parameter $t$ -- equal\nto $\\left( 12\\tau \\right) ^{-1}$ in Baxter's original notation \\cite%\n{Baxter68} -- measures the strength of surface adhesion relatively to the\nthermal energy $k_{B}T$ ($k_{B}$ being the Boltzmann constant, $T$ the\nabsolute temperature) and increases with decreasing temperature.\n\nIf we adopt an `inter-molecular reference frame' (with both polar axis and\ncartesian $z$-axis taken along $\\mathbf{r}_{12}$), then the cartesian\ncomponents of $\\hat{\\mathbf{r}}$ and $\\mathbf{u}_{i}$ are $(0,0,1)$ and $%\n(\\sin \\theta _{i}\\cos \\varphi _{i}$, $\\sin \\theta _{i}\\sin \\varphi _{i}$, $%\n\\cos \\theta _{i})$, respectively, and thus%\n\\begin{equation}\nD(1,2)=2\\cos \\theta _{1}\\cos \\theta _{2}-\\sin \\theta _{1}\\sin \\theta\n_{2}\\cos \\left( \\varphi _{1}-\\varphi _{2}\\right) .  \\label{eq4b}\n\\end{equation}\n\nThe strength of adhesion between two particles $1$ and $2$ at contact\ndepends -- in a continuous way -- on the relative orientation of $\\mathbf{u}%\n_{1}$ and $\\mathbf{u}_{2}$ as well as on the versor $\\widehat{\\mathbf{r}}%\n_{12}$ of the intermolecular distance. We shall call \\textit{parallel} any\nconfiguration with $\\mathbf{u}_{1}\\cdot \\mathbf{u}_{2}=1$, while \\textit{%\nantiparallel} configurations are those with $\\mathbf{u}_{1}\\cdot \\mathbf{u}%\n_{2}=-1$ (see Figure 1). For all configurations with $D(1,2)>0$, the\nanisotropic part of adhesion is attractive and adds to the isotropic one.\nThus, the surface adhesion is maximum, and larger than in the isotropic\ncase, when $\\mathbf{u}_{1}=\\mathbf{u}_{2}=$ $\\widehat{\\mathbf{r}}_{12}$ and\nthus $\\epsilon (1,2)=1+2\\alpha $ (head-to-tail parallel configuration, shown\nin Figure 1b). On the contrary, when $D(1,2)<0$ the anisotropic contribution\nis repulsive and subtracts from the isotropic one, so that the total sticky\ninteraction still remains attractive. Then, the stickiness is minimum, and\nmay even vanish for $\\alpha =1\/2$, when $\\mathbf{u}_{1\\text{ }}=-$ $\\mathbf{u%\n}_{2}=\\widehat{\\mathbf{r}}_{12}$ and thus $\\epsilon (1,2)=1-2\\alpha $\n(head-to-head or tail-to-tail antiparallel configurations, reported in\nFigure 1c). The intermediate case of \\textit{orthogonal }configuration ($%\n\\mathbf{u}_{2}$ perpendicolar to $\\mathbf{u}_{1}$) corresponds to $D(1,2)=0,$\nwhich is equivalent to the isotropic SHS interaction.\n\nIt proves convenient to `split' $f^{\\mathrm{SHS}}(1,2)$ as \n\\begin{equation}\n\\text{\\ }f^{\\mathrm{SHS}}(1,2)=f_{0}(r)+f_{\\mathrm{ex}}(1,2),  \\label{eq6a}\n\\end{equation}%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nf_{0}(r)=f^{\\mathrm{HS}}(r)+t\\ \\sigma \\delta \\left( r-\\sigma \\right) \\equiv\nf^{\\mathrm{isoSHS}}(r) \\\\ \nf_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ \\sigma \\delta \\left( r-\\sigma\n\\right) \\ D(1,2),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{eq6b}\n\\end{equation}%\nwhere the spherically symmetric $f_{0}(r)$ corresponds to the `reference'\nsystem with isotropic background adhesion, while $f_{\\mathrm{ex}}(1,2)$ is\nthe orientation-dependent `excess' term.\n\nWe remark that, as shown in Ref. I (see also Table I in Appendix A of the\npresent paper), convolutions of $f^{\\mathrm{SHS}}$-functions generate\ncorrelation functions with a more complex angular dependence. Therefore, in\naddition to $D(1,2)$, it is necessary to consider also \n\\begin{equation}\n\\Delta (1,2)=\\mathbf{u}_{1}\\cdot \\mathbf{u}_{2}\\text{ }=\\cos \\theta _{1}\\cos\n\\theta _{2}+\\sin \\theta _{1}\\sin \\theta _{2}\\cos \\left( \\varphi _{1}-\\varphi\n_{2}\\right) ,\\ \n\\end{equation}%\nwhere the last equality holds true in the inter-molecular frame. The limits\nof variation for $\\Delta (1,2)$ are clearly $-1\\leq \\Delta (1,2)\\leq 1$.\n\n\\section{BASIC FORMALISM}\n\nThis section, complemented by Appendix A, presents the main steps of \\\nWertheim's formalism, as well as its extension to our model.\n\n\\subsection{Molecular Ornstein-Zernike equation}\n\nThe \\textit{molecular OZ integral equation} for a pure and homogeneous fluid\nof molecules interacting via non-spherical pair potentials is\n\n\\begin{equation}\nh(1,2)=c(1,2)+\\rho \\int d\\mathbf{r}_{3}\\ \\left\\langle \\ c(1,3)\\ h(3,2)\\\n\\right\\rangle _{\\Omega _{3}}\\ ,  \\label{oz4}\n\\end{equation}%\nwhere $h(1,2)$ and $c(1,2)$ are the total and direct correlation functions,\nrespectively, $\\rho $ is the number density, and $g(1,2)=1+h(1,2)$ is the\npair distribution function \\cite{Friedman85,Lee88,Hansen06}. Moreover, the\nangular brackets with subscript $\\Omega $ denote an average over the\norientations, i.e. $\\left\\langle \\cdots \\right\\rangle _{\\Omega }=\\left( 4\\pi\n\\right) ^{-1}\\int d\\Omega \\ \\cdots .$\n\nThe presence of convolution makes convenient to Fourier transform (FT) this\nequation, by integrating with respect to the space variable $\\mathbf{r}$\nalone according to\n\n\\begin{equation}\n\\widehat{F}\\left( \\mathbf{k},\\Omega _{1},\\Omega _{2}\\right) =\\int d\\mathbf{r}%\n\\ F(\\mathbf{r},\\Omega _{1},\\Omega _{2})\\ \\exp (i\\mathbf{k\\cdot r}).\n\\label{oz5}\n\\end{equation}%\nThe $\\mathbf{r}$-space convolution becomes a product in $\\mathbf{k}$-space,\nthus leading to%\n\\begin{equation}\n\\widehat{h}(\\mathbf{k},\\Omega _{1},\\Omega _{2})=\\widehat{c}(\\mathbf{k}%\n,\\Omega _{1},\\Omega _{2})+\\rho \\ \\left\\langle \\widehat{c}(\\mathbf{k},\\Omega\n_{1},\\Omega _{3})\\ \\widehat{h}(\\mathbf{k},\\Omega _{3},\\Omega\n_{2})\\right\\rangle _{\\Omega _{3}}\\ .  \\label{oz6}\n\\end{equation}\n\nAs usual the OZ equation involves two unknown functions, $h$ and $c$, and\ncan be solved only after adding a closure, that is a second (approximate)\nrelationship among $c$, $h$ and the potential.\n\n\\subsection{Splitting of the OZ equation: reference and excess part}\n\nThe particular form of our potential, as defined by the Mayer function of\nEq. (\\ref{eq2}), gives rise to a remarkable exact splitting of the original\nOZ equation. Using diagrammatic methods \\cite{Friedman85,Lee88,Hansen06} it\nis easy to see that both $c$ and $h$ can be expressed as graphical series\ncontaining the Mayer function $f$ as \\textit{bond function}. If $\\ f^{%\n\\mathrm{SHS}}=f_{0}+f_{\\mathrm{ex}}$ is substituted into all graphs of the\nabove series, each diagram with $n$ $f$-bonds will generate $2^{n}$ new\ngraphs. In the cluster expansion of $c$, the sum of all graphs having only $%\nf_{0}$-bonds will yield $c_{0}(r)=c^{\\mathrm{isoSHS}}(r)$, i.e. the the\ndirect correlation function (DCF) of the reference fluid with isotropic\nadhesion. On the other hand, all remaining diagrams have \\textit{at least one%\n} $f_{\\mathrm{ex}}$-bond, whose expression is given by Eq. (\\ref{eq6b}).\nThus, in the sum of this second subset of graphs it is possible to factorize \n$\\alpha t$, and we can write \n\\begin{equation}\n\\text{\\ }c^{\\mathrm{SHS}}(1,2)=c_{0}(r)+c_{\\mathrm{ex}}(1,2),  \\label{oz7a}\n\\end{equation}%\n\\begin{equation}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\left\\{ \n\\begin{array}{c}\nc_{0}(r)=c^{\\mathrm{isoSHS}}(r),\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\n\\\\ \nc_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ c^{\\dagger }(1,2).\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{oz7b}\n\\end{equation}%\nSimilarly, for $h$ we get\n\n\\begin{equation}\n\\text{\\ }h^{\\mathrm{SHS}}(1,2)=h_{0}(r)+h_{\\mathrm{ex}}(1,2),  \\label{oz8a}\n\\end{equation}%\n\\begin{equation}\n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\\left\\{ \n\\begin{array}{c}\nh_{0}(r)=h^{\\mathrm{isoSHS}}(r),\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }\n\\\\ \nh_{\\mathrm{ex}}(1,2)=\\left( \\alpha t\\right) \\ h^{\\dagger }(1,2).\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{oz8b}\n\\end{equation}%\n\\ \\ \\ \\ \\ \n\nNote that this useful separation into reference and excess part may also be\nextended to other correlation functions, such as $\\gamma (1,2)\\equiv\nh(1,2)-c(1,2)$, $g(1,2)=1+h(1,2)$, and the `cavity' function $%\ny(1,2)=g(1,2)\/e(1,2)$. The function $\\gamma $ coincides with the OZ\nconvolution integral, without singular $\\delta $-terms. Similarly $y$ is\nalso `regular', and its exact expression reads $y\\left( 1,2\\right) =\\exp %\n\\left[ \\ \\gamma \\left( 1,2\\right) +B(1,2)\\ \\right] $, where the `bridge'\nfunction $B$ is defined by a complicated cluster expansion \\cite%\n{Friedman85,Lee88,Hansen06}.\n\nFrom Eqs. (\\ref{oz7a})-(\\ref{oz8b}), which are merely a consequence of the\nparticular form of $f_{\\mathrm{ex}}$ in the splitting of $f^{\\mathrm{SHS}}$,\none immediately sees that, if the anisotropy degree\\textit{\\ }$\\alpha $\ntends to zero, then%\n\\begin{equation}\n\\lim_{\\alpha \\rightarrow 0}c_{\\mathrm{ex}}(1,2)=\\lim_{\\alpha \\rightarrow\n0}h_{\\mathrm{ex}}(1,2)=\\lim_{\\alpha \\rightarrow 0}y_{\\mathrm{ex}}(1,2)=0.\n\\label{oz9}\n\\end{equation}\n\nNote that the spherically symmetric parts $c_{0}$ and $h_{0}$ must be\nrelated through the OZ equation for the reference fluid with isotropic\nadhesion (\\textit{reference OZ equation})\n\n\\begin{equation}\nh_{0}(r)=c_{0}(r)+\\rho \\int d\\mathbf{r}_{3}\\ c_{0}(r_{13})\\ h_{0}(r_{32}).\\ \n\\label{ozeq1}\n\\end{equation}%\nThus, substituting $c$ and $h$ of Eq. (\\ref{oz4}) with $c_{0}+c_{\\mathrm{ex}%\n} $ and $h_{0}+h_{\\mathrm{ex}}$, respectively, and subtracting Eq. (\\ref%\n{ozeq1}), we find that $c_{\\mathrm{ex}}$ and $h_{\\mathrm{ex}}$ must obey the\nfollowing relation\n\n\\begin{eqnarray*}\nh_{\\mathrm{ex}}(1,2) &=&c_{\\mathrm{ex}}(1,2)+\\rho \\int d\\mathbf{r}_{3}\\ %\n\\left[ \\ c_{0}(r_{13})\\ \\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle\n_{\\Omega _{3}}\\right. \\\\\n&&\\left. +\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ \\right\\rangle _{\\Omega\n_{3}}h_{0}(r_{32})+\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ h_{\\mathrm{ex}%\n}(3,2)\\ \\right\\rangle _{\\Omega _{3}}\\ \\right] \\ .\n\\end{eqnarray*}%\nand when \n\\begin{equation}\n\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ \\right\\rangle _{\\Omega\n_{3}}=\\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle _{\\Omega _{3}}=0\n\\label{eq_cond}\n\\end{equation}%\nthe orientation-dependent excess parts $c_{\\mathrm{ex}}$ and $h_{\\mathrm{ex}%\n} $ satisfy the equality \n\\begin{equation}\nh_{\\mathrm{ex}}(1,2)=c_{\\mathrm{ex}}(1,2)+\\rho \\int d\\mathbf{r}_{3}\\\n\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\ h_{\\mathrm{ex}}(3,2)\\ \\right\\rangle\n_{\\Omega _{3}}\\ ,  \\label{ozeq2}\n\\end{equation}%\nwhich is decoupled from that of the reference fluid and may be regarded as\nan OZ equation for the excess part (\\textit{excess OZ equation}). As we\nshall see, condition (\\ref{eq_cond}) is satisfied in our scheme.\n\nWe stress that, in principle, the closures for Eq. (\\ref{ozeq1}) and Eq. (%\n\\ref{ozeq2}), respectively, might be \\textit{different}. In addition,\nalthough the two OZ equations\\ are decoupled, a suitably selected closure\nmight establish a relationship between $F_{0}$ and $F$ $(F=c,h).$\n\n\\subsection{Percus-Yevick closure with orientational linearization}\n\nFor hard-core fluids, $h$ and $c$ inside the core are given by \n\\begin{equation}\n\\left\\{ \n\\begin{array}{ccc}\nh(1,2)=-1\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } &  & \\text{for \\ }0<r<\\sigma ,\n\\\\ \nc(1,2)=-\\left[ 1+\\gamma (1,2)\\right] &  & \\text{ for \\ }0<r<\\sigma .%\n\\end{array}%\n\\right.  \\label{oz6a}\n\\end{equation}%\nAt the same time, we have the following \\textit{exact} relations \n\\begin{eqnarray*}\nh(1,2) &=&g(1,2)-1=e(1,2)y\\left( 1,2\\right) -1, \\\\\nc(1,2) &=&f(1,2)\\left[ 1+\\gamma (1,2)\\right] +e(1,2)\\left[ y\\left(\n1,2\\right) -1-\\gamma (1,2)\\right] .\n\\end{eqnarray*}\n\nSince $c$, $h$ and $g$ are discontinuous for hard-core fluids and involve $%\n\\delta $-terms for sticky particles, it is more convenient to define\nclosures in terms of $y$ and $\\gamma $, which are still continuous and\nwithout $\\delta $-singularities. The \\textit{Percus-Yevick} approximation\nfor molecular fluids with orientation-dependent interactions corresponds to\nassuming%\n\\begin{equation}\ny^{\\mathrm{PY}}\\left( 1,2\\right) =1+\\gamma (1,2)\\text{ \\ \\ \\ everywhere,}\n\\label{w3a}\n\\end{equation}%\nand thus, for the DCF,%\n\\begin{equation}\nc^{\\mathrm{PY}}\\left( 1,2\\right) =f(1,2)\\ \\left[ \\ 1+\\gamma (1,2)\\ \\right] \n\\text{,}  \\label{w3b}\n\\end{equation}%\nwhich implies that $c$ vanishes beyond the range of the potential.\n\nHowever, the dependence of $\\gamma (1,2)$ on angles may still be very\ncomplex. A possible procedure is to perform a series expansion of all\ncorrelation functions in terms of an \\textit{infinite} set of \\textit{%\nrotational invariants}, which are angular basis functions -- related to the\nspherical harmonics -- having the property of rotational invariance valid\nfor homogeneous fluids \\cite{Blum71}. Unfortunately, the full PY\napproximation requires an infinite number of expansion coefficients for both \n$c(1,2)$ and $h(1,2)$. This approach is usually impracticable, but sometimes\neven unnecessary, as it is possible that the most significant angular basis\nfunctions are included in a small closed subset of that infinite set. Indeed\nthis happens, for instance, in the DHS model within the MSA \\cite{Wertheim71}%\n, where the set $\\{1,\\Delta ,D\\}$ is the required subset. Although this does \n\\textit{not} happen in our model, we shall argue that the same truncation is\nsufficient due to the dipolar symmetry of the anisotropic adhesion.\n\nIndeed, a natural assumption is that the only nonzero harmonics in $c(1,2)$\nand $h(1,2)$ are those contained in $f(1,2)$ and those which can be obtained\nfrom that set by convolution \\cite{Cummings86}. Now, the angular basis\nfunctions included in our $f$-bond are only $1$ and $D$, but the convolution\nof two $f$-bonds involves the angular average of two $D^{\\prime }$s, which\nyields \\cite{Wertheim71}%\n\\begin{equation*}\n\\ \\left\\langle \\ D(\\mathbf{k},\\Omega _{1},\\Omega _{3})\\ D(\\mathbf{k},\\Omega\n_{3},\\Omega _{2})\\ \\right\\rangle _{\\Omega _{3}}=\\frac{1}{3}\\left[ D(\\mathbf{k%\n},\\Omega _{1},\\Omega _{2})+2\\Delta (\\mathbf{k},\\Omega _{1},\\Omega _{2})%\n\\right]\n\\end{equation*}%\nin $\\mathbf{k}$-space, and thus generates also $\\Delta $. Consequently, we\nwill expand any angle-dependent correlation function $F$ as%\n\\begin{equation}\nF(1,2)=F_{0}(r)+F_{\\Delta }(r)\\Delta (1,2)+F_{D}(r)D(1,2)+\\cdots \\ ,\n\\label{oz7}\n\\end{equation}%\nneglecting all higher order terms. In other words, we assume that all\nangular series expansions can be \\textit{truncated} after these first three\nterms, \\textit{linear} with respect to the angular basis functions. Using\nthis spirit in the PY approximation, given by Eq. (\\ref{w3b}), we obtain the\nfollowing PY correlation functions with \\textit{orientational linearization}\n(OL):%\n\\begin{eqnarray}\nc^{\\mathrm{PY-OL}}(1,2) &=&c_{0}(r)+c_{\\Delta }(r)\\Delta (1,2)+c_{D}(r)D(1,2)\n\\notag \\\\\n&=&c_{0}(r)+\\left( \\alpha t\\right) \\ \\left[ \\ c_{\\Delta }^{\\dagger\n}(r)\\Delta (1,2)+c_{D}^{\\dagger }(r)D(1,2)\\ \\right] ,  \\label{oz10aa}\n\\end{eqnarray}%\nand\n\n\\begin{eqnarray}\nh^{\\mathrm{PY-OL}}(1,2) &=&h_{0}(r)+h_{\\Delta }(r)\\Delta (1,2)+h_{D}(r)D(1,2)\n\\notag \\\\\n&=&h_{0}(r)+\\left( \\alpha t\\right) \\ \\left[ \\ h_{\\Delta }^{\\dagger\n}(r)\\Delta (1,2)+h_{D}^{\\dagger }(r)D(1,2)\\ \\right] .  \\label{oz10bb}\n\\end{eqnarray}%\nwhere%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nc_{0}(r)=\\Lambda _{0}\\text{\\ }\\sigma \\delta \\left( r-\\sigma \\right) \\\\ \nc_{\\Delta }(r)=\\Lambda _{\\Delta }\\text{\\ }\\sigma \\delta \\left( r-\\sigma\n\\right) \\\\ \nc_{D}(r)=\\Lambda _{D}\\ \\sigma \\delta \\left( r-\\sigma \\right)%\n\\end{array}%\n\\right\\} \\qquad \\text{for \\ }r\\geq \\sigma ,  \\label{closure}\n\\end{equation}%\nwith%\n\\begin{equation}\n\\text{\\ \\ }\\left\\{ \n\\begin{array}{l}\n\\Lambda _{0}=y_{0}^{\\mathrm{PY}}(\\sigma )\\ t\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{\\Delta }=y_{\\Delta }^{\\mathrm{PY}}(\\sigma )\\ t\\text{\\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{D}=\\left[ \\ y_{D}^{\\mathrm{PY}}(\\sigma )+\\alpha \\ y_{0}^{\\mathrm{PY%\n}}(\\sigma )\\right] \\ t = y_{D}^{\\mathrm{PY}}(\\sigma )\\ t+\\alpha \\ \\Lambda\n_{0} ,%\n\\end{array}%\n\\right.  \\label{oz21}\n\\end{equation}%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\ny_{0}^{\\mathrm{PY}}\\left( r\\right) =1+\\gamma _{0}(r)\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \ny_{\\Delta }^{\\mathrm{PY}}(r)=\\gamma _{\\Delta }(r)=\\left( \\alpha t\\right) \\\ny_{\\Delta }^{\\dagger }(r)\\text{ \\ \\ } \\\\ \ny_{D}^{\\mathrm{PY}}(r)=\\gamma _{D}(r)=\\left( \\alpha t\\right) \\\ny_{D}^{\\dagger }(r).\\text{ \\ }%\n\\end{array}%\n\\right.  \\label{ysigma}\n\\end{equation}\n\nClearly for $f(1,2)$ no truncation is required, as the expansion \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nf_{0}(r)=f^{\\mathrm{isoSHS}}(r)=f^{\\mathrm{HS}}(r)+t\\text{\\ }\\sigma \\delta\n\\left( r-\\sigma \\right) \\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nf_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nf_{D}(r)=\\left( \\alpha t\\right) \\ \\sigma \\delta \\left( r-\\sigma \\right) \n\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.\n\\end{equation}%\nis exact.\n\nIt can be shown that $c(1,2)$ and $h(1,2)$ must have the same approximate\nform in view of the OZ equation, Eq. (\\ref{oz4}).\n\nThe solution of the original OZ equation (\\ref{oz4}) is then equivalent to\nthe calculation of the radial coefficients $c_{0}(r),c_{\\Delta }(r),c_{D}(r)$\nand $h_{0}(r),h_{\\Delta }(r),h_{D}(r)$, which are the projections of $c(1,2)$\nand $h(1,2)$ onto the angular basis $\\left\\{ 1,\\Delta ,D\\right\\} $. The \n\\textit{core condition} on $h$, Eq. (\\ref{oz6a}), becomes%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nh_{0}(r)=-1 \\\\ \nh_{\\Delta }(r)=0 \\\\ \nh_{D}(r)=0%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma .  \\label{oz10a}\n\\end{equation}\n\nNote that in the zero density limit $\\gamma (1,2)=\\rho \\int d\\mathbf{r}_{3}\\\n\\left\\langle \\ c(1,3)\\ h(3,2)\\ \\right\\rangle _{\\Omega _{3}}$ must vanish,\nand thus $y^{\\mathrm{PY}}(1,2)\\rightarrow 1$, i.e.\n\n\\begin{equation*}\n\\lim_{\\rho \\rightarrow 0}y_{0}^{\\mathrm{PY}}(r)=1\\text{, \\ \\ \\ \\ \\ \\ \\ }%\n\\lim_{\\rho \\rightarrow 0}y_{\\Delta }^{\\mathrm{PY}}(r)=\\lim_{\\rho \\rightarrow\n0}y_{D}^{\\mathrm{PY}}(r)=0,\n\\end{equation*}%\nwhile both $c\\left( 1,2\\right) $ and $h\\left( 1,2\\right) $ must reduce to $%\nf(1,2)$: \n\\begin{equation*}\n\\begin{array}{c}\n\\lim_{\\rho \\rightarrow 0}F_{0}(r)=f_{0}(r)\\text{ } \\\\ \n\\text{\\ }\\lim_{\\rho \\rightarrow 0}F_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\lim_{\\rho \\rightarrow 0}F_{D}(r)=f_{D}(r)\\text{ }%\n\\end{array}%\n\\text{\\ \\ \\ \\ \\ \\ }(F=c,h),\n\\end{equation*}%\nand%\n\\begin{equation}\n\\lim_{\\rho \\rightarrow 0}\\Lambda _{0}=t\\text{, \\ \\ \\ \\ \\ \\ \\ }\\lim_{\\rho\n\\rightarrow 0}\\Lambda _{\\Delta }=0\\text{, \\ \\ \\ \\ \\ \\ }\\lim_{\\rho\n\\rightarrow 0}\\Lambda _{D}=\\alpha t\\text{.}\n\\end{equation}\n\nMoreover as $\\alpha \\rightarrow 0$ all $\\Delta $- and $D$-coefficients of $%\nc,h$ and $y$ vanish, so that the isotropic adhesion case is recovered.\nFinally, it is also worth stressing that the \\textit{same} $\\delta $-term\narises in $c,$ $h$ and $g$, that is\n\n\\begin{equation*}\nF(1,2)=F_{\\mathrm{reg}}(1,2)+F_{\\text{sing}}(1,2)\\text{ \\ \\ \\ \\ \\ \\ (}%\nF=c,h,g),\n\\end{equation*}%\nwhere $F_{\\mathrm{reg}}$ is the `regular' part (i.e., the part with no $%\n\\delta $-singularity, and -- at most -- some step discontinuities), while $%\nF_{\\text{sing}}(1,2)=\\ \\sigma \\delta \\left( r-\\sigma \\right) \\Lambda (1,2)$\nis the singular term representing the anisotropic surface adhesion ( with$\\\n\\Lambda (1,2)=\\Lambda _{0}+\\Lambda _{\\Delta }\\Delta (1,2)+\\Lambda _{D}D(1,2)$\n\\ ).\n\n\\subsection{Integral equations for the projections of $c$ and $h$}\n\nIn the following, we extend Wertheim theory \\cite{Wertheim71} to our model,\nin order to obtain the radial projections of $c$ and $h$.\n\n\\ \\ The PY-OL approximation to the excess anisotropic part of the\ncorrelation functions is \n\\begin{equation}\n\\begin{array}{c}\nc_{\\mathrm{ex}}^{\\mathrm{PY-OL}}(1,2)=c_{\\Delta }(r)\\Delta\n(1,2)+c_{D}(r)D(1,2)\\text{ } \\\\ \nh_{\\mathrm{ex}}^{\\mathrm{PY-OL}}(1,2)=h_{\\Delta }(r)\\Delta\n(1,2)+h_{D}(r)D(1,2),%\n\\end{array}%\n\\end{equation}%\nthus verifying the required property $\\left\\langle \\ c_{\\mathrm{ex}}(1,3)\\\n\\right\\rangle _{\\Omega _{3}}=\\left\\langle \\ h_{\\mathrm{ex}}(3,2)\\\n\\right\\rangle _{\\Omega _{3}}=0$ described in Section III, and allowing the\nsplitting of the molecular OZ equation into a reference and an excess part.\n\nThe first part is the \\textit{reference} PY equation, and coincides with\nthat solved by Baxter for the fluid with isotropic adhesion \\cite%\n{Baxter68,Baxter71}:\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{cc}\nh_{0}(r)=c_{0}(r)+\\rho \\ (h_{0}\\star c_{0})\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ } &  \\\\ \nh_{0}(r)=-1\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ } & 0<r<\\sigma \\\\ \nc_{0}(r)=\\Lambda _{0}\\ \\sigma \\delta \\left( r-\\sigma \\right) \\text{ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } & r\\geq \\sigma ,%\n\\end{array}%\n\\right.  \\label{w1}\n\\end{equation}%\nwhere the symbol $\\star $ denotes spatial convolution, i.e. $(A\\star\nB)(r_{12})=\\int A(r_{13})B(r_{32})\\;d\\mathbf{r}_{3}$.\n\nThe second part is the \\textit{excess PY-OL equation}, given by Eq. (\\ref%\n{ozeq2}) coupled with the PY-OL closure. Following an extension of\nWertheim's approach, as described in detail in Appendix A, Eq. (\\ref{ozeq2})\ncan be splitted into the following system for the $\\Delta -$ and $D-$%\nprojections of $c$ and $h:$ \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nh_{\\Delta }(r)=c_{\\Delta }(r)+\\frac{1}{3}\\rho ~\\left( \\ c_{\\Delta }\\star\nh_{\\Delta }+2\\ c_{D}^{0}\\star h_{D}^{0}\\ \\right) \\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n} \\\\ \nh_{D}^{0}(r)=c_{D}^{0}(r)+\\frac{1}{3}\\rho ~\\left( \\ c_{\\Delta }\\star\nh_{D}^{0}+c_{D}^{0}\\star h_{\\Delta }+c_{D}^{0}\\star h_{D}^{0}\\ \\right) ,%\n\\end{array}%\n\\right.\n\\end{equation}%\nwhere $c_{D}^{0}(r)$ and $h_{D}^{0}(r)$ are defined by the relationship%\n\\begin{equation}\nF_{D}^{0}(r)=F_{D}(r)-3\\int_{r}^{\\infty }\\frac{F_{D}(x)}{x}\\ dx\\text{ \\ \\ \\\n\\ \\ \\ \\ \\ }(F=c,h),  \\label{oz12}\n\\end{equation}%\nwhose inverse is \\cite{Wertheim71} \n\\begin{equation}\nF_{D}(r)=F_{D}^{0}(r)-\\frac{3}{r^{3}}\\int_{0}^{r}F_{D}^{0}(x)\\ x^{2}\\ dx.\n\\label{oz12a}\n\\end{equation}\n\nThe core conditions become%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\n\\text{ \\ }h_{\\Delta }(r)=0\\text{ \\ \\ \\ \\ \\ \\ } \\\\ \nh_{D}^{0}(r)=-3K%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma ,  \\label{oz14a}\n\\end{equation}%\nwith \n\\begin{equation}\nK=\\int_{\\sigma ^{-}}^{\\infty }\\frac{h_{D}(x)}{x}\\ dx=K_{\\mathrm{reg}%\n}+\\Lambda _{D}\\ ,  \\label{oz14b}\n\\end{equation}%\n\\begin{equation}\nK_{\\mathrm{reg}}=\\int_{\\sigma }^{\\infty }\\frac{h_{D,\\text{\\textrm{reg}}}(r)}{%\nr}\\ dr.  \\label{oz14bb}\n\\end{equation}%\nNote that the presence of the $\\delta $-singularity in $h_{D}(x)$ requires\nthe specification of $\\sigma ^{-}$ as lower integration limit, unlike the\ncase of Ref. I where only the regular part $K_{\\mathrm{reg}}$ is present.\nMoreover, since $h_{D}(r)=\\alpha t\\ \\ h_{D}^{\\dagger }(r)$, from Eq. (\\ref%\n{oz14b}) one could also write%\n\\begin{equation}\nK=\\alpha t\\ \\mathcal{K}\\text{ ,}\n\\end{equation}%\nwhich shows that $K$ is related to the anisotropy degree, and vanishes both\nin the symmetric adhesion case ($\\alpha =0$) and in the HS limit ($t=0$).\nSince $h_{D}(r)\\rightarrow f_{D}(r)=\\left( \\alpha t\\right) \\sigma \\delta\n(r-\\sigma )$ in the zero density limit, one then finds that\n\n\\begin{equation}\n\\lim_{\\eta \\rightarrow 0}K=\\alpha t.\n\\end{equation}\n\nFinally, the PY-OL closure for the new DCFs reads%\n\\begin{equation}\n\\left. \n\\begin{array}{c}\nc_{\\Delta }(r)=\\Lambda _{\\Delta }\\ \\sigma \\delta \\left( r-\\sigma \\right) \\\\ \nc_{D}^{0}(r)=\\Lambda _{D}\\ \\sigma \\delta \\left( r-\\sigma \\right)%\n\\end{array}%\n\\right\\} \\text{ \\ \\ \\ \\ \\ }r\\geq \\sigma  \\label{oz14c}\n\\end{equation}%\n(for simplicity, here and in the following we omit the superscript PY-OL).\n\n\\subsection{Decoupling of the integral equations}\n\nIt is possible to \\textit{decouple} the two equations for $\\Delta $- and $D$%\n-coefficients, by introducing two new unknown functions, which are linear\ncombinations of the previous ones. As shown in Appendix A, if we define $%\nF_{1}(r)$ and $F_{2}(r)$ $\\left( F=c,h\\right) $ through the relations\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\nF_{1}\\left( r\\right) =\\left( 3\\mathcal{L}_{1}\\right) ^{-1}\\left[ F_{\\Delta\n}(r)-F_{D}^{0}(r)\\right] \\text{ } \\\\ \nF_{2}\\left( r\\right) =\\left( 3\\mathcal{L}_{2}\\right) ^{-1}\\left[ F_{\\Delta\n}(r)+2F_{D}^{0}(r)\\right]%\n\\end{array}%\n\\right. \\text{\\ \\ \\ \\ }\\left( F=c,h\\right) ,\n\\end{equation*}%\nthen we get the OZ equations\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\nh_{1}(r)=c_{1}(r)+\\rho _{1}\\ (h_{1}\\star c_{1})\\text{ } \\\\ \nh_{2}(r)=c_{2}(r)+\\rho _{2}\\ (h_{2}\\star c_{2}),%\n\\end{array}%\n\\right.\n\\end{equation*}%\nwith the following densities and core conditions\n\n\\begin{equation*}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{1}=\\mathcal{L}_{1}\\ \\rho \\text{ } \\\\ \n\\rho _{2}=\\mathcal{L}_{2}\\ \\rho ,%\n\\end{array}%\n\\right. \\qquad \\qquad \\left\\{ \n\\begin{array}{c}\nh_{1}(r)=K\/\\mathcal{L}_{1}\\text{ \\ \\ \\ } \\\\ \nh_{2}(r)=-2K\/\\mathcal{L}_{2}%\n\\end{array}%\n\\right. \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma .\n\\end{equation*}%\n\\ \\ \n\nThe decoupling of the three different projections of $c$ and $h$ is\nremarkable: the molecular anisotropic OZ equation reduces to a set of three\nradial integral relations, which may be regarded as OZ equations for three\n`hypothetical' fluids (labelled as $0,1,2$) with \\textit{spherically\nsymmetric} interactions. We stress that there is not a unique solution to\nthe decoupling problem, since - in principle - there exist infinite possible\nchoices for $\\left( \\mathcal{L}_{1},\\mathcal{L}_{2}\\right) $. The final\nresults are clearly independent of the values of $\\left( \\mathcal{L}_{1},%\n\\mathcal{L}_{2}\\right) $.\n\nIn the present paper, we adopt Wertheim's choice, i.e. $\\mathcal{L}_{1}=-K$\nand $\\mathcal{L}_{2}=2K$, which leads to%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{1}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D}^{0}(r)-F_{\\Delta }(r)\\right] \n\\text{ } \\\\ \nF_{2}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D}^{0}(r)+\\frac{1}{2}F_{\\Delta\n}(r)\\right]%\n\\end{array}%\n\\right. \\text{\\ \\ \\ \\ }\\left( F=c,h\\right) ,\n\\end{equation}\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{1}=-K\\rho \\text{ } \\\\ \n\\rho _{2}=2K\\rho ,%\n\\end{array}%\n\\right. \\qquad \\qquad \\left\\{ \n\\begin{array}{c}\nh_{1}(r)=-1 \\\\ \nh_{2}(r)=-1%\n\\end{array}%\n\\right. \\text{ \\ \\ \\ \\ \\ for \\ }0<r<\\sigma\n\\end{equation}%\n(in Ref. I, $F_{1}$ and $F_{2}$ were denoted as $F_{-}$ and $F_{+}$,\nrespectively).\n\nNote that the auxiliary fluids have densities different from that of the\nreference fluid (the negative sign of $\\rho _{1}$ poses no special\ndifficulty).\n\nWe can also write\n\n\\begin{equation}\nF_{m}\\left( r\\right) =F_{m,\\mathrm{reg}}(r)+\\Lambda _{m}\\ \\sigma \\delta\n\\left( r-\\sigma \\right) ,  \\label{oz15a}\n\\end{equation}%\nwith\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{1,\\mathrm{reg}}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D,\\mathrm{reg}%\n}^{0}(r)-F_{\\Delta ,\\mathrm{reg}}(r)\\right] \\text{ } \\\\ \nF_{2,\\mathrm{reg}}\\left( r\\right) =\\frac{1}{3K}\\left[ F_{D,\\mathrm{reg}%\n}^{0}(r)+\\frac{1}{2}F_{\\Delta ,\\mathrm{reg}}(r)\\right]%\n\\end{array}%\n\\right.  \\label{oz15b}\n\\end{equation}%\nand%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\Lambda _{1}=\\frac{1}{3K}\\left( \\Lambda _{D}-\\Lambda _{\\Delta }\\text{\\ }%\n\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n=\\frac{1}{3K}\\left[ h_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})-h_{\\Delta ,\\text{%\n\\textrm{reg}}}(\\sigma ^{+})\\right] \\ t\\text{ }+\\alpha \\ \\frac{1}{3K}\\\n\\Lambda _{0}\\text{ \\ } \\\\ \n\\text{\\ \\ \\ \\ \\ \\ } \\\\ \n\\Lambda _{2}=\\frac{1}{3K}\\left( \\Lambda _{D}+\\frac{1}{2}\\Lambda _{\\Delta\n}\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\text{ \\ \\ }=\\frac{1}{3K}\\left[ h_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})+\\frac{%\n1}{2}h_{\\Delta ,\\text{\\textrm{reg}}}(\\sigma ^{+})\\right] \\ t\\text{ }+\\alpha\n\\ \\frac{1}{3K}\\ \\Lambda _{0}\\text{.\\ \\ }%\n\\end{array}%\n\\right.  \\label{oz16}\n\\end{equation}%\n(since $\\gamma _{\\ldots }(\\sigma )=h_{\\ldots ,\\text{\\textrm{reg}}}(\\sigma\n^{+})-c_{\\ldots ,\\text{\\textrm{reg}}}(\\sigma ^{+})$, and $c_{\\ldots ,\\text{%\n\\textrm{reg}}}(\\sigma ^{+})=0$ within the PY-OL closure).\n\nKnowing the correlation functions $F_{1}(r)$ and $F_{2}(r)$ $\\left( \\text{%\nwith }F=c,h\\right) $, one can derive $F_{\\Delta }\\left( r\\right)\n,F_{D}^{0}\\left( r\\right) $, i.e. \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{\\Delta }\\left( r\\right) =2K\\left[ F_{2}\\left( r\\right) -F_{1}\\left(\nr\\right) \\right] \\ \\text{\\ \\ \\ \\ } \\\\ \nF_{D}^{0}\\left( r\\right) =2K\\left[ F_{2}\\left( r\\right) +\\frac{1}{2}%\nF_{1}\\left( r\\right) \\right] \\ ,%\n\\end{array}%\n\\right.  \\label{so11}\n\\end{equation}%\nand \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\Lambda _{\\Delta }=2K\\left( \\Lambda _{2}-\\Lambda _{1}\\right) \\text{\\ \\ \\ \\ }\n\\\\ \n\\Lambda _{D}=K\\left( 2\\Lambda _{2}+\\Lambda _{1}\\right) \\text{ .\\ \\ }%\n\\end{array}%\n\\right.  \\label{c10}\n\\end{equation}\n\nFinally, from $F_{\\Delta }\\left( r\\right) ,F_{D}^{0}\\left( r\\right) $ one\nhas to evaluate $F_{\\Delta }\\left( r\\right) ,F_{D}\\left( r\\right) $, by\nemploying Eq. (\\ref{oz12a}). We note the following points:\n\ni) Insertion of $h_{D}^{0}(r)=h_{D,\\text{\\textrm{reg}}}^{0}(r)+\\Lambda _{D}\\\n\\sigma \\delta (r-\\sigma )$ into Eq. (\\ref{oz12a}) yields $h_{D}(r)=h_{D,%\n\\text{\\textrm{reg}}}(r)+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma ),$ with%\n\\begin{equation}\nh_{D,\\text{\\textrm{reg}}}(r)=\\left\\{ \n\\begin{array}{cc}\n0\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }0<r<\\sigma ,\\text{\\ \\ \\ \\ \\ \\ } &  \\\\ \nh_{D,\\text{\\textrm{reg}}}^{0}(r)+3r^{-3}\\left[ K_{\\mathrm{reg}}\\sigma\n^{3}-\\int_{\\sigma }^{r}h_{D,\\text{\\textrm{reg}}}^{0}(x)\\ x^{2}\\ dx\\right] \n\\text{ \\ \\ \\ \\ \\ \\ \\ } & r>\\sigma .%\n\\end{array}%\n\\right.  \\label{c11}\n\\end{equation}%\nAt $r=2\\sigma $ $h_{D,\\text{\\textrm{reg}}}$ and $h_{D,\\text{\\textrm{reg}}%\n}^{0}$ have the same discontinuity. We also get\n\n\\begin{equation}\nh_{D,\\text{\\textrm{reg}}}(\\sigma ^{+})=h_{D,\\text{\\textrm{reg}}}^{0}(\\sigma\n^{+})+3K_{\\mathrm{reg}}.  \\label{f7}\n\\end{equation}%\nClearly, these results must agree with those obtained from Eq. (\\ref{oz12}),\ni.e. \n\\begin{equation*}\n\\begin{array}{c}\nh_{D}^{0}(r)=h_{D}(r)-3\\psi (r),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\psi (r)\\equiv \\int_{r}^{\\infty }h_{D}(x)\\ x^{-1}\\ dx=\\Lambda _{D}\\ \\theta\n\\left( \\sigma -r\\right) +\\int_{r}^{\\infty }h_{D,\\text{\\textrm{reg}}}(x)\\\nx^{-1}\\ dx.%\n\\end{array}%\n\\text{\\ }\n\\end{equation*}%\nIn order to recover Eq. (\\ref{f7}) along this second route, note that $\\psi\n(r)$ is not continuous at $r=\\sigma $. In fact, from Eqs. (\\ref{oz14b}) and (%\n\\ref{oz14bb}) follows $\\psi (\\sigma ^{-})=K$ whereas $\\psi (\\sigma ^{+})=K_{%\n\\mathrm{reg}}.$\n\nii) Similarly, for $c_{D}(r)$ we obtain $c_{D}(r)=c_{D,\\text{\\textrm{reg}}%\n}(r)+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma )$, with\n\n\\begin{equation}\nc_{D,\\text{\\textrm{reg}}}(r)=c_{D,\\text{\\textrm{reg}}}^{0}(r)-3r^{-3}\\left[\n\\int_{0}^{r}\\ c_{D,\\text{\\textrm{reg}}}^{0}(x)x^{2}\\ dx+\\Lambda _{D}\\sigma\n^{3}\\ \\theta (r-\\sigma )\\right] ,  \\label{c12}\n\\end{equation}%\nsince $\\int_{0}^{r}\\ \\delta (x-\\sigma )x^{2}dx=\\sigma ^{2}\\theta (r-\\sigma )$%\n. On the other hand, from Eq. (\\ref{oz12}) one easily finds that%\n\\begin{equation}  \\label{cDcD0}\nc_{D}(r)=c_{D}^{0}(r)\\text{ \\ \\ \\ for \\ }r\\geq \\sigma .\n\\end{equation}%\n\\ \\ \\ \n\niii) By applying the relationship (\\ref{oz12a}) to $c_{D}(r)$, using Eq. (%\n\\ref{cDcD0}) and noticing that $c_{D}(x)=0$ for $r>\\sigma $ within the PY-OL\napproximation, leads to a \\textit{sum rule}: \n\\begin{equation}\n\\int_{0}^{\\infty }c_{D}^{0}(x)\\ x^{2}\\ dx=\\int_{0}^{\\sigma }\\ c_{D,\\text{%\n\\textrm{reg}}}^{0}(x)x^{2}\\ dx+\\Lambda _{D}\\sigma ^{3}\\ =0,  \\label{f4b}\n\\end{equation}%\nthat we will exploit later.\n\n\\section{ANALYTIC\\ SOLUTION}\n\nWe have seen that the molecular PY-OL integral equation (IE) for our \\textit{%\nanisotropic}-SHS model splits into three IE's \n\\begin{equation}\n\\left\\{ \n\\begin{array}{cc}\nh_{m}(r)=c_{m}(r)+\\rho _{m}\\ (h_{m}\\star c_{m})\\text{\\ \\ } &  \\\\ \nh_{m}(r)=-1\\text{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } & \n0<r<\\sigma \\\\ \nc_{m}(r)=\\Lambda _{m}\\ \\sigma \\delta \\left( r-\\sigma \\right) \\text{ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ } & r\\geq \\sigma%\n\\end{array}%\n\\right. \\qquad \\left( m=0,1,2\\right) ,  \\label{gpy1}\n\\end{equation}%\nwhere\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\rho _{0}=\\rho \\text{ \\ \\ \\ \\ \\ } \\\\ \n\\rho _{1}=-K\\rho \\text{ } \\\\ \n\\rho _{2}=2K\\rho ,%\n\\end{array}%\n\\right.  \\label{gpy2}\n\\end{equation}%\nand the `amplitudes' of the adhesive $\\delta $-terms are \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\text{ }\\Lambda _{0}=\\left[ 1+h_{0,\\text{\\textrm{reg}}}(\\sigma ^{+})\\right]\n\\ t=y_{0}^{\\mathrm{PY}}(\\sigma )t\\text{ }\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\text{ }\\Lambda _{m}=h_{m,\\text{\\textrm{reg}}}(\\sigma ^{+})\\ t\\text{ }+%\n\\mathcal{P} = y_{m}^{\\mathrm{PY}}(\\sigma )t+\\mathcal{P}\\text{ \\ \\ \\ \\ \\ }%\n\\left( m=1,2\\right) ,\\text{ }%\n\\end{array}%\n\\right.  \\label{gpy3}\n\\end{equation}%\nwith%\n\\begin{equation}\n\\mathcal{P}=\\frac{1}{3}\\ \\frac{\\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\ }{K}+%\n\\frac{K_{\\mathrm{reg}}}{K}\\ t=\\frac{1}{3}\\ \\frac{\\alpha \\ \\Lambda _{0}\\ }{K}+%\n\\frac{K_{\\mathrm{reg}}}{K}\\ t.  \\label{gpy4}\n\\end{equation}%\nHere, the new expressions of $\\Lambda _{1}$ and $\\Lambda _{2}$ have been\nobtained from Eqs. (\\ref{oz16}) with the help of Eqs. (\\ref{f7}) and (\\ref%\n{oz15b}).\n\nThe essential difference with respect to Ref. I lies in the closure, which\nis -- of course -- related to the model potential. While Wertheim's paper on\nDHS \\cite{Wertheim71} employed the MSA closure, which performs properly for\nlong-ranged electrostatic potentials at low strength of interaction, our\nPY-OL closure is more appropriate for the short-ranged potential of the\npresent model.\n\nThe first integral equation IE0 is fully independent, whereas IE1 and IE2\ndepend on the solution of IE0 (unlike the case of Ref. I), because of the\npresence of $\\Lambda _{0}$ inside $\\Lambda _{1\\text{ }}$and $\\Lambda _{2}$.\nWhile IE0 is exactly the PY equation for the \\textit{reference} SHS with \n\\textit{isotropic} adhesion solved by Baxter \\cite{Baxter68,Baxter71}, IE1\nand IE2 are different from both Wertheim's MSA solution for DHS and Baxter's\nPY solution for SHS. We remark that the closures for IE1 and IE2 are not PY\nas $\\Lambda _{1\\text{ }}$and $\\Lambda _{2}$ -- given by Eq. (\\ref{gpy3}) --\ndiffer, by the term $\\mathcal{P}$, from those appropriate for the PY choice,\ncorresponding to $\\Lambda _{m}^{\\mathrm{PY}}=y_{m}^{\\mathrm{PY}}(\\sigma )t$.\n\nConsequently, IE1 and IE2 can be rekoned as belonging to a class of \\textit{%\ngeneralized PY }\\ (GPY) \\textit{approximations}, introduced in Ref. [39],\nwhich admit an analytic solution. Thus, the PY-OL closure for $c(1,2)$ leads\nto a PY integral equation for $c_{0}(r)$, coupled to a two GPY integral\nequations for $c_{1}(r)$ and $c_{2}(r)$ (which are linear combinations of $%\nc_{\\Delta }(r)$ and $c_{D}^{0}(r)$).\n\nOn comparing the three IE's and their closures given by Eq. (\\ref{gpy1}), it\nis apparent that they have exactly the same form, but differ by the density $%\n\\rho _{m}$ and the expression for $\\Lambda _{m}$. The first integral\nequation IE0 corresponds to an isotropic SHS fluid with density $\\rho $. On\nthe other hand, IE1 and IE2 refer to `auxiliary' isotropic SHS fluids with\ndensities $\\rho _{1}$ and $\\rho _{2}$, and adhesion parameters $\\Lambda _{1}$\nand $\\Lambda _{2}$, respectively. Note that, according to Eqs. (\\ref{oz16})\n, $\\Lambda _{m}$ is not evaluated at the actual density $\\rho _{m}$ of the\nauxiliary fluid, but at the real density $\\rho $ . These remarks strongly\nsuggests that the solutions of IE0, IE1 and IE2 can be expressed in terms of\nan \\textit{unique} solution -- the PY one for isotropic SHS -- by changing\nonly $\\rho _{m}$ and $\\Lambda _{m}$. This can be achieved by the formal\nmapping \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nF_{0}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{0},\\Lambda _{0})\\text{ } \\\\ \nF_{1}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{1},\\Lambda _{1})\\text{ } \\\\ \nF_{2}(r)=F^{\\mathrm{isoSHS}}(r;\\eta _{2},\\Lambda _{2}),%\n\\end{array}%\n\\right. \\qquad \\quad (F=q,c,h)  \\label{gpy5}\n\\end{equation}%\nwhere $\\eta _{0}=\\eta $ is the \\textit{real} volume fraction, while $\\eta\n_{1}$ and $\\eta _{2}$ are `modified volume fractions' of the `auxiliary'\nfluids $1$ and $2$, i.e.,\\ \n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\eta _{0}=\\eta \\equiv \\left( \\pi \/6\\right) \\rho \\sigma ^{3}\\text{ \\ \\ } \\\\ \n\\eta _{1}=-K\\eta \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \n\\eta _{2}=2K\\eta \\text{\\ .\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{gpy6}\n\\end{equation}%\nIn Eqs. (\\ref{gpy5}) $q(r)$ denotes the Baxter factor correlation function,\nintroduced in the next Subsection.\n\nIt is worth noting that this result for SHS mirrors the analog of the MSA\nsolution for DHS \\cite{Wertheim71} where all the three harmonic coefficients\nof can be expressed similarly, in terms of a single PY solution for the\nreference HS fluid.\n\n\\bigskip\n\n\\subsection{Baxter factorization}\n\nWe shall now solve Eqs. ($\\ref{gpy1}$) by using the Wiener-Hopf\nfactorization due to Baxter \\cite{Baxter71}. Let us recall its basic steps.\nAfter Fourier transforming the OZ equation for a one-component fluid with\nspherically symmetric interactions, one assumes the following factorization: \n\\begin{equation}\n\\begin{array}{c}\n1-\\rho \\widetilde{c}\\left( k\\right) =Q(k)Q(-k), \\\\ \nQ(k)=1-2\\pi \\rho \\int_{0}^{\\infty }q(r)\\ e^{ikr}dr.%\n\\end{array}\n\\label{bf1}\n\\end{equation}%\nThen it can be shown that the introduction of the `factor correlation\nfunction' $q(r)$ allows the OZ equation to be cast into the form \\cite%\n{Baxter71}\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nrc\\left( r\\right) =-q^{\\prime }(r)+2\\pi \\rho \\int_{0}^{\\infty }du\\ q\\left(\nu\\right) q^{\\prime }\\left( r+u\\right) ,\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nrh\\left( r\\right) =-q^{\\prime }(r)+2\\pi \\rho \\int_{0}^{\\infty }du\\ q\\left(\nu\\right) \\left( r-u\\right) h\\left( |r-u|\\right) ,%\n\\end{array}%\n\\right.  \\label{ie2b}\n\\end{equation}%\nwhere the prime denotes differentiation with respect to $r$. Solving these\nBaxter equations is tantamount to determining -- within a chosen closure --\nthe function $q(r)$, from which $c\\left( r\\right) $ and $h\\left( r\\right) $\ncan be easily calculated. It is also necessary to remember that, for \\textit{%\nall} closures leading to $c\\left( r\\right) =0$ for $r>\\sigma $, one finds $%\nq\\left( r\\right) =0$ for $r>\\sigma $ \\cite{Gazzillo04}.\n\nOn applying Baxter's factorization to Eqs. ($\\ref{gpy1}$), we get \n\\begin{equation}\nrh_{m}\\left( r\\right) =-q_{m}^{\\prime }(r)+2\\pi \\rho _{m}\\int_{0}^{\\sigma\n}du\\ q_{m}\\left( u\\right) \\left( r-u\\right) h_{m}\\left( |r-u|\\right) .\n\\label{ie3}\n\\end{equation}%\nwith $m=0,1,2$. Now the closure $c_{m}(r)=\\Lambda _{m}\\ \\sigma \\delta \\left(\nr-\\sigma \\right) $ for \\ $r\\geq \\sigma $ implies that the same $\\delta $%\n-term must appear in $h_{m}\\left( r\\right) $. Thus, for $0\\leq r\\leq \\sigma $%\n, using $h_{m}(r)=-1+\\Lambda _{m}\\ \\sigma \\delta \\left( r-\\sigma \\right) $,\nwe find\n\n\\begin{equation*}\nq_{m}^{\\prime }(r)=a_{m}r+b_{m}\\sigma -\\Lambda _{m}\\ \\sigma ^{2}\\delta\n\\left( r-\\sigma \\right) ,\n\\end{equation*}%\nwith%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\na_{m}=\\ 1-2\\pi \\rho _{m}\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ , \\\\ \nb_{m}\\sigma =\\ 2\\pi \\rho _{m}\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ u\\\n.\\text{ \\ \\ }%\n\\end{array}%\n\\right.  \\label{ie4}\n\\end{equation}\n\nThe $\\delta $-term of $q_{m}^{\\prime }(r)$ means that $q_{m}(r)$ has a\ndiscontinuity $q_{m}(\\sigma ^{+})-q_{m}(\\sigma ^{-})=-\\Lambda _{m}\\sigma\n^{2} $, with $q_{m}(\\sigma ^{+})=0.$ Integrating $q_{m}^{\\prime }(r)$,\nsubstituting this result into Eqs. ($\\ref{ie4}$), and solving the\ncorresponding algebraic system, we find the following solution\n\n\\begin{equation}\nq_{m}(r)=\\left\\{ \n\\begin{array}{cc}\n\\ \\frac{1}{2}a_{m}(r-\\sigma )^{2}+\\left( a_{m}+b_{m}\\right) \\sigma (r-\\sigma\n)\\ +\\Lambda _{m}\\ \\sigma ^{2} & \\text{ \\ \\ \\ \\ }0\\leq r\\leq \\sigma , \\\\ \n0 & \\text{ \\ \\ \\ otherwise,}%\n\\end{array}%\n\\right.  \\label{so1}\n\\end{equation}%\n\\begin{eqnarray}\na_{m} &=&\\ a^{\\mathrm{HS}}(\\eta _{m})-\\frac{12\\eta _{m}\\ \\Lambda _{m}\\ }{%\n1-\\eta _{m}}\\   \\label{so2} \\\\\n&&  \\notag \\\\\nb_{m} &=&\\ b^{\\mathrm{HS}}(\\eta _{m})+\\frac{6\\eta _{m}\\ \\Lambda _{m}\\ }{%\n1-\\eta _{m}}\\ \\   \\label{so3} \\\\\n&&  \\notag \\\\\n\\text{\\ }\\eta _{m} &=&\\left( \\pi \/6\\right) \\rho _{m}\\sigma ^{3}\\text{ \\ \\ \\ }\n\\\\\n&&  \\notag \\\\\na^{\\mathrm{HS}}(x) &=&\\frac{1+2x}{\\left( 1-x\\right) ^{2}},\\text{ \\ \\ \\ \\ \\ \\\n\\ }b^{\\mathrm{HS}}(x)=-\\frac{3x}{2\\left( 1-x\\right) ^{2}},\n\\end{eqnarray}\n\nFrom the first of Eqs. ($\\ref{ie2b}$) we get the DCFs $c_{m}(r)=c_{m,\\text{%\n\\textrm{reg}}}(r)+\\ \\Lambda _{m}\\ \\sigma \\delta (r-\\sigma )$, where $c_{m,%\n\\text{\\textrm{reg}}}(r)=0$ for $r\\geq \\sigma ,$ and for $0<r<\\sigma $\n\n\\begin{eqnarray}\nc_{m,\\text{\\textrm{reg}}}(r) &=&-\\frac{1}{2}\\eta _{m}\\ a_{m}^{2}\\left( \\frac{%\nr}{\\sigma }\\right) ^{3}+6\\eta _{m}\\ \\left[ \\ (a_{m}+b_{m})^{2}-2a_{m}\\\n\\Lambda _{m}\\ \\right] \\left( \\frac{r}{\\sigma }\\right)  \\notag \\\\\n&&-a_{m}^{2}-12\\eta _{m}\\ \\Lambda _{m}^{2}\\left( \\frac{r}{\\sigma }\\right)\n^{-1}.\n\\end{eqnarray}\n\nThe second of Eqs. ($\\ref{ie2b}$) yields the total correlation functions $%\nh_{m}(r)=h_{m,\\text{\\textrm{reg}}}(r)+\\ \\Lambda _{m}\\ \\sigma \\delta\n(r-\\sigma ).$ For $r>\\sigma $, Eqs. ($\\ref{ie3}$) becomes%\n\\begin{equation}\nH_{m,\\text{\\textrm{reg}}}\\left( r\\right) =12\\eta _{m}\\ \\sigma ^{-3}\\left\\{ \n\\begin{array}{cc}\n\\begin{array}{c}\n\\int_{0}^{r-\\sigma }du\\ q_{m}\\left( u\\right) \\ H_{m,\\text{\\textrm{reg}}%\n}\\left( r-u\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ } \\\\ \n+\\ \\int_{r-\\sigma }^{\\sigma }du\\ q_{m}\\left( u\\right) \\left( u-r\\right)\n+\\Lambda _{m}\\sigma ^{2}\\ q_{m}\\left( r-\\sigma \\right) \\text{ \\ \\ \\ }%\n\\end{array}\n& \\sigma <r<2\\sigma , \\\\ \n&  \\\\ \n\\int_{0}^{\\sigma }du\\ q_{m}\\left( u\\right) \\ H_{m,\\text{\\textrm{reg}}}\\left(\nr-u\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }r>2\\sigma ,\\text{\\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ } & \n\\end{array}%\n\\right.  \\label{so7}\n\\end{equation}%\nwhere $H_{m}\\left( r\\right) \\equiv rh_{m}\\left( r\\right) $. Due to the last\nterm of Eq. ($\\ref{so7}$) and the discontinuity of $q_{m}\\left( r\\right) $\nat $r=\\sigma $, $h_{m,\\text{\\textrm{reg}}}(r)$ has a jump of at $r=2\\sigma $ \n\\cite{Kranendonk88,Miller04}: $h_{m,\\text{\\textrm{reg}}}(2\\sigma ^{+})-h_{m,%\n\\text{\\textrm{reg}}}(2\\sigma ^{-})=-6\\eta _{m}\\ \\Lambda _{m}^{2}.$\n\n\\bigskip\n\n\\subsection{An important relationship}\n\nIn Appendix B it is shown that a remarkable consequence of the sum rule (\\ref%\n{f4b}) is the condition%\n\\begin{equation}\na_{2}=a_{1}\\text{ ,}  \\label{so10}\n\\end{equation}%\nthat will play a significant role in the determination of the unknown\nparameters $\\Lambda _{1},$ $\\Lambda _{2}$ and $K$ \\ (see Appendix B).\n\n\\bigskip\n\n\\subsection{Reference fluid coefficients}\n\nThe $m=0$ case corresponds to Baxter's PY results for the reference fluid of\nisotropic SHS particles \\cite{Baxter68,Baxter71}. We have: $q_{0}(r)=q^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0}),$ and%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{0}(r)=c^{\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})=c_{\\text{\\textrm{reg}}}^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})+\\Lambda _{0}\\ \\sigma \\delta (r-\\sigma\n)\\text{ } \\\\ \nh_{0}(r)=h^{\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})=h_{\\text{\\textrm{reg}}}^{%\n\\mathrm{isoSHS}}(r;\\eta ,\\Lambda _{0})+\\Lambda _{0}\\ \\sigma \\delta (r-\\sigma\n)%\n\\end{array}%\n\\right.  \\label{f1}\n\\end{equation}%\n(for simplicity, we omit -- here and in the following -- the superscript PY).\n\n\\subsection{$\\Delta -$ and $D-$coefficients}\n\nWe write $q_{m}(r)=q^{\\mathrm{isoSHS}}(r;\\eta _{m},\\Lambda _{m})$ with $%\nm=1,2 $. Then,\n\ni) For the $\\Delta $\\textit{-coefficients}, after recalling Eq. (\\ref{oz15a}%\n) and exploiting Eqs. (\\ref{c10}), we end up with:%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{\\Delta }(r)=2K\\left[ c_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda\n_{2})-c_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] +\\Lambda\n_{\\Delta }\\ \\sigma \\delta (r-\\sigma )\\text{\\ \\ \\ } \\\\ \nh_{\\Delta }(r)=2K\\left[ h_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda\n_{2})-h_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] +\\Lambda\n_{\\Delta }\\ \\sigma \\delta (r-\\sigma ).\\text{ \\ }%\n\\end{array}%\n\\right.  \\label{f2}\n\\end{equation}\n\nii) For the $D$\\textit{-coefficients}, we get\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nc_{D}^{0}(r)=2K\\left[ c_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda _{2})+%\n\\frac{1}{2}c_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] \\\n+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma )\\text{ \\ } \\\\ \nh_{D}^{0}(r)=2K\\left[ h_{0,\\text{\\textrm{reg}}}(r;2K\\eta ,\\Lambda _{2})+%\n\\frac{1}{2}h_{0,\\text{\\textrm{reg}}}(r;-K\\eta ,\\Lambda _{1})\\right] \\\n+\\Lambda _{D}\\ \\sigma \\delta (r-\\sigma ).\\text{ }%\n\\end{array}%\n\\right.  \\label{f3}\n\\end{equation}%\nFinally, from $c_{D}^{0}(r)$ and $h_{D}^{0}(r)$ we can calculate $c_{D}(r)$\nand $h_{D}(r)$, as described by Eqs. (\\ref{c12}) and (\\ref{c11}),\nrespectively.\n\nIn short, a) our PY-OL solution -- $\\left\\{ c_{0},c_{\\Delta },c_{D}\\right\\} $\nand $\\left\\{ h_{0},h_{\\Delta },h_{D}\\right\\} $ -- satisfies both the PY\nclosures and the core conditions; b) all\\textit{\\ }coefficients contain a\nsurface adhesive $\\delta -$term; c) $\\left\\{ h_{0},h_{\\Delta },h_{D}\\right\\} \n$ all exhibit a step discontinuity at $r=2\\sigma $.\n\n\\bigskip\n\n\\section{EVALUATION OF THE PARAMETERS $K$, $\\Lambda _{1}$ AND $\\Lambda _{2}$}\n\nThe calculation of the Baxter functions $q_{m}s$ ($m=0,1,2$) requires the\nevaluation of $K,$ $\\Lambda _{1}$, and $\\Lambda _{2}$, for a given set of $%\n\\alpha ,\\eta $ and $t$ values, a task that we address next.\n\n\\bigskip\n\n\\subsection{Exact expressions}\n\nFour equations are needed to find the three quantities $\\Lambda\n_{m}=q_{m}(\\sigma ^{-})\/\\sigma ^{2}$ $(m=0,1,2)$, as well as the parameter $%\nK\\left( \\eta ,t,\\alpha \\right) $. We stress that the \\textit{almost fully\nanalytical} determination of these unknown parameters was lacking in Ref.\n[32] and represents an important part of the present work. Our detailed\nanalysis is given in Appendix B, and we quote here the main results.\n\ni) For $\\Lambda _{0}$, the same PY equation found by Baxter for isotropic\nSHS \\cite{Baxter68,Baxter71} \n\\begin{equation}\n12\\eta t\\ \\Lambda _{0}^{2}-\\left( 1+\\frac{12\\eta }{1-\\eta }t\\right) \\Lambda\n_{0}+y_{\\sigma }^{\\mathrm{HS}}(\\eta )t=0.  \\label{b5}\n\\end{equation}%\nOnly the smaller of the two real solutions (when they exist) is physically\nsignificant \\cite{Baxter68,Baxter71}, and reads\n\n\\begin{equation}\n\\Lambda _{0}=y_{0}^{\\mathrm{PY}}(\\sigma )t=\\frac{y_{\\sigma }^{\\mathrm{HS}%\n}(\\eta )t}{\\frac{1}{2}\\left[ 1+\\frac{12\\eta }{1-\\eta }t+\\sqrt{\\left( 1+\\frac{%\n12\\eta }{1-\\eta }t\\right) ^{2}-48\\eta \\ y_{\\sigma }^{\\mathrm{HS}}(\\eta )\\\nt^{2}}\\right] },  \\label{p1}\n\\end{equation}\n\nii) For $\\Lambda _{1}$ and $\\Lambda _{2}$, two other quadratic equations,\ni.e. \\ \\ \\ \\ \n\\begin{equation}\n12\\eta _{m}t\\ \\Lambda _{m}^{2}-\\left( 1+\\frac{12\\eta _{m}}{1-\\eta _{m}}%\nt\\right) \\Lambda _{m}+h_{\\sigma }^{\\mathrm{HS}}(\\eta _{m})t=-\\mathcal{P}%\n\\text{\\quad ~~}(m=1,2).  \\label{p1b}\n\\end{equation}\n\niii) The fourth equation is the following linear relationship between $%\n\\Lambda _{1}$ and $\\Lambda _{2}$%\n\\begin{equation}\n\\frac{12\\eta _{2}\\ \\Lambda _{2}\\ }{1-\\eta _{2}}-\\frac{12\\eta _{1}\\ \\Lambda\n_{1}\\ }{1-\\eta _{1}}=\\frac{\\eta _{2}\\left( 4-\\eta _{2}\\right) \\ }{\\left(\n1-\\eta _{2}\\right) ^{2}}-\\frac{\\eta _{1}\\left( 4-\\eta _{1}\\right) \\ }{\\left(\n1-\\eta _{1}\\right) ^{2}},  \\label{p1c}\n\\end{equation}%\n$\\ $which stems from the condition $a_{2}=a_{1}$.\n\nThe analysis of Appendix B gives\n\n\\begin{equation}\n\\Lambda _{2}\\left( \\eta _{1},\\eta _{2},t,\\alpha \\right) =\\Lambda _{1}\\left(\n\\eta _{2},\\eta _{1},t,\\alpha \\right)\n\\end{equation}%\nwith\n\n\\begin{equation}\n\\Lambda _{m}=\\Lambda +\\Lambda _{m}^{\\mathrm{ex}}\\text{ \\ \\ \\ \\ \\ \\ \\ }(m=1,2)\n\\label{p2a}\n\\end{equation}%\n\\begin{equation}\n\\Lambda =\\frac{1}{3}+\\frac{1}{4}\\left( \\frac{\\eta _{1}}{1-\\eta _{1}}+\\frac{%\n\\eta _{2}}{1-\\eta _{2}}\\right) =\\frac{1}{3}+\\allowbreak \\frac{x(1+4x)}{%\n4\\left( 1+x\\right) \\left( 1-2x\\right) }  \\label{p2b}\n\\end{equation}%\n\\begin{equation}\n\\Lambda _{1}^{\\mathrm{ex}}=\\frac{\\eta _{2}}{4\\left( 1-\\eta _{2}\\right) }%\nW_{0}^{\\mathrm{ex}},\\qquad \\Lambda _{2}^{\\mathrm{ex}}=\\frac{\\eta _{1}}{%\n4\\left( 1-\\eta _{1}\\right) }W_{0}^{\\mathrm{ex}},  \\label{p2c}\n\\end{equation}%\nwhere we have introduced $\\eta _{1}=-x$, $\\eta _{2}=2x$ \\ ( $x\\equiv K\\eta $\n), and $W_{0}^{\\mathrm{ex}}$ is defined in Appendix B. All these quantites\nare analytic functions of $x=K\\eta $. Thus, to complete the solution, we\nneed an equation for $K$, which can be written as\n\n\\begin{equation}\nK=\\alpha t\\ \\mathcal{K},\\text{ }\\ \\ \\ \\ \\ \\text{with \\ \\ \\ }\\ \\mathcal{K}%\n\\text{ }=\\frac{y_{0}^{\\mathrm{PY}}(\\sigma )}{Z(\\eta _{1},\\eta _{2},t)},\n\\label{p3}\n\\end{equation}%\n\\begin{equation}\nZ=\\frac{3}{2}\\left( \\Lambda _{1}+\\Lambda _{2}\\right) -3\\left\\{ \\frac{1}{2}%\n\\sum_{m=1}^{2}\\ \\left[ 12\\eta _{m}\\ \\Lambda _{m}^{2}-\\frac{12\\eta\n_{m}\\Lambda _{m}}{1-\\eta _{m}}+h_{\\sigma }^{\\mathrm{HS}}(\\eta _{m})\\right] +%\n\\frac{K_{\\mathrm{reg}}}{K}\\right\\} t  \\label{p4}\n\\end{equation}%\nand $\\lim_{\\eta \\rightarrow 0}Z(\\eta _{1},\\eta _{2},t)=1$. Insertion of\nfound expressions for $\\Lambda _{1},$ $\\Lambda _{2}$ and $K_{\\mathrm{reg}}$\n(see Appendix B) into Eq. (\\ref{p3}) yields a single equation for $K$ that\nwe have solved numerically, although some further analytic simplifications\nare probably possible.\n\nOur solution is then almost fully analytical, as only the final equation for \n$K$ is left to be solved numerically.\n\n\\subsection{Approximate expressions}\n\nFor practical use we next derive very accurate analytical approximations to $%\nK$, $\\Lambda _{1}$ and $\\Lambda _{2}$, which provide an useful tool for\nfully analytical calculations. Since in all cases of our interest we always\nfind $x=K\\eta \\ll 1$, a serie expansion leads to:\n\n\\begin{equation}\nW_{0}^{\\mathrm{ex}}=\\frac{2}{3}\\allowbreak \\left( 1+5x\\right) t+\\mathcal{O}%\n\\left( x^{2}\\right) ,\n\\end{equation}%\nand, consequently,\n\n\\begin{equation}\n\\Lambda _{1}^{\\mathrm{ex}}=\\frac{x\\left( 1+5x\\right) }{3\\left( 1-2x\\right) }%\nt+\\mathcal{O}\\left( x^{3}\\right) ,\\qquad \\Lambda _{2}^{\\mathrm{ex}}=-\\frac{%\nx\\left( 1+5x\\right) }{6\\left( 1+x\\right) }t+\\mathcal{O}\\left( x^{3}\\right) .\n\\label{r5}\n\\end{equation}%\nSimilarly we can expand $Z$ in Eq. (\\ref{p4}) as\n\n\\begin{equation}\nZ(x,t)=1+z_{1}(t)x+z_{2}(t)x^{2}+O\\left( x^{3}\\right) ,\n\\end{equation}%\nwith%\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\nz_{1}(t)=\\frac{1}{4}\\left( 3+11t\\right) \\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \nz_{2}(t)=\\frac{1}{4}\\left( 15+61t-4t^{2}\\right) .%\n\\end{array}%\n\\right.  \\label{r6}\n\\end{equation}%\nInsertion of this result into Eq. (\\ref{p3}) yields a cubic equation for $K,$%\n\\begin{equation*}\nz_{2}(t)\\eta ^{2}K^{3}+z_{1}(t)\\eta K^{2}+K-\\alpha t\\ y_{0}^{\\mathrm{PY}%\n}(\\sigma )=0,\n\\end{equation*}%\nwhich, again with the help of Eq. (\\ref{p3}), is equivalent to a cubic\nequation for $Z$ \n\\begin{equation}\nZ^{3}-Z^{2}+z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta \\right]\nZ+z_{2}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta \\right] ^{2}=0.\n\\label{r7}\n\\end{equation}%\nThe physically acceptable solution then reads%\n\\begin{equation}\nZ(\\eta ,t)=\\frac{1}{3}\\left( 1+\\sqrt[3]{\\mathcal{B}+\\sqrt{\\mathcal{B}^{2}-%\n\\mathcal{C}^{3}}}+\\sqrt[3]{\\mathcal{B}-\\sqrt{\\mathcal{B}^{2}-\\mathcal{C}^{3}}%\n}\\right) ,  \\label{r8}\n\\end{equation}%\nwhere\n\n\\begin{equation}\n\\left\\{ \n\\begin{array}{c}\n\\mathcal{B}=1+\\frac{9}{2}z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma\n)\\eta \\right] +\\frac{27}{2}z_{2}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}%\n}(\\sigma )\\eta \\right] ^{2} \\\\ \n\\mathcal{C}=1+3z_{1}(t)\\left[ \\alpha t\\ y_{0}^{\\mathrm{PY}}(\\sigma )\\eta %\n\\right] .\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }%\n\\end{array}%\n\\right.  \\label{r9}\n\\end{equation}%\nIn conclusion, our approximate analytic solution for $K$, $\\Lambda _{1}$ and \n$\\Lambda _{2}$ includes three simple steps: i) calculate $K$ by using Eqs. (%\n\\ref{p3}), (\\ref{r8})-(\\ref{r9}), (\\ref{r6}); ii) evaluate $x=K\\eta $; iii)\nsolve for $\\Lambda _{1}$ and $\\Lambda _{2}$ by means of Eqs. (\\ref{p2b}) and\n(\\ref{r5}).\n\n\\bigskip\n\n\\subsection{Numerical comparison}\n\nIn order to assess the precision of previous approximations, we have\ncalculated $K$, $\\Lambda _{1}$ and $\\Lambda _{2}$ by two methods: i) solving\nnumerically Eqs. (\\ref{s2}), and ii) using our analytic approximations.\nAfter fixing $\\alpha =1\/2,$ we have increased the adhesion strength (or\ndecreased the temperature) from $t=0$ (HS limit) up to $t=0.8$, for some\nrepresentative values of the volume fraction ($\\eta =0.01$, $0.1$, $0.2$ and \n$0.4)$. The maximum value of $t$ corresponds to $\\tau =1\/(12t)\\simeq 0.1$,\nwhich lies close to the critical temperature of the isotropic SHS fluid. On\nthe other hand, $\\eta =0.01$ has been chosen to illustrate the fact that, as \n$\\eta \\rightarrow 0$, the parameter $K$ tends to $\\alpha t$. The linear\ndependence of $K$ on $t$ in this case is clearly visible in the top panel of\nFigure 2.\n\nIn Figures 2 and 3 the exact and approximate results for $K$, $\\Lambda _{1}$\nand $\\Lambda _{2}$ are compared. The agreement is excellent: at $\\eta =0.1$, \n$0.2$ and $0.4$, the relative error on $K$ does not exceed $0.1\\%$, $0.4\\%$\nand $1\\%$, respectively, while the maximum of the absolute relative errors\non $\\Lambda _{1}$ and $\\Lambda _{2}$ always remain less than $0.05$, $0.2$\nand $0.6~\\%$ in the three above-mentioned cases. It is worth noting that, as \n$\\eta $ increases, the variations of $\\Lambda _{1}$ and $\\Lambda _{2}$ are\nalways relatively small; on the contrary, $K$ experiences a marked change,\nwith a progressive lowering of the relevant curve.\n\n\\bigskip\n\n\\section{Some illustrative results on the local orientational structure}\n\nArmed with the knowledge of the analytic expression for the $q_{m}s$ a rapid\nnumerical calculation of the three harmonic coefficients $\\left\\{\nh_{0},h_{\\Delta },h_{D}\\right\\} $ appearing in\n\n\\begin{equation}\ng^{\\mathrm{PY-OL}}(1,2)=1+h_{0}(r)+h_{\\Delta }(r)\\Delta (1,2)+h_{D}(r)D(1,2).\n\\label{g1}\n\\end{equation}%\ncan be easily obtained as follows. From the second Baxter IE ($\\ref{ie2b}$),\none can generate $h(r)$ directly from $q(r)$, avoiding the passage through $%\nc(r)$. From $\\left\\{ q_{0},q_{1},q_{2}\\right\\} $ one first obtains $\\left\\{\nh_{0},h_{1},h_{2}\\right\\} $, by applying a slight extension of Perram's\nnumerical method \\cite{Perram75} and then derive $\\left\\{ h_{0},h_{\\Delta\n},h_{D}\\right\\} $, according to the above-mentioned recipes.\n\nThe main aim of the present paper was to present the necessary mathematical\nmachinery to investigate thermophysical properties. We now illustrate the\ninterest of the model by reporting some preliminary numerical results on the\norientational dependence of $g^{\\mathrm{PY-OL}}(1,2) $ -- i.e. on the local\norientational structure -- as a consequence of the anisotropic adhesion. A\nmore detailed analysis will be reported in a forthcoming paper.\n\nConsider the configuration depicted in Figure 4. Let a generic particle $1$\nbe fixed at a position $\\mathbf{r}_{1\\text{ }}$in the fluid with orientation \n$\\mathbf{u}_{1\\text{ }}$, and consider another particle $2$ located along\nthe straigth half-line which originates from the center of $1$ and with\ndirection $\\mathbf{u}_{1\\text{ }}$. This second particle has then a fixed\ndistance $r$ from $1$, but can assume all possible orientations $\\mathbf{u}%\n_{2\\text{ }}$, which -- by axial symmetry -- can be described by a single\npolar angle $\\theta \\equiv \\theta _{2}$ (i.e., the angle between $\\mathbf{u}%\n_{1\\text{ }}$and $\\mathbf{u}_{2\\text{ }}$) with respect to the\nintermolecular reference frame. Within this geometry, we have $\\left( \\theta\n_{1},\\varphi _{1}\\right) =(0,0)$ and $\\varphi _{2}=0$, obtaining $\\Delta\n(1,2)=\\cos \\theta $, $D(1,2)=2\\cos \\theta $. Consequently, $%\ng(1,2)=g(r,\\theta _{1},\\varphi _{1},\\theta _{2},\\varphi _{2})$ reduces to \n\\begin{equation}\ng(r,\\theta )=g_{0}(r)+\\left[ h_{\\Delta }(r)+2h_{D}(r)\\right] \\cos \\theta ,\n\\label{eq6}\n\\end{equation}%\nwhere $\\theta \\equiv \\theta _{2}$, and $g_{0}(r)=1+h_{0}(r)$ is the radial\ndistribution function of the reference isotropic SHS fluid.\n\nClearly, $g(r,\\theta )$ is proportional to the probability of finding, at a\ndistance $r$ from a given molecule $1$, a molecule $2$ having a \\textit{%\nrelative} orientation $\\theta $. We consider the three most significant\nvalues of this angle: i) $\\theta =0$, which corresponds to the `parallel'\nconfiguration of $\\mathbf{u}_{1}$ and $\\mathbf{u}_{2}$; ii) $\\theta =\\pi \/2$%\n, for the `orthogonal' configuration; and $\\theta =\\pi $, for the two\n`antiparallel' (head-to-head and tail-to-tail) configurations. From Eq. (\\ref%\n{eq6}) it follows that%\n\\begin{equation}\n\\begin{array}{c}\ng^{\\mathrm{par}}(r)=g(r,0)=g_{0}(r)+\\left[ h_{\\Delta }(r)+2h_{D}(r)\\right] ,%\n\\text{ \\ \\ \\ } \\\\ \ng^{\\mathrm{ortho}}(r)=g(r,\\pi \/2)=g_{0}(r),\\text{ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ } \\\\ \ng^{\\mathrm{antipar}}(r)=g(r,\\pi )=g_{0}(r)-\\left[ h_{\\Delta }(r)+2h_{D}(r)%\n\\right] .\\text{ \\ }%\n\\end{array}\n\\label{g2}\n\\end{equation}%\nNote that $g^{\\mathrm{ortho}}(r)$ coincides with the isotropic result $%\ng_{0}(r)$.\n\nIn Figures 5 we depict the above sections through the three-dimensional\nsurface corresponding to $g(r,\\theta )$, i.e., $g^{\\mathrm{par}}(r)$, $g^{%\n\\mathrm{ortho}}(r)$ and $g^{\\mathrm{antipar}}(r)$, for $\\eta =0.3$ with $%\nt=0.2$ and $t=0.6$, respectively, at the highest asymmetry value admissible\nin the present model, i.e. $\\alpha =1\/2$. The most significant features from\nthese plots are: i) $g^{\\mathrm{antipar}}(\\sigma ^{+})>g^{\\mathrm{par}%\n}(\\sigma ^{+})$; ii) for $r>2\\sigma $ $g^{\\mathrm{antipar}}(r)\\approx g^{%\n\\mathrm{par}}(r)\\approx g_{0}(r)$, i.e., the anisotropic adhesion seems to\naffect only the first coordination layer, $\\sigma <r<2\\sigma $, around each\nparticle.\n\nThe interpretation of these results is the following. In view of i) we see\nthat the parallel configuration is less probable than the antiparallel one\nat contact. Such a finding, together with ii), means that chain formation\ncharacteristic of polymerization is inhibited by the short-ranged\nanisotropic adhesion exploited here. This strictly contrasts with the case\nof long-ranged DHS fluids, where it is believed \\cite{Camp00,Tlusty00} that\nchaining phenomena might preempt the gas-liquid transition. This specific\nfeature of the present model is extremely interesting and we plan a\nthroughout investigation on this topic in a future publication.\n\n\\bigskip\n\n\\section{Phase stability}\n\nIn view of previous findings, a very natural question is whether the\naddition of our anisotropic sticky term to the potential changes phase\nstability and phase transition curves with respect to the corresponding\nisotropic case. We believe the answer to be positive. This is strongly\nsuggested by results obtained for similar anisotropic models, such as hard\nspheres with `sticky points'\\ \\cite%\n{Sciortino05,Bianchi06,Michele06,Lomakin99,Starr03,Zhang04,Glotzer04a,Glotzer04b,Sciortino07}\nor `sticky patches'\\ \\cite%\n{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07}.\n\nWe now briefly comment on this issue. Within our formalism, this problem of\nstability can be conveniently analyzed using standard formalism\ndevised to this aim \\cite{Gray84,Stecki81,Chen92,Klapp97}.\n\nWe start from the stability condition with respect to small but arbitrary\nfluctuations of the one-particle density $\\rho(1)$ from the equilibrium configuration,\ndenoted as 'eq' \\cite{Stecki81,Chen92,Klapp97}, \n\n\\begin{eqnarray}\n\\int d(1) \\int d(2) \\left [\\frac{\\delta(1,2)}{\\rho(1)} - c\\left(1,2\\right) \n\\right]_{eq} \\delta \\rho(1) \\delta \\rho(2) > 0.\n\\label{stability}\n\\end{eqnarray}\n\nHere $d(i)$ stands for $d \\mathbf{r}_i ~d \\Omega_i$, $i=1,2$, and we assume the\nequilibrium one-particle density to be $\\rho\/4\\pi$ \\cite{Stecki81,Chen92,Klapp97}. \n\nWe expand the fluctuations both in Fourier modes and in spherical harmonics\n\\cite{Gray84}\n\n\\begin{eqnarray}\n\\delta \\rho \\left(j\\right) \\equiv \\delta \\rho \\left(\\mathbf{r}_j,\\Omega_j\\right)\n&=& \\int \\frac{d \\mathbf{k}}{\\left(2\\pi\\right)^3} ~\\mathrm{e}^{\\mathrm{i} \\mathbf{k} \\cdot\n\\mathbf{r}_j} \\sum_{l=0}^{+\\infty} \\sum_{m=-l}^{+l} \\delta \\tilde{\\rho} \\left(\\mathbf{k}\n\\right) Y_{lm} \\left(\\Omega_j\\right). \n\\label{expansion}\n\\end{eqnarray}\n\nUsing the orthogonality relation \\cite{Gray84}\n\n\\begin{eqnarray}\n\\int d \\Omega ~Y_{lm}^{*} \\left(\\Omega\\right) Y_{l'm'}\\left(\\Omega\\right) &=&\n\\delta_{l l'} \\delta_{m m'},\n\\label{orthogonality}\n\\end{eqnarray}\n\nstandard manipulations \\cite{Klapp97} show that condition (\\ref{stability}) can\nbe recast into the form\n\n\\begin{eqnarray}\n\\sum_{l_{1},l_{2}=0}^{+\\infty} \\sum_{m_{1}=-l_{1}}^{+l_{1}} \\sum_{m_{2}=-l_{2}}^{+l_{2}}\n\\int \\frac{d \\mathbf{k}}{\\left(2 \\pi\\right)^3} ~ \\delta \\tilde{\\rho}_{l_{1} m_{1}} \n\\left(\\mathbf{k} \\right) \\delta \\tilde{\\rho}_{l_{1} m_{1}}^{*} \n\\left(\\mathbf{k} \\right) \\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &>0&,\n\\label{stability2}\n\\end{eqnarray}\n\nwhere the matrix elements $\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right)$\nare given by\n\n\\begin{eqnarray} \\label{matrix}\n\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &=& \\left(-1\\right)^{m_{1}}\n\\frac{4 \\pi}{\\rho} \\delta_{l_{1} l_{2}} \\delta_{m_{1},-m_{2}} -\n\\int d \\Omega_1 \\int d ~ \\Omega_{2} Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) \nY_{l_{2} m_{2}} \\left(\\Omega_2\\right) \\\\ \\nonumber\n&\\times&  \\int d\\mathbf{r} ~\\mathrm{e}^{\\mathrm{i} \\mathbf{k}\n\\cdot \\mathbf{r}} c\\left(\\mathbf{r},\\Omega_1,\\Omega_2\\right).\n\\end{eqnarray}\n\nThe problem of the stability has been reported to the character \nof the eigenvalues of matrix (\\ref{matrix}). This turns out to be particularly\nsimple in our case. Using the results (\\ref{wertheim_int}) it is easy to\nsee that\n\n\\begin{eqnarray}\n\\int d \\mathbf{r}~ \\mathrm{e}^{\\mathrm{i} \\mathbf{k} \\cdot \\mathbf{r}}\nc\\left(\\mathbf{r},\\Omega_1,\\Omega_2 \\right) &=& \\tilde{c}_{0} \\left(k\\right)\n+\\tilde{c}_{\\Delta} \\left(k\\right) \\Delta\\left(\\Omega_1,\\Omega_2\\right) +\n\\overline{c}_{D} \\left(k\\right) D\\left(\\Omega_1,\\Omega_2,\\Omega_k\\right).\n\\label{integral_c}\n\\end{eqnarray}\n\nInsertion of Eq.(\\ref{integral_c}) into Eq.(\\ref{matrix}) leads to\n\n\\begin{eqnarray} \\label{matrix2}\n\\tilde{A}_{l_{1} m_{1} l_{2} m_{2}} \\left(\\mathbf{k} \\right) &=& \n\\left(-1\\right)^{m_{1}}\n\\frac{4 \\pi}{\\rho} \\delta_{l_{1} l_{2}} \\delta_{m_{1},-m_{2}} \\\\ \\nonumber\n&-& \\left[\n\\tilde{c}_{0}\\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(0)}\n+\\tilde{c}_{\\Delta} \\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(\\Delta)}\n+\\tilde{c}_{D} \\left(k\\right) I_{l_{1} m_{1} l_{2} m_{2}}^{(D)}\n\\right],\n\\end{eqnarray}\n\n\\noindent where we have introduced the following integrals, which can be evaluated\nin the intermolecular frame, using standard properties of the\nspherical harmonics \\cite{Gray84}\n\n\\begin{eqnarray} \\label{integrals}\nI_{l_{1} m_{1} l_{2} m_{2}}^{(0)} &\\equiv& \\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) = 4 \\pi \\delta_{l_{1} 0,} \\delta_{l_{2},0} \\delta_{m_{1} 0}\n\\delta_{m_{2}} \\delta_{0} \\\\ \\nonumber\nI_{l_{1} m_{1} l_{2} m_{2}}^{(\\Delta)} &\\equiv& \\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) ~\\Delta\\left(\\Omega_1,\\Omega_2\\right)= \\frac{4}{3} \\pi \\delta_{l_{1} 1} \n\\delta_{l_{2},1} \\delta_{m_{1} 0} \\delta_{m_{2},0} \\\\ \\nonumber\nI_{l_{1} m_{1} l_{2} m_{2}}^{(D)} \\left(\\cos \\theta \\right) &\\equiv& \n\\int d \\Omega_1 \\int d \\Omega_2\n~Y_{l_{1} m_{1}} \\left(\\Omega_1\\right) Y_{l_{2} m_{2}} \\left(\\Omega_2\n\\right) D\\left(\\Omega_1,\\Omega_2,\\Omega_k\\right) \\\\ \\nonumber \n&=& \\frac{4}{3} \\pi \\delta_{l_{1},1} \n\\delta_{l_{2},1} \\delta_{m_{1}, 0}\n\\delta_{m_{2},0} ~2 ~P_{2}\\left(\\cos \\theta \\right) \n\\end{eqnarray}\n\nand where $P_2(x)=(3 x^2 -1)\/2$ is the second Legendre polynomial.\n\nHence, the matrix (\\ref{matrix}) is diagonal and the relevant terms\nare\n\n\\begin{eqnarray}\n\\tilde{A}_{0000} \\left(k\\right) &=& 4 \\pi\n\\left[ \\frac{1}{\\rho}-\\tilde{c}_{0} \\left(k\\right) \\right],\n\\label{element00}\n\\end{eqnarray}\nwhose positiveness is recognized as the isotropic stability condition,\nand\n\\begin{eqnarray}\n\\tilde{A}_{1010}\\left(\\mathbf{k}\\right) &=& 4 \\pi \\left\\{\n\\frac{1}{\\rho} -\\frac{1}{3} \\left[\\tilde{c}_{\\Delta}\\left(k\\right)\n+ 2 P_{2} \\left(\\cos \\theta\\right) \\overline{c}_{D}\\left(k\\right)\n\\right] \\right\\}.\n\\label{element11}\n\\end{eqnarray} \n\n\\noindent All remaining diagonal terms have the form $\\tilde{A}_{l0l0}=\n4 \\pi\/\\rho>0$. \n\nIn order to test for possible angular instabilities, we consider\nthe limit $k \\to 0$ of Eq.~(\\ref{element11}) namely\n\n\\begin{eqnarray}\n\\tilde{A}_{1010}\\left(0\\right) &=& \\frac{4 \\pi}{\\rho} \\left\\{\n1 -\\frac{\\rho}{3} \\left[\\tilde{c}_{\\Delta}\\left(0\\right)\n+ 2 P_{2} \\left(\\cos \\theta\\right) \\overline{c}_{D}\\left(0\\right)\n\\right] \\right\\}.\n\\label{element11k0}\n\\end{eqnarray} \n\\noindent\nThis can be quickly computed with the aid of Eqs.~(\\ref{so11}), (\\ref{bf1}),\nthe fact that $\\bar{c}_{D}(0)=\\tilde{c}_{D}^{0}(0)$ and the identity\n(\\ref{so10}). We find\n\n\\begin{eqnarray}\n\\label{element11_res}\n\\tilde{A}_{1010}\\left(0\\right) &=& \\frac{4 \\pi}{\\rho} a_1^2,\n\\end{eqnarray}\nwhich is independent of the angle $\\theta$. This value is found to be always\npositive as $a_1>0$ (see Fig.\\ref{fig6}).\nWithin this first-order approximation, therefore the only instability\nin the system stems from the isotropic compressibility. \nThe reason for this can be clearly traced back to the first-order\napproximation to the angular dependence of the correlation functions. If\nquadratic terms in $\\Delta $ and $D$ were included into the series expansion\nfor correlation functions, the particular combination leading to a cancellation\nof the angular dependence in the stability matrix \n$\\tilde{A}_{l_{1}m_{1}l_{2}m_{2}}\\left(0\\right)$ would not occur, leading to\na different result. \n\nThis fact is consistent with the more general statement that, in any\napproximate theory, thermodynamics usually requires a higher degree of\ntheoretical accuracy than the one sufficient for obtaining significant\nstructural data. Conceptually, the need of distinguishing structural results\nfrom thermodynamical ones is rather common. For instance, in statistical\nmechanics of liquids it is known that approximating the model potential only\nwith its repulsive part (for instance, the hard sphere term) can account for\nall essential features of the structure, but yields unsatisfactory\nthermodynamics. On the other hand, the present paper refers to a \\textit{%\nsimplified} statistical-mechanical tool, i.e. the OZ equation within our\nPY-OL closure, which has been explicitly selected to allow an analytical\nsolution. Our results however indicate that the first-order expansion used\nin the PY-OL closure can give reasonable information about structure, but\nnot on thermodynamics, where a higher level of sophistication is required.\n\n\\section{Concluding remarks}\n\nIn this paper we have discussed an anisotropic variation of the original\nBaxter model of hard spheres with surface adhesion. In addition to the HS\npotential, molecules of the fluid interact via an isotropic sticky\nattraction plus an additional anisotropic sticky correction, whose strength\ndepends on the orientations of the particles in dipolar way. By varying the\nvalue of a parameter $\\alpha $, the anisotropy degree can be changed.\nConsequenly, the strength of the total sticky potential can vary from twice\nthe isotropic one down to the limit of no adhesion (HS limit). These\nparticles may be regarded as having two non-uniform, hemispherical,\n`dipolar-like patches', thus providing a link with uniformly adhesive\npatches \\cite{Jackson88,Ghonasgi95,Sear99,Mileva00,Kern03,Zhang05,Fantoni07}.\n\nWe have obtained a full analytic solution of the molecular OZ equation,\nwithin the PY-OL approximation, by using Wertheim's technique \\cite%\n{Wertheim71}. Our PY-OL approximation should be tested against exact\ncomputer simulations, in order to assess its reliability. Nevertheless, we\nmay reasonably expect the results to be reliable even at experimentally\nsignificant densities, notwithstanding the truncation of the higher-orders\nterms in the angular expansion. Only one equation, for the parameter $K$,\nhas to be solved numerically. In additon, we have provided analytic\napproximations to $K$, $\\Lambda _{1}$ and $\\Lambda _{2}$ so accurate that,\nin practice, the whole solution can really be regarded as fully analytical.\nFrom this point of view, the present paper complements the above-mentioned\nprevious work by Blum \\textit{et al. }\\cite{Blum90}.\n\nWe have also seen that thermophysical properties require a more detailed\ntreatment of the angular part than the PY-OL closure. Nonetheless, \neven within the PY-OL oversimplified framework, our findings\nare suggestive of a dependence of the fluid-fluid coexistence line on\nanisotropy.\n\nOur analysis envisions a number of interesting perspectives, already hinted\nby the preliminary numerical results reported here. It would be very\ninteresting to compare the structural and thermodynamical properties of this\nmodel with those stemming from truly dipolar hard spheres \\cite{Stecki81,\nChen92,Klapp97}.\nThe possibility of local orientational ordering can be assessed by computing\nthe pair correlation function $g(1,2)$ for the most significant\ninterparticle orientations. We have shown that this task can be easily\nperformed within our scheme. This should provide important information about\npossible chain formation and its subtle interplay with the location of the\nfluid-fluid transition line. The latter bears a particular interest in view\nof the fact that computer simulations on DHS are notoriously difficult and\ntheir predictions regarding the location of such a transition line have\nproven so far unconclusive \\cite{Frenkel02}. The long-range nature of DHS\ninteractions may in fact promote polymerization preempting the usual\nliquid-gas transition \\cite{Tlusty00}. Our preliminary results on the\npresent model strongly suggest that this is not the case for sufficiently\nshort-ranged interactions, thus allowing the location of such a transition\nline to be studied as a function of the anisotropy degree of the model. Our\nsticky interactions have only attractive adhesion, the only repulsive part\nbeing that pertinent to hard spheres, whereas the DHS potential is both\nattractive and repulsive, depending on the orientations.\n\nFinally, information about the structural ordering in the present model\nwould neatly complement those obtained by us in a recent parallel study on a\nSHS fluid with one or two uniform circular patches \\cite{Fantoni07}. Work\nalong this line is in progress and will be reported elsewhere.\n\n\\acknowledgments \nWe acknowledge financial support from PRIN 2005027330. It is our pleasure to\nthank Giorgio Pastore and Mark Miller for enlighting discussions on the\nsubject.\n\n\\bigskip ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nMillisecond pulsars are commonly believed to be descendants of normal neutron stars that have been spun-up and recycled back as radio pulsars by acquiring angular momentum from their companion during the low-mass X-ray binary (LMXB) phase \\citep{ACR82,RS82}.\n\nThere are about $\\sim$20 high confidence nuclear or accretion powered (see Table~1) millisecond X-ray pulsars (MSXPs) which are thought to be the progenitors of millisecond radio pulsars (MSRPs) \\citep{WK98}. These MSXPs may become observable in radio wavelengths once accretion ceases, or the column density of the plasma from the fossil disk around the neutron star becomes thin enough to allow vacuum gap formation that leads to the production of coherent radio emission. Towards the end of the secular LMXB evolution, as accretion rates fall below a critical value above which detection presumably may be hampered due to absorption or dispersion \\citep{TBE94}, the neutron star can re-appear as a MSRP.\n\nAlthough the connection between LMXBs and MSRPs has been significantly strengthened after the discovery of quasi-periodic kHz oscillations and X-ray pulsations in some transient X-ray sources \\citep{WK98, MSS02, GCM02, GMM05}, no radio pulsations from MSXPs have been detected so far \\citep{BBP03}. \n\nAt the end of the recycling process the neutron star will reach an equilibrium period \\citep{BH91} which is approximated by the Keplerian orbital period at the Alfven radius \\citep{GL92}:\n\n\\begin{equation}\nP_{eq} \\sim 1.9\\, ms\\, B_{9}^{6\/7} \\biggl (\\frac{M}{1.4 \\,M_{\\sun}}\\biggr)^{-5\/7} \\biggl(\\frac{\\dot{m}}{\\dot{M}_{Edd}}\\biggr)^{-3\/7} R^{16\/7}_{6}\n\\end{equation}\nwhere $B_{9}$ and $R_{6}$ are the neutron star surface magnetic dipole field and radius in units of $10^{9}$ G and $10^{6}$ cm respectively. The Eddington limited accretion rate $\\dot{M}_{Edd}$ for a neutron star typically is $\\sim10^{-8}\\,M_{\\sun}\\,yr^{-1}$ above which the radiation pressure generated by accretion will stop the accretion flow. This equilibrium period combined with the dominant mechanism for energy loss delineates the subsequent kinematics of the spun-up millisecond pulsar. The magnetic dipole model then implies a ``spin-up region'' ($\\dot{P}\\, \\sim\\, P_{0}^{4\/3} $) \\citep[see][]{ACW99} on which the recycled neutron stars will be reborn as MSRPs. At the end of the active phase, MSXPs accreting with $\\dot{m}$ and  spinning with $P_{eq}$, presumably transition into a MSRPs with an initial spin period of $P_{0} \\sim P_{eq}$.\n\nIn the standard spin-down model, the MSRP evolution is driven by pure magnetic dipole radiation, i.e. braking index $n=3$ in vacuum \\citep[see][]{MT77, LK04}.  Alternative energy loss mechanisms such as multipole radiation or gravitational wave emission, especially during the initial phases of the reborn millisecond pulsars, have been suggested by several authors  \\citep{K91,B98} but have yet to be observationally corroborated. Advanced Laser Interferometer Gravitational Wave Observatory (LIGO) will be able to probe the frequency space at which millisecond pulsars are expected to radiate gravitational waves, thereby putting stringent constraints on the micro physics of millisecond pulsars.\n\nThe advances in radio observations, increased sky coverage with deep exposures of current surveys combined with robust post-bayesian statistical techniques that incorporate minimal assumptions, give us unprecedented predictive power on the joint period-spindown ($P-\\dot{P}$) and implied magnetic field ($B$) distributions.\n\nIn this {\\it Letter}, we attempt to go beyond phenomenological arguments and test whether MSXPs can produce the characteristics of the observed MSRPs within the framework of the standard model \\citep[and references therein]{BH91}.\n\t\n\t\\section{\\label{sec:dist}The Joint Period - Spindown ($P-\\dot{P}$) Distribution}\n\t\t\t\t\t\t\t\nThe evolution of millisecond pulsars can be consistently described in terms of {\\bf i)} the equilibrium period distribution ($D$) of MSXPs at the end of the LMXB evolution {\\bf ii)} the mass accretion rates ($\\dot{M}$) of the progenitor population during the recycling process  {\\bf iii)} Galactic birth rates ($R$), and {\\bf iv)} the dominant energy loss mechanism after the onset of radio emission.  \n\n\t\t\t\\subsection{\\label{sec:stat}Statistics}\n\nWe devise a semi-analytical evolution function $\\mathcal{E}$ to parametrize the evolution of millisecond pulsars after the accretion phase, which can be described in closed form as:\n\\begin{eqnarray}\n\\displaystyle\\sum_{i=0}^{r} \\mathcal{E}(D_{i},\\dot{M}_{i},R_{i}\\, | \\, \\alpha_{i}^{k},\\beta_{i}^{k})  \\xrightarrow{n=3} \\mathcal{PDF}(P,\\dot{P})  \\label{stat.eq} \n\\end{eqnarray}\nwhere $\\mathcal{PDF}$ is the probability distribution function. The shape parameters $\\alpha$ and $\\beta$ define the distributions (i.e., $D,\\dot{M}, R$ for k=1,2,3) for the Beta functions\\footnote{Beta functions are commonly preferred in Bayesian statistics as the least restrictive and most flexible prior distributions. It can take the form of an uninformative (e.g. uniform) prior, a monotonic line, concave, convex, unimodal (e.g. normal) or any extreme combinations of these shapes.}\\citep{EHP00} inferred from observations at each Monte-Carlo realization ``r''. \n\nThe evolution function $\\mathcal{E}$ is built by randomly choosing initiation seeds from a period distribution $D$, which is then convolved via the standard model to consequently sample the $P-\\dot{P}$ parameter space. For the observed MSXPs, the period distribution which seeds will be randomly chosen from is the observed $P_{MSXP}$ distribution (table~1). We uniquely construct a ``relaxed multidimensional Kolmogorov-Smirnov (K-S) filter'' (fig.~\\ref{fig:probdist}) to check population consistencies by calculating the 2D K-S \\citep{FF87} probabilities ($P_{2DK-S}$) between observed MSRPs and the synthetic population that is formed by these properly evolved progenitor seeds. The filtering is reiterated for each realization to obtain synthetic populations with consistent distributions as:\n\\begin{eqnarray}\nD\\,(\\alpha_{i}^{1},\\beta_{i}^{1})  \\xrightarrow{filter}  D_{i} \\label{statD.eq} \\\\\n\\dot{M}(\\alpha_{i}^{2},\\beta_{i}^{2})   \\xrightarrow{filter}  \\dot{M}_{i}   \\label{statM.eq} \\\\\nR\\,(\\alpha_{i}^{3},\\beta_{i}^{3})  \\xrightarrow{filter}  R_{i}  \\label{statR.eq}\n\\end{eqnarray}\nwhich is then used to construct the $\\mathcal{PDF}$ in Equation ~\\protect\\ref{stat.eq}.\n\nNominally any $P_{2DK-S} > 0.2$ value would imply consistent populations in a 2D K-S test. By allowing $0.005 < P_{2DK-S} < 0.2$ with lower fractions (see fig.~\\ref{fig:probdist}), we oversample outliers to compensate for possible statistical fluctuations and contaminations. A peak sampling rate around the nominal acceptance value of $P_{2DK-S}\\sim0.2$ is the most optimal scheme that prevents strong biases due to over or under-sampling. The main goal for oversampling outliers and relaxing the K-S filter is to test whether the standard model can at least marginally produce very fast millisecond pulsars with relatively high magnetic fields like PSR B1937+21.\n\nThe predictive significance of the $P-\\dot{P}$  distribution for the probability map (fig~\\protect\\ref{fig:MSPs}) is obtained from a Monte-Carlo run with $r=10^{7}$ valid realizations that produce consistent synthetic samples. Whilst sampling the $P-\\dot{P}$ space, no assumptions were made regarding the progenitor period distribution ($D$), the accretion ($\\dot{M}$), or the Galactic birth ($R$) rates. The filter  (eq.~\\protect\\ref{statD.eq}, \\protect\\ref{statM.eq}, \\protect\\ref{statR.eq}) is implicitly driven by the observed MSRPs.\n\t\nFig \\protect\\ref{fig:MSPs} shows the expected $P-\\dot{P}$ distribution for the standard model assuming that MSRPs have evolved from a progenitor population similar to the observed MSXPs. We do not include MSRPs in globular clusters because the $P-\\dot{P}$ values in these cases may not necessarily be the sole imprint of the binary evolution, but can be significantly changed by possible gravitational interactions due to the crowded field. To explore the extend of the effects of an unevenly sampled progenitor population, we also show the region in the $P-\\dot{P}$ space that is sensitive to alternative $P_{MSXP}$ distributions. The probability map is overlaid with the observed MSRPs. \n\n\t\\section{\\label{sec:dis}Discussion and Conclusions}\n\nThe discovery of millisecond pulsations from neutron stars in LMXBs has substantiated the theoretical prediction that links MSRPs and LMXBs. Since then, the recycling process that produces MSRPs on a spin-up region from LMXBs, followed by spin-down due to dipole radiation has been conceived as the ``standard evolution'' of millisecond pulsars. However, the question whether all observed MSRPs could be produced within this framework has not been quantitatively addressed until now.\n\nThe standard evolutionary process produces millisecond pulsars with periods ($P$) and spin-downs ($\\dot{P}$) that are not entirely independent. The possible $P-\\dot{P}$ values that MSRPs can attain are {\\it jointly} constrained by the equilibrium period distribution ($D$) of the progenitor population, the mass accretion rates ($\\dot{M}$) during the recycling process and the dominant energy loss mechanism after the onset of radio emission. \n\nIn order to test whether the observed MSRPs can be reconciled with a single coherent progenitor population that evolves via magnetic dipole braking after the spin-up process, we have produced the predictive joint $P-\\dot{P}$ distribution of MSRPs for the standard model. We did not put restrictions on any of the parameters that drive the evolution. Acceptable $D,\\dot{M}$ and $R$ values were implicitly filtered.  We have relaxed the K-S filter (see fig.~\\ref{fig:probdist}) in order to oversample outliers and see whether it is even remotely feasible to produce young millisecond pulsars, like PSR B1937+21 or J0218+4232, that have higher B fields. The color contours in Figure~\\ref{fig:MSPs} represent the $P-\\dot{P}$ densities for MSRPs that are direct descendants of observed MSXPs (i.e. initial spin periods $P_{0}\\sim P_{MSXP}$). \n\nThe standard evolutionary model is able to successfully produce the general demographics of older MSRPs. It fails, however, to predict the younger and fastest MSRP sub-population that have higher $B$ fields.\n\nAccretion rates that MSRPs have experienced during their accretion phase deduced from observed $P-\\dot{P}$ values, combined with the observed MSXP period distribution ($D \\equiv  P_{MSXP}$) produces mostly older MSRPs, including MSRPs with spin-down ages $\\tau_{c} > 10^{10}$ yrs. Figure ~\\ref{fig:MSPs} shows clearly that the apparent enigma of millisecond pulsars with spin-down ages older than the age of the Galaxy is mainly a manifestation of very low accretion rates during the late stages of the LMXB evolution. \n\nOn the other hand, no physically motivated $P_{MSXP}$ distribution has been able to produce the whole MSRP population consistently. The observed period distribution of MSXPs is likely to be under-sampled due to observational selection effects. It is also possible that some neutron stars in LMXBs simply do not produce observable pulses. In order to understand how the predicted $P-\\dot{P}$ distribution is affected by different MSXP period distributions, we have estimated the whole extend of the $P-\\dot{P}$ region that is sensitive to the prior. The values that may be produced for different $P_{MSXP}$ distributions are shown by the shaded areas in Figure ~\\ref{fig:MSPs}. No MSXP period distribution could mimic the observed relative ratios of young\/old pulsars with high B fields. The fraction of the observed young\/old MSRPs with high $B$ fields is higher than what the standard model predicts by several orders of magnitude. This may further be exacerbated by strong selection effects that limit our ability to observe very fast millisecond pulsars \\citep{HRS07}. The choice of a standard K-S test instead of the relaxed 2D K-S only increases the statistical significance. Hence, we argue that young millisecond pulsars with higher magnetic fields (e.g. PSR B1937+21) are inconsistent with the standard model. \n\nTherefore, it is tempting to suggest that the fastest spinning millisecond pulsars, in particular PSR B1937+21, may originate from a different evolutionary channel.  While it appears that ordinary magnetic-dipole spin down from a source population similar to the observed MSXPs is adequate to explain the great majority of observed MSRPs, the low final accretion rates that are required cannot be reconciled with the high accretion rates needed to produce the fastest, youngest pulsars. We believe that it is necessary to posit the existence of a separate class of progenitors, most likely with a different distribution of magnetic fields, accretion rates and equilibrium spin periods, presumably among the LMXBs that have not been revealed as MSXPs.  Understanding this additional channel is clearly critical to developing a natural solution to the long lasting ``birth rate problem'' \\citep[see, e.g.][]{KN88}.\n\nIt is also possible that the standard evolutionary model fails at another point. For example, if MSRPs during some portion of their evolution lose energy through a dominant mechanism other than magnetic dipole radiation (e.g. multipole radiation, gravitational wave or neutrino emission), then the evolution of pulsars through the $P-\\dot{P}$ diagram could be complex. \n\nA combination of the above mentioned factors (i.e. alternative progenitors and subsequent non-standard radiation) are then likely to play a role in millisecond pulsar evolution. A MSXP period distribution that has sharp multimodal features coupled with non-standard energy loss mechanisms may be able to reconcile for the joint $P-\\dot{P}$ distribution of millisecond pulsars.\n\n\\acknowledgements \nThe research presented here has made use of the August 2008 version of the ATNF Pulsar Catalogue \\citep{MHT93}. The authors acknowledge NSF grant AST-0506453.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nFor many interpretations  of the modal operators -- e.g., for deontic, epistemic,  game-theoretic, and high-probability interpretations -- it is necessary to adopt logics that are weaker than the normal ones; e.g., deontic paradoxes, see \\cite{G13,M06},  are one of the main motivations for adopting a non-normal  deontic logic. Non-normal  logics, see \\cite{C80} for naming conventions,  are quite well understood from a semantic point of view  by means of neighbourhood semantics  \\cite{H09,P17}.\n  Nevertheless, until recent years their proof theory has been rather limited  since it was mostly confined to Hilbert-style axiomatic systems. This situation seems to be rather unsatisfactory since it is difficult to find derivations in axiomatic systems. When the aim is to find derivations and to analyse their structural properties, sequent calculi are to be preferred to  axiomatic systems. Recently  different kinds of sequent calculi for non-normal logics have been proposed: Gentzen-style calculi  \\cite{I05,I11,L00,O15}; labelled \\cite{GM14,CO18} and display \\cite{P19} calculi based on  translations into normal modal logics; labelled calculi based on the internalisation of neighbourhood  \\cite{N17,NO19} and  bi-neighbourhood  \\cite{DON} semantics;   and, finally, \\mbox{linear nested sequents  \\cite{L17}.}\n \n  \n This paper, which extends the results  presented in \\cite{O15},  concentrates on Gentzen-style calculi since they are better suited than labelled calculi, display calculi, and  nested sequents to give  decision procedures (computationally well-behaved) and constructive proofs of interpolation theorems. We consider cut- and contraction-free G3-style sequent calculi for all the logics in the cube of non-normal modalities and for their  extensions with the deontic axioms $D^\\Diamond:=\\Box A\\supset\\Diamond A$ and $D^\\bot:=\\neg\\Box\\bot$.  The calculi we present have the subformula property and  allow for a straightforward decision procedure by a terminating loop-free proof search. Moreover, with the  exception of the calculi for {\\bf EC(N)} and its deontic extensions, they are \\emph{standard} \\cite{G16} -- i.e., each operator is handled by a finite number of rules with a finite number of premisses -- and they admit of a Maehara-style constructive proof of Craig's interpolation  theorem.\n \n This work improves on previous ones  on Gentzen-style calculi for non-normal logics  in that we  prove cut admissibility   for non-normal modal and deontic logics, and not only for the modal ones  \\cite{L00,I05,I11}. Moreover, we prove height-preserving admissibility of weakening and contraction, whereas neither weakening nor contraction is admissible in \\cite{L00,I05} and weakening but not contraction is admissible in \\cite{I11}. The admissibility of contraction is a major improvement since, as it is well known, contraction can be as bad as cut for  proof search: we may continue to duplicate some formula forever  and, therefore,  we need a (computationally expensive) loop-checker to ensure termination.  Proof search  procedures based on contraction-free calculi terminate because the height of derivations is bounded by a number depending on the complexity of the end-sequent and, therefore, we avoid the need of  loop-checkers.  To illustrate, the introduction  of contraction-free calculi has allowed to give computationally optimal decision procedures for propositional intuitionistic logic  ({$\\mathbf{IL_p}$) \\cite{H93} and for the normal modal  logics {\\bf K} and {\\bf T} \\cite{B97,H95}.   The existence of a loop-free terminating decision procedure has also allowed  to give a constructive proof of uniform interpolation for $\\mathbf{IL_p}$ \\cite{P92} as well as for  {\\bf K} and {\\bf T} \\cite{B07}.  The cut- and contraction-free calculi for non-normal logics considered here are such that the height of each derivation is bounded by the weight of its end-sequent and, therefore, we easily obtain a polynomial space upper complexity bound for proof search. This upper bound  is  optimal for the  logics  having $C$  as theorem (the satisfiability problem for non-normal modal logics without $C$   is in  {\\sc NP}, see \\cite{V89}).\n \n  Moreover,  the introduction of well-behaved calculi for non-normal deontic logics is interesting since  proof analysis can be applied to  the deontic paradoxes \\cite{M06} that are one of the central topics of deontic reasoning. We illustrate this in Section  \\ref{forrester} by considering Forrester's Paradox \\cite{F84} and by showing that proof analysis cast  doubts on the widespread opinion \\cite{M06,P17,T97} that Forrester's argument provides evidence against rule $RM$ (see Table \\ref{rulesinf}).\nIf   Forrester's argument  is formalized as in \\cite{M06} then  it does not compel us to adopt a deontic logic weaker than {\\bf KD}. If, instead, it is formalised as in \\cite{T97} then it forces the adoption of a logic where $RM$ fails, but the formal derivation  differs substantially from Forrester's informal argument. \n \n It is also given a constructive proof of interpolation for all logics having a standard calculus. To our knowledge in the literature there is no other constructive  study of interpolation in non-normal  logics.  In \\cite[Chap(s). 3.8 and 6.6]{F83} a constructive proof of Craig's (and Lyndon's)  interpolation theorem is given for the modal logics {\\bf K} and {\\bf R}, and for some of their extensions, including the deontic ones, but the proof makes use of model-theoretic notions. A proof of interpolation by the  Maehara-technique for {\\bf KD} is given in \\cite{V93}. For a thorough study of  interpolation in modal logics we refer the reader to \\cite{G05}. A model-theoretic proof of interpolation for {\\bf E} is given in \\cite{H09}, and a coalgebraic proof  of (uniform) interpolation for all the logics considered here, as well as all other rank-1 modal logics (see below), is given in \\cite{P13}. As it is explained in Example \\ref{prob}, we have not been able to prove interpolation for calculi containing the non-standard rule $LR$-$C$ (see Table \\ref{Modal rules}) and, as far as we know, it is still an open problem whether it is possible to give a constructive proof of interpolation for these logics.\n \n \\paragraph{Related Work.}\\label{related}\n The modal rules of inference  presented in Table \\ref{Modal rules} are obtained from the rules presented in \\cite{L00} by adding weakening contexts to the conclusion of the rules. This minor modification, used also in \\cite{I11,P13,PS10} for several modal rules, allows us to shift from set-based sequents to multiset-based ones  and to prove not only that cut is admissible, as it is done in \\cite{I05,I11,L00}, but also that weakening and contraction are height-preserving admissible. Given that implicit contraction is not eliminable from set-based sequents, the decision procedure for non-normal logics given in \\cite{L00} is based on  a model-theoretic inversion technique so that it is possible to define a  procedure that  outputs a derivation for all valid sequents and a finite   countermodel  for all invalid ones. One weakness of this decision  procedure is that  it does not respect the subformula property for logics without rule $RM$ (the procedure  adds instances of the excluded middle).\n \n\n \nThe paper \\cite{I05} considers multiset-based calculi for the non-normal logic  {\\bf M(N)} and for its extensions with axioms $D^\\Diamond,T, 4, 5$, and $B$. Nevertheless,  neither weakening nor contraction is eliminable  because there are no weakening contexts in the conclusion of the modal rules. In \\cite{I11} multiset-based sequent calculi for the non-normal  logic {\\bf E(N)} and  for its extensions with  axioms $D^\\Diamond,T$, 4, 5, and $B$ are given. The rules $LR$-$E$ and $R$-$N$ are  as in Table \\ref{Modal rules}, but the deontic axiom  $D^\\Diamond$ is expressed  by the following  rule:\n\n$$\n\\infer[\\infrule D\\text{-}2]{\\Box A,\\Box B,\\Gamma\\Rightarrow\\Delta}{A,B\\Rightarrow&(\\Rightarrow A,B)}\n$$\nwhere the right premiss  is present when we are working over $LR$-$E$ and  it has to be omitted when we work over $LR$-$M$. In the calculi in \\cite{I05,I11} weakening and contraction are taken as primitive rules and not as admissible ones as in the present approach. Even if it is easy to show that weakening is eliminable from the calculi in \\cite{I11}, contraction cannot be eliminated because rule \\emph{D-2} has exactly two principal formulas and, therefore, it is not possible to permute contraction up with respect to instances of  rule \\emph{D-2} (see Theorem \\ref{contr}). \nThe presence of a non-eliminable rule of contraction makes the elimination of cut more problematic: in most cases we cannot  eliminate the cut directly, but we have to consider the rule known as multicut \\cite[p. 88]{NP01}.  Moreover, cut is not eliminable from the calculus given in \\cite{I11} for the deontic  logic {\\bf END}. The formula  $D^\\bot:= \\neg\\Box\\bot$ is a theorem of this logic, but it can be derived only with a non-eliminable instance of  cut  as in:\n\n$$\n\\infer[\\infrule R\\neg]{\\Rightarrow \\neg\\Box\\bot}{\n\\infer[\\infrule Cut]{\\Box\\bot\\Rightarrow}{\n\\infer[\\infrule R\\mbox{-}N]{\\Rightarrow\\Box\\top}{\\Rightarrow\\top}&\n\\infer[\\infrule D\\mbox{-}2]{\\Box\\top,\\Box\\bot\\Rightarrow}{\\bot,\\top\\Rightarrow&\\Rightarrow\\bot,\\top}}}\n$$ \n\nFinally, it is worth noticing that all the non-normal logics we consider here are  \\emph{rank-1} logics in the sense of \\cite{P13,PS10,PS09} -- i.e., logics whose modal axioms are propositional combinations of formulas of shape $\\Box\\phi$, where  $\\phi$ is purely propositional -- and the calculi we give for the  modal logics {\\bf E}, {\\bf M}, {\\bf K} and {\\bf KD} are explicitly considered in \\cite{P13,PS09}.  Thus, they are part of the family of modal coalgebraic logics  \\cite{P13,PS10,PS09} and most of the results in this paper can be seen as  instances of general results that hold for rank-1 (coalgebraic) logics. If, in particular, we consider cut-elimination for coalgebraic logics \\cite{PS10} then all our calculi  absorb congruence  and Theorem \\ref{contr} and case 3 of Theorem \\ref{cut} show that they absorb  contraction and cut. Hence, \\cite[Thm. 5.7]{PS10} entails  that cut and contraction are   admissible in these calculi; moreover, \\cite[Props. 5.8 and 5.11]{PS10} entail that they are one-step cut free complete w.r.t. coalgebraic semantics. This latter result gives a semantic proof of cut admissibility  in the calculi considered here. Analogously, if we consider decidability, the polynomial space upper bound we find in Section \\ref{decision} coincides with that found in \\cite{PS09} for rank-1 modal logics. \n\n\n\\paragraph{Synopsis. }\nSection \\ref{secaxiom} summarizes the basic notions of axiomatic systems and of neighbourhood semantics for  non-normal  logics.  Section \\ref{seccalculi} presents G3-style  sequent calculi for these logics and then shows that weakening and contraction are height-preserving admissible and that cut is (syntactically) admissible.  Section \\ref{secdecax} describes a  terminating proof-search decision procedure for all calculi, it shows that each calculus is equivalent to the  corresponding axiomatic system, and  it applies proof search to Forrester's paradox.  Finally, Section \\ref{secinterpol} gives a Maehara-style    constructive proof of Craig's interpolation theorem for the logics having a standard calculus.\n\\section{Non-normal Logics}\\label{secaxiom}\n\\subsection{Axiomatic Systems}\nWe introduce, following \\cite{C80},  the basic notions  of non-normal  logics. Given a countable set of propositional variables $\\{p_n\\,|\\,n\\in \\mathbb{N}\\}$, the formulas of the modal language $\\mathcal{L}$  are generated by:\n\n$$\nA::= \\;p_n\\;|\\;\\bot\\;|\\;A\\wedge A\\;|\\;A\\lor A\\;|\\;A\\supset A\\;|\\;\\Box A\n$$\nWe remark that $\\bot$ is a 0-ary logical symbol. This will be extremely important in the proof of Craig's interpolation theorem. As usual $\\neg A$ is a shorthand for $A\\supset\\bot$, $\\top$ for $\\bot\\supset\\bot$, $A \\leftrightarrow B$  for $(A\\supset B)\\wedge(B\\supset A)$, and $\\Diamond A$ for $\\neg\\Box\\neg A$. We follow the usual conventions for parentheses. \n\n\n\n\\begin{table}\n\\caption{ Rules of inference}\\label{rulesinf}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hline\\hline\\noalign{\\smallskip}\n\\infer[\\infrule RE]{\\Box A\\leftrightarrow\\Box B}{A\\leftrightarrow B}\n&$\\quad$&\n\\infer[\\infrule RM]{\\Box A\\supset\\Box B}{A\\supset B}\n\\\\\\noalign{\\smallskip\\smallskip\\smallskip}\n\\infer[\\infrule RR,\\; n\\geq 1]{(\\Box A_1\\wedge\\dots\\wedge\\Box A_n)\\supset\\Box B}{(A_1\\wedge\\dots\\wedge A_n)\\supset B}\n&$\\quad$&\n\\infer[\\infrule RK,\\; n\\geq 0]{(\\Box A_1\\wedge\\dots\\wedge\\Box A_n)\\supset\\Box B}{(A_1\\wedge\\dots\\wedge A_n)\\supset B}\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{table}\n\\caption{ Axioms}\\label{axioms}\n\\begin{center}\n\\begin{tabular}{cclccclcccl}\n\\hline\\hline\\noalign{\\smallskip}\n$M$)&$\\quad$&$\\Box (A\\wedge B)\\supset(\\Box A\\wedge\\Box B)$&${}$\\qquad{}\\qquad{}\\qquad&$C$)&$\\quad$&$(\\Box A\\wedge\\Box B)\\supset\\Box(A\\wedge B)$ \\\\\\noalign{\\smallskip\\smallskip\\smallskip}$N$)&$\\quad$&$\\Box \\top$&${}$\\qquad{}\\qquad{}\\qquad&\n$D^\\bot$)&$\\quad$&$\\neg\\Box\\bot$ \\\\\\noalign{\\smallskip\\smallskip\\smallskip}$D^\\Diamond$)&$\\quad$&$ \\Box A\\supset\\Diamond A$\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nLet {\\bf L}  be the logic containing  all $\\mathcal{L}$-instances of propositional tautologies as axioms, and   modus ponens ($MP$) as inference rule.  The minimal non-normal modal logic {\\bf E} is the logic {\\bf L} plus the rule $RE$ of Table \\ref{rulesinf}. We will consider all the logics that are obtained by extending {\\bf E} with some set of axioms from Table \\ref{axioms}. We will denote the logics according to the axioms that define them, e.g. {\\bf EC} is the logic  {\\bf E}$\\,\\oplus \\,C$, and {\\bf EMD$^\\bot$} is {\\bf E}$\\,\\oplus\\, M\\oplus D^\\bot$.  By {\\bf X} we denote any of these logics and we  write \\mbox{{\\bf X} $\\vdash A$} whenever $A$ is a theorem of {\\bf X}. We will call \\emph{modal} the logics containing neither $D^\\bot$ nor $D^\\Diamond$, and \\emph{deontic} those containing at least one of them. We have followed the usual naming conventions for the modal axioms, but we have introduced new  conventions for the deontic ones: $D^\\bot$ is usually called either  $CON$ or $P$ and $D^\\Diamond$ is usually called $D$, cf. \\cite{G00,G13,M06}.\n \n It is also possible to give an equivalent  rule-based axiomatization of some of these logics. In particular, the logic {\\bf EM}, also called {\\bf M}, can be axiomatixed as {\\bf L} plus the rule $RM$ of Table \\ref{rulesinf}.  The logic  {\\bf EMC}, also called {\\bf R}, can be axiomatized as {\\bf L} plus the rule  $RR$ of Table \\ref{rulesinf}. Finally,  the logic {\\bf EMCN}, i.e. the smallest normal modal logic {\\bf K},  can be axiomatized as {\\bf L} plus the rule $RK$ of Table \\ref{rulesinf}. These rule-based axiomatizations will be useful later on since they simplify the proof of the equivalence between  axiomatic systems and  sequent calculi (Theorem \\ref{comp}).\n\n\n\n\n\n\n\nThe following proposition states the well-known relations between the theorems of non-normal modal  logics. For a proof  the reader is referred to \\cite{C80}.\n\n\n\\begin{proposition} For any formula $A\\in \\mathcal{L}$ we have that {\\bf E} $\\vdash A$ implies {\\bf M}  $\\vdash A$; {\\bf M} $\\vdash A$   implies {\\bf R} $\\vdash A$; {\\bf R} $\\vdash A$  implies {\\bf K} $\\vdash A$. Analogously for the logics containing axiom $N$ and\/or axiom $C$.\\end{proposition}\n\n\nAxiom $D^\\bot$ is {\\bf K}-equivalent to $D^\\Diamond$, but the correctness of $D^\\Diamond$ has been a big issue in the literature on deontic logic. This fact urges the study of  logics weaker than {\\bf KD}, where $D^\\bot$ and $D^\\Diamond$ are no more equivalent \\cite{C80}. \nThe deontic  formulas $D^\\bot$ and $D^\\Diamond$ have the following relations in the logics we are considering.\n\\begin{proposition} $D^\\bot$ and $D^\\Diamond$ are  independent  in {\\bf E}; $D^\\bot$ is derivable from $D^\\Diamond$ in  non-normal logics containing at least one of the axioms  $M$  and $N$; $D^\\Diamond$ is derivable from $D^\\bot$ in non-normal logics containing  axiom $C$.\n\\end{proposition}\nIn Figure  \\ref{cube} the reader finds the lattice of non-normal modal logics, see \\cite[p. 237]{C80}, and in Figure \\ref{cubed}  the lattice of non-normal deontic logics. \n \\begin{figure}[t\n\\begin{center}\n\\begin{tikzpicture}\n\\node at (1,-2)  {EM={\\bf M}};\n\\node at (4,-4)  {{\\bf E}};\n\\node at (4,-2)  {{\\bf EC}};\n\\node at (7,-2)  {{\\bf EN}};\n\\node at (1,0)  {EMC={\\bf R}};\n\\node at (4,2)  {EMCN={\\bf K}};\n\\node at (4,0)  {{\\bf EMN}};\n\\node at (7,0)  {{\\bf ECN}};\n\n\\draw (3.8,-3.8) -- (1,-2.2);\n\\draw (4,-3.8) -- (4,-2.2); \n\\draw (4.2,-3.8) -- (7,-2.2);\n\n\\draw (1,-1.8) -- (1,-0.2);\n\\draw (4,0.2) -- (4,1.8); \n\\draw (7,-1.8) -- (7,-0.2);\n\n\\draw (1.2,0.2) -- (3.8,1.8);\n\\draw (6.8,0.2) -- (4.2,1.8);\n\\draw (1.2,-1.8) -- (3.8,-0.2);\n\\draw (6.8,-1.8) -- (4.2,-0.2);\n\n\\draw (3.8,-1.8) -- (1.2,-0.2);\n\\draw (4.2,-1.8) -- (6.8,-0.2);\n\\end{tikzpicture}\n\\caption{Lattice of non-normal modal logics}\\label{cube}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[t]\n\\begin{center}\n\\scalebox{0.80000}{\\begin{tikzpicture}\n\\filldraw  [black] (5,-4.2) circle (4pt) \n\t\t     (6.7,-4.2) circle (4pt)\n\t\t     (5,-3) circle (4pt);\n\\node[below] at (4.8,-4.3) {\\small{\\bf ED$^\\bot$}};\n\\node[below] at (6.7,-4.3)  {\\small{\\bf ED$^\\Diamond$}};\n\\node[left] at (4.9,-3) {\\small{ ED$^\\bot$D$^\\Diamond$}=};\n\\node[left] at (4.9,-3.3) {\\small{\\bf ED}};\n\\draw (5,-4.2) -- (1,-1.7)\n\\draw (5,-3) -- (1,-0.5)\n\\draw (5,-4.2) -- (5,-3)\n\\draw (6.7,-4.2) -- (5,-3)\n\\draw  (5,-3) -- (5,-0.5)\n\\draw (6.7,-4.2) -- (6.7,-1.6)\n\\draw (5,-4.2) -- (9,-1.7)\n\\draw (5,-3) -- (9,-0.5)\n\n\n\n\\filldraw  [black] (6.7,-1.5) circle (4pt) \n\t\t     (5,-0.5) circle (4pt);\n\\node[left] at (6.5,-1.5)  {\\small{\\bf ECD$^\\Diamond$}};\n\\node[left] at (4.9,-0.5)  {\\small{ ECD$^\\bot$=}};\n\\node[left] at (4.9,-0.8)  {\\small{ \\bf ECD}};\n\\draw (5,-0.5) -- (9,2)\n\\draw (6.7,-1.5)-- (5,-0.5)\n\\draw (5,-0.5) -- (1,2)\n\n\n\\filldraw  [black] (9,-1.7) circle (4pt) \n\t\t   \n\t\t     (9,-0.5) circle (4pt);\n\\node[right] at (9.2,-1.7)  {\\small{\\bf END$^\\bot$}};\n\\node[right] at (9.1,-0.5)  {\\small{ END$^\\Diamond$}= {\\bf END}};\n\\draw (9,-0.5) -- (9,2)\n\\draw (9,-1.7) -- (9,-0.5)\n\\draw (9,-1.7) -- (5,0.8)\n\\draw (9,-0.5) -- (5,2)\n\n\n\\filldraw  [black] (1,2) circle (4pt)  \n\\node[left] at (0.9,2)  {\\small{{ RD$^\\bot$}= {RD$^\\Diamond$}= {\\bf RD}}};\n\\draw (1,2) -- (5,4.5);\n\n\\filldraw  [black] (5,4.5) circle (4pt)  \n\\node[above] at (5,4.6)  {\\small{{KD$^\\bot$}= {KD$^\\Diamond$}= {\\bf KD}}};\n\n\\filldraw  [black] (1,-1.7) circle (4pt) \n\t\t     (1,-0.5) circle (4pt);\n\\node[left] at (0.9,-1.7)  {\\small{\\bf MD$^\\bot$}};\n\\node[left] at (0.9,-0.5) {\\small{ MD$^\\Diamond$}= {\\bf MD}};\n\\draw (1,-1.7) -- (5,0.8)\n\\draw (1,-0.5) -- (5,2)\n\\draw (1,-1.7) -- (1,-0.5)\n\t\t     (5,2) circle (4pt);\n\\draw (1,-0.5) -- (1,2)\n\n\\filldraw  [black] (5,0.8) circle (4pt) \n\t\t    (5,2) circle (4pt); \n\\node[right] at (5.2,0.8)  {\\small{\\bf MND$^\\bot$}};\n\\node[right] at (5.1,2)  {\\small{ MND$^\\Diamond$}= {\\bf MND}};\n\\draw (5,0.8) -- (5,2);\n\\draw (5,2) -- (5,4.5);\n\n\\filldraw  [black]\n\t\t     (9,2) circle (4pt);\n\\node[right] at (9.1,2)  {\\small{ECND$^\\bot$} = {ECND$^\\Diamond$}= {\\bf ECND}};\n\\draw (9,2) -- (5,4.5)\n\\end{tikzpicture}}\n\\caption{Lattice of non-normal deontic logics}\\label{cubed}\n\\end{center}\n\\end{figure\n\n\n\n\\subsection{Semantics}\\label{semantics}\nThe most widely known semantics for non-normal logics is neighbourhood semantics. We sketch its main tenets following \\cite{C80}, where neighbourhood models are called \\emph{minimal models}. \n\n\\begin{definition}A \\emph{neighbourhood model} is a triple $\\mathcal{M}:=\\langle W,\\, N,\\,P\\rangle$, where $W$ is a non-empty set of possible worlds; $N:W\\longrightarrow 2^{2^W}$ is a neighbourhood function that associates to each possible world $w$ a set $N(w)$ of subsets   of $W$; and $P$ gives a truth value to each propositional variable at each world. \\end{definition}\n\n\\noindent The definition of truth of a formula $A$ at a world $w$ of a neighbourhood model $\\mathcal{M}$ -- $\\models_w^\\mathcal{M}A$ -- is the standard one for the classical connectives with the addition of\n\n$$\n\\models_w^\\mathcal{M}\\Box A\\qquad \\textnormal{iff}\\qquad || A||^\\mathcal{M}\\in N(w)\n$$\nwhere $||A||^\\mathcal{M}$ is the truth set of $A$ -- i.e., $||A||^\\mathcal{M}=\\{w\\,|\\,\\models_w^\\mathcal{M} A\\}$. We say that a formula $A$  is \\emph{valid} in a class $\\mathcal{C}$ of neighbourhood models iff it is true in every world of every $\\mathcal{M}\\in\\mathcal{C}$.\n\nIn order to give  soundness and completeness results for non-normal modal and deontic  logics with respect to (classes of) neighbourhood models, we introduce the following definition.\n\\begin{definition} Let $\\mathcal{M}=\\langle W,\\, N,\\,P\\rangle$ be a neighbourhood model, $X,Y \\in 2^W$, and $w\\in W$, we say that: \n\\begin{itemize}\n\\item $\\mathcal{M}$ is \\emph{supplemented}    if $X\\cap Y\\in N(w)$ imples $X\\in N(w)$ and $Y\\in N(w)$;\n\\item$\\mathcal{M}$ is \\emph{closed under finite intersection}  if $X\\in N(w)$ and $Y\\in N(w)$ imply $X\\cap Y\\in N(w)$;\n\\item $\\mathcal{M}$ \\emph{contains the unit} if   $W\\in N(w)$;\n\\item $\\mathcal{M}$ is \\emph{non-blind}  if $X\\in N(w)$ implies $X\\neq \\emptyset$;\n\\item $\\mathcal{M}$ is \\emph{complement-free} if $X\\in N(w)$ implies $W-X\\not\\in N(w)$.            \n\\end{itemize}\\end{definition}\n\n\n\\begin{proposition}\\label{corrax} We have the following correspondence results between $\\mathcal{L}$-formulas and the properties of the neighbourhood function defined above:\n\\begin{itemize}\n\\item Axiom $M$ corresponds to supplementation;\n\\item Axiom $C$ corresponds to closure under finite intersection;\n\\item Axiom $N$ corresponds to containment of the unit;\n\\item Axiom $D^\\bot$ corresponds to non-blindness;\n\\item Axiom $D^\\Diamond$ corresponds to complement-freeness.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{theorem}\\label{compax} {\\bf E} is sound and complete with respect to the class of all neighbourhood models. Any logic {\\bf X} which is obtained by extending {\\bf E} with some axioms from Table \\ref{axioms} is sound and complete with respect to the class of all neighbourhood models which satisfies all the properties corresponding to the axioms of {\\bf X}.\n\\end{theorem}\n\nSee \\cite{C80} for the proof of Proposition \\ref{corrax} and of Theorem \\ref{compax}.\n\n\\section{Sequent Calculi}\\label{seccalculi}\n\n We introduce  sequent calculi for non-normal logics that extend the multiset-based  sequent calculus  {\\bf G3cp} \\cite{NP01,NP11,T00} for classical propositional logic  -- see Table \\ref{G3cp} -- by adding some modal and deontic rules from Table \\ref{Modal rules}.   In particular, we  consider the modal sequent calculi given in Table \\ref{modcalculi}, which will be shown to capture the modal logics of Figure \\ref{cube}, and their deontic extensions given in Table \\ref{deoncalculi}, which will be shown to capture all deontic logics of Figure \\ref{cubed}.\nWe adopt the following notational conventions: we use {\\bf G3X} to denote a generic calculus from either Table \\ref{modcalculi} or Table \\ref{deoncalculi}, and we use {\\bf G3Y(Z)}  to denote both {\\bf G3Y} and {\\bf GRYZ}.  All the rules in Tables \\ref{G3cp} and  \\ref{Modal rules} but  $LR$-$C$ and $L$-$D^{\\Diamond_C}$ are standard rules in the sense of \\cite{G16}: each of them is a  single rule with a fixed number of premisses; $LR$-$C$ and $L$-$D^{\\Diamond_C}$, instead, stand for a recursively enumerable set of rules with a variable number of premisses.   \n\n For an introduction to {\\bf G3cp} and  the relevant notions, the reader is referred to \\cite[Chapter 3]{NP01}.\n   We sketch here the main notions that will be used in this paper.   A \\emph{sequent} is an expression  $\\Gamma\\Rightarrow \\Delta$, where $\\Gamma$ and $\\Delta$ are finite, possibly empty, multisets of formulas. If $\\Pi$ is the (possibly empty) multiset $A_1,\\dots,A_m$ then $\\Box\\Pi$ is  the (possibly empty) multiset $\\Box A_1,\\dots,\\Box A_m$. A \\emph{derivation} of a sequent $\\Gamma\\Rightarrow\\Delta$ in {\\bf G3X} is an upward growing  tree of sequents having $\\Gamma\\Rightarrow\\Delta$ as root, initial sequents or instances of rule $L\\bot$ as leaves, and such that each non-initial node is the conclusion of an instance of one  rule of {\\bf G3X} whose premisses are its children. In the rules in Tables \\ref{G3cp} and \\ref{Modal rules}, the multisets $\\Gamma$ and $\\Delta$ are called \\emph{contexts}, the other formulas occurring in the conclusion (premiss(es), resp.) are called \\emph{principal} (\\emph{active}). In a sequent the \\emph{antecedent} (\\emph{succedent}) is the multiset occurring to the left (right) of the sequent arrow $\\Rightarrow$.  As for {\\bf G3cp}, a sequent $\\Gamma\\Rightarrow\\Delta$ has the following  \\emph{denotational interpretation}: the conjunction of the formulas in $\\Gamma$ implies the disjunction of the formulas in $\\Delta$.\n \nAs measures for inductive proofs we  use the weight of a formula and the height of a derivation. The \\emph{weight} of a formula $A$, $w(A)$, is defined inductively as follows: $w(\\bot)=w(p_i)=0$; $w(\\Box A)=w(A)+1$; $w(A\\circ B)=w(A)+w(B)+1$ (where    $\\circ$ is one of the binary connectives $\\wedge,\\,\\lor,\\,\\supset$). The \\emph{weight} of a sequent is the sum of the weight of the formulas occurring in that sequent.  The \\emph{height} of a derivation is the length of its longest branch minus one. A rule of inference is said to be (\\emph{height-preserving}) \\emph{admissible} in {\\bf G3X} if, whenever its premisses are derivable in {\\bf G3X}, then also its conclusion is derivable (with at most the same derivation height) in {\\bf G3X}. The \\emph{modal depth} of a formula (sequent) is the maximal number of nested modal operators occurring in it(s members).\n  \n\n   \\begin{table}\n\\caption{ The sequent calculus {\\bf G3cp}}\\label{G3cp}\n\\begin{center}\n\\scalebox{0.85000}{\\begin{tabular}{cccc}\n\\hline\\hline\\noalign{\\smallskip}\n Initial sequents: &$\\qquad p_n,\\Gamma\\Rightarrow\\Delta,p_n$&& $p_n$ propositional variable \\\\\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  Propositional rules:&\n  \\infer[\\infrule L\\wedge]{A\\wedge B,\\g\\To\\d}{A,B,\\g\\To\\d}\n\n&$\\qquad$&\n\\infer[\\infrule R\\wedge]{\\g\\To\\d,A\\wedge B}{\\g\\To\\d,A\\quad&\\g\\To\\d,B}\n\n\\\\\\noalign{\\smallskip\\smallskip}\n\\infer[\\infrule L\\bot]{\\bot,\\g\\To\\d}{}&\n\\infer[\\infrule L\\lor]{A\\lor B,\\g\\To\\d}{A,\\g\\To\\d\\quad&B,\\g\\To\\d}\n\n&&\n\\infer[\\infrule R\\lor]{\\g\\To\\d,A\\lor B}{\\g\\To\\d,A,B}\n\n\\\\\\noalign{\\smallskip\\smallskip}&\n\\infer[\\infrule L\\supset]{A\\supset B,\\g\\To\\d}{\\g\\To\\d,A\\quad&B,\\g\\To\\d}\n\n&&\n\\infer[\\infrule R\\supset]{\\g\\To\\d,A\\supset B}{A,\\g\\To\\d,B}\n\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}}\n\\end{center}\n\\caption{Modal and deontic rules}\\label{Modal rules}\n\\begin{center}\n\\scalebox{0.87000}{\\begin{tabular}{llll}\n\\hline\\hline\\noalign{\\smallskip}\n\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\g\\To\\d,\\Box B}{A\\Rightarrow B\\quad&B\\Rightarrow A}\n&\n\\infer[\\infrule LR\\mbox{-}M ]{\\Box A,\\g\\To\\d,\\Box B}{A\\Rightarrow B}\n&\n\\multicolumn{2}{c}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box \\Pi,\\g\\To\\d,\\Box B}{A,\\Pi\\Rightarrow B}}\n\n\\\\\\noalign{\\smallskip\\smallskip}\n\\multicolumn{2}{c}{\\infer[\\infrule LR\\mbox{-}C ]{\\Box A_1,\\dots,\\Box A_n,\\g\\To\\d,\\Box B}{A_1,\\dots,A_n\\Rightarrow B&B\\Rightarrow A_1&{}^{\\dots}&B\\Rightarrow A_n}\n}\n&\n\\infer[\\infrule LR\\mbox{-}K]{\\Box\\Pi,\\g\\To\\d,\\Box B}{\\Pi\\Rightarrow B}&\n\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box B}{\\Rightarrow B}\n\\\\\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\end{tabular}}\n\n\\scalebox{0.8700}{\\begin{tabular}{llllll}\n\n\n&\n\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box A,\\g\\To\\d}{A\\Rightarrow}\n&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}, \\,|\\Pi|\\leq 2]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow&\\Rightarrow\\Pi}\n&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_M}, \\,|\\Pi|\\leq 2]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow}\n\\\\\\noalign{\\smallskip\\smallskip}\n\\phantom{aaaaaaaa}&\n\\multicolumn{2}{c}{\\infer[\\infrule L\\mbox{-}D^{\\Diamond_C}]{\\Box \\Pi,\\Box\\Sigma,\\g\\To\\d}{\\Pi,\\Sigma\\Rightarrow&\\{ \\Rightarrow A, B|\\; A\\in\\Pi, B\\in \\Sigma \\}}\n}\n&\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box\\Pi,\\g\\To\\d}{\\Pi\\Rightarrow}&\\phantom{aaaaaaa}\n\\\\\n\\noalign{\\smallskip}\\hline\\hline\n\\end{tabular}}\\end{center}\n \\caption{Modal  sequent calculi (\\checkmark= rule of the calculus, $\\star$ = admissible rule, $-$ = neither)}\\label{modcalculi}\n \n \\begin{center}\n\\scalebox{0.66000}{ \\begin{tabular}{r|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}  \n\\noalign{\\smallskip} \\hline\\hline\\noalign{\\smallskip\\smallskip}\n&{\\bf G3E}&{\\bf G3EN}&{\\bf G3M}& {\\bf G3MN}&{\\bf G3C}&{\\bf G3CN}& {\\bf G3R}&{\\bf G3K}\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$E$&\\checkmark&\\checkmark&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$M$&$-$&$-$&\\checkmark&\\checkmark&$-$&$-$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$C$&$-$&$-$&$-$&$-$&$\\checkmark$&$\\checkmark$&$\\star$&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$R$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$LR$-$K$&$-$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark\\\\\n\n \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n$R$-$N$&$-$&$\\checkmark$&$-$&$\\checkmark$&$-$&$\\checkmark$&$-$&$\\star$\\\\\\noalign{\\smallskip}\\hline\\hline\n  \\end{tabular}}\\end{center}\n\\caption{Deontic sequent calculi (\\checkmark= rule of the calculus, $\\star$ = admissible rule, $-$ = neither)}\\label{deoncalculi}\n\\begin{center}\n\\scalebox{0.66000}{\\begin{tabular}{l|c|c|c|c|c|c|c|c|c|c|}\n\n\\hline\\hline\\noalign{\\smallskip\\smallskip\\smallskip}\n  &{\\bf G3E(N)D$^\\bot$}&{\\bf G3ED$^\\Diamond$}&{\\bf G3E(N)D}&{\\bf G3M(N)D$^\\bot$}&{\\bf G3M(N)D}&{\\bf G3CD$^\\Diamond$}&{\\bf G3C(N)D}&{\\bf G3RD}&{\\bf G3KD}\\\\\n  \n     \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^\\bot$&\\checkmark&$-$&\\checkmark&\\checkmark&$\\star$&$-$&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_E}$&$-$&\\checkmark&\\checkmark&$-$&$\\star$&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n  \n    \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_M}$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^{\\Diamond_C}$&$-$&$-$&$-$&$-$&$-$&\\checkmark&$\\star$&$\\star$&$\\star$\\\\\n  \n   \\noalign{\\smallskip}\\hline\\noalign{\\smallskip\\smallskip}\n  $L$-$D^*$&$-$&$-$&$-$&$-$&$-$&$-$&\\checkmark&\\checkmark&\\checkmark\n   \n\\\\\n\\noalign{\\smallskip}\\hline\\hlin\n\\end{tabular}}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Structural rules of inference}\nWe are now going to prove that the calculi {\\bf G3X} have the same good structural properties of {\\bf G3cp}: weakening and contraction are height-preserving admissible  and  cut is admissible. All proofs are extension of those for {\\bf G3cp}, see  \\cite[Chapter 3]{NP01}; in most cases, the modal rules have to be treated differently from the propositional ones  because of the presence of empty contexts in the premiss(es) of the modal ones. We adopt the following notational convention: given a derivation tree $\\mathcal{D}_k$, the derivation tree of the $n$-th leftmost premiss of its last step is  denoted by $\\mathcal{D}_{kn}$. We begin by showing that the restriction to atomic initial sequents, which is needed to have the propositional rules invertible, is not limitative in that initial  sequents with arbitrary principal  formula are derivable in {\\bf G3X}.\n\n\\begin{proposition}\\label{genax} Every instance of $A,\\Gamma\\Rightarrow\\Delta, A$ is derivable in {\\bf G3X}.\\end{proposition}\n\n\\begin{proof} By induction on the weight of $A$. If $w(A)=0$ -- i.e., $A$ is atomic or $\\bot$ -- then we have an instance of an initial sequent or of a conclusion of $L\\bot$ and there is nothing to prove. If $w(A)\\geq1$,  we  argue by cases according to the construction of $A$. In each case we apply, root-first, the appropriate rule(s) in order to obtain sequents where some proper subformula of $A$ occurs both in the antecedent and in the succedent. The claim then holds by the inductive hypothesis (IH). To illustrate, if $A\\equiv\\Box B$ and we are in {\\bf G3M(ND)}, we have:\n\n$$\n\\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\g\\To\\d,\\Box B}{\\infer[\\infrule IH]{B\\Rightarrow B}{}}\n$$\n  \\end{proof}\n\n\\begin{theorem}\\label{weak} The left and right rules of weakening are height-preserving admissible in {\\bf G3X}\\\n\n$$\n\\infer[\\infrule LW]{A,\\g\\To\\d}{\\g\\To\\d}\n\\qquad\n\\infer[\\infrule RW]{\\g\\To\\d,A}{\\g\\To\\d}\n$$\n  \\end{theorem}\n\n\\begin{proof}The proof is a straightforward induction on the height of the derivation $\\mathcal{D}$ of $\\Gamma\\Rightarrow \\Delta$.\n If the last step of $\\mathcal{D}$ is by a propositional rule, we have to apply the same rule to the weakened premiss(es), which are derivable by IH, see \\cite[Thm. 2.3.4]{NP01}. If it is by a modal or deontic rule, we proceed by adding $A$ to the appropriate weakening context of the conclusion of that rule  instance. To illustrate, if the last rule is $LR$-$E$, we transform \n \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box B,\\g\\To\\d,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_2}  }\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{\\Box B,A,\\g\\To\\d,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_2}  }\n$$\n\n{}\\end{proof}\n\n\n\n\\noindent Before considering contraction, we recall some facts that will be useful later on. \n\\begin{lemma} In {\\bf G3X} the rules$\\quad$\n\\infer[\\infrule L\\neg]{\\neg A,\\g\\To\\d}{\\g\\To\\d,A}\nand$\\quad$\n\\infer[\\infrule R\\neg]{\\g\\To\\d,\\neg A}{A,\\g\\To\\d}\nare admissible.\n\\end{lemma}\n\\begin{proof}\nWe have the following derivations (the step by $RW$ is admissible thanks to Theorem \\ref{weak}):\n\n$$\\infer[\\infrule L\\supset]{A\\supset\\bot,\\g\\To\\d}{\\g\\To\\d,A&\\infer[\\infrule L\\bot]{\\bot,\\g\\To\\d}{}}\n\\qquad\\qquad\n\\infer[\\infrule R\\supset]{\\g\\To\\d,A\\supset\\bot}{\\infer[\\infrule RW]{A,\\g\\To\\d,\\bot}{A,\\g\\To\\d}}\n$$\n\\end{proof}\n\n\\begin{lemma}\\label{inv}All propositional rules are height-preserving invertible in {\\bf G3X}, that is the derivability of (a possible instance of) a conclusion of a propositional rule entails the derivability, with at most the same derivation height, of its premiss(es).\\end{lemma}\n\n\\begin{proof} We have only to extend the proof  for {\\bf G3cp}, see  \\cite[Thm. 3.1.1]{NP01}, with new cases for the modal and deontic rules. If $A\\circ B$ occurs in the antecedent (succedent) of the conclusion of an instance of a  modal or deontic rule then it must be a member of the weakening context $\\Gamma$ ($\\Delta$)  of this rule instance, and we have only to change the weakening context according to the rule we are inverting.\\end{proof}\n\n\\begin{theorem}\\label{contr}The left and right rules of contraction are height-preserving admissible in {\\bf G3X}\\vspace{0.3cm}\n\n$$\n\\infer[\\infrule LC]{A,\\g\\To\\d}{A,A,\\g\\To\\d}\n\\qquad\n\\infer[\\infrule RC]{\\g\\To\\d,A}{\\g\\To\\d,A,A}\n$$\n \\end{theorem}\n \n \\begin{proof} The proof is by simultaneous  induction on the height  of the derivation $\\mathcal{D}$  of the  premiss for left and right  contraction. The base case is straightforward. For the inductive steps, we have different strategies according to whether the last step in $\\mathcal{D}$ is by a propositional  rule or not. If the last step in $\\mathcal{D}$ is  by a propositional rule, we have two subcases: if the contraction formula is not principal in that step, we apply the inductive hypothesis and then the rule. Else  we start by using the height-preserving invertibility --  Lemma \\ref{inv} -- of that rule, and then we apply the inductive hypothesis and the rule, see \\cite[Thm. 3.2.2]{NP01} for details. \n \n If the last step in $\\mathcal{D}$ is by a modal or deontic rule, we have two subcases: either (the last step is by one  of $LR$-$C$, $LR$-$R$, $LR$-$K$, $L$-$D^{\\Diamond_E}$, $L$-$D^{\\Diamond_M}$, $L$-$D^{\\Diamond_C}$ and $L$-$D^*$ and) both occurrences of the contraction formula $A$ of $LC$ are principal in the last step or  some  instance of the contraction formula  is    introduced in the appropriate  weakening context of the conclusion. In the first subcase, we apply the inductive hypothesis to the premiss and then the rule. An interesting example is when the last step in $\\mathcal{D}$ is by $L$-$D^{\\Diamond_E}$. We transform\n \n$$\n\\infer[\\infrule LC]{\\Box B,\\Gamma\\Rightarrow\\Delta}{\\infer[\\infrule L\\mbox{-}D^{\\Diamond}]{\\Box B,\\Box B,\\Gamma\\Rightarrow\\Delta}{\\deduce{B,B\\Rightarrow\\quad}{\\vdots\\;\\mathcal{D}_1}&\\deduce{\\Rightarrow B,B}{\\vdots\\;\\mathcal{D}_2}}}\n\\qquad\\textrm{ into }\\qquad\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond}]{\\Box B,\\Gamma\\Rightarrow\\Delta}{\\deduce{B\\Rightarrow\\qquad}{\\vdots\\;IH(\\mathcal{D}_1)}&\\deduce{\\Rightarrow B\\qquad}{\\vdots\\;IH(\\mathcal{D}_2)}}\n$$\n \n\\noindent where $IH(\\mathcal{D}_1)$ is obtained  by applying the inductive hypothesis for the left rule of contraction to $\\mathcal{D}_1$ and $IH(\\mathcal{D}_2)$ is obtained by applying the inductive hypothesis for the right rule of contraction to $\\mathcal{D}_2$.\n\nIn the second subcase,  we apply an instance of the same   modal or deontic rule which introduces one less occurrence of $A$ in the appropriate context of the conclusion. Let's consider $RC$.  If the last step is by $LR$-$M$ and no  instance of $A$ is principal in the last rule, we transform\\vspace{0.3cm}\n$$\n\\infer[\\infrule  RC]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,\\Box C}{ \\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,A,\\Box C}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_1}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\Box B,\\Gamma'\\Rightarrow\\Delta',A,\\Box C}{\\deduce{B\\Rightarrow C\\quad}{\\vdots\\;\\mathcal{D}_1}}\n$$\n\n\\end{proof}\n \n \\begin{theorem}\\label{cut}The rule of cut is admissible in {\\bf G3X}\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\deduce{\\g\\To\\d,D}{\\vdots\\;\\mathcal{D}_1}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n$$ \n\\end{theorem}\n \n \\begin{proof}\n We consider an uppermost application of $Cut$ and we show that either it is eliminable, or it can be permuted upward in the derivation until we reach sequents where it is eliminable. The proofs, one for each calculus, are by induction on the weight of the cut formula $D$ with a sub-induction on the sum of the heights of the derivations of the two premisses (cut-height for shortness). The proof can be organized in 3 exhaustive cases:\n \n \\begin{enumerate}\n \\item At least one of the premisses of cut  is an initial sequent or a conclusion of $L\\bot$;\n\\item The cut formula in not principal in the last step of at least one of the two premisses;\n \\item The cut formula is principal in both premisses.\n \\end{enumerate}\n \n\n\n\\medskip\n\\noindent {\\bf{\\large\\textbullet}$\\quad$ Case (1).}$\\quad$ Same as for {\\bf G3cp}, see \\cite[Thm. 3.2.3]{NP01} for the details.\n \\medskip\n \n\n \\medskip\n\\noindent {\\bf{\\large\\textbullet}$\\quad$ Case (2).}$\\quad$ We have many subcases according to the last rule applied in the derivation  ($\\mathcal{D}^\\star$)  of the  premiss where the cut formula is not principal. For the propositional rules, we refer the reader to \\cite[Thm. 3.2.3]{NP01}, where it is given a procedure that allows to reduce the cut-height. If the last rule applied in $\\mathcal{D}^\\star$ is a modal or deontic one, we  can transform the derivation into a cut-free one because the conclusion of \\mbox{\\it{Cut}} is derivable  by replacing the last step of  $\\mathcal{D}^\\star$ with  the appropriate instance of the same modal or deontic rule. We present explicitly only the cases where the last step of the left premiss is by $LR$-$E$ and $L$-$D^\\bot$ and the cut formula is not principal in it, all other transformations being similar.\n \\medskip\n\n \n\n\n\\noindent $\\mathbf{LR\\textrm{-}E}:\\quad$ \\, If the left premiss  is by rule $LR$-$E$  (and $\\Gamma\\equiv \\Box A,\\Gamma'$ and $\\Delta\\equiv \\Delta',\\Box B$), we  transform\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta',\\Box B,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta',\\Box B,D}{\\deduce{A\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{B\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}E ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta',\\Box B,\\Sigma}{\\deduce{A\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{B\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}\n$$\n\n\n\n\\noindent $\\mathbf{L}${\\bf-}$\\mathbf{D^\\bot}:\\quad$ \\, If the left premiss  is by rule $L$-$D^\\bot$, we  transform\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box A,\\Gamma'\\Rightarrow\\Delta,D}{\\deduce{A\\Rightarrow }{\\vdots\\;\\mathcal{D}_{11}}}&\\deduce{D,\\Pi\\Rightarrow\\Sigma}{\\vdots\\;\\mathcal{D}_2}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule L\\mbox{-}D^\\bot ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\deduce{A\\Rightarrow }{\\vdots\\;\\mathcal{D}_{11}}}\n$$\n\n\n\n\n \n\n\n\\medskip\n\n\\noindent {\\bf {\\large\\textbullet}$\\quad$ Case (3).}$\\quad$ If the cut formula $D$  is principal in both premisses, we have cases according to the principal operator of $D$. In each case we have a procedure that allows to reduce the weight of the cut formula, possibly increasing the cut-height. For the propositional cases, which are the same for all the logics considered here, see \\cite[Thm. 3.2.3]{NP01}.\n\n If $D\\equiv\\Box C$, we consider the different logics one by one, without repeating the common cases. \\\\\n\n\\noindent\\textbullet$\\quad${\\bf G3E(ND).}$\\quad$ Both premisses are by rule $LR$-$E$, we have\n\n$$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box B}{\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}}}\n$$\n\n\\noindent and we transform it into the following derivation that has two  cuts with cut formulas of lesser weight, which are admissible by IH.\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{ A\\Rightarrow B}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}&\\infer[\\infrule Cut]{B\\Rightarrow A}{\\deduce{B\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3EN(D).}$\\quad$ Left premiss by $R$-$N$ and right one by $LR$-$E$. We transfor\n\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box A}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box A}{\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule R\\mbox{-}N ]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box A}{\\infer[\\infrule Cut]{\\Rightarrow A}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\\noindent \\textbullet$\\quad${\\bf G3E(N)D$^\\bot$.}$\\quad$\nLeft premiss is by $LR$-$E$, and  right one by $L$-$D^\\bot$. We transform \n\n $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box C,\\Pi\\Rightarrow\\Sigma}{\\deduce{C\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule L\\mbox{-}D^\\bot ]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{A\\Rightarrow }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\\noindent \\textbullet$\\quad${\\bf G3E(N)D$^\\Diamond$.}$\\quad$\nLeft premiss is by $LR$-$E$, and  right one by $L$-$D^{\\Diamond_E}$. We transform ($|\\Xi|\\leq 1$\n\n{\\small $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box C,\\Box\\Xi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Xi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}&\\deduce{\\Rightarrow C,\\Xi}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\text{into} \n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box A,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{ \\Rightarrow\\Xi,A}{\\deduce{\\Rightarrow \\Xi, C}{\\vdots\\;\\mathcal{D}_{22}}&\\deduce{C\\Rightarrow A}{\\vdots\\;\\mathcal{D}_{12}}}&\\infer[\\infrule Cut]{A,\\Xi\\Rightarrow }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Xi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n$$}\n\n\\noindent\\textbullet$\\quad${\\bf G3E(N)D.}$\\quad$ Left premiss by $LR$-$E$ and right one by   $L$-$D^\\bot$ or $L$-$D^{\\Diamond_E}$. Same as  above.\\vspace{0.3cm}\n\n\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3END$^\\bot$.}$\\quad$ Left premiss by $R$-$N$ and right one by $L$-$D^\\bot$. We transform\n\n $$\n\\infer[\\infrule Cut]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box C,\\Pi\\Rightarrow\\Sigma}{\\deduce{C\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer=[\\infrule LWs\\mbox{ and }RWs ]{\\Gamma,\\Pi\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{\\phantom{C}\\Rightarrow^{} }{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3END.}$\\quad$ Left premiss by $R$-$N$ and right one by $L$-$D^{\\Diamond_E}$. We transform ($|\\Xi|\\leq 1$)\n\n$$\n\\infer[\\infrule Cut]{\\Box \\Xi,\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[\\infrule R\\mbox{-}N]{\\Gamma\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\n\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\Box C,\\Box \\Xi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Xi}{\\vdots\\;\\mathcal{D}_{21}}\\Rightarrow&\\Rightarrow \\deduce{C,\\Xi}{\\vdots\\;\\mathcal{D}_{22}}}}\n\\qquad\\text{into}\\qquad\n\\infer[\\infrule (\\star)]{\\Box \\Xi,\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{\\Xi\\Rightarrow}{\n\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\n\\deduce{C,\\Xi\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}}}\n$$\nwhere $(\\star)$ is an instance of $L$-$D^\\bot$ if $|\\Xi|=1$, else ($|\\Xi|=0$ and) it is some instances\\mbox{ of $LW$ and $RW$.} \n\n\\noindent \\textbullet$\\quad${\\bf G3M(ND).}$\\quad$\n Both premisses are by rule $LR$-$M$, we transfor\n $$\n\\infer[\\infrule Cut]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}M]{\\Box C,\\Pi\\Rightarrow\\Sigma',\\Box B}{\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n\\qquad\\mbox{into}\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma',\\Pi\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{A\\Rightarrow B }{\\deduce{A\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3MN(D).}$\\quad$ Left premiss by $R$-$N$ and right one by $LR$-$M$. Similar to the case with left premiss by $R$-$N$ and right one by $LR$-$E$.\\vspace{0.3cm}\n\n\\noindent \\textbullet$\\quad${\\bf G3M(N)D$^\\bot$} and {\\bf G3M(N)D.}$\\quad$\nLeft premiss is by $LR$-$M$, and  right one by  $L$-$D^\\bot$ or $L$-$D^{\\Diamond_M}$. Similar to the  case with left premiss by $LR$-$E$ and right by $L$-$D^\\bot$ or $L$-$D^{\\Diamond_E}$, respectively.\\vspace{0.3cm}\n\n\n\n\\noindent\\textbullet$\\quad${\\bf G3MND$^\\bot$} and {\\bf G3MND.}$\\quad$ The cases with left premiss by $R$-$N$ and right one by  a deontic rule are like the analogous ones we have already considered.\n\n\n\\medskip\n\\noindent \\textbullet$\\quad${\\bf G3C(ND).} Both premisses are by rule $LR$-$C$. Let us agree to use $\\Lambda$ to denote the non-empty  multiset $A_1,\\dots,A_n$, and $\\Xi$ for the (possibly empty) multiset $B_2,\\dots B_m$.  The derivation\n\n$$\n\\infer[\\infrule Cut]{\\Box\\Lambda,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box E}{\\infer[\\infrule LR\\mbox{-}C]{\\Box\\Lambda,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Lambda\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{11}}& \\deduce{C\\Rightarrow A_1}{\\vdots \\;\\mathcal{D}_{A_1}}&{}^{\\dots}&\\deduce{C\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}}&\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C,\\Box\\Xi,\\Pi'\\Rightarrow\\Sigma',\\Box E}{\\deduce{C,\\Xi\\Rightarrow E}{\\vdots \\;\\mathcal{D}_{21}}&\\deduce{ E\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{C}}&{}^{\\dots}& \\deduce{E\\Rightarrow B_m}{\\vdots \\;\\mathcal{D}_{B_m}}}}\n$$\n\n\\noindent is transformed into the following derivation having  $n+1$ cuts on formulas of lesser weight\n\\noindent\\scalebox{0.8000}{$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Box\\Lambda,\\Gamma',\\Box\\Xi,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box E}{\\infer[\\infrule Cut]{\\Lambda,\\Xi\\Rightarrow E}{\\deduce{\\Lambda\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C,\\Xi\\Rightarrow E}{\\vdots\\;\\mathcal{D}_{21}}}&\n\\infer[\\infrule{Cut}\\;\\dots]{E\\Rightarrow A_1}{\\deduce{E\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{C}}&\\deduce{C\\Rightarrow A_1}{\\vdots\\;\\mathcal{D}_{A_n}}}&\n\\infer[\\infrule Cut]{E\\Rightarrow A_n}{\\deduce{E\\Rightarrow C}{\\vdots\\:\\mathcal{D}_{C}}&\\deduce{C\\Rightarrow A_n}{\\vdots\\;\\mathcal{D}_{A_n}}}&\n \\deduce{E\\Rightarrow B_1}{\\vdots \\;\\mathcal{D}_{B_1}}\n &{}^{\\dots}& \\deduce{E\\Rightarrow B_m}{\\vdots \\;\\mathcal{D}_{B_m}}\n }\n$$}\\vspace{0.3cm}\n\n\\noindent \\textbullet$\\quad${\\bf G3CN(D).} Left premiss by $R$-$N$ and right premiss by $LR$-$C$. We have\n\n$$\n\\infer[\\infrule Cut]{\\Gamma,\\Box A_1,\\dots,\\Box A_n,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box B}{\\infer[\\infrule R\\mbox{-}N]{\\g\\To\\d,\\Box C}{\\deduce{\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C,\\Box A_1,\\dots,\\Box A_n,\\Pi'\\Rightarrow\\Sigma',\\Box B}{\\deduce{C,A_1,\\dots,A_n\\Rightarrow B}{\\vdots \\;\\mathcal{D}_{21}}&\\deduce{ B\\Rightarrow C}{\\vdots \\;\\mathcal{D}_{C}}&{}^{\\dots}& \\deduce{B\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}}}\n$$\n where $A_1,\\dots,A_n$ (and thus also $\\Box A_1,\\dots,\\Box A_n$)  may or may not be the empty multiset. If $A_1,\\dots,A_n$ is not empty, we transform it  into the following derivation having one cut with cut formula of lesser weigh\n \n$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Gamma,\\Box A_1,\\dots\\Box A_n,\\Pi'\\Rightarrow\\Delta,\\Sigma', \\Box B}{\n\\infer[\\infrule Cut]{A_1,\\dots,A_n\\Rightarrow B}{\\deduce{\\Rightarrow C\\quad}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C,A_1,\\dots, A_n\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}&\n \\deduce{B\\Rightarrow A_1}{\\vdots \\;\\mathcal{D}_{A_1}}\n &{}^{\\dots}& \\deduce{B\\Rightarrow A_n}{\\vdots \\;\\mathcal{D}_{A_n}}\n }\n$$\nIf, instead, $A_1,\\dots,A_n$ is empty, we transform it into\n\n$$\n\\infer[\\infrule R\\mbox{-}N]{\\Gamma,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{\\Rightarrow B}{\\deduce{\\Rightarrow C\\qquad}{\\vdots\\:\\mathcal{D}_{11}}&\\deduce{C\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n$$\n\n\\noindent \\textbullet$\\quad${\\bf G3CD$^{\\Diamond}$.} Left premiss by $LR$-$C$ and right premiss by $L$-$D^{\\Diamond_C}$. We transform  (we assume $\\Xi= A_1,\\dots A_k$, $\\Theta= C, B_2,\\dots, B_m$ and $\\Lambda= D_1,\\dots, D_n$) \n\n$$\n\\infer[\\infrule Cut ]{\\Box\\Xi,\\Box B_1,\\dots, \\Box B_m,\\Box \\Lambda,\\Gamma',\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[LR\\mbox{-}C]{\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{\\{ C\\Rightarrow A_i\\,|\\, A_i\\in\\Xi\\}}{\\vdots\\;\\mathcal{D}_{1A_i}}}&\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_C}]{\\Box C,\\Box B_2,\\dots,\\Box B_m,\\Box\\Lambda,\\Pi'\\Rightarrow\\Sigma}{\n\\deduce{\\Theta,\\Lambda\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}&\n\\deduce{\\{ \\Rightarrow E,D_j\\,|\\, E\\in\\Theta\\text{ and } D_j\\in\\ \\Lambda\\}}{\\vdots\\;\\mathcal{D}_{\\Theta_{i}\\Lambda_j}}}}\n$$\ninto the following derivation having $1+(k\\times n)$ cuts on formulas of lesser weight\n\n$$\\scalebox{0.830}{\n\\infer[\\infrule{ L\\mbox{-}D^{\\Diamond_C}}]{\\Box\\Xi,\\Box B_1,\\dots, \\Box B_m,\\Box \\Lambda,\\Gamma',\\Pi'\\Rightarrow\\Delta,\\Sigma}{\n\\infer[\\infrule Cut]{\\Xi,B_2,\\dots, B_m,\\Lambda\\Rightarrow}{\n\\deduce{\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,B_2,\\dots,B_m,\\Lambda\\Rightarrow}{\\vdots\\;\\mathcal{D}_{21}}}\n&\n\\infer[\\infrule Cut]{\\{\\Rightarrow A_i,D_j| A_i\\in\\Xi,\\, D_j\\in\\Lambda\\}}{\\deduce{\\Rightarrow D_j,C}{\\vdots\\;\\mathcal{D}_{\\Theta_1,\\Lambda_j}}&\\deduce{C\\Rightarrow A_i}{\\vdots\\;\\mathcal{D}_{1A_i}}}\n&\n\\deduce{\\{\\Rightarrow B_i,D_j| B_i\\in\\Theta-C,\\, D_j\\in \\Lambda\\}}{\\vdots\\;\\mathcal{D}_{\\Theta_i\\Lambda_j}}\n}\n}$$\n\n\n\n\n\n\n\n\n\n\n\\noindent \\textbullet$\\quad${\\bf G3C(N)D.}$\\quad$ Left premiss by $LR$-$C$ and right one by $L$-$D^*$. It is straightforward to transform the derivation into another one having one cut with cut formula of lesser weight. \\medskip\n\n\n\n\n\\noindent \\textbullet$\\quad${\\bf G3R(D).}$\\quad$ Both premisses are by rule $LR$-$R$, we transform\n\n\\noindent\\scalebox{0.88000}{ $$\n\\infer[\\infrule Cut]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule LR\\mbox{-}R]{\\Box C,\\Box\\Psi,\\Pi'\\Rightarrow\\Sigma',\\Box B}{\\deduce{C,\\Psi\\Rightarrow B}{\\vdots\\;\\mathcal{D}_{21}}}}\n\\quad\\mbox{into}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma',\\Box B}{\\infer[\\infrule Cut]{A,\\Xi,\\Psi\\Rightarrow B }{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Psi\\Rightarrow B}{\\vdots\\,\\mathcal{D}_{21}}}}\n$$}\n\\medskip\n\n\\noindent\\noindent \\textbullet$\\quad${\\bf G3RD$^\\star$.}$\\quad$Left premiss is by $LR$-$R$, and  right one by $L$-$D^\\star$, we transform\n\n\\noindent$$\n\\infer[\\infrule Cut]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box\\Xi,\\Gamma'\\Rightarrow\\Delta,\\Box C}{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}}&\\infer[\\infrule L\\mbox{-}D^*]{\\Box C,\\Box\\Psi,\\Pi'\\Rightarrow\\Sigma}{\\deduce{C,\\Psi\\Rightarrow }{\\vdots\\;\\mathcal{D}_{21}}}}\n\\quad\\mbox{into}\\quad\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box A,\\Box\\Xi,\\Gamma',\\Box\\Psi,\\Pi'\\Rightarrow\\Delta,\\Sigma}{\\infer[\\infrule Cut]{A,\\Xi,\\Psi\\Rightarrow  }{\\deduce{A,\\Xi\\Rightarrow C}{\\vdots\\;\\mathcal{D}_{11}}&\\deduce{C,\\Psi\\Rightarrow }{\\vdots\\,\\mathcal{D}_{21}}}}\n$$ \n\n\n\\noindent \\textbullet$\\quad${\\bf G3K(D).}$\\quad$\nThe new cases with respect to {\\bf G3R(D)} are those with left premiss by an instance of $LR$-$K$ that has no principal formula in the antecedent. These cases can be treated like cases with left premiss by $R$-$N$.\\end{proof}\n\n\n\n\n\n\\section{Decidability and syntactic completeness}\\label{secdecax}\n\\subsection{Decision procedure for {\\bf G3X}}\\label{decision}\nEach calculus {\\bf G3X} has the strong subformula property since all  active formulas of  each rule  in Tables \\ref{G3cp} and \\ref{Modal rules} are  proper subformulas of the the principal formulas and no formula  disappears in moving from premiss(es) to conclusion. As usual, this gives us a syntactic proof of consistency.\n\\begin{proposition}\\label{cons}\\\n\\begin{enumerate}\n\\item Each  premiss of each rule of {\\bf G3X} has smaller weight than its conclusion;\n\\item Each premiss of each modal or deontic rule of {\\bf G3X} has smaller modal depth than its conclusion;\n\\item The calculus {\\bf G3X} has the subformula property:  a {\\bf G3X}-derivation of a sequent $\\mathcal{S}$ contains only sequents composed of subformulas of $\\mathcal{S}$;\n\\item The empty sequent is not {\\bf G3X}-derivable.\n\\end{enumerate}\\end{proposition} \nWe also have an effective method to decide the derivability of a sequent in {\\bf G3X}: we start from the desired sequent $\\Gamma\\Rightarrow\\Delta$ and we construct all possible {\\bf G3X}-derivation trees until either we find a tree where each leaf is an initial sequent or a conclusion of $L\\bot$ -- we have found a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ -- or we have checked all possible {\\bf G3X}-derivations   and we have found none --  $\\Gamma\\Rightarrow\\Delta$ is not {\\bf G3X}-derivable. \n\nMore in details, we present  a depth-first procedure that tests {\\bf G3X}-derivability in polynomial space. As it is usual in decision procedures involving non-invertible rules, we have trees involving two kinds of branching. Application of a rule with more than one premiss produce an \\emph{AND-branching} point, where all branches have to be derivable to obtain a derivation. Application of a non-invertible rule to a sequent that can be the conclusion of different instances of non-invertible rules produces an \\emph{OR-branching} point, where only one branch need be derivable to obtain a derivation.\nIn the procedure below we will assume that, given a calculus {\\bf G3X} and given a sequent $\\Box\\Pi,\\Gamma^p\\Rightarrow\\Delta^p,\\Box\\Sigma$ (where $\\Gamma^p$ and $\\Delta^p$ are multisets of propositional variables), there is some fixed way of ordering the finite (see below) set of instances of modal and deontic rules of {\\bf G3X} (\\emph{{\\bf X}-instances}, for shortness) having that sequent as conclusion. Moreover, we will represent the root of branches above an OR-branching point  by nodes of shape $\\Box_i$, where $\\Box_i$ is the name of the $i$-th {\\bf X}-instance applied (in the order of  all {\\bf X}-instances having that conclusion).  To illustrate, if we are in {\\bf G3EN} and we have to consider  $\\Box A,\\Box B,\\Gamma^p\\Rightarrow\\Delta^p,\\Box C$ then we obtain (fixing one way of ordering the three {\\bf X}-instances having that sequent as conclusion):\n \n$$\n\\infer{\\Box A,\\Box B,\\Gamma^p\\Rightarrow\\Delta^p,\\Box C}{\n\\infer[\\qquad]{LR\\mbox{-}E}{A\\Rightarrow C& C\\Rightarrow A}&\n\\infer[\\qquad]{LR\\mbox{-}E}{B\\Rightarrow C&C\\Rightarrow B}&\n\\infer{R\\mbox{-}N}{\\Rightarrow B}}\n$$\nwhere the lowermost sequent is an  OR-branching point  and the two nodes   $LR$-$E_1$ and $LR$-$E_2$ are AND-branching points. Finally, Given an AND(OR)-branching point \n\n$$\\infer{\\mathcal{S}}{\\mathcal{S}_1&\\dots&\\mathcal{S}_n}$$ we say that the branch above $\\mathcal{S}_i$ is an  \\emph{unexplored AND(OR)-branch} if no one  of its nodes has already been active.\n\n\n\\begin{definition}[{\\bf G3X}-decision procedure]\\label{decisiontree}\\\n\\begin{description}\n\\item[Stage 1.] We write the one node tree $\n\\Gamma\\Rightarrow\\Delta$\nand we label $\\Gamma\\Rightarrow\\Delta$ as active.\n\\item[Stage n+1.] Let $\\mathcal{T}_n$ be the tree constructed at stage $n$, let $\\mathcal{S}\\equiv\\Pi\\Rightarrow\\Sigma$ be its active sequent, and let $\\mathcal{B}$ be the branch  going from   the root  of $\\mathcal{T}_n$ to $\\mathcal{S}$. \n\\begin{description}\n\\item[Closed.] If $\\mathcal{S}$ is such that $p\\in\\Pi\\cap\\Sigma$ (for some propositional variable $p$) or $\\bot\\in\\Pi$, then we label $\\mathcal{S}$ as closed and\n\\begin{description}\n\\item[Derivable.] If $\\mathcal{B}$ contains no  unexplored AND-branch, the procedure ends and  \\mbox{$\\Gamma\\Rightarrow\\Delta$} is {\\bf G3X}-derivable;\n\\item[AND-backtrack.]If, instead, $\\mathcal{B}$ contains  unexplored AND-branches, we choose the topmost one and we label as active its leftmost unexplored leaf. {\\bf Else}\n\\end{description}\n\\item[Propositional.] if $\\mathcal{S}$ can be the conclusion of some instances $\\circ_1,\\dots,\\circ_m$ of the invertible propositional rules, we extend  $\\mathcal{B}$ by applying  one of such instances: \n\n$$\\infer[\\infrule{\\circ_i\\quad 1\\leq i\\leq m}]{\\mathcal{S}}{\\mathcal{S}_1&(\\mathcal{S}_2)}$$ \\noindent where, if  $\\mathcal{S}_2$, if present,  $\\mathcal{S}$ is an AND-branching point. {\\bf Else}\n\\item[Modal.] If $\\mathcal{S}$ can be the conclusion of the following  canonically ordered list of   {\\bf X}-instances:\n\n\n$$\n\\infer[\\infrule \\Box_1]{\\mathcal{S}}{\\mathcal{S}_1^1&\\dots&\\mathcal{S}^1_k }\n\\qquad\\deduce[\\dots]{\\phantom{a}}{}\\qquad\n\\infer[\\infrule \\Box_m]{\\mathcal{S}}{\\mathcal{S}_1^m&\\dots&\\mathcal{S}^m_l }\n$$ \nthen we extend  $\\mathcal{B}$ as follows:\n\n\n$$\n\\infer{\\mathcal{S}}{\\infer{\\Box_1\\phantom{^1}}{\\mathcal{S}_1^1&\\dots&\\mathcal{S}^1_k }&\\deduce[\\dots]{\\phantom{a}}{}&\\infer{\\Box_m\\phantom{^1}}{\\mathcal{S}_1^m&\\dots&\\mathcal{S}^m_l}\n}\n$$\nwhere, if $m\\geq 2$,  $\\mathcal{S}$ is  OR-branching  and, if $\\Box_i$ is a rule with more than one premiss, $\\Box_i$ is AND-branching. Moreover, we label $\\mathcal{S}_1^1$ as active. {\\bf Else} \n\\item[Open.] No rule of {\\bf G3x} can be applied to $\\mathcal{S}$, then  we label $\\mathcal{S}$ as open and\n\\begin{description}\n\\item[Underivable.] If $\\mathcal{B}$ contains no unexplored  OR-branch, the procedure ends and  \\mbox{$\\Gamma\\Rightarrow\\Delta$} is not  {\\bf G3X}-derivable;\n\\item[OR-backtrack.]If, instead, $\\mathcal{B}$ contains unexplored OR-branches, we choose the topmost one and we label as active its leftmost unexplored leaf. \n\\end{description}\n\\end{description}\n\\end{description}\n\\end{definition}\n \nTermination  can be shown as follows. Proposition \\ref{cons}.1 entails that the height of each branch of the tree $\\mathcal{T}$  constructed in a {\\bf G3X}-decision procedure for a sequent $\\Gamma\\Rightarrow\\Delta$  is bounded by the weight of  $\\Gamma\\Rightarrow\\Delta$  (in particular, given Proposition \\ref{cons}.2, the number of OR-branching points occurring in a branch is bounded by the modal depth of $\\Gamma\\Rightarrow\\Delta$). Moreover, $\\mathcal{T}$ is  finitary branching  since all rules of {\\bf G3X} are finitary branching rules, and since each sequent can be the conclusion of a finite number  $k$ of  {\\bf X}-instances (for each {\\bf G3X} $k$ is bounded by a function of $|\\Gamma|$ and $|\\Delta|$)}.   Hence, after a finite number of stages we are either in case {\\bf Derivable} or in case {\\bf Underivable} and, in both cases, the procedure ends. In the first case we can easily extract a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ from $\\mathcal{T}$ (we just have to delete  all unexplored branches as well as all underivable sub-trees   above an OR-branching point). In the latter case, thanks to Proposition \\ref{cons}.3, we know that (modulo the order of the invertible propositional rules)  we have explored the whole search space for a {\\bf G3X}-derivation of $\\Gamma\\Rightarrow\\Delta$ and we have found none. \n\n\n\nWe  prove that it is possible to test {\\bf G3X}-derivability in polynomial space by showing how it is possible to store only the active node together with a stack containing  information sufficient to reconstruct unexplored branches. For the propositional part of the calculi, we proceed as in \\cite{B97,H93,H95}: each entry of the stack is a triple containing the name of the rule applied, an index recording which of its premisses  is active, and its principal formula. For the  {\\bf X}-instances two complications arise: we need to record which OR-branches are unexplored yet, and we have to keep track of the weakening contexts of the conclusion in the premisses of {\\bf X}-instances. The first problem has already been  solved by having assumed that the {\\bf X}-instances applicable to a given sequent have a fixed canonical order.  The second problem is solved by adding a numerical superscript to the formulas occurring in a sequent and by imposing that:\\\\\n- All formulas in the end-sequent have 1 as superscript;\\\\\n - The superscript $k$ of the principal formulas of rules and of initial sequents are maximal in that sequent;\\\\- Active formulas of {\\bf X}-instances (propositional rules) have $k+2$ ($k$, respectively) as superscript;\\\\-  \n  Contexts are copied in the premisses of each rule.\\\\ By doing so, the contexts of the conclusion are copied in the premisses in each rule of {\\bf G3X},  but they cannot be principal in the trees above the premisses of the  {\\bf X}-instances because their superscript is never maximal therein.  It is immediate to see the the superscripts occurring in a derivation are bounded by (twice) the modal depth of the end-sequent.\n  \n  Instances of all modal and deontic rules in Table \\ref{Modal rules} but $LR$-$C$ and $L$-$D^{\\Diamond_C}$ are such that there is no need to  record their principal formulas in the stack entry: they are the boxed version of the formulas  having maximal superscript in the active premiss; moreover, the name of the rule and the number of the premiss  allow to reconstruct the position of the principal formulas (for the right premiss of  $LR$-$E$ and $L$-$D^{\\Diamond_E}$, we have to switch the two formulas). In instances of rules $LR$-$C$ and $L$-$D^{\\Diamond_C}$, instead,  this  doesn't hold since in all premisses but the leftmost one  there is no subformula of some principal formulas. We can overcome this problem by copying in each premiss all principal formulas having no  active subformula in that premiss and by adding one to  their superscript. We also keep fixed the position of all formulas (modulo the swapping of the two active formulas). To illustrate, one such instance is:\n  \n  $$ \\scalebox{0.9}{\\infer[\\infrule LR\\mbox{-}C]{\\Box A^k_1,\\Box A^k_2,\\Gamma\\Rightarrow\\Delta,\\Box B^k}{A^{k+2}_1,A^{k+2}_2,\\Gamma\\Rightarrow\\Delta, B^{k+2}\\quad&\n B^{k+2},\\Box A^{k+1}_2,\\Gamma\\Rightarrow\\Delta, A_1^{k+2}\n \\quad&\n \\Box A^{k+1}_1,B^{k+2},\\Gamma\\Rightarrow\\Delta, A_2^{k+2}\n }\n}$$\n In this way, given  the name of the modal or deontic rule applied, any premiss of this rule instance, and its position among the premisses of this rule, we can reconstruct both the conclusion of this rule instance and  its position  in the fixed order of {\\bf X}-instances concluding that sequent (thus we know which OR-branches are unexplored yet). In doing so, we use the hp-admissibility of contraction to ensure that no formula has more than one occurrence in the antecedent or in the succedent of the conclusion of {\\bf X}-instances (otherwise we might be unable to reconstruct which of two identical {\\bf X}-instances we are considering). Hence, for {\\bf X}-instances each stack entry records the name of the rule applied and an index recording which premiss we are considering. \n \n\nThe decision procedure is like in Definition \\ref{decisiontree}. The only novelty is that at each stage, instead of storing the full tree constructed so far, we store only the active node and the stack, we push an entry  in the stack and, if we are in a backtracking case, we pop stack entries (and we  use them  to reconstruct the corresponding active sequent) until we reach an entry recording unexplored branches of the appropriate kind, if any occurs. \n\n\\begin{theorem}\n{\\bf G3X}-derivability is decidable in $\\mathcal{O}(n\\, \\log{} n)$-{\\sc space}, where $n$ is the weight of the end-sequent.\n\\end{theorem}\n\n\\begin{proof}\nWe have already argued that proof search terminates.\nAs   in \\cite{B97,H93,H95}, Proposition \\ref{cons}.1 entails that the  stack depth is bounded by $\\mathcal{O}(n)$ and, by storing the principal formulas of propositional rules as indexes into the end-sequent,  each entry requires $\\mathcal{O}(\\log{}n)$ space. Hence we have an $\\mathcal{O}(n\\, \\log{} n)$  space bound for the stack. Moreover, the active sequent contains at most $\\mathcal{O}(n)$ subformulas of the end-sequent and their numerical superscripts. Each such subformula requires $\\mathcal{O}(\\log{}n)$ space since it can be recorded as an index into the end-sequent;  its numerical superscript requires $\\mathcal{O}(\\log{}n)$ too since there are at most $\\mathcal{O}(n)$ superscripts. Hence also the active sequent requires $\\mathcal{O}(n\\log{}n)$ space\n\\end{proof}\n\n\n\\subsection{Equivalence with the axiomatic systems}\nIt is now time to show that the sequent calculi introduced are equivalent to the non-normal logics of Section \\ref{secaxiom}.  We write {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$ if the sequent $\\Gamma\\Rightarrow\\Delta$ is  derivable in {\\bf G3X}, and we say that $A$ is derivable in {\\bf G3X} whenever {\\bf G3X} $\\vdash \\;\\Rightarrow A$. We begin by proving the following \n\n\\begin{lemma}\\label{ax} All the axioms of the  axiomatic system {\\bf X} are derivable in {\\bf G3X}.\\end{lemma}\n\n\\begin{proof} A straightforward application of the rules of the appropriate sequent calculus, possibly using Proposition \\ref{genax}.  As an example, we show that the deontic axiom $D^\\bot$ is derivable by means of rule $L$-$D^\\bot$ and that axiom  $C$ is derivable by means of  $LR$-$C$.\n\n$$\n\\infer[\\infrule R\\neg]{\\Rightarrow\\neg\\Box\\bot}{\\infer[\\infrule L\\mbox{-}D^\\bot]{\\Box\\bot\\Rightarrow}{\\infer[\\infrule L\\bot]{\\bot\\Rightarrow}{}}}\n\\qquad\n\\infer[\\infrule R\\supset]{\\Rightarrow \\Box A\\wedge\\Box B\\supset\\Box(A\\wedge B)}{\\infer[\\infrule L\\wedge]{\\Box A\\wedge \\Box B\\Rightarrow \\Box (A\\wedge B)}{\\infer[\\infrule LR\\mbox{-}C]{\\Box A,\\Box B\\Rightarrow \\Box(A\\wedge B)}{\\infer[\\infrule R\\wedge]{A,B\\Rightarrow A\\wedge B}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow A}{}&\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow B}{}}&\n\\infer[\\infrule L\\wedge  ]{A\\wedge B\\Rightarrow A}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow A}{}}&\n\\infer[\\infrule L\\wedge]{A\\wedge B\\Rightarrow B}{\\infer[\\infrule{\\ref{genax}}]{A,B\\Rightarrow B}{}}}}}\n$$\n\n\\end{proof}\n\nNext we prove  the equivalence of the sequent calculi for non-normal logics with the corresponding axiomatic systems in the sense that  a sequent $\\Gamma\\Rightarrow\\Delta$ is derivable in {\\bf G3X} if and only if  its characteristic formula $\\bigwedge\\Gamma\\supset\\bigvee \\Delta$ is derivable in {\\bf X} (where the empty antecedent stands for $\\top$ and the empty succedent for $\\bot$).  As a consequence  each calculus is sound and complete with respect to the appropriate class of neighbourhood models (see  Section \\ref{semantics}).\n\n\\begin{theorem}\\label{comp} Derivability in  the sequent system {\\bf G3X} and in the axiomatic system {\\bf X} are equivalent, i.e.\n\n\\begin{center}\n{\\bf G3X} $\\vdash\\;\\Gamma\\Rightarrow\\Delta \\qquad$iff$\\qquad${\\bf X} $\\vdash\\bigwedge\\Gamma\\supset\\bigvee\\Delta$\n\\end{center}\\end{theorem}\n\n\\begin{proof}\nTo prove the right-to-left implication, we argue by induction on the height of the  axiomatic derivation in  {\\bf X}. The base case is covered by Lemma \\ref{ax}. For the inductive steps, the case of $MP$ follows by the admissibility of Cut and the invertibility of rule $R\\supset$.  If the last step is by $RE$, then $\\Gamma=\\emptyset$ and  $\\Delta$ is $\\Box C\\leftrightarrow \\Box D$. We know that (in {\\bf X}) we have derived $\\Box C\\leftrightarrow\\Box D$ from $C\\leftrightarrow D$. Remember that $C\\leftrightarrow D$ is defined as $(C\\supset D)\\wedge (D\\supset C)$.  Thus we assume, by inductive hypothesis (IH) , that {\\bf G3ED} $\\vdash\\;  \\Rightarrow C\\supset D\\wedge D\\supset C$. From this, by invertibility of  $R\\wedge$ and $R\\supset$ (Lemma \\ref{inv}), we obtain that {\\bf G3ED} $\\vdash\\;  C\\Rightarrow D$ and {\\bf G3ED} $\\vdash\\;  D\\Rightarrow C$.    We can thus  proceed as follows \\vspace{0.3cm}\n\n$$\n\\infer[\\infrule R\\wedge]{\\Rightarrow (\\Box C\\supset \\Box D)\\wedge(\\Box D\\supset \\Box C)}{\\infer[\\infrule R\\supset]{\\Rightarrow \\Box C\\supset\\Box D}{\\infer[\\infrule LR\\mbox{-}E]{\\Box C\\Rightarrow \\Box D}{\\infer[IH+\\ref{inv}]{C\\Rightarrow D}{}&\\infer[IH+\\ref{inv}]{D\\Rightarrow C}{}}\n}&\n\\infer[\\infrule R\\supset]{\\Rightarrow \\Box D\\supset\\Box C}{\\infer[\\infrule LR\\mbox{-}E]{\\Box D\\Rightarrow \\Box C}{\\infer[IH+\\ref{inv}]{D\\Rightarrow C}{}&\\infer[IH+\\ref{inv}]{C\\Rightarrow D}{}}}}\n$$\n\n\nFor the converse implication, we assume {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$, and show, by induction on the height of the derivation in sequent calculus, that {\\bf X} \\mbox{$\\vdash \\bigwedge \\Gamma\\supset\\bigvee\\Delta$.} If the derivation has height 0,  we have an initial sequent -- so $\\Gamma\\cap\\Delta\\neq\\emptyset$ -- or an instance on $L\\bot$ -- thus $\\bot\\in\\Gamma$. In both cases the claim holds. If the height is $n+1$, we consider the last rule applied in the derivation. If it is a propositional one, the proof is straightforward. If it is a modal rule, we argue by cases. \n\n\n\nIf the last step of a derivation in {\\bf G3E(ND)} is by  $LR$-$E$, we have derived $\\Box C,\\Gamma'\\Rightarrow\\Delta',\\Box D$ from $C\\Rightarrow D$ and $D\\Rightarrow C$. By IH and propositional reasoning, {\\bf ED} $\\vdash C\\leftrightarrow D$, thus {\\bf ED} $\\vdash \\Box C\\supset \\Box D$. By some propositional steps we conclude {\\bf ED} $\\vdash (\\Box C\\wedge\\bigwedge\\Gamma')\\supset (\\bigvee\\Delta'\\lor\\Box D).$  The cases of $LR$-$M$,  $LR$-$R$, and $LR$-$K$  can be treated in a similar manner (thanks, respectively, to the rule $RM$, $RR$,  and $RK$ from Table \\ref{rulesinf}).\n\nIf we are in {\\bf G3C(ND)}, suppose the last step is the following instance of $LR$-$C$:\n$$\n\\infer[\\infrule LR\\mbox{-}C]{\\Box C_1,\\dots,\\Box C_k,\\Gamma'\\Rightarrow \\Delta',\\Box D}{C_1,\\dots C_k\\Rightarrow D&D\\Rightarrow C_1&\\dots& D\\Rightarrow C_k}\n$$\nBy IH, we have  that  {\\bf C(ND)} $\\vdash D\\supset C_i$ for all $i\\leq k$, and, by propositional reasoning, we have that  {\\bf C(ND)} $\\vdash D\\supset C_1\\wedge\\dots\\wedge C_k$. We also know, by IH, that  {\\bf C(ND)} $\\vdash C_1\\wedge\\dots\\wedge C_k\\supset D$. By applying $RE$ to these two theorems we get that \n\\begin{equation}\\label{3}\n\\mathbf{ C(ND)} \\vdash \\Box(C_1\\wedge\\dots\\wedge C_k)\\supset \\Box D\n\\end{equation}\n By using axiom $C$ and propositional reasoning, we know that \n \\begin{equation}\\label{4}\n \\mathbf{ C(ND)} \\vdash \\Box C_1\\wedge\\dots\\wedge\\Box C_k\\supset\\Box(C_1\\wedge\\dots\\wedge C_k)\n \\end{equation}\n By applying transitivity to (\\ref{4}) and (\\ref{3}) and some propositional steps, we conclude that \n $$\n  \\mathbf{ C(ND)} \\vdash (\\Box C_1\\wedge\\dots\\wedge\\Box C_k\\wedge \\bigwedge\\Gamma')\\supset (\\bigvee\\Delta'\\lor\\Box D)\n  $$\n  \n  Let's now consider rule $L$-$D^\\bot$.  Suppose we are in {\\bf G3XD$^\\bot$}  and  we have derived  $\\Box C,\\Gamma'\\Rightarrow\\Delta$ from $C\\Rightarrow$. By IH,  {\\bf XD$^\\bot$} $\\vdash C\\supset\\bot$, and we know that {\\bf xD$^\\bot$} $\\vdash \\bot \\supset C$. Thus by $RE$ (or $RM$), we get {\\bf XD$^\\bot$} $\\vdash \\Box C\\supset \\Box \\bot$. By  contraposing it and then applying a $MP$ with the axiom $D^\\bot$, we get that {\\bf XD$^\\bot$} $\\vdash \\neg\\Box C$. By some easy propositional steps we  conclude  {\\bf XD$^\\bot$} $\\vdash ( \\Box C\\wedge\\bigwedge\\Gamma')\\supset \\bigvee\\Delta$. The case $R$-$N$ is similar.\n\nLet's consider rules $L$-$D^{\\Diamond_E}$. Suppose we are in {\\bf G3ED$^\\Diamond$}  and we have derived \\mbox{$\\Box A,\\Box B,\\Gamma'\\Rightarrow\\Delta$} from the premisses $A,B\\Rightarrow$ and $\\Rightarrow A,B$. By induction we get that {\\bf ED$^\\Diamond$}$\\vdash A\\wedge B\\supset\\bot$ and {\\bf ED$^\\Diamond$}$\\vdash A\\lor B$. Hence, {\\bf ED$^\\Diamond$}$\\vdash B\\supset \\neg A$ and {\\bf ED$^\\Diamond$}$\\vdash\\neg A\\supset B$. By applying $RE$ we get that  \n\n$$ \\mathbf{ED^\\Diamond}\\vdash \\Box B\\supset\\Box \\neg A$$\n which, thanks to  axiom $D^\\Diamond$, entails that  \n \n $$\\mathbf{ ED^\\Diamond}\\vdash \\Box B\\supset\\neg\\Box A$$ \n  By some propositional steps we conclude   \n  \n $$\\mathbf{ ED^\\Diamond}\\vdash (\\Box A\\wedge\\Box B\\wedge \\bigwedge\\Gamma')\\supset\\bigvee\\Delta$$\n   Notice that, thanks to Proposition \\ref{cons}.4 and Theorem \\ref{cut},  we can assume that instances of rule $L$-$D^\\Diamond$ always  have   two principal formulas. Otherwise the calculus would prove the empty sequent (we will also assume that neither $\\Pi$ nor $\\Sigma$ is empty in instances of rule $L$-$D^{\\Diamond_C}$).\n   \n    The case of $L$-$D^{\\Diamond_M}$ is analogous to that of $L$-$D^\\bot$ for instances with one principal formula and to that of $L$-$D^{\\Diamond_E}$ for instances with two principal formulas.\n   \n\nLet's consider rule $L$-$D^{\\Diamond_C}$. Suppose we have a {\\bf G3CD$^\\Diamond$}-derivation whose last step is:\n$$\n\\infer{\\Box\\Pi,\\Box\\Sigma,\\Gamma'\\Rightarrow \\Delta'}{\\Pi,\\Sigma\\Rightarrow &\\{\\Rightarrow A,B| \\,A\\in\\Pi\\text{ and }B\\in \\Sigma\\}}\n$$\nBy induction and by some easy propositional steps we   know that {\\bf ECD$^\\Diamond$} $\\vdash \\bigwedge\\Pi\\leftrightarrow\\neg\\bigwedge\\Sigma$. By rule $RE$ we derive {\\bf ECD$^\\Diamond$} $\\vdash\\Box\\bigwedge\\Pi\\supset\\Box\\neg\\bigwedge\\Sigma$, which, thanks to axiom $D^\\Diamond$, entails that {\\bf ECD$^\\Diamond$} $\\vdash\\Box\\bigwedge\\Pi\\supset\\neg\\Box\\bigwedge\\Sigma$. By transitivity with two (generalized) instances of axiom $C$ we obtain {\\bf ECD$^\\Diamond$} $\\vdash \\bigwedge\\Box\\Pi\\supset \\neg\\bigwedge\\Box\\Sigma$. By some easy  propositional steps we conclude that  \\mbox{{\\bf ECD$^\\Diamond$} $\\vdash (\\bigwedge\\Box\\Pi\\wedge\\bigwedge\\Box\\Sigma\\wedge\\bigwedge\\Gamma')\\supset\\bigvee\\Delta$.}\n\nThe admissibility of $L$-$D^*$ in {\\bf EC(N)D}, {\\bf RD}, and {\\bf KD} is similar to that of $LR$-$C$:  in (\\ref{3}) we replace $\\Box D$ with $\\Box \\bot$ and then we use theorem $D^\\bot$ to transform it into $\\bot$.\n\\end{proof}\nBy combining this and Theorem \\ref{compax} we have the following result.\n\\begin{corollary}\nThe calculus {\\bf G3X} is sound and complete with respect to the class of all neighbourhood models for {\\bf X}.\n\\end{corollary}\n\n\n\\subsection{Forrester's Paradox}\\label{forrester}\n  As an application of our decision procedure, we use it to analyse two formal reconstructions of Forrester's paradox \\cite{F84}, which   is one of the many paradoxes that endanger the normal deontic logic {\\bf KD} \\cite{M06}. Forrester's informal argument goes as follows:\n\n\\begin{quote}\n\nConsider the following three statements:\n\\begin{enumerate}\n\\item Jones murders Smith.\n\\item Jones ought not murder Smith.\n\\item  If Jones murders Smith, then Jones ought to murder Smith gently.\n\\end{enumerate}\nIntuitively, these sentences appear to be consistent. However 1 and 3 together imply that\n\\begin{itemize}\n\\item[4.] Jones ought to murder Smith gently.\n\\end{itemize}\nAlso we accept the following conditional:\n\\begin{itemize}\n\\item[5.]  If Jones murders Smith gently, then Jones  murder Smith.\n\\end{itemize}\nOf course, this is \\emph{not} a logical validity but, rather, a fact about the world we live in. Now, if we assume that the monotonicity rule is valid, then statement 5 entails\n\\begin{itemize}\n\\item[6.]  If Jones ought to murder Smith gently, then Jones  ought to murder Smith.\n\\end{itemize}\nAnd so, statements 4 and 6 together imply\n\\begin{itemize}\n\\item[7.] Jones ought to murder Smith.\n\\end{itemize}\nBut [given the validity of $D^\\Diamond$] this contradicts statement 2. The above argument suggests  that classical deontic logic should \\emph{not} validate the monotonicity rule [$RM$] \\cite[p. 16]{P17}  \n\\end{quote} \n \n\n We  show that Forrester's paradox is not a valid argument in deontic logics by presenting, in Figure \\ref{fig}, a failed {\\bf G3KD}-proof search  of the sequent that expresses it\n\\begin{equation}\ng\\supset m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow\n\\end{equation}\n where $m$ stands for 'John \\underline{m}urders Smith' and $g$ for `John murders Smith \\underline{g}ently' \\cite[pp. 87--91]{M06}.\nNote that, by Theorem \\ref{comp}, if Forrester's paradox  is not {\\bf G3KD}-derivable, then it is not valid  in all the weaker  deontic logics we have considered.\n\n\\begin{figure}\n$$\\infer[\\infrule L\\supset]{g\\supset m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow}{\n\\infer[\\infrule L\\supset]{m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow g}{\n\\deduce{m,\\Box \\neg m\\Rightarrow g,m^{\\phantom{a}}}{\\textnormal{closed}}&\n\\infer{\\Box g,\\Box\\neg m,m\\Rightarrow g^{\\phantom{a}}}{\n\\infer{L\\mbox{-}D^\\star}{\n\\infer[\\infrule L\\neg]{g,\\neg m\\Rightarrow}{\\deduce{g\\Rightarrow m}{\\textnormal{open}}}}\n&\n\\infer{L\\mbox{-}D^\\star}{\\deduce{g\\Rightarrow }{\\textnormal{open}}}\n&\n\\infer{L\\mbox{-}D^\\star}{\\infer[\\infrule L\\neg]{\\neg m\\Rightarrow}{\\deduce{\\Rightarrow m}{\\textnormal{open}}}}}\n}&\n\\deduce{m,m\\supset\\Box g,\\Box\\neg m,m\\Rightarrow^{\\phantom{A}}}{\\vdots}}$$\n\\caption{Failed {\\bf G3KD}-proof search of Forrester's paradox \\cite{M06}}\\label{fig}\n\\end{figure}\n\n To make our failed proof search into a derivation of Forrester's paradox, we would have to add (to {\\bf G3MD$^\\Diamond$} or stronger calculi) a non-logical axiom $\\Rightarrow g\\supset m$, and to have cut as a primitive -- and  ineliminable -- rule of inference.  An  Hilbert-style axiomatization of Forrester's argument -- e.g., \\cite[p. 88]{M06} -- hides this cut with a non-logical axiom in the  step where   $\\Box g\\supset\\Box m$ is derived from $ g\\supset m$,  by one of $RM$, $R R$ or $RK$. This step -- i.e., the step from 5 to 6 in the informal argument above -- is not acceptable because none of these rules allows to infer its conclusion when the premiss is  an assumption  and not a  theorem. We have here an instance of the same  problem that has led many authors to conclude that the deduction theorem fails in modal logics, conclusion that has been  shown to be wrong in \\cite{NH12}.\n \n An alternative formulation of Forrester's argument is given in  \\cite{T97}, where the sentence `John murders Smith gently' is expressed by the complex formula $g\\wedge m$ instead of by the atomic $g$. In this case Forrester's argument becomes valid whenever the monotonicity rule is valid as it shown in Figure \\ref{figfig}. Nevertheless, whereas it was an essential ingredient of the informal version, under this formalization premiss 5 becomes dispensable. Hence it is disputable that this is an acceptable way of formalising Forrester's argument.\n \\begin{figure}\n$$ \\infer[\\infrule L\\supset]{m\\supset\\Box(m\\wedge g),\\Box\\neg m,m\\Rightarrow}{\n \\deduce{\\Box\\neg m,m\\Rightarrow m^{\\phantom{a}}}{\\textnormal{closed}}&\n\\infer{\\Box(g\\wedge m),\\Box\\neg m,m\\Rightarrow}{ \\infer{L\\mbox{-}D^\\star}{\n\\infer[\\infrule L\\wedge]{g\\wedge m,\\neg m\\Rightarrow}{\n\\infer[\\infrule L\\neg]{g, m,\\neg m\\Rightarrow}{\n\\deduce{g,m\\Rightarrow m}{\\textnormal{closed}}}}}&\n  \\infer{L\\mbox{-}D^\\star}{g\\wedge m\\Rightarrow}&\n   \\infer{L\\mbox{-}D^\\star}{\\neg m\\Rightarrow}}\n  }$$\n \\caption{ Succesfull {\\bf G3MD}-proof search for the alternative version of  Forrester's paradox \\cite{T97}}\\label{figfig}\n \\end{figure}\n  \n This is not the place to discuss at length  the correctness of formal representation of Forrester's argument  and their implications for deontic logics. We just wanted to illustrate how  the calculi {\\bf G3XD} can be  used to analyse formal representations of the  deontic paradoxes. If Forrester's argument  is formalised as in \\cite{M06} then  it does not force to adopt a deontic logic weaker than {\\bf KD}. If, instead, it is formalised as in \\cite{T97} then it forces the adoption of a logic where $RM$ fails, but the formal derivation  differs substantially from Forrester's informal argument \\cite{F84}.\n \n\\section{Craig's Interpolation Theorem}\\label{secinterpol}\nIn this section we use Maehara's \\cite{M60,M61} technique to prove Craig's interpolation theorem for each modal or deontic logic {\\bf X} which has  $C$  as theorem only if it has also $M$  (Example \\ref{prob} illustrates the problem with the non-standard rule $LR$-$C$).\n\n\\begin{theorem}[Craig's interpolation theorem] \\label{Craig}\nLet $A\\supset B$ be a theorem of a  logic {\\bf X}  that differs from {\\bf EC(N)} and its deontic extensions {\\bf EC(N)D} and {\\bf ECD$^\\Diamond$}, then \nthere is a formula $I$, which contains  propositional variables common to $A$ and $B$ only, such that both $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X}.\n\n\\end{theorem}\n\n\\noindent In order to prove this theorem, we use the following notions.\n\n\\begin{definition}\n\nA \\emph{partition} of a sequent $\\Gamma\\Rightarrow \\Delta$  is any pair of sequents \\\\$\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$ such that $\\Gamma_1,\\Gamma_2=\\Gamma$ and $\\Delta_1,\\Delta_2=\\Delta$.\\\\\nA \\emph{{\\bf G3X}-interpolant of a partition} $\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$ is any formula $I$  such that:\n\\begin{enumerate}  \n\\item All propositional variables in $I$ are    in $(\\Gamma_1\\cup\\Delta_1)\\cap(\\Gamma_2\\cup\\Delta_2)$;\n \\item {\\bf G3X} $\\vdash\\Gamma_1\\Rightarrow\\Delta_1,I$ and {\\bf G3X} $\\vdash I,\\Gamma_2\\Rightarrow\\Delta_2$.\n \\end{enumerate}\n\n\\end{definition}\n\nIf $I$ is a {\\bf G3X}-interpolant of the partition $\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>$, we write\n\n$$(\\textrm{{\\bf G3X}}\\vdash)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{I}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n$$\n\n\\noindent where one or more of the multisets $\\Gamma_1,\\Gamma_2,\\Delta_1,\\Delta_2$ may be empty. When the set of propositional variables in $(\\Gamma_1\\cup\\Delta_1)\\cap(\\Gamma_2\\cup\\Delta_2)$ is empty, the {\\bf X}-interpolant has to be constructed from $\\bot$ (and $\\top$).\nThe proof of  Theorem \\ref{Craig} is by the following lemma, originally due to Maehara \\cite{M60,M61} for (an extension of) classical logic.\n\n\\begin{lemma}[Maehara's lemma]\\label{Maehara} If {\\bf G3X} $\\vdash \\Gamma\\Rightarrow\\Delta$ and $LR$-$C$ (and $L$-$D^{\\Diamond_C}$) is not a  rule of {\\bf G3X} (see Tables \\ref{modcalculi} and \\ref{deoncalculi}), every partition   of $\\Gamma\\Rightarrow\\Delta$ has a {\\bf G3X}-interpolant.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is  by induction on the height of the derivation $\\mathcal{D}$ of $\\Gamma\\Rightarrow\\Delta$. We have to show that each partition of an initial sequent (or of a conclusion of a 0-premiss rule) has a  {\\bf G3X}-interpolant and that for each rule of {\\bf G3X} (but $LR$-$C$ and $L$-$D^{\\Diamond_C}$) we have an effective procedure that outputs a {\\bf G3X}-interpolant for any partition of its conclusion from the interpolant(s) of suitable partition(s) of its premiss(es). The proof is modular and, hence, we can consider the modal rules without having to reconsider them in the different calculi.\n\nFor the base case of initial sequents with $p$ principal formula, we have four possible partitions, whose interpolants are:\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<p,\\Gamma_1'\\Rightarrow\\Delta_1',p\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<p,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{p}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,p\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1',p\\;\\stackrel{\\neg p}{||}\\;p,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(4)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;p,\\Gamma'_2\\Rightarrow\\Delta'_2,p\\>$\n\\end{tabular}\\end{center} \n\n\\noindent and for the base case of rule $L\\bot$, we have:\n\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<\\bot,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;\\bot,\\Gamma'_2\\Rightarrow\\Delta_2,\\>$\n\\end{tabular}\\end{center}\n\n For the proof of (some of) the propositional  cases the reader is referred to \\cite[pp. 117-118]{T00}. Thus, we have only to prove that all the modal and deontic  rules of Table \\ref{Modal rules} (modulo  $LR$-$C$ and $L$-$D^{\\Diamond_C}$)  behave as desired.\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-E}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n A\\Rightarrow B\\qquad B\\Rightarrow A\n \\justifies\n \\Box A,\\Gamma\\Rightarrow\\Delta,\\Box B\n \\using LR\\textrm{-}E\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  four kinds of partitions of the conclusion:\n\n\\begin{center}\\begin{tabular}{ccc}\n$(1)\\;\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(2)\\;\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\qquad$&$\\qquad(4)\\;\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\n\\end{tabular}\\end{center}\n\n\\noindent In each  case we have to choose  partitions of the premisses that permit to construct a {\\bf G3E(ND)}-interpolant for the partition under consideration.\n\n In case {\\bf(1)} we have \n $$\n\\framebox{ \\infer[\\infrule LR\\mbox{-}E]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>}{ \\<A\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Rightarrow\\>&\\<B\\Rightarrow A\\;\\stackrel{D}{||}\\;\\Rightarrow\\>}\n} $$\n  \n\\noindent   This can be shown as follows. By  IH there is some $C$ ($D$) that is a {\\bf G3E(ND)}-interpolant of the given partition of the left (right) premiss.  Thus both $C$ and $D$ contains only propositional variables common to $A$ and $B$; and (i) $\\vdash A\\Rightarrow B,C\\;$ (ii) $\\vdash C\\Rightarrow\\;$ (iii) $\\vdash B\\Rightarrow A,D\\;$ and (iv) $\\vdash D\\Rightarrow\\;$. Since the common language of the partitions of the premisses is empty,   no propositional variable can occur in $C$ nor in $D$. Here is a proof that $C$ is a {\\bf G3E(ND)}-interpolant of the partition under consideration (the sequents $A\\Rightarrow B$ and $B\\Rightarrow A$ are derivable since they are the premisses of the instance of $LR$-$E$ we are considering):\n  \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma'_1\\Rightarrow\\Delta_1',\\Box B,C}{\\infer{A\\Rightarrow B}{}&\\infer{B\\Rightarrow A}{}}\n\\qquad\\qquad\n\\infer[\\infrule LWs+RWs]{C,\\Gamma_2\\Rightarrow\\Delta_2}{\\infer[\\infruler{(ii)}]{C\\Rightarrow}{}}\n$$\n\n\n\n\\noindent In case {\\bf(2)} we have\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}E]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>}{ \\<A\\Rightarrow \\;\\stackrel{C}{||}\\;\\Rightarrow B\\> &  \\<B\\Rightarrow \\;\\stackrel{D}{||}\\;\\Rightarrow A\\>}\n}$$\n  \n\\noindent By IH it holds that  some $C$ and $D$ are {\\bf G3E(ND)}-interpolants of the given partitions of the premisses.  Thus, (i) $\\vdash A\\Rightarrow C\\;$ (ii) $\\vdash C\\Rightarrow B\\;$ (iii) $\\vdash B\\Rightarrow D\\;$ (iv) $\\vdash D\\Rightarrow A\\;$ and (v)  all propositional variables in $C\\cup D$ are in $A\\cap B$.\n Here is a proof that $\\Box C$ is a {\\bf G3E(ND)}-interpolant of the given partition (the language condition is satisfied thanks to (v)\\,):\n \n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infruler{(i)}]{A\\Rightarrow C}{}&\n\\infer[\\infrule Cut]{C\\Rightarrow A}{\\infer[\\infrule Cut]{C\\Rightarrow D}{\\infer[\\infruler{(ii)}]{C\\Rightarrow B}{}&\\infer[\\infruler{(iii)}]{B\\Rightarrow D}{}}&\n\\infer[\\infruler{(iv)}]{D\\Rightarrow A}{}}}\n$$\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box C,\\Gamma_2\\Rightarrow\\Delta_2',\\Box B}{\\infer[\\infruler{(ii)}]{C\\Rightarrow B}{}&\\infer[\\infrule Cut]{B\\Rightarrow C}{\\infer[\\infruler{(iii)}]{B\\Rightarrow D}{}&\\infer[\\infrule Cut]{D\\Rightarrow C}{\\infer[\\infruler{(iv)}]{D\\Rightarrow A}{}&\\infer[\\infruler{(i)}]{A\\Rightarrow C}{}}}}\n$$\n\n\\noindent  In case {\\bf(3)} we have \n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}E]{\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{ \\<\\Rightarrow B\\;\\stackrel{C}{||}\\;A\\Rightarrow \\> &  \\<\\Rightarrow A\\;\\stackrel{D}{||}\\;B\\Rightarrow \\>}\n}$$\n\\noindent  By IH, there are $C$ and $D$ that are {\\bf G3E(ND)}-interpolants of the partitions of the premisses. Thus (i) $\\vdash \\Rightarrow B,C\\;$ (ii) $\\vdash C,A\\Rightarrow\\;$ (iii) $\\vdash \\Rightarrow A,D\\;$ and (iv) $\\vdash D,B\\Rightarrow\\;$. We prove that $\\Diamond C$  is a {\\bf G3E(ND)}-interpolant of the (given partition of the) conclusion as follows:\n\n$$\n\\infer[\\infrule R\\neg]{\\Gamma_1\\Rightarrow\\Delta_1',\\Box B,\\neg\\Box\\neg C}{\\infer[\\infrule LR\\mbox{-}E]{\\Box\\neg C,\\Gamma_1\\Rightarrow\\Delta_1',\\Box B}{\n\\infer[\\infrule L\\neg]{\\neg C\\Rightarrow B}{\\infer[\\infruler{(i)}]{\\Rightarrow B,C}{}}&\n\\infer[\\infrule R\\neg]{B\\Rightarrow\\neg C}{\\infer[\\infrule Cut]{B,C\\Rightarrow}{\n\\infer[\\infrule Cut]{C\\Rightarrow D}{\\infer[\\infruler{(iii)}]{\\Rightarrow D,A}{}&\\infer[\\infruler{(ii)}]{A,C\\Rightarrow}{}}&\n\\infer[\\infruler{(iv)}]{D,B\\Rightarrow}{}}}}}\n$$\n\n$$\n\\infer[\\infrule L\\neg]{\\neg\\Box\\neg C,\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2}{\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2,\\Box\\neg C}{\\infer[\\infrule R\\neg]{A\\Rightarrow \\neg C}{\\infer[\\infruler{(ii)}]{C,A\\Rightarrow}{}}&\n\\infer[\\infrule L\\neg]{\\neg C\\Rightarrow A}{\\infer[\\infrule Cut]{\\Rightarrow A,C}{\n\\infer[\\infruler{(iii)}]{\\Rightarrow A,D}{}&\n\\infer[\\infrule Cut]{D\\Rightarrow C}{\\infer[\\infruler{(ii)}]{\\Rightarrow C,B}{}&\\infer[\\infruler{(iv)}]{B,D\\Rightarrow}{}}}}}}\n$$\n\\noindent  In case {\\bf(4)} we have \n\n$$\n\\framebox{\n\\infer[\\infrule LR-E]{\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\<\\Rightarrow \\;\\stackrel{C}{||}\\;A\\Rightarrow B\\>&\\<\\Rightarrow \\;\\stackrel{D}{||}\\;B\\Rightarrow A \\>}\n}\n$$\n \\noindent By IH, there are  {\\bf G3E(ND)}-interpolants $C$ and $D$  of the partitions of the premisses. Thus (i) $\\vdash \\Rightarrow C\\;$ (ii) $\\vdash C,A\\Rightarrow B\\;$ (iii) $\\vdash \\Rightarrow D\\;$ and (iv) $\\vdash D,B\\Rightarrow A\\;$.  Since the common language of the partitions of the premisses is empty,  no propositional variable  occurs in $C$ nor in $D$. We  show that $C$ is a {\\bf G3E(ND)}-interpolant of the partition under consideration as follows (as in case {\\bf (1)}, $A\\Rightarrow B$ and $B\\Rightarrow A$, being the premisses of the instance of $LR$-$E$ under consideration, are derivable):\n \n$$\n\\infer[\\infrule LWs+RWs]{\\Gamma_1\\Rightarrow\\Delta_1,C}{\\infer[\\infruler{(i)} ]{\\Rightarrow C}{}}\n\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{C,\\Box A,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B}{\\infer{A\\Rightarrow B}{}&\\infer{B\\Rightarrow A}{}}\n$$\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-M}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}M]{\\Box A,\\Gamma\\Rightarrow\\Delta,\\Box B}{A\\Rightarrow B}\n$$\n\n\\noindent we give directly the {\\bf G3M(ND)}-interpolants of the possible  partitions  of the conclusion (and of the appropriate partition of the  premiss). The proofs are parallel to those for $LR$-$E$.\n\n$$\n\\framebox{ \\infer[\\infrule LR\\mbox{-}M]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>}{\\<A\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Rightarrow\\> } \n\\qquad\n \\infer[\\infrule LR\\mbox{-}M]{\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Gamma_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\<A\\Rightarrow \\;\\stackrel{C}{||}\\;\\Rightarrow B\\>}\n }\n$$\n$$\\framebox{\n\\infer[\\infrule LR\\mbox{-}M]{\\<\\Gamma_1\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{\\<\\Rightarrow B\\;\\stackrel{C}{||}\\;A\\Rightarrow \\>} \n\\qquad\n\\infer[\\infrule LR\\mbox{-}M]{\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Box A,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{ \\<\\Rightarrow \\;\\stackrel{C}{||}\\;A\\Rightarrow B\\> }\n}$$\n  \n  \n\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-R}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}R]{\\Box A,\\Box \\Pi,\\Gamma\\Rightarrow\\Delta,\\Box B}{A,\\Pi\\Rightarrow B}\n$$\n  \\noindent we have  four kinds of partitions of the conclusion:\n\n\n\\begin{center}\\begin{tabular}{ll}\n$(1)\\qquad$&$\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(2)$&$\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1\\;||\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)$&$\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1',\\Box B\\;||\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(4)$&$\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>$\n\\end{tabular}\\end{center}\n\n\n In case {\\bf(1)} we have two subcases according to whether $\\Pi_2$ is empty or not. If it is not empty we have \n\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow \\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\; \\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2\\> }{\\<A,\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; \\Pi_2\\Rightarrow\\;\\> }\n}$$\n  \n\\noindent  By IH, there is  a {\\bf G3R(D$^\\star$)}-interpolant $C$ of the chosen partition of the premiss. Thus (i) $\\vdash A,\\Pi_1\\Rightarrow B,C$ and (ii) $\\vdash C,\\Pi_2\\Rightarrow$, and we have the following derivations\n\n$$\n\\infer[\\infrule R\\neg]{\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B,\\neg\\Box\\neg C}{\\infer[\\infrule LR\\mbox{-}R]{\\Box\\neg C,\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B}{\\infer[\\infrule L\\neg]{\\neg C,A,\\Pi_1\\Rightarrow B}{\\infer[\\infruler{(i)}]{A,\\Pi_1\\Rightarrow B,C}{}}}}\n\\qquad\n\\infer[\\infrule L\\neg]{\\neg\\Box\\neg C,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2}{\\infer[\\infrule LR\\mbox{-}R]{\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2,\\Box \\neg C}{\\infer[\\infrule R\\neg]{\\Pi_2\\Rightarrow \\neg C}{\\infer[\\infruler{(ii)}]{C,\\Pi_2\\Rightarrow}{}}}}\n$$\n\n\\noindent When $\\Pi_2$ (and $\\Box \\Pi_2$)  is empty we cannot proceed as above since we cannot apply $LR$-$R$  in the right derivation. But in this case, reasoning like in case {\\bf (1)} for rule $LR$-$E$, we can show that\n\n$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma_1'\\Rightarrow \\Delta_1',\\Box B\\;\\stackrel{C}{||}\\; \\Gamma_2'\\Rightarrow\\Delta_2\\> }{\\<A,\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> }\n}$\n\n\n\n\\noindent Cases {\\bf(2)} and {\\bf(3)} are similar to  the corresponding cases for rule $LR$-$E$:\n{\\footnotesize$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box A,\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta'_2,\\Box B\\>}{\\< A,\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow B\\>}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta'_1\\,\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box A,\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>}{\\<\\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\; A,\\Pi_2\\Rightarrow\\;\\>}\n}$$}\n  \n\n\n In case {\\bf(4)} we have two subcases according to whether $\\Pi_1$ is empty or not:\n{\\small$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}R]{\\<\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||} \\Box A,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\;\\Rightarrow\\;\\stackrel{C}{||}A,\\Pi_2\\Rightarrow B\\>}\\quad\n\\infer[\\infrule LR\\mbox{-}R]{\\<\\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||} \\Box A,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}A,\\Pi_2\\Rightarrow B\\>}\n}$$ }\nThe proofs are similar to those for case {\\bf (1)}.\n  \n\n\n\n\\noindent\\textbullet$\\quad$ {\\bf LR-K}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule LR\\mbox{-}K]{\\Box \\Pi,\\Gamma\\Rightarrow\\Delta,\\Box B}{\\Pi\\Rightarrow B}\n$$\n\\noindent we give directly the {\\bf G3K(D)}-interpolants of the two  possible  partitions  of the conclusion:\n\n{\\small$$\n\\framebox{\\infer[\\infrule LR\\mbox{-}K]{\\<\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2',\\Box B\\>}{\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow B\\>}\\qquad\n\\infer[\\infrule LR\\mbox{-}K]{\\<\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1',\\Box B\\;\\stackrel{\\Diamond C}{||}\\;\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2\\>}{\\< \\Pi_1\\Rightarrow B\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>}\n}$$}\nThe proofs are, respectively, parallel to those for cases {\\bf (2)} and {\\bf(3)} of $LR$-$E$ (when $\\Pi=\\emptyset$, we can proceed as for rule $R$-$N$ and use $C$ instead of $\\Box C$ and of $\\Diamond C$, respectively).\n  \n\n\n  \n\n\n\\noindent\\textbullet$\\quad$ {\\bf L-D}$^\\bot)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n A\\Rightarrow \n \\justifies\n \\Box A,\\Gamma\\Rightarrow\\Delta\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  two kinds of partitions of the conclusion, whose {\\bf G3XD$^\\bot$}-interpolants are, respectively:\n\n \\begin{center}\\framebox{ \\begin{prooftree}\n\\<A\\Rightarrow\\;\\stackrel{C}{||}\\;\\Rightarrow\\;\\>\n \\justifies\n\\<\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree}\\qquad\n   \\begin{prooftree}\n\\<\\;\\Rightarrow\\;\\stackrel{C}{||}\\; A\\Rightarrow\\;\\>\n \\justifies\n\\<\\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{C}{||}\\; \\Box A,\\Gamma_2'\\Rightarrow\\Delta_2\\>\n \\using \\infrule{L\\mbox{-}D^\\bot}\n  \\end{prooftree} }\n  \\end{center}\n  \n\n\\noindent\\textbullet$\\quad$ {\\bf L-D$^\\Diamond$}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n$$\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_E}]{\\Box A,\\Box B,\\Gamma\\Rightarrow \\Delta}{A,B\\Rightarrow\\qquad&\\Rightarrow A,B }\\qquad\n\\textnormal{or}\\qquad\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_M}]{\\Box A,\\Box B,\\Gamma\\Rightarrow \\Delta}{A,B\\Rightarrow}\n$$\n  \\noindent we have  three kinds of partitions of the conclusion:\n\n\n\\begin{center}\\begin{tabular}{ll}\n$(1)\\qquad$&$\\<\\Box A,\\Box B,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n$(2)$&$\\<\\Gamma_1\\Rightarrow\\Delta_1\\;||\\;\\Box A,\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\>$\\\\\\noalign{\\smallskip\\smallskip}\n$(3)$&$\\<\\Box A,\\Gamma'_1\\Rightarrow\\Delta_1\\;||\\;\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\>\\quad$\\\\\\noalign{\\smallskip\\smallskip}\n\\end{tabular}\\end{center}\n\n In cases {\\bf (1)} and {\\bf (2)} we have, respectively (omitting the right premiss for $L$-$D^{\\Diamond_M}$):\n \n $$\n\\framebox{\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Box A,\\Box B,\\Gamma_1'\\Rightarrow \\Delta_1\\;\\stackrel{ C}{||}\\; \\Gamma_2\\Rightarrow\\Delta_2\\> }{\\<A,B\\Rightarrow \\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\;\\stackrel{D}{||}\\; \\Rightarrow \\;A,B\\>}\n\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Gamma_1\\Rightarrow \\Delta_1\\;\\stackrel{ C}{||}\\; \\Box A,\\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\> }{\\<A,B\\Rightarrow \\;\\stackrel{C}{||}\\; \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\;\\stackrel{D}{||}\\; \\Rightarrow \\;A,B\\>}\n}$$\n\nFinally, in case {\\bf (3)} we have:\n\n $$\n\\framebox{\\infer[\\infrule L\\mbox{-}D^\\Diamond]{\\<\\Box A,\\Gamma_1'\\Rightarrow \\Delta_1\\;\\stackrel{ \\Box C}{||}\\; \\Box B,\\Gamma'_2\\Rightarrow\\Delta_2\\> }{\\<A\\Rightarrow \\;\\stackrel{C}{||}\\; B \\Rightarrow\\;\\> \\qquad \\<\\Rightarrow \\; A\\stackrel{D}{||}\\; \\Rightarrow \\;B\\>}\n}$$\nBy IH, we can assume that $C$ is an interpolant of the partition of the left premiss and $D$ of the right one.\nWe have the following {\\bf G3YD$^\\Diamond$}-derivations ({\\bf Y} $\\in\\{$ {\\bf E,M}$\\}$):\n\n$$\n\\infer[\\infrule LR\\mbox{-}E]{\\Box A,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infrule IH]{A\\Rightarrow C}{}&\n\\infer[\\infrule Cut]{C\\Rightarrow A}{\\infer[\\infrule IH]{\\Rightarrow A,D}{}&\n\\infer[\\infrule Cut]{D,C\\Rightarrow}{\n\\infer[\\infrule IH]{D\\Rightarrow B}{}&\\infer[\\infrule IH]{B,C\\Rightarrow}{}\n}}}\n$$\n\n$$\n\\infer[\\infrule L\\mbox{-}D^{\\Diamond_{E}}]{\\Box C,\\Box B,\\Gamma_2'\\Rightarrow\\Delta_2}{\n\\infer[\\infrule IH]{C,B\\Rightarrow}{}&\\infer[\\infrule Cut]{\\Rightarrow C,B}{\n\\infer[\\infrule Cut]{\\Rightarrow C,D}{\n\\infer[\\infrule IH]{\\Rightarrow D,A}{}&\n\\infer[\\infrule IH]{A\\Rightarrow C}{}}&\n\\infer[\\infrule IH]{D\\Rightarrow B}{}}\n}\n$$\nIt is also immediate to notice that  $\\Box C$ satisfies the language condition for being a {\\bf G3YD$^\\Diamond$}-interpolant of the conclusion since, by IH, we know that each propositional variable occurring in $C$ occurs   in $A\\cap B$.\n\n\\noindent\\textbullet$\\quad$ {\\bf L-D$^\\star$}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n \\Pi\\Rightarrow \n \\justifies\n \\Box\\Pi, \\Gamma\\Rightarrow\\Delta\n \\using \\infrule{L\\mbox{-}D^\\star}\n  \\end{prooftree} \n  \\end{center}\n  \\noindent we have  the following kind of  partition:\n\\quad$\\< \\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>$\n\n\nIf $\\Pi_1$ is not empty we have: \n\\begin{center} \\framebox{\\begin{prooftree}\n\\<\\Pi_1\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>\n \\justifies\n\\< \\Box\\Pi_1,\\Gamma'_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Box C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\n \\using  \\infrule{L\\mbox{-}D^\\star}\n  \\end{prooftree} }\n  \\end{center}\n\n\\noindent By IH, there is some  $C$ that is  an interpolant of the premiss. It holds  that $\\vdash \\Pi_1\\Rightarrow C$ and $\\vdash C,\\Pi_2\\Rightarrow\\;$. We  show that $\\Box C$ is a {\\bf G3YD}-interpolant ({\\bf Y} $\\in\\{${\\bf R,K}$\\}$) of the partition of the conclusion as follows:\n\n$$\n\\infer[\\infrule LR\\mbox{-}Y]{\\Box\\Pi_1,\\Gamma_1'\\Rightarrow\\Delta_1,\\Box C}{\\infer[\\infruler{IH}]{\\Pi_1\\Rightarrow C}{}}\n\\qquad\n\\infer[\\infrule L\\mbox{-}D^*]{\\Box C,\\Box \\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2}{\\infer[\\infruler{IH}]{C,\\Pi_2\\Rightarrow}{}}\n$$\nIf, instead, $\\Pi_1$ is empty then $\\Pi_2$ cannot be empty and we have\n\\begin{center} \\framebox{\\begin{prooftree}\n\\<\\;\\Rightarrow\\;\\stackrel{C}{||}\\;\\Pi_2\\Rightarrow\\;\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\Diamond C}{||}\\;\\Box\\Pi_2,\\Gamma'_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{ L\\mbox{-}D^\\star}\n  \\end{prooftree} }\n  \\end{center} \n  \n\\noindent   By IH there is a formula $C$, containing no propositional variable, such that  $\\vdash \\;\\Rightarrow C$ and $\\vdash C,\\Pi_2\\Rightarrow\\;$ . Thus, {\\bf G3YD} $\\vdash\\Gamma_1\\Rightarrow\\Delta_1,\\Diamond C$ ($L$-$D^*$ makes $\\Rightarrow\\Diamond C$ derivable from $\\Rightarrow C$) and  {\\bf G3YD} $\\vdash\\Diamond\\top,\\Box\\Pi_2,\\Gamma_2'\\Rightarrow\\Delta_2$ ($LR$-$Y$ makes  $\\Diamond C,\\Box\\Pi_2\\Rightarrow$ derivable from $C,\\Pi_2\\Rightarrow$ when $\\Pi_2\\neq \\emptyset$).\n \n \n\n \\noindent\\textbullet$\\quad$ {\\bf R-N}$)\\quad$ If the last rule applied in $\\mathcal{D}$ is \n\n\\begin{center} \\begin{prooftree}\n \\Rightarrow A \n \\justifies\n\\Gamma\\Rightarrow\\Delta,\\Box A\n \\using \\infrule{R\\mbox{-}N}\n  \\end{prooftree} \n  \\end{center}\n\n\\noindent The interpolants for the two possible partitions are\n\n\\noindent\\framebox{ \\begin{tabular}{cccc}\n $(1)\\;$& \\begin{prooftree}\n\\<\\;\\Rightarrow A\\stackrel{\\bot}{||}\\;\\Rightarrow\\;\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1',\\Box A\\;\\stackrel{\\bot}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2\\>\n \\using \\infrule{R\\mbox{-}N}\\quad\n  \\end{prooftree} &\n \n  \n\n$(2)\\;$& \\begin{prooftree}\n\\<\\;\\Rightarrow \\;\\stackrel{\\top}{||}\\;\\Rightarrow A\\>\n \\justifies\n\\< \\Gamma_1\\Rightarrow\\Delta_1\\;\\stackrel{\\top}{||}\\;\\Gamma_2\\Rightarrow\\Delta_2',\\Box A\\>\n \\using \\infrule{R\\mbox{-}N}\n  \\end{prooftree} \n  \\end{tabular}}\\vspace{0.3cm}\n \n \n\n\\noindent This completes the proof.{}\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{Craig}]\nAssume that $A\\supset B$ is a theorem of {\\bf X}. By Theorem \\ref{comp} and Lemma \\ref{inv} we have that {\\bf G3X} $\\vdash A\\Rightarrow B$. By Lemma \\ref{Maehara}  (taking $A$ as $\\Gamma_1$ and  $B$ as $\\Delta_2$ and $\\Gamma_2,\\Delta_1$ empty) and Theorem \\ref{comp} there exists  a formula $I$ that is an interpolant of $A\\supset B$ -- i.e. $I$ is such such that  all propositional variables occurring in $I$, if any, occur in both  $A$ and $B$ and such that  $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X}.{}\n\\end{proof}\n\nObserve that the proof is constructive in that  Lemma \\ref{Maehara} gives a procedure to  extract  an interpolant for $A\\supset B$ from a given derivation of $A\\Rightarrow B$. Furthermore the proof is purely proof-theoretic in that it makes no use of model-theoretic notions.\n\nCraig's theorem is often -- e.g., in \\cite{M61} for an extension of  classical   logic -- stated in the following stronger  version:\n\\begin{quote}\nIf $A\\supset B$ is a theorem of the logic {\\bf X}, then \n\\begin{enumerate}\n\\item If $A$ and $B$ share some propositional variable, there is a formula $I$, which contains  propositional variables common to $A$ and $B$ only, such that both $A\\supset I$ and $I\\supset B$ are theorems of {\\bf X};\n\\item Else, either $\\neg A$  or $B$ is a theorem of {\\bf X}.\n\\end{enumerate}\n\\end{quote}\n\n\\noindent But the second condition  doesn't hold for   modal and deontic logics where at least one of $N:=\\Box\\top$ and $D^\\bot:=\\Diamond\\top$ is not a theorem.\nTo illustrate,  it holds that  $ \\Box\\top\\supset\\Box\\top$ is a theorem of {\\bf E} and its interpolant is $\\Box\\bot$ (see Figure \\ref{fig}), but neither  $\\neg \\Box\\top$ nor $\\Box\\top$ is a theorem of {\\bf E}. Analogously, we have that $\\Box\\bot\\supset\\Box\\bot$ is a theorem of  {\\bf E} and its interpolant is $\\Box\\bot$ (see Figure \\ref{fig}), but neither $\\neg \\Box\\bot$ nor $\\Box\\bot$ is a  theorem of {\\bf E}. These counterexamples work in all extensions of {\\bf E} that don't have both $N$ and $D^\\bot$ as theorems: to prove the stronger version of Craig's theorem we need $N$ and $D^\\bot$, respectively.\n\n\\begin{figure}\n\\scalebox{0.9900}{\\infer[\\infrule LR\\mbox{-}E]{\\< \\;\\Box\\top\\Rightarrow\\;\\stackrel{\\Box\\top}{||}\\;\\Rightarrow\\Box\\top\\;\\>}{\n\\< \\;\\top\\Rightarrow\\;\\stackrel{\\top}{||}\\;\\Rightarrow\\top\\;\\>& \\< \\;\\top\\Rightarrow\\;\\stackrel{\\top}{||}\\;\\Rightarrow\\top\\;\\>}\\qquad\n\\infer[\\infrule LR\\mbox{-}E]{\\< \\;\\Box\\bot\\Rightarrow\\;\\stackrel{\\Box\\bot}{||}\\;\\Rightarrow\\Box\\bot\\;\\>}{\n\\< \\;\\bot\\Rightarrow\\;\\stackrel{\\bot}{||}\\;\\Rightarrow\\bot\\;\\>& \\< \\;\\bot\\Rightarrow\\;\\stackrel{\\bot}{||}\\;\\Rightarrow\\bot\\;\\>}\n\n\n\\caption{Construction of an {\\bf ED}-interpolant for   $\\Box\\top\\supset\\Box\\top$ and for $\\Box\\bot\\supset\\Box\\bot$}\\label{fig}\n\\end{figure}\n\n\nAmong the deontic logics considered here, the stronger version of Craig's theorem holds only for {\\bf END$^{\\bot(\\Diamond)}$},  {\\bf MND$^{\\bot(\\Diamond)}$}, and {\\bf KD}, as shown by the following \n\n\\begin{corollary}\\label{cor} Let {\\bf XD} be one of {\\bf END$^{\\bot(\\Diamond)}$},  {\\bf MND$^{\\bot(\\Diamond)}$}, and {\\bf KD}. If $A\\supset B$ is a theorem of {\\bf XD} and $A$ and $B$ share no propositional variable, then either $\\neg A$ or $B$ is a theorem of {\\bf XD}.\n\\end{corollary}\n\n\\begin{proof}\nSuppose that {\\bf XD} $\\vdash A\\supset B$ and that $A$ and $B$ share no propositional variable, then the interpolant $I$  is constructed from $\\bot $ and $\\top$ by means of classical and deontic operators.  Whenever  $D^\\bot$ and $N$ are theorems of {\\bf XD}, we have that $\\Diamond\\top\\leftrightarrow \\top$, $\\Box\\top\\leftrightarrow \\top$,  $\\Diamond\\bot\\leftrightarrow \\bot$, and $\\Box\\bot\\leftrightarrow \\bot$ are theorems of {\\bf XD}.  Hence, the interpolant $I$ is (equivalent to) either $\\bot$ or $\\top$. In the first case {\\bf XD} $\\vdash\\neg A$ and in the second one {\\bf XD} $\\vdash B$.{}\n\\end{proof}\n\n\\noindent As noted in \\cite[p. 298]{F83}, Corollary \\ref{cor} is a Halld\\'en-completeness  result. A logic {\\bf X} is \\emph{Halld\\'en-complete} if, for every formulas $A$ and $B$ that share no propositional variable, {\\bf X} $\\vdash A\\lor B$ if and only if {\\bf X} $\\vdash A$ or {\\bf X} $\\vdash B$. All the modal and deontic logics considered here, being based on classical logic, are such that $A\\supset B$ is equivalent to $\\neg A \\lor B$. Thus  the deontic logics considered in Corollary \\ref{cor} are Halld\\'en-complete, whereas all other non-normal logics for which we have proved interpolation  are Halld\\'en-incomplete since they don't satisfy Corollary \\ref{cor}.\n\n\n\\begin{example}[Maehara's lemma and rule $LR$-$C$]\\label{prob}\n We have not been able to prove Maehara's Lemma \\ref{Maehara} for rule $LR$-$C$ because of the cases where the principal formulas of the antecedent are splitted in the two elements of the partition. In particular, if we have two principal formulas in the antecedent, the problematic partitions  are (omitting the weakening contexts):\n \\begin{center}\n (1)\\quad $\\<\\Box A_1\\Rightarrow\\;||\\;\\Box A_2\\Rightarrow \\Box B\\>$\\qquad\\qquad\n (2)\\quad $\\<\\Box A_1\\Rightarrow\\Box B\\;||\\;\\Box A_2\\Rightarrow\\>$\n \\end{center}\nTo illustrate, an interpolant of the first partition would be a formula $I$ such that:\n\n$$\n (i)\\quad \\vdash \\Box A_1\\Rightarrow I\\qquad (ii)\\quad\\vdash I,\\Box A_2\\Rightarrow\\Box B\\qquad(iii)\\quad p\\in I\\textnormal{ only if }p\\in (A_1)\\cap(A_2,B)\n$$\nBut we have not been able to find partitions of the premisses allowing to find such  $I$. More in details, for the first premiss it is natural to   consider the partition $\\< A_1\\Rightarrow\\;\\stackrel{C}{||}\\; A_2\\Rightarrow  B\\>$ in order to find an $I$ that satisfies $(iii)$.  But, for any combination of the partitions of the other two premisses that is compatible with $(iii)$, we can prove  that   $(ii)$ is satisfied (by $\\Box C$) but we have not been able to prove that also $(i)$ is satisfied.\n\\end{example}\n\n\\section{Conclusion}\\label{conc}\nWe presented cut- and contraction-free sequent calculi for non-normal modal and deontic logics. We have proved that these calculi have good structural properties in that weakening and contraction are height-preserving admissible and cut is (syntactically) admissible. Moreover, we have shown that these calculi allow for a terminating decision procedure whose complexity is in {\\sc Pspace}. Finally, we have given a constructive proof of Craig's interpolation property for all the logics that do not contain  rule $LR$-$C$. As far as we know, it is still an open problem whether it is possible to give a constructive proof of interpolation for these logics. Another open question is whether the calculi given here can be used to give a constructive proof of  the uniform interpolation property for non-normal logics as it is done in \\cite{P92} for $\\mathbf{IL_p}$ and in \\cite{B07} for  {\\bf K} and {\\bf T}. \n\n\n\n\n\n\n\\vspace{0.3cm}\n\n\\noindent {\\bf Thanks.} Thanks are due to Tiziano Dalmonte, Simone Martini,  and two anonymous referees for many helpful suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sxn0intrdxn}\nMRI is a household name as a diagnostic tool in the medical field\n\\cite{vp,tey},\nwith an impressive resume in many other fields, including the\nstudy of materials \\cite{callaghan,blumch,bjj,rugar},\ncorrosion of metals, monitoring batteries and supercapacitors\n\\cite{britton2010,ilott2014}.\n\nHowever, historically, MRI of bulk metals has been very rare, limited to\nspecialized cells using r.f. gradients (with limited control) \\cite{si0grld},\ninstead of the magnetic field gradients employed in conventional MRI.\nIn other studies involving bulk metals, the MRI targeted the surrounding\ndielectric (electrolyte in electrochemical and fuel cells with metallic\nelectrodes, or tissues with embedded metallic implants)\n\\cite{si0pines,si0camacho,si0bennet,si0olsen,si0alm,si0viano,si0shafiei,si0graf,\nsi0koch,si0jfz,si0garwood2011,si0garwood}.\nNotwithstanding, these studies addressed issues that can cause distortions and\nlimit sensitivity of the MRI images, such as bulk magnetic susceptibility (BMS) \neffects and eddy currents (produced on bulk metal surface due to\ngradient switching).\n\nThe dearth of mainstream bulk metal MRI is rooted in unique challenges posed by \nthe physics of propagation of r.f. EM fields in bulk metals\n(MRI employs r.f. pulses to generate the MR signal leading to the images).\nAll along, it has been known that the incident r.f. field decays rapidly and\nexponentially inside the metallic conductor (Fig.\\ref{figSI0dltaEff}), a\nphenomenon known as skin effect \\cite{jackson,si0griffiths,si0ulaby}.\nThe characteristic length of decay ($\\delta$, the {\\em skin depth}), typically\nof the order of several microns (Eq.(\\ref{eq:eq0sknDpth})),\ncharacterizes the limited r.f. penetration into the metal. This in turn, results\nin attenuated MR signal intensity for bulk metals \\cite{si0pines,rangeet}.\nTurning the tables, Bhattacharyya et. al.,\\cite{rangeet} exploited the skin\neffect to separate and quantify bulk and non-bulk metal NMR signals in Li ion\nbatteries to monitor the growth of dendritic metallic structures.\nSubsequently, for bulk metal MRI, yet another impediment was correctly diagnosed\n\\cite{chandrashekar}.\nIt was found that the orientation of the bulk metal surface, relative to \\mbox{$\\bf B_1$}\n(the r.f. magnetic field),\ncritically affected the outcome.\nUsing optimal alignment of the bulk metal (electrodes), relative to \\mbox{$\\bf B_1$},\nrecent studies have successfully demonstrated and highlighted bulk metal MRI\nalbeit, primarily applied to batteries and electrochemical cells\n\\cite{chandrashekar,ilott,romanenko,britton2014,hjc2015}.\n\nThough unanticipated at the time, the recent bulk metal MRI findings eased the\nimplementation of MRI of liquid electrolyte, by helping mitigate adverse effects\ndue to the metal in the vicinity of lithium, zinc and titanium electrodes,\nyielding fresh insights\n\\cite{romanenko,britton2014,si0britton2013,si0klett,si0furo}.\nSimilar benefits may be expected to accrue for MRI based radiology of soft\ntissues with embedded metallic implants (pacemeakers, prosthetics, dental\nimplants, etc.).\n\\\\\n\n\n\nHere, we present several key findings on bulk metal MRI and CSI:\n\\begin{itemize}\n\\item\nDuring a systematic noninvasive thickness measurement of bulk metal strips\nby MRI \\cite{romanenko}\n(section S\\ref{sxnSI0thcknss}),\nwe come across unexpected regions of intensity, and assign them to two mutually \northogonal pairs of faces of the strip.\n\n\\item\nTo explain the peculiar ratios of intensities from these different regions in\nbulk metal MRI and CSI, we derive formulae from first principles, unveling\na surprising underlying reason: \n{\\em differing effective elemental volumes for these different regions}.\n\n\\item\nIn the process, the images enable a visualization of a virtual EM vacuum\ninside the bulk metal via an {\\em MRI tunnel}.\n\n\\item\nAdditionally, we demonstrate that the bulk metal CSI distinguishes different\nfaces (surfaces) of a metal block according to their distinct NMR chemical\nshifts.\n\n\\end{itemize}\n\nWe attained these by employing three {\\em phantoms} (samples)\n{\\bf P0, P1, P3}, depicted schematically in Fig.\\ref{fig0phntms} and described\nin Methods section \\ref{subsxn0mthdsPhntms}.\nAll phantoms were derived from the same stock of 0.75 $mm$ thick lithium (Li)\nstrip. Phantom P3 is a super strip composed of three Li strips pressed together,\nforming an {\\em effective} single strip three times thicker than the\nindividual strips in phantoms P0 and P1.\n\nThe setup of phantoms, r.f. coil and the gradient assembly ensures that the\nimaging directions $x,y,z$ fulfill the condition that \n$x \\parallel a \\parallel \\mbox{$\\bf B_1$}$ and  $z \\parallel \\mbox{$\\bf B_0$}$\n(the static main magnetic field), with the possibility to reorient the phantom\nabout the $x$-axis; $a,b,c$  are the three sides of the strips.\n\nSince all MRI and CSI images to follow were acquired with the given phantom's\n$bc$ faces {\\em normal} ($\\perp$, {\\em perpendicular}) to \\mbox{$\\bf B_1$}, these images\nbear the imprint of having no contribution from these faces to the magnetic\nresonance (MR) signal \\cite{chandrashekar,ilott,romanenko,britton2014}, since\n\\mbox{$\\bf B_1$}\\ penetration into the metal is maximal when it is {\\em parallel}\n($\\parallel$) to the metal surface, and minimal when $\\perp$ metal surface\n\\cite{jackson,chandrashekar,ilott}.\n\nFor details on the MRI experiments, including the nomenclature, the reader is\nreferred to \nMethods section \\ref{subsxn0mthdsMri}.\n\n\n\\section{MRI}\n\\label{sxn0mri}\nFig.\\ref{fig0xy0yz} furnishes stackplots (intensity along the vertical axis) of\n\\mbox{$^7$Li} 2d MRI (without slice selection) of phantom P3.\nPanel (a) displays MRI($xy$). Panel (b) displays MRI($yz$).\n\nIt is straightforward to infer that the {\\em walls} of high intensity regions in\neither image emanate from $ac$ faces of the P3 strip\n\\cite{chandrashekar,ilott,romanenko}, as we did while measuring the\nthickness of metal strips (Figs.\\ref{fig0twoStrps} and \\ref{fig0xy},\nsection S\\ref{sxnSI0thcknss}).\nIn either image, contributions along the non imaged axis sum up to yield the\nhigh intensity walls.\n\nHowever, the unexpected intensity between the two $ac$ faces of the super strip,\nin both the images, is perplexing.\nThe 2d MRI($xy$) in Fig.\\ref{fig0xy0yz}a exhibits a low intensity {\\em plateau}\nspanning the walls.\nThe 2d MRI($yz$) in Fig.\\ref{fig0xy0yz}b displays low intensity {\\em ridges}  \nbridging the walls.\n\\\\\n\n\\subsection{Visualizing a virtual eletromagnetic vacuum by MRI}\nTo understand better these unexpected regions of intensity, we acquired\n\\mbox{$^7$Li} 3d MRI($xyz$) of phantom P3, shown in Fig.\\ref{fig0xyz}.\nIn addition to the $ac$ faces (separated along $y$),\nthe $ab$ faces (separated along $z$) are revealed for the first time.\n\nAs noted earlier, $bc$ faces (being $\\perp$ to \\mbox{$\\bf B_1$}) are absent.\nThe hollow region in MRI($xyz$) arises from the skin depth phenomenon\n\\cite{rangeet,chandrashekar,ilott,jackson,si0griffiths,si0ulaby},\nrestricting the EM fields to effectively access only a limited\nsubsurface underneath the $ac$ and $ab$ faces\n(section S\\ref{sxnSI0subSurface} and Fig.\\ref{figSI0dltaEff}).\nThe presence of faces $\\parallel$ \\mbox{$\\bf B_1$}, coupled with the conspicuous absence\nof faces $\\perp$ \\mbox{$\\bf B_1$}, in combination with the hollow region, imparts the 3d\nimage an appearance of an {\\em MRI tunnel}, supplying a compelling visualization\nof a virtual {\\em EM vacuum} in the interior of a metallic conductor\n(hitherto depicted only schematically in literature\n(for e.g. Ref.\\cite{rangeet})).\n\\\\\n\nWith the aid of 3d MRI in Fig.\\ref{fig0xyz}, the intensity regions in\n2d MRI($xy$) and 2d MRI($yz$) images of Fig.\\ref{fig0xy0yz}, can be easily\ninterpreted as simply regions resulting respectively from projections\nalong $z$ and $x$ axes of the 3d image.\nIt is convincingly clear that the intensity between $ac$ faces\n(either the plateau or the ridges), is due to the pair of $ab$ faces\nof the superstrip P3.\n\nYet, the basis for the relative intensity values remains elusive at\nthis stage.\n\nFor the 2d MRI($xy$) in Fig.\\ref{fig0xy0yz}a, it can be argued that,\nfor the $ac$ face the entire length of side $c=$7 $mm$\n(Fig.\\ref{fig0phntms}) contributes to the signal,\nwhile for the $ab$ face, only a subsurface depth\n$ \\mbox{$\\delta_{\\text{eff}}$} \\approx$ 9.49 $\\mu m$ contributes\n(Eq.(\\ref{eq:eq0sknDpth}), Eq.(\\ref{eq:eq0dltaEff}),\nsection S\\ref{sxnSI0subSurface} and Fig.\\ref{figSI0dltaEff}).\nThis would lead to a ratio of the corresponding intensities, $S_{ac}\/S_{ab}$,\nto be of the order of $c\/(2 \\mbox{$\\delta_{\\text{eff}}$}) \\approx 368 $ (Fig.\\ref{fig0xy0yz0sim}a),\nin obvious and jarring disagreement with the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) of 6.6.\n\nFor the 2d MRI($yz$) in Fig.\\ref{fig0xy0yz}b, the expected intensity pattern in \na stack plot would be one with equal intensities from $ab$ and $ac$ faces,\nsince they share the same side, $a$, along $x$ (non imaged) axis\n(Fig.\\ref{fig0xy0yz0sim}b).\nThis again, is in stark contrast with the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) of 10.\n\nFor the MRI($xyz$), naively, uniform intensity would be expected from both $ab$ \nand $ac$ faces, resulting in a ratio of unity.  Instead, the observed ratio\n(of maxima of $S_{ac}$ and $S_{ab}$) is found to be 3.8.\n \nThus, the MRI images bear peculiar intensity ratios from comfortably identified \n(from 3d MRI) regions of the bulk metal.\nWe will return to this topic later.\n\\\\\n\n\\section{CSI}\n\\label{sxn0csi}\nThe \\mbox{$^7$Li} NMR spectrum of phantom P3 (Fig.\\ref{fig0csi} inset)\ncontains two distinct peaks in the Knight shift region for metallic \\mbox{$^7$Li}\n(see Methods section \\ref{subsxn0mthdsMri}),\ncentered at $\\delta_1$= 256.4 and $\\delta_2$= 266.3 ppm.\nAt first sight it might seem odd that a metallic strip, of regular geometry and\nuniform density, that is entirely composed of identical Li atoms, gives\nrise to two NMR peaks instead of the expected single peak.\n\nTo gain additional insight as to the spatial distribution of the Li metal\nspecies with different NMR shifts, we turn to CSI, which combines an NMR\nchemical shift (CS) dimension with one or more imaging(I) dimensions\n\\cite{callaghan,haacke,si0spnglr,si0kwf}.\n\nThe 2d CSI($y$) shown in Fig.\\ref{figSI0csi} comprises of two bands separated\nalong $y$ located at $\\delta_2$, along the CS dimension,\nwhile a low intensity band spans them along $y$ at a CS of $\\delta_1$, strongly\nhinting at, the two bands (at $\\delta_2$) being associated with $ac$ faces.\n\nThis observation called for adding an additional\nimaging dimension along $z$, leading to 3d CSI($yz$), which is realized in\nFig.\\ref{fig0csi},\nwhere $y$ and $z$ are the imaging dimensions, accompanied by the CS dimension.\nThe bands separated along $z$, occur at $\\delta_1$.\nThe bands separated along $y$, occur at $\\delta_2$.\nIn conjunction with the 3d MRI image in Fig.\\ref{fig0xyz},\nit is evident that the pairs of bands at $\\delta_1$ and $\\delta_2$ arise from\n$ab$ and $ac$ faces respectively, completing the spatio-chemical assignment.\n\nThese assignments readily carry over to 2d CSI($y$) in Fig.\\ref{figSI0csi},\nwith the pair of bands at $\\delta_2$ and the low ridge spanning them at \n$\\delta_1$, being respectively identified with $ac$ and $ab$ faces.  Similarly,\nin the NMR spectrum, the short and tall peaks\nrespectively at $\\delta_{1,2}$ are assigned to $ab$ and $ac$ faces, consistent\nwith the reported \\cite{rangeet,ilott} experiments and simulations.\n\nThat different types of faces of the bulk metal strip suffer different NMR\n(Knight) shifts according to their orientations {\\em relative} to \\mbox{$\\bf B_0$},\nis consistent with previous observations and simulations \\cite{rangeet,ilott},\nand has been traced to bulk magnetic susceptibility (BMS) effect\n\\cite{rangeet,chandrashekar,ilott,hjc2015,hoffman,lina,hjc}.\n\nInterestingly, the 3d CSI sheds new light on previous bulk metal NMR studies.\nFor instance, in an earlier study \\cite{rangeet}, a similar shift\ndifference between NMR peaks was observed at $\\parallel$ and $\\perp$\norientations (relative to \\mbox{$\\bf B_0$}) of the major faces of a thinner metal strip,\nby carrying out two {\\em separate} experiments.\nHere, phantom P3 furnishes these two orientations in a {\\em single} experiment, \nvia $ac$ and $ab$ faces (Fig.\\ref{fig0phntms}).\nThe present work provides physical insight into another previous \\cite{ilott}\nobservation. It was found that the intensity of NMR peak arising from $ab$\nfaces, unlike that from the $ac$ faces, was invariant under rotation about\n$z \\parallel c \\parallel B_0$ axis.\nOur 3d MRI (Fig.\\ref{fig0xyz}) and 3d CSI (Fig.\\ref{fig0csi}), directly\ndemonstrate that such a rotation leaves the orientation of \\mbox{$\\bf B_1$}\\ relative to\n$ab$ face (but not the $ac$ face) the same (signal intensity from a given face\ndepends on its orientation relative to \\mbox{$\\bf B_1$} \\cite{chandrashekar,ilott}).\nNote that the shifts themselves remain unshifted since they depend on the\norientation of the faces relative to \\mbox{$\\bf B_0$}, which does not change under this\nrotation ($ac$ and $bc$ faces remain $\\parallel \\mbox{$\\bf B_0$}$, whilst $ab$ faces\nremain $\\perp \\mbox{$\\bf B_0$}$).\n\nThus, bulk metal CSI supplies direct\nevidence, that the bulk metal chemical (Knight) shifts resulting from BMS are\ncorrelated with the differing orientations (relative to \\mbox{$\\bf B_0$}) of different parts \nof the bulk metal.\\\\\n\nLike for MRI, the basis for the ratio of intensities from the $ac$ and $ab$\nfaces ($S_{ac}\/S_{ab} \\approx$ 2.8), is not immediately intuitively obvious\nand will be explored next.\n\n\n\n\\section{Intensity ratio formulae for bulk metal MRI and CSI}\n\\label{sxn0formulae}\nThe peculiar intensity ratios in MRI and CSI, of signals $S_{ab}$ and $S_{ac}$,\narising respectively from $ab$ and $ac$ faces of phantom P3\n(Fig.\\ref{fig0xy0yz}, sections \\ref{sxn0mri} and \\ref{sxn0csi}),\ncould be due to gradient switching involved in the MRI experiments (the\nresultant eddy currents could be different for $ab$ and $ac$ faces).\nHowever, as shown in section S\\ref{sxnSI0mri2dNtnst}, this can be ruled out on\nthe basis of 2d MRI($yz$) and MRI($zy$) at mutually orthogonal orientations\n(related by a rotation about $x \\parallel a \\parallel \\mbox{$\\bf B_1$}$),\nshown in Fig.\\ref{fig0p3hrzntl}.\n\nAnd yet, it is possible to derive, from elementary considerations\nand first principles, and arrive at expressions for the {\\em ratios} of MRI\nand CSI intensities from $ab$ and $ac$ faces.\\\\\n\nFor the 2d MRI($xy$), in Fig.\\ref{fig0xy0yz}a, the signal intensity from the\n$ab$ faces can be written as (see Eq.(\\ref{eq:eq1dltaEff}))\n\\begin{equation}\n S_{ab} (x,y)\\propto dx\\ dy\\ \\int dz =  dx\\ dy\\ 2\\delta_{\\text{eff}}\n\\label{eq:eq0sabxy}\n\\end{equation}\nwith  $dx\\ dy\\ dz$ denoting {\\em elemental} volume of the metal,\nand \\mbox{$\\delta_{\\text{eff}}$}\\ is the {\\em effective} subsurface {\\em depth} that would account\nfor the MR signal in the {\\em absence} of \\mbox{$\\bf B_1$}\\ decay\n(see Eq.(\\ref{eq:eq0dltaEff})\nand Fig.\\ref{figSI0dltaEff}).\nAbove, the integral over $z$, is replaced by \\mbox{$\\delta_{\\text{eff}}$}\\, underneath the two $ab$\nfaces separated along $z$.\n\nSimilarly, for the signal intensity from {\\em either} of the $ac$ faces,\n\\begin{equation}\n S_{ac} (x,y)\\propto dx\\ dy\\ \\int dz = dx\\ \\delta_{\\text{eff}}\\ c\n\\label{eq:eq0sacxy}\n\\end{equation}\nsince the subsurface now is $\\perp y$.\n\nEq.(\\ref{eq:eq0sabxy}) and Eq.(\\ref{eq:eq0sacxy}), reveal {\\em differing\neffective elemental volumes}(voxels) underneath these faces:\n\\begin{equation}\ndV_{\\text{eff}}^{\\text{ab}} = dx\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0dVeffab}\n\\end{equation}\n\\begin{equation}\ndV_{\\text{eff}}^{\\text{ac}} = dx\\ \\delta_{\\text{eff}}\\ dz \n\\label{eq:eq0dVeffac}\n\\end{equation}\nFrom Eq.(\\ref{eq:eq0sabxy}) and Eq.(\\ref{eq:eq0sacxy}),\n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}} =\\frac{c}{2\\Delta y}\n\\label{eq:eq0ratioxy}\n\\end{equation}\nwhere we have replaced $dy$ by $\\Delta y$, the resolution along\n$y\\ \\parallel b$.\nConsulting the Methods section \\ref{subsxn0mthdsMri} and Fig.\\ref{fig0phntms},\n$c=7$ $mm$, $\\Delta y$=0.25 $mm$ and Eq.(\\ref{eq:eq0ratioxy}) yields a\ncalculated ratio of $S_{ac}\/S_{ab}$= 14\n(as illustrated in Fig.\\ref{fig0xy0yz0drvd}a),\nwithin an order of magnitude of the observed ratio (section \\ref{sxn0mri},\nFig.\\ref{fig0xy0yz}a) and a 25 fold improvement relative to the expected ratio\n(Fig.\\ref{fig0xy0yz0sim}a).\n\nAlso, Eq.(\\ref{eq:eq0ratioxy}) reveals that $S_{ac}\/S_{ab}$ increases with\nincreasing resolution along $y$, as shown in the three images in\nFig.\\ref{figSI0td20td40td80},\nwith relative resolutions increasing by factors of 1, 2 and 4,\nyielding calculated  $S_{ac} \/ S_{ab}$ ratios of 7, 14 and 28 respectively.\nThe corresponding observed ratios (of maxima of $S_{ac}$ and $S_{ab}$) of\n3.3, 6.6, and 11.6, are within an order of magnitude of the calculated values.\nRemarkably, these observed ratios themselves increase by factors of 1, 2 and\n3.5, mimicking closely the corresponding factors of resolution increase.\n\\\\\n\nContinuing in the same vein, for the 2d MRI($yz$), in Fig.\\ref{fig0xy0yz}b,\n\\begin{equation}\n S_{ab} (y,z) \\propto dy\\ dz \\int dx = a\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0sabyz}\n\\end{equation}\nwhile,\n\\begin{equation}\n S_{ac} (y,z) \\propto dy\\ dz \\int dx = a\\ \\delta_{\\text{eff}}\\ dz\n\\label{eq:eq0sacyz}\n\\end{equation}\nleading to\n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}}= \\frac{\\Delta z}{\\Delta y}\n\\label{eq:eq0ratioyz}\n\\end{equation}\nonce again, replacing $dy,\\ dz$ by $\\Delta y,\\ \\Delta z$, the respective\nresolutions along $y,\\ z$.\nUsing the values of $\\Delta y, \\Delta z$= 0.0357, 1 $mm$\nin Eq.(\\ref{eq:eq0ratioyz}), ensues\na calculated ratio of $S_{ac} \/ S_{ab} \\approx$ 28\n(as illustrated in Fig.\\ref{fig0xy0yz0drvd}b),\nwithin an order of magnitude of the observed ratio (section \\ref{sxn0mri},\nFig.\\ref{fig0xy0yz}b), and 3.5 fold better than the expected ratio\n(Fig.\\ref{fig0xy0yz0sim}b). More importantly, the expected intensity pattern is \neven {\\em qualitatively} (visually) different from the experiment, unlike the\nderived pattern.\n\n\nSimilarly, for the 2d MRI($zy$) in Fig.\\ref{fig0p3hrzntl}b, of phantom P3 in\n{\\em horizontal} orientation, it can be easily shown that, \n\\begin{equation}\n \\frac{S_{ac}}{S_{ab}}= \\frac{\\Delta y}{\\Delta z}\n\\label{eq:eq0ratiozy}\n\\end{equation}\nUsing the values of $\\Delta z, \\Delta y$= 0.0357, 1 $mm$\nin Eq.(\\ref{eq:eq0ratiozy}), results in a calculated ratio of\n$S_{ac} \/ S_{ab} \\approx$ 28, within an order of magnitude of the observed\nratio (section S\\ref{sxnSI0mri2dNtnst}).\\\\\n\nProceeding along the same lines, for the MRI($xyz$) in Fig.\\ref{fig0xyz},\n\\begin{equation}\n S_{ab} (x,y,z) \\propto dx\\ dy\\ dz =  dx\\ dy\\ \\delta_{\\text{eff}}\n\\label{eq:eq0sabxyz}\n\\end{equation}\nwhile,\n\\begin{equation}\n S_{ac} (x,y,z) \\propto dx\\ dy\\ dz =  dx\\ \\delta_{\\text{eff}}\\ dz\n\\label{eq:eq0sacxyz}\n\\end{equation}\nAs usual by now, replacing $dy,\\ dz$ by $\\Delta y,\\ \\Delta z$, the respective\nresolutions along $y,\\ z$, we obtain again Eq.(\\ref{eq:eq0ratioyz}).\nConsulting the Methods section \\ref{subsxn0mthdsMri}, \n$\\Delta y, \\Delta z$= 0.25, 1 $mm$, respectively.  Using these values in \nEq.(\\ref{eq:eq0ratioyz}) yields a calculated ratio of $S_{ac} \/ S_{ab}$= 4,\nwithin an order of magnitude of the observed ratio\n(section \\ref{sxn0mri}).\n\nFig.\\ref{fig0xyzSlc25} shows (in stack plot representation, with vertical axis\ndenoting intensity) $xy$ slices (along $z$) from the 3d MRI($xyz$).\nThe central slice contains no signal between the walls of intensity (from $ac$\nfaces) as expected.\nHowever, the slice from the top $ab$ face exhibits a {\\em plateau} of intensity\nbetween the $ac$ faces, visually demonstrating that $S_{ab} \\neq S_{ac}$ in the \nnon-hollow regions of the 3d image, in place of the expected uniform intensity.\n\\\\\n\nSimilarly, for the 3d CSI($yz$) in Fig.\\ref{fig0csi}, it can be shown that\nthe ratio $S_{ac} \/ S_{ab}$ is given by Eq.(\\ref{eq:eq0ratioyz}), which along\nwith the relevant experimental parameters for this image,\nyields a calculated value of 2, within an order of magnitude of the observed\nratio (of maxima of $S_{ac}$ and $S_{ab}$) of 2.8.\n\nOn the other hand, for the 2d CSI($y$), in Fig.\\ref{figSI0csi}, it can be shown \nthat the ratio $S_{ac} \/ S_{ab}$ is given by Eq.(\\ref{eq:eq0ratioxy}), from\nwhich we obtain a calculated value of 14, using the experimental parameters in\nsection \\ref{subsxn0mthdsMri}.  The measured ratio (of maxima of $S_{ac}$ and\n$S_{ab}$) of 9.5, is again within an order of magnitude of the calculated value.\n\\\\\n\nThus, the $S_{ac} \/ S_{ab}$ ratios calculated from\nEqs.(\\ref{eq:eq0ratioxy}), (\\ref{eq:eq0ratioyz}) and (\\ref{eq:eq0ratiozy}),\nagree with the observed values within an order of magnitude\nfor 2d MRI($xy$), 2d MRI($yz$), 2d MRI($zy$), 3d MRI($xyz$), 3d CSI($yz$) and\n2d CSI($y$).\nIn fact, discrepancies between observed and derived $S_{ac}\/S_{ab}$ ratios range\nonly by factors of 0.7 to 2.8 across various MRI and CSI images\n(see Table.\\ref{table:tbl0ratio}).\nMore importantly, the derived patterns {\\em resemble} the observed patterns,\nunlike the expected patterns, which differ even visually from the observed\npatterns (for e.g., see\nFigs.\\ref{fig0xy0yz}, \\ref{fig0xy0yz0sim} and \\ref{fig0xy0yz0drvd}).\n\n\\begin{table}[h]\n \\caption{\n  The observed, derived (from\n  Eqs.(\\ref{eq:eq0ratioxy}), (\\ref{eq:eq0ratioyz}), (\\ref{eq:eq0ratiozy})),\n  and expected (from skin depth arguments alone) $S_{ac} \/ S_{ab}$ ratios.\n }\n \\begin{tabular}{|c| c c c|}\n  \\hline\n    & & $S_{ac} \/ S_{ab}$    &  \\\\\n  \\hline\n   Experiment & Observed & Derived & Expected \\\\\n\\hline\nMRI($xy$) & & & \\\\\nFig.\\ref{figSI0td20td40td80}a &  3.3 &  7 & 368 \\\\\nFig.\\ref{figSI0td20td40td80}b &  6.6 & 14 & 368 \\\\\nFig.\\ref{figSI0td20td40td80}c & 11.6 & 28 & 368 \\\\\n & & & \\\\\nMRI($yz$)    & 10  & 28  &   1 \\\\\nMRI($zy$)    & 10  & 28  &   1 \\\\\nMRI($xyz$)   & 3.8 &  4  &   1 \\\\\n2d CSI($y$)  & 9.5 & 14  & 368 \\\\\n3d CSI($yz$) & 2.8 &  2  &   1 \\\\\n\\hline\n \\end{tabular}\n \\label{table:tbl0ratio}\n\\end{table}\n\nIn summary, the formulae unveil the underlying reason for the significant\ndeparture of observed $S_{ac}\/S_{ab}$ from expected values:\n{\\em differing effective elemental volumes underneath these faces},\nas revealed by Eqs.(\\ref{eq:eq0dVeffab}) and (\\ref{eq:eq0dVeffac}).\nThe derived patterns bear closer resemblance to\nexperiment, than what is expected from skin depth consideratons alone, or from\nconventional specifications of the voxel= $\\Delta x \\Delta y \\Delta z$ (see for\ne.g.  Figs.\\ref{fig0xy0yz}, \\ref{fig0xy0yz0sim} and \\ref{fig0xy0yz0drvd}).\n\nOn a practical note, these formulae can guide experimental strategies to\nrelatively enhance MRI and CSI signals from different regions of the bulk metal.\n\n\n\\section{Conclusions}\n\\label{sxn0cnclusn}\nIn conclusion, the unexpected findings presented here may impact bulk metal MRI \nand CSI studies in general,\nvia fresh insights\nfor data collection, analysis and interpretation.\nThe bulk metal MRI and CSI\n(correlating different bulk metal surfaces with distinct chemical shifts)\nresults in this study have the noninvasive diagnostic potential in other fields\nsuch as\nstructure  of metals and alloys \\cite{rdrgz,flyn},\nmetallurgy (metal fatigue, fracture, strain)\n\\cite{mtllurg,bppag,mvrkks},\ncatalysis\n\\cite{zhong,yuan},\nbulk metal surface science and surface chemistry\n\\cite{tlptr,whttn,rgg},\nmetallic medical implants, dielectric MRI in the vicinity of bulk metals\netc.\n(section \\ref{sxn0intrdxn}).\n\nThe findings may also lead to as yet unforeseen applications\n(section \\ref{sxn0intrdxn}) since,\n(i) they are of a fundamental nature,\n(ii) there are no inherent limitations to the approach employed\n(scalability, different metals, systems other than batteries, etc., are all\npossible),\n(iii) the study utilizes only standard MRI tools (hardware, pulse sequences,\ndata acquisition and processing), ensuring ease of implementation and\nreproducibility. Thus it is likely to benefit from advances made in the\nmainstream (medical) MRI field.\n\n\n\n\n\\section{Methods}\n\\label{sxn0mthds}\n\\subsection{Phantoms}\n\\label{subsxn0mthdsPhntms}\nAll phantoms were assembled and sealed in an argon filled glove box.\nAll three phantoms, P0, P1 and P3, shown in Fig.\\ref{fig0phntms} \n(of dimensions $a \\times b \\times c$),\nwere derived from a (0.75 $mm$ thick) stock Li strip (Alfa Aesar 99.9\\%).\nThe Li strips were mounted on a 2.3 $mm$ thick teflon strip and the resulting\nsandwich bound together with Kapton tape.\nEach phantom was placed in a flat bottom glass tube (9.75 $mm$ inner diameter\n(I.D.), 11.5 $mm$ outer diameter (O.D.) and  5 $cm$ long), with\nthe longest side, $a$, $\\parallel$ to the tube axis and to the axis of the\nhome built horizontal loop gap resonator (LGR)  r.f. coil (32 $mm$ long, 15 $mm$\nO.D.), thus guaranteeing $\\mbox{$\\bf B_1$} \\parallel a$.\nThe phantom containing glass tubes were wrapped with Scotch tape\nto snugly fit into the r.f. coil.\n\nIn  Fig.\\ref{fig0phntms}, $x,y,z$ specify the imaging (gradient) directions,\nwith $z \\parallel \\mbox{$\\bf B_0$}$ (the main magnetic field).\nFor our horizontal LGR r.f. coil (the MR resonator) and the gradient assembly\nsystem, $\\mbox{$\\bf B_1$} \\parallel x$, resulting in $\\mbox{$\\bf B_1$} \\parallel x \\parallel a$.\n\\begin{itemize}\n\\item {\\bf P0}: Pair of Li strips separated by a teflon strip;\nfor each Li strip,\n$a \\times b \\times c=$ 20 x 0.75 x 7 $mm^3$.\n\\item\n{\\bf P1}: Single Li strip.\n$a \\times b \\times c=$ 15 x 0.75 x 7 $mm^3$.\n\\item\n{\\bf P3}: Three  Li strips pressed together to yield a single\ncomposite super strip.\n$a \\times b \\times c=$ 15 x 2.25 x 7 $mm^3$.\n\\end{itemize}\n\n\n\\subsection{MRI and CSI}\n\\label{subsxn0mthdsMri}\nMagnetic resonance experiments were conducted on a $B_0$=21$T$ magnet\n(corresponding to \\mbox{$^7$Li} Larmor frequency of 350 MHz)\noperating under Bruker Avance III system with Topspin spectrometer control and\ndata acquisition, and equipped with a triple axes ($x,y,z$) gradient amplifier\nassembly, using a multinulcear MRI probe\n(for a triple axes 63 $mm$ I.D. gradient stack by Resonance Research Inc.), \nemploying the LGR r.f. coil (resonating at 350 MHz) desribed above.\n\n\nThe MRI and CSI data were acquired using spin-echo imaging pulse sequence\nwithout slice selection \\cite{callaghan,haacke} (yielding sum total of signal\nconributions from the non imaged dimensions).\nFrequency encoding gradient was\nemployed for the directly detected dimension and phase encoding gradients for\nthe indirect dimensions \\cite{callaghan,haacke}.\nThe CSI experiments were carried out with the NMR chemical shift as the directly\ndetected dimension, with phase encoding gradients along the indirectly detected \nimaging dimensions.\nThe r.f. pulses were applied at a carrier frequency of 261 ppm (to excite the\nmetallic \\mbox{$^7$Li} nuclear spins in the Knight shift region\n\\cite{rangeet,abragam}),\ntypically with a strength of 12.5 $kHz$, with a recycle (relaxation) delay of\n0.5 $s$.\nThe gradient dephasing delay and phase encoding gradient duration were 0.5 $ms$.\n\nThroughout this manuscript,\nthe first axis label ($x,y,z$) describing an MRI experiment stands for frequency\nencoding dimension and the remaining ones correspond to phase econded\ndimensions.   For e.g., MRI($xyz$) implies frequency encoding along $x$ axis,\nand phase encoding along the remaining directions.\n\n$G_x,G_y,G_z$ and $N_x,N_y,N_z$ denote respectively the gradient strengths in\nunits of $T\/m$ and number of data points in $k$-space ($^*$ denoting complex\nnumber of points acquired in quadrature) \\cite{callaghan,haacke}, along\n$x,y,z$ axes.\n$L_x,L_y,L_z$ and $\\Delta x,\\Delta y,\\Delta z$ are respectively the resultant\nnominal field of view (FOV) and resolution, in units of $mm$, along $x,y,z$ axes\n\\cite{callaghan,haacke}.\n\nAlso, $n$ is the number of transients accumulated for signal averaging and\n$SW$ is the spectral width (in units of $kHz$) for the directly detected\ndimension in MRI and CSI.\n\\\\ \\\\\n{\\bf 1d MRI({\\em y}):} \\\\\n$n=64,\\ SW=50$ \\\\\n$G_y=0.42,\\ N_y^*=200,\\ L_y=7.143,\\ \\Delta y= 0.0357$\n\\\\\n{\\bf 2d MRI({\\em xy}):} \\\\\n$n=32,\\ SW=100$ \\\\\n$G_x=0.24,\\ N_x^*=200,\\ L_x=25,\\ \\Delta x= 0.125$ \\\\\n$L_y=10$ \\\\\n(1) $G_y=0.12,\\ N_y=20,\\  \\Delta y= 0.500$\n     (Figs.\\ref{fig0twoStrps}a,\\ref{figSI0td20td40td80}a) \\\\\n(2) $G_y=0.24,\\ N_y=40,\\  \\Delta y= 0.250$\n     (Figs.\\ref{fig0twoStrps}b,\\ref{fig0xy}b,\\ref{fig0xy0yz}a,\n     \\ref{figSI0td20td40td80}b) \\\\\n(3) $G_y=0.48,\\ N_y=80,\\  \\Delta y= 0.125$ (Fig.\\ref{figSI0td20td40td80}c) \\\\\n{\\bf 2d MRI({\\em yz}):} \\\\\n$n=32$ \n$SW=50$ \\\\\n$G_y=0.42,\\ N_y^*=200,\\ L_y=7.143,\\ \\Delta y= 0.0357$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$\n\\\\\n{\\bf 2d MRI({\\em zy}):} \\\\\n$n=32,\\ SW=50$ \\\\\n$G_z=0.42,\\ N_z^*=200,\\ L_z=7.143,\\ \\Delta z= 0.0357$ \\\\\n$G_y=0.06,\\  N_y=16,\\    L_y=16,\\ \\Delta y= 1.000$\n\\\\\n{\\bf MRI({\\em xyz}):} \\\\\n$n=16,\\ SW=100$ \\\\ \n$G_x=0.24,\\ N_x^*=200,\\ L_x=25,\\ \\Delta x= 0.125$ \\\\\n$G_y=0.24,\\ N_y=40,\\    L_y=10,\\ \\Delta y= 0.250$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$\n\\\\\n{\\bf 2d CSI({\\em y}):} \\\\\n$n=8,\\  SW=100$, number of data points(complex)=$1024$ \\\\ \n$G_y=0.24,\\ N_y=40,\\    L_y=10,\\ \\Delta y= 0.250$\n\\\\\n{\\bf 3d CSI({\\em yz}):} \\\\\n$n=24,\\ SW=100$, number of data points(complex)=$1024$ \\\\ \n$G_y=0.12,\\ N_y=20,\\    L_y=10,\\ \\Delta y= 0.500$ \\\\\n$G_z=0.06,\\  N_z=16,\\    L_z=16,\\ \\Delta z= 1.000$ \\\\\n\\\\\nAll data were processed in Bruker's Topspin, with one zero fill prior to complex\nfast Fourier Transform (FFT) along each dimension either without any window\nfunction or with sine-bell window function.\nAll data were 'normalized' (to $\\approx 10$, for plotting convenience) to aid\ncomparing {\\em relative} intensities from different regions {\\em within} a given\nimage.\nFor the purpose of determining the ratios of signal intensities associated\nwith different regions of the bulk metal, the intensity values were measured\ndirectly from the processed images either in Topspin or Matlab (for e.g.,\n'datatip' utility in Matlab, yields the coordinates and the 'value' (intensity) \nof a data point by clicking on it, in 1d, 2d and 3d plots). \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nApplication of external pressure is a powerful method to tune the intricate interplay of competing energy scales in correlated materials and the emergence of novel unconventional phases in a clean fashion. It offers significant advantages as a control parameter compared with chemical substitution and application of magnetic fields, because it does not introduce additional disorder as in the case of substitution of one element by an other or polarize the electrons as a magnetic field does.\n\nA large variety of experimental setups has been developed to probe physical properties under hydrostatic pressure.\\cite{Nicklas2015} In contrast, experiments under uniaxial pressure appeared to be limited to low pressure and only few experimental probes. Recently, however, the development of piezoelectric-driven pressure devices opened a new perspective.\\cite{Hicks2014a,Barber2019} These devices allow the application of large positive and negative pressures and the amplitude of the applied pressure can be easily changed at low temperatures. In a short period of time experimental stages to access a large number of physical properties of materials have been developed. These include electrical transport,\\cite{Stern2017,Steppke2017} magnetic susceptibility,\\cite{Hicks2014b} nuclear-magnetic resonance,\\cite{Kissikov2017,Luo2019,Pustogow2019} muon-spin resonance,\\cite{Grinenko2020} and angle-resolved photo emission, for which mechanically or thermally activated cells have also been introduced.\\cite{Flototto2018,Ricco2018,Pfau2019a,Pfau2019b,Sunko2019}\n\nAn important quantity to characterize a material is the specific heat, which is the fundamental thermodynamic property giving information on the internal degrees of freedom of a material and the entropy related with them.\nTo address the experimental challenge of studying the heat capacity under large uniaxial pressures, we employ a variation of known AC heat-capacity measurement techniques.\\cite{Sullivan1968} Heat capacity measurement has been combined with uniaxial pressure previously,\\cite{Jin1992,Reinders1994,Miclea2002,Zieve2004,Dix2009} but with traditional, anvil-based uniaxial pressure cells. Samples have been thermally isolated by using low thermal conductivity materials, such as stainless steel or superconducting NbTi, as piston or additional spacer.  However in previous anvil-based uniaxial-pressure measurements, e.g.\\ on the unconventional superconductor Sr$_2$RuO$_4$, it did not prove practical to maintain high stress homogeneity,\\cite{Kittaka2010,Taniguchi2015} which is one of the main challenges in carrying out this kind of experiments. Furthermore, under applied uniaxial pressure the samples even may deform plastically. To reduce these effects we apply force to the sample through a layer of epoxy,\\cite{Hicks2014a} which acts as a conformal layer that dramatically improves stress homogeneity. However, it also makes heat-capacity measurement more challenging, because the epoxy layer provides an unavoidable strong thermal link to the pressure cell.\n\nFor our study we have used Sr$_2$RuO$_4$ that provides a demanding test of our new apparatus. Sr$_2$RuO$_4$ is an unconventional superconductor with a superconducting transition temperature up to $T_c=1.5$~K in the best crystals.\\cite{Mackenzie1998,Mackenzie2003,Mackenzie2017} From resistivity and magnetic susceptibility experiments it is known that $T_c$ shows a pronounced dependence on the applied uniaxial pressure.\\cite{Hicks2014b,Steppke2017,Barber2018} This and the sharp superconducting transition anomaly make it an ideal material to demonstrate the potential of our technique for the study of correlated materials. A successful experiment on Sr$_2$RuO$_4$ can only be done using a technique that introduces no disorder or plastic deformations, and that probes a region in which the strain induced in the sample is highly homogeneous.\n\n\n\n\\section{METHOD}\n\nFor a setup in which the sample is strongly coupled to the environment as it is in a pressure cell, whether hydrostatic or uniaxial, standard quasi-adiabatic or relaxation techniques are limited to cases where the heat capacity of the whole pressure cell including the sample is measured and the heat capacity of the sample can then be separated from the (large) addenda. That implies restrictions on the materials which can be investigated and limits experiments to low temperatures. The advantage of such a technique is that one obtains absolute values of the heat capacity, but the resolution and the pressure regime are limited. In heat-capacity measurements at higher pressure, where anvil-type cells are used, or for uniaxial pressure experiments, the application of this technique is not possible anymore. The mass of sample is negligible with respect to that of the pressure apparatus. In these cases the heat capacity can only be determined using an AC heat capacity measurement technique.\\cite{Sullivan1968} With the AC technique it is possible to record heat capacity data in a wide range of parameter space on a sample which is not well thermally isolated from its environment by adjusting the measurement frequency. The drawback is that it is generally challenging to obtain absolute values of the heat capacity and one usually has to be content with data having arbitrary units. As we demonstrate, however, it can still yield a wealth of useful information.\n\n\\subsection{AC Heat Capacity}\\label{AC Heat Capacity}\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.9\\linewidth]{Scheme_HC}\n\\centering\n\\caption{\n(a) Schematic drawing of the thermal couplings of a sample in an AC heat-capacity setup.\n(b) A schematic diagram of the frequency response curve $F$ against $\\omega$. $\\omega$ is the angular frequency of the temperature oscillation. The curve can be divided into three regions separated by $\\omega_1$ and $\\omega_2$.\n }\n\\label{Scheme_HC}\n\\end{figure}\n\nIn the AC heat capacity measurement technique an alternating current is applied at frequency $\\omega\/2$ to the heater, leading to an AC heat power at frequency $\\omega$ to determine the heat capacity $C_{AC}$. Here $\\omega=2\\pi f$ is the angular frequency.\nThe governing relationship for measurements of the AC heat capacity is\n\\begin{equation}\n  C_{AC}=  \\frac{P}{\\omega T_{AC} } F(\\omega).\n\\label{Cac}\n\\end{equation}\n$P$ is the average power and $F(\\omega)$ is a frequency response curve that characterizes the thermalization of the sample, and differs from sample to sample, because it depends on time constants determined by thermal conductances and heat capacities of the system.\n$F(\\omega)$ depends on the time constants $\\tau_1$ and $\\tau_2$:\n\\begin{equation}\\label{F_omega}\n  F(\\omega)=\\left[ 1+\\frac{1}{\\omega^2\\tau_1^2}+\\omega^2\\tau_2^2\\right] ^{-1\/2}.\n\\end{equation}\n$\\tau_1=C_{AC}\/k_b$ describes the time scale of the applied heat power decaying to the environment, whereas  $\\tau_2=\\sqrt{\\tau_h^2+\\tau_{\\theta}^2+\\tau_{\\rm int}^2}$ describes the internal thermal time scale within the system itself.\nHere $\\tau_h=C_h\/k_h$,$\\tau_{\\theta}=C_{\\theta}\/k_{\\theta}$  and $\\tau_{\\rm int}=C_s\/k_s$ (see also Fig.\\ \\ref{Scheme_HC}a). The time constants $\\tau_h$ $\\tau_{\\theta}$ and $\\tau_{\\rm int}$ describe the time scales for the heater and, thermometer and sample to be thermalized, respectively. $C_h$, $C_{\\theta}$, and $C_s$ are the heat capacity of the heater, thermometer, and sample, respectively. For a good design, the responses of heater and thermometer need to be fast so one should aim at $\\tau_{\\rm int}\\gg \\tau_h$ and $\\tau_{\\theta}$.\n\nA schematic diagram  of $F(\\omega)$ is shown in Fig.\\ \\ref{Scheme_HC}b. At low frequencies, indicated as regime I, $\\omega\\ll\\omega_1= 1\/\\tau_1=k_b\/C_{AC}$, $F(\\omega)$ is reduced due to dissipation of temperature oscillations into the environment and at high frequencies, $\\omega\\gg\\omega_2= 1\/\\tau_2\\cong k_s\/C_s$, because the heater-sample-temperature sensor system does not thermalize, marked as regime III.\nIn the plateau region between these limits $F(\\omega)\\approx 1$ and\n\\begin{equation}\\label{C_ac_prox}\nC_{AC}\\approx  \\frac{P}{\\omega T_{AC} }.\n\\end{equation}\n\nIn addition to the temperature oscillations the application of the oscillatory heating power leads to an temperature offset $T_{DC}$ in the sample, which can be determined in the low frequency limit $\\omega\\ll\\omega_1$. Here $F(\\omega)=\\omega\\tau_1$ and the temperature offset can be estimated as\n\\begin{equation}\\label{T_DC}\nT_{DC}\\approx  \\frac{P}{k_b}.\n\\end{equation}\n\n\n\n\n\\subsection{Experimental Setup}\n\n\nThe general considerations in the section above show that it should be possible to measure the heat capacity in an uniaxial pressure apparatus as shown in Fig.\\ \\ref{Scheme_thermalConductivity}a by choosing the correct set of experimental parameters. In the following we will explain the experimental setup and describe the details of the preparation process using the example of a Sr$_2$RuO$_4$ single crystal.\n\n\nThe sample is marked by a red circle in Fig.\\ \\ref{Scheme_thermalConductivity}a and shown in detail in Fig.\\ \\ref{Scheme_thermalConductivity}b. In this setup the applied force results in a normal strain\n\\begin{equation}\\label{strain}\n\\varepsilon_{xx}=\\frac{l-l_0}{l_0}\n\\end{equation}\nin the sample. Here $l_0$ is the length of the unstrained sample and $l$ the length of the strained sample. The length change is measured capacitively and can be controlled. \\MN{}{The applied strain can go beyond $1\\%$. In Sr$_2$RuO$_4$ the Young's modulus is about 180~GPa and correspondingly the applied uniaxial pressure can reliably reach up to about 2~GPa. However, the maximum uniaxial pressure depends strongly on the mechanical properties of the investigated material.} Further details can be found in Ref.\\ \\onlinecite{Hicks2014a}.\nThe present AC heat-capacity technique can be adapted to different types of uniaxial pressure devices, e.g.\\ to a stress-controlled apparatus \\cite{Barber2019} \\MN{}{and  is fully compatible with experiments in magnetic fields}\n\n\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.9\\linewidth]{Scheme_thermalConductivity}\n\\centering\n\\caption{\n(a) Photograph of the uniaxial pressure apparatus used in the present study. The red circle marks the sample region.\n(b) Photograph of the setup of the heat capacity measurements under strain including heater and thermometer. The sample is glued between the jaws of the uniaxial pressure device.  \\MN{}{The exposed length, width and thickness of the shown sample are 2~mm, 200~$\\mu$m and 150~$\\mu$m, respectively. The device} allows the application of compressive and tensile strains. The red, yellow, and white rectangles represent the (quasi)homogeneous, inhomogeneous, and unstrained regions, respectively, see text for details.\n(c) Schematic diagram of the setup illustrating the photograph in (b).}\n\\label{Scheme_thermalConductivity}\n\\end{figure}\n\n\nIn Fig.\\ \\ref{Scheme_thermalConductivity}b we show a photograph of the bar-shaped sample that has been carefully cut, polished, and then mounted within the jaws of the uniaxial pressure rig.\nThe nature of the apparatus means that only the central part of the sample is homogeneously strained. Force is transferred to the sample through the epoxy layer around the sample. The sample ends which are protruding beyond it are unstrained, and there are intermediate regions, marked in yellow in Fig.\\ \\ref{Scheme_thermalConductivity}b, where the strain is built up. Therefore, we have to choose the measurement conditions in a way that we only probe the homogeneous part of the sample. On the example of a Sr$_2$RuO$_4$ single crystal we will demonstrate that this is in principle possible by varying the excitation frequency $f_{\\rm exc}=f\/2$ of the heater, if the characteristic parameters of the setup, such as the different thermal conductances, have been chosen in the appropriate range.\n\nFor the experiments single crystalline Sr$_2$RuO$_4$ was aligned using a bespoke Laue x-ray camera, and cut using a wire saw into thin bars with whose long axis aligned with the [100] direction of the crystal. For the best results these bars were polished using home-made apparatus based on diamond impregnated paper with a minimum grit size of 1~$\\mu$m. The bar was then mounted within the jaws of the uniaxial pressure rig using Stycast 2850FT epoxy with Catalyst 23LV (Henkel Loctide). A resistive thin film resistor chip (State of the Art, Inc., Series No.:\\ S0202DS1001FKW) as heater and a Au-AuFe(0.07\\%) thermocouple are fixed to opposite sides of the sample using Dupont 6838 single component silver-filled epoxy. \\MN{}{The resistance of heater is about 640~$\\Omega$ and the applied power is in the range of $\\mu$W. The heater is connected electrically using manganin wires providing a low thermal conductance to the bath. At 1~K the thermal conductance of the Stycast layers and the manganin wires is about $10^{-4}$ and $10^{-7}$~ W\/K, respectively. Thus, the heat loss is largely dominated by the Stycast layers.} The thermocouple was spot-welded in-house and its calibration fixed by reference to that of a calibrated RuO$_2$ thermometer.\\footnote{In the temperature range between 0.15 and 4.5~K the thermopower $S$ of the Au-AuFe(0.07\\%) thermocouple is described by $S(T)=[10.1483\\cdot T\/{\\rm K}-8.75772\\cdot (T\/{\\rm K})^2+4.00231\\cdot (T\/{\\rm K})^3-0.838741\\cdot (T\/{\\rm K})^4+0.0667604\\cdot (T\/{\\rm K})^5]{\\rm ~\\mu V\/K}$ .} Special care was taken when epoxying to the pressure cell to minimize tilt and ensure an as homogeneous strain field as possible.\n\n\nThe uniaxial pressure apparatus was mounted on a dilution refrigerator (Oxford Instruments), with thermal coupling to the mixing chamber via a high purity silver wire. The data were acquired between 500~mK and 4.2~K, with operation above 1.5~K achieved by circulating a small fraction of the mixture.  The extremely low noise level of 20~${\\rm pV\/\\sqrt{Hz}}$ on the thermocouple readout was achieved by the combination of an EG\\&G 7265 lock-in amplifier and a \\MN{}{high frequency} low temperature transformer (\\MN{}{LTT-h from} CMR direct) mounted on the 1~K pot of the dilution refrigerator, operating at a gain of 300. \\MN{}{The input impedance of the transformer is about $0.1~\\Omega$, which ensured a flat frequency response from several hundred Hz to several tens of kHz.} A Keithley 6221 low-noise current source was used to drive the heater. The piezo-electric actuators were driven at up to $\\pm400$ V using a bespoke high-voltage amplifier.\n\n\n\n\\subsection{Strain inhomogeneity}\n\nThe nature of our setup is that the strain profile along the direction of the application of the uniaxial pressure is not homogeneous.\nAs we will describe below, by adjusting the excitation frequency $f_{exc}$ to an appropriate value the actual heat-capacity measurement can be confined to the quasi-homogeneously strained region of the sample. Besides this source of strain inhomogeneity there are other sources which can be reduced in the preparation and mounting process of the sample in the apparatus.\n\n\n\\subsubsection*{Imperfections of sample surface and geometry}\n\nThe bar-shaped needles cut from crystals have typically terraces and irregular shapes on their surface which can induce inhomogeneous strain fields when they are under uniaxial pressure. Imperfections may also lead to an early failure of pressurized samples reducing the maximum achievable pressure.\nA perfect sample is a cuboid, i.e.\\ each surface is parallel to the opposite one, and has a smooth surface roughness. Therefore, we carefully polish our samples and inspect the shape and the surface quality under a microscope before mounting in the uniaxial pressure apparatus.\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{heater_mounting}\n\\centering\n\\caption{(a) Heater fixed with a silver foil to the sample. The contact to the sample is on the whole plane.\n(b) The simulation of the strain $\\varepsilon_{xx}$ pattern corresponding to the setup in (a). One of the silver-filled epoxy blocks was set to be invisible, as indicated by the\ndash-dotted lines, such that the strain profile on the edge of the sample is visible.\n(c) Heater fixed to the sample by four thin silver wires on the edges of the sample.\n(d) Corresponding simulation to (c). The strain inhomogeneity is reduced in the center compared with the setup shown in (a).\n }\n\\label{heater_mounting}\n\\end{figure}\n\n\\subsubsection*{Bending}\n\nAsymmetric mounting of a sample leads to bending.\\cite{Barber2017} An ideal sample mounting is a sample mounted between two plates with symmetrical epoxy layers on top and at the bottom. However, the sample might end up with a small offset in height. To reduce inhomogeneity in preparing the sample we aim for an aspect ratio $l_s\/t >10$,  where $l_s$ is the exposed length and $t$ the thickness of the sample\n\n\\subsubsection*{Mounting of the heater}\n\nOne of the main sources of inhomogeneous strain fields originates from the sample configuration in the AC heat-capacity setup. In order to transmit the heating power from the heater resistance to the sample, we use thermal contacts made by silver wires glued to the sample using silver-filled epoxy. Since the Young's modulus of silver and the sample are generally very different, as in our example of Sr$_2$RuO$_4$, the contacts create inhomogeneous strain fields. We tried to minimize this effect. We realized two different types of silver contacts to the sample. Figures \\ref{heater_mounting}a and \\ref{heater_mounting}c show photographs of the setups. In the first one a silver strip was glued on a contact length of about 300~$\\mu$m on both edges to the sample using silver-filled epoxy. In the second, the thermal contact is divided into 4 smaller areas instead of a large one, by gluing 8 silver wires with diameter of 50~$\\mu$m on both edges. The total contact area in both cases is almost the same.\nThe experiments on our test sample Sr$_2$RuO$_4$ showed indeed a significant sharpening of the superconducting transition anomaly in the latter case.\n\n\nIn addition to the experiments we simulated the strain fields in the sample by a finite element method using a commercial software package.\\footnote{Autodesk Inventor 2015, Autodesk Inc..}\nFor the simulation we set the Young's modulus of the sample to 180 GPa and the Poisson's ratio to 0.33. For the dimensions of the sample we used the values from the experiment, a thickness 100~$\\mu$m, width 300~$\\mu$m, and length 2~mm. One of the sample ends was set to be fixed and the other end was subjected to a pressure of 0.18~GPa, leading to $\\varepsilon_{xx}=0.1$\\%. The silver and silver-filled epoxy were set to have Poisson's ratio of 0.35. The Young's modulus for the silver epoxy was set to be $1\/3$ of that of the silver, which is 110~GPa. The results for both configurations are shown in Figs.\\ \\ref{heater_mounting}b and \\ref{heater_mounting}d. The color bar shows the strain scale, ranging from 0.07 to 0.13\\%. In the first configuration, with silver strip glued with silver-filled epoxy on both edges to the sample, the strain inhomogeneity on the sample is greater than 60\\% in the center (see Fig.\\ \\ref{heater_mounting}b).\nIn the second design using 8 silver wires with diameter of 50~$\\mu$m on both edges for the thermal contact, the strain inhomogeneity is strongly reduced in the bulk, except in the regions very close to the contact surfaces. Since heat capacity is a bulk-sensitive measurement, the inhomogeneity near the surface is negligible. The strain inhomogeneity in this configuration is only about $10$\\% in the center region of the sample. This shows that it is highly desirable to have separated smaller contact areas to transmit the heat to the sample in order to reduce strain inhomogeneities in accordance with the experimental results. In the following we continued with the second configuration.\n\n\n\n\n\\section{RESULTS}\n\nWe demonstrate the capabilities of our setup and discuss its advantages and limitations by showing representative data from experiments on Sr$_{2}$RuO$_{4}$. The first step in an AC heat-capacity experiment is to find a suitable measurement frequency in the plateau region of the frequency response curve $F(\\omega)$. We note that the existence of this plateau depends on the respective characteristics of the setup as discussed in Sec.\\ \\ref{AC Heat Capacity}. If a suitable frequency has been found, temperature \\MN{}{sweeps, also in applied magnetic field, or pressure\/magnetic field ramps} can be conducted and the heat capacity recorded. According to Eq.\\ \\ref{C_ac_prox} we will plot our results on Sr$_{2}$RuO$_{4}$ as $P\/[\\omega T_{AC}(T)]$. As we will discuss in Sec.\\ \\ref{Cv} $C_{AC}\\approx P\/(\\omega T_{AC})$ is not strictly valid in our setup and has to be treated with caution.\n\n\n\\subsection{Measuring frequency}\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{frequency_effect}\n\\centering\n\\caption{(a) Frequency sweeps at 1 and 4.23~K.\n(b) Data  recorded at 313~Hz for zero and small strains up to $\\varepsilon_{xx}=-0.19$\\%.\n(c) Data on  Sr$_2$RuO$_4$ in the region around its superconducting transition at $\\varepsilon_{xx}=-0.19$\\% for different frequencies.\n }\n\\label{frequency_effect}\n\\end{figure}\n\n\n\nFigure \\ref{frequency_effect}a shows the frequency response at 1 and 4.23~K in case of our example Sr$_2$RuO$_4$ crystal. It shows a broad plateau between a few hundred hertz and several kilohertz at both temperatures, attesting that in principle heat-capacity measurements should be possible in the desired temperature range. By raising the temperature from 1 to 4.23~K the plateau narrows slightly but remains well-defined.\n\nIn the lower frequency part of the plateau in Fig.\\ \\ref{frequency_effect}a, temperature oscillations extend throughout the sample and all three regions the homogenously strained in the center, the unstrained portions at the ends and the regions where strain builds up are probed in a measurement (see Fig.\\ \\ref{Scheme_thermalConductivity}b). Figure \\ref{frequency_effect}b shows $P\/[\\omega T_{AC}(T)]$ recorded at $f_{exc}=313$~Hz for different $\\varepsilon_{xx}$. At zero strain we see a single sharp transition anomaly at $T_c\\approx1.45$~K. Upon increasing $|\\varepsilon_{xx}|$ the step-like feature moves to higher temperatures, consistent with the increase in $T_c$ with strain,\\cite{Hicks2014b} but a second feature remains at the original zero-strain transition. This latter feature stems from the unstrained part of the sample.\n\nTo reduce the size of the probed part of the sample and restrict it to the homogenously strained region in the center, we increased the measurement frequency. We note that we still stay in the plateau region of frequency response curve. To demonstrate the importance of this increase in measurement frequency, we applied modest strain $\\varepsilon_{xx}=-0.19$~\\% and increased $f_{exc}$ from 313~Hz in steps to 2503~Hz. The data are displayed in Fig.\\ \\ref{frequency_effect}c.\nAt 313 and 613~Hz, in addition to the peak at $\\approx 1.65$~K corresponding to the transition in the central, strained, portion of the sample, a smaller peak is visible at $\\approx 1.45$~K, corresponding to the transition in the end portions.  This feature shows that temperature oscillations extend into the sample ends at these frequencies.\nTo avoid this, one has to work at the high end of the feasible range of frequencies. For this particular sample, a measurement frequency above $\\sim1.5$~kHz was required.\nWorking at high frequencies with low enough power to avoid heating gives a very small signal, an r.m.s.\\ thermocouple voltage of only $1 - 2$~nV. Therefore, the described low temperature passive amplification was employed to achieve an r.m.s.\\ noise level of 20~pVHz$^{-1\/2}$, ensuring a signal-to-noise ratio in excess of 50.\n\n\n\n\n\\subsection{Heat-capacity results on Sr$_2$RuO$_4$}\n\nBased on considerations outlined in the previous section we selected an excitation frequency of $f_{exc}=1503$~Hz to measure the heat capacity of Sr$_2$RuO$_4$. The results for three different strains $\\varepsilon_{xx}=0$\\%, $-0.25$\\%, and $-0.37$\\% are presented in Fig.\\ \\ref{HC_Sr2RuO4} as $P\/[\\omega T_{AC}(T)]$. Additionally the inset shows the results from a standard relaxation-type heat-capacity measurement from a piece of sample cut from the same crystal. It is qualitatively similar to the results in the uniaxial pressure cell at zero strain.\n\n\\begin{figure}[tb!]\n\\includegraphics[width=0.95\\linewidth]{HC_Sr2RuO4}\n\\centering\n\\caption{Recorded signal $P\/[\\omega T_{AC}(T)]$ of Sr$_2$RuO$_4$ as function of temperature for three different strains $\\varepsilon_{xx}=0$\\%, $-0.25$\\%, and $-0.37$\\%. The inset shows a specific-heat experiment on a piece from the same crystal using a standard relaxation time method.\n}\n\\label{HC_Sr2RuO4}\n\\end{figure}\n\nAccording to Eq.\\ \\ref{C_ac_prox} we find $C_{AC}(T)\\approx P\/[\\omega T_{AC}(T)]$. However, this relation has to be taken with caution since the probed sample volume is not constant as function of temperature. We have selected $f_{exc}$ in order to probe the homogenously strained portion of the sample, but we have to notice that the thermal conductivity $\\kappa$ of any studied material varies as function of temperature and strain, and as a consequence the probed sample volume also changes. To obtain absolute values of the volume specific heat $c_v(T)$ at a certain strain the temperature dependence of the thermal conductivity $\\kappa(T)$ has to be known at that strain too.\n\n\n\n\\subsection{Determination of the volume specific heat}\\label{Cv}\n\nThe conversion between the measured signal and the volume or molar specific heat is trivial in a conventional setup, because the volume (or mass) of the sample is constant. In our measurements, the probed sample volume varies since the thermal diffusion length $l_d$, which depends on thermal conductivity, specific heat, and frequency, changes as a function of temperature. Therefore, it is nontrivial to convert our data $P\/[\\omega T_{AC}(T)]$ to volume specific heat $c_v$. We start with an ideal case to demonstrate the relation between $P\/[\\omega T_{AC}(T)]$ and $c_v$ in case of our experimental setup.\nSuppose that the heater contact is point-like in the center of a very narrow sample such that the heat flow is one-dimensional propagating in the left and right direction. The probed volume $V$ is equal to the cross-sectional area $A$ times twice the diffusion length $l_d$, which is a function of the angular frequency $\\omega$, the volume specific heat $c_v$ and the thermal conductivity $\\kappa$.\n\\begin{equation}\n  l_d=\\sqrt{\\frac{2\\kappa(T)}{\\omega c_v(T)}}\t\n  \\label{S1}\n\\end{equation}\n$C_{AC}$ can be expressed as follows:\n\\begin{equation}\\label{S2}\n  C_{AC}=c_v \\times V = c_v \\times A\\times l_d = \\frac{2A}{\\sqrt{\\omega}} \\sqrt{2\\kappa(T) c_v(T)}.\n\\end{equation}\nBy using Eq.\\ \\ref{Cac} and \\ref{S2} we finally obtain the volume specific heat $c_v$:\n\\begin{equation}\\label{S3}\n  c_{v}(T)=\\left (\\frac{P \\times F(\\omega)}{2A}\\right )^2 \\times \\frac{1}{\\omega\\times 2\\kappa(T)} \\times \\frac{1}{[T_{AC}(T)]^2}.\n\\end{equation}\t\nThis exemplifies the reciprocal dependence of $c_v(T)$ on the thermal conductivity and the square of the temperature-oscillation amplitude in case of a simplified one dimensional model.\n\nWe further note that the excitation frequency in our current measurement is not too far away from the upper cut-off frequency, which describes the time scale for the heat propagating from the heater to the thermocouple. At this excitation frequency $F(\\omega)<1$ and depends on temperature adding a further uncertainty on the determination of $c_v(T)$.\n\n\nThe validity of the Eqs.\\ \\ref{S2} and \\ref{S3} is based on the above-mentioned assumptions that the heater contact is point-like and the heat flow is one-dimensional. In reality, both the sample width and the heater contact size are finite. This implies for the experimental setup to satisfy the assumptions of the examined model system, the exposed sample length ($l_s$) must be far longer than the heater length ($l_h$) and the sample width ($w_s$), $l_s \\gg l_h,w_s$. Our present setup is already a good approximation to an ideal configuration but could in principle be further optimized.\n\nIn spite of the above caveats, we note that in some cases quantitative statements on the evolution of the specific heat on varying uniaxial pressure are possible based on the presently accessible data. For example, in superconductors, as in the case of Sr$_2$RuO$_4$, it is possible to obtain information on the evolution of the size of the superconducting transition anomaly with pressure, which is an important quantity characterizing superconductivity. In that case the thermal conductivity does not show any abrupt change across the transition and close to $T_c$\n\\begin{equation}\\label{S4}\n  \\frac{c_{v}^s}{c_{v}^n}=\\frac{\\kappa_n}{\\kappa_s} \\times \\left(\\frac{T_{AC}^n}{T_{AC}^s}\\right)^2 \\approx \\left(\\frac{T_{AC}^n}{T_{AC}^s}\\right)^2\n\\end{equation}\t\nwith $\\kappa_n\\approx\\kappa_s$.\nThe indices $s$ and $n$ indicate the corresponding values in the superconducting and in the normal state, respectively.\n\n\n\\section{CONCLUSION}\n\nWe have developed a new experimental setup using piezoelectric-driven uniaxial pressure cells for probing heat capacity at low temperatures. By optimizing our preparation and measuring processes we achieve an extremely high resolution and a high strain homogeneity in the probed sample volume.\nThe technique can be easily extended to different temperature regions. In addition to temperature sweeps, heat capacity can be recorded as function of applied pressure, and our apparatus is also fully compatible with work in magnetic fields.\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank A.\\ S.\\ Gibbs, F.\\ Jerzembeck, N.\\ Kikugawa, Y.\\ Maeno, D.\\ A.\\ Sokolov for providing and characterizing the samples and M.\\ Brando and U.\\ Stockert for experimental support.\n\\end{acknowledgments}\n\n\\section*{DATA AVAILABILITY}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\\section*{REFERENCES}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe ubiquitous appearance of translation symmetry in physical systems signals the importance of having a complete picture of the complex role it may play. In particular, although the ground state energy (associated with time-translation symmetry) of a many-body quantum system or a quantum field theory is frequently studied, the ground state \\textit{momentum} (associated with space-translation symmetry) is rarely discussed. Rather, in most cases one focuses on the momentum difference between excited states and the ground state. In this work we reveal a connection between the momentum and the entanglement structure of a quantum state, in the context of lattice spin (boson) systems:\n\\begin{theorem}\nIf a quantum state $|\\Psi\\rangle$ in a lattice spin (boson) system is an eigenstate of the lattice translation operator $T:|\\Psi\\rangle\\to e^{iP}|\\Psi\\rangle$ with a non-trivial momentum $e^{iP}\\neq1$, then $|\\Psi\\rangle$ must be long-range entangled, namely $|\\Psi\\rangle$ cannot be transformed to an un-entangled product state $|000...\\rangle$ through an adiabatic evolution or a finite-depth quantum circuit (local unitary). \n\n\n\\label{Thm}\n\\end{theorem}\n\nThe intuition behind this statement follows from the sharp difference between translation $T$ and an ordinary onsite symmetry $G$ that is defined as a tensor product of operators acting on each lattice-site (such as the electromagnetic $U(1)$). A product state may recreate any total symmetry charge $Q$ under $G$ by simply assigning individual local Hilbert space states to carry some charge $Q_\\alpha$ such that $Q=\\sum_{\\alpha}Q_\\alpha$. However in the case of non-onsite translation symmetry, all translation-symmetric product states, which take the form $|\\alpha\\rangle^{\\otimes L}$, can only carry trivial charge (lattice momentum). This suggests that non-trivial momentum is an inherently non-local quantity that cannot be reproduced without faraway regions still retaining some entanglement knowledge of each other, i.e. the state must be long-range entangled.\n\nIn condensed matter physics, we are often interested in ground states of translational-invariant local Hamiltonians. If the ground state is short-range entangled~\\cite{PhysRevB.82.155138} (SRE) in the sense that it is connected to a product state through a finite-depth (FD) quantum circuit, then we expect the ground state to be unique, with a finite gap separating it from the excited states. In contrast for long-range entangled~\\cite{PhysRevB.82.155138} (LRE) ground states, we expect certain ``exotic'' features: possible options include spontaneous symmetry-breaking cat states (e.g. GHZ-like states), topological orders (e.g. fractional quantum Hall states), and gapless states (e.g. metallic or quantum critical states). Theorem~\\ref{Thm} provides us an opportunity to explore the interplay between translation symmetry and the above modern notions. An immediate corollary is\n\\begin{corollary}\nIf a non-zero momentum state $|\\Psi\\rangle$ is realized as a ground state of a local spin Hamiltonian, then the ground state cannot be simultaneously unique and gapped. Possible options include (1) gapless spectrum, (2) intrinsic topological order and (3)  spontaneous translation symmetry breaking.\n\\end{corollary}\nIn fact, we show in Sec.~\\ref{sec:wCDW} that option (2) is a special subset of option (3) through the mechanism of ``weak symmetry-breaking\"~\\cite{Kitaev06}.\n\nOur result is reminiscent of the celebrated Lieb-Schultz-Mattis-Oshikawa-Hastings (LSMOH) theorems \\cite{LSM,Oshikawa1999,Hastings04}, which state that in systems with charge $U(1)$ and translation symmetries, a ground state with fractional $U(1)$ charge filling (per unit cell) cannot be SRE. In our case the non-trivial lattice momentum $e^{iP}\\neq1$ plays a very similar role as the fractional charge density in LSMOH. In fact, as we discuss in Sec.~\\ref{sec: LSMOH}, our theorem can be viewed as a more basic version of LSMOH that only involves translation symmetry, from which the standard LSMOH can be easily derived. As a by-product, we also discover a previously unknown version of LSMOH constraint that involves an onsite $\\mathbb{Z}_n$ symmetry and lattice translations. \n\nThe rest of this paper will be structured as follows: in Sec.~\\ref{sec:proof} we provide a proof of Theorem~\\ref{Thm} via a quantum circuit approach, and generalize it to fermion systems. Three consequences of Theorem~\\ref{Thm} are discussed in Sec.~\\ref{sec:consequences}: in Sec.~\\ref{sec: LSMOH} we discuss several LSMOH-type theorems; in Sec.~\\ref{sec:wCDW} we show that a gapped topological order must \\textit{weakly} break translation symmetry if one of its ground states on torus has nonzero momentum -- this is a generalization of the Tao-Thouless physics in fractional quantum Hall effect~\\cite{PhysRevB.28.1142,PhysRevB.77.155308}; in Sec.~\\ref{sec:cSPT} we discuss the implication of Theorem~\\ref{Thm} for the classification of crystalline symmetry-protected topological (SPT) phases. We end with some discussions in Sec.~\\ref{sec:discussions}.\n\n\n\\section{Proof}\n\\label{sec:proof}\n\nIn this section we prove that SRE states necessarily possess trivial momentum, conversely implying that all non-trivial momentum ground states must be LRE. The approach that we take utilizes the quantum circuit formalism, which is equivalent to the usual adiabatic Hamiltonian evolution formulation~\\cite{doi:10.1126\/science.273.5278.1073,PhysRevB.82.155138} but conceptually cleaner. In particular we will harness the causal structure of quantum circuits, which will allow us to `cut and paste' existing circuits to create useful new ones.\n\nWe shall first prove Theorem~\\ref{Thm} in one space dimension, from which the higher-dimensional version follows immediately.\n\n\\subsection{Proof in $1$d}\n\\label{sec:1dproof}\n\nFirst let us specify our setup more carefully. We consider a spin (boson) system with a local tensor product Hilbert space $\\mathcal{H}=\\otimes_i\\mathcal{H}_i$ where $\\mathcal{H}_i$ is the local Hilbert space at unit cell $i$. The system is put on a periodic ring with $L$ unit cells so $i\\in\\{1,2...L\\}$. In each unit cell the Hilbert space $\\mathcal{H}_i$ is $q$-dimensional ($q$ does not depend on $i$), with a basis labeled by $\\{|a_i\\rangle_i\\}$ ($a_i\\in\\{0,1...q-1\\}$). The translation symmetry is implemented by a unitary operator that is uniquely defined through its action on the tensor product basis \n\\begin{eqnarray}\n\\label{eq:Tboson}\n    T:\\hspace{5pt} & & |a_1\\rangle_1\\otimes|a_2\\rangle_2\\otimes...|a_{L-1}\\rangle_{L-1}\\otimes|a_L\\rangle_L \\nonumber \\\\ & &\\longrightarrow |a_L\\rangle_1\\otimes|a_1\\rangle_2\\otimes...|a_{L-2}\\rangle_{L-1}\\otimes|a_{L-1}\\rangle_L.\n\\end{eqnarray}\nUnder this definition of translation symmetry (which is the usual definition), we have\\footnote{Importantly, we are not dealing with translation under twisted boundary condition, in which case $T^L=g$ for some global symmetry $g$. Many of our conclusions in this work need to be rephrased or reexamined for such twisted translations.} $T^L=1$ and any translational-symmetric product state $|\\varphi\\rangle^{\\otimes L}$ has trivial lattice momentum $e^{iP}=1$.\n\nNow consider a SRE state $|\\Psi_{P(L)}\\rangle$ with momentum $P(L)$. By SRE we mean that there is a quantum circuit $U$ with depth $\\xi\\ll L$ that sends $|\\Psi_{P(L)}\\rangle$ to the product state $|\\mathbf{0}\\rangle\\equiv |0\\rangle^{\\otimes L}$ (we do not assume $U$ to commute with translation). The depth $\\xi$ will be roughly the correlation length of $|\\Psi_{P(L)}\\rangle$. Our task is to prove that $P(L)=0$ mod $2\\pi$ as long as $\\xi\\ll L$. Notice that this statement is in fact stronger than that for FD circuit which requires $\\xi\\sim O(1)$ as $L\\to \\infty$. For example, our result holds even if $\\xi\\sim {\\rm PolyLog}(L)$, which is relevant if we want the quantum circuit to simulate an adiabatic evolution more accurately~\\cite{Haah2018}. Our result is also applicable if the existence of $U$ requires extra ancilla degrees of freedom (DOF) that enlarges the onsite Hilbert space to $\\tilde{\\mathcal{H}}_i$ with dimension $\\tilde{q}>q$ (for example see Ref.~\\cite{ElsePoWatanabe2019}), since ancilla DOFs by definition come in product states and therefore cannot change the momentum.\n\nThe proof will be split into two steps where in \\textit{Step 1} we first prove that the momentum is trivial for all $L=mn$ where $m,n\\in\\mathbb{Z}^+$ are mutually coprime satisfying $m,n\\gg\\xi$. In \\textit{Step 2} we use the results of \\textit{Step 1} to show that this may be extended to all other lengths.\n\n\n\n\\underline{\\textit{Step 1:}} A key ingredient of the proof is to recognize that the entanglement structure of the SRE state $|\\Psi_{P(L)}\\rangle$ on system size $L=mn$, where $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$, is adiabatically connected to that of $m$ identical unentangled length $n$ SRE systems. The existence of such an adiabatic deformation, which is of a similar flavour to those presented in Refs.~\\cite{PhysRevX.7.011020} and \\cite{PhysRevB.96.205106}, is due to the finite correlation length of SRE systems, and will be explained in the following paragraph.\n\n\n\nTake the SRE state $|\\Psi_{P(L)}\\rangle$ placed on a periodic chain of length $L=mn$ with $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$. Let us try to decouple this system at some point (say between site $i$ and $i+1$) via an adiabatic evolution, creating an `open' chain.\nTo show that such a decoupling cut exists, we use the fact that SRE states always have a FD quantum circuit $U$ that sends the ground state to the $|\\mathbf{0}\\rangle\\equiv |0\\rangle^{\\otimes L}$ product state (see Fig.~\\ref{fig:unitaries}(a)). The appropriate cut is then created by modifying this circuit to form a new lightcone-like FD quantum circuit $\\tilde{U}$ with all unitaries outside the `lightcone', i.e. those that do not affect the transformation that sends the two sites $i$ and $i+1$ to $|0\\rangle$, set to identity (see Fig.~\\ref{fig:unitaries}(b)). Such a modified circuit would span $\\sim\\xi$ qudits on either side of the cut and by construction takes the two sites on either side of the cut to $|0\\rangle$, thus completely removing any entanglements across the link\\footnote{This can be better understood in reverse: consider the state constructed by $\\tilde{U} U^\\dag|\\mathbf{0}\\rangle$ ($=\\tilde{U}|\\Psi_{P(L)}\\rangle$) which never directly couples qudits on either side of the cut. Thus $\\tilde{U}$ can be understood as completely removing the entanglement across the applied link.}.\nLet us concretely take $\\tilde{U}^{[0]}$ to denote the appropriate lightcone cut between the last and first qudit (recall that we are on a ring), and define the shifted adiabatic cut between the $x-1$ and $x$th qudits to be $\\tilde{U}^{[x]}\\equiv T^x \\tilde{U}^{[0]}T^{-x}$. If the ground state is translation-symmetric we have $\\tilde{U}^{[x]}|\\Psi_{P(L)}\\rangle=e^{-i x P(L)}T^x\\tilde{U}^{[0]}|\\Psi_{P(L)}\\rangle$\nso we see that $\\tilde{U}^{[x]}$ performs the same cut (up to a phase factor) at any link. By construction this means that the local density matrices of a region surrounding the cut obeys $\\rho_{lr}=\\rho_l\\otimes|00\\rangle\\langle00|\\otimes\\rho_r$, where the left (right) region to the cut is denoted $l$ ($r$), which in turn implies that the operations $\\tilde{U}^{[x]}$ disentangles the system along that cut.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.98\\columnwidth]{unitaries.pdf}\n    \\caption{\\label{fig:unitaries} (Color online) Depiction of finite-depth quantum circuits applied on $|\\Psi_P\\rangle$. Here qudits are depicted as solid circles while unitaries are depicted as rectangles. (a) A SRE state $|\\Psi_P\\rangle$ is always connected to the $|\\mathbf{0}\\rangle$ trivial state via a FD quantum circuit $U$. From $U$ a lightcone-like `adiabatic cut' $\\tilde{U}$ can be created (framed in blue). (b) $\\tilde{U}$ connects $|\\Psi_P\\rangle$ to a state that is completely decoupled across the cut.}\n\\end{figure}\n\nThe cutting procedure may be simultaneously applied to two separate links, as long as they are separated by a distance much greater than the correlation length. With this in mind, let us identically apply the cut on an $L=mn$ length system with a cut after every $n$th qudit, as depicted in Fig.~\\ref{fig:adiabaticcutting}, via the FD quantum circuit $\\tilde{U}^{[0]}\\tilde{U}^{[n]}...\\tilde{U}^{[(m-1)n]}$. Since the adiabatic deformation fully disentangles the system across the cuts, the resulting state should take the form $|\\tilde{\\Psi}_1\\rangle\\otimes|\\tilde{\\Psi}_2\\rangle\\otimes...|\\tilde{\\Psi}_m\\rangle$ where each $|\\tilde{\\Psi}_i\\rangle$ is an $n$-block SRE state. \n\nNow let us examine the symmetries of this resultant system. The original $\\mathbb{Z}_{mn}$ translation symmetry, generated by operator $T$, of the original system is broken by the adiabatic deformation. However the $\\mathbb{Z}_m$ translation symmetry subgroup, generated by operator $T^n$, is preserved since by construction identical cuts occurs at every $n$th junction. This immediately implies that all the $n$-block states are identical $|\\tilde{\\Psi}_i\\rangle=|\\tilde{\\Psi}\\rangle$ and the total state after the cut is simply $|\\tilde{\\Psi}\\rangle^{\\otimes m}$.\nThus we know that the original $\\mathbb{Z}_m$ quantum number is the same as the final one which must be trivial since we are dealing with an $n$-block product state $|\\tilde{\\Psi}\\rangle^{\\otimes m}$. This implies\n\\begin{align}\nnP(L)=0\\mod 2\\pi\\quad,\n\\label{eq:LP}\n\\end{align}\n$\\forall L=mn$ with $m,n\\in\\mathbb{Z}^+$ and $n\\gg\\xi$.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=0.95\\columnwidth]{cutting_chains.pdf}\n    \\caption{\\label{fig:adiabaticcutting} (Color online) Illustration of the adiabatic cutting procedure on a periodic length $L=mn$ chain. Here we take $m=4$ example to demonstrate how four identical cuts, applied by $\\tilde{U}$ (blue rectangle) at every $n$th link, on a length $L=4n$ state $|\\Psi_{P(L)}\\rangle$ (purple circle) produces four decoupled length $n$ SRE states.}\n\\end{figure}\n\nUsing this relation on a general system length $L=p_1^{q_1}p_2^{q_2}...p_d^{q_d}$ (here we are using prime factorisation notation) we arrive at the condition\n\\begin{align}\nP(L)=0\\mod \\frac{2\\pi}{p_1^{r_1}p_2^{r_2}...p_d^{r_d}}\\quad,\n\\end{align}\n$\\forall{r_i\\in\\{1,...,q_i\\}}$ such that $p_1^{r_1}p_2^{r_2}...p_d^{r_d}\\gg\\xi$. If $L$ factorises into at least two mutually coprime numbers $m,n$ with $m,n\\gg\\xi$ then these conditions can only be satisfied if \n\\begin{align}\nP(L)=0\\mod 2\\pi\\quad,\n\\label{eq:zeroP}\n\\end{align}\nwhich is satisfied for almost all large enough $L$.\n\n\\underline{\\textit{Step 2:}} There are a sparse set of cases for which \\textit{Step 1} does not enforce trivial momentum, the most notable case being when $\\tilde{L}=p^{q}$ with $p$ prime and $q\\in\\mathbb{Z}^+$. Factorisations such as $\\tilde{L}=p_1^{q_1}p_2$ are also not covered if $p_1^{q_1}\\not\\gg \\xi$. \n\nTo show that these cases also possess trivial momentum, once again take a SRE state $|\\Psi_{P(L)}\\rangle$ on a general length $L$ system with momentum $P(L)\\mod 2\\pi$. By the definition of a SRE state, there exists a FD quantum circuit $V_{L}$ such that $|\\Psi_{P(L)}\\rangle=V_L|\\mathbf{0}\\rangle$. This circuit obeys $TV_LT^\\dag|\\mathbf{0}\\rangle=e^{iP(L)}V_L|\\mathbf{0}\\rangle$, meaning that it boosts the trivial momentum of the $|\\mathbf{0}\\rangle$ state by $P(L)\\mod 2\\pi$. Consider the composition of a circuit\n\\begin{align}\n    \\left(TV_L^\\dag T^\\dag\\right) V_L|\\mathbf{0}\\rangle=e^{-iP(L)}|\\mathbf{0}\\rangle\\quad.\n    \\label{eq:boostL}\n\\end{align}\nAs may be understood via the causality structure the phase $e^{-iP(L)}$ will come piecewise from lightcone circuits. Let us understand this in detail: split $\\tilde{V}_L\\equiv TV_L^\\dag T^\\dag V_L$ into a light-cone circuit $\\tilde{V}_{L,1}$ and reverse lightcone circuit $\\tilde{V}_{L,2}$ such that $\\tilde{V}_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$, as depicted in Fig.~\\ref{fig:UgeneralL}. The causal structure of the light cone guarantees that a gate $U_1$ in $\\tilde{V}_{L,1}$ and a gate $U_2$ in $\\tilde{V}_{L,1}$ must commute if $U_2$ appears in a layer after $U_1$, which then allows for the decomposition $\\tilde{V}_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$.\nAlthough the exact form of this decomposition is quite malleable, for concreteness let us define $\\tilde{V}_{L,1}$ to be constructed causally such that the 1st (lowest) layer consists of a single 2-qudit gate (as seen in Fig.~\\ref{fig:UgeneralL}).\n$\\tilde{V}_{L,1}$ will have support over qudits in the range $[L-\\eta,L]$, where by the SRE nature $\\eta\\ll L$. Due to Eq.~\\ref{eq:boostL} we see that\n\\begin{align}\n    \\tilde{V}_{L,2} |\\mathbf{0}\\rangle=|0...0\\rangle^{[1,L-\\eta-1]}\\otimes|\\alpha\\rangle^{[L-\\eta,L]}\n\\end{align}\nfor some $|\\alpha\\rangle$. By construction, we have\n\\begin{align}\n    \\tilde{V}_{L,1} |\\alpha\\rangle=e^{-iP(L)}|0...0\\rangle^{[L-\\eta,L]}\\quad,\n    \\label{eq:circ1tildeU}\n\\end{align}\nsuch that we satisfy Eq.~\\ref{eq:boostL}.\n\n\\begin{figure}[t]\n    \\centering\n        \\includegraphics[width=\\columnwidth]{UgeneralLf.pdf}\n    \\caption{\\label{fig:UgeneralL} (Color online) Illustration of splitting $TV_L^\\dag T^\\dag V_L=\\tilde{V}_{L,1}\\tilde{V}_{L,2}$ with $\\tilde{V}_{L,1}\\tilde{V}_{L,2}|\\mathbf{0}\\rangle=e^{-iP(L)}|\\mathbf{0}\\rangle$. Here we have taken a snapshot of the circuit to focus on $\\tilde{V}_{L,1}$ (framed in blue), however the support of $\\tilde{V}_{L,1}$ (in the depicted example 16 qudits) is actually much smaller than the system length. Recall that the circuit is periodic such that the orange arrows, corresponding to components of $\\tilde{V}_{L,2}$ (framed in orange), eventually connect on the far side of the ring.\n    }\n\\end{figure}\n\nNow we will extend the circuit $V_L$ from length $L$ to $nL$ for some $n\\in\\mathbb{Z}^+$, where $n,L$ are coprime and $\\gg\\xi$, and denote this extended circuit $V_{nL}$. To do this we simply unstitch the circuit $V_L$ at some link and reconnect the ends of $n$ consecutive copies of this unstitched $V_L$ circuit to create a FD quantum circuit $V_{nL}$. Let us see what happens to $\\tilde{V}_{nL}\\equiv TV_{nL}^\\dag T^\\dag V_{nL}$ by once again splitting the circuit into two $\\tilde{V}_{nL}=\\tilde{V}_{nL,1}\\tilde{V}_{nL,2}$, where $\\tilde{V}_{nL,k}=\\prod_{j=0}^{n-1} T^{jL}\\tilde{V}_{L,k}(T^\\dag)^{jL}$ with $k\\in\\{1,2\\}$. By construction and due to the SRE nature of state construction\n\\begin{align}\n    \\tilde{V}_{nL,2} |\\mathbf{0}\\rangle^{\\otimes n}=\\left(|0...0\\rangle^{[1,L-\\eta-1]}\\otimes|\\alpha\\rangle^{[L-\\eta,L]}\\right)^{\\otimes n}\\quad.\n\\end{align}\nHowever, by Eq.~\\ref{eq:circ1tildeU}, we have\n\\begin{align}\n    \\tilde{V}_{nL,1} \\tilde{V}_{nL,2} |\\mathbf{0}\\rangle^{\\otimes n}=e^{-inP(L)}|\\mathbf{0}\\rangle^{\\otimes n}\\quad,\n    \\label{eq:circ1tildeUnL}\n\\end{align}\nso this implies\n\\begin{align}\n    TV_{nL} T^\\dag|\\mathbf{0}\\rangle^{\\otimes n} =e^{inP(L)}V_{nL}|\\mathbf{0}\\rangle^{\\otimes n}\\quad,\n    \\label{eq:boostnL}\n\\end{align}\nwhich means that $V_{nL}$ boosts the momentum of $|\\mathbf{0}\\rangle$ on a length $nL$ system to a state with momentum $P(nL)=nP(L)\\mod 2\\pi$. In \\textit{Step 1} we showed that $P(nL)=0\\mod 2\\pi$, so this implies $nP(L)=0\\mod2\\pi$. Since this holds for two mutually coprime values of $n$, one concludes that $1$d SRE translation-symmetric states have $P(L)=0\\mod2\\pi$ for all $L\\gg\\xi$.\n\n\n\n\n\n\\subsection{Higher-dimensional extension}\n\nOur result can be extended to higher dimensions. Consider a $d$-dimensional lattice system and a state $|\\Psi\\rangle$ that has nontrivial momentum $P$ along, say, $\\hat{x}$ direction. We can view the state as a 1d state along the $\\hat{x}$ axis, with an enlarged Hilbert space per unit cell (generally exponentially large in $\\prod_i L_i$ with $i$ denoting the transverse directions). A finite-depth quantum circuit of the $d$-dimensional system will also be a finite-depth quantum circuit when viewed as a $1$d circuit along the $\\hat{x}$-direction (a proof and a somewhat subtle example are presented in Appendix~\\ref{app:higherdSRE}; the converse is not true but that does not concern us here). This immediately implies that a SRE state on the $d$-dimensional system must also be SRE when viewed as a $1$d state along $\\hat{x}$. What we proved in Sec.~\\ref{sec:1dproof} thus implies that the non-trivial momentum state $|\\Psi\\rangle$ must be long-range entangled. In particular, imposing locality in the transverse directions will only further restrict possible FD circuit, and will certainly not lead to possibilities beyond the $1$d proof. This completes the proof of Theorem~\\ref{Thm}.\\hfill$\\blacksquare$\n\n\n\\subsection{Fermion systems}\n\\label{sec:Fermions}\n\n\n\nIt is not difficult to generalize our Theorem~\\ref{Thm} to fermionic system. The only subtlety is that the usual definition of translation symmetry in fermion systems has an extra $\\mathbb{Z}_2$ sign structure compared to the naive implementation in Eq.~\\ref{eq:Tboson}. Instead of specifying the sign structure in the tensor product basis as in Eq.~\\ref{eq:Tboson}, it is more convenient to define translation operator through $Tc_{i,\\alpha}T^{-1}=c_{i+1,\\alpha}$ where $c_{i,\\alpha}$ is a fermion operator in unit cell $i$ with some internal index $\\alpha$, and $c_{L+1,\\alpha}=c_{1,\\alpha}$. This operator relation, together with $T|\\mathbf{0}\\rangle=|\\mathbf{0}\\rangle$ for the fermion vacuum, uniquely determines the action of $T$ on any state. Now consider a product state $|\\varphi\\rangle^{\\otimes L}$, it is easy to verify that the momentum is $e^{iP}=1$ for odd $L$ and $e^{iP}=\\pm1$ for even $L$, where the sign is the fermion parity on each site $\\langle\\varphi|(-1)^{\\sum_\\alpha c^{\\dagger}_{\\alpha}c_{\\alpha}}|\\varphi\\rangle$. We can then go through the proof in Sec.~\\ref{sec:proof}, but now with fermion parity preserving FD quantum circuits, and conclude the following:\n\\begin{theorem}\nAny short-range entangled translation eigenstate $|\\Psi\\rangle$ in a lattice fermion system must have momentum (say in the $x$-direction) $e^{iP_x}=1$ if $L_x$ is odd, and $e^{iP_x}=\\pm1$ if $L_x$ is even. States violating this condition must in turn be long-range entangled.\n\\label{FermionThm}\n\\end{theorem}\nThe details of the proof are presented in Appendix~\\ref{app:fermionproof}.\n\nUsing the same proof technique, we can extend the above result further in various directions. We mention two such extensions without going into the details: (1) for $L_x$ even, the option of $e^{iP_x}=-1$ is possible for a SRE state only if $V\/L_x$ is odd ($V=L_xL_y...$ being the volume); (2) if the total fermion parity is odd in a system with even $V$, then any translation eigenstate must be LRE.\n\n\n\\section{Consequences}\n\\label{sec:consequences}\n\nOne of the beauties of Theorem~\\ref{Thm} lies in the non-trivial consequences that easily follow. For this section, it is useful to introduce an alternative, but equivalent, formulation of Theorem~\\ref{Thm}\n\\begin{manualtheorem}{1}[Equivalent]\nIf there exists a finite-depth local unitary that boosts a state's momentum to a different value (mod $2\\pi$), then the state is necessarily long-range entangled.\n\\end{manualtheorem}\nThe equivalence of this new formulation with the one introduced in Sec.~\\ref{sec:intro} can be understood as follows: if all translation-symmetric SRE states possess trivial momentum then non-trivial momentum states must be LRE. Thus if there exists a finite-depth local unitary that can boost a state's momentum to a different value then at least one of either the original or final state possesses non-trivial momentum and must be LRE. The other state is connected to the LRE state via a finite-depth local unitary and thus must also be LRE. The converse follows by contradiction: assume there exists a SRE state that has non-trivial momentum. Such a state (by definition of SRE) is connected via a FD local unitary to the translation-symmetric direct-product state $|\\alpha\\rangle^{\\otimes L}$ which in turn has trivial momentum. Since there now exists a FD local unitary that boosts the momentum to a different value, this implies that the original state was LRE which leads to the contradiction.\n\nThis equivalent formulation allows for a direct test for long-range entanglement that we will demonstrate on known and previously unknown LSM theories, and topological orders. In the following discussions we will mostly focus on spin (boson) systems for simplicity, but the results can be generalized quite readily to fermion systems as well.\n\n\n\\subsection{LSMOH constraints}\n\\label{sec: LSMOH}\n\n\nThe original Lieb-Schultz-Mattis (LSM) theory~\\cite{LSM} along with the extensions by Oshikawa~\\cite{Oshikawa1999} and Hastings~\\cite{Hastings04}, collectively referred to as LSMOH, and their descendants are powerful tools for understanding the low-energy nature of lattice systems. In one of its most potent forms the theorem states that systems with $U(1)$ and translational symmetry that have non-commensurate $U(1)$ charge filling must be `exotic', meaning that they cannot be SRE states. Since the conception of the original LSM theory the field has flourished rapidly with many extensions that impose similar simple constraints based on symmetry~\\cite{Nachtergaele2007,Watanabe14551,Po2017,LU2020168060,lu2017liebschultzmattis,PhysRevB.98.125120,PhysRevB.99.075143}, and connections to various fields of physics such as symmetry-protected topological (SPT) order and t'Hooft anomaly in quantum field theory~\\cite{Cheng2015,Jian2017,Cho2017,Metlitski2017,PhysRevB.101.224437,Ye2021}. These sort of constraints also have immediate experimental consequences, as they provide general constraints in determining candidate materials of exotic states such as quantum spin liquids~\\cite{Balents2010}. Thus, unsurprisingly, there has been a lot of interest in generating more LSMOH-like theorems that provide simple rules to find exotic states. In the following section we provide simple proofs of some known and previously unknown LSMOH theorems.\n\n\n\nThe first example we consider is the aforementioned non-commensurate 1d $U(1)\\times T$ LSM ($T$ denotes the translation symmetry). In this case there exists a local unitary momentum boost that is the large gauge transformation $U=e^{i\\frac{2\\pi }{L}\\sum x\\hat{n}_x}$, where $\\hat{n}_x$ is the local number operator at $x$. Notice that this transformation is an on-site phase transformation and thus a FD quantum circuit of depth 1. The commutation relation with translation is $TUT^\\dag=e^{i 2\\pi \\frac{\\hat{N}}{L}}U$ ($\\hat{N}$ being the total charge) which means that for non-commensurate filling $\\frac{\\hat{N}}{L}\\notin\\mathbb{Z}$, we may always boost the momentum by a non-trivial value $2\\pi \\frac{\\hat{N}}{L}$. Via the equivalent formulation of Theorem~\\ref{Thm}, this immediately implies that non-commensurate filling leads to a LRE state. This observation may be summarised as\n\\begin{corollary}\n{\\normalfont($U(1)\\times T$ LSM) A $1$d translation and $U(1)$ symmetric state that possesses non-commensurate $U(1)$ charge filling must be long-range entangled.}\n\\end{corollary}\nThe standard LSM theorem follows from this statement since we may now apply it to a \\textit{ground} state of a $1$d translation and $U(1)$ symmetric local spin Hamiltonian to show that the state must be either gapless or a spontaneously symmetry-broken cat state.\nNotice that, strictly speaking, the statement we proved differs slightly from the standard LSM theorem, in that we did not directly prove the vanishing of the energy gap. Rather we showed that any simultaneous eigenstate of translation and $\\hat{N}$ such that $\\langle\\hat{N}\\rangle\/L\\notin\\mathbb{Z}$ must be LRE. In principle we do not even need to assume the parent Hamiltonian to be translation or $U(1)$ symmetric, just that the state itself be translation and $U(1)$ symmetric. In fact the statement encompasses all states, not just the ground state, which is perhaps unsurprising since LRE is fundamentally a property of a state and not the Hamiltonian.\n\nThe higher-dimensional $U(1)\\times T$ LSMOH theorem may be proved following the same logic if $\\langle\\hat{N}\\rangle\/L_i\\notin\\mathbb{Z}$ for some direction $i$ (similar to what was done in Ref.~\\cite{Oshikawa1999}). For generic values of $L_i$ the above condition may not hold, and more elaborate arguments are needed (for example see Ref.~\\cite{YaoOshikawa2020}) which we will not discuss here. \n\nOur proof of the LSM theorem has an appealing feature compared to the classic proof~\\cite{LSM}: we did not need to show that the state $|\\Omega'\\rangle=U|\\Omega\\rangle$ had excitation energy $\\sim O(1\/L)$ (relative to the ground state $|\\Omega\\rangle$); rather it suffices for us to show that $|\\Omega'\\rangle$ has a different lattice momentum compared to $|\\Omega\\rangle$, which is enough to establish the LRE nature of $|\\Omega\\rangle$. Next we shall use this simplifying feature to generalize the $U(1)\\times T$ LSM theorem to a new constraint involving only discrete $\\mathbb{Z}_n\\times T$ symmetries.\n\n\n\nLet us consider a spin chain ($1$d) with translation symmetry and an onsite $\\mathbb{Z}_n$ symmetry generated by $Z\\equiv\\otimes_iZ_i$ ($Z_i^n=1$). We consider the case when the system size $L=nM$ for some $M\\in\\mathbb{N}$, and study simultaneous eigenstates of the translation and $\\mathbb{Z}_n$ symmetries. If such a state is an unentangled product state $\\otimes_i|\\varphi\\rangle_i$, then by definition $Z=1$ when acting on this state, namely the state carries trivial $\\mathbb{Z}_n$ charge. This turns out to be true for any symmetric SRE state, which we now prove. Suppose a translation eigenstate $|\\Psi\\rangle$ has $Z|\\Psi_P\\rangle=e^{i2\\pi Q\/n}|\\Psi\\rangle$ for some $Q\\neq0$ (mod $n$). We can construct a local unitary which is an $\\mathbb{Z}_n$-analogue of the large gauge transform\n\\begin{equation}\nU=\\otimes_iZ_i^i,\n\\end{equation}\nwhere $i$ is the unit cell index. For system size $L=nM$ one can verify that $TUT^{-1}U^{\\dagger}=Z^{\\dagger}$. This means that the momentum of the twisted state $U|\\Psi\\rangle$ will differ from that of the untwisted $|\\Psi\\rangle$ by $\\langle\\Psi|Z^{\\dagger}|\\Psi\\rangle=e^{-i2\\pi Q\/n}\\neq1$. By the equivalent form of Theorem~\\ref{Thm} $|\\Psi\\rangle$ must be LRE. We therefore have\n\\begin{corollary}\n\\label{ZnLSM}\n{\\normalfont($\\mathbb{Z}_n\\times T$ LSM)} A $1$d translation and $\\mathbb{Z}_n$ symmetric ground state that possesses non-trivial $\\mathbb{Z}_n$ charge on system lengths $L=nM$ for some $M\\in\\mathbb{N}$ cannot be short-range entangled, and thus is either gapless or spontaneously symmetry-broken cat state.\n\\end{corollary}\n\nThe above statement also generalizes to higher dimensions if $L_i=nM$ for some direction $i$. For systems with $U(1)$ symmetry, we can choose to consider a $Z_L$ subgroup of the $U(1)$, and the above $\\mathbb{Z}_n\\times T$ LSM theorem leads to the familiar $U(1)\\times T$ LSMOH theorem.\n\n\nThe two LSM-type theorems discussed so far, together with our Theorem~\\ref{Thm}, can all be viewed as ``filling-type\" LSM theorems, in the sense that these theorems constraint a symmetric many-body state $|\\Psi\\rangle$ to be LRE when $|\\Psi\\rangle$ carries certain non-trivial quantum numbers, such as lattice momentum $e^{iP}\\neq1$, total $U(1)$ charge $Q\\notin L\\mathbb{Z}$ or total $\\mathbb{Z}_n$ charge $Q\\notin L\\mathbb{Z}\/n\\mathbb{Z}$.\n\nThere is another type of LSM theorems that involve projective symmetry representations in the onsite Hilbert space, the most familiar example being the spin-$1\/2$ chain with $SO(3)$ symmetry. Our Theorem~\\ref{Thm} can also be used to understand some (but possibly not all) of the projective symmetry types of LSM. Here we discuss one illuminating example with onsite $\\mathbb{Z}_2\\times\\mathbb{Z}_2$ symmetry in one dimension~\\cite{PhysRevB.83.035107,Ogata2019,Ogata2021}, such that the generators of the two $\\mathbb{Z}_2$ group anti-commutes when acting on the local Hilbert space: $X_iZ_i=-Z_iX_i$ (this can simply be represented by the Pauli matrices $\\sigma_x$, $\\sigma_z$). Now set the length $L=2N$ with odd $N$, and consider the three local unitaries $U_x=(\\mathds{1}\\otimes\\sigma_x)^{\\otimes N}$, $U_z=(\\mathds{1}\\otimes\\sigma_z)^{\\otimes N}$, and $U_{xz}=(\\sigma_x\\otimes\\sigma_z)^{\\otimes N}$. One can verify the commutation relations $TU_{x}T^\\dag=(-1)^{Q_x}U_x$, $TU_{z}T^\\dag=(-1)^{Q_z}U_z$, and $TU_{xz}T^\\dag=(-1)^{N+Q_x+Q_z} U_{xz}$. These commutation relations imply that the momentum of any symmetric state $|\\Psi\\rangle$ will be boosted by $\\Delta P=\\pi$ by at least one of the three unitaries, therefore $|\\Psi\\rangle$ must be LRE by Theorem~\\ref{Thm}.\n\n\n\n\n\\subsection{Topological orders: weak CDW}\n\\label{sec:wCDW}\n\n\nWe now consider an intrinsic (bosonic) topological order on a $d$-dimensional torus. By definition there will be multiple degenerate ground states, separated from the excitation continuum by a finite energy gap. If one of the ground states $|\\Psi_a\\rangle$ has a non-trivial momentum, say along the $\\hat{x}$ direction, then according to Theorem 1 this state should be LRE even when viewed as a one-dimensional system in $\\hat{x}$ direction (with the other dimensions $y,z...$ viewed as internal indices). Since there is no intrinsic topological order in one dimension, the only mechanism for the LRE ground state is spontaneous symmetry breaking. The lattice translation symmetry is the only relevant symmetry here -- all the other symmetries can be explicitly broken without affecting the LRE nature of $|\\Psi_a\\rangle$, since the state will still have nontrivial momentum. Therefore $|\\Psi_a\\rangle$ must be a cat state that spontaneously breaks the $\\hat{x}$-translation symmetry~\\footnote{Another way to see this is to note that a cat state is composed of individual SRE states. Since we have proven that translation symmetric SRE states possess trivial momentum, it follows that the cat state may only achieve non-trivial momentum when the individual SRE states break translation symmetry, i.e. the cat state must correspond to translation symmetry breaking.}, also known as a charge density wave (CDW) state~\\cite{RevModPhys.60.1129}. Furthermore, any other ground state $|\\Psi_{b\\neq a}\\rangle$ can be obtained from $|\\Psi_a\\rangle$ by a unitary operator $U_{ba}$ that is non-local in the directions transverse to $\\hat{x}$, but crucially is local in $\\hat{x}$ -- for example in two dimensions $U_{ba}$ corresponds to moving an anyon around the transverse cycle. By Theorem~\\ref{Thm} we then conclude that $|\\Psi_b\\rangle$ is also a CDW in $\\hat{x}$. \n\nPerhaps the most familiar example of the above statement is the fractional quantum Hall effect. It is known that the $1\/k$ Laughlin state on the torus is adiabatically connected to a quasi-one-dimensional CDW state in the Landau gauge, also known as the Tao-Thouless state~\\cite{PhysRevB.28.1142,PhysRevB.77.155308}. For example for $k=2$ the Tao-Thouless state with momentum $P=\\pi n$, in the Landau orbit occupation number basis, reads\n\\begin{equation}\n    |101010...\\rangle+e^{i\\pi n}|010101...\\rangle.\n\\end{equation}\n\n\nThe CDW nature of the ground states is perfectly compatible with the topological order being a symmetric state, since there is no \\textit{local} CDW order parameters with nonzero expectation value. The CDW order parameter in this case is non-local in the directions transverse to $\\hat{x}$. For example, in two-dimensions the CDW order parameter is defined on a large loop that wraps around the cycle transverse to $\\hat{x}$. This phenomenon is dubbed \\textit{weak} symmetry breaking in Ref.~\\cite{Kitaev06}. The weak spontaneous symmetry breaking requires a certain degeneracy for the ground state. This degeneracy is naturally accommodated by the ground state manifold of the topological order. For example for the above Tao-Thouless state at $k=2$ the CDW order requires a two-fold ground state degeneracy, which is nothing but the two degenerate Laughlin states on torus.\n\n\nThe above results can be summarized as follows:\n\\begin{corollary}\n\\label{cor:toporderCDW}\nIf a ground state of a gapped topological order on a $d$-dimensional torus ($d>1$) has a non-trivial momentum in $\\hat{x}$, then any ground state of this topological order must \\textit{weakly} break translation symmetry in $\\hat{x}$.\n\\end{corollary}\nA further example of these results, alongside the effects of anyon condensation, applied upon the $\\mathbb{Z}_2$ topologically ordered Toric code is demonstrated in the Appendix~\\ref{app:Toriccode}. The above result also implies the following constraint on possible momentum carried by a topologically ordered ground state:\n\\begin{corollary}\nIf a gapped topological order has $q$ degenerate ground states on torus, then the momentum of any ground state in any direction is quantized: \n\\begin{equation}\nP^{(a)}_i=2\\pi N^{(a)}_i\/q,\n\\end{equation}\nwhere $N^{(a)}_i$ is an integer depending on the ground state (labeled by $a$) and direction $i$.\n\\end{corollary}\nThis is simply because for other values of the momentum, the ground state degeneracy required by the spontaneous translation-symmetry-breaking order will be larger than the ground state degeneracy from the topological order, which results in an inconsistency. An immediate consequence of the above corollary is that invertible topological orders (higher-dimensional states that are LRE by our definition but has only a unique gapped ground state on closed manifolds), such as the chiral $E_8$ state\\cite{Kitaev06}, cannot have nontrivial momentum on a closed manifolds since $q=1$.\n\n\n\nThe above statement immediately implies that the momenta of topological ordered ground states are robust under adiabatic deformations, as long as the gap remains open and translation symmetries remain unbroken. For the Tao-Thouless states this conclusion can also be drawn from the LSM theorem if the $U(1)$ symmetry is unbroken. Our result implies that the momenta of Laughlin-Tao-Thouless states are robustly quantized even if the $U(1)$ symmetry is explicitly broken.\n\n\n\n\n\\subsection{Crystalline symmetry-protected topological phases}\n\\label{sec:cSPT}\n\n\nThere has been growing interest and successes in understanding the symmetry-protected topological (SPT) phases associated with crystalline symmetries~\\cite{PhysRevX.7.011020,ThorngrenElse,PhysRevB.96.205106,Shiozaki2018,SongFangQi2018,Else2018}. When the protecting symmetry involves lattice translation, a crucial ``smoothness'' assumption~\\cite{ThorngrenElse,PhysRevB.96.205106} is used. Essentially one assumes that for such SPT phases the inter unit-cell entanglement can be adiabatically removed, possibly with the help of additional ancilla degrees of freedom. This allows one to formally ``gauge'' the translation symmetry~\\cite{ThorngrenElse} and build crystalline topological phases out of lower-dimensional states~\\cite{PhysRevB.96.205106,SongFangQi2018,Else2018}. \n\nOur result, namely Theorem~\\ref{Thm}, serves as a non-trivial check on the smoothness assumption in the following sense. If there were SRE states with non-trivial lattice momenta, such states would have irremovable inter-unit cell entanglement since unentangled states cannot have non-trivial momentum. Equivalently the correlation length $\\xi$ cannot be tuned to be smaller than the unit cell size $a$. In fact, if such states exist, they would by definition be non-trivial SPT states protected solely by translation symmetry -- such SPT states would be beyond all the recent classifications. \n\nWe note that our result is a necessary condition, but not a proof, for the smoothness assumption, as there may be other ways to violate the assumption without involving a ground state momentum. It will be interesting to see if the arguments used in this work can be extended to fully justify the smoothness assumption.\n\n\n\\section{Discussions}\n\\label{sec:discussions}\n\nIn this paper we have shown that a quantum many-body state with non-trivial lattice momentum is necessarily long-range entangled, hence establishing a simple yet intriguing connection between two extremely familiar concepts in physics: translation symmetry and quantum entanglement. Many directions can be further explored, which we briefly comment on in the remainder of this Section.\n\nOne important aspect that we have so far skipped over is that LSM theory is in fact intimately connected to quantum anomalies~\\cite{Cheng2015,Jian2017,Cho2017,Metlitski2017,PhysRevB.101.224437,Ye2021}. This is natural since they both provide UV conditions that constraint the low-energy behaviours. For the ``projective symmetry\" type of LSM theorems, this connection has been precisely established and it is known that such LSM constraints correspond to certain discrete (quantized) t'Hooft anomalies. For the ``partial filling'' type of LSM such as the familiar $U(1)\\times T$ constraint, however, the connection has been discussed~\\cite{Song2021Polarization,Else2021FL,Wang20,PhysRevResearch.3.043067} but has yet to be fully developed. As we discussed in Sec.~\\ref{sec: LSMOH}, our main result (Theorem~\\ref{Thm}) can be viewed as a ``partial filling'' type of LSM that only involves translation symmetry. It is therefore natural to ask whether Theorem~\\ref{Thm} can be understood from an anomaly perspective. To achieve this goal, it is clear that the standard quantized t'Hooft anomaly is insufficient (a point which was also emphasized in Ref.~\\cite{Else2021FL} for the $U(1)\\times T$ LSM) -- for example, the toric code discussed in Appendix~\\ref{app:Toriccode} has no t'Hooft anomaly since one can condense the $e$ particle to obtain a trivial symmetric state. One would therefore need to expand the notion of anomaly to accommodate the partial-filling type of LSM constraints including the one discussed in this work, possibly along the line of the ``unquantized anomaly\" discussed in Ref.~\\cite{PhysRevResearch.3.043067}. We leave this aspect to future work.\n\nAnother powerful consequence of the traditional $U(1)\\times T$ LSM theorem is on the stability of the LRE ground states (with partial charge filling) under symmetric perturbations: assuming the charge compressibility is finite (could be zero), then a small perturbation will not change the charge filling discontinuously, so the system remains LRE under small symmetric perturbations (unless the perturbation leads to spontaneous symmetry-breaking like the BCS attraction). It is natural to ask whether the other ``partial filling\" types of LSM theorems can serve similar purposes. In fact Ref.~\\cite{PhysRevResearch.3.043067} discussed precisely this point under the notion of ``unquantized anomaly''. The unquantized anomalies are very similar to Theorem~\\ref{Thm} and \\ref{FermionThm} and Corollary~\\ref{ZnLSM}, except that the key quantity is not the discrete charges (lattice momentum or $\\mathbb{Z}_n$ charges) on a specific systems size $L$, but the charge densities (momentum density or $\\mathbb{Z}_n$ charge density). Such discrete charge densities can not be defined for a fixed $L$, but may be defined for a sequence of systems with $L\\to \\infty$. Ref.~\\cite{PhysRevResearch.3.043067} argued that, in the context of Weyl and Dirac semimetals, as long as these discrete charge densities are well behaved in the $L\\to\\infty$ limit, the unquantized anomalies will protect the LRE nature of the states under symmetric perturbations. Our work here can be viewed as a rigorous justification of the unquantized anomalies in Ref.~\\cite{PhysRevResearch.3.043067} on fixed system sizes.\n\nAssuming a well-behaving momentum density in the thermodynamic limit, we can also apply our results to a Fermi liquid with a generic Fermi surface shape, such that the ground state from the filled Fermi sea has a non-vanishing momentum density (this requires breaking of time-reversal, inversion and reflection symmetries). This can be viewed as a non-perturbative explanation for the stability of such low-symmetry Fermi surface, even in the absence of the charge $U(1)$ symmetry. (Recall that perturbatively the stability comes from the fact that the Cooper pairing terms no longer connect opposite points on the Fermi surface).\n\nAnother question one may ask is whether a broader group of non-onsite symmetries obey similar charge and entanglement restrictions. It is easy to see that exactly the same constraint holds for glide reflections and screw rotations, since when the system is viewed as $1$d there is no difference between glide reflection, screw rotation and translation. It is also easy to see that the constraint does \\textit{not} hold for point group symmetries (rotations and reflections), because such symmetries will be onsite at some points in space (the fixed points of point groups). It is therefore important that translation symmetry is \\textit{everywhere} non-onsite. The question becomes even more intriguing if we consider more general unitary operators (such as quantum cellular automata~\\cite{GNVW}).\n\nThere are many more natural avenues for further exploration. The interplay between the non-local nature of translation symmetry with crystalline symmetry anomalies is not yet well understood and requires more concrete mathematical grounding such as a rigorous proof of when the smoothness condition is valid. Relatedly it remains to be determined whether translation symmetry may be truly treated as an onsite symmetry and gauged, or whether its non-locality and non-trivial momentum may hinder or require modifications to the usual gauging process. Implications of our results on the ``emergibility\" of phases may also provide fruitful insights to achievable and unachievable states on the lattice~\\cite{PhysRevX.11.031043,Ye2021}. Our work has shown without a doubt that translation symmetry is many-faceted and plays a crucial role in the entanglement properties of crystalline materials.\n\n\n\n\\begin{acknowledgments}\nWe acknowledge insightful discussions with Yin-Chen He, Timothy Hsieh, and especially Liujun Zou. We thank Anton Burkov for a previous collaboration that inspired this work. We thank the anonymous referees for their careful reading of our manuscript and their illuminating comments and questions. LG was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by a Vanier Canada Graduate Scholarship. \nResearch at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nIn many models of set theory, Souslin trees offer a variety of different\nhomogeneity or rigidity properties.\nProbably the most prominent homogeneity property for Souslin trees is\n\\emph{strong homogeneity} (cf. Section \\ref{sec:str_hom} for the definition)\nwhich implies that the tree is in a certain sense minimal with respect to its\nautomorphism group.\nOn the other hand, a great number of rigidity notions\n(i.e. absence of nontrivial automorphisms) for Souslin trees\nand an array of implications between most of them are known.\nIn this paper, which resulted out of a part of the authors PhD thesis\n\\cite{diss}, we present some interrelations between the class\nof strongly homogeneous Souslin trees and that of free trees,\nthe latter consisting of those Souslin trees which have the strongest known\nrigidity properties.\n\nThe key result which leads to these correspondences\nis a certain method for decomposing a strongly homogeneous Souslin tree\ninto $n$ free factors\n(Theorem \\ref{thm:str_hom=free_x_free}, which is a strengthening of a known\nthough unpublished result).\nThis decomposition uses an elementary,\nbut apparently new combinatorial tool,\nan $n$-optimal matrix of partitions,\nwhich we introduce in the first section.\nAs will be seen in Section 2, there are several ways to decompose a strongly\nhomogeneous Souslin tree into $n$ free trees.\nBut the construction we give using an $n$-optimal matrix of partitions\nenables us to prove strong consequences about the behaviour of the factors\nwhich finally are used in the third section to separate certain notions\nof parametrized rigidity for Souslin trees (which are all weakenings of\nfreeness) in Corollaries \\ref{cor:n-free} and \\ref{cor:n-free_not_UBP}.\n\nA few words on the structure of the paper and the assumed background\nwhich differs strongly from section to section.\nThe first section\nis about the very elementary notion of $n$-optimal matrices of\npartitions and does not assume any prerequisites.\nThe other two sections treat Souslin trees and their structural properties.\nIn Section 2 we review strong homogeneity and freeness for Souslin trees\nand prove two decomposition theorems for strongly homogeneous\nSouslin trees.\nThe final section collects several rigidity notions for Souslin trees\n(most of them taken from \\cite{degrigST}) and gives the aforementioned\nseparation results.\nSome definitions and proofs in Section 3 refer to the technique of forcing\nwhich we do not review here.\nAnd even though we give the necessary definitions concerning Souslin trees\nat the beginning of Section 2, some acquaintance with this subject will\ncertainly enhance the reader's understanding\nof the constructions in Section 2\n(very good references, also on forcing, are,\ne.g. \\cite{devlin-johnsbraten,kunen,jechneu}).\nAnyway, we have made an effort to write a paper that is accessible to an\naudience exceeding the circle of experts on Souslin trees.\n\n\\section{Optimal matrices of partitions}\n\nThe main idea is as follows:\nConsider an infinite matrix with $\\omega$  rows and $n$ columns\nwhere $n$ is a natural number larger than 1:\n$$\n\\begin{pmatrix}\nP_{0,0}&\\ldots&P_{0,m}&\\ldots&P_{0,n-1}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\nP_{k,0}&\\ldots&P_{k,m}&\\ldots&P_{k,n-1}\\\\\n\\vdots&\\vdots&\\vdots&\\vdots&\\vdots\\\\\n\\end{pmatrix}\n$$\nSuppose that the entries of this matrix are partitions of the set $\\omega$\nof natural numbers.\nWe want to choose these partitions in a way such that\n(i) we get an infinite set whenever we intersect a finite family of subsets\nof $\\omega$ coming from (distinct) partitions of a single column and\n(ii) we get a singleton whenever we intersect $n$ sets belonging\nto partitions each coming from different columns.\nIn the following definition the latter requirement is stated\nin a slightly stronger form: we want to obtain a singleton whenever we\nintersect $n$ sets not all coming from the same column.\nThe construction in the proof of Lemma \\ref{lm:opt_mtx_prt} actually yields\nmatrices that satisfy this stronger condition,\nand we will use it in the proof of Proposition \\ref{prp:decomp}\nto derive an additional result.\n\n\\begin{defi}\n\\label{defi:opt_mtx_prt}\nFor $n\\in\\omega$,\nan $n$\\emph{-optimal matrix of partitions}\nis a family $(P_{k,m}\\mid k\\in\\omega,\\,m<n)$\nof infinite partitions $P_{k,m}=(a_i^{k,m}\\mid i\\in\\omega)$ of $\\omega$\nwith the following properties.\n\\begin{enumerate}[(i)]\n\\item \\emph{Column-wise agreement:}\nFor all $m<n$ and all $i:k\\to\\omega$ where $k\\in\\omega$,\nthe intersection $\\bigcap_{\\ell<k}a_{i(\\ell)}^{\\ell,m}$ is infinite.\n\\item $n$\\emph{-optimality:}\nFor all maps $(i,k,m):n\\to\\omega\\times\\omega\\times n$\nwith $(k(j),m(j))\\neq(k(\\ell),m(\\ell))$ for all $j<\\ell<n$\nand $m(j)\\neq m(\\ell)$ for at least one pair $j,\\ell<n$\nthe intersection\n$$\\bigcap_{j<n} a^{k(j),m(j)}_{i(j)}\\quad\\text{ is a singleton.}$$\n\\end{enumerate}\n\\end{defi}\n\nNote that if in (ii) the domain of $(i,k,m)$ is restricted\nto a proper subset of $n$,\ni.e., if we intersect over less than $n$ sets,\nthen the corresponding intersection has to be infinite as well.\n\n\\begin{lm}\n\\label{lm:opt_mtx_prt}\nThere is an $n$-optimal matrix of partitions for every natural number $n>1$.\n\\end{lm}\n\\begin{proof}\nTo start we fix a bijective enumeration\n$h=(h_0,\\ldots,h_{n-1}):\\omega\\to\\omega^n$ and define $a_i^{0,m}$\nto be the pre-image of $i$ under $h_m$.\nLet $P_{0,m}:=\\{a_i^{0,m}\\mid i\\in\\omega\\}$.\n\nThe rest of the construction consists of a three-fold recursion.\nThe outer loop is indexed with $(k,m)\\in\\omega\\times n$,\nand goes row by row, from the left to the right.\nOne could also say that the progression of the indices\nfollows the lexicographic order of $\\omega\\times n$, i.e.,\n$m$ grows up to $n-1$ and then drops down to 0 while $k$ increases to $k+1$.\n(The first $n$ stages of the outer loop, where $k=0$,\nhave been included in the recursive anchor in the first line of the proof.)\n\nThe inner recursion loops are common $\\omega$-recursions.\nIn each stage of the middle one we define one element $a^{k,m}_i$\nof the partition $P_{k,m}$,\nand the innermost consists of a choice procedure for the elements\nof that set $a^{k,m}_i$.\n\nSo assume that the partitions $P_{\\ell,m}=\\{a_i^{\\ell,m}\\mid i\\in\\omega\\}$\nhave already been defined for $(\\ell,m)<_\\mathrm{lex}(k,n)$\nand also the $i$ first sets\n$a^{k,m}_0=a_0,\\ldots,a^{k,m}_{i-1}=a_{i-1}$ of $P_{k,m}$\nhave been fixed.\nAssume also,\nthat the family constructed so far\nhas the properties (i) and (ii) from Definition~\\ref{defi:opt_mtx_prt}.\nWe inductively choose three sequences\n$x_\\ell,\\,y_\\ell$ and $z_\\ell$ of members of $\\omega\\setminus\\bigcup_{h<i}a_h$\nand afterwards set $a_i:=\\{x_\\ell,y_\\ell\\mid\\ell\\in\\omega\\}$.\nThe members of the $x$ sequence will make the matrix satisfy\ncolumn-wise agreement (requirement (i))\nwhile the $y_\\ell$ guarantee that the intersections\nfor (ii) ($n$-optimality) are non-empty.\nThe elements $z_\\ell$ go back to the stack\nand build the pool for the construction of the further members of $P_{k,m}$.\n\nWe need the following objects $f,\\,b,\\,I,\\,c,\\,\\tau$ and $d$\nfor book-keeping and assume that they\nhave been fixed at the start of the definition\nof the members of the partition $P_{m,k}$, before the construction of $a_0$.\n\nLet $f:\\omega\\to{^k \\omega}$ be onto and $\\aleph_0$-to-1\nand set for $\\ell\\in\\omega$\n$$b(\\ell):=\\bigcap_{j<k} a^{j,m}_{f(\\ell)(j)}.$$\nThese sets $b(\\ell)$ have to be met by $a_i$ infinitely many times.\nSo we will choose $x_\\ell$ from $b(\\ell)$.\nLet $I$ be the set of subsets $\\sigma$ of\n$(\\omega\\times k\\times n)\\cup(\\omega\\times(k+1)\\times m)$ of\ncardinality $n-1$ such that $\\mathrm{pr}_{2,3}\\!\\!\\upharpoonright\\!\\!\\sigma$ is injective and \n$\\mathrm{pr}_3\\!\\!\\upharpoonright\\!\\!\\sigma$ is not equal to to the constant function whose only\nvalue is $m$,\nwhere $\\mathrm{pr}_3$ and $\\mathrm{pr}_{2,3}$ are the projections to the third\nand to the second and third component of a triple respectively.\nSo if $\\sigma\\in I$ then the elements of $\\sigma$\nare indices for $n-1$ members of pairwise distinct partitions and these \npartitions lie in at least two distinct columns of the $n$-optimal matrix.\nFor $\\sigma\\in I$ let\n$$c(\\sigma):=\\bigcap_{(j,p,q)\\in\\sigma} a^{p,q}_j$$\nand note that this set is infinite,\nbecause if $P_{r,s}$ is a partition which is not involved in $\\sigma$\nthen $c(\\sigma)$ meets every element of $P_{r,s}$ in exactly one natural\nnumber by $n$-optimality.\n\nAlso the sets $c(\\sigma)$ have to be met by $a_i$.\nSo we fix a bijective enumeration $\\tau$ of $I$ and\nchoose $y_\\ell$ from $c(\\tau(\\ell))$\nunless that set has already been hit by earlier members of $a_i$.\n\nCondition (ii) imposes that each set $c(\\sigma)$ be met in only one element.\nSo once the intersection of $a_i$ and $c(\\sigma)$ is non-empty,\nthat particular set $c(\\sigma)$ has to be avoided in later choices\nof members of $a_i$. \nWe thus define for every natural number $x$ the set\n$$d(x):=\\bigcup\\left\\{ c(\\sigma)\\mid \\, \\sigma\\in I\\text{ and }\nx\\in c(\\sigma)\n\\right\\}$$\nand choose $x_\\ell$ and $y_\\ell$ from outside\n$\\bigcup_{j<\\ell} d(x_j)\\cup d(y_j)$.\n\nWe now turn to the formal definition of our three sequences\n$x_\\ell,\\,y_\\ell$ and $z_\\ell$\nand argue afterwards, why these choices are always possible.\nSet $e=\\bigcup_{h<i}a_h$ and let inductively\n\\begin{eqnarray*}\nx_\\ell&:=&\n\\min\n \\left(\n  b(\\ell)\n  \\setminus\n  \\left(\n   e\\cup\\bigcup_{j<\\ell}(d(x_j)\\cup d(y_j)\\cup\\{z_j\\})\n  \\right)\n \\right),\\\\\ny_\\ell&:=&\n\\begin{cases}\nx_\\ell,&\nc(\\tau(\\ell))\n\\cap\n\\bigcup_{j<\\ell}\\{x_j,y_j,x_{j+1}\\}\n \\neq\n\\emptyset\\\\\n\\min \n c(\\tau(\\ell))\n \\setminus \n  (e\\cup\\{z_0,..,z_{\\ell-1}\\}),&\n\\text{otherwise,}\n\\end{cases}\\\\\nz_\\ell&:=&\n\\min\n \\left(\n  b(\\ell)\n  \\setminus\n  \\left(\n   e\\cup\\bigcup_{j<\\ell}\\{x_j,y_j,z_j,x_{j+1},y_{j+1}\\}\n  \\right)\n \\right).\n\\end{eqnarray*}\nWe finally show that this construction does not break down, i.e., that the \nsets, from which $x_\\ell$ or $z_\\ell$ are picked, are non-empty for all\n$\\ell$.  (A variation shows that the choice $y_\\ell$ is always possible, as well.)\nThe argument splits depending on whether $k$ is less than or at least $n-1$.\nSo let $k<n-1$ and $\\ell\\in\\omega$, set\n$\\rho=\\rho(\\ell)=\\{(f(\\ell)(j),j,m)\\,:\\,j<k\\}$ and fix $\\sigma\\in I$ such that $\\rho\\subset\\sigma$,\nsay $\\sigma=\\rho\\cup\\{(j_1,p_1,q_1),\\ldots,(j_r,p_r,q_r)\\}$ for $r=n-1-k$.\nNow let\n$$F:=F(\\ell,\\sigma):=\n\\left\\{\n\\{(g_1,p_1,q_1),\\ldots,(g_r,p_r,q_r)\\}\n\\,:\\,\n(g_1,\\ldots,g_r)\\in\\omega^r\n\\right\\}$$\nand note that $b(\\ell)$ is the disjoint union of the family\n$\\{c(\\rho\\cup\\tau)\\,:\\,\\tau\\in F\\}$:\n$$b(\\ell)=\\dot{\\bigcup_{\\tau\\in F}}c(\\rho\\cup\\tau).$$\nIn order to prove that we always can choose our new element of the $x$-, $y$-\nor $z$-sequence, we first state two consequences of $n$-optimality:\n\\begin{enumerate}\n  \\item For every $\\tau\\in F$ and $h<i$ the intersection $c(\\rho\\cup\\tau)\\cap a_h$\n    is a singleton.\n  \\item Given any $\\sigma'\\in I$, for every $\\tau\\in F$ with\n    $\\sigma'\\ne\\rho\\cup\\tau$ the intersection $c(\\rho\\cup\\tau)\\cap c(\\sigma') $ has at\n    most one element.\n  \\end{enumerate}\nNow note that in order to define $x_\\ell$ or $z_\\ell$,\nwe subtract only finitely many sets of the form\n$a_h$ or $c(\\sigma')$ from $b(\\ell)$ .\nSo in any case, infintely many natural numbers remain in\nthe pool.\n\nNext let $k\\ge n-1$. Then any set of the form $c(\\sigma)$ meets $b(\\ell)$ in\nat most one member by a direct application of $n$-optimality.\nFurthermore the $z$-part of the construction yields by induction on\n$i$ that $b(\\ell)\\setminus e=b(\\ell)\\setminus(\\bigcup_{h<i}a_h)$ is an\ninfinite set.\n\n\n\nWe leave the verification that this construction indeed yields an\n$n$-optimal matrix of partitions for the reader.\n\\end{proof}\n\nThere is also a proof of Lemma \\ref{lm:opt_mtx_prt}\nwhich uses a more sophisticated\nforcing style argument in the construction of the partition $P_{k,m}$,\nbut since it does not significantly decrease the length of the proof\nwe stuck to the elementary recursive construction given above.\n\n\n\\section{Strongly homogeneous and free Souslin trees}\n\\label{sec:str_hom_+_free}\n\n\\subsection{Preliminaries on Souslin trees}\n\\label{sec:prel_ST}\n\nA \\emph{tree} is a partial order $(T,<_T)$ where for all $t\\in T$\nthe set of predecessors $\\{s\\mid s<_T t\\}$\nis well-ordered by $<_T$.\nThe elements of a tree are called \\emph{nodes}.\nFor a node $t\\in T$ we let $\\NF(t)$ be the set of $t$'s immediate successors.\nThe \\emph{height} of the node $t$ in $T$ is the order\ntype of the set of its predecessors under the ordering of $T$,\n$\\hgt_T(t):=\\ot(\\{s\\mid s<_T t\\},<_T)$.\nFor an ordinal $\\alpha$ we let $T_\\alpha$ denote\nthe set of nodes of $T$ with height $\\alpha$.\nIf $\\hgt_T(s)>\\alpha$ we let $s\\!\\!\\upharpoonright\\!\\!\\alpha$ be the unique\npredecessor of $s$ in level $\\alpha$.\n\nThe \\emph{height of a tree} $T$, $\\hgt T$,\nis the minimal ordinal $\\alpha$ such that $T_\\alpha$ is empty.\nAn \\emph{antichain} is a set of pairwise incomparable nodes of $T$,\nso for $\\alpha<\\hgt T$,\nthe level $T_\\alpha$ is an antichain of $T$.\n\nNodes, that do not have $<_T$-successors, are called \\emph{leaves}, and\n$T$ is called \\emph{$\\kappa$-splitting} or \\emph{$\\kappa$-branching},\n$\\kappa$ a cardinal, if all nodes of $T$\nhave exactly $\\kappa$ immediate successors, except for the leaves.\n\nA \\emph{branch} is a subset $b$ of $T$ that is linearly ordered by $<_T$ and\nclosed downwards, i.e. if $s<_T t\\in b$ then $s\\in b$.\nUnder the notion of a \\emph{normal} tree we subsume the following\nfour conditions:\n\\begin{enumerate}[a)]\n\\item there is a single minimal node called the \\emph{root};\n\\item each node $s$ with $\\hgt(s)+1<\\hgt T$ has at least two immediate successors;\n\\item each node has successors in every higher non-empty level;\n\\item branches of limit length have unique limits (if they are extended in the tree),\ni.e., if $s,t$ are nodes of $T$ of limit height\nwhose sets of predecessors coincide, then $s=t$.\n\\end{enumerate}\nNote that by condition c) leaves can\nonly appear in the top level of a normal tree.\n\nFor a node $t\\in T$ we denote by $T(t)$ the set $\\{s\\in T:t\\leq_T s\\}$\nof nodes above (and including) $t$ which becomes a tree when equipped\nwith the ordering inherited from $T$.\nA tree $T$ is said to be \\emph{homogeneous},\nif for all pairs $s,t\\in T$ of the same height there is\na tree isomorphism (of partial orders) between $T(s)$ and $T(t)$,\nthe trees of nodes in $T$ above $s$ and $t$ respectively.\nFor many classes of trees, such as Souslin trees, this is equivalent to \nthe condition that for each pair $s,t\\in T$ of nodes of the same height there\nis an automorphism of $T$ mapping $s$ to $t$.\nA tree is \\emph{rigid} if it does not admit any non-trivial automorphism.\n\nWe will consider two operations on the class of trees: sum and product.\nGiven trees $(T^i,<_i)$ for $i\\in I$, the \\emph{tree sum} of this family,\ndenoted by $\\bigoplus_{i\\in I}T^i$ is the disjoint union of the sets $T^i$\nplus a common root $r\\notin \\bigcup T^i$.\nThe tree order $<$ on $\\bigoplus T^i$ is\ngiven by the (disjoint) union of the tree orders of summands as well as\nthe relation $r< t$ for all $t\\in \\bigcup T^i$.\nThe height of $\\bigoplus T^i$ is given by the ordinal\n$1 + \\sup\\{\\hgt T^i: i\\in I\\}$.\n\nLet now all trees $T^i$ be of height $\\mu$.\nThe \\emph{tree product} $\\bigotimes_{i\\in I}T^i$\nover the family $(T^i)_{i\\in I}$\nis given by the union over the cartesian products of the levels\n$T^i_\\alpha$:\n$$\\bigotimes_{i\\in I}T^i := \\bigcup_{\\alpha<\\mu}\\prod_{i\\in I} T^i_\\alpha.$$\nThe product tree order is simply the conjunction of the relations $<_i$.\n\nIn order to make a decomposition of a tree into a product feasible\nwe also introduce the notion of a nice tree equivalence relation.\nLet $T$ be a normal and $\\aleph_0$-splitting tree and $\\equiv$\nan equivalence relation on $T$.\nThen we say that $\\equiv$ is a \\emph{nice tree equivalence relation (nice\n  t.e.r.)} if $\\equiv$ respects levels (i.e., it refines $T\\otimes T$),\nis compatible with the tree order (i.e., $\\hgt(s)=\\hgt(r) $ and\n$s<t\\equiv u > r$ imply $s\\equiv r$), the quotient partial order\n$T\/\\!\\!\\equiv$ of $\\equiv$-classes ordered by the inherited partial order, i.e.\n$$[s]<_\\equiv [t]\\quad\\iff\\quad s< t\\,,$$\nis a normal and $\\aleph_0$-splitting tree and the relation is nice,\nby which we mean that for all triples of nodes $s,r,t$ such that\n$s\\equiv r$ and $t$ is above $s$ there is a node $u\\equiv t$, $u$ above\n$r$.\nAnother way to formulate this last property ``niceness'' associates to each\nbranch $b$ through $T$ a subtree $T^{b}_\\equiv:=\\bigcup_{s\\in\n  b}s\/\\!\\!\\equiv$ of $T$ and requires that it satisfies point c) in our\ndefinition of normal trees, i.e., every node $t\\in T^{b}_\\equiv$ has\nsuccessors in every higher level of $T^b_\\equiv$. \n\nNow consider the case that a tree $T$ carries nice tree equivalence relations\n$\\equiv_i$ for $i<n$ such that for every level $\\alpha$\nand every $n$-sequence of equivalence classes\n$c_i\\in (T\/\\!\\!\\equiv_i)_\\alpha$, $i<n$\nthe intersection $\\bigcap c_i$ is a singleton $\\{t\\}$ with $t\\in T_\\alpha$.\nThen we have a natural isomorphism\nbetween the tree $T$ and the product of its quotient trees,\n$\\bigotimes_{i<n} T\/\\!\\!\\equiv_i$ given by\n$t\\mapsto (t\/\\!\\!\\equiv_i : i< n)$.\n\nWe finally come to Souslin trees.\nIn general,\na \\emph{Souslin tree} is a tree $T$ of height $\\omega_1$\nsuch that\nevery family of pairwise incomparable nodes\nand also every branch of $T$ is at most countable.\n\\emph{Unless stated otherwise,\nwe will only consider normal and $\\aleph_0$-splitting Souslin trees.}\nIn this case the sole absence of uncountable\nantichains -- referred to as \\emph{the contable chain condition (c.c.c.)} --\nalready implies that the tree has no cofinal branch.\nA main elementary feature of Souslin trees is that their square is\nno longer Souslin:\n$T\\otimes T$ violates the c.c.c. for every Souslin tree $T$.\n\nThe naming \\emph{Souslin} of these trees stems from their tight connection\nto the famous question of\nMikhail Yakovlevich Souslin that was published as \\emph{Probl\\`{e}me 3)}\non page 223 of the first issue of Fundamenta Mathematicae in 1920\n(cf. also \\cite{devlin-johnsbraten} or \\cite[II.4]{kunen};\nthe latter transliterates the name as \\emph{Suslin}). \nIt is well known that the existence of Souslin trees is independent of ZFC,\nso whenever we assume that there is some Souslin tree, we make an extra\nassumption beyond the realm of standard set theory.\n\n\\subsection{Strongly homogeneous and free trees}\n\\label{sec:str_hom}\n\nWe take a closer look at two classes of Souslin trees,\nthat are widely known among set theorists, although\noften under different names, of which we try to state as many as possible.\n\nStrongly homogeneous Souslin trees occur quite often\nin set theoretic literature.\nIn \\cite{larson-todorcevic} they are called\ncoherent Souslin trees and play a central\nrole in the solution of Katetov's Problem\non the metrizability of certain compact spaces.\nShelah and Zapletal show in \\cite[Theorem 4.12]{shelah-zapletal} that\nTodorcevic's term for a Souslin tree\nin one Cohen real is strongly homogeneous,\nLarson gives a direct $\\diamondsuit$-construction (\\cite[Lemma 1.2]{larson}),\nand also Jensen's construc\\-tion (under the same hypothesis)\nof a 2-splitting, homogeneous tree,\nas carried out in \\cite[Chapter IV]{devlin-johnsbraten},\nis easily seen to yield a strongly homogeneous tree.\n\n\\begin{defi}\nA Souslin tree $T$ is called \\emph{strongly homogeneous} if there is a family\n$(\\psi_{st}\\mid s,t\\in T,\\,\\hgt s=\\hgt t)$ which has the following properties:\n\\begin{enumerate}[1)]\n\\item $\\psi_{st}$ is an isomorphism between\nthe tree $T(s)$ of nodes above $s$ and the tree $T(t)$ of nodes\nabove $t$ and $\\psi_{ss}$ is the identity.\n\\item (commutativity) For all nodes $r,s,t$ of the same level of $T$\nwe have $\\psi_{rt}=\\psi_{st}\\circ\\psi_{rs}$.\n\\item (coherence) For nodes $r,s$ from the same level,\n$t$ above $r$ and $u=\\psi_{rs}(t)$ we require that $\\psi_{tu}$ is\nthe restriction of $\\psi_{rs}$ to the tree $T(t)\\subset T(r)$.\n\\item (transitivity) If $t$ and $u$ are nodes\non the same limit level $T_\\alpha$, then there is a level $T_\\gamma$\nbelow such that for the corresponding predecessors $r$ of $t$ and $s$ of $u$\nwe have $\\psi_{rs}(t)=u$.\n\\end{enumerate}\n\\end{defi}\n\nSome authors call such a family of tree isomorphisms associated to a strongly\nhomogeneous tree a \\emph{coherent family}.\n\nGiven any homogeneous tree,\nit is easy to define a family on $T$ with the properties 1-3) above.\nThe crucial property of a coherent family is that of \\emph{transitivity},\nwhich means that every limit level is a minimal extension of the initial\nsegment below with respect to the coherent family on that initial segment. \nAlso the automorphism group of a strongly homogeneous Souslin tree $T$\nis in a sense minimal, as shown in the following proposition.\n\n\\begin{prp}\n\\label{prp:str_hom_aut}\nEvery automorphism $\\varphi$ of a strongly homogeneous Souslin tree $T$ is\neventually equal to the union of a subset of the coherent family\n$(\\psi_{st})$, i.e.,\nthere is a countable ordinal $\\alpha$ such that for all\nnodes $t$ of height greater than $\\alpha$ we have\n$$\\varphi(t) =\n\\psi_{t\\upharpoonright\\alpha,\\varphi(t\\upharpoonright\\alpha)}(t).$$\n\\end{prp}\nThis implies that the automorphism group of such a tree has exactly\n$2^{\\aleph_0}$ elements.\n\\begin{proof}\nIt suffices to show that above every node $r\\in T$ there is a node\n$s$, such that all $t>s$ are mapped by the automorphism\n$\\varphi$ according to the rule\nstated above with $\\alpha=\\hgt(s)$.\n\nTo reach a statement contradicting the transitivity\nof the family $(\\psi_{st})$,\nwe assume that there is a node $r\\in T$,\nsuch that for each successor $s$ of $r$ there is\na node $t\\geq s$, such that $\\varphi(t)\\neq\\psi_{s\\varphi(s)}(t)$.\nWe can inductively choose an increasing sequence of ordinals\n$\\alpha_n$ such that for all nodes $t\\in T_{\\alpha_{n+1}}$ we have\n$$\\varphi(t)\\neq\n\\psi_{t\\upharpoonright\\alpha_n \\varphi(t\\upharpoonright\\alpha_n)}(t).$$\nLet $\\alpha$ be the supremum of the $\\alpha_n$ and pick any node\n$t\\in T_\\alpha$.\nSince $\\alpha$ is a limit ordinal and by transitivity of the coherent family\nwe find an $n\\in\\omega$ such that\n$\\varphi(t)=\\psi_{t\\upharpoonright\\alpha_n \\varphi(t\\upharpoonright\\alpha_n)}(t)$\nwhich is of course impossible by the choice of the $\\alpha_n$.\n\\end{proof}\n\nNow we come to \\emph{free} trees. Also this property has several different\nnames, e.g. full\n(Jensen, Todor\\c{c}evi\\'{c}, \\cite{GKH,todorcevic_trees_and_orders}) or\n'Souslin and all derived trees Souslin'\n(Abraham and Shelah, \\cite{abraham-shelah_aronszajn_trees,abraham-shelah}).\nIn the context of \\cite{degrigST}\n(cf. Section \\ref{sec:free} of the present article)\nfree trees could also be called\n'$<\\!\\!\\omega$-fold Souslin off the generic branch'.\n\n\\begin{defi}\nA normal tree $T$ of height $\\omega_1$ is \\emph{free}\nif for every finite (and non-empty) set of nodes\n$s_0,\\ldots,s_n$ of $T$ of the same height,\nthe tree product $\\bigotimes_{i=0}^n T(s_i)$ satisfies the c.c.c.\n\\end{defi}\n\nFree trees are easily seen to be rigid Souslin trees\nas the product of two isomorphic relative trees $T(s)$ and $T(t)$\nwould clearly not be Souslin.\nIn Section \\ref{sec:separating} we will also consider weaker,\nparametrized forms of freeness.\n\n\\subsection{Decompositions of strongly homogeneous Souslin trees}\n\\label{sec:dec_free}\n\nWe now come to the key result of this paper.\nThe following theorem is stated in \\cite[p.246]{shelah-zapletal}\nin the case $n=2$ without proof.\nLarson gives the construction of a single free subalgebra\nof a strongly homogeneous Souslin algebra in terms of trees\nin the proof of Theorem 8.5 in his paper \\cite{larson}.\nSome ideas in the following proof are borrowed from that construction.\n\n\\begin{thm}\\label{thm:str_hom=free_x_free}\nFor every natural number $n>1$ and every\n$\\aleph_0$-branching, strongly homogeneous Souslin tree $T$\nthere are free Souslin trees $S_0,\\ldots,S_{n-1}$\nsuch that $T\\cong \\bigotimes_{m<n} S_m$.\n\\end{thm}\n\n\\begin{proof}\nLet $T$ be a strongly homogeneous Souslin tree and\ndenote by $\\psi_{s,t}$ the members of the coherent family of $T$.\nWe inductively (level by level) define $n$ nice t.e.r.s\n$\\equiv_0,\\ldots,\\equiv_{n-1}$ with the following properties:\n\\begin{itemize}\n\\item $T\/\\!\\!\\equiv_m$ is a free Souslin tree for $m<n$.\n\\item For any sequence $(s_0,\\ldots,s_{n-1})\\in T_\\alpha$\nthe intersection of the classes $s_m\/\\!\\!\\equiv_m$ for $m<n$ is\na singleton: $\\bigcap_{m<n} (s_m\/\\!\\!\\equiv_m)=\\{r\\}$ for some $r\\in T_\\alpha$.\n\\end{itemize}\nThe second claim entails the existence of the isomorphism between\n$T$ and $\\bigotimes_{m<n} T\/\\!\\!\\equiv_m$.\n\nLet $(P_{k,m}\\,:\\, m<n,k\\in\\omega)$ be an $n$-optimal matrix of partitions,\nwhere we view each $P_{k,m}$ as enumerated by $a_i^{k,m}$, $i\\in\\omega$.\nIn order to define t.e.r.s we transfer the whole\nmatrix of the $P_{k,m}$ to every set $\\NF(s)$ for $s\\in T$ in a coherent way:\nChoose for every $\\alpha<\\omega_1$\nan anchor node $r_\\alpha\\in T_\\alpha$\nand a bijection $\\sigma_\\alpha:\\omega\\to\\NF(r_\\alpha)$,\nand define for $s\\in T_\\alpha$ and all indices $i,k,m$ the sets\n$$a_i^{k,m}(s):=(\\psi_{r_\\alpha,s}\\circ\\sigma_\\alpha)''a_i^{k,m}.$$\nThen clearly for every $s\\in T$, $k\\in\\omega$ and $m<n$,\nthe set $P_{k,m}(s):=\\{a_i^{k,m}(s)\\mid i\\in\\omega\\}$ forms\na partition of $\\NF(s)$, and these partitions are linked to each other\nby the coherent family in a coherent way, i.e., \n$\\psi_{s,t}$ transfers $P_{k,m}(s)$ to $P_{k,m}(t)$.\n\nFix $m<n$ in order to define $\\equiv_m$ on $T$ by recursion on the height.\nWe will also enumerate the $\\equiv_m$-classes of each level in\norder type $\\omega$, i.e.,\nwe will fix an onto mapping $h:T\\to\\omega$,\nsuch that for $s,t\\in T_\\alpha$ we have\n$s\\equiv_m t$ if and only if $h(s)=h(t)$.\n\nChoose $P_{0,m}(\\text{root})$ as the partition\nof the set $T_1=\\NF(\\text{root})$ and let $\\equiv_m$ on level $T_1$\nbe the equivalence relation with classes\n$a_i^{0,m}(\\text{root})$ for $i\\in\\omega$.\nLet $h(\\text{root})=0$ and choose $h$ on $T_1$ in a way,\nsuch that nodes $s$ and $t$ are $\\equiv_m$-equivalent\njust in case that their $h$-values coincide.\n\nNext we consider the case where $\\alpha$ is a successor ordinal,\n$\\alpha=\\gamma+1$ for some $\\gamma<\\omega_1$.\nLet $s,t\\in T_\\alpha$ and let $s^-<_T s$ and $t^-<_T t$ be\ntheir direct predecessors on level $\\gamma$.\nWe let  $s\\equiv_m t$ if and only if their direct predecessors\nare $\\equiv_m$-equivalent, $s^-\\equiv_m t^-$\n(so in particular $h(s^-,m)=h(t^-,m)$),\nand if there is $i\\in\\omega$ such that\n$$s\\in a_i^{h(s^-),m}(s^-)\n\\text{ and }t\\in a_i^{h(t^-),m}(t^-).$$\nIn words, the $\\equiv_m$-equivalence of the direct predecessors\ngives us a natural number $h(s^-)$ and we apply\n$P_{h(s^-),m}$ on level $\\alpha$ to decide\nwhether or not $s$ and $t$ are $\\equiv_m$-equivalent. \nExtend $h$ to level $T_\\alpha$ as described above.\n\nOn limit stages $\\lambda$ the relation $\\equiv_m$ is already determined\nby its behaviour below,\nand we choose the $h\\!\\!\\upharpoonright\\!\\! T_\\lambda$ once more in any way such that\n$h(s)=h(t)$ is equivalent to\n$\\equiv_m$-equivalence for nodes $s,t\\in T_\\lambda$.\n\nHaving finished the construction of the relation $\\equiv_m$,\nwe show that it produces a nice t.e.r,\nwhere all properties but niceness follow rather easily from the construction.\nSo we only deduce niceness.\nLetting $s\\equiv_m r$ on level $\\alpha$ and $t$ above $s$ we claim \nthat $\\psi_{s,r}(t)\\equiv_m t$ and show this\nby induction on the height of $t$ above $s$.\nFor successor stages the claim follows directly\nfrom the construction and the inductive hypothesis,\nsince the relevant partition $P_{j,m}$\nis transferred via $\\psi_{s,r}$ by the coherence of the coherent family.\nThe limit case follows directly from the inductive assumption.\n(This property of $\\equiv_m$,\nthat $\\equiv_m$-equivalence lifts from $s$ and $r$\nto preimages and images under\n$\\psi_{s,r}$ will be used again in the proof of the Claim below.)\n\nIt remains to prove the two properties stated before the construction.\nWe start with the freeness of $T\/\\!\\!\\equiv_m$.\nLet $s_0,\\ldots,s_{k-1}$ be pairwise non-$m$-equivalent nodes\nof the same height $\\alpha$ for some natural number $k$.\nWe write $S_i$ for $(T\/\\!\\!\\equiv_m)(s_i\/\\!\\!\\equiv_m)$ and try to find\nfor every antichain $A$ of $\\bigotimes_{i<k}S_i$\nan antichain $B$ of $T$ of the same cardinality.\nThis would prove that $T\/\\!\\!\\equiv_m$ is a free tree.\nWe get a hint about where to look for the members of such an antichain\n$B$ from the following\\\\\n{\\bf Claim.} \nFix $m<n$ and pairwise non-$m$-equivalent nodes\n$s_0,\\ldots,s_{k-1}\\in T_\\alpha$.\nFor any sequence $(t_0,\\ldots,t_{k-1})$ of nodes in $T_\\beta$,\nwith $\\alpha<\\beta$ and $s_i<t_i$ for $i<k$,\nthe intersection of the classes\n$t_i\/\\!\\!\\equiv_m\\cap\\, T(s_i)$ above the nodes $s_i$,\nshifted above $s_0$ by $\\psi_{s_i,s_0}$, i.e., the set\n$$\\bigcap_{i<k} {\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$$\nis infinite and therefore non-empty.\n\\begin{proof}[Proof of the Claim.]\nBy induction on the height $\\beta$ of the nodes $t_i$,\nstarting with $\\beta=\\alpha+1$.\nIn this minimal case we have $t_i^-=s_i$.\nSo the sets ${\\psi_{s_i,s_0}}\"(t_i\/\\!\\!\\equiv_m)$\nbelong to distinct partitions $P_{h(s_i),m}(s_0)$, $i<k$\nand therefore have an infinite intersection\nby property (i) of the $n$-optimal matrix.\n\nFor the higher successor case $\\beta=\\gamma+1,\\,\\alpha<\\gamma$,\nwe simulate this initial situation.\nBy the inductive hypothesis pick a node\n$$r_0\\in\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(t^-_i\/\\!\\!\\equiv_m)>s_0,$$\nand let $r_i:=\\psi_{s_0,s_i}(r_0)>s_i$ for $i<k$.\nWe then know that $r_i\\equiv_m t_i^-$,\nso $t_i\/\\!\\!\\equiv_m$ has elements above $r_i$.\nAs a consequence $\\bigcup_{i<k}{\\psi_{r_i,r_0}}''(t_i\/\\!\\!\\equiv_m)$\nis infinite by the same argument as above and furthermore\na subset of $\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$.\n\nFor the case where $\\beta$ is a limit ordinal\nwe choose $\\gamma<\\beta$ large enough,\nsuch that letting $q_i=t_i\\!\\!\\upharpoonright\\!\\!\\gamma$ for all $i,j<k$ we have\n$\\psi_{q_i,q_j}(t_i)=t_j$.\nThis is possible due to the transitivity of the coherent family.\nWe also require $\\alpha<\\gamma$.\nThe inductive hypothesis gives us a node\n$$r_0\\in\\bigcup_{i<k}{\\psi_{s_i,s_0}}''(q_i\/\\!\\!\\equiv_m)\\subset T_\\gamma,$$\nwhich  we copy to $r_i:=\\psi_{s_0,s_i}(r_0)$.\nBy this choice, we also have $r_i\\equiv_m q_i$.\nWe consider $u=\\psi_{q_i,r_0}(t_i)$.\nBy the commutativity of the coherent family this definition of $u$\nis independent from the choice of $i<k$.\nBut then\n$$\\psi_{s_0,s_i}(u)=\\psi_{r_0,r_i}(u)=\n\\psi_{r_0,r_i}\\circ\\psi_{q_i,r_0}(t_i)=\\psi_{q_i,r_i}(t_i)$$\nwhere the first equation follows from coherence,\nthe second from the definition of $u$ and the third one from commutativity.\nSo the property stated above right after the construction of $\\equiv_m$\nimplies that $\\psi_{s_i,s_0}(t_i)\\equiv_m u$\nsince $r_i\\equiv_m t_i$ for all $i<k$.\nThis completes the proof of the Claim.\n\\end{proof}\nBy virtue of the Claim we can pick for every tuple\n$(t_0\/\\!\\!\\equiv_m,\\ldots,t_{k-1}\/\\!\\!\\equiv_m)$\nof our antichain $A\\subset \\bigotimes_{i<k} S_i$\na node $u\\in\\bigcap_{i<k}{\\psi_{s_i,s_0}}''(t_i\/\\!\\!\\equiv_m)$ and\ncollect all these nodes in a set $B$.\nThen $B$ is clearly an antichain of $T$ with the same cardinality as $A$.\nSo we have shown that $T\/\\!\\!\\equiv_m$ is indeed a free tree.\n\nNow for the second property.\nLet $(s_0,\\ldots,s_{n-1})$ be any sequence of nodes of some $T_\\alpha$.\nWe need to show that $\\bigcap_{m<n}s_m\/\\!\\!\\equiv_m$ has a unique element.\nThis is done by induction on $\\alpha>0$.\nStarting with $\\alpha=1$ we know that\n$(s_m\/\\!\\!\\equiv_m)=a_{i_m}^{k_m,m}(\\mathrm{root})$\nfor some $i_m$ and $k_m$.\nSo property (ii) of our $n$-optimal matrix is all we need here.\nFor $\\alpha=\\gamma+1$ we assume that the classes $s_m^-\/\\!\\!\\equiv_m$\nmeet in a single node, say $r\\in T_\\gamma$.\nThe set of elements of $s_m\/\\!\\!\\equiv_m$ which lie above $r$\nis then just $a_{i_m}^{h_m(r),m}(r)$ \nand again property (ii) of the matrix proves the claim.\nIn the limit case we once more use the transitivity of the coherent family.\nSo let $\\alpha$ be a limit and $\\gamma<\\alpha$ large enough such that\n$\\psi_{q_m,q_\\ell}(s_m)=s_\\ell$ where we abbreviate $s_m\\!\\!\\upharpoonright\\!\\!\\gamma=q_m$.\nFor a last time in this proof we use the commutativity of the\ncoherent family:\nLet $r$ be the unique element of the intersection\nof the classes $q_m\/\\!\\!\\equiv_m$.\nThen $t=\\psi_{q_m,r}(s_m)$ is well defined and\nindependent from the choice of $m<n$.\nBy the lifting property of the equivalence relations stated above,\nit follows from $q_m\\equiv_m r$, that $s_m\\equiv_m t$.\n\\end{proof}\n\nWe now state an algebraic feature which distinguishes our\nmethod of decomposition as just carried out from other decompositions,\nnamely that partial products of our decomposition are always \\emph{rigid},\ncf. Remark \\ref{rem:decomp}.\n\n\\begin{prp}\n\\label{prp:decomp}\nLet $T$ be a strongly homogeneous Souslin tree and assume that\nit has been decomposed into a product of $n$ free trees $S_0,\\ldots, S_{n-1}$\nby the procedure presented in the last proof.\nThen the product of less than $n$ pairwise distinct trees from the \nsequence $S_0,\\ldots,S_{n-1}$ is a rigid Souslin tree.\n\\end{prp}\n\\begin{proof}\nIt is clear that the product tree is Souslin and that it\nis sufficient to show rigidity only for the case of $n-1$ factors, where $n>2$.\nSo assume that $R:=\\bigotimes_{i<n-1}S_i$ admits the automorphism\n$\\varphi'\\ne \\id$ and derive a contradiction as follows.\n\nIdentifying $T$ and $R\\otimes S_{n-1}$ we can lift $\\varphi'$\nto an automorphism $\\varphi = \\varphi'\\otimes \\id$ of $T$.\nBy Proposition \\ref{prp:str_hom_aut} there is a countable ordinal $\\alpha$\nsuch that above level $T_\\alpha$ the mapping $\\varphi$ is given by\na subfamily of the coherent family of $T$.\nAs $\\varphi' \\ne \\id$ there must be a node\n$\\bar{s}=(s\/\\!\\!\\equiv_0,\\ldots,s\/\\!\\!\\equiv_{n-1})\\in R$ such that \n$\\varphi'(\\bar{s})\\ne \\bar{s}$.\nWe certainly can assume that $\\hgt_R(\\bar{s})=\\alpha$.\n\nPick an $\\ell<n-1$ such that the $\\ell$th component $\\bar{s}_\\ell$\nof $\\bar{s}$ is not mapped to itself under $\\varphi'$.\nWe fix a representative $s\\in \\bar{s}_\\ell=s\/\\!\\!\\equiv_\\ell\\subset T_\\alpha$\nand let $q:=\\varphi(s)$ in order to get\n$\\varphi(r)=\\psi_{sq}(r)$ for all (immediate) successors of $s$.\nThis is where we will find a contradiction.\n\nIn the construction above we have associated to class $s\/\\!\\!\\equiv_\\ell$\na natural number $h(s,\\ell)$ which defined the index of the partition \nthat was used to extend $\\equiv_\\ell$ to the successors of the members of\nthe class $s\/\\!\\!\\equiv_\\ell$.\nThe same holds for $q=\\varphi(s)$, but as $q\\not\\equiv_\\ell s$\nwe have $h(q,\\ell)\\ne h(s,\\ell)$.\nThis implies that $\\equiv_\\ell$ for immediate succesors of $s$ and of $q$\nis formed by virtue of different partitions of column $\\ell$ of our\npartition matrix.\nFix any immediate successor $r$ of $s$.\nSince $\\varphi'$ is assumed to be a well-defined mapping\n$\\varphi$ should map the intersection $\\bigcap_{i<n-1} r\/\\!\\!\\equiv_i$\nonto the set $\\bigcap_{i<n-1} \\varphi(r)\/\\!\\!\\equiv_i$. \nIn particular, the equation\n$${\\psi_{sq}}''\\left(T(s)\\cap\\bigcap_{i<n-1} r\/\\!\\!\\equiv_i\\right)\n= \\left( T(q) \\cap \\bigcap_{i<n-1} \\psi_{sq}(r)\/\\!\\!\\equiv_i\\right)$$\nshould be true.\nBut this is not the case, because, by $n$-optimality of the matrix\nof partitions, the left-hand-side of the above equation intersects\nevery $\\equiv_\\ell$-class of the immediate successors of $q$\nin exactly one element, while the right-hand-side is a subset\nof such an $\\equiv_\\ell$-class and at the same time an infinite set.\n\\end{proof}\n\nIn the next section we will find corollaries of\nTheorem \\ref{thm:str_hom=free_x_free} in a similar vein.\nI especially mention Lemma \\ref{lm:free_to_str_hom} which states that\nforcing with such a partial product (as above) turns the complementary\npartial product, which was rigid in the ground model,\ninto a strongly homogeneous Souslin tree in the generic extension.\n\nThe following complements Theorem \\ref{thm:str_hom=free_x_free}.\n\n\\begin{thm}\\label{thm:str_hom=str_hom_x_str_hom}\nEvery $\\aleph_0$-branching, strongly homogeneous Souslin tree $T$\nis (iso\\-morphic to) the tree product\nof $n$ strongly homogeneous Souslin trees for any given natural number $n>0$.\n\\end{thm}\n\\begin{proof}\nThis is just a simpler variant of the construction in the proof\nof Theorem \\ref{thm:str_hom=free_x_free} where\nwe use only the first row of the matrix of partitions\n(or just any bijection between $\\omega^n$ and $\\omega$).\nIt is then easy to verify that the coherent family of $T$ descends\nto the thus obtained factor trees and renders them strongly homogeneous.\n\\end{proof}\n\n\\begin{rem}\n\\label{rem:decomp}\n\\begin{enumerate}[(i)]\n\\item Though, of course, not every tree product\nof two strongly homogeneous Souslin trees is Souslin again\n(e.g.\\ take $T\\otimes T$),\nthere is a converse to the last theorem:\nIf $S$ and $T$ are strongly homogeneous Souslin trees and\nthe tree product $S\\otimes T$ satisfies the c.c.c.,\nthen $S\\otimes T$ is a strongly homogeneous Souslin tree as well.\n\\item We see that there are two essentially distinct ways to decompose\na strongly homogeneous tree into (at least three) free factors.\nAn application of Theorem \\ref{thm:str_hom=str_hom_x_str_hom}\nto decompose a given strongly homogeneous Souslin tree $T$ into $\\ell$\nstrongly homogenous factors $S_0,\\ldots,S_{\\ell-1}$\nfollowed by an $\\ell$-fold application of the procedure used in the proof of\nTheorem \\ref{thm:str_hom=free_x_free} to decompose the tree $S_k$ into $m_k$\nfree trees $R^k_i$ for $0\\leq i < m_k$\nnever results in the same decomposition\nas directly using the proof of Theorem\n\\ref{thm:str_hom=free_x_free} to decompose $T$ into\n$\\sum_{k=0}^{\\ell-1} m_k$ free factors.\nThe partial products of the latter decomposition are all rigid by\nProposition \\ref{prp:decomp} while\nthe first also has partial products that are strongly homogeneous.\n\\end{enumerate}\n\\end{rem}\n\n\\section{Separating high degrees of rigidity}\n\\label{sec:separating}\n\nIn this chapter we review several families of rigidity notions\nfor Souslin trees, all of them weaker than freeness.\nThese definitions (except for that of an \\emph{$n$-free} Souslin tree)\nare all taken from \\cite{degrigST}.\nMost of these definitions refer to the technique of forcing applied\nwith a Souslin tree as the forcing partial order.\nWe do not review forcing here.\nBut recall,\nthat forcing with a Souslin tree always assumes the inverse order on the tree\n(i.e., trees grow downwards when considered as forcing partial orders,\nthe root is the maximal element, etc.) and adjoins a cofinal branch.\n\nThis section is divided in five short subsections.\nThe first two introduce the rigidity notions to be considered\nand the last three state many and prove some separations between them.\nWe only give proofs that either are elementary or use the\nproof of the Decomposition\nTheorem \\ref{thm:str_hom=free_x_free}.\n\n\\subsection{Parametrized freeness}\n\\label{sec:free}\n\nConsidering the definition of the property \\emph{free} for Souslin trees\nit is natural to ask whether or not it makes any difference\nif the number of the factors in the tree products,\nthat are required to be Souslin, is bounded.\nThis leads to the following definition which we rightaway connect\nto the definition of \\emph{being $n$-fold Souslin off the generic branch}\nmet in \\cite{degrigST}.\n\n\\begin{defi}\nLet $n$ be a positive natural number.\n\\begin{enumerate}[a)]\n\\item We say that a Souslin tree $T$ is \\emph{$n$-free} if for every subset\n$P$ of size $n$ of some level $T_\\alpha$, $\\alpha<\\omega_1$, the\ntree product $\\bigotimes_{s\\in P} T(s)$ satisfies the c.c.c.\n\\item A Souslin tree is said to be\n$n$\\emph{-fold Souslin off the generic branch},\nif for any sequence $\\vec{b}=(b_0,\\ldots,b_{n-1})$\ngeneric for the $n$-fold forcing product of (the inverse partial order\nof) $T$ and any node $s\\in T\\setminus\\bigcup_{i\\in n}b_i$,\nthe subtree $T(s)$ of all nodes of $T$ above $s$ is a Souslin tree\nin the generic extension $M[\\vec{b}]$ (which amounts to requiring that the\nadjunction of $\\vec{b}$ does not collapse $\\omega_1$ and preserves the\nc.c.c. of the $T(s)$, $s\\notin\\bigcup b_i$).\n\\end{enumerate}\n\\end{defi}\nIt is easy to see that a 2-free Souslin tree or a tree which is Souslin\noff the generic branch cannot be decomposed as the product of two Souslin\ntrees. And this common feature is no coincidence.\n\n\\begin{prp}\\label{prp:free_Sotgb}\nFor a positive natural number $n$ and a normal Souslin tree $T$\nthe following statements are equivalent.\n\\begin{enumerate}[a)]\n\\item $T$ is $n$-fold Souslin off the generic branch.\n\\item $T$ is $(n\\!+\\!1)$-free.\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nWe start with the implication (b$\\to$a).\nAssume that $T$ is $n+1$-free and let $\\vec{b}=(b_0,\\ldots,b_{n-1})$ be\ngeneric for $T^{\\otimes n}$, the $n$-fold tree product of $T$ with itself.\nChoose $\\alpha<\\omega_1$ large enough,\nsuch that the nodes $t_i:=b_i(\\alpha)$ are pairwise incompatible.\nFinally, pick a node $t_n\\in T_\\alpha$ distinct from all the $b_i(\\alpha)$.\nBy our freeness assumption on $T$,\nthe product tree $\\bigotimes_{i\\in{n+1}}T(t_i)$\nsatisfies the countable chain condition.\nBut then $M[\\vec{b}]\\vDash$''$T(t_n)$ is Souslin''\nby a standard argument concerning chain conditions in forcing iterations.\nNow it is easy to see that $T$ is $n$-fold Souslin off the generic branch.\n\nFor the other direction we inductively show that\n$T$ is $m$-free for $m\\leq n+1$,\nassuming that $T$ is $n$-fold Souslin off the generic branch.\nThe inductive claim is trivial for $m=1$. \nSo let $m\\geq1$ and let $s_0,\\ldots,s_m$ be pairwise distinct nodes\nof the same height.\nThen for any generic sequence\n$\\vec{b}=(b_0,\\ldots,b_{m-1})$ for $\\bigotimes_{i\\in m}T(s_i)$\nwe know that $T(s_m)$ is Souslin in the  generic extension $M[\\vec{b}]$.\nFinally the two-step iteration $\\bigotimes_{i\\in m}T(s_i)\\ast \\check{T}(s_m)$\nis isomorphic to $\\bigotimes_{i\\in m+1}T(s_i)$\nand satisfies the countable chain condition.\n\\end{proof}\n\nThis proposition implies that a free tree $T$ is also\n\\emph{free off the generic branch} in the sense that\nin the generic extension obtained by adjoining\na cofinal branch $b$ through $T$, for every node $t\\in T\\setminus b$,\nthe tree $T(t)$ is still free.\n\n\\subsection{Further types of rigidity}\n\nIn Sections 1-4 of \\cite{degrigST} different notions of rigidity\nfor Souslin trees are collected:\n(ordinary) rigidity, total rigidity and the unique branch property and their\nabsolute counterparts,\nwhere absoluteness refers to forcing extensions obtained by\nadjoining a generic branch to the Souslin tree under consideration.\nIn this context also the stronger notion of being\n($n$-fold) Souslin off the generic branch is introduced \nwhich we already considered in the last section.\n\n\\begin{defi}\n\\begin{enumerate}[a)]\n\\item A Souslin tree $T$ is called \\emph{$n$-absolutely rigid}, if $T$ is a rigid tree in the generic extension\nobtained by forcing with $T^n$ (or equivalently $T^{\\otimes n}$).\n\\item A Souslin tree is \\emph{totally rigid}, if the trees $T(s)$ and $T(t)$ are non-isomorphic for all pairs\nof distinct nodes $s$ and $t$ of $T$. It is \\emph{$n$-absolutely totally rigid} if it is totally rigid\nafter forcing with $T^n$.\n\\item A Souslin tree $T$ has the \\emph{unique branch property (UBP)}, if forcing with $T$ adjoins only a single\ncofinal branch to $T$. For $n>0$ we say, that $T$ has the $n$-\\emph{absolute UBP}, if forcing with\n$T^{n+1}$ adjoins exactly $n+1$ cofinal branches to $T$.\n\\end{enumerate}\n\\end{defi}\n\n\nFuchs and Hamkins prove implications as well as some independencies\nbetween these rigidity notions.\nThey also give in \\cite[Section 4]{degrigST}\na diagram of implications between the\ndegrees of rigidity that we have approximately\nreconstructed here for the convenience of the reader.\n\n\\begin{table}[htb]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{ccccccc}\n2-free&$\\longleftarrow$&\\makebox[2.5cm][c]{3-free}&$\\longleftarrow$&\\makebox[2.5cm][c]{4-free}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\nUBP&$\\longleftarrow$&\\makebox[2.5cm][c]{absolutely UBP}&$\\longleftarrow$&\\makebox[2.5cm][c]{2-absolutely UBP}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\ntotally rigid&$\\longleftarrow$&\\makebox[3cm][c]{abs. totally rigid}&$\\longleftarrow$&\n\\makebox[3cm][c]{2-abs. totally rigid}&$\\longleftarrow$&$\\ldots$\\\\\n$\\downarrow$&&$\\downarrow$&&$\\downarrow$&&\\\\\nrigid&$\\longleftarrow$&\\makebox[2.5cm][c]{absolutely rigid}&$\\longleftarrow$&\\makebox[2.5cm][c]{2-absolutely rigid}&$\\longleftarrow$&$\\ldots$\\\\\n&&&&&&\n\\end{tabular} \\end{small}\n\\end{center}\n\\caption{Implications between degrees of rigidity for Souslin trees.}\n\\label{diagram}\n\\end{table}\n\nFuchs and Hamkins show that\nthe part of the diagram to the left and below ``absolutely UBP''\nis complete in the sense that there are no further\ngeneral implications between these rigidity properties.\nThey ask whether the rest of the diagram\nis complete as well, cf.~\\cite[Question 4.1]{degrigST}.\nWe will show (resp. state) below that there are\nneither implications from left to the right\n(including downwards diagonals,\ncf.~Corollaries \\ref{cor:n-free} and \\ref{cor:n-free_not_UBP}),\nnor from the second to the upper row (Theorem \\ref{thm:not_simple_UBP}).\n\n\\begin{rem}\n\\label{rem:diagram}\n\\item Using a standard $\\diamondsuit$-construction\nscheme for a Souslin tree (e.g., cf. \\cite[Section 2]{degrigST})\nit is not hard to construct a Souslin tree $T$ with the following two\nfeatures:\n\\begin{itemize}\n\\item On each level $T_\\alpha$ no two distinct nodes\nhave the same number of immediate successors.\nSo in particular $T$ is $n$-absolutely totally rigid\nfor every $n\\in \\omega$.\n\\item The substructure $R$ of $T$ obtained by\nrestricting the supporting set to the nodes on the limit levels of $T$\nplus the root, is a homogeneous Souslin tree.\nThen in a generic extension obtained by forcing with $T$ there are\nmany cofinal branches in $R$ and each of them gives rise to a cofinal branch\nof $T$, which is thus not a UBP tree.\n(In fact, every $\\aleph_0$-spiltting Souslin tree can be extended to an\n$n$-absolutely totally rigid Souslin tree by inserting new successor levels \nsuch that every two nodes of the same height have a different number of\nimmediate successors.)\n\\end{itemize}\nThis shows that in Diagram \\ref{diagram} there can be no arrows that\npoint upwards from the two lower rows.\nSo the only question left open is whether there should be any more arrows\nbetween the two lower rows, but a similar construction as the one alluded to\nabove should also eliminate those.\n\\end{rem}\n\n\\subsection{Distinct degrees of freeness}\n\nOur next corollary of (the proof of)\nTheorem~\\ref{thm:str_hom=free_x_free} gives the separation of\nthe finite degrees of freeness, i.e.,\nit shows that the family of parametrized freeness\nconditions is properly increasing in strength.\n\n\\begin{cor}\\label{cor:n-free}\nIf there is a strongly homogeneous Souslin tree,\nthen there is an $n$-free, but not $n+1$-free tree.\n\\end{cor}\n\\begin{proof}\nLet the strongly homogeneous Souslin tree $T$ be decomposed\nas the tree product of $n$ free trees $S_i$ for $i<n$\nas carried out in the proof of Theorem~\\ref{thm:str_hom=free_x_free}.\nWe show, that the tree sum of the factors,\n$$R:=\\bigoplus_{i<n}S_i$$\nis an $n$-free but not $n+1$-free Souslin tree.\nThe Claim used in the proof of Theorem~\\ref{thm:str_hom=free_x_free}\nremains true in the following variant:\\\\\n{\\bf Claim'.} For any pair of sequences $(s_0,\\ldots,s_{n-1})$ in $T_\\alpha$ and $t_i>s_i$ in $T_\\beta$ and\nany sequence $m:n\\to n$ the intersection\n$$\\bigcap_{i<n}{\\psi_{s_i,s_0}}''t_i\/\\!\\!\\equiv_{m(i)}$$\nis not empty.\n\nModulo the obvious changes in the notation,\nthe proof of the Claim' remains completely the same as before,\nexploiting the $n$-optimality of the matrix.\nAnd also with the same argument as above we can construct\nan antichain of $T$ from any given antichain of $R$\nmaintaining the cardinality. So $R$ is $n$-free.\n\nWe argue that $R$ is not $n$-fold Souslin off the generic branch.\n\nIf $b_i$ is a cofinal branch through $S_i$\nthen in the generic extension obtained by adjoining\n$\\vec{b}=(b_0,\\ldots,b_{n-1})$, the strongly homogeneous\ntree $T$ has a cofinal branch as well,\nthus destroying the Souslinity of all subtrees of $R$.\n\\end{proof}\n\n\\subsection{Freeness and absolute rigidity}\n\nIn this section we improve upon the result of the last one\nby showing that $n$-freeness of a tree does not even imply\n$(n\\!-\\!1)$-absolute rigidity.\n\n\\begin{lm}\\label{lm:free_to_str_hom}\nLet the strongly homogeneous Souslin tree $T$ be\ndecomposed as the tree product of $n$ free trees $S_i$ for $i<n$\nas carried out in the proof of Theorem~\\ref{thm:str_hom=free_x_free}.\nLet $\\{a,b\\}$ be a partition of the set $n$ with $a,b\\neq\\varnothing$\nand set $\\mathbb{P}:=\\bigotimes_{i\\in a}S_i$ and $R:=\\bigotimes_{i\\in b}S_i$.\nThen\n$$\\vDash_{\\mathbb{P}}\\text{``}\\check{R}\\text{ is strongly homogeneous.''}$$\n\\end{lm}\n\\begin{proof}\nWe adopt the notation from the statement of the lemma and\nargue inside the generic extension obtained by adjoining a generic branch\n$c$ to the Souslin tree $\\mathbb{P}$.\nThen we have the natural isomorphism $\\rho:R\\cong c\\otimes R\\subset T$.\nDenote the canonical projection $T\\to R$ by $\\pi$.\n\nWe define the tree isomorphisms $\\varphi_{{r}{s}}$\n(members of the coherent family of $R$ in the generic extension)\nfor nodes ${r}$ and ${s}$ of $R_\\alpha$, $\\alpha<\\omega_1$\nfrom the members $\\psi_{\\rho({r})\\rho({s})}$ of the coherent family of $T$.\nFor this, we refer to the maps $h(\\cdot,j):T\\to\\omega$\nused in the construction of the t.e.r.s $\\equiv_j$ in the\nproof of Theorem~\\ref{thm:str_hom=free_x_free}.\nWe collect them and define $h:T\\to\\omega^{|b|}$ by\nsimply concatenating the values $h(t,j)$ for $t\\in T$ and $j\\in b$.\n\nIf ${r},\\,{s}\\in R_\\alpha$ and $h(\\rho({r}))=h(\\rho({s}))$,\nthen we let\n$$\\varphi_{{r}{s}}:=\n\\pi\\circ\\psi_{\\rho({r})\\rho({s})}\\circ\\rho:R({r})\\to R({s}).$$\nIt follows from the fact that the $n$-optimal partition matrices\nare transported between the (sets of immediate successors of the)\nnodes by the members $\\psi_{tu}$ of the coherent family of $T$,\nthat this definition is sound and indeed yields an isomorphism.\n\nNow let $r,s\\in R_\\alpha$ with $h(\\rho({r}))\\neq h(\\rho({s}))$.\nIn order to define $\\varphi_{{r}{s}}$ we compose the tree isomorphisms,\nthat we have already defined for the immediate successors of ${r}$ and ${s}$.\nFor every direct successor ${u}\\in\\NF({r})$\nthere is exactly one ${v}\\in\\NF({s})$ with $h(\\rho({u}))=h(\\rho({v}))$.\nThis follows from the $n$-optimality of the partition matrix.\nLet $\\varphi_{{r}{s}}({u})$ be just this ${v}$.\nIf ${x}$ is a non-immediate successor of ${r}$,\nthen first find the immediate successor ${u}$ of ${r}$\nbelow $x$ and the image $v=\\varphi_{rs}(u)$,\nand set\n$$\\varphi_{{r}{s}}({x}):=\\varphi_{{u}{v}}({x}).$$\n\nIt remains to prove, that the family of tree isomorphisms just defined\nis coherent, commutative and transitive.\nCommutativity and coherence are inherited from the coherent family of $T$.\n(Note that $\\varphi_{rs}(x)=y$ implies that $h(\\rho(x))=h(\\rho(y))$,\nso the two cases do not interfere.)\nAs for transitivity,\nlet $x,y\\in R_\\lambda$ for some countable limit ordinal $\\lambda$.\nThen by the transitivity of the family of the $\\psi_{tu}$ for $T$\nthere are $t<\\rho(x)$ and $u<\\rho(y)$ with $\\psi_{tu}(\\rho(x))=\\rho(y)$.\nBut then $t$ and $u$ lie in $b\\otimes R$,\nso there are $r,s\\in R$ such that\n$\\rho(r)=t$ and $\\rho(s)=u$ and thus $\\varphi_{rs}(x)=y$.\n\\end{proof}\n\nSo, e.g.\\ in the case $n=2$,\nforcing with one free tree does not only destroy the freeness of another one,\nbut even turns the latter into a strongly homogeneous Souslin tree,\ni.e., it adjoins many generic automorphisms.\n\n\n\\begin{cor}\n\\label{cor:n-free_not_UBP}\nLet $n>1$.\nIf there is a strongly homogeneous Souslin tree,\nthen there is an $n$-free tree\nwhich is not $(n\\!-\\!1)$-absolutely rigid.\n\\end{cor}\n\\begin{proof}\nWe fix $n>1$ and\nuse the tree $R$ from the proof of Corollary~\\ref{cor:n-free}\nobtained from a strongly homogeneous tree $T$\nas the tree sum $R=\\bigoplus_{i<n}S_i$\nof the free factors $S_i$, $i<n$ of $T$.\nFrom Corollary~\\ref{cor:n-free} we know, that $R$ is $n$-free.\n\nTo show that $R$ is not $(n\\!-\\!1)$-absolutely rigid we refer\nto Lemma~\\ref{lm:free_to_str_hom}.\nIt follows directly from the case that $a=n\\setminus\\{i\\}$ for some $i<n$\nthat $R$ is not rigid in the generic extension obtained by adjoining\na cofinal branch through the trees $S_j$ for $j<n$ and $j\\neq i$.\nBut this generic extension can also be reached by forcing with\n$R^{\\otimes n-1}$.\n\\end{proof}\n\n\\subsection{Freeness and the unique branch property}\n\nWe start with an easy result deduced from the elementary properties\nof finitely free trees for the second column of the diagram.\n\n\\begin{prp}\n\\label{prp:3-free+UBP}\nIf there is a 3-free Souslin tree,\nthen there is also a Souslin tree which has the UBP and is not 2-free.\n\\end{prp}\n\\begin{proof}\nLet $T$ be $3$-free and pick distinct nodes $s,t\\in T$ of the same height.\nWe show, that the Souslin tree $S=T(s)\\otimes T(t)$ has the UBP\\@.\nLet $b\\otimes c$ be a generic,\ncofinal branch in $S$ (viewing $b$ and $c$ as trees).\nBy the 2-fold Souslinity off the generic branch of $T$,\nevery tree of the form $T(r)$ with $r\\in T\\setminus (b\\cup c)$\nis Souslin in the generic extension by $b\\otimes c$.\nOn the other hand,\nif there was a second cofinal branch through $S$ in the generic extension,\nthen one of its components would have to pass\nthrough such a node $r\\notin b\\cup c$, which yields a contradiction.\n\nTo prove that $S$ is not $2$-free,\nlet $u>s$ and $v,w>t$ be of the same height, where $v\\ne w$.\nThen\n$$S(u,v)\\otimes S(u,w) \\cong T(u)\\otimes T(v)\\otimes T(u)\\otimes T(w)$$\nhas an uncountable antichain,\nbecause it has the square of the Souslin tree $T(u)$ as a factor.\n\\end{proof}\n\nThis result cannot be improved by simply requiring $T$ to be free, because\nby iterating the forcing with a tree product of two factors\n$n\\!+\\!1$ times, we adjoin at least $2^n$ cofinal branches.\n\nWe do have the following non-implication result for the $n$-absolute UBP\nand $2$-freeness under the stronger assumption of $\\diamondsuit$.\n\n\\begin{thm}\n\\label{thm:not_simple_UBP}\nAssume $\\diamondsuit$.\nThen there is a Souslin tree\nwhich is not 2-free but has the $n$-absolute UBP for all $n\\in\\omega$.\n\\end{thm}\n\nThe methods of proof for this theorem lie beyond the scope of this paper.\nIt uses ideas from \\cite{degrigST} and \\cite{SAE}.\nA proof-sketch can be found in \\cite[Theorem 1.6.3]{diss}.\n\n\\subsection{Further directions}\n\nAs a closing remark we mention how Diagram \\ref{diagram},\nwhich captures the implications between four families of rigidity notions and\nimplications between them, could possibly be extended.\n\n\\begin{description}\n\\item[Real rigidity] \nIn \\cite{abraham-shelah_aronszajn_trees} two Aronszajn trees are called\n\\emph{really different} if there is no isomorphism between any of their\nrestrictions to some club set of levels. In this vein, we could call a Souslin\ntree \n\\emph{really rigid} if all of its restrictions to club sets of levels are\nrigid. This property is clearly stronger than ordinary rigidity yet\nindependent of total rigidity (cf. Remark \\ref{rem:diagram}) and is implied\nby the unique branching property. \nAlso the variant of \\emph{real, total\n  rigidity} and the $n$-absolute versions of real and of real, total rigidity\ncould be considered. \n\n\\item[Self-specializing trees] A normal tree $T$ of height $\\omega_1$ is called \\emph{special} if there is a\ncountable family $(A_n)_{n\\in\\omega}$ of antichains of $T$ that covers all of\n$T$. As $T$ is uncountable, one of the $A_n$ has to be uncountable as well,\nso a special tree $T$ is not Souslin.\nOn the other hand, every branch of $T$ meets each antichain $A_n$\nin at most one node and is therefore countable.\n\nA \\emph{self-specializing tree} is a Souslin tree $T$ that specializes itself\nby forcing a generic branch $b$ through it, i.e., in the generic extension\nobtained by adjoining $b$ to the universe, the tree $T\\setminus b$ is special.\nSelf-specializing trees can be found in models of $\\diamondsuit$.\nThey are UBP: a second cofinal branch in $T$ would prevent $T\\setminus b$\nfrom being special. But of course they are not Souslin off the generic branch, \nand they can neither be 2-absolutely really rigid nor absolutely UBP,\nbecause forcing with a special tree collapses $\\omega_1$,\nand in this second generic extension the limit levels of $T$ form an\n$\\aleph_0$-splitting tree of countable height which must be homogeneous\nby a result of Kurepa (cf.\\cite[p.102]{kurepa}). \n\nNow let us call a Souslin tree $T$ \\emph{$n$-self-specializing} if it is\n$n$-free (i.e. $(n\\!-\\!1)$-fold Souslin off the generic branch) and\nforcing a generic branch $\\vec{b}$ through $T^n$ makes $T\\setminus\\tilde{b}$\nspecial where $\\tilde{b}$ is the set of components of the elements of\n$\\vec{b}$. It is not yet verified but seems quite plausible\nthat one can construct an $n$-self-specializing tree under $\\diamondsuit$.\nIn the implication diagram its place could be between $n$-free\nand $(n\\!-\\!1)$-absolutely UBP, yet it is stronger than both of these\nproperties. And there would be no horizontal implications, for\nan $n$-self-specializing tree is neither $(n\\!-\\!1)$-self-specializing nor\n$(n\\!+\\!1)$-self-specializing.\n\n\n\\end{description}\nAs is clear from the outset, adding these families to Diagram\n\\ref{diagram} results in a far more complicated directed graph which is in\nparticular non-planar.\nWe leave such considerations for future work.\n\n\\subsection*{Acknowledgements}\nThanks are due to Piet Rodenburg for pointing out a flaw in the proof of Lemma\n\\ref{lm:opt_mtx_prt}.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nGravitational lensing is the process in which light from background galaxies is deflected as it travels towards us. The deflection is a result of the gravitation of the intervening mass.\nMeasuring the deformations in a large sample of galaxies offers a direct probe of the matter distribution in the Universe (including dark matter) and can thus be directly compared to theoretical models of structure formation. The statistical properties of the weak-lensing field can be assessed by a statistical analysis of either the shear field or the convergence field.\nOn the one hand, convergence is a direct tracer of the total matter distribution integrated along the line of sight, and is therefore directly linked with the theory. On the other hand, the shear (or more exactly, the reduced shear) is a direct observable and usually preferred for simplicity reasons.\n\nAccordingly, the most common method for characterising the weak-lensing field distribution is the shear two-point correlation function. It is followed very closely by the mass-aperture two-point correlation functions, which are the result of convolving the shear two-point correlation functions by a compensated filter \\citep{2pcf:schneider02} that is able to separate the E and B modes of the two-point correlation functions \\citep{wl:crittenden02}.\nHowever, gravitational clustering is a non-linear process, and in particular, the mass distribution is highly non-Gaussian at small scales. For this reason, several estimators of the three-point correlation functions have been proposed, either in the shear field \\citep{wl:bernardeau02,wl:benabed06} or using the mass-aperture filter \\citep{map:kilbinger05}. The three-point correlation functions are the lowest order statistics to quantify non-Gaussianity in the weak-lensing field and thus provide additional information on structure formation models.\n\nThe convergence field can also be used to measure the two- and three-point correlation functions and other higher-order statistics. When we assume that the mass inversion (the \ncomputation of the convergence map from the measured shear field) is properly conducted, the shear field contains the same information as the convergence maps \\citep[e.g.][]{2pcf:schneider02,3pcf:shi11}. While it carries the same information, the lensing signal is more compressed in the convergence maps than in the shear field, which makes it easier to extract and computationally less expensive.\nThe convergence maps becomes a new tool that might bring additional constraints complementary to those that we can obtain from the shear field.\nHowever, the weak-lensing signal being highly non-Gaussian at small scales, mass-inversion methods using smoothing or de-noising to regularise the problem are not optimal.\n\nReconstructing convergence maps from weak lensing is a difficult task because of shape noise, irregular sampling, complex survey geometry, and the fact that the shear is not a direct observable.\nThis is an ill-posed inverse problem and requires regularisation to avoid pollution from spurious B modes.\nSeveral methods have been derived to reconstruct the projected mass distribution from the observed shear field. The first non-parametric mass reconstruction was proposed by \\cite{wl:kaiser93} and was further improved by \\cite{wl:bartelmann95}, \\cite{wl:kaiser95}, \\cite{wl:schneider95}, and \\cite{wl:squires96}. These linear inversion methods are based on smoothing with a fixed kernel, which acts as a regularisation of the inverse problem. Non-linear reconstruction methods were also proposed using different sets of priors and noise-regularisation techniques \\citep{wl:bridle98,wl:seitz98,wl:marshall02,wl:pires09,wl:jullo14, wl:lanusse16}.\nConvergence mass maps have been built from many surveys, including\nthe COSMOS Survey \\citep{cosmos:massey07}, \nthe Canada France Hawa\\\"i Telescope Lensing Survey CFHTLenS \\citep{cfhtlens:vanwaerbeke13},  the CFHT\/MegaCam Stripe-82 Survey \\citep{cs82:shan14}, \nthe Dark Energy Survey Science Verification DES SV \\citep{dessv:chang15,dessv:vikram15,dessv:jeffrey18}, \nthe Red Cluster Sequence Lensing Survey RCSLenS \\citep{rcslens:hildebrandt16}, and the Hyper SuprimeCam Survey \\citep{hsc:oguri18}.\nWith the exception of \\cite{dessv:jeffrey18}, who used the non-linear reconstruction proposed by \\cite{wl:lanusse16}, all these methods are based on the standard Kaiser \\& Squires method.\n \n \nIn the near future, several wide and deep weak-lensing surveys are planned: Euclid \\citep{euclid:laureijs11}, Large Synoptic Survey Telescope LSST \\citep{lsst:abell09}, and Wide Field Infrared Survey Telescope WFIRST \\citep{wfirst:green12}.\nIn particular, the Euclid satellite will survey 15 000 deg$^2$ of the sky to map the geometry of the dark Universe.\nOne of the goals of the Euclid mission is to produce convergence maps for non-Gaussianity studies and constrain cosmological parameters.\nTo do this, two different mass inversion methods are being included into the official Euclid data processing pipeline. \nThe first method is the standard Kaiser \\& Squires method (hereafter KS). Although it is well known that the KS method has several shortcomings, it is taken as the reference for cross-checking the results. The second method is a new non-linear mass-inversion method (hereafter KS+) based on the formalism developed in \\cite{wl:pires09}. The KS+ method aims at performing the mass inversion with minimum information loss. This is done by performing the mass inversion with no other regularisation than binning while controlling systematic effects. \n  \n\nIn this paper, the performance of these two mass-inversion methods is investigated using the Euclid Flagship mock galaxy catalogue (version 1.3.3, Castander F. et al. in prep) with realistic observational effects (i.e. shape noise, missing data, and the reduced shear). The effect of intrinsic alignments is not studied in this paper because we lack simulations that would properly model  intrinsic alignments.\nHowever, intrinsic alignments also need to be considered seriously because they affect second- and higher-order statistics. A contribution of several percent is expected to two-point statistics  \\citep[see e.g.][]{ia:joachimi13}.\n\nWe compare the results obtained with the KS+ method to those obtained with a version of the KS method in which no smoothing step is performed other than binning. \nWe quantify the quality of the reconstruction using two-point correlation functions and moments of the convergence.\nOur tests illustrate the efficacy of the different mass-inversion methods in preserving the second-order statistics and higher-order moments.\n\n\n\n\nThe paper is organised as follows.\nIn Sect. 2 we present the weak-lensing mass-inversion problem and the standard KS method.\nSection 3 presents the KS+ method we used to correct for the different systematic effects.\nIn Sect. 4 we explain the method with which we compared these two mass-inversion methods.\nIn Sect. 5 we use the Euclid Flagship mock galaxy catalogue with realistic observational effects such as shape noise and complex survey geometry and consider the reduced shear to investigate the performance of the two mass-inversion methods. First, we derive simulations including only one issue at a time to test each systematic effect independently. Then we derive realistic simulations that include them all and study the systematic effects simultaneously.\nWe conclude in Sect. 6.\n\n\\section{Weak-lensing mass inversion}\n\\label{inversion}\n\n\\subsection{Weak gravitational lensing formalism}\n\\label{formalism}\nIn weak-lensing surveys, the shear field $\\gamma({\\vec{\\theta}})$ is derived from the ellipticities of the background galaxies at position {\\vec{\\theta}} in the image. The two components of the shear can be written in terms of the lensing potential $\\psi({\\vec{\\theta}})$ as \\citep[see e.g.][]{wl:bartelmann01}\n\\begin{eqnarray}\n\\label{eq:gamma_psi} \n\\gamma_1 & = & \\frac{1}{2}\\left( \\partial_1^2 - \\partial_2^2 \\right) \\psi, \\nonumber \\\\\n\\gamma_2 & = & \\partial_1 \\partial_2 \\psi,\n\\end{eqnarray}\nwhere the partial derivatives $\\partial_i$ are with respect to the angular coordinates $\\theta_i$, $i = 1,2$ representing the two dimensions of sky coordinates. \nThe convergence $\\kappa({\\vec{\\theta}})$ can also be\nexpressed in terms of the lensing potential as\n\\begin{eqnarray}\n\\label{eq:kappa_psi}\n\\kappa =  \\frac{1}{2}\\left(\\partial_1^2 + \\partial_2^2 \\right) \\psi.\n\\end{eqnarray}\nFor large-scale structure lensing, assuming a spatially flat Universe, the convergence at a sky position ${\\vec{\\theta}}$ from sources at comoving distance $r$ is defined by \n\\begin{eqnarray}\n\\label{eq:kappa_r}\n\\kappa(\\vec{\\theta}, r) =\\frac{3H^2_0\\Omega_{\\rm m}}{2 c^2}\\int_0^r {\\rm d}r' \\frac{r'(r-r')}{r} \\frac{\\delta(\\vec{\\theta}, r')}{a(r')},\n\\end{eqnarray}\nwhere $H_0$ is the Hubble constant, $\\Omega_{\\rm m}$ is the matter density, $a$ is the scale factor, and $\\delta \\equiv (\\rho-\\bar\\rho)\/\\bar\\rho$ is the density contrast (where $\\rho$ and $\\bar\\rho$ are the 3D density and the mean 3D density, respectively).\nIn practice, the expression for $\\kappa$ can be generalised to sources with a distribution in redshift, or equivalently, in comoving distance $f(r)$, yielding\n\\begin{eqnarray}\n\\label{eq:kappa}\n\\kappa(\\vec{\\theta}) =\\frac{3H^2_0\\Omega_{\\rm m}}{2 c^2}\\int_0^{r_{\\rm H}} {\\rm d}r'p(r')r' \\frac{\\delta(\\vec{\\theta}, r')}{a(r')},\n\\end{eqnarray}\nwhere $r_{\\rm H}$ is the comoving distance to the horizon.\nThe convergence map reconstructed over a region on the sky gives us the integrated mass-density fluctuation weighted by the lensing-weight function $p(r')$,\n\\begin{eqnarray}\n\\label{eq:kappa_r}\np(r') =\\int_{r'}^{r_{\\rm H}} {\\rm d}r f(r)\\frac{r-r'}{r}.\n\\end{eqnarray}\n\n\n\n\n\\subsection{Kaiser \\& Squires method (KS)}  \n\\label{ks}\n\n\n\\subsubsection{KS mass-inversion problem}  \n\n\nThe weak lensing mass inversion problem consists of reconstructing the convergence $\\kappa$ from the measured shear field $\\gamma$. We can use complex notation to represent the shear field, $\\gamma = \\gamma_1 + {\\rm i} \\gamma_2$, and the convergence field, $\\kappa = \\kappa_{\\rm E} + {\\rm i} \\kappa_{\\rm B}$, with $\\kappa_{\\rm E}$ corresponding to the curl-free component and $\\kappa_{\\rm B}$ to the gradient-free component of the field, called E and B modes by analogy with the electromagnetic field.\nThen, from Eq.~(\\ref{eq:gamma_psi}) and Eq.~(\\ref{eq:kappa_psi}), we can derive the relation between the shear field $\\gamma$  and the convergence field $\\kappa$.\nFor this purpose, we take the Fourier transform of these equations and obtain\n\\begin{eqnarray}\n\\label{eq:kappa2gamma}\n\\hat \\gamma = \\hat P \\, \\hat \\kappa,\n\\end{eqnarray}\nwhere the hat symbol denotes Fourier transforms, $\\hat P = \\hat P_1 + {\\rm i} \\hat P_2$,\n\\begin{eqnarray}\n\\hat{P_1}(\\vec{\\ell}) & = & \\frac{\\ell_1^2 - \\ell_2^2}{\\ell^2}, \\nonumber \\\\\n\\hat{P_2}(\\vec{\\ell}) & = & \\frac{2 \\ell_1 \\ell_2}{\\ell^2},\n\\end{eqnarray}\nwith $\\ell^2 \\equiv \\ell_1^2 + \\ell_2^2$ and $\\ell_i$ the wave numbers corresponding to the angular coordinates $\\theta_i$. \n\n$\\hat P$ is a unitary operator. The inverse operator is its complex conjuguate $\\hat P^* = \\hat P_1 - {\\rm i} \\hat P_2$ , as shown by \\cite{wl:kaiser93},\n\\begin{eqnarray}\n\\label{eq:gamma2kappa}\n\\hat \\kappa =  \\hat P^* \\, \\hat \\gamma.\n\\end{eqnarray}\nWe note that to recover $\\kappa$ from $\\gamma,$ there is a degeneracy when $\\ell_1 = \\ell_2 = 0$. Therefore the mean value of $\\kappa$ cannot be recovered if only shear information is available. This is the so-called mass-sheet degeneracy \\citep[see e.g.][for a discussion]{wl:bartelmann95}.\nIn practice, we impose that the mean convergence vanishes across the survey by setting the reconstructed $\\ell = 0$ mode to zero. This is a reasonable assumption for large-field reconstruction \\citep[e.g.][]{wl:seljak98}.\n\n\nWe can easily derive an estimator of the E-mode and B-mode convergence in the Fourier domain,\n\\begin{eqnarray}\n\\label{eq:fourier}\n\\hat{\\tilde \\kappa}_{\\rm E} &=& \\hat P_1 \\hat \\gamma_1  +  \\hat P_2 \\hat \\gamma_2,\\\\ \\nonumber \n\\hat{\\tilde \\kappa}_{\\rm B} &=& - \\hat P_2 \\hat \\gamma_1 +  \\hat P_1 \\hat \\gamma_2.\n\\end{eqnarray}\nBecause the weak lensing arises from a scalar potential (the lensing potential $\\psi$), it can be shown that weak lensing only produces E modes. However, intrinsic alignments and imperfect corrections of the point spread function (PSF) generally generate both E and B modes. The presence of B modes can thus be used to test for residual systematic effects in current weak-lensing surveys.\n\n\\subsubsection{Missing-data problem in weak lensing}\nThe shear is only sampled at the discrete positions of the galaxies where the ellipticity is measured. \nThe first step of the mass map-inversion method therefore is to bin the observed ellipticities of galaxies on a regular pixel grid to create what we refer to as the observed shear maps $\\gamma^{\\rm{obs}}$.\nSome regions remain empty because various masks were applied to the data, such as the masking-out of bright stars or camera CCD defects. In such cases, the shear is set to zero in the original KS method,\n\\begin{eqnarray}\n\\label{eq:mask}\n\\gamma^{\\rm{obs}} &=& M \\gamma^{\\rm n},\n\\end{eqnarray}\nwith $M$ the binary mask (i.e. $M = 1$ when we have information at the pixel, $M = 0$ otherwise) and $\\gamma^{\\rm n}$ the noisy shear maps.\nAs the shear at any sky position is non-zero in general, this introduces errors in the reconstructed convergence maps.\nSome specific methods address this problem by discarding masked pixels at the noise-regularisation step \\cite[e.g.][]{cfhtlens:vanwaerbeke13}. However, as explained previously, this intrinsic filtering results in subtantial signal loss at small scales. Instead, inpainting techniques are used in the KS+ method to fill the masked regions (see Appendix A).\n\n\n\n\\subsubsection{Weak-lensing shape noise}\n\\label{sect_shape_noise}\nThe gravitational shear is derived from the ellipticities of the background galaxies.\nHowever, the galaxies are not intrinsically circular, therefore their measured ellipticity is a combination of their intrinsic ellipticity and the gravitational lensing shear. \nThe shear is also subject to measurement noise and uncertainties in the PSF correction. All these effects can be modelled as an additive noise, $N = N_1 + {\\rm i} N_2$,\n\\begin{eqnarray}\n\\gamma^{\\rm n} &=& \\gamma+ N\n\\label{eq:noise1}\n\\end{eqnarray}\nThe noise terms $N_1$ and $N_2$ are assumed to be Gaussian and uncorrelated with zero mean and standard deviation,\n\\begin{eqnarray}\n\\sigma_{\\rm n}^i = \\frac{\\sigma_{\\rm \\epsilon}}{\\sqrt{N_{\\rm g}^i}}, \n\\label{eq:noise2}\n \\end{eqnarray}\nwhere $N_{\\rm g}^i$ is the number of galaxies in pixel $i$.\nThe root-mean-square shear dispersion per galaxy, $\\sigma_{\\rm \\epsilon}$, arises both from the measurement uncertainties and the intrinsic shape dispersion of galaxies. The Gaussian assumption is a reasonable assumption, and $\\sigma_{\\rm \\epsilon}$ is set to 0.3 for each component as is generally found in weak-lensing surveys \\citep[e.g.][]{sigmae:leauthaud07, sigmae:schrabback15, sigmae:schrabback18}. The surface density of usable galaxies is expected to be around $n_{\\rm g} = 30$ gal. arcmin$^{-2}$  for the Euclid Wide survey \\citep{euclid:cropper13}.\n\n\nThe derived convergence map is also subject to an additive noise,\n\\begin{eqnarray}\n\\hat{\\tilde \\kappa}^{\\rm n} = \\hat P^* \\, \\hat{ \\gamma}^{\\rm n} = \\hat \\kappa + \\hat P^* \\, \\hat{N}.\n\\label{kappan}\n\\end{eqnarray}\nIn particular, the E component of the convergence noise is\n\\begin{eqnarray}\nN_{\\rm E}= N_1* P_1 + N_2 * P_2 , \n\\label{eq:noise3}\n\\end{eqnarray}\nwhere the asterisk denotes the convolution operator, and $P_1$ and $P_2$ are the inverse Fourier transforms of $\\hat{P_1}$ and $\\hat{P_2}$.\nWhen the shear noise terms $N_1$ and $N_2$ are Gaussian, uncorrelated, and with a constant standard deviation across the field, the convergence noise is also Gaussian and uncorrelated.\nIn practice, the number of galaxies varies slightly across the field. The variances of $N_1$ and $N_2$ might also be slightly different, which can be modelled by different values of $\\sigma_{\\epsilon}$ for each component. These effects introduce noise correlations in the convergence noise maps, but they were found to remain negligible compared to other effects studied in this paper.\n\n\nIn the KS method, a smoothing by a Gaussian filter is frequently applied to the background ellipticities before mass inversion to regularise the solution.\nAlthough performed in most applications of the KS method, this noise regularisation step is not mandatory. It was introduced to avoid infinite noise and divergence at very small scales.\nHowever, the pixelisation already provides an intrinsic regularisation. This means that there is no need for an additional noise regularisation prior to the inversion. Nonetheless, for specific applications that require denoising in any case, the filtering step can be performed before or after the mass inversion.\n\n\n\\section{Improved Kaiser \\& Squires method (KS+)}  \n\\label{iks}\n\nSystematic effects in mass-inversion techniques must be fully controlled in order to use convergence maps as cosmological probes for future wide-field weak-lensing experiments such as \\textit{Euclid}\\xspace. \nWe introduce the KS+ method based on the formalism developed in \\cite{wl:pires09} and \\cite{wl:jullo14}, which integrates the necessary corrections for imperfect and realistic measurements.\nWe summarise its improvements over KS in this section and evaluate its performance in Sect. \\ref{results_1}.\n\n\nIn this paper, the mass-mapping formalism is developed in the plane. \nThe mass inversion can also be performed on the sphere, as proposed in \\cite{sphere:pichon09} and \\cite{sphere:chang18}, and the extension of the KS+ method to the curved sky is being investigated. \nHowever, the computation time and memory required to process the spherical mass inversion means limitations in terms of convergence maps resolution and\/or complexity of the algorithm.\nThus, planar mass inversions remain important for reconstructing convergence maps with a good resolution and probing the non-Gaussian features of the weak-lensing field (e.g. for peak-count studies).\n\n\\subsection{Missing data}\n\\label{KS+}\n\nWhen the weak-lensing shear field $\\gamma$ is sampled on a grid of $N \\times N$ pixels, we can describe the complex shear and convergence fields by their respective matrices. In the remaining paper, the notations $\\bm \\gamma$ and $\\bm \\kappa$ stand for the matrix quantities.\n\n\n\n\nIn the standard version of the KS method, the pixels with no galaxies are set to zero.\nFig.~\\ref{shear_mask} shows an example of simulated shear maps without shape noise derived from the Euclid Flagship mock galaxy catalogue (see Sect. \\ref{sect_simu} for more details). The upper panels of Fig.~\\ref{shear_mask} show the two components of the shear with zero values (displayed in black) corresponding to the mask of the missing data.\nThese zero values generate an important leakage during the mass inversion.\n\\begin{figure*}[h]\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=g1_mask,height=6.cm,width=7.cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=g2_mask,height=6.cm,width=7.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=g1_inp_mask,height=6.cm,width=7.cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=g2_inp_mask,height=6.cm,width=7.cm,clip=}\n}}\n}\n\\caption{Simulated shear maps with missing data covering a field of $5^\\circ \\times 5 ^\\circ$. The left panels show the first component of the shear $\\gamma_1$ , and the right panels present the second component of the shear $\\gamma_2$. The upper panels show the incomplete shear maps, where the pixels with no galaxies are set to zero (displayed in black). The lower panels show the result of the inpainting method that allows us to fill the gaps judiciously.}\n\\label{shear_mask}\n\\end{figure*}\n\n\nWith KS+, the problem is reformulated by including additional assumptions to regularise the problem. \nThe convergence $\\bm{\\kappa}$ can be analysed using a transformation $\\mathbf \\Phi, $ which yields a set of coefficients $\\bm \\alpha = \\mathbf \\Phi^{\\rm T} \\bm{\\kappa}$ ($\\mathbf \\Phi$ is an orthogonal matrix operator, and $\\mathbf \\Phi^{\\rm T}$ represents the transpose matrix of $\\mathbf \\Phi$).\nIn the case of the Fourier transformation, $\\mathbf \\Phi^{\\rm T}$ would correspond to the discrete Fourier transform (DFT) matrix, and $\\bm \\alpha$ would be the Fourier coefficients of $\\bm{\\kappa}$.\nThe KS+ method uses a prior of sparsity, that is, it assumes that there is a transformation $\\mathbf \\Phi$ where the convergence $\\bm{\\kappa}$ can be decomposed into a set of coefficients $\\bm \\alpha$, where most of its coefficients are close to zero. \nIn this paper, $\\mathbf \\Phi$ was chosen to be the discrete cosine transform (DCT) following \\cite{wl:pires09}. The DCT expresses a signal in terms of a sum of cosine functions with different frequencies and amplitudes. It is similar to the DFT, but uses smoother boundary conditions. This provides a sparser representation. Hence the use of the DCT for JPEG compression.\n\nWe can rewrite the relation between the observed shear $\\bm{ \\gamma}^{\\rm{obs} }$ and the noisy convergence $\\bm{\\kappa}^{\\rm n}$ as\n\\begin{eqnarray}\n\\bm{\\gamma}^{\\rm{obs}} =\\mathbf M \\mathbf{P}  \\bm{\\kappa}^{\\rm n},\n\\label{miss}\n\\end{eqnarray}\nwith $\\mathbf M$ being the mask operator and $\\mathbf P$ the KS mass-inversion operator.\nThere is an infinite number of convergence $\\bm{\\kappa}^{\\rm n}$ that can fit the observed shear $\\bm \\gamma^{\\rm{obs}}$.\nWith KS+, we first impose that the mean convergence vanishes across the survey, as in the KS method. \nThen, among all possible solutions, KS+ searches for the sparsest solution $\\tilde{\\bm{\\kappa}}^{\\rm n}$ in the DCT $\\mathbf \\Phi$ (i.e. the convergence $\\bm{\\kappa}^{\\rm n}$ that can be represented with the fewest large coefficients). \nThe solution of this mass-inversion problem is obtained by solving\n\\begin{equation}\n\\min_{\\bm{\\tilde{\\kappa}}^{\\rm n}}  \\| \\mathbf \\Phi^{\\rm T} \\bm{\\tilde{\\kappa}}^{\\rm n}  \\|_0    \\textrm{ subject to }  \\parallel  \\bm{\\gamma}^{\\rm{obs}} - \\mathbf M  \\mathbf P   \\bm{\\tilde{\\kappa}}^{\\rm n}   \\parallel^2 \\le \\sigma^2,\n\\label{eq1}\n\\end{equation}\nwhere  $|| z ||_0$ the pseudo-norm, that is, the number of non-zero entries in $z$,  $|| z ||$ the classical $l_2$ norm (i.e. $|| z || =\\smash{\\sqrt{ \\sum_{k}(z_{k})^2}}$), and $\\sigma$ stands for the  standard deviation of the input shear map measured outside the mask.\nThe solution of this optimisation task can be obtained through an iterative thresholding algorithm called morphological component analysis (MCA), which was introduced by \\cite{mca:elad05} and was adapted to the weak-lensing problem in \\cite{wl:pires09}. \n\n\\cite{wl:pires09} used an additional constraint to force the B modes to zero. This is optimal when the shear maps have no B modes. However, any real observation has some residual B modes as a result of intrinsic alignments, imperfect PSF correction, etc. The B-mode power is then transferred to the E modes, which degrades the E-mode convergence reconstruction. We here instead let the B modes free, and an additional constraint was set on the power spectrum of the convergence map. \nTo this end, we used a wavelet transform to decompose the convergence maps into a set of aperture mass maps using the starlet transform algorithm \\citep{starck:book98,starck:book02}. \nThen, the constraint consists of renormalising the standard deviation (or equivalently, the variance) of each aperture mass map inside the mask regions to the values measured in the data, outside the masks, and then reconstructing the convergence through the inverse wavelet transform.\nThe variance per scale corresponding to the power spectrum at these scales allows us to constrain a broadband power spectrum of the convergence $\\bm \\kappa$ inside the gaps.\n\n\nAdding the power spectrum constraints yields the final sparse optimisation problem,\n\\begin{equation}\n\\min_{\\bm{\\tilde{\\kappa}}^{\\rm n}}  \\| \\mathbf \\Phi^{\\rm T} \\bm{\\tilde{\\kappa}}^{\\rm n}  \\|_0    \\textrm{ s.t. }  \\parallel \\bm{\\gamma}^{\\rm{obs}} - \\mathbf M \\mathbf P \\mathbf{W^{\\rm T}} \\mathbf Q  \\mathbf{W} \\bm{\\tilde{\\kappa}}^{\\rm n} \\parallel^2 \\le \\sigma^2,\n\\label{eq2}\n\\end{equation}\nwhere $\\mathbf{W}$ is the forward wavelet transform and $\\mathbf{W^{\\rm T}}$ its inverse transform, and $\\mathbf Q$ is the linear operator used to impose the power spectrum constraint.\nMore details about the KS+ algorithm are given in Appendix A.\n\nThe KS+ method allows us to reconstruct the in-painted convergence maps and the corresponding in-painted shear maps, where the empty pixels are replaced by non-zero values. These interpolated values preserve the continuity of the signal and reduce the signal leakage during the mass inversion (see lower panels of Fig.~\\ref{shear_mask}).  \nThe quality of the convergence maps reconstruction with respect to missing data is evaluated in Sect. \\ref{results_1}.\nAdditionally, the new constraint allows us to use the residual B modes of the reconstructed maps to test for the presence of residual systematic effects and possibly validate the shear measurement processing chain.\n\n\n\n\n\\subsection{Field border effects}\n\\label{border}\n\n\n\n\\begin{figure*}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=g1_border,height=6.cm,width=7.cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=g2_border,height=6.cm,width=7.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=g1_inp_border,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=g2_inp_border,height=6.cm,width=7cm,clip=}\n}}\n}\n\\caption{Upper panels: Simulated shear maps covering a field of $5^\\circ \\times 5 ^\\circ$ ,  extended to a field of $10^\\circ \\times 10^\\circ$ by zero padding (zero values are displayed in black). Lower panels: Result of the inpainting method that allows us to extrapolate the shear on the borders. The left panels show the first component of the shear $\\gamma_1$, and the right panels present the second component of the shear $\\gamma_2$.}\n\\label{shear_border}\n\\end{figure*}\n\n\n\nThe KS and KS+ mass-inversion methods relate the convergence and the shear fields in Fourier space.\nHowever, the discrete Fourier transform implicitly assumes that the image is periodic along both dimensions. Because there is no reason for opposite borders to be alike, the periodic image generally presents strong discontinuities across the frame border. These discontinuities cause several artefacts at the borders of the reconstructed convergence maps. The field border effects can be addressed by removing the image borders, which throws away a large fraction of the data.\nDirect finite-field mass-inversion methods have also been proposed \\citep[e.g.][]{wl:seitz96,wl:seitz01}. Although unbiased, convergence maps reconstructed using these methods are noisier than those obtained with the KS method.\nIn the KS+ method, the problem of borders is solved by taking larger support for the image and by considering the borders as masked regions to be in-painted.\nThe upper panels of Fig.~\\ref{shear_border} show the two components of a shear map covering $5^{\\circ} \\times 5^{\\circ}$ degrees and extending to a field of $10^{\\circ} \\times 10^{\\circ}$.\nThe inpainting method is then used to recover the shear at the field boundaries, as shown in the lower panels of Fig.~\\ref{shear_border}. \nAfter the mass inversion is performed, the additional borders are removed. This technique reduces the field border effects by pushing the border discontinuities farther away. \n\n\n\\subsection{Reduced shear}\n\\label{reduced}\nIn Sect. \\ref{ks} we assumed knowledge of the shear, in which case the mass inversion is linear.\nIn practice, the observed galaxy ellipticity is not induced by the shear $\\gamma,$ but by the reduced shear $g$ that depends on the convergence $\\kappa$ corresponding to that particular line of sight,\n\\begin{eqnarray}\ng \\equiv \\frac{\\gamma}{1-\\kappa}.\n\\label{reducedshear}\n\\end{eqnarray}\nWhile the difference between the shear $\\gamma$ and the reduced shear $g$ is small in the regime of cosmic shear ($\\kappa \\ll 1$), neglecting it might nevertheless cause a measurable bias at small angular scales \\citep[see e.g.][]{wl:white05, wl:shapiro09}.\nIn the standard version of KS, the Fourier estimators are only valid when the convergence is small ($\\kappa \\ll 1$),\nand they no longer hold near the centre of massive galaxy clusters.\nThe mass-inversion problem becomes non-linear, and it is therefore important to properly account for reduced shear.\n\nIn the KS+ method, an iterative scheme is used to recover the E-mode convergence map, as proposed in \\cite{wl:seitz95}. The method consists of solving the linear inverse problem iteratively (see Eq.~\\ref{eq:fourier}), using at each iteration the previous estimate of the E-mode convergence to correct the reduced shear using Eq.~(\\ref{reducedshear}). \nEach iteration then provides a better estimate of the shear. This iterative algorithm was found in \\cite{wl:jullo14} to quickly converge to the solution (about three iterations). The KS+ method uses the same iterative scheme to correct for reduced shear, and we find that it is a reasonable assumption in the case of large-scale structure lensing.\n\n\\subsection{Shape noise}\n\n\\label{section:shearnoise}\n\n\n\n\\begin{figure*}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=kappa_512,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=kappa_noise,height=6.cm,width=7.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kappa_g5,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=kappa_mrlens,height=6.cm,width=7cm,clip=}\n}}\n\n}\n\\caption{Shape-noise effect. The upper panels show the original E-mode convergence $\\kappa$ map (left) and the noisy convergence map with $n_{\\rm g} = 30$ gal. arcmin$^{-2}$ (right).  The lower panels show the reconstructed maps using a linear Gaussian filter with a kernel size of $\\sigma = 3\\arcmin$ (left) and the non-linear MRLens filtering using $\\alpha_{\\rm FDR} = 0.05$ (right). The field is $5^\\circ \\times 5^\\circ$ downsampled to $512 \\times 512$ pixels.}\n\n\\label{kappa_noise}\n\\end{figure*}\n\n\nIn the original implementation of KS, the shear maps are first regularised with a smoothing window (i.e. a low-pass filter) to obtain a smoothed version of the shear field. Then, Eq.~(\\ref{eq:fourier}) is applied to derive the convergence maps. \nIn contrast, the KS+ method aims at producing very general convergence maps for many applications. In particular, it produces noisy maps with minimum information loss. \n\nHowever, for specific applications (e.g. galaxy cluster detection and characterisation), it can be useful to add an additional de-noising step, using any of the many regularisation techniques that have been proposed \\citep{wl:bridle98,wl:seitz98,wl:marshall02, wl:starck06, wl:lanusse16}. \nTo compare the results of the KS and KS+ methods on noisy maps, we used a linear Gaussian\nand the non-linear MRLens filter \\citep{wl:starck06} for noise suppression.\nFig.~\\ref{kappa_noise} illustrates the effect of shape noise on reconstructing the convergence map.\nThe upper panels show one E-mode convergence map reconstructed from noise-free (left) and noisy (right) shear data. The convergence map is dominated by the noise. The lower panels show the results of the Gaussian filter (left) and MRLens filter (right). The Gaussian filter gives a smoothed version of the noisy convergence map, whose level of smoothness is set by the width of the Gaussian ($\\sigma$). Thus, the amplitude of the over-densities (in blue) are systematically lowered by the Gaussian filter. \nIn contrast, \nthe MRLens filter uses a prior of sparsity to better recover the amplitude of the structures and uses a parameter, the false-discovery rate ($\\alpha_{\\rm FDR}$), to control the average fraction of false detections (i.e. the number of pixels that is truly inactive, declared positive) made over the total number of detections \\citep{benjamini95}. For some other applications (e.g. two- or three-point correlation), the integrity of the reconstructed noisy convergence maps might be essential and this denoising step can be avoided.\n\n\n\\section{Method}\n\\label{method}\n\n\n\n\\subsection{Comparing second-order statistics}\n\\label{2pcf}\nThe most common tools for constraining cosmological parameters in weak-lensing studies are the shear two-point correlation functions.\nFollowing \\cite{wl:bartelmann01}, they are defined by considering pairs of positions $\\vec{\\vartheta}$ and $\\vec{\\theta+\\vartheta}$, and defining the tangential and cross-component of the shear $\\gamma_{\\rm t}$ and $\\gamma_{\\times}$ at position $\\vec{\\vartheta}$ for this pair as\n\\begin{eqnarray}\n\\gamma_{\\rm t} &=& -\\operatorname{\\mathcal{R}e}(\\gamma \\operatorname{e}^{-2{\\rm i}\\varphi}),\\\\\n\\gamma_{\\times} &=& -\\operatorname{\\mathcal{I}m}(\\gamma \\operatorname{e}^{-2{\\rm i}\\varphi}),\n\\end{eqnarray}\nwhere $\\varphi$ is the polar angle of the separation vector $\\vec{\\theta}$.\nThen we define the two independent shear correlation functions\n\\begin{eqnarray}\n\\xi_\\pm(\\theta) &:=& \\langle \\gamma_{\\rm t} \\gamma_{\\rm t}' \\rangle \\pm  \\langle \\gamma_\\times \\gamma_\\times' \\rangle \\\\\n&=& \\frac{1}{2\\pi} \\int_0^{\\infty} d\\ell \\, \\ell \\, P_{\\rm \\kappa}(\\ell) \\,{\\rm J}_{0,4}(\\ell \\theta) ,\n\\end{eqnarray}\nwhere the Bessel function ${\\rm J}_0$ $({\\rm J}_4)$ corresponds to the plus (minus) correlation function, $P_{\\kappa}(\\ell)$ is the power spectrum of the projected matter density, and $\\ell$ is the Fourier variable on the sky.\nWe can also compute the two-point correlation functions of the convergence ($\\kappa = \\kappa_{\\rm E} + \\rm i \\kappa_{\\rm B}$), defined as\n\\begin{eqnarray}\n\\xi_{\\kappa_{\\rm E}}(\\theta) = \\langle \\kappa_{\\rm E} \\kappa_{\\rm E}' \\rangle,\\nonumber \\\\\n\\xi_{\\kappa_{\\rm B}}(\\theta) = \\langle \\kappa_{\\rm B} \\kappa_{\\rm B}' \\rangle.\n\\end{eqnarray}\nWe can verify that these two quantities agree \\citep{2pcf:schneider02}:\n\\begin{eqnarray}\n \\xi_+(\\theta) = \\xi_{\\kappa_{\\rm E}}(\\theta) + \\xi_{\\kappa_{\\rm B}}(\\theta).\n \\end{eqnarray}\nWhen the B modes in the shear field are consistent with zero, the two-point correlation of the shear ($\\xi_+$) is equal to the two-point correlation of the convergence $(\\xi_{\\kappa_{\\rm E}})$. Then the differences between the two are due to the errors introduced by the mass inversion to go from shear to convergence.\n\nWe computed these two-point correlation functions using the tree code \\texttt{athena}  \\citep{athena:kilbinger14}. The shear two-point correlation functions were computed by averaging over pairs of galaxies of the mock galaxy catalogue, whereas the convergence two-point correlation functions were computed by averaging over pairs of pixels in the convergence map. The convergence two-point correlation functions can only be computed for separation vectors $\\vec{\\theta}$ allowed by the binning of the convergence map. \n\n\n\\subsection{Comparing higher-order statistics}\n\\label{hos}\n\nTwo-point statistics cannot fully characterise the weak-lensing field at small scales where it becomes non-Gaussian \\citep[e.g.][]{pt:bernardeau02}. Because the small-scale features carry important cosmological information, we computed the third-order moment, $\\langle \\kappa_{\\rm E}^3 \\rangle$, and the fourth-order moment,  $\\langle \\kappa_{\\rm E}^4\\rangle$, of the convergence. Computations were performed on the original convergence maps provided by the Flagship simulation, as well as on the convergence maps reconstructed from the shear field with the KS and KS+ methods. \nWe evaluated the moments of convergence at various scales by computing aperture mass maps \\citep{map:schneider96, map:schneider97}. Aperture mass maps are typically obtained by convolving the convergence maps with a filter function of a specific scale (i.e. aperture radii). We performed this here by means of a wavelet transform using the starlet transform algorithm \\citep{starck:book98,starck:book02}, which simultaneously produces a set of aperture mass maps on dyadic (powers of two) scales (see Appendix A for more details). Leonard et al. (2012) demonstrated that the aperture mass is formally identical to a wavelet transform at a specific scale and the aperture mass filter corresponding to this transform is derived. The wavelet transform offers significant advantages over the usual aperture mass algorithm in terms of computation time, providing speed-up factors of about 5 to 1200 depending on the scale.\n\n\n\\subsection{Numerical simulations}\n\\label{sect_simu}\n\n\n\nWe used the Euclid Flagship mock galaxy catalogue version 1.3.3\n(Castander F. et al., in prep) derived from N-body cosmological simulation \\citep{flagship:potter17} with parameters $\\Omega_{\\rm m}=0.319$, $\\Omega_{\\rm b} = 0.049$, $\\Omega_{\\Lambda} = 0.681$, $\\sigma_8=0.83$, $n_{\\rm s}=0.96$, $h=0.67$, and the particle mass was \\smash{$m_{\\rm p} \\sim 2.398 \\times10^9\\,\\text{\\ensuremath{\\textup{M}_{\\odot}}} h^{-1}$}. The galaxy light-cone catalogue contains 2.6 billion galaxies over $5000\\,\\deg^2$ , and it extends up to $z=2.3$. It has been built using a hybrid halo occupation distribution and halo abundance matching (HOD+HAM) technique, whose galaxy-clustering properties were discussed in detail in \\cite{flagship:crocce15}.\nThe lensing properties were computed using the Born approximation and projected mass density maps (in \\texttt{HEALPix} format with $N_{\\rm side}=8192$) generated from the particle light-cone of the Flagship simulation.\nMore details on the lensing properties of the Flagship mock galaxy catalogue can be found in \\cite{flagship:fosalba15,flagship:fosalba18}.\n\nIn order to evaluate the errors introduced by the mass-mapping methods, we extracted ten contiguous shear and convergence fields of $10^\\circ \\times 10^\\circ$ from the Flagship mock galaxy catalogue, yielding a total area of 1000 deg$^2$. The fields correspond to galaxies that lie in the range of \\smash{$15^\\circ < \\alpha < 75^\\circ$} and \\smash{$15^\\circ < \\delta < 35^\\circ$} , where $\\alpha$ and $\\delta$ are the right ascension and declination, respectively.\nIn order to obtain the density of 30 galaxies per arcmin$^2$ foreseen for the Euclid Wide survey, we randomly selected one quarter of all galaxies in the catalogue. \nThen projected shear and convergence maps were constructed by combining all the redshifts of the selected galaxies.\nMore sophisticated selection methods based on galaxy magnitude would produce slightly different maps. However, they would not change the performances of the two methods we studied here.\nThe fields were down-sampled to $1024 \\times 1024$ pixels, which corresponds to a pixel size of about \\ang[angle-symbol-over-decimal]{;0.6;}. Throughout all the paper, the shaded regions stand for the uncertainties on the mean estimated from the total 1000 deg$^2$ of the ten fields. Because the Euclid Wide survey is expected to be 15 000 deg$^2$, the sky coverage will be 15 times larger than the current mock. Thus, the uncertainties will be smaller by a factor of about 4.\n\n\n\\begin{figure*}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=diff_kappa_mask_ks_m20,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=diff_kappa_mask_ki_m20,height=6.cm,width=7cm,clip=}\n}}\n}\n\\caption{Missing data effects: Pixel difference outside the mask between the original E-mode convergence $\\kappa$ map and the map reconstructed from the incomplete simulated noise-free shear maps using the KS method (left) and the KS+ method (right). The field is $10^\\circ \\times 10^\\circ$ downsampled to $1024 \\times 1024$ pixels. The missing data represent roughly 20\\% of the data.}\n\\label{kappa_missing}\n\\end{figure*}\n\n\n\\subsection{Shear field projection}\n\\label{projections}\n\nWe considered fields of $10^\\circ \\times10^\\circ$. The fields were taken to be sufficiently small to be approximated by a tangent plane. \nWe used a gnomonic projection to project the points of the celestial sphere onto a tangent plane, following \\cite{cmb:pires12}, who found that this preserves the two-point statistics. We note, however, that higher-order statistics may behave differently under different projections.\n\nThe shear field projection is obtained by projecting the galaxy positions from the sphere ($\\alpha$, $\\delta$) in the catalogue onto a tangent plane ($x$, $y$).\nThe projection of a non-zero spin field such as the shear field requires a projection of both the galaxy positions and their orientations.\nProjections of the shear do not preserve the spin orientation, which can generate substantial B modes (depending on the declination) if not corrected for.  \nTwo problems must be considered because of the orientation. First, the projection of the meridians are not parallel, so that north is not the same everywhere in the same projected field of view. Second, the projection of the meridians and great circles is not perpendicular, so that the system is locally non-Cartesian.\nBecause we properly correct for the other effects (e.g. shape noise, missing data, or border effects) and consider large fields of view ($10^{\\circ} \\times 10^{\\circ}$) possibly at high latitudes, these effects need to be considered.\nThe first effect is dominant and generates substantial B modes (increasing with latitude) if not corrected for. This can be easily corrected \nfor by measuring the shear orientation with respect to local north. We find that this correction is sufficient for the residual errors due to projection to become negligible compared to errors due to other effects.\n\n\n\n\n\\section{Systematic effects on the mass-map inversion}\n\\label{results_1}\nIn this section, we quantify the effect of field borders, missing data, shape noise, and the approximation of shear by reduced shear on the KS and KS+ mass-inversion methods. The quality of the reconstruction is assessed by comparing the two-point correlation functions, third- and fourth-order moments.\n\n\n\\subsection{Missing data effects}\n\\label{sect_missing}\n\nWe used the ten noise-free shear fields of $10^\\circ \\times 10^\\circ$ described in Sect.~\\ref{sect_simu} and the corresponding noise-free convergence maps.\nWe converted the shear fields into planar convergence maps using the KS and KS+ methods, masking 20\\% of the data as expected for the \\textit{Euclid}\\xspace survey. \nThe mask was derived from the Data Challenge 2 catalogues produced by the Euclid collaboration using the code \\texttt{FLASK} \\citep{flask:xavier16}.\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=Residual_PDF_mask.pdf,height=6.5cm,clip=}\n}\n\\caption{Missing data effects: PDF of the residual errors between the original E-mode convergence map and the reconstructed maps using KS (blue) and KS+ (red), measured outside the mask.}\n\\label{missing_residual_PDF}\n\\end{figure}\n\n\n\nFig.~\\ref{kappa_missing} compares the results of the KS and KS+ methods in presence of missing data.\nThe figure shows the residual maps, that is, the pixel difference between the original E-mode convergence map and the reconstructed maps.\nThe amplitude of the residuals is larger with the KS method. Detailed investigation shows that the excess error is essentially localised around the gaps. \nBecause the mass inversion operator $\\mathbf P$ is intrinsically non-local, it generates artefacts around the gaps.\nIn order to quantify the average errors,\nFig.~\\ref{missing_residual_PDF} shows the probability distribution function (PDF) of the residual maps, estimated outside the mask. \nThe standard deviation is 0.0080 with KS and 0.0062 with KS+. The residual errors obtained with KS are then 30\\% larger than those obtained with KS+.\n\n\n\\begin{figure}[h!]\n\\vbox{\n\\centerline{\n\\psfig{figure=correlation_missing_m20.pdf,width=9.cm,clip=}\n}\n\\caption{Missing data effects: Mean shear two-point correlation function $\\xi_+$ (black) and corresponding mean convergence two-point correlation function $\\xi_{\\kappa_{\\rm E}}$ reconstructed using the KS method (blue) and using the KS+ method (red) from incomplete shear maps. The estimation is only made outside the mask $M$. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative two-point correlation errors introduced by missing data effects, that is, the normalised difference between the upper curves.}\n\\label{missing_corr}\n}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=skewness_mask.pdf,width=9.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kurtosis_mask.pdf,width=9.cm,clip=}\n}}\n}\n\\caption{Missing data effects: Third-order (upper panel) and fourth-order (lower panel) moments estimated on seven wavelet bands of the original E-mode convergence map (black) compared to the moments estimated on the KS (blue) and KS+ (red) convergence maps at the same scales. The KS and KS+ convergence maps are reconstructed from incomplete noise-free shear maps. The estimation of the third- and fourth-order moments is made outside the mask. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative higher-order moment errors introduced by missing data effects, that is, the normalised difference between the upper curves.}\n\\label{missing_hos}\n\\end{figure}\n\n\n\nThe quality of the mass inversion at different scales can be estimated using the two-point correlation function and higher-order moments computed at different scales.\nFig.~\\ref{missing_corr} compares the two-point correlation functions computed on the convergence and shear maps outside the mask.\nBecause the B mode is consistent with zero in the simulations, we expect that these two quantities are equal within the precision of the simulations (see Sect.~\\ref{2pcf}).\nThe KS method systematically underestimates the original two-point correlation function by a factor of about 2 on arcminute scales, but can reach factors of 5 at larger scales.\nThe mass-inversion operator $\\mathbf P$ being unitary, the signal energy is conserved by the transformation (i.e. $\\sum(\\gamma_1^2+ \\gamma_2^2) = \\sum(\\kappa_{\\rm E}^2+ \\kappa_{\\rm B}^2)$, where the summation is performed over all the pixels of the maps). We found that about 10\\% of the total energy leaks into the gaps and about 15\\% into the B-mode component.\nIn contrast, the errors of the KS+ method are of the order of a few percent at scales smaller than $1^\\circ$. At any scale, the KS+ errors are about 5-10 times smaller than the KS errors, remaining in the $1\\sigma$ uncertainty of the original two-point correlation function.\n\nFig.~\\ref{missing_hos} shows the third-order (upper panel) and fourth-order (lower panel) moments estimated at six different wavelet scales ( \\ang[angle-symbol-over-decimal]{;2.34;},  \\ang[angle-symbol-over-decimal]{;4.68;},  \\ang[angle-symbol-over-decimal]{;9.37;},  \\ang[angle-symbol-over-decimal]{;18.75;},  \\ang[angle-symbol-over-decimal]{;37.5;}, and \\ang[angle-symbol-over-decimal]{;75.0;}) using the KS and KS+methods. For this purpose, the pixels inside the mask were set to zero in the reconstructed convergence maps. The aperture mass maps corresponding to each wavelet scale were computed, and the moments were calculated outside the masks.\n\nThe KS method systematically underestimates the third- and fourth-order moments at all scales.\nBelow 10$\\arcmin$, the errors on the moments remain smaller than 50\\%, and they increase with scale up to a factor 3. In comparison, the KS+ errors remain much smaller at all scales, and remain within the 1$\\sigma$ uncertainty.\n\n\n\\subsection{Field border effects}\nFig.~\\ref{kappa_borders} compares the results of the KS (left) and KS+ (right) methods for border effects. It shows the residual error maps corresponding to the pixel difference between the original E-mode convergence map and the reconstructed maps.\nWith KS, as expected, the pixel difference shows errors at the border of the field. With KS+, there are also some low-level boundary effects, but these errors are considerably reduced and do not show any significant structure at the field border.\nIn KS+, the image is extended to reduce the border effects. The effect of borders decreases when the size of the borders increases. A border size of 512 pixels has been selected for \\textit{Euclid}\\xspace as a good compromise between precision and computational speed. It corresponds to extending the image to be in-painted to 2048 $\\times$ 2048 pixels.\nAgain, the PDF of these residuals can be compared to quantify the errors. For the two methods, Fig.~\\ref{border_residual_PDF} shows the residuals PDFs computed at the boundaries (as dotted lines) and in the remaining central part of the image (as solid lines). The border width used to compute the residual PDF is 100 pixels, which corresponds to about one degree. \nWith the KS method, the standard deviation of the residuals in the centre of the field is 0.0062.\n In the outer regions, the border effect causes errors of 0.0076 (i.e. 25\\% larger than at the centre). Away from the borders, the KS+ method gives results similar to the KS method (0.0060). However, it performs much better at the border, where the error only reaches 0.0061. The small and uniform residuals of the KS+ method show how efficiently it corrects for borders effects. \n \n\n\\begin{figure*}\n\\centerline{\n\\hbox{\n\\psfig{figure=diff_kappa_nomask_ks,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=diff_kappa_nomask_ki,height=6.cm,width=7cm,clip=}\n}}\n\\caption{Field border effects: Pixel difference between the original E-mode convergence $\\kappa$ map and the map reconstructed from the corresponding simulated shear maps using the KS method (left) and the KS+ method (right). The field is $10^\\circ \\times 10^\\circ$ downsampled to $1024 \\times 1024$ pixels.}\n\\label{kappa_borders}\n\\end{figure*}\n\n\n\\begin{figure}\n\\centerline{\n\\psfig{figure=Residual_PDF_nomask.pdf,height=6.5cm,clip=}\n}\n\\caption{Field border effects: PDF of the residual errors between the original E-mode convergence map and the convergence maps reconstructed using KS (blue) and KS+ (red). The dotted lines correspond to the PDF of the residual errors measured at the boundaries of the field, and the solid lines show the PDF of the residual errors measured in the centre of the field. The borders are 100 pixels wide.}\n\\label{border_residual_PDF}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\psfig{figure=correlation_border.pdf,width=9.cm,clip=}\n}\n\\caption{Field border effects: Mean shear two-point correlation function $\\xi_+$ (black) compared to the corresponding mean convergence two-point correlation function $\\xi_{\\kappa_{\\rm E}}$ reconstructed using the KS method (blue) and the KS+ method (red). The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative two-point correlation error introduced by border effects.}\n\\label{border_corr}\n}\n\\end{figure}\n\nAs before, the scale dependence of the errors can be estimated using the two-point correlation function and higher-order moments computed at different scales. \nFig.~\\ref{border_corr} shows the two-point correlation functions.\nFor both methods, the errors increase with angular scale because the fraction of pairs of pixels that include boundaries increase with scale. The loss of amplitude at the image border is responsible for significant errors in the two-point correlation function of the KS convergence maps. \nIn contrast, the errors are about five to ten times smaller with the KS+ method and remain in the $1\\sigma$ uncertainty range of the original two-point correlation function.\n\nFig.~\\ref{border_hos} shows field borders effects on the third-order (upper panel) and fourth-order (lower panel) moments of the convergence maps at different scales.\nAs was observed earlier for the two-point correlation estimation, the KS method introduces errors at large scales on the third- and fourth-order moment estimation. With KS+, the discrepancy is about $1\\%$ and within the $1\\sigma$ uncertainty.\n\nWhen the two-point correlation functions and higher-order moments are computed far from the borders, the errors of the KS method decrease, as expected. In contrast, we observe no significant improvement when the statistics are computed similarly on the KS+ maps, indicating that KS+ corrects for borders properly.\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=skewness_border.pdf,width=9.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kurtosis_border.pdf,width=9.cm,clip=}\n}}}\n\\caption{Field border effects: Third-order (upper panel) and fourth-order (lower panel) moments estimated on seven wavelet bands of the original convergence (black) compared to the moments estimated on the KS (blue) and KS+ (red) convergence maps reconstructed from noise-free shear maps. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative higher-order moment errors introduced by border effects.}\n\\label{border_hos}\n\\end{figure}\n\n\n\n\\subsection{Reduced shear}\nIn this section we quantify the errors due to the approximation of shear ($\\gamma$) by the reduced shear ($g$).\nTo this end, we used the noise-free shear fields described in Sect.~\\ref{sect_simu} and computed the reduced shear fields using Eq.~(\\ref{reducedshear}) and the convergence provided by the catalogue. We then derived the reconstructed convergence maps using the KS and KS+ methods.\n\nFor both methods, the errors on the convergence maps are dominated by field border effects. \nWe did not find any estimator able to separate these two effects and then identify the reduced shear effect in the convergence maps. \nThe errors introduced by the reduced shear can be assessed by comparing the shear and reduced shear two-point correlation functions (see Fig.~\\ref{reduced_corr}), however. While the differences are negligible at large scales, they reach the percent level on arcminute scales \\citep[in agreement with][]{wl:white05}, where they become comparable or larger than the KS+ errors due to border effects.\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\psfig{figure=correlation_reduced_2.pdf,width=9.cm,clip=}\n}\n\\caption{Reduced shear effects: Relative two-point correlation error between the mean two-point correlation functions \\smash{$\\xi_+^{\\gamma}$} estimated from the shear fields and corresponding mean two-point correlation function \\smash{$\\xi_+^{\\rm g}$} estimated from the reduced shear fields without any correction.}\n\\label{reduced_corr}\n}\n\\end{figure}\n\n\n\\subsection{Shape noise}\n\nIn this section we study the effect of the shape noise on convergence maps.\nWe derived noisy shear maps, assuming a Gaussian noise ($\\sigma_{\\epsilon} = 0.3$).\nThen, we compared the two mass-inversion methods. \nThe pixel difference cannot be used in this case because the convergence maps are noise dominated (see Fig.~\\ref{kappa_noise}, upper right panel).\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\psfig{figure=correlation_border_noise.pdf,width=9.cm,clip=}\n}\n\\caption{Shape noise effects: Mean shear two-point correlation function $\\xi_+$ (black) and corresponding mean convergence two-point correlation function $\\xi_{\\kappa_{\\rm E}}$ estimated from complete noisy shear fields. The convergence maps have been estimated using the KS method (blue) and using the KS+ method (red). The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative two-point correlation error introduced by shape noise.}\n\\label{border_noise_corr}\n}\n\\end{figure}\nHowever, we can still assess the quality of the convergence maps using two-point correlation functions because the ellipticity correlation is an unbiased estimate of the shear correlation, and similarly, the convergence two-point correlation functions is unbiased by the shape noise.\n\n \nFig.~\\ref{border_noise_corr} compares the results of the KS and KS+ methods when shape noise is included.\nCompared to Fig.~\\ref{border_corr}, the two-point correlation of the noisy maps is less smooth because the noise fluctuations do not completely average out. However, the amplitude of the errors introduced by the mass inversion remain remarkably similar to the errors computed without shape noise for the KS and KS+ methods. The same conclusions then hold: the errors are about five times smaller with the KS+ method.\n\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=skewness_border_noise.pdf,width=9.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kurtosis_border_noise.pdf,width=9.cm,clip=}\n}}}\n\\caption{Shape noise effects: Third-order (upper panel) and fourth-order (lower panel) moments estimated on seven wavelet bands of the original convergence with realistic shape noise (black) compared to the moments estimated on the KS (blue) and KS+ (red) convergence reconstructed from noisy shear maps. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative higher-order moment errors introduced by shape noise.}\n\\label{noise_border_hos}\n\\end{figure}\n\nMoments of noisy maps are biased and potentially dominated by the shape noise contribution. For instance, the total variance in the noisy convergence map is expected to be the sum of the variance in the noise-free convergence map and the noise variance. Therefore moments of the noisy KS and KS+ convergence maps cannot be directly compared to moments of the original noise-free convergence maps. \nInstead, Fig.~\\ref{noise_border_hos} compares them to the moments of the original convergence maps where noise was added with properties similar to the noise expected in the convergence maps. For this purpose, we generated noise maps $N_1$ and $N_2$ for each field using Eq.~(\\ref{eq:noise1}) and (\\ref{eq:noise2}), and we derived the noise to be added in the convergence using Eq.~(\\ref{eq:noise3}).\n\nThe comparison of Fig.~\\ref{noise_border_hos} to Fig.~\\ref{border_hos} shows that the third-order moment of the convergence is not affected by shape noise. In contrast, the fourth-order moment is biased for scales smaller than 10$\\arcmin$. The two methods slightly underestimate the third- and fourth-order moments at large scales. However, with KS+, the errors are reduced by a factor of 2 and remain roughly within the $1\\sigma$ uncertainty.\n\n\n\n\n\n\n\\subsection{All systematic effects taken into account simultaneously}\nIn this section, we assess the performance of KS and KS+ for realistic data sets by combining the effects of shape noise, reduced shear, borders, and missing data.\n\n\n\\begin{figure*}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=kappa_1024_m20,height=6.cm,width=6.7cm,clip=}\n\\hspace{0.6cm}\n\\psfig{figure=mask_m20,height=6.cm,width=6.4cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kappa_mask_g5_m20,height=6.cm,width=7cm,clip=}\n\\hspace{0.2cm}\n\\psfig{figure=kappa_mask_mrlens_m20,height=6.cm,width=7cm,clip=}\n}}\n\n}\n\\caption{All systematic effects: The upper panels show the original E-mode convergence $\\kappa$ map (left) and the mask that is applied to the shear maps (right). The lower panels show the convergence map reconstructed from an incomplete noisy shear field using the KS method (left) and using the KS+ method (right) applying a nonlinear MRLens filtering with $\\alpha_{\\rm FDR} = 0.05$. The field is $10^\\circ \\times 10^\\circ$ downsampled to $1024 \\times 1024$ pixels.}\n\n\\label{kappa_complete}\n\\end{figure*}\n\n\nFig.~\\ref{kappa_complete} compares the results of the KS method and the KS+ method combined with a filtering step to correct for all systematic effects in one field.\nWe used the nonlinear MRLens filter to reduce the noise in the KS and KS+ convergence maps because it is particularly well suited for the detection of isotropic structures \\citep{stat:pires09a, wl:pires12, wl:lin16}. Again, KS+ better recovers the over-densities because it reduces the signal leakage during the mass inversion compared to KS.\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\psfig{figure=correlation_missing_noise_m20.pdf,width=9.cm,clip=}\n}\n\\caption{All systematic effects: Mean shear two-point correlation function $\\xi_+$ (black) and corresponding mean convergence two-point correlation function $\\xi_{\\kappa_{\\rm E}}$ estimated from incomplete noisy shear fields. The convergence maps have been estimated using KS (blue) and KS+ (red). The convergence two-point correlations were estimated outside the mask. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the normalised difference between the two upper curves.}\n\\label{missing_noise_corr} \n}\n\\end{figure}\n\n\n\nFig.~\\ref{missing_noise_corr} shows the two-point correlation computed with the two methods. \nThe masked regions were excluded from the two-point correlation computation, resulting in fewer pairs and higher noise than in Fig.~\\ref{border_noise_corr}.\nAgain, the strong leakage due to missing data is clearly observed with the KS method. \nThe results obtained with the KS+ method reduce the errors in the mean convergence two-point correlation function by a factor of about 5, and the errors remain roughly within the 1$\\sigma$ uncertainty.\n\n\n\n\n\\begin{figure}\n\\vbox{\n\\centerline{\n\\hbox{\n\\psfig{figure=skewness_mask_noise.pdf,width=9.cm,clip=}\n}}\n\\vspace{0.3cm}\n\\centerline{\n\\hbox{\n\\psfig{figure=kurtosis_mask_noise.pdf,width=9.cm,clip=}\n}}}\n\\caption{All systematic effects: Third-order (upper panel) and fourth-order (lower panel) moments estimated on seven wavelet bands of the original convergence with realistic noise (black) compared to the moments estimated using KS (blue) and KS+ (red) obtained from incomplete noisy shear maps. The third- and fourth-order moments are estimated outside the mask. The shaded area represents the uncertainties on the mean estimated on 1000 deg$^{2}$. The lower panel shows the relative higher-order moment errors.}\n\\label{noise_missing_hos}\n\\end{figure}\n\n\n\n\nIn Fig.~\\ref{noise_missing_hos} we test the efficacy of the mass-inversion methods in preserving higher-order moments of the convergence maps in a realistic setting. As before, realistic noise was added to the original convergence maps for comparison.\nAs was observed earlier in the noise-free case, the KS method systematically underestimates the third- and fourth-order moments at all scales. \nWith KS+, the errors are significantly reduced, by a factor of about 2 in the third-order moment and by a factor of about 10 in the fourth-order moment estimation, at all scales.\nAlthough reduced, the errors of the KS+ method on the third-order moment cannot be neglected.\nThese errors might result from noise correlations introduced by the inpainting method in the shear maps. Inside the gaps, the noise is indeed correlated because it is interpolated from the remaining data. These noise correlations propagate into the convergence maps and can explain the bias in the moment estimation.\n\nWe note that the two-point correlation functions and higher-order moments are here only used to probe the accuracy of the reconstruction methods. For specific applications, the small residuals of the KS+ method can be reduced even more using additional treatment such as down-weighting the region around the mask when the moments are computed \\citep[e.g.][]{cfhtlens:vanwaerbeke13}.\n\n\n\n\n\n\\section{Conclusion}\n\\label{sect_cl}\n\nThis paper was motivated by the use of convergence maps in \\textit{Euclid}\\xspace to constrain cosmological parameters and to assess other physical constraints. \nConvergence maps encode the lensing information in a different manner, allowing more optimised computations than shear.\nHowever, the mass-inversion process is subject to field border effects, missing data, reduced shear, intrinsic alignments, and shape noise. This requires accurate control of the systematic effects during the mass inversion to reduce the information loss as much as possible. \nWe presented and compared the two mass-inversion methods that are included in the official Euclid data-processing pipeline: the standard Kaiser \\& Squires (KS) method, and an improved Kaiser \\& Squires (KS+) mass-inversion technique that integrates corrections for the mass-mapping systematic effects.\n The systematic effects on the reconstructed convergence maps were studied using the Euclid Flagship mock galaxy catalogue.\n\n\nIn a first step, we analysed and quantified one by one the systematic effects on reconstructed convergence maps using two-point correlation functions and moments of the convergence.\nIn this manner, we quantified the contribution of each effect to the error budget to better understand the error distribution in the convergence maps. With KS, missing data are the dominant effect at all scales. Field border effects also have a strong effect, but only at the map borders. These two effects are significantly reduced with KS+. The reduced shear is the smallest effect in terms of contribution and only affects small angular scales. \nThe study also showed that pixellisation provides an intrinsic regularisation and that no additional smoothing step is required to avoid infinite noise in the convergence maps.\n\n\nIn a second step, we quantified the errors introduced by the KS and KS+ methods in a realistic setting that included the systematic effects. \nWe showed that the KS+ method reduces the errors on the two-point correlation functions and on the moments of the convergence compared to the KS method. \nThe errors introduced by the mass inversion on the two-point correlation of the convergence maps are reduced by a factor of about 5. \nThe errors on the third-order and fourth-order moment estimates are reduced by factors of about 2 and 10, respectively.\nSome errors remain in the third-order moment that remain within the $2\\sigma$ uncertainty. They might result from noise correlations introduced by the inpainting method inside the gaps.\n\nOur study was conducted on a mock of 1000 deg$^2$ divided into ten fields of 10$^{\\circ}$ $\\times$ 10$^{\\circ}$ to remain in the flat-sky approximation. \\textit{Euclid}\\xspace will observe a field of 15 000 deg$^2$. As long as KS+ has not been extended to the curved sky, it is not possible to apply the method to larger fields without introducing significant projection effects. However, the \\textit{Euclid}\\xspace survey can be divided into small fields, which allows reducing the uncertainties in the statistics that are estimated on the convergence maps. Moreover, we can expect that part of the errors will average out.\n\nRecent studies have shown that combining the shear two-point statistics with higher-order statistics of the convergence such as higher-order moments \\citep{hos:vicinanza18}, Minkowski functionals \\citep{hos:vicinanza19}, or peak counts \\citep{hos:liu15,hos:martinet18} allows breaking common degeneracies. \nThe precision of the KS+ mass inversion makes the E-mode convergence maps a promising tool for such cosmological studies.\nIn future work, we plan to propagate these errors into cosmological parameter constraints using higher-order moments and peak counts.\n\n\n\\label{sect_cl}\n\n\n\n\n\n\\section*{Acknowledgments}\n{This study has been carried inside the Mass Mapping Work Package of the Weak Lensing Science Working Group of the \\textit{Euclid}\\xspace project to better understand the impact of the mass inversion systematic effects on the convergence maps. \nThe authors would like to thank the referees and editors for their valuable comments, which helped to improve the manuscript.\nS. Pires thanks F. Sureau, J. Bobin, M. Kilbinger, A. Peel and J.-L. Starck for useful discussions. \n\\AckEC\n\n\\section*{Appendix A: KS+ inpainting algorithm}\n\n\nThis appendix describes the KS+ method presented in Sect.~\\ref{iks} in more detail.\nThe solution of the KS+ mass inversion is obtained through the iterative algorithm described in Algorithm~\\ref{algo}.\n\nThe outer loop starting at step 5 is used to correct for the reduced shear using the iterative scheme described in Sect.~\\ref{reduced}. The inner loop starting at step 7 is used to solve the optimisation problem defined by Eq.~(\\ref{eq2}). \n$\\bm \\Phi$ is the discrete cosine transform operator matrix.\nIf the convergence ${\\bm \\kappa}$ is sparse in $\\bm \\Phi$, most of the signal is contained in the strongest DCT coefficients. The smallest coefficients result from missing data, border effects, and shape noise. Thus, the algorithm is based on an iterative algorithm with a threshold that decreases exponentially (at each iteration) from a maximum value to zero, following the decreasing law $F$ described in \\cite{wl:pires09}. \nBy accumulating increasingly more high DCT coefficients through each iteration, the gaps in $\\bm{\\tilde \\gamma}$ fill up steadily, and the power of the spurious B modes due to the gaps decreases.\nThe algorithm uses the fast Fourier transform at each iteration to compute the shear maps $\\bm \\gamma$ from the convergence maps $\\bm \\kappa$ (step 14) and the inverse relation (step 16). \n\n\n\nA data-driven power spectrum prior is introduced at steps 11-13. To do so, the KS+ algorithm uses the undecimated isotropic wavelet transform that decomposes an image $\\bm \\kappa$ into a set of coefficients $\\{ \\bm{w_1}, \\bm{w_2}, ..., \\bm{w_J}, \\bm{c_J}\\}$, as a superposition of the form\n\\begin{eqnarray}\n\\bm \\kappa[i_1, i_2]= \\bm{c_{J}}[i_1, i_2] + \\sum_{j=1}^{J} \\bm{w_{j}}[i_1,i_2], \n  \\label{wavelet}\n \\end{eqnarray}\nwhere $\\bm{c_{J}}$ is a smoothed version of the image, and $\\bm \\kappa$ and $\\bm{w_{j}}$ are a set of aperture mass maps (usually called wavelet bands) at scale $\\theta = 2^{j}$. Then, we estimate the variance on each wavelet band $\\bm{w_j}$. The variance per scale estimated in this way can be directly compared to the power spectrum. This provides a way to estimate a broadband power spectrum of the convergence $\\bm \\kappa$ from incomplete data. \nThe power spectrum is then enforced by multiplying each wavelet coefficient by the factor \\smash{$\\sigma_j^{\\rm{out}}\/\\sigma_j^{\\rm{in}}$} inside the gaps, where \\smash{$\\sigma_j^{\\rm{in}}$} and \\smash{$\\sigma_j^{\\rm{out}}$} are the standard deviation estimated in the wavelet band $\\bm{w_j}$ inside and outside the mask, respectively.\nThis normalisation can be described by a linear operator $\\mathbf Q$ as used in Eq.~(\\ref{eq2}). \nThe constraint is applied on the E- and B-mode components before reconstructing the convergence $\\bm \\kappa$ by backward wavelet transform.\n\\begin{algorithm}[H]\n\\caption{KS+ algorithm}     \n\\label{algo}  \n\\begin{enumerate}\n\\item[1.] Project the shear from the celestial sphere onto a tangent plane by projecting the galaxy positions and applying a local rotation to the shear field.\n\\item[2.] Bin the projected shear onto a grid and define $\\bm{\\tilde{\\gamma}}$ as the average shear in each pixel.\n\\item[3.] Set the mask $\\mathbf M$: $M[i_1, i_2] = 1$ for pixels where we have information and $M[i_1, i_2] = 0$ for pixels with no galaxies, and take a support twice larger for the shear maps and include the borders in the masked region (see Fig.~\\ref{shear_mask}).\n\\item[4.] Set the maximum number of iterations to $I_{\\rm max}=100$, the maximum threshold $\\lambda_{\\rm max} = \\max(\\mid \\bm \\Phi^{\\rm T} \\mathbf P^* \\bm{\\tilde{\\gamma}} \\mid),$ and the minimum threshold $\\lambda_{\\rm min} = 0$.\n\\item[5.] Set $k = 0$, $\\bm{\\kappa_{\\rm E}^{k}}=0$ and iterate:\n\\begin{enumerate}\n\\item[6.] Update the shear  $\\bm{\\tilde{\\gamma}^{k}}  = \\bm{\\tilde{\\gamma}} \\, (1-\\bm{\\kappa_{\\rm E}^{k}})$ and initialise the solution to $\\bm{\\kappa^{k}} = \\mathbf P^*  \\bm{\\tilde{\\gamma}^{k}}$.\n\\item[7.] Set $i = 0$, $\\lambda^{0}=\\lambda_{\\rm max}$, $\\bm{\\kappa^{i}}=\\bm{\\kappa^{k}}$ and iterate: \n   \\begin{enumerate}\n    \\item[8.] Compute the forward transform: $\\bm \\alpha = \\bm{\\Phi^{\\rm T}} \\bm{\\kappa^{i}}$.\n    \\item[9.] Compute $\\bm{\\tilde \\alpha}$ by setting to zero the coefficients $\\bm \\alpha$ below the threshold $\\lambda^{i}$.\n    \\item[10.] Reconstruct $\\bm{\\kappa^{i}}$ from $\\bm{\\tilde \\alpha}$: $\\bm{\\kappa^{i}} = \\bm{\\Phi} \\bm{ \\tilde\\alpha}$.\n    \\item[11.] Decompose $\\bm{\\kappa^{i}}$ into its wavelet coefficients $\\{ \\bm{w_1}, \\bm{w_2}, ..., \\bm{w_J}, \\bm{c_J}\\}$.\n    \\item[12.] Renormalise the wavelet coefficients $\\bm{w_j}$ by a factor $\\sigma_j^{\\rm{out}}\/\\sigma_j^{\\rm{in}}$ inside the gaps.\n    \\item[13.] Reconstruct  $\\bm{\\kappa^{i}}$ by performing the backward wavelet transform from the normalised coefficients. \n    \\item[14.] Perform the inverse mass relation: $\\bm{\\gamma^{i}} = \\mathbf P  \\bm{\\kappa^{i}}$.\n    \\item[15.] Enforce the observed shear $\\bm{\\tilde\\gamma}$ outside the gaps:\n     $\\bm{\\gamma^{i}} = (1-\\mathbf M) \\, \\bm{\\gamma^{i}} + \\mathbf M \\bm{\\tilde{ \\gamma}^{k}}$.\n     \\item[16.] Perform the direct mass inversion: $\\bm{\\kappa^{i}} = \\mathbf P^* \\bm{\\gamma^{i}}$.\n     \\item[17.] Update the threshold: $\\lambda^i =  F(i, \\rm \\lambda_{min}, \\lambda_{max})$.\n     \\item[18.] Set $i=i+1$. If $i<I_{\\rm max}$, return to step 8.\n        \\end{enumerate}\n \\item[19.] Set $k=k+1$, $\\kappa^{k}=\\kappa^{i}$. If $k < 3$, return to step 6.\n\\end{enumerate}\n\\end{enumerate}\n\\end{algorithm}\n\n\n\\section*{Appendix B: Institutions}\n\n\n$^{1}$ Universit\\'e Paris Diderot, AIM, Sorbonne Paris Cit\\'e, CEA, CNRS F-91191 Gif-sur-Yvette Cedex, France\\\\\n$^{2}$ Universit\\\"ats-Sternwarte M\\\"unchen, Fakult\\\"at f\\\"ur Physik, Ludwig-Maximilians-Universit\\\"at M\\\"unchen, Scheinerstrasse 1, 81679 M\\\"unchen, Germany\\\\\n$^{3}$ Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, D-85748 Garching, Germany\\\\\n$^{4}$ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino Torinese (TO), Italy\\\\\n$^{5}$ INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025 Pino Torinese (TO), Italy\\\\\n$^{6}$ APC, AstroParticule et Cosmologie, Universit\\'e Paris Diderot, CNRS\/IN2P3, CEA\/lrfu, Observatoire de Paris, Sorbonne Paris Cit\\'e, 10 rue Alice Domon et L\\'eonie Duquet, 75205, Paris Cedex 13, France\\\\\n$^{7}$ Instituto de Astrof\\'isica e Ci\\^encias do Espa\\c{c}o, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal\\\\\n$^{8}$ Institut de F\\'isica d'Altes Energies IFAE, 08193 Bellaterra, Barcelona, Spain\\\\\n$^{9}$ INAF-Osservatorio Astronomico di Roma, Via Frascati 33, I-00078 Monteporzio Catone, Italy\\\\\n$^{10}$ Department of Physics \"E. Pancini\", University Federico II, Via Cinthia 6, I-80126, Napoli, Italy\\\\\n$^{11}$ INFN section of Naples, Via Cinthia 6, I-80126, Napoli, Italy\\\\\n$^{12}$ INAF-Osservatorio Astronomico di Capodimonte, Via Moiariello 16, I-80131 Napoli, Italy\\\\\n$^{13}$ Centre National d'Etudes Spatiales, Toulouse, France\\\\\n$^{14}$ Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK\\\\\n$^{15}$ ESAC\/ESA, Camino Bajo del Castillo, s\/n., Urb. Villafranca del Castillo, 28692 Villanueva de la Ca\\~nada, Madrid, Spain\\\\\n$^{16}$ Department of Astronomy, University of Geneva, ch. d'\\'Ecogia 16, CH-1290 Versoix, Switzerland\\\\\n$^{17}$ Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s\/n, 08193 Barcelona, Spain\\\\\n$^{18}$ Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain\\\\\n$^{19}$ INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, I-34131 Trieste, Italy\\\\\n$^{20}$ INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Piero Gobetti 93\/3, I-40129 Bologna, Italy\\\\\n$^{21}$ INAF-IASF Milano, Via Alfonso Corti 12, I-20133 Milano, Italy\\\\\n$^{22}$ von Hoerner \\& Sulger GmbH, Schlo{\\ss}Platz 8, D-68723 Schwetzingen, Germany\\\\\n$^{23}$ Aix-Marseille Univ, CNRS\/IN2P3, CPPM, Marseille, France\\\\\n$^{24}$ Institute for Computational Science, University of Zurich, Winterthurerstrasse 190, 8057 Zurich, Switzerland\\\\\n$^{25}$ Institut de Physique Nucl\\'eaire de Lyon, 4, rue Enrico Fermi, 69622, Villeurbanne cedex, France\\\\\n$^{26}$ Universit\\'e de Gen\\`eve, D\\'epartement de Physique Th\\'eorique and Centre for Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Gen\\`eve 4, Switzerland\\\\\n$^{27}$ Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway\\\\\n$^{28}$ Argelander-Institut f\\\"ur Astronomie, Universit\\\"at Bonn, Auf dem H\\\"ugel 71, 53121 Bonn, Germany\\\\\n$^{29}$ Centre for Extragalactic Astronomy, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK\\\\\n$^{30}$ Observatoire de Sauverny, Ecole Polytechnique F\\'ed\\'erale de Lau- sanne, CH-1290 Versoix, Switzerland\\\\\n$^{31}$ Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA, 91109, USA\\\\\n$^{32}$ Dipartimento di Fisica e Astronomia, Universit\\'a di Bologna, Via Gobetti 93\/2, I-40129 Bologna, Italy\\\\\n$^{33}$ Institute of Space Sciences (IEEC-CSIC), c\/Can Magrans s\/n, 08193 Cerdanyola del Vall\\'es, Barcelona, Spain\\\\\n$^{34}$ Centro de Investigaciones Energ\\'eticas, Medioambientales y Tecnol\\'ogicas (CIEMAT), Avenida Complutense 40, 28040 Madrid, Spain\\\\\n$^{35}$ Instituto de Astrof\\'isica e Ci\\^encias do Espa\\c{c}o, Faculdade de Ci\\^encias, Universidade de Lisboa, Tapada da Ajuda, PT-1349-018 Lisboa, Portugal\\\\\n$^{36}$ Departamento de F\\'isica, Faculdade de Ci\\^encias, Universidade de Lisboa, Edif\\'icio C8, Campo Grande, PT1749-016 Lisboa, Portugal\\\\\n$^{37}$ Universidad Polit\\'ecnica de Cartagena, Departamento de Electr\\'onica y Tecnolog\\'ia de Computadoras, 30202 Cartagena, Spain\\\\\n$^{38}$ Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA\\\\\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sect:1}\n\nHARPS is a fiber-fed, cross-dispersed echelle spectrograph. The instrument is fed by two fibers, one for the target and the other for a reference lamp or sky. The spectrograph re-images the two fibers on a mosaic of two 2k4 CCDs, where two echelle spectra of 72 orders are formed. The resolution of the spectrograph is R$=$115,000 with a resolution element oversampling of 3.2 CCD pixels \\citep[for more information, see][]{Mayor-2003,Pepe-2002}.\n\n{\\bf To prevent any large drift in radial velocity (RV), HARPS has been designed to reach an extreme long-term stability of 0.01 mbar in pressure and 0.01 K in temperature. The tiny RV instrumental drifts related to residual pressure and temperature changes are measured by a Thorium-Argon reference lamp \\citep[recently upgraded to a Fabry-Perot interferometer calibration source,][]{Wildi-2011}.}\n\nDue to those exceptional characteristics, HARPS, {\\bf and now its copy HARPS-North \\citep[][]{Cosentino-2012},} has been shown to be the most precise spectrographs to search for extra-solar planets. As an example, the HARPS RV of the magnetically quiet stars $\\tau$\\,Ceti (\\object{HD10700}) and \\object{HD85512} show a standard deviation of 0.92\\ms and 1.05\\ms over more than six years of data, respectively (see Table 2 in \\citet{Pepe-2011} for more examples).\n\nThe precision of the HARPS spectrograph might be even below the \\ms level, as we know that RV measurements are contaminated by different kinds of stellar signals that induce variations that can be as high as a few \\ms. To our current knowledge, these stellar signals can be differentiated in four categories: \n\\begin{itemize}\n\\item stellar oscillations produced by pressure waves propagating in the convective zone of the star \\citep[][]{Dumusque-2011a,Arentoft-2008,Kjeldsen-2005}\n\\item stellar granulation that is the result of convection at the stellar surface \\citep[][]{Dumusque-2011a}\n\\item short-term stellar activity induced by stellar rotation in the presence of stellar active regions \\citep[][]{Dumusque-2014a,Boisse-2012b,Meunier-2010a,Saar-1997b}\n\\item long-term stellar activity induced by stellar magnetic cycles \\citep[][]{Meunier-2013,Dumusque-2011c}\n\\end{itemize}\n\nIn this paper, we present a new type of perturbing signal that is not induced by the stars themselves, but by the design of the CCD detector. In Section \\ref{sect:2} we highlight the presence of this signal in the RVs of several stars observed with HARPS. We then demonstrate the origin of this signal in Section \\ref{sect:3} and present a solution to correct for this signal in Section \\ref{sect:4}.  Section \\ref{sect:5} concludes the paper.\n\n\n\n\\section{A one-year signal present in HARPS RV data} \\label{sect:2}\n\nSeveral stars that have been intensively observed with HARPS during several years show a low-amplitude signal at a period of one year. This is for example the case of \\object{HD128621} ($\\alpha$\\,Cen\\,B), \\object{HD1461}, \\object{HD154088}, or \\object{HD31527}. Note that other stars are also affected by this annual effect.\n\n\\subsection{HD128621} \\label{sect:2-0}\n\nIn \\citet{Dumusque-2012}, the authors present a detailed analysis of the RV of HD128621 and detect the presence of an Earth-mass planet orbiting at a short period of 3.24 days. In addition to this planetary signal, a detection of the binary RV drift induced by HD128620 ($\\alpha$\\,Cen\\,A), a RV variation induced by active regions, and a long-term drift due to the stellar magnetic cycle are reported. As explained in \\citet{Lindegren-2003} and shown in \\citet{Dumusque-2011c} and \\citet{Meunier-2013}, a stellar magnetic cycle induces a correlated effect in RV because active regions strongly suppress stellar convection. By fitting a 3rd order polynomial to the raw RVs between 2008 and 2011 to remove simultaneously the binary and the magnetic cycle effect, a yearly signal is detected in the residuals (Figure \\ref{fig:2-0} top left panel). This one-year signal cannot be detected in the \\citet[][]{Dumusque-2012} published RV data, because this effect was discovered at the time, and was corrected by using the method described in Sec. \\ref{sect:4}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{periodogram_raw_RVs.pdf}\n\\caption{{\\bf Generalized Lomb Scargle (GLS) periodograms \\citep[][]{Zechmeister-2009}} of the RV residuals of our four stars after removing their planetary signals and their magnetic cycle effect (see text in Sec \\ref{sect:2} for more details about the fitted models). The red horizontal lines represent a false alarm probability level of 10\\%. For each star the one-year signal is significant and is highlighted by the red regions.}\n\\label{fig:2-0}\n\\end{center}\n\\end{figure}\n\nThe yearly signal is extracted from the data by fitting a sinusoid with a 365.25-day period, in addition to the 3rd order polynomial. This yearly signal that has an amplitude of 2.83 $\\pm$ 0.47\\ms is shown in the top panel of Figure \\ref{fig:2-1}. The bottom panel of the same figure shows the barycentric Earth radial velocity (BERV), i.e. the velocity of Earth in the direction of HD128621. The stellar velocity is the {\\bf relativistic} sum of the raw RV as measured by the spectrograph and the BERV. The BERV varies with a period of one year because Earth orbits the Sun, and its maximum amplitude depends on the position of the star in the sky, but can be as large as $\\pm$30\\kms. This velocity variation can be translated to a shift of the spectrum on the HARPS CCD of $\\pm$37 pixels\\footnote{one pixel on the HARPS CCD represent 820\\ms}. {\\bf Although the strongest signal in the BERV is the one with the one-year period, other signals with different periods are relevant for precise Doppler work at the \\ms precision. Examples of these other signals are the rotation of the Earth around its axis, the revolution around the Sun of the solar system planets or relativistic effects \\citep[see e.g.][for a detailed review]{Wright-2014}. In this study, we focus on the yearly signal because it is the only one that seem to be detected in the data.}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8cm]{HD128621_RV_BERV_corr}\n\\caption{\\emph{Top: }{\\bf RV residuals of HD128621 in blue after fitting a 3rd order polynomial. The red curve corresponds to the fitted one-year signal}. \\emph{Bottom: }Barycentric Earth radial velocity (BERV) of HD128621, which is the velocity of the Earth in the direction of HD128621.}\n\\label{fig:2-1}\n\\end{center}\n\\end{figure}\n\nThe fact that an anti-correlation is observed between the BERV for HD128621 and the one-year signal detected in the RVs (Figure \\ref{fig:2-2}, top left panel) indicates that the two yearly signals are in opposition of phase. The BERV is the velocity of the Earth in the direction of the star, and the RV is the measure of the stellar velocity in the direction of Earth. Therefore this opposition in phase {\\bf suggests} that the one year signal detected in the RVs is induced by the orbit of Earth around the Sun and is not corresponding to an extrasolar planet.\n\n\n\\subsection{HD1461} \\label{sect:2-1}\n\nDiaz et al. 2015 (in prep.) analyzed the HARPS RV data of HD1461 and found the presence of two planets orbiting at 5.77 and 13.50 days with 0.05 and 0.09 of eccentricity, respectively. In addition, the star also shows a long-term magnetic cycle in the variation of its calcium activity index. The effect of the magnetic cycle is seen in the RVs of HD1461, and Diaz et al. 2015 (in prep.) fit it using a Keplerian with a period of 3508 days and an eccentricity of 0.10. \n\nAfter fitting a three Keplerian model to the RVs of HD1461 with parameters fixed to these previous values, the RV residuals show a signal at a one-year period (Figure \\ref{fig:2-0} top right panel). This signal has an amplitude of 1.31 $\\pm$ 0.20\\ms when a model with three Keplerians plus a one year sinusoid is fitted to the data.\n\nAs for HD128621, the anti-correlation between the BERV for HD1461 and the one year signal seen in the RVs of the star shows that the two signals are in opposition of phase (Figure \\ref{fig:2-2} top right panel). Therefore the revolution of Earth around the Sun {\\bf seems to be} detected in the RVs of HD1461.\n\n\\subsection{HD154088} \\label{sect:2-2}\n\n\\citet{Mayor-2011} announced the detection of a super-Earth orbiting around HD154088. After fitting a Keplerian model with the planet parameters reported in \\citet{Mayor-2011} and a 2nd order polynomial drift to account for the magnetic cycle, the RV residuals show a significant signal at a one-year period (Figure \\ref{fig:2-0} bottom left panel). This signal has an amplitude of 0.76 $\\pm$ 0.16\\ms when a model with one Keplerian, a 2nd order polynomial, plus a one year sinusoid is fitted to the data.\n\nAs for HD128621 and HD1461, the anti-correlation between the BERV for HD154088 and the one year signal seen in the RVs {\\bf suggests} that the signal of Earth orbiting the Sun is detected (Figure \\ref{fig:2-2} bottom left panel).\n\n\\subsection{HD31527} \\label{sect:2-3}\n\nA planetary system of three planets orbiting HD31527 has been announced by \\citet{Mayor-2011}. The activity index variation for the star is very low, with an average log(R'$_{HK}$) level of -4.96. We therefore do not expect any impact of activity on the RVs. When fitting the three planets with the parameters reported in \\citet{Mayor-2011}, a signal with a period close to one year is seen in the RV residuals (Figure \\ref{fig:2-0} bottom right panel). This signal has an amplitude of 0.59 $\\pm$ 0.15\\ms when a model with three Keplerians plus a one year sinusoid is fitted to the data.\n\nAs for the other stars studied in this paper, the opposition in phase between the one-year signal detected in the RV residuals and the BERV for HD31527 {\\bf suggests} that the signal of Earth is detected in the stellar RVs (Figure \\ref{fig:2-2} bottom right panel).\n\n\\begin{figure*}[!th]\n\\begin{center}\n\\includegraphics[width=16cm]{RV_BERV_corr_all_stars}\n\\caption{Correlations between the RV induced by the one year signal and the barycentric Earth radial velocity (BERV) for our four stars {\\bf in red. The blue points represent the underlying data with error bars.}}\n\\label{fig:2-2}\n\\end{center}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Origin of the one-year signal} \\label{sect:3}\n\nWe show in the preceding section that HARPS RV data of several stars that have been intensively observed during several years are affected by a one-year signal that is in opposition of phase with the yearly variation of the BERV of the target. {\\bf The same opposition in phase observed in the RV of several stars cannot be a coincidence. It suggests that the signal that we are detecting is due to the Earth orbiting the Sun. Several different effects could be responsible for this annual signal:\n\\begin{itemize}\n\\item imprecision in the code to calculate the BERV \\citep[][]{Wright-2014,Kopeikin-1999},\n\\item incorrect stellar coordinates,\n\\item telluric absorption line contamination,\n\\item incorrect observatory coordinates,\n\\item incorrect reduction of observing times,\n\\item uncorrected systematics in seasonal variations at the telescope,\n\\end{itemize}\nand others. It is difficult to rule out an effect by studying a group of stars, because each star will have a different BERV. However, this is not the goal here and we will demonstrate that the yearly signal is, in the presented example, mostly created by imperfections of the CCD that induce a deformation of spectral lines when the stellar spectrum is red- and blue-shifted on the CCD due to Earth orbiting the Sun.}\n\n\n\\subsection{Measuring the radial velocity of individual spectral lines}  \\label{sect:3-1}\n\n\nWhen Earth is revolving around the Sun, the BERV of a target is changing as a function of time, and this translates into a shift of the recorded spectrum on the CCD due to the Doppler effect. If a spectral line passes over an imperfection of the CCD, this spectral line can undergo a deformation that will introduce a RV variation correlated or anti-correlated with the BERV.\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=8cm]{HD128621_RV_order_61}\n\\caption{RV residuals as a function of time for HD128621 derived from the HARPS spectral order 61, after fitting a 3rd order polynomial to the data. The one-year signal is significant in the corresponding GLS periodogram (red region).}\n\\label{fig:3-0}\n\\end{center}\n\\end{figure}\n\nTo test this hypothesis, we first looked at the RV of each spectral order individually, as some spectral orders might be more affected than others. In Figure \\ref{fig:3-0}, we can see the RVs derived using only the spectral order 61. This spectral order spans a wavelength between 6145 and 6215 \\AA. {\\bf Despite the absence of relevant telluric lines, the RV of this spectral order is nevertheless} strongly affected by a one-year signal.\n\nTo go a step further and detect which lines in spectral order 61 are responsible for the yearly signal, we fitted each individual line stronger than 20\\% in depth with a Gaussian and selected the center of the Gaussian as the average wavelength of the line. This procedure is performed on all the stellar spectra to get the average wavelength position of each line as a function of time. The variation in average wavelength position as a function of time is then translated into a RV variation. Figure \\ref{fig:3-1} shows the RV variation of four spectral lines as a function of time. The top two lines are very close in wavelength, {\\bf however one of them shows a strong one-year signal, while the other does not}. The same behavior is observed for the two spectral lines at the bottom of Figure \\ref{fig:3-1}. The two lines not affected by the yearly signal show no correlation between RV and BERV (see Figure. \\ref{fig:3-2}), while the two other lines show a strong anti-correlation. This anti-correlation is similar to the one observed when using the RV derived with all the spectral lines in the spectrum (see Figure \\ref{fig:2-2}), {\\bf and therefore those two lines are examples of lines contributing to the detected yearly signal.}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=8cm]{HD128621_RV_spectral_line_6154_68}\n\\includegraphics[width=8cm]{HD128621_RV_spectral_line_6155_56}\n\\includegraphics[width=8cm]{HD128621_RV_spectral_line_6198_74}\n\\includegraphics[width=8cm]{HD128621_RV_spectral_line_6199_86}\n\\caption{RV residuals as a function of time for HD128621 derived from different spectral lines present in the HARPS spectral order 61, after fitting a third order polynomial to the RV of each line. The corresponding GLS periodograms are also shown and highlight in red the {\\bf presence of the one-year signal for spectral lines at 6155.56 and 6199.86 \\AA\\,, and the absence of the signal for the nearby lines at 6154.68 and 6198.74 \\AA.}}\n\\label{fig:3-1}\n\\end{center}\n\\end{figure*}\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=8cm]{HD128621_RV_BERV_corr_spectral_line_6154_68}\n\\includegraphics[width=8cm]{HD128621_RV_BERV_corr_spectral_line_6155_56}\n\\includegraphics[width=8cm]{HD128621_RV_BERV_corr_spectral_line_6198_74}\n\\includegraphics[width=8cm]{HD128621_RV_BERV_corr_spectral_line_6199_86}\n\\caption{Correlations between the RV derived from different spectral lines present in the HARPS spectral order 61 and the BERV for HD128621. The red lines represent a linear fit to the data, and the corresponding correlation coefficient is reported in the legend. A strong anti-correlation is observed for spectral lines at 6155.56 and 6199.86 \\AA\\,}\n\\label{fig:3-2}\n\\end{center}\n\\end{figure*}\n\n\n\\subsection{Origin of the CCD imperfections deforming spectral lines}  \\label{sect:3-2}\n\nThe HARPS CCD is composed of a mosaic of two 2x4K EEV44-82 CCDs (\\href{http:\/\/www.eso.org\/sci\/php\/optdet\/instruments\/harps\/FDR-Harps-detectors.pdf}{HARPS detector final design review\\footnote{http:\/\/www.eso.org\/sci\/php\/optdet\/instruments\/harps\/FDR-Harps-detectors.pdf}}). Each CCD device is {\\bf subdivided in 1024x512 blocks of pixels} (1024 in the cross-dispersion direction and 512 in the spectral direction) with a pixel size of 15\\,$\\mu$m within each block. The block boundaries show sometimes a 1\\% quantum efficiency (QE) variation over 1 row or column due to photolithography stepper mismatches. This variation in QE is induced by a variation in pixel size at the area boundaries, due to difficulties to perform the stitching of all the blocks together. These boundaries will be called block stitchings hereafter.\n\nTo get a wavelength solution for each pixel illuminated by both spectrograph fibers, a Thorium-Argon lamp is used to illuminate each fiber. The emission lines in the Thorium-Argon lamp create a finger print on the CCD, and by knowing precisely the wavelength of each individual line, it is possible to derive a wavelength for all the pixels in each spectral order. Because Thorium-Argon emission lines are sparse on the detector, a {\\bf polynomial fit is performed on each order} to get a wavelength for all pixels. {\\bf The wavelength solution is correct if all pixels have the same size, which is not the case for the boundaries between the 1024x512 blocks constituting the CCD. This polynomial fit therefore introduces an error in wavelength solution at those boundaries}. This effect has been shown by \\citet{Wilken-2010} (see Figure 4 of their paper), where the authors compare wavelength solutions obtained with a laser frequency comb and a Thorium-Argon lamp. Every 512 pixels in the spectral direction, a step as high as 40 \\ms (1\/20th of a pixel on HARPS) is observed when comparing the two wavelength solutions. These differences can be explained by the {\\bf larger number of regular emission peaks} present in the laser frequency comb that can resolve the difference in pixel size at the block stitchings, which the spatially sparse Thorium-Argon emission lines can not do.\n\nIn Figure \\ref{fig:3-3}, we show the part of the spectrum around both sets of spectral lines studied in the preceding section (6154.68 and 6155.56 \\AA\\,, and 6198.74 and 6199.86 \\AA\\,). The two different spectra in blue and red correspond to observations done at the extremes of the BERV, highlighting the shift of the spectrum on the CCD. As we can see, the spectral lines at 6155.56 and 6199.86 \\AA\\, affected by the yearly signal cross the block stitchings represented by the green vertical dashed line when Earth orbits the Sun. The two other lines do not cross those boundaries and therefore do not show the one-year variation.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=8cm]{HD128621_opposite_BERV_1st_stitching_order_61_modif}\n\\includegraphics[width=8cm]{HD128621_opposite_BERV_6th_stitching_order_61_modif}\n\\caption{Zoom on small spectral regions of spectral order 61. The blue spectrum corresponds to an observation of HD128621 when the BERV was maximum and the red spectrum when the BERV was minimum. The green dashed vertical lines corresponds to the position of the block stitchings. Spectral lines at 6155.56 and 6199.86 \\AA\\, showing the yearly effect cross these block stitchings, while the other lines, not affected, do not.}\n\\label{fig:3-3}\n\\end{center}\n\\end{figure*}\n\n\n\\section{Correcting for the one-year signal} \\label{sect:4}\n\nIn the preceding section, we demonstrated that the boundaries between the blocks constituting the CCD, so-called block stitchings, are responsible for the yearly signal detected in HD128621, HD1461, HD154088 and HD31527. A simple solution to get rid of this yearly signal is to remove the spectral lines passing over these boundaries from the correlation mask used to derive the RVs. Because each star has its own gamma velocity, the position of the spectral lines on the CCD will differ from one star to the other, and therefore a different correlation mask has to be generated for each star. \n\nTo select the spectral lines at wavelength $\\lambda_i$ that cross the block stitching when Earth is revolving around the Sun, the first step is to shift their wavelength to the stellar gamma velocity $\\gamma_{\\star}$. The new wavelength in the stellar rest frame $\\lambda_{\\star,i}$ can be computed using the Doppler effect formalism $\\lambda_{\\star,i}= \\lambda_i \\left(\\sqrt{\\frac{1+\\gamma_{\\star}\/c}{1-\\gamma_{\\star}\/c}} - 1\\right)$. The second step is to identify the wavelength $\\lambda_{BS,j}$ of the block stitchings, which can be done using the wavelength solution and keeping in mind that the block stitchings appear every 512 pixels in the spectral direction. Finally, all the spectral lines in the correlation mask that have a wavelength in the stellar rest frame obeying to the relation\n\\begin{eqnarray}\n\\lambda_{BS,j} - min_{BERV} - W \\le \\lambda_{\\star,i} \\le \\lambda_{BS,j} + max_{BERV} + W\n\\end{eqnarray}\nshould be excluded. Here $W$ corresponds to the width of a spectral line. We fixed this value to 10\\kms\\, to exclude the wings of each spectral line \\footnote{The average full width at half maximum of a line is a few \\kms for a solar like star rotating slowly \\citep[][]{Gray-2008}}. \n\nIn Figure \\ref{fig:4-0}, we illustrate the process of excluding the spectral lines from the correlation mask taking the example of spectral order 61. The top panel shows two spectra of HD128621 taken at the extremes of the BERV. Each block stitching is localized in the wavelength domain, and the blue region across each block stitching correspond to the region where spectral lines should be excluded. The bottom panel shows a spectrum of HD128621 taken when the BERV was close to zero. All the spectral lines appearing in the correlation mask and shifted to the stellar gamma velocity are shown, and the ones falling close to the block stitching are excluded (red lines). Note that on the blue side of the spectral order, a line is exclude while not falling on a block stitching. There is some overlap between the spectral orders, and this line is rejected in spectral order 60.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=16cm]{wavelength_solution_order_61}\n\\caption{\\emph{Top: }The blue spectrum corresponds to spectral order 61 of an observation of HD128621 when the BERV was maximum and the red spectrum, spectral order 61 when the BERV was minimum. The green dashed vertical lines correspond to the position of the block stitchings and the blue regions to the zone where spectral lines should be excluded. \\emph{Bottom: } Spectrum corresponding to order 61 of an observation of HD128621 when the BERV was close to zero. The dashed vertical lines are the spectral lines present in the correlation mask, shifted to the stellar gamma velocity. The spectral line falling inside the excluding regions are highlighted in red and should be excluded from the correlation mask when deriving RVs to prevent the one-year block stitching effects.}\n\\label{fig:4-0}\n\\end{center}\n\\end{figure*}\n\nFor each star, two different correlation masks are generated. One with only the lines falling close to the block stitchings, and another one without those lines. The RVs measured with both correlation masks can be seen in Figure \\ref{fig:4-1}. It is clear that the spectral lines crossing the block stitching are the origin of the one-year signal (see left panel). Once those spectral lines are removed from the correlation mask, the yearly signal disappears (see right panel). Note that this Figure should be compared with Figure \\ref{fig:2-0}.\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=8cm]{periodogram_only_CCD_gaps_RVs}\n\\includegraphics[width=8cm]{periodogram_without_CCD_gaps_RVs}\n\\caption{\\emph{Left: }GLS periodograms of the RV residuals of our four stars after removing their planetary signals and their magnetic cycle effect (see text in Sec \\ref{sect:2} for more details about the fitted models). The RVs are derived using the correlation masks with only the spectral lines crossing the block stitchings. The periods close to one year are highlighted by the red regions, and are significantly above the 10\\% false alarm probability level (red horizontal lines). \\emph{Right: }Same plots but for the correlation masks cleaned from spectral lines crossing the block stitchings.}\n\\label{fig:4-1}\n\\end{center}\n\\end{figure*}\n\n\\section{Conclusion}  \\label{sect:5}\n\nWe have detected a one year signal in the data of several stars intensively observed with HARPS during several years. This signal is induced by spectral lines crossing the block stitchings of the HARPS CCD when the stellar spectrum is shifted back and forth on the detector due to the Earth's orbit around the Sun. At the position of each block stitching, the wavelength solution is not correct because the sparse emission spectrum of the Thorium-Argon lamp cannot resolve the different pixel size. Spectral lines crossing block stitchings therefore undergo a deformation when pixel position is translated into wavelength using the wavelength solution, which induces a RV variation. Removing those spectral lines from the correlation mask used to derive the RV with the cross-correlation technique allows to get rid of this yearly signal. Note that a different correlation mask for each star has to be generated due to a different stellar gamma velocity, and therefore a different position of the spectral lines on the HARPS CCD.\n\nThe amplitude of the one-year signal is different for each star, because of different BERV and different spectral lines crossing the block stitchings of the CCD (see Figure \\ref{fig:2-2}). If the amplitude of the BERV is only a few \\kms (maximum is 30\\kms), then the stellar spectra will only be shifted on the CCD by a few pixels, which will not deform enough the spectral lines crossing the block stitchings and therefore not induce a significant signal at one year. In addition, if only weak spectral lines, that only have a small weight in the final RV computation as described in \\citet{Pepe-2003}, cross block stitchings, the induced  RV variation will also be negligible. {\\bf The combination of these two effects explains why some stars are strongly affected by the one-year signal, while others are effected to a lesser degree.}\n\nThe number of spectral lines rejected in the correlation mask is of the order of 20\\%. This can therefore be a problem when observing faint targets, for which the signal-to-noise-ratio of the spectra is low. A possible way of removing the yearly signal without rejecting spectral lines would be to include as free parameters the pixel sizes at all block stitching, and thus derive a corrected wavelength solution (Coffinet et al. 2015, in prep.).\n\n\n{\\bf For calibration sources such as Fabry-Perot etalons or laser frequency-combs} \\citep[][]{Langellier-2014,Wilken-2012} that have a higher density of emission lines on the detector, it is possible to detect the CCD block stitchings and therefore directly obtain a correct wavelength solution. In addition to a better RV precision, these calibration sources should produce data that are not affected by the one-year signal described in this paper.\n\nThis yearly signal is extremely problematic when looking for long-period small-amplitude planets as it will inject signal at 365.25 days and the first corresponding harmonics, generally 182.63 and 121.75 days. It is therefore mandatory to correct this yearly signal from the HARPS RV data if such planet want to be detected. {\\bf For a user detecting a spurious signal at one year in HARPS RV data, two different methods can be used to suppress this signal. Using the pipeline, it is possible to recalculate a new correlation mask based on the technique described in Section 4 and reprocess the CCF routine. Otherwise fitting a sinusoidal signal with a one-year period is precise enough to a few dozens of \\cms. A careful check of the anti-correlation between the fitted signal and the BERV should however be done, to prevent removing one-year planetary signals that could exist in the data.}\n\n\n\\acknowledgments\n{\\bf We thanks the referee for his valuable comments and suggestions that improved the first version of the paper. X.D. thanks NASA's Transiting Exoplanet Survey Satellite (TESS) mission for partial support via subaward 5710003554 from MIT to SAO.} We are grateful to all technical and scientific collaborators of the HARPS Consortium, ESO Headquarters and ESO La Silla who have contributed with their extraordinary passion and valuable work to the success of the HARPS project.\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Adams spectral sequence is an important tool in stable homotopy theory. Given finite spectra $X$ and $Y$, the classical Adams spectral sequence is\n\\[\nE_2^{s,t} = \\Ext_{\\mathcal{A}}^{s,t}\\left( H^*Y, H^*X \\right) \\Ra [\\Sigma^{t-s}X, Y^{\\wedge}_p] ,\n\\]\nwhere $H^*X := H^*(X;\\mathbb{F}_p)$ denotes mod $p$ cohomology and $\\mathcal{A} = H\\mathbb{F}_p^* H\\mathbb{F}_p$ denotes the mod $p$ Steenrod algebra.\nDetermining the differentials in the Adams spectral sequence generally requires \na combination of techniques and much ingenuity.\nThe approach that provides a basis for our work is found in~\\cite{Maunder64},\nwhere Maunder showed that the differential $d_r$ in this spectral sequence is\ndetermined by $r^{\\text{th}}$ order cohomology operations, which we now review.\n\nA primary cohomology operation in this context is simply an element of the Steenrod algebra,\nand it is immediate from the construction of the Adams spectral sequence \nthat the differential $d_1$ is given by primary cohomology operations.\nA secondary cohomology operation corresponds to a relation among primary operations,\nand is partially defined and multi-valued: it is defined on the kernel of\na primary operation and takes values in the cokernel of another primary operation.\nTertiary operations correspond to relations between relations, and have correspondingly\nmore complicated domains and codomains.\nThe pattern continues for higher operations.\nUsing that cohomology classes are representable,\nsecondary cohomology operations can also be expressed using $3$-fold Toda brackets\ninvolving the cohomology class and two operations whose composite is null.\nHowever, what one obtains in general is a \\emph{subset} of the Toda bracket with less indeterminacy.\nThis observation will be the key to our generalization of Maunder's result to \nother Adams spectral sequences in other categories.\n\nThe starting point of this paper is the following observation.\nOn the one hand, the Adams spectral sequence can be constructed in any \ntriangulated category equipped with a projective class or an injective class~\\cite{Christensen98}.\nFor example, the classical Adams spectral sequence is constructed in the stable homotopy category\nwith the injective class consisting of retracts of products $\\prod_i \\Sigma^{n_i} H\\mathbb{F}_p$.\nOn the other hand, higher Toda brackets can also be defined in an arbitrary triangulated category.\nThis was done by Shipley in~\\cite{Shipley02}, based on Cohen's construction for spaces and spectra~\\cite{Cohen68},\nand was studied further in~\\cite{Sagave08}.\nThe goal of this paper is to describe precisely how the Adams $d_r$ can be described as\na particular subset of an $(r+1)$-fold Toda bracket which can be viewed as an $r^{\\text{th}}$ order cohomology operation,\nall in the context of a general triangulated category.\n\nTriangulated categories arise throughout mathematics,\nso our work applies in various situations.\nAs an example, we give calculations involving the Adams spectral sequence in \nthe stable module category of a group algebra.\nEven in stable homotopy theory, there are a variety of Adams spectral sequences,\nsuch as the Adams--Novikov \nspectral sequence or the motivic Adams spectral sequence, and our results apply to all of them.\nMoreover, by working with minimal structure, our approach gains a certain elegance.\n\n\\subsection*{Organization and main results}\n\nIn Section~\\ref{se:AdamsSS}, we review the construction of the Adams spectral sequence in a triangulated category equipped with a projective class or an injective class. In Section~\\ref{se:3-fold-Toda-brackets}, we review the construction of $3$-fold Toda brackets in a triangulated category and some of their basic properties. Section~\\ref{se:AdamsD2} describes how the Adams $d_2$ is given by $3$-fold Toda brackets. This section serves as a warm-up for Section~\\ref{se:AdamsDr}.\n\nIn Section~\\ref{se:HigherBrackets}, we recall the construction of higher Toda brackets in a triangulated category via filtered objects. We provide a family of alternate constructions, and prove that they are all equivalent. The main result is Theorem~\\ref{th:n-fold}, which says roughly the following.\n\n\\begin{thm*}\nThere is an inductive way of computing an $n$-fold Toda bracket $\\left\\langle f_n, \\ldots, f_1 \\right\\rangle \\subseteq \\cat{T}(\\Sigma^{n-2} X_0, X_n)$, where the inductive step picks three consecutive maps and reduces the length by one. The $(n-2)!$ ways of doing this yield the same subset, up to an explicit sign.   \n\\end{thm*}\n\nAs a byproduct, we obtain Corollary~\\ref{co:SelfDual}, which would be tricky to prove directly from the filtered object definition.\n\n\\begin{cor*}\nToda brackets are self-dual up to suspension: $\\left\\langle f_n, \\ldots, f_1 \\right\\rangle \\subseteq \\cat{T}(\\Sigma^{n-2} X_0, X_n)$ corresponds to the Toda bracket computed in the opposite category\n\\[\n\\left\\langle f_1, \\ldots, f_n \\right\\rangle \\subseteq \\cat{T}^{\\mathrm{op}}\\left( \\Sigma^{-(n-2)} X_n, X_0 \\right) = \\cat{T}(X_0, \\Sigma^{-(n-2)} X_n).\n\\] \n\\end{cor*}\n\nSection~\\ref{se:AdamsDr} establishes how the Adams $d_r$ is given by $(r+1)$-fold Toda brackets. Our main results are Theorems~\\ref{th:d1pdex} and \\ref{th:AdamsDrCohomOp}, which say roughly the following.\n\n\\begin{thm*}\nLet $[x] \\in E_r^{s,t}$ be a class in the $E_r$ term of the Adams spectral sequence. %\nAs subsets of $E_1^{s+r,t+r-1}$, we have\n\\begin{align*}\nd_r [x] &= \\left\\langle \\Sigma^{r-1} d_1 , \\ldots , \\Sigma^2 d_1, \\Sigma d_1 , \\Sigma p_{s+1} ,  \\delta_s x \\right\\rangle \\\\\n&= \\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle .\n\\end{align*}\n\\end{thm*}\n\nHere, $d_1$, $p_{s+1}$, and $\\delta_s$ are maps appearing in the Adams resolution of $Y$, where each $d_1$ is a primary cohomology operation. The first expression of $d_r [x]$ is an $(r+1)$-fold Toda bracket. The second expression denotes an appropriate subset of the bracket $\\left\\langle \\Sigma^{r-1} d_1, \\ldots, \\Sigma d_1, d_1 , x \\right\\rangle$ with some choices dictated by the Adams resolution of $Y$. This description exhibits $d_r [x]$ as an $r^{\\text{th}}$ order cohomology operation applied to $x$.\nIn Section~\\ref{se:sparse}, we show that when certain sparseness assumptions are made, the subset\n$\\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle$ coincides with the full Toda bracket,\nand give examples of this phenomenon.\n\nIn Appendix~\\ref{se:StableMod}, we compute examples of Toda brackets in stable module categories. In particular, Proposition~\\ref{pr:proper-inclusion-C4} provides an example where the inclusion $d_2 [x] \\subseteq \\left\\langle \\Sigma d_1, d_1, x \\right\\rangle$ is proper. Appendix~\\ref{se:Heller} provides for the record a short, simple proof of a theorem due to Heller, that $3$-fold Toda brackets determine the triangulated structure. As a corollary, we note that the $3$-fold Toda brackets indirectly determine the higher Toda brackets.\n\n\n\\subsection*{Related work}\n\nDetailed treatments of secondary operations can be found in \\cite{Adams60}*{\\S 3.6},\nwhere Adams used secondary cohomology operations to solve the Hopf invariant one problem,\n\\cite{MosherT68}*{Chapter 16}, and \\cite{Harper02}*{Chapter~4}.\n\nThere are various approaches to higher order cohomology operations and higher Toda brackets in the literature, many of which use some form of enrichment in spaces, chain complexes, or groupoids; see for instance \\cite{Spanier63}, \\cite{Maunder63}, \\cite{Kochman80}, and \\cite{Klaus01}. In this paper, we work solely with the triangulated structure, without enhancement, and provide no comparison to those other approaches.\n\nIn \\cite{BauesJ06} and \\cite{BauesJ11}, Baues and Jibladze express the Adams $d_2$ in terms of\nsecondary cohomology operations, and this is generalized to higher differentials by Baues and Blanc in~\\cite{BauesB15}.\nTheir approach starts with an injective resolution as in Diagram~\\eqref{eq:InjResol},\nand witnesses the equations $d_1 d_1 = 0$ by providing suitably coherent null-homotopies,\ndescribed using mapping spaces.\nUsing this coherence data, the authors express a representative of $d_r [x]$ as a\nspecific element of the Toda bracket $\\left\\langle \\Sigma^{r-1} d_1, \\ldots, \\Sigma d_1, d_1, x \\right\\rangle$.\nWhile this approach makes use of an enrichment, \nwe suspect that by translating the (higher dimensional) null-homotopies into lifts to fibers or extensions to cofibers,\none could relate their expression for $d_r [x]$ to ours.\n\n\n\n\\subsection*{Acknowledgments}\n\nWe thank Robert Bruner, Dan Isaksen, \nPeter Jorgensen, \nFernando Muro, \nIrakli Patchkoria, \nSteffen Sagave, and Dylan Wilson for helpful conversations, as well as the referee for their useful comments.  The second author also thanks the Max-Planck-Institut f\\\"ur Mathematik for its hospitality.\n\n\n\n\\section{The Adams spectral sequence}\\label{se:AdamsSS}\n\nIn this section, we recall the construction of the Adams spectral sequence in a triangulated category, along with some of its features. We follow~\\cite{Christensen98}*{\\S 4}, or rather its dual. Some references for the classical Adams spectral sequence are~\\cite{Adams74}*{\\S III.15}, \\cite{Margolis83}*{Chapter 16}, and~\\cite{Bruner09}. %\nBackground material on triangulated categories can be found in \\cite{Neeman01}*{Chapter 1}, \n\\cite{Margolis83}*{Appendix 2}, and \\cite{Weibel94}*{Chapter 10}.\nWe assume that the suspension functor $\\Sigma$ is an equivalence, with chosen inverse $\\Sigma^{-1}$.\nMoreover, we assume we have chosen natural isomorphisms $\\Sigma \\Sigma^{-1} \\cong \\mathrm{id}$ and $\\Sigma^{-1} \\Sigma \\cong \\mathrm{id}$\nmaking $\\Sigma$ and $\\Sigma^{-1}$ into an adjoint equivalence.\nWe silently use these isomorphisms when needed, e.g., when we say that a triangle\nof the form $\\Sigma^{-1} Z \\to X \\to Y \\to Z$ is distinguished.\n\n\\begin{defn}[{\\cite{Christensen98}*{Proposition 2.6}}]\nA \\textbf{projective class} in a triangulated category $\\cat{T}$ is a pair $(\\cat{P}, \\cat{N})$ where $\\cat{P}$ is a class of objects and $\\cat{N}$ is a class of maps satisfying the following properties.\n\\begin{enumerate}\n\\item A map $f \\colon X \\to Y$ is in $\\cat{N}$ if and only if the induced map\n\\[\nf_* \\colon \\cat{T}(P,X) \\to \\cat{T}(P,Y)\n\\]\nis zero for all $P$ in $\\cat{P}$. In other words, $\\cat{N}$ consists of the \\textbf{$\\cat{P}$-null} maps.\n\\item An object $P$ is in $\\cat{P}$ if and only if the induced map\n\\[\nf_* \\colon \\cat{T}(P,X) \\to \\cat{T}(P,Y)\n\\]\nis zero for all $f$ in $\\cat{N}$.\n\\item For every object $X$, there is a distinguished triangle $P \\to X \\ral{f} Y \\to \\Sigma P$, where $P$ is in $\\cat{P}$ and $f$ is in $\\cat{N}$.\n\\end{enumerate}\n\\end{defn}\n\nIn particular, the class $\\cat{P}$ is closed under arbitrary coproducts and retracts. The objects in $\\cat{P}$ are called \\textbf{projective}.\n\n\\begin{defn}\nA projective class is \\textbf{stable} if it is closed under shifts, i.e., $P \\in \\cat{P}$ implies $\\Sigma^n P \\in \\cat{P}$ for all $n \\in \\mathbb{Z}$.%\n\\end{defn}\n\nWe will assume for convenience that our projective class is stable. We suspect that many of the results can be adapted to unstable projective classes, with a careful treatment of shifts. \n\n\\begin{defn}\nLet $\\cat{P}$ be a projective class. A map $f \\colon X \\to Y$ is called\n\\begin{enumerate}\n\\item \\textbf{$\\cat{P}$-epic} if the map\n\\[\nf_* \\colon \\cat{T}(P,X) \\to \\cat{T}(P,Y)\n\\]\nis surjective for all $P \\in \\cat{P}$. Equivalently, the map to the cofiber $Y \\to C_f$ is $\\cat{P}$-null.\n\\item \\textbf{$\\cat{P}$-monic} if the map\n\\[\nf_* \\colon \\cat{T}(P,X) \\to \\cat{T}(P,Y)\n\\]\nis injective for all $P \\in \\cat{P}$. Equivalently, the map from the fiber $\\Sigma^{-1} C_f \\to X$ is $\\cat{P}$-null.\n\\end{enumerate}\n\\end{defn}\n\n\n\\begin{ex}\\label{ex:GhostSphere}\nLet $\\cat{T}$ be the stable homotopy category and $\\cat{P}$ the projective class consisting of retracts of wedges of spheres $\\bigvee_i S^{n_i}$. This is called the \\emph{ghost projective class}, studied for instance in \\cite{Christensen98}*{\\S 7}.%\n\\end{ex}\n\nNow we dualize everything.\n\n\\begin{defn}\nAn \\textbf{injective class} in a triangulated category $\\cat{T}$ is a projective class in the opposite category $\\cat{T}^{\\mathrm{op}}$. Explicitly, it is a pair $(\\cat{I}, \\cat{N})$ where $\\cat{I}$ is a class of objects and $\\cat{N}$ is a class of maps satisfying the following properties.\n\\begin{enumerate}\n\\item A map $f \\colon X \\to Y$ is in $\\cat{N}$ if and only if the induced map\n\\[\nf^* \\colon \\cat{T}(Y,I) \\to \\cat{T}(X,I)\n\\]\nis zero for all $I$ in $\\cat{I}$.\n\\item An object $I$ is in $\\cat{I}$ if and only if the induced map\n\\[\nf^* \\colon \\cat{T}(Y,I) \\to \\cat{T}(X,I)\n\\]\nis zero for all $f$ in $\\cat{N}$.\n\\item For every object $X$, there is a distinguished triangle $\\Sigma^{-1} I \\to W \\ral{f} X \\to I$, where $I$ is in $\\cat{I}$ and $f$ is in $\\cat{N}$.\n\\end{enumerate}\n\\end{defn}\n\nIn particular, the class $\\cat{I}$ is closed under arbitrary products and retracts. The objects in $\\cat{I}$ are called \\textbf{injective}.\nJust as for projective classes, we will assume for convenience that our injective class is stable.\n\n\n\\begin{ex}\\label{ex:InjClass}\nLet $\\cat{T}$ be the stable homotopy category. Take $\\cat{N}$ to be the class of maps inducing zero on mod $p$ cohomology and $\\cat{I}$ to be the retracts of (arbitrary) products $\\prod_i \\Sigma^{n_i} H\\mathbb{F}_p$ with $n_i \\in \\mathbb{Z}$. One can generalize this example with any cohomology theory (spectrum) $E$ instead of $H\\mathbb{F}_p$, letting $\\cat{I}_E$ denote the injective class consisting of retracts of products $\\prod_i \\Sigma^{n_i} E$.\n\\end{ex}\n\n\\begin{defn}\nLet $\\cat{I}$ be an injective class. A map $f \\colon X \\to Y$ is called\n\\begin{enumerate}\n\\item \\textbf{$\\cat{I}$-monic} if the map\n\\[\nf^* \\colon \\cat{T}(Y,I) \\to \\cat{T}(X,I)\n\\]\nis surjective for all $I \\in \\cat{I}$. Equivalently, the map from the fiber $\\Sigma^{-1} C_f \\to X$ is $\\cat{I}$-null.\n\\item \\textbf{$\\cat{I}$-epic} if the map\n\\[\nf^* \\colon \\cat{T}(Y,I) \\to \\cat{T}(X,I)\n\\]\nis injective for all $I \\in \\cat{I}$. Equivalently, the map to the cofiber $Y \\to C_f$ is $\\cat{I}$-null.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{rem}\nThe projectives and $\\cat{P}$-epic maps determine each other via the lifting property\n\\[\n\\xymatrix{\n& X \\ar@{->>}[d]^f \\\\\nP \\ar@{-->}[ur] \\ar@{->}[r] & Y. \\\\\n}\n\\]\nDually, the injectives and $\\cat{I}$-monic maps determine each other via the extension property\n\\[\n\\xymatrix{\nX \\ar[r] \\ar@{ >->}[d]_f & I \\\\\nY . \\ar@{-->}[ur] &  \\\\\n}\n\\]\nThis is part of the equivalent definition of a projective (resp.\\ injective) class described in \\cite{Christensen98}*{Proposition 2.4}. \n\\end{rem}\n\n\n\n\\begin{convention}\\label{co:SuspensionIso}\nWe will implicitly use the natural isomorphism $\\cat{T}(A,B) \\cong \\cat{T}(\\Sigma^k A, \\Sigma^k B)$ sending a map $f$ to $\\Sigma^k f$. \n\\end{convention}\n\n\\begin{defn}\\label{def:AdamsResol}\nAn \\textbf{Adams resolution} of an object $X$ in $\\cat{T}$ with respect to a projective class $(\\cat{P}, \\cat{N})$ is a diagram\n\\begin{equation} \\label{eq:AdamsResolProj}\n\\cxymatrix{\n\\mathllap{X =\\ }X_0 \\ar[rr]^{i_0} & & X_1 \\circar[dl]^{\\delta_0} \\ar[rr]^{i_1} & & X_2 \\circar[dl]^{\\delta_1} \\ar[rr]^{i_2} & & X_3 \\circar[dl]^{\\delta_2} \\ar[r] & \\cdots \\\\\n& P_0 \\ar@{->>}[ul]^{p_0} & & P_1 \\ar@{->>}[ul]^{p_1} & & P_2 \\ar@{->>}[ul]^{p_2} & & \\\\\n}\n\\end{equation}\nwhere every $P_s$ is projective, every map $i_s$ is in $\\cat{N}$, and every triangle $P_s \\ral{p_s} X_s \\ral{i_s} X_{s+1} \\ral{\\delta_s} \\Sigma P_s$ is distinguished. Here the arrows $\\delta_s \\colon X_{s+1} {\\ooalign{$\\longrightarrow$\\cr\\hidewidth$\\circ$\\hidewidth\\cr}} P_{s}$ denote degree-shifting maps, namely, maps $\\delta_s \\colon X_{s+1} \\to \\Sigma P_{s}$.\n\nDually, an \\textbf{Adams resolution} of an object $Y$ in $\\cat{T}$ with respect to an injective class $(\\cat{I}, \\cat{N})$ is a diagram\n\\begin{equation}\\label{eq:AdamsResolInj}\n\\cxymatrix{\n\\mathllap{Y =\\ }Y_0 \\ar@{ >->}[dr]_{p_0} & & Y_1 \\ar@{ >->}[dr]_{p_1} \\ar[ll]_{i_0} & & Y_2 \\ar@{ >->}[dr]_{p_2} \\ar[ll]_{i_1} & & Y_3 \\ar[ll]_{i_2} & \\cdots \\ar[l] \\\\\n& I_0 \\circar[ur]_{\\delta_0} & & I_1 \\circar[ur]_{\\delta_1} & & I_2 \\circar[ur]_{\\delta_2} & & \\\\\n}\n\\end{equation}\nwhere every $I_s$ is injective, every map $i_s$ is in $\\cat{N}$, and every triangle $\\Sigma^{-1} I_s \\ral{\\Sigma^{-1} \\delta_s} Y_{s+1} \\ral{i_s} Y_s \\ral{p_s} I_s$ is distinguished.%\n\\end{defn}\nFrom now on, fix a triangulated category $\\cat{T}$ and a (stable) injective class $(\\cat{I}, \\cat{N})$ in $\\cat{T}$. \n\n\\begin{lem}\nEvery object $Y$ of $\\cat{T}$ admits an Adams resolution.\n\\end{lem}\n\nGiven an object $X$ and an Adams resolution of $Y$, applying $\\cat{T}(X,-)$ yields an exact couple\n\\[\n\\xymatrix{\n\\bigoplus_{s,t} \\cat{T}(\\Sigma^{t-s} X, Y_s) \\ar[rr]^-{i = \\oplus (i_s)_*} & & \n\\bigoplus_{s,t} \\cat{T}(\\Sigma^{t-s} X, Y_s) \\ar[dl]^-{p = \\oplus (p_s)_*} \\\\\n& \\bigoplus_{s,t} \\cat{T}(\\Sigma^{t-s} X, I_s) \\ar[ul]^-{\\delta = \\oplus (\\delta_s)_*}\n}\n\\]\nand thus a spectral sequence with $E_1$ term\n\\[\nE_1^{s,t} %\n        = \\cat{T}\\left( \\Sigma^{t-s} X, I_s \\right)\n        \\cong \\cat{T}\\left( \\Sigma^t X, \\Sigma^s I_s \\right)\n\\]\nand differentials\n\\[\nd_r \\colon E_r^{s,t} \\to E_r^{s+r, t+r-1}\n\\]\ngiven by $d_r = p \\circ i^{-(r-1)} \\circ \\delta$, where $i^{-1}$ means choosing an $i$-preimage.\nThis is called the \\textbf{Adams spectral sequence} with respect to the injective class $\\cat{I}$\nabutting to $\\cat{T}(\\Sigma^{t-s} X, Y)$.\n\n\n\\begin{lem}\nThe $E_2$ term is given by\n\\[\nE_2^{s,t} = \\Ext_{\\cat{I}}^{s,t}(X,Y) := \\Ext_{\\cat{I}}^{s}(\\Sigma^t X,Y)\n\\]\nwhere $\\Ext_{\\cat{I}}^{s}(X,Y)$ denotes the $s^{\\text{th}}$ derived functor of $\\cat{T}(X,-)$ (relative to the injective class $\\cat{I}$) applied to the object $Y$.\n\\end{lem}\n\n\\begin{proof}\nThe Adams resolution of $Y$ yields an $\\cat{I}$-injective resolution of $Y$\n\\begin{equation}\\label{eq:InjResol}\n\\xymatrix @C=3.3pc {\n0 \\ar[r] & Y \\ar[r]^-{p_0} & I_0 \\ar[r]^-{(\\Sigma p_1) \\delta_0} & \\Sigma I_1 \\ar[r]^-{(\\Sigma^2 p_2) (\\Sigma \\delta_1)} & \\Sigma^2 I_2 \\ar[r] & \\cdots \\\\\n} \\qedhere\n\\end{equation}\n\\end{proof}\n\n\\begin{rem}\\label{re:NotGenerate}\nWe do not assume that the injective class $\\cat{I}$ generates, \ni.e., that every non-zero object $X$ admits a non-zero map $X \\to I$ to an injective. \nHence, we do not expect the Adams spectral sequence to be conditionally convergent in general; c.f.~\\cite{Christensen98}*{Proposition~4.4}.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\\begin{ex}\\label{ex:EBased}\nLet $E$ be a commutative (homotopy) ring spectrum. \nA spectrum is called \\textbf{$E$-injective} if it is a retract of $E \\sm W$ for some $W$ \\cite{HoveyS99}*{Definition 2.22}. \nA map of spectra $f \\colon X \\to Y$ is called \\textbf{$E$-monic} if the map $E \\sm f \\colon E \\sm X \\to E \\sm Y$ is a split monomorphism. \nThe $E$-injective objects and $E$-monic maps form an injective class in the stable homotopy category. \nThe Adams spectral sequence associated to \nthis injective class\nis the \\emph{Adams spectral sequence based on $E$-homology}, as described in \\cite{Ravenel04}*{Definition~2.2.4}, also called the \\emph{unmodified Adams spectral sequence} in \\cite{HoveyS99}*{\\S 2.2}. Further assumptions are needed in order to identify the $E_2$ term as $\\Ext$ groups in $E_*E$-comodules.\n\\end{ex}\n\n\n\n\n\\begin{defn}\nThe \\textbf{$\\cat{I}$-cohomology} of an object $X$ is the family of abelian groups\\break $H^I(X) := \\cat{T}(X,I)$ indexed by the injective objects $I \\in \\cat{I}$.\n\nA \\textbf{primary operation} in $\\cat{I}$-cohomology is a natural transformation $H^I(X) \\to H^J(X)$ of functors $\\cat{T}^{\\mathrm{op}} \\to \\mathrm{Ab}$. Equivalently, by the (additive) Yoneda lemma, a primary operation is a map $I \\to J$ in $\\cat{T}$.\n\\end{defn}\n\n\\begin{ex}\nThe differential $d_1$ is given by primary operations. More precisely, let $x \\in E_1^{s,t}$ be a map $x \\colon \\Sigma^{t-s} X \\to I_s$. Then $d_1(x) \\in E_1^{s+1,t}$ is the composite\n\\[\n\\xymatrix{\n\\Sigma^{t-s} X \\ar[r]^-{x} & I_s \\ar[r]^-{\\delta_s} & \\Sigma Y_{s+1} \\ar[r]^-{\\Sigma p_{s+1}} & \\Sigma I_{s+1}. \\\\\n}\n\\]\nIn other words, $d_1(x)$ is obtained by applying the primary operation $d_1 := (\\Sigma p_{s+1}) \\delta_s \\colon I_s \\to \\Sigma I_{s+1}$ to $x$.\n\\end{ex}\n\n\\begin{prop}\nA primary operation $\\theta \\colon I \\to J$ appears as $d_1 \\colon I_s {\\ooalign{$\\longrightarrow$\\cr\\hidewidth$\\circ$\\hidewidth\\cr}} I_{s+1}$ in some Adams resolution if and only if $\\theta$ admits an $\\cat{I}$-epi -- $\\cat{I}$-mono factorization.\n\\end{prop}\n\n\\begin{proof}\nThe condition is necessary by construction. In the factorization $d_1 = (\\Sigma p_{s+1}) \\delta_s$, the map $\\delta_s$ is $\\cat{I}$-epic while $p_{s+1}$ is $\\cat{I}$-monic.\n\nTo prove sufficiency, assume given a factorization $\\theta = iq \\colon I \\to W \\to J$, where $q \\colon I \\twoheadrightarrow W$ is $\\cat{I}$-epic and $i \\colon W \\hookrightarrow J$ is $\\cat{I}$-monic. Taking the fiber of $q$ twice yields the distinguished triangle\n\\[\n\\xymatrix{\n\\Sigma^{-1} W \\ar[r] & Y_0 \\ar@{ >->}[r] & I \\ar@{->>}[r]^q & W \\\\\n}\n\\]\nwhich we relabel\n\\[\n\\xymatrix{\nY_1 \\ar[r]^-{i_0} & Y_0 \\ar@{ >->}[r]^-{p_0} & I \\ar@{->>}[r]^-{\\delta_0} & \\Sigma Y_1. \\\\\n}\n\\]\nRelabeling the given map $i \\colon W \\hookrightarrow J$ as $\\Sigma p_1 \\colon \\Sigma Y_1 \\hookrightarrow \\Sigma I_1$, we can continue the usual construction of an Adams resolution of $Y_0$ as illustrated in Diagram~\\eqref{eq:AdamsResolInj}, in which $\\theta = iq$ appears as the composite $(\\Sigma p_1) \\delta_0$. Note that by the same argument, for any $s \\geq 0$, $\\theta$ appears as $d_1 \\colon I_s {\\ooalign{$\\longrightarrow$\\cr\\hidewidth$\\circ$\\hidewidth\\cr}} I_{s+1}$ in some (other) Adams resolution.\n\\end{proof}\n\n\\begin{ex}\nNot every primary operation appears as $d_1$ in an Adams resolution. For example, consider the stable homotopy category with the projective class $\\cat{P}$ generated by the sphere spectrum $S = S^0$, that is, $\\cat{P}$ consists of retracts of wedges of spheres. The $\\cat{P}$-epis (resp. $\\cat{P}$-monos) consist of the maps which are surjective (resp. injective) on homotopy groups. The primary  operation $2 \\colon S \\to S$ does \\emph{not} admit a $\\cat{P}$-epi -- $\\cat{P}$-mono factorization.\n\nIndeed, assume that $2 = iq \\colon S \\twoheadrightarrow W \\hookrightarrow S$ is such a factorization. We will show that this implies $\\pi_2 (S\/2) = \\mathbb{Z}\/2 \\oplus \\mathbb{Z}\/2$, contradicting the known fact $\\pi_2 (S\/2) = \\mathbb{Z}\/4$. Here $S\/2$ denotes the mod $2$ Moore spectrum, sitting in the cofiber sequence $S \\ral{2} S \\to S\/2$.\n\nBy the octahedral axiom applied to the factorization $2 = iq$, there is a diagram\n\\[\n\\xymatrix{\nS \\ar@{=}[d] \\ar@{->>}[r]^-{q} & W \\ar@{ >->}[d]^{i} \\ar[r] & C_q \\ar[d]^{\\alpha} \\ar@{ >->}[r]^{\\delta'} & S^1 \\ar@{=}[d] \\\\\nS \\ar[r]^2 & S \\ar@{->>}[d]^{j} \\ar[r] & S\/2 \\ar[d]^{\\beta} \\ar[r]^{\\delta} & S^1 \\\\\n& C_i \\ar@{=}[r] & C_i & \\\\\n}\n\\]\nwith distinguished rows and columns. The long exact sequence in homotopy yields $\\pi_n C_q = \\mbox{}_2 \\pi_{n-1} S$, \nwhere the induced map $\\pi_n(\\delta') \\colon \\pi_n C_q \\to \\pi_n S^1$ corresponds to the inclusion $\\mbox{}_2 \\pi_{n-1} S \\hookrightarrow \\pi_{n-1} S$. Likewise, we have $\\pi_n C_i = \\left( \\pi_{n} S \\right) \/ 2$, \nwhere the induced map $\\pi_n(j) \\colon \\pi_n S \\to \\pi_n C_i$ corresponds to the quotient map $\\pi_{n} S \\twoheadrightarrow \\left( \\pi_{n} S \\right) \/ 2$. The defining cofiber sequence $S \\ral{2} S \\to S\/2$ yields the exact sequence\n\\[\n\\xymatrix{\n\\pi_n S \\ar[r]^2 & \\pi_n S \\ar[r] & \\pi_n (S\/2) \\ar[r]^{\\pi_n \\delta} & \\pi_{n-1} S \\ar[r]^2 & \\pi_{n-1} S \\\\\n}\n\\] \nwhich in turn yields the short exact sequence\n\\begin{equation*}%\n\\xymatrix{\n0 \\ar[r] & \\left( \\pi_n S \\right) \/ 2 \\ar[r] & \\pi_n (S\/2) \\ar[r]^{\\pi_n \\delta} & \\mbox{}_2 \\pi_{n-1} S \\ar[r] & 0. \\\\\n}\n\\end{equation*}\nThe map $\\pi_n (\\alpha) \\colon \\mbox{}_2 \\pi_{n-1} S \\to \\pi_n(S\/2)$ is a splitting of this sequence, because of the equality $\\pi_n(\\delta) \\pi_n (\\alpha) = \\pi_n(\\delta \\alpha) = \\pi_n(\\delta')$. However, the short exact sequence does not split in the case $n=2$, by the isomorphism $\\pi_2(S\/2) = \\mathbb{Z}\/4$. \nFor references, see~\\cite{Schwede12}*{Proposition II.6.48}, \\cite{Schwede10}*{Proposition 4},\nand~\\cite{MO100272}.\n\\end{ex}\n\n\n\\section{\\texorpdfstring{$3$}{3}-fold Toda brackets}\\label{se:3-fold-Toda-brackets}\n\n\nIn this section, we review different constructions of $3$-fold Toda brackets and some of their properties.\n\n\\enlargethispage{3pt}\n\\begin{defn} \\label{def:TodaBracket}\nLet $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3$ be a diagram in a triangulated category $\\cat{T}$. We define subsets of $\\cat{T}(\\Sigma X_0, X_3)$ as follows.\n\\begin{itemize}\n\\item The \\textbf{iterated cofiber Toda bracket} $\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\cc} \\subseteq \\cat{T}(\\Sigma X_0, X_3)$ consists of all maps $\\psi \\colon \\Sigma X_0 \\to X_3$ that appear in a commutative diagram\n\\begin{equation} \\label{eq:CofCof}\n\\cxymatrix{\nX_0 \\ar@{=}[d] \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] \\ar[r] & C_{f_1} \\ar[d]^{\\varphi} \\ar[r] & \\Sigma X_0 \\ar[d]^{\\psi} \\\\\nX_0 \\ar[r]^-{f_1} & X_1 \\ar[r]^-{f_2} & X_2 \\ar[r]^-{f_3} & X_3 \\\\\n}\n\\end{equation}\nwhere the top row is distinguished.\n\\item The \\textbf{fiber-cofiber Toda bracket} $\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc} \\subseteq \\cat{T}(\\Sigma X_0, X_3)$ consists of all composites $\\beta \\circ \\Sigma \\alpha \\colon \\Sigma X_0 \\to X_3$, where $\\alpha$ and $\\beta$ appear in a commutative diagram\n\\begin{equation} \\label{eq:FibCof}\n\\vcenter{\n\\xymatrix @C=3.3pc {\nX_0 \\ar[d]_-{\\alpha} \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] & & \\\\\n\\Sigma^{-1} C_{f_2} \\ar[r] & X_1 \\ar[r]^-{f_2} & X_2 \\ar@{=}[d] \\ar[r] & C_{f_2} \\ar[d]^-{\\beta}  \\\\\n& & X_2  \\ar[r]^-{f_3} & X_3 \\\\\n}\n}\n\\end{equation}\nwhere the middle row is distinguished.\n\\item The \\textbf{iterated fiber Toda bracket} $\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\ff} \\subseteq \\cat{T}(\\Sigma X_0, X_3)$ consists of all maps $\\Sigma \\delta \\colon \\Sigma X_0 \\to X_3$ where $\\delta$ appears in a commutative diagram\n\\begin{equation} \\label{eq:FibFib}\n\\vcenter{\n\\xymatrix @C=3.3pc {\nX_0 \\ar[d]_{\\delta} \\ar[r]^-{f_1} & X_1 \\ar[d]_{\\gamma} \\ar[r]^-{f_2} & X_2 \\ar@{=}[d] \\ar[r]^-{f_3} & X_3 \\ar@{=}[d] \\\\\n\\Sigma^{-1} X_3 \\ar[r] & \\Sigma^{-1} C_{f_3} \\ar[r] & X_2 \\ar[r]^-{f_3} & X_3 \\\\\n}\n}\n\\end{equation}\nwhere the bottom row is distinguished.\n\\end{itemize}\n\\end{defn}\n\n\\begin{rem}\\label{re:3-fold-negation}\nIn the literature, there are variations of these definitions, which sometimes\ndiffer by a sign.\nWith the notion of cofiber sequence implicitly used in~\\cite{Toda62},\nour definitions agree with Toda's.\nThe Toda bracket also depends on the choice of triangulation.\nGiven a triangulation, there is an associated negative triangulation whose\ndistinguished triangles are those triangles whose negatives are distinguished\nin the original triangulation (see~\\cite{Balmer02}).\nNegating a triangulation negates the $3$-fold Toda brackets.\nDan Isaksen has pointed out to us that in the stable homotopy category\nthere are $3$-fold Toda brackets which are not equal to their own \nnegatives.\nFor example, Toda showed in~\\cite{Toda62}*{Section~VI.v, and Theorems~7.4\nand~14.1} that the Toda bracket $\\left\\langle 2 \\sigma, 8, \\nu \\right\\rangle$\nhas no indeterminacy and contains an element $\\zeta$ of order $8$.\nWe give another example in Example~\\ref{ex:negative}.\n\\end{rem}\n\nThe following proposition can be found in \\cite{Sagave08}*{Remark 4.5 and Figure 2} and was kindly pointed out by Fernando Muro. It is also proved in \\cite{Meier12}*{\\S 4.6}. We provide a different proof more in the spirit of what we do later. In the case of spaces, it was originally proved by Toda \\cite{Toda62}*{Proposition 1.7}. \n\n\\begin{prop} \\label{TodaBracketsAgree}\nThe iterated cofiber, fiber-cofiber, and iterated fiber definitions of Toda brackets coincide. More precisely, for any diagram $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3$ in $\\cat{T}$, the following subsets of $\\cat{T}(\\Sigma X_0, X_3)$ are equal:\n\\[\n\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\cc} = \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc} = \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\ff}.\n\\]\n\\end{prop}\n\n\\begin{proof}\nWe will prove the first equality; the second equality is dual.\n\n($\\supseteq$) Let $\\beta (\\Sigma \\alpha) \\in \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc}$ be obtained from maps $\\alpha$ and $\\beta$ as in Diagram \\eqref{eq:FibCof}. Now consider the diagram with distinguished rows\n\\[\n\\xymatrix{\nX_0 \\ar[d]^{\\alpha} \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] \\ar[r] & C_{f_1} \\ar@{-->}[d]^{\\varphi} \\ar[r] & \\Sigma X_0  \\ar[d]^{\\Sigma \\alpha} \\\\\n\\Sigma^{-1} C_{f_2} \\ar[r] & X_1 \\ar[r]^-{f_2} & X_2 \\ar@{=}[d] \\ar[r] & C_{f_2} \\ar[d]^{\\beta} \\\\\n& & X_2 \\ar[r]^-{f_3} & X_3 \\\\\n}\n\\]\nwhere there exists a filler $\\varphi \\colon C_{f_1} \\to X_2$. The commutativity of the tall rectangle on the right exhibits the membership $\\beta (\\Sigma \\alpha) \\in \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\cc}$.\n\n($\\subseteq$) Let $\\psi \\in \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\cc}$ be as in Diagram \\eqref{eq:CofCof}. The octahedral axiom comparing the cofibers of $q_1$, $\\varphi$, and $\\varphi \\circ q_1 = f_2$ yields a commutative diagram\n\\[\n\\xymatrix @C=1.1pc @R=0.88pc {\n&& && \\Sigma^{-1} C_{\\varphi} \\ar[dd]_{- \\Sigma^{-1} \\iota} \\ar@{=}[rr] & & \\Sigma^{-1} C_{\\varphi} \\ar[dd]_{- \\Sigma^{-1} \\eta} & \\\\ \\\\\nX_0 \\ar[dd]_{\\alpha} \\ar[rr]^-{f_1} && X_1 \\ar@{=}[dd] \\ar[rr]^-{q_1} && C_{f_1} \\ar[dd]_{\\varphi} \\ar[rr]^-{\\iota_1} & & \\Sigma X_0  \\ar[dd]_{\\Sigma \\alpha} \\ar@\/_1pc\/[dddl]_(0.35){\\psi} \\ar[rr]^-{- \\Sigma f_1} && \\Sigma X_1 \\ar@{=}[dd] \\\\ \\\\\n\\Sigma^{-1} C_{f_2} \\ar[rr]^(0.52){- \\Sigma^{-1} \\iota_2} && X_1 \\ar[rr]^-{f_2} && X_2 \\ar[dd]_{q} \\ar[dr]_{f_3} \\ar[rr]^(0.4){q_2} & & C_{f_2} \\ar[dd]^{\\xi} \\ar@{-->}[dl]^{\\!\\beta} \\ar[rr]^{\\iota_2} && \\Sigma X_1 \\\\\n&& && & X_3 & & \\\\\n&& && C_{\\varphi} \\ar@{-->}[ur]^-{\\theta} \\ar@{=}[rr] & & C_{\\varphi}, & \\\\\n}\n\\]\nwhere the rows and columns are distinguished. By exactness of the sequence\n\\[\n\\xymatrix @C=3.3pc {\n\\cat{T}(C_{f_2}, X_3) \\ar[r]^-{(\\Sigma \\alpha)^*} & \\cat{T}(\\Sigma X_0, X_3) \\ar[r]^-{(- \\Sigma^{-1} \\eta)^*} & \\cat{T}(\\Sigma^{-1} C_{\\varphi}, X_3)\n}\n\\]\nthere exists a map $\\beta \\colon C_{f_2} \\to X_3$ satisfying $\\psi = \\beta (\\Sigma \\alpha)$ if and only if the restriction of $\\psi$ to the fiber $\\Sigma^{-1} C_{\\varphi}$ of $\\Sigma \\alpha$ is zero. That condition does hold: one readily checks the equality $\\psi (- \\Sigma^{-1} \\eta) = 0$.\nThe chosen map $\\beta \\colon C_{f_2} \\to X_3$ might \\emph{not} satisfy the equation $\\beta q_2 = f_3$, but we will correct it to another map $\\beta'$ which does. The error term $f_3 - \\beta q_2$ is killed by restriction along $\\varphi$, %\nand therefore factors through the cofiber of $\\varphi$, i.e., there exists a factorization\n\\[\nf_3 - \\beta q_2 = \\theta \\iota\n\\]\nfor some $\\theta \\colon C_{\\varphi} \\to X_3$. The corrected map $\\beta' := \\beta + \\theta \\xi \\colon C_{f_2} \\to X_3$ satisfies $\\beta' q_2 = f_3$. \nMoreover, this corrected map $\\beta'$ still satisfies $\\beta' (\\Sigma \\alpha) = \\psi = \\beta (\\Sigma \\alpha)$, since the correction term satisfies $\\theta \\xi (\\Sigma \\alpha) = 0$. %\n\\end{proof}\n\nThanks to the proposition, we can write $\\left\\langle f_3, f_2, f_1 \\right\\rangle$ if we \ndo not need to specify a particular definition of the Toda bracket.\n\nWe also recall this well-known fact, and leave the proof as an exercise:\n\n\\begin{lem}\\label{le:indeterminacy}\nFor any diagram $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3$ in $\\cat{T}$,\nthe subset $\\left\\langle f_3, f_2, f_1 \\right\\rangle$ of $\\cat{T}(\\Sigma X_0, X_3)$ is a coset of\nthe subgroup\n\\[\n  (f_3)_* \\, \\cat{T}(\\Sigma X_0, X_2) + (\\Sigma f_1)^* \\, \\cat{T}(\\Sigma X_1, X_3) .\n\\vspace{-18pt}\n\\]\n\\qed\n\\end{lem}\n\nThe displayed subgroup is called the \\textbf{indeterminacy}, and when\nit is trivial, we say that the Toda bracket \\textbf{has no indeterminacy}.\n\n\\begin{lem}\\label{le:MoveAround}\nConsider maps $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3 \\ral{f_4} X_4$. Then the following inclusions of subsets of $\\cat{T}(\\Sigma X_0, X_4)$ hold.\n\\begin{enumerate}\n\\item\n\\[\nf_4 \\left\\langle f_3, f_2, f_1 \\right\\rangle \\subseteq \\left\\langle f_4 f_3, f_2, f_1 \\right\\rangle\n\\]\n\\item\n\\[\n\\left\\langle f_4, f_3, f_2 \\right\\rangle f_1 \\subseteq \\left\\langle f_4, f_3, f_2 f_1 \\right\\rangle\n\\]\n\\item\n\\[\n\\left\\langle f_4 f_3, f_2, f_1 \\right\\rangle \\subseteq \\left\\langle f_4, f_3 f_2, f_1 \\right\\rangle\n\\]\n\\item\n\\[\n\\left\\langle f_4, f_3, f_2 f_1 \\right\\rangle \\subseteq \\left\\langle f_4, f_3 f_2, f_1 \\right\\rangle.\n\\]\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n(1)-(2) These inclusions %\nare straightforward.\n\n(3)-(4) Using the iterated cofiber definition, the subset $\\left\\langle f_4 f_3, f_2, f_1 \\right\\rangle_{\\cc}$ consists of the maps $\\psi \\colon \\Sigma X_0 \\to X_4$ appearing in a commutative diagram\n\\[\n\\xymatrix{\nX_0 \\ar@{=}[d] \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] \\ar[r] & C_{f_1} \\ar[d]^{\\varphi} \\ar[rr] & & \\Sigma X_0 \\ar[d]^{\\psi} \\\\\nX_0 \\ar[r]^-{f_1} & X_1 \\ar[r]^-{f_2} & X_2 \\ar[r]^-{f_3} & X_3 \\ar[r]^-{f_4} & X_4 \\\\\n}\n\\]\nwhere the top row is distinguished. Given such a diagram, the diagram\n\\[\n\\xymatrix{\nX_0 \\ar@{=}[d] \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] \\ar[r] & C_{f_1} \\ar[dr]^{f_3 \\varphi} \\ar[rr] & & \\Sigma X_0 \\ar[d]^{\\psi} \\\\\nX_0 \\ar[r]^-{f_1} & X_1 \\ar[r]^-{f_2} & X_2 \\ar[r]^-{f_3} & X_3 \\ar[r]^-{f_4} & X_4 \\\\\n}\n\\]\nexhibits the membership $\\psi \\in \\left\\langle f_4, f_3 f_2, f_1 \\right\\rangle_{\\cc}$. A similar argument can be used to prove the inclusion $\\left\\langle f_4, f_3, f_2 f_1 \\right\\rangle_{\\ff} \\subseteq \\left\\langle f_4, f_3 f_2, f_1 \\right\\rangle_{\\ff}$.\n\\end{proof}\n\n\\begin{ex}\nThe inclusion $\\left\\langle f_4 f_3, f_2, f_1 \\right\\rangle \\subseteq \\left\\langle f_4, f_3 f_2, f_1 \\right\\rangle$ need not be an equality. For example, consider the maps $X \\ral{0} Y \\ral{1} Y \\ral{0} Z \\ral{1} Z$. The Toda brackets being compared are\n\\begin{align*}\n\\left\\langle 1_Z 0, 1_Y, 0 \\right\\rangle &= \\left\\langle 0, 1_Y, 0 \\right\\rangle \\\\\n&= \\left\\{ 0 \\right\\} \\\\\n\\left\\langle 1_Z, 0 1_Y, 0 \\right\\rangle &= \\left\\langle 1_Z, 0, 0 \\right\\rangle \\\\\n&= \\cat{T}(\\Sigma X, Z).\n\\end{align*}\n\\end{ex}\n\n\\begin{defn}\nIn the setup of Definition \\ref{def:TodaBracket}, the \\textbf{restricted Toda brackets} are the subsets of the Toda bracket\n\\begin{align*}\n&\\left\\langle f_3, \\stackrel{\\alpha}{f_2, f_1} \\right\\rangle_{\\fc} \\subseteq \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc} \\\\\n&\\left\\langle \\stackrel{\\beta}{f_3, f_2}, f_1 \\right\\rangle_{\\fc} \\subseteq \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc}\n\\end{align*}\nconsisting of all composites $\\beta (\\Sigma \\alpha) \\colon \\Sigma X_0 \\to X_3$, where $\\alpha$ and $\\beta$ appear in a commutative diagram \\eqref{eq:FibCof} where the middle row is distinguished, with the prescribed map $\\alpha \\colon X_0 \\to \\Sigma^{-1} C_{f_2} $ (resp. $\\beta \\colon C_{f_2} \\to X_3$).\n\\end{defn}\n\n\n\n\nThe lift to the fiber $\\alpha \\colon X_0 \\to \\Sigma^{-1} C_{f_2}$ is a witness of the equality $f_2 f_1 = 0$. Dually, the extension to the cofiber $\\beta \\colon C_{f_2} \\to X_3$ is a witness of the equality $f_3 f_2 = 0$.\n\n\\begin{rem} \\label{ComposeWitness}\nLet $X_1 \\ral{f_2} X_2 \\ral{q_2} C_{f_2} \\ral{\\iota_2} \\Sigma X_1$ be a distinguished triangle. By definition, we have equalities of subsets\n\\begin{align*}\n&\\left\\langle f_3, \\stackrel{\\alpha}{f_2, f_1} \\right\\rangle_{\\fc} = \\left\\langle f_3, \\stackrel{1}{f_2, {-}}\\!\\! \\Sigma^{-1} \\iota_2 \\right\\rangle_{\\fc} (\\Sigma \\alpha) \\\\\n&\\left\\langle \\stackrel{\\beta}{f_3, f_2}, f_1 \\right\\rangle_{\\fc}  = \\beta \\left\\langle \\stackrel{1}{q_2, f_2}, f_1 \\right\\rangle_{\\fc}.\n\\end{align*}\n\\end{rem}\n\n\n\n\n\n\\section{Adams \\texorpdfstring{$d_2$}{d2} in terms of \\texorpdfstring{$3$}{3}-fold Toda brackets}\\label{se:AdamsD2}\n\nIn this section, we show that the Adams differential $d_r$ can be expressed in several ways \nusing $3$-fold Toda brackets.  One of these expressions is as a secondary cohomology operation.\n\nGiven an injective class $\\cat{I}$,\nan Adams resolution of an object $Y$ as in Diagram~\\eqref{eq:AdamsResolInj}, and an object $X$, consider a class $[x] \\in E_2^{s,t}$ represented by a cycle $x \\in E_1^{s,t} = \\cat{T}(\\Sigma^{t-s} X, I_s)$. Recall that $d_2 [x] \\in E_2^{s+2,t+1} $ is obtained as illustrated in the diagram \n\\begin{equation*}\n\\xymatrix{\n\\cdots & Y_s \\ar[l] \\ar@{ >->}[dr]_{p_s} & & Y_{s+1} \\ar@{ >->}[dr]_{p_{s+1}} \\ar[ll]_{i_s} & & Y_{s+2} \\ar@{ >->}[dr]_{p_{s+2}} \\ar[ll]_{i_{s+1}} & & Y_{s+3} \\ar[ll]_{i_{s+2}} & \\cdots \\ar[l] \\\\\n& & I_s \\circar[ur]_{\\delta_s} & & I_{s+1} \\circar[ur]_{\\delta_{s+1}} & & I_{s+2} \\circar[ur]_{\\delta_{s+2}} & & \\\\\n& & \\Sigma^{t-s} X \\ar[u]_x \\ar@\/^0.5pc\/@{-->}[uurrr]_(0.4){\\widetilde{x}} \\ar@\/_0.5pc\/[urrrr]_{d_2(x)} & & & & & & \\\\\n}\n\\end{equation*}%\nExplicitly, since $x$ satisfies $d_1(x) = (\\Sigma p_{s+1}) \\delta_s x = 0$, we can choose a lift $\\widetilde{x} \\colon \\Sigma^{t-s} X {\\ooalign{$\\longrightarrow$\\cr\\hidewidth$\\circ$\\hidewidth\\cr}} \\Sigma Y_{s+2}$ of $\\delta_s x$ to the fiber of $\\Sigma p_{s+1}$. Then the differential $d_2$ is given by \n\\[\nd_2 [x] = \\left[ (\\Sigma p_{s+2}) \\widetilde{x} \\right].\n\\]\n\nFrom now on, we will unroll the distinguished triangles and keep track of the suspensions. %\nFollowing Convention \\ref{co:SuspensionIso}, we will use the identifications\n\\[\nE_1^{s+2,t+1} = \\cat{T}(\\Sigma^{t-s-1} X, I_{s+2}) \\cong \\cat{T}(\\Sigma^{t-s} X, \\Sigma I_{s+2}) \\cong \\cat{T}(\\Sigma^{t-s+1} X, \\Sigma^2 I_{s+2}).\n\\]\n\n\\begin{prop} \\label{pr:DifferentD2}\nDenote by $d_2 [x] \\subseteq E_1^{s+2,t+1}$ the subset of all representatives of the class $d_2 [x] \\in E_2^{s+2,t+1}$. Then the following equalities hold:\n\\begin{enumerate}\n\\item\\label{it:d1pdex}\n\\begin{align*}\nd_2 [x] &= \\left\\langle \\stackrel{\\Sigma^2 p_{s+2}}{\\Sigma d_1,\\strut \\Sigma p_{s+1}}, \\delta_s x \\right\\rangle_{\\fc} \\\\\n&= \\left\\langle \\Sigma d_1, \\Sigma p_{s+1}, \\delta_s x \\right\\rangle\n\\end{align*}\n\\item\n\\begin{align*}\nd_2 [x] &= (\\Sigma^2 p_{s+2}) \\left\\langle \\stackrel{\\!\\!1}{\\Sigma \\delta_{s+1}, \\Sigma p_{s+1}}, \\delta_s x \\right\\rangle_{\\fc} \\\\\n&= (\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, \\Sigma p_{s+1}, \\delta_s x \\right\\rangle\n\\end{align*}\n\\item\\label{it:d1d1x}\n\\[\nd_2 [x] = \\left\\langle \\stackrel{\\ \\beta}{\\Sigma d_1, d_1}, x \\right\\rangle_{\\fc} ,\n\\]\nwhere $\\beta$ is the composite $C \\ral{\\widetilde{\\beta}} \\Sigma^{2} Y_{s+2} \\ral{\\Sigma^2 p_{s+2}} \\Sigma^{2} I_{s+2}$ and $\\widetilde{\\beta}$ is obtained from the octahedral axiom applied to the factorization $d_1 = (\\Sigma p_{s+1}) \\delta_{s} \\colon I_s \\to \\Sigma Y_{s+1} \\to \\Sigma I_{s+1}$.\n\\end{enumerate}\n\\end{prop}\n\nIn~\\eqref{it:d1d1x}, $\\beta$ is a witness to the fact that the composite $(\\Sigma d_1) d_1$ \nof primary operations is zero, and so the restricted Toda bracket is a secondary operation.\n\n\\begin{proof}\nNote that $t$ plays no role in the statement, so we will assume without loss of generality that $t=s$ holds.\n\n(1) The first equality holds by definition of $d_2 [x]$, namely choosing a lift of $\\delta_s x$ to the fiber of $\\Sigma p_{s+1}$. \nThe second equality follows from the fact that $\\Sigma^2 p_{s+2}$ is the \\emph{unique} extension of $\\Sigma d_1 = (\\Sigma^2 p_{s+2}) (\\Sigma \\delta_{s+1})$ to the cofiber of $\\Sigma p_{s+1}$. Indeed, $\\Sigma \\delta_{s+1}$ is $\\cat{I}$-epic and $\\Sigma I_{s+2}$ is injective, so that the restriction map\n\\[\n(\\Sigma \\delta_{s+1})^* \\colon \\cat{T}(\\Sigma^2 Y_{s+2}, \\Sigma^2 I_{s+2}) \\to \\cat{T}(\\Sigma I_{s+1}, \\Sigma^2 I_{s+2})\n\\]\nis injective.\n\n(2) The first equality holds by Remark \\ref{ComposeWitness}. The second equality holds because $\\Sigma \\delta_{s+1}$ is $\\cat{I}$-epic and $\\Sigma I_{s+2}$ is injective, as in part (1).\n\n\n(3) The map $d_1 \\colon I_s \\to \\Sigma I_{s+1}$ is the composite $I_s \\ral{\\delta_s} \\Sigma Y_{s+1} \\ral{\\Sigma p_{s+1}} \\Sigma I_{s+1}$. The octahedral axiom applied to this factorization yields the dotted arrows in a commutative diagram\n\\[\n\\cxymatrix{\nI_s \\ar@{=}[d] \\ar[r]^-{\\delta_s} & \\Sigma Y_{s+1} \\ar[d]_{\\Sigma p_{s+1}} \\ar[r]^-{\\Sigma i_s} & \\Sigma Y_s \\ar@{-->}[d]^{\\widetilde{\\alpha}} \\ar[r]^-{-\\Sigma p_s} & \\Sigma I_s \\ar@{=}[d] \\\\\nI_s \\ar[r]^-{d_1} & \\Sigma I_{s+1} \\ar[d]_{\\Sigma \\delta_{s+1}} \\ar@{-->}[r]^-{q} & C_{d_1} \\ar@{-->}[d]^{\\widetilde{\\beta}} \\ar@{-->}[r]^-{\\iota} & \\Sigma I_s \\\\\n& \\Sigma^2 Y_{s+2} \\ar[d]_{-\\Sigma^{2} i_{s+1}} \\ar@{=}[r] & \\Sigma^2 Y_{s+2} \\ar[d] & \\\\\n& \\Sigma^2 Y_{s+1} \\ar[r]^{\\Sigma^{2} i_s} & \\Sigma^2 Y_{s} & \\\\\n}\n\\]\nwhere the rows and columns are distinguished and the equation $(-\\Sigma^2 i_{s+1}) \\widetilde{\\beta} = (\\Sigma \\delta_s) \\iota$ holds. The restricted bracket $\\left\\langle \\stackrel{\\ \\beta}{\\Sigma d_1, d_1}, x \\right\\rangle_{\\fc}$ consists of the maps $\\Sigma X \\to \\Sigma^2 I_{s+2}$ appearing as downward composites in the commutative diagram\n\\[\n\\xymatrix@C-7pt@R-3pt{\n& & & & \\Sigma X \\ar@{-->}[d]_-{\\Sigma \\alpha} \\ar[rr]^{- \\Sigma x} & & \\Sigma I_s \\ar@{=}[d] \\\\\nI_s \\ar[rr]^-{d_1} & & \\Sigma I_{s+1} \\ar@{=}[dd] \\ar[rr]^-{q} & & C_{d_1} \\ar[dl]_(0.55){\\widetilde{\\beta}\\!\\!} \\ar[dd]^{\\beta} \\ar[rr]^-{\\iota} && \\Sigma I_s \\\\\n& & & \\Sigma^2 Y_{s+2} \\ar[dr]^-{\\!\\Sigma^2 p_{s+2}} & & \\\\\n& & \\Sigma I_{s+1} \\ar[rr]_-{\\Sigma d_1} \\ar[ur]^-{\\Sigma \\delta_{s+1}\\!} & & \\Sigma^2 I_{s+2} & \\\\\n}\n\\]\n\n($\\supseteq$) Let $\\beta (\\Sigma \\alpha) \\in \\left\\langle \\stackrel{\\beta}{d_1, d_1}, x \\right\\rangle_{\\fc}$. By definition of $\\beta$, we have $\\beta (\\Sigma \\alpha) = (\\Sigma^2 p_{s+2}) \\widetilde{\\beta} (\\Sigma \\alpha)$. Then $\\widetilde{\\beta} (\\Sigma \\alpha) \\colon \\Sigma X \\to \\Sigma^{2} Y_{s+2}$ is a valid choice of the lift $\\widetilde{x}$ in the definition of $d_2[x]$:\n\\begin{align*}\n(\\Sigma^2 i_{s+1}) \\widetilde{\\beta} (\\Sigma \\alpha) &= -(\\Sigma \\delta_s) \\iota (\\Sigma \\alpha) \\\\%By the equation -i_{s+1} \\widetilde{\\beta} = \\delta_s \\iota in the octahedron\n&= - (\\Sigma \\delta_s) (-\\Sigma x) \\\\\n&= \\Sigma (\\delta_s x).\n\\end{align*}\n($\\subseteq$) Given a representative $(\\Sigma p_{s+2}) \\widetilde{x} \\in d_2 [x]$, let us show that $\\Sigma \\widetilde{x} \\colon \\Sigma X \\to \\Sigma^2 Y_{s+2}$ factors as $\\Sigma X \\ral{\\Sigma \\alpha} C_{d_1} \\ral{\\widetilde{\\beta}} \\Sigma^{2} Y_{s+2}$ for some $\\Sigma \\alpha$, yielding a factorization of the desired form\n\\begin{align*}\n(\\Sigma^2 p_{s+2}) (\\Sigma \\widetilde{x}) &= (\\Sigma^2 p_{s+2}) \\widetilde{\\beta} (\\Sigma \\alpha) \\\\\n&= \\beta (\\Sigma \\alpha).\n\\end{align*}\nBy construction, the map $(\\Sigma^2 i_s) (-\\Sigma^2 i_{s+1}) \\colon \\Sigma^2 Y_{s+2} \\to \\Sigma^2 Y_{s}$ is a cofiber of $\\widetilde{\\beta}$. The condition\n\\[\n(\\Sigma^2 i_s) (\\Sigma^2 i_{s+1}) (\\Sigma \\widetilde{x}) = (\\Sigma^2 i_s) \\Sigma (\\delta_s x) = 0\n\\]\nguarantees the existence of some lift $\\Sigma \\alpha \\colon \\Sigma X \\to C_{d_1}$ of $\\Sigma \\widetilde{x}$. The chosen lift $\\Sigma \\alpha$ might \\emph{not} satisfy $\\iota (\\Sigma \\alpha) = - \\Sigma x$, but we will correct it to a lift $\\Sigma \\alpha'$ which does. The two sides of the equation become equal after applying $-\\Sigma \\delta_s$, i.e., $(-\\Sigma \\delta_s) (-\\Sigma x) = (-\\Sigma \\delta_s) \\iota (\\Sigma \\alpha)$ holds.\nHence, the error term factors as\n\\[\n-\\Sigma x - \\iota \\Sigma \\alpha = (-\\Sigma p_s)(\\Sigma \\theta)\n\\]\nfor some $\\Sigma \\theta \\colon \\Sigma X \\to \\Sigma Y_s$, since $-\\Sigma p_s$ is a fiber of $-\\Sigma \\delta_s$. The corrected map $\\Sigma \\alpha' := \\Sigma \\alpha + \\widetilde{\\alpha} (\\Sigma \\theta) \\colon \\Sigma X \\to C_{d_1}$ satisfies $\\iota (\\Sigma \\alpha') = - \\Sigma x$ \nand still satisfies $\\widetilde{\\beta} (\\Sigma \\alpha') = \\widetilde{\\beta} (\\Sigma \\alpha) = \\Sigma \\widetilde{x}$, since the correction term $\\widetilde{\\alpha} (\\Sigma \\theta)$ satisfies $\\widetilde{\\beta} \\widetilde{\\alpha} (\\Sigma \\theta) = 0$. \n\\end{proof}\n\n\\begin{prop}\\label{pr:inclusions}\nThe following inclusions of subsets hold in $E_1^{s+2,t+1}$: %\n\\[\nd_2 [x] \\subseteq (\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, d_1, x \\right\\rangle \\subseteq \\left\\langle \\Sigma d_1, d_1, x \\right\\rangle .\n\\]\n\\end{prop}\n\n\\begin{proof}\nThe first inclusion is \n\\[\nd_2 [x] = (\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, \\Sigma p_{s+1}, \\delta_s x \\right\\rangle \\subseteq (\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, (\\Sigma p_{s+1}) \\delta_s, x \\right\\rangle,\n\\]\nwhereas the second inclusion is\n\\[\n(\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, d_1, x \\right\\rangle \\subseteq \\left\\langle (\\Sigma^2 p_{s+2}) (\\Sigma \\delta_{s+1}), d_1, x \\right\\rangle,\n\\] \nboth using Lemma~\\ref{le:MoveAround}.\n\\end{proof}\n\n\\begin{prop}\\label{pr:proper-inclusion}\nThe inclusion $(\\Sigma^2 p_{s+2}) \\left\\langle \\Sigma \\delta_{s+1}, d_1, x \\right\\rangle \\subseteq \\left\\langle \\Sigma d_1, d_1, x \\right\\rangle$ need \\emph{not} be an equality in general.\n\\end{prop}\n\nIt was pointed out to us by Robert Bruner that this can happen in principle.\nWe give an explicit example in Proposition~\\ref{pr:proper-inclusion-C4}.\n\n\n\n\\section{Higher Toda brackets}\\label{se:HigherBrackets}\n\n\n\n\n\n\nWe saw in Section~\\ref{se:3-fold-Toda-brackets} that there are several equivalent ways\nto define $3$-fold Toda brackets.\nFollowing the approach given in~\\cite{McKeown12}, we show that\nthe fiber-cofiber definition generalizes nicely to $n$-fold Toda brackets.\nThere are $(n-2)!$ ways to make this generalization, and we prove\nthat they are all the same up to a specified sign.\nWe also show that this Toda bracket is self-dual.\n\nOther sources that discuss higher Toda brackets in triangulated categories\nare~\\cite{Shipley02}*{Appendix A}, \\cite{Gelfand03}*{IV \\S 2} and~\\cite{Sagave08}*{\\S 4},\nwhich all give definitions that follow Cohen's approach for spectra or spaces~\\cite{Cohen68}.\nWe show that our definition agrees with those of~\\cite{Shipley02} and~\\cite{Sagave08}.\n(We believe that it sometimes differs in sign from~\\cite{Cohen68}.  We have not compared carefully with~\\cite{Gelfand03}.)\n\n\\begin{defn}\\label{def:TodaFamily}\nLet $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3$ be a diagram\nin a triangulated category $\\cat{T}$.\nWe define the \\textbf{Toda family}\nof this sequence to be the collection $\\mathrm{T}(f_3, f_2, f_1)$\nconsisting of all pairs $(\\beta, \\Sigma \\alpha)$, where $\\alpha$ and $\\beta$ appear in a commutative diagram\n\\[\n\\xymatrix @C=3.3pc {\nX_0 \\ar[d]_-{\\alpha} \\ar[r]^-{f_1} & X_1 \\ar@{=}[d] & & \\\\\n\\Sigma^{-1} C_{f_2} \\ar[r] & X_1 \\ar[r]^-{f_2} & X_2 \\ar@{=}[d] \\ar[r] & C_{f_2} \\ar[d]^-{\\beta}  \\\\\n& & X_2  \\ar[r]^-{f_3} & X_3 \\\\\n}\n\\]\nwith distinguished middle row.\nEquivalently,\n\\[\n\\xymatrix @C=3.3pc {\n& & & \\Sigma X_0 \\ar[d]_-{\\Sigma \\alpha} \\ar[r]^-{-\\Sigma f_1} & \\Sigma X_1 \\ar@{=}[d] \\\\\n& X_1 \\ar[r]^-{f_2} & X_2 \\ar@{=}[d] \\ar[r] & C_{f_2} \\ar[d]^-{\\beta} \\ar[r] & \\Sigma X_1 \\\\\n& & X_2  \\ar[r]^-{f_3} & X_3 , \\\\\n}\n\\]\nwhere the middle row is again distinguished.  (The negative of $\\Sigma f_1$\nappears, since when a triangle is rotated, a sign is introduced.)\nNote that the maps in each pair form a composable sequence\n$\\Sigma X_0 \\ral{\\Sigma \\alpha} C_{f_2} \\ral{\\beta} X_3$,\nwith varying intermediate object,\nand that the collection of composites $\\beta \\circ \\Sigma \\alpha$ is exactly the\nToda bracket $\\langle f_3, f_2, f_1 \\rangle$, using the fiber-cofiber definition\n(see Diagram~\\eqref{eq:FibCof}).\n(Also note that the Toda family is generally a proper class,\nbut this is only because the intermediate object can be varied up to isomorphism,\nand so we will ignore this.)\n\nMore generally, if $S$ is a set of composable triples of maps,\nstarting at $X_0$ and ending at $X_3$, we define $\\mathrm{T}(S)$ to\nbe the union of $\\mathrm{T}(f_3, f_2, f_1)$ for each triple\n$(f_3, f_2, f_1)$ in $S$.\n\\end{defn}\n\n\\begin{defn}\\label{def:HigherToda}\nLet\n$X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} \\cdots \\ral{f_n} X_n$\nbe a diagram in a triangulated category $\\cat{T}$.\nWe define the \\textbf{Toda bracket} $\\langle f_n, \\ldots, f_1 \\rangle$\ninductively as follows.\nIf $n = 2$, it is the set consisting of just the composite $f_2 f_1$.\nIf $n > 2$, it is the union of the sets\n$\\langle \\beta, \\Sigma \\alpha, \\Sigma f_{n-3}, \\ldots, \\Sigma f_1 \\rangle$,\nwhere $(\\beta, \\Sigma \\alpha)$ is in $T(f_n, f_{n-1}, f_{n-2})$.\n\\end{defn}\n\n\nIn fact, there are $(n-2)!$ such definitions, depending on a\nsequence of choices of which triple of consecutive maps to apply\nthe Toda family construction to.\nIn Theorem~\\ref{th:n-fold} we will enumerate these choices\nand show that they all agree up to sign.\n\n\\begin{ex}\\label{ex:4FoldBracket}\nLet us describe $4$-fold Toda brackets in more detail. We have\n\\[\n\\left\\langle f_4, f_3, f_2, f_1 \\right\\rangle\n = \\bigcup_{\\beta, \\alpha} \\left\\langle \\beta, \\Sigma \\alpha, \\Sigma f_1 \\right\\rangle\n = \\bigcup_{\\beta, \\alpha} \\bigcup_{\\beta', \\alpha'} \\{ \\beta' \\circ \\Sigma \\alpha' \\}\n\\]\nwith $(\\beta, \\Sigma \\alpha) \\in T(f_4, f_3, f_2)$ and $(\\beta', \\Sigma \\alpha') \\in T(\\beta, \\Sigma \\alpha, \\Sigma f_1)$. These maps fit into a commutative diagram\n\\[\n  \\xymatrix{\n    \\Sigma^2 X_0 \\ar[r]^-{\\Sigma \\alpha'} &\tC_{\\Sigma \\alpha} \\ar[r] \\ar[ddr]^(0.3){\\beta'} &\t\\Sigma^2 X_1 &  \\text{row = $\\mathrlap{-\\Sigma^2 f_1}$} \\\\\n    \\Sigma X_1 \\ar[r]^{\\Sigma \\alpha} &\tC_{f_3} \\ar[r] \\ar[dr]_(0.45){\\beta} \\ar[u] &\t\\Sigma X_2 &\t\\text{row = $\\mathrlap{-\\Sigma f_2}$} \\\\\n    X_2 \\ar[r]^{f_3} &\tX_3 \\ar[u] \\ar[r]_{f_4} &\tX_4 \\\\\n  & 0 \\ar[u] \\\\\n  }\n\\]\nwhere the horizontal composites are specified as above, and each ``snake''\n\\[\n\\xymatrix{\n& \\cdot \\ar[r] & \\cdot \\\\\n\\cdot \\ar[r] & \\cdot \\ar[u] & \\\\\n}\n\\]\nis a distinguished triangle.\nThe middle column is an example of a \\emph{$3$-filtered object} as defined below.\n\\end{ex}\n\nNext, we will show that Definition \\ref{def:HigherToda} coincides with the definitions of higher Toda brackets in~\\cite{Shipley02}*{Appendix A} and~\\cite{Sagave08}*{\\S 4}, which we recall here.\n\n\\begin{defn}\\label{def:NFiltered}\nLet $n \\geq 1$ and consider a diagram in $\\cat{T}$\n\\[\n\\xymatrix{\nY_0 \\ar[r]^-{\\lambda_1} &\tY_1 \\ar[r]^-{\\lambda_2} &\tY_2 \\ar[r] &\t\\cdots \\ar[r]^-{\\lambda_{n-1}} &\tY_{n-1} \\\\\n}  \n\\]\nconsisting of $n-1$ composable maps. An \\textbf{$n$-filtered object} $Y$ based on $(\\lambda_{n-1}, \\ldots, \\lambda_1)$ consists of a sequence of maps\n\\[\n\\xymatrix{\n0 = F_0 Y \\ar[r]^-{i_{0}} &\tF_{1} Y \\ar[r]^-{i_{1}} &\t\\cdots \\ar[r]^-{i_{n-1}} &\tF_n Y = Y \\\\ \n}\n\\]\ntogether with distinguished triangles\n\\[\n\\xymatrix{\nF_{j} Y \\ar[r]^-{i_j} &\tF_{j+1} Y \\ar[r]^-{q_{j+1}} & \\Sigma^{j} Y_{n-1-j} \\ar[r]^-{e_j} &\t\\Sigma F_j Y \\\\\t\n}\n\\]\nfor $0 \\leq j \\leq n-1$, such that for $1 \\leq j \\leq n-1$, the composite\n\\[\n\\xymatrix{\n\\Sigma^j Y_{n-1-j} \\ar[r]^-{e_j} &\t\\Sigma F_j Y \\ar[r]^-{\\Sigma q_j} &\t\\Sigma^{j} Y_{n-j} \\\\\n}\n\\]\nis equal to $\\Sigma^j \\lambda_{n-j}$. In particular, the $n$-filtered object $Y$ comes equipped with maps\n\\[\n\\sigma'_Y \\colon Y_{n-1} \\cong F_1 Y \\to Y \n\\]\n\\[\n\\sigma_Y \\colon Y = F_n Y \\to \\Sigma^{n-1} Y_0.\n\\]\n\\end{defn}\n\n\\begin{defn}\\label{def:HigherTodaSS}\nLet $X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} \\cdots \\ral{f_n} X_n$\nbe a diagram in a triangulated category $\\cat{T}$.\nThe \\textbf{Toda bracket} in the sense of Shipley--Sagave $\\langle f_n, \\ldots, f_1 \\rangle_{\\Ship} \\subseteq \\cat{T}(\\Sigma^{n-2} X_0, X_n)$ is the set of all composites appearing in the middle row of a commutative diagram\n\\[\n\\xymatrix{\n& X_{n-1} \\ar[d]_{\\sigma'_X} \\ar[dr]^-{f_n} & \\\\\n\\Sigma^{n-2} X_0 \\ar[dr]_{\\Sigma^{n-2} f_1} \\ar@{-->}[r] & X \\ar[d]^{\\sigma_X} \\ar@{-->}[r] & X_n \\\\\n& \\Sigma^{n-2} X_1 , & \\\\\n}\n\\]\nwhere $X$ is an $(n-1)$-filtered object based on $(f_{n-1}, \\ldots, f_3, f_2)$.\n\\end{defn}\n\n\\begin{ex}\nFor a $3$-fold Toda bracket $\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\Ship}$, a $2$-filtered object $X$ based on $f_2$ amounts to a cofiber of $-f_2$, more precisely, a distinguished triangle\n\\[\n\\xymatrix{\nX_2 \\ar[r]^-{\\sigma'_X} & X \\ar[r]^-{\\sigma_X} & \\Sigma X_1 \\ar[r]^-{\\Sigma f_2} & \\Sigma X_2.\n}\n\\]\nUsing this, one readily checks the equality $\\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\Ship} = \\left\\langle f_3, f_2, f_1 \\right\\rangle_{\\fc}$, as noted in \\cite{Sagave08}*{Definition 4.5}.\n\\end{ex}\n\n\\begin{ex}\\label{ex:4FoldBracketSS}\nFor a $4$-fold Toda bracket $\\left\\langle f_4, f_3, f_2, f_1 \\right\\rangle_{\\Ship}$, a $3$-filtered object $X$ based on $(f_3, f_2)$ consists of the data displayed in the diagram\n\\[\n  \\xymatrix{\n    & F_3 X = X \\ar[r]^-{q_3 = \\sigma_X} & \\Sigma^2 X_1 & \\\\\n    \\Sigma X_1 \\ar[r]^-{- \\Sigma^{-1} e_2} & F_2 X \\ar[r]^-{q_2} \\ar[u]_{i_2} & \\Sigma X_2 & \\text{row = $\\mathrlap{-\\Sigma f_2}$} \\\\\n    X_2 \\ar[r]^-{- \\Sigma^{-1} e_1} & F_1 X \\ar[u]_{i_1} \\ar[r]^-{q_1}_-{\\cong} & X_3 & \\text{row = $\\mathrlap{-f_3}$} \\\\\n  & F_0 X = 0 , \\ar[u]_{i_0} \\\\\n  }\n\\]\nwhere the two snakes are distinguished. The bracket consists of the maps $\\Sigma^2 X_0 \\to X_4$ appearing as composites of the dotted arrows in a commutative diagram\n\\[\n  \\xymatrix{\n    \\Sigma^2 X_0 \\ar@{-->}[r] & X \\ar[r]^-{\\sigma_X} \\ar@\/^1pc\/@{-->}[ddr] & \\Sigma^2 X_1 & \\text{row = $\\mathrlap{\\Sigma^2 f_1}$} \\\\\n    \\Sigma X_1 \\ar[r]^-{- \\Sigma^{-1} e_2} & F_2 X \\ar[r]^-{q_2} \\ar[u] & \\Sigma X_2 & \\text{row = $\\mathrlap{-\\Sigma f_2}$} \\\\\n    X_2 \\ar[r]^-{- f_3} & X_3 \\ar[u] \\ar[r]^-{f_4} & X_4 & \\\\\n  & 0 , \\ar[u] \\\\\n  }\n\\]\nwhere the two snakes are distinguished. By negating the first and third map in each snake,\nthis recovers the description in Example \\ref{ex:4FoldBracket}, thus proving the equality of subsets\n\\[\n\\left\\langle f_4, f_3, f_2, f_1 \\right\\rangle_{\\Ship} = \\left\\langle f_4, f_3, f_2, f_1 \\right\\rangle.\n\\]\n\\end{ex}\n\n\\begin{prop}\nDefinitions \\ref{def:HigherToda} and \\ref{def:HigherTodaSS} agree. In other words, we have the equality\n\\[\n\\left\\langle f_n, \\ldots, f_1 \\right\\rangle_{\\Ship} = \\left\\langle f_n, \\ldots, f_1 \\right\\rangle\n\\]\nof subsets of $\\cat{T}(\\Sigma^{n-2} X_0, X_n)$.%\n\\end{prop}\n\n\\begin{proof}\nThis is a straightforward generalization of Example \\ref{ex:4FoldBracketSS}.\n\\end{proof}\n\nWe define the \\textbf{negative} of a Toda family $T(f_3, f_2, f_1)$\nto consist of pairs $(\\beta, -\\Sigma \\alpha)$ for $(\\beta, \\Sigma \\alpha) \\in T(f_3, f_2, f_1)$.\n(Since changing the sign of two maps in a triangle doesn't affect\nwhether it is distinguished, it would be equivalent to put the\nminus sign with the $\\beta$.)\n\n\\begin{lem}\\label{le:four-fold}\nLet\n$X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3 \\ral{f_4} X_4$\nbe a diagram in a triangulated category $\\cat{T}$.\nThen the two sets of pairs\n$T(T(f_4, f_3, f_2), \\Sigma f_1)$ and\n$T(f_4, T(f_3, f_2, f_1))$ are negatives of each other.\n\\end{lem}\n\nThis is stronger than saying that the two ways of computing the Toda bracket\n$\\langle f_4, f_3, f_2, f_1 \\rangle$ are negatives, and the stronger statement\nwill be used inductively to prove Theorem~\\ref{th:n-fold}.\n\n\\begin{proof}\nWe will show that the negative of\n$T(T(f_4, f_3, f_2), \\Sigma f_1)$\nis contained in the family\n$T(f_4, T(f_3, f_2, f_1))$.\nThe reverse inclusion is proved dually.\n\nSuppose $(\\beta, \\Sigma \\alpha)$ is in $T(T(f_4, f_3, f_2), \\Sigma f_1)$,\nthat is, $(\\beta, \\Sigma \\alpha)$ is in $T(\\beta', \\Sigma \\alpha', \\Sigma f_1)$\nfor some $(\\beta', \\Sigma \\alpha')$ in $T(f_4, f_3, f_2)$.\nThis means that we have the following commutative diagram\n\\[\n  \\xymatrix@!@C=-0.2ex@R=-0.2ex{\n                 &&                           && \\Sigma X_1 \\ar[rr]^{-\\Sigma f_2} \\ar@{-->}[dd]^{\\Sigma \\alpha'} && \\Sigma X_2 \\ar@{=}[dd] \\\\\n                                                                                                                           \\\\\nX_2 \\ar[rr]^{f_3} && X_3 \\ar[dr]_-{f_4} \\ar[rr] && C_{f_3} \\ar@{-->}[dl]^{\\!\\beta'} \\ar[rr] \\ar[dd]        && \\Sigma X_2 \\\\\n                 &&             & X_4                                                                                      \\\\\n                 &&                         && C_{\\Sigma \\alpha'} \\ar@{-->}[ul]_{\\!\\beta} \\ar[dd] \\\\\n                 &&             & \\Sigma^2 X_0 \\ar@{-->}[ur]^{\\Sigma \\alpha} \\ar[dr]_-{- \\Sigma^2 f_1}                                  \\\\\n                 &&                         && \\Sigma^2 X_1 && \\\\\n  }\n\\]\nin which the long row and column are distinguished triangles.\n\nUsing the octahedral axiom, there exists a map $\\delta : C_{f_2} \\to X_3$\nin the following diagram making the two squares commute\nand such that the diagram can be extended as shown,\nwith all rows and columns distinguished:\n\\[\n  \\xymatrix@!@C=-0.7ex@R=-0.7ex{\n                        &&        & \\Sigma X_0 \\ar@{-->}[dl]_{\\gamma\\!} \\ar[dr]^{\\!-\\Sigma f_1}\\\\\nX_2 \\ar[rr] \\ar@{=}[dd] && C_{f_2} \\ar[rr] \\ar@{-->}[dd]^{\\delta} && \\Sigma X_1 \\ar[rr]^{-\\Sigma f_2} \\ar@{-->}[dd]^{\\Sigma \\alpha'} && \\Sigma X_2 \\ar@{=}[dd] \\\\\n                                                                                                                           \\\\\nX_2 \\ar[rr]^{f_3}  && X_3 \\ar[dr]^-{f_4} \\ar[rr] \\ar[dd] && C_{f_3} \\ar@{-->}[dl]^{\\!\\!\\beta'} \\ar[rr] \\ar[dd]        && \\Sigma X_2          \\\\\n                 &&             & X_4                                                                                       \\\\\n                 && C_{\\delta} \\ar@{=}[rr] \\ar[dd] && C_{\\Sigma \\alpha'} \\ar@{-->}[ul]_{\\!\\!\\beta} \\ar[dd]     &&       \\\\\n                 &&             & \\Sigma^2 X_0 \\ar@{-->}[ur]^(0.4){\\Sigma \\alpha\\!} \\ar[dr]_(0.4){-\\Sigma^2 \\!f_1\\!\\!\\!\\!} \\ar@{-->}[dl]_{\\Sigma \\gamma\\!\\!\\!}     \\\\\n                 && \\Sigma C_{f_2} \\ar[rr]   && \\Sigma^2 X_1   .               &&        \\\\\n  }\n\\]\nDefine $\\Sigma \\gamma$ to be the composite \n$\\Sigma^2 X_0 \\to C_{\\Sigma \\alpha'} = C_{\\delta} \\to \\Sigma C_{f_2}$, where the first map is $\\Sigma \\alpha$.\nThen the small triangles at the top and bottom of the last diagram commute as well.\nTherefore, $(\\delta, \\gamma)$ is in $T(f_3, f_2, f_1)$.\nMoreover, this diagram shows that\n$(\\beta, - \\Sigma \\alpha)$ is in $T(f_4, \\delta, \\gamma)$,\ncompleting the argument.\n\\end{proof}\n\n\nTo concisely describe different ways of computing higher Toda\nbrackets, we introduce the following notation.\nFor $0 \\leq j \\leq n-3$, write $T_j(f_n, f_{n-1}, \\ldots, f_1)$ for the set of tuples\n\\[\n  \\{ (f_n, f_{n-1}, \\ldots, f_{n-j+1}, \\beta, \\Sigma \\alpha, \\Sigma f_{n-j-3}, \\ldots, \\Sigma f_1) \\},\n\\]\nwhere $(\\beta, \\Sigma \\alpha)$ is in $T(f_{n-j}, f_{n-j-1}, f_{n-j-2})$.\n(There are $j$ maps to the left of the three\\break used for the Toda family.)\nIf $S$ is a set of $n$-tuples of composable maps, we define\\break\n$T_j(S)$ to be the union of the sets $T_j(f_n, f_{n-1}, \\ldots, f_1)$\nfor $(f_n, f_{n-1}, \\ldots, f_1)$ in $S$.\nWith this\\break notation, the standard Toda bracket $\\left\\langle f_n, \\ldots, f_1 \\right\\rangle$ consists of the composites of all the pairs\\break occurring in the iterated Toda family\n\\[\n\\mathrm{T}(f_n, \\ldots, f_1) := T_0(T_0(T_0(\\cdots T_0(f_n, \\ldots, f_1) \\cdots ))).\n\\]\nA general Toda bracket is of the form \n$T_{j_1}(T_{j_2}(T_{j_3}(\\cdots T_{j_{n-2}}(f_n, \\ldots, f_1) \\cdots )))$,\nwhere\\break $j_1, j_2, \\ldots, j_{n-2}$ is a sequence of natural numbers\nwith $0 \\leq j_i < i$ for each $i$.\nThere are $(n-2)!$ such sequences.\n\n\\begin{rem}\nThere are six ways to compute the five-fold Toda bracket \n$\\langle f_5, f_4, f_3, f_2, f_1 \\rangle$, as the set of composites\nof the pairs of maps in one of the following sets:\n\\begin{align*}\n&T_0(T_0(T_0(f_5, f_4, f_3, f_2, f_1))) =  T(T(T(f_5, f_4, f_3), \\Sigma f_2), \\Sigma^2 f_1),\\\\\n&T_0(T_0(T_1(f_5, f_4, f_3, f_2, f_1))) =  T(T(f_5, T(f_4, f_3, f_2)), \\Sigma^2 f_1),\\\\\n&T_0(T_1(T_1(f_5, f_4, f_3, f_2, f_1))) =  T(f_5, T(T(f_4, f_3, f_2), \\Sigma f_1)),\\\\\n&T_0(T_1(T_2(f_5, f_4, f_3, f_2, f_1))) =  T(f_5, T(f_4, T(f_3, f_2, f_1))),\\\\\n&T_0(T_0(T_2(f_5, f_4, f_3, f_2, f_1))), \\quad\\text{and}\\\\\n&T_0(T_1(T_0(f_5, f_4, f_3, f_2, f_1))).\n\\end{align*}\nThe last two cannot be expressed directly just using $T$.\n\\end{rem}\n\nNow we can prove the main result of this section.\n\n\\begin{thm}\\label{th:n-fold}\nThe Toda bracket computed using the sequence $j_1, j_2, \\ldots, j_{n-2}$\nequals the standard Toda bracket up to the sign $(-1)^{\\sum j_i}$.\n\\end{thm}\n\n\\begin{proof}\nLet $j_1, j_2, \\ldots, j_{n-2}$ be a sequence with $0 \\leq j_i < i$ for each $i$.\nLemma~\\ref{le:four-fold} tells us that if we replace consecutive entries \n$k, k+1$ with $k, k$ in any such sequence, the two Toda brackets agree up to a sign.\nTo begin with, we ignore the signs.\nWe will prove by induction on $\\ell$ that the initial portion\n$j_1, \\ldots, j_\\ell$ of such a sequence can be converted into\nany other sequence, using just the move allowed by Lemma~\\ref{le:four-fold} and its inverse,\nand without changing $j_i$ for $i > \\ell$.\nFor $\\ell = 1$, there is only one sequence $0$.\nFor $\\ell = 2$, there are two sequences, $0, 0$ and $0, 1$, and Lemma~\\ref{le:four-fold} applies.\nFor $\\ell > 2$, suppose our goal is to produce the sequence $j'_1, \\ldots, j'_\\ell$.\nWe break the argument into three cases:\n\\medskip\n\n\\noindent\nCase 1: $j'_\\ell = j_\\ell$.  We can directly use the induction hypothesis to\nadjust the entries in the first $\\ell - 1$ positions.\n\\medskip\n\n\\noindent\nCase 2: $j'_\\ell > j_\\ell$.  By induction, we can change the first $\\ell - 1$\nentries in the sequence $j$ so that the entry in position $\\ell - 1$ is $j_\\ell$,\nsince $j_{\\ell} < j'_{\\ell} \\leq \\ell - 1$.\nThen, using Lemma~\\ref{le:four-fold}, we can change the entry in position $\\ell$ to $j_\\ell + 1$.\nContinuing in this way, we get $j'_\\ell$ in position $\\ell$, and then we\nare in Case 1.\n\\medskip\n\n\\noindent\nCase 3: $j'_\\ell < j_\\ell$.  Since the moves are reversible, this is equivalent to Case 2.\n\nTo handle the sign, first note that signs propagate through the Toda family construction.\nMore precisely, suppose $S$ is a set of $n$-tuples of maps, and let $S'$ be a set obtained\nby negating the $k^{\\text{th}}$ map in each $n$-tuple for some fixed $k$.\nThen $T_j(S)$ has the same relationship to $T_j(S')$, possibly for a different value of $k$.\n\nAs a result, applying the move of Lemma~\\ref{le:four-fold} changes the resulting\nToda bracket by a sign.\nThat move also changes the parity of $\\sum_i j_i$.\nSince we get a plus sign when each $j_i$ is zero, it follows that the\ndifference in sign in general is $(-1)^{\\sum_i j_i}$.\n\\end{proof}\n\nAn animation of this argument is available at~\\cite{Canim}.\nIt was pointed out by Dylan Wilson that the combinatorial part of the above proof \nis equivalent to the well-known fact that if a binary operation is associative on triples,\nthen it is associative on $n$-tuples.\n\n\\medskip\n\nIn order to compare our Toda brackets to the Toda brackets in the opposite\ncategory, we need one lemma.\n\n\\begin{lem}\\label{le:suspension-of-Toda-family}\nLet\n$X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} X_3$\nbe a diagram in a triangulated category $\\cat{T}$.\nThen the Toda family $T(\\Sigma f_3, \\Sigma f_2, \\Sigma f_1)$ is the negative\nof the suspension of $T(f_3, f_2, f_1)$.\nThat is, it consists of $(\\Sigma \\beta, -\\Sigma^{2} \\alpha)$ for $(\\beta, \\Sigma \\alpha)$ \nin $T(f_3, f_2, f_1)$.\n\\end{lem}\n\n\\begin{proof}\nGiven a distinguished triangle $\\Sigma^{-1} C_{f_2} \\ral{k} X_1 \\ral{f_2} X_2 \\ral{\\iota} C_{f_2}$,\na distinguished triangle involving $\\Sigma f_2$ is\n\\[\nC_{f_2} \\ral{-\\Sigma k} \\Sigma X_1 \\ral{\\Sigma f_2} \\Sigma X_2 \\ral{\\Sigma \\iota} \\Sigma C_{f_2} .\n\\]\nBecause of the minus sign at the left, the maps that arise in the Toda\nfamily based on this triangle are $-\\Sigma^2 \\alpha$ and $\\Sigma \\beta$,\nwhere $\\Sigma \\alpha$ and $\\beta$ arise in the Toda family based on the starting triangle.\n\\end{proof}\n\nGiven a triangulated category $\\cat{T}$, the opposite category $\\cat{T}^{\\mathrm{op}}$\nis triangulated in a natural way.  The suspension in $\\cat{T}^{\\mathrm{op}}$ is $\\Sigma^{-1}$\nand a triangle\n\\[\n\\xymatrix{Y_0 \\ar[r]^{g_1} & Y_1 \\ar[r]^{g_2} & Y_2 \\ar[r]^-{g_3} & \\Sigma^{-1} Y_0}\n\\]\nin $\\cat{T}^{\\mathrm{op}}$ is distinguished if and only if the triangle\n\\[\n\\xymatrix{\\Sigma \\Sigma^{-1} Y_0 & Y_1 \\ar[l]_-{g_1'} & Y_2 \\ar[l]_-{g_2} & \\Sigma^{-1} Y_0 \\ar[l]_-{g_3} }\n\\]\nin $\\cat{T}$ is distinguished, where $g_1'$ is the composite of $g_1$ with\nthe natural isomorphism $Y_0 \\cong \\Sigma \\Sigma^{-1} Y_0$.\n\n\\begin{cor}\\label{co:SelfDual}\nThe Toda bracket is self-dual up to suspension.\nMore precisely, let\n$X_0 \\ral{f_1} X_1 \\ral{f_2} X_2 \\ral{f_3} \\cdots \\ral{f_n} X_n$\nbe a diagram in a triangulated category $\\cat{T}$.\nThen the subset\n\\[\n  \\left\\langle f_1, \\ldots, f_n \\right\\rangle^{\\cat{T}^{\\mathrm{op}}} \\subseteq \\cat{T}^{\\mathrm{op}}(\\Sigma^{-(n-2)} X_n, X_0)\n= \\cat{T}(X_0, \\Sigma^{-(n-2)} X_n)\n\\]\ndefined by taking the Toda bracket in $\\cat{T}^{\\mathrm{op}}$ is sent to the subset\n\\[\n  \\left\\langle f_n, \\ldots, f_1 \\right\\rangle^{\\cat{T}} \\subseteq \\cat{T}(\\Sigma^{n-2} X_0, X_n)\n\\]\ndefined by taking the Toda bracket in $\\cat{T}$ under the bijection\n$\\Sigma^{n-2} : \\cat{T}(X_0, \\Sigma^{-(n-2)} X_n) \\to \\cat{T}(\\Sigma^{n-2} X_0, X_n)$.\n\\end{cor}\n\n\\begin{proof}\nFirst we compare Toda families in $\\cat{T}$ and $\\cat{T}^{\\mathrm{op}}$.\nIt is easy to see that the Toda family\n$T^{\\cat{T}^{\\mathrm{op}}}(f_1, f_2, f_3)$\ncomputed in $\\cat{T}^{\\mathrm{op}}$ consists of the pairs\n$(\\alpha, \\Sigma^{-1} \\beta)$ for $(\\Sigma \\alpha, \\beta)$ in the Toda family\n$T^{\\cat{T}}(f_3, f_2, f_1)$ computed in $\\cat{T}$.\nIn short, one has to desuspend and transpose the pairs.\n\nUsing this, one can see that the iterated Toda family\n\\[\nT^{\\cat{T}^{\\mathrm{op}}}(T^{\\cat{T}^{\\mathrm{op}}} \\cdots T^{\\cat{T}^{\\mathrm{op}}}(f_1, f_2, f_3), \\ldots, \\Sigma^{-(n-3)} f_n)\n\\]\nis equal to the transpose of\n\\[\n\\Sigma^{-1} T^{\\cat{T}}(\\Sigma^{-(n-3)} f_n, \\Sigma^{-1} T^{\\cat{T}}(\\Sigma^{-(n-4)} f_{n-1}, \\Sigma^{-1} T^{\\cat{T}} \\cdots \\Sigma^{-1} T^{\\cat{T}}(f_3, f_2, f_1) \\cdots ))\n\\]\nBy Lemma~\\ref{le:suspension-of-Toda-family}, the desuspensions pass\nthrough all of the Toda family constructions, introducing an overall\nsign of $(-1)^{1+2+3+\\cdots+(n-3)}$, and producing\n\\[\n\\Sigma^{-(n-2)} T^{\\cat{T}}(f_n, T^{\\cat{T}}(f_{n-1}, T^{\\cat{T}} \\cdots T^{\\cat{T}}(f_3, f_2, f_1) \\cdots ))\n\\]\nBy Theorem~\\ref{th:n-fold}, composing the pairs gives the usual\nToda bracket up to the sign\\break $(-1)^{0+1+2+\\cdots+(n-3)}$.\nThe two signs cancel, yielding the result.\n\\end{proof}\n\nWe do not know a direct proof of this corollary.\nTo summarize, our insight is that\nby generalizing the corollary to all $(n-2)!$ methods of computing the\nToda bracket, we were able to reduce the argument to the $4$-fold case (Lemma~\\ref{le:four-fold}) and some combinatorics.\n\n\\begin{rem}\nAs with the $3$-fold Toda brackets (see Remark~\\ref{re:3-fold-negation}),\nthe higher Toda brackets depend on the triangulation.\nIf the triangulation is negated, the $n$-fold Toda brackets change\nby the sign $(-1)^n$.\n\\end{rem}\n\n\n\n\\section{Higher order operations determine \\texorpdfstring{$d_r$}{dr}}\\label{se:AdamsDr}\n\nIn this section, we show that the higher Adams differentials can be\nexpressed in terms of higher Toda brackets, in two ways.\nOne of these expressions is as an $r^{\\text{th}}$ order cohomology operation.\n\nGiven an injective class $\\cat{I}$,\nan Adams resolution of an object $Y$ as in Diagram~\\eqref{eq:AdamsResolInj}, and an object $X$, consider a class $[x] \\in E_r^{s,t}$ represented by an element $x \\in E_1^{s,t} = \\cat{T}(\\Sigma^{t-s} X, I_s)$.\nThe class $d_r[x]$ is the set of all $(\\Sigma p_{s+r}) \\widetilde{x}$, where $\\widetilde x$\nruns over lifts of $\\delta_s x$ through the $(r-1)$-fold composite $\\Sigma(i_{s+1} \\cdots i_{s+r-1})$\nwhich appears across the top edge of the Adams resolution.\n\nOur first result will be a generalization of\nProposition~\\ref{pr:DifferentD2}\\eqref{it:d1pdex},\nexpressing $d_r$ in terms of an $(r+1)$-fold Toda bracket.\n\n\\begin{thm}\\label{th:d1pdex}\nAs subsets of $E_1^{s+r,t+r-1}$, we have\n\\[\nd_r [x] = \\left\\langle \\Sigma^{r-1} d_1 , \\ldots , \\Sigma^2 d_1, \\Sigma d_1 , \\Sigma p_{s+1} ,  \\delta_s x \\right\\rangle .\n\\]\n\\end{thm}\n\n\\begin{proof}\nWe compute the Toda bracket, applying the Toda family construction starting from\nthe right, which introduces a sign of $(-1)^{1+2+\\cdots+(r-2)}$, by Theorem~\\ref{th:n-fold}.\nWe begin with the Toda family $T(\\Sigma d_1, \\Sigma p_{s+1}, \\delta_s x)$.\nThere is a distinguished triangle\n\\[\n  \\cxymatrix{\\Sigma Y_{s+2} \\ar[r]^-{\\Sigma i_{s+1}} & \\Sigma Y_{s+1} \\ar[r]^-{\\Sigma p_{s+1}} & \\Sigma I_{s+1} \\ar[r]^-{\\Sigma \\delta_{s+1}} & \\Sigma^2 Y_{s+2},}\n\\]\nwith no needed signs.\nThe map $\\Sigma d_1$ factors through $\\Sigma \\delta_{s+1}$ as $\\Sigma^2 p_{s+2}$, and this\nfactorization is unique because $\\Sigma \\delta_{s+1}$ is $\\cat{I}$-epic and $\\Sigma^2 I_{s+2}$ is injective.\nThe other maps in the Toda family are $\\Sigma x_1$, where $x_1$ is a lift \nof $\\delta_s x$ through $\\Sigma i_{s+1}$.\nSo \n\\[\n  T(\\Sigma d_1, \\Sigma p_{s+1}, \\delta_s x) = \\{ (\\Sigma^2 p_{s+2} , \\, \\Sigma x_1) \\mid x_1 \\text{ a lift of $\\delta_s x$ through $\\Sigma i_{s+1}$}  \\}.\n\\]\n(The Toda family also includes $(\\Sigma^2 p_{s+2} \\, \\phi, \\, \\phi^{-1} (\\Sigma x_1))$, where $\\phi$\nis any isomorphism, but these contribute nothing additional to the later computations.)\nThe composites of such pairs give $d_2[x]$, up to suspension, recovering\nProposition~\\ref{pr:DifferentD2}\\eqref{it:d1pdex}.\n\nContinuing, for each such pair we compute\n\\[\n\\begin{aligned}\n  T(\\Sigma^2 d_1, \\Sigma^2 p_{s+2}, \\Sigma x_1)\n&= -\\Sigma T(\\Sigma d_1, \\Sigma p_{s+2}, x_1) \\\\\n&= -\\Sigma \\{ (\\Sigma^2 p_{s+3} , \\, \\Sigma x_2) \\mid x_2 \\text{ a lift of $x_1$ through $\\Sigma i_{s+2}$}  \\}.\n\\end{aligned}\n\\]\nThe first equality is Lemma~\\ref{le:suspension-of-Toda-family}, and the second reuses\nthe work done in the previous paragraph, with $s$ increased by $1$.\nComposing these pairs gives $-d_3[x]$.\nThe sign which is needed to produce the standard Toda bracket is $(-1)^1$,\nand so the signs cancel.\n\nAt the next step, we compute\n\\[\n\\begin{aligned}\n  T(\\Sigma^3 d_1, \\Sigma^3 p_{s+3}, -\\Sigma^2 x_2)\n&= -\\Sigma^2 T(\\Sigma d_1, \\Sigma p_{s+3}, x_2) \\\\\n&= -\\Sigma^2 \\{ (\\Sigma^2 p_{s+4} , \\, \\Sigma x_3) \\mid x_3 \\text{ a lift of $x_2$ through $\\Sigma i_{s+3}$}  \\}.\n\\end{aligned}\n\\]\nAgain, the composites give $-d_4[x]$.\nSince it was a double suspension that passed through the Toda family, no additional\nsign was introduced.\nSimilarly, the sign to convert to the standard Toda bracket is $(-1)^{1+2}$,\nand since $2$ is even, no additional sign was introduced.\nTherefore, the signs still cancel.\n\nThe pattern continues.  \nIn total, there are $1+2+\\cdots+(r-2)$ suspensions that pass through the Toda\nfamily, and the sign to convert to the standard Toda bracket is also based on\nthat number, so the signs cancel.\n\\end{proof}\n\n\\begin{rem}\nTheorem~\\ref{th:d1pdex} can also be proved using the definition Toda\nbrackets based on $r$-filtered objects, \nas in Definitions~\\ref{def:NFiltered} and~\\ref{def:HigherTodaSS}.\nHowever, one must work in the opposite category $\\cat{T}^{\\mathrm{op}}$.\nIn that category, there is a unique $r$-filtered object, up to isomorphism,\nbased on the maps in the Toda bracket.\nOne of the dashed arrows in the diagram from Definition~\\ref{def:HigherTodaSS}\nis also unique, and the other corresponds naturally to the choice of lift\nin the Adams differential.\n\\end{rem}\n\n\\medskip\n\nIn the remainder of this section, we describe the analog of\nProposition~\\ref{pr:DifferentD2}\\eqref{it:d1d1x}.\nWe begin by defining restricted higher Toda brackets, in terms of\nrestricted Toda families.\n\nConsider a Toda family $T(g h_1, g_1 h_0, g_0 h)$, where the maps\nfactor as shown, there are distinguished triangles\n\\begin{equation}\\label{eq:tri}\n  \\cxymatrix{Z_i \\ar[r]^{g_i} & J_i \\ar[r]^{h_i} & Z_{i+1} \\ar[r]^{k_i} & \\Sigma Z_i}\n\\end{equation}\nfor $i = 0, 1$,\nand $g$ and $h$ are arbitrary maps $Z_2 \\to A$ and $B \\to Z_0$, respectively.\nThis information determines an essentially unique element of the Toda family in the following way.\nThe octahedral axiom applied to the factorization $g_1 h_0$\nyields the dotted arrows in a commutative diagram\n\\[\n\\cxymatrix{\nJ_0 \\ar@{=}[d] \\ar[r]^-{h_0} & Z_1 \\ar[d]_(0.45){g_1} \\ar[r]^-{k_0} & \\Sigma Z_0 \\ar@{-->}[d]^{\\alpha_2} \\ar[r]^-{-\\Sigma g_0} & \\Sigma J_0 \\ar@{=}[d] \\\\\nJ_0 \\ar[r]^-{g_1 h_0} & J_1 \\ar[d]_{h_1} \\ar@{-->}[r]^-{q} & W_2 \\ar@{-->}[d]^{\\beta_2} \\ar@{-->}[r]^-{\\iota} & \\Sigma J_0 \\\\\n& Z_2 \\ar[d]_{k_1} \\ar@{=}[r] & Z_2 \\ar[d]^{\\gamma_2} & \\\\\n& \\Sigma Z_1 \\ar[r]^{\\Sigma k_0} & \\Sigma^2 Z_0 , & \\\\\n}\n\\]\nwhere the rows and columns are distinguished and $\\gamma_2 := (\\Sigma k_0) k_1$.\nIt is easy to see that $-\\Sigma(g_0 h)$ lifts through $\\iota$ as $\\alpha_2 (\\Sigma h)$,\nand that $g h_1$ extends over $q$ as $g \\beta_2$.\nWe define the \\textbf{restricted Toda family} to be the set \n$T(g h_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_1 h_0 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_0 h)$ consisting of the pairs $(g \\beta_2, \\, \\alpha_2 (\\Sigma h))$\nthat arise in this way.\nSince $\\alpha_2$ and $\\beta_2$ come from a distinguished triangle involving a fixed map $\\gamma_2$,\nsuch pairs are unique up to the usual ambiguity of replacing\nthe pair with $(g \\beta_2 \\phi, \\, \\phi^{-1} \\alpha_2 (\\Sigma h))$, where $\\phi$ is an isomorphism.\nSimilarly, given any map $x : B \\to J_0$,\nwe define $T(g h_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_1 h_0 , x)$ to be the set \nconsisting of the pairs  $(g \\beta_2, \\, \\Sigma \\alpha)$,\nwhere $\\beta_2$ arises as above and $\\Sigma \\alpha$ is any lift of $-\\Sigma x$ through $\\iota$.\n\n\\begin{defn}\nGiven distinguished triangles as in Equation~\\eqref{eq:tri}, for $i = 1, \\ldots, n-1$,\nand maps $g : Z_n \\to A$ and $x : B \\to J_1$, we define the \\textbf{restricted Toda bracket}\n\\[\n\\left\\langle g h_{n-1} \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_{n-1} h_{n-2} \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_3 h_2 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_2 h_1 , x \\right\\rangle\n\\]\ninductively as follows:\nIf $n = 2$, it is the set consisting of just the composite $g h_1 x$.\nIf $n = 3$, it is the set of composites of the pairs in $T(g h_2 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_2 h_1 , x)$.\nIf $n > 3$, it is the union of the sets\n\\[\n\\left\\langle g \\beta_2 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}}\\, \\alpha_2 (\\Sigma h_{n-3}) \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}}\\, \\Sigma (g_{n-3} h_{n-4}) \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots , \\Sigma x \\right\\rangle ,\n\\]\nwhere $(g \\beta_2, \\alpha_2 (\\Sigma h_{n-3}))$ is in $T(g h_{n-1} \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_{n-1} h_{n-2} \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} g_{n-2} h_{n-3})$.\n\\end{defn}\n\n\\begin{rem}\nDespite the notation, we want to make it clear that these restricted\nToda families and restricted Toda brackets depend on the choice of\nfactorizations and on the distinguished triangles in Equation~\\eqref{eq:tri}.\nMoreover, the elements of the restricted Toda families are not simply pairs,\nbut also include the factorizations of the maps in those pairs, and\nthe distinguished triangle involving $\\alpha_2$ and $\\beta_2$.\nThis information is used in the $(n-1)$-fold restricted Toda bracket\nthat is used to define the $n$-fold restricted Toda bracket.\n\\end{rem}\n\nRecall that the maps $d_1$ are defined to be $(\\Sigma p_{s+1}) \\delta_s$, and that we\nhave distinguished triangles\n\\[\n  \\cxymatrix{Y_s \\ar[r]^{p_s} & I_s \\ar[r]^-{\\delta_s} & \\Sigma Y_{s+1} \\ar[r]^{\\Sigma i_s} & \\Sigma Y_s}\n\\]\nfor each $s$.\nThe same holds for suspensions of $d_1$, with the last map\nchanging sign each time it is suspended.  \nThus for $x : \\Sigma^{t-s} X \\to I_s$ in the $E_1$ term, the $(r+1)$-fold restricted Toda bracket\n$\\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle$\nmakes sense for each $r$, where we are implicitly using the defining factorizations\nand the triangles from the Adams resolution.\n\n\\begin{thm}\\label{th:AdamsDrCohomOp}\nAs subsets of $E_1^{s+r,t+r-1}$, we have\n\\[\nd_r [x] = \\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle .\n\\]\n\\end{thm}\n\nThis is a generalization of Proposition~\\ref{pr:DifferentD2}\\eqref{it:d1d1x}.\nThe data in the Adams resolution is the witness that the composites of\nthe primary operations are zero in a sufficiently coherent way to permit\nan $r^{\\text{th}}$ order cohomology operation to be defined.\n\n\\begin{proof}\nThe restricted Toda bracket $\\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle$\nis defined recursively, working from the left.\nEach of the $r-2$ doubly restricted Toda families has essentially one element.\nThe first one involves maps $\\alpha_2$, $\\beta_2$ and $\\gamma_2$ that form a distinguished\ntriangle, and $\\gamma_2$ is equal to $[(-1)^r \\Sigma^r i_{s+r-2}][-(-1)^r \\Sigma^r i_{s+r-1}]$.\nWe will denote the corresponding maps in the following octahedra $\\alpha_k$,  $\\beta_k$ and $\\gamma_k$,\nwhere each $\\gamma_k$ equals $[(-1)^r \\Sigma^r i_{s+r-k}] \\, \\gamma_{k-1}$,\nand so $\\gamma_k = -(-1)^{rk} \\Sigma^r (i_{s+r-k} \\cdots i_{s+r-1})$.\nOne is left to compute the singly restricted Toda family\n$\\left\\langle \\Sigma^r p_{s+r} \\beta_{r-1} \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}}\\, \\alpha_{r-1} \\Sigma^{r-2} \\delta_s ,\\, \\Sigma^{r-2} x \\right\\rangle$,\nwhere $\\alpha_{r-1}$ and $\\beta_{r-1}$ fit into a distinguished triangle\n\\[\n  \\cxymatrix{\\Sigma^{r-1 }Y_{s+1} \\ar[r]^-{\\alpha_{r-1}} & W_{r-1} \\ar[r]^-{\\beta_{r-1}} & \\Sigma^{r} Y_{s+r} \\ar[r]^-{\\gamma_{r-1}} & \\Sigma^r Y_{s+1} ,}\n\\]\nand $\\gamma_{r-1} = - \\Sigma^r (i_{s+1} \\cdots i_{s+r-1})$.\nThus, to compute the last restricted Toda bracket, one uses the following diagram,\nobtained as usual from the octahedral axiom:\n\\[\n\\xymatrix@C+24pt{\n& & & \\Sigma^{t-s+r-1} X \\ar[d]^(0.45){-\\Sigma^{r-1} x} \\\\\n\\Sigma^{r-2} I_{s} \\ar@{=}[d] \\ar[r]^-{\\Sigma^{r-2} \\delta_{s}} & \\Sigma^{r-1} Y_{s+1} \\ar[d]_{\\alpha_{r-1}} \\ar[r]^-{\\!(-1)^r \\, \\Sigma^{r-1} i_{s}} & \\Sigma^{r-1} Y_{s} \\ar@{-->}[d]^{\\alpha_r} \\ar[r]^-{- \\Sigma^{r-1} p_{s}} & \\Sigma^{r-1} I_{s} \\ar@{=}[d] \\\\\n\\Sigma^{r-2} I_{s} \\ar[r]^-{} & W_{r-1} \\ar[d]_{\\beta_{r-1}} \\ar@{-->}[r]^-{q_{r-1}} & W_r \\ar@{-->}[d]^{\\beta_r} \\ar@{-->}[r]^-{\\iota_{r-1}} & \\Sigma^{r-1} I_{s} \\\\\n\\Sigma^r I_{s+r} & \\Sigma^r Y_{s+r} \\ar[l]_{\\Sigma^r p_{s+r}} \\ar[d]_{\\gamma_{r-1}} \\ar@{=}[r] & \\Sigma^r Y_{s+r} \\ar[d]^{\\gamma_r} & \\\\\n& \\Sigma^r Y_{s+1} \\ar[r]^{(-1)^r \\, \\Sigma^r i_{s}} & \\Sigma^r Y_{s} . & \\\\\n}\n\\]\nUp to suspension, both $d_r[x]$ and the last restricted Toda bracket are computed by\ncomposing certain maps $\\widetilde{x} : \\Sigma^{t-s+r-2} X \\to \\Sigma^r Y_{s+r}$ with $\\Sigma^r p_{s+r}$.\nFor $d_r[x]$, the maps $\\widetilde{x}$ must lift $\\Sigma^{r-1} (\\delta_s x)$ through $- \\gamma_{r-1}$.\nFor the last bracket, the maps $\\widetilde{x}$ are of the form $\\beta_r y$,\nwhere $y : \\Sigma^{t-s+r-1} X \\to W_r$ is a lift of $-\\Sigma^{r-1} x$ through $\\iota_{r-1}$.\nAs in the proof of Proposition~\\ref{pr:DifferentD2}\\eqref{it:d1d1x}, one can\nsee that the possible choices of $\\widetilde{x}$ coincide.\n\\end{proof}\n\nWe next give a description of $d_r[x]$ using higher Toda brackets defined\nusing filtered objects, as in Definitions~\\ref{def:NFiltered} and~\\ref{def:HigherTodaSS}.\nThe computation of the restricted Toda bracket above produces a sequence \n\\begin{equation}\\label{eq:W}\n  \\cxymatrix{0 = W_0 \\ar[r]^-{q_0} & W_1 \\ar[r]^{q_1} & \\cdots  \\ar[r]^{q_{r-1}} & W_r , }\n\\end{equation}\nwhere $W_k$ is the fibre of the $k$-fold composite $\\Sigma^r (i_{s+r-k} \\cdots i_{s+r-1})$.\n(The map $\\gamma_k$ may differ in sign from this composite, but that doesn't affect the fibre.)\nFor each $k$, we have a distinguished triangle\n\\[\n  \\xymatrix@C+3pt{W_k \\ar[r]^-{q_k} & W_{k+1} \\ar[r]^-{\\iota_k}\n & \\Sigma^{r-1} I_{s+r-k-1} \\ar[rrr]^-{-(\\Sigma \\alpha_k)(\\Sigma^{r-1} \\delta_{s+r-k-1})} &&& \\Sigma W_k ,}\n\\]\nwhere we extend downwards to $k=0$ by defining $W_1 = \\Sigma^{r-1} I_{s+r-1}$\nand using the non-obvious triangle\n\\[\n  \\xymatrix@C+7pt{W_0 \\ar[r]^-{q_0 = 0} & W_{1} \\ar[r]^-{\\iota_0 = -1}\n & \\Sigma^{r-1} I_{s+r-1} \\ar[r]^-{0} & \\Sigma W_0 .}\n\\]\nOne can check that \n\\[\n(\\Sigma \\iota_{k-1})(-\\Sigma \\alpha_k)(\\Sigma^{r-1} \\delta_{s+r-k-1}) = (\\Sigma^r p_{s+r-k})(\\Sigma^{r-1} \\delta_{s+r-k-1})\n= \\Sigma^{r-1} d_1 = \\Sigma^k (\\Sigma^{r-k-1} d_1) ,\n\\]\nwhere $\\Sigma^{r-k-1} d_1$ is the map appearing in the $(k+1)$st spot of the Toda bracket.\nIn other words, the sequence~\\eqref{eq:W} is an $r$-filtered object based on\n$(\\Sigma^{r-2} d_1, \\ldots, d_1)$.\n\nThe natural map $\\sigma_W : W_r \\to \\Sigma^{r-1} I_s$ is $\\iota_{r-1}$,\nand the natural map $\\sigma_W' : \\Sigma^{r-1} I_{s+r-1} \\cong W_1 \\to W_r$ is the composite\n$q_{r-1} \\cdots q_1 \\iota_0 = -q_{r-1} \\cdots q_1$.\nThe Toda bracket computed using the filtered object $W$ consists of all\ncomposites appearing in the middle row of this commutative diagram:\n\\begin{equation}\\label{eq:Wlift}\n\\cxymatrix{\n& \\Sigma^{r-1} I_{s+r-1} \\ar[d]_{\\sigma'_W} \\ar[dr]^-{\\Sigma^{r-1} d_1}  & \\\\\n\\Sigma^{t-s+r-1} X \\ar[dr]_{\\Sigma^{r-1} x} \\ar@{-->}[r]^-a & W_r \\ar[d]^{\\sigma_W} \\ar@{-->}[r]_-b & \\Sigma^r I_{s+r} \\\\\n& \\Sigma^{r-1} I_s . & \\\\\n}\n\\end{equation}\nWe claim that there is a natural choice of extension $b$.\nSince $\\Sigma^{r-1} d_1 = (\\Sigma^r p_{s+r})(\\Sigma^{r-1} \\delta_{s+r-1})$, it suffices\nto extend $\\Sigma^{r-1} \\delta_{s+r-1}$ over $\\sigma_W'$.\nWell, $\\beta_2$ by definition is an extension of $\\Sigma^{r-1} \\delta_{s+r-1}$ over $q_1$,\nand each subsequent $\\beta_k$ gives a further extension.\nBecause $\\iota_0 = -1$, $-(\\Sigma^r p_{s+r}) \\beta_r$ is a valid choice for $b$.\n\nOn the other hand, as described at the end of the previous proof,\nthe lifts $a$ of $\\Sigma^{r-1} x$ through $\\sigma_W = \\iota_{r-1}$, when \ncomposed with $-(\\Sigma^r p_{s+r}) \\beta_r$, give exactly\nthe Toda bracket computed there.\n\nIn summary, we have:\n\n\\begin{thm}\nGiven an Adams resolution of $Y$ and $r \\geq 2$, there is an associated\n$r$-filtered object $W$ and a choice of a map $b$ in Diagram~\\eqref{eq:Wlift},\nsuch that for any $X$ and class $[x] \\in E_r^{s,t}$,\nwe have\n\\[\nd_r [x] = \\left\\langle \\Sigma^{r-1} d_1 , \\ldots , \\Sigma d_1 , d_1 , x \\right\\rangle ,\n\\]\nwhere the Toda bracket is computed only using the $r$-filtered object $W$\nand the chosen extension $b$.\n\\end{thm}\n\n\n\\section{Sparse rings of operations}\\label{se:sparse}\n\nIn this section, we focus on injective and projective classes which \nare generated by an object with a ``sparse'' endomorphism ring.\nIn this context, we can give conditions under which the restricted Toda bracket\nappearing in Theorem~\\ref{th:AdamsDrCohomOp} is equal to the unrestricted Toda bracket,\nproducing a cleaner correspondence between Adams differentials and Toda brackets.\nWe begin in Subsection~\\ref{ss:sparse-injective} by giving the results in the case\nof an injective class, and then briefly summarize the dual results in Subsection~\\ref{ss:sparse-projective}.\nSubsection~\\ref{ss:sparse-examples} gives examples.\n\nLet us fix some notation and terminology, also discussed in \\cite{Sagave08}, \\cite{Patchkoria12}, \n\\cite{SchwedeS03}*{\\S 2}, \nand \\cite{BensonKS04}.\n\n\\begin{defn}\nLet $N$ be a natural number. A graded abelian group $R_*$ is \\textbf{$N$-sparse} if $R_*$ is concentrated in degrees which are multiples of $N$, i.e., $R_i = 0$ whenever $i \\not\\equiv 0 \\pmod{N}$.\n\\end{defn}\n\n\n\\subsection{Injective case}\\label{ss:sparse-injective}\n\n\\begin{nota}\nLet $E$ be an object of the triangulated category $\\cat{T}$. Define the \\textbf{$E$-cohomology} of an object $X$ to be the graded abelian group $E^*X$ given by $E^n X := \\cat{T}(X,\\Sigma^n E)$. Postcomposition makes $E^*X$ into a left module over the graded endomorphism ring $E^*E$.%\n\\end{nota}\n\n\\begin{assum}\nFor the remainder of this subsection, we assume the following.\n\\begin{enumerate}\n\\item The triangulated category $\\cat{T}$ has infinite %\nproducts.\n\\item The graded ring $E^* E$ is $N$-sparse for some $N \\geq 2$.%\n\\end{enumerate}\n\\end{assum}\n\nLet $\\cat{I}_E$ denote the injective class generated by $E$, as in Example~\\ref{ex:InjClass}. Explicitly, $\\cat{I}_E$ consists of retracts of (arbitrary) products $\\prod_i \\Sigma^{n_i} E$.\n\n\\begin{lem}\\label{le:SparseInj}\nWith this setup, we have:\n\\begin{enumerate}\n\\item Let $I$ be an injective object such that $E^* I$ is $N$-sparse. Then $I$ is a retract of a product $\\prod_i \\Sigma^{m_i N} E$.\n\\item If, moreover, $W$ is an object such that $E^*W$ is $N$-sparse, then we have $\\cat{T}(W,\\Sigma^t I) = 0$ for $t \\not\\equiv 0 \\pmod{N}$. \n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n(1) $I$ is a retract of a product $P = \\prod_i \\Sigma^{n_i} E$, with a map $\\iota \\colon I \\hookrightarrow P$ and retraction $\\pi \\colon P \\twoheadrightarrow I$. Consider the subproduct $P' = \\prod_{N \\mid n_i} \\Sigma^{n_i} E$, with inclusion $\\iota' \\colon P' \\hookrightarrow P$ (via the zero map into the missing factors) and projection $\\pi' \\colon P \\twoheadrightarrow P'$. Then the equality\n\\[\n\\iota' \\pi' \\iota = \\iota \\colon I \\to P\n\\]\nholds, using the fact that $E^*I$ is $N$-sparse. %\nTherefore, we obtain $\\pi \\iota' \\pi' \\iota = \\pi \\iota = 1_I$, so that $I$ is a retract of $P'$. %\n\n(2) By the first part, $\\cat{T}(W,\\Sigma^t I)$ is a retract of\n\\begin{align*}\n\\cat{T}(W,\\Sigma^t \\prod_i \\Sigma^{m_i N} E) &= \\cat{T}(W,\\prod_i \\Sigma^{m_i N+ t} E) \\\\\n&= \\prod_i \\cat{T}(W,\\Sigma^{m_i N+ t} E) \\\\ \n&= \\prod_i E^{m_i N+ t}W \\\\\n&= 0 ,\n\\end{align*}\nusing the assumption that $E^*W$ is $N$-sparse.\n\\end{proof}\n\n\n\\begin{lem}\\label{le:SparseBracket}\nLet $I_0 \\ral{f_1} I_1 \\ral{f_2} I_2 \\to \\cdots \\ral{f_r} I_r$ be a diagram\nin $\\cat{T}$, with $r \\leq N+1$. Assume that each object $I_j$ is injective and \nthat each $E^*(I_j)$ is $N$-sparse. Then the iterated Toda family $\\mathrm{T}(f_r, f_{r-1}, \\ldots, f_1)$ is either empty or consists of a single composable pair $\\Sigma^{r-2} I_0 \\to C \\to I_r$, up to automorphism of $C$.\n\\end{lem}\n\n\\begin{proof}\nIn the case $r=2$, there is nothing to prove, so we may assume $r \\geq 3$. The iterated Toda family is obtained by $r-2$ iterations of the $3$-fold Toda family construction. The first iteration computes the Toda family of the diagram\n\\[\n\\xymatrix{\nI_{r-3} \\ar[r]^-{f_{r-2}} & I_{r-2} \\ar[r]^-{f_{r-1}}  & I_{r-1} \\ar[r]^-{f_{r}} & I_{r}. \\\\\n}\n\\]\nChoose a cofiber of $f_{r-1}$, i.e., a distinguished triangle $I_{r-2} \\ral{f_{r-1}} I_{r-1} \\to C_1 \\to \\Sigma I_{r-2}$.\nA lift of $f_{r-2}$ to the fiber $\\Sigma^{-1} C_1$, if it exists, is determined up to\n\\[\n\\cat{T}(I_{r-3}, \\Sigma^{-1} I_{r-1}) = \\cat{T}(\\Sigma I_{r-3}, I_{r-1}),\n\\]\nwhich is zero by Lemma~\\ref{le:SparseInj}(2).\nLikewise, an extension of $f_{r}$ to the cofiber $C_1$, if it exists, is determined up to\n\\[\n\\cat{T}(\\Sigma I_{r-2}, I_{r}) = 0.\n\\]\nHence, $\\mathrm{T}(f_r, f_{r-1}, f_{r-2})$ is either empty or consists of a single pair $(\\beta_1, \\Sigma \\alpha_1)$,\nup to automorphisms of $C_1$.\nIt is easy to see that the object $C_1$ has the following property:\n\\begin{equation}\\label{eq:SparseCohom1}\n\\text{If $E^*W = 0$ for $\\ast \\equiv 0,1 \\;(\\bmod\\; N)$, then $\\cat{T}(W,C_1) = 0$.}\n\\end{equation}\nFor $r \\geq 4$, the next iteration computes the Toda family of the diagram\n\\[\n\\xymatrix{\n\\Sigma I_{r-4} \\ar[r]^-{\\Sigma f_{r-3}} & \\Sigma I_{r-3} \\ar[r]^-{\\Sigma \\alpha_1} & C_1 \\ar[r]^-{\\beta_1} & I_{r}. \\\\\n}\n\\]\nThe respective indeterminacies are\n\\[\n\\cat{T}(\\Sigma^2 I_{r-4}, C_1),\n\\]\nwhich is zero by Property~\\eqref{eq:SparseCohom1}, and \n\\[\n\\cat{T}(\\Sigma^2 I_{r-3}, I_{r}),\n\\]\nwhich is zero by Lemma~\\ref{le:SparseInj}(2), since $N \\geq 3$ in this case.\nHence, $\\mathrm{T}(\\beta_1, \\Sigma \\alpha_1, \\Sigma f_{r-3})$ is either empty or consists of a single pair $(\\beta_2, \\Sigma \\alpha_2)$,\nup to automorphism of the cofiber $C_2$ of $\\Sigma \\alpha_1$.\nRepeating the argument inductively, the successive iterations compute the Toda family of a diagram\n\\[\n\\xymatrix @C=3.3pc {\n\\Sigma^j I_{r-3-j} \\ar[r]^-{\\Sigma^j f_{r-2-j}} & \\Sigma^j I_{r-2-j} \\ar[r]^-{\\Sigma \\alpha_j} & C_j \\ar[r]^-{\\beta_j} & I_{r} \\\\\n}\n\\]\nfor $0 \\leq j \\leq r-3$, where $C_j$ has the following property:\n\\begin{equation}\\label{eq:SparseCohomj}\n\\text{If $E^*W = 0$ for $\\ast \\equiv 0,1,\\ldots,j \\;(\\bmod\\; N)$, then $\\cat{T}(W,C_j) = 0$.}\n\\end{equation}\nThe indeterminacies $\\cat{T}(\\Sigma^{j+1} I_{r-3-j}, C_j)$ and $\\cat{T}(\\Sigma^{j+1} I_{r-2-j}, I_{r})$\nagain vanish.\nHence,\\break $\\mathrm{T}(\\beta_j, \\Sigma \\alpha_j, \\Sigma^j f_{r-2-j})$ is either empty or consists of a single pair \n$(\\beta_{j+1}, \\Sigma \\alpha_{j+1})$, up to automorphism of $C_{j+1}$.\nNote that the argument works until the last iteration $j = r-3$, by the assumption $r-2 < N$.\n\\end{proof}\n\nWe will need the following condition on an object $Y$:\n\\begin{condition}\\label{co:InjSparse}\n$Y$ admits an $\\cat{I}_E$-Adams resolution $Y_{\\bullet}$ (see \\eqref{eq:AdamsResolInj}) such that for each injective $I_j$ in the resolution,\n$E^* (\\Sigma^j I_j)$ is $N$-sparse.\n\\end{condition}\n\n\\pagebreak[2]\n\\begin{rem}\\leavevmode\n\\begin{enumerate}\n\\item Condition~\\ref{co:InjSparse} implies that $E^* Y$ is itself $N$-sparse, because of the surjection $E^* I_0 \\twoheadrightarrow E^* Y$.\n\\item The condition can be generalized to: there is an integer $m$ such that for each $j$, $E^* (\\Sigma^j I_j)$ is concentrated in degrees $\\ast \\equiv m \\pmod{N}$. We take $m=0$ for notational convenience.\n\\item We will see in Propositions~\\ref{pr:CohomProduct} and~\\ref{pr:CoherentRing} situations in which\nCondition~\\ref{co:InjSparse} holds.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{thm}\\label{th:SparseDr}\nLet $X$ and $Y$ be objects in $\\cat{T}$ and consider the Adams spectral sequence abutting to $\\cat{T}(X,Y)$ with respect to the injective class $\\cat{I}_E$. Assume that $Y$ satisfies Condition~\\ref{co:InjSparse}. Then for all $r \\leq N$, the Adams differential is given, as subsets of $E_1^{s+r,t+r-1}$, by\n\\[\nd_r [x] = \\left\\langle \\Sigma^{r-1} d_1 , \\ldots, \\Sigma d_1, d_1 , x \\right\\rangle.\n\\]\nIn other words, the restricted bracket appearing in Theorem~\\ref{th:AdamsDrCohomOp} coincides with the full Toda bracket.\n\\end{thm}\n\n\\begin{proof}\nWe will show that \n\\[\n  \\left\\langle \\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1 , x \\right\\rangle = \\left\\langle \\Sigma^{r-1} d_1 , \\ldots, \\Sigma d_1, d_1 , x \\right\\rangle.\n\\]\nConsider the diagram \n\\[\n\\xymatrix{\nI_s \\ar[r]^-{d_1} & \\Sigma I_{s+1} \\ar[r]^-{\\Sigma d_1} & \\Sigma^2 I_{s+2} \\ar[r] & \\cdots \\ar[r] & \\Sigma^{r-1} I_{r-1} \\ar[r]^-{\\Sigma^{r-1} d_1} & \\Sigma^{r} I_{s+r} \\\\\nX \\ar[u]^{x} & & & & & \\\\\n}\n\\]\nwhose Toda bracket is being computed. The corresponding Toda family is\n\\[\n\\mathrm{T}(\\Sigma^{r-1} d_1 , \\ldots, \\Sigma d_1, d_1 , x) = \\mathrm{T} \\left( \\mathrm{T}(\\Sigma^{r-1} d_1 , \\ldots, \\Sigma d_1, d_1), \\Sigma^{r-2} x \\right).\n\\]\nWe know that\n\\[\n  \\mathrm{T}(\\Sigma^{r-1} d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\ldots \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} \\Sigma d_1 \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} d_1) \\subseteq \\mathrm{T}(\\Sigma^{r-1} d_1 , \\ldots, \\Sigma d_1, d_1).\n\\]\nBy Lemma~\\ref{le:SparseBracket}, %\nthe Toda family on the right has at most one element, up to automorphism.\nBut fully-restricted Toda families are always non-empty, so the inclusion must be an equality.\nWrite $\\Sigma^{r-2} I_{s} \\ral{f} C \\ral{g} \\Sigma^r I_{s+r}$ for an element of these families.\nIt remains to show that the inclusion\n\\[\n  \\left\\langle g \\overset{!\\hspace*{1.5pt}}{,\\rule{0pt}{3pt}} f, \\Sigma^{r-2} x \\right\\rangle \\subseteq \\left\\langle g, f, \\Sigma^{r-2} x \\right\\rangle \n\\]\nis an equality, i.e., that the extension of $g$ to the cofiber of $f$ is unique.\nThis follows from the equality $\\cat{T}(\\Sigma^{r-1} I_{s}, \\Sigma^r I_{s+r}) = 0$, which uses the assumption on the injective objects $I_j$ and that $r-1 < N$.\n\\end{proof}\n\nNext, we describe situations in which Theorem~\\ref{th:SparseDr} applies.\n\n\\begin{prop}\\label{pr:CohomProduct}\nAssume that every product of the form $\\prod_i \\Sigma^{m_i N} E$ has cohomology\\break $E^* \\left( \\prod_i \\Sigma^{m_i N} E \\right)$ which is $N$-sparse. Then every object $Y$ such that $E^* Y$ is $N$-sparse also satisfies Condition~\\ref{co:InjSparse}.\n\\end{prop}\n\n\\begin{proof}\nLet $(y_i)$ be a set of non-zero generators of $E^* Y$ as an $E^* E$-module. %\nThen the corresponding map $Y \\to \\prod_i \\Sigma^{\\abs{y_i}} E$ is $\\cat{I}_E$-monic into an injective object;\nwe take this map as the first step $p_0 \\colon Y_0 \\to I_0$, with cofiber $\\Sigma Y_1$. By our assumption on $Y$, each degree $\\abs{y_i}$ is a multiple of $N$, and thus $E^* I_0$ is $N$-sparse, by the assumption on $E$. The distinguished triangle $Y_1 \\to Y_0 \\ral{p_0} I_0 \\to \\Sigma Y_1$ induces a long exact sequence on $E$-cohomology which implies that the map $I_0 \\to \\Sigma Y_1$ is injective on $E$-cohomology.  It follows that $E^*(\\Sigma Y_1)$ is $N$-sparse as well. Repeating this process, we obtain an $\\cat{I}_E$-Adams resolution of $Y$ such that for every $j$, $E^* (\\Sigma^j Y_j)$ and $E^* (\\Sigma^j I_j)$ are $N$-sparse.\n\\end{proof}\n\nThe condition on $E$ is discussed in Example \\ref{ex:Compact}.\n\n\\begin{prop}\\label{pr:CoherentRing}\nAssume that the ring $E^* E$ is left coherent, and that $E^* Y$ is $N$-sparse and finitely presented as a left $E^*E$-module. Then $Y$ satisfies Condition~\\ref{co:InjSparse}.\n\\end{prop}\n\n\\begin{proof}\nSince $E^*Y$ is finitely generated over $E^*E$, the map $p_0 \\colon Y \\to I_0$ can be chosen so that $I_0 = \\prod_i \\Sigma^{m_i N} E \\cong \\oplus_i \\Sigma^{m_i N} E$ is a finite product. \nIt follows that $E^* I_0$ is $N$-sparse and finitely presented.\nWe have that $E^{*-1}Y_1 = \\ker \\left( p_0^* \\colon E^*I_0 \\twoheadrightarrow E^*Y \\right)$.\nThis is $N$-sparse, since $E^* I_0$ is, and is finitely presented over $E^*E$, since both $E^* I_0$ and $E^*Y$ are, and $E^*E$ is coherent \\cite{Bourbaki98}*{\\S I.2, Exercises 11--12}. %\nRepeating this process, we obtain an $\\cat{I}_E$-Adams resolution of $Y$ such that for every $j$, $\\Sigma^j I_j$ is a finite product of the form $\\prod_i \\Sigma^{m_i N} E$.\n\\end{proof}\n\n\n\n\\subsection{Projective case}\\label{ss:sparse-projective}\n\nThe main applications of Theorem~\\ref{th:SparseDr} are to projective classes instead of injective classes. For future reference, we state here the dual statements of the previous subsection and adopt a notation inspired from stable homotopy theory.\n\n\\begin{nota}\nLet $R$ be an object of the triangulated category $\\cat{T}$. Define the \\textbf{homotopy} (with respect to $R$) of an object $X$ as the graded abelian group $\\pi_* X$ given by $\\pi_n X := \\cat{T}(\\Sigma^n R,X)$. Precomposition makes $\\pi_*X$ into a right module over the graded endomorphism ring $\\pi_* R$.%\n\\end{nota}\n\n\\begin{assum}\nFor the remainder of this subsection, we assume the following.\n\\begin{enumerate}\n\\item The triangulated category $\\cat{T}$ has infinite %\ncoproducts.\n\\item The graded ring $\\pi_* R$ is $N$-sparse for some $N \\geq 2$.\n\\end{enumerate}\n\\end{assum}\n\nLet $\\cat{P}_R$ denote the stable projective class spanned by $R$, as in Example~\\ref{ex:GhostSphere}. Explicitly, $\\cat{P}_R$ consists of retracts of (arbitrary) coproducts $\\oplus_i \\Sigma^{n_i} R$.\n\n\\begin{condition}\\label{co:ProjSparse}\n$X$ admits an $\\cat{P}_R$-Adams resolution $X_{\\bullet}$ as in Diagram~\\eqref{eq:AdamsResolProj} such that each projective $P_j$ satisfies that $\\pi_* (\\Sigma^{-j} P_j)$ is $N$-sparse.\n\\end{condition}\n\n\\begin{thm}\\label{th:SparseDrProj}\nLet $X$ and $Y$ be objects in $\\cat{T}$ and consider the Adams spectral sequence abutting to $\\cat{T}(X,Y)$ with respect to the projective class $\\cat{P}_R$. Assume that $X$ satisfies Condition~\\ref{co:ProjSparse}. Let $[y] \\in E_r^{s,t}$ be a class represented by $y \\in E_1^{s,t} = \\cat{T}(\\Sigma^{t-s} P_s, Y)$. Then for all $r \\leq N$, the Adams differential is given, as subsets of $E_1^{s+r,t+r-1}$, by\n\\[\nd_r [y] = \\left\\langle y, d_1, \\Sigma^{-1} d_1, \\ldots, \\Sigma^{-(r-1)} d_1 \\right\\rangle.\n\\]\n\\end{thm}\n\nNote that we used Corollary~\\ref{co:SelfDual} to ensure that the equality holds as stated, not merely up to sign.\n\n\\begin{prop}\\label{pr:HomotCoproduct}\nAssume that every coproduct of the form $\\oplus_i \\Sigma^{m_i N} R$ has homotopy\\break $\\pi_* \\left( \\oplus_i \\Sigma^{m_i N} R \\right)$ which is $N$-sparse. Then every object $X$ such that $\\pi_* X$ is $N$-sparse also satisfies Condition~\\ref{co:ProjSparse}.\n\\end{prop}\n\nRecall the following terminology:\n\n\\begin{defn}\nAn object $X$ of $\\cat{T}$ is \\textbf{compact} if the functor $\\cat{T}(X,-)$ preserves infinite coproducts.%\n\\end{defn}\n\n\\begin{ex}\\label{ex:Compact}\nIf $R$ is compact in $\\cat{T}$, then $R$ satisfies the assumption of Proposition~\\ref{pr:HomotCoproduct}. This follows from the isomorphism\n\\[\n\\pi_* \\left( \\oplus_i \\Sigma^{m_i N} R \\right) \\cong \\bigoplus_i \\pi_* (\\Sigma^{m_i N} R) = \\bigoplus_i \\Sigma^{m_i N} \\pi_* R  \n\\]\nand the assumption that $\\pi_* R$ is $N$-sparse. The same argument works if $R$ is a retract of a coproduct of compact objects.\n\nDually, if $E$ is cocompact in $\\cat{T}$, then $E$ satisfies the assumption of Proposition~\\ref{pr:CohomProduct}. \nThis holds more generally if $E$ is a retract of a product of cocompact objects.\n\\end{ex}\n\n\\begin{rem}\nSome of the related literature deals with compactly generated triangulated categories. As noted in Remark~\\ref{re:NotGenerate}, we do \\emph{not} assume that the object $R$ is a generator, i.e., that the condition $\\pi_* X = 0$ implies $X=0$.\n\\end{rem}\n\n\\begin{prop}\\label{pr:CoherentRingProj}\nAssume that the ring $\\pi_* R$ is right coherent, and that $\\pi_* X$ is $N$-sparse and finitely presented as a right $\\pi_* R$-module. Then $X$ satisfies Condition~\\ref{co:ProjSparse}.\n\\end{prop}\n\nThe following is a variant of \\cite{Patchkoria12}*{Lemma 2.2.2}, where we do not assume that $R$ is a generator.\nIt identifies the $E_2$ term of the spectral sequence associated to the projective class $\\cat{P}_R$.\nThe proof is straightforward.\n\n\\begin{prop}\\label{pr:ExtEndoRing}\nAssume that the object $R$ is compact.\n\\begin{enumerate}\n\\item Let $P$ be in the projective class $\\cat{P}_R$. Then the map of abelian groups\n\\[ %\n\\cat{T}(P,Y) \\to \\Hom_{\\pi_* R} (\\pi_* P, \\pi_* Y)\n\\] %\nis an isomorphism for every object $Y$.\n\\item There is an isomorphism\n\\[\n\\Ext^{s}_{\\cat{P}_R}(X,Y) \\cong \\Ext^{s}_{\\pi_* R}(\\pi_* X, \\pi_* Y)\n\\]\nwhich is natural in $X$ and $Y$. \n\\end{enumerate}\n\\end{prop}\n\n\n\n\n\\subsection{Examples}\\label{ss:sparse-examples}\n\nTheorem~\\ref{th:SparseDrProj} applies to modules over certain ring spectra. We describe some examples, along the lines of \\cite{Patchkoria12}*{Examples 2.4.6 and 2.4.7}.\n\n\\begin{ex}\\label{ex:RingSpectrum}\nLet $R$ be an $A_{\\infty}$ ring spectrum, and let $h\\Mod{R}$ denote the homotopy category of the stable model category of (right) $R$-modules %\n\\cite{SchwedeS03}*{Example 2.3(ii)} \n\\cite{EKMM97}*{\\S III}. Then $R$ itself, the free $R$-module of rank $1$, is a compact generator for $h\\Mod{R}$. The $R$-homotopy of an $R$-module spectrum $X$ is the usual homotopy of $X$, as suggested by the notation:\n\\[\nh\\Mod{R}(\\Sigma^n R, X) \\cong h\\Mod{S}(S^n, X) = \\pi_n X.\n\\]\nIn particular, the graded endomorphism ring $\\pi_* R$ is the usual coefficient ring of $R$.\n\nThe projective class $\\cat{P}_R$ is the ghost projective class \\cite{Christensen98}*{\\S 7.3}, generalizing  Example~\\ref{ex:GhostSphere}, where $R$ was the sphere spectrum $S$. The Adams spectral sequence relative to $\\cat{P}_R$ is \nthe universal coefficient spectral sequence\n\\[\n\\Ext_{\\pi_* R}^{s}(\\Sigma^t \\pi_* X, \\pi_* Y) \\Ra h\\Mod{R}(\\Sigma^{t-s} X,Y)\n\\]\nas described in \\cite{EKMM97}*{\\S IV.4} and~\\cite{Christensen98}*{Corollary 7.12}. %\nWe used Proposition~\\ref{pr:ExtEndoRing} to identify the $E_2$ term.\n\nSome $A_{\\infty}$ ring spectra $R$ with sparse homotopy $\\pi_* R$ are discussed in \\cite{Patchkoria12}*{\\S 4.3, 5.3, 6.4}. In view of Proposition~\\ref{pr:ExtEndoRing}, the Adams spectral sequence in $h\\Mod{R}$ collapses at the $E_2$ page if $\\pi_* R$ has (right) global dimension less than $2$. %\n\nThe Johnson--Wilson spectrum $E(n)$ has coefficient ring\n\\[\n\\pi_* E(n) = \\mathbb{Z}_{(p)}[v_1, \\ldots, v_n, v_n^{-1}], \\quad \\abs{v_i} = 2(p^i - 1),\n\\]\nwhich has global dimension $n$ and is $2(p-1)$-sparse. Hence, Theorem~\\ref{th:SparseDrProj} applies in this case to the differentials $d_r$ with $r \\leq 2(p-1)$, while $d_r$ is zero for $r > n$.\nLikewise, connective complex $K$-theory $ku$ has coefficient ring\n\\[\n\\pi_* ku = \\mathbb{Z}[u], \\quad \\abs{u} = 2,\n\\]\nwhich has global dimension $2$ and is $2$-sparse.\n\\end{ex}\n\n\\begin{ex}\\label{ex:DGA}\nLet $R$ be a differential graded (\\emph{dg} for short) algebra over a commutative ring $k$, and consider the category of dg $R$-modules $\\dgMod{R}$. The homology $H_* X$ of a dg $R$-module is a (graded) $H_* R$-module. %\nThe derived category $D(R)$ is defined as the localization of $\\dgMod{R}$ with respect to quasi-isomorphisms. \nThe free dg $R$-module $R$ is a compact generator of $D(R)$. The $R$-homotopy of an object $X$ of $D(R)$ is its homology $\\pi_* X = H_* X$. In particular, the graded endomorphism ring of $R$ in $D(R)$ is the graded $k$-algebra $H_* R$.\n\nThe Adams spectral sequence relative to $\\cat{P}_R$ is \nan Eilenberg--Moore spectral sequence\n\\[\n\\Ext_{H_* R}^{s} \\left( \\Sigma^t H_* X, H_* Y \\right) \\Ra D(R)(\\Sigma^{t-s} X, Y)\n\\]\nfrom ordinary $\\Ext$ to differential $\\Ext$, as described in \\cite{BarthelMR14}*{\\S 8, 10}. See also \\cite{KrizM95}*{\\S III.4}, \\cite{HoveyPS97}*{Example 10.2(b)}, \nand \\cite{EilenbergM66}.\n\\end{ex}\n\n\\begin{rem}\nExample~\\ref{ex:DGA} can be viewed as a special case of Example~\\ref{ex:RingSpectrum}. Letting $HR$ denote the Eilenberg--MacLane spectrum associated to $R$, the categories $\\Mod{HR}$ and $\\dgMod{R}$ are Quillen equivalent, by \\cite{SchwedeS03}*{Example 2.4(i)} \\cite{Shipley07HZ}*{Corollary 2.15}, yielding a triangulated equivalence $h\\Mod{HR} \\cong D(R)$. The generator $HR$ corresponds to the generator $R$ via this equivalence.\n\\end{rem}\n\n\\begin{ex}\\label{ex:Ring}\nLet $R$ be a ring, viewed as a dg algebra concentrated in degree $0$. Then Example~\\ref{ex:DGA} yields the ordinary derived category $D(R)$. The graded endomorphism ring of $R$ in $D(R)$ is $H_* R$, which is $R$ concentrated in degree $0$.\nThis is $N$-sparse for any $N \\geq 2$.  \n\nThe Adams spectral sequence relative to $\\cat{P}_R$ is the hyperderived functor spectral sequence\n\\[\n\\Ext_{H_* R}^{s} \\left( \\Sigma^t H_* X, H_* Y \\right) = \\prod_{i \\in \\mathbb{Z}} \\Ext_{R}^{s} \\left( H_{i-t} X, H_{i} Y \\right) \\Ra D(R)(\\Sigma^{t-s}X, Y) = \\mathbf{Ext}_{R}^{s-t}(X,Y)\n\\]\nfrom ordinary $\\Ext$ to hyper-$\\Ext$, as described in \\cite{Weibel94}*{\\S 5.7, 10.7}.%\n\\end{ex}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRogers \\cite{Rog58} introduced the notion of computable numbering\n(of all partial computable functions) and the notion of reducibility.\nHe showed that the set of all equivalence classes of computable numberings\nforms an upper semilattice with respect to reducibility which is called\nthe \\emph{Rogers semilattice}. He asked whether this semilattice is\na lattice, and if not, whether any two elements have a lower bound.\nFriedberg \\cite{Fri58} constructed an injective computable numbering\n(called a Friedberg numbering) by a finite-injury priority argument.\nPour-El \\cite{Pou64} showed that every Friedberg numberings of is\nminimal. She also showed that there are two incomparable Friedberg\nnumberings through modifying Friedberg's construction. These numberings\nare non-equivalent minimal elements, and therefore they have no lower\nbound. Thus Rogers' questions were negatively answered.\n\nShen \\cite{She12} gave some examples of game-theoretic proofs of\ntheorems in computability theory and algorithmic information theory.\nIn particular, he gave the game-theoretic proof of the theorem of\nFriedberg. The game representation of Friedberg's construction is\nclear and intuitional, and can be used to prove other existence theorems\nof Friedberg numberings as we will demonstrate in the paper.\n\nIn \\prettyref{sec:Friedberg's construction} we present Shen's proof\nto use later. We provide game proofs of certain well-known existence\ncriteria of Friedberg numberings for general classes of partial computable\nfunctions. In \\prettyref{sec:Modifications}, we give two proofs of\nthe theorem of Pour-El using two games. Also we give the proof of\nthe existence of an infinite c.e. sequence and an independent sequence\nof Friedberg numberings. These are essentially modifications of Shen's\nproof.\n\n\n\\section{Notations and Definitions}\n\nWe denote by $\\mathcal{P}^{\\left(1\\right)}$ the set of all partial\ncomputable functions from $\\mathbb{N}$ to $\\mathbb{N}$. $\\braket{\\cdot,\\cdot}$\nis a computable pairing function which is a computable bijection between\n$\\mathbb{N}^{2}$ and $\\mathbb{N}$. Let $\\mathcal{A}$ be any set.\nA surjective map $\\nu:\\mathbb{N}\\to\\mathcal{A}$ is called a \\emph{numbering}\nof $\\mathcal{A}$. Let $\\nu$ and $\\mu$ be numberings of $\\mathcal{A}$.\nWe say that $\\nu$ is \\emph{reducible to} $\\mu$, denoted by $\\nu\\leq\\mu$,\nif there is a total computable function $f:\\mathbb{N}\\to\\mathbb{N}$\nsuch that $\\nu=\\mu\\circ f$. We say that $\\nu$ and $\\mu$ are \\emph{equivalent}\nif they are reducible to each other. We say that $\\nu$ and $\\mu$\nare \\emph{incomparable} if they are not reducible to each other. In\nthis paper, we often identify a numbering $\\nu$ of a set of partial\nmaps from $X$ to $Y$ with the partial map $\\nu\\left(i,x\\right)=\\nu\\left(i\\right)\\left(x\\right)$\nfrom $\\mathbb{N}\\times X$ to $Y$. A numbering $\\nu$ of a subset\nof $\\mathcal{P}^{\\left(1\\right)}$ is said to be \\emph{computable}\nif it is computable as a partial function from $\\mathbb{N}^{2}$ to\n$\\mathbb{N}$. A computable injective numbering is called a \\emph{Friedberg\nnumbering}. A sequence $\\set{\\nu_{i}}_{i\\in\\mathbb{N}}$ of numberings\nof a subset of $\\mathcal{P}^{\\left(1\\right)}$ is said to be \\emph{uniformly\nc.e.} if it is uniformly c.e. as a sequence of partial functions from\n$\\mathbb{N}^{2}$ to $\\mathbb{N}$, or equivalently, if it is computable\nas a partial function from $\\mathbb{N}^{3}$ to $\\mathbb{N}$. We\nsay that a sequence $\\set{\\nu_{i}}_{i\\in\\mathbb{N}}$ of numberings\nof a set $\\mathcal{A}$ is \\emph{independent} if $\\nu_{i}\\nleq\\bigoplus_{j\\neq i}\\nu_{j}$\nfor all $i\\in\\mathbb{N}$, where $\\bigoplus_{i\\in\\mathbb{N}}\\nu_{j}$\nis the direct sum of $\\set{\\nu_{i}}_{i\\in\\mathbb{N}}$ defined by\n$\\bigoplus_{i\\in\\mathbb{N}}\\nu_{i}\\left(\\braket{j,k}\\right)=\\nu_{j}\\left(k\\right)$.\n\n\n\\section{\\label{sec:Friedberg's construction}Friedberg's construction and\nthe infinite game with two boards}\n\\begin{thm}[{Friedberg \\cite[Corollary to Theorem 3]{Fri58}}]\n\\label{thm:Fri58-Corollary-to-Theorem3}$\\mathcal{P}^{\\left(1\\right)}$\nhas a Friedberg numbering.\\end{thm}\n\\begin{proof}[Proof (Shen \\cite{She12})]\nFirst, we consider an infinite game $\\mathcal{G}_{0}$ and prove\nthat the existence of a computable winning strategy of $\\mathcal{G}_{0}$\nfor one of the players implies the existence of a Friedberg numbering\nof $\\mathcal{P}^{\\left(1\\right)}$. The game $\\mathcal{G}_{0}$ is\nas follows:\n\\begin{description}\n\\item [{Players}] Alice, Bob.\n\\item [{Protocol}] FOR $s=0,1,2,\\ldots$:\n\n\nAlice announces a finite partial function $A_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\nBob announces a finite partial function $B_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\\item [{Collateral duties}] $A_{s}\\subseteq A_{s+1}$ and $B_{s}\\subseteq B_{s+1}$\nfor all $s\\in\\mathbb{N}$.\n\\item [{Winner}] Let $A=\\bigcup_{s\\in\\mathbb{N}}A_{s}$ and $B=\\bigcup_{s\\in\\mathbb{N}}B_{s}$.\nBob wins if\n\n\\begin{enumerate}\n\\item for each $i\\in\\mathbb{N}$, there is a $j\\in\\mathbb{N}$ such that\n$A\\left(i,\\cdot\\right)=B\\left(j,\\cdot\\right)$;\n\\item for any $i,j\\in\\mathbb{N}$, if $i\\neq j$, then $B\\left(i,\\cdot\\right)\\neq B\\left(j,\\cdot\\right)$.\n\\end{enumerate}\n\\end{description}\n\nWe consider $A$ and $B$ as two boards, $A$-table and $B$-table.\nEach board is a table with an infinite number of rows and columns.\nEach player plays on its board. At each move player can fill finitely\nmany cells with any natural numbers. The collateral duties prohibit\nplayers from erasing cells.\n\n\nA strategy is a map that determines the next action based on the previous\nactions of the opponent. Since any action in this game is a finitary\nobject, we can define the computability of strategies via g\\\"odelization.\nSuppose that there is a computable winning strategy for Bob. Let Alice\nfill $A$-table with the values of some computable numbering of $\\mathcal{P}^{\\left(1\\right)}$\nby using its finite approximation, and let Bob use some computable\nwinning strategy. Clearly $B$ is a Friedberg numbering of $\\mathcal{P}^{\\left(1\\right)}$.\n\n\nSecond, we consider an infinite game $\\mathcal{G}_{1}$, which is\na simplified version of $\\mathcal{G}_{0}$, and describe a computable\nwinning strategy of $\\mathcal{G}_{1}$. The game $\\mathcal{G}_{1}$\nis as follows:\n\\begin{description}\n\\item [{Players}] Alice, Bob.\n\\item [{Protocol}] FOR $s=0,1,2,\\ldots$:\n\n\nAlice announces a finite partial function $A_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\nBob announces a finite partial function $B_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$\nand a finite set $K_{s}\\subseteq\\mathbb{N}$.\n\n\\item [{Collateral duties}] $A_{s}\\subseteq A_{s+1}$, $B_{s}\\subseteq B_{s+1}$\nand $K_{s}\\subseteq K_{s+1}$ for all $s\\in\\mathbb{N}$.\n\\item [{Winner}] Let $A=\\bigcup_{s\\in\\mathbb{N}}A_{s}$, $B=\\bigcup_{s\\in\\mathbb{N}}B_{s}$\nand $K=\\bigcup_{s\\in\\mathbb{N}}K_{s}$. Bob wins if\n\n\\begin{enumerate}\n\\item for each $i\\in\\mathbb{N}$, there is a $j\\in\\mathbb{N}\\setminus K$\nsuch that $A\\left(i,\\cdot\\right)=B\\left(j,\\cdot\\right)$;\n\\item for any $i,j\\in\\mathbb{N}\\setminus K$, if $i\\neq j$, then $B\\left(i,\\cdot\\right)\\neq B\\left(j,\\cdot\\right)$.\n\\end{enumerate}\n\\end{description}\n\nWe consider that in this game Bob can \\emph{invalidate} some rows\nand that we ignore invalid rows when we decide the winner. Bob cannot\nvalidate invalid rows again.\n\n\nTo win this game, Bob hires a countable number of assistants who guarantee\nthat each of the rows in $A$-table appears in $B$-table exactly\nonce. At each move, the assistants work one by one. The $i$-th assistant\nstarts working at move $i$. She can reserve a row in $B$-table exclusively,\nfill her reserved row, and invalidate her reserved row. The instruction\nfor the $i$-th assistant: \\textit{if you have no reserved row, reserve\na new row. Let $k$ be the number of rows such that you have already\ninvalidated. If in the current state of $A$-table the first $k$\npositions of the $i$-th row are identical to the first $k$ positions\nof some previous row, invalidate your reserved row. If you have a\nreserved row, copy the current contents of the $i$-th row of $A$-table\ninto your reserved row.} These instructions guarantee in the limit\nthat\n\\begin{itemize}\n\\item if the $i$-th row in $A$-table is identical to some previous row,\nthen the $i$-th assistant invalidates her reserved row infinitely\nmany times, so she has no permanently reserved row;\n\\item if not, the $i$-th assistant invalidates her reserved row only finitely\nmany times, so she has a permanently reserved row.\n\\end{itemize}\n\n\\noindent In the second case, she faithfully copies the contents of\nthe $i$-th row of $A$-table into her permanently reserved row. We\ncan assume that each of the rows in $B$-table has been reserved or\ninvalidated in the limit: when some assistant reserves a row let her\nselect the first unused row. Then Bob wins the simplified game. Now\nwe prove the above properties. Suppose that the $i$-th row in $A$-table\nis not identical to any previous row in the limit. For each of the\nprevious rows, select some column witnessing that this row is not\nidentical to the $i$-th row. Let $k$ be the maximum of the selected\ncolumns. Wait for convergence of the rectangular area $\\left[0,i\\right]\\times\\left[0,k\\right]$\nof $A$-table. After that, the first $k$ positions of the $i$-th\nrow in $A$-table are not identical to the first $k$ positions of\nany previous row, and hence the $i$-th assistant invalidates her\nreserved row at most $k$ times. Conversely, suppose that the $i$-th\nassistant invalidates her reserved row only finitely many times. Let\n$k$ be the number of invalidations. After the $k$-th invalidation,\nthe $i$-th row in $A$-table is not identical to any previous row,\nand the same is true in the limit.\n\n\nFinally, we describe a computable winning strategy of $\\mathcal{G}_{0}$\nthrough modifying the winning strategy of $\\mathcal{G}_{1}$ described\nabove. We say that a row is odd if it contains a finite odd number\nof non-empty cells. We can assume without loss of generality that\nodd rows never appear in $A$-table: if Alice fills some cells in\na row making this row odd, Bob ignores one of these cells until Alice\nfills other cells in this row. We replace invalidation to \\emph{odd-ification}:\ninstead of invalidating a row, fill some cells in this row making\nit new and odd. Bob consider that odd-ified rows in $B$-table are\ninvalid. This modification guarantees that each of the non-odd rows\nof $A$-table appears in $B$-table exactly once. Bob hires an additional\nassistant who guarantees that each odd row appears in $B$-table exactly\nonce. At each move, the additional assistant reserves some row exclusively\nand fills some cells in this row making it new and odd so that all\nodd rows are exhausted in the limit. Thus Bob wins this game, and\nthe theorem is proved.\n\n\\end{proof}\nKummer \\cite{Kum90a} gave a priority-free proof of the existence\nof a Friedberg numbering of $\\mathcal{P}^{\\left(1\\right)}$. The key\npoint of his proof is to split $\\mathcal{P}^{\\left(1\\right)}$ into\n$\\mathcal{P}^{\\left(1\\right)}\\setminus\\mathcal{O}$ and $\\mathcal{O}$,\nwhere $\\mathcal{O}$ is the set of all \\emph{odd} partial functions.\nObserve that $\\mathcal{P}^{\\left(1\\right)}\\setminus\\mathcal{O}$ has\na computable numbering, $\\mathcal{O}$ has a Friedberg numbering,\nand any finite subfunction of a partial function in $\\mathcal{P}^{\\left(1\\right)}\\setminus\\mathcal{O}$\nhas infinitely many extensions in $\\mathcal{O}$. He provided the\nfollowing useful criterion.\n\\begin{cor}[{Kummer \\cite[Extension Lemma]{Kum89}}]\n\\label{cor:Kum89-Extension-Lemma}Let $\\mathcal{A}$ and $\\mathcal{B}$\nbe disjoint subsets of $\\mathcal{P}^{\\left(1\\right)}$. If $\\mathcal{A}$\nhas a computable numbering, $\\mathcal{B}$ has a Friedberg numbering,\nand every finite subfunction of a member of $\\mathcal{A}$ has infinitely\nmany extensions in $\\mathcal{B}$, then $\\mathcal{A}\\cup\\mathcal{B}$\nhas a Friedberg numbering.\\end{cor}\n\\begin{proof}[Proof sketch]\nLet us play the game $\\mathcal{G}_{0}$ where Alice fills $A$-table\nwith the values of some computable numbering of $\\mathcal{P}^{\\left(1\\right)}$,\nand Bob uses the strategy which is obtained by modifying the winning\nstrategy of $\\mathcal{G}_{0}$ as follows. In this strategy, we do\nnot assume that odd rows never appear in $A$-table, and Bob does\nnot ignore cells in $A$-table. Assistants use partial functions in\n$\\mathcal{B}$ instead of odd partial functions. Replace odd-ification\nto \\emph{$\\mathcal{B}$-ification}: instead of odd-ificating a row,\nfill some cells in this row making it an unused member of $\\mathcal{B}$\nin the limit. The additional assistant guarantees that each member\nof $\\mathcal{B}$ appears in $B$-table exactly once. These actions\nare possible since $\\mathcal{B}$ has a Friedberg numbering. Then,\nBob wins, and $B$ becomes a Friedberg numbering of $\\mathcal{A}\\cup\\mathcal{B}$.\\end{proof}\n\\begin{cor}[{Pour-El and Putnam \\cite[Theorem 1]{PP65}}]\nLet $\\mathcal{A}$ be a subset of $\\mathcal{P}^{\\left(1\\right)}$\nand $f$ be a member of $\\mathcal{P}^{\\left(1\\right)}$ with an infinite\ndomain. If $\\mathcal{A}$ has a computable numbering, then there is\na subset $\\mathcal{B}$ of $\\mathcal{P}^{\\left(1\\right)}$ such that\n\\begin{enumerate}\n\\item $\\mathcal{A}\\subseteq\\mathcal{B}$,\n\\item the domain of every member of $\\mathcal{B}\\setminus\\mathcal{A}$ is\nfinite,\n\\item for any $g\\in\\mathcal{B}\\setminus\\mathcal{A}$, there is an $h\\in\\mathcal{A}$\nwith $g\\subseteq f\\cup h$,\n\\item $\\mathcal{B}$ has a Friedberg numbering.\n\\end{enumerate}\n\\end{cor}\n\\begin{proof}[Proof sketch]\nLet us play the game $\\mathcal{G}_{0}$ where Alice fills $A$-table\nwith the values of some computable numbering of $\\mathcal{P}^{\\left(1\\right)}$,\nand Bob uses the strategy which is obtained by modifying the winning\nstrategy of $\\mathcal{G}_{0}$ as follows. When some assistant fills\na cell in the $j$-th column making this row odd, she must use $f\\left(j\\right)$\nfor filling. The instruction for the additional assistant: \\textit{if\nthere is an odd row $i$ in $A$-table such that this row is not identical\nto any row in $B$-table, reserve a new row, copy the current contents\nof the $i$-th row of $A$-table into your reserved row, and release\nyour reserved row. }Released rows cannot be used forever. Note that\nthe additional assistant exceptionally does not ignore cells in $A$-table.\nThen, Bob wins, $\\mathcal{B}=\\set{B\\left(i,\\cdot\\right)|i\\in\\mathbb{N}}$\nhas the desired properties, and $B$ becomes a Friedberg numbering\nof $\\mathcal{B}$.\n\\end{proof}\n\n\\section{\\label{sec:Modifications}Modifications}\n\\begin{thm}[{Pour-El \\cite[Theorem 2]{Pou64}}]\n\\label{thm:Pou64-Theorem2}There are two incomparable Friedberg numberings\nof $\\mathcal{P}^{\\left(1\\right)}$.\n\\end{thm}\nThe first proof is obtained from the proof of \\prettyref{thm:Fri58-Corollary-to-Theorem3}\nthrough modifying in the same way done by Pour-El.\n\\begin{proof}[Proof (asymmetric version)]\nWe consider the following game $\\mathcal{G}_{2}$:\n\\begin{description}\n\\item [{Players}] Alice, Bob.\n\\item [{Protocol}] FOR $s=0,1,2,\\ldots$:\n\n\nAlice announces a finite partial function $A_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\nBob announces a finite partial function $B_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\\item [{Collateral duties}] $A_{s}\\subseteq A_{s+1}$ and $B_{s}\\subseteq B_{s+1}$\nfor all $s\\in\\mathbb{N}$.\n\\item [{Winner}] Let $A=\\bigcup_{s\\in\\mathbb{N}}A_{s}$ and $B=\\bigcup_{s\\in\\mathbb{N}}B_{s}$.\nBob wins if\n\n\\begin{enumerate}\n\\item for each $i\\in\\mathbb{N}$, there is a $j\\in\\mathbb{N}$ such that\n$A\\left(i,\\cdot\\right)=B\\left(j,\\cdot\\right)$;\n\\item for any $i,j\\in\\mathbb{N}$, if $i\\neq j$, then $B\\left(i,\\cdot\\right)\\neq B\\left(j,\\cdot\\right)$;\n\\item for each $i\\in\\mathbb{N}$, if $A\\left(i,\\cdot\\right)$ is total,\nthen there is a $j\\in\\mathbb{N}$ such that $B\\left(A\\left(i,j\\right),\\cdot\\right)\\neq A\\left(j,\\cdot\\right)$.\n\\end{enumerate}\n\\end{description}\n\nSuppose that there is a computable winning strategy of $\\mathcal{G}_{2}$\nfor Bob. Let Alice fill $A$-table with the values of some Friedberg\nnumbering of $\\mathcal{P}^{\\left(1\\right)}$, and let Bob use some\ncomputable winning strategy. Then, $A$ is a Friedberg numbering of\n$\\mathcal{P}^{\\left(1\\right)}$, and $B$ is a Friedberg numbering\nof $\\mathcal{P}^{\\left(1\\right)}$ to which $A$ is not reducible.\nSince $A$ is minimal, $B$ is also not reducible to $A$.\n\n\nWe describe a computable winning strategy of $\\mathcal{G}_{2}$. Bob\nuses the winning strategy of $\\mathcal{G}_{0}$ described in the proof\nof \\prettyref{thm:Fri58-Corollary-to-Theorem3}, which guarantees\nthat the first two winning conditions are satisfied. For the third\nwinning condition, Bob adds the following instruction for the $i$-th\nassistant: \\textit{if the $i$-th row $i$-th column in $A$-table\nhas been filled, and you have reserved the $A\\left(i,i\\right)$-th\nrow, then odd-ify the $A\\left(i,i\\right)$-th row.} Each of these\ninstructions is done at most once because after doing this the corresponding\nrequirement is permanently satisfied. Hence they do not interrupt\nsatisfying the first two winning conditions. It remains to show that\nthe third winning condition is also satisfied. Suppose that $A\\left(i,\\cdot\\right)$\nis total. We can assume without loss of generality that $i$ is the\nleast index of $A\\left(i,\\cdot\\right)$, i.e., there is no $j<i$\nwith $A\\left(j,\\cdot\\right)=A\\left(i,\\cdot\\right)$. Since $A\\left(i,\\cdot\\right)$\nis not either odd or even, the $i$-th assistant has a permanently\nreserved row $j$. The additional instruction guarantees that $j\\neq A\\left(i,i\\right)$.\nBy the second winning condition, we have that $B\\left(A\\left(i,i\\right),\\cdot\\right)\\neq B\\left(j,\\cdot\\right)=A\\left(i,\\cdot\\right)$.\nThus Bob wins this game.\n\n\\end{proof}\nThe second proof is more symmetric and parameterized. In the second\nproof, we consider an infinite game where Alice constructs two partial\nfunctions $A$ and $R$, and Bob constructs two partial functions\n$B$ and $C$. The aim of Bob is to guarantee that each of $B$ and\n$C$ contains all the partial functions from $A$ exactly once, and\n$B$ and $C$ are not reducible to each other by any total function\nfrom $R$. A sketch of the winning strategy is as follows: let Bob\nfix a computable injective numbering $\\varphi$ of non-odd partial\nfunctions which will be used as a witness of incomparability of $B$\nand $C$ relative to $R$. The assistant responsible for $\\varphi_{2i}$\nguarantees that $R\\left(i,\\cdot\\right)$ is not a reduction function\nfrom $C$ to $B$. The assistant responsible for $\\varphi_{2i+1}$\nguarantees that $R\\left(i,\\cdot\\right)$ is not a reduction function\nfrom $B$ to $C$. We now describe the details.\n\\begin{proof}[Proof (symmetric version)]\nWe consider the following game $\\mathcal{G}_{3}$:\n\\begin{description}\n\\item [{Players}] Alice, Bob.\n\\item [{Protocol}] FOR $s=0,1,2,\\ldots$:\n\n\nAlice announces two finite partial functions $A_{s},R_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\nBob announces two finite partial functions $B_{s},C_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\\item [{Collateral duties}] $A_{s}\\subseteq A_{s+1}$, $R_{s}\\subseteq R_{s+1}$,\n$B_{s}\\subseteq B_{s+1}$ and $C_{s}\\subseteq C_{s+1}$ for all $s\\in\\mathbb{N}$.\n\\item [{Winner}] Let $A=\\bigcup_{s\\in\\mathbb{N}}A_{s}$, $R=\\bigcup_{s\\in\\mathbb{N}}R_{s}$,\n$B=\\bigcup_{s\\in\\mathbb{N}}B_{s}$ and $C=\\bigcup_{s\\in\\mathbb{N}}C_{s}$.\nBob wins if\n\n\\begin{enumerate}\n\\item for each $i\\in\\mathbb{N}$, there is a $j\\in\\mathbb{N}$ such that\n$A\\left(i,\\cdot\\right)=B\\left(j,\\cdot\\right)$;\n\\item for each $i\\in\\mathbb{N}$, there is a $j\\in\\mathbb{N}$ such that\n$A\\left(i,\\cdot\\right)=C\\left(j,\\cdot\\right)$;\n\\item for any $i,j\\in\\mathbb{N}$, if $i\\neq j$, then $B\\left(i,\\cdot\\right)\\neq B\\left(j,\\cdot\\right)$;\n\\item for any $i,j\\in\\mathbb{N}$, if $i\\neq j$, then $C\\left(i,\\cdot\\right)\\neq C\\left(j,\\cdot\\right)$;\n\\item for each $i\\in\\mathbb{N}$, if $R\\left(i,\\cdot\\right)$ is total,\nthen there is a $j\\in\\mathbb{N}$ such that $B\\left(R\\left(i,j\\right),\\cdot\\right)\\neq C\\left(j,\\cdot\\right)$;\n\\item for each $i\\in\\mathbb{N}$, if $R\\left(i,\\cdot\\right)$ is total,\nthen there is a $j\\in\\mathbb{N}$ such that $C\\left(R\\left(i,j\\right),\\cdot\\right)\\neq B\\left(j,\\cdot\\right)$.\n\\end{enumerate}\n\\end{description}\n\nFirst, to consider the constant functions separately, we modify the\nwinning strategy of $\\mathcal{G}_{0}$ described in the proof of \\prettyref{thm:Fri58-Corollary-to-Theorem3}.\nBob adds the following instruction for the $i$-th assistant: \\textit{let\n$k$ be the number of rows such that you have already odd-ified. If\nin the current state of $A$-table the first $k$ positions of the\n$i$-th row are constant, odd-ify your reserved row.} Next, Bob hires\na countable number of additional assistants who guarantee that each\nof the constant functions appears in $B$-table exactly once. At each\nmove, all assistants work one by one. The $i$-th additional assistant\nstarts working at move $i$. She can reserve a row in $B$-table exclusively,\nfill her reserved row, and odd-ify her reserved row. The instruction\nfor the $i$-th additional assistant: \\textit{if you have no reserved\nrow, reserve a new row. If you have a reserved row, fill your reserved\nrow so that this row becomes the constant function $i$ in the limit.}\nThe modified strategy is still a computable winning strategy of $\\mathcal{G}_{0}$.\n\n\nSecond, we describe a computable winning strategy of $\\mathcal{G}_{3}$.\nTo guarantee that the first four winning conditions are satisfied,\nBob uses two copies of the winning strategy of $\\mathcal{G}_{0}$\ndescribed in the previous paragraph. At each move, the strategies\nwork one by one. For the fifth winning condition, Bob adds the following\ninstruction for the $2i$-th additional assistant: \\textit{if you\nhave reserved the $j$-th row in $C$-table, the $i$-th row $j$-th\ncolumn in $R$-table has been filled, and you have reserved the $R\\left(i,j\\right)$-th\nrow in $B$-table, then odd-ify the $R\\left(i,j\\right)$-th row in\n$B$-table.} Symmetrically, for the sixth condition, Bob adds the\nfollowing instruction for the $2i+1$-st additional assistant: \\textit{if\nyou have reserved the $j$-th row in $B$-table, the $i$-th row $j$-th\ncolumn in $R$-table has been filled, and you have reserved the $R\\left(i,j\\right)$-th\nrow in $C$-table, then odd-ify the $R\\left(i,j\\right)$-th row in\n$C$-table.} Since each of these instructions is done at most once,\nthe first four winning conditions are still satisfied. We now prove\nthat the last two winning conditions are also satisfied. Suppose that\n$R\\left(i,\\cdot\\right)$ is total. The $2i$-th additional assistant\nhas a permanently reserved row $c$ in $C$-table. The additional\ninstruction guarantees that she does not permanently reserve the $R\\left(i,c\\right)$-th\nrow in $B$-table. Hence $B\\left(R\\left(i,c\\right),\\cdot\\right)$\nis not identical to the constant function $2i$. It follows that $B\\left(R\\left(i,c\\right),\\cdot\\right)\\neq C\\left(c,\\cdot\\right)$.\nSimilarly, we have that $C\\left(R\\left(i,b\\right),\\cdot\\right)\\neq B\\left(b,\\cdot\\right)$\nfor some $b$. Thus Bob wins this game.\n\n\\end{proof}\nThe following can be proved in a way similar to the second proof of\n\\prettyref{thm:Pou64-Theorem2}. It suffices to consider an infinite\ngame where Bob constructs a countable number of partial functions\n$B^{0},B^{1},B^{2},\\ldots$ simultaneously. To ensure finiteness of\nactions, Bob shall start constructing the $i$-th numbering $B^{i}$\nat move $i$.\n\\begin{cor}[{Khutoretskii \\cite[Corollary 2]{Khu69}}]\n\\label{cor:Khu69-Corollary2}There is a uniformly c.e. sequence of\npairwise incomparable Friedberg numberings of $\\mathcal{P}^{\\left(1\\right)}$.\n\\end{cor}\nWe can also obtain the following criterion.\n\\begin{cor}\n\\label{cor:another-Extension-Lemma}Let $\\mathcal{A}$, $\\mathcal{B}$\nand $\\mathcal{C}$ be disjoint subsets of $\\mathcal{P}^{\\left(1\\right)}$.\nIf $\\mathcal{A}$ has a computable numbering, $\\mathcal{B}$ and $\\mathcal{C}$\nhave Friedberg numberings, and every finite subfunction of a member\nof $\\mathcal{A}\\cup\\mathcal{C}$ has infinitely many extensions in\n$\\mathcal{B}$, then there is a uniformly c.e. sequence of pairwise\nincomparable Friedberg numberings of $\\mathcal{A}\\cup\\mathcal{B}\\cup\\mathcal{C}$.\n\\end{cor}\n\\prettyref{cor:Khu69-Corollary2} is an instance of \\prettyref{cor:another-Extension-Lemma}\nwhere $\\mathcal{B}$ is the set of all odd partial functions, $\\mathcal{C}$\nis the set of all constant functions, and $\\mathcal{A}$ is the complement\nof $\\mathcal{B}\\cup\\mathcal{C}$.\n\\begin{cor}[{Kummer \\cite[Lemma]{Kum90b}}]\n\\label{cor:Kum90b-Lemma}Let $\\mathcal{A}$ and $\\mathcal{B}$ be\ndisjoint subsets of $\\mathcal{P}^{\\left(1\\right)}$. If $\\mathcal{A}$\nand $\\mathcal{B}$ have Friedberg numberings, and every finite subfunction\nof a member of $\\mathcal{A}$ has infinitely many extensions in $\\mathcal{B}$,\nthen there is a uniformly c.e. sequence of pairwise incomparable Friedberg\nnumberings of $\\mathcal{A}\\cup\\mathcal{B}$.\\end{cor}\n\\begin{proof}\nFix a Friedberg numbering $\\nu$ of $\\mathcal{A}$. Apply \\prettyref{cor:another-Extension-Lemma}\nto $\\mathcal{A}'=\\set{\\nu\\left(2i\\right)|i\\in\\mathbb{N}}$, $\\mathcal{B}$\nand $\\mathcal{C}=\\set{\\nu\\left(2i+1\\right)|i\\in\\mathbb{N}}$.\n\\end{proof}\nConversely, we can prove \\prettyref{cor:another-Extension-Lemma}\nby using \\prettyref{cor:Kum90b-Lemma} as follows. Applying \\prettyref{cor:Kum89-Extension-Lemma}\nto $\\mathcal{A}$ and $\\mathcal{C}$, we have a Friedberg numbering\nof $\\mathcal{A}\\cup\\mathcal{C}$. By \\prettyref{cor:Kum90b-Lemma},\n$\\left(\\mathcal{A}\\cup\\mathcal{C}\\right)\\cup\\mathcal{B}$ has a uniformly\nc.e. sequence of pairwise incomparable Friedberg numberings.\n\\begin{cor}[{Kummer \\cite[Theorem 1]{Kum90b}}]\n\\label{cor:Kum90b-Theorem1}Let $\\mathcal{A}$ be a subset of $\\mathcal{P}^{\\left(1\\right)}$.\nIf $\\mathcal{A}$ has a Friedberg numbering, and every finite subfunction\nof a member of $\\mathcal{A}$ has infinitely many extensions in $\\mathcal{A}$,\nthen there is a uniformly c.e. sequence of pairwise incomparable Friedberg\nnumberings of $\\mathcal{A}$.\\end{cor}\n\\begin{proof}\nSee \\cite{Kum90b}.\n\\end{proof}\nIt is immediate from \\prettyref{cor:Khu69-Corollary2} that the Rogers\nsemilattice of $\\mathcal{P}^{\\left(1\\right)}$ has a countable antichain.\nNaturally, we can ask whether the Rogers semilattice of $\\mathcal{P}^{\\left(1\\right)}$\nhas a countable chain. This, in fact, is true. This was proved by\nKhutoretskii \\cite[Proof of Corollary 1]{Khu71}. Moreover, there\nis a uniformly c.e. independent sequence of Friedberg numberings of\n$\\mathcal{P}^{\\left(1\\right)}$. Consequently, the Rogers semilattice\nof $\\mathcal{P}^{\\left(1\\right)}$ is a universal countable partial\norder. We provide a game-theoretic proof of this theorem.\n\\begin{thm}\nThere is a uniformly c.e. independent sequence of Friedberg numberings\nof $\\mathcal{P}^{\\left(1\\right)}$.\\end{thm}\n\\begin{proof}\nLet us consider the following game $\\mathcal{G}_{4}$:\n\\begin{description}\n\\item [{Players}] Alice, Bob.\n\\item [{Protocol}] FOR $s=0,1,2,\\ldots$:\n\n\nAlice announces two finite partial functions $A_{s},R_{s}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}$.\n\n\nBob announces $s+1$ finite partial functions $B_{s}^{k}:\\mathbb{N}^{2}\\rightharpoonup\\mathbb{N}\\ \\left(k\\leq s\\right)$.\n\n\\item [{Collateral duties}] $A_{s}\\subseteq A_{s+1}$, $R_{s}\\subseteq R_{s+1}$\nand $B_{s}^{k}\\subseteq B_{s+1}^{k}\\ \\left(k\\leq s\\right)$ for all\n$s\\in\\mathbb{N}$.\n\\item [{Winner}] Let $A=\\bigcup_{s\\in\\mathbb{N}}A_{s}$, $R=\\bigcup_{s\\in\\mathbb{N}}R_{s}$\nand $B^{k}=\\bigcup_{k\\leq s}B_{s}^{k}\\ \\left(k\\in\\mathbb{N}\\right)$.\nBob wins if\n\n\\begin{enumerate}\n\\item for any $k\\in\\mathbb{N}$ and for each $i\\in\\mathbb{N}$, there is\na $j\\in\\mathbb{N}$ such that $A\\left(i,\\cdot\\right)=B^{k}\\left(j,\\cdot\\right)$;\n\\item for any $i,j,k\\in\\mathbb{N}$, if $i\\neq j$, then $B^{k}\\left(i,\\cdot\\right)\\neq B^{k}\\left(j,\\cdot\\right)$;\n\\item for any $k\\in\\mathbb{N}$ and for each $i\\in\\mathbb{N}$, if $R\\left(i,\\cdot\\right)$\nis total, then there is a $j\\in\\mathbb{N}$ such that $\\bigoplus_{l\\neq k}B^{l}\\left(R\\left(i,j\\right),\\cdot\\right)\\neq B^{k}\\left(j,\\cdot\\right)$.\n\\end{enumerate}\n\\end{description}\n\nLet us explain how Bob wins this game by an effective way. To guarantee\nthat the first two winning conditions are satisfied, Bob uses countable\ncopies of the winning strategy of $\\mathcal{G}_{0}$ described in\nthe second proof of \\prettyref{thm:Pou64-Theorem2}. At each move,\nthe strategies work one by one. The $i$-th strategy starts running\nat move $i$. For the third winning condition, Bob adds the following\ninstruction for the $\\braket{i,k}$-th additional assistant:\\textit{\nif you have reserved the $j$-th row in $B^{k}$-table, and the $i$-th\nrow $j$-th column in $R$-table has been filled, and, in addition,\nyou have reserved the $R\\left(i,j\\right)$-th row in $\\bigoplus_{l\\neq k}B^{l}$-table,\nodd-ify the $R\\left(i,j\\right)$-th row in $\\bigoplus_{l\\neq k}B^{l}$-table.}\nEach instruction is done at most once. The first two winning conditions\nare satisfied. Finally, we prove that the third winning condition\nis also satisfied. Suppose that $R\\left(i,\\cdot\\right)$ is total.\nThe $\\braket{i,k}$-th additional assistant has a permanently reserved\nrow $j$ in $B^{k}$-table. Then, by the additional instruction, she\ndoes not permanently reserve the $R\\left(i,j\\right)$-th row in $\\bigoplus_{l\\neq k}B^{l}$-table,\nso $\\bigoplus_{l\\neq k}B^{l}\\left(R\\left(i,j\\right),\\cdot\\right)\\neq B^{k}\\left(j,\\cdot\\right)$.\nThus Bob wins this game.\n\n\\end{proof}\n\\begin{cor}\nIf $\\mathcal{A}$, $\\mathcal{B}$ and $\\mathcal{C}$ are subsets of\n$\\mathcal{P}^{\\left(1\\right)}$ satisfying the hypotheses of \\prettyref{cor:another-Extension-Lemma},\nthen there is a uniformly c.e. independent sequence of Friedberg numberings\nof $\\mathcal{A}\\cup\\mathcal{B}\\cup\\mathcal{C}$.\n\\end{cor}\n\n\\begin{cor}\nIf $\\mathcal{A}$ and $\\mathcal{B}$ are subsets of $\\mathcal{P}^{\\left(1\\right)}$\nsatisfying the hypotheses of \\prettyref{cor:Kum90b-Lemma}, then there\nis a uniformly c.e. independent sequence of Friedberg numberings of\n$\\mathcal{A}\\cup\\mathcal{B}$.\n\\end{cor}\n\n\\begin{cor}\nIf $\\mathcal{A}$ is a subset of $\\mathcal{P}^{\\left(1\\right)}$ satisfying\nthe hypotheses of \\prettyref{cor:Kum90b-Theorem1}, then there is\na uniformly c.e. independent sequence of Friedberg numberings of $\\mathcal{A}$.\n\\end{cor}\n\n\\section*{Acknowledgement}\n\nThe author is grateful to Alexander Shen for discussion and helpful\ncomments. He recommended that we consider the symmetric game $\\mathcal{G}_{3}$\ninstead of the asymmetric game $\\mathcal{G}_{2}$, and he gave the\nsketch of the winning strategy of $\\mathcal{G}_{3}$.\n\n\\bibliographystyle{amsplain-url}\n\\nocite{*}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe purpose of a visual system is to sample light in such a way as\nto provide an animal with actionable knowledge of its surroundings\nthat will permit it to survive and reproduce \\cite{ref19}. In most\ncases, this goal is achieved most effectively by a highly regular\ntwo-dimensional (2D) array of photoreceptors that evenly sample\nincoming light to produce an accurate representation of the visual\nscene. Classical sampling theory \\cite{ref1, ref20} as well as\nmore recent studies \\cite{ref2, ref3, ref21} have demonstrated\nthat the optimal arrangement of a 2D array of detectors is a\ntriangular lattice (i.e., a hexagonal array). Indeed, modeling\nstudies suggest that any deviation from a perfectly regular\narrangement of photoreceptors will cause deterioration in the\nquality of the image produced by a retina \\cite{ref4}.\nAccordingly, many species have evolved an optimal sampling\narrangement of their photoreceptors. For example, the insect\ncompound eye consists of a perfect hexagonal array of\nphotoreceptive ommatidia \\cite{ref5, ref6}. In addition, many\nteleost fish \\cite{ref7, ref8, ref9} and some reptiles\n\\cite{ref10} possess nearly crystalline arrangements of\nphotoreceptors. These and other examples attest that a perfect or\nnearly perfectly ordered arrangement of photoreceptors can be\nrealized in a biological system.\n\n\nDiurnal birds have one of the most sophisticated cone visual\nsystems of any vertebrate, consisting of four types of single cone\n(violet, blue, green and red) which mediate color vision and\ndouble cones involved in luminance detection \\cite{ref22, ref23,\nref24}; see Fig. \\ref{fig1_cone}. Despite the presence of numerous\nevolutionary specializations in the avian eye, the overall\narrangement of bird cone photoreceptors is not perfectly ordered\nbut rather is irregular \\cite{ref11, ref12}. The five avian cone\ntypes exist as five independent, spatial patterns, all embedded\nwithin a single monolayered epithelium. The individual cone\npatterns in the bird's retina are arranged such that cones of one\ntype almost never occur in the near vicinity of other cones of the\nsame type \\cite{ref12}. In this way, the bird achieves a much more\nuniform arrangement of each of the cone types than would exist in\na random (Poisson) pattern of points.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=12.5cm,keepaspectratio]{fig1.eps}\\\\\n\\end{center}\n\\caption{Spatial arrangements of chicken cone photoreceptors. The\nleftmost panel shows a flatmount preparation of a post-hatch day\n15 chicken retina with colored oil droplets within the inner\nsegments of the five cone photoreceptor types. Size bar = 10\n$\\mu$m. The additional panels (from left to right) depict the same\nfield of view under illumination with ultraviolet, blue and green\nlight, respectively. Oil droplet auto-fluorescence permits\nsub-type classification of the individual photoreceptor cells and\ndetermination of their spatial coordinates.  These figures derive\nfrom Ref. \\cite{ref12}.} \\label{fig1_cone}\n\\end{figure*}\n\nGiven the utility of the perfect triangular-lattice arrangement of\nphotoreceptors for vision \\cite{ref4}, the presence of disorder in\nthe spatial arrangement of avian cone patterns is puzzling. It is\ncrucial to ascertain whether the apparent ``disordered''\nphotoreceptor arrangements correspond to a different optimal\nsolution because of constraints, such as cell size polydispersity,\nthat are not present in the aforementioned insect retinas. By\nanalyzing the chicken cone photoreceptor system using a variety of\nsensitive microstructural descriptors that arise in statistical\nmechanics and particle-packing theory \\cite{ref27}, we show here\nthat the avian system possesses a remarkable type of correlated\ndisorder at large length scales known as hyperuniformity\n\\cite{ref13}, which has heretofore not been observed in a living\norganism. Moreover, in a departure from any known physical system,\nthe patterns of both the total population and the individual cell\ntypes are simultaneously hyperuniform, which we term {\\it\nmulti-hyperuniformity}. We subsequently determine effective\ninteractions between the photoreceptors on multiple length scales\nthat could possibly explain their unusual disordered hyperuniform\nstate. Specifically, we consider two types of interactions that\nhave competing effects in determining the degree of order in the\nsystem. Indeed, we show via computer simulations that the\nlocal-energy minimizing configurations of such a many-particle\ninteracting system quantitatively capture, with high accuracy, the\nunique spatial characteristics, including multi-hyperuniformity.\nThe fact that the aforementioned competing interactions lead to\ndisordered hyperuniform systems suggests that the photoreceptor\npatterns may represent the most uniform sampling arrangement\nattainable in the avian system due to intrinsic packing\nconstraints associated with the photoreceptor cells.\n\n\nThe rest of the paper is organized as follows: In Sec. II, we\nprovide definitions of fundamental concepts used in our analysis\nincluding various statistical microstructural descriptors, order\nmetrics as well as the concept of hyperuniformity. In Sec. III, we\nquantitatively investigate structural characteristics of avian\nphotoreceptor patterns containing multiple cell species and show\nthat both the overall pattern and the arrangements of individual\nspecies are hyperuniform. In Sec. IV, we determine the effective\ninteractions between the photoreceptors and devise a unique\nmultiscale packing model incorporating such interactions. We show\nthat our multiscale packing model can lead to point configurations\nthat are virtually indistinguishable from the actual photoreceptor\narrangements. In Sec. V, we provide concluding remarks.\n\n\\section{Definitions and fundamental concepts}\n\nBefore presenting our analysis of the avian photoreceptor system,\nwe first briefly review the ``hyperuniformity'' concept and its\nquantification, which plays a central role in this paper. In\naddition, we introduce the order metrics that will be employed to\ncharacterize the avian patterns.\n\n\\subsection{Hyperuniform Systems}\n\n\nThe ensemble-averaged structure factor of infinite point configurations\nin $d$-dimensional Euclidean space at number density $\\rho$ is defined via\n\\begin{equation}\n\\label{eq_Sk} S({\\bf k})= 1+ \\rho {\\tilde h}({\\bf k}),\n\\end{equation}\nwhere ${\\tilde h}({\\bf k})$ is the Fourier transform of the total\ncorrelation function $h({\\bf r}) = g_2({\\bf r})-1$ and $g_2({\\bf\nr})$ is the pair correlation function of the system. Note that definition (\\ref{eq_Sk}) implies  that the forward\nscattering contribution to the diffraction pattern is omitted. For statistically homogeneous\nand isotropic systems, the focus of this paper, $g_2$ depends on\nthe radial distance $r \\equiv |{\\bf r}|$ between the points (cell centers) as well\nas the number density $\\rho$. In two dimensions, the quantity $\\rho g_2(r)2\\pi r dr$\nis proportional to the conditional probability of finding a cell\ncenter at a distance between $r$ and $r+dr$ given that a cell\ncenter is at the origin, where $\\rho$ is\nthe number of cell centers per unit area.\n\n\nThe small-$k$ behavior of the structure factor $S(k)$ encodes information\nabout large-scale spatial correlations in the system and in the limit\n$k \\rightarrow 0$ determines whether the system is hyperuniform.\nSpecifically, an infinite point configuration in $d$-dimensional Euclidean space\nis {\\it hyperuniform} if\n\\begin{equation}\n\\label{eq_Sk2} \\lim_{k\\rightarrow 0}S(k) = 0,\n\\end{equation}\nwhich implies that the infinite-wavelength density fluctuations of the system\n(when appropriately scaled) vanish \\cite{ref13}.\n\nFor computational purposes, the structure factor $S({\\bf k})$ for\na given finite point configuration can be obtained directly from\nthe positions of the points ${\\bf r}_j$ \\cite{chase}, i.e.,\n\\begin{equation}\nS({\\bf k}) = \\frac{1}{N} \\left |{\\sum_{j=1}^N \\exp(i {\\bf k} \\cdot\n{\\bf r}_j)}\\right |^2 \\quad ({\\bf k} \\neq {\\bf 0}),\n\\label{coll}\n\\end{equation}\nwhere $N$ is the total number points in the system (under periodic boundary\nconditions) and ${\\bf k}$\nis the wave vector. Note that the forward scattering contribution (${\\bf k} = 0$)\nin (\\ref{coll}) is\nomitted, which makes relation (\\ref{coll})  completely consistent with the definition\n(\\ref{eq_Sk}) in the ergodic infinite-system limit. For statistically homogeneous\nand isotropic systems, the focus of this\npaper, the structure factor $S(k)$ only depends on the magnitude\nof the scalar wavenumber $k = |{\\bf k}| = 2\\pi n\/L$, where $n = 0,\n1, 2\\ldots$ and $L$ is the linear size of the system.\n\n\nA hyperuniform point configuration has the property that the\nvariance in the number of points in an observation window $\\Omega$\ngrows more slowly than the volume of that window \\cite{ref13}.  In\nthe case of a spherical observation window of radius $R$, this definition\nimplies that the local number variance $\\sigma^2(R)$ grows more\nslowly than $R^d$ in $d$ dimensions, The expression for the local number\nvariance of a statistically homogeneous point configuration in a\nspherical observation window is given exactly by\n\\begin{equation}\\label{numvar}\n\\sigma^2(R) = \\rho v(R) \\left[1+\\rho \\int_{\\mathbb{R}^d}\nh(\\mathbf{r}) \\alpha(\\mathbf{r}; R) d\\mathbf{r}\\right],\n\\end{equation}\nwhere $v(R)$ is the volume of a spherical window of radius $R$ and\n$\\alpha(r; R)$ is the \\emph{scaled intersection volume}, i.e.,\nthe intersection volume of two spheres of\nradius $R$ separated by a distance $r$ divided by the volume of\na sphere $v(R)$. We remark that the average number of points in an\nobservation window is $\\langle N(R)\\rangle = \\rho v(R)$ for any\nstatistically homogeneous point configuration.\n\nIt has been shown that the number variance \\eqref{numvar}, under\ncertain conditions, admits the following asymptotic scaling\n\\cite{ref13}:\n\\begin{equation}\\label{numasymp}\n\\sigma^2(R) = 2^d \\phi\\left\\{A\\left(\\frac{R}{D}\\right)^d +\nB\\left(\\frac{R}{D}\\right)^{d-1} + {\\cal\nO}\\left[\\left(\\frac{R}{D}\\right)^{d-3}\\right]\\right\\},\n\\end{equation}\nwhere\n\\begin{align}\nA &= 1+\\rho\\int_{\\mathbb{R}^d} h(\\mathbf{r}) d\\mathbf{r} =\n\\lim_{\\lVert\\mathbf{k}\\rVert\\rightarrow 0}\nS(\\mathbf{k})\\label{An},\n\\end{align}\nand $D$ is a characteristic microscopic length associated with the\npoint configuration (e.g., the average nearest-neighbor distance\nbetween the points). Clearly, when the coefficient $A = 0$, i.e.,\n$\\lim_{{\\bf k}\\rightarrow 0}S({\\bf k}) = 0$ satisfies the\nrequirements for hyperuniformity. The relation in \\eqref{An} then\nimplies that hyperuniform point patterns do not possess\ninfinite-wavelength density fluctuations (when appropriately\nscaled) and hence from (\\ref{numasymp}) the number variance scales\nas the surface area of the window for large $R$, i.e.,\n$\\sigma^2(R) \\sim R^{d-1}$ in the large-$R$ limit. This result is\nvalid for all periodic point patterns (including perfect\ncrystals), quasicrystals, and disordered systems in which the pair\ncorrelation function $g_2$ decays to unity exponentially fast\n\\cite{ref13}. The degree to which large-scale density fluctuations\nare suppressed enables one to rank order crystals, quasicrystals\nand special disordered systems \\cite{ref13, chase}. Disordered\nhyperuniform structures can be regarded as new states of\ndisordered matter in that they behave more like perfect crystals\nor quasicrystals in the manner in which they suppress density\nfluctuations on large length scales, and yet are also like liquids\nand glasses in that they are statistically isotropic structures\nwith no Bragg peaks. Thus, hyperuniform disordered materials\npossess a ``hidden order'' that is not apparent on short length\nscales.\n\nFor disordered hyperuniform systems with a total correlation\nfunction $h(r)$ that does not decay to zero exponentially fast,\nother dependencies of the number variance on $R$ may be observed.\nFor example, it is known that if $S(k) \\sim k$ for $k \\rightarrow\n0$ or, equivalently, if the total correlation function $h \\sim -\nr^{-(d+1)}$ for large $r$, then $\\sigma^2(R) \\sim (a_0\\ln R + a_1)\nR^{d-1}$. More generally, for any reciprocal power law,\n\\begin{equation}\nS(k) \\sim k^{\\alpha} \\quad (k \\rightarrow 0)\n\\end{equation}\nor, equivalently,\n\\begin{equation}\nh(r)\\sim-\\frac{1}{r^{d+\\alpha}} \\quad (r \\rightarrow +\\infty),\n\\end{equation}\none can observe a number of different kinds of dependencies of the\nasymptotic number variance $\\sigma^2$ on the window radius $R$ for\n$R \\rightarrow \\infty$ \\cite{ref13, chase, chase2}:\n\\begin{equation}\n\\label{eq_S0} \\sigma^2(R) \\sim \\left \\{\n\\begin{array}{c@{\\hspace{0.3cm}}c@{\\hspace{0.3cm}}c}\nR^{d-1}\\ln R, & \\alpha = 1, &\n\\\\ R^{d-\\alpha}, & \\alpha<1, &\n\\\\ R^{d-1}, & \\alpha>1, & \\end{array} \\right .\n\\end{equation}\nNote that in all cases, the number variance of a hyperuniform\npoint pattern grows more slowly than $R^d$.\n\n\\subsection{Order metrics}\n\nThe local bond-orientational-order metric $q_6$ is defined as\n\\cite{ref25}\n\\begin{equation}\nq_6 = \\left |{\\frac{1}{N_b}\\sum_j\\sum_k \\exp(6i\n\\theta_{jk})}\\right |,\n\\end{equation}\nwhere $j$ runs over all cells in the system, $k$ runs over all\nneighbors of cell $j$, $\\theta_{jk}$ is the angle between some\nfixed reference axis in the system and the bond connecting the\ncenters of cells $j$ and $k$, and $N_b$ is the total number of\nsuch bonds in the system. This quantity indicates the degree of\norientational order in the local arrangement of the immediate\nneighbors of a cell and it is maximized (i.e., $q_6=1$) for the\nperfect hexagonal arrangement.\n\nTo characterize translational order of a configuration, we use the\nfollowing translation order metric $T$ introduced in Ref.\n\\cite{order_T} and further applied in Ref. \\cite{ref26},\n\\begin{equation}\nT = \\frac{1}{\\eta_c}\\int_0^{\\eta_c}|g_2(r)-1|dr =\n\\frac{1}{\\eta_c}\\int_0^{\\eta_c}|h(r)|dr\n\\end{equation}\nwhere $g_2(r)$ is the pair correlation function, $h(r) = g_2(r)-1$\nis the total correlation function and $\\eta_c$ is a numerical\ncutoff determined by the linear size of the system. The\ntranslational order metric measures the deviation of the spatial\narrangement of cell centers in a pattern from that of a totally\ndisordered system (i.e., a Poisson distribution of points). The\ngreater the deviation from zero, the more ordered is the point\nconfiguration.\n\n\n\\section{Structural properties of experimentally obtained photoreceptor\npatterns}\n\nThe chicken retina contains five different cone  cell types of\ndifferent sizes: violet, blue, green, red and double. Each cell\ntype of this {\\it multicomponent} system is maximally sensitive to\nvisible light of a different wavelength. The spatial coordinates\nof each cell can be determined by the presence of a colored oil\ndroplet in the cell's inner segment (Fig. \\ref{fig1_cone}). Since\nthe oil droplets used to identify the locations of individual\nphotoreceptors are not always in exactly the same plane\n\\cite{ref12}, pairs of real photoreceptors sometimes appear to be\ncloser to one another than they are in actuality and in the\nsimulations. In addition, the original slightly curved retina\nepithelium was flattened for imaging purposes \\cite{ref12}. These\neffects introduce small errors in the intercell small-distance\nbehavior but do not affect the overall statistics, especially on\nlarge length scales. The spatial coordinate datasets of post-hatch\nday 15 chicken (Gallus gallus) cone photoreceptors were obtained\nfrom a published study \\cite{ref12}. Each dataset contains\napproximately 4430 photoreceptors, and the average numbers of\nviolet, blue, green, red and double species are respectively 350,\n590, 880, 670 and 1840. To clearly illustrate the photoreceptor\npatterns of different species, only a portion of the entire system\nis shown in Fig. \\ref{fig_cellpacking}. We compute a variety of\nthe associated statistical structural descriptors and order\nmetrics to quantify the degree of spatial regularity (or disorder)\nof the cell arrangements.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=10.5cm,keepaspectratio]{fig3.eps}\\\\\n\\end{center}\n\\caption{Experimentally obtained configurations representing the\nspatial arrangements of chicken cone photoreceptors. Upper panels:\nThe configurations shown from left to right respectively\ncorrespond to violet, blue, green species. Lower panels: The\nconfigurations shown from left to right respectively correspond to\nred, double species and the overall pattern.}\n\\label{fig_cellpacking}\n\\end{figure*}\n\n\n\\subsection{Disordered Hyperuniformity}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=10cm,keepaspectratio]{fig4.eps} \\\\\n\\end{center}\n\\caption{Structure factors $S(k)$ of the experimentally obtained\npoint configurations representing the spatial arrangements of\nchicken cone photoreceptors. The experimental data were obtained\nby averaging 14 independent patterns. The estimated values of\n$S(k=0)$ by extrapolation for violet, blue, green, red, double and\nthe overall population in the actual pattern are respectively\ngiven by $2.11\\times 10^{-3}$, $6.10\\times10^{-4}$,\n$1.06\\times10^{-3}$, $5.72\\times10^{-4}$, $1.38\\times10^{-4}$,\n$1.13\\times10^{-3}$.} \\label{fig_Sk}\n\\end{figure*}\n\n\nAs discussed in Sec. IIB, a point pattern is hyperuniform if the\nnumber variance $\\sigma^2(R)$ within a spherical sampling window\nof radius $R$ (in $d$ dimensions) grows more slowly than the\nwindow volume for large $R$, i.e., more slowly than $R^d$\n\\cite{ref13}. The property of hyperuniformity can also be\nascertained from the small wavenumber behavior of the structure\nfactor, i.e., $S(k=0)=0$ of the pattern \\cite{ref13}, which\nencodes information about large-scale spatial correlations (see\nSec. IIB for details). We find that $S(k)$ for the cell\nconfigurations associated with both the total population and the\nindividual photoreceptor species are hyperuniform and each of\nthese structure factors vanishes linearly with k as k tends to\nzero, i.e., $S(k) \\sim k$ ($k \\rightarrow 0$) (see Fig.\n\\ref{fig_Sk}). As discussed in Sec. IIB [cf. Eq. (\\ref{eq_S0})],\nsuch a linear behavior indicates a power-law decay for large $r$\nvalues in the pair correlation function (i.e., $g_2(r)-1 \\sim\n-1\/r^{3}$) instead of an exponential decay and therefore\nquasi-long-range correlations in the system. We will elaborate on\nthis point in the ensuing discussion.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=8.5cm,keepaspectratio]{fig5.eps} \\\\\n\\end{center}\n\\caption{The number variance $\\sigma^2(R)$ associated with the\nphotoreceptor patterns in chicken retina as well as the associated\nfitting function of the form $\\sigma^2(R) = A R^2 + B R\\ln(R) + C\nR$. We found that the values of the parameter $A$ are several\norders of magnitude smaller than the other two parameters,\nindicating that the associated patterns are effectively\nhyperuniform. Also shown in each plot is the ``surface term'' $C\nR$ for purposes of comparison. The window radius $R$ is normalized\nwith respect to the mean nearest neighbor distance $d_0$ of the\ncorresponding point configurations.} \\label{fig_sigma}\n\\end{figure*}\n\n\nWe have directly computed the number variance $\\sigma^2(R)$ for\nthe individual and overall patterns and verified that they are\nalso consistent with hyperuniformity, i.e., the ``volume term'' in\n$\\sigma^2(R)$ is several orders of magnitude smaller than the\nother terms [c.f. Eq.~\\eqref{numasymp}]. Specifically, for each\n$R$ value, 2500 windows are randomly placed in the system without\noverlapping the system boundary. The finite system size $L$\nimposes an upper limit on the largest window size, which is chosen\nto be $R_{max} = L\/2$ here. Figure \\ref{fig_sigma} shows the\nexperimental data as well as the associated fitting functions of\nthe form\n\\begin{equation}\n\\sigma^2(R) = A R^2 + B R\\ln(R) + C R,\n\\end{equation}\nwhere $A = S(k=0)$ and $B, C>0$. Note that in the plots, the\nwindow size $R$ is normalized by the corresponding\nnearest-neighbor distance $d_0$ for each species. Also shown in\neach plot is the corresponding ``surface term'' $C R$ for purposes\nof comparison. The numerical values of the fitting parameters for\nboth the overall pattern and the individual species are given in\nTable \\ref{tab_1}. It can be clearly seen that the values of the\nparameter $A$ are several orders of magnitude smaller than the\nother two parameters, indicating that the associated patterns are\neffectively hyperuniform. These values are also consistent with\nthe numerical values of $S(k=0)$ obtained by directly fitting\n$S(k)$ for small $k$ values \\cite{footnote0}.\n\n\\begin{table*}[h]\n\\caption{The numerical values of the fitting parameters for both\nthe overall pattern and the individual species.}\n\\begin{tabular}{c|@{\\hspace{0.35cm}}c@{\\hspace{0.35cm}}c@{\\hspace{0.35cm}}c@{\\hspace{0.35cm}}c@{\\hspace{0.35cm}}c@{\\hspace{0.35cm}}c}\n\\hline\n&  Violet & Blue &   Green  & Red & Double & Overall \\\\\n\\hline $A$ &  $2.53\\times 10^{-4}$ &   $9.24\\times 10^{-4}$ &\n$1.07\\times 10^{-3}$ & $1.77\\times 10^{-3}$ & $4.46\\times 10^{-3}$\n& $1.93\\times 10^{-3}$ \\\\\n$B$ &  0.203 &  0.198 &  0.169 &  0.146 &  0.122 &  0.127\\\\\n$C$ & 1.22 & 1.14 &  1.03 &  1.09  &  1.17  &   1.06 \\\\\n\\hline\n\\end{tabular}\n\\label{tab_1}\n\\end{table*}\n\nThe fact that the photoreceptor patterns display both overall\nhyperuniformity and homotypic hyperuniformity implies that if any\nsubpopulation of the individual species is removed from the\noverall population, the remaining pattern is still hyperuniform.\nWe term such patterns {\\it multi-hyperuniform} because distinct\nmultiple subsets of the overall point pattern are themselves\nhyperuniform. These are highly unusual and unique structural\nattributes. Until now, the property of \\textit{overall}\nhyperuniformity was identified only in a special subset of\ndisordered physical systems \\cite{ref15, ref16, ref18, berthier,\nweeks, aleksPRL, jiaoPRE, helium, plasma,universe, fermion,\nref17}. The chicken photoreceptor patterns provides the first\nexample of a disordered hyperuniform biological system. In\naddition, the photoreceptor patterns possess quasi-long-range\n(QLR) correlations as indicated by the linear small-$k$ behavior\nin $S(k)$. We will elaborate on these points in Sec. V.\n\n\\subsection{Pair Correlation Functions}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=10cm,keepaspectratio]{fig6.eps} \\\\\n\\end{center}\n\\caption{Pair correlation functions $g_2(r)$ of the experimentally\nobtained point configurations representing the spatial\narrangements of chicken cone photoreceptors. The experimental data\nwere obtained by averaging 14 independent patterns. The distance\nis rescaled by the average nearest neighbor distance $d_n$ in the\nsystem.} \\label{fig_g2}\n\\end{figure*}\n\nWe find that each cell is associated with an effective exclusion\nregion (i.e., an area in 2D) with respect to any other cells,\nregardless of the cell types. The size of these exclusion regions\nroughly corresponds to the size of the cells themselves\n\\cite{ref12}. In addition, cells belonging to the same subtype\n(i.e., like-cells) are found to be mutually separated from one\nanother almost as far as possible, leading to a larger effective\nexclusion region associated with like-cells of each species. The\nexclusion effects are quantitatively captured by the associated\npair-correlation functions (Fig. \\ref{fig_g2}). The hard-core\nexclusion effect is manifested in $g_2(r)$ as an interval of $r$\nfor which $g_2(r) = 0$ (i.e., an ``exclusion gap'') and $g_2(r)$\napproaches its large-$r$ asymptotic value of unity very quickly,\nindicating the absence of any long-range spatial ordering. This is\nto be contrasted with ordered systems, such as crystals, whose\npair correlation functions are composed of individual Dirac delta\nfunctions at specific $r$ values.\n\n\\subsection{Order Metrics}\n\n\\begin{table*}[h]\n\\caption{Bond-orientational and translational order metrics, $q_6$\nand $T$, respectively, of the chicken photoreceptor patterns. The\nexperimental data were obtained by averaging 14 independent\npatterns.}\n\\begin{tabular}{@{\\vrule height 10.5pt depth4pt  width0pt}c|c|c}\n\\hline\n~Species~ &  ~$q_6$~ & ~$T$~ \\\\\n\\hline\nViolet & ~0.150~  & ~0.304~ \\\\\nBlue & ~0.158~  & ~0.411~  \\\\\nGreen & ~0.130~  & ~0.278~  \\\\\nRed & ~0.147~  & ~0.254~  \\\\\nDouble & ~0.184~  & ~0.390~ \\\\\nAll & ~0.058~  & ~0.096~  \\\\\n\\hline\n\\end{tabular}\n\\label{tab_2}\n\\end{table*}\n\n\nA bond-orientational order metric $q_6$ \\cite{ref25} and a\ntranslational order metric $T$ \\cite{ref26} were  used next to\nquantify the degree of spatial regularity in the photoreceptor\npatterns (see Tab. \\ref{tab_2}), each of which are maximized by\nthe triangular lattice and minimized by a spatially uncorrelated\npoint pattern. Interestingly, the $q_6$ and $T$ values for the\ntotal population are close to the corresponding values for\npolydisperse hard-disk packings we obtained, implying that local\ncell exclusion effect plays a primary role in determining the\noverall pattern. In contrast, the higher $q_6$ and $T$ values for\nindividual cell species suggest that like-cells interact with one\nanother on a length scale larger than the size of a single cell,\nwhich tends to increase the degree of order in the arrangements of\nlike-cells.\n\nFrom a functional point of view, photoreceptor cells of a given\ntype maximize their sampling efficiency when arranged on an\nordered triangular lattice, as in the case of the compound eye of\ninsects \\cite{ref5, ref6}. Importantly, the triangular lattice has\nbeen shown to be the most hyperuniform pattern \\cite{ref13}, i.e.,\nit minimizes the large-scale density fluctuations among all 2D\npatterns. However, this most hyperuniform pattern may not be\nachieved if other constraints  (e.g., cell size polydispersity)\nare operable. We therefore hypothesize that the disordered\nhyperuniformity of avian photoreceptor patterns represents a\ncompromise between the tendency of the individual cell types to\nmaximize their spatial regularity and the countervailing effects\nof packing heterotypic cell types within a single epithelium,\nwhich inhibits the spatial regularity of the individual cell\ntypes. In other words, the avian photoreceptors are driven to\nachieve the most ``uniform'' spatial distribution subject to\nheterotypic cell packing constraints.\n\n\n\\section{Computational Model That Yields Multi-Hyperuniform Patterns}\n\nOur initial attempt to model the avian photoreceptor cell patterns\nemployed classic packing models of polydisperse hard disks that\nare driven to their ``jammed states'' \\cite{ref27}. However, these\nmodels failed to generate patterns with multi-hyperuniformity. Such standard\njamming models involving interactions on a single length scale are\ninsufficient to represent the two competing effects leading to the\nphotoreceptor patterns and motivated us to develop a unique\nmultiscale packing model as described below.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=10.5cm,keepaspectratio]{fig7.eps}\n\\end{center} \\caption{Illustration of the hard-core and soft-core\ninteractions in a two-species system containing black and red\ncells. The left panel shows the exclusion regions (circular disks\nwith two distinct sizes) associated with the two types of cells,\nwhich is proportional to the actual sizes of the cells. The black\ncells have a larger exclusion region than the red cells. The\nmiddle panel illustrates the soft-core repulsive interaction\n(large concentric overlapping circles of the solid black disks)\nbetween the black cells. Such a repulsive interaction will drive\nthe black cells to arrange themselves in a perfect triangular\nlattice in the absence of other species. The right panel\nillustrates the soft-core repulsive interaction (large concentric\noverlapping circles of the solid red disks) between the red\ncells.} \\label{fig_packing}\n\\end{figure*}\n\nIn the experimental data representing the spatial arrangements of\nchicken cone photoreceptors, each cell is represented by a point.\nWe refer to these points as ``cell centers'', although they may\nnot correspond to the actual geometrical centers of the cells.\n\nIn order to modify a simple hard-core interaction, we consider two\ntypes of effective cell-cell interactions: isotropic short-range\nhard-core repulsions between any pair of cells and isotropic\nlong-range soft-core repulsions between pairs of like-cells (i.e.,\ncells of the same subtype). The multiscale nature of the model\nresults from the multiple length scales involved in these\ninteractions for different species, as we discuss now. The\nstrength of the hard-core repulsion is characterized by the radius\n$R^i_h$ of a hard-disk exclusion region associated with a cell\ntype $i$. This interaction imposes a nonoverlap constraint such\nthat the distance between the cells $i$ and $j$ cannot be smaller\nthan $(R^i_h + R^j_h)$, which mimics the physical cell packing\nconstraint. In this regard, $R^i_h$ will also be referred to as\nthe radius of a cell $i$ in the ensuing discussions. The relative\nmagnitudes of $R^i_h$ are estimated from an electron micrograph\nshowing photoreceptor cell packing at the level of the inner\nsegment (see discussion below) \\cite{ref11}. The characteristic\nradius $R_s$ of the soft-core repulsion is associated with the\nmean nearest-neighbor distance of the cells of the same type.\nSpecifically, the pair potential between two like-cells is given\nby\n\\begin{equation}\n\\label{potential} E(r) =\n\\left\\{{\\begin{array}{c@{\\hspace{0.8cm}}c}\n\\displaystyle{\\frac{\\alpha}{\\beta+1}}(2R_s-r)^{\\beta+1} & r\\le\n2R_s,\n\\\\\\\\ 0 & r>2R_s, \\end{array}} \\right .\n\\end{equation}\nwhere the parameters $\\alpha>0$ and $\\beta>0$ set the scale of the\ninteraction energy \\cite{footnote2}. In our simulations, we\nrequire that value of $R_s$ be uniquely determined by the\nassociated cell number density $\\rho$, i.e., $R_s =\n\\frac{1}{2}\\sqrt{2\/(\\sqrt(3)\\rho)}$. This implies that a system\ncomposed of cells of the same type (i.e., a single-component\nsystem) interacting via a pair potential given by Eq.\n\\eqref{potential} at number density $\\rho$ (i.e., the number of\ncells per unit area) possesses the triangular-lattice ground\nstate, i.e., an arrangement associated with a minimal total energy\n(sum of the total interaction energy between any pairs of\nlike-cells). In other words, when the total energy in a\nsingle-component system is reduced to its minimal value (e.g.,\nzero), sufficiently slowly from an arbitrary initial\nconfiguration, the cells will reorganize themselves into a\ntriangular-lattice arrangement.\n\n\n\nWhen the system contains multiple species, the hard-core and\nsoft-core interactions represent two competing effects in\ndetermining the packing arrangement of the cells; see Fig.\n\\ref{fig_packing}. Specifically, the polydisperse hard-disk\nexclusion regions induce geometrical frustration in the packing,\ni.e., in this five-component system, it is not possible for the\nsubset of disks with the same size, surrounded by disks with\ndifferent sizes to be arranged on a perfect triangular lattice. On\nthe other hand, the long-range soft interaction between like\nspecies tends to drive the cells of the same type to arrange\nthemselves on a perfect triangular lattice. Note that although the\nrelative magnitudes of $R^i_h$ for different species (i.e., the\nratio between any two $R^i_h$) are fixed, the actual values of\n$R^i_h$ are variable and used as a tuning parameter in our model.\nAs stated above, the ratios between $R^i_h$ are estimated from a\npreviously published study \\cite{ref11}. Specifically, the\nrelative sizes of the violet, blue, green, red and double species\nare 1.00, 1.19, 1.13, 1.06 and 1.50, respectively. Given the\nnumber of cells of each species, the values of $R^i_h$ can be\nuniquely determined from the packing fraction $\\phi$ of the cells\n(i.e., the fraction of space covered by the cells) and vice versa,\n\\begin{equation}\n\\label{eq_phi}\n\\phi = \\displaystyle{\\frac{1}{A}\\sum_i N_i \\pi (R^i_h)^2},\n\\end{equation}\nwhere $N_i$ is the number of cells of species $i$ and $A$ is the\narea of the system.\n\nOur Monte Carlo algorithm, which involves iterating ``growth'' and ``relaxation'' steps,\nworks as follows:\n\n\\begin{itemize}\n\n\\item{(1) Initialization. In the beginning of the simulation, cell centers of each species are generated in a\nsimulation box using the random-sequential-addition (RSA) process\n\\cite{ref27}. Specifically, for each species $i$, $N_i$ cell\ncenters are randomly generated such that these cell centers are\nmutually separated by a minimal distance $\\mu R_s$ $(0<\\mu<1)$. In addition,\nthe newly added cell cannot overlap any existing cells in the box\n(determined by the hard-core radius $R_s$), regardless of cell types. The initial covering fraction $\\phi_I$\nassociated with the hard-core exclusion regions is determined by\n$R^i_h$ via Eq. (\\ref{eq_phi}), and is about $80\\%$ of the RSA saturation density \\cite{ref27}.}\n\n\n\\item{(2) Growth step. At each stage $n$, the cells are allowed to randomly move a prescribed\nmaximal distance ($\\sim 0.25 R^i_h$) and direction such that no\npairs of cells overlap. After a certain number ($\\approx$1,000) of\nsuch random movements for each cell, the radius $R^i_h$ of each\ncell is increased by a small amount such that the size ratios of\nthe cells remain the same. This leads to an increase of the\npacking fraction $\\phi_n$ at this stage by an amount of about $1\\% - 3\\%$.\nNote that in this ``growth'' step, the long-range\nsoft interactions between the like-cells are turned off.}\n\n\\item{(3) Relaxation step. At the end of the ``growth'' step,\nthe soft interactions are then turned on, and the cells are\nallowed to relax from their current positions to reduce the total\nsystem energy subject to the nonoverlap condition. The steepest\ndecent method is used to drive the system to the closest local\nenergy minimum (i.e., the inherent structure \\cite{ref27}) associated with the starting configuration. This is\nreferred to as the ``relaxation'' process.}\n\n\n\\item{(4) Statistics. After the relaxation process, structural statistics of the\nresulting configuration of cell centers are obtained and compared\nto the corresponding experimental data. To ensure that the simulations\nmatch the data for the pair statistics to the best extent possible,\nwe introduce a deviation metric $\\Delta$. Specifically, $\\Delta$ is the\nnormalized sums of the squared differences between the simulated and experimental $S(k)$\nand $g_2(r)$ associated with the simulated and actual patterns, i.e.,\n\\begin{equation}\n\\label{eq_delta}\n\\Delta = \\frac{1}{n_S}\\sum_i^{n_S}\\sum_r[g^{(i)}_2(r) - \\bar{g}^{(i)}_2(r)]^2\n +\\frac{1}{n_S}\\sum_i^{n_S}\\sum_k[S^{(i)}(k) - \\bar{S}^{(i)}(k)]^2,\n\\end{equation}\nwhere $n_S = 6$ is the total number of species including both the 5 individual species\nand the overall pattern, $g_2^{(i)}(r)$ and $S^{(i)}(k)$ are the simulated functions associated with\nspecies $i$, and $\\bar{g}_2^{(i)}(r)$ and $\\bar{S}^{(i)}(k)$ are the corresponding\nexperimentally measured functions.}\n\n\n\\item{(5) The growth and relaxation steps described in the bullet items (2) and (3),\nrespectively, are repeated until $\\phi_n$ reaches a prescribed\nvalue $\\phi_F$. Specifically, the configuration obtained by relaxation at stage $n$ is used\nas the starting point for the growth step at stage $n+1$. The best simulated pattern (i.e., that with the smallest\ndeviation metric $\\Delta_{min}$) and the associated $\\phi^*$ value are\nthen identified.}\n\n\\end{itemize}\n\nAt a given packing fraction $\\phi$ (or equivalently a set of\n$R^i_h$), the polydispersity of the exclusion regions associated\nwith different species and the resulting nonoverlap constraints\nfrustrate the spatial order in the system. For example, the\nlong-range soft interaction drives a single-species system to the\ntriangular-lattice arrangement in the absence of other species. On\nthe other hand, for any $\\phi >0$, it is impossible for cells of a\nparticular species, surrounded by cells of other species to sit on\na perfect triangular lattice \\cite{ref12}. Therefore, the\ndisordered point configurations obtained by minimizing the energy\nassociated with the soft repulsive interactions subject to the\nhard-core packing constraints are the local energy minima (i.e., inherent structures)\nof the system. The extent to which the structure deviates from that of a\nperfect triangular lattice (i.e., global energy minimum) is\ndetermined by the parameter $\\phi$ (or, equivalently, $R^i_h$).\nTherefore, by tuning this parameter in our algorithm, one can, in\nprinciple, generate a continuous spectrum of configurations of\ncell centers with varying degrees of spatial order (see Appendix).\nNote that in the limit $R^i_h \\rightarrow 0$, triangular-lattice\narrangements for individual species are accessible again and the\nresulting configuration is a superposition of five\ntriangular-lattice arrangements of the cell centers.\n\n\nWe note that the order of the aforementioned growth and relaxation\nsteps can be interchanged without affecting the final\nconfiguration. In addition, instead of starting from a disordered\nRSA arrangement of cell centers as described above, we have also\nused ordered initial configurations (i.e., superposition of\ntriangular-lattice arrangements), leading to the same\nconfiguration at a given number density $\\rho$. However, the\ninitial packing density $\\phi_I$ associated with ordered initial\nconfigurations is very low and thus, it is computationally\ninefficient to start from such initial configurations. By tuning\nthe ``strength'' of the hard-core interactions via the packing\nfraction associated with the exclusion regions, our multiscale\npacking model enables us to produce disordered point configuration\nwith various degrees of hyperuniformity, examples of which are\nprovided in the Appendix for a three-component system for\nillustrative purposes.\n\n\n\\subsection{Modeling Avian Photoreceptor System via Multiscale\nParticle Packing}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=11.5cm,keepaspectratio]{fig8.eps} \\\\\n\\end{center}\n\\caption{Left panel: The bond-orientational order metric $q_6$ of\nthe individual species as a function of the packing fraction\n$\\phi$ associated with the exclusion regions. Right panel: The\ntranslational order metric $T$ of the individual species as a\nfunction of the packing fraction $\\phi$ associated with the\nexclusion regions.} \\label{fig_order}\n\\end{figure*}\n\nBy using the multiscale packing model, we were able to accurately\nreproduce the unique features of the native avian photoreceptors.\nWe modeled the aforementioned two competing effects as two types\nof effective interactions between the cells: a long-range\nsoft-core repulsion between the cells of the same type (that would\nlead to an ordered triangular-lattice arrangement in the absence\nof packing constraints) and a short-range hard-core repulsion\n(with polydisperse exclusion regions associated with different\ncell species) between any pair of cells that frustrates spatial\nordering in the system. Given the sizes of the hard-core exclusion\nregions associated with each cell species (or equivalently the\npacking fraction $\\phi$ of the exclusion regions), the system is\nallowed to relax to a state that is a local energy minimum for the\nlong-range soft-core repulsive interactions between like-species.\nSuch long-range interactions would drive each of the five cell\nspecies in the multicomponent system to the associated\ntriangular-lattice arrangement (global energy minimum) in the\nabsence of the hard-core repulsions. As we increase the strength\nof the hard-core repulsions by increasing $\\phi$, the degree of\norder in the system, which is quantified by the order metrics\n$q_6$ and $T$, decreases (see Fig. \\ref{fig_order}). It is\nimportant to emphasize that these disordered hyperuniform avian\nphotoreceptor patterns are {\\it not} simple random perturbations\nof a triangular-lattice pattern. Statistically equivalent\ndisordered hyperuniform patterns have also been obtained from\ndisordered initial configurations (e.g., RSA packings). Thus, the\nunique structural features in these patterns are not attributed to\nparticular initial configurations but rather arise from the two\ncompeting effects, which are well captured by our multiscale\npacking model.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=10.5cm,keepaspectratio]{fig9.eps}\\\\\n\\end{center}\n\\caption{Simulated point configurations representing the spatial\narrangements of chicken cone photoreceptors. Upper panels: The\nconfigurations shown from left to right respectively correspond to\nviolet, blue, green species. Lower panels: The configurations\nshown from left to right respectively correspond to red, double\nspecies and the overall pattern. The simulated patterns for\nindividual photoreceptor species are virtually indistinguishable\nfrom the actual patterns obtained from experimental measurements.}\n\\label{fig_simupacking}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=10cm,keepaspectratio]{fig10.eps} \\\\\n\\end{center}\n\\caption{Comparison of the structure factors $S(k)$ of the\nexperimentally obtained and simulated point configurations\nrepresenting the spatial arrangements of chicken cone\nphotoreceptors. The simulation data were obtained by averaging 50\nindependent configurations.} \\label{fig_simuSk}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[height=10cm,keepaspectratio]{fig11.eps} \\\\\n\\end{center}\n\\caption{Comparison of the pair correlation functions $g_2(r)$ of\nthe experimentally obtained and simulated point configurations\nrepresenting the spatial arrangements of chicken cone\nphotoreceptors. The simulation data were obtained by averaging 50\nindependent configurations. The distance is rescaled by the\naverage nearest neighbor distance $d_n$ in the system.}\n\\label{fig_simug2}\n\\end{figure*}\n\n\n\\begin{table*}[h]\n\\caption{Comparison of the bond-orientational and translational\norder metrics, $q_6$ and $T$, of the experimental and simulated\npoint configurations. The simulation data were obtained by\naveraging 50 independent configurations.}\n\\begin{tabular}{@{\\vrule height 10.5pt depth4pt  width0pt}c|c|c|c|c }\n\\hline\n&  \\multicolumn{2}{|c|}{$q_6$} & \\multicolumn{2}{|c}{$T$} \\\\\n~Species~& ~Exp.~ & ~Sim.~ & ~Exp.~ & ~Sim.~ \\\\\n\\hline\nViolet & 0.150 &  0.148 & 0.304 &  0.327 \\\\\nBlue & 0.158 &  0.164 & 0.411  & 0.395 \\\\\nGreen & 0.130 &  0.134 & 0.278 &  0.266 \\\\\nRed & 0.147 &  0.149 & 0.254  & 0.263 \\\\\nDouble & 0.184 &  0.189 & 0.390 &  0.363\\\\\nAll & 0.058 &  0.063 & 0.096  & 0.108 \\\\\n\\hline\n\\end{tabular}\n\\label{tab_simu}\n\\end{table*}\n\n\n\nThe simulation box contains 2600 cell centers and the\nnumbers of violet, blue, green, red and double species are\nrespectively 210, 355, 530, 405, and 1100. The\nrelative sizes of the violet, blue, green, red and double species\nare 1.00, 1.19, 1.13, 1.06 and 1.50, respectively.\nThe initial packing fraction associated with the hard cores\nis $\\phi_I = 0.45$ and the simulation stops at $\\phi_F = 0.7$.\nAt $\\phi \\approx 0.58$, the resulting\nconfigurations (see Fig. \\ref{fig_simupacking}) are virtually\nindistinguishable from the actual photoreceptor patterns, as\nquantified using a variety of descriptors. Specifically, the\nassociated structure factors (see Fig. \\ref{fig_simuSk}) and pair\ncorrelation functions (see Fig. \\ref{fig_simug2}) match the\nexperimental data very well, as quantified by the minimum\ndeviation metric value of $\\Delta_{min} \\approx 0.4$ [c.f. Eq.(\\ref{eq_delta})].\nWe note that the major contributions to $\\Delta_{min}$ are the large\nfluctuations in the experimental data due to a limited number of samples.\n(The initial value of $\\Delta$ is roughly $3.16$.)\nThe order metrics $q_6$ and $T$ of the simulated pattern also match\nthose of the experimental data very well (see Tab. \\ref{tab_simu}).\nThis is a stringent test for the simulations to pass. The success\nof the simulations strongly suggests that the disordered\nhyperuniform photoreceptor patterns indeed arise from the\ncompetition between cell packing constraints and the tendency to\nmaximize the degree of regularity for efficient light sampling,\nsuggesting that the individual photoreceptor types are as uniform\nas they can be, given the packing constraints within the\nphotoreceptor epithelium.\n\n\n\\section{Conclusions and Discussion}\n\nBy analyzing the chicken cone photoreceptor patterns using a\nvariety of sensitive microstructural descriptors arising in\nstatistical mechanics and particle-packing theory, we found that\nthese disordered patterns display both overall and homotypic\nhyperuniformity, i.e., the system is multi-hyperuniform. This\nsingular property implies that if any subset of the individual\nspecies is removed from the overall population, the remaining\npattern is still hyperuniform. Importantly, it is highly\nnontrivial to devise an algorithm that would remove a large\nfraction of the points from a disordered hyperuniform system while\nleaving the remaining point pattern hyperuniform, and yet Nature\nhas found such a design.\n\nUntil now, the property of \\textit{overall} hyperuniformity was\nidentified only in a special subset of disordered physical\nsystems, including ground-state liquid helium \\cite{helium},\none-component plasmas \\cite{plasma}, Harrison-Zeldovich power\nspectrum of the density fluctuations of the early Universe\n\\cite{universe}, fermionic ground states \\cite{fermion}, classical\ndisordered ground states \\cite{ref17}, and maximally random jammed\npackings of equal-sized hard particles \\cite{aleksPRL, jiaoPRE}.\nAll of these examples involve single-component systems. More\nrecently, disordered multicomponent physical systems such as\nmaximally random jammed (MRJ) hard-particle packings \\cite{ref16,\nref18, ref15} have been identified that possess an appropriately\ngeneralized hyperuniformity property ascertained from the local\nvolume fraction fluctuations. However, the multicomponent\nphotoreceptor avian system pattern, which represents the first\nexample of a disordered hyperuniform system in a living organism,\nis singularly different from any of these hyperuniform physical\nsystems in that in the pattern each species and the total\npopulation are hyperuniform, i.e., the avian patterns are\nmulti-hyperuniform. Although it is not very difficult to construct\na overall hyperuniform system by superposing subsystems that are\nindividually hyperuniform, the reverse process (i.e., decomposing\na hyperuniform system into individually hyperuniform subsets) is\nhighly nontrivial. It will be of interest to identify other\ndisordered hyperuniform biological systems. It is likely that some\nother epithelial tissues and phyllotactic systems \\cite{ref27}\npossess such attributes. Interestingly, it has been shown that the\nlarge-scale  number-density fluctuations associated with the\nmalignant cells in brain tumors are significantly suppressed,\nalthough the cell patterns in such brain tumors are not\nhyperuniform \\cite{plos}.\n\n\nIn addition, the photoreceptor patterns possess quasi-long-range\n(QLR) correlations as indicated by the linear small-$k$ behavior\nin $S(k)$. Such QLR correlations are also observed in the\nground-state liquid helium \\cite{helium}, the density fluctuations\nof the early Universe \\cite{universe}, fermionic ground states\n\\cite{fermion} and MRJ packings of hard particles \\cite{ref16,\nref18, ref15}. In the MRJ particle packings, it is believe that\nthe QLR correlations arise from the competition between the\nrequirement of jamming and maximal disorder in the system\n\\cite{ref16, ref18, ref15}. As we showed employing the unique\nmultiscale packing model, the multicomponent avian system that is\nboth homotypic and overall hyperuniform, i.e., multi-hyperuniform,\ncan result from two competing interactions between the\nphotoreceptors.\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[width=10.5cm,keepaspectratio]{fig2.eps}\\\\\n\\end{center}\n\\caption{Left panel: A random-sequential-addition (RSA) packing of\nhard, identical circular disks in two-dimensions with a packing\nfraction $\\phi = 0.54$, which is close to the saturation state.\nRight panel: An equilibrium system of hard, identical disks at\n$\\phi = 0.54$. The fact that neither of these systems is\nhyperuniform, as discussed in the text, indicates that hard-core\nexclusion effects alone are not sufficient to induce\nhyperuniformity.} \\label{fig2_rsa}\n\\end{figure*}\n\nIt is noteworthy that while hard-core exclusion and high density\nin a disordered particle packing are necessary conditions to\nachieve a hyperuniform state, these are not sufficient conditions.\nFigure \\ref{fig2_rsa} shows a nonequilibrium\nrandom-sequential-addition (RSA) packing of hard circular disks in\ntwo-dimensions with a packing fraction $\\phi = 0.54$ (left panel),\nwhich is generated by randomly and sequentially placing hard disks\nin a domain without overlapping existing disks, until there is no\nroom for additional disks \\cite{rsa}. The right panel of Fig.\n\\ref{fig2_rsa} shows an equilibrium system of hard disks at $\\phi\n= 0.54$ (right panel). The structure factor values at $k=0$ for\nthe RSA and equilibrium systems are respectively given by $S(0)=\n0.059$ \\cite{rsa} and $S(0) = 0.063$ \\cite{salbook, newref43,\nfootnote}. Although hard-core exclusion play a central role in\nthese two distinct high-density packings, neither packing is\nhyperuniform, as indicated by the relatively large positive values\nof the corresponding $S(0)$.\n\n\n\n\nTo understand the origin of the unique spatial features of the\navian photoreceptor patterns, we have devised a unique multiscale\ncell packing model that suggests that photoreceptor types interact\nwith both short- and long-ranged repulsive forces and that the\nresultant competition between the types gives rise to the singular\ncell patterns. The fact that a disordered hyperuniform pattern\ncorresponds to a local optimum associated with the multiscale\npacking problem indicates that such a pattern may represent the\nmost uniform sampling arrangement attainable in the avian system,\ninstead of the theoretical optimal solution of a regular hexagonal\narray. Specifically, our studies show how fundamental physical\nconstraints can change the course of a biological optimization\nprocess. Although it is clear that physical cell packing\nconstraints are the likely cause of the short-range hard-core\nrepulsion, the origin of the effective longer-range soft-core\nrepulsion is less obvious. We hypothesize that repulsive forces of\nthis type occur during retinal development and may be secondary to\ncell-cell interactions during photoreceptor neurogenesis. However,\na comprehensive test of this hypothesis is beyond the scope of\nthis investigation, and therefore its resolution represents a\nfascinating avenue for future research.\n\n\nRecent studies have shown that disordered hyperuniform materials\ncan be created that possess unique optical properties, such as\nbeing ``stealthy'' (i.e., transparent to incident radiation at\ncertain wavelengths) \\cite{ref17}. Moreover, such disordered\nhyperuniform point patterns have been employed to design isotropic\ndisordered network materials that possess complete photonic band\ngaps (blocking all directions and polarizations of light)\ncomparable in size to those in photonic crystals \\cite{ref28,\nman13}. While the physics of these systems are not directly\nrelated to the avian photoreceptor patterns, such investigations\nand our present findings demonstrate that a class of disordered\nhyperuniform materials are endowed with novel photonic properties.\n\nBesides capturing the unusual structural features of photoreceptor\npatterns, our multiscale packing model represents a unique\nalgorithm that allows one to generate multi-hyperuniform\nmulticomponent systems with varying degrees of order by tuning the\npacking fraction $\\phi$ of the hard-core exclusion regions (see\nAppendix for additional examples). This knowledge could now be\nexploited to  produce multi-hyperuniform disordered structures for\napplications in condensed matter physics and materials science. For example, it would be of\ninterest to explore whether colloidal systems can be synthesized\nto have such repulsive interactions in order  to self assemble\ninto the aforementioned unique disordered arrangements and to study\nthe resulting optical properties. It is noteworthy that it has\nalready been demonstrated that three-dimensional disordered hyperuniform\npolymer networks can be fabricated for photonic applications using direct\nlaser writing \\cite{polymer}.\n\n\n\n\n\n\n\\begin{acknowledgments}\nThe authors are grateful to Paul Steinhardt for useful\ndiscussions. Y. J. and S. T. were supported by the National Cancer\nInstitute under Award NO. U54CA143803 and by the Division of\nMathematical Sciences at the National Science Foundation under\nAward No. DMS-1211087. J.C.C. was supported by NIH grants\n(EY018826, HG006346 and HG006790) and J.C.C., M.M.-H. and H.H.\nwere supported by a grant from the Human Frontier Science Program.\nH.H. also acknowledges the support of the German Research\nFoundation (DFG) within the Cluster of Excellence, ``Center for\nAdvancing Electronics Dresden''. This work was partially supported\nby a grant from the Simons Foundation (Grant No. 231015 to\nSalvatore Torquato).\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe Ap star $\\gamma$ Equ (HD 201601, BS 8097) is one of the brightest\nobjects of this class, with the apparent luminosity $V=4.66$ mag. The exact \nspectral type of this object\nis A9p (SrCrEu subclass). The magnetic field of $\\gamma$ Equ has been\nstudied for more than 50 years, starting from October 1946 (see Babcock\n1958). The longitudinal magnetic field $B_e$ of this star does not exhibit\nperiodic variations in time scales typical of stellar rotation,\n$0.5 - 30$ days. Such a variability of the $B_e$ field was observed in most\nAp stars. The above effect is commonly interpreted as the result of\nstellar rotation (oblique dipole model).\n\nThe first measurements by Babcock (1958) showed that the value of the\nlongitudinal magnetic field $B_e$ of $\\gamma$ Equ was positive in 1946--52,\nand approached nine hundred G. From that time on the value of $B_e$ slowly\ndecreased and even changed sign in 1970\/71. One could interpret the magnetic \nbehavior of $\\gamma$ Equ either as secular variations, or variations\ncaused by extremely slow rotation. If the latter picture is correct,\nthen the corresponding magnetic and rotational periods are in the range\nfrom 72 to 110 years (Bonsack \\& Pilachowski 1974; Leroy et al. 1994; \nBychkov \\& Shtol' 1997; Scholz et al. 1997).\n\nThe behavior of the $B_e$ field in $\\gamma$ Equ was investigated by many\nauthors in the second half of the twenty{\\sl th} century. For this research\nwe compiled $B_e$ observations published by Bonsack \\& Pilachowski (1974),\nScholz (1975; 1979), Borra \\& Landstreet (1980), Zverko et al. (1989), \nMathys (1991), Bychkov et al. (1991), Bychkov \\& Shtol' (1997), Scholz et\nal. (1997), Mathys \\& Hubrig (1997), Hildebrandt et al. (2000),\nLeone \\& Kurtz (2003) and Hubrig et al. (2004).\n\nWe included in this paper our unpublished magnetic $B_e$ measurements which\nwere obtained during the past seven years. All the new magnetic observations\nshowed, that the slow decrease of the $B_e$ field in $\\gamma$ Equ apparently\nreached the minimum in 1996--2002 and has actually started to increase. \n\nIn this paper we determined the accurate parameters of secular\nvariability of $\\gamma$ Equ: the period $P_{mag}$, the amplitude and the\ntime of zero phase for $B_e$ variations, which were approximated by a sine\nwave. We support the hypothesis that the long-term $B_e$ \nvariation in $\\gamma$ Equ is a periodic feature. Possible origin of this\nvariation cannot be uniquely determined, see discussion in \nSection ~\\ref{sec:discussion} of this paper.\n\n\\section{ Observations and data processing }\n\nWe have performed spectropolarimetric observations of Zeeman line splitting\nfor $\\gamma$ Equ at the Coude focus of the 1-m optical telescope (Special\nAstrophysical Observatory, Russian Academy of Sciences).\nZeeman spectra were obtained with the echelle spectrograph GECS (Musaev\n1996). We have put the achromatic analyser of circularly polarised light\nin front of the spectrometer slit. Images of the Zeeman echelle spectra\nwere recorded from CCD detectors in standard FITS format.\nFinal reduction of the archived spectra was performed with the standard\nMIDAS software (Monin 1999).\n\nEffects of instrumental polarisation on $B_e$ measurements obtained with\nthis instrument were investigated by Bychkov et al. (1998, 2000).\n\nTable~\\ref{tab:saores} presents the full set of our $B_e$ measurements\nof $\\gamma$ Equ (total 33 $B_e$ points). \nThe meaning of the first 3 columns is obvious. The fourth column\ngives the number $N$ of spectral lines which were\nused for the measurement of $B_e$ for a given exposure. \nTime length $\\Delta t$ of the exposure (in min) is given in the last column\nof Table~\\ref{tab:saores}.\n\nOn average, the value of a single $B_e$ number listed in \nTable~\\ref{tab:saores} was obtained after averaging of $B_e$ measurements\nobtained in 500-1300 spectral lines. Standard deviation $\\sigma_{B_e}$\nfor the resulting value of $B_e$ was computed in the standard manner as\nthe error of an arithmetic mean value.\n\nErrors $\\sigma_{B_e}$ determined in the above way reached rather low values\nin several observations listed in Table~\\ref{tab:saores}. In 2005\/2006\nwe plan to verify the reality of such $\\sigma_{B_e}$ by a special program\nof $B_e$ observations. Actually we accept these errors {\\sl bona fide}\nand note the following properties of our $B_e$ measurements.\n\nThe referee pointed out that a few pairs of $B_e$ measurements of one night\nin Table~\\ref{tab:saores} differ by only a few G, which is substantially\nless than the corresponding standard deviation $\\sigma_{B_e}$. \nWe can explain this only as a purely random effect, and do not see\nany reason for it either in the acquisition of observational data or\ntheir reduction.\n\nSecondly, series of measurements taken within a few nights generally \nshow a scatter of the order of 100 G, which is much higher than the \nstandard errors $\\sigma_{B_e}$ in Table~\\ref{tab:saores}. The latter are \nof the order of $20-30$ G, and such a discrepancy suggests that our\nstandard deviations are systematically underestimated, and are in fact\nof the order of 100 G. On the other hand,\nsuch a scatter of $\\approx$ 100 G is not inconsistent with the\nshort-term variability of light and the longitudinal magnetic field\n$B_e$ in $\\gamma$ Equ in time scales of minutes or above it. \n\nLeone \\& Kurtz (2003) recently discovered periodic variations of the\nlongitudinal magnetic field $B_e$ in $\\gamma$ Equ over the pulsation\nperiod of this star, $P_{puls} = 12.1$ min. The estimated amplitude\n$\\Delta B_e = 240$ G for this period, therefore, these variations \nat least can contribute to the scatter of our $B_e$ points collected\nin Table~\\ref{tab:saores}. \n\nStudy of the rapid periodic $B_e$ variations on a time scale of minutes \nwas also presented in Bychkov et al. (2005b) for $\\gamma$ Equ. They\ndid not found conclusive evidence of such variations above the noise\nlevel at $\\approx 240$ G.   \n\nWe also performed spectral analysis of the full set of 298 $B_e$ time\nseries from years 1946--2004. We concluded that there are no short-period\nfield variations with periods above ca. 1 day, but were not able to extend\nour analysis for shorter periods, see Section 4 of this paper.\n\n\n{\n\\newdimen\\digitwidth\n\\setbox0=\\hbox{\\rm0}\n\\digitwidth=\\wd0\n\\catcode`?=\\active\n\\def?{\\kern\\digitwidth}\n\n\\newcommand{\\displaystyle}{\\displaystyle}\n\\begin{table}\n\\caption{Measurements of $B_e$ in $\\gamma$ Equ (HD 201601). }\n\\label{tab:saores}\n\\renewcommand{\\arraystretch}{1.1}\n\\begin{tabular}{|l|r|c|r|c|}\n\\noalign{\\vskip 2 mm}\n\\hline\nJD \\hskip1mm 2400000.+  & $B_e$ (G) & $\\sigma_{B_e}$ (G)&\\hskip-2mm $N$ &\n\\hskip2mm $\\Delta t$ (min)\\\\\n\\hline\n49648.323   & --1045   & 21 & 706 & 30\\\\[-1.0pt]\n49648.345   & --1315   & 26 & 755 & 30\\\\[-1.0pt]\n49649.229   & --1463   & 37 & 576 & 30\\\\[-1.0pt]\n49649.257   & --1159   & 31 & 656 & 30\\\\[-1.0pt]\n49932.424   & --1317   & 26 & 691 & 60\\\\[-1.0pt]\n49932.469   & --1317   & 26 & 675 & 60\\\\[-1.0pt]\n49933.460   & --1316   & 26 & 700 & 60\\\\[-1.0pt]\n49933.507   & --1317   & 29 & 704 & 60\\\\[-1.0pt]\n50023.158   & --1291   & 22 & 501 & 40\\\\[-1.0pt]\n50023.189   & --1380   & 23 & 650 & 40\\\\[-1.0pt]\n50066.128   & --1539   & 26 & 718 & 40\\\\[-1.0pt]\n50066.157   & --1611   & 62 & 532 & 40\\\\[-1.0pt]\n51533.1229  & --1014   & 16 & 966 & 30\\\\[-1.0pt]\n51533.1451  & --1011   & 14 & 701 & 30\\\\[-1.0pt]\n51535.1847  & -- 902   & 16 & 955 & 40\\\\[-1.0pt]\n51535.2153  & -- 901   & 19 & 855 & 40\\\\[-1.0pt]\n51536.1069  & -- 670   & 18 & 821 & 30\\\\[-1.0pt]\n51536.1285  & -- 642   & 24 & 508 & 30\\\\[-1.0pt]\n51888.166   & --1069   & 18 & 847 & 30\\\\[-1.0pt]\n51888.190   & --1092   & 20 &1353 & 30\\\\[-1.0pt]\n51889.103   & -- 890   & 20 & 847 & 30\\\\[-1.0pt]\n51889.126   & -- 865   & 20 & 817 & 30\\\\[-1.0pt]\n51890.142   & -- 742   & 21 & 770 & 30\\\\[-1.0pt]\n52163.3000  & -- 845   & 19 & 833 & 30\\\\[-1.0pt]\n52163.3201  & -- 855   & 19 & 732 & 30\\\\[-1.0pt]\n52164.2861  & -- 956   & 16 & 947 & 30\\\\[-1.0pt]\n52164.3076  & -- 967   & 16 & 914 & 30\\\\[-1.0pt]\n52165.2812  & --1061   & 17 & 835 & 40\\\\[-1.0pt]\n52165.3111  & --1029   & 16 & 991 & 40\\\\[-1.0pt]\n52186.2229  & -- 922   & 17 &1085 & 30\\\\[-1.0pt]\n52186.2451  & -- 942   & 17 &1055 & 30\\\\[-1.0pt]\n52187.2673  & -- 882   & 16 &1072 & 30\\\\[-1.0pt]\n52188.2395  & -- 908   & 18 & 838 & 30\\\\\n\\hline\n\\end{tabular}\n\\end{table}  }\n\n\\section{ Magnetic period of $\\gamma$ Equ }\n\\label{sec:magnet}\n\nMagnetic observations presented in Table~\\ref{tab:saores} represent \ncompletely new data. They cover time span of ca. 7 years \nand include the\nphase when the effective magnetic field $B_e$ in $\\gamma$ Equ apparently\nreached its minimum value, and then the slow decrease of $B_e$ observed\nin the recent $\\approx$ 50 years has been reversed. This fact is of \nextraordinary importance, because it allows one for a fairly accurate \ndetermination of the magnetic period and the amplitude of $B_e$ variations\nin $\\gamma$ Equ.\n\nWe have compiled the set of 298 observations of the $B_e$ field in\n$\\gamma$ Equ, scattered in the literature, and appended our measurements.\nThese data cover the time period 1946--2004 (58 years). They are\ndisplayed in Fig.~\\ref{fig:long}. \nNote, that the $B_e$ measurements obtained by Babcock (1958) apparently\ncover the phase of the maximum longitudinal magnetic field in $\\gamma$ Equ.\n\nThe set of $B_e$ measurements analysed in this paper is rather \nheterogeneous. The data have been obtained by several different observers\nover a long time period using various instruments and techniques, and it\nis impossible to estimate or test credibly their systematic and random errors,\nparticularly for the earliest observations of the longitudinal magnetic\nfield in $\\gamma$ Equ.\n\nTherefore, we arbitrarily assumed that systematic errors of the $B_e$\nobservations are equal to zero. In other words, all the $B_e$ points for\n$\\gamma$ which were found in the literature are fully compatible.\n\nRandom errors of individual $B_e$ points frequently were given in the\nsource papers, and are denoted by vertical bars in Fig.~\\ref{fig:long}.\nThese errors\nwere not directly available for the earliest \nphotographic measurements by H.W. Babcock (1958) and Bonsack \\& Pilachowski \n(1974). We adopted here the estimated error for Babcock's data equal\n238 G, and 151 G for Bonsack \\& Pilachowski. These numbers were obtained \nin our thorough reanalysis of the earliest papers dealing with measurements\nof stellar magnetic fields, cf. Section 3.1 in Bychkov et al. (2003).\n\nDetermination of the period and other parameters of the apparent magnetic\nvariability for $\\gamma$ Equ was performed in the following manner.\nAssuming that the run of the observed longitudinal field $B_e$ with time\n$T$ can be approximated by a sine wave \n\\begin{equation}\n  B_e (T)=B_0+B_1 \\sin\\left[{2\\pi (T-T_0)\\over P}-{\\pi\\over 2}\\right] \\, ,\n\\label{equ:sigma1}\n\\end{equation}    \nwe determined all four parameters: the period $P$, the average field $B_0$,\nthe amplitude $B_1$ and the time of zero phase $T_0$\nusing the iterative technique of nonlinear fitting.\n\nStarting values of $P$, $B_0$, $B_1$, $T_0$ and their standard deviations\nwere found by our computer code for the nonlinear least squares method\n(Bychkov et al. 2003). \nThe final values and their errors were then computed with the public domain\ncode ``nlfit.f'', which is designed for curve and surface fitting with the \nLevenberg-Marquardt procedure ({\\sc ODRPACK v. 2.01} subroutines). The code\nis available at the site {\\tt www.netlib.org}.\n\n\nFitting of a sine wave to all the 298 $B_e$ points with errors as in\nFig.~\\ref{fig:long} gave very poor results with the $\\chi^2$ for\na single degree of freedom $\\chi^2\/\\nu = 18.0420$. Such fits are unacceptable,\nand in case of $\\gamma$ Equ the poor fit is the result of underestimated\nerrors of many $B_e$ points. Many $B_e$ observations presented in \nFig.~\\ref{fig:long} have very low errors, which sometimes are less than 20 G.\nOur new $B_e$ points, which are collected in Table~\\ref{tab:saores}, also\nare of such a high formal accuracy.\n\nWe cannot judge, whether an apparent scatter of $B_e$ points in\nFig.~\\ref{tab:saores} is due to unrealistic error estimates or the intrinsic\nshort-term variability of the longitudinal magnetic field in $\\gamma$ Equ.\nThe estimated random error of $B_e$ points about the starting sine wave\nequals to 213 G. For the final fitting of a sine we assumed that all the\n298 points have identical errors of 213 G. \n\nFinal\nvalues of the fitted parameters and their standard deviations $\\sigma$\nfor the sine phase curve are given below. \n\\halign {\\hskip 1 cm #\\hfil\\hskip1mm &#\\hfil \\cr\n\\noalign{\\vskip 5 mm}\n   $P_{mag}$  & = $33278 \\pm 1327$ days $= 91.1 \\pm 3.6$ years \\cr\n   $T_0 $     & = JD $2417795.0 \\pm 1057. $      \\cr\n   $B_0 $     & = $-\\, 262 \\pm 22.4 $ G          \\cr\n   $B_1 $     & = $+\\, 839  \\pm 22.1 $ G         \\cr\n   $r   $     & = $-\\, 0.524 \\pm 0.043$          \\cr\n\\noalign{\\vskip 5 mm}\n}\n\\noindent\nIn other words, a parameter range from $-\\sigma $ to $+\\sigma$ is just\nthe true 68\\% confidence interval for this parameter. \n\nThe above fit of a sine wave with uniform errors of 213 G is very good, with\n$\\chi^2\/\\nu = 1.0134$. The effect of inhomogeneity in the $B_e$ time series\nplus the possible existence of rapid magnetic variability in $\\gamma$ Equ\nwere compensated by the increase of the random error, and neither should\ninfluence the above parameters of secular magnetic variability in \n$\\gamma$ Equ.\n\nThe standard parameter $r$ was defined for the oblique rotator model\nof an Ap star. It is related to the angle $\\beta$ between the magnetic\ndipole axis and the rotational axis, and the angle $i$ between the rotational\naxis and the line of sight (Preston 1967):\n\\begin{equation}\nr = {{\\cos\\beta \\cos i - \\sin\\beta \\sin i} \\over\n     {\\cos\\beta \\cos i + \\sin\\beta \\sin i}} \n  = {B_e (\\min) \\over {B_e (\\max)}}  \\, .\n\\label{eqn:rrr}\n\\end{equation}\nParameters $B_e (\\min)$ and $B_e (\\max)$ of the $B_e$ sine wave for \n$\\gamma$ Equ are given by\n\\halign {\\hskip 1 cm #\\hfil\\hskip1mm &#\\hfil\\hskip1mm &# \\hfil \\cr\n\\noalign{\\vskip 2 mm}\n   $B_e({\\rm max}) $ & $=B_0+B_1$ & = $+\\,\\,\\, 577 \\pm 31.4$ G   \\cr\n   $B_e({\\rm min}) $ & $=B_0-B_1$ & = $ -1101 \\pm 31.4$ G        \\cr\n\\noalign{\\vskip 2 mm}\n}\nNote, that the meaning of $B_e({\\rm max}) $ and $B_e({\\rm min}) $ for\nuse in Eq.~\\ref{eqn:rrr} is different: $B_e({\\rm max}) $ denotes there\nthe value of magnetic intensity which has the higher absolute value, and\n$B_e({\\rm min})$ has the lower absolute value. In this way we obtained the\nvalue of $r$ for $\\gamma$ Equ equal to $r=577 \/ (-1101) = -0.524 $.\n\nBychkov et al. (2005a) presented an extensive catalog of the magnetic\nphase curves and their parameters for 136 stars on the main sequence and\nabove it. We quoted there the \npreviously estimated period for $\\gamma$ Equ, $P_{mag}=27027^d$,\nwhich was obtained on the basis of a shorter series of $B_e$ data.\nThis paper and the new, more accurate $P_{mag} = 33278^d$ represents a \nmajor revision of the previously known magnetic period of $\\gamma$ Equ.\n\n\\begin{figure}\n\\resizebox{\\hsize}{0.8\\hsize}{\\rotatebox{0}{\\includegraphics{fig1.eps}}}\n\\caption[]{The longitudinal magnetic field $B_e$ for $\\gamma$ Equ in years\n   1946--2004. }\n\\label{fig:long}\n\\end{figure}\n\n      \n\\section{Search for additional magnetic periods in $\\gamma$ Equ}\n\nSignificant scatter of the observed points in the long-term run of $B_e (T)$\nin Fig.~\\ref{fig:long} suggests the search for short-term periodicities.\nWe applied the strategy of prewhitening to the set of available $B_e$\nmeasurements, and removed the principal sine-wave variations from the data.\nPrewhitened data were then analysed with the method developed by Kurtz \n(1985), and with his Fortran code (Kurtz 2004). \n\nSuch a search for peaks in the $B_e$ amplitude spectrum of $\\gamma$ Equ\nin this paper was restricted to trial periods higher than 1 day. \nThis is because many of the earlier magnetic observations for this star \neither have poorly determined the time of measurement, or have \nlong times of exposure (see e.g. Babcock 1958). \nThe star $\\gamma$ Equ exhibits rapid nonradial pulsations and the\ncorresponding $B_e$ with the period $P_{mag}=12.1$ min (Leone \\& Kurtz\n2003) and, possibly, with simultaneous shorter periods \n(Bychkov et al. 2005b). None of them were analysed in this paper.\n\nWe have identified two additional periods of statistically low significance\nin the range $P_{mag} > 1^d$, see Fig.~\\ref{fig:short}:\n\n\\vskip 3 mm\n$P_1 = 348.07$ days, amplitude $=122$ G   \\par\n$P_2 =  23.44$ days, amplitude $=110$ G   \\par\n\\vskip 3 mm\n\n\\noindent\nBoth peaks in the amplitude spectrum in Fig.~\\ref{fig:short} exhibit low\nsignal to noise ratio, with noise level at ca. 80 G. The period $P_1$ \nis close to 1 year. Since most of the existing $B_e$ observations for $\\gamma$\nEqu were performed in months July-November, then the peak $P_1$ in the\namplitude spectrum represents a false period which most likely reflects the\naverage 1-year repetition time in the acquisition of the existing magnetic\nmeasurements.\n\nWe believe that the peak $P_2$ in the amplitude spectrum of the $B_e$ field\nof $\\gamma$ Equ is the random effect of a pure noise. The peak is very\nnarrow, in fact, it only appears in a single bin of a very dense discrete\nfrequency mesh.\n\nKurtz (1983) discussed the possible existence of the period of $\\approx 38$\ndays in his photometric observations of $\\gamma$ Equ\nin 1981. That period was of low probability, but  \npossibly could be identified with the real rotational period in\nthis star. We do not confirm the existence of the 38 day period in \nlong-term $B_e$ observations of $\\gamma$ Equ, see Fig.~\\ref{fig:short}.\n\n\n\n\\begin{figure}\n\\resizebox{\\hsize}{0.8\\hsize}{\\rotatebox{0}{\\includegraphics{fig2.eps}}}\n\\caption[]{Amplitude spectrum of the $B_e$ time series for \n   $\\gamma$ Equ, years 1946--2004. }\n\\label{fig:short}\n\\end{figure}\n\n\n\\section{ Discussion }\n\\label{sec:discussion}\n\nThere exist three possible explanations for the observed long-term behavior \nof the longitudinal magnetic field in $\\gamma$ Equ:\n\n\\vskip 3 mm\n\\noindent\n{\\bf 1.} Precession of the rotational axis (Lehmann 1987).   \\par\n\\noindent\n{\\bf 2.} Solar-like magnetic cycle (Krause \\& Scholz 1981),  \\par\n\\noindent\n{\\bf 3.} Rotation with the period of 91.2 years.   \n\\vskip 3 mm\n\nThe Ap star $\\gamma =$ HD 201601 in fact is a binary system. One can\nassume, that the gravitational force from the secondary companion can cause\nprecession of the Ap star. As the result, the angle between the rotational\naxis and the direction towards the Earth varies periodically. Therefore,\nchanges of the aspect can in principle cause apparent variations of the\nlongitudinal magnetic field $B_e$ or the amplitude of its variations.\n\nEffects of precession in long-period Ap stars were studied by Lehmann \n(1987), who showed that the oblateness of stars caused by the rotational\nor magnetic flattening is not adequate to produce observable precession \neffects. The only exception was 52 Her, where the observed behavior of \nthe star could be interpreted as a precessional motion. \n\nThe above considerations indicate that the precession theory does not\nconvincingly explain $B_e$ variations in this star. \n\nThe idea by Krause \\& Scholz (1981) that we actually observe the solar-like\nmagnetic cycle in $\\gamma$ Equ in which the global magnetic field\nreverses its polarity, cannot be easily verified by the existing \nobservations of the global longitudinal magnetic field $B_e$. Moreover,\none can note that such an idea requires the existence of a mechanism in the\ninterior of $\\gamma$ Equ which ensures the transfer of huge magnetic\nenergy into electric currents and vice versa. Note that the required\nefficiency of such a mechanism and the amplitude of magnetic field \nvariations in $\\gamma$ Equ is ca. four orders of magnitude larger than\nthat in the Sun in a similar timescale.\n\nFollowing the widely accepted picture of an Ap\nstar, we believe that the magnetic field of $\\gamma$\nEqu can be approximated by a dipole located in the center\nof the star. The dipole is inclined to the rotational axis of $\\gamma$ Equ.\nWe assume that the magnetic field is stable and remains frozen in the \ninterior of a rotating star at the time of observations, i.e. during at\nleast of 58 years. Therefore, slow variations of the $B_e$ field\nin $\\gamma$ Equ are caused by an extremely slow rotation, in which case our\n$P_{mag} = P_{rot} = 33278^d$. Such an explanation is supported to some \nextent by polarimetric measurements by Leroy et al. (1994).\n\nWe plan to perform high accuracy polarimetric \nmeasurements of $\\gamma$ Equ with the new version of MINIPOL. The device\nwas constructed to measure the angles and the degree of linear polarisation\nof stellar radiation, and will be operational at the Special\nAstrophysical Observatory in 2006. We also expect that shall be able to\nverify the extremely slow rotation of $\\gamma$ Equ measuring the rate of \nchange for the polarisation angle of stellar radiation.\n\n\n\\section{Summary }\n\nThe Ap star $\\gamma$ Equ (HD 201601) exhibited slow and systematic\ndecrease of the longitudinal magnetic field $B_e$ starting from 1946,\nwhen the global magnetic field of this star was discovered (Babcock 1958).\nWe have compiled the full set of 298 existing $B_e$ measurements, which\nconsists of the $B_e$ data published in the literature and our observations\nobtained during recent 7 years. The latter magnetic data (33 $B_e$ points)\nwere measured with the echelle spectrograph in the Coude focus of the 1-m\ntelescope at the Special Astrophysical Observatory.\nOur newest observations showed that the longitudinal magnetic field $B_e$\nof $\\gamma$ Equ reached its local minimum and started to rise in 1998-2004.\n\nAll the available data cover the time period of 58 years (1946-2004) and\ninclude both phases of the maximum and minimum $B_e$. \nAssuming that the secular variability of the $B_e$ field is a periodic\nfeature, we determined parameters of the magnetic field curve in\n$\\gamma$ Equ and give the value of its period, $P=91.1 \\pm 3.6$ years,\nwith the zero phase (maximum of $B_e$) at \n$T_0 =$ JD $2417795.0 \\pm 1057$. Sine-wave fit to the $B_e$ phase curve \nyields $B_e({\\rm max}) =+577 \\pm 31$ G and $B_e({\\rm min}) =-1101 \\pm 31$ G.\n\nSpectral analysis of the 58-year long $B_e$ time series essentially do not\nshow the existence of shorter periods, down to trial periods of $\\approx$\n1 day. More specifically, there are no real shorter periods in the run of the\nlongitudinal magnetic field $B_e$ with amplitudes exceeding the noise\nlevel of 80 G.\n\n\n\\section*{Acknowledgments}\n\nWe are grateful to John D. Landstreet, the referee, for his criticism\nand suggestions regarding our computations and the manuscript. \nWe thank Don Kurtz for providing his Fortran software used\nhere to compute the amplitude spectrum of $\\gamma$ Equ.  \nThis research was supported by the Polish Committee for Scientific\nResearch grant No. 1 P03D 001 26.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Appendix}\n\n\\renewcommand{\\thesubsection}{\\Alph{subsection}}\n\n\\subsection{Results under more attacks}\n\nIn order to verify the effectiveness of the proposed method, in this section, we further evaluate the robustness of our method under a broader range of powerful attacks: $1)$ AutoAttack~\\cite{croce2020reliable} (an ensemble of four strong diverse attacks, which is widely considered as the strongest attack for robustness evaluation), $2)$ CW attack~\\cite{carlini2017towards} (CW-200), $3)$ PGD attack with restart~\\cite{madry2018towards} (PGD-200), $4)$ One-pixel attack~\\cite{su2019one}, $5)$ Spatial Transformation attack~\\cite{xiao2018spatially}, as well as $6)$ Color Channel attack~\\cite{kantipudi2020color}. PGD-200 and CW-200 both restart 5 times with 40 optimization steps each restart.\n\nIn Table~\\ref{tab:other_attacks-1}, we report the robust accuracy under these attacks with AdvCL serving as baseline on CIFAR100.\nThe results show that our methods can improve robustness under all different attacks across almost all settings, \\textit{e.g.}, 21.43\\% vs. 19.57\\% under AutoAttack and 29.56\\% vs. 27.13\\% under PGD-200 attack, with loss function $\\mathcal{L}^{IP+HN}$ (Equation~\\ref{eq:ip+hn}), under Linear Probing.\n\n\\begin{table}[h]\n\\fontsize{8}{9}\\selectfont \n\\renewcommand\\arraystretch{1.5}\n\\centering\n\\caption{Robustness evaluation under diverse attacks on CIFAR100 with AdvCL as baseline.}\n\\vspace{5pt}\n\\label{tab:other_attacks-1}\n\\scalebox{0.88}{\n\\begin{tabular}{clcccccc}\n\\toprule\n\\multicolumn{2}{c}{\\begin{tabular}[c]{@{}c@{}} Training Methods\\end{tabular}} & PGD-200 & CW-200 & AA & One-pix. & Spatial-Tr. & Color-Ch. \\\\ \\hline\\hline\n\\multirow{4}{*}{Linear Probing} & AdvCL & 27.13 & 21.85 & 19.57 & 72.10 & 47.94 & 25.62 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & 27.87 & 22.10 & 19.80 & 69.60 & 49.31 & 25.88 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & 29.43 & 23.10 & 21.23 & \\textbf{73.20} & 51.57 & 28.01 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{29.56} & \\textbf{23.60} & \\textbf{21.43} & 73.00 & \\textbf{52.62} & \\textbf{28.94} \\\\ \\hline\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Linear\\\\ Finetuning\\end{tabular}} & AdvCL & 27.29 & 22.01 & 20.09 & \\textbf{72.80} & 47.31 & 24.98 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & 27.84 & 22.37 & 20.06 & 71.60 & 46.22 & 24.23 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & \\textbf{29.79} & \\textbf{23.79} & 21.52 & 70.80 & \\textbf{51.04} & \\textbf{27.84} \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & 29.58 & 23.64 & \\textbf{21.66} & 71.70 & 49.87 & 27.14 \\\\ \\hline\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Full\\\\ Finetuning\\end{tabular}} & AdvCL & 29.48 & 25.73 & 24.46 & \\textbf{72.20} & 57.86 & 25.12 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & 30.10 & 26.05 & 24.73 & 71.00 & 58.95 & 25.55 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & \\textbf{30.46} & \\textbf{26.60} & \\textbf{25.22} & 69.30 & 59.04 & \\textbf{26.02} \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{30.46} & 26.54 & 25.06 & 69.00 & \\textbf{59.33} & 25.70 \\\\ \\toprule\n\\end{tabular}}\n\\end{table}\n\nTable~\\ref{tab:other_attacks-2} provides results on CIFAR10 under canonical optimization-based attack methods: PGD-200, CW-200 and AutoAttack. Our methods also yield robustness gain in almost all settings. \n\n\\begin{table}[h]\n\\fontsize{8}{9}\\selectfont \n\\renewcommand\\arraystretch{1.5}\n\\centering\n\\vspace{10pt}\n\\caption{Robustness evaluation under optimization-based attacks on CIFAR10, with AdvCL as baseline.}\n\\vspace{5pt}\n\\label{tab:other_attacks-2}\n\\begin{tabular}{clccc}\n\\toprule\n\\multicolumn{2}{c}{\\begin{tabular}[c]{@{}c@{}}Training Methods\\end{tabular}} & PGD-200 & CW-200 & AutoAttack \\\\ \\hline\\hline\n\\multirow{4}{*}{Linear Probing} & AdvCL & 51.05 & 45.65 & 43.48 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & 51.99 & 46.02 & 43.57 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & \\textbf{52.36} & \\textbf{46.09} & \\textbf{43.68} \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & 52.01 & 45.35 & 42.92 \\\\ \\hline\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Linear\\\\ Finetuning\\end{tabular}} & AdvCL & 52.30 & 46.04 & 43.93 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & 52.77 & 46.60 & \\textbf{44.22} \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & \\textbf{53.22} & \\textbf{46.44} & 44.15 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & 52.77 & 45.55 & 43.01 \\\\ \\hline\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Full\\\\ Finetuning\\end{tabular}} & AdvCL & 52.90 & 50.92 & 49.58 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP}$ & \\textbf{53.61} & 51.25 & 49.90 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{HN}$ & 53.25 & 51.11 & 49.93 \\\\\n & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & 53.51 & \\textbf{51.46} & \\textbf{50.28} \\\\ \\toprule\n\\end{tabular}\n\\end{table}\n\\vspace{5pt}\n\nBesides, we also report results compared with RoCL under PGD-200, CW-200 and AutoAttack in Table~\\ref{tab:rocl-other_attacks}, which further validate the effectiveness of the proposed methods. For instance, 25.09\\% vs. 23.51\\% under CW-200 attack, Adversarial Full Finetuning scheme, on CIFAR100.\n\n\\begin{table}[h]\n\\fontsize{8}{9}\\selectfont \n\\renewcommand\\arraystretch{1.7}\n\\centering\n\\vspace{10pt}\n\\caption{Robustness evaluation under optimization-based attacks, with RoCL as baseline, on CIFAR-10 and CIFAR-100.}\n\\vspace{5pt}\n\\label{tab:rocl-other_attacks}\n\\begin{tabular}{cclccc}\n\\toprule\n\\multicolumn{1}{l}{Dataset} & \\multicolumn{2}{c}{\\begin{tabular}[c]{@{}c@{}}Training Methods\\end{tabular}} & PGD-200 & CW-200 & AutoAttack \\\\ \\hline\\hline\n\\multirow{6}{*}{CIFAR10} & \\multirow{2}{*}{Linear Probing} & RoCL & 32.47 & 33.33 & 24.11 \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{34.13} & \\textbf{34.59} & \\textbf{24.58} \\\\ \\cline{3-6} \n & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Linear\\\\ Finetuning\\end{tabular}} & RoCL & 42.58 & 40.21 & \\textbf{31.81} \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{43.54} & \\textbf{41.26} & 30.37 \\\\ \\cline{3-6} \n & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Full\\\\ Finetuning\\end{tabular}} & RoCL & 50.33 & 47.57 & 46.69 \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{51.47} & \\textbf{48.26} & \\textbf{47.05} \\\\ \\hline\n\\multirow{6}{*}{CIFAR100} & \\multirow{2}{*}{Linear Probing} & RoCL & 14.93 & 14.75 & 7.58 \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{17.95} & \\textbf{16.57} & \\textbf{8.58} \\\\ \\cline{3-6} \n & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Linear\\\\ Finetuning\\end{tabular}} & RoCL & 22.59 & 18.99 & \\textbf{11.93} \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{24.46} & \\textbf{20.69} & 11.69 \\\\ \\cline{3-6} \n & \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}Adversarial Full\\\\ Finetuning\\end{tabular}} & RoCL & 27.95 & 23.51 & 22.70 \\\\\n &  & \\ \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{29.37} & \\textbf{25.09} & \\textbf{24.01} \\\\ \\toprule\n\\end{tabular}\n\\end{table}\n\\vspace{5pt}\n\\end{document}\n\n\\section{Introduction}\n\n\nThough Deep Neural Networks (DNNs) have exhibited highly expressive performance and even surpass humans on many tasks, they are rigorously challenged by their vulnerability to \\textit{adversarial examples} \\cite{szegedy2014intriguing,goodfellow2014explaining}, which are artificially crafted to mislead the state-of-the-art models into incorrect predictions.\nDNN's weakness in the face of adversarial examples poses severe threats to their applications in security-critical systems, e.g. autonomous driving, surveillance and face recognition~\\cite{akhtar2018threat,kurakin2018adversarial,cao2019adversarial,vakhshiteh2020threat}.\nOne of the most powerful defense strategies against adversarial attacks is adversarial training (AT)~\\cite{kannan2018adversarial,shafahi2019adversarial,zhang2019you,wong2020fast,zhang2019theoretically,zhu2019freelb}, built upon a min-max optimization problem where the inner one constructs adversaries by maximizing the loss and the outer one fits the network's parameters to them by minimizing the expected loss~\\cite{madry2018towards}. AT is effective yet highly depends on labeled data~\\cite{chen2020adversarial,kim2020adversarial}.\n\n\nHow to leverage unlabeled data to perform AT is an important and worth-exploring problem~\\cite{gowal2020self,chen2020adversarial}.\nIn real-world practice, we can easily acquire huge amounts of domain-related unlabeled data, while it is often highly expensive to get them labeled.\nMoreover, to emphasize the value of unlabeled data, some recent works claim that robustness should not inherently require labels because we essentially ask predictors to be stable around naturally occurring inputs~\\cite{carmon2019unlabeled}, and they resort to semi-supervised learning to boost model's robustness \\cite{alayrac2019labels,carmon2019unlabeled}. \nNevertheless, these methods still require labeled images to generate pseudo-supervision for consequent adversarial training, and can be in essence regarded as fully-supervised adversarial training. How to train robust models totally from unlabeled data remains an important but less explored question.\nMost recently, some works attempted to combine contrastive and adversarial training to perform AT on unlabeled data. The main idea is to generate adversaries against the contrastive loss first, then maximize the similarity between clean views and their adversarial counterparts~\\cite{kim2020adversarial,fan2021does,jiang2020robust}.\nRoCL\\cite{kim2020adversarial} first proposed the above unlabeled AT scheme against contrastive loss and AdvCL\\cite{fan2021does} propose to minimize the learning task gap between unlabeled contrast and labeled fintuning by introducing pseudo-supervision in pretraining stage, achiving the state-of-the-art performance.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.41]{images\/intro\u56fe4.png}\n\\caption{\nThe schematic illustration for the conflict brought by adversarial contrastive learning and our proposed new views on adversaries to reduce such conflicts.\n}\n\\label{fig:intro}\n\\end{figure}\n\n\n\n\nAlthough existing \\textit{Adversarial Contrastive Learning} works have achieved satisfactory robustness in unlabeled settings, we argue that current simple\nextension from contrastive learning to adversarial\ndoesn't treat adversaries correctly and brings severe conflicts in pretraining objectives. The adversaries in ACL are generated against the contrastive loss, i.e., they are away from the anchor as far as possible and pushed close to other datapoints, contrary to the CL's objective. Conventional CL pushes away an anchor with other data points, when we introduce adversarial learning, the contrast between adversaries and their anchors attract the anchors towards other data points that the adversaries are close to, which is a totally opposite direction against CL's exertion from the geometry perspective, as shown in Figure \\ref{fig:intro}(b), engendering absolutely conflicts in training objective. Moreover, we will show by a limit case that current introduction of adversaries actually sets a potential objective to shrink all images into one point. The problem is rooted in that current adversarial contrastive learning treats adversaries the same as other clean augmentations and asks the anchors to head towards them.\n\nSo how to take adversaries correctly in adversarial contrastive learning? We present two new treatments for adversaries from the perspectives of both positives and negatives to alleviate this conflict. Firstly, as positives, we propose to view adversaries as \\textit{inferior positives} that have asymmetric similarity with normal positives (clean augmentations) to directly reduce the conflict, as shown in Figure~\\ref{fig:intro}(c). On the other hand, we propose to treat them as \\textit{hard negatives} that should be upweighted when pushed away from other data points to further help relieve the conflict and make the model more robustness-aware, as shown in Figure~\\ref{fig:intro}(d).\n\nWhen viewed as inferior positives, we propose that the similarity between clean views and adversarial views shouldn't be symmetric, i.e., we hope adversarial ones similar to clean ones and to be pulled back, but actually don't hope the model learns to represent clean images as adversaries which are intentionally perturbed. We put forward an adaptive gradient stopping strategy to model this asymmetry.\nMoreover, we argue that adversarial views have excellent intrinsic properties to become hard negatives in which case they should be upweighted to make the model more robustness-aware. As proposed in \\cite{robinson2020contrastive}, two principles to make a good negative sample for $x$ are: 1. \"True negative\" $x^-$ whose label differs from that of the anchor $x$; 2. The embedding currently believes to be similar to $x$.\nAdversaries, as stated above, are often close to some other data points that could have different labels, satisfying our desire for hard negatives perfectly, what we need to do is to distinguish each data point from its surrounding other classes' adversarial views. Here we combine ideas of positive-unlabeled learning~\\cite{du2014analysis,elkan2008learning} with adversarial and resort to the prevalent practice for reweighting negatives in contrastive learning \\cite{robinson2020contrastive,chuang2020debiased}, to effectively sample true adversarial negatives and reweight each sample according to the current similarity.\n\nTo sum up, our contributions are as follows: \n\\begin{itemize}\n    \\item We are the first to consider modeling the asymmetric property of \\textit{Adversarial Contrastive Learning} and propose to perform asymmetric gradient stopping, which can boost both standard accuracy and robustness.\n    \\item We provide a new perspective for adversaies in contrastive learning, i.e., view them as hard negatives, which can also be extended to other adversarial scenarios.\n    \\item We present a generalized asymmetric InfoNCE loss: A-InfoNCE that unifies all current contrastive learning methods and can integrate our two proposed new views conveniently.\n    \\item Our methods are compatible with current adversarial contrastive learning methods, outperform the chosen baselines and achieve new state-of-the-art performance.\n   \n\\end{itemize}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\nIn this work, we study enhancing model robustness using unlabeled data and investigate the \\textit{identity confusion} issue in Adversarial CL, \\textit{i.e.}, adversaries with different identities attract their anchors together, contradicting to the objective of CL. \nWe present a generic asymmetric objective \\textit{A-InfoNCE}, and treat adversaries discriminatingly as \\textit{inferior positives} or \\textit{hard negatives}, which can overcome the identify confusion challenge.\nComprehensive experiments with quantitative and qualitative analysis show that our methods can enhance existing Adversarial CL methods effectively.\nFurther, it lies in our future work to extend the proposed asymmetric form to other CL settings to take into consideration the asymmetric characteristics between different views.\n\n\\section*{Acknowledgement}\n\nThis work was supported in part by the National Key R\\&D Program of China under Grant 2021ZD0112100, partly by Baidu Inc. through Apollo-AIR Joint Research Center. We would also like to thank the anonymous reviewers for their insightful comments.\n\\section{Asymmetric InfoNCE}\n\n\\subsection{Notations}\n\n\n\\paragraph{Contrastive Learning (CL) }\n\nCL aims to learn generalizable features by maximizing agreement between self-created positive samples while contrasting to negative samples.\nIn typical contrastive learning, each instance $x$ will be randomly transformed into two views $(x_1, x_2)$, then fed into a feature encoder $f$ with parameters $\\theta$ to acquire normalized projected features, \\textit{i.e.}, $z_i = f(x_i;\\theta)$.\nLet $\\mathcal{P}(i)$ denote the set of positive views of $x_i$, containing the views transformed from $x$ with the same instance-level \\textit{identity} (\\textit{e.g.}, augmentations of the original image $x_i$); $\\mathcal{N}(i)$ denotes the set of negative views of $x_i$, containing all the views from other instances.\nThe conventional InfoNCE loss function~\\cite{oord2018representation} used in CL for a positive pair $(x_i,x_j)$ is defined as:\n\\begin{align}\n    \\mathcal{L}_{\\rm CL}(x_i,x_j) \n    =\n    - \\log \\frac\n    {\\exp({\\rm sim}(z_i, z_j)\/t)}\n    {\n        \\exp({\\rm sim}(z_i, z_j)\/t) + \n        \\sum_{k\\in \\mathcal{N}(i)} \\exp({\\rm sim}(z_i, z_k)\/t)\n    }\n   \n\\end{align}\nwhere $x_i$ serves as the anchor, ${\\rm sim}(z_i, z_j)$ denotes a similarity metric (\\textit{e.g.}, cosine similarity) between $z_i$ and $z_j$, and $t$ is a temperature parameter.\nThe final loss of the CL problem is averaged over all positive pairs of instances.\n\n\\paragraph{Adversarial CL}\nAdversarial CL can be regarded as an extension of CL by adding adversarial samples into the positive sets $\\mathcal{P}(\\cdot)$ to contrast. Adversarial CL is typically modeled as the following min-max optimization formulation to incorporate instance-wise attack~\\cite{madry2018towards,fan2021does}:\n\\begin{align}\n    \\min_\\theta \\mathbb{E}_{x\\in \\mathcal{X}} \\max_{||\\delta||_\\infty \\leq \\epsilon} \\sum_i\\sum_{j\\in \\mathcal{P}(i)}\n    \\mathcal{L}_{\\rm CL}(x_i, x_j),\\quad \\mathcal{P}(i)\\leftarrow \\mathcal{P}(i) \\cup \\{\\hat{x}_i+\\delta\\}\n\\end{align}\nwhere $\\hat{x}_i$ is the view of $x_i$ used to generate adversarial samples, $\\delta$ is the adversarial perturbation whose infinity norm is constrained as less than $\\epsilon$.\nIn the above formulation, the inner maximization problem constructs adversarial samples by maximizing the contrastive loss, and the outer minimization problem optimizes the expected worst-case loss w.r.t. the feature encoder $f$. \n\n\\subsection{Asymmetric InfoNCE: A Generic Learning Objective}\nCurrent Adversarial CL frameworks directly inherit CL's conventional contrastive loss (\\textit{e.g.}, InfoNCE) to evaluate the similarity between adversarial and clean views in a symmetric fashion. This can result in ineffective or even conflicting updates during CL training as aforementioned.\nTo address this challenge, we propose a generic Asymmetric InfoNCE loss (\\textit{A-InfoNCE}) to incorporate the asymmetric influences between different contrast instances, given by:\n\\begin{equation} \\label{eq:1}\n\\resizebox{\\textwidth}{!}{\n    $\\mathcal{L}^{\\rm asym}_{\\rm CL} (x_i, x_j;{\\alpha},\\lambda^p, \\lambda^n) \n    = \n    - \\log \\frac\n    {\\lambda^p_j \\cdot \\exp({\\rm sim^{\\alpha}}(z_i, z_j)\/t)}\n    {\\lambda^p_j \\cdot \\exp({\\rm sim^{\\alpha}}(z_i, z_j)\/t) + {\\sum_{k\\in \\mathcal{N}(i)} \\lambda^n_k\\cdot\\exp({\\rm sim^{\\alpha}}(z_i, z_k)\/t)}}\n   \n    $\n}\n\\end{equation}\nwhere $\\rm sim^{\\alpha}(\\cdot)$ is a generalized similarity metric that enables the incorporation of asymmetric relationships (a concrete instantiation is described in the next section); $\\lambda^p$ and $\\lambda^n$ are asymmetric weighting factors for positive and negative pairs, respectively.\nIt is worth noting that although A-InfoNCE is proposed to address the \\textit{identity confusion} issue in Adversarial CL, it can be easily extended to other CL settings when the asymmetric characteristics between different views need to be captured.\nA-InfoNCE can also generalized to many existing CL methods, for example, $\\mathcal{P}(i)$ and $\\mathcal{N}(i)$ can be altered to different choices of positive and negative views; ${\\rm sim^{\\alpha}}(z_i, z_j)$ is also changeable to a symmetric similarity metric for $z_i$ and $z_j$. $\\lambda^p$ and $\\lambda^n$ control the weights of different positive\/negative pairs. Generalization strategies are itemized below:\n\\begin{itemize}\n    \\item If ${\\rm sim^{\\alpha}}(z_i, z_j)$ is a symmetric similarity metric and $\\lambda^p, \\lambda^n = 1$, it degrades to the conventional InfoNCE loss used in CL~\\cite{pmlr-v119-chen20j}.\n    \\item If $\\mathcal{P}(i)$ is altered, it corresponds to positives sampling~\\cite{tian2020contrastive,bachman2019learning,tian2020makes}\n    . When we add adversaries into $\\mathcal{P}(i)$, it degenerates to the conventional Adversarial CL objectives, where $\\lambda^p, \\lambda^n = 1$ with symmetric ${\\rm sim^{\\alpha}}(z_i, z_j)$~\\cite{kim2020adversarial,jiang2020robust,fan2021does}.\n    \\item If we seek better $\\mathcal{N}(i)$, it echos negative sampling methods~\\cite{robinson2020contrastive,kalantidis2020hard} such as Moco~\\cite{he2020momentum}, which maintains a queue of consistent negatives; or mimics DCL~\\cite{chuang2020debiased} that debiases $\\mathcal{N}(i)$ into true negatives. \n    \\item If we change $\\lambda^p$ and $\\lambda^n$, it mirrors the pair reweighting works~\\cite{chuang2020debiased,robinson2020contrastive} that assign different weights to each pair according to a heuristic measure of importance such as similarity.\n\\end{itemize}\nWhile most existing methods adopt a symmetric similarity metric, we claim that in some scenarios the asymmetric similarity perspective needs to be taken into account, especially when the quality and property of different views vary significantly.\nIn this paper, we focus on the study of Adversarial CL, and demonstrate the benefits of \ncapturing the asymmetric relationships between adversaries and clean views.\nSpecifically, we design two instantiations to model the asymmetric relationships between adversarial and clean samples, as detailed in next section.\nBoth instantiations \ncan be integrated into the proposed \\textit{A-InfoNCE} framework.\n\n\n\n\n\n\n\\section{Related Work}\n\n\\paragraph{Contrastive Learning}\n\nCL has been widely applied to learn generalizable features from unlabeled data~\\cite{pmlr-v119-chen20j,he2020momentum,tian2020contrastive,grill2020bootstrap,chen2020improved,caron2020unsupervised,bachman2019learning,oord2018representation,chen2020big,chen2021large,khosla2020supervised}. The basic idea is instance discrimination~\\cite{wu2018unsupervised}.\nRepresentative works include CMC~\\cite{tian2020contrastive}, SimCLR\\cite{pmlr-v119-chen20j}\n, MoCo\\cite{he2020momentum}, SwAV\\cite{caron2020unsupervised}, BYOL\\cite{grill2020bootstrap}.\nThere is also a stream of work focusing on refined sampling on different views for improved performance ~\\cite{tian2020contrastive,kalantidis2020hard,chuang2020debiased,robinson2020contrastive,tao2021clustering}.\nFor example, DCL\\cite{chuang2020debiased} proposed to \\textit{debias} the assumption that all negative pairs are true negatives. HCL\\cite{robinson2020contrastive} extended DCL and proposed to mine hard negatives for contrastive learning, whose embeddings are uneasy to discriminate. \n\n\\paragraph{Adversarial Training}\n\nAdversarial training (AT) stems from \\cite{goodfellow2014explaining} and adopts a min-max training regime that optimizes the objective over adversaries generated by maximizing the loss~\\cite{madry2018towards,zhang2019theoretically,shafahi2019adversarial,zhang2019you,wong2020fast,zhu2019freelb,gan2020large,pang2019rethinking}. Some recent work introduced unlabeled data into AT~\\cite{hendrycks2019using,chen2020adversarial,carmon2019unlabeled,alayrac2019labels,kim2020adversarial}. By leveraging a large amount of unlabeled data, \\cite{carmon2019unlabeled,alayrac2019labels} performed semi-supervised self-training to first generate pseudo-supervisions, then conducted conventional supervised AT. Our work explores how to learn robust models without any class labels.\n\n\n\\paragraph{Adversarial Contrastive Learning}\n\nSome recent studies applied CL on adversarial training~\\cite{kim2020adversarial,jiang2020robust,fan2021does,gowal2020self}, by considering adversaries as positive views for contrasting, such that the learned encoder renders robust data representations. RoCL~\\cite{kim2020adversarial} was the first to successfully show robust models can be learned in an unsupervised manner.\nAdvCL~\\cite{fan2021does} proposed to empower CL with pseudo-supervision stimulus.\nSame as CL, these Adversarial CL methods perform symmetric contrast for all pairs, which could potentially induces conflicts in CL and AT training objectives. We are the first to investigate the asymmetric properties of Adversarial CL, by treating adversaries discriminatingly.\n\\section{Adversarial Asymmetric Contrastive Learning}\n\n\nThis section explains the instantiations of the \\textit{A-InfoNCE} loss for Adversarial CL. From the \\textit{inferior-positive} perspective, to reduce the impact of identity confusion, we first design a new asymmetric similarity metric ${\\rm sim^{\\alpha}}(z_i, z_j^{adv})$ for modeling the asymmetric relationships and weakening the learning signals from adversarial examples. From the \\textit{hard-negative} perspective, we view adversaries as hard negatives for other negative samples, and reweight each negative pairs by assigning similarity-dependent weights to ease the identity confusion. \n\n\n\\subsection{Adversarial Samples as Inferior Positives}\n\n\nAdversarial samples with different identities may attract their anchors (clean samples) in a contradicting manner to the exertion of CL. By weakening the learning signal from these adversarial examples in positive contrast (as \\textit{inferior positives} that attract the anchors less), we can effectively mitigate the undesired pull from clean samples via an adaptive gradient stopping strategy.\n\n\\subsubsection{Asymmetric Similarity Function.}\n\nAs the symmetric nature of InfoNCE can bring conflicts in Adversarial CL, we design a new asymmetric similarity function ${\\rm sim^{\\alpha}}(z_i, z_j)$ for \\textit{A-InfoNCE}, by manipulating the scale of gradient for each contrasted branch. We decompose it into two parts for each branch:\n\\begin{align} \\label{eq:2}\n    {\\rm sim^{\\alpha}}(z_i, z_j) \n    = \n    \\alpha \\cdot {\\rm \\overline{sim}}(z_i, z_j) \n    +\n    (1 - \\alpha) \\cdot {\\rm \\overline{sim}}(z_j, z_i)\n\\end{align}\nwhere ${\\overline{\\rm sim}(a, b)}$ means the one-sided similarity of $a$ to $b$, \\textit{i.e.}, when maximizing ${\\overline{\\rm sim}(a, b)}$, we freeze $b$ and only move $a$ towards $b$. This can be implemented by stopping the gradient back-propagation for $b$ and only optimizing $a$.\n\nWe use a hyperparameter $\\alpha$ to control how much $z_i$ and $z_j$ head towards each other. For a clean sample and an adversarial sample, we let $\\alpha$ denote the coefficient of the clean branch's movement. If $\\alpha$ is 0, it performs total gradient freezing on the clean branch and only adversarial representations are optimized through training. Our empirical analysis finds that $\\alpha$ is relatively easy to tune for boosted performance. We show that any value lower than 0.5 brings reasonable performance boost (see Figure~\\ref{fig:ablation1}), when clean samples move less towards adversaries, following the intrinsic asymmetric property of Adversarial CL.\n\n\\subsubsection{Adaptive $\\alpha$-annealing.}\n\n\nWhen the \\textit{identity confusion} is at play, it is necessary to treat adversarial samples inferior to ensure model robustness. But as training progresses, when model learns robust representations and the negative identity-changing impact of adversarial perturbation wanes, we consider adversarial perturbation as strong augmentations, equal to other typical transformations~\\cite{pmlr-v119-chen20j}. \n\nThe question is how to measure the reduction of instance confusion effect. Here we take a geometry perspective and propose to adaptively tune the proportional coefficient $\\alpha$ on-the-fly based on Euclidean distance. Let $d_{i,j} = ||z_i - z_j||_2$ denote the distance between an original image and its adversarial view in the representation space.\nGiven $\\alpha_{min}$, $d_{max}$, $\\alpha_{max}$, $d_{min}$,\nthe goal is for $\\alpha$ to be $\\alpha_{max}$ when the distance approximates $d_{min}$, and $\\alpha_{min}$ to be close to $d_{max}$. During training, we first compute the current representation distance $d$, then use a simple linear annealing strategy to compute $\\alpha$:\n\\begin{align}\n    \\alpha = \\alpha_{min} + (d_{max}-d)\\frac{\\alpha_{max}-\\alpha_{min}}{d_{max}-d_{min}}\n\\end{align}\n$d_{min}$ and $\\alpha_{min}$ can be treated as hyperparameters. $\\alpha_{max}$ is 0.5, indicating adversarial perturbation is equal to other transformations and ${\\rm sim}^\\alpha(z_i, z_j)$ degrades to the symmetric similarity. Moreover, we use the first $N$ epochs as a warm-up to compute the average distance as $d_{max}$, in which period $\\alpha$ is fixed.\n\n\\subsubsection{Adversarial CL Loss with Inferior Positives.}\nWith the above asymmetric similarity function $\\rm sim^\\alpha(\\cdot)$ and the \\textit{A-InfoNCE} loss function $\\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j;{\\alpha},\\lambda^p, \\lambda^n)$, the complete Adversarial CL loss with \\textit{inferior positives} (IP) can be written as: \n\\begin{small}\n\\begin{align} \\label{eq:ip}\n        \\mathcal{L}^{\\rm IP} \n    = \n        \\sum_i\\sum_{j\\in \\mathcal{P}(i)} \\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j; 0.5, 1, 1) \n    +\n        \\gamma \\cdot \\sum_i\\sum_{j\\in \\mathcal{P}(i)} \\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j^{adv}; \\alpha, 1, 1)\n\\end{align}\n\\end{small}\nwhere the first part stands for standard CL loss that maximizes the similarity between two clean views, which is symmetric ($\\alpha=0.5$) with $\\lambda^p = \\lambda^n = 1$, degrading to the conventional InfoNCE loss. The second part is a robust CL loss that maximizes the agreement between clean and adversarial views, but uses the asymmetric similarity function (\\ref{eq:2}) with a hyperparameter $\\alpha$ that gives weaker learning signals to the counterparts of inferior adversarial samples. The hyperparameter $\\gamma$ balances the robustness and accuracy objectives.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Adversarial Samples as Hard Negatives}\n\nBesides inferior positives, we also propose an alternative view of adversaries as \\textit{hard negatives}~\\cite{robinson2020contrastive} that be pushed away from surrounding data points with higher weights. This can potentially assuage the confusion brought by adversarial samples of the current instance residing too close to the negative samples of the same instance (as illustrated in Figure 1 (d)). Furthermore, this strategy encourages the model towards more robustness-aware, by giving adversarial samples possessing undiscriminating features higher weights in the pretraining stage, further enhancing Adversarial CL.\n\n\nIn practice, we assign a weight of similarity to each pair. To set a basis for weight assigning, we adopt a simple and adaptive weighting strategy used in~\\cite{robinson2020contrastive}, \\textit{i.e.}, taking each pair's similarity as its weight, with $w_{i,j} = \\exp({\\rm {sim}}(z_i, z_j)\/t)$. By doing so, the adversaries with bad instance-level identity (greater similarity to negative samples) can be automatically assigned with higher weights. The weights can adaptively decay as the instance identity recovers during training.\n\nHowever, as the commonly-used $\\mathcal{N}(i)$ is uniformly sampled from the entire data distribution $p(x)$~\\cite{chuang2020debiased} (\\textit{e.g.}, SimCLR~\\cite{pmlr-v119-chen20j} uses other instances in the current batch as negative samples), simply taking similarities as weights may heavily repel semantically-similar instances whose embeddings should be close. To estimate the true negatives distribution $p^-(x)$ , we take advantage of PU-learning~\\cite{du2014analysis,elkan2008learning} and resort to DCL,HCL~\\cite{chuang2020debiased,robinson2020contrastive} to debias negative sampling.\n\nPU-learning~\\cite{du2014analysis} decomposes the data distribution as: $p(x) = \\tau p^+ (x) + (1-\\tau) p^- (x)$, where $p^+(x), p^-(x)$ denote the distribution of data from the same or different class of $x$, and $\\tau$ is the class prior. Thus\n$p^{-}(x)$ can be rearranged as $p^{-}(x)=\\big(p(x) - \\tau p^+ (x)\\big)\/(1-\\tau)$. We can use all instances and positive augmentations containing adversarial samples of $x$ to estimate $p(x)$ and $p^+(x)$, respectively. Following~\\cite{chuang2020debiased}, we debias the negative contrast part in (\\ref{eq:1}) as:\n\\begin{small}\n\\begin{align}\n    \\frac{1}{1-\\tau}\n    \\Big(\n        \\sum_{k\\in \\mathcal{N}(i)} w_{i,k}^n \\cdot \\exp({\\rm sim^{\\alpha}}(z_i, z_k)\/t)\n    -\n        \\frac{N}{M} \\cdot \\tau \n        \\sum_{j\\in \\mathcal{P}(i)} w_{i,j}^p \\cdot \\exp({\\rm sim^{\\alpha}}(z_i, z_j)\/t)\n    \\Big)\n\\end{align}\n\\end{small}\nwhere $M, N$ are the numbers of postives and negatives, $w_{i,k}^n$ is the aforementioned weights for negatives, $w_{i,j}^p$ is a expandable weight for positives (set as 1 in our implementation, other choices can be further explored in the future work).\n\n\\subsubsection{Adversarial CL Loss with Hard Negatives.}\nWe substitute (7) into the \\textit{A-InfoNCE} loss function (\\ref{eq:1}) and rearrange it, acquiring the instantiation of \\textit{A-InfoNCE} loss with \\textit{hard negatives} (HN), with concrete forms of $\\lambda^p$ and $\\lambda^n$ as:\n\\begin{small}\n\\begin{align} \\label{eq:hn}\n        \\mathcal{L}^{HN} \n    = \n        \\sum_i\\sum_{j\\in \\mathcal{P}(i)}\n        \\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j; \\alpha, \\frac{M-(M+N)\\tau}{M-M\\tau}w_{i,j}^p, \\frac{1}{1-\\tau}w_{i,k}^n),\\quad k\\in\\mathcal{N}(i)\n\\end{align}\n\\end{small}\nDue to the lack of class information, we treat $\\tau$ as a hyperparameter and set as~\\cite{chuang2020debiased} suggested. \n\n\\subsubsection{Combined Adversarial CL Loss.}\nFinally, we can view adversaries both as inferior positives and hard negatives for other negative samples. This leads to following combined Adversarial CL loss:\n\\begin{small}\n\\begin{align} \\label{eq:ip+hn}\n        \\mathcal{L}^{IP+HN}\n    &=\n        \\sum_i\\sum_{j\\in \\mathcal{P}(i)} \\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j; 0.5, \\frac{M-(M+N)\\tau}{M-M\\tau}w_{i,j}^p, \\frac{1}{1-\\tau}w_{i,k}^n) \\ \n    + \\nonumber \\\\\n        & \\gamma \\cdot \\sum_i\\sum_{j\\in \\mathcal{P}(i)} \\mathcal{L}^{\\rm asym}_{\\rm CL}(x_i, x_j^{adv}; \\alpha, \\frac{M-(M+N)\\tau}{M-M\\tau}w_{i,j}^p, \\frac{1}{1-\\tau}w_{i,k}^n),\\quad k\\in\\mathcal{N}(i)\n\\end{align}\n\\end{small}\n\\section{Experiments}\n\nTo demonstrate the effectiveness and generalizability of the proposed approach, we present experimental results across different datasets and model training strategies. Our methods are compatible with existing Adversarial CL frameworks, and can be easily incorporated by replacing their CL loss. \nWe choose two baselines and replace their loss with $\\mathcal{L}^{IP}$(in Equation~\\ref{eq:ip}), $\\mathcal{L}^{HN}$(\\ref{eq:hn}) and $\\mathcal{L}^{IP+HN}$(\\ref{eq:ip+hn}) for evaluation.\n\n\\textbf{Datasets.} We mainly use CIFAR-10 and CIFAR-100 for our experiments. Each dataset has 50,000 images for training and 10,000 for test. STL-10 is also used for transferability experiments. Following previous work~\\cite{fan2021does}, we use ResNet-18~\\cite{he2016deep} as the encoder architecture in all experiments.\n\n\\setlength{\\tabcolsep}{0pt}\n\\setlength{\\arrayrulewidth}{0.2mm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[t]\n\\caption{Results for replacing the objectives of the two baselines with $\\mathcal{L}^{IP}$, $\\mathcal{L}^{HN}$ and $\\mathcal{L}^{IP+HN}$, in Standard Accuracy (SA) and Robust Accuracy (RA).\nThe pre-trained methods are evaluated under the Linear Probing (LP), Adversarial Linear Finetuning (ALF) and Adversarial Full Finetuning (AFF) strategies.\nSupervised methods are trained under conventional adversarial training scheme\n}\n\\centering\n\\fontsize{8.5}{9}\\selectfont \n\\scalebox{0.9}{\n\\begin{tabularx}{1\\textwidth}\n{  m{1.2cm}\n   m{1.6cm}\n   m{2.2cm}\n   \n   \n   \n   \n   \n   \n    P{1.1cm}\n    P{1.2cm}\n    P{1.1cm}\n    P{1.3cm}\n    P{1.1cm}\n    P{1.1cm}\n   \n   \n   \n   \n   \n   \n    P{0.5cm}\n    }\n\\toprule\n\\multicolumn{1}{l}{\\multirow{3}{*}{Dataset}} & \\multicolumn{2}{c}{\\multirow{3}{*}{\\makecell[c]{Pre-training \\\\ Methods}}} & \\multicolumn{6}{c}{Finetuning Strategies} \\\\ \\cmidrule(l){4-9} \n\\multicolumn{1}{l}{} & \\multicolumn{2}{c}{} & \\multicolumn{2}{c}{Linear Probing} & \\multicolumn{2}{l}{\\begin{tabular}[c]{@{}c@{}}Adversarial Linear\\\\ Finetuning\\end{tabular}} & \\multicolumn{2}{c}{\\begin{tabular}[c]{@{}c@{}}Adversarial Full\\\\ Finetuning\\end{tabular}} \\\\ \\cmidrule(l){4-9} \n\\multicolumn{1}{c}{} & \\multicolumn{2}{c}{} & \\multicolumn{1}{c}{SA} & \\multicolumn{1}{c}{RA} & \\multicolumn{1}{c}{SA} & \\multicolumn{1}{c}{RA} & \\multicolumn{1}{c}{SA} & \\multicolumn{1}{c}{RA} \\\\\n\\hline\\hline\n\\multirow{10}{*}{\\makecell[c]{CIFAR \\\\ 10}} & \\multirow{2}{*}{Supervised} & AT~\\cite{madry2018towards} &-&-&-&-& 78.99 & 47.41 & \\scriptsize{1}\\\\\n &  & TRADES~\\cite{zhang2019theoretically} &-&-&-&-& 81.00 & 53.27 & \\scriptsize{2} \\\\ \\cmidrule(l){2-9} \n & \\multirow{8}{*}{\\begin{tabular}[c]{@{}c@{}}Self-\\\\ Supervised \\end{tabular}} & RoCL~\\cite{kim2020adversarial} & 83.84 & 38.98 & 79.23 & 47.82 & 77.83 & 50.54 & \\scriptsize{3} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP}$ & \\textbf{87.63} & 41.46 & \\textbf{84.15} & 50.08 & 78.97 & 50.29 & \\scriptsize{4} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{HN}$ & 84.14 & 40.00 & 79.40 & 48.31 & 78.84 & 51.73 & \\scriptsize{5} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP+HN}$ & 85.69 & \\textbf{42.96} & 81.91 & \\textbf{50.90} & \\textbf{80.06} & \\textbf{52.95} & \\scriptsize{6} \\\\ \\cmidrule(l){3-9}\n &  & AdvCL~\\cite{fan2021does} & 81.35 & 51.00 & 79.24 & 52.38 & 83.67 & 53.35 & \\scriptsize{7} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP}$ & 82.37 & 52.33 & 80.05 & \\textbf{53.22} & \\textbf{84.12} & 53.56 & \\scriptsize{8} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{HN}$ & 81.34 & 52.61 & 78.69 & 53.20 & 83.44 & \\textbf{54.07} & \\scriptsize{9} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{83.15} & \\textbf{52.65} & \\textbf{80.41} & 53.19 & 83.93 & 53.74 & \\scriptsize{10} \\\\ \\midrule\n\\multirow{10}{*}{\\makecell[c]{CIFAR \\\\ 100}} & \\multirow{2}{*}{Supervised} & AT~\\cite{madry2018towards} & -&-&-&- & 49.49 & 23.00 & \\scriptsize{11} \\\\\n &  & TRADES~\\cite{zhang2019theoretically} & -&-&-&- & 54.59 & 28.43 & \\scriptsize{12} \\\\ \\cmidrule(l){2-9} \n & \\multirow{8}{*}{\\begin{tabular}[c]{@{}c@{}}Self-\\\\ Supervised\\end{tabular}} & RoCL~\\cite{kim2020adversarial} & 55.71 & 18.49 & 49.30 & 25.84 & 51.19 & 26.69 & \\scriptsize{13} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP}$ & 59.30 & 21.34 & 54.49 & \\textbf{30.33} & 52.39 & 27.84 & \\scriptsize{14} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{HN}$ & 58.77 & 21.17 & 56.38 & 28.03 & \\textbf{55.85} & 29.57 & \\scriptsize{15} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{59.74} & \\textbf{22.54} & \\textbf{57.57} & 29.22 & 55.79 & \\textbf{29.92} & \\scriptsize{16} \\\\ \\cmidrule(l){3-9}\n &  & AdvCL~\\cite{fan2021does} & 47.98 & 27.99 & \\textbf{47.45} & 28.29 & 57.87 & 29.48 & \\scriptsize{17} \\\\ \n &  & \\   \\ w\/ $\\mathcal{L}^{IP}$ & 49.48 & 28.84 & 45.39 & 28.40 & \\textbf{59.44} & 30.49 & \\scriptsize{18} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{HN}$ & 49.44 & 29.01 & 47.32 & \\textbf{28.69} & 58.41 & 29.93 & \\scriptsize{19} \\\\\n &  & \\   \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{50.59} & \\textbf{29.12} & 45.72 & 28.45 & 58.70 & \\textbf{30.66} & \\scriptsize{20} \\\\ \\bottomrule\n\\end{tabularx}}\n\\label{table:1}\n\\end{table}\n\n\\textbf{Baselines.} We compare with two baselines: RoCL~\\cite{kim2020adversarial}, the first method to combine CL and AL; and AdvCL~\\cite{fan2021does}, the current state-of-the-art framework. During experiments, we observe severe overfitting of AdvCL when training 1000 epochs (experiment setting in the original paper), with performance inferior to training for 400 epochs.\nThus, we pre-train 400 epochs on AdvCL at its best-performance setting. All other settings are the same as original papers except for some hyperparameter tuning. Our methods are also compatible with some recent work like SwARo~\\cite{wahed2022adversarial} and CLAF~\\cite{rahamim2022robustness}, by modeling the asymmetry between clean and adversarial views as aforementioned.\n\n\\textbf{Evaluation.}\nFollowing \\cite{jiang2020robust} and \\cite{fan2021does}, we adopt three finetuning strategies to evaluate the effectiveness of contrastive pre-training: $1)$ Linear Probing (LP): fix the encoder and train the linear classifier; $2)$ Adversarial Linear Finetuning (ALF): adversarially train the linear classifier; $3)$ Adversarial Full Finetuning (AFF): adversarially train the full model. We consider two evaluation metrics: $1)$ Standard Accuracy (SA): classification accuracy over clean images; $2)$ Robust Accuracy (RA): classification accuracy over adversaries via PGD-20 attacks~\\cite{madry2018towards}. \nRobustness evaluation under more diverse attacks is provided in the appendix.\n\n\n\\subsection{Main Results}\nIn Table \\ref{table:1}, we report standard accuracy and robust accuracy of each model, learned by different pre-training methods over CIFAR-10 and CIFAR-100. Following previous works~\\cite{kim2020adversarial,jiang2020robust,fan2021does} and common practice in contrastive learning~\\cite{pmlr-v119-chen20j,he2020momentum}, we first use unlabeled images in CIFAR-10\/-100 to pre-train, then introduce labels to finetune the model. As shown in Table \\ref{table:1}, our methods achieve noticeable performance improvement over baselines in almost all scenarios, when replacing the original loss with our proposed adversarial CL loss.\n\n\nIn comparison with RoCL, $\\mathcal{L}^{IP}$ brings significant performance boost on both standard and robust accuracy consistently across different training methods (row 4 vs. 3, row 14 vs. 13) (except for RA of AFF on CIFAR10).\nComparing to AdvCL,  $\\mathcal{L}^{IP}$  also brings noticeable margin (row 8 vs. 7, row 18 vs. 17). This can be attributed to that $\\mathcal{L}^{IP}$ aims to lower the priority of adversaries and prevent clean samples moving towards other instances, which results in better instance discrimination and improves clean~\\cite{wu2018unsupervised} and robust accuracy.\n$\\mathcal{L}^{HN}$ also yields substantial boost on robust and standard accuracy (\\textit{e.g.}, row 15 vs. 13).\nWe hypothesize this is due to that  $\\mathcal{L}^{HN}$ helps\nalert the model to adversarial samples by assigning higher weights for adversaries in negative contrast. \nWhen combined together, in most settings both standard and robust accuracy are further boosted, especially for Linear Probing. This is because directly mitigating the negative impact of \\textit{identity confusion} by $\\mathcal{L}^{IP}$ and helping adversarial get rid of false identities by $\\mathcal{L}^{HN}$ can complement each other, bringing further performance boost.\n\\subsection{Transferring Robust Features}\n\n\\setlength{\\tabcolsep}{0pt}\n\\setlength{\\arrayrulewidth}{0.2mm}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[t]\n\\fontsize{8.5}{10}\\selectfont  \n\\centering\n\\caption{Transferring results from CIFAR-10\/100 to STL-10, compared with AdvCL~\\cite{fan2021does}, evaluated in Standard accuracy (SA) and Robust accuracy (RA) across different finetuning methods with ResNet-18\n}\n\\scalebox{0.9}{\n\\begin{tabularx}{1\\textwidth}\n{\n   P{2cm}\n   m{2.4cm}\n   \n   \n   \n   \n   \n   \n    P{1.2cm}\n    P{1.3cm}\n    P{1.2cm}\n    P{1.4cm}\n    P{1.2cm}\n    P{1.3cm}\n}\n\\toprule\n\\multirow{3}{*}{Dataset} & \\multirow{3}{*}{\\makecell[c]{Pre-training \\\\ Methods}} & \\multicolumn{6}{c}{Finetuning Strategies} \\\\ \\cline{3-8} \n &  & \\multicolumn{2}{c}{Linear Probing} & \\multicolumn{2}{c}{\\makecell[c]{Adversarial Linear \\\\ Finetuning}} & \\multicolumn{2}{c}{\\makecell[c]{Adversarial Full \\\\ Finetuning}} \\\\ \\cline{3-8} \n &  & SA & RA & SA & RA & SA & RA \\\\\n \\hline\\hline\n\\multirow{4}{*}{\\makecell[c]{CIFAR10\\\\$\\downarrow$\\\\STL10}} & AdvCL~\\cite{fan2021does} & 64.45 & 37.25 & 60.86 & 38.84 & 67.89 & 38.78 \\\\\n & \\   \\ w\/ $\\mathcal{L}^{IP}$ & 64.83 & 37.30 & 61.95 & 38.90 & \\textbf{68.25} & 39.03 \\\\\n & \\   \\  w\/ $\\mathcal{L}^{HN}$ & 65.24 & \\textbf{38.18} & \\textbf{62.83} & \\textbf{39.70} & 67.88 & \\textbf{39.75} \\\\\n & \\   \\  w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{67.19} & 37.00 & 61.34 & 39.35 & 67.95 & 39.12 \\\\ \\hline\n\\multirow{4}{*}{\\makecell[c]{CIFAR100\\\\$\\downarrow$\\\\STL10}} & AdvCL~\\cite{fan2021does} & 52.28 & 30.01 & 49.84 & 32.14 & 63.13 & 35.24 \\\\\n & \\   \\ w\/ $\\mathcal{L}^{IP}$ & 52.65 & \\textbf{31.33} & 50.18 & 33.15 & 63.26 & \\textbf{35.34} \\\\\n & \\   \\ w\/ $\\mathcal{L}^{HN}$ & 51.88 & 31.29 & \\textbf{50.73} & \\textbf{33.62} & 62.91 & 34.88 \\\\\n & \\   \\ w\/ $\\mathcal{L}^{IP+HN}$ & \\textbf{53.41} & 31.30 & 51.10 & 33.23 & \\textbf{63.69} & 35.09 \\\\ \\bottomrule\n\\end{tabularx}}\n\\label{table:2}\n\\end{table}\n\nLearning robust features that are transferable is a main goal in self-supervised adversarial learning. It is of great significance if models pre-trained with a huge amount of unlabeled data possess good transferability by merely light-weight finetuning. For example, Linear Probing is often 10$\\times$ quicker than conventional adversarial training, with only a linear classifier trained.\n\nHere we evaluate the robust transferability of the proposed approach, by transfering CIFAR-10 and CIFAR-100 to STL-10, \\textit{i.e.}, use unlabeled images in CIFAR-10\/-100 to pretrain, then use STL-10 to finetune and evaluate the learned models. As shown in Table~\\ref{table:2}, our methods yield both clean and robust accuracy gains in most settings, up to 1.48\\% (33.62\\% vs. 32.14\\%) in robust accuracy and 2.74\\% (67.19\\% vs. 64.45\\%) in clean accuracy.\n\n\n\\subsection{Ablation studies}\n\nWe design a basic adversarial contrastive model, named CoreACL, to study the effect of each component in our proposed methods.\nCoreACL only contains the contrastive component with three positive views: two clean augmented views and one adversarial view of the original image.\n\n\\subsubsection{Fixed $\\alpha$ for Asymmetric Similarity Function.}\nWe first use fixed $\\alpha$ without adaptive annealing to explore the effectiveness of \\textit{inferior positives}. Figure~\\ref{fig:ablation1}\n\\setlength{\\intextsep}{12pt}\n\\begin{wrapfigure}[14]{r}{8.0cm}\n    \\centering\n    \\includegraphics[scale=0.27]{images\/ab2.png}\n    \\caption{Deep probing for asymmetric similarity function with different $\\alpha$.}\n    \\label{fig:ablation1}\n\\end{wrapfigure}\npresents the results with different $\\alpha$ values when training models for 200 epochs.\nRecall that $\\alpha$ represents the tendency of the clean sample heading towards the adversarial sample. $\\alpha < 0.5$ means clean samples move less toward the adversaries\n(vice versa for $\\alpha > 0.5$), and $\\alpha = 0.5$ degenerates to the original symmetric similarity function form. \n\n\nCompared with symmetric CoreACL ($\\alpha=0.5$), our approach achieves better robustness and accuracy when $\\alpha<0.5$ (adversarial examples are treated as \\textit{inferior positives}). Intriguingly, when $\\alpha=1.0$, the extreme case when only clean samples are attracted by adversaries, we observe the presence of a trivial solution~\\cite{chen2021exploring}, that is all images collapse into one point. This validates our observation that adversaries with false identities are indeed pulling their positives towards other instances in the positive contrasts, with the risk of drawing all samples together.\nIt is also worth noting that when $\\alpha < 0.2$, performance begins to drop, showing that a small but non-zero $\\alpha$ is the optimal setting empirically.\n\n\\subsubsection{Fixed $\\alpha$ vs. $\\alpha$-Annealing.}\nAs shown in Table 3, compared to CoreACL, fixed $\\alpha$ obtains higher clean accuracy (81.29\\% vs. 78.90\\%) but with no gain on robust accuracy. Adaptive annealing $\\alpha$ achieves both higher robust accuracy (50.24\\% vs. 51.27\\%) and better clean accuracy (79.46\\% vs. 78.90\\%).\n\n\\setlength{\\intextsep}{3pt} \n\\begin{wraptable}[12]{r}{5.5cm\n\t\\centering  \n\t\\fontsize{8.5}{9}\\selectfont  \n\t\\begin{threeparttable}\n\t\t\\caption{Ablation studies, evaluated in SA, RA and time cost. Trained for 400 epochs on 2 Tesla V100 GPUs.}\n\t\t\\label{tab:headings}  \n\t\t\\begin{tabular}\n        {\n        m{1.95cm}\n        P{1cm}\n        P{1cm}\n        P{1.5cm}\n        }\n\t\t\t\\toprule  \n\t\t\t\\makecell[c]{Methods} & SA & RA & Time Cost (s\/epoch)\\cr  \n\t\t\t\\midrule  \n\t\t\t\\noalign{\\smallskip}\n\t\t\tCoreACL & 78.90 & 50.27 & 96 \\\\\n            \\ w\/fixed $\\alpha$ & 81.29 & 50.24 & 96\\\\\n            \\ w\/annealing $\\alpha$ & 79.46 & 51.37 & 101 \\\\ \\hline\n            \\ w\/$\\mathcal{L}^{IP+HN}$ & 81.19 & 51.31 & 101 \\\\\n            AdvCL & 81.35 & 51.00 & 182 \\\\\n\t\t\t\\bottomrule  \n\t\t\\end{tabular}  \n\t\\end{threeparttable} \n\t\\label{table:3}\n\\end{wraptable}\n\\subsubsection{Comparison with AdvCL.}\nTable 3 reports the performance and computation cost comparisons with AdvCL.\nCoreACL with $\\mathcal{L}^{IP+HN}$ achieves similar performance to AdvCL, which is equivalent to integrate additional components (high frequency view and pseudo-supervision) into CoreACL. The computation time of AdvCL is almost twice than that of $\\rm w\/\\mathcal{L}^{IP+HN}$, which could due to extra computation on contrasting high frequency views and the pseudo-labeled adversarial training.\nOur methods only need to compute pair-wise Euclidean distance for $\\alpha$-annealing in $\\mathcal{L}^{IP}$, and no extra cost introduced in $\\mathcal{L}^{HN}$. \n\n\\subsubsection{Effect of Hard Negatives.}\nTo investigate the effect of hard negatives, we\n\\setlength{\\tabcolsep}{4pt}\n\\setlength{\\intextsep}{6pt} \n\\begin{wraptable}[12]{r}{8cm}\n\\renewcommand\\arraystretch{1.5}\n\t\\centering  \n\t\\fontsize{9}{9}\\selectfont \n\t\\scalebox{0.9}{\n\t\\begin{threeparttable}\n\t\t\\caption{Ablation studies for AdvCL with hard negatives (AdvCL-HN), evaluated under Linear Probing (LP), Adversarial Linear Finetuning (ALF) and Adversarial Full Finetuning (AFF)}  \n\t\t\\label{tab:performance_comparison}  \n\t\t\\begin{tabular}{ccccccc} \n\t\t\t\\toprule  \n\t\t\t\\multirow{2}{*}{Methods}&  \n\t\t\t\\multicolumn{2}{c}{LP}&\\multicolumn{2}{c}{ALF}&\\multicolumn{2}{c}{AFF}\\cr  \n\t\t\t\\cmidrule(lr){2-3} \\cmidrule(lr){4-5}  \\cmidrule(lr){6-7}  \n\t\t\t&SA&RA&SA&RA&SA&RA\\cr  \n\t\t\t\\midrule  \n             \n\t\t\t{AdvCL-HN}&{81.34}&{\\bf 52.96}&{78.69}&{\\bf 53.20}&83.44&{\\bf 54.07}\\cr  \n\t\t\t{w\/o debias}&{\\bf 81.52}&51.61&{\\bf 78.89}&52.34&{\\bf 83.73}&{54.01}\\cr \n\t\t{w\/o reweight}&76.93&50.01&73.49&49.86&81.74&52.60\\cr \n\t\t\t\\bottomrule  \n\t\t\\end{tabular}  \n\t\\end{threeparttable}}  \n\\end{wraptable}\nevaluate each component (negatives debiasing~\\cite{chuang2020debiased}, reweighting~\\cite{robinson2020contrastive}) as shown in Table 4. With negatives-debiasing removed, we observe decrease in robust accuracy, with slightly increased standard accuracy. We hypothesize that without debiasing, semantically similar adversarial representations that should be mapped closely are pushed away instead. \nIn addition, the removal of negatives reweighting results in a sharp performance drop, showing that viewing adversarial views as \\textit{hard negatives} with higher weights plays a key role in discriminating adversarial samples.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.42]{images\/hist-2.png}\n    \\caption{Histograms of Euclidean distance (normalized) distribution of all negative pairs learned by different objectives in (a) CIFAR10 (first row) and (b) CIFAR100 (second row). Baseline is AdvCL~\\cite{fan2021does}; IP: baseline with Inferior Positives; HN: baseline with Hard Negatives. On each dataset, our methods are better at differentiating different instances (with larger distance between negative pairs)\n    }\n    \\label{fig:ablation2}\n\\end{figure}\n\n\n\\begin{figure}[b]\n    \\centering\n    \\includegraphics[scale=0.39]{images\/rocl-tsne3.png}\n    \\caption{t-SNE visualizations in a global view on CIFAR-10 validation set. The embeddings are learned by different self-supervised pre-training methods (SimCLR(a), RoCL(b) and RoCL-IP(c)) (\\textit{colored figure})\n    }\n    \\label{fig:global tsne}\n\\end{figure}\n\n\\subsection{Qualitative Analysis}\n\nFigure \\ref{fig:ablation2} shows the distribution of normalized Euclidean distance over all negative pairs. We take AdvCL~\\cite{fan2021does} as the baseline and compare it with its enhanced versions with our methods.\nGenerally, our methods can shift the original distribution curve right (larger distance), meaning that treating adversaries as inferior positives or hard negatives encourages the model to separate negative pairs further apart and induce better instance discrimination. This suggests that our proposed methods effectively mitigate the negative impacts of \\textit{identity confusion}.\n\nFigure \\ref{fig:global tsne} provides 2-D visualization (t-SNE~\\cite{van2008visualizing} on CIFAR-10) for the embeddings learnt by SimCLR~\\cite{pmlr-v119-chen20j}, RoCL~\\cite{kim2020adversarial} and RoCL enhanced by $\\mathcal{L}^{IP}$ (RoCL-IP). Each class is represented in one color.\nCompared to SimCLR, RoCL representations are corrupted by adversaries and exhibit poor class discrimination. RoCL-IP yields better class separation compared with RoCL.\nThis shows that asymmetric similarity consideration eases instance-level identity confusion.\n\n\n\\section{Introduction}\n\nWell-performed models trained on clean data can suffer miserably when exposed to simply-crafted adversarial samples~\\cite{szegedy2014intriguing,goodfellow2014explaining,carlini2017towards,dong2018boosting}.\nThere has been many adversarial defense mechanisms designed to boost model robustness using labeled data~\\cite{kannan2018adversarial,shafahi2019adversarial,zhang2019you,wong2020fast,zhang2019theoretically,zhu2019freelb,athalye2018obfuscated}. In practice, however, obtaining large-scale annotated data can be far more difficult and costly than acquiring unlabeled data. Leveraging easily-acquired unlabeled data for adversarial learning, thus becomes particularly attractive.\n\nContrastive Learning (CL)~\\cite{hadsell2006dimensionality}, which performs instance discrimination~\\cite{wu2018unsupervised} (Figure 1 (a)) by maximizing agreement between augmentations of the same instance in the learned latent features while minimizing the agreement between different instances,\nhas made encouraging progress in self-supervised learning~\\cite{pmlr-v119-chen20j,he2020momentum,chen2020improved,grill2020bootstrap}. Due to its effectiveness in learning rich representations and competitive performance over fully-supervised methods, CL has seen a surge of research in recent years, such as\npositive sampling~\\cite{pmlr-v119-chen20j,tian2020contrastive,bachman2019learning,tian2020makes}, negative sampling~\\cite{he2020momentum,kalantidis2020hard,chuang2020debiased,wu2018unsupervised}, \npair reweighting~\\cite{chuang2020debiased,robinson2020contrastive}, and different contrast methods~\\cite{grill2020bootstrap,caron2020unsupervised,li2020prototypical}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.43]{images\/intro-18.png}\n\\caption{\nIllustrations of ($a$) Contrastive Learning; ($b$) Adversarial Contrastive Learning; and our proposed methods for viewing adversarial samples asymmetrically as: ($c$) Inferior Positives (asymmetric contrast), and ($d$) Hard Negatives. In each circle, data points are augmentations of the same instance, sharing the same \\textit{Identity}. In ($b$), the Adversarial sample (\\textit{A}) shares the same Identity (\\textit{ID:2}) as the current Instance (\\textit{I}), but resides close to a different Identity (\\textit{ID:1}), thus \\textit{Identity Confusion} problem occurs. Specifically, the Adversarial sample (\\textit{A}) of Instance \\textit{(I)} exhibits similar representations to the Negative sample (\\textit{N}) of (\\textit{I}), which makes the positive contrast (\\textit{A}$\\leftrightarrow$\\textit{I}) and negative contrast (\\textit{N}$\\leftrightarrow$\\textit{I}) undermine each other in the training process (\\textit{colored figure}). \n}\n\\label{fig:intro}\n\\end{figure}\n\nRecently, contrastive learning has been extended to adversarial learning tasks in a self-supervised manner, leading to a new area of \\textit{adversarial contrastive learning} (Adversarial CL) ~\\cite{kim2020adversarial,fan2021does,jiang2020robust,gowal2020self}. \nThe main idea is to generate adversarial samples as additional positives of the same instance~\\cite{kim2020adversarial,fan2021does,jiang2020robust} for instance-wise attack, and maximize the similarity between clean views of the instance and their adversarial counterparts as in CL, while also solving the min-max optimization problem following canonical adversarial learning objective~\\cite{madry2018towards,shafahi2019adversarial,zhang2019you,wong2020fast,zhang2019theoretically,zhu2019freelb}.\nFor example, RoCL\\cite{kim2020adversarial} first proposed an attack mechanism against contrastive loss to confuse the model on instance-level identity, in a self-supervised adversarial training framework. AdvCL\\cite{fan2021does} proposed to minimize the gap between unlabeled contrast and labeled finetuning by introducing pseudo-supervision in the pre-training stage.\n\n\nAlthough these Adversarial CL methods showed improvement on model robustness, we observe that a direct\nextension from CL to adversarial learning (AL) can introduce ineffective CL updates during training.\nThe core problem lies in that they add worst-case perturbations $\\delta$ that no longer guarantee the preservation of instance-level identity~\\cite{kim2020adversarial} (\\textit{i.e.}, different from other data augmentation methods,  adversarial samples can reside faraway from the current instance in the feature space after several attack iterations, because the attack objective is to make adversaries away from the current instance while approximating other instances, against the CL objective). \nAs illustrated in Figure~\\ref{fig:intro}(b), when the adversarial sample (\\textit{A}) of the current instance (\\textit{I}) are in close proximity to negative samples (\\textit{N}), CL objective minimizes the agreement between negative samples and current instance (\\textit{I} and \\textit{N} are pushed away from each other), while AL objective maximizes the agreement between adversarial samples and current instance (\\textit{A} and \\textit{I} are pulled together as \\textit{A} is considered as an augmented view of \\textit{I}). Meanwhile, \\textit{A} and \\textit{N} share similar representations, which renders the two objectives contradicting to each other. We term this conflict as ``\\textit{identity confusion}'', it means $A$ attracts and `confuses' $I$ with a false identity induced by $N$, which impedes both CL and AL from achieving their respective best performance.\n\nTo address this issue of \\textit{identity confusion}, we propose to treat adversarial samples unequally and discriminatingly, and design a generic asymmetric InfoNCE objective (\\textit{A-InfoNCE}), in order to model the asymmetric contrast strengths between positive\/negative samples.\nFirstly, to mitigate the direct pull between adversarial sample (\\textit{A}) and current instance (\\textit{I}) (Figure~\\ref{fig:intro} (c)) that might dampen the effectiveness of CL, we propose to treat adversarial samples as \\textit{inferior positives} that induce weaker learning signals to attract their counterparts in a lower degree when performing positive contrasts.\nThis asymmetric consideration in AL promises a trade-off and reduces conflicting impact on the CL loss.\n\nSecondly, to encourage adversarial samples (\\textit{A}) to escape from false identities induced by negative samples (\\textit{N}) that share similar representations to (\\textit{A}) (pushing \\textit{A} away from \\textit{N}) (Figure~\\ref{fig:intro}(d)), we consider adversarial samples (\\textit{A}) as \\textit{hard negatives}~\\cite{robinson2020contrastive} of other negative samples (\\textit{N}), by strengthening the negative contrast between \\textit{A} and \\textit{N} in CL computation.\nTo effectively sample true adversarial negatives and re-weight each sample, we follow positive-unlabeled learning~\\cite{du2014analysis,elkan2008learning} and contrastive negatives reweighting~\\cite{robinson2020contrastive,chuang2020debiased} practice. \n\nOur contributions are summarized as follows: \n$1)$\n  We propose an generic asymmetric InfoNCE loss, \\textit{A-InfoNCE}, to address the \\textit{identity confusion} problem in Adversarial CL, by viewing adversarial samples \n     as \\textit{inferior positives} or \\textit{hard negatives}.\n   \n   \n   \n  2) Our approach is compatible to existing Adversarial CL methods, by simply replacing standard CL loss with \\textit{A-InfoNCE}. \n   3) Experiments on CIFAR-10, CIFAR-100 and STL-10 show that our approach consistently outperforms existing Adversarial CL methods.\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*{\\large\\textsf{\\refname}}%\n      \\@mkboth{\\MakeUppercase\\refname}{\\MakeUppercase\\refname}%\n      \\list{\\@biblabel{\\@arabic\\c@enumiv}}%\n           {\\settowidth\\labelwidth{\\@biblabel{#1}}%\n            \\leftmargin\\labelwidth\n            \\advance\\leftmargin\\labelsep\n            \\@openbib@code\n            \\usecounter{enumiv}%\n            \\let\\p@enumiv\\@empty\n            \\renewcommand\\theenumiv{\\@arabic\\c@enumiv}}%\n      \\sloppy\n      \\clubpenalty4000\n      \\@clubpenalty \\clubpenalty\n      \\widowpenalty4000%\n      \\sfcode`\\.\\@m}\n     {\\def\\@noitemerr\n       {\\@latex@warning{Empty `thebibliography' environment}}%\n      \\endlist}\n\\makeatother\n\n\\topmargin -2.0cm\n\\oddsidemargin -1.0cm\n\\textwidth 18.5cm\n\\textheight 24cm\n\\footskip 1.0cm\n\n\n\n\\newenvironment{sciabstract}{%\n\\begin{quote} \\bf}\n{\\end{quote}}\n\n\\title{\\textsf{\\textbf{A quantum Fredkin gate}}}\n\n\n\\author\n{Raj B. Patel,$^{1,\\ast}$ Joseph Ho,$^{1}$ Franck Ferreyrol,$^{1,2}$ Timothy C. Ralph,$^{3}$ \\\\\n\\& Geoff J. Pryde,$^{1,\\ast}$\\\\\n\\\\\n\\normalsize{$^{1}$CQC2T and Centre for Quantum Dynamics, Griffith University,}\\\\\n\\normalsize{Brisbane 4111, Australia}\\\\\n\\normalsize{$^{2}$ Laboratoire Photonique, Numerique et Nanosciences, Institut d'Optique,}\\\\\n\\normalsize{CNRS and Universit\\'{e} de Bordeaux, Talence, France}\\\\\n\\normalsize{$^{3}$ CQC2T and School of Mathematics and Physics, University of Queensland, }\\\\\n\\normalsize{Brisbane 4072, Australia}\\\\\n\\normalsize{$^\\ast$To whom correspondence should be addressed; E-mail:  r.patel@griffith.edu.au}\\\\\n\\normalsize{or g.pryde@griffith.edu.au}\n}\n\n\n\\date{}\n\n\n\n\n\n\\begin{document}\n\n\n\\baselineskip12pt\n\n\\twocolumn[\n  \\begin{@twocolumnfalse}\n\\maketitle\n\n\\begin{sciabstract}\nKey to realising quantum computers is minimising the resources required to build logic gates into useful processing circuits. While the salient features of a quantum computer have been shown in proof-of-principle experiments, difficulties in scaling quantum systems have made more complex operations intractable. This is exemplified in the classical Fredkin (controlled-SWAP) gate for which, despite theoretical proposals, no quantum analogue has been realised. By adding control to the SWAP unitary, we use photonic qubit logic to demonstrate the first quantum Fredkin gate, which promises many applications in quantum information and measurement. We implement example algorithms and generate the highest-fidelity three-photon GHZ states to-date. The technique we use allows one to add a control operation to a black-box unitary, something impossible in the standard circuit model. Our experiment represents the first use of this technique to control a two-qubit operation and paves the way for larger controlled circuits to be realised efficiently.\n\\end{sciabstract}\n\\end{@twocolumnfalse}\n]\n\n\\section*{\\large\\textsf{Introduction}}\nOne of the greatest challenges in modern science is the realisation of quantum computers\\cite{Kok2007,OBrien2009,Ladd2010} which, as they increase in scale, will allow enhanced performance of tasks in secure networking, simulations, distributed computing and other key tasks where exponential speedups are available. Processing circuits to realise these applications are built up from logic gates that harness quantum effects such as superposition and entanglement. At present, even small-scale and medium-scale quantum computer circuits are hard to realise because of the requirement to control enough quantum systems sufficiently well in order to chain together many gates into circuits. One example of this is the quantum Fredkin gate, which requires at least five two-qubit gates\\cite{Smolin1996} to be implemented in the standard circuit model. Thus, despite featuring prominently in schemes for quantum computation\\cite{Vandersypen2001,Lopez2012,Lanyon2007}, error-correction\\cite{Chuang1996,Barenco1997}, cryptography\\cite{Buhrman2001,Horn2005,Gottesman2001}, and measurement\\cite{Ekert2002,FuiasekFilip2002}, no such gate has been realised to date.\n\nThe quantum Fredkin gate, shown in Fig. 1A, is a three-qubit gate whereby, conditioned on the state of the control qubit, the quantum states of the two target qubits are swapped. The original, classical version of the gate first proposed by Fredkin \\cite{Fredkin1982} also serves as one of the first examples of a reversible logic operation where the number of bits are conserved and no energy is dissipated as a result of erasure. In the framework of universal quantum computation,  gates are also reversible, so it may seem natural to ask whether it is possible to construct a quantum version of the Fredkin gate. The first design of the quantum Fredkin gate was proposed by Milburn \\cite{Milburn1989} and was to use single photons as qubits and cross-Kerr nonlinearities to produce the necessary coherent interactions. Further schemes utilising linear optics developed these ideas further \\cite{Chau1995,Smolin1996,Fiurasek2006,Fiurasek2008,Gong2008} by using ancilla photons, interference, and multiple two-qubit\\cite{OBrien2003,Pooley2012} and single-qubit gates. However concatenating multiple probabilistic gates in this fashion typically leads to a multiplicative reduction in the overall probability of success of $<1\/100$. Hence it would be desirable to be able to construct a quantum Fredkin gate directly without decomposition and avoid the associated resource overhead.\n\nWe begin by describing the concept of our experiment. We perform the controlled-SWAP operation by adding control to the SWAP unitary $U_{SWAP}$ applying the technique in Zhou \\textit{et al.} \\cite{Zhou2011}, to greatly reduce the complexity of quantum circuits. The notion of adding control to a black-box unitary is forbidden or difficult in many architectures\\cite{Araujo2014,Thompson2013} -- optics lends itself well to this approach because the optical implementation of the unitary leaves the vacuum state unchanged. Here we utilise this method to simplify a controlled multi-qubit operation.\n   \\begin{figure*}\n   \\begin{center}\n      \\includegraphics[width=\\textwidth]{Fig1_S.pdf}\n   \\end{center}\n\\noindent\\small{{\\bf Fig. 1.} Experimental arrangement and truth table measurements. \\textbf{A}, The quantum Fredkin gate circuit. The states of the target qubits are either swapped or not swapped depending on the state of the control qubit. \\textbf{B}, Concept of our experiment. Two SPDC photon sources allow production of path entanglement such that modes $R$ and $Y$ are entangled with modes $B$ and $G$. The SWAP operation is carried out on the path modes, depending on the control photon's state,  such that arrival of the control photon indicates a system state of $\\alpha|H\\rangle^{C}|\\psi\\rangle^{T1}|\\varphi\\rangle^{T2} + \\beta|V\\rangle^{C}|\\varphi\\rangle^{T1}|\\psi\\rangle^{T2}$. \\textbf{C}, The experimental arrangement. Entangled photons are produced via SPDC  (see Materials and Methods). Entering the gate via single-mode fiber, the two target photons are sent through a PBS. The path-entangled state in Eq. \\ref{4GHZ} is produced after each target photon enters a displaced Sagnac interferometer and the which-path information is erased on an NPBS. QWPs and HWPs encode the polarisation state in Eq. \\ref{4GHZU}. The control consists of a polarisation beam displacer interferometer. The desired control state is encoded onto modes $1R$ and $1B$ and coherently recombined. A tilted HWP is used to set the phase of the output state. Successful operation is heralded by four-fold coincidence events between the control, target, and trigger detectors. \\textbf{D}, Ideal (transparent bars) and measured (solid bars) truth table data for our gate. A total of 620 four-fold events were measured for each of the eight measurements, giving $\\left\\langle\\mathcal{O}\\right\\rangle = 96\\pm4\\%$.}\n   \\end{figure*}\nA key idea in our demonstration is to use entanglement in a non-qubit degree of freedom (we use the photon's path mode) to drive the operation of the gate. This path entanglement can be produced in different ways. In our demonstration (Fig. 1B), it is generated from spontaneous parametric down-conversion (SPDC). Given the physical arrangement of the circuit and that we only accept detection events where a single photon is counted at each of the four outputs simultaneously, the optical quantum state produced by SPDC is converted to the required four-photon path-mode entangled state (see Materials and Methods) and has the form\n\\begin{equation}\\label{4GHZ}\n \\left(|11\\rangle_{B}|11\\rangle_{G}|00\\rangle_{R}|00\\rangle_{Y} + |00\\rangle_{B}|00\\rangle_{G}|11\\rangle_{R}|11\\rangle_{Y}\\right)\/\\sqrt{2}\n\\end{equation}\n where $B$, $R$, $Y$, and $G$ refer to path-modes and, for example, $|11\\rangle_{B}$ indicates a photon occupying mode $1B$ and another occupying $2B$. The path-modes are distributed throughout the circuit such that $U_{SWAP}$ is applied only to the $B$ and $G$ modes.  The qubit state is encoded on the polarisation of the photon.  Because the photons are in a spatial superposition, polarisation preparation optics must be applied to both path-modes of each photon. Hence, an arbitrary, separable, three-qubit state $|\\xi\\rangle|\\psi\\rangle|\\varphi\\rangle$ can be prepared as an input to the gate. In particular, the control qubit is encoded on modes $1R$ and $1B$, target 1 is encoded on modes $2R$ and $2B$, and target 2 is encoded on modes $1G$ and $1Y$,  yielding\n\\begin{equation}\\label{4GHZU}\n  \\left(|\\xi\\rangle^{C}_{1B}|\\psi\\rangle^{T1}_{2B}|\\varphi\\rangle^{T2}_{1G}|H\\rangle^{Tr}_{2G} + |\\xi\\rangle^{C}_{1R}|\\psi\\rangle^{T1}_{2R}|\\varphi\\rangle^{T2}_{1Y}|V\\rangle^{Tr}_{2Y}\\right)\/\\sqrt{2}\n\\end{equation}\nThe two control modes $1R$ and $1B$ are mixed on a polarising beam splitter (PBS), wheras a 50:50 non-polarising beam splitter (NPBS) is used to erase the path information in the target and trigger arms. The SWAP is implemented via rearrangement of the path-modes such that the target modes $2B$ and $1G$ are swapped wheras $2R$ and $1Y$ are not. Successful operation of the gate occurs when photons are detected at the control, target 1, and target 2 detectors (simultaneously with a photon detection at either trigger detector).  The polarisation state of the three-qubit system, given the required modes are occupied, is $\\alpha|H\\rangle^{C}|\\psi\\rangle^{T1}|\\varphi\\rangle^{T2} + \\beta|V\\rangle^{C}|\\varphi\\rangle^{T1}|\\psi\\rangle^{T2}$ as expected from application of the Fredkin gate on the state  $|\\xi\\rangle^{C}|\\psi\\rangle^{T1}|\\varphi\\rangle^{T2}$ where $|\\xi\\rangle = \\alpha|H\\rangle + \\beta|V\\rangle$. Taking into consideration the probability of recording a four-fold coincidence, successful execution of the gate occurs one-sixteenth of\n the time, on average. This can be increased to one-fourth of the time by collecting the target photons from both NPBS outputs.\n\n \\section*{\\large\\textsf{Results}}\nThe experimental arrangement of the quantum Fredkin gate is shown in Fig. 1C and consists of three interferometers designed to be inherently phase-stable. Pairs of polarisation entangled photons, produced by two SPDC crystals (see Materials and Methods), impinge on a PBS. Two orthogonally polarised photons, one from each source, are sent to separate displaced Sagnac interferometers. Initially, they are incident on a beam splitter where one half of the interface acts as a PBS and the other half acts as an NPBS. Entering at the PBS side, photons may travel along counterpropagating path modes where the polarisation state $|\\psi\\rangle$ is encoded onto one mode and the state $|\\varphi\\rangle$ is encoded on the other. The two paths are then recombined on the NPBS side of the beam splitter where the path information is erased (see Methods and Materials), giving the path-mode entangled state in equation \\ref{4GHZ} whilst the polarisation encoding procedure leads to the state in Eq. \\ref{4GHZU}. The control of the gate is realised in a polarisation interferometer consisting of two calcite beam displacers. The desired polarisation state of the control is encoded onto modes $1R$ and $1B$, which are coherently recombined in the second beam displacer. Given successful operation (arrival of a photon at the control detector), the preparation of the control photon in $|H\\rangle = |1\\rangle$ projects the target photons onto path modes $1G$ and $2B$ which undergo SWAP; conversely, preparing $|V\\rangle = |0\\rangle$ projects the target photons onto path modes $2R$ and $1Y$, which undergo the identity operation. In practice, the trigger arm consists of a half-wave plate (HWP) whose optic axis (OA) is set to $22.5^{\\circ}$, producing diagonal $|D\\rangle=\\frac{1}{\\sqrt{2}}(|H\\rangle + |V\\rangle)$ or anti-diagonal $|A\\rangle=\\frac{1}{\\sqrt{2}}(|H\\rangle - |V\\rangle)$ polarised photons, and a PBS. Successful operation is heralded by measuring four-fold coincidences across the trigger, control and two target detectors.\n\nThe logical operation of the gate was measured by performing eight measurements, one for each of the possible logical inputs. For each input we measure a total of 620 four-fold events distributed across the eight possible output states. Under ideal operation, for a given input, there is a single output. The solid bars in Fig. 1D depict the experimentally measured truth table data, $M_{exp}$ , whereas the transparent bars represent the ideal truth table  $M_{ideal}$. To quantify the mean overlap between $M_{exp}$ and $M_{ideal}$, we calculate $\\left\\langle\\mathcal{O}\\right\\rangle = Tr \\left(M_{exp}M_{ideal}^{T}\/M_{ideal}M_{ideal}^{T}\\right)= 96\\pm4\\%$ which confirms excellent performance in the logical basis.  The slight reduction in fidelity is most likely due to the imperfect extinction of our polarisation optics.\n  \\begin{figure*}\n   \\begin{center}\n      \\includegraphics{Fig2_S.pdf}\n   \\end{center}\n   \\noindent\\small{{\\bf Fig. 2.} Real (left) and imaginary (right) parts of the reconstructed density matrices for our four GHZ states. Fidelity and purity were calculated for each state. \\textbf{A}, $|GHZ_{1}^{+}\\rangle$:  $F = 0.88.\\pm 0.01$ and $P = 0.79 \\pm 0.02$. \\textbf{B},  $|GHZ_{1}^{-}\\rangle$: $F = 0.90 \\pm 0.01$ and $P = 0.83 \\pm 0.02$. \\textbf{C},  $|GHZ_{2}^{+}\\rangle$: $F = 0.93 \\pm 0.01$ and $P = 0.87 \\pm 0.02$. \\textbf{D}, $|GHZ_{2}^{-}\\rangle$: $F = 0.92 \\pm 0.01$ and $P = 0.85 \\pm 0.02$.}\n   \\end{figure*}\n\nWe demonstrate the full quantum nature of our gate by preparing the control in a superposition $|\\xi\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle \\pm |1\\rangle)$ which places the gate in a superposition of the SWAP and identity operations. Using our gate, we produce four of the eight maximally entangled three-photon Greenberger-Horne-Zeilinger (GHZ) states, namely\n\\begin{align}\\label{3GHZa}\n \\frac{1}{\\sqrt{2}}(|0\\rangle &\\pm |1\\rangle)^{C}|1\\rangle^{T1}|0\\rangle^{T2}\\rightarrow |GHZ_1^{\\pm}\\rangle \\nonumber\\\\\n &= \\frac{1}{\\sqrt{2}}\\left(|0\\rangle^{C}|1\\rangle^{T1}|0\\rangle^{T2} \\pm e^{i(\\phi + \\theta(\\vartheta))}|1\\rangle^{C}|0\\rangle^{T1}|1\\rangle^{T2}\\right)\n\\end{align}\nand,\n \\begin{align}\\label{3GHZb}\n \\frac{1}{\\sqrt{2}}(|0\\rangle &\\pm |1\\rangle)^{C}|0\\rangle^{T1}|1\\rangle^{T2}\\rightarrow |GHZ_2^{\\pm}\\rangle \\nonumber\\\\\n &= \\frac{1}{\\sqrt{2}}\\left(|0\\rangle^{C}|0\\rangle^{T1}|1\\rangle^{T2} \\pm e^{i(\\phi + \\theta(\\vartheta))}|1\\rangle^{C}|1\\rangle^{T1}|0\\rangle^{T2}\\right)\n\\end{align}\nHere $\\phi$ is a phase shift intrinsic to the gate, and $ \\theta(\\vartheta)$ is a corrective phase shift that can be applied by tilting a HWP at OA by an angle $\\vartheta$, such that $\\phi + \\theta(\\vartheta) = 2n\\pi$ (see Materials and Methods). In doing so, we are able to test the coherent interaction of all three qubits in the gate, which is a key requirement for constructing universal quantum computers. For each of the four states in Eqs. \\ref{3GHZa} and \\ref{3GHZb}, we perform three-qubit quantum state tomography (QST) to fully characterise the state. The control and target qubits are measured independently in the $D\/A$ basis, which we denote as $\\sigma_x$; in the $R\/L$ basis $(\\sigma_y)$, where  $|R\\rangle=\\frac{1}{\\sqrt{2}}(|H\\rangle + i|V\\rangle)$ and  $|L\\rangle=\\frac{1}{\\sqrt{2}}(|H\\rangle - i|V\\rangle)$; and in the $H\/V$ basis $(\\sigma_z)$. Therefore full state reconstruction can be carried out by a set of 27 measurements settings $(\\sigma_x\\sigma_x\\sigma_x, \\sigma_x\\sigma_x\\sigma_y...)$ effectively resulting in an over-complete set of 216 projective measurements as each measurement setting has eight possible outcomes. Figure 2 shows the real (left) and imaginary (right) parts of the reconstructed density matrices of the four GHZ states, each of which was calculated from $\\sim5000$ four-fold events using a maximum-likelihood algorithm. We measure fidelities and purities of $F = 0.88 \\pm 0.01$ and $P = 0.79 \\pm 0.02$ for $|GHZ_{1}^{+}\\rangle$, $F = 0.90 \\pm 0.01$ and $P = 0.83 \\pm 0.02$ for $|GHZ_{1}^{-}\\rangle$, $F = 0.93 \\pm 0.01$, and $P = 0.87 \\pm 0.02$ for $|GHZ_{2}^{+}\\rangle$, and $F = 0.92\\pm 0.01$ and $P = 0.85 \\pm 0.02$ for $|GHZ_{2}^{-}\\rangle$. The errors were calculated from 500 samples of a Monte-Carlo simulation. These values are most likely limited by imperfect mode overlap at the NPBS in each displaced Sagnac interferometer. Nevertheless, to the best of our knowledge, these values are the highest reported for photonic GHZ states surpassing the previous values reported in Hamel \\textit{et al.} \\cite{Hamel2014}.\n  \\begin{figure}[t]\n   \\begin{center}\n   \\includegraphics[width=\\linewidth]{Fig3_S.pdf}\n   \\end{center}\n   \\noindent\\small{{\\bf Fig. 3.} Measured correlations for violations of Mermin's and Svetlichny's inequalities. \\textbf{A},  Mermin's inequality resulting in $S_M = 3.58 \\pm 0.06$, a violation by 24 standard deviations. \\textbf{B}, Svetlichny's inequality with  $S_{Sv} = 4.88 \\pm 0.13$, a violation by 7 standard deviations. Error bars were calculated from Poissonian counting statistics.}\n   \\end{figure}\n\nWe perform further measurements to characterise the quality of the $|GHZ_{2}^{+}\\rangle$ state. GHZ states can show a strong contradiction between local hidden-variable theories and quantum mechanics \\cite{Pan2000}. Mermin \\cite{Mermin1990} derived a Bell-like inequality by imposing locality and realism for three particles, which holds for any local hidden-variable theory\n\\begin{align}\\label{Mermin}\nS_M &= |E(a',b,c') + E(a,b',c') + E(a,b,c) - E(a',b',c)|\\nonumber\\\\\n &\\leq 2\n\\end{align}\n\\begin{figure}[H]\n   \\begin{center}\n      \\includegraphics[width=\\linewidth]{Fig4_S.pdf}\n   \\end{center}\n   \\noindent\\small{{\\bf Fig. 4.} Estimations of nonlinear functionals of a single-qubit state with the quantum Fredkin gate. \\textbf{A},  Circuit diagram of the network. \\textbf{B}, Measurements of the overlap of two single qubit states, $|\\langle{T1}|{T2}\\rangle|^2$.  The fringe visibility or overlap was measured for states $|0\\rangle^{T1}|0\\rangle^{T2}$ (black), $\\frac{1}{\\sqrt{2}}\\left(|0\\rangle \\pm 1\\rangle\\right)^{T1}|0\\rangle^{T2}$ (red), and $|0\\rangle^{T1}|1\\rangle^{T2}$ (blue) with values $0.82 \\pm 0.02$, $0.52 \\pm 0.02$, and $0.05 \\pm 0.01$, respectively. \\textbf{C}, Measurements of the state purity.  We measure a visibilities ranging from  $0.82 \\pm 0.02$ for a pure state to $0.03 \\pm 0.02$ for a maximally mixed state.}\n   \\end{figure}\nThis inequality can be violated by performing measurements with settings $a = b = c = \\sigma_x$ and $a' = b' = c' = \\sigma_y$ with a maximum violation of $S_M = 4$. From the QST of $|GHZ_{2}^{+}\\rangle$, 747 of the total 5029 four-fold events can be used to calculate the correlation functions $E$ in Eq. \\ref{Mermin}; these results are shown in Fig. 3A. This leads to $S_M = 3.58 \\pm 0.06$ which is a violation by 24 standard deviations. The implication of using these particular measurement settings is that the state exhibits genuine tripartite entanglement.\n\nAn additional test, namely, the violation of Svetlichny's inequality, is required to test whether the state is capable of displaying tripartite non-locality \\cite{Svetlichny1987,Lavoie2009}. Non-local hidden variable theories cannot be ruled out with Mermin's inequality, as they can be violated for arbitrarily strong correlations between two of the three particles. Svetlichny's inequality takes the form\n\\begin{align}\\label{Svet}\nS_{Sv} =& |E(a,b,c) + E(a,b,c') + E(a,b',c) - E(a,b',c')\\nonumber\\\\\n+& E(a',b,c) - E(a',b,c') - E(a',b',c) - E(a',b',c')|\\nonumber\\\\\n&\\leq 4\n\\end{align}\nwith settings $a = Sv1_\\pm$ (where $|Sv1_\\pm\\rangle = \\frac{1}{\\sqrt{2}}(|H\\rangle \\pm e^\\frac{i3\\pi}{4}|V\\rangle)$), $a' = Sv2_\\pm$ (where $|Sv2_\\pm\\rangle = \\frac{1}{\\sqrt{2}}(|H\\rangle \\pm e^\\frac{i\\pi}{4}|V\\rangle)$), $b' = c =\\sigma_x$, and $b = c' = \\sigma_y$. The maximum violation allowed by quantum mechanics is $S_{Sv} = 4\\sqrt{2}$. Figure 3B shows the correlations calculated from 2348 four-fold events leading to $S_{Sv} = 4.88 \\pm 0.13$, which is a violation by 7 standard deviations.\n\nAn application of the quantum Fredkin gate is the direct estimation of non-linear functionals\\cite{Ekert2002} of a quantum state, described by a density matrix $\\rho$, without recourse to QST. Here $\\rho = \\varrho_{T1} \\otimes \\varrho_{T2}$ is the density matrix of two separable subsystems. The circuit we employ is shown in Fig. 4A, where an interferometer is formed using two Hadamard gates and a variable phase shift $\\theta(\\vartheta)$. This interferometer is coupled to the controlled-SWAP operation of our quantum Fredkin gate such that measuring the control in the logical basis leads to an interference pattern given by $\\textrm{Tr}[U_{SWAP}\\varrho_{T1} \\otimes \\varrho_{T2}] = \\textrm{Tr}[\\varrho_{T1}\\varrho_{T2}] = ve^{i\\theta(\\vartheta)}$. If $\\varrho_{T1} \\neq \\varrho_{T2}$ then measurement of the fringe visibility provides, for pure states, a direct measure of the state overlap $|\\langle{T1}|{T2}\\rangle|^2$, where $\\varrho_{T1} = |{T1}\\rangle\\langle{T1}|$ and $\\varrho_{T2} = |{T2}\\rangle\\langle{T2}|$. Conversely, if $\\varrho_{T1} = \\varrho_{T2}$ then the fringe visibility provides an estimate of the length of the Bloch vector ( that is, the purity $P = \\textrm{Tr}[\\varrho^2]$). We realise the Hadamard operations in Fig. 4A by setting the quarter wave plate (QWP) and HWP combinations to prepare or measure $\\sigma_x$.\n\nFigure 4B shows the results of preparing the target qubits in the states $|0\\rangle^{T1}|0\\rangle^{T2}$, $\\frac{1}{\\sqrt{2}}\\left(|0\\rangle + 1\\rangle\\right)^{T1}|0\\rangle^{T2}$, and $|0\\rangle^{T1}|1\\rangle^{T2}$, corresponding to ideal (measured) overlaps and visibilities of 1 ($0.82 \\pm 0.02$), 0.5 ($0.52 \\pm 0.02$), and 0 ($0.05 \\pm 0.01$), respectively. Although the maximum visibility we are able to measure is limited by the performance of the three interferometers in the circuit, our measurements show a clear reduction in visibility as the single qubit states are made orthogonal. Figure 4C shows the result of setting  $\\varrho_{T1} = \\varrho_{T2}$. As we increase the degree of mixture (see Materials and Methods), we observe a reduction in visibility from $0.82 \\pm 0.02$ for a pure state to $0.03 \\pm 0.02$ for a maximally mixed state.\n\n\\section*{\\large\\textsf{Discussion}}\nIn conclusion, we have used linear optics to perform the first demonstration of the quantum Fredkin gate.  This is achieved by exploiting path-mode entanglement to add control to the SWAP operation. Our implementation has an improved success rate of more than one order of magnitude compared to previous proposals and does not require ancilla photons or decomposition into two-qubit gates. Our gate performs with high accuracy in the logical basis and operates coherently on superposition states. We have used the gate to generate genuine tripartite entanglement with the highest fidelities to date for photonic GHZ states and have implemented a small-scale algorithm to characterise quantum states without QST.\n\nAn alternative method for generating the polarisation-path entanglement that drives the gate is the use of C-path gates\\cite{Zhou2011} at the input. Our implementation varies from a fully heralded quantum Fredkin gate (see Materials and Methods), which does not require preexisting entanglement; however it demonstrates the key properties of a quantum Fredkin gate. For completely general quantum circuits that incorporate Fredkin (or similar controlled-arbitrary-unitary) gates at arbitrary circuit locations, the C-path methodology may be necessary at the cost of some additional resources and success probability (see Materials and Methods), though we  conjecture that specific circuits comprising multiple Fredkin gates might be optimised using similar techniques to those that allow us to simplify the Fredkin down from a circuit of five two-qubit gates.  Nevertheless, for small algorithms or operations and whenever possible, it is significantly favourable to directly generate path entanglement.\n\nThe quantum Fredkin gate has many applications across quantum information processing. Our demonstration should stimulate the design and implementation of even more complex quantum logic circuits. Later we became aware of related work carried out by Takeuchi\\cite{Takeuchi2015}.\n\n\\section*{\\large\\textsf{Materials and Methods}}\n\n\\paragraph*{Source\\\\}\nOur source consisted of a $150\\textrm{ fs}$ pulsed Ti-Sapphire laser operating at a rate of  $80\\textrm{ MHz}$ and at a wavelength of $780\\textrm{ nm}$, which was frequency doubled using a $2\\textrm{ mm}$ LBO crystal. Two dispersion-compensating ultrafast prisms spatially filter any residual  $780\\textrm{ nm}$ laser light. The frequency- doubled light (with power $100\\textrm{ mW}$) pumped two $2\\textrm{ mm}$ type-II $\\beta$ barium borate (BBO) crystals in succession. Entangled photons, generated via SPDC, were collected at the intersection of each set of emission cones. They then encountered an HWP with its OA at $45^{\\circ}$ and an additional $1\\textrm{ mm}$ type-II BBO crystal used to compensate for spatial and temporal walk-offs. The single photons were coupled into single-mode fiber and delivered to the gate. This configuration gave, on average, a four-fold coincidence rate of 2.2 per minute at the output of the gate.\n\\paragraph*{Entangled state preparation\\\\}\nEach SPDC source emitted pairs of entangled photons of the form $|\\psi^{+}_{1}\\rangle = \\frac{1}{\\sqrt{2}}\\left(|H\\rangle_{1B}|V\\rangle_{2B}+|V\\rangle_{1R}|H\\rangle_{2R}\\right)$ and $|\\psi^{+}_{2}\\rangle = \\frac{1}{\\sqrt{2}}\\left(|H\\rangle_{1Y}|V\\rangle_{2Y}+|V\\rangle_{1G}|H\\rangle_{2G}\\right)$. Polarisation optics were used to distribute the path-modes throughout the circuit and thus convert this state into the path-entangled states $|\\psi^{+}_{1}\\rangle = \\frac{1}{\\sqrt{2}}\\left(|1\\rangle_{1B}|1\\rangle_{2B}|0\\rangle_{1R}|0\\rangle_{2R}+|0\\rangle_{1B}|0\\rangle_{2B}|1\\rangle_{1R}|1\\rangle_{2R}\\right)$ and \\\\$|\\psi^{+}_{2}\\rangle = \\frac{1}{\\sqrt{2}}\\left(|1\\rangle_{1Y}|1\\rangle_{2Y}|0\\rangle_{1G}|0\\rangle_{2G}+|0\\rangle_{1Y}|0\\rangle_{2Y}|1\\rangle_{1G}|1\\rangle_{2G}\\right)$. Path modes from $|\\psi^{+}_{1}\\rangle$ and $|\\psi^{+}_{2}\\rangle$ were combined on a PBS (Fig 1C, PBS with outputs $2R$, $1G$, $1Y$, and $2B$); along with post-selection of four-fold coincidence events at the outputs of the control, target, and trigger outputs, this led to Eq. \\eqref{4GHZ} in the main text. Each qubit was encoded using photon polarisation: using Eq. \\eqref{4GHZ}, considering that each photon exists in a superposition of path-modes and omitting the unoccupied modes, an arbitrary polarisation state can be encoded onto each qubit by performing a local unitary operation on each mode, giving equation \\eqref{4GHZU}. The state encoding was performed inside the beam displacer (control qubit) and displaced Sagnac (target qubits) interferometers.\n\\paragraph*{Tuning the phase\\\\}\nThe phase was tuned by tilting an HWP set to its OA. To set the correct phase for each of the four GHZ states, we varied the tilt of the HWP and measured fringes in the four-fold coincidences with our measurement apparatus in the $\\sigma_x\\sigma_y\\sigma_y$ basis. For the $|GHZ_{1,2}^{+}\\rangle$ $\\left(|GHZ_{1,2}^{-}\\rangle\\right)$ we set the tilt to maximise (minimise) the occurrence of the $|DRR\\rangle$, $|DLL\\rangle$, $|ARL\\rangle$, and $|ALR\\rangle$ events.\n\\paragraph*{Mixed state preparation\\\\}\nThe mixed states of the form $\\varrho = m|0\\rangle\\langle{0}| + \\frac{(1-m)}{2}\\left(|0\\rangle\\langle{0}| + |1\\rangle\\langle{1}|\\right)$ were obtained by measuring output statistics for a combination of pure input states. The input states of the target were prepared, in varying proportions given by the parameter $m$, as  $0.25 (1 + m)^2|0\\rangle^{T1}|0\\rangle^{T2}$, $0.25(1 - m^2)|0\\rangle^{T1}|1\\rangle^{T2}$, $0.25(1 - m^2)|1\\rangle^{T1}|0\\rangle^{T2}$, and $0.25 (1 - m)^2|1\\rangle^{T1}|1\\rangle^{T2}$.  The aggregated data resulted in a fringe pattern which reflects the purity of the mixed single-qubit state.\n\\paragraph*{Erasing the which-path information\\\\}\nGeneration of path-mode entanglement and successful operation of the gate in the quantum regime relied on the erasure of the which-path information in the two displaced Sagnac interferometers. We tested this by performing a Hong-Ou-Mandel (HOM) two-photon interference measurement after each interferometer. After overlapping path modes $2R$ and $1G$ on an NPBS, an HWP with its OA set to $22.5^{\\circ}$ rotated the polarisation of the photons to $|D\\rangle$ and $|A\\rangle$, respectively. Sending these photons into the same port of a PBS led to bunching at the output if the path-modes were indistinguishable. Doing the same for modes $2B$ and $1Y$ gave two separate HOM dips (see Materials and Methods) with visibilities of $90 \\pm 5\\%$ and $91 \\pm 6\\%$.\n\\paragraph*{Heralding the quantum Fredkin gate\\\\}\nIn order to use a quantum Fredkin gate as part of a much larger quantum circuit (with gates in series), it is preferable for the gate to be heralded. Realising our gate in this manner involves adding C-path gates\\cite{Zhou2011} to each input. For the best probability of success $P_{success}$, each C-path gate requires two heralded C-NOT gates\\cite{Pittman2001} which, in turn, require two entangled pair ancillae. Execution of the C-path gate succeeds with $P_{success} = (1\/4)^2$\\cite{Zhou2011,Pittman2001}. C-path gates are not a necessity at the output if successful execution is heralded by non-detections at the relevant NPBS ports, at an additional probability cost of factor $1\/4$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDue to the increasing growth of 3D point cloud data, point cloud semantic segmentation has been receiving more and more attention in the 3D computer vision community. Most of these segmentation methods focus on fully supervised segmentation with manually annotated points \\cite{hu2020randla, thomas2019kpconv, lei2020seggcn, zhao2019pointweb, wang2019graph}. However, annotating large-scale 3D point clouds is a cumbersome process, which is costly in labor and time. Particularly, the number of point clouds in some real scenes such as the indoor scene can often reach the order of magnitude to millions. Therefore, it is difficult to obtain the accurate labels of these million points for full-supervised segmentation.\n\n\nDifferent from full-supervised point cloud segmentation, semi-supervised segmentation aims to learn a good label prediction for point clouds with partially annotated points. Recent works have been dedicated to the semi-supervised point cloud segmentation task.\nGuinard~\\emph{et al.}~\\cite{guinard2017weakly} propose a weakly supervised conditional random field classifier for 3D LiDAR point cloud segmentation.\nHowever, it converts the segmentation task into an optimization problem, and the contextual information in point clouds is ignored. Mei~\\emph{et al.} propose a semi-supervised 3D LiDAR point cloud segmentation method~\\cite{mei2019semantic}, where the 3D data is projected to range images for feature embedding, and the inter-frame constraints are combined with some labeled samples to encourage feature consistency. Nonetheless, the constraints along the LiDAR sequential frames are not available in general 3D segmentation datasets. Lately,~\\cite{XuLee_CVPR20} proposes a semi-supervised point cloud segmentation method, which employs three constraints to enhance the feature learning of unlabeled points, including block-level label penalization, data augmentation with rotation and flipping for prediction consistency, and a spatial and color smoothness constraint in local regions. Although it can obtain effective segmentation results, the long-range relations are ignored in this method.\n\n\nAlthough some efforts have been made on semi-supervised point cloud segmentation, how to accurately predict the labels of unannotated points for segmentation is still a challenging problem. Particularly, since point clouds are irregular, it is difficult to exploit the geometry structures of point clouds to accurately infer pseudo labels of unannotated points for label propagation. In addition, the uncertainty of inferred pseudo labels of unannotated points hinders the network from learning discriminative features of point clouds, leading to inaccurate label prediction.\n\nAiming at the aforementioned two problems, in this paper, we propose a novel semi-supervised semantic point cloud segmentation network, named SSPC-Net. We first divide the point clouds into superpoints and build the superpoint graph, where the superpoint is a set of points with isotropically geometric features. Thus, we can convert the point-level label prediction problem in the point cloud segmentation task into the superpoint-level label prediction problem. Following the method in ~\\cite{landrieu2018large}, we employ the gated graph neural network (GNN)~\\cite{li2015gated} for superpoint feature embedding.\nIn order to fully exploit the local geometry structure of the constructed superpoint graph, we then develop a dynamic label propagation method to accurately infer pseudo labels for unsupervised superpoints. Specifically, the labels of supervised superpoints are gradually extended to the adjacent superpoints with high semantic similarity along the edges of the superpoint graph.\nWe also adopt a superpoint dropout strategy to obtain the high-quality pseudo labels during the label propagation process, where the extended superpoints with low confidences are dynamically pruned.\nFurthermore, we propose a coupled attention mechanism to learn the discriminative context features of superpoints. We alternatively perform attention on the supervised and extended superpoints so that the discrimination of the features of the supervised and extended superpoints can be boosted each other, alleviating the uncertainty of the inferred pseudo labels of the unsupervised superpoints.\nFinally, we employ a combined cross-entropy loss to train the segmentation network.\nExtensive results on various indoor and outdoor datasets demonstrate that our method can yield good performance with only few point-level annotations.\n\n\nThe main contributions of this paper are summarized as: \\textbf{(1)} We develop a dynamic superpoint label propagation method to accurately infer the pseudo labels of unsupervised superpoints. We also present a superpoint dropout strategy to select the high-quality pseudo labels. \\textbf{(2)} We propose a coupled attention mechanism on the supervised and extended superpoints to learn the discriminative features of the superpoints. \\textbf{(3)} Our proposed method can yield better performance than the current semi-supervised point cloud semantic segmentation method with fewer labels.\n\n\\section{Related Work}\n\\textbf{Deep learning on 3D point clouds.}\nRecently, many deep learning methods are proposed to tackle point cloud classification and segmentation.\nSome methods~\\cite{wu20153d,maturana2015voxnet,sedaghat2016,qi2016volumetric} voxelize point clouds and employ 3D CNNs for feature embedding. However, the voxel-based methods suffer from the large memory cost due to the high-resolution voxels. By projecting point clouds into 2D views, \\cite{su15mvcnn,boulch2017unstructured,tatarchenko2018tangent} use classic CNNs to extract features from point clouds. However, the view-based methods are sensitive to the density of 3D data.\nTo reduce memory cost and additional preprocessing, Qi~\\emph{et al.} propose PointNet, which directly processes the unordered point clouds and uses multi-layer perceptrons (MLPs) and the maxpooling function for feature embedding. Following PointNet, many efforts~\\cite{qi2017pointnet++,klokov2017escape,wang2019graph,hua2018pointwise,li2018pointcnn,zhao2019pointweb,wang2019dynamic,thomas2019kpconv,wu2019point,liu2019point2sequence,han2020point2node,zhao2020jsnet,feng2018gvcnn,ma2018learning} are proposed for point cloud processing. Although these methods have achieved decent performance, their models depend on fully annotated 3D point clouds for training.\nHowever, in this paper, we focus on the semi-supervised point cloud semantic segmentation.\n\n\\textbf{Semi-\/Weakly supervised deep learning on 3D point clouds.}\nMany efforts~\\cite{mei2019semantic,wei2020multi,XuLee_CVPR20} have been proposed to tackle semi-\/weakly supervised point cloud semantic segmentation. In \\cite{mei2019semantic}, Mei~\\emph{et al.} introduce a semi-supervised 3D LiDAR data segmentation method. It first converts the 3D data to depth maps and then applies CNNs for feature embedding. In addition to a small part of supervised data, it also leverages the temporal constraints along the LiDAR scans sequence to boost feature consistency. Therefore, it is not practicable for general point cloud segmentation cases. Inspired by CAM~\\cite{zhou2016learning}, Wei~\\emph{et al.} propose MPRM~\\cite{wei2020multi} with scene-level and subcloud-level labels for weakly supervised segmentation. Specifically, it leverages a point class activation map (PCAM) to obtain the localization of each class and then generates point-wise pseudo labels with a multi-path region mining module. In this way, the segmentation network can be trained in a fully supervised manner. However, in practice, generating the subcloud-level annotation is still time-consuming.\nLately, in~\\cite{XuLee_CVPR20}, Xu~\\emph{et al.} propose a semi-supervised algorithm, which uses three constraints on the unlabeled points, $i.e.$, the block level labels for penalizing the negative categories in point clouds, data augmentation with random in-plane rotation and flipping for feature consistency and a spatial and color smoothness constraint in point clouds. \n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.9\\linewidth]{.\/img\/framework_final_v2.pdf}\n\t\\end{center}\n\t\\caption{Overview of the proposed semi-supervised semantic point cloud segmentation network (SSPC-Net). We first leverage the gated GNN to extract superpoints features. Then based on the superpoint graph, we conduct the dynamic label propagation strategy to generate pseudo labels. Next, based on the supervised superpoints and the extended superpoints, we perform a coupled attention mechanism to further boost the extraction of discriminative contextual features in the point cloud.}\n\t\\label{fig_outline}\n\\end{figure*}\n\n\n\\section{Our Method}\nIn this section, we present our semi-supervised point cloud segmentation network and the outline of our framework is shown in Fig. \\ref{fig_outline}. We first introduce the superpoint graph embedding module. Then we propose a dynamic label propagation approach combined with a superpoint dropout strategy. Next, we propose a coupled attention mechanism to learn discriminative contextual features of superpoints. Finally, we depict the framework of our method.\n\n\n\\subsection{Superpoint Graph Embedding}\\label{sec_graph_embeddding}\n\nTo obtain the superpoints and learn the superpoint features, following \\cite{landrieu2018large}, we perform an unsupervised superpoints partition approach to generate superpoints and then build superpoint graphs combined with graph neural network (GNN) for superpoints feature embedding.\nDenote $\\mathcal{G=(V, E)}$ as the superpoint graph built upon superpoints, where $\\mathcal{V}$ is the node set and $\\mathcal{E}$ is the edge set. Edge $(i,j) \\in \\mathcal{E}$ links node $i \\in \\mathcal{V}$ with $j \\in \\mathcal{V}$. \nWe first perform a lightweight PointNet-like structure on the superpoints to obtain superpoints features. After that, we learn the superpoint embedding with the gated GNN used in~\\cite{li2015gated}.\nGiven the superpoint embeddings and the semi-supervision, we can penalize the model with incomplete supervision. For a point cloud consists of $N$ superpoints, we define $a_i \\in \\{0,1\\}^{N}$ to indicate whether the $i$-th superpoint has supervision. Then the segmentation loss $\\mathcal{L}_{s}$ on the superpoint graph embedding module can be formulated as:\n\\begin{equation}\n\\mathcal{L}_{s} = \\frac{1}{A} \\sum \\nolimits _{i=1}^{N}a_i \\cdot \\mathcal{F}_{loss}\\left(z_i, \\bm{y}_i\\right)\n\\end{equation}\nwhere $\\mathcal{F}_{loss}$ is the loss function and we choose the cross-entropy loss in experiments, $A=\\sum \\nolimits _{i=1}^{N} a_i$ is adopted for normalization, $z_i$ represents the superpoint-level label of $i$-superpoint and $\\bm{y}_i$ is the prediction logit.\n\nThe reason why we choose the superpoint graph as the representation of point cloud is at two points. On the one hand, the superpoint is geometrically isotropic and therefore we can directly extend the point-level label to the superpoint-level label, which alleviates the lack of supervision. On the other hand, since the superpoint graph is rooted in the geometric structure of the point cloud, where the linking edges between the superpoints greatly facilitate the feature propagation. Thus we can obtain more discriminative contextual features of superpoints. \n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.95\\linewidth]{.\/img\/extension_v3.pdf}\n\t\\end{center}\n\t\\caption{The procedure of our dynamic label propagation. We progressively propagate the superpoint-level label and discard the extended superpoint with low confidence.}\n\t\\label{fig_extension}\n\\end{figure}\n\n\n\\subsection{Dynamic Label Propagation}\\label{sec_extension_growing}\nTo propagate superpoint labels, we propose a dynamic label propagation strategy to generate pseudo labels. \nSuppose we have constructed three sets: the supervised superpoints set $S$, unsupervised superpoints set $U$, and extended superpoints set $E$. Note that at the beginning we set $E=\\varnothing$. Besides, elements in each set indicate the index of superpoints.  \n\n\nFor $\\forall ~i \\in T, T=S\\cup E$, we use the adjacent superpoints to construct candidate set $\\mathcal{N}_i$, where we consider propagating labels in it. Suppose $z_i$ is the label of $i$-th superpoint. Note that $\\forall ~j \\in\\mathcal{N}_i$ must satisfy two constraints: $j\\in U$ and the predicted category of the $j$-th superpoint should be the same as that of the $i$-th superpoint, that is, $z_i$. Compared with other unsupervised superpoints, elements in $\\mathcal{N}_i$ are  with higher possibilities to be assigned with pseudo labels, due to the close geometric relations and the close distances to the $i$-th superpoint. To generate high-quality pseudo labels, we assess the confidence scores of the superpoints in $\\mathcal{N}_i$ and denote the scores as $\\bm{m}_i\\in\\mathbb{R}^{|\\mathcal{N}_i|}$. Then, we enumerate all the superpoints in $\\mathcal{N}_i$ and select the superpoint with the highest confidence score. The operation can be formulated as:\n\\begin{equation}\nj^*=\\mathop{\\arg\\max}\\limits_{j=1,2,\\ldots, |\\mathcal{N}_i|}({m}_{i,j})\n\\label{eqn:constraint}\n\\end{equation}\nwhere $j^*$ represents the index of the superpoint with the highest confidence score in $\\mathcal{N}_i$. To further ensure the high quality of pseudo labels, we set the threshold $\\tau$ to filter the selected superpoints with dissatisfactory confidence values. When the confidence score $m_{i,j^*}\\geqslant \\tau$, the $j^*$-th superpoint is selected and assigned with pseudo label $z_i$. Then $j^*$ will be removed from the unsupervised superpoints set $U$ and added to the extended superpoints set $E$. On the contrary, if there is no superpoint satisfying the constraint, no superpoint will be extended from $\\mathcal{N}_{i}$. In the experiments, $\\tau$ is empirically set to 0.9. Note that for each extension procedure, we merge the supervised superpoints set $S$ and the extended superpoints set $E$ to the new set $T=S\\cup E$ for further extension. Because the extended superpoints with pseudo labels can also be treated as the superpoints with supervision for further label propagation. In this way, we can progressively propagate the labels of the supervised superpoints and generate more high-quality pseudo labels for unsupervised superpoints in $U$. What's more, Algorithm \\ref{algorithm_extension} shows the details of the graph-based supervision extension procedure.\n\n\n\\begin{algorithm}[t]\n\t\\DontPrintSemicolon\n\t\\KwIn{Supervised superpoints set $S$, unsupervised superpoints set $U$, extended superpoints set $E$, threshold $\\tau$}\n\t\\KwOut{Updated sets $U$ and $E$}\n\t$T = S\\cup E$ \\\\\n\t\\For {$~i \\in T$}\n\t{\n\t\tDenote $z_i$ as the label of $i$-th superpoint \\\\\n\t\tConstruct the candidate supeproints set $\\mathcal{N}_i$ \\\\\n\t\t\\If{$\\mathcal{N}_i\\neq \\varnothing $}\n\t\t{\n\t\t\tGenerate the confidence scores $\\bm{m}_i$ \\\\\n\t\t\t$j^*=\\mathop{\\arg\\max}\\limits_{j=1,2,\\ldots, |\\mathcal{N}_i|}({m}_{i,j})$ \\\\\n\t\t\t\\If{$m_i^{j^*}\\geqslant \\tau$}\n\t\t\t{\n\t\t\t\tAssign pseudo label $z_i$ to the $j^*$-th superpoint \\\\\n\t\t\t\t$U := U \\setminus \\{j^*\\} \\quad E := E \\cup \\{j^*\\}$\n\t\t\t}\n\t\t}\n\t\t\n\t}\n\t\\caption{Graph-based supervision extension}\\label{algorithm_extension}\n\\end{algorithm}\n\n\nSince our extension strategy is performed progressively, we consider removing the low-confidence superpoints in the extended superpoints set $E$. Hence, we propose a superpoint dropout strategy assessing the reliability of the extended superpoints in the embedding space. In the superpoints set $T=S\\cup E$, we cluster the superpoints into $c$ classes according to the superpoints labels or pseudo labels, where $c$ is the number of categories. Suppose $\\mathcal{C}_i$ is the $i$-th cluster set that contains the index of the superpoints belonging to the $i$-th category. In addition, we denote $\\bm{v}_i$ as the feature of the cluster center of $\\mathcal{C}_i$, which is computed by averaging the features of all the superpoints in $\\mathcal{C}_i$. We assess the confidence of the extended superpoints by considering its distance to the corresponding cluster center in the feature space. For $\\forall ~j\\in E\\cap \\mathcal{C}_i$, its Euclidean distance to the cluster center in the feature space is formulated as: \n\\begin{equation}\nd_i^j = \\left\\|\\bm{f}_{j}-\\bm{v}_i \\right \\|_2\n\\end{equation} \nwhere $\\bm{f}_j\\in\\mathbb{R}^{D}$ is the feature of $j$-th superpoint, and $\\bm{v}_i\\in\\mathbb{R}^{D}$ is the feature of cluster center. Smaller distance indicates the higher reliability of extended superpoints, whereas the larger distance means the higher uncertainty. Therefore, in each cluster, we discard $k$ extended superpoints that are furthest from the cluster center, where $k$ is set to 0.05$*|E\\cap\\mathcal{C}_i|$. In other words, we retain the most reliable 95\\% superpoints and drop the 5\\% unreliable superpoints in the set $E\\cap\\mathcal{C}_i$. Our superpoint dropout strategy is explained in Algorithm \\ref{algorithm_dropout}.\n\nConcretely, as shown in Fig. \\ref{fig_extension}, we perform our graph-based dynamic label propagation strategy every ${M}$ epochs, therefore, the extended superpoints are gradually ``growing'' on the graph from the supervised superpoints. The reason why we conduct the extension operation in a multi-stage manner instead of every epoch is that our extension strategy is a cumulative one, which means that too much extension operations will cause redundant extended superpoints and aggravate the memory cost. Meanwhile, the model is not stable at the beginning, which is not conducive to generating extended superpoints. \n\n\\begin{algorithm}[t]\n\t\\DontPrintSemicolon\n\t\\KwIn{Number of classes $c$, supervised superpoints set $S$, unsupervised superpoints set $U$, extended superpoints set $E$}\n\t\\KwOut{Updated sets $U$ and $E$}\n\t$T = S\\cup E$ \\\\\n\tCluster on $T$ and obtain $c$ cluster sets: $\\mathcal{C}_1,\\mathcal{C}_2, \\ldots,\\mathcal{C}_c$ \\\\\n\t\\For {$i = 1:c $}\n\t{\t\t\t\n\t\tCompute the feature $\\bm{v}_i$ of the cluster center of $\\mathcal{C}_i$ \\\\\n\t\t\\For{each $j \\in E\\cap\\mathcal{C}_i$}\n\t\t{\n\t\t\tGenerate the feature $\\bm{f}_j$ of the $j$-th superpoint \\\\\n\t\t\tCompute the distance $d_i^j = \\left\\|\\bm{f}_{j}-\\bm{v}_i \\right \\|_2$\n\t\t}\n\t\tFind the farthest 5\\% superpoints (set as $\\mathcal{C}_{drop}$) in $E\\cap\\mathcal{C}_i$ from the cluster center according to the distance $\\bm{d}_i$ \\\\\n\t\t$E := E \\setminus \\mathcal{C}_{drop}\\quad U := U \\cup \\mathcal{C}_{drop}$\n\t}\n\t\\caption{Superpoint dropout strategy}\\label{algorithm_dropout}\n\\end{algorithm}\n\n\n\n\\subsection{Coupled Attention for Feature Enhancement}\\label{sec_feature_enhance}\nAiming to learn more discriminative contextual features in point clouds, we propose a coupled attention mechanism.\nFor $\\forall ~i\\in S$, we denote the corresponding embedding as $\\bm{h}_i \\in \\mathbb{R}^{D}$. Similarly, for $\\forall ~j\\in E$, we denote the corresponding embedding as $\\bm{h}_j$. By weighing all the extended superpoints, we extract the novel contextual feature of $i$-th superpoint with attention mechanism:\n\\begin{equation}\n\\bm{x}_i=\\sum \\nolimits_{j\\in E} g\\left(\\phi(\\bm{h}_i, \\bm{h}_j) \\right)\\odot \\alpha(\\bm{h}_j)\n\\end{equation}\nwhere $\\phi(\\bm{h}_i, \\bm{h}_j)=\\mathop{MLP}(\\bm{h}_i-\\bm{h}_j)$ embeds the channel-wise relations between superpoints, $\\alpha(\\bm{h}_j)=\\mathop{MLP}(\\bm{h}_j)$ is a unary function for individual superpoint embedding, $\\phi(\\cdot,\\cdot):\\mathbb{R}^{D} \\rightarrow \\mathbb{R}^{D}$ and $\\alpha:\\mathbb{R}^{D} \\rightarrow \\mathbb{R}^{D}$, $\\odot$ is the Hadamard product. $g$ is a normalization function and is defined as:\n\\begin{equation}\ng\\left(\\phi_l(\\bm{h}_i, \\bm{h}_j) \\right) = \\frac{\\exp(\\phi_{l}({\\bm h}_{i}, {\\bm h}_{j}))}{\\sum\\nolimits_{r \\in E}\\exp(\\phi_{l}({\\bm h}_{i}, {\\bm h}_{r}))}\n\\end{equation}\nwhere $l=1,2,\\ldots,D$, represents $l$-th element of embedding $\\phi_{l}(\\cdot,\\cdot)$. Consequently, the matrix representation of the attention operation on the supervised superpoints in $S$ can be formulated as:\n\\begin{equation}\n\\bm{X}_s = \\sum \\nolimits_{j\\in E}\\bm{W}_{es,j} \\odot \\bm{H}_{e,j}\n\\end{equation}\nwhere $\\bm{X}_s\\in\\mathbb{R}^{|S|\\times D}$, $\\bm{W}_{es,j}\\in\\mathbb{R}^{|S|\\times D}$, $\\bm{H}_{e,j} \\in \\mathbb{R}^{|S|\\times D}$ and $j$ enumerates the extended superpoints in $E$. Note that $\\bm{W}_{es}\\in\\mathbb{R}^{|S|\\times|E|\\times D}$ represents the channel-wise weights from the extended superpoints to the supervised superpoints.\n\n\nOnce we obtain the attention embedding $\\bm{X}_s\\in\\mathbb{R}^{|S|\\times D}$, we can derive new segmentation logits of supervised superpoints and formulate the loss as:\n\\begin{equation}\n\\mathcal{L}_{es} = \\frac{1}{|S|} \\sum \\nolimits _{i \\in S}\\mathcal{F}_{loss}\\left(z_i, \\mathop{FC} \\left(\\bm{X}_{s,i}\\right)\\right)\n\\end{equation}\nwhere $z_i$ is the superpoint-level label, $\\bm{X}_{s,i}$ is the attention feature of the corresponding supervised superpoint, and $\\mathcal{F}_{loss}$ is the cross-entropy loss adopted in experiments. Note that $\\mathop\n{FC}$ is the fully connected layer, which maps $\\bm{X}_{s,i}\\in\\mathbb{R}^{|S|\\times D}$ from the $D$-dim to the dimension of the categories.\n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.95\\linewidth]{.\/img\/coupled_attention_v6.pdf}\n\t\\end{center}\n\t\\caption{The coupled attention for feature enhancement.\n\t}\n\t\\label{fig_attention}\n\\end{figure}\n\n\n\nSimilarly, to promote the feature characterization of the extended superpoints, we then perform attention on the extended superpoints in reverse. By weighting the new features enhanced by the attention operation of the supervised superpoints, we boost the context feature propagation and thus enhance the robustness of the features of the extended superpoints. Thus, for $\\forall j \\in E$, the new embedding of the corresponding superpoint can be calculated as:\n\\begin{equation}\n\\bm{y}_j=\\sum \\nolimits_{i\\in S} g\\left(\\psi(\\bm{h}_j, \\bm{x}_i) \\right)\\odot \\beta(\\bm{x}_i)\n\\end{equation}\nwhere $\\psi(\\bm{h}_j, \\bm{x}_i)=\\mathop{MLP}(\\bm{h}_j-\\bm{x}_i)$ characterizes the dependencies of the extended superpoints on the attention embeddings of the supervised superpoints. $\\beta(\\bm{x}_i)=\\mathop{MLP}(\\bm{x}_i)$ is a unary function similar to $\\alpha$, $\\psi(\\cdot,\\cdot):\\mathbb{R}^{D} \\rightarrow \\mathbb{R}^{D}$ and $\\beta:\\mathbb{R}^{D} \\rightarrow \\mathbb{R}^{D}$. $g$ is a normalization function defined as:\n\\begin{equation}\ng\\left(\\psi_{l}(\\bm{h}_j, \\bm{x}_i) \\right) = \\frac{\\exp(\\psi_{l}({\\bm h}_{j}, {\\bm x}_{i}))}{\\sum\\nolimits_{r \\in S}\\exp(\\psi_{l}({\\bm h}_{j}, {\\bm x}_{r}))}\n\\end{equation}\nwhere $l=1,2,\\ldots,D$, denotes $l$-th element of embedding $\\psi(\\cdot,\\cdot)$. Then the matrix representation of the attention operation on the extended superpoints in $E$ can be defined as:\n\\begin{equation}\n\\bm{Y}_e = \\sum \\nolimits_{i\\in S}\\bm{W}_{ese,i} \\odot \\bm{\\mathcal{X}}_{s,i}\n\\end{equation}\nwhere $\\bm{Y}_e\\in\\mathbb{R}^{|E|\\times D}$, $\\bm{W}_{ese,i}\\in\\mathbb{R}^{|E|\\times D}$, $\\bm{\\mathcal{X}}_{s,i}\\in\\mathbb{R}^{|E|\\times D}$ and $i$ enumerates superpoints in $S$. Note that $\\bm{\\mathcal{X}}_{s}$ is the feature after employing function $\\beta(\\cdot)$ on attention feature $\\bm{X}_{s}$. In this way, we develop the coupled attention, $i.e.$, $\\bm{W}_{ese}\\in\\mathbb{R}^{|S|\\times|E|\\times D}$ denotes the channel-wise weights from the attentional supervised superpoints to extended superpoints.\n\nThen the loss $\\mathcal{L}_{ese}$ on the extended superpoints with enhanced attention features can be formulated as:\n\\begin{equation}\n\\mathcal{L}_{ese} = \\frac{1}{|E|} \\sum \\nolimits _{j \\in E}\\mathcal{F}_{loss}\\left(z^{p}_j, \\mathop{FC} \\left(\\bm{Y}_{e,j}\\right)\\right)\n\\end{equation}\nwhere $z^{p}_j$ is the pseudo label and $\\mathcal{F}_{loss}$ is the cross-entropy loss as well. $\\mathop{FC}$ maps the feature to the category space.\n\nSpecifically, as shown in Fig. \\ref{fig_attention}, our coupled attention considers the intra- and inter-relations concurrently. To encourage the feature consistency in different point clouds, \nwe integrate the supervised superpoints and extended superpoints in various point clouds into sets $S$ and $E$, respectively.\nThe connections between $S$ and $E$ are constructed within and across various point cloud samples, and superpoints with the same labels are encouraged to have more similar semantic embeddings compared to those with diverse classes. \nAs a result, by alternatively performing attention on the supervised and extended superpoints, more long-range dependencies between superpoints are built. Hence, the model learns more discriminative and robust contextual features of the supervised and unsupervised superpoints. \n\n\n\n\n\\subsection{Framework}\\label{sec_framework}\nThe framework of our model is illustrated in Fig. \\ref{fig_outline}. In our framework, the superpoint graph embedding module is the basis of our point cloud feature embedding. Based on this module, the dynamic label propagation method assesses the semantic similarity between the superpoints and propagates the superpoint-level supervision along the edges of the superpoint graph. Then, with the extended superpoints searched by the dynamic label propagation module, we propose a coupled attention mechanism to boost the contextual feature learning of the point cloud. \n\nThe final objective function is a combination of the three objectives $\\mathcal{L}_{final}=\\mathcal{L}_{s} + \\lambda_1\\cdot \\mathcal{L}_{es} +\\lambda_2\\cdot \\mathcal{L}_{ese}$ and we empirically set $\\lambda_1, \\lambda_2$ to 1.\nAs shown in Fig. \\ref{fig_outline}, the dynamic label propagation module and coupled attention module are only conducted in the training stage. For testing, we obtain the inferred prediction directly from the superpoint graph embedding module. \n\n\n\\section{Experiments}\n\\subsection{Implementation Details}\nTo train our model, we adopt Adam optimizer with a base learning rate of 0.01. For the S3DIS~\\cite{armeni20163d}, ScanNet~\\cite{dai2017scannet} and vKITTI~\\cite{Gaidon2016Virtual} dataset, we employ the mini-batch size of 4, 8, 8, respectively. We empirically implement the dynamic label propagation module every $M=40$ epochs. \n\n\n\n\\textbf{Semi-supervision generation.}\nTo produce the semi-supervision of point clouds, we randomly select a part of the points with annotations in each class. For example, given a point cloud containing $n$ points with $c$ classes, suppose the supervision rate be $r$, then we evenly distribute the supervision budget $r\\cdot n$ and randomly sample $ (r\\cdot n)\/c$ points in each category as the supervised points.\nThe label of superpoint is the category with the most annotated points.\nIf there is no supervised point contained, then the superpoint will be unsupervised. \nNote that compared with the sampling strategy of random sampling annotated points directly in point clouds, our labeling mechanism is more in coincident with the human annotation behavior, since the random sampling strategy will result in that most of the supervised points will be occupied by the areas with simple geometric structure but more points, $e.g.$, walls, roads, etc.\nFor evaluation, all the quantitative results are computed at the point level.\n\n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\begin{adjustbox}{width=0.88\\linewidth}\n\t\t\\large\n\t\t\\begin{tabular}{c|l|c|ccc}\n\t\t\n\t\t\t\\toprule\n\t\t\t\\multicolumn{2}{c|}{{\\bf Method}} &\\multicolumn{1}{c|}{Rate}&mIoU&mAcc&OA\\\\\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multicolumn{6}{c}{6-fold cross validation} \\\\\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{7}{*}{Full}\n\t\t\t&PointNet&100\\%&47.6&66.2&78.5\\\\\t\t\t\n\t\t\t&SPGraph &100\\%& 62.1 &73.0 &85.5 \\\\\n\t\t\t&PointCNN &100\\%&{\\bf65.3} &{\\bf75.6} &{\\bf88.1} \\\\\n\t\t\n\t\t\n\t\t\t&RSNet&100\\%&56.4&66.4&-\\\\\n\t\t\n\t\t\t& G+RCU2 &100\\%&49.7 &66.4 &81.1 \\\\\n\t\t\t& 3P-RNN &100\\%&56.3 &73.6 &86.9 \\\\\n\t\t\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{3}{*}{Semi-}\n\t\t\n\t\t\n\t\t\n\t\t\t&Baseline &0.002\\% &45.1 &63.7 &73.9 \\\\\n\t\t\t&{\\bf SSPC-Net} &0.002\\% &48.5 &68.3 &79.1\\\\\t\n\t\t\t&{\\bf SSPC-Net} &0.01\\%&{\\bf54.5}&{\\bf70.8} &{\\bf 80.4}\\\\\t\t\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multicolumn{6}{c}{Fold 5} \\\\\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{5}{*}{Full}\n\t\t\t&PointNet&100\\%&41.1&49.0&-\\\\\n\t\t\t&PointNet++&100\\%&47.8&-&-\\\\\n\t\t\t&SPGraph &100\\%&{\\bf58.0} &{\\bf66.5} &{\\bf86.3} \\\\\n\t\t\n\t\t\t&SegCloud&100\\%&48.9&57.3&-\\\\\n\t\t\t&PointCNN&100\\%&57.2&63.8&85.9\\\\\n\t\t\n\t\t\t\\midrule\n\t\t\t\\multirow{6}{*}{Semi-}\n\t\t\n\t\t\t&Semi-Seg&1pt &44.5&-&-\\\\\n\t\t\t&Semi-Seg&10\\% &48.0&-&-\\\\\t\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t&Baseline &0.002\\% &39.6 &52.1 &72.4 \\\\\t \t\n\t\t\n\t\t\t&{\\bf SSPC-Net}&0.002\\% &43.0 &56.4 &76.2\\\\\n\t\t\t&{\\bf SSPC-Net}&0.01\\%&51.5 &63.8 &82.0\\\\\n\t\t\t&{\\bf SSPC-Net}&1pt &{\\bf53.8} &{\\bf63.9} &{\\bf83.8}\\\\\n\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Evaluation on the S3DIS dataset.}\n\t\\label{tab_results_s3dis}\n\\end{table}\n\n\n\\subsection{Semi-supervised Semantic Segmentation}\n\\textbf{S3DIS.}\nS3DIS~\\cite{armeni20163d} dataset is an indoor 3D dataset including 6 areas and 13 categories. Three metrics are adopted for quantitative evaluation: mean IoU (mIoU), mean class accuracy (mAcc), and overall accuracy (OA).\n\nThe quantitative and visual results are shown in Tab. \\ref{tab_results_s3dis} and Fig. \\ref{fig_visual_results}, respectively. For a fair comparison, we test our framework with the ``1pt'' labeling strategy adopted in \\cite{XuLee_CVPR20} (dubbed ``Semi-Seg'' in Tab. \\ref{tab_results_s3dis}) as well, which samples one point in each category of each block as the supervised point. It can be seen that our SSPC-Net achieves a significant gain of 9.3\\% in terms of mIoU with the ``1pt'' labeling strategy. In \\cite{XuLee_CVPR20}, Xu {\\em et~al.} split the point cloud into blocks and then train and test their model on each block separately. Nonetheless, our model learns the embeddings of superpoints in the whole point cloud, therefore we can obtain more discriminative contextual features and yield better performance. Note that in Tab.~\\ref{tab_results_s3dis}, ``Baseline'' represents our method without the label propagation strategy and coupled attention mechanism. One can see that our SSPC-Net improves the performance from 39.6\\% to 43.0\\% in terms of mIoU with the supervision rate of 0.002\\% on Area 5 of the S3DIS dataset, benefiting from the pseudo labels generated from the label propagation and the discriminative contextual features extracted by the coupled attention mechanism.\n\n\\textbf{ScanNet.}\nScanNet~\\cite{dai2017scannet} is an indoor scene dataset containing 1513 point clouds with 20 categories. We split the dataset into a training set with 1201 scenes and a testing set with 312 scenes following \\cite{qi2017pointnet++}. We adopt overall semantic voxel labeling accuracy (OA) and mean IoU (mIoU) for evaluation.\n\n\nWe list the quantitative results on the testing set in Tab.~\\ref{tab_scannet_vkitti}. Similar to S3DIS, ScanNet is also an indoor dataset, but the point cloud of ScanNet is much sparser than that of S3DIS. This brings greater challenges to the propagation of supervised labels. However, the proposed model can still achieve good segmentation results and even outperform some fully supervised methods like PointNet \\cite{qi2017pointnet} with semi-supervision. Furthermore, the performance of the proposed model is much better than the baseline method, which further validates the effectiveness of our method.\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\LARGE\n\t\\begin{adjustbox}{width=0.99\\linewidth}\n\t\t\\begin{tabular}{c|l|c|cc|ccc}\n\t\t\t\\toprule\n\t\t\t\\multicolumn{2}{c|}{\\multirow{2}{*}{{\\bf Method}}} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Rate}} &\n\t\t\t\\multicolumn{2}{c|}{ScanNet} & \\multicolumn{3}{c}{vKITTI}  \\\\\n\t\t\t\\multicolumn{2}{c|}{}                       & & mIoU & OA                    & mIoU & mAcc & OA            \\\\ \n\t\t\t\\midrule\n\t\t\t\\multirow{7}{*}{Full}\n\t\t\t&PointNet&100\\%\t&-&73.9&34.4&47.0&79.7\\\\\n\t\t\t&PointNet++&100\\%&-&{\\bf84.5}  &-&-&-\\\\\n\t\t\t\n\t\t\t&SSP + SPG&100\\% &- &- &{\\bf52.0} &{\\bf67.3} &84.3    \\\\\n\t\t\n\t\t\t&G+RCU&100\\% &-&-      &35.6&57.6&79.7\\\\\n\t\t\n\t\t\n\t\t\t&RSNet&100\\% &{\\bf39.3} &79.2 &-&-&-    \\\\\n\t\t\t&3P-RNN&100\\%&-&- &41.6&54.1&{\\bf87.8} \\\\\n\t\t\n\t\t\n\t\t\t&3DCNN&100\\%&-&73.0 &-&-&- \\\\\n\t\t\t\\midrule\n\t\t\t\\multirow{3}{*}{Semi-}\n\t\t\n\t\t\n\t\t\t&Baseline&0.01\\% &24.1&38.2 &35.7 &53.4 &79.2 \\\\\n\t\t\t&{\\bf SSPC-Net}&0.01\\% &27.1&66.6 &41.0 &55.7 &81.2 \\\\\n\t\t\t&{\\bf SSPC-Net}&0.05\\% &{\\bf39.3}&{\\bf77.1} &{\\bf50.6} &{\\bf64.8} &{\\bf85.4} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Evaluation on the ScanNet and vKITTI datasets.}\n\t\\label{tab_scannet_vkitti} \t \n\\end{table}\n\n\n\\begin{figure}[t]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.87\\linewidth]{.\/img\/visual_v4.pdf}\n\t\\end{center}\n\t\\caption{The visual results on the S3DIS dataset with supervision rate of 0.002\\%.}\n\t\\label{fig_visual_results}\n\\end{figure}\n\n\n\\textbf{vKITTI.}\nvKITTI~\\cite{Gaidon2016Virtual} dataset mimics the real-world KITTI dataset and contains the synthetic outdoor scenes with 13 classes (including road, tree, terrain, car, etc.). For evaluation, we split the dataset into 6 non-overlapping sub-sequences and employ 6-fold cross validation following \\cite{ye20183d}. Mean IoU (mIoU), mean class accuracy (mAcc)\nand overall accuracy (OA) are employed for evaluation.\n\n\nThe quantitative results are presented in Tab.~\\ref{tab_scannet_vkitti}. With the 0.01\\% point-level annotations, compared with the baseline method, our model achieves better segmentation results due to the dynamic label propagation strategy and the discriminative contextual features generated from the coupled attention module. In addition, our model can achieve better or comparable performance than some fully supervised methods with only 0.01\\% and 0.05\\% of the supervised points.\n\n\n\\subsection{Ablation Study}\n\\textbf{Contribution of individual components.}\nIn this section, we investigate the contribution of the proposed components to model performance. The evaluation results on Area5 of the S3DIS dataset of different components with the supervision ratio of 0.002\\% and 0.01\\% are shown in Tab. \\ref{tab_components}, where the components are the graph embedding (Graph Emb.), dynamic label propagation (Label Prop.), coupled attention for feature enhancement (Coup. Attn.). It can be observed that there is an obvious promotion on the performance with the addition of dynamic label propagation and coupled attention module, which further demonstrates the effectiveness of these strategies for the semi-supervision.\n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\huge\n\t\\begin{adjustbox}{width=1.0\\linewidth}\n\t\n\t\n\t\t\\begin{tabular}{ccc|ccc|ccc}\n\t\t\t\\toprule\n\t\t\t\\multicolumn{3}{c|}{{\\bf Components}} & \\multicolumn{3}{c|}{Rate$=$0.002\\%}                                           & \\multicolumn{3}{c}{Rate$=$0.01\\%}                                            \\\\\n\t\t\t\\midrule\n\t\t\n\t\t\tGraph     & Label    & Coup.    & \\multirow{2}{*}{mIoU} & \\multirow{2}{*}{mAcc} & \\multirow{2}{*}{OA} & \\multirow{2}{*}{mIoU} & \\multirow{2}{*}{mAcc} & \\multirow{2}{*}{OA} \\\\\n\t\t\tEmb. & Prop. & Attn. & & & & &  &   \\\\\t\t\n\t\t\t\\midrule\n\t\t\t\\checkmark\t&\t\t\t&\t\t\t&39.6 &52.1 &72.4 &48.5 &61.2 &80.3 \\\\\n\t\t\t\\checkmark\t&\\checkmark &\t        &40.9 &55.8 &73.6 &50.0 &60.6 &80.8 \\\\\n\t\t\t\\checkmark\t&\\checkmark\t&\\checkmark\t&{\\bf 43.0} &{\\bf 56.4}  &{\\bf 76.2} &{\\bf 51.5} &{\\bf 63.8}  &{\\bf 82.0}\\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{The contribution of different components on Area5 of the S3DIS dataset with different annotation rates.}\n\t\\label{tab_components}\t\n\\end{table}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.78\\linewidth]{.\/img\/curve_v4_cropped.pdf}\n\t\\caption{The percentage of supervised superpoints (ss) and extended superpoints (es) during training. Note that ``all'' means the overall superpoints.}\n\t\\label{fig_ext_ablation}\n\\end{figure}\n\n\n\n\n\\textbf{Supervision rate.} The number of supervised points plays an important role in the segmentation performance. The more labeled points, the smaller gap of data distribution between the semi-supervision and full supervision. To discuss the effect of various labeling rates on model performance, we test our method on Area5 of the S3DIS dataset. The results are shown in Tab. \\ref{tab_comparison}. Combined with Tab. \\ref{tab_results_s3dis}, it can be observed that with only few labeled points, our model has already achieved effective segmentation results. With the growth of supervision, the performance of our model further increases. It is worth noting that we pay more attention to the cases of extremely few supervision signals, which is more challenging for the point cloud segmentation task.\n\n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\large\n\t\\begin{adjustbox}{width=0.83\\linewidth}\n\t\t\\begin{tabular}{c|ccc|c}\n\t\t\t\\toprule\n\t\t\t\\;\\;\\;\\;\\;{\\bf Rate}\\;\\;\\;\\;\\;&\\;\\;mIoU & mAcc&OA\\;\\;&\\;OA of es\\; \\\\\n\t\t\t\\midrule\n\t\t\t0.002\\%      &43.0 &56.4 &76.2     & 87.3 \\\\\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\t0.01\\%      &51.5 &63.8 &82.0      &90.9 \\\\\n\t\t\n\t\t\n\t\t\t0.1\\%       &56.2 &66.1 &84.6      & 91.0 \\\\\n\t\t\n\t\t\n\t\t\t1.0\\%       &58.3 &66.5 &85.7      &90.1 \\\\\n\t\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Comparison of various supervision rates on Area5 of the S3DIS dataset, where ``es'' represents the extended superpoints.} \n\t\\label{tab_comparison}\n\\end{table}\n\n\n\n\\begin{table}[t]\n\t\\centering\n\t\\LARGE\n\t\\begin{adjustbox}{width=0.82\\linewidth}\n\t\t\\begin{tabular}{c|ccc}\n\t\t\t\\toprule\n\t\t\t\\;\\;\\;\\;{\\bf Interval} \\bm{$M$}\\;\\;\\;\\; &\\;\\;\\;mIoU\\;\\;\\; &\\;\\;\\;mAcc\\;\\;\\; &\\;\\;\\;OA\\;\\;\\; \\\\\n\t\t\t\\midrule\n\t\t\t20 &50.2 &61.1 &81.2 \\\\\n\t\t\t30 &50.8 &63.3 &81.5 \\\\\t\t\t\n\t\t\t40 &{\\bf 51.5} &{\\bf 63.8} &{\\bf 82.0} \\\\\n\t\t\t50 &49.6 &61.5 &80.7 \\\\\n\t\t\t60 &49.9 &62.2 &81.0 \\\\\t\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\caption{Comparison of segmentation results with various interval $M$ of the dynamic label propagation method in the case of the supervision rate of 0.01\\%.} \n\t\\label{tab_extension_M}\n\\end{table}\n\n\n\\textbf{Number of the extended superpoints.} The dynamic label propagation strategy plays an important role in our model. As shown in Fig. \\ref{fig_ext_ablation}, we show the proportion of the supervised superpoints and extended superpoints in the training set when testing on Area 5 of the S3DIS dataset. With the increase of the annotated points, the proportion of the supervised superpoints increases rapidly. Because the probability of a superpoint containing a supervised point is getting higher as well. However, when there are fewer supervised points, the percentage of extended superpoints is obviously larger.\nThis demonstrates the importance of pseudo labels facing extremely few point annotations.\n\n\n\n\\textbf{Quality of the extended superpoints.} To analyze the quality of the extended superpoints, we evaluate the overall accuracy of the extended superpoints (OA of es) in Tab. \\ref{tab_comparison}. Noted that, similar to the aforementioned metrics, the quantitative results of the extended superpoints are conducted at the point level as well. \nFrom Tab. \\ref{tab_comparison}, one can see that the overall accuracy of extended superpoints is around 90\\%, which demonstrates the high quality of extended superpoints. This further proves the effectiveness of our label propagation strategy which generates high-quality pseudo labels. In addition, the high quality of pseudo labels of the extended superpoints further reveals the reason for the improved performance based on the label propagation module.\n\n\n\\textbf{Epoch interval in dynamic label propagation.}\nDuring the training, we perform the dynamic label propagation method every $M$ epochs. For comparison, we train our model with various interval $M$ while keeping other parameters unchanged with the supervision rate of 0.01\\%. The evaluation results on Area 5 of the S3DIS dataset are shown in Tab. \\ref{tab_extension_M}. It can be observed that when $M=40$, our model achieves the best performance.\n\n\\section{Conclusion}\nIn this paper, we proposed a semi-supervised point cloud segmentation network. We first partitioned the point cloud into superpoints and built superpoint graphs to explore the long-range relations in the point cloud. Then based on superpoint graphs, we proposed a dynamic label propagation method combined with a superpoint dropout strategy to generate high-quality pseudo labels for the unsupervised superpoints. Next, we proposed a coupled attention module to learn discriminative contextual features of superpoints and fully exploit the generated pseudo labels. Our method can achieve better performance than the current semi-supervised point cloud segmentation methods with fewer labels.\n\n\\section{Acknowledgments}\nThis work was supported by the National Science Fund of China (Grant Nos.\nU1713208, 61876084), Program for Changjiang Scholars.\n\n{\n\t\\bibliographystyle{IEEEtran}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Relative systoles}\n\n\nWe prove a systolic inequality for a~$\\phi$--relative systole of a\n\\mbox{$\\phi$--essential}\n$2$--complex~$X$, where~$\\phi \\colon\\thinspace \\pi_1(X) \\to G$ is a homomorphism to a\nfinitely presented group~$G$.  Thus, we show that universally for\nany~$\\phi$--essential Riemannian~$2$--complex~$X$, and any~$G$, we\nhave~$\\sys(X, \\phi)^2 \\leq 8 \\, \\area(X)$.  Combining our results with\na method of L.~Guth, we obtain new quantitative results for\ncertain~$3$--manifolds: in particular for the Poincar\\'e\nhomology sphere~$\\Sigma$, we have~$\\sys(\\Sigma)^3 \\leq 24 \\, \\vol(\\Sigma)$.  To\nstate the results more precisely, we need the following definition.\n\nLet~$X$ be a finite connected~$2$--complex.  Let~$\\phi \\colon\\thinspace \\pi_{1}(X) \\to\nG$ be a group homomorphism.  Recall that~$\\phi$ induces a classifying\nmap (defined up to homotopy)~$X \\to K(G,1)$.\n\n\\begin{definition}\nThe complex~$X$ is called~$\\phi$--{\\em essential\\\/} if the classifying\nmap~$X \\to K(G,1)$ cannot be homotoped into the~$1$--skeleton\nof~$K(G,1)$.\n\\end{definition}\n\n\n\n\\begin{definition}\nGiven a piecewise smooth Riemannian metric on~$X$, the~$\\phi$--relative\nsystole of~$X$, denoted~$\\sys(X,\\phi)$, is the least length of a loop\nof~$X$ whose free homotopy class is mapped by~$\\phi$ to a nontrivial\nclass.\n\\end{definition}\n\nWhen~$\\phi$ is the identity homomorphism of the fundamental group, the\nrelative systole is simply called the systole, and denoted~$\\sys(X)$.\n\n\\begin{definition} \n\\label{def:sigma}\nThe~$\\phi$--systolic area~$\\sigma_\\phi(X)$ of~$X$ is defined as\n\\begin{equation*}\n\\sigma_{\\phi}(X) = \\frac{\\area(X)}{\\sys(X,\\phi)^{2}}.\n\\end{equation*}\nFurthermore, we set\n\\begin{equation*}\n\\sigma_{*}(G) = \\inf_{X, \\phi} \\sigma_{\\phi}(X),\n\\end{equation*}\nwhere the infimum is over all~$\\phi$--essential piecewise Riemannian\nfinite connected \\mbox{$2$--complexes}~$X$, and homomorphisms~$\\phi$\nwith values in~$G$.\n\\end{definition}\n\nIn the present text, we prove a systolic inequality for\nthe~$\\phi$--relative systole of a~$\\phi$--essential~$2$--complex~$X$.\nMore precisely, in the spirit of Guth's text \\cite{Gu09}, we prove a\nstronger, {\\em local\\\/} version of such an inequality, for almost\nextremal complexes with minimal first Betti number.  Namely, if~$X$\nhas a minimal first Betti number among all~$\\phi$--essential piecewise\nRiemannian \\mbox{$2$--complexes} satisfying~$\\sigma_{\\phi}(X) \\leq\n\\sigma_*(G) +\\varepsilon$ for an~$\\varepsilon>0$, then the area of a\nsuitable disk of~$X$ is comparable to the area of a Euclidean disk of\nthe same radius, in the sense of the following result.\n\n\n\\begin{theorem} \n\\label{13}\nLet~$\\varepsilon >0$.  Suppose~$X$ has a minimal first Betti number\namong all~$\\phi$--essential piecewise Riemannian\n$2$--complexes satisfying~$\\sigma_{\\phi}(X) \\leq \\sigma_*(G)\n+\\varepsilon$.  Then each ball centered at a point~$x$ on\na~$\\phi$--systolic loop in~$X$ satisfies the area lower bound\n\\begin{equation*}\n\\area \\, B(x,r) \\geq\n\\frac{\\left(r-\\varepsilon^{1\/3}\\right)^2}{2+\\varepsilon^{1\/3}}\n\\end{equation*}\nwhenever~$r$ satisfies~$\\varepsilon^{1\/3} \\leq r \\leq\n\\frac{1}{2}\\sys(X,\\phi)$.\n\\end{theorem}\n\nA more detailed statement appears in~Proposition~\\ref{prop:minB}.  The\ntheorem immediately implies the following systolic inequality.\n\n\\begin{corollary} \n\\label{coro:A}\nEvery finitely presented group~$G$ satisfies\n\\begin{equation*}\n\\sigma_*(G) \\geq \\frac{1}{8}, \n\\end{equation*}\nso that every piecewise Riemannian~$\\phi$--essential~$2$--complex~$X$\nsatisfies the inequality\n\\begin{equation*}\n\\sys(X,\\phi)^{2} \\leq 8 \\, \\area(X).\n\\end{equation*}\n\\end{corollary}\n\nIn the case of the absolute systole, we prove a similar lower bound\nwith a Euclidean exponent for the area of a suitable disk, when the\nradius is smaller than half the systole, without the assumption of\nnear-minimality.  Namely, we will prove the following theorem.\n\n\\begin{theorem} \n\\label{theo:B}\nEvery piecewise Riemannian essential~$2$--complex~$X$ admits a\npoint~$x\\in X$ such that the area of the~$r$--ball centered at~$x$ is\nat least~$r^2$, that is,\n\\begin{equation}\n\\label{11c}\n\\area ( B(x,r)) \\geq r^2,\n\\end{equation}\nfor all~$r \\leq \\frac{1}{2} \\sys(X)$.\n\\end{theorem}\n\nWe conjecture a bound analogous to \\eqref{11c} for the area of a\nsuitable disk of a \\mbox{$\\phi$--essential}~$2$--complex~$X$, with\nthe~$\\phi$--relative systole replacing the systole, {\\it cf.}~the GG-property\nbelow.  The application we have in mind is in the case\nwhen~$\\phi \\colon\\thinspace \\pi_1(X)\\to \\Z_p$ is a homomorphism from the fundamental\ngroup of~$X$ to a finite cyclic group.  Note that the conjecture is\ntrue in the case when~$\\phi$ is a homomorphism to~$\\Z_2$, by Guth's\nresult \\cite{Gu09}.\n\n\n\\begin{definition}[GG-property%\n\\footnote{GG-property stands for the property analyzed by M.~Gromov\nand L.~Guth}\n] \n\\label{def:GG}\nLet~$C>0$.  Let~$X$ be a finite connected~$2$--complex,\nand~$\\phi \\colon\\thinspace \\pi_1(X) \\to G$, a group homomorphism.  We say that~$X$ has\nthe~$\\rm{GG}_{C}$-property for~$\\phi$ if\nevery piecewise smooth Riemannian metric on~$X$ admits a point~$x \\in\nX$ such that the~$r$--ball of~$X$ centered at~$x$ satisfies the bound\n\\begin{equation} \n\\label{eq:ball}\n\\area \\, B(x,r) \\geq C r^2,\n\\end{equation}\nfor every~$r \\leq \\frac{1}{2} \\sys(X,\\phi)$.\n\\end{definition}\n\nNote that if the~$2$--complex~$X$ is~$\\varepsilon$--almost minimal,\ni.e., satisfies the bound $\\sigma_{\\phi}(X) \\leq G_*(G) +\n\\varepsilon$, and has least first Betti number among all such\ncomplexes, then it satisfies~\\eqref{eq:ball} for some~$C>0$ and\nfor~$r\\geq \\varepsilon^{1\/3}$ by Theorem~\\ref{13}.\n\nModulo such a conjectured bound, we prove a systolic inequality for\nclosed~$3$--manifolds with finite fundamental group.\n\n\n\\begin{theorem}\n\\label{theo:main}\nLet~$p\\geq 2$ be a prime.  Assume that\nevery~$\\phi$--essential~$2$--complex has the~$\\rm{GG}_C$-property\n\\eqref{eq:ball} for each homomorphism~$\\phi$ into~$\\Z_p$ and for some\nuniversal constant~$C>0$.  Then every orientable closed\nRiemannian~$3$--manifold~$M$ with finite fundamental group of order\ndivisible by~$p$, satisfies the bound\n\\begin{equation*}\n\\sys(M)^3 \\leq 24 \\, C^{-1} \\; \\vol(M).\n\\end{equation*}\nMore precisely, there is a point~$x\\in M$ such that the volume of\nevery~$r$--ball centered at~$x$ is at least~$\\frac{C}{3}r^{3}$, for all\n$r \\leq \\frac{1}{2} \\sys(M)$.\n\\end{theorem}\n\n\nA slightly weaker bound can be obtained modulo a weaker GG-property,\nwhere the point~$x$ is allowed to depend on the radius~$r$.\n\nSince the GG-property is available for~$p=2$ and~$C=1$ by Guth's article\n\\cite{Gu09}, we obtain the following corollary.\n\n\\begin{corollary}\nEvery closed Riemannian~$3$--manifold~$M$ with fundamental group of\neven order satisfies\n\\begin{equation}\n\\label{Poincare}\n\\sys(M)^3 \\leq 24 \\; \\vol(M).\n\\end{equation}\n\\end{corollary}\n\nFor example, the Poincar\\'e homology~$3$--sphere satisfies the systolic\ninequality \\eqref{Poincare}. \\\\\n\nIn the next section, we present related developments in systolic\ngeometry and compare some of our arguments in the proof of\nTheorem~\\ref{theo:main} to Guth's in~\\cite{Gu09},\n{\\it cf.}~Remark~\\ref{rem:compare}.  Additional recent developments in\nsystolic geomety include \\cite{AK, BB10, Bal08, e7, BT, Be08, Bru,\nBru2, Bru3, DKR, DR09, Elm10, EL, Gu09, KK, KK2, Ka4, KR2, KSh, NR,\nPar10, Ro, RS08, Sa08, Sa10}.\n\n\n\n\n\\section{Recent progress on Gromov's inequality}\n\nM.~Gromov's upper bound for the~$1$--systole of an essential\nmanifold~$M$ \\cite{Gr1} is a central result of systolic geometry.\nGromov's proof exploits the Kuratowski imbedding of~$M$ in the Banach\nspace~$L^\\infty$ of bounded functions on~$M$.  A complete analytic\nproof of Gromov's inequality \\cite{Gr1}, but still using the\nKuratowski imbedding in~$L^\\infty$, was recently developed by\nL.~Ambrosio and the second-named author~\\cite{AK}.  See\nalso~\\cite{AW}.\n\nS.~Wenger~\\cite{wen} gave a complete analytic proof of an\nisoperimetric inequality between the volume of a manifold~$M$, and its\nfilling volume, a result of considerable independent interest.  On the\nother hand, his result does not directly improve or simplify the proof\nof Gromov's main filling inequality for the filling radius.  Note that\nboth the filling inequality and the isoperimetric inequality are\nproved simultaneously by Gromov, so that proving the isoperimetric\ninequality by an independent technique does not directly simplify the\nproof of either the filling radius inequality, or the systolic\ninequality.\n\nL.~Guth \\cite{Gu11} gave a new proof of Gromov's systolic inequality\nin a strengthened {\\em local\\\/} form.  Namely, he proved Gromov's\nconjecture that every essential manifold with unit systole contains a\nball of unit radius with volume uniformly bounded away from zero.\n\nMost recently, Guth \\cite {Gu09} re-proved a significant case of\nGromov's systolic inequality \\cite{Gr1} for essential manifolds,\nwithout using Gromov's filling invariants.\n\nActually, in the case of surfaces, Gromov himself had proved better\nestimates, without using filling invariants, by sharpening a technique\nindependently due to Y.~Burago and V.~Zalgaller \\cite[p.~43]{BZ}, and\nJ.~Hebda \\cite{Hebda}.  Here the essential idea is the following.\n\nLet~$\\gamma(s)$ be a minimizing non-contractible closed geodesic of\nlength~$L$ in a surface~$S$, where the arclength parameter~$s$ varies\nthrough the interval~$[-\\frac{L}{2}, \\frac{L}{2}]$.  We consider\nmetric balls (metric disks)~$B(p,r) \\subset S$ of radius~$r<\n\\frac{L}{2}$ centered at~$p=\\gamma(0)$.  The two points~$\\gamma(r)$\nand~$\\gamma(-r)$ lie on the boundary sphere (boundary curve)~$\\partial\nB(p,r)$ of the disk.  If the points lie in a common connected\ncomponent of the boundary (which is necessarily the case if~$S$ is a\nsurface and~$L=\\sys(S)$, but may fail if~$S$ is a more\ngeneral~$2$--complex), then the boundary curve has length at\nleast~$2r$.  Applying the coarea formula\n\\begin{equation}\n\\label{11b}\n\\area \\, B(p,r)=\\int_0^r \\length \\, \\partial B(p,\\rho) \\, d\\rho,\n\\end{equation}\nwe obtain a lower bound for the area which is quadratic in~$r$.\n\nGuth's idea is essentially a higher-dimensional analogue of Hebda's,\nwhere the minimizing geodesic is replaced by a minimizing\nhypersurface.  Some of Guth's ideas go back to the even earlier texts\nby Schoen and Yau \\cite{SY78, SY79}.\n\nThe case handled in \\cite{Gu09} is that of~$n$--dimensional manifolds\nof maximal~$\\Z_2$--cuplength, namely~$n$.  Thus, Guth's theorem covers\nboth tori and real projective spaces, directly generalizing the\nsystolic inequalities of Loewner and Pu, see \\cite{Pu} and \\cite{SGT}\nfor details.\n\n\n\n\n\\begin{remark} \\label{rem:compare}\nTo compare Guth's argument in his text~\\cite{Gu09} and our proof of\nTheorem~\\ref{theo:main}, we observe that the topological ingredient of\nGuth's technique exploits the multiplicative structure of the\ncohomology ring~$H^*(\\Z_2;\\Z_2)=H^*(\\R {\\mathbb P}^\\infty; \\Z_2)$.\nThis ring is generated by the~$1$--dimensional class.  Thus, every\n$n$--dimensional cohomology class decomposes into the cup product\nof~$1$--dimensional classes.  This feature enables a proof by\ninduction on~$n$.\n\nMeanwhile, for~$p$ odd, the cohomology ring~$H^*(\\Z_p;\\Z_p)$ is not\ngenerated by the~$1$--dimensional class; see Proposition~\\ref{42} for\na description of its structure.  Actually, the square of\nthe~$1$--dimensional class is zero, which seems to yield no useful\ngeometric information.\n\nAnother crucial topological tool used in the proof of~\\cite{Gu09} is\nPoincar\\'e duality which can be applied to the manifolds representing\nthe homology classes in~$H_*(\\Z_2;\\Z_2)$.  For~$p$ odd, the homology\nclasses of~$H_{2k}(\\Z_p;\\Z_p)$ cannot be represented by manifolds.\nOne could use D.~Sullivan's notion of~$\\Z_p$--manifolds,\n{\\it cf.}~\\cite{Su,MS}, to represent these homology class, but they do not\nsatisfy Poincar\\'e duality.\n\nFinally, we mention that, when working with cycles representing\nhomology classes with torsion coefficients in~$\\Z_p$, we exploit a\nnotion of volume which ignores the multiplicities in~$\\Z_p$,\n{\\it cf.}~Definition~\\ref{def:Vol}.  This is a crucial feature in our proof. \nNote that minimal cycles with torsion coefficients were studied by\nB.~White \\cite{Wh2}.\n\\end{remark}\n\n\n\n\\section{Area of balls in~$2$--complexes}\n\nIt was proved in~\\cite{Gr1} and~\\cite{KRS} that a finite~$2$--complex\nadmits a systolic inequality if and only if its fundamental group is\nnonfree, or equivalently, if it is~$\\phi$--essential for~$\\phi= {\\rm\nId}$.\n\n\n\nIn~\\cite{KRS}, we used an argument by contradiction, relying on an\ninvariant called {\\em tree energy\\\/}, to prove a bound for the\nsystolic ratio of a~$2$--complex.  We present an alternative short proof\nwhich yields a stronger result and simplifies the original argument.\n\n\\begin{theorem} \n\\label{theo:r2}\nLet~$X$ be a piecewise Riemannian finite essential~$2$--complex.  There\nexists~$x \\in X$ such that the area of every~$r$--ball centered at~$x$\nis at least~$r^2$ for every~$r \\leq \\frac{1}{2} \\sys(X)$.\n\\end{theorem}\n\nAs mentioned in the introduction, we conjecture that this result still\nholds for~$\\phi$--essential complexes and with the~$\\phi$--relative\nsystole in place of~$\\sys$.\n\n\\begin{proof}\nWe can write the Grushko decomposition of the fundamental group of~$X$\nas\n\\begin{equation*}\n\\pi_1(X) = G_1*\\cdots*G_r*F,\n\\end{equation*}\nwhere~$F$ is free, while each group~$G_i$ is nontrivial,\nnon-isomorphic to~$\\Z$, and not decomposable as a nontrivial free\nproduct.\n\nConsider the equivalence class~$[G_1]$ of~$G_1$ under external\nconjugation in~$\\pi_1(X)$.  Let~$\\gamma$ be a loop of least length\nrepresenting a nontrivial class~$[\\gamma]$ in~$[G_1]$.  Fix~$x \\in\n\\gamma$ and a copy of~$G_1 \\subset \\pi_1(X,x)$ containing the homotopy\nclass of~$\\gamma$.  Let~$\\overline{X}$ be the cover of~$X$ with\nfundamental group~$G_1$.\n\n\\begin{lemma}\nWe have~$\\sys(\\overline{X}) = \\length(\\gamma)$.\n\\end{lemma}\n\n\\begin{proof}\nThe loop~$\\gamma$ lifts to~$\\overline{X}$ by construction of the subgroup\n$G_1$.  Thus,~$\\sys(\\overline{X}) \\leq \\length(\\gamma)$.  Now, the\ncover~$\\overline{X}$ does not contain noncontractible loops~$\\delta$\nshorter than~$\\gamma$, because such loops would project to~$X$ so that\nthe nontrivial class~$[\\delta]$ maps into~$[G_1]$, contradicting our\nchoice of~$\\gamma$.\n\\end{proof}\n\nContinuing with the proof of the theorem, let~$\\bar{x} \\in \\overline{X}$\nbe a lift of~$x$.  Consider the level curves of the distance function\nfrom~$\\bar{x}$.  Note that such curves are necessarily connected, for\notherwise one could split off a free-product-factor~$\\Z$\nin~$\\pi_1(\\overline{X})=G_1$, {\\it cf.}~\\cite[Proposition 7.5]{KRS},\ncontradicting our choice of~$G_1$.  In particular, the\npoints~$\\gamma(r)$ and~$\\gamma(-r)$ can be joined by a path contained\nin the curve at level~$r$.  Applying the coarea formula~\\eqref{11b},\nwe obtain a lower bound~$\\area \\, B(\\bar{x},r)\\geq r^2$ for the area\nof an~$r$--ball~$B(\\bar{x},r) \\subset \\overline{X}$, for all~$r \\leq\n\\frac{1}{2} \\length(\\gamma)=\\frac{1}{2} \\sys(\\overline{X})$.\n\nIf, in addition, we have~$r \\leq \\frac{1}{2}\\sys(X)$ (which apriori\nmight be smaller than~$\\frac{1}{2} \\sys(\\overline{X})$), then the ball\nprojects injectively to~$X$, proving that\n\\begin{equation*}\n\\area(B(x,r)\\subset X) \\geq r^2\n\\end{equation*}\nfor all~$r\\leq \\frac{1}{2}\\sys(X)$.\n\\end{proof}\n\n\n\\section{Outline of argument for relative systole}\n\\label{three}\n\nLet~$X$ be a piecewise Riemannian connected~$2$--complex, and\nassume~$X$ is~$\\phi$--essential for a group homomorphism\n$\\phi \\colon\\thinspace \\pi_1(X)\\to G$.  We would like to prove an area lower bound\nfor~$X$, in terms of the~$\\phi$--relative systole as in\nTheorem~\\ref{theo:r2}.  Let~$x \\in X$.  Denote by~$B=B(x,r)$ and\n$S=S(x,r)$ the open ball and the sphere (level curve) of radius~$r$\ncentered at~$x$ with~$r < \\frac{1}{2} \\sys(X,\\phi)$.  Consider the\ninterval~$I=[0,\\frac{L}{2}]$, where~$L=\\length(S)$.\n\n\\begin{definition} \n\\label{def:Y}\nWe consider the complement~$X \\setminus B$, and attach to it a buffer\ncylinder along each connected component~$S_i$ of~$S$.  Here a buffer\ncylinder with base~$S_i$ is the quotient\n\\begin{equation*}\nS_i \\times I\/ \\!\\! \\sim\n\\end{equation*}\nwhere the relation~$\\sim$ collapses each subset~$S_i \\times \\{ 0 \\}$\nto a point~$x_i$.  We thus obtain the space\n\\[\n\\left( S_i \\times I\/ \\!\\! \\sim \\right) \\cup_f \\left( X \\setminus B\n\\right),\n\\]\nwhere the attaching map~$f$ identifies\n$S_i\\times\\left\\{\\tfrac{L}{2}\\right\\}$ with~$S_i\\subset X\\setminus B$.\nTo ensure the connectedness of the resulting space, we attach a\ncone~$CA$ over the set of points~$A=\\{ x_i \\}$.  We set the length of the\nedges of the cone~$CA$ equal to~$\\sys(X,\\phi)$.  We will denote by\n\\begin{equation} \\label{eq:Y}\nY=Y(x,r)\n\\end{equation}\nthe resulting~$2$--complex.  The natural metrics on~$X \\setminus B$ and\non the buffer cylinders induce a metric on~$Y$.\n\\end{definition}\n\nIn the next section, we will show that~$Y$ is~$\\psi$--essential for\nsome homomorphism~$\\psi \\colon\\thinspace \\pi_1(Y) \\to G$ derived from~$\\phi$.  The\npurpose of the buffer cylinder is to ensure that the relative systole\nof~$Y$ is at least as large as the relative systole of~$X$.  Note that\nthe area of the buffer cylinder is~$L^2\/2$.\n\n\nWe normalize~$X$ to unit relative systole and take a point~$x$ on a\nrelative systolic loop of~$X$.\nSuppose~$X$ has a minimal first Betti number among the complexes\nessential in~$K(G,1)$ with almost minimal systolic area (up to\nepsilon).  We sketch below the proof of the local relative systolic\ninequality satisfied by~$X$.\n\nIf for every~$r$, the space~$Y=Y(x,r)$ has a greater area than~$X$,\nthen\n\\begin{equation*}\n\\area \\, B(r) \\leq \\tfrac{1}{2}(\\length \\, S(r))^2\n\\end{equation*}\nfor every~$r < \\frac{1}{2} \\sys(X,\\phi)$.  Using the coarea\ninequality, this leads to the differential inequality~$y(r) \\leq\n\\tfrac{1}{2} y'(r)^2$.  Integrating this relation shows that the area\nof~$B(r)$ is at least~$\\frac{r^2}{2}$, and the conclusion follows.\n\nIf for some~$r$, the space~$Y$ has a smaller area than~$X$, we argue\nby contradiction.  We show that a~$\\phi$--relative systolic loop of~$X$\n(passing through~$x$) meets at least two connected components of the\nlevel curve~$S(r)$.  These two connected components project to two\nendpoints of the cone~$CA$ connected by an arc of~$Y \\setminus CA$.\nUnder this condition, we can remove an edge~$e$ from~$CA$ so that the\nspace~$Y'=Y \\setminus e$ has a smaller first Betti number than~$X$.\nHere~$Y'$ is still essential in~$K(G,1)$, and its relative systolic\narea is better than the relative systolic area of~$X$, contradicting\nthe definition of~$X$.\n\n\n\\section{First Betti number and essentialness of~$Y$} \\label{sec:remov}\n\nLet~$G$ be a fixed finitely presented group.  We are mostly interested\nin the case of a finite group~$G=\\Z_p$.  Unless specified otherwise,\nall group homomorphisms have values in~$G$, and all complexes are\nassumed to be finite.  Consider a homomorphism~$\\phi \\colon\\thinspace \\pi_1(X) \\to\nG$ from the fundamental group of a piecewise Riemannian finite\nconnected~$2$--complex~$X$ to~$G$.\n\n\n\\begin{definition}\nA loop~$\\gamma$ in~$X$ is said to be~$\\phi$--contractible if the image\nof the homotopy class of~$\\gamma$ by~$\\phi$ is trivial, and\n$\\phi$--noncontractible otherwise.  Thus, the~$\\phi$--systole of~$X$,\ndenoted by~$\\sys(X,\\phi)$, is defined as the least length of a\n$\\phi$--noncontractible loop in~$X$.  Similarly, the\n\\mbox{$\\phi$--systole} based at a point~$x$ of~$X$, denoted\nby~$\\sys(X,\\phi,x)$, is defined as the least length of\na~$\\phi$--noncontractible loop based at~$x$.\n\\end{definition}\n\n\\forget\n\\begin{definition}\n\\label{42b}\nThe~$\\phi$--systolic area of~$X$ is defined as\n$$\n\\sigma_{\\phi}(X) = \\frac{\\area(X)}{\\sys(X,\\phi)^{2}}.\n$$\n\\end{definition}\n\\forgotten\n\nThe following elementary result will be used repeatedly in the sequel.\n\n\\begin{lemma} \n\\label{lem:trivial}\nIf~$r < \\frac{1}{2} \\sys(X,\\phi,x)$, then\nthe~$\\pi_{1}$--homomorphism~$i_{*}$ induced by the inclusion~$B(x,r)\n\\subset X$ is trivial when composed with~$\\phi$, that is~$\\phi \\circ\ni_{*}=0$.  More specifically, every loop in~$B(x,r)$ is homotopic to a\ncomposition of loops based at~$x$ of length at most~$2r+\\varepsilon$, for\nevery~$\\varepsilon>0$.\n\\end{lemma}\n\n\n\nWithout loss of generality, we may assume that the piecewise\nRiemannian metric on~$X$ is piecewise flat.  Let~$x_{0} \\in X$.  The\npiecewise flat~$2$--complex~$X$ can be embedded into some~$\\R^N$ as a\nsemialgebraic set and the distance function~$f$ from~$x_0$ is a\ncontinuous semialgebraic function on~$X$, {\\it cf.}~\\cite{BCR98}.\nThus,~$(X,B)$ is a CW-pair when~$B$ is a ball centered at~$x_0$ (see\nalso \\cite[Corollary~6.8]{KRS}).  Furthermore, for almost every~$r$,\nthere exists a~$\\eta >0$ such that the set\n\\begin{equation*}\n\\{ x \\in X \\mid r-\\eta < f(x) < r+\\eta \\}\n\\end{equation*}\nis homeomorphic to~$S(x_0,r) \\times (r-\\eta,r+\\eta)$ where~$S(x_0,r)$\nis the~$r$--sphere centered at~$x_{0}$ and the~$t$--level curve of~$f$\ncorresponds to~$S(x_0,r) \\times \\{t\\}$, {\\it cf.}~\\cite[\\S~9.3]{BCR98}\nand~\\cite{KRS} for a precise description of level curves on~$X$.  \nIn such case, we say that~$r$ is a \\emph{regular value} of~$f$. \\\\\n\n\\forget \n\nSince the function~$\\ell(r) = \\length \\, f^{-1}(r)$ is piecewise\ncontinuous, {\\it cf.}~\\cite[\\S~9.3]{BCR98}, the condition~$\\area \\, B >\n\\lambda \\, (\\length \\, S)^{2}$ is open (see ~\\eqref{eq:lambda} below).\nTherefore, slightly changing the value of~$r$ if necessary, we can\nassume that~$r$ is regular. \n\n\\forgotten\n\nConsider the connected~$2$--complex~$Y=Y(x_0,r)$ introduced in\nDefinition~\\ref{def:Y}, with~$r < \\frac{1}{2} \\sys(X,\\phi)$ and~$r$\nregular.  Since~$r$ is a regular value, there exists~$r_- \\in (0,r)$\nsuch that~$B \\setminus B(x_0,r_-)$ is homeomorphic to the product\n\\[\nS \\times [r_-,r) = \\coprod_i S_i \\times [r_-,r).\n\\]\nConsider the map\n\\begin{equation} \n\\label{eq:XY}\n\\pi \\colon\\thinspace X \\to Y\n\\end{equation}\nwhich leaves~$X \\setminus B$ fixed, takes~$B(x_0,r_-)$ to the vertex\nof the cone~$CA$, and sends~$B \\setminus B(x_0,r_-)$ to the union of\nthe buffer cylinders and~$CA$.\nThis map induces an epimorphism between the first\nhomology groups.  In particular,\n\\begin{equation} \\label{eq:b1}\nb_1(Y) \\leq b_1(X).\n\\end{equation}\n\n\\medskip\n\n\\forget\n\\begin{lemma}\n\\label{lem:betti}\nWe have\n\\[\nb_1(Y) \\leq b_1(X).\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSince~$r$ is a regular value, there exists~$r_- \\in (0,r)$ such that~$B \\setminus B(x,r_-)$ is homeomorphic to~$S \\times [r_-,r) = \\coprod_i S_i \\times  [r_-,r)$.\nThe map~$X \\to Z$ which leaves~$X \\setminus B$ fixed and takes~$B(x,r_-)$ to the vertex of the cone~$C$ of~$Z$ induces an epimorphism between the first homology groups.\nHence the result.\n\\end{proof}\n\\forgotten\n\n\n\\forget\n\\begin{proof}\nLet~$f$ be the distance function from~$x_0$.  It is convenient to\nintroduce the Reeb space~$\\widehat{X}$ obtained from~$X$ by collapsing to\npoints the connected components of the level curves~$f^{-1}(t)$, for\nevery~$t \\in [0,r]$ (level curves for~$t>r$ are unaffected).  Note\nthat the lower bound for the systole no longer holds for~$\\widehat{X}$ due\nto possible ``shortcuts'' created in the graph~$T\\subset\\widehat{X}$\ncorresponding to~$t\\leq r$.\n\nSince the fibers of the map~$X\\to \\widehat{X}$ are connected, by the\ncovering homotopy property, we obtain that every closed path in\n$\\widehat{X}$ lifts to a closed path in~$X$, proving the surjectivity of\n$\\pi_1(X)\\to \\pi_1(\\widehat{X})$.\n\nWe first assume that~$X\\setminus B$ is connected.  Then the Reeb\nspace~$\\widehat{X}$ is homotopy equivalent%\n\\footnote{The fact that the ``Reeb graph'' is indeed a finite graph\nfollows from semialgebraicity; see \\cite{KRS} for a detailed\ndiscussion.}\nto the union~$Y \\cup T$ obtained by attaching a finite graph~$T$ to\nthe finite set~$\\{x_1,\\ldots,x_n\\} \\subset Y$,\ncf.~Definition~\\eqref{def:Y}.  By van Kampen's theorem, the removal of\nthe graph leads to a further decrease in the Betti number.  The\nnon-closed path~$\\alpha$ closes up to a loop in~$X$ but not in~$Y$.\n\nIf~$X\\setminus B$ is not connected, our space~$Y$ is homotopy\nequivalent to a connected component of~$\\widehat{X}$ with the graph~$T$\nremoved, proving the lemma.\n\\end{proof}\n\\forgotten\n\n\n\\forget\nLet~$A=\\{x_1,\\ldots, x_n\\}$ be the finite set formed by the\npoints~$x_i$.  Let~$Y \\cup CA$ be the space obtained by attaching a\ncone over~$A$ to~$Y$.  Consider the map\n\\[\n\\widehat{X} \\to Y \\cup CA\n\\]\nwhich leaves~$Y$ fixed and takes~$T \\setminus (\\cup_i e_i)$ to the\nvertex of~$CA$, where the~$e_i$ are the semi-edges of~$T$ with\nendpoints~$x_i$.  The composite\n\\[\nX \\to \\widehat{X} \\to Y \\cup CA,\n\\]\nwhere~$X \\to \\widehat{X}$ is the quotient map, leaves~$X \\setminus\n\\overline{B}$ fixed and induces an epimorphism between the first\nhomology groups.  Hence,\n$$\nb_1(Y) \\leq b_1(Y \\cup CA) \\leq b_1(X).\n$$\n\nNow, suppose that the projection of some arc~$\\alpha$ of~$X \\setminus\nB$ to~$Y$ connects two points of~$A$.  Then the space~$Y \\cup CA$ is\nhomotopy equivalent to~$(Y \\cup CA') \\vee S^{1}$, where~$A'\n\\subset A$.  That is,\n$$\nY \\cup CA \\simeq (Y \\cup CA') \\vee S^{1}.\n$$\nWe deduce that\n$$\nb_1(Y) < b_1(Y \\cup CA) \\leq b_1(X). \n$$\n\\forgotten\n\n\n\\begin{lemma} \\label{lem:class}\nIf~$r < \\frac{1}{2} \\sys(X,\\phi)$, then~$Y$ is~$\\psi$--essential for\nsome homomorphism~$\\psi \\colon\\thinspace \\pi_{1}(Y) \\to G$ such that\n\\begin{equation} \\label{eq:circ}\n\\psi \\circ \\pi_* =\\phi\n\\end{equation}\nwhere~$\\pi_*$ is the~$\\pi_1$--homomorphism induced by \\mbox{$\\pi \\colon\\thinspace X \\to Y$}.\n\\end{lemma}\n\n\\begin{proof}\nConsider the CW-pair~$(X,B)$ where~$B=B(x_0,r)$.  By\nLemma~\\ref{lem:trivial}, the restriction of the classifying\nmap~$\\varphi \\colon\\thinspace X \\to K(G,1)$ induced by~$\\phi$ to~$B$ is homotopic to a\nconstant map.  Thus, the classifying map~$\\varphi$ extends to~$X \\cup\nCB$ and splits into\n\\[\nX \\hookrightarrow X \\cup CB \\to K(G,1),\n\\]\nwhere~$CB$ is a cone over~$B \\subset X$ and the first map is the\ninclusion map.  Since~$X \\cup CB$ is homotopy equivalent to the\nquotient~$X\/B$, {\\it cf.}~\\cite[Example~0.13]{Hat}, we obtain the following\ndecomposition of~$\\varphi$ up to homotopy:\n\\begin{equation} \\label{eq:XB}\nX \\stackrel{\\pi}{\\longrightarrow} Y \\to X\/B \\to K(G,1).\n\\end{equation}\n\nHence,~$\\psi \\circ \\pi_* = \\phi$ for the~$\\pi_1$--homomorphism~$\\psi \\colon\\thinspace \\pi_1(Y) \\to G$ induced by the map~$Y \\to K(G,1)$.  \nIf the map~$Y \\to K(G,1)$ can be homotoped into the~$1$--skeleton of~$K(G,1)$, the same\nis true for\n\\[\nX \\to Y \\to K(G,1)\n\\]\nand so for the homotopy equivalent map~$\\varphi$, which contradicts\nthe~$\\phi$--essentialness of~$X$.\n\\end{proof}\n\n\n\\section{Exploiting a ``fat\" ball}\n\nWe normalize the~$\\phi$--relative systole of~$X$ to one, i.e.~$\\sys(X,\\phi)=1$.  \nChoose a fixed~$\\delta \\in (0,\\frac{1}{2})$\n(close to~$0$) and a real parameter~$\\lambda > \\frac{1}{2}$ (close\nto~$\\frac{1}{2}$).\n\n\\begin{proposition} \n\\label{prop:reeb}\nSuppose there exist a point~$x_{0} \\in X$ and a value~$r_{0} \\in\n(\\delta,\\frac{1}{2})$ regular for~$f$ such that\n\\begin{equation} \n\\label{eq:lambda}\n\\area \\, B > \\lambda \\, (\\length \\, S)^{2}\n\\end{equation}\nwhere~$B=B(x_{0},r_{0})$ and~$S=S(x_{0},r_{0})$. \nThen there exists a\npiecewise flat metric on~$Y=Y(x_{0},r_{0})$\nsuch that the systolic areas ({\\it cf.}~Definition~\\ref{def:sigma}) satisfy\n$$\n\\sigma_{\\psi}(Y) \\leq \\sigma_{\\phi}(X).\n$$\n\\end{proposition}\n\n\\begin{proof}\nConsider the metric on~$Y$ described in Definition~\\ref{def:Y}.\nStrictly speaking, the metric on~$Y$ is not piecewise flat since the\nconnected components of~$S$ are collapsed to points, but it can be\napproximated by piecewise flat metrics.\n\nDue to the presence of the buffer cylinders, every loop of~$Y$ of\nlength less than~$\\sys(X,\\phi)$ can be deformed into a loop of~$X\n\\setminus B$ without increasing its length.  Thus, by~\\eqref{eq:circ},\none obtains\n\\begin{equation*}\n\\sys(Y,\\psi) \\geq \\sys(X,\\phi) = 1.\n\\end{equation*}\nFurthermore, we have\n\\begin{equation*}\n\\area \\, Y \\leq \\area \\, X - \\area \\, B + \\tfrac{1}{2} (\\length \\,\nS)^{2}.\n\\end{equation*}\nCombined with the inequality~\\eqref{eq:lambda}, this leads to\n\\begin{equation} \\label{eq:wh}\n\\sigma_{\\psi}(Y) < \\sigma_{\\phi}(X) - \\left( \\lambda - \\tfrac{1}{2}\n\\right) (\\length \\, S)^{2}.\n\\end{equation}\nHence,~$\\sigma_{\\psi}(Y) \\leq \\sigma_{\\phi}(X)$,\nsince~$\\lambda > \\frac{1}{2}$.\n\\end{proof}\n\n\n\n\\section{An integration by separation of variables}\n\nLet~$X$ be a piecewise Riemannian finite connected~$2$--complex.  Let\n\\mbox{$\\phi \\colon\\thinspace \\pi_{1}(X) \\to G$} be a nontrivial homomorphism to a\ngroup~$G$.  We normalize the metric to unit relative systole:\n$\\sys(X,\\phi)=1$.  The following area lower bound appeared\nin~\\cite[Lemma~7.3]{RS08}.\n\n\\begin{lemma} \\label{lem:BS}\nLet~$x \\in X$,~$\\lambda >0$ and~$\\delta \\in (0,\\frac{1}{2})$.\nIf \n\\begin{equation} \\label{eq:BS}\n\\area \\, B(x,r) \\leq \\lambda \\, (\\length \\, S(x,r))^{2}\n\\end{equation}\nfor almost every~$r \\in (\\delta,\\frac{1}{2})$, then \n$$\n\\area \\, B(x,r) \\geq \\frac{1}{4\\lambda} (r-\\delta)^{2}\n$$\nfor every~$r \\in (\\delta,\\frac{1}{2})$.\n\nIn particular,~$\\displaystyle \\area(X) \\geq \\frac{1}{16 \\lambda} \\,\n\\sys(X,\\phi)^{2}$.\n\\end{lemma}\n\n\\begin{proof}\nBy the coarea formula, we have\n\\begin{equation*}\na(r) := \\area \\, B(x,r) = \\int_0^r \\ell(s) \\, ds\n\\end{equation*}\nwhere~$\\ell(s)=\\length \\, S(x,s)$.  Since the function~$\\ell(r)$ is\npiecewise continuous, the function~$a(r)$ is continuously\ndifferentiable for all but finitely many~$r$ in~$(0,\\frac{1}{2})$\nand~$a'(r)=\\ell(r)$ for all but finitely many~$r$\nin~$(0,\\frac{1}{2})$.  By hypothesis, we have\n$$\na(r) \\leq \\lambda \\, a'(r)^2\n$$\nfor all but finitely many~$r$ in~$(\\delta,\\frac{1}{2})$.\nThat is,\n$$ \\left( \\sqrt{a(r)} \\right)' = \\frac{a'(r)}{2 \\sqrt{a(r)}} \\geq\n\\frac{1}{2\\sqrt{\\lambda}}.~$$\nWe now integrate this differential inequality from~$\\delta$ to~$r$, to\nobtain\n$$\n \\sqrt{a(r)} \\geq \\frac{1}{2\\sqrt{\\lambda}} (r-\\delta).\n$$\nHence, for every~$r \\in (\\delta, \\frac{1}{2})$, we obtain\n\\[\n a(r) \\geq \\frac{1}{4 \\lambda} (r-\\delta)^{2},\n\\]\ncompleting the proof.\n\\end{proof}\n\\forget\n, or\n\\begin{equation*}\ndr \\leq \\lambda^{1\/2} a^{-1\/2} da.\n\\end{equation*}\nWe now integrate this differential inequality from~$\\delta$ to~$r$, to\nobtain\n\\begin{equation*}\nr-\\delta \\leq \\int_{a(\\delta)}^{a(r)} \\lambda^{1\/2} a^{-1\/2} da,\n\\end{equation*}\nand hence\n\\begin{equation*}\nr-\\delta \\leq 2 \\lambda^{1\/2} \\left( a(r)^{1\/2} - a(\\delta^{1\/2})\n\\right) \\leq 2 \\lambda^{1\/2} a(r)^{1\/2},\n\\end{equation*}\nproving the result so long as we have the inequality for all the\nintermediate values of~$r$.  \n\nWhy is that?\n\\forgotten\n\n\n\n\\section{Proof of relative systolic inequality}\n\nWe prove that if~$X$ is a~$\\phi$--essential piecewise\nRiemannian~$2$--complex which is almost minimal (up to~$\\varepsilon$),\nand has least first Betti number among such complexes, then~$X$ possesses\nan~$r$--ball of large area for each~$r< \\tfrac{1}{2} \\sys(X, \\phi)$.\nWe have not been able to find such a ball for an\narbitrary~$\\phi$--essential complex (without the assumption of almost\nminimality), but at any rate the area lower bound for almost minimal\ncomplexes suffices to prove the~$\\phi$--systolic inequality for\nall~$\\phi$--essential complexes, as shown below.\n\n\\forget\n\\begin{definition}\nLet~$G$ be a group.  We set\n\\begin{equation*}\n\\sigma_{*}(G) = \\inf_{X} \\sigma_{\\phi}(X),\n\\end{equation*}\nwhere the infimum is over all~$\\phi$--essential piecewise Riemannian\nfinite~$2$--complexes~$X$, where the homomorphism~$\\phi$ has values\nin~$G$.\n\\end{definition}\n\\forgotten\n\n\\begin{remark}\nWe do not assume at this point that~$\\sigma_{*}(G)$ is nonzero,\n{\\it cf.}~Definition~\\ref{def:sigma}.  In fact, the proof of\n$\\sigma_{*}(G)>0$ does not seem to be any easier than the explicit\nbound of Corollary~\\ref{coro:A}.\n\\end{remark}\n\nTheorem~\\ref{13} and Corollary~\\ref{coro:A} are consequences of the\nfollowing result.\n\n\\begin{proposition} \n\\label{prop:minB}\nLet~$\\varepsilon >0$.  Suppose~$X$ has a minimal first Betti number\namong all~$\\phi$--essential piecewise Riemannian~$2$--complexes\nsatisfying \n\\begin{equation} \\label{eq:eps}\n\\sigma_{\\phi}(X) \\leq \\sigma_*(G) +\\varepsilon.\n\\end{equation}  \nThen each ball centered at a point~$x$ on a~$\\phi$--systolic loop in~$X$\nsatisfies the area lower bound\n\\begin{equation*}\n\\area \\, B(x,r) \\geq\n\\frac{(r-\\delta)^2}{2+\\frac{\\varepsilon}{\\delta^2}}\n\\end{equation*}\nfor every~$r \\in \\left(\\delta,\\frac{1}{2}\\sys(X,\\phi) \\right)$, where\n$\\delta \\in \\left(0,\\frac{1}{2}\\sys(X,\\phi)\\right)$.  In particular,\nwe obtain the bound\n\\begin{equation*}\n\\sigma_*(G) \\geq \\frac{1}{8}.\n\\end{equation*}\n\\end{proposition}\n\n\\forget\n\\begin{proof}\nIf for each~$r$ we have~$a(r) \\leq a'(r)$ then we separate variables\nas in the previous section to obtain the area lower bound.\n\nIf for some~$r$, we have~$a(r) > a'(r)$, then there are two\npossibilities: either~$S(r)$ is connected, and we obtain a lower bound\nof~$r^2$ for the area by Hebda's trick, or~$S(r)$ is disconnected.\nBut the latter case is impossible by the hypothesis if minimality of\nBetti number.\n\\end{proof}\n\nWe can now proceed with the proof of the relative systolic inequality\nfor essential~$2$--complexes.\n\\forgotten\n\n\\begin{proof\nWe will use the notation and results of the previous sections.\nChoose~$\\lambda > 0$ such that\n\\begin{equation}\n\\label{52}\n\\varepsilon < 4 \\left(\\lambda - \\tfrac{1}{2} \\right) \\delta^{2}.\n\\end{equation}\nThat is,\n\\[\n\\lambda > \\frac{1}{2} + \\frac{\\varepsilon}{4 \\delta^2} \\quad \\mbox{\n(close to } \\frac{1}{2} + \\frac{\\varepsilon}{4 \\delta^2}).\n\\]\nWe normalize the metric on~$X$ so that its~$\\phi$--systole is equal to one.\nChoose a point~$x_{0} \\in X$ on a~$\\phi$--systolic loop~$\\gamma$ of~$X$.  \n\nIf the balls centered at~$x_0$ are too ``thin'',\ni.e., the inequality~\\eqref{eq:BS} is satisfied for~$x_{0}$ and almost\nevery~$r \\in (\\delta,\\frac{1}{2})$, then the result follows from\nLemma~\\ref{lem:BS}.  \n\nWe can therefore assume that there exists a ``fat'' ball centered at~$x_0$, i.e., the hypothesis of Proposition~\\ref{prop:reeb} holds\nfor~$x_{0}$ and some regular~$f$--value~$r_{0} \\in\n(\\delta,\\frac{1}{2})$, where~$f$ is the distance function from~$x_0$.\n(Indeed, almost every~$r$ is regular for~$f$.)\nArguing by contradiction, we show that the assumption on the minimality of the first Betti number rules out this case.\n\nWe would like to construct a~$\\psi$--essential piecewise\nflat~$2$--complex~$Y'$ with~$b_1(Y') < b_1(X)$ such that\n$\\sigma_{\\psi}(Y') \\leq \\sigma_{\\phi}(X)$ and therefore\n\\begin{equation}\n\\sigma_{\\psi}(Y') \\leq \\sigma_{*}(G) + \\varepsilon\n\\end{equation}\nfor some homomorphism~$\\psi \\colon\\thinspace \\pi_{1}(Y') \\to G$.\n\nBy Lemma~\\ref{lem:class} and Proposition~\\ref{prop:reeb}, the space~$Y = Y(x_{0},r_{0})$, endowed with the piecewise Riemannian metric of Proposition~\\ref{prop:reeb}, satisfies\n\\begin{equation*}\n\\sigma_{*}(G) \\leq \\sigma_{\\psi}(Y) \\leq \\sigma_{\\phi}(X).\n\\end{equation*}\nCombined with the inequalities~\\eqref{eq:wh} in the proof of\nProposition~\\ref{prop:reeb} and~\\eqref{eq:eps}, this yields\n\\begin{equation*}\n\\left( \\lambda - \\frac{1}{2} \\right) (\\length \\, S)^{2} < \\varepsilon.\n\\end{equation*}\nFrom~$\\varepsilon < 4 (\\lambda - \\frac{1}{2}) \\delta^{2}$\nand~$\\delta \\leq r_{0}$, we deduce that\n$$\n\\length \\, S < 2 r_{0}.\n$$\n\nNow, by Lemma~\\ref{lem:trivial}, the~$\\phi$--systolic\nloop~$\\gamma\\subset X$ does not entirely lie in~$B$.  Therefore, there\nexists an arc~$\\alpha_0$ of~$\\gamma$ passing through~$x_{0}$ and lying\nin~$B$ with endpoints in~$S$.  We have\n\\[\n\\length(\\alpha_0) \\geq 2r_{0}.\n\\]\nIf the endpoints of~$\\alpha_0$ lie in the same connected component\nof~$S$, then we can join them by an arc~$\\alpha_1 \\subset S$ of length\nless than~$2r_{0}$.  By Lemma~\\ref{lem:trivial}, the loop~$\\alpha_0\n\\cup \\alpha_1$, lying in~$B$, is~$\\phi$--contractible.  Therefore, the\nloop~$\\alpha_1 \\cup (\\gamma \\setminus \\alpha_0)$, which is shorter\nthan~$\\gamma$, is~$\\phi$--noncontractible.  Hence a contradiction.\n\nThis shows that the~$\\phi$--systolic loop~$\\gamma$ of~$X$ meets two\nconnected components of~$S$. \n\nSince a~$\\phi$--systolic loop is length-minimizing, the loop~$\\gamma$\nintersects~$S$ exactly twice.  Therefore, the complementary\narc~$\\alpha=\\gamma \\setminus \\alpha_0$, joining two connected\ncomponents of~$S$, lies in~$X \\setminus B$.\nThe two endpoints of~$\\alpha$ are connected by a length-minimizing arc\nof~$Y \\setminus (X \\setminus \\overline{B})$ passing exactly through\ntwo edges of the cone~$CA$.\n\nLet~$Y'$ be the~$2$--complex obtained by removing the interior of one\nof these two edges from~$Y$.  The complex~$Y'=Y \\setminus e$ is\nclearly connected and the space~$Y$, obtained by gluing back the\nedge~$e$ to~$Y$, is homotopy equivalent to~$Y' \\vee S^1$.  That is,\n\\begin{equation} \\label{eq:Y'}\nY \\simeq Y' \\vee S^1.\n\\end{equation}\nThus,~$Y'$ is~$\\psi$--essential if we still denote by~$\\psi$ the\nrestriction of the homomorphism~$\\psi \\colon\\thinspace \\pi_1(Y) \\to G$\nto~$\\pi_1(Y')$.  Furthermore, we clearly have\n\\[\n\\sigma_{\\psi}(Y') = \\sigma_{\\psi}(Y) \\leq \\sigma_{\\phi}(X).\n\\]\nCombined with~\\eqref{eq:b1}, the homotopy equivalence~\\eqref{eq:Y'}\nalso implies\n$$\nb_1(Y') < b_1(Y) \\leq b_1(X).\n$$\nHence the result.\n\\end{proof}\n\n\n\\begin{remark}\nWe could use round metrics (of constant positive Gaussian curvature)\non the ``buffer cylinders\" of the space~$Y$ in the proof of\nProposition~\\ref{prop:reeb}.  This would allow us to choose~$\\lambda$\nclose to~$\\frac{1}{2 \\pi}$ and to derive the lower bound\nof~$\\frac{\\pi}{8}$ for~$\\sigma_{\\phi}(X)$ in Corollary~\\ref{coro:A}.\nWe chose to use flat metrics for the sake of simplicity.\n\\end{remark}\n\n\n\\section{Cohomology of Lens spaces}\n\nLet~$p$ be a prime number.  The group~$G=\\Z_p$ acts freely on the\ncontractible sphere~$S^{2\\infty+1}$ yielding a model for the\nclassifying space\n\\begin{equation*}\nK = K(\\Z_{p},1) = S^{2\\infty+1}\/\\Z_{p}.\n\\end{equation*}\nThe following facts are well-known, {\\it cf.}~\\cite{Hat}.\n\n\\begin{proposition}\n\\label{42}\n\n\nThe cohomology ring~$H^*(\\Z_p;\\Z_p)$ for~$p$ an odd prime is the\nalgebra~$\\Z_p(\\alpha)[\\beta]$ which is exterior on one\ngenerator~$\\alpha$ of degree~$1$, and polynomial with one\ngenerator~$\\beta$ of degree~$2$.  Thus,\n\\begin{itemize}\n\\item\n$\\alpha$ is a generator of~$H^1(\\Z_p;\\Z_p)\\simeq \\Z_p$,\nsatisfying~$\\alpha^2=0$;\n\\item\n$\\beta$ is a generator of~$H^2(\\Z_p;\\Z_p)\\simeq \\Z_p$.\n\\end{itemize}\n\\end{proposition}\n\nHere the~$2$--dimensional class is the image under the Bockstein\nhomomorphism of the~$1$--dimensional class.  The cohomology of the\ncyclic group is generated by these two classes.  The cohomology is\nperiodic with period~$2$ by Tate's theorem.  Every even-dimensional\nclass is proportional to~$\\beta^n$.  Every odd-dimensional class is\nproportional to~$\\alpha \\cup \\beta^n$.\n\nFurthermore, the reduced integral homology is~$\\Z_p$ in odd dimensions\nand vanishes in even dimensions.  The integral cohomology is~$\\Z_p$ in\neven positive dimensions, generated by a lift of the class~$\\beta$\nabove to~$H^2(\\Z_p;\\Z)$.\n\n\n\\begin{proposition}\n\\label{41}\n\\label{33}\nLet~$M$ be a closed~$3$--manifold~$M$ with~$\\pi_1(M)=\\Z_{p}$.  Then its\nclassifying map~$\\varphi \\colon\\thinspace M \\to K$ induces an\nisomorphism\n\\[\n\\varphi_i \\colon\\thinspace H_i(M;\\Z_p)\\simeq H_i(K;\\Z_p)\n\\]\nfor~$i=1,2,3$.\n\\end{proposition}\n\n\\begin{proof}\nSince~$M$ is covered by the sphere, for~$i=2$ the isomorphism is a\nspecial case of Whitehead's theorem.  Now consider the exact sequence\n(of Hopf type)\n\\begin{equation*}\n\\pi_3(M) \\overset{\\times p}{\\longrightarrow}H_3(M;\\Z)\\to\nH_3(\\Z_p;\\Z)\\to 0\n\\end{equation*}\nsince~$\\pi_2(M)=0$.  Since the homomorphism~$H_3(M;\\Z) \\to\nH_3(\\Z_p;\\Z)$ is onto, the result follows by reduction modulo~$p$.\n\\end{proof}\n\n\n\n\n\\section{Volume of a ball}\n\nOur Theorem \\ref{theo:main} is a consequence of the following result.\n\n\\begin{theorem} \n\\label{theo:ball}\nAssume the~$\\rm{GG}_C$-property~\\eqref{eq:ball} is satisfied for some universal constant\n$C>0$ and every homomorphism~$\\phi$ into a finite group~$G$. \nThen every closed\nRiemannian~$3$--manifold~$M$ with fundamental group~$G$ contains a\nmetric ball~$B(R)$ of radius~$R$ satisfying\n\\begin{equation}\n\\label{24}\n\\vol \\, B(R) \\geq \\frac{C}{3} R^3,\n\\end{equation}\nfor every~$R\\leq\\frac{1}{2}\\sys(M)$.\n\\end{theorem}\n\n\\forget\nRecall the following result.\n\n\\begin{proposition}\n\\label{006}\nIn an orientable~$3$--manifold, cup product on~$H^1\\otimes H^2$ in\ncohomology with~$\\Z_p$ coefficients is dual to intersection between\na~$2$--cycle and a ~$1$--cycle with coefficients in~$\\Z_p$.\n\\end{proposition}\n\nHere the global orientation allows one to count an integer\nintersection index, which is then reduced modulo~$p$. \\\\\n\\forgotten\n\nWe will first prove Theorem~\\ref{theo:ball} for a\nclosed~$3$--manifold~$M$ of fundamental group~$\\Z_{p}$, with~$p$\nprime.  We assume that~$p$ is odd (the case~$p=2$ was treated by\nL.~Guth).  In particular,~$M$ is orientable.  Let~$D$ be a~$2$--cycle\nrepresenting a nonzero class~$[D]$ in\n\\begin{equation*}\nH_2(M;\\Z_p) \\simeq H_{1}(M;\\Z_{p}) \\simeq \\Z_p.\n\\end{equation*}\nDenote by~$D_0$ the finite~$2$--complex of~$M$ given by the support\nof~$D$.  Without loss of generality, we can assume that~$D_0$ is\nconnected.  The restriction of the classifying map~$\\varphi \\colon\\thinspace M \\to\nK$ to~$D_0$ induces a homomorphism~$\\phi \\colon\\thinspace \\pi_{1}(D_0) \\to \\Z_{p}$.\n\n\\begin{lemma} \\label{lem:DB}\nThe cycle~$D$ induces a trivial relative class in the homology of\nevery metric~$R$--ball~$B$ in~$M$ relative to its boundary, with~$R <\n\\frac{1}{2} \\sys(M)$.  That is,\n$$\n[D \\cap B] = 0 \\in H_{2}(B,\\partial B;\\Z_{p}).\n$$\n\\end{lemma}\n\n\\begin{proof}\nSuppose the contrary.  By the Lefschetz-Poincar\\'e duality theorem,\nthe relative~$2$--cycle~$D \\cap B$ in~$B$ has a nonzero intersection\nwith an (absolute)~$1$--cycle~$c$ of~$B$.  Thus, the intersection\nbetween the~$2$--cycle~$D$ and the~$1$--cycle~$c$ is nontrivial\nin~$M$.  Now, by Lemma~\\ref{lem:trivial}, the~$1$--cycle~$c$ is\nhomotopically trivial in~$M$.  Hence a contradiction.\n\\end{proof}\n\nWe will exploit the following notion of volume for cycles with torsion\ncoefficients.\n\n\\begin{definition} \n\\label{def:Vol}\nLet~$D$ be a~$k$--cycle with coefficients in~$\\Z_p$ in a Riemannian\nmanifold~$M$.  We have\n\\begin{equation}\n\\label{11}\nD= \\sum_i n_i \\sigma_i\n\\end{equation}\nwhere each~$\\sigma_i$ is a~$k$--simplex, and each~$n_i\\in \\Z_p^*$ is\nassumed nonzero.  We define the notion of~$k$--area~$\\area$ for cycles\nas in \\eqref{11} by setting\n\\begin{equation}\n\\label{12}\n\\area(D)= \\sum_i |\\sigma_i|,\n\\end{equation}\nwhere~$|\\sigma_i|$ is the~$k$--area induced by the Riemannian metric\nof~$M$.\n\\end{definition}\n\n\\begin{remark}\nThe non-zero coefficients~$n_i$ in \\eqref{11} are ignored in defining\nthis notion of volume.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem~\\ref{theo:ball}]\nWe continue the proof of Theorem~\\ref{theo:ball} when the fundamental\ngroup of~$M$ is isomorphic to~$\\Z_p$, with~$p$ an odd prime.  We will\nuse the notation introduced earlier.  Suppose now that~$D$ is a\npiecewise smooth~$2$--cycle area minimizing in its homology\nclass~$[D]\\not=0\\in H_2(M;\\Z_p)$ up to an arbitrarily small error\nterm~$\\varepsilon>0$, for the notion of volume (area) as defined\nin~\\eqref{12}.\n\n\nRecall that~$\\phi \\colon\\thinspace \\pi_1(D_0)\\to \\Z_p$ is the homomorphism induced\nby the restriction of the classifying map~$\\varphi \\colon\\thinspace K \\to M$ to the\nsupport~$D_0$ of~$D$.  By Proposition~\\ref{33}, the~$2$--complex~$D_0$\nis~$\\phi$--essential.  Thus, by hypothesis of Theorem~\\ref{theo:ball},\nwe can choose a point~$x \\in D_0$ satisfying\nthe~$\\rm{GG}_C$-property~\\eqref{eq:ball}, i.e., the area of~$R$--balls\nin~$D_0$ centered at~$x$ grows at least as~$C R^2$ for~$R <\n\\frac{1}{2} \\sys(D_0,\\phi)$.\nTherefore, the intersection of~$D_0$ with the~$R$--balls of$M$\ncentered at~$x$ satisfies\n\\begin{equation}\n\\label{111}\n\\area(D_0\\cap B(x,R)) \\geq CR^2\n\\end{equation}\nfor every~$R < \\frac{1}{2} \\sys(D_0,\\phi)$.\nThe idea of the proof is to control the area of distance spheres\n(level surfaces of the distance function) in~$M$, in terms of the\nareas of the distance disks in~$D_0$.\n\nLet~$B=B(x,R)$ be the metric~$R$--ball in~$M$ centered at~$x$\nwith~$R<\\frac{1}{2} \\sys(M)$.  We subdivide and slightly perturb~$D$\nfirst, to make sure that~$D \\cap \\bar B$ is a subchain of~$D$.  Write\n\\[\nD=D_- + D_+,\n\\]\nwhere~$D_-$ is a relative~$2$--cycle of~$\\bar B$, and~$D_+$ is a\nrelative~$2$--cycle of~$M\\setminus B$.  By Lemma~\\ref{lem:DB},~$D_-$\nis homologous to a~$2$--chain~$\\mathcal{C}$ contained in the distance\nsphere~$\\partial B = S(x,R)$ with\n\\[\n\\partial \\mathcal{C} = \\partial D_- = - \\partial D_+.\n\\]\nWe subdivide and perturb~$\\mathcal{C}$ in~$S(x,R)$ so that the\ninteriors of its~$2$--simplices either agree or have an empty\nintersection.  Here the simplices of the~$2$--chain~$\\mathcal{C}$ may\nhave nontrivial multiplicities.\nSuch multiplicities necessarily affect the volume of a chain if one\nworks with integer coefficients.\nHowever, these multiplicities are ignored for the notion\nof~$2$--volume~\\eqref{12}.  This special feature allows us to derive\nthe following: the~$2$--volume~\\eqref{12} of the chain~$\\mathcal{C}$\nis a lower bound for the usual area of the distance sphere~$S(x,R)$.\n\nNote that the homology class~$[\\mathcal{C}+D_+]=[D] \\in H_2(M;\\Z_p)$\nstays the same.  We chose~$D$ to be area minimizing up\nto~$\\varepsilon$ in its homology class in~$M$ for the notion of\nvolume~\\eqref{12}.  Hence we have the following bound:\n\\begin{equation}\n\\label{112}\n\\area(S(x,R)) \\geq \\area(\\mathcal{C}) \\geq \\area(D_-) - \\varepsilon\n\\geq \\area (D_0 \\cap B) - \\varepsilon.\n\\end{equation}\nNow, clearly~$\\sys(M) \\leq \\sys(D_0,\\phi)$.  Combining the\nestimates~\\eqref{111} and~\\eqref{112}, we obtain\n\\begin{equation}\n\\label{113}\n\\area ( S(x,R)) \\geq C R^2 - \\varepsilon\n\\end{equation}\nfor every~$R<\\frac{1}{2} \\sys(M)$.  Integrating the\nestimate~\\eqref{113} with respect to~$R$ and letting~$\\varepsilon$ go\nto zero, we obtain a lower bound of~$\\frac{C}{3} R^3$ for\nthe~$3$--volume of some~$R$--ball in the closed manifold~$M$, proving\nTheorem~\\ref{theo:ball} for closed~$3$--manifolds with fundamental\ngroup~$\\Z_{p}$. \\\\\n\nSuppose now that~$M$ is a closed~$3$--manifold with finite (nontrivial)\nfundamental group.  Choose a prime~$p$ dividing the\norder~$|\\pi_1(M)|$ and consider a cover~$N$ of~$M$ with fundamental group cyclic\nof order~$p$.\nThis cover satisfies~$\\sys(N) \\geq \\sys(M)$, and we apply the\nprevious argument to~$N$.\n\nNote that the reduction to a cover could not have been done in the\ncontext of M.~Gromov's formulation of the inequality in terms of the\nglobal volume of the manifold.  Meanwhile, in our formulation using a\nmetric ball, following L.~Guth, we can project injectively the ball of\nsufficient volume, from the cover to the original manifold.  Namely,\nthe proof above exhibits a point~$x \\in N$ such that the volume of\nthe~$R$--ball~$B(x,R)$ centered at~$x$ is at least~$\\frac{C}{3} R^3$\nfor every~$R < \\frac{1}{2} \\sys(M)$.  Since~$R$ is less than half the\nsystole of~$M$, the ball~$B(x,R)$ of~$N$ projects injectively to an\n$R$--ball in~$M$ of the required volume, completing the proof of\nTheorem~\\ref{theo:ball}.\n\\end{proof}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:1}\nThe understanding of diffusion and transport of passive \ntracers in a given velocity field has both theoretical and practical \nrelevance in many fields of science and engineering, \ne.g. mass and heat transport in geophysical flows \n(for a review see \\cite{davis,davis2}),\ncombustion and chemical engineering \\cite{Moffatt}.\n\n\nOne common interest is the study of the mechanisms\nwhich lead to transport enhancement as a fluid is driven \nfarther from the motionless state. \nThis is related to the fact that the Lagrangian motion of \nindividual tracers can be rather complex even in simple laminar flows\n\\cite{Ottino,lagran}.\n\nThe dispersion of passive scalars in a given velocity field is the result,\nusually highly nontrivial, of two different contributions:\nmolecular diffusion and advection.\nIn particular, one can have rather fast transport, even without\nmolecular diffusion, in presence of {\\it Lagrangian chaos \\\/}, \nwhich is the sensitivity to initial conditions of\nLagrangian trajectories.\nIn addition, also for a 2D stationary velocity field, \nwhere one cannot have Lagrangian chaos \\cite{Licht}, in presence of \na particular geometry of the streamlines the diffusion can \nbe much larger than the one due only to the molecular \ncontribution, as in the case of spatially periodic stationary\nflows \\cite{Rosenbluth,Shraiman}.\n\nTaking into account the molecular diffusion, the motion of \na test particle (the tracer) is described by the following Langevin\nequation:\n\\begin{equation}\n\\frac{d{\\bf x}}{dt}= {\\bf u}({\\bf x},t)+\n\\mbox{\\boldmath $\\eta$}(t),\n\\label{eq:langevin}\n\\end{equation}\nwhere \n${\\bf u}({\\bf x},t)$ \nis the Eulerian \nincompressible velocity field at the point ${\\bf x}$ \nand time $t$, $\\mbox{\\boldmath $\\eta$}(t)$ is a Gaussian white noise \nwith zero mean and\n\\begin{equation}\n<\\eta_{i}(t) \\eta_{j}(t^{'}) >= 2 D_{0} \\delta_{ij} \\delta(t-t^{'})\\,,\n\\label{eq:whitenoise}\n\\end{equation}\nwhere $D_{0}$ is the (bare) molecular diffusivity.\n\nDenoting \n$\\Theta({\\bf x},t)$ \nthe concentration of tracers,\none has:\n\\begin{equation}\n{\\partial}_{t} \\Theta+ \n\\left( {\\bf u} \\cdot \\mbox{\\boldmath $\\nabla$} \\right) \\Theta=\nD_{0} \\,\\Delta \\Theta \\,.\n\\label{eq:fokker}\n\\end{equation}\n\nFor an Eulerian velocity field  \nperiodic in space, or anyway  defined in infinite domains, \nthe long-time, large-distance behavior of the diffusion process\nis described by the effective diffusion tensor $D_{ij}^{E}$\n({\\it eddy-diffusivity tensor}\\\/):\n\\begin{equation}\nD_{ij}^{E}= \\lim_{t\\rightarrow \\infty} \\frac {1}{2t}\n<(x_{i}(t)-<x_{i}>)(x_{j}(t)-<x_{j}>)>\\,,\n\\label{def:eddydiff}\n\\end{equation}\nwhere now ${\\bf x}(t)$ is the position of the the tracer \nat time $t$, $i,j=1,\\cdots,d$ (being $d$ the spatial dimension) , and\nthe average is taken over the initial positions\nor, equivalently, over an ensemble of test particles.\nThe tensor $D^{E}_{ij}$ gives the long-time, large-distance\nequation for $< \\! \\Theta \\!>$ i.e. the concentration field locally averaged\n over a volume of linear distance much larger than the\ntypical length $l_{u}$ of the velocity field, according to\n\\begin{equation}\n{\\partial}_{t} <\\Theta>= \\sum_{i,j=1}^{d} D_{ij}^{E}\\, \n\\frac{{\\partial}^{2}}{\\partial x_{i} \\partial x_{j}} <\\Theta>\\,.\n\\label{eq:eddydiff}\n\\end{equation}\nThe above case, with finite $D_{ij}^{E}$, is\nthe typical situation where the diffusion, for very large \ntimes, is a standard diffusion process. \nHowever there are also cases \nshowing the so-called {\\it anomalous diffusion}\\\/: the spreading \nof the particles does not behave linearly with time but\nhas a power law $t^{2\\nu}$ with $\\nu \\neq 1\/2$.\nTransport anomalies are, in general, indicators of the \npresence of strong \ncorrelation in the dynamics, even at large time and space scales\n\\cite{georges}.\n\nIn the case of infinite spatial domains and periodic \nEulerian fields the powerful multiscale technique \n(also known as homogenization in mathematical literature) \ngives a useful tool for studying standard diffusion\nand, with some precautions, also the anomalous situations \n\\cite{BCVV}.\n\nOn the other hand we have to stress the fact that \ndiffusivity tensor (\\ref{def:eddydiff}) is \nmathematically well defined only in the limit of infinite times, \ntherefore it gives \na sensible result only if the characteristic length\n$l_{u}$ of the velocity field is much smaller than the size \n$L$ of the domain. \n\nThe case when $l_{u}$ and $L$ are not well separated\nis rather common in many geophysical problems, e.g.\nspreading of pollutants in Mediterranean or Baltic sea, \nand also in plasma physics.\nTherefore it is important to introduce some other \ncharacterizations of the diffusion properties\nwhich can be used also in non ideal cases.\nFor instance, \\cite{Zambia} propose to employ exit times \nfor the study of transport in basins with complicated geometry.\n\nIn Section \\ref{sec:2} we introduce a characterization of\nthe diffusion behavior in terms of the typical time\n$\\tau(\\delta)$ at scale $\\delta$;\nthis allows us to define a finite size diffusion \ncoefficient \n$D(\\delta) \\sim \\delta^{2}\/\\tau(\\delta)$.\n>From the shape of $\\tau(\\delta)$ \nas a function of $\\delta$, one can distinguish different\nspreading regimes.\n\nIn Section \\ref{sec:3} we present the results of numerical experiments\nin closed basins and present new results\nrelative to the behavior of the diffusion coefficient \nnear the boundary (a detailed discussion \nis in the appendix).\n\nIn Section \\ref{sec:4} we summarize our results and\npresent conclusions and we discuss \nthe possibility of treatment of experimental data \naccording to the method introduced in Section \\ref{sec:2}.\n\n\\section{Finite size diffusion coefficient}\n\\label{sec:2}\nBefore a general discussion let us start with a simple example.\nConsider the relative diffusion of a cloud of N test particles\nin a smooth, spatially periodic velocity field with characteristic\nlength $l_{u}$.  We assume that the Lagrangian motion is chaotic \ni.e. the maximum Lyapunov exponent $\\lambda$ is positive.\nDenoting with $R^{2}(t)$ the square of the typical radius of\nthe cloud\n\\begin{equation}\nR^{2}(t)= \n\\ll|{\\bf x}_{i}(t)-\\ll{\\bf x}_{i}(t)\\gg|^{2}\\gg\\,,\n\\label{def:disprel}\n\\end{equation}\nwhere\n\\begin{equation}\n\\ll{\\bf x}_i(t)\\gg={1 \\over N} \\sum_{i=1}^N {\\bf x}_i(t)\n\\end{equation}\nwe expect the following regimes to hold\n\\begin{equation}\n\\overline{R^{2}(t)} \\simeq \\left\\{ \n\\begin{array}{ll}\nR^{2}(0)\\exp(L(2)t) & \\;\\;\\;\\;\n{\\mbox {if    $\\overline{R^{2}(t)}^{1\/2} \\ll l_{u}$}}\n \\\\\n2 D t & \\;\\;\\;\\;\n{\\mbox {if    $\\overline{R^{2}(t)}^{1\/2} \\gg l_{u}$}}\n\\end{array}\n\\label{eq:regimiperR}\n\\right.\n\\,,\n\\label{example1} \n\\end{equation}\nwhere $L(2) \\geq 2\\lambda$ is the generalized Lyapunov exponent\n\\cite{BPPV85,PV87}, $D$ is the diffusion coefficient and \nthe overbar denotes the average over initial conditions.\n\nIn this paper we prefer to study the relative diffusion \n(\\ref{def:disprel}) instead of the usual absolute diffusion.\nFor spatially infinite cases, without mean drift\nthere is no difference;\nfor closed basins the relative dispersion is,\nfor many aspects, more interesting than the absolute one\nand, in addition, the latter is dominated by \nthe sweeping induced by large scale flow.\n\nFurthermore we underline \nthat although the dynamics of the ocean circulation is dominated\nby large mesoscale gyres, the smaller scales \nactivities within the gyres\ncontrol important local phenomena as deep water \nformation in North Atlantic and in Mediterranean \nbasin \\cite{marshal}.\nTherefore the study of relative diffusion could be \nrelevant to describe this small-scale motion\nand can give crucial informations on the way \nto parameterize the subgrid scales \nin ocean numerical global model \\cite{garret}. \n\nAnother, at first sight rather artificial, way to describe\nthe above behavior is by introducing the ``doubling \ntime\\\/'' $\\tau(\\delta)$ at scale $\\delta$ as follows:\nwe define a series of thresholds $\\delta^{(n)}= r^{n} \\delta^{(0)}$,\nwhere $\\delta^{(0)}$ is the initial size of the cloud, defined according\nto (\\ref{def:disprel}), and then we measure the time $T(\\delta^{(0)})$ \nit takes for the growth\nfrom $\\delta^{(0)}$ to $\\delta^{(1)}= r \\delta^{(0)}$, and so on\nfor $T(\\delta^{(1)})\\,,\\;T(\\delta^{(2)})\\,,\\ldots$\nup to the largest scale under consideration.\nFor the threshold rate $r$ any value can be chosen but too large ones\nmight not separate different scale contributions, \nthough strictly speaking the term ``doubling time''\nrefers to the threshold rate $r=2$.\n\nPerforming ${\\cal N} \\gg 1$ experiments with\ndifferent initial conditions for the cloud, we define the \ntypical doubling time $\\tau(\\delta)$ at scale \n$\\delta$ as\n\\begin{equation}\n\\tau(\\delta) = < T(\\delta) >_e =\\frac{1}{{\\cal N}}\n \\sum_{i=1}^{{\\cal N}} T_{i}(\\delta)\\,.\n\\label{def:taudelta}\n\\end{equation}\nLet us stress the fact that the average \nin (\\ref{def:taudelta}) is different from the usual\ntime average.\n\nFrom the average doubling time we can define the finite size \nLagrangian Lyapunov exponent as\n\\begin{equation}\n\\lambda(\\delta)=\\frac{\\ln r}{\\tau(\\delta)}\\,,\n\\end{equation}\nwhich is a measure of the average rate of separation of two\nparticles at a distance $\\delta$. Let us remark that $\\lambda(\\delta)$\nis independent of $r$, for $r \\rightarrow 1^{+}$. \nFor very small separations (i.e. $\\delta \\ll l_u$) one recovers the standard \nLagrangian Lyapunov exponent $\\lambda$,\n\\begin{equation}\n\\lambda=\\lim_{\\delta \\rightarrow 0} \\frac{1}{\\tau(\\delta)}\n\\ln r\\,.\n\\label{def:liapfromtau}\n\\end{equation}\nSee \\cite{ABCPV} for a detailed discussion about \nthese points.\nIn this framework the\nfinite size diffusion coefficient $D(\\delta)$ dimensionally turns out to be\n\\begin{equation}\nD(\\delta)=\\delta^{2}\\lambda(\\delta)\\,.\n\\label{def:fsd}\n\\end{equation}\nNote the absence of the factor $2$, as one can expect by\nthe definition (\\ref{def:eddydiff}), in the denominator of\n$D(\\delta)$ in equation (\\ref{def:fsd}); this is due \nto the fact that $\\tau(\\delta)$ is a difference of times.\nFor a standard diffusion process $D(\\delta)$ approaches the diffusion\ncoefficient $D$ (see eq. (\\ref{eq:regimiperR})) in the limit of \nvery large separations ($\\delta \\gg l_u$). This result stems from \nthe scaling of the doubling times $\\tau(\\delta) \\sim \\delta^2$ for \nnormal diffusion. \n\nThus the finite size Lagrangian Lyapunov exponent $\\lambda(\\delta)$, or\nits counterpart $D(\\delta)$, embody the asymptotic behaviors \n\\begin{equation}\n\\lambda(\\delta) \\sim \\left\\{ \n\\begin{array}{ll}\n\\lambda & \\;\\;\\;\\;\n{\\mbox {if    $\\delta \\ll l_{u}$}}\n \\\\\nD\/\\delta^{2} & \\;\\;\\;\\;\n{\\mbox {if    $\\delta \\gg l_{u}$}}\n\\end{array}\n\\right.\n\\,,\n\\label{eq:regimipertau} \n\\end{equation}\nOne could naively conclude, matching the behaviors \nat $\\delta \\sim l_{u}$, that $D \\sim \\lambda l_{u}^{2}$.\nThis is not always true, since one can have a rather large range\nfor the crossover due to the \nfact that nontrivial correlations can be present in \nthe Lagrangian dynamics \\cite{FV89}.\n\nAnother case where the \nbehavior of $\\tau(\\delta)$ as a function of $\\delta$ \nis essentially well understood\nis 3D fully developed turbulence. \nFor sake of simplicity we neglect intermittency \neffects. \nThere are then three different ranges:\n\\begin{enumerate}\n\\item \n$\\delta \\ll \\eta ={\\mbox {Kolmogorov length}}$ :\n$1\/\\tau(\\delta) \\sim \\lambda$;\n\\item\n$\\eta \\ll \\delta \\ll l={\\mbox{ typical size of the \nenergy containing eddies}}$: \nfrom the Richardson law \n$\\overline{R^{2}(t)} \\sim t^{3}$ \none has \n$1\/\\tau(\\delta) \\sim \\delta^{-2\/3}$;\n\\item\n$\\delta \\gg l$ : usual diffusion behavior \n$1\/\\tau(\\delta) \\sim \\delta^{-2}\\,.$\n\\end{enumerate}\n\nOne might wonder that the proposal to introduce\nthe time $\\tau(\\delta)$ is just another way \nto look at\n$\\overline{R^{2}(t)}$ as a function of $t$.\nThis is true only in limiting cases, when\nthe different characteristic lengths are \nwell separated and intermittency is weak.\nIn \\cite{previouswork1,previouswork2,sabot} rather \nclose techniques are used for the computation of\nthe diffusion coefficient in nontrivial cases.\n\nThe method of working at fixed scale $\\delta$,\nallows us to extract the physical information at that spatial\nscale avoiding unpleasant troubles of the method of\nworking at a fixed delay time $t$.\nFor instance, if one has a strong intermittency, and this is a rather\nusual situation, $R^{2}(t)$ as a function of \n$t$ can appear very different in each realization.\nTypically one can have, see figure \\ref{fig1}a,\ndifferent exponential rates of growth for different\nrealizations, producing a rather odd behavior\nof the average $\\overline{R^{2}(t)}$ without\nany physical meaning. For instance in figure \\ref{fig1}b we show\nthe average  $\\overline{R^{2}(t)}$ versus time $t$; at large times we\nrecover the diffusive behavior but at intermediate times \nappears an apparent anomalous regime which is only due to \nthe superposition of exponential and diffusive contributions\nby different samples at the same time.\nOn the other hand exploiting the tool of doubling times one has \nan unambiguous result (see figure \\ref{fig1}c).\n\nOf course the interesting situations are those where\nthe different characteristic lengths ($\\eta\\,,\\;l\\,,\\;L$) \nare not very different and therefore each \nscaling regime for $\\overline{R^2(t)}$ is not well evident.\n\n\\section{Numerical results}\n\\label{sec:3}\nHere we present some numerical experiments \nin simple models with\nLagrangian chaos in the zero molecular diffusion limit.\nBefore showing the results, we describe the numerical \nmethod adopted.\n\nWe choose a passive tracers trajectory having a chaotic behavior, \ni.e. with a positive maximum Lyapunov \nexponent, computed by using standard algorithms \\cite{BeneGalg}.\nThen we place $N-1$ passive tracers around the first one\nin a cloud of initial size \n\\begin{eqnarray}\nR(0)=\\delta(0)=\\delta^{(0)}\\,,\n\\nonumber\n\\end{eqnarray} \nwith $R(0)$ defined by equation (\\ref{def:disprel}).  \nIn order to have average properties we repeat this procedure \nreconstructing the passive cloud around the last\nposition reached by the reference chaotic tracer in the previous \nexpansion.\nThis ensures that the initial expansion of the cloud \nis exponential\nin time, with typical exponential rate equal to the\nLyapunov exponent.\n\nFurther we define a series of thresholds $\\delta^{(n)}=r^{n}\\delta^{(0)}$\n(as described in Section 2) \n$n=1,\\cdots,n_{max}$ and we measure the time $T_{n}$ \nspent in  expanding from $\\delta^{(n)}$ to $\\delta^{(n+1)}\\,$.\nThe value of $n_{max}$ has to be chosen in such a way that\n$\\delta^{(n_{max})}\\sim \\delta_{max}$, where $\\delta_{max}$\ncorresponds to the uniform distribution of the tracers in the basin\n (see forthcoming discussion and the Appendix). Each realization stops\nwhen $\\delta(t)=\\delta^{(n_{max})}$.\n\nTherefore following \\cite{ABCPV} we define a scale dependent\nLagrangian Lyapunov exponent as:\n\\begin{equation}\n\\lambda(\\delta^{(n)}) = \\frac{1}{<{T_{n}}>_e} \\ln r =\n\\frac{1}{\\tau(\\delta^{(n)})} \\, \\ln r.\n\\label{def:lambdadidelta}\n\\end{equation}\nIn equation (\\ref{def:lambdadidelta}) we have implicitly assumed that\nthe evolution of the size $\\delta(t)$ of the cloud is continuous in time.\nThis is not true in the case of discontinuous processes such as maps or\nin the analysis of experimental data taken at \nfixed delay times.\nDenoting $T_{n}$ the time to reach size \n$\\tilde{\\delta} \\geq \\delta^{(n+1)}$ from $\\delta^{(n)}$ ,\nnow $\\tilde{\\delta}$ is a fluctuating quantity,\n equation (\\ref{def:lambdadidelta}) has to be modified as follows \n\\cite{ABCPV}:\n\\begin{equation}\n\\lambda(\\delta^{(n)}) = \\frac{1}{<{T_{n}}>_e}   \n\\left< \\ln \\left( \\frac{\\tilde{\\delta}}{\\delta^{(n)}}\\right) \\right>_e\n\\,.\n\\label{def:lambdadidelta1}\n\\end{equation}\n\nIn our numerical experiments we have the regimes \ndescribed in sect. 2: exponential regime\n, i.e. $\\lambda(\\delta)=\\lambda$, and diffusion-like regime\ni.e. $\\lambda(\\delta)=D\/\\delta^{2}$, at least if the size $L$ \nof the basin is large enough.\n\nFor cloud sizes close to the saturation value $\\delta_{max}$\nwe expect the following behavior to hold for a broad class \nof systems:\n\\begin{equation}\n \\lambda(\\delta)=\\frac{D(\\delta)}{\\delta^{2}} \\propto\n\\frac{(\\delta_{max}-\\delta)}{\\delta} \\,.\n\\label{eq:nearbound}\n\\end{equation}\nThe constant of proportionality\nis given by  the second eigenvalue of the \nPerron-Frobenius operator which is related to the typical time \nof exponential relaxation of tracers' density to the uniform distribution\nActually, the analytical evaluation of this eigenvalue can be \nperformed only for extremely simple dynamical systems\n(for instance random walkers, as shown in the Appendix).\nAs a consequence the range of validity for (\\ref{eq:nearbound})\ncan be assessed only by numerical simulation.\n\n\\subsection{A model for transport in Rayleigh-B\\'enard convection}\nThe advection in two dimensional incompressible flows is described,\nin absence of molecular diffusion, by Hamiltonian equation of motion  \nwhere the Hamilton function is the stream function $\\psi$:\n\\begin{equation}\n\\frac{dx}{dt}=\\frac{\\partial \\psi}{\\partial y}\\,, \\;\\;\\;\n\\frac{dy}{dt}=-\\frac{\\partial \\psi}{\\partial x}\\,.\n\\label{eq:hamilton}\n\\end{equation}\nIf $\\psi$ is time-dependent the system (\\ref{eq:hamilton})\nis non-autonomous and in general non-integrable, then\nchaotic trajectories may exist. \n\nOne example is the model introduced in \\cite{gollub}\nto describe the chaotic advection\nin the time-periodic Rayleigh-B\\'enard convection.\nIt is defined by the stream function:\n\\begin{equation}\n\\psi(x,y,t)=\\frac{A}{k} \n\\sin\\left\\{ k \\left[ x+B \\sin(\\omega t)\\right]\\right\\}\nW(y)\\,,\n\\label{eq:gollubinf}\n\\end{equation}\nwhere $W(y)$ is a function that satisfies rigid \nboundary conditions on the surfaces $y=0$ and $y=a$ \n(we use $W(y)=\\sin(\\pi y\/a)$).\nThe direction $y$ is identified with the vertical direction\nand the two surfaces $y=a$ and $y=0$ are the top and bottom\nsurfaces of the convection cell.\nThe time dependent term $B\\sin(\\omega t)$ represents \nlateral oscillations of the roll pattern  \nwhich mimic the even oscillatory instability \\cite{gollub}.\n\nTrajectories starting near the roll\nseparatrices could have positive Lyapunov exponent and thus\ndisplay chaotic motion and diffusion in the x direction. \nIt is remarkable that in spite of the simplicity of the model,\nthe agreement of the numerical results with experimental ones is quite\ngood \\cite{gollub}.\n\nDefining a passive cloud in the $x$ direction (i.e. a\nsegment) and performing the expansion experiment described \nin the previous section\nwe have that, until $\\delta$ is below a fraction of the\ndimension of the cell, $\\lambda(\\delta)=\\lambda$ (figure \\ref{fig2}a).\nFor larger values of $\\delta$ we have \nthe standard diffusion $\\lambda(\\delta)=D\/\\delta^{2}$ \nwith good quantitative agreement with the value of the \ndiffusion coefficient evaluated by the standard technique, i.e.\nusing $\\overline{R^{2}(t)}$ as a function of time $t$\n(compare figure \\ref{fig2}a with figure \\ref{fig2}b).\n\nTo confine the motion of tracers in a closed domain,\ni.e. $x \\in [-L,L]$, we must slightly modify the streamfunction\n(\\ref{eq:gollubinf}). \nWe have modulated the oscillating term\nin such a way that for $|x|=L$ the amplitude of the oscillation\nis zero, i.e. $B \\rightarrow B \\sin(\\pi x\/L)$ with $L=2\\,\\pi n\/k$\n($n$ is the number of convective cells).\nIn this way\nthe motion is confined in $[-L,L]$.\n\nIn figure \\ref{fig3} we show $\\lambda(\\delta)$ for two values of $L$. \nIf $L$ is large enough one can well see the three regimes, \nthe exponential one, the diffusive one and the saturation given\nby equation (\\ref{eq:nearbound}).\nDecreasing $L$ the range\nof the diffusive regime decreases, and for small values of\n$L$ it disappears. \n\n\\subsection{Modified Standard Map}\nOne of the simplest deterministic dynamical system displaying both\nexponential growth of separation for close trajectories \nand asymptotic diffusive behavior \nis the standard (Chirikov - Taylor) mapping \\cite{Chi79}.\nIt is customarily defined as \n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\nx_{n+1}=x_n+K \\sin y_n & \\\\\n y_{n+1}=y_n+x_{n+1}& \\mbox{ mod $2 \\pi $}\n\\end{array}\n\\right.\n\\label{eq:standard}\n\\end{equation}\nThis mapping conserves the area in the phase space.\nIt is widely known that for large enough values of the \nnonlinearity strength parameter $K \\gg K_c \\simeq 1$ the motion\nis strongly chaotic in almost all the phase space.\nIn this case the standard map, in the $x$-direction\n mimics the behavior of a one-dimensional random walker,\nstill being deterministic, and so one expects the behavior of\n$\\lambda(\\delta)$ to be quite similar to the one already \nencountered in the model for Rayleigh-B\\'enard convection\nwithout boundaries. \nNumerical iteration of (\\ref{eq:standard}) for a cloud of particles\nclearly shows the two regimes described in \n(\\ref{eq:regimipertau}), similar to that showed for the \nmodel discussed in the previous section.\n\nWe turn now to the more interesting case in which the domain is limited\nby boundaries reflecting back the particle. \nTo achieve the confinement of the trajectory\ninside a bounded region we modify the standard map in the following\nway\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\nx_{n+1}=x_n+K f(x_{n+1})\\sin y_n & \\\\ \ny_{n+1}=y_n+x_{n+1}-K f'(x_{n+1}) \\cos y_n & \\mbox{ mod $ 2 \\pi$}.\n\\end{array}\n\\right.\n\\label{eq:modified}\n\\end{equation}\nwhere $f(x)$ is a function which has its only zeros in $\\pm L$.  \nSince the mapping is defined in implicit form, \nthe shape of $f$ must be chosen in such a way to assure\na unique definition for $(x_{n+1},y_{n+1})$ given $(x_n,y_n)$. \nFor any $f$ fulfilling this request the mapping \n(\\ref{eq:modified}) conserves the area.\nA trial choice could be\n\\begin{equation}\nf(x)=\n\\left\\{\n\\begin{array}{ll}\n1 & |x|<\\ell \\\\\n\\begin{displaystyle}\n{L-|x| \\over L-\\ell}\n\\end{displaystyle}\n & \\ell<|x|<L\n\\end{array}\n\\right.\n\\label{eq:f}\n\\end{equation}\nStrictly speaking this is not quite an appropriate choice, since it renders\nthe map discontinuous at $|x| = \\ell$, but \nthis is an irrelevant point and it is easy to bypass this\nobstacle by assuming a suitably smoothed version of (\\ref{eq:f}).\n\nPerforming the doubling times computation (\\ref{def:taudelta})\none recovers both\nthe exponential and diffusive regimes for \n$\\lambda(\\delta)$, and in addition one has the saturation regime\n(\\ref{eq:nearbound}). Figure \\ref{fig4} shows the behavior of\nthe scale dependent diffusion coefficient $D(\\delta)$ (\\ref{def:fsd}).\nApproaching the saturation value $\\delta_{max}$ the diffusion \ncoefficient quickly drops to zero, following the asymptotic law \n(\\ref{eq:nearbound}) derived in the appendix.\nThe qualitative behaviors in figure \\ref{fig4} do not depend on the details \nof the function $f$. \n                   \n\\subsection{Point vortices in a Disk}\nAs another example, we consider the  two-dimensional  \ntime-dependent flow generated by the motion of $N$ point vortices \nin a closed domain \\cite{Aref}.\nFor a unitary disk the positions of the vortices $(x_i=r_i \\cos \\,\\theta_i\n,y_i=r_i \\sin \\,\\theta_i)$, with circulation\n$\\Gamma_i$, evolve according to the Hamiltonian dynamics \n\\begin{equation}\n\\dot{x_i} = {1 \\over \\Gamma_i} {\\partial H \\over \\partial y_i}, \\: \\: \\:\n\\dot{y_i} = - {1 \\over \\Gamma_i} {\\partial H \\over \\partial x_i}\n\\end{equation}\nwhere the Hamiltonian is\n\\begin{equation}\nH = - {1 \\over 4\\pi} \\sum_{i>j} \\Gamma_i \\Gamma_j \\log \\left[\n{r_i^2+r_j^2-2 r_i r_j \\cos (\\theta_i-\\theta_j) \\over 1 + r_i^2 r_j^2 -\n2 r_i r_j \\cos (\\theta_i-\\theta_j)} \\right] +\n{1 \\over 4 \\pi} \\sum_{i=1}^N \\Gamma_i^2 \\log (1-r_i^2)\n\\label{}\n\\end{equation}\n\nPassive tracers evolve according to (\\ref{eq:hamilton}) with $\\psi$ given\nby\n\\begin{equation}\n\\psi(x,y) = - {1 \\over 4\\pi} \\sum_{i}^{N} \\Gamma_i\n\\log \\left[{r^2+r_i^2-2 r r_i \\cos(\\theta-\\theta_i) \\over\n1 + r^2 r_i^2 - 2 r r_i \\cos(\\theta-\\theta_i)} \\right]\n\\label{}\n\\end{equation}\nwhere $(x=r \\cos \\,\\theta,y=r \\sin \\,\\theta)$ denote the tracer\nposition.\n\nFigure \\ref{fig5} shows the relative diffusion as a function of \ntime in a system with 4 vortices. \nApparently there is an intermediate regime of anomalous diffusion.\nOn the other hand from figure \\ref{fig6} one can see rather clearly \nthat, with the method of working at fixed scale, only\ntwo regimes survive: the exponential one and that one \ndue to the saturation. \nComparing figure \\ref{fig5} and figure \\ref{fig6} one understands that\nthe mechanism described in Section 2 \nhas to be held for responsible of this spurious anomalous diffusion.\nWe stress the fact that these misleading behaviors are \ndue to the superposition of different regimes and that \nthe method of working at fixed scale has the advantage \nto eliminate this trouble.\n\nThe absence of the diffusive range \n$\\lambda(\\delta) \\sim \\delta^{-2}$\nis due to the fact that the characteristic \nlength of the velocity field, which is comparable with \nthe typical distance between two close vortices, is not \nmuch smaller than the size of the basin.\n\n\\section{Conclusions}\n\\label{sec:4}\nIn this paper we investigated the relative dispersion of passive tracers\nin closed basins. Instead of the customary approach based on \nthe average size of the cloud of tracers as a function of time,\nwe introduced a typical inverse time $\\lambda(\\delta)$ which \ncharacterizes the diffusive process at fixed scale $\\delta$.\n\nFor very small values of $\\delta$, $\\lambda(\\delta)$ coincides with the\nmaximum Lagrangian Lyapunov exponent which is positive in\nthe case of chaotic Lagrangian motion.\nFor larger $\\delta$ the shape of $\\lambda(\\delta)$ \ndepends on the detailed mechanism of spreading which is given\nby the structure of the advecting flow, which is in turn conditioned \nby the presence of boundaries. In the case of diffusive regime, one\nexpects the scaling $\\lambda(\\delta) \\simeq \\delta^{-2}$, which leads to a\nnatural generalization of the diffusion coefficient as\n$D(\\delta)=\\lambda(\\delta) \\delta^2$. \n\nThe effectiveness of finite size quantities $\\lambda(\\delta)$ \nor $D(\\delta)$ in characterizing the dispersion properties of\na cloud of particles is demonstrated by several numerical examples.\n\nFurthermore, when $\\delta$ gets close to its saturation value\n(i.e. the characteristic size of the basin), a simple argument gives \nthe shape of $\\lambda(\\delta)$ which is expected to be universal\nwith respect to a wide class of dynamical systems.\n\nIn the limiting case when the characteristic length of\nthe Eulerian velocity $l_u$ and the size of the basin $L$ are\nwell separated, the customary approach and the proposed method\ngive the same information. \nIn presence of strongly intermittent Lagrangian motion, or when\n$l_u\/L$ is not much smaller than one, the traditional method\ncan give misleading results, for instance apparent anomalous\nscaling over a rather wide time interval, as demonstrated by\na simple example.\n\nWe want to stress out that our method is very \npowerful in separating the different scales acting on diffusion \nand consequently it could give improvement about the parameterization \nof small-scale motions of complex flows.\nThe proposed method could be also relevant in the analysis of\ndrifter experimental data or in numerical models for Lagrangian\ntransport, in particular for addressing the question about the\nexistence of low dimensional chaotic flows.\n\n\\section{Acknowledgments}\nWe thank E. Aurell and A. Crisanti for useful suggestions and \nfirst reading of the paper. G.B. and A.C. thank the Istituto di\nCosmogeofisica del CNR, Torino, for hospitality.\nThis work was partially supported by INFN {\\it Iniziativa specifica\nMeccanica Statistica FI11}, by CNR (Progetto speciale coordinato\n{\\it Variabilit\\`a e Predicibilit\\`a del Clima}) and by EC-Mast contract\nMAS3-CT95-0043 (CLIVAMP).\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nRecently, the HADES collaboration has measured $\\Lambda(1405)$ production in proton-proton reactions at a beam kinetic energy of 3.5~GeV \\cite{Agakishiev:2012xk}, where the $\\Lambda(1405)$ hyperon has been reconstructed in the charged $\\Sigma^{\\pm}\\pi^{\\mp}$ decay channels. By investigating the $\\Sigma\\pi$ invariant mass distributions, clear peak structures below 1400~MeV\/$c^2$ were observed. These structures were interpreted by a small contribution of $\\Sigma(1385)^0$ and a large contribution of a low mass $\\Lambda(1405)$ signal (see also \\cite{Agakishiev:2012ja}). Besides this, the spectra also showed a considerable contribution by $\\Lambda(1520)$ production and by non$\\--$resonant $\\Sigma\\pi$ production, resulting in a phase space like background below the $\\Lambda(1405)$. The experimental data were finally described by an incoherent sum of Monte Carlo simulations, where the $\\Lambda(1405)$ was simulated to follow a Breit-Wigner type distribution with a Breit-Wigner mass of 1385~MeV\/$c^2$ and a width of 50~MeV\/$c^2$. With help of these simulations the experimental data were corrected for the effects of acceptance and efficiency. These corrected data allow to compare any (more advanced) model to the obtained $\\Sigma\\pi$ invariant mass distributions. \\\\\nThe experimental data, where the maximum of the $\\Sigma\\pi$ missing mass (see Fig.~\\ref{fig:LA1405_HADES_Thomas_Comp}) lies below the nominal value of $1405\\,\\mathrm{MeV\/}c^2$ associated to the $\\Lambda(1405)$, suggest a shift of this resonance towards lower masses.\nIn this paper we want to address different possible explanations for the observed mass shift of the $\\Lambda(1405)$ in the new HADES data and aim to stimulate theorists to further investigate the production of this resonance in $p+p$ reactions.\nThe $\\Lambda(1405)$ spectral function measured in $p+p$ collisions by the HADES collaboration differs from the predictions by theoretical models and also from some of the experimental observations. These differences reside in the complex nature of the resonance.\n\n From the theoretical point of view, the $\\Lambda(1405)$ is normally treated in a unitarized coupled channel framework based on chiral SU(3) dynamics \\cite{Hyodo:2007jq,Borasoy:2006sr,Kaiser:1995eg,Oset:1997it}, where this resonance is generated dynamically as a coherent sum of two different poles. The first pole, $z_1$, is located at higher energies of around 1420~MeV and is mainly associated with a narrow quasi-bound $\\bar{K}N$ state. The second pole, $z_2$, is found at lower energies of about 1390~MeV and this pole strongly couples to a broad $\\Sigma\\pi$ resonance. As the relative contribution of these two states depends on the entrance channel, also the observed properties of the $\\Lambda(1405)$ could differ for different reactions. Therefore, in order to understand the complex formation process of the $\\Lambda(1405)$, it is important that experiments measure this resonance in different collision systems, and that, at the same time, theory provides appropriate models for each of those systems. \n \nFirst we refer to the measurement presented by the ANKE collaboration in \\cite{Zychor:2007gf}, where the $\\Lambda(1405)$ spectral shape is reconstructed in $p+p$ collisions at 2.83~GeV beam kinetic energy out of the decay into $\\Sigma^0$ and $\\pi^0$ pairs.\nThese data are within the systematic errors and the statistical significance of the mass bins around 1400~MeV\/$c^2$ consistent with the HADES results.\\\\\nFrom the theory side, the authors of \\cite{Geng:2007vm} followed a unitarized coupled channel approach based on the chiral Lagrangian in order to \npredict the $\\Lambda(1405)$ line shape in $p+p$ reactions for the $\\Sigma^0\\pi^0$ decay channel. In their Ansatz the $\\Lambda(1405)$ was \ngenerated from pion, kaon and $\\rho$ meson exchange mechanisms, all of them leading to a different coupling to the two $\\Lambda(1405)$ poles. The \ncoherent sum of all contributions results in a $\\Lambda(1405)$ line shape with a maximum in the $\\Sigma\\pi$ mass distribution at around \n$1410\\,\\mathrm{MeV\/}c^2$. With this approach the authors of \\cite{Geng:2007vm} delivered a result compatible with the $\\Lambda(1405)$ signal, measured by the ANKE collaboration in the $\\Sigma^0\\pi^0$ decay channel \\cite{Zychor:2007gf}. \nThis calculation is the only one available for $p+p$ reactions but since the HADES data refer to the charged decay channels\na quantitative comparison results difficult. Nevertheless, the results by \\cite{Geng:2007vm} have been used in this work as a starting point to model the $\\Lambda(1405)$ as the combination of two Breit-Wigner functions.\\\\\nIn general, the HADES data show a larger contribution by the non$\\--$resonant $\\Sigma\\pi$ production in comparison with ANKE. In particular, it has \nbeen shown in \\cite{Agakishiev:2012qx} that the $\\Delta^{++}$ is strongly coupling to the $\\Sigma^- \\pi^+ p K^+$ final state via the reaction $p+p\n\\rightarrow \\Sigma^- + \\Delta^{++} + K^+$ and it cannot be excluded that $N^*\/\\Delta^0$ resonances contribute to the $\\Sigma^+\\pi^-pK^+$ channel\n as well. These contributions might appear in the $I=0$ channels and hence interfere with the resonant amplitude.\n\nAside from the predictions by models that consider the $\\Lambda(1405)$ as the combination of two main poles, one has to consider those models where \na single pole is associated to the formation of the resonance. \nIn \\cite{Hassanvand:2012dn} a phenomenological  ${\\mathrm{\\bar{K}}p}$ interaction was employed to derive the mass distribution of $\n\\Lambda(1405)$.\nThis approach was used to fit the HADES data. For this purpose all simulated contributions, mentioned in\n\\cite{Agakishiev:2012xk}, were subtracted so that only the pure $\\Lambda(1405)$ signal was left. The fit within this phenomenological approach\n results into a good description of the experimental data and allowed to extract the mass and width of the $\\Lambda(1405)$ to\n$1405^{+11} _{-9}\\mathrm{ MeV\/c^2}$ and $62\\pm 10 \\mathrm{MeV\/}c^2$, which is in good agreement with the PDG values \\cite{PDG} and\ndoes not imply any evident shift of the spectral function.\n\nIn the sector of $\\gamma$-induced reactions, the CLAS collaboration has recently published new high quality data on the $\\Lambda(1405)$ \\cite{Moriya:2013eb,Moriya:2013hwg,Schumacher:2013vma}. In these reports, all three $\\Sigma\\pi$ decay channels have been investigated simultaneously for different incident photon energies. The observed $\\Lambda(1405)$ spectral shape partially appears at higher masses, clearly above 1400 MeV\/$c^2$. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{LA1405_HADES_Thomas_Comp}\n\t\\caption{Comparison between the HADES data and the data of \\cite{Thomas:1973uh} and \\cite{Engler:1965zz}, measured in $\\pi^-+p$ reactions. All three spectra show the sum of the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ data samples.}\n\t\\label{fig:LA1405_HADES_Thomas_Comp}\n\\end{figure}\nThe structures measured by CLAS show a dependency on the photon incident energy and also differ among the three $\\Sigma\\pi$ decay channels. \nIndeed, one has to consider that for photon-induced reactions the interference between the $I=0$ and $I=1$  channels is not negligible as for proton- and pion-induced reactions. \nRecent theoretical works \\cite{PhysRevC.87.055201,Roca:2013cca} employ parameters fitted to the CLAS experimental data which \nallow for a small SU(3) breaking. This study shows that more precise calculations including higher order corrections could be needed in this sector \nand also suggests the possible existence of a $I=1$  bound state in the vicinity of the  \n$\\bar{\\mathrm{K}N}$ threshold. Additionally, the CLAS collaboration reported recently on the first observation of the $\\Lambda(1405)$ in electron-\ninduced reactions \\cite{Lu:2013nza} showing very different features respect to the photon-induced results. \n\nIn the sector of pion-induced reactions, Thomas et al. \\cite{Thomas:1973uh} and Engler et al. \\cite{Engler:1965zz} have measured $\\pi^-+p$ \ncollisions at a beam momentum of 1.69~GeV\/$c$ and have reconstructed the $\\Lambda(1405)$ from its decay into $\\Sigma^{\\pm}\\pi^{\\mp}$. The \nresults for the $\\Sigma^+\\pi^-+\\Sigma^-\\pi^+$ invariant mass spectra are shown in the black and open data points in Fig.~\n\\ref{fig:LA1405_HADES_Thomas_Comp}, respectively.\nAccording to \\cite{Thomas:1973uh}, the spectra consist of several contributions, namely 46\\% $\\Lambda(1405)$, 8\\% $\\Sigma(1385)^0$, 3\\% $\\Lambda(1520)$ and 43\\% non$\\--$resonant $\\Sigma\\pi$ production. The broad peak structure around 1400~MeV\/$c^2$ is mainly identified with the $\\Lambda(1405)$ signal. \nThis experimental spectrum, however, is not fully understood from the theoretical side, which expects a large contribution from the first $\\Lambda(1405)$ pole, shifting the expected $\\Lambda(1405)$ distribution to higher mass values \\cite{Hyodo:2003jw}. Therefore, the question arises if, in case of $\\pi^-$-induced reactions, the coupling to the second, broad pole at $\\approx 1390$~MeV is underestimated by theory.\\\\\nIt is interesting to compare the results from $\\pi^-$-induced reactions to the new HADES data. The gray histogram in Fig.~\\ref{fig:LA1405_HADES_Thomas_Comp} shows the summed invariant mass spectrum of $\\Sigma^+\\pi^-+\\Sigma^-\\pi^+$ of \\cite{Agakishiev:2012xk}. As the relative contributions from $\\Lambda(1405)$, $\\Sigma(1385)^0$, $\\Lambda(1520)$ and non$\\--$resonant $\\Sigma\\pi$ channels in these data are quite similar to the ones in the considered $\\pi^-+p$ reactions, a comparison between the data sets is justified. In order to allow such a comparison, the data from \\cite{Thomas:1973uh} and \\cite{Engler:1965zz} have been scaled appropriately. The agreement between the three spectra in the region around 1400~MeV\/$c^2$ is excellent, indicating that all measurements observe a similar low mass $\\Lambda(1405)$ signal. Furthermore, this suggests that in both, $p+p$ and $\\pi^-+p$ reactions, the broad $\\Sigma\\pi$ pole might be dominant in the coupling to the $\\Lambda(1405)$ state. \n\\noindent However, since the measured $\\Sigma\\pi$ final state does not contain the pure signature of the $\\Lambda(1405)$, but also contains a \nconsiderable contribution of non$\\--$resonant background, the interpretation of this result is not straight forward.\n \\section{Influence of interference effects}\n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{OsetFitted}\n\t\\caption{The $\\Lambda(1405)$ spectral shape calculated in \\cite{Geng:2007vm} (gray histogram). The spectrum is fitted with Eq.~(\\ref{equ:DoubleBW}). The two phase space modified Breit-Wigner functions $C_{\\mathrm{p.s.}}(m)\\left|BW_1(m)\\right|^2q_{\\mathrm{c.m.}}$ and $C_{\\mathrm{p.s.}}(m)\\left|BW_2(m)\\right|^2q_{\\mathrm{c.m.}}$ are shown in the dotted and dashed lines.}\n\t\\label{fig:OsetFitted}\n\\end{figure}  \nInterference between the resonant and the non$\\--$resonant amplitudes can affect the observed mass distribution considerably. In order to evaluate this scenario, we take the result of the chiral Ansatz \\cite{Geng:2007vm} as a starting point to develop a simple model, which we use to parametrize the $\\Lambda(1405)$ amplitude and to interpret the HADES data. The predicted $\\Sigma\\pi$ spectrum of \\cite{Geng:2007vm} is shown in the gray band of Fig.~\\ref{fig:OsetFitted}, with a peak at 1410~MeV and with the typical $\\Lambda(1405)$ shape, having a sharp drop to the $\\bar{K}N$ threshold. The idea is to reconstruct  this spectrum by a coherent sum of two Breit-Wigner functions (BW), where each of these BW amplitudes represents one of the two $\\Lambda(1405)$ poles. This approach is similar to what has been proposed in \\cite{Jido:2003cb} to represent via Breit-Wigner distributions the results of the unitarized coupled channel calculations. It is clear that any variation of the parameters associated to the two poles does not fulfill unitarity anymore and hence our procedure is not equivalent to a full-fledged calculation. Still it is interesting to see how much the parameters have to be modified to fit the experimental data.\n Within this approach the  total $\\Lambda(1405)$ amplitude reads then as follows:\n\\begin{eqnarray}\n\\frac{d \\sigma}{d m}&=&\\left|T_{\\Lambda(1405)}\\right|^2= \\nonumber \\\\\n&=&C_{\\mathrm{p.s.}}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{\\mathrm{c.m.}} \\label{equ:DoubleBW} \\\\\n&&\\mbox{with } BW_i=A_i\\frac{1}{\\left(m-m_{0,i}\\right)^2+im_{0,i}\\Gamma_{0,i}} \\nonumber\n\\end{eqnarray}\n$C_{\\mathrm{p.s.}}(m)$ is a dimensionless weight function, normalized to unity in the mass range of 1280-1730~MeV\/$c^2$. This function considers the limited production phase space of the $\\Lambda(1405)$ in $p+p$ reactions. $q_{\\mathrm{c.m.}}$ is the decay momentum of $\\Sigma$ and $\\pi$ in the $\\Lambda(1405)$ rest frame (in units of [MeV\/c]). The Breit-Wigner function is a simple relativistic parametrization with amplitude $A_i$ in units of $\\left[\\sqrt{\\mu b\/\\mbox{c}}\\cdot \\mbox{MeV\/c}^2\\right]$, mass $m_{0,i}$ and width $\\Gamma_{0,i}$ in units of [MeV\/$c^2$]. Thus, the whole expression has dimensions of $\\left[\\frac{\\mu b}{\\mbox{MeV\/c}^2}\\right]$. We also introduce a free phase $e^{i\\varphi_1}$, which determines the interference between the two Breit-Wigner functions. Furthermore, we make use of the recent coupled channel calculations by Ikeda et al. \\cite{Ikeda:2012au}, which are constrained by the new SIDDARTHA data on kaonic hydrogen \\cite{Bazzi:2011zj}. In this work the $\\mathrm{\\bar{K}N}$ pole was found at $z_1=1424^{+7}_{-23}+i26^{+3}_{-14}$~MeV, while the $\\Sigma\\pi$ pole was found at $z_2=1381^{+18}_{-6}+i81^{+19}_{-8}$~MeV. With these values we constrain the Breit-Wigner mass $m_ic^2=Re(z_i)$ and the Breit-Wigner width $\\Gamma_{0,i}c^2=2Im(z_i)$ of our model to vary only within the given ranges.\\\\\nAlthough the parametrization of Eq.~(\\ref{equ:DoubleBW}) is simplified compared to the advanced calculations in \\cite{Geng:2007vm}, it still allows us to reconstruct the spectral shape in Fig.~\\ref{fig:OsetFitted} (gray band).\nBy fitting the Eq.~(\\ref{equ:DoubleBW}) to the theoretical prediction,  we obtain the black distribution with the fit parameters listed in Table~\\ref{tab:tableInter}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:tableInter}%\nTable with fit parameters obtained by fitting Eq.~(\\ref{equ:DoubleBW}) to the theoretical prediction of \\cite{Geng:2007vm} shown in Fig.~\\ref{fig:OsetFitted}.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_1$}&\n\\textrm{$\\Gamma_1$}&\n\\textrm{$m_2$}&\n\\textrm{$\\Gamma_2$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n1426 & 28 & 1375 & 147 & 0.23 & 205$^{\\circ}$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table} \nThis distribution is consistent with the gray band in Fig.~\\ref{fig:OsetFitted} . Especially the peak structure around 1410~MeV\/$c^2$ and the drop to the $\\bar{K}N$ threshold is reproduced correctly. Additionally included in the figure are the absolute contributions of the two Breit-Wigner functions $C_{p.s.}(m)\\left|BW_1(m)\\right|^2q_{c.m.}$ and $C_{p.s.}(m)\\left|BW_2(m)\\right|^2q_{c.m.}$ (dotted and dashed lines).\\\\ \nIn this way we have fixed the parametrization of the $\\Lambda(1405)$ and can now study the maximal interference effects with the non$\\--$resonant background. \nFor this purpose, we fit the HADES results for the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ invariant mass distributions simultaneously with the following two functions:\n\\begin{widetext}\n\\begin{eqnarray}\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^+\\pi^-}=\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}+e^{i\\alpha}A_{\\Sigma^+\\pi^-}T_{\\Sigma^+\\pi^-}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSpPm} \\\\\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^-\\pi^+}=\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}+e^{i\\beta}A_{\\Sigma^-\\pi^+}T_{\\Sigma^-\\pi^+}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSmPp}\n\\end{eqnarray} \n\\end{widetext}\nThe contributions from $\\Sigma(1385)^0$ and $\\Lambda(1520)$ are parameterized as Breit-Wigner functions so that they match the extracted shapes and yields reported in \\cite{Agakishiev:2012xk}. We assume here that they do not interfere with the other contributions to the $\\Sigma\\pi$ invariant mass spectra and thus add them incoherently in the Eq.s~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}). \\\\\nThe $\\Lambda(1405)$ is parameterized as described above. $A_{\\Lambda(1405)}$ is a free fit parameter which determines the absolute yield of $\\Lambda(1405)$.\\\\      \nThe non$\\--$resonant background shapes for the $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ channels ($T_{\\Sigma^+\\pi^-}$ and $T_{\\Sigma^-\\pi^+}$) are described by modified polynomials of 4th order, which read as follows: \n\\begin{eqnarray}\nT_{\\Sigma\\pi}(m)=\\left[C_{p.s.}(m)q_{c.m.}\\sum^{4}_{n=0}a_nm^n\\right]^{\\frac{1}{2}} \\label{equ:NonRes}\n\\end{eqnarray}\nThis parametrization has no physical meaning but it was chosen such to describes the simulated $\\Sigma\\pi$ invariant mass distributions in \\cite{Agakishiev:2012xk}. The parameters $a_n$ are given in units of $\\left[\\left(\\mbox{MeV\/c}^2\\right)^{-n}\\right]$ and their values are listed in Table~\\ref{tab:table1}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:table1}%\nTable with coefficients for the description of the non$\\--$resonant background according to Eq.~(\\ref{equ:NonRes}).}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n\\textrm{$T_{\\Sigma\\pi}$}&\n\\textrm{$a_0$}&\n\\textrm{$a_1$}&\n\\textrm{$a_2$}\\\\\n\\colrule\n$T_{\\Sigma^+\\pi^-}$ & $7.949\\cdot10^{-3}$ & $-4.412\\cdot10^{-7}$ & $-1.558\\cdot10^{-9}$    \\\\\n$T_{\\Sigma^-\\pi^+}$ & $-2.387\\cdot10^{0}$ & $2.512\\cdot10^{-3}$ & $1.315\\cdot10^{-6}$   \\\\\n\\colrule\\\\\n\\textrm{$T_{\\Sigma\\pi}$}&\n\\textrm{$a_3$}&\n\\textrm{$a_4$}\\\\\n\\colrule\n$T_{\\Sigma^+\\pi^-}$ & $2.942\\cdot10^{-12}$ & $-1.121\\cdot10^{-15}$ \\\\\n$T_{\\Sigma^-\\pi^+}$ & $-2.254\\cdot10^{-9}$ & $6.481\\cdot10^{-13}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nThe modified polynomials are multiplied by constant factors $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ (see~Eq. (\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp})), which have dimensions of $\\left[\\frac{\\sqrt{\\mu b\/\\mbox{c}}}{\\mbox{MeV\/c}^2}\\right]$. These are again free fit parameters, determining the absolute yields of the non$\\--$resonant channels, where a value of $A_{\\Sigma\\pi}=1$ corresponds to the yield extracted in \\cite{Agakishiev:2012xk}. Furthermore, complex phases $e^{i\\alpha}$ and $e^{i\\beta}$ have been included so that the modeled background can interfere with the $\\Lambda(1405)$ amplitude. The values of these phases are determined by the fitting procedure as well. Hence, the simultaneous fit of the two functions (\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}) to the experimentally determined $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ invariant mass distributions is characterized by five free parameters.\\\\\nWe consider here a scenario of maximal interference, which means that the whole non$\\--$resonant background interferes with the $\\Lambda(1405)$ amplitude.\n The best results of the fitting procedure (gray lines) are illustrated together with the experimental data in Fig.~\\ref{fig:Interferences} panel a) and b). The black lines show the amplitude of the $\\Lambda(1405)$, the red lines the contribution by the non$\\--$resonant $\\Sigma\\pi$ channels and the gray lines correspond to the coherent sum of all contributions. The fit parameters are listed in Table~\\ref{tab:table2}.  \n\\begin{table}[tbp]\n\\caption{\\label{tab:table2}%\nObtained fit parameters after fitting Eq.s~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}) to the experimental data points in Fig.~\\ref{fig:Interferences}.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n\\textrm{$A_{\\Lambda(1405)}$}&\n\\textrm{$A_{\\Sigma^+\\pi^-}$}&\n\\textrm{$A_{\\Sigma^-\\pi^+}$}&\n\\textrm{$\\alpha$}&\n\\textrm{$\\beta$}\\\\\n\\colrule\n$1.06$ & $0.93$ & $1.04$ & $67^{\\circ}$ & $109^{\\circ}$  \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Interferences_1}\n\t\\caption{ Missing mass spectrum to proton and $K^+$. The black data points are the measurements of \\cite{Agakishiev:2012xk}, for the $\\Lambda(1405)$ in the $\\Sigma^+\\pi^-$ (a) and $\\Sigma^-\\pi^+$ (b) decay channel. Panel c) shows the summed spectrum of a) and b). The black lines in a) and b) are the results from the simultaneous fit with Eq.~(\\ref{equ:DistSpPm}) and (\\ref{equ:DistSmPp}), the gray line represents the sum of all fitted functions. In c) the fit functions corresponding to the  $\\Sigma^+\\pi^-$ and $\\Sigma^-\\pi^+$ channels respectively are shown in gray and the sum of both functions is shown in the black line.}\n\t\\label{fig:Interferences}\n\\end{figure}\nA reasonable description of the data is achieved, expressed in a normalized $\\chi^2$ value of 1.23. In panel c) of Fig.~\\ref{fig:Interferences} we show the sum of the two invariant mass distributions and compare it to the sum of our model (black line). \\\\\nThe main message of the three pictures is that, assuming a maximal interference between the $\\Lambda(1405)$ and the non$\\--$resonant contributions and the large phases given in Table~\\ref{tab:table2}, one obtains a shift of the maximum to lower masses. This is the case although the mass maximum of the initial $\\Lambda(1405)$ amplitude is located around 1400~MeV\/$c^2$. By integrating $\\left|A_{\\Lambda(1405)}T_{\\Lambda(1405)}\\right|^2$ (black lines in panel a) and b)), the total cross section of the reaction $p+p\\rightarrow\\Lambda(1405)+p+K^+$ is determined to 3.3 $\\mu b$. This is considerably smaller than the value extracted in \\cite{Agakishiev:2012xk}, where an incoherent approach was used with a low mass $\\Lambda(1405)$ signal.\\\\    \nAt this point one has to emphasize some details of the model used in this work. First, the parametrization of the $\\Lambda(1405)$ as a simple sum of two Breit-Wigner amplitudes is not equivalent to the full coupled channel calculation of \\cite{Geng:2007vm}. A second point is the description of the non$\\--$resonant background $T_{\\Sigma\\pi}$ as polynomial functions. This is certainly a simplification. Also the assumption that the non$\\--$resonant channels just have a constant complex phase is oversimplified.\nFurthermore, we assume that the whole $\\Sigma\\pi$ non$\\--$resonant background appears in the s-wave channel with I=0, testing here the maximal possible interference.\nHowever, this maximal interference scenario turns out to be unlikely since no comparable mass shifts have been observed in the spectral shape of the $\\Sigma(1385)^+$ in the $\\mathrm{\\Lambda-p}$ final state \\cite{Agakishiev:2011qw} or for the $\\Lambda(1520)$ in the $\\Sigma\\pi$ decay channel. Moreover, it seems to be rather peculiar that interferences between the $\\Lambda(1405)$ and the non$\\--$resonant background should result in the same mass shift for both, the $\\Sigma^+\\pi^-$ and the $\\Sigma^-\\pi^+$ invariant mass distributions, where in both cases the physical origin of the non$\\--$resonant background is quite different. According to \\cite{Agakishiev:2012xk}, the $\\Sigma^-\\pi^+$ non$\\--$resonant background arises from a strong contribution of a $\\Delta^{++}$, whereas other mechanisms, e.g. $N^*\/\\Delta^0$ production via the reaction $p+p\\rightarrow\\Sigma^+ + N^*\/\\Delta^0+K^+\\rightarrow\\Sigma^++(\\pi^-+p)+K^+$, could contribute to the $\\Sigma^+\\pi^-$ non$\\--$resonant spectrum.\\\\\nOn the other hand, the presented results shall just emphasize that interference effects can play a significant role. Indeed, our model, even though it is not a full-fledged theoretical approach, has shown that interference effects can significantly shift the observed mass peak in the experimental spectra. We cannot prove that these effects are indeed responsible for the observed low mass $\\Lambda(1405)$ signal. We just aim to illustrate the importance of a serious treatment of the non$\\--$resonant background in any theoretical approach.\n\\section{Contributions of the two poles}\nHaving illustrated the maximal possible influence of interference effects between the $\\Lambda(1405)$ resonance and the non$\\--$resonant background, we now consider the second extreme case, where this particular interference term is neglected. The observed low mass peaks in the HADES data are then completely attributed to the \"pure'' $\\Lambda(1405)$ signal. With this assumption one can try to determine the parameters and the relative contribution of the two $\\Lambda(1405)$ poles. \nAs a starting point, we again parameterize the $\\Lambda(1405)$ as a coherent sum of the two Breit-Wigner amplitudes (see Eq.~(\\ref{equ:DoubleBW})). This time, however, $m_{0,1}$, $\\Gamma_{0,1}$, $m_{0,2}$ and $\\Gamma_{0,2}$ as well as $A_1$ and $A_2$ shall be determined directly from the HADES data. As before, the real and imaginary part of both poles are constrained by the results of \\cite{Ikeda:2012au} to $z_1=1424^{+7}_{-23}+i26^{+3}_{-14}$~MeV and $z_2=1381^{+18}_{-6}+i81^{+19}_{-8}$~MeV. Now the two functions~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) are used to described the experimental data points of Fig.~\\ref{fig:Interferences} a) and b). \n\\begin{widetext}\n\\begin{eqnarray}\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^+\\pi^-}=C_{p.s.}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{c.m.}+\\left|A_{\\Sigma^+\\pi^-}T_{\\Sigma^+\\pi^-}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSpPm2} \\\\\n\\left(\\frac{d\\sigma}{dm}\\right)_{\\Sigma^-\\pi^+}=C_{p.s.}(m)\\left|BW_1(m)e^{i\\varphi_1}+BW_2(m) \\right|^2q_{c.m.}+\\left|A_{\\Sigma^-\\pi^+}T_{\\Sigma^-\\pi^+}\\right|^2+\\left|BW_{\\Sigma(1385)^0}\\right|^2 + \\left|BW_{\\Lambda(1520)}\\right|^2 \\label{equ:DistSmPp2}\n\\end{eqnarray} \n\\end{widetext}\nIn these Eq.s all individual contributions besides the two $\\Lambda(1405)$ amplitudes sum up incoherently. The non$\\--$resonant background is fixed in yield to the HADES results \\cite{Agakishiev:2012xk}, by setting $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ to 1. In this way only $m_{0,1}$, $\\Gamma_{0,1}$, $m_{0,2}$, $\\Gamma_{0,2}$, $A_1$, $A_2$ and $\\varphi_1$ are the free fit parameters in Eq.~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}). The results for the best fit ($\\chi^2\/ndf=1.04$) are shown in Fig.~\\ref{fig:Fit_Weise} and the obtained fit parameters are listed in Table~\\ref{tab:table3}.     \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Fit_Weise_1}\n\t\\caption{(Color online) Same as Fig.~\\ref{fig:Interferences} but now the data are fitted with the Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}). See text for details.}\n\t\\label{fig:Fit_Weise}\n\\end{figure}\n\\begin{table}[tbp]\n\\caption{\\label{tab:table3}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}).}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n$1418$ & $58$ & $1375$ & $146$ & $0.395$ & $178^{\\circ}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\nA very good description of the data is achieved with the maximum in the $\\Lambda(1405)$ distribution now appearing at around $1385$~MeV\/$c^2$. Integrating this signal results in a total cross section of $\\sigma=9.0$~$\\mu b$ for the reaction $p+p\\rightarrow\\Lambda(1405)+p+K^+$, which is in good agreement with the result quoted in \\cite{Agakishiev:2012xk}. The composition of the $\\Lambda(1405)$ signal itself is illustrated in Fig.~\\ref{fig:L1405_Weise}. The resulting $\\Lambda(1405)$ amplitude differs strongly from the results reported in \\cite{Geng:2007vm} and shown in Fig. \\ref{fig:OsetFitted}. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{L1405_Weise}\n\t\\caption{$\\Lambda(1405)$ signal (black line) obtained by fitting Eq.~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) to the experimental data in Fig.~\\ref{fig:Fit_Weise}. The dotted and dashed lines show the contributions of $C_{p.s.}(m)\\left|BW_1(m)\\right|^2q_{c.m.}$ and $C_{p.s.}(m)\\left|BW_2(m)\\right|^2q_{c.m.}$.}\n\t\\label{fig:L1405_Weise}\n\\end{figure}\nThe $z_2$ pole (dashed line) has the dominant contribution with mass and width values at the edge of the allowed fit range ($m_{0,2}=1375$~MeV, $\\Gamma_{0,2}=146$~MeV). This very broad signal interferes with the $z_1$ pole (dotted line) in a way that the high mass region of the $\\Lambda(1405)$ is strongly suppressed, creating a peak below 1400~MeV\/$c^2$ (solid black line). However, with the large amount of free parameters, with the broad ranges in which the mass and width values are constrained and with the limited number of data points, it is difficult to derive strict conclusions about the relative contributions and the positions of the two poles. It can just be claimed that a significant part of the $\\Lambda(1405)$ amplitude could be associated to the second, broad pole. One should also notice here that the $\\Lambda(1405)$ signal shows a tail for masses above 1440~MeV\/$c^2$. This tail was not considered by the authors of \\cite{Agakishiev:2012xk}, who assumed the $\\Lambda(1405)$ to be only located at lower energies and who therefore used the high mass range of the $\\Sigma\\pi$ spectra to determine the contribution of the $\\Sigma\\pi$ non$\\--$resonant channels. From the results in Fig.~\\ref{fig:Fit_Weise} and \\ref{fig:L1405_Weise} one sees, however, that this assumption might be too simple. Thus, a further possibility is to not fix the non$\\--$resonant background but to treat $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ as two additional fit parameters.\nApplying this new fit to the data, results in the fit values listed in Table~\\ref{tab:table4}.\n\\begin{table}[tbp]\n\\caption{\\label{tab:table4}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) with $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ as additional free parameters.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$} &\n\\textrm{$A_{\\Sigma^+\\pi^-}$} & \n\\textrm{$A_{\\Sigma^-\\pi^+}$} \\\\\n\\colrule\n$1431$ & $58$ & $1375$ & $146$ & $0.204$ & $164^{\\circ}$ & $0.857$ & $0.887$\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table} \nThe fit quality of $\\chi^2\/ndf=1.03$ is as good as before, but the contribution of the $z_1$ pole is further reduced as seen by comparing the $A_1\/A_2$ ratios in Table~\\ref{tab:table3} and \\ref{tab:table4}. However, as the number of free parameters has further increased, the fit to the data is not robust anymore so that the results of Table~\\ref{tab:table4} are not very reliable.\\\\ \nIn summary, it can be concluded that it is rather difficult to precisely determine the relative contribution between $z_1$ and $z_2$ and simultaneously to determine their exact positions in the complex energy plane just by a fit to the HADES data alone. In fact, an appropriate theory model for proton-proton reactions with a serious treatment of all possible background contributions is required to make further conclusions. Nevertheless, the results obtained in this work clearly show that, in order to describe the new HADES data, a rather large contribution of the broad, low mass $\\Lambda(1405)$ pole is needed, provided that no interference with the non$\\--$resonant background is present.\\\\\nIn this context, it would also be important to have more restrictive constraints on the real and imaginary part of $z_1$ and $z_2$. The values quoted above allow a rather large variation of these parameters. However, the situation is even more unclear. In a recent work by Mai and Meissner \\cite{Mai:2012dt}, who also used the latest SIDDARTHA results, the positions of the two poles were derived to $z_1=1428^{+2}_{-1}+i8^{+2}_{-2}$~MeV and $z_2=1497^{+11}_{-7}+i75^{+9}_{-9}$~MeV. \nThe imaginary part of the first pole is much smaller than the one of \\cite{Ikeda:2012au}, but even more spectacular is the totally different value in the real part of $z_2$, which is shifted by about 100~MeV to higher energies. We can also take these values to fit the HADES data. For this purpose, the non$\\--$resonant background amplitudes $A_{\\Sigma^+\\pi^-}$ and $A_{\\Sigma^-\\pi^+}$ are again fixed to 1 and the mass and width values are allowed to vary within the given ranges. The best fit result is shown in Fig.~\\ref{fig:Fit_Meissner} and the obtained fit parameters are listed in Table~\\ref{tab:tableMeissner}. The corresponding decomposition of the $\\Lambda(1405)$ spectrum is shown in Fig.~\\ref{fig:L1405_Meissner1}. \n\\begin{figure}[tbp]\n\t\\centering\n\t\t\\includegraphics[width=0.46\\textwidth]{Fit_Meissner_1}\n\t\\caption{ Same as Fig.~\\ref{fig:Fit_Weise} but the mass and width constraints for the two $\\Lambda(1405)$ poles are taken from \\cite{Mai:2012dt}. See text for details.}\n\t\\label{fig:Fit_Meissner}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=0.45\\textwidth]{L1405_Meissner}\n\t\\caption{Same as Fig.~\\ref{fig:L1405_Weise} but the constraints of \\cite{Mai:2012dt} for the mass and width values of the two $\\Lambda(1405)$ poles were used in the fits. See text for details.}\n\t\\label{fig:L1405_Meissner1}\n\\end{figure}\n\n\\begin{table}[tbp]\n\\caption{\\label{tab:tableMeissner}%\nObtained free fit parameters after fit with Eq.s~(\\ref{equ:DistSpPm2}) and (\\ref{equ:DistSmPp2}) and by using the constraints of \\cite{Mai:2012dt} for the mass and width values of the two $\\Lambda(1405)$ poles.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n\\textrm{$m_{0,1}$}&\n\\textrm{$\\Gamma_{0,1}$}&\n\\textrm{$m_{0,2}$}&\n\\textrm{$\\Gamma_{0,2}$}&\n\\textrm{$A_1\/A_2$}&\n\\textrm{$\\varphi_1$}\\\\\n\\colrule\n$1427$ & $20$ & $1490$ & $168$ & $0.17$ & $264^{\\circ}$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}      \nThe fit result is very poor, which is also expressed in a normalized $\\chi^2$-value of $\\chi^2\/ndf=3.4$. With the $z_2$ pole having such a large real part, it becomes impossible to create a peak structure at around 1385~MeV\/$c^2$ like it was observed by the HADES collaboration. Within our simplified model, we can thus conclude that the pole positions extracted in \\cite{Mai:2012dt} are not compatible with the new HADES data.\n\n\\section{Summary} \n\nWe have presented different interpretations of the low mass $\\Lambda(1405)$ signal, measured by the HADES collaboration in $p+p$ reactions. It was shown that the obtained signal is very similar to the results obtained in $\\pi^-+p$ reactions. One possible explanation for the observed mass shift is based on interference effects. For that purpose we have developed a simple model, where we assumed the $\\Lambda(1405)$ to consist of two poles, parameterized as Breit-Wigner amplitudes. The contribution and position of these poles were first chosen such to match the  $\\Lambda(1405)$ line shape of \\cite{Geng:2007vm} with the mass peak appearing at $\\approx 1410$~MeV\/$c^2$. With this model we could show that maximum interference between the $\\Lambda(1405)$ amplitude and the non$\\--$resonant $\\Sigma\\pi$ background can indeed create a low mass signal and thus describe the HADES data. However, it was argued that this explanation is probably very unrealistic. In a second approach we have neglected interference effects with the non$\\--$resonant background and have used the HADES data themselves to determine the position and the relative contribution of the two $\\Lambda(1405)$ poles. With a relatively large contribution of the second, broad $\\Lambda(1405)$ pole, $z_2$, we could achieve a very good description of the HADES data. However, the limited statistic in the data did not allow to draw firm conclusions on the precise contribution of the two poles. Also their position in the complex energy plane could not be determined precisely just by a fit to this single data set. Nevertheless, it was shown that, within our simple model, a high energy pole position of $z_2=1497^{+11}_{-7}+i75^{+9}_{-9}$~MeV, like it was obtained in \\cite{Mai:2012dt}, is not compatible with the new HADES data. \\\\            \n\n\nThe authors would like to thank Wolfram Weise, Wolfgang Koenig and Piotr Salabura for very useful discussions. This work is supported by the Munich funding agency, BMBF 06DR9059D,05P12WOGHH, TU M\\\"unchen, Garching\n(Germany) MLL M\\\"unchen: DFG EClust 153, VH-NG-330 BMBF 06MT9156 TP6 GSI.     \n     \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nLet $S$ be a set of rectangles in the plane, with vertical and horizontal sides, whose interiors do not intersect.  We say that two rectangles $A$ and $B$ in $S$ \\textit{\\textbf{see each other}} if there is a vertical or horizontal line segment intersecting the interiors of both $A$ and $B$ and intersecting no other (closed) rectangles in $S$, like the dotted lines in Figure~\\ref{fig:RVG-example}. We refer to such segments as \\textit{\\textbf{lines of sight}}, and under this definition we may consider them to have small positive width. For example, there is no line of sight between rectangles $B$ and $F$ in Figure~\\ref{fig:RVG-example}, since a line of sight needs positive width.  \n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.6\\textwidth]{RVG-example.pdf}\n\\caption{A rectangle visibility graph and a corresponding RV-representation with integer rectangles.}\n\\label{fig:RVG-example} \n\\end{figure} \n\nWe construct a graph $G$ with a vertex for every rectangle in $S$, and an edge between two vertices if and only if their corresponding rectangles see each other.  We say that $S$ is a \\textbf{\\textit{rectangle visibility}} (\\textit{\\textbf{RV-}}) \\textit{\\textbf{representation}} of $G$, and $G$ is a \\textit{\\textbf{rectangle visibility graph}} or \\textit{\\textbf{RVG}}.  We allow rectangles in $S$ to share edges. \n\nA similar notion of rectangle visibility graph was first introduced in 1976 by Garey, Johnson and So \\cite{garey76} as a tool to study the design of printed circuit boards.  Their RVGs have only $1\\times 1$ squares, located at a set of lattice points in a grid.  Hutchinson continued work on this problem in 1993 \\cite{hutchinson93}.  In \\cite{bose1996rectangle}, Bose, Dean, Hutchinson and Shermer described the problem of \\textit{two-layer routing} in ``very large-scale integration'' (VLSI) design as follows: \n\\begin{quote} {\\small\nIn two-layer routing, one embeds processing components and their connections (sometimes called \\textit{wires}) in two layers of silicon (or other VLSI material).  The components are embedded in both layers.  The wires are also embedded in both layers, but one layer holds only horizontal connections, and the other holds only vertical ones.  If a connection must be made between two components that are not cohorizontal or covertical, then new components (called \\textit{vias}) are added to connect horizontal and vertical wires together, resulting in bent wires that alternate between the layers.  However, vias are large compared to wires and their use should be minimized.  In this setting, asking if a graph is a rectangle-visibility graph is the same as asking if a set of components can be embedded so that there is a two-layer routing of their connections that uses no vias.  Our requirement that visibility bands have positive width is motivated by the physical constraint that wires must have some minimum width.  A similar problem arises in printed-circuit board design, as printed-circuit boards naturally have two sides, and connecting wires from one side to the other (the equivalent of making vias) is relatively expensive.}\n\\end{quote}\nIn 1995, Dean and Hutchinson began the study of RVGs in their own right.  They focused on bipartite RVGs, and showed that $K_{p,q}$ is an RVG if and only if $p \\leq 4$, and that every bipartite RVG with $n$ vertices has at most $4n-12$ edges \\cite{Dean95}.  In 1996 Bose, Dean, Hutchinson, and Shermer sought to characterize families of graphs that are RVGs.  They proved that every graph with maximum degree four is an RVG, and every graph that can be decomposed into two caterpillar forests is an RVG, among other results \\cite{bose1996rectangle}.  In 1999 Hutchinson, Shermer, and Vince \\cite{hutchinson1999representations} proved that every RVG with $n$ vertices has at most $6n-20$ edges, and this bound is tight for $n\\geq 8$.  RVGs have since been studied by many other authors \\cite{angelini18,biedl16,  Cahit98, GD2014, Dean97,Dean98,dean08,Dean2010,Streinu03}, including generalizations to 3-dimensional boxes \\cite{bose99,develin03,Fekete99,gethner11}, rectilinear polygons with more than four edges \\cite{digiacomo18, liotta21}, and other variations.\n\nIn 1997, Kant, Liotta, Tamassia, and Tollis considered the minimum area, height, and width required to represent a tree as an RVG, as measured by the smallest bounding box containing all of the rectangles in the RV-representation \\cite{kant97}.  They obtained asymptotic bounds on the area, width, and height of these representations and found a linear-time algorithm to construct them.  In this paper we consider a similar problem, but seek exact bounds on the area, width, height, and perimeter of an RV-representation of any graph with $n$ vertices.  We say that $\\area(G)$, $\\perimeter(G)$, $\\height(G)$, and $\\width(G)$ are the minimum area, perimeter, height, and width, respectively, of the bounding box of any integer rectangle visibility representation of the graph $G$. These are the objects of study in this paper.\n\nIn Section~\\ref{Definitions} we specify the rectangle visibility graphs we consider and provide definitions and notation needed for the paper.  We finish the section with lemmas we will use in later sections.\n\nIn Section~\\ref{SeparatingExamples} we show that these four measures of size of a rectangle visibility graph are all distinct, in the sense that there exist two graphs $G_1$ and $G_2$ with $\\area(G_1)<\\area(G_2)$ but $\\perimeter(G_2)<\\perimeter(G_1)$, and analogously for all other combinations of these parameters. \n\nIn Section~\\ref{SeparatingRepresentations} we show that these measures are not necessarily all attained by the same representation; i.e., there is a graph $G_3$ with two RVG representations $S_1$ and $S_2$ with $\\area(G_3)=\\area(S_1)<\\area(S_2)$ but $\\perimeter(G_3)=\\perimeter(S_2)<\\perimeter(S_1)$, and analogously for all other combinations of these parameters.\n\nIn Section \\ref{SmallParameters} we characterize the graphs that have the smallest height, width, area, and perimeter among all graphs with $n$ vertices.\n\nIn Section~\\ref{LargeParameters} we investigate the graphs with largest height, width, area, and perimeter.  We show that, among graphs with $n \\leq 6$ vertices, the empty graph $E_n$ has largest area, and for graphs with 7 or 8 vertices, the complete graphs $K_7$ and $K_8$ have larger area than $E_7$ and $E_8$, respectively. Using this, we show that for all $n \\geq 7$, the empty graph $E_n$ does not have largest area among all RVGs on $n$ vertices.  The graphs with more than 6 vertices that maximize these parameters are still unknown.\n\nIn Section~\\ref{Conclusions}, we conclude with a number of open questions.\n\n\\section{Basic Definitions and Results} \\label{Definitions}\n\nA rectangle with horizontal and vertical sides whose corners are integer lattice points is said to be an \\textit{\\textbf{integer rectangle}}.  We consider only integer rectangles for the remainder of the paper. Each rectangle is specified by the two $x$-coordinates and two $y$-coordinates of its corners.  For a set $S$ of rectangles, the smallest rectangle with horizontal and vertical sides containing $S$ is the \\textit{\\textbf{bounding box}} of $S$.\n\nSuppose $G$ is an RVG with RV-representation $S$ contained in the bounding box $R$, and say $R$ has corners with $x$-coordinates $0$ and $u$ and $y$-coordinates $0$ and $v$, with $u,v\\in \\mathbb{Z}$.  We can view $R$ as a $u \\times v$ grid, with $v$ rows and $u$ columns, and with rectangles in $S$ each contained in a consecutive set of rows and columns.  For example, in the representation shown in Figure~\\ref{fig:RVG-example}, rectangle $A$ is contained in rows 3, 4, 5, and 6, and column 1.\n\nWe use the convention that lower case letters are vertices of the graph $G$, and the corresponding upper case letters are rectangles in the RV-representation $S$ of $G$; e.g., $a$ is a vertex in $G$, and $A$ is its corresponding rectangle in $S$.  For a given rectangle $A$ in $S$, we denote the $x$-coordinates of its vertical sides by $x_1^A$ and $x_2^A$ with $x_1^A < x_2^A$, and the $y$-coordinates of its horizontal sides by $y_1^A$ and $y_2^A$, with $y_1^A < y_2^A$. In other words, as a Cartesian product of intervals, we have $$A=[x_1^A,x_2^A] \\times [y_1^A,y_2^A].$$\n\nWe also introduce notation to refer to the set of rectangles in $S$ that are above (respectively, below, to the left of, or to the right of) a given rectangle $A$. Specifically, let the set of rectangles above (north of) $A$ be denoted by\n$$ {\\mathcal{N}}(A) = \\{ X \\in S :  y_1^X \\geq y_2^A \\mbox{ and } (x_1^X,x_2^X) \\cap (x_1^A,x_2^A) \\not= \\emptyset \\}.$$  Similarly define ${\\mathcal{S}}(A)$, ${\\mathcal{W}}(A)$, and ${\\mathcal{E}}(A)$ (rectangles south, west, and east of $A$, respectively). For example, in Figure \\ref{fig:RVG-example}, $\\mathcal{N}(D)=\\{B,E\\},$ while $\\mathcal{E}(A)=\\{B, E, D\\}$ and $\\mathcal{S}(A)=\\varnothing.$  Note that $A$ might not see every rectangle in ${\\mathcal{N}}(A)$ if there are other rectangles obstructing the view (and similarly for rectangles in the other three sets). \n\nLet $R$ be the smallest bounding box having horizontal and vertical sides and containing all the rectangles in a set of integer rectangles $S$.  For the remainder of the paper, we turn $R$ so that $\\height(R) \\leq \\width(R)$. \nGiven a graph $G$, the \\textit{\\textbf{area}}, \\textit{\\textbf{perimeter}}, \\textit{\\textbf{height}}, and \\textit{\\textbf{width}} of $G$ are the minimums of the corresponding parameters taken over all bounding boxes of RV-representations of $G$ with height less than or equal to width.\n\nWe conclude this section with some preliminary results. \nFirst we explore the extent to which we can focus on the parameters of connected graphs, and in what ways the values for disconnected graphs are determined or bounded by the parameters of their connected components.\n\nFor convenience in stating the next result, we introduce the following notation.\nFor any positive integers $h$ and $w$, let ${\\mathcal{F}}_{h,w}$ denote the (finite) set of graphs that have RV-representations in an $h \\times w$ bounding box.\n\n\n\n\n\n\n\\begin{lemma}\\label{NewDisjointLem}\nIf $G$ is the disjoint union of graphs $H$ and $J$, then:\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n\\item $\\height(G)=\\height(H)+\\height(J),$\n\\item $\\perimeter(G)=\\perimeter(H)+\\perimeter(J),$ \n\\item $\\width(G)=\\min \\{  \\max \\{x+b, y+a\\} \\, | \\, H \\in {\\mathcal{F}}_{x,y}, J \\in {\\mathcal{F}}_{a,b} \\}$,\n\\item $\\area(G)=\\min \\{ (x+a)(y+b) \\, | \\, H \\in {\\mathcal{F}}_{x,y}, J \\in {\\mathcal{F}}_{a,b} \\}$.\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proof}\nSuppose $G$ is the disjoint union of graphs $H$ and $J$.  Given any RV-representations $S_1$ and $S_2$ of $H$ and $J$, we construct two RV-representations of $G$. As indicated in Figure~\\ref{Glue}, we identify the upper right corner of $S_1$ with the lower left corner of either $S_2$ or  $S_2^T$, where $S_2^T$ denotes the RV-representation of $J$ formed by transposing $S_2$ across its main (top left to lower right) diagonal.  If $S_1$ is $x \\times y$ and $S_2$ is $a \\times b$, then it follows that\n$ G \\in {\\mathcal{F}}_{x+a,y+b} \\cap {\\mathcal{F}}_{x+b,y+a}.$\nThis implies each of the expressions in (i)-(iv) are upper bounds for the parameters of $G$.\n\nTo prove these expressions are also lower bounds, note that \n any RV-representation $S$ of $G$ must have the rectangles corresponding to vertices of $H$ in separate rows and columns from the rectangles corresponding to vertices of $J$.  If $S$ has height smaller than $\\height(H)+\\height(J)$, then either the rectangles in $S$ corresponding to vertices of $H$ must form an RV-representation of $H$ with height less than $\\height(H)$ or the rectangles in $S$ corresponding to vertices of $J$ must form an RV-representation of $J$ with height less than $\\height(J)$.  Therefore no such representation is possible.  Similar arguments apply to the other parameters.\n\\end{proof}\n\n\\begin{remark}\nThe following example illustrates a subtlety captured by the formula in Lemma~\\ref{NewDisjointLem}(iv).  Consider the graphs $P_4+K_{1,4}$ and $K_{1,4}+K_{1,4}$.  We have $\\area(P_4+K_{1,4})=27$, obtained from a $1 \\times 4$ RV-representation of $P_4$ and a $2 \\times 5$ RV-representation of $K_{1,4}$.  But we also have $\\area(K_{1,4}+K_{1,4})=36$, obtained from two copies of a $3 \\times 3$ RV-representation of $K_{1,4}$ (both representations of $K_{1,4}$ are given in Figure~\\ref{fig:width-examples2a}).  So the minimum area of $H + K_{1,4}$ uses different representations of $K_{1,4}$ depending on $H$.\n\\end{remark}\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.7\\textwidth]{disjoint-union.pdf}\n\\caption{Two options for a combined representation of a disjoint union of two RVGs}\n\\label{Glue} \n\\end{figure} \n\n\\begin{corollary}\\label{NewDisjointCor}\nIf $G$ is the disjoint union of graphs $H$ and $J$, then:\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n\\item $\\width(G) \\leq \\width(H) + \\width(J)$,\n\\item $\\area(G) \\leq (\\width(H) + \\width(J))^2$.\n\\end{enumerate} \n\\end{corollary}\n\n\\begin{proof}\nThese both follow immediately from the \nconstruction in the proof of Lemma~\\ref{NewDisjointLem} shown in Figure~\\ref{Glue}. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\begin{lemma}\nSuppose $G$ is a graph with RV-representation $S$ with bounding box $R$.  If $\\height(G)=\\height(R)$ and $\\width(G)=\\width(R)$, then $\\area(G)=\\area(R)$ and $\\perimeter(G)=\\perimeter(R)$.  In other words, if $S$ realizes the height and width of $G$, then $S$ also realizes the area and perimeter of $G$.  \n\\label{lemma-same-height-width}\n\\end{lemma}\n\n\\begin{proof}\nAny representation with smaller area or perimeter must have smaller height or smaller width, which is impossible by hypothesis.\n\\end{proof}\n\nLater (in Table~\\ref{table-separating2}), we will see that the hypotheses of Lemma~\\ref{lemma-same-height-width} are necessary.  In particular, $G_4$ has two different representations for minimizing area and perimeter.\n\n\n\\section{Height, Width, Area, and Perimeter induce distinct orderings of RVGs} \\label{SeparatingExamples}\n\nIn this section, we consider the various notions of height, width, perimeter, and area of RVGs. We show that these parameters represent independent measures of RVGs, in the sense that they do not always give identical orderings of the sets of graphs on a given number of vertices. \nExamples to illustrate these results are summarized in Tables~\\ref{table-separating1}  and \\ref{table-separating-new}. Graphs $G_1$, $G_2$, $G_3$, and $G_4$ in these tables are shown in Figures~\\ref{fig:width-examplesBB} and~\\ref{fig:width-examplesCC}.  For each pair of parameters, there exists a pair of graphs with an equal number of vertices that are oppositely ordered by those parameters. We have verified the height, width, area, and perimeter of all connected graphs with 6 or fewer vertices by computer search   \\cite{WebList} and the claims regarding $P_6$, $C_6$, $G_1$, and $G_2$ in Tables~\\ref{table-separating1}  and \\ref{table-separating-new} follow easily, see Figures~\\ref{fig:width-examplesAA} and \\ref{fig:width-examplesBB}.\nThe claims regarding $G_3$ and $G_4$, each with 15 vertices, are proved in Theorems~\\ref{width-theorem} and \\ref{width-theorem2}.  \n\n \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|c|cccc} \n Graph & Vertices & Height & Width & Area & Perimeter \\\\ \\hline \\hline\n  $P_6$  & 6 & {\\bf 1} & 4 & {\\bf 6} & 14 \\\\ \n  $C_6$  & 6 & 2 & {\\bf 3} & 8 & {\\bf 12} \\\\ \\hline\n  $G_1$ & 6 & {\\bf 2} & -- & 10 & -- \\\\\n  $G_2$ & 6 & 3 & -- & {\\bf 9} & -- \\\\ \\hline\n $G_3$ &  15 & -- & 6 & -- & {\\bf 18} \\\\\n $G_4$ & 15 & -- & {\\bf 5} & -- & 20 \\\\   \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Separating examples for height, width, area, and perimeter.}\n\\label{table-separating1}\n\\end{table}\n\n\n\\begin{table}[ht!]\n\\begin{tabular}{c|ccc}\n & Perimeter & Height & Width \\\\ \\hline \\hline\n  Area & $P_6,C_6$ & $G_1,G_2$ & $P_6,C_6$ \\\\ \n  Perimeter & -- & $P_6,C_6$ & $G_3,G_4$ \\\\ \n  Height & -- & -- & $P_6,C_6$ \\\\\n  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graph pairs that are oppositely ordered by each of the parameter pairs.}\n\\label{table-separating-new}\n\\end{table}\n \n\n\\begin{figure}[ht!]\n    \\centering \\includegraphics[width=.9\\textwidth]{P6-and-C6.pdf}\n    \\caption{Graphs $P_6$ and $C_6$ show that height and area order graphs differently than  width and perimeter.}\n    \\label{fig:width-examplesAA}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n    \\includegraphics[width=.8\\textwidth]{G1-and-G2-new.pdf}\n    \\caption{Graphs $G_1$ and $G_2$ show that height orders graphs differently than area.}\n    \\label{fig:width-examplesBB}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{width-example1.pdf} \\hspace{1em}\n    \\includegraphics[width=.4\\textwidth]{width-example2.pdf}\n    \\caption{Graphs $G_3$ and $G_4$ show that width orders graphs differently than perimeter.}\n    \\label{fig:width-examplesCC}\n\\end{figure}\n\n\n\n \n\n\n\n\n\n\\begin{theorem}\nThe graph $G_3$ shown on the left in Figure~\\ref{fig:width-examplesCC} has perimeter 18 and width 6. \\label{G1-proof}\n\\label{width-theorem}\n\\end{theorem}\n\n\\begin{proof}\n\\noindent To find the perimeter of $G_3$, we first consider its area. Let $v$ be the vertex of degree 10 in $G_3$. In any RV-representation of $G_3$, the corresponding rectangle $V$ must have perimeter at least 10, so its area is at least 4. Together with the 14 other vertices, we see $\\area(G_3)\\ge 18.$  Now suppose $G_3$ can be represented in a bounding box of height $h$, width $w$, and perimeter $p=2h+2w$. Such a rectangle has maximum area when $h=w=p\/4$, so $\\area(R) \\leq p^2\/16$. But $\\area(R) \\geq 18$, so $p \\geq \\sqrt{18 \\cdot 16} > 16.$  Since $p$ must be even,  $\\perimeter(G_3)=18$ by Figure \\ref{fig:width-examplesCC}.\n\nNext we consider $\\width(G_3)$.  If $\\width(G_3)<6$, then $G_3$ can be represented in a $5\\times 5$ box $R$.   In a $5 \\times 5$ box with 14 other vertices, $V$ has area at most 11, and hence $V$ must be $1 \\times 4$, $1 \\times 5$, $2 \\times 3$, $2 \\times 4$, $2 \\times 5$, or $3 \\times 3$.  We rule out each possibility below.\n \n \n\\begin{figure}[ht!]\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{G1-cases.pdf}\n    \\caption{Diagrams illustrating several of the cases in the proof of Theorem~\\ref{G1-proof}.}\n    \\label{fig:width-proof}\n\\end{figure} \n \n  $\\bullet \\;$ If  $V$ is $1 \\times 4$, at least one unit of the perimeter of $V$ is on the boundary of $R$, and therefore $V$ cannot represent a degree-10 vertex. \n\n  $\\bullet \\;$ If $V$ is $1 \\times 5$ or $2 \\times 5$, it must touch opposite sides of $R$; thus $V$ must see 5 rectangles on each of its other two sides in order to have degree 10. But then $v$ is a cut vertex, and $G_3$ has none. \n\n  $\\bullet \\;$ If  $V$ is $3\\times 3$, it cannot have an edge on the boundary of $R$ and represent a degree-10 vertex. In this case, $V$ occupies the middle of $R$ as shown on the left in Figure~\\ref{fig:width-proof}. The four vertices not adjacent to $v$ must be represented by $1\\times 1$ squares in the corners of $R$.  Among these four, two (disjoint) pairs have a unique common neighbor in $G_3$.  Locating the rectangles for these common neighbors in their respective  $3 \\times 1$ blocks of $R$, we see there now remain only 6 locations for the other 8 vertices.   \n\n  $\\bullet \\;$ If $V$ is $2\\times 4$, it must (by symmetry) appear as in the middle of Figure~\\ref{fig:width-proof}. To have degree 10, $V$ must see a $1\\times 1$ square at $W$. But $v$ has no neighbor of degree less than $3$.   \n\n  $\\bullet \\;$ If $V$ is $2\\times 3$, it must (by symmetry) appear as on the right of Figure~\\ref{fig:width-proof}.  To have degree 10, $V$ must see distinct rectangles on each unit of its perimeter.  Numbering the locations in $R$ as in the right of Figure~\\ref{fig:width-proof}, if location 3 is empty, location 2 must contain a $1\\times 1$ square, but $v$ has no neighbor of degree less than $3$. If a $2 \\times 1$ rectangle $X$ covers location 3, the rectangles in locations 1 through 6 form a path $P_5$ in the neighborhood of $V$, but $G_3$ has no such subgraph.  Thus, locations 3 and (by symmetry) 7 contain $1 \\times 1$ squares. But $G_3 -v$ has no vertices of degree 2 that are at distance 4.\n \\end{proof}           \n \n\n\\begin{theorem}\nThe graph $G_4$ shown on the right in Figure~\\ref{fig:width-examplesCC} has perimeter 20 and width 5. \\label{G2-proof}\n\\label{width-theorem2}\n\\end{theorem}\n               \n\\begin{proof} \nWe use the labeling in Figure~\\ref{fig:width-examplesCC}.  We begin with perimeter.  Note that since $G_4$ has 15 vertices, every RV-representation of $G_4$ has area at least 15.  Then if $\\perimeter(G_4)<20$, it must have a representation that fits in a $3 \\times 6$ or $4 \\times 5$ bounding box. But a $3 \\times 6$ box has area 18, so the 3 vertices of degree more than four in $G_4$ must be represented with $1 \\times 2$ rectangles. But then all the vertices of $G_4$ must have degree $\\leq 6$, a contradiction.\n\nThus $G_4$ must be representable in a $4 \\times 5$ box. The two vertices of degree 5 in $G_4$ must each be represented with a rectangle of area at least 2. In a $4 \\times 5$ box $R$, $U$ then has area 3 or 4 and cannot lie in a corner. All vertices of $G_4$ have degree at least 3, so no corner of $R$ has a $1 \\times 1$ square. Therefore, each corner of $R$ is either empty or occupied by a rectangle of area at least 2. With 15 vertices, this exceeds the total area of 20 in $R$: each vertex contributes at least one, the four corners require at least one additional unit of area each, and $U$ needs two or three additional units of area not lying in a corner. Now $\\perimeter(G_4)=20$ by Figure~\\ref{fig:width-examplesCC}.\n\nWe next consider width. If $\\width(G_4)<5$, then $G_4$ can be represented in a $4 \\times 4$ box. But then it also has a $4 \\times 5$ representation, which we just proved impossible. By Figure~\\ref{fig:width-examplesCC}, we see that $\\width(G_4) =5$. \n\\end{proof}\n\n\n\\section{Minimizing Height, Width, Area, and Perimeter can require distinct representations of an RVG} \\label{area} \\label{SeparatingRepresentations}\n\nIn this section, we further explore our four parameters and we observe that, even for a single RVG, it is possible that the set of representations minimizing one of them may be disjoint from the set of representations minimizing another.  \nExamples to illustrate these results are summarized in Tables~\\ref{table-separating2} and \\ref{table-separating2b}. For each pair of parameters, there exists a graph that requires distinct representations to separately minimize each parameter in that pair. Specifically, the star $K_{1,4}$ has representations minimizing height that are distinct from those minimizing width, area, and perimeter; the graph $G_5$ shown in Figure~\\ref{fig:width-examples2a} has representations minimizing width that are distinct from those minimizing the other parameters; and the graph $G_6$ shown in Figure~\\ref{fig:width-examples2b} has distinct representations minimizing area and perimeter. Since $K_{1,4}$ has fewer than 7 vertices, the claims regarding it are verified by our computer search \\cite{WebList}. The claims regarding $G_5$ and $G_6$, each with 7 vertices, are proved in Theorems~\\ref{G5-theorem} and \\ref{G6-theorem} below.  \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|c|cccc} \nGraph (representation) & Vertices & Height & Width  & Area & Perimeter\\\\ \\hline \\hline \n  $K_{1,4}$ ($S_1$) & 5 & {\\bf 2} & 5  & 10 & 14\\\\\n  $K_{1,4}$ ($S_2$) & 5 & 3 & {\\bf 3}  & {\\bf 9} & {\\bf 12}\\\\ \\hline\n  $G_5$ ($S_1$) & 7 & {\\bf 2} & 5  & {\\bf 10} & {\\bf 14}\\\\\n  $G_5$ ($S_2$) & 7 & 4 & {\\bf 4}  & 16 & 16\\\\ \\hline\n  $G_6$ ($S_1$) & 7 & {\\bf 2} & 7  & {\\bf 14} & 18\\\\\n  $G_6$ ($S_2$) & 7 & 4 & {\\bf 4}  & 16 & {\\bf 16}\\\\  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graphs representations used to minimize height, width, area, and perimeter.}\n\\label{table-separating2}\n\\end{table} \n\n\\begin{table}[ht!]\n\\begin{tabular}{c|ccc}\n & Perimeter & Height & Width \\\\ \\hline \\hline\n  Area & $G_6$ & $K_{1,4}$ & $G_5$ \\\\ \n  Perimeter & -- & $K_{1,4}$ & $G_5$ \\\\ \n  Height & --& --& $G_5$ \\\\\n  \\hline\n\\end{tabular} \n\\smallskip\n\\caption{Graphs requiring different representations to minimize area, perimeter, height, and width.}\n\\label{table-separating2b}\n\\end{table}\n\n\\begin{theorem}\nThe graph $G_5$ shown in Figure~\\ref{fig:width-examples2a} has height 2, width 4, area 10, and perimeter 14.  Furthermore, minimizing width requires a different RV-representation from height, area, or perimeter. \\label{G5-theorem}\n\\end{theorem}\n\n\n\n\\begin{figure}[ht!]\n    \\centering\n   \\includegraphics[width=\\textwidth]{K14-and-G5-new.pdf}\n    \\caption{Graphs $K_{1,4}$ and $G_5$ are shown.  For $K_{1,4}$, one representation ($S_1$) minimizes height and another ($S_2$) minimizes area, perimeter, and width.  For $G_5$, one representation ($S_1$) minimizes height, area, and perimeter, and another  ($S_2$) minimizes width.}\n    \\label{fig:width-examples2a}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n    \\centering\n   \\includegraphics[width=.6\\textwidth]{G6-new.pdf}\n    \\caption{Graph $G_6$ with distinct representations minimizing area and perimeter.}\n    \\label{fig:width-examples2b}\n\\end{figure}\n \n\\begin{proof}\n The representation $S_1$ in Figure \\ref{fig:width-examples2a} has height 2, width 5, area 10, and perimeter 14. The representation $S_2$ has height 4, width 4, area 16, and perimeter 16. \n \nWe claim $G_5$ has no $3 \\times 4$ representation. Each of the 3-cycles in $G_5$ must have at least 2 rows that occupy at least 2 units of area. This implies that some row in a $3 \\times 4$ box has 2 units of each 3-cycle, which therefore must see each other. Since $G_5$ has no edges joining these 3-cycles, this is impossible.\n\nIt follows that $G_5$ has no $3 \\times 3$ or $2 \\times 4$ representation. Since $G_5$ is not a path, $\\height(G_5)\\geq 2$ and these facts together imply that $G_5$ has height 2, width 4, area 10, and perimeter 14.  \nSince the only representations of $G_5$ with width 4 are $4 \\times 4$, any representation minimizing width does not minimize height, area, or perimeter.\n\\end{proof}\n  \n \n\n\n\\begin{theorem}  \nThe graph $G_6$ shown in Figure~\\ref{fig:width-examples2b} has perimeter 16 and area 14.  Furthermore, minimizing perimeter requires a\ndifferent RV-representation than area. \\label{G6-theorem}\n\\end{theorem}\n\n\n\n\n\\begin{proof}\n The first representation in Figure \\ref{fig:width-examples2b} has perimeter 18 and area 14. The second has perimeter 16 and area 16. \n \n The 4 vertices of degree 1  each require at least 3 units of length on the perimeter with free lines of sight. The degree 2 vertex requires at least 2 units, and the 2 degree 3 vertices each require at least 1 unit. So $\\perimeter(G_6)=16$.  \n \n If $\\area(G_6)<14,$ then $G_6$ can be represented in a rectangle of area 7, 8, ..., or 13. Since $G_6$ is not a path, prime areas are not possible, so $G_6$ must have height $>1$. Since $\\perimeter(G_6)=16$, the only possible bounding box must be $2 \\times 6$. We claim this is impossible. Since $G_6$ has 7 vertices, a box with 6 columns would force some column to contain portions of 2 rectangles. Neither of these rectangles can represent a vertex of degree 1, or else the remaining graph must be a path, so they must have degrees 2 and 3, which implies that $D$ has an empty horizontal line of sight. But now, taken together, the 5 vertices of degree 1 or 2 require 5 empty horizontal lines of sight, while the bounding box only has height 2, with at most 4 such lines.   \n \n The only rectangle with height $>1$ and area 14 is $2 \\times 7$, so the perimeter and area must be achieved with distinct representations.\n\\end{proof}\n\n\n\\section{Graphs with small area, perimeter, height, and width} \\label{SmallParameters} \n\nIn this section, we address the question of which graphs on a given number of vertices minimize each parameter. \n\nFor any real number $x$, we use $\\lceil x \\rceil$ and $\\lfloor x \\rfloor$ to denote the integer ceiling and floor of $x$, respectively. We use $[[ x ]] = \\lfloor x + 1\/2 \\rfloor$ to denote $x$ rounded to the nearest integer. Recall that, for any positive integers $h$ and $w$, we let ${\\mathcal{F}}_{h,w}$ denote the (finite) set of graphs that have RV-representations in an $h \\times w$ bounding box.\n\n\\begin{theorem}\\label{lowerbds}\nLet $G$ be any graph with $n$ vertices and suppose $G$ has an RV-representation. Then the following hold.\n\\begin{enumerate}[label=\\rm{(\\roman*).}]\n    \\item The height of $G$ satisfies\n    $$\\height(G) \\geq 1.$$\n     Equality holds if and only if $G \\cong P_n$, the path on $n$ vertices.\n    \\item The area of $G$ satisfies\n    $$\\area(G) \\geq n.$$  \n     Equality holds if and only if $G \\cong P_h \\openbox P_w$, for some positive integers $h$ and $w$, where $n=h \\cdot w$. \n    \\item The width of $G$ satisfies\n    $$\\width(G) \\geq \\lceil \\sqrt{n} \\rceil.$$ \n    Equality holds if and only if $G \\in {\\mathcal{F}}_{w,w}$ where $w = \\lceil \\sqrt{n} \\rceil$.\n    \\item The perimeter of $G$ satisfies\n    $$\\perimeter(G) \\geq 2 \\cdot [[\\sqrt{n}]] + 2 \\cdot \\lceil \\sqrt{n} \\rceil.$$ \n    Equality holds if and only if $G \\in {\\mathcal{F}}_{h,w}$ for some positive integers $h$ and $w$, where $h+w = [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$ and $hw \\geq n$.\n\\end{enumerate} \n\\end{theorem}\n\n\\begin{proof}\n To see (i) and (ii), note that every RV-representation of a graph must have height at least 1 and area at least $n$, since each vertex requires at least a $1\\times 1$ rectangle. For $G$ to have height exactly 1, every rectangle in such a representation must be on the same row in the representation, so $G$ is a path.  For $G$ to have area exactly $n$,  every rectangle in such a representation must be $1 \\times 1$, with no empty space in the bounding box $R$. Thus $G$ is the grid $P_h \\openbox P_w$, where $h$ is the height of $R$ and $w$ is the width of $R$.  \n\n \n We show (iii) by way of contradiction.  Suppose $G$ can be represented in an $a \\times b$ bounding box $R$, where $a \\leq b < \\lceil \\sqrt{n} \\rceil$. Then $ b \\leq \\lceil \\sqrt{n} \\rceil -1$, so $b <  \\sqrt{n}$. But now the area of $R$ is $ab <  n$, which is impossible since $G$ has $n$ vertices.  It follows  that $\\width(G) = \\lceil \\sqrt{n} \\rceil$ if and only if $G \\in {\\mathcal{F}}_{w,w}$ where $w = \\lceil \\sqrt{n} \\rceil$. \n \n We also show (iv) by way of contradiction.  Suppose $G$ can be represented in an $a \\times b$ bounding box $R$, where $a + b < [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$. By (ii), we know $ab \\geq n$. We consider cases for whether $\\sqrt{n}$ rounds up or down. In each case, we find that\n $$ (a+b)^2 < 4n \\leq 4ab,$$\n which implies that $(a-b)^2 <0,$ a contradiction.  \n It follows that $\\perimeter(G) = 2 \\cdot [[\\sqrt{n}]] + 2 \\cdot \\lceil \\sqrt{n} \\rceil$ if and only if $G \\in {\\mathcal{F}}_{h,w}$ for some positive integers $h$ and $w$, where $h+w = [[\\sqrt{n}]] + \\lceil \\sqrt{n} \\rceil$ and $hw \\geq n$. \n\n\\end{proof}\n\n\\begin{remark}\nThe condition for equality in Theorem~\\ref{lowerbds}(iv) restricts the bounding box to be very nearly square. Specifically, we can say the following. \nLet $k$ denote the integer such that $k^2 < n \\leq (k+1)^2$.\n\nIf $k^2 < n \\leq k(k+1)$, then equality holds in Theorem~\\ref{lowerbds}(iv) if and only if $G \\in {\\mathcal{F}}_{k-t,k+1+t}$ for some integer $t$ where $$0 \\leq t \\leq \\sqrt{(k+{\\textstyle\\frac{1}{2}})^2-n}-{\\textstyle\\frac{1}{2}}.$$\n\nIf $k(k+1) < n \\leq (k+1)^2$, then equality holds in Theorem~\\ref{lowerbds}(iv) if and only if $G \\in {\\mathcal{F}}_{k+1-t,k+1+t}$ for some integer $t$ where $$0 \\leq t \\leq \\sqrt{(k+1)^2-n}.$$\n\\end{remark}\n\nFor example, any RVG with $n=70$ vertices must have perimeter at least 34, with equality only when $G$ has a $7 \\times 10$ or $8 \\times 9$ representation.\nSimilarly, any RVG with $n=120$ vertices must have perimeter at least 44, with equality only when $G$ has a $10 \\times 12$ or $11 \\times 11$ representation. \n\n\n\\section{Graphs with large area, perimeter, height, and width} \\label{CompleteArea} \\label{LargeParameters}\n\nIn this section we turn to the question of which graphs on a given number of vertices maximize our four parameters.  \n\nRecall that the \\textit{\\textbf{empty graph}} $E_n$ is the graph with $n$ vertices and no edges. Among small graphs (at most 6 vertices), the empty graphs maximize each of the four parameters. When the number of vertices is larger than 7, however, we will see that the empty graph no longer reigns supreme. Our proof is constructive, as we will provide specific graphs that we will prove are larger than $E_n$ in each parameter. But our results here leave open, perhaps for future work, the more difficult question of which graphs on $n$ vertices actually achieve the maximum values for the four parameters.\n\nWe begin with the small graphs.\n\n\\begin{theorem}\nFor $1 \\leq n \\leq 6$, among all graphs with $n$ vertices, the empty graph $E_n$ has largest height, width, area, and perimeter.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma~\\ref{NewDisjointLem}, the empty graph $E_n$ has height $n$, width $n$, area $n^2$, and perimeter $4n$.  Figures~\\ref{fig:small representations} and \\ref{fig:small representations6} show RV-representations of all connected graphs with at most 6 vertices.  These figures show that no other connected graphs with $2 \\leq n \\leq 6$ vertices exceed any of these values.  For a disconnected graph $G$, Lemma~\\ref{NewDisjointLem} implies that, as long as $G$ has at least one component with more than one vertex, we can combine the representations of the components as in Figure~\\ref{Glue} to obtain height less than $n$, area less than $n^2$, and perimeter less than $4n$. Because $K_2$ has width 2, the graph $K_2 + E_{n-2}$ has width $n$, but no graph has larger width than $E_n$ for $n \\leq 6$.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.7\\textwidth]{small-representations.pdf}\n\\caption{RV-representations of all connected graphs with between 1 and 5 vertices.}\n\\label{fig:small representations}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=.9\\textwidth]{small-representations6-labeled-updated.pdf}\n\\caption{RV-representations of all connected graphs with 6 vertices, using the labeling from~\\cite{small-graphs-website}.  These representations have smallest area, by computer search.}\n\\label{fig:small representations6}\n\\end{figure} \n\n\nOur next results focus heavily on the RV-representations of the complete graph $K_n$.  Let $S$ be any set of rectangles representing $K_n$ and let $R = [0,u] \\times [0,v]$ denote the smallest bounding box containing them. \nWe define the set ${\\mathcal{T}}_S$ of {\\bf top rectangles of $S$} as follows:\n$$\n{\\mathcal{T}}_S = \\{ X \\in S :  y_1^X \\geq y_1^Y \\mbox{ for all } Y \\in S\\}.$$\nWe now prove that for $K_n$, ${\\mathcal{T}}_S$ contains a single rectangle when $n\\geq 6$.\n\n\\medskip\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[height=2in]{top-box-proof.pdf}\n    \\caption{An example to illustrate the proof of Lemma~\\ref{lonely}}\n    \\label{fig:lonely rectangle}\n\\end{figure}\n \n \n\\begin{lemma}\\label{lonely}  Suppose $n\\geq 6$ and let $S$ be any rectangle visibility representation of $K_n$.\nThen $|{\\mathcal{T}}_S  | =1.$\n\\end{lemma}\n \n\\begin{proof} By way of contradiction, suppose $|{\\mathcal{T}}_S  | \\geq 2.$ Fix any distinct rectangles $A, B \\in {\\mathcal{T}}_S$, and note that ${\\mathcal{N}}(A)$ and ${\\mathcal{N}}(B)$ are empty, as illustrated by the example in Figure~\\ref{fig:lonely rectangle}. Without loss of generality, assume that $x_1^A \\leq x_1^B$ and $y_2^A \\geq y_2^B$ (i.e., the taller rectangle is on the left). We observe the following:\n\\medskip\n \n$\\bullet \\;$ \n\n${\\mathcal{W}}(A)$ is empty:\nIf $F \\in {\\mathcal{W}}(A)$, then $F$ cannot see $B.$\n\\medskip\n \n$\\bullet \\;$ \n$|{\\mathcal{E}}(B)| \\leq 1$:\nOtherwise, fix $I$, $J \\in {\\mathcal{E}}(B)$ with $x_1^I < x_1^J$, and note that $y_1^I, y_1^J\\le y_1^B$. Then $y_2^I > y_2^B$ since $I$ must see $A$. But now $J$ cannot see $B$, a contradiction.\n\\medskip \n \n$\\bullet \\;$ \n${\\mathcal{S}}(A)={\\mathcal{S}}(B)$:\nIf $C \\in {\\mathcal{S}}(A)$ then $y_2^C \\leq y_1^A = y_1^B$. But $C$ sees $B$, so $C \\in {\\mathcal{S}}(B)$. Thus ${\\mathcal{S}}(A) \\subseteq {\\mathcal{S}}(B)$ and, similarly, ${\\mathcal{S}}(B) \\subseteq {\\mathcal{S}}(A)$. \n\\medskip\n \n  \n$\\bullet \\;$ \n$|{\\mathcal{S}}(A)| >1$:  \nOtherwise $|{\\mathcal{E}}(A)| \\geq 4$, since $n \\geq 6$ and ${\\mathcal{N}}(A)$ and ${\\mathcal{W}}(A)$ are empty.\nSince $|{\\mathcal{E}}(B)|\\leq 1$, this implies $|{\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)| \\geq 2$. Fix distinct $G$, $H \\in {\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)$ with $x_1^G  \\leq x_1^H$.  Now $y_2^H > y_2^G$ since $H$ sees $A$. But then $G$ cannot see $B$, a contradiction.\n\\medskip\n \n$\\bullet \\;$ \n$|{\\mathcal{S}}(A)| \\leq 1$:  \nOtherwise, fix distinct $C$,$D \\in {\\mathcal{S}}(A)$ with $y_1^C  \\geq y_1^D$. \n  Since ${\\mathcal{S}}(A)={\\mathcal{S}}(B),$ both $C$, $D$ see $A$ and $B$ from below. Now ${\\mathcal{E}}(A)\\cap {\\mathcal{W}}(B)$ is empty, since if $G \\in {\\mathcal{E}}(A) \\cap {\\mathcal{W}}(B)$, then $D$ cannot see $G$. Since ${\\mathcal{N}}(A)$ and ${\\mathcal{W}}(A)$ are empty and $n \\geq 6$, it follows that\n  $|{\\mathcal{E}}(A)| \\geq 3$ and thus ${\\mathcal{E}}(B) \\geq 2$, a contradiction.\n\\medskip\n\nHaving shown that  $|{\\mathcal{S}}(A)| > 1$ and $|{\\mathcal{S}}(A)| \\leq 1$, we have arrived at a contradiction, and we conclude that $|{\\mathcal{T}}_S  | =1.$\n\\end{proof}\n\n\\begin{figure}\n    \\includegraphics[height=1.9in]{extracting-operation.pdf}\n   \n    \\caption{Applying the extracting operation $S \\uparrow A$.}\n    \\label{fig:my_label}\n\\end{figure}\n\nAnother operation that will be useful is an extraction operation that can move a certain rectangle to the top row of the bounding box. Specifically, for a rectangle visibility representation $S$ of $K_n$ with bounding box $R = [0,u] \\times [0,v]$ and a rectangle $A \\in \\mathcal{T}_S$, we define $S \\uparrow A$ to be the set of rectangles in $R$ given by\t$$ S \\uparrow A = \\{f(X) : X \\in S\\},$$\nwhere $f(A) = [0,u] \\times [v-1,v],$ and where, for every $X \\not= A$, $$ f(X) = [x_1^X,x_2^X] \\times [y_1^X, \\min\\{y_2^X,v-1\\}].$$   As illustrated in Figure~\\ref{fig:my_label}, the function $f$ maps the rectangle $A$ to the top row of $R$ and maps no other rectangle to that row. Furthermore, $f(X) \\subseteq X$ for all $X \\not= A$, so the rectangles of $ S \\uparrow A$ do not overlap.\n\n\\begin{lemma}\\label{extract} For any $n \\geq 1$ let $S$ be a rectangle visibility representation of $K_n$ with bounding box $R = [0,u] \\times [0,v]$. If ${\\mathcal{T}}_S =\\{A\\}$ then $S \\uparrow A$ also represents $K_n$ and has bounding box $R$.\n\\end{lemma}\n \n \\begin{proof} Since $f$ bijectively maps the $n$ rectangles of $S$ to the $n$ rectangles of $S\\uparrow A$, and since these representations share the same bounding box $R$, it remains only to show that $f$ preserves adjacency.  \n \t\nSince ${\\mathcal{N}}(A)$ is empty and the graph is complete, $S$ is partitioned as $$ S = \\{A\\} \\cup {\\mathcal{E}}(A) \\cup {\\mathcal{W}}(A) \\cup {\\mathcal{S}}(A).$$ For any distinct $X$ and $Y$ in $S$, we claim  $f(X)$ and $f(Y)$ see each other: \n\n\\smallskip\n\n\\textbf{Case 1. $X=A$ and $Y \\in {\\mathcal{E}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, note ${\\mathcal{N}}(Y)$ is empty. So ${\\mathcal{N}}(f(Y))=\\{f(A)\\}$, and $f(Y)$ sees $f(X)$ vertically. \n\n\\smallskip\n\n\\textbf{Case 2. $X=A$ and $Y \\in {\\mathcal{S}}(A)$.} Note\n${\\mathcal{S}}(f(A)) \\supseteq {\\mathcal{S}}(A)$\nand so  $f(Y)$ sees $f(X)$ vertically.\n\n\\smallskip\n\n\\textbf{Case 3. $X\\not=A$ and $Y \\in {\\mathcal{S}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, any line of sight between $X$ and $Y$ must be contained in the region below the top row of $R$. The only change to this region in $S \\uparrow A$ is the removal of $A$, so $f(X)$ still sees $f(Y)$ along the original line of sight.\n\n\\smallskip\n\n\\textbf{Case 4. $\\{X,Y\\} \\subseteq {\\mathcal{E}}(A)$.} If $X$ sees $Y$ vertically, say with $X$ above $Y$, then $y_1(X) > y_1(A)$ so that $Y$ can see $A$.  Since ${\\mathcal{T}}_S=\\{A\\}$, $X$ must see $Y$ horizontally. If the only line of visibility from $X$ to $Y$ were in the top row of $R$, then $y_2^X =y_2^Y = v$ and one of $X$,$Y$ could not see $A$. Therefore, $X$ must see $Y$ in a lower row, and so $f(X)$ still sees $f(Y)$ horizontally.\n\n\\smallskip\n\n\\textbf{Case 5. $X \\in {\\mathcal{E}}(A)$ and $Y \\in {\\mathcal{W}}(A)$.} Since ${\\mathcal{T}}_S=\\{A\\}$, $X$ must see $Y$ horizontally. If $X$ sees $Y$ in any row below the top row of $R$, then $f(X)$ still sees $f(Y)$ in that same row. If $X$ {\\em only} sees $Y$ in the top row of $R$, then $y_2^A < v$. Since ${\\mathcal{T}}_S=\\{A\\}$, both $X$,$Y$ see $A$ horizontally in the top row of $A$. Since the bottom row of $f(A)$ is above the top row of $A$, now $f(X)$ sees $f(Y)$ horizontally in what was the top row of $A$.\n\n\\smallskip\n\nBy symmetry, these cover all possible cases.\n\\end{proof}\n\n\\begin{figure}[ht!]\n \\centerline{\\includegraphics[height=1.9in]{K7-height.pdf} }\n \\caption{The seven rectangles in the proof of Theorem~\\ref{htk7}.}\n\\label{fig:K7proof}\n \\end{figure}\n \n\\begin{lemma}\\label{4out} Assume $n\\geq 6$ and $K_n$ has a rectangle visibility representation with bounding box $R = [0,u] \\times [0,v]$. Then $K_n$ has a rectangle visibility representation with bounding box $R$ in which the boundary of $R$ is covered by 4 rectangles of height or width 1.\n\\end{lemma}\n\n\\begin{proof}\n\tApply\n\tLemma \\ref{extract} successively in each of the four directions. Each time, a rectangle is brought to the corresponding boundary without changing the bounding box $R$.\n\\end{proof}\n\nRecall that a \\textit{\\textbf{bar visibility graph}} $G$ is a graph representable with a set of disjoint horizontal bars in the plane, with edges between bars that have vertical lines of sight between them.  All bar visibility graphs are planar \\cite{Duchet83, Tamassia86, Wismath85}.\n\n\\begin{lemma}\nSuppose $S$ is an RV-representation for a graph $G$.  If we partition the edges of $G$ into those with vertical lines of sight and horizontal lines of sight in $S$, then the subgraphs $G_V(S)$ and $G_H(S)$ of $G$ with these edges are bar visibility graphs, and hence planar graphs. \\label{bar-planar}\n\\end{lemma}\n\n\\begin{proof}\nReplace each rectangle $A$ in $S$ with a horizontal line segment at the top edge of $A$.  This is a bar visibility representation of $G_V$.  Rotating $S$ by 90 degrees and then replacing each rectangle by its new top edge yields a bar visibility representation of $G_H$.\n\\end{proof}\n\n\\begin{theorem} \\label{htk7} The complete graph $K_7$ has $\\height(K_7)=7$.\n\\end{theorem}\n\\begin{proof}\nSuppose $S$ is an RV-representation of $K_7$ with minimum height. By Lemma \\ref{4out}, we may assume the boundary of the bounding box $R$ is covered by 4 rectangles of height or width 1, as in Figure \\ref{fig:K7proof}.  Label these $A$, $B$, $C$, and $D$ clockwise from the top.\n\nThe remaining 3 rectangles $E$, $F$, and $G$ in the interior induce a 3-clique. If the 3 edges of this clique all correspond to horizontal lines of sight, then, together with rectangles $B$ and $D$, the edges of a 5-clique are represented entirely by horizontal lines of sight.  This is a 5-clique in $G_H$, which is impossible by Lemma~\\ref{bar-planar}.  Similarly, the edges among $E$, $F$, and $G$ cannot all be vertical lines of sight. Rotating and renaming if necessary, assume $E$ sees $F$ and $G$ vertically and $F$ sees $G$ horizontally. Then $F$ and $G$ must be on the same side of $E$ and we may assume $F,G \\in {\\mathcal{S}}(E)$ and $G \\in {\\mathcal{E}}(F)$, as shown in Figure~\\ref{fig:K7proof}. \n\nThe following five horizontal lines of sight must occupy five distinct rows in $R$:\n $BD$, $BE$,  $FG$, $BF$, and $DG$. To see why, notice first that edge $BD$ must have its own row to reach all the way across the representation. The row for $BE$ must be distinct from $FG$, $BF$, and $DG$ since $F,G \\in {\\mathcal{S}}(E)$. Rows for $FG$ and $BF$ are distinct since $B,G \\in {\\mathcal{E}}(F)$. Rows for $FG$ and $DG$ are distinct since $F, D \\in {\\mathcal{W}}(G)$. Rows for $BF$ and $DG$ are distinct since $G \\in {\\mathcal{E}}(F)$.\n \nSince $A$ and $C$ each take their own row by construction, it follows that $R$ has height at least 7. The representation of $K_7$ in Figure \\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{theorem}\\label{wk7} The complete graph $K_7$ has $\\width(K_7)=8$.\n\\end{theorem}\n\\begin{proof}  If $\\width(K_7)=7$, then Lemma~\\ref{lowerbds}(iii) would guarantee an RV-representation $S$ in a $7 \\times 7$ box $R$.\nBy Lemma \\ref{4out} and Theorem \\ref{htk7}, we may assume the boundary of $R$ is covered by 4 rectangles of height or width 1. Label these $A,B,C,D$ clockwise from the top.  As before, assume $F,G \\in {\\mathcal{S}}(E)$ and $G \\in {\\mathcal{E}}(F)$. \n\nThe following six vertical lines of sight must occupy six distinct columns in $R$:\n $AC,AF,EF,EG,AG,CE$. To see why, note that edge $AC$ must have its own column to reach all the way across the representation. Any column meeting $F$ cannot meet $G$, since $G \\in {\\mathcal{E}}(F)$. Columns for $CE$,$EF$,$EG$ are distinct since $C,F,G \\in {\\mathcal{S}}(E)$. Columns for $EG$,$AG$ are distinct since $A,E \\in {\\mathcal{N}}(G)$. Columns for $AF$,$EF$ are distinct since $A,E \\in {\\mathcal{N}}(F)$. Columns for $AF$,$CE$ are distinct since $E \\in {\\mathcal{N}}(F)$. Columns for $AG$,$CE$ are distinct since $E \\in {\\mathcal{N}}(G)$.\n \n Thus $R$ has width at least 8.  By Figure \\ref{fig:complete-graph-examples}, $\\text{width}(K_7)=8$. \n\\end{proof}\n\n\\begin{corollary}\n$\\area(K_7)=56$ and $\\perimeter(K_7)=30$.\n\\end{corollary}\n\n\\begin{proof}\nTheorems~\\ref{htk7} and \\ref{wk7} show that any representation of $K_7$ requires height at least 7 and width at least 8.  Figure~\\ref{fig:complete-graph-examples} shows a representation of $K_7$ with height 7 and width 8 exactly.  Therefore this representation also has smallest area and perimeter by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\n\\begin{theorem} \\label{htk8}\nThe complete graph $K_8$ has $\\rm{height}(K_8)=10$.\n\\end{theorem} \n\n\\begin{proof} Suppose $S$ is an RV-representation of $K_8$ with minimum height. By Lemma \\ref{4out}, we may assume the boundary of the bounding box $R$ is covered by 4 rectangles of height or width 1, as in Figure \\ref{4out}. Label these $A$, $B$, $C$, and $D$ clockwise from the top.\n\nRectangles $E$, $F$, $G$, and $H$ in the interior must induce a 4-clique. By Lemma~\\ref{bar-planar}, the edges in this clique that correspond to vertical lines of sight must form a triangle-free subgraph (and similarly for the horizontal edges).  Up to isomorphism, there are only two decompositions of $K_4$ into a pair of triangle-free graphs, shown in Figure~\\ref{fig:K8proof}.\n \n \\begin{figure}[ht!]\n \\centerline{\\includegraphics[height=1in]{K8-coloring.pdf}}\n \\caption{The partition of the edges of $K_8$ in Theorem~\\ref{htk8}.}\n\\label{fig:K8proof} \n \\end{figure}\n\nFirst we claim that the graph on the right of Figure~\\ref{fig:K8proof} is impossible. To see why, let the solid edges denote vertical lines of sight. Since $FG$ is horizontal and $EF$ and $EG$ are vertical, $F$ and $G$ are on the same side of $E$. Assume $F,G \\in {\\mathcal{S}}(E)$. Since $EH$ is horizontal and $FH$ and $GH$ are vertical, $F, G \\in {\\mathcal{S}}(H)$. Renaming if necessary, $E \\in {\\mathcal{E}}(H)$ and $F \\in {\\mathcal{E}}(G)$. So $x_2^G \\leq x_1^F$. But $x_1^E < x_2^G$ and $x_1^F < x_2^H$. It follows that $x_1^E < x_2^H$, contradicting that $E \\in {\\mathcal{E}}(H)$. A similar argument elimates the case when the solid edges denote horizontal lines of sight.\n\nNext we claim that the graph on the left requires $R$ to contain at least 10 rows. The graph is symmetric in solid and dotted edges, so assume the solid edges denote vertical lines of sight. Since $FG$ is horizontal and $EF$ and $EG$ are vertical, $F$ and $G$ are on the same side of $E$. Assume $F,G \\in {\\mathcal{S}}(E)$.\nSince $EH$ is horizontal and $EF$ and $FH$ are vertical, $E$ and $H$ are on the same side of $F$. So $H \\in {\\mathcal{N}}(F)$.\nSince $FH$ is vertical and $GH$ and $FG$ are horizontal, $F$ and $H$ are on the same side of $G$. Assume $F,H \\in {\\mathcal{W}}(G)$.\nSince $EG$ is vertical and $GH$ and $EH$ are horizontal, $E$ and $G$ are on the same side of $H$. So $H \\in {\\mathcal{W}}(E)$. \n\nThe following 8 horizontal lines of sight occupy distinct rows in $R$: \n $$BD, DE, EH, BH, GH, DG, FG, BF. $$\nEdge $BD$ must have its own row to reach all the way across the representation. Any row meeting $E$ cannot meet $F$ or $G$, since $F,G \\in {\\mathcal{S}}(E)$. Any row meeting $F$ cannot meet $H$, since $H \\in {\\mathcal{N}}(F)$. \nRows $DE$ and $EH$ are distinct since $D,H \\in {\\mathcal{W}}(E)$.\nRows $EH$ and $BH$ are distinct since $B,E \\in {\\mathcal{E}}(H)$.\nRows $BH$ and $GH$ are distinct since $B,G \\in {\\mathcal{E}}(H)$.\nRows $GH$ and $DG$ are distinct since $D,H \\in {\\mathcal{W}}(G)$.\nRows $DG$ and $FG$ are distinct since $F,D \\in {\\mathcal{W}}(G)$.\nRows $FG$ and $BF$ are distinct since $B,G \\in {\\mathcal{E}}(F)$.\nRows $DE$ and $BH$ are distinct since $H \\in {\\mathcal{W}}(E)$.\nRows $BH$ and $DG$ are distinct since $H \\in {\\mathcal{W}}(G)$.\nRows $DG$ and $BF$ are distinct since $F \\in {\\mathcal{W}}(G)$.\n \nSince $A$ and $C$ each take their own row by construction, $R$ has height $\\geq 10$. The representation of $K_8$ shown in Figure~\\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.3\\textwidth]{complete7-rectangles1.pdf} \\hspace{1cm} \\includegraphics[width=.4\\textwidth]{complete8-rectangles1.pdf}\n\\caption{A representation of $K_7$ in a $7 \\times 8$ bounding box, and a representation of $K_8$ in a $10 \\times 10$ bounding box.}\n\\label{fig:complete-graph-examples}\n\\end{figure}\n\n\\begin{theorem} \\label{wk8} \nThe complete graph $K_8$ has $\\width(K_8)=10$.\n\\end{theorem} \n\n\\begin{proof} Since, by definition, $\\width(K_8) \\geq \\rm{height}(K_8) =10$, the representation of $K_8$ shown in Figure \\ref{fig:complete-graph-examples} proves equality.\n\\end{proof}\n\n\\begin{corollary}\nThe complete graph $K_8$ has $\\area(K_8)=100$ and $\\perimeter(K_8)=40$.\n\\end{corollary}\n\n\\begin{proof}\nTheorems~\\ref{htk8} and \\ref{wk8} show that any representation of $K_8$ requires height at least 10 and width at least 10.  Figure~\\ref{fig:complete-graph-examples} shows a representation of $K_8$ with height 10 and width 10 exactly.  Therefore this representation also has smallest area and perimeter by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\nNote that $K_8$ is the largest complete RVG \\cite{hutchinson1999representations}, so we can't investigate the size of RV-representations of larger complete graphs.  But using the disjoint unions of complete graphs and Lemma~\\ref{NewDisjointLem}, we can construct graphs on $n$ vertices whose RV-representations are larger than the empty graph for all $n \\geq 8$, as follows.\n\n\\begin{corollary} \\label{disjoint complete graphs}\nFix any positive integer $n$ and write $n=8q+r$ for integers $q$ and $r$ with  $0 \\leq r < 8$.  Define the graph $G_n = q K_8 + E_r$, which has $q$ disjoint copies of $K_8$ and $r$ isolated vertices.  Then $G_n$ has $n$ vertices, $\\height(G_n) = \\width(G_n)=n+2q$, $\\area(G_n)=(n+2q)^2$ and $\\perimeter(G_n)=4n+8q$.\n\\end{corollary}\n\n\\begin{proof}\nBy Lemma~\\ref{NewDisjointLem}, $\\height(G_n)= q \\cdot \\height(K_8)+ \\height(E_r)$.  By Theorem~\\ref{htk8}, $\\height(K_8)=10$, and again by Lemma~\\ref{NewDisjointLem}, $\\height(E_r)=r$.  So $\\height(G_n)=10q+r=n+2q$.\n\nSince the bounding boxes of the representations of $K_8$ and $E_r$ are square, the same argument holds for $\\width(G_n)$.  Since the representations of $G_n$ with minimum height and width are the same, these representations also yield the minimum area and perimeter of $G_n$ by Lemma~\\ref{lemma-same-height-width}.\n\\end{proof}\n\n\\begin{corollary}\nAmong all rectangle visibility graphs with $n \\geq 7$ vertices, the empty graph $E_n$ does not have the largest width, area, or perimeter.\nAmong all rectangle visibility graphs with $n \\geq 8$ vertices, the empty graph $E_n$ does not have the largest height.\n\\end{corollary}\n\n\n\\section{Directions for Further Research} \\label{Conclusions}\n\nWe conclude with a number of open problems and questions that could further this line of research.\n\n\\begin{enumerate}[wide]\n\\item We have established that for $n=7,8$ the complete graph exceeds the empty graph in area, perimeter, height, and width. We have also shown that for $n>8$, the empty graph does not maximize any of these parameters. Accordingly, it is natural to ask, in general, which rectangle visibility graph(s) with $n$ vertices have largest height, perimeter, width, and area?  Note that for $n>8$, $K_n$ is not a rectangle visibility graph \\cite{hutchinson1999representations}.  Furthermore, when $r=7$ in Corollary~\\ref{disjoint complete graphs}, we can replace $E_7$ by $K_7$ to obtain a graph with width $n+2q+1$. Are there any other graphs that beat the graph in Corollary~\\ref{disjoint complete graphs}?\n\n\\item Tables~\\ref{table-separating1} and~\\ref{table-separating-new} show that, for each pair of parameters of area, perimeter, height, and width, there are pairs of graphs that share the same number of vertices, but that are ordered oppositely by that pair of parameters.  However, it does not consider triples and quadruples of parameters.  For example, is there a pair of graphs $G_1$ and $G_2$ for which $\\area(G_1)<\\area(G_2)$ but both $\\height(G_2)<\\height(G_1)$ and $\\width(G_2)<\\width(G_1)$?\n\n\\item Tables~\\ref{table-separating2} and~\\ref{table-separating2b} show that, for each pair of parameters of area, perimeter, height, and width, there is a graph with two representations that are ordered oppositely by that pair of parameters.  However, it does not consider triples and quadruples of parameters.  For example, is there a graph $G$ that requires three distinct RV-representations to minimize its area, perimeter, and height?\n\n\\item Say that an RV-representation $S$ is \\textit{compressible} if we can delete a row or column of $S$ and still have a representation of the same graph. For a given number of vertices $n$, which graphs have the largest incompressible representations, in terms of area, perimeter, height, or width?  How large are these values?\n\n\\item We might consider additional measures of size in terms of the rectangles in an RV-representation, rather than the bounding box.  For example, for an RV-representation $S$, say $\\recarea(S)$ is the area of the largest rectangle in $S$.  Then $\\recarea(G)$ is the smallest value of $\\recarea(S)$ for any RV-representation of $G$.  Which graphs $G$ with $n$ vertices have largest $\\recarea(G)$?\n\n\\item We conjecture that if $\\height(G)=2$ then $G$ is outerplanar.  Is this true?  Can we characterize other families of graphs of specific area, perimeter, height, or width?\n\n\\item We can consider other dimensions. For dimension 1, what is the minimum length of an integer bar visibility graph on $n$ vertices?  For 3-dimensional box visibility graphs, there are many parameters measuring the size of an integer box visibility representation.  Which graphs require the largest 3-dimensional representation, as measured by these parameters?  Note that in \\cite{Fekete99} Fekete and Meijer proved that $K_{56}$ is a 3-dimensional box visibility graph.\n \n \\end{enumerate}\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{NGC3516: X-ray and UV Observations}\n\n NGC3516 contains the strongest UV absorption system known in a\nSeyfert 1 galaxy. This system contains at least two distinct\ncomponents: a broad (FWHM$\\sim$2000 km\/s) variable component and a\nnarrow ($\\sim$500 km\/s) non-variable component.  Both the broad and\nthe narrow systems contain high as well as low ionization absorption\nlines. Recent observations have shown that the broad high ionization\nabsorption lines have {\\bf \\it disappeared} since $\\sim$1992 (Koratkar\net al.  1996 and references therein, Kriss et al. 1996)\n\n We analyze a high signal-to-noise (S\/N) ROSAT PSPC archival spectrum\nof NGC3516 obtained in 1992.  The high S\/N allows the strong detection\nof both OVII and OVIII edges independently, in spite of the limited\nspectral resolution of the PSPC. A warm absorber fit to the data shows\nthat the absorber is highly ionized (U$=10^{+2.6}_{-2.1}$), and has a\nlarge column density N$_H \\sim 10^{22}$ cm$^{-2}$.\n\n\\section{The XUV Absorber}\n\n In several AGN, the X-ray and the UV absorbers were found to be one\nand the same (the `XUV' absorbers, Mathur et al. 1994, 1995). Is it\nalso true for NGC3516? The absorption systems in NGC3516 are clearly\ncomplex with multiple components (Kriss et al. ~1996). It is the {\\it\nhigh ionization, broad} absorption system that is most likely to be\nassociated with the X-ray warm absorber.  Investigation of this\nquestion is tricky, however, because the broad absorption lines have\ndisappeared. Here we argue that the XUV absorption picture is {\\it\nconsistent} with the presence of a highly ionized X-ray absorber and the\ncurrent non-detection of CIV and NV broad absorption lines (see Fig. 1).\nThe X-ray absorber {\\it MUST} have a UV signature showing OVI\nabsorption lines (Fig. 1). Since there were no simultaneous\nROSAT \\& far-UV observations in 1992, this cannot be directly determined.\nHowever, we note that in the 1995 HUT observations, OVI doublets are\nunresolved, but consistent with being  broad\n(FWHM=1076$\\pm$146 km\/s, Kriss et al. 1996).\n\\begin{figure}\n\\vspace*{-0.9in}\n\\centerline{\n\\epsscale{.75}\n\\plotone{mathurs.eps}\n}\n\\vspace*{-0.7in}\n\\caption{Ionization fractions f of OVI, OVII, OVIII, CIV and NV as a\nfunction of ionization parameter, U (with CLOUDY, Ferland 1991). The\nvertical lines define the range of U for which the ratio\nf$_{\\mbox{OVII}}$\/f$_{\\mbox{OVIII}}$ lies within the observed ROSAT\nrange. The arrows on the CIV and NV curves indicate the lower limits\nof f$_{\\mbox{CIV}}$${_>\\atop^{\\sim}}$ $3\\times 10^{-4}$ and f$_{\\mbox{NV}}$${_>\\atop^{\\sim}}$\n$3.1\\times 10^{-4}$ based on the published IUE data.  The + mark\ncorresponds to the HUT data in Kriss et al. 1996.}\n\\end{figure}\n\n We argue that the XUV absorber in NGC3516 has evolved with time (Fig.\n1). {\\bf Pre-1992:} It showed broad, high ionization CIV and NV\nabsorption lines and an X-ray ionized absorber (U ${_<\\atop^{\\sim}}$ 7). As it\nevolves, outflowing and expanding, the density falls and the\nionization parameter increases. {\\bf 1992:} CIV and NV absorption lines\ndisappeared; X-ray absorber is still present with OVI lines in the UV\n(U$\\sim$10) (No UV data available to verify). {\\bf 1995:} CIV and NV\nabsorption line remain absent; X-ray absorber is present. OVI lines\nare present, and detected with HUT (U$\\sim$13.5). {\\bf Post-1996:} We\npredict that the OVI absorption lines will disappear as the ionization\nparameter increases further (U${_>\\atop^{\\sim}}$ 20). The OVIII edge will continue\nto strengthen relative to the OVII edge.  Eventually, even the X-ray\nabsorber will also disappear.\n\n\\acknowledgments\nSM gratefully acknowledges the financial support of NASA grant\nNAGW-4490 (LTSA) and BW, TA of NASA contract NAS8-39073 (ASC).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn ordinary heavy baryon constitutes a pair of light quarks and a\nheavy quark. Since the charm and bottom quarks are very heavy in\ncomparison with the light quarks, it is plausible to take the limit of\nthe infinitely heavy mass of the heavy quark, i.e. $m_Q\\to \\infty$. In\nthis limit, the physics of heavy baryons become simple. The spin of\nthe heavy quark is conserved, because of its infinitely heavy mass. It\nresults in the conservation of the total spin of light quarks:\n$\\bm{J}_L \\equiv \\bm{J}-\\bm{J}_Q$, where $\\bm{J}_L$, $\\bm{J}_Q$, and\n$\\bm{J}$ denote the spin of the light-quark pair, that of the\nheavy quark, and the total spin of the heavy baryon. This is called \nthe heavy-quark spin symmetry that allows $\\bm{J}_L$ to be a\ngood quantum number. Moreover, the physics is kept intact under the\nplacement of heavy quark flavors. This is called the heavy-quark\nflavor symmetry~\\cite{Isgur:1989vq, Isgur:1991wq,\n  Georgi:1990um, Manohar:2000dt}. Then a heavy quark becomes static,\nso that it can be considered as a static color source. Its importance\nis only found in making the heavy baryon a color singlet, and\nin giving higher-order contributions arising from $1\/m_Q$ \ncorrections. Consequently, the dynamics inside a heavy baryon is\nmainly governed by the light quarks. \n\nThe flavor structure of the heavy baryon is also determined by\nthem. Since there are two light quarks inside the heavy baryon, we\nhave two different flavor  $SU_{\\mathrm{f}}(3)$ irreducible\nrepresentations, i.e. $\\bm{3}\\otimes \\bm{3}=\\overline{\\bm{3}} \\oplus\n\\bm{6}$.  In the language of a quark model, the spatial part of the  \nheavy-baryon ground state is symmetric due to the zero orbital angular\nmomentum, and the color part is totally antisymmetric. Since the\nflavor anti-triplet ($\\overline{\\bm{3}}$) is antisymmrtric, the spin\nstate corresponding to {$\\overline{\\bm{3}}$} should be antisymmetric. Thus, the\nbaryons belonging to the anti-triplet should be $J_L=0$. Similarly, the\nflavor-symmetric sextet ($\\bm{6}$) should be symmetric in spin space,\ni.e. $J_L=1$. This leads to the fact that the baryon antitriplet has\nspin $J=1\/2$, while the baryon sextet carries spin $J=1\/2$ or $J=3\/2$,\nwith the spin of the light-quark pair being coupled with the heavy\nquark spin $J_Q=1\/2$.  So, we can classify 15 different lowest-lying\nheavy baryons as shown in Fig.~\\ref{fig:1} in the case of charmed\nbaryons.   \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.35]{fig1.pdf}\\qquad\n\\caption{The anti-triplet ($\\overline{\\bm{3}}$) and sextet ($\\bm{6}$)\n  representations of the lowest-lying heavy baryons. The left panel\n  draws the weight diagram for the anti-triplet with the total spin\n  $\\frac{1}{2}$. The centered panel corresponds to that for the sextet\n  with the total spin $1\/2$ and the right panel depicts that\n  for the sextet with the total spin $3\/2$.} \n\\label{fig:1}\n\\end{figure}\n\nRecently, there has been a series of new experimental data on the\nspectra of heavy baryons~\\cite{Aaltonen:2007ar, Chatrchyan:2012ni,\n  Abazov:2008qm, Kuhr:2011up, Aaij:2012da, Aaij:2013qja, Aaij:2014esa,\n  Aaij:2014lxa, Aaij:2014yka}, which renewed interest\nin the physics of the heavy baryons. The lowest-lying singly heavy baryons\nare now almost classified except for $\\Omega_b ^\\ast$. In the meanwhile, the\nLHCb Collaboration has announced the first finding of two heavy\npentaquarks, $P_c(4380)$ and $P_c(4450)$ ~\\cite{Aaij:2015tga,\n  Aaij:2016phn, Aaij:2016ymb, Aaij:2016iza}. Very recently, the\nfive excited $\\Omega_c$ baryons were reported~\\cite{Aaij:2017nav}, among\nwhich the four of them was confirmed by the Belle\nexperiment~\\cite{Yelton:2017qxg}. Interestingly the two of the excited\n$\\Omega_c$s, i.e. $\\Omega_c(3050)$ and $\\Omega_c(3119)$, have very narrow\nwidths: $\\Gamma_{\\Omega_c(3050)}=(0.8\\pm0.2\\pm0.1)\\,\\mathrm{MeV}$ and\n$\\Gamma_{\\Omega_c(3119)}=(1.1\\pm0.8\\pm0.4)\\,\\mathrm{MeV}$.   \n\nWhile there is a great deal of theoretical approaches for the\ndescription of heavy baryons, we will focus on a pion mean-field\napproach in the present short review. This mean-field \napproach was first proposed by E. Witten in this seminal\npapers~\\cite{Witten:1979kh,Witten:1983}, where he asserted that in the\nlimit of the large number of colors ($N_c$) the nucleon can be\nregarded as a bound state of $N_c$ \\textit{valence} quarks in a pion\nmean field with a hedgehog symmetry~\\cite{Pauli:1942kwa,\n  Skyrme:1961vq}. Since a baryon mass \nis proportional to $N_c$ whereas the quantum fluctuation around the\nsaddle point of the pion field is suppressed by $1\/N_c$, the\nmean-field approach is a rather plausible method for explaining\nproperties of baryons. The presence of $N_c$ \\textit{valence} quarks\nin this large $N_c$ limit, which consist of the lowest-lying baryons,    \nproduce the pion mean fields by which they are influenced\n\\emph{self-consistently}. This picture is very similar to a Hartree \napproximation in many-body theories. Witten also showed how to\nconstruct the mean-field theory for the baryon schematically in\ntwo-dimensional quantum chromodynamics (QCD). Though his idea was\ncriticized sometimes ago by S. Coleman~\\cite{Coleman} because of its\ntechnical difficulties, it is worthwhile to pursue it to see how far\nwe can describe the structure of the baryon in the pion mean-field\napproach.   \n\nThe chiral quark-soliton model ($\\chi$QSM)~\\cite{Diakonov:1987ty,\n  Christov:1995vm, Diakonov:1997sj} has been\nconstructed based on Witten's argument. The $\\chi$QSM starts from the\neffective chiral action (E$\\chi$A) that was derived from the instanton\nvacuum~\\cite{Diakonov:1983hh, Diakonov:1985eg}. The E$\\chi$A respects\nchiral symmetry and its spontaneous breakdown, in which the essential\nphysics of the lowest-lying hadrons consists. One can derive the\nclassical energy of the nucleon by computing the nucleon correlation\nfunction in Euclidean space, taking the Euclidean time to go to\ninfinity. Minimizing the classical energy self-consistently in the\nlarge $N_c$ limit with the $1\/N_c$ meson quantum fluctuations\nsuppressed, we obtain the classical mass and the self-consistent\nprofile function of the chiral soliton. While we ignore the $1\/N_c$\nquantum fluctuations around the saddle point of the soliton field, we\nneed to take into account the zero modes that do not change the\nsoliton energy. Since the soliton with hedgehog symmetry is not\ninvariant under translational, rotational and isotopic\ntransformations, we impose these symmetry properties on the\nsoliton and obtain a completely new solution with the same classical\nenergy. Because of the hedgehog symmetry, an  \n$\\mathrm{SU(2)}$ soliton needs to be embedded into the isospin\nsubgroup of the flavor\n$\\mathrm{SU(3)}_{\\mathrm{f}}$~\\cite{Witten:1983}, which was   \nalready utilized by various chiral soliton\nmodels~\\cite{Guadagnini:1983uv, Mazur:1984yf, Jain:1984gp}. This\ncollective quantization of the chiral soliton leads to the collective\nHamiltonian with effects of flavor $\\mathrm{SU(3)}_{\\mathrm{f}}$\nsymmetry breaking. The $\\chi$QSM has one salient feature: the  \nright hypercharge is constrained to be $Y'=N_c\/3$ imposed by the $N_c$ \nvalence quarks. This right hypercharge selects allowed representations\nof light baryons such as the baryon octet ($\\bm{8}$), the decuplet\n($\\bm{10}$), etc. The $\\chi$QSM was successfully applied to the\nproperties of the lowest-lying light baryons such as the mass\nsplittings~\\cite{Blotz:1992pw, Yang:2010fm}, the form\nfactors~\\cite{Kim:1995mr, Silva:2001st, Ledwig:2010tu},\nthe magnetic moments~\\cite{Kim:1995ha, Wakamatsu:1996xm, Kim:1997ip,\n  Kim:2005gz}, hyperon \nsemileptonic decays~\\cite{Ledwig:2008ku, Yang:2015era}, \nparton distributions~\\cite{Diakonov:1996sr, Wakamatsu:2003wg},\ntransversities of the nucleon~\\cite{Kim:1995bq, Kim:1996vk,\n  Schweitzer:2001sr}, generalized parton\ndistributions~\\cite{Goeke:2001tz}, and so on.     \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.25]{fig2.pdf}\n\\caption{Schematic picture of a heavy baryon. The $N_c-1$ valence\n  quarks are filled in the lowest-lying valence level $K^P=0^+$ with\n  the heavy quark stripped off. $K^P$ denotes the grand spin which we\n  will explain later and  $P$ is the corresponding parity of the\n  level. The presence of the valence quarks will interact with the sea\n  quarks filled in the Dirac sea each other. This interaction will\n  bring about the pion mean field.} \n\\label{fig:2}\n\\end{figure}\nVery recently, Ref.~\\cite{Yang:2016qdz} extended a mean-field \napproach to describe the masses of singly heavy baryons, being  \nmotivated by Ref.~\\cite{Diakonov:2010tf}. A singly heavy baryon\nconstitutes a heavy baryon and $N_c-1$ light valence quarks (see\nFig.~\\ref{fig:2}). In the limit of $m_Q\\to\\infty$, the heavy quark can\nbe considered as a static color source. Thus, the dynamics inside a\nheavy baryon is governed by the $N_c-1$ valence quarks. The presence\nof the $N_c-1$ valence quarks will produce the pion mean fields as in\nthe case of the light baryons. However, there is one very significant\ndifference.   \nthe constraint right hyper charge is taken to be $Y' =(N_c-1)\/3$ and\nallows the lowest-lying representations: the baryon anti-triplet\n($\\overline{\\bm{3}}$), the baryon sextet ($\\bm{6}$), the baryon\nanti-decapentaplet ($\\overline{\\bm{15}}$). The model reproduced\nsuccessfully the mass splitting of the baryon anti-triplet and sextet\nin both the charm and bottom sectors. In addition, the mass of the\n$\\Omega_b^\\ast $ baryon, which has not yet found, was predicted. The model\nwas further extended by including the second-order perturbative\ncorrections of flavor $SU_{\\mathrm{f}}(3)$ symmetry\nbreaking~\\cite{Kim:2018xlc}. The magnetic moments \nbaryons~\\cite{Yang:2018uoj} and electromagnetic form\nfactors~\\cite{Kim:2018nqf} of the singly heavy baryons were also\nstudied within the same framework. The $\\chi$QSM was also used to\ninterpret the five $\\Omega_c$ baryons newly found by the LHCb\nCollaboration~\\cite{Kim:2017jpx, Kim:2017khv}. Within the present\nframework, two of the $\\Omega_c$s with the smaller widths are\nclassified as the members of the baryon $\\overline{\\bm{15}}$, whereas\nall other $\\Omega_c$'s belong to the excited baryon sextet. The widths\nwere quantitatively well reproduced without any free parameter. In the\npresent work, we will review briefly these recent investigations on the singly\nheavy baryons. \n\nWe sketch the present work as follows: In Section\nII, we review the general formalism of the $\\chi$QSM for singly heavy\nbaryons. In Section III, we examine the mass splittings of the heavy\nbaryons, emphasizing the discussion of the effects of\n$\\mathrm{SU(3)}_{\\mathrm{f}}$ breaking. In Section IV, we discuss the\nrecent results of the magnetic moments and electromagnetic form\nfactors of the heavy baryons. In Section V, we briefly introduce a\ntheoretical interpretation of the excited $\\Omega_c$ baryons found by the\nLHCb, based on the present mean-field approach. The final Section is\ndevoted to the conclusions and outlook.\n\n\\section{The chiral quark-soliton model for singly heavy baryons} \nIn the present approach, a heavy baryon is considered as a bound state\nof the $N_c-1$ valence quarks in the pion mean field with a heavy\nquark stripped off from the valence level. Thus, the correlation\nfunction of the heavy baryon can be expressed in terms of the $N_c-1$\nvalence quarks\n\\begin{align}\n\\Pi_{B}(0, T) = \\langle J_B (0, T\/2) J_B^\\dagger\n  (0,-T\/2) \\rangle_0 = \\frac{1}{Z}\\int  \\mathcal{D} U\n  \\mathcal{D}\\psi^\\dagger \n  \\mathcal{D}\\psi J_B(0,T\/2) J_B^\\dagger (0,-T\/2)\n  e^{\\int d^4 x\\,\\psi^\\dagger (i\\rlap{\/}{\\partial} + i\n  MU^{\\gamma_5}+ i \\hat{m})\\psi}  ,   \n\\label{eq:corr1}\n\\end{align}\nwhere $J_B$ denotes the light-quark current with the $N_c-1$ \nlight quarks for a heavy baryon $B$\n\\begin{align}\nJ_B(\\bm{x}, t) = \\frac1{(N_c-1)!}\n  \\varepsilon^{\\beta_1\\cdots\\beta_{N_c-1}} \\Gamma_{J'J_3',TT_3}^{\\{f\\}}\n  \\Psi_{\\beta_1f_1}(\\bm{x}, t) \\cdots \\Psi_{\\beta_{N_c-1}f_{N_c-1}}\n  (\\bm{x}, t).      \n\\end{align}\n$\\beta_i$ stand for color indices and\n$\\Gamma_{J'J_3',TT_3}^{\\{f_1\\cdots f_{N_c-1}\\}}$ represents a matrix \nwith both flavor and spin indices. $J'$ and $T$ are the spin and\nisospin of the heavy baryon, respectively. $J_3'$ and $T_3$ are their\nthird components, respectively. The notation $\\langle \\cdots \\rangle_0$ in\nEq.~(\\ref{eq:corr1}) is the vacuum expectation value, $M$\nthe dynamical quark mass, and the chiral field $U^{\\gamma_5}$ is\ndefined as    \n\\begin{align}\nU^{\\gamma_5} = U\\frac{1+\\gamma_5}{2} + U^\\dagger \\frac{1-\\gamma_5}{2}   \n\\end{align}\nwith \n\\begin{align}\nU = \\exp(i\\pi^a \\lambda^a).  \n\\end{align}\nHere, $\\pi^a$ represents the pseudo-Goldstone boson field and\n$\\hat{m}$ denotes the flavor matrix of the current quarks, written as\n$\\hat{m}=\\mathrm{diag}(m_{\\mathrm{u}},\\,m_{\\mathrm{d}},\\,m_{\\mathrm{s}})$. We \nassume isospin symmetry, i.e. $m_{\\mathrm{u}}=m_{\\mathrm{d}}$. Since\nthe strange current quark mass is small enough, we will treat it\nperturbatively.   \n\nIntegrating over the quark fields, we derive the correlation function\nas \n\\begin{align}\n\\Pi_{B}(0, T) =\n  \\frac{1}{Z}\\Gamma_{J'J_3',TT_3}^{\\{f\\}}\\Gamma_{J'J_3',TT_3}^{\\{g\\}*} \\int\n  \\mathcal{D} U \\prod_{i=1}^{N_c-1}  \\left\\langle 0,T\/2\\left|\n  \\frac1{D(U)} \\right|0,-T\/2\\right\\rangle\n  e^{-S_{\\mathrm{eff}}(U)}, \n\\label{eq:corr2}\n\\end{align}  \nwhere the single-particle Dirac operator $D(U)$ is defined as \n\\begin{align}\nD(U) = i\\gamma_4 \\partial_4 + i\\gamma_k \\partial_k + i MU^{\\gamma_5} +\n  i \\hat{m}   \n\\end{align}\nand $S_{\\mathrm{eff}}$ is the effective chiral action written\nas \n\\begin{align}\nS_{\\mathrm{eff}} = -N_c \\mathrm{Tr}\\log D(U).  \n\\label{eq:effecXac}\n\\end{align}\nEquation~(\\ref{eq:corr2}) can be schematically depicted as\nFig.~\\ref{fig:2}. It consists of two different terms: The first and\nsecond ones are respectively called the \\textit{valence-quark\n  contribution} and \\textit{sea-quark contribution} within the\n$\\chi$QSM.  \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig3.pdf}\\qquad\n\\caption{Correlation function for a heavy baryon} \n\\label{fig:3}\n\\end{figure}\nWhen the Euclidean time $T$ is taken from $-\\infty$ to $\\infty$, \nthe correlation function picks up the ground-state\nenergy~\\cite{Diakonov:1987ty, Christov:1995vm} \n\\begin{align}\n\\lim_{T\\to\\infty} \\Pi_B(T) \\sim \\exp[-\\left\\{(N_c-1) E_{\\mathrm{val}} +\n  E_{\\mathrm{sea}}\\right\\} T],   \n\\end{align}\nwhere $E_{\\mathrm{val}}$ and $E_{\\mathrm{sea}}$ the valence and\nsea quark energies. Minimizing self-consistently the energies around\nthe saddle point of the chiral field $U$\n\\begin{align}\\label{eq:sol}\n\\left.\\frac{\\delta}{\\delta U}[ (N_c-1) E_{\\mathrm{val}} +\n  E_{\\mathrm{sea}}]\\right|_{U_c} = 0,    \n\\end{align}\nwe get the classical soliton mass \n\\begin{align}\\label{eq:solnc}\nM_{\\mathrm{sol}} = (N_c-1) E_{\\mathrm{val}}(U_c) + E_{\\mathrm{sea}}(U_c).  \n\\end{align}\nNote that a singly heavy baryon has a heavy quark, so its classical\nis expressed as the sum of the classical and heavy-quark masses\n\\begin{align}\nM_{\\mathrm{cl}} = M_{\\mathrm{sol}} + m_Q.  \n\\label{eq:classical_mass}\n\\end{align}\nWe want to mention that $m_Q$ is the \\textit{effective} heavy quark\nmass that is different from that of QCD and will be absorbed\nin the center mass of each representation. \n\nThe rotational excitations of the soliton with $N_c-1$ valence quarks\nwill produce the lowest-lying heavy baryons. To keep the hedgehog\nsymmetry, the SU(2) soliton $U_c(\\bm{r})$ will be embedded into\nSU(3)~\\cite{Witten:1983}  \n\\begin{align}\nU(\\bm{r}) = \\begin{pmatrix}\nU_c (\\bm{r}) & 0 \\\\\n0 & 1\n\\end{pmatrix}.\n\\label{eq:embed}\n\\end{align}\nAs mentioned in Introduction, we consider explicitly the rotational zero\nmodes.  Assuming that the soliton $U(\\bm{r})$ in Eq.(\\ref{eq:embed})\nrotates slowly, we apply the rotation matrix $A(t)$ in\n$\\mathrm{SU}_{\\mathrm{f}}(3)$ space    \n\\begin{align}\nU(\\bm{r},\\,t) = A(t) U(\\bm{r}) A^\\dagger (t).  \n\\end{align}\nThen, we can derive the collective Hamiltonian for heavy\nbaryons\n\\begin{align}\nH =& H_{\\mathrm{sym}} + H^{(1)}_{\\mathrm{sb}} +  H^{(2)}_{\\mathrm{sb}},\n\\end{align}\nwhere $H_{\\mathrm{sym}}$ represents the flavor SU(3) symmetric part, \n$H^{(1)}_{\\mathrm{sb}}$ and $H^{(2)}_{\\mathrm{sb}}$ the\nSU(3) symmetry-breaking parts respectively to the first and second\norders. $H_{\\mathrm{sym}}$ is expressed\nas  \n\\begin{align} \nH_{\\mathrm{sym}}=M_{\\mathrm{cl}}+\\frac{1}{2I_{1}}\\sum_{i=1}^{3}\n\\hat{J}^{2}_{i} +\\frac{1}{2I_{2}}\\sum_{a=4}^{7}\\hat{J}^{2}_{a},  \n\\end{align} \nwhere $I_{1}$ and $I_{2}$ are the moments of inertia of the\nsoliton and the operators  $\\hat{J}_{i}$ denote the SU(3) \ngenerators. We get the eigenvalue of the quadratic Casimir operator\n$\\sum_{i=1}^8 J_i^2$ in the $(p,\\,q)$ representation, given as    \n\\begin{align}\nC_2(p,\\,q) = \\frac13 \\left[p^2 +q^2 + pq + 3(p+q)\\right],   \n\\label{eq:Casimir}\n\\end{align}\nwhich leads to the eigenvalues of $H_{\\mathrm{sym}}$ \n\\begin{align} \nE_{\\mathrm{sym}}(p,q) = M_{\\mathrm{cl}}+ \\frac{1}{2I_{1}} J_L(J_L+1) \n+\\frac{1}{2I_{2}}\\left[C_2(p,\\,q) - J_L(J_L+1)\\right] \n-\\frac{3}{8I_{2}} {Y'}^{2}.\n\\label{eq:RotEn}\n\\end{align} \nThe right hypercharge $Y'$ is constrained by the $N_c-1$ valence\nquarks inside a singly heavy baryon, i.e. $Y'=(N_c-1)\/3$. The\ncorresponding collective wave functions of the singly heavy baryon is\nthen obtained as  \n\\begin{align} \n\\psi_B^{({\\mathcal{R}})}(JJ_3,J_L;A)=\n\\sum_{m_{3}=\\pm1\/2}C^{J J_3}_{J_{Q} m_{3} J_L \nJ_{L3}} \\chi_{m_{3}} \\sqrt{\\mathrm{dim}(p,\\,q)}\n(-1)^{-\\frac{ Y'  }{2}+{J}_{L3}}\n  D^{(\\mathcal{R})\\ast}_{(Y,T,T_3)(Y'  ,J_L,-J_{L3})}(A),  \n\\label{eq:waveftn}\n\\end{align} \nwhere \n\\begin{align}\n\\mathrm{dim}(p,\\,q) = (p+1)(q+1)\\left(1+\\frac{p+q}{2}\\right).  \n\\end{align}\n $J$ and $J_3$ in Eq.~(\\ref{eq:waveftn}) are the spin angular\n momentum and its third component of the heavy baryon, respectively.   \n$J_L$ and $J_{Q}$ represent the soliton spin and\nheavy-quark spin, respectively. ${J_{L3}}$ and ${m_{3}}$ are the\ncorresponding third components, respectively. Since the spin operator\nfor the heavy baryon is given as \n\\begin{align}\n\\label{eq:quantum}\n\\bm{J}=\\bm{J}_{Q}+\\bm{J}_L,\n\\end{align} \nthe relevant Clebsch-Gordan coefficients appear in\nEq.(\\ref{eq:waveftn}). The SU(3) Wigner $D$ function in\nEq.(\\ref{eq:waveftn}) means just the wave-function for the quantized\nsoliton with the $N_c-1$ valence quarks, and $\\chi_{m_3}$\nis the Pauli spinor for the heavy quark. $\\mathcal{R}$ designates a\nSU(3) irreducible representation corresponding to $(p,\\,q)$. \nSince the soliton is coupled to the heavy quark, we finally obtain the\nthree lowest-lying representations illustrated in\nFig.~\\ref{fig:1}. In the limit of $m_Q\\to\\infty$, the two sextet\nrepresentations are degenerate. One needs to introduce a \nhyperfine spin-spin interaction to lift this degeneracy. As will be\ndiscussed soon, this hyperfine interaction will be determined by using\nthe experimental data on the masses of heavy baryons.    \n\nIn the present zero-mode quantization scheme, we find the following\nthe two important selection rule. The allowed SU(3) representations\nmust contain states with $Y'=(N_c-1)\/3$ and the isospin $\\bm{T}$ of\nthe states with $Y'=(N_c-1)$\/3 are coupled with the soliton so that we\nhave a singlet $\\bm{K}=\\bm{T}+\\bm{J}_{L}=\\bm{0}$, where $\\bm{K}$ is called \nthe grand spin. The lowest-lying heavy baryons have the grand spin\n$K=0$, that is, we must have always $J_{L}=T$ with $Y'=(N_c-1)\/3$ for the\nground-state heavy baryons as shown in fig.~\\ref{fig:4}. \n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig4.pdf}\\qquad\n\\caption{The baryon anti-triplet has the  $J_{L}=T=0$ state with $Y'=2\/3$ whereas\n  the baryon sextet contains the  $J_{L}=T=1$ state with $Y'=2\/3$.}\n\\label{fig:4}\n\\end{figure}   \n\nAn observable of the heavy baryon can be expressed in general as a\nthree-point correlation function\n\\begin{align}\n\\langle B,\\,p'| J_\\mu(0) |B,\\,p\\rangle  &= \\frac1{\\mathcal{Z}}\n  \\lim_{T\\to\\infty} \\exp\\left(i p_4\\frac{T}{2} - i p_4'\n  \\frac{T}{2}\\right) \\int d^3x d^3y \\exp(-i \\bm{p}'\\cdot \\bm{y} + i\n  \\bm{p}\\cdot \\bm{x}) \\cr\n& \\hspace{-1cm} \\times \\int \\mathcal{D}U\\int \\mathcal{D} \\psi \\int\n  \\mathcal{D} \\psi^\\dagger J_{B}(\\bm{y},\\,T\/2) \\psi^\\dagger(0)\n  \\gamma_4 \\Gamma \\mathcal{O} \\psi(0) J_B^\\dagger (\\bm{x},\\,-T\/2)\n  \\exp\\left[-\\int d^4 z   \\psi^\\dagger iD(U) \\psi\\right],\n  \\label{eq:3corrftn}\n\\end{align}\nwhere $\\Gamma$ and $\\mathcal{O}$ represent respectively generic Dirac\nspin and flavor matrices. Computing Eq.~(\\ref{eq:3corrftn}), one can  \nstudy heavy baryonic observables such as form factors, magnetic moments,\naxial-vector constants, etc. For the detailed formalism, we refer to\nRefs.~\\cite{Kim:1995mr, Christov:1995vm}. \n\n\\section{Mass splittings  of the singly heavy baryons}\nWe first discuss the mass splittings of the singly heavy baryons. In\norder to obtain the mass splittings, one should include the\nsymmetry-breaking part of the collective\nHamiltonian~\\cite{Blotz:1992pw, Christov:1995vm} \n\\begin{align} \nH^{(1)}_{\\mathrm{sb}} \n&= \\overline{\\alpha} D^{(8)}_{88}+ \\beta \\hat{Y}\n+ \\frac{\\gamma}{\\sqrt{3}}\\sum_{i=1}^{3}D^{(8)}_{8i}\n\\hat{J}_{i},\n\\label{sb}\n\\end{align}\nwhere\n\\begin{align} \n\\overline{\\alpha} = \\left (-\\frac{\\overline{\\Sigma}_{\\pi N}}{3m_0}+\\frac{\n  K_{2}}{I_{2}}Y'  \n\\right )m_{\\mathrm{s}},\n \\;\\;\\;  \\beta=-\\frac{ K_{2}}{I_{2}}m_{\\mathrm{s}}, \n\\;\\;\\;  \\gamma = 2\\left (\n  \\frac{K_{1}}{I_{1}}-\\frac{K_{2}}{I_{2}}   \\right ) m_{\\mathrm{s}}.\n\\label{eq:alphaetc}\n\\end{align}\nThe parameters $\\overline{\\alpha}$, $\\beta$, and\n$\\gamma$ are the essential ones in determining the masses of the\nlowest-lying singly heavy baryons, which are expressed in terms of the\nmoments of inertia $I_{1,\\,2}$ and $K_{1,\\,2}$. However, we do not\nneed to fit them, since they are related to $\\alpha$,\n$\\beta$, and $\\gamma$ in the light-baryon sector. \nThe valence parts are only different from those in the light baryon\nsector by the color factor $N_c-1$. So, we need to replace $N_c$ by $N_c-1$\nin the valence parts of all the relevant dynamical parameters\ndetermined in the light-baryon sector. The valence part of\n$\\overline{\\Sigma}_{\\pi N}$ is just the $\\pi N$ sigma term with\ndifferent $N_c$ factor: $\\overline{\\Sigma}_{\\pi N} = (N_c-1)N_c^{-1}\n\\Sigma_{\\pi    N}$, where $\\Sigma_{\\pi N} = (m_u+m_d)\\langle N|\n\\bar{u} u +  \\bar{d} d|N\\rangle = (m_u+m_d) \\sigma$. On the other\nhand, the sea parts should be kept intact as in the light baryon\nsector. \n\nThe dynamical parameters $\\alpha$, $\\beta$ and $\\gamma$ have been\nfixed by using the experimental data on the baryon octet masses and\na part of the baryon decuplet and anti-decuplet masses with  \nisospin symmetry breaking effects~\\cite{Yang:2010id}. The\nvalues of $\\alpha$, $\\beta$, and $\\gamma$ have been obtained by\nthe $\\chi^2$ fit~\\cite{Yang:2010fm}\n\\begin{align}\n\\alpha  =  -255.03\\pm5.82 \\;{\\rm MeV},\\;\\;\\;\n\\beta  =  -140.04\\pm3.20  \\;{\\rm MeV},\\;\\;\\;\n\\gamma  =  -101.08\\pm2.33 \\;{\\rm MeV},\n\\label{eq:abrNumber}\n\\end{align}\nWhile $\\beta$ and $\\gamma$ are not required to be changed in the\nheavy-baryon sector, $\\alpha$ should be modified by\n\\begin{align}\n  \\label{eq:alpha}\n\\overline{\\alpha} = \\rho \\alpha,\n\\end{align}\nwhere $\\rho=(N_c-1)\/N_c$.\nHowever, there is a caveat when one uses the values of\nEq.~\\eqref{eq:abrNumber}. As mentioned above, only the valence parts\nshould be modified, while the scaling in Eq.~\\eqref{eq:alpha} changes\nthe sea part too.  To compensate this we choose $\\rho \\approx 0.9$. If\none computes the parameters $\\overline{\\alpha}$, $\\beta$, and $\\gamma$\nin a self-consistent way, we do not have this problem~\\cite{Kim:2018xlc}.\n\nConsidering the first-order perturbative corrections of $m_{\\mathrm{s}}$, \none can express the masses of the singly heavy baryons in\nrepresentation $\\mathcal{R}$ as  \n\\begin{align}\nM_{B,\\mathcal{R}}^Q = M_{\\mathcal{R}}^Q + M_{B,\\mathcal{R}}^{(1)}  \n\\label{eq:FirstOrderMass}\n\\end{align}\nwith\n\\begin{align}\nM_{\\mathcal{R}}^Q = m_Q + E_{\\mathrm{sym}}(p,q).  \n\\end{align}\nHere, $M_{\\mathcal{R}}^Q$ is the center mass of a heavy baryon in\nrepresentation $\\mathcal{R}$. $E_{\\mathrm{sym}}(p,q)$ is the\neigenvalue energy of the symmetric part of the collective Hamiltonian\ndefined in Eq.~\\eqref{eq:RotEn}. Note that the lower index $B$\ndesignates a certain baryon in a specific representation\n$\\mathcal{R}$. The upper index $Q$ denotes either the charm sector\n($Q=c$) or the bottom sector ($Q=b$). \nThen the center masses for the anti-triplet and sextet representations\nare obtained as \n\\begin{align}\nM_{\\overline{\\bm{3}}}^Q = M_{\\mathrm{cl}} + \\frac1{2I_2}, \\;\\;\\; \nM_{\\bm{6}}^Q = M_{\\overline{\\bm{3}}}^Q +  \\frac1{I_1},\n\\end{align}\nwhere $M_{\\mathrm{cl}}$ was defined in Eq.~\\eqref{eq:classical_mass}.\nThe second term in Eq.~(\\ref{eq:FirstOrderMass}), which arises from\nthe linear-order $m_{\\mathrm{s}}$ corrections, is proportional to the\nhypercharge of the soliton with the light-quark pair \n\\begin{align}\nM^{(1)}_{B,{\\cal{R}}} = \\langle B, {\\cal{R}} | H_{\\mathrm{sb}}^{(1)} \n| B, {\\cal{R}} \\rangle  = Y\\delta_{{\\cal{R}}},\n\\end{align}\nwhere\n \\begin{align} \n\\delta_{\\overline{\\bm{3}}}=\\frac{3}{8}\\overline{\\alpha}+\\beta, \\;\\;\\;\\;\n\\delta_{\\bm{6}}=\\frac{3}{20}\n  \\overline{\\alpha}+\\beta-\\frac{3}{10}\\gamma.  \n\\label{eq:deltas}\n\\end{align}\nFinally, we arrive at the expressions for the masses of the\nlowest-lying baryon anti-triplet and sextet as follows \n\\begin{align}\nM_{B,\\overline{\\bm{3}}}^Q = M_{\\overline{\\bm{3}}}^Q  +\n                     Y \\delta_{\\overline{\\bm{3}}} ,\\;\\;\\;\nM_{B,\\bm{6}}^Q =M_{\\bm{6}}^Q  + Y  \\delta_{\\bm{6}}, \n  \\label{eq:firstms}\n\\end{align}\nwith the linear-order $m_{\\mathrm{s}}$ corrections taken into account. \n\nSince the baryon sextet with spin 1\/2 and 3\/2 are degenerate, we need\nto remove the degeneracy by introducing the hyperfine spin-spin\ninteraction Hamiltonian~\\cite{Zeldovich}. Typically, the hyperfine\nHamiltonian is written as \n\\begin{align}\nH_{LQ} = \\frac{2}{3}\\frac{\\kappa}{m_{Q}\\,M_{\\mathrm{sol}}}\\bm{J}_L\n\\cdot  \\bm{J}_{Q}  \n= \\frac{2}{3}\\frac{\\varkappa}{m_{Q}}\n\\bm{J}_{L} \\cdot \\bm{J}_{Q}, \n\\label{eq:ssinter}\n\\end{align}\nwhere $\\kappa$ stands for the flavor-independent hyperfine coupling.\n$M_{\\mathrm{sol}}$ has been incorporated into an unknown \ncoefficient $\\varkappa$ that will be fixed by using the experimental\ndata. . The Hamiltonian $H_{LQ}$ does not affect the \n$\\overline{\\bm{3}}$ states with $J_L=0$. On the other hand, \nthe baryon sextet acquire additional contribution from $H_{LQ}$ which\nbring about the splitting between different spin states \n\\begin{align}\nM_{B,{\\bm{6}}_{1\/2}}^{Q} =  \nM_{B,\\bm{6}}^Q\\;-\\;\\frac{2}{3}\\frac{\\varkappa}{m_{Q}}, \n\\;\\;\\;\nM_{B,{\\bm{6}}_{3\/2}}^{Q}  = \nM_{B,\\bm{6}}^Q\\;+\\;\\frac{1}{3} \\frac{\\varkappa}{m_{Q}}, \n\\label{eq:Csextet}\n\\end{align}\nwhich leads to the splitting \n\\begin{align}\nM_{B,{\\bm{6}}_{3\/2}}^{Q}\\;-\\; M_{B,{\\bm{6}}_{1\/2}}^{Q}  =\n  \\frac{\\varkappa}{m_{Q}}  . \n\\label{eq:DCsextet}\n\\end{align}\nThe numerical values of $\\varkappa\/m_Q$ were determined by using the center\nvalues of the masses of the baryon sextet~\\cite{Yang:2016qdz} \n\\begin{align}\n\\frac{\\varkappa}{m_c} = (68.1\\pm 1.1)\\,\\mathrm{MeV},\\;\\;\\;\n\\frac{\\varkappa}{m_b} = (20.3\\pm 1.0)\\,\\mathrm{MeV}.\n  \\label{eq:kappavalue}\n\\end{align}\nNote that $\\varkappa$ is flavor-independent. So, knowing the ratio\n$m_c\/m_b$, one can extract the value of $\\varkappa$ from\nEq.~\\eqref{eq:kappavalue}.\n\nWe now present the numerical results of the masses of the heavy\nbaryons~\\cite{Yang:2016qdz}. Using the values of $\\overline{\\alpha}$,\n$\\beta$, and $\\gamma$, we can immediately determine the values of\n$\\delta_{\\overline{\\bm{3}}}$ and $\\delta_{\\bm{6}}$ defined in\nEq.~\\eqref{eq:deltas} \n\\begin{align}\n\\delta_{\\overline{\\bm{3}}} = (-203.8\\pm 3.5)\\,\\mathrm{MeV},\\;\\;\\;\n\\delta_{\\bm{6}} = (-135.2\\pm 3.3)\\,\\mathrm{MeV}.\n\\end{align}\nIncluding the results of $\\varkappa\/m_c$ and $\\varkappa\/m_b$, we can\nobtain the numerical results of the heavy baryon masses. \nIn Table~\\ref{tab:1} and Table~\\ref{tab:2} the numerical results of\nthe charmed and bottom baryon masses are presented, respectively. They\nare in good agreement with the experimental data taken from \nRef.~\\cite{PDG2017}. The mass of $\\Omega_b^*$ is still experimentally\nunknown. Thus, the prediction of its mass is given as \n\\begin{align}\nM_{\\Omega_b^*} = (6095.0\\pm 4.4)\\,\\mathrm{MeV}.  \n\\end{align}\nThe uncertainties in Tables~\\ref{tab:1} and \\ref{tab:2} are due to\nthose in $\\overline{\\alpha}$, $\\beta$, $\\gamma$, and $\\varkappa\/m_Q$. \n\\begin{table}[htp]\n\\begin{centering}\n\\begin{tabular}{c|ccc}\n\\hline \\hline\n$\\mathbf{\\mathcal{R}}_{J}^{Q}$ \n& $B_{c}$ \n& Mass \n& Experiment\n\\tabularnewline[0.1em]\n\\hline \n\\multirow{2}{*}{$\\mathbf{\\overline{3}}_{1\/2}^{c}$} \n& $\\Lambda_{c}$ \n& $2272.5 \\pm 2.3$\n& $2286.5 \\pm 0.1$\n\\tabularnewline\n& $\\Xi_{c}$ \n& $2476.3 \\pm 1.2$\n& $2469.4 \\pm 0.3$\n\\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{1\/2}^{c}$} \n& $\\Sigma_{c}$ \n& $2445.3 \\pm 2.5$\n& $2453.5 \\pm 0.1$\n\\tabularnewline\n& $\\Xi_{c}^{\\prime}$ \n& $2580.5 \\pm 1.6$\n& $2576.8 \\pm 2.1$\n\\tabularnewline\n& $\\Omega_{c}$ \n& $2715.7 \\pm 4.5$\n& $2695.2 \\pm 1.7$\n\\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{3\/2}^{c}$} \n& $\\Sigma_{c}^{\\ast}$ \n& $2513.4 \\pm 2.3$\n& $2518.1 \\pm 0.8$\n\\tabularnewline\n& $\\Xi_{c}^{\\ast}$ \n& $2648.6 \\pm 1.3$\n& $2645.9 \\pm 0.4$\n\\tabularnewline\n& $\\Omega_{c}^{\\ast}$ \n& $2783.8 \\pm 4.5$\n& $2765.9 \\pm 2.0$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\par\\end{centering}\n\\caption{The numerical results of the charmed baryon masses in \n  comparison with the experimental data~\\cite{PDG2017}.}\n\\label{tab:1}\n\\end{table}\n\n\\begin{table}[htp]\n\\centering{}\\vspace{3em}%\n\\begin{tabular}{c|ccc}\n\\hline\\hline \n$\\mathbf{\\mathcal{R}}_{J}^{Q}$ \n& $B_{b}$ \n& Mass\n& Experiment\n\\\\\n\\hline \n\\multirow{2}{*}{$\\mathbf{\\overline{3}}_{1\/2}^{b}$} \n& \\textcolor{black}{$\\Lambda_{b}$} \n& $5599.3 \\pm 2.4 $\n& $5619.5 \\pm 0.2$ \n \\tabularnewline\n& \\textcolor{black}{$\\Xi_{b}$} \n& $5803.1 \\pm 1.2 $\n& $5793.1 \\pm 0.7 $  \n \\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{1\/2}^{b}$} \n& \\textcolor{black}{$\\Sigma_{b}$} \n& $5804.3 \\pm 2.4 $\n& $5813.4 \\pm 1.3$  \n\\tabularnewline\n& \\textcolor{black}{$\\Xi_{b}^{\\prime}$} \n& $5939.5 \\pm 1.5 $\n& $5935.0 \\pm 0.05$ \n\\tabularnewline\n& \\textcolor{black}{$\\Omega_{b}$} \n& $6074.7 \\pm 4.5 $\n& $6048.0 \\pm 1.9$  \n \\tabularnewline\n\\hline \n\\multirow{3}{*}{$\\mathbf{6}_{3\/2}^{b}$} \n& \\textcolor{black}{$\\Sigma_{b}^{\\ast}$} \n& $5824.6 \\pm 2.3 $\n& $5833.6 \\pm 1.3$  \n\\tabularnewline\n&\\textcolor{black}{$\\Xi_{b}^{\\ast}$} \n& $5959.8 \\pm 1.2 $\n& $5955.3 \\pm 0.1$ \n \\tabularnewline\n& \\textcolor{black}{$\\Omega_{b}^{\\ast}$} \n& $6095.0 \\pm 4.4 $\n& $-$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\caption{The results of the masses of the bottom baryons in comparison\nwith the experimental data~\\cite{PDG2017}.}\n\\label{tab:2}\n\\end{table}\n\n\\section{Magnetic moments of heavy baryons} \nIn this Section, we briefly summarize a recent work on the magnetic\nmoments of the heavy baryons~\\cite{Yang:2018uoj}.\nStarting from Eq.~\\eqref{eq:3corrftn}, one can derive the general\nexpressions of the collective operator for the magnetic moments   \n\\begin{align}\n  \\label{eq:MagMomOp}\n \\hat{\\mu} = \\hat{\\mu}^{(0)}  + \\hat{\\mu}^{(1)}, \n\\end{align}\nwhere $\\hat{\\mu}^{(0)}$ and $\\hat{\\mu}^{(1)}$ denote the leading\nand rotational $1\/N_c$ contributions, and the linear $m_{\\mathrm{s}}$\ncorrections respectively \n\\begin{align}\n\\hat{\\mu}^{(0)} & =  \n\\;\\;w_{1}D_{\\mathcal{Q}3}^{(8)}\n\\;+\\;w_{2}d_{pq3}D_{\\mathcal{Q}p}^{(8)}\\cdot\\hat{J}_{q}\n\\;+\\;\\frac{w_{3}}{\\sqrt{3}}D_{\\mathcal{Q}8}^{(8)}\\hat{J}_{3},\\cr\n\\hat{\\mu}^{(1)} & =  \n\\;\\;\\frac{w_{4}}{\\sqrt{3}}d_{pq3}D_{\\mathcal{Q}p}^{(8)}D_{8q}^{(8)}\n+w_{5}\\left(D_{\\mathcal{Q}3}^{(8)}D_{88}^{(8)}+D_{\\mathcal{Q}8}^{(8)}D_{83}^{(8)}\\right)\n\\;+\\;w_{6}\\left(D_{\\mathcal{Q}3}^{(8)}D_{88}^{(8)}-D_{\\mathcal{Q}8}^{(8)}D_{83}^{(8)}\\right).\n\\label{eq:magop}\n\\end{align}\n $d_{pq3}$ is the SU(3) symmetric tensor of which the indices run over \n$p=4,\\cdots,\\,7$. $\\hat{J_3}$ and $\\hat{J}_{p}$ denote the third\nand the $p$th components of the spin operator acting on the soliton\nwith the light-quark pair. $D_{\\mathcal{Q}3}^{(8)}$ arises from the\nrotation of the electromagnetic current  \n\\begin{align}\nD_{\\mathcal{Q}3}^{(8)} = \\frac12 \\left( D_{33}^{(8)} + \\frac1{\\sqrt{3}}\n  D_{83}^{(8)}\\right).\n\\end{align}\nThe coefficients $w_i$ in Eq.~\\eqref{eq:magop} are independent of\nbaryons involved, which encode the interaction of light quarks with\nthe electromagnetic current. Each term has a physical meaning: $w_1$\nrepresents the leading-order contribution, a part of the rotational\n$1\/N_c$ corrections, and linear $m_{\\mathrm{s}}$ corrections, whereas\n$w_2$ and $w_3$ describe the rest of the rotational $1\/N_c$\ncorrections. $w_1$ includes the $m_s$-dependent term, which is not\nexplicitly involved in the breaking of flavor SU(3) symmetry. So, we\nneed to treat $w_1$ as if it had contained the SU(3) symmetric part.\n On the other hand, $w_4$, $w_5$, and $w_6$ are the\nSU(3) symmetry breaking terms. There are yet another $m_{\\mathrm{s}}$\ncorrections, which arise from the collective wave functions. Though \n$w_i$ can be determined within a specific chiral solitonic model such\nas the $\\chi$QSM~\\cite{Kim:1995mr, Kim:1995ha}, we will use the values\nof $w_i$, which have been already fixed from the experimental data on\nthe magnetic moments of the baryon octet. \n\nThe baryon wave function given in Eq.~\\eqref{eq:waveftn} is not enough\nto compute the magnetic moments, because the collective wave functions\nshould be revised when the perturbation coming from the strange\ncurrent quark mass is considered. In this case, the baryon is no more\nin a pure state but is mixed with higher representations. \nIn Ref.~\\cite{Kim:2018nqf}, the collective baryon wave functions for\nthe heavy baryons have been already derived. Those for the baryon\nanti-triplet ($J_{L}=0$) and the sextet ($J_{L}=1$) are expressed respectively\nas~\\cite{Kim:2018nqf}    \n\\begin{align}\n&|B_{\\overline{\\bm3}_{0}}\\rangle = |\\overline{\\bm3}_{0},B\\rangle + \np^{B}_{\\overline{15}}|\\overline{\\bm{15}}_{0},B\\rangle, \\cr\n&|B_{\\bm6_{1}}\\rangle = |{\\bm6}_{1},B\\rangle +\n  q^{B}_{\\overline{15}}|{\\overline{\\bm{15}}}_{1},B \n\\rangle + q^{B}_{\\overline{24}}|{\n{\\overline{\\bm{24}}}_{1}},B\\rangle,\n\\label{eq:mixedWF1}\n\\end{align}\nwith the mixing coefficients\n\\begin{eqnarray}\np_{\\overline{15}}^{B}\n\\;\\;=\\;\\;\np_{\\overline{15}}\\left[\\begin{array}{c}\n-\\sqrt{15}\/10\\\\\n-3\\sqrt{5}\/20\n\\end{array}\\right], \n& \nq_{\\overline{15}}^{B}\n\\;\\;=\\;\\;\nq_{\\overline{15}}\\left[\\begin{array}{c}\n\\sqrt{5}\/5\\\\\n\\sqrt{30}\/20\\\\\n0\n\\end{array}\\right], \n& \nq_{\\overline{24}}^{B}\n\\;\\;=\\;\\;\nq_{\\overline{24}}\\left[\\begin{array}{c}\n-\\sqrt{10}\/10\\\\\n-\\sqrt{15}\/10\\\\\n-\\sqrt{15}\/10\n\\end{array}\\right],\n\\label{eq:pqmix}\n\\end{eqnarray}\nrespectively, in the basis $\\left[\\Lambda_{Q},\\;\\Xi_{Q}\\right]$ for\nthe anti-triplet and $\\left[\\Sigma_{Q}\\left(\\Sigma_{Q}^{\\ast}\\right),\\;\n  \\Xi_{Q}^{\\prime}\\left(\\Xi_{Q}^{\\ast}\\right),\\;\\Omega_{Q}\n  \\left(\\Omega_{Q}^{\\ast}\\right)\\right]$ for the sextets. The\nparameters $p_{\\overline{15}}$, $q_{\\overline{15}}$, and\n$q_{\\overline{24}}$ are written by \n\\begin{eqnarray}\np_{\\overline{15}}\n\\;\\;=\\;\\;\n\\frac{3}{4\\sqrt{3}}\\overline{\\alpha}{I}_{2}, \n& \nq_{\\overline{15}}\n\\;\\;=\\;\\;\n{\\displaystyle -\\frac{1}{\\sqrt{2}}\n\\left(\\overline{\\alpha}+\\frac{2}{3}\\gamma\\right)\nI_{2}}, \n& \nq_{\\overline{24}}\\;\\;=\\;\\;\n\\frac{4}{5\\sqrt{10}}\n\\left(\\overline{\\alpha}-\\frac{1}{3}\\gamma\\right)\nI_{2}.\n\\label{eq:pqmix2}\n\\end{eqnarray}\n Combining\nEq.~\\eqref{eq:mixedWF1} with the heavy-quark spinor as in\nEq.~\\eqref{eq:waveftn}, one can construct the collective wave\nfunctions for the heavy baryon states~\\cite{Yang:2018uoj}. \n\nComputing the baryon matrix elements of $\\hat{\\mu}$ in\nEq.~\\eqref{eq:MagMomOp}, we get the magnetic moments of the \nheavy baryons  \n\\begin{equation}\n\\mu_{B}=\\mu_{B}^{(0)}+\\mu_{B}^{(\\mathrm{op})}+\\mu_{B}^{(\\mathrm{wf})}\n\\label{eq:mu_B}\n\\end{equation}\nwhere $\\mu_{B}^{(0)}$ is the part of the magnetic moment in\nthe chiral limit and $\\mu_{B}^{(\\mathrm{op})}$ comes from\n$\\hat{\\mu}^{(1)}$ in Eq.~\\eqref{eq:MagMomOp}, which include $w_4$,\n$w_5$, and $w_6$. $\\mu_{B}^{(\\mathrm{wf})}$ is derived from the\ninterference between the $\\mathcal{O}(m_{\\mathrm{s}})$ and\n$\\mathcal{O}(1)$ parts of the collective wave functions in\nEq.~\\eqref{eq:mixedWF1}.   \n\nSince the soliton with the light-quark pair for the baryon\nanti-triplet has spin $J_L=0$, , the magnetic moments of the baryon\nanti-triplet vanish. In this case $1\/m_Q$ contributions are the\nleading ones. However, we will not include them, since we need to go\nbeyond the mean-field approximation to consider the $1\/m_Q$\ncontributions within the present framework. \n\nSince $w_1$ contains both the leading-order contributions and the \n$1\/N_c$ rotational corrections, we have to decompose them. Following\nthe argument of Ref.~\\cite{Kim:2017khv}, we can separately consider\neach contribution. The coefficients $w_1$, $w_2$, and $w_3$ are\nexpressed in terms of the model dynamical parameters  \n\\begin{align}\n  \\label{eq:w123}\nw_{1}  =  \nM_{0}\\;-\\;\\frac{M_{1}^{\\left(-\\right)}}{I_{1}^{\\left(+\\right)}},\\;\\;\\;\nw_{2}  = -2\\frac{M_{2}^{\\left(-\\right)}}{I_{2}^{\\left(+\\right)}},\\;\\;\\;\nw_{3}  = -2\\frac{M_{1}^{\\left(+\\right)}}{I_{1}^{\\left(+\\right)}},\n\\end{align}\nwhere the explicit forms of $M_0$, $M_1^{(\\pm)}$, $M_2^{(-)}$ are\ngiven in Refs.~\\cite{Kim:1995mr, Praszalowicz:1998j}. $I_1^{(+)}$ and \n$I_2^{(+)}$ are the moments of inertia with the notation of\nRef.~\\cite{Praszalowicz:1998j} taken. In the limit of the small soliton\nsize, the parameters in Eq.~\\eqref{eq:w123} can be simplified as  \n\\begin{align}\nM_{0}\\;\\rightarrow\\;-2N_{c}K,\n\\;\\;\\;\n\\frac{M_{1}^{\\left(-\\right)}}{I_{1}^{\\left(+\\right)}}\n\\;\\rightarrow\\;\\frac{4}{3}K, \\;\\;\\;\n   \\frac{M_{1}^{\\left(+\\right)}}{I_{1}^{\\left(+\\right)}}\n  \\;\\rightarrow\\;-\\frac{2}{3}K,\\;\\;\\;\n\\frac{M_{2}^{\\left(-\\right)}}{I_{2}^{\\left(+\\right)}}\n\\;\\rightarrow\\;-\\frac{4}{3}K.\n\\label{eq:sss}  \n\\end{align} \nThese results yield the expressions of the magnetic moments in the\nnonrelativistic (NR) quark model. For example, the ratio of the proton\nand magnetic moments can be correctly obtained as\n$\\mu_p\/\\mu_n=-3\/2$. In the NR limit, we also derive the relation\n$M_{1}^{\\left(-\\right)}\\;=\\;-2M_{1}^{\\left(+\\right)}$. Furthermore, we\nhave to assume that this relation can be also applied to the \ncase of the realistic soliton size. Then, we can write the \nleading-order contribution $M_0$ in terms of $w_1$ and $w_3$\n\\begin{align}\n  \\label{eq:4}\nM_0= w_1 + w_3.  \n\\end{align}\nSince a heavy baryon constitutes $N_c-1$ valence quarks, the original\n$M_0$ is modified by introducing $(N_c-1)\/N_c$. As mentioned\npreviously, only the valence part of $M_0$ should be changed by this\nscaling factor. Since, however, we have determined the values of\n$w_i$ using the experimental data, we can not fix separately the\nvalence and sea parts. Thus, we introduce an additional scaling factor\n$\\sigma$ to express a new coefficient $\\tilde{w}_1$ \n\\begin{align}\n\\label{eq:w1tilde}\n\\tilde{w}_1 = \\left[\\frac{N_c-1}{N_c} (w_1+w_3) - w_3\\right] \\sigma. \n\\end{align}\n$\\sigma$ compensates also possible deviations from the NR relation\n$M_{1}^{\\left(-\\right)}\\;=\\;-2M_{1}^{\\left(+\\right)}$ assumed to be \nvalid in the realistic soliton case. The value of $\\sigma$ is taken to\nbe $\\sigma\\sim0.85$. \n\nConsidering the scaling parameters, we are able to determine the\nfollowing values for $w_i$\n\\begin{align}\n\\tilde{w_{1}} \n& =  \n-10.08\\pm0.24,\n\\cr\nw_{2} & =  4.15\\pm0.93,\n\\cr\nw_{3} & =  8.54\\pm0.86,\n\\cr\n\\overline{w}_{4} & =  -2.53\\pm0.14,\n\\cr\n\\overline{w}_{5} & =  -3.29\\pm0.57,\n\\cr\n\\overline{w}_{6} & =  -1.34\\pm0.56.\n\\label{eq:numW}\n\\end{align}\n\nBefore we carry on the calculation of the magnetic moments, we examine\nthe general relations between them. First, we find the generalized\nColeman and Glashow relations~\\cite{Coleman:1961jn}, which arise from \nthe isospin invariance\n\\begin{align}\n\\mu(\\Sigma_{c}^{++})\\;-\\;\\mu(\\Sigma_{c}^{+})\n& =  \n\\mu(\\Sigma_{c}^{+})\\;-\\;\\mu(\\Sigma_{c}^{0}),\n\\cr\n\\mu(\\Sigma_{c}^{0})\\;-\\;\\mu(\\Xi_{c}^{\\prime0}) \n& =  \n\\mu(\\Xi_{c}^{\\prime0})\\;-\\;\\mu(\\Omega_{c}^{0}),\n\\cr\n2 [\\mu(\\Sigma_{c}^{+})\\,-\\,\\mu(\\Xi_{c}^{\\prime0})]\n& =  \n\\mu(\\Sigma_{c}^{++})\\,-\\,\\mu(\\Omega_{c}^{0}).\n\\label{eq:coleman}  \n\\end{align}\nSimilar relations were also found in Ref.~\\cite{Banuls:1999mu}.\nHowever, there is one very important difference. While the \nColeman-Glashow relations are known to be valid in the chiral \nlimit, the relations in Eq.~\\eqref{eq:coleman} are justified \neven when the effects of SU(3) flavor symmetry breaking are\nconsidered. We also find the relation according to\nthe $U$-spin symmetry\n\\begin{align}\n\\mu(\\Sigma_{c}^{0})\\;=\\;\n\\mu(\\Xi_{c}^{\\prime0})\\;=\\;\n\\mu(\\Omega_{c}^{0})\\;=\\;\n-2\\mu(\\Sigma_{c}^{+})\\;=\\;\n-2\\mu(\\Xi_{c}^{\\prime+})\\;=\\;\n-\\frac{1}{2}\\mu (\\Omega_{c}^{0}),\n\\label{eq:Usym}\n\\end{align}\nwhich are only valid in the SU(3) symmetric case. \nWe derive also the sum rule given as    \n\\begin{align}\n\\sum_{B_c\\in\\mathrm{sextet}}\\mu(B_c)\\;=\\;0\n\\label{eq:sum}\n\\end{align}\nin the SU(3) symmetric case. \n\n\\begin{table}[htp]\n\\caption{Numerical results of the magnetic moments for the charmed\n  baryon sextet with $J=1\/2$ in units of the nuclear magneton $\\mu_N$.} \n\\renewcommand{\\arraystretch}{1.3\n\\begin{tabular}{ccc}\n\\hline \\hline\n$\\mu\\left[6_{1}^{1\/2},\\;B_{c}\\right]$ \n& $\\mu^{(0)}$ \n& $\\mu^{(\\text{total})}$ \n\\tabularnewline \\hline\n$\\Sigma_{c}^{\\text{++}}$ \n& $2.00\\pm0.09$\n& $2.15\\pm0.1$ \n\\tabularnewline\n$\\Sigma_{c}^{\\text{+}}$ \n& $0.50\\pm0.02$ \n& $0.46\\pm0.03$ \n\\tabularnewline\n$\\Sigma_{c}^{0}$ \n& -$1.00\\pm0.05$ \n& -$1.24\\pm0.05$ \n\\tabularnewline\n\\hline \n$\\Xi_{c}^{\\prime+}$ \n& $0.50\\pm0.02$ \n& $0.60\\pm0.02$ \n\\tabularnewline\n$\\Xi_{c}^{\\prime0}$ \n& -$1.00\\pm0.05$ \n& -$1.05\\pm0.04$ \n\\tabularnewline\n\\hline \n$\\Omega_{c}^{0}$ \n& -$1.00\\pm0.05$ \n& -$0.85\\pm0.05$ \n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:3}\n\\end{table}\n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Numerical results of magnetic moments for charmed baryon\n  sextet with $J=3\/2$ in units of the nuclear magneton $\\mu_N$.} \n\\begin{tabular}{ccc}\n\\hline \\hline \n$\\mu\\left[6_{1}^{3\/2},\\;B_{c}\\right]$ \n& $\\mu^{(0)}$ \n& $\\mu^{(\\text{total})}$ \n\\tabularnewline\n\\hline \n$\\Sigma_{c}^{\\ast\\text{++}}$ \n& $3.00\\pm0.14$ \n& $3.22\\pm0.15$ \n\\tabularnewline\n$\\Sigma_{c}^{\\ast\\text{+}}$ \n& $0.75\\pm0.04$ \n& $0.68\\pm0.04$ \n\\tabularnewline\n$\\Sigma_{c}^{\\ast0}$ \n& $-1.50\\pm0.07$ \n& $-1.86\\pm0.07$ \n\\tabularnewline\n\\hline \n$\\Xi_{c}^{\\ast+}$ \n& $0.75\\pm0.04$ \n& $0.90\\pm0.04$ \n\\tabularnewline\n$\\Xi_{c}^{\\ast0}$ \n& $-1.50\\pm0.07$ \n& $-1.57\\pm0.06$ \n\\tabularnewline\n\\hline \n$\\Omega_{c}^{\\ast0}$ \n& -$1.50\\pm0.07$ \n& -$1.28\\pm0.08$ \n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:4}\n\\end{table}\nIn Tables~\\ref{tab:3} and \\ref{tab:4}, we list the numerical results\nof the charmed baryon sextet with spin 1\/2 and 3\/2, respectively. We\nobtain exactly the same results for the bottom baryons because of the\nheavy-quark symmetry in the $m_Q\\to\\infty$ limit. In\nRef.~\\cite{Yang:2018uoj}, a detailed discussion can be found, the\npresent results being compared with those from many other models. \n\\section{Excited $\\Omega_c$ baryons}\nThe present mean-field approach was applied to the classification of\nthe excited $\\Omega_c^0$'s that were recently reported by the LHCb\nCollaboration~\\cite{Aaij:2017nav}. The masses and decay widths of the\n$\\Omega_c^0$'s, which were reported by the LHCb Collaboration, are\nlisted in Table~\\ref{tab:5}. The Belle Collaboration has confirmed the\nfour of them~\\cite{Yelton:2017qxg} (see Table~\\ref{tab:6}). The Belle\ndata unambiguously confirmed the existence of the $\\Omega_c(3066)$ and\n$\\Omega_c(3090)$, and $\\Omega_c(3000)$ and $\\Omega_c(3050)$ are also\nconfirmed with reasonable significance. On the other hand the narrow\nresonance $\\Omega_c(3119)$ was not seen in the Belle experiment but\nthe nonobservation of $\\Omega_c(3119)$ is not in disagreement because\nit is due to the small yield. \n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Experimental data on the five $\\Omega_c^0$ baryons reported by the\n  LHCb Collaboration~\\cite{Aaij:2017nav}.} \n\\begin{tabular}{ccc}\n\\hline \\hline \nResonance\n& Mass (MeV)\n& Decay width  (MeV)\n\\tabularnewline\n\\hline \n $\\Omega_c(3000)^0$\n& $3000.4\\pm 0.2\\pm0.1_{-0.5}^{+0.3}$ \n& $4.5\\pm 0.6\\pm 0.3$\n\\tabularnewline\n $\\Omega_c(3050)^0$\n& $3050.2\\pm0.1\\pm0.1_{-0.5}^{+0.3}$\n& $0.8\\pm 0.2\\pm 0.1$\n\\tabularnewline\n $\\Omega_c(3066)^0$\n& $3065.6\\pm 0.1\\pm 0.3_{-0.5}^{+0.3}$ \n& $3.5\\pm0.4\\pm0.2$\n\\tabularnewline\n $\\Omega_c(3090)^0$\n& $3090.2\\pm 0.3\\pm 0.5_{-0.5}^{+0.3}$\n& $8.7\\pm 1.0 \\pm 0.8$\n\\tabularnewline\n $\\Omega_c(3119)^0$\n& $3119.1\\pm 0.3 \\pm 0.9_{-0.5}^{+0.3}$\n& $1.1 \\pm 0.8 \\pm 0.4$\n\\tabularnewline\n$\\Omega_{c}(3188)$ \n& $3188\\pm 5 \\pm 13$\n& $60\\pm 15 \\pm 11$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:5}\n\\end{table}\n\n\\begin{table}[htp]\n\\renewcommand{\\arraystretch}{1.3\n\\caption{Experimental data on the four $\\Omega_c^0$ baryons reported by the\n  Belle Collaboration~\\cite{Yelton:2017qxg}.}\n\\begin{tabular}{cc}\n\\hline \\hline \nResonance\n& Mass (MeV)\n\\tabularnewline\n\\hline \n $\\Omega_c(3000)^0$\n& $3000.7\\pm 1.0\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3050)^0$\n& $3050.2\\pm0.4\\pm0.2$\n\\tabularnewline\n $\\Omega_c(3066)^0$\n& $3064.9\\pm 0.6\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3090)^0$\n& $3089.3\\pm 1.2\\pm 0.2$\n\\tabularnewline\n $\\Omega_c(3119)^0$\n& --\n\\tabularnewline\n$\\Omega_{c}(3188)$ \n& $3199\\pm 9 \\pm 4$\n\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\label{tab:6}\n\\end{table} \n\nWhen one examines the excited heavy baryons in the present work, we\nneed to consider states with the grand spin $K=1$. Since we have the\nquantization rule $\\bm{K}=\\bm{J}_L+\\bm{T}$, the possible values of the\nspin are determined by \n\\begin{align}\nJ_L = |K-T|,\\cdots, K+T.  \n\\end{align}\nThus, In the case of $T=0$ which corresponds to the anti-triplet with\n$Y'=2\/3$, we must have $J_L=1$ because of $K=1$. Combining it with the\nheavy-quark spin 1\/2, we have \\textit{two} excited baryon\nanti-triplet. Similarly, $T=1$ corresponds to the sextet. In this\ncase $J_L$ can have the values of 0, 1, and 2. Being coupled with the\nheavy-quark spin $1\/2$, we get \\textit{five} excited baryon sextets:\n$(1\/2)$, $(1\/2,\\,3\/2)$, and $(3\/2,\\,5\/2)$, corresponding to $J_L=0$, and\n$J_L=1$, and $J_L=2$. In each sextet representation, we have a\nisosinglet $\\Omega_c^0$. Thus, is is natural to think that the newly\nfound five $\\Omega_c^0$'s are those in the excited baryon sextets.   \nNote that the representations for each value of $J$ are degenerate in\nthe limit of $m_Q\\to\\infty$. So, we need to introduce an additional\nhyperfine spin-spin interaction as done for the ground-state baryon\nsextet\n\\begin{align}\nH_{LQ}  = \\frac23 \\frac{\\varkappa'}{m_Q} \\bm{J}_L \\cdot \\bm{J}_Q,\n\\end{align}\nwhich is very similar to Eq.~\\eqref{eq:ssinter}. $\\varkappa'$ can be\nfixed by using the experimental data on the masses of the excited\nbaryon anti-triplet.\n\nFollowing Refs.~\\cite{Diakonov:2012zz, Diakonov:2013qta}, we revise the\neigenvalues of the symmetric Hamiltonian for the excited baryons\n($K\\neq 0$) as \nfollows \n\\begin{align}\nM_{\\mathcal{R}}^{(K)\\prime} &=  M_{\\mathrm{cl}}^{(K)\\prime} + \\frac1{2I_2} \\left[\n  C_2(\\mathcal{R}) - T (T+1) - \\frac34 Y^{\\prime 2} \\right]  \\cr\n& + \\frac1{2I_1} \\left[(1-a_K) T(T+1) + a_K J_L(J_L+1) - a_K(1-a_K) K(K+1) \n  \\right], \n\\label{eq:symmass}\n\\end{align}\nwhere $C_2(\\mathcal{R})$ is the eigenvalue of the SU(3) Casimir\noperator, which was already defined in Eq.~\\eqref{eq:Casimir}.  The\nparameter $a_K$ is related to one-quark excitation. \nThe collective wave functions for the soliton are derived as  \n\\begin{align}\n\\Phi_{B,J_L, J_{L3},(T,K)}^{\\mathcal{R}} = \\sqrt{\\frac{2J_L+1}{2K+1}} \\sum_{T_3\n  J_{L3}' K_3'} C_{TT_3J_LJ_{L3}'}^{KK_3}  (-1)^{(T+T_3)}\n  \\Psi_{(\\mathcal{R^*};-Y'TT_3)}^{(\\mathcal{R};B)} D_{J_{L3}'J_{L3}}^{(J_L)*}(S)\n  \\chi_{K_3'}, \n\\end{align}\nwhere index $(\\mathcal{R};YTT_3)$ denotes the SU(3) quantum numbers of\na corresponding baryon in representation $\\mathcal{R}$, and\n$(\\mathcal{R}^*;-Y'TT_3)$ is attached to a fixed value of $Y'$ and\nis formally given in a conjugate representation to $\\mathcal{R}$. The\nfunction $D^{(J_L)}$ represents the SU(2) Wigner $D$ function and\n$\\chi_{K_3}$ is the spinor corresponding to $K$ and $K_3$. The \nwave function for the excited baryons can be constructed by coupling\n$\\Phi_{B,J_L, J_{L3},(T,K)}^{\\mathcal{R}}$ with the heavy-quark\nspinor.  \n\nThe SU(3) symmetry-breaking Hamiltonian in Eq.~\\eqref{sb} also needs to\nbe extended to describe the mass splittings of the excited heavy\nbaryons \n\\begin{align}\nH_{\\mathrm{sb}}^{(K)} = \\overline{\\alpha} D_{88}^{(8)} + \\beta \\hat{Y}\n  + \\frac{\\gamma}{\\sqrt{3}} \\sum_{i=1}^3 D_{8i}^{(8)} \\hat{T}_i +\n  \\frac{\\delta}{\\sqrt{3}} \\sum_{i=1}^3 D_{8i}^{(8)} \\hat{K}_i. \n\\label{eq:excitedsu3br}\n\\end{align}\nThe additional parameter $\\delta$ can be determined by using the mass\nspectrum of excited baryons. \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig5.pdf}\\qquad\n\\caption{Schematic picture of the first excited heavy baryons. A\n  possible excitation of a quark from the Dirac sea to the valence\n  level might have $K^P=1^-$.} \n\\label{fig:5}\n\\end{figure}\nAs shown in Fig.~\\ref{fig:5}, the transition from a $K^P=1^-$\nDirac-sea level to an unoccupied $K^P=0^+$ state may correspond to the\nfirst excited heavy baryons~\\cite{Diakonov:2010tf}. Note that such a transition\nis only allowed in the heavy-baryon sector, not in the light-baryon\nsector. As discussed already, there are two baryon anti-triplets and\nfive baryon sextets. From Eq.~\\eqref{eq:symmass}, we can derive the\nfollowing expressions\n\\begin{align}\nM_{\\overline{\\bm{3}}}^{\\prime} &= M_{\\mathrm{cl}}^{\\prime} +\n  \\frac1{2I_2} + \\frac1{I_1} (a_1^2),\\cr\nM_{\\bm{6}}^{\\prime} &= M_{\\overline{\\bm{3}}}^{\\prime} +\n                           \\frac{1-a_1}{I_1} + \\frac{a_1}{I_1}\\times\n                          \\left\\{\n                           \\begin{array}{ll} \n-1 & \\mbox{ for $J_{L}=0$} \\\\\n0 & \\mbox{ for $J_{L}=1$} \\\\\n2 & \\mbox{ for $J_{L}=2$}\n                           \\end{array} \\right. .\n  \\label{eq:excitedM3-6}\n\\end{align}\nConsidering the SU(3) symmetry breaking from\nEq.~\\eqref{eq:excitedsu3br}, we find the splitting parameters for the\n$\\overline{\\bm{3}}$ and $\\bm{6}$ \n\\begin{align}\n\\delta_{\\overline{\\bm{3}}}' &= \\frac38 \\overline{\\alpha} + \\beta =\n  \\delta_{\\overline{\\bm{3}}} = -180\\,\\mathrm{MeV},\\cr\n \\delta_{\\bm{6}J_{L}}' &= \\delta_{\\bm{6}} -\\frac{3}{20}\\delta \\times\n                     \\left\\{\n \\begin{array}{ll} \n-1 & \\mbox{ for $J_{L}=0$} \\\\\n0 & \\mbox{ for $J_{L}=1$} \\\\\n2 & \\mbox{ for $J_{L}=2$}\n                           \\end{array} \\right.,\n\\label{eq:excitedeltas}\n\\end{align}\nwhere we see that $\\delta_{\\overline{\\bm{3}}}'$ is just the same as\n$\\delta_{\\overline{\\bm{3}}}$ given in\nEq.~\\eqref{eq:deltas}. $\\delta_{\\bm{6}}$ is given as $-120$\nMeV. Though we do not know the numerical value of the new parameter\n$\\delta$, we still can analyze the mass splittings of the newly found\n$\\Omega_c$'s, using the splittings between the states with different\nvalues of $J_{L}$. \n\nWe now turn to the hyperfine splittings. The two anti-triplets of spin\n1\/2 and 3\/2 and the two sextets of spin 1\/2 and 3\/2 are split by \n\\begin{align}\n\\Delta_{\\overline{\\bm{3}}}^{\\mathrm{hf}} =\n  \\Delta_{\\bm{6}J_{L}=1}^{\\mathrm{hf}} = \\frac{\\varkappa'}{m_c},    \n\\end{align}\nwhereas another two sextets of spin 3\/2 and 5\/2 are split by\n\\begin{align}\n  \\Delta_{\\bm{6}J_{L}=2}^{\\mathrm{hf}} = \\frac53 \\frac{\\varkappa'}{m_c}.    \n\\label{eq:hf6}\n\\end{align}\nOne sextet of spin 1\/2 from the $J_{L}=0$ case has no hyperfine\nsplitting. The results are depicted in Fig.~\\ref{fig:6}.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.2]{fig6.pdf}\\qquad\n\\caption{Mass splitting of the five excited sextets.}\n\\label{fig:6}\n\\end{figure}\nNote that the $\\Delta_1$ represent the splittings\nbetween the $J_{L}=0$ state and the degenerate $J_{L}=1$ state, whereas\n$\\Delta_2$ denote those between degenerate $J_{L}=1$ and $J_{L}=2$ states \n\\begin{align}\n\\Delta_1 = \\frac{a_1}{I_1} + \\frac{3}{20} \\delta,\\;\\;\\; \\Delta_2 =\n  2\\Delta_1.  \n\\label{eq:Jsplit}\n\\end{align}\nWe will soon see that the relation $\\Delta_1=2\\Delta_2$ will play an\ncritical role \nin identifying the excited $\\Omega_c$'s within the $\\chi$QSM. \n\nIf one identifies $\\Lambda_c(2592)$ and $\\Xi_c(2790)$ as the members\nof the excited baryon anti-triplet of spin $(1\/2)^-$ with negative\nparity, and $\\Lambda_c(2592)$ and $\\Xi_c(2790)$ as those of the\nexcited baryon anti-triplet of spin $(3\/2)^-$, then we find\n$\\delta_{\\overline{\\bm{3}}}=-198$ and $-190$ MeV, which are more or\nless in agreement with the value given in\nEq.~\\eqref{eq:excitedeltas}. The $\\varkappa'\/m_c$ can be also\ndetermined as \n\\begin{align}\n\\frac{\\varkappa'}{m_c} = \\frac13 (M_{\\Lambda_c(2628)} + 2\n  M_{\\Xi_c(2818)}) - \\frac13 (M_{\\Lambda_c(2592)} + 2 M_{\\Xi_c(2790)})\n  = 30\\,\\mathrm{MeV}, \n\\label{eq:exhfvalue}\n\\end{align}\nand $M_{\\overline{\\bm{3}}}$ is also fixed by \n\\begin{align}\nM_{\\overline{\\bm{3}}} = \\frac29 (M_{\\Lambda_c(2628)} + 2\n  M_{\\Xi_c(2818)}) + \\frac19 (M_{\\Lambda_c(2592)} + 2 M_{\\Xi_c(2790)})\n  = 2744\\,\\mathrm{MeV}.  \n\\end{align}\n\nWe now assert that as a minimal scenario the newly found $\\Omega_c$ baryons\nby the LHCb Collaboration belong to the five excited sextets. Then\n$\\Omega_c(3000)$ can be identified as the state with $(J_{L}=0,\\,1\/2^-)$,\nwhich corresponds to the lightest state in Fig.~\\ref{fig:6}. All other\nfour states can be consequently identified as depicted in\nFig.~\\ref{fig:6}. Including the hyperfine interactions, we get the\nresults as summarized in Table~\\ref{tab:7}.\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[thp]\n\\caption{Scenario 1: All five LHCb\n  $\\Omega_{c}$ states are assigned to the excited baryon sextets.\n\\label{tab:7}%\n\\begin{center}%\n\\begin{tabular}\n[c]{ccccc}\\hline\\hline\n$J_{L}$ & $S^{P}$ & $M$~[MeV] & $\\varkappa^{\\prime}\/m_{c}$~[MeV] &\n                                                            $\\Delta_{J_{L}}$~[MeV]\\\\ \\hline\n0 & $\\frac{1}{2}^{-}$ & 3000 & not applicable & not applicable\\\\ \n\\multirow{2}{*}{1} & $\\frac{1}{2}^{-}$ & 3050 & \\multirow{2}{*}{16} &\n\\multirow{2}{*}{61}\\\\\n~ & $\\frac{3}{2}^{-}$ & 3066 &  & \\\\\n\\multirow{2}{*}{2} & $\\frac{3}{2}^{-}$ & 3090 & \\multirow{2}{*}{17} &\n\\multirow{2}{*}{47}\\\\\n& $\\frac{5}{2}^{-}$ & 3119 &  & \\\\\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\par\n\\end{table}\n\\renewcommand{\\arraystretch}{1}\nWe find at least three different contradictions arising from the\nassignment of these $\\Omega_c$ states as the members of the\nexcited sextets within the $\\chi$QSM.  Firstly, this assignment\nrequires that the hyperfine splitting should be almost as twice as \nsmaller than in the $\\overline{\\bm{3}}$ case. Secondly,  the robust\nrelation $\\Delta_2=2\\Delta_1$ given in Eq.~\\eqref{eq:Jsplit}  is badly\nbroken. Finally, there are two orthogonal sum rules\n$\\sigma_1=\\sigma_2=0$ derived from the $\\chi$QSM  \n\\begin{align}\n\\sigma_1&=6\\; \\Omega_c(J_{L}=0,1\/2^-)- \\Omega_c(J_{L}=1,1\/2^-)-8 \\;\n          \\Omega_c(J_{L}=1,3\/2^-)+3\\; \\Omega_c(J_{L}=2,5\/2^-) , \\label{sr}\\\\ \n\\sigma_2&=-4\\; \\Omega_c(J_{L}=0,1\/2^-)+9\\; \\Omega_c(J_{L}=1,1\/2^-)-3 \\;\n          \\Omega_c(J_{L}=1,3\/2^-)-5 \\; \\Omega_c(J_{L}=2,3\/2^-)  \n+3\\; \\Omega_c(J_{L}=2,5\/2^-),  \\notag\n\\end{align}\nwhich are also badly broken. Thus, we come to the conclusion that the\nthe five $\\Omega_c$ baryons is unlikely to belong to the excited\nsextets. A similar conclusion was drawn by\nRef.~\\cite{Karliner:2017kfm} in a different theoretical framework.  \nMoreover, the computed decay widths of the excited $\\Omega_c$'s do not\nmatch with the experimental data. Therefore, the first scenario is\nunrealistic in the present mean-field approach. \n\nSince the first scenario is not suitable for identifying the five\nexcited $\\Omega_c$ baryons, we have to come up with another\nscenario. Observing that two of them have rather narrower decay widths\nthan other three $\\Omega_c$'s, we assert that these narrow\n$\\Omega_c(3050)$ and $\\Omega_c(3119)$ belong to the possible exotic\nanti-decapentaplet ($\\overline{\\bm{15}}$) which is yet another\nlowest-lying allowed representation, whereas three of them belong to\nthe excited sextet. We find in this scenario that two other members of\nthe excited baryon sextet with $J_{L}=2$ have masses above the $\\Xi D$\nthreshold at 3185 MeV. Since they have rather broad widths, they are\nnot clearly seen in the LHCb data and may fall into the bump\nstructures appearing in the LHCb data.  \n\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[thp]\n\\caption{Scenario 2. Only three LHCb states are\nassigned to {the} sextets. }%\n\\label{tab:8}%\n\\begin{center}%\n\\begin{tabular}\n[c]{ccccc}\\hline \\hline\n$J_{L}$& $S^{P}$ & $M$~[MeV] & $\\varkappa^{\\prime}\/m_{c}$~[MeV] & $\\Delta_{J_{L}}%\n$~[MeV]\\\\\\hline\n0 & $\\frac{1}{2}^{-}$ & 3000 & not applicable & not applicable\\\\\n\\multirow{2}{*}{1} & $\\frac{1}{2}^{-}$ & 3066 & \\multirow{2}{*}{24} &\n\\multirow{2}{*}{82}\\\\\n~ & $\\frac{3}{2}^{-}$ & 3090 &  & \\\\\n\\multirow{2}{*}{2} & $\\frac{3}{2}^{-}$ & \\emph{3222} & input & input\\\\\n& $\\frac{5}{2}^{-}$ & \\emph{3262} & 24 & 164\\\\\\hline \\hline\n\\end{tabular}\n\\end{center}\n\\par\n\\end{table}\n\\renewcommand{\\arraystretch}{1}\nThe results of the second scenario are summarized in Table~\\ref{tab:8}\nexcept for the $\\Omega_c(3050)$ and $\\Omega_c(3119)$ which will be\ndiscussed separately.  The italic numbers correspond to the bump\nstructures from which $\\Omega_c(3222)$ used as input. Scenario 2\nprovides a much more plausible prediction than scenario 1\ndoes. Interestingly, the value of $\\varkappa'\/m_c\\approx 24$ MeV is\ncloser to that determined from the excited baryon anti-triplets, given\nin Eq.~\\eqref{eq:exhfvalue}. Moreover, the relation\n$\\Delta_1=2\\Delta_2$ is nicely satisfied in this scenario.  \n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig7.pdf}\\qquad\n\\caption{Representation of the anti-decapentaplet\n  ($\\overline{\\bm{15}}$). As in the case of the baryon sextet, there\n  are two baryon anti-decapentaplets with spin 1\/2 and 3\/2. The\n  $\\Omega_c$s belong to the isotriplet in the $\\overline{\\bm{15}}$plet.}\n\\label{fig:7}\n\\end{figure}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.3]{fig8.pdf}\\qquad\n\\caption{The allowed representations for the lowest-lying heavy\n  baryons. }\n\\label{fig:8}\n\\end{figure}\nThe anti-decapentaplet ($\\overline{\\bm{15}}$) was first suggested by\nDiakonov~\\cite{Diakonov:2010tf}. Figure~\\ref{fig:7} illustrates the\nrepresentation of the $\\overline{\\bm{15}}$.  Since the\n$\\overline{\\bm{15}}$ belongs to the allowed representations for the\nground-state heavy baryons, it satisfies the quantization rule\n$\\bm{J}_{L}+\\bm{T}=\\bm{0}$, so $T=J_{L}=1$ (see Fig.~\\ref{fig:8}).  When the\nlight-quark pair with $J_{L}=1$ is coupled to the heavy-quark spin, there\nare two possible $\\overline{\\bm{15}}$ representations that are\ndegenerate in the limit of $m_Q\\to \\infty$. It means that one needs to\nconsider the hyperfine interaction defined in Eq.~\\eqref{eq:ssinter}.\nAs given in Eq.~\\eqref{eq:kappavalue}, the value of $\\varkappa\/m_c$ is\naround 68 MeV. Surprisingly, the mass difference between the\n$\\Omega_c(3050)$ and the $\\Omega_c(3119)$ is  \n\\begin{align}\nM_{\\Omega_c(3\/2^+)} (3119) -  M_{\\Omega_c(1\/2^+)} (3050) =\n  \\frac{\\varkappa}{m_c} \\approx   69\\,\\mathrm{MeV} \n\\end{align}\nwhich is almost the same as what was determined from the lowest-lying\nsextet baryons. The decay widths of the excited $\\Omega_c$ baryons\npredicted within the present framework further support the\nplausibility of scenario 2~\\cite{Kim:2017khv}.  The decay widths for\nthe $\\Omega_c(3050)$ and $\\Omega_c(3119)$ are predicted to be \n\\begin{align}\n\\Gamma_{\\Omega_c(3050)(\\overline{\\bm{15}},1\/2^+)} =\n  0.48\\,\\mathrm{MeV}, \\;\\;\\; \n\\Gamma_{\\Omega_c(3119)(\\overline{\\bm{15}},3\/2^+)} = 1.12\\,\\mathrm{MeV},  \n\\end{align}\nwhich are in good agreement with the LHCb data\n$\\Gamma_{\\Omega_c(3050)}=(0.8\\pm0.2\\pm 0.1)$ MeV and\n$\\Gamma_{\\Omega_c(3119)}=(1.1\\pm0.8\\pm 0.4)$ MeV. For detailed\ndiscussion related to the decay widths of $\\Omega_c$, we refer to\nRef.~\\cite{Kim:2017khv}.   \n\nIn addition to scenarios 1 and 2, we also tried to examine\nseveral other scenarios but find that they all turned out to be\ninconsistent with the experimental data. Finally, we want to emphasize\nthat the $\\Omega_c(3050)$ and $\\Omega_c(3119)$ assigned to the members\nof the $\\overline{\\bm{15}}$ are isotriplets. It implies that if they\nindeed belong to the $\\overline{\\bm{15}}$, charged $\\Omega_c^{\\pm}$\nshould exist. Knowing that the excited $\\Omega_c^0$'s have been\nmeasured in the $\\Xi_c^+K_c^-$ channel, we propose that the $\\Xi_c^+\nK^0$ and $\\Xi_c^0 K^-$ channels need to be scanned in the range of the\ninvariant mass between 3000 MeV and 3200 MeV to find an isovector\n$\\Omega_c$'s. If they do not exist, this will falsify the present\npredictions. \n\\section{Conclusion and outlook}\nIn the present short review, we briefly summarized a series of recent\nworks on the properties of the singly heavy baryons within a pion\nmean-field approach, also known as the chiral quark-soliton model.\nIn the limit of the infinitely heavy quark mass ($m_Q\\to \\infty$), \nthe heavy quark inside a heavy baryon can be treated as a mere static\ncolor source. Then a heavy baryon is portrayed as a state of $N_c-1$\nvalence quarks bound by the pion mean field with a heavy stripped off\nfrom the valence level.  This mean-field approach has a certain\nvirtue since both the light and heavy baryons can be dealt with on an\nequal footing. It means that we can bring all dynamical parameters\nwhich have been already determined in the light-baryon sector to\ndescribe the heavy baryons. Indeed we can simply replace the\n$N_c$-counting prefactor by $N_c-1$ for the valence contributions to\nthe heavy baryons. Accordingly, we were able to explain the masses of\nthe lowest-lying heavy baryons and the magnetic moments of them\nwithout introducing additional parameters except for the hyperfine\nspin-spin interactions. We have employed the same framework to\nidentify the newly found excited $\\Omega_c$ baryons reported by the\nLHCb Collaboration. Assigning the three of them to the excited baryon\nsextets and the two of them with narrower decay widths to the possible\nexotic baryon anti-decapentaplet, we were able to classify the\n$\\Omega_c$'s successfully.  Since the $\\Omega_c$ baryons in the\nanti-decapentaplet are the isovector baryons, we anticipate that\ncharged $\\Omega_c$'s might be found in other channels such as the $\\Xi_c^+\nK^0$ and $\\Xi_c^0 K^-$. \n\nThe present model can be further applied to future investigations on\nvarious properties and form factors of heavy baryons. As already shown\nin Ref.~\\cite{Kim:2018nqf}, the electric form factor of the charged\nheavy baryon indicates that a heavy baryon is an electrically compact\nobject. Transition form factors of heavy baryons will further reveal\ntheir internal structure. Understanding excited heavy baryons is\nanother crucial issue that should be investigated. Related studies are\nunder way. \n\n\\begin{acknowledgments}\nI am very grateful to M. V. Polyakov, M. Prasza{\\l}owicz, and\nGh.-S. Yang for fruitful collaborations and discussions over decades.  \nI am thankful to J.-Y. Kim for the discussion related to the\nelectromagnetic form factors of heavy baryons. \nI want to express the gratitude to the editors of the Journal of the\nKorean Physical Society (JKPS) for giving me an opportunity to join\nthe very special 50th anniversary celebration of the JKPS.   \nThe present work was supported by the Inha University Grant in 2017.\n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{intro}\n\n\nOne of the first, and most important, problems to be tackled by the theory of linear elasticity is that of the buckling of a column under an axial load. \nUsing Bernoulli's beam equations, Euler found the critical load of compression $N_\\text{cr}$ leading to the buckling of a slender cylindrical column of radius $B$ and length $L$.\nAs recalled by Timoshenko and Gere \\cite{tige61}, Euler looked at the case of an ideal column with (built in)-(free) end conditions. \nWhat is now called ``Euler buckling'', or the ``\\emph{fundamental case} of buckling of a prismatic bar'' \\cite{tige61} corresponds to the case of a bar with hinged-hinged end conditions. \nThe corresponding critical load is given by\n\\begin{equation} \\label{euler}\n\\dfrac{N_\\text{cr}}{\\pi^3B^2} = \\frac{E}{4}\\left(\\frac{B}{L}\\right)^2,\n\\end{equation}\nwhere $E$ is the Young modulus.\nThe extension of this formula to the case of a thick column is a non-trivial, even sophisticated, problem of non-linear three-dimensional elasticity. \nIn general, progress can be only made by using reductive (rod, shells, etc.) theories. \nHowever, there is another choice of boundary conditions for which the criterion~(\\ref{euler}) is valid: namely, the case where both ends are ``guided'' or ''sliding'' (the difference between the two cases lies in the shape of the deflected bar, which is according to the half-period of a sine in one case and of a cosine in the other case.) \nIn exact non-linear elasticity, there exists a remarkable three-dimensional analytical solution to this problem (due to Wilkes \\cite{Wilk55}) which describes a small-amplitude (incremental) deflection superimposed upon the large homogeneous deformation of a cylinder compressed between two lubricated platens. \nIn this case, the Euler formula can be extended to the case of a column with finite dimensions, for arbitrary constitutive law.\n\nThe exact incremental solution allows for an explicit derivation of Euler's formula at the \\emph{onset of non-linearity}, which combines third-order elastic constants with a term in $(B\/L)^4$.\nIn Goriely et al. \\cite{Goriely08}, we showed that for an  incompressible cylinder, \n\\begin{equation}\\label{incompressible_case}\n\\frac{N_\\text{cr}}{\\pi^3B^2} = \\frac{E}{4}\\left(\\frac{B}{L}\\right)^2\n - \\frac{\\pi^2}{96}\\left(\\frac{20}{3}E + 9 \\mathcal{A}\\right)\\left(\\frac{B}{L}\\right)^4,\n\\end{equation}\nwhere $\\mathcal{A}$ is Landau's third-order elasticity constant.\nThis formula clearly shows that \\emph{geometrical non-linearities} (term in $(B\/L)^4$) are intrinsically coupled to \\emph{physical non-linearities} (term in $\\mathcal{A}$) for this problem.\n(For connections between Euler's theory of buckling and incremental incompressible nonlinear elasticity, see the early works of Wilkes \\cite{Wilk55}, Biot \\cite{Biot63}, Fosdick and Shield \\cite{FoSh63}, and the references collected in  \\cite{Goriely08}.)\n\nNow, in third-order \\emph{in}compressible elasticity, there are two elastic constants \\cite{Ogden74b}: the shear modulus $\\mu$ ($=E\/3$) and $\\mathcal{A}$.\nIn third-order \\emph{compressible} elasticity, there are five elastic constants:\n$\\lambda$ and $\\mu$, the (second-order) Lam\\'e constants (or equivalently, $E$ and $\\nu$, Young's modulus and Poisson's ratio), and $\\mathcal{A}$, $\\mathcal{B}$, and $\\mathcal{C}$, the (third-order) Landau constants. \nEuler's formula at order $(B\/L)^2$, equation (\\ref{euler}), involves only one elastic constant, $E$. \nIt is thus natural to ask whether Poisson's ratio, $\\nu$, plays a role in the non-linear correction to Euler formula  of $(B\/L)^4$, the next-order term.\nThe final answer is found as formula \\eqref{correc} below, which shows that the non-linear correction involves all five elastic constants.\n\n\n\\section{Finite compression and buckling}\n\n\nIn this section, we recall the equations governing the homogeneous compression of a cylinder in the theory of exact (finite) elasticity. \nWe also recall the form of some incremental solutions that is, of some small-amplitude static deformations which may be superimposed upon the large compression, and which indicate the onset of instability for the cylinder.\n\n\n\n\\subsection{Large deformation}\n\n\nWe take a cylinder made of a hyperelastic, compressible, isotropic solid with strain-energy density $W$ say, with radius $B$ and length $L$ in its undeformed configuration.\nWe denote by ($R$, $\\Theta$, $Z$) the coordinates of a particle in the cylinder at rest, in a cylindrical system.\n\nThen we consider that the cylinder is subject to the following deformation,\n\\begin{equation}  \\label{deformation}\n{r} = \\lambda_1R, \\qquad {\\theta}= \\Theta, \\qquad {z}=\\lambda_3 Z,\n\\end{equation}\nwhere ($r$, $\\theta$, $z$) are the current coordinates of the particle initially at  ($R$, $\\Theta$, $Z$),   $\\lambda_1$ is the radial stretch ratio and $\\lambda_3$ is the axial stretch ratio.\nExplicitly, $\\lambda_1 = b\/B$ and $\\lambda_3 = l\/L$, where $b$ and $l$ are the radius and length of the deformed cylinder, respectively.\nThe physical components of  $\\tens{F}$, the corresponding deformation gradient, are: $\\tens{F} = \\textrm{Diag} \\left(\\lambda_1, \\lambda_1, \\lambda_3 \\right)$, showing that the deformation is equi-biaxial and homogeneous (and thus, universal). \n\nThe (constant) Cauchy stresses required to maintain the large homogeneous compression are \n(see, for instance, Ogden \\cite{Ogden84}),\n\\begin{equation} \\label{Cauchy stress tensor}\n\\sigma_i = \\left( \\det \\tens{F} \\right)^{-1}\\lambda_i W_i, \\qquad i=1,2,3 \\quad \\textrm{(no sum)},\n\\end{equation}\nwhere $W_i \\equiv \\partial W\/ \\partial \\lambda_i$.\nIn our case, $\\sigma_1=\\sigma_2$ because the deformation is equi-biaxial, and  $\\sigma_1=\\sigma_2=0$ because the outer face of the cylinder is free of traction. \nHence\n\\begin{equation}\n\\sigma_1=\\sigma_2= \\lambda_1^{-1}\\lambda_3^{-1}W_1=0, \\qquad \n\\sigma_3= \\lambda_1^{-2} W_3. \\label{sigma}\n\\end{equation}\nNote that we may use the first equality to express one principal stretch in terms of the other (provided, of course, that inverses can be performed).\n\n\n\\subsection{Incremental equations}\n\n\nNow we recall the equations governing the equilibrium of incremental solutions, in the neighborhood of the finite compression.\nThey read in general as \\cite{Ogden84}\n\\begin{equation} \\label{Incremental_equations_equilibrium}\n\\textrm{div}\\ \\tens{s}= \\mathbf{0},\n\\end{equation}\nwhere $\\tens{s}$ is the incremental nominal stress tensor.\nIt is related to $\\mathbf{u}$, the incremental mechanical displacement, through the incremental constitutive law,\n\\begin{equation} \\label{component_form}\n\\mathbf{s} = \\tens{B} \\left(\\text{grad} \\ \\mathbf{u}\\right)^T,\n\\end{equation}\nwhere $\\tens{B}$ is the fourth-order tensor of incremental elastic moduli and the gradient is computed in the current cylindrical coordinates \\cite{DoHa06}. \nThe non-zero components of $\\tens{B}$, in a coordinate system aligned with the principal axes associated with the deformation \\eqref{deformation}, are given in general by \\cite{Ogden84}\n\\begin{align} \n& JB_{iijj}= \\lambda_i\\lambda_j W_{ij}, \\nonumber \\\\\n&JB_{ijij}= (\\lambda_i W_i - \\lambda_j W_j)\\lambda_i^2\/(\\lambda_i^2-\\lambda_j^2), &  i\\neq j,\\ \\lambda_i\\neq \\lambda_j, \\nonumber \\\\\n&JB_{ijji}= (\\lambda_j W_i - \\lambda_i W_j)\\lambda_i \\lambda_j\/(\\lambda_i^2-\\lambda_j^2), &  i\\neq j,\\ \\lambda_i\\neq \\lambda_j, \\nonumber \\\\\n& JB_{ijij}=(B_{iiii}-B_{iijj} + \\lambda_i W_i)\/2, & i\\neq j,\\ \\lambda_i= \\lambda_j, \\nonumber \\\\\n& JB_{ijji}=B_{jiij}=B_{ijij} - \\lambda_i W_i, &  i\\neq j,\\ \\lambda_i= \\lambda_j, \n\\end{align}\n(no sums), where $W_{ij}\\equiv \\partial^2 W\/(\\partial \\lambda_i\\partial \\lambda_j)$. \nNote that here, some of these components are not independent one from another because $\\lambda_1=\\lambda_2$ and $\\sigma_1=\\sigma_2=0$. \nIn particular, we find that \n\\begin{align}\n& B_{1212} = B_{2121} = B_{1221}, \\qquad \n   B_{2323}=B_{1313}=B_{1331}=B_{2332},\n   \\notag \\\\\n& B_{2222}=B_{1111}, \\quad \n   B_{2233}=B_{1133},\n   \\quad\n    B_{3232}=B_{3131},  \\quad \n   B_{1122}=B_{1111}-2B_{1212}. \n\\end{align}\n\n\n\\subsection{Incremental solutions}\n\n\nWe look for incremental static solutions that are periodic along the circumferential and axial directions, and have yet unknown radial variations. \nThus our ansatz for the components of the mechanical displacement is the same as Wilkes's \\cite{Wilk55}:\n\\begin{equation} \\label{slon}\nu_r = U_r(r) \\cos n\\theta \\cos kz, \\quad\nu_\\theta = U_{\\theta}(r) \\sin n \\theta \\cos kz, \\quad\nu_z = U_z(r) \\cos n\\theta \\sin k z, \n\\end{equation}\nwhere $n=0,1,2,\\ldots$ is the \\textit{circumferential mode number}; $k$ is the \\textit{axial wavenumber}; the subscripts $(r,\\theta,z)$ refer to components within the cylindrical coordinates $(r,\\theta,z)$; and all upper-case functions are functions of $r$ alone. \n\nDorfmann and Haughton \\cite{DoHa06} show that the following displacements $\\mathbf{U}^{(1)}$,  $\\mathbf{U}^{(2)}$, and  $\\mathbf{U}^{(3)}$ are solutions to the incremental equations \\eqref{Incremental_equations_equilibrium}, \n\\begin{equation} \\label{U12}\n\\mathbf{U}^{(1)}(r), \\, \\mathbf{U}^{(2)}(r) = \n\\left[ I'_{n}(qkr), -\\frac{n}{qkr}I_n(qkr), -\\dfrac{(B_{1111}q^2-B_{3131})}{q(B_{1313}+B_{1133})}I_n(qkr)\\right]^T,\n\\end{equation}\nand \n\\begin{equation} \\label{U3}\n\\mathbf{U}^{(3)}(r) = \\left[\\dfrac{1}{r}I_n(q_3kr), - \\dfrac{q_3k}{n}I'_{n}(q_3kr), 0\\right]^{T},\n\\end{equation}\nwhere $q=q_1, q_2$ and $I_n$ is the modified Bessel function of order $n$.\nHere $q_1$, $q_2$, and $q_3$ are the square roots of the roots $q_1^2$, $q_2^2$ of the following quadratic in $q^2$: \n\\begin{equation} \nB_{1313} B_{1111} q^4+[(B_{1133}+B_{1313})^2-B_{1313}B_{3131}-B_{3333}B_{1111}]q^2+B_{3333}B_{3131}=0,\n\\end{equation}\nand of the root of the following linear equation in $q^2$\n\\begin{equation}\nB_{1212}q^2-B_{3131}=0,\n\\end{equation}\nrespectively.\n\n\n\nFrom \\eqref{component_form} we find that the incremental traction on planes normal to the axial direction has components of the same form as that of the displacements, namely \n\\begin{align} \\label{soln}\n& s_{r r} = S_{r r}(r) \\cos n\\theta \\cos k z, \\notag \\\\\n& s_{r\\theta} = S_{r\\theta}(r) \\sin n \\theta \\cos k z, \\notag \\\\\n& s_{r z} = S_{r z}(r) \\cos n\\theta \\sin k z, \n\\end{align}\nsay, where again all upper-case functions are functions of $r$ alone. \nThen we find that the traction solutions corresponding to the solutions \\eqref{U12}-\\eqref{U3} are given by\n\\begin{multline}\nr \\mathbf{S}^{(1)}(r), \\ \\ r \\mathbf{S}^{(2)}(r)= \n\\left[ \n2B_{1212} I'_n(q k r) \\right.\n\\\\ \\left . - \\left(2B_{1212}  \\dfrac{n^2}{q k r} + q k r B_{1111} - \\dfrac{k r B_{1133}\\left(B_{1111}q^2 - B_{3131}\\right)}{q\\left(B_{1313}+B_{1133}\\right)}\\right) I_n(q k r), \\right.\n \\\\ \n\\left. 2  n B_{1212}\\left(\\frac{I_n(q k r)}{q k r}-I'_n(q k r)\\right),\n -  k r B_{1313}\\left(\\frac{B_{1111}q^2-B_{3131}}{B_{1313}+B_{1133}}+1\\right)I'_{n}(q k r)\n\\right]^T,\n\\end{multline}\nand\n\\begin{multline}\nr \\mathbf{S}^{(3)}(r) = \\left[\n 2 B_{1212}\\left(\\frac{I_n(q_3kr)}{r}-q_3 k I'_n(q_3kr)\\right), \\right.\n \\\\\n\\left. B_{1212}\\left(\\frac{2q_3k}{n}I'_n(q_3kr)-\\left(\\frac{2n}{r}+\\frac{q_3^2k^2r}{n}\\right)I_n(q_3kr)\\right),\n-  k B_{1313}I_n(q_3kr)\\right]^{T}.\n\\end{multline}\n\nThe general solution to the incremental equations of equilibrium is thus of the form\n\\begin{equation} \nr \\mathbf{S}(r)  = \n \\begin{bmatrix} [c|c|c]\nr \\mathbf{S}^{(1)}(r) & r \\mathbf{S}^{(2)}(r) & r \\mathbf{S}^{(3)}(r)\n\\end{bmatrix}\n\\mathbf{c},\n\\end{equation}\nwhere $\\mathbf{S} \\equiv [S_{r r}, S_{r\\theta}, S_{r z}]^T$ and $\\mathbf{c}$ is a constant three-component vector.\nNote that we use the quantity $r \\mathbf{S}$ for the traction (instead of $\\mathbf{S}$), because it is the Hamiltonian conjugate to the displacement in cylindrical coordinates \\cite{Shuv03}.\n\nNow when the cylinder is compressed (by platens say), its end faces should stay in full contact with the platens so that the first incremental boundary condition is \n\\begin{equation}\nu_{z} = 0, \\qquad \\text{on} \\quad z=0, l, \n\\end{equation}\nwhich leads to \\cite{DoHa06,Goriely08}\n\\begin{equation}\nk = m \\pi\/l,\\end{equation}\nfor some integer $m$, the \\emph{axial mode number}.\nFrom \\eqref{soln}, we now see that on the thrust faces, we have\n\\begin{equation}\ns_{r z} = 0, \\qquad \\text{on} \\quad z=0, l, \n\\end{equation}\nwhich means that the end faces of the column are in sliding contact with the thrusting platens.\nIn other words, in the limit of a slender column, we recover the Euler strut with \\emph{sliding-sliding}, or \\emph{guided-guided} end conditions.  \nIn Figure \\ref{fig4}, we show the first two axi-symmetric and two asymmetric modes of incremental buckling.\n\\begin{figure}[!ht]\n\\begin{center}\n\\epsfig{figure=modes.pdf, width=.6\\textwidth}\n\\end{center}\n \\caption{First two axi-symmetric and two asymmetric modes of buckling for a compressed strut with guided-guided end conditions: $n$ is the circumferential mode number and $m$ the axial mode number. For slender enough cylinders, the $n=1$, $m=1$ mode is the first mode of buckling.}\n \\label{fig4}\n\\end{figure}\n\nThe other boundary condition is that the cylindrical face is free of incremental traction: $\\mathbf{S}(b) = \\mathbf{0}$.\nThis gives \n\\begin{equation}\n\\Delta \\equiv \\det \\begin{bmatrix} [c|c|c]\nb \\mathbf{S}^{(1)}(b) & b \\mathbf{S}^{(2)}(b) & b \\mathbf{S}^{(3)}(b)\n\\end{bmatrix} = 0.\n\\end{equation}\n\n\n\\section{Euler buckling}\n\n\n\n\n\\subsection{Asymptotic expansions}\n\\label{Asymptotic Euler buckling}\n\n\nWe now specialize the analysis to the asymmetric buckling mode $n=1$, $m=1$, corresponding to the Euler buckling with guided-guided end conditions, in the limit where the axial compressive stretch $\\lambda_3$ is close to 1 (the other modes are not reached for slender enough cylinders). \nTo this end, we only need to consider the so-called \\emph{third-order elasticity} expansion of the strain energy density, for example that of Landau and Lifshitz \\cite{LaLi86}, \n\\begin{equation}\nW = \n\\dfrac{\\lambda}{2}\\left(\\textrm{tr}\\tens{E}\\right)^2 +\\mu \\ \\textrm{tr}(\\tens{E}^2) + \\dfrac{\\mathcal{A}}{3}\\textrm{tr}(\\tens{E}^3) + \\mathcal{B}\\left(\\textrm{tr}\\tens{E}\\right)\\textrm{tr}(\\tens{E}^2)\n+  \\dfrac{\\mathcal{C}}{3}\\left(\\textrm{tr}  \\tens{E}\\right)^3,\n\\end{equation}\nwhere $\\tens{E}=\\tens{E}^T$ is the Lagrange, or Green, strain tensor defined as $\\tens{E}=\\left(\\tens{F}^T\\tens{F}-\\tens{I}\\right)\/2$,\n$\\lambda$ and $\\mu$ are the Lam\\'e moduli, and $\\mathcal{A}$, $\\mathcal{B}$, $\\mathcal{C}$ are the Landau third-order elastic constants\n(Note that there are other, equivalent, expansions based on other invariants, such as the ones proposed by Murnaghan \\cite{Murn51}, Toupin and Bernstein \\cite{ToBe61},  Bland \\cite{Blan69}, or Eringen and Suhubi \\cite{ErSu74}, see Norris \\cite{Norr99} for the connections.)\n\nTo measure how close $\\lambda_3$ is to 1, we introduce $\\epsilon$, a small parameter proportional to the slenderness of the  deformed cylinder,\n\\begin{equation} \\label{epsilon}\n\\epsilon=k b=\\pi b\/l.\n\\end{equation}\nThen we expand the radial stretch $\\lambda_1$ and the critical buckling stretch $\\lambda_3$ in terms of $\\epsilon$ up to order $M$,\n\\begin{equation}\n\\lambda_1=\\lambda_1(\\epsilon)=1+\\sum_{p=1}^{M}\\alpha_p\\epsilon^p + \\mathcal{O}(\\epsilon^{M+1}),\n\\qquad\n\\lambda_3=\\lambda_3(\\epsilon)=1+\\sum_{p=1}^{M}\\beta_p\\epsilon^p + \\mathcal{O}(\\epsilon^{M+1}),\n\\end{equation}\nsay, where the $\\alpha$'s and $\\beta$'s are to be determined shortly. \nSimilarly, we expand $\\Delta$ in powers of $\\epsilon$,\n\\begin{equation}\\nonumber\n\\Delta = \\sum_{p=1}^{M_d}d_p\\epsilon^p + \\mathcal{O}(\\epsilon^{M_d+1}),\n\\end{equation}\nsay, and solve each order $d_p=0$ for the coefficients $\\alpha_p$ and $\\beta_p$, making use of the condition $\\sigma_1=0$.\nWe find that $\\alpha_p$ and $\\beta_p$ vanish identically for all odd values of $p$, and that $\\lambda_1$ and $\\lambda_3$, up to the fourth-order in $\\epsilon$, are given by \n\\begin{equation}\\label{lambda1_3}\n\\lambda_1=1+\\alpha_{2}\\epsilon^2+\\alpha_{4}\\epsilon^4+O(\\epsilon^6),\n\\qquad\n\\lambda_3=1+\\beta_{2}\\epsilon^2+\\beta_{4}\\epsilon^4+O(\\epsilon^6),\n\\end{equation}\nwith $\\alpha_2$ and $\\alpha_4$ given by \n\\begin{align} \\label{alpha}\n \\alpha_2 = & \\dfrac{\\nu}{4}, \\notag  \\\\\n \\alpha_4 = & -  \\dfrac{\\nu(1+\\nu)}{32} \\notag \\\\ \n&  - \\dfrac{(1+\\nu)(1-2\\nu)}{16E} \\left[\\nu^2\\mathcal{A} + (1-2\\nu + 6\\nu^2)\\mathcal{B} + (1 - 2\\nu)^2 \\mathcal{C} \\right] - \\nu \\beta_4,  \n\\end{align}\nwherein \n\\begin{align} \\label{beta}\n \\beta_2 = & -\\dfrac{1}{4},  \\notag \\\\\n \\beta_4  = & \\dfrac{29 + 39 \\nu + 8 \\nu^2}{96(1+\\nu)}\\notag \\\\\n & + \\dfrac{1}{16E} \\left[( 1 - 2\\nu^3)\\mathcal{A} + 3(1-2\\nu)(1+2\\nu^2)\\mathcal{B} + (1 - 2\\nu)^3 \\mathcal{C} \\right]. \n\\end{align}\nNote that we switched from Lam\\'e constants to Poisson's ratio and Young's modulus for these expressions, using the  connections $\\nu = \\lambda\/(2\\lambda+2\\mu)$ and $E = \\mu(3\\lambda+2\\mu)\/(\\lambda+\\mu)$.\n\n\n\\subsection{Onset of nonlinear Euler buckling}\n\n\nThe analytical results presented above are formulated in terms of the \\emph{current} geometrical parameter $\\epsilon$, defined in \\eqref{epsilon}. \nIn order to relate these results to the classical form of Euler buckling, we introduce the \\emph{initial} geometric slenderness $B\/L$. \nRecalling that $\\epsilon=\\pi b\/l$, $\\lambda_3=l\/L$, and $b=\\lambda_1B$, we find that \n\\begin{equation}\\label{condition}\n\\epsilon \\lambda_3 = \\pi  \\lambda_1 (B\/L).\n\\end{equation} \nWe expand $\\epsilon$ in powers of $B\/L$, and solve \\eqref{condition} to obtain \n\\begin{eqnarray}\\label{epsilon2}\n\\nonumber \\epsilon \n&=& \\pi (B\/L)  + (\\alpha_2-\\beta_2) \\pi^3 (B\/L)^3 \n+ \\mathcal{O}\\left((B\/L)^4\\right) \\\\\n&=& \\pi (B\/L)  + (1 + \\nu)(\\pi^3\/4)(B\/L)^3 + \\mathcal{O}\\left((B\/L)^4\\right).\n\\end{eqnarray}\n\nSecond, we wish to relate the axial compression to the current axial load $N$. \nTo do so, we integrate the axial stress over the faces of the cylinder, \n\\begin{equation} \\label{axialforce}\n N = -2\\pi\\int_0^b r \\sigma_3 \\text{d}r = -\\pi b^2 \\sigma_3=-\\pi \\lambda_1^2 B^2 \\sigma_3,\n\\end{equation}\nbecause $\\sigma_3$ is constant, given by (\\ref{sigma})$_{2}$.\n\nFinally, in order to write the nonlinear buckling formula, we expand $\\lambda_1$ and $\\lambda_3$ in  \\eqref{axialforce}, using \\eqref{lambda1_3}, and then expand $\\epsilon$ in powers of the slenderness ($B\/L$), using \\eqref{epsilon2}.\nIt gives the desired expression for the first \\emph{non-linear correction to the Euler formula},\n\\begin{equation} \\label{correc}\n\\dfrac{N_\\text{cr}}{\\pi^3 B^2} = \\dfrac{E}{4}  \\left(\\dfrac{B}{L}\\right)^2 - \\dfrac{\\pi^2}{96} \\delta_\\text{\\ NL} \\left(\\dfrac{B}{L}\\right)^4,\n\\end{equation}\nwhere\n\\begin{multline}\n\\delta_\\text{\\ NL} = 2\\dfrac{13 + 12 \\nu - 2 \\nu^2}{(1+\\nu)} E \\\\\n + 12\\left[(1 - 2\\nu^3)\\mathcal{A} + 3 (1 - 2\\nu)(1 + 2\\nu^2)\\mathcal{B} + (1 - 2\\nu)^3\\mathcal{C}\\right]. \n\\end{multline}\n\nWe now check this equation against its incompressible counterpart \\eqref{incompressible_case}.\nTheoretical considerations and experimental measurements \\cite{Ogden74b}, \\cite{WHIZ08}, \\cite{CaGF03}, \\cite{DeOg10}, show that in the incompressible limit, $E$ and $\\mathcal{A}$ remain finite, $\\nu \\rightarrow 1\/2$, $(1-2\\nu)\\mathcal{B} \\rightarrow  -E\/3$, and $(1-2\\nu)^3 \\mathcal{C} \\rightarrow 0$.\nIt is then a simple exercise to verify that \\eqref{correc} is indeed consistent with \\eqref{incompressible_case} in those limits.\n\n\n\n\\subsection{Examples}\n\n\n\\begin{table}[!ht]\n\\caption{Lam\\'e constants and Landau third-order elastic moduli for five solids ($10^{9}$ N$\\cdot$ m$^{-2}$)}\n\\label{tab:1}      \n\\begin{tabular}{lrrrrr}\n\\hline\\noalign{\\smallskip}\nmaterial & $\\lambda$ & $\\mu$ & $\\mathcal{A}$ & $\\mathcal{B}$ & $\\mathcal{C}$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n      Polystyrene & $ 1.71$ & $0.95$ & $-10$ & $-8.3$ & $-10.6$ \\\\ \n\t\t\tSteel Hecla 37 & $111$ & $82.1$ & $-358$ & $-282$ & $-177$ \\\\\n\t\t\tAluminium 2S  & $57$ & $27.6$ & $-228$ & $-197$ & $-102$ \\\\\n\t\t\tPyrex glass & $13.5$ & $27.5$ & $420$ & $-118$ & $132$ \\\\\n\t\t\tSiO$_2$ melted & $15.9$ & $31.3$ & $-44$ & $93$ & $36$ \\\\ \n\t\t\t\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\nTo evaluate the importance of the non-linear correction, we computed the critical axial stretch ratio of column buckling for two solids.\nIn Table \\ref{tab:1}, we list the second- and third-order elastic constants of five compressible solids, as collected by Porubov \\cite{Poru03} (in the Table we converted the ``Murnaghan constants'' given by Porubov to the Landau constants $\\mathcal{A,B,C}$).\nFigure \\ref{fig5} shows the variations of $\\lambda_3$ with the squared slenderness $(B\/L)^2$, for pyrex and silica (two last lines of Table 1).\n\\begin{figure}[!ht]\n\\centering \\mbox{\\subfigure{\\epsfig{figure=Pyrex.pdf, width=.45\\textwidth}}}\n  \\quad \\quad\n     \\subfigure{\\epsfig{figure=SiO2.pdf, width=.45\\textwidth}}\n \\caption{Comparison of the different Euler formulas obtained by expanding the exact solution to order 2 (classical Euler buckling formula, plot labeled ``Euler$_2$'') and to order 4 (plot labeled ``Euler$_4$''), for pyrex (figure on the left) and for silica (figure on the right).}\n \\label{fig5}\n\\end{figure}\n\n\n\\section{Conclusions}\n\n\nThe present analysis provides an asymptotic formula for the critical value of the load for the Euler buckling problem, with guided-guided (sliding-sliding) end conditions. This formula was checked both in the incompressible limit and on particular cases against the exact value of the buckling obtained from the exact solutions. Not surprisingly it reinforces the universal and generic nature of the Euler buckling formula as the correction is small for most systems even when nonlinear elastic effects and nonlinear geometric effects are taken into account. It would be of great interest to see if these effects could be observed experimentally.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Method, results of modeling, and application} \n \nThe most important and useful sources for the aim of neutron stars (NSs)  $M$ and $R$ finding \nare X-ray bursting NSs with photospheric radius expansion \\cite{LvPT:93}. \n The relation between observed normalization $K$ (for blackbody fit of spectra) and the \nreal ratio of NS radius $R$ to distance on late outburst phases is: \n\\begin{equation} \\label{u1} \n   K^{1\/2} = \\frac{R_{\\rm BB}{\\rm (km)}}{D_{10}} = \\frac{R{\\rm (km)}}{\\fc^2~D_{10}}(1+z), \n\\end{equation} \nwhere $D_{10}$ is the distance in units of10 kpc, and $\\fc = T_{\\rm c}\/T_{\\rm eff}$ is a color correction factor. \nTherefore, on these phases $K(t)$ dependence reflects $\\fc(t)$ dependence only.  \nWe suggest to fit the observed $K^{-1\/4}$ -- $F$ relation by a theoretical $\\fc$ -- $l\\equiv L\/L_{\\rm Edd}$ relation, where $F$ is  \nthe integral observed flux. From this fit we can obtain two independent values:  $R{\\rm (km)}\\times(1+z)\/D_{10}$ \nand $F_{\\rm Edd} \\sim L_{\\rm Edd}\/((1+z)D^2_{10})$. Combining these values, we can obtain an observed $M\/R$ relation, \nwhich is independent on the distance and physically corresponds to a maximum possible effective temperature on  \nthe NS surface. \nIf the distance is known  \n we can find $M$ and $R$ \nsimultaneously. \nFor this method extended theoretical $\\fc(l)$ calculations are necessary. \n \nWe computed model \natmospheres of X-ray bursting NSs subject to the constraints \nof hydrostatic and radiative equilibrium assuming planar geometry \nin LTE approximation with Compton scattering taken into account (see details  \nof the code in \\cite{SP:06,SW:07}). \n \nWe calculated an extended set of NS model \natmospheres with 6 chemical compositions (pure H, He, and solar H\/He mixture \nwith $Z$ = 1, 0.3, 0.1 and 0.01 $Z_\\odot$), 3 surface gravities: $\\log~g$ = 14.0, 14.3 and \n14.6, and 20 luminosities $L$: 0.001, 0.003, 0.01, 0.03, 0.05, 0.07, 0.1, 0.15, 0.2, 0.3, 0.4, \n0.5, 0.6, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, and 0.98 $L_{\\rm Edd}$. Corresponding $T_{\\rm eff}$ were \ncalculated from $L$ using $\\log g$ and chemical composition. \nThe model emergent redshifted spectra were fitted \nby diluted blackbody spectra $F_{\\rm E} = wB_{\\rm E}(\\fc T_{\\rm eff})$  \nin the {\\it RXTE}\/PCA energy band $3 - 20$ keV. Here $w \\approx \\fc^{-4}$ is the dilution factor. \nThe accepted redshifts were calculated from $\\log g$ assuming $M = 1.4 M_\\odot$. \nResults are partially presented in Fig.\\,\\ref{v1sfig1}, left panel. \n \n\\begin{figure} \n  \\includegraphics[height=.185\\textheight]{sv1_01.eps} \n  \\includegraphics[height=.175\\textheight]{sv1_02.eps} \n  \\includegraphics[height=.18\\textheight]{sv1_03.eps} \n  \\caption{ \\label{v1sfig1} \n{\\it Left:} Dependence of color correction factors on the relative luminosity  \nfor low gravity and various chemical compositions in NS atmosphere models. \n {\\it Middle:} Comparison of the observed dependence of $K^{-1\/4} - F$ for 4U\\,1724$-$307 (croses) to the best fit theoretical models \n$\\fc - l$.  \n {\\it Right:} Constraints on  mass and radius of the neutron star 4U\\,1724$-$307.  \n} \n\\end{figure} \n \n \n We fitted the observed $K^{-1\/4}$ -- $F$ relation, obtained for the extremely long outburst of \n4U\\,1724$-$307~  in November 8, 1996 ({\\it RXTE}, \\cite{molkov:00}) by computed $\\fc$ -- $l$ relations. \nWe obtained limitations on $R$ and $M$ for the adopted \ndistance $D = 5.3 \\pm 0.6$ kpc \\cite{ort:97} and various chemical compositions, see Fig.\\,\\ref{v1sfig1}. \nThe values of $M$ and $R$ obtained for H-rich atmospheres  correspond to a stiff Equation of State in the inner NS core. \nHelium atmospheres are not acceptable. More details can be found in \\cite{SPRW:10}. \n \n\\begin{theacknowledgments} \n  \nThe work is supported by the DFG grant SFB \/ Transregio 7 ``Gravitational Wave Astronomy'' (V.S.),  \nRussian Foundation for Basic Research (grant  09-02-97013-p-povolzhe-a, V.S.), and \nthe Academy of Finland (grant 127512, J.P.). \n\\end{theacknowledgments} \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nStochastic optimization problems of the following form\n\\begin{align}\\label{eq-main01}\n\\inf_{\\boldsymbol{\\beta} \\in \\mathcal D} \\rho^F(Y \\cdot \\boldsymbol{\\beta}^\\top \\mathbf X)\n\\end{align}\narise naturally from many machine learning (ML) and operations research (OR) applications,\nwhere $F$ represents the joint distribution of random variables $(Y,\\mathbf X) \\in \\{-1,1\\} \\times {\\mathbb R}^n$, and $\\boldsymbol{\\beta}^\\top \\mathbf X$ is a random variable linearly depending on the decision variable $\\boldsymbol{\\beta}$ and can be interpreted generally as an affine decision rule. The binary random variable $Y$, taking values from $\\{-1,1\\}$, occurs most often in ML to represent binary outcomes\nfor a classification problem. In the setup where $Y$ is constant, the formulation \\eqref{eq-main01} covers a wide array of regression problems in ML and risk minimization problems in OR. The function $\\rho^F$ generally represents a measure of risk that maps a given random variable to a real value that quantifies the riskiness of the random variable. The notion of risk in this paper is broadly defined as the undesirability of a random variable $Z$, i.e., $Z$ with a larger value of $\\rho^F(Z)$ is less preferable. In ML, the measure $\\rho^F$ typically takes the form of an expected function, i.e.\n$ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$,\nwhere $\\ell$ represents a loss function used to penalize regression or classification errors and the expected penalty $\\mathbb{E}^F[\\ell(Z)]$ is considered as the risk of prediction errors. In OR applications, the function $\\ell$ often represents a disutility function and the expected disutility is considered as a measure of risk. In some important ML and OR applications, however, the measure $\\rho^F$ can go beyond an expected function. This is the case for example of $\\nu$-support vector machine (\\cite{SSWB00}) in ML, or Conditional Value-at-Risk (CVaR) minimization in portfolio optimization, where the measure $\\rho^F$ is a non-expected function, i.e. nonlinear in distributions.\n\nIn this paper, we study the distributionally robust counterpart of  \\eqref{eq-main01}, i.e.\n\\begin{align}\\label{eq-main02}\n\\inf_{\\boldsymbol{\\beta} \\in \\mathcal D} \\sup_{F\\in {\\cal B}(F_0,\\epsilon)} \\rho^F(Y \\cdot \\boldsymbol{\\beta}^\\top \\mathbf X),\n\\end{align}\nfor a broad class of measures $\\rho^F$, where ${\\cal B}(F_0,\\epsilon)$ denotes a ball of distributions centred at a reference distribution $F_0$ with a radius $\\epsilon$. In particular, we consider the case where the distance between a distribution $F$ and the reference distribution $F_0$ is measured according to the optimal transportation cost of moving the probability mass from $F_0$ to $F$, also known as the Wasserstein distance (\\cite{K42}, \\cite{V09}). The transportation cost of moving a unit mass between any two points $\\xi_1, \\xi_2 \\in \\mathbb{R}^n$ is most often calculated by the norm $\\Vert \\xi_1-\\xi_2\\Vert^p$ for some order $p \\geq 1$, and the corresponding optimal transportation cost is called the type-$p$ Wasserstein distance (\\cite{K19}). Throughout this paper, a ball ${\\cal B}_p(F_0,\\epsilon)$ defined based on the type-$p$ Wasserstein distance is called type-$p$ Wasserstein ball. While there are several other ways to measure the distance between two distributions and define the ball ${\\cal B}(F_0,\\epsilon)$, such as $\\phi$-divergence (\\cite{B13}, \\cite{HH18}, \\cite{JG16}), the optimal transportation cost has become an increasingly popular distance measure in both OR and ML, given its many desirable theoretical properties (see e.g. \\cite{GK16}, \\cite{EK18}). The distributionally robust optimization problem \\eqref{eq-main02} with a type-$p$ Wasserstein ball ${\\cal B}_p(F_0,\\epsilon)$ has been studied extensively for the case where $\\rho^F$ is an expected function, i.e. $ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ (see e.g. \\cite{K19}, \\cite{SKE19}, \\cite{GCK17}, \\cite{G22}).\n\nThere are two key findings in this stream of works. First, in the case where the empirical distribution is chosen as the\nreference distribution $F_0$, the Wasserstein distributionally robust optimization model \\eqref{eq-main02}, with ${\\cal B}(F_0,\\epsilon):={\\cal B}_p(F_0,\\epsilon)$ and $\\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$, can enjoy in some settings generalization bounds, i.e.  upper confidence bounds on the out-of-sample performances (see e.g. \\cite{SKE19}, \\cite{G22}). In particular, \\cite{G22} recently shows that in the case where the Wasserstein ball ${\\cal B}_p(F_0,\\epsilon)$ is of type-1 or 2, i.e. $p=1,2$, the bounds can be established with the radius $\\epsilon$ of the Wasserstein ball ${\\cal B}_p(F_0,\\epsilon)$ chosen in the square-root order $N^{-1\/2}$, where $N$ denotes the sample size. These bounds are in sharp contrast with, and much less conservative than, the bounds derived directly from the measure concentration property of Wasserstein distance (\\cite{EK18}). The latter require the radius $\\epsilon$ to be chosen in the order of $N^{-1\/\\max\\{2,n\\}}$, where $n$ denotes the dimension of the random vector in \\eqref{eq-main02}, and is suffered from the curse of dimensionality (\\cite{SKE19}). Second, in the case where the Wasserstein ball is of type-1, i.e. $p=1$, the Wasserstein distributionally robust optimization model \\eqref{eq-main02} with $\\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ has been found equivalent to the nominal problem \\eqref{eq-main01} with the addition of a regularizer on the decision variable $\\boldsymbol{\\beta}$ when the loss function $\\ell$ is Lipschitz continuous (\\cite{SKE19}). \\cite{BKM19} shows a similar equivalence relation holds when the Wasserstein ball is of type-2, i.e. $p=2$, and the loss function $\\ell$ is a square function. This relation to the classical regularization scheme, commonly applied in ML, provides a powerful interpretation of the distributionally robust optimization model and has stimulated considerable interest in its applications in ML and OR (see e.g. \\cite{BK21}, \\cite{BKM19}, \\cite{CP18}, \\cite{GCK17}, \\cite{GCK22}).\n\n\nIt remains largely unclear, however, whether these findings, particularly the generalization bounds, can be carried over to the general setting of  \\eqref{eq-main02}, where the ball ${\\cal B}(F_0,\\epsilon)$ is a general type-$p$ Wasserstein ball ${\\cal B}_p(F_0,\\epsilon)$, $p \\in [1,\\infty]$ and $\\rho^F$ is a general measure of risk, i.e. potentially a non-expected function. The question of how to bound out-of-sample risk for a general measure of risk $\\rho^F$ arises naturally from many risk minimization problems (see e.g. \\cite{PDB16}, \\cite{L18}, and references therein) or regression\/classification problems that entail the use of a risk measure (see e.g. \\cite{RU13}, \\cite{RUZ08}, \\cite{GU17}). To date, however, even the notion of generalization bounds has not been well defined in the literature for a general measure of risk $\\rho^F$. It is perhaps doubtful also whether generalization bounds can be established for a general measure of risk $\\rho^F$ without suffering from the curse of dimensionality, given that such bounds are only known for the special case of type-1 and 2 Wasserstein ball and an expected function $\\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ (\\cite{G22}), and that a general measure $\\rho^F$ can have distinctly different properties than an expected function. In fact, even in the literature of ML, the question of how to obtain generalization bounds, possibly through the classical regularization scheme, for the nominal problem \\eqref{eq-main01} in general has been largely left open (see e.g. \\cite{SB14}). This may have to do with the fact that existing analyses in Wasserstein DRO or ML for building generalization bounds rely heavily on exploiting the properties of expected function. As a key contribution of this paper, we show how to establish generalization bounds for the general model \\eqref{eq-main02} and bypass the curse of dimensionality by leveraging the structure of affine decision rules. Our result may be viewed as a generalization of the result in \\cite{SKE19}, who study the special case of \\eqref{eq-main02} with the type-1 Wasserstein ball and an expected function as the measure $\\rho^F$. Their approach to derive \ngeneralization bounds with the $N^{-1\/2}$-rate, i.e. with the radius $\\epsilon$ chosen in the square-root order, is fundamentally different from ours and can hardly be applicable beyond this special case, since it relies heavily on the equivalency of this special case to a regularized model. \\cite{G22} is a recent effort on studying how to break the curse of dimensionality in studying generalization bounds for a more general setting of Wasserstein DRO. While the setting in  \\cite{G22} is more general than ours in that it does not impose the structure of affine decision rules, it is more restrictive in that the measure $\\rho^F$ is largely limited to an expected function. The approach of \\cite{G22} not only relies on the properties of expected function, but also applicable only to type-1 and 2 Wasserstein balls. Given the reliance of existing approaches on the properties of expected function, one may suspect the availability of $N^{-1\/2}$-rate bounds may hinge heavily on the functional form of $\\rho^F$. Yet, as shown in this paper, it turns out that the bounds can be established (almost) independently from the form of $\\rho^F$ and thus for the model \\eqref{eq-main02} in great generality. Our finding that the Wasserstein DRO model \\eqref{eq-main02} can break the curse of dimensionality virtually for any problem with affine decision rules offers a solid theoretical underpinning for justifying the power of Wasserstein DRO in general out-of-sample tests.\n\nAnother focus of this work is on investigating whether the other key finding of Wasserstein DRO, namely its equivalency to the classical regularization scheme in ML, can also be carried over to a more general setting in \\eqref{eq-main02}.\nTo answer this, we pay particular attention first to the setting where the Wasserstein ball ${\\cal B}(F_0,\\epsilon)$ is of any type, i.e. $p \\in [1,\\infty]$, and the measure $\\rho^F$ is an expected function $ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ with a loss function $\\ell$ having a growth order $p$. The setting includes, as a special case, the distributionally robust least square regression problem studied in \\cite{BKM19} when $p=2$ and the loss function $\\ell$ is a square function, and many other importance instances of classification, regression, and risk minimization problems (see Section \\ref{rcr}).  We show that for many loss functions $\\ell$ arising from practical applications, it is possible to build an exact relation between the Wasserstein DRO model \\eqref{eq-main02} and the classical regularization scheme in more general terms. To demonstrate the generality of our result, we prove something stronger - there is no loss function $\\ell$, other than the ones we identify, under which the exact relation can hold. This gives a definite answer as to how far one can interpret the Wasserstein DRO model from a classical regularization perspective. Our regularization results reveal also the tractability of solving the Wasserstein DRO model for many non-Lipschitz continuous loss functions $\\ell$ and higher-order Wasserstein balls, i.e. $p>1$. It came to our attention that the recent work of \\cite{SZZ21} proposes to solve Wasserstein distributionally robust classification and regression problems for non-Lipschitz continuous loss functions $\\ell$ and higher order Wasserstein balls. Yet, their focus is on solving the problems by an approximation approach, given the challenge of identifying tractable reformulations for the Wasserstein DRO problems. Our results, on the other hand, show the cases that can be solved exactly via regularization reformulations. The results also enable us to discover the equivalency relation in the setting where the measure $\\rho^F$ is a non-expected function. To the best of our knowledge, in this setting, the exact relation between the Wasserstein DRO model \\eqref{eq-main02} and classical regularization has been known only for the case where the measure $\\rho^F$ is variance (\\cite{BCZ22}) or a distortion risk measure (\\cite{W14}). We show that the equivalency relation exists for a larger family of measures, including for instance higher-order risk measures (\\cite{K07}) and other measures emerging more recently from the literature of OR and ML (\\cite{RU13}, \\cite{RUZ08}, \\cite{GU17}).\n\n\n\\subsection*{Related work}\n\n\\emph{From the perspective of generalization bounds}.\nA series of works done by \\cite{BKM19}, \\cite{BK21}, \\cite{BMS22} take a different approach to tackle the curse of dimensionality. They study the classical setting of Wasserstein DRO, where the measure $\\rho^F$ is an expected function, and propose a radius selection rule for Wasserstein balls. They show the rule can be applied to build a confidence region of the optimal solution, and the radius can be chosen in the square-root order as the sample size goes to infinity. Although this allows for bypassing the curse of dimensionality, the bounds derived from the rule are only valid in the asymptotic sense. \\cite{BCZ22} also takes this approach to obtain generalization bounds for mean-variance portfolio selection problems. On the other hand, the generalization bounds established in this paper, like those in \\cite{SKE19}, \\cite{CP18}, and \\cite{G22}, break the curse of dimensionality in a non-asymptotic sense, i.e. applicable to any finite sample size and dimension.\n\n\n\n\n\n\\emph{From the perspective of the equivalency between Wasserstein DRO and regularization}. While the focus of this work is on studying the exact equivalency between Wasserstein DRO and regularization, there is an active stream of works studying the asymptotic equivalence in the setting where the measure $\\rho^F$ is an expected function\n(see \\cite{GCK17, BKM19, BMS22, VNSDMS18, BDSW20}). In particular, \\cite{GCK22} introduce the notion of variation regularization and show that for a broad class of loss functions, Wasserstein DRO is asymptotically equivalent to a variation regularization problem.\n\n\n\n\nThe rest of the paper is organized as follows.  In Section \\ref{WDRO}, we start with some preliminaries regarding Wasserstein distributionally robust optimization and then provide a list of examples for different measures $\\rho^F$. In Section \\ref{Gen}, we provide generalization bounds for the general formulation \\eqref{eq-main02}. We perform in Section \\ref{reg} a systematic study of the equivalency between the distributionally robust optimization model \\eqref{eq-main02} and the regularized versions of the problem \\eqref{eq-main01}. Section \\ref{sec:conclusion} concludes the paper.\n\n\\section{Wasserstein Distributionally Robust Optimization Model} \\label{WDRO}\n\\subsection{Preliminaries}\\label{sec:preliminaries}\nLet  $(\\Omega, \\mathcal A,\\mathbb{P})$ be an atomless probability space.  A random vector $\\xi$ is a measurable mapping from $\\Omega$ to $\\mathbb{R}^{n+1}$, $n\\in\\mathbb{N}$. Denote by $F_{\\xi}$ the distribution of  $\\xi$ under $\\mathbb{P}$.\nFor $p\\ge 1$, let $\\mathcal{M}_p:=\\mathcal{M}_p(\\Xi)$ be the set of all distributions on $\\Xi\\subseteq\\mathbb{R}^{n+1}$ with finite $p$th moment in each component. Denote $q$ as the H\\\"{o}lder conjugate of $p$, i.e. $1\/p+1\/q=1$.\nRecall that given any two distributions $F_1 \\in \\mathcal{M}_p$ and $F_2 \\in \\mathcal{M}_p$, the type-$p$ Wasserstein metric is defined as\n\\begin{align} \\label{eq:dwasser}\nW_{d,p}\\left(F_1, F_2\\right):= \\left( \\inf_{\\pi \\in \\Pi(F_1,F_2)}\n\\mathbb{E}^\\pi [d(\\xi_1, \\xi_2)^p] \\right)^{ {1}\/{p}},\n\\end{align}\nwhere  $d(\\cdot,\\cdot):\\Xi\\times\\Xi\\to\\mathbb{R}_{\\geq 0}\\cup\\{+\\infty\\}$ is a metric on $\\Xi$.\nThe set $\\Pi(F_1,F_2)$ denotes the set of all joint distributions of $\\xi_1 \\in \\Xi$ and $\\xi_2 \\in \\Xi$ with marginals $F_1$ and $F_2$ respectively. The metric is often interpreted as the minimal transportation cost of moving the mass from the distribution $F_1$ to the distribution $F_2$ with the cost calculated according to  the chosen function $d( \\xi_1, \\xi_2)^p$.\n\nConsider now the distributionally robust optimization problem \\eqref{eq-main02}, where a ball of distributions\nneeds to be defined for random variables $\\xi=(Y, \\mathbf X) \\in \\Xi$ with $\\Xi=\\{-1,1\\} \\times {\\mathbb R}^n \\subseteq \\mathbb{R}^{n+1}$. We apply the type-$p$ Wasserstein metric \\eqref{eq:dwasser} with $d( \\xi_1 ,\\xi_2 )= d ((Y_1,\\mathbf X_1),(Y_2,\\mathbf X_2)) $, defined by the following additively separable form\n\\begin{align}\\label{eq-classificationmetric}\nd((Y_1,\\mathbf X_1), (Y_2,\\mathbf X_2)) :=\n\\Vert \\mathbf X_1-\\mathbf X_2 \\Vert + \\Theta(Y_1-Y_2),\n\\end{align}\nwhere $\\Vert \\cdot \\Vert$ is any given norm on $\\mathbb{R}^n$ with its dual norm $\\|\\cdot\\|_*$ defined by $\\|\\mathbf y\\|_*=\\sup_{\\|\\mathbf x\\|\\le 1}\\mathbf x^\\top \\mathbf y$, and $\\Theta: \\mathbb{R} \\rightarrow \\{0,\\infty\\}$ satisfies $\\Theta(s) = 0$ if $s=0$ and $\\Theta(s) = \\infty$ otherwise. That is, the function \\eqref{eq-classificationmetric} assigns an infinitely large cost to any discrepancy in $Y$, i.e. $Y_1-Y_2 \\neq 0$, and reduces to a general norm on $\\mathbf X$ when there is no discrepancy in $Y$, i.e. $Y_1-Y_2=0$. With this form of norm, we define the ball of distribution $\\overline{{\\cal B}}_p(F_0,\\epsilon)$ by\n\\begin{align}\\label{eq-WU-multi}\n\t\\overline{{\\cal B}}_{p}(F_0,\\epsilon) =\\left\\{F\\in \\mathcal{M}_p: W_{d,p}(F,F_0) \\le\\epsilon\\right\\},\n\\end{align}\nand call it type-$p$ Wasserstein ball throughout this paper. In the remainder of this paper, we will show that the distributionally robust optimization problem \\eqref{eq-main02} with the above definition of Wasserstein ball, i.e.\n\\begin{align}\\label{eq-main05}\n\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\sup_{F\\in \\overline{{\\cal B}}_{p}(F_0,\\epsilon)} \\rho^F(Y \\cdot \\boldsymbol{\\beta}^\\top \\mathbf X)\n\\end{align}\ncan enjoy generalization guarantees for a general measure $\\rho^F$ and any type-$p$ Wasserstein ball. Thereafter, we will shed light on this general guarantee by drawing the connection between the distributionally robust optimization model \\eqref{eq-main05} and the classical regularization scheme in Section \\ref{reg}.\n\nWe first highlight in the next section the generality of the model \\eqref{eq-main05} to accommodate a wide array of measures $\\rho^F$ arising from machine learning and operations research applications.\n\n\\subsection{Classification, Regression, and Risk Minimization} \\label{rcr}\n\n\n\\subsubsection{Classification}\nThe distributionally robust optimization model \\eqref{eq-main05} can be directly applied to classification problems in ML, where the random variable $Y \\in \\{-1,1\\}$ is often termed the output and the random variables $\\mathbf X \\in {\\mathbb R}^n$ are considered as the input. The use of a linear scoring function $\\boldsymbol{\\beta}^{\\top}\\mathbf X$ remains the most popular approach in classification to predict the sign of the output $Y$, and the product $Y \\cdot \\boldsymbol{\\beta}^\\top \\mathbf X$ would be positive if the prediction is correct and negative otherwise. The set $\\mathcal D$ encodes the prior knowledge of the decision variables $\\boldsymbol{\\beta}$, also called the weight variables. Distributionally robust classification problems  \\eqref{eq-main05} that minimize the worst-case expected prediction error, i.e.  $ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$, have been studied in \\cite{SKE19} for the case where $p=1$, i.e. the type-1 Wasserstein ball, and the loss function $\\ell$ is convex Lipschitz continuous. We present several examples of $\\rho^F$ below for classification, where either the loss function $\\ell$ is not\nLipschitz continuous or the measure $\\rho^F$ is not an expected function.\n\n\\paragraph{Example (i): Higher-order hinge loss}\n$\\rho(Z) = \\mathbb{E}[(1-Z)_+^s]$, $s \\geq 1$.\n~\n\nIn the case $s=2$, the classification model is also known as smooth support vector machine (SSVM), first proposed in \\cite{LM01}.\n\n\\paragraph{Example (ii): Higher-order SVM}\n$\\rho(Z) = \\mathbb{E}[|1-Z|^s]$, $s \\geq 1$.\n~\n\nIn the case $s=2$, the classification model is widely known as least-squares support vector machine (LS-SVM) (\\cite{SV99}).\n\n\\paragraph{Example (iii): Sum-Exp}\n$\\rho(Z) = \\frac{1}{t}\\left( \\mathbb{E}[e^{-tZ}]\\right)$, $t > 0$.\n~\n\nThis measure appears in the popular classification model, AdaBoost (\\cite{FS97}).\n\n\\paragraph{Example (iv): $\\nu$-support vector machine}\n$\\rho(Z) = {\\rm CVaR}_\\alpha(-Z)$, $\\alpha\\in(0,1)$, where CVaR denotes the conditional value-at-risk (\\cite{RU02}) defined by\n\\begin{align*}\n\t {\\rm CVaR}_\\alpha(Z)=\\frac{1}{1-\\alpha}\\int_\\alpha^1 F^{-1}_Z(s)\\mathrm{d} s,\n\\end{align*}\nwhere $F_Z^{-1}$ is the quantile function of $Z$.  It is known that $\\nu$-support vector machine, introduced in the seminal work of \\cite{SSWB00}, employs CVaR as the measure (see e.g. \\cite{GU17}).\n\n\n\n\\subsubsection{Regression}\nThe distributionally robust optimization model \\eqref{eq-main05} can also be applied to regression problems in ML. Namely, without loss of generality, the output in regression problems can be represented by the first random variable, denoted by\n $X_{1}$, in $\\mathbf X$ given that the output in regression takes real values, whereas the input can be represented by the rest of the random variables, denoted by $\\mathbf X_{(2,:)}$, in $\\mathbf X$. By setting $\\boldsymbol{\\beta}:= (1, - \\boldsymbol{\\beta}_r)$, $\\boldsymbol{\\beta}_r \\in \\mathcal D$, and $F_0$ be a reference distribution satisfying $F_0(\\{Y=1\\})=1$\nin \\eqref{eq-main05}, we arrive at the following distributionally robust regression model\n\\begin{align}\\label{regression}\n\\inf_{\\boldsymbol{\\beta}_r\\in \\mathcal D}\\sup_{F\\in \\overline{{\\cal B}}_{p}(F_0,\\epsilon)} \\rho^F((1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X).\n\\end{align}\nThe above model seeks a linear regressor $\\boldsymbol{\\beta}_r^{\\top} \\mathbf X_{(2,:)}$ to predict the output $X_1 \\in \\mathbb{R}$ from the input\n $X_{(2,:)}$. It has been studied in \\cite{SKE19} for the case where $p=1$, i.e. the type-1 Wasserstein ball, and $ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ with a convex Lipschitz continuous loss function $\\ell$. Several other examples of $\\rho^F$ are presented below.\n\n\n\n\n\n\n\n\\paragraph{Example (i): Higher-order regression}\n$\\rho(Z) = \\mathbb{E}[|Z|^s]$, $s\\ge 1$.\n~\n\nIn the case $s=2$, the regression model is the well-known least-square regression.\n\n\n\\paragraph{Example (ii): Higher-order $c$-insensitive regression}\n$\\rho(Z) = \\mathbb{E}[(|Z|-c)_+^s]$, $s\\ge 1$ and $c\\ge 0$.\n~\n\nThe regression model is the well-known $c$-insensitive support vector regression ($c$-SVR) (\\cite{DBKSV97}) in the case $s=1$ and $c$-smooth support vector regression ($c$-SSVR) (\\cite{LHH05}) in the case $s=2$.\n\n\\paragraph{Example (iii): $\\nu$-support vector regression }\n$\\rho(Z) = {\\rm CVaR}_\\alpha(|Z|)$, $\\alpha\\in(0,1)$.\n~\n\n$\\nu$-support vector regression (\\cite{S98}) is a popular alternative to the $c$-insensitive support vector regression. It allows for bypassing the difficulty of specifying the insensitivity parameter $c$ in the $c$-insensitive support vector regression.\n\n\n\\subsubsection{Risk minimization}\nThe distributionally robust optimization model \\eqref{eq-main05} can also accommodate a general risk minimization problem, taking the form of\n\\begin{align}\\label{eq-mainp}\n\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\sup_{F\\in \\overline{{\\cal B}}_{p}(F_0,\\epsilon)} \\rho^F(\\boldsymbol{\\beta}^\\top \\mathbf X).\n\\end{align}\nThe most classical example of a risk minimization problem is portfolio optimization, in which case the variable $\\mathbf X$ represents a random vector of losses from $d$ different financial assets and $\\bm \\beta$ denotes a portfolio vector.\n\n\\paragraph{Example (i): Lower partial moments (LPM)} $\\rho(Z) = \\mathbb{E}[(Z-c)_+^s]$, $s\\ge 1$ and $c\\in\\mathbb{R}$.\n~\n\nLower partial moments represent an important class of downside risk measure, first introduced by \\cite{B75} and \\cite{F77} and more recently studied by \\cite{CSS11} in DRO.\n\n\\paragraph{Example (ii): CVaR-Deviation} $\\rho(Z) = {\\rm CVaR}_\\alpha(Z-\\mathbb{E}[Z])$, $\\alpha\\in(0,1)$.\n\nThis represents an important example of deviation measures built upon risk measures (\\cite{RUZ08}). Its generalization can be found in Section \\ref{reg}.\n\n\n\n\\paragraph{Example (iii): Higher moment coherent risk measures} $\\rho(Z) = \\inf_{t\\in\\mathbb{R}}\\{t+c(\\mathbb{E}[(Z-t)_+^s])^{1\/s}\\}$, $s,c\\ge 1$.\n\nThis measure is a well-known generalization of CVaR that is closely related to lower partial moments (LPM) and\ncompatible with the second order stochastic dominance and utility theory (\\cite{K07}).\n\n\n\n~\n\n\n\n\nWe summarize in Table \\ref{tab-R} a list of measures $\\rho^F$ considered in this paper.\n\n\n\n\n\n\n\n\\begin{table}\n\\caption{Risk functions analyzed in this paper.}\\label{tab-R}\\def2{2}\n\\begin{center}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{llc}\n\\hline\n \\textbf{Applications} &    \\textbf{Risk function}   & \\textbf{Formulation}\\\\[8pt]\n\\hline\n\\textbf{Classification} & \\makecell[l]{Higher-order hinge loss\\\\[5pt] Higher-order SVM\\\\\n[5pt] Sum-Exp \\\\\n[5pt] $\\nu$-support vector machine} & \\makecell{$\\mathbb{E}[(1-Z)_+^s]$, $s\\ge 1$\\\\[5pt] $\\mathbb{E}[|1-Z|^s]$, $s\\ge 1$\\\\[5pt] $\\frac1t \\mathbb{E}[e^{-tZ}]$, $t>0$\\\\[5pt] ${\\rm CVaR}_\\alpha(-Z)$, $\\alpha\\in(0,1)$}\\\\\n\\hline\n\\textbf{Regression} & \\makecell[l]{Higher-order regression\\\\[5pt] Higher-order $c$-insensitive\\\\[5pt] $\\nu$-support vector regression}      &\\makecell{$\\mathbb{E}[|Z|^s]$, $s\\ge 1$\\\\[5pt] $\\mathbb{E}[(|Z|-c)^s_+]$, $s\\ge 1$, $c\\ge 0$\\\\[5pt] ${\\rm CVaR}_\\alpha(|Z|)$, $\\alpha\\in(0,1)$}\\\\[5pt]\n\\hline\n\\textbf{Risk minimization} & \\makecell[l]{Lower partial moments\\\\[5pt] CVaR-Deviation\\\\[5pt] Higher moment risk measure\n}  &  \\makecell{$\\mathbb{E}[(Z-c)_+^s]$, $s\\ge 1$, $c\\in\\mathbb{R}$\\\\[5pt]  ${\\rm CVaR}_\\alpha(Z-\\mathbb{E}[Z])$, $\\alpha\\in(0,1)$\\\\[5pt] $\\inf_{t\\in\\mathbb{R}}\\{t+c(\\mathbb{E}[(Z-t)_+^s])^{1\/s}\\}$, $s,c\\ge 1$\n}  \\\\\n\\hline\n\\end{tabular}\n}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\n\n\\section{Generalization bounds} \\label{Gen}\nWasserstein distributionally robust optimization is most commonly applied in a data-driven setting, where the joint distribution $F$ of random variables $(Y,\\mathbf X) \\in \\{-1,1\\} \\times {\\mathbb R}^n$ can only be partially observed through  sample data $(\\widehat{y}_i,\\widehat{\\mathbf x}_i)$, $i=1,...,N$, independently drawn from $F$. In this setting, the empirical distribution,\ni.e. $\\widehat{F}_N:= \\frac{1}{N} \\sum_{i=1}^N \\delta_{(\\widehat{y}_i,\\widehat{\\mathbf x}_i)}$, is chosen as the reference distribution $F_0$ in Wasserstein DRO where $\\delta_{\\mathbf x}$ denotes the point-mass at $\\mathbf x$. The key question underlying Wasserstein DRO is whether there exist upper confidence bounds on the out-of-sample performances, i.e. generalization bounds, that can scale gracefully in the dimensionality of the random vector $(Y,\\mathbf X)$, i.e. breaking the curse of dimensionality. Such out-of-sample performance guarantees are only known for the case where the Wasserstein ball is of order $p \\in [1,2]$ and the measure $\\rho$ is an expected function, i.e.  $ \\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ for some $\\ell$ (\\cite{G22}, \\cite{SKE19} and \\cite{CP18}). Existing analyses hinge on the specific choice of the order $p$ and loss functions $\\ell$.\n\nIn this section, we seek to answer the question of whether generalization bounds exist for a more general setting of the Wasserstein DRO model \\eqref{eq-main05}. Let $\\rho^F$ be a distribution-invariant measure, i.e. $\\rho^F(Z_1)=\\rho^F(Z_2)$ for any $Z_1 \\equiv_{F} Z_2$\nand $F^*$ denote the true distribution of $(Y,\\mathbf X)$. We say that generalization bounds exist if the following condition can hold with high probability\n\\begin{align*}\n\t\\rho^{F^*}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\le \\sup_{F\\in \\overline{{\\cal B}}_p(\\widehat{F}_N,\\epsilon_N)}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X)+\\tau_N,~~\\forall~\\bm \\beta\\in\\mathcal D,\n\\end{align*}\nwhere the radius $\\epsilon_N$ can decrease in the order of $1\/\\sqrt{N}$ (or $\\sqrt{(\\log N)\/N}$) and the residual $\\tau_N$ diminishes in the order of $1\/N$. The bounds, if exist, would break the curse of dimensionality. The order $1\/\\sqrt{N}$ for the radius follows closely the square root law.\n\nWe take two steps to study generalization bounds. First, in Section  \\ref{out-of-sample-risk} we study how to build a confidence bound on the out-of-sample performance for a single solution $\\bm \\beta$. Then in Section \\ref{union}, we extend the result to all $\\bm \\beta \\in\\mathcal D$ by building a union bound on the out-of-sample performances for all $\\bm \\beta \\in\\mathcal D$, i.e. the generalization bound. The first step is critical. As shown below, quite surprisingly, a confidence bound can be built for any measure $\\rho$ and a Wasserstein ball of any type.\n\n\n\n\n\\subsection{Confidence bounds on out-of-sample risks} \\label{out-of-sample-risk}\nThe notion of out-of-sample performance has not been well defined and studied for a general measure $\\rho$. Throughout this section, we adopt the following definitions.\n\\begin{definition}\nGiven a solution $\\bm \\beta \\in\\mathcal D$, its (Wasserstein) in-sample risk is defined by\n\\begin{equation} \\label{in-sample}\n\\sup_{F\\in \\overline{{\\cal B}}_p(\\widehat{F}_N,\\epsilon)}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X),\n\\end{equation}\nwhereas its out-of-sample risk is defined by\n$$\\rho^{F^*}(Y\\cdot {\\bm \\beta}^\\top \\mathbf X).$$\nLet $\\widehat{J}_N$ denote the in-sample risk achieved by the solution of the DRO model \\eqref{eq-main05}, i.e.\n\\begin{align}\\label{eq-JN}\n\\widehat{J}_N=\\inf_{\\bm \\beta\\in \\mathcal D}\\sup_{F\\in \\overline{{\\cal B}}_p(\\widehat{F}_N,\\epsilon)}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X),\n\\end{align}\nand $J_{oos}$ denote the out-of-sample risk of the solution, i.e.\n\\begin{align}\\label{eq-Joos}\nJ_{oos}=\\rho^{F^*}(Y\\cdot\\widehat{\\bm \\beta}_N^\\top \\mathbf X),\n\\end{align}\nwhere $\\widehat{\\bm \\beta}_N$ is the optimal solution of the problem \\eqref{eq-JN}.\n\\end{definition}\n\nWe show that under the following assumptions, the out-of-sample risk of a solution $\\bm \\beta \\in \\mathcal D$ can be bounded by its in-sample risk with high probability.\n\n\\begin{assumption}[Light-tailed distribution]\\label{assum-1}\nThere exists an exponent $a>p$ such that\n$$\nA:=\\mathbb{E}^{F^*}[\\exp({\\|\\mathbf X\\|^a})]<+\\infty.\n$$\n\\end{assumption}\n\n\\begin{assumption}\\label{assum-2}\nThe feasible set $\\mathcal D\\subseteq \\mathbb{R}^n$ is bounded, that is $U_{\\mathcal D}:=\\sup_{\\bm \\beta\\in \\mathcal D}\\|\\bm \\beta\\|_*<+\\infty$.\n\\end{assumption}\n\n\\begin{assumption}\\label{assum-3}\nThe feasible set ${\\mathcal D}\\subseteq \\mathbb{R}^n$ is away from the origin, that is $L_{\\mathcal D}:=\\inf_{\\bm \\beta\\in \\mathcal D}\\|\\bm \\beta\\|_*>0$.\n\\end{assumption}\n\nAssumption \\ref{assum-1} requires the tail of the distribution $F^*$ decays at an exponential rate, which is a common assumption in Wasserstein DRO. It enables invoking the measure concentration property of Wasserstein metrics (\\cite{FG15}).  Assumptions \\ref{assum-2} and \\ref{assum-3} impose some restrictions on the decision space ${\\mathcal D}$.\nIt is interesting to note that these two assumptions appeared also in the earlier work of  \\cite{SKE19}, even though the approach taken in \\cite{SKE19} to build $1\/\\sqrt{N}$-rate confidence bounds is fundamentally different from, and more restrictive than, our approach. Their approach applies only to the case where the Wasserstein ball is of type-1 and the measure $\\rho^F$ is an expected function  $\\rho^F(Z) = \\mathbb{E}^F[\\ell(Z)]$ with the loss function $\\ell$ assumed to be convex Lipschitz continuous. Our approach, on the other hand, allows for building $1\/\\sqrt{N}$-rate confidence bounds for any type of Wasserstein ball and measure $\\rho$.\n\n\n\nBefore delving into the details of our approach, we present first the confidence bounds obtained from our approach, as the main result of this section.\n\\begin{theorem}\\label{th-FSG}\nSuppose that Assumptions \\ref{assum-1}, \\ref{assum-2} and \\ref{assum-3} hold and $\\eta\\in(0,1)$.  By setting the radius $\\epsilon$ in problem \\eqref{in-sample} as $\\epsilon_{p,N}(\\eta)\/L_{\\mathcal D}$, where\n\\begin{align}\\label{eq-epsilon}\n\\epsilon_{p,N}(\\eta)=\\begin{cases}\n\\left(\\frac{\\log(c_1\\eta^{-1})}{c_2 N}\\right)^{1\/2},~~&\\text{if}~N\\ge \\frac{\\log(c_1\\eta^{-1})}{c_2},\\\\\n\\left(\\frac{\\log(c_1\\eta^{-1})}{c_2 N}\\right)^{p\/a},~~&\\text{if}~N< \\frac{\\log(c_1\\eta^{-1})}{c_2},\n\\end{cases}\n\\end{align}\nand $c_1,c_2$ are positive constants that only depend on $U_{\\mathcal D},$ $a$, $A$, and $p$\\footnote{Theorem 2 of \\cite{FG15} (in the one-dimension case) shows that for a distribution $F$ on $\\mathbb{R}$ satisfying $\nA:=\\mathbb{E}^{F}[\\exp({\\gamma \\|X\\|^a})]<+\\infty\n$, $a>p$,\nit holds that  \n\\begin{align*}\n\t\\mathbb{P}\\left(W_p\\left(\\widehat{F}_{N},F\\right)\\ge \\epsilon\\right)\\le \\begin{cases}\n\t\tc_1\\exp(-c_2 N\\epsilon^2)~~&\\text{if}~\\epsilon\\le 1,\\\\\n\t\tc_1\\exp(-c_2 N\\epsilon^{a\/p})~~&\\text{if}~\\epsilon> 1,\n\t\\end{cases}\n\\end{align*}\nwhere $\\widehat{F}_{N}$ is the empirical distribution based on the independent sample drawn from $F$, and  $c_1$ and $c_2$ are constants depending on $\\gamma, a,A,p$; see more details in Theorem 2 of \\cite{FG15}. The constants $c_1$ and $c_2$ in \\eqref{eq-epsilon} are exactly the same as those in Theorem 2 of \\cite{FG15} in one-dimension case with $\\gamma= U_{\\mathcal D}^{-a}$. See the more details in Appendix.}, we have\n\\begin{align*}\n\\mathbb{P}\\left(\\rho^{F^*}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\le \\sup_{F\\in \\overline{{\\cal B}}_p(\\widehat{F}_N,\\epsilon_{p,N}(\\eta)\/L_{\\mathcal D})}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\right)\\ge 1-\\eta,~~\\forall \\bm \\beta\\in \\mathcal D.\n\\end{align*}\nIn particular, letting $\\bm \\beta=\\widehat{\\bm \\beta}_N$ in the above equation yields\n\\begin{align} \\label{perf}\n\\mathbb{P}(J_{oos}\\le \\widehat{J}_N)\\ge 1-\\eta.\n\\end{align}\n\\end{theorem}\nOne can see that the above confidence bounds are essentially dimension-independent, and for any $N\\ge {\\log(c_1\\eta^{-1})}\/{c_2}$, the radius decreases in the order of $1\/\\sqrt{N}$. They provide also out-of-sample guarantee for the solution of the DRO model \\eqref{eq-JN}, i.e. \\eqref{perf}, for any choice of the type-$p$ Wasserstein ball and measure $\\rho$.\n\nThe approach we take to obtain the above general {dimension-independent} bounds consists of two steps. First, we reduce the Wasserstein ball defined over $(Y,\\mathbf X) \\sim F_{(Y,\\mathbf X)}$, with dimensionality $n+1$, to a Wasserstein ball defined over $Y\\cdot\\bm\\beta^{\\top}{\\bf X} \\sim F_{Y\\cdot\\bm\\beta^{\\top}\\bf X}$, with dimensionality $1$. Then, we derive confidence bounds by applying a measure concentration property of the Wasserstein metric to the one-dimensional Wasserstein ball. Our first step is formally summarized as follows. {To state it, we introduce the Wasserstein ball on $\\mathbb{R}^n$ with the metric $d(\\cdot,\\cdot)$ being a norm:\n\\begin{align}\\label{eq-WU-multi-d}\n\t{\\cal B}_{p}(F_0,\\epsilon) =\\left\\{F\\in \\mathcal{M}_p(\\mathbb{R}^n): W_{p}(F,F_0) \\le\\epsilon\\right\\},\n\\end{align}\nwhere  \\begin{align} \\label{eq:dwasser-d}\nW_{p}\\left(F_1, F_2\\right):= \\left( \\inf_{\\pi \\in \\Pi(F_1,F_2)}\n\\mathbb{E}^\\pi [\\|\\xi_1- \\xi_2\\|^p] \\right)^{ {1}\/{p}},\n\\end{align}\nand $\\Vert \\cdot \\Vert$ is the norm of \\eqref{eq-classificationmetric} on $\\mathbb{R}^n$ with its dual norm $\\|\\cdot\\|_*$. Without loss of generality, for $n=1$, i.e., on $\\mathbb{R}$, assume that $\\|\\cdot\\|=|\\cdot|$ is the absolute-value norm.\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem}\\label{thm:221010-1}\n\tFor $p\\in[1,+\\infty]$, $\\epsilon\\ge0$, $\\bm \\beta\\in\\mathbb{R}^n$, a distribution $F_0$ on $\\{-1,1\\} \\times {\\mathbb R}^n$, and $(Y_0,\\mathbf X_0)\\sim F_0$, we have\n\\begin{align}\n\t\\label{eq:221010-1}\n\t\\{ F_{Y\\cdot\\bf X}: F_{(Y,{\\bf X})} \\in \\overline{\\cal B}_p (F_0,\\epsilon)\\}\n\n\t={\\cal B}_p (F_{Y_0\\cdot\\bf X_0},\\epsilon)\n\\end{align}\nand\n\\begin{align}\n\t\t\\label{eq:221010-2}\n\t\t\t\\{F_{Y \\cdot \\bm \\beta^\\top \\mathbf X}: F_{(Y,\\mathbf X)}\\in \\overline{\\cal B}_p (F_0,\\epsilon)\\}\n\t\n\t\t=\\mathcal B_p\\left(F_{Y_0\\cdot\\bm\\beta^{\\top}\\bf X_0},\\epsilon\\|\\bm\\beta\\|_*\\right).\n\\end{align}\n\\end{theorem}\n\nNote that the above theorem will be invoked also in Section \\ref{reg} for studying the Wasserstein DRO model \\eqref{eq-main05} from a regularization perspective.\nBy Theorem \\ref{thm:221010-1}, we know that\n\\begin{align*}\n\\{F_{Y \\cdot \\bm \\beta^\\top \\mathbf X}: F_{(Y,\\mathbf X)}\\in \\overline{\\cal B}_p (\\widehat{F}_N,\\epsilon)\\}\n= \\mathcal B_p(\\widehat{F}_{N,\\bm \\beta},\\epsilon\\|\\bm \\beta\\|_*),\n\\end{align*}\nwhere $\\widehat{F}_{N,\\bm\\beta}=\\frac{1}{N}\\sum_{i=1}^N \\delta_{\\widehat{y}_i\\cdot \\bm\\beta^\\top\\widehat{\\mathbf x}_i}$.  If the one-dimensional Wasserstein ball $\\mathcal B_p(\\widehat{F}_{N,\\bm \\beta},\\epsilon\\|\\bm \\beta\\|_*)$ can contain the distribution $F^*_{\\bm\\beta} :=F_{Y^*\\cdot \\bm\\beta^\\top \\mathbf X^*}$, i.e. the distribution of a random variable projected from $(Y^*,\\mathbf X^*) \\sim F^*$, with high probability, this will then allow us to derive the confidence bounds in Theorem \\ref{th-FSG}. Our second step achieves this by applying the following measure concentration property of one-dimensional Wasserstein metric, built upon Theorem 2 of \\cite{FG15}.\n\n\n\n\\begin{lemma}\\label{th-MC}\nIf Assumptions \\ref{assum-1} and \\ref{assum-2} hold, then for any $\\bm \\beta\\in \\mathcal D$, $\\eta\\in(0,1)$ and $N\\ge 1$, we have\n\\begin{align*}\n\\mathbb{P}\\left(W_p\\left(\\widehat{F}_{N,\\bm\\beta},F^*_{\\bm\\beta}\\right)\\le \\epsilon_{p,N}(\\eta)\\right)\\ge1-\\eta,\n\\end{align*}\nwhere $\\widehat{F}_{N,\\bm\\beta}=\\frac{1}{N}\\sum_{i=1}^N \\delta_{\\widehat{y}_i\\cdot \\bm\\beta^\\top\\widehat{\\mathbf x}_i}$, $F^*_{\\bm\\beta} :=F_{Y^*\\cdot \\bm\\beta^\\top \\mathbf X^*}$ with $(Y^*,\\mathbf X^*) \\sim F^*$, and $\\epsilon_{p,N}$ is defined in \\eqref{eq-epsilon}.\n\\end{lemma}\n\nCombining the two steps, we obtain the confidence bounds in Theorem \\ref{th-FSG}.\n\n\n\n\\paragraph{Proof of Theorem \\ref{th-FSG}}\n \tDenote by $\\epsilon_N:=\\epsilon_{p,N}(\\eta)\/L_{\\mathcal D}$.\n\tThe result follows by noting that\n\t\\begin{align}\n  \\mathbb{P}\\left(\\rho^{F^*}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\le\n\\sup_{F\\in \\overline{\\cal B}_p(\\widehat{F}_N,\\epsilon_N)}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\right)\n&=  \\mathbb{P}\\left(\\rho^{F^*_{\\bm\\beta}}(Z)\\le\n\\sup_{F\\in \\{F_{Y \\cdot \\bm \\beta^\\top \\mathbf X}: F_{(Y,\\mathbf X)}\\in \\overline{\\cal B}_p(\\widehat{F}_N,\\epsilon_N)\\} }\\rho^{F}(Z)\\right) \\notag \\\\\n&\\ge \\mathbb{P}\\left(F^*_{\\bm\\beta}\\in\\{F_{Y \\cdot \\bm \\beta^\\top \\mathbf X}: F_{(Y,\\mathbf X)}\\in \\overline{\\cal B}_p (\\widehat{F}_N,\\epsilon_N)\\}\\right)\n\\notag\\\\\n\t&=\\mathbb{P}\\left(F^*_{\\bm\\beta}\\in \\mathcal  B_p\\left(\\widehat{F}_{N,\\bm\\beta},\\|\\bm \\beta\\|_*\\epsilon_{p,N}(\\eta)\/L_{\\mathcal D}\\right)\\right)  \\notag\\\\\n\t&=\\mathbb{P}\\left(W_p\\left(\\widehat{F}_{N,\\bm\\beta},F^*_{\\bm\\beta}\\right)\\le \\|\\bm \\beta\\|_*\\epsilon_{p,N}(\\eta)\/L_{\\mathcal D}\\right) \\nonumber\\\\\n\t&\\ge \\mathbb{P}\\left(W_p\\left(\\widehat{F}_{N,\\bm\\beta}\n\t,F^*_{\\bm\\beta}\\right)\\le \\epsilon_{p,N}(\\eta)\\right)\\notag\\\\\n\t&\\ge 1-\\eta\\notag,\n\t\\end{align}\nwhere the  second equality and the last inequality follow from Theorem \\ref{thm:221010-1} and Lemma \\ref{th-MC}, respectively.\\qed\n\nIt should be clear from above that one can also obtain naive confidence bounds by applying measure concentration results ({\\cite{FG15}) directly to {the probability $\\mathbb{P}(F^*\\in \\overline{\\mathcal  B}_{p}(\\widehat{F}_N,\\epsilon_N))$.}\nThis naive approach, unfortunately, suffers from the curse of dimensionality, with the radius scaled in the order of $O(N^{-\\min\\{p\/n,1\/2\\}})$ (see e.g., \\cite{EK18}).\n\n\n\n\n\n\n\n\\begin{remark}\nSuppose that the metric $d(\\cdot,\\cdot)$ on $\\{-1,1\\}\\times\\mathbb{R}^n$ is defined by\n\\begin{align} \\label{theta}\nd((Y_1,\\mathbf X_1),(Y_2,\\mathbf X_2)):=\\|\\mathbf X_1-\\mathbf X_2\\|+\\theta|Y_1-Y_2|\n\\end{align}\nfor some $\\theta\\in(0,\\infty]$, and we use the convention that $0\\cdot \\infty=0$.\nObviously, the metric in \\eqref{eq-classificationmetric} is a special case of \\eqref{theta} with $\\theta=\\infty$. Note that\nthe size of $\\overline{\\cal B}_p(F_0,\\epsilon)$ is smaller when $\\theta$ is larger. Therefore, the result of finite sample guarantee in Theorem \\ref{th-FSG} is also valid if one chooses to apply the metric \\eqref{theta} with $\\theta\\in(0,\\infty)$.\n\\end{remark}\n\n\\subsection{Union bounds} \\label{union}\nWith Theorem \\ref{th-FSG}, we are now ready to move on to building generalization bounds. The confidence bounds in Theorem \\ref{th-FSG} apply to any fixed decision $\\bm \\beta\\in \\mathcal D$. Generalization bounds require further establishing that the in-sample risk can bound the out-of-sample risk uniformly for all $\\bm \\beta\\in\\mathcal D$ with high probability. In the case where the set $\\mathcal D$ is finite, one can easily build generalization bounds for any type of Wasserstein ball and measure $\\rho$ by applying the union bound to Theorem \\ref{th-FSG}. Clearly, these union bounds remain to be dimension-independent and the radius required to reach any probability level remains in the same order.\n\nThe focus of this section is on the case where the set $\\mathcal D$ is not finite. We build generalization bounds from Theorem \\ref{th-FSG} by applying the standard covering number argument (see e.g., \\cite{G22}). Recall that for $\\tau>0$, an $\\tau$-cover of $\\mathcal D$, denoted by $\\mathcal D_\\tau$, is a subset of $\\mathcal D$ which satisfies that\nfor each $\\bm \\beta\\in \\mathcal D$, there exists $\\widetilde{\\bm \\beta}\\in\\mathcal D_\\tau$ such that $\\|\\widetilde{\\bm \\beta}-\\bm \\beta\\|_\\mathcal D\\le \\tau$, where $\\|\\cdot\\|_\\mathcal D$ is a norm on $\\mathcal D$.\nThe covering number $\\mathcal N(\\tau;\\mathcal D,\\|\\cdot\\|_\\mathcal D)$ of $\\mathcal D$ with respect to $\\|\\cdot\\|_\\mathcal D$ is defined as the smallest cardinality of an $\\tau$-cover of $\\mathcal D$.\n\nWe show that under the following assumption on the measure $\\rho$, one can obtain $1\/\\sqrt{N}$-rate generalization bounds for any type of Wasserstein ball and measure $\\rho$.\n\n\\begin{assumption}\\label{assum-4}\nLet $p\\in[1,+\\infty]$ and $\\rho: L^p\\to \\mathbb{R}$. Assume that there exist $M>0$ and $k\\in[1,p]$ such that\n\\begin{align*}\n\t|\\rho(Z_1)-\\rho(Z_2)|\\le M \\left(\\mathbb{E}[|Z_1-Z_2|^k]\\right)^{1\/k},~~\\forall Z_1,Z_2\\in L^p.\n\\end{align*}\n\\end{assumption}\n\n\\begin{theorem}\\label{th-unionrho}\nGiven that Assumptions \\ref{assum-1}-\\ref{assum-4} hold,  $\\eta\\in(0,1)$, and\n\t\\begin{align}\\label{eq-epsilonunion}\n\t\t\\epsilon_N:=\\epsilon_{p,N}\\left(\\frac{\\eta}{\\mathcal N(1\/N;\\mathcal D,\\|\\cdot\\|_*)}\\right)\\big\/L_{\\mathcal D},\n\t\\end{align}\nwith $\\epsilon_{p,N}(\\cdot)$ defined in \\eqref{eq-epsilon}, then\n\\begin{align*}\n\\mathbb{P}\\left(\t\\rho^{F^*}(Y\\cdot\\bm \\beta^\\top\\mathbf X)\\le \\sup_{F\\in \\overline{\\cal B}_p(\\widehat{F}_N,\\epsilon_N)}\\rho^{F}(Y\\cdot\\bm \\beta^\\top\\mathbf X)+\\tau_N,~~\\forall~\\bm \\beta\\in\\mathcal D \\right) \\ge 1-\\eta-\\frac1N,\n\\end{align*}\n where\n$\n\\tau_N= {M[2\\left(\\mathbb{E}^{F^*}[\\|\\mathbf X\\|^k]\\right)^{1\/k}+\\left({\\rm\\mathbb{V}ar}^{F^*}(\\|\\mathbf X\\|^k)\\right)^{1\/2k}+\\epsilon_N]}\/{N}.\n$\n\\end{theorem}\n\nIt is known that the covering number in \\eqref{eq-epsilonunion} satisfies $\\log(\\mathcal N(\\tau; \\mathcal D,\\|\\cdot\\|_*))\\le n\\log(1+2B\/\\tau)$, where  $B$ is the diameter of $\\mathcal D$, i.e. $B=\\sup_{\\boldsymbol{\\beta},\\widetilde{\\boldsymbol{\\beta}}\\in\\mathcal D}\\|\\boldsymbol{\\beta}-\\widetilde{\\boldsymbol{\\beta}}\\|_*$ (see Example 5.8 of \\cite{W19}). Moreover, we have $B\\le 2 U_{\\mathcal D}$, which follows from Assumption \\ref{assum-2}. With these facts, Theorem \\ref{th-unionrho} implies that the radius $\\epsilon_N$ can be chosen in the order of $\\sqrt{(\\log N)\/N}$ such that the in-sample risk can be an upper bound of the \nout-of-sample risk, up to a residual in the order of  $1\/N$, for all $\\boldsymbol{\\beta}\\in\\mathcal D$. That is, $1\/\\sqrt{N}$-rate generalization bounds exist. Assumption \\ref{assum-4} holds for a large class of measures $\\rho$ but does not hold for expected functions $\\rho(Z)=\\mathbb{E}[\\ell(Z)]$ when the loss function $\\ell$ is not Lipschitz continuous. We show below that if only expected functions are considered, one can still obtain $1\/\\sqrt{N}$-rate generalization bounds even if the loss function is not Lipschitz continuous, by relaxing Assumption \\ref{assum-4} to the following.\n\\begin{assumption}\\label{assum-EU}\nLet $p\\in[1,+\\infty]$ and $\\rho(Z)=\\mathbb{E}[\\ell(Z)]$, $Z\\in L^p$ for some $\\ell:\\mathbb{R}\\to\\mathbb{R}$.\nAssume that there exists a function $f:\\mathbb{R}^n\\to\\mathbb{R}_+$ and constant $a_1,a_2\\ge 0$ and $k\\in[1,p]$ satisfying $f(\\mathbf x)\\le a_1\\|\\mathbf x\\|^k+a_2$ for all $\\mathbf x\\in \\mathbb{R}^n$, such that\n\\begin{align*}\n\\left|\\ell(\\bm \\beta^\\top \\mathbf x)-\\ell({\\widetilde{\\bm \\beta}}^\\top \\mathbf x)\\right|\\le f(\\mathbf x)\\|\\bm \\beta-{\\widetilde{\\bm \\beta}}\\|_*,~~\\forall \\bm \\beta,\n{\\widetilde{\\bm \\beta}}\\in \\mathcal D.\n\\end{align*}\n\\end{assumption}\n\n\\begin{theorem}\\label{th-unionEL}\nGiven that Assumptions \\ref{assum-1}-\\ref{assum-3} and \\ref{assum-EU} hold and $\\eta\\in(0,1)$, then\n\\begin{align*}\n\\mathbb{P}\\left(\t\t\\mathbb{E}^{F^*}[\\ell(Y\\cdot\\bm \\beta^\\top\\mathbf X)]\\le \\sup_{F\\in \\overline{\\cal B}_p(\\widehat{F}_N,\\epsilon_N)}\\mathbb{E}^{F}[\\ell(Y\\cdot\\bm \\beta^\\top\\mathbf X)]+\\tau_N,~\\forall~\\bm \\beta\\in\\mathcal D\\right) \\ge 1-\\eta-\\frac1N,\n\t\\end{align*}\nwhere\n\t$\n\t\t\\tau_N=[{a_1(2^{k-1}+1)\\mathbb{E}^{F^*}[\\|\\mathbf X\\|^k]+a_1 2^{k-1}(\\sqrt{{\\rm\\mathbb{V}ar}^{F^*}(\\|\\mathbf X\\|^k)}+\\epsilon_N^k)+2a_2}]\/{N}\n\t$\nand $\\epsilon_N$ is defined by \\eqref{eq-epsilonunion}.\n\\end{theorem}\n\nBy the same reasoning as for Theorem \\ref{th-unionrho}, we can conclude from above that $1\/\\sqrt{N}$-rate generalization bounds exist. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A Regularization Perspective} \\label{reg}\nRegularization generally refers to any means that can be applied to avoid overfitting and enhance generalization. In this regard, with the generalization guarantees established in the previous section, the Wasserstein DRO model \\eqref{eq-main05} can be particularly well justified as a general regularization model. In this section, we seek to provide further insights as to the regularization effect of the model \\eqref{eq-main05} on the decision variable $\\boldsymbol{\\beta}$, which offers an alternative, and practically useful, interpretation to the model. In particular, it is known that in some settings Wasserstein DRO is equivalent to a regularized empirical optimization model in ML, with a norm regularizer capturing the regularization effect on the decision variable $\\boldsymbol{\\beta}$. We show that a similar equivalence relationship can be established more generally for many instances of the model \\eqref{eq-main05}, such as those discussed in Section \\ref{rcr}. In addition to explaining the regularization effect, the equivalence relationship reveals also how these instances, previously unknown for their tractability, can be solved efficiently via regularized empirical optimization models. Our results are general in that not only do they unify all the previously known equivalence relationships, they also provide necessary conditions under which such an equivalence relationship can exist. Theorem \\ref{thm:221010-1} greatly facilitates our analysis. Note first that by \\eqref{eq:221010-1} in the theorem, the Wasserstein DRO model \\eqref{eq-main05} can be recast as\n\\begin{align}\\label{eq-main5}\n\\inf_{\\boldsymbol{\\beta} \\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)}\\ \\rho^F(\\boldsymbol{\\beta}^\\top \\mathbf Z).\n\\end{align}\n\n\nIn the case where $\\rho^F$ is an expected function, the Wasserstein distributionally robust optimization model \\eqref{eq-main5}, i.e.\n\\begin{align}\\label{eq-main-ef}\n\\inf_{\\boldsymbol{\\beta} \\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\mathbb{E}^F[\\ell(\\boldsymbol{\\beta}^\\top \\mathbf Z)],\n\\end{align}\nis known to be equivalent to a regularized model of the form\n\\begin{align}\\label{eq-main21}\n\\inf_{\\boldsymbol{\\beta} \\in \\mathcal D} \\left\\{\\mathbb{E}^{F_0}[\\ell(\\boldsymbol{\\beta}^\\top \\mathbf Z)] + {\\rm Lip(\\ell)}\\epsilon\\Vert\\bm\\beta\\Vert_*\\right\\},\n\\end{align}\nwhen $p=1$, i.e. the type-1 Wasserstein ball, and ${\\rm Lip(\\ell)}$ is the Lipschitz constant of $\\ell$.\n\n\nIn the case where the Wasserstein ball is of a higher order, i.e. $p>1$, and\/or the loss function $\\ell$ is not Lipschitz continuous, the relationship between the model \\eqref{eq-main-ef} and a regularized model is largely unknown except the special case of $p=2$ and $\\ell(y) = y^2$ (\\cite{BKM19}). It remains largely open also whether \\eqref{eq-main-ef} in this case can be tractably solved. Higher-order Wasserstein balls can be attractive from a practical standpoint, given that they are less conservative than the type-1 Wasserstein ball. It is natural to wonder whether there exists an equivalence relationship between the model \\eqref{eq-main-ef} and a regularized model of the form \\eqref{eq-main21} in the case of a higher-order Wasserstein ball, i.e. $p>1$. We show below that such an equivalence relationship exists if and only if the loss function $\\ell$ is linear or takes the form of an absolute function.\n\n\\begin{theorem}\\label{th-RegularizationEL}\n\tLet $\\ell:\\mathbb{R}\\to\\mathbb{R}$ be a convex function. For $p\\in(1,\\infty]$, suppose that $\\mathbb{E}[|\\ell(Z)|]<+\\infty$ for all $Z\\in L^p$.\n\tThen the following statements are equivalent.\n\n\n\n\t\n\t\n\t\n\t%\n\n\n\n\n\n\t\\begin{itemize}\n\t\t\\item[(i)] For any $F_0\\in\\mathcal M_p(\\mathbb{R}^n)$, $\\epsilon\\ge 0$ and $\\mathcal D\\subseteq \\mathbb{R}^n$, we have\n\t\t$$\n\t\t\\inf_{\\bm\\beta\\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\mathbb{E}^F[\\ell(\\boldsymbol{\\beta}^\\top \\mathbf Z)]=\\inf_{\\bm\\beta\\in \\mathcal D} \\left\\{\\mathbb{E}^{F_0}[\\ell(\\boldsymbol{\\beta}^\\top \\mathbf Z)] + C\\epsilon \\Vert\\bm\\beta\\Vert_*\\right\\}\n\t\t$$\n\t\twith some $C >0$.\n\t\n\t\t\n\t\t\\item[(ii)] $\\ell$ has one of the following two forms:\n\t\t\n\t\t-$\\ell_1(x)=C x+b$ or $\\ell_1(x)=-C x+b$ with some $b\\in\\mathbb{R}$;\n\t\t\n\t\t-$\\ell_2(x)=C|x-m|+b$  with some $m,b\\in \\mathbb{R}$.\n\t\t\n\t\n\t\\end{itemize}\n\n\n\\end{theorem}\n\nThe above result, which applies to any type-$p$ Wasserstein ball, is perhaps surprising in that the Wasserstein DRO model is equivalent to the same regularized model, regardless of the order $p$. This turns out to be the case when the slope of the loss function $\\ell$ takes values only from a constant and its negative. The result indicates also that it would not be possible to obtain a regularized model in the form of \\eqref{eq-main21} for any other loss function $\\ell$. In other words, if there does exist a certain equivalence relationship between the Wasserstein DRO model \\eqref{eq-main-ef} and a regularized model for some other loss function $\\ell$, the regularized model must take some form other than \\eqref{eq-main21}. Before moving on to discussing other forms of regularization, it is worth noting first the following application of the above result in regression.\n\n\\begin{example} {\\bf (Regression)}\n~\n\n{\\bf - (Least absolute deviation (LAD) regression)}\nApplying $\\ell_2(x)=C|x-m|+b$ in Theorem \\ref{th-RegularizationEL} and setting $m=0$, $b=0$, and $C=1$, we arrive at the distributionally robust counterpart of the least absolute deviation regression, i.e.\n\\begin{align}\n\\inf_{\\boldsymbol{\\beta}_r\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)}\\mathbb{E}^F[|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|].\n\\end{align}\nIt is equivalent to\n\\begin{align*}\n\\inf_{\\bm\\beta_r\\in \\mathcal D}\\left\\{\\mathbb{E}^{F_0}[|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|] + \\epsilon \\Vert (1, -\\boldsymbol{\\beta}_r)\\Vert_*\\right\\},\n\\end{align*}\nfor any $p \\geq 1$.\n\\end{example}\n\nWe now turn our attention to exploring whether there exists any other form of regularization equivalent to the Wasserstein DRO model \\eqref{eq-main-ef}. In particular, we seek to address the case where the loss function $\\ell$ is not Lipschitz continuous, such as those loss functions of a high order arising from the applications discussed in Section \\ref{rcr}. It is known that the worst-case expectation problem defined based on the type-1 Wasserstein ball can be unbounded if the loss function is not Lipschitz continuous. To study the case that goes beyond Lipschitz continuous functions, we consider the following formulation of the Wasserstein DRO model \\eqref{eq-main-ef}\n\\begin{align}\\label{eq-main12}\n\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\mathbb{E}^F[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z)],\n\\end{align}\nwhere $\\ell^p$ denotes a Lipschitz continuous function $\\ell$ raised to the power of $p>1$, with the same order as that of the type-$p$ Wasserstein ball.\n\n\n\n\n\n\nAs the main result, we show below that for several loss functions $\\ell$ of practical interest, the Wasserstein DRO model \\eqref{eq-main12} is equivalent to an alternative form of regularization.\n\n\\begin{theorem}\\label{th-highorderloss}\nSuppose that $\\ell$ has one of the following four forms:\n\n-$\\ell_1(x)=(x-m)_+$ with some $m\\in\\mathbb{R}$;\n\n-$\\ell_2(x)=(x-m)_-$ with some $m\\in\\mathbb{R}$;\n\n-$\\ell_3(x)=(|x-m_1|-m_2)_+$ with some $m_1\\in\\mathbb{R}$ and $m_2\\ge 0$;\n\n-$\\ell_4(x)=|x-m|+b$ with some $m\\in\\mathbb{R}$ and $b>0$.\\\\\nThen for any $p\\in(1,\\infty)$, $F_0\\in\\mathcal M_p(\\mathbb{R}^n)$, $\\epsilon\\ge 0$ and $\\mathcal D\\subseteq \\mathbb{R}^n$, we have\n$$\n\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\mathbb{E}^F[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z)]=\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\left(\\left(\\mathbb{E}^{F_0}[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z)]\\right)^{1\/p} + \\epsilon||\\boldsymbol{\\beta}||_*\\right)^p.\n$$\n\\end{theorem}\n\nThe above result covers, quite remarkably, all the instances of expected functions in Section \\ref{rcr}, except Sum-Exp in classification.\n\n\\begin{example} {\\bf (Classification)} \\label{ex1}\n~\n\n{\\bf - (Higher-order hinge loss)}\nApplying $\\ell_2(x)=(x-m)_-$ and setting $m=1$, we have that the following classification problem with a higher-order hinge loss\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in \\overline{\\cal B}_{p}(F_0,\\epsilon)} \\mathbb{E}^F[(1-Y\\cdot \\bm \\beta^{\\top}\\mathbf X)_+^p]\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D} \\left(\\left(\\mathbb{E}^{F_0}[(1-Y\\cdot \\bm \\beta^{\\top}\\mathbf X)_+^p]\\right)^{1\/p}+ \\varepsilon ||\\bm \\beta||_* \\right)^p.\n\\end{align*}\n\n{\\bf - (Higher-order SVM)}\nApplying $\\ell_3(x)=(|x-m_1|-m_2)_+$ and setting $m_1=1$ and $m_2=0$, we have that the higher-order SVM classification problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in \\overline{\\cal B}_{p}(F_0,\\epsilon)} \\mathbb{E}^F[|1-Y\\cdot \\bm \\beta^{\\top}\\mathbf X|^p]\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D} \\left( (\\mathbb{E}^{F_0}[|1-Y\\cdot \\bm \\beta^{\\top}\\mathbf X|^p])^{1\/p}+ \\varepsilon ||\\bm \\beta||_* \\right)^p.\n\\end{align*}\n\\end{example}\n\n\\begin{example} {\\bf (Regression)} \\label{ex2}\n~\n\n{\\bf - (Higher-order error measure)}\nApplying $\\ell_3(x)=(|x-m_1|-m_2)_+$ and setting $m_1=0$ and $m_2=0$, we have that the regression with a higher order error measure\n\\begin{align*}\n\\inf_{\\bm\\beta_r\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)} \\mathbb{E}^F[|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|^p],\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta_r\\in \\mathcal D} \\left( \\left(\\mathbb{E}^{F_0}[|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|^p]\\right)^{1\/p}+ \\varepsilon \\|(1,-\\bm \\beta_r)\\|_* \\right)^p.\n\\end{align*}\n\n{\\bf - (Higher order $c$-insensitive measure)}\nApplying $\\ell_3(x)=(|x-m_1|-m_2)_+$ and setting $m_1=0$ and $m_2=c$, we have that the following regression problem with a higher order $c$-insensitive measure\n\\begin{align*}\n\\inf_{\\bm\\beta_r\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)} \\mathbb{E}^F[(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|-c)_+^p]\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta_r\\in \\mathcal D} \\left( \\left(\\mathbb{E}^{F_0}[(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|-c)_+^p]\\right)^{1\/p}+ \\varepsilon ||(1,-\\bm \\beta_r)||_* \\right)^p.\n\\end{align*}\n\\end{example}\n\n\\begin{example} {\\bf (Risk minimization)} \\label{ex3}\n~\n\n{\\bf - (Lower partial moments)}\nApplying $\\ell_1(x)=(x-m)_+$ and setting $m=c$, we have that the risk minimization with lower partial moments\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)} \\mathbb{E}^F[(\\bm\\beta^{\\top}\\mathbf X-c)_+^p]\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D} \\left( \\left(\\mathbb{E}^{F_0}[(\\bm\\beta^{\\top}\\mathbf X-c)_+^p]\\right)^{1\/p}+ \\varepsilon ||\\bm\\beta||_* \\right)^p.\n\\end{align*}\n\\end{example}\n\nWith these many examples covered by the theorem, it is perhaps tempting to conjecture that the equivalence relationship may hold more broadly for other loss functions $\\ell$. We provide a negative answer below.\n\n\\begin{theorem}\\label{th-necessity-highorderloss}\nLet $\\ell:\\mathbb{R}\\to\\mathbb{R}$ be a  nonnegative, Lipschitz continuous and convex function. For an integer $p\\in(1,\\infty)$, suppose that for any $F_0\\in\\mathcal M_p(\\mathbb{R}^n)$, $\\epsilon\\ge 0$ and $\\mathcal D\\subseteq \\mathbb{R}^n$, we have\n\\begin{equation} \\label{imposs}\n\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\mathbb{E}^F[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z)]=\\inf_{\\boldsymbol{\\beta}\\in \\mathcal D} \\left(\\left(\\mathbb{E}^{F_0}[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z)]\\right)^{1\/p} + C\\epsilon||\\boldsymbol{\\beta}||_*\\right)^p\n\\end{equation}\nwith some $C>0$. Then $\\ell$ must be one of the four forms in Theorem \\ref{th-highorderloss} multiplying a constant $C$.\n\\end{theorem}\n\nThis is an ``impossibility\" theorem, which puts to rest any effort attempting to draw the equivalence relation for other convex Lipschitz continuous functions $\\ell : \\mathbb{R} \\rightarrow \\mathbb{R}_{\\geq 0}$. This theorem should be of fundamental importance to the study of Wasserstein DRO, given the continuous interests and efforts in motivating Wasserstein DRO from a classical regularization perspective. It shows {\\it exactly} how far one can take this perspective.\n\n\nEven though Theorem \\ref{th-necessity-highorderloss} points out the impossibility to draw the equivalence relation\n\\eqref{imposs} for a more general loss function $\\ell$, Theorem \\ref{th-highorderloss} can in fact be applied more broadly, as a powerful basis, to derive alternative equivalence relations for a richer family of measure $\\rho$. In particular, there is a large family of measures $\\rho$ that can be expressed generally by the following two forms\n\\begin{align}\\label{eq-infEU\n\t\\rho^F(Z) = \\inf_{t \\in \\mathbb{R}} \\left\\{t+\\left(\\mathbb{E}^F[\\ell^p(Z,t)]\\right)^{1\/p}\\right\\}~~~\n {\\rm and}~~~\n\t\\mathcal V^F(Z) = \\inf_{t \\in \\mathbb{R}} \\mathbb{E}^F[\\ell^p(Z,t)]\n\\end{align}\nfor some loss functions $\\ell$,  such as the measure applied in $\\nu$-support vector regression, higher moment coherent risk measures in Section \\ref{rcr}, and variance.\n\n\nWe show in the appendix (see Lemma \\ref{lm-minmaxeq}) that for a wide range of loss functions $\\ell$, the following switching of $\\sup$ and $\\inf$ is valid\n$$\\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\inf_{t \\in \\mathbb{R}} \\pi_{i,\\ell}(F,t)=\n\\inf_{t \\in \\mathbb{R}} \\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)}\\pi_{i,\\ell}(F,t),~~i=1,2,$$\nwhere\n$$\n\\pi_{1,\\ell}(F,t)=t+\\left(\\mathbb{E}^F[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z,t)]\\right)^{1\/p} ~~{\\rm and}~~\\pi_{2,\\ell}(F,t)=\\mathbb{E}^F[\\ell^p(\\boldsymbol{\\beta}^\\top \\mathbf Z,t)].\n$$\nThis, combined with Theorem \\ref{th-highorderloss}, leads to the following.\n\n\n\n\\begin{corollary}\\label{co-1125-1}\nFor any $p\\in [1,\\infty)$ and $c>1$, let $\\rho^F$ be defined by \\eqref{eq-infEU}, where  $\\ell(z,t):=c\\ell(z-t)$, and $\\ell$ is one of  $\\ell_1,\\ell_3,\\ell_4$ in Theorem \\ref{th-highorderloss} or $\\ell(z,t)=c(|z|-t)_+$. \n\nIt holds that for $F_0\\in\\mathcal M_p(\\mathbb{R}^n)$ and $\\epsilon\\ge 0$\n$$\n \\inf_{\\boldsymbol{\\beta}\\in\\mathcal D}\\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)} \\rho^F (\\boldsymbol{\\beta}^\\top \\mathbf Z)=   \\inf_{\\boldsymbol{\\beta}\\in\\mathcal D}\\left\\{\\rho^{F_0}(\\boldsymbol{\\beta}^\\top \\mathbf Z) +  c \\epsilon||\\boldsymbol{\\beta}||_*\\right\\}.\n$$\n\\end{corollary}\n\n\n\n\\begin{example} {\\bf (Classification)} \\label{ex5}\nSetting $\\ell(z,t)=(z-t)_+$ in Corollary \\ref{co-1125-1} and $p=1$, we have\nthat the following $\\nu$-support vector machine problem\n\t\\begin{align*}\n\t\t\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in \\overline{{\\cal B}}_{1}(F_0,\\epsilon)} {\\rm CVaR}_\\alpha^F(-Y \\cdot \\boldsymbol{\\beta}^\\top\n\t\\mathbf X)\n\t\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\t\t\\inf_{\\bm\\beta\\in \\mathcal D} \\left\\{{\\rm CVaR}_\\alpha^{F_0}(-Y \\cdot \\boldsymbol{\\beta}^\\top\n\t\\mathbf X)+ \\frac{1}{1-\\alpha}\\varepsilon ||\\bm \\beta||_* \\right\\}.\n\\end{align*}\n\\end{example}\n\n\n\\begin{example} {\\bf (Regression)} \\label{ex5}\nSetting $\\ell(z,t)=(|z|-t)_+$ in Corollary \\ref{co-1125-1} and $p=1$, we have that the following $\\nu$-support vector regression problem\n\t\\begin{align*}\n\t\t\\inf_{\\bm\\beta_r\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{1}(F_0,\\epsilon)} {\\rm CVaR}_\\alpha^F(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|)\n\t\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\t\t\\inf_{\\bm\\beta_r\\in \\mathcal D} \\left\\{{\\rm CVaR}_\\alpha^{F_0}(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|)+ \\frac{1}{1-\\alpha}\\varepsilon ||(1,-\\bm \\beta_r)||_* \\right\\}.\n\\end{align*}\n\\end{example}\n\n\n\n\n\n\n\\begin{example} {\\bf (Risk minimization)} \\label{ex6}\n{Setting $\\ell(z,t)=\\ell_1(z-t)=(z-t)_+$ in Corollary \\ref{co-1125-1},} we have that the following problem of minimizing higher moment risk measures\n\\begin{align*}\n\\inf_{\\bm\\beta\\in  \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)}\n\\inf_{t\\in\\mathbb{R}}\\left\\{t+c\\left(\\mathbb{E}^F[(\\bm\\beta^{\\top}\\mathbf X-t)_+^p]\\right)^{1\/p}\\right\\}\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in  \\mathcal D, t\\in\\mathbb{R}}\\left\\{t+c\\left(\\mathbb{E}^{F_0}[(\\bm\\beta^{\\top}\\mathbf X-t)_+^p]\\right)^{1\/p}\n+ c\\varepsilon   ||\\bm\\beta||_*\\right\\}.\n\\end{align*}\n\\end{example}\n\n\n\\begin{corollary} \\label{cor1}\nFor any $p\\in [1,\\infty)$, let $\\mathcal V^F$ be defined by \\eqref{eq-infEU}, where  $\\ell(z,t):=c\\ell(z-t)$  and $\\ell$ is  $\\ell_3$ or $\\ell_4$ in Theorem \\ref{th-highorderloss}.\nIt holds that for $F_0\\in\\mathcal M_p(\\mathbb{R}^n)$ and $\\epsilon\\ge 0$\n$$\n  \\inf_{\\boldsymbol{\\beta}\\in\\mathcal D}\\sup_{F\\in \\mathcal  B_p(F_0,\\epsilon)}  \t\\mathcal V^F (\\boldsymbol{\\beta}^\\top \\mathbf Z)= \\inf_{\\boldsymbol{\\beta}\\in\\mathcal D}\\left( \\left(  \t\\mathcal V^{F_0}(\\boldsymbol{\\beta}^\\top \\mathbf Z) \\right)^{1\/p} +  \\epsilon||\\boldsymbol{\\beta}||_* \\right) ^p.\n$$\n\\end{corollary}\n\nThe above corollary can accommodate, for example, the case where variance is applied as the measure. Variance is not listed as an example in Section \\ref{rcr} because it has been studied in \\cite{BCZ22}. Yet, our approach to derive the equivalency relation is different from, and significantly more general than, the approach taken in \\cite{BCZ22}. We give the following example to highlight the generality of our approach.\n\n\\begin{example}  {(\\cite{BCZ22})} \\label{ex7}\nWhen $p=2$ and $\\ell(z,t) =\\ell_3(z-t) = |z-t|$ (with $m_1=0$ and  $m_2=0$),\n$\\mathcal V^F = {\\rm\\mathbb{V}ar}^F$. That is,\n\\begin{align*}\n\\sup_{F\\in \\mathcal B_{2}(F_0,\\epsilon)}{\\rm\\mathbb{V}ar}^F(\\bm\\beta^\\top\\mathbf Z)\n=\\sup_{F\\in \\mathcal B_{2}(F_0,\\epsilon)}\\inf_{t\\in\\mathbb{R}} \\mathbb{E}^F[(\\bm\\beta^\\top\\mathbf Z-t)^2].\n\\end{align*}\nApplying Corollary \\ref{cor1} yields\n\\begin{align*}\n\t\\sup_{F\\in \\mathcal B_{2}(F_0,\\epsilon)}{\\rm\\mathbb{V}ar}^F(\\bm\\beta^\\top\\mathbf Z)\n\t&=\\left(\\sqrt{{\\rm\\mathbb{V}ar}^{F_0}(\\bm\\beta^\\top\\mathbf Z)}\n\t+\\epsilon\\|\\bm\\beta\\|_*\\right)^2.\n\\end{align*}\n\\end{example}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{The case of exponential functions}\nSum-Exp for classification in Section \\ref{rcr}, i.e. $\\rho(Z):=\\frac1t \\mathbb{E}[e^{-tZ}]$, is a special case that cannot be covered by the theorems above. We proceed by assuming that the Wasserstein ball in \\eqref{eq-main5} is of type-$\\infty$, since otherwise the problem may be unbounded. We can make the following observation and identify the regularization counterpart of the Sum-Exp  classification problems.\n\n\\begin{proposition}\\label{ex-MMinfinity}\nFor a monotonic function $\\rho: L^\\infty\\to\\mathbb{R}$, that is,  $\\rho(Z_1)\\le\\rho(Z_2)$ whenever $Z_1\\le Z_2$ a.s.,\nwe have\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in {\\cal B}_\\infty(F_0,\\epsilon)}\\rho^F(\\bm\\beta^{\\top}\\mathbf Z)=\\inf_{\\bm\\beta\\in \\mathcal D} \\rho^{F_0}(\\bm\\beta^{\\top}\\mathbf Z+\\epsilon\\|\\bm\\beta\\|_*).\n\\end{align*}\n\\end{proposition}\n\n\\begin{example}  {\\bf (Classification)} \\label{ex9}\n~\n\n{\\bf - (Sum-Exp)}\nApplying $\\rho(Z)=\\frac1t \\mathbb{E}[e^{tZ}]$, we have that the Sum-Exp classification problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D}\\sup_{F\\in \\overline{\\cal B}_{\\infty}(F_0,\\epsilon)} \\frac1t \\mathbb{E}^F[e^{-tY\\cdot \\bm \\beta^{\\top}\\mathbf X}]\n\\end{align*}\nis equivalent to the regularization problem\n\\begin{align*}\n\\inf_{\\bm\\beta\\in \\mathcal D} \\frac1t \\mathbb{E}^{F_0}\\left[e^{-t(Y\\cdot \\bm \\beta^{\\top}\\mathbf X - \\epsilon\\|\\bm\\beta\\|_*)}\\right].\n\\end{align*}\n\\end{example}\n\n\n\n\\subsection*{The case of distortion functionals}\n $\\nu$-support vector machine (regression) and CVaR-deviation in Section \\ref{rcr} are two examples of how quantile-based risk measures such as CVaR play the role of building blocks for other measures. It is known in the literature of risk measures that CVaR is a special case of distortion functional,\n\n defined by\n\\begin{align*}\n\t\\rho_h^F(Z)=\\int_0^1 F^{-1}(s)\\mathrm{d} h(s),\n\\end{align*}\nwhere $h:[0,1]\\to\\mathbb{R}$ is called a distortion function and $F^{-1}$ is left-quantile function of $F$, i.e. $F^{-1}(s)=\\inf\\{x: F(x)\\ge s\\}$ for $s\\in(0,1]$, and $F^{-1}(0)=\\inf\\{x: F(x)>0\\}$.\nDistortion functionals allow for taking into account a spectrum of quantiles  with respect to different probability levels.  In particular, when the distortion function $h$ is a convex function, the distortion functional $\\rho_h^F$ is a mixture of CVaRs with the mixture weights determined by the derivative of $h$ (see e.g. \\cite{RU13}). The following generalization of CVaR-deviation appears in \\cite{RUZ08}\n\\begin{equation} \\label{gencvar}\n\\inf_{\\bm\\beta\\in  \\mathcal D}\\sup_{F\\in {\\cal B}_{p}\n(F_0,\\epsilon)} \\rho^F_h(\\bm\\beta^{\\top}\\mathbf X\n-\\mathbb{E}^F[\\bm\\beta^{\\top}\\mathbf X]\n),\n\\end{equation}\nwhere $h$ is convex. Similarly, one can consider the following generalization of $\\nu$-support vector regression\n\\begin{equation} \\label{gen-nu}\n\\inf_{\\bm\\beta_r\\in \\mathcal D}\\sup_{F\\in {\\cal B}_{p}(F_0,\\epsilon)} \\rho^F_h(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|),\n\\end{equation}\nwhere $h$ is increasing and convex. We show below that for both cases, there exists an exact equivalence relation between the Wasserstein DRO model and regularization.\n\n\\begin{proposition}\\label{prop:disabsolute} \\label{prop-DRdeviation}\n\\begin{itemize}\n\\item [(i)] Let $h: [0,1]\\to\\mathbb{R}$ be a convex function. For $p\\in[1,\\infty]$,\nthe problem \\eqref{gencvar} is equivalent to\n\\begin{equation}\n\\inf_{\\bm\\beta\\in \\mathcal D} \\left\\{\\rho^{F_0}_h(\\bm\\beta^{\\top}\\mathbf X\n-\\mathbb{E}^{F_0}[\\bm\\beta^{\\top}\\mathbf X])\n+  \\|h'_-+h(0)-h(1)\\|_q  \\varepsilon||\\bm\\beta||_*\\right\\},\n\\end{equation}\nwhere $\\|g\\|_q=(\\int_0^1 |g(s)|^q \\mathrm{d} s)^{1\/q}$, and $h_-'$ is the left derivative of $h$.\n\\item [(ii)]Let $h:[0,1]\\to\\mathbb{R}$ be an increasing and convex distortion function. For $p\\in[1,\\infty]$,\nthe problem \\eqref{gen-nu} is equivalent to\n\\begin{equation}\n\\inf_{\\bm\\beta_r\\in \\mathcal D} \\left\\{\\rho^{F_0}_h(|(1, - \\boldsymbol{\\beta}_r)^{\\top}\\mathbf X|)+ \\|h'_-\\|_q  \\varepsilon ||(1,-\\bm \\beta_r)||_*\\right\\}.\n\\end{equation}\n\\end{itemize}\n\\end{proposition}\n\nOne can easily apply the above proposition with $h(s)= (s-\\alpha)_+\/(1-\\alpha)$, where $(s)_+=\\max\\{s, 0\\}$, to derive the regularization counterpart for the example of CVaR-deviation in risk minimization (in Section \\ref{rcr}).\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we first show that Wasserstein DRO provides generalization guarantees for virtually any problem with affine decision rules - namely, for any type-$p$ Wasserstein ball and (almost) any measure of risk. This justifies the use and reveals the strength of Wasserstein DRO beyond the classical expectation optimization setting. Our approach sheds light on why generalization guarantees can be obtained (almost) independently of the choice of a Wasserstein ball and the measure of risk. Although our approach is limited to problems with affine decision rules, such problems are very common in practical ML and OR applications given the popularity of affine decision rules. Such problems also constitute an important basis for studying more sophisticated forms of problems. We further show how to draw exact equivalency relations between Wasserstein DRO and the classical regularization scheme for a wide range of regression, classification, and risk minimization problems that have not been considered to date using Wasserstein DRO. Our result broadens considerably the class of Wasserstein DRO problems that can be solved exactly via their regularization formulations. We also prove that our equivalency results are, in some sense, the most general that one can possibly obtain, in the case where the measure is an expected function (Theorems \\ref{th-RegularizationEL} and \\ref{th-necessity-highorderloss}). This provides valuable insights into how far the classical regularization interpretation can be applied to Wasserstein DRO.\nThe generalization and regularization results presented in this paper should be of high interest to ML and OR researchers\/practitioners who seek to gain confidence in Wasserstein DRO for a broader set of applications.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWhilst the Cold Dark Matter (CDM) paradigm is remarkably successful at explaining the universe's large-scale structure there are many known issues pertaining to structure formation on small-length scales. These include the ``missing satellites'' \\cite{MissingSat1, MissingSat2}, ``Too Big to Fail'' \\cite{TBTF1, TBTF2} and ``core-cusp'' \\cite{CuspCore1, CuspCore2, CuspCore3} problems all of which can broadly be thought of as an overprediction of density at small-scales by simulations of CDM when compared with observation. \n\nConsequently, solutions to these problems generally try to replicate the gross characteristics of CDM on large length scales where it performs well, whilst suppressing structure formation at small-scales where the CDM paradigm fares poorly. Examples of these solutions include Warm Dark Matter (WDM) \\cite{WDM, WDM1, WDM2, WDM3} in which the dark matter is endowed with a velocity dispersion allowing it to stream freely and wash out density perturbations below some scale. Another example is Late Kinetic Decoupling (LKD) \\cite{InteractingDM, InteractingDM2, LKD, LKD2, LKD3} in which the DM is allowed to couple to a relativistic ``heat bath'' (i.e. photons or standard model neutrinos) via elastic scattering until fairly late in the universe's evolution. The coupling allows small-scale modes (i.e. modes which entered the Hubble horizon early and therefore experience the strong effects of the coupling) to experience acoustic oscillations in their density while large scale modes enter the horizon after decoupling and therefore experience little or no suppression of power. \n\nBoth of these solutions lead to, all other things considered, a sharp cutoff in the matter power spectrum and as a result can alleviate some of the problems for CDM at small-length scales. Given that these scenarios are identical at large length scales and possess the same phenomenological characteristics on the small, the question arises as to whether a cosmological observable exists which will allow for these scenarios to be distinguished. \n\nThis manuscript is intended as a short summary of \\cite{Diacoumis:2017hff}, in which we investigated CMB spectral distortions as a probe of these scenarios, and is organised as follows. In Section 2 we give a brief overview of CMB spectral distortions. In sections 3 and 4 we examine the details of DM--neutrino and DM--photon interactions respectively and comment on the effect that these would have on spectral distortions. In section 5 we discuss the implications of our results for future experiments such as PRISM. For further details the reader is encouraged to refer to \\cite{Diacoumis:2017hff}.\n\n\\section{Spectral Distortions}\n\nCMB Spectral Distortions are defined as any deviation in the energy spectrum of CMB photons from a perfect ``blackbody'' shape, they occur during periods where thermal equilibrium is not maintained in the photon fluid i.e. periods of significant energy injection or energy release. One of the most well-known mechanisms for producing CMB Spectral Distortions is through the dissipation of standing sound waves in the photon-baryon fluid whereby energy is transferred from the sound wave to the energy spectrum of the photons due to diffusion damping \\cite{Hu:1993gc, ChlubaRev}. Since diffusion damping occurs at extremely small length scales \\(1 \\, \\textrm{Mpc}^{-1} \\, \\leq k \\leq 10^{4} \\, \\textrm{Mpc}^{-1}\\),  roughly corresponding to the scales at which LKD and WDM offer solutions to the small-scale structure problems of CDM, we expect the details of the Dark Matter microphysics to have a significant impact on the form of the distortions.\n\nWe emphasise that spectral distortions are expected to occur via this mechanism even in CDM cosmology, the point is that DM microphysics can significantly alter the form of the perturbations which in turn leads to distortion patterns which deviate from CDM. We will refer throughout this manuscript to the \\(\\mu\\)-parameter which characterises the ``effective chemical potential'' in the photon energy spectrum created as a result of the distortion.\n\n\\section{DM -- neutrino interactions}\nIn the absence of interactions with DM, neutrinos stream freely and therefore carry away some of the power of the perturbation without sourcing a spectral distortion. This is reflected in the form of the photon temperature transfer function which has the form \n\\begin{equation} \\label{eq:Theta}\n\\Theta_{1} \\approx A \\, c_{s} \\sin\\left(k r_{s}\\right)\\exp\\left(-\\frac{k^{2}}{k^{2}_{D}}\\right).\n\\end{equation}\nHere the acoustically oscillating part of the perturbation depends on the sound horizon \\(r_{s}\\), the diffusion damping is characterised by the diffusion scale \\(k_{D}\\), and the WKB amplitude is given by\n\\begin{equation} \\label{eq:amp}\nA \\simeq \\left(1 +\\frac{4}{15}f_{\\nu}\\right)^{-1},\n\\end{equation}\nwhere \\(f_{\\nu} = \\rho_{\\nu}\/(\\rho_{\\nu} + \\rho_{\\gamma}) \\simeq 0.41\\) is the ratio of the neutrino energy density to the total energy density of the relativistic species. The factor of \\(f_{\\nu}\\) amounts to a small correction to the amplitude as a result of the anisotropic stress of the neutrino fluid. \n\nA coupled DM--neutrino system exhibits similar characteristics to the familiar photon--baryon system in that the DM--neutrino fluid experiences acoustic oscillations damped by neutrino diffusion in the tight-coupling limit. The amplitude of the DM density perturbations is reduced on small length scales which serves as the basis for the solution to the small-scale problems of CDM cosmology. At the same time, coupling the neutrinos to DM prevents them from free-streaming and allows them to behave more like a `perfect' relativistic fluid with vanishing anisotropic stress \\cite{GravCluster}.\n\nThe absence of anisotropic stress allows the neutrinos to participate in acoustic oscillations of the perturbations in the same way as photons as they are no longer free-streaming. In the tightly coupled limit the correction factor \\(f_{\\nu}\\) vanishes completely from the amplitude Eq. (\\ref{eq:amp}) and the oscillation amplitude is enhanced relative to CDM. The net result is an \\textit{enhanced} \\(\\mu\\)-parameter as the amplitude of the perturbation has increased and there is more power that can potentially be dissipated leading to larger spectral distortions.\nWe emphasise that this effect is purely gravitational and arises because the neutrinos anisotropic stress vanishes in the presence of the interaction. Figure \\ref{NeutrinoMu} shows clearly that this is a saturated effect and causes a maximum ~\\(20\\%\\) increase in the \\(\\mu\\)-parameter when the DM--neutrino coupling is tight.\n\n\\begin{figure}[h]\n\\includegraphics[width=22pc]{NeutrinoMu.png}\\hspace{2pc}%\n\\begin{minipage}[b]{10pc}\\caption{\\label{NeutrinoMu}Expected chemical potential \\(\\mu\\)-distortion as a function of DM--neutrino scattering cross section \\(\\sigma_{DM-\\nu}\\) Blue denotes time independant cross-sections and red denotes cross sections proportional to temperature squared.\\\\ \\\\ \\\\}\n\\end{minipage}\n\\end{figure}\n\n\\section{DM -- photon interactions} \nIn the photon--baryon fluid the diffusion damping scale is set by \n\\begin{equation} \\label{eq:diffusion}\n\\partial_{z}k_{D}^{-2} = - \\frac{c^{2}_{s}a}{2\\mathcal{H}\\dot{\\kappa}}\\left(\\frac{16}{15} + \\frac{R^{2}}{R+1}\\right),\n\\end{equation}\nwhere \\(c_{s}\\) is the sound speed of the photon--baryon fluid, \\(\\dot{\\kappa}\\) is the interaction rate of photons and baryons and \\(R = (3\/4)\\rho_{b}\/\\rho_{\\gamma}\\) is the baryon-to-photon energy density ratio. Note that the first term in Eq. (\\ref{eq:diffusion}) arises due to the viscosity of the photon fluid and the second term (proportional to \\(R^{2} << 1\\)) arises due to the heat conduction between photons and baryons, here the heat conduction term is extremely subdominant as the photons are tightly coupled to the baryons for the entire time period of interest and do not allow for efficient heat exchange between the two fluids \\cite{Weinberg:1972kfs, Hu:1996mn}.\n\nThe case of DM--photon elastic scattering is distinctly non-trivial as the DM is not tightly coupled to the photons for the entirety of it's evolution and therefore enters a \\textit{weak coupling regime} in which diffusion damping due to heat conduction can be significant or even the dominant mode of dissipation. Here the diffusion damping scale is given by\n\\begin{equation} \\label{analytic}\n\\partial_{z}k_{D}^{-2} \\simeq - \\frac{c^{2}_{s}a}{2\\mathcal{H}} \\left[ \\frac{1}{\\dot{\\kappa} + \\dot{\\mu}_{\\gamma}}\\frac{16}{15} + \\frac{3\\dot{\\mu}}{k^{2}}\\left(\\frac{k^{2}}{k^{2} +3 S_{\\gamma}^{-2}\\dot{\\mu_{\\gamma}}^{2}}\\right)  \\right]\n\\end{equation}\nwhere \\(\\dot{\\mu}_{\\gamma}\\) is the DM--photon interaction rate and \\(S = (3\/4)\\rho_{c}\/\\rho_{\\gamma}\\) is the DM-to-photon energy density ratio. When tight-coupling is satisfied (\\(S^{-1}_{\\gamma}\\dot{\\mu}_{\\gamma} > k\\)) the heat conduction term in this expression reduces to a form proportional to \\(S_{\\gamma}^{2}\\) analogous to the heat conduction term in Eq. (\\ref{eq:diffusion}). \n\nThe effect of the heat conduction damping is twofold: firstly, it affects the evolution of the photon temperature transfer functions through additional damping by Eq. (\\ref{eq:Theta}) which thereby damps the amount of perturbation power able to be dissipated and secondly, it directly affects the heating rate creating a temporally localised `burst' of energy. This second effect arises because the DM--photon coupling can enter a weak coupling regime during the \\(\\mu\\)-era, in contrast with the photon--baryon coupling which remains tight for the entire duration of the \\(\\mu\\)-era. To understand the `burst' shape consider that extremely tightly coupled fluids will be almost comoving and therefore not have a significant amount of heat conduction whereas fluid which are completely decoupled cannot communicate with each other efficiently enough to allow heat conduction to take place. The `burst' shape is therefore a result of the DM and photons slipping past each other during the weak couping regime and contributing to a large heat conduction between the two fluids. Both effects are demonstrated in Figure 2 which shows the heating rate of the CMB photons in which the `burst' shape at intermediate times and the suppression of power at late times are both visible.\n\n\\begin{figure}[h]\n\\begin{minipage}[c]{0.6\\textwidth}\n\\includegraphics[width=22pc]{Photon_Heating.png}%\n\\end{minipage}\\hspace{2pc}\n\\begin{minipage}[b]{0.3\\textwidth}\\caption{\\label{PhotonHeatingConst}Heating rate of CMB photons as a function of the scale factor \\(a\\) for differerent values of the parameter \\(u_{\\gamma} \\equiv \\sigma_{\\textrm{DM}-\\gamma}\/\\sigma_{\\textrm{T}}\\left(100 \\, \\textrm{GeV}\/m_{\\textrm{DM}}\\right)\\). \\\\}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\begin{minipage}[c]{0.6\\textwidth}\n    \\includegraphics[width=22pc]{Sigma_Photon_Const.png}\n    \\includegraphics[width=22pc]{Sigma_Photon_T2.png}\n  \\end{minipage}\\hspace{2pc}\n  \\begin{minipage}[c]{0.3\\textwidth}\n    \\caption{\\label{blah}\\textit{Top}:  Upper bounds for DM-X scattering time independent cross-sections as a function of DM mass. Existing constraints from Planck + WP \\cite{Wilknu, Wilkphoton}(dotted), MW satellites \\cite{Boehm:2014vja} (small-dashed) and Lyman-\\(\\alpha\\) forest \\cite{Wilknu} (dot-dashed) shown alongside projected constraints from PRISM in \\cite{ChlubaKam} (long-dashed) and this work (solid). Blue, orange denotes DM-\\(\\gamma\\) and DM-\\(\\nu\\) scattering respectively. \\\\ \\\\\n       \\textit{Bottom}: Same as top but for DM-X scattering cross-sections proportional to temperature squared as a function of DM mass. Existing constraints from Lyman-\\(\\alpha\\) forest \\cite{Wilknu} (dot-dashed) \\\\ \\\\ } \\label{Constraints}\n  \\end{minipage}\n\\end{figure}\n\n\\section{Results}\nIn the case of DM--neutrino elastic scattering the overall amplitude of the perturbations is enhanced by \\(\\sim 10\\%\\) relative to \\(\\Lambda\\)CDM in the limit of tight coupling. As a result, the expected \\(\\mu\\) distortion is enhanced significantly (see Figure \\ref{NeutrinoMu}) and may be distinguished from the \\(\\Lambda\\)CDM predicition if the present-day value of the scattering cross section is at least as large as \\(\\sigma_{\\textrm{DM}-\\nu} \\sim 4.8 \\times 10^{-32}  \\left(m_{\\textrm{DM}}\/\\textrm{GeV}\\right) \\, \\textrm{cm}^{2}\\) for time-independent cross sections, and \\(\\sigma^{0}_{\\textrm{DM}-\\nu} \\sim 2.5 \\times 10^{-47}  \\left(m_{\\textrm{DM}}\/\\textrm{GeV}\\right) \\, \\textrm{cm}^{2}\\) for $\\sigma_{{\\rm DM}-\\gamma} \\propto T^2$.   In the latter case, it is interesting to note that the constraining power of PRISM on dark matter--neutrino elastic scattering may potentially exceed current limits from the Lyman-\\(\\alpha\\) forest (see bottom of Figure \\ref{Constraints}).\n\nIn the case of DM--photon elastic scattering we derive a new analytical expression Eq. (\\ref{analytic}) for the diffusion damping scale in the presence of a DM--photon elastic scattering interaction which acccounts for the fact that the DM--photon fluid enters a weak coupling regime while the photons are still tightly coupled to the baryons. This expression shows that dissipation due to heat conduction is significant in the case of DM--photon elastic scattering and can actually be the dominant mode of dissipation during the \\(\\mu\\)-era for certain parameter values. We find that future experiments such as PRISM may be sensitive to DM--photon elastic scattering if the cross section is at least as large as \\(\\sigma_{\\textrm{DM}-\\gamma} \\sim 1.1 \\times 10^{-30}  \\left(m_{\\textrm{DM}}\/\\textrm{GeV}\\right) \\, \\textrm{cm}^{2}\\) for time-independent cross sections, and \\(\\sigma^{0}_{\\textrm{DM}-\\nu} \\sim 1.8 \\times 10^{-40}  \\left(m_{\\textrm{DM}}\/\\textrm{GeV}\\right) \\, \\textrm{cm}^{2}\\) for $\\sigma_{{\\rm DM}-\\gamma} \\propto T^2$. This method is not as constraining as the case of DM--neutrino elastic scattering due to competing effects dominating the heating rate of the photons at different times (see Section 3 for further discussion.)\n\nIn summary, we have identified a number of physical features of DM--neutrino and DM--photon interacting models which influence the evolution of photon perturbations in the early universe. These physical features produce different CMB Spectral Distortions and can be probed with a high degree of sensitivity by future experiments such as PRISM.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{table*}\n\\caption{Details of the observations.\\label{observations}}\n\\begin{center}\n\\tiny\n\\begin{tabular}{ccccccccc}\n\\hline\nObservation & MJD &  \\multicolumn{2}{c}{Map peak} & \\multicolumn{2}{c}{Beam} & \\multicolumn{2}{c}{1$\\sigma$ rms} \n& Notes  \\\\\ndate  & & \\multicolumn{2}{c}{(mJy\/beam)} & \\multicolumn{2}{c}{(mas $\\times$ mas, $^\\circ$)} &\n\\multicolumn{2}{c}{(mJy\/beam)}  &  \\\\\n     & & 15\\,GHz & 24\\,GHz & 15\\,GHz & 24\\,GHz& 15\\,GHz & 24\\,GHz&   \\\\\n\\hline\n\\hline\n2011\/01\/14 & 55575 & 348 & 319 &\\ \\ 1.05 $\\times$ 0.65, 15.3  \\ \\ &\\ \\ 0.79 $\\times$ 0.47, 8.84 \\ \\ & 0.19 & 0.18 & No MK, no NL    \\\\\n2011\/02\/25 & 55617 & 391 & 338 & 1.16 $\\times$ 0.74, 14.4 & 0.64 $\\times$ 0.39, $-$6.52 & 0.35 & 0.18 & NL snowing \\\\\n2011\/03\/29 & 55649 & 386 & 359 & 1.06 $\\times$ 0.66, $-$5.37 & 0.65 $\\times$ 0.39, $-$4.26 & 0.17 & 0.24 & No HK \\\\\n2011\/04\/25 & 55675 & 367 & 308 & 0.92 $\\times$ 0.50, $-$3.79 & 0.61 $\\times$ 0.34, $-$3.88 & 0.28 & 0.33 & -  \\\\\n2011\/05\/31 & 55712 & 355 & 297 & 0.93 $\\times$ 0.51, $-$5.56 & 0.64 $\\times$ 0.35, $-$8.41 & 0.25 & 0.29 & - \\\\\n2011\/06\/29 & 55741 & 262 & 208 & 0.89 $\\times$ 0.50, $-$7.17 & 0.56 $\\times$ 0.32, $-$12.3 & 0.17 & 0.37 & No LA   \\\\\n2011\/07\/28 & 55770 & 220 & 197 & 0.91 $\\times$ 0.56, $-$0.89 & 0.60 $\\times$ 0.37, $-$1.14 & 0.20 & 0.30 & -  \\\\\n2011\/08\/29 & 55802 & 275 & 200 & 0.97 $\\times$ 0.55, 0.26 & 0.63 $\\times$ 0.35, $-$2.70 & 0.18 & 0.27 & No HK  \\\\ \n2011\/09\/28 & 55832 & 264 & 238 & 1.06 $\\times$ 0.67, 16.2 & 0.72 $\\times$ 0.47, 18.4 & 0.26 & 0.24 & No MK  \\\\ \n2011\/10\/29 & 55863 & 261 & 167 & 1.06 $\\times$ 0.69, 0.09 & 0.73 $\\times$ 0.44, $-$3.73 & 0.26 & 0.14 & HK snowing  \\\\ \n2011\/11\/28 & 55893 & 283 & 201 & 1.02 $\\times$ 0.59, 18.0 & 0.70 $\\times$ 0.42, 14.3 & 0.28 & 0.17 & No PT, FD, MK  \\\\ \n2011\/12\/23 & 55918 & 295 & 287 & 0.89 $\\times$ 0.49, 1.82 & 0.61 $\\times$ 0.35, $-$5.74 & 0.15 & 0.21 & No HK  \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=7.5cm]{15ghz_stacked.eps}\n \\includegraphics[width=7.5cm]{24ghz_stacked.eps}\n \\caption{Images of Mrk 421 at 15\\,GHz (left panel) and 24\\,GHz (right panel). These two images were obtained by stacking all of the images of the twelve epochs, at the respective frequency. The restoring beam for the 15\\,GHz image is 0.9 mas$\\times$0.55 mas and the peak flux density is 307.2 mJy\/beam. For the 24\\,GHz image, the restoring beam is 0.6 mas$\\times$0.35 mas and the peak is 251.9 mJy\/beam. The first contour is 0.35 mJy\/beam, which corresponds to three times the off-source noise level. Contour levels are drawn at ($-$1, 1, 1.4, 2, 2.8, 4...) in steps of $\\sqrt{2}$.}\n\\label{maps}\n \\end{figure*}\n\n\nMarkarian 421 (R.A.=$11^h$\\ $04^m$\\ $27.313943^s$, Dec.=$+38^\\circ$\\ 12\\arcmin\\ 31.79906\\arcsec, J2000) is one of the nearest ($z=0.031$) and brightest BL~Lac objects in the sky. It was the first extragalactic source detected at \nTeV energies by the Cherenkov telescope at Whipple Observatory \\citep{Punch1992}. The spectral energy \ndistribution (SED) of this object, dominated by non-thermal emission, has two\nsmooth broad components: one at lower energies, from radio band to the soft X-ray domain, and another at higher\nenergies peaking at $\\gamma$-ray energies \\citep{Abdo2011}. \nThe low-frequency peak is certainly due to synchrotron emission from relativistic electrons in the jet interacting \nwith the magnetic field, whereas the high-frequency peak is probably due to the inverse Compton scattering of the same population of \nrelativistic electrons with synchrotron low-energy photons \\citep[synchrotron-self-compton model, SSC, see][]{Abdo2011, Tavecchio2001}. \nIn this framework, multiwavelength (MWL) coordinated campaigns are a fundamental tool for understanding the physical properties of the source, e.g. by studying variability, which is present at all frequencies, but particularly TeV energies where \\citet{Gaidos1996} measured a doubling time of $\\sim15$ minutes. The accurate MWL study and SED modeling performed by \\citet{Abdo2011} revealed some interesting results, such as the size of the emitting region $R$ and the magnetic field $B$, which in the context of the leptonic scenario they constrained to be $R \\lesssim 10^4 R_g$ and $B \\sim$ 0.05 G. \nHowever, the details of the physical processes responsible for the observed emission are still poorly constrained.\nBecause of the considerable variability and the broadband spectrum, multiwavelength  long-term observations are required for a good comprehension of the emission mechanisms.\n \nThis study is part of a new multi-epoch and multi-instrument campaign, which also involves observations in the sub-mm (SMA), optical\/IR (GASP), UV\/X-ray (Swift, RXTE, MAXI), and $\\gamma$ rays (Fermi-LAT, MAGIC, VERITAS), as well as at the cm wavelengths with low resolution observations (e.g. F-GAMMA, Medicina). \nThe aim of this observational effort is to shed light on fundamental questions such \nas the nature of the radiating particles, the connection between the radio and $\\gamma$-ray emission, \nthe location of the emitting regions, and the origin of the flux variability.\nVery long baseline interferometry (VLBI) plays an important role in addressing these scientific questions because it is the only technique that can resolve (at least partially) the inner structure of the jet.\nTherefore, cross-correlation studies of Very Long Baseline Array (VLBA) data with data from other energy ranges (in particular $\\gamma$ rays) \ncan provide us with important information about the structure of the jet and the location of the blazar emission.\n\nAt radio frequencies, Mrk 421 clearly shows a one-sided jet structure aligned at a small angle with respect \nto the line of sight \\citep{Giroletti2006}.\nIn this work, we present new VLBA observations to study in detail the inner jet\nstructure on parsec scales. We are able to investigate the evolution of shocks that arise in the jet, \nby means of the model-fitting technique. In earlier works \\citep{Piner1999, Piner2004}, the jet\ncomponents show only subluminal apparent motion, which seems to be a common\ncharacteristic of TeV blazars. Thanks to accurate measurements of changes on parsec scales, by the VLBA, we can find\nvalid constraints on the geometry and kinematics of the jet. \n\nThis paper is structured as follows: in Section 2 we introduce the dataset, in Section 3 we report the results of this work (model fits, flux density variations, apparent speeds, jet sidedness, spectral index), and in Section 4 we discuss results giving our own interpretation in the astrophysical context. We used the following conventions for cosmological parameters:  $H_0=70$ km sec$^{-1}$ Mpc$^{-1}$, $\\Omega_M=0.25$ and $\\Omega_\\Lambda=0.75$, in a flat Universe. We defined the spectral index $\\alpha$ such that $S_{\\nu}\\propto\\nu^\\alpha$.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[angle=-90, clip, width=0.9\\columnwidth]{gen15.ps} \n\\includegraphics[angle=-90, clip, width=0.9\\columnwidth]{gen24.ps} \\\\\n\\caption{Images of Mrk 421 with model fit components for the first epoch at 15\\,GHz (left panel) and at 24\\,GHz (right panel). Levels are drawn at $(-1, 1, 2, 4...) \\times$ the lowest contour, that is at 1.0 mJy\/beam for both images, in steps of 2. The restoring beam is shown in the bottom left corner; its size is given in Table~\\ref{observations}.} \n\\label{components}\n\\end{figure*}\n\n\\section{Observations}\nWe observed Mrk 421 throughout 2011 with the VLBA. \nThe source was observed once per month, for a total of 12 epochs, at three \nfrequencies: 15, 24, and 43 GHz. In this paper, we present the complete analysis of the whole 15 and 24\\,GHz datasets.\nWe also observed, at regular intervals, three other sources \n(J0854+2006, J1310+3220, and J0927+3902) used as fringe finders and calibrators for the band pass, the instrumental (feed) polarization, and the electric-vector position angle.\nAt each epoch, Mrk 421 was observed for nearly 40 minutes at each frequency, spread into several scans of about 3 minutes each, interspersed  with calibrator sources in order to improve the (u,v)-coverage. Calibrators were observed for about 10 minutes each, generally spread into three scans of 3 minutes. \n\nFor calibration and fringe-fitting, we used the AIPS software package \\citep{Greisen2003}, and for\nimage production the standard self-calibration procedures included in the DIFMAP software\npackage \\citep{Shepherd1997}, which uses the CLEAN algorithm proposed by \\citet{Hogbom1974}. \nIn some epochs, one or more antennas did not work properly because of technical\nproblems; for a complete report (see Table~\\ref{observations}).\n\n\\section{Results}\n\\subsection{Images}\nWe show in Fig.~\\ref{maps} two images of Mrk 421 at 15 and 24\\,GHz, which were produced by stacking all the images respectively at 15 and 24\\,GHz created with DIFMAP. The alignment of the images was checked by comparing the pixel position of the peak. In both images, we set the lowest contour equal to about three times the off-source residual rms noise level.\n\nAll the 12 images at each frequency show a similar structure, consisting of a well-defined and well-collimated one-sided jet structure emerging from a compact nuclear region (core-dominated source). This is the typical structure of a BL~Lac object\n\\citep{Giroletti2004a}. \nThe jet extends for roughly 4.5 mas (2.67 pc)\\footnote{1 mas corresponds to 0.59 pc.}, \nwith a position angle (PA) of $\\sim-35^\\circ$ (measured from north through east). This morphology agrees with the results of other studies of similar angular resolution \\citep{Marscher1999}.\n\nSince the morphology was very stable from epoch to epoch, the stacking did not smear any details of the structure. A sample couple of 15 and 24\\,GHz images is shown in Fig.~\\ref{components}.\n\\subsection{Model fits}\nFor each epoch, we used the model-fitting routine in DIFMAP to fit the visibility data of the source in the $(u,v)$ plane with either elliptical or circular Gaussian components. In this way, we were able to investigate in detail the inner jet structure and its evolution. \nFor all epochs at 15\\,GHz, a good fit was obtained with five Gaussian components, while at 24\\,GHz we needed six components. At both frequencies, we identified the core with the brightest, innermost, and most compact feature.\nWe label the other components C1, C2, C3, and C4, starting from the outermost (C1) to the innermost (C4). \nThe higher angular resolution achieved at 24\\,GHz resolves the second innermost 15\\,GHz component (C4 located at $\\sim$0.45 mas from the core) into two features (C4b at $\\sim$0.3 mas and C4a $\\sim$0.7 mas from the core) (see Fig.~\\ref{components}).\n\nThanks to the extremely fine time-sampling, we were able to make an attempt to identify the same component in each epoch.\nOverall, the components extend out to a region of about 5 mas. In this way, with a limited number of components, it was possible to analyze the proper motions and flux density levels at various times. \nFrom Fig.~\\ref{modelfit}, we can clearly see that the data occupy well-defined regions in the radius vs. time plot, and that this behavior helps us to identify the individual components across epochs. \n\nAll details of the model fit analysis are shown in Table~\\ref{gaussian}.\nWe calculated the uncertainties in the position (error bars in Fig.~\\ref{modelfit}) using the ratio of the size of each component to the signal-to-noise ratio (S\/N). In the case of very bright, compact components, the nominal error value is too small and replaced by a conservative value equal to 10\\% of the beam size \\citep{Orienti2011}. On the other hand, when the calculated error is very large (i.e. comparable to the component radius, as in the case of a very extended component with a very low flux density), we replace it with a threshold value equal to the maximum value of the calculated errors for the same component at different epochs.\n\nBy comparing with \\citet{Piner2010} and \\citet{Piner2005}, we can argue that the components C2, C3, C4a, and C4b of the present analysis quite likely correspond respectively to the components C5, C6, C7, and C8 of the aforementioned works. We also suggest that our component C1 corresponds to their component C4a. We will consider a more accurate identification of our components with those of previous analyses in a forthcoming paper, where the 43\\,GHz data analysis will also be presented.\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[clip,width=\\columnwidth]{modelfit.eps}\n\\caption{Results of model-fit analysis. Circles and triangles refer, respectively, to positions of the Gaussian components at 15 and 24\\,GHz.} \n\\label{modelfit}\n\\end{figure}\n\n\\addtocounter{table}{1}\n\n\\subsection{Flux density variability} \\label{flux}\nUsing the results of the model-fit technique, we analyzed the temporal evolution of\nthe flux density for each component of the source. The brightest component represents the core; at 15\\,GHz, it has a mean value of around 350 mJy, which decreases along the jet until values of about 10 mJy. By comparing the flux density of each component at the various epochs, it emerges that there are no significant variations in the flux densities of the C1-C4 components. The flux density of each component remains roughly constant\nat various times within the uncertainties calculated; in any case, there is no indication of flaring activity. \nAny small variations may be artifacts brought about by our fitting procedures: for instance, the flux density of the inner components may be underestimated in some cases, because part of it was incorporated into the core component flux, or the flux density of the most extended features may be underestimated at epochs missing some short baseline data because of telescope failures.\nThis is a remarkable result because it also allows us to identify components based on their flux density values and confirms our choice of components based on their positions.\n\nIn Fig.~\\ref{lightcurves}, we show the light curve for Mrk 421 during 2011 at 15\\,GHz (upper panel) and at 24\\,GHz (lower panel), considering the total flux density (squares) and the core flux density (triangles). The light curve reveals an interesting feature: in the second part of the year (starting at MJD $\\sim$55700), we clearly note a decrease in the total flux density. From the complete flux-density analysis, we found that the core is the component responsible for the decrease, while the extended region does not display any significant variations. To further exclude calibration effects, we performed the same analysis on the three calibrators. In Fig.~\\ref{lightcurves}, we present the light curves of the calibrator J1310+3220 (diamonds) at 15\\,GHz (upper panel) and 24\\,GHz (lower panel). From this comparison, we clearly see that the trend of the light curves for the two sources is very different. We can assert that the flux density decrease observed for the core of Mrk 421 is a real feature. Error bars were calculated by considering a calibration error of about $10\\%$ of the flux density and a statistical error equal to three times the map rms noise.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{lightcurve15.eps} \\\\\n\\includegraphics[width=\\columnwidth]{lightcurve24.eps}\n\\caption{Light curves for Mrk421 (squares represents the total flux density and triangles represent the core flux density) and the calibrator J1310+3220 (diamonds represent the total flux density, with scale given on the right hand $y$-axis). The upper and lower panels refer respectively to the 15\\,GHz and 24\\,GHz data.} \n\\label{lightcurves}\n\\end{figure}\n\n\\subsection{Spectral index analysis}\nThanks to our multi-frequency data set, we were able to carry out a detailed analysis of the spectral index distribution on the parsec scale. We conducted a quantitative assessment of the spectral index of the components by comparing the flux density at the two frequencies at each epoch and averaging the results to reduce the statistical fluctuations. For the core, we obtained a value of $\\alpha\\sim-0.3\\pm 0.2$, and for the outermost components (C1, C2, and C3, considered altogether), we obtained $\\alpha\\sim-1.2\\pm 0.5$. The uncertainties were calculated from the theory of the propagation of errors using the formula\n\\begin{equation}\n\\Delta \\alpha = \\frac{1}{\\log(24\/15)} \\sqrt{\\left(\\frac{\\Delta S_{15}}{S_{15}}\\right)^2 + \\left(\\frac{\\Delta S_{24}}{S_{24}}\\right)^2},\n\\end{equation}\nwhere $S$ and $\\Delta S$ represent the flux density and the respective uncertainty (see the last paragraph of section~\\ref{flux}).\nThese values agree with the results obtained from the spectral index image, which we produced with the following procedure: we first produced new images for each epoch and frequency using the same $(u,v)$ range and the same restoring beam ($0.4$ mas $\\times$ $0.7$ mas). We then produced the spectral index image by combining the two images, resulting respectively from the stacking of all images at 15\\,GHz and at 24\\,GHz, using the AIPS task COMB, clipping the pixels with a S\/N $<$ 3 using the input images. By summing images we were able to increase the S\/N, and results are more reliable; the stability of the individual features guarantees that we do not lose structural information in the averaging process. The alignment of the images at the two frequencies was checked by comparing the pixel position of the peak.\nWhen a shift was present, we used the LGEOM task in AIPS to align the images.\nThe resulting image presents the typical flat spectrum in the core region, with a steepening along the jet radius. However, despite the higher S\/N obtained with image stacking, a significant spread in the spectral index values is present in the diffuse jet region, hence we do not show the image here.\n\n\n\\subsection{Apparent speeds}\nFrom our model-fit, we infer a small or no displacement for the jet components. To verify and quantify this statement, we determined the speeds of each component by means of linear fits to the separation of the individual features from the core at different epochs. For the \nthree outer components (C1, C2, and C3), we used combined data at 15 and 24 GHz, since the positions of each component at the two frequencies are generally consistent within the error bars.\nFor the two inner components (C4a and C4b), we only used data at 24\\,GHz.\nResults are shown in Table~\\ref{speeds}.\n\nWe found low values for the apparent speeds, in agreement with previous studies \\citep[e.g.][]{Piner2005}.\nThe two innermost components (C4a and C4b) are essentially stationary, with an upper limit to their separation velocity of $\\sim0.1$c. In addition C2 and C3 are consistent with being stationary,  while the outermost component C1 has a low-significance (1.5$\\sigma$) subluminal motion$\\sim0.3$c. \nIf this trend of increasing velocity at larger radii were real and if the apparent speeds shown in Table~\\ref{speeds} represented the bulk apparent speed of the plasma in the jet, we could speculate that some mechanism involving an acceleration acts in the outer region of the jet.\n\n\\begin{table}[t]\n\\begin{center}\n\\footnotesize\n\\caption{Apparent speeds from the linear fit analysis.\\label{speeds}}\n\\begin{tabular}{lcc}\n\\hline\nComponent & Apparent speed & $\\beta_{\\rm app}$ \\\\\n& (mas\/yr) &  \\\\\n\\hline\n\\hline\nC1   &$0.17\\pm0.12$ & $0.34\\pm0.24$ \\\\\nC2   &$0.08\\pm0.10$ & $0.16\\pm0.20$ \\\\\nC3  &$0.05\\pm0.06$ & $0.10\\pm0.11$ \\\\\nC4a  &$-0.01\\pm0.09$ & $-0.02\\pm0.17$ \\\\\nC4b   &$-0.02\\pm0.06$ & $-0.03\\pm0.11$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Jet\/counter-jet ratio}\nWe estimated the ranges of viewing angles $\\theta$ and of $\\beta$ from the jet\/counter-jet brightness ratio. Assuming that the source has two symmetrical jets of the same intrinsic power, we used the equation\n$$\\frac{B_J}{B_{cJ}}=R=\\left(\\frac{1+\\beta \\cos\\theta}{1-\\beta \\cos\\theta}\\right)^{2-\\alpha},$$\nwhere $B_J$ and $B_{cJ}$ are, respectively, the jet and counter-jet brightnesses and $\\alpha$ represents the spectral index defined in the introduction; we adopted the $(2-\\alpha)$ exponent, since the jet is smooth and does not contain well-defined compact {\\it blobs}.\nFor the jet brightness, we used $B_J\\sim28.4$ mJy\/beam, measured at 24\\,GHz, in the image resulting from the stacking of all the 12 epoch images, in the jet region located at $\\sim1$ mas from the core. For the counter-jet, which is not visible, we used an upper limit provided by the $3\\sigma$ rms noise level measured in the image, which resulted in $B_{cJ}=0.11$ mJy\/beam; this consequently yields a lower limit to  both $R$ and $\\beta\\cos\\theta$. With a value of $\\alpha=-0.4$, in agreement with our spectral index images, we obtained $R>254.8$ and then $\\beta\\cos\\theta>0.82$. Therefore, the minimum allowed jet bulk velocity is $\\beta_{\\rm min}=0.82$ (corresponding to a bulk Lorentz factor $\\gamma>1.74$) and the maximum viewing angle is $\\theta_{\\rm max} = 35.0^\\circ$.\n\\vspace{-0.25 cm}\n\\section{Discussion and conclusions}\n\nThe Doppler factor defined as\n\n$$\\delta=\\frac{1}{\\gamma(1-\\beta \\cos\\theta)}$$\n\n\\noindent is a key element in the study of blazars, since it affects various parameters such as the observed brightness, the SED peak frequency, the variability timescale, and more. Modeling of the SED and study of the variability in different wavebands generally require large values of the blazar Doppler factors; in the case of Mrk421, \\citet{Gaidos1996} estimated $\\delta>9$, from the observed TeV variability time of about 30 min and \\citet{Abdo2011} required a Doppler factor between 20 and 50 to reproduce the broadband SED. In turn, VLBI observations can also constrain $\\delta$ by posing limits on $\\beta$ and $\\theta$, as provided by the various arguments discussed in this work.\n\nWhen closely spaced repeated observations are available, the study of the proper motion is a useful tool in determining the ranges for $\\beta$ and $\\theta$. Surprisingly, several works \\citep[e.g.][]{Giroletti2004b, Piner2008, Piner2010} have reported subluminal motions, sometimes consistent with the component being stationary, in the jets of all the $\\sim$10 TeV blazars for which \nproper motion studies have been performed in the literature. Thanks to the large number of dual-frequency observations, the fine time-sampling, and the high quality of the data provided by the good $(u,v)$-coverage, we performed a robust identification of the Gaussian components and constrained their motion to be consistent with no displacement at all. At the same time, the high sensitivity and in particular the stacked image place significant constraints on the jet\/counter-jet ratio.\n\nThe first immediate consequence is that we can reject the hypothesis that the small $\\beta_{\\rm app}$ is solely due to a projection effect, since it would require an unrealistically narrow viewing angle: in the case of component C4b, the upper limit to the observed motion implies a viewing angle $<1.3^\\circ$ to reproduce the observed jet\/counter-jet ratio (and even smaller to agree with the high energy limits). If the jets' distribution is isotropic on the sky, the real number of misaligned sources (parent population) is incompatible with these very small values of $\\theta$. For example, in the Bologna Complete Sample selected by \\citet{Giovannini2005} at low frequency (and thus free from Doppler favoritism bias), one would expect fewer than 0.03 sources with $\\theta<1.3^\\circ$ \\citep[see also][]{Tavecchio2005,Henri2006}. \n\nOn the other hand, since larger values of the viewing angle capable of reproducing the observed lack of proper motion are incompatible with the jet\/counter-jet ratio, we conclude that the pattern velocity cannot be representative of the bulk flow velocity. The Gaussian components obtained in our model fit provide a good description of the visibility data but do not represent well-defined, high-contrast jet features \\citep[see][]{Lyutikov2010}. In our interpretation, the low apparent speeds found imply that the proper motion of Mrk421 does not provide any information about the jet bulk velocity; even on the basis of the sole jet brightness ratio, untenable viewing angles would be necessary to match the pattern and bulk velocities. \n\nWhat then are the real values of the viewing angle and the jet bulk velocity in Mrk421? To reproduce the observed jet asymmetry, we needed to consider a range of velocities $0.82<\\beta<1$ and angles $0^\\circ<\\theta<35.0^\\circ$. \nWe could exclude the upper range of the $\\theta$ values, since this would not reproduce the high Doppler factors required by high energy observations \\citep{Gaidos1996, Abdo2011}; in particular, we were unable to achieve $\\delta>20, 10, 5, 3$ when $\\theta>3.0^\\circ, 5.7^\\circ, 11.5^\\circ, 19.4^\\circ$, respectively.\n\nSmaller angles thus seem to be favored since they were the only ones consistent with high values of $\\delta$; however, such small angles would still represent a challenge to the observed radio properties. We were able to estimate the intrinsic power of the radio core $P_c^{\\rm intr}$, by debeaming the observed monochromatic luminosity  of the core $P_c^{\\rm obs}$ with the equation\n\n$$P_c^{\\rm obs}=P_c^{\\rm intr} \\times \\delta_c^{2-\\alpha}.$$\nWith a value of $P_c^{\\rm obs}\\sim6.8\\times10^{23}$ W\\,Hz$^{-1}$ at 15\\,GHz,  $\\alpha=-0.3$, and $\\delta=20$, we obtained for the intrinsic power of the core $P_c^{\\rm intr}\\sim5.8\\times10^{20}$ W\\,Hz$^{-1}$; this value was at the very low end of the typical range of intrinsic power found for different samples of radio galaxies \\cite[e.g.][]{Liuzzo2011}, suggesting that lower values of $\\delta$ provide a more typical core power. \n\nMoreover, the 12 monthly observations have not revealed any dramatic flux-density variability in the core of the source, which further points to a lower Doppler factor for the radio jet.\nWe estimated the variability brightness temperature of the core ($T_{\\rm B,var}$) with the formula proposed by \\citet{Hovatta2009}\n$$ T_{\\rm B,var}= 1.548 \\times 10^{-32} \\frac{\\Delta S_{\\rm max}d_L^2}{\\nu^2\\tau^2(1+z)},$$\nwhere $\\nu$ is the observed frequency in GHz, $z$ is the redshift, $d_L$ is\nthe luminosity distance in meters, $\\Delta S_{\\rm max}$ is the difference between the maximum value for the core flux density and the minimum value, and $\\tau$ is the variability time. With the values provided by our observations and $\\tau$ $\\sim$ 90 days, we obtaind a value of $T_{\\rm B,var} \\sim 2.1\\times 10^{10}$~K, which does not require any significant beaming.\n\nWe similarly calculated $T_{\\rm B}$ for the most compact component following the standard formula \\citep{Piner1999,Tingay1998}\n$$T_{\\rm B}=1.22 \\times 10^{12} \\frac{S(1+z)}{ab\\nu^2},$$\nwhere $S$ is the flux density of the component measured in Jy, $a$ and $b$ are the full widths at half maximum of the major and minor axes respectively of the component measured in mas, $z$ is the redshift, and $\\nu$ is the observation frequency in GHz. The resulting $T_{\\rm B}$'s are on the order of a few $\\times 10^{11}$ K, only slightly exceeding the limit derived by \\citet{Readhead1994} from equipartition arguments.\n\nTaken together, the lack of superluminal features, the low core dominance, and the weak variability suggest a scenario in which no strong beaming is required in the radio jet. This is not uncommon in TeV blazars \\citep{Piner2004, Piner2008}, but unprecedentedly firm observational support for it has been provided by our intensive campaign. Low values of the Doppler factor, e.g. $\\delta \\sim 3$, can reproduce the observational radio properties, including the jet brightness asymmetry. \n\nWe conclude that the Doppler factor must be different in the radio band than the $\\gamma$-ray band. Since we do not expect that the viewing angle changes significantly, this leads us to the necessity of a velocity structure in the jet, as previously discussed by e.g.\\ \\citet{Chiaberge2000},  \\citet{Georganopoulos2003}, and \\citet{Ghisellini2005}. Our images do not provide strong evidence in favour of either a radial or transverse velocity structure, although previous works have revealed a limb brightening in Mrk421, on both a milliarcsecond scale at 43 GHz \\citep{Piner2010} and at $d>10$ mas at 5 GHz \\citep{Giroletti2006}. This would favor the presence of a transverse velocity structure across the jet axis. This structure consists of two components: a fast inner \\textit{spine} and a slower outer \\textit{layer}. Different Doppler factors were obtained depending on whether we measured the speed of the spine or the layer.  \n\nA viable scenario for Mrk421 is that the viewing angle is between $2^\\circ$ and $5^\\circ$, which is consistent with the statistical counts of low-power radio sources and the possibility of reaching the high Doppler factors required by SED modeling and high-energy variability. The jet velocity is structured, with a typical Lorentz factor of $\\gamma\\sim 1.8$ in the radio region (yielding $\\delta \\sim 3$), and $\\gamma \\sim \\delta \\sim 20$ in the high-energy emission region. For example, assuming  $\\theta=4^\\circ$, $\\beta_{\\rm h.e.}=0.998$, and $\\beta_{\\rm r}=0.82$, we obtained a value of $\\delta_{\\rm h.e.}=14.3$ and $\\delta_{\\rm r}=3.2$ and successfully reproduced all the observational properties of the source.\n\nIn summary, the detailed analysis presented in this paper has largely confirmed with improved quality expectations based on the knowledge so far achieved for TeV blazars. We have also estimated with a good level of significance some important and fundamental parameters ($\\delta$, $\\theta$, $\\beta$, $\\alpha$) that characterize the physical processes in blazars. \nHowever, there is still much to be understood and we expect to obtain other\nsignificant results from the analysis extended to other wavelengths, particularly in the $\\gamma$-ray domain. Additional works on the dataset presented in this paper are planned, and will deal with, e.g., the 43\\,GHz images and the polarization properties. Moreover, we intend to combine our dataset  with those of other works (e.g.\\ \\citealt{Piner2005} or the MOJAVE survey, \\citealt{Lister2009}) to increase the temporal coverage of the observations and obtain even tighter constraints over a longer time frame.\n\\vspace{-0.2 cm}\n\\begin{acknowledgements}\nThis work is based on observations obtained through the BG207 VLBA project, which makes use of the Swinburne University of Technology software correlator, developed as part of the Australian Major National Research Facilities Programme and operated under licence \\citep{Deller2011}. The\nNational Radio Astronomy Observatory is a facility of the National Science Foundation operated\nunder cooperative agreement by Associated Universities, Inc. For this paper we made use of the NASA\/IPAC Extragalactic Database NED which is operated by the JPL, Californian Institute of Technology, under contract with the National Aeronautics and Space Administration. We acknowledge financial contribution from grant PRIN-INAF-2011. This research is partially supported by KAKENHI (24540240). KVS and YYK are partly supported by the Russian Foundation for Basic Research (project 11-02-00368), and the basic research program ``Active processes in galactic and extragalactic objects'' of the Physical\nSciences Division of the Russian Academy of Sciences. YYK is also supported by the Dynasty Foundation. The research at Boston University was supported in part by NASA through Fermi grants NNX08AV65G, NNX08AV61G, NNX09AT99G, NNX09AU10G, and NNX11AQ03G, and by US National Science Foundation grant AST-0907893. We thank Dr. Claire Halliday for the language editing work which improved the text of the present manuscript.\n\\end{acknowledgements}\n\\vspace{-0.8 cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{INTRODUCTION} \nThe life cycles of dwarf galaxies with present-day stellar masses $\\lesssim10^{9}~M_{\\odot}$ illustrate many challenges to our current understanding of galaxy formation and evolution. While we see that starbursts do not play an important role in the global star-formation at the present epoch \\citep{lee}, it is likely that the star formation histories of dwarf galaxies are complex and varied \\citep{mateo98} and that their typical star-formation rates were higher in the past \\cite[e.g.][]{gallagher}.  That star formation in low-mass galaxies may be very burst-like is predicted by  hydrodynamical simulations \\cite[e.g.,][]{pelupessy,stinson,shensims}.  In general, star formation appears to be regulated by stellar feedback in the form of supernovae and winds that heat and deplete the central reservoirs of cold gas required for continued star formation.  In simulations of lower mass systems, feedback is predicted to eject gas out of the galaxy and into the halo, resulting in an episodic star formation history \nacross the entire galaxy \\citep{stinson}. Repeated bursts have been cited as the \ndriving \nforce behind intense feedback mechanisms that can change the dynamical profile of the systems by driving baryons out of the center of the halo on short time scales, which also displaces the dark matter from the center and creates a cored dark matter profile \\citep{n96,pontzen,governato,zolotov,amorisco}, potentially addressing one of the principal challenges to the standard $\\Lambda$CDM cosmology. \n\n\nUntil recently, however, the progenitors of contemporary dwarf galaxies at high-redshift could not be studied directly.  ``Archaeological'' studies of resolved stellar populations in local dwarfs \\cite[e.g.,][]{grebel, weisz} have confirmed that their star-formation histories are indeed not smooth, and have also found that a large fraction of their stars formed at early epochs ($z > 1$). For such old stellar populations ($>7$ Gyr), the age resolution achieved by the archaeological approach is insufficient to distinguish star formation events with durations of $10^7$ and $10^8$ years, leaving the actual ``burstiness'' of the star formation history unconstrained. \n\n\n\n\nLook-back studies, directly observing the star formation activity in very low-mass systems at $z > 1$, were impractical until recently.  But, near-infrared spectroscopy and deep imaging from the Wide-field Camera 3 (WFC3) on the \\textit{Hubble Space Telescope} (HST) has provided a new opportunity. An abundant population of galaxies at $z\\sim 1.7$ with extremely high equivalent widths (EWs) was found by \\citet{vdw} using data from the Cosmic Assembly Near-infrared DEep Legacy Survey\\footnote{\\url{http:\/\/candels.ucolick.org\/}}  \\cite[CANDELS,][]{candels1,candels2}.  They find a large number of objects with unusual $I-J$ and $J-H$ colors which imply the presence of a bright emission line in the $J$-band, likely to be [O III] after considering other photometric constraints.  Slitless grism spectroscopy confirms the redshifts and emission-line interpretation of four of these objects.  These are the high-EW tail of the emission line galaxies found in \\citet{straughn} via slitless grism spectroscopy at the HST, a \ntail which is also probed by \\citet{atek}.  \nOther objects with similarly low masses and metallicities at these redshifts have been discovered in \\citeauthor{gb2} (2012b; $EW_{[O III],rest}$=1499 \\AA) and \\citeauthor{vdw13} (2013; $EW_{[O III],rest}$=1200 \\AA) assisted by strong gravitational lensing, in \\citeauthor{masters} (2014; $EW_{[O III],rest}$=154 \\AA), in \\citeauthor{erb10} (2010; $EW_{[O III],rest}$=285 \\AA), and in \\citeauthor{maseda} (2013; discussed further below).  All of these objects are emission-line dominated systems with low metallicities and high equivalent widths, the so-called ``Extreme Emission Line Galaxies'' (EELGs).  \n\nThese systems are likely the high-EW tail of the distribution of high-redshift dwarf galaxies, and resemble the class of blue compact dwarf galaxies \\cite[BCDs,][]{sargent} observed locally in several ways: low masses, high SFRs relative to their mass, and strong emission lines. However, the EELGs are indeed ``extreme,'' with sSFR values an order of magnitude higher than the BCDs, similar to the strong [O III] emitters (``green peas'') discovered photometrically in the SDSS by \\citet{greenpea}, as well as spectroscopically by \\citet{amorin} and \\citet{izotovSDSS}.  As suggested by those authors, the star-formation mode exhibited in the ``green peas'' is likely a relic of a mode that was much more prevalent in the earlier Universe: their comoving number density is one to two orders of magnitude lower than the value of $3.7\\times10^{-4}~Mpc^{-3}$ for EELGs at $z\\sim 1.7$ \\citep{vdw}.  \n\nThe implication of strong emission lines and an extremely faint continuum are that these systems have low masses and are undergoing an intense burst of star formation.  Equivalent widths $>$ 100~\\AA$ $ and stellar masses of $10^8-10^9 ~M_{\\odot}$ imply specific star formation rates (sSFR) in excess of 10 Gyr$^{-1}$, which is more than an order of magnitude higher than star-forming systems of equivalent masses at lower redshifts \\citep{karim}.  These low stellar and dynamical masses are confirmed in \\citet{maseda}, who also rule-out significant contributions to the total mass from older stellar populations for objects with restframe [O III]$\\lambda5007$ equivalent widths $> 500$ \\AA$ $ and intimate that the bursts are intense and have low metallicities. The implied star formation rates and masses have only been reproduced recently in hydrodynamical simulations \\cite[e.g. in][]{shensims}.  \n\nAlthough these previous observational studies have placed constraints on various quantities, many uncertainties remain.  In the case of \\citet{maseda} and \\citet{vdw}, all of the observed [O III], H$\\beta$, and H$\\alpha$ emission is attributed to star formation and not AGN.  In \\citet{vdw}, upper-limits are placed on the black hole masses from the UV-continuum slopes, but their starbursting nature is merely plausible given the lack of knowledge about low-metallicity AGN \\citep{izotovAGN, kewley13}.  Low metallicies are simply inferred from the consistency of the SED fits using low-Z (0.2 $Z_{\\odot}$) templates with the observed photometry.  \n\nThese objects represent a field of growing importance, both for studies of low-mass dwarf galaxies and for the highest-z galaxies observed.  Depending on the strength, duration, and initial mass of these bursts, the descendants could display a wide range of masses, from present-day $\\sim 10^9~M_{\\odot}$ dwarf galaxies to potentially Milky Way-like systems, particularly if merging is important.  In any case, these galaxies are also significant contaminants in searches for higher-z dropout sources \\citep{atek}, as in the case of \\textit{UDFj-30546284} \\citep{ellis,gb13,bouwens}.  As mentioned in \\citet{atek} and \\citet{coe}, great care must be taken in the interpretation of high-z candidates, as EELGs can potentially reproduce their observed colors (although the implied EWs in the latter case exceeded those that can be produced from star formation alone).\n\nSome issues still remain in our understanding of such systems, such as stringent limits to the low masses and metallicities, including those with EWs $<$ 500 \\AA, and the starburst origin of their strong emission lines.  Here we \ncombine both high- and low-resolution near-IR spectroscopy with broadband \nphotometry.  With the low-resolution spectra, we select candidates for follow-up high-resolution spectroscopy and obtain emission line fluxes.  The high-resolution spectra constrain various emission line ratios, some of which are useful diagnostics of AGN activity, as well as line widths, which are themselves a probe of the dynamical masses of the systems.  This provides strong evidence for their low masses and low metallicities, as well as confirming their starbursting nature.  Sophisticated modeling of the broadband SEDs  constrains the stellar masses and ages, as well as providing information on the dust content and metallicities. Together, this tells us about the strength and duration of the star-forming event.\n\n\n\n\n\nThe remainder of this paper is organized as follows.  In $\\S$ 2 we describe the near-infrared spectroscopy and multi-band photometry used in the initial candidate selection process and the subsequent follow-up observations.  $\\S$ 3 presents the results of the spectroscopic study, including emission-line widths, physical sizes, and masses.  In $\\S$ 4 we confirm their low metallicities and rule out AGN as a significant source of contamination.  $\\S$ 5 presents the implications of this work for the gas content of these systems, and $\\S$ 6 summarizes our findings and puts the results into the overall context of the formation history of dwarf galaxies.  We adopt a flat\n$\\Lambda$CDM cosmology with $\\Omega_m=0.3$ and H$_0=70 ~$km s$^{-1}$\nMpc$^{-1}$ and a \\citet{chabrier03} IMF throughout.\n\n\\section{DATA}\n\n\n\n\\subsection{Candidate Selection}\n\\label{sec:sel}\nIn order to search for and investigate these ``starbursting dwarf galaxies,'' we take a multi-faceted approach.  Our preliminary search utilizes data from the 3D-HST survey\\footnote{\\url{http:\/\/3dhst.research.yale.edu\/Home.html}} \\citep{vd,gb}, a near-infrared spectroscopic Treasury program utilizing the WFC3.  This program provides WFC3\/IR primary and Advanced Camera for Surveys (ACS) parallel imaging and grism spectroscopy over approximately three-quarters (625 square arcminutes) of the CANDELS fields.  The main source of spectroscopic data comes from the WFC3 G141 grism, with an effective wavelength coverage of 1.1 to 1.65 $\\mu$m.\n\n\\begin{figure}\n\n\n \\includegraphics[width=0.475\\textwidth]{f1.eps}\n\n\n\\caption{Restframe equivalent widths as a function of redshift for our entire sample.  Circles represent equivalent widths from [O III]$_{5007}$ and squares represent equivalent widths from H$\\alpha$.  The equivalent widths were determined from photometry and grism spectroscopy, see Section \\ref{sec:sel}.  For objects in which both [O III] and H$\\alpha$ are visible in the grism spectrum, both equivalent widths are plotted.  Open symbols show objects without LUCI1 line detections, due to intrinsic faintness or skyline contamination.  Our emission-line detection rate is close to 100\\% for objects with EW $>$ 300 \\AA.}\n\\label{fig:ewz}\n\\end{figure}\n\n\n\n\n\nThe grism data allow us to select and confirm strong line emitters spectroscopically.  Photometric cuts, such as the $iJH$ cut of \\citet{vdw} and a similar $ViJH$ selection (excess in $H$ compared to a blue continuum, all from CANDELS), are used to preselect objects with strong features in their SEDs.  The G141 grism data are reduced according to \\citet{gb} for the UKIDSS Ultra-Deep Survey (UDS), GOODS-South (GOODS-S), and COSMOS fields, and are then used to confirm bright lines with little or no associated continuum.  While we find numerous examples of these objects at $z > 1$ (Maseda et al. in prep.), we focus here on objects where the emission lines do not fall in the wavelength range between the $J$-, $H$-, and $K$-bands, enhancing the chances for detectability from the ground. The low-resolution WFC3 grism data ($R\\sim 130$) provide redshift information such that targets can be selected with [O III] in the redshift range  1.15 $< z <$ 2.40 and H$\\alpha$ in the redshift range 0.64 $< z <$ 1.59 to $\\delta$z\/z $\\sim$ 0.005.  However, our photometric preselection relies on flux excesses such that we mostly see objects at 1.3 $\\lesssim z \\lesssim$ 1.8 and 2.1 $\\lesssim z \\lesssim$ 2.3.  Note that we do not resolve the continuum in our ground-based observations (discussed presently), so all EW values are calculated from the grism spectra directly.  In total, we select 31 objects for ground-based spectroscopic observations using this method.  An additional five candidates are taken from the sample of \\citet{vdw} and one from \\citet{straughn11}.  Their equivalent widths as a function of redshift are shown in Figure \\ref{fig:ewz}.  Non-detections are due to either intrinsic faintness in the lines (we are sensitive to line flux in our ground-based spectra, not EW) or contamination from OH-skylines.\n\n\n\n\n\n\n\n\\subsection{LBT\/LUCI1 Spectroscopy}\n\nWe observed our grism-selected sample with the LUCI1 multi-object spectrograph \\citep{seifert} on the 8.4 m \\textit{Large Binocular Telescope} (LBT).  We use LUCI1 in MOS mode, splitting our 31 candidates between four masks, during April 2012 (two masks in the COSMOS field), October 2012 (one mask in the UDS field), and March 2013 (an additional mask in the COSMOS field).  Approximately two hours are spent observing in each of the $J$- and $H$-bands for the first two COSMOS masks (A and B) using 1$''$ slits, two hours in each of the $H$- and $K$-bands for the UDS mask (C) with 0.6$''$ slits, and three total hours on the final COSMOS mask (D) in the $J$- and $H$-bands using 0.6$''$ slits.  All data are taken using the high-resolution $210\\_zJHK$ grating ($R_J=8460$, $R_H=7838$, $R_K=6687$). The exposures are dithered by 3$''$ and are of varying durations, depending on the band: $J$-band data using 600s exposures, $H$-band data with 300s exposures, and $K$-band data with 120s exposures.  The shorter \nintegrations in the $H$- and $K$-bands lead to lower signal-to-noise (S\/N) due to additional readnoise, but are necessary so as not to saturate the detector in the regions with the brightest OH sky lines.  Seeing was generally good (between 0.5$''$ and 1$''$ in the optical) during the COSMOS observations, with good transparency.  For the UDS observations, seeing was generally better than 1$''$.  Table \\ref{tab:obs} details the observations of individual objects, including IDs (from the CANDELS catalogs, discussed in Section \\ref{sec:sed}), coordinates, and the main line detections.\n\\begin{deluxetable}{lcccc}\n\\tabletypesize{\\scriptsize} \n\\tablecaption{Summary of Near-IR Observations\\label{tab:obs}}\n\\tablewidth{0pt}\n\\tablehead{ \\colhead{ID} & \\colhead{RA} & \\colhead{Dec} & \\colhead{Mask} & \\colhead{Observed} \\\\\n    & (deg) & (deg) & &Lines}\\\\\n    \\startdata\nGOODS-S-7892&53.17194 &-27.75915&...&[O III], H$\\alpha$\\\\\nGOODS-S-43693&53.07129&-27.70580&...&H$\\alpha$\\\\\nGOODS-S-43928&53.05158&-27.70476&...&[O III]\\\\\nUDS-6195&34.42648&-5.25577&...&[O III], H$\\alpha$\\\\\nUDS-6377&34.42857&-5.25532&...&[O III], H$\\alpha$\\\\\nUDS-7665&34.39076 &-5.25080 &C&[O III]\\tablenotemark{a}\\\\\nUDS-10138&34.42336 &-5.24226&C&[O III]\\tablenotemark{a}\\\\\nUDS-12435&34.41087 &-5.23481&C&H$\\alpha$\\tablenotemark{a}\\\\\nUDS-12539&34.47389&-5.23423&...&[O III], H$\\alpha$\\\\\nUDS-12920&34.39870 &-5.23320&C&...\\\\\nUDS-15319&34.40516& -5.22493&C&...\\\\\nUDS-19167&34.43140 &-5.21212&C&[O III]\\\\\nUDS-24154&34.39137 &-5.19531&C&[O III], H$\\alpha$\\\\    \nCOSMOS-8509&150.09837 &2.26596 &B&...\\\\    \nCOSMOS-8700&150.09740 &2.26848 &B&...\\\\    \nCOSMOS-10599&150.09535 &2.28725 &B&[O III]\\\\\nCOSMOS-11530&150.11931 &2.29688 &B&...\\\\  \nCOSMOS-12102&150.09728 &2.30252 &B&[O III], H$\\alpha$\\\\\nCOSMOS-13184&150.12424 &2.31367 &B&[O III]\\\\    \nCOSMOS-14249&150.11011 &2.32459 &B&...\\\\\nCOSMOS-14435&150.16232&2.32602&D&...\\\\\nCOSMOS-15091&150.15955&2.33330&D&[O III]\\\\\nCOSMOS-16152&150.18762 &2.34469&A&...\\\\\nCOSMOS-16286&150.17699&2.34539&D&[O III]\\tablenotemark{b}\\\\\nCOSMOS-16566&150.17067&2.34830&D&[O III], H$\\alpha$\\\\\nCOSMOS-17118&150.15114&2.35410&D&[O III]\\\\\nCOSMOS-17162&150.15134 &2.35482&A&...\\\\\nCOSMOS-17295&150.18318&2.35537&D&...\\\\%H$\\beta$, [O III]& Frame-Frame\\\\\nCOSMOS-17539&150.12814 &2.35810&A&...\\\\\nCOSMOS-17839&150.15677 &2.36080&A&[O III]\\\\\nCOSMOS-18299&150.17098 &2.36536&A&...\\\\\nCOSMOS-18358&150.16719 &2.36689&A&[O III]\\\\\nCOSMOS-18582&150.13281 &2.36878&A&...\\\\\nCOSMOS-18777&150.18628 &2.37054&A&...\\\\\nCOSMOS-19049&150.13886 &2.37340&A&[O III]\\tablenotemark{b}, H$\\alpha$\\\\\nCOSMOS-19077&150.18309 &2.37295&A&[O III]\\\\\nCOSMOS-20589&150.18056&2.38822&D&... \\enddata\n\\tablecomments{All IDs refer to the CANDELS catalog for that particular field.  Mask A was observed on 21 April 2012, mask B on 22 April 2012, mask C on 10 and 11 October 2012, and mask D on 12 March 2013, all at the LBT; \\textit{GOODS-S-7892} was observed on 15 December 2012, \\textit{GOODS-S-43693} on 1 October 2012, \\textit{GOODS-S-43928} on 15 October 2012, \\textit{UDS-6195} and \\textit{UDS-6377} on 27 August 2012, and \\textit{UDS-12539} on 2 and 27 August 2012 (120 minutes in the NIR), all at the VLT.  Line detections are at least 1$\\sigma$.}\n\\tablenotetext{a}{Due to technical problems during the first night of observations, the total exposure time used for these line extractions is only 3600s in $H$.}\n\\tablenotetext{b}{Only the [O III]$\\lambda$5007 component.}\n\\end{deluxetable}\n\\subsubsection{LUCI1 Data Reduction}\nWe first mask regions of the spectra that are affected by persistence due to the acquisition and alignment exposures.  This effect is reduced with each readout, so we only mask the regions if they are 2$\\sigma$ higher than the background level.  We then create flat-field images from lamp-illuminated exposures and remove cosmic rays using a median-stacking technique.  The most important cosmetic step is the removal of bad pixels, which are identified in the lamp-illuminated exposures, and the hot pixels in the spectra, which are identified in dark exposures.  Additionally, for an as-yet unknown reason, the first exposure of every series has small ``halos'' around the hot pixels.  As such, we remove slightly larger regions around these hot pixels in the first exposure of every series.  Our wavelength calibration is done using the OH sky lines, with a code based on \\texttt{XIDL} routines\\footnote{\\url{http:\/\/www.ucolick.org\/\n~xavier\/IDL\/}}.  We also use \\texttt{XIDL} for the final sky subtraction, which uses a spline-fitting algorithm to measure and remove the lines.  To maximize S\/N, we do not use frame-frame subtraction and instead measure the sky from the individual frames, with the exception being some objects with particularly bad skyline contamination, where frame-frame subtraction better removes the skylines but adds noise to the spectrum.  Dithering the exposures by 3$''$ ensures a decreased dependence on the pixel-to-pixel variations in the detector.\n\nSince our objects of interest have virtually no visible continuum, one-dimensional spectral extraction must be done carefully.  We must visually search for the lines in the stacked reduced spectra.  Since we know the wavelengths of the brightest lines from the grism data, this exercise is straightforward.  We isolate the line region according to a signal-to-noise cut, and then collapse that region in the wavelength direction to create a slice containing the spatial line profile.  This profile is fit with a standard Gaussian function.  The width of the distribution, $\\sigma$, and the center, $\\mu$, are then used in the full spectral extraction: a Gaussian function with these same $\\sigma$ and $\\mu$ values is fit for each pixel row in the spatial direction tracing a constant distance from the edge of the (curved) slit with the amplitude as the only free parameter, reflecting the electron counts at that particular wavelength.  All sets of observations were reduced and analyzed separately and the resolution is incorporated into the calculation of the intrinsic line widths to remove systematics in the velocity dispersion measurements.  Flux calibration is done by comparing the integrated counts from the LUCI1 spectrum to the integrated line flux from the 3D-HST grism spectrum when possible.\n\n\n\n\\subsection{VLT\/X-Shooter Spectroscopy}\n\n\nFor an additional six sources, we obtained near-IR and visible spectra using the X-Shooter spectrograph \\citep{vernet} on the 8.2 m \\textit{ESO Very Large Telescope} (VLT).  We observed five of the objects initially found in \\citet{vdw}:  three with previous grism-spectroscopic confirmation in \\citet{straughn11} and Weiner et al. (in prep.), and the two remaining candidates with the largest photometrically inferred line fluxes.  A sixth candidate was selected from the \\citet{straughn11} sample.  Observations were done in long-slit mode from August to December 2012 with 40 minute integrations using 1$''$\/0.9$''$\/0.9$''$ (UVB\/VIS\/NIR) slits and the 100k\/1pt\/hg\/1$\\times$2 readout mode.  See Table \\ref{tab:obs} for the targets and observing dates.  The proximity of objects \\textit{UDS-6377} and \\textit{UDS-6195} allowed for them to be observed in the same slit.  \n\n\n\nAlthough the X-Shooter spectrograph also observes in the UV-Blue, four of our six objects were not observed during dark time, rendering the data unusable.  The near-IR region of X-Shooter spans the combined $YJHK$ region from 1024$-$2048 nm, while the visible region spans 559.5$-$1024 nm.  Reduction of the X-Shooter data is performed using version 2.0.0 of the ESO XSHOOTER pipeline\\footnote{\\url{http:\/\/www.eso.org\/sci\/software\/pipelines\/xshooter\/xsh-pipe-recipes.html}}, which provides merged, 2D near-IR and visible spectra.  Extraction is performed in a similar manner to the LUCI1 data.\n\n\n\n\n\n\n\\section{Dynamical and Stellar Masses}\nIn order to confirm our hypothesis that these systems represent starbursting dwarf galaxies, we must confirm their low stellar masses, low metallicities, and high star formation rates.  Stellar masses can be constrained through SED fits to broadband photometry, and metallicities can be constrained by observing emission-line ratios, such as [O III]\/H$\\beta$.  Any star formation rate is contingent on the nature of the emission lines, since AGN can also produce very high excitations.\n\\subsection{Methods and Results}\n\n\\label{sec:lines}\n\n\n\n\\begin{deluxetable*}{lccccccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Sample of Emission Line Galaxies\\label{tab:lines}}\n\\tablewidth{0pt}\n\\tablehead{ \\colhead{ID} & \\colhead{m$_{F140W}$} & \\colhead{r$_{eff}$\\tablenotemark{a}} & \\colhead{f$_{H\\alpha}$} &\\colhead{EW$_{H\\alpha}$}&\\colhead{f$_{[O III]}$}&\\colhead{EW$_{[O III],5007}$}&\\colhead{z$_{spec}$}&\\colhead{$\\sigma_{H\\alpha}$}&\\colhead{$\\sigma_{[O III]}$} \\\\\n    & (AB) & (kpc) & &(\\AA) &&(\\AA)& &($\\>{\\rm km}\\,{\\rm s}^{-1}$) &($\\>{\\rm km}\\,{\\rm s}^{-1}$)}\n\\startdata\n GOODS-S-33131 &  23.66$\\pm$0.08&  0.68$\\pm$0.62& ... & ... &  7.36$\\pm$2.99 &  693$\\pm$47&  1.687 & 48.7$\\pm$4.3 &   52.3$\\pm$5.7 \\\\\n GOODS-S-43693 &  24.36$\\pm$0.12&  0.35$\\pm$0.06& ... & ... &  16.9$\\pm$0.88 &  861$\\pm$66&  1.738 & 54.4$\\pm$4.5 &... \\\\\n GOODS-S-43928 &  24.59$\\pm$0.15&  1.9$\\pm$0.48& 4.5$\\pm$1.2\\tablenotemark{b} & 199\\tablenotemark{b} &  3.7$\\pm$1.6\\tablenotemark{b} &  176\\tablenotemark{b} &  1.472 &...&  31.4$\\pm$8.2  \\\\\n \n UDS-6195 &  24.26$\\pm$0.13&  1.4$\\pm$0.42& ... & ... &  ... &  701$\\pm$95\\tablenotemark{c}&  1.687 & 69.9$\\pm$4.9 &  54.7$\\pm$6.1 \\\\\n UDS-6377 &  24.53$\\pm$0.17&  0.67$\\pm$0.04& ... & ... &  ... &  731$\\pm$86\\tablenotemark{c}&  1.664 & 54.5$\\pm$4.5 &  48.2$\\pm$5.9 \\\\\n  UDS-7665 &  25.40$\\pm$0.14&  0.51$\\pm$0.08& ... & ... &  13.7$\\pm$2.76 &  803$\\pm$162&  2.298 & ...&  57.8$\\pm$9.7 \\\\\n   UDS-10138 &  23.77$\\pm$0.03&  0.75$\\pm$0.08&  ... &  ...&  10.9$\\pm$0.57 &  322$\\pm$17 &  2.151 & ...&  80.9$\\pm$10.0 \\\\\n    UDS-12435 &  23.42$\\pm$0.03&  1.0$\\pm$0.06&  ...&  ... &  12.2$\\pm$0.94 &  263$\\pm$31 &  1.611 &  65.2$\\pm$11.3 & ...\\\\\n UDS-12539 &  23.39$\\pm$0.06&  1.3$\\pm$0.07& ... & ... &  35.5$\\pm$2.73 &  713$\\pm$42&  1.621 & 81.3$\\pm$4.3 &  71.1$\\pm$5.7\\\\\n  UDS-19167 &  23.99$\\pm$0.04&  1.1$\\pm$0.20 &  ...&  ...&  21.5$\\pm$2.82&  723$\\pm$95 &  2.185 & ...&  54.2$\\pm$9.4 \\\\\n  UDS-24154 &  23.78$\\pm$0.04 &  1.8$\\pm$0.16 &  ...&  ... &  21.9$\\pm$3.09 &  503$\\pm$34 &  2.297 & 72.5$\\pm$13.1 &  61.0$\\pm$10.8 \\\\\nCOSMOS-10599 &  24.47$\\pm$0.26  &  0.67$\\pm$0.16&...& ...&  11.8$\\pm$0.89 &  714$\\pm$85 &  2.220 & ...&  30.9$\\pm$9.0 \\\\\nCOSMOS-12102 &  22.82$\\pm$0.06 & 1.6$\\pm$0.08 &  19.9$\\pm$1.0 &  360$\\pm$18 &  49.4$\\pm$2.28 &  630$\\pm$29 &  1.463 &  230.8$\\pm$14.7 &  241.3$\\pm$12.7 \\\\\nCOSMOS-13184 &  23.91$\\pm$0.16 &  0.53$\\pm$0.10 &... & ... &  11.5$\\pm$2.94 &  598$\\pm$189 &  2.199 & ...&  40.3$\\pm$8.9 \\\\\nCOSMOS-15091 &  25.46$\\pm$0.13& 0.75$\\pm$0.11& ... & ... &  5.99$\\pm$3.61 &  628$\\pm$152&  1.583 & ...&  38.2$\\pm$10.0 \\\\\nCOSMOS-16286 &  24.64$\\pm$0.11& 1.1$\\pm$0.13& 0.39$\\pm$3.32 & 41$\\pm$345 &  8.76$\\pm$3.46 &  888$\\pm$351&  1.444 & ...&  46.7$\\pm$14.4 \\\\\nCOSMOS-16566 & 24.60$\\pm$0.09&  1.4$\\pm$0.09& 4.22$\\pm$3.26 & 560$\\pm$432 &  7.28$\\pm$3.58 &  388$\\pm$191&  1.437 & 25.5$\\pm$14.0 &  32.8$\\pm$8.4 \\\\\nCOSMOS-17118 & 24.16$\\pm$0.13 & 4.8$\\pm$0.33& ... & ... &  12.3$\\pm$4.22 &  493$\\pm$169&  1.656 & ...&  46.5$\\pm$8.8 \\\\\nCOSMOS-17839& 24.36$\\pm$0.24 &  0.99$\\pm$0.06 & 3.27$\\pm$3.20 &  325$\\pm$230 &  16.3$\\pm$3.58 &  1126$\\pm$247 &  1.412 & ...&  43.3$\\pm$8.9 \\\\\nCOSMOS-18358 &  22.84$\\pm$0.04 &  1.6$\\pm$0.02& ... & ... &  30.2$\\pm$0.97 &  316$\\pm$26 &  1.645 & ...&  55.9$\\pm$9.0 \\\\\n COSMOS-19049 &  22.93$\\pm$0.05 &  2.3$\\pm$0.06 &  16.4$\\pm$1.2 &  368$\\pm$28 &  20.9$\\pm$2.20 &  330$\\pm$35 &  1.370 &  81.9$\\pm$50.2 &  122.0$\\pm$11.0\\\\\n COSMOS-19077 &  23.69$\\pm$0.12 &  1.6$\\pm$0.05 & ...& ... &  20.5$\\pm$0.77 &  536$\\pm$20 &  1.649 & ...&  47.7$\\pm$9.5\\enddata\n\\tablecomments{All fluxes are given in units of 10$^{-17}$ erg s$^{-1}$ cm$^{-2}$.  Equivalent widths are quoted in the rest-frame.  A description of the size measurements is given in Section \\ref{sec:lines}.}\n\\tablenotetext{a}{\\citet{vdw12}}\n\\tablenotetext{b}{\\citet{straughn11}}\n\\tablenotetext{c}{\\citet{vdw}}\n\\end{deluxetable*}\n\n\n\nIn our near-IR spectra, the most prominent lines seen are [O III]$\\lambda$5007 and H$\\alpha$, along with their associated complexes ([O III]$\\lambda$4959+H$\\beta$, and [N II]$\\lambda\\lambda$6548,6584, respectively).  With the exception of \\textit{COSMOS-12102}, the emission lines can be well-fit by a Gaussian function, as described in \\citet{maseda}.  \\textit{COSMOS-12102} has the broadest emission lines of the sample.  In addition to their broadness, they display some degree of asymmetry and are thus not well-fit by Gaussian functions.  \n  \nThe skewness could be caused by several processes, such as the presence of strong outflows.  Such interpretations are beyond the scope of this paper and demand additional observations.\n\n \nThe best-fitting redshifts, velocity dispersions, and line ratios are given in Table \\ref{tab:lines}.\n\n\n\n\n\nAs described in \\citet{maseda}, velocity dispersions of the strong emission lines can be used to estimate the dynamical masses, assuming the line width comes entirely from gravitational motion in a virialized system such that\n\\begin{equation}\n M_{dyn} = 3\\frac{r_{\\rm{eff}}\\sigma^2}{G}.\n\\label{eqn:dyn}\n\\end{equation}\nHere, $\\sigma$ is the observed line width from our NIR spectrum and $r_{eff}$ is the effective radius of the galaxy from the public CANDELS catalog released in \\citet{vdw12}.  Typical objects are 1.0$\\pm$0.1 kpc in both the $J_{F125W}$- and $H_{F160W}$-bands.\n\nThese dynamical masses are listed in Table \\ref{tab:dp}, ranging from $10^{8.39}$ to $10^{10.6} ~M_{\\odot}$, with a median mass of $10^{9.13}~M_{\\odot}$.  The uncertainty in the dynamical mass estimate comes primarily from the systematic uncertainty in the proportionality constant of 3, which relates the intrinsic velocity $v$ to the observed velocity dispersion $\\sigma$, which we assume to be the same factor of 33\\% as \\citet{rix}, since in most cases our observed line widths and physical sizes are well-constrained.  Further details can be found in \\citet{maseda}.  \\citet{amorin12}, in a study of ``green peas,'' observe multiple star-forming regions and gas flows which is seen as asymmetries and broad, low-intensity wings.  While we do not see such clear evidence for outflows via asymmetric line profiles (\\textit{COSMOS-12102} excepted) or broad wings, we can not currently rule-out contributions of non-gravitational motions to the observed line widths since we do not resolve the continuum and can only observe the bright, central line regions.  Any such contributions would mean that the intrinsic dispersion is lower and that our dynamical mass estimates are upper-limits.  \n \n\n\n\n\\label{sec:sed}\nMulti-band photometry is obtained from 3D-HST \\citep{skelton}, covering 0.3-24 $\\mu$m for the GOODS-S (23 bands), UDS (17 bands), and COSMOS (31 bands) fields.  Visual inspection of the Spitzer-IRAC frames show contamination from nearby objects in \\textit{UDS-7665}, \\textit{COSMOS-13184}, and \\textit{COSMOS-15091}, so their SEDs do not include the contaminated points.  For the same reason, we do not include any the data from the 5.8 and 8$~\\mu$m IRAC channels for this sample, even though it is available as part of the publicly-released catalogs.  Only three objects have detections at 24 $\\mu$m with Spitzer-MIPS (discussed in Section \\ref{sec:starburstagn}), and an upper limit of 10 $\\mu$Jy is adopted for the remainder of the sample.\n\nWe fit the broadband spectral energy distributions of our galaxies in the same manner as \\citet{maseda} using a custom version of the \\texttt{MAGPHYS} code\\footnote{\\url{http:\/\/www.iap.fr\/magphys}} \\citep{magphys}, which computes the emission by stellar populations and the attenuation by dust in a two-component ISM, and includes nebular line emission computed self-consistently using the \\citet{pacifici12} model (\\citeauthor{cl} \\citeyear{cl}; C. Pacifici et al in prep).  The broad-band fluxes computed with this model include the contamination by emission lines, so they can be directly and robustly compared with the observed fluxes that we know are likely emission-line contaminated for these galaxies \\cite[at these and higher redshifts, it has been shown that an improper treatment of nebular contamination to broadband magnitudes results in an overestimate of stellar masses and hence an underestimate in sSFRs, e.g.][]{atek,curtislake,stark}.  \\texttt{MAGPHYS} compares the input photometry to an extensive library of SED templates spanning a wide range in parameters such as star formation history, metallicity, age, and dust optical depth using a Bayesian method.  As such, all results quoted are the medians of the posterior probability distributions \nfor each parameter, with \nuncertainties corresponding to the 16$^{th}$ and 84$^{th}$ percentiles for the distribution.  In cases where the output probabilities are not well constrained, typically due to the (potentially sytematic) uncertainties in the photometry, we adopt an uncertainty of 0.3 dex in the relevant parameter: a formal error of 0 is indicative of the models not fitting the data well.  0.3 dex is the typical uncertainty in determining stellar masses from fits to broadband photometry \\citep{conroy}.  An example probability distribution for some of the various parameters is shown in the Appendix, Figure \\ref{fig:pdf}. Results of the SED fitting are given in Table \\ref{tab:dp}, showing high sSFRs, low stellar masses, low metallicities, young ages, and low dust extinction.  Since the NIR sizes represent the restframe optical and hence the stellar continuum at these redshifts, we are probing the same physical region in both mass estimates.\n\nThe median $\\tau_V$ ($V$-band optical depth seen by young stars in the birth clouds) is 0.2, consistent with the very blue observed SEDs.  Even with the lack of infrared data to directly probe the dust content of these systems, we can place a limit on the dust mass based on the total dust attenuation and luminosity inferred from the SED fits and a prior on the dust temperature as in \\cite{dacunha}.  Resulting limits are $\\lesssim 10^7~M_{\\odot}$ and hence negligible compared to the stellar masses.\n\nAs mentioned before, the critical piece of additional information that we include to perform these fits is the line fluxes.  We see a median SFR of $\\sim$9 $M_{\\odot}~yr^{-1}$ which, combined with the low stellar masses, justifies our emission line criteria for the selection of starbursts. By separating the emission lines from the stellar continuum light, we are better able to trace the gas-phase metallicities.  This results in metallicity estimates consistent with direct probes of the oxygen abundance using emission-line ratios (see Section \\ref{sec:metals}).  In addition, it allows for better estimates of the extinction in the HII regions, which produce the aforementioned $\\tau_V$ values.\n\nOur model library of stellar population SEDs contains a broad range of\ncomplex star formation histories (SFHs), including bursts on top of\nextended SFHs with a variety of evolutionary trends (rising, falling,\nand constant).  Despite these efforts, which far exceed the still\ncommon use of exclusively exponentially-declining SFHs, systematic\nuncertainties remain.  In particular for galaxies with significant\nstar formation activity in the past $\\sim 50$~Myr, as is the case\nhere, red supergiants with individual luminosities of $\\sim\n10^5~L_{\\odot}$ can easily outshine more massive populations of stars\nwith any age $>50$~Myr, especially in the near-infrared.  Prior\nknowledge of the SFH is needed to address this issue, producing a\ndegree of circularity in the problem of stellar mass determinations.\nKeeping this in mind, we proceed and note where necessary that for\ngalaxies with estimated ages $\\lesssim 50$~Myr the mass (and age)\nestimates must be lower limits, as seen in tails to higher masses and \nmass-weighted ages, e.g. Figure \\ref{fig:pdf}.\n\nExtracted near-IR spectra and SED fitting results for \\textit{GOODS-S-33131} are shown in Figure \\ref{fig:caspec}; all remaining objects in our sample are shown in Figures 2a, 2b, and \\ref{fig:seds}.  Telluric corrections are applied as needed.\n\n\\begin{figure}\n \\includegraphics[width=.475\\textwidth]{ca22115.eps}\n\n\n\\caption{Example of a broadband SED including best-fitting model (top), WFC3 grism image\/spectrum (middle), and X-Shooter spectrum (bottom) for an object in our sample.  The NIR spectra for the remaining objects can be seen in the Figures 2a for [O III] and 2b for H$\\alpha$; the SEDs for the remaining objects can be seen in the Appendix, Figure \\ref{fig:seds}.  SED fits are performed as described in Section \\ref{sec:sed}, with red points denoting the measured photometry (open points are upper limits), the blue curve denoting the non-attenuated SED, and the black curve denoting the observed SED including dust attenuation.  The direct F140W image is shown on the left and the dispersed G141 grism image is shown to the right, with important spectral lines labeled and contamination subtracted as described in Momcheva et al. (in prep.).  The X-Shooter spectrum is smoothed by 3 pixels and is flux-calibrated according to the grism line flux.  The shaded gray area represents the +\/$-$ 1$\\sigma$ flux uncertainties and the red curve shows the best-\nfitting model of \nthe emission lines.}\n\\label{fig:caspec}\n\n\\end{figure}\n\n\\begin{deluxetable*}{lcccccc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{Derived Parameters\\label{tab:dp}}\n\\tablewidth{0pt}\n \\tablehead{ \\colhead{ID} & \\colhead{log $M_{dyn}$} & \\colhead{log $M_{\\star}$} & \\colhead{log Age} & \\colhead{Z} & \\colhead{log SFR} & \\colhead{$\\tau_{V}$} \\\\\n &($M_{\\odot}$)&($M_{\\odot}$)&(yr)&($Z_{\\odot})$&($M_{\\odot}~$yr$^{-1}$)&}\n\\startdata\nGOODS-S-33131 & 9.05$\\pm$0.30&8.91$_{-0.075}^{+0.050}$&7.78$_{-0.065}^{+0.190}$&0.321$_{-0.132}^{+0.180}$&0.892$_{-0.080}^{+0.030}$&0.242$_{-0.085}^{+0.030}$\\\\\nGOODS-S-43693 & 8.86$\\pm$0.31&8.28$_{-0.050}^{+0.195}$&7.50$_{-0.045}^{+0.240}$&0.147$_{-0.098}^{+0.160}$&0.557$_{-0.065}^{+0.090}$&0.137$_{-0.090}^{+0.050}$\\\\\nGOODS-S-43928 & 9.12$\\pm$0.38&8.83$_{-0.065}^{+0.070}$&8.39$_{-0.260}^{+0.175}$&0.169$_{-0.120}^{+0.272}$&0.262$_{-0.080}^{+0.190}$&0.327$_{-0.145}^{+0.505}$\\\\\nUDS-6195 & 9.47$\\pm$0.33&7.60$_{-0.300}^{+0.320}$&6.85$_{-0.300}^{+0.385}$&0.299$_{-0.130}^{+0.300}$&0.447$_{-0.065}^{+0.300}$&0.027$_{-0.300}^{+0.085}$\\\\\nUDS-6377 & 9.04$\\pm$0.31&8.32$_{-0.300}^{+0.140}$&6.96$_{-0.300}^{+0.735}$&0.081$_{-0.022}^{+0.096}$&1.08$_{-0.580}^{+0.300}$&1.12$_{-0.835}^{+0.025}$\\\\\nUDS-7665& 9.07$\\pm$0.33& 8.52$_{-0.300}^{+0.300}$& 7.04$_{-0.300}^{+0.300}$& 0.155$_{-0.300}^{+0.300}$& 1.21$_{-0.300}^{+0.300}$& 0.862$_{-0.300}^{+0.300}$\\\\\nUDS-10138 & 9.53$\\pm$0.31&8.43$_{-0.300}^{+0.270}$&7.10$_{-0.300}^{+0.430}$&0.177$_{-0.300}^{+0.010}$& 1.05$_{-0.135}^{+0.300}$&0.112$_{-0.040}^{+0.300}$\\\\\nUDS-12435 & 9.47$\\pm$0.33&9.42$_{-0.055}^{+0.045}$&8.60$_{-0.130}^{+0.050}$&0.189$_{-0.138}^{+0.392}$& 0.677$_{-0.030}^{+0.070}$&0.082$_{-0.040}^{+0.085}$\\\\\nUDS-12539& 9.66$\\pm$0.30&8.67$_{-0.300}^{+0.300}$&7.05$_{-0.300}^{+0.300}$&0.581$_{-0.300}^{+0.300}$&1.29$_{-0.300}^{+0.300}$&0.667$_{-0.300}^{+0.300}$\\\\\nUDS-19167& 9.35$\\pm$0.34& 8.96$_{-0.300}^{+0.300}$ & 8.58$_{-0.300}^{+0.300}$& 0.085$_{-0.300}^{+0.300}$& 1.02$_{-0.300}^{+0.300}$& 0.192$_{-0.300}^{+0.300}$\\\\\nUDS-24154 & 9.67$\\pm$0.33&9.10$_{-0.300}^{+0.100}$&7.49$_{-0.300}^{+0.165}$&0.531$_{-0.030}^{+0.300}$& 1.38$_{-0.050}^{+0.300}$&0.462$_{-0.005}^{+0.105}$\\\\\nCOSMOS-10599 & 8.65$\\pm$0.40&8.72$_{-0.300}^{+0.240}$& 7.49$_{-0.300}^{+0.245}$ & 0.531$_{-0.362}^{+0.300}$& 1.00$_{-0.020}^{+0.015}$&0.462$_{-0.300}^{+0.080}$\\\\\nCOSMOS-12102& 10.8$\\pm$0.30&9.57$_{-0.300}^{+0.300}$& 8.40$_{-0.300}^{+0.300}$& 0.155$_{-0.300}^{+0.300}$& 1.63$_{-0.300}^{+0.300}$&1.77$_{-0.300}^{+0.300}$\\\\\nCOSMOS-13184 & 8.78$\\pm$0.36&9.00$_{-0.220}^{+0.070}$& 7.28$_{-0.095}^{+0.365}$ & 0.087$_{-0.040}^{+0.068}$& 1.29$_{-0.120}^{+0.230}$&0.407$_{-0.110}^{+0.245}$\\\\\nCOSMOS-15091& 8.88$\\pm$0.37&7.92$_{-0.120}^{+0.280}$& 7.61$_{-0.105}^{+0.965}$ & 0.169$_{-0.084}^{+0.042}$& 0.062$_{-0.020}^{+0.200}$&0.077$_{-0.300}^{+0.115}$\\\\\nCOSMOS-16286 & 9.22$\\pm$0.40&8.44$_{-0.125}^{+0.160}$& 7.92$_{-0.650}^{+0.525}$ & 0.171$_{-0.054}^{+0.044}$& 0.317$_{-0.325~}^{+0.440~}$&1.05$_{-0.850}^{+0.500}$\\\\\nCOSMOS-16566 & 9.02$\\pm$0.37&8.60$_{-0.045}^{+0.040}$& 8.29$_{-0.110}^{+0.065}$ & 0.137$_{-0.074}^{+0.324}$& 0.147$_{-0.045}^{+0.035}$&0.037$_{-0.020}^{+0.050}$\\\\\nCOSMOS-17118& 9.86$\\pm$0.33&8.47$_{-0.315}^{+0.210}$& 7.66$_{-0.425}^{+0.920}$ & 0.155$_{-0.070}^{+0.136}$& 0.752$_{-0.085}^{+0.300}$&0.192$_{-0.010}^{+0.105}$\\\\\nCOSMOS-17839 & 9.11$\\pm$0.34&8.19$_{-0.190}^{+0.135}$&7.43$_{-0.195}^{+0.335}$& 0.091$_{-0.018}^{+0.300}$& 0.392$_{-0.035}^{+0.105}$&0.037$_{-0.300}^{+0.005}$\\\\\nCOSMOS-18358& 9.54$\\pm$0.32& 9.43$_{-0.300}^{+0.300}$& 8.12$_{-0.300}^{+0.300}$ & 0.629$_{-0.300}^{+0.300}$ & 1.11$_{-0.300}^{+0.300}$&0.037$_{-0.300}^{+0.300}$\\\\\nCOSMOS-19049 & 10.4$\\pm$0.30&9.85$_{-0.300}^{+0.168}$ &8.36$_{-0.300}^{+0.165}$& 0.081$_{-0.300}^{+0.046}$&1.31$_{-0.300}^{+0.120}$&1.72$_{-0.300}^{+0.200}$\\\\\nCOSMOS-19077 & 9.40$\\pm$0.34&8.92$_{-0.225}^{+0.300}$& 8.05$_{-0.555}^{+0.300}$ & 0.173$_{-0.300}^{+0.358}$& 0.692$_{-0.300}^{+0.285}$&0.197$_{-0.300}^{+0.265}$\\enddata\n\\tablecomments{Quoted values for $M_{\\star}$, Age (mass-weighted), Z, SFR, and $\\tau_{V}$ are the medians of the probability distributions from \\texttt{MAGPHYS} with associated +\/-- 1$\\sigma$ values.  Cases where we have an uncertainty of 0 occur when the data cannot be well-explained by the models and not when the models constrain the output parameters well, which manifests itself as a large $\\chi^2$ value.  As such, we will adopt an uncertainty of 0.3 dex \\cite[the typical uncertainty for stellar masses obtained from fitting broadband photometry,][]{conroy} in those cases to be used in the subsequent analysis. }\n\n\\end{deluxetable*}\n\n\n\\begin{figure*}\n\\label{fig:2a}\n\\epsscale{.98}\n\\figurenum{2a}\n \\plotone{f2_1.eps}\n \\caption{WFC3 G141 grism and LUCI1 or X-Shooter spectrum.  The spectra are smoothed by 3 pixels and are flux-calibrated according to the grism line fluxes.  The shaded gray area represents the +\/$-$ 1$\\sigma$ flux uncertainties and the red curve shows the best-fitting model of the emission lines.  The dotted lines show the position of the [O III]$\\lambda\\lambda$5007, 4959 and H$\\beta$ emission lines.}\n\\end{figure*}\n\n \\begin{figure}\n \\label{fig:2b}\n \\figurenum{2b}\n \\epsscale{.98}\n  \\plotone{f2_2.eps}\n   \\caption{WFC3 G141 grism and LUCI1 or X-Shooter spectrum, same as Figure \\ref{fig:2a} but for the detected H$\\alpha$ lines.  The positions of the [N II] lines as well as the H$\\alpha$ line are denoted by the dotted lines.}\n \\end{figure}\n\n\n\\begin{figure}\n\n\\includegraphics[width=0.475\\textwidth]{f3.eps}\n\n\\caption{Ratio of stellar mass to total dynamical mass versus age (mass-weighted) for the sample, with dynamical masses based on line measurements coming from the ground-based NIR spectra and stellar masses and ages from \\texttt{MAGPHYS}.}\n\\label{fig:dynmass}\n\\end{figure}\n\n\\begin{figure}\n \\includegraphics[width=0.475\\textwidth]{masshist.eps}\n \\caption{Histograms of stellar mass for our sample compared to the larger H$\\alpha$ samples of \\citet{fs09}, \\citet{erb06}, \\citet{mancini11}, and \\citet{contini} at $z>1$.  While variations in the stellar templates and the IMF used can alter stellar mass estimates by $\\sim$0.3 dex, our sample still lies at considerably lower masses (up to 2 orders of magnitude lower) than the other samples.}\n \\label{fig:masshist}\n\\end{figure}\n\n\n\\begin{figure}\n\n\\includegraphics[width=0.475\\textwidth]{f4.eps}\n\n\\caption{Stellar mass versus observed line width for the sample.  There is no clear trend of increasing line width with increasing stellar mass.  The overplotted dashed line corresponds to $M_{\\star} \\sim M_{virial}$ for a fixed $r_{\\rm{eff}}$ of 1 kpc.}\n\\label{fig:sigmass}\n\\end{figure}\n\nAs demonstrated in \\citet{maseda}, there is a tight, linear relation between the total dynamical and stellar mass estimates.  Figure \\ref{fig:dynmass} may show that the stellar-to-dynamical mass ratio correlates with the (mass-weighted) age as well.  This qualitatively agrees with the model in which a non-replenished reservoir of gas is turned into stars, such that the older systems have formed a proportionally larger number of stars.  For further discussion, see Section \\ref{sec:discussion}.  To date, these systems represent the lowest dynamical masses ever measured at this epoch, and typically have lower stellar masses than those galaxies in similar star-forming spectroscopic surveys \\cite[e.g.][see also Figure \\ref{fig:masshist} and Section \\ref{sec:comp}]{masters}. Note that the systematic errors in $M_{dyn}$ dominate the size of the error bar.  \\textit{COSMOS-12102} shows a broad and asymmetric line profile for which the line width most likely does not trace the underlying gravitational potential.  The lack of a clear trend in the relationship \nbetween the two main ``observed'' quantities, $M_{\\star}$ and $\\sigma$, can be seen in Figure \\ref{fig:sigmass}, which suggests that any relationship between $M_{\\star}$ and $M_{dyn}$ must be driven by a relationship between the size of the galaxy and $M_{\\star}$.  Figure \\ref{fig:sizemass} shows this relationship, as our sample lies on or below the observed size-mass relation for late-type galaxies at $2 < z < 2.25$ as found in \\cite{vdw14} in 3D-HST and CANDELS.  \n\n\n\\begin{figure}\n\n\\includegraphics[width=0.475\\textwidth]{f5.eps}\n\\caption{Effective radius versus stellar mass for the sample.  The overplotted lines are the low-mass extrapolation (and instrinsic scatter) to the relationship for star-forming galaxies at $2 < z < 2.25$ as found in \\citet{vdw14}.  The emission line galaxies in our sample fall on or somewhat below the size-mass relation for normal star-forming galaxies at similar redshifts.}\n\\label{fig:sizemass}\n\\end{figure}\n\n\\begin{figure}\n\n \\includegraphics[width=0.475\\textwidth]{f6.eps}\n\\caption{sSFR versus stellar mass.  Black points are the results from this study, orange points are the values from \\citet{vdw}, the red point is from \\citet{erb10}, the green point is from \\citet{gb2}, the blue lines are the results for the high\/low-z bins of star-forming galaxies in \\citet{fumagalli}, and the green lines are the results for the characteristic sSFR (SFR*\/M$_{\\star}$, see the text for details) for narrow-band selected star-forming galaxies in \\citet{sobral}.  The diagonal dotted line represents the nominal detection limit from the 3D-HST survey of 2.8 $M_{\\odot}$ yr$^{-1}$ at $z \\sim 1.5$. The sSFRs are averaged over 10 Myr.}\n\\label{fig:ssfr}\n\\end{figure}\n\nThe extreme nature of the star-formation in these systems is clearly seen in Figure \\ref{fig:ssfr}.  The stellar masses measured here are beginning to probe the same regime as \\citet{vdw} and \\citet{gb2}, with similar sSFR values.  Note that the sSFR values obtained from \\texttt{MAGPHYS} are averaged over the last 10 Myr: given the ephemeral nature of the bursts, the current (s)SFR may not reflect the most vigorous period of star formation in these systems. These are more than an order of magnitude in excess of the sSFR values characteristic of the star-forming population of massive galaxies at similar redshifts, measured in H$\\alpha$ from \\citet{fumagalli}.\n\n\n\n\n\\subsection{Comparison to Other Studies}\n\\label{sec:comp}\nAt lower redshifts 0.11 $\\leq z \\leq$ 0.93, \\citet{amorin14a,amorin14b} have isolated a sample of EELGs selected on [O III] flux that show remarkable similarities to our sample with sizes $r_{1\/2} \\sim$ 1.3 kpc, masses $10^7-10^{10} M_{\\odot}$, sSFR values $10^{-7}-10^{-9}$ yr$^{-1}$, and metallicity estimates of 0.05 $-$ 0.6 $Z_{\\odot}$ (some determined ``directly\" using the [O III]$\\lambda$4363 line as well, see Section \\ref{sec:metals}).  Deep HST-ACS $I$-band images reveal that most (80\\%) of their EELG sample show non-axisymmetric morphologies indicative of recent mergers or interactions.  Only with samples that are complete over the redshift ranges in question will allow for a direct comparison between the two populations, which are currently only split artificially according to either optical or near-IR observations.  However, caution must be taken in any interpretation: the much higher number density at $z\\sim$1.7 from \\citet{vdw} than at very low-z from \\citet{greenpea} implies that there could be a very different mechanism involved to trigger the bursts.  Connecting the two populations in a qualitative sense is the subject of ongoing work.\n\nOur current observations do not allow us to make strong conclusions about the internal dynamics of individual systems.  Currently, the only opportunities to study the internal dynamics of such small systems at high-redshift is with gravitational lensing: \\citet{jones10} note that their $z\\sim2-3$ strongly-lensed galaxies would resemble mergers or dispersion-dominated systems without the additional spatial resolution provided by the lensing.  Two objects similar to those presented here are \\textit{MACS J2135-0102} ($M_{\\star}$=9.8 $M_{\\odot}$, z=3, $r_{1\/2}$=1.75 kpc, SFR=40 $M_{\\odot}$ yr$^{-1}$, $\\sigma_{H\\alpha}$=54$\\>{\\rm km}\\,{\\rm s}^{-1}$, $V_c$=67$\\>{\\rm km}\\,{\\rm s}^{-1}$) from \\citet{jones10} and  \\textit{SHIZELS-10} ($M_{\\star}$=9.4 $M_{\\odot}$, z=1.45, $r_{1\/2}$=2.3 kpc, SFR=10 $M_{\\odot}$ yr$^{-1}$, $\\sigma_{H\\alpha}$=64$\\>{\\rm km}\\,{\\rm s}^{-1}$, $V_c$=26$\\>{\\rm km}\\,{\\rm s}^{-1}$) from \\citet{swinbank12}, which both appear to have a (relatively weak) rotational component to their dynamics.  Further discussion of the dynamics of our present sample is deferred to Section \\ref{sec:discussion}.\n\nSeveral other studies have begun to characterize the starforming properties of high-z EELGs using various techniques.  The most obvious comparable study to this work is the WFC3 Infrared Spectroscopic Parallels survey \\cite[WISP,][]{wisp}, and specifically the study of \\citet{masters}.  As a similar WFC3 grism survey, they are also able to select galaxies based on emission lines instead of photometric techniques, and thus can isolate a sample of high-EW ELGs.  \\citet{masters} present a sample of 26 such ELGs with a median restframe [O III] EW of 154 \\AA $ $ (our median is 629 \\AA).  These galaxies show similarly-low velocity dispersions ($\\sim$ 70 $\\>{\\rm km}\\,{\\rm s}^{-1}$) and hence also have dynamical masses $\\lesssim$ 10$^{10} M_{\\odot}$.  They derive stellar masses using an assumed M\/L ratio and star-formation history in a similar fashion to \\citet{vdw}, which have been shown to generally agree with SED-derived stellar masses \\citep{maseda}, and have also begun to probe the $M_{\\star} \\lesssim 10^9 M_{\\odot}$ regime.  Specific discussion of their metallicity estimates is deferred to Section \\ref{sec:metals}.\n\nIn addition to WISP, narrowband studies have also begun to uncover the general starforming population of galaxies at $z > 1$, probing stellar masses below 10$^{9.5} M_{\\odot}$.  \\citet{sobral} use data from the HiZELS survey to study H$\\alpha$ emitters at redshifts $z = 0.84, 1.47$, and $2.23$.  The size of their survey area ($\\sim$ 2 deg$^2$) and the depth of the imaging allows them to isolate large and pure samples of H$\\alpha$ emitters down to a restframe H$\\alpha$+[NII] EW of 25 \\AA, constraining the stellar mass function of star-forming galaxies down to 10$^{9.5} M_{\\odot}$ at these redshifts.  Indeed, their sample also includes a number of objects with restframe H$\\alpha$+[NII] EW values in excess of 300 \\AA $ $ and with stellar masses from SED-fitting below 10$^{9} M_{\\odot}$.  Their results for the characteristic sSFR (i.e. the typical SFR for a galaxy at a given mass and redshift divided by its mass) as a function of mass and redshift is shown in Figure \\ref{fig:ssfr}.  While some of our objects can be considered ``normal\" at these redshifts according to this determination, a majority of them still have higher sSFRs than expected, albeit typically within 1 dex.  This reinforces the notion that these objects are the high-EW tail of the total distribution of star-forming galaxies at these redshifts and do not comprise a genuinely separate population \\citep{vdw}.\n\n\\section{Emission-Line Ratios}\n\\subsection{Starbursts or AGN?}\n\\label{sec:starburstagn}\nSo far, the main caveat is that we have assumed that star formation is primarily responsible for the strong line emission.  The most compelling evidence to support this assumption is the relation between stellar mass and dynamical mass, and that the dynamical masses, with two exceptions, do not exceed $10^{10}~M_{\\odot}$ \\citep{maseda}.  Such a result would be entirely coincidental in the case that the emission lines are powered by AGN since the width of AGN emission lines is not coupled to the stellar mass of the host galaxy.  In other words, we observe narrow emission lines in these small ($\\sim$ 1 kpc) systems, while typical AGN narrow line regions have much larget emission-line widths $\\sigma > 200 \\>{\\rm km}\\,{\\rm s}^{-1}$ \\citep{osterbrock}.  However, it is useful to look for evidence of AGN contributions.\n\nAlthough low-metallicity, low-mass AGN are exceedingly rare in the local Universe \\citep{izotovAGN}, there is some evidence that they may be more common at higher-z \\citep{trump11,xue,reines}. Such AGN could cause large observed line fluxes.  At $z> 1$, AGN identification with a simple diagnostic such a high [O III]$\\lambda$5007\/H$\\beta$ ratio becomes insufficient by itself, as shown by \\citet{trump}.  We thus utilize the ``BPT'' diagnostics of \\citet{bpt} for the objects in our sample with measurements of [N II] and\/or [S II] in addition to H$\\alpha$.  Given that these lines are typically quite weak in star forming galaxies and the strong influence of OH skylines in our NIR spectra, in some instances we can only place an upper-limit on the ratios of those lines with H$\\alpha$.  These two BPT diagrams are shown in the left and central panels of Figure \\ref{fig:mex} with contours showing galaxies from the SDSS MPA-JHU value-added DR7 catalog.\n\n \nMost of our points plausibly lie on an extension of the starforming region of the BPT diagrams and not on the extension of the AGN region, or at least they lie far from the main locus of AGN-powered emission lines.  However, low-metallicity AGN can lie in the starforming region as well \\citep{groves,kewley13}, preventing us from completely ruling out the contribution of AGN to our sample with these diagnostics.  In some cases we observe higher [O III]\/H$\\beta$ ratios compared to the other starforming galaxies, but this can be explained simply by higher ionizations and lower metallicities in these systems compared to the low-z sample of SDSS galaxies.  \\citet{kewley13} find that starforming galaxies at $z > 1$ are consistent with models that have more extreme ISM conditions than those in the local Universe\\footnote{While beyond the scope of this paper, we would like to point out the large uncertainties in assumptions about the ISM conditions in galaxies at high redshift given the lack of knowledge about ionizing spectrum of hot stars at these early times and low metallicities.}.  High electron temperatures (discussed in \nthe next section) support such a hypothesis.  This would be a further, unexplained coincidence if they are AGN in addition to\nthe low dynamical masses described previously.  While \\citet{trump11} suggest that AGN are widespread in low-mass $1.3<z<2.4$ galaxies, the emission lines are not actually dominated by the AGN.  This is most evident in the $L_{0.5-10~keV}$\/$L_{[O III]}$ relationship: the [O III] lines have some AGN contribution, but (on average) less than 50\\%.\n\nAdditionally, we utilize the Mass-Excitation (MEx) diagnostic of \\citet{juneau}, which combines the [O III]$\\lambda$5007\/H$\\beta$ ratio with the stellar mass.  \\citet{trump} conclude that this diagnostic also gives a meaningful probabilistic constraint on the AGN\/SF nature of galaxies at $z > 1$ using a combination of the BPT diagnostics and X-ray data.  While this diagnostic is easily applicable to our data, we can not constrain the MEx AGN\/starforming \\textit{probabilities} for most of our sample given that it is not properly calibrated for objects with such low metallicities and high sSFRs, AGN or otherwise: the five objects with a probability of star formation ($P_{SF}$, compared to the probability that they are AGN; \\textit{COSMOS-18358}, \\textit{COSMOS-13184}, \\textit{COSMOS-10599}, \\textit{UDS-24154}, and \\textit{UDS-12435}) have a median $P_{SF}$ value of 0.940, firmly placing them in the starforming regime.  The remaining objects, while probabilistically unconstrained, still lie far from the  \npopulation of AGN in the \nMEx diagnostic plot, as shown in Figure \\ref{fig:mex}.\n\n\nWe can place other constraints on the AGN nature of these systems in much the same way as \\cite{vdw}, i.e. independent of any measured emission line ratio(s).  Most objects in our sample do not have strong 24 $\\mu$m detections using Spitzer-MIPS: \\textit{COSMOS-18358} has a MIPS 24$\\mu$m flux of 21.5$\\pm$9.0 $\\mu$Jy and clearly appears to be a merger; \\textit{COSMOS-19049} has a flux of 14.1$\\pm$8.6 $\\mu$Jy, is physically large with $r_{eff}=2.3$ kpc, and also has broad lines with $\\sigma_{[O III]}=122 \\>{\\rm km}\\,{\\rm s}^{-1}$; \\textit{COSMOS-12102} has a flux of 55.3$\\pm$8.7 $\\mu$Jy and is further discussed below.  While none of our objects have X-ray detections, GOODS-S is also the only field with sufficient depth to find $z\\sim2$ AGN which are not quasars.  That being said, it is possible to hide even a rapidly accreting X-ray AGN in a low mass galaxy \\citep{aird}.  The consistent (and resolved) $J_{F125W}$- and $H_{F160W}$-sizes, as well as the sizes of the emission lines in the grism spectra, rule out the presence of a strong point source dominating the emission.  \n\nAn exception could be \\textit{COSMOS-12102}, illustrated by the star in Figure \\ref{fig:mex}, which is also has the largest line width in our sample.  \\citet{reines} find active black holes in similar mass dwarf galaxies with broad H$\\alpha$ emission in the local Universe with $M_{BH} \\lesssim 10^6~M_{\\odot}$.  Using their relation (Equation 5) of $L_{H\\alpha}$ and FWHM$_{H\\alpha}$ to $M_{BH}$, \\textit{COSMOS-12102} has a black hole mass of $\\sim 10^{6.2}~M_{\\odot}$, which is in their observed range.  The mass determination is fraught with systematic uncertainties, such as variations in the geometry of the broad-line region and that at least some of the H$\\alpha$ luminosity comes from star formation, but we cannot conclusively rule-out some AGN contribution for this galaxy.\n\n\\begin{figure}\n\n\\includegraphics[scale=.46]{f7.eps}\n\n\\caption{AGN\/SF emission line diagnostic plots.  From left to right, the BPT1 diagnostic of [N II]$\\lambda$6584\/H$\\alpha$, the BPT2 diagnostic of [S II]$\\lambda$6718+6731\/H$\\alpha$, and the MEx diagnostic \\citep{juneau} of stellar mass.  Divisions between the AGN and the star-forming regions in the BPT diagrams come from \\citet{kewley}.  In all cases, the gray contours represent data from the SDSS MPA-JHU value-added DR7 catalog: these emission line and stellar mass measurements are described by \\citet{tremonti} and \\citet{kauffmann}.  Arrows denote 3$\\sigma$ upper limits.  Each of the diagnostics point to star formation as the ionizing source with at most mild AGN contribution, the possible exception being \\textit{COSMOS-12102} (star symbol).  The large uncertainties in [O III]\/H$\\beta$ are caused by very low and uncertain H$\\beta$ fluxes, as [O III] is robustly detected in all of these cases; the true ratio could be even higher than the values plotted here.}\n\\label{fig:mex}\n\\end{figure}\n\n\\subsection{Metallicity}\n\\label{sec:metals}\n\\begin{figure*}\n\\begin{center}\n \\includegraphics[scale=.65]{f8a.eps}\\includegraphics[scale=.65]{f8b.eps}\n\\end{center}\n\\caption{Left panel: Oxygen abundances as a function of stellar mass. Overplotted are the MZ relations of \\citet{kewleyellison}, \\citet{henry13}, and \\citet{erb06} with low-mass extensions given as the dotted lines, using the \\citet{maiolino} parameterization.  The dashed line is the \\citet{amorin} relation for luminous compact ``green pea'' galaxies at $0.11 < z < 0.35$. Right panel: Oxygen abundances for our sample as a function of $\\mu_{32}$ as defined in \\citet{mannucci}, with the ``FMRs\" from \\citet{mannucci} and the high-SFR bin from \\citet{zahid} overplotted.  In both panels, filled points denote abundances obtained from the ``direct'' $T_e$ method and open points denote other methods.  The red point shows the result from \\citet{erb10} and the green point shows the result from \\citet{gb2}.  Overall we observe that our objects lie on or below the low-mass\/high-SFR extrapolations to these observed relationships at similar redshifts.}  \n\\label{fig:metallicity}\n\n\\end{figure*}\n\nIn order to measure the gas-phase oxygen abundances of these galaxies as a proxy for the metallicity, we first implement the so-called ``direct'' or $T_e$ method which requires a detection of the auroral [O III]$\\lambda$4363 line as well as the [O III]$\\lambda\\lambda$4959, 5007 and [O II]$\\lambda$3727 lines.  Some of these lines lie in the UV\/Visible at these redshifts, so we can only apply this method to the X-Shooter sample.  Using the calibrations outlined in \\citet{izotov06}, we convert the [O III] emission-line ratios into an electron temperature (with the electron density constrained by the ratio of [O II]$\\lambda$3729 to $\\lambda$3726 or [S II]$\\lambda$6717 to $\\lambda$6731) in the O$^{++}$ region.  The total oxygen abundance in the galaxy is  O\/H = O$^+$\/H$^+$ + O$^{++}$\/H$^+$, assuming $\\log T_e($O$^+) = 0.7\\log T_e($O$^{++})+0.3$.  Two objects, \\textit{UDS-12539} and \\textit{UDS-6377}, have detections of $\\lambda$4363 and an upper limit can be obtained for a third, \\textit{GOODS-S-43928}.  This object also lacks a detection of [O II]$\n\\lambda$3727 due to \nskyline \ncontamination, so the O$^+$\/H$^+$ component cannot be constrained directly.  The relative contribution of the O$^+$\/H$^+$ component to the total oxygen abundance is 1.8\\% and 7.3\\% for the other two objects, so we neglect its contribution to the abundance of the third object.  The N2 method of \\cite[][PP04]{pp04}, which uses the (log) ratio [N II]$\\lambda$6584\/H$\\alpha$, verifies this result.  All derived temperatures are in excess of 20,000 K.\n\n\\begin{deluxetable}{lcc}\n\\tablecaption{Metallicity Estimates\\label{tab:z}}\n\\tablehead{ \\colhead{ID} & \\colhead{12 + log(O\/H)} & \\colhead{Method}}\n\\startdata\nGOODS-S-43693&8.10$\\pm$0.34&N2\\\\\nGOODS-S-43928&7.70$\\pm$0.53&$T_e$\/N2\\\\\nUDS-6195&8.11$\\pm$0.23&O3N2\\\\\nUDS-6377&7.52$\\pm$0.37&$T_e$\\\\\nUDS-12539&7.45$\\pm$0.09&$T_e$\\\\\nUDS-24154&8.25$\\pm$0.32&O3N2\\\\\nCOSMOS-19049&8.20$\\pm$0.20&N2\\enddata\n\\end{deluxetable}\n\nFor the remainder of the sample, we must resort to other methods, namely the aforementioned N2 method and also the O3N2 method, both from PP04; O3N2 is the (log) ratio [O III]$\\lambda\\lambda$4959, 5007\/H$\\beta$\/N2.  For both methods, uncertainties include contributions from the error in the line ratios as well as the systematic scatter in the relations.  If H$\\beta$ is not detected at more than 1$\\sigma$ in our LUCI1 or X-Shooter spectrum, we estimate the [O III]\/H$\\beta$ ratio from the grism spectrum.  Also note that in the two instances where the N2 method is used the \\citet{maiolino} calibration results in metallicities consistent with the PP04 values.  Results are displayed in Table \\ref{tab:z}, with a median $12 + \\log(O\/H)$ value of 7.90 (0.15 $Z_{\\odot}$).  This value agrees well with the median \\texttt{MAGPHYS}-derived metallicity of 0.17 $Z_{\\odot}$ for the full sample.\n\nGiven the relatively small number of direct measurements of the oxygen abundance in high-z galaxies, this provides an important piece of information.  Since the standard R23 diagnostic \\citep{r23} using [O III]$\\lambda\\lambda$ 4959, 5007 + [O II] $\\lambda\\lambda$ 3726, 3729 +  H$\\beta$ is double-valued, it is important whether higher-z galaxies belong to the metal-rich upper branch or to the metal-poor lower branch.  \\citet{henry13a} argue in favor of the upper-branch for galaxies with log ($M_{\\star}$\/$M_{\\odot}$) $>$ 8.2 at $z\\sim0.7$, and \\citet{henry13} favor the upper-branch for galaxies with log ($M_{\\star}$\/$M_{\\odot}$) $>$ 8.8 at $1.3<z<2.3$.  Our measurements suggest that these systems plausibly lie on the lower branch, even at log ($M_{\\star}$\/$M_{\\odot}$) $<$ 9.2.  \n\nIn the left panel of Figure \\ref{fig:metallicity} we show these results as well as those of \\citet{gb2} and \\citet{erb10} with the low-mass extrapolations to the mass-metallicity (MZ) relations from \\citet{kewleyellison} for $z \\sim 0.1$, \\citet{henry13} for emission line galaxies at $z \\sim 1.8$, and \\citet{erb06} for starforming galaxies at $z \\sim 2.2$ utilizing the metallicity calibrations and functional parameterization of \\citet{maiolino}, as well as the results from \\citet{amorin}.  The right panel of Figure \\ref{fig:metallicity} shows the same data plotted against the low-mass extrapolation of the ``fundamental metallicity relation'' (FMR) of \\citet{mannucci} and the relation for high-SFR galaxies at $z\\sim1.6$ from \\citet{zahid}.  Neither method is directly calibrated below  $\\sim 10^9~M_{\\odot}$, which is precisely the range we probe here.\n\nIn the case of the MZ relation, our results generally agree with the \\citet{erb06} and \\citet{henry13} extrapolations and typically have lower metallicities than objects with similar masses and star formation rates at lower redshifts.  We see good agreement with distribution in $12 + \\log(O\/H)$ with $M_{\\star}$ from \\citet{amorin} as well.  The slight overall trend observed where higher-mass objects have higher metallicities supports the idea that the lowest-mass objects are the youngest and therefore have formed less mass of heavy elements.  The youngest objects should also have the highest sSFRs, which is known to drive much of the scatter in the MZ relation.  \\citet{henry13a} also explain that the steepness of the O\/H versus $M_{\\star}$ relation has a direct theoretical impact \non the role of outflows in these systems.\n\nIn the FMR relation the solid points, denoting $T_e$-determined abundances, are considered more reliable at low metallicities \\citep{maiolino} and lie somewhat below the FMR, perhaps more in line with an extrapolation to the $z\\sim1.6$ findings of \\citet{zahid}.  Since \\citet{mannucci} do not see the higher-mass turnoff in the relation until $z > 2.5$, we may be observing the evolution of the FMR on the low-mass and\/or high-SFR end, although we do not have the number statistics yet to quantify any offset.  At the very least, we can conclusively rule-out that these objects lie above the FMR \\cite[cf.][who claim this relation is driven by the higher average SFRs of the systems probed at higher redshifts]{stott}.\n\n\n\\section{Constraints on the Gas Fraction and the Star Formation Efficiency}\n\\label{sec:discussion}\nIn this section we use the observed velocity dispersions $\\sigma$ and\nsizes $r_{\\rm{eff}}$, combined with dynamical stability criteria, to\nconstrain the gas fraction and its implications.  We assume that the\nsystems consist entirely of stars and gas: we neglect the contribution\nof dark matter to the total dynamical mass as measured within the\ncentral kpc.  We also assume that these systems are isolated and not embedded in larger (gaseous) structures that exert pressure.\n\nWe do not know the geometry of the systems, and therefore consider two\nextreme cases: for the case of a sphere with uniform density we calculate the\nJeans mass $M_J$; for the case of a thin rotating disk we calculate\nthe Toomre parameter $Q$.  In both cases we assume that the gaseous\nbody has the same extent as the stellar body, and that the velocity\nwidth of the nebular lines traces the total gas kinematics.\n\nFor a uniform sphere the Jeans mass \\citep{binney} is given by\n\\begin{equation}\n\\label{eqn:jeans}\nM_J = \\frac{4\\pi}{3} \\rho_0\\bigg(\\frac{\\pi\\sigma^2}{4G\\rho_0}\\bigg)^{3\/2},\n\\end{equation}\nwhere we have equated the sound speed with observed velocity\ndispersion.  This velocity represents the combined effect of all\nsources of pressure that act against collapse, which include thermal\nmotions (associated with the physical sound speed) as well as\nturbulence and streaming motions.\n\n$\\rho_0$ is the density of the gas, which is given by the gas mass\n($f_{gas}\\times M_{dyn}$) and the size:\n\\begin{equation}\n\\label{eqn:rho}\n\\rho_0=\\frac{f_{gas}M_{dyn}}{(4\/3)\\pi r_{eff}^3},\n\\end{equation}\nwhere $f_{gas}$ is the gas fraction.  The total dynamical mass\n$M_{dyn}$ is taken from Equation \\ref{eqn:dyn}, with a value of 5\n for the proportionality constant for consistency with\nthe case under consideration here: that of a sphere with uniform\ndensity\\footnote{Elsewhere in this paper we use a value of 3, which corresponds to other, more realistic geometries such as inclined disks and radially-concentrated density profiles (e.g., isothermal).}.\n\nFor typical values of $r_{eff}$ (1 kpc) and $\\sigma$ (50 $\\>{\\rm km}\\,{\\rm s}^{-1}$)\nwe find that $M_J \\simeq M_{gas} (\\equiv f_{gas} M_{dyn})$ for\n$f_{gas}=0.66$.  Given that substantial star formation in these\nsystems the gaseous body must be unstable: we conclude, assuming a\nhomogeneous gaseous sphere, that $f_{gas} \\gtrsim 0.66$.\n\nIn order to address the question to what extent this conclusion is\naffected by the chosen geometry, we now consider the other (opposite)\ncase, and assume that these systems are rotating disks, where instability can be described by the Toomre parameter $Q$\n\\citep{toomre,binney}\n\\begin{equation}\n Q_{gas} = \\frac{\\sigma_z \\kappa}{\\pi G \\Sigma_{gas}}\n\\end{equation}\nwhere $\\sigma_z$ is the velocity dispersion perpendicular to the disk,\nand $\\Sigma_{gas}$ is the average gas-mass surface density, given by\n\\begin{equation}\n\\label{eqn:sigma}\n \\Sigma_{gas} = \\frac{M_{gas}}{2 \\pi r_{eff}^2} = \\frac{M_{dyn} f_{gas}}{2 \\pi r_{eff}^2},\n\\end{equation}\nwhere $f_{gas}$ is the gas fraction.  The epicyclic frequency, $\\kappa$, in a rotating exponential disk is \n\\begin{equation}\n\\label{eqn:kappa}\n \\kappa = \\sqrt{2}(v_t\/r_{eff}),\n\\end{equation}\nwhere $v_t$ is the circular velocity of the disk.\n\n\nWhile $\\sigma_z$ and $v_t$ are not observed directly, both contribute to the observed linewidth, $\\sigma$.  The contribution of rotation to $\\sigma$, assuming an average inclination of 60 degrees and an exponential disk, is $\\sim v_{rot}\/\\sqrt{2}$, such that we have\n\\begin{equation}\n\\label{eqn:sigmaobs}\n \\sigma_{obs}^2 = \\sigma_z^2 + v_t^2\/2,\n\\end{equation}\nwhich is empirically supported \\citep{kassin}.\n\nCombining the above, we solve for $v_t$ as a function of $\\sigma_{obs}$:\n\\begin{equation}\n\\label{eqn:vt}\n v_t^2 = \\sigma_{obs}^2\\left(1 \\pm \\sqrt{1-(9\/4)f_{gas}^2Q_{gas}^2 } \\right),\n\\end{equation}\n\nsuch that an unstable system ($Q < 1$) requires\n\\begin{equation}\n f_{gas} > \\frac{2}{3}\n\\end{equation}\nin order to produce a unique, physical solution.\n\nRemarkably, regardless of whether we assume a homogeneous sphere or a\nrotating disk for the gas, we find that a high gas fraction is needed\nto explain the observed star formation activity.  At the same time,\nthe non-negligible contribution of the stellar mass to the total mass\nimplies that $f_{gas}$ cannot be arbitrarily close to unity and should\nbe $\\lesssim 0.9$ (see Figure \\ref{fig:dynmass}).\n\nA significant caveat is that \nthe proportionality constants in Equations \\ref{eqn:jeans},\n\\ref{eqn:rho}, \\ref{eqn:dyn}, \\ref{eqn:kappa}, \\ref{eqn:sigmaobs}, and\n\\ref{eqn:vt} depend on the details of the assumed geometry and\ndynamical structure; their variation can alter the threshold value of\n$f_{gas}$.  In addition, we ignore the stabilizing effect of the\nstellar disk on the gas disk, however this increases the implied gas\nfraction further.\n\nThe median gas fraction in our sample is 72\\% (i.e. the y-axis in\nFigure \\ref{fig:dynmass} is a proxy for $f_{gas}$), in agreement with\nthe theoretical calculation.  While some objects are observed to have\nlower gas fractions, 18 out of our 22 objects are consistent with\n$f_{gas} >$ 2\/3 within 1$\\sigma$.\n\nIn the following we assume $f_{gas}$=2\/3 in order to constrain the\nstar formation efficiency. In Figure \\ref{fig:gas} we show the star\nformation rate surface density, assuming $\\Sigma_{SFR} = SFR \/ (2\\pi\n\\times r_{eff})$, versus the gas surface mass density $\\Sigma_{gas}$\n(from Equation \\ref{eqn:sigma}). The implied gas depletion time scale\nranges from $\\tau_{depl} = 10^{7.5}$ to $10^9$ yr, with a median of $3\n\\times 10^8$ yr.  \n\nCompared to normal present-day galaxies, gas is efficiently\ntransformed into stars but, with the exception of a few objects, not\nas efficiently as observed in starbursting regions in the Milky Way\nand starbursting galaxies in the present-day or high-redshift universe\n\\citep[i.e.][]{kennicutt07,daddi}.  This mostly reflects the large gas\nfractions needed to produce systems that are unstable against star\nformation, rather than a modest level of star formation: the inverse\nsSFR (stellar mass growth time scale) of $1\/sSFR = 5 \\times 10^7$ yr\nis very short, among the fastest ever measured.  Hence, the stellar\nmass grows at a dynamical time scale ($\\tau_{dyn}\\sim3 \\times 10^7$\nyr).\n\nThe physical interpretation is that the star formation rate is not\nlimited by availability of fuel but by the dynamical time scale.\nHowever, we have to keep in mind that we selected galaxies based on\ntheir sSFR: we are biased against objects that are older and\/or have\nlonger star formation time scales.  Further empirical investigation of\nlower levels of star formation in similarly massive galaxies is needed\nto address this issue.\n\nHowever, we propose that star formation will halt within $\\sim\n50$~Myr, long before the gas reservoir is depleted.  First, the\nstability arguments given above imply that the gas fraction only needs\nto be reduced by a small amount (from the assumed $f_{gas}=2\/3$ to,\nsay, $f_{gas}<0.5$).  Given the observed SFR, this takes ~$\\sim$50\nMyr.  Second, feedback should play an important role in these low-mass\nsystems with small escape velocities (several 100 $\\>{\\rm km}\\,{\\rm s}^{-1}$) and high\nSFR: gas is easily transported out of the galaxy, and perhaps even out\nof the halo, preventing recycling.  This paradigm is supported by \\citet{law09,law12},\nwho see that higher mass starforming galaxies at z $\\sim$ 2 can support more extended rotationally-supported disks\nand are less efficient at driving outflows than their lower mass counterparts.\n\nIn the above we have ignored the infall of cold gas, which could\ncontinue to feed and maintain the starburst.  Assuming that these\ngalaxies reside in relatively low-mass halos ($\\sim10^{11} M_{\\odot}$) the\ntypical accretion rate onto the halo is several $M_{\\odot}$\/yr\n\\citep{mcbride}, somewhat lower than the SFR.  The accretion rate onto\nthe halo is not necessarily equal to the accretion rate onto the\ngalaxy, and the latter is likely not constant.  A period of\nabove-average accretion for several 100 Myr could, in principle,\nsustain the starburst.  However, above-average accretion events are\nmore likely to be of short duration, such that one such event can\nignite the observed starburst by pushing the gas mass surface density\nabove the threshold for star formation or by disturbing the\nalready-present gas, creating an instability.  Hence, enhanced accretion could cause the star formation\nactivity but not maintain it.\n\nBased on the Jeans and Toomre instability arguments presented above,\npurely gaseous systems with velocity dispersions and sizes as observed\nbecome unstable once they reach a total mass of a few times\n$10^9~M_{\\odot}$, close to the observed masses of the objects in our\nsample.  \n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.475\\textwidth]{f9.eps}\n\n\\end{center}\n\\caption{SFR surface density versus gas-mass surface density.  The gas masses are estimated according to $M_{gas} = (2\/3) \\times M_{dyn}$, which is inferred as a probable gas fraction given the star formation rates and $M_{\\star}$\/$M_{dyn}$ ratios (which imply $0.5 \\lesssim f_{gas} \\lesssim 0.9$) in these systems.  The solid black line shows the relation for local spiral galaxies from \\citet{kennicutt07} and the dotted line shows the result for local (U)LIRG and high-z SMGs\/QSOs from \\citet{daddi}.  The location of the points suggests that these objects spend at least some of the time forming stars more efficiently than the normal, present-day spiral galaxies.  Our constraints are not stringent enough to confirm or rule out gas depletion timescales that are on par with or even shorter than more massive, starbursting systems.}\n\\label{fig:gas}\n\\end{figure}\n\n\n\n\n\\section{Summary}\nWe presented near-infrared spectroscopy from the LBT\/LUCI1\nmulti-object near-IR spectrograph and the VLT\/X-Shooter wide band spectrograph\nfor a sample of HST\/WFC3 grism-selected\nemission line objects with restframe equivalent widths of $EW = 200 - 1100$ \\AA$ $ for [O III]$\\lambda$5007\nand\/or H$\\alpha$, and located in the redshift range $1.3 < z < 2.3$. The observed emission lines are narrow,\nwith measured velocity dispersions down to $\\sigma = 30$ km s$^{-1}$, implying low dynamical masses\nof $\\sim10^9~M_{\\odot}$, even for the lower-EW objects not included in \\cite{maseda}.  Stellar masses determined\nusing sophisticated \\texttt{MAGPHYS} SED fitting to broadband magnitudes and the inclusion of line fluxes results in low stellar masses\nas well, $\\sim3\\times10^{8}~M_{\\odot}$.  Ratios of $M_{\\star}$ to\n$M_{dyn}$ range from 1\/10 to 1, which makes AGN-dominated SEDs unlikely.  Emission-line ratios and the narrow line widths also suggest that AGN do not significantly contribute to our sample, and therefore we conclude that the main ionizing source is hot, massive stars.\n\n\n\nDirect probes of the oxygen abundances within these galaxies and [O III]\/H$\\beta$ line ratios of typically $\\gtrsim$ 5 corroborate the expectation that these low mass systems have low metallicities, between 0.05 and 0.3 $Z_{\\odot}$.  They lie on or below the (extrapolated) mass-metallicity relationships for these redshifts \\citep{henry13,erb06} which, combined with their young SED-derived ages, \nreinforces the notion that these are nascent galaxies undergoing their first major episode of star formation.  \n\nMeasured sSFR values of $\\sim10^{-8}$ yr$^{-1}$ for these galaxies are up to two orders of magnitude larger than those of typical $10^{10}~M_{\\odot}$ starforming galaxies at $z\\gtrsim1$ \\citep{fumagalli}, as well as comparable to or greater than the values from other high-EW systems as discovered in deep narrowband searches \\citep{sobral} and in deep spectroscopic studies at both similar \\citep{masters} and lower redshifts \\citep{amorin14a,amorin14b}.  Such high sSFR values have been difficult to reproduce in hydrodynamical simulations, but recently \\citet{shensims} made significant progress by combining a high gas density threshold for star formation and a blastwave scheme for supernova feedback in their simulations of low-mass galaxies.\n\nSuch low mass systems, with observed velocity dispersions of $\\sigma\\sim50\\>{\\rm km}\\,{\\rm s}^{-1}$ and sizes of $\\sim$ 1 kpc are only unstable against star formation if their gas fractions are high (above 2\/3), in agreement with the observed $M_{\\star}\/M_{dyn}$ relation.  The bursts are likely to be short-lived ($\\sim$50 Myr), as, even in the absence of feedback, their intense star formation will rapidly build up stellar mass and lower their sSFR well before the gas depletion timescale ($\\sim$100 Myr).\n\nThese results strengthen the conclusions from \\citet{vdw}, who argued that EELGs represent\nlow-mass, starbursting galaxies. Additionally, the existence of (at least) two strong galaxy-galaxy lenses in the CANDELS\/3D-HST fields where the background galaxy is an EELG at $z=1.85$ and $3.42$ \\cite[][respectively]{gb2,vdw13} suggests that this type of object is common. The ubiquity of EELGs may be even more pronounced at high redshifts \\cite[$>$6;][]{smit}.  Such systems at $z=$ 1 $-$ 2 thus may present an opportunity to study how star formation proceeded in the early Universe before the advent of the next generation of observatories, such as the \\textit{James Webb Space Telescope}.\n\n\nThe new generation of submillimeter observatories, such as ALMA, can provide direct estimates of the gas masses. Searching for the presence of outflowing material would provide valuable clues about the feedback processes going on in these systems, which is especially relevant to support the hypothesis that these bursts can create the cored dark matter profiles observed in local dwarf galaxies \\cite[e.g.][]{amorisco}.  \n\n\n\n\n\\acknowledgements{M.V.M. is a member of the International Max Planck Research School for Astronomy\nand Cosmic Physics at the University of Heidelberg, IMPRS-HD, Germany.  C.P. acknowledges support by the KASI-Yonsei Joint Research Program for the Frontiers of Astronomy and Space Science funded by the Korea Astronomy and Space Science Institute.  D.C.K. acknowledges funding from NSF grant AST-08-08133.}\n\n{\\it Facilities:} \\facility{LBT}, \\facility{VLT:Melipal}, \\facility{HST}.\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\\Large\\bit Introduction}\n\nImpulsive pp-waves prominently arise as ultrarelativistic limits of \nblack-hole geometries \\cite{AS,BaNa3,BaNa4,LouSa}. Since their curvature is\n\ncompletely concentrated on a null hyperplane they are necessarily\ndistributional in nature. Penrose \\cite{Pen} gave an intrinsic \ndescription of these geometries in terms of his ``cut and paste''\napproach. A different approach to derive the AS-geometry, which\nrepresents the ultrarelativistic limit of Schwarzschild, was given by 't Hooft\nand Dray \\cite{tHo}. They made use of the limit of geodesics in\nSchwarzschild to derive Penroses junction conditions. We will somehow\ntake the opposite point of view in this work and \nconsider the trajectories of massive and massless particles in \ngeometries generated by an impulsive gravitational wave in their own right. \nIt turns out that the calculation inevitably requires the\ndefinition of the point value $\\theta(0)$ at the location of the pulse,\nthus leaving the usual framework of distribution theory. \n>From a physical point of view this would induce a loss of predictability \nsince the above point value is completely arbitrary. The ambiguity\nis however resolved if one imposes the physically sensible requirement\nthat a freely falling (geodetic) observer experiences a constant flow\nof eigentime, i.e. that the norm of the tangent vector remains constant.\nOur result may be reinterpreted in the sense of Colombeau's theory\nof multiplication of distributions if one replaces distributional equality by\nweak equivalence of Coloumbeau functions. \\par\nIn section one we calculate the geodesics for an arbitrary impulsive\nwave profile. The ambiguity that arises from $\\theta(0)$ is resolved\nby taking the covariant constancy of the tangent vector into account.\nSection two gives a brief account of the basic notions of Colombeau's\nnew generalized functions together with an extension to arbitrary \n$C^\\infty$ manifolds. Finally, in section three we look at the calculations\nof section one from the point of view of Colombeau theory thereby justifying\nthe previously obtained results. \n\n\\section*{\\Large\\bit 1) Geodesics in (impulsive) pp-wave geometries }\n\nA pp-wave is usually characterized as\nbeing a vacuum geometry which admits a covariantly constant null-vector \n$p^a$ \\cite{JEK}. However, since we do not want to restrict ourselves\nto vacuum geometries, we require that\nthe Ricci-tensor is proportional to the tensor square of $p^a$.\n>From these requirements one shows \\cite{SiGo,AiBa2} that the metric \nmay be written as\n\\begin{equation}\\label{impwave}\n  g_{ab} = \\eta_{ab} + f p_a p_b \\quad \\mbox{together with}\\quad \n  (p\\partial)f=0,\n\\end{equation}\nwhere $f$ denotes the so called wave profile and $\\eta_{ab}$ is the\nflat part of the decomposition. Using the covariant \nderivative $\\partial_a$\nassociated with $\\eta_{ab}$ one obtains the difference tensor\n\\begin{equation}\n  C^a{}_{bc} = \\frac{1}{2}\\left( p^a \\partial_bf p_c + \n    p^a \\partial_c f p_b  - \\partial^a f p_b p_c \\right)\n\\end{equation}\nChoosing a conjugate null vector $\\bar{p}^a$ (i.e. $\\bar{p}\\cdot p=-1$) \nallows us to decompose\n\\begin{displaymath}\n  \\partial_af = - \\bar{p}_a \\underbrace{(p\\partial)f}_{=0} \n  - p_a (\\bar{p}\\partial)f + \\tilde{\\partial}_a f,\n\\end{displaymath}\nwhere the tilde refers to quantities which are projected in the\northogonal two space of $p^a$ and $\\bar{p}^a$.\nThis decomposition in turn gives rise to\n\\begin{equation}\n  \\label{diff}\n  C^a{}_{bc} = \\frac{1}{2}\\left( p^a \\tilde{\\partial}_bf p_c + \n    p^a \\tilde{\\partial}_c f p_b  - \\tilde{\\partial}^a f p_b p_c \n    - p^a p_b p_c (\\bar{p}\\partial)f \\right).\n  \\end{equation}\nWith respect to an affine parameterization the geodesic equation \nbecomes\n\n\\begin{displaymath}\n  \\ddot{x}^a + C^a{}_{bc}\\dot{x}^b  \\dot{x}^c = 0.\n\\end{displaymath}\nDecomposition with respect to the conjugate null directions yields\n\\begin{eqnarray}\\label{geo}\n  &&p\\ddot{x} =0,\\nonumber\\\\\n  &&\\bar{p}\\ddot{x} + \\frac{1}{2}\\left( \n    -2(p\\dot{x}) (\\dot{\\tilde{x}}\\tilde{\\partial})f + \n    (p\\dot{x})^2(\\bar{p}\\partial)f \n    \\right)=0,\\nonumber\\\\  \n  &&\\ddot{\\tilde{x}} - \\frac{1}{2} (p\\dot{x})^2 \\tilde{\\partial}f =0.\n\\end{eqnarray}\nThe first line of (\\ref{geo}) allows us to choose $px$ as affine parameter\nunless we are moving in a $px=const$ surface. This choice leaves us with two\nequations\n\\begin{eqnarray}\n  \\label{ngeo}\n  &&(\\bar{p}x)'' + \\left(-x'\\tilde{\\partial}f + \n    \\frac{1}{2}(\\bar{p}\\partial)f \\right) =0,\\nonumber\\\\\n  &&\\tilde{x}'' - \\frac{1}{2} \\tilde{\\partial} f = 0,\n\\end{eqnarray}\nwhere the prime denotes differentiation with respect to $px$.\nUp to now our derivation did not make use of the fact that\nwe are considering impulsive wave profiles, i.e. \n$f(px,\\tilde{x}) = \\tilde{f}(\\tilde{x}) \\delta(px)$.\nBy a slight abuse of notation we will drop the tilde on the \n$f(\\tilde{x})$ in the following.\nPhysically the above form implies that the geodesics are straight lines\nabove and below the null hyperplane of the pulse\n\\begin{eqnarray}\\label{geoans}\n  \\bar{p}x &=& a(px) + b + \\theta(px)\\left( a^+ (px) + b^+ \\right),\\nonumber\\\\ \n  \\tilde{x} &=& \\tilde{a}(px) + \\tilde{b} + \\theta(px)\n  \\left( \\tilde{a}^+ (px) + \\tilde{b}^+ \\right).\n\\end{eqnarray}\nPlugging (\\ref{geoans}) into the second line of (\\ref{ngeo}) gives\n$$\n\\delta(px)\\tilde{a}^+ + \\delta'(px)\\tilde{b}^+ = \n\\frac{1}{2}\\delta(px)\\tilde{\\partial} f,\n$$\nwhich requires \n\n$$\\tilde{b}^+=0 \\quad\\mbox{and}\\quad \n\\tilde{a}^+=\\frac{1}{2}\\tilde{\\partial} f(\\tilde{b}),\n$$\nwhereas the first line \n$$\na^+\\delta(px) + b^+ \\delta'(px) = \\frac{1}{2}\\tilde{x}'\\tilde{\\partial} f\n\\delta(px) + \\frac{1}{2} \\delta'(px) f(\\tilde{b})\n$$\nleaves us with \n$$\na^+=\\frac{1}{2}\\left( \\tilde{a} + \\theta(0)\\tilde{a}^+\\right)\n\\tilde{\\partial} f(\\tilde{b})\\quad\\mbox{and}\\quad\nb^+=\\frac{1}{2}f(\\tilde{b}).\n$$\nSo the geodesic becomes\n\\begin{eqnarray}\n  \\label{geores}\n  \\bar{p}x &=& a(px) + b + \\theta(px)\\left( \\frac{1}{2}(\n    \\tilde{a} + \\theta(0)\\tilde{a}^+ )\\tilde{\\partial} f(\\tilde{b})(px) +\n   \\frac{1}{2}f(\\tilde{b})\\right),\\nonumber\\\\\n   \\tilde{x} &=& \\tilde{a} (px) + \\tilde{b} + \\theta(px)\\frac{1}{2}\n   \\tilde{\\partial} f(\\tilde{b}) (px).\n\\end{eqnarray}\nAt first glance it seems as if we were done, would it not be for the\nominous factor $\\theta(0)$ which destroys the predictability of the\nscattering process.\nHowever the physical requirement of the covariantly constant \nnorm of the tangent vector will save the day, i.~e.~\n$$\n(\\dot{x}\\nabla)(g_{ab}\\dot{x}^a\\dot{x}^b) = \n2g_{ab}(\\dot{x}\\nabla)\\dot{x}^a\\dot{x}^b = 0 \\quad\\Rightarrow\\quad \ng_{ab}\\dot{x}^a\\dot{x}^b = const\n$$ \nFor a general pp-wave this gives\n\\begin{eqnarray}\n  \\label{covconst}\n&&\\dot{x}^a\\dot{x}^b(\\eta_{ab} + fp_a p_b) = \n-2(p\\dot{x})(\\bar{p}\\dot{x}) + \\dot{\\tilde{x}}^2 + f (p\\dot{x})^2 = \nconst.\\nonumber\\\\\n&&\\quad\\Rightarrow\\quad -2(\\bar{p}x') + (\\tilde{x}')^2 + f = const\n\\end{eqnarray}\nInserting (\\ref{geores}) into (\\ref{covconst}) we obtain immediately\n\\begin{equation}\n  -2a + \\tilde{a}^2 + \\theta(px) \n  \\frac{1}{4}(\\tilde{\\partial}f)^2(\\tilde{b})\\left(1-2\\theta(0)\\right) \n = const,\n\\end{equation}\nwhich is only constant if we choose $\\theta(0)=1\/2$, thereby\nrestoring predictability.\n\n\\section*{\\Large\\bit 2) Weak equality and Colombeau's product of \ndistributions}\n\nAlthough the result of the previous section is physically \nsatisfactory  it suffers from mathematical deficiencies. This is\ndue to the fact that  we implicitly multiplied discontinuous functions\nwith (singular) distributions resulting in the ambiguous point value\n  $\\theta(0)$. Therefore this section is devoted to a brief summary of \nColombeau's new generalized functions \\cite{Col1,Col2,ArBi}, \ntogether with an extension to arbitrary manifolds, since they\n provide a natural framework for the multiplication\nof distributions on a mathematically rigorous basis. From the \nphysical point of view Colombeau objects may be regarded as \n(arbitrary) regularisations of singular distributions. More \nprecisely\\footnote{For a motivation of the growth condition see \\cite{Col1}} \none considers one-parameter families $(f_\\epsilon)_{0<\\epsilon<1}$ \nof $C^\\infty$ functions with moderate growth in the parameter $\\epsilon$,\nnamely\n\\begin{eqnarray}\\label{Mod}\n  && {\\mathcal E}_M = \\{ (f_\\epsilon)\\vert f_\\epsilon \\in \n    C^\\infty({\\mathbb R}^n)\\quad \\forall K \\subset {\\mathbb R}^n compact, \n    \\forall \\alpha\\in {\\mathbb N}^n\\quad  \\\\\n  &&\\hspace*{1cm}\\exists\\, N\\in {\\mathbb N},\\exists\\> \\eta > 0,\\exists\\> c>0 \n    \\quad s.t. \\sup_{x\\in K}\\vert D^\\alpha f_\\epsilon(x)\\vert \\leq \n  \\frac{c}{\\epsilon^N}\\quad\\forall 0<\\epsilon< \\eta\\}.\\nonumber\n\\end{eqnarray}\nAddition, multiplication and differentiation are defined as pointwise\noperations, turning ${\\mathcal E}_M$ into an algebra.\n$C^\\infty$ functions $f$ are naturally embedded into ${\\mathcal E}_M$ as \nconstant families, i.e. $f_\\epsilon=f$, whereas continuous functions or \nelements of \n${\\mathcal D}'$ require the use of a so called ``mollifier'' \n$\\varphi$ for their embedding\n\\begin{equation}\\label{moll}\n f_\\epsilon(x) = \\int\\!\\! d^n\\!y \\frac{1}{\\epsilon^n}\n \\varphi\\fracwithdelims(){y-x}{\\epsilon}f(y)\\qquad \\int \nd^n\\!y\\varphi(y) =1,\n\\end{equation}\ntogether with additional conditions on the momenta of $\\varphi$ \n\\cite{Col1,Col2}. \nWith regard to distributions the above convolution integral is formal. \nIn order to identify the different embeddings of $C^\\infty$ functions  \none takes the quotient of the algebra ${\\mathcal E}_M$ with respect to\nthe ideal \n\\begin{eqnarray}\n  && {\\mathcal N} = \\{ (f_\\epsilon)\\vert f_\\epsilon \\in \n  C^\\infty({\\mathbb R}^n)\\quad\n  \\forall K \\subset {\\mathbb R}^n compact, \\forall \\alpha\\in {\\mathbb N}^n,  \n   \\forall N\\in {\\mathbb N}\\\\\n  &&\\hspace*{1cm}\\exists\\> \\eta > 0,\\exists\\> c>0,\\quad \n  s.t. \\sup_{x\\in K}\\vert D^\\alpha f_\\epsilon(x)\\vert \\leq \n  c\\epsilon^N\\quad\\forall\\> 0<\\epsilon< \\eta\\}.\\nonumber\n\\end{eqnarray}\nthereby obtaining the Colombeau algebra $\\mathcal G$.\nElements of $\\mathcal G$ are therefore equivalence classes of \none-parameter families of $C^\\infty$ functions. In order to make contact\nwith distribution theory one considers a coarser equivalence relation which \nis usually called weak equality or association. It is intuitively \nmotivated by the fact that the limit $\\epsilon\\to 0$ (if it exists) should \nreproduce the corresponding distributional object. More precisely, two \nColombeau-objects $(f_\\epsilon)$ and $(g_\\epsilon)$ are associated if\n\\begin{equation}\\label{assoc}\n  \\lim_{\\epsilon\\to 0} \\int\\! d^n\\!x\\left( f_\\epsilon(x)-g_\\epsilon(x) \\right)\n  \\varphi(x)=0\\qquad \\forall \\varphi\\in C^\\infty_0.\n\\end{equation}\nAssociation behaves like equality on the level of distributions.\nIt is compatible with addition and differentiation and it allows\nmultiplication with $C^\\infty$ functions. However, it does not respect\nmultiplication of Colombeau objects, as might have been expected.\nThe classical example in this context is provided by the powers of the \n$\\theta$ function, which upon naive multiplication would lead to \ncontradictions about the point value of \n$\\theta(0)$. Specifically\n$$\n(\\theta(x))^n = \\theta(x) \\Rightarrow n\\theta(0)\\theta'(x) = \\theta'(x)\n$$\nwhich cannot hold for arbitrary $n$ since $\\theta(0)$ is $n$-independent.\nLooking at the above equality from the Colombeau point of view\nwe have\n$$\n(\\theta(x))^n \\approx \\theta(x) \\Rightarrow n (\\theta(x))^{n-1}\\theta'(x)\n\\approx \\theta'(x)\n$$\nwhich holds separately for each $n$ and thereby allows us to replace \n$(\\theta)^{n-1}\\theta'$ by $(1\/n)\\theta'$. The contradiction is\navoided because multiplication by theta would break the association. \n\nThe above concepts made explicit use of the additive (group) structure of \n${\\mathbb R}^n$. Specifically the convolution integral used for the\nembedding of continuous functions and distributions in general lacks\ngeneralization to arbitrary manifolds. In the following we will \ngive a coordinate--independent formulation of the Colombeau \nalgebra\\footnote{A more detailed version will be given in \\cite{Bal2}}.\nTo begin with let us briefly recall the definition of distributions\non arbitrary (paracompact) $C^\\infty$ manifolds.\nThe natural generalization of distribution space as\nthe dual of a suitable test-function space \\cite{GeSh} is achieved by \nreplacing \ntest functions by test forms. That is we define a distribution as (continuous)\na linear functional on the space of $C^\\infty$ n-forms with compact \nsupport together with the usual locally convex vector space topology\n\\cite{ChoBr}.\nLocally integrable functions $f$ give rise to regular functionals via\n$$\n\\tilde\\varphi \\mapsto (f,\\tilde{\\varphi}) := \n\\int\\limits_{M} f\\tilde{\\varphi} \\qquad \n\\forall\\tilde{\\varphi}\\in C_0^\\infty(M)\n$$\nThe natural generalization of the derivative of a distribution\nis given by the notion of the Lie-derivative with respect to\nan arbitrary $C^\\infty$ vector-field $X$, i.~e.~\n$$\n({\\mathcal L}_X f,\\tilde{\\varphi}) := (-1) (f,{\\mathcal L}_X\\tilde{\\varphi}).\n$$\nThe above definition shows that distribution space does not require\nadditional structure but is purely a concept\nwhich depends on the differentiable structure of the manifold.\nIt reduces to the usual notion of distribution space on ${\\mathbb R}^n$\nif one makes use of the natural volume form $d^n x$ and decomposes\nevery test-form $\\tilde{\\varphi}=\\varphi(x)d^n\\! x$.\nLet us now try to do the same for the Colombeau algebra.\nThe $C^\\infty$ functions of moderate growth in $\\epsilon$ are easily \ngeneralized\n\\begin{eqnarray*}\n{\\mathcal E}_M(M) &=& \\{ (f_\\epsilon) \\in C^\\infty(M) | \\forall\\>\nK \\subset M\\>\\mbox{compact}, \n\\forall\\>\\{X_1,\\dots,X_p\\}\\>\\\\\n&&\\hspace*{0.5cm}X_i\\in\\Gamma(TM),[X_i,X_j ]=0,\\exists N\\in{\\mathbb N},\\exists \n\\eta> 0, \n\\exists c>0\\> \\\\\n&&\\hspace*{0.5cm} s.t. \\sup_{x\\in K}|{\\mathcal L}_{X_1}\\dots{\\mathcal L}_{X_p}\nf_\\epsilon(x)|\n\\leq \\frac{c}{\\epsilon^N}\\quad 0<\\epsilon<\\eta\\}\n\\end{eqnarray*}\nIn the same manner we may extend the ideal ${\\mathcal N}$ \n\\begin{eqnarray*}\n{\\mathcal N}(M) &=& \\{ (f_\\epsilon) \\in C^\\infty(M) | \\forall\\>\nK \\subset M\\>\\mbox{compact},\\forall\\>\\{X_1,\\dots,X_p\\}\\> \\\\\n&&\\hspace*{0.5cm}X_i\\in\\Gamma(TM),\n [X_i,X_j]=0,\\forall q\\in{\\mathbb N},\\exists \\eta> 0, \\exists c>0\\>\\\\\n&&\\hspace*{0.5cm} s.t. \\sup_{x\\in K}|{\\mathcal L}_{X_1}\\dots\n{\\mathcal L}_{X_p}f_\\epsilon(x)|\n\\leq c\\epsilon^q\\quad 0<\\epsilon<\\eta\\},\n\\end{eqnarray*}\nAs usual the Colombeau-algebra ${\\mathcal G}(M)$ is defined to be the \nquotient of ${\\mathcal E}_M(M)$ with respect to ${\\mathcal N}(M)$. \nIn order to generalize the embedding of continuous functions \nand more generally of distributions, we have to find an analogue \nof the smoothing kernel $\\varphi$. The immediate problem we are facing\nis that the expression used in ${\\mathbb R}^n$\n\\begin{equation}\\label{falt}\nf_\\epsilon(x) = \\int f(y)\\varphi\\fracwithdelims(){y-x}{\\epsilon}\n\\frac{1}{\\epsilon^n}d^n\\! y\n\\end{equation}\nmakes explicit use of the linear structure of ${\\mathbb R}^n$ as may\nbe seen from the argument of $\\varphi$ in (\\ref{falt}). Moreover, in\norder to allow the identification of the two types of embeddings of\n$C^\\infty$ functions, as discussed at the beginning of this section, one \nhas to require\n$$\n\\int y^\\alpha \\varphi(y) d^n\\! y = 0 \\qquad 1\\leq |\\alpha | \\leq q,\n$$\nfor some finite but arbitrary $q\\in {\\mathbb N}$. Once again this expression\ndepends  explicitly on the chosen coordinates. At first glance it seems \nthat the above conditions have to be modified \\cite{ColMer} to allow a \ngeneralization to an\narbitrary manifold. However, if one makes use of the tangent bundle $TM$\nof the manifold $M$ the above conditions may be taken as they are.\nIn order to show how this can be done let us consider local \ncoordinates on $TM$ that are induced by coordinates on $M$,\ni.~e.~ a bundle chart.\nLet $(U,x)$ denote a local coordinate system of $M$ then $(\\pi^{-1}(U),\n(x,\\lambda))$ will denote the corresponding bundle chart, where\n$TM@>\\pi>>M$ refers to the canonical projection and $\\lambda$ to the \ncoordinates along the fibres. Diffeomorphisms $M @>\\mu>>M$ induce\nfibre-preserving diffeomorphisms on \n$TM@>(\\mu,(\\partial\\mu\/\\partial x))>>TM$, namely \nso called bundle morphisms. The smoothing kernels will now be taken to\nbe $C^\\infty$ n-forms on $TM$, which allow a local ADM-like \nrepresentation\n$$\n\\tilde{\\varphi} = \\varphi(x,\\lambda)(d\\lambda^i+N^i{}_jdx^j)^n.\n$$\nMaking use of the embedding map \n$$\ni_x :T_x M@>>> TM\\qquad \\lambda \\mapsto (x,\\lambda)\n$$\nwe require\n$\\tilde{\\varphi}$ to obey\n\\begin{eqnarray}\\label{smoothk}\n  &&\\int\\limits_{T_xM}\\!\\!i_x^*\\tilde{\\varphi} = \n  \\int\\limits_{T_xM}\\!\\!\\varphi(x,\\lambda)d^n\\lambda=1,\\qquad\n  i_x^*\\tilde{\\varphi} \\in \\Omega_0(T_xM),\\nonumber\\\\\n  &&\\int\\limits_{T_xM}\\!\\!\\lambda^\\alpha i_x^*\\tilde{\\varphi} = \n  \\int\\limits_{T_xM}\\!\\!\\lambda^\\alpha\\varphi(x,\\lambda)d^n\\lambda=0\n  \\qquad\\forall\\>1\\leq |\\alpha|\\leq q.\n\\end{eqnarray}\nAll of the conditions (\\ref{smoothk}) are invariant under \n$M$-diffeomorphisms, since the latter act linearly on the fibres.\nMoreover, rescaling the smoothing kernel is a well defined concept, \nsince the transformation\n$$\n\\phi_\\epsilon: (x,\\lambda) \\mapsto (x,\\frac{1}{\\epsilon}\\lambda)\\qquad\n\\tilde{\\varphi}_\\epsilon := \\phi_\\epsilon^*\\tilde{\\varphi}\n$$\nis a specific case of the left action of the structure group \n$GL(n,{\\mathbb R})$ of $TM$.\nHaving solved the first part of the problem we now have to decide\nhow to generalize the convolution integral necessary for the embedding \nof the continuous functions in the Colombeau algebra. Let us therefore \nchoose a local coordinate system on $M$, which at the same time induces\ncoordinates on every tangent space in its domain. Let us, moreover, denote \nthe representative of a $C^\\infty(M)$ function with respect to the above \ncoordinates by $f(x)$. Identification of the local coordinates\non $M$ with those of the fibre attached at $x$ allows us to ``lift''\n$f$ to $T_xM$ by defining the value of the lift at $\\lambda$ to\nbe $f(x+\\lambda)$. The corresponding smoothened function $f_\\epsilon$ \non $M$ will then be defined to be\n$$   \nf_\\epsilon(x):=\\int\\limits_{T_xM}\\!\\!f(x+\\lambda)\\varphi(x,\n\\frac{\\lambda}{\\epsilon})\\frac{1}{\\epsilon^n}d^n\\!\\lambda\n=\\int\\limits_{T_xM}\\!\\!f(x+\\epsilon\\lambda)\\varphi(x,\\lambda)d^n\\!\\lambda\n$$\nThis definition is not coordinate invariant because the action of an\n$M$-diffeomorphism $\\mu$ yields\n\\begin{eqnarray}\\label{coinv1}\n  \\bar{f}_\\epsilon(\\bar{x}) &=& \\int\\limits_{T_{\\bar{x}}M}\\!\\!\\bar{f}\n  (\\bar{x}+\\bar{\\lambda})\\bar{\\varphi}(\\bar{x},\n  \\frac{\\bar{\\lambda}}{\\epsilon})\\frac{1}{\\epsilon^n}\n  d^n\\!\\bar{\\lambda}\n  =\\int\\limits_{T_{\\bar{x}}M}\\!\\!\\bar{f}(\\bar{x}+\\epsilon\\bar{\\lambda})\n  \\bar{\\varphi}(\\bar{x},\\bar{\\lambda})d^n\\!\\bar{\\lambda}\\nonumber\\\\\n  &=&\\int\\limits_{T_xM}\\!\\!f(\\mu^{-1}(\\mu(x)+\\epsilon\n  \\frac{\\partial\\mu}{\\partial x}\\lambda)\\varphi(x,\\lambda)d^n\\!\\lambda,\n\\end{eqnarray}\nwhich obviously differs from the definition in the unbarred system \n\\begin{equation}\\label{coinv2}\n  f_\\epsilon(x)=\\int\\limits_{T_xM}\\!\\!f(x+\\epsilon\\lambda)\n  \\varphi(x,\\lambda)d^n\\!\\lambda\n\\end{equation}\nThe difference of (\\ref{coinv1}) and (\\ref{coinv2}) is of order\n$\\epsilon^{q+1}$ if all the momenta of $\\varphi$ up to order $q$ vanish,\nwhich is a necessary condition to belong to the ideal $\\mathcal N$.\nThus the corresponding Colombeau object is well-defined since the \ncoordinate change only generates a motion within its equivalence class. \nFinally, the concept of association is also without problems, i.~e.~\n$$\n(f_\\epsilon) \\approx (g_\\epsilon)\\>\\mbox{iff}\\> \n\\lim_{\\epsilon\\to 0}\\int\\limits_M(f_\\epsilon - g_\\epsilon)\\tilde{\\varphi}=0\n\\>\\forall \\tilde{\\varphi}\\in\\Omega_0(M).\n$$ \nWe will say that the Colombeau object $(f_\\epsilon)$ gives rise to a\ndistribution $T$ if \n$$\n(T,\\tilde{\\varphi}):=\\lim_{\\epsilon\\to 0} \\int\\limits_M f_\\epsilon \n\\tilde{\\varphi}\n$$\ndefines a (continuous) linear functional over the space of test-forms.\nThe correspondence will in general be many to one. It is easy\nto see that two different distributions with associated Colombeau\nobjects require the latter to be also different. \nAs in the case of ${\\mathbb R}^n$, association of Colombeau-objects\nbecomes equality on the level of distributions and all vector space\noperations together with Lie-derivatives do not break the association.\n\n\\section*{\\Large\\bit 3) Geodesic equation and Colombeau theory}\nLet us now look at the geodesic equation from the point of view \nof Colombeau theory. That is, replace the delta-function\nappearing in (\\ref{impwave}) by an arbitrary regularisation \n$\\delta$ \nand do the same for the $\\theta$'s that enter in (\\ref{geoans}).\nTaking into account the three Killing-vectors of the general \nimpulsive wave-profile \\cite{AiBa2,AiBa3}\n\\begin{equation}  \\label{killings}\n  \\xi_1^a = p^a\\quad \\xi_{\\tilde{t}}^a = (\\tilde{t}x )p^a - \n  (px)\\tilde{t}^a ,\n\\end{equation}\nwhere $\\tilde{t}$ denotes an arbitrary vector in the subspace transversal \nto $p^a,\\bar{p}^a$, yields\n\\begin{eqnarray}\\label{coconst}\n  && p\\dot{x} \\approx const\\nonumber\\\\\n  && (\\tilde{t}x )p\\dot{x} - (px)(\\tilde{t}\\dot{x})\\approx const.\n\\end{eqnarray}\nThe first line of (\\ref{coconst}) just tells us that $px$ is an \naffine parameter, while\nthe second line fixes the transversal shift $\\tilde{b}^+$ in (\\ref{geoans}) \nto be zero.\nThe requirement that the length of tangent vector is covariantly\nconstant along the geodesic becomes\n\\begin{equation}\n  -2a + \\tilde{a}^2 + (-2a^+ + (\\tilde{a}^+)^2)\\theta +  \n  ( -2b^+ + f(\\tilde{b}) )\\theta' \\approx const,\n\\end{equation}\nwhich implies that the coefficients of $\\theta$ and $\\theta'$\nhave to be zero \n\\begin{eqnarray}\n  a^+ &=& \\tilde{a}\\tilde{a}^+ + \\frac{1}{2}(\\tilde{a}^+)^2,\\nonumber\\\\\n  b^+ &=& \\frac{1}{2} f(\\tilde{b}).\n\\end{eqnarray}\n>From the projection of the geodesic onto the orthogonal complement\nof $p^a$ and $\\bar{p}^a$\n\\begin{equation}\n  \\tilde{x}'' - \\frac{1}{2}\\tilde{\\partial}f \\delta \\approx 0\n\\end{equation}\ntogether with the above results we obtain\n$$\n\\tilde{a}^+ = \\frac{1}{2}\\tilde{\\partial}f(\\tilde{b})\n$$\nSince all the plus-parameters are determined in terms of the\ninitial data, the remaining equation has to be satisfied consistently.\n\\begin{eqnarray}\\label{consist}\n  &&(\\bar{p}x)'' -  (x'\\tilde{\\partial})f\\>\\delta - \n   \\frac{1}{2} f\\>\\delta'\\approx 0\\nonumber\\\\\n  &&(a^+(px)+ b^+)\\theta'' + 2 a^+\\theta' \n  - \\frac{1}{2}(\\tilde{a} + ((px)\\theta)' (\\tilde{a}^+\n  \\tilde{\\partial})f)\\>\\delta - \\frac{1}{2} f(\\tilde{b})\\>\\delta' \n  \\approx 0\\nonumber\\\\\n  &&b^+\\theta'' +  a^+\\theta' -\\frac{1}{2} (\\tilde{a}\\tilde{\\partial})\n  f(\\tilde{b})\\theta' -\\frac{1}{2}(\\tilde{a}^+\n  \\tilde{\\partial})f\\>((px)\\theta)'\\delta - \\frac{1}{2} f(\\tilde{b})\\theta''\n  \\approx 0\\nonumber\\\\\n  &&\\frac{1}{8}(\\tilde{\\partial}f)^2(\\tilde{b})\\theta'\n  -\\frac{1}{4} (\\tilde{\\partial}f(\\tilde{b})\n  \\tilde{\\partial})f\\>((px)\\theta)'\\delta \\approx 0.\n\\end{eqnarray}\nTaking into account that $((px)\\theta)'$ is also a Theta-function,\nwhich will be denoted by $\\hat{\\theta}$ one has \n$\\hat{\\theta}\\delta \\approx C \\theta'$ as may be seen from \n\\begin{eqnarray}\\label{ambi}\n  (\\hat{\\theta}\\delta,\\varphi) &=&\\int\\limits_{-\\infty}^\\infty\\!du\n  \\frac{1}{\\epsilon}\\phi\\fracwithdelims(){u}{\\epsilon}\n  \\int\\limits_{-\\infty}^u\\!dv\\frac{1}{\\epsilon}\n  \\psi\\fracwithdelims(){v}{\\epsilon}\\varphi(u)\n  \\qquad\\phi,\\psi\\in C^\\infty_0\\nonumber\\\\\n&=&\\int\\limits_{-\\infty}^\\infty\\!du\\phi(u)\\int\\limits_{-\\infty}^u\\!dv\\psi(v)\n\\varphi(\\epsilon u) = C\\varphi(0) + {\\mathcal O}(\\epsilon).\n\\end{eqnarray}\nIn (\\ref{ambi}) $\\phi$ and $\\psi$ respectively\ndenote the regularisations of $\\delta$ and $\\hat{\\theta}'$.\nTherefore (\\ref{consist}) becomes\n$$\n\\theta' \\frac{1}{8}(\\tilde{\\partial}f)^2(\\tilde{b})(1-2C) \\approx 0,\n$$\nwhich requires $C$ to be $1\/2$. However, this is precisely the type of \ncondition we encountered in section one. It restricts the regularisations\none is allowed to choose for $\\delta$ and $\\theta$ in order to obtain\na consistent (regularisation--independent) result.\n\n\\vfill\n\\noindent\n{\\bf Acknowledgement:} The author wants to thank Peter Aichelburg for \nmany useful discussions and the critical reading of the manuscript, and \nMichael Oberguggenberger and Michael Kunzinger for introducing him\nto Colombeau theory during his stay in Innsbruck in February 1996.  \n\\newpage\n\\section*{\\Large\\bit Conclusion}\n\nIn this work we solved the geodesic equation for arbitrary impulsive \npp-wave geometries. Due to the singular character of the wave profile,\nwhich contains a delta-function, one inevitably leaves the framework of\n(classical) distribution theory. This fact manifests itself in the\nappearance of the ambiguous pointvalue $\\theta(0)$. However, since\nwe made use of an affine parametrization, the length of the tangent vector \nremains covariantly constant along the geodesic. This requirement fixed\n$\\theta(0)$ to be $1\/2$. In order to justify our result mathematically \nwe made use of the recently developed framework of Colombeau's new \ngeneralized functions, which is designed to allow a (consistent) \nmultiplication of distributions. Although the original definition of the \nColombeau algebra $\\mathcal G$ made use of ${\\mathbb R}^n$-specific concepts \nwe showed that a generalisation to an arbitrary manifold $M$ is possible by \nemploying the corresponding tangent bundle $TM$. Finally we showed that our \ncondition on $\\theta(0)$ may be justified from the point of view of\nColombeau-theory as condition on the ``regularizations'' used for the pulse \nand the geodetic trajectory, respectively. \\par \nAs a next step we will investigate semiclassical scattering,\nnamely various wave equations in a general impulsive pp-background with\nspecific emphasis on those geometries arising as ultrarelativistic \nlimits of black-hole geometries.\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSupernovae are bright stellar explosions which can now be observed up to\nredshifts $z \\sim 1$. Several programmes (Perlmutter et al. 1997, 1998;\nSchmidt et al. 1997; Garnavich et al. 1997) are presently devoted to discover\nhigh--redshift supernovae for cosmological purposes. The goal is to determine\nthe geometry of the Universe through the effect that $\\Omega_{M}$,\n$\\Omega_{\\Lambda}$ have in the magnitude--redshift relationship of Type Ia\nsupernovae (SNe Ia). The same programmes provide rates of explosion up to\nlarge redshifts through proper evaluation of control time and efficiency of\ndetection. Indeed, the Supernova Cosmology Project has already achieved the\nfirst results after evaluating the data collected in its 1995 and 1996\ndiscovery runs (Pain et al. 1996, 1997).\n \nIn this ${\\it Letter}$ we outline the theoretical background for the\nprediction of those number counts for different geometries of the Universe\nand for various SNe Ia progenitors.\n\nThe pace at which stars formed in the past (the SFR) and the evolutionary\nclock to explosion of those binary supernovae is reflected in the counts. The\nevolutionary clock spans, in the case of SNe Ia ---as opposed to SNe II---,\nan important fraction of the age of the Universe.\n\nSNe Ia counts extending up to apparent red magnitudes $m_{R}\\sim 23-25$ could\nprovide a good test of the SNe Ia binary progenitor systems. We also find\nthat SNe Ia counts should be sensitive to a $\\Lambda$--dominancy in our\nUniverse through a larger increase at $z \\sim 1$. Refinements in the\ndetermination of the global SFR and continued SNe Ia searches at large\nredshifts should furnish, in the near future, an accurate enough basis for\nboth tests.\n\n\\section{Modeling}\n\nThe global SFR recently derived for high redshift galaxies in the Hubble Deep\nField (HDF) by Madau et al. (1996) has been extended to lower redshifts\n(Madau 1997) in accordance with the results of Lilly et al. (1995, 1996). The\nvariation of the SFR with $z$ is similar to that of the space density of\nquasars (Shaver et al. 1996), peaking at $z\\simeq 2$, the latter showing,\nhowever, a steeper slope both prior and after maximum. The above $SFR(z)$ at\n$z > 2$ has been obtained from the UV luminosity density along redshift.\nRowan--Robinson et al. (1997) have also derived a $SFR(z)$ from ISO infrared\nobservations of galaxies in the HDF. The general shape agrees well with\nMadau's (1996) data but the inferred values are higher, maybe implying that\nabout 2\/3 of star formation at high $z$ has taken place shrouded by dust. In\na different approach, Pei \\& Fall (1995) have dealt with the determination of\nthe global star formation history by tracking the evolution of the global HI\ncontents of the Universe as measured from $Ly\\alpha$ QSO absorption--line\nsystems. The results of Madau et al. (1996) agree with those from that very\ndifferent approach. On the other hand, Lilly et al. (1996) provide, from the\nCanada--France--Hawaii Redshift Survey, an estimate of the comoving\nluminosity density of the Universe over the redshift range $0 < z < 1$, which\ncan be interpreted in terms of a $SFR(z)$ which agrees with previous\nestimates of a significant increase with $z$ up to $z\\sim 1.5-2$. Prospects\nto further explore the global SFR include  the number counts of SNe II\ndiscovered at high redshift, which should trace the activity of star\nformation in the Universe along $z$. Such numbers on SN II are obtainable in\ncurrent high--$z$ SNe searches.  Given the continuous improvement in the\ndetermination of the global SFR, we base the present calculation of SNe Ia\nnumber counts on the most recent empirical results. We evaluate the rate of\nexplosion of SNe Ia by convolving their time to explosion with the SFR. The\nrate can then be calculated as:\n\n$$r_{SNeIa}(t) = \\int_{0}^{t} R(t - \\tau)SFR(\\tau) d\\tau\\eqno(1)$$\n\n\\noindent\nwhere $R(t)$ is the SNe Ia rate after an instantaneous outburst, and $t$ and\n$\\tau$ are in the SN rest frame. Depending on the progenitor systems, in some\ncases we would have  SNe Ia with a peak rate of explosion at 10$^{9}$ yr. In\nother cases, there can be some SNe Ia exploding even only a few $10^{7}$ yr\nafter star formation already, thus becoming the brightest optical events at\n$z  >> 1$. $SFR(\\tau)$ is derived from the global $SFR(z)$ (Madau et al.\n1996; Madau 1997). The adopted $SFR(z)$ (for $H_{0} = 50\\ km\\ s^{-1}\\\nMpc^{-1}$, $\\Omega_{M} = 1.0$, $\\Omega_{\\Lambda}$ = 0) is shown in the top\npanel of Figure 1, along with the measured data points and their error bars.\nNote that the data points from Connolly et al. (1997) for the $1\\hbox{\\lower .8ex\\hbox{$\\,\\buildrel < \\over\\sim\\,$}}\nz\\hbox{\\lower .8ex\\hbox{$\\,\\buildrel < \\over\\sim\\,$}} 2$ range fall somewhat below whereas those from Rowan--Robinson et\nal. (1997) for the $0.6\\hbox{\\lower .8ex\\hbox{$\\,\\buildrel < \\over\\sim\\,$}} z\\hbox{\\lower .8ex\\hbox{$\\,\\buildrel < \\over\\sim\\,$}} 1$ range are significantly above\nthe adopted curve. The $SFR(z)$ is transformed according to the geometries of\nthe model universes considered. Here we restrict this presentation to three\nfavored models of the Universe: {\\it Model A}, a flat universe with $H_{0} =\n65\\ km\\ s^{-1}$, $\\Omega_{M} = 1$, $\\Omega_{\\Lambda} = 0$; {\\it Model B}, an\nopen universe with $H_{0} = 65\\ km\\ s^{-1}\\ Mpc^{-1}$, $\\Omega_{M} = 0.3$,\n$\\Omega_{\\Lambda} = 0$, and {\\it Model C}, a flat universe with $H_{0} = 65\\\nkm\\ s^{-1}\\ Mpc^{-1}$, $\\Omega_{M} = 0.3$, $\\Omega_{\\Lambda} = 0.7$.\n\nThe $SFR(z)$  derived for each case is next transformed into $SFR(t)$ (in the\ncomoving frame). The derived global $SFR(t)$, when time--integrated over the\nwhole history of the corresponding universe (model $A$: $t_{0} = 10.0\\ Gyr$;\nmodel $B$: $t_{0} = 12.2\\ Gyr$; model $C$: $t_{0} = 13.6\\ Gyr$), produces the\nobserved stars today (Guzm\\'an et al. 1997).\n\nThe global SNe Ia rates, $r_{SNe Ia}(t)$ ($yr^{-1}\\ Mpc^{-3}$), for each\nmodel universe and family of SNe Ia progenitor systems, are calculated in the\ncomoving frame according to (1), and integrated over comoving volume to\nobtain the expected SNe Ia counts ($yr^{-1} sq. deg^{-1}$) as a function of\nz. We integrate over $dV$:\n\n$$dV = {d_{M}^{2}\\over (1 + \\Omega_{k}H_{0}^{2}d_{M}^{2})^{1\/2}}\\\nd(d_{M})d\\Omega\\eqno(2)$$\n\n\\noindent\nwhere $d_{M}$ is the proper motion distance and $\\Omega_{k} = 1 - \\Omega_{M}\n- \\Omega_{\\Lambda}$ is the fractional contribution of the curvature to the\nexpansion (Carroll, Press, \\& Turner 1992). $d_{M}$ is related to the\nluminosity distance $d_{L}$ through $d_{L} = (1+z)\\ d_{M}$ (Weinberg 1972).\n$d_{L}$ is calculated as:\n\n$$d_{L} = { (1 + z) \\over H_{0}|\\Omega_{k}|^{1\/2}}\\\nsinn\\left\\{|\\Omega_{k}|^{1\/2}\\int_{0}^{z_{1}}\\left[(1+z)^{2} (1+\\Omega_{M}z)\n- z(2+z)\\Omega_{\\Lambda}\\right]^{-1\/2}dz\\right\\} \\eqno(3)$$\n\n\\noindent\nwhere $sinn$ stands for $sinh$ if $\\Omega_{k} > 0$ and for $sin$ if\n$\\Omega_{k} < 0$ (both $sinn$ and the $\\Omega_{k}$ terms disappear from (3)\nif $\\Omega_{k} = 0$, leaving only the integral times $(1 + z)\/H_{0}$).\n\n\\noindent\nThe dependence on cosmological parameters of the comoving volume derivative\n$dV\/dz d\\Omega$ differs from one model of Universe to another. A different\ncosmological effect is the age--redshift (t(z)) relationship for different\nmodel Universes, which changes the z at which the SNe Ia rates peak.\n\nThe results are tightly linked to the reliability of the global SFR. An\nestimate of its current uncertainty can be obtained by looking at the data\npoints in the top panel of Figure 1. Increasingly accurate SFRs should\nsteadily improve the SNe Ia rate predictions. There is, as well, increasing\nevidence of universality in the slope of the initial mass function (IMF) from\ntests in different metallicity and age environments. Such a ``universal''\nmass function would be well represented by the Salpeter (1955) power law with\n$x = 0.86 \\pm 0.23$. Both the overall trend to convergence in the estimate of\nthe global $SFR(z)$ and the almost constancy of the IMF favor the possibility\nof deriving global values for the SNe Ia explosion rates.\n\n\\noindent{\\it Stellar clock: binary progenitors}. All evolutionary models for\nSNe Ia progenitors involve the accretion of mass by a C+O WD from a companion\nin a close binary system. Current research on SNe Ia has discarded some\nformerly proposed SNe Ia scenarios. The two classes of binary systems\nconsidered here as the most likely systems giving SNe Ia are: (1) the merging\nof two white dwarfs ---also called double degenerate scenario (DD)---, and\n(2) the accretion by the WD of H  from a less evolved star in a close binary\nsystem ---a single degenerate scenario (SD)---. We call this second case a\ncataclysmic--like system (CLS). The time evolution of the SNe Ia rates after\nstar formation differs from one family of progenitor systems to another, and\nthere are variations within each family.  We have modeled the SNe Ia rates by\nmeans of the same Monte Carlo code as in previous works (Ruiz--Lapuente,\nBurkert, \\& Canal 1995,1997;  Canal, Ruiz--Lapuente, \\& Burkert 1996,\nhereafter RCB95,97 and CRB96),  and adopting different  prescriptions for\neach evolutionary path and for the initial binary parameters.  We have also\ncompared our predictions with those of other authors:  (1) The physical input\nfor the merging of two C+O white dwarfs (WDs) is modeled both as in Iben \\&\nTutukov (1984)  and in RBC95. We allow  for a number of different physical\ndescriptions of the binary  evolution, and we try different values of the\ncommon envelope  parameter $\\alpha$ (a measure of the efficiency with which\norbital energy is used in envelope ejection).  (2) The physical modeling of\nthe explosion of a C+O WD when it reaches the Chandrasekhar mass by accretion\nand burning of H from a main--sequence, subgiant, or giant  companion is\nmodeled as in the early work by Iben \\& Tutukov (1984), as in CRB96,  and\nfinally including the most recently proposed variation of the binary\nevolution by Hachisu, Kato, \\& Nomoto (1996), who find a ``wind'' solution\nfor fast hydrogen accretion by a WD from a red--(sub)giant companion, but we\napply the same evolutionary constraints as Yungelson et al. (1996).\nParticularly relevant to the modeling are the slope of the IMF, the\ndistribution of mass ratios $q$ of the secondary to the primary, and the\ndistribution of initial separations $A_{0}$ in the progenitor binary sytems,\ntogether with the prescriptions for mass transfer.  By trying various\nphysical approaches adopted by different modelers and exploring different\nvalues of the initial binary parameters (IMF, $q$, $A_{0}$ distributions) and\nassumptions as to the mass transfer rates, we obtain the characteristic\nbehaviour of each stellar clock and the uncertainties in the absolute scale\nof the rates.\n\nIn Figure 1, a comparison of $R(t)$ in (1) with those from other authors\n(middle  panels: different IMF, $q$ and $A_{0}$ distributions; bottom panel:\ndifferent assumptions as to the allowed mass transfer rates) is made. From\nsuch comparison we concluded in our previous work that the evolution of the\nSNe Ia rates (rise, peak, decline) along cosmic time, for a given class of\nsystems,  have  broad common features shared by the predictions from\ndifferent authors (Iben \\& Tutukov 1984; Tutukov \\& Yungelson 1994; Yungelson\net al. 1996; our own modeling), which basically describe  the clock of the\ncorresponding systems. The Hachisu, Kato \\& Nomoto (1996) solution, here\ncalled CLS(W), would allow to have some relatively young SNe Ia in the CLS\nscenario, and the SNe Ia rate would increase fast with redshift. On the other\nhand, the DD scenario predicts a flatter increase of the rate from z $\\simeq$\n0 to z$\\simeq$ 1, in all model universes. In all cases, the $R(t)$ shown here\nwould give, for our own Galaxy and the present time, SNe Ia rates which are\nwithin the current error bars of the observational estimates.\n\nVersions CLS and CLS(W) in the bottom panel of Figure 1 illustrate the range\nof uncertainty in the time of start of the explosions in the SD scenarios.\nFor the DD progenitors we have chosen the most favorable physical\nassumptions, those that enhance the numbers of SNe Ia at high $z$, in order\nto best reproduce the first observational results. Any other choices would\ngive lower numbers. In fact, the observed numbers seem to be above the\npredictions for this type of binary progenitor system (but one should stress\nhere the uncertainty in the global SFR). A clear feature of the DD scenario\nis not only that the SNe Ia rates  do not increase fast towards higher\nredshifts but also that they are lower than those predicted for the SD\nprogenitors. We should note here that we are assuming the universality of the\ndistributions of binary parameters determined for the solar neighbourhood\n(Duquennoy \\& Mayor 1991). We have explored, however, the effects that\nplausible changes in the initial $q$ and $A_{0}$ distributions would have in\nthe outcome and found them to be only moderate (see below).\n\n\\noindent{\\it Counts as a function of magnitude}. We compare the model counts\nwith the observations by deriving the dependence of the number counts on\n$m_{R}$, the apparent magnitude of the SNe Ia in the $R$ photometric band. In\nfact, in this transformation  the intrinsic dispersion in brightness of the\nSNe Ia is not a key factor, since we are using the $N-mag$  relationship\nmeasured in intervals of 0.2 to 0.5 mag for our tests (we could also use\n$N-z$, thus avoiding the transformation to magnitude). The intrinsic\ndispersion in magnitude is, in contrast, very important for cosmological\ntests using $m(z)$.\n\nWe assume the SNe Ia to have an average absolute blue magnitude:\n\n$$M_{B} = -18.52 + 5\\ log(h\/0.85)\\eqno(4)$$\n\n\\noindent\nwhere $h\\equiv H_{0}\/100$  (Perlmutter et al. 1997).\n\nThe distance modulus for each $z$ is calculated from the luminosity distance\n$d_{L}$. The apparent $m_{R}$ at maximum as a function of $z$ is determined\nfrom:\n\n$$m_{R} = M_{B} + 5\\ log (d_{L}(z)) - 5\\  + K_{corr}\\eqno(5)$$\n\n\\noindent\nwhere $K_{corr}$ is taken from Kim, Goobar, \\& Perlmutter (1996) (it includes\nthe full transformation from $B$ into $R$ magnitudes). We finally calculate\nthe variation of the SNe Ia rate ($yr^{-1}\\ sq. deg^{-1}$ $\\Delta mag^{-1}$)\nwith apparent red magnitude $m_{R}$ (we take $\\Delta m_{R}=0.5\\ mag$).\n\nThe results are displayed in the four panels of Figure 2. Shown in the Figure\nare the data points currently available (Pain et al. 1996; Pain, private\ncommunication). The dips in the curves are entirely due to the shape of the\n$K$--corrections. The different slopes of the curves from one family of\nprogenitors to another mainly reflect how fast the SNe Ia rate declines after\nreaching its peak value plus the time elapsed between star formation and the\npeak (steeper decline and longer time delay for the CLS and CLS(W) models\nthan for the DD model: see the middle and bottom panels of Figure 1), and the\nabsolute values of the rates at low redshift ($m_{R} \\simeq 20-21$) are\nsensitive to the ages of the corresponding model universes ($t_{0}(A)<\nt_{0}(B)< t_{0}(C)$). The slopes of the rates when comparing different models\nof the Universe are sensitive to the contribution of a non--zero $\\Lambda$.\nThe faster increase of the rates with magnitude (redshift) along the model\nsequence A--C corresponds to the increasing comoving volume derivatives\n($dV\/dz d\\Omega$) in the respective model universes. Such derivative is large\nif $\\Omega_{\\Lambda}$ gives a significant contribution to $\\Omega$. As a\ncheck of the sensitivity of the results to the choices of the binary model\nparameters, we have also calculated the $dN\/dm_{R}$--$m_{R}$ relationship,\nfor the DD  progenitor, adopting  different $q$ and $A_{0}$ distributions\n(dotted  line in the middle panel of Figure 1): the final curve is not very\nsensitive to the various choices of the distributions explored in this work\n(the model is shown  by the continuous line  in the four panels of Figure 2).\nTaking at their face values the SFR and parameters for the SNe Ia progenitor\nevolution adopted, and also the two data points, the results for both the CLS\nand CLS(W) systems give better fits to the data than those for the DD\nsystems, model universe A being most favored and model C giving the worst fit\nfor any kind of system. But those conclusions are preliminary and we must\nawait the reduction of the uncertainties in the observed counts and in the\nglobal SFR for the intermediate--mass stars leading to SNe Ia. As we see from\nFigure 2, at present the uncertainty in the SNe Ia counts at $m_{R}\\simeq 22$\nis of the order of a factor of 2. That is the same as the difference between\nthe prediction for the DD model and the lowest one for the SD model, and also\nsimilar to the range covered by the two extreme predictions (CLS and CLS(W))\nfor the SD model. Any increase in the SFR would almost homologously shift\nupwards all the count predictions. At higher magnitudes, the range of\npredictions for the SD models becomes narrower while the differences with the\nDD model increase. Thus, if we knew both the SNe Ia counts at $m_{R}\\simeq\n23$ and the global SFR to better than a factor 1.5, we could already\ndiscriminate between the SD and the DD models. The sensitivity to the\ncosmological parameters $\\Omega_{M}$ and $\\Omega_{\\Lambda}$ is still low at\nthose magnitudes. If we had determined, for instance, the DD model to be the\nright one, then the SNe Ia counts at $m_{R} = 24.5$ (corresponding to\n$z\\simeq 1$) in the C universe ($\\Omega_{M} = 0.3$, $\\Omega_{\\Lambda} = 0.7$)\nshould be twice those in the A universe ($\\Omega_{M} = 1$, $\\Omega_{\\Lambda}\n= 0$), the factor of increase from $m_{R} = 22$ to $m_{R} = 24.5$ being also\ntwice as large in case C as compared with case A (both contrasts would be\nsharper if the SD model were the right one). The high--$z$ SN searches to\nmeasure $\\Omega_{M}$, $\\Omega_{\\Lambda}$ from the variation of the apparent\nmagnitudes of SNe Ia are now almost reaching $z = 1$ (Perlmutter et al. 1998;\nGarnavich et al. 1997). From samples large enough to derive the SNe Ia counts\nat $z\\simeq 1$, the constraints on both parameters should be tighter than\nthose obtainable from the counts alone, but the latter could provide a\nsupplementary test, since it is based on a different approach: the use of a\nvolume effect instead of a redshift--magnitude effect. Counts extending to\neven higher magnitudes would more clearly reveal the geometry of the\nUniverse. Suggested SN searches up to infrared magnitudes $K\\sim 26-27$\n(Miralda--Escud\\'e \\& Rees 1997) would extend well beyond that point.\n\n\\section{Conclusions and future prospects}\n\nThe theoretical results presented here are intended to show the potential of\nthe comparison between model predictions and the results of undergoing and\nfuture SNe Ia searches. The slopes of the $dN\/dm_{R}$--$m_{R}$ curves are\nmainly sensitive to the general characteristics of each SNe Ia model while\nthe absolute values depend more on model parameters and on the SFR. Hence the\ninterest of data covering a broader $m_{R}$ (or $z$) range than those\ncurrently available. On the other hand, the differences in the predictions\nfrom any SNe Ia progenitor assumed, for $\\Omega_{\\Lambda}$--dominated\nuniverses as compared with matter--dominated and open universes, become\nsignificant at large enough values of $m_{R}$. In a Universe dominated by\n$\\Omega_{\\Lambda}$  the number counts of SNe Ia show a larger increase at $z\n= 1$ due to the large volume encompassed at that redshift. Useful information\nwould be obtained by the measurement of the evolution of the SNe Ia counts up\nto $z = 1$ and beyond (possibly in the K--band for higher redshifts).\n\nThe first data obtained by Pain et al. (1996) gave a value of\n$34.4^{+23.9}_{-16.2}$ SNe Ia $yr^{-1}\\ sq. deg.^{-1}$ in a magnitude range\nof 21.3 $<$ R $<$ 22.3.  These first results were obtained from three SNe\ndiscovered at redshifts 0.374, 0.420, and 0.354 (SN 1994H, SN 1994al \\& SN\n1994F) and  the small number statistics dominates this very first\nmeasurement. A larger bulk of data, already obtained, will now reduce the\nstatistical uncertainties, and prospects to extend the observations up to\nhigher $z$ are on the way. It is thus too soon to extract firm conclusions\nfrom the data. Our purpose here is to propose a new useful test to be\ncompleted in the near future.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:1}\n\nThe CFT bootstrap program started in the 70's with the works \\cite{FERRARA1973161,Polyakov:1974gs}. Using the Virasoro algebra they solved every unitary 2D CFT with central charge $c<1$. However they were unable to extend their methods to higher dimensions or larger central charge. \n\nThe recent revival of the bootstrap program caused by \\cite{Rattazzi_2008} was due to its novel ways of analyzing crossing symmetry. Not only it was able to be replicated in arbitrary dimensions but was also extended to similar feats of the original in special cases, as the $3D$ Ising and the $O\\left(N\\right)$ models, \\cite{El_Showk_2012,Kos_2014,Kos_2015}. With their new formulation we are able to show for which values of the CFT data there is no unitary CFT.\n\nOur method is dual to theirs, we demonstrated how to construct simple unitary 4-point functions using the generalized polynomials. These will be points that cannot be excluded by their methods. An important distinction between the two is that this is purely analytical. We won't be able to find every unitary CFT. However, as will be shown in section \\ref{sec:4}, the generalized polynomials cover a substantial region of the space for small scaling dimension.\n\nThis work is divided as follows, in section \\ref{sec:2} we review the numerical bootstrap approach to motivate why starting with crossing symmetric expressions is interesting. In section \\ref{sec:4} we introduce and analyze the generalized polynomials. Lastly, in \\ref{sec:5} we present a computational method to find large scalar gaps.\n\nThe appendices cover the details not present in the main text, in \\ref{ap:A} we discuss a special case of generalized polynomials and the relation between crossing symmetry and a more general basis. Appendix \\ref{ap:B} presents the properties of the hypergeometric functions in the OPE expansion and proves the important statements from the main text.\n\n\\section{Reviewing the bootstrap equation and crossing symmetry} \\label{sec:2}\n\nBefore starting it would be good to review some points. For a more in depth review check \\cite{Poland:2018epd,Rychkov:2016iqz,simmonsduffin2016tasi}. The two main elements in the CFT bootstrap program are the OPE and crossing symmetry. We can use the OPE to write all correlation functions in terms of the CFT data of primary operators, $\\left\\{\\Delta_{i},\\lambda_{ijk}\\right\\}$, and we can use crossing symmetry to relate different orderings in a correlator. A 4-point function of identical scalars $\\phi$ is give by\n\n\\begin{align}\n    &\\mathcal{G}\\left(u,v\\right)=\\underset{J,\\Delta}{\\sum} \\left(\\lambda_{\\phi\\phi\\mathcal{O}}\\right)^{2} G^{\\left(D\\right)}_{J_{\\mathcal{O}},\\Delta_{\\mathcal{O}}}\\left(u,v\\right), \\\\\n    \\label{eq:02} & \\mathcal{G}\\left(u,v\\right)=\\mathcal{G}\\left(\\frac{u}{v},\\frac{1}{v}\\right)=\\left(\\frac{u}{v}\\right)^{\\Delta_\\phi}\\mathcal{G}\\left(v,u\\right).\n\\end{align}\n\nIn the numerical bootstrap the conformal block decomposition is applied to the crossing relations to obtain the bootstrap equation. This new manifestly unitary constraint is analyzed numerically with functionals to bound the CFT data\n\n\\begin{align} \\label{eq:06}\n    \\underset{J,\\Delta}{\\sum}\\left(\\lambda_{\\phi \\phi \\mathcal{O}}\\right)^{2}\\left( \\frac{G^{\\left(D\\right)}_{J_\\mathcal{O},\\Delta_{\\mathcal{O}}}\\left(u,v\\right)}{u^{\\Delta_{\\phi}}}-\\frac{G^{\\left(D\\right)}_{J_\\mathcal{O},\\Delta_{\\mathcal{O}}}\\left(v,u\\right)}{v^{\\Delta_{\\phi}}}\\right)=\\underset{J,\\Delta}{\\sum}\\left(\\lambda_{\\phi \\phi \\mathcal{O}}\\right)^{2} F_{J,\\Delta}^{\\Delta_{\\phi}}\\left(u,v\\right)=0, ~~~~ \\left(\\lambda_{\\phi \\phi \\mathcal{O}}\\right)^{2}\\geq0.\n\\end{align}\n\nIn practice we cannot expect to find an exact unitary 4-point function from this equation, but we can from the crossing relations $\\left(\\ref{eq:02}\\right)$. If we construct a crossing symmetric expression and then show that it satisfy unitarity we would have a point in the data space that cannot be excluded without new constrains. In the optimal case we would be able to pinch the boundary between the excluded and allowed region, similar to \\cite{2020}, however this is not what we have now. Although not optimal this still allow us to find interesting results.\n\nAssuming we already have a crossing symmetric expression, whether a closed form or an approximation, the challenge becomes finding when they are unitary. The conformal blocks are closely related to the analytic structure of 4-point functions at the point $u=0$, however crossing symmetry is related to series at the point $u=v=1$. This loss of information about $u=0$ is where most of the difficulty resides. In appendix \\ref{ap:A} we show this in a simple example.\n\nTo bypass this loss we search for solutions of crossing symmetry that behaves similar to a conformal block close to $u=0$. The conformal blocks behave as $G^{\\left(D\\right)}_{J,\\Delta} \\sim \\mathcal{N}_{J,\\Delta} u^{\\frac{\\Delta-J}{2}}$, $\\left(u\\xrightarrow[]{}0 ~~ v\\xrightarrow[]{}1\\right)$, with our normalization shown in \\ref{ap:B}. We use a basis of functions $\\left\\{f_{i}\\left(u,v\\right)\\right\\}$ that for small $u$ it behaves as a sum of power laws, $f_{i}\\left(u,v\\right) \\sim \\sum_{j} \\mathcal{C}_{i, j}\\left(v\\right) u^{p_{i, j}}$ and are flexible enough to work for any scaling dimension in $\\left(\\ref{eq:02}\\right)$. We will start with the simplest basis, the generalized polynomials, and show that it already has interesting results.\n\n\\section{Generalized polynomials} \\label{sec:4}\n\nThe generalized polynomials are polynomial-like functions with arbitrary powers of $u$ and $v$ and with a finite number of terms to keep crossing symmetry under control. They are given by the ansatz\n\n\\begin{align} \\label{eq:12}\n    & \\mathcal{G}\\left(u,v\\right) = \\frac{1}{v^{\\Delta_{\\phi}}}\\underset{i=1}{\\overset{N<\\infty}{\\sum}}~ \\xi_{i} ~ u^{a_{i}} v^{b_{i}} = \\underset{i=1}{\\overset{M}{\\sum}}~ E_{i} \\left(a_{i},b_{i},c_{i}\\right), \\\\\n    \\label{eq:13} \\left(a,b,c\\right) &= \\frac{u^{a} v^{b}+u^{a}v^{c}+u^{b}v^{a}+u^{b}v^{c}+u^{c}v^{a}+u^{c}v^{b}}{N_{a,b,c} ~ v^{\\Delta_{\\phi}}}, ~~ a+b+c=2\\Delta_{\\phi}, \\\\\n     N_{a,b,c} &= \\left\\{\\begin{tabular}{c}\n         1 if $a\\neq b$, $b\\neq c$, $c\\neq a$, \\\\\n         2 if $a=b\\neq c$, $b=c\\neq a$, $c=a\\neq b$, \\\\\n         6 if $a=b=c$.\n    \\end{tabular}\\right.\n\\end{align}\n\nFrom the polynomial-like basis $\\left\\{u^{a_{i}} v^{b_{i}-\\Delta_{\\phi}}\\right\\}$ we created a new crossing symmetric basis $\\left\\{\\left(a_{i},b_{i},c_{i}\\right)\\right\\}$ which we can use to construct our correlators. Using the power law behavior at $u=0$ we can predict what is the form of their conformal block decomposition. By matching the powers in the conformal blocks with those in the polynomial basis we find that the correct series should have the following structure\n\n\\begin{align} \\label{eq:15}\n    u^{p} {v^{q}} \\sim \\underset{J,n}{\\sum} A^{\\left(D\\right)}_{J,n} ~ G_{J,2p+J+n}^{\\left(D\\right)}\\left(u,v\\right).\n\\end{align}\n\nWe can see that the series generated by two power laws $u^{p}$ and $u^{q}$ won't overlap unless their powers differ by an integer or half-integer.\n\nUnlike unitarity, crossing symmetry do not depend on the number of dimensions, if $D\\geq2$, as such expressions $\\left(\\ref{eq:12}\\right)$ and $\\left(\\ref{eq:15}\\right)$ are general results. In\n \\ref{ap:B} we give all the properties needed to show that the correct conformal block decomposition for general $D$ has the form\n\n\\begin{align} \\label{eq:16}\n    \\frac{u^{p}\\left(v^{q}+v^{r}\\right)}{v^{\\Delta_{\\phi}}} = \\underset{m\\geq n}{\\sum} \\frac{1+\\left(-1\\right)^{n+m}}{n! m!} F^{n,m}_{p,D}\\left(\\left|\\frac{q-r}{2}\\right|\\right) G^{\\left(D\\right)}_{m-n,2p+n+m}\\left(u,v\\right),\n\\end{align}\n\nwhere $F_{p,D}^{n,m}\\left(t\\right)$ is a polynomial in $t$. For the rest of the main text we set $D=2$.\n\nThe relation between the $2D$ and $1D$ conformal blocks, $G_{J,\\Delta}\\sim k_{\\frac{\\Delta+J}{2}}k_{\\frac{\\Delta-J}{2}}$, allow us to use the results found in \\cite{Hogervorst_2017} to calculate $F_{p,2}^{n,m}\\left(t\\right)$ exactly. Both the result and the relation between conformal blocks are written in the Dolan-Osborn variables, so we rewrite the functions $u^{p}v^{q}$ as $ \\left(z^{p}\\left(1-z\\right)^{q}\\right)\\left(\\bar{z}^{p}\\left(1-\\bar{z}\\right)^{q}\\right)$ and use \\cite{Hogervorst_2017} to obtain\n\n\\begin{align} \\label{eq:18}\n    z^{p}\\left(1-z\\right)^{q} = \\underset{n=0}{\\overset{\\infty}{\\sum}} \\dfrac{\\left(-1\\right)^{n} \\left(p\\right)^{2}_{n}}{n! \\left(2p-1+n\\right)_{n}} {}_3F_{2} \\left(\\left.\\begin{array}{c}\n    \\begin{array}{ccc}\n        -n, & 2p-1+n, & -q \\\\\n    \\end{array}{}\\\\ \\begin{array}{cc}\n        p, & p \\\\\n    \\end{array}{} \\end{array}{} \\right| 1\\right) k_{p+n}\\left(z\\right).\n\\end{align}\n\nThese hypergeometric function have multiple properties we can exploit, to make them simpler to express we'll rewrite them as\n\n\\begin{align} \\label{eq:19}\n    f_{p}^{n}\\left(t\\right) &= \\dfrac{\\left(p\\right)^{2}_{n}}{\\left(2p-1+n\\right)_{n}} {}_3F_{2} \\left(\\left.\\begin{array}{c}\n    \\begin{array}{ccc}\n        -n, & 2p-1+n, & \\frac{p}{2}-t \\\\\n    \\end{array}{}\\\\ \\begin{array}{cc}\n        p, & p \\\\\n    \\end{array}{} \\end{array}{} \\right| 1\\right),\n\\end{align}\n\nwhich yields\n\n\\begin{align}\n    z^{p}\\left(1-z\\right)^{q} &= \\underset{n=0}{\\overset{\\infty}{\\sum}} \\dfrac{\\left(-1\\right)^{n} }{n!} f_{p}^{n}\\left(\\frac{p}{2}+q\\right) k_{p+n}\\left(z\\right),\\\\ \n    \\label{eq:20} F_{p,2}^{n,m}\\left(t\\right) &= 2^{m-n}f_{p}^{n}\\left(t\\right)f_{p}^{m}\\left(t\\right).\n\\end{align}\n\nThe first main property of $f_{p}^{n}\\left(t\\right)$ is its parity, $f_{p}^{n}\\left(-t\\right)=\\left(-1\\right)^{n}f_{p}^{n}\\left(t\\right)$, so we only need to analyze the region $t\\geq0$. Second $f$ is a continuous Hahn polynomial, meaning they have a simple recurrence relation\n\n\\begin{align} \\label{eq:22}\n    f_{p}^{n+1}\\left(t\\right)=t f_{p}^{n}\\left(t\\right)+\\dfrac{n\\left(n+p-1\\right)^{2} \\left(n+2p-2\\right)}{4\\left(2n+2p-1\\right)\\left(2n+2p-3\\right)} f_{p}^{n-1}\\left(t\\right).\n\\end{align}\n\nUsing induction together with the initial conditions $f_{p}^{0}\\left(t\\right)=1$ and $f_{p}^{1}\\left(t\\right)=t$ and $p\\geq0$ we prove that $f_{p}^{n}\\left(t\\right)\\geq0$. The only step left is to prove $\\left(\\ref{eq:16}\\right)$ for $D=2$ and $a+b+c=2\\Delta_{\\phi}$\n\n\\begin{align} \\label{eq:23}\n    & \\nonumber \\frac{u^{p}\\left(v^{q}+v^{r}\\right)}{v^{\\frac{p+q+r}{2}}} = \\underset{m\\geq n}{\\sum} \\frac{\\left(-1\\right)^{n+m}}{n! m!} \\left( F_{p,2}^{n,m}\\left(\\frac{q-r}{2}\\right)+F_{p,2}^{n,m}\\left(\\frac{r-q}{2}\\right) \\right) G^{\\left(2\\right)}_{m-n,2p+n+m}\\left(u,v\\right) \\\\\n    & \\nonumber = \\underset{m\\geq n}{\\sum} \\frac{\\left(-1\\right)^{n+m}}{n! m!} \\left( F_{p,2}^{n,m}\\left(\\left|\\frac{q-r}{2}\\right|\\right)+F_{p,2}^{n,m}\\left(-\\left|\\frac{q-r}{2}\\right|\\right) \\right) G^{\\left(2\\right)}_{m-n,2p+n+m}\\left(u,v\\right) \\\\\n    & = \\underset{m\\geq n}{\\sum} \\frac{\\left(-1\\right)^{n+m}}{n! m!}\\left(1+\\left(-1\\right)^{n+m}\\right)  F_{p,2}^{n,m}\\left(\\left|\\frac{q-r}{2}\\right|\\right)  G^{\\left(2\\right)}_{m-n,2p+n+m}\\left(u,v\\right).\n\\end{align}\n\nAlthough all these OPE coefficients are positive they are not sufficient to guarantee unitarity.\n\n\n\\subsection{Unitary generalized polynomials and the scalar gap}\n\nFor a 4-point function of identical real scalars we expect the identity to appear in the OPE, but according to $\\left(\\ref{eq:23}\\right)$ this is not the case when all parameters in $\\left(a,b,c\\right)$ are non-zero. Since we've only assumed identical scalars we don't know whether $\\phi$ is null or it is charged, in any case, we cannot call $\\mathcal{G}$ unitary. Still it is possible for any $\\left(a,b,c\\right)$ to generate an unitary 4-point function. Because of the positivity of $\\left(\\ref{eq:23}\\right)$ the sum $\\mathcal{G}_{\\mathbf{1}}+\\left(a,b,c\\right)$ will be unitary if $\\mathcal{G}_{\\mathbf{1}}$ is unitary. For us $\\mathcal{G}_{\\mathbf{1}}$ will be just another generalized polynomial. Bellow we plot the scalar gap of every possible unitary 4-point function we can guarantee exists from $\\left(\\ref{eq:23}\\right)$.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.5]{g2-.png}\n    \\caption{The blue curve shows an approximation to the bound given by the numerical bootstrap, the other curves and regions are the places we can find unitary solutions using only $\\left(\\ref{eq:23}\\right)$.}\n    \\label{fig:01}\n\\end{figure}\n\nThere are two cases that stand out in the plot, these correspond to the generalized free boson, $\\left(0,\\Delta_{\\phi},\\Delta_{\\phi}\\right)$, as the black line and the cosine operators in the free boson, $\\left(0,0,2\\Delta_{\\phi}\\right)$, as the red curve.\n\nUntil now all we've analyzed is equivalent to a basis with a single element $\\left\\{\\left(a,b,c\\right)\\right\\}$. For a general basis we cannot find a complete set of solutions like we did previously, however we can still focus in searching for interesting 4-point functions. In our case functions with large scalar gap.\n\nNon-trivial solutions, those containing negative basis coefficients, form the majority of the interesting 4-point functions, yet most basis cannot create them. This is the result of two properties of $\\left(\\ref{eq:23}\\right)$, first, all OPE coefficients are positive and second, the operators in the OPE have twist $\\tau=2p+2n$. The first means that whenever a basis coefficient becomes negative there are up to 3 potential series of only negative OPE coefficients. The second severely limits which kinds of terms could be used to counteract the first.\n\nChoosing a basis is, then, an extremely important first step when trying to analyze any property of a CFT. Once one is chosen we need to know which tools we have to say with certain confidence whether an expression is unitary or not. If the number of elements is small we can still use analytical methods to prove unitarity, but for larger basis we will usually rely in the asymptotic expansion of the $f_{p}^{n}\\left(t\\right)$ polynomials. To test this method we start by analyzing small scaling dimensions. Since we didn't need a large basis to get close to the boundary we used analytical methods to prove their existence.\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[scale=0.5]{g3-.png}\n    \\caption{Regions we could prove the existence of unitary solution analytically.}\n    \\label{fig:02}\n\\end{figure}\n\n\\section{Searching for large gaps} \\label{sec:5}\n\nDespite having exact expression for the OPE coefficients there still is a small, but important, hassle in $\\left(\\ref{eq:19}\\right)$, the function $f_{0}^{n}\\left(t\\right)$ is not well defined. For $n\\geq2$ the following redefinition is enough\n\n\\begin{align}\n    f_{p}^{n}\\left(t\\right) &= \\dfrac{\\left(\\Gamma\\left(p+n\\right)\\right)^{2}}{\\left(2p-1+n\\right)_{n}} {}_3 \\overset{\\sim}{F}_{2} \\left(\\left.\\begin{array}{c}\n    \\begin{array}{ccc}\n        -n, & 2p-1+n, & \\frac{p}{2}-t \\\\\n    \\end{array}{}\\\\ \\begin{array}{cc}\n        p, & p \\\\\n    \\end{array}{} \\end{array}{} \\right| 1\\right).\n\\end{align}\n\nBy analyzing $\\left(1-z\\right)^{t}=1-t~z+\\mathcal{O}\\left(z^{2}\\right)$ it becomes clear that $f_{0}^{0}\\left(t\\right)=1$ and $f_{0}^{1}\\left(t\\right)=t$. With every value of $f_{p}^{n}\\left(t\\right)$ defined it will be useful to think of $\\frac{f_{p}^{n}}{n!}\\equiv0$ for $n<0$. As was said, the operators in the OPE for a generalized polynomials have twist $\\tau=2p+2n$, this means that these series overlap only when the parameter differ by an integer. We will represent this overlapping series by $\\mathcal{E}_{\\left[p\\right]}$, $\\left[p\\right]=p-\\lfloor p \\rfloor$\n\n\\begin{align}\n    & \\mathcal{G} = \\underset{\\left[p\\right]}{\\sum} \\underset{m\\geq n}{\\sum} \\left(1+\\left(-1\\right)^{n+m}\\right)\\mathcal{E}_{\\left[p\\right]}^{n,m} G^{\\left(2\\right)}_{m-n,2\\left[p\\right]+n+m}, \\\\ \n    \\label{eq:28} & \\mathcal{E}_{\\left[p\\right]}^{n,m} = \\underset{i}{\\sum} 2^{m-n}E_{i}\\frac{ f_{\\left[p\\right]+\\lfloor p_{i}\\rfloor}^{n-\\lfloor p_{i}\\rfloor}\\left(t_{i}\\right)}{\\left(n-\\lfloor p_{i}\\rfloor\\right)!}  \\frac{f_{\\left[p\\right]+\\lfloor p_{i}\\rfloor}^{m-\\lfloor p_{i}\\rfloor}\\left(t_{i}\\right) }{\\left(m-\\lfloor p_{i}\\rfloor\\right)!}.\n\\end{align}\n\nNo matter which property we try to explore using this method the code should always search for a set of coefficients $E_{i}$ that makes every $\\mathcal{E}_{\\left[p\\right]}^{n,m}$ non-negative. To deal with the infinite number of inequalities we can use the asymptotic form of the $f$ polynomials. By making sure that the asymptotic expression is positive and that $\\mathcal{E}_{\\left[p\\right]}^{n,m}\\geq0$ for a sufficient large $n$ and $m$ we would know with confidence whether the expression is unitary.\n\nFor our search for large scalar gaps this can be coded by introducing two types of linear constrains, the positivity plus normalization, $\\mathcal{E}_{0}^{0,0}=1$ and $\\mathcal{E}_{\\left[p\\right]}^{n,n+J}\\geq0$ for $n\\leq n_{min}$ and $J\\leq J_{min}$, and the gap condition, $\\mathcal{E}_{\\left[p\\right]}^{n,n}=0$ if $0<2\\left[p\\right]+2n<\\Delta_{1}$. Any set of $E_{i}$ that satisfy these conditions could be further tested for $\\mathcal{E}_{\\left[p\\right]}^{n,n+J}\\geq0$ when $n\\leq n_{max}$ and $J\\leq J_{max}$.\n\nFor the green points in the plots bellow we've used the basis\n $\\mathcal{B}=\\left\\{\\left(a,b,c\\right)\\left|a,b,c\\in \\mathbb{Z}_{\\geq0} \\right\\}\\right.$. In this special basis the value of $n_{max}$ and $J_{max}$ can be calculated using some analytical results from the continuous Hahn polynomials. This way we've proved that there are no negative OPE coefficients. This is not the biggest possible gap, as is shown in the second plot bellow when adding half-integers to the basis the gap increases.\n\n\\begin{figure}[h]\n\n\\begin{minipage}[b]{0.48\\linewidth}\n\n    \\centering\n    \\includegraphics[scale=0.5]{plot1----.png}\n    \\caption{The green dots are new unitary solutions, compared to the last plot, the light blue are possible unitary, we tested until twist and spin 200, the black are points we cannot reach with our current basis, the black line is the generalized free boson and the dashed line is $\\frac{8\\Delta}{3}$.}\n    \\label{fig:03}\n\n\\end{minipage} \n\\hfill\n\\begin{minipage}[b]{0.48\\linewidth}\n\n    \\centering\n    \\includegraphics[scale=0.5]{plot2--.png}\n    \\caption{By introducing half-integers parameters to the basis we were able to reach the red points, the black ones are points we cannot reach with this new basis.}\n    \\label{fig:04}\n\n\\end{minipage} \n\n\\end{figure}\n\nIn AdS\/CFT the large radius limit is a way to recover the flat space-time physics, \\cite{Paulos_2017}. In this limit theories like the generalized free boson are interpreted as free particles with their masses proportional to the scaling dimension $\\Delta_{\\phi}$. The scalar gap of $2\\Delta_{\\phi}$ can be interpreted the 2-particle bound state have twice the particle mass. Figure \\ref{fig:03} shows that for large $\\Delta_{\\phi}$ there are theories with scalar gap close to $\\frac{8\\Delta_{\\phi}}{3}$. Any theories with the same type of interpretation in this region would have $\\frac{8m}{3}$ as the bound state mass of particles of mass $m$. This behavior is not restricted to integers scaling dimension, with a less robust program it was possible to find possible solutions with non-integer scaling dimension connecting these points.\n\n\\section{Conclusion}\n\nIn this work we presented a method to construct simple unitary 4-point functions. Unlike the dual formalism from \\cite{Rattazzi_2008} this is completely analytical, as such we are not able to create every possible 4-point functions. We've analyzed the generalized polynomials trying to maximize the scalar gap.\n\nWe used the properties of the hypergeometric polynomials $f_{p}^{n}\\left(t\\right)$ to study their OPE decomposition and to create a computational method to search for unitary solutions. The main results are given by figure \\ref{fig:02} and \\ref{fig:03}. Figure \\ref{fig:02} shows that despite its simple form the generalized polynomials seems to cover a significant portion of the allowed physical space. Figure \\ref{fig:03} shows that even for relative large scaling dimensions it still possible to find unitary 4-point functions with significantly large scalar gap than the generalized free boson. In the large radius limit of AdS\/CFT this can be related to the mass of the 2-particle bound state, that is, their bound state would have a mass significantly larger than twice the particle mass.\n\nThe results from the program, which can be found at \\href{https:\/\/github.com\/rgnato\/Generalized_polynomials}{GitHub}, are valid only for $D=2$, however in \\ref{ap:B} we used the results of \\cite{Hogervorst_2016,Kaviraj_19} to find recursion relations for the $F_{p,D}^{n,m}\\left(t\\right)$ polynomials in $D$. Most of the code could be re-purposed to analyze higher dimensional CFTs or other observables, but we may need another way to prove, or at least be confident about unitarity.\n\n\\section*{Acknowledgements}\n\nI would like to thank my advisor Pedro Vieira for helping me during this project, for the discussions and for the ideas exchanged. I like to thank both Pedro, Alexandre Homrich and Matheus Fabri for helping check this work. I would also like to thanks the CNPq for the partial financial support with the grant No. 132286\/2018-1 during my stay in the IFT-UNESP\/ICTP-SAIFR. And during my stay at PI. The work was supported in part by the Natural Sciences and Engineering Research Council of Canada. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper we consider the problem of finding an invariant Riemannian metric for a mapping preserving a given volume form on a Riemannian manifold. The setup of the problem is the following.\n\nLet $M$ be a smooth, closed and oriented finite dimensional manifold. Then the set of all Riemannian metrics $\\mathcal{M}$ can be considered as an infinite dimensional manifold. Its tangent vectors can be given an $L^2$ inner product~\\cite{Ebin, Clarke} and thus computing the curvature of $\\mathcal{M}$ makes sense. The sectional curvature of $\\mathcal{M}$ is nonpositive, but $\\mathcal{M}$ is not geodesically complete~\\cite{Freed, Clarke}.\n\nInstead of $\\mathcal{M}$ we consider a submanifold $\\mathcal{M}_\\mu$ of $\\mathcal{M}$ consisting of Riemannian metrics which all have the same given volume form $\\mu$. The submanifold $\\mathcal{M}_\\mu$ is an infinite dimensional globally symmetric space. The exponential mapping is a diffeomorphism, it has nonpositive curvature and any two points can be joined by a unique geodesic. For details of these statements, see~\\cite{Ebin, Freed, Clarke}.\n\nDiffeomorphisms preserving a given volume form are called volumorphisms. They act naturally by pullback on $\\mathcal{M}_\\mu$ and the action is isometric. Let $g$ be some Riemannian metric on $M$ whose induced volume form is $\\mu$ and $\\phi$ a volumorphism of $M$. Tangent spheres defined by $g$ are mapped to tangent ellipsoids by the pushforward of $\\phi$. The sphere and its image under the pushforward of $\\phi$, the ellipsoid, have the same volume due to the fact that $\\phi$ is a volumorphism, but we have no control on how much the sphere is distorted.\n\nWe ask the following question. If the tangent spheres are boundedly distorted under the iterations of the pushforward of $\\phi$, is there another Riemannian metric, say $h$, such that tangent spheres defined by $h$ are mapped to spheres? Since in this case the mapping $\\phi$ considered as a mapping $(M,h)\\to (M,h)$ would be an isometry, the Riemannian metric $h$ is in this sense an optimal Riemannian metric for $\\phi$. A similar idea has been used in the study of quasiconformal mappings in~\\cite{Tukia, Iwaniec}.\n\nWe formalise the idea explained above and ask: ``If the action of a volumorphism on $\\mathcal{M}_\\mu$ has a bounded orbit, is there a fixed point of this action?''. We consider the space of Riemannian metrics of Sobolev class $H^s$ and assume that the mapping $\\phi$ is of Sobolev class $H^{s+1}$, $s>n\/2$. We show that the answer to the question above is affirmative if we allow the possibility that the fixed point is of lower regularity than $H^s$. \n\nA fixed point of the action of $\\phi$ on $\\mathcal{M}_\\mu^s$, if it exists, will belong to a metric space $(X,\\d)$ of $\\mu$-a.e. positive definite symmetric $(0,2)$-tensor fields with volume form agreeing with $\\mu$ a.e. The elements of $X$ are also assumed to satisfy a certain natural integrability condition. As a metric space, $(X,\\d)$ is a complete global Alexandrov nonpositive curvature space containing $(\\mathcal{M}_\\mu^s,d)$ as an isometrically embedded subset. Here $d$ is the distance metric induced by the weak Riemannian metric on $\\mathcal{M}_\\mu^s$. The space $(X,\\d)$ is defined in Theorem~\\ref{X}, and Theorem~\\ref{vol_fp} is the fixed point theorem for the action of volumorphisms on $\\mathcal{M}_\\mu^s$. We expect $(X,\\d)$ to be the metric completion of $(\\mathcal{M}_\\mu^s,d)$.\n\nTo find a fixed point for the action of a volumorphism on $\\mathcal{M}_\\mu$, we generalize a mean ergodic theorem and a fixed point theorem to suit our nonlinear setting. These are Theorems~\\ref{main_thm} and~\\ref{fp_thm}. Mean ergodic theorems consider the convergence of averages of the iterates of the points under the action. In a nonlinear setting there is no obvious notion of average, but on nonpositively curved spaces, such as $\\mathcal{M}_\\mu^s$, there is a natural generalization of averages~\\cite{Jost, Jostb, Karcher}. Averages are also called means or centers of mass in the literature.\n\nThe class of metric spaces where we formulate our mean ergodic theorem and fixed point theorem is that of complete global Alexandrov nonpositive curvature spaces, which are also known as (complete) $\\mbox{CAT}(0)$ spaces. Alexandrov nonpositive curvature spaces have been studied in a setting similar to ours in~\\cite{Jost, Jostb}. In both of the above-mentioned theorems, we assume that the mapping considered is nonexpansive. The mean ergodic theorem additionally assumes that the means of the iterates of the mapping satisfy a certain convexity property.\n\n\\section{Mean ergodic theorem and fixed point theorem in a global Alexandrov nonpositive curvature space}\nIn this section, we formulate and prove a mean ergodic theorem and a fixed point theorem in a nonlinear setting. First we give a brief review of the class of global Alexandrov nonpositive curvature (NPC) spaces we are working on. For details and examples of Alexandrov NPC spaces we refer to~\\cite{Jost,Jostb}.\n\nA metric space $(\\mathcal{N},d)$ is said to be a geodesic length space if for any two points $p,q\\in N$ there exists a rectifiable curve $\\gamma:[0,1]\\to \\mathcal{N}$ with $\\gamma(0)=p$ and $\\gamma(1)=q$ and length equal to $d(p,q)$. Such a curve is called a \\emph{geodesic}.\n\n\\begin{definition}\n A geodesic length space $(\\mathcal{N},d)$ is said to be a global Alexandrov nonpositive curvature (NPC) space if for any three points $p,q,r$ of $\\mathcal{N}$ and any geodesic $\\gamma:[0,1]\\to \\mathcal{N}$ with $\\gamma(0)=p$ and $\\gamma(1)=r$, we have for $0\\leq t\\leq 1$\n\\begin{equation*}\n d^2(q,\\gamma(t))\\leq (1-t) d^2(q,\\gamma(0))+td^2(q,\\gamma(1))-t(1-t)l(\\gamma)^2.\n\\end{equation*}\nHere $l(\\gamma)$ is the length of the geodesic $\\gamma$. \n\\end{definition}\nThe inequality above is called the Alexandrov NPC inequality. We remark that global Alexandrov NPC spaces are simply connected and that for any two given points there is a unique geodesic connecting the points. See Lemma 2.2.1 of~\\cite{Jostb} and the discussion that follows the lemma.\n\nWe will also need the concept of convex sets in geodesic length spaces. A subset of a geodesic length space is \\emph{convex} if any two points of the subset can be joined by a geodesic whose image is contained in that subset. The convex hull $co(S)$ of a subset $S$ of a geodesic length space is the smallest convex subset of $\\mathcal{N}$ containing $S$. \n\nThe convex hull of an arbitrary subset of a geodesic length space need not exist, but global Alexandrov NPC spaces have the property that the convex hull of any set exists~\\cite[Lemma 3.3.1]{Jostb}. By that same lemma, we can express the convex hull of a subset $S\\subset \\mathcal{N}$ as follows. Set $C_0=S$ and define $C_k$ to be the union of all geodesic arcs between points of $C_{k-1}$. We have\n\\begin{equation}\\label{cohull}\n co(S)=\\bigcup_{k=0}^\\infty C_k.\n\\end{equation}\n\nWe record for future reference that the diameter of a set $S$ and its convex hull $co(S)$ are the same. It holds trivially that $\\mbox{diam}(co(S))\\geq \\mbox{diam}(S)$. Let $\\epsilon>0$ and choose $p,q\\in co(S)$ such that $d(p,q)+\\epsilon=\\mbox{diam}(co(S))$. The Alexandrov NPC inequality implies that\n\\begin{equation}\n d^2(q,\\gamma(t))\\leq \\max\\{d^2(q,\\gamma(0)),d^2(q,\\gamma(1))\\}\n\\end{equation}\nfor any $q\\in \\mathcal{N}$, $\\gamma:[0,1]\\to\\mathcal{N}$ a geodesic and $t\\in [0,1]$. By this inequality, it can be seen from the definition of the sets $C_k$ that \n\\begin{equation*}\n d^2(p,q)\\leq \\max_{i\\in I}\\{d^2(p_i,q_i)\\},\n\\end{equation*}\nwhere $I$ is a finite index set and $p_i,q_i\\in C_0=S$. It follows that $d(p,q)$ is bounded from above by $\\mbox{diam}(S)$. Thus we also have $\\mbox{diam}(S)\\geq \\mbox{diam}(co(S))$. \n\nMean ergodic theorems, in general, are convergence theorems for means of iterates of points under a given action to a limit invariant under the action. We need the concept of mean to proceed. The mean in a vector space is just the arithmetic average of the vectors, but the concept of mean generalizes to many other spaces as follows.\n\n\\begin{definition}[Mean]\\label{mean_map}\n Let $S=\\{S_0,S_1,\\ldots, S_{n-1}\\}$ be a finite subset of a metric space $\\mathcal{N}$ with a metric $d$. The \\emph{mean function} of $S$ is the function $F_S$ on $\\mathcal{N}$ given by\n\\begin{equation*}\n F_S(p)=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(p,S_i).\n\\end{equation*}\nIf there exists a unique minimizer of $F_S$, then we denote \n\\begin{equation*}\nm(S)=\\mbox{ the unique minimizer of } F_S\n\\end{equation*}\nand call $m(S)$ the \\emph{mean} of $S$.\n\\end{definition}\nIn a complete global Alexandrov NPC space the unique minimizer for $F_S$ exists for all finite subsets $S$ of $\\mathcal{N}$ and belongs to the closure $\\overline{co}(S)$ of $co(S)$; see Theorem 3.2.1 and Lemma 3.3.4 of~\\cite{Jostb}. Thus, in this case, we can also consider the mean as a mapping $S\\to \\overline{co}(S)$.\n\nWe are mainly interested in the means of iterations of points by a given mapping $T$. In this case, we denote the mean function of $n$ iterates of $p\\in \\mathcal{N}$ by\n\\begin{equation}\\label{F_n}\n F_n(r,p)=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(r,T^ip)\n\\end{equation}\nand the mean of $n$ iterates of $p$ is denoted by\n\\begin{equation*}\n m_n(p)=\\mbox{ the unique minimizer of } F_n(\\cdot,p).\n\\end{equation*}\n\n\nAn important class of mappings on $(\\mathcal{N},d)$ we are going to consider is that of nonexpansive mappings. A mapping $T$ from a metric space $(\\mathcal{N},d)$ to itself is called \\emph{nonexpansive} if\n\\begin{equation*}\n d(Tp,Tq)\\leq d(p,q)\n\\end{equation*}\nfor all $p,q\\in \\mathcal{N}$. Thus, the class of nonexpansive mappings contains not only contractions, but also isometries.\n\nConvexity plays a crucial role in the formulation and proof of our main theorem. A function $F:\\mathcal{N}\\to \\mathbb{R}$ is said to be convex if for every geodesic $\\gamma:[0,1]\\to \\mathcal{N}$ the function $F\\circ \\gamma:[0,1]\\to\\mathbb{R}$ is convex. We say that a mapping $T:\\mathcal{N}\\to \\mathcal{N}$ is \\emph{distance convex} if for all $n \\in \\mathbb{N}$ and $q\\in \\mathcal{N}$ the mapping \n\\begin{equation}\\label{T_convex}\n d^2(m_n(\\cdot),q):\\mathcal{N}\\to \\mathbb{R}^+\n\\end{equation}\nis convex. In a normed vector space, any linear mapping is distance convex, yet this definition seems to be new. However, the proof of our main theorem naturally employs the definition, which suggests that the class of distance convex mappings might be of further interest.\n\nWe are now ready to state our main theorem.\n\\begin{thm}[Mean Ergodic Theorem]\\label{main_thm} Let $(\\mathcal{N},d)$ be a complete global Alexandrov NPC space and $T:\\mathcal{N}\\to\\mathcal{N}$ a nonexpansive distance convex mapping. Then, for any $p\\in \\mathcal{N}$ whose orbit is bounded, and any $q\\in \\mathcal{N}$, the following are equivalent:\n\\begin{align*}\n  (i) \\qquad &Tq=q \\mbox{ and } q\\in \\overline{co}\\{p,Tp,T^2p,\\ldots\\}, \\\\\n (ii) \\qquad  &q=\\lim_{n}m_n(p), \\\\\n (iii) \\qquad &q=\\mbox{w-}\\lim_n m_n(p), \\\\\n  (iv) \\qquad &q \\mbox{ is a weak cluster point of the sequence } (m_n(p)).\n\\end{align*}\n\\end{thm}\n\nHere $\\mbox{w-}\\lim_n m_n(p)$ refers to \\emph{weak convergence} defined in terms of projections as follows. For any $p\\in \\mathcal{N}$ and any geodesic arc $\\gamma$ in $\\mathcal{N}$, there exists a unique point $\\pi(p,\\gamma)$ on $\\gamma$ that is closest to $p$. We call $\\pi(p,\\gamma)$ the \\emph{projection} of $p$ onto $\\gamma$. A point $q\\in \\mathcal{N}$ is the weak limit of a sequence $(p_n)\\subset \\mathcal{N}$ if for every geodesic arc through $q$ the sequence $\\pi(p_n,\\gamma)$ converges to $q$. Similarly, a point $q\\in \\mathcal{N}$ is a weak cluster point of a sequence $(p_n)\\subset \\mathcal{N}$ if, for every neighborhood $U$ of $q$, there are infinitely many natural numbers $n\\in \\mathbb{N}$ such that $\\pi(p_n,\\gamma)\\in U$ for every geodesic arc $\\gamma$ through $q$. See Definitions 2.5 and 2.7 of~\\cite{Jost} for details on weak convergence. \n\nWithout the assumption that the mapping is distance convex, we still get an interesting weaker version of the theorem.\n\n\\begin{thm}[Fixed point theorem]\\label{fp_thm} \nLet $(\\mathcal{N},d)$ be a complete global Alexandrov NPC space and $T:\\mathcal{N}\\to\\mathcal{N}$ a nonexpansive mapping. Then, for any $p\\in \\mathcal{N}$ whose orbit is bounded there exists a fixed point $q$ of $T$ in the subset $\\overline{co}\\{p,Tp,T^2p,\\ldots\\}$ of $\\mathcal{N}$. \n\\end{thm}\n\nThe proof of our main theorem is quite lengthy due to the nonstandard framework we are working in. Once the mean ergodic theorem is proved, the fixed point theorem follows easily by using the same techniques. The outline of the proof of Theorem~\\ref{main_thm} follows Krengel's proof of a mean ergodic theorem for Ces\\'aro bounded operators in Banach spaces~\\cite[Theorem 1.1. p.72]{Krengel}. \n\nBefore the proofs of the theorems, we give several auxiliary results to clarify the proofs. The first two statements are general convexity results in global Alexandrov NPC spaces. The statements that follow consider the behavior of minimizers of sequences of convex functions. Then, the results achieved so far are applied to study the behavior of means of iterates of distance convex nonexpansive mappings. The first of the last two auxiliary results shows that projections to convex sets are continuous. The last auxiliary result gives a sufficient condition for the existence of a fixed point of a nonexpansive mapping. \n\nWe begin with a definition.\n\n\\begin{definition}[Uniform convexity]\nA nonnegative lower semicontinuous function $\\psi:\\mathcal{N}\\to\\mathbb{R}^+$ on a geodesic length space $(\\mathcal{N},d)$ is said to be \\emph{uniformly convex} if the following quantitative strict convexity condition holds:\n\nFor any geodesic $\\gamma:[0,1]\\to \\mathcal{N}$ and $\\epsilon>0$ there exists $\\d>0$ such that if\n\\begin{equation*}\n \\psi(\\gamma(\\partial))\\geq \\partial\\psi(\\gamma(0))+\\partial\\psi(\\gamma(1))-\\d\n\\end{equation*}\nthen\n\\begin{equation*}\n d(\\gamma(0),\\gamma(1))<\\epsilon.\n\\end{equation*}\n\nA family $\\mathcal{F}$ of nonnegative lower semicontinuous functions $\\mathcal{N}\\to\\mathbb{R}^+$ is said to be \\emph{equiconvex} if there is a positive number $\\d(\\epsilon)$ such that the above holds for all $F\\in\\mathcal{F}$ for any $\\d$ smaller than $\\d(\\epsilon)$. In this case, we call $\\d(\\epsilon)$ the \\emph{modulus of convexity} of $\\mathcal{F}$.\n\\end{definition}\n \nWe have the following.\n\\begin{lemma}\\label{d2_uni_c}\n Let $(\\mathcal{N},d)$ be a global Alexandrov NPC space. The family of functions $\\mathcal{F}=\\{d^2(\\cdot,p):\\mathcal{N}\\to \\mathbb{R}^+ : \\ p\\in \\mathcal{N}\\}$ is equiconvex with $\\d(\\epsilon)=\\epsilon^2\/4$.\n\\end{lemma}\n\\begin{proof}\nLet $p\\in\\mathcal{N}$, $\\epsilon>0$ and $\\gamma:[0,1]\\to \\mathcal{N}$ be a geodesic. Let $\\d<\\epsilon^2\/4$ and assume that the inequality\n\\begin{equation}\\label{str_con}\n d^2(\\gamma(\\partial),p)\\geq \\partial d^2(\\gamma(0),p)+\\partial d^2(\\gamma(1),p)-\\d\n\\end{equation}\nholds. In a global Alexandrov NPC the distance between two points is given by the length of the unique geodesic joining the points. Thus the NPC inequality reads \n\\begin{equation*}\n \\frac{1}{4}d^2(\\gamma(0),\\gamma(1))\\leq -d^2(\\gamma(\\partial),p)+ \\partial d^2(\\gamma(0),p)+\\partial d^2(\\gamma(1),p).\n\\end{equation*}\nTogether with~\\eqref{str_con} we have\n\\begin{equation*}\n \\frac{1}{4}d^2(\\gamma(0),\\gamma(1))\\leq \\d<\\epsilon^2\/4.\n\\end{equation*}\n\\end{proof}\n\n\\begin{cor}\\label{set_mean_str_conv}\nLet $(\\mathcal{N},d)$ be a global Alexandrov NPC space. The family $\\mathcal{F}=\\{F_S:  S \\mbox{ a finite subset of } \\mathcal{N} \\}$ of mean functions $F_S$ is equiconvex with $\\d(\\epsilon)=\\epsilon^2\/4$.\n\\end{cor}\n\\begin{proof}\n Let $S=\\{S_0,\\ldots,S_{n-1}\\}$ be a finite set, let $\\epsilon>0$ and $\\gamma:[0,1]\\to \\mathcal{N}$ be a geodesic. Let $\\d<\\epsilon^2\/4$ and assume that the inequality \n\\begin{equation}\\label{F_str_con}\nF_S(\\gamma(\\partial))\\geq \\partial F_S(\\gamma(0))+\\partial F_S(\\gamma(1))-\\d\n\\end{equation}\nholds. If the inequality\n\\begin{equation*}\n d^2(\\gamma(\\partial),S_i)\\geq \\partial d^2(\\gamma(0),S_i)+\\partial d^2(\\gamma(1),S_i)-\\d\n\\end{equation*}\nis false for all $i=0,\\ldots, n-1$, then the inequality~\\eqref{F_str_con} is also false. Thus we have for some $0\\leq i_0\\leq n-1$,\n\\begin{equation*}\n d^2(\\gamma(\\partial),S_{i_0})\\geq \\partial d^2(\\gamma(0),S_{i_0})+\\partial d^2(\\gamma(1),S_{i_0})-\\d.\n\\end{equation*}\nHence we have\n\\begin{equation*}\n d(\\gamma(0),\\gamma(1))<\\epsilon\n\\end{equation*}\nby the previous lemma.\n\\end{proof}\n\nThe proof of our main theorem considers the asymptotic behavior of the mean functions of the iterates of points of a given mapping. In this case, we wish to analyze the asymptotic behavior of the minimizers of these mean functions.  A criterion for the asymptotic minimizers to be close is given by the lemma below.\n\n\n\\begin{lemma}\\label{uni_strc_appr}\nLet $(\\mathcal{N},d)$ be a geodesic length space. Let $(F_n)$ and $(G_n)$ be two sequences of functions $\\mathcal{N}\\to \\mathbb{R}^+$, for which there exist unique minimizers, $(f_n)$ and $(g_n)$ respectively. Assume also that $(f_n)$ and $(g_n)$ belong to some subset $S$ of $\\mathcal{N}$ and that $(F_n)$ is equiconvex with modulus of convexity of $\\d(\\epsilon)$.\n\nLet $\\epsilon>0$ and assume that there exists an $N\\in\\mathbb{N}$ such that the inequality\n\\begin{equation*}\n \\sup_{p\\in S}|F_n(p)-G_n(p)|<\\d(\\epsilon)\n\\end{equation*}\nholds for all $n\\geq N$. Then\n\\begin{equation*}\n d(f_n,g_n)<\\epsilon\n\\end{equation*}\nfor all $n\\geq N$.\n\\end{lemma}\n\n\\begin{proof}\nDenote $\\widetilde{F}_n=F_n-F_n(f_n)$ and $\\widetilde{G}_n=G_n-G_n(g_n)$. Now, the unique minima of $\\widetilde{F}_n$ and $\\widetilde{G}_n$ are zero and the sequence of functions $(\\widetilde{F}_n)$ is equiconvex with modulus of convexity $\\d(\\epsilon)$. \n\nLet $\\epsilon>0$. By assumption, there is an $N=N(\\epsilon)\\in\\mathbb{N}$ such that for all $n\\geq N$\n\\begin{equation*}\n \\sup_{p\\in S}|F_n(p)-G_n(p)|<  \\d(\\epsilon).\n\\end{equation*}\nIf $G_n(g_n)-F_n(f_n)\\geq 0$, we have\n\\begin{align*}\n |G_n(g_n)-F_n(f_n)|&=G_n(g_n)-F_n(f_n)\\leq G_n(f_n)-F_n(f_n) \\\\\n  &\\leq|G_n(f_n)-F_n(f_n)| < \\d(\\epsilon).\n\\end{align*}\nThis implies that\n\\begin{equation*}\n \\sup_{p\\in S}|\\widetilde{F}_n(p)-\\widetilde{G}_n(p)|\\leq \\sup_{p\\in S}|F_n(p)-G_n(p)|+ |G_n(g_n)-F_n(f_n)| < 2\\d(\\epsilon).\n\\end{equation*}\nIf $G_n(g_n)-F_n(f_n)\\leq 0$, an analogous proof shows that the same conclusion still holds.\n\nSince $\\widetilde{G}_n(g_n)=0$, we get the following inequality\n\\begin{equation*}\n 2\\d(\\epsilon)> \\sup_{p\\in S}|\\widetilde{F}_n(p)-\\widetilde{G}_n(p)|_S \\geq  |\\widetilde{F}_n(g_n)-\\widetilde{G}_n(g_n)|=\\widetilde{F}_n(g_n).\n\\end{equation*}\nLet $\\gamma_n: [0,1]\\to \\mathcal{N}$ be a geodesic with $\\gamma_n(0)=f_n$ and $\\gamma_n(1)= g_n$. Then, at the midpoint of the geodesic, we have the following estimate \n\\begin{equation*}\n \\widetilde{F}_n(\\gamma_n(\\partial))+\\d(\\epsilon)\\geq \\d(\\epsilon) > \\partial(\\widetilde{F}_n(\\gamma_n(0))+\\widetilde{F}_n(\\gamma_n(1))).\n\\end{equation*}\nThe equiconvexity of the family $(\\widetilde{F}_n)$ implies\n\\begin{equation*}\nd(f_n,g_n) <\\epsilon\n\\end{equation*}\nfor all $n\\geq N$.\n\\end{proof}\n\n\\begin{lemma}\\label{Tdc_impl_close}\n Let $T:\\mathcal{N}\\to \\mathcal{N} $ be a distance convex nonexpansive mapping on an Alexandrov NPC space $(\\mathcal{N},d)$. Assume that $T$ has a bounded orbit $\\{p,Tp,T^2p,\\ldots\\}$ and that $s\\in co\\{p,Tp,T^2p,\\ldots\\}$. Then there is an $N\\in \\mathbb{N}$ such that the means of the first $n$ iterates of $p$ and $s$ satisfy \n\\begin{equation*}\n d(m_n(p),m_n(s))<\\epsilon\n\\end{equation*}\nfor all $n\\geq N$.\n\\end{lemma}\n\\begin{proof}\nDenote the orbit of $p$ by $S_p$. Assume that $s\\in co(S_p)$ and let $F_n:\\mathcal{N}\\times\\mathcal{N} \\to \\mathbb{R}$ be the mean function\n\\begin{equation*}\n F_n(r,q)=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(r,T^iq)\n\\end{equation*}\nof the first $n$ iterates of the second argument. The family $\\{F_n(\\cdot,s): n\\in \\mathcal{N}\\}$ is equiconvex with modulus of convexity $\\d(\\epsilon)=\\epsilon^2\/4$ by Corollary~\\ref{set_mean_str_conv}. \n\nSince $s$ belongs to the bounded set $co(S_p)$ and $T$ is nonexpansive, the orbit of $s$ is bounded: For $i\\in \\mathbb{N}$, we have \n\\begin{align}\\label{bded_orbits}\nd(s,T^is)&\\leq d(s,p)+d(p,T^ip)+ d(T^ip,T^is) \\nonumber \\\\\n  &\\leq 2d(p,s)+d(p,T^ip)\\leq 3~\\mbox{diam}(co(S_p)).\n\\end{align}\nThus the unique minimizers of the functions $\\{F_n(\\cdot,s)\\}$ belong to the bounded set $\\overline{co}(S_s) \\subset \\mathcal{N}$ by the remarks following Definition~\\ref{mean_map}.\n\nRecall from Eq.~\\ref{cohull} that the convex hull of $S_p$ can be expressed as $\\cup_{k=0}^{\\infty}C_k$. We prove the claim of the lemma by induction on the index $k$. Assume $s\\in C_0=S_p$. Now $s=T^{i_0}p$ for some $i_0$ and we have the following estimate (for $n> i_0$):\n\\begin{align*}\n F_n(r,s)&=F_n(r,T^{i_0}p)=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(r,T^iT^{i_0}p)=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(r,T^ip) \\\\\n &-\\frac{1}{n}\\sum_{i=0}^{i_0-1}d^2(r,T^ip)+\\frac{1}{n}\\sum_{i=n}^{i_0+n-1}d^2(r,T^ip).\n\\end{align*}\nBy the remarks above, $\\overline{co}(S_s)$ is bounded and $\\overline{co}(S_p)$ is bounded by assumption. Thus we have that\n\\begin{equation*}\n \\sup_{r\\in \\overline{co}(S_s)\\cup\\overline{co}(S_p)}|F_n(r,s)-F_n(r,p)|< \\epsilon^2\/4\n\\end{equation*}\nfor all $n\\geq N$ for some sufficiently large $N=N(s)\\in \\mathbb{N}$. By Lemma~\\ref{uni_strc_appr}, there holds\n\\begin{equation*}\n d(m_n(s),m_n(p))<\\epsilon\n\\end{equation*}\nfor all $n\\geq N(s)$. We have shown that the claim holds for $k=0$ for any $s\\in C_0$.\n\nAssume then that the claim holds for some $k>0$ and that $s\\in C_{k+1}$. By the definition of $C_{k+1}$, the point $s$ is of the form $s=\\gamma(t_0)$, where $\\gamma:[0,1]\\to \\mathcal{N}$ is a geodesic with $\\gamma(0), \\gamma(1)\\in C_k$ and $t_0\\in [0,1]$. The induction assumption for $\\gamma(0)$ and $\\gamma(1)$ together with the assumption that $T$ is distance convex yields\n\\begin{align*}\n d^2(&m_n(s),m_n(p))\\leq t_0 d^2(m_n(\\gamma(0)),m_n(p)) \\\\\n  &+(1-t_0) d^2(m_n(\\gamma(1)),m_n(p))<\\epsilon^2\n\\end{align*}\nfor all $n\\geq \\max\\{N(\\gamma(0)),N(\\gamma(1))\\}$, where $N(\\gamma(0))$ and $N(\\gamma(0))$ are defined by the induction assumption. This completes the induction step.\n\\end{proof}\n\nProjections to closed convex sets in a complete global Alexandrov NPC space exist by Lemma 2.5. of~\\cite{Jost}. The continuity of the projections is given by the following. \n\\begin{lemma}\n Let $(\\mathcal{N},d)$ be a complete global Alexandrov NPC space and $C\\subset \\mathcal{N}$ a closed convex set. Then the projection to $C$ is a continuous mapping.\n\\end{lemma}\n\\begin{proof}\nLet $\\epsilon>0$, and denote by $\\pi$ the projection to $C$. Let $p,q\\in \\mathcal{N}$ and let $\\gamma$ be a geodesic with $\\gamma(0)=\\pi(p)$ and $\\gamma(1)=\\pi(q)$. Since $C$ is convex, we have $\\gamma(\\partial)\\in C$. Thus, by the definition of $\\pi$ there holds\n\\begin{equation*}\n d(\\pi(p),p)\\leq d(\\gamma(\\partial),p) \\mbox{ and } d(\\pi(q),q)\\leq d(\\gamma(\\partial),q).\n\\end{equation*}\nWe also have, by the convexity of $d(\\cdot,p)$,\n\\begin{equation*}\n d(\\gamma(\\partial),p)\\leq \\max\\{d(\\pi(p),p),d(\\pi(q),p)\\}= d(\\pi(q),p),\n\\end{equation*}\nsince $d(\\pi(p),p)\\leq d(\\pi(q),p)$. The above inequalities together with the triangle inequality give\n\\begin{align*}\n d(\\gamma(\\partial),p)&\\leq d(\\pi(q),p)\\leq d(\\pi(q),q)+d(q,p) \\\\ \n  &\\leq d(\\gamma(\\partial),q)+d(q,p)\\leq d(\\gamma(\\partial),p)+2d(q,p).\n\\end{align*}\nThus, for any $p$ close enough to $q$, we have $d^2(\\pi(q),p)\\leq d^2(\\gamma(\\partial),p)+\\epsilon^2\/4$ and therefore\n\\begin{equation*}\n \\partial d^2(\\pi(p),p)+ \\partial d^2(\\pi(q),p) \\leq d^2(\\gamma(\\partial),p)+\\epsilon^2\/4.\n\\end{equation*}\nThe NPC inequality gives\n\\begin{equation*}\n d(\\pi(p),\\pi(q))\\leq \\epsilon\n\\end{equation*}\nfor any $p$ close enough to $q$. Thus $\\pi$ is continuous.\n\\end{proof}\n\nThe last lemma before the proofs of the main theorems is the following one giving a sufficient condition for a nonexpansive mapping to have a fixed point.\n\\begin{lemma}\\label{cond_for_fpoint}\n Let $(\\mathcal{N},d)$ be a complete global Alexandrov NPC space and let $T:\\mathcal{N}\\to\\mathcal{N}$ be nonexpansive. If a sequence $p_n$ converges weakly to $q$ and $d(p_n,Tp_n)\\to 0$, $n\\to \\infty$, then $Tq=q$.\n\\end{lemma}\n\n\\begin{figure}\n \\begin{center}\n  \\includegraphics[scale=0.6]{.\/fig1.pdf}\n\\caption{\\label{jotai} The dashed line is a geodesic arc connecting $p_n$ and $c_n$. The geodesic arc connecting $Tp_n$ and $T(\\pi(p_n))$ has length at most $d(p_n,\\pi(p_n))$ and it will approach the dashed line as $n$ tends to infinity.}\n \\end{center}\n\\end{figure}\n\\begin{proof}\nThe claim follows from geometric considerations illustrated in Figure~\\ref{jotai}. Let $C$ be the maximal geodesic containing $q$ and $Tq$ and $\\pi:\\mathcal{N}\\to C$ the projection to this arc. Let us denote the point $\\pi (T(\\pi(p_n)))\\in C$ by $c_n$ and let $\\triangle_n$ be the geodesic triangle, whose vertices are $p_n, \\pi(p_n)$ and $c_n$. We say that the angle between the geodesic arcs $\\overrightarrow{\\pi(p_n)p_n}$ and $\\overrightarrow{\\pi(p_n)c_n}$ is a right angle, because $\\pi(p_n)$ minimizes the distance from $p_n$ to $C$ and since $\\overrightarrow{\\pi(p_n)c_n}\\subset C$. We will show, in a sense which will be made precise, that, for any $n$ large enough, the angle between the geodesic arcs $\\overrightarrow{c_n p_n}$ and $\\overrightarrow{c_n\\pi(p_n)}$ is arbitrarily close to a right angle. Thus, as in Euclidean geometry, we will be able to deduce that the length of the side $\\overrightarrow{\\pi(p_n)c_n}$ of the geodesic triangles $\\triangle_n$ will tend to zero as $n\\to\\infty$. This observation will yield the claim $Tq=q$.\n\nWe have $c_n=\\pi (T(\\pi(p_n)))\\to Tq$, $n\\to\\infty$, because $p_n$ converges weakly to $q$, $\\pi$ and $T$ are continuous and $Tq\\in C$. The sequences $(T\\pi(p_n))$ and $(c_n)$ both converge to $Tq$. Thus\n\\begin{equation*}\n |d(T(\\pi(p_n)),p_n)-d(c_n,p_n)|\\to 0, \\ n\\to\\infty.\n\\end{equation*}\nBecause $d(Tp_n,p_n)\\to 0$, by assumption, we moreover have\n\\begin{equation}\\label{dist_est}\n |d(T(\\pi(p_n)),Tp_n)-d(c_n,p_n)|\\to 0, \\ n\\to\\infty.\n\\end{equation}\n\nLet $\\gamma_n$ be a geodesic with $\\gamma_n(0)=\\pi(p_n)$ and $\\gamma_n(1)=c_n$. Since $\\pi(p_n)$ and $c_n$ belong to $C$, $\\gamma_n(t)\\in C$ for all $t\\in [0,1]$. The nonexpansiveness of $T$ yields an estimate\n\\begin{align*}\n d^2&(\\gamma_n(\\partial),p_n)\\geq \\partial d^2(\\pi(p_n),p_n) + \\partial d^2(Tp_n,T(\\pi(p_n)) \\\\\n  &=\\partial d^2(\\gamma_n(0),p_n) + \\partial d^2(Tp_n,T(\\pi(p_n))\n\\end{align*}\nHere we have also used the fact that $\\pi(p_n)$ minimizes the distance from $p_n$ to $C$ and therefore also to the arc of $\\gamma_n$. By the estimate~\\eqref{dist_est}, we can choose an $N\\in \\mathbb{N}$ such that for all $n\\geq N$ there holds\n\\begin{equation*}\n d^2(T(p_n),T(\\pi(p_n)))> d^2(c_n,p_n)-\\epsilon^2\/4=d^2(\\gamma_n(1),p_n)-\\epsilon^2\/4,\n\\end{equation*}\nwhich shows that\n\\begin{equation*}\nd^2(\\gamma_n(\\partial),p_n)> \\partial d^2(\\gamma_n(0),p_n)+ \\partial d^2(\\gamma_n(1),p_n)-\\epsilon^2\/4.\n\\end{equation*}\nNow the equiconvexity of the family $\\{d^2(\\cdot,p): p\\in \\mathcal{N}\\}$ of functions shows that the length of the geodesic arcs $\\overrightarrow{\\pi(p_n)c_n}$ of the triangles $\\triangle_n$ will tend to zero,\n\\begin{equation*}\n d(\\pi(p_n),c_n)<\\epsilon,\n\\end{equation*}\nwhenever $n\\geq N$. Since $\\pi(p_n)\\to q$ and $c_n\\to Tq$, we have shown that $Tq=q$.\n\\end{proof}\n\n\n\n\nWe are finally set up for the proof of our main theorem.\n\\begin{proof}[Proof of Theorem~\\ref{main_thm}]\n$(i)\\Rightarrow (ii)$: Let $\\epsilon>0$. Since by assumption $q\\in \\overline{co}\\{p,Tp,T^2p,\\ldots\\}$, we can find a point $s\\in co\\{p,Tp,T^2p,\\ldots\\}$ such that\n\\begin{equation}\\label{approximity}\n 2D d(s,q)+ 3d^2(s,q)< \\epsilon^2\/5,\n\\end{equation}\nwhere $D$ is the finite diameter of the union of the convex hulls of the orbits $S_q$ and $S_s$ of $q$ and $s$. The fact that $D$ is finite follows from considerations similar to those in~\\eqref{bded_orbits}. By Lemma~\\ref{Tdc_impl_close}, we have\n\\begin{equation*}\n d(m_n(s),m_n(p))<\\epsilon,\n\\end{equation*}\nwhenever $n$ is large enough. \n\nLet us then estimate the distance\n\\begin{equation*}\n d(m_n(s),m_n(q)).\n\\end{equation*}\nFor this, we estimate the corresponding functions $F_n(\\cdot,s)$ and $F_n(\\cdot,q)$ (see Eq.~\\ref{F_n}). Let $\\d>0$. For fixed $r\\in \\overline{co}(S_q)\\cup\\overline{co}(S_s)$ we get the following estimate:\n\\begin{align*}\n F_n(r,s) & =\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(r,T^i s)\\leq \\frac{1}{n}\\sum_{i=0}^{n-1}(d(r,T^i q) + d(T^i q,T^i s))^2 \\\\\n  & =\\frac{1}{n}\\sum_{i=0}^{n-1}\\l(d^2(r,T^i q) + 2 d(r,T^iq)d(T^i q,T^i s)+d^2(T^i q,T^i s)\\r)  \\\\\n  & \\leq \\frac{1}{n}\\sum_{i=0}^{n-1}\\l(d^2(r,T^i q) + 2Dd(s,q) +d^2(s,q)\\r) < F_n(r,q)+ \\epsilon^2\/5. \n\\end{align*}\nHere we have used~\\eqref{approximity} and the fact that $T$ is nonexpansive. A similar calculation shows that \n\\begin{equation*}\n F_n(r,q)\\leq F_n(r,s) + 2Dd(s,q)+3d^2(s,q)< F_n(r,s)+ \\epsilon^2\/5.\n\\end{equation*}\nThus we have\n\\begin{equation*}\n \\sup_{r\\in \\overline{co}(S_q)\\cup\\overline{co}(S_s)}|F_n(r,s)-F_n(r,q)|\\leq \\epsilon^2\/5<\\epsilon^2\/4\n\\end{equation*}\nand Lemma~\\ref{uni_strc_appr} yields\n\\begin{equation*}\n d(m_n(s),m_n(q))<\\epsilon\n\\end{equation*}\nfor all $n$ large enough.\n\nCombining the estimates above shows that\n\\begin{equation*}\nd(q,m_n(p))=d(m_n(q),m_n(p))\\leq d(m_n(q),m_n(s))+d(m_n(s),m_n(p))<2\\epsilon,\n\\end{equation*}\nwhenever $n$ is large enough. Here we have used the assumption $q=Tq$ to deduce $q=m_n(q)$.\n\n$(ii)\\Rightarrow (iii)$: Strong convergence implies weak convergence by the continuity of the projections. $(iii)\\Rightarrow (iv)$: This is obvious.\n\n$(iv)\\Rightarrow (i)$: First we show that the weak cluster point of the sequence $(m_n(p))$ belongs to $\\overline{co}\\{p,Tp,T^2p,\\ldots\\}=\\overline{co}(S_p)$. Since $q$ is a weak cluster point of a sequence $(m_n(p))$, there exists a subsequence $(m_{n_k}(p))$ converging weakly to $q$ as $k\\to\\infty$. The sequence $(m_{n_k}(p))$ belongs to $\\overline{co}(S_p)$ and in particularly is bounded~\\cite[Lemma 3.3.4]{Jostb}. By the version of Mazur's lemma by Jost~\\cite[Thm. 2.2]{Jost}, the bounded sequence $(m_{n_k}(p))$ contains a subsequence such that its mean values converge to $q$. But now the mean values of elements of any subsequence of $(m_{n_k}(p))$ belong to the closed set $\\overline{co}\\{m_{n_k}(p): k\\in \\mathbb{N}\\}\\subset \\overline{co}(S_p)$. Thus $q\\in\\overline{co}(S_p)$.\n\nIt remains to prove that $q=Tq$. For this let $\\epsilon>0$. We use the nonexpansivity of $T$ to show first that, for all $n$ large enough,\n\\begin{equation*}\n d(Tm_n(p),m_n(p))<\\epsilon.\n\\end{equation*}\nWe have\n\\begin{align*}\n F_n(Tm_n(p),p)&=\\frac{1}{n}\\sum_{i=0}^{n-1}d^2(Tm_n(p),T^ip)\\leq \\frac{1}{n}\\sum_{i=0}^{n-1}d^2(m_n(p),T^ip) \\\\ \n  &+\\frac{1}{n}d^2(T m_n(p),p)-\\frac{1}{n}d^2(m_n(p),T^{n-1}p)= F_n(m_n(p),p) \\\\\n  &+\\frac{1}{n}d^2(Tm_n(p),p)-\\frac{1}{n}d^2(m_n(p),T^{n-1}p).\n\\end{align*}\nBy the boundedness of the orbit of $p$, we can choose $N\\in\\mathbb{N}$ such that, for all $n\\geq N$,\n\\begin{equation*}\n F_n(m_n(p),p)> F_n(Tm_n(p),p)-\\epsilon^2\/2.\n\\end{equation*}\nFor a geodesic $\\gamma:[0,1]\\to \\mathcal{N}$ with $\\gamma(0)=m_n(p)$ and $\\gamma(1)=Tm_n(p)$, we now have\n\\begin{equation*}\n F_n(\\gamma(\\partial),p)\\geq F_n(m_n(p),p)> \\partial F_n(\\gamma(0),p) + \\partial F_n(\\gamma(1),p) -\\partial \\epsilon^2\/2.\n\\end{equation*}\nHere the first inequality follows from the fact that $m_n(p)$ is the minimizer of the function $F_n(\\cdot,p)$. The equiconvexity of the family of functions $\\{F_n(\\cdot,p)\\}$ now yields\n\\begin{equation*}\n d(Tm_n(p),m_n(p))<\\epsilon\n\\end{equation*}\nwhenever $n$ is large enough. \n\nWe have seen that the sequence $(m_{n_k}(p))$ converges weakly to $q$ and \n\\begin{equation*}\n d(Tm_{n_k}(p),m_{n_k}(p))\\to 0, \\ k\\to \\infty.\n\\end{equation*}\nThus, by Lemma~\\ref{cond_for_fpoint}, we deduce that $Tq=q$.\n\\end{proof}\n\nFrom the proof of the theorem we can see that the distance convexity of $T$ was only used to prove (via Lemma~\\ref{Tdc_impl_close}) that the means converge strongly to the fixed point. We use this observation to prove Theorem~\\ref{fp_thm}.\n\n\\begin{proof}[Proof of Theorem~\\ref{fp_thm}]\nLet $p\\in \\mathcal{N}$ have bounded orbit. By Theorem 2.1 of~\\cite{Jost}, the bounded sequence $(m_n(p))$ contains a subsequence converging weakly to some element $q\\in \\mathcal{N}$. Thus $q$ is a weak cluster point of the sequence $(m_n(p))$. Now, the exact same argument as in the part $(iv)\\Rightarrow (i)$ of the proof of Theorem~\\ref{main_thm} concludes the proof.\n\\end{proof}\n\n\\section{Fixed point for volumorphisms on the space of Riemannian metrics with fixed volume form}\nIn this section we apply the fixed point theorem to volumorphisms acting on the space of all Sobolev Riemannian metrics having a fixed volume form. We give a natural condition for a fixed point to exist if we allow that the fixed point satisfies only mild regularity assumptions.  We begin by describing the space of Riemannian metrics with a fixed volume form. We refer to~\\cite{Ebin,Clarke} for basic results and properties of this space and to~\\cite{Freed} for calculations of its geodesics and curvature.\n\nLet $M$ be a smooth compact oriented finite dimensional manifold. We denote by $\\mathcal{M}^s$ the set of all Riemannian metrics on $M$, which are of Sobolev class $H^s$. Throughout this section, we will assume $s>n\/2$. The space $\\mathcal{M}^s$ is an infinite dimensional manifold with a weak Riemannian structure given by\n\\begin{equation*}\n \u00a0\\langle U,V \\rangle_g=\\int_M \\tr{g^{-1}Ug^{-1}V}dV_g,\n\\end{equation*}\nwhere $U,V$\u00a0are tangent vectors at $g\\in\\mathcal{M}^s$ and $dV_g$ is the volume form induced by $g$. Tangent vectors of $\\mathcal{M}^s$ are symmetric $(0,2)$-tensor fields of Sobolev class $H^s$.\n\nLet $\\mu$ be a volume form on $M$. Consider next the subset $\\mathcal{M}_\\mu^s$ of $\\mathcal{M}^s$ consisting of the elements of $\\mathcal{M}^s$ whose induced volume form is $\\mu$. This subset is an infinite dimensional submanifold of $\\mathcal{M}^s$ with the induced inner product\n\\begin{equation*}\n \\langle U,V \\rangle_g=\\int_M \\tr{g^{-1}Ug^{-1}V}d\\mu.\n\\end{equation*}\nHere $U$ and $V$, tangent vectors at the point $g$, are traceless (with respect to $g$) symmetric $(0,2)$-tensor fields.  \n\nThe geodesics of $\\mathcal{M}_\\mu^s$ can be given explicitly:\n\\begin{equation*}\\label{can_path}\n g(t)=g\\exp{t(g^{-1}A)}.\n\\end{equation*}\nHere $g(0)=g\\in \\mathcal{M}_\\mu^s$ and $\\dot{g}(0)=A\\in T_g\\mathcal{M}_\\mu^s$. The geodesic $g(t)$ is of constant speed\n\\begin{equation*}\n ||\\dot{g}(t)||^2_{g(t)}=\\int_M\\tr{(g^{-1}A)^2}d\\mu.\n\\end{equation*}\nGeodesics $g(t)$ exist for all times $t$ and we can easily see that the unique solution $A$ of the equation\n\\begin{equation*}\n g(t)|_{t=1}=h\n\\end{equation*}\nis\n\\begin{equation*}\n A=g\\log{(g^{-1}h)}.\n\\end{equation*}\nThus the distance, at least formally, is given by\n\\begin{equation*}\n d(g,h)=\\int_0^1 ||\\dot{g}(t)||_{g(t)}dt=\\l(\\int_M\\tr{(g^{-1}A)^2}d\\mu\\r)^{1\/2}.\n\\end{equation*}\nThat is,\n\\begin{equation}\\label{P_distance}\n d^2(g,h)=\\int_M\\tr{(\\log{(g^{-1}h)})^2}d\\mu.\n\\end{equation}\n\nThe calculation above is formal in the sense that for general infinite dimensional weak Riemannian manifolds it is nontrivial how geodesics relate to the distance function. However, in our case, the exponential mapping $T_g\\mathcal{M}_\\mu^s\\to \\mathcal{M}_\\mu^s$ is a diffeomorphism onto $\\mathcal{M}_\\mu^s$ for any $g\\in \\mathcal{M}_\\mu^s$, and therefore the distance between $g$ and $h$ is given by the norm of the tangent vector $A=g\\log{(g^{-1}h)}\\in T_g\\mathcal{M}_\\mu^s$ justifying~\\eqref{P_distance}. See~\\cite[Prop. 2.23, Prop. 2.46]{Clarke} for details. \n\nWe record that $\\mathcal{M}_\\mu^s$ is indeed a global Alexandrov NPC space. We omit the proof since it is essentially the same as the proof of Theorem~\\ref{X} below.\n\\begin{thm}\\label{M_mu_NPC}\n The space $\\mathcal{M}_\\mu^s$ of Riemannian metrics of Sobolev class $H^s$, $s>n\/2$, with a fixed volume form $\\mu$ on a compact orientable manifold is a global Alexandrov NPC space.\n\\end{thm}\n\nA volumorphism is a diffeomorphism preserving a given volume form. We denote by $\\mathcal{D}_\\mu^{s+1}$ the space of Sobolev $H^{s+1}$ volumorphisms on $M$. The natural action of $\\mathcal{D}_\\mu^{s+1}$ on $\\mathcal{M}_\\mu^s$ is given by pullback. A straightforward calculation shows that the action is actually an isometry in the sense of Riemannian geometry and Eq.~\\ref{P_distance} shows that it is also an isometry in the metric sense. In particular, the action is nonexpansive and we are, almost, in the setup of our fixed point Theorem~\\ref{fp_thm}.\n\nThe last needed assumption to apply Theorem~\\ref{fp_thm} would be the completeness of $\\mathcal{M}_\\mu^s$ as a metric space $(\\mathcal{M}_\\mu^s,d)$. However, $(\\mathcal{M}_\\mu^s,d)$ is not metrically complete, which is quite expected since we are in a sense considering an $L^2$ inner product in the subset of Sobolev $H^s$ Riemannian metrics on $M$.\n\nThe metric completion of the manifold of all Riemannian metrics $\\mathcal{M}^s$ with respect to its distance metric is characterized in~\\cite{Clarke}. The elements of the completion of $\\mathcal{M}^s$ can be identified with measurable semimetrics with finite volume. See~\\cite[Thm 5.25.]{Clarke} for details. As $\\mathcal{M}_\\mu^s$ is a submanifold of $\\mathcal{M}^s$, it follows that the $d$-metric completion $\\overline{\\mathcal{M}}_\\mu^s$ of $\\mathcal{M}_\\mu^s$ is a subset of the metric completion of $\\mathcal{M}^s$. The next theorem implies that actually more is true. \n\n\\begin{thm}\\label{X}\n Let $\\mu$ be a volume form on a smooth oriented compact manifold $M$. Let $X$ be the set of $\\mu$-measurable a.e. positive definite symmetric $(0,2)$-tensor fields $g$ on $M$ with volume form agreeing with $\\mu$ a.e. and\n\\begin{equation*}\n \\int_M\\tr{(\\log{(g^{-1}h)})^2}d\\mu <\\infty,\n\\end{equation*}\nfor all $h\\in \\mathcal{M}_\\mu^s$. Then, if $X$ is equipped with the metric\n\\begin{equation}\\label{metric}\n \\d(g,h)=\\l(\\int_M\\tr{(\\log{(g^{-1}h)})^2}d\\mu\\r)^{1\/2},\n\\end{equation}\n$(X,\\d)$ is a complete global Alexandrov NPC space.\n\nThe geodesics of $(X,\\d)$ are given by the formula\n\\begin{equation}\\label{olP_geo}\n g(t)=g \\exp{tg^{-1}A}.\n\\end{equation}\nHere $g\\in X$ and $A$ belongs to the set of $(L^2,|\\cdot|_g,\\mu)$-integrable symmetric $(0,2)$-tensor fields on $M$ satisfying $\\Trg{A}{g}=0$ a.e. Moreover, the space of mappings $\\mathcal{D}_\\mu^{s+1}$ acts by pullback isometrically on $(X,\\d)$.\n\\end{thm}\n\n\n\\begin{proof}\nWe first show that $X$ equipped with the mapping $\\d:X\\times X\\to [0,\\infty)$ is a metric space. This follows from noting that the integrand in the formula~\\eqref{metric} of the metric $\\d$ is, pointwise in any local coordinates, the square of the distance of matrices in a space isometric to $S:=SL(n,\\mathbb{R})\/SO(n,\\mathbb{R})$. This is because all elements of $X$ have the same volume form a.e., and therefore the coordinate representations of the elements of $X$ are pointwise positive definite symmetric matrices with \\emph{equal} determinants a.e.\n\nThe observation above yields the following. If $\\d(g,h)=0$, then the integrand of~\\eqref{metric} equals zero a.e. and consequently $g=h$ a.e. The triangle inequality, and the fact that $\\d$ is finite, follow from the Minkowski inequality for $L^2(M,\\mu)$ and the triangle inequality on $(S,d_S)$. We conclude that $(X,\\d)$ is a metric space. We also see that $\\mathcal{D}_\\mu^{s+1}$ acts isometrically on $X$.\n\nThe metric $d_S$ of $S$ is given by\n\\begin{align*}\n d^2_S(G,H)&=||\\log (H^{1\/2}G^{-1}H^{1\/2})||^2=\\tr{\\l(\\log(H^{1\/2}G^{-1}H^{1\/2})\\r)^2} \\\\\n  &=\\tr{(\\log(G^{-1}H))^2}=\\tr{(\\log(H^{-1}G))^2}\n\\end{align*}\nfor $G,H\\in S$, see~\\cite[Ch. 20]{Iwaniec}. A straightforward calculation shows that a path $\\Gamma:\\mathbb{R} \\to S$ of the form\n\\begin{equation}\\label{S_geo}\n \\Gamma(t)=G\\exp{t\\log{(G^{-1}H)}}\n\\end{equation}\nhas on the interval $[0,1]$ the length $l(\\Gamma)$,\n\\begin{equation*}\n l(\\Gamma):=\\sup\\l\\{\\sum_{i=0}^n d_S(\\Gamma(t_i),\\Gamma(t_{i-1})): 0=t_0<t_1<\\cdots < t_n=1, n\\in \\mathbb{N} \\r\\},\n\\end{equation*}\nequal to $d_S(G,H)$. This implies that paths of the form~\\eqref{S_geo} are geodesics in $S$. As a Riemannian manifold, $S$ is a complete globally symmetric space of nonpositive sectional curvature and therefore a complete global Alexandrov NPC space, see~\\cite[p.11-18, p.55]{Jostb}.\n\nSimilarly, if we calculate the length of a path\n\\begin{equation*}\n \\gamma(t)=g \\exp{t\\log{(g^{-1}h)}}\n\\end{equation*}\non an interval $[0,1]$, we see that it equals $\\d(g,h)$. Thus, there is a geodesic connecting any two points in $X$. The NPC inequality for $(X,\\d)$ is inherited from $(S,d_S)$ via integration of the NPC inequality on $(S,d_S)$. Since $(X,\\d)$ is simply connected, it follows that $(X,\\d)$ is actually a global Alexandrov NPC space~\\cite[Cor. 2.3.2]{Jostb}. More generally we see that paths of the form~\\eqref{olP_geo} are geodesics in $(X,\\d)$. \n\nIt remains to prove that $(X,\\d)$ is complete. The proof of this fact is analogous to showing that $L^p$ spaces are complete, cf.~\\cite[Ch. 7.3]{Royden}. For this, let $(g_n)$ be a Cauchy sequence in $(X,\\d)$. There is a subsequence of $(g_n)$, which we still denote by $(g_n)$, such that\n\\begin{align*}\n \\sum_{n=0}^{\\infty}\\d(g_{n},g_{n+1})=s<\\infty.\n\\end{align*}\nThus, by Fatou's lemma and by the Cauchy-Schwarz inequality for $L^2(M,\\mu)$, we have that\n\\begin{align*}\n \\int_M &\\sum_{n=0}^{\\infty}\\l(\\tr{(\\log(g_n^{-1}g_{n+1}))^2}\\r)^{1\/2}d\\mu \\\\\n  &\\leq \\liminf_{k\\to\\infty}\\sum_{n=0}^{k}\\int_M \\l(\\tr{(\\log(g_n^{-1}g_{n+1}))^2}\\r)^{1\/2}d\\mu  \\\\\n  &\\leq \\liminf_{k\\to\\infty} \\sum_{n=0}^{k} \\l( \\int_M \\tr{(\\log(g_n^{-1}g_{n+1}))^2}d\\mu \\r)^{1\/2}\\mbox{Vol}_\\mu(M)^{1\/2} \\\\\n  &=\\sum_{n=0}^{\\infty}\\d(g_n,g_{n+1}) \\mbox{Vol}_\\mu(M)^{1\/2}<\\infty.\n\\end{align*}\nTherefore, the integrand $\\sum_{n=0}^{\\infty}\\l(\\tr{(\\log(g_n^{-1}g_{n+1}))^2}\\r)^{1\/2}$ above is finite a.e. \n\nLet $\\epsilon>0$. It follows from what we observed that, for a.e. $x\\in M$, there is an index $N=N_x\\in \\mathbb{N}$ such that\n\\begin{align*}\n d_S&(g_{n}(x),g_{n+k}(x))\\leq \\sum_{j=n}^{n+k-1}d_S(g_{j}(x),g_{j+1}(x)) \\leq \\sum_{j=n}^{\\infty}d_S(g_{j}(x),g_{j+1}(x)) \\\\\n  &=\\sum_{j=n}^{\\infty}\\l(\\tr{(\\log(g_j^{-1}(x)g_{j+1}(x)))^2}\\r)^{1\/2}<\\epsilon\n\\end{align*}\nwhenever $n\\geq N_x$ and for all $k\\in \\mathbb{N}$. Thus, $(g_n(x))$ is Cauchy a.e. By the metric completeness of $(S,d_S)$, it follows that, for a.e. $x\\in M$, the sequence $g_{n}(x)$ tends to some $g(x)$. \n\nThe limit is a symmetric positive semi-definite $(0,2)$ tensor field $g$, defined a.e., and has volume form equalling $\\mu$ a.e. The a.e. positiveness of the volume form $\\mu$ implies that $g$ is actually positive definite a.e. The fact that $\\d(g,h)<\\infty$, for all $h\\in \\mathcal{M}_\\mu^s$, follows from Fatou's lemma and the boundedness of Cauchy sequences. We also have $\\d(g_n,g)\\to 0$ as $n\\to\\infty$.\n\\end{proof}\n\nBy the discussion of this section we have that $(\\mathcal{M}_\\mu^s,d)$ is isometrically embedded in $(X,\\d)$. It is reasonable to expect that actually $(X,\\d)$ is the metric completion of $(\\mathcal{M}_\\mu^s,d)$. The proof of this, which is equivalent to proving that $\\mathcal{M}_\\mu^s$ is dense in $(X,\\d)$, does not seem to be straightforward however.\n\nWe have the main theorem of this section.\n\\begin{thm}\\label{vol_fp}\nLet $(X,\\d)$ be as in Theorem~\\ref{X}. If the action of a volumorphism $\\phi\\in\\mathcal{D}_\\mu^{s+1}$ has a bounded orbit in $(X,\\d)$ for some $p\\in X$, then there exists a fixed point $g$ for the action in the $\\d$-closure of the subset $co\\{p,Tp,T^2p,\\ldots\\}$ of $X$. With respect to this fixed point $\\phi$ is a Riemannian isometry,\n\\begin{equation*}\n \\phi^*g=g.\n\\end{equation*}\n\\end{thm}\n\n\\subsection*{Acknowledgements}\n\nThe author is supported by the Finnish National Graduate School in Mathematics and its Applications. This work was inspired by the international scientific workshop in honor of Steven Rosenberg's 60th birthday. The author also wishes to thank the referee for valuable comments and interest in this work.\n\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{sec:level1}First-level heading:\\protect\\\\ The line\n\\section{\\label{sec:level1}INTRODUCTION}\n\nImage-guided radiotherapy (IGRT) is commonly used in commercial stereotactic radiotherapy systems for focusing irradiation on a tumour subject to motion. The technology to locate lung tumour position in real time during irradiation is important because of the trend towards real-time adaptive treatment. Some commercial stereotactic radiotherapy systems support real-time tracking of gold fiducial markers\\cite{1SyncTraX}\\cite{2Harada}\\cite{3Shiinoki}. However, marker implantation is invasive and carries the risks of pneumothorax and marker migration\\cite{4Bhagat}. Markerless tracking is being investigated for the next generation of IGRT\\cite{5Bahig}\\cite{6Yang}\\cite{7Dhont}. In most cases, template matching\\cite{8Patel}\\cite{9Shiinoki}\\cite{10Teske} or a correlation model\\cite{11Shieh} with X-ray images as a training data set are used. Other cases use a correlation model\\cite{12Li} with digitally reconstructed radiographs (DRRs) as training data sets. These methodologies involve the generation of a small personalized training data set for each patient. However, conventional template matching and correlation models often cause robustness problems due to inter- and intra-fractional change or induced artefacts in computed tomography.\n\nImage recognition by machine learning, ng trackespecially deep learning (DL), is another strategy to improve the robustness of markerless tracking\\cite{13Terunuma}. DL is a de facto standard for robust image recognition methods. However, DL for medical imaging usually requires multiple-subject data sets for training. Further, data collection is challenging and does not always work well due to the heterogeneity of patient data.\n\nHere, we propose a different strategy using patient-specific DL with large personalized training data sets generated from individual patients for real-time markerless lung tumour tracking, avoiding the need for collection of a large patient data set. Our strategy uses a personalized training data set generated from each patient's 4-dimensional treatment planning computed tomography (4D-CT). \\textbf{FIG. \\ref{fig1}} shows tumour tracking by patient-specific DL in treatment workflow. The personalized data generation and training process could be done end-to-end automatically using treatment planning with 4D-CT data. Moreover, these processes could be done in the treatment facility without taking patient data out of the facility, avoiding privacy problems. The tracking is performed during treatment, with the possibility of pretreatment rehearsal.\n\nWe validated the feasibility of our strategy by evaluatiing accuracy in both a digital phantom simulation study and an epoxy phantom study.\n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\includegraphics[width=8cm]{fig1.png}\n    \\caption{\\label{fig:epsart}Procedure for markerless tumour tracking by patient-specific DL in treatment workflow.}\n    \\label{fig1}\n  \\end{center}\n\\end{figure}\n\n\n\\section{\\label{sec:level1}MATERIALS AND METHODS}\n\n\\subsection{\\label{sec:level2}Deep learning for markerless tracking}\n\nWe designed a neural network model (\\textbf{FIG. \\ref{fig2}}) for 2-class semantic segmentation based on a fully convolutional neural network (FCN)\\cite{14FCN}. Our model classifies an input image to tumour area or background area pixel-wise for markerless tumour tracking. We combined the FCN with a pixel shuffle layer\\cite{15PixShuf} instead of deconvolution layers for \\textgreater 15 $\\mathrm{fps}$ (\\textless 66.7 $\\mathrm{ms\/frame}$) real-time processing. We used wider convolution sizes in our model than typical FCN using deconvolution layers such as the original FCN and U-net\\cite{16U-Net}, because we needed not only local textures but also non-local features such as tumour contours to recognize the tumour. A pixel shuffle layer is faster than deconvolution layers, making it suitable for real-time processing.\n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\includegraphics[width=8cm]{fig2.png}\n    \\caption{\\label{fig:epsart}The neural network model based on FCN combined with a pixel shuffle layer. F: Filter size. S: Stride width. P: Padding size. The left-side image is an input image and the right-side image is a labelled image classified to tumour area (white) or background area (black).}\n    \\label{fig2}\n  \\end{center}\n\\end{figure}\n\nIn the training process, we used DRRs generated from each phantom's 4D-CT as input images. Also, we used labelled images classified to tumour area or background area as teacher images. We explained the detail of training data set, namely, pairs of a training DRR and a labelled image, in the simulation study section and the phantom study section (below). We trained models by using softmax cross entropy as a loss function and using Adam (adaptive moment estimation)\\cite{17Adam} as an optimization algorithm in 200 mini batches and 10 epochs. We consider that Adam optimization is suitable for stable and fast training for a variety of data sets without tuning training parameters for each data set, because Adam automatically updates a learning rate parameter in its internal algorithm. Each projection view was trained independently by using each training data set.\n\n\\subsection{\\label{sec:level2}The simulation study with a digital 4D-CT phantom}\n\\subsubsection{Digital 4D-CT phantom}\n\nIn the simulation study, a digital respiratory motion phantom including ribs and vertebrae in the form of a 4D-CT (XCAT\\textregistered, Duke University, Durham NC, USA)\\cite{18XCAT}\\cite{19XCAT_HP} was used. The XCAT consists of $512 \\times 512 \\times 400$ voxels (1 $\\mathrm{mm^3}$ voxel) and features 200 phases between peak inhalation and peak exhalation.\n\nWe created a digital tumour motion phantom synchronized with the XCAT respiratory motion with the same spatial resolution. The digital tumour phantom motion range was 40 $\\mathrm{mm}$ in the superior-inferior (SI) direction, the main component of respiratory motion. We created 3-$\\mathrm{cm}$ spherical and $1.5 \\times 2.25 \\times 3$-$\\mathrm{cm}$ ovoid digital tumour phantoms with CT values of $100$ Hounsfield units (HU), which are large enough to be overlapped by ribs.\n\n\\subsubsection{Training data sets}\n\nWe used only ten phases with the same phase interval with the XCAT digital phantom, because 4D-CT generally consists of ten phases. DRRs were generated by projecting each 4D-CT phase with the same projection geometry as the two-view fluoroscopic tracking systems in the epoxy phantom study. DRRs were $768 \\times 768$ pixels ($388 \\times 388$ $\\mu m^2$ pixels), identical to the flat panel detector (FPD) in the epoxy phantom study (below). The DRR generation algorithm was based on a ray tracing algorithm\\cite{20Siddon} and was also used for patient positioning with image registration\\cite{21Mori_DRR}\\cite{22Mori_first}.\n\nWe coupled an XCAT DRR and a tumour DRR using 4D-CT with the same projection geometry, allowing us to acquire an XCAT DRR with a tumour as a training DRR (\\textbf{FIG. \\ref{fig3}}). \n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\includegraphics[width=8cm]{fig3.png}\n    \\caption{\\label{fig:epsart}Generating XCAT digitally reconstructed radiographs (DRRs) with a tumour as training DRRs from XCAT and a digital motion tumour phantom on 4D-CT.}\n    \\label{fig3}\n  \\end{center}\n\\end{figure}\n\nWe solved the problem of overlapping bone by using large numbers of training DRRs generated with displacing projection geometries to simulate a range of tumour motion. These training DRRs used the digital tumour phantom with overlapping bone. This solution also works as data augmentation.\n\nTo generate the DRRs, we displaced projection geometry with 3-dimensional translation ($x$, $y$, $z$) and rotation ($\\psi$, $\\varphi$, $\\theta$) of an X-ray tube\/FPD, by $-6$ to $+6$ $\\mathrm{mm}$ with 3-$\\mathrm{mm}$ steps and $-2^\\circ$ to $+2^\\circ$ with $1^\\circ$ steps. We then acquired $5^6 - 1 = 15,624$ DRRs per 4D-CT phase (\\textbf{FIG. \\ref{fig4}}). Finally, we acquired $156,240$ DRRs in ten 4D-CT phases. Because we can interpolate and extrapolate tumour motion in discrete 4D-CT phases, the geometric displacements were calculated from the tumour phantom motion range between 4D-CT phases.\n\n\\begin{figure*}\n\\begin{ruledtabular}\n  \\begin{tabular}{ccc}\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig4_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig4_upper_right.png}}\n    \\end{minipage} \\\\\n    & $(x, y, z) = (+6, +6, +6),$ & $(x, y, z) = (-6, -6, -6),$ \\\\\n    & $(\\psi, \\varphi, \\theta) = (+2, +2, +2)$ & $(\\psi, \\varphi, \\theta) = (-2, -2, -2)$\\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig4_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig4_lower_right.png}}\n    \\end{minipage} \\\\\n    & $(x, y, z) = (+6, +6, +6),$ & $(x, y, z) = (-6, -6, -6),$ \\\\\n    & $(\\psi, \\varphi, \\theta) = (+2, +2, +2)$ & $(\\psi, \\varphi, \\theta) = (-2, -2, -2)$\\\\\n  \\end{tabular}\n  \n  \\caption{\\label{fig:epsart}Examples of training digitally reconstructed radiographs (DRRs) of a spherical tumour phantom with displaced projection geometry using digital phantoms. ($x$, $y$, $z$) and ($\\psi$, $\\varphi$, $\\theta$) are displacements from the original projection geometry.}\n  \\label{fig4}\n\n  \\vspace{4mm}\n\n  \\begin{tabular}{ccc}\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig5_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig5_upper_right.png}}\n    \\end{minipage} \\\\\n    & $(x, y, z) = (+6, +6, +6),$ & $(x, y, z) = (-6, -6, -6),$ \\\\\n    & $(\\psi, \\varphi, \\theta) = (+2, +2, +2)$ & $(\\psi, \\varphi, \\theta) = (-2, -2, -2)$\\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig5_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.25}{\\includegraphics{fig5_lower_right.png}}\n    \\end{minipage} \\\\\n    & $(x, y, z) = (+6, +6, +6),$ & $(x, y, z) = (-6, -6, -6),$ \\\\\n    & $(\\psi, \\varphi, \\theta) = (+2, +2, +2)$ & $(\\psi, \\varphi, \\theta) = (-2, -2, -2)$\\\\\n  \\end{tabular}\n  \\caption{\\label{fig:epsart}Examples of training digitally reconstructed radiographs (DRRs) of a 3-cm tumour phantom with displaced projection geometry, random contrast transformation, and random noise. ($x$, $y$, $z$) and ($\\psi$, $\\varphi$, $\\theta$) are displacements from the base projection geometry.}\n  \\label{fig5}\n\n\\end{ruledtabular}\n\\end{figure*}\n\nWe also generated large numbers of labelled images corresponding to training DRRs. In a labelled image, tumour area was calculated as the projection area of a tumour in 4D-CT. The other area was labelled as background area. This allowed us to automatically generate phantom-specific training data sets, namely pairs of a training DRR and a labelled image.\n\n\\subsubsection{Markerless tracking}\n\nDRRs for all 4D-CT phases with the same projection geometry, without displacements, were used as tracking images. These tracking images consisted of 200 frames and were not included in the training data set. \n\nA tumour coordinate was calculated from semantic segmentation of a tracking image by using a phantom-specific trained model for every frame. Trained models output the `tumour score' (0.0-1.0) pixel-wise. Therefore, we were able to calculate a tumour coordinate as the density centre \nweighted by pixel-wise tumour score thresholding \\textgreater 0.5. Using the density centre potentially allows acquisition of a tumour coordinate with sub-pixel accuracy in a tracking image. Misdetections of a few pixels had little effect on the density centre coordinate. Therefore, we expected higher accuracy than that of object detection methods outputting an object coordinate directly, such as Faster R-CNN\\cite{23Ren}.\n\n\\subsubsection{Tracking accuracy evaluation}\n\nThe ground truth of a tumour coordinate is the density centre of a tumour DRR weighted by pixel values with sub-pixel accuracy. We evaluated `tracking error' as the 2-dimensional distance between a tracking coordinate and a ground truth coordinate in each view. We evaluated not only absolute error on the isocentre in mm but also relative error on FPD by pixel units, because tracking errors may be affected by pixel resolution. We defined `tracking accuracy' as the ratio of frames satisfying \\textless 1 $\\mathrm{mm}$ tracking error on the isocentre. Tracking accuracy corresponds to gating accuracy in typical isotropic PTV-to-CTV margins of 1 $\\mathrm{mm}$\\cite{24Mori_track}. In case with no pixels having a \\textgreater 0.5 tumour score, we considered that tumour detection did not succeed in this frame.\n\n\\subsection{\\label{sec:level2}The phantom study with an epoxy respiratory motion phantom}\n\\subsubsection{Epoxy respiratory motion phantom}\n\nIn the phantom study, we validated multi-modality DL, which uses DRRs for training and X-ray images for tracking, using an epoxy respiratory motion phantom. A programmable respiratory motion phantom (Dynamic Thorax Phantom MODEL 008A\\textregistered, Computerized Imaging Reference Systems, Inc., Norfolk VA, USA) \\cite{25CIRS} was used. This is an epoxy chest phantom with vertebrae but no ribs. It allows the operator to change spherical tumour size (1 $\\mathrm{cm}$, 2 $\\mathrm{cm}$ and 3 $\\mathrm{cm}$ diameter). The CT values of the epoxy `tumours' are approximately 30 $\\mathrm{HU}$.\n\nA 4D-CT of the MODEL 008A phantom was acquired (Aquilion ONE\\textregistered, Canon Medical Systems Corporation, Otawara City, Japan) with different sized tumour phantoms. We marked phantom position with the CT scanner's laser markers to reposition the chest phantom for tracking in fluoroscopy. We then guided a tumour phantom on a rod in a sinusoidal motion ($\\pm20$ $\\mathrm{mm}$ amplitude and 4 second cycle) in the SI direction. 4D-CT phases were synchronized with tumour motion. This 4D-CT consisted of $512 \\times 512 \\times 204$ voxels ($0.625 \\times 0.625 \\times 1$ $\\mathrm{mm}$ voxel) and ten phases for a full single respiratory cycle (T00T90). Because 4D-CT was acquired in the same manner as during treatment workflow, in discrete phases, the 4D-CT imaging range of tumour motion was approximately 38 $\\mathrm{mm}$, which was narrower than the actual tumour motion range. Also, motion artefacts, which gave the tumours a more strongly differentiated appearance than most cases of inter- and intra-fractional changes, were detected in some 4D-CT phases.\n\n\\subsubsection{Training data sets}\n\nWe generated large numbers of training DRRs with displacing projection geometry from 4D-CT data of the chest phantom to simulate a variety of tumour motion patterns. We displaced the projection geometry by $-6$ to $+6$ $\\mathrm{mm}$ with 3-$\\mathrm{mm}$ steps and $-2^\\circ$ to $+2^\\circ$ with $1^\\circ$ steps, respectively. We then acquired $5^6 = 15,625$ DRRs per 4D-CT phase (\\textbf{FIG. \\ref{fig5}}). Finally, we acquired $156,250$ DRRs in all ten 4D-CT phases. Because we can interpolate and extrapolate tumour motion in discrete 4D-CT phases, the geometry displacement range was decided by the tumour motion range between 4D-CT phases. \n\nIn multi-modality DL using DRRs for training and X-ray images for tracking, we faced the specific problem of differences in image quality between DRRs and X-ray images, caused mainly by beam hardening effect, scattering, sensitivity characteristics of FPDs, and shot noise. Therefore, a DRR cannot be converted to an X-ray image precisely, because pixel values of a DRR do not correspond to pixel values of an X-ray image one-to-one. We solved this problem by using large numbers of training DRRs with random contrast transformation and random noise addition.\n\nThe contrast variation range was decided by analysing differences in contrast between DRRs and X-ray images. We applied gamma transformation with random $\\gamma$ values ($\\gamma = 0.4\u20132.5$) to each DRR as random contrast transformation. We also added Gaussian noise ($\\sigma = 25$) as random noise such that the noise variation range was $\\pm1$ standard deviation of the X-ray image noise.\n\nJust as in the simulation study, we also generated large numbers of labelled images corresponding to training DRRs.\n\n\\subsubsection{Markerless tracking}\n\nTwo-view X-ray fluoroscopy images of the chest phantom were simultaneously taken and marked with the laser positioning system (SyncTraX FX4\\textregistered, Shimadzu Corporation, Inc., Kyoto, Japan) \\cite{1SyncTraX} (\\textbf{FIG. \\ref{fig6}}). We manually positioned the phantom as close as possible to its location in the 4D-CT using the laser markers. Because we had not yet constructed a synchronizing system between X-ray fluoroscopy frame and measurement of tumour position, we acquired X-ray images at the 25 fixed tumour positions on the same trajectory taken by the 4D-CT with a tumour phantom inserted. The other structures of the phantom except the `tumour' were fixed at the same position. The X-ray settings were 110 $\\mathrm{kV}$, 50 $\\mathrm{mA}$, 4 $\\mathrm{ms}$, the same as in our gold marker tracking. These X-ray images consisted of $768 \\times 768$ pixels ($388 \\times 388$ $\\mu m^2$ pixel). They were taken in slightly different positions than in the 4D-CT because of differences in phase intervals.\n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\includegraphics[width=8cm]{fig6.png}\n    \\caption{\\label{fig:epsart}Geometry of the two-view X-ray fluoroscopy imaging system SyncTraX FX4\\textregistered with a radiotherapy system. Two views cross at the isocentre without being blocked by the gantry head. Source-object distance (SOD) = 2353 $\\mathrm{mm}$. Source-image distance (SID) = 4172 $\\mathrm{mm}$.}\n    \\label{fig6}\n  \\end{center}\n\\end{figure}\n\nWe applied a Gaussian blur filter to each X-ray image to simulate the same spatial resolution as training DRRs, which was lower than that of X-ray images, because DRR spatial resolution was limited by the CT spatial resolution (image quality of DRRs and X-ray images was close to each other for multi-modality DL). These blurred X-ray images were used as tracking images.\n\nAs in the simulation study, a tumour coordinate was calculated from semantic segmentation of a tracking image by using a phantom-specific trained model for every frame. A tumour coordinate was calculated as the density centre of the tumour score with sub-pixel accuracy, which was essentially unaffected by the Gaussian blur filter.\n\n\\subsubsection{Tracking accuracy evaluation}\n\nWe measured displacement of a motion tumour rod of MODEL 008A phantom as ground truth of the tumour coordinate using a laser displacement meter (LK-G155\\textregistered, Keyence Inc., Osaka, Japan, \\textless 20 $\\mu m$ accuracy) outside of the X-ray fields of view (\\textbf{FIG. \\ref{fig7}}). We had previously calibrated between displacements of a tumour phantom and tumour coordinates on X-ray images. We then acquired tumour coordinates from displacements of the tumour phantom at 25 different positions. \n\n\\begin{figure}[htb]\n  \\begin{center}\n    \\includegraphics[width=8cm]{fig7.png}\n    \\caption{\\label{fig:epsart}Tracking accuracy evaluation system using a MODEL 008A phantom with a laser displacement meter (a). We measured displacement of a motion tumour rod of the MODEL 008A phantom outside both X-ray fields of view.}\n    \\label{fig7}\n  \\end{center}\n\\end{figure}\n\nAs in the simulation study, we evaluated not only absolute error on the isocentre in mm but also relative error on FPD by pixel units. Also, we evaluated `tracking accuracy'.\n\n\\subsection{\\label{sec:level2}Implementation}\n\nThe personalized data generation process was programmed with C++ and CUDA\\textregistered (NVIDIA Corporation, Santa Clara CA, USA) running on the GPU. The training process and the tracking process were programmed with Python with the open-source DL framework Chainer (Preferred Networks, Inc., Tokyo, Japan), running on the GPU.\n\nAll processes (i.e., personalized data generation, training, and tracking) were done with a single computer (Windows\\textregistered 7 64 bit, Intel Xeon CPU E5-2637 dual, 64GB RAM, Microsoft, Inc. Redmond WA, USA; NVIDIA Quadro M6000 24GB GPU dual). Personalized data generation, training, and tracking were run as two-view processes simultaneously by using dual GPUs.\n\n\n\\section{\\label{sec:level1}RESULTS}\n\\subsection{\\label{sec:level2}Digital simulation study}\n\n\\textbf{FIG. \\ref{fig8}} shows an example of a tracking image. There were no false detections, extra-detections, outside the tumour area, in spite of many similar rib structures in the tracking image.\n\n\\textbf{FIG. \\ref{fig9}} shows the tracking error distribution for all frames. The tracking error distribution was \\textless 1 $\\mathrm{mm}$ on the isocentre and almost \\textless 1 pixel on the FPD without effects of pixel resolution in both views and with both tumour shapes.\n\n\\textbf{TABLE \\ref{tab1}} shows a summary of tracking accuracy. We achieved 100\\% tracking accuracy in both views and with both tumour shapes in spite of overlying bone.\n\n\\begin{ruledtabular}\n\\begin{figure*}\n  \\begin{tabular}{ccc}\n     & \\textbf{Spherical tumour} & \\textbf{Ovoid tumour}\\\\ \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig8_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig8_upper_right.png}}\n    \\end{minipage} \\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig8_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig8_lower_right.png}}\n    \\end{minipage} \\\\\n  \\end{tabular}\n  \\caption{\\label{fig:epsart}An example of a tracking image in the digital simulation study. A calculated tumour coordinate is the centre of the green rectangle in the tracking image.}\n  \\label{fig8}\n\\end{figure*}\n  \n\\begin{figure*}\n  \\begin{tabular}{ccc}\n     & \\textbf{Spherical tumour} & \\textbf{Ovoid tumou}r\\\\ \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig9_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig9_upper_right.png}}\n    \\end{minipage} \\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig9_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.49\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig9_lower_right.png}}\n    \\end{minipage} \\\\\n  \\end{tabular}\n  \\caption{\\label{fig:epsart}The 2-dimensional tracking error distribution on FPD for all frames in the digital simulation study. The centre of a graph, the origin, means zero error. The scale units are one pixel on the FPD. The red circle means 1 mm error on the isocentre, the criteria for tracking accuracy.}\n  \\label{fig9}\n\\end{figure*}\n\n\\begin{table*}\n  \\caption{\\label{tab:table3}Summary of tracking accuracy in the digital simulation study. Tracking error shows `mean error' $\\pm$ `standard deviation' with (maximum error) for all frames.}\n\\begin{tabular}{cccccc}\n  & &\\multicolumn{2}{c}{\\textbf{View 1}} & \\multicolumn{2}{c}{\\textbf{View2}}\\\\\n \\multicolumn{2}{c}{\\textbf{Tumour shape}} & \\textbf{Spherical} & \\textbf{Ovoid} & \\textbf{Spherical} & \\textbf{Ovoid}\\\\ \\hline\n \\multicolumn{2}{c}{\\textbf{Tracking accuracy}} & 100\\% & 100\\% & 100\\% & 100\\%\\\\ \\hline\n  & \\textbf{Isocentre} & 0.05 $\\pm$ 0.02 & 0.05 $\\pm$ 0.03 & 0.06 $\\pm$ 0.03 & 0.07 $\\pm$ 0.04\\\\\n \\textbf{Tracking error} & [$\\mathrm{mm}$] & (0.12) & (0.13) & (0.16) & (0.25)\\\\ \\cline{2-6}\n \\textbf{(Maximum error)} & \\textbf{FPD} & 0.22 $\\pm$ 0.11 & 0.23 $\\pm$ 0.13 & 0.27 $\\pm$ 0.15 & 0.32 $\\pm$ 0.20\\\\\n  & [$\\mathrm{pixel}$] & (0.56) & (0.60) & (0.72) & (1.13)\\\\\n\\end{tabular}\n\\label{tab1}\n\\end{table*}\n\\end{ruledtabular}\n\n\\subsection{\\label{sec:level2}Epoxy phantom study}\n\n\\textbf{FIG. \\ref{fig10}} shows an example of a tracking image. There were no false detections in the search area using the 2- and 3-$\\mathrm{cm}$ tumours despite the tumour overlapping the spine in view 2. However, there were false detections in the search area in some frames using the 1-$\\mathrm{cm}$ tumours.\n\n\\textbf{FIG. \\ref{fig11}} shows the tracking error distribution for all frames. In the cases of the 2- and 3-$\\mathrm{cm}$ tumours, the tracking error distribution was almost \\textless 1 mm on the isocentre and almost \\textless 5 pixels on the FPD without effects of pixel resolution in both views. However, with the 1-$\\mathrm{cm}$ tumours, tracking error distribution was \\textgreater 1 mm, and was worse than with the 2- and 3-$\\mathrm{cm}$ tumours. A common bias error for all tumour sizes by manual positioning of the phantom was not detected.\n\n\\textbf{TABLE \\ref{tab2}} shows a summary of tracking accuracy. We achieved \\textgreater 94.7\\% tracking accuracy in 2- and 3-$\\mathrm{cm}$ tumours in spite of multi-modality DL using DRRs for training and X-ray images for tracking. However, tracking accuracy with 1-$\\mathrm{cm}$ tumours was only 40.3\\%.\n\n\\begin{ruledtabular}\n\\begin{figure*}\n  \\begin{tabular}{cccc}\n     & \\textbf{3-cm tumour} & \\textbf{2-cm tumour} & \\textbf{1-cm tumour}\\\\ \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_upper_middle.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_upper_right.png}}\n    \\end{minipage} \\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_lower_middle.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig10_lower_right.png}}\n    \\end{minipage} \\\\\n  \\end{tabular}\n  \n  \\caption{\\label{fig:epsart}An example of a tracking image in an epoxy phantom study. The calculated tumour coordinate is in the centre of the green rectangle. The red rectangle is the search area for a tumour.}\n  \\label{fig10}\n\\end{figure*}\n\n\\begin{figure*}\n  \\begin{tabular}{cccc}\n     & \\textbf{3-cm tumour} & \\textbf{2-cm tumour} & \\textbf{1-cm tumour}\\\\ \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 1}} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_upper_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_upper_middle.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_upper_right.png}}\n    \\end{minipage} \\\\ \\hline \\hline\n    \\rotatebox[origin=c]{90}{\\textbf{View 2}} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_lower_left.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_lower_middle.png}}\n    \\end{minipage} &\n    \\begin{minipage}{0.33\\linewidth}\n      \\centering\n      \\scalebox{0.2}{\\includegraphics{fig11_lower_right.png}}\n    \\end{minipage} \\\\\n  \\end{tabular}\n  \n  \\caption{\\label{fig:epsart}The 2-dimensional tracking error distribution on flat panel detectors (FPD) for all frames (except \\textgreater 6-pixel errors) in the epoxy phantom study. The centre of a graph, the origin, means zero error. The scale units are one pixel on FPD. The red circle means 1 $\\mathrm{mm}$ error on the isocentre, the criteria of tracking accuracy.}\n  \\label{fig11}\n\\end{figure*}\n\n\\begin{table*}\n\\caption{\\label{tab:table3}Summary of tracking accuracy in the epoxy phantom study. Tracking error shows `mean error' $\\pm$ `standard deviation' with (maximum error) for all frames.}\n\\begin{tabular}{cccccccc}\n  & &\\multicolumn{3}{c}{\\textbf{View 1}} & \\multicolumn{3}{c}{\\textbf{View2}}\\\\\n \\multicolumn{2}{c}{\\textbf{Tumour size}} & \\textbf{3 cm} & \\textbf{2 cm} & \\textbf{1 cm} & \\textbf{3 cm} & \\textbf{2 cm} & \\textbf{1 cm}\\\\ \\hline\n \\multicolumn{2}{c}{\\textbf{Tracking accuracy}} & 100\\% & 100\\% & 73.1\\% & 100\\% & 94.7\\% & 40.3\\%\\\\ \\hline\n  & \\textbf{Isocentre} & 0.3 $\\pm$ 0.1 & 0.3 $\\pm$ 0.1 & 0.6 $\\pm$ 0.4 & 0.3 $\\pm$ 0.2 & 0.5 $\\pm$ 0.2 & 1.8 $\\pm$ 4.7\\\\\n \\textbf{Tracking error} & [$\\mathrm{mm}$] & (0.5) & (0.9) & (1.8) & (0.8) & (1.3) & (30.4)\\\\ \\cline{2-8}\n \\textbf{(Maximum error)} & \\textbf{FPD} & 1.3 $\\pm$ 0.4 & 1.5 $\\pm$ 0.6 & 2.9 $\\pm$ 1.9 & 1.5 $\\pm$ 0.8 & 2.4 $\\pm$ 1.1 & 8.4 $\\pm$ 21\\\\\n  & [$\\mathrm{pixel}$] & (2.4) & (4.0) & (8.3) & (3.6) & (5.8) & (139.8)\\\\\n\\end{tabular}\n \\label{tab2}\n\\end{table*}\n\\end{ruledtabular}\n\n\\subsection{\\label{sec:level2}Processing time}\n\nThe personalized data generation processing took about 9 hours to output $156,250$ DRRs into a hard disk drive (HDD). This processing time is rate-limited by HDD access. The training process took about 17 hours, and is also rate-limited by HDD access. The tracking process (except image reading) took $32.5 \\pm 4.7$ $\\mathrm{ms\/frame}$ (30.8 $\\mathrm{fps}$) for real-time processing.\n\n\n\\section{\\label{sec:level1}DISCUSSION}\n\nWe proved the potential feasibility of a real-time markerless tumour tracking framework for stereotactic lung radiotherapy based on patient-specific DL with a personalized data generation strategy using a digital phantom simulation study and an epoxy phantom study. The personalized data generation and training process could be done end-to-end automatically between treatment planning and treatment, without manual procedures such as template creation just before the treatment process while a patient is positioned and fixed. Also, our framework does not require cone beam CT (CBCT) data taken while a patient is positioned. Therefore, our framework has potential to improve treatment throughput and reduce patient burden, compared to some conventional markerless tracking frameworks which require manual procedures just before the treatment process or require CBCT\\cite{8Patel}\\cite{9Shiinoki}\\cite{10Teske}\\cite{11Shieh}. Also, our framework using DL with a large personalized training data set has potential to improve the robustness and accuracy of markerless tracking, compared to frameworks using conventional machine learning with a small personalized training data set\\cite{12Li}.\n\nThe digital simulation study had 100\\% tracking accuracy in spite of overlying bone. Also, as the tracking errors were significantly smaller than the displacements for training DRRs, they could be easily distinguished from each other. These results indicate that our method was able to learn tumour shape while ignoring overlying bone by using training data sets with a variety of overlying bone patterns. Also, we succeeded in tracking both spherical and ovoid tumour phantoms. This result indicates that our method has the potential to track all varieties of tumour shape.\n\nIn the epoxy phantom study, we achieved \\textgreater 94.7\\% tracking accuracy in 2- and 3-$\\mathrm{cm}$ tumours in spite of multi-modality DL using DRRs for training and X-ray images for tracking. This result indicates that we were able to allow it to learn modality invariance characteristics by using training data sets with a range of contrast and noise. This was achieved despite discrete 4D-CT phases and a narrower 4D-CT imaging range of tumour motion. This result indicates its success at interpolation and extrapolation of tumour motion. Also, this was achieved without being affected by manual positioning error despite the fixed bone structures. This result indicates that our model did not learn only relative positional relationships between bone structures and a tumour. However, we need additional evaluation of tumour trajectories in all three dimensions, not only in the SI direction, in order to validate its robustness for irregular tumour motion compared with training data. We achieved this accuracy despite motion artefact from a tumour in 4D-CT. We consider that our method has the potential to detect tumours regardless of motion in 4D-CT or inter- and intra-fractional changes in tumour shape. However, we need additional evaluation with a retrospective clinical study to validate its robustness. In 1-$\\mathrm{cm}$ tumours, tracking was difficult because of the low contrast of the X-ray images due to scattered radiation. We required additional evaluation with 1-$\\mathrm{cm}$ `tumours' in the digital simulation study, which ignores scattered radiation. The tracking accuracy could be improved in these small masses by improving the contrast variation algorithm for training DRRs to simulate scattered radiation. Currently, we have calculated a tumour coordinate using only a present frame without temporal information such as tumour coordinates in past frames. We may be able to reduce tumour detection inaccuracies by using temporal information. In this case, we can evaluate tracking accuracy by using irregular respiratory waveforms. \n\nIn the labelled image, tumour area was labelled as the projection area of a tumour on the 4D-CT. But, in actual treatment workflow, tumour area can be labelled as the planning target volume (PTV) or clinical target volume (CTV) generated by treatment planning by projecting PTV or CTV to a training DRR. In this case, tumour contours also can be acquired by positioning PTV or CTV on the tumour centre coordinate during tracking.\n\nWe consider that patient-specific DL has the potential to provide better accuracy for each patient than standard DL using multiple-patient data sets. Patient-specific DL can be considered overfitting for a specific patient. Generally speaking, standard DL provides good accuracy for all patients. Conversely, overfitting provides better accuracy for a specific patient than standard DL.\n\nThe personalized data generation process and training process took about 26 hours in total. These processes have to be done between treatment planning and treatment in the workflow. The training time of 26 hours might not be problematic because, generally, treatment commences about one week after treatment planning. However, more than 24 hours may not be sufficient if our method is applied for retraining or fine tuning during the course of treatment, or for instance at an MRI linac. We can substantially shorten these processing times by generating DRRs and training data sets in memory without rate-limiting HDD accessibility.\n\nThe tracking process requires \\textgreater 15 $\\mathrm{fps}$ (\\textless 66.7 $\\mathrm{ms\/frame}$) real-time processing, so our tracking process of 32.5 $\\mathrm{ms\/frame}$ (30.8 $\\mathrm{fps}$) is sufficient.\n\n\n\\section{\\label{sec:level1}CONCLUSIONS}\n\nWe proved the potential feasibility of a real-time markerless tumour tracking framework for stereotactic lung radiotherapy based on patient-specific DL with personalized data generation by evaluating the tracking accuracy in both digital phantom and epoxy phantom studies.\n\n\n\\section*{\\label{sec:level1}CONFLICT OF INTEREST STATEMENT}\n\nWataru Takahashi and Shota Oshikawa are employees of Shimadzu Corporation, Japan.\n\n\n\\section*{\\label{sec:level1}FUNDING}\n\nThis work was supported by Shimadzu Corporation, Kyoto, Japan and National Institute of Radiological Sciences, Chiba, Japan.\n\n\n\\begin{acknowledgments}\nConstruction of the tracking accuracy evaluation system was supported by Ryuichi Ito, Seiji Yamanaka, and Yui Torigoe of Shimadzu Corporation. We thank Libby Cone, MD, MA, from DMC Corp. (\\url{http:\/\/www.dmed.co.jp\/}) for editing drafts of this manuscript.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Conclusion}\nThis work proposes a comprehensive analyze on how a DNN can describe emotional states. To this purpose, we first studied how many dimensions are sufficient to accurately represent an emotion resulting from a facial expression. We then conclude that three dimensions are a good trade-off between accuracy and compactness, agreeing with the arousal-valence-dominance~\\cite{russell_circumplex_1980}\\cite{mehrabian1996pleasure} psychologist model.\nThereby, we came up with a DNN providing a 3-dimensional compact representation of emotion, learned in a multi-domain fashion on RAF~\\cite{li_reliable_2017}, SFEW~\\cite{dhall_static_2011} and AffecNet~\\cite{mollahosseini_affectnet:_2017}. We set up a comparison with the state-of-the-arts and showed that our model can compete with models having much larger feature sizes. It proves that bigger representations are not necessary for emotion recognition. In addition, we implemented a visualization process enabling to qualitatively evaluate the consistency of the compact features extracted from emotion faces by our model. We thus showed that DNN trained on emotion recognition are naturally learning an arousal-valence-like~\\cite{russell_circumplex_1980} encoding of the emotion. As a future work we plan to also apply state-of-the-art techniques -- as Deep Locality Preserving Loss~\\cite{li_reliable_2017} or Covariance Pooling~\\cite{acharya_covariance_2018} -- to enhance our compact representation. In addition, nothing warranty that the learned CAKE bears the same semantic meanings as arousal-valence-dominance does: further interpreting the perceived semantic of the dimensions would therefore be an interesting piece of work.\n\n\n\\section{Introduction}\n\\begin{figure}\n\\begin{floatrow}\n\\ffigbox[0.45\\textwidth]{\\includegraphics[width=\\linewidth]{Visualizations\/EmotionArouval_TEST_bigger-cropped.pdf}}{\\caption{Comparison of the discrete and continuous (arousal-valence) representations using AffectNet's annotations~\\cite{mollahosseini_affectnet:_2017}.}\\label{fig:affecnet_arouval}}\n\\ffigbox[.45\\textwidth]{\\input{plots\/courbeAVdim.tex}}{\\caption{Influence of adding supplementary dimensions to arousal-valence when predicting emotion on AffectNet~\\cite{mollahosseini_affectnet:_2017}.}\\label{fig:courbeAVdim}}\n\\end{floatrow}\n\\end{figure}\n\nFacial expression is one of the most used human means of communication after language. Thus, the automated recognition of facial expressions -- such as emotions -- has a key role in affective computing, and its development could benefit human-machine interactions. \n\nDifferent models are used to represent human emotion states. Ekman~\\textit{et al.}~\\cite{ekman_constants_1971} propose to classify the human facial expression resulting from an emotion into six classes (\\textit{resp.} happiness, sadness, anger, disgust, surprise and fear) supposed to be independent across the cultures. This model has the benefit of simplicity but could be not sufficient to address the whole complexity of human affect. Moreover it suffers from serious intra-class variations as, for instance, soft smile and laughing equally belong to \\textit{happiness}. That is why Ekman's emotion classes are sometimes assembled into compound emotions~\\cite{du2014compound} (\\textit{e.g.} happily surprised).\nOthers have chosen to represent emotion with an n-dimensional continuous space, as opposite to the Ekman's discrete classes. Russel has built the \\textit{Circumplex Model of Affect}~\\cite{russell_circumplex_1980} in which emotion states are described by two values: arousal and valence.\n\\textit{Arousal} represents the excitation rate -- the higher the arousal is, the more intense the emotion is -- and \\textit{valence} defines whether the emotion has a positive or a negative impact on the subject. Russels suggests in \\cite{russell_circumplex_1980} that all Ekman's emotions~\\cite{ekman_constants_1971} and compound emotions could be mapped in the \\textit{circumplex model of affect}. Furthermore, this two-dimensional approach allows a more accurate specification of the emotional state, especially by taking its intensity into account.\n\nA third dimension has been added by Mehrabian~\\textit{et al.}~\\cite{mehrabian1996pleasure} -- the \\textit{dominance} -- which depends on the degree of control exerted by a stimulus. Last, Ekman and Friesen~\\cite{ekman_measuring_1976} have come up with the \\textit{Facial Action Code System} (FACS) using anatomically based action units. Developed for measuring facial movements, FACS is well suited for classifying facial expressions resulting from an affect. \n\nBased on these emotion representations, several large databases of face images have been collected and annotated according to emotion. EmotioNet~\\cite{benitez-quiroz_emotionet:_2016} gathers faces annotated with Action Units~\\cite{ekman_measuring_1976}; SFEW~\\cite{dhall_static_2011}, FER-13~\\cite{goodfellow_challenges_2013} and RAF~\\cite{li_reliable_2017} propose images in the wild annotated in basic emotions; AffecNet~\\cite{mollahosseini_affectnet:_2017} is a database annotated in both discrete emotion~\\cite{ekman_constants_1971} and arousal-valence~\\cite{russell_circumplex_1980}.\n\nThe emergence of these large databases has allowed to develop automatic emotion recognition systems, such as the recent approaches based on Deep Neural Networks (DNN). AffectNet's authors~\\cite{mollahosseini_affectnet:_2017} use three AlexNet~\\cite{krizhevsky_imagenet_2017} to learn respectively emotion classes, arousal and valence. In \\cite{ng_deep_2015}, the authors make use of transfer learning to counteract the smallness of the SFEW~\\cite{dhall_static_2011} dataset, by pre-training their model on ImageNet~\\cite{deng_imagenet:_2009} and FER~\\cite{goodfellow_challenges_2013}. In \\cite{acharya_covariance_2018} authors implement {\\em Covariance Pooling} using second order statistics when training on emotion recognition (on RAF~\\cite{li_reliable_2017} and SFEW~\\cite{dhall_static_2011}). \n\nEmotion labels, FACS and continuous representations have their own benefits -- simplicity of the emotion classes, accuracy of the arousal-valence, objectivity of the FACS, \\textit{etc.} -- but also their own drawbacks --  imprecision, complexity, ambiguity, \\textit{etc}. \nTherefore several authors have tried to leverage the benefits of all these representations. Khorrami \\textit{et al.}~\\cite{khorrami_deep_2015} first showed that neural networks trained for expression recognition implicitly learn facial action units.\nContributing to highlighting the close relation between emotion and Action Units, Pons \\textit{et al.}~\\cite{pons_multi-task_2018} learned a multitask and multi-domain ResNet~\\cite{he_deep_2015} on both discrete emotion classes (SFEW~\\cite{dhall_static_2011}) and Action Units (EmotioNet~\\cite{benitez-quiroz_emotionet:_2016}). \nFinally, Li \\textit{et al.}~\\cite{li_reliable_2017} proposed a \"\\textit{Deep Locality-Preserving Learning}\" to handle the variability inside an emotion class, by making classes as compact as possible. \n\nIn this context, this paper focuses on the links between arousal-valence and discrete emotion representations for image-based emotion recognition. More specifically, the paper proposes a methodology for learning very compact embedding, with not more than 3 dimensions, performing very well on emotion classification task, making the visualization of emotions easy, and bearing similarity with the arousal-valence representation. \n\n\n\\section{Learning Very Compact Emotion Embeddings}\n\\label{methods}\n\\subsection{Some Intuitions About Emotion Representations}\n\\label{subsec:preliminary_study}\n\nWe first want to experimentally measure the dependence between emotion and arousal-valence as yielded in~\\cite{russell_circumplex_1980}. We thus display each sample of the AffectNet~\\cite{mollahosseini_affectnet:_2017} validation subset in the arousal-valence space and color them according to their emotion label (Figure~\\ref{fig:affecnet_arouval}). For instance, a face image labelled as \\textit{neutral} with an arousal and a valence of zero is located at the center of Figure~\\ref{fig:affecnet_arouval} and colored in blue. It clearly appears that a strong dependence exists between discrete emotion classes and arousal-valence. Obviously, it is due in part to the annotations of the AffectNet~\\cite{mollahosseini_affectnet:_2017} dataset, as the arousal-valence have been constrained to lie in a predefined confidence area based on the emotion annotation. Nevertheless, this dependence agrees with the \\textit{Circumplex Model of Affect}~\\cite{russell_circumplex_1980}.\n\nTo evaluate further how arousal-valence representation is linked to emotion labels, we train a classifier made of one  fully connected layer\\footnote{By \"fully connected layer\" we denote a linear layer with biases and without activation function.} (fc-layer) to infer emotion classes from arousal-valence values provided by AffectNet~\\cite{mollahosseini_affectnet:_2017} dataset. We obtain the accuracy of 83\\%, confirming that arousal-valence can be an excellent \\textit{2-d} compact emotion representation.  \n\nThis raises the question of the optimality of this 2-\\textit{d} representation. Would adding a third dimension to arousal-valence make the classification performance better? To address this question, we used the 512-\\textit{d} hidden representation of a ResNet-18~\\cite{he_deep_2015} trained to predict  discrete emotions on the AffectNet dataset~\\cite{mollahosseini_affectnet:_2017}. This representation is then projected into a more compact space using a fc-layer outputting $k$ dimensions, which are concatenated with the arousal-valence values. On top of this representation, we add another fc-layer predicting emotion classes. The two fc-layers are finally trained using Adam optimizer~\\cite{kingma2014adam}.\nAdding 1 dimension to arousal-valence gives a gain of +3 points on the accuracy. It agrees with the assumption that a three-dimensional representation is more meaningful than a two-dimensional one~\\cite{mehrabian1996pleasure}. The benefit of adding more than 1 dimension is exponentially decreasing; with +512 dimensions, the gain is only of +0.6 points compared to adding 1 dimension, as shown in Figure~\\ref{fig:courbeAVdim}.\n\nFrom these observations, the use of a compact representation seems to be consistent with discrete emotion classes, as it enables an accuracy of 83\\% and 86\\% -- respectively for a 2-\\textit{d} and a 3-\\textit{d} representation -- and it even may allow to describe affect states with more contrast and accuracy. \nEven if arousal-valence is a good representation for emotion recognition, the question of its optimality has not been answered by these preliminary experiments. In other words, is it possible to learn 2-\\textit{d} (or 3-\\textit{d}) embedding better than those built on arousal-valence? We positively answer this question in Section~\\ref{subsec:method}. \n\n\\subsection{Learning Compact and Accurate Representations of Emotions}\n\\label{subsec:method}\nBased on the previous observations, this section proposes a methodology for learning a compact embedding for emotion recognition from images.\n\\paragraph{Features extraction}\nThe basic input of our model is an image containing one face displaying a given emotion. We first extract 512-\\textit{d} features specialized in emotion recognition.\nSo as to, we detect the face, align its landmarks by applying an affine transform and crop the face region. The so-obtained face is then resized into $224\\times224$ and fed to a ResNet-18~\\cite{he_deep_2015} network (Figure~\\ref{fig:cross_arch}, \\textit{Features extraction}). The face image is augmented (\\textit{e.g.} jittering, rotation), mostly to take the face detector noise into account. We also use cutout~\\cite{devries_improved_2017} -- consisting in randomly cutting a $45\\times45$ pixels sized patch from the image -- to regularize and improve the robustness of our model to facial occlusions.\nOur ResNet outputs 512-\\textit{d} features, on top of which a fc-layer can be added. At training time, we also use dropout~\\cite{srivastava2014dropout} regularization.\nThe neural network can be learned from scratch on two given tasks: discrete emotion classification or arousal-valence regression. \n\n\\paragraph{Compact emotion encoding}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth]{Diagrams\/model_schema.pdf}\n    \\caption{Our approach's overview. Left: we use a ResNet-18 previously trained for discrete emotion recognition or arousal valence regression to extract 512-d hidden representations from face images. Center: using these hidden representations, CAKE or AVk representations (center) are learned to predict discrete emotions. Right: the learning process is multi-domain, predicting emotions on three different datasets with three different classifiers. Gray blocks are non-trainable weights while blue blocks are optimized weights.}\n    \\label{fig:cross_arch}\n\\end{figure}\nCompact embedding is obtained by projecting the 512-\\textit{d} features provided by the ResNet-18 (pretrained on discrete emotion recognition) into smaller k-dimensional spaces (Figure~\\ref{fig:cross_arch}, \\textit{Emotion Encoding}) in which the final classification is done.\nThe $k$ features may be seen as a compact representation of the emotion, and the performance of the classifier can be measured for different values of $k$. CAKE-2, CAKE-3, \\textit{etc.}, denote such classifiers with $k=2$, $k=3$, \\textit{etc}.\n\nIn the same fashion we can train the ResNet-18 using arousal-valence regression. In this case, the so-obtained arousal-valence regressor can be used to infer arousal-valence values from novel images and concatenate them to the $k$ features of the embedding. Thus we reproduce here the exact experiment done in Section~\\ref{subsec:preliminary_study} in order to assess the benefit of a third (or more) dimension. The difference is that arousal-valence are not ground truth values but predicted ones. These methods are denoted as AV1, AV2, AV3, \\textit{etc.} for the different values of $k$.\n\n\\paragraph{Domain independent embedding}\nAs we want to ensure a generic compact enough representation, independent of the datasets, we learn the previously described model jointly on several datasets, without any further fine-tuning.\n\nOur corpus is composed of AffectNet~\\cite{mollahosseini_affectnet:_2017}, RAF~\\cite{li_reliable_2017} and SFEW~\\cite{dhall_static_2011}, labelled with seven discrete emotion classes: \\textit{neutral}, \\textit{happiness}, \\textit{sad}, \\textit{surprise}, \\textit{fear}, \\textit{disgust} and \\textit{anger}.\nOur training subset is composed of those of AffectNet (283901 elts., \\textit{95.9\\%} of total), RAF (11271 elts., \\textit{3.81\\%} of total) and SFEW (871 elts., \\textit{0.29\\%} of total). Our testing subset is composed of the subsets commonly used for evaluation in the literature (\\textit{validation} of SFEW and AffecNet, \\textit{test} of RAF).\n\nTo ease the multi-domain training, we first pre-train our features extractor model on AffectNet and freeze its weights. Then we apply the same architectures as described before, but duplicate the last fc-layer in charge of emotion classification in three dataset-specific layers (Figure~\\ref{fig:cross_arch}, \\textit{multi-domain learning}). The whole model loss is a modified softmax cross entropy defined as follows:\n\\begin{equation}\nLoss=\\frac{1}{N} \\sum_{i=1}^{N} \\sum_{j=1}^{3} w_{class}^{i,j} w_{dataset}^{j} \\ E(y^i,\\hat{y}^{i,j})\n\\end{equation}\nwhere $j$ is ranging in [AffectNet, RAF, SFEW],  $y^i$ is the label of $i^{th}$ element, $\\hat{y}^{i,j}$ is the prediction of the $j^{th}$ classifier on the $i^{th}$ element, E is the softmax cross entropy loss,  $N$ is the number of elements in the batch, $w_{class}^i$ is a weight given to the $i^{th}$ element of the batch depending on its emotion class and $w_{dataset}^{j}$ is a weight given to the $j^{th}$ classifier prediction.\nEach sample of the multi-domain dataset is identified according to its original database, allowing to choose the correct classifier's output when computing the softmax cross entropy.\n\nThe $ w_{class}$ weight is defined as:\n$\n    w_{class}^{i,j} = \\frac{N_{total}^j}{N_{class}^{i,j} \\times nbclass}\n$\nwhere $N_{total}^j$ is the number of elements in the $j^{th}$ dataset, $N_{class}^{i,j}$ is the number of elements in the class of the $i^{th}$ element of the $j^{th}$ dataset and $nbclass$ is the number of classes (7 in our case). The goal here is to fix the important class imbalance in the dataset by forcing to fit the uniform distribution, as previously done by \\cite{mollahosseini_affectnet:_2017}.\n\nThe $w_{dataset}$ weight permits to take the imbalance between dataset's sizes into account.\n\\begin{align}\n w_{dataset}^{j}= \\left\\{\n \\begin{array}{cl}\n      \\frac{1}{\\log N_{total}^{j}}  & sample \\in j^{th} dataset\\\\\n      0       & sample \\notin j^{th} dataset\n\\end{array}\n\\right.\n\\end{align}\n\nWe thus define a global loss enabling to optimize the last two layers of our model (namely \\textit{Emotion Encoding} and \\textit{Multi-domain Learning}  in Figure~\\ref{fig:cross_arch}) on the three datasets at the same time. The dimension $k$ (or $k+2$ in the case of the arousal-valence approach) can easily be changed and help to evaluate the interest of supplementary dimensions for emotion representation. \n\n\n\n\\section{Experiments}\n\\label{results}\n\\subsection{Evaluation Metrics}\nWe measure the classification performance with the \\textit{accuracy} and the \\textit{\\textbf{macro} F1 Score}~\\eqref{eq:f1}. \\textit{Accuracy} measures the number of correctly classified samples. Instead of accuracy, we prefer \\textit{macro F1 score} which gives the same importance to each class:\n\\begin{equation}\\begin{aligned}\nF_{1macro}=\\frac{1}{N_c}\\sum_{i}^{N_c}F_{1i} \\quad\nF_{1i}=2\\frac{prec_i \\cdot rec_i}{prec_i+rec_i} \\quad\nprec_i=\\frac{tp_i}{tp_i+fp_i} \\quad\nrec_i=\\frac{tp_i}{tp_i+fn_i} \\quad\n\\label{eq:f1}\n\\end{aligned}\n\\end{equation}\nwhere $i$ is the class index; $prec_i$, $rec_i$  and $F_{1i}$ are the precision, the recall and the F1-score of class $i$; $N_c$ is the number of classes; $tp$, $fp$ and $fn$ are the true positives, false positives and false negatives rates. All scores are averaged on 10 runs, with different initializations, and given with associated standard deviations, on our multi-domain testing subset.\n\n\\subsection{Compactness of the Representation}\n\\label{subsec:compactness}\n\\begin{figure}\n\\begin{floatrow}\n\\ffigbox[.5\\textwidth]{\\input{plots\/compare_dim.tex}\n}{\\caption{Influence of representation size on the multi-domain F1 score.}\\label{fig:repsize}}\n\n\\capbtabbox{\\input{plots\/multiDomainRes.tex}}{\\caption{Evaluation of compact representations on AffectNet, SFEW, RAF.}\n\\label{table:compact}}\n\\end{floatrow}\n\\end{figure}\n\nWe first evaluate the quality of the representations in a multi-domain setting. Table~\\ref{table:compact} reports the  F1-score of CAKE-2, AV, CAKE-3 and AV1 trained on three datasets with three different classifiers, each one being specialized on a dataset as explained in Section~\\ref{methods}. Among the 2-\\textit{d} models (AV and CAKE-2), AV is better, taking benefits from the knowledge transferred from the AffectNet dataset. This is not true anymore for the 3D models, where CAKE-3 is better than AV1, probably because of its greater number of trainable parameters.\n\nTo validate the hypothesis of the important gain brought by adding a third dimension, we run the \"CAKE\" and \"AVk\" experiments with different representation sizes. To simplify the analysis of the results, we plot in Figure~\\ref{fig:repsize} a multi-domain F1-score, \\textit{i.e.} the weighted average of the F1-scores according to the respective validation set sizes.\nWe observe that the gain in multi-domain F1-score is exponentially decreasing for both representations -- note that the representation size axis is in log scale -- and thus the performance gap between a representation of size 2 and size 3 is the more important. We also observe that \"CAKE\" representations still seem to yield better results than \"AVk\" when the representation size is greater than 2. \n\nThese first experiment shows that a very compact representation can yield good performances for emotion recognition. It also is in line with the \"dominance\" dimension hypothesis, as a third dimension brought the more significant gain in performance. After 3 dimensions, the gain is much less significant.\n\n\\subsection{Accuracy of the Representation}\nTo evaluate the efficiency of the CAKE-3 compact representation, we compare its accuracy with \nstate-of-the-art approaches (Table~\\ref{table:comparison}) on the public datasets commonly used in the literature for evaluation (\\textit{validation} of SFEW and AffecNet, \\textit{test} of RAF). In order to get a fair comparison, we add a \\textit{\"Rep. Dim.\"} column corresponding to the size of the last hidden representation -- concretely, we take the penultimate fully connected output size.\nWe report the scores under the literature's metrics, namely the mean of the per class recall for RAF~\\cite{li_reliable_2017} and the accuracy for SFEW~\\cite{dhall_static_2011} and AffectNet~\\cite{mollahosseini_affectnet:_2017}. To the best of the author's knowledge no other model has been evaluated before on the AffectNet's seven classes.\n\nCAKE-3 is outperformed by Covariance Pooling~\\cite{acharya_covariance_2018} and Deep Locality Preserving~\\cite{li_reliable_2017}.\nNevertheless, it is still competitive as the emotion representation is far more compact -- 3-\\textit{d} \\textit{versus} 2000-\\textit{d} -- and learned in a multi-domain fashion. Moreover, we gain 1 point on RAF when we compare to models of same size (2 millions parameters), \\textit{e.g.} \\textit{Compact Model}~\\cite{kuo2018compact}. These results support the conclusion made in \\ref{subsec:compactness}, as we show that a compact representation of the emotion learned by small models is competitive with larger representations. This finally underlines that facial expressions may be encoded efficiently into a 3-\\textit{d} vector and that using a large embedding on small datasets may lead to exploit biases of the dataset more than to learn emotion recognition.\n\n\\begin{table}\n\\begin{tabular}{c|c|ccc|}\n\\cline{2-5}\n    & Rep. Dim. & \\multicolumn{1}{c}{RAF~\\cite{li_reliable_2017}} & \\multicolumn{1}{c}{SFEW~\\cite{dhall_static_2011}} & AffectNet~\\cite{mollahosseini_affectnet:_2017}    \\\\ \\hline\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{Covariance Pooling~\\cite{acharya_covariance_2018}}}    & 2000                & 79.4                     & -                         & -              \\\\ \\cline{2-5} \n\\multicolumn{1}{|c|}{}                                                                      & 512                 & -                        & 58.1                     & -              \\\\ \\hline\n\\multicolumn{1}{|c|}{Deep Locality Preserving~\\cite{li_reliable_2017}}                      & 2000                & 74.2                    & 51.0                     & -              \\\\ \\hline\n\\multicolumn{1}{|c|}{Compact Model~\\cite{kuo2018compact}}                                   & 64                  & 67.6                    & -                         & -              \\\\ \\hline\n\\multicolumn{1}{|c|}{VGG\\cite{li_reliable_2017}}                                            & 2000                & 58.2                    & -                         & -              \\\\ \\hline\n\\multicolumn{1}{|c|}{Transfer Learning~\\cite{ng_deep_2015}}                                 & 4096                & -                        & 48.5                      & -              \\\\ \\hline\\hline\n\\multicolumn{1}{|c|}{{ours (CAKE-3)}}                                                      & {3}          & {68.9}           & {44.7}            & {58.2} \\\\ \\hline\n\\multicolumn{1}{|c|}{{ours (Baseline)}}                                                & {512}        & {71.7}           & {48.7}            & {61.7} \\\\ \\hline\n\\end{tabular}\n\\caption{Accuracy of our model regarding state-of-the-art methods. The size of the representation is taken into account. Metrics are the average of per class recall for RAF and accuracy for SFEW and AffectNet.}\n\\label{table:comparison}\n\\end{table}\n\nOur experiments also allow to perform a cross-database study as done in \\cite{li_reliable_2017}. This study consists in evaluating a model trained on dataset B on a dataset A. Thereby we obtain Table~\\ref{table:cross_f1} with the evaluation of each classifier on each dataset.\nResults on SFEW~\\cite{dhall_static_2011} -- trained or evaluated -- are constantly lower than others, with a higher standard deviation. This could be due to the insufficient number of samples in the SFEW training set or more probably to the possible ambiguity in the annotation of SFEW compared to AffectNet and RAF. Confirming this last hypothesis, the \\textit{RAF classifier} has the better generalization among the datasets. It is in line with the claim of Li~\\textit{et al.}~\\cite{li_reliable_2017} that RAF has a really reliable annotation with a large consensus between different annotators. Finally, it also underlines the difficulty to find a reliable evaluation of an emotion recognition system because of the important differences between datasets annotations.\n\n\\begin{table}\n\\begin{tabular}{cl|l|l|l|}\n\\cline{3-5}\n\\multicolumn{1}{l}{}                                       &           & \\multicolumn{3}{c|}{Dataset}                     \\\\ \\cline{3-5} \n\\multicolumn{1}{l}{}                                       &           & AffectNet         & SFEW              & RAF      \\\\ \\hline\n\\multicolumn{1}{|c|}{\\multirow{3}{*}{Classifier}} & AffectNet & \\textbf{58.1 } \\textit{($\\pm$ 0.5)} & 27.6 \\textit{($\\pm$ 2.6)}          & 53.8  \\textit{($\\pm$ 0.6)} \\\\ \\cline{2-5} \n\\multicolumn{1}{|c|}{}                                     & SFEW      & 35.1  \\textit{($\\pm$ 2.1)}          & \\textbf{34.1 } \\textit{($\\pm$ 1.0)} & 47.3  \\textit{($\\pm$ 1.2)} \\\\ \\cline{2-5} \n\\multicolumn{1}{|c|}{}                                     & RAF       & 51.8  \\textit{($\\pm$ 0.4)}          & 31.5  \\textit{($\\pm$ 1.7)}         & \\textbf{64.4 } \\textit{($\\pm$ 0.6)} \\\\ \\hline\n\\end{tabular}\n\\caption{Cross-database evaluation on CAKE-3 model (F1-Score).}\n\\label{table:cross_f1}\n\\end{table}\n\\subsection{Visualizing Emotion Maps}\nVisualizations are essential to better appreciate how DNN performs classifications, as well as to visualize emotion boundaries and their variations across datasets. Our visualization method consists in densely sampling the compact representation space -- 2-\\textit{d} or 3-\\textit{d} -- into a mesh grid, and feeding it to a formerly trained model -- AV, CAKE-2 or CAKE-3 -- in order to compute a dense map of the predicted emotions. Not all the coordinates of the mesh grid belong to real emotions and some of them would never happen in real applications. \n\nThe construction of the mesh grid depends on the model to be used. \nFor the AV and the CAKE-2 models, we have simply built it using 2d vectors with all values ranging in intervals containing maximum and minimum values of the coordinates observed with real images. As the CAKE-3 model is dealing with a three-dimensional representation, it is not possible to visualize it directly on a plane figure.\nTo overcome this issue we modify CAKE-3 into a CAKE-3-Norm representation where all the coordinates are constrained to be on the surface of the unit sphere, and visualize spherical coordinates. \nEven if CAKE-3-Norm shows lower performances (about 2 points less than CAKE-3), the visualization is still interesting, bringing some incentives about what has really been learned.\n\nFigure~\\ref{fig:visu} shows the visualization results  for CAKE-3-Norm, AV and CAKE-2 representations (\\textit{resp.} from top to down).\nEach dot is located by the coordinates of its compact representation -- \\((arousal, valence)\\) for AV, \\((k_1, k_2)\\) for CAKE-2 and spherical coordinates ($\\phi$ and $\\theta$) for CAKE-3-Norm -- and colored according to the classifier prediction. The per class macro F1-score is displayed inside each emotion area.\n\nFirst, each compact representation -- CAKE-2, CAKE-3-Norm and AV -- exhibits a strong consistency across the datasets (in Figure~\\ref{fig:visu}, compare visualizations on the same row). Indeed, the three classifiers show a very similar organization of the emotion classes, which is demonstrating the reliability of the learned representation. Thereby, the \\textit{neutral} class -- in blue -- is always placed at the origin and tends to neighbor all other classes. It is in line with the idea of neutral as an emotion with a very low intensity. Nevertheless, we can witness small inter-dataset variations, especially on SFEW~\\cite{dhall_static_2011} (in Figure~\\ref{fig:visu}, middle column) with \\textit{disgust} and \\textit{fear} -- \\textit{resp.} brown and purple -- which are almost missing. This underlines the disparities of annotations across the datasets and confirms the need of multi-domain frameworks when wishing to achieve a more general emotion recognition model.\n\nSecond, we can analyze variations between the different representations for a given dataset (in Figure~\\ref{fig:visu}, compare visualizations on the same column). As AV is based on arousal-valence, we observe the same emotion organization as in Figure \\ref{fig:affecnet_arouval}. Especially, as the majority of the AffectNet's training (and validation) samples have a positive arousal, the classifier do not use the whole space (in Figure~\\ref{fig:visu}, second row: see green, blue and orange areas) unlike CAKE-2 and CAKE-3 which are not constrained by arousal-valence. \n\nWe can find many similarities between these three representations, but the most impressive come across when comparing CAKE-2 and AV. Despite the inequality of scaling -- which causes the \\textit{neutral} area (blue) to be smaller in CAKE-2 -- AV and CAKE-2 compact representations are very close. Indeed, the area classes are organized exactly in the same fashion. The only difference is that for AV they are disposed in a clockwise order around \\textit{neutral} whereas for CAKE-2 they are disposed in an anticlockwise order. This observation shows that a DNN trained on the emotion recognition classification is able to learn an arousal-valence-like representation of the emotion. It contributes -- along with Khorrami~\\cite{khorrami_deep_2015} who points that DNNs trained to recognize emotions are learning action units~\\cite{ekman_measuring_1976} -- to bring the dependence across the emotion representations in the forefront.\n\n\\begin{figure}[hb]\n    \\begin{minipage}{.86\\linewidth}\n    \\includegraphics[width=\\linewidth]{Visualizations\/gridEval_K3_nolegend-cropped.pdf}\n    \\end{minipage}\n    \\begin{minipage}{.86\\linewidth}\n    \\includegraphics[width=\\linewidth]{Visualizations\/gridEval_AV_nolegend-cropped.pdf}\n    \\end{minipage}\n    \\begin{minipage}{.86\\linewidth}\n    \\includegraphics[width=\\linewidth]{Visualizations\/gridEval_K2-cropped_v2-cropped.pdf}\n    \\end{minipage}\n  \\caption{Visualization of CAKE-3-Norm, AV and CAKE-2. Rows indicate evaluated representation -- \\textit{resp.} from top to down: CAKE-3-Norm, AV, CAKE-2 -- and columns indicate datasets -- \\textit{resp.} from left to right: AffectNet~\\cite{mollahosseini_affectnet:_2017}, SFEW~\\cite{dhall_static_2011} and RAF~\\cite{li_reliable_2017}.}\n  \\label{fig:visu}\n\\end{figure}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn recent years there has been a renewed interested in the spectral \ntheory of non-self-adjoint operators where, as opposed to the self-adjoint \ncase, the norm of the resolvent can be very large even far away from \nthe spectrum. Equivalently the spectrum of such operators can be \nhighly unstable even under very small perturbations of the operator. \n\\par\nEmphasized by the works of L.N. Trefethen and M. Embree, see for example \n\\cite{TrEm05}, E.B. Davies, M. Zworski and many others \n\\cite{Da97,Da99,NSjZw04,ZwChrist10,DaHa09}, \nthe phenomenon of spectral instability of non-self-adjoint operators has become \na popular and vital subject of study. In view of this it is very natural to add \nsmall random perturbations.\n\\par\nOne line of recent research concerns the case of elliptic (pseudo)\\-differential \noperators subject to small random perturbations, cf. \n\\cite{BM,Ha06,Ha06b,HaSj08,SjAX1002,Vo14}.\n\\subsection{Perturbations of Jordan blocks}\nIn this paper we shall study the spectrum of a random perturbation of\nthe large Jordan block $A_0$ :\n\\begin{equation}\\label{jb}\nA_0=\\begin{pmatrix}0 &1 &0 &0 &...&0\\\\\n                   0 &0 &1 &0 &...&0\\\\\n                   0 &0 &0 &1 &...&0\\\\\n                   . &. &. &. &...&.\\\\\n                   0 &0 &0 &0 &...&1\\\\\n                   0 &0 &0 &0 &...&0\n\\end{pmatrix}: {\\mathds{C}}^N\\to {\\mathds{C}}^N.\n\\end{equation}\nPerturbations of a large Jordan block have already been studied, \ncf. \\cite{SjZw07,Zw02,DaHa09,GuMaZe14}.\n\\begin{itemize}\n\\item M.~Zworski \\cite{Zw02} noticed that for every $z\\in D(0,1)$,\n  there are associated exponentially accurate quasi-modes when $N\\to\n  \\infty $. Hence the open unit disc is a region of spectral \n  instability.\n\\item We have spectral stability (a good resolvent estimate) in\n${\\mathds{C}}\\setminus \\overline{D(0,1)}$, since $\\| A_0\\| =1$.\n\\item $\\sigma (A_0)=\\{0 \\}$.  \n\\end{itemize}\nThus, if $A_\\delta =A_0+\\delta Q$ is a small (random) perturbation of\n$A_0$ we expect the eigenvalues to move inside a small neighborhood of $\\overline{D(0,1)}$.\n\\par\nIn the special case when $Qu=(u|e_1)e_N$, where $(e_j)_1^N$ is the canonical\nbasis in ${\\mathds{C}}^N$, the eigenvalues of $A_\\delta $ are of the form \n$$\n\\delta ^{1\/N}e^{2\\pi ik\/N},\\ k\\in {\\mathds{Z}}\/N{\\mathds{Z}},\n$$\nso if we fix $0<\\delta \\ll 1$ and let $N\\to \\infty $, the spectrum\n``will converge to a uniform distribution on $S^1$''.\n\\par E.B.~Davies and M.~Hager \\cite{DaHa09} studied random\nperturbations of $A_0$. They showed that with probability close to 1,\nmost of the eigenvalues are close to a circle:\n\\begin{thm}\\label{th1}\nLet $A=A_0+\\delta Q$, $Q=(q_{j,k}(\\omega ))$ where $q_{j,k}$ are\nindependent and identically distributed random variables \n$\\sim {\\mathcal{N}}_{{\\mathds{C}}}(0,1)$.\nIf $0<\\delta \\le N^{-7}$, $R=\\delta\n^{1\/N}$, $\\sigma >0$, then with probability $\\ge 1-2N^{-2}$, we have\n$\\sigma (A_\\delta )\\subset D(0,RN^{3\/N})$ and\n$$\n\\# (\\sigma (A_\\delta )\\cap D(0,Re^{-\\sigma }))\\le \\frac{2}{\\sigma\n}+\\frac{4}{\\sigma }\\ln N.\n$$\n\\end{thm}\n  A recent result by A.~Guionnet, P.~Matched Wood and\n  O.~Zeitouni \\cite{GuMaZe14} implies that when $\\delta $ is\n  bounded from above by $N^{-\\kappa -1\/2}$ for some $\\kappa >0$ and\n  from below by some negative power of $N$, then \n  %\n$$\n\\frac{1}{N}\\sum_{\\mu \\in \\sigma (A_{\\delta})}\\delta (z-\\mu )\\to \\hbox{the\n  uniform measure on }S^1,\n$$\nweakly in probability.\n \\\\\n \\par\n  The main purpose of this paper is to obtain, for a small coupling \n  constant $\\delta$, more information about the distribution of \n  eigenvalues of $A_\\delta$ in the interior of a disc, where the \n  result of Davies and Hager only yields a logarithmic upper bound \n  on the number of eigenvalues; see Theorem \\ref{ed5} below.\n  %\n  \\par\n  %\n  In order to obtain more information in this\n  region, we will study the expected eigenvalue density, adapting the\n  approach of \\cite{Vo14}. (For random polynomials and Gaussian analytic \n  functions such results are more classical, \n  \\cite{Kac43,SZ03,HoKrPeVi09,So00,ShZe98,Sh08}.) \n  \\\\\n  \\par \n  \\textit{Acknowledgments.}\nWe would like to thank St\\'ephane Nonnenmacher for his observation \non the relation of the density in Theorem \\ref{ed5} with the Poincar\\'e \nmetric. The first author was partially supported by the project ANR \nNOSEVOL $2011$ BS $010119$ $01$.\n\\section{Main result}\\label{mres}\nLet $0< \\delta \\ll 1$ and consider the following random \nperturbation of $A_0$ as in \\eqref{jb}: \n\\begin{equation}\\label{jb.p}\n A_{\\delta} = A_0 + \\delta Q, \\quad Q=(q_{j,k})_{1 \\leq j,k\\leq N},\n\\end{equation}\nwhere $q_{j,k}$ are independent and identically distributed \ncomplex random variables, following the complex Gaussian law \n$\\mathcal{N}_{\\mathds{C}}(0,1)$.\n\\par\nIt has been observed by Bordeaux-Montrieux \\cite{BM} the we have the following \nresult.\n\\begin{prop}\nThere exists a $C_0>0$ such that the following holds: Let \n$X_j\\sim \\mathcal{N}_{\\mathds{C}}(0,\\sigma_j^2)$, $1\\leq j\\leq N <\\infty$ be \nindependent and identically distributed complex Gaussian random \nvariables. Put $s_1=\\max \\sigma_j^2$. Then, for every $x>0$, we \nhave \n %\n \\begin{equation*}\n  \\mathds{P}\\left[\\sum_{j=1}^N |X_j|^2 \\geq x\\right]\n    \\leq \\exp\\left( \n    \\frac{C_0}{2s_1} \\sum_{j=1}^N \\sigma_j^2 - \\frac{x}{2s_1}\n    \\right).\n \\end{equation*}\n\\end{prop}\nAccording to this result we have \n$$\nP(\\Vert Q\\Vert_\\mathrm{HS}^2\\ge x)\\le \\exp \\left(\\frac{C_0}{2}N^2-\\frac{x}{2} \\right)\n$$\nand hence if $C_1>0$ is large enough,\n\\begin{equation}\\label{pj.28}\n\\Vert Q\\Vert_\\mathrm{HS}^2\\le C_1^2N^2,\\hbox{ with probability }\\ge 1-e^{-N^2}.\n\\end{equation}\nIn particular (\\ref{pj.28}) holds for the ordinary operator norm of $Q$. \nWe now state the principal result of this work.\n\\begin{thm}\\label{ed5}\nLet $A_{\\delta}$ be the $N\\times N$-matrix in \\eqref{jb.p} and restrict the\nattention to the parameter range\n$e^{-N\/{\\mathcal{O}}(1)}\\le \\delta \\ll 1$, $N\\gg 1$. Let $r_0$ belong to a\nparameter range,\n$$\n\\frac{1}{{\\mathcal{O}}(1)}\\le r_0\\le 1-\\frac{1}{N},\n$$\n\\begin{equation}\\label{ed.60}\n\\frac{r_0^{N-1}N}{\\delta}(1-r_0)^2+\\delta N^3\\ll 1,\n\\end{equation}\nso that $\\delta \\ll N^{-3}$. Then, for all $\\varphi\\in\\mathcal{C}_0(D(0,r_0-1\/N))$\n %\n  \\begin{equation*}\\label{ed.61}\n   \\mathds{E}\\left[ \n      \\mathds{1}_{B_{\\mathds{C}^{N^2}}(0,C_1N)}(Q) \n      \\sum_{\\lambda\\in\\sigma(A_{\\delta})}\n      \\varphi(\\lambda)\n      \\right]\n   =\n    \\frac{1}{2\\pi}\\int\\varphi(z)\\Xi(z) L(dz),\n  \\end{equation*}\n %\n where \n %\n \\begin{equation*}\n  \\Xi(z) =  \\frac{4}{(1-|z|^2)^2}\n    \\left(\n    1 + \n    \\mathcal{O}\\!\\left(\\frac{|z|^{N-1}N}{\\delta}(1-|z|)^2\n\t      +\\delta N^3\\right)\n    \\right).\n \\end{equation*}\nis a continuous function independent of $r_0$. \n$C_1>0$ is the constant in (\\ref{pj.28}).\n\\end{thm}\nCondition \\eqref{ed.60} is equivalent to\n\\begin{equation*}\n r_0^{N-1}(1-r_0)^2 \\ll \\frac{\\delta}{N}\\left( 1 - \\delta N^3\\right).\n\\end{equation*}\nIt is necessary that $r_0 < 1 - 2(N+1)^{-1}$ for this inequality to \nbe satisfied. For such $r_0$ the function $[0,r_0]\\ni r \\mapsto r^{N-1}(1-r)^2$ \nis increasing, and so inequality \\eqref{ed.60} is preserved if we replace $r_0$ by \n$|z|\\leq r_0$.\n\\\\\n\\par\nThe leading contribution of the density $\\Xi(z)$ is \nindependent of $N$ and is equal to the Lebesgue density \nof the volume form induced by the Poincar\\'e metric on the \ndisc $D(0,1)$. This yields a very small density of \neigenvalues close to the center of the disc $D(0,1)$ which is, \nhowever, growing towards the boundary of $D(0,1)$.\n\\\\\n\\par\nA similar result has been obtained by M. Sodin and B. Tsirelson \nin \\cite{SoTs04} for the distribution of zeros of a certain class \nof random analytic functions with domain $D(0,1)$ linking the \nfact that the density is given by the volume form induced by \nthe Poincar\\'e metric on $D(0,1)$ to its invariance under \nthe action of $SL_2(\\mathds{R})$.\n\\subsection{Numerical Simulations}\nTo illustrate the result of Theorem \\ref{ed5}, we present the \nfollowing numerical calculations (Figure \\ref{fig:e5_e4} and \n\\ref{fig:e3_e2}) for the eigenvalues of the \n$N\\times N$-matrix in \\eqref{jb.p}, where $N=500$ and the coupling \nconstant $\\delta$ varies from $10^{-5}$ to $10^{-2}$. \n\\begin{figure}[ht]\n \\begin{minipage}[b]{0.49\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig_e5.pdf}\n \\end{minipage}\n \\hspace{0cm}\n \\begin{minipage}[b]{0.49\\linewidth}\n  \\centering \n  \\includegraphics[width=\\textwidth]{fig_e4.pdf}\n \\end{minipage}\n \\caption{On the left hand side $\\delta=10^{-5}$ and \n  on the right hand side $\\delta=10^{-4}$.}\n  \\label{fig:e5_e4}\n\\end{figure}\n\\begin{figure}[ht]\n \\begin{minipage}[b]{0.44\\linewidth}\n  \\centering\n  \\includegraphics[width=\\textwidth]{fig_e3.pdf}\n \\end{minipage}\n \\hspace{0cm}\n \\begin{minipage}[b]{0.54\\linewidth}\n  \\centering \n  \\includegraphics[width=\\textwidth]{fig_e2.pdf}\n \\end{minipage}\n \\caption{On the left hand side $\\delta=10^{-3}$ and \n  on the right hand side $\\delta= 10^{-2}$.}\n  \\label{fig:e3_e2}\n\\end{figure}\n\\section{A general formula}\n\\setcounter{equation}{0}\nTo start with, we shall obtain a general formula (due to \\cite{Vo14}\nin a similar context). Our treatment is slightly different in that we\navoid the use of approximations of the delta function and also that we\nhave more holomorphy available.\n\\par\nLet $g(z,Q)$ be a holomorphic function on $\\Omega \\times W\\subset {\\mathds{C}}\n\\times {\\mathds{C}}^{N^2}$, where $\\Omega\\subset\\mathds{C}$, $W\\subset {\\mathds{C}}^{N^2}$ \nare open bounded and connected. Assume that\n\\begin{equation}\\label{ed.1} \\hbox{for every }Q\\in W,\\\n  g(\\cdot ,Q)\\not\\equiv 0.\n\\end{equation}\nTo start with, we also assume that \n\\begin{equation}\\label{ed.2} \n\\hbox{for almost all }Q\\in W,\\ g(\\cdot ,Q) \\hbox{ has only simple zeros.}\n\\end{equation}\nLet $\\phi \\in C_0^\\infty (\\Omega )$ and let $m\\in C_0(W)$. We are interested in \n\\begin{equation}\\label{ed.3}\nK_\\phi =\\int \\left(\\sum_{z;\\, g(z,Q)=0} \\phi (z)\\right) m(Q)L(dQ),\n\\end{equation}\nwhere we frequently identify the Lebesgue measure with a differential\nform,\n\\begin{equation*}\nL(dQ)\\simeq (2i)^{-{N^2}}d\\overline{Q}_1\\wedge dQ_1\\wedge ... \\wedge\nd\\overline{Q}_{N^2}\\wedge dQ_{N^2} =:(2i)^{-{N^2}}d\\overline{Q}\\wedge dQ.\n\\end{equation*}\nIn (\\ref{ed.3}) we\ncount the zeros of $g(\\cdot ,Q)$ with their multiplicity and notice\nthat the integral is finite: For every compact set $K\\subset W$ the\nnumber of zeros of $g(\\cdot ,Q)$ in $\\mathrm{supp\\,}\\phi $, counted \nwith their multiplicity, is uniformly bounded, for $Q\\in K$. This \nfollows from Jensen's formula.\n\\par \nNow assume,\n\\begin{equation}\\label{ed.4}\ng(z,Q)=0 \\Rightarrow d_Qg\\ne 0.\n\\end{equation}\nThen \n$$\n\\Sigma :=\\{(z,Q)\\in \\Omega \\times W;\\, g(z,Q)=0 \\}\n$$ \nis a smooth complex hypersurface in $\\Omega \\times W$ and from\n(\\ref{ed.2}) we see that \n\\begin{equation}\\label{ed.5}\nK_\\phi = \\int_\\Sigma \\phi (z)m(Q)(2i)^{-{N^2}}d\\overline{Q}\\wedge dQ,\n\\end{equation}\nwhere we view $(2i)^{-{N^2}}d\\overline{Q}\\wedge dQ$ as a complex $({N^2},{N^2})$-form on\n$\\Omega \\times W$, restricted to $\\Sigma $, which yields a non-negative differential form \nof maximal degree on $\\Sigma$. \n\\par \nBefore continuing, let us eliminate the assumption\n(\\ref{ed.2}). Without that assumption, the integral in (\\ref{ed.3}) is\nstill well-defined. It suffices to show \\eqref{ed.5} for all \n$\\phi\\in\\mathcal{C}_0^{\\infty}(\\Omega_0\\times W_0)$ when $\\Omega_0\\times W_0$ \nis a sufficiently small open neighborhood of any given point \n$(z_0,Q_0)\\in \\Omega\\times W$. When $g(z_0,Q_0)\\neq 0$ or \n$\\partial_z g(z_0,\\Omega_0)\\neq 0$ we already know that this holds, so we \nassume that for some $m\\geq 2$, $\\partial_z^kg(z_0,Q_0)=0$ for $0\\leq k \\leq m-1$, \n$\\partial_z^mg(z_0,Q_0)\\neq 0$. \n\\par\nPut $g_{\\varepsilon}(z,Q)=g(z,Q)+\\varepsilon$, $\\varepsilon\\in\\mathrm{neigh}(0,\\mathds{C})$. \nBy Weierstrass' preparation theorem, if $\\Omega_0,W_0$ and $r>0$ are small \nenough,\n\\begin{equation*}\n g_{\\varepsilon}(z,Q) = k(z,Q,\\varepsilon)p(z,Q,\\varepsilon) \\quad \n \\text{in } \\Omega_0\\times W_0\\times D(0,r),\n\\end{equation*}\nwhere $k$ is holomorphic and non-vanishing, and \n\\begin{equation*}\n p(z,Q,\\varepsilon)= z^m + p_1(Q,\\varepsilon)z^{m-1} + \\dots \n\t\t     + p_m(Q,\\varepsilon).\n\\end{equation*}\nHere, $p_j(Q,\\varepsilon)$ are holomorphic, and $p_j(0,0)=0$. \n\\par\nThe discriminant $D(Q,\\varepsilon)$ of the polynomial $p(\\cdot,Q,\\varepsilon)$ \nis holomorphic on $W_0\\times D(0,r)$. It vanishes precisely when \n$p(\\cdot,Q,\\varepsilon)$ - or equivalently $g_{\\varepsilon}(\\cdot,Q)$ - has a \nmultiple root in $\\Omega_0$.\n\\par\nNow for $0 < |\\varepsilon| \\ll 1$, the $m$ roots of $g_{\\varepsilon}(\\cdot,Q_0)$ \nare simple, so $D(Q_0,\\varepsilon)\\neq 0$. Thus, $D(\\cdot,\\varepsilon)$ is not \nidentically zero, so the zero set of $D(\\cdot,\\varepsilon)$ in $W_0$ is of measure \n$0$ (assuming that we have chosen $W_0$ connected). This means that for \n$0<|\\varepsilon|\\ll 1$, the function $g_{\\varepsilon}(\\cdot,Q)$ has only simple \nroots in $\\Omega$ for almost all $Q\\in W_0$.\n\\par\nLet $\\Sigma _\\epsilon $ be the zero set of\n$g_\\epsilon $, so that $\\Sigma _\\epsilon \\to \\Sigma $ in the natural\nsense. We have \n$$\n\\int \\left(\\sum_{z;\\, g_\\epsilon (z,Q)=0 }\\phi\n(z)\\right) m(Q)(2i)^{-{N^2}}d\\overline{Q}\\wedge dQ=\\int_{\\Sigma _\\epsilon }\\phi\n(z)m(Q)(2i)^{-{N^2}}d\\overline{Q}\\wedge dQ\n$$ \nfor $\\phi\\in\\mathcal{C}_0^{\\infty}(\\Omega_0\\times W_0)$, \nwhen $\\epsilon >0$ is small enough, depending on $\\phi $, $m$. Passing\nto the limit $\\epsilon =0$ we get (\\ref{ed.5}) under the assumptions\n(\\ref{ed.1}), (\\ref{ed.4}), first for \n$\\phi\\in\\mathcal{C}_0^{\\infty}(\\Omega_0\\times W_0)$, and then by partition \nof unity for all $\\phi\\in\\mathcal{C}_0^{\\infty}(\\Omega\\times W)$. Notice that \nthe result remains valid if we replace $m(Q)$ by $m(Q)1_B(Q)$ where $B$ is \na ball in $W$.\n\\par\nNow we strengthen the assumption (\\ref{ed.4}) by assuming that we have\na non-zero $Z(z)\\in {\\mathds{C}}^{N^2}$ depending smoothly on $z\\in \\Omega $\n(the dependence will actually be holomorphic in the application below) \nsuch that\n\\begin{equation}\\label{ed.6}\ng(z,Q)=0 \\Rightarrow \\left(\\overline{Z}(z)\\cdot \\partial _Q\n\\right)g(z,Q)\\ne 0.\n\\end{equation}\nWe have the corresponding orthogonal decomposition\n$$\nQ=Q(\\alpha )=\\alpha _1\\overline{Z}(z)+\\alpha ',\\quad \\alpha '\\in\n\\overline{Z}(z) ^\\perp , \\ \\alpha_1\\in\\mathds{C},\n$$ \nand if we identify unitarily $\\overline{Z}(z)^\\perp$ with ${\\mathds{C}}^{{N^2}-1}$ \nby means of an orthonormal basis $e_2(z),...,e_{N^2}(z)$, so\nthat $\\alpha '=\\sum_2^{N^2} \\alpha _je_j(z)$ we get global coordinates\n$\\alpha _1,\\alpha _2,...,\\alpha _{N^2}$ on $Q$-space.\n\\par \nBy the implicit function theorem, at least locally near any given\npoint in $\\Sigma $, we can represent $\\Sigma $ by $\\alpha\n_1=f(z,\\alpha ')$, $\\alpha '\\in \\overline{Z}(z)^\\perp\\simeq {\\mathds{C}}^{{N^2}-1}$, where $f$ is\nsmooth. (In the specific situation below, this will be valid\nglobally.) \nClearly, since $z,\\alpha _2,...,\\alpha _{N^2}$ are complex coordinates on\n$\\Sigma $, we have on $\\Sigma$ that\n\\begin{equation*}\n  \\frac{1}{(2i)^{N^2}}d\\overline{Q}\\wedge dQ\n  =\n  J(f)\\frac{d\\overline{z}\\wedge dz}{2i} \n  (2i)^{1-N^2}d\\overline{\\alpha}_2\\wedge d\\alpha_2 \\wedge ... \n  \\wedge d\\overline{\\alpha}_{N^2}\\wedge d\\alpha_{N^2}\n\\end{equation*}\nwith the convention that \n\\begin{equation*}\n J(f)\\frac{d\\overline{z}\\wedge dz}{2i}  \\geq 0, \\quad\n (2i)^{1-N^2}d\\overline{\\alpha}_2\\wedge d\\alpha_2 \\wedge ... \n\\wedge d\\overline{\\alpha}_{N^2}\\wedge d\\alpha_{N^2} >0.\n\\end{equation*}\nThus \n\\begin{equation}\\label{ed.7}\n\\begin{split}\nK_\\phi =\\int \\phi (z)m\\left(f(z,\\alpha ')\\overline{Z}(z)+\\alpha\n  '\\right)J(f)(z,\\alpha _2,...,\\alpha _{N^2}) \\times \\\\\n(2i)^{-{N^2}}d\\overline{z}\\wedge dz\\wedge d\\overline{\\alpha}_2\\wedge d\n\\alpha_2 \\wedge ... \\wedge d\\overline{\\alpha}_{N^2}\\wedge\nd\\alpha_{N^2} .\n\\end{split}\n\\end{equation}\nThe Jacobian $J(f)$ is invariant under any $z$-dependent unitary change\nof variables, $\\alpha _2,...,\\alpha _{N^2}\\mapsto \\widetilde{\\alpha\n}_2,...,\\widetilde{\\alpha }_{N^2}$, so for the calculation of $J(f)$ at a\ngiven point $(z_0,\\alpha _0')$, we are free to choose the most\nappropriate orthonormal basis $e_2(z),...,e_{N^2}(z)$ in\n$\\overline{Z}(z)^\\perp$ depending smoothly on $z$. We write\n(\\ref{ed.7}) as \n\\begin{equation}\\label{ed.8}\nK_\\phi =\\int \\phi (z) \\Xi (z) \\frac{d\\overline{z}\\wedge dz}{2i},\n\\end{equation}\nwhere the density $\\Xi (z)$ is given by \n\\begin{equation}\\label{ed.9}\\begin{split}\n\\Xi (z)=\\int_{\\alpha '=\\sum_2^{N^2} \\alpha _je_j(z)}m(f(z,\\alpha\n')\\overline{Z}(z)+\\alpha ')J(f)(z,\\alpha _2,...,\\alpha _{N^2})\\times \\\\\n(2i)^{1-{N^2}}d\\overline{\\alpha}_2\\wedge d\\alpha_2\\wedge\n... \\wedge d\\overline{\\alpha}_{N^2}\\wedge d\\alpha_{N^2}.\n\\end{split}\n\\end{equation}\nBefore continuing, let us give a brief overview on the organization \nof following sections: \n\\par\nIn Section \\ref{gru} we will set up an auxiliary Grushin problem \nyielding the effective function $g$ as above. Section \\ref{chvar} \ndeals with the appropriate choice of coordinates $Q$ and the \ncalculation of the Jacobian $J(f)$. Finally, in Section \\ref{pfThm} \nwe complete the proof of Theorem \\ref{ed5}.\n\n\\section{Grushin problem for the perturbed Jordan block}\\label{gru}\n\\subsection{Setting up an auxiliary problem}\nFollowing \\cite{SjZw07}, we introduce an auxiliary Grushin problem.\nDefine $R_+:{\\mathds{C}}^N\\to {\\mathds{C}}$ by\n\\begin{equation}\\label{pj.5}\nR_+u=u_1,\\ u=(u_1\\ ...\\ u_N)^\\mathrm{t}\\in {\\mathds{C}}^N. \n\\end{equation} \nLet $R_-:\\, {\\mathds{C}}\\to {\\mathds{C}}^N$ be defined by\n\\begin{equation}\\label{pj.6}\nR_-u_-=(0\\ 0\\ ...\\ u_-)^\\mathrm{t}\\in {\\mathds{C}}^N.\n\\end{equation}\nHere, we identify vectors in ${\\mathds{C}}^N$ with column matrices. Then\nfor $|z|<1$, the operator\n\\begin{equation}\\label{pj.7}\n{\\mathcal{A}}_0=\\begin{pmatrix}A_0-z &R_-\\\\ R_+ &0\\end{pmatrix}: \n{\\mathds{C}}^{N+1}\\to {\\mathds{C}}^{N+1}\n\\end{equation}\nis bijective. In fact, identifying \n\\begin{equation*}\n{\\mathds{C}}^{N+1}\\simeq\n\\ell^2([1,2,...,N+1])\\simeq\\ell^2({\\mathds{Z}}\/(N+1){\\mathds{Z}}), \n\\end{equation*}\nwe have\n${\\mathcal{A}}_0=\\tau ^{-1}-z\\Pi _N$, where $\\tau u(j)=u(j-1)$ (translation\nby 1 step to the right) and $\\Pi _Nu=1_{[1,N]}u$. Then ${\\mathcal{A}}_0=\\tau\n^{-1}(1-z\\tau \\Pi _N)$, $(\\tau \\Pi _N)^{N+1}=0$,\n$$\n{\\mathcal{A}}_0^{-1}=(1+z\\tau \\Pi _N+(z\\tau \\Pi _N)^2+...+(z\\tau \\Pi\n_N)^N)\\circ \\tau .\n$$\nWrite\n$$\n{\\mathcal{E}}_0:={\\mathcal{A}}_0^{-1}=:\\begin{pmatrix}E^0 &E_+^0\\\\\nE_-^0 &E_{-+}^0\n\\end{pmatrix}.\n$$\nThen \n\\begin{equation}\\label{pj.8}\nE^0\\simeq \\Pi _N(1+z\\tau \\Pi _N+...(z\\tau \\Pi _N)^{N-1})\\tau \\Pi _N,\n\\end{equation}\n\\begin{equation}\\label{pj.9}\nE_+^0=\\begin{pmatrix}1\\\\z\\\\ ..\\\\z^{N-1}\\end{pmatrix},\\\nE_-^0=\\begin{pmatrix}z^{N-1} & z^{N-2} &... &1\\end{pmatrix},\n\\end{equation}\n\\begin{equation}\\label{pj.10}\nE_{-+}^0=z^N.\n\\end{equation}\nA quick way to check \\eqref{pj.9}, \\eqref{pj.10} is to write \n$\\mathcal{A}_0$ as an $(N+1)\\times (N+1)$-matrix where we moved \nthe last line to the top, with the lines labeled from \n$0$ ($\\equiv N+1 \\mod (N+1)\\mathds{Z}$) to $N$ and the columns from \n$1$ to $N+1$.\n\\par\nContinuing, we see that\n\\begin{equation}\\label{pj.10.2}\n\\Vert E^0\\Vert\\le G(|z|),\\ \\Vert E_\\pm^0\\Vert\\le\nG(|z|)^{\\frac{1}{2}},\\ \\Vert E_{-+}^0\\Vert\\le 1 ,\n\\end{equation}\nwhere $\\Vert \\cdot \\Vert$ denote the natural operator norms and\n\\begin{equation}\\label{pj.10.4}\nG(|z|):=\\min \\left( N,\\frac{1}{1-|z|} \\right)\\asymp\n1+|z|+|z|^2+...+|z|^{N-1} .\n\\end{equation}\nNext, consider the natural Grushin problem for $A_\\delta $. If\n$\\delta \\Vert Q\\Vert G(|z|)<1$, we see that\n\\begin{equation}\\label{pj.11}\n{\\mathcal{A}}_\\delta =\\begin{pmatrix}A_\\delta -z  &R_-\\\\ R_+ &0\\end{pmatrix}\n\\end{equation}\nis bijective with inverse $${\\mathcal{E}}_\\delta =\\begin{pmatrix}E^\\delta\n  & E_+^\\delta \\\\ E_+^\\delta  &E_{-+}^\\delta \\end{pmatrix},$$\nwhere\n\\begin{equation}\\label{pj.11.5}\n\\begin{split}\n  E^\\delta =&E^0-E^0\\delta QE^0+E^0(\\delta QE^0)^2-...=E^0(1+\\delta\n  QE^0)^{-1},\\\\\n  E_+^\\delta =&E_+^0-E^0\\delta QE_+^0+(E^0\\delta\n  Q)^2E_+^0-...=(1+E^0\\delta\n  Q)^{-1}E^0_+,\\\\\n  E_-^\\delta =&E_-^0-E_-^0\\delta QE^0+E_-^0(\\delta\n  QE^0)^2-...=E_-^0(1+\\delta\n  QE^0)^{-1},\\\\\n  E^\\delta _{-+}=&E^0_{-+}-E_-^0\\delta QE_+^0+E_-^0\\delta QE^0\\delta\n  QE_+^0-...\\\\\n  &=E_{-+}^0-E_-^0\\delta Q(1+E^0\\delta Q)^{-1}E_+^0.\n\\end{split}\n\\end{equation}\nWe get\n\\begin{equation}\\label{pj.12}\n\\begin{split}\n&\\Vert E^\\delta \\Vert\\le \\frac{G(|z|)}{1-\\delta \\Vert Q\\Vert G(|z|)},\\\n\\Vert E_\\pm^\\delta \\Vert\\le \\frac{G(|z|)^{\\frac{1}{2}}}{1-\\delta \\Vert\n  Q\\Vert G(|z|)},\\\\\n&| E_{-+}^\\delta - E_{-+}^0 |\\le \\frac{\\delta \\Vert Q\\Vert G(|z|)}{1-\\delta \\Vert Q\\Vert G(|z|)}.\n\\end{split}\n\\end{equation}\nIndicating derivatives with respect to $\\delta $ with dots and\nomitting sometimes the super\/sub-script $\\delta $, we have\n\\begin{equation}\\label{pj.13}\n\\dot{{\\mathcal{E}}}=-{\\mathcal{E}}\\dot{{\\mathcal{A}}}{\\mathcal{E}}=-\\begin{pmatrix}EQE &\n  EQE_+\\\\ E_-QE &E_-QE_+.\n\\end{pmatrix}\n\\end{equation}\nIntegrating this from 0 to $\\delta $ yields\n\\begin{equation}\\label{pj.14}\n\\Vert  E^\\delta -E^0\\Vert \\le \\frac{G(|z|)^2 \\delta \\Vert Q \\Vert}{(1-\\delta \\Vert Q\\Vert\n    G(|z|))^2} ,\\quad \n    \\Vert  E_\\pm^\\delta -E_\\pm^0\\Vert \\le \n    \\frac{G(|z|)^{\\frac{3}{2}}\\delta \\Vert Q \\Vert}{(1-\\delta \\Vert Q\\Vert G(|z|))^2}.\n\\end{equation}\nWe now sharpen the assumption that $\\delta \\Vert Q\\Vert G(|z|)<1$ to \n\\begin{equation}\\label{pj.15}\n\\delta \\Vert Q\\Vert G(|z|)<1\/2. \n\\end{equation}\nThen \n\\begin{equation}\\label{pj.16}\n\\begin{split}\n&\\Vert E^\\delta \\Vert\\le 2G(|z|),\\\n\\Vert E_\\pm^\\delta \\Vert\\le 2 G(|z|)^{\\frac{1}{2}},\\\\\n&| E_{-+}^\\delta - E_{-+}^0 |\\le 2\\delta \\Vert Q\\Vert G(|z|).\n\\end{split}\n\\end{equation}\nCombining this with the identity $\\dot{E}_{-+}=-E_-QE_+$ that follows\nfrom (\\ref{pj.13}), we get\n\\begin{equation}\\label{pj.17}\n\\Vert \\dot{E}_{-+}+E_-^0QE_+^0\\Vert \\le 16 G(|z|)^2\\delta\n  \\Vert Q\\Vert ^2,\n\\end{equation}\nand after integration from $0$ to $\\delta $,\n\\begin{equation}\\label{pj.18}\nE_{-+}^\\delta =E_{-+}^0-\\delta E_-^0QE_+^0 +{\\mathcal{O}}(1)G(|z|)^2(\\delta\n\\Vert Q\\Vert)^2 . \n\\end{equation}\nUsing (\\ref{pj.9}), (\\ref{pj.10}) we get with $Q=(q_{j,k})$,\n\\begin{equation}\\label{pj.19}\nE_{-+}^\\delta =z^N-\\delta \\sum_{j,k=1}^N q_{j,k}z^{N-j+k-1}\n  +{\\mathcal{O}}(1)G(|z|)^2(\\delta\n\\Vert Q\\Vert)^2,\n\\end{equation}\nstill under the assumption (\\ref{pj.15}).\n\\subsection{Estimates for the effective Hamiltonian}\n\n\\par We now consider the situation at the beginning of Section\n\\ref{mres}:\n$$\nA_\\delta =A_0+\\delta Q,\\ Q=(q_{j,k}(\\omega ))_{j,k=1}^N,\\\nq_{j,k}(\\omega )\\sim {\\mathcal{N}}_{\\mathds{C}}(0,1)\\hbox{ independent.}\n$$\nIn the following, we often write $|\\cdot|$ for the Hilbert-Schmidt norm \n$\\lVert\\cdot\\rVert_{\\mathrm{HS}}$. As we recalled in (\\ref{pj.28}), we have \n\\begin{equation}\\label{ed.10}\n|Q|\\le C_1N\\hbox{ with probability }\\ge 1-e^{-N^2},\n\\end{equation}\nand we shall work under the assumption that $|Q|\\le C_1N$.\nWe let $|z|<1$ and assume:\n\\begin{equation}\\label{ed.11}\n\\delta NG(|z|)\\ll 1.\n\\end{equation}\nThen with probability $\\ge 1-e^{-N^2}$, we have (\\ref{pj.15}),\n(\\ref{pj.19}) which give for $g(z,Q):=E_{-+}^\\delta $, \n\\begin{equation}\\label{ed.12}\ng(z,Q)=z^N+\\delta (Q|\\overline{Z}(z))+{\\mathcal{O}}(1)(G(|z|)\\delta N)^2.\n\\end{equation}\nHere, $Z$ is given by\n\\begin{equation}\\label{ed.13}\nZ=\\left( z^{N-j+k-1}\\right)_{j,k=1}^N.\n\\end{equation}\nA straight forward calculation shows that\n\\begin{equation}\\label{ed.14}\n  | Z | =\\sum_0^{N-1}|z|^{2\\nu }=\\frac{1-|z|^{2N}}{1-|z|^2}=\n\\frac{1-|z|^N}{1-|z|}\\,\\,\\frac{1+|z|^N}{1+|z|},\n\\end{equation}\nand in particular,\n\\begin{equation}\\label{ed.15}\n\\frac{G(|z|)}{2}\\le | Z| \\le G(|z|).\n\\end{equation}\n\\par \nThe middle term in (\\ref{ed.12}) is bounded in modulus by $\\delta\n|Q||Z|\\le \\delta C_1NG(|z|)$ and we assume that $|z|^N$ is much\nsmaller than this bound:\n\\begin{equation}\\label{ed.16}\n|z|^N\\ll \\delta C_1NG(|z|).\n\\end{equation}\nMore precisely, we work in a disc $D(0,r_0)$, where \n\\begin{equation}\\label{ed.25}\nr_0^N\\le C^{-1}\\delta C_1NG(r_0)\\le C^{-2},\\quad r_0\\le 1-N^{-1}\n\\end{equation}\nand $C\\gg 1$. In fact, the first inequality in \\eqref{ed.25} \ncan be written $m(r_0) \\leq C^{-1}\\delta C_1 N$ and \n$m(r)= r^N(1-r)$ is increasing on $[0,1-N^{-1}]$ so the inequality \nis preserved if we replace $r_0$ by $|z|\\leq r_0$. Similarly, the \nsecond inequality holds after the same replacement since $G$ is \nincreasing. \n\\par \nIn view of (\\ref{ed.11}), we see that\n$$\n\\left(G(|z|)\\delta N \\right)^2\\ll \\delta G(|z|)N\n$$\nis also much smaller than the upper bound on the middle term. \n\n\\par By the Cauchy inequalities,\n\\begin{equation}\\label{ed.17}\nd_Qg=\\delta Z\\cdot dQ+{\\mathcal{O}}(1)G(|z|)^2\\delta ^2N.\n\\end{equation}\nThe norm of the first term is $\\asymp \\delta G\\gg G^2\\delta ^2N$,\nsince $G\\delta N\\ll 1$. (When applying the Cauchy inequalities, we\nshould shrink the radius $R=C_1N$ by a factor $\\theta <1$, but we have\nroom for that, if we let $C_1$ be a little larger than necessary to\nstart with.)\n\nWriting \n$$\nQ=\\alpha _1\\overline{Z}(z)+\\alpha ',\\ \\alpha '\\in\n\\overline{Z}(z)^\\perp\\simeq {\\mathds{C}}^{{N^2}-1},\n$$\nwe identify $g(z,Q)$ with a function $\\widetilde{g}(z,\\alpha )$ which is\nholomorphic in $\\alpha $ for every fixed $z$ and satisfies\n\\begin{equation}\\label{ed.18}\n\\widetilde{g}(z,\\alpha )=z^N+\\delta |Z(z)|^2\\alpha _1+{\\mathcal{O}}(1)\nG(|z|)^2\\delta ^2N^2,\n\\end{equation}\nwhile (\\ref{ed.17}) gives\n\\begin{equation}\\label{ed.19}\n\\partial _{\\alpha _1}\\widetilde{g}(z,\\alpha )=\\delta |Z(z)|^2\n+{\\mathcal{O}}(1)G(|z|)^3\\delta ^2N,\n\\end{equation}\nand in particular,\n$$\n\\left| \\partial _{\\alpha _1}\\widetilde{g} \\right|\\asymp \\delta G(|z|)^2.\n$$\nThis derivative does not depend on the choice of unitary\nidentification $\\overline{Z}^\\perp\\simeq {\\mathds{C}}^{{N^2}-1}$. Notice that\nthe remainder in (\\ref{ed.18}) is the same as in (\\ref{ed.12}) and\nhence a holomorphic function of $(z,Q)$. In particular it is a\nholomorphic function of $\\alpha _1,...,\\alpha _{N^2}$ for every fixed $z$\nand we can also get (\\ref{ed.19}) from this and the Cauchy\ninequalities. In the same way, we get from (\\ref{ed.18}) that\n\\begin{equation}\\label{ed.19.5}\n\\partial _{\\alpha _j}\\widetilde{g}(z,\\alpha )={\\mathcal{O}}(1)G(|z|)^2\\delta ^2N,\\ j=2,...,{N^2}.\n\\end{equation}\n\\par \nThe Cauchy inequalities applied to (\\ref{ed.12}) give,\n\\begin{equation}\\label{ed.26}\n\\partial _zg(z,Q)=Nz^{N-1}+\\delta Q\\cdot \\partial _zZ(z)\n+{\\mathcal{O}}(1)\\frac{(G(|z|)\\delta N)^2}{r_0-|z|}.\n\\end{equation}\nThen, for $\\widetilde{g}(z,\\alpha _1,\\alpha ')=g(z,\\alpha\n_1\\overline{Z}(z)+\\alpha ')$, $\\alpha '= \\sum_2^{N^2} \\alpha _je_j$ \nwe shall see that\n\\begin{equation}\\label{ed.27}\\begin{split}\n\\partial _z\\widetilde{g}=\n Nz^{N-1}+\\delta \\alpha _1\\partial _z\\left( |Z|^2 \\right)\n +{\\mathcal{O}}(1)\\frac{(G\\delta N)^2}{r_0-|z|}+{\\mathcal{O}}(1)G^2\\delta\n^2N\\left| \\sum_2^{N^2} \\alpha _j\\partial _ze_j \\right|,\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{ed.28}\\begin{split}\n\\partial _{\\overline{z}}\\widetilde{g}\n=& \\delta \\alpha _1\\partial _{\\overline{z}}\\left( |Z|^2 \\right) \n+{\\mathcal{O}}(1)G^2\\delta ^2N\\left| \\alpha _1\\overline{\\partial\n    _zZ}+\\sum_2^{N^2} \\alpha _j\\partial _{\\overline{z}}e_j \\right| .\n\\end{split}\\end{equation}\n\\par \nThe leading terms in (\\ref{ed.27}), (\\ref{ed.28}) can be obtained\nformally from (\\ref{ed.18}) by applying $\\partial _z$, $\\partial\n_{\\overline{z}}$ and we also notice that \n$$\n\\partial _z|Z|^2=\\overline{Z}\\cdot \\partial _zZ,\\ \\\n\\partial _{\\overline{z}}|Z|^2=Z\\cdot \\overline{\\partial _zZ }.\n$$\nHowever it is not clear how to handle the remainder in (\\ref{ed.18}),\nso we verify (\\ref{ed.27}), (\\ref{ed.28}), using (\\ref{ed.17}),\n(\\ref{ed.26}):\n\\[\n\\begin{split}\n&\\partial _z\\widetilde{g}=\\partial _zg+d_Qg\\cdot \\sum_2^{N^2} \\alpha\n_j\\partial _ze_j=\\\\\n&Nz^{N-1}+\\delta Q\\cdot \\partial _zZ+{\\mathcal{O}}(1)\\frac{(G\\delta\n  N)^2}{r_0-|z|}+(\\delta Z\\cdot dQ +{\\mathcal{O}}(1)G^2\\delta ^2N)\\cdot\n\\sum_2^{N^2} \\alpha _j\\partial _ze_j\\\\\n&= Nz^{N-1}+\\delta \\alpha _1\\partial _z\\left(|Z|^2 \\right) +\\delta\n\\sum_2^{N^2} \\alpha _je_j\\cdot \\partial _zZ +\\delta Z\\cdot \\sum_2^{N^2}\\alpha\n_j\\partial _ze_j\\\\\n&\\hskip 5cm +\\hbox{ the remainders in (\\ref{ed.27})}.\n\\end{split}\n\\]\nThe 3d and the 4th terms in the last expression add up to \n$$\n\\delta \\partial _z\\left(\\sum_2^{N^2} \\alpha _je_j\\cdot Z\n\\right)=\\delta \\partial _z(0)=0,\n$$\nand we get (\\ref{ed.27}).\n\n\\par Similarly,\n\\[\n\\begin{split}\n&\\partial _{\\overline{z}}\\widetilde{g}=d_Qg\\cdot \\left(\\alpha\n  _1\\overline{\\partial _zZ}+\\sum_2^{N^2} \\alpha _j\\partial\n  _{\\overline{z}}e_j \\right)\\\\\n&=\\left(\\delta Z\\cdot dQ+{\\mathcal{O}}(1)G^2\\delta ^2N  \\right)\\cdot \\left(\\alpha\n_1\\overline{\\partial _zZ}+\\sum_2^{N^2} \\alpha _j\\partial\n_{\\overline{z}}e_j\\right) .\n\\end{split}\n\\]\nUp to remainders as in (\\ref{ed.28}), this is equal to \n\\begin{align*}\n\\delta \\alpha _1Z\\cdot \\overline{\\partial _zZ}+\\delta \\sum_2^{N^2} \\alpha\n_jZ\\cdot \\partial _{\\overline{z}}e_j &=\n\\delta \\alpha _1\\partial _{\\overline{z}}\\left(|Z|^2 \\right) +\\delta\n\\sum_2^{N^2} \\alpha _j\\partial _{\\overline{z}}\\left(Z\\cdot e_j\n\\right)\n \\notag \\\\\n&=\\delta \\alpha _1 \\partial _{\\overline{z}}\\left(|Z|^2 \\right).\n\\end{align*}\n\\par \nHere, we know that \n$$\n|Z(z)|=\\sum_0^{N-1}(z\\overline{z})^\\nu =:K(z\\overline{z}),\n$$\n\\begin{equation}\\label{ed.29}\n\\begin{split}\n\\partial _z\\left(|Z(z)|^2 \\right)&=2KK'\\overline{z},\\\\\n\\partial _{\\overline{z}}\\left(|Z(z)|^2 \\right)&=2KK'z.\n\\end{split}\n\\end{equation}\nObserve also that $K(t)\\asymp G(t)$ and that $G(|z|)\\asymp G(|z|^2)$.\n\nThe following result implies that $K'(t)$ and $K(t)^2$ are of the same\norder of magnitude.\n\\begin{prop}\\label{ed1}\n  For $k\\in {\\mathds{N}}$, $2\\le N\\in {\\mathds{N}}\\cup \\{+\\infty \\}$, $0\\le t<\n  1$, we put\n\\begin{equation}\\label{ed.29.1}\nM_{N,k}(t)=\\sum _{\\nu =1}^{N-1} \\nu ^kt^\\nu ,\n\\end{equation}\nso that $K(t)=K_N(t)=M_{N,0}(t)+1$, $K'(t)\\asymp M_{N-1,1}(t)+1$. For each fixed $k\\in {\\mathds{N}}$, we\nhave uniformly with respect to $N$, $t$:\n\\begin{equation}\\label{ed.29.2}\nM_{\\infty ,k}(t)\\asymp \\frac{t}{(1-t)^{k+1}},\n\\end{equation}\n\\begin{equation}\\label{ed.29.3}\nM_{\\infty ,k}(t)-M_{N,k}(t)\\asymp \\frac{t^N}{1-t}\\left(N+\\frac{1}{1-t} \\right)^k.\n\\end{equation}\nFor all fixed $C>0$ and $k\\in {\\mathds{N}}$, we have uniformly,\n\\begin{equation}\\label{ed.29.4}\nM_{N,k}(t)\\asymp M_{\\infty ,k}(t),\\hbox{ for }0\\le t\\le\n1-\\frac{1}{CN},\\ N\\ge 2.\n\\end{equation}\n\\end{prop}\n\nNotice that under the assumption in (\\ref{ed.29.4}), the estimate\n(\\ref{ed.29.3}) becomes\n$$\nM_{\\infty ,k}(t)-M_{N,k}(t)\\asymp \\frac{t^NN^k}{1-t}.\n$$\nWe also see that in any region $1-{\\mathcal{O}}(1)\/N\\le t<1$, we have\n$$\nM_{N,k}(t)\\asymp N^{k+1},\n$$\nso together with (\\ref{ed.29.4}), \\eqref{ed.29.2}, this shows that\n\\begin{equation}\\label{ed.29.4.5}\nM_{N,k}(t)\\asymp t\\min \\left(\\frac{1}{1-t},N \\right)^{k+1}.\n\\end{equation}\n\\begin{proof}\nThe statements are easy to verify when $0\\le t\\le 1-1\/{\\mathcal{O}}(1)$ and the\n$N$-dependent statements (\\ref{ed.29.3}), (\\ref{ed.29.4}) are clearly\ntrue when $N\\le {\\mathcal{O}}(1)$. Thus we can assume that $1\/2\\le t<1$ and\n$N\\gg 1$. \n\n\\par Write $t=e^{-s}$ so that $0<s\\le 1\/{\\mathcal{O}}(1)$ and notice\nthat $s\\asymp 1-t$. For $N\\in {\\mathds{N}}$, we put\n\\begin{equation}\\label{ed.29.5}\nP_{N,k}(s)=\\sum_{\\nu =N}^\\infty \\nu ^ke^{-\\nu s},\n\\end{equation}\nso that\n\\begin{equation}\\label{ed.29.6}\nP_{N,k}(s)=\\begin{cases}M_{\\infty ,k}(t)\\hbox{ when }N=1,\\\\\nM_{\\infty ,k}(t)-M_{N,k}(t)\\hbox{ when }N\\ge 2.\n\\end{cases}\n\\end{equation}\nWe regroup the terms in (\\ref{ed.29.5}) into sums with $\\asymp 1\/s$\nterms where $e^{-\\nu s}$ has constant order of magnitude:\n$$\nP_{N,k}(s)=\\sum_{\\mu =1}^{\\infty }\\Sigma (\\mu ),\\quad \\Sigma (\\mu\n)=\\sum_{N+\\frac{\\mu -1}{s}\\le\\nu < N+\\frac{\\mu }{s}}\\nu ^ke^{-\\nu s}.\n$$\nHere, since the sum $\\Sigma (\\mu )$ consists of $\\asymp 1\/s$ terms of the order\n$\\nu ^k e^{-(Ns+\\mu )} $,\n$$\n\\Sigma (\\mu )\\asymp e^{-(Ns+\\mu )}\\sum _{N+\\frac{\\mu -1}{s}\\le\\nu <\n  N+\\frac{\\mu }{s}}\\nu ^k \\asymp e^{-(Ns+\\mu )}\\frac{(Ns+\\mu )^k}{s^{k+1}}.\n$$\nHence,\n\\[\\begin{split}\n  P_{N,k}(s)&\\asymp \\frac{e^{-Ns}}{s^{k+1}}\\sum_{\\mu =1}^\\infty e^{-\\mu\n  }\n  (Ns+\\mu )^k\\\\\n  &\\asymp \\frac{e^{-Ns}}{s^{k+1}}\n  (Ns+1)^k=\\frac{e^{-Ns}}{s}\\left(N+\\frac{1}{s}\\right)^k.\n\\end{split}\\]\nRecalling (\\ref{ed.29.6}) and the fact that $s\\asymp 1-t$, $1\/2\\le\nt<1$, we get\n(\\ref{ed.29.2}) when $N=1$ and (\\ref{ed.29.3}) when $N\\ge 2$. \n\n\\par It remains to show (\\ref{ed.29.4}) and it suffices to do so for\n$1\/2\\le t\\le 1-C\/N$, $N\\gg 1$ and for $C\\ge 1$ sufficiently large but\nindependent of $N$. Indeed, for $1-C\/N\\le t\\le 1-1\/{\\mathcal{O}}(N)$, both\n$M_{N,k}(t)$ and $M_{\\infty ,k}(t)$ are $\\asymp N^{1+k}$. We can also\nexclude the case $k=0$ where we have explicit formulae. \n\n\\par To get the equivalence (\\ref{ed.29.4}) for $1\/2\\le t\\le 1-C\/N$,\n$k\\ge 1$,\nit suffices, in view of (\\ref{ed.29.2}), (\\ref{ed.29.3}), to show that\nfor such $t$ and for $N\\gg 1$, we have \n$$\n\\frac{N^kt^N}{1-t}\\le \\frac{1}{D}\\frac{1}{(1-t)^{k+1}},\n$$\nfor any given $D\\ge 1$, provided that $C$ is large enough. In other\nterms, we need\n$$\nt^N(1-t)^k\\le \\frac{1}{D}N^{-k},\\hbox{ for }\\frac{1}{2}\\le t\\le 1-\\frac{C}{N},\n$$\nwhen $C=C(D)$ is large enough and $N\\ge N(C)\\gg 1$. The left hand side\nin this inequality is an increasing function of $t$ on the interval\n$[0,1\/\\left( 1+k\/N \\right)]$. If $t\\le\n1-C\/N \\le 1\/(1+k\/N)$ (which is fulfilled when $C\\ge 2k$ and $N\\gg N(C)$)\nit is \n$$\n\\le \\left(1-\\frac{C}{N} \\right)^N\\left(\\frac{C}{N} \\right)^k=\\left( 1\n+{\\mathcal{O}}_C\\left(\\frac{1}{N} \\right)\\right)e^{-C}C^kN^{-k}.\n$$\nThis is $\\le N^{-k}\/D$ if $C\\ge C(D)$, $N\\ge N(C)$.\n\\end{proof}\nFor simplicity we will restrict the attention to the region\n\\begin{equation}\\label{ed.30}\n|z|\\le r_0-1\/N,\n\\end{equation}\nwhere $G\\asymp (1-|z|)^{-1}$, $G'\\asymp (1-|z|)^{-2}$.\n\\par \nIt follows from the calculation (\\ref{ed.35.5}) below, that \n$$\n|\\partial _zZ|^2=\\left(\\frac{2}{t}\\left(K(t\\partial _t)^2K\n\t\t  +(t\\partial _tK)^2 \\right) \\right)_{t=|z|^2}.\n$$\nThis is $\\asymp 1$ for $|z|\\le 1\/2$ and for $1\/2\\le |z|<1-1\/N $ it is in \nview of Proposition \\ref{ed1} and the subsequent observation\n$$\\asymp M_{N,0}M_{N,2}+M_{N,1}^2\\asymp \\frac{1}{(1-t)^4},\\ t=|z|^2.$$ \nIn the region (\\ref{ed.30}) we get:\n\\begin{equation}\\label{ed.31}\n|Z'(z)|\\asymp G(|z|)^2.\n\\end{equation}\n(\\ref{ed.29}), (\\ref{ed.30}), (\\ref{ed.31}) will be used in\n(\\ref{ed.27}), (\\ref{ed.28}). \n\\par \nCombining the implicit function theorem and Rouch\\'e's theorem to \n\\eqref{ed.18},we see that for \n$|\\alpha '|<C_1N$, $\\alpha '=\\sum_2^N \\alpha _je_j\\in \n\\overline{Z}(z)^\\perp$, the equation\n\\begin{equation}\\label{ed.20}\n\\widetilde{g}(z,\\alpha _1,\\alpha ')=0\n\\end{equation}\nhas a unique solution\n\\begin{equation}\\label{ed.21}\n\\alpha _1=f(z,\\alpha ')\\in D(0,C_1N\/G(|z|)).\n\\end{equation}\nHere, we also use \\eqref{ed.11}, \\eqref{ed.16}. Moreover, $f$ satisfies\n\\begin{equation}\\label{ed.22}f(z,\\alpha ')=-\\frac{z^N}{\\delta\n    |Z|^2}+{\\mathcal{O}}(1)\\delta N^2={\\mathcal{O}}(1)\\left(\\frac{|z|^N}{\\delta\n      G^2}+\\delta N^2 \\right).\\end{equation}\n\n\\par Differentiating the equation (\\ref{ed.20}) (where $\\alpha _1=f$)\nwe get\n$$\n\\partial _z\\widetilde{g}+\\partial _\\alpha \\widetilde{g}\\partial\n_zf=0,\\ \\partial _{\\overline{z}}\\widetilde{g}+\\partial _\\alpha \\widetilde{g}\\partial\n_{\\overline{z}}f=0.\n$$\nHence,\n\\begin{equation}\\label{ed.23}\n\\begin{cases}\n\\partial _zf=-\\left(\\partial _{\\alpha _1}\\widetilde{g}\n\\right)^{-1}\\partial _z\\widetilde{g},\\\\\n\\partial _{\\overline{z}}f=-\\left(\\partial _{\\alpha _1}\\widetilde{g}\n\\right)^{-1}\\partial _{\\overline{z}}\\widetilde{g}.\n\\end{cases}\n\\end{equation}\nSince $\\widetilde{g}$ is holomorphic in $\\alpha _1,\\alpha '$ and in\n$\\alpha _1,\\alpha _2,...,\\alpha _{N^2}$, we see that $f$ is holomorphic in\n$\\alpha '$ and in $\\alpha _2,...,\\alpha _{N^2}$ Applying $\\partial\n_{\\alpha _2},...,\\partial _{\\alpha _{N^2}}$ to (\\ref{ed.20}), we get\n\\begin{equation}\\label{ed.24}\n\\partial _{\\alpha _j}f=-\\left(\\partial _{\\alpha _1}\\widetilde{g}\n\\right)^{-1} \\partial _{\\alpha _j}\\widetilde{g},\\ 2\\le j\\le {N^2}.\n\\end{equation}\n\\par \nCombining (\\ref{ed.19}) in the form,\n$$\n\\partial _{\\alpha _1}\\widetilde{g}(z,\\alpha )=(1+{\\mathcal{O}}(G(|z|)\\delta\nN))\\delta |Z(z)|^2,\n$$\n(\\ref{ed.19.5}), (\\ref{ed.27}), (\\ref{ed.28}) with (\\ref{ed.23}) and\n(\\ref{ed.24}), we get\n\\begin{equation}\\label{ed'.1}\n\\begin{split}\n&\\partial _zf=-\\frac{(1+{\\mathcal{O}}(G\\delta N))}{\\delta |Z(z)|^2}\\times\n\\\\ & \\left(Nz^{N-1}+\\delta\nf\\partial _z\\left(|Z|^2 \\right)\n+{\\mathcal{O}}\\left( G^2\\delta ^2N\\right)\\left|\\sum_2^{N^2} \\alpha _j\\partial _ze_j\n\\right|\n+{\\mathcal{O}}(1)\\frac{(G\\delta N)^2}{r_0-|z|}\n\\right).\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{ed'.2}\n\\begin{split}\n\\partial _{\\overline{z}}f=&-\\frac{(1+{\\mathcal{O}}(G\\delta N))}{\\delta\n  |Z(z)|^2}\\times \\\\\n&\\left(\\delta f\\partial _{\\overline{z}}\\left(|Z|^2 \\right)\n+{\\mathcal{O}}\\left( G^2\\delta ^2N\\right)\\left| f\\overline{\\partial _zZ}\n  +\\sum_2^{N^2} \\alpha _j\\partial _{\\overline{z}}e_j \\right|\n \\right),\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{ed'.3}\n\\partial _{\\alpha _j}f={\\mathcal{O}}(1)\\frac{G^2\\delta ^2N}{\\delta\n  G^2}={\\mathcal{O}}(\\delta N),\\ 2\\le j\\le {N^2}.\n\\end{equation}\nFrom (\\ref{ed.29}) and the observation prior to Proposition \\ref{ed1}\nwe know that \n$$\n\\partial _z\\left( |Z|^2 \\right),\\ \\partial _{\\overline{z}}\\left( |Z|^2\n\\right) \\asymp G(|z|)^3|z|.\n$$\nRecall also that $|Z|\\asymp G(|z|)$. Using this in (\\ref{ed'.1}),\n(\\ref{ed'.2}), we get\n\\begin{equation}\\label{ed'.4}\n\\begin{split}\n\\partial _zf=&\\frac{{\\mathcal{O}}(1)}{\\delta G^2}\\times \\\\\n&\\left(N|z|^{N-1}+\\delta |f|G^3|z|\n+{\\mathcal{O}}\\left(G^2\\delta ^2N \\right)\\left| \\sum_2^{N^2} \\alpha _j\\partial\n    _ze_j \\right| +{\\mathcal{O}}(1)\\frac{G^2\\delta ^2N^2}{r_0-|z|})\n\\right) .\n\\end{split}\n\\end{equation}\n\\section{Choosing appropriate coordinates}\\label{chvar}\n\\par \nThe next task will be to choose an orthonormal basis\n$e_1(z),e_2,...,e_{N^2}(z)$ in ${\\mathds{C}}^{N^2}$ with\n$e_1(z)=|Z(z)|^{-1}\\overline{Z}(z)$ such that we get a nice control\nover $\\sum_2^{N^2}\\alpha _j\\partial _ze_j$, $\\sum_2^{N^2}\\alpha _j\\partial\n_{\\overline{z}}e_j$ and such that \n$$\n{{dQ_1\\wedge ...\\wedge dQ_{N^2}}_\\vert}_{\\alpha _1=f(z,\\alpha ')}\n$$\ncan be expressed easily up to small errors. Consider a point $z_0\\in\nD(0,r_0-N^{-1})$. We shall see\nbelow that the vectors $\\overline{Z}(z)$, $\\overline{\\partial _zZ}(z)$\nare linearly independent for every $z\\in D(0,1)$\n\\begin{prop}\\label{ed2}\nThere exists an orthonormal basis $e_1(z),e_2(z),...,e_{N^2}(z)$ in ${\\mathds{C}}^{N^2}$, \ndepending smoothly on $z\\in \\mathrm{neigh\\,}(z_0)$ such that\n\\begin{equation}\\label{ed'.6}\ne_1(z)=|Z(z)|^{-1}\\overline{Z}(z),\n\\end{equation}\n\\begin{equation}\\label{ed'.7}\n{\\mathds{C}}e_1(z_0)\\oplus{\\mathds{C}}e_2(z_0)={\\mathds{C}}\\overline{Z}(z_0)\\oplus\n\\overline{\\partial _zZ}(z_0),\n\\end{equation}\n\\begin{equation}\\label{ed'.8}\ne_j(z)-e_j(z_0)={\\mathcal{O}}((z-z_0)^2),\\ j\\ge 3.\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe choose $e_1(z)$ as in (\\ref{ed'.6}). Let $e_3(z_0),...,e_{N^2}(z_0)$ be\nan orthonormal basis in $\\left({\\mathds{C}}\\overline{Z}(z_0)\\oplus {\\mathds{C}}\n\\overline{\\partial _zZ}(z_0) \\right)^\\perp$. Then we get an\northonormal family $e_3(z),...,e_{N^2}(z)$ in $e_1(z)^\\perp$ in the\nfollowing way:\n\n\\par Let $V_0$ be the isometry ${\\mathds{C}}^{{N^2}-2}\\to {\\mathds{C}}^{N^2}$, defined\nby $V_0\\nu _j^0=e_j(z_0)$, $j=3,...,{N^2}$, where $\\nu _3^0,...,\\nu _{N^2}^0$\nis the canonical basis in ${\\mathds{C}}^{{N^2}-2}$ with a non-canonical\nlabeling. Let $\\pi (z)u=(u|e_1(z))e_1(z)$ be the orthogonal\nprojection onto ${\\mathds{C}}e_1(z)$. For $z\\in \\mathrm{neigh\\,}(z_0,{\\mathds{C}})$, \nlet $V(z)=(1-\\pi (z))V_0$. Then $f_j(z)=V(z)\\nu _j^0$,\n$j=3,...,{{N^2}}$ form a linearly independent system in $e_1(z)^\\perp$ and\nwe get an orthonormal system of vectors that span the same hyperplane\nin $e_1(z)^\\perp$ by Gram orthonormalization,\n$$\ne_j(z)=V(z)(V^*(z)V(z))^{-\\frac{1}{2}}\\nu _j^0,\\ 3\\le j\\le {N^2}.\n$$\n\n\\par We have \n$$\nV(z)\\nu _j^0=(1-\\pi (z))e_j(z_0)=e_j(z_0)-(e_j(z_0)|e_1(z))e_1(z),\n$$\n$$\n(e_j(z_0)|e_1(z))=\\frac{(e_j(z_0)|\\overline{Z}(z))}{|Z(z)|}={\\mathcal{O}}((z-z_0)^2),\n$$\nsince $(e_j(z_0)|\\overline{Z}(z))=e_j(z_0)\\cdot Z(z)=:k(z)$ is a\nholomorphic function of $z$ with\n$k(z_0)=(e_j(z_0)|\\overline{Z}(z_0))=0$,\n$k'(z_0)=(e_j(z_0)|\\overline{\\partial _zZ}(z_0))=0$.\nThus, $V(z)=V(z_0)+{\\mathcal{O}}(z-z_0)^2)$ and we conclude that\n(\\ref{ed'.8}) holds. Let $e_2(z)$ be a normalized vector in\n$(e_1(z),e_3(z),e_4(z),...,e_{N^2}(z))^\\perp$ depending smoothly on\n$z$. Then $e_1(z),e_2(z),...,e_{N^2}(z)$ is an orthonormal basis and since\n$e_3(z_0),...,e_{N^2}(z_0)$ are orthogonal to\n$\\overline{Z}(z_0),\\overline{\\partial Z}(z_0)$ by construction, we get (\\ref{ed'.7}).\n\\end{proof}\n\\par \nWe can make the following explicit choice:\n\\begin{equation}\\label{ed'.9}\ne_2(z)=|f_2|^{-1}f_2,\\ f_2=\\overline{\\partial _zZ}(z)-\\sum_{j\\ne\n  2}(\\overline{\\partial _zZ}(z)|e_j(z))e_j(z),\n\\end{equation}\nso that for $z=z_0$,\n\\begin{equation}\\label{ed'.10}\ne_2(z_0)=|f_2(z_0)|^{-1}f_2(z_0),\\ f_2(z_0)= \n\\overline{\\partial _zZ}(z_0)-(\\overline{\\partial _zZ}(z_0)|e_1(z_0))e_1(z_0).\n\\end{equation}\n\\par \nWe next compute some scalar products and norms with $Z$ and\n$\\partial _zZ$. Recall that $Z(z)=\\left(z^{N-j+k-1} \\right)_{j,k=1}^N$\nand that $|Z(z)|=K(|z|^2)$, $K(t)=\\sum_0^{N-1}t^\\nu $. Repeating\nbasically the same computation, we get\n$$\nz\\partial _zZ=\\left((N-j+k-1)z^{N-j+k-1} \\right)_{j,k=1}^N,\n$$\nand\n\\begin{equation}\\label{ed.35.5}\\begin{split}\n&|z\\partial _zZ|^2=\\sum_{j,k=1}^N (N-j+k-1)^2|z|^{2(N-j+k-1)}\n=\\sum_{\\nu ,\\mu =0 }^{N-1}(\\nu +\\mu )^2|z|^{2(\\nu +\\mu )}\\\\\n&=\\sum_0^{N-1}\\nu ^2|z|^{2\\nu }\\sum_0^{N-1}|z|^{2\\mu }+2\\sum_0^{N-1}\\nu\n|z|^{2\\nu }\\sum_0^{N-1}\\mu |z|^{2\\mu }+\\sum_0^{N-1}|z|^{2\\nu\n}\\sum_0^{N-1}\\mu ^2|z|^{2\\mu }\\\\\n&=2\\left(K(t\\partial _t)^2K + (t\\partial _tK)^2 \\right)_{t=|z|^2}.\n\\end{split}\n\\end{equation}\nSimilarly,\n\\[\n\\begin{split}\n(z\\partial _zZ|Z)=&\\sum_{j,k=1}^N (N-j+k-1)|z|^{2(N-j+k-1)}\\\\\n=&\\sum_{\\nu =0}^{N-1}\\sum_{\\mu =1}^{N-1} (\\nu +\\mu )|z|^{2(\\nu +\\mu\n  )}\\\\\n=&2(Kt\\partial _tK)_{t=|z|^2}.\n\\end{split}\n\\]\nThen, by a straight forward calculation,\n\\begin{equation}\\label{ed.36}\n|\\partial _zZ|^2-\\frac{|(\\partial\n  _zZ|Z)|^2}{|Z|^2}=\\left(\\frac{2}{t}\\left(K(t\\partial _t)^2K -(t\\partial _tK)^2 \\right)\\right)_{t=|z|^2}\n\\end{equation}\n\\par \nHere,\n\\[\\begin{split}\n\\frac{2}{t}\\left(K(t\\partial _t)^2K-(t\\partial _tK)^2 \\right)\n&=\\frac{2}{t}\\sum_0^{N-1}t^\\nu \\sum_0^{N-1}\\nu ^2t^\\nu\n-\\frac{2}{t}\\left(\\sum_0^{N-1} \\nu t^\\nu  \\right)^2\\\\\n=\\sum_{\\nu ,\\mu =0}^{N-1} \\left(\\nu ^2+\\mu ^2-2\\nu \\mu\n  \\right)t^{\\nu +\\mu -1}  \n&=\\sum_{\\nu ,\\mu =0}^{N-1}(\\nu -\\mu )^2t^{\\nu +\\mu -1}\\\\\n&=\\sum_{k=0}^{2N-3}a_{k,N}t^k, \n\\end{split}\n\\]\nwhere\n$$\na_{k,N}=\\sum_{\\substack{\\nu +\\mu -1=k \\\\ 0\\le \\nu ,\\mu \\le N-1}}(\\nu -\\mu )^2.\n$$\nWe observe that\n$$\na_{k,N}\\le {\\mathcal{O}}(1)(1+k)^3\\hbox{ uniformly with respect to }N,\n$$\n$$a_{k,N}=a_{k,\\infty }\\hbox{ is independent of }N\\hbox{ for }k\\le N-2,$$\n$$\na_{k,\\infty }\\ge (1+k)^3\/{\\mathcal{O}}(1).\n$$\nWe conclude that\n$$\n\\frac{1}{C}\\left(1+M_{N-1,3} \\right)\\le\n\\frac{2}{t}\\left(K(t\\partial _t)^2K-(t\\partial _tK)^2 \\right)\n \\le C\\left(1+M_{2N-2,3} \\right)\n$$\nand \\eqref{ed.29.4.5} shows that the first and third members are of the same\norder of magnitude,\n$$\n\\asymp 1+M_{N,3}(t)\\asymp \\min \\left(\\frac{1}{1-t},N \\right)^4\n$$\nwhich is $\\asymp 1+ M_{\\infty ,3}(t)$, for $0\\le t\\le 1-1\/N$. From this\nand Proposition \\ref{ed1} we get:\n\\begin{prop}\\label{ed3}\nWe have\n\\begin{equation}\\label{ed.37}\n\\frac{2}{t}(K(t\\partial _t)^2K-(t\\partial _tK)^2)\\asymp K^4,\\ 0<t\\le 1-1\/N,\n\\end{equation}\nwhere we recall that $K=K_N$ depends on $N$ and that \n$$\nK_N=K_\\infty -\\frac{t^N}{1-t}.\n$$\nWe have\n\\begin{equation}\\label{ed.38}\n\\begin{cases}\nt\\partial _tK_N=t\\partial _tK_\\infty +{\\mathcal{O}}\\left(\\frac{Nt^N}{1-t}\n\\right),\\ &t\\le 1-\\frac{1}{N},\\\\\n(t\\partial _t)^2K_N=(t\\partial _t)^2K_\\infty +{\\mathcal{O}}\\left(\\frac{N^2t^N}{1-t}\n\\right),\\ &t\\le 1-\\frac{1}{N},\n\\end{cases}\n\\end{equation}\nand it follows that\n\\begin{align}\\label{ed.39}\n  &\\frac{2}{t}\\left( K_N(t\\partial _t) ^2K_N-(t\\partial _tK_N) ^2\\right)-\n  \\frac{2}{t}\\left( K_\\infty (t\\partial _t) ^2K_\\infty -(t\\partial _t\n    K_\\infty) ^2 \\right)\n    \\notag \\\\\n  &= {\\mathcal{O}}\\left(\\frac{N^2t^N}{(1-t)^2} \\right),\n\\end{align}\nfor $t\\le 1-1\/N$.\n\\end{prop}\nProposition \\ref{ed3} and (\\ref{ed.36}) give\n\\begin{equation}\\label{ed.40}\n|\\partial _zZ|^2-\\frac{|(\\partial _zZ|Z)|^2}{|Z|^2}\\asymp K(|z|^2)^4.\n\\end{equation}\nThis implies that $\\partial _zZ$, $Z$ are linearly independent.\n\\par Assume that $$|\\nabla _ze_1(z)|={\\mathcal{O}}(m)$$\nfor some weight $m\\ge 1$. We shall see below that this holds when\n$m=K(|z|^2)$. Then $\\| \\nabla _z\\Pi \\|={\\mathcal{O}}(m)$ and hence\n$\\|\\nabla_zV \\|={\\mathcal{O}}(m)$. It follows that $\\| \\nabla\n_z(V^*(z)V(z))\\|={\\mathcal{O}}(m)$. By standard (Cauchy-Riesz)\nfunctional calculus, using also that $\\| V(z)^{-1}\\|={\\mathcal{O}}(1)$, we\nget $\\| \\nabla _z(V^*(z)V(z))^{-\\frac{1}{2}}\\| = {\\mathcal{O}}(m)$. Hence\n$\\| \\nabla _zU(z)\\|={\\mathcal{O}}(m)$, where $U(z)=V(z)(V^*(z)V(z))^{-1\/2}$\nis the isometry appearing in the proof of Proposition \\ref{ed2}. Since $\\nabla _ze_j=(\\nabla\n_zU(z))\\nu _j^0$, we conclude that $\\| \\nabla _zU(z)\\| ={\\mathcal{O}}(m)$, so \n\\begin{equation}\\label{ed.41}\n\\left| \\sum_3^{N^2} \\alpha _j\\nabla _ze_j \\right| \\le {\\mathcal{O}}(m)\\|\\alpha\n\\|_{{\\mathds{C}}^{{N^2}-2}}.\n\\end{equation}\nWe next show that we can take $m=K(|z|^2)$. We have\n\\begin{equation}\\label{ed.42}\n\\nabla _ze_1=\\frac{\\nabla _z\\overline{Z}}{|Z|}-\\frac{\\nabla\n  _z|Z|}{|Z|^2}\\overline{Z}=\\frac{\\nabla\n  _z \\overline{Z}}{K}-\\frac{K'\\nabla _z(z\\overline{z})}{K^2}\\overline{Z}.\n\\end{equation}\nBy \\eqref{ed.35.5},\n$$\n|\\partial _z Z|=\\left(\\frac{2}{t}\\left(K(t\\partial\n    _t)^2K+(t\\partial _tK)^2 \\right)\n\\right)^{\\frac{1}{2}}_{t=|z|^2}={\\mathcal{O}}(K^2).\n$$\nSince $Z$ is holomorphic, this leads to the same estimates for\n$|\\nabla _zZ|$ and $|\\nabla _z\\overline{Z}|$, and \n$|\\partial_z^2 Z| = \\mathcal{O}(K^3)$, for $|z|< 1-N^{-1}$, by the Cauchy \ninequalities. Using this in\n(\\ref{ed.42}), we get\n\\begin{equation}\\label{ed.43}\n|\\nabla _ze_1|={\\mathcal{O}}(K).\n\\end{equation}\nThus we can take $m=K(|z|^2)$ in (\\ref{ed.41}). Let $f_2$ be the\nvector in (\\ref{ed'.9}) so that $e_2(z)=|f_2|^{-1}f_2$. Recall that\n$e_j=U(z)\\nu _j^0$, where we now know that $\\| \\nabla _zU(z)\\| ={\\mathcal{O}}(K)$. \nWrite,\n$$\n\\nabla _zf_2=\\nabla _z\\overline{\\partial _zZ}-\\sum_{j\\ne 2}\\left(\n(\\nabla _z\\overline{\\partial _zZ}|e_j)e_j+(\\overline{\\partial\n  _zZ}|\\nabla _ze_j)e_j+(\\overline{\\partial _zZ}|e_j)\\nabla _ze_j\n \\right).\n$$\n\n\\par Here, $|\\nabla _z\\overline{\\partial _zZ}|={\\mathcal{O}}(K^3)$, as we\nhave just seen. It is also clear that the term for $j=1$ in the sum\nabove is ${\\mathcal{O}}(K^3)$. It remains to study\n$|\\mathrm{I}+\\mathrm{II}+\\mathrm{III}|\\le\n|\\mathrm{I}|+|\\mathrm{II}|+|\\mathrm{III}|$, where\n\\[\\begin{split}\n\\mathrm{I}&=\\sum_3^{N^2}(\\nabla _z\\overline{\\partial _zZ}|e_j)e_j,\\\\\n\\mathrm{II}&=\\sum_3^{N^2}(\\overline{\\partial\n  _zZ}|\\nabla _ze_j)e_j,\\\\\n\\mathrm{III}&=\\sum_3^{N^2}(\\overline{\\partial _zZ}|e_j)\\nabla _ze_j.\n\\end{split}\n\\]\n\n\\par Here, $|\\mathrm{I}|\\le |\\nabla _z\\overline{\\partial _zZ}|=\n{\\mathcal{O}}(K^3)$ and by (\\ref{ed.41}) we have \n$|\\mathrm{III}|\\le {\\mathcal{O}}(K)|\\overline{\\partial _zZ}|={\\mathcal{O}}(K^3)$. Further,\n\\[\n\\begin{split}\n\\mathrm{II}&=\\sum_3^{N^2} (\\overline{\\partial _zZ}|(\\nabla _zU(z))\\nu\n_j^0)e_j\\\\\n&=\\sum _3^{N^2}((\\nabla _zU(z))^*\\overline{\\partial _zZ}|\\nu _j^0)e_j,\n\\end{split}\n\\]\nso\n$$\n|\\mathrm{II}|=|(\\nabla _zU(z))^*\\overline{\\partial _zZ}|={\\mathcal{O}}(K)K^2={\\mathcal{O}}(K^3).\n$$\nThus, \n\\begin{equation}\\label{ed.43.5}\n|\\nabla _zf_2|={\\mathcal{O}}(K^3).\n\\end{equation}\nRecall from (\\ref{ed'.10}) that for $z=z_0$,\n$$\nf_2=\\overline{\\partial _zZ}-(\\overline{\\partial _zZ}|e_1)e_1,\n$$\n$$\n|f_2|^2=|\\partial _zZ|^2-\\frac{|(\\partial _zZ|Z)|^2}{|Z|},\n$$\nso by (\\ref{ed.40}),\n$$\n|f_2(z_0)|\\asymp K(|z_0|^2)^2,\n$$\nHence,\n$$\n|f_2(z)|\\asymp K^2,\\ z\\in \\mathrm{neigh\\,}(z_0).\n$$\nFrom this, (\\ref{ed'.9}) and (\\ref{ed.40}), we conclude first that\n$\\nabla_z|f_2|=\\mathcal{O}(K^3)$ and then that \n\\begin{equation}\\label{ed.44}\n|\\nabla _ze_2|={\\mathcal{O}}(K).\n\\end{equation}\nThis completes the proof of the fact that we can take $m=K$ above. In\nparticular (\\ref{ed.41}) holds with $m=K(|z|^2)\\asymp G(|z|)$, so \n\\begin{equation}\\label{ed.45}\n\\left| \\sum_2^{N^2} \\alpha _j\\partial _ze_j \\right| \\le \n{\\mathcal{O}}(1)G|\\alpha |\\le {\\mathcal{O}}(1)GN,\n\\end{equation}\nwhere we used the assumption that $|Q|\\le C_1N$ in the last step. \n\nCombining this with (\\ref{ed'.4}), (\\ref{ed'.3}), (\\ref{ed.22}), (\\ref{ed.29}) and\nthe observation prior to Proposition \\ref{ed1}, we get \n\\[\n\\begin{split}\n\\partial _zf&=\\frac{{\\mathcal{O}}(1)}{\\delta G^2}\\left(N|z|^{N-1} +\\delta\n  \\left( \\frac{|z|^N}{\\delta G^2}+\\delta N^2 \\right)G^3+G^2\\delta\n  ^2NGN+\\frac{G^2\\delta ^2N^2}{r_0-|z|} \\right)\\\\\n&={\\mathcal{O}}(1)\\left( \\frac{N|z|^{N-1}}{\\delta G^2}+\\frac{|z|^N}{\\delta\n    G}+G\\delta N^2+\\frac{\\delta N^2}{r_0-|z|} \\right).\n\\end{split}\n\\]\nIn the last parenthesis the second term is dominated by the first one and\nthe third term is dominated by the fourth. If we\nrecall that $r_0-|z|\\ge 1\/N$, we get\n\\begin{equation}\\label{ed.46}\n\\partial _zf={\\mathcal{O}}(1)\\left(\\frac{N|z|^{N-1}}{\\delta G^2} +\\delta N^3 \\right).\n\\end{equation}\nSimilarly, from (\\ref{ed'.2}), (\\ref{ed.31}) we get\n\\[\n\\begin{split}\n  \\partial _{\\overline{z}}f&=\\frac{{\\mathcal{O}}(1)}{\\delta\n    G^2}\\left(\\delta \\left(\\frac{|z|^N}{\\delta G^2}+\\delta N^2 \\right)\n    G^3+G^2\\delta ^2N\\left(\\left(\\frac{|z|^N}{\\delta G^2}+\\delta\n        N^2 \\right)G^2+GN \\right) \\right)\\\\\n&={\\mathcal{O}}(1)\\left( \\frac{|z|^N}{\\delta G}+\\delta N^2G +\n\t  N|z|^N +G^2\\delta ^2N^3+G\\delta N^2\n\\right).\n\\end{split}\n\\]\nUsing (\\ref{ed.11}), we get\n\\begin{equation}\\label{ed.47}\n\\partial _{\\overline{z}}f={\\mathcal{O}}(1)\\left(\\frac{|z|^N}{\\delta\n    G}+\\delta N^2G \\right),\n\\end{equation}\nsee \\eqref{ed.22}. This will be used together with the estimates $\\partial _{\\alpha\n  _j}f={\\mathcal{O}}(\\delta N)$ in (\\ref{ed'.3}).\n\nThe differential form $dQ_1\\wedge dQ_2\\wedge ...\\wedge dQ_{N^2}$ will\nchange only by a factor of modulus one if we express $Q$ in another\nfixed orthonormal basis and we will choose for that the basis\n$e_1(z_0),...,e_{N^2}(z_0)$:\n\\[\nQ=\\sum_1^{N^2} Q_k e_k(z_0),\\ \\ Q_k=(Q|e_k(z_0)).\n\\]\nWrite \n$$Q=\\alpha _1\\underbrace{\\overline{Z}(z)}_{|Z(z)|e_1(z)}+\\sum_2^{N^2} \\alpha _ke_k(z)\n$$\nand restrict to $\\alpha _1=f(z,\\alpha _2,...,\\alpha _{N^2})$, where we\nsometimes identify $\\alpha '\\in \\overline{Z}(z)^\\perp$ with $(\\alpha\n_2,...,\\alpha _{N^2})$:\n$$\n{{Q}_\\vert}_{\\alpha _1=f(z,\\alpha ')}=\nf(z,\\alpha ')\\overline{Z}(z)+\\sum_2^{N^2} \\alpha _ke_k(z).\n$$\nThen,\n$$\nQ_j=f(\\overline{Z}(z)|e_j(z_0))+\\sum_{k=2}^{N^2 }\\alpha _k(e_k(z)|e_j(z_0)),\n$$\n\\[\n\\begin{split}\ndQ_j=&(d_zf+d_{\\alpha\n  '}f)(\\overline{Z}(z)|e_j(z_0))+f(d_z\\overline{Z}(z)|e_j(z_0))\\\\\n&+\\sum_{k=2}^{N^2}\\alpha _k(d_ze_k(z)|e_j(z_0)) +\\sum_{k=2}^{N^2}d\\alpha _k(e_k(z)|e_j(z_0)).\n\\end{split}\n\\]\nTaking $z=z_0$ until further notice, we get with $\\alpha '=(\\alpha\n_2,...,\\alpha _{N^2})$:\n$$\ndQ_j=(d_zf+d_{\\alpha '}f)(\\overline{Z}|e_j)+f(\\overline{\\partial\n  _zZ}|e_j)d\\overline{z}+\\alpha _2(d_ze_2|e_j)+\\begin{cases}d\\alpha\n  _j,\\ j\\ge 2,\\\\ 0,\\ j=1\\end{cases}.\n$$\nHere, we used (\\ref{ed'.8}). The first term to the right is equal to\n$(d_zf+d_{\\alpha '}f)|\\overline{Z}|$ when $j=1$ and it vanishes when\n$j\\ge 2$. The second term vanishes for $j\\ge 3$, by (\\ref{ed'.7}). The\nthird term is equal to $-\\alpha _2(e_2|d_ze_j)$ (by differentiation of\nthe identity $(e_2|e_j)=\\delta _{2,j}$) and it vanishes for $j\\ge 3$\n(remember that we take $z=z_0$). Thus, for $z=z_0$:\n\\[\n\\begin{split}\n  dQ_1&=|\\overline{Z}| (d_zf+d_{\\alpha '}f)+f(\\overline{\\partial\n    _zZ}|e_1)d\\overline{z}-\\alpha _2(e_2|d_ze_1),\\\\\n  dQ_2&=d\\alpha _2+f(\\overline{\\partial _zZ}|e_2)d\\overline{z}-\\alpha\n  _2(e_2|d_ze_2),\\\\\ndQ_j&=d\\alpha _j,\\ j\\ge 3.\n\\end{split}\n\\]\nWhen forming $dQ_1\\wedge d\\overline{Q}_1\\wedge ...\\wedge\ndQ_{N^2}\\wedge d\\overline{Q}_{N^2}$ we see that the terms in $d\\alpha\n_j$ for $j\\ge 3$ in the expression for $dQ_1$ will not contribute, so in that expression\nwe can replace $d_{\\alpha '}f$ by $\\partial _{\\alpha _2}fd\\alpha\n_2$. Using (\\ref{ed.46}), (\\ref{ed.47}), (\\ref{ed'.3}), (\\ref{ed.22}),\n(\\ref{ed.31}) this gives, where ``$\\equiv$'' means equivalence up to\nterms that do not influence the $2{N^2}$ form above:\n\\[\\begin{split}\ndQ_1\\equiv &-\\alpha _2(e_2|d_ze_1) \n+{\\mathcal{O}}(1)\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3 \\right)dz\\\\\n& +{\\mathcal{O}}(1)\\left(\\frac{|z|^N}{\\delta }+G^2\\delta N^2\n\\right)d\\overline{z}+{\\mathcal{O}}(\\delta NG)d\\alpha _2.\n\\end{split}\n\\]\nSimilarly, using also (\\ref{ed.44}),\n\\[\n\\begin{split}\ndQ_2=d\\alpha _2+{\\mathcal{O}}\\left(\\frac{|z|^N}{\\delta }+\\delta\n  N^2G^2+|\\alpha _2|G \\right)d\\overline{z}+{\\mathcal{O}}\\left(|\\alpha _2|G \\right)dz.\n\\end{split}\n\\]\n\\par \nWhen computing $dQ_1\\wedge dQ_2$ we notice that the terms in $dz\n\\wedge d\\overline{z}$ will not contribute to the $2N^2$-form\n$dQ_1\\wedge d\\overline{Q}_1\\wedge ...\\wedge dQ_{N^2}\\wedge\nd\\overline{Q}_{N^2}$. We get \n\\begin{equation}\\label{ed.48}\n\\begin{split}\ndQ_1\\wedge d Q_2\\equiv & -\\alpha _2(e_2|d_ze_1)\\wedge d\\alpha _2 \\\\\n& +{\\mathcal{O}}(1)\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3+|\\alpha\n  _2|\\delta NG^2 \\right)dz\\wedge d\\alpha _2\\\\\n&+ {\\mathcal{O}}(1)\\left(\\frac{|z|^N}{\\delta }+G^2\\delta N^2+|\\alpha\n  _2|\\delta NG^2 \\right)d\\overline{z}\\wedge d\\alpha _2 .\n\\end{split}\n\\end{equation}\nHere,\n\\[\n\\begin{split}\n(e_2|d_ze_1)&=\\left( e_2|d_z \\left( |Z|^{-1} \\right)\\overline{Z}\\right)\n=\\left(e_2||Z|^{-1}d_z\\overline{Z} \\right)+\\left(e_2|\n  d_z\\left(|Z|^{-1} \\right) \\overline{Z} \\right)\\\\\n&=|Z|^{-1}\\left(e_2 |\\overline{\\partial _zZ} d\\overline{z} \\right)\n+0=|Z|^{-1} (e_2| \\overline{\\partial _zZ}) dz,\n\\end{split}\n\\]\nso the first term in (\\ref{ed.48}) is equal to \n$$\n-\\frac{\\alpha _2}{|Z|}(e_2|\\overline{\\partial\n  _zZ})dz\\wedge d\\alpha _2 ={\\mathcal{O}}(1)\\alpha _2 G\nd z\\wedge d\\alpha _2.\n$$\nNotice that \n$dQ_1\\wedge d\\overline{Q}_1 \\wedge dQ_2 \\wedge d\\overline{Q}_2=-dQ_1\n\\wedge dQ_2 \\wedge d\\overline{Q}_1 \\wedge d\\overline{Q}_2$. From\n(\\ref{ed.48}) and its complex conjugate we get \n\\[\\begin{split}\n&dQ_1\\wedge d\\overline{Q}_1\\wedge dQ_2\\wedge d\\overline{Q}_2\\\\\n&\\equiv\\left( - \\frac{|\\alpha _2|^2}{|Z|^2}\\left| \\left(e_2 |\n      \\overline{\\partial _zZ} \\right) \\right|^2\n      +{\\mathcal{O}}(1)\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3+|\\alpha\n    _2|\\delta NG^2 \\right)^2\\right.\\\\\n& \\left. +{\\mathcal{O}}(1)|\\alpha _2|G\\left(\\frac{N|z|^{N-1}}{\\delta\n      G}+G\\delta N^3+|\\alpha _2|G^2\\delta N \\right)\n \\right)dz\\wedge d\\overline{z}\\wedge d\\alpha _2\\wedge\n d\\overline{\\alpha }_2 .\n\\end{split}\n\\]\n\\begin{prop}\\label{ed4}\n  We express $Q$ in the canonical basis in ${\\mathds{C}}^{N^2}$ or in any\n  other fixed orthonormal basis. Let $e_1(z),...,e_{N^2}(z)$ be an\n  orthonormal basis in ${\\mathds{C}}^{N^2}$ depending smoothly on $z$ and\n  with $e_1(z)=|Z(z)|^{-1}\\overline{Z}(z)$, ${\\mathds{C}}e_1(z)\\oplus{\\mathds{C}}\n  e_2(z)={\\mathds{C}}\\overline{Z}(z)\\oplus \\overline{\\partial\n    _zZ}(z)$. Write $Q=\\alpha _1\\overline{Z}(z)+\\sum_2^{N^2}\\alpha\n  _je_j(z)$, and recall that the hypersurface \n  %\n  \\begin{equation*}\n   \\{(z,Q)\\in D(0,r_0-1\/N)\\times B(0,C_1N);\\, E_{-+}^\\delta (z,Q)=0\\}\n  \\end{equation*}\n  %\n  is given by (\\ref{ed.21}) with $f$ as in (\\ref{ed.22}). Then the\nrestriction of $dQ\\wedge d\\overline{Q}$ to this hypersurface, is given\nby \n\\begin{equation}\\label{ed.49}\n\\begin{split}\n&dQ\\wedge d\\overline{Q}=J(f)dz\\wedge d\\overline{z}\\wedge d\\alpha '\n\\wedge d\\overline{\\alpha }',\\\\\n&J(f)= - \\frac{|\\alpha _2|^2}{|Z|^2}\\left| \\left(e_2 |\n      \\overline{\\partial _zZ} \\right) \\right|^2\n      +{\\mathcal{O}}(1)\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3+|\\alpha\n    _2|\\delta NG^2 \\right)^2\\\\\n&\\phantom{J(f)=}  +{\\mathcal{O}}(1)|\\alpha _2|G\\left(\\frac{N|z|^{N-1}}{\\delta\n      G}+G\\delta N^3+|\\alpha _2|G^2\\delta N \\right).\n\\end{split}\n\\end{equation}\nHere $\\alpha '=(\\alpha _2,...,\\alpha _{N^2})$, $d\\alpha '\\wedge\nd\\overline{\\alpha }'=d\\alpha _2\\wedge d\\overline{\\alpha }_2\\wedge\n...\\wedge d\\alpha _{N^2}\\wedge d\\overline{\\alpha }_{N^2}$.\n\\end{prop}\n\\section{Proof of Theorem \\ref{ed5}}\\label{pfThm}\nLet $Q\\in {\\mathds{C}}^{N^2}$ be an $N\\times N$ matrix whose entries are\nindependent random variables $\\sim {\\mathcal{N}}_{\\mathds{C}}(0,1)$, so that the\ncorresponding probability measure is\n$$\n\\pi ^{-N^2}e^{-|Q|^2}(2i)^{-N^2}d\\overline{Q} _1\\wedge dQ_1\\wedge\n...\\wedge d\\overline{Q}_{N^2}\\wedge dQ_{N^2}=\\frac{1}{(2\\pi\n  i)^{N^2}}e^{-|Q|^2}d\\overline{Q}\\wedge dQ.\n$$\nWe are interested in \n\\begin{equation}\\label{ed.50}\nK_\\phi ={\\mathds{E}}\\left(1_{B_{{\\mathds{C}}^{N^2}}}(0,1)\\sum _{\\lambda \\in\n    \\sigma (A_0+\\delta Q)}\\phi (\\lambda ) \\right),\\ \\ \\phi \\in C_0(D(0,r_0-1\/N),\n\\end{equation}\nwhich is of the form (\\ref{ed.3}) with\n\\begin{equation}\\label{ed.51}\nm(Q)=1_{B_{{\\mathds{C}}^{N^2}}}(Q)\\pi ^{-N^2}e^{-|Q|^2},\n\\end{equation}\nso we have (\\ref{ed.8}), (\\ref{ed.9}) with $J(f)$ as in (\\ref{ed.49})\nand $f$ as in (\\ref{ed.21}). More explicitly,\n$$\n\\Xi (z)=\\int_{|f|^2|Z(z)|^2+|\\alpha '|^2\\le C_1^2N^2} \\pi\n^{-N^2}e^{-|f(z,\\alpha ')|^2|Z(z)|^2-|\\alpha '|^2}J(f)(z,\\alpha ')L(d\\alpha ').\n$$\nBy \\eqref{ed.22}, \\eqref{ed.11}, \\eqref{ed.16} : \n\\begin{equation*}\n |f| \\leq \\mathcal{O}(1)\\frac{N}{G}\\left( \\frac{|z|^N}{\\delta NG} + \\delta NG\\right)\n  \\ll \\frac{N}{G}.\n\\end{equation*}\nWe now strengthen \\eqref{ed.11}, \\eqref{ed.16} to the assumption\n\\begin{equation}\\label{ed.52}\n\\frac{|z|^N}{\\delta N G}+\\delta NG \\ll \\frac{1}{N},\\hbox{ for all }z\\in D(0,r_0),\n\\end{equation}\nimplying that $|f|G \\ll 1$, for all $z\\in D(0,r_0)$. Equivalently, \nby the same reasoning as after \\eqref{ed.25}, $r_0$ should satisfy \n\\begin{equation}\\label{ed.52.5}\n \\frac{r_0^N}{\\delta N G(r_0)} + \\delta NG(r_0) \\ll \\frac{1}{N}.\n\\end{equation}\nThen \n$$\ne^{-|f(z,\\alpha ')|^2|Z(z)|^2}\n  = 1+{\\mathcal{O}}(1)N^2\n    \\left(\\frac{|z|^N}{\\delta NG}+\\delta NG\\right)^2,\n$$\nand using (\\ref{ed.49}), we get\n\\[\n\\begin{split}\n\\Xi (z)=&\\left(1+ {\\mathcal{O}}(1) N^2\\left(\\frac{|z|^N}{\\delta NG}\n  +\\delta NG\\right)^2 \\right)\\times \\\\ & \\frac{|(e_2|\\overline{\\partial\n_zZ})|^2}{|Z|^2}\\int _{|(f|Z|,\\alpha ')|\\le C_1N}|\\alpha _2|^2\ne^{-|\\alpha '|^2}\\pi ^{1-N^2}L(d\\alpha ')\\\\\n&+{\\mathcal{O}}(1)\\int e^{-|\\alpha '|^2}\\left(\\frac{N|z|^{N-1}}{\\delta\n    G}+G\\delta N^3+|\\alpha _2|\\delta NG^2\n\\right)^2\\pi^{1-N^2}L(d\\alpha ')\\\\\n+&{\\mathcal{O}}(1)\\int e^{-|\\alpha '|^2}|\\alpha _2|G\\left(\\frac{N|z|^{N-1}}{\\delta\n    G}+G\\delta N^3+|\\alpha _2|\\delta NG^2\n\\right)\\pi^{1-N^2}L(d\\alpha ').\n\\end{split}\n\\]\n\n\\par Since $|f||Z|\\ll N$, the first integral is equal to\n$$\n\\int_{\\mathds{C}} \\frac{1}{\\pi }|w|^2e^{-|w|^2}L(dw)+{\\mathcal{O}}\n\\left(e^{-N^2\/{\\mathcal{O}}(1)} \\right) =1+{\\mathcal{O}}\\left(e^{-N^2\/{\\mathcal{O}}(1)} \\right).\n$$\nThe sum of the other two integrals is equal to \n\\[\n\\begin{split}\n&{\\mathcal{O}}(1)\\left(\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3+\\delta\n  NG^2\\right)^2+G \\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta N^3+\\delta\n  NG^2\\right)\\right)\\\\\n&={\\mathcal{O}}(1)\\left(\\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta\n    N^3\\right)^2+G \\left(\\frac{N|z|^{N-1}}{\\delta G}+G\\delta\n    N^3\\right)\\right) .\n\\end{split}\n\\]\nNoticing that\n$$\n\\frac{|(e_2|\\overline{\\partial _zZ})|^2}{|Z|^2} ={\\mathcal{O}}(G^2),\n$$\nwe deduce that\n\\begin{equation}\\label{ed.53}\n\\begin{split}\n\\Xi(z)=& \\frac{|(e_2|\\overline{\\partial _zZ})|^2}{|Z|^2} \\\\\n&+ {\\mathcal{O}}(1)\\left(G^2N^2\\left(\\frac{|z|^{N-1}}{\\delta G^2}+\\delta\n    N^2\\right)^2+G^2N \\left(\\frac{|z|^{N-1}}{\\delta G^2}+\\delta\n    N^2\\right)\\right) .\n\\end{split}\n\\end{equation}\nWe next study the leading term in (\\ref{ed.53}), given by\n\\begin{equation}\\label{ed.54}\n\\frac{|(\\overline{\\partial _zZ}|e_2)|^2}{\\pi |Z|^2}.\n\\end{equation}\nSince $\\overline{\\partial _zZ}$ belongs to the span of\n$e_1=\\overline{\\partial _zZ}\/|Z|$ and $e_2$, we have\n$$\n|(\\overline{\\partial _zZ}|e_2)|^2=|\\overline{\\partial\n  _zZ}|^2-|(\\overline{\\partial _zZ}|e_1)|^2,\n$$ \nso the leading term (\\ref{ed.54}) is \n$$\\frac{1}{\\pi |Z|^2}\\left(|\\overline{\\partial\n    _zZ}|^2-\\frac{|(\\overline{\\partial _zZ}|\\overline{Z})|^2}{|Z|^2}\n\\right),$$\nwhich by (\\ref{ed.36}) is equal to \n\\begin{equation}\\label{ed.55}\n{\\frac{2}{\\pi t}\\left(\\frac{(t\\partial _t)^2K}{K}-\\frac{(t\\partial _tK)^2}{K^2} \\right)}_{t=|z|^2}.\n\\end{equation}\nHere, $K=K_N(t)=\\sum_0^{N-1}t^\\nu $ is the function appearing in\nProposition \\ref{ed3}. Let us first compute the limiting quantity\nobtained by replacing $K=K_N$ in (\\ref{ed.55}) by $K_\\infty\n=1\/(1-t)$. Since $\\partial _tK_\\infty =K_\\infty ^2$, we get\n$$\nt\\partial _tK_\\infty =tK_\\infty ^2,\\ \\ (t\\partial _t)^2K_\\infty\n=tK_\\infty ^2+2t^2K_\\infty ^3,\n$$\nand\n\\begin{equation}\\label{ed.56}\n\\frac{2}{\\pi t}\\left(\\frac{(t\\partial _t)^2K_\\infty }{K_\\infty\n  }-\\frac{(t\\partial _tK_\\infty )^2}{K_\\infty ^2} \\right)=\\frac{2}{\\pi\n}K_\\infty ^2=\\frac{2}{\\pi }\\frac{1}{(1-t)^2}.\n\\end{equation}\n\n\\par We next approximate the expression (\\ref{ed.55}) with\n(\\ref{ed.56}), using (\\ref{ed.39}) and the fact that $K=(1+{\\mathcal{O}}(t^N))K_\\infty $ \n(uniformly with respect to $N$). The expression\n(\\ref{ed.55}) is equal to\n\\[\\begin{split}\n&\\frac{2}{\\pi tK^2}(K(t\\partial _t)^2K-(t\\partial _tK)^2)\\\\\n&= \\frac{2(1+{\\mathcal{O}}(t^N))}{\\pi tK_\\infty ^2}\\left(K_\\infty\n  (t\\partial _t)^2K_\\infty -(t\\partial _tK_\\infty )^2 +{\\mathcal{O}}\n  (N^2t^NK_\\infty ^2) \\right).\n\\end{split}\n\\]\nHere,\n$$\n(t\\partial _tK_\\infty )^2={\\mathcal{O}}(t^2K_\\infty ^4),\\ \\ K_\\infty\n(t\\partial _t)^2K_\\infty ={\\mathcal{O}}(tK_\\infty ^3+t^2K_\\infty ^4),\n$$\nso the last expression becomes,\n$$\n\\frac{2}{\\pi t}\\left(\\frac{(t\\partial _t)^2K_\\infty }{K_\\infty\n  }-\\frac{(t\\partial _tK_\\infty )^2}{K_\\infty ^2} \\right)+{\\mathcal{O}}\n  (t^NK_\\infty +t^{N+1}K_\\infty ^2+t^{N-1}N^2),\n$$\nwhere the first two terms in the remainder are dominated by the last one.\nWe conclude that the difference\nbetween the expressions (\\ref{ed.55}) and (\\ref{ed.56}) is ${\\mathcal{O}}(t^{N-1}N^2)$, \nand using also (\\ref{ed.53}), we get,\n\\begin{equation}\\label{ed.57}\n\\begin{split}\n\\Xi(z)=& \\frac{2}{\\pi (1-|z|^2)^2}+{\\mathcal{O}}(|z|^{2(N-1)}N^2) \\\\\n&+ {\\mathcal{O}}(1)\\left(G^2N^2\\left(\\frac{|z|^{N-1}}{\\delta G^2}+\\delta\n    N^2\\right)^2+G^2N \\left(\\frac{|z|^{N-1}}{\\delta G^2}+\\delta\n    N^2\\right)\\right) .\n\\end{split}\n\\end{equation}\nThe remainder term can be written\n\\[\n{\\mathcal{O}}(G^2)\\left(\\frac{|z|^{2(N-1)}N^2}{G^2}+\\frac{|z|^{2(N-1)}N^2}{\\delta\n  ^2G^4}+\\delta ^2N^6 +\\frac{|z|^{N-1}N}{\\delta G^2}+\\delta N^3 \\right).\n\\]\nBy (\\ref{ed.52}), $\\frac{1}{\\delta G}\\gg N^2$, so the second term is \n$$\n\\gg \\frac{|z|^{2(N-1)}N^2}{G^2}N^4,\n$$\nwhich is much larger than the first term. We now strengthen\n(\\ref{ed.52}) to\n$$\n\\frac{|z|^{N-1}}{\\delta G^2}+\\delta N^2\\ll \\frac{1}{N},\n$$\nor equivalently to\n\\begin{equation}\\label{ed.58}\n\\frac{|z|^{N-1}N}{\\delta G^2}+\\delta N^3\\ll 1 .\n\\end{equation}\nThen remainder in (\\ref{ed.57}) becomes\n$$\n{\\mathcal{O}}(G^2)\\left(\\frac{|z|^{N-1}N}{\\delta G^2}+\\delta N^3 \\right),\n$$\nand (\\ref{ed.57}) becomes\n\\begin{equation}\\label{ed.59}\n\\Xi (z)=\\frac{2}{\\pi (1-|z|^2)^2}\\left(1+{\\mathcal{O}}\n\\left(\\frac{|z|^{N-1}N}{\\delta G^2} +\\delta N^3 \\right) \\right),\n\\end{equation}\nwhich concludes the proof of Theorem \\ref{ed5}.\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nState and actuator constraints are often encountered in real-world systems but systematic feedback controller design remains challenging.\nThe main difficulty arises from needing to \\emph{predict} whether the system will remain in the feasible set when selecting a control input.\nRepeatedly solving a constrained finite-horizon optimal control problem (i.e., model predictive control) is one way to ensure feasibility, \nbut solving a nonlinear optimization in real-time can be difficult.\nAlternatively, one can avoid trajectory optimization entirely by constructing safe invariant sets, i.e., a set of states that guarantee feasibility indefinitely.\nThrough the development of control barrier functions (CBFs)~\\cite{ames2016control}, safe stabilizing controllers can be synthesized by simply solving a quadratic program (QP), and has recently been used in several applications~\\cite{ames2019control}.\nHowever, model error can significantly degrade the performance of these controllers to the extent that safety may no longer be guaranteed.\nWe develop a general framework that guarantees safety through parameter adaptation and online model estimation for uncertain nonlinear systems.\n\nControl barrier functions heavily rely on a model so it is critical to develop methodologies that maintain safety for uncertain systems.\nIn~\\cite{xu2015robustness} the so-called zeroing CBFs were shown to be Input-to-State stable, a property that was used to prove a superset of a safe set is forward invariant.\nThe size of the superset was characterized in~\\cite{kolathaya2018input} by introducing Input-to-State safety.\nStronger safety guarantees can be obtained via robust optimization as demonstrated in~\\cite{gurriet2018towards}.\nLearning-based methods~\\cite{fan2019bayesian,taylor2019learning} have been developed to address the conservatism of the robust strategies, but these can require extensive offline training to substantially improve the model.\nAdaptive CBFs (aCBFs)~\\cite{taylor2019adaptive} use ideas from adaptive control theory to ensure a safe set is forward invariant with online parameter adaptation.\nHowever, aCBFs have a much more restrictive invariance condition that limits the system to remain in sublevel sets of the safety set, ultimately leading to conservative behavior.\nTo address conservatism, \\cite{taylor2019adaptive} limited parameter adaptation to a region near the boundary of the safe set.\nHowever, the resulting safety controller is not necessarily Lipschitz and can exhibit closed-loop chattering.\n\n\nThe contributions of this work are threefold.\nFirst, robust aCBFs (RaCBFs) are defined and shown to guarantee safety for uncertain nonlinear systems.\nWhen combined with parameter adaptation, RaCBFs ensure forward invariance of a \\emph{tightened set} where the degree of tightening can be selected based on the desired conservatism.\nRaCBFs are far less conservative than aCBFs and results in a locally Lipschitz safety controller.\nSecond, RaCBFs are combined with the data-driven method set membership identification \\cite{tanaskovic2014adaptive} to safely reduce modeling error and expand the set of allowable states. \nThis is the first work to utilize both parameter adaptation and data-driven model estimation within the context of safety-critical control. \nAnd third, RaCBFs are merged with a recent direct adaptive controller \\cite{lopez2019contraction} based on the contraction metric framework \\cite{lohmiller1998contraction,manchester2017control}.\nContraction expresses distance-like functions \\emph{differentially} rather than explicitly, and the existence of a metric only requires stabilizability -- a far weaker condition than those needed for feedback linearization or backstepping -- making the unification more general than existing methods.\nThe approach is demonstrated on the pitch dynamics of an aircraft with uncertain nonlinear aerodynamics.\nThe system is non-invertible, not in strict-feedback form, and has non-polynomial dynamics highlighting the generality of the proposed method.\n\n\\section{Problem Formulation \\& Preliminaries}\nConsider the nonlinear system\n\\protect\\begin{equation}\n\\dot{x} = f(x) - \\Delta(x)^\\top \\theta + B(x)u,\n\\label{eq:unc_dyn}\n\\end{equation}\nwith unknown parameters $\\theta \\in \\mathbb{R}^p$ and known dynamics $\\Delta : \\mathbb{R}^n \\rightarrow \\mathbb{R}^{p\\times n} $, state $x \\in \\mathbb{R}^n$, control input $u \\in \\mathbb{R}^m$, nominal dynamics $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n}$, and control input matrix $B : \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n\\times m}$ with columns $b_i(x)$ $i=1,\\dots,m$.\nLet $\\mathcal{S}^n_+$ be the set of all $n\\times n$ symmetric positive definite matrices. \nA smooth Riemannian manifold $\\mathcal{M}$ is equipped with a smooth Riemannian metric $M: \\mathbb{R}^n\\times\\mathbb{R} \\rightarrow \\mathcal{S}^n_+$ that defines an inner product $\\left< \\cdot, \\cdot \\right>_x$ on the tangent space $T_x \\mathcal{M}$ at every point $x$.\nThe metric $M(x,t)$ defines local geometric notions such as angles, length, and orthogonality.\nThe directional derivative of metric $M(x,t)$ along vector $v$ is $\\partial_v M = \\sum_i\\frac{\\partial M}{\\partial x_i} v_i$.\nA parameterized differentiable curve $c: \\left[0,1\\right] \\rightarrow \\mathcal{M}$ is a regular if $\\frac{\\partial c}{\\partial s} = c_s \\neq 0$ $\\forall s \\in \\left[0~1\\right]$.\nLet $\\Upsilon(p,q)$ be the family of curves connecting $p,~q\\in \\mathcal{M}$, then a \\emph{geodesic} $\\gamma: \\left[0~1\\right] \\rightarrow \\mathcal{M}$ is the extremum of the energy functional\n\\begin{equation*}\n    \\gamma(s) = \\underset{c(s)\\in\\Upsilon(p,q)}{\\argmin}~ E(c,t) = \\int_0^1 c_s^\\top M(c,t) c_s ds,\n\\end{equation*}\nwhere $E$ is the Riemannian energy. \nIf the manifold $\\mathcal{M}$ is a complete metric space, such as $\\mathbb{R}^n$, $n$-sphere $\\mathbb{S}^n$, or any of their respective closed subsets, then a geodesic is guaranteed to exist by the Hopf-Rinow theorem \\cite{carmo1992riemannian}.\nIn the sequel, the time argument in $M(c,t)$ and $E(c,t)$ is dropped for clarity.\n\n\n\\section{Adaptive Safety}\n\\label{sec:acbf}\n\\subsection{Background}\nFirst consider the nominal dynamics of \\cref{eq:unc_dyn}, i.e., $\\Delta(x) = 0$.\nLet a closed convex set $\\mathcal{C} \\subset \\mathbb{R}^n$ be a 0-superlevel set of a continuously differentiable function $h: \\mathbb{R}^n \\rightarrow \\mathbb{R}$ where\n\\begin{equation*}\n    \\begin{aligned}\n    \\mathcal{C} & = \\left\\{ x \\in \\mathbb{R}^n : h(x) \\geq 0  \\right\\} \\\\\n    \\partial \\mathcal{C} & = \\left\\{ x \\in \\mathbb{R}^n : h(x) = 0  \\right\\} \\\\\n    \\text{Int}\\left(\\mathcal{C}\\right) & = \\left\\{ x \\in \\mathbb{R}^n : h(x) > 0  \\right\\}.\n    \\end{aligned}\n\\end{equation*}\nIf the nominal dynamics are locally Lipschitz, then given an initial condition $x_0$, there exists a maximum time interval $I(x_0) = [t_0,~T)$ such that $x(t)$ is a unique solution on $I(x_0)$. \nThe following definitions are largely taken from \\cite{ames2016control,ames2019control}.\n\\begin{definition}\n\\label{def:fi}\nThe set $\\mathcal{C}$ is \\emph{forward invariant} if for every $x_0\\in \\mathcal{C}$, $x(t) \\in \\mathcal{C}$ for all $t\\in I(x_0)$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def:safety}\nThe nominal system is \\emph{safe} with respect to set $\\mathcal{C}$ if the set $\\mathcal{C}$ is forward invariant.\n\\end{definition}\n\n\\begin{definition}\nA continuous function $\\alpha : \\mathbb{R} \\rightarrow \\mathbb{R}$ is an \\emph{extended class $\\mathcal{K}_\\infty$ function} if it is strictly increasing, $\\alpha(0) = 0$, and is defined on the entire real line.\n\\end{definition}\n\n\\begin{definition}\nLet $\\mathcal{C}$ be a 0-superlevel set for a continuously differentiable function $h:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, then $h$ is a \\emph{control barrier function} if there exists an extended class $\\mathcal{K}_\\infty$ function $\\alpha$ such that\n\\begin{equation}\n\\label{eq:cbf}\n    \\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[\\frac{\\partial h}{\\partial x}(x)\\left(f(x) + B(x) u\\right)\\right] \\geq - \\alpha(h(x)).\n\\end{equation}\n\\end{definition}\n\n\\begin{theorem}\nLet $\\mathcal{C}\\subset\\mathbb{R}^n$ be a 0-superlevel set of a continuously differentiable function $h:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, if $h$ is a CBF on $\\mathcal{C}$ then any locally Lipschitz continuous controller satisfying \\cref{eq:cbf} renders the set $\\mathcal{C}$ safe for the nominal system. \n\\end{theorem}\n\n\\subsection{Adaptive CBFs}\nAdaptive CBFs (aCBFs) \\cite{taylor2019adaptive} provide a general framework to guarantee safety through parameter adaptation for systems with structured uncertainties.\nThe notion of safety for uncertain systems must be extended to a family of safe sets $\\mathcal{C}_{\\theta}$ parameterized by $\\theta$. \nMore precisely, the family of safe sets are 0-superlevel sets of a continuously differentiable function $h_a: \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$.\nIf the uncertain dynamics in \\cref{eq:unc_dyn} are locally Lipschitz then the definitions of forward invariance and safety can be directly extended to $\\mathcal{C}_\\theta$.\n\n\\begin{definition}[\\cite{taylor2019adaptive}]\n\\label{def:acbf}\nLet $\\mathcal{C}_\\theta$ be a family of 0-superlevel sets parameterized by $\\theta$ for a continuously differentiable function $h_a:\\mathbb{R}^n\\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, then $h_a$ is an \\emph{adaptive control barrier function} if for all $\\theta$\n\\begin{equation}\n\\label{eq:acbf}\n    \\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[ \\frac{\\partial h_a}{\\partial x}(x,\\theta)\\left( f(x) - \\Delta(x)^\\top \\Lambda(x,\\theta) + B(x) u\\right)\\right] \\geq 0,\n\\end{equation}\nwhere $\\Lambda(x,\\theta) := \\theta - \\Gamma \\left(\\frac{\\partial h_a}{\\partial \\theta}(x,\\theta)\\right)^\\top$ and $\\Gamma \\in \\mathcal{S}^p_+$ is a symmetric positive definite matrix\n\\end{definition}\n\nA controller that satisfies \\cref{eq:acbf} can be combined with an adaptation law to render the uncertain systems safe with respect to $\\mathcal{C}_{{\\theta}}$ \\cite{taylor2019adaptive}.\nHowever, \\cref{eq:acbf} makes the level sets of $h_a$ forward invariant so it is a much stricter condition than \\cref{eq:cbf}.\nMore precisely, the distance to the boundary of the safe set must monotonically increase, i.e., $\\dot{h}_a(x,\\theta) \\geq 0$ for all time (\\cref{fig:acbf}).\nThis can lead to extremely conservative behavior as the system only operates in a set that is monotonically shrinking.\nIn \\cite{taylor2019adaptive}, a modified aCBF was proposed\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:acbf_m}\n    \\bar{h}_a(x,\\theta) = \\begin{cases} \n    \\sigma^2 ~ &\\text{if} ~ h_a(x,\\theta)\\geq\\sigma \\\\\n    \\sigma^2 - (h_a(x,\\theta)-\\sigma)^2 ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwhich satisfies \\cref{eq:acbf} if $h_a$ is a valid aCBF.\nThis modification expands the set of allowable states but the resulting controller is not necessarily Lipschitz and can exhibit high-frequency oscillations in closed-loop, as shown in \\cref{ex:lipschitz}. \n\n\\begin{example}\n\\label{ex:lipschitz}\nConsider the uncertain system $\\dot{x} = - \\theta + u$ with $\\theta>0$ and aCBF $h_a(x) = x - \\ubar{x}$.\nLet the controller $\\kappa$ be the solution to $\\kappa = {\\argmin}~\\frac{1}{2} u^2$ subject to $\\dot{\\bar{h}}_a(x) \\geq 0$.\nThen\n\\begin{equation}\n\\label{eq:u_ex}\n\\begin{aligned}\n    \\kappa = \\begin{cases}\n        0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n        \\mathrm{max}(0,\\hat{\\theta}) ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwhere $\\hat{\\theta}$ is the estimate of $\\theta$ and is modified based on \\cite{taylor2019adaptive}\n\\begin{equation}\n\\label{eq:ha_ex}\n\\begin{aligned}\n    \\dot{\\hat{\\theta}} = \\begin{cases}\n        0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n        - \\Gamma \\left[h_a(x)-\\sigma\\right] \\frac{\\partial h_a}{\\partial x} ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwith $\\Gamma > 0$. \nFor $\\hat{\\theta}(0) = \\hat{\\theta}_0 \\leq 0$ then $\\kappa = 0$ so the closed-loop response is $x_{cl}(t) = - \\theta (t-t_0) + x_0$ where $x_{cl}(t_0) = x_0 > \\ubar{x}$ and $t \\geq t_0$.\nFrom the adaptation law \\cref{eq:ha_ex}, it is easy to see that $\\dot{\\hat{\\theta}} \\geq 0 $ which will necessarily lead to $\\hat{\\theta} > 0$ since $h_a \\leq \\sigma$ until $\\kappa > \\theta$.\nFor $\\hat{\\theta} > 0$, the closed-loop response becomes\n\\begin{equation}\n\\label{eq:cl_ex}\n    \\begin{aligned}\n        x_{cl}(t) = \\begin{cases}\n            -\\theta (t-t_0) + x_0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n            \\hspace{0.28cm}\\tilde{\\theta} (t-t_0') + x_0' ~~ &\\text{otherwise},\n        \\end{cases}\n    \\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\theta}:=\\hat{\\theta} - \\theta$ and $x_{cl}(t_0') = x_0'$.\nSince $\\tilde{\\theta} > 0$, \\cref{eq:cl_ex} will continuously switch between its two solutions.\nThe control policy $\\kappa$ must then also switch between $0$ and $\\hat{\\theta}$ based on \\cref{eq:u_ex}.\nFurthermore, $\\kappa$ is not locally Lipschitz continuous and will exhibit high-frequency oscillations of magnitude $\\hat{\\theta}$ in closed-loop.\n\\end{example}\n\nThe intuition behind \\cref{ex:lipschitz} is that chatter arises due to the barrier condition switching between being trivially satisfied, i.e., $\\dot{\\bar{h}}_a = 0 \\geq 0$ for all $u$, to satisfied only for a particular $u$, i.e., a $u$ so that $\\dot{\\bar{h}}_a \\geq 0$.\nThe approach developed in this work addresses the conservatism of aCBFs and results in a locally Lipschitz continuous controller.\n\n\n\\subsection{Robust aCBFs}\n\\label{sub:racbf}\nThis section will show that a \\emph{tightened} set can be made forward invariant if the unknown model parameters are bounded and the parameter adaptation rate is an \\emph{admissible} (to be defined) symmetric positive definite matrix.\n\n\\begin{assumption}\nThe unknown parameters $\\theta$ belong to a known closed convex set $\\Theta$.\nThe parameter estimation error $\\tilde{\\theta}:= \\hat{\\theta} - \\theta$ then also belongs to a known closed convex set $\\tilde{\\Theta}$ and the maximum possible parameter error is $\\tilde{\\vartheta}$.\n\\end{assumption}\n\nLet $\\mathcal{C}^r_\\theta$ be a family of superlevel sets parameterized by $\\theta$ for a continuously differentiable function ${h}_r: \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$  \n\\begin{equation*}\n    \\begin{aligned}\n    \\mathcal{C}^r_\\theta & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) \\geq \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}  \\right\\} \\\\\n    \\partial \\mathcal{C}^r_\\theta & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) = \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta} \\right\\} \\\\\n    \\text{Int}\\left(\\mathcal{C}^r_\\theta\\right) & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) > \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta} \\right\\},\n    \\end{aligned}\n\\end{equation*}\nwhere $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix that will dictate the parameter adaptation rate.\nThe set $\\mathcal{C}^r_\\theta$ can be viewed as a \\emph{tightened} set with respect to $\\mathcal{C}_\\theta$, i.e., $\\mathcal{C}^r_\\theta \\subset \\mathcal{C}_\\theta$, shown in \\cref{fig:racbf}.\nOne can select the desired subset $\\mathcal{C}^r_\\theta$ to be made forward invariant \\textit{a priori}  by choosing $h_r(x_r,\\theta_r) >0 $ for appropriate $x_r,~\\theta_r$ so ${h}_r(x_r,\\theta_r) = \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}$.\nTo reduce conservatism, one can either 1) have fast parameter adaptation or 2) reduce model error.\nThe first scenario can lead to well-known undesirable effects in practice so the second scenario is the most viable, and will be explored more in \\cref{sec:smid}.\n\\cref{eq:unc_dyn} is again assumed to be locally Lipschitz so \\cref{def:fi,def:safety} hold.\n\n\\begin{definition}\n\\label{definition:racbf}\nLet $\\mathcal{C}^r_\\theta$ be a family of superlevel sets parameterized by $\\theta$ for a continuously differentiable function $h_r:\\mathbb{R}^n\\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, then $h_r$ is a \\emph{robust adaptive control barrier function}  if there exists an extended class $\\mathcal{K}_\\infty$ function $\\alpha$ such that for all $\\theta\\in\\Theta$ \n\\begin{equation}\n\\label{eq:racbf}\n\\begin{aligned}\n    &\\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[ \\frac{\\partial h_r}{\\partial x}(x,\\theta)\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\theta) + B(x) u\\right]\\right] \\\\ \n   \n    & \\hspace{3cm} \\geq -\\alpha\\left(h_r(x,\\theta) - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}\\right),\n\\end{aligned}\n\\end{equation}\nwhere $\\Lambda(x,\\theta) := \\theta - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\theta}(x,\\theta)\\right)^\\top$, $\\tilde{\\vartheta}$ is the maximum possible parameter error, and $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix.\n\\end{definition}\n\nThe invariance condition \\cref{eq:racbf} is reminiscent of that in \\cref{eq:cbf} and is less conservative than that in \\cref{eq:acbf} because the system is allowed to approach the boundary of $\\mathcal{C}^r_\\theta$.\n\\cref{thm:racbf} shows the existence of a RaCBF, coupled with an adaptation law, renders the set $\\mathcal{C}^r_\\theta$ forward invariant and hence safe.\n\n\\begin{theorem}\n\\label{thm:racbf}\nLet $\\mathcal{C}^r_{\\hat{\\theta}}\\subset\\mathbb{R}^n$ be a superlevel set of a continuously differentiable function $h_r:\\mathbb{R}^n \\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, if $h_r$ is a RaCBF on $\\mathcal{C}^r_{\\hat{\\theta}}$, then any locally Lipschitz continuous controller satisfying \\cref{eq:racbf} renders the set $\\mathcal{C}^r_{\\hat{\\theta}}$ safe for the uncertainty system with adaptation law and adaptation gain\n\\begin{equation*}\n\\dot{\\hat{\\theta}} = \\Gamma \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}) \\right)^\\top, ~~ \\lambda_{\\min}(\\Gamma) \\geq \\frac{\\|\\tilde{\\vartheta}\\|^2}{2h_r(x_r,{\\theta}_r)},\n\\end{equation*}\nwhere $\\tilde{\\vartheta}$ is the maximum possible parameter error, $h_r(x_r,\\theta_r)>0$ can be chosen freely based on the desired conservatism, and $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix.\nFurthermore, the original set $\\mathcal{C}_{\\hat{\\theta}}$ is also safe for the uncertain system.\n\\end{theorem}\n\n\n\n\\begin{remark}\n\\label{remark:proj}\nThe projection operator \\cite{slotine1991applied} can be used to enforce parameter bounds by modifying the above adaptation law as opposed to capturing them explicitly with $h_r$. \nThis can simplify the design of $h_r$ without forfeiting safety.\nThe proof is omitted but one can show that a positive semi-definite term appears in the same composite candidate CBF used in \\cref{thm:racbf} when adaptation is temporarily stopped.\n\\end{remark}\n\n\n\\begin{figure}[t!]\n\\vskip 0.1in\n    \\begin{subfigure}{.48\\columnwidth}\n         \\centering\n         \\includegraphics[trim=100 0 100 0, clip,width=1\\linewidth]{figures\/acbf.pdf}\n         \\caption{{Safe set with adaptive control barrier functions (aCBFs).}}\n         \\label{fig:acbf}\n     \\end{subfigure}\n    \\hspace{0.3em}\n     \\begin{subfigure}{.48\\columnwidth}\n         \\centering\n         \\includegraphics[trim=100 0 100 0, clip,width=1\\linewidth]{figures\/racbf.pdf}\n         \\caption{{Safe set with robust adaptive control barrier functions (RaCBFs).}}\n         \\label{fig:racbf}\n     \\end{subfigure}\n    \\caption{Visual comparison of safe sets with adaptive and robust adaptive control barrier functions. (a): System is restricted to level sets (black dashed lines) of aCBF $h_a$. (b): System allowed to operate in larger set with RaCBF $h_r$ reducing conservatism.}\n    \\vskip -0.25in\n\\end{figure}\n\nSeveral remarks can be made about \\cref{thm:racbf}.\nFirst, safety is \\emph{guaranteed} for all possible parameter realizations through adaptation with minimal conservatism.\nHence, RaCBFs expand and improve the \\emph{adaptive safety} paradigm.\nSecond, the minimum eigenvalue condition for the adaptation rate depends on the desired conservatism, i.e., the degree of tightening by choice of $h_r(x_r,\\theta_r)$.\nFor low conservatism, i.e., a small $h_r(x_r,\\theta_r)$ value, the adaptation rate must be large so the parameter estimates can change quickly to ensure forward invariance of $\\mathcal{C}^r_\\theta$.\nThere is thus a fundamental trade-off between conservatism and parameter adaptation rate that must be weighed carefully given the well-known undesirable effects of high-gain adaptation.\nThird, the RaCBF condition in \\cref{eq:racbf} can be used as a safety filter for an existing tracking controller or as a constraint within an optimization.\n\\cref{sec:result} will show the latter but with a contraction-based controller.\nLastly, if the adaptation gain must be small (or the maximum parameter error is large) then RaCBFs can be conservative albeit not to the same extent as aCBFs.\nBetter performance can be obtained if the model parameters can be robustly and accurately estimated.\nInstead of obtaining a point-estimate of the parameters, this work will instead identify the \\emph{set} of possible parameter values.\n\n\\subsection{RaCBFs with Set Membership Identification}\n\\label{sec:smid}\nSet membership identification (SMID) is a model estimation technique that constructs an unfalsified set of model parameters.\nSMID was originally developed to identify transfer functions for uncertain linear systems \\cite{pararrieter1992set}, but has been more recently applied to linear \\cite{tanaskovic2014adaptive,lorenzen2017adaptive} and nonlinear adaptive MPC \\cite{lopez2019adaptive}.\nAssume that the true parameters $\\theta^*$ belong to an initial set of possible parameters $\\Theta^0$, i.e., $\\theta^* \\in \\Theta^0$.\nGiven $k$ state, input, and rate measurements (denoted as $x_{1:k}$ and so forth), a set $\\Xi$ can be constructed such that\n\\begin{equation*}\n   \n   \n\\Xi = \\left\\{ \\varrho :  |\\dot{x}_{1:k} - f_{1:k} + \\Delta_{1:k}^\\top \\varrho - B_{1:k}u_{1:k}| \\leq D  \\right\\},\n\\end{equation*}\nwhere $D$ can be treated as a tuning parameter that dictates the conservativeness of SMID. \nIt can also represent a disturbance or noise bound \\cite{tanaskovic2014adaptive,lopez2019adaptive}.\nThe set of possible parameter values can then be updated via $\\Theta^{j+1} = \\Theta^j \\cap \\Xi$ for all $j\\geq 0$.\nIn practice, $\\Xi$ can be found by solving a linear program and set intersection can be efficiently done through a combination of min and max operations.\nRestricting $\\hat{\\theta} \\in \\Theta^j$ then $\\tilde{\\theta} \\in \\Tilde{\\Theta}^j$ where $\\Tilde{\\Theta}^j$ is the set of possible parameter errors.\nThe following lemma shows the advantage of performing set identification over point-estimation techniques.\n\\begin{lemma}\n\\label{lemma:smid}\nModel uncertainty monotonically decreases with set membership identification, i.e., $\\tilde{\\Theta}^{j+1} \\subseteq \\tilde{\\Theta}^j$ for all $j\\geq 0$.\n\\end{lemma}\n\\begin{proof}\nSince $\\theta^* \\in \\Theta^{j+1}$ then $\\theta^* \\in \\Theta^{j} \\cap \\Xi$ which is true if $\\theta^* \\in \\Theta^j$ so $\\Theta^{j+1}\\subseteq\\Theta^j$ and $\\tilde{\\Theta}^{j+1} \\subseteq \\tilde{\\Theta}^j$.\n\\end{proof}\n\nThe motivation to combine SMID with RaCBFs is to enlarge the tightened set $\\mathcal{C}_\\theta^r$.\nTo do so, one must ensure $\\mathcal{C}^r_\\theta$ remains forward invariant as the set of model parameters is updated.\nIn general this is non-trivial to prove since the maximum possible parameter error is now time varying.\nHowever, \\cref{thm:racbf_smid} shows that safety is maintained if the model uncertainty monotonically decreases.\n\n\\begin{theorem}\n\\label{thm:racbf_smid}\nLet $\\mathcal{C}^r_\\theta$ be a superlevel set of a continuously differentiable function ${h}_r : \\mathbb{R}^n\\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$. If the system is safe on $\\mathcal{C}^r_\\theta$ then it remains safe if the maximum allowable model parameter error $\\tilde{\\vartheta}$ monotonically decreases. \nMoreover, the tightened set $\\mathcal{C}^r_\\theta$ converges to $\\mathcal{C}_\\theta$ monotonically.\n\\end{theorem}\n\n\nCombining RaCBFs and SMID provides a mechanism to 1) modify parameters via adaptation to achieve safety and 2) update the model to reduce uncertainty and conservatism.\nSafety is \\emph{guaranteed} even as the model parameters are modified online, and the system's performance will only improve as more data is collected.\nThis \\emph{adaptive data-driven safety} paradigm can be merged with a stabilizing adaptive controller for safe reference tracking.\nTo maximize the generality of the proposed unification, the adaptive controller must be applicable to a broad class of nonlinear systems.\n\n\\section{Adaptive Control with Contraction Metrics}\n\\label{sec:accm}\nSeveral adaptive control techniques have been proposed for nonlinear systems, including methods based on feedback linearization, variable structure, and backstepping (see \\cite{slotine1991applied,miroslav1995nonlinear}).\nThese methods are limited to certain classes of systems because they rely on \\emph{explicitly} constructing a control Lyapunov function (CLF) to prove stability.\nThis work will instead utilize a \\emph{differential} approach based on contraction analysis that can be applied to a broad class of systems.\n\n\\begin{figure}[t!]\n\\vskip 0.05in\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/geodesic.pdf}\n         \\caption{{Geodesic and arbitrary curve connecting current and desired state.}}\n         \\label{fig:geodesig}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/manifold.pdf}\n         \\caption{{Differential CLFs along geodesic connecting current and desired state.}}\n         \\label{fig:manifold}\n     \\end{subfigure}\n    \\caption{Geodesic and differential CLF visualization. (a): Geodesic (grey) connecting current $x$ (red) and desired $x_d$ (blue) state. (b): Differential CLFs are integrated along geodesic to achieve exponential convergence.}\n    \\label{fig:dclf}\n    \\vskip -0.2in\n\\end{figure}\n\n\\subsection{Contraction Metrics}\nThe nominal differential dynamics of \\cref{eq:unc_dyn} are $\\dot{\\delta}_x  = A(x,u)\\delta_x + B(x)\\delta_u$ where $A(x,u) = \\frac{\\partial f}{\\partial x} + \\sum_{i=1}^m \\frac{\\partial b_i}{\\partial x}u_i$.\nContraction analysis searches for a \\emph{control contraction metric} (CCM) $M(x)$ such that a \\emph{differential} CLF $\\delta V = \\delta_x^\\top M(x) \\delta_x$ satisfies $\\delta \\dot{V}\\leq - 2 \\lambda \\delta V$ for all $x$.\nA global CLF can be obtained by integrating along a geodesic $\\gamma(s)$, illustrated in \\cref{fig:dclf}, with $\\gamma(0) = x_d$ and $\\gamma(1) = x$ where $x_d$ and $x$ are the desired and current state.\nThe Riemannian energy and tracking error both converge to zero exponentially, i.e., $\\dot{E} \\leq - 2 \\lambda E$.\nLet $W(x) = M(x)^{-1}$, $M(x)$ is CCM if \\cite{manchester2017control}\n\\begin{gather}\n     B^\\top_{\\perp}\\left( W A ^\\top + A W - \\dot{W} + 2 \\lambda W \\right)B_{\\perp} \\preceq 0 \\tag{C1} \\label{eq:ccm_c1} \\\\\n    \\partial_{b_i}W - W \\frac{\\partial b_i}{\\partial x}^\\top  -  \\frac{\\partial b_i}{\\partial x} W = 0 ~~ i=1,\\dots,m \\tag{C2} \\label{eq:ccm_c2}\n\\end{gather}\nwhere $B_\\perp(x)$ is the annihilator matrix of $B(x)$, i.e., $B^\\top_\\perp B = 0$.\n\\cref{eq:ccm_c1} ensures the dynamics orthogonal to $u$ are contracting and is a stabilizability condition.\n\\cref{eq:ccm_c2} requires the column vectors of $B(x)$ form a Killing vector for the dual metric $W(x)$ leading to simpler controllers \\cite{manchester2017control}. \n\\cref{eq:ccm_c1} and \\cref{eq:ccm_c2} will be referred to as the \\emph{strong CCM conditions}.\n\n\n\n\n\n\\subsection{Adaptive Control \\& Contraction}\nA novel adaptive control method was developed in \\cite{lopez2019contraction} for closed-loop contracting systems with \\emph{extended matched uncertainties}, i.e., $\\Delta(x)^\\top \\theta \\in \\text{span}\\{B,ad_fB\\}$ were $ad_fB$ is the Lie bracket of the vector fields $f(x)$ and $B(x)$.\nTo stabilize such systems, the \\emph{parameter-dependent} metric $M(x,\\hat{\\theta})$ was introduced and must satisfy the strong CCM conditions \\emph{for all} possible $\\hat{\\theta}$.\nThis led to the following result.\n\n\\begin{theorem}[\\cite{lopez2019contraction}]\n\\label{thm:exMatched}\nIf a parameter-dependent metric can be computed for \\cref{eq:unc_dyn} with extended matched uncertainties, then the closed-loop systems is asymptotically stable with\n\\begin{equation}\n    \\dot{\\hat{\\theta}} = - \\Gamma \\Delta(x) M(x,\\hat{\\theta})\\gamma_s(1)\n    \\label{eq:accm}\n\\end{equation}\nwhere $\\gamma_s(s):=\\frac{\\partial \\gamma}{\\partial s}$ is the geodesic speed and $\\Gamma \\in \\mathcal{S}^p_+$ is a symmetric positive definite matrix.\n\\end{theorem}\n\n\\begin{remark}\nFor \\emph{matched} uncertainties, the metric is \\emph{independent} of the unknown parameters $\\hat{\\theta}$ \\cite[Lemma 1]{lopez2019contraction} simplifying its computation.\nOtherwise, sum-of-square or robust optimization must be utilized to compute $M(x,\\hat{\\theta})$.\n\\end{remark}\n\n\\begin{remark}\nSeveral modifications can be made to \\cref{eq:accm} that improve transients or robustness including the projection operator discussed in \\cref{sub:racbf} (see \\cite{lopez2019contraction}).\n\\end{remark} \n\n\n\\subsection{Offline Design \\& Online Computation}\n\\label{sub:computation}\nA contraction metric is computed offline via sum-of-square programming for polynomial systems \\cite{manchester2017control} or by imposing \\cref{eq:ccm_c1,eq:ccm_c2} at sampled points in the state space; a process called gridding.\nGeodesics are computed online at each time step by solving a nonlinear program (NLP) with the current state.\nGeodesics are often guaranteed to exist by Hopf-Rinow and are less expensive to compute than solving nonlinear MPC.\nGiven a geodesic $\\gamma(s)$, the Riemannian energy can be interpreted as a CLF so a pointwise min-norm controller similar to that in \\cite{primbs2000receding} can found by solving the QP\n\\begin{equation*}\n    \\begin{aligned}\n        & u^* = ~\\underset{u\\in \\mathcal{U}}\\argmin ~\\frac{1}{2} u^\\top u \\\\\n        & \\text{s.t.} ~\\gamma_s(1)^\\top M(x,\\hat{\\theta})\\dot{\\hat{x}} - \\gamma_s(0)^\\top M(x_d,\\hat{\\theta})\\dot{x}_d  \\leq - \\lambda E(\\gamma(s),\\hat{\\theta}) \n    \\end{aligned}\n\\end{equation*}\nwhere $\\hat{\\theta}$ is the current parameter estimate, $\\gamma_s$ is the geodesic speed, $\\dot{\\hat{x}}$ is \\cref{eq:unc_dyn} but with $\\hat{\\theta}$, $\\dot{x}_d $ is the desired dynamics for desired state $x_d$.\nThe stability constraint imposes $\\dot{E} \\leq - 2 \\lambda E$ where $\\dot{E}$ is the first variation of the Riemannian energy, which has a known form \\cite{carmo1992riemannian}, and results in exponential convergence of the tracking error.\nIt is more general than the traditional Lyapunov stability as a distance-like function does not need to computed explicitly.\nSafety can be directly embedded in the above QP, resulting in a single optimization for a safe stabilizing controller.\n\n\n\\begin{figure*}[t]\n\\vskip 0.1in\n    \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30, clip,width=1\\textwidth]{figures\/q_c_i.pdf}\n         \\caption{Pitch rate $q$.}\n         \\label{fig:q_i}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/u_c_i.pdf}\n         \\caption{Control input $u$.}\n         \\label{fig:u_i}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/h_c_i.pdf}\n         \\caption{Barrier function $h$.}\n         \\label{fig:h_i}\n     \\end{subfigure}\n     \\caption{Comparison of modified aCBFs \\cref{eq:acbf_m}, RaCBFs, and RaCBFs \\& SMID for desired terminal state $x_d = [180^\\circ~0~0]^\\top$ (Immelmann turn) and maximum pitch rate $q_m$. (a): Pitch rate tracking where RaCBFs and the modified aCBFs exhibit similar conservatism due to model error.\n     RaCBF \\& SMID allows the aircraft to utilize 97.9\\% of the maximum allowable pitch rate. `\n     (b): Control chatter is observed with the modified aCBFs while RaCBFs generate continuous control inputs. (c): Safety is maintained but RaCBFs and the modified aCBFs are conservative due to potential model error. RaCBFs \\& SMID permit states closer to the boundary of the safe set without losing safety guarantees.\n     For tests $k_q^* = 0.2$, $\\ell^*_\\alpha = -1$, $\\Gamma_B = 20$, $\\Gamma_C = 50$, $\\alpha(r) = 10r$, and $D = 0.1$.}\n     \\label{fig:results_immelmann}\n\\vskip -0.2in\n\\end{figure*}\n\n\\section{Adaptive \\& Data-Driven Safety}\n\\label{sec:result}\nA safe and stabilizing controller can be computed by unifying RaCBFs, SMID, and adaptive control with contraction.\nThe individual components of the controller are summarized below with their respective computational complexity.\n\\\\[4pt]\n\\noindent \\textbf{1) Compute geodesic (NLP)}\n\\begin{equation*}\n   \n    \\gamma(s) = \\underset{c(s)\\in\\Upsilon(x,x_d)}{\\argmin}~ E(c,\\hat{\\theta}_C) = \\int_0^1 c_s^\\top M(c,\\hat{\\theta}_C) c_s ds\n\\end{equation*}\n\n\\noindent \\textbf{2) Compute controller (QP \\& Quadrature)}\n\\begin{align*}\n    & \\kappa  = ~\\underset{u \\in \\mathcal{U}}{\\argmin} ~ \\frac{1}{2} u^\\top u + r \\epsilon^2 \\\\\n    &\\text{s.t.} ~\\gamma_s(1)^\\top M(x,\\hat{\\theta}_C)\\dot{\\hat{x}} - \\gamma_s(0)^\\top M(x_d,\\hat{\\theta}_C)\\dot{x}_d  \\\\\n    & \\hspace{4.6cm} \\leq - \\lambda E(\\gamma(s),\\hat{\\theta}_C)+ \\epsilon \\nonumber  \\\\\n    & \\hspace{.4cm} \\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}_B)\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}_B) + B(x) u\\right] \\\\\n    & \\hspace{3.1cm} \\geq -\\alpha\\left({h}_r(x,\\hat{\\theta}_B) - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}\\right) \\nonumber\n\\end{align*}\n\\noindent \\textbf{3) Update parameters (Quadrature)} \\label{item:quad}\n\\begin{equation*}\n\\begin{aligned}\n    & \\dot{\\hat{\\theta}}_C = - \\Gamma_C \\Delta(x) M(x,\\hat{\\theta}_C) \\gamma_s(1) \\\\\n    & \\dot{\\hat{\\theta}}_B =  \\Gamma_B \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}_B)\\right)^\\top\n\\end{aligned}\n\\end{equation*}\n\\vskip -.03in\n\\noindent \\textbf{4) Update parameter error bounds (LP)} \\label{item:lp}\n\\begin{align*}\n        & \\Xi = \\left\\{ \\varrho : ~ |\\dot{x}_{1:k} - f_{1:k} + \\Delta_{1:k}^\\top \\varrho - B_{1:k}\\kappa_{1:k}| \\leq D  \\right\\} \\\\\n         &\\Theta^{j+1} = \\Theta^{j} \\cap \\Xi, ~~ \\tilde{\\vartheta} = \\underset{\\varrho_i,\\forall i}{\\text{sup}} ~\\Theta^{j+1} - \\underset{\\varrho_i,\\forall i}{\\vphantom{\\text{sup}}\\text{inf}}~ \\Theta^{j+1} \\nonumber\n\\end{align*}\n\\vskip -.02in\n\nThe NLP in Step~1) can be efficiently solved by parameterizing geodesics with a set of polynomial basis functions. \nWe adopt the same strategy as in \\cite{leung2017nonlinear} and utilize the the Chebychev Pseudospectral method and Clenshaw-Curtis quadrature to compute a geodesic at each time step.\nUsing the geodesic computed in Step~1), the QP in Step~2) is solved to generate a safe and stabilizing controller $\\kappa$.\nThe QP is similar to that in \\cite{ames2016control} but the stability constraint is replaced with the first variation of the Riemannian energy \\cite{carmo1992riemannian}.\nUnder the premise that \\cref{eq:unc_dyn} is locally Lipschitz, one can show that $\\kappa$ is guaranteed to be locally Lipschitz from \\cite[Theorem 3]{ames2016control} as both the geodesic speed $\\gamma_s$ and metric $M(x)$ are also locally Lipschitz from their definitions.\nNote that $\\dot{x}_d$ is the desired dynamics and $\\dot{\\hat{x}}$ is \\cref{eq:unc_dyn} but with $\\hat{\\theta}_C$.\nStep~3) is simple quadrature and is not computationally expensive.\nNote that the parameter adaptation $\\dot{\\hat{\\theta}}_C$ for the controller should be temporarily stopped when the safety constraint is active to prevent undesirable transients.\nOtherwise, the parameter estimates will windup as the tracking error may increase to ensure safety.\n\nThe LP in Step~4) has $2k$ constraints and is solved $2p$ times at every time step for the upper and lower bound of each parameter. \nSet intersection is done by taking the appropriate minimum or maximum of the newest and current bounds.\nThe complexity of Step~4) can be bounded by either removing redundant constraints or terminating when $\\tilde{\\vartheta} \\leq \\varepsilon$ where $\\varepsilon$ is a predefined threshold.\nStep~4) can also be done outside the control loop since stability and safety do not rely on real-time updates of the parameter bounds; although real-time bounds are desirable to quickly eliminate conservatism.\nConsequently, non-causal filtering can be used to accurately estimate $\\dot{x}$ if necessary.\nMoreover, the right hand side of the inequality can be replaced by $D + \\mathcal{E}$ where $\\mathcal{E}$ is the maximum estimation error of the rate vector, i.e., $|\\dot{\\hat{x}}| \\leq \\mathcal{E}$.\nThe proposed method was tested in MATLAB R2018B with the built-in solvers without any code optimization on a 1.6GHz Intel i5 processor.\nThe NLP was initialized with a linear curve $c(s) = (x-x_d)s+x_d$ at each time step.\n\n\n\n\\section{Illustrative Example}\n\\label{sec:example}\nConsider the simplified pitch dynamics of an aircraft \\cite{mccormick1995aerodynamics}\n\\begin{equation*}\n    \\left[ \\begin{array}{c} \\dot{\\theta} \\\\ \\dot{\\alpha} \\\\ \\dot{q} \\end{array} \\right] = \\left[ \\begin{array}{c}  q \\\\ q - \\bar{L}(\\alpha)  \\\\ -k_q q + \\bar{M}(\\alpha) \\end{array} \\right] + \\left[ \\begin{array}{c} 0 \\\\ 0 \\\\ 1 \\end{array} \\right] u,\n\\end{equation*}\nwhere $\\theta$, $\\alpha$, and $q$ are the pitch angle, angle of attack, and pitch rate.\n$\\bar{L}(\\alpha)$ and $\\bar{M}(\\alpha)$ are the aerodynamic lift and moment.\nThe system is not feedback linearizable as the controllability matrix drops rank at $\\bar{L}'(\\alpha)=0$ and is not in strict-feedback form.\nUtilizing flat plat theory \\cite{mccormick1995aerodynamics}, the aerodynamics of a high-performance aircraft are approximately $\\bar{L}(\\alpha) = 0.8\\mathrm{sin}(2\\alpha)$ and $\\bar{M}(\\alpha) = -\\ell_{\\alpha} \\bar{L}(\\alpha)$.\nThe parameters $k_q$ and $\\ell_\\alpha$ are unknown but $k_q \\in [0.1~0.8]$ and $\\ell_\\alpha \\in [-3~1]$.\nA metric quadratic in $\\alpha$ was synthesized via gridding for $\\alpha \\in [-5^{\\circ}~50{^\\circ}]$ and $q \\in [-10^{\\circ\/s}~50^{\\circ\/s}]$.\nNote that $\\bar{L}'(\\alpha)=0$ is in the chosen grid range.\nThe function $h_r(q) = q_m - q$ where $q_m = 50^{\\circ\/s}$ can be easily shown to be a valid RaCBF that enforces $q \\leq q_m$.\n\n\nThe desired terminal state $x_d = [180^\\circ~0~0]^\\top$ corresponds to the first portion of the aerobatic maneuver known as the Immelmann turn.\nThe vehicle executes a half loop before executing a half roll (not considered here) resulting in level flight in the opposite direction.\n\\cref{fig:q_i} shows modified aCBFs and RaCBFs exhibit similar behavior in terms of conservatism as they do not utilize the maximum allowable pitch rate.\nHowever, modified aCBFs exhibit high-frequency oscillations due to the chatter in the control input, seen in \\cref{fig:u_i}; a result of the formulation as the safety constraint continuously switches between active and inactive. \nHigh-frequency oscillations are also seen in the barrier function in \\cref{fig:h_i} but are absent with RaCBFs.\n\\cref{fig:q_i} shows RaCBFs with SMID results in less conservatism as 97.9\\% of the maximum allowable pitch rate is utilized.\nMoreover, the set of allowable states is considerably larger as is evident by the small value of the barrier function in \\cref{fig:h_i}.\nThe parameter error bounds, shown in \\cref{fig:smid_i}, were reduced by 63.0\\% for $k_q$ and 90.5\\% for ${\\ell}_\\alpha$.\n\\cref{fig:comp_time_i} shows the computation time is within real-time constraints and can be easily reduced by utilizing faster solvers or a lighter programming language.\n\n\\begin{figure}[t!]\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/smid_i.png}\n         \\caption{{Parameter bounds.}}\n         \\label{fig:smid_i}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/comp_time_i.png}\n         \\caption{{Computation time.}}\n         \\label{fig:comp_time_i}\n     \\end{subfigure}\n    \\caption{Parameter bounds and computation time. (a): Parameter bounds monotonically approach the true parameter values $\\ell_\\alpha^*$ and $k_q^*$. (b): Computation time for the proposed controller is well within real-time constraints.}\n    \\label{fig:results_comp_i}\n    \\vskip -0.2in\n\\end{figure}\n\n\\begin{figure*}[t]\n\\vskip 0.1in\n    \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30, clip,width=1\\textwidth]{figures\/q_c.pdf}\n         \\caption{Pitch rate $q$.}\n         \\label{fig:q}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/u_c.pdf}\n         \\caption{Control input $u$.}\n         \\label{fig:u}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/h_c.pdf}\n         \\caption{Barrier function $h$.}\n         \\label{fig:h}\n     \\end{subfigure}\n     \\caption{Comparison of modified aCBFs \\cref{eq:acbf_m}, RaCBFs, and RaCBFs \\& SMID with a full desired trajectory corresponding to $\\theta_d = -20^\\circ \\mathrm{cos}(t)$. (a): Pitch rate tracking where RaCBFs and aCBFs exhibit similar conservativeness due to model error.\n     RaCBF \\& SMID achieves the best performance because model uncertainty is reduced via online estimation. \n     (b): Control chatter is observed with aCBFs while RaCBFs generate continuous control inputs. (c): Safety maintained but RaCBFs and aCBFs are conservative due to potential model error. RaCBFs \\& SMID is the least conservative since model is estimated online.\n     For tests $k^* = 0.2$, $\\ell^*_\\alpha = -1$, $\\Gamma_B = 20$, $\\Gamma_C = 50$, $\\alpha(r) = 10r$, and $D = 0.1$.}\n     \\label{fig:results}\n\\vskip -0.2in\n\\end{figure*}\n\nNow consider the scenario where a full desired trajectory described by $\\theta_d = - 20^\\circ \\mathrm{cos}(t)$ is available.\nA metric was synthesized for a new grid range $\\alpha \\in [-60^{\\circ}~60{^\\circ}]$ and $q \\in [-20^{\\circ\/s}~20^{\\circ\/s}]$; a metric quadratic in $\\alpha$ was again found to be valid over the grid range.\nThe function $h_r(q) = 1 - (\\nicefrac{q}{q_m})^2$ where $q_m = 20^{\\circ\/s}$ is a valid RaCBF that enforces $|q| \\leq q_m$.\nThe results in \\cref{fig:results} show the same exact behavior as in \\cref{fig:results_immelmann}: control input chattering occurs with the modified aCBFs \\cref{fig:u} resulting in high-frequency oscillations in both the pitch rate \\cref{fig:q} and barrier function \\cref{fig:h}. \nChattering does not occur with RaCBFs.\nAdditionally, RaCBFs with SMID again has the best tracking performance in \\cref{fig:q} and is the least conservative in \\cref{fig:h}.\nThe parameter bounds at different time instances are shown in \\cref{fig:smid_sine}.\nThe bounds again monotonically decrease resulting in a reduction of 16.5\\% and 77.3\\% for ${k}_q$ and ${\\ell}_\\alpha$, respectively.\nThe reduction is less than that in the Immelmann turn due to the trajectory not sufficiently exciting $q$ relative to $\\bar{L}(\\alpha)$.\nThe largest reduction occurred at $t=3s$ which is when the barrier function in \\cref{fig:h} becomes less conservative.\nThe computation time shown in \\cref{fig:comp_time_sine} is comparable to that in \\cref{fig:comp_time_i} with the NLP solve time being slightly less as the linear geodesic initialization is a better initial guess for small tracking error. \nThis further confirms that the proposed approach can be run in real-time.\n\n\\begin{figure}[t!]\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/smid_sine.png}\n         \\caption{{Parameter bounds.}}\n         \\label{fig:smid_sine}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/comp_time_sine.png}\n         \\caption{{Computation time.}}\n         \\label{fig:comp_time_sine}\n     \\end{subfigure}\n    \\caption{Parameter bounds and computation time for desired trajectory. (a): Parameter bounds monotonically approach the true parameter values $\\ell_\\alpha^*$ and $k_q^*$. (b): Computation time for the proposed controller is well within real-time constraints.}\n    \\label{fig:results_comp_sine}\n    \\vskip -0.2in\n\\end{figure}\n\n\\section{Conclusion}\nThis work presented a framework that guarantees safety for uncertain nonlinear systems through parameter adaptation and data-driven model estimation.\nThe unification with a contraction-based adaptive controller allows the approach be applied to a broad class of systems.\nExtending to systems with probabilistic model bounds, non-parametric uncertainties, and external disturbances is future work.\n\n\\section*{Appendix}\n\\label{sec:appendix}\n\n\\begin{proof}[Proof of \\cref{thm:racbf}]\nConsider the composite candidate CBF $h = h_r(x,\\hat{\\theta}) - \\frac{1}{2} \\tilde{\\theta}^\\top\\Gamma^{-1}\\tilde{\\theta}$,\nwhere the minimum eigenvalue of $\\Gamma$ must satisfy $\\lambda_{\\min}(\\Gamma) \\geq \\frac{\\|\\tilde{\\vartheta}\\|^2}{2h_r(x_r,{\\theta}_r)}$ for any $h_r(x_r,\\theta_r)>0$.\nDifferentiating $h$ with respect to \\cref{eq:unc_dyn},\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} & = \\dot{h}_r(x,\\hat{\\theta}) -  \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}} \\\\\n    &= \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\theta + B(x) u\\right] + \\frac{\\partial h_r}{\\partial \\hat{\\theta}} \\dot{\\hat{\\theta}} - \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}}  \\\\\n   \n   \n\\end{aligned}\n\\end{equation*}\nAdding and subtracting $\\frac{\\partial h_r}{\\partial x}\\Delta(x)^\\top\\left[\\hat{\\theta} - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\right)^\\top \\right]$ and using the definition of $\\Lambda(x,\\hat{\\theta})$,\n\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} & = \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}) + B(x) u\\right] + \\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\dot{\\hat{\\theta}} \\\\\n    & \\hspace{1.5cm} - \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}} + \\frac{\\partial h_r}{\\partial x}\\Delta(x)^\\top\\left[\\tilde{\\theta} - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\right)^\\top \\right].\n\\end{aligned}\n\\end{equation*}\nChoosing $\\dot{\\hat{\\theta}} = \\Gamma \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}\\right)^\\top$, then\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} &= \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}) + B(x) u\\right] \\\\\n    & \\geq - \\alpha\\left( h_r - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}  \\right) \\geq -\\alpha(h),\n\\end{aligned}\n\\end{equation*}\nwhere the first inequality is obtained via the definition of a RaCBF and the second by noting $|\\tilde{\\theta}| \\leq \\tilde{\\vartheta}$ so $h = h_r - \\frac{1}{2}\\tilde{\\theta^\\top}\\Gamma^{-1}\\tilde{\\theta} \\geq h_r - \\frac{1}{2}\\tilde{\\vartheta^\\top}\\Gamma^{-1}\\tilde{\\vartheta}$. \nSince $h\\geq0$ and $h_r \\geq h$ $\\forall t$, then $h_r \\geq \\frac{1}{2}\\tilde{\\vartheta^\\top}\\Gamma^{-1}\\tilde{\\vartheta}  \\geq 0$ and $\\mathcal{C}^r_\\theta$ is forward invariant.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:racbf_smid}]\nSince the model uncertainty is changing via estimation, the maximum allowable parameter error is time varying, i.e., $\\tilde{\\vartheta}(t)$.\nFrom \\cref{lemma:smid}, $\\tilde{\\Theta}$ monotonically decreases so $\\dot{\\tilde{\\vartheta}} \\leq 0$. \nLet $h_r$ be a candidate RaCBF, then $\\dot{{h}} = \\dot{h}_{r} - \\tilde{\\vartheta}\\Gamma^{-1}\\dot{\\tilde{\\vartheta}}  \\geq \\dot{h}_{r}$ since $\\dot{\\tilde{\\vartheta}} \\leq 0$ for all $t$.\nInequality \\cref{eq:racbf} in Definition~\\ref{definition:racbf} is then still satisfied for $\\dot{\\tilde{\\vartheta}} \\leq 0$.\nUsing the steps in \\cref{thm:racbf}, the system is safe with respect to $\\mathcal{C}^r_{\\hat{\\theta}}$.\nMoreover, since $\\dot{\\tilde{\\vartheta}} \\leq 0$ then $\\tilde{\\vartheta} \\rightarrow 0 $ so $\\mathcal{C}^r_{\\hat{\\theta}} \\rightarrow \\mathcal{C}_{\\hat{\\theta}}$.\n\\end{proof}\n\n\n\n\n\\noindent \\textbf{Acknowledgements}  \nWe thank David Fan for stimulating discussions.\nThis work was supported by the NSF Graduate Research Fellowship Grant No. 1122374.\n\n\n\\balance\n\\bibliographystyle{ieeetr}\n\n\\section{Introduction}\nState and actuator constraints are often encountered in real-world systems but systematic feedback controller design remains challenging.\nThe main difficulty arises from needing to \\emph{predict} whether the system will remain in the feasible set when selecting a control input.\nRepeatedly solving a constrained finite-horizon optimal control problem (i.e., model predictive control) is one way to ensure feasibility, \nbut solving a nonlinear optimization in real-time can be difficult.\nAlternatively, one can avoid trajectory optimization entirely by constructing safe invariant sets, i.e., a set of states that guarantee feasibility indefinitely.\nThrough the development of control barrier functions (CBFs)~\\cite{ames2016control}, safe stabilizing controllers can be synthesized by simply solving a quadratic program (QP), and has recently been used in several applications~\\cite{ames2019control}.\nHowever, model error can significantly degrade the performance of these controllers to the extent that safety may no longer be guaranteed.\nWe develop a general framework that guarantees safety through parameter adaptation and online model estimation for uncertain nonlinear systems.\n\nControl barrier functions heavily rely on a model so it is critical to develop methodologies that maintain safety for uncertain systems.\nIn~\\cite{xu2015robustness} the so-called zeroing CBFs were shown to be Input-to-State stable, a property that was used to prove a superset of a safe set is forward invariant.\nThe size of the superset was characterized in~\\cite{kolathaya2018input} by introducing Input-to-State safety.\nStronger safety guarantees can be obtained via robust optimization as demonstrated in~\\cite{gurriet2018towards}.\nLearning-based methods~\\cite{fan2019bayesian,taylor2019learning} have been developed to address the conservatism of the robust strategies, but these can require extensive offline training to substantially improve the model.\nAdaptive CBFs (aCBFs)~\\cite{taylor2019adaptive} use ideas from adaptive control theory to ensure a safe set is forward invariant with online parameter adaptation.\nHowever, aCBFs have a much more restrictive invariance condition that limits the system to remain in sublevel sets of the safety set, ultimately leading to conservative behavior.\nTo address conservatism, \\cite{taylor2019adaptive} limited parameter adaptation to a region near the boundary of the safe set.\nHowever, the resulting safety controller is not necessarily Lipschitz and can exhibit closed-loop chattering.\n\n\nThe contributions of this work are threefold.\nFirst, robust aCBFs (RaCBFs) are defined and shown to guarantee safety for uncertain nonlinear systems.\nWhen combined with parameter adaptation, RaCBFs ensure forward invariance of a \\emph{tightened set} where the degree of tightening can be selected based on the desired conservatism.\nRaCBFs are far less conservative than aCBFs and results in a locally Lipschitz safety controller.\nSecond, RaCBFs are combined with the data-driven method set membership identification \\cite{tanaskovic2014adaptive} to safely reduce modeling error and expand the set of allowable states. \nThis is the first work to utilize both parameter adaptation and data-driven model estimation within the context of safety-critical control. \nAnd third, RaCBFs are merged with a recent direct adaptive controller \\cite{lopez2019contraction} based on the contraction metric framework \\cite{lohmiller1998contraction,manchester2017control}.\nContraction expresses distance-like functions \\emph{differentially} rather than explicitly, and the existence of a metric only requires stabilizability -- a far weaker condition than those needed for feedback linearization or backstepping -- making the unification more general than existing methods.\nThe approach is demonstrated on the pitch dynamics of an aircraft with uncertain nonlinear aerodynamics.\nThe system is non-invertible, not in strict-feedback form, and has non-polynomial dynamics highlighting the generality of the proposed method.\n\n\\section{Problem Formulation \\& Preliminaries}\nConsider the nonlinear system\n\\protect\\begin{equation}\n\\dot{x} = f(x) - \\Delta(x)^\\top \\theta + B(x)u,\n\\label{eq:unc_dyn}\n\\end{equation}\nwith unknown parameters $\\theta \\in \\mathbb{R}^p$ and known dynamics $\\Delta : \\mathbb{R}^n \\rightarrow \\mathbb{R}^{p\\times n} $, state $x \\in \\mathbb{R}^n$, control input $u \\in \\mathbb{R}^m$, nominal dynamics $f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n}$, and control input matrix $B : \\mathbb{R}^n \\rightarrow \\mathbb{R}^{n\\times m}$ with columns $b_i(x)$ $i=1,\\dots,m$.\nLet $\\mathcal{S}^n_+$ be the set of all $n\\times n$ symmetric positive definite matrices. \nA smooth Riemannian manifold $\\mathcal{M}$ is equipped with a smooth Riemannian metric $M: \\mathbb{R}^n\\times\\mathbb{R} \\rightarrow \\mathcal{S}^n_+$ that defines an inner product $\\left< \\cdot, \\cdot \\right>_x$ on the tangent space $T_x \\mathcal{M}$ at every point $x$.\nThe metric $M(x,t)$ defines local geometric notions such as angles, length, and orthogonality.\nThe directional derivative of metric $M(x,t)$ along vector $v$ is $\\partial_v M = \\sum_i\\frac{\\partial M}{\\partial x_i} v_i$.\nA parameterized differentiable curve $c: \\left[0,1\\right] \\rightarrow \\mathcal{M}$ is a regular if $\\frac{\\partial c}{\\partial s} = c_s \\neq 0$ $\\forall s \\in \\left[0~1\\right]$.\nLet $\\Upsilon(p,q)$ be the family of curves connecting $p,~q\\in \\mathcal{M}$, then a \\emph{geodesic} $\\gamma: \\left[0~1\\right] \\rightarrow \\mathcal{M}$ is the extremum of the energy functional\n\\begin{equation*}\n    \\gamma(s) = \\underset{c(s)\\in\\Upsilon(p,q)}{\\argmin}~ E(c,t) = \\int_0^1 c_s^\\top M(c,t) c_s ds,\n\\end{equation*}\nwhere $E$ is the Riemannian energy. \nIf the manifold $\\mathcal{M}$ is a complete metric space, such as $\\mathbb{R}^n$, $n$-sphere $\\mathbb{S}^n$, or any of their respective closed subsets, then a geodesic is guaranteed to exist by the Hopf-Rinow theorem \\cite{carmo1992riemannian}.\nIn the sequel, the time argument in $M(c,t)$ and $E(c,t)$ is dropped for clarity.\n\n\n\\section{Adaptive Safety}\n\\label{sec:acbf}\n\\subsection{Background}\nFirst consider the nominal dynamics of \\cref{eq:unc_dyn}, i.e., $\\Delta(x) = 0$.\nLet a closed convex set $\\mathcal{C} \\subset \\mathbb{R}^n$ be a 0-superlevel set of a continuously differentiable function $h: \\mathbb{R}^n \\rightarrow \\mathbb{R}$ where\n\\begin{equation*}\n    \\begin{aligned}\n    \\mathcal{C} & = \\left\\{ x \\in \\mathbb{R}^n : h(x) \\geq 0  \\right\\} \\\\\n    \\partial \\mathcal{C} & = \\left\\{ x \\in \\mathbb{R}^n : h(x) = 0  \\right\\} \\\\\n    \\text{Int}\\left(\\mathcal{C}\\right) & = \\left\\{ x \\in \\mathbb{R}^n : h(x) > 0  \\right\\}.\n    \\end{aligned}\n\\end{equation*}\nIf the nominal dynamics are locally Lipschitz, then given an initial condition $x_0$, there exists a maximum time interval $I(x_0) = [t_0,~T)$ such that $x(t)$ is a unique solution on $I(x_0)$. \nThe following definitions are largely taken from \\cite{ames2016control,ames2019control}.\n\\begin{definition}\n\\label{def:fi}\nThe set $\\mathcal{C}$ is \\emph{forward invariant} if for every $x_0\\in \\mathcal{C}$, $x(t) \\in \\mathcal{C}$ for all $t\\in I(x_0)$.\n\\end{definition}\n\n\\begin{definition}\n\\label{def:safety}\nThe nominal system is \\emph{safe} with respect to set $\\mathcal{C}$ if the set $\\mathcal{C}$ is forward invariant.\n\\end{definition}\n\n\\begin{definition}\nA continuous function $\\alpha : \\mathbb{R} \\rightarrow \\mathbb{R}$ is an \\emph{extended class $\\mathcal{K}_\\infty$ function} if it is strictly increasing, $\\alpha(0) = 0$, and is defined on the entire real line.\n\\end{definition}\n\n\\begin{definition}\nLet $\\mathcal{C}$ be a 0-superlevel set for a continuously differentiable function $h:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, then $h$ is a \\emph{control barrier function} if there exists an extended class $\\mathcal{K}_\\infty$ function $\\alpha$ such that\n\\begin{equation}\n\\label{eq:cbf}\n    \\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[\\frac{\\partial h}{\\partial x}(x)\\left(f(x) + B(x) u\\right)\\right] \\geq - \\alpha(h(x)).\n\\end{equation}\n\\end{definition}\n\n\\begin{theorem}\nLet $\\mathcal{C}\\subset\\mathbb{R}^n$ be a 0-superlevel set of a continuously differentiable function $h:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, if $h$ is a CBF on $\\mathcal{C}$ then any locally Lipschitz continuous controller satisfying \\cref{eq:cbf} renders the set $\\mathcal{C}$ safe for the nominal system. \n\\end{theorem}\n\n\\subsection{Adaptive CBFs}\nAdaptive CBFs (aCBFs) \\cite{taylor2019adaptive} provide a general framework to guarantee safety through parameter adaptation for systems with structured uncertainties.\nThe notion of safety for uncertain systems must be extended to a family of safe sets $\\mathcal{C}_{\\theta}$ parameterized by $\\theta$. \nMore precisely, the family of safe sets are 0-superlevel sets of a continuously differentiable function $h_a: \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$.\nIf the uncertain dynamics in \\cref{eq:unc_dyn} are locally Lipschitz then the definitions of forward invariance and safety can be directly extended to $\\mathcal{C}_\\theta$.\n\n\\begin{definition}[\\cite{taylor2019adaptive}]\n\\label{def:acbf}\nLet $\\mathcal{C}_\\theta$ be a family of 0-superlevel sets parameterized by $\\theta$ for a continuously differentiable function $h_a:\\mathbb{R}^n\\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, then $h_a$ is an \\emph{adaptive control barrier function} if for all $\\theta$\n\\begin{equation}\n\\label{eq:acbf}\n    \\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[ \\frac{\\partial h_a}{\\partial x}(x,\\theta)\\left( f(x) - \\Delta(x)^\\top \\Lambda(x,\\theta) + B(x) u\\right)\\right] \\geq 0,\n\\end{equation}\nwhere $\\Lambda(x,\\theta) := \\theta - \\Gamma \\left(\\frac{\\partial h_a}{\\partial \\theta}(x,\\theta)\\right)^\\top$ and $\\Gamma \\in \\mathcal{S}^p_+$ is a symmetric positive definite matrix\n\\end{definition}\n\nA controller that satisfies \\cref{eq:acbf} can be combined with an adaptation law to render the uncertain systems safe with respect to $\\mathcal{C}_{{\\theta}}$ \\cite{taylor2019adaptive}.\nHowever, \\cref{eq:acbf} makes the level sets of $h_a$ forward invariant so it is a much stricter condition than \\cref{eq:cbf}.\nMore precisely, the distance to the boundary of the safe set must monotonically increase, i.e., $\\dot{h}_a(x,\\theta) \\geq 0$ for all time (\\cref{fig:acbf}).\nThis can lead to extremely conservative behavior as the system only operates in a set that is monotonically shrinking.\nIn \\cite{taylor2019adaptive}, a modified aCBF was proposed\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:acbf_m}\n    \\bar{h}_a(x,\\theta) = \\begin{cases} \n    \\sigma^2 ~ &\\text{if} ~ h_a(x,\\theta)\\geq\\sigma \\\\\n    \\sigma^2 - (h_a(x,\\theta)-\\sigma)^2 ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwhich satisfies \\cref{eq:acbf} if $h_a$ is a valid aCBF.\nThis modification expands the set of allowable states but the resulting controller is not necessarily Lipschitz and can exhibit high-frequency oscillations in closed-loop, as shown in \\cref{ex:lipschitz}. \n\n\\begin{example}\n\\label{ex:lipschitz}\nConsider the uncertain system $\\dot{x} = - \\theta + u$ with $\\theta>0$ and aCBF $h_a(x) = x - \\ubar{x}$.\nLet the controller $\\kappa$ be the solution to $\\kappa = {\\argmin}~\\frac{1}{2} u^2$ subject to $\\dot{\\bar{h}}_a(x) \\geq 0$.\nThen\n\\begin{equation}\n\\label{eq:u_ex}\n\\begin{aligned}\n    \\kappa = \\begin{cases}\n        0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n        \\mathrm{max}(0,\\hat{\\theta}) ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwhere $\\hat{\\theta}$ is the estimate of $\\theta$ and is modified based on \\cite{taylor2019adaptive}\n\\begin{equation}\n\\label{eq:ha_ex}\n\\begin{aligned}\n    \\dot{\\hat{\\theta}} = \\begin{cases}\n        0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n        - \\Gamma \\left[h_a(x)-\\sigma\\right] \\frac{\\partial h_a}{\\partial x} ~~ &\\text{otherwise},\n    \\end{cases}\n\\end{aligned}\n\\end{equation}\nwith $\\Gamma > 0$. \nFor $\\hat{\\theta}(0) = \\hat{\\theta}_0 \\leq 0$ then $\\kappa = 0$ so the closed-loop response is $x_{cl}(t) = - \\theta (t-t_0) + x_0$ where $x_{cl}(t_0) = x_0 > \\ubar{x}$ and $t \\geq t_0$.\nFrom the adaptation law \\cref{eq:ha_ex}, it is easy to see that $\\dot{\\hat{\\theta}} \\geq 0 $ which will necessarily lead to $\\hat{\\theta} > 0$ since $h_a \\leq \\sigma$ until $\\kappa > \\theta$.\nFor $\\hat{\\theta} > 0$, the closed-loop response becomes\n\\begin{equation}\n\\label{eq:cl_ex}\n    \\begin{aligned}\n        x_{cl}(t) = \\begin{cases}\n            -\\theta (t-t_0) + x_0  ~ &\\text{if} ~ h_a(x)\\geq\\sigma \\\\\n            \\hspace{0.28cm}\\tilde{\\theta} (t-t_0') + x_0' ~~ &\\text{otherwise},\n        \\end{cases}\n    \\end{aligned}\n\\end{equation}\nwhere $\\tilde{\\theta}:=\\hat{\\theta} - \\theta$ and $x_{cl}(t_0') = x_0'$.\nSince $\\tilde{\\theta} > 0$, \\cref{eq:cl_ex} will continuously switch between its two solutions.\nThe control policy $\\kappa$ must then also switch between $0$ and $\\hat{\\theta}$ based on \\cref{eq:u_ex}.\nFurthermore, $\\kappa$ is not locally Lipschitz continuous and will exhibit high-frequency oscillations of magnitude $\\hat{\\theta}$ in closed-loop.\n\\end{example}\n\nThe intuition behind \\cref{ex:lipschitz} is that chatter arises due to the barrier condition switching between being trivially satisfied, i.e., $\\dot{\\bar{h}}_a = 0 \\geq 0$ for all $u$, to satisfied only for a particular $u$, i.e., a $u$ so that $\\dot{\\bar{h}}_a \\geq 0$.\nThe approach developed in this work addresses the conservatism of aCBFs and results in a locally Lipschitz continuous controller.\n\n\n\\subsection{Robust aCBFs}\n\\label{sub:racbf}\nThis section will show that a \\emph{tightened} set can be made forward invariant if the unknown model parameters are bounded and the parameter adaptation rate is an \\emph{admissible} (to be defined) symmetric positive definite matrix.\n\n\\begin{assumption}\nThe unknown parameters $\\theta$ belong to a known closed convex set $\\Theta$.\nThe parameter estimation error $\\tilde{\\theta}:= \\hat{\\theta} - \\theta$ then also belongs to a known closed convex set $\\tilde{\\Theta}$ and the maximum possible parameter error is $\\tilde{\\vartheta}$.\n\\end{assumption}\n\nLet $\\mathcal{C}^r_\\theta$ be a family of superlevel sets parameterized by $\\theta$ for a continuously differentiable function ${h}_r: \\mathbb{R}^n \\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$  \n\\begin{equation*}\n    \\begin{aligned}\n    \\mathcal{C}^r_\\theta & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) \\geq \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}  \\right\\} \\\\\n    \\partial \\mathcal{C}^r_\\theta & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) = \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta} \\right\\} \\\\\n    \\text{Int}\\left(\\mathcal{C}^r_\\theta\\right) & = \\left\\{ x \\in \\mathbb{R}^n : {h}_r(x,\\theta) > \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta} \\right\\},\n    \\end{aligned}\n\\end{equation*}\nwhere $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix that will dictate the parameter adaptation rate.\nThe set $\\mathcal{C}^r_\\theta$ can be viewed as a \\emph{tightened} set with respect to $\\mathcal{C}_\\theta$, i.e., $\\mathcal{C}^r_\\theta \\subset \\mathcal{C}_\\theta$, shown in \\cref{fig:racbf}.\nOne can select the desired subset $\\mathcal{C}^r_\\theta$ to be made forward invariant \\textit{a priori}  by choosing $h_r(x_r,\\theta_r) >0 $ for appropriate $x_r,~\\theta_r$ so ${h}_r(x_r,\\theta_r) = \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}$.\nTo reduce conservatism, one can either 1) have fast parameter adaptation or 2) reduce model error.\nThe first scenario can lead to well-known undesirable effects in practice so the second scenario is the most viable, and will be explored more in \\cref{sec:smid}.\n\\cref{eq:unc_dyn} is again assumed to be locally Lipschitz so \\cref{def:fi,def:safety} hold.\n\n\\begin{definition}\n\\label{definition:racbf}\nLet $\\mathcal{C}^r_\\theta$ be a family of superlevel sets parameterized by $\\theta$ for a continuously differentiable function $h_r:\\mathbb{R}^n\\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, then $h_r$ is a \\emph{robust adaptive control barrier function}  if there exists an extended class $\\mathcal{K}_\\infty$ function $\\alpha$ such that for all $\\theta\\in\\Theta$ \n\\begin{equation}\n\\label{eq:racbf}\n\\begin{aligned}\n    &\\underset{u \\in \\mathcal{U}}{\\text{sup}}~ \\left[ \\frac{\\partial h_r}{\\partial x}(x,\\theta)\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\theta) + B(x) u\\right]\\right] \\\\ \n   \n    & \\hspace{3cm} \\geq -\\alpha\\left(h_r(x,\\theta) - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}\\right),\n\\end{aligned}\n\\end{equation}\nwhere $\\Lambda(x,\\theta) := \\theta - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\theta}(x,\\theta)\\right)^\\top$, $\\tilde{\\vartheta}$ is the maximum possible parameter error, and $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix.\n\\end{definition}\n\nThe invariance condition \\cref{eq:racbf} is reminiscent of that in \\cref{eq:cbf} and is less conservative than that in \\cref{eq:acbf} because the system is allowed to approach the boundary of $\\mathcal{C}^r_\\theta$.\n\\cref{thm:racbf} shows the existence of a RaCBF, coupled with an adaptation law, renders the set $\\mathcal{C}^r_\\theta$ forward invariant and hence safe.\n\n\\begin{theorem}\n\\label{thm:racbf}\nLet $\\mathcal{C}^r_{\\hat{\\theta}}\\subset\\mathbb{R}^n$ be a superlevel set of a continuously differentiable function $h_r:\\mathbb{R}^n \\times \\mathbb{R}^p\\rightarrow\\mathbb{R}$, if $h_r$ is a RaCBF on $\\mathcal{C}^r_{\\hat{\\theta}}$, then any locally Lipschitz continuous controller satisfying \\cref{eq:racbf} renders the set $\\mathcal{C}^r_{\\hat{\\theta}}$ safe for the uncertainty system with adaptation law and adaptation gain\n\\begin{equation*}\n\\dot{\\hat{\\theta}} = \\Gamma \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}) \\right)^\\top, ~~ \\lambda_{\\min}(\\Gamma) \\geq \\frac{\\|\\tilde{\\vartheta}\\|^2}{2h_r(x_r,{\\theta}_r)},\n\\end{equation*}\nwhere $\\tilde{\\vartheta}$ is the maximum possible parameter error, $h_r(x_r,\\theta_r)>0$ can be chosen freely based on the desired conservatism, and $\\Gamma \\in \\mathscr{S}^p_+ \\subset \\mathcal{S}^p_+$ is an admissible symmetric positive definite matrix.\nFurthermore, the original set $\\mathcal{C}_{\\hat{\\theta}}$ is also safe for the uncertain system.\n\\end{theorem}\n\n\n\n\\begin{remark}\n\\label{remark:proj}\nThe projection operator \\cite{slotine1991applied} can be used to enforce parameter bounds by modifying the above adaptation law as opposed to capturing them explicitly with $h_r$. \nThis can simplify the design of $h_r$ without forfeiting safety.\nThe proof is omitted but one can show that a positive semi-definite term appears in the same composite candidate CBF used in \\cref{thm:racbf} when adaptation is temporarily stopped.\n\\end{remark}\n\n\n\\begin{figure}[t!]\n\\vskip 0.1in\n    \\begin{subfigure}{.48\\columnwidth}\n         \\centering\n         \\includegraphics[trim=100 0 100 0, clip,width=1\\linewidth]{figures\/acbf.pdf}\n         \\caption{{Safe set with adaptive control barrier functions (aCBFs).}}\n         \\label{fig:acbf}\n     \\end{subfigure}\n    \\hspace{0.3em}\n     \\begin{subfigure}{.48\\columnwidth}\n         \\centering\n         \\includegraphics[trim=100 0 100 0, clip,width=1\\linewidth]{figures\/racbf.pdf}\n         \\caption{{Safe set with robust adaptive control barrier functions (RaCBFs).}}\n         \\label{fig:racbf}\n     \\end{subfigure}\n    \\caption{Visual comparison of safe sets with adaptive and robust adaptive control barrier functions. (a): System is restricted to level sets (black dashed lines) of aCBF $h_a$. (b): System allowed to operate in larger set with RaCBF $h_r$ reducing conservatism.}\n    \\vskip -0.25in\n\\end{figure}\n\nSeveral remarks can be made about \\cref{thm:racbf}.\nFirst, safety is \\emph{guaranteed} for all possible parameter realizations through adaptation with minimal conservatism.\nHence, RaCBFs expand and improve the \\emph{adaptive safety} paradigm.\nSecond, the minimum eigenvalue condition for the adaptation rate depends on the desired conservatism, i.e., the degree of tightening by choice of $h_r(x_r,\\theta_r)$.\nFor low conservatism, i.e., a small $h_r(x_r,\\theta_r)$ value, the adaptation rate must be large so the parameter estimates can change quickly to ensure forward invariance of $\\mathcal{C}^r_\\theta$.\nThere is thus a fundamental trade-off between conservatism and parameter adaptation rate that must be weighed carefully given the well-known undesirable effects of high-gain adaptation.\nThird, the RaCBF condition in \\cref{eq:racbf} can be used as a safety filter for an existing tracking controller or as a constraint within an optimization.\n\\cref{sec:result} will show the latter but with a contraction-based controller.\nLastly, if the adaptation gain must be small (or the maximum parameter error is large) then RaCBFs can be conservative albeit not to the same extent as aCBFs.\nBetter performance can be obtained if the model parameters can be robustly and accurately estimated.\nInstead of obtaining a point-estimate of the parameters, this work will instead identify the \\emph{set} of possible parameter values.\n\n\\subsection{RaCBFs with Set Membership Identification}\n\\label{sec:smid}\nSet membership identification (SMID) is a model estimation technique that constructs an unfalsified set of model parameters.\nSMID was originally developed to identify transfer functions for uncertain linear systems \\cite{pararrieter1992set}, but has been more recently applied to linear \\cite{tanaskovic2014adaptive,lorenzen2017adaptive} and nonlinear adaptive MPC \\cite{lopez2019adaptive}.\nAssume that the true parameters $\\theta^*$ belong to an initial set of possible parameters $\\Theta^0$, i.e., $\\theta^* \\in \\Theta^0$.\nGiven $k$ state, input, and rate measurements (denoted as $x_{1:k}$ and so forth), a set $\\Xi$ can be constructed such that\n\\begin{equation*}\n   \n   \n\\Xi = \\left\\{ \\varrho :  |\\dot{x}_{1:k} - f_{1:k} + \\Delta_{1:k}^\\top \\varrho - B_{1:k}u_{1:k}| \\leq D  \\right\\},\n\\end{equation*}\nwhere $D$ can be treated as a tuning parameter that dictates the conservativeness of SMID. \nIt can also represent a disturbance or noise bound \\cite{tanaskovic2014adaptive,lopez2019adaptive}.\nThe set of possible parameter values can then be updated via $\\Theta^{j+1} = \\Theta^j \\cap \\Xi$ for all $j\\geq 0$.\nIn practice, $\\Xi$ can be found by solving a linear program and set intersection can be efficiently done through a combination of min and max operations.\nRestricting $\\hat{\\theta} \\in \\Theta^j$ then $\\tilde{\\theta} \\in \\Tilde{\\Theta}^j$ where $\\Tilde{\\Theta}^j$ is the set of possible parameter errors.\nThe following lemma shows the advantage of performing set identification over point-estimation techniques.\n\\begin{lemma}\n\\label{lemma:smid}\nModel uncertainty monotonically decreases with set membership identification, i.e., $\\tilde{\\Theta}^{j+1} \\subseteq \\tilde{\\Theta}^j$ for all $j\\geq 0$.\n\\end{lemma}\n\\begin{proof}\nSince $\\theta^* \\in \\Theta^{j+1}$ then $\\theta^* \\in \\Theta^{j} \\cap \\Xi$ which is true if $\\theta^* \\in \\Theta^j$ so $\\Theta^{j+1}\\subseteq\\Theta^j$ and $\\tilde{\\Theta}^{j+1} \\subseteq \\tilde{\\Theta}^j$.\n\\end{proof}\n\nThe motivation to combine SMID with RaCBFs is to enlarge the tightened set $\\mathcal{C}_\\theta^r$.\nTo do so, one must ensure $\\mathcal{C}^r_\\theta$ remains forward invariant as the set of model parameters is updated.\nIn general this is non-trivial to prove since the maximum possible parameter error is now time varying.\nHowever, \\cref{thm:racbf_smid} shows that safety is maintained if the model uncertainty monotonically decreases.\n\n\\begin{theorem}\n\\label{thm:racbf_smid}\nLet $\\mathcal{C}^r_\\theta$ be a superlevel set of a continuously differentiable function ${h}_r : \\mathbb{R}^n\\times \\mathbb{R}^p \\rightarrow \\mathbb{R}$. If the system is safe on $\\mathcal{C}^r_\\theta$ then it remains safe if the maximum allowable model parameter error $\\tilde{\\vartheta}$ monotonically decreases. \nMoreover, the tightened set $\\mathcal{C}^r_\\theta$ converges to $\\mathcal{C}_\\theta$ monotonically.\n\\end{theorem}\n\n\nCombining RaCBFs and SMID provides a mechanism to 1) modify parameters via adaptation to achieve safety and 2) update the model to reduce uncertainty and conservatism.\nSafety is \\emph{guaranteed} even as the model parameters are modified online, and the system's performance will only improve as more data is collected.\nThis \\emph{adaptive data-driven safety} paradigm can be merged with a stabilizing adaptive controller for safe reference tracking.\nTo maximize the generality of the proposed unification, the adaptive controller must be applicable to a broad class of nonlinear systems.\n\n\\section{Adaptive Control with Contraction Metrics}\n\\label{sec:accm}\nSeveral adaptive control techniques have been proposed for nonlinear systems, including methods based on feedback linearization, variable structure, and backstepping (see \\cite{slotine1991applied,miroslav1995nonlinear}).\nThese methods are limited to certain classes of systems because they rely on \\emph{explicitly} constructing a control Lyapunov function (CLF) to prove stability.\nThis work will instead utilize a \\emph{differential} approach based on contraction analysis that can be applied to a broad class of systems.\n\n\\begin{figure}[t!]\n\\vskip 0.05in\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/geodesic.pdf}\n         \\caption{{Geodesic and arbitrary curve connecting current and desired state.}}\n         \\label{fig:geodesig}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/manifold.pdf}\n         \\caption{{Differential CLFs along geodesic connecting current and desired state.}}\n         \\label{fig:manifold}\n     \\end{subfigure}\n    \\caption{Geodesic and differential CLF visualization. (a): Geodesic (grey) connecting current $x$ (red) and desired $x_d$ (blue) state. (b): Differential CLFs are integrated along geodesic to achieve exponential convergence.}\n    \\label{fig:dclf}\n    \\vskip -0.2in\n\\end{figure}\n\n\\subsection{Contraction Metrics}\nThe nominal differential dynamics of \\cref{eq:unc_dyn} are $\\dot{\\delta}_x  = A(x,u)\\delta_x + B(x)\\delta_u$ where $A(x,u) = \\frac{\\partial f}{\\partial x} + \\sum_{i=1}^m \\frac{\\partial b_i}{\\partial x}u_i$.\nContraction analysis searches for a \\emph{control contraction metric} (CCM) $M(x)$ such that a \\emph{differential} CLF $\\delta V = \\delta_x^\\top M(x) \\delta_x$ satisfies $\\delta \\dot{V}\\leq - 2 \\lambda \\delta V$ for all $x$.\nA global CLF can be obtained by integrating along a geodesic $\\gamma(s)$, illustrated in \\cref{fig:dclf}, with $\\gamma(0) = x_d$ and $\\gamma(1) = x$ where $x_d$ and $x$ are the desired and current state.\nThe Riemannian energy and tracking error both converge to zero exponentially, i.e., $\\dot{E} \\leq - 2 \\lambda E$.\nLet $W(x) = M(x)^{-1}$, $M(x)$ is CCM if \\cite{manchester2017control}\n\\begin{gather}\n     B^\\top_{\\perp}\\left( W A ^\\top + A W - \\dot{W} + 2 \\lambda W \\right)B_{\\perp} \\preceq 0 \\tag{C1} \\label{eq:ccm_c1} \\\\\n    \\partial_{b_i}W - W \\frac{\\partial b_i}{\\partial x}^\\top  -  \\frac{\\partial b_i}{\\partial x} W = 0 ~~ i=1,\\dots,m \\tag{C2} \\label{eq:ccm_c2}\n\\end{gather}\nwhere $B_\\perp(x)$ is the annihilator matrix of $B(x)$, i.e., $B^\\top_\\perp B = 0$.\n\\cref{eq:ccm_c1} ensures the dynamics orthogonal to $u$ are contracting and is a stabilizability condition.\n\\cref{eq:ccm_c2} requires the column vectors of $B(x)$ form a Killing vector for the dual metric $W(x)$ leading to simpler controllers \\cite{manchester2017control}. \n\\cref{eq:ccm_c1} and \\cref{eq:ccm_c2} will be referred to as the \\emph{strong CCM conditions}.\n\n\n\n\n\n\\subsection{Adaptive Control \\& Contraction}\nA novel adaptive control method was developed in \\cite{lopez2019contraction} for closed-loop contracting systems with \\emph{extended matched uncertainties}, i.e., $\\Delta(x)^\\top \\theta \\in \\text{span}\\{B,ad_fB\\}$ were $ad_fB$ is the Lie bracket of the vector fields $f(x)$ and $B(x)$.\nTo stabilize such systems, the \\emph{parameter-dependent} metric $M(x,\\hat{\\theta})$ was introduced and must satisfy the strong CCM conditions \\emph{for all} possible $\\hat{\\theta}$.\nThis led to the following result.\n\n\\begin{theorem}[\\cite{lopez2019contraction}]\n\\label{thm:exMatched}\nIf a parameter-dependent metric can be computed for \\cref{eq:unc_dyn} with extended matched uncertainties, then the closed-loop systems is asymptotically stable with\n\\begin{equation}\n    \\dot{\\hat{\\theta}} = - \\Gamma \\Delta(x) M(x,\\hat{\\theta})\\gamma_s(1)\n    \\label{eq:accm}\n\\end{equation}\nwhere $\\gamma_s(s):=\\frac{\\partial \\gamma}{\\partial s}$ is the geodesic speed and $\\Gamma \\in \\mathcal{S}^p_+$ is a symmetric positive definite matrix.\n\\end{theorem}\n\n\\begin{remark}\nFor \\emph{matched} uncertainties, the metric is \\emph{independent} of the unknown parameters $\\hat{\\theta}$ \\cite[Lemma 1]{lopez2019contraction} simplifying its computation.\nOtherwise, sum-of-square or robust optimization must be utilized to compute $M(x,\\hat{\\theta})$.\n\\end{remark}\n\n\\begin{remark}\nSeveral modifications can be made to \\cref{eq:accm} that improve transients or robustness including the projection operator discussed in \\cref{sub:racbf} (see \\cite{lopez2019contraction}).\n\\end{remark} \n\n\n\\subsection{Offline Design \\& Online Computation}\n\\label{sub:computation}\nA contraction metric is computed offline via sum-of-square programming for polynomial systems \\cite{manchester2017control} or by imposing \\cref{eq:ccm_c1,eq:ccm_c2} at sampled points in the state space; a process called gridding.\nGeodesics are computed online at each time step by solving a nonlinear program (NLP) with the current state.\nGeodesics are often guaranteed to exist by Hopf-Rinow and are less expensive to compute than solving nonlinear MPC.\nGiven a geodesic $\\gamma(s)$, the Riemannian energy can be interpreted as a CLF so a pointwise min-norm controller similar to that in \\cite{primbs2000receding} can found by solving the QP\n\\begin{equation*}\n    \\begin{aligned}\n        & u^* = ~\\underset{u\\in \\mathcal{U}}\\argmin ~\\frac{1}{2} u^\\top u \\\\\n        & \\text{s.t.} ~\\gamma_s(1)^\\top M(x,\\hat{\\theta})\\dot{\\hat{x}} - \\gamma_s(0)^\\top M(x_d,\\hat{\\theta})\\dot{x}_d  \\leq - \\lambda E(\\gamma(s),\\hat{\\theta}) \n    \\end{aligned}\n\\end{equation*}\nwhere $\\hat{\\theta}$ is the current parameter estimate, $\\gamma_s$ is the geodesic speed, $\\dot{\\hat{x}}$ is \\cref{eq:unc_dyn} but with $\\hat{\\theta}$, $\\dot{x}_d $ is the desired dynamics for desired state $x_d$.\nThe stability constraint imposes $\\dot{E} \\leq - 2 \\lambda E$ where $\\dot{E}$ is the first variation of the Riemannian energy, which has a known form \\cite{carmo1992riemannian}, and results in exponential convergence of the tracking error.\nIt is more general than the traditional Lyapunov stability as a distance-like function does not need to computed explicitly.\nSafety can be directly embedded in the above QP, resulting in a single optimization for a safe stabilizing controller.\n\n\n\\begin{figure*}[t]\n\\vskip 0.1in\n    \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30, clip,width=1\\textwidth]{figures\/q_c_i.pdf}\n         \\caption{Pitch rate $q$.}\n         \\label{fig:q_i}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/u_c_i.pdf}\n         \\caption{Control input $u$.}\n         \\label{fig:u_i}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/h_c_i.pdf}\n         \\caption{Barrier function $h$.}\n         \\label{fig:h_i}\n     \\end{subfigure}\n     \\caption{Comparison of modified aCBFs \\cref{eq:acbf_m}, RaCBFs, and RaCBFs \\& SMID for desired terminal state $x_d = [180^\\circ~0~0]^\\top$ (Immelmann turn) and maximum pitch rate $q_m$. (a): Pitch rate tracking where RaCBFs and the modified aCBFs exhibit similar conservatism due to model error.\n     RaCBF \\& SMID allows the aircraft to utilize 97.9\\% of the maximum allowable pitch rate. `\n     (b): Control chatter is observed with the modified aCBFs while RaCBFs generate continuous control inputs. (c): Safety is maintained but RaCBFs and the modified aCBFs are conservative due to potential model error. RaCBFs \\& SMID permit states closer to the boundary of the safe set without losing safety guarantees.\n     For tests $k_q^* = 0.2$, $\\ell^*_\\alpha = -1$, $\\Gamma_B = 20$, $\\Gamma_C = 50$, $\\alpha(r) = 10r$, and $D = 0.1$.}\n     \\label{fig:results_immelmann}\n\\vskip -0.2in\n\\end{figure*}\n\n\\section{Adaptive \\& Data-Driven Safety}\n\\label{sec:result}\nA safe and stabilizing controller can be computed by unifying RaCBFs, SMID, and adaptive control with contraction.\nThe individual components of the controller are summarized below with their respective computational complexity.\n\\\\[4pt]\n\\noindent \\textbf{1) Compute geodesic (NLP)}\n\\begin{equation*}\n   \n    \\gamma(s) = \\underset{c(s)\\in\\Upsilon(x,x_d)}{\\argmin}~ E(c,\\hat{\\theta}_C) = \\int_0^1 c_s^\\top M(c,\\hat{\\theta}_C) c_s ds\n\\end{equation*}\n\n\\noindent \\textbf{2) Compute controller (QP \\& Quadrature)}\n\\begin{align*}\n    & \\kappa  = ~\\underset{u \\in \\mathcal{U}}{\\argmin} ~ \\frac{1}{2} u^\\top u + r \\epsilon^2 \\\\\n    &\\text{s.t.} ~\\gamma_s(1)^\\top M(x,\\hat{\\theta}_C)\\dot{\\hat{x}} - \\gamma_s(0)^\\top M(x_d,\\hat{\\theta}_C)\\dot{x}_d  \\\\\n    & \\hspace{4.6cm} \\leq - \\lambda E(\\gamma(s),\\hat{\\theta}_C)+ \\epsilon \\nonumber  \\\\\n    & \\hspace{.4cm} \\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}_B)\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}_B) + B(x) u\\right] \\\\\n    & \\hspace{3.1cm} \\geq -\\alpha\\left({h}_r(x,\\hat{\\theta}_B) - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}\\right) \\nonumber\n\\end{align*}\n\\noindent \\textbf{3) Update parameters (Quadrature)} \\label{item:quad}\n\\begin{equation*}\n\\begin{aligned}\n    & \\dot{\\hat{\\theta}}_C = - \\Gamma_C \\Delta(x) M(x,\\hat{\\theta}_C) \\gamma_s(1) \\\\\n    & \\dot{\\hat{\\theta}}_B =  \\Gamma_B \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}(x,\\hat{\\theta}_B)\\right)^\\top\n\\end{aligned}\n\\end{equation*}\n\\vskip -.03in\n\\noindent \\textbf{4) Update parameter error bounds (LP)} \\label{item:lp}\n\\begin{align*}\n        & \\Xi = \\left\\{ \\varrho : ~ |\\dot{x}_{1:k} - f_{1:k} + \\Delta_{1:k}^\\top \\varrho - B_{1:k}\\kappa_{1:k}| \\leq D  \\right\\} \\\\\n         &\\Theta^{j+1} = \\Theta^{j} \\cap \\Xi, ~~ \\tilde{\\vartheta} = \\underset{\\varrho_i,\\forall i}{\\text{sup}} ~\\Theta^{j+1} - \\underset{\\varrho_i,\\forall i}{\\vphantom{\\text{sup}}\\text{inf}}~ \\Theta^{j+1} \\nonumber\n\\end{align*}\n\\vskip -.02in\n\nThe NLP in Step~1) can be efficiently solved by parameterizing geodesics with a set of polynomial basis functions. \nWe adopt the same strategy as in \\cite{leung2017nonlinear} and utilize the the Chebychev Pseudospectral method and Clenshaw-Curtis quadrature to compute a geodesic at each time step.\nUsing the geodesic computed in Step~1), the QP in Step~2) is solved to generate a safe and stabilizing controller $\\kappa$.\nThe QP is similar to that in \\cite{ames2016control} but the stability constraint is replaced with the first variation of the Riemannian energy \\cite{carmo1992riemannian}.\nUnder the premise that \\cref{eq:unc_dyn} is locally Lipschitz, one can show that $\\kappa$ is guaranteed to be locally Lipschitz from \\cite[Theorem 3]{ames2016control} as both the geodesic speed $\\gamma_s$ and metric $M(x)$ are also locally Lipschitz from their definitions.\nNote that $\\dot{x}_d$ is the desired dynamics and $\\dot{\\hat{x}}$ is \\cref{eq:unc_dyn} but with $\\hat{\\theta}_C$.\nStep~3) is simple quadrature and is not computationally expensive.\nNote that the parameter adaptation $\\dot{\\hat{\\theta}}_C$ for the controller should be temporarily stopped when the safety constraint is active to prevent undesirable transients.\nOtherwise, the parameter estimates will windup as the tracking error may increase to ensure safety.\n\nThe LP in Step~4) has $2k$ constraints and is solved $2p$ times at every time step for the upper and lower bound of each parameter. \nSet intersection is done by taking the appropriate minimum or maximum of the newest and current bounds.\nThe complexity of Step~4) can be bounded by either removing redundant constraints or terminating when $\\tilde{\\vartheta} \\leq \\varepsilon$ where $\\varepsilon$ is a predefined threshold.\nStep~4) can also be done outside the control loop since stability and safety do not rely on real-time updates of the parameter bounds; although real-time bounds are desirable to quickly eliminate conservatism.\nConsequently, non-causal filtering can be used to accurately estimate $\\dot{x}$ if necessary.\nMoreover, the right hand side of the inequality can be replaced by $D + \\mathcal{E}$ where $\\mathcal{E}$ is the maximum estimation error of the rate vector, i.e., $|\\dot{\\hat{x}}| \\leq \\mathcal{E}$.\nThe proposed method was tested in MATLAB R2018B with the built-in solvers without any code optimization on a 1.6GHz Intel i5 processor.\nThe NLP was initialized with a linear curve $c(s) = (x-x_d)s+x_d$ at each time step.\n\n\n\n\\section{Illustrative Example}\n\\label{sec:example}\nConsider the simplified pitch dynamics of an aircraft \\cite{mccormick1995aerodynamics}\n\\begin{equation*}\n    \\left[ \\begin{array}{c} \\dot{\\theta} \\\\ \\dot{\\alpha} \\\\ \\dot{q} \\end{array} \\right] = \\left[ \\begin{array}{c}  q \\\\ q - \\bar{L}(\\alpha)  \\\\ -k_q q + \\bar{M}(\\alpha) \\end{array} \\right] + \\left[ \\begin{array}{c} 0 \\\\ 0 \\\\ 1 \\end{array} \\right] u,\n\\end{equation*}\nwhere $\\theta$, $\\alpha$, and $q$ are the pitch angle, angle of attack, and pitch rate.\n$\\bar{L}(\\alpha)$ and $\\bar{M}(\\alpha)$ are the aerodynamic lift and moment.\nThe system is not feedback linearizable as the controllability matrix drops rank at $\\bar{L}'(\\alpha)=0$ and is not in strict-feedback form.\nUtilizing flat plat theory \\cite{mccormick1995aerodynamics}, the aerodynamics of a high-performance aircraft are approximately $\\bar{L}(\\alpha) = 0.8\\mathrm{sin}(2\\alpha)$ and $\\bar{M}(\\alpha) = -\\ell_{\\alpha} \\bar{L}(\\alpha)$.\nThe parameters $k_q$ and $\\ell_\\alpha$ are unknown but $k_q \\in [0.1~0.8]$ and $\\ell_\\alpha \\in [-3~1]$.\nA metric quadratic in $\\alpha$ was synthesized via gridding for $\\alpha \\in [-5^{\\circ}~50{^\\circ}]$ and $q \\in [-10^{\\circ\/s}~50^{\\circ\/s}]$.\nNote that $\\bar{L}'(\\alpha)=0$ is in the chosen grid range.\nThe function $h_r(q) = q_m - q$ where $q_m = 50^{\\circ\/s}$ can be easily shown to be a valid RaCBF that enforces $q \\leq q_m$.\n\n\nThe desired terminal state $x_d = [180^\\circ~0~0]^\\top$ corresponds to the first portion of the aerobatic maneuver known as the Immelmann turn.\nThe vehicle executes a half loop before executing a half roll (not considered here) resulting in level flight in the opposite direction.\n\\cref{fig:q_i} shows modified aCBFs and RaCBFs exhibit similar behavior in terms of conservatism as they do not utilize the maximum allowable pitch rate.\nHowever, modified aCBFs exhibit high-frequency oscillations due to the chatter in the control input, seen in \\cref{fig:u_i}; a result of the formulation as the safety constraint continuously switches between active and inactive. \nHigh-frequency oscillations are also seen in the barrier function in \\cref{fig:h_i} but are absent with RaCBFs.\n\\cref{fig:q_i} shows RaCBFs with SMID results in less conservatism as 97.9\\% of the maximum allowable pitch rate is utilized.\nMoreover, the set of allowable states is considerably larger as is evident by the small value of the barrier function in \\cref{fig:h_i}.\nThe parameter error bounds, shown in \\cref{fig:smid_i}, were reduced by 63.0\\% for $k_q$ and 90.5\\% for ${\\ell}_\\alpha$.\n\\cref{fig:comp_time_i} shows the computation time is within real-time constraints and can be easily reduced by utilizing faster solvers or a lighter programming language.\n\n\\begin{figure}[t!]\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/smid_i.png}\n         \\caption{{Parameter bounds.}}\n         \\label{fig:smid_i}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/comp_time_i.png}\n         \\caption{{Computation time.}}\n         \\label{fig:comp_time_i}\n     \\end{subfigure}\n    \\caption{Parameter bounds and computation time. (a): Parameter bounds monotonically approach the true parameter values $\\ell_\\alpha^*$ and $k_q^*$. (b): Computation time for the proposed controller is well within real-time constraints.}\n    \\label{fig:results_comp_i}\n    \\vskip -0.2in\n\\end{figure}\n\n\\begin{figure*}[t]\n\\vskip 0.1in\n    \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30, clip,width=1\\textwidth]{figures\/q_c.pdf}\n         \\caption{Pitch rate $q$.}\n         \\label{fig:q}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/u_c.pdf}\n         \\caption{Control input $u$.}\n         \\label{fig:u}\n     \\end{subfigure}\n     \\hfill{}\n     \\begin{subfigure}{.66\\columnwidth}\n         \\centering\n         \\includegraphics[trim=150 30 150 30,clip, width=1\\textwidth]{figures\/h_c.pdf}\n         \\caption{Barrier function $h$.}\n         \\label{fig:h}\n     \\end{subfigure}\n     \\caption{Comparison of modified aCBFs \\cref{eq:acbf_m}, RaCBFs, and RaCBFs \\& SMID with a full desired trajectory corresponding to $\\theta_d = -20^\\circ \\mathrm{cos}(t)$. (a): Pitch rate tracking where RaCBFs and aCBFs exhibit similar conservativeness due to model error.\n     RaCBF \\& SMID achieves the best performance because model uncertainty is reduced via online estimation. \n     (b): Control chatter is observed with aCBFs while RaCBFs generate continuous control inputs. (c): Safety maintained but RaCBFs and aCBFs are conservative due to potential model error. RaCBFs \\& SMID is the least conservative since model is estimated online.\n     For tests $k^* = 0.2$, $\\ell^*_\\alpha = -1$, $\\Gamma_B = 20$, $\\Gamma_C = 50$, $\\alpha(r) = 10r$, and $D = 0.1$.}\n     \\label{fig:results}\n\\vskip -0.2in\n\\end{figure*}\n\nNow consider the scenario where a full desired trajectory described by $\\theta_d = - 20^\\circ \\mathrm{cos}(t)$ is available.\nA metric was synthesized for a new grid range $\\alpha \\in [-60^{\\circ}~60{^\\circ}]$ and $q \\in [-20^{\\circ\/s}~20^{\\circ\/s}]$; a metric quadratic in $\\alpha$ was again found to be valid over the grid range.\nThe function $h_r(q) = 1 - (\\nicefrac{q}{q_m})^2$ where $q_m = 20^{\\circ\/s}$ is a valid RaCBF that enforces $|q| \\leq q_m$.\nThe results in \\cref{fig:results} show the same exact behavior as in \\cref{fig:results_immelmann}: control input chattering occurs with the modified aCBFs \\cref{fig:u} resulting in high-frequency oscillations in both the pitch rate \\cref{fig:q} and barrier function \\cref{fig:h}. \nChattering does not occur with RaCBFs.\nAdditionally, RaCBFs with SMID again has the best tracking performance in \\cref{fig:q} and is the least conservative in \\cref{fig:h}.\nThe parameter bounds at different time instances are shown in \\cref{fig:smid_sine}.\nThe bounds again monotonically decrease resulting in a reduction of 16.5\\% and 77.3\\% for ${k}_q$ and ${\\ell}_\\alpha$, respectively.\nThe reduction is less than that in the Immelmann turn due to the trajectory not sufficiently exciting $q$ relative to $\\bar{L}(\\alpha)$.\nThe largest reduction occurred at $t=3s$ which is when the barrier function in \\cref{fig:h} becomes less conservative.\nThe computation time shown in \\cref{fig:comp_time_sine} is comparable to that in \\cref{fig:comp_time_i} with the NLP solve time being slightly less as the linear geodesic initialization is a better initial guess for small tracking error. \nThis further confirms that the proposed approach can be run in real-time.\n\n\\begin{figure}[t!]\n    \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/smid_sine.png}\n         \\caption{{Parameter bounds.}}\n         \\label{fig:smid_sine}\n     \\end{subfigure}\n    \\hspace{0.6em}\n     \\begin{subfigure}{.47\\columnwidth}\n         \\centering\n         \\includegraphics[width=1\\linewidth]{figures\/comp_time_sine.png}\n         \\caption{{Computation time.}}\n         \\label{fig:comp_time_sine}\n     \\end{subfigure}\n    \\caption{Parameter bounds and computation time for desired trajectory. (a): Parameter bounds monotonically approach the true parameter values $\\ell_\\alpha^*$ and $k_q^*$. (b): Computation time for the proposed controller is well within real-time constraints.}\n    \\label{fig:results_comp_sine}\n    \\vskip -0.2in\n\\end{figure}\n\n\\section{Conclusion}\nThis work presented a framework that guarantees safety for uncertain nonlinear systems through parameter adaptation and data-driven model estimation.\nThe unification with a contraction-based adaptive controller allows the approach be applied to a broad class of systems.\nExtending to systems with probabilistic model bounds, non-parametric uncertainties, and external disturbances is future work.\n\n\\section*{Appendix}\n\\label{sec:appendix}\n\n\\begin{proof}[Proof of \\cref{thm:racbf}]\nConsider the composite candidate CBF $h = h_r(x,\\hat{\\theta}) - \\frac{1}{2} \\tilde{\\theta}^\\top\\Gamma^{-1}\\tilde{\\theta}$,\nwhere the minimum eigenvalue of $\\Gamma$ must satisfy $\\lambda_{\\min}(\\Gamma) \\geq \\frac{\\|\\tilde{\\vartheta}\\|^2}{2h_r(x_r,{\\theta}_r)}$ for any $h_r(x_r,\\theta_r)>0$.\nDifferentiating $h$ with respect to \\cref{eq:unc_dyn},\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} & = \\dot{h}_r(x,\\hat{\\theta}) -  \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}} \\\\\n    &= \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\theta + B(x) u\\right] + \\frac{\\partial h_r}{\\partial \\hat{\\theta}} \\dot{\\hat{\\theta}} - \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}}  \\\\\n   \n   \n\\end{aligned}\n\\end{equation*}\nAdding and subtracting $\\frac{\\partial h_r}{\\partial x}\\Delta(x)^\\top\\left[\\hat{\\theta} - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\right)^\\top \\right]$ and using the definition of $\\Lambda(x,\\hat{\\theta})$,\n\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} & = \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}) + B(x) u\\right] + \\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\dot{\\hat{\\theta}} \\\\\n    & \\hspace{1.5cm} - \\tilde{\\theta}^\\top\\Gamma^{-1}\\dot{\\hat{\\theta}} + \\frac{\\partial h_r}{\\partial x}\\Delta(x)^\\top\\left[\\tilde{\\theta} - \\Gamma \\left(\\frac{\\partial h_r}{\\partial \\hat{\\theta}}\\right)^\\top \\right].\n\\end{aligned}\n\\end{equation*}\nChoosing $\\dot{\\hat{\\theta}} = \\Gamma \\Delta(x) \\left(\\frac{\\partial h_r}{\\partial x}\\right)^\\top$, then\n\\begin{equation*}\n\\begin{aligned}\n    \\dot{h} &= \\frac{\\partial h_r}{\\partial x}\\left[ f(x) - \\Delta(x)^\\top \\Lambda(x,\\hat{\\theta}) + B(x) u\\right] \\\\\n    & \\geq - \\alpha\\left( h_r - \\frac{1}{2}\\tilde{\\vartheta}^\\top \\Gamma^{-1} \\tilde{\\vartheta}  \\right) \\geq -\\alpha(h),\n\\end{aligned}\n\\end{equation*}\nwhere the first inequality is obtained via the definition of a RaCBF and the second by noting $|\\tilde{\\theta}| \\leq \\tilde{\\vartheta}$ so $h = h_r - \\frac{1}{2}\\tilde{\\theta^\\top}\\Gamma^{-1}\\tilde{\\theta} \\geq h_r - \\frac{1}{2}\\tilde{\\vartheta^\\top}\\Gamma^{-1}\\tilde{\\vartheta}$. \nSince $h\\geq0$ and $h_r \\geq h$ $\\forall t$, then $h_r \\geq \\frac{1}{2}\\tilde{\\vartheta^\\top}\\Gamma^{-1}\\tilde{\\vartheta}  \\geq 0$ and $\\mathcal{C}^r_\\theta$ is forward invariant.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{thm:racbf_smid}]\nSince the model uncertainty is changing via estimation, the maximum allowable parameter error is time varying, i.e., $\\tilde{\\vartheta}(t)$.\nFrom \\cref{lemma:smid}, $\\tilde{\\Theta}$ monotonically decreases so $\\dot{\\tilde{\\vartheta}} \\leq 0$. \nLet $h_r$ be a candidate RaCBF, then $\\dot{{h}} = \\dot{h}_{r} - \\tilde{\\vartheta}\\Gamma^{-1}\\dot{\\tilde{\\vartheta}}  \\geq \\dot{h}_{r}$ since $\\dot{\\tilde{\\vartheta}} \\leq 0$ for all $t$.\nInequality \\cref{eq:racbf} in Definition~\\ref{definition:racbf} is then still satisfied for $\\dot{\\tilde{\\vartheta}} \\leq 0$.\nUsing the steps in \\cref{thm:racbf}, the system is safe with respect to $\\mathcal{C}^r_{\\hat{\\theta}}$.\nMoreover, since $\\dot{\\tilde{\\vartheta}} \\leq 0$ then $\\tilde{\\vartheta} \\rightarrow 0 $ so $\\mathcal{C}^r_{\\hat{\\theta}} \\rightarrow \\mathcal{C}_{\\hat{\\theta}}$.\n\\end{proof}\n\n\n\n\n\\noindent \\textbf{Acknowledgements}  \nWe thank David Fan for stimulating discussions.\nThis work was supported by the NSF Graduate Research Fellowship Grant No. 1122374.\n\n\n\\balance\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTwo-dimensional (2D) superconductivity in thin films and Josephson junction arrays (JJAs) reveal complex classical and quantum phase transitions and rich dynamics that depend on competing energy scales and coherence, including the extensively studied superconductor-insulator transition (SIT) \\cite{Jaeger1989, Fisher1991, Lee1990, Goldman2010, Gantmakher2010, Vladimir2012}, which fits into a framework of quantum critical phenomena exhibiting universal scaling \\cite{Mason1999, Yazdani1995, Steiner2005, Bollinger2011, Schneider2012}. New classes of materials have extended SIT studies to include strictly 2D materials \\cite{Allain2012,Tamir2019,Fatemi2018}, hybrid super-semi heterostructures \\cite{Boettcher2018,Vaitieknas2020,Tosato.2022,Boettcher.2022}, and high-temperature superconductors \\cite{Yang2019}, including a transition to a topological insulating phase \\cite{Wu2018}. An apparent metallic phase, with saturating low-temperature finite resistance, is observed in many of these systems \\cite{Kapitulnik2019}, which is incompatible with simple universal scaling.\n\nJJAs enrich the landscape by allowing controlled Coulomb interaction and frustration due to magnetic flux commensuration \\cite{Newrock1999, Fazio2001}. Coulomb charging of Josephson-coupled islands makes the classical XY spin system into a quantum problem, with phase and charge on the islands acting as conjugate variables with an uncertainty relation. Charging also introduces a new type of disorder in the form of a random offset on the island. The periodicity of the array adds complexity in the form of a Hofstadter-like spectrum \\cite{Teitel.1983, Lankhorst.2018b}, and possible spin glass phases \\cite{Vinokur.1987,Spivak.1991,Phillips.2003}. The discrete structure of a JJA also results in periodic pinning of vortices and antivortices, whose unbinding and free motion at finite temperature is described at zero magnetic field by a  Berzinskii-Kosterlitz-Thouless (BKT) phase transition \\cite{Halperin1979, Lobb.1983, Newrock1999, Fazio2001, Gantmakher2010}, as investigated recently in this system \\cite{Boettcher.2022}. \n\n Collective pinning of vortices resulting in a zero resistance state, which has been mapped onto Mott insulator of frozen vortices \\cite{Nelson1993}, was investigated experimentally \\cite{Poccia2015, Lankhorst2018, Mironov.2020, Rezvani.2020, Rezvani.2020b, Pei.2022}, showing good agreement with theory, including scaling \\cite{Granato.2018,Granato.2019}. In this picture, the zero-resistance state of the JJAs near nonzero integer and half integer flux quanta per plaquette, denoted frustration, $f$, is described as vortices pinned by a combination of the periodic array potential and, importantly, {\\it collective} pinning due to other vortices. Individual (noncollective) vortex pinning in the binding potential of the array has been thoroughly modelled in metallic JJAs \\cite{Lobb.1983, Rzchowski1990} and for intentional pinning \\cite{Berdiyorov.2005} and antipinning sites \\cite{Berdiyorov.2008}, patterned to prevent dissipation in superconducting films \\cite{Eley.2021}. We note that the Mott insulator in JJAs is closely related to the theoretical Mott insulator state at weak disorder in the 2D Bose Hubbard model. For stronger disorder the Bose Hubbard model shows a glass phase associated with individual vortex pinning \\cite{POLLET2013,Yao2014}.\n\nHere, we investigate a dynamic transition from the superconducting to the resistive state driven by an applied dc current, including scaling analysis of differential resistance at the transition. Previous experiments ~\\cite{Poccia2015, Lankhorst2018, Mironov.2020, Rezvani.2020, Rezvani.2020b, Pei.2022} and theory \\cite{Granato.2018,Granato.2019} interpreted the flux-dependent dynamic transition in terms of a dynamic Mott transition based on a scaling analysis that yielded exponents consistent with a Mott transition. For the gate-tuned semiconductor-based JJA investigated here, both the Josephson coupling, $E_J$ and the vortex pinning potential, $E_B$, are tuned by a gate voltage \\cite{Boettcher2018, Boettcher.2022}. \n\nExamining these transition as a dynamical phase transition, we find reasonable scaling, though with nonuniversal scaling exponents that differ from previous experiments \\cite{Poccia2015, Lankhorst2018, Pei.2022}. We also find different exponents on the low-field and high-field sides of the transitions. This is discussed in  (Sec.~\\ref{scaling}). We conclude that our transition is not a simple Mott transition in the same universality class as previously reported. This may be due to the geometry of our structure or weaker vortex pinning in the InAs\/Al system. We note that earlier studies of similar dynamic transitions were interpreted in terms of depinning of vortices from the array potential rather than as a Mott transition \\cite{Benz.1990, Jiang.2004}. \n\nBeyond scaling, several new features of dynamical vortex transitions are presented, elucidating the underlying vortex structure. First, the dip-to-peak transition from the fully superconducting state is {\\it split} at zero magnetic field, $f=0$, unlike the cases of $f=$~1\/2, 1, and other integer $f$. We interpret the split peak as reflecting absent or sparse vortices at $f=0$, unlike at nonzero $f$, consistent with a BKT picture at $f=0$ where vortex-antivortex pairs annihilate at low temperature. \n\nImportantly, we find that when the system is tuned (by gate voltage) into the anomalous metal phase \\cite{Boettcher2018}, an {\\it unsplit} peak at $f=0$ is instead observed. This suggests that remnant unpaired vortices and antivortices, absent in the low-temperature low-current superconducting state, are abundant at $f=0$ in the anomalous metal under otherwise similar conditions. The size of the splitting at $f=0$ in the superconducting state is found to depend on $I_{\\rm dc}$, reflecting the vortex density needed to support a dynamical transition. This is discussed in Sec.~\\ref{split}. \n\nSecond, the evolution of the dip-to-peak transition  with increasing $I_{\\rm dc}$ is found to be symmetric about $f=1\/2$ but highly asymmetric about $f=1$ and larger integers. The right-left symmetry around $f=1\/2$ follows from the underlying checkerboard vortex configuration \\cite{Franz.1995, Lankhorst.2018b}: excess vortices are attracted to unfilled sites while deficit vortices, equivalent to antivortices, are attracted to filled sites. The situation at $f=1$ is quite different and naturally asymmetric: the array is full, contains one vortex per site. Any excess vortex is repelled at each site while any deficit, or antivortex, is attracted to each site, where it can annihilate. Higher integer $f$ values mirror the asymmetry at $f=1$ about the intervening half-integer symmetry point. This is discussed in Sec.~\\ref{evenodd}. \n\n\\section{Hybrid Array and Dynamical Vortex Transition}\n\\label{device}\n\n\\begin{figure*}[t]\n    \\centering\n\t\\includegraphics[width= 6.4 in]{Figure1.pdf}\n\t\\caption{\\textbf{Hybrid array and frustration-dependent dynamical vortex transitions}. \\textbf{a,b)} Schematic of device with $W=100$ by $L=400$ epitaxial $a = 1\\, \\mu$m Al squares (gray), separated by $b=350$~nm strips (blue) of exposed InAs heterostructure, with ac + dc current bias, $I$, and voltage $V$ measured using side probes.  False-color micrograph taken before top gate was deposited. Gate voltage, $V_g$, controls carrier density in InAs strips between Al squares \\cite{Boettcher2018}. \\textbf{c)} Sheet resistance, $R_s \\equiv (W\/L) V\/I$, as a function of dc current, $I_{\\rm dc}$, and perpendicular magnetic field, $B_\\perp$, shows $R_{s}=0$ for small $I_{\\rm dc}$ with enhanced critical current, $I_{0}$ at $f=$~0, 1\/4, 1\/3, 1\/2, 2\/3, and 1. \\textbf{d)} Line cuts of \\textbf{c} shows dips in $R_s$ at  $f=$~0, 1\/2, and 1, going to $R_{s} \\sim 0$ for low $I_{\\rm dc}$. \\textbf{e-f)} Evolution of dips in differential resistance, $dV\/dI_{s} \\equiv (W\/L)\\ dV_{ac}\/dI_{\\rm ac}$, as a function of $B_{\\perp}$  into a split peak at $f=0$, a symmetric peak at $f=1\/2$, and an asymmetric peak at $f=1$. Each case is discussed in the text. }\n\t\\label{fig1}\n\\end{figure*}\n\nThe device we investigated, shown in Fig.~\\ref{fig1}, is based on an epitaxial Al\/InAs heterostructure patterned by wet etching to form a square array of $1 \\times 1\\: \\mu {\\rm m}^{2}$ islands separated by 350~nm strips of exposed semiconductor \\cite{Boettcher2018}. A Ti\/Au top gate, separated from the array by 40 nm of atomic-layer deposited HfO$_2$ insulator, was used to control the carrier density in the strips between islands. The gated Hall bar has four side-probes for voltage measurements and two wide contacts at the ends for applying current, and is $W = 100$ islands wide with $L = 400$ between voltage probes. The sample is measured in a dilution refrigerator with a base temperature of 30 mK. All lines are filtered using QDevil rf and low-pass filters. \n\n\n\nThe top gate voltage, $V_g$, controls the Josephson coupling, $E_J$, between islands, driving the system from a superconducting state at $V_g \\sim -3$~V to an insulating state at $V_g \\sim -4$~V \\cite{Boettcher2018}. The top gate also controls the barrier height, $E_B$, (energy saddle) between vortex pinning sites (energy minima) at the corners of the square islands. Numerical studies for metallic islands suggest $E_B \\propto E_J$ \\cite{Rzchowski1990}. Similar numerical studies have not been done for the semiconducting junctions. At intermediate gate voltages, an anomalous metallic phase was previously investigated, showing saturating gate-dependent sheet resistance at low temperature \\cite{Boettcher2018, Boettcher.2022}. In this work, $V_g$ is set to yield a superconducting state at base temperature in the absence of applied dc current and was only modified slightly toward the anomalous metal phase where indicated.\n\n We probe the device in the nonlinear regime by measuring both the ac and dc parts of the total voltage, $V$ in a pair of side probe contacts, when total current $I$, consisting of a varied dc part, $I_{\\rm dc}$ and ac part, $I_{\\rm ac}=5$~nA, was applied to the end contacts. A magnetic field, $B_\\perp$, was applied perpendicular to the plane of the array using an external solenoid controlled by a Keithley 2400 source-measurement unit. The area, $A=(a+b)^2$, of one plaquette of the array,  with $a=1\\,\\mu$m and $b=350$~nm, gives a characteristic magnetic field $\\Phi_0\/A = 1.4$~mT, where $\\Phi_0 = h\/2e$, corresponding to frustration $f=1$. Features associated with integer $f$ values are seen in Figs.~\\ref{fig1}(c,d).\n\nFor small dc currents ($I_{\\rm dc}\\lesssim 2\\, \\mu$A), large dips in both sheet resistance, $R_s \\equiv (W\/L)\\,V\/I$, and differential sheet resistance, $dV\/dI_s \\equiv (W\/L)\\,dV\/dI$, reach zero at $f=$~0, 1\/2, and 1, with moderate dips at $f=$~1\/4, 1\/3 and 2\/3, as seen in Fig.~\\ref{fig1}(e). Increasing $I_{\\rm dc}$ beyond an $f$-dependent critical value, $I_0(f)$, results in dip-to-peak transitions  in $dV\/I_s$ while minima in $R_s$ remain minima, consistent with previous experiments~\\cite{Poccia2015, Lankhorst2018, Mironov.2020, Rezvani.2020, Rezvani.2020b, Pei.2022}. The dip-to-peak transition in $dV\/dI_s$, visible at $f=1\/2$ and $f=1$, marks the onset of differential vortex motion without complete dissipative vortex flow. \n\n\\section{Zero-field transitions from \\\\ superconductor and anomalous metal}\n\\label{split}\n\nAt zero magnetic field, $f=0$, the dynamical transition appears split, so that $dV\/dI_s$ remains a minimum as a function of $B_\\perp$, unlike $f=1\/2$ and $f=1$, as seen in Figs.~\\ref{fig1}(e,f).  The absence of a dip-to-peak transition at $f=0$, unlike for $f=$~1\/2 and 1, is consistent with some previous results \\cite{Poccia2015, Pei.2022} but not others \\cite{Jiang.2004, Mironov.2020, Rezvani.2020b}, as discussed below.\n\n\nWe interpret the split peak as indicating that vortices are absent or sparse at $f=0$, consistent with a BKT picture in which unbound vortices pair and annihilate below a critical temperature, $T_{\\rm BKT}$. \nThis suggests that the zero-resistance state at $f=0$, where vortices are absent or sparse, is qualitatively different from the zero-resistance states at $f=1\/2$ and $f=1$, where vortices of a single sign are abundant but frozen. \n\nThe splitting of the transition around $f=0$ is found to increase at lower $I_{\\rm dc}$, as seen in Figs.~\\ref{fig1}(e,f). This is seen most clearly as the downward orientation of the bright features on either side of zero field in Fig.~\\ref{fig1}(e), indicating that $I_0$ decreases with increasing vortex density, a signature of the role of vortex interaction in the dynamical transition.\nThis dependence is similar the vortex-density ($\\propto f$) dependence of the vortex melting temperature \\cite{Obaidat.2008}.\n\nTuning the gate voltage from $V_g=-2.00$~V  to $V_g=-3.23$~V drives the system from the superconducting state, where $R_{s}$ falls below measurement resolution, $\\lesssim 0.1 \\Omega$, at low temperature, to the anomalous metal phase, where $R_s$ saturates at a gate-voltage dependent value, up to $\\sim h\/4e^{2}$, at low temperature \\cite{Boettcher2018, Boettcher.2022}. \n\nAs shown in Fig.~\\ref{fig2}, in the anomalous metal phase a dip-to peak-transition of $dV\/dI_s$ as a function of $I_{\\rm dc}$ {\\it is} observed at $f=0$, while $R_s$ remains a minimum, similar to the dynamical transition observed at  $f=1\/2$ and other integer $f$ values.\n\nThe {\\it unsplit} transition in the anomalous metal phase suggests that vortices and antivortices (in equal number) are present at $f=0$, unlike in the superconducting phase. The nonvanishing $R_s$, which characterizes the anomalous metal phase, further suggests that at least some of the remnant vortices and antivortices are mobile even at $I_{\\rm dc} < I_0$ and lowest temperatures. \n\nA picture of residual unpaired vortices at $f=0$ in the anomalous metal but not the superconductor is consistent with the observed vanishing of $T_{\\rm BKT}$ in the anomalous metal \\cite{Boettcher.2022}. One would anticipate that in the superconducting phase at higher temperatures, $T>T_{\\rm BKT}$, an unsplit transition would occur at $f=0$. Reported dynamical transitions at $f=0$ are either at elevated temperatures \\cite{Jiang.2004, Rezvani.2020b} or in the anomalous metal phase \\cite{Mironov.2020}, consistent with our observations.\n\n\n\\begin{figure}[t]\n\t\\includegraphics[width= 2.6 in]{Figure2.pdf}\n\t\t\\caption{\\textbf{Zero-field dynamical transition from the anomalous metal phase}. \\textbf{a)} Sheet resistance, $R_{s}$, as a function of perpendicular magnetic field, $B_\\perp$ over a range of dc currents, $I_{\\rm dc}$. For all dc currents, zero field remains a resistance minimum, similar to the superconducting regime, Fig.~\\ref{fig1}(d) \\textbf{b)} Dip-to-peak transition at $f=0$ over the same range of dc currents. This behavior contrasts the superconducting regime in Fig.~\\ref{fig1}(f), which shows a split peak at $B_\\perp = 0$. The dip-to-peak transition suggests that vortices and antivortices are absent at $f=0$ in the superconducting phase but present in the anomalous metal phase.\n\t\t\t}\n\t\\label{fig2}\n\\end{figure}\n\n\\section{Scaling and critical exponents}\n\\label{scaling}\n\nThe dip-to-peak transition at commensurate frustration can be described as melting of a frozen vortex lattice and has been analyzed as a dynamical Mott transition~\\cite{Nelson1993, Poccia2015, Lankhorst2018, Granato.2018, Granato.2019, Mironov.2020, Rezvani.2020, Rezvani.2020b, Pei.2022}. Within this interpretation, one expects scaling at the transition among relevant parameters controlling the transition, namely the dc current bias, $I_{\\rm dc}$, and the distance, $b \\equiv f-f_{c}$, from the commensurate frustration values, in this case $f_{c}=$~1\/2 or 1. \n\n\n\nFollowing Refs.~\\cite{Poccia2015, Lankhorst2018}, we introduce a scaling form for differential sheet resistance across the dip-to-peak transition, \n\\begin{align}\ndV(f, I)\/dI_s - dV(f, I)\/dI_s|_{I=I_{0}} = \\mathcal{F}(|I - I_0|\/|b|^{\\varepsilon}),\n\\end{align}\n\\noindent where $\\mathcal{F}(x)$ is a function of the scaled variable $ x \\equiv |I- I_0|\/|b|^{\\varepsilon}$ and $\\varepsilon$ is the scaling exponent. \n\nA scaling exponent $\\varepsilon \\sim 2\/3$ was measured at $f=1$ \\cite{Poccia2015, Lankhorst2018} and $f=2$ \\cite{Poccia2015}, in a square array of Nb islands on Au, consistent with theory for a dynamical Mott transition. At $f=1\/2$ in the same system, the value $\\varepsilon \\sim 1\/2$ was found experimentally \\cite{Poccia2015, Lankhorst2018}, and later confirmed numerically ~\\cite{Granato.2018, Granato.2019}. Other experiments investigating the dynamic transition at commensurate $f$ found different exponents~\\cite{Pei.2022} in the triangular lattice, or did not pursue scaling analysis~\\cite{Mironov.2020, Rezvani.2020, Rezvani.2020b}. \n\nThe dip-to-peak transition at $f=1\/2$ was found to be right-left symmetric, while the transition at $f=1$ was asymmetric, as discussed in Sec.~\\ref{evenodd}. A consequence of the asymmetry is that the value of the commensurate frustration, $f_{c}$, depended on $I_{\\rm dc}$ [dots in Figs.~\\ref{fig3}(a,b)]. \n The asymmetry at $f=1$ also yielded different separatrices on the left and right, $I_0^L = 2.5\\: \\mu\\text{A}$, $I_0^R=2.35 \\: \\mu\\text{A}$, while a single separatrix value was found for $f=1\/2$ (solid and dashed curves in Figs.~\\ref{fig3}(a,b). Similar asymmetries are also found at $f=$~2, 3 and 4, as seen in Fig.~\\ref{fig4}.  \n \n \\begin{figure*}\n\t\\includegraphics[width= 6\n\tin]{Figure3.pdf}\n\t\t\\caption{\\textbf{Dynamical transitions, scaling, and critical exponents}. \\textbf{a-b)} Differential sheet resistance, $dV\/dI_s$, shows a dip-to-peak transition is a function of dc current, $I_{\\rm dc}$, at commensurate frustration. ($f=1\/2$) and full ($f=1$). Insets: the transitions on each side, denoted as the left and right branches of $f=1\/2$ and $f=1$, are represented as superconductor-insulator transitions: down-bending curves are transitions to the pinned vortex state, while upward-bending curves are transitions to a state of vortex flow. The horizontal field-independent curves separating each state are marked as separatrices used in the scaling analysis (Left: black solid line. Right: black dotted line). Scaling plots of $f=1\/2$ (\\textbf{c-d}) and $f=1$ (\\textbf{e-f}), showing that left and right branches yield different scaling exponents, i.e. there is an asymmetry around each critical frustration field. Exponents extracted for $f=1\/2$ are $\\epsilon =2.1$ (left) and 1.2 (right), while at $f=1$ we extract $\\epsilon = 1.5$ (left) and 2.5 (right).\n\t\t\t}\n\t\\label{fig3}\n\\end{figure*}\n\nScaling exponents $\\varepsilon$ were obtained by fitting the slope on a log-log plot of differential resistance after subtracting the separatrix curve, $\\frac{d}{dI}[dV\/dI_{s}-dV\/dI_{s}|_{I_0^{L\/R}}]$ versus $1\/b$ (see Methods). Scaled data collapse reasonably well, as seen in Figs.~\\ref{fig3}(c-f), but yield different exponents on the left and right sides of the peaks, $\\varepsilon = 2.1$ (left) and $\\varepsilon =1.2$ (right) for $f=1\/2$, and $\\varepsilon = 1.5$ (left) and $\\varepsilon =2.5$ (right)  for $f=1$, inconsistent with \\cite{Poccia2015, Lankhorst2018}.\n\n\nWe speculate that different values for $\\varepsilon$ could be caused by stronger pinning potential, $E_{B}$, in the Nb devices in \\cite{Poccia2015,Lankhorst2018} compared to the Al array studied here, leading to a different regime where dynamics may be influenced by competing effects associated with binding potential and vortex-vortex interactions. A model of vortex pinning in metallic arrays yielded, $E_B = 0.2\\,E_J$, where $E_J=(\\hbar\/2e)\\,i_c$ is the Josephson coupling, and $i_c$ is the single-island critical current \\cite{Rzchowski1990}. Taking $i_c\\sim 5 \\,\\mu$A from \\cite{Poccia2015} compared to $i_{c}\\sim 0.5\\,\\mu$A in our Al array suggests roughly an order of magnitude difference in $E_B$.\n\n\n\\begin{figure}\n\t\\includegraphics[width= 2.6 in]{Figure4.pdf}\n\t\t\\caption{\\textbf{Even, odd and zero flux states}. \\textbf{a)} Differential sheet resistance curves for different values of dc current as a function of perpendicular magnetic field, $B_\\perp$, at gate voltage $V_g=-2.8$~V, showing the evolution of vortex states for half-integer frustration $f=1\/2$ and integers $f=1,2,3$ and 4. Odd minima (black dots) move down in field with increasing dc current, while even minima (green dots) move up in field. Minima at $f=1\/2$ is roughly insenstive to dc current, consistent with the overall symmetry in $f$ of the $f=1\/2$ transition. \\textbf{b)} Differential sheet resistance (color) as a function of $B_\\perp$ and $I_{\\rm dc}$ shows commensurate features at factional and integer frustration, $f$. Bright features at the tips of the zero-resistance spikes are the dip-to-peak signature of the dynamical transition. Note the alternating, even-odd asymmetric bright features at integer $f$ values, the symmetric bright feature at $f=\\pm 1\/2$, and the vanishing differential sheet resistance remaining at large $I_{\\rm dc}$ at $f=0$.\n\t\t\t}\n\t\\label{fig4}\n\\end{figure}\n\n\\section{Skewed transitions and Even-odd structure}\n\\label{evenodd}\nFigures \\ref{fig1}(e) and \\ref{fig2} reveal a striking difference between the dip-to-peak transitions at $f=1\/2$ and $f=1$. Throughout the transition at $f=1\/2$, each curve, representing a different $I_{\\rm dc}$ value, is symmetric in $f$ about the transition point.  In contrast, for the $f=1$ transition, the curves are highly asymmetric, and in fact in the crossover from dip to peak are nearly antisymmetric lying above the high-current saturation on the right side and below it on the left side. This difference can also be seen in Fig.~\\ref{fig1}(e). The bright feature at the top of the $f=1\/2$ peak, corresponding to the peak in $dV(f)\/dI_{s}$, is flat, while the bright feature at $f = 1$ is tilted, indicating that the peak in $dV(f)\/dI_{s}$ occurs at lower $I_{\\rm dc}$ at higher $f$. \n\nSymmetry in $f$ around $f=1\/2$ follows from the presumed checkerboard vortex configuration near $f=1\/2$ \\cite{Teitel.1983}, which is symmetric with respect to the addition or subtraction of vortices. Excess or deficit vortices slightly above or below $f=1\/2$ are expected to form a low-density superlattice on top of a base checkerboard \\cite{Franz.1995, Lankhorst.2018b}. The symmetry between excess and deficit vortices should persist for weak disorder, which can then pin the superlattice. In contrast, $f=1$ is not symmetric with respect to the addition or subtraction of dilute vortices. Excess vortices above $f=1$ are repelled by each site in the full lattice, while deficit vortices, or antivortices, are attracted to each site, where they can annihilate with a vortex. Within this picture, excess vortices should be weakly pinned and contribute to low-current melting \\cite{Franz.1995}, while vacancies resulting from annihilation with deficit vortices should more readily pin, lowering vortex mobility. The asymmetry between excess and deficit at $f=1$ is a signature of vortex interaction. \n\nLooking at integer transitions above $f=1$, an even-odd structure is evident in Fig.~\\ref{fig4}, where the $f=3$ transition is skewed in the same direction as $f=1$, while transitions at $f=2$ and $f=4$ are skewed in the opposite direction. Even-odd behavior is also visible below the transition, in the current-dependent position of the minima of $dV\/dI_{s}$, marked as dots in Fig.~\\ref{fig4}(a). Some understanding of this structure follows from the arguments above, that half-filling of the array is a symmetry point, so one might expect $f=2$ to be show a reflection symmetry of the $f=1$ behavior about $f=3\/2$, continuing by reflection about half-integers to higher integers.  \n\nWhile the overall even-odd pattern at integer $f$ presumably reflects the square potential of the array, the difference between $f=1\/2$ (symmetric) and $f=1$ (asymmetric) is also seen in triangular arrays \\cite{Pei.2022}, and reflects a more basic difference between half and full filling. Even-odd structure of dynamical transition in triangular lattices has not been reported. It will be interesting to investigate vortex filling in various lattices experimentally and numerically.\n\n\n\\section{Discussion}\n\\label{discussion}\n\nWe have investigated a dynamical transition from frozen to mobile vortices in a gate-tunable superconductor-semiconductor Josephson junction array. Tuning the gate into the superconducting phase, where a zero-resistance state is observed at low temperature and current, we see dip-to-peak transitions in differential resistance near frustration $f=$~1\/2, 1, 2, 3, and 4, similar to previous studies in metallic arrays. Motivated by the mapping of this transition to a Mott melting transition of frozen vortices, found good scaling at the transitions but not the Mott exponent found previously, perhaps due to a weaker binding potential in our Al arrays. \n\nThe split transition at $f=0$ in the superconducting phase suggests that vortices are absent, not frozen, at $f=0$, consistent with a BKT model in which vortices and antivortices annihilate at $f=0$ below a critical temperature. When the array is tuned to the anomalous metal phase, a simple unsplit transition is observed at $f=0$ suggesting that frozen and perhaps some unfrozen vortices are present. These observations are consistent with previously reported experiments.\n\nThe transition at $f=1\/2$ is symmetric in $f$ around the transition, reflecting the symmetry of the underlying half-filled checkerboard lattice. At $f = 1$, on the other hand, the transition is strongly asymmetric, suggesting that excess vortices on top of an underlying full lattice melt easily, but deficit vortices (antivortices) do not.  We find that the asymmetry of the $f=1$ transition persists to higher integers, mirrored about half integers, giving an overall even-odd pattern to transitions at higher integers. Further work is needed to understand this asymmetry, and how it depends on island and lattice geometry. This could also depend on the energetic of vortex configurations \\cite{Berdiyorov.2005, Berdiyorov.2008} and binding multiple or giant vortices in the potential minima \\cite{Baelus2002,Baelus2006}.\n\nWe emphasize an general connection between these highly tunable Josephson arrays and Bose-Hubbard systems \\cite{Bruder.2005, Arovas.2021}. In these arrays, complexity is controlled by frustration and quantum by charging energy of the islands, relevant as the superconductor-insulator transition is approached. \n\n\\emph{Acknowledgments} \nWe thank A. Kapitulnik, S. Kivelson, B. Spivak, and V. Vinokur for useful discussions. \nResearch supported by Microsoft Station Q, the Danish National Research Foundation, and a research grant (Project 43951) from VILLUM FONDEN.\n\n\\section*{Methods}\n\\label{methods}\n\\subsection*{Extracting scaling exponents}\nThis subsection provides details of the scaling analysis shown in Fig.~\\ref{fig3}. The scaling exponent, $\\varepsilon$, defined in Eq.~1 was  extracted from the slope of a log-log plot of $d\/dI(dV\/dI_s-dV\/dI_s|_{I=I_0})$ versus $1\/b$, allowing separate left and right critical currents, $I^L_0$ and $I^R_0$.  \n\nFigure~\\ref{figS} shows the spread of measured values. For each value of $b$ a mean of all data was calculated. A curve through the means is show as a solid black curve. Then, a line with offset was fit to the means. The linear fit is shown as a green dashed line. The slope of the linear fit yields $b$. The process is repeated for each value of $f$ and on the left and right sides for $f=1\/2$ and $f=1$.\nThe value of $f$ extracted from the fit to $b$ takes into account the dependence of $f_c$ on $I_{\\rm dc}$. Values of $f_c(I_{\\rm dc})$ are shown as green and back dots in Fig.~\\ref{fig4} \n\nWe note that only points near the separatrix (above and below) were included in the analysis of extracting the scaling exponent for the right branch of $f_c=1$ due to the large asymmetry, see Fig.~\\ref{figS}(d). \n\n\\setcounter{figure}{0}\n\\makeatletter \n\\renewcommand{\\thefigure}{S\\@arabic\\c@figure}\n\\makeatother\n\\begin{figure}[t]\n\t\\includegraphics[width= 3.4 in]{supplemental.pdf}\n\t\t\\caption{\\textbf{Extracting scaling exponents}. \\textbf{a-d)} Log-log plots of the slope of the differential resistance, with the separatrix subtracted, constructed to extract of scaling exponents, $\\varepsilon$, separately for left and right branches around $f=1\/2$ and $f=1$. Means of logs are calculated for each value of $b$ (solid black curve) then means fit to a linear function (dashed green line). }\n\t\\label{figS}\n\\end{figure}\n\n\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\vspace{1mm}\n\\noindent\n\nThe spin properties of hadrons inclusively produced in high energy \ninteractions are related to the fundamental properties of quarks and \ngluons and to their elementary interactions in a much more subtle way \nthan unpolarized quantities. They test unusual basic dynamical \nproperties and reveal how the usual  models for quark distribution and\nhadronization -- successful in predicting unpolarized cross-sections -- may \nnot be adequate to describe spin effects. We consider here such two cases,\nsingle spin asymmetries in Deep Inelastic Scattering and the polarization\nof mesons produced in the fragmentation of polarized quarks at LEP.\n\n\\section{Single spin asymmetries in DIS}\n\n\\vspace{1mm}\n\\noindent\n\nWe discuss first single spin asymmetries in DIS processes. \nWe start by reminding that single spin asymmetries in large $p_{_T}$ inclusive \nhadronic reactions are forbidden in leading-twist perturbative QCD, \nreflecting the fact that single spin asymmetries are zero at the partonic \nlevel and that collinear parton configurations inside hadrons do not allow \nsingle spin dependences. However, experiments tell us in several cases, \n\\cite{ada1,ada2} that single spin asymmetries are large and indeed non \nnegligible.\n\nThe usual arguments to explain this apparent disagreement between pQCD \nand experiment invoke the moderate $p_{_T}$ values of the data -- a few \nGeV, not quite yet in the true perturbative regime -- and the importance \nof higher-twist effects. Several phenomenological models have recently\nattempted to explain the large single spin asymmetries observed in\n$p^\\uparrow p \\to \\pi X$ as twist-3 effects which might be due to intrinsic \npartonic $\\mbox{\\boldmath $k$}_\\perp$ in the fragmentation \\cite{col1} and\/or distribution \nfunctions \\cite{siv1}-\\cite{noi}. \n\nLet us consider a process in which one has convincing evidence \nthat partons and perturbative QCD work well and successfully describe the \nunpolarized and leading-twist spin data, namely Deep Inelastic Scattering \n(DIS). In particular we shall discuss single spin asymmetries in the inclusive,\n$\\ell N^\\uparrow \\to \\ell + jets$ and $\\ell N^\\uparrow \\to hX$, \nreactions looking at possible origins of such asymmetries and \ndevising strategies to isolate and discriminate among them \\cite{alm}.\n\nAccording to the QCD hard scattering picture and the factorization theorem \n\\cite{col1,col2,col3} the cross-section for the $\\ell N^\\uparrow \\to hX$ \nreaction is given by\n\\begin{eqnarray}\n& & \\frac{E_h \\, d^3\\sigma^{\\ell + N,S \\to h + X}} {d^{3} \\mbox{\\boldmath $p$}_h} = \n\\sum_{q; \\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}, \\lambda^{\\,}_h} \n\\int \\frac {dx \\, d^2\\mbox{\\boldmath $k$}_\\perp d^2\\mbox{\\boldmath $k$}_\\perp^\\prime} {\\pi z} \\label{gen} \\\\  \n& & \\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\n\\frac{d\\hat\\sigma^{q,P_q}}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\>\n\\rho^{q^{\\prime}}_{\\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}}\n(x, \\mbox{\\boldmath $k$}_\\perp, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\> \\widetilde \nD_{\\lambda^{\\,}_h, \\lambda^{\\,}_h}^{\\lambda^{\\,}_{q^{\\prime}}, \\lambda^{\\prime}_{q^{\\prime}}}\n(z, \\mbox{\\boldmath $k$}_\\perp^\\prime) \\,. \\nonumber\n\\end{eqnarray}\n\nLet us briefly discuss the different quantities appearing in the above \nequation; more details can be found in Ref. \\cite{alm}.\n$\\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp)$ is the quark distribution function,\nthat is the total number density of quarks $q$  with momentum fraction \n$x$ and intrinsic transverse momentum $\\mbox{\\boldmath $k$}_\\perp$ inside a polarized nucleon\n$N$ with spin four-vector $S$. \n\n$d\\hat\\sigma^{q,P_q} \/ d\\hat t$ is the cross-section for the $\\ell \nq^\\uparrow \\to \\ell q$ process, with an unpolarized lepton and an initial \nquark with polarization $P_q$, while the final quark and lepton polarization\nare summed over. Notice that for helicity conserving elementary \ninteractions $d\\hat\\sigma^{q,P_q} \/ d\\hat t$ equals the unpolarized \ncross-section $d\\hat\\sigma^q \/ d\\hat t$.\n$\\rho^{q^\\prime}$ is the helicity density matrix of the final quark \nproduced in the $\\ell q^\\uparrow$ interaction and \n\\begin{equation}\n\\widetilde D^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) = \\sum_{\\lambda^{\\,}_q, \\lambda^{\\prime}_q} \n\\rho^q_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} \\> \\widetilde D_{\\lambda^{\\,}_h, \n\\lambda^{\\,}_h}^{\\lambda^{\\,}_q, \\lambda^{\\prime}_q}(z, \\mbox{\\boldmath $k$}_\\perp)\n\\end{equation}\ndescribes the fragmentation process of a polarized quark $q$ \nwith spin $s_q$ into a hadron $h$ with helicity $\\lambda^{\\,}_h$, \nmomentum fraction $z$ and intrinsic transverse momentum $\\mbox{\\boldmath $k$}_\\perp$ with \nrespect to the jet axis. It is simply the inclusive cross-section for\nthe $q^\\uparrow \\to hX$ fragmentation process. As it will be shown in the \nsecond part we can safely neglect the coherent interactions of the \nfragmenting quark, as we are not looking at the spin state of the final \nhadron. The usual unpolarized fragmentation function is given by\n\\begin{equation}\nD^h_q(z) = {1\\over 2} \\sum_{\\lambda^{\\,}_q, \\lambda^{\\,}_h}\n\\int d^2\\mbox{\\boldmath $k$}_\\perp \\>\n\\widetilde D^{\\lambda^{\\,}_q,\\lambda^{\\,}_q}_{\\lambda^{\\,}_h,\\lambda^{\\,}_h}\n(z, \\mbox{\\boldmath $k$}_\\perp) \\,.\n\\label{fr}\n\\end{equation}\n\nSimilar formulae hold also when the elementary interaction\nis $\\ell q \\to \\ell q g$ rather than $\\ell q \\to \\ell q$: in the latter \ncase two jets are observed in the final state -- the target jet and the \ncurrent quark jet -- and in the former case three -- the target jet and \n$q$ + $g$ current jets.\n\nIn Eq. (\\ref{gen}) we have taken into account intrinsic transverse momenta \nboth in the distribution and the fragmentation process;\nthe $\\mbox{\\boldmath $k$}_\\perp$ dependences are expected to have negligible effects on \nunpolarized variables for which they are indeed usually neglected, but \nthey can be of crucial importance for some single spin observables, as \ndiscussed in Refs. \\cite{col1}-\\cite{noi}.\n\nWe discuss now possible sources of single spin effects in Eq. (\\ref{gen}).\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $k_\\perp$}} {\\bf effects in fragmentation process}\n\\cite{col1}\n\\vskip 6pt\n\\nobreak\n \nThe fragmentation process of a transversely polarized quark into a hadron \n$h$ (whose polarization in not observed) with fixed $z$ and $\\mbox{\\boldmath $k$}_\\perp$ \nmay depend on the quark spin orientation, provided the quark spin \n$s_q$ has a component perpendicular to the hadron-quark plane (otherwise \nany spin dependence would be forbidden by parity conservation). That is, \nthere might be a non zero {\\it quark analysing power} \\cite{col1}:\n\\begin{equation}\nA_q^h \\equiv {\\widetilde D^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) \n- \\widetilde D^h_{q,-s_q}(z, \\mbox{\\boldmath $k$}_\\perp) \\over \\widetilde \nD^h_{q, s_q}(z, \\mbox{\\boldmath $k$}_\\perp) + \\widetilde D^h_{q,-s_q}(z,\\mbox{\\boldmath $k$}_\\perp)} \\,\\cdot\n\\label{qap}\n\\end{equation}\nBy rotational invariance $\\widetilde D^h_{q,-s_q}(z,\\mbox{\\boldmath $k$}_\\perp) =\n\\widetilde D^h_{q, s_q}(z,-\\mbox{\\boldmath $k$}_\\perp)$, which shows immediately how the \nquark analysing power vanishes for $\\mbox{\\boldmath $k$}_\\perp = 0$. \n \n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $k_\\perp$}} {\\bf effects in distribution functions}\n\\cite{siv1}-\\cite{noi}\n\\vskip 6pt\n\\nobreak\n\nA similar idea had been previously proposed in Refs. \\cite{siv1, siv2}\nand later rediscovered in Ref. \\cite{noi}, concerning the distribution\nfunctions: that is, the number of quarks (whose spin is not observed)\nwith fixed $x$ and $\\mbox{\\boldmath $k$}_\\perp$ inside a transversely polarized nucleon\nmay depend on the nucleon spin orientation and the function\n\\begin{equation}\n\\Delta\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) \\equiv\n\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N^\\downarrow}(x, \\mbox{\\boldmath $k$}_\\perp)\n= \\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N,^\\uparrow}(x,-\\mbox{\\boldmath $k$}_\\perp)\n\\label{nap}\n\\end{equation}\nmay be different from zero. $\\uparrow$ and $\\downarrow$ refer to the \nnucleon spin up or down with respect to the quark-nucleon plane. \n\nIn terms of the usual light-cone operator definition of structure \nfunctions one has \\cite{col1,dra}\n\\begin{eqnarray}\n\\Delta\\tilde f_{q\/N}^{N^\\uparrow}\n&=& 2 \\, {\\rm Im} \\int {dy^- d\\mbox{\\boldmath $y$}_\\perp \\over(2 \\pi)^3}\ne^{-i x p^+ y^- + i \\mbox{\\boldmath $k$}_\\perp\\cdot\\mbox{\\boldmath $y$}_\\perp} \\nonumber\\\\\n&&\\quad\\langle p, -|\\bar\\psi_a(0,y^-,y_\\perp)\n{\\gamma^+\\over 2}\\psi_a(0)|p, + \\rangle \\,. \n\\label{del+-}\n\\end{eqnarray}\n\nIn Ref. \\cite{col1} it is argued that such off-diagonal (in the helicity\nbasis) matrix elements are zero due to the time-reversal invariance of QCD, \nand indeed this is proven by exploiting the time-reversal and parity \ntransformation properties of free Dirac spinors. However, in Ref. \\cite{dra} \nit has been shown that this need not be so in chiral models with quark moving \nin a background of chiral fields.\n\nBoth $\\widetilde D^h_{q^\\uparrow}(z, \\mbox{\\boldmath $k$}_\\perp) \n- \\widetilde D^h_{q^\\downarrow}(z, \\mbox{\\boldmath $k$}_\\perp)$ and \n$\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) - \n\\tilde f_{q\/N}^{N^\\downarrow}(x, \\mbox{\\boldmath $k$}_\\perp)$ can be considered as \nnew fundamental spin and $\\mbox{\\boldmath $k$}_\\perp$ dependent non perturbative \nfunctions describing respectively quark fragmentation and distribution\nproperties. In the sequel we shall devise strategies to test their\nrelevance. \n\n\\vskip 12pt\n\\noindent\n{\\bf Single spin effects in the elementary interactions} \n\\vskip 6pt\n\nAs we already mentioned both perturbative QED and QCD at high energy \ndo not allow single helicity flips in the $\\ell q$ interactions, so that \nthere cannot be any dependence on the quark polarization in \n$d\\hat\\sigma^{q,P_q}\/d\\hat t$. Similarly, the perturbative QCD evolution\nof the distribution and fragmentation functions is not expected to\nintroduce any single spin dependence. We must conclude that the hard \nelementary interactions are unlikely to introduce any single\nspin effect: however, this basic QED and QCD property should also\nbe tested. \n\n\\vskip 6pt\nLet us now describe a set of possible measurements which could single out\nsome of the above mechanisms and test them. \n\n\\goodbreak\n\\vskip 6pt\n\\noindent\n$a) \\> \\ell N^\\uparrow \\to \\ell + 2\\,jets$\n\\vskip 4pt\n\\nobreak\n\nHere one avoids any fragmentation effect by looking at the fully \ninclusive cross-section for the process $\\ell N^\\uparrow \\to \\ell + 2\\, jets$,\nthe 2 jets being the target and current ones; this is the usual DIS, \nand Eq. (\\ref{gen}) becomes  \n\\begin{equation}\n\\frac{d^2\\sigma^{\\ell + N,S \\to \\ell + X}} {dx \\, dQ^2} = \\sum_q\n\\int d^2\\mbox{\\boldmath $k$}_\\perp \\> \\tilde f_{q\/N}^{N,S}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\n\\frac{d\\hat\\sigma^{q,P_q}}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp) \\,. \n\\label{gena}\n\\end{equation}\n\nIn this case the elementary interaction is a pure QED, \nhelicity conserving one, $\\ell q \\to \\ell q$, and $d\\hat\\sigma^{q,P_q}\/d\\hat t$\ncannot depend on the quark polarization. Some spin dependence might only\nremain in the distribution function, due to intrinsic $\\mbox{\\boldmath $k$}_\\perp$ effects\n\\cite{siv1}-\\cite{noi}, \\cite{dra} and we have\n\\begin{eqnarray}\n\\frac{d^2\\sigma^{\\ell N^\\uparrow \\to \\ell + X}} {dx \\, dQ^2} \n&-& \\frac{d^2\\sigma^{\\ell N^\\downarrow \\to \\ell + X}} {dx \\, dQ^2} \n= \\sum_q \\int d^2\\mbox{\\boldmath $k$}_\\perp \\nonumber \\\\\n&\\times& \\Delta\\tilde f_{q\/N}^{N^\\uparrow}(x, \\mbox{\\boldmath $k$}_\\perp) \\>\\>\n\\frac{d\\hat\\sigma^q}{d\\hat t}(x, \\mbox{\\boldmath $k$}_\\perp) \\,. \n\\label{asymg}\n\\end{eqnarray}\nDespite the fact that $\\Delta\\tilde f_{q\/N}^{N^\\uparrow}$ is an odd\nfunction of $\\mbox{\\boldmath $k$}_\\perp$ a non zero value of the above difference \n-- of ${\\cal O}(k_\\perp\/\\sqrt {Q^2})$, twist 3 -- \nmight remain even after integration on $d^2\\mbox{\\boldmath $k$}_\\perp$ because of\nthe $\\mbox{\\boldmath $k$}_\\perp$ dependence of $d\\hat\\sigma^q\/d\\hat t$, similarly \nto what happens in $pp^\\uparrow \\to \\pi X$ \\cite{noi}. The observation\nof a non vanishing value of the single spin effect of Eq. (\\ref{asymg})\nwould be a decisive test in favour of the mechanism suggested in \nRefs. \\cite{siv1}-\\cite{noi} and would allow an estimate of the\nnew function (\\ref{nap}).\n\n\\vskip 6pt\n\\noindent\n$b) \\> \\ell N^\\uparrow \\to h + X \\> (2\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp \\not= 0)$\n\\vskip 4pt\n\nOne looks for a hadron $h$, with transverse momentum $\\mbox{\\boldmath $k$}_\\perp$, inside \nthe quark current jet; the final lepton may or may not be observed.\nThe elementary subprocess is $\\ell q \\to \\ell q$ and Eq. (\\ref{gen}) yields\n\\begin{eqnarray}\n& & \\frac{E_h \\, d^5\\sigma^{\\ell + N^\\uparrow \\to h + X}} \n{d^{3} \\mbox{\\boldmath $p$}_h d^2 \\mbox{\\boldmath $k$}_\\perp} \n- \\frac{E_h \\, d^5\\sigma^{\\ell + N^\\downarrow \\to h + X}} \n{d^{3} \\mbox{\\boldmath $p$}_h d^2 \\mbox{\\boldmath $k$}_\\perp} \\label{coll} \\\\\n&=& \\sum_q \\int \\frac {dx} {\\pi z} \\>    \n\\Delta_{_T} q(x) \\> \\Delta_{_N} \\hat\\sigma^q (x, \\mbox{\\boldmath $k$}_\\perp) \\,\n\\left[ \\tilde D^h_{q^\\uparrow}(z, \\mbox{\\boldmath $k$}_\\perp)\n- \\tilde D^h_{q^\\uparrow}(z, - \\mbox{\\boldmath $k$}_\\perp) \\right]\n\\nonumber\n\\end{eqnarray}\nwhere $\\Delta_{_T}q$ is the polarized number density for transversely spinning \nquarks $q$ and $\\Delta_{_N} \\hat\\sigma^q$ is the elementary cross-section \ndouble spin asymmetry\n\\begin{equation}\n\\Delta_{_N} \\hat\\sigma^q = {d\\hat \\sigma^{\\ell q^\\uparrow \\to \n\\ell q^\\uparrow} \\over d\\hat t} - {d\\hat \\sigma^{\\ell q^\\uparrow \\to \n\\ell q^\\downarrow} \\over d\\hat t} \\,\\cdot\n\\label{del}\n\\end{equation}\n\nIn Eq. (\\ref{coll}) we have neglected the $\\mbox{\\boldmath $k$}_\\perp$ effect in the \ndistribution function, which can be done once the asymmetry discussed\nin $a)$ turns out to be negligible. We are then testing directly the \nmechanism suggested in Ref. \\cite{col1} and a non zero value of\nthe l.h.s. of Eq. (\\ref{coll}) would be a decisive test in its favour \nand would allow an estimate of the new function appearing in the\nnumerator of Eq. (\\ref{qap}). Notice again that even upon integration over\n$d^2\\mbox{\\boldmath $k$}_\\perp$ the spin asymmetry of Eq. (\\ref{coll}) might survive,\nat higher twist order $k_\\perp\/p_{_T}$, due to some $\\mbox{\\boldmath $k$}_\\perp$ dependence \nin $\\Delta_{_N} \\hat\\sigma^q$.\n\n\\goodbreak\n\\vskip 6pt\n\\noindent\n$c) \\> \\ell N^\\uparrow \\to h + X \\> (2\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp = 0)$\n\\vskip 4pt\n\\nobreak\n\nBy selecting events with the final hadron collinear to the jet axis\n($\\mbox{\\boldmath $k$}_\\perp = 0$) one forbids any single spin effect in the fragmentation\nprocess. As in the fully inclusive case $a)$ the observation of a\nsingle spin asymmetry in such a case would imply single, $\\mbox{\\boldmath $k$}_\\perp$\ndependent, spin effects in the distribution functions. \n\n\\vskip 6pt\n\\noindent\n$d) \\> \\ell N^\\uparrow \\to h + X \\> (3\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp \\not= 0)$\n\\vskip 4pt\n \nThe elementary process is now $\\ell q \\to \\ell qg$ and one looks at \nhadrons with $\\mbox{\\boldmath $k$}_\\perp \\not= 0$ inside the $q$ current jet. Single\nspin asymmetries can originate either from single spin effects in the \nfragmentation process or distribution functions. One should not, in \nprinciple, forget possible spin effects in the elementary QCD interaction.\n\n\\vskip 6pt\n\\noindent\n$e) \\> \\ell N^\\uparrow \\to \\ell + 3\\,jets$ or \n$\\ell N^\\uparrow \\to h + X \\> (3\\,jets, \\> \\mbox{\\boldmath $k$}_\\perp = 0)$\n\\vskip 4pt\n\nThese cases are analogous to $a)$ and $c)$ respectively: the measurement \neliminates spin effects arising from the fragmentation functions. \nThe only possible origin of a single spin asymmetry would reside \nin the distribution function. However, if no effect is observed in cases\n$a)$ and $c)$, but some effect is observed here, then one has to \nconclude that there must be some single spin effect in the elementary \nQCD interaction. Utterly unexpectedly, this would question the\nvalidity of quark helicity conservation, a fundamental property of\npQCD which has never been directly tested. \n\n\\vskip 6pt\nIn summary, a study of single transverse spin asymmetries in DIS could\nprovide a series of profound tests of our understanding of large $p_{_T}$\nQCD-controlled reactions.\n\n\\section{{\\mbox{\\boldmath $\\rho^{\\,}_{1,-1}(V)$}} in the process \n{\\mbox{\\boldmath $e^- e^+ \\to q\\bar q \\to V + X$}}}\n\n\\vspace{1mm}\n\\noindent\n\nWe consider now the spin properties of hadrons produced at LEP. It was pointed \nout in Refs. \\cite{akp} and \\cite{aamr} that final state interactions \nbetween the $q$ and $\\bar q$ created in $e^+ e^-$ annihilations \n-- usually neglected, but indeed necessary -- might give origin to non \nzero spin observables which would otherwise be forced to vanish: \nthe off-diagonal element $\\rho^{\\,}_{1,-1}$ of the helicity density matrix of \nvector mesons may be sizeably different from zero \\cite{akp} due to a \ncoherent fragmentation process which takes into account $q \\bar q$ \ninteractions. The incoherent fragmentation of a single independent quark \nleads instead to zero values for such off-diagonal elements. \n\nWe present predictions \\cite{abmq} for $\\rho^{\\,}_{1,-1}$ of several vector \nmesons $V$ provided they are produced in two jet events, carry\na large momentum or energy fraction $z=2E_{_V}\/\\sqrt s$, and have a small\ntransverse momentum $p_{_T}$ inside the jet. Our estimates are in agreement\nwith the existing data and are crucially related both to the presence \nof final state interactions and to the Standard Model couplings of the \nelementary $e^- e^+ \\to q \\bar q$ interaction. \n\nThe helicity density matrix of a hadron $h$ inclusively produced in the \ntwo jet event $e^- e^+ \\to q\\bar q \\to h + X$ can be written as \n\\cite{akp, aamr}\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\prime}_h}(h) \n= {1\\over N_h} \\sum_{q,X,\\lambda^{\\,}_X,\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q},\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q) \\>\\> \nD^*_{\\lambda^{\\prime}_h \\lambda^{\\,}_X; \\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \\,,\n\\label{rhoh}\n\\end{equation}\nwhere $\\rho(q\\bar q)$ is the helicity density matrix of the $q\\bar q$ state \ncreated in the annihilation of the unpolarized $e^+$ and $e^-$,\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q)\n= {1\\over 4N_{q\\bar q}} \\sum_{\\lambda^{\\,}_{-}, \\lambda^{\\,}_{+}}\nM^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_{\\bar q};\\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\>\nM^*_{\\lambda^{\\prime}_q \\lambda^{\\prime}_{\\bar q}; \\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\,.\n\\label{rhoqq}\n\\end{equation}\nThe $M$'s are the helicity amplitudes for the $e^-e^+ \\to q\\bar q$ process and\nthe $D$'s are the fragmentation amplitudes, {\\it i.e.} the helicity\namplitudes for the process $q\\bar q \\to h+X$; the $\\sum_{X,\\lambda_X}$ stands \nfor the phase space integration and the sum over spins of all the unobserved \nparticles, grouped into a state $X$. The normalization factors $N_h$ and\n$N_{q\\bar q}$ are given by:\n\\begin{equation}\nN_h = \\sum_{q,X; \\lambda^{\\,}_h, \\lambda^{\\,}_X, \\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q},\n\\lambda^{\\prime}_q, \\lambda^{\\prime}_{\\bar q}} \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q};\n\\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}}\\,(q\\bar q) \\>\\> \nD^*_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\prime}_q,\\lambda^{\\prime}_{\\bar q}} \\,\n= \\sum_q D^h_q \\,,\n\\label{nh}\n\\end{equation}\nwhere $D^h_q$ is the usual fragmentation function of quark $q$ into \nhadron $h$, and \n\\begin{equation}\nN_{q\\bar q} = {1\\over 4} \n\\sum_{\\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q}; \\lambda^{\\,}_{-}, \\lambda^{\\,}_{+}} \\vert \nM^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_{\\bar q}; \\lambda^{\\,}_{-} \\lambda^{\\,}_{+}} \\vert^2 \\,.\n\\label{nqq}\n\\end{equation}\n\nThe helicity density matrix for the $q\\bar q$ state can be computed \nin the Standard Model and its non zero elements are given by\n\\begin{eqnarray}\n\\rho^{\\,}_{+-;+-}(q\\bar q) &=& 1 - \\rho^{\\,}_{-+;-+}(q\\bar q) \\> \\simeq \\> \n{1\\over 2}\\,{(g_{_V} - g_{_A})^2_q \\over (g^2_{_V} + g^2_{_A})_q}\n\\label{rhoqqd} \\\\\n\\rho^{\\,}_{+-;-+}(q\\bar q) &=& \\rho^*_{+-;-+}(q\\bar q) \\> \\simeq \\> \n{1\\over 2}\\,{(g^2_{_V} - g^2_{_A})_q \\over (g^2_{_V} + g^2_{_A})_q} \n\\, {\\sin^2\\theta \\over 1+ \\cos^2\\theta} \\, \\cdot\n\\label{rhoqqap}\n\\end{eqnarray}\nThese expressions are simple but approximate and hold at the $Z_0$ pole, \nneglecting electromagnetic contributions, masses and terms proportional \nto $g_{_V}^l$; the full correct expressions can be found in Ref. \\cite{abmq}.\n\nNotice that, inserting the values of the coupling constants\n\\begin{equation}\ng_{_V}^{u,c,t} =  \\>\\> {1\\over 2} - {4\\over 3}\\sin^2\\theta_{_W} \\quad\\quad\ng_{_V}^{d,s,b} = -{1\\over 2} + {2\\over 3}\\sin^2\\theta_{_W} \\quad\\quad\ng_{_A}^{u,c,t} = - g_{_A}^{d,s,b} = {1\\over 2} \\label{cc}\n\\nonumber\n\\end{equation} \none has\n\\begin{equation}\n\\rho^{\\,}_{+-;-+}(u\\bar u, c\\bar c, t\\bar t) \n\\simeq -0.36 {\\sin^2\\theta \\over 1 + \\cos^2\\theta} \n\\quad\\quad \n\\rho^{\\,}_{+-;-+}(d\\bar d, s\\bar s, b\\bar b) \n\\simeq -0.17 {\\sin^2\\theta \\over 1 + \\cos^2\\theta} \n\\, \\cdot \\label{rho+-ap}\n\\end{equation}\n\nEq. (\\ref{rho+-ap}) clearly shows the $\\theta$ dependence of \n$\\rho^{\\,}_{+-;-+}(q\\bar q)$. In case of pure electromagnetic interactions\n($\\sqrt s \\ll M_{_Z}$) one has exactly:\n\\begin{equation}\n\\rho^{\\gamma}_{+-;-+}(q\\bar q) = {1\\over 2}\n\\,{\\sin^2\\theta \\over 1+ \\cos^2\\theta} \\, \\cdot\n\\label{rhoqqelm}\n\\end{equation}\nNotice that Eqs. (\\ref{rho+-ap}) and (\\ref{rhoqqelm}) have the same\nangular dependence, but a different sign for the coefficient in front,\nwhich is negative for the $Z$ contribution.\n\nBy using the above equations for $\\rho(q\\bar q)$ into Eq. (\\ref{rhoh}) one \nobtains the most general expression of $\\rho(h)$ in terms of the $q\\bar q$ spin \nstate and the unknown fragmentation amplitudes.\n\nDespite the ignorance of the fragmentation process some predictions\ncan be made \\cite{abmq} by considering the production of hadrons almost \ncollinear with the parent jet: the $q \\bar q \\to h + X$ fragmentation is \nthen essentially a c.m. forward process and the unknown $D$ amplitudes must \nsatisfy the angular momentum conservation relation \\cite{bls}\n\\begin{equation}\nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q,\\lambda^{\\,}_{\\bar q}} \n\\sim \\left( \\sin{\\theta_h\\over 2} \\right)^{\n\\vert \\lambda^{\\,}_h - \\lambda^{\\,}_X - \\lambda^{\\,}_q + \\lambda^{\\,}_{\\bar q} \\vert} \n\\simeq \\left( {p_{_T}\\over z\\sqrt s} \\right)^{\n\\vert \\lambda^{\\,}_h - \\lambda^{\\,}_X - \\lambda^{\\,}_q + \\lambda^{\\,}_{\\bar q} \\vert} \\,,\n\\label{frd}\n\\end{equation}\nwith $\\theta_h$ the angle between the hadron momentum, \n$\\mbox{\\boldmath $h$} = z \\mbox{\\boldmath $q$} + \\mbox{\\boldmath $p$}_{_T}$, and the quark momentum $\\mbox{\\boldmath $q$}$.\n\nThe bilinear combinations of fragmentation amplitudes contributing to\n$\\rho(h)$ are then not suppressed by powers of $(p_{_T}\/z\\sqrt s)$\nonly if the exponent in Eq. (\\ref{frd}) is zero, which greatly reduces the \nnumber of relevant helicity configurations.\n\nThe fragmentation process is a parity conserving one and the fragmentation\namplitudes must then also satisfy the forward parity relationship\n\\begin{equation}\nD^{\\,}_{-\\lambda^{\\,}_h -\\lambda^{\\,}_X; -+} = (-1)^{S^{\\,}_h + S^{\\,}_X + \n\\lambda^{\\,}_h - \\lambda^{\\,}_X} \\> \nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; +-} \\,.\n\\label{par}\n\\end{equation}\n\nBefore presenting analytical and numerical results for the coherent quark \nfragmentation let us remember that in case of incoherent single quark \nfragmentation Eq. (\\ref{rhoh}) becomes\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\prime}_h}(h) \n= {1\\over N_h} \\sum_{q,X,\\lambda^{\\,}_X,\\lambda^{\\,}_q,\\lambda^{\\prime}_q}\nD^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q} \\>\\>\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} \\>\\>\nD^*_{\\lambda^{\\,}_h \\lambda^{\\,}_X; \\lambda^{\\,}_q} \\,,\n\\label{rhohp1}\n\\end{equation}\nwhere $\\rho(q)$ is the quark $q$ helicity density matrix related to\n$\\rho(q\\bar q)$ by\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\prime}_q} = \\sum_{\\lambda^{\\,}_{\\bar q}}\n\\rho^{\\,}_{\\lambda^{\\,}_q, \\lambda^{\\,}_{\\bar q}; \\lambda^{\\prime}_q,\n\\lambda^{\\,}_{\\bar q}} (q\\bar q) \\,.\n\\end{equation}\n\nIn such a case angular momentum conservation for the collinear quark \nfragmentation requires $\\lambda^{\\,}_q = \\lambda^{\\,}_h + \\lambda^{\\,}_X$;\nthe Standard Model computation of $\\rho(q)$ gives only diagonal\nterms [$\\rho^{\\,}_{++}(q) = \\rho^{\\,}_{+-;+-}(q\\bar q)$, \n$\\rho^{\\,}_{--}(q) = \\rho^{\\,}_{-+;-+}(q\\bar q)$], and one ends up\nwith the usual probabilistic expression\n\\begin{equation}\n\\rho^{\\,}_{\\lambda^{\\,}_h \\lambda^{\\,}_h}(h) \n= {1\\over N_h} \\sum_{q, \\lambda^{\\,}_q}\n\\rho^{\\,}_{\\lambda^{\\,}_q \\lambda^{\\,}_q} \\>\\>\nD_{q,\\lambda^{\\,}_q}^{h, \\lambda^{\\,}_h} \\,,\n\\label{rhohp2}\n\\end{equation}\nwhere $D_{q,\\lambda^{\\,}_q}^{h, \\lambda^{\\,}_h}$ is the polarized fragmentation \nfunction of a $q$ with helicity $\\lambda^{\\,}_q$ into a hadron $h$ with \nhelicity $\\lambda^{\\,}_h$. Off-diagonal elements of $\\rho(h)$ are all zero.\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $e^- e^+ \\to BX, \\> (S_{_B} = 1\/2, \\> p_{_T}\/\\sqrt s \\to 0)$}}\n\\vskip 6pt\n\\nobreak\nLet us consider first the case in which $h$ is a spin 1\/2 baryon. It was\nshown in Ref. \\cite{aamr} that in such a case the coherent quark \nfragmentation only induces small corrections to the usual incoherent \ndescription \n\\begin{eqnarray}\n\\rho^{\\,}_{++}(B) &=& {1 \\over N_{_B}} \\sum_q \n\\left[ \\rho^{\\,}_{+-;+-}(q\\bar q) \\> D_{q,+}^{B,+} + \\rho^{\\,}_{-+;-+}(q\\bar q) \\> \nD_{q,-}^{B,+} \\right] \\\\\n\\rho^{\\,}_{+-}(B) &=& {\\cal O} \\left[ \\left(\n{p_{_T} \\over z \\sqrt s} \\right) \\right] \\label{rho+-b}\\,.\n\\end{eqnarray}\n\nThat is, the diagonal elements of $\\rho(B)$ are the same as those given by \nthe usual probabilistic formula (\\ref{rhohp2}), with small corrections\nof the order of $(p_{_T}\/z\\sqrt s)^2$, while off-diagonal elements are \nof the order $(p_{_T}\/z\\sqrt s)$ and vanish in the $p_{_T}\/\\sqrt s \\to 0$\nlimit.\n\nThe matrix elements of $\\rho(B)$ are related to the longitudinal ($P_z$)\nand transverse ($P_y$) polarization of the baryon:\n\\begin{equation}\nP_z = 2\\rho_{++} - 1, \\quad\\quad\\quad\\quad P_y = -2\\,{\\rm Im} \\rho_{+-} \\,.\n\\end{equation}\nSome data are available on $\\Lambda$ polarization, both longitudinal and\ntransverse, from ALEPH Collaboration \\cite{aleph} and they do agree with the\nabove equations. In particular the transverse polarization, at \n$\\sqrt s = M_{_Z}$, $p_{_T} \\simeq 0.5$ GeV\/$c$ and $z \\simeq 0.5$ is \nindeed of the order 1\\%, as expected from Eq. (\\ref{rho+-b}).\n\n\\goodbreak\n\\vskip 12pt\n\\noindent\n{\\mbox{\\boldmath $e^- e^+ \\to VX, \\> (S_{_V} = 1, \\> p_{_T}\/\\sqrt s \\to 0)$}}\n\\vskip 6pt\n\\nobreak\nIn case of final spin 1 vector mesons one has, always in the limit of small\n$p_{_T}$ \\cite{akp}, \\cite{abmq} \n\\begin{eqnarray}\n\\rho^{\\,}_{00}(V) &=& {1 \\over N_{_V}} \\sum_q D_{q,+}^{V,0} \\\\\n\\rho^{\\,}_{11}(V) &=& {1 \\over N_{_V}} \\sum_q\n\\left[ \\rho_{+-;+-}(q\\bar q) D_{q,+}^{V,1} + \\rho_{-+;-+}(q\\bar q) D_{q,-}^{V,1}\n\\right] \\\\\n\\rho^{\\,}_{1,-1}(V) &=& {1 \\over N_{_V}} \\sum_{q,X}\nD^{\\,}_{10;+-} \\> D^*_{-10;-+} \\> \\rho^{\\,}_{+-;+-}(q\\bar q) \\,.\n\\end{eqnarray}\n\nAgain, the diagonal elements have the usual probabilistic expression; \nhowever, there is now an off-diagonal element, $\\rho^{\\,}_{1,-1}$, \nwhich may survive even in the $p_{_T}\/\\sqrt s \\to 0$ limit. In the sequel\nwe shall concentrate on it. Let us first notice that, in the collinear limit,\none has\n\\begin{eqnarray}\nD^{V,0}_{q,+} &=& \\sum_X \\vert D^{\\,}_{0-1;+-} \\vert^2 = D^{V,0}_{q,-} \\\\\nD^{V,1}_{q,+} &=& \\sum_X \\vert D^{\\,}_{10;+-} \\vert^2 = D^{V,-1}_{q,-} \\\\\nD^{V,1}_{q,-} &=& \\sum_X \\vert D^{\\,}_{12;+-} \\vert^2 = D^{V,-1}_{q,+} \\,,\n\\end{eqnarray}\nwith $ D_q^V = D^{V,0}_{q,+} + D^{V,1}_{q,+} + D^{V,-1}_{q,+}$ and \n$N_{_V} = \\sum_q D_q^V$. We also notice that the two fragmentation \namplitudes appearing in Eq. (30) are related by parity and their product\nis always real. $\\rho^{\\,}_{00}$ and $\\rho^{\\,}_{1,-1}$ can be measured\nthrough the angular distribution of two body decays of $V$. \n\nIn order to give numerical estimates of $\\rho^{\\,}_{1,-1}$ we make some\nplausible assumptions\n\\begin{equation}\nD^{h,1}_{q,-} = D^{h,-1}_{q,+} = 0 \\quad\\quad\\quad\nD^{h,0}_{q,+} = \\alpha^V_q \\> D^{h,1}_{q,+} \\label{ass} \\,.\n\\end{equation}\nThe first of these assumptions simply means that quarks with helicity\n1\/2 ($-1\/2$) cannot fragment into vector mesons with helicity $-1$ ($+1$).\nThis is true for valence quarks assuming vector meson wave functions \nwith no orbital angular momentum, like in $SU(6)$. The second assumption \nis also true in $SU(6)$ with $\\alpha^V_q = 1\/2$ for \nany valence $q$ and $V$. Rather than taking  \n$\\alpha^V_q = 1\/2$ we prefer to relate the value of $\\alpha^V_q$ to the \nvalue of $\\rho^{\\,}_{00}(V)$ which can be or has been measured.\nIn fact, always in the $p_{_T} \\to 0$ limit, one has \\cite{abmq}\n\\begin{equation}\n\\rho^{\\,}_{00}(V) = {\\sum_q \\alpha^V_q \\, D^{h,1}_{q,+}\n\\over \\sum_q \\> (1+\\alpha^V_q) \\, D^{h,1}_{q,+}} \\,\\cdot\n\\label{rho00}\n\\end{equation}\nIf $\\alpha^V_q$ is the same for all valence quarks in $V$ \n($\\alpha^V_q = \\alpha^V$) \none has, for the valence quark contribution:\n\\begin{equation}\n\\alpha^V = {\\rho^{\\,}_{00}(V) \\over 1 - \\rho^{\\,}_{00}(V)} \\,\\cdot\n\\label{alrho}\n\\end{equation}\n\nFinally, one obtains \\cite{abmq}\n\\begin{equation}\n\\rho^{\\,}_{1,-1}(V) \\simeq [1 - \\rho^{\\,}_{0,0}(V)] \\,\n{\\sum_q \\, D^{V,1}_{q,+} \\> \\rho_{+-;-+}(q\\bar q) \n\\over \\sum_q \\, D^{V,1}_{q,+}} \\,\\cdot\n\\label{rho1-1tss}\n\\end{equation}\n\nWe shall now consider some specific cases in which we expect \nEq. (\\ref{rho1-1tss}) to hold; let us remind once more that our \nconclusions apply to spin 1 vector mesons produced in \n$e^- e^+ \\to q \\bar q \\to V+X$ processes in the limit of small $p_{_T}$\nand large $z$, {\\it i.e.}, to vector mesons produced in two jet events\n($e^- e^+ \\to q\\bar q$) and collinear with one of them ($p_{_T} = 0$), \nwhich is the jet generated by a quark which is a valence quark for the\nobserved vector meson (large $z$). These conditions should be met \nmore easily in the production of heavy vector mesons. \n\nAmong other results one obtains \\cite{abmq}:\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^{*}) &\\simeq& [1 - \\rho^{\\,}_{0,0}(D^{*})] \\>\n\\rho_{+-;-+}(c \\bar c) \\label{rhoda} \\\\\n\\rho^{\\,}_{1,-1}(\\phi)  &\\simeq& [1 - \\rho^{\\,}_{0,0}(\\phi)] \\>\n\\rho_{+-;-+}(s\\bar s) \\label{rhopa} \\\\\n\\rho^{\\,}_{1,-1}(K^{*0}) &\\simeq& {1\\over 2} \\> [1 - \\rho^{\\,}_{0,0}(K^{*0})] \n\\> [\\rho_{+-;-+}(d\\bar d) + \\rho_{+-;-+}(s\\bar s)] \\,. \\label{rhok0a} \n\\end{eqnarray}\n\nEqs. (\\ref{rhoda})-(\\ref{rhok0a}) show how the value of $\\rho^{\\,}_{1,-1}(V)$\nare simply related to the off-diagonal helicity density matrix element \n$\\rho_{+-;-+}(q\\bar q)$ of the $q\\bar q$ pair created in the elementary \n$e^- e^+ \\to q\\bar q$ process; such off-diagonal elements would not appear \nin the incoherent independent fragmentation of a single quark, yielding \n$\\rho^{\\,}_{1,-1}(V)=0$.\n\nBy inserting into the above equations the value of $\\rho^{\\,}_{00}$ when\navailable \\cite{opal} and the expressions of $\\rho^{\\,}_{+-;-+}$,\nEq. (18), one has:\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^{*}) &\\simeq& -(0.216 \\pm 0.007) \\ \n{\\sin^2\\theta \\over 1 + \\cos^2\\theta} \\\\\n\\rho^{\\,}_{1,-1}(\\phi) &\\simeq& -(0.078 \\pm 0.014) \\ \n{\\sin^2\\theta \\over 1 + \\cos^2\\theta} \\\\\n\\rho^{\\,}_{1,-1}(K^{*0}) &\\simeq& -0.170 \\ \n[1- \\rho^{\\,}_{0,0}(K^{*0})] \\ {\\sin^2\\theta \\over 1 + \\cos^2\\theta}  \\,\\cdot\n\\end{eqnarray}\nFinally, in case one collects all meson produced at different angles in\nthe full available $\\theta$ range (say $\\alpha < \\theta < \\pi -\\alpha, \n\\> |\\cos\\theta| < \\cos\\alpha$) an average should be taken in $\\theta$, \nweighting the different values of $\\rho^{\\,}_{1,-1}(\\theta)$ with the \ncross-section for the $e^-e^+ \\to V+X$ process; this gives \\cite{abmq}:\n\\begin{eqnarray}\n\\langle \\rho^{\\,}_{1,-1}(D^{*}) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -(0.216 \\pm 0.007) \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\label{rhodn} \\\\\n\\langle \\rho^{\\,}_{1,-1}(\\phi) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -(0.078 \\pm 0.014) \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\label{rhopn} \\\\\n\\langle \\rho^{\\,}_{1,-1}(K^{*0}) \\rangle_{[\\alpha, \\pi-\\alpha]}\n&\\simeq& -0.170 \\ [1- \\rho^{\\,}_{0,0}(K^*)] \\ \n{3 - \\cos^2\\alpha \\over 3 + \\cos^2\\alpha} \\,\\cdot \\label{rhok0n} \n\\end{eqnarray}\n\nThese results have to be compared with data \\cite{opal}\n\\begin{eqnarray}\n\\rho^{\\,}_{1,-1}(D^*) &=& -0.039 \\pm 0.016 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.5 \\quad\\quad \\cos\\alpha = 0.9 \\\\ \n\\rho^{\\,}_{1,-1}(\\phi) &=& -0.110 \\pm 0.070 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.7 \\quad\\quad \\cos\\alpha = 0.9 \\\\ \n\\rho^{\\,}_{1,-1}(K^{*0}) &=& -0.090 \\pm 0.030 \\quad\\quad {\\rm for}\n\\quad\\quad z > 0.3 \\quad\\quad \\cos\\alpha = 0.9  \n\\end{eqnarray}\nwhich shows a good qualitative agreement with the theoretical\npredictions. We notice that while the mere fact that $\\rho_{1,-1}$ differs \nfrom zero is due to a coherent fragmentation of the $q\\bar q$ pair, the actual \nnumerical values depend on the Standard Model coupling constants; for example,\n$\\rho_{1,-1}$ would be positive at smaller energies, at which the one gamma\nexchange dominates, while it is negative at LEP energy where the one $Z$\nexchange dominates. $\\rho_{1,-1}$ has also a peculiar dependence on \nthe meson production angle, being small at small and large angles\nand maximum at $\\theta = \\pi\/2$. Such angular dependence has been tested\nin case of $K^{*0}$ production, assuming no dependence of $\\rho_{00}$\non $\\theta$, and indeed one has \\cite{opal}, in agreement \nwith Eqs. (\\ref{rhok0a}) and (\\ref{rho+-ap})\n\\begin{equation}\n\\left[ {\\rho^{\\,}_{1,-1} \\over 1- \\rho^{\\,}_{00}} \\right]_{|\\cos\\theta|<0.5} \n\\cdot \n\\left[ {\\rho^{\\,}_{1,-1} \\over 1- \\rho^{\\,}_{00}} \\right]^{-1}\n_{|\\cos\\theta|>0.5} = 1.5 \\pm 0.7\n\\end{equation}\n\nThese results are encouraging; it would be interesting to have more and \nmore detailed data, possibly with a selection of final hadrons with the \nrequired features for our results to hold. It would also be interesting to \ntest the coherent fragmentation of quarks in other processes, like\n$\\gamma\\gamma \\to VX$, $pp \\to D^*X$ and $\\ell p \\to VX$. The first two\nprocesses are similar to $e^-e^+ \\to VX$ in that a $q\\bar q$ pair is created \nwhich then fragments coherently into the observed vector meson; one assumes\nthat the dominating elementary process in $pp \\to D^*X$ is $gg \\to c \\bar c$.\nIn both these cases one has for $\\rho^{\\,}_{+-;-+}(q\\bar q)$ the same value\nas in Eq. (\\ref{rhoqqelm}), so that one expects a {\\it positive} value\nof $\\rho^{\\,}_{1,-1}(V)$. \n\nIn the last process, the production of vector mesons in DIS, the quark \nfragmentation is in general a more complicated interaction of the quark \nwith the remnants of the proton and it might be more difficult to obtain \nnumerical predictions. However, if one observes $D^*$ mesons one can \nassume or select kinematical regions for which the underlying elementary \ninteraction is $\\gamma^* g \\to c\\bar c$: again, one would have the same \n$\\rho^{\\,}_{+-;-+}(c\\bar c)$ as in Eq. (\\ref{rhoqqelm}), and one would \nexpect a positive value of $\\rho^{\\,}_{1,-1}(D^*)$. It would indeed be\ninteresting to perform these simple tests of coherent fragmentation \neffects.\n\n\\vskip 18pt\n\\noindent\n{\\bf Acknowledgements}\n\\vskip 6pt\n\\noindent\nI would like to thank the organizers of the Workshop and DESY for financial\nsupport\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\bigskip\n\nGiven an open bounded connected domain $\\Omega \\subset \\mathbb{R}^{N}$ with\na sufficiently regular (locally Lipschitz) boundary, $\\partial \\Omega $, let\nus consider the integra\n\\begin{equation}\nI\\left( u\\right) :=\\int_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right) \\,dx\n\\label{Functional}\n\\end{equation\nto be minimized on a class of Sobolev functions $u:\\Omega \\rightarrow \n\\mathbb{R}^{d}$ with a kind of boundary conditions to be described later.\nAll over the paper we assume the integrand $W:\\mathbb{R}^{d\\times\nN}\\rightarrow \\mathbb{R\\cup }\\left\\{ +\\infty \\right\\} $ to be \\emph\npolyconvex}. This means that the representatio\n\\begin{equation*}\nW\\left( \\xi \\right) =g\\left( \\mathbb{T}\\left( \\xi \\right) \\right) \n,\\;\\; \\xi \\in \\mathbb{R}^{d\\times N},\n\\end{equation*}\nholds for some convex function $g:\\mathbb{R}^{\\tau \\left( d,N\\right)\n}\\rightarrow \\mathbb{R\\cup }\\left\\{ +\\infty \\right\\} $\n\\begin{equation*}\n\\tau( d,N) :=\\underset{s=1}{\\overset{d\\wedge N}{\\dsum }}\n\\varkappa(s) ,\\;\\; \\varkappa(s) :=\\dbinom{\n}{d}\\dbinom{s}{N}=\\frac{d!N!}{(s!) ^{2}( d-s)!(\nN-s) !},\n\\end{equation*}\nwher\n\\begin{equation*}\n\\mathbb{T}\\left(\\xi \\right) :=\\left( \\mathrm{Adj}_{1}\\xi ,\\mathrm{Adj\n_{2}\\xi ,\\mathrm{Adj}_{3}\\xi ,\\dots ,\\mathrm{Adj}_{d\\wedge N}\\xi \\right) \n,\\;\\; \\xi \\in \\mathbb{R}^{d\\times N},\n\\end{equation*\nand $\\mathrm{Adj}_{k}\\xi $ is the vector of all \\emph{minors} of the matrix \n\\xi $ of order $k=1,2,\\dots ,d\\wedge N$, respectively. In particular, \n\\mathrm{Adj}_{1}\\xi =\\xi $ and $\\mathrm{Adj}_{d}\\xi =\\det \\,\\xi $ whenever \nd=N$.\n\nIt is known that, under strong coercivity assumptions on $W$ to assure weak convergence of the minors of gradients for the minimizing sequence, the functional $I$ attains its minimum on $\\bar{u}\\left( \\cdot \\right) +\\mathbf{\n}_{0}^{1,p}\\left( \\Omega ;\\mathbb{R}^{d}\\right) $, $p\\geq 1$. We refer to the fundamental work by J. Ball \\cite{B} motivated by problems coming from nonlinear elasticity and to \\cite{ADM, MQY, My} for further improvements.\n\n\nThe lower semicontinuity for general polyconvex integrands with respect to the weak convergence in $W^{1,p}(\\Omega;\\mathbb{R}^{d}),$       $\\Omega\\subset \\mathbb{R}^{N},$ has been the subject of many investigations.\nNamely, Marcellini showed in \\cite{M} that this property holds whenever $p<N$. Later, this result was improved by Dacorogna and Marcellini in \\cite{DM} who proved the lower semicontinuity for $p>N-1$ while Mal\\'{y} in \\cite{My} exhibited a counterexample for $p<N-1$. The limit case $p=N-1$ was adressed in \\cite{ADM, DMS, CDM, FH}. Very recently (see \\cite{FFLMMV} and \\cite{DPMF})  the limit case was studied for polyconvex integrands depending on $x$ and\/or on $u$.\n\nBesides the Dirichlet boundary condition $u\n\\bar{u}$ for the displacement one considered, for instance, boundary condition on traction, which somehow depends on the normal\nderivatives of $u\\left( \\cdot \\right) $ (generalizing the Neumann boundary\ndata). Observe that these conditions (displacement, traction or a mixed one)\ncan be applied either to the whole boundary $\\partial \\Omega $, or to some\nof its subsets of positive Hausdorff measure leaving the rest free.\nMoreover, some restrictions on the \\emph{jacobian} may be relevant and\npractically justified. For example, the constraint $\\det \\,\\nabla u\\left(\nx\\right) >0$ means that the minimum is searched among the deformations\npreserving orientation, while $\\det \\,\\nabla u\\left( x\\right) =1$ refers to\nthe case of incompressible elastic body.\n\nOne of the possible applications of the above variational problem is\nregarded to plastic surgery, namely, in the woman breast reduction, where we\ndeal with a sort of very elastic and soft tissue. Some recent publications\n(see, e.g., \\cite{Ay, D, S1, PY, S2}) were devoted to mathematical setting\nof the related problems and to their numerical simulations. Medical\nexaminations allow to consider the involved tissue as a neo-Hookean\ncompressible material (see \\cite{Z}). We have a more precise model when the strain energy is  defined\nby the integral (\\ref{Functional}) with the density $W:\\mathbb{R}^{3\\times\n3}\\rightarrow \\mathbb{R}$\n\\begin{eqnarray}\n&&W\\left( \\xi \\right) :=\\mu \\left( \\limfunc{tr}\\left( \\xi \\cdot \\xi\n^{T}\\right) -3-2\\ln \\left( \\det \\xi \\right) \\right)   \\notag \\\\\n&&+\\lambda \\left( \\det \\xi -1\\right) ^{2}+\\beta \\limfunc{tr}\\mathrm{\\,\n\\left( \\mathrm{Adj\\,}\\xi \\cdot \\mathrm{Adj\\,}\\xi ^{T}\\right) \\,,\n\\label{Int_ex}\n\\end{eqnarray\nwhere \"$\\limfunc{tr}$\" means the trace of a matrix, $\\mathrm{Adj\\,}\\xi :\n\\mathrm{Adj}_{2}\\xi $, and the symbol \"$T$\" stands for the matrix\ntransposition.  One of the steps of the\n(breast reduction) surgery is the suturing, which mathematically can\nbe seen as an identification of points of some surface piece $\\Gamma\n^{+}\\subset \\partial \\Omega $ with points of another one $\\Gamma ^{-}\\subset\n\\partial \\Omega $. Denoting the respective correspondence between the points\nof $\\Gamma ^{+}$ and $\\Gamma ^{-}$ by $\\sigma $, we are led to a new type of constrain\n\\begin{equation}\nu(x) =(u\\circ \\sigma)(x), \\;x\\in\n\\Gamma ^{+},  \\label{sutur_cond}\n\\end{equation\ncalled the \\emph{knitting boundary condition}. Let us note that the one-to-one\nmapping $\\sigma $ is not \\emph{a priori} given and should be chosen to\nguarantee the minimum value to the functional (\\ref{Functional}). In other\nwords, a minimizer of (\\ref{Functional}) (if any) should be a pair $\\left(\nu,\\sigma \\right) $ where $u\\in \\mathbf{W}^{1,p}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $, $p\\geq 1$, and $\\sigma :\\Gamma ^{+}\\rightarrow \\Gamma ^{-}$\nis sufficiently regular. We set the natural hypothesis that $\\sigma $ and\nits inverse $\\sigma ^{-1}$ are Lipschitz transformations (with the same\nLipschitz constant $L>0$). Practically this means that the sutured tissue\ncan not be extended nor compressed too much.\n\n\n\nMotivated by the problem coming from the plastic surgery we will consider\njust the case $p=2$ and $d=N=3$, although the results remain\ntrue in the case $p>2$ and arbitrary $d=N\\geq 2$ as well.\n\nThe paper is organized as follows. In the next section we give the exact\nsetting of the variational problem together with the main hypotheses on the\nintegrand $W$. For simplicity of references we put here also some important\nfacts regarded with the Sobolev functions. In Section 3 we justify first the\nwell-posedness of the problem by showing that the composed function from the\nknitting condition (\\ref{sutur_cond}) belongs to the respective Lebesgue\nclass. Afterwards, we prove existence of a minimizer as an accumulation\npoint of an arbitrary minimizing sequence (the so called \\emph{direct method\n, see \\cite{Dac}). The paper is concluded with a necessary optimality\ncondition for the given problem (see Section 4) allowing to construct\neffective numerical algorithms, which can be successively applied in the\nmedical practice.\n\n\\bigskip\n\n\\section{\\protect\\bigskip Main hypotheses and auxiliary results}\n\nIn what follows we fix a nonempty open bounded and connected set $\\Omega\n\\subset \\mathbb{R}^{3}$ whose boundary $\\partial \\Omega $ is assumed to be\nlocally Lipschitz (see, e.g., \\cite[p. 354]{L}). By the symbol $\\mathcal{L\n^{m}$ ($dx$) we denote the Lebesgue measure in the space $\\mathbb{R}^{m}$, \nm=2,3$, while $\\mathcal{H}^{2}$ means the two-dimensional \\emph{Hausdorff\nmeasure} (see, e.g., \\cite{F}).\n\nLet us divide the surface $\\partial \\Omega $ into several parts $\\Gamma _{i}\n, $i=1,2,3,4$, in such a way that $\\mathcal{H}^{2}\\left( \\Gamma _{i}\\cap\n\\Gamma _{j}\\right) =0$ for $i\\neq j$. Moreover, we set $\\Gamma _{4}:=\\Gamma\n^{+}\\cup \\Gamma ^{-}$ where $\\Gamma ^{\\pm }\\subset \\Gamma $ with $\\mathcal{H\n^{2}\\left( \\Gamma ^{\\pm }\\right) >0$ and $\\mathcal{H}^{2}\\left( \\Gamma\n^{+}\\cap \\Gamma ^{-}\\right) =0$ are also given.\n\nSuppose that $W:\\mathbb{R}^{3\\times 3}\\rightarrow \\mathbb{R}$ is a \\emph\npolyconvex function} satisfying the \\emph{growth\\ assumption}\n\\begin{equation}\nW(\\xi) \\geq c_{0}+c_{1}\\left\\vert \\xi \\right\\vert\n^{2}+c_{2}\\left\\vert \\mathrm{Adj\\,}\\xi \\right\\vert ^{2}+c_{3}\\left(\\det \\xi\n\\right)^{2}, \\;\\; \\xi \\in \\mathbb{R}^{3\\times 3},\n\\label{growth_cond}\n\\end{equation\nwhere $c_{0}\\in \\mathbb{R}$ and $c_{i}>0$, $i=1,2,3$, are some given\nconstants. Here and in what follows by $\\left\\vert \\cdot \\right\\vert $ we\ndenote the norm of both a vector in $\\mathbb{R}^{n}$ and a $3\\times 3\n-matrix.\n\nTaking into account that $\\limfunc{tr}\\left( \\xi \\cdot \\xi ^{T}\\right)\n=\\left\\vert \\xi \\right\\vert ^{2}$ for each matrix $\\xi \\in \\mathbb{R\n^{3\\times 3}$, we see that the integrand (\\ref{Int_ex}) satisfies the above\nproperties. Indeed, it is convex as a function of $\\mathbb{T}\\left( \\xi\n\\right) $ being represented as a sum of three terms, which are convex w.r.t. \n$\\xi $, $\\det \\xi $ and $\\mathrm{Adj\\,}\\xi $, respectively. Furthermore\n\\begin{equation*}\nW(\\xi) =-3\\mu +\\lambda +\\mu \\left\\vert \\xi \\right\\vert\n^{2}+\\beta \\left\\vert \\mathrm{Adj\\,}\\xi \\right\\vert ^{2}+f( \\det \\xi), \\;\\xi \\in \\mathbb{R}^{3\\times 3},\n\\end{equation*\nwhere the functio\n\\begin{equation*}\nf\\left( t\\right) :=\\frac{\\lambda }{2}t^{2}-2\\lambda t-2\\mu \\ln t, \\;t>\n,\n\\end{equation*\nis lower bounded by some (negative) constant.\n\nSince on various pieces of the surface $\\partial \\Omega $ the boundary conditions\nare structurally different (some part of $\\partial \\Omega $ can be left even\nfree), to set the problem we use the notion of the \\emph{trace operator},\nwhich associates to each $u\\in $ $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ a function $\\limfunc{Tr}u$ defined on the boundary, $\\partial\n\\Omega,$ which can be interpreted as the \"\\emph{boundary values}\" of $u$. We refer to \\cite[pp. 465-474]{L}, where\nthe existence and uniqueness of the trace operator were proved for scalar Sobolev functions $u \\in \\mathbf{W}^{1,p}(\\Omega),   p>1.$ For vector-valued functions $u: \\Omega\\rightarrow \\mathbb{R}^{3}$ instead, we can argue componentwise. So, applying \\cite[Theorem 15.23]{L}, we define the trace as the linear and bounded operator\n$\\limfunc{Tr}:\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) \\rightarrow \\mathbf{L}^{2}(\\partial \\Omega ;\\mathbb{R\n^{3})$ satisfying the following properties:\n\n\\begin{enumerate}\n\\item $\\limfunc{Tr}u\\left( x\\right) =u\\left( x\\right) $, $x\\in \\partial\n\\Omega $, whenever $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) \\cap \\mathbf{C}\\left( \\overline{\\Omega };\\mathbb{R}^{3}\\right) $;\n\n\\item for each $u\\in \\mathbf{W}^{1,2}(\\Omega;\\mathbb{R}^3)$\nand any test function $\\varphi \\in \\mathbf{C}^{1}(\\overline{\\Omega };\\mathbb{R}^{3})$ the equalitie\n\\begin{equation*}\n\\int_{\\Omega}u_{j}\\frac{\\partial \\varphi_{j}}{\\partial x_{i}}dx=-\\int_{\\Omega}\\varphi_{j}\\frac{\\partial u_{j}}{\\partial x_{i}}dx+\\int_{\\partial \\Omega }\\varphi_{j}\n\\limfunc{Tr}(u_j)\\nu_i d\\mathcal{H}^{2}\n\\end{equation*\nhold for each $ i, j=1,2,3$ where $\\nu:=(\\nu_1,\\nu_2,\\nu_3)^{T}$ means the unit outward normal to \n\\partial \\Omega $.\n\\end{enumerate}\n \nIn addition to the properties above, observe that the trace operator gives a compact embedding into the space $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R}^{3}\\right)$ that will be crucial to obtain the main result in Section 3. Namely, the following proposition takes place.\n\n\\begin{proposition}\n\\label{traceconv}Let $\\Omega \\subset \\mathbb{R}^{3}$ be an open bounded set\nwith locally Lipschitz boundary. Then for each $\\left\\{ u_{n}\\right\\}\n\\subset \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ converging to \n$u$ weakly in $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ the\nsequence of traces $\\left\\{ \\limfunc{Tr}u_{n}\\right\\} $ converges to \n\\limfunc{Tr}u$ strongly in $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R\n^{3}\\right)$.\n\\end{proposition}\n\nThe proof is based essentially on the following lemma giving a nice estimate\nfor the surface integral of the trace operator.\n\n\\begin{lemma}\n\\label{Lemmatrace}Let $\\Omega \\subset \\mathbb{R}^{3}$ be as in Proposition \\ref{traceconv}. Then there exists a constant\n$C>0$ such that \n\\begin{equation}\n\\int_{\\partial \\Omega }\\left\\vert \\limfunc{Tr}u\\right\\vert ^{2}d\\mathcal{H\n^{2}\\leq C\\left( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\left\\vert u\\right\\vert\n^{2}dx+\\varepsilon \\int_{\\Omega }\\left\\vert \\nabla u\\right\\vert\n^{2}dx\\right)   \\label{Estimate_tr}\n\\end{equation\nfor any $\\varepsilon >0$ and any $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$\n\\end{lemma}\n\n\\begin{proof}\nGiven $x\\in \\partial \\Omega ,$ due to the Lipschitz hypothesis there exists\na neighborhood $U_{x}$ of $x$ such that $U_{x}\\cap \\partial \\Omega $ can be\nrepresented as the graph of a Lipschitz function w.r.t. some (local)\ncoordinates. Without loss of generality, we may assume that \n\\begin{equation*}\nU_{x}\\cap \\Omega =\\left\\{ \\left( y^{\\prime },y_{3}\\right) :f_{x}\\left(\ny^{\\prime }\\right) <y_{3}\\leq f_{x}\\left( y^{\\prime }\\right) +\\delta\n_{x},~y^{\\prime }\\in G_{x}\\right\\} \n\\end{equation*\nan\n\\begin{equation*}\nU_{x}\\cap \\partial \\Omega =\\left\\{ \\left( y^{\\prime },f_{x}\\left( y^{\\prime\n}\\right) \\right) :y^{\\prime }\\in G_{x}\\right\\} ,\n\\end{equation*\nwhere $\\delta _{x}>0,$ $G_{x}\\subset \\mathbb{R}^{2}$ is an open set in the\nspace of the first two coordinates and $f_{x}:G_{x}\\rightarrow \\mathbb{R}$\nis a Lipschitz function. By compactness there exists a finite number of\npoints $x^{i}\\in \\partial \\Omega \\,,~i=1,\\dots ,q,$ such that \n\\begin{equation*}\n\\partial \\Omega =\\tbigcup_{i=1}^{q}\\left( U_{x^{i}}\\cap \\partial \\Omega\n\\right) .\n\\end{equation*\nSet $\\delta _{i}:=\\delta _{x^{i}},~U_{i}:=U_{x^{i}},~G_{i}:=G_{x^{i}}$ and~ \nf_{i}:=f_{x^{i}}$, $i=1,\\dots ,q$. Denote by $L>0$ the biggest Lipschitz\nconstant of the functions $f_{i}.$\n\nLet us choose $\\varepsilon <\\min \\left\\{ \\delta _{i}:i=1,\\dots ,q\\right\\} $\nand consider first the function $u\\in \\mathbf{C}^{1}\\left( \\overline{\\Omega \n\\right) \\cap \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$ Given \ni\\in \\left\\{ 1,\\dots ,q\\right\\} ,$ by the Newton-Leibniz formula, for each \nx^{\\prime }\\in G_{i}$ and $\\ x_{3}\\in \\mathbb{R}$ with $f_{i}\\left(\nx^{\\prime }\\right) \\leq x_{3}<f_{i}\\left( x^{\\prime }\\right) +\\varepsilon $\nwe have \n\\begin{equation*}\nu\\left( x^{\\prime },f_{i}\\left( x^{\\prime }\\right) \\right) =u\\left(\nx^{\\prime },x_{3}\\right) -\\int_{f_{i}\\left( x^{\\prime }\\right) }^{x_{3}\n\\frac{\\partial u}{\\partial x_{3}}\\left( x^{\\prime },s\\right) ~ds.\n\\end{equation*\nIn turn, by Cauchy-Schwartz inequality\n\\begin{eqnarray*}\n\\left\\vert u\\left( x^{\\prime },f_{i}\\left( x^{\\prime }\\right) \\right)\n\\right\\vert ^{2} &\\leq &2\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right)\n\\right\\vert ^{2}+\\left( \\int_{f_{i}\\left( x^{\\prime }\\right) }^{f_{i}\\left(\nx^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}{\\partial x_{3}\n\\left( x^{\\prime },s\\right) \\right\\vert ds\\right) ^{2}\\right)  \\\\\n&\\leq &2\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right) \\right\\vert\n^{2}+\\varepsilon \\int_{f_{i}\\left( x^{\\prime }\\right) }^{f_{i}\\left(\nx^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}{\\partial x_{3}\n\\left( x^{\\prime },s\\right) \\right\\vert ^{2}ds\\right) .\n\\end{eqnarray*\nHence,\n\n\\begin{eqnarray*}\n&&\\left\\vert u\\left( x^{\\prime },f_{i}(x^{\\prime })\\right) \\right\\vert ^{2}\\sqrt\n1+\\left\\vert \\nabla f_{i}\\left( x^{\\prime }\\right) \\right\\vert ^{2}} \\\\\n&\\leq &2\\sqrt{1+L^{2}}\\left( \\left\\vert u\\left( x^{\\prime },x_{3}\\right)\n\\right\\vert ^{2}+\\varepsilon \\int_{f_{i}\\left( x^{\\prime }\\right)\n}^{f_{i}\\left( x^{\\prime }\\right) +\\varepsilon }\\left\\vert \\frac{\\partial u}\n\\partial x_{3}}\\left( x^{\\prime },s\\right) \\right\\vert ^{2}ds\\right) .\n\\end{eqnarray*\nIntegrating both parts of the previous inequality in the cylinder \n\\begin{equation*}\n\\Omega _{i}:=\\left\\{ \\left( x^{\\prime },x_{3}\\right) :f\\left( x^{\\prime\n}\\right) \\leq x_{3}\\leq f\\left( x^{\\prime }\\right) +\\varepsilon ,~x^{\\prime\n}\\in G_{i}\\right\\} \\subset U_{i}\\cap \\Omega ,\n\\end{equation*\ngive\n\\begin{eqnarray*}\n\\varepsilon \\int_{U_{i}\\cap \\partial \\Omega }\\left\\vert u\\left( x\\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  &\\leq &2\\sqrt{1+L^{2}\n\\left( \\int_{\\Omega _{i}}\\left\\vert u\\left( x\\right) \\right\\vert\n^{2}dx+\\varepsilon ^{2}\\int_{\\Omega _{i}}\\left\\vert \\frac{\\partial u}\n\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right)  \\\\\n&\\leq &2\\sqrt{1+L^{2}}\\left( \\int_{\\Omega }\\left\\vert u\\left( x\\right)\n\\right\\vert ^{2}dx+\\varepsilon ^{2}\\int_{\\Omega }\\left\\vert \\frac{\\partial \n}{\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right) .\n\\end{eqnarray*}\n\nSumming in $i=1,\\dots ,q$ we hav\n\\begin{eqnarray}\n&&\\int_{\\partial \\Omega }\\left\\vert u\\left( x\\right) \\right\\vert ^{2}\n\\mathcal{H}^{2}\\left( x\\right)   \\label{traceu} \\\\\n&\\leq &2q\\sqrt{1+L^{2}}\\left( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\left\\vert\nu\\left( x\\right) \\right\\vert ^{2}dx+\\varepsilon \\int_{\\Omega }\\left\\vert \n\\frac{\\partial u}{\\partial x_{3}}\\left( x\\right) \\right\\vert ^{2}dx\\right) .\n\\notag\n\\end{eqnarray\nThe inequality (\\ref{Estimate_tr}) for each $u \\in \\mathbf{W}^{1,2}(\\Omega ;\\mathbb{R}^{3})$ follows now from (\\ref{traceu}) by the\nproperties of the trace operator and by the density of smooth functions.\n\\end{proof}\n\n\\medskip\n\n\\begin{proof}[Proof of Proposition 1]\nGiven a sequence $\\left\\{ u_{n}\\right\\} \\subset \\mathbf{W}^{1,2}\\left(\n\\Omega ;\\mathbb{R}^{3}\\right) $ converging weakly to $u$ in $\\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $, there exists $M>0$ with \n\\begin{equation*}\n\\int_{\\Omega }\\left\\vert \\nabla u_{n}\\left( x\\right) \\right\\vert ^{2}dx\\leq M\n\\end{equation*\nfor all $n\\in \\mathbb{N}.$ Then, by the \\emph{Rellich-Kondrachov theorem}\n(see \\cite[Theorem 11.21, p. 326]{L}), $u_{n}\\rightarrow u$ $\\ $strongly$\\ \nin$~\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) .$ Applying Lemma \\re\n{Lemmatrace} to $\\limfunc{Tr}\\left( u_{n}-u\\right) =\\limfunc{Tr}u_{n}\n\\limfunc{Tr}u,$ we hav\n\\begin{align*}\n\\int_{\\partial \\Omega }\\vert \\limfunc{Tr}u_{n}-\\limfunc{Tr}\nu\\vert ^{2}d\\mathcal{H}^{2}&\\leq C(\\frac{1}{\\varepsilon }\\int_{\\Omega }\\vert\nu_{n}-u\\vert ^{2}dx+\\varepsilon \\int_{\\Omega }\\vert \\nabla\nu_{n}-\\nabla u\\vert ^{2}dx)\\\\\n&\\leq C( \\frac{1}{\\varepsilon }\\int_{\\Omega }\\vert\nu_{n}-u\\vert ^{2}dx+4\\varepsilon M) .\n\\end{align*}\nHenc\n\\begin{equation*}\n\\underset{n\\rightarrow \\infty }{\\lim \\sup }\\int_{\\partial \\Omega }\\left\\vert \n\\limfunc{Tr}u_{n}-\\limfunc{Tr}u\\right\\vert ^{2}d\\mathcal{H}^{2}\\leq\n4\\varepsilon CM.\n\\end{equation*\nLetting $\\varepsilon \\rightarrow 0^{+}$ concludes the proof.\\medskip \n\\end{proof}\n\nWe will use also the so called \\emph{generalized\nPoincar\\'{e} inequality} (see \\cite[Theorem 6.1-8, p. 281]{C}).\n\n\\begin{proposition}\n\\label{poincare}Given an open bounded domain $\\Omega \\subset \\mathbb{R}^{3}$\nwith locally Lipschitz boundary and a measurable subset $\\Gamma \\subset\n\\partial \\Omega $ with $\\mathcal{H}^{2}\\left( \\Gamma \\right) >0$ there\nexists a constant $C>0$ such that \n\\begin{equation}\n\\underset{\\Omega }{\\dint }\\left\\vert u\\left( x\\right) \\right\\vert ^{2}dx\\leq\nC\\left[ \\underset{\\Omega }{\\dint }\\left\\vert \\nabla u\\left( x\\right)\n\\right\\vert ^{2}\\,dx+\\left\\vert \\underset{\\Gamma }{\\dint }\\limfunc{Tr\nu\\left( x\\right) \\,d\\mathcal{H}^{2}\\left( x\\right) \\right\\vert ^{2}\\right] \n\\label{poincare_ineq}\n\\end{equation\nfor each $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $.\n\\end{proposition}\n\n\\medskip\nLet us formulate now the boundary conditions in terms of the trace operator.\nConsider first a surface $\\mathcal{S}\\subset \\mathbb{R}^{3}$ defined by some\ncontinuous function $h:\\mathbb{R}^{3}\\rightarrow \\mathbb{R}$\n\\begin{equation*}\n\\mathcal{S}:=\\left\\{ u\\in \\mathbb{R}^{3}:h\\left( u\\right) =0\\right\\} .\n\\end{equation*}\nThen, given $L\\geq 1$ we denote by $\\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma\n^{-}\\right) $ the set of all functions $\\sigma :\\Gamma ^{+}\\rightarrow\n\\Gamma ^{-}$ satisfying the inequalitie\n\\begin{equation}\n\\frac{1}{L}\\left\\vert x-y\\right\\vert \\leq \\left\\vert \\sigma \\left( x\\right)\n-\\sigma \\left( y\\right) \\right\\vert \\leq L\\left\\vert x-y\\right\\vert \n\\label{Lipschitz}\n\\end{equation\nfor all $x,y\\in \\Gamma ^{+}$, and introduce the set $\\mathcal{W}_{L}\\subset \n\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) \\times \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ of all pairs $\\left( u,\\sigma\n\\right) $ satisfying the (boundary) conditions:\n\n\\begin{enumerate}\n\\item[(C$_{1}$)] $\\limfunc{Tr}u\\left( x\\right) =x$ for $\\mathcal{H}^{2}\n-a.e. $x\\in \\Gamma _{1}$;\n\n\\item[(C$_{2}$)] $h\\left( \\limfunc{Tr}u\\left( x\\right) \\right) =0$ for \n\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma _{2}$;\n\n\\item[(C$_{3}$)] $\\limfunc{Tr}u\\left( x\\right) =\\limfunc{Tr}u\\left( \\sigma\n\\left( x\\right) \\right) $ for $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$.\n\\end{enumerate}\nThus, we can write the \\emph{knitting variational problem} in the for\n\\begin{equation}\n\\min \\left\\{ \\int_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right)dx:\\left( u,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\}.  \\label{varproblem}\n\\end{equation\n\n\\begin{figure}[htb]\t\n\\begin{center}\n\\begin{tikzpicture}\n\\node[label=below:~]  (x1) at (2,2)  {$\\bullet$};\n\\node[label=below:~ ]  (x2) at (2,0)  {$\\bullet$};\n\\node[label=below:~ ]  (x3) at (0,0)  {$\\bullet$};\n\\node[label=below:~ ]  (x4) at (1,1)  {$\\bullet$};\n  \\node[label=below:~ ]  (x5) at (0,1)  {$\\bullet$};\n\n  \\draw (x1) .. controls (3,1)  .. (x2);\n \\draw (x2) .. controls (1,-0.3)  .. (x3);\n \\draw (x3) -- (x4);\n \\draw (x4) -- (x5);\n  \\draw (x5) .. controls (1,3)  .. (x1);\n  \n  \n\\node[label=below:~]  (z1) at (8,2)  {$\\bullet$};\n\\node[label=below:~ ]  (z2) at (8,0)  {$\\bullet$};\n\\node[label=below:~ ]  (z3) at (6,0)  {$\\bullet$};\n\\node[label=below:~ ]  (z4) at (7,1)  {$\\bullet$};\n\n\n  \\draw (z1) .. controls (9,1)  .. (z2);\n \\draw (z2) -- (z3);\n \\draw[very thick] (z3) -- (z4);\n\n  \\draw (z3) .. controls (6,2)  .. (z1);  \n  \n\\draw[->] (3.5,1) -- (5.3,1);  \n  \n\\node [above] at (4.4,1.2) {$u$};\n\n\\node [above] at (3,1.2) {$\\Gamma_1$};\n\\node [above] at (10,1.2) {$\\Gamma_1=u(\\Gamma_1)$};\n\n\\node [below] at (1,-0.3) {$\\Gamma_2$};\n\\node [below] at (7.5,-0.3) {$u(\\Gamma_2),\\;\\; h(x)=0$};\n\n\\node [above] at (0.2,1.8) {$\\Gamma_3$};\n\\node [above] at (6.2,1.9) {$u(\\Gamma_3)$};\n\n\\node [above] at (0.8,1.1) {$\\Gamma_+$};\n\n\n\\node [below] at (0.8,0.6) {$\\Gamma_-$};\n\\node [below] at (7,0.65) {$u(\\Gamma_{\\pm})$};\n\n\\draw[->] (0.2,0.9) -- (0.3,0.4);  \n\\node [below] at (0,0.8) {$\\sigma$};\n\n\\node [below] at (2,1.3) {$\\Omega$};\n\\node [below] at (8,1.3) {$u ({\\Omega})$};\n  \n\\end{tikzpicture}\n\\end{center}\n\\caption{General scheme of plastic surgery.}\n\\end{figure}\n\nLet us emphasize the difference between the boundary conditions (C$_{1}$)$-\n(C$_{3}$) above (see Fig. 1). The first condition (C$_{1}$) means that the surface \n\\Gamma _{1}$ is a part of the elastic body (breast) that remains fixed. The condition (C$_{2}$),\ninstead, can be interpreted as the knitting of a part of the incised  breast \n(surface $\\Gamma _{2}$) to the fixed surface $\\mathcal{S}$ of the woman's\nchest, while (C$_{3}$) means the knitting of the cut breast surface $\\Gamma\n_{4}:=\\Gamma ^{+}\\cup \\Gamma ^{-},$ of $\\Gamma ^{+}$ into $\\Gamma ^{-}$.\nFinally, the piece $\\Gamma _{3}$ of the boundary $\\partial \\Omega $ remains\nfree and admits an arbitrary configuration depending on the knitting process.\n\n\\bigskip\n\n\\section{\\protect\\bigskip Existence of minimizers}\n\nBefore proving the main existence theorem let us justify that for each \n\\sigma \\in \\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ the boundary\ncondition (C$_{3}$) makes sense.\n\n\\begin{lemma}\n\\label{measurability_comp}Let $\\Omega \\subset \\mathbb{R}^{3}$ be an open\nbounded connected domain with locally Lipschitz boundary and $\\Gamma ^{\\pm\n}\\subset \\partial \\Omega $ be $\\mathcal{H}^{2}$-measurable sets with \n\\mathcal{H}^{2}\\left( \\Gamma ^{\\pm }\\right) >0$ and $\\mathcal{H}^{2}\\left(\n\\Gamma ^{+}\\cap \\Gamma ^{-}\\right) =0$. Then for each $u\\in \\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ and $\\sigma \\in \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $, $L\\geq 1$, the composed\nfunction $\\limfunc{Tr}u\\circ \\sigma :\\Gamma ^{+}\\rightarrow \\mathbb{R}^{3}$\nis measurable w.r.t. the measure $\\mathcal{H}^{2}$ on $\\Gamma ^{+}$.\n\\end{lemma}\n\n\\begin{proof}\nGiven $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ by the\ndensity argument there exists a sequence of continuous functions $v_{n}\n\\overline{\\Omega }\\rightarrow \\mathbb{R}^{3}$ converging to $u$ in the space \n$\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $. Consequently (see the properties $1$ and $2$ of traces), \nv_{n}=\\limfunc{Tr}v_{n}\\rightarrow \\limfunc{Tr}u$, as $\\ n\\rightarrow \\infty $,\nin $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb{R}^{3}\\right) $. Then, up\nto a subsequence, we hav\n\\begin{equation}\nv_{n}\\left( y\\right) \\rightarrow \\limfunc{Tr}u\\left( y\\right) \\;\\;\n\\forall y\\in \\Gamma ^{-}\\setminus E_{0}^{-}  \\label{appr_cont_a.e.}\n\\end{equation\nwhere $E_{0}^{-}\\subset \\Gamma ^{-}$ is some set with null Hausdorff\nmeasure. So, it remains to prove tha\n\\begin{equation}\n\\mathcal{H}^{2}\\left( \\sigma ^{-1}\\left( E_{0}^{-}\\right) \\right) =0\n\\label{measure_0}\n\\end{equation\n$\\,$because in such case we deduce from (\\ref{appr_cont_a.e.}) that the\nsequence of (continuous) functions $\\left\\{ v_{n}\\left( \\sigma \\left(\nx\\right) \\right) \\right\\} $ converges to $\\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) $ for all $x\\in \\Gamma ^{+}$ up to the negligible set of\npoints $E_{0}^{+}:=\\sigma ^{-1}\\left( E_{0}^{-}\\right) $.\n\nOn the other hand, (\\ref{measure_0}) follows easily from the left inequality\nin (\\ref{Lipschitz}) and from the definition of \\emph{Hausdorff measure}\n(see \\cite[p. 171]{F})\n\\begin{equation*}\n\\mathcal{H}^{2}\\left( E\\right) :=\\frac{\\pi }{4}\\lim_{\\varepsilon \\rightarrow\n0}\\,\\inf \\,\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}A_{i}\\right) ^{2}\n\\end{equation*\nwhere the infimum is taken over all coverings $\\left\\{ A_{i}\\right\\}\n_{i=1}^{\\infty }$ of $E$ with the diameters $\\limfunc{diam}A_{i}\\leq\n\\varepsilon $. In fact, given $\\eta >0$ we find $\\varepsilon >0$ and a\nfamily $\\left\\{ A_{i}\\right\\} _{i=1}^{\\infty }$ with $\\tbigcup_{i=1}^{\\infty\n}A_{i}\\supset E_{0}^{-}$ , $\\limfunc{diam}A_{i}\\leq \\varepsilon \/L$ an\n\\begin{equation*}\n\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}A_{i}\\right) ^{2}\\leq \\frac{\\eta }\nL^{2}}.\n\\end{equation*\nSince due to (\\ref{Lipschitz}) obviously $\\limfunc{diam}\\,\\left( \\sigma\n^{-1}\\left( A_{i}\\right) \\right) \\leq L\\limfunc{diam}\\,A_{i}\\leq \\varepsilon \n$, $i=1,2,\\dots $; \n\\\\the family $\\left\\{ \\sigma ^{-1}\\left( A_{i}\\right) \\right\\}\n_{i=1}^{\\infty }$ is a covering of $E_{0}^{+}$ an\n\\begin{equation*}\n\\sum_{i=1}^{\\infty }\\left( \\limfunc{diam}\\,\\left( \\sigma ^{-1}\\left(\nA_{i}\\right) \\right) \\right) ^{2}\\leq L^{2}\\sum_{i=1}^{\\infty }\\left( \n\\limfunc{diam}A_{i}\\right) ^{2}\\leq \\eta ,\n\\end{equation*\nwe conclude that $\\mathcal{H}^{2}\\left( E_{0}^{+}\\right) =0$, and the \n\\mathcal{H}^{2}$-measurability of $x\\mapsto \\limfunc{Tr}u\\left( \\sigma\n\\left( x\\right) \\right) $ on $\\Gamma ^{+}$ follows.\n\\end{proof}\n\nLet us give now an \\emph{a priori} estimate for the \"weighted\" integra\n\\begin{equation*}\n\\dint\\limits_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) ,\n\\end{equation*\nimplying, in particular, that the composed function $\\limfunc{Tr}u\\circ\n\\sigma $ belongs to the class $\\mathbf{L}^{2}\\left( \\partial \\Omega ;\\mathbb\nR}^{3}\\right) $.\n\n\\begin{lemma}\n\\label{estimate_w}Let $\\Omega \\subset \\mathbb{R}^{3}$ and $\\Gamma ^{\\pm\n}\\subset \\partial \\Omega $ be such as in Lemma \\ref{measurability_comp}. Then\ngiven $L\\geq 1$ there exists a constant $\\mathfrak{L}_{L}>0$ such that the inequalit\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathfrak{L}_{L\n\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr}u\\left( y\\right) \\right\\vert ^{2}\n\\mathcal{H}^{2}\\left( y\\right)   \\label{estimate_traces}\n\\end{equation\nholds whenever $u\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $\nand $\\sigma \\in \\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $.\n\\end{lemma}\n\n\\begin{proof}\nTo prove the estimate (\\ref{estimate_traces}) we employ the local\nlipschitzianity of the surfaces $\\Gamma ^{\\pm }$. Namely, given $y\\in \\Gamma\n^{-}$ we choose an (open) ball $B\\left( y,\\varepsilon _{y}\\right) $, \n\\varepsilon _{y}>0$, such that $\\Gamma ^{-}\\cap B\\left( y,\\varepsilon\n_{y}\\right) $ can be represented as the graph of a Lipschitz continuous\nfunction with respect to some (local) coordinates. Without loss of\ngenerality we can suppose that this function (say $f_{y}^{-}$) is defined on\nan open set $D_{y}^{-}$ from the cartesian product of the first two\ncomponents $x^{\\prime }:=\\left( x_{1},x_{2}\\right) $ and admits as values\nthe component $x_{3}$, i.e., \n\\begin{equation}\n\\label{Gamma_-cap}\n\\Gamma ^{-}\\cap B\\left( y,\\varepsilon _{y}\\right) =\\left\\{ \\left( z^{\\prime\n},f_{y}^{-}\\left( z^{\\prime }\\right) \\right) :z^{\\prime }\\in\nD_{y}^{-}\\right\\}.\n\\end{equation\nBy the compactness of $\\Gamma^{-}$ one can find a finite number of points \ny^{1},\\dots ,y^{q}\\in \\Gamma^{-}$ with \n\\begin{equation}\n\\Gamma ^{-}=\\Gamma ^{-}\\cap \\tbigcup\\limits_{j=1}^{q}B\\left( y^{j},\\frac\n\\varepsilon_{_{y^{j}}}}{2}\\right) .  \\label{repres.Gamma_-}\n\\end{equation\nSet $\\varepsilon _{j}:=\\varepsilon _{y^{j}}$, $D_{j}^{-}:=D_{y^{j}}^{-}$ and \n$f_{j}^{-}\\left( z^{\\prime }\\right) :=f_{y^{j}}^{-}\\left( z^{\\prime }\\right) \n$, $z^{\\prime }\\in D_{j}^{-}$, $j=1,\\dots ,q$.\n\nSimilarly, for any $x\\in \\Gamma ^{+}$ there exist $\\delta _{x}>0$, an open\ndomain $D_{x}^{+}\\subset \\mathbb{R}^{2}$ and a Lipschitz function \nf_{x}^{+}:D_{x}^{+}\\rightarrow \\mathbb{R}$ such that \n\\begin{equation*}\n\\Gamma ^{+}\\cap B\\left( x,\\delta _{x}\\right) =\\left\\{ \\left( z^{\\prime\n},f_{x}^{+}\\left( z^{\\prime }\\right) \\right) :z^{\\prime }\\in\nD_{x}^{+}\\right\\} .\n\\end{equation*\nWe do not loose generality assuming that the value of $f_{x}^{+}$ is the\nlast component of the vector $z\\in \\Gamma ^{+}$ (as in (\\ref{Gamma_-cap\n)). Again due to the compactness argument there exists a finite number of\npoints $x^{1},\\dots ,x^{r}\\in \\Gamma ^{+}$ such tha\n\\begin{equation*}\n\\Gamma ^{+}=\\Gamma ^{+}\\cap \\tbigcup\\limits_{i=1}^{r}B\\left( x^{i},\\delta\n_{i}\\right) \n\\end{equation*\nwher\n\\begin{equation}\n\\delta _{i}:=\\underset{1\\leq j\\leq q}{\\min }\\left( \\delta _{x^{i}},\\tfrac\n\\varepsilon _{j}}{2L}\\right) ,\\;\\; i=1,\\dots ,r.\n\\label{def_delta}\n\\end{equation\nSet also $D_{i}^{+}:=D_{x^{i}}^{+}$ and $f_{i}^{+}\\left( z^{\\prime }\\right)\n:=f_{x^{i}}^{+}\\left( z^{\\prime }\\right) $, $z^{\\prime }\\in D_{i}^{+}$, \ni=1,\\dots ,r$, and denote by $L_{\\Gamma }$ the maximal Lipschitz constant of\nthe functions $f_{j}^{-}:D_{j}^{-}\\rightarrow \\mathbb{R}$, $j=1,\\dots ,q$,\nand $f_{i}^{+}:D_{i}^{+}\\rightarrow \\mathbb{R}$, $i=1,\\dots ,r$.\n\nWe claim that given $i=1,\\dots ,r$ and $\\sigma \\in \\Sigma _{L}$, the image \n\\sigma \\left( \\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) \\right) $ is\ncontained in $\\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon _{j}\\right) $ for\nsome $j=1,\\dots ,q$. Indeed, let us choose $j$ such that $\\sigma \\left(\nx^{i}\\right) \\in B\\left( y^{j},\\varepsilon _{j}\/2\\right) $ (see (\\re\n{repres.Gamma_-})). Taking an arbitrary $z\\in \\Gamma ^{+}\\cap B\\left(\nx^{i},\\delta _{i}\\right) $ we, in particular, have $\\left\\vert\nz-x^{i}\\right\\vert <\\frac{\\varepsilon _{j}}{2L}$ (see (\\ref{def_delta})) and\nby (\\ref{Lipschitz}) $\\left\\vert \\sigma \\left( z\\right) -\\sigma \\left(\nx^{i}\\right) \\right\\vert <\\varepsilon _{j}\/2$. However, assuming that \n\\sigma \\left( z\\right) \\notin \\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon\n_{j}\\right) $ we have $\\left\\vert \\sigma \\left( z\\right) -y^{j}\\right\\vert\n\\geq \\varepsilon _{j}$ and henc\n\\begin{equation*}\n\\left\\vert \\sigma \\left( z\\right) -\\sigma \\left( x^{i}\\right) \\right\\vert\n\\geq \\left\\vert \\sigma \\left( z\\right) -y^{j}\\right\\vert -\\left\\vert \\sigma\n\\left( x^{i}\\right) -y^{j}\\right\\vert \\geq \\varepsilon _{j}-\\frac\n\\varepsilon _{_{j}}}{2}=\\frac{\\varepsilon _{_{j}}}{2},\n\\end{equation*\nwhich is a contradiction. In what follows we associate to each $\\sigma \\in \\Sigma _{L}$ and to each $i=1,\\dots ,r$ an index \nj=j\\left( \\sigma ,i\\right) \\in \\left\\{ 1,\\dots ,q\\right\\} $ such tha\n\\begin{equation*}\n\\sigma \\left( \\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) \\right)\n\\subset \\Gamma ^{-}\\cap B\\left( y^{j},\\varepsilon _{j}\\right) .\n\\end{equation*}\n\nThe latter inclusion allows us to define correctly the (injective) mapping \n\\psi _{\\sigma }^{i}:D_{i}^{+}\\rightarrow D_{j\\left( \\sigma ,i\\right) }^{-}$\nsuch tha\n\\begin{equation}\n\\sigma \\left( x\\right) =\\sigma \\left( x^{\\prime },f_{i}^{+}\\left( x^{\\prime\n}\\right) \\right) =\\left( \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right)\n,f_{j}^{-}\\left( \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) \\right)\n\\right) ,x^{\\prime }\\in D_{i}^{+}.  \\label{def_psi}\n\\end{equation\n>From (\\ref{Lipschitz}) it follows immediately that $\\psi _{\\sigma }^{i}$ is\nLipschitz\n\\begin{eqnarray}\n\\left\\vert \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) -\\psi _{\\sigma\n}^{i}\\left( z^{\\prime }\\right) \\right\\vert  &\\leq &\\left\\vert \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) -\\sigma \\left(\nz^{\\prime },f_{i}^{+}\\left( z^{\\prime }\\right) \\right) \\right\\vert   \\notag\n\\\\\n&\\leq &L\\left( \\left\\vert x^{\\prime }-z^{\\prime }\\right\\vert ^{2}+\\left(\nf_{i}^{+}\\left( x^{\\prime }\\right) -f_{i}^{+}\\left( z^{\\prime }\\right)\n\\right) ^{2}\\right) ^{1\/2}  \\label{Lip_psi} \\\\\n&\\leq &L\\sqrt{1+L_{\\Gamma }^{2}}\\left\\vert x^{\\prime }-z^{\\prime\n}\\right\\vert ,\\;\\; x^{\\prime },z^{\\prime }\\in D_{i}^{+}. \n\\notag\n\\end{eqnarray\nSo, by \\emph{Rademacher's Theorem,} $\\psi _{\\sigma }^{i}$ is $\\mathcal{L}^{2}\n-$a.e. differentiable on $D_{i}^{+}$ with $\\mathcal{L}^{2}$-measurable\ngradient, and the inequalit\n\\begin{equation*}\n\\left\\vert \\nabla \\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) \\right\\vert\n\\leq M:=L\\sqrt{1+L_{\\Gamma }^{2}}  \\label{bound_grad_psi}\n\\end{equation*\nholds for a.e. $x^{\\prime }\\in D_{i}^{+}$. Notice that the inverse mapping \n\\left( \\psi _{\\sigma }^{i}\\right) ^{-1}$ is well defined on $G_{\\sigma\n}^{i}:=\\psi _{\\sigma }^{i}\\left( D_{i}^{+}\\right) \\subset D_{j\\left( \\sigma\n,i\\right) }^{-}$ by the formula similar to (\\ref{def_psi}), namely\n\\begin{equation*}\n\\sigma ^{-1}\\left( y^{\\prime },f_{j}^{-}\\left( y^{\\prime }\\right) \\right)\n=\\left( \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime }\\right)\n,f_{i}^{+}\\left( \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime\n}\\right) \\right) \\right) ,y^{\\prime }\\in G_{\\sigma }^{i}.\n\\end{equation*\nIn the same way as (\\ref{Lip_psi}) we deduce, from (\\ref{Lipschitz}), that \n\\left( \\psi _{\\sigma }^{i}\\right) ^{-1}$ is Lipschitz on the (open) set \nG_{\\sigma }^{i}$ and, the estimat\n\\begin{equation}\n\\left\\vert \\nabla \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime\n}\\right) \\right\\vert \\leq M  \\label{bound_grad_psi_inv}\n\\end{equation\nholds for $\\mathcal{L}^{2}$-a.e. $y^{\\prime }\\in G_{\\sigma }^{i}$.\n\nIntegrating the function $\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx\\right) \\right) \\right\\vert ^{2}$ on the surface piece $\\Gamma ^{+}\\cap\nB\\left( x^{i},\\delta _{i}\\right) $ we pass first to the double integra\n\\begin{eqnarray}\n&&\\underset{\\Gamma ^{+}\\cap B\\left( x^{i},\\delta _{i}\\right) }{\\int \n\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right)   \\notag \\\\\n&=&\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) \\right) \\right\\vert\n^{2}\\sqrt{1+\\left\\vert \\nabla f_{i}^{+}\\left( x^{\\prime }\\right) \\right\\vert\n^{2}}dx^{\\prime }  \\label{int_change_var} \\\\\n&\\leq &\\sqrt{1+L_{\\Gamma }^{2}}\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc\nTr}u\\left( \\sigma \\left( x^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right)\n\\right) \\right) \\right\\vert ^{2}\\,dx^{\\prime }.  \\notag\n\\end{eqnarray\nDue to the representation (\\ref{def_psi}) we can make the change of\nvariables $y^{\\prime }=\\psi _{\\sigma }^{i}\\left( x^{\\prime }\\right) $, \nx^{\\prime }\\in D_{i}^{+}$, in the integral (\\ref{int_change_var}), and\nreturning then to the surface integral on a piece of $\\Gamma ^{-}$, we hav\n\\begin{eqnarray}\n&&\\diint\\limits_{D_{i}^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left(\nx^{\\prime },f_{i}^{+}\\left( x^{\\prime }\\right) \\right) \\right) \\right\\vert\n^{2}\\,dx^{\\prime }  \\notag \\\\\n&=&\\diint\\limits_{G_{\\sigma }^{i}}\\left\\vert \\limfunc{Tr}u\\left( y^{\\prime\n},f_{j}^{-}\\left( y^{\\prime }\\right) \\right) \\right\\vert ^{2}\\left\\vert \\det\n\\nabla \\left( \\psi _{\\sigma }^{i}\\right) ^{-1}\\left( y^{\\prime }\\right)\n\\right\\vert \\,dy^{\\prime }  \\notag \\\\\n&\\leq &6M^{3}\\diint\\limits_{G_{\\sigma }^{i}}\\left\\vert \\limfunc{Tr}u\\left(\ny^{\\prime },f_{j}^{-}\\left( y^{\\prime }\\right) \\right) \\right\\vert ^{2}\\sqrt\n1+\\left\\vert \\nabla f_{j}^{-}\\left( y^{\\prime }\\right) \\right\\vert ^{2}\ndy^{\\prime }  \\label{change_var1} \\\\\n&\\leq &6M^{3}\\underset{\\Gamma ^{-}}{\\int }\\left\\vert \\limfunc{Tr}u\\left(\ny\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) .  \\notag\n\\end{eqnarray\nHere we used the estimate (\\ref{bound_grad_psi_inv}) and the obvious\ninequality $\\left\\vert \\det A\\right\\vert \\leq 6\\left\\Vert A\\right\\Vert ^{3}$\n($A$ is an arbitrary $3\\times 3$-matrix). Since the sets $\\Gamma ^{+}\\cap\nB\\left( x^{i},\\delta _{i}\\right) $, $i=1,\\dots ,r$, cover the surface \n\\Gamma ^{+}$, taking into account (\\ref{int_change_var}) and (\\re\n{change_var1}) we conclude tha\n\\begin{eqnarray*}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  &\\leq &\\underset{i=\n}{\\overset{r}{\\dsum }}\\underset{\\Gamma ^{+}\\cap B\\left( x^{i},\\delta\n_{i}\\right) }{\\int }\\left\\vert \\limfunc{Tr}u\\left( \\sigma \\left( x\\right)\n\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  \\\\\n&\\leq &\\mathfrak{L}\\underset{\\Gamma ^{-}}{\\int }\\left\\vert \\limfunc{Tr\nu\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) \n\\end{eqnarray*\nwhere $\\mathfrak{L}:=6rM^{3}\\sqrt{1+L_{\\Gamma }^{2}}>0$ depends just on the\nLipschitz constant $L\\geq 1$ and on the properties of the domain $\\Omega $\n(namely, of its boundary).\\bigskip \n\\end{proof}\n\n\\medskip\n\nProving the existence theorem we pay the main attention to the validity of\nthe boundary condition (C$_{3}$) where Lemma \\ref{estimate_w} is crucial.\n\n\\begin{theorem}\n\\label{Th_exist} Let $W:\\mathbb{R}^{3\\times 3}\\rightarrow \\mathbb{R\\cup \n\\left\\{ +\\infty \\right\\} $ be a polyconvex function satisfying the growth\nassumption (\\ref{growth_cond}). Then problem (\\ref{varproblem}) admits a\nminimizer whenever there exists at least one pair $\\omega :=\\left( u,\\sigma\n\\right) \\in \\mathcal{W}_{L}$ wit\n\\begin{equation*}\nI\\left( \\omega \\right) :=\\dint_{\\Omega }W\\left( \\nabla u\\left( x\\right)\n\\right) \\,\\,dx<+\\infty .\n\\end{equation*}\n\\end{theorem}\n\n\\begin{proof}\nLet us consider a minimizing sequence $\\left\\{ \\left( u_{n},\\sigma\n_{n}\\right) \\right\\} \\subset \\mathcal{W}_{L}$ of the functional (\\re\n{Functional}), e.g., such a\n\\begin{equation}\n\\dint_{\\Omega }W\\left( \\nabla u_{n}\\left( x\\right) \\right) \\,dx\\leq \\inf\n\\left\\{ \\dint_{\\Omega }W\\left( \\nabla u\\left( x\\right) \\right) \\,dx:\\left(\nu,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\} +\\frac{1}{n}<+\\infty .\n\\label{min_seq}\n\\end{equation\nTaking into account the estimate (\\ref{growth_cond}) we deduce from (\\re\n{min_seq}) that the sequences $\\left\\{ \\nabla u_{n}\\right\\} $, $\\left\\{ \n\\mathrm{Adj\\,}\\nabla u_{n}\\right\\} $ and $\\left\\{ \\det \\,\\nabla\nu_{n}\\right\\} $ are bounded in $\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R\n^{3\\times 3}\\right) $ and in $\\mathbf{L}^{2}\\left( \\Omega ;\\mathbb{R}\\right) \n$, respectively. Applying Proposition \\ref{poincare} and the boundary\ncondition (C$_{1}$) we find a constant $C>0$ such that the inequalit\n\\begin{equation*}\n\\underset{\\Omega }{\\dint }\\left\\vert u_{n}\\left( x\\right) \\right\\vert\n^{2}dx\\leq C\\left[ \\underset{\\Omega }{\\dint }\\left\\vert \\nabla u_{n}\\left(\nx\\right) \\right\\vert ^{2}\\,dx+\\left\\vert \\dint\\limits_{\\Gamma _{1}}x\\,\n\\mathcal{H}^{2}\\left( x\\right) \\right\\vert \\right] \n\\end{equation*\nholds for each  $n\\geq 1$. So, the sequence $\\left\\{ u_{n}\\right\\}$ is bounded in $\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ and by the \\emph{Banach-Alaoglu theorem}, up to a subsequence,\nconverges weakly to some function $\\bar{u}\\in \\mathbf{W}^{1,2}\\left( \\Omega \n\\mathbb{R}^{3}\\right) $. Without loss of generality, we can also assume that \n$\\left\\{ \\mathrm{Adj\\,}\\nabla u_{n}\\right\\} $ and $\\left\\{ \\mathrm{\\det \\,\n\\nabla u_{n}\\right\\} $ converge weakly to some functions $\\xi \\in \\mathbf{L\n^{2}\\left( \\Omega ;\\mathbb{R}^{3\\times 3}\\right) $ and $\\eta \\in \\mathbf{L\n^{2}\\left( \\Omega ;\\mathbb{R}\\right) $, respectively. Now, by Theorem 8.20 \n\\cite[pp. 395-396]{Dac} due to the uniqueness of the limit we deduce that \n\\xi \\left( x\\right) =\\mathrm{Adj\\,}\\nabla u\\left( x\\right) $ and $\\eta\n\\left( x\\right) =\\mathrm{\\det \\,}\\nabla u\\left( x\\right) $ for almost all \nx\\in \\Omega $. Thus we have the weak convergence\\ of the sequence $\\left\\{ \n\\mathbb{T}\\left( \\nabla u_{n}\\right) \\right\\} $ to the vector-function \n\\mathbb{T}\\left( \\nabla u\\right) $ in the space $\\mathbf{L}^{2}\\left( \\Omega\n;\\mathbb{R}^{\\tau \\left( 3,3\\right) }\\right) $.\n\nOn the other hand, recalling that $\\left\\{ \\sigma _{n}\\right\\} \\subset\n\\Sigma _{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $ (see (\\ref{Lipschitz}))\nby \\emph{Ascoli's theorem} up to a subsequence, not relabeled, $\\left\\{\n\\sigma _{n}\\right\\} $ converges uniformly to $\\overline{\\sigma }\\in \\Sigma\n_{L}\\left( \\Gamma ^{+};\\Gamma ^{-}\\right) $.\n\nSince the integrand $W$ is polyconvex, it can be represented as $W\\left( \\xi\n\\right) =g\\left( \\mathbb{T}\\left( \\xi \\right) \\right) $, $\\xi \\in \\mathbb{R\n^{3\\times 3}$, with some convex function $g:\\mathbb{R}^{\\tau \\left(\n3,3\\right) }\\rightarrow \\mathbb{R}$, and, therefore\n\\begin{eqnarray*}\n\\underset{\\Omega }{\\dint }W\\left( \\nabla \\overline{u}\\left( x\\right) \\right)\n\\,\\,dx &=&\\underset{\\Omega }{\\dint }g\\left( \\mathbb{T}\\left( \\nabla \n\\overline{u}\\left( x\\right) \\right) \\right) \\,\\,dx\\leq \\underset\nn\\rightarrow \\infty }{\\lim \\inf }\\,\\underset{\\Omega }{\\dint }g\\left( \\mathbb\nT}\\left( \\nabla u_{n}\\left( x\\right) \\right) \\right) \\,\\,dx \\\\\n&\\leq &\\inf \\left\\{ \\underset{\\Omega }{\\dint }W\\left( \\nabla u\\left(\nx\\right) \\right) \\,\\,dx:\\left( u,\\sigma \\right) \\in \\mathcal{W}_{L}\\right\\} \n.\n\\end{eqnarray*\nThus, it remains just to prove that $\\bar{\\omega}:=\\left( \\overline{u}\n\\overline{\\sigma }\\right) \\in \\mathcal{W}_{L}$ (i.e., that the Sobolev\nfunction $\\overline{u}$ satisfies the boundary conditions (C$_{1}$)$-$(C$_{3}\n$) above with the transformation $\\overline{\\sigma }$). The validity of (C\n_{1}$) and (C$_{2}$) follows immediately from Proposition \\ref{traceconv}.\nIn fact, the weak convergence of $\\left\\{ u_{n}\\right\\} $ in $\\mathbf{W\n^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $ implies the strong convergence\nof traces $\\left\\{ \\limfunc{Tr}u_{n}\\right\\} $ in $\\mathbf{L}^{2}\\left(\n\\partial \\Omega ;\\mathbb{R}^{3}\\right) $. So, up to a subsequence, $\\limfunc\nTr}u_{n}\\left( x\\right) \\rightarrow \\limfunc{Tr}\\bar{u}\\left( x\\right) $ for \n$\\mathcal{H}^{2}$-a.e. $x\\in \\partial \\Omega $. In particular, $\\limfunc{Tr\n\\bar{u}\\left( x\\right) =x$ and $h\\left( \\limfunc{Tr}\\bar{u}\\left( x\\right)\n\\right) =0$ almost everywhere on $\\Gamma _{1}$ and on $\\Gamma _{2}$,\nrespectively (w.r.t. the Hausdorff measure).\n\nIn order to verify the condition (C$_{3}$) we observe first tha\n\\begin{equation}\n\\limfunc{Tr}{u}_{n}(x) =\\limfunc{Tr}{u}_{n}({\n\\sigma}_{n}(x)) ,n=1,2,\\dots ,\n\\label{cond_C3_n}\n\\end{equation\nfor $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$ and consider the surface\nintegra\n\\begin{equation*}\n\\mathcal{J}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( x\\right)\n-\\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right) .\n\\end{equation*\nBy the Minkowski's inequality we hav\n\\begin{equation}\n\\mathcal{J}^{1\/2}\\leq \\left( \\mathcal{J}_{1}^{n}\\right) ^{1\/2}+\\left( \n\\mathcal{J}_{2}^{n}\\right) ^{1\/2}+\\left( \\mathcal{J}_{3}^{n}\\right) ^{1\/2}\n\\label{estimate_int}\n\\end{equation\nwher\n\\begin{eqnarray*}\n&&\\mathcal{J}_{1}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u\n\\left( x\\right) -\\limfunc{Tr}u_{n}\\left( \\sigma _{n}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)  {;} \\\\\n&&\\mathcal{J}_{2}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}u_{n}\\left(\n\\sigma _{n}\\left( x\\right) \\right) -\\limfunc{Tr}\\bar{u}\\left( \\sigma\n_{n}\\left( x\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \n {;} \\\\\n&&\\mathcal{J}_{3}^{n}:=\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u\n\\left( \\sigma _{n}\\left( x\\right) \\right) -\\limfunc{Tr}\\bar{u}\\left( \\bar\n\\sigma}\\left( x\\right) \\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left(\nx\\right) .\n\\end{eqnarray*}\n\nTaking into account the equalities (\\ref{cond_C3_n}) by using Proposition\n\\ref{traceconv} we immediately obtain that $\\mathcal{J}_{1}^{n}\\rightarrow 0$\nas $n\\rightarrow \\infty $.\n\nDue to the linearity of the trace operator, applying Lemma \\ref{estimate_w}\nand again Proposition \\ref{traceconv} we arrive a\n\\begin{equation*}\n\\mathcal{J}_{2}^{n}\\leq \\mathfrak{L}_L\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr\n\\left( u_{n}-\\bar{u}\\right) \\left( x\\right) \\right\\vert ^{2}d\\mathcal{H\n^{2}\\left( x\\right) \\rightarrow 0 { \\ \\ as \\ \\ }n\\rightarrow \\infty \n.\n\\end{equation*}\n\nLet us approximate now $\\bar{u}\\in \\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R\n^{3}\\right) $ by a sequence of continuous functions $v_{k}:\\overline{\\Omega \n\\rightarrow \\mathbb{R}^{3}$, $k=1,2,\\dots $ (with respect to the norm of \n\\mathbf{W}^{1,2}\\left( \\Omega ;\\mathbb{R}^{3}\\right) $). Then (see\nProposition \\ref{traceconv}) $v_{k}=\\limfunc{Tr}v_{k}\\rightarrow \\limfunc{Tr\n\\bar{u}$ as $k\\rightarrow \\infty$ in $\\mathbf{L}^{2}\\left( \\partial \\Omega\n;\\mathbb{R}^{3}\\right) $. In particular, given $\\varepsilon >0$ there exists\nan index $k^{\\ast }\\geq 1$ such tha\n\\begin{equation*}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( y\\right) -v_{k^{\\ast\n}}\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H}^{2}\\left( y\\right) \\leq\n\\varepsilon .\n\\end{equation*\nBy using Lemma \\ref{estimate_w} similarly as was done to estimate the\nintegral $\\mathcal{J}_{2}^{n}$ we hav\n\\begin{eqnarray}\n&&\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( \\sigma _{n}\\left(\nx\\right) \\right) -v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right)   \\notag \\\\\n&\\leq &\\mathfrak{L}_{L}\\int_{\\Gamma ^{-}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left(\ny\\right) -v_{k^{\\ast }}\\left( y\\right) \\right\\vert ^{2}d\\mathcal{H\n^{2}\\left( y\\right) \\leq \\mathfrak{L}_{L}\\varepsilon ,\\;\\; n=1,2,\\dots \n,  \\label{Int_3_1}\n\\end{eqnarray\nand similarly\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert \\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left(\nx\\right) \\right) -v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right)\n\\right\\vert ^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathfrak{L}_{L}\\varepsilon \n.  \\label{Int_3_2}\n\\end{equation\nOn the other hand, by the uniform continuity of $v_{k^{\\ast }}$ and the uniform\nconvergence $\\sigma _{n}\\rightarrow \\overline{\\sigma }$ as $n\\rightarrow\n\\infty $, we find a number $n^{\\ast }\\geq 1$ such tha\n\\begin{equation*}\n\\left\\vert v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right) \\right)\n-v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert \\leq\n\\varepsilon \n\\end{equation*\nfor all $n\\geq n^{\\ast }$ and all $x\\in \\Gamma ^{+}$, and, consequently\n\\begin{equation}\n\\int_{\\Gamma ^{+}}\\left\\vert v_{k^{\\ast }}\\left( \\sigma _{n}\\left( x\\right)\n\\right) -v_{k^{\\ast }}\\left( \\bar{\\sigma}\\left( x\\right) \\right) \\right\\vert\n^{2}d\\mathcal{H}^{2}\\left( x\\right) \\leq \\mathcal{H}^{2}\\left( \\Gamma\n^{+}\\right) \\varepsilon ^{2},n\\geq n^{\\ast }.  \\label{Int_3_3}\n\\end{equation\nJoining together the inequalities (\\ref{Int_3_1}), (\\ref{Int_3_2}) and (\\re\n{Int_3_3}) we obtain tha\n\\begin{equation*}\n\\left( \\mathcal{J}_{3}^{n}\\right) ^{1\/2}\\leq \\left( \\mathfrak{L}_L\\varepsilon\n\\right) ^{1\/2}+\\left( \\mathfrak{L}_L\\varepsilon \\right) ^{1\/2}+\\left( \\mathcal\nH}^{2}\\left( \\Gamma ^{+}\\right) \\varepsilon ^{2}\\right) ^{1\/2},n\\geq\nn^{\\ast }.\n\\end{equation*\nSince $\\varepsilon >0$ is arbitrary and the constant $\\mathfrak{L}_L$ does not\ndepend on $n=1,2,\\dots $, we conclude that all the three integrals in the\nright-hand side of (\\ref{estimate_int}) tend to zero as $n\\rightarrow \\infty \n$. Thus $\\mathcal{J}=0$, or, in other words, $\\limfunc{Tr}\\bar{u}\\left(\nx\\right) -\\limfunc{Tr}\\bar{u}\\left( \\bar{\\sigma}\\left( x\\right) \\right) =0$\nfor $\\mathcal{H}^{2}$-a.e. $x\\in \\Gamma ^{+}$, and the theorem is proved.\n\\end{proof}\n\n\n\\section{Necessary conditions of optimality}\n\nIn this section, under some additional hypotheses, we deduce necessary conditions of optimality for problem (\\ref{varproblem}).\n\nTo simplify, assume that the function $W$ is twice continuously differentiable  and $h$ is continuously differentiable.\nMoreover, suppose that the surfaces $\\Gamma_1, \\Gamma_2, \\Gamma^+, \\Gamma^-, \\Gamma_4$ are sufficiently smooth. \n\nGiven $\\Gamma \\subset \\partial\\Omega,$ with $\\mathcal{H}^{2}(\\Gamma)>0,$ in what follows we denote by $\\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3})$ the family of restrictions to $\\Gamma$ of all functions $u:\\Omega\\rightarrow \\mathbb{R}^{3},$ whose gradients are continuous up to the boundary. Let us supply $\\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})$ with the natural sup-norm. \n\nWe consider the problem (\\ref{varproblem}) defined in the space $\\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3})$.\n\n\\begin{theorem}\nLet $(\\bar{u},\\bar{\\sigma})\\in \\mathbf{C}^{1}({\\Omega};\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})$ be a minimizer of problem (\\ref{varproblem}). Assume that $\\nabla h(\\bar{u}(x))\\neq 0$, $x\\in \\Gamma_2$ and ${\\rm det}\\nabla\\bar{u}(\\bar{\\sigma}(x))\\neq 0$, $x\\in\\Omega$. Then the following conditions are satisfied:\n\\begin{eqnarray}\n&& {\\rm Div}(\\nabla W)(\\nabla\\bar{u}(x))=0,\\;\\; x\\in\\Omega; \\label{f1}\\\\\n\\smallskip\n&& \\nabla W(\\nabla\\bar{u}(x))\\nu(x)=0,\\;\\; x\\in \\Gamma_3; \\label{f2}\\\\\n\\smallskip\n&& \\nabla W(\\nabla\\bar{u}(x))\\nu(x)\\times\\nabla h(\\bar{u}(x))=0,\\;\\; x\\in \\Gamma_2; \\label{f3}\\\\\n\\smallskip\n&&\\nabla W(\\nabla\\bar{u}(x))\\nu(x)=0,\\;\\; x\\in \\Gamma^{\\pm}. \\label{f4}\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{proof} Let us write the constraints in the minimization problem (\\ref{varproblem}) as $F(u,\\sigma)=0$ where the map  \n$\nF:\\mathbf{C}^{1}({\\Omega};\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})\\rightarrow \\mathbf{C}^{1}({\\Gamma}_1;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}({\\Gamma}_2;\\mathbb{R})\\times \\mathbf{C}^{1}({\\Gamma}^+;\\mathbb{R}^{3})$ is given by\n$$\nF(u,\\sigma ):=(u(x)-x,h(u(x)),u(x)-u(\\sigma(x))).\n$$\nUnder our assumptions the map $F$ and the functional $I$ are both Fr\\'{e}chet differentiable. In particular, for the (Fr\\'{e}chet) derivative of $F$ at the point $(\\bar{u},\\bar{\\sigma})$ we have\n$$\nDF(\\bar{u},\\bar{\\sigma})(\\tilde{u},\\tilde{\\sigma})(x)=\n\\left(\n\\begin{array}{c}\n\\tilde{u}(x)\\\\\n\\langle\\nabla h(\\bar{u}(x)),\\tilde{u}(x)\\rangle\\\\\n\\tilde{u}(x)-\\tilde{u}(\\bar{\\sigma}(x))-\\nabla \\bar{u}(\\bar{\\sigma}(x))\\tilde{\\sigma}(x)\n\\end{array}\n\\right).\n$$\nHere, and in what follows, $\\langle\\cdot,\\cdot\\rangle$ denotes the inner product in $\\mathbb{R}^{3}.$ Taking into account that $\\nabla h(\\bar{u}(x))\\neq0$ on $\\Gamma_2$ and that the jacobian matrix $\\nabla\\bar{u}(\\bar{\\sigma}(x))$ is not degenerated, we have that the linear operator $DF(\\bar{u},\\bar{\\sigma})$ is onto the space $\\mathbf{C}^{1}({\\Gamma}_1;\\mathbb{R}^{3})\\times \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})\\times \\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3}).$\nBy the Lagrange multipliers rule (see, e.g., \\cite{IT}) there exist linear continuous functionals $\\lambda_1,\\;\\lambda_2,\\;\\lambda^+$ on $\\mathbf{C}^{1}(\\Gamma_1;\\mathbb{R}^{3})$,\\; $\\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})$ and $\\mathbf{C}^{1}(\\Gamma^+;\\mathbb{R}^{3}),$ respectively, such that\n\\begin{equation*}\n\\int_{\\Omega}\\sum_{i,j=1}^{3}\\frac{\\partial W(\\nabla\\bar{u}(x))} {\\partial\\xi_{ij}}\\frac{\\partial\\tilde{u}_i(x)}{\\partial x_j}dx\n+\\lambda_1(\\tilde{u})+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)\n+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma})=0.\n\\end{equation*}\n\n\\noindent Applying the Divergence theorem we get\n\\begin{eqnarray}\n&&-\\int_{\\Omega}\\left\\langle {\\rm Div}{(\\nabla W(\\nabla\\bar{u}(x)))},\\tilde{u}(x)\\right\\rangle dx\n+\\int_{\\partial\\Omega}\\left\\langle{\\nabla W(\\nabla\\bar{u}(x))\\nu(x),\\tilde{u}(x)}\\right\\rangle d\\mathcal{H}^{2}(x) \\nonumber \\\\\n&&+\\lambda_1(\\tilde{u})+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma}))=0. \\label{f5} \n\\end{eqnarray}\nHere $\\nu(x)$ is the unit outer normal to the boundary. Varying $\\tilde{u}$ in (\\ref{f5}) such that $\\tilde{u}(x)=0$ on $\\partial{\\Omega}$, we obtain  (\\ref{f1}). Taking then $\\tilde{u} \\in \\mathbf{C}^{1}(\\Omega;\\mathbb{R}^{3})$ with $\\tilde{u}(x)=0$ on $\\partial\\Omega\\setminus \\Gamma_3$ we arrive at (\\ref{f2}).\nFurthermore, choosing appropriate functions $\\tilde{u}$ in (\\ref{f5}) we obtain\n\\begin{equation}\n\\label{f6}\n\\int_{\\Gamma_2}\\left\\langle\\nabla W(\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^{2}(x)\n+\\lambda_2(\\langle\\nabla h(\\bar{u}),\\tilde{u}\\rangle)=0 \n\\end{equation}\nwhenever $\\tilde{u}(x)=0,\\; x\\in\\partial\\Omega\\setminus \\Gamma_2$, and\n\\begin{equation}\n\\label{f7}\n\\int_{\\Gamma^+\\cup\\Gamma^-}\\left\\langle{\\nabla W(\\nabla\\bar{u}(x)) }\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^{2}(x)\n+\\lambda^+(\\tilde{u}-\\tilde{u}(\\bar{\\sigma})-\\nabla \\bar{u}(\\bar{\\sigma})\\tilde{\\sigma})=0\\;\\; \n\\end{equation}\nwhenever $\\tilde{u}(x)=0,\\;\\; x\\in\\partial\\Omega\\setminus (\\Gamma^+\\cup\\Gamma^-)$.\n\n\nDenote by $\\Gamma_2^0$ the part of $\\Gamma_2$ where the vectors $a(x):=\\nabla W(\\nabla\\bar{u}(x)) \\nu (x)$ and\n$b(x):=\\nabla h(\\bar{u})(x)$ are co-linear. Taking an arbitrary $c \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R})$ such that $c(x)=0$ and $\\nabla c(x)=0$ on $\\Gamma_{2}^0$, let us define\n$$\n\\hat{u}(x):=\n\\left\\{\n\\begin{array}{cl}\n\\frac{a(x)\\langle a(x),b(x)\\rangle-b(x)|a(x)|^2}{\\langle a(x),b(x)\\rangle^2-|a(x)|^2|b(x)|^2}c(x),& x\\in\\Gamma_2\\setminus\\Gamma_2^0,\\\\\n\\smallskip\n0, & x\\in\\Gamma_2^0. \n\\end{array}\n\\right.\n$$\nObviously, $\\hat{u} \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R}^3)$, $\\langle\\hat{u}(x), a(x)\\rangle=0$ and $\\langle\\hat{u}(x),b(x)\\rangle=c(x)$ for $x\\in \\Gamma_2$. From (\\ref{f6}) we\nget $\\lambda_2(c)=\\lambda_2(\\langle b,\\hat{u}\\rangle)=0$. Hence, varying $\\tilde{u} \\in \\mathbf{C}^{1}(\\Gamma_2;\\mathbb{R}^3)$ in (\\ref{f6}) in a suitable way (in particular, setting $\\hat u(x)=0$ on $\\Gamma_2^0$) we get $a(x)=0$ in $\\Gamma_2\\setminus\\Gamma_2^0.$ Thus the equality (\\ref{f3}) follows.\n\n\nFinally, taking $\\tilde{u}=0$, from (\\ref{f7}) we get $\\lambda^+=0$, and, as a consequence, \n$$\n\\int_{\\Gamma^-}\\left\\langle\\nabla W (\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^2(x)\n+\\int_{\\Gamma^+}\\left\\langle\\nabla W (\\nabla\\bar{u}(x))\\nu (x),\\tilde{u}(x)\\right\\rangle d\\mathcal{H}^2(x)=0,\n$$\nwhich implies (\\ref{f4}). \n\\end{proof}\n\n\\bigskip\n\n\\bigskip\n\n\\noindent  {\\bf Acknowledgements}\n\n\\bigskip \n\nThe authors are grateful to Hor\\'{a}cio Costa and Augusta Cardoso for fruitful discussion \nof medical aspects of the problem and also to Giovanni Leoni, who kindly communicate the proof of Lemma \\ref\n{Lemmatrace}.\n\nThis research was supported by Funda\\c{c}\\~{a}o para a Ci\\^{e}ncia e\nTecnologia (FCT), Portuguese Operational Programme for Competitiveness\nFactors (COMPETE), Portuguese National Strategic Reference Framework (QREN)\nand European Regional Development Fund (FEDER) through Project VAPS\n(EXPL\/MAT-NAN\/0606\/2013).\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nAlthough we have known since 2012 that there exists a neutral spin-0 resonance with properties (mass and couplings) that are compatible within the experimental error, with those of the scalar SM-Higgs boson~ \\cite{Aad:2012tfa, Chatrchyan:2012xdj}, these data do not exclude the existence of more fields of this sort.\nIn fact, almost all the extensions of the standard model (SM) include extra scalar \nmultiplets: complex~\\cite{Chikashige:1980ui} or real singlets~\\cite{Hill:1987ea,Davoudiasl:2004be,vanderBij:2006ne}, two~\\cite{Branco:2011iw} or more doublets~\\cite{Machado:2012ed}, and Hermitian~\\cite{Brdar:2013iea} and\/or non-Hermitian triplets~\\cite{Konetschny:1977bn,Magg:1980ut,Cheng:1980qt,Escobar:1982dp}. Moreover, extra scalar multiplets usually are introduced in a given model just in order to give masses to the neutrino and\/or charged leptons. Hence, they may have small vacuum expectation values (VEV). \nHowever, this usually implies that there might be light neutral scalars which can be easily ruled out by phenomenology. In two-Higgs doublets this is not the case when there is a positive quadratic term $\\mu^2>0$, which behaves like a positive mass square term in the scalar potential. In this case such parameters may dominate the contributions to the masses of the multiplets' members, which are almost mass degenerated, i.e. in the context of models with one or more inert doublets, see~\\cite{Machado:2012ed} and \\cite{Ma:2006km}. \n\nThe 3-3-1 models are intrinsically multi-Higgs models. For instance,\nin the minimal 3-3-1 model (here denoted by m331 for short), the charged leptons gain mass from a triplet and a sextet and the neutrino gain Majorana masses only through  the sextet ~\\cite{Pisano:1991ee,Foot:1992rh,Frampton:1992wt}. \nOn the other hand, in the model with heavy charged~\\cite{Pleitez:1992xh} or neutral leptons~\\cite{Montero:1992jk}, only the triplets are needed, if right-handed neutrinos are introduced and the type-I seesaw mechanism is implemented. \n\nBecause of the sextet, the scalar potential in the m331 becomes more complicated, for this reason it was pointed out in Ref.~\\cite{Montero:2001tq}  that the sextet can be omitted if a dimension-five effective operator, involving only triplets, is in charge of the mass generation of the charged leptons and neutrinos.  The m331 model without the sextet was called the \"reduced\" m331 model in Ref.~\\cite{Ferreira:2011hm} because only the triplets $\\rho$ and $\\chi$ are introduced. However, there are important differences between our model and those in Refs.~\\cite{Montero:2001tq,Ferreira:2011hm} as we will discuss in Sec.~\\ref{sec:con}. Moreover, as in the case of the SM, the question now is, how does this effective operator arise at tree and\/or loop level~\\cite{Ma:1998dn,Bonnet:2012kz,Sierra:2014rxa}? In the context of the m331 model, mechanisms for generating effective dimension-five operators for the case of the neutrino masses were given in Ref.~\\cite{Montero:2001ts}. \nHowever, in those works thePontecorvo-Maki-Nakawaga-Sakata (PMNS) matrix was not considered. It is far from obvious that the same parameters that allow to obtain the correct lepton masses also accommodate a realistic PMNS matrix. We show that this is possible in the m331 model with a heavy sextet which implements a sort of type-II seesawlike mechanism in the charged lepton sector and, by introducing right-handed neutrinos, neutrino masses arise from a type-I seesaw mechanism. \n\nIn fact, we show that the sextet is just a way to generate, at the tree level, the effective five-dimensional operator proposed in Ref.~\\cite{Montero:2001tq} in order to give the charged leptons their correct masses. This happens if all fields in this multiplet are heavy and its neutral components gain a small ($s^0_2$) or a zero ($s^0_1$) VEV. We also study in this model the conditions upon the dimensionless coupling constants that imply a scalar potential bounded from below, with a global minimum as well. We obtain a realistic PMNS mixing matrix as well. \n\nThe outline of this paper is the following. In Sec.~\\ref{sec:scalars} we give the scalar representation content of the m331 model and the scalar potential of the model. In Sec.~\\ref{sec:masses} we obtain the scalar mass spectra of the model under the conditions of $Z_7\\otimes Z_3$ discrete symmetries. In Sec.~\\ref{sec:pmns} we obtain the charged lepton and neutrino masses and the corresponding PMNS matrix.\nOur conclusions appear in Sec.~\\ref{sec:con}. The conditions for a stable minimum, at tree level, of the scalar potential are obtained in Appendix~\\ref{sec:appendixa}. In Appendix B we consider the Goldstone bosons in the model with exact mass matrices. \n\n\\section{The scalar sector in the m331 model}\n\\label{sec:scalars}\n\nThe scalar potential in several 3-3-1 models was considered in Refs.~\\cite{Diaz:2003dk,Nguyen:1998ui,Giraldo:2011gd,Hernandez:2014vta}. Here, however, we\nwill study only the  m331 model in the situation in which the sextet gain a small VEV, the extra scalars in the sextet are heavy,  and there is no explicit total lepton number violation in the scalar potential, which is avoided by an appropriate discrete symmetry.\n\nIn the m331 model the scalar sector is composed of a sextet $S\\sim(\\textbf{6},0)$ \nand three  triplets: $\\eta=(\\eta^0\\,\\;-\\!\\eta^{-}_1\\,\\eta^+_2)^T\\sim({\\bf3},0)$,\n$\\rho=(\\rho^+\\,\\rho^0\\,\\rho^{++})^T\\sim({\\bf3},1)$, and\n$\\chi=(\\chi^-\\,\\chi^{--}\\,\\chi^0)^T\\sim({\\bf3},-1)$, where $(x,y)$ refer to the $(SU(3)_L,U(1)_X)$ transformations. Only the triplet $\\eta$ and the sextet~$S$,\n\\begin{equation}\nS=\\left(\n\\begin{array}{ccc}\ns^0_1& \\frac{s^-_1}{\\sqrt2} & \\frac{s^+_2}{\\sqrt2}\\\\\n\\frac{s^-_1}{\\sqrt2}& S^{--}_1&\\frac{s^0_2}{\\sqrt2}\\\\\n\\frac{s^+_2}{\\sqrt2}&\\frac{s^0_2}{\\sqrt2}&S^{++}_2\n\\end{array}\n\\right),\n\\label{sextet1}\n\\end{equation}\ncouple to the leptons through the Yukawa interactions $\\overline{(\\Psi_L)^c}\\Psi_LS^*$ and $\\overline{(\\Psi_L)^c}\\Psi_L\\eta$. \n\nWe can write the $SU(3)$ multiplets above in terms of the $SU(2)$ ones. For the triplets we write\n\\begin{equation}\n\\eta=\n\\left(\n\\begin{array}{c}\n\\Phi_\\eta \\\\ \\eta^+_2\\end{array}\\right)\\sim({\\bf3},0),\\quad\n\\rho=\\left(\\begin{array}{c}\n\\Phi_\\rho \\\\ \\rho^{++}\\end{array} \\right)\\sim({\\bf3},1),\\quad\n\\chi=\\left(\n\\begin{array}{c}\n\\Phi_\\chi\\\\ \\chi^0\\end{array} \\right)\\sim({\\bf3},-1).\n\\label{tripletos}\n\\end{equation}\nThe sextet in Eq.~(\\ref{sextet1}) can be written as\n\\begin{equation}\nS=\\left(\n\\begin{array}{cc}\nT & \\frac{\\Phi_s}{\\sqrt2} \\\\\n\\frac{\\Phi^t_s}{\\sqrt2}  & S^{--}_2\n\\end{array}\n\\right),\\quad S^*=\\left(\n\\begin{array}{cc}\nT^* & \\frac{\\Phi^*_s}{\\sqrt2} \\\\\n\\frac{\\Phi^\\dagger_s}{\\sqrt2}  & S^{++}_2\n\\end{array}\n\\right),\n\\label{sextet2}\n\\end{equation}\nwhere $\\Phi^t_s$ means the transpose of the doublet $\\Phi_s$. Under the $SU(2)\\otimes U(1)_Y$ group the multiplets $\\Phi_{\\eta,\\rho,\\chi,s}$  in Eqs.~(\\ref{tripletos}) and (\\ref{sextet2}) transform as\n\\begin{equation}\n\\Phi_\\eta=\\left(\n\\begin{array}{c}\n\\eta^0\\\\ -\\eta^-_1\n\\end{array}\n\\right),\\;  \\Phi_\\rho=\\left(\n\\begin{array}{c}\n\\rho^+\\\\ \\rho^0\n\\end{array}\n\\right),\\; \\Phi_\\chi=\\left(\n\\begin{array}{c}\n\\chi^-\\\\ \\chi^{--}\n\\end{array}\n\\right), \\; \\Phi_s=\\left(\n\\begin{array}{c}\ns^+_2\\\\ s^0_2\n\\end{array}\n\\right),\n\\label{dubletos}\n\\end{equation}\nwhere these are doublets with weak hypercharge $Y=-1,+1,-3,+1$,  and $T$ in Eq.~(\\ref{sextet2})\n\\begin{equation}\nT=\\left( \\begin{array}{cc}\ns^0_1 &\\frac{s^+_1}{\\sqrt2} \\\\\n\\frac{s^+_1}{\\sqrt2} & S^{--}_1\n\\end{array}\n\\right),\n\\label{triplet}\n\\end{equation}\nis a triplet with $Y=2$. The $SU(2)$ singlets $\\eta^+_2,\\rho^{++},\\chi^0,S^{--}_2$ have $Y=+2,+4,0,+4$, respectively. \n\nThe total lepton number assignment in the scalar sector is~\\cite{Liu:1993gy}\n\\begin{equation} \nL(T^*,\\eta^-_2,\\Phi_\\chi,\\rho^{--},S^{--}_2)=+2,\\quad L(\\Phi_{\\eta,\\rho,s},\\chi^0)=0.\n\\label{ln}\n\\end{equation}\nNotice that the  only scalar doublet carrying lepton number is $\\Phi_\\chi$, and both members of the doublet have electric charge; for this reason, $\\langle \\Phi_\\chi\\rangle=0$ always. The existence of scalars carrying lepton number implies the possibility of explicit breaking of this quantum number in the scalar potential. It is possible to avoid such terms by imposing an appropriate discrete symmetry. We show one possibility in Table~\\ref{z7}.\nIn the table, $Q_{1,2}$ denote the quark triplets and $Q_3$, the quark antitriplet, $j_{mR}$ and $J$ are exotic quarks carrying electric charge of -4\/3 and 5\/3, in units of the positron charge $e$. For more details, see Ref.~\\cite{Machado:2013jca}.  \n\nSince the complex triplet  $T$ and the singlet (under $SU(2)$) $S^{++}_2$   carry lepton number they do not mix with  $\\Phi_{\\eta,\\rho,\\chi}$ if there are no lepton number violating terms in the scalar potential. As we will show below, there is some range of the parameter space that allows $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle\/v_W\\ll1$, where $v_W=246$ GeV is the electroweak energy scale.  In this situation the neutral scalar $s^0_1$ does not participate in the spontaneous symmetry breaking and $s^0_2$ has a small effect on the vector and charged lepton masses. At this stage, active neutrinos are massless and the charged leptons gain a rather small mass. However, these scalar fields are heavy and the charged leptons gain the appropriate mass through the interaction with the triplet $\\eta$ and an effective interaction involving the triplets $\\rho$ and $\\chi$. Similar to the standard model a non-Hermitian scalar triplet generates, at tree level, the neutrino masses by the interaction $(1\/\\Lambda)\\phi^0\\phi^0\\nu\\nu$ through the exchange of a complex triplet~\\cite{Ma:1998dn} (see  Fig.~\\ref{fig1}).\n\n\\begin{table}\n\n\t\\centering\n\t\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\\hline\\hline\n\t\t& $Q_{(1,2)L}$ & $Q_{3L}$ & $U_{aR}$ & $D_{aR}$ & $\\Psi_{aL}$ & $\\nu_{aR}$ & $\\eta$ & $\\chi$ & $\\rho$ & $S$ & $j_{mR}$ & $J_{R}$ \\\\ \\hline\n\t\t$Z_{7}$ & 1 & $\\omega^6$ & $\\omega^5$ & $\\omega^3$ & $\\omega^2$ & $\\omega^6$ & $\\omega^3$ & $\\omega^6$\n\t\t& $\\omega^5$ & $\\omega^4$ & $\\omega^6$ & $\\omega^2$ \\\\ \\hline\n\t\t$Z_{3}$ & 1 & w$^2$ & w  & w & w  & w  & w  & 1 \n\t\t& w & w  & 1 & w \\\\ \\hline \\hline\n\t\\end{tabular}\n\t\\caption{Transformation properties of the fermion and scalar fields under $Z_{7} \\otimes Z_{3} $. Here $\\omega=e^{i2\\pi\/7}$ and w$=e^{i2\\pi\/3}$. } \n\t\\label{z7}\n\n\\end{table}\n\nThe most general scalar potential involving the three triplets and the sextet is~\\cite{Liu:1993gy}\n\\begin{equation}\nV(\\eta,\\rho,\\chi,S)=V^{(2)}+V^{(3)}+V^{(4a)}+\\cdots +V^{(4e)},\n\\label{potential}\n\\end{equation}\nwhere \n\\begin{eqnarray}\nV^{(2)}&=&\\sum_{X=\\eta,\\rho,\\chi,S}\\mu^2_X\\textrm{Tr}(X^\\dagger X),\\nonumber \\\\\nV^{(3)}&=& \\frac{1}{3!} \\,f_1\\epsilon_{ijk} \\eta_i\\rho_j\\chi_k+f_2 (\\chi^T S^* \\rho+\\rho^T S^*\\chi) +f_3 \\eta^T S^* \\eta+\n\\frac{f_4}{3!}\\,\\epsilon_{ijk}\\epsilon_{mnl}\\;S^*_{im}S^*_{jn}S^*_{kl} ,\n\\nonumber \\\\\nV^{(4a)}&=&a_1(\\eta^\\dagger \\eta)^2+a_2(\\rho^\\dagger\\rho)^2+ a_3(\\chi^\\dagger \\chi)^2+\n\\chi^\\dagger\\chi(a_4\\eta^\\dagger\\eta+a_5\\rho^\\dagger\\rho)+a_6(\\eta^\\dagger\\eta)(\\rho^\\dagger \\rho)\\nonumber\\\\&& \na_7(\\chi^\\dagger \\eta)(\\eta^\\dagger\\chi)+a_8(\\chi^\\dagger\\rho)(\\rho^\\dagger\\chi)+\na_9(\\eta^\\dagger\\rho)(\\rho^\\dagger\\eta)+[a_{10}(\\chi^\\dagger\\eta)(\\rho^\\dagger\\eta)+H.c.],\\nonumber \\\\\nV^{(4b)}&=& b_1\\chi^\\dagger S\\hat{\\chi}\\eta+b_2\\rho^\\dagger S\\hat{\\rho}\\eta+b_3\\eta^\\dagger S[\\hat{\\chi}\\rho-\\hat{\\rho}\\chi]+H.c.,\\nonumber \\\\\nV^{(4c)}&=&   c_1\\textrm{Tr}[\\hat{\\eta}S\\hat{\\eta}S]+c_2\\textrm{Tr}[\\hat{\\rho}S\\hat{\\chi}S]+H.c.,\\nonumber \\\\\nV^{(4d)}&=&d_1(\\chi^\\dagger \\chi)\\textrm{Tr}S S^*+d_2 [(\\chi^\\dagger S)(S^* \\chi)]+d_3(\\eta^\\dagger\\eta)\\textrm{Tr}(S S^*)+d_4\\textrm{Tr}[(S^*\\eta)(\\eta^\\dagger S)]\\nonumber \\\\\n&&+d_5(\\rho^\\dagger\\rho)\\textrm{Tr}S S^*+d_6 \\textrm{Tr}[(S^* \\rho)(\\rho^\\dagger S)],\\nonumber \\\\\nV^{(4e)}&=& e_1(\\textrm{Tr}S S^*)^2+e_2\\textrm{Tr}(S S^*S S^*),\n\\label{potential2}\n\\end{eqnarray} \nand we have defined in the $V^{(b)}$ and $V^{(c)}$ terms $\\hat{x}_{ij}= \\epsilon_{ijk}x_k$, with $x=\\eta,\\rho,\\chi$. Notice also that $S^\\dagger=S^*$ since $S$ is a symmetric matrix. The conditions for having a potential bounded from below in Eq.~(\\ref{potential2}), under the conditions in Table~\\ref{z7}, are given in Appendix~\\ref{sec:appendixa}.\n\nConcerning the vacuum alignment and the conservation of the lepton number $L$, five possibilities can be considered (see also Ref.~\\cite{Liu:1993gy}) \n\\begin{enumerate}\n\\item[a)] Explicit $L$ violation and $\\langle s^0_{1,2}\\rangle\\not=0$ and arbitrary. This is the most general case and it has not been consider in the literature.  \n\\item[b)] Explicit lepton number violation in the scalar potential and  $\\langle s^0_1\\rangle = 0$ and  $\\langle s^0_2\\rangle\\not=0$ at tree level, but $\\langle s^0_1\\rangle\\not=0$  by loop corrections  \\cite{Pleitez:1992xh,Frampton:1993wu}.\n\\item[c)] No explicit $L$ violation and $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle=0$. Notwithstanding, the latter condition is not stable under radiative corrections unless a fine-tuning\nis imposed. \n \n\\item[d)] No explicit lepton number violation but $\\langle s^0_1\\rangle\\not=0$ and $\\langle s^0_2\\rangle\\not=0$. In this case there is a triplet Majoron that has been ruled out by the $Z$ invisible width~\\cite{GonzalezGarcia:1989zh}.\n\n\\item[e)] No explicit $L$ violation and $\\langle s^ 0_1\\rangle=0$ but $\\langle  s^0_2\\rangle\\not=0$. Although $L$ is conserved, there is violation of the family numbers $L_{e,\\mu,\\tau}$. In this case $\\langle s^0_1\\rangle=0$ is stable at tree and higher-order level~\\cite{Foot:1992rh,Pleitez:1992xh}. \n\\end{enumerate}\n\nHere we will consider the last case, (e), with $\\langle s^0_1\\rangle=0$ and $\\langle s^0_2\\rangle\/v_W \\ll 1$.  \nThis case occurs if the constraint $v^2_W=\\sum v^2_i=(246\\,\\textrm{GeV})^2$ (note that $i = \\rho, \\eta, s_1, s_2$) is saturated with the $v^2_\\eta$ and $v^2_\\rho$ as in Ref.~\\cite{Machado:2013jca}. \nMoreover, as we said before, in order to simplify the scalar potential, we impose a $Z_7$ discrete symmetry which forbids the $L$ violating terms, $f_3,f_4,a_{10},b_3,c_2=0$, but also the terms $c_1$ and $b_{1,2}$ are forbidden if we impose an additional discrete $Z_3$ symmetry. See Table~\\ref{z7}.\n\n\\section{Scalar mass spectra in the model}\n\\label{sec:masses}\n \nLet us consider the scalar potential in Eq.~(\\ref{potential2}) with the $Z_{7}\\otimes Z_3$ symmetries given in Table~\\ref{z7}. We make as usual $y^0=(1\/\\sqrt{2})(v_y+X^0_y+iI^0_y)$, where $y=\\eta,\\rho,\\chi$ and $s_2$.\nThe constraint equations, obtained by imposing that $\\partial V\/\\partial v_y=0$, being $V$ the potential in (\\ref{potential2}), under the conditions of the item e) above and considering all VEVs real, are given by\n\\begin{eqnarray}\n&& v_\\eta\\left[\\mu^2_\\eta+a_1v^2_\\eta+\\frac{a_6}{2}v^2_\\rho+\\frac{a_4}{2}v^2_\\chi+ d_3v^2_{s_2}+\\frac{f_1v_\\rho v_\\chi}{2\\sqrt{2}v_\\eta}\\right]=0,\\nonumber \\\\&&\nv_\\rho\\left[\\mu^2_\\rho+\\frac{a_6}{2}v^2_\\eta+ a_2v^2_\\rho+\\frac{a_5}{2}v^2_\\chi+ \\frac{d_{56}}{2} v^2_{s_2}+\\frac{(f_1v_\\eta+f_2v_{s_2})v_\\chi} {2\\sqrt{2}v_\\rho}\\right]=0,\\nonumber \\\\&&\nv_\\chi\\left[\\mu^2_\\chi+\\frac{a_4}{2}v^2_\\eta+\\frac{a_5}{2}v^2_\\rho+a_3v^2_\\chi+ d_{12}v^2_{s_2}+\\frac{(f_1v_\\eta v_\\rho+f_2v_{s_2})v_\\rho}{2\\sqrt{2}v_\\chi}\\right]=0,\\nonumber \\\\ && \nv_{s_2}\\left[2\\mu^2_S+d_3 v^2_\\eta+\\frac{d_{56}}{2} v^2_\\rho+\\frac{d_{12}}{2}v^2_\\chi+2e_{12}v^2_{s_2}+\\frac{f_2v_\\rho v_\\chi}{2\\sqrt{2}v_{s_2}}\\right]=0,\n\\label{ce}\n\\end{eqnarray} \nwhere we have defined $d_{56}=2d_5+d_6$, $d_{12}=2d_1+d_2$, and $e_{12}=2e_1+e_2$.\nNotice that no VEV can be zero unless $f_1=f_2=0$. However, if this were the case, the scalar potential has a non-Abelian symmetry larger than the rest of the Lagrangian. Hence, $f_1,f_2\\not=0$ in order that the gauge symmetry of the scalar potential is the same as the other terms of the Lagrangian. \nIn the case of the sextet, even if we had begun with $\\mu^2_S>0$ and $v_{s_2}=0$, the term $f_2\\not=0$ induces a tadpole which implies a counterterm leaving this VEV arbitrary. Hence, we assume $\\mu_S^2<0$ and $v_{s_2}\\not=0$ but is small in the sense that $v_{s_2}\/v_W\\ll1$. In Appendix~\\ref{sec:appendixb} we show explicitly the Goldstone bosons. \n\nHere, we will consider the mass matrices of the $C\\!P$-even scalars and other mass matrices in Appendix~\\ref{sec:appendixb}, assuming that $v_{s_2}\/v_\\chi<<1$. We also disregard some off-diagonal terms besides the ones from the assumption just mentioned, assuming that the respective diagonal elements are much bigger, to further simplify the matrices. In this approximation it is possible to obtain exact eigenvectors, but the case of the $CP$-even neutral scalars is more complicated and we will not consider here in detail. We show only that at least the two neutral scalars in the sextet are heavy. Analytical expressions, within the approximation above, of the masses and the eigenstates of the neutral $C\\!P$-odd and the charged sectors are given.\n \n\\subsection{Neutral $C\\!P$-even scalars}\n\\label{subsec:cpeven2}\n\nIn this sector the mass matrix $m^2$ is $5\\times5$ decompose, in the approximation used, into $4\\times4$ + $1\\times1$,  where \nthe $4\\times4$ matrix, in the basis $(X_\\eta^0,X_\\rho^0, X_\\chi^0, X_{s2}^0 )$, is given by\n\\begin{equation}\n\\left(\n\\begin{array}{cccc}\n2 a_1 v_\\eta^2-\\frac{f_1 v_\\rho v_\\chi}{\\sqrt{2} v_\\eta} & \t a_6v_\\eta v_\\rho+\\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\rho}{\\sqrt{2}}+a_4 v_\\eta v_\\chi & d_3 v_\\eta v_{s_2} \\\\\n& \\frac{4 a_2 v_\\rho^3-(\\sqrt{2} f_1 v_\\eta -f_2 v_{s_2}) v_\\chi}{2 v_\\rho} & \\frac{f_1 v_\\eta}{\\sqrt{2}}+\\frac{f_2 v_{s_2}}{2}+a_5 v_\\rho v_\\chi & \\frac{1}{2} [(2 d_5+d_6) v_\\rho v_{s_2}+f_2 v_\\chi] \\\\\n& & \\frac{4 a_3 v_\\chi^3-\\sqrt{2} f_1 v_\\eta v_\\rho-f_2 v_\\rho v_{s_2}}{2 v_\\chi} & \\frac{1}{2} [f_2 v_\\rho+(2 d_1+d_2) v_{s_2} v_\\chi] \\\\\n&  &  & (2 e_1+e_2) v_{s_2}^2-\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\label{cpeven1}\n\\end{equation}\nand the $1\\times1$ part implies the eigenvalue is $m^2_5=\\frac{(d_2  v_\\chi^2+2 d_4 v_\\eta^2 -d_6 v_\\rho^2 -2 e_2 v_{s_2}^2)v_{s_2}-2 f_2 v_\\rho v_\\chi}{4 v_{s_2}}>0$, which implies\n$d_2v^2_\\chi\/4-2 f_2 v_\\rho v_\\chi\/v_{s_2}>(2d_4v^2_\\eta- d_6 v_\\rho^2 -2 e_2 v_{s_2}^2)\/4$. Recall that $f_2<0$.\nHere the mass eigenstates are denoted by $m^2_i,\\;i=1,\\cdots,5$.\nIn fact, we can see that $m^2_5\\approx (1\/4)d_2v^2_\\chi-f_2v_\\rho v_\\chi\/v_{s_2}$; hence, this is a large mass.  One of them must be mainly that of the 125 GeV discovery at the LHC. Although we have not given details, according to the results in Ref.~\\cite{Machado:2013jca}, if $Re\\,\\rho^0=0.42 h+\\cdots$ this scalar has the same coupling with the top quarks that is numerically equal to that from the Higgs boson in the SM.  \n\n\\subsection{Neutral CP-odd scalars}\n\\label{subsec:cpodd2}\n\nThe mass matrix in Eq.~(\\ref{i2}), in the approximation of subsection ~\\ref{subsec:cpeven2},\nalso decomposes in  $3\\times3$ and  diagonal $2\\times2$ matrices. The $3\\times3$ matrix,\nin the basis $(I_\\eta^0,I_\\rho^0, I_\\chi^0) $, is\n\\begin{equation}\nM^2\\approx \\frac{1}{2} \\left(\n\\begin{array}{ccc}\n\\frac{f_1 v_\\rho v_\\chi}{\\sqrt{2} v_\\eta} & \\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\rho}{\\sqrt{2}} \\\\\n\\frac{f_1 v_\\chi}{\\sqrt{2}} & \\frac{f_1 v_\\eta v_\\chi}{\\sqrt{2} v_\\rho} & \\frac{f_1 v_\\eta}{\\sqrt{2}} \\\\\n\\frac{f_1 v_\\rho}{\\sqrt{2}} & \\frac{f_1 v_\\eta}{\\sqrt{2}} & \\frac{f_1 v_\\eta v_\\rho}{\\sqrt{2} v_\\chi} \\\\\n\\end{array}\n\\right).\n\\label{i1}\n\\end{equation}\n\nThis matrix has two zero eigenvalues and another nonzero one: \n\\begin{equation}\n\\begin{array}{cl}\nM^2_1&=M^2_2=0 \\\\\nM^2_3&= -\\frac{f_1 \\left(v_\\eta^2 \\left(v_\\rho^2+v_\\chi^2\\right)+v_\\rho^2 v_\\chi^2\\right)}{\\sqrt{2} v_\\eta v_\\rho v_\\chi}\\approx -\\frac{f_1v_\\eta v_\\chi}{\\sqrt{2}v_\\rho}>0,\\;\\;f_1<0,\n\\label{a1}\n\\end{array}\n\\end{equation}\nwith respective eigenvectors given by the columns of the matrix:\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n-\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} & -\\frac{v_\\eta v_\\chi^2}{\\sqrt{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2\\right)}} & \\frac{v_\\chi}{v_\\eta \\sqrt{\\left(\\frac{1}{v_\\rho^2}+\\frac{1}{v_\\eta^2}\\right) v_\\chi^2+1}} \\\\\n0 & v_\\rho \\sqrt{\\frac{v_\\eta^2+v_\\chi^2}{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} & \\frac{v_\\eta v_\\chi}{\\sqrt{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} \\\\\n\\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} & -\\frac{v_\\eta^2 v_\\chi}{\\sqrt{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2\\right)}} & \\frac{v_\\eta v_\\rho}{\\sqrt{\\left(v_\\rho^2+v_\\chi^2\\right) v_\\eta^2+v_\\rho^2 v_\\chi^2}} \\\\\n\\end{array}\n\\right).\n\\label{e1}\n\\end{equation}\n\nThe eigenvalues for the $2\\times2$ part are $M^2_4$ and $M^2_5$:\n\\begin{equation}\n\\begin{array}{cl}\nM^2_4&=-\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}}>0,\\;\\;f_2<0 \\\\\nM^2_5&=-\\frac{d_2 v_{s_2} v_\\chi^2+2 f_2 v_\\rho v_\\chi}{4 v_{s_2}}>0,\\;\\; 2\\vert f_2\\vert v_\\rho>d_2v_{s_2}.\n\\label{a2} \n\\end{array}\n\\end{equation}\nWe see from Eqs.~(\\ref{a1}) and (\\ref{a2}), that the three physical pseudoscalar fields are heavy and also induce the type-II seesawlike mechanism in the charged lepton sector.\n\n\\subsection{Singly charged scalars 1}\n\\label{subsec:charged12}\n\nIf the lepton number is conserved in the scalar potential (\\ref{potential}) under the conditions in Table~\\ref{z7}, as we are assuming, the charged scalar mass matrix splits in two sectors $(\\rho^+,\\eta_1^+,h^+)$ and $(\\chi^+, \\eta_2^+, h_2^+)$. \n\nIn the basis $(\\rho^+,\\eta_1^+,h^+)$, the mass matrix is\n\\begin{equation}\nm^2_+\\approx \\frac{1}{2}\\left(\n\\begin{array}{ccc}\n-\\frac{2 \\sqrt{2} f_1 v_\\eta v_\\chi-2 a_9 v_\\eta^2 v_\\rho}{4 v_\\rho} & \\frac{f_1 v_\\chi}{\\sqrt{2}}-\\frac{a_9 v_\\eta v_\\rho}{2} & 0 \\\\\n\\frac{f_1 v_\\chi}{\\sqrt{2}}-\\frac{a_9 v_\\eta v_\\rho}{2} & \\frac{1}{4} \\left(2 a_9 v_\\rho^2-\\frac{2 \\sqrt{2} f_1 v_\\rho v_\\chi}{v_\\eta}\\right) & 0 \\\\\n0 & 0 & -\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nwith the following eigenvalues,\n\\begin{equation}\n\\begin{array}{cl}\nm^2_{+1}&=0 \\\\\nm^2_{+2}&= -\\frac{f_2 v_\\rho v_\\chi}{2 v_{s_2}}>0,\\;\\;f_2<0 \\\\\nm^2_{+3}&= \\frac{\\left(v_\\eta^2+v_\\rho^2\\right) \\left(a_9 v_\\eta v_\\rho-\\sqrt{2} f_1 v_\\chi\\right)}{2 v_\\eta v_\\rho}\\approx -\\frac{1}{\\sqrt2}\\frac{f_1 v_\\eta v_\\chi}{v_\\rho}>0,\n\\end{array}\n\\label{a3}\n\\end{equation}\nand the eigenvectors are given by the column of the matrix\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n\\frac{v_\\rho}{\\sqrt{v_\\eta^2+v_\\rho^2}} & 0 & -\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\rho^2}} \\\\\n\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\rho^2}} & 0 & \\frac{v_\\rho}{\\sqrt{v_\\eta^2+v_\\rho^2}} \\\\\n0 & 1 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nAccording to (\\ref{a3}), both physical charged scalar in this sector are very heavy. \n\n\\subsection{Singly charged scalars 2}\n\\label{subsec:charged22}\n\nIn the other singly charged sector the mass matrix in the basis\n$(\\chi^+, \\eta_2^+, h_2^+)$ is \n\\begin{equation}\nM_+^2 \\approx \\frac{1}{2} \\left(\n\\begin{array}{ccc}\n-\\frac{2 \\sqrt{2} f_1 v_\\eta v_\\rho-2 a_7 v_\\eta^2 v_\\chi}{4 v_\\chi} & \\frac{1}{2} \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right) & 0 \\\\\n\\frac{1}{2} \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right) & \\frac{1}{2} v_\\chi \\left(a_7 v_\\chi-\\frac{\\sqrt{2} f_1 v_\\rho}{v_\\eta}\\right) & 0 \\\\\n0 & 0 & -\\frac{v_\\chi (2 f_2 v_\\rho+d_2 v_{s_2} v_\\chi)}{4 v_{s_2}} \\\\\n\\end{array}\n\\right)\n\\end{equation}\n\nthe mass eigenvalues are,\n\n\\begin{equation}\n\\begin{array}{cl}\nM^2_{+1}&=0 \\\\\nM^2_{+2}&= -\\frac{v_\\chi (d_2 v_{s_2} v_\\chi+2 f_2 v_\\rho)}{4 v_{s_2}}>0, \\;\\;2\\vert f_2\\vert v_\\rho>d_2v_{s_2}v_\\chi\\\\\nM^2_{+3}&= \\frac{\\left(v_\\eta^2+v_\\chi^2\\right) \\left(a_7 v_\\eta v_\\chi-\\sqrt{2} f_1 v_\\rho\\right)}{2 v_\\eta v_\\chi}>0,\n\\label{a4}\n\\end{array}\n\\end{equation}\nwith the following eigenvectors:\n\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\n-\\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} & 0 & \\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} \\\\\n\\frac{v_\\eta}{\\sqrt{v_\\eta^2+v_\\chi^2}} & 0 & \\frac{v_\\chi}{\\sqrt{v_\\eta^2+v_\\chi^2}} \\\\\n0 & 1 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nAgain, the charged scalar masses in this sector might be heavy [see Eq.~(\\ref{a4})].\n\n\\subsection{Double charge scalars}\n\\label{subsec:2charges}\n\nThe mass matrix in the basis $(\\chi^{++},\\rho^{++},S_1^{++},S_2^{++})$ is \n\\begin{equation}\nM_{++}^2 \\approx \\frac{1}{2}\\left(\n\\begin{array}{cccc}\n\\frac{v_\\rho \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\chi} & \\frac{1}{2} \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right) & 0 & 0 \\\\\n\\frac{1}{2} \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right) & \\frac{v_\\chi \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\rho} & 0 & 0 \\\\\n0 & 0 & -\\frac{v_\\chi (2 f_2 v_\\rho+d_2 v_{s_2} v_\\chi)}{4 v_{s_2}} & 0 \\\\\n0 & 0 & 0 & \\frac{v_\\chi (d_2 v_{s_2} v_\\chi-2 f_2 v_\\rho)}{4 v_{s_2}} \\\\\n\\end{array}\n\\right),\n\\end{equation}\nand the masses squared of the fields are,\n\n\\begin{equation}\n\\begin{array}{cl}\nM^2_{++1}&=0 \\\\\nM^2_{++2}&= \\frac{v_\\chi (d_2 v_{s_2} v_\\chi-2 f_2 v_\\rho)}{4 v_{s_2}}>0,\\;\\; \\\\\nM^2_{++3} &= -\\frac{v_\\chi (d_2 v_{s_2} v_\\chi+2 f_2 v_\\rho)}{4 v_{s_2}}>0 \\\\\nM^2_{++4}&= \\frac{\\left(v_\\rho^2+v_\\chi^2\\right) \\left(a_8 v_\\rho v_\\chi-\\sqrt{2} f_1 v_\\eta\\right)}{2 v_\\rho v_\\chi}>0,\n\\end{array}\n\\end{equation}\nwith the respective eigenvectors:\n\\begin{equation}\n\\left(\n\\begin{array}{cccc}\n-\\frac{v_\\chi}{\\sqrt{v_\\rho^2+v_\\chi^2}} & 0 & 0 & \\frac{v_\\rho}{\\sqrt{v_\\rho^2+v_\\chi^2}} \\\\\n\\frac{v_\\rho}{\\sqrt{v_\\rho^2+v_\\chi^2}} & 0 & 0 & \\frac{v_\\chi}{\\sqrt{v_\\rho^2+v_\\chi^2}} \\\\\n0 & 0 & 1 & 0 \\\\\n0 & 1 & 0 & 0 \\\\\n\\end{array}\n\\right).\n\\end{equation}\nIn this doubly charged sector, all physical scalars might be heavy. \n\n\\section{The lepton masses and the PMNS matrix}\n\\label{sec:pmns}\n\nThe problem of the leptonic mixing matrix has already been considered in the context of some 331 models without quarks with exotic charge, right-handed neutrinos  transforming nontrivially under $SU(3)_L$, three right-handed Majorana leptons, and with flavor  symmetries like $T_7$ \\cite{Hernandez:2015cra}, $A_4$~\\cite{Hernandez:2015tna} and extra scalars transforming as singlets under $SU(3)$. In these sort of models an $S_3$~\\cite{Hernandez:2014lpa} symmetry is also used. Here, however, we will consider the m331 using only the gauge symmetries and also right-handed neutrinos as the extra degrees of freedom. \n\nAs said before, the possibility that the lepton masses are generated by an effective dimensional operator was presented in Ref.~\\cite{Montero:2001tq}. However, here we will consider the case when the sextet is introduced but its degrees of freedom are heavy enough to generate an effective nonrenormalizable interaction. This implements a type-II seesaw mechanism in the charged lepton sector. The latter situation arises when the members of the sextet are very heavy and one of the neutral scalars gains a zero VEV and the other one a small VEV. \n\nThe Yukawa interactions are given by\n\\begin{eqnarray}\n-\\mathcal{L}^{lep}_1&=&-\\frac{1}{2}\\epsilon_{ijk}\\,\\overline{(\\Psi_{ia})^c}G^\\eta_{ab} \\Psi_{jb}\\eta_k+\\frac{1}{2\\Lambda_l}\\,\n\\overline{(\\Psi_{a})^c} \\tilde{G}^s_{ab} (\\chi^*\\rho^\\dagger+ \\rho^*\\chi^\\dagger)\\Psi_{b} \\nonumber \\\\ &+&\n\\overline{(\\Psi_{aL})}(G^\\nu)_{ab}\\nu_{aR}\\eta+ \\overline{(\\nu_{aR})^c}(M_R)_{ab}\\nu_{bR}+\\overline{(\\Psi_{ia})^c}G^s_{ab} \\Psi_{jb}S_{ij}+H.c.,\n\\label{effective1}\n\\end{eqnarray}\nwhere $\\Lambda_l$ is a mass scale related to the origin of the dimension five interaction.\nThe second term in the first line of Eq.~(\\ref{effective1}), the dimension-five operator, is generated by the loop in Fig.~\\ref{fig1}. Notice that in Eq.~(\\ref{effective1}) the interactions with the sextet appear and, although they do not contribute significantly to the charged lepton masses, the degrees of freedom in this multiplet might be exited at high energies.\n\nWe will assume, for the sake of simplicity, that $M_R$ is diagonal and that $m_{3R}\\equiv M>m_{R1},m_{R2}$, and $M^{-1}_R=(1\/M)\\bar{M}_R$, where \n$\\bar{M}=\\textrm{diag}(r_{1},r_{2},1))$ and $r_1\\equiv M\/m_{R1},r_2\\equiv M\/m_{R2}$. In this case we have the mass matrices in the lepton sector\n\\begin{equation}\nM^\\nu\\approx -\\frac{v^2_\\eta}{2}G^\\nu\\frac{\\bar{M}_R}{M}G^{\\nu T},\\quad\nM^l_{ab}=G^\\eta_{ab}\\frac{v_\\eta}{\\sqrt2}+\\frac{1}{\\Lambda_l}\\tilde{G}^s_{ab}v_\\rho v_\\chi.\n\\label{mmassa3}\n\\end{equation}\nIf it is the sextet, through the interaction $\\overline{(\\Psi_{ai})^c} G^s_{ab} \\Psi_{jb}S_{ij}$ and the $f_2$ trilinear term in the scalar potential involving $\\rho,\\chi$ and $S$, then  we have $1\/\\Lambda_l=f_2\/m^2_{s_2} $ and\n$\\tilde{G}^s=G^s$. \n\nThese mass matrices are diagonalized as follows: \n\\begin{equation} \n\\hat{M}^\\nu=V^{\\nu T}_LM^\\nu V^\\nu_L,\\;\\; \\hat{M}^l=V^{l\\dagger}_L M^l V^l_R,\n\\label{def}\n\\end{equation} \nwhere $\\hat{M}^\\nu = diag (m_1, m_2, m_3),\\;\\hat{M}^l = diag (m_e, m_\\mu, m_\\tau)$. The relation between symmetry eigenstates (primed) and mass eigenstates (unprimed) are $l^\\prime_{L,R}=V^l_{L,R}l_{L,R}$\nand $\\nu^\\prime_L=V^\\nu_L \\nu_L$, where\n$l^\\prime_{L,R}=(e^\\prime,\\mu^\\prime,\\tau^\\prime)^T_{L,R}$, $l_{L,R}=(e,\\mu,\\tau)^T_{L,R}$  and\n$\\nu ^\\prime_L=(\\nu_e\\,\\nu_\\mu\\,\\nu_\\tau)^T_L$\nand $\\nu_L=(\\nu_1\\,\\nu_2\\,\\nu_3)_L$. \n\nIn the following we assume  $v_\\chi\\approx \\Lambda_l$ and, as in Ref.~\\cite{Machado:2013jca}, $v_\\rho\\sim 54,v_\\eta\\sim 240$ GeV. \nThe neutrino mass matrix is as in Eq.~(\\ref{mmassa3}). Solving simultaneously the following equations:\n\\begin{equation}\n\\hat{M}^\\nu_{ab}=V^{\\nu T}_L M^\\nu_{ab}V^\\nu_L,\\quad \nV^{l\\dagger}_L M^l M^{l\\dagger}V^l_L=V^{l\\dagger}_R M^{l\\dagger}  M^lV^l_R=(\\hat{M}^l)^2,\\quad V_{PMNS}=V^{l\\dagger}_LV^\\nu_L,\n\\end{equation}\nwhere $M^\\nu$ and $M^l$ are defined in Eq.~(\\ref{mmassa3}), and $V_{PMNS}$ is the mixing matrix in the lepton sector (PMNS), the values for the charged lepton masses obtained are (in MeV) $(m_e,m_\\mu,m_\\tau)=(0.509648,105.541,1775.87)$\nand for the neutrinos masses (in eV) $(m_1,m_2,m_3)=(0.051,-0.0194,0.0174)$ which are consistent with \n$\\Delta m^2_{23}=2.219\\times10^{-3}\\,(\\textrm{eV})^2$ and $\\Delta m^2_{21}=7.5\\times 10^{-5}\\,(\\textrm{eV})^2$. \nThese values for the masses arise from the following values for the Yukawa matrices: $v^2_\\eta\/M=0.33$ eV, \n$G^\\nu_{11}=0.109$, $G^\\nu_{12}=0.097$, \n$G^\\nu_{13}=0.101$, $G^\\nu_{22}=0.09$, $G^\\nu_{23}=-0.02$, $G^\\nu_{33}=0.0106$ in the neutrino sector; and \n$G^s_{11}=-0.0453,G^s_{12}=-0.0076,G^s_{13}=-0.0008,G^s_{22}=0.0015,\nG^s_{23}=0.0001,G^s_{33}=1.84\\times10^{-5}$,\n$G^\\eta_{12}=,G^\\eta_{13}=G^\\eta_{13}=-0.00001$ in the charged lepton sector. The only way to avoid the latter fine-tunning is \nto consider $v_\\eta$ smaller, but in the context of the Ref.~\\cite{Machado:2013jca} this VEV is already fixed.\n\nWe obtain for the diagonalization matrices\n\\begin{equation}\nV^\\nu_L\\approx\\left(\\begin{array}{ccc}\n-0.24825& -0.57732& 0.77786\\\\\n0.73980 &-0.40539  &0.53698 \\\\\n-0.62535 &-0.70877 &0.32647 \\\\\n\\end{array}\\right)\n\\label{lep1}\n\\end{equation}\nand\n\\begin{equation}\nV^l_L\\approx\\left(\\begin{array}{ccc}\n-0.00985 &0.01457 &-0.99984 \\\\\n-0.31848& -0.94787 &-0.01067 \\\\\n0.94788&-0.31833 & -0.01398\\\\\n\\end{array}\\right),\\quad \nV^l_R\\approx\\left(\\begin{array}{ccc}\n0.00501&0.00716 & 0.99996\\\\\n0.00261&0.9910  & -0.00717\\\\\n0.99998 &-0.00265 & -0.00499\\\\\n\\end{array}\\right)\n\\label{lep2}\n\\end{equation}\n\nNotice that we have defined the lepton mixing matrix as $V_{PMNS}=V^{l\\dagger}_LV^\\nu_L$, which means that this matrix appears in the charged currents coupled to $W^-$.  We obtain from Eqs.~(\\ref{lep1}) and (\\ref{lep2}) the following values for the PMNS matrix:\n\\begin{equation}\n\\vert V_{PMNS}\\vert \\approx\\left(\\begin{array}{ccc}\n0.826& 0.548 & 0.130\\\\\n0.506&0.618  &0.602 \\\\\n0.249 &0.563 &0.788 \\\\\n\\end{array}\\right),\n\\label{pmns}\n\\end{equation}\nwhich is in agreement, within 3$\\sigma$, with the experimental data given in Ref.~\\cite{GonzalezGarcia:2012sz},\n\\begin{equation}\n\\vert V_{PMNS}\\vert \\approx\\left(\\begin{array}{ccc}\n0.795-0.846& 0.513-0.585 & 0.126-0.178\\\\\n0.205-0.543&0.416-0.730  &0.579 - 0.808 \\\\\n0.215 - 0.548 &0.409 - 0.725 &0.567 -0.800 \\\\\n\\end{array}\\right),\n\\label{pmnsexp}\n\\end{equation}\nand we see that it is possible to accommodate all lepton masses and the PMNS matrix. We do not consider $CP$ violation here.\n\n\\section{Conclusions}\n\\label{sec:con}\n\nWe have shown that even if we introduce the sextet in such a way that it practically does not contribute to the lepton masses because of the small VEVs, and since its components are very heavy, it might generate the dimension-five operator involving only the triplets $\\rho$ and $\\chi$ in Eq.~(\\ref{effective1}) trough a process like the one shown in Fig.~\\ref{fig1}. A similar operator can be obtained for the neutrinos as in Ref.~\\cite{Montero:2001tq}, but here we prefer to introduce right-handed neutrinos in order to implement a type-I seesaw mechanism. Moreover, in the charged lepton sector the mass generation is similar to the type-II seesaw mechanism inducing small masses for neutrinos through the exchange of a heavy non-Hermitian triplet~\\cite{Cheng:1980qt,Konetschny:1977bn,Magg:1980ut}.\nThe existence of several mechanisms to generate this interaction in the context of the standard model have been shown in the literature~\\cite{Ma:1998dn,Bonnet:2012kz,Sierra:2014rxa}.  Notwithstanding, the effective operator in Eq.~(\\ref{effective1}) can be originated by the effects of higher-dimension operators. \n\nIn our case, the sextet is introduced in the model as in the m331 model, and through its interactions with the other scalars in the scalar potential and with leptons in the Yukawa interactions, their degrees of freedom can be exited at high energies mainly in lepton colliders.\nNotice, that all extra scalars in the model are heavy except for two neutral scalars that correspond to the fields in a two-Higgs-doublet extension of the SM.\nWe also show the conditions under which we have a weak copositivity of the scalar potential and generate the vacuum stability at tree level and a global minimum as well. It is interesting to study the same problem at the one-loop level. For instance, in a model with two doublets (one of them inert) taking into account a neutral scalar with a mass of 125 GeV, the stability of the vacuum was shown in Ref.~\\cite{Goudelis:2013uca}; however, such analysis is beyond the scope of our paper.\n\nIn order to obtain the correct mass for the charged leptons, besides the effective interaction, it is necessary to consider the interactions with the triplet $\\eta$. For neutrinos, as we are considering that $v_{s_1}=0$, we have introduced right-handed components to generate the type-I seesaw mechanism. With the unitary (orthogonal if we neglect phases) matrices that diagonalize the mass matrices in the lepton sector, it is possible to accommodate a realistic Pontecorvo-Maki-Nakawaga-Sakata matrix. The constraints on the masses of the extra particles in the m331 model coming from lepton violation processes will be considered elsewhere.\nIn the present context we recall that it is the neutral scalar $\\rho^0$ which has the larger projection on the neutral scalar with a mass of near 125 GeV. For details see Ref.~\\cite{Machado:2013jca}.\n\nFinally, we would like to discuss the differences between our present model and those in Refs.~\\cite{Montero:2001tq,Ferreira:2011hm}. Our model is the usual minimal 3-3-1 (in the sense that the lepton sector consists only of the known leptons) and the four scalar multiplets, three triplets, and one sextet \\textit{plus} right-handed neutrinos. Although the sextet has interactions mediated by scalars [see Eq.~(\\ref{effective1})], its neutral components do not contribute significantly to the lepton masses.\nThe degrees of freedom in the sextet decouple at low energies; however, its interactions have to be taken into account for some phenomenology at sufficiently high energy. For instance, if this mechanism is implemented at the 100 GeV--1TeV scale, the interactions of the charged leptons with the left-handed neutrinos could have some signature at the LHC or other colliders. (See \\cite{Ng:2015hba} and references therein.)\n\nIn  Ref.~\\cite{Montero:2001tq}, only three triplets $\\eta,\\rho$ and $\\chi$ were considered, being the sextet avoided at all, with the charged leptons and neutrinos gain mass only through nonrenormalizable interactions. The model of Ref.~\\cite{Ferreira:2011hm} is  considered the so called \"reduced\" 3-3-1 model with only the triplets $\\rho$ and $\\chi$ (with no $\\eta$ and $S$ at all). Therefore, for generating all the fermion masses they need, besides the usual Yukawa interactions with these triplets, nonrenormalizable interactions involving the same triplets. Although this is an interesting situation, the model with only the triplets $\\rho$ and $\\chi$ has experimental troubles if we accept the existence of the Landau-like pole. See Ref.~\\cite{Dong:2014bha} for a discussion of the 3-3-1 models with only two triplets, in particular the troubles with the ``reduced\" minimal 3-3-1 model~\\cite{Ferreira:2011hm}.   \n\nThus, in the limit of a heavy sextet, our model is a 3-3-1 model with three triplets and the sextet, but the latter one does not contribute to the lepton masses and almost not at all to the spontaneous symmetry breaking since its VEVs are zero or very small in the sense of $v_{s_2}\/v_W\\ll1$. The lepton interactions with the sextet and the interactions among all the scalars \nbecome important only at high energies. Our case is more similar to the type-II seesaw mechanism in which a complex heavy triplet is in charge of generating the neutrino masses~\\cite{Ma:1998dn}.\n\n\n\\acknowledgments\n\nThe authors would like to thank for full support to CNPq  (GDC), CAPES (ACBM) and for partial\nsupport to CNPq (VP).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider the problem of online learning in Markov decision processes (MDPs) where a learner sequentially interacts with an environment \nby repeatedly taking actions that influence the future states of the environment while incurring some immediate costs. The goal of the \nlearner is to choose its actions in a way that the accumulated costs are as small as possible. Several variants of this problem have been \nwell-studied in the literature, primarily in the case where the costs are assumed to be independent and identically distributed \n\\citep{sutton,Puterman1994,ndp,Sze10}. In the current paper, we consider the case where the costs are generated by an arbitrary external \nprocess and the learner aims to minimize its total loss during the learning procedure---conforming to the learning paradigm known as \n\\emph{online learning} \\citep{CBLu06:Book,SS12}. In the online-learning framework, the performance of the learner is measured in terms of \nthe \\emph{regret} defined as the gap between the total costs incurred by the learner and the total costs of the best comparator chosen\nfrom a pre-specified class of strategies. In the case of online learning in MDPs, a natural class of strategies is the set of all \nstate-feedback policies: several works studied minimizing regret against this class both in the \nstationary-cost~\\citep{bartlett09regal,jaksch10ucrl,AYSze11} and the non-stochastic \nsetting~\\citep{even-dar09OnlineMDP,yu09ArbitraryRewards,neu10o-ssp,neu12ssp-trans,ZiNe13,DGS14,neu14o-mdp-full,AYBK14}. In the \nnon-stochastic setting, most works \nconsider MDPs with unstructured, finite state spaces and guarantee that the regret increases no faster than~$O(\\sqrt{T})$ as the number of \ninteraction rounds $T$ grows large. A notable exception is the work of \\citet{AYBK14}, who consider the special \ncase of (continuous-state) linear-quadratic control with arbitrarily changing target states, and propose an algorithm that guarantees a \nregret bound of $O(\\log^2 T)$.\n\nIn the present paper, we study another special class of MDPs that turns out to allow fast rates. Specifically, we consider the class of \nso-called \\emph{linearly solvable MDPs} (in short, LMDPs), first proposed and named so by \\citet{Tod06}.\nThis class takes its name \nafter the special property that the Bellman optimality equations characterizing\nthe optimal behavior policy take the form of a system of linear equations,\nwhich makes optimization remarkably straightforward in such problems. The\ncontinuous formulation (in both space and time) was discovered independently\nby~\\citet{Kap05} and is known as \\emph{path integral control}.\nLMDPs have many interesting properties. For example, optimal control laws for\nLMDPs can be linearly combined to derive composite optimal control laws\nefficiently~\\citep{Tod09}. Also, the inverse optimal control problem in LMDPs can be expressed as a convex optimization problem~\\citep{dvijotham2010inverse}.\nLMDPs generalize an existing duality between optimal control computation and Bayesian inference~\\citep{todorov2008general}.\nIndeed, the popular belief propagation algorithm used in dynamic probabilistic graphical models is\nequivalent the the power iteration method used to solve LMDPs~\\citep{KGO12}.\n\nThe LMDP framework has found applications in robotics~\\citep{latentkl,ariki}, crowdsourcing~\\citep{Abbasi}, and controlling the growth \ndynamics of complex networks~\\citep{klnetwork}. The related path integral control framework of \\citet{Kap05} has been applied in several \nreal-world tasks, including robot navigation~\\citep{KinjoFN2013}, motor skill \nreinforcement learning~\\citep{TheodorouJMLR10a,rombokas2013reinforcement,pireps}, aggressive car maneuvering~\\citep{aggressive} or \nautonomous flight of teams of quadrotors~\\citep{pi_uavs}.\n\n\nIn the present paper, we show that besides the aforementioned properties, the structure of LMDPs also enables constructing efficient online \nlearning procedures with very low regret. In particular, we show that, under some mild assumptions on the structure of the LMDP, the \n(conceptually) simplest online learning strategy of \\emph{following the leader} guarantees a regret of order $\\log^2 T$, vastly improving \nover the best known previous result by \\citet*{GRW12}, who prove a regret bound of order $T^{3\/4+\\epsilon}$ for arbitrarily small \n$\\epsilon>0$ under the same assumptions. Our approach is based on the observation that the optimal control law arising from the LMDP \nstructure is a smooth function of the underlying cost function, enabling rapid learning without any regularization whatsoever.\n\nThe rest of the paper is organized as follows. Section~\\ref{sec:bck} introduces the formalism of LMDPs and summarizes some basic facts that \nour technical content is going to rely on. Section~\\ref{sec:online} describes our online learning model. Our learning algorithm is \ndescribed in Section~\\ref{sec:ftl} and analyzed in Section~\\ref{sec:analysis}. Finally, we draw conclusions in Section~\\ref{sec:discussion}.\n\n\\paragraph{Notation.} We will consider several real-valued functions over a finite state-space $\\mathcal{X}$, and we will often treat these functions \nas finite-dimensional (column) vectors endowed with the usual definitions of the $\\ell_p$ norms. The set of probability distributions over \n$\\mathcal{X}$ will be denoted as $\\Delta(\\mathcal{X})$. Indefinite sums with running variables $x,y$ or $s$ are understood to run through all $\\mathcal{X}$.\n\n\\section{Background on linearly solvable MDPs}\\label{sec:bck}\nThis section serves as a quick introduction into the formalism of linearly solvable MDPs (LMDPs, \\cite{Tod06}). These decision processes \nare defined by the tuple $\\ev{\\mathcal{X},P,c}$, where $\\mathcal{X}$ is a finite set of \\emph{states}, $P:\\mathcal{X} \\rightarrow \\Delta(\\mathcal{X})$ is a transition kernel called \nthe \\emph{passive dynamics} (with $P(x'|x)$ being the probability of the process moving to state $x'$ given the previous state $x$) and \n$c:\\mathcal{X}\\rightarrow[0,1]$ is the \\emph{state-cost function}. Our Markov decision process is a sequential decision-making problem where the initial \nstate $X_0$ is drawn from some distribution $\\mu_0$, and the following steps are repeated for an indefinite number of rounds $t=1,2,\\dots$:\n\\begin{enumerate}[leftmargin=.7cm]\n \\item The learner chooses a transition kernel $Q_t:\\mathcal{X}\\rightarrow \\Delta(\\mathcal{X})$ satisfying $\\mathop{supp}Q_t(\\cdot|x) \\subseteq \n\\mathop{supp}P(\\cdot|x)$ \nfor all $x\\in\\mathcal{X}$.\n \\item The learner observes $X_t\\in\\mathcal{X}$ and draws the next state $X_{t+1}\\sim Q(\\cdot|X_t)$.\n \\item The learner incurs the cost\n \\[\n  \\ell(X_t,Q_t) = c(X_t) + \\kl{Q_t(\\cdot|X_t)}{P(\\cdot|X_t)},\n \\]\n where $\\kl{q}{p}$ is the relative entropy (or Kullback-Leibler divergence) between the probability distributions $p$ \nand $q$ defined as $\\kl{q}{p}=\\sum_x q(x)\\log \\frac{q(x)}{p(x)}$.\n\\end{enumerate}\n\nThe state-cost function $c$ should be thought of as specifying the objective for the learner in the MDP, while the relative-entropy term \ngoverns the costs associated with significant deviations from the passive dynamics. Accordingly, we refer to this component as the \n\\emph{control \ncost}. A central question in the theory of Markov decision problems is finding a behavior policy that minimizes (some notion of) the \nlong-term total costs. In this paper, we consider the problem of \\emph{minimizing the long-term average cost-per-stage} \n$\\lim\\sup_{T\\rightarrow\\infty} \\frac 1T \\sum_{t=1}^T \\ell(X_t,Q_t)$. Assuming that the passive dynamics $P$ is aperiodic and irreducible, this \nlimit is minimized by a \\emph{stationary} policy $Q$ (see, e.g., \\citet[Sec.~8.4.4]{Puterman1994}). Below, we provide two distinct \nderivations for the optimal stationary policy that minimizes the average costs under this assumption.\n\n\\subsection{The Bellman equations}\\label{sec:bellman}\nWe first take an approach rooted in dynamic programming \\citep{Ber07:DPbookVol2}, following \\citet{Tod06}. Under our assumptions, the \noptimal stationary policy minimizing the average cost is given by finding the solution to the Bellman optimality equation\n\\begin{equation}\\label{eq:bellman_V}\n v(x) = c(x) - \\lambda + \\min_{q\\in\\Delta(\\mathcal{X})} \\ev{\\kl{q}{P(\\cdot|x)} + \\sum_{x'} q(x') v(x')}\n\\end{equation}\nfor all $x\\in\\mathcal{X}$, where $v$ is called the \\emph{optimal value function} and $\\lambda\\in\\mathbb{R}$ is the average cost associated with the \noptimal policy\\footnote{This solution is guaranteed to be unique up to a constant shift of the values: if $v$ is a solution, then so is \n$v + a$ for any $a\\in \\mathbb{R}$. Unless stated otherwise, we will assume that $v$ is such that $v(x_0) = 0$ holds for a fixed state \n$x_0\\in\\mathcal{X}$.}.  Linearly solvable MDPs get their name from the fact that the Bellman optimality \nequation can be rewritten in a simple \nlinear form. To see this, observe that by elementary calculations involving Lagrange multipliers, we have\n\\begin{align*}\n \\min_{q\\in\\Delta(\\mathcal{X})} \\ev{\\kl{q}{P(\\cdot|x)} + \\sum_{x'} q(x') v(x')} =& -\\log \\sum_{x'} P(x'|x) e^{-v(x')},\n\\end{align*}\nso, after defining the exponentiated value function $z(x) = e^{-v(x)}$ for all $x$, plugging into Equation~\\eqref{eq:bellman_V} and \nexponentiating both sides gives\n\\begin{equation}\\label{eq:bellman_Z}\n z(x) = e^{\\lambda - c(x)} \\sum_{x'} P(x'|x) z(x').\n\\end{equation}\nRewriting the above set of equations in matrix form, we obtain the linear equations\n\\[\n e^{-\\lambda} z = GPz,\n\\]\nwhere $G$ is a diagonal matrix with $G_{ii} = e^{-c(i)}$. By the \\emph{Perron-Frobenius theorem} (see, e.g., Chapter~8 of \\cite{Mey00}) \nconcerning positive matrices, the above system of linear equations has a unique\\footnote{As in the case of the Bellman equations, this \nsolution is unique up to a \\emph{scaling} of $z$.} solution satisfying $z(x)\\ge 0$ for all $x$, and this \neigenvector corresponds to the largest eigenvalue $e^{-\\lambda}$ of $GP$. Since the solution of the Bellman optimality equation \n\\eqref{eq:bellman_V} is unique (up to a constant shift corresponding to a constant scaling of $z$), we obtain that $\\lambda$ is the average \ncost of the optimal policy. In summary, the Bellman optimality \nequation takes the form of a \\emph{Perron--Frobenius eigenvalue problem}, which can be efficiently solved by iterative methods such as the \nwell-known power method for finding top eigenvectors. Finally, getting back to the basic form~\\eqref{eq:bellman_V} of the Bellman equations, \nwe can conclude after simple calculations that the optimal policy can be computed  for all $x,x'$ as\n\\[\n Q(x'|x) = \\frac{P(x'|x) z(x')}{\\sum_y P(y|x) z(y)}.\n\\]\n\n\\subsection{The convex optimization view}\\label{sec:convex}\nWe also provide an alternative (and, to our knowledge, yet unpublished) view of the optimal control problem in LMDPs, based on convex \noptimization. For the purposes of this paper, we find this form to be more insightful, as it enables us to study our \nlearning problem in the framework of online convex optimization \\citep{Haz11,Haz16,SS12}. To derive this form, observe that under \nour assumptions, every feasible policy $Q$ induces a stationary distribution $\\mu_Q$ over the state space $\\mathcal{X}$ satisfying $\\mu_Q^\\mathsf{\\scriptscriptstyle T} \n= \\mu_Q^\\mathsf{\\scriptscriptstyle T} Q$. This stationary distribution and the policy together induce a distribution $\\pi_Q$ over $\\mathcal{X}^2$ defined for all $x,x'$ \nas $\\pi_Q(x,x') = \\mu_Q(x) Q(x'|x)$. We will call $\\pi_Q$ as the \\emph{stationary transition measure} induced by $Q$, which is motivated by \nthe observation that $\\pi_Q(x,x')$ corresponds to the probability of observing the transition $x\\rightarrow x'$ in the equilibrium state: \n$\\pi_Q(x,x') = \\lim_{T\\rightarrow \\infty} \\frac 1T \\sum_{t=1}^T \\PP{X_t = x, X_{t+1}=x'}$. Notice that, with this notation, the average \ncost-per-stage of policy $Q$ can be rewritten in the form\n\\[\n\\begin{split}\n &\\lim_{T\\rightarrow\\infty} \\frac 1T \\sum_{t=1}^T \\EE{\\ell(X_t,Q)} = \\sum_{x} \\mu_Q(x) \\Bpa{c(x) + \\kl{Q(\\cdot|x)}{P(\\cdot|x)}}\n \\\\\n &\\qquad\\qquad= \\sum_{x,x'} \\pi_Q(x,x') \\pa{c(x) + \\log\\frac{\\pi_Q(x,x')}{P(x'|x)\\sum_y \\pi_Q(x,y)}}\n \\\\\n &\\qquad\\qquad= \\sum_{x,x'} \\pi_Q(x,x') \\log\\frac{\\pi_Q(x,x')}{\\sum_y \\pi_Q(x,y)} + \\sum_{x,x'} \\pi_Q(x,x') \\pa{c(x) - \\log\\pa{P(x'|x)}}.\n\\end{split}\n\\]\nThe first term in the final expression above is the \\emph{negative conditional entropy} of $X'$ relative to $X$, where $(X,X')$ is a pair \nof random states drawn from $\\pi_Q$. Since the negative conditional entropy is convex in $\\pi_Q$ (for a proof, see \nAppendix~\\ref{app:convexity}) and the second term in the expression is linear in $\\pi_Q$, we can see that $\\lambda$ is a convex function of \n$\\pi_Q$. This suggests that we can \nview the optimal control problem as having to find a feasible stationary transition measure $\\pi$ that minimizes the expected costs. \nIn short, defining \n\\begin{equation}\\label{eq:statcost}\n f(\\pi;c) = \\sum_{x,x'} \\pi(x,x') \\pa{c(x) + \\log\\frac{\\pi(x,x')}{P(x'|x)\\sum_y \\pi(x,y)}}\n\\end{equation}\nand $\\Delta(M)$ as the (convex) set of feasible stationary transition measures $\\pi$ satisfying\n\\begin{equation}\\label{eq:statdist}\n\\begin{split}\n \\sum_{x'} \\pi(x,x') &= \\sum_{x''} \\pi(x'',x) \\;\\;\\;\\;\\;\\,\\qquad (\\forall x),\n \\\\\n \\sum_{x,x'} \\pi(x,x')&=1,\n \\\\\n \\pi(x,x') &\\ge 0 \\qquad\\qquad\\qquad\\qquad (\\forall x,x'),\n \\\\\n \\pi(x,x') &= 0 \\qquad\\qquad\\qquad\\qquad (\\forall x,x': P(x'|x)=0),\n\\end{split}\n\\end{equation}\nthe optimization problem can be succinctly written as $\\min_{\\pi\\in\\Delta(M)} f(\\pi;c)$. In Appendix~\\ref{app:dual}, we provide a \nderivation of the optimal control given by Equation~\\eqref{eq:bellman_Z} starting from the formulation given above. We also remark that our \nanalysis will heavily rely on the fact that $f(\\pi;c)$ is affine in $c$. \n\n\n\n\n\n\n\\section{Online learning in linearly solvable MDPs}\\label{sec:online}\nWe now present the precise learning setting that we consider in the present paper.\nWe will study an online learning scheme where for each round $t=1,2,\\dots,T$, the following steps are repeated:\n\\begin{enumerate}[leftmargin=.7cm]\n \\item The learner chooses a transition kernel $Q_t:\\mathcal{X}\\rightarrow \\Delta(\\mathcal{X})$ satisfying $\\mathop{supp}Q_t(\\cdot|x) \\subseteq \n\\mathop{supp}P(\\cdot|x)$ for all $x\\in\\mathcal{X}$.\n \\item The learner observes $X_t\\in\\mathcal{X}$ and draws the next state $X_{t+1}\\sim Q_t(\\cdot|X_t)$.\n \\item Obliviously to the learner's choice, the environment chooses state-cost function $c_t:\\mathcal{X}\\rightarrow[0,1]$.\n \\item The learner incurs the cost\n \\[\n  \\ell_t(X_t,Q_t) = c_t(X_t) + \\kl{Q_t(\\cdot|X_t)}{P(\\cdot|X_t)}.\n \\]\n \\item The environment reveals the state-cost function $c_t$.\n\\end{enumerate}\nThe key change from the stationary setting described in the previous section is that the state-cost function now may \\emph{change \narbitrarily} between each round, and the learner is only allowed to observe the costs \\emph{after it has made its decision}. We stress \nthat we assume that the learner \\emph{fully knows} the passive dynamics, so the only difficulty comes from having to deal with the \nchanging costs. As usual in the online-learning literature, our goal is to do nearly as well as the best \\emph{stationary} policy \nchosen in hindsight after observing the entire sequence of cost functions. To define our precise performance measure, we first define the \naverage reward of a policy $Q$ as\n\\[\n \\mathcal{L}_T(Q) = \\EE{\\sum_{t=1}^T \\ell_t(X_t',Q)},\n\\]\nwhere the state trajectory $X_t'$ is generated sequentially as $X_t'\\sim Q(\\cdot|X_{t-1}')$ and the expectation integrates over the \nrandomness of the \ntransitions. Having this definition in place, we can specify the best stationary policy\\footnote{The existence of the minimum is warranted \nby the fact that $\\mathcal{L}_T$ is a continuous function bounded from below on its compact domain.} $Q^*_T = \\mathop{\\mbox{ \\rm arg\\,min}}_{Q} \\mathcal{L}_T(Q)$ and define \nour \nperformance measure as the (total expected) \\emph{regret} against $Q^*_T$:\n\\[\n R_T = \\EE{\\sum_{t=1}^T \\ell_t(X_t,Q_t)} - \\mathcal{L}_T(Q^*_T),\n\\]\nwhere the expectation integrates over both the randomness of the state transitions and the potential randomization used by the \nlearning algorithm. Having access to this definition, we can now formally define the goal of the learner as having to come up \nwith a sequence of policies $Q_1,Q_2,...$ that guarantee that the total regret grows sublinearly, that is, that the average per-round \nregret asymptotically converges to zero.\n\nFor our analysis, it will be useful to define an idealized version of the above online optimization problem, where the learner is \nallowed to \\emph{immediately switch} between the stationary distributions of the chosen policies. By making use of the convex-optimization \nview given in Section~\\ref{sec:convex}, we define an auxiliary online convex optimization (or, in short, OCO, see, e.g., \n\\citealp{Haz11,SS12}) problem called the \n\\emph{idealized OCO problem} where in each round $t$, the following steps are repeated:\n\\begin{enumerate}\n \\item The learner chooses the stationary transition measure $\\pi_t\\in\\Delta(M)$.\n \\item Obliviously to the learner's choice, the environment chooses the loss function $\\widetilde{\\ell}_t = f(\\cdot;c_t)$.\n \\item The learner incurs a loss of $\\widetilde{\\ell}_t(\\pi_t)$. \n \\item The environment reveals the loss function $\\widetilde{\\ell}_t$.\n\\end{enumerate}\nThe performance of the learner \nin this setting is measured by the \\emph{idealized regret}\n\\[\n \\overline{R}_T = \\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi_t) - \\min_{\\pi\\in\\Delta(M)}\\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi).\n\\]\n\nThroughout the paper, we will consider \\emph{oblivious environments} that choose the sequence of state-cost functions without taking into \naccount the states visited by the learner. This assumption will enable us to simultaneously reason about the expected costs under any \nsequence of state distributions, and thus to make a connection between the idealized regret $\\overline{R}_T$ and the true regret $R_T$. \nThis technique was first used by \\citet{even-dar09OnlineMDP} and was shown to be essentially inevitable by \\citet{yu09ArbitraryRewards}: As \ndiscussed in their Section~3.1, no learning algorithm can avoid linear regret if the environment is not oblivious.\n\n\n\n\\section{Algorithm and main result}\\label{sec:ftl}\nIn this section, we propose a simple algorithm for online learning in LMDPs based on the ``follow-the-leader'' (FTL) strategy. On a high \nlevel, the idea of this algorithm is greedily betting on the policy that seems to have been optimal for the total costs observed so \nfar. While this strategy is known to fail catastrophically in several simple learning problems (see, e.g., \\citealt{CBLu06:Book}), it is \nknown to perform well in several important scenarios such as sequential prediction under the logarithmic loss \\citep{MF92} or prediction \nwith expert advice under bounded losses, given that losses are stationary \\citep{Kot16} and often serves as a strong benchmark \nstrategy \\citep{REGK14,SNL14}. In our learning problem, following the leader is a very natural choice of algorithm, as the convex \nformulation of Section~\\ref{sec:convex} suggests that we can effectively build on the analysis of Follow-the-Regularized-Leader-type \nalgorithms without having to explicitly regularize the objective.\n\nIn precise terms, our algorithm computes the sequence of policies $Q_1,Q_2,\\dots,Q_T$ by running FTL \\emph{in the idealized setting}: in \nround $t$, the algorithm chooses the stationary transition measure \n\\[\n\\begin{split}\n \\pi_t &= \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \\sum_{s=1}^{t-1} \\widetilde{\\ell}_s(\\pi) = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \\sum_{s=1}^{t-1} f(\\pi;c_s)\n \\\\\n &= \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} (t-1)\\cdot f\\pa{\\pi;\\frac{1}{t-1}\\sum_{s=1}^{t-1}c_s} = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \nf\\pa{\\pi;\\overline{c}_t},\n\\end{split}\n\\]\nwhere the third equality uses the fact that $f$ is affine in its second argument and the last step introduces the average state-cost \nfunction $\\overline{c}_t = \\frac {1}{t-1} \\sum_{s=1}^{t-1} c_s$. This form implies that $\\pi_t$ can be computed as the optimal control for the \nstate-cost function $\\overline{c}_t$, which can be done by following the procedure described in Section~\\ref{sec:bellman}. Precisely, we define the  \ndiagonal matrix $G_t$ with its $i$\\th~diagonal element $e^{-\\overline{c}_t(i)}$, let $\\gamma_t$ be the largest eigenvalue of $G_tP$ and $z_t$ be the \ncorresponding (unit-norm) right eigenvector. Also, let $v_t = -\\log z_t$ and $\\lambda_t = -\\log\\gamma_t$, and note that $\\lambda_t = \nf(\\pi_t;\\overline{c}_t)$ is the optimal average-cost-per-stage of $\\pi_t$ given the cost function $\\overline{c}_t$.\nFinally, we define the policy used in round $t$ as\n\\begin{equation}\\label{eq:optQ}\n Q_t(x'|x) = \\frac{P(x'|x) z_t(x')}{\\sum_y P(y|x) z_t(y)}\n\\end{equation}\nfor all $x'$ and $x$. We denote the induced stationary distribution by $\\mu_t$. The algorithm is presented as Algorithm~\\ref{alg:ftl}.\n\n\\begin{algorithm}\n \\textbf{Input:} Passive dynamics $P$.\n \\\\\\textbf{Initialization:} $\\overline{c}_1(x) = 0$ for all $x\\in\\mathcal{X}$.\n \\\\\\textbf{For $t=1,2,\\dots,T$, repeat}\n \\begin{enumerate}[leftmargin=.7cm]\n  \\item Construct $G_t = \\left[\\mbox{diag}(e^{-\\overline{c}_t})\\right]$.\n  \\item Find the right eigenvector $z_t$ of $G_t P$ corresponding to the largest eigenvalue.\n  \\item Compute the policy\n  \\[\n Q_t(x'|x) = \\frac{P(x'|x) z_t(x')}{\\sum_y P(y|x) z_t(y)}.\n\\]\n  \\item Observe state $X_{t}$ and draw $X_{t+1}\\sim Q_t(\\cdot|X_{t})$.\n  \\item Observe state-cost function $c_t$ and update $\\overline{c}_{t+1} = \\frac{\\pa{t-1}\\overline{c}_t + c_t}{t}$.\n \\end{enumerate}\n  \\caption{Follow The Leader in LMDPs} \\label{alg:ftl}\n\\end{algorithm}\n\n\nNow we present our main result. First, we state two key assumptions about the underlying passive dynamics; both of these assumptions are  \nalso made by \\citet{GRW12}.\n\\begin{assumption}\\label{ass:irred}\n The passive dynamics $P$ is irreducible and aperiodic. In particular, there exists a natural number $H>0$ such that $\\pa{P^n}(y|x)>0$ for \nall $n\\ge H$ and all $x,y\\in\\mathcal{X}$. We will refer to $H$ as the (worst-case) \\emph{hitting time}.\n\\end{assumption}\n\\begin{assumption}\\label{ass:ergod}\n The passive dynamics $P$ is ergodic in the sense that its Markov--Dobrushin ergodicity coefficient is strictly less than $1$:\n \\[\n  \\alpha(P) = \\max_{x,y\\in\\mathcal{X}} \\onenorm{P(\\cdot|x)-P(\\cdot|y)} < 1.\n \\]\n\\end{assumption}\nA standard consequence (see, e.g., \\citealt{Sen2006}) of Assumption~\\ref{ass:ergod} is that the passive dynamics mixes quickly: for any \ndistributions $\\mu,\\mu'\\in\\Delta(\\mathcal{X})$, we have\n\\[\n \\onenorm{\\pa{\\mu-\\mu'}^\\mathsf{\\scriptscriptstyle T} P}\\le \\alpha(P)\\onenorm{\\mu-\\mu'}.\n\\]\nWe will sometimes refer to $\\tau(P) = \\pa{\\log\\pa{1\/\\alpha(P)}}^{-1}$ as the \\emph{mixing time} associated with $P$. \nNow we are ready to state our main result:\n\\begin{theorem}\\label{thm:main}\n Suppose that the passive dynamics satisfies Assumptions~\\ref{ass:irred} and~\\ref{ass:ergod}. Then, the regret of Algorithm~\\ref{alg:ftl} \nsatisfies $R_T = O(\\log^2 T)$.\n\\end{theorem}\nThe asymptotic notation used in the theorem hides a number of factors that depend only on the \npassive dynamics $P$. In particular, the bound scales polynomially with the worst-case mixing time $\\tau$ of \nany optimal policy, and shows no \\emph{explicit} dependence on the number of states.\\footnote{Of course, the mixing time time does depend \non the size of the state space in general.} We explicitly state the bound at the end of the proof \npresented in the next section as Equation~\\eqref{eq:fullbound}, when all terms are formally defined.\n\n\\section{Analysis}\\label{sec:analysis}\nIn this section, we provide a series of lemmas paving the way towards proving Theorem~\\ref{thm:main}. The attentive reader may find some of \nthese lemmas familiar from related work: indeed, we build on several technical results from \\citet{even-dar09OnlineMDP,neu14o-mdp-full} and \n\\citet{GRW12}. Our main technical contribution is an efficient combination of these tools that enables us to go way beyond the \nbest known bounds for our problem, proved by \\citet{GRW12}. Throughout the section, we will assume that the conditions of \nTheorem~\\ref{thm:main} are satisfied.\n \nBefore diving into the analysis, we state some technical results that we will use several times. We defer all proofs to \nAppendix~\\ref{sec:app}. First, we present some important facts regarding LMDPs with bounded state-costs. In particular, we define $Q^*(c)$ \nas the optimal policy with respect to an arbitrary state-cost function $c$ and let $\\mathcal{C}$ be the set of all state-costs bounded \nin $[0,1]$. We define $\\mathcal{Q}^*$ as the set of optimal policies induced by state-cost functions in $\\mathcal{C}$: $\\mathcal{Q}^* = \nQ^*(\\mathcal{C})$. Observe that $Q_t\\in\\mathcal{Q}^*$ for all $t$, as $Q_t = Q^*(\\overline{c}_t)$ and $\\overline{c}_t \\in \\mathcal{C}$ for all $t$. Below, we \ngive several useful results concerning policies in $\\mathcal{Q}^*$.\nFor stating these results, let $c\\in\\mathcal{C}$ and $Q = Q^*(c)$. We first note that the average cost $\\lambda$ of $Q$ is bounded in \n$[0,1]$: By the Perron-Frobenius theorem (see, e.g., \\citealp[Chapter~8]{Mey00}), we have that the largest eigenvalue of $GP$ is \nbounded by the maximal and minimal row sums of $GP$: $e^{-\\lambda}\\in[e^{-\\max_x c(x)},e^{-c(x)}]$, which translates to \nhaving $\\lambda\\in[0,1]$ under our assumptions. The next key result bounds the value functions and the control costs in terms of the \nhitting time:\n\\begin{lemma}\\label{lem:vbound}\n For all $x,y$ and $t$, the value functions satisfy $v_t(x)-v_t(y) \\le H$. Furthermore, all policies $Q\\in\\mathcal{Q}^*$ satisfy\n\\[\n \\max_x \\kl{Q(\\cdot|x)}{P(\\cdot|x)} \\le H+1.\n\\]\n\\end{lemma}\nThe proof is loosely based on ideas from \\citet{bartlett09regal}.\nThe second statement guarantees that the mixing time $\\tau(Q) = \\pa{\\log(1\/\\alpha(Q))}^{-1}$ is finite for all policies in $\\mathcal{Q}^*$:\n\\begin{lemma}\\label{lem:mixing}\n The Markov--Dobrushin coefficient $\\alpha(Q)$ of any policy $Q\\in\\mathcal{Q}^*$ is bounded as\n \\[\n  \\alpha(Q) \\le \\alpha(P) + \\pa{1-\\alpha(P)} \\pa{1-e^{-H-2}}<1.\n \\]\n\\end{lemma}\nThe proof builds on the previous lemma and uses standard ideas from Markov-chain theory.\nIn what follows, we will use $\\tau = \\max_{Q\\in\\mathcal{Q}^*} \\tau(Q)$ and $\\alpha = \\max_{Q\\in\\mathcal{Q}^*} \\alpha(Q)$ to denote the worst-case \nmixing time and ergodicity coefficient, respectively. With this notation, we can state the following lemma that establishes that the value \nfunctions are $2\\tau$-Lipschitz with respect to the state-cost function. For pronouncing and proving the statement, it is useful to define \nthe \\emph{span seminorm} $\\spannorm{c} = \\max_x c(x) - \\min_y c(y)$. Note \nthat it is easy to show that $\\spannorm{\\cdot}$ is indeed a seminorm as it satisfies all the requirements to be a norm except that it maps \nall constant vectors (and not just zero) to zero. \n\\begin{lemma}\\label{lem:c_to_v}\n Let $f$ and $g$ be two state-cost functions taking values in the interval $[0,1]$ and let $v_f$ and $v_g$ be the corresponding optimal \nvalue \nfunctions. Then,\n\\[\n \\spannorm{v_f - v_g} \\le 2\\tau \\infnorm{f-g}.\n\\]\n\\end{lemma}\n The proof roughly follows the proof of Proposition~3 of \\citet{GRW12}, with the slight difference that we make the constant factor in the \nbound explicit. A consequence of this result is our final key lemma in this section that actually makes our fast rates possible: a bound on \nthe change-rate of the policies chosen by the algorithm.\n\\begin{lemma}\\label{lem:change} \n$\\max_x \\onenorm{Q_t(\\cdot|x) - Q_{t+1}(\\cdot|x)} \\le \\frac{\\tau}{t}$.\n\\end{lemma}\nThe proof is based on ideas by \\citet{GRW12}. As for the proof of Theorem~\\ref{thm:main}, we follow the path of \n\\citet{even-dar09OnlineMDP,neu14o-mdp-full,GRW12}, and first analyze the idealized setting where the learner is allowed to directly pick \nstationary distributions instead of policies. Then, we show how to relate the idealized regret of FTL to its true regret in the original \nproblem.\n\n\\subsection{Regret in the idealized OCO problem}\nLet us now consider the idealized online convex optimization problem described at the end of Section~\\ref{sec:online}. \nIn this setting, our algorithm can be formally stated as choosing the stationary transition measure $\\pi_t = \\mathop{\\mbox{ \\rm arg\\,min}}_{\\pi\\in\\Delta(M)} \nf(\\pi;\\overline{c}_t)$. This view enables us to follow a standard proof technique for analyzing online convex optimization algorithms, going back to \nat least \\citet{MF92}.\nThe first ingredient of our proof is the so-called ``follow-the-leader\/be-the-leader'' lemma \\citet[Lemma~3.1]{CBLu06:Book}:\n\\begin{lemma} \\label{lem:btl}\n$\\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi_{t+1}) \\le \\min_\\pi \\sum_{t=1}^T \\widetilde{\\ell}_t(\\pi)$.\n\\end{lemma}\nThe second step exploits the bound on the change rate of the policies to show that looking one step into the future does not buy much \nadvantage. Note however that controlling the change rate is not sufficient by itself, as our loss functions are effectively unbounded.\n\\begin{lemma}\\label{lem:price}\n$\\sum_{t=1}^T \\pa{\\widetilde{\\ell}_t(\\pi_{t}) - \\widetilde{\\ell}_t(\\pi_{t+1})} \\le 2 \\pa{\\tau^2+1} \\pa{1+\\log T}$.\n\\end{lemma}\nIn the interest of space, we only provide a proof sketch here and defer the full proof to Appendix~\\ref{app:price}.\n\\begin{proofsketch}\n Let us define $\\Delta_t = \\overline{c}_{t+1} - \\overline{c}_{t}$. By exploiting the affinity of $f$ in its second argument, we can start by proving\n$\\lambda_t - \\lambda_{t+1} \\le \\infnorm{\\Delta_t}$. Furthermore, by using the form of the optimal policy $Q_t$ given in Eq.~\\eqref{eq:optQ} \nand the form of $f$ given in Eq.~\\eqref{eq:statcost}, we can obtain\n\\begin{align*}\n \\widetilde{\\ell}_t(\\pi_{t}) - \\widetilde{\\ell}_t(\\pi_{t+1}) &= \\pa{\\mu_{t} - \\mu_{t+1}}^\\mathsf{\\scriptscriptstyle T} \\pa{c_t + \\overline{c}_{t}} + \\mu_{t+1}^\\mathsf{\\scriptscriptstyle T} \\pa{\\overline{c}_{t} - \n\\overline{c}_{t+1}}\n + \\lambda_{t} - \\lambda_{t+1}\n \\\\\n &\\le 2 \\onenorm{\\mu_{t+1} - \\mu_t} + 2\\infnorm{\\Delta_t}.\n\\end{align*}\nThe first term can be bounded by a simple argument (see, e.g., Lemma~4 of \\citealt{neu14o-mdp-full}) that leads to\n\\[\n\\onenorm{\\mu_{t+1} - \\mu_t} \\le \\max\\ev{\\tau(Q_t),\\tau(Q_{t+1})} \\max_x \\onenorm{Q_{t+1}(\\cdot|x)-Q_t(\\cdot|x)}.\n\\]\nNow, the first factor can be bounded by $\\tau$ and the second by appealing to Lemma~\\ref{lem:change}. The proof is \nconcluded by plugging the above bounds into Equation~\\eqref{eq:lpidiff}, using $\\infnorm{\\Delta_t} \\le 1\/t$, summing up both sides, and \nnoting that $\\sum_{t=1}^T 1\/t \\le 1 + \\log T$.\n\\end{proofsketch}\nPutting Lemmas~\\ref{lem:btl} and~\\ref{lem:price} together, we obtain the following bound on the idealized regret of FTL:\n\\begin{lemma}\\label{lem:ideal}\n$\\overline{R}_T \\le 2\\pa{\\tau^2+1}\\pa{1+\\log T}$.\n\\end{lemma}\n\n\n\\subsection{Regret in the reactive setting}\nWe first show that the advantage of the true best policy $Q^*_T$ over our final policy $Q_{T+1}$ is bounded. \n\\begin{lemma} Let $p^* = \\min_{x,x':P(x'|x)>0} P(x'|x)$ be the smallest non-zero transition probability under the passive dynamics and $B = \n-\\log p^*$. Then,\n$ \\sum_{t=1}^T \\overline{\\loss}_t(\\pi_{T+1}) - \\mathcal{L}_T(Q_T^*) \\le \\pa{2\\tau + 2}\\pa{B+1}$.\n\\end{lemma}\nThe proof follows from applying Lemma~1 from \\citet{neu14o-mdp-full} and observing that $\\ell_t(X_t,Q_T^*) \\le B+1$ holds for all $t$.\nIt remains to relate the total cost of FTL to the total idealized cost of the algorithm. This is done in the following lemma:\n\\begin{lemma} \\label{lem:trueloss}\n$ \\sum_{t=1}^T \\pa{\\EE{\\ell_t(Q_t,X_t)} - \\overline{\\loss}_t(\\pi_{t})} \\le \\pa{\\tau+1}^3 \\pa{1+\\log T}^2 + 2\\pa{\\tau + 1}\\pa{3+\\log T}$.\n\\end{lemma}\n\\begin{proof}\n  Let $p_t(x) = \\PP{X_t = x}$. Similarly to the proof of Lemma~\\ref{lem:price}, we rewrite $\\overline{\\loss}_t(\\pi_t)$ using \nEquation~\\eqref{eq:lpiform} to obtain\n \\[\n \\begin{split}\n  \\EE{\\ell_t(Q_t,X_t) - \\overline{\\loss}_t(\\pi_{t})}\n  &=\n  \\sum_{x} \\pa{p_t(x) - \\mu_t(x)} \\pa{c_t(x) + v_t(x) + \\lambda_{t} - \\overline{c}_{t}(x) - \\sum_{x'} Q_t(x'|x) v_t(x')}\n  \\\\\n  &\\le  \\sum_{x} p_t(x) \\pa{v_t(x) - \\sum_{x'} Q_t(x'|x) v_{t}(x')} + \\onenorm{p_t- \\mu_t},\n \\end{split}\n \\]\n where the last step uses $\\sum_{x} \\mu_t(x) Q_t(x'|x) = \\mu_t(x')$ and $\\infnorm{c_t - \\overline{c}_t}\\le 1$.\n Now, noticing that $\\sum_{x} p_t(x) Q_t(x'|x) = p_{t+1}(x')$, we obtain\n \\[\n \\begin{split}\n  &\\sum_{t=1}^T \\EE{\\ell_t(Q_t,X_t) - \\overline{\\loss}_t(\\pi_{t})}\n  \\le \\sum_{t=1}^T \\pa{p_t-p_{t+1}}^\\mathsf{\\scriptscriptstyle T} v_t + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t}\n  \\\\\n  &\\quad= \\sum_{t=1}^{T} p_t^\\mathsf{\\scriptscriptstyle T} \\pa{v_t - v_{t-1}} + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t} - p_{T+1}^\\mathsf{\\scriptscriptstyle T} v_T\n  \\le \\sum_{t=1}^T \\frac{2\\tau}{t} + \\sum_{t=1}^T \\onenorm{\\mu_t - p_t} - p_{T+1}^\\mathsf{\\scriptscriptstyle T} v_T,\n \\end{split}\n \\]\n where the last inequality uses Lemma~\\ref{lem:c_to_v} and $\\infnorm{\\overline{c}_{t} - \\overline{c}_{t-1}}\\le 1\/t$ to bound the first term. By \nLemma~\\ref{lem:c_to_v}, this last term can be bounded by $\\norm{v_T}_s = \\norm{v_T - v_0}_s\\le 2\\tau\\infnorm{\\overline{c}_T} \\le 2\\tau$, where $v_0$ \nis the value function corresponding to the all-zero state-cost function.\n \n In the rest of the proof, we are going to prove the inequality\n \\begin{equation}\\label{eq:pmudiff}\n \\onenorm{\\mu_t - p_t} \\le 2 e^{-\\pa{t-1}\/\\tau} +  \\frac{2(\\tau+1)^3\\pa{1+ \\log t}}{t}.\n \\end{equation}\n It is easy to see that this trivially holds for $\\pa{2 \\tau \\log t}\/t \\ge \n1$, so we will assume that the contrary holds in the following derivations.\nTo prove Equation~\\eqref{eq:pmudiff} for larger values of $t$, we can follow the proofs \nof Lemma~5 of \\citet{neu14o-mdp-full} or Lemma~5.2 of \\citet{even-dar09OnlineMDP} to obtain\n\\begin{equation}\\label{eq:mu_to_p}\n \\onenorm{\\mu_t - p_t} \\le 2 e^{-\\pa{t-1}\/\\tau} +  \\tau \\pa{\\tau+1} \\sum_{n=1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n}.\n\\end{equation}\nFor completeness, we include a proof in Appendix~\\ref{app:mu_to_p}.\nFor bounding the last term, we split the sum at $B=\\left\\lfloor t - \\tau\\log t\\right\\rfloor$:\n\\[\n \\begin{split}\n  \\sum_{n=1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n} &= \n  \\sum_{n=1}^{B} \\frac{e^{-(t-n)\/\\tau}}{n} + \\sum_{n=B+1}^{t-1} \\frac{e^{-(t-n)\/\\tau}}{n}\n \\\\\n &= e^{-(t-B)\/\\tau} \\sum_{n=1}^{B} \\frac{e^{-(B-n)\/\\tau}}{n} + \\sum_{n=B+1}^{t} \\frac{e^{-(t-n)\/\\tau}}{n}\n \\\\\n &\\le \\frac{1}{t} \\cdot \\frac{1}{1-e^{-1\/\\tau}} + \\frac{\\tau\\log t}{t-\\tau\\log t} \\le \n \\frac{\\tau }{t} + \\frac{\\tau\\log t}{t} \\cdot \\frac{1}{1-\\pa{\\tau\\log t}\/ t}\n \\\\\n &\\le \\frac{\\tau}{t} + \\frac{2\\tau \\log t}{t} \\le \\frac{2\\tau\\pa{1 + \\log t}}{t},\n \\end{split}\n\\]\nwhere the first inequality follows from bounding the $1\/n$ factors by $1$ and $1\/B$, respectively, and bounding the sums by \nthe full geometric sums. The second-to-last inequality follows from our assumption that $(2\\tau\\log t)\/t \\le 1$.  That is, we have \nsuccessfully proved Equation~\\eqref{eq:pmudiff}. Now the statement of the lemma follows from summing up for all $t$ and noting that \n$\\sum_{t=1}^T \\frac{1}{t} \\le 1 + \\log T$ and $\\sum_{t=1}^T e^{-\\pa{t-1}\/\\tau} \\le \\tau + 1$.\n\\end{proof}\nNow the proof of Theorem~\\ref{thm:main} follows easily from combining the bounds of Lemmas~\\ref{lem:ideal}--\\ref{lem:trueloss}. The result \nis\n\\begin{equation}\\label{eq:fullbound}\n R_T \\le \n 2\\pa{\\tau+1}^3 \\pa{1+\\log T}^2 + 2\\pa{\\tau^2 + \\tau + 2}  (3+\\log T) + \\pa{2\\tau+2}\\pa{B+2}.\n\\end{equation}\nThus, we can see that the bound indeed demonstrates a polynomial dependence on the mixing time $\\tau$, \nand depends logarithmically on the smallest non-zero transition probability $p^*$ via $B = -\\log p^*$.\n\n\n\n\\section{Discussion}\\label{sec:discussion}\nIn this paper, we have shown that, besides the well-established computational advantages, linearly solvable MDPs also admit a remarkable\ninformation-theoretic advantage: fast learnability in the online setting. In particular, we show that achieving a regret of $O(\\log^2 T)$ \nis achievable by the simple algorithm of following the leader, thus greatly improving on the best previously known regret bounds of \n$O(T^{3\/4})$. At first sight, our improvement may appear dramatic: in their paper, \\citet{GRW12} pose the possibility of improving their \nbounds to $O(\\sqrt{T})$ as an important open question (Sec.~VII.). In light of our results, these conjectured improvements are also grossly \nsuboptimal. On the other hand, our new results can be also seen to complement well-known results on fast rates in online learning (see, \ne.g., \\citealt{EGMRW15} for an excellent summary). Indeed, our learning setting can be seen as a generalized variant of sequential \nprediction under the  relative-entropy loss (see, e.g., \\citealp[Sec.~3.6]{CBLu06:Book}), which is known to be \\emph{exp-concave}. Such \nexp-concave losses are well-studied in the online learning literature, and are known to allow logarithmic regret bounds \\citep{KW99,HAK07}.\n\nInspired by these related results, we ask the question: Is the loss function $f$ defined in Section~\\ref{sec:convex} exp-concave? While our \nderivations Appendix~\\ref{app:convexity} indicate that $f$ has curvature in certain directions, we were not able to prove its \nexp-concavity. Similarly to the approach of \\citet{MF92}, our analysis in the current paper merely exploits the Lipschitzness of the optimal \npolicies with respect to the cost functions, but otherwise does not explicitly make use of the curvature of $f$. We hope that our work \npresented in this paper will inspire future studies that will clarify the exact role of the LMDP structure in efficient online learnability, \npotentially also leading to a better understanding of policy gradient algorithms for LMDPs \\citep{Tod10}.\n\nFinally, let us comment on the tightness of our bounds. Regardless of whether the loss function $f$ is exp-concave or not, \nwe are almost certain that our rates can be improved to at least $O(\\log T)$ by using a more sophisticated algorithm. While our focus in \nthis paper was on improving the asymptotic regret guarantees, we also slightly improve on the results \\citet{GRW12} in that we make the \nleading constants more explicit. However, we expect that the dependence on these constants may also be improved in future work. \nNote however that the potential \nlooseness of our bounds does not impact the performance of the algorithm itself, as it never makes use of any problem-dependent constants.\n\n\\paragraph{Acknowledgements}\nThis work was supported by the UPFellows Fellowship (Marie Curie COFUND program n${^\\circ}$ \n600387) and the Ramon y Cajal program RYC-2015-18878 (AEI\/MINEICO\/FSE, UE). \nThe authors wish to thank the three anonymous reviewers for their valuable comments that helped to improve the paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDeriving the light hadron spectrum from the first principles of\nQCD has been a major subject of lattice QCD\nsimulations\\cite{ref:reviews}.\nA precise determination of the known hadron spectrum\nwould lead us to a fundamental verification of QCD.\nWe should also clarify the nature of observed hadrons,\nprovide predictions for hadrons not in the quark model,\nand give informations for quantities of phenomenological\nimportance.\n\nIn order to achieve these goals, understanding and control\nof various systematic errors are required.\nOne of major sources of systematic errors is that of\na finite lattice spacing.\nRecent progress in reducing this systematic error\nhas been made in two ways.\nFor the quenched QCD spectrum,\ndevelopment of computer power has enabled to push simulations \ntoward smaller lattice spacings on physically larger lattices with \nhigher statistics than the previous attempts.\nAs a result we are now in a status to discuss the problem of how well\nquenched QCD describes the experimental spectrum.\nAnother progress in reducing scaling violation is brought \nwith the use of improved quark actions.\nTests of improvement, previously made mainly in quenched QCD, \nhave been extended this year to full QCD.\n\nFinite size effects and chiral extrapolations have been\nstudied extensively in the past.\nSeveral studies to investigate these systematic errors were \nalso reported at the Symposium.\n\nIn this review we attempt to describe the present status of\nspectroscopic studies.\nProgress in quenched QCD spectrum is summarized in\nsec.~\\ref{sec:progress}, emphasizing results in the continuum limit.\nDiscussions on several issues in spectroscopic studies follow\nin sec.~\\ref{sec:issues},\nwhich include study of finite size effects, chiral extrapolations,\nand quenching error in meson decay constants.\nAfter discussions on improvement of quark actions\nin sec.~\\ref{sec:improve},\nattempts toward a realistic calculation in full QCD are presented\nin sec.~\\ref{sec:fullQCD}.\nSec.~\\ref{sec:other} is devoted to results for masses of glueballs\nand exotics.  Our conclusions are given in sec.~\\ref{sec:conclusions}.\n\n\n\\section{Progress in Quenched QCD Spectrum}\\label{sec:progress}\n\\subsection{major simulations}\n\nRecent quenched simulations made with the plaquette gauge action\nare compiled in Table \\ref{tab:table-q}. \nSee sec.~\\ref{sec:improve} for those with improved \ngauge actions. \n\nDeriving precise quenched results in the continuum limit \nis a first step toward understanding the light hadron spectrum. \nThe GF11 collaboration\\cite{ref:GF11mass} carried out\nthe first systematic effort to achieve this goal \nwith the Wilson quark action \nusing three lattices with $a^{-1} = 1.4-2.8$ GeV  and \nthe spatial size $La \\approx 2.3$ fm.\n \nThis year the CP-PACS collaboration reported further effort \nin this direction\\cite{ref:CPPACS}.\nThey made high statistics simulations\non four lattices with $a^{-1} = 2.0 - 4.2$ GeV and $La \\approx 3$ fm.\nHadron masses are calculated for five quark masses\ncorresponding to $m_\\pi\/m_\\rho$ = 0.75, 0.7, 0.6, 0.5\nand 0.4, the last point being closer to the chiral limit\nthan ever attempted for the Wilson action. \nThey reported continuum values of hadron masses with a statistical \nerror of 0.5 \\% for mesons and 1--3 \\% for baryons.\n\nAnother trend in this year's simulations is a pursuit \nof reduction of scaling violation with the use of \nthe Sheikholeslami-Wohlert\\cite{ref:SW} or clover action.\nEfforts in this direction were made by \nthe UKQCD\\cite{ref:UKQCDlat96,ref:UKQCDb57,ref:UKQCDlat97} \nand JLQCD \\cite{ref:JLQCD-fB} collaborations\nfor the tadpole-improved\\cite{ref:LM} clover action\nand by the UKQCD, QCDSF\\cite{ref:QCDSFlatest} and APETOV\\cite{ref:APETOV}\ncollaborations for the non-perturbatively $O(a)$-improved\\cite{ref:NPI} \nclover action (see also Ref.\\cite{ref:Wittig} on this subject).\nThese studies have not yet reached the level of simulations with the Wilson \naction, being restricted to the parameter range \n$m_\\pi\/m_\\rho \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.5$, $a^{-1} \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 3$ GeV, and\n$La \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 2.0$ fm.\n\nFor the Kogut-Susskind (KS) quark action, \nthe MILC collaboration\\cite{ref:MILClat96} last year \nreported a result of nucleon mass in the continuum limit based on\nsimulations on four lattices with $a^{-1}=0.6 - 2.4$ GeV and \n$La \\approx 2.7$ fm.\nNot much progress has been made this \nyear\\cite{ref:MILC-Tsukuba-Gottlieb,ref:KimOhta}.\n\n\\begin{table*}[t]\n\\caption{Recent spectrum runs in quenched QCD\nwith the standard gauge action. \nNew results since Lattice 96 are marked by double asterisks and\nthose with increased statistics by asterisks.\nQuark actions are denoted in parentheses by\nW: Wilson, C: clover, and KS: Kogut-Susskind. \nClover coefficients are denoted by 1: tree level, TP: tadpole improved,\nTP1: one-loop tadpole improved, and NP: non-perturbatively improved.}\n\\label{tab:table-q}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n        &  $\\beta$   & size &  (fm) & \\#conf. & \\ \\ \\ $m_\\pi\/m_\\rho$   & \\#m & ref.\\\\\n\\hline\n\\hline\nMILC (W)** & 5.70 & $(12-24)^3\\times48$ & 1.7--3.4 & 404-170 & 0.90-0.50 & 6  & \n\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat97} \\\\\n\\hline\nCP-PACS (W)**  & 5.90 & $32^3\\times56$ & 3.21 & 800 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.10 & $40^3\\times70$ & 3.04 & 600 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.25 & $48^3\\times84$ & 3.03 & 420 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\nCP-PACS (W)**  & 6.47 & $64^3\\times112$& 3.03 &  91 & 0.75-0.40 & 5 & \\cite{ref:CPPACS}\\\\\n\\hline\n\\hline\nUKQCD (C=TP)  & 5.70 & $(12,16)^3\\times24$ & (2.1,2.8) & (482,145) & 0.78,0.65 & 2 & \n\\cite{ref:UKQCDlat96,ref:UKQCDb57} \\\\\nUKQCD (C=TP)  & 6.00 & $16^3\\times48$ & 1.6  & 499 & 0.76-0.62 & 3 & \n\\cite{ref:UKQCDlat96,ref:UKQCDlat97}\\\\\nUKQCD (C=TP)* & 6.20 & $24^3\\times48$ & 1.8  & 218 & 0.75-0.49 & 3 & \n\\cite{ref:UKQCDlat96,ref:UKQCDlat97} \\\\\nUKQCD (C=NP)** & 6.00 & $(16,32)^3\\times48$ & (1.7,3.3)  & (497,70) & 0.77-0.50 & 3 & \n\\cite{ref:UKQCDlat97}\\\\\nUKQCD (C=NP)** & 6.20 & $24^3\\times48$ & 1.7  & 251 & 0.71-0.54 & 3 & \n\\cite{ref:UKQCDlat97} \\\\\n\\hline\nQCDSF (C=1)** & 5.70& $16^3\\times32$ & 2.4 & & 0.66-0.44 & 3 & \n\\cite{ref:QCDSFlatest} \\\\\nQCDSF (C=NP)** & 5.70& $16^3\\times32$ & 3.3& & 0.77-0.56 & 6 & \n\\cite{ref:QCDSFlatest} \\\\\nQCDSF (W)*     & 6.00& $(16,24)^3\\times32$ & (1.4,2.0) & O(5000,100)& 0.93-0.50 \n& (4,3) & \\cite{ref:QCDSFlatest,ref:QCDSFb60}\\\\\nQCDSF (C=NP)* & 6.00& $(16,24)^3\\times32$ & (1.7,2.6) & O(1000,200) & 0.90-0.41 & \n(6,3) & \\cite{ref:QCDSFlatest,ref:QCDSFb60} \\\\\nQCDSF (W)**    & 6.20& $24^3\\times48$ & 1.6 & O(100) & 0.94-0.61 & 5 & \n\\cite{ref:QCDSFlatest}\\\\\nQCDSF (C=NP)** & 6.20& $24^3\\times48$ & 1.8 & O(300) & 0.90-0.59  & 5 & \n\\cite{ref:QCDSFlatest}\\\\\nQCDSF (C=NP)** & 6.20& $32^3\\times64$ & 2.4 & O(40) & 0.55-0.39  & 3 &\n\\cite{ref:QCDSFlat97-2} \\\\\n\\hline\nAPETOV (W)**& 6.20 & $24^3\\times48$ & 1.7 & 50 & & 7 & \n\\cite{ref:APETOV} \\\\\nAPETOV (C=NP)**&6.20 & $24^3\\times48$ & 1.9 & 50 & 0.98-0.56 & 7 & \n\\cite{ref:APETOV} \\\\\n\\hline\nJLQCD (C=TP1)** & 5.90& $16^3\\times40$ & 2.0 & 400 & 0.76-0.56 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\nJLQCD (C=TP1)** & 6.10& $24^3\\times64$ & 2.1 & 200 & 0.77-0.50 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\nJLQCD (C=TP1)** & 6.30& $32^3\\times80$ & 2.2 & 100 & 0.81-0.52 & 4 & \n\\cite{ref:JLQCD-fB}\\\\\n\\hline\n\\hline\nKim-Ohta (KS)* & 6.50 & $48^3\\times64$ & 2.6 & 350 & 0.65-0.28& 4 & \n\\cite{ref:KimOhta}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-0.5cm}\n\\end{table*}\n\n\n\\subsection{quenched spectrum in the continuum limit}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{spectrum.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Quenched light hadron spectrum in the continuum limit \nreported by GF11\\protect\\cite{ref:GF11mass} and\nCP-PACS\\protect\\cite{ref:CPPACS} as compared to experiment.}\n\\label{fig:spectrum}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:spectrum} we plot the result for the quenched \nlight hadron spectrum reported by the CP-PACS collaboration \nas compared to the GF11 result and experiment.  \nThe quenched spectrum depends on the choice of hadron masses to set \nthe lattice scale and light quark masses.  Results for two choices are shown \nin Fig.~\\ref{fig:spectrum}, one employing $m_\\pi, m_\\rho$ and $m_K$ and \nthe other replacing $m_K$ with $m_\\phi$.\nThe disagreement of about 5--10\\%\nobserved for strange hadrons between the two choices represent a \nmanifestation of quenching error.\n\nThe GF11 result, albeit not covering the entire spectrum, \nshowed agreement with experiment within the quoted error \nof 2\\% for mesons and 4--8\\% for baryons.\nComparing their result with the CP-PACS result obtained with \nthe same input (filled circles), \none finds a sizable difference for $K^*, \\phi, \\Xi^*$ and $\\Omega$. \nIn fact the CP-PACS result with significantly reduced errors \nexhibits a clear systematic deviation from experiment both for \nmesons and baryons.  \n\n\\subsection{meson spectrum}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{Kstar-phi.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_{K^{*}}$ and $m_\\phi$ with $m_K$ as input\nfor the Wilson (filled symbols) and clover (open symbols) \nactions. In parentheses of legends are physical lattice sizes \nin fm. Lines are continuum extrapolation adopted by GF11 and CP-PACS. \nLeft-most triangles are GF11 estimates for infinite volume.\n}\n\\label{fig:Kstar-phi}\n\\vspace{-7mm}\n\\end{figure}\n\nThe CP-PACS result in the continuum shows that the value of $m_{K^*}$ is \n3\\%(6$\\sigma$) smaller than experiment and $m_\\phi$ by \n5\\% (7$\\sigma$) if $m_K$ is used as input.  Alternatively, with \n$m_\\phi$ as input, they find that $m_{K^{*}}$ agrees with\nexperiment to 0.6\\%, but $m_K$ is larger by 9\\%(7$\\sigma$).\nThis means that a small value of hyperfine splitting, previously\nobserved at finite lattice spacings\\cite{ref:GF11mass,ref:LANLmass},\nremains in the continuum limit, which is different from the conclusion \nof the GF11 collaboration after the continuum extrapolation. \n\nThe origin of the discrepancy is clearly seen in Fig.\\ref{fig:Kstar-phi}\nwhere the continuum extrapolations of $m_{K^{*}}$ and $m_\\phi$ are plotted.\nThe CP-PACS data (filled circles) show very small scaling violation, \nin contrast to an increase exhibited by the GF11 results. \nThe continuum extrapolation of GF11 strongly\ndepends on the small values of results at $\\beta=5.7$ obtained on a \nlattice of size $La \\approx 2.3$~fm ($L=16$).  Their additional \nresults for a larger lattice with $La \\approx 3.4$~fm ($L=24$), also \nshown in Fig.~\\ref{fig:Kstar-phi},  are higher \nby 2--3\\%, and are more compatible with the CP-PACS results.  \nWhether one can attribute the difference of the GF11 results \nbetween $L=16$ and 24 to finite-size effects \nis not clear since values of the two groups \nfor smaller lattice spacings are consistent.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{hyperfine.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Meson hyperfine splitting obtained with $m_K$ as input. }\n\\label{fig:hyperfine}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:hyperfine} we plot the meson hyperfine splitting as \na function of the pseudo-scalar meson mass squared where $m_K$ is used \nas input.  \nThe CP-PACS data at four values of $\\beta$ (filled symbols) \nscale well and do not reproduce the experimental \nvalue of $K$--$K^{*}$ mass splitting.\n\nIn Figs.~\\ref{fig:Kstar-phi} and \\ref{fig:hyperfine}, the clover results \nhave also been plotted with open symbols. \nWe observe that they lie slightly above the Wilson results.\nThis agrees with the expectation that the clover term should increase \nthe hyperfine splitting compared to that of the Wilson action. \nHowever, there is a problematical feature that the difference \nof results for the two actions increases toward the continuum \nlimit rather than decreasing as $O(a)$. \nIn fact the UKQCD collaboration\\cite{ref:UKQCDlat97}\nconcluded this year \nthat $m_{K^{*}}$ linearly extrapolated to the continuum limit is\nconsistent with experiment using either $m_K$ or $m_\\phi$ as input. \n\nWe should emphasize that the difference of meson masses for the two actions\nis tiny(1--2\\%) and no more than a 3$\\sigma$ effect at finite $\\beta$.\nLattice sizes of $La\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$<$} 2$~fm employed in the clover studies may be \ntoo small to avoid finite-size errors at this level of precision.\nStatistical errors of the clover results,  \nwhich are larger by a factor 2--3 compared to those of the Wilson action, \nalso need to be reduced to resolve the discrepancy. \n\nWe compile results for the $J$ parameter\\cite{ref:J} in Fig.~\\ref{fig:J}.  \nAs has been known, results for the Wilson action and its improved ones\nconsistently lie below the experimental value for a wide range\nof lattice spacing. Results for the KS action\\cite{ref:JLQCD-BKKS}\nalso converge to a similar value from above.\n \nA small value of $J$ is equivalent to a small hyperfine splitting if the \nlatter is a linear function of quark mass.\nThis correspondence is satisfied for the Wilson results, \nwhile it is apparently not for the clover case.  This represents \nanother problem which needs to be understood in the quenched meson \nspectrum.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{J.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Results for $J$ parameter. \nData are taken from CP-PACS\\protect\\cite{ref:CPPACS},\nJLQCD\\protect\\cite{ref:JLQCD1000,ref:JLQCD-BKKS}\nfor the Wilson and KS actions, respectively,\nLANL\\protect\\cite{ref:LANLmass},\nSCRI\\protect\\cite{ref:SCRIlat96},\nAlford {\\it et al.}\\protect\\cite{ref:Alfordlat95,ref:Alfordlat96}\nfor the D234 and D234(2\/3) actions, respectively.\nLines are fits to the CP-PACS results and the KS results.}\n\\label{fig:J}\n\\vspace{-7mm}\n\\end{figure}\n\n\\subsection{baryon spectrum}\n\nIn Fig.~\\ref{fig:ndE} we plot the continuum extrapolation of\nrepresentative baryon masses reported by the GF11 and CP-PACS \ncollaborations.  \nThe quenched value of nucleon mass has been a long debated issue.  \nPrevious high statistics results\\cite{ref:APE-OLD,ref:QCDPAX,ref:LANLmass}\n(see also Ref.\\cite{ref:UKQCDlat97}) at $\\beta\\approx 5.7-6.2$ \nobtained by a chiral extrapolation \nfrom $m_\\pi\/m_\\rho\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.5$ yielded a value higher than \nexperiment. The GF11 results also shared this feature, and agreement \nwith experiment in the continuum limit was obtained only after \na finite-size correction.\n\nThe CP-PACS data down to $m_\\pi\/m_\\rho\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 0.4$ show \nthat the nucleon and $\\Lambda$ masses have a negative\ncurvature in terms of $1\/K$ toward the chiral limit.  \nThe bending significantly lowers the nucleon mass even at finite $\\beta$\nas shown in Fig.~\\ref{fig:ndE}, and a linear continuum extrapolation \nleads to a value 2.3\\% lower than experiment, albeit consistent \nwithin a 3\\% statistical error.\nThe nucleon mass for the KS action from the MILC \ncollaboration\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat96}\nis also consistent with experiment.\nSee Sec.~\\ref{sec:chiral-N} for further discussion on the chiral \nextrapolation.\n\nFor $\\Delta$ and $\\Omega$ masses, the GF11 and CP-PACS results \nare reasonably consistent at similar lattice spacings. \nThe continuum extrapolation is different, especially for $\\Omega$, \nwith the GF11 case strongly \naffected by the results at $\\beta=5.7$ on an $L=16$ lattice.  \n\nIn the continuum limit, the CP-PACS results \nshow a systematic deviation from experiment.  \nFor the octet, the non-strange nucleon mass is consistent with \nexperiment, while strange baryon masses are lower \nby 5--8\\% (3--5\\%) with $m_K$ ($m_\\phi$) as input.\nHowever, the Gell-Mann-Okubo (GMO) relation is well\nsatisfied at a 1\\% level.\n\nThe GMO relation is also well satisfied for the decuplet, where \nit takes the form of an equal spacing rule, \nwith at most 10\\% deviations.\nHowever, the average spacing is too small by 30\\% (20\\%) with\n$m_K$ ($m_\\phi$) as input.\n\nBaryon mass splittings were extensively studied at $\\beta=6.0$ on \na $32^3\\times 64$ lattice in Ref.~\\cite{ref:LANLmass}, which \nreported the validity of the GMO relations and \nthe smallness of the decuplet mass splitting. \nThe CP-PACS data confirm these results and extend them as the \nproperty of the quenched baryon spectrum in the continuum.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\vspace*{-8mm}\n\\epsfxsize=7.8cm \\epsfbox{ndE.ps}\n\\end{center}\n\\vspace{-15mm}\n\\caption{Continuum extrapolations of baryon masses\nfrom CP-PACS (filled symbols) and GF11 (open symbols). } \n\\label{fig:ndE}\n\\vspace{-6mm}\n\\end{figure}\n\n\\subsection{quark mass for the Wilson action}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{mstrange.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Comparison of strange quark masses obtained \nfrom the Ward identity and perturbation.\nMasses are in $\\overline {\\rm MS}$ scheme at $\\mu=2$ GeV.}\n\\label{fig:strange-mass}\n\\vspace{-8mm}\n\\end{figure}\n\nThe Wilson action explicitly breaks chiral symmetry at finite lattice spacing.\nOne of its manifestations is that quark mass $m_q^{WI}$ defined \nby the Ward identity \\cite{ref:Bo,ref:Itoh86,ref:MM} \ndoes not agree with quark mass $m_q^P$ defined perturbatively\nat finite lattice spacings\\cite{ref:Itoh86,ref:LANLmass}.\n\nThis problem was examined by four groups this year. \nThe CP-PACS collaboration compared the two definitions\nfor the Wilson action, and reported that they linearly extrapolate to \na consistent value in the continuum limit\\cite{ref:CPPACS}.  \nThe JLQCD collaboration employed an extended current and found indications\nthat scaling violation for $m_q^{WI}$ becomes smaller than that for the local \ncurrent\\cite{ref:JLQCD-Kura}.  \nThe QCDSF collaboration\\cite{ref:QCDSFlatest} reported that\nthe two definitions give consistent results in the continuum limit\nalso for the non-perturbatively $O(a)$ improved clover action.\nThe Ape collaboration\\cite{ref:Ape-QM} reported that $m_q^{WI}$\nare compatible with $m_q^{P}$ at each $\\beta$ when\nrenormalization factors determined non-perturbatively are used.\n\nWe summarize results for the strange quark mass in \nFig.~\\ref{fig:strange-mass}.  \nThe agreement of $m_q^{WI}$ with $m_q^{P}$ \nin the continuum limit supports our expectation that \nchiral symmetry of the Wilson and clover actions is recovered\nin the continuum limit.\nThe disagreement of the values \n$m_s\\approx 135$ MeV obtained with $m_\\phi$ as input and \n$m_s\\approx 110$ MeV found with $m_K$ as input originates \nfrom the small meson hyperfine splitting, and hence represents a \nquenching uncertainty.\nFurther results on quark masses are reviewed in Ref.~\\cite{ref:Gupta}.\n\n\\section{Issues in Spectroscopic Studies}\\label{sec:issues}\n\n\n\\subsection{finite size effects in quenched QCD}\\label{sec:FS}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=6.5cm \\epsfbox{finite.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Nucleon mass for various source\/sink and lattice sizes \nat $\\beta=5.7$ and $m_\\pi\/m_\\rho \\approx 0.5$.\nData are slightly shifted in horizontal axis for clarity.\n}\n\\label{fig:finite}\n\\vspace{-7mm}\n\\end{figure}\n\nIn quenched QCD finite-size effects of hadron masses are expected to \nbe smaller than in full QCD due to Z(3) symmetry. \nFor the nucleon mass with the KS action, the magnitude has been \nestimated to be less than 2\\% at $m_\\pi\/m_\\rho \\approx 0.5$\nfor $La\\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 2$ fm\\cite{ref:Aoki-FS,ref:MILC-FS-Q}.\nOn the other hand, the GF11 result\\cite{ref:GF11mass} for the Wilson \naction at $\\beta=5.7$ showed \na larger effect of 5\\% between the sizes $L=16$ (2.3~fm) to 24 (3.4~fm).\n \nThe MILC collaboration carried out extensive runs \nat $\\beta=5.7$ with the Wilson action for the sizes $L=12-24$, \nand we reproduce their results for \nthe nucleon mass\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat97} \ntogether with those of GF11 in Fig.~\\ref{fig:finite}.\n\nThe GF11 result for $L=16$ significantly depends on the source\/sink size, \nwith the value for the size 4 consistent with those for $L=24$. \nThe MILC results for $L=16$ \ndo not show a source size dependence.  Their values for the sizes \n$L=12-24$ mutually agree within the statistical error of about 2\\%, \nand are also consistent with the GF11 results for $L=24$.\n\nThese comparisons strongly suggest that finite-size effect at \n$La\\approx 2$~fm is already 2\\% or less also for the Wilson action, \nrather than 5\\% estimated by GF11.  This implies that finite-size \neffects are negligible for $La \\approx 3$ fm as employed by \nthe CP-PACS collaboration.\n\n\n\\subsection{chiral extrapolation of nucleon mass}\\label{sec:chiral-N}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{chiral.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{(a) Chiral extrapolations of the CP-PACS nucleon masses \nat $\\beta=5.9$. (b) Continuum extrapolations of the nucleon masses\nobtained by various chiral extrapolations.\n}\n\\label{fig:chiral}\n\\vspace{-7mm}\n\\end{figure}\n\nLast year the MILC \ncollaboration\\cite{ref:MILC-Tsukuba-Gottlieb,ref:MILClat96} \nemphasized the difficulties in reliable chiral extrapolation for the \nnucleon mass using their high precision data with the \nKS action.  The results obtained for light quarks down to \n$m_\\pi\/m_\\rho \\approx 0.3-0.4$ exhibit a negative curvature, \nand the mass in the chiral limit is sensitive to the choice of \nfitting functions.\n \nThe CP-PACS data for the nucleon mass for the Wilson action measured \ndown to $m_\\pi\/m_\\rho \\approx 0.4$ also show a negative curvature.\nThey tried to fit their data using four fitting\nfunctions; a cubic function in quark mass,\na form predicted by chiral perturbation theory ($\\chi$PT)\nin full QCD\\cite{ref:xPT} given by  \n$m_N = c_0 + c_1 m_\\pi^2 + c_2 m_\\pi^3$, and two forms \nin quenched QCD (Q$\\chi$PT)\\cite{ref:QxPT-S,ref:QxPT-BG,ref:LSc1} given by \n$m_N = c_0 + c_1 m_\\pi + c_2 m_\\pi^2$ and \n$m_N = c_0 - 0.53 m_\\pi + c_1 m_\\pi^2 + c_2 m_\\pi^3$\nwhere in the latter the coefficient of the linear term  \nis fixed to a value estimated from experiment. \nAs shown in Fig.~\\ref{fig:chiral}(a),\nthe four fitting functions describe data equally well, but deviate \nsignificantly toward the chiral limit. \n\n\\begin{figure}[t]\n\\vspace*{-3mm}\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{decay-ratio.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$f_K\/f_\\pi-1$. \nData are from CP-PACS\\protect\\cite{ref:CPPACS},\nGF11\\protect\\cite{ref:GF11decay},\nLANL\\protect\\cite{ref:LANLdecay},\nUKQCD\\protect\\cite{ref:UKQCDlat97}, \nQCDSF\\protect\\cite{ref:QCDSFlatest},\nJLQCD\\protect\\cite{ref:JLQCD-fB}, and\nOSU\\protect\\cite{ref:OSU}.}\n\\label{fig:decay-ratio}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.~\\ref{fig:chiral}(b) we show how the choice of chiral extrapolations \naffects the nucleon mass in the continuum limit.  \nHaving precision results \ndown to $m_\\pi\/m_\\rho=0.4$ at each $\\beta$ helped to constrain the \nuncertainty in the continuum limit almost within the statistical error \nof 3\\%.  \n\nA major difficulty in exploring the chiral limit in quenched QCD simulations \nis the presence of exceptional configurations.\nA method has recently been proposed to avoid this \ndifficulty\\cite{ref:Eichten}.  It would be very interesting to see \nif the method allows to obtain reliable results near the chiral limit as \nclose as $m_\\pi\/m_\\rho \\approx 0.2$, \nwhich would be needed to control the chiral extrapolation at a few \\% \nprecision level.\n\n\nThe APETOV collaboration\\cite{ref:APETOV} studied quark mass\ndependence of octet baryon masses for the non-perturbatively $O(a)$ \nimproved action\nfor the range of $m_\\pi\/m_\\rho = 0.98-0.56$.\nThey found that linearity is better \nif one includes the $O(m_qa)$ improvement term in the \ndefinition of quark mass.\n\n\n\n\\subsection{decay constants and quenching error}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{decay.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Results for $f_\\pi$.\nSee the caption of Fig.~\\protect\\ref{fig:decay-ratio} for\nreferences except JLQCD\\protect\\cite{ref:JLQCD-Kura}\nfor the Wilson action.}\n\\label{fig:decay}\n\\vspace{-7mm}\n\\end{figure}\n\nIt has been observed for the Wilson action that\n$f_K\/f_\\pi-1$ in quenched QCD is much smaller than experiment,\nwhich is considered to be a quenching error \n(see Ref.~\\cite{ref:Sharpelat96} for a recent review).\nIn Fig.~\\ref{fig:decay-ratio} we compile recent results for the \nratio.  Small values in the range 0.1--0.15 are also obtained for the clover \nand KS actions.  A discrepancy of 30--40\\% with experiment \nroughly agrees with estimates based on quenched chiral perturbation \ntheory\\cite{ref:QxPT-BG}.\n\nIn Fig.~\\ref{fig:decay} we summarize the status with the determination \nof the pion decay constant.  Continuum values for the Wilson action \nreported by various groups are consistent with each other, and \nare slightly smaller than experiment, while the situation with the clover \nresults is very unsatisfactory, suffering from a large discrepancy among \ngroups. \n\n\\section{Improvement of Quark Actions}\\label{sec:improve}\n\\begin{table*}[t]\n\\caption{Tests of improved quark actions with improved gauge actions.\nAbbreviations for gauge actions in brackets are \nTILW: tadpole-improved \nL\\\"uscher-Weisz\\protect\\cite{ref:TILW-LW,ref:LM,ref:TILW-Al},\nTISY: tadpole-improved Symanzik\\protect\\cite{ref:Symanzik,ref:LM},\nSY: Symanzik\\protect\\cite{ref:Symanzik}.}\n\\label{tab:table-I}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n    &  $\\beta_{pl}$   & size &  (fm) & \\#conf. & \\ \\ \\ $m_\\pi\/m_\\rho$    & \\#m \\ \n& ref. \\\\\n\\hline\n\\hline\nSCRI (C=NP)[TILW]   & 7.75-12 &  $8^3\\times15$ & & O(1000) & & 1 &  \\cite{ref:SCRIlat97} \\\\\n\\hline\nAlford {\\it et al.} (D234c,C)[TISY] & 1.157 & $5^3\\times18$ & 2.0 & & 0.76,0.70 & 2 & \n\\cite{ref:Alfordlat97}\\\\\nAlford {\\it et al.} (D234c,C)[TISY] & 1.719 & $8^3\\times20$ & 2.0 & & 0.76,0.70 & 2 & \n\\cite{ref:Alfordlat97}\\\\\n\\hline\nDeGrand          & \\multicolumn{6}{c}{Fixed point actions} & \\cite{ref:DeGrand} \\\\\n\\hline\n\\hline\nMILC  (KS,Naik)[TILW] & 7.60 & $16^3\\times32$ & & 100 &0.82-0.3 & 5 & \n\\cite{ref:MILClat97}\\\\\nMILC  (KS,Naik)[TILW] & 7.75 & $16^3\\times32$ & & 200 &0.76-0.33 & 5 & \n\\cite{ref:MILClat97}\\\\\nMILC  (KS,Naik)[TILW] & 7.90 & $16^3\\times32$ & & 200 &0.80-0.27 & 6 & \n\\cite{ref:MILClat97}\\\\\n\\hline\nBielefeld  (fat)[SY] & 4.1 & $16^3\\times30$ & & 57 & $\\approx 0.65$ &  & \n\\cite{ref:Bielefeld}\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace{-7mm}\n\\end{table*}\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{rho07-Q.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Comparison of $m_N\/m_\\rho$ at $m_\\pi\/m_\\rho=0.7$\nfor various quark actions.\nC-ML and D234c-ML employ mean link for the tadpole factor. \nGauge actions are denoted in brackets.\nLattice spacings are set with the string tension \n($\\protect\\sqrt\\sigma=427$ MeV)\nexcept for results with TISY gauge action which use \nthe charmonium spectrum.} \n\\label{fig:Rho07-Q}\n\\vspace{-7mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\vspace*{-3mm}\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{mv07.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_V$ at $m_\\pi\/m_\\rho=0.7$. Symbols are the \nsame as in Fig.~\\protect\\ref{fig:Rho07-Q}.\nSold lines are extrapolation of SCRI \ndata\\protect\\cite{ref:SCRIlat96}.\nLattice spacings for the C-ML and D234c-ML actions are \nrecalibrated by us to those given by $\\protect\\sqrt\\sigma$.\n}\n\\label{fig:mv07}\n\\vspace{-7mm}\n\\end{figure}\n\nSeveral groups have been testing improved quark actions with improved \ngauge actions.  In this section we discuss quenched results in this category.  \nNew simulations since Lattice 96 are listed in Table \\ref{tab:table-I}.\n\n\\subsection{improvement of the Wilson action}\n\nImprovement of the Wilson quark action by adding the clover term \nhas been extensively investigated both\nwith the standard gauge \naction\\cite{ref:UKQCDlat96,ref:UKQCDb57,ref:UKQCDlat97,ref:QCDSFlatest,ref:QCDSFb60,ref:QCDSFlat97-2,ref:APETOV,ref:JLQCD-fB}\nand with improved gauge \nactions\\cite{ref:Bock,ref:SCRIlat96,ref:Alfordlat96,ref:Alfordlat97}.\nWe plot in Fig.~\\ref{fig:Rho07-Q} the mass ratio $m_N\/m_\\rho$ at \n$m_\\pi\/m_\\rho =0.7$.  We clearly observe that the clover term \nsignificantly reduces scaling violation so that the ratio agrees \nwith the phenomenological value\\cite{ref:Ono} within 5\\% \nalready at $a\\approx 0.4$~fm.\n\nThe D234 action\\cite{ref:D234} is designed to achieve improvement  \nbeyond the clover action.  \nResults\\cite{ref:Alfordlat95,ref:Alfordlat96,ref:Alfordlat97,ref:Bock}\nfor a class of D234 actions, however, do not show clear improvement\nfor the mass ratio compared with those for the clover action.\n\n\n\nScaling test of hadron masses themselves at a fixed $m_\\pi\/m_\\rho$\nis useful to examine the functional dependence of scaling violation \non the lattice spacing.\nUsing the tadpole-improved L\\\"uscher-Weisz (TILW) gauge action \nfor which we expect only small scaling violation, the \nSCRI group\\cite{ref:SCRIlat96} showed last year that mass results for\nthe tadpole-improved clover action are consistent with an \n$O(a^2)$ scaling behavior,  \nwhile Wilson data need both $O(a)$ and $O(a^2)$ terms.\n \nIn Fig.\\ref{fig:mv07} we reproduce their figure for the vector \nmeson mass at $m_\\pi\/m_\\rho=0.7$,\nadding new results for the Wilson\\cite{ref:CPPACS}(open circles) \nand clover\\cite{ref:UKQCDlat97} (filled circles) actions on\nthe standard plaquette gauge action.\nThe results for the two actions lie on the respective \nextrapolation curves of the SCRI results, showing a reduction of \nscaling violation with the clover action also for the plaquette \ngauge action. \n\nThe Cornell group\\cite{ref:Alfordlat97}\ntested improvement using mean value of link in the Landau \ngauge rather than plaquette for the tadpole factor(right triangles).  \nThey reported that the mean link is superior in reducing scaling\nviolation effects over plaquette.\n\nLet us also mention that\nnon-perturbative determinations of\nthe clover coefficient with improved gauge actions \nhave been attempted\\cite{ref:SCRIlat97,ref:Klassen}.\nSpectrum calculations are in progress.\n\n\n\n\n\n\\subsection{improvement of the KS action}\n\nThe MILC\ncollaboration\\cite{ref:MILClat96}\nstudied the KS and Naik\\cite{ref:Naik} three-link actions\nusing the TILW gauge action, and compared them with those for\nthe KS action on the standard gauge acton.\nThey found that $m_N\/m_\\rho$ is improved by the use of the\nimproved gauge action, but the Naik improvement has a relatively\nsmall effect on the mass ratio. \nPushing the calculation toward higher $\\beta$\\cite{ref:MILClat97}, \nthey found little difference between the Naik and KS actions.\n\nAnother direction of improvement tested by the MILC \ncollaboration\\cite{ref:MILCfat} \nis the use of fat link, \nin which one replaces a link variable with a weighted sum of \nthe link and staples.  \nThis is expected to improve flavor symmetry,\nand indeed they found a substantial reduction \nin the mass difference between the Goldstone and non-Goldstone pions. \n\nThe Bielefeld group\\cite{ref:Bielefeld} studied the fat link \nimprovement with the Symanzik gauge action.\nThey also observed improvement of flavor symmetry for this quark \naction, while $O(p^2)$ and $O(p^4)$ improved actions \nwhich include many link paths do not show any significant \nimprovement of flavor symmetry. \n\n\n\\section{Toward Full QCD Spectrum}\\label{sec:fullQCD}\n\\begin{table*}[t]\n\\caption{Recent spectrum runs in full QCD for $N_f$=2.\nNew results since Lattice 96 are marked by double asterisks and\nthose with increased statistics by asterisks.}\n\\label{tab:table-F}\n\\begin{center}\n\\begin{tabular}{lrrrrrrr}\n   &       $\\beta$    &  size   &  (fm) & traj.    &  $m_\\pi\/m_\\rho$ & \\#m  & ref.\\\\\n\\hline\n\\hline\nSESAM (W)*    &  5.6   &  $16^3\\times32$ &  1.4  & $200\\times 25$ &  0.84-0.7 & 3 &\n\\cite{ref:SESAMlat96,ref:Hoeber}\\\\\nT$\\chi$L (W)* &  5.6   &  $24^3\\times40$ &  2.0  & O(3000)     &  0.7,0.55 & 2 &\n\\cite{ref:TxLlat96,ref:Hoeber} \\\\\n\\hline\nUKQCD (C=1.76)**  &  5.2  & $12^3\\times24$ &     & 50 conf. & 0.85-0.75 & 4 &\n\\cite{ref:Talevi}\\\\\n\\hline\nCP-PACS (W,C=1,TP)** &  & $(12,16)^3\\times32$ & \\multicolumn{4}{c}\n{study of action improvement} & \\cite{ref:CPPACS-F} \\\\\n\\hline\n\\hline\nMILC (KS)  & 5.30 & $12^3\\times32$ & 3.7 &1000-5000 & 0.8-0.3   & 8 &\n\\cite{ref:MILClat96} \\\\\nMILC (KS)* & 5.415& $16^3\\times32$ & 3.2 &1000-2000  & 0.77-0.44 & 6 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\nMILC (KS)* & 5.415& $12^3\\times24$ & 2.4 &2000      & 0.46       & 1 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\nMILC (KS)  & 5.50 & $24^3\\times64$ & 3.6 &1000-2000 & 0.69-0.63 & 2 &\n\\cite{ref:MILClat96}\\\\\nMILC (KS)  & 5.50 & $20^3\\times48$ & 3.0 &2000      & 0.56-0.48 & 2 &\n\\cite{ref:MILClat96}\\\\\nMILC (KS)* & 5.60 & $24^3\\times64$ & 2.6 &1500-2000 & 0.75-0.53 & 4 &\n\\cite{ref:MILClat96,ref:MILClat97-F}\\\\\n\\hline\nColumbia(KS,$N_f$=2)*& 5.70 & $16^3\\times32(40)$& 1.5 & 1400-4900 &0.70-0.57& 4 & \n\\cite{ref:Columbialat96,ref:Columbialat97} \\\\\nColumbia(KS,$N_f$=4)*& 5.40 & $16^3\\times32$& 1.5 & 2700-4500&0.72-0.67& 2 &\n \\cite{ref:Columbialat96,ref:Columbialat97} \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\vspace*{-3mm}\n\\end{table*}\n\nWith progress of our understanding of the quenched spectrum, \nincreasingly larger efforts are beginning to be spent in simulations of \nfull QCD.  Here we summarize recent work listed in \nTable \\ref{tab:table-F}.\n\n\\subsection{progress with the KS action}\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.0cm \\epsfbox{edplot.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{Edinburgh plot for $N_f$=2 KS quarks reported by \nMILC\\protect\\cite{ref:MILClat97-F} together with those \nfrom HEMCGC\\protect\\cite{ref:HEMCGC},\nColumbia\\protect\\cite{ref:Columbialat97,ref:Columbia32},\nand Kyoto-Tsukuba\\protect\\cite{ref:K-T}.\nIn parentheses are lattice sizes.}\n\\label{fig:Ed-KS-Nf2}\n\\vspace{-7mm}\n\\end{figure}\n\nThe MILC collaboration\\cite{ref:MILClat96} continued their study \nof the $N_f=2$ KS spectrum for $\\beta=5.3-5.6$ employing large lattices \nof a size $La \\rlap{\\lower 3.5 pt\\hbox{$\\mathchar \\sim$}}\\raise 1pt \\hbox {$>$} 2.6$~fm. In Fig.~\\ref{fig:Ed-KS-Nf2} we show their  \nresults in the Edinburgh plot together with those of previous \nstudies\\cite{ref:Columbialat97,ref:HEMCGC,ref:Columbia32,ref:K-T}. \n\nThe ratio $m_N\/m_\\rho$ decreases toward weak coupling. \nTaking advantage of improved precision of their results \nas is clear from Fig.~\\ref{fig:Ed-KS-Nf2}, \nthe MILC collaboration attempted a continuum extrapolation of \n$m_N\/m_\\rho$ for a fixed value of $m_\\pi\/m_\\rho$.  They find \n$m_N\/m_\\rho=1.252(37)$ at the physical point in the continuum limit. \n\nAlso of interest is the problem of\nhow the KS spectrum depends on the number of dynamical \nquark flavors.\nColumbia group\\cite{ref:Columbialat97} showed that\nthe four flavor hadron spectrum is nearly parity doubled on \na $16^3\\times 32$ lattice at $\\beta=5.4$.\nChiral symmetry breaking effects are smaller for four\nflavors than for two or zero flavors.\n\n\\subsection{progress with the Wilson action}\nSimulations of full QCD with the Wilson quark action for $N_f$=2\nhave been pushed forward by the SESAM\\cite{ref:SESAMlat96,ref:Hoeber}\nand T$\\chi$L\\cite{ref:TxLlat96,ref:Hoeber} collaborations.  \nSimulations were initially made at $\\beta=5.6$ on a $16^3$ \nspatial lattice ($La \\approx 1.4$ fm) for $m_\\pi\/m_\\rho=0.85-0.7$ (SESAM), \nwhich have been extended to those on a larger lattice $24^3$ \n($La \\approx 2.0$ fm) and closer to the chiral limit with \n$m_\\pi\/m_\\rho =$0.7 and 0.55 (T$\\chi$L).\n\nAn important aspect of their study is a careful examination of \nvarious algorithmic issues of full QCD simulation, including\ndevelopment and tuning of efficient Wilson marix \ninverter\\cite{ref:Frommer}\nand a detailed autocorrelation study. \n\nFor the spectrum, they observed 3\\% (5\\%) finite-size effects\nfor $\\rho$-meson (nucleon) at $m_\\pi\/m_\\rho \\approx 0.7$.\nThe magnitude is comparable to that for the KS \naction\\cite{ref:K-T,ref:MILC-FS-F}.\nThey estimated strange hadron masses,\ntreating the strange quark as a valence quark\nin the presence of two light dynamical quarks.\nThe $K-K^{*}$ mass splitting is smaller than experiment by \n15\\%, contrary to the expectation that dynamical sea quark \neffects alleviate the small hyperfine splitting of quenched \nQCD.  It is possible that dynamical quarks employed is still \ntoo heavy to improve the splitting significantly.\n\nSESAM and T$\\chi$L also studied the static potential and several \nhadron matrix elements to explore effects of sea quarks. See \nRef.~\\cite{ref:Gusken} for a review.\n\n\\subsection{full QCD with improved actions}\n\nTill last year there were only sporadic attempts toward full \nQCD simulations of the light hadron spectrum with improved \nactions\\cite{ref:SCRIlat96-F}.\nThis year the CP-PACS collaboration\\cite{ref:CPPACS-F} \nand the UKQCD collaboration\\cite{ref:Talevi} presented \npreliminary results of a systematic attempt in this direction. \n\nThe CP-PACS collaboration made a comparative study of improvement \nat a coarse lattice $a^{-1} \\approx 0.9-1.5$ GeV employing \nthe plaquette and an RG-improved action\\cite{ref:Iwasaki} for gluons and\nthe Wilson and tadpole-improved clover action for quarks.\nFor one action combination, they also explored the chiral limit \ndown to $m_\\pi\/m_\\rho\\approx 0.4$ with simulations on a $16^3\\times 32$ \nlattice.\nThe UKQCD collaboration employed the plaquette action at $\\beta=5.2$ and \nthe clover action with a clover coefficient of 1.76.\nSimulations were made for four values of sea quark masses\nand the spectrum is calculated for four values of valence quark \nmasses on each dynamical quark ensemble.\n\n\\begin{figure}[t]\n\\begin{center} \\leavevmode\n\\epsfxsize=7.5cm \\epsfbox{rho07.ps}\n\\end{center}\n\\vspace{-13mm}\n\\caption{$m_N\/m_\\rho$ in full QCD with $N_f=2$ as a function of \n$m_\\rho a$ both calculated at $m_\\pi\/m_\\rho =0.7$. \nAbbreviations for gauge actions are\nP: plaquette and  R: RG-improved\\protect\\cite{ref:Iwasaki}, and \nfor quark actions W: Wilson and C: clover.\nData are taken from CP-PACS\\protect\\cite{ref:CPPACS-F,ref:CPPACS},\nSCRI\\protect\\cite{ref:SCRI-W2}, and\nSESAM\\protect\\cite{ref:Hoeber}.\n}\n\\label{fig:rho07}\n\\vspace{-7mm}\n\\end{figure}\n\nIn Fig.\\ref{fig:rho07} we compile full QCD results for $m_N\/m_\\rho$ \nas a function of $m_\\rho a$, both calculated at $m_\\pi\/m_\\rho =$ 0.7.\nResults for the Wilson quark action have large scaling violation \nand approximately lie on a single \ncurve, irrespective of the choice of gauge actions. \nIn contrast the lattice spacing dependence is much \nweaker for the clover actions, again irrespective of the gauge action, \nand  the value of the ratio is close to a phenomenological estimate\neven on a very coase lattice of $a^{-1} \\approx 1.0$ GeV. \nThese results show that a significant improvement of $m_N\/m_\\rho$ \ndue to the clover term observed for the quenched case also \nholds in full QCD. \n\nAnother interesting question in full QCD is to what extent \nthe lattice scale obtained from the hadron spectrum agrees with \nthat from the static potential.\nThe clover term is important also in this regard. \nA mismatch of the scale determined from $m_\\rho$ in the chiral limit \nand that with the string tension observed for the Wilson action\nat $a^{-1} \\approx 1.0$ GeV is much reduced by the use of the clover \naction\\cite{ref:CPPACS-F}.\nThe UKQCD collaboration reported that the scale determined from $m_{K^{*}}$ \napproximately agrees with that from $r_0$ for each value of \ndynamical quark.\n\nFor an effect of improvement of gauge actions, \nrotational symmetry of the potential is improved to a great extent\nalso in full QCD\\cite{ref:CPPACS-F}.\n\nThe effects of improvement summarized here are parallel to those \nobserved in quenched QCD, and come mainly from valence quarks rather than \ndynamical sea quarks.  \nNovertheless, they are important since they show that \nrealistic full QCD simulations are possible without having \nto reduce the lattice spacing below $a^{-1}\\approx 2$ GeV which is  \nneeded with the standard action.\n\n\\section{Other Topics}\\label{sec:other}\nCalculation of glueball masses in quenched QCD\nhas reached a stage to pinpoint the mass\nranges at least for the scalar glueball. \nThe GF11 collaboration\\cite{ref:GF11lat97} \nreported $m_{0^{++}}=1710(63)$ MeV as the infinite volume value \nin the continuum from a reanalysis of their data\\cite{ref:GF11gb}.\nThis value is consistent or slightly higher than the previous\nresults by other groups\\cite{ref:UKQCDgb,ref:MPn,ref:Luo}.\n\nThe central effort of the GF11 collaboration has been a \ncalculation of the mass of the $s\\bar s$ scalar \nmeson\\cite{ref:GF11lat97,ref:GF11lat96}, for which they \nfound values below $m_{s\\bar s} < 1500$ MeV.\nThey conclude that the observed meson $f_J(1710)$ is mainly a\nscalar glueball, while $f_0(1500)$ is mainly an $s\\bar s$\nquarkonium.  \n\nThe SESAM collaboration\\cite{ref:SESAMgb}\nmade a glueball mass measurement with their \nfull QCD runs.\nNo clear dynamical quark effects are seen in the glueball\nmasses. Instead, they observed strong finite size effects\nin the scalar glueball mass, which may be an indication of\nthe presence of mixing between the glueball and \nthe $s\\bar s$ scalar meson. \n\nTwo groups have contributions for spin exotic meson masses. \nThe UKQCD collaboration\\cite{ref:UKQCDlat97-ex} increased\nstatistics since last year. \nCalculating masses at one combination of $\\beta$\nand the quark mass and employing a model to estimate masses at\nthe strange quark,  \nthey obtained $m_{1^{-+}}(s\\bar s) =2000(200)$ MeV.\nThe MILC collaboration\\cite{ref:MILClat97-ex}\nmade simulations at $\\beta$=5.85 and 6.15. \nExtrapolation to the strange quark mass was made to \nobtain $m_{1^{-+}}(s\\bar s)=2170(80)$ MeV.\nThe two results are consistent within 10\\%.\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nA number of interesting studies have been made this year, making a step\nforward toward a precise determination of the light hadron spectrum.\n\nFor the quenched spectrum, a systematic deviation from experiment\nhas been uncovered both in the meson and baryon sectors.\nQuantitative results have been accumulated with improved actions \nboth for quenched and full QCD, clarifiing to what extent improving \nactions reduce scaling violations\nin the light hadron spectrum.\nQuenched clover simulations are moving toward high precision determination \nof physical quantities exploiting the improved scaling behavior,  and  \nsimilar effort should be pursued with other improved actions.\n\nAnd finally, attempts toward a realistic simulation in full QCD have begun.\nIn my opinion, there is real hope that such a calculation could be achieved \nwith the current generation of computers through application\nof improved actions.\n\n\\ \\\\\nI am deeply indebted to all the colleagues who made their results available\nto me before the conference.\nI also would like to thank Y.~Iwasaki and A.~Ukawa for \ncritical comments and suggestions on the manuscript.\nThis work is in part supported by the Grant-in-Aid of Ministry of\nEducation, Science and Culture (Nos. 08NP0101 and 09304029). \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n It is fair to say that tetrad calculations are generally considered superior \nto classical coordinate methods for the calculation of curvature in spacetime. \nExperiments by Campbell and Wainwright \\cite{campbell\/wainwright:1977}\nnow dating back many years showed that tetrad methods are faster than \ncoordinate methods by factors of $2 \\sim 4$. \nEven larger factors have been obtained by MacCallum \\cite{maccallum:1989}\nin a Euclidean context. \nWithin the well known system SHEEP, for example, the advice to the \nbeginner is to always use frame versions of the metric (e.g. MacCallum\nand Skea \\cite{maccallum\/skea:1994} p. 23). On the commercial side, within the\nsystem MACSYMA2 \\cite{macsyma} the demonstration \nCTENSOR4 begins with an explanation that ``frame fields'' (orthonormal bases) \nallow the computations to run much more quickly. The demonstration calculates \nthe bases components of the Ricci tensor for the Kerr-Newman spacetime in \nBoyer-Lindquist coordinates, and is a good place to begin our discussion.\\\\ \n\nIn Table 1 we have reproduced this demonstration within the system \nGRTensorII \\cite{musgrave\/pollney\/lake:1994}\\cite{grhome} running under\nMapleV Release 3 \\cite{maple}, and have included the \ncalculation of the Weyl tensor.\nThe Table demonstrates some interesting \nproperties. The theoretical advantage of the frame approach is clearly \ndemonstrated in the Boyer-Lindquist coordinates (Column BKN). However, \nunder the elementary coordinate transformation $u=a \\cos\\theta$ this advantage \nfails to deliver superior performance (Column BKNU). The importance of\nstrategic application of simplification at intermediate steps is\nillustrated in Column BKNS. For this test simplification of components\nhas been carried out only after the components of the Ricci and Weyl tensors\nare calculated. It is worth noting that without some optimization in the \nsimplification strategy (e.g. post-calculation simplification only) this \ncalculation cannot be executed in MapleV on a 32 bit machine.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrr}\n\\hline\\hline\nCalculation             & BKN        & BKNU  & BKNS \\\\ \\hline\n$R_{(a)(b)}$            & 6.7              & 5.6         & 30.0 \\\\\n$C_{(a)(b)(c)(d)}$      & 1.4              & 0.8         &  9.5 \\\\ \\hline\nTotal                   & 8.1              & 6.4         & 39.5 \\\\ \\hline\n                        &                  &             &      \\\\\n$R_{ab}$                & 8.4              & 2.1         & 18.9 \\\\\n$C_{abcd}$              & 21.1             & 4.3         & 67.5 \\\\ \\hline\nTotal                   & 29.5             & 6.4         & 86.4 \\\\ \n\\hline\\hline\n\\end{tabular}\\\\\n\\vspace{\\baselineskip}\n\\begin{minipage}{.9\\linewidth}\nTable~1: Average CPU time\\footnotemark~in seconds for the calculation and \nsimplification of the bases components of the Ricci and Weyl tensors\n($R_{(a)(b)}$ and $C_{(a)(b)(c)(d)}$) compared to the same for the coordinate \ncomponents ($R_{ab}$ and  $C_{abcd}$). BKN refers to the Kerr-Newman\nspacetime in Boyer-Lindquist coordinates, and BKNU the same but with the \ntransformation $u=a\\cos\\theta$. For both BKN and BKNU the \nsimplification procedures have been structured for optimum \nperformance\\footnotemark $^,$\\footnotemark. For the Column BKNS the tetrad\ncomponents of BKN are used, but simplification procedures are not applied\nto intermediate calculation steps, only to the final results.\n\\addtocounter{footnote}{-2}\n\\end{minipage}\\\\\n\\end{table}\nClearly one could dismiss the findings in Table 1 if the implementation \nof the bases algorithms in GRTensorII were particularly inefficient compared \nto coordinate methods. We do not believe this to be the case\n\\cite{lake\/musgrave\/pollney:1995b}. Rather, as we attempt to show in what\nfollows, we believe that Table 1 reflects the \nfact that bases methods are not fundamentally superior to classical \ncoordinate methods. We find that the most important criterion for speed is \nthe minimization of the time spent on simplification. This underlines the \nimportance of the choice of coordinates or tetrad in a computer algebra \ncalculation and, more importantly,  points out the fact that the user must be \nable to select the style of simplification which is most appropriate for the \nparticular problem.\n\\footnotetext{All times are in seconds as\nreturned by the MapleV {\\tt status} function and are\nthe average of four runs on a Sun Sparc 5 (see Section \\ref{sec:2.4}). The\nmaximum deviation from the average is less than 5\\% for times exceeding \n2 seconds and about 10\\% for shorter times.}\n\\addtocounter{footnote}{1}\n\\footnotetext{We consider a worksheet (a sequence of calculation \nand simplification procedures) to be optimized when the execution time\nhas reached a minimum.}\n\\addtocounter{footnote}{1}\n\\footnotetext{Due to their length, the complete text\nof worksheets used for these tests have not been included in this\nreport (except for an example in Appendix \\ref{app:C}), however they\nhave been made publicly available \\cite{worksheets}.}\n\n\\section{Protocol for comparisons}\n\\subsection{Choice of algorithms}\\label{sec:2.1}\n Within the framework of tetrad methods, the formalism of Newman and Penrose \n\\cite{newman\/penrose:1962} has proven most useful for calculations in spacetime\n(see e.g. \\cite{campbell\/wainwright:1977}). Some \nof the earliest applications of computer algebra to relativity stressed the \nefficiency of this formalism (e.g. \\cite{campbell\/wainwright:1977}). \nMcLenaghan \\cite{mclenaghan:1994} (see also Allen et al. \\cite{allenetal:1994})\nhas emphasized two distinct approaches within this formalism.\nThese are distinguished as the methods of {\\em Cartan} and {\\em Debever} in \n\\cite{mclenaghan:1994}, and as Methods A and B in \\cite{allenetal:1994}, a \nnotation which we adopt here. The methods are outlined in\n\\cite{mclenaghan:1994} and \\cite{allenetal:1994} with references and we do not\nrepeat this material here\\footnote{\nThe curvature component $\\Phi_{12}$ is consistently incorrect in\n\\cite{mclenaghan:1994} and \\cite{allenetal:1994}. In particular, the\ncoefficients of $\\mu\\tau$ and $\\nu\\sigma$ are -1 and 1 respectively,\n{\\em not} -2 and +2. This error is also present in the {\\tt debever} package\nin MapleV Releases 2 and 3.}.\nWe simply note that Method A uses the definitions of Newman and \nPenrose explicitly, while Method B essentially uses definitions constructed \nso as to avoid inversion of coordinate indicies. \n\nIn this paper we compare these two approaches to classical coordinate methods\n(suitably optimized).\n\n\\subsection{Basis for comparison}\nThe null tetrad formalism is sufficiently distinct from the classical \ncoordinate approach that a basis for the comparison of the two methods is \nnot clearly defined. In the classical approach the ``curvature'' of a spacetime\nis usually considered evaluated when the coordinate components of the Ricci \nand Weyl tensors have been evaluated. In the Newman-Penrose (NP)\nformalism it is the tetrad\ncomponents of these tensors (the $\\Phi$s and $\\Psi$s) that display the \n``curvature''. The complication that arises in a comparison of such different\nmethods is the fact that the natural output of each method is distinct. Now\ngiven the coordinate components, and the null tetrad, the tetrad components \nfollow in the usual way \\cite{newman\/penrose:1962}. One could then form a\nbasis for comparison \nby defining the ``curvature'' as the tetrad components of the Ricci and Weyl \ntensors . This is the comparison used in \\cite{campbell\/wainwright:1977}.\nNaturally, this puts the \nclassical component method at a disadvantage since the extra sums involved \nare not a natural part of the method. In this paper we have tried to cover \nall possibilities by having both NP approaches output the tetrad components \nof the Ricci and Weyl tensors, and the coordinate approach output both the \ncoordinate and tetrad components of these tensors.\n\n\\subsection{Choice of spacetimes}\n To compare the performance of algorithms for spacetime calculations it is \nessential that a variety of spacetimes be considered, and that within a given\nspacetime different tetrads (coordinates) be examined. Campbell and \nWainwright \\cite{campbell\/wainwright:1977} chose to examine the spacetimes of \nGriffiths \\cite{griffiths:1975}, Lewis-Papapetrou \\cite{ernst:1968},\nBondi \\cite{bondi\/vanderburg\/metzner:1962}, and Debever \n\\cite{debever\/mclenaghan\/tariq:1979}.\nMore recently, the examination by Allen et al. \\cite{allenetal:1994} \n(which is a comparison of the NP approaches) included these spacetimes \n(with a more general form of the Debever metric, the \nDebever-McLenaghan-Tariq metric \\cite{debever\/mclenaghan\/tariq:1979}) and\nalso the plane wave, 2x2 decomposable, and static spherically symmetric\nspacetimes \\cite{krameretal:1980}. We have found these last \nthree spacetimes to be too simple  since the associated calculation times\nare too short to form a reliable basis for comparison. They also include \none form of the Kerr-Newman metric\\footnote{\nCampbell and Wainwright \\cite{campbell\/wainwright:1977} also examined the\nKerr-Newman metric. However, their description involves intermediate hand\nsimplification in this case and does not include comparison to the component\nmethod. In this Section we take the position that a fair comparison of computer\nalgorithm efficiency can only be carried out if no hand simplification of\ntensor components is permitted.} and a\ngeneral tetrad (their Case 9). Here we examine the Kerr-Newman\nmetric in a variety of forms. We do not include the tetrad 9 given in \n\\cite{allenetal:1994} since it does not conform to the requirements of a null\ntetrad in the Newman-Penrose formalism \\cite{krameretal:1980}.\n\n\\subsection{Method of comparison}\\label{sec:2.4}\n The comparisons were made by way of 26 re-executable Maple worksheets, and \nthe calculations were performed with GRTensorII\n\\cite{musgrave\/pollney\/lake:1994} under MapleV \nRelease 3 \\cite{maple} (in the X-Windows interface) with patchlevel 3 on a \nSun SPARC5 running SunOS 4.1.4\nand equipped with a 75MHz CPU and 64Mb of RAM\\footnote{This configuration was\nchosen for its reliability and reproducibility of CPU times, not its speed. \nBy way of a comparison, a Pentium 133 is about 1.5 times faster, however the \nDOS\/Windows implementation of MapleV Release 3  reports only integer CPU\ntimes.}. \n\nThe Maple worksheets and associated input files used in these tests are\nsummarized \nin Appendix \\ref{app:A}. The associated tetrads, and line elements are shown\nin Appendix \\ref{app:B}. The reproduction of tetrads is prone to errors, and\nreferences \\cite{campbell\/wainwright:1977}, \\cite{mclenaghan:1994} and \n\\cite{allenetal:1994} all contain misprints in the tetrad components. Appendix \n\\ref{app:B} has been produced directly from the input files and so an error\nconstitutes not a misprint, but an actual error in the input which would \ninvalidate our conclusions in the case concerned. The worksheets are \navailable for detailed examination and execution \\cite{worksheets}.\n\nThe worksheets are constructed in the following way. Either the contravariant\nor covariant tetrad is loaded into GRTensorII from an input file. The choice \ndetermines NP algorithm to be used for the tetrad part of the calculation. \nThe tetrad components of the Ricci and Weyl tensors are then evaluated, \nusually in simplified form\\footnote{A tensor component is considered to\nbe fully simplified when its size, as measured in Maple `words', reaches\na minimum.} (see Section \\ref{sec:3}). The metric is then \ngenerated from the tetrad, and the covariant coordinate components of the\nRicci and Weyl tensors are evaluated and simplified when necessary (again\nsee Section \\ref{sec:3}). We have been careful to \nensure that the metric, though generated from the tetrad, is presented in \noptimal form. Assuming that the tetrads are given, the worksheets then \nevaluate the tetrad components from the covariant coordinate \ncomponents. These are then simplified to the exact form of the tetrad \ncalculation.\n\nWe believe that each worksheet has been fully optimized\\footnotemark[2].\nThat is, the procedures (e.g. precalculation of the spin coefficients) and\nsimplification procedures (type and order of simplification) at each step \nof the calculation have been constructed so as to present each approach in \nits best performance mode. It should be pointed out that partial optimization \nof a worksheet is straightforward (see Appendix \\ref{app:D}), but the full \noptimization of a worksheet is a somewhat involved task. Full optimization is\nnecessary if a real comparison of approaches is to be given.\n\nA sample worksheet is given in Appendix \\ref{app:C} along with the\nassociated input file. In \nGRTensorII the input files contain no information beyond the components of the\nbasis or metric. In particular, no simplification information is contained in\nthe input file. The simplification strategy used is read from the worksheet.\n\n\\section{Comparisons}\\label{sec:3}\n\nSummarized in Table 2 are the results of our tests\\footnotemark[1]\nand the results\navailable from the previous tests in \\cite{campbell\/wainwright:1977} and \n\\cite{allenetal:1994}. Before we discuss the \ncomparisons it is appropriate to emphasize the importance of simplification \nprocedures. The fastest calculation in the Kerr-Newman metric is the \ncovariant NP tetrad calculation (Method B) in Row 9. If even only part of the \nsimplification strategy is altered (e.g. the background simplification\nprocedure used before the components are more fully simplified) the \nexecution time can increase by a factor well over two orders of magnitude. \nThe global simplification strategy is of paramount importance.\n\n\\begin{table}\n\\vspace*{-2\\baselineskip}\n\\centering\n\\begin{tabular}{rlrrrrrrrrrrrrr}\\hline\\hline\n1. & Coord's & $A$    & $B$   & $C$     & $D$    & $E$     & $F$     & $A\/D$\n   & \\cite{allenetal:1994}  & $A\/B$   & $A\/C$   & \n\\cite{campbell\/wainwright:1977}   & $D\/E$   & $D\/F$ \\\\ \\hline\n 2. & {\\bf Grif}     & 2.0  & 1.9 &  2.9  & 1.6 & 2.3  &  3.5  & 1.25  & 1.65\n  & 1.05  & 0.69  & 0.39  & 0.70  & 0.46 \\\\\n 3. & {\\bf L-P}     & 1.9  & 2.8 &  4.9  & 2.6 & 2.9  &  6.1  & 0.73  & 1.44\n  & 0.68  & 0.39  & 0.40  & 0.90  & 0.43 \\\\\n 4. & {\\bf Bondi1}   & 3.5  & 6.2 & 10.5  & 4.4 & 6.2  & 10.4  & 0.80  & 0.83\n  & 0.56  & 0.33  & 0.27  & 0.71  & 0.42 \\\\\n 5. & {\\bf Bondi2}   & 1.5  & 1.8 &  4.7  & 2.6 & 1.8  &  4.6  & 0.58  &  ?  \n  & 0.83  & 0.32  &  ?    & 1.44  & 0.57 \\\\\n 6. & {\\bf Deb}      &11.1  &13.3 & 39.8  &24.0 &13.6  & 39.7  & 0.46  &      \n & 0.84  & 0.28  &       & 1.76  & 0.60 \\\\\n 7. & {\\bf DMT1}     &10.9  &13.5 & 40.6  &22.3 &13.5  & 40.7  & 0.49  &      \n & 0.81  & 0.27  &       & 1.65  & 0.55 \\\\\n 8. & {\\bf DMT2}     &35.3  &32.1 &109.0  &55.0 &32.6  &105.0  & 0.64  &0.93  \n & 1.10  & 0.32  &       & 1.69  & 0.52 \\\\\n 9. & {\\bf KN-Euc1}  &22.7  &13.1 & 19.9  & 6.3 &13.7  & 20.0  & 3.60  &5.80 \n  & 1.68  & 1.14  &       & 0.46  & 0.32 \\\\\n10. & {\\bf KN-Euc2}  &54.4  &13.2 & 17.2  &24.4 &13.1  & 16.2  & 2.23  &      \n & 4.12  & 3.16  &       & 1.86  & 1.51 \\\\\n11. & {\\bf KN-BL1}   &26.6  &27.5 & 38.4  &30.0 &30.0  & 34.6  & 0.89  &      \n & 0.97  & 0.69  &       & 1.00  & 0.87 \\\\\n12. & {\\bf KN-EF1}   &32.5  &22.7 & 30.2  & 7.8 &22.7  & 25.4  & 4.17  &      \n & 1.43  & 1.08  &       & 0.34  & 0.31 \\\\\n13. & {\\bf KN-BL2}   &12.6  & 6.7 & 14.9  &18.2 & 9.1  & 15.8  & 0.69  &      \n & 1.88  & 0.84  &       & 2.00  & 1.15 \\\\\n14. & {\\bf KN-EF2}   &41.0  &22.9 & 38.9  &20.8 &22.8  & 26.0  & 1.97  &      \n & 1.79  & 1.05  &       & 0.91  & 0.80 \\\\\n\\hline\\hline\n\\end{tabular}\\\\\n\\vspace{\\baselineskip}\n\\begin{minipage}{.9\\linewidth}\nTable 2: CPU times and comparisons for optimized calculations. Columns $A$\nthrough $F$ give CPU times in seconds\\protect\\footnotemark[1].\\\\\n\\vspace*{-2\\baselineskip}\n\\begin{center}\n\\begin{tabular}{cl}\nColumn & \\\\\n$A$ & -- total CPU time to generate and simplify the curvature \ncomponents ($\\Phi$s and $\\Psi$s)\\\\\n&  \\hspace*{5mm}from a contravariant tetrad using the standard NP approach \n(`Method A').\\\\\n$B$ & -- time for the simplified (covariant) coordinate components of\nthe Ricci and Weyl\\\\\n&  \\hspace*{5mm}tensors for the metric generated from the tetrad $A$.\\\\\n$C$ & -- time (including $B$) to generate and simplify the tetrad\ncurvature components from\\\\\n&  \\hspace*{5mm}the coordinate components calculated in $B$, given the tetrad \n$A$.\\\\\n$D$ & -- the time to generate and simplify the curvature components from\na covariant\\\\\n&  \\hspace*{5mm}tetrad using the modified NP approach of \\cite{allenetal:1994} \n(`Method B'). \\\\\n$E$ & -- time for the simplified (covariant) coordinate components of the\nRicci and Weyl\\\\\n&  \\hspace*{5mm}tensors for the metric generated from the covariant tetrad \n$D$.\\\\\n$F$ & -- time (including $E$) to generate and simplify the tetrad curvature\ncomponents from\\\\\n&  \\hspace*{5mm}the coordinate components calculated in $E$ given the tetrad \n$D$.\\\\\n\\end{tabular}\n\\end{center}\nThe differences in Columns $B$ and $E$ are due to differences in the exact\nform of the metric generated from the tetrad. The spacetimes and form of the\noutput are distinguished by the following abbreviations:\n\\begin{center}\n\\begin{tabular}{ll}\n{\\bf Grif} & Griffiths metric \\cite{griffiths:1975}. Output in factored form.\\\\\n{\\bf L-P} & Lewis-Papapetrou metric \\cite{ernst:1968}. Output in factored \n\tform.\\\\\n{\\bf Bondi1} & Bondi metric \\cite{bondi\/vanderburg\/metzner:1962}. Output in \n\tfactored form.\\\\\n{\\bf Bondi2} & Bondi metric. Output in expanded form.\\\\\n{\\bf Deb} & Debever metric \\cite{debever\/mclenaghan\/tariq:1979}. Output in\n\tnormal form (does not simplify further).\\\\\n{\\bf DMT1} & Debever-McLenaghan-Tariq metric \n\t\\cite{debever\/mclenaghan\/tariq:1979}. Output in normal form (does not\\\\\n\t& simplify further).\\\\\n{\\bf DMT2} & Debever-McLenaghan-Tariq metric in general form\n\t\\cite{debever\/mclenaghan\/tariq:1979}. Output in normal\\\\\n\t& form (does not simplify further).\\\\\n{\\bf KN-Euc1} & Kerr-Newman metric \\cite{allenetal:1994}. Output in factored\n\tform.\\\\\n{\\bf KN-Euc2} & Modified form of 9.  Output in factored form.\\\\\n{\\bf KN-BL1} & Kerr-Newman metric in Boyer-Lindquist coordinates\n\t\\cite{boyer\/lindquist:1967}. Output in\\\\\n\t& factored form.\\\\\n{\\bf KN-EF1} & Kerr-Newman metric in advanced Eddington-Finkelstein coordinates\n\t\\cite{mtw}.\\\\\n\t& Output in factored form.\\\\\n{\\bf KN-BL2} & Kerr-Newman metric in modified Boyer-Lindquist coordinates\n\tusing\\\\\n\t& $u=a\\cos\\theta$. Output in factored form.\\\\\n{\\bf KN-EF2} & Modified form of 12. Output in factored form.\\\\\n\\end{tabular}\n\\end{center}\n\\end{minipage}\\\\\n\\vspace{\\baselineskip}\n\\end{table}\n\nA number of interesting points emerge from Table 2.\n\n{\\bf i)} It is appropriate to begin by comparing our results with previously \npublished tests.  Starting with the work of Campbell and Wainwright \n\\cite{campbell\/wainwright:1977}, although the exact\nform of their output is unavailable, we observe a notable agreement for\nthe ratio $A\/C$ and their results for the Lewis-Papapetrou and Bondi metrics\nas shown in Table 2. For the Griffiths metric our component times \nare somewhat faster\\footnote{The fourth component of $m^a$ for this metric\ngiven in \\cite{campbell\/wainwright:1977} is wrong. We \nbelieve this to be a misprint which would not alter the time reported.}.\nQuite naturally, a concern at the time was the storage requirements for the  \ncalculations. They report storage requirements for the component method a \nfactor $2 \\sim 5$ times that of the tetrad approach. Whereas storage is no \nlonger the concern that it once was, we note that we have observed storage \nrequirements for the component calculations only about $1.1 \\sim 1.5$ times\nthat of the tetrad method.\n\nThe paper by Allen et al. \\cite{allenetal:1994} is concerned with the ratio\n$A\/D$, \nthat is, the relative performance of the two null tetrad methods. The exact \nform of their output is unavailable. In general we find that the performance\nof the {\\em Debever} approach (Method B) is overestimated in \n\\cite{allenetal:1994}\\footnote{The errors in $\\Phi_{12}$ within the MapleV\n{\\tt debever} package affects only their result for the \nDebever-McLenaghan-Tariq metric (See footnote 4).}.\nAlthough the central thesis of \\cite{allenetal:1994} is the superiority of the \n{\\em Debever} approach, this rests principally on their analysis of the \nKerr-Newman metric. It is clear from Table 2 that this superior\nperformance is tetrad (coordinate) dependent.\n\n{\\bf ii)} For metrics of a general type (Rows 3 through 8 in Table\n2) we find that the standard NP approach is faster than the\nalternative proposed in \\cite{mclenaghan:1994} and \\cite{allenetal:1994}.\nWe find that this superior performance is not uniformly maintained \nif the general functions are replaced by specific ones. We have used the \nKerr-Newman metric as an example, and as can be seen from Rows 9 through 14 \nof Table 2, the relative performance of the two tetrad approaches\nis highly dependent on the tetrads (coordinates). However, whereas the\n{\\em Debever} approach can significantly outpace the NP approach (Rows 9 \nand 12), it is never far behind.\n\n{\\bf iii)} For metrics of a general type (Rows 3 through 8 in Table\n2) both tetrad approaches are superior to a calculation of the\ntetrad components from the coordinate components. This is exactly as one\nwould expect. The extra sums involved slow the coordinate approach down \n(compare Columns $B$ and $C$ as well as $E$ and $F$ ). Again, however, this \nsuperior performance is not \nuniformly maintained if the general functions are replaced by specific ones. \nColumns $A\/C$ and $D\/F$ for Rows 9 through 14 show that the classical approach \ncan rival the tetrad approaches even for a calculation of the tetrad \ncomponents.\n\n{\\bf iv)} It could be argued that the natural output of the coordinate\napproach, as \nregards the calculation of curvature,  is simply the coordinate components of \nthe Ricci and Weyl tensors. For metrics of a general type the standard NP \napproach retains its superiority over the coordinate calculation (Column \n$A\/B$). This is not true for the modified tetrad approach (Column $D\/E$). \nColumns $A\/B$ and $D\/E$ indicate that at least in the Kerr-Newman metric the \ncalculation of the coordinate components of the Ricci and Weyl tensors is \nusually faster than the calculation of the tetrad components.\n\n\\section{Discussion}\nOur central conclusion is that the best algorithm, as regards speed, for the\ncomputer algebra calculation of curvature in spacetime is the one that \nminimizes the amount of time spent on simplification. This underlines the \nimportance of the careful choice of coordinates or tetrad in a computer \ncalculation and, more importantly, demonstrates that the user must be able to\nstyle the global simplification strategy in a manner most appropriate for the \nparticular problem being studied. An appropriate simplification strategy \ncan change an untractable problem into one which can be solved essentially \ninstantaneously\\footnote{Some simplification strategies appropriate to \nGRTensorII in MapleV are given in Appendix \\ref{app:D}.}. Our comparisons \n(Table 2) indicate that there is no\nuniformly superior algorithm. In the development of these comparisons we\nobserved that the differences between procedures optimized with respect to the\nglobal simplification strategy for the procedure were less than the variations\nwithin a given procedure for different simplification strategies.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lrrrr}\\hline\\hline\nCalculation        & {\\bf Mix} & {\\bf Mix1} & {\\bf Mix2} & {\\bf Mix3}\\\\ \\hline\n$R_{(a)(b)}$       & 8.7       & 7.5        & 4.2        & 2.5 \\\\\n$C_{(a)(b)(c)(d)}$ & 8.5       & 1.1        & 7.4        & 1.1 \\\\ \\hline\nTotal              &17.2       & 8.6        &11.6        & 3.6 \\\\ \\hline\n                   &           &            &            &     \\\\\n$R_{ab}$           & 8.5       & 8.5        & 8.5        & 8.5 \\\\\n$C_{abcd}$         &52.5       &13.7        &52.5        &13.7 \\\\ \\hline\nTotal              &61.0       &22.2        &61.0        &22.2 \\\\\n\\hline\\hline\n\\end{tabular}\\\\\n\\vspace{\\baselineskip}\n\\begin{minipage}{.9\\linewidth}\nTable 3: Average CPU time in seconds for the calculation and \nsimplification of the bases components of the Ricci and Weyl tensors\n($R_{(a)(b)}$ and $C_{(a)(b)(c)(d)}$) compared to the same for the coordinate\ncomponents ($R_{ab}$ and $C_{abcd}$). {\\bf Mix} refers to the standard\n1-forms with trigonometric functions \\cite{mtw2} and a time dependent basis\ninner product. The same inner product is used for {\\bf Mix1} but the \ntrigonometric functions have been transformed away. For {\\bf Mix2} a\nconstant basis inner product has been used,\nand in {\\bf Mix3} the trigonometric functions have been transformed away. \nIn all cases the simplification procedures have been structured for optimum \nperformance.\n\\end{minipage}\n\\end{table}\nAlthough we have not found any algorithm to be uniformly superior, there\ncertainly are cases where the appropriate algorithm stands out. A case in\npoint is the mixmaster spacetime \\cite{mtw}\\footnote{This\nexample was suggested to us by Prof. C. W. Misner.}.\nHere an appropriate choice of basis\nremoves any angular dependence in the bases components of both the Ricci and\nWeyl tensors. The tetrad approach (not a null tetrad in this case) would\ncertainly be expected to outperform a coordinate calculation in this case.\nThis is confirmed in Table 3. We have considered both a constant\nbasis inner product (for {\\bf Mix1} and {\\bf Mix3}) and a time dependent one\n(for {\\bf Mix} and {\\bf Mix2}). The coordinate transformations are simply \n$\\Theta=\\cos\\theta$ and \n$\\Psi=\\sin\\psi$. It is clear that the classical coordinate calculation is no \nmatch for the basis approach exactly as one would guess. Interestingly, it is \nthe Weyl tensor calculation that improves under coordinate transformation, \nand the Ricci tensor under change in the basis inner product.\n\n\\subsection*{Acknowledgements}\nIt is a pleasure to thank Malcolm MacCallum for a number of useful\ndiscussions and suggestions. This work was supported in part by grants\n(to KL) from the Natural Sciences and Engineering Research Council of Canada\nand the Advisory Research Committee of Queen's University. PM and DP would\nlike to thank the Ministry of Colleges and Universities of Ontario for\nfinancial support through Ontario Graduate Scholarships.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\begin{figure}[bth]\n\\centering\n\\includegraphics[width=56mm,clip]{Fig\/phen-pot_new.eps}\n\\caption{Three examples of the modern $NN$ potential in $^1S_0$ (spin singlet and $S$-wave) channel: Bonn\\cite{Bonn}, Reid93\\cite{Reid93} and AV18\\cite{AV18}. }\n\\label{fig:potential}\n\\end{figure}\nIn 1935, in order to explain the origin of the nuclear force which binds protons and neutrons (nucleons) inside nuclei, Yukawa  introduced virtual particles, $\\pi$ mesons, an exchange of which between nucleons produces the famous Yukawa potential\\cite{Yukawa}. Since then, both theoretically and experimentally, enormous efforts have been devoted to understand the nucleon-nucleon ($NN$) potential, recent examples of which are displayed in Fig.\\ref{fig:potential}.\nThese modern $NN$ potentials are characterized as follows\\cite{Taketani, Machleidt}.\nAt long distances ($r\\ge 2$ fm) there exists weak attraction generated by the one pion exchange potential(OPEP),\nand contributions from the exchange of multi-pions and\/or  heavy mesons such as $\\rho$ makes an overall attraction a little stronger at medium distances ( 1 fm $\\le r \\le $ 2 fm). \nAt short distances ($r \\le$ 1 fm), on the other hand, attraction turns into repulsion, and it becomes stronger as $r$ becomes smaller, forming the strong repulsive core\\cite{Jastrow}.\nAlthough the repulsive core is essential not only for describing the $NN$ scattering data but also for the stability of atomic nuclei, its origin remains one of the most fundamental problems in nuclear physics for a long time\\cite{OSY}. \n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=50mm,angle=270, clip]{Fig\/Fig-potential.ps} \n\\caption{The central (effective central) $NN$ potential (MeV) as a function of $r$ (fm) for the singlet (triplet) at $m_\\pi\\simeq 530$ MeV in quenched QCD\\cite{IAH1}. }\n\\label{fig:potential_zero}\n\\end{figure}\n\nIn a recent paper\\cite{IAH1}, using lattice QCD simulations, three of the present authors have calculated the $NN$ potential, which possesses the above three features of the modern $NN$ potentials, as shown in Fig.\\ref{fig:potential_zero} .  This result has received general recognition\\cite{nature}.\n\nThe above potentials have been extracted from the Schr\\\"odinger equation as\n\\begin{eqnarray}\nV_E({\\bf x} ) \\varphi_E(x) &=& \\left( E +\\frac{\\nabla^2}{2m}\\right)\\varphi_E({\\bf x})\n\\end{eqnarray}\nwith the reduced mass $m = m_N\/2$,\nusing the equal-time Bethe-Salpeter wave function $\\varphi_E({\\bf x})$, defined by\n\\begin{eqnarray}\n\\varphi_E({\\bf x}) &=& \\langle 0 \\vert N({\\bf x},0) N({\\bf 0}, 0)\\vert NN; E\\rangle ,\n\\end{eqnarray}\nwhere $ \\vert NN; E\\rangle $ is an eigen-state of two nucleons with energy $E$ and $N({\\bf x},t)$ is an interpolating operator for the nucleon.  The potentials in Fig.\\ref{fig:potential_zero} are obtained at $E\\simeq 0$.\nFrom this definition it is clear that the potential $V_E({\\bf x})$ may depend on the value of energy $E$ and\/or the choice of the operator $N({\\bf x},t)$. In this talk, we focus on the energy dependence of the potential $V_E({\\bf x})$. In Sect.\\ref{sec:Ising},  $V_E({\\bf x})$ from an integrable model in 2 dimensions is considered\\cite{ABW}. The $NN$ potential calculated at $E\\not= 0$ in quenched QCD is presented in Sect.\\ref{sec:QCD}. Our discussions are given in Sect.\\ref{sec:discussion}. \n  \n\\section{Potentials from an integrable model }\n\\label{sec:Ising}\nIn this section we consider the Ising field theory in 2 dimensions, where \nthe one-particle state of mass $M$ and the rapidity $\\theta$, denoted by $\\vert \\theta \\rangle$,\nhas momentum ${\\bf p}= M(\\cosh\\theta, \\sinh\\theta)$ with state normalization\n\\begin{eqnarray}\n\\langle \\theta^\\prime \\vert \\theta \\rangle &=& 4\\pi \\delta(\\theta - \\theta^\\prime) .\n\\end{eqnarray}\nThe Bethe-Salpeter wave function is defined by\n\\begin{eqnarray}\n\\Psi (r, \\theta) &=& i \\langle 0 \\vert \\sigma(x,0)\\sigma(0,0)\\vert \\theta, -\\theta\\rangle^{\\rm in}\n\\end{eqnarray}\nwhere $ \\vert \\theta, -\\theta \\rangle^{\\rm in}$ is the 2-particle in-state and $r = M x$. The spin field $\\sigma({\\bf x})$ is normalized as\n\\begin{eqnarray}\n\\langle 0 \\vert \\sigma({\\bf x})\\vert \\theta \\rangle = e^{-i {\\bf p}\\cdot{\\bf x}}.\n\\end{eqnarray}\nThe explicit form of this  wave function has been calculated by Fonseca and Zamolodchikov\\cite{FZ} as\n\\begin{eqnarray}\n\\Psi(r,\\theta) &=& \\frac{e^{\\chi(r)\/2}}{\\cosh\\theta}\\left[\n\\Phi_+(r,\\theta)^2 \\cosh\\left(\\frac{\\varphi(r)}{2} -\\theta\\right) -\\Phi_-(r,\\theta)^2\\cosh\\left(\\frac{\\varphi(r)}{2}+\\theta\\right)\\right]\n\\end{eqnarray}\nwhere $\\Phi_\\pm$, $\\varphi$ and $\\chi$ satisfy\n\\begin{eqnarray}\n\\Phi_\\pm^\\prime (r,\\theta) &=& \\frac{1}{2}\\sinh\\left(\\varphi(r) \\pm\\theta\\right) \\Phi_\\mp(r,\\theta) , \\label{eq:Phi}\\\\\n\\frac{1}{r}\\left[ r\\varphi^\\prime(r)\\right]^\\prime &=& \\frac{1}{2}\\sinh\\left( 2\\varphi(r)\\right)  ,\n\\label{eq:phi} \\\\\n\\frac{1}{r}\\left[ r\\chi^\\prime(r)\\right]^\\prime &=& \\frac{1}{2}\\left[1-\\cosh\\left( 2\\varphi(r)\\right)\\right] . \\label{eq:chi}\n\\end{eqnarray}\nIn the limit that $r\\rightarrow 0$, \nthe wave function has the expansion\n\\begin{eqnarray}\n\\Psi(r,\\theta) \\sim C r^{3\/4} \\sinh(\\theta) + O(r^{7\/4}),\n\\label{eq:short}\n\\end{eqnarray}\nwhich is expected from the operator product expansion (OPE),\n\\begin{eqnarray}\n\\sigma(x,0) \\sigma(0,0) \\sim G(r){\\bf 1} + c r^{3\/4}{\\cal E}(0) + \\cdots,\n\\end{eqnarray}\nwhere ${\\cal E}(x)$ is the mass operator of dimension 1.\n\nWe can solve the coupled equations (\\ref{eq:Phi}), (\\ref{eq:phi}) and (\\ref{eq:chi}) for $\\Phi_\\pm, \\varphi, \\chi$ numerically with their boundary conditions at $r=0$\\cite{ABW}. From the wave function, a rapidity-dependent potential can be obtained by\n\\begin{eqnarray}\nV_\\theta (r) &=& \\frac{\\Psi^{\\prime\\prime}(r,\\theta) + \\sinh^2\\theta\\, \\Psi(r,\\theta)}{\\Psi(r,\\theta)} .\n\\end{eqnarray}\nAs $r\\rightarrow 0$, however, from  (\\ref{eq:short}),\nthe potential $V_\\theta$ becomes rapidity independent: \n\\begin{eqnarray}\nV_\\theta(r,\\theta) &\\sim&  -\\frac{3}{16}\\frac{1}{r^2},\n\\end{eqnarray}\nwhere not only the power of $r$ is universally -2 but also \nthe overall coefficient is determined as $ 3\/4 \\times (3\/4-1) $ from the $r^{3\/4}$ behaviour of the wave function.\nIn Fig.\\ref{fig:potential_ising}, $r^2 V_\\theta(r)$, the potential multiplied by $r^2$, is plotted as a function of $r$ for several values of $\\theta$.\nWe observe that an energy(rapidity)-dependence of potentials is small at $\\theta \\le 0.6$\\footnote{\nNote that the singularity of the potential for $\\theta=1.0$ is caused by the vanishing of the corresponding wave function at this point.}. In particular, potentials are almost identical between $\\theta=0$ and $\\theta=0.3$.  The energy dependence of the Ising potential seems weak at low energy. Although the physics in the Ising model is vastly different from QCD,\nwe hope that a similar property holds for the $NN$ potential. In the next section we investigate an energy dependence of the $NN$ potentials in quenched QCD.\n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=56mm, angle=270, clip]{Fig\/Ising_pot1.eps} \n\\caption{The Ising potential (multiplied by $r^2$) $r^2 V_\\theta (r)$ for $\\theta=1.0$ (dotted), $\\theta=0.6$ (dot-dashed), $\\theta = 0.3$ (dashed) and $\\theta = 0$ (solid).  }\n\\label{fig:potential_ising}\n\\end{figure} \n\n\\section{Nucleon-nucleon potentials at non-zero energy in quenched QCD}\n\\label{sec:QCD}\n\\begin{figure}[bth]\n\\centering\n\\includegraphics[width=65mm, clip]{Fig\/W1S0_APBC_PBC.t11.r13_16.eps} \n\\caption{ The $NN$ wave function with APBC (red bars) at $t=11 a$,  together with the fit by the Green's function at large distances (green crosses).}\n\\label{fig:wave_APBC}\n\\end{figure} \n\nWe follow the strategy in Ref.\\cite{IAH1,IAH2} to define the wave function and to calculate the potential through it. Let us explain our set-up of numerical simulations. Gauge configurations are generated in quenched QCD on a $32^3\\times 48$ lattice with the plaquette gauge action at $\\beta = 5.7$, which corresponds to $a\\simeq 0.137$ fm. We employ the Wilson quark action with anti-periodic boundary condition (APBC) in space at the hopping parameter $K=0.1665$, corresponding to $m_\\pi\\simeq 530$ MeV and $m_N \\simeq 1330$ MeV. The minimum momentum is given by ${\\bf p}_{\\rm min} = \\displaystyle\\frac{\\pi}{32a}(1,1,1)$, which leads to $\\vert {\\bf p}_{\\rm min}\\vert \\simeq 240$ MeV and\n$E =\\displaystyle \\frac{k^2}{m_N} \\simeq 50$ MeV, where $E$ is the non-relativistic energy in the center of mass system.  2000 configurations are accumulated to obtain our result.\n\nWe first determine a value of $k^2$, by fitting the wave function at large distance ($13 a \\le \\vert {\\bf x} \\vert \\le 16 a$) \nwith the Green's function of the Helmholtz equation on an $L^3$ box, given by \n\\begin{eqnarray}\nG({\\bf x}; k^2) &=& \\frac{1}{L^{3}}\\sum_{{\\bf n}\\in \\Gamma} \\frac{e^{i(2\\pi\/L) {\\bf n}\\cdot{\\bf x}}}{(2\\pi\/L)^2 {\\bf n}^2 -k^2}, \\quad\n\\Gamma =\\left\\{ \\left. \\left(n_x+\\frac{1}{2},n_y+\\frac{1}{2},n_z+\\frac{1}{2}\\right)\\right\\vert  n_x,n_y,n_z\\in {\\bf Z}\\right\\},\n\\end{eqnarray}\nas plotted in Fig.\\ref{fig:wave_APBC}. The fit gives %\n$ k^2 a^2=0.030(4)$ with %\n$\\chi^2\/{\\rm dof}  \\simeq 1.8$ at $t=11 a$ , \nwhich corresponds to $E\\simeq 50 $ MeV in physical unit.\n\n\\begin{figure}[bt]\n\\centering\n\\includegraphics[width=63mm,clip]{Fig\/V1S0_APBC_PBC.t9.r13_16.eps} \n\\includegraphics[width=63mm,clip]{Fig\/V1S0_APBC_PBC.t9.r13_16.s.eps} \n\\caption{ Left: The central $NN$ potentials for the $^1S_0$ state with APBC (red bars) and PBC (blue crosses) in quenched QCD at $t=9a$.\nRight: Its zoom-in.} \n\\label{fig:potential_APBC}\n\\end{figure} \n\nIn Fig.\\ref{fig:potential_APBC}, the central $NN$ potential for the $^1S_0$ state with  APBC ( $E\\simeq 50$ MeV) is plotted as a function of $r$ at $t=9a$, together with the one with PBC ( $E\\simeq 0$ ). \nFluctuations of data with APBC at large distances ( $ r\\ge 1.5$ fm ) are mainly caused by \ncontaminations from excited states, together with statistical noises.  Data at larger $t$ are needed to reduce such contaminations from excited states, though statistical errors also become larger.\nA non-trivial part of potential at $ r < 1.5$ fm, on the other hand, are less affected by such contaminations.\nAs seen from Fig. \\ref{fig:potential_APBC}, the $NN$ potentials are almost identical between $E\\simeq 0$ and $E\\simeq 50$ MeV.\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nAs discussed in the introduction, the potential defined from the Bethe-Salpeter wave function depends on the energy:\n\\begin{eqnarray}\nV_E({\\bf x})\\varphi_E({\\bf x}) &=& \\left( E + \\frac{\\nabla^2}{2m}\\right) \\varphi_E({\\bf x}) .\n\\end{eqnarray}\nIn \\cite{IAH2,Aoki1}, it is shown that the energy-dependent potential can be converted to the energy-independent but non-local potential as\n\\begin{eqnarray}\n\\int d^3 y\\,  U({\\bf x},{\\bf y}) \\varphi_E({\\bf y}) &=& \\left( E + \\frac{\\nabla^2}{2m}\\right) \\varphi_E({\\bf x}) = V_E({\\bf x})\\varphi_E({\\bf x}) .\n\\end{eqnarray}\nWe then apply the derivative expansion to this non-local potential\\cite{IAH2} as\n\\begin{eqnarray}\nU({\\bf x}, {\\bf y} ) &=& V({\\bf x}, \\nabla)\\delta({\\bf x} - {\\bf y}) \\\\\nV({\\bf x}, \\nabla) &=& V_0(r) + V_{\\sigma}(r) ({\\bf \\sigma}_1\\cdot {\\bf \\sigma}_2) + V_T(r) \\, S_{12} + O(\\nabla)\n\\end{eqnarray}\nwhere $r = \\vert {\\bf x}\\vert$, $\\sigma_{1,2}$ represents the spin of nucleons, and \n\\begin{eqnarray}\nS_{12} &=&\\frac{3}{r^2} ({\\bf\\sigma}_1\\cdot{\\bf x}) ({\\bf\\sigma}_2\\cdot{\\bf x})\n-({\\bf \\sigma}_1\\cdot {\\bf \\sigma}_2) \n\\end{eqnarray}\nis the tensor operator. Our result in the previous section indicates that non-locality is very weak.\n\nThe analysis for the potentials in the Ising field theory in 2 dimensions suggests an interesting possibility that the universality of potentials at short distance can be understood from a point of view of the operator product expansion (OPE).   If this is the case, the origin of the repulsive core might be explained by the OPE. We are currently working on this problem.  \n\nBefore closing this talk,\nwe consider an alternative possibility  to construct the energy-independent local potential\\cite{Aoki1}. The inverse scattering theory suggests that there exists an unique energy independent potential, which gives the correct phase shift at all energies.\nHere we propose a new method to construct the energy-independent local potential from $V_E$.\nFor simplicity , the 1 dimensional case is considered. Suppose that $\\Phi(x) \\equiv \\Lambda_E(x) \\varphi_E(x)$ satisfies the Schr\\\"odinger equation with the energy-independent  local potential $V(x)$,\n\\begin{eqnarray}\n\\left( -\\frac{d^2}{d x^2} + V(x)\\right) ( \\Lambda_E(x) \\varphi_E(x) ) &=& E (\\Lambda_E (x) \\varphi_E(x) ),\n\\end{eqnarray}\nwe obtain the following differential equation,\n\\begin{eqnarray}\nV(x) \\Lambda_E(x) &=& V_E (x)\\Lambda_E(x) + \\Lambda_E^{\\prime\\prime}(x) +2\\Lambda_E^\\prime(x)(\\log\\varphi_E(x) )^\\prime .\n\\end{eqnarray}\n(Here we set $2m = 1$ .)\nIf $V(x)$ is given, $\\Lambda_E(x)$ can be easily obtained from this equation. \nWe first consider a finite box with size $L$, which allows only discrete momenta, $k_n \\simeq 2\\pi n\/L$, $n=0,1,2,\\cdots .$ Once $E_n = k_n^2$ is given, $\\varphi_E(x) $ becomes zero\nat $n+1$ points $\\Omega_n =\\{x_0,x_1, \\cdots, x_n \\} $. We then have\n\\begin{eqnarray}\n0 &=& K_E (x_i) \\Lambda_E (x_i) + 2 \\Lambda_E^\\prime (x_i)\\varphi_E^\\prime(x_i)\n\\end{eqnarray}\nfor $x_i \\in \\Omega_n$, where $K_E(x) \\equiv (E + d^2\/d x^2 ) \\varphi_E(x) $.\nSince $\\Omega_n$ becomes dense in $[0,L]$ in the $n\\rightarrow\\infty$ limit, $V(x)$ can be constructed as\n\\begin{eqnarray}\nV(x) &=& \\lim_{E\\rightarrow\\infty} \\left\\{ V_E(x) - 2 X_E(x) (\\log\\varphi_E(x) )^\\prime - X_E^\\prime(x) + X_E(x)^2\\right\\}\n\\label{eq:local}\n\\end{eqnarray}\nwhere $X_E(x)$ is an interpolations of $X_E(x_i) =\\displaystyle\\frac{K_E(x_i)}{2\\varphi_E^\\prime(x_i)}$\nwith $x_i\\in \\Omega_n$. If the limit (\\ref{eq:local}) exists,\nthe energy-independent local potential $V(x)$ can be obtained.\nIn the 3 dimensional case, we first introduce the polar coordinate, and then apply the above procedure in 1 dimension to the radial variable $r$ with the fixed angular momentum $l$.\n\n\\section*{ Acknowledgements }\nOur simulations have been performed with IBM Blue Gene\/L at KEK under a support of its Large Scale simulation Program, Nos. 06-21, 07-07, 08-19.\nWe are grateful for authors and maintainers of {\\tt CPS++}\\cite{cps},\nof which a modified version is used for measurement done in this work.\nJ.B. and S.A. are grateful to the Max-Planck-Institut f\\\"ur Physik for its hospitality.\nThis work was supported in part by the Hungarian National Science Fund OTKA (under T049495) and the Grant-in-Aid of the Japanese Ministry of Education, Science, Sports and Culture (Nos. 18540253, 19540261, 20028013, 20340047 ). \n  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nLeveraging feedback to obtain the channel state information at the transmitter (CSIT) enables a wireless system to adapt its transmission\nstrategy to the varying wireless environment. The growing number of wireless users, as well as their increasing demands for higher data rate\nservices impose a significant burden on the feedback link. In particular for OFDMA systems, which have emerged as the core technology in 4G and\nfuture wireless systems, full CSIT feedback may become prohibitive because of the large number of resource blocks. This motivates more efficient\nfeedback design approaches in order to achieve performance comparable to a full CSIT system with reduced feedback. In the recent years,\nconsiderable work and effort has been focused on limited or partial feedback design, e.g., see \\cite{love08} and the references therein. To the\nbest of our knowledge, most of the existing partial feedback designs are homogeneous, i.e., users' feedback consumptions do not adapt to the\nunderlying channel statistics. In this paper, we propose and analyze a heterogeneous feedback design, which aligns users' feedback needs to the\nstatistical properties of their wireless environments.\n\nCurrent homogeneous feedback design in OFDMA systems groups the resource blocks into subband \\cite{zhu09} which forms the basic scheduling and\nfeedback unit. Since the subband granularity is determined by the frequency selectivity, or the coherence bandwidth of the underlying channel,\nit would be beneficial to adjust the subband size of different users according to their channel statistics. Empirical measurements and analysis\nfrom the channel modeling field have shown that the root mean square (RMS) delay spread which is closely related to the coherence bandwidth, is\nboth location and environment dependent \\cite{asplund06, huang12}. The typical RMS delay spread for an indoor environment in WLAN does not\nexceed hundreds of nanoseconds; whereas in the outdoor environment of a cellular system, it can be up to several microseconds. Intuitively,\nusers with lower RMS delay spread could model their channel with a larger subband size and require less feedback resource than the users with\nhigher RMS delay spread. Herein, we investigate this heterogeneous feedback design in a multiuser opportunistic scheduling framework where the\nsystem favors the user with the best channel condition to exploit multiuser diversity \\cite{knopp95, viswanath02}. There are two major existing\npartial feedback strategies for opportunistic scheduling, one is based on thresholding where each user provides one bit of feedback per subband\nto indicate whether or not the particular channel gain exceeds a predetermined or optimized threshold \\cite{sanayei07, hassel07, chen08,\npugh10}. The other promising strategy currently considered in practical systems such as LTE \\cite{sesia11} is the best-M strategy, where the\nreceivers order and convey the M best channels \\cite{jung07, ko07, choi07, choi08, pedersen09, leinonen09, donthi11, hur11}. The best-M partial\nfeedback strategy is embedded in the proposed heterogeneous feedback framework. Apart from the requirement of partial feedback to save feedback\nresource, the study of imperfections is also important to understand the effect of channel estimation error and feedback delay on the\nheterogeneous feedback framework. These imperfections are also considered in our work.\n\n\\subsection{Focus and Contributions of the Paper}\nAn important step towards heterogeneous feedback design is leveraging the ``match\" among coherence bandwidth, subband size and partial feedback.\nUnder a given amount of partial feedback, if the subband size is much larger than the coherence bandwidth, then multiple independent channels\ncould exist within a subband and the subband-based feedback could only be a coarse representative of the channels. On the other hand, if the\nsubband size is much smaller than the coherence bandwidth, then channels in adjacent subbands are likely to be highly correlated and requiring\nfeedback on adjacent subbands could be a waste of resource; or a small amount of subband-based partial feedback may not be enough to reflect the\nchannel quality. In order to support this heterogeneous framework, we first consider the scenario of a general correlated channel model with one\ncluster of users with the same coherence bandwidth. The subband size is adjustable and each user employs the best-M partial feedback strategy to\nconvey the M best channel quality information (CQI) which is defined to be the subband average rate. The simulation result  shows that a\nsuitable chosen subband size yields higher average sum rate under partial feedback conforming the aforementioned intuition. This motivates the\ndesign of heterogeneous feedback to ``match\" the subband size to the coherence bandwidth. The above-mentioned study, though closely reflects the\nrelevant mechanism, is not analytically tractable due to two main reasons. Firstly, the general correlated channel model complicates the\nstatistical analysis of the CQI. Secondly, the use of subband average rate as CQI makes it difficult to analyze the multi-cluster scenario.\nTherefore, a simplified generic channel model is needed that balances the competing needs of analytical tractability and practical relevance.\n\nIn order to facilitate analysis, a subband fading channel model is developed that generalizes the widely used frequency domain block fading\nchannel model. The subband fading model is suited for the multi-cluster analysis. According to the subband fading model, the channel frequency\nselectivity is flat within each subband, and independent across subbands. Since the subband sizes are different across different clusters, the\nnumber of independent channels are heterogeneous across clusters and this yields heterogeneous partial feedback design. Another benefit of the\nsubband fading model is that the CQI becomes the channel gain and thus facilitate further statistical analysis. Under the multi-cluster subband\nfading model\\footnote[1]{An initial treatment of a two-cluster scenario was first presented in \\cite{huang11}.} and the assumption of perfect\nfeedback, we derive a closed form expression for the average sum rate. Additionally, we approximate the sum rate ratio for heterogeneous design,\ni.e., the ratio of the average sum rate obtained by a partial feedback scheme to that achieved by a full feedback scheme, in order to choose\ndifferent best-M for users with different coherence bandwidth. We also compare and demonstrate the potential of the proposed heterogeneous\nfeedback design against the homogeneous case under the same feedback constraint in our simulation study.\n\nThe average sum rate helps in understanding the system performance with perfect feedback. In practical feedback systems, imperfections occur\nsuch as channel estimation error and feedback delay. These inevitable factors degrade the system performance by causing outage\n\\cite{piantanida09, isukapalli10}. Therefore, rather than using average sum rate as the performance metric, we employ the notion of average\ngoodput \\cite{lau08, wu10, akoum10} to incorporate outage probability. Under the multi-cluster subband fading model, we perform analysis on the\naverage goodput and the average outage probability with heterogeneous partial feedback. In addition to examining the impact of\nimperfect feedback on multiuser diversity \\cite{ma05, kuhne08}, we also investigate how to adapt and optimize the average goodput in the presence of\nthese imperfections. We consider both the fixed rate and the variable rate scenarios, and utilize bounding technique and an efficient\napproximation to derive near-optimal strategies.\n\nTo summarize, the contributions of this paper are threefold: a conceptual heterogeneous feedback design framework to adapt feedback amount to\nthe underlying channel statistics, a thorough analysis of both perfect and imperfect feedback systems under the multi-cluster subband fading\nmodel, and the development of approximations and near-optimal approaches to adapt and optimize the system performance. The rest of the paper is\norganized as follows. The motivation under the general correlated channel model and the development of system model is presented in Section\n\\ref{system}. Section \\ref{perfect} deals with perfect feedback, and Section \\ref{imperfect} examines imperfect feedback due to channel\nestimation error and feedback delay. Numerical results are presented in Section \\ref{numerical}. Finally, Section \\ref{conclusion} concludes the\npaper.\n\n\n\\section{System Model}\\label{system}\n\n\\subsection{Motivation for Heterogeneous Partial Feedback}\\label{motivation}\nThis part provides justification for the adaptation of subband size with one cluster of users under the general correlated channel model, and\nmotivates the design of heterogeneous partial feedback for the multi-cluster scenario in Section \\ref{multicluster}. Consider a downlink\nmultiuser OFDMA system with one base station and $K$ users. One cluster of user is assumed in this part and users in this cluster are assumed to\nexperience the same frequency selectivity. The system consists of $N_\\mathsf{c}$ subcarriers. $H_{k,n}$, the frequency domain channel transfer\nfunction between transmitter and user $k$ at subcarrier $n$,  can be written as:\n\\begin{equation} \\label{system:eq_1}\nH_{k,n}=\\sum_{l=0}^{L-1}\\sigma_lF_{k,l}\\exp\\left(-\\frac{j2\\pi ln}{N_\\mathsf{c}}\\right),\n\\end{equation}\nwhere $L$ is the number of channel taps, $\\sigma_l$ for $l=0,\\ldots,L-1$ represents the channel power delay profile and is normalized, i.e.,\n$\\sum_{l=0}^{L-1}\\sigma_l^2=1$, $F_{k,l}$ denotes the discrete time channel impulse response, which is modeled as complex Gaussian distributed\nrandom processes with zero mean and unit variance $\\mathcal{CN}(0,1)$ and is i.i.d. across $k$ and $l$. Only fast fading effect is considered in\nthis paper, i.e., the effects of path loss and shadowing are assumed to be ideally compensated by power control\\footnote[2]{This assumption has\nbeen employed in \\cite{kuhne08, leinonen09, pugh10} to simplify the scheduling policy. With the same average SNR, the opportunistic scheduling\npolicy is also long-term fair. When different average SNR is assumed, the proportional-fair scheduling policy \\cite{viswanath02} can be\nutilized.}. The received signal of user $k$ at subcarrier $n$ can be written as:\n\\begin{equation} \\label{system:eq_2}\nu_{k,n}=\\sqrt{P_{\\mathsf{c}}}H_{k,n}s_{k,n}+v_{k,n},\n\\end{equation}\nwhere $P_{\\mathsf{c}}$ is the average received power per subcarrier, $s_{k,n}$ is the transmitted symbol and $v_{k,n}$ is the additive white\nnoise distributed as $\\mathcal{CN}(0,\\sigma_{n_{\\mathsf{c}}}^2)$. From (\\ref{system:eq_1}), it can be shown that $H_{k,n}$ is distributed as\n$\\mathcal{CN}(0,1)$. The channels at different subcarriers are correlated, and the correlation coefficient between subcarriers $n_1$ and $n_2$\ncan be described as follows:\n\\begin{equation} \\label{system:eq_3}\n\\mathrm{cov}(H_{k,n_1},H_{k,n_2})=\\sum_{l=0}^{L-1}\\sigma_l^2\\exp\\left(-\\frac{j2\\pi l(n_2-n_1)}{N_\\mathsf{c}}\\right).\n\\end{equation}\n\nIn general, adjacent subcarriers are highly correlated. In order to reduce feedback needs, $R_\\mathsf{c}$ subcarriers are formed as one resource\nblock, and $\\eta$ resource blocks are grouped into one subband\\footnote[3]{E.g., in LTE, one resource block consists of $12$ subcarriers, and\none subband can contain $1$ to $8$ resource blocks \\cite{dahlman11}.}. Thus, there are $N=\\frac{N_\\mathsf{c}}{R_\\mathsf{c}}$ resource blocks and\n$\\frac{N}{\\eta}$ subbands\\footnote[4]{Throughout the paper, $N_\\mathsf{c}$, $N$ and $\\eta$ are assumed to be a radix $2$ number. A more general\ntreatment is possible but this will result in edge effects making for more complex notation without much insight.}. In this manner, each user\nperforms subband-based feedback to enable opportunistic scheduling at the transmitter. Since the channels are correlated and there is one CQI to\nrepresent a given subband, the CQI is a function of the all the individual channels within that subband. Herein, we employ the following subband\n(aggregate) average rate $S_{k,r}$ as the functional form\\footnote[5]{This functional form employs the capacity formula and the resulting\neffective SNR has a geometric mean interpretation. Other functional forms of the CQI exist in practical systems such as exponential effective\nSNR mapping (EESM) \\cite{ericsson03, song11, donthi11j} and mutual information per bit (MMIB) \\cite{wan06, fan11} to map the effective SNR to\nthe block-error-rate (BLER) curve. The intuitions are similar: to obtain a representative CQI as a single performance measure corresponding to\nthe rate performance.} \\cite{forney98, al96} of the CQI for user $k$ at subband $r$:\n\\begin{equation} \\label{system:eq_4}\nS_{k,r}\\triangleq\\frac{1}{\\eta R_{\\mathsf{c}}}\\sum_{n=(r-1)\\eta R_{\\mathsf{c}}+1}^{r\\eta\nR_{\\mathsf{c}}}\\log_2\\left(1+\\frac{P_{\\mathsf{c}}|H_{k,n}|^2}{\\sigma_{n_{\\mathsf{c}}}^2}\\right).\n\\end{equation}\n\nEach user employs the best-M partial feedback strategy and conveys back the $M$ best CQI values selected from $S_{k,r}, 1\\leq r\\leq\n\\frac{N}{\\eta}$. A detailed description of the best-M strategy can be found in \\cite{choi08, leinonen09, hur11}. After the base station receives\nfeedback, it performs opportunistic scheduling and selects the user $k$ for transmission at subband $r$ if user $k$ has the largest CQI at\nsubband $r$. Also, it is assumed that if no user reports CQI for a certain subband, scheduling outage happens and the transmitter does not\nutilize it for transmission.\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.55\\linewidth]{Figure\/fig1.eps}\n\\caption{Comparison of average sum rate for different subband sizes ($\\eta=1,2,4$) and partial feedback ($M=2,4$) with respect to the number of\nusers. A general correlated channel model is assumed with an exponential power delay profile. ($N_{\\mathsf{c}}=256$, $N=32$, $L=16$, $\\delta=4$,\n$\\frac{P_{\\mathsf{c}}}{\\sigma_{n_{\\mathsf{c}}}^2}=10$ dB)} \\label{fig_1}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.8\\linewidth]{Figure\/fig2.eps}\n\\caption{Illustration of the multi-cluster subband fading channel model for two different clusters with $16$ resource blocks. The subband sizes\nequal $2$ and $4$ for the two different clusters respectively. According to the subband fading model, the channel frequency selectivity is flat\nwithin each subband, and independent across subbands. The subband sizes can be heterogeneous across clusters, and this leads to heterogeneous\nchannel frequency selectivity across clusters. The subband fading model approximates the general correlated channel model, and is useful for\nstatistical analysis.} \\label{fig_2}\n\\end{figure}\n\nNow we demonstrate the need to adapt the subband size to achieve the potential ``match\" among coherence bandwidth, subband size and partial\nfeedback through a simulation example. The channel is modeled according to the exponential power delay profile \\cite{weinfurtner02, mckay08,\neslami11}: $\\sigma_l^2=\\frac{1-\\exp(-1\/\\delta)}{1-\\exp(-L\/\\delta)}\\exp\\left(-\\frac{l}{\\delta}\\right)$ for $0\\leq l<L$, where the parameter\n$\\delta$ is related to the RMS delay spread. The simulation parameters are: $N_{\\mathsf{c}}=256$, $N=32$, $L=16$, $\\delta=4$,\n$\\frac{P_{\\mathsf{c}}}{\\sigma_{n_{\\mathsf{c}}}^2}=10$ dB. The subband size $\\eta$ can be adjusted and ranges from $1$ to $4$ resource blocks. We\nconsider partial feedback with $M=2$ and $M=4$. The average sum rate of the system for different subband sizes and partial feedback with respect\nto the number of users is shown in Fig. \\ref{fig_1}. Under the given coherence bandwidth, several observations can be made. Firstly, the curves\nwith $\\eta=4$ has the smallest increasing rate because a larger subband size gives a poor representation of the channel. Secondly, the curve\nwith $\\eta=1,M=2$ has the smallest average sum rate because a small amount of partial feedback is not enough to reflect the channel quality.\nThirdly, the two curves $\\eta=1,M=4$ and $\\eta=2,M=2$ possess similar increasing rate. This is because the underlying channel is highly\ncorrelated within $2$ resource blocks and thus having $M$-best feedback with $\\eta=2$ yields similar effect as having $2M$-best feedback with\n$\\eta=1$. From the above observations, $\\eta=2$ matches the frequency selectivity and there would be performance loss or waste of feedback\nresource when a subband size is blindly chosen. In a multi-cluster scenario where users in different clusters experience diverse coherence\nbandwidth, this advocates heterogeneous subband size and heterogeneous feedback.\n\nThe general correlated channel model as well as the non-linearity of the CQI, though useful to demonstrate the need for heterogeneous feedback,\ndoes not lend itself to tractable statistical analysis. To develop a tractable analytical framework, an approximated channel model is needed. A\nwidely used model is the block fading model in the frequency domain \\cite{mceliece84, medard02} due to its simplicity and capability to provide\na good approximation to actual physical channels. According to the block fading model, the channel frequency selectivity is flat within each\nblock, and independent across blocks \\cite{chen08, leinonen09, hur11}. Herein, we generalize the block fading model to the subband fading model\nfor the multi-cluster scenario. We assume that users possessing similar frequency selectivity are grouped into a cluster and the subband size is\nperfectly matched to the coherence bandwidth for a given cluster\\footnote[6]{In practical systems, since the coherence bandwidth is determined\nby the channel statistics which vary on the order of tens of seconds or more, the cluster information can be learned and updated through\ninfrequent user feedback. Therefore, the cluster is formed dynamically but in a slow way compared to the time variation of the fast fading\neffect which is on the order of milliseconds.}. According to the subband fading model, for a given cluster with a perfectly matched subband\nsize, the channel frequency selectivity is flat within each subband, and independent across subbands. Fig. \\ref{fig_2} demonstrates the subband\nfading model for two different clusters with different subband sizes under a given number of resource blocks.\n\n\n\\subsection{Multi-Cluster Subband Fading Model}\\label{multicluster}\nWe now present the multi-cluster subband fading model. Consider a downlink multiuser OFDMA system with one base station and $G$ clusters of\nusers. The system consists of $N$ resource blocks and the total number of users equals $K$. Users in cluster $\\mathcal{K}_g$ are indexed by the\nset $\\mathcal{K}_g=\\{1,\\ldots,k,\\ldots,K_g\\}$ for $1\\leq g\\leq G$, with $|\\mathcal{K}_g|=K_g$ and $\\sum_{g=1}^G K_g=K$. In our framework, users\nin the same cluster group their resource blocks into subbands in the same manner while each cluster can potentially employ a different grouping\nwhich enables the subband size to be heterogeneous between clusters. Denote $\\eta_g$ as the subband size for cluster $\\mathcal{K}_g$, and\n$\\eta_g \\in \\{2^0,2^1,\\ldots,N\\}$. The $\\eta_g$'s are ordered such that $\\eta_1<\\cdots<\\eta_G$. Based on the assumption for $\\eta_g$, the number\nof subbands in cluster $\\mathcal{K}_g$ equals $\\frac{N}{\\eta_g}$.\n\nLet $H_{k,r}^{(g)}$ be the frequency domain channel transfer function between transmitter and user $k$ in cluster $\\mathcal{K}_g$ at subband\n$r$,  where $1\\leq k \\leq K_g, 1\\leq r \\leq \\frac{N}{\\eta_g}$. $H_{k,r}^{(g)}$ is distributed as $\\mathcal{CN}(0,1)$. According to the subband\nfading model, $H_{k,r}^{(g)}$ is assumed to be independent across users and subbands in cluster $\\mathcal{K}_g$. The feedback for different\nclusters is at different granularity, and so to model the channel for the different clusters of users at the same basic resource block level,\nsome additional notation is needed. Let $\\tilde{H}_{k,n}^{(g)}=H_{k,\\lceil \\frac{n}{\\eta_g}\\rceil}^{(g)}$ with $1\\leq n \\leq N$ denote the\nresource block based channel transfer function. Then the received signals of user $k$ in cluster $\\mathcal{K}_g$ at resource block $n$ can be\nrepresented by:\n\\begin{equation} \\label{system:eq_5}\nu_{k,n}^{(g)}= \\sqrt{P}\\tilde{H}_{k,n}^{(g)}s_{k,n}^{(g)}+v_{k,n}^{(g)},\n\\end{equation}\nwhere $P$ is the average received power per resource block, $s_{k,n}^{(g)}$ is the transmitted symbol and $v_{k,n}^{(g)}$ is additive white\nnoise distributed with $\\mathcal{CN}(0,\\sigma_n^2)$.\n\nLet $Z_{k,r}^{(g)}\\triangleq |H_{k,r}^{(g)}|^2$ denote the CQI for user $k$ in cluster $\\mathcal{K}_g$ at subband $r$. In order to reduce the\nfeedback load, users employ the best-M strategy to feed back their CQI. In the basic best-M feedback policy, users measure CQI for each resource\nblock at their receiver and feed back the CQI values of the $M$ best resource blocks chosen from the total $N$ values. For each resource block,\nthe scheduling policy selects the user with the largest CQI among the users who fed back CQI to the transmitter for that resource block.\nHowever, in our heterogeneous partial feedback framework, since the number of independent CQI for cluster $\\mathcal{K}_g$ is $\\frac{N}{\\eta_g}$,\na fair and reasonable way to allocate the feedback resource is to linearly scale the feedback amount for users in cluster $\\mathcal{K}_g$. To be\nspecific, user $k$ in $\\mathcal{K}_G$ (i.e., the cluster with the largest subband size) is assumed to feed back the $M$ best CQI selected from\n$\\{Z_{k,r}^{(G)}\\}, 1\\leq r \\leq \\frac{N}{\\eta_G}$, whereas user $k$ in $\\mathcal{K}_g$ conveys the $\\frac{\\eta_G}{\\eta_g}M$ best CQI selected\nfrom $\\{Z_{k,r}^{(g)}\\}, 1\\leq r \\leq \\frac{N}{\\eta_g}$. After receiving feedback from all the clusters, for each resource block the system\nschedules the user for transmission with the largest CQI. It is useful to note that the user feedback is based on the subband level, while the\nbase station schedules transmission at the resource block level.\n\n\n\\section{Perfect Feedback}\\label{perfect}\nIn this section, the CQI are assumed to be fed back without any errors and the average sum rate is employed as the performance metric for system\nevaluation. We derive a closed form expression for the average sum rate in Section \\ref{spectral_efficiency} for the multi-cluster heterogeneous\nfeedback system. In Section \\ref{ratio} we analyze the relationship between the sum rate ratio and the choice of the best-M.\n\n\\subsection{Derivation of Average Sum Rate}\\label{spectral_efficiency}\nAccording to the assumption, the CQI $Z_{k,r}^{(g)}$ is i.i.d. across subbands and users, and thus let $F_Z$ denote the CDF. Because only a\nsubset of the ordered CQI are fed back, from the transmitter's perspective, if it receives feedback on a certain resource block from a user, it\nis likely to be any one of the CQI from the ordered subset. We now aim to find the CDF of the CQI seen at the transmitter side as a consequence\nof partial feedback. Let $\\tilde{Y}_{k,n}^{(g)}$ denote the reported CQI viewed at the transmitter for user $k$ in $\\mathcal{K}_g$ at resource\nblock $n$. Also, let $Y_{k,r}^{(g)}$ represent the subband-based CQI seen at the transmitter for user $k$ in $\\mathcal{K}_g$ at subband $r$,\nthen $\\tilde{Y}_{k,n}^{(g)}=Y_{k,\\lceil \\frac{n}{\\eta_g}\\rceil}^{(g)}$. The following lemma describes the CDF of $\\tilde{Y}_{k}^{(g)}$ (the\nindex $n$ is dropped for notational simplicity), which is denoted by $F_{\\tilde{Y}_{k}^{(g)}}$.\n\n\\begin{lemma} \\label{lemma_1}\nThe CDF of $\\tilde{Y}_{k}^{(g)}$ is given by:\n\\begin{equation} \\label{perfect:eq_1}\nF_{\\tilde{Y}_{k}^{(g)}}(x)=\\sum_{m=0}^{\\frac{\\eta_G}{\\eta_g}M-1}\\xi_g(N,M,\\boldsymbol\\eta,m)(F_Z(x))^{\\frac{N}{\\eta_g}-m},\n\\end{equation}\nwhere the vector $\\boldsymbol\\eta\\triangleq(\\eta_1,\\cdots,\\eta_g,\\cdots,\\eta_G)$ and\n\\begin{equation} \\label{perfect:eq_2}\n\\xi_g(N,M,\\boldsymbol\\eta,m)=\\sum_{i=m}^{\\frac{\\eta_G}{\\eta_g}M-1}\\frac{\\frac{\\eta_G}{\\eta_g}M-i}{\\frac{\\eta_G}{\\eta_g}M}{\\frac{N}{\\eta_g}\\choose\ni}{i\\choose m}\\left(-1\\right)^{i-m}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe proof is provided in Appendix \\ref{appenA}.\n\\end{proof}\n\nLet $k_n^*$ demote the selected user at resource block $n$, then according to the scheduling policy:\n\\begin{equation} \\label{perfect:eq_3}\nk_n^*=\\arg\\max_{k\\in \\mathcal{U}_n}\\;\\{\\tilde{Y}_{k,n}^{(1)},\\cdots,\\tilde{Y}_{k,n}^{(g)},\\cdots,\\tilde{Y}_{k,n}^{(G)}\\},\n\\end{equation}\nwhere $\\mathcal{U}_n\\triangleq\\{\\mathcal{U}_n^{(1)},\\cdots,\\mathcal{U}_n^{(g)},\\cdots,\\mathcal{U}_n^{(G)}\\}$ is the set of users who convey\nfeedback for resource block $n$, with $|\\mathcal{U}_n^{(g)}|=\\tau_g$ representing the number of users belonging to $\\mathcal{U}_n$ in cluster\n$\\mathcal{K}_g$. It can be easily seen that in the full feedback case, i.e., $M=M_F\\triangleq\\frac{N}{\\eta_G}$, $|\\mathcal{U}_n^{(g)}|=K_g$. For\nthe general case when $1\\leq M<M_F$, the probability mass function (PMF) of $\\mathcal{U}_n$ is given by:\n\\begin{equation} \\label{perfect:eq_4}\n\\mathbb{P}(\\mathcal{U}_n)=\\left(\\prod_{g=1}^G{K_g\\choose \\tau_g}\\right)\\left(\\frac{\\eta_G M}{N}\\right) ^{\\sum_{g=1}^G\\tau_g}\\left(1-\\frac{\\eta_G\nM}{N}\\right)^{K-\\sum_{g=1}^G\\tau_g}, \\quad 0\\leq \\tau_g\\leq K_g.\n\\end{equation}\n\\textit{Remark:} Only the largest subband size $\\eta_G$ appears in the expression of $\\mathbb{P}(\\mathcal{U}_n)$ instead of the vector\n$\\boldsymbol\\eta$. This is due to our heterogeneous partial feedback design to let users in cluster $\\mathcal{K}_g$ convey back the\n$\\frac{\\eta_G}{\\eta_g}M$ best CQI out of $\\frac{N}{\\eta_g}$ values.\n\nNow we turn to determine the conditional CDF of the CQI for the selected user at resource block $n$, conditioned on the set of users providing\nCQI for that resource block. Since users are equiprobable to be scheduled according to the fair scheduling policy, the condition on $k_n^*$ is\nnot described explicitly, and so we denote the conditional CDF as $F_{X_n|\\mathcal{U}_n}$, where $X_n|\\mathcal{U}_n$ is the conditional CQI of\nthe selected user at resource block $n$. Notice from Lemma \\ref{lemma_1} that $\\tilde{Y}_{k}^{(g)}$ possess a different distribution for\ndifferent $g$ due to the heterogeneous feedback from different clusters. Using order statistics \\cite{david03} yields $F_{X_n|\\mathcal{U}_n}$\nas:\n\\begin{equation} \\label{perfect:eq_5}\nF_{X_n|\\mathcal{U}_n}(x)=\\prod_{g=1}^G(F_{\\tilde{Y}_{k}^{(g)}}(x))^{\\tau_g}.\n\\end{equation}\nThen the polynomial form of $F_{X_n|\\mathcal{U}_n}$ can be obtained, which is stated in the following theorem.\n\n\\begin{theorem} \\label{theorem_1}\nThe CDF of $F_{X_n|\\mathcal{U}_n}$ is given by:\n\\begin{equation} \\label{perfect:eq_6}\nF_{X_n|\\mathcal{U}_n}(x)=\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)(F_Z(x))^{\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m},\n\\end{equation}\nwhere the vector $\\boldsymbol\\tau\\triangleq(\\tau_1,\\cdots,\\tau_g,\\cdots,\\tau_G)$,\n$\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)\\triangleq\\sum_{g=1}^G\\tau_g\\left(\\frac{\\eta_G}{\\eta_g}M-1\\right)$,\n\\begin{equation} \\label{perfect:eq_7}\n\\Theta_{g}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)=\\left\\{\n\\begin{array}{l} \\mathop{\\sum}\\limits_{i=0}^m\\Lambda_1(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,i)\\Lambda_2(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m-i),\\quad g=1\\\\\n\\mathop{\\sum}\\limits_{i=0}^m\\Theta_{g-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,i)\\Lambda_{g+1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m-i),\\quad 2\\leq g<G\\\\\n\\end{array} \\right.\n\\end{equation}\n\\begin{equation} \\label{perfect:eq_8}\n\\Lambda_g(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)=\\left\\{\n\\begin{array}{l} (\\xi_g(N,M,\\boldsymbol\\eta,0))^{\\tau_g},\\quad m=0\\\\ \\frac{1}{m\\xi_g(N,M,\\boldsymbol\\eta,0)}\\sum_{\\ell=1}^{\\min\\left(m,\\frac{\\eta_G}{\\eta_g}\nM-1\\right)}((\\tau_g+1)\\ell-m)\\\\\n\\times\\xi_g(N,M,\\boldsymbol\\eta,\\ell)\\Lambda_g(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m-\\ell),\\quad 1\\leq m<\\tau_g(\\frac{\\eta_G}{\\eta_g}M-1)\\\\\n(\\xi_g(N,M,\\boldsymbol\\eta,\\frac{\\eta_G}{\\eta_g} M-1))^{\\tau_g},\\quad m=\\tau_g(\\frac{\\eta_G}{\\eta_g} M-1).\\\\ \\end{array} \\right.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe proof is provided in Appendix \\ref{appenA}.\n\\end{proof}\n\nAfter obtaining the conditional CDF $F_{X_n|\\mathcal{U}_n}$, let $C_P(M)$ denote the average sum rate and it can be computed using the following\nprocedure.\n\\begin{align}\nC_P(M)&=\\frac{1}{N}\\sum_{n=1}^N\\mathbb{E}[\\log_2(1+X_n)]\\notag\\\\\n&\\mathop{=}\\limits^{(a)}\\mathbb{E}_{\\mathcal{U}}\\left[\\int_0^\\infty \\log_2(1+\\rho\nx)d(F_{X|\\mathcal{U}}(x))\\right]\\notag\\\\\n&\\mathop{=}\\limits^{(b)}\\mathbb{E}_{\\mathcal{U}}\\left[\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\int_0^\\infty\n\\log_2(1+\\rho\nx)d(F_Z(x))^{\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m}\\right]\\notag\\\\\n\\label{perfect:eq_9}&\\mathop{=}\\limits^{(c)}\\sum_{\\boldsymbol\\tau\\neq\\mathbf{0}}\\mathbb{P}(\\mathcal{U})\n\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\mathcal{I}_1\\left(\\rho,\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m\\right),\n\\end{align}\nwhere $\\rho\\triangleq \\frac{P}{\\sigma_n^2}$ and $\\mathbb{P}(\\mathcal{U})$ is given by (\\ref{perfect:eq_4}). (a) follows from the conditional\nexpectation of $X_n|\\mathcal{U}_n$ and the identically distributed property (let $X$ and $\\mathcal{U}$ represent $X_n$ and $\\mathcal{U}_n$\nrespectively), (b) follows from (\\ref{perfect:eq_6}) in Theorem \\ref{theorem_1}, (c) follows from (\\ref{perfect:eq_4}), and define\n$\\mathcal{I}_1(a,b)\\triangleq \\int_0^\\infty \\log_2(1+ax)d(F_Z(x))^b$. $\\mathcal{I}_1(a,b)$ is computed in Appendix \\ref{appenA} to be:\n\\begin{equation}\n\\label{perfect:eq_10}\n\\mathcal{I}_1(a,b)=\\frac{b}{\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}\\frac{(-1)^{\\ell}}{\\ell+1}\\exp\\left(\\frac{\\ell+1}{a}\\right)E_1\\left(\\frac{\\ell+1}{a}\\right),\n\\end{equation}\nwhere $E_1(x)=\\int_x^\\infty \\exp(-t)t^{-1}dt$ is the exponential integral function \\cite{abramowitz72}.\n\nThe average sum rate for the full feedback is a special case and is given by:\n\\begin{equation}\n\\label{perfect:eq_11} C_P(M_F)=\\int_0^\\infty \\log_2(1+\\rho x)d(F_Z(x))^K=\\mathcal{I}_1(\\rho,K).\n\\end{equation}\n\n\\textit{Remark:} It is noteworthy to mention that the functional form of $C_P(M)$ in (\\ref{perfect:eq_9}) consists of two main parts. The first\npart, which involves $\\mathbb{P}(\\mathcal{U})$ and $\\Theta_{G-1}(\\cdot,\\cdot,\\cdot,\\cdot,\\cdot)$, accounts for the randomness of the set of\nusers who convey feedback as well as the scheduling policy. This part is inherent to the heterogeneous partial feedback strategy, and is\nindependent of the system metric for evaluation, such as the average sum rate employed in this paper. The second part\n$\\mathcal{I}_1(\\cdot,\\cdot)$ depends on statistical assumption of the underlying channel and the system metric, and it is impacted by partial\nfeedback as well.\n\n\n\\subsection{Sum Rate Ratio and Best-M}\\label{ratio}\nWe now examine how to determine the smallest $M$ that results in almost the same performance, in terms of average sum rate, as the full feedback\ncase. Applying the same technique as in \\cite{choi08, hur11}, define $\\gamma_P$ as the spectral efficiency ratio and the problem can be\nformulated as:\n\\begin{equation}\n\\label{perfect:eq_12} \\mathrm{Find\\; the\\; minimum\\;} M^{\\ast},\\quad s.t.\\; \\gamma_P=\\frac{C_P(M^{\\ast})}{C_P(M_F)}\\geq \\gamma.\n\\end{equation}\nThe above problem can be numerically solved by substituting the expressions for $C_P(M)$ and $C_P(M_F)$. In order to obtain a simpler and\ntractable relationship between $M$ and $K$ given $\\boldsymbol\\eta$, i.e., the tradeoff between the amount of partial feedback and the number of\nusers given existing heterogeneity of channel statistics in frequency domain, an approximation is utilized similar to that in \\cite{hur11}, by\nobserving that $\\mathcal{I}_1(a,b)$ in (\\ref{perfect:eq_10}) is slowly increasing in $b$ with fixed $a$ (This phenomenon is due to the\nsaturation of multiuser diversity \\cite{sharif05}). Observing\n$\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)=1$ and employing the binomial theorem\nyields the approximation for the spectral efficiency ratio as:\n\\begin{equation}\n\\label{perfect:eq_13} \\gamma_P\\simeq 1-\\left(1-\\frac{\\eta_G M^{\\ast}}{N}\\right)^K.\n\\end{equation}\nFrom (\\ref{perfect:eq_12}) and (\\ref{perfect:eq_13}), the minimum required $M^{\\ast}$ can be obtained as follows:\n\\begin{equation} \\label{perfect:eq_14}\nM^{\\ast}\\geq \\frac{N}{\\eta_G}\\left(1-\\left(1-\\gamma\\right)^{\\frac{1}{K}}\\right).\n\\end{equation}\n\\textit{Remark:} It can be seen that $M^*$ depends on the system parameters ($N,K,\\gamma$) as well on the largest subband size $\\eta_G$. It is\nalso a consequence of our heterogeneous partial feedback assumption to let users in cluster $\\mathcal{K}_g$ convey back the\n$\\frac{\\eta_G}{\\eta_g}M$ best CQI out of $\\frac{N}{\\eta_g}$ values. This results in the fact that obtaining feedback information from users\nbelonging to different clusters have almost the same statistical influence on scheduling performance.\n\n\n\n\\section{Imperfect Feedback}\\label{imperfect}\nAfter analyzing the heterogeneous partial feedback design with perfect feedback, we turn to examine the impact of feedback imperfections in this\nsection. We develop the imperfect feedback model due to channel estimation error and feedback delay in Section \\ref{imperfect_model}, and\ninvestigate the influence of imperfections on two different transmission strategies in Section \\ref{fix_rate} and \\ref{vari_rate}. Then we\npropose how to optimize the system performance to adapt to the imperfections in Section \\ref{optimize}.\n\n\n\\subsection{Imperfect Feedback Model}\\label{imperfect_model}\nThe imperfect feedback model is built upon the subband fading model for the perfect feedback case. To differentiate from the notation for the\nperfect feedback case and focus on the imperfect feedback model, the resource block index is dropped. Let $h_k$ denote the frequency domain\nchannel transfer function of user $k$ (users in different clusters are not temporally distinguished to avoid notational overload). Due to\nchannel estimation error, the user only has its estimated version $\\hat{h}_k$, and the relationship between $h_k$ and $\\hat{h}_k$ can be modeled\nas:\n\\begin{equation}\n\\label{imperfect:eq_1} h_k=\\hat{h}_k+w_k,\n\\end{equation}\nwhere $w_k\\sim \\mathcal{CN}(0,\\sigma_{w_k}^2)$ is the channel estimation error. The channel of each resource block is assumed to be estimated\nindependently, which yields the channel estimation errors $w_k$ i.i.d. across users and resource blocks, i.e., $w_k\\sim\n\\mathcal{CN}(0,\\sigma_{w}^2)$. It is clear that the base station makes decision on scheduling and adaptive transmission depending on CQI, a\nfunction of $\\hat{h}_k$. Thus this information can be outdated due to delay between the instant CQI is measured and the actual instant of use\nfor data transmission to the selected user. Let $\\tilde{h}_k$ be the actual channel transfer function and we employ a first-order\nGaussian-Markov model \\cite{kuhne08, isukapalli10, akoum10} to describe the time evolution and to capture the relationship with the delayed\nversion as follows:\n\\begin{equation}\n\\label{imperfect:eq_2} \\tilde{h}_k =\n\\alpha_k(\\hat{h}_k+w_k)+\\sqrt{1-\\alpha_k^2}\\varepsilon_k,\n\\end{equation}\nwhere $\\varepsilon_k$ accounts for the innovation noise and is distributed as $\\mathcal{CN}(0,1)$. The delay time between $\\tilde{h}_k$ and\n$\\hat{h}_k$ is not explicitly written for notational simplicity, and $\\alpha_k\\in [0,1]$ is used to model the correlation coefficient. Since the\nfeedback delay is mainly caused by the periodic feedback interval and processing complexity \\cite{akoum10}, the innovation noise $\\varepsilon_k$\nare i.i.d. across users and a common $\\alpha$ is assumed. Moreover, $w_k$ and $\\varepsilon_k$ are assumed independent. Therefore, for notational\nsimplicity, the user index $k$ in the aforementioned parameters is dropped and $\\hat{Z}\\triangleq|\\hat{h}|^2$ is denoted as CQI.\n\nLet $\\tilde{\\chi},\\chi$ and $\\hat{\\chi}$ represent: the actual CQI of the selected user for transmission, its outdated version, and its outdated\nestimate respectively ($\\hat{\\chi}$ corresponds to $X$ for the perfect feedback case in Section \\ref{spectral_efficiency}). Notice that the PDF\nof the outdated estimate $\\hat{\\chi}$ depends on the heterogeneous feedback design and the scheduling strategy, whereas the conditional PDF of\n$\\tilde{\\chi}|\\hat{\\chi}$ only depends on $\\alpha$ and $\\sigma_w^2$. Employing the same method in \\cite{ma05, kuhne08}, the conditional PDF is\nobtained as follows:\n\\begin{equation}\n\\label{imperfect:eq_3}\nf_{\\tilde{\\chi}|\\hat{\\chi}}(x|\\hat{\\chi})=\\frac{\\alpha_w^2}{2}\\exp\\left(-\\frac{\\alpha_w^2x+\\alpha_w^2\\alpha^2\\hat{\\chi}}{2}\\right)I_0(\\alpha_w^2\\alpha\\sqrt{\\hat{\\chi}x}),\n\\end{equation}\nwhere $\\alpha_w=\\sqrt{\\frac{2}{\\alpha^2\\sigma_w^2+1-\\alpha^2}}$, and $I_0(\\cdot)$ is the zeroth-order modified Bessel function of the first kind\n\\cite{abramowitz72}.\n\nSince the feedback is imperfect, there are two types of issues that arise. The first is the choice of the incorrect user to serve. However,\nbecause of the i.i.d nature of the errors this does not compromise the fairness and also does not complicate the determination of the CDF. The\nsecond problem is that of outage because the rate adaptation is made by the base station based on the erroneous CQI. Because of the error in the\nCQI, the rate chosen may exceed the rate that the channel can support and so the base station has to take steps to mitigate this effect of\noutage. A conservative strategy will result in less outage but under utilization of the channel while an aggressive strategy will result in good\nutilization of the channel but only for a small fraction of the time. We now present two transmission strategies to address the outage issue.\n\n\\subsection{Fix Rate Strategy}\\label{fix_rate}\nIn the fix rate conservative scenario, a system parameter $\\beta_0$ is chosen for rate adaptation, and outage results under the following condition:\n\\begin{equation}\n\\label{imperfect:eq_4} \\mathrm{Declare\\; outage\\; if:}\\quad \\{\\tilde{\\chi}\\leq\\beta_0|\\hat{\\chi}\\}.\n\\end{equation}\nThe system average goodput is defined as the total average bps\/Hz successfully transmitted \\cite{lau08}. We derive the average goodput and\naverage outage probability for a given choice of system parameter $\\beta_0$ in the following procedure.\n\nFirstly the conditional outage probability is expressed as:\n\\begin{equation}\n\\label{imperfect:eq_5} \\mathbb{P}(\\tilde{\\chi}\\leq\\beta_0|\\hat{\\chi})=1-\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\hat{\\chi}},\\alpha_w\\sqrt{\\beta_0}),\n\\end{equation}\nwhere $\\mathcal{Q}_1(a,b)=\\int_b^{\\infty}t\\exp(-\\frac{t^2+a^2}{2})I_0(at)dt$ is the first-order Marcum-Q function \\cite{nuttall72}. Denote\n$R_0(\\beta_0,M)$ as the average goodput for the heterogeneous partial feedback system, which is written according to definition:\n\\begin{equation}\n\\label{imperfect:eq_6}\nR_0(\\beta_0,M)=\\mathbb{E}_{\\mathcal{U}}\\left[\\mathbb{E}_{\\hat{\\chi}|\\mathcal{U}}\\left[\\mathbb{P}(\\tilde{\\chi}\\geq\\beta_0|\\hat{\\chi})\\log_2(1+\\rho\\beta_0)\\right]\\right].\n\\end{equation}\nThen, from (\\ref{perfect:eq_4}) and (\\ref{perfect:eq_9}), $R_0(\\beta_0,M)$ can be computed as:\n\\begin{align}\n&R_0(\\beta_0,M)\\notag\\\\\n&=\\mathbb{E}_{\\mathcal{U}}\\left[\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\int_0^\\infty\n\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{\\beta_0})\\log_2(1+\\rho\\beta_0)d(F_{\\hat{Z}}(x))^{\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m}\\right]\\notag\\\\\n\\label{imperfect:eq_7}&=\\sum_{\\boldsymbol\\tau\\neq\\mathbf{0}}\\mathbb{P}(\\mathcal{U})\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\log_2(1+\\rho\\beta_0)\\mathcal{I}_2\\left(\\beta_0,\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m\\right),\n\\end{align}\nwhere $\\mathcal{I}_2(a,b)\\triangleq\\int_0^\\infty \\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{a})d(F_{\\hat{Z}}(x))^b$.\n$\\mathcal{I}_2(a,b)$ is computed in Appendix \\ref{appenB} to be:\n\\begin{equation}\n\\label{imperfect:eq_8}\n\\mathcal{I}_2(a,b)=\\frac{2b}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}}\\left(\\exp(-\\frac{\\vartheta^2}{2})+\\exp(-\\frac{\\zeta_{\\ell}\\vartheta^2}{2(\\varpi^2+\\zeta_{\\ell})})(1-\\exp(-\\frac{\\varpi^2\\vartheta^2}{2(\\varpi^2+\\zeta_{\\ell})}))\\right),\n\\end{equation}\nwhere $\\varpi=\\alpha_w\\alpha$, $\\vartheta=\\alpha_w\\sqrt{a}$,\n$\\zeta_{\\ell}=\\frac{2(\\ell+1)}{1-\\sigma_w^2}$.\n\nThe average outage probability $P_0(\\beta_0,M)$ for the heterogeneous partial feedback design can be directly calculated from definition and\n(\\ref{imperfect:eq_7}) as follows:\n\\begin{align}\nP_0(\\beta_0,M)&=\\mathbb{E}_{\\mathcal{U}}\\left[\\mathbb{E}_{\\hat{\\chi}|\\mathcal{U}}\\left[\\mathbb{P}(\\tilde{\\chi}\\leq\\beta_0|\\hat{\\chi})\\right]\\right]\\notag\\\\\n\\label{imperfect:eq_9}&=\\sum_{\\boldsymbol\\tau\\neq\\mathbf{0}}\\mathbb{P}(\\mathcal{U})\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\left(1-\\mathcal{I}_2\\left(\\beta_0,\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m\\right)\\right).\n\\end{align}\n\nThe average goodput and average outage probability for the full feedback scenario is a special case and is given by:\n\\begin{align}\nR_0(\\beta_0,M_F)&=\\log_2(1+\\rho\\beta_0)\\mathcal{I}_2(\\beta_0,K),\\notag\\\\\n\\label{imperfect:eq_10}P_0(\\beta_0,M_F)&=1-\\mathcal{I}_2(\\beta_0,K).\n\\end{align}\n\n\n\\subsection{Variable Rate Strategy}\\label{vari_rate}\nInstead of choosing a conservative system parameter to account for the fix rate scenario as in the previous subsection, we consider an approach\nwe refer to as the variable rate strategy. In the variable rate scenario, a system parameter $\\beta_1$ is chosen and outage results under the\nfollowing condition:\n\\begin{equation}\n\\label{imperfect:eq_11} \\mathrm{Declare\\; outage\\; if:}\\quad \\{\\tilde{\\chi}\\leq\\beta_1\\hat{\\chi}|\\hat{\\chi}\\},\n\\end{equation}\nwhere $\\beta_1$ can be regarded as the backoff factor. The system average goodput and average outage probability can be derived utilizing the\nfollowing procedure.\n\nNow under the variable rate scenario, the conditional outage probability is expressed as:\n\\begin{equation}\n\\label{imperfect:eq_12}\n\\mathbb{P}(\\tilde{\\chi}\\leq\\beta_1\\hat{\\chi}|\\hat{\\chi})=1-\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\hat{\\chi}},\\alpha_w\\sqrt{\\beta_1\\hat{\\chi}}).\n\\end{equation}\nUsing the same method as (\\ref{imperfect:eq_6}) and (\\ref{imperfect:eq_7}), let $R_1(\\beta_1,M)$ denote the average goodput for the variable\nrate scenario whose expression can be written as follows:\n\\begin{align}\n\\label{imperfect:eq_13}\nR_1(\\beta_1,M)&=\\mathbb{E}_{\\mathcal{U}}\\left[\\mathbb{E}_{\\hat{\\chi}|\\mathcal{U}}\\left[\\mathbb{P}(\\tilde{\\chi}\\geq\\beta_1\\hat{\\chi}|\\hat{\\chi})\\log_2(1+\\rho\\beta_1\\hat{\\chi})\\right]\\right]\\notag\\\\\n&=\\sum_{\\boldsymbol\\tau\\neq\\mathbf{0}}\\mathbb{P}(\\mathcal{U})\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\mathcal{I}_3\\left(\\beta_1,\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m\\right),\n\\end{align}\nwhere $\\mathcal{I}_3(a,b)\\triangleq\\int_0^\\infty \\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{ax})\\log_2(1+\\rho ax)d(F_{\\hat{Z}}(x))^b$.\n\nFor the full feedback case, the average goodput is given by:\n\\begin{equation}\n\\label{imperfect:eq_14} R_1(\\beta_1,M_F)=\\mathcal{I}_3(\\beta_1,K).\n\\end{equation}\n\nNote that unlike $\\mathcal{I}_2(a,b)$, $\\mathcal{I}_3(a,b)$ can not be written in closed form. Therefore, bounding technique and suitable\napproximation are attractive to find closed form alternatives for $\\mathcal{I}_3(a,b)$. The following proposition presents a valid closed form\nupper bound for $\\mathcal{I}_3(a,b)$ in the low $\\mathsf{SNR}$ regime.\n\\begin{proposition}\n\\label{proposition_1} In the low $\\mathsf{SNR}$ regime, $\\mathcal{I}_3(a,b)$ can be efficiently upper bounded by:\n\\begin{align}\n\\mathcal{I}_3^{\\mathrm{UB}}(a,b)=&\\frac{4\\rho ab}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}^2}\\Bigg(1+\\frac{\\vartheta^2}{\\varphi_{\\ell}}\\Bigg(\\frac{\\varpi^2}{\\varphi_{\\ell}}{}_2F_1\\left(1,\\frac{3}{2};2;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)-{}_2F_1\\left(\\frac{1}{2},1;1;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)\\notag\\\\\n\\label{imperfect:eq_15}&+\\frac{2\\zeta_{\\ell}}{\\varphi_{\\ell}}\\Bigg(\\frac{\\varpi^2}{\\varphi_{\\ell}}{}_2F_1\\left(\\frac{3}{2},2;2;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)-\\frac{1}{2}{}_2F_1\\left(1,\\frac{3}{2};1;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)\\Bigg)\\Bigg)\\Bigg),\n\\end{align}\nwhere $\\varpi=\\alpha_w\\alpha$, $\\vartheta=\\alpha_w\\sqrt{a}$, $\\zeta_{\\ell}=\\frac{2(\\ell+1)}{1-\\sigma_w^2}$,\n$\\varphi_{\\ell}=\\varpi^2+\\vartheta^2+\\zeta_{\\ell}$, and ${}_2F_1(\\cdot,\\cdot;\\cdot;\\cdot)$ is the Gaussian hypergeometric function\n\\cite{abramowitz72}.\n\\end{proposition}\n\\begin{proof}\nThe proof is provided in Appendix \\ref{appenB}.\n\\end{proof}\n$\\mathcal{I}_3^{\\mathrm{UB}}(a,b)$ is valid and tight especially for the low $\\mathsf{SNR}$ regime. In order to track $\\mathcal{I}_3(a,b)$ over\nthe whole $\\mathsf{SNR}$ regimes, we propose the following approximation method by leveraging Jensen's inequality \\cite{boyd04}. Recall the\ndefinition of $\\mathcal{I}_3(a,b)=\\mathbb{E}[\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\check{\\chi}},\\alpha_w\\sqrt{a\\check{\\chi}})\\log_2(1+\\rho\na\\check{\\chi})]$, where the random variable $\\check{\\chi}$ is defined to have CDF $(F_{\\hat{Z}}(x))^b$. Firstly, $\\mathbb{E}[\\check{\\chi}]$ can\nbe computed and is given by:\n\\begin{align}\n\\mathbb{E}[\\check{\\chi}]&=\\int_0^\\infty\nx\\frac{b}{1-\\sigma_w^2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\exp\\left({-\\frac{(\\ell+1)x}{1-\\sigma_w^2}}\\right)dx\\notag\\\\\n\\label{imperfect:eq_16}&=\\frac{b}{1-\\sigma_w^2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\left(\\frac{1-\\sigma_w^2}{\\ell+1}\\right)^2.\n\\end{align}\nThen plugging (\\ref{imperfect:eq_16}) into $\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{ax})\\log_2(1+\\rho ax)$ yields:\n\\begin{equation}\n\\label{imperfect:eq_17}\n\\mathcal{I}_3^{A}(a,b)=\\mathcal{Q}_1\\left(\\alpha_w\\alpha\\sqrt{\\mathbb{E}[\\check{\\chi}]},\\alpha_w\\sqrt{a\\mathbb{E}[\\check{\\chi}]}\\right)\\log_2(1+\\rho\na\\mathbb{E}[\\check{\\chi}]).\n\\end{equation}\n\nNote that $\\mathcal{I}_3^{A}(a,b)$ would serve as an upper bound from Jensen's inequality if the function of interest\n$\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{ax})\\log_2(1+\\rho ax)$ was concave in $x$. Properties of this function such as monotonicity\nand concavity are of interest and lead to rigorous arguments in support of this bound. If outage does not occur, extensive analysis can be\ncarried out due to the well known properties of the $\\log(\\cdot)$ function. However, the concavity (or log-concavity) of\n$\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{\\beta_1x})$ in $x$ (notice that $x$ appears in both entries of $\\mathcal{Q}_1(\\cdot,\\cdot)$)\nstill remains an important open problem \\cite{yu11}. Our numerical evidence suggests that\n$\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{\\beta_1x})\\log_2(1+\\rho \\beta_1x)$ is concave and monotonically increasing in $x$ for\npractical choices of $\\beta_1$.  For any given $\\beta_1$ preserving the aforementioned property, Jensen's inequality yields an upper bound,\nwhose tightness is of interest and discussed in the following proposition. The word practical is used to exclude the situation when $\\beta_1$\napproaches its maximum $1$ which in turn enables $\\mathcal{Q}_1(\\cdot,\\cdot)$ to dominate the goodput to incur extreme outage. This makes\nintuitive sense according to the definition of average goodput.\n\\begin{proposition}\n\\label{proposition_2} Let $\\{\\check{\\chi}_b\\}$ be the family of positive i.i.d. random variables. If\n$\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{\\beta_1x})\\log_2(1+\\rho \\beta_1x)$ is concave and monotonically increasing in $x$ for any\ngiven $\\beta_1$, then the Jensen bound is asymptotically tight, i.e., as $b\\rightarrow\\infty$,\n\\begin{equation}\n\\label{imperfect:eq_18} \\frac{\\mathbb{E}[\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\check{\\chi}_b},\\alpha_w\\sqrt{\\beta_1\\check{\\chi}_b})\\log_2(1+\\rho\n\\beta_1\\check{\\chi}_b)]}{\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\mathbb{E}[\\check{\\chi}_b]},\\alpha_w\\sqrt{\\beta_1\\mathbb{E}[\\check{\\chi}_b]})\\log_2(1+\\rho\n\\beta_1\\mathbb{E}[\\check{\\chi}_b])}\\rightarrow 1.\n\\end{equation}\n\\begin{proof}\nThe proof is provided in Appendix \\ref{appenB}.\n\\end{proof}\n\\end{proposition}\nNonetheless, when the aforementioned property is not preserved (e.g., $\\beta_1$ approaches $1$), Jensen's inequality does not hold but the\nexpression has been experimentally found to be a good approximation and so can still be used. Therefore, (\\ref{imperfect:eq_17}) is denoted as\nJensen approximation. We conduct a numerical study and demonstrate the tightness of Jensen approximation in Fig. \\ref{fig_3}. It is observed\nthat the approximation method is very tight for moderate (even small) number of users and for all values of $\\beta_1\\in[0,1]$, which shows its\npotential in accurately tracking the performance of average goodput.\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.68\\linewidth]{Figure\/fig3.eps}\n\\caption{Calculating the average goodput from numerical evaluation ($M=2,4,16$) and Jensen approximation ($M=16$) for the variable rate scenario\nunder different $\\rho$. ($N=64$, $\\boldsymbol\\eta=(1,4)$, $K_1=K_2=K\/2=10$, $\\alpha=0.98$, $\\sigma_w^2=0.01$, $\\rho=10$ dB, and $20$ dB)}\n\\label{fig_3}\n\\end{figure}\n\nNow we calculate the average outage probability. Since it does not involve the $\\log(\\cdot)$ function, it can be computed into closed form as\nfollows:\n\\begin{align}\nP_1(\\beta_1,M)&=\\mathbb{E}_{\\mathcal{U}}\\left[\\mathbb{E}_{\\hat{\\chi}|\\mathcal{U}}\\left[\\mathbb{P}(\\tilde{\\chi}\\leq\\beta_1\\hat{\\chi}|\\hat{\\chi})\\right]\\right]\\notag\\\\\n\\label{imperfect:eq_19}&=\\sum_{\\boldsymbol\\tau\\neq\\mathbf{0}}\\mathbb{P}(\\mathcal{U})\\sum_{m=0}^{\\Phi(M,\\boldsymbol\\eta,\\boldsymbol\\tau)}\\Theta_{G-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)\\left(1-\\mathcal{I}_4\\left(\\beta_1,\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g-m\\right)\\right),\n\\end{align}\nwhere \\begin{align}\n\\mathcal{I}_4(a,b)&\\triangleq\\int_0^\\infty\n\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{x},\\alpha_w\\sqrt{ax})d(F_{\\hat{Z}}(x))^b\\notag\\\\\n&\\mathop{=}\\limits^{(a)}\\frac{2b}{(1-\\sigma_w^2)}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\int_0^\\infty\\mathcal{Q}_1(\\alpha_w\\alpha x,\\alpha_w\\sqrt{a}x)\\exp\\left(-\\frac{(\\ell+1)x^2}{1-\\sigma_w^2}\\right)xdx\\notag\\\\\n\\label{imperfect:eq_20}&\\mathop{=}\\limits^{(b)}\\frac{b}{(1-\\sigma_w^2)}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}}\\left(1+\\frac{\\psi_{\\ell}}{\\varsigma_{\\ell}}\\right),\n\\end{align}\n$\\varpi=\\alpha_w\\alpha$, $\\vartheta=\\alpha_w\\sqrt{a}$, $\\zeta_{\\ell}=\\frac{2(\\ell+1)}{1-\\sigma_w^2}$,\n$\\varphi_{\\ell}=\\varpi^2+\\vartheta^2+\\zeta_{\\ell}$, $\\psi_{\\ell}=\\varpi^2-\\vartheta^2+\\zeta_{\\ell}$,\n$\\varsigma_{\\ell}=\\sqrt{\\varphi_{\\ell}^2-4\\varpi^2\\vartheta^2}$. (a) follows from change of variables; (b) follows from applying\n\\cite[B.48]{simon02}.\n\nIn the case of full feedback, the average outage probability $P_1(\\beta_1,M_F)$ becomes:\n\\begin{equation}\n\\label{imperfect:eq_21} P_1(\\beta_1,M_F)=1-\\mathcal{I}_4(\\beta_1,K).\n\\end{equation}\n\n\n\\subsection{Optimization and Adaptation to Imperfections}\\label{optimize}\nWe have obtained the relationship between the system parameter ($\\beta_0$ or $\\beta_1$) and the system average goodput, and we now aim to\nmaximize the average goodput by adapting the system parameters.\n\nConsider the optimization of $R_1(\\beta_1,M)$ to obtain the optimal backoff factor $\\beta_1^{\\ast}$. It is observed from (\\ref{imperfect:eq_13})\nthat directly optimizing $R_1(\\beta_1,M)$ is tedious, and a near-optimal method is now proposed to obtain $\\beta_1^{\\ast}$. This method is\ninspired by the results in Section \\ref{ratio}, which show that the minimum required $M^{\\ast}$ can be chosen to achieve almost the same\nperformance as a system with full feedback. Thus an optimal $\\beta_1^{\\ast}$ for the full feedback scenario can be optimized first, and then\n$M^{\\ast}$ is obtained to ``match\" the system performance. Looking again at Fig. \\ref{fig_3} with emphasis on different number $M$ of partial\nfeedback, as $M$ gets larger, the optimal $\\beta_1$ converges to the full feedback case. In this example, $M^{\\ast}=4$ is adequate to match the\nsystem performance. It is noteworthy to mention that this adaptation philosophy can be applied to partial feedback systems wherein system\nparameters are optimized according to full feedback assumption first and minimum required partial feedback is chosen subsequently.\n\nNote that a closed form approximation has been obtained to track $R_1(\\beta_1,M_F)$ in Section \\ref{vari_rate}, which is denoted as\n$R_1^A(\\beta_1,M_F)\\triangleq\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)$. The following proposition demonstrates the optimal property of $\\beta_1$\nwhen optimizing $R_1^A(\\beta_1,M_F)$.\n\\begin{proposition}\n\\label{proposition_3} There exists a unique global optimal $\\beta_1$ that maximizes $R_1^A(\\beta_1,M_F)$.\n\\end{proposition}\n\\begin{proof}\nThe proof is provided in Appendix \\ref{appenB}.\n\\end{proof}\nFrom the above analysis, the optimization strategy can be described as:\n\\begin{equation}\n\\label{imperfect:eq_22} \\beta_1^{\\ast}=\\arg\\mathop{\\max}\\limits_{0\\leq\\beta_1\\leq1} R_1^A(\\beta_1,M_F)\\simeq\n\\arg\\mathop{\\max}\\limits_{0\\leq\\beta_1\\leq1} R_1(\\beta_1,M_F).\n\\end{equation}\nSince it is proved in Proposition \\ref{proposition_3} that $R_1^A(\\beta_1,M_F)$ is quasiconcave \\cite{boyd04} in $\\beta_1$, numerical approach\nsuch as Newton-Raphson method can be applied to obtain $\\beta_1^{\\ast}$. As discussed before, once $\\beta_1^*$ is found, the minimum required\n$M^*$ can be obtained by solving (\\ref{perfect:eq_12}) or relying on (\\ref{perfect:eq_14}).\n\nThe same strategy can be carried over to the optimization of $\\beta_0$, which is presented as follows:\n\\begin{equation}\n\\label{imperfect:eq_23} \\beta_0^{\\ast}=\\arg\\mathop{\\max}\\limits_{\\beta_0} R_0(\\beta_0,M_F).\n\\end{equation}\nThe impact of imperfections on system parameter adaptation, and the comparison between the fixed rate and variable rate strategies will be\nexamined through simulations in Section \\ref{numerical_imperfect}.\n\n\n\n\\section{Numerical Results}\\label{numerical}\nIn this section, we conduct a numerical study to verify the results developed and to draw some insight.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig4_a.eps}&\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig4_b.eps}\\\\\n    \\scriptsize{(a)}&\n    \\scriptsize{(b)}\\\\\n\\end{tabular}\n\\caption{The required minimum $M$ for heterogeneous perfect feedback design: (a) Comparison of the required minimum $M$ between numerically\nsolving (\\ref{perfect:eq_12}) and using approximation (\\ref{perfect:eq_14}) under different $\\gamma$ with respect to the number of users;\n($N=64$, $\\boldsymbol\\eta=(1,4)$, $K_1=K_2=K\/2$, $\\rho=10$ dB) (b) Computing the required minimum $M$ with respect to different number of users\nwhen varying the ratio of the number of users in cluster $\\mathcal{K}_1$. ($N=64$, $\\boldsymbol\\eta=(1,4)$, $\\rho=10$ dB, $\\gamma=0.99$)}\n\\label{fig_4}\n\\end{figure}\n\n\n\n\\subsection{Perfect Feedback Scenario}\\label{numerical_perfect}\nThe number of resource blocks $N$ is assumed to be $64$ for simulations throughout this section. We first consider a $2$-cluster system. Fig.\n\\ref{fig_4} (a) plots the minimum required $M$ obtained by directly solving (\\ref{perfect:eq_12}) and alternatively by the approximation\n(\\ref{perfect:eq_14}) for two thresholds: $\\gamma=0.99$ and $0.9$. Note that the result from (\\ref{perfect:eq_14}) is rounded with the ceiling\nfunction since the required $M$ is an integer. The other simulation parameters are $\\boldsymbol\\eta=(1,4)$ (i.e., $M_F=16$), and $\\rho=10$ dB.\nIt is observed that the results from the approximate expression matches quite well with the exact computation. The question of whether the\nrequired $M$ is sensitive to the partition of users in the system is examined in Fig. \\ref{fig_4} (b) wherein the ratio of the number of users\nin cluster $\\mathcal{K}_1$ is changed and the minimum required $M$ is depicted for different total number of users with threshold $\\gamma=0.99$.\nInterestingly, the result turns out to be ``uniform\". As discussed in Section \\ref{perfect}, it is due to the heterogenous feedback design\nassumption to let users in cluster $\\mathcal{K}_1$ consume $\\frac{\\eta_G}{\\eta_1} M$ ($4M$ in this simulation) feedback which results in the\nfact that obtaining feedback information from users belonging to different clusters have almost the same influence on scheduling performance.\nTherefore, the representative simulation setup $K_g=K\/G$ can be employed when the system performance metric is investigated with respect to the\ntotal number of users.\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.7\\linewidth]{Figure\/fig5.eps}\n\\caption{Comparison of the average sum rate for a $4$-cluster system under different feedback strategies with respect to the number of users\n($N=64$, $\\boldsymbol\\eta=(1,2,4,8)$, $K_1=K_2=K_3=K_4=K\/4$, $M=2,4$, $\\rho=10$ dB)} \\label{fig_5}\n\\end{figure}\n\nWe now consider a $4$-cluster system with subband size vector $\\boldsymbol\\eta=(1,2,4,8)$ (i.e., $M_F=8$). Fig. \\ref{fig_5} demonstrates the\nbenefit of using heterogeneous feedback design. One of the competing strategies is also heterogeneous, but treats users from each cluster\nseparately. In particular, the system firstly clusters the users based on their channel statistics, and then serves the clusters one by one\nrequiring feedback only from the served cluster of users. In this way, the feedback amount is varying over time depending on the partition of\nusers. This strategy is denoted as \\emph{separate} heterogeneous feedback compared to our \\emph{joint} heterogeneous feedback design. The other\ncompeting strategies are homogeneous without taking advantage of the channel statistics of different users. To maintain at least the same\nfeedback amount for fair comparison, each user in the homogeneous case is assumed to feed back\n$\\lceil\\sum_{g=1}^G\\frac{\\eta_G}{\\eta_g}\\frac{M}{G}\\rceil$ CQI values. Two subband sizes are assumed for the homogeneous feedback. It is clear\nthat for the homogeneous case, users in cluster $\\mathcal{K}_1$ have more independent feedback while users in cluster $\\mathcal{K}_4$ suffer\nfrom redundant feedback. The average sum rate for two different values of $M$ are shown in Fig. \\ref{fig_5}. The separate heterogenous feedback\nis observed to have the worst performance from a sum rate perspective because it does not fully exploit multiuser diversity, but it consumes the\nleast feedback. Our joint heterogenous feedback design is shown to perform much better than the two homogeneous strategies for the $4$-cluster\nsystem. It is due to the fact that by considering the existing heterogeneity among users, the proposed heterogeneous design can make the best\nuse of the degrees of freedom in the frequency domain in order to enhance the system performance as well as reduce feedback needs.\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig6_a.eps}&\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig6_b.eps}\\\\\n    \\scriptsize{(a)}&\n    \\scriptsize{(b)}\\\\\n\\end{tabular}\n\\caption{Comparison of fixed rate and variable rate strategies under normalized parameter $\\beta$ ($\\beta=\\beta_1=\\beta_0\/10$) for different\nnumber of users $K$ ($N=64$, $\\alpha=0.98$, $\\sigma_w^2=0.01$, $\\rho=10$ dB): (a) Comparison of average goodput; (b) Comparison of average\noutage probability.} \\label{fig_6}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\begin{tabular}{cc}\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig7_a.eps}&\n    \\includegraphics[width=0.5\\linewidth]{Figure\/fig7_b.eps}\\\\\n    \\scriptsize{(a)}&\n    \\scriptsize{(b)}\\\\\n\\end{tabular}\n\\caption{The effect of channel estimation error ($\\sigma_w^2$) and feedback delay ($\\alpha$) on the optimal value of $\\beta_0$ and $\\beta_1$\n($\\rho=10$ dB, $K=10$, $\\sigma_w^2=0:0.005:0.1$, $\\alpha=0.9:0.005:0.99$): (a) The optimal fix rate parameter $\\beta_0$ with respect to\n$\\sigma_w^2$ and $\\alpha$; (b) The optimal variable rate parameter $\\beta_1$ with respect to $\\sigma_w^2$ and $\\alpha$.} \\label{fig_7}\n\\end{figure}\n\n\n\\subsection{Imperfect Feedback Scenario}\\label{numerical_imperfect}\nFig. \\ref{fig_6} exhibits the comparison between the fix rate and variable rate outage scenarios as well as the effect of the number of users on\nthe optimization of $\\beta_0$ and $\\beta_1$. In order to show the system performance of the two scenarios in one figure, a normalized parameter\n$\\beta$ is defined. While examining the variable rate plots $\\beta=\\beta_1,$ and when considering the fixed rate plots $\\beta=\\beta_0\/10$. The\nsystem parameters are: $\\alpha=0.98$, $\\sigma_w^2=0.01$, and $\\rho=10$ dB. It can be seen that for both scenarios, larger number of users $K$\nyields better system performance, i.e., higher average goodput and lower average outage probability. This is a consequence of increased\nmultiuser diversity gain to combat the imperfections in the feedback system.\n\n\\begin{figure}[t]\n\\centering\n    \\includegraphics[width=0.6\\linewidth]{Figure\/fig8.eps}\n\\caption{Comparison of the average goodput for a $4$-cluster system with fix rate and variable rate strategies using optimized $\\beta_0^*$ and\n$\\beta_1^*$. The average goodput is calculated using the best-M partial feedback scheme when the minimum required $M^*$ is computed after\nobtaining $\\beta_0^*$ or $\\beta_1^*$. ($N=64$, $\\boldsymbol\\eta=(1,2,4,8)$, $K_1=K_2=K_3=K_4=K\/4$, $\\alpha=0.98$, $\\sigma_w^2=0.01$, $\\rho=10$\ndB)} \\label{fig_8}\n\\end{figure}\n\nFig. \\ref{fig_7} illustrates the effect of channel estimation error $\\sigma_w^2$ and feedback delay $\\alpha$ on the optimal value of $\\beta_0$\nand $\\beta_1$. Here $\\sigma_w^2$ is varied from $0$ to $0.1$, and $\\alpha$ is varied from $0.9$ to $0.99$ in steps of $0.005$. It can be\nobserved from the changing profiles that both the optimal values of $\\beta_0$ and $\\beta_1$ get smaller as the imperfections become worse.\nTherefore, the system should adjust the system parameters to adapt to the encountered imperfections.\n\nNow we consider the adaptation of system parameters ($\\beta_0$ or $\\beta_1$) and partial feedback in a $4$-cluster heterogeneous feedback\nsystem. The system parameters are: $\\boldsymbol\\eta=(1,2,4,8)$, $\\alpha=0.98$, $\\sigma_w^2=0.01$, and $\\rho=10$ dB. For both transmission\nstrategies and for a given number of users $K$, the optimal value of $\\beta_0^*$ or $\\beta_1^*$ is first optimized according to the full\nfeedback case discussed in Section \\ref{optimize}. Then, a minimum required $M^*$ is obtained by matching the system performance to that in the\nfull feedback case. Fig. \\ref{fig_8} demonstrates the average goodput for both transmission strategies with $M^*$ and $\\beta_0^*$ (or\n$\\beta_1^*$). We observe that there is almost a constant performance gain for the variable rate strategy compared with the fix rate one. This is\ndue to the fact that for the variable rate scenario, the system is adapting the transmission parameters conditioned on the past memory even if\nit is the outdated one. If the channel estimation error and feedback delay are not severe, the imperfections can be compensated by multiplying\nwith the backoff factor and relying on the past feedback.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion}\\label{conclusion}\nIn this paper, we propose and analyze a heterogeneous feedback design adapting the feedback resource according to users' frequency domain\nchannel statistics. Under the general correlated channel model, we demonstrate the gain by achieving the potential match among coherence\nbandwidth, subband size and partial feedback. To facilitate statistical analysis, we employ the subband fading model for the multi-cluster\nheterogeneous feedback system. We derive a closed form expression of the average sum rate under perfect partial feedback assumption, and provide\na method to obtain the minimum heterogeneous partial feedback required to obtain performance comparable to a scheme using full feedback. We also\nanalyze the effect of imperfections on the heterogeneous partial feedback system. We obtain a closed form expression for the average goodput of\nthe fix rate scenario, and utilize a bounding technique and tight approximation to track the performance of the variable rate scenario. Methods\nadapting the system parameters to maximize the average system goodput are proposed. The heterogeneous feedback design is shown to outperform the\nhomogeneous one with the same feedback resource. With imperfections, the system adjusting the transmission strategy and the amount of partial\nfeedback is shown to yield better performance. The developed analysis provides a theoretical reference to understand the approximate behavior of\nthe proposed heterogeneous feedback system and its interplay with practical imperfections. Dealing with the general channel correlation and the\ncorresponding nonlinear nature of the CQI are interesting directions for the heterogeneous feedback system.\n\n\n\n\n\n\n\n\n\n\n\n\n\\appendices\n\\section{}\\label{appenA}\n\\textit{Proof (sketch) of Lemma \\ref{lemma_1}:} The methodology is an extension of the work in \\cite{hur11} which deals with the homogeneous\nfeedback case with one cluster of users and one specific subband size.\n\nLet $F_{Y_{k}^{(g)}}$ denote the CDF of $Y_{k}^{(g)}\\triangleq Y_{k,r}^{(g)}$. Substituting the subband size $\\frac{N}{\\eta_g}$ and the number\nof reported CQI $\\frac{\\eta_G}{\\eta_g} M$ for user $k$ in cluster $\\mathcal{K}_g$ makes $F_{Y_{k}^{(g)}}$ satisfy (\\ref{perfect:eq_1}). It can\nbe shown that $F_{\\tilde{Y}_{k}^{(g)}}(x)=\\mathbb{P}(\\tilde{Y}_{k,n}^{(g)}\\leq x)=\\mathbb{P}(Y_{k,\\lceil\\frac{n}{\\eta_g}\\rceil}^{(g)}\\leq\nx)=F_{Y_{k}^{(g)}}(x)$, which concludes the proof.\n\n\\medskip\n\n\\textit{Proof of Theorem \\ref{theorem_1}:} Substituting the expressions of $F_{\\tilde{Y}_{k}^{(g)}}$ from Lemma \\ref{lemma_1} and combining\n(\\ref{perfect:eq_5}) yield:\n\\begin{equation} \\label{appendix:eq_1}\nF_{X_n|\\mathcal{U}_n}(x)=(F_Z(x))^{\\sum_{g=1}^G\\frac{N}{\\eta_g}\\tau_g}\\prod_{g=1}^G\\left(\\sum_{m=0}^{\\frac{\\eta_G}{\\eta_g}M-1}\\frac{\\xi_g(N,M,\\boldsymbol\\eta,0)}{(F_Z(x))^m}\\right)^{\\tau_g}.\n\\end{equation}\n\nApplying \\cite[0.314]{gradshteyn07} to a finite-order power series in (\\ref{appendix:eq_1}),\n$\\left(\\sum_{m=0}^{\\frac{\\eta_G}{\\eta_g}M-1}\\frac{\\xi_g(N,M,\\boldsymbol\\eta,0)}{(F_Z(x))^m}\\right)^{\\tau_g}$ can be expressed as\n$\\sum_{m=0}^{\\tau_g(\\frac{\\eta_G}{\\eta_g}M-1)}\\frac{\\Lambda_g(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)}{(F_Z(x))^m}$, where the expression for\n$\\Lambda_g(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$ is described in Theorem \\ref{theorem_1}. Note that the coefficients of $\\frac{1}{(F_Z(x))^m}$\ncan be computed in a recursive manner.\n\nThen we employ \\cite[0.316]{gradshteyn07} for the multiplication of power series. For $g=1$, $\\Theta_1(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$\ncan be calculated from $\\Lambda_1(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$ and $\\Lambda_2(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$ as:\n\\begin{equation*}\n\\Theta_1(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)=\\mathop{\\sum}\\limits_{i=0}^m\\Lambda_1(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,i)\\Lambda_2(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m-i).\n\\end{equation*}\nFor $2\\leq g<G$, $\\Theta_{g}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$ can be computed from $\\Theta_{g-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$\nand $\\Lambda_{g+1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)$ in the following manner:\n\\begin{equation*}\n\\Theta_{g}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m)=\\mathop{\\sum}\\limits_{i=0}^m\\Theta_{g-1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,i)\\Lambda_{g+1}(N,M,\\boldsymbol\\eta,\\boldsymbol\\tau,m-i).\n\\end{equation*}\nThis concludes the proof.\n\n\\medskip\n\n\\textit{Derivation of $\\mathcal{I}_1(a,b)$:} From the definition of $Z$, $F_Z(x)=1-\\exp(-x)$ and $f_Z(x)=\\exp(-x)$. Thus\n$d(F_Z(x))^b=b(F_Z(x))^{b-1}f_Z(x)dx=b\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\exp(-(\\ell+1)x)dx$, where the last equality follows from\nthe binomial theorem.\n\nTherefore, $\\int_0^\\infty\n\\log_2(1+ax)d(F_Z(x))^b=\\frac{b}{\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\int_0^\\infty\\ln(1+ax)\\exp(-(\\ell+1)x)dx$. Applying\n\\cite[4.337.2]{gradshteyn07} yields (\\ref{perfect:eq_10}).\n\n\\section{}\\label{appenB}\n\n\\textit{Derivation of $\\mathcal{I}_2(a,b)$:} From the definition of $\\hat{Z}$, it can be shown that\n$F_{\\hat{Z}}(x)=1-\\exp(-\\frac{1}{1-\\sigma_w^2}x)$ and $f_{\\hat{Z}}(x)=\\frac{1}{1-\\sigma_w^2}\\exp(-\\frac{1}{1-\\sigma_w^2}x)$. Then\n$\\mathcal{I}_2(a,b)$ can be calculated as:\n\\begin{align}\n\\mathcal{I}_2(a,b)&\\mathop{=}\\limits^{(a)}\\frac{2b}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\int_0^\\infty\\mathcal{Q}_1(\\alpha_w\\alpha\nx,\\alpha_w\\sqrt{a})\\exp\\left(-\\frac{(\\ell+1)x^2}{1-\\sigma_w^2}\\right)xdx\\notag\\\\\n&\\mathop{=}\\limits^{(b)}\\frac{2b}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}}\\left(\\mathcal{Q}_1(0,\\vartheta)+\\exp(-\\frac{\\zeta_{\\ell}\\vartheta^2}{2(\\varpi^2+\\zeta_{\\ell})})(1-\\mathcal{Q}_1(0,\\frac{\\varpi\\vartheta}{\\sqrt{\\varpi^2+\\zeta_{\\ell}}}))\\right)\\notag\\\\\n\\label{appendix:eq_2}&\\mathop{=}\\limits^{(c)}\\frac{2b}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}}\\left(\\exp(-\\frac{\\vartheta^2}{2})+\\exp(-\\frac{\\zeta_{\\ell}\\vartheta^2}{2(\\varpi^2+\\zeta_{\\ell})})(1-\\exp(-\\frac{\\varpi^2\\vartheta^2}{2(\\varpi^2+\\zeta_{\\ell})}))\\right),\n\\end{align}\nwhere $\\varpi=\\alpha_w\\alpha$, $\\vartheta=\\alpha_w\\sqrt{a}$, $\\zeta_{\\ell}=\\frac{2(\\ell+1)}{1-\\sigma_w^2}$. (a) is obtained by substituting the\nexpression of $d(F_{\\hat{Z}}(x))^b$ and using change of variables; (b) follows from applying \\cite[B.18]{simon02}; (c) follows from using the\nfact that $\\mathcal{Q}_1(0,\\vartheta)=\\exp(-\\frac{\\vartheta^2}{2})$.\n\n\\medskip\n\n\\textit{Proof of Proposition \\ref{proposition_1}:}\n\\begin{align}\n\\mathcal{I}_3(a,b)&\\mathop{=}\\limits^{(a)}\\frac{b}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\int_0^\\infty\\mathcal{Q}_1(\\alpha_w\\alpha\n\\sqrt{x},\\alpha_w\\sqrt{ax})\\ln(1+\\rho ax)\\exp\\left(-\\frac{(\\ell+1)x}{1-\\sigma_w^2}\\right)dx\\notag\\\\\n&\\mathop{<}\\limits^{(b)}\\frac{2\\rho\nab}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}\\binom{b-1}{\\ell}(-1)^{\\ell}\\int_0^\\infty\\mathcal{Q}_1(\\alpha_w\\alpha\nx,\\alpha_w\\sqrt{a}x)\\exp\\left(-\\frac{(\\ell+1)x^2}{1-\\sigma_w^2}\\right)x^3dx\\notag\\\\\n&\\mathop{=}\\limits^{(c)}\\frac{4\\rho ab}{(1-\\sigma_w^2)\\ln2}\\sum_{\\ell=0}^{b-1}(-1)^{\\ell}\\frac{1}{\\zeta_{\\ell}^2}\\Bigg(1+\\frac{\\vartheta^2}{\\varphi_{\\ell}}\\Bigg(\\frac{\\varpi^2}{\\varphi_{\\ell}}{}_2F_1\\left(1,\\frac{3}{2};2;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)-{}_2F_1\\left(\\frac{1}{2},1;1;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)\\notag\\\\\n\\label{appendix:eq_3}&\\quad+\\frac{2\\zeta_{\\ell}}{\\varphi_{\\ell}}\\Bigg(\\frac{\\varpi^2}{\\varphi_{\\ell}}{}_2F_1\\left(\\frac{3}{2},2;2;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)-\\frac{1}{2}{}_2F_1\\left(1,\\frac{3}{2};1;\\frac{4\\varpi^2\\vartheta^2}{\\varphi_{\\ell}^2}\\right)\\Bigg)\\Bigg)\\Bigg),\n\\end{align}\nwhere $\\varpi=\\alpha_w\\alpha$, $\\vartheta=\\alpha_w\\sqrt{a}$, $\\zeta_{\\ell}=\\frac{2(\\ell+1)}{1-\\sigma_w^2}$,\n$\\varphi_{\\ell}=\\varpi^2+\\vartheta^2+\\zeta_{\\ell}$, and ${}_2F_1(\\cdot,\\cdot;\\cdot;\\cdot)$ is the Gaussian hypergeometric function\n\\cite{abramowitz72}. (a) is obtained by substituting the expression of $d(F_{\\hat{Z}}(x))^b$; (b) follows from the fact that when $\\rho\\ll1$,\n$\\rho ax$ is a tight upper bound for $\\ln(1+\\rho ax)$; note that change of variables are used; (c) follows from applying \\cite[B.60]{simon02}.\n\n\\medskip\n\n\\textit{Proof (sketch) of Proposition \\ref{proposition_2}:} Define\n$s(\\check{\\chi}_b)\\triangleq\\mathcal{Q}_1(\\alpha_w\\alpha\\sqrt{\\check{\\chi}_b},\\alpha_w\\sqrt{\\beta_1\\check{\\chi}_b})\\log_2(1+\\rho\n\\beta_1\\check{\\chi}_b)]$. Firstly it must be shown that $\\frac{s(\\check{\\chi}_b)}{s(\\mathbb{E}[\\check{\\chi}_b])}$ converges to $1$ in\nprobability. For $\\forall\\epsilon>0$, it is now shown that:\n\\begin{align}\n\\mathbb{P}\\left(\\left|\\frac{s(\\check{\\chi}_b)}{s(\\mathbb{E}[\\check{\\chi}_b])}-1\\right|\\geq\\epsilon\\right)=&\\mathbb{P}\\left(\\left|\\frac{s(\\check{\\chi}_b)-s(\\mathbb{E}[\\check{\\chi}_b])}{s(\\mathbb{E}[\\check{\\chi}_b])}\\right|\\geq\\epsilon\\right)\\notag\\\\\n\\label{appendix:eq_6}&\\mathop{\\leq}\\limits^{(a)}\\mathbb{P}\\left(\\frac{s(|\\check{\\chi}_b-\\mathbb{E}[\\check{\\chi}_b]|)}{s(\\mathbb{E}[\\check{\\chi}_b])}\\geq\\epsilon\\right)\\mathop{\\rightarrow}\\limits^{(b)}0,\n\\end{align}\nwhere (a) follows from the concave and monotonically increasing property of $s(\\cdot)$: $|s(x)-s(y)|<s(|x-y|)$; (b) follows from the asymptotic\nscaling rate of $\\mathbb{E}[\\check{\\chi}_b]$ and $|\\check{\\chi}_b-\\mathbb{E}[\\check{\\chi}_b]|$, and the utilization of the Chebyshev's\ninequality. From extreme value theory and asymptotic analysis of order statistics \\cite{david03, sharif05}, it is known that the tail behavior\nof $\\check{\\chi}_b$ converges to type $3$ Gumbel distribution, which enables $\\mathbb{E}[\\check{\\chi}_b]$ to scale as $\\log b$ and\n$|\\check{\\chi}_b-\\mathbb{E}[\\check{\\chi}_b]|$ to scale as $\\log\\log b$.\n\nThen a method similar to that in \\cite{sanayei07} can be employed to prove the uniformly integrable property \\cite{billingsley95} of\n$\\frac{s(\\check{\\chi}_b)}{s(\\mathbb{E}[\\check{\\chi}_b])}$. By combining the above property along with the convergence in probability leads to\nconvergence in the mean \\cite{billingsley95}, which concludes the proof.\n\n\n\\medskip\n\n\\textit{Proof of Proposition \\ref{proposition_3}:} It must be shown that\n$\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)=\\mathcal{Q}_1\\left(\\alpha_w\\alpha\\sqrt{\\mathbb{E}[\\check{\\chi}]},\\alpha_w\\sqrt{\\beta_1\\mathbb{E}[\\check{\\chi}]}\\right)\\log_2(1+\\rho\n\\beta_1\\mathbb{E}[\\check{\\chi}])$ is strictly quasiconcave in $\\beta_1$.\n\nThis property can be proved by log-concavity \\cite{boyd04}. It is shown in \\cite{finner97, yu11} that $\\mathcal{Q}_1(\\sqrt{a},\\sqrt{b})$ is\nlog-concave in $b\\in[0,\\infty)$ for $a\\geq 0$. Also, $\\log(1+b)$ is concave thus log-concave in $b\\in[0,\\infty)$. Since log-concavity is\nmaintained in multiplication, $\\mathcal{Q}_1(\\sqrt{a},\\sqrt{b})\\log(1+b)$ is log-concave in $b\\in[0,\\infty)$. From the definition of\n$\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)$, it is now proved to be log-concave in $\\beta_1\\in[0,\\infty)$ since $\\mathbb{E}[\\check{\\chi}]$ is\nirrelevant to $\\beta_1$. Therefore, it is quasiconcave in $\\beta_1\\in[0,\\infty)$ because log-concave functions are also quasiconcave.\n\nIn addition, it is clear that $\\mathop{\\lim}\\limits_{\\beta_1\\rightarrow 0}\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)=0$. Also, it is now shown that:\n\\begin{align}\n0&\\leq \\mathop{\\lim}\\limits_{\\beta_1\\rightarrow\\infty}\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)\\notag\\\\\n&\\mathop{\\leq}\\limits^{(a)}\\mathop{\\lim}\\limits_{\\beta_1\\rightarrow\\infty}\\exp\\left(-\\frac{(\\alpha_w\\sqrt{\\beta_1\\mathbb{E}[\\check{\\chi}]}-\\alpha_w\\alpha\\sqrt{\\mathbb{E}[\\check{\\chi}]})^2}{2}\\right)\\log_2(1+\\rho\\beta_1\\mathbb{E}[\\check{\\chi}])\\notag\\\\\n\\label{appendix:eq_5}&\\mathop{=}\\limits^{(b)}\\mathop{\\lim}\\limits_{\\beta_1\\rightarrow\\infty}\\frac{\\rho}{2\\alpha_w^2\\ln2}\\frac{1}{\\left(1+\\rho\\beta_1\\mathbb{E}[\\check{\\chi}]\\right)\\left(1-\\frac{\\alpha}{\\sqrt{\\beta_1}}\\right)}\\exp\\left(-\\frac{(\\alpha_w\\sqrt{\\beta_1\\mathbb{E}[\\check{\\chi}]}-\\alpha_w\\alpha\\sqrt{\\mathbb{E}[\\check{\\chi}]})^2}{2}\\right)=0,\n\\end{align}\nwhere (a) follows from the upper bound $\\mathcal{Q}_1(a,b)\\leq \\exp\\left(-\\frac{(b-a)^2}{2}\\right)$ for $b>a\\geq0$ \\cite{simon02}; (b) follows\nfrom applying L'Hospital's rule. Therefore, there exists a unique global optimal $\\beta_1$ which maximizes\n$\\mathcal{I}_3^{\\mathrm{A}}(\\beta_1,K)$.\n\n\n\n\n\n\\section*{Acknowledgment}\nThe authors want to express their deep appreciation to the anonymous reviewers and the Associated Editor for their many valuable comments and\nsuggestions, which have greatly helped to improve this paper.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Stable Matching Problem with Ties: Strongly stable Matching}\n\\section{Strongly Stable Matching}\n\nA matching $M$ of men and women is {\\em strongly stable} if there is no blocking pair $(m,w)$ such that they are not married in $M$\nbut either (1) both of them prefer each other to their partners in $M$, or (2) one of them prefers the other to his\/her partner in $M$ and the other one is\nindifferent.\nFormally, \na pair of man and woman $(m,w)$ is {\\em blocking for a strongly stable matching} $M$ if they are\nnot matched in $M$ and\\\\\n\n\\ifdefined\\ISBOOK\n$((mrank[m][w] \\leq mrank[m][M(m)])  \\wedge $\n$ (wrank[w][m] < wrank[w][M(w)]))$\\\\\n$ \\vee ((mrank[m][w] < mrank[m][M(m)])  \\wedge $\n$(wrank[w][m] \\leq wrank[w][M(w)])). $\n\\else\n\\hspace*{0.2in} $((mrank[m][w] \\leq mrank[m][M(m)])  \\wedge $\\\\\n\\hspace*{0.2in} $ (wrank[w][m] < wrank[w][M(w)]))$\\\\ \n$ \\vee ((mrank[m][w] < mrank[m][M(m)])  \\wedge $\\\\\n\\hspace*{0.2in} $(wrank[w][m] \\leq wrank[w][M(w)])). $\n\\fi\n\n\nAs in superstable matching algorithm, we let $mpref[i][k]$ denote the set of women ranked $k$ by man $i$.\nAs before, we will use $G[i]$ to denote the $mrank$ that the man $i$ is currently considering. Initially, $G[i]$ is $1$ for all $i$, i.e.,\neach man proposes to all his top choices. We define a bipartite graph $Y(G)$ on the set of men and women with respect to any $G$ as follows.\nIf a woman does not get any proposal in $G$, then she is unmatched. If she receives multiple proposals then there is an edge\nfrom that woman to all men in the most preferred rank. For superstable matching, we required $Y(G)$ to be a perfect matching.\nFor strongly stable matching, we only require $Y(G)$ to contain a perfect matching.\n\nWe first note that a strongly stable matching may not exist.\nThe following example is taken from \\cite{IRVING1994261}. \\\\\n\\\\\n$m1: w1, w2$\\\\\n$m2:$ both choices are ties\\\\\n\\\\\n$w1: m2, m1$\\\\\n$w2: m2, m1$\n\n\nThe matching $\\{(m1, w1), (m2, w2)\\}$ is blocked by the pair $(m2, w1)$: $w1$ strictly prefers $m2$ and $m2$ is indifferent between $w1$ and $w2$.\nThe only other matching is $\\{(m1, w2), (m2, w1)\\}$.  This matching is blocked by $(m2, w2)$: $w2$ strictly prefers $m2$ and $m2$ is indifferent between\n$w1$ and $w2$.\n\n\\remove{\nFor strongly stable matchings, we will focus only on finding the least proposal vector $G$ such that $Y(G)$ contains a perfect matching.\nTo that end, we define the predicate $B(G)$ to be true if $G$ is the least vector such that $Y(G)$ has a perfect matching.\nWe show that whenever there is a perfect matching in $G_1$ and $G_2$ and there is no perfect matching in $G_1 \\cap G_2$, then\nthere exists $G_3 < G_1 \\cap G_2$ that has a perfect matching.\n\n}\n\nConsider any bipartite graph with an equal number of men and women. If there is no perfect matching in the graph,\nthen by Hall's theorem there exists a set of men of size $r$ who collectively are adjacent to fewer than $r$ women.\nWe define {\\em deficiency} of a subset $Z$ of men as $|Z| - N(Z)$ where $N(Z)$ is the {\\em neighborhood} of $Z$ (the set of vertices that are adjacent to\nat least one vertex in $Z$). The deficiency $\\delta(G)$ is the maximum deficiency taken over all subsets of men.\nWe call a subset of men $Z$ {\\em critical} if it is maximally deficient and does not contain any maximally deficient proper subset.\nOur algorithm to find a strongly stable matching is simple. \nWe start with $G$ as the global state vector with top choices for all men.\nIf $Y(G)$ has a perfect matching, \nwe are done. The perfect matching in $Y(G)$ is a strongly stable matching.\nOtherwise, there must be a critical subset of men with maximum deficiency. These set of men must then advance on their proposal number, if possible.\nIf these men cannot advance, then there does not exist a strongly stable marriage and the algorithm terminates.\n\n\\begin{algorithm}\n \\SetAlgoRefName{LLP-ManOptimalStronglyStableMarriage}\n$P_j$: Code for thread $j$\\\\\n {\\bf input}: $mpref[i,k]$: set of int for all $i,k$; $wrank[k][i]$: int for all $k,i$;\\\\\n  {\\bf init}:   $G[j] := 1$;\\\\\n{\\bf always}:  $Y(j) = mpref[j][G[j]];$\\\\\n \\BlankLine\n  {\\bf forbidden($j$)}:\\\\\n  \\hspace*{0.2in} $j$ is a member of the critical subset of men in the graph $Y(G)$\\\\\n  \\hspace*{0.2in}  {\\bf advance}: $G[j] := G[j]+1; $\n\\caption{A Parallel Algorithm for Man-Optimal Strongly Stable Matching \\label{fig:strongly-stable}}\n\\end{algorithm}\n\n\\ref{fig:strongly-stable} is the LLP version of the algorithm proposed by Irving and the interested reader is referred to \n\\cite{IRVING1994261}  for the details and the proof of correctness.\nSimilar to superstable marriages, \nwe also get the following result.\n\\begin{theorem}\nThe set of strongly stable marriages, $L_{stronglystable}$, is a sublattice of the lattice $L$. \n\\end{theorem}\nObserve that each element in $L_{stronglystable}$ is not a single marriage but a set of marriages. This is in contrast to \n$L_{superstable}$, where each element corresponds to a single marriage.\n\n\\remove {\n\\begin{algorithm}\n \\SetAlgoRefName{LLP-ManOptimalStronglyStableMarriage}\n {\\bf input}: $mpref[i,k]$: set of int for all $i,k$; $rank[k][i]$: int for all $k,i$;\\\\\n  {\\bf init}:   $\\forall j: G[j] := 1$;\\\\\n{\\bf always}: \\\\\n\\> Y(G) =  bipartite graph between men and women such that there is an edge between a man $m$ and a woman $w$\\\\ if\n$w$ is one of the top choices for $m$ according to $G$ and $m$ is one of the top choices for $w$ of all proposals received in $G$\\\\\n\\> $\\delta(G)$ = maximum deficiency of any subset of men in $Y(G)$\\\\\n\\> \\>  $maximallyDeficient(J) \\equiv |J - N(J)| = \\delta(G)$\\\\\n\\> \\>  $critical(J) \\equiv maximallyDeficient(J) \\wedge \\forall K \\subsetneq J: \\neg maximallyDeficient(K)$\\\\\n\\\\\n\\> \\myblue{\\bf forbidden(j)}:  $\\exists J: critical(J) \\wedge (j \\in J) $\\\\\n\\>  \\myblue{\\bf advance}: $G[j] := G[j]+1; $\\\\\n\\caption{A Parallel Algorithm for Man-Optimal Strongly Stable Matching \\label{fig:strong-stable}}\n\\end{algorithm}\n\n}\n\n\n\n\n\\remove {\n\n\n\\begin{theorem}\nThere is a strongly stable matching with $G$ as the proposal vector iff $Y(G)$ has a perfect matching.\n\\end{theorem}\n\\begin{proof}\nIt is clear that if $Y(G)$ does not have a perfect matching, then there cannot be a strongly stable matching corresponding to $G$.\nWe show the converse: \nif $Y(G)$ has a perfect matching then\nthere is a strongly stable matching with $G$ as the proposal vector. \nIf not, there must be a pair $(m,w)$ such that\nat least one of them can improve their partner while the other one either improves the partner or is indifferent.\nSuppose that man $m$ can improve his partner. This means that man $m$ proposed to $w$ in the proposal vector\nstrictly less than $G$. Since $G$ has a perfect matching, the man $m$ has proposed to a less preferable woman $w'$.\nThe woman $w$ is matched to a man $m'$ in the perfect matching in $Y(G)$.\nSince the choices for women can only improve or stay the same as the proposal vector increases, we know that\n$rank[w][m'] \\leq rank[w][m]$.\nIf $rank[w][m'] < rank[w][m]$, then $(m,w)$ is not a blocking pair because the woman prefers $m'$ to $m$.\n\nBy induction, we can assume that there is no strongly stable matching in any vector less than $G$.\nSince all men have women in their top rank, they cannot improve their rank. \nSince women have edges only for their most preferred rank, \nthey can also not improve their rank by going with an edge that in not in the matching.\n\\end{proof}\n\nWe now claim that the predicate $B(G) \\equiv Y(G) ~ \\mbox{contains a perfect matching}$ is a lattice-linear predicate.\n\n\\begin{lemma}\nIf $Y(G)$ in not a perfect matching, then at least one index in $G$ is forbidden.\n \\end{lemma}\n\\begin{proof}\nIf there is no perfect matching in $Y(G)$, then there must be a minimal set of \nmen, $Z$\nsuch that the set of neighbors of $Z$ is strictly smaller in size than $Z$. We claim that every man $i \\in Z$ is forbidden.\nConsider any $H \\geq G$ such that $G[i]$ equals $H[i]$. Suppose, if possible $H$ has a strongly stable matching $M_H$.\nIn $M_H$, man $i$ is matched with a woman in $w \\in mpref[i][H[i]]$. The woman must rank man $i$ as the most preferred until $G$; otherwise, there\nwould not be any edge between $w$ and $i$ in $G$ and therefore in $H$. There has to be at least one more edge from $w$ to some other man in $Y(G)$; otherwise,\nwe can remove man $i$ from $Z$ and $w$ from $N(Z)$ and we have a set $Z'$ strictly smaller than $Z$ such that the set of neighbors of $Z'$ is smaller than $Z'$, contradicting\nthat $Z$ is a {\\em minimal} underdemanded set.\nSuppose $w$ has an edge to man $j$ in $Y(G)$. \n\nIf $H[j] > G[j]$, then $(j,w)$ is a blocking pair to the matching in $H$ because man $j$ prefers $w$ to his match in $H$ and $w$ is indifferent between\n$i$ and $j$. \n\nIf $H[j] = G[j]$, then  we do the similar analysis for $j$. The woman who is matched with man $j$ must have at least one edge outside of $\\{i,j\\}$; otherwise, \nwe can remove both $i$ and $j$ and their neighbors from $Z$ and get a smaller underdemanded set. Continuing in this manner, we either find a blocking pair for $H$ or\ncontradict minimality of $Z$.\n\\end{proof}\n\n\n}\n\n\n\n\n\n\\section{Stable Matching Problem with Ties: Superstable Matching}\n\\section{Superstable Matching}\n\nIn many applications, agents (men and women for the stable marriage problem) may not totally order all their choices. Instead, they\nmay be indifferent to some choices \\cite{IRVING1994261,manlove2002structure}.\nWe generalize $mpref[i][k]$\n to a set of women instead of a single woman. Therefore, \n $mrank$ function is not 1-1 anymore. Multiple women may have the same rank.\nSimilarly, $wrank$ function is not 1-1 anymore. Multiple men may have the same rank.\nWe now define the notion of blocking pairs for a matching $M$ with ties \\cite{IRVING1994261}.\nWe let $M(m)$ denote the woman matched with the man $m$ and $M(w)$ denote the man\nmatched with the woman $w$.\nIn the version, called {\\em weakly stable} matching $M$, there is no blocking pair of \nman and woman $(m,w)$ who are not married in $M$ but strictly prefer\neach other to their partners in $M$. Formally,\na pair of man and woman $(m,w)$ is {\\em blocking for a weakly stable matching} $M$ if they are\nnot matched in $M$ and\\\\\n\\ifdefined\\ISBOOK\n$(mrank[m][w] < mrank[m][M(m)])  \\wedge$\n$ (wrank[w][m] < wrank[w][M(w)]. $\n\\else\n$(mrank[m][w] < mrank[m][M(m)])  \\wedge$\\\\\n$ (wrank[w][m] < wrank[w][M(w)]. $\n\\fi\n\nFor the weakly stable matching, ties can be broken arbitrarily and any matching that is stable in the resulting instance is also\nweakly stable for the original problem.\nTherefore, Gale-Shapley algorithm is applicable for the weakly stable matching \\cite{IRVING1994261}.\nWe focus on other forms of stable matching --- superstable and strongly stable matchings.\n\nA matching $M$ of men and women is {\\em superstable} if there is no blocking pair $(m,w)$ such that they are not married in $M$\nbut they either prefer each other to their partners in $M$ or are indifferent with their partners in $M$.\nFormally, \na pair of man and woman $(m,w)$ is {\\em blocking for a super stable matching} $M$ if they are\nnot matched in $M$ and\\\\\n\\ifdefined\\ISBOOK\n$(mrank[m][w] \\leq mrank[m][M(m)])  \\wedge$\n$ (wrank[w][m] \\leq wrank[w][M(w)]. $\n\\else\n$(mrank[m][w] \\leq mrank[m][M(m)])  \\wedge$\\\\\n$ (wrank[w][m] \\leq wrank[w][M(w)]. $\n\\fi\n\n\nThe algorithms for superstable marriage have been proposed in \\cite{IRVING1994261,manlove2002structure}.\nOur goal is to show that LLP algorithm is applicable to this problem as well.\nAs before, we will use $G[i]$ to denote the $mrank$ that the man $i$ is currently considering. Initially, $G[i]$ is $1$ for all $i$, i.e.,\neach man proposes to all his top choices. We say that $G$ has a superstable matching if there exist $n$ women $w_1, w_2, \\ldots w_n$ such that\n$\\forall i: w_i \\in mpref[i][G[i]]]$ and the set $(m_i, w_i)$ is a superstable matching.\n\nWe define a bipartite graph $Y(G)$ on the set of men and women with respect to any $G$ as follows.\nIf a woman does not get any proposal in $G$, then she is unmatched. If she receives multiple proposals then there is an edge\nfrom that woman to all men in the most preferred rank. We say that $Y(G)$ is a perfect matching if \nevery man and woman has exactly one adjacent edge in $Y(G)$, \n\nWe claim \n\\begin{lemma}\nIf $Y(G)$ is not a perfect matching, then\nthere is no superstable matching with $G$ as the proposal vector. \n\\end{lemma}\n\\begin{proof}\nIf there is a man with no adjacent edge in $Y(G)$  then it is clear that $G$ cannot have a superstable matching.\nNow consider the case when a man has at least two adjacent edges. If all the adjacent women for this man have degree one, then\nexactly one of them can be matched with this man and other women will remain unmatched.\nTherefore, there is at least one woman $w$ who is also adjacent to  another man $m'$. If $w$ is matched with $m$, then $(m',w)$ is a blocking pair.\nIf $w$ is matched with $m'$, then $(m,w)$ is a blocking pair.\n\\end{proof}\n\nWe now claim that the predicate $B(G) \\equiv Y(G) ~ \\mbox{is a perfect matching}$ is a lattice-linear predicate.\n\n\\begin{lemma}\\label{lem:superstable}\nIf $Y(G)$ is not a perfect matching, then at least one index in $G$ is forbidden.\n \\end{lemma}\n \\begin{proof}\nConsider any man $i$ such that there is no edge adjacent to $i$ in $Y(G)$. This happens when all women that man $i$ has proposed in state $G$ have rejected him.\nConsider any $H$ such that\n$H[i]$ equals $G[i]$. All the women had rejected man $i$ in $G$. As $H$ is greater than $G$, these women can only have\nmore choices and will reject man $i$ in $H$ as well.\n\nNow suppose that every man has at least one adjacent edge. \nLet $Z(G)$ be the set of women with degree exactly one. \nIf every woman is in $Z(G)$, then we have that $Y(G)$ is a perfect matching because every man has at least one adjacent edge.\nIf not, consider any man $i$ who is not matched to a woman in $Z(G)$. This means that all the women he is adjacent to have\ndegrees strictly greater than one. In $H$ all these women would have either better ranked proposals or equally ranked proposals. In either case,\nman $i$ would not be matched with any of these women. Hence, $i$ is forbidden.\n\\end{proof}\n\nWe are now ready to present \\ref{fig:super-stable}. \nIn \\ref{fig:super-stable}, we start with the proposal vector $G$ with all components $G[j]$ as $1$. \nWhenever a woman receives multiple proposals, she rejects proposals by men who are ranked lower than anyone who has proposed to her.\nWe say that a man $j$ is forbidden in $G$, if\nevery woman $z$ that man $j$ proposes in $G$ is either engaged to or proposed by someone who she prefers to $j$ or is indifferent with respect to $j$.\n \\ref{fig:super-stable} is a parallel algorithm because all processes $_j$ such that forbidden($j$) is true can advance in parallel.\n\n\n\n\\begin{algorithm}\n \\SetAlgoRefName{LLP-ManOptimalSuperStableMarriage}\n$P_j$: Code for thread $j$\\\\\n {\\bf input}: $mpref[i,k]$: set of int for all $i,k$; $wrank[k][i]$: int for all $k,i$;\\\\\n  {\\bf init}:   $G[j] := 1$;\\\\\n{\\bf always}:  $Y(j) = mpref[j][G[j]];$\\\\\n \\BlankLine\n  {\\bf forbidden($j$)}:\\\\\n  \\hspace*{0.2in}  $\\forall z \\in Y(j): \\exists i \\neq j:  \\exists k \\leq G[i]: (z \\in mpref[i][k]) \\wedge (wrank[z][i] \\leq wrank[z][j]))$\\\\\n  \/\/ all women $z$ in the current proposals from $j$ have been proposed by someone who either they prefer or are indifferent over $j$.\\\\\n \\hspace*{0.2in}  {\\bf advance}: $G[j] := G[j]+1; $\n\\caption{A Parallel Algorithm for Man-Optimal Super Stable Matching \\label{fig:super-stable}}\n\\end{algorithm}\n\nLet us verify that this algorithm indeed generalizes the standard stable marriage algorithm. For the standard stable marriage problem, $mpref[i,k]$ is singleton for all\n$i$ and $k$. Hence, $Y(j)$ is also singleton. Using $z$ for the singleton value in $Y(j)$, we get the expression\n$ \\exists i \\neq j:  \\exists k \\leq G[i]: (z = mpref[i][k]) \\wedge (wrank[z][i] < wrank[z][j]))$ which is identical to the stable marriage problem once we\nsubstitute $<$ for $\\leq$ for comparing the $wrank$ of man $i$ and man $j$.\n\nWhen the preference list has a singleton element for each rank as in the classical stable marriage problem, we know that there always exists at least one stable marriage. However, in presence of ties there is no guarantee of existence of a superstable marriage.\nConsider the case with two men and women where each one of them does not have any strict preference.\nClearly, for this case there is no superstable marriage.\n\nBy symmetry of the problem, one can also get woman-optimal superstable marriage by switching the roles of men and women.\nLet $mpref[i].length()$ denote the number of equivalence classes of women for man $i$. When all women are tied for the man $i$, the number of equivalence classes is equal to $1$, and when there\nare no ties then it is equal to $n$. \nConsider the distributive lattice $L$ defined as the cross product of $mpref[i]$ for each $i$. \nWe now have the following result.\n\\begin{theorem}\nThe set of superstable marriages, $L_{superstable}$, is a sublattice of the lattice $L$. \n\\end{theorem}\n\\begin{proof}\nFrom Lemma \\ref{lem:superstable}, the set of superstable marriages is closed under meet.\nBy symmetry of men and women, the set is also closed under join.\n\\end{proof}\n\nIt is already known that the set of superstable marriages forms a distributive lattice \\cite{spieker1995set}.\nThe set of join-irreducible elements of the lattice $L_{superstable}$ forms a partial order\n(analogous to the rotation poset \\cite{gusfield1989stable}) that can be used to generate all superstable marriages.\nVarious posets to generate all superstable marriages are discussed in \\cite{scott2005study,DBLP:conf\/wads\/HuG21}\n\n\nWe note that the algorithm \\ref{fig:super-stable} can also be used to find the constrained superstable marriage. In particular, the following\npredicates are lattice-linear:\n\\begin{enumerate}\n\\item\nRegret of man $i$ is at most regret of man $j$.\n\\item\nThe proposal vector is at least $I$.\n\\end{enumerate}\n\n\\remove{\n\n\\begin{figure}[htbp]\\begin{center}\n\\begin{tabbing}\nx\\=xxx\\=xxx\\=xxx\\=xxx\\=xxx\\= \\kill\n${\\bf P_i}$:: \/\/ Process for Man $i$\\\\\n\\> {\\bf input}\\\\\n\\> \\>   $mpref$: array[$1$..$n$]  of set of $1..n;$ \\\\\n\\> \\> \/\/ men's preferences: each a set of women\\\\\n\\> {\\bf var}\\\\\n\\> \\>   $G_i:1..n$ initially $1$; \/\/ proposal number by $P_i$ \\\\\n\\> \\>  $numrejects: 0..n$ initially $0$; \/\/ number of women who have rejected this man\\\\\n\\\\\n\\> Upon receiving a message ``initiate'' from environment;\\\\\n\\> \\>  $acceptedProps := \\emptyset;$\\\\\n\\> \\>  $rejectedProps := \\emptyset;$\\\\\n\\> \\>    for each $w \\in mpref[G_i]$\\\\\n\\> \\> \\>  send $(``proposal\", i)$ to woman $w$;\\\\\n\\\\\n\\> Upon receiving a message $(``reject\", j)$:\\\\\n\\> \\>   {\\bf if} $(j \\in mpref[G_i])$ {\\bf then} \/\/ rejected by one of the current partners\\\\\n\\> \\>  \\>$rejectedProps := rejectedProps \\cup \\{j\\}$;\\\\\n\\> \\> \\> {\\bf if} $|rejectedProps| = mpref[G_i]$ {\\bf then}\\\\\n\\> \\> \\> \\> {\\bf if} $(G_i = mpref.length)$ \/\/no more choices left\\\\\n\\> \\> \\> \\> \\>  announce ``no super stable marriage possible\" ;\\\\\n\\> \\> \\>  \\> {\\bf else } \/\/ move to the next ranked women\\\\\n\\> \\> \\> \\> \\> $G_i := G_i+1$;\\\\\n\\> \\> \\> \\> \\> $acceptedProps := \\emptyset;$\\\\\n\\> \\> \\> \\> \\>  $rejectedProps := \\emptyset;$\\\\\n\\> \\> \\> \\>   for each $w \\in mpref[G_i]$\\\\\n\\> \\> \\> \\> \\>  send $(``proposal\", i)$ to woman $w$;\\\\\n\\\\\n\\> Upon receiving a message $(``accept\", q)$:\\\\\n\\> \\> {\\bf if} $(q \\in mpref[G_i])$\\\\\n\\> \\>  $acceptedProps := acceptedProps \\cup \\{q\\};$\\\\\n\\\\\n${\\bf Q_i}$:: \/\/ Process for Woman $i$\\\\\n\\> {\\bf input}\\\\\n\\> \\>   $wrank$: array[$1$..$n$]  of $1..n;$ \/\/ rank of each man by the woman \\\\\n\\> {\\bf var}\\\\\n\\> \\>   $partner$: $0..n;$ initially $0$ \/\/ current partner \\\\\n\\\\\n\\> Upon receiving a message $(``proposal'', j)$:\\\\\n\\> \\>   {\\bf if} $(partner = 0)$ {\\bf then}\\\\\n\\> \\> \\>  $partner := j$;\\\\\n\\> \\>   {\\bf else if} $(wrank[j] < wrank[partner]) $ {\\bf then}\\\\\n\\> \\> \\> send $(``reject\", i)$ to $P_{partner}$;\\\\\n\\> \\> \\>  $partner := j$;\\\\\n\\> \\> {\\bf else} \/\/ $wrank[j] \\geq wrank[partner]\n\\\\\n\\\\\n{\\bf Environment}::\\\\\n Process that (1) initiates the diffusing computation and \\\\\n (2) detects Termination\\\\\n \\\\\n \\> send ``initiate'' message to all $P_i$\\\\\n\\> Upon Detecting Termination of Diffusing Computation\\\\\n \\> \\> Announce the current assignment as a stable marriage \\\\\n \\> \\> satisfying external constraints. Halt\\\\\n\\end{tabbing}\n\n\\end{center}\n\\caption{{A diffusing distributed computation algorithm for constrained SMP} for men $P_i$ and women $Q_i$\\label{fig:CSMP}}\n\\end{figure}\n\n\n}\n\n\n\\section{Introduction}\nThe Lattice-Linear Predicate (LLP) algorithm \\cite{DBLP:conf\/spaa\/Garg20} is a general technique for designing parallel algorithms for combinatorial\noptimization problems. In \\cite{DBLP:conf\/spaa\/Garg20}, it is shown that the stable marriage problem, the shortest path problem in a graph, and \nthe assignment problem can all be solved using the LLP algorithm.\nIn \\cite{Garg:ICDCN22},  many dynamic programming problems, in \\cite{DBLP:conf\/sss\/Garg21}, the housing problem, and\nin \\cite{AlvGar22}, it is shown that the minimum spanning tree problem can be solved using the LLP\nalgorithm.\nIn \\cite{DBLP:conf\/sss\/GuptaK21}, Gupta and Kulkarni extend LLP algorithms for deriving self-stabilizing algorithms.\nIn this paper, we show that many generalizations of the stable matching problem can also be solved using\nthe LLP algorithm. A forthcoming book on parallel algorithms \\cite{Garg23} gives a uniform description of these and other problems\nthat can be solved using the LLP algorithm.\n\nThe Stable Matching Problem (SMP) \\cite{gale1962college} has wide applications in economics, distributed computing, resource allocation and many other fields \\cite{maggs2015algorithmic,iwama2008survey}.\nIn the standard SMP, there are $n$ men and $n$ women each with their totally ordered preference list. \nThe goal is to find \na matching between men and women such that there is no instability, i.e., there is no pair of a woman and a man such that they are not married to each other but\nprefer each other over their partners. \nIn this paper, we show that LLP algorithm can be used to derive solutions to a more general problem than SMP, called {\\em constrained SMP}. In our formulation, in addition to men's preferences and women's preferences, there may be a set of\n{\\em lattice-linear} \nconstraints \non the set of marriages consistent with men's preferences. \nFor example, we may state that Peter's regret  \\cite{gusfield1989stable} should be less than that of Paul,\nwhere the {\\em regret} of a man in a matching is the choice number he is assigned. As another example, we may require\nthe matching must contain some pairs called  {\\em forced pairs}, or must not contain some pairs called {\\em forbidden pairs} \\cite{Dias2003}.\nWe call such constraints {\\em external} constraints. Any algorithm to solve constrained SMP can solve standard SMP by setting (external) constraints to the empty set.\n\nIn this paper, \nwe also present a distributed algorithm to solve the constrained SMP in an asynchronous system. One of the goals is to show how a parallel LLP algorithm\ncan be converted into a distributed asynchronous algorithm.\nOur distributed algorithm uses a diffusing computation whose\ntermination is detected using a standard algorithm such as the Dijkstra-Scholten algorithm. The algorithm uses\n$O(n^2)$ messages each of size $O(\\log n)$.\nKipnis and Patt-Shamir \\cite{KipnisP10} have given a distributed algorithm for stable matching in a synchronous system. There are many differences with their work. First, they do not\nconsider external constraints and their work is not easily extensible for incorporating external constraints.\nSecond, for termination detection, they require each rejected node to broadcast the fact that the protocol has not terminated on a shortest-path tree. This step\nrequires the assumption of synchrony for termination detection and incurs additional message overhead. Our algorithm avoids such broadcasts and works for asynchronous systems.\nTheir paper suggests use of $\\alpha$ synchronizer \\cite{JACM::Awerbuch1985} for simulating in asynchronous systems. However, each round adds $O(n^2)$ messages for using\n$\\alpha$ synchronizer. Thus, our algorithm not only solves a more general problem, it is also more efficient for running the traditional SMP in an asynchronous system.\n\nWe also consider the generalizations of the stable matching problem to the case when the preference lists may have ties.\nThe problem of stable marriage with ties is clearly more general than the standard stable matching problem and has also been\nextensively studied \\cite{IRVING1994261,gusfield1989stable,david2013algorithmics}. \nWe consider three versions of matching with ties. \nIn the first version, called {\\em weakly stable} matching $M$, there is no blocking pair of \nman and woman $(m,w)$ who are not married in $M$ but strictly prefer\neach other to their partners in $M$. In the second version, called {\\em superstable} matching $M$, we require\nthat there is no blocking pair of man and woman $(m,w)$ who are not married in $M$\nbut either (1) both of them prefer each other to their partners in $M$, or (2) one of them prefers the other over his\/her partner in $M$ and the other one is\nindifferent, or (3) both of them are indifferent to their spouses.\nThe third version, called {\\em strongly stable matching}, we require that\nif there is no blocking pair $(m,w)$ such that they are not married in $M$\nbut either (1) both of them prefer each other to their partners in $M$, or (2) one of them prefers the other over his\/her partner in $M$ and the other one is\nindifferent. Algorithms for these problems are well-known; our goal is to \npresent LLP algorithms for these problems.\n\n\n\n\n\n\\section{Background: Lattice-Linear Predicate Detection Algorithm}\nIn this section, we give a self-contained description of the Lattice-Linear Predicate detection algorithm.\nThe reader should consult \\cite{DBLP:conf\/spaa\/Garg20} for more details.\nLet ${L}$ be the lattice of all $n$-dimensional vectors of reals greater than or equal to zero vector and less than or equal to a given vector $T$\nwhere the order on the vectors is defined by the component-wise natural $\\leq$.\nThe lattice is used to model the search space of the combinatorial optimization problem.\nThe combinatorial optimization problem is modeled as finding the minimum element in ${L}$ that satisfies a boolean {\\em predicate} $B$, where\n$B$ models {\\em feasible} (or acceptable solutions).\nWe are interested in parallel algorithms to solve the combinatorial optimization problem with $n$ processes.\nWe will assume that the systems maintains as its state the current candidate vector $G \\in {L}$ in the search lattice, \nwhere $G[ i]$ is maintained at process $i$. We call $G$, the global state, and $G[ i]$, the state of process $i$.\n\nFig. \\ref{fig:poset-lattice} shows a finite poset corresponding to $n$ processes ($n$ equals two in the figure), and the corresponding lattice of all eleven global states.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=2.5in,height=1.0in]{lattice2-eps-converted-to.pdf}\n\\caption{A poset and its corresponding distributive lattice $L$ \\label{fig:poset-lattice}}\n\\end{center}\n\\end{figure}\n\nFinding an element in lattice that satisfies the given predicate $B$, is called the {\\em predicate detection} problem.\nFinding the {\\em minimum} element that satisfies $B$ (whenever it exists) is the combinatorial optimization problem.\nA key concept in deriving an efficient  predicate detection algorithm is that of a {\\em\nforbidden} state.  \nGiven a predicate $B$, and a vector $G \\in {L}$, a state $G[j]$ is {\\em forbidden} (or equivalently, the index $j$ is forbidden) if \nfor any vector $H \\in {L}$ , where $G \\leq H$, if $H[j]$ equals $G[ j]$, then $B$ is false for $H$.\nFormally,\n\\begin{definition}[Forbidden State  \\cite{chase1998detection}]\n  Given any distributive lattice ${L}$ of  $n$-dimensional vectors of $\\mathbf{R}_{\\ge 0}$, and a predicate $B$, we define\n  $ \\forbidden(G,j,B) \\equiv \\forall H \\in {L} : G \\leq H : (G[j] = H[j]) \\Rightarrow\n  \\neg B(H).$\n\\end{definition}\n\nWe define a predicate $B$ to\nbe {\\em lattice-linear} with respect to a lattice ${L}$\n if for any global state $G$,  $B$ is false in $G$ implies that $G$ contains a\n{\\em forbidden state}. Formally,\n\\begin{definition}[lattice-linear Predicate  \\cite{chase1998detection}]\nA boolean predicate $B$ is {\\em {lattice-linear}} with respect to a lattice ${L}$\niff\n$\\forall G \\in {L}: \\neg B(G) \\Rightarrow (\\exists j: \\forbidden(G,j,B))$.\n\\end{definition}\n\n\nOnce we determine $j$ such that $forbidden(G,j,B)$, \nwe also need to determine how to advance along index $j$.\nTo that end, we extend the definition of forbidden as follows.\n\\begin{definition}[$\\alpha$-forbidden]\n Let $B$ be any boolean predicate on the lattice ${L}$ of all assignment vectors.\n For any $G$, $j$ and positive real $\\alpha > G[ j]$, we define $\\mbox{forbidden}(G,j, B, \\alpha)$ iff\n $$  \\forall H \\in {L}:H \\geq G: (H[j] < \\alpha) \\Rightarrow \\neg B(H).  $$\n\\end{definition}\n\nGiven any lattice-linear predicate $B$, suppose $\\neg B(G)$. This means that $G$ must be advanced on all\nindices $j$ such that $\\forbidden(G,j,B)$.  We use a function $\\alpha(G,j, B)$ such that $\\forbidden(G, j, B, \\alpha(G,j, B))$ holds\nwhenever $\\forbidden(G,j,B)$ is true.  With the notion of $\\alpha(G, j, B)$, we have the Algorithm $LLP$.\nThe algorithm $LLP$ has two inputs --- the predicate $B$ and the top element of the lattice $T$. It returns the least vector $G$ which is less than or equal to $T$\nand satisfies $B$ (if it exists). Whenever $B$ is not true in the current vector $G$, the algorithm advances on all forbidden indices $j$\nin parallel. This simple parallel algorithm can be used to solve a large variety of combinatorial optimization problems\nby instantiating different $\\forbidden(G,j,B)$ and $\\alpha(G,j,B)$.\n\n\\begin{algorithm}\n\\SetAlgoRefName{LLP}\n vector {\\bf function} getLeastFeasible($T$: vector, $B$: predicate)\\\\\n {\\bf var} $G$: vector of reals initially $\\forall i: G[ i] = 0$;\\\\\n {\\bf while} $\\exists j: \\forbidden(G,j,B)$ {\\bf do}\\\\\n    \\hspace*{0.2in}    {\\bf for all} $j$ such that $\\forbidden(G,j,B)$  {\\bf in parallel}:\\\\\n    \\hspace*{0.2in}\\h {\\bf if} $(\\alpha(G,j,B) > T[j])$ then return null; \\\\\n   \n   \\hspace*{0.2in}\\h    {\\bf else} $G[ j] := \\alpha(G,j,B)$;\\\\\n       {\\bf endwhile};\\\\\n     {\\bf return} $G$; \/\/ the optimal solution\n\\caption{Find the minimum vector at most $T$ that satisfies $B$\\label{fig:alg-llp}}\n\\end{algorithm}\n\nThe following Lemma is useful in proving lattice-linearity of predicates.\n\\begin{lemma}\\label{lem:basic-LLP}  \\cite{DBLP:conf\/spaa\/Garg20,chase1998detection}\nLet $B$ be any boolean predicate defined on a lattice ${L}$ of vectors. \\\\\n(a) Let $f:{L} \\rightarrow \\mathbf{R}_{\\ge 0}$ be any monotone function  defined on the lattice ${L}$ of vectors of $\\mathbf{R}_{\\ge 0}$.\nConsider the predicate\n$B \\equiv G[ i] \\geq f(G)$ for some fixed $i$. Then, $B$ is lattice-linear.\\\\\n(b) If $B_1$ and $B_2$ are lattice-linear then $B_1 \\wedge B_2$ is also lattice-linear.\n\\end{lemma}\n\nWe now give an example of lattice-linear predicates for scheduling of $n$ jobs. Each job $j$ requires time $t_j$ for completion and has a set of\n prerequisite jobs, denoted by $pre(j)$, such that it can be started only after all its prerequisite jobs\nhave been completed. Our goal is to find the minimum completion time for each job.\nWe let our lattice ${L}$ be the set of all possible completion times. A completion vector $G \\in {L}$ is feasible iff $B_{jobs}(G)$ holds where\n$B_{jobs}(G) \\equiv \\forall j: (G[ j] \\geq t_j) \\wedge (\\forall i \\in pre(j): G[ j] \\geq G[ i] + t_j)$.\n$B_{jobs}$ is lattice-linear because if it is false, then there exists $j$ such that \neither $G[ j] < t_j$ or $\\exists i \\in pre(j): G[ j] < G[ i]+t_j$. We claim that $\\forbidden(G, j, B_{jobs})$. Indeed, any vector $H \\geq G$ cannot\nbe feasible with $G[ j]$ equal to $H[j]$. The minimum of all vectors that satisfy feasibility corresponds to the minimum completion time.\n\nAs an example of a predicate that is not lattice-linear, consider the predicate $B \\equiv \\sum_j G[ j] \\geq 1$ defined on the space of \ntwo dimensional vectors. Consider the vector $G$ equal to $(0,0)$. The vector $G$ does not satisfy $B$. For $B$ to be lattice-linear\neither the first index or the second index should be forbidden. However, \nnone of the indices are\nforbidden in $(0,0)$. The index $0$ is not\nforbidden because the vector $H = (0,1)$ is greater than $G$, has $H[0]$ equal to $G[ 0]$ but it still satisfies $B$.\nThe index $1$ is also not forbidden\nbecause $H =(1,0)$ is greater than $G$, has $H[1]$ equal to $G[ 1]$ but it satisfies $B$.\n\n \\label{sec:prog}\n \n We now go over the notation used in description of our parallel algorithms.\nFig. \\ref{fig:examples} shows a parallel algorithm for the job-scheduling problems.\n\nThe {\\bf var} section gives the variables of the problem.\nWe have a single variable $G$ in the example shown in Fig. \\ref{fig:examples}. \n\n $G$ is an array of objects such that\n $G[j]$ is the state of thread $j$ for a parallel program.\n\nThe {\\bf input} section gives all the inputs to the problem. These inputs are constant in the program and do not change during execution.\n\n The {\\bf init} section is used to initialize the state of the program.\n All the parts of the program\n are applicable to all values of $j$. For example, the {\\em init} section of the job scheduling program in Fig. \\ref{fig:examples}\n specifies that $G[j]$ is initially $t[j]$. Every thread $j$ would initialize $G[j]$.\n\n The {\\bf always} section defines additional variables which are derived from $G$.\n The actual implementation of these variables are left to the system. They can be viewed as\n macros. We will show its use later.\n\n\n\n The LLP algorithm gives the desirable predicate either by using the {\\bf forbidden} predicate or {\\bf ensure} predicate.\n The {\\em forbidden} predicate has an associated {\\em advance} clause that specifies how $G[j]$ must be advanced\n whenever the forbidden predicate is true.\n For many problems, it is more convenient to use the complement of the forbidden predicate.\n The {\\em ensure} section specifies the desirable predicates of the form $(G[j] \\geq expr)$ or \n $(G[j] \\leq expr)$. \n\n\n\n\n The statement {\\em ensure} $G[j] \\geq expr$  simply means that whenever thread $j$ finds $G[j]$ to be less than \n $expr$; it can advance $G[j]$ to $expr$.\n\n\n\n  Since $expr$ may refer to $G$, just by setting $G[j]$ equal to $expr$, there is no guarantee \n  that $G[j]$ continues to be  equal to $expr$ --- the value of $expr$ may change because of changes in other components.\n  We use {\\em ensure} statement whenever $expr$ is a monotonic function of $G$ and therefore the predicate\n  is lattice-linear. \n\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\small {\n\\fbox{\\begin{minipage}  {\\textwidth}\\sf\n\\begin{tabbing}\nxxxx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\= \\kill\n$P_j$: Code for thread $j$\\\\\n{\\bf var} $G$: array[$1$..$n$] of $0..maxint$;\/\/ shared among all threads\\\\\n{\\bf input}: $t[j]: int$, $pre(j)$: list of $1..n$;\\\\\n{\\bf init}: $G[j] := t[j]$;\\\\\n\\\\\n{\\bf \\underline{job-scheduling}}:\\\\\n\\> {\\bf forbidden}: $G[j] < \\max \\{G[i] + t[j] ~|~ i \\in pre(j)\\}$;\\\\\n\\> \\> {\\bf advance}: $G[j] := \\max \\{G[i] + t[j] ~|~ i \\in pre(j)\\}$;\\\\\n\\\\\n{\\bf \\underline{job-scheduling}}:\\\\\n\\> {\\bf ensure}: $G[j] \\geq \\max \\{G[i] + t[j] ~|~ i \\in pre(j)\\}$;\\\\\n\\\\\n{\\bf \\underline{shortest path from node $s$: Parallel Bellman-Ford}} \\\\\n\\> {\\bf input}: $pre(j)$: list of $1..n$; $w[i,j]$: int for all $i \\in pre(j)$\\\\\n\\> {\\bf init}: if $(j=s)$ then $G[j] := 0$ else $G[j]$ := maxint;\\\\\n\\> {\\bf ensure}:  $ G[j] \\leq \\min \\{ G[i] + w[i,j] ~|~ i \\in pre(j) \\}$\n\\end{tabbing}\n\\end{minipage}\n }\n }\n\\caption{LLP Parallel Program for (a) job scheduling problem using forbidden predicate (b) job scheduling problem using ensure clause and (c) the shortest path problem\n\\label{fig:examples}}\n\\end{center}\n\\vspace*{-0.2in}\n\\end{figure}\n\n\\section{A Parallel Algorithm for the Constrained Stable Matching Problem}\n We now derive the algorithm for the stable matching problem using Lattice-Linear Predicates \\cite{DBLP:conf\/wdag\/Garg17}.\nWe let $G[i]$ be the choice number that man $i$ has proposed to. Initially, $G[i]$ is $1$ for all men. \n \n \n \\begin{definition}\n An assignment $G$ is feasible for the stable marriage problem if (1) it corresponds to a perfect matching (all men are paired with different women) \n and (2) it has no blocking pairs. \n \\end{definition}\n \n The predicate ``$G$ is a stable marriage'' is a lattice-linear predicate \\cite{DBLP:conf\/spaa\/Garg20}  which\nimmediately gives us \\ref{fig:GS}.\n The {\\bf always} section defines variables which are derived from $G$.\n These variables can be viewed as\nmacros. For example, for any thread  $z = mpref[j][G[j]]$. This means that\nwhenever $G[j]$ changes, so does $z$.\nIf man $j$ is forbidden, it is clear that any vector in which man $j$ is matched with $z$ and the other man $i$ is matched\n with his current or a worse choice can never be a stable marriage. Thus, it is safe for man $j$ to advance to the next choice.\n \\begin{algorithm}\n \\SetAlgoRefName{LLP-ManOptimalStableMarriage}\n$P_j$: Code for thread $j$\\\\\n {\\bf input}: $mpref[i,k]$: int for all $i,k$; $wrank[k][i]$: int for all $k,i$;\\\\\n  {\\bf init}:   $G[j] := 1$;\\\\\n{\\bf always}:  $z = mpref[j][G[j]];$\\\\\n {\\bf forbidden}: \\\\\n  $\\exists i:\\exists k \\leq G[i]:(z = mpref[i][k]) \\wedge (wrank[z][i] < wrank[z][j])$\\\\\n \\hspace*{0.2in}  {\\bf advance}: $G[j] := G[j]+1; $\n\\caption{A Parallel Algorithm for Stable Matching \\label{fig:GS}}\n\\end{algorithm}\n\nWe now generalize \\ref{fig:GS} algorithm to solve the constrained stable marriage problem.\nIn the standard stable matching problem, there are no constraints\non the order of proposals made by different men. \nLet $E$ be the set\nof proposals made by men to women.\nWe also call these proposals {\\em events}\nwhich are executed by\n$n$ processes corresponding to $n$ men denoted by $\\{P_1 \\ldots P_n\\}$.\nEach of the events can be characterized by a tuple $(i,j)$ that corresponds to\nthe proposal made by man $i$ to woman $j$.\nWe impose a partial order $\\rightarrow_p$ on this set of events to model the order in which\nthese proposals can be made. In the standard SMP, every man $P_i$ has its preference list $mpref[i]$ such that\n$mpref[i][k]$ gives the $k^{th}$ most preferred woman for $P_i$. We model $mpref$ using\n$\\rightarrow_p$; if $P_i$ prefers woman $j$ to woman $k$, then\nthere is an edge from the event $(i,j)$ to the event $(i,k)$. As in SMP, we assume that\nevery man gives a total order on all women.\nEach process makes proposals to women in the decreasing order of preferences (similar to Gale-Shapley algorithm).\n\nIn the standard stable matching problem, there are no constraints\non the order of proposals made by different men, and $\\rightarrow_p$ can be visualized as a partial order $(E, \\rightarrow_p)$ with\n$n$ disjoint chains.\nWe generalize the SMP problem to include external constraints\non the set of proposals.\nIn the constrained SMP, $\\rightarrow_p$  can relate proposals made by different men\nand therefore $\\rightarrow_p$ forms a general poset $(E, \\rightarrow_p)$.\nFor example, the constraint that Peter's regret is less than or equal to John can be modeled by adding $\\rightarrow_p$ edges\nas follows. For any regret $r$, we add an $\\rightarrow_p$ edge from the proposal by John with regret $r$ to the\nproposal by Peter with regret $r$.\nWe draw $\\rightarrow_p$ edges in solid \nedges as shown in Fig. \\ref{fig:csmp-model}.\n\n\n\n\nLet $G \\subseteq E$ denote the  global state of the system. A global state $G$ is simply the subset of events executed in the computation\nsuch that it preserves the order of events within each $P_i$. \nSince all events executed by a process $P_i$ are totally ordered,\n it is sufficient to record the number of events executed \nby each process in a global state. \nLet $G[i]$ be the number of proposal made by $P_i$. \nInitially, $G[i]$ is $1$ for all men.\nIf $P_i$ has made $G[i]>0$ proposals, then \n$mpref[i][G[i]]$ gives the identity of the woman last proposed by $P_i$. \nWe let $event(i, G[i])$ denote the event in which $P_i$ makes a proposal to $mpref[i][G[i]]$.\nWe also use $succ(event(i, G[i]))$ to denote the next proposal made by $P_i$, if any.\n\n\n\nFor the constrained SMP, we have $\\rightarrow_p$ edges that relate proposals of different processes.\nThe graph in Fig. \\ref{fig:csmp-model} shows an example of using $\\rightarrow_p$ edges in the constrained SMP.\nFor this problem, we work with {\\em consistent global states} (or order ideals \\cite{davey,Gar:2015:bk}).\nA global state $G \\subseteq E$ is {\\em consistent} if\n$ \\forall e,f \\in E: (e \\rightarrow_p f) \\wedge (f \\in G) \\Rightarrow (e \\in G).$\nIn the context of constrained SMP, it is easy to verify that $G$ is consistent iff \nfor all $j$, there does not exist $i$ such that $$succ(event(j, G[j])) \\rightarrow_p event(i, G[i]).$$\nIt is well known that the set of all consistent global states of a finite poset forms a finite\ndistributive lattice \\cite{davey,Gar:2015:bk}. We use the lattice of all consistent global states as ${L}$ for\nthe predicate detection.\n\nIn the standard SMP, women's preferences\nare specified by preference lists $wpref$ such that $wpref[i][k]$ gives the $k^{th}$ most preferred man for woman $i$.\nIt is also convenient to define $wrank$ such that $wrank[i][j]$ gives the choice number $k$ for which $wpref[i][k]$ equals $j$, i.e.,\n$wpref[i][k] = j$ iff $wrank[i][j] = k$.\nWe model these preferences using edges on the computation graph as follows. If an event $e$ \ncorresponds to a proposal by $P_i$ to woman $q$ and she prefers $P_j$, then we add a dashed\n edge\nfrom $e$ to the event $f$ that corresponds to $P_j$ proposing to woman $q$.\nThe set $E$ along with the dashed edges also forms a partial order $(E, \\rightarrow_w)$ where \n$e \\rightarrow_w f$ iff both proposals are to the same woman and that woman prefers the proposal\n$f$ to $e$. \nWith $((E, \\rightarrow_p), \\rightarrow_w)$ we can model any SMP specified using $mpref$ and $wpref$.\n\nFigure \\ref{fig:cuts} gives an example of a standard SMP problem in Fig. \\ref{fig:SMP} in our model. To avoid cluttering the\nfigure, we have shown preferences of all men but preferences of only two of the women.\nFig \\ref{fig:csmp-model} gives an example of a constrained SMP. Since both $\\rightarrow_p$ and $\\rightarrow_w$ are\ntransitive relations, we draw only the transitively reduced diagrams.\n\n\\begin{figure}\n\\begin{tabular}{l | l l l l |   l l   l | l l l l |}\nmpref & & & &  & & & wpref \\\\\nP1   &   w4 & w1 & w2 & w3  &  & & w1 & P4 & P1 & P3 & P2\\\\\nP2   &   w2 & w3 & w1 & w4 &  & & w2 & P1 & P4 & P2 & P3\\\\\nP3   &   w3 & w1 & w4 & w2 &  & & w3 & P1 & P2 & P4 & P3\\\\\nP4   &   w2 & w4 & w3 & w1 &  & & w4 & P3 & P1 & P4 & P2\\\\\n\\end{tabular}\n\\caption{\\label{fig:SMP}  Stable Matching Problem specified using men preference list (mpref) and\nwomen preference list (wpref).}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[htbp]\n\\vspace*{-0.3in}\n\\begin{center}\n\\includegraphics[height=3.5in]{mexample2.pdf}\n\\caption{\\label{fig:cuts}  Men preferences are shown in blue solid edges. Preferences of women 1 and 2  are shown in dashed green edges.\nIn the standard SMP graph, there are no blue edges from any event in $P_i$ to any event in $P_j$ for distinct $i$ and $j$.}\n\\end{center}\n\\vspace*{-0.2in}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\vspace*{-0.3in}\n\\begin{center}\n\\includegraphics[height=3.5in]{mCS.pdf}\n\\caption{\\label{fig:csmp-model}  Constrained SMP Graph corresponding to constraint that the {\\em regret} for $P_2$ is less than or equal to that of $P_1$.\nIt also shows the preference of $w3$ of $P4$ over $P3$.}\n\\end{center}\n\\vspace*{-0.2in}\n\\end{figure}\n\n\n\nThe above discussion motivates the following definition.\n\\begin{definition}[Constrained SMP Graph]\nLet $E = \\{(i,j) | i \\in [1..n] \\mbox{ and } j \\in [1..n] \\}$.\nA Constrained SMP Graph $((E, \\rightarrow_p), \\rightarrow_w)$ is a directed graph on $E$ with two sets of edges $\\rightarrow_p$ and $\\rightarrow_w$ with the following\nproperties: (1) $(E, \\rightarrow_p)$ is a poset such that the set $P_i = \\{ (i,j) | j \\in [1..n] \\}$ is a chain for all $i$, and\n(2) $(E, \\rightarrow_w)$ is a poset such that the set $Q_j = \\{ (i,j) | i \\in [1..n] \\}$ is a chain for all $j$  and there is no $\\rightarrow_w$ edge\nbetween proposals to different women, i.e.,\nfor all $i,j,k,l: (i,j) \\rightarrow_w (k,l) \\Rightarrow (j=l)$.\n\\end{definition}\n\nGiven a global state $G$, we define the {\\em frontier} of $G$ as the set of maximal events executed by \nany process. The frontier includes only the last event executed by $P_i$ (if any). Formally,\n$\\frontier(G) = \\{ e \\in G ~|~ \\forall f \\in G$ such that  $f \\neq e$, $f$ and $e$ are executed by $P_i$ implies $f \\rightarrow_p e$ \\}.\nWe call the events in $G$ that are not in $\\frontier(G)$ as pre-frontier events.\n\nWe now define the feasible predicate on  global states as follows.\n\\begin{definition}[feasibility for marriage]\nA global state $G$ is  feasible for marriage iff (1)  $G$ is a consistent global state, and (2) there is no dashed edge ($\\rightarrow_w$) from a frontier event to any event of $G$ (frontier or pre-frontier).\nFormally,\n$B_{marriage}(G) \\equiv$\\\\\n$ consistent(G) \\wedge  (\\forall e \\in \\frontier(G), \\forall g \\in G: \\neg (e \\rightarrow_w g).$\n\n\n\\end{definition}\n\\remove{\nA green edge from a frontier event $(i,j)$ to another frontier event $(k,l)$ would imply that $j$ is equal to $l$\nand both men $i$ and $k$ are assigned the same woman which violates the matching property.\nFor example, in Fig. \\ref{fig:csmp-model}, $(P_3,w_3)$ and $(P_4, w_3)$ cannot both be frontier events of an feasible global state.\nA green edge from a frontier event $(i,j)$ to a pre-frontier event $(k,l)$ in a global state $G$ implies that woman $j$ prefers man $k$ to her partner $i$ and man $k$ prefers $j$ to his partner in $G$.\nFor example, in Fig. \\ref{fig:csmp-model}, $(P_3,w_3)$ and $(P_4, w_1)$ cannot both be frontier events of an feasible global state because there is\na green edge from $(P_3, w_3)$ to a pre-frontier event $(P_4, w_3)$.\n}\n\n\n\nIt is easy to verify that the problem of finding a stable matching is the same as finding a global state that \nsatisfies the predicate $B_{marriage}$ which is defined purely in graph-theoretic terms on the\nconstrained SMP graph.\nThe next task is to show that $B_{marriage}$ is lattice-linear.\n\n\n\n\\begin{theorem}\\label{lem:CSMP-forbid}\nFor any global state $G$ that is not a constrained stable matching,\nthere exists  $i$ such that $\\forbidden(G,i,B_{marriage})$.\n\\end{theorem}\n\\begin{proof}\n\nFirst suppose that $G$ is not consistent, i.e., there exists $f \\in G$ such that\nthere exists $e \\not \\in G$ and $e \\rightarrow_p f$. Suppose that $e$ is on $P_i$. Then, \n$\\forbidden(G,i,B)$ holds because any global state $H$ that is greater than $G$ cannot be consistent\nunless $e$ is included.\n\nNext, suppose that $G$ is a consistent global state but the assignment for $G$ is not a matching.\nThis means that for some distinct $i$ and $j$, both $G[i]$ and $G[j]$ refer to the same woman, say $w$.\nSuppose that $w$ prefers $j$ to $i$, then we claim $\\forbidden(G, i, B)$.\nConsider any $H$ such that $H[i] = G[i]$ and $H[j] \\geq G[j]$.\nFirst consider the case $H[j] = G[j]$. In this case, the same woman $w$ is still assigned to \ntwo men and hence $H$ is not a stable matching. \nNow consider the case $H[j] > G[j]$. In this case, \nthe woman $w$ prefers man $j$ to $i$,\nand the man $j$ prefers $w$ to the woman assigned in $H[j]$ violating stability.\n\nNow suppose that the assignment for $G$ is a constrained matching but not stable. \nSuppose that $(j,w)$ is a blocking pair in $G$. Let $i$ be assigned to $w$ in $G$\n (i.e., the woman corresponding to $G[i]$ prefers man $j$ to $i$,\nand the man $j$ also prefers her to his assignment).\nWe claim that $\\forbidden(G, i, B)$.\nConsider any $H$ such that $H[i] = G[i]$ and $H[j] \\geq G[j]$.\nIn this case, $(j,w)$ continues to be blocking in $H$.\nThe woman $w$ prefers man $j$ to $i$,\nand the man $j$ prefers $w$ to the woman assigned in $H[j]$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\nWe now apply the detection of lattice-linear global predicates for the constrained stable matching. \n\n\n\n \\begin{algorithm}\n \\SetAlgoRefName{LLP-ConstrainedStableMarriage}\n$P_j$: Code for thread $j$\\\\\n {\\bf input}: $mpref[i,k]$: int for all $i,k$; $wrank[k][i]$: int for all $k,i$;\\\\\n  {\\bf init}:   $G[j] := 1$;\\\\\n{\\bf always}:  $z = mpref[j][G[j]];$\\\\\n {\\bf forbidden}: \\\\\n  $\\exists i: \\exists k \\leq G[i]: (z = mpref[i][k]) \\wedge (wrank[z][i] < wrank[z][j])$ $\\vee (\\exists i: succ(event(j, G[j])) \\rightarrow_p event(i, G[i]]))$ \\\\\n\\hspace*{0.2in}  {\\bf advance}: {\\bf if} $(G[j] < n)$ then $G[j] := G[j]+1; $\\\\\n\\hspace*{0.2in} \\hspace*{0.2in} {\\bf else} print(``no constrained stable marriage'')\n\\caption{A Parallel Algorithm for the Constrained Stable Matching \\label{fig:alg}}\n\\end{algorithm}\n\n\\remove {\n\\begin{figure}[htb]\n\\begin{center}\n\\fbox{\\begin{minipage}  {\\textwidth}\\sf\n\\begin{tabbing}\nxx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\= \\kill\n{\\bf Algorithm Constrained-Stable-Matching}:\nUse Algorithm $LLP$ where\\\\\n$T$ = $(n,n,...,n)$; \/\/maximum number of proposals at $P_i$\\\\\n$z = mpref[j][G[j]]$; \/\/current woman assigned to man $j$\\\\\n$\\forbidden(G, j, B_{marriage}) \\equiv   (G[j] = 0)$ \\\\\n \\>  $\\vee (\\exists i: \\exists k \\leq G[i]: (z = mpref[i][k]) \\wedge (wrank[z][i] < wrank[z][j]))$\\\\\n \\> $\\vee (\\exists i: succ(event(j, G[j])) \\rightarrow_p event(i, G[i]]))$ \\\\\n\n$\\alpha(G,j,B_{marriage}) = (G[j]+1)$;\n\\end{tabbing}\n\\end{minipage}\n}\n\\end{center}\n\\vspace*{-0.2in}\n\\caption{An efficient algorithm to find the man-optimal constrained stable matching  \\label{fig:alg}}\n\\vspace*{-0.1in}\n\\end{figure}\n}\n\n\nThe algorithm to find the man-optimal constrained stable marriage is shown in Fig. \\ref{fig:alg}.\nFrom the proof of Theorem \\ref{lem:CSMP-forbid}, we get the\nfollowing implementation of $\\forbidden(G, j, B_{marriage})$ in Fig. \\ref{fig:alg}.\nThe first disjunct holds when the woman $z$ assigned to man $j$ is such that there exists a man $i$\nwho is either (1) currently assigned to $z$ and woman $z$ prefers man $i$ , or (2) currently assigned\nto another woman but he prefers $z$ to the current assignment. The first case holds when $k=G[i]$ and the\nsecond case holds when $k < G[i]$.\nThe first case is equivalent to checking if a dashed edge exists from $(j, z)$ to a frontier event.\nThe second case is equivalent to checking if a dashed edge exists to a pre-frontier event.\nThe second disjunct checks that the assignment for $G$ satisfies all external constraints with respect to $j$.\n\nOur algorithm generalizes the Gale-Shapley algorithm in that it allows specification of external constraints.\n\n\n\nWe now show an execution of the algorithm on the CSMP in Fig. \\ref{fig:csmp-model}.\nSince every $P_i$ must make at least one proposal, we start with \nthe first proposal for every $P_i$. The corresponding assignment is\n$[w_4, w_2, w_3, w_2]$, i.e., $P_1$ is assigned $w_4$, $P_2$ is assigned $w_2$ and so on.\nIn this global state $G$, the second component is forbidden. This is because\n$w_2$ prefers $P_4$ over $P_2$. We advance on  $P_2$ to get the global state\n$[w_4, w_3, w_3, w_2]$. Now, \nbecause $w_3$ prefers $P_2$ over $P_3$, $P_3$ must advance.\nWe get the global state\n$[w_4, w_3, w_1, w_2]$. This is a stable matching. However, it does not satisfy\n the constraint that the regret of $P_2$ is less than or equal to that of  $P_1$.\n Here, $P_1$ is forbidden and $P_1$ must advance. We now get the global state\n$[w_1, w_3, w_1, w_2]$ which is not a matching. \nSince $w_1$ prefers $P_1$ over  $P_3$, $P_3$ must advance.\nWe reach the global state\n$[w_1, w_3, w_4, w_2]$ which satisfies the constrained stable matching.\n\n\n\n\n\nWe have discussed man-oriented constrained stable marriage problem. \nOne can also get an LLP algorithm the for woman-oriented constrained stable marriage problem.\nThe paper \\cite{GarHu20} gives an algorithm $\\beta$ that does the downward traversal in the proposal lattice in search of a stable marriage.\nWhen men and women are equal then such a traversal\ncan be accomplished by switching the roles of men and women. However, in \\cite{GarHu20} is is assumed that\nthe number of men $n_m$ may be much smaller than the number of women $n_w$.  It has the\ntime complexity of $O(n_m^2 + n_w)$. Switching the roles of men and women is not feasible\nwithout increasing the complexity of the algorithm. \n\n\n\\section{A Distributed Algorithm for the Constrained Stable Matching Problem}\nAlthough the standard SMP has been studied in a distributed system setting (e.g., \\cite{brito2005distributed,kipnis2009note}), \nwe study the constrained SMP in a distributed system setting. \nOur goal is to show how a parallel LLP algorithm can be converted to a distributed program.\nWe assume an asynchronous system in which all channels are FIFO and reliable and\nthat processes do not crash.\n\n\nWe assume that each man and woman knows only his or her preference lists. \n$P_i$ corresponds to the computation at man  $i$ and $Q_i$ corresponds to the computation at woman $i$.\nEach process $P_i$ is responsible for updating its own component in $G[i]$.\nFor the LLP algorithm, we will assume that the only variable at $P_i$ is $G$ and\nall other variables such as $mpref$ are constants.\nIn addition, each man is given\na list of prerequisite proposals for each of the women that he can propose to.\nIn terms of the constrained-SMP graph, this corresponds to every man knowing the\nincoming solid\nedges for the chain that corresponds to that man in the graph.\nFrom $mpref$, one can also derive $mrank$, the rank $P_i$ assigns to each woman.\n\nThe process $Q_i$ has $wpref$, preferences of woman $i$. However, it is more convenient to\nkeep $wrank$, the rank $Q_i$ assigns to each man. This information is input to $Q_i$.\nThe only variable a woman $Q_i$\nmaintains is the {\\em partner}. Note that given $G$, the partner for each woman can be derived.\nHowever, in a distributed system setting it is more efficient to maintain the partner at each woman.\n\nWhenever $G[i]$ is updated by $P_i$, we will assume that $P_i$ sends a message\nto other relevant processes informing them about the update. Each process keeps\nenough information to be able to evaluate its forbidden predicate. Since the message\ntransfer takes time, the data structures are not necessarily up to date at each process.\nIn particular $P_j$ may have an old value of $G[i]$ maintained at $P_i$.\nWe show that the LLP algorithm has the advantage that it works correctly\ndespite the fact that processes use old values of $G$. Each process evaluates\nits forbidden predicate and advances its state whenever the forbidden predicate is true.\nThe algorithm terminates when no process is forbidden.  In a distributed system setting, \nwe need some process to determine that the system has reached such a state.\nA possible solution for running LLP algorithms in a distributed environment is to \nrun it as a diffusing computation\\cite{dijkstra1980termination}  and use a termination detection algorithm\nalong with the LLP algorithm.\n\nWe now present a diffusing computation for\nsolving the constrained SMP.\nWe adopt the standard rules of a diffusing computation.\nA {\\em passive} process can become {\\em active} only on receiving messages, and only an active process can send a message.\nWe assume the existence of a process called environment that starts the algorithm by\nsending {\\em initiate} messages to all men. In our algorithm shown in Fig. \\ref{fig:CSMP},\n\nThere are four types of messages used in this algorithm. There are exactly $n$ {\\em initiate} messages sent\nby the environment to all men. Each man can send two types of messages. He sends {\\em propose} messages\nto women one at a time in the order given by $mpref$. These messages are sent whenever the current state of the man\nis forbidden and he needs to advance to the next woman.\nA man may sometimes skip proposing some women as\nexplained later. A man also sends {\\em advance} messages to other men which may force other men to skip\ncertain proposals to satisfy external constraints. \n\nA woman acts only when she receives a {\\em propose} message \nfrom a man $j$.\nOn receiving a {\\em propose} message, if she is currently not engaged, she \ngets engaged to man $j$. If she is engaged to a man and the new proposal is preferable to her current\npartner then she sends a {\\em reject} message to the current partner. If the new proposal is less preferable, then\nshe sends a {\\em reject} message to the proposer. The variable {\\em partner} indicates her partner at any point.\nIf the value of {\\em partner} is zero, then that woman is free; otherwise, she is engaged.\nNote that a woman never sends any {\\em accept} message.\nThe algorithm is based on the assumption that if a woman has received a proposal and not rejected it, then\nshe has accepted the message (the algorithm assumes that no messages are lost).\n\nWe now explain the behavior of men for each message type he receives as shown in Fig. \\ref{fig:CSMP}.\nOn receiving an {\\em initiate} message from the environment,  we know that any assignment must have\nat least one proposal from that man. To satisfy external constraints, all proposals that are prerequisite must also be made.\nHence, the man sends an {\\em advance} message to all men with prerequisite proposals. He then sends a proposal to his top choice.\nOn receiving a {\\em reject} message, he first checks if the {\\em reject} message is from his current partner. Since a man\nmay have advanced to a different proposal, there is no need for any action if the {\\em reject} message is from an earlier proposal.\nIf the {\\em reject} message is for the current proposal, then\nthe man knows that he must make another proposal. If he is out of proposals, then\nhe announces that there is no stable marriage with external constraints. Otherwise, he moves on to the next best proposal after\nsending out {\\em advance} messages to all men with prerequisite proposals.\nOn receiving an {\\em advance} message with woman $w$, the man must ensure that he has made a proposal to woman $w$.\nIf he has already made a proposal to $w$, then there is nothing to be done; otherwise, \nhe skips all proposals till\nhe gets to his choice which corresponds to $w$. Next, he makes a proposal to $w$ thereby satisfying external constraints.\n\n\n\n\n\n\\begin{figure}[htbp]\\begin{center}\n\\begin{tabbing}\nx\\=xxx\\=xxx\\=xxx\\=xxx\\=xxx\\= \\kill\n${\\bf P_i}$:: \/\/ Process for Man $i$\\\\\n\\> {\\bf input}\\\\\n\\> \\>   $mpref$: array[$1$..$n$]  of $1..n;$ \/\/ men's preferences\\\\\n\\> \\>   $mrank$: array[$1$..$n$]  of $1..n;$ \/\/ rank of each of the women by man\\\\\n\\> \\>  \/\/ $mrank$ can be derived from $mpref$\\\\\n\\> \\>   $prerequisite$: array[$1$..$n$] of list of proposals; \\\\\n\\> \\> \/\/ list of proposals that must be executed before $mpref[i]$\\\\ \n\\> {\\bf var}\\\\\n\\> \\>   $G_i:1..n$ initially $1$; \/\/ proposal number by $P_i$ \\\\\n\\\\\n\\> Upon receiving a message ``initiate'' from environment;\\\\\n\\> \\>    for each $(m,w) \\in prerequisite[G_i]$\\\\\n\\> \\> \\>  send $(``advance\" , w)$ to $P_m$;\\\\\n\\> \\>  send $(``proposal\", i)$ to woman $mpref[G_i]$;\\\\\n\\\\\n\\> Upon receiving a message $(``reject\", j)$:\\\\\n\\> \\>   {\\bf if} $(mpref[G_i] = j)$ {\\bf then} \/\/ rejected by current partner\\\\\n\\> \\> \\>   {\\bf if} $(G_i = n)$ {\\bf then}\\\\\n\\> \\> \\> \\>  Announce ``no constrained stable marriage possible\" ;\\\\\n\\> \\> \\>   {\\bf else} \\\\\n\\> \\> \\> \\>  $G_i := G_i+1$;\\\\\n\\> \\> \\> \\>   for each $(m,w) \\in prerequisite[G_i]$\\\\\n\\> \\> \\> \\>  \\> send $(``advance\" , w)$ to $P_m$;\\\\\n\\> \\> \\> \\>  send $(``proposal\", i)$ to woman $mpref[G_i]$;\\\\\n\n\\\\\n\\> Upon receiving a message $(``advance\", q)$:\\\\\n\\> \\>        {\\bf while} $(mrank[q] > G_i)$ \\\\\n\\> \\> \\>        $G_i := G_i+1$\\\\\n\\> \\> \\>         for each $(m,w) \\in prerequisite[G_i]$\\\\\n\\> \\> \\>  \\>       send $(``advance\" , w)$ to $P_m$;\\\\\n\\> \\>        {\\bf endwhile};\\\\\n\\> \\>  send $(``proposal\", i)$ to woman $mpref[G_i]$;\\\\\n\\\\\n${\\bf Q_i}$:: \/\/ Process for Woman $i$\\\\\n\\> {\\bf input}\\\\\n\\> \\>   $wrank$: array[$1$..$n$]  of $1..n;$ \/\/ rank of each man by the woman \\\\\n\\> {\\bf var}\\\\\n\\> \\>   $partner$: $0..n;$ initially $0$ \/\/ current partner \\\\\n\\\\\n\\> Upon receiving a message $(``proposal'', j)$:\\\\\n\\> \\>   {\\bf if} $(partner = 0)$ {\\bf then}\\\\\n\\> \\> \\>  $partner := j$;\\\\\n\\> \\>   {\\bf else if} $(wrank[j] < wrank[partner]) $ {\\bf then}\\\\\n\\> \\> \\> send $(``reject\", i)$ to $P_{partner}$;\\\\\n\\> \\> \\>  $partner := j$;\\\\\n\\\\\n\\\\\n{\\bf Environment}::\\\\\n Process that (1) initiates the diffusing computation and \\\\\n (2) detects Termination\\\\\n \\\\\n \\> send ``initiate'' message to all $P_i$\\\\\n\\> Upon Detecting Termination of Diffusing Computation\\\\\n \\> \\> Announce the current assignment as a stable marriage \\\\\n \\> \\> satisfying external constraints. Halt\\\\\n\\end{tabbing}\n\n\\end{center}\n\\caption{{A diffusing distributed computation algorithm for constrained SMP} for men $P_i$ and women $Q_i$\\label{fig:CSMP}}\n\\end{figure}\n\n\n\nObserve that when a man $P_i$ advances, he does not inform his existing partner, if any. Since the number of men and women are same, his partner\nwill eventually get a proposal from someone who she prefers to $P_i$ if there exists a constrained stable matching.\nHis partner $q$ can never be matched with $P_j$ such that $q$ prefers $P_i$ over $P_j$. Otherwise, \nwe have a blocking pair: both $q$ and $P_i$ prefer each other over their partners.\n\nIf there are no external constraints, then there are no {\\em advance} messages, and the algorithm is \na distributed version of the Gale-Shapley algorithm. \nEven in the presence of external constraints, the algorithm shares the following properties with the Gale-Shapley\nalgorithm. As the algorithms progress, the partner for a man can only get worse and the partner for a woman can only get better.\nBoth these properties are direct results of the way men send their proposals and the way women respond to proposals.\n\nThere are also some crucial differences from the Gale-Shapley algorithm.\nIn the Gale-Shapley algorithm, once a woman is engaged she continues to be engaged.\nFor any woman $w$, the predicate that there exists a man such that he is assigned to $w$ is a stable predicate.\nAs a result, the termination of Gale-Shapley (sequential or distributed version) is easy to detect. When all women have been proposed to, the system\nhas reached a stable matching. However, due to external constraints, it is not true in CSMP that once a woman is engaged she\ncontinues to stay engaged. The man who she was engaged to, may be required to advance on receiving an {\\em advance} message\nand then that woman is no longer logically assigned to that man. For the constrained SMP algorithm, we need additional messages to detect termination.\nIt is the environment process that initiates the computation and detects termination \nof the computation. We assume that a termination detection algorithm such as that of Dijkstra and Scholten \\cite{dijkstra1980termination} \nis running in conjunction with the CSMP algorithm. Termination in a diffusing computation corresponds to the condition \nthat all processes are passive and there are no messages in-transit. \n\nWe now show that the algorithm in Fig. \\ref{fig:CSMP} correctly finds the least assignment (or man-optimal) constrained stable matching whenever it exists.\nThe correctness follows from the following invariants.\n\\begin{lemma}\nAny assignment $M$ in which $M[i] < G_i$ for any $P_i$ cannot be a constrained stable marriage.\n\\end{lemma}\n\\begin{proof}\nInitially, the invariant is true because $G_i$ is initialized to $1$ and $M[i]<1$ implies that $P_i$ has not proposed to any one.\nThere are only two reasons the  $G_i$ variable is incremented.\nEither the woman corresponding to the current proposal has sent a {\\em reject} or\na man has sent a message to {\\em advance} beyond the current woman.\nWe first consider the case when the current proposal was rejected by the woman $q$.\nIt is sufficient to show that any assignment in which this man is assigned $q$ cannot be a stable marriage.\nSuppose $q$ rejected $P_i$ in favor of $P_j$. If $P_j$ is also assigned to $q$ in $G$, then\nit is not a matching. If $P_j$ is assigned to a woman that he proposes to later, then we have that $q$ assigned to\n$P_i$ prefers $P_j$ and $P_j$ prefers $q$ to the woman he is assigned.\nIf $G_i$ is advanced because of an {\\em advance} message from $P_j$, then any assignment in which $M[i] < G_i$ does\nnot satisfy prerequisite constraints due to $\\rightarrow_p$.\n\\end{proof}\n\nTo show that the algorithm gives a stable matching on termination, if it exists, we show that\nthe number of successful proposals is equal to $n$ on termination. \nA proposal is defined to be successful\nif it is not rejected by a woman and not advanced over by a man and thereby rejected by the man.\nWe start the algorithm by each process sending out a proposal. Thus, there are $n$ proposals to start with.\nAny proposal that is rejected by a woman leads to another proposal if the reject message is not in transit.\nAny proposal that is skipped due to prerequisite constraints also leads to another proposal. \nSo either a man\nruns out of proposals, or the computation has not terminated until every man has made a successful proposal.\nThis assertion gives us\n\\begin{lemma}\nIf the algorithm announces that the current assignment denotes stable marriage, then the assignment given by $G$\nis a stable matching satisfying external constraints, i.e., if $P_i$ is paired with $mpref[i][G_i]$, then \nthe assignment satisfies constrained stable matching.\n\\end{lemma}\n\\begin{proof}\nSince there are no {\\em reject} messages, {\\em advance} messages, or {\\em propose} messages in transit,  we know that\nthere are $n$ successful proposals. Each successful proposal has the property that\nthe value of current for $P_i$ equals $j$ iff the value of partner for $Q_j$ equals $i$.\nSince any proposal that violates stability is rejected and any proposal that violates external constraints is advanced\nwe get that the assignment on termination is a stable matching satisfying external constraints.\n\\end{proof}\n\n\n\n\n\nWe now analyze the message complexity of the algorithm.\nSuppose that there are $e$ external constraints, $n$ men, $n$ women\nand $m$ unsuccessful proposals.\nThere are $n$ initiate messages.\nFor every unsuccessful proposal, the algorithm uses at most one {\\em reject} message.\nThere are exactly $n$ final successful proposals resulting in one message per proposal in the\ndiffusing computation.\nIf there are $e$ external constraints (solid edges) across processes), then there are at most $e$ advance messages.\nThus, the messages in the diffusing computation are at most\n$n$ {\\em initiate messages}, $m$  unsuccessful {\\em propose} messages, $m$ {\\em reject} messages, $n$ successful {\\em propose} messages,\nand $e$ {\\em advance} messages.\nThus, the total number of messages in the diffusing computation is at most\n$2m+2n+e$.\n\nTermination detection algorithms such as Dijkstra and Scholten's requires as many messages as the\napplication messages in the worst case giving us the overall message complexity of $4m+4n+2e$ messages.\nWe note here that this message complexity can be reduced by various optimizations such as \ncombining the {\\em signal\/ack} messages of Dijkstra and Scholten's algorithm with application messages.\nFor example, a {\\em reject} message can also serve as an {\\em ack} message for a {\\em propose} message.\nFor simplicity, we do not consider these optimizations in the paper.\nSince both $m$ and $e$ are $O(n^2)$, we get $O(n^2)$ overall message complexity.\nAlthough the number of unsuccessful proposals can be $O(n^2)$ in the worst case,\nthey are $O(n \\log n)$ on an average for the standard SMP \\cite{knuth1997stable} .\nNote that each message carries only $O(\\log n)$ bits.\n\n\\remove {\n\\section{A Token-based Algorithm for Constrained SMP}\n\n\\label{sec:vdist}\nThe diffusing computation based distributed algorithm requires $4(m+n)+e$ messages of size $O(\\log n)$ bits.\nWe now show a token-based algorithm that requires $2(m+n)$ messages although each message is of size $O(n \\log n)$ bits..\n\n\n\nThe distributed WCP detection algorithm uses a unique token.  The\ntoken contains two vectors.  The first vector is labeled {$G$}.\nThis vector defines the current candidate cut.  If {$G[i]$} has\nthe value $k$, then the proposal $k$ from process $P_i$ is part of the\ncurrent assignment.  Note that the current assignment may not be a matching, i.e., a woman may be assigned to multiple men.\nAlso, the current assignment may not satisfy the external constraints.\nThe token is initialized with $\\forall i : G[i] = 0$.\n\nThe second vector is labeled $color$, where $color[i]$\nindicates the color for the candidate proposal from process\n$P_i$.  The color of a state can be $red$, $green$ or $yellow$.\nIf $color[i]$ equals $red$, then the proposal $(i,G[i])$ and all its\npredecessors have been eliminated and can never satisfy the constrained stable matching.\nIf $color[i] = green$, then there is no state in $G$\nsuch that $(i,G[i])$ happened before that state and the woman assigned to $P_i$ has not been \nassigned to any other man. If $color[i] = yellow$, then the proposal $(i,G[i])$ satisfies external constraints but the woman \nassigned to $i$ is assigned to another man $j$. In this case, the woman would decide if $P_i$ is preferable to $P_j$ or not.\nWhen the token goes to that woman, she would convert the $yellow$ color to either $red$ or $green$ depending upon her \npreference.\n The token is\ninitialized with $\\forall i : color[i] = red$. Whenever the token satisfies is green for all $P_i$, the algorithm terminates and\nthe constrained stable matching is given by $G$. If a process $P_i$ finds that its color is red in the token and it has run out of proposals,\nthen the algorithm terminates with the messages that ``no constrained stable marriage exists.''\n\n\nThe token is sent to process $P_i$ only when $color[i] = red$.\nWhen it receives the token, $P_i$ moves to the next proposal in its $mpref[i]$ and then checks for violations of\nconsistency conditions with this new proposal.  \nThis activity is\nrepeated until the candidate proposal satisfies all external constraints.\nIt then checks if its proposal is to a woman who is already engaged.\nIn that case, the color is labeled yellow; otherwise, \nit can be labeled green. \n\nNext, $P_i$ examines the token to see if any other man violates\nexternal constraints.  If it finds any $j$ such that $(j, G[j])$ happened\nbefore $(i, G[i])$, then it makes $color[j]$ red. \n\nNow $P_i$ is ready to ship the token.\nIf all states in\n$G$ are green, that is, $G$ satisfies all external constraints, no woman\nis assigned to multiple men, and all men are assigned,  then $P_i$\nhas detected the constrained stable matching. Otherwise, if its color is yellow, the token is sent to\nthe corresponding woman.  If the color is green, then the token is sent to a man whose color is red.\n\nThe algorithm for these actions\nis given in Fig. \\ref{f:monitor-dwcp}.  Note that the token can start on any\nprocess.\n\n\\begin{figure}[htbp]\\begin{center}\n\\fbox{\\begin{minipage}  {\\textwidth}\\sf\n\\begin{tabbing}\nx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\=xxxx\\= \\kill\n$P_i$:\\\\\n\\>  {\\bf var}\\\\\n\\> \\>  \/\/ vector clock from the candidate state\\\\\n\\> \\>  $candidate$: array[$1$..$n$] of integer initially $0$; \\\\\n\\\\\n\\>   Upon receiving the token $(G, D, color)$ \\\\\n\\>  \\>    {\\bf while} $(color[i] = red)$ {\\bf do}\\\\\n\\> \\> \\>      {\\bf if} $(G[i] < n) G[i]++$; \/\/ move to the next proposal if any;\\\\\n\\> \\> \\>      else { announce ``no constrained stable marriage''; halt;} \\\\\n\\> \\> \\>     $D := max (D, G[i].v);$\\\\\n\\> \\> \\>      {\\bf if} $(D[i] < G[i])$ {\\bf then}\\\\\n\\> \\> \\> \\>      $color[i]:=green;$\\\\\n\\> \\>    {\\bf endwhile};\\\\\n\\> \\>      {\\bf for} $j := 1$ {\\bf to } $n$, $(j \\neq i)$  {\\bf do}\\\\\n\\> \\> \\>       {\\bf if} $(D[j] >= G[j])$ {\\bf then}\\\\\n\\> \\> \\> \\>         $color[j] := red$;\\\\\n\\> \\>      {\\bf endfor}\\\\\n\\> \\>      {\\bf for} $j := 1$ {\\bf to } $n$, $(j \\neq i)$  {\\bf do}\\\\\n\\> \\> \\>       {\\bf if} $(color[j] = green) \\wedge (G[i].w = G[j].w)$ {\\bf then}\\\\\n\\> \\> \\> \\>         $color[i] := yellow$;\\\\\n\\> \\>      {\\bf endfor}\\\\\n\\> \\>      {\\bf if} ($\\exists j: color[j] = yellow)$ {\\bf then} send token\nto woman $G[i].w$;\\\\\n\\> \\>      {\\bf else if} ($\\exists j: color[j] = red)$ {\\bf then} send token\nto woman $P_j$;\\\\\n\\> \\>      {\\bf else} {announce \"constrained stable marriage found\"; return $G$;}\\\\\n\\\\\n\\\\\nWoman $wi$:\\\\\n\\>   Upon receiving the token $(G, D, color)$ from $P_j$\\\\\n\\>   $cur$ := man $k \\neq j$ such that $(G[k].w = i) $ and $(G[k].color = green)$;\\\\\n\\> \\> {\\bf if} $(wrank[i][j] > wrank[i][cur])$ \\\\\n\\> \\> \\>  $color[j] := red; $ send token to $P_j$\\\\\n\\> \\> {\\bf else}\\\\\n\\> \\> \\>  $color[cur] := red; $ send token to $P_{cur}$\n\\end{tabbing}\n\\end{minipage}\n}\n\\end{center}\n\\caption{Monitor process algorithm \\label{f:monitor-dwcp}}\n\\end{figure}\n\n\nWe first analyze the time complexity for computation. It is easy to\nsee that whenever a man receives the token, it advances by at least one proposal.\n Every time a proposal is advanced, $O(n)$ work\nis performed by the process with the token.  There are at most\n$n^2$ proposals; therefore, the total computation time for all processes\nis $O(n^3)$.  The work for any process in the distributed\nalgorithm is at most $O(n^2)$. Upon receiving a token, either that man sends the token to another man\nor the woman who he proposes next. A woman can send token only to a man.\nTherefore, the total number of messages is at most twice the number of proposals\nexplored before the algorithm terminates. Each message carries $O(n)$ integers (assuming that an integer is sufficient to encode $n$).\n}\n\n\n\\input{bxx-superStable.tex}\n\n\\input{bxx-strongStable.tex}\n\n\\section{Conclusions and Future Work}\nWe have shown that the Lattice-Linear Parallel Algorithm can solve many problems in the stable marriage literature.\nWe have shown that the LLP Algorithm can also be converted into an asynchronous distributed algorithm.\n\n\nIn the constrained SMP formulation, we have assumed that $(E, \\rightarrow_p)$ is a poset for simplicity.\nOur algorithms are applicable when $(E, \\rightarrow_p)$ may have cycles. For the general graph $(E, \\rightarrow_p)$ we can consider\nthe graph on strongly connected components which is guaranteed to be acyclic. By viewing each strongly\nconnected component as a super-proposal in which multiple proposals are made simultaneously, the same\nanalysis and algorithms can be applied.\n\nWe have also derived parallel LLP algorithms for stable matching problems with ties.\nOur technique gives an easy derivation of algorithms to find the man-optimal matchings as well as the\nsublattice representation of superstable and strongly stable matchings.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\subsection{Basic notation}\n\\label{subsec-1.1}\nWe use the same basic notation as in \\cite{Hed3}. So, we\nwrite $\\mathbb{R}$ for the real line, $\\mathbb{C}$ for the complex plane, \n$\\mathbb{C}_\\infty:=\\mathbb{C}\\cup\\{\\infty\\}$ for the extended complex plane \n(the Riemann sphere). We use $\\mathrm{d} s$ for normalized arc length\nmeasure, and $\\mathrm{d} A$ for normalized area measure, while $\\partial_z$,\n$\\bar\\partial_z$ are the standard Wirtinger derivatives, and\n$\\hDelta_z=\\partial_z\\bar\\partial_z$. \nWe let $\\mathbb{D}$ denote the open unit disk, $\\mathbb{T}:=\\partial\\mathbb{D}$ the unit circle, \nand $\\mathbb{D}_e$ the exterior disk. We use the sesquilinear forms $\\langle\\cdot,\n\\cdot\\rangle_\\mathbb{T}$ and $\\langle \\cdot,\\cdot\\rangle_\\mathbb{D}$ where the measure is\n$\\mathrm{d} s$ and $\\mathrm{d} A$, respectively.\nWe write $1_E$ for the indicator function of a subset $E$. \n\n\n\\subsection{The Bloch space and the Bloch seminorm}\nThe \\emph{Bloch space} consists of those holomorphic functions \n$g:\\mathbb{D}\\to\\mathbb{C}$ that are subject to the seminorm boundedness condition\n\\begin{equation}\n\\|g\\|_{\\mathcal{B}(\\mathbb{D})}:=\\sup_{z\\in\\mathbb{D}}(1-|z|^2)|g'(z)|<+\\infty.\n\\label{eq-Blochnorm}\n\\end{equation}\nLet $\\mathrm{aut}(\\mathbb{D})$ denote the group of sense-preserving M\\\"obius \nautomorphism of $\\mathbb{D}$. By direct calculation,\n\\[\n\\|g\\circ\\gamma\\|_{\\mathcal{B}(\\mathbb{D})}=\\|g\\|_{\\mathcal{B}(\\mathbb{D})},\\qquad\n\\gamma\\in\\mathrm{aut}(\\mathbb{D}),\n\\]  \nwhich says that the Bloch seminorm is invariant under all M\\\"obius \nautomorphisms of $\\mathbb{D}$.\nAn immediate observation we can make at this point is that provided that \n$g(0)=0$, we have the estimate\n\\begin{equation*}\n|g(z)|\\le\\|g\\|_{\\mathcal{B}(\\mathbb{D})}\\int_0^{|z|}\\frac{\\mathrm{d} t}{1-t^2}\n=\\frac12\\,\\|g\\|_{\\mathcal{B}(\\mathbb{D})}\\log\\frac{1+|z|}{1-|z|},\\qquad z\\in\\mathbb{D},\n\\end{equation*}\nwhich is sharp pointwise. This estimate is related with the interpretation\nof $f\\in\\mathcal{B}(\\mathbb{D})$ as the hyperbolically Lipschitz continuous functions.\n\n\n\\subsection{The Bergman projection of bounded functions}\n\\label{subsec-PLinfty}\n\nFor $\\mu\\in L^1(\\mathbb{D})$, let  \n\\[\n\\mathbf{P} \\mu(z):=\\int_\\mathbb{D}\\frac{\\mu(w)}{(1-z\\bar w)^2}\\,\\mathrm{d} A(w),\\qquad z\\in\\mathbb{D},\n\\]\nbe its \\emph{Bergman projection}. Restricted to $L^2(\\mathbb{D})$, it is the \northogonal projection onto the subspace of holomorphic functions. In addition, \nit acts boundedly on $L^p(\\mathbb{D})$ for each $p$ in the interval $1<p<+\\infty$ \n(see, e.g., \\cite{HKZ}).\nThe derivative of $\\mathbf{P} \\mu$ is given by\n\\[\n(\\mathbf{P} \\mu)'(z):=2\\int_\\mathbb{D}\\frac{\\bar w \\mu(w)}{(1-z\\bar w)^3}\\,\\mathrm{d} A(w),\n\\qquad z\\in\\mathbb{D},\n\\]\nso that if $\\mu\\in L^\\infty(\\mathbb{D})$ with $\\|\\mu\\|_{L^\\infty(\\mathbb{D})}\\le1$, we get\nthat (see Per\\\"al\\\"a \\cite{Per})\n\\begin{multline}\n|(\\mathbf{P} \\mu)'(z)|\\le 2\n\\int_\\mathbb{D}\\frac{|w|}{|1-z\\bar w|^3}\\,\\mathrm{d} A(w)\n\\\\\n=2\n\\sum_{k=0}^{+\\infty}\n\\frac{[(3\/2)_k]^2}{(\\frac32+k)(k!)^2}|z|^{2k}\\le 2\n\\Gamma(\\tfrac32)^{-2}\n(1-|z|^2)^{-1}=\\frac{8\\pi^{-1}}{1-|z|^2}.\n\\label{eq:Peralalaest0.0}\n\\end{multline}\nBut what happens if $m$ vanishes on a big enough disk? Can we then\nget a better estimate? By the change-of-variables formula,\n\\[\n(\\mathbf{P}\\mu)'(z)=2\\int_\\mathbb{D}\\frac{\\bar\\phi(\\xi)\\mu\\circ\\phi(\\xi)}\n{(1-z\\bar\\phi(\\xi))^3}|\\phi'(\\xi)|^2\\mathrm{d} A(\\xi),\n\\]\nso that with\n\\[\n\\phi(\\xi)=\\frac{z-\\xi}{1-\\bar z\\xi},\n\\]\nwe find that\n\\[\n(\\mathbf{P}\\mu)'(z)=\\frac{2}{1-|z|^2}\\int_\\mathbb{D}\\frac{\\bar z-\\bar\\xi}\n{(1-\\bar z\\xi)^2}\\mu\\circ\\phi(\\xi)\\mathrm{d} A(\\xi). \n\\]\n\n\\begin{lem} If $\\mu\\circ\\phi=0$ holds on $\\mathbb{D}(0,\\varrho)$ for some $\\varrho$ with\n$0<\\varrho<1$, then \n\\begin{equation}\n(1-|z|^2)|(\\mathbf{P}\\mu)'(z)|\\le 2\\int_{\\mathbb{D}\\setminus\\mathbb{D}(0,\\varrho)}\\frac{1}\n{|1-\\bar z\\xi|}\\mathrm{d} A(\\xi)=\\mathrm{O}\\bigg((1-\\varrho)\\log\\frac{1}{1-\\varrho}\\bigg).\n\\end{equation}\n\\label{lem:logestimate}\n\\end{lem}\n\nThe details of the estimate of the integral are left to the interested reader.\n\n\n\\subsection{Exponential integrability}\n\nSuppose $g=\\mathbf{P}\\mu$, where $\\mu\\in L^\\infty(\\mathbb{D})$ with $\\|\\mu\\|_{L^\\infty(\\mathbb{D})}\\le1$.\nLet $g_r$ denotes the dilate $g_r(\\zeta):=g(r\\zeta)$.\nIt is of importance to control the integrals\n\\[\n\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s\n\\]  \nfor $t\\in\\mathbb{C}$ which are close to $0$. The following estimate was offered by Ivrii \\cite{Ivrii}.\n\n\\begin{thm} {\\rm (Ivrii)}\nIn the above setting, we have the estimate\n\\[\n\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s=\\mathrm{O}\\bigg(\\frac{1}{(1-r^2)^{(\\Sigma^2+\\mathrm{o}(1))|t|^2\/4}}\\bigg)\n\\]  \nuniformly as $|t|\\to0$. Here, $\\Sigma^2$ is an absolute constant with $0<\\Sigma^2\\le1$.\n\\label{thm:Ivr}\n\\end{thm}\n\nThe constant $\\Sigma^2$ was introduced by Astala, Ivrii, Per\\\"al\\\"a, and Prause \\cite{AIPP}.\nLater, Hedenmalm \\cite{Hed3} found another interpretation  of the constant in terms of a zero\npacking problem.\n\n\\begin{thm} {\\rm (Hedenmalm)}\nIt holds that \n\\[\n\\Sigma^2=1-\\rho_{\\mathbb{H}}<1,\n\\]\nwhere\n\\begin{equation}\n\\rho_{\\mathbb{H}}:=\n\\liminf_{r\\to1^-}\\inf_f\\frac{\\int_{\\mathbb{D}(0,r)}\\Phi_f(z)\\frac{\\mathrm{d} A(z)}{1-|z|^2}}\n{\\int_{\\mathbb{D}(0,r)}\\frac{\\mathrm{d} A(z)}{1-|z|^2}}=\n\\liminf_{r\\to1^-}\\inf_f\\frac{\\int_{\\mathbb{D}(0,r)}\\Phi_f(z)\\frac{\\mathrm{d} A(z)}{1-|z|^2}}\n{\\log\\frac{1}{1-r^2}}.\n\\end{equation}\nHere, the infimum runs over all polynomials $f$, and we use the notation\n\\[\n\\Phi_f(z):=\\big((1-|z|^2)|f(z)|-1\\big)^2,\\qquad z\\in\\mathbb{D}.\n\\]\n\\label{thm:Hed}\n\\end{thm}\n\nIvrii's arXiv preprint \\cite{Ivrii} has unfortunately not been published. However, this\ndoes not mean that Ivrii's contribution is flawed. Here we shed light the various aspects\nof Ivrii's argument which we believe are correct or at least may be fixed so that they are\ncorrect.\n\n\\section{Exponential integrability}\n\\subsection{The Littlewood-Paley identity}\n\nWe use a variant of\nthe Littlewood-Paley identity, for $f\\in H^2$, where $H^2$ is the usual Hardy space of\nTaylor series with square summable coefficients:\n\\begin{multline}\n\\int_\\mathbb{T} |f|^2\\mathrm{d} s=|f(0)|^2+\\int_\\mathbb{D}|f'(z)|^2\\log\\frac{1}{|z|^2}\n\\mathrm{d} A(z)\n\\\\\n=\\int_\\mathbb{D}|f(z)|^2\\mathrm{d} A(z)+\\int_\\mathbb{D}|f'(z)|^2(1-|z|^2)\\mathrm{d} A(z).\n\\label{eq:LittlewoodPaley}\n\\end{multline}\nWe apply this identity to $f(z)=\\exp(\\frac12 tg_r(z))$, where $g_r(z)=g(rz)$ is shorthand\nfor the dilated function, with $0<r<1$. \n\\begin{multline}\n\\label{eq:LP2}\n\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s=\\int_\\mathbb{D}|\\mathrm{e}^{t g_r}|\\mathrm{d} A+\n\\frac{|t|^2}{4}\\int_\\mathbb{D}(1-|z|^2)|(g_r)'|^2|\\mathrm{e}^{t g_r}|\n\\,\\mathrm{d} A(z)\n\\\\\n\\le\nr^{-2}\\int_{\\mathbb{D}(0,r)}\\big|\\mathrm{e}^{t g}\\big|\\mathrm{d} A+\\frac{|t|^2}{4}\n\\int_{\\mathbb{D}(0,r)}(1-|z|^2)^2|g'|^2|\\mathrm{e}^{t g}|\\,\n\\frac{\\mathrm{d} A(z)}{1-|z|^2}.\n\\end{multline}\nWe may estimate the first integral on the right-hand side uniformly:\n\\begin{multline*}\nr^{-2}\\int_{\\mathbb{D}(0,r)}|\\mathrm{e}^{t g}|\\mathrm{d} A\\le\n\\int_{\\mathbb{D}}|\\mathrm{e}^{t g}|\\mathrm{d} A\\le \\int_\\mathbb{D}\n\\bigg(\\frac{1+|z|}{1-|z|}\\bigg)^{8\\pi^{-1}|t|}\\mathrm{d} A(z)\n\\\\\n=\\int_0^1\n\\bigg(\\frac{1+r}{1-r}\\bigg)^{8\\pi^{-1}|t|}2r\\mathrm{d} r\\le\n\\frac{4^{8\\pi^{-1}|t|}}{1-8\\pi^{-1}|t|},\n\\end{multline*}\nprovided that $|t|<\\pi\/8$. In particular, we find that\n\\begin{equation*}\nr^{-2}\\int_{\\mathbb{D}(0,r)}\\big|\\mathrm{e}^{t g}\\big|\\mathrm{d} A\\le\n\\int_{\\mathbb{D}}\\big|\\mathrm{e}^{t g}\\big|\\mathrm{d} A\\le 4,\\qquad |t|\\le\\frac{\\pi}{16}.\n\\end{equation*}\nWe implement this estimate into \\eqref{eq:LP2}, and obtain that\n\\begin{equation}\n\\label{eq:LP3}\n\\int_\\mathbb{T} \\big|\\mathrm{e}^{t g_r}\\big|\\mathrm{d} s\\le\n4+\\frac{|t|^2}{4}\n\\int_{\\mathbb{D}(0,r)}(1-|z|^2)^2|g'|^2\\big|\\mathrm{e}^{t g}\\big|\\,\n\\frac{\\mathrm{d} A(z)}{1-|z|^2},\\qquad |t|\\le\\frac{\\pi}{16}.\n\\end{equation}\nBy Per\\\"al\\\"a's estimate \\eqref{eq:Peralalaest0.0}, the function\n\\begin{equation}\nN_g(z):=(1-|z|^2)^2|g'(z)|^2\n\\label{eq:functionN_gdef}\n\\end{equation}\nis uniformly bounded,\n\\begin{equation}\n0\\le N_g\\le \\frac{64}{\\pi^2},\n\\label{eq:Peralaest0.1}\n\\end{equation}\nand the estimate \\eqref{eq:LP3} simplifies:\n\\begin{equation}\n\\label{eq:LP4}\n\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s\\le\n4+\\frac{|t|^2}{4}\n\\int_{\\mathbb{D}(0,r)}N_g|\\mathrm{e}^{t g}|\\,\n\\frac{\\mathrm{d} A(z)}{1-|z|^2},\\qquad |t|\\le\\frac{\\pi}{16}.\n\\end{equation}\nWe proceed to analyze this inequality.\n\n\\subsection{A toy calculation}\nLet us suppose for the moment that $N_g\\le\\eta$ holds for some positive\n$\\eta$. This is of course true for $\\eta=64\\pi^{-2}$, but we are now\ninterested in smaller values of $\\eta$. Under this hypothesis, we see from\n\\eqref{eq:LP4} that\n\\begin{equation}\n\\label{eq:LP5}\n\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s\\le\n4+\\eta\\frac{|t|^2}{4}\n\\int_{\\mathbb{D}(0,r)}|\\mathrm{e}^{t g}|\\,\n\\frac{\\mathrm{d} A(z)}{1-|z|^2},\\qquad |t|\\le\\frac{\\pi}{16}.\n\\end{equation}\nLet us write\n\\[\nI(r,t):=\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{T}(0,r)}:=\\int_\\mathbb{T} |\\mathrm{e}^{t g_r}|\\mathrm{d} s,\n\\]\nand observe that\n\\begin{equation}\n\\label{eq:LP6}\n\\int_{\\mathbb{D}(0,r)}|\\mathrm{e}^{t g}|\\,\n\\frac{\\mathrm{d} A(z)}{1-|z|^2}=\\int_0^r \\frac{2u\\mathrm{d} u}{1-u^2}\n\\int_\\mathbb{T}|\\mathrm{e}^{t g_{u}}|\\mathrm{d} s=\\int_0^r I(u,t)\\,\\frac{2u\\mathrm{d} u}{1-u^2},\n\\end{equation}\nso that \\eqref{eq:LP5} would assert that for $|t|\\le\\frac{\\pi}{16}$, \n\\begin{equation}\n\\label{eq:LP7}\nI(r,t)\\le\n4+\\eta\\frac{|t|^2}{4}\n\\int_0^r I(u,t)\\,\\frac{2u\\mathrm{d} u}{1-u^2}.\n\\end{equation}\nIt is possible to argue (by iteration, or in terms of differential\ninequalities) that the estimate \\eqref{eq:LP7} leads to the upper bound  \n\\begin{equation}\n\\label{eq:Iest0}\nI(r,t)\\le 4\\,(1-r^2)^{-\\eta|t|^2\/4}.\n\\end{equation}\nIf this is true with some fixed $\\eta<1$, say $\\eta=\\frac12$, then we would\nget the desired growth estimate of $I(r,t)$ automatically. To be able to\ndefeat the estimate $N_g\\le\\eta$ the function $g'$ should exhibit some\ngrowth. If we instead want $N_g\\ge\\eta$ to hold on a reasonably fat set,\nthe natural way to achieve the growth is by having zeros in $g'$. \nWe now discuss the rather opposite behavior of the functions $\\mathrm{e}^{tg}$\nand $N_g$, respectively. Here, $\\mathrm{e}^{tg}$ is basically hyperbolically \nlocally constant, while $N_g$ is instead oscillatory.  \n\n\\subsection{The scale associated with $\\mathrm{e}^{tg}$}\nThe function $g$ is in the Bloch space, and hence Lipschitz continuous\nin the hyperbolic metric. Then $tg$ has a Lipschitz constant which\nis $\\mathrm{O}(|t|)$, which makes it asymptotically constant on hyperbolic disks\nof any fixed radius (in fact, it is basically constant over hyperbolic\ndistances that are $\\mathrm{o}(|t|^{-1})$). On the other hand, the function $N_g$\ncan be much more oscillatory. \n\nLet us discretize the argument in the toy calculation, but without making\nany fixed assertion regarding the size of $N_g$. We let $L$ be a large\npositive real, which we think of as fixed, and consider successive radii\n\\[\nr_k=\\frac{1-\\mathrm{e}^{-Lk}}{1+\\mathrm{e}^{-Lk}},\\qquad k=0,1,2,\\ldots,\n\\]\nwhich tend to $1$ quickly. We consider the annuli\n\\[\n\\mathbb{A}_k:=\\mathbb{D}(0,r_k)\\setminus\\bar\\mathbb{D}(0,r_{k-1}),\\qquad k=1,2,3,\\ldots.\n\\]\nThe hyperbolic width of the annulus $\\mathbb{A}_k$ equals\n\\[\n\\int_{r_{k-1}}^{r_k}\\frac{2\\mathrm{d} u}{1-u^2}=\\log\\frac{1+r_k}{1-r_{k}}-\n\\log\\frac{1+r_{k-1}}{1-r_{k-1}}=L,  \n\\]\nwhile the weighted area is\n\\[\n|\\mathbb{A}_k|_{A_{-1}}:=\\int_{\\mathbb{A}_k}\\frac{\\mathrm{d} A(z)}{1-|z|^2}=\\log\n\\frac{1-r_{k-1}^2}{1-r_k^2}=L+\\mathrm{O}(\\mathrm{e}^{-L(k-1)}).\n\\]\nWe introduce the notation\n\\[\n\\langle h\\rangle_{\\mathbb{A}_k}=\n\\frac{\\int_{\\mathbb{A}_k}h\\frac{\\mathrm{d} A}{1-|z|^2}}{\\int_{\\mathbb{A}_k}\n\\frac{\\mathrm{d} A}{1-|z|^2}}=\\frac{1}{|\\mathbb{A}_k|_{A_{-1}}}\n\\int_{\\mathbb{A}_k}h(z)\\frac{\\mathrm{d} A(z)}{1-|z|^2}\n\\]\nfor the weighted average of $h$ on the annulus $\\mathbb{A}_k$. The object that\nwe will focus on are the annular averages\n\\[\n\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k},\n\\]\nfor $t$ close to $0$. This quantity models $I(r_k,t)$. Indeed, since $\\mathrm{e}^{tg}$\nis essentially constant in the radial direction inside $\\mathbb{A}_k$ for small\n$|t|$, the integral is mainly over the angular direction, which gives \nthat\n\\[\n\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}=\\mathrm{e}^{\\mathrm{O}(L|t|)}\nI(r_k,t)=\\mathrm{e}^{\\mathrm{O}(L|t|)}\n\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{T}(0,r_k)}.\n\\]\n\n\\subsection{The scale associated with $N_g$}\nThe function $N_g$ is bounded, but it can be highly oscillatory. It varies\nat much shorter distances. We are interested in maximizing the average quantity\n\\[\n\\langle N_g |\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}.\n\\]\nSince $N_g$ and $|\\mathrm{e}^{tg}|$ vary at different scales, it is tempting to think\nlike a probabilist \nand suggest that we have independent stochastic variables, so that \n\\[\n\\langle N_g |\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}\\sim\n\\langle N_g \\rangle_{\\mathbb{A}_k}\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}.\n\\]\nLet us carry on and see where this line of thinking would lead us.  \nWe obtain from \\eqref{eq:LP4} and the assumed independence that\n\\begin{multline*}\n\\mathrm{e}^{\\mathrm{O}(L|t|)}\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}\\le 4+\\frac{|t|^2}{4}\n\\int_{\\mathbb{D}(0,r_k)}N_g|\\mathrm{e}^{tg}|\\frac{\\mathrm{d} A}{1-|z|^2}\n\\\\\n\\sim\n4+\\frac{|t|^2}{4}\\sum_{j=1}^{k}\n|\\mathbb{A}_j|_{A_{-1}}\\langle N_g\\rangle_{\\mathbb{A}_j}\\langle |\\mathrm{e}^{tg}|\n\\rangle_{\\mathbb{A}_j}.\n\\end{multline*}\nMoreover, let us suppose that for big $k$,\n\\begin{equation}\n\\langle N_g\\rangle_{\\mathbb{A}_j}\\le \\sigma_g(L)^2,\n\\label{eq:asymptvar99}\n\\end{equation}\nwhere $\\sigma_g(L)^2=\\sigma_g^2+\\mathrm{o}(1)$ as $L\\to+\\infty$.  The nonnegative\nparameter $\\sigma_g^2$ would express how big $N_g$ can be, on average, in\na rather precise sense. Then\nwe would have that\n\\begin{multline*}\n\\mathrm{e}^{\\mathrm{O}(L|t|)}\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}\\lesssim\n4+\\frac{|t|^2}{4}\\sum_{j=1}^{k}\n|\\mathbb{A}_j|_{A_{-1}}\\langle N_g\\rangle_{\\mathbb{A}_j}\\langle |\\mathrm{e}^{tg}|\n\\rangle_{\\mathbb{A}_j}\n\\\\\n\\le\nM+\\frac{|t|^2}{4}\\sigma_g(L)^2\\sum_{j=1}^k|\\mathbb{A}_j|_{A_{-1}}\n\\langle |\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_j}.\n\\end{multline*}\nMoreover, iteration would give an estimate of the type\n\\begin{multline}\n\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}=\\mathrm{O}\\bigg(\\exp\\bigg(\n\\mathrm{e}^{\\mathrm{O}(L|t|)}\\frac{|t|^2\\sigma_g(L)^2Lk}{4}\\bigg)\\bigg)\n\\\\\n=\\mathrm{O}\\bigg(\\frac{1}{(1-r_k^2)^{\\mathrm{e}^{\\mathrm{O}(L|t|)}|t|^2\\sigma_g(L)^2\/4}}\\bigg).\n\\label{eq:iteration0.99}\n\\end{multline}\nThis estimate along the radii $r_k$ may be extended (with minor losses) \nto all radii $r$ if we take into account the hyperbolic Lipschitz property\nwith small Lipschitz constant.\n\n\\section{Grid of boxes}\n\n\\subsection{Boxes of hyperbolic size $\\asymp L$}\nIt is actually not so difficult to make the above reasoning more rigorous.\nWe split the annulus $\\mathbb{A}_k$ into into equal ``boxes'', by cutting\nin the radial direction. Each box $Q_{k,l}$ is of the type\n$\\theta_{l-1}\\le\\theta\\le\\theta_l$\nand $r_{k-1}\\le r\\le r_{k}$, so that $\\theta_l-\\theta_{l-1}$ is independent of\n$l$. We will need each box to be of the angular width $\\mathrm{O}(L\\mathrm{e}^{-Lk})$, so\nthere should be on the order of magnitude $\\asymp L^{-1}\\mathrm{e}^{Lk}$ boxes.\nIn each box $Q=Q_{k,l}$ we would try to maximize the average\n$\\langle N_g\\rangle_Q$. In each such box $Q$, $|e^{tg}|$ is roughly constant,\nso that\n\\begin{equation}\n\\langle N_g|\\mathrm{e}^{tg}|\\rangle_Q\\sim \\langle N_g\\rangle_{Q}\\langle |e^{tg}|\\rangle_Q\n\\label{eq:indepSV}\n\\end{equation}\nholds. \nAll we need now is to have an estimate of the form\n\\begin{equation}\n\\langle N_g\\rangle_{Q}\\le \\sigma_g(L)^2,\n\\label{eq:localN_g}\n\\end{equation}\nwith $\\sigma_g(L)^2\\le \\Sigma^2+\\mathrm{o}(1)$, where $\\Sigma^2<1$ is independent of\n$g$. Indeed,  then we obtain from \\eqref{eq:indepSV} that\n\\begin{multline}\n\\langle N_g|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}=\\frac{1}{\\#(Q)}\n\\sum_{Q}\\langle N_g|\\mathrm{e}^{tg}|\\rangle_{Q}\n\\\\\n\\le \\mathrm{e}^{\\mathrm{O}(L|t|)}\\sigma_g(L)^2\\sum_Q\n\\langle N_g|\\mathrm{e}^{tg}|\\rangle_{Q}=\\mathrm{e}^{\\mathrm{O}(L|t|)}\\sigma_g(L)^2\n\\langle|\\mathrm{e}^{tg}|\\rangle_{\\mathbb{A}_k}.\n\\end{multline}\nThis would permit us to apply the iteration argument which leads to\n\\eqref{eq:iteration0.99}, as desired. But what kind of argument would be\nable to give us \\eqref{eq:localN_g}?\n\n\\subsection{Localization of $\\mu$ to boxes}\nHere, we recall the formula \\eqref{eq:functionN_gdef}, and split\n$\\mu=\\mu_Q+\\mu_{Q^c}$, where $\\mu_Q=1_Q\\mu$ and $\\mu_{Q^c}=1_{\\mathbb{D}\\setminus Q}\\mu$.\nThen $g=\\mathbf{P}\\mu=g_Q+g_{Q^c}$, where $g_Q=\\mathbf{P}\\mu_Q$ and $g_{Q^c}=\\mathbf{P}\\mu_{Q^c}$.\nWe estimate\n\\begin{equation*}\n(1-|z|^2)|(\\mathbf{P} \\mu_{Q^c})'(z)|\\le 2(1-|z|^2)\n\\int_{\\mathbb{D}\\setminus Q}\\frac{|w|}{|1-z\\bar w|^3}\\,\\mathrm{d} A(w)=\\mathrm{O}(L\\,\\mathrm{e}^{-L\/2})\n\\end{equation*}\nfor $z\\in\\frac12Q$, if we invoke Lemma \\ref{lem:logestimate}, assuming that the\ncorresponding hyperbolic distance between $\\frac12Q$ and $\\mathbb{D}\\setminus Q$ is at least $L\/2$.\nThis means that for the size of $N_g$ in $\\frac12Q$, only $\\mu_Q$ matters.\nHere, it is actually better for us to work with bigger subboxes $(1-\\epsilon)Q$ in place\nof $\\frac12Q$, which we can if $L$ is assumed big enough so that $\\epsilon L$ is also big.\nHere, we think of $(1-\\epsilon)Q$ as a correspondingly smaller box with the same mass \ncenter point and all other sizes shrunk by the factor $1-\\epsilon$. This procedure of going from the\nbox $Q$ to the slighly smaller box $(1-\\epsilon)Q$ is explained in some detail\nin the subsection below.\nThe relevance of $\\epsilon L$ is that it should measure the\nhyperbolic distance between $(1-\\epsilon)Q$ and $\\mathbb{D}\\setminus Q$, and we need $\\epsilon L$ big to\nbe able to ensure localization in terms of Lemma \\ref{lem:logestimate}.\n\nWe aim to show that $\\langle N_g\\rangle_{(1-\\epsilon)Q}\\le \\sigma_g(L)^2$ holds, with\n$\\sigma_g(L)^2=\\sigma_g^2+\\mathrm{o}(1)$, and $\\sigma_g^2\\le\\Sigma^2$.\nAs a first step, we find the box $Q=Q_{k,l}$ for fixed $k$ such that\n$\\langle N_g\\rangle_{(1-\\epsilon)Q}$ is maximal (we maximize over $l$), say\n$\\langle N_g\\rangle_{(1-\\epsilon)Q}=\\alpha$.\nThen for that $Q=Q_{\\text{max}}$, $\\mu_Q$ must a rather optimal, and we may\nuse a periodization procedure to replace the original $\\mu$ in the entire\nannulus by copies of $\\mu_{Q_\\text{max}}$ so as to make\n$\\langle N_g\\rangle_{(1-\\epsilon)Q}\\approx\\alpha$ for all the other boxes\nin the annulus. This would make\n$\\langle N_g\\rangle_{\\mathbb{A}_k}\\approx\\alpha+\\mathrm{O}(\\epsilon)$.\nWe would now like to work with optimization along all the annuli at the\nsame time. In fact, is better to have scale invariance as well, which\nis only approximately true for the disk. The point is that if we have found\nan optimal box $Q=Q_\\textrm{max}$, we would like to use $\\mu_Q$ to replace\n$\\mu$ along all the other annuli as well, by just scaling the optimal\n$\\mu_Q$ (like we did for rotations before). \nNeglecting the technicalities, we should then have\n$\\langle N_g\\rangle_{\\mathbb{A}_j}\\approx \\alpha+\\mathrm{O}(\\epsilon)$ for all\nbig $j$. We should think of the correct value of $\\alpha$ to appear in the limit\nas we zoom in at the circle $\\mathbb{T}$, when the circle begins to look like a straight line.\n\n\\subsection{Summing over boxes (or annuli)}\nIt remains to find out how big can be this optimal value $\\alpha$.\nBy summing over successive annuli we obtain\n\\[\n\\int_{\\mathbb{D}(0,r_k)}|g'(z)|^2(1-|z|^2)\\mathrm{d} A=\\sum_{j=1}^{k}|\\mathbb{A}_j|_{A_{-1}}\n\\langle N_g\\rangle_{\\mathbb{A}_j}\\approx(\\alpha+\\mathrm{O}(\\epsilon))(Lk+\\mathrm{O}(1)),\n\\]\nso that\n\\[\n\\frac{\\int_{\\mathbb{D}(0,r_k)}|g'(z)|^2(1-|z|^2)\\mathrm{d} A}\n{\\int_{\\mathbb{D}(0,r_k)}(1-|z|^2)^{-1}\\mathrm{d} A(z)}\\approx\\alpha+\\mathrm{O}(\\epsilon).\n\\]\nThe left-hand side expression equals the\naverage of $N_g$ on the disk $\\mathbb{D}(0,r_k)$. That average would be trivially\ndominated by the maximal local average $\\langle N_g\\rangle_Q$, but our argument\n(which needs to be made more technically sophisticated) gives that the\nlocal maximal average over big boxes is the same as the maximal average over\nthe disks $\\mathbb{D}(0,r_k)$. \nBy the Littlewood-Paley identity \\eqref{eq:LittlewoodPaley},\n\\[\n\\int_{\\mathbb{D}(0,r)}|g'(z)|^2(1-|z|^2)\\mathrm{d} A=r\\int_{\\mathbb{T}(0,r)}|g|^2\\mathrm{d} s\n-\\int_{\\mathbb{D}(0,r)}|g|^2\\mathrm{d} A\n\\]\nit follows that the optimal value of $\\alpha$ equals\nMcMullen's \\emph{asymptotic variance}: \n\\[\n\\sigma_g^2:=\\limsup_{r\\to1^-}\\frac{\\int_\\mathbb{T}|g_r|^2\\mathrm{d} s}{\\log\\frac{1}{1-r^2}}.\n\\]\nMoreover, by Theorem \\ref{thm:Hed},\n\\[\n\\sup_{g}\\sigma_g^2=\\Sigma^2=1-\\rho_{\\mathbb{H}}<1,\n\\]\nwhere the supremum runs over all $g=\\mathbf{P}\\mu$ with $\\|\\mu\\|_{L^\\infty(\\mathbb{D})}$.\n\n\\subsection{Wrapping up the proof} \nWe return to the various boxes $Q$. We need to keep $L$ big, $\\epsilon$ small, but\n$\\epsilon L$ big too. Then the averages $\\langle N_g\\rangle_{(1-\\epsilon)Q}$\napproximate well $\\langle N_g\\rangle_Q$, given that $N_g$ enjoys the bound\n\\eqref{eq:Peralaest0.1}. \n\nThe careful analysis of the boxes would require a better way to represent the disk which\nis stable under dilatation, not just rotation. We may model $\\mathbb{D}$ in terms of \nlogarithmic coordinates. We write $z=\\mathrm{e}^{\\mathrm{i}\\pi\\zeta}$, where $\\mathrm{Im}\\, \\zeta>0$ and\n$\\mathrm{Re}\\, \\zeta\\in\\mathbb{R}\/2\\mathbb{Z}$. The boxes are now just rectangles that form a grid. The Bergman \nprojection may be expressed in these coordinates, if  $\\mathcal{D}$ is the \nfundamental domain in the upper half-plane with\nreal part between $-1$ and $1$:\n\\[\n\\mathbf{P} \\mu (\\mathrm{e}^{\\mathrm{i}\\pi\\zeta})=\\int_{\\mathcal{D}}\n\\frac{1}{(1-\\mathrm{e}^{\\mathrm{i}\\pi(\\zeta-\\bar\\xi)})^2}\\mu(\\mathrm{e}^{\\mathrm{i}\\pi\\xi})\\,\\mathrm{e}^{-2\\pi\\mathrm{Im} \\xi}\\mathrm{d} A(\\xi).\n\\]\nSimilarly, we find that\n\\begin{multline}\n(1-|z|^2)(\\mathbf{P}\\mu)'(z)\\big|_{z:=\\mathrm{e}^{\\mathrm{i}\\pi\\zeta}}\n\\\\=\n2(1-\\mathrm{e}^{-2\\pi\\mathrm{Im}\\zeta})\\int_{\\mathcal{D}}\n\\frac{\\mathrm{e}^{-\\mathrm{i}\\pi\\bar\\xi}}{(1-\\mathrm{e}^{\\mathrm{i}\\pi(\\zeta-\\bar\\xi)})^3}\\,\n\\mu(\\mathrm{e}^{\\mathrm{i}\\pi\\xi})\\,\\mathrm{e}^{-2\\pi\\mathrm{Im} \\xi}\\mathrm{d} A(\\xi),\n\\label{eq:Ngbasic}\n\\end{multline}\nand we observe that for $\\zeta,\\xi\\in\\mathcal{D}$, the formula\n\\begin{multline}\n\\label{eq:singres0}\n\\frac{1}{(\\mathrm{e}^{\\mathrm{i}\\pi(\\zeta-\\bar\\xi)}-1)^3}=\n\\\\\n\\sum_{j=-1}^1\\bigg\\{(\\mathrm{i}\\pi(\\zeta-\\bar\\xi+2j))^{-3}-\n\\frac32((\\mathrm{i}\\pi(\\zeta-\\bar\\xi+2j))^{-2}+(\\mathrm{i}\\pi(\\zeta-\\bar\\xi+2j))^{-1}\\bigg\\}+\\mathrm{O}(1)\n\\end{multline}\nallows us to control the singularity. The sum over $j\\in\\{-1,0,1\\}$ is due to the fact that the\nleft-hand side expression is $2$-periodic in the variable $\\zeta-\\bar\\xi$. But we may as well \nfocus our attention to the\ncontribution with $j=0$, and observe that the contributions associated with \nthe second a third term on the right-hand side of \n\\eqref{eq:singres0} are negligible, i.e. correspond to asymptotically vanishing contributions\nin \\eqref{eq:Ngbasic} as $\\mathrm{Im}\\,\\zeta\\to0$ (i.e., as $|z|\\to1$). Similarly,\n\\[\n\\mathrm{e}^{-\\mathrm{i}\\pi\\bar\\xi}\\,\\mathrm{e}^{-2\\pi\\mathrm{Im}\\, \\xi}=1-\\mathrm{i}\\pi\\bar\\xi -2\\pi\\mathrm{Im}\\,\\xi+\\mathrm{O}(|\\xi|^2),\n\\]\nwhere only the constant $1$ makes a significant contribution. For $\\zeta\\in\\mathcal{D}$ with\n$|\\zeta|\\le\\frac12$, we see from\n\\eqref{eq:Ngbasic} that\n\\begin{multline}\n(1-|z|^2)(\\mathbf{P}\\mu)'(z)\\big|_{z:=\\mathrm{e}^{\\mathrm{i}\\pi\\zeta}}\n\\\\=\n-4\\pi(\\mathrm{Im}\\,\\zeta+\\mathrm{O}(\\mathrm{Im}\\,\\zeta)^2)\\int_{\\mathcal{D}\\cap\\mathbb{D}}\n\\frac{1}{{(\\mathrm{i}\\pi(\\zeta-\\bar\\xi)})^3}\\,\n\\mu(\\mathrm{e}^{\\mathrm{i}\\pi\\xi})\\,\\mathrm{d} A(\\xi)+\\mathrm{o}(1),\n\\label{eq:Ngbasic2}\n\\end{multline}\nas $\\mathrm{Im}\\,\\zeta\\to0$.\nEffectively this reduces to the study of the the derivative of the Bergman projection on \nthe upper half-plane $\\mathbb{H}$, since the main kernel in \\eqref{eq:Ngbasic2} appears\nthis way. Moreover, in $\\mathbb{H}$, we have access to both translation and dilatation \ninvariance. This is key to making the argument with boxes effective.\n\nSome clarifying words should perhaps be added regarding the boxes $(1-\\epsilon)Q$ compared\nwith $Q$. More precisely, how should the size reduction be made? It is easier to explain these\nmatters in the context of the upper half-plane. We work with the hyperbolic metric \n$\\mathrm{d} s_{\\mathbb{H}}(\\zeta)=(\\mathrm{Im} \\zeta)^{-1}|\\mathrm{d} \\zeta|$ and weighted area element \n$\\mathrm{d} A_{-1}(\\zeta)=(\\mathrm{Im}\\zeta)^{-1}\\mathrm{d} A(\\zeta)$. A prototypical box $Q$ of size $L$ would look like \nthis:\n\\[\nQ=\\big\\{\\zeta\\in\\mathbb{H}: \\,0<\\mathrm{Re}\\,\\zeta<L,\\,\\,\\mathrm{e}^{-L}<\\mathrm{Im}\\,\\zeta<1\\big\\},\n\\]\nOther boxes could be a horizontal translate of this one, or a dilate of such a translate. The weighted\narea of $Q$ is\n\\[\n|Q|_{A_{-1}}=\\int_Q\\frac{\\mathrm{d} A(\\zeta)}{\\mathrm{Im}\\zeta}=\\pi^{-1}L^2.\n\\]\nWe choose $(1-\\epsilon)Q$ to be the smaller box\n\\[\n(1-\\epsilon)Q=\\big\\{\\zeta\\in\\mathbb{H}: \\,\\epsilon L<\\mathrm{Re}\\,\\zeta<(1-\\epsilon)L,\\,\\,\n\\mathrm{e}^{-(1-\\epsilon)L}<\\mathrm{Im}\\,\\zeta<\\mathrm{e}^{-\\epsilon L}\\big\\}.\n\\]\nThen the weighted area of the box $(1-\\epsilon)Q$ is given by\n\\[\n|(1-\\epsilon)Q|_{A_{-1}}=\\int_{(1-\\epsilon)Q}\\frac{\\mathrm{d} A(\\zeta)}{\\mathrm{Im}\\zeta}=\\pi^{-1}(1-2\\epsilon)^2L^2.\n\\]\nMoreover, the hyperbolic distance from any point of $(1-\\epsilon)Q$ to any point of the complement\n$\\mathbb{H}\\setminus Q$ is at least $\\ge\\epsilon L$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe large-scale halo of hot gas  provides\na unique way to measure the baryonic and gravitational\nmass of galaxy clusters.\nThe baryonic mass can be measured directly from\nthe observation of the hot X-ray emitting intra-cluster medium (ICM), and \nof the associated stellar component \\citep[e.g.][]{giodini2009,gonzales2007},\nwhile measurements of the gravitational mass require\nthe assumption of hydrostatic equilibrium between the gas and\ndark matter. \nCluster cores are subject to a variety of non-gravitational\nheating and cooling processes that may result in deviations\nfrom hydrostatic equilibrium, and in inner regions beyond the\ncore the ICM is expected to be in hydrostatic equilibrium\nwith the dark matter potential. At the\noutskirts, the low-density ICM and the proximity to the\nsources of accretion results in the onset\nof new physical processes\nsuch as  departure from hydrostatic equilibrium \\citep[e.g.,][]{lau2009},\nclumping of the gas \\citep{simionescu2011},\ndifferent temperature between electrons and ions \\citep[e.g.,][]{akamatsu2011},\nand flattening of the entropy profile \\citep{sato2012}, leading\nto possible sources of systematic uncertainties in the measurement of masses.\n\n\nThe detection of hot gas at large radii is limited\nprimarily by its intrinsic low surface brightness,  uncertainties\nassociated with the subtraction of background (and foreground) emission,\nand the ability to remove contamination from\ncompact sources unrelated to the cluster.\nThanks to its low detector background, \\it Suzaku\\rm\\ reported the\nmeasurement of ICM temperatures  to $r_{200}$\\ and beyond\nfor a few nearby clusters\n\\citep[e.g.][]{akamatsu2011,walker2012a,walker2012b,simionescu2011,burns2010,kawaharada2010,\nbautz2009,george2009}; to date \\emph{Abell~1835}\\\nhas not  been the target of a \\it Suzaku\\rm\\ observation.\n\nIn this paper we report the \\it Chandra\\rm\\ detection of X-ray emission\n  in \\emph{Abell~1835}\\ beyond $r_{200}$, using three observations \nfor a total of 193~ksec exposure time, extending the analysis\nof these \\it Chandra\\rm\\ data performed by \\cite{sanders2010}.\nThe radius $r_{\\Delta}$ is defined as the radius within which\nthe average mass density is $\\Delta$ times the critical density of\nthe universe at the cluster's redshift for our choice of \ncosmological parameters. The virial radius of a cluster\nis defined as the equilibrium radius of the collapsed\nhalo, approximately equivalent to one half of its turnaround radius \n\\cite[e.g.][]{lacey1993, eke1998}.\nFor an $\\Omega_{\\Lambda}$-dominated universe,\nthe virial radius is approximately $r_{100}$ \\citep[e.g.][]{eke1998}.\n\\emph{Abell~1835}\\ is the most luminous cluster in the \n\\cite{dahle2006} sample of clusters at $z=0.15-0.3$ selected\nfrom the \\emph{Bright Cluster Survey}.\nThe combination of high luminosity and availability of\n deep \\it Chandra\\rm\\ observations with local background make \\emph{Abell~1835}\\\nand ideal candidate to study its emission to the virial radius. \n\\emph{Abell~1835}\\ has a redshift of $z=0.253$,\nwhich for $H_{0}=70.2$~km~s$^{-1}$~Mpc$^{-1}$, $\\Omega_{\\Lambda}=0.73$,\n$\\Omega_M=0.27$ cosmology \\citep{komatsu2011} corresponds\nto an angular-size distance of $D_A=816.3$~Mpc, and a scale of \n237.48 kpc per arcmin.\n\n\n\\section{Chandra and ROSAT observations of Abell~1835 and the detection\nof cluster emission beyond $r_{200}$}\n\\label{sec:Sx}\n\\subsection{Chandra observations} \n\\it Chandra\\rm\\ observed \\emph{Abell~1835}\\ three times between December 2005 and August 2006\n(observations ID 6880, 6881 and 7370), with a combined clean exposure time\nof 193~ks. The three observations had similar aimpoint towards the\ncenter of the cluster (R.A. 14h01m02s, Dec. +02d51.5m J2000) and different\nroll angles. All observations were taken with the ACIS-I detector configuration,\nwhich consists of four ACIS front-illuminated chips in a two-by-two square,\nplus a fifth identical chip that may be used to measure \nthe \\emph{in situ} soft X-ray background.\nFigure~\\ref{fig:a1835} is an image from the longest observation (ID 6880, 118ks)\nin the soft X-ray band (0.7-2 keV). In addition to a large number of \ncompact X-ray sources that were excluded from further analysis, the data\nshow a clear detection of diffuse X-ray emission associated with two\nadditional low-mass clusters identified from the \\emph{Sloan Digital Sky Survey}, \nMAXBCG J210.31728+02.75364 and WHL J140031.8+025443.\nThe cluster MAXBCG J210.31728+02.75364 is the only cluster in the vicinity of \\emph{Abell~1835}\\\nreported in the MAXBCG catalog of \\cite{koester2007}, and it has\na measured  photo-$z$\nof 0.238, while the catalog of \\cite{wen2009} reports a photo-$z$\nof 0.269 for the same source; given the uncertainties associated\nwith photometric redshifts, it is likely that the cluster is\nin physical association with \\emph{Abell~1835}\\ ($z=0.253$).\nThe \\cite{wen2009} catalog also reports another optically-identified cluster \nin the area, WHL J140031.8+025443, with a spectroscopic redshift of $z=0.2505$.\nThe association of these two groups with \\emph{Abell~1835}\\ is confirmed by redshift\ndata provided by C. Haines (personal communication), who measures\na redshift of $z=0.250$ for WHL J140031.8+025443, and $z=0.245$ for MAXBCG J210.31728+02.75364.\n\nSince the goal of this paper is to study the\ndiffuse emission associated with \\emph{Abell~1835}, we excise a region of radius 90~arcsec\naround the position of the two clusters (black circles in \nFigure~\\ref{fig:a1835}), and study their emission separately from that of \\emph{Abell~1835}\\\n(see Section~\\ref{sec:low-mass-clusters}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.1in,angle=-90]{a1835Chandra.ps}\n\\caption{Image of \\emph{Abell~1835}\\ from observation 6880, in the 0.7-2 keV band.\nThe data were smoothed with a Gaussian kernel of $\\sim$6 arcsec standard error.\nThe dashed circles correspond to radial distances of approximately $r_{500}$\\ and $r_{200}$,\nand the full black circles mark the position of the two low-mass clusters associated with\n\\emph{Abell~1835}.}\n\\label{fig:a1835}\n\\end{figure}\n\n\n\\subsection{Chandra data analysis}\nThe reduction of the \\it Chandra\\rm\\ observations follow the procedure\ndescribed in \\cite{bonamente2006} and \\cite{bonamente2011}, which consists of filtering the\nobservations for possible periods of flaring background,\nand applying the latest calibration; no significant flares\nwere present in these observations. The reduction was\nperformed in CIAO~4.2, using CALDB 4.3; in Sec.~\\ref{sec:robustness} we discuss the impact of calibration\nchanges on our results. One of the calibration issues\nthat can affect the measurement of cluster emission is the uncertainty in\nthe contamination of the optical blocking filter, which causes\na reduction in the low energy quantum efficiency of the \\it Chandra\\rm\\ detectors.\nThe spatial and time dependence of this contaminant affects primarily\nthe effective area at $\\leq$0.7 keV~\\footnote{See \\cite{marshall2004} and \n\\it Chandra\\rm\\  calibration memos at cxc.harvard.edu.}, with an estimated residual\nerror of $\\leq$ 3\\% at higher energy. We therefore limit our spatial and spectral \nanalysis to the $\\geq$0.7~keV band.\nThe superior angular resolution of the \\it Chandra\\rm\\ mirrors \\citep{weisskopf2000}\nresults in a point-spread function with a 0.5~arcsec FWHM, and therefore \nthere is negligible contribution from the bright cluster core to \nthe emission in the outer annuli, and from secondary scatter (stray light) by sources outside\nthe field of view.\n\nThe subtraction of particle and sky background is one of the\nmost crucial aspects of the analysis of low surface brightness cluster regions.\nWe use \\it Chandra\\rm\\ blank-sky background observations,\nrescaled according to the high-\nenergy flux of the cluster, to ensure a correct subtraction\nof the particle background that is dominant at $E\\geq9.5$~keV, \nwhere the Chandra detectors have no effective area.\nThe temporal and spatial variability of the soft X-ray background at $E<2$~keV\nalso requires that a peripheral region free of cluster emission\nis used to measure any local enhancement (or deficit) of soft X-ray\nemission relative to that of the blank-sky fields, and account for this\ndifference in the analysis. This method \nis accurate for the determination of the temperature profile, but may result\nin small errors in the measurement of the surface brightness profile.\nIn fact, the blank-sky background is a combination of a particle component \nthat is not vignetted, and a sky component that is vignetted. To determine the\nsurface brightness of the cluster and of the local soft X-ray background,\na more accurate procedure consists of subtracting the non-vignetted\nparticle component as measured from \\it Chandra\\rm\\ observations in which\nthe ACIS detector was stowed \\cite[e.g.,][]{hickox2007}, after rescaling the\nstowed background to match the $E\\geq9.5$~keV cluster count rate, as in \nthe case of the blank-sky background.\n\nPoint sources are identified and removed using a wavelet detection method\nthat correlates the cluster observation with wavelet functions of \ndifferent scale sizes (\\emph{wavdetect} in CIAO). Subtraction\nof point soures from the blank-sky observations were \nperformed by eye, with results that closely match those\nof the wavelet method.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in, angle=-90]{Sx_stow_07-2.ps}\n\\caption{Exposure corrected surface brightness profile of \\emph{Abell~1835}\\ in the soft X-ray band (0.7-2 keV),\nobtained by subtraction of the particle background from the ACIS stowed\nobservations. The radii $r_{500}$, $r_{200}$\\ and the virial radius ($\\sim r_{100}$) are \nestimated from the data in Section~\\ref{sec:r500} (see Table~\\ref{tab:vikh-masses}).\nThe dashed red line is the average background level in the region $\\geq$ 700 arcsec.}\n\\label{fig:Sx}\n\\end{figure}\n\n\\subsection{Measurement of the surface brightness profile with Chandra}\nThe surface brightness profile obtained using this background subtraction\nis shown in Figure~\\ref{fig:Sx}, in which the red line represents\nthe average value of the background at radii $\\geq$~700 arcsec, where\nthe surface brightness profile is consistent with a constant level.\nTo determine the outer radius at which \\it Chandra\\rm\\ has a significant detection\nof the cluster, we also include sources of systematic errors in our analysis.\nOne source of uncertainty is the error in the measurement of the background level,\nshown in Figure~\\ref{fig:Sx-closeup} as the solid red lines.\nThe error is given by the standard deviation of the weighted mean of the datapoints\nat radii greater than 700~arcsec,\n to illustrate that\neach bin in the surface brightness profile beyond this radius\nis consistent with a constant level of the background. \n\nAnother source of uncertainty is the amount by which the stowed background is to\nbe rescaled to match the cluster count rate at high energy. The stowed background\ndataset applicable to the dates of observation of \\emph{Abell~1835}\\ has an exposure\ntime of 367~ksec, and the relative error in the rescaling of the background to match\nthe cluster count rate at high energy is 0.7\\%, as determined by the Poisson\nerror in the photon counts at high energy. \nMoreover, \\cite{hickox2006}\nhas shown that the spectral distribution of the particle background is remarkably stable,\neven in the presence of changes in the overall flux, and that \nthe ratio of soft-to-hard (2-7 keV to 9.5-12 keV) count rates remains constant to within $\\leq$2~\\%. We therefore\napply a systematic error of 2~\\% in the stowed background flux, to account for this\npossible source of uncertainty, in addition to the 0.7\\% error\ndue to the uncertainty in the rescaling of the background. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in, angle=-90]{Sx_stow_07-2_closeup.ps}\n\\caption{Close-up view of Figure~\\ref{fig:Sx}, in which the red lines represent the\n1-$\\sigma$ confidence in the background level as determined from the $\\geq$700~arcsec region,\nand the green error bars combine the statistical and systematic errors in the determination\nof the surface brightness.\n}\n\\label{fig:Sx-closeup}\n\\end{figure}\n\n\nIn Figure~\\ref{fig:Sx-closeup} the green error\nbars represent the cumulative effect of the statistical error due to the counting\nstatistic,  and the\nsources of errors associated with the use of the stowed background; the systematic\nerrors were added linearly to the statistical error as a conservative measure.\nThis error analysis shows that the emission from \\emph{Abell~1835}\\ remains significantly \nabove the background beyond $r_{200}$\\ and\nuntil approximately a radius of 600 arcsec, or approximately 2.4~Mpc. \nThe significance of the detection in the region 450-600\" (the\nfive datapoints in Figure~\\ref{fig:Sx-closeup} after the $r_{200}$ marker)\nis calculated as 5.5$\\sigma$, and is obtained by using the larger systematic\nerror bars for the surface brightness profile (in green in Figure~\\ref{fig:Sx-closeup}),\nadded in quadrature to the error in the determination of the background level\nfrom the $\\geq 700$\" region (red lines in Figure~\\ref{fig:Sx-closeup}).\n\nTo further test the effect of the background subtraction, we repeat our\nbackround subtraction process using the $\\geq 600$\" region \n(instead of the $\\geq 700$\" region) . The background level\nincreases by less than 1$\\sigma$ of the value previously determined (e.g., the\ntwo levels are statistically indistinguishable), and the significance of detection in\nthe region 450-600\" is 4.7$\\sigma$. Therefore we conclude that it is unlikely that\nthe excess of emission beyond $r_{200}$\\ and out to the virial radius is due to errors in the background\nsubtraction process.\nA similar result can be obtained including the 2-7 keV band,\nbut the signal-to-noise is reduced because at large radii this band is \ndominated by the background due to the softening of the cluster emission. \nWe estimate $r_{200}$\\ and the virial radius ($\\sim r_{100}$) from\nthe \\it Chandra\\rm\\ data in Section~\\ref{sec:r500}.\n\n\\subsection{Measurement of the\nsurface brightness profile with the ROSAT Position Sensitive Proportional Counter}\n\n\\it ROSAT\\rm\\ observed \\emph{Abell~1835}\\ on July 3--4 2003 for 6~ks with the Position\nSensitive Proportional Counter (PSPC), observation ID was 800569.\nThe PSPC has a 99.9\\% rejection of particle background in the 0.2-2~keV band\n\\citep{plucinsky1993} and an average angular resolution of $\\sim$30~arcsec that makes it \nvery suitable for observations of low surface brightness objects such as\nthe outskirts of galaxy clusters \\citep[e.g.][]{bonamente2001,bonamente2002,bonamente2003}.\nWe reduce the event file following the procedure described in  \\cite{snowden1994}\nand \\cite{bonamente2002}, which consists of corrections for detector gain fluctuations, and\nremoval of periods with a \\emph{master veto} rate of $\\leq$170 counts~s$^{-1}$ in order\nto discard periods of high background. These filters result in a clean exposure time of\n5.9~ks.\n\nSince the PSPC background is given only by the photon background, we generate an\nimage in the 0.2-2 keV band and use the exposure map to correct for the position--dependent \nvariations in the detector response and mirror vignetting.\nWe masked out the two low-mass cluster regions as we did for the \\it Chandra\\rm\\ data and \nall visible point sources, and obtained and exposure-corrected surface brightness profile\nout to a radial distance of $\\sim$20 arcmin, which corresponds to the location of the\ninner support structure of the PSPC detector. The \\it ROSAT\\rm\\ surface brightness profile therefore\ncovers the entire azimuthal range.\nIn Figure~\\ref{fig:rosat} we  show the radial profile of the surface brightness in the 0.2-2 keV\nband, showing a $\\sim$2~$\\sigma$ excess of emission in the 400-600\" region using the background\nlevel calculated from the region $\\geq$700\", as done for the \\it Chandra\\rm\\ data.\nThe \\it ROSAT\\rm\\ data therefore provide additional evidence of emission beyond $r_{500}$\\ and out to the\nvirial radius, although the short \\it ROSAT\\rm\\ exposure does not have sufficient number of counts\nto provide a detection with the same significance as in the \\it Chandra\\rm\\ data.\n\n\\begin{figure}\n\\includegraphics[width=2.4in,angle=-90]{.\/Sx-rosat.ps}\n\\caption{Surface brightness profile in 0.2-2 keV band from a 6~ks\nobservation with ROSAT PSPC. The background level is determined from \nthe data at radii $\\geq$~700\", as in the \\it Chandra\\rm\\ data.}\n\\label{fig:rosat}\n\\centering\n\\end{figure}\n\n\n\\section{Analysis of the Chandra spectra}\n\\label{sec:kT}\n\\subsection{Measurement of the temperature profile of Abell~1835}\n\\label{sec:spectral-fits}\nWe measure the temperature profile of \\emph{Abell~1835}\\ following\nthe background subtraction method described in Sec.~\\ref{sec:Sx},\nwhich makes use of the blank-sky background dataset and a\nmeasurement of\nthe local enhancement of the soft X-ray background, as\nis commonly done for \\it Chandra\\rm\\ data \\citep[e.g.][]{vikhlinin2006, maughan2008, bulbul2010}.\nIn Figure~\\ref{fig:soft-back} we show the spectral distribution\nof the local soft X-ray background enhancement, as determined from\na region beyond the virial radius ($\\geq$700~arcsec);\nthis emission was modelled with an APEC emission\nmodel of $kT\\sim 0.25$~KeV and of Solar abundance, consistent with\nGalactic emission, and then subtracted from all\nspectra. The spectra were fit in the 0.7-7 keV band using the \nminimum $\\chi^2$ statistic, after binning to ensure that there are\nat least 25 counts per bin. We use XSPEC version 12.6.0s\nfor the spectral analysys.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.3in,angle=-90]{backSpectrum.ps}\n\\caption{Spectrum of the local enhancement of the soft X-ray\nbackground from observation 6880. The other two exposures have\nsimilar levels of soft X-ray fluxes above the blank-sky emission,\nwhich is modeled as an unabsorbed $\\sim0.25$~keV thermal plasma at $z=0$.\nThe best-fit model has a $\\chi^2_{min}=73.9$ for 78 degrees of freedom,\nfor a null hypothesis probability of 61\\%.}\n\\label{fig:soft-back}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.5in,angle=-90]{330-450.ps}\n\\includegraphics[width=2.5in,angle=-90]{450-600.ps}\n\\caption{Blank-sky background subtracted spectra of\nregions 330-450\" and 450-600\" from observation 6880.\nThe solid lines are the best-fit model of the local\nsoft X-ray enhancement of Figure~\\ref{fig:soft-back} (red),\nand its 90\\% upper limit (green).}\n\\label{fig:spectra}\n\\end{figure*}\n\nIn Figure~\\ref{fig:spectra} we show the spectra of\nthe outermost two regions, to show the impact of the \nsoft X-ray residuals in the background subtraction.\nThe importance of background systematics in the detection\nof emission and measurement of cluster temperatures for\nregions of low surface brightness was\nrecently addressed by \\cite{leccardi2008} using \\it XMM-Newton\\rm\\ data.\nFor our \\it Chandra\\rm\\ observations, the two main sources of uncertainty when determining the temperature\nof the outer regions are the subtraction of the blank-sky background, and\nthe subtraction of the locally-determined soft X-ray background.\nTable~\\ref{tab:background} reports the statistics of the background relevant\nto the outer regions of the cluster, with  both regions $\\sim$~10-20\\% above\nthe blank-sky background,  determined with a precision of 1-2\\%.\nThe additional soft X-ray background accounts for a significant portion\nof the remaining signal, as shown in Figure~\\ref{fig:spectra}; the 90\\% \nupper limit to the measurement  of this background is shown as the\ngreen lines, and emission from the cluster is still detected with\nhigh statistical significance. Both sources of error are \nincluded in the temperature measurements at large radii.\n\n\n\\begin{table*}\n\\centering\n\\caption{Background levels in outer regions}\n\\label{tab:background}\n\\begin{tabular}{lccc}\n\\hline\n \t\t& \\multicolumn{3}{c}{Observation ID}\\\\\n\t \t& 6880 \t& 6881\t&7370\\\\ \n\\hline\nExposure time (ks) & 114.1 & 36.0 & 39.5 \\\\\nCorrection to Blank-sky Subtraction$^{a}$ & -0.04$\\pm$0.01 & -0.125$\\pm$0.015 & -0.04$\\pm$0.015 \\\\\n\\hline\n\\multicolumn{4}{c}{Region 330-450\"}\\\\\nTotal Counts\t& 18,124 & 4,938 & 5,686 \\\\\nCount rate (c s$^{-1}$) & 0.158$\\pm$0.001 & 0.137$\\pm$0.002 & 0.144$\\pm$0.002\\\\\nNet count rate$^{b}$ ($10^{-2}$ c s$^{-1}$) \t& $2.75\\pm0.10$ & $2.54\\pm0.20$ & $2.36\\pm0.20$ \\\\\nPercent above back. & 17.4$\\pm$0.6 & 18.2$\\pm$1.4 & 16.4$\\pm$1.4 \\\\\nSXB count rate ($10^{-3}$ c s$^{-1}$) \t&  3.34$\\pm$0.77 & 7.20$\\pm$0.94 & 7.10$\\pm$0.71 \\\\\n\\hline\n\\multicolumn{4}{c}{Region 450-600\"} \\\\\nTotal Counts    & 15,811 & 4,901 & 5,483 \\\\\nCount rate (c s$^{-1}$) & 0.139$\\pm$0.001 & 0.136$\\pm$0.002 & 0.139$\\pm$0.002 \\\\\nNet count rate$^{b}$ ($10^{-2}$ c s$^{-1}$)     &1.02$\\pm$0.10 & 1.58$\\pm$0.20 & 1.23$\\pm$0.20 \\\\\nPercent above back. & 7.3$\\pm$0.7 & 11.6$\\pm$1.5 & 8.8$\\pm$1.4 \\\\\nSXB count rate ($10^{-3}$ c s$^{-1}$)   & 3.06$\\pm$0.70 & 7.50$\\pm$0.98 & 7.32$\\pm$0.73\\\\\n\\hline\n\\end{tabular}\n\n\\flushleft\n$a$: This is the fractional correction of the blank-sky data, to match the high-energy flux\nin the cluster observation.   \\\\\n$b$ This is the background-subtracted count rate, including cluster and soft X-ray background \n(SXB) signal.\n\\end{table*}\n\nWe use the APEC code \\citep[][code version 1.3.1]{smith2001} to model the \\it Chandra\\rm\\ spectra, with a\nfixed Galactic HI column density of $N_H=2.04\\times10^{20}$~cm$^{-2}$ \\citep{kalberla2005}.\nThe region at radii $\\leq 330$\" have a variable metal abundance, while the outer\nregion have a fixed abundance of $A=0.3$. In addition to the statistical errors\nobtained from the XSPEC fits, we add a systematic error of 10\\% in the\ntemperature measured in the core and a 5\\% error to the other region,\nto account for possible systematic uncertainties due to the \\it Chandra\\rm\\\ncalibration \\citep[see, e.g.,][]{bulbul2010}. \nOne possible source of systematic uncertainty in our results is indicated \nby the systematic difference between the \\it Chandra\\rm\/ACIS and\\it XMM-Newton\\rm\/EPIC \ntemperature measurements of galaxy clusters \\citep{nevalainen2010}\nThis amounts to a $\\pm$10\\% bias in the calibration of the effective area at 0.5 keV, \nwhich decreases roughly linearly towards 0\\% bias at 2 keV. \nAssuming that \\it XMM-Newton\\rm\/pn has a more accurately calibrated effective area, \nwe reduced the \\it Chandra\\rm\\ effective area by multiplying it with a linear function \nas indicated by the \\it Chandra\\rm\/\\it XMM-Newton\\rm\\ comparison. As a result, the temperature at the \noutermost radial bin decreases by $\\sim$ 5\\%. Thus, the cross-calibration \nuncertainties between \\it Chandra\\rm\\ and \\it XMM-Newton\\rm\\ do not explain the low temperature we measure in the outermost radial bin.\nUncertainties in the Galactic column density of HI do not impact significantly our results.\nChanging the value of  $N_H$ by $\\pm$10\\%, consistent with the variations between the \\cite{kalberla2005}\nand the \\cite{dickey1990} measurements, results in a change of best-fit temperature\nin each bin by less than 2\\%.\n\n\nGiven the emphasis of this paper on the detection of\nemission at large radii, we investigate\nthe sources of uncertainty caused by the background subtraction\nin the outer region at $\\geq$330\". We report the results of this \nerror analysis in Table~\\ref{tab:kT-err}, where \\emph{cornorm} refers\nto the normalization of the blank-sky background, and \\emph{soft residuals}\nrefers to the normalization of the soft X-ray residual model, as \nreported in Table~\\ref{tab:background}. In the analysis that follows,\nwe add the systematic errors caused by these sources linearly to the\nstatistical error.\nOur data do not constrain well the metal abundance of the plasma\nin the outer regions.\nUsing an abundance of $A=0.5$ instead of the nominal $A=0.3$\nleads to negligible changes in the best-fit temperature for both of\nthe outer annuli.\nIn the extreme case of an $A=0.0$ metal abundance, \nboth regions have an acceptable fit with the best-fit temperatures\nchange respectively by $+6$\\% for the 330-450\" region ($\\Delta \\chi^2=+1.3$), and\nby $-22$\\% for the 450-600\" region ($\\Delta \\chi^2=+9.2$,\nbest fit decreases from 1.26 to 0.98 keV). \nWe therefore find that, in the case of exceptionally low metalllicity, the\ntemperature profile we measure from these \\it Chandra\\rm\\ data would be even significantly steeper\nthan indicated by the result in Table~\\ref{tab:kT-err}.\nGiven that these data do not provide direct indication that the plasma in the outer regions \nmay have null metal content, we do not fold in this source of systematic error in the\nanalysis that follows.\n\n\n\n\\begin{table}\n\\caption{Temperature measurement and error analysis from\nthe \\it Chandra\\rm\\ data.}\n\\centering\n\\label{tab:kT-err}\n\\begin{tabular}{lcc}\n\\hline\nRegion & \\multicolumn{2}{c}{Projected Temperature (keV)}\\\\\n\\hline\n       & Measurement$^{a}$  & Calibration error$^{b}$  \\\\\n0-10\"  & 4.78$\\pm$0.06 & $\\pm$0.48 \\\\\n10-20\" & 7.09$\\pm$0.14 & $\\pm$0.71 \\\\\n20-30\" & 8.72$\\pm$0.27 & $\\pm$0.87 \\\\\n30-60\" & 9.47$\\pm$0.21 & $\\pm$0.47 \\\\ \n60-90\" & 10.57$\\pm$0.33 & $\\pm$0.53 \\\\\n90-120\" & 9.97$\\pm$0.44 & $\\pm$0.50 \\\\\n120-180\"& 9.68$\\pm$0.49 & $\\pm$0.48 \\\\\n180-240\" & 7.85$\\pm$0.65 &$\\pm$0.39 \\\\\n240-330\" & 6.02$\\pm$0.65 &$\\pm$0.30 \\\\\n330-450\" & 3.75$\\pm$0.72 & $\\pm$0.19\\\\\n450-600\" & 1.26$\\pm$0.16 & $\\pm$0.06 \\\\\n\\hline\n\t\\multicolumn{3}{c}{Measurement of Temperature Using} \\\\\n\t \\multicolumn{3}{c}{Background Systematic Errors (keV)}\\\\\n         & $+1\\sigma$ cornorm$^{c}$ & $-1\\sigma$ cornorm \\\\\n330-450\" & 3.02$\\pm$0.54 & 4.67$\\pm$1.00 \\\\\n450-600\" & 1.09$\\pm$0.10 & 1.31$\\pm$0.18 \\\\\n\t & $+1\\sigma$ soft res.$^{d}$ & $-1\\sigma$ soft res.\\\\\n450-600\" & 4.53$\\pm$1.03 & 3.05$\\pm$0.54\\\\\n450-600\" & 1.37$\\pm$0.25 & 1.16$\\pm$0.12\\\\\n \\multicolumn{3}{c}{Summary of Background Systematic Errors$^{e}$} \\\\\n330-450\" & \\multicolumn{2}{c}{$\\pm 0.83\\pm 0.74$ keV} \\\\ \n450-600\" & \\multicolumn{2}{c}{$\\pm 0.11\\pm 0.10$ keV} \\\\\n\\hline\n\\end{tabular}\n\n\\flushleft\n$a$: Uncertainty is 1$\\sigma$ statistical error from counting statistics only.\\\\\n$b$: Includes \\it XMM-Newton\\rm\/\\it Chandra\\rm\\ cross-calibration uncertainty of the effective area \\citep{nevalainen2010}.\\\\\n$c$: This is temperature obtained by varying by $\\pm 1\\sigma$ the fractional\ncorrection of the blank-sky data, to match the high-energy flux in the cluster observations.\\\\\n$d$: This is the temperature obtained by varying by $\\pm 1\\sigma$ the normalization\nof the best-fit model to the soft X-ray background residuals.\\\\\n$e$: Obtained from the average deviation of the $\\pm 1\\sigma $ `cornorm' and 'soft. res'\nmeasurements from the measurement with nominal values of these parameters.\n\\end{table}\n\n\\cite{sanders2010} measured temperature profiles for \\emph{Abell~1835}\\ out\nto approximately $r_{500}$\\ with \\it Chandra\\rm\\ and \\it XMM-Newton\\rm. Using the same \\it Chandra\\rm\\ \nobservations we analyze in this paper, their temperature profile\nhas a similar drop from the peak value to their outermost\nannulus ($322\\pm42$\"), where they measure a temperature of kT=$4.67\\pm^{0.82}_{0.52}$~keV\nthat is consistent with our measurements. Likewise, from the \\it XMM-Newton\\rm\\ data their outermost\nradial bin ($300\\pm10$\") has a temperature of kT=$5.2\\pm^{1.2}_{0.7}$~keV, also\nin agreement with our results. \nThe only measurement of the \\emph{Abell~1835}\\ temperature to the virial radius\navailable in the literature is that of \\cite{snowden2008}, who\ndoes report a temperature profile out to a distance of 12~arcmin\nfrom a long \\it XMM-Newton\\rm\\ observation (and out to 7' from a shorter observation).\nIn particular,\nthey report a temperature of $kT=3.14\\pm0.93$ for the region\n420-540\", which straddles our measurements at 330-450\" ($3.75\\pm0.72$~keV)\nand at 450-600\" ($1.26\\pm0.16$, statistical errors only). The same paper\nalso reports a measurement of $kT=3.33\\pm1.75$~keV for the region 540-720\",\ni.e., beyond our outer annulus. Their temperature is somewhat higher that\nours, although the large error bars cannot exclude that the \\it Chandra\\rm\\\nand \\it XMM-Newton\\rm\\ measurements\nare consistent. Therefore our results confirm and extend the\nearlier \\it XMM-Newton\\rm\\ analysis of \\cite{snowden2008}.\n\n\\subsection{Measurement of the average temperature of \nMAXBCG J210.31728+02.75364\nand WHL J140031.8+025443}\n\\label{sec:low-mass-clusters}\nWe also measure the temperature of the two SDSS clusters detected\nin our \\it Chandra\\rm\\ images, \nMAXBCG J210.31728+02.75364 \nand WHL J140031.8+025443. The two clusters  are located between a distance of $\\sim$380-650\"\nfrom the cluster center, and therefore we start by extracting a spectrum for this annulus\nexcluding two regions of 1.5' radius centered at the two clusters.\nThis radius was determined by visual inspection, after smoothing of the \\it Chandra\\rm\\ image with a\nGaussian kernel of $\\sigma=6$ arcsec. \nFor this annulus, we measure a temperature of $kT=1.85\\pm0.36$~keV for a fixed\nabundance of $A=0.3$ Solar. We then use this spectrum as the local background for the\ntwo cluster regions, and measure a temperature $kT=2.73\\pm^{0.93}_{0.54}$~keV i\nfor MAXBCG J210.31728+02.75364\n(357 source photons, 19\\% above the average emission of the annulus),\nand $kT=2.09\\pm^{4.6}_{0.55}$~keV for WHL J140031.8+025443 (538 photons, 27\\% above background). \nFor both clusters, we assumed the same Galactic $HI$ column density as for \\emph{Abell~1835}, and\na fixed metal abundance of $A=0.3$ Solar. For both clusters we also extract spectra in\nregions larger than 1.5', and determine that no additional source photons are present\nfrom these two clusters beyond this radius.\n\n\n\\subsection{Tests of robustness of the temperature measurement at large radii}\n\\label{sec:robustness}\nTo further test the measurement of temperatures especially at large radii,\nwhere\nthe background subtraction is especially important,\nwe also measure the temperature profile using the same stowed\nbackground data that was used for the surface brightness measurement\nof Figures~\\ref{fig:Sx} and \\ref{fig:Sx-closeup}. As in the case\nof the blank-sky background, we first rescale the stowed data to match\nthe high-energy count rate of the cluster observation, and\nuse a region at large radii ($\\geq$ 700 arcsec) to measure\nthe local X-ray background. We model the background using\nan APEC plus a power-law model, the latter component necessary\nto model the harder emission due to unresolved AGNs that is typically removed when\nthe blank-sky background is used instead, and apply this\nmodel to all cluster regions. We find that the temperature profile\nis consistent within the 1~$\\sigma$ statistical errors \nof the values provided in Table~\\ref{tab:kT-err} for each region,\nand therefore conclude that the temperature drop at large radii, and\nespecially in the outermost region, is not sensitive\nto the background subtraction method.\n\nThe temperature measurement is also dependent on an accurate subtraction\nof background (and foreground) sources of emission. \nPoint sources in the field of view are detetected using the CIAO\ntool  \\emph{wavdetect},  which correlates the image with wavelets\nat small angular scales (2 and 4 pixels, one pixel is 1.96\"), searches the results for 3-$\\sigma$ correlations, and\nreturns a list of elliptical regions to be excluded from the analysis. \nWe study in particular the\neffect of background sources on the measurement of the temperature in the outermost\nannulus (450-600\"). In this region,  \\emph{wavdetect} finds\n24 point sources, plus portions of the two low-mass galaxy clusters\ndescribed in Section~\\ref{sec:low-mass-clusters}.\nWe extract a spectrum for this region from the longest observation (ID 6880),\nand now include in the spectrum all\npoint sources excluded in the previous analysis.\nWe find a count rate of $3.20\\pm0.13\\times 10^{-2}$ counts~s$^{-1}$, compared\nto the point source-subtracted rate of $1.02\\pm0.10\\times 10^{-2}$ counts~s$^{-1}$,\ncorresponding to an increase in background-subtracted flux by a factor of three.\nWe then fit the spectrum with the same APEC model as described in Section~\\ref{sec:spectral-fits},\nand find a best-fit temperature of $kT=1.96\\pm0.17$~keV for a best-fit goodness statistic\nof $\\chi^2=537$ for 429 degrees of freedom (or $\\chi^2_{red}$=1.25), compared\nto the temperature of $1.22\\pm0.19$~keV for a $\\chi^2=415$ for 389 degrees of freedom (or $\\chi^2_{red}$=1.08).\nWe therefore conclude that an accurate subtraction of point sources and unrelated\nsources of diffuse emission is crucial to obtain an accurate measurement of the\ntemperature profile, especially in regions of low-surface brightness such as those\nnear the virial radius.\n\nChanges in the instrument calibration affect the measurement of temperatures.\nWe therefore repeat the same data reduction and spectral analysis  using the latest \nsoftware and calibration database available at time of writing (CIAO 4.4 and CALDB 4.5.1) for\nthe longest observation (ID 6880),\nand obtain a new temperature profile for the same regions as reported in Table~\\ref{tab:kT-err}.\nIn the outermost two regions, we measure a temperature of 3.04$\\pm0.69$~keV (330-450\") and \n1.23$\\pm$0.21~keV (450-600\"), well within the 1-$\\sigma$ confidence intervals of the measurements\nusing the older calibration (3.40$\\pm$0.76 and 1.22$\\pm$0.19 respectively, also in agreement with the\nvalues of Table ~\\ref{tab:kT-err} obtained from the combination of all exposures). \nThe temperature of the inner regions are also always within 1-$\\sigma$\nof the results obtained with the earlier calibration, and we therefore conclude that changes in the instrument\ncalibration do not affect significantly our results.\n\n\n\n\\section{Measurement of masses and gas mass fraction}\n\nWe fit the surface brightness and the temperature profiles with\nthe \\cite{vikhlinin2006} model. The electron density is modelled\nwith a double-$\\beta$ profile modified by a cuspy core component and an exponential cutoff at large radii,\nfor a total of eleven model parameters;\n the temperature has both a cool-core component to follow\nthe cooler gas in the core, and a decreasing profile at large radii, for an additional\nnine parameters. For our analysis, we follow  \\cite{vikhlinin2006} and fix the\n$\\gamma$=3.0 parameter, and do not use the cuspy-core component ($\\alpha=0$)\nor the second $\\beta$-model component, so that the density is modelled by just \none $\\beta$-model with an exponential cutoff, for just four free parameters\n(core radius $r_c$, exponent $\\beta$, scale radius $r_s$ and exponential cutoff\nexponent $\\epsilon$, see Table~\\ref{tab:vikh-fit}). \nFor the temperature profile, we fix the parameter $a=0$, and the\nremaining eight parameters are reported in Table~\\ref{tab:vikh-fit}.\n\n\n\nWe use a Monte Carlo Markov chain (MCMC) method that we used in\nprevious papers \\citep[e.g.,][]{bonamente2004,bonamente2006}.\nThe MCMC analysis consists of a projection of the three-dimensional\nmodels and a comparison \nof the projected surface brightness and temperature profiles,\nand results in simultaneous estimation\nof the posterior distributions of all model paramters. Uncertainties in the  parameters are obtained\nfrom the posterior distributions, with 1-$\\sigma$ errors  assigned using the 68.3\\%\nconfidence interval around the median of the distribution. \n\nThe gas mass is directly calculated from the electron density\nmodel parameters via\n\\begin{equation}\nM_{gas}(r) = m_p \\mu_e \\int_0^r n_e(r) 4 \\pi r^2 dr\n\\end{equation}\nand the total gravitational mass via the equation of\nhydrostatic equilibrium,\n\\begin{equation}\nM(r) = - \\frac{kT(r) r}{\\mu_e m_p G} \\left(\\frac{d \\ln n_e}{d \\ln r} + \\frac{d \\ln kT}{d \\ln r} \\right),\n\\label{eq:hse}\n\\end{equation}\nwhere $m_p$ is the proton mass, $\\mu_e \\simeq 1.17$ the mean electron molecular weight, and \n$G$ the gravitational constant. The total density of matter is simply obtained via\n\\begin{equation*}\n\\rho(r) = \\frac{1}{4 \\pi r^2} \\frac{d M(r)}{dr}\n\\end{equation*}\nand therefore can be obtained via a derivative of the mass profile.\nIn Equation~\\ref{eq:hse}, the term $A = d \\ln n_e\/d \\ln r + d \\ln kT\/d \\ln r$\nand its first derivative are always negative, as is $d kT(r)\/d r$ at large radii.\nTherefore, the density can be rewritten as\n\\begin{equation}\n\\rho(r) = - \\frac{1}{4 \\pi r^2 \\mu_e m_p G} \\left[ kT \\left( A +r \\frac{dA}{dr} \\right)  + r A \\frac{d kT}{dr} \\right] \n\\label{eq:density}\n\\end{equation}\nin which the only negative term  is the one containing $A \\cdot d kT(r)\/d r$, while\nthe other two terms remain positive out to large radii.\n\n\\subsection{Modelling of the Chandra data out to the virial radius}\n\\label{sec:hse}\nThe \\cite{vikhlinin2006} model provides a satisfactory fit\nout to the outermost radius of 600\";\nFigure~\\ref{fig:kt-0-600-fit} shows the best-fit models to the temperature \nand surface brightness profiles,\nbest-fit parameters\nof the model are reported in Table~\\ref{tab:vikh-fit}.\nThe temperature profile measured by \\it Chandra\\rm\\ in Figure~\\ref{fig:kt-0-600-fit}\nis so steep that it causes the total matter density $\\rho(r)$ to become \\emph{negative}\nat approximately 400\", indicating that the temperature profile \ncannot originate from gas in hydrostatic equilibrium. \nThe situation is illustrated in Figure~\\ref{fig:mass-0-600}, where the relevant terms\nof Equation~\\ref{eq:density} are plotted individually; \n the density inferred from hydrostatic equilibrium becomes negative \nwhere the\nnegative term crosses the positive ones,\n and the mass profile has a negative slope beyond that point. These fit parameters\ntherefore lead to an unacceptable situation, and responsibility for this inconsistency\ncan be attributed to an overly steep temperature profile, with a drop by a factor of\nten between approximately 1.5' to 10'. \n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.25in, angle=-90]{vikh_temp_0-600chain_90CI.ps}\n\\includegraphics[width=2.25in, angle=-90]{SB_vikh_0-600chain_0-600_600-1100.ps}\n\\caption{Left: Best-fit Vikhlinin model for the projected temperature\nprofile out to 600\", with 90\\% confidence intervals. Right:  Best-fit\nVikhlinin model to the 0.7-2 keV surface brightness (model+background) profile.\nEmission beyond 600\" is statistically\nconsistent with the background, in  blue is the extrapolation out to 1100\". \nPrior removal of the stowed background\ncaused the lower backgroud  level in Figure~\\ref{fig:Sx}.\n}\n\\label{fig:kt-0-600-fit}\n\\end{figure*}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Best-fit parameters for the Vikhlinin model using \\it Chandra\\rm\\ data out to 330\"}\n\\label{tab:vikh-fit}\n\\begin{tabular}{cccccccccc}\n\\hline\n$n_{e0}$ & $r_{c}$ & $\\beta$ & $r_{s}$ & $\\epsilon$ & $n_{e02}$ & $\\gamma$ & $\\alpha$ & $\\chi^{2}_{tot} \\textrm{(d.o.f.)}$\\\\\n(10$^{-2} $cm$^{-3}$) & (arcsec) & & (arcsec) & & & & & &\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 330\"}\\\\\n$9.602\\pm^{0.488}_{0.415}$ & $6.743\\pm^{0.373}_{0.403}$ & $0.498\\pm^{0.009}_{0.009}$ & $119.8\\pm^{13.3}_{13.4}$ & $1.226\\pm^{0.098}_{0.097}$ \n& 0.0 & 3.0 & 0.0 & \\nodata\\\\ \n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 600\"}\\\\\n$9.763\\pm^{0.447}_{0.450}$ & $6.346\\pm^{0.385}_{0.343}$ & $0.488\\pm^{0.009}_{0.009}$ & $96.44\\pm^{9.55}_{8.67}$ & $1.067\\pm^{0.075}_{0.079}$\n& 0.0 & 3.0 & 0.0 & \\nodata\\\\\n\\hline\n\\hline\n$T_{0}$ & $T_{min}$ & $r_{cool}$ & $a_{cool}$ & $r_{t}$ & $a_{t}$ & $b_{t}$ & $c_{t}$ &\\\\\n(keV) & (keV) & (arcsec) & & (arcsec) & & & & &\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 330\"}\\\\\n$38.25\\pm^{19.63}_{17.23}$ & 3.0 & $92.48\\pm^{52.63}_{40.52}$ & 1.0 & $257.5\\pm^{143.0}_{66.72}$ & 0.0 & $1.024\\pm^{0.426}_{0.283}$ & 2.0 &  39.0 (83)\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 600\"}\\\\\n$10.17\\pm^{0.85}_{0.60}$ & 3.0 & $11.82\\pm^{3.61}_{2.29}$ & $1.924\\pm^{0.802}_{0.568}$ & 600.0 & 0.0 & $2.800\\pm^{0.224}_{0.210}$ & 10.0 & 106.4 (154)\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Masses Calculated using \\it Chandra\\rm\\ data out to 330\", and Extrapolated out to $r_{100}$}\n\\label{tab:vikh-masses}\n\\begin{tabular}{ccccc}\n\\hline\n$\\Delta$ & $r_{\\Delta}$ & $M_{gas}$ & $M_{total}$ & $f_{gas}$\\\\\n         & (arcsec)     & $\\times 10^{13}~M_{\\odot}$ & $\\times 10^{14}~M_{\\odot}$ & \\\\\n\\hline\n2500 & $164.9\\pm^{4.1}_{3.9}$ & $4.70\\pm^{0.15}_{0.14}$ & $5.03\\pm^{0.38}_{0.35}$ & $0.093\\pm^{0.004}_{0.004}$\\\\\n500  & $326.6\\pm^{7.1}_{6.9}$ & $10.75\\pm^{0.23}_{0.23}$ & $7.80\\pm^{0.52}_{0.49}$ & $0.138\\pm^{0.006}_{0.006}$\\\\\n200  & $453.3\\pm^{15.2}_{15.1}$ & $15.36\\pm^{0.48}_{0.48}$ & $8.35\\pm^{0.86}_{0.81}$ & $0.184\\pm^{0.014}_{0.012}$\\\\\n100  & $570.9\\pm^{26.6}_{25.3}$ & $19.53\\pm^{0.84}_{0.82}$ & $8.34\\pm^{1.22}_{1.06}$ & $0.234\\pm^{0.024}_{0.022}$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{mass-0-600.ps}\n\\includegraphics[width=2.3in, angle=-90]{terms-0-600.ps}\n\\caption{Mass profile using data out to 600\" and the temperature fit of Figure~\\ref{fig:kt-0-600-fit},\nand the radial distribution of the positive and negative terms in the density\nequation (Equation~\\ref{eq:density}).}\n\\label{fig:mass-0-600}\n\\end{figure*}\n\n\nThe results presented in this section provide\nevidence  that the gas detected by \\it Chandra\\rm\\ near the virial radius is\n\\emph{not} in hydrostatic equilibrium, and a number of theoretical\nstudies do in fact suggest that  beyond $r_{500}$\\ the intergalactic\nplasma is not supported solely by thermal pressure \\citep[e.g.][]{lau2009}.\n\\it Suzaku\\rm\\ has reported the measurement of emission near the\nvirial radius for several clusters, including \\emph{Abell~1413},\n\\emph{Hydra~A}, \\emph{Perseus}, \\emph{PKS0745-191}, \\emph{Abell~1795}, \\emph{Abell~1689}\nand \\emph{Abell~2029}\n\\citep{hoshino2010,sato2012,simionescu2011,george2009, bautz2009,kawaharada2010,\nwalker2012a,walker2012b}. Some of these results do in fact report an apparent\ndecrease in total mass with radius \\citep[e.g.]{george2009,kawaharada2010} \nand lack of hydrostatic equilibrium at large radii \\citep[e.g.][]{bautz2009},  similar\nto the results presented in this paper.\nTemperature profiles measured by \\it Suzaku\\rm\\ typically do not feature as extreme a temperature drop\nas the one reported in Figure~\\ref{fig:kt-0-600-fit}, i.e., a factor of nearly 10\nfrom peak to outer radius, although in some cases the drop of temperature from the peak\nvalue to that at $r_{200}$\\ is consistent with the one reported in this paper.\n\n\n\\subsection{Modelling of the Chandra data out to $r_{500}$}\n\\label{sec:r500}\nThe steepening of the radial profile beyond 400\" is driven by the\ntemperature of the last datapoint beyond $r_{200}$. \nWe also model the surface brightness and temperature profiles of the\n\\it Chandra\\rm\\ data out to only 330\", or approximately $r_{500}$, and find the\nbest-fit \\cite{vikhlinin2006} model for the temperature profile\nreported  in Figure~\\ref{fig:kT-0-330} and Table~\\ref{tab:vikh-fit}.\nWe measure a gas mass fraction of $f_{gas}(r_{500})=0.138\\pm0.006$;\nif we add the mean stellar fraction as measured by either \\cite{giodini2009}\n($f_{\\star}=0.019\\pm0.002$) or by \\cite{gonzales2007} ($f_{\\star}\\simeq0.012$)\nassuming $M(r_{500})=7.1\\times 10^{14}$ $M_{\\odot}$,\nwe find that \\emph{Abell~1835}\\ has an average baryon content within  $r_{500}$\\\nthat is consistent with the cosmic abundance of $\\Omega_b\/\\Omega_M=0.167\\pm0.007$ \\citep{komatsu2011}\nat the 2-$\\sigma$ level. As is the case in most clusters, especially relaxed\nones, the radial distribution of the gas mass fraction increases with radius\n\\citep[e.g.,][]{vikhlinin2006}.\n\nWe use this modelling of the data to measure $r_{500}$, and to provide estimates\nfor $r_{200}$\\ and the virial radius.\nThe extrapolation of this model to 600\" now falls above the measured temperature profile,\nand the mass profile using hydrostatic equilibrium is monotonic.\nThis best-fit model is marginally compatible with the assumption of hydrostatic equilibrium.\nIn fact, Table~\\ref{tab:vikh-masses} shows that the extrapolated  mass profile\nflattens around $r_{200}$, with virtually no additional mass being necessary\nbeyond this radius to sustain the hot gas in hydrostatic equilibrium.\nMoreover, between $r_{500}$\\ and $r_{200}$, all of the gravitational mass is accounted\nby the hot gas mass, i.e., \\emph{no} dark matter is required beyond $r_{500}$.\nThis extrapolation of the $\\leq$~$r_{500}$\\ data to the virial radius therefore\nleads to a dark matter halo that is much more concentrated than\nthe hot gas.\n\n\n\\section{Entropy profile and convective instability at\nlarge radii}\n\\label{sec:entropy}\nThe Schwarzschild\ncriterion for the onset of convective instability is given by the\ncondition of buoyancy of an infinitesimal blob of gas that is displaced by an\namount $dr$, $d \\rho_{blob} < d \\rho$,\nwhere $\\rho_{blob}$ is the density of the displaced blob, assumed to attain pressure\nequilibrium with the surrounding, and $\\rho$ is the density of ambient medium.\nIf the blob is displaced adiabatically, using pressure $P$ and entropy $s$ as the\nindependent thermodynamic variables in the derivatives of $\\rho_{blob}$ and $\\rho$, \nthe buoyancy condition gives \n\\begin{equation}\n\\left. \\frac{\\partial \\rho}{\\partial s} \\right|_{p} ds > 0\n\\label{eq:buoyancy}\n\\end{equation}\nas condition for convective instability, i.e., a blob that is displaced radially outward will\nfind itself in a medium of higher density and continue to rise to larger radii. Since\n$({\\partial \\rho}\/{\\partial s})_P=-\\rho^2 (\\partial T \/ \\partial P)_s<0$ (material\nis heated upon adiabatic compression),  Equation~\\ref{eq:buoyancy} simply\nreads that \\emph{a radially decreasing entropy profile is convective unstable}.\n\nAn ideal gas has an entropy of\n\\begin{equation}\nS = \\nu R \\left( \\frac{3}{2} \\ln T - \\ln \\rho + C\\right)\n\\end{equation}\nwhere $\\nu$ is the number of moles, $R$ is the gas constant, and $C$ is a constant.\nIn astrophysical applications, it is customary \\citep[e.g.][]{cavagnolo2009} to use a definition\nof entropy that is related to the thermodynamic entropy by an operation of\nexponential and a constant offset,\n\\begin{equation}\nS = \\frac{kT}{n_e^{2\/3}},\n\\label{eq:entropy}\n\\end{equation}\nThe entropy $S$ defined by Equation~\\ref{eq:entropy} has\nunits\nof keV cm$^{2}$, and it is required to be radially increasing to maintain convective equilibrium.\nNumerical simulations\nindicate that entropy outside the core is predicted to increase with radius approximately \nas $r^{1.1}$ or $r^{1.2}$  \\citep{voit2005,tozzi2001}.\nIn Figure~\\ref{fig:entropy-profile} we show the radial profile of the entropy out to\nthe outer radius of 10 arcmin, with a significant decrease at large radii that indicates\nan incompatibility of the best-fit model with convective equilibrium. For comparison,\nwe also show the entropy profile measured using the modelling of the data\nout to only $r_{500}$, as described in Sec.~\\ref{sec:r500}. This entropy profile\nuses the shallower temperature profile of Figure~\\ref{fig:kT-0-330}, and its\nextrapolation to larger radii remains non-decreasing, i.e., marginally consistent\nwith convective equilibrium.\n\nThe Schwarzschild criterion \ndoes not apply in the presence of a magnetic field. For typical\nvalues of the thermodynamic quantities of the ICM, the electron and ion gyroradii are\nseveral orders of magnitude smaller than the mean free path for Coulomb collisions\n\\citep[e.g.][]{sarazin1988}, even for a magnetic field of order 1 $\\mu G$, and therefore\ndiffusion takes place primarily along field lines \\citep[e.g.][]{chandran2007}.\nThere is strong evidence of magnetic\nfields in the central regions of clusters \\citep[e.g., radio halos, ][]{venturi2008,cassano2006},\nthough it is not clear whether magnetic fields are ubiquitous\nnear the virial radius, as in the case of Abell~3376 \\citep{bagchi2006}.\nIn the presence of magnetic fields, \\cite{chandran2007} has shown that the\ncondition for convective instability is simply $dT\/dR<0$.\n\n  \nThe \\it Chandra\\rm\\ data out to the virial radius therefore indicate\nthat the ICM is convectively unstable, regardless of the\npresence of a magnetic field. In fact, in the absence of magnetic\nfields near the virial radius, Figure~\\ref{fig:entropy-profile} shows that \\emph{Abell~1835}\\\nfails the standard Schwarzschild criterion, i.e., the entropy decreases with radius;\nin the presence of magnetic fields, the negative gradient in the temperature profile alone\nis sufficient for the onset of convective instability \n\\citep[e.g., as discussed by ][]{chandran2007}.\n  Convective instabilities would carry hotter\ngas from the inner regions towards the outer region within a few sound crossing\ntimes. As shown by \\cite{sarazin1988}, the sound crossing time for a 10~keV\ngas is $\\sim 0.7$~Gyr for a 1~Mpc distance,\nand an unstable temperature gradient such as that of Figure~\\ref{fig:kt-0-600-fit}\nwould be flattened by convection within a few Gyrs.\nConvection could in principle also result in an additional pressure gradient\ndue to the flow of hot plasma to large radii, which can in turn help support the gas\nagainst gravitational forces. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{vikh_temp_0-330chain_extrapolated_to_600_90CI.ps}\n\\includegraphics[width=2.3in, angle=-90]{fgas_profile.ps}\n\\caption{\nTemperature and gas mass fraction profiles measured from a fit to the \\it Chandra\\rm\\ data out to 330\", and extrapolation of the\nbest-fit model out to 600\".}\n\\label{fig:kT-0-330}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{entropy_0-600.ps}\n\\includegraphics[width=2.3in, angle=-90]{entropy_0-330.ps}\n\\caption{\nDeprojected entropy profiles using the full \\it Chandra\\rm\\ data out to 600\" (left, see Section~\\ref{sec:hse}),\nand using only data out to $r_{500}$\\ (right, see Section~\\ref{sec:r500}).}\n\\label{fig:entropy-profile}\n\\end{figure*}\n\n\n\\section{Discussion and interpretation}\nIn this paper we have reported the detection of X-ray emission in \\emph{Abell~1835}\\ with \\it Chandra\\rm\\ that extends out to approximately\nthe cluster's virial radius. The emission can be explained by the presence of a cooler\nphase of the plasma that is dominant at large radii, possibly linked to the infall\nof gas from large-scale filamentary structures. We also investigate the effects of clumping of the gas \nat large radii, and conclude that in principle a radial gradient in the clumping factor of the hot ICM\ncan explain the apparent flattening of the entropy profile and the turn-over of the mass profile.\n\\subsection{Detection of X-ray emission out to the virial radius}\nThe detection of X-ray emission out to a radial distance of 10 arcmin, or approximately 2.4~Mpc,\nindicates the presence of diffuse gas out to the cluster's virial radius.\nThis is the first detection of gas out to the virial radius with \\it Chandra\\rm, matching\nother detections obtained with \\it Suzaku\\rm\\ for nearby clusters \n\\citep[e.g.][]{akamatsu2011,walker2012a,walker2012b,simionescu2011,burns2010,kawaharada2010,\nbautz2009,george2009}.\nDespite its higher background, \\it Chandra\\rm\\ provides a superior angular resolution to image and remove emission from unrelated sources.\nAs can be seen from Figure~\\ref{fig:a1835}, there are approximately 100 point-like sources that were automatically\ndetected and removed, and we were also able to identify two low-mass clusters that are likely associated with \\emph{Abell~1835}.\n\\it Chandra\\rm\\ therefore has the ability to constrain the emission of clusters to the virial radius, especially for higher-redshift\ncool-core clusters for which the \\it Suzaku\\rm\\ point-spread function would cause significant contamination from the \ncentral signal to large radii.\n\n\nIt is not easy to interpret the emission at the outskirts as an extension \nof the hot gas detected at radii $\\leq$~$r_{500}$. In fact, as shown in Section~\\ref{sec:hse}, the steepening of the\ntemperature profile is incompatible with the assumption of\nhydrostatic equilibrium at large radii. \nWe also showed in Section~\\ref{sec:entropy} that\nthe gas has a negative entropy gradient beyond this radius, rendering it convectively unstable. \nTherefore, if the temperature profile of Figure~\\ref{fig:kt-0-600-fit} originates from\na single phase of the ICM, convection would transport hotter gas towards the outskirts, flattening\nthe temperature profile within a few Gyrs. Cooling of the gas by thermal radiation cannot be\nresponsible for off-setting the heating by convection, since the cooling time\n ($t_{cool} \\sim kT^{1\/2} n_e^{-1}$) is longer at the outskirts than in the peak-temperature regions\ndue to the higher density.\n\n\\subsection{Warm-hot gas towards the cluster outskirts}\nA possible interpretation for the detection of emission near the virial radius and its\nsteep temperature profile is the presence of a separate phase at the cluster outskirts\nthat is not in hydrostatic equilibrum with the cluster's potential.\nIn this case, cooler gas may be the result of infall from filamentary structures that\nfeed gas into the cluster, and the temperature of this \\emph{warm-hot} gas may in fact be\nlower than that shown in Figure~\\ref{fig:kt-0-600-fit} \n(i.e., $kT \\sim 1.25$~keV for the region $\\geq 450$\") if\nthis gas lies in projection against the tail end of the hotter ICM.\n\nWe estimate the mass of this putative warm-hot gas assuming that all of the\nemission from the outermost region is from a uniform density gas\nseen in projection. This assumption may result in an overestimate\nof the emission measure; in fact,  the extrapolation of the gas density profile \nin the hydrostatic or convective scenarios may yield a significant amount of\nemission in the last radial bin. \nWe were unable to perform a self-consistent modelling\nof the emission  in the full radial range, since the low signal-to-noise\nratio does not allow a two-phase modelling in the last radial bin.\nIn this simple uniform density warm-hot gas scenario, \nthe gas is in a filamentary structure\nof length $L$ and area $A=\\pi(R_{out}^2-R_{in}^2)$, where\n$R_{out}=600$\" and $R_{in}=450\"$; this is the same model\nalso considered in \\cite{bonamente2005} for the cluster \\emph{Abell~S1101}.\nSince the length $L$ of the filament along the sightline is unknown,\nwe must either assume $L$ or the electron density $n_e$, and \nestimate the mass implied by the detected emission.\nThe emission integral for this region is proportional to\n\\begin{equation}\nK = \\frac{10^{-14}}{4 \\pi D_A^2 (1+z)^2} n_e^2 V,\n\\end{equation}\nwhere $K$ is measured in XSPEC from a fit to the spectrum, $D_A$ is\nthe angular distance in cm, $z$ is the cluster redshift, and \nthe volume is $V=A \\times L$. For this estimate we assume \nfor simplicity that\nthe mean atomic weights of hydrogen and of the electrons are\nthe same, $\\mu_e=\\mu_H$.\nUsing the best-fit spectral model with $kT=1.26\\pm0.16$ keV,\nwe measure $K=1.05\\pm 0.13 \\times 10^{-4}$. If we assume a filament of\nlength $L=10$~Mpc, then the average density is $n_e=2.4\\pm0.3$~cm$^{-3}$,\nand the filament mass is $4.6\\pm0.6 \\times 10^{13}$~$M_{\\odot}$.\nAlternatively, a more diffuse filament gas of $n_e=10^{-5}$~cm$^{-3}$\nwould require a filament of length $L=58\\pm8$~Mpc, with\na mass of $1.1\\pm0.2\\times 10^{14}$~$M_{\\odot}$, comparable to the\nentire hot gas mass within $r_{200}$. The fact that a lower density gas\nyields a higher mass is given by the fact that, for a measured value of $K$\nwe obtain $n_e \\propto L^{-1\/2}$, and therefore the mass is proportional to $L^{1\/2}$.\nFor comparison, the gas mass for this shell inferred from the standard\nanalysis, i.e.,  assuming that the gas is in the shell itself,\nis $\\sim 3\\times 10^{13}$~$M_{\\odot}$, as can be also seen from Table~\\ref{tab:vikh-masses}.\n\nIf the gas is cooler, then the mass budget would increase further.\nIn fact, the bulk of the emission from cooler gas falls outside of the \\it Chandra\\rm\\ bandpass,\nand for a fixed number of detected counts the required emission integral increases.\nWe illustrate this situation by fitting the annulus to an emission\nmodel with a fixed value of $KT=0.5$~keV, which result in a value\nof $K=1.88\\pm 0.24 \\times 10^{-4}$ (the fit is significantly poorer, with\n$\\Delta \\chi^2=+10$ for one fewer degree of freedom). \nAccordingly, the filament mass estimates would be increased \napproximately by a factor of two. \n\n\nA warm-hot phase at $T\\leq 10^7$~K is expected to be a significant reservoir of baryons\nin the universe \\citep[e.g.][]{cen1999,dave2001}. Using the \\it ROSAT\\rm\\ soft X-ray \nPosition Sensitive Proportional Counter (PSPC)  detector --  better suited to\ndetect the emission from sub-keV plasma --  we\nhave already reported \nthe detection of a large-scale halo of emission around the \\emph{Coma} cluster out to $\\sim$~5 Mpc, well beyond the\ncluster's virial radius \\citep{bonamente2003,bonamente2009}. \nIt is possible to speculate that the high mass of \\emph{Abell~1835}, one\nof the most luminous and massive clusters on the \\emph{Bright Cluster Survey} sample \\citep{ebeling1998},\nis responsible for the heating of the infalling gas to temperatures that makes it\ndetectable by \\it Chandra\\rm, and that other massive clusters may therefore provide\nevidence of emission to the virial radius with the \\it Chandra\\rm\\ ACIS detectors.  \nThe infall scenario is supported by the \\emph{Herschel} observations\nof \\cite{pereira2010}, who measure a galaxy velocity distribution for \\emph{Abell~1835}\\\nthat does not appear to decline at large radii as in most of the other clusters\nin their sample. A possible interpretation for their data is the presence of a \nsurrounding filamentary structure  that is infalling into the cluster.\n\n\\subsection{Effects of gas clumping at large radii}\nMasses and entropy measured in this paper assume that the gas has a uniform density\nat each radius.  To quantify the effect of departures from uniform density, we\n   define the clumping factor $C$ \nas the ratio of density averages over a large region,\n\\begin{align}\nC & = \\frac{\\langle n_e^2 \\rangle}{\\langle n_e \\rangle^2}\n\\end{align}\nwith $C \\geqslant 1$.\nClumped gas emits more efficiently than gas of uniform\ndensity, \nand the same surface brightness $I$ results in a lower estimate for the gas density and mass,\n\\begin{equation}\nI \\propto \\int <n_e^2> dl = \\int <n_e>^2 C dl,\n\\end{equation}\nwhere $l$ is a distance along the sightline.\nFrom Figure~\\ref{fig:entropy-profile} we see that the entropy drop from\napproximately 400\" to 600\" would be offset by a decrease in $n_e^{2\/3}$ by a factor\nof 3, or a decrease in $n_e$ by a factor of 5. We therefore suggest that a clumping\nfactor of $C \\simeq 25$ at 600\" would in principle be able to provide\na flat entropy profile, and even higher clumping factors would provide\nan increasing entropy profile in better agreement with theory \\citep[e.g.][]{voit2005,tozzi2001}.\nNumerical simulations by \\cite{nagai2011} suggest  values of the clumping factor\n$C \\leq 3$ near $r_{200}$, with significantly higher clumping possible at larger radii. \nUse of the \\cite{nagai2011}  model in the analysis of a large sample of galaxy clusters by \\cite{eckert2012}\nresults in better agreement of observations with numerical simulations. \n\nClumping can also affect the measurement of hydrostatic masses.\nIn particular, gas with an increasing radial profile of the clumping factor\ncould result in a steeper gradient of the density profile, when compared with what is measured assuming\na uniform density. According to Equation~\\ref{eq:hse}, this \nwould result in larger estimates of the hydrostatic mass, in principle able to reduce or entirely\noffset the apparent decreas of $M(r)$ reported in Figure~\\ref{fig:mass-0-600}.\nWe therefore conclude that a radial increase in the clumping of the gas can in principle\naccount for the apparent decrease of the mass profile and of the entropy profile\nreported in this paper (Figures~\\ref{fig:mass-0-600} and \\ref{fig:entropy-profile}), and therefore\nit is a viable scenario to interpret our \\it Chandra\\rm\\ observations.\nClumping of the gas at large radii has also been suggested based on \\it Suzaku\\rm\\ observations\n\\citep[e.g.,][]{simionescu2011}.\n\n\\section{Conclusions}\nIn this paper we have reported the detection of emission from \\emph{Abell~1835}\\ with \\it Chandra\\rm\\\nout to the cluster's virial radius. The cluster's surface brightness\nis significantly above the background level out to a radius of\napproximately 10 arcminutes, which correspond to $\\sim$2.4 Mpc at the\ncluster's redshift. We have investigated several sources of systematic\nerrors in the background subtraction process, and determined that the\nsignificance of the detection in the outer region (450-600\") is\n$\\geq 4.7$~$\\sigma$, and the emission cannot be explained\nby fluctuations in the background. Detection out to the virial\nradius is also implied by the \\it XMM-Newton\\rm\\ temperature profile\nreported by \\cite{snowden2008}.\n\nThe \\it Chandra\\rm\\ superior angular resolution made it straightforward to\nidentify and subtract sources of X-ray emission that are unrelated to the cluster. \nIn addition to a large number of point sources, we have identified X-ray emission\nfrom two low-mass clusters that were selected from the SDSS data,\nMAXBCG J210.31728+02.75364 \\citep{koester2007}\nand WHL J140031.8+025443 \\citep{wen2009}.\nThe two clusters have photometric and spectroscopic redshifts that make them\nlikely associated with \\emph{Abell~1835}. These are the only two\nSDSS-selected clusters that are in the vicinity of \\emph{Abell~1835}.\n\n\nThe outer regions of the \\emph{Abell~1835}\\ cluster have a sharp drop in the temperature\nprofile, a factor of about ten from the peak temperature. The sharp drop\nin temperature implies that the hot gas cannot be in hydrostatic equilibrium, and\nthat the hot gas would be convectively unstable. A possible scenario to\nexplain the observations is the presence of \\emph{warm-hot} gas\nnear the virial radius that is not in hydrostatic equilibrium with\nthe cluster's potential, and with a mass budget comparable to that\nof the entire ICM. The data  are also consistent with an alternative scenario\nin which a significant clumping of the gas at large radii is responsible\nfor the apparent negative gradients of the mass and entropy profiles\nat large radii.\n\n\n\n\n\\bibliographystyle{mn2e}\n\\bibliographystyle{apj}\n\\input{ms.bbl}\n\n\n\n\\label{lastpage}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe advent of the {\\it Einstein} observatory changed the belief that\nearly-type galaxies contain little interstellar gas by revealing hot\nX-ray emitting halos associated with many of them (e.g.  Forman et\nal. 1979).  Subsequent X-ray observations led to the conclusion that\nthese galaxies can retain large amounts (up to $\\sim 10 ^{11}\nM_{\\odot}$) of hot ($T \\sim 10^{7}$ K) interstellar medium\n(Forman, Jones \\& Tucker 1985; Trinchieri \\& Fabbiano 1985; Canizares,\nFabbiano \\& Trinchieri 1987).\n\n\\begin{figure*}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.6\\hsize\n \\epsffile{xray_fig.eps}\n \\caption{Grey-scale {\\it ROSAT} HRI image of the core of the cluster\nAbell~2634. The image has been smoothed using a Gaussian kernel with a\ndispersion of 8 arcsec. The positions of the cluster galaxies with\nmeasured redshift are marked with crosses.} \n\\end{center}\n\\label{grayscale}\n\\end{figure*}\n\n\nHowever, this picture might be expected to be different for galaxies\nthat reside near the centres of rich clusters of galaxies, since their\nproperties must be affected by their dense environment.  For example,\nwe might expect interstellar medium (ISM) gas to be stripped from the\ngalaxy by the ram pressure resulting from the passage of the galaxy\nthrough the intracluster medium (ICM) (Gunn \\& Gott 1972).  Stripping\nof the ISM can also result from tidal interactions with other nearby\ngalaxies (Richstone 1975; Merritt 1983, 1984).  The most dramatic and\nwell-studied example of a galaxy which appears to be in the process of\nbeing stripped of its ISM is the elliptical galaxy M86 in the Virgo\ncluster, which shows a `plume' of X-ray emission emanating from it\n(Forman et al. 1979; White et al. 1991; Rangarajan et al. 1995).\n\nIn addition to the mechanisms which remove the ISM of a galaxy, gas\ncan also be replenished.  The gravitational pull of a galaxy attracts\nthe surrounding ICM.  This gas ends up being concentrated in or behind\nthe galaxy, depending on the velocity of the galaxy relative to the\nICM (see, for example, Sakelliou, Merrifield \\& McHardy 1996). Stellar\nwinds can also replenish the hot gas in a galaxy's ISM.\n\nAll the processes mentioned above take place simultaneously. The\nrelative importance of each process depends on: the galaxies'\nvelocities; the local density of the ICM; the number density of\ngalaxies; their orbits in the cluster; and the gravitational potential\nof each galaxy.  It is therefore {\\it a priori} difficult to say which\nmechanism dominates in the cores of rich clusters of galaxies, and\nhence whether cluster galaxies are surrounded by the extensive X-ray\nemitting halos that we see associated with galaxies in the field.\n\nUnfortunately, X-ray observations of rich clusters have generally not\nbeen of high enough quality to answer this question, since any\nemission from the galaxies is hard to detect against the high X-ray\nbackground produced by the cluster's ICM (Canizares \\& Blizzard 1991;\nVikhlinin, Forman \\& Jones 1994; Bechtold et al. 1983; Grebenev et\nal. 1995; Soltan \\& Fabricant 1990; Mahdavi et al. 1996). In the cases\nwhere galaxy X-ray emission has been reported, the studies have been\nrestricted to a few bright cluster galaxies, and it has not proved\npossible to investigate the general galaxy population in a\nstatistically complete manner.\n\nIn order to search for X-ray emission from galaxies in a moderately\nrich environment, we obtained a deep {\\it ROSAT} HRI observation of\nthe core of the rich cluster Abell~2634, which is a nearby (z=0.0312)\ncentrally-concentrated cluster of richness class I.  In \\S2.1 we\ndescribe the analysis by which the X-ray emission from the galaxies in\nthis cluster was detected.  In \\S2.2 we explore the properties of this\nX-ray emission, and show that the galaxies in this cluster lack the\nextensive gaseous halos of similar galaxies in poorer environments.\nIn \\S3, we show how this difference can be attributed to ram\npressure stripping.\n\n\n\\section{X-ray observations and analysis}\n\nThe core of Abell~2634 was observed with the {\\it ROSAT} HRI in two\npointings, in January and June 1995, for a total of $62.5\\,{\\rm\nksec}$.  The analysis of these data was performed with the IRAF\/PROS\nsoftware.\n\nInspection of the emission from the cD galaxy and other bright X-ray\nsources in the images from the two separate observations indicates\nthat the two sets of observations do not register exactly and that a\ncorrection to the nominal {\\it ROSAT} pointing position is\nrequired. Therefore, the second set of observations was shifted by\n$\\sim$ 2.0 arcsec to the east and $\\sim$ 0.8 arcsec to the south; such\na displacement is consistent with typical {\\it ROSAT} pointing\nuncertainties (Briel et al. 1996).  Both images were then registered\nwith the optical reference frame to better than an arcsecond.  A\ngrey-scale image of the total exposure is shown in Fig. 1.  The image\nhas been smoothed with a Gaussian kernel of 8 arcseconds dispersion.\nAt the distance of Abell~2634, $1\\,{\\rm arcsec}$ corresponds to\n$900\\,{\\rm pc}$.\\footnote{Here, as throughout this paper, we have\nadopted a Hubble constant of $H_{0}=50\\; {\\rm km\\:s^{-1}\\:Mpc^{-1}}$.}\n\n\\begin{table}\n \\caption{Bright Sources}\n \\begin{tabular}{cccc}\n\\hline \\hline\nSource & $\\alpha$(J2000) & $\\delta$(J2000) & ID\/notes \\\\\n     & $^{\\rm h} \\; ^{\\rm m} \\; ^{\\rm s} $ & $\\degr \\; \\arcmin \\;\n\\arcsec$ & \\\\\n\\hline\n1 & 23 38 29.1 & 27 01 53.5 & cD galaxy \\\\\n2 & 23 37 56.1 & 27 11 31.3 & cluster \\\\\n3 & 23 39 01.6 & 27 05 35.9 & star \\\\\n4 & 23 39 00.5 & 27 00 27.9 & star\\\\\n5 & 23 38 31.7 & 27 00 30.5 & nothing \\\\\n6 & 23 38 41.5 & 26 48 04.1 & star \\\\\n7 & 23 38 19.8 & 26 56 41.5 & ? \\\\\n8 & 23 38 07.4 & 26 55 52.8 & star \\\\\n9 & 23 37 57.5 & 26 57 30.1 & galaxy ?\\\\\n10 & 23 37 45.3 & 26 57 53.1 & two objects \\\\\n11 & 23 37 26.2 & 27 08 14.6 & ? \\\\\n\n\\hline\n \\end{tabular}\n\\end{table}\n\nThis deep image of Abell~2634 reveals the largescale X-ray emission\nfrom the hot ICM of the cluster and a few bright sources, which are\nnumbered on Fig.~1.  Source 1 is the cD galaxy NGC~7720, located near\nthe centre of Abell~2634. It hosts the prototype wide-angle tailed\nradio source 3C~465 (e.g.~Eilek et al.  1984).  Source 2 is a\nbackground cluster at a redshift of $cz \\simeq 37,000 \\ {\\rm km \\\ns^{-1}}$ (Pinkney et al.  1993; Scodeggio et al.  1995).  For the rest\nof the X-ray bright sources, the Automatic Plate Measuring \nmachine, run by the Royal Greenwich Observatory in Cambridge, was used\nto obtain optical identifications.  Table 1. gives the positions of\nthese sources as determined from the X-ray image, and the class of\ntheir optical counterparts.  The position of source 7 coincides with a\nfaint object in the Palomar sky survey, but there is also a nearby\nstar, and source 11 does not seem to have a discernible optical\ncounterpart.  All these sources were masked out in the subsequent\nanalysis.  \n\nThe positions of galaxies that are members of Abell~2634 are also\nindicated on Fig.~1.  Pinkney et al.\\ (1993) collected the redshifts\nof $\\sim$150 galaxies that are probable members Abell~2634 (on the\nbasis that their redshifts lie in the range $6,000 < cz < 14,000\\,\n{\\rm km\\,s^{-1}}$), and Scodeggio et al. (1995) have increased the\nnumber of galaxies whose redshifts confirm that they are cluster\nmembers up to $\\sim 200$.  The sample of redshifts is complete to a\nmagnitude limit of 16.5, and from this magnitude-limited sample we\nhave selected those galaxies that appear projected within a circle of\n15 arcmin radius, centered on the cD galaxy.  This selection yields 62\ngalaxies, of which the vast majority are of type E and S0 -- only 10\nare classified as spirals or irregular.  The positions of these\ngalaxies are taken from the CCD photometry of Pinkney (1995) and\nScodeggio et al. (1995), and are accurate to $\\sim 1\\,{\\rm arcsec}$.\nThey are marked as crosses on Fig.~1.\n\nInspection of Fig.~1 reveals several cases where the location of a\ngalaxy seems to coincide with an enhancement in the cluster's X-ray\nemission, and it is tempting to interpret such enhancements as the\nemission from the galaxy's ISM.  However, it is also clear from Fig.~1\nthat the X-ray emission in this cluster contains significant\nsmall-scale fluctuations and non-uniformities.  We must therefore\nconsider the possibility that the apparent associations between galaxy\nlocations and local excesses in the X-ray emission may be\nchance superpositions.  We therefore now present a more objective\napproach to searching for the X-ray emission from cluster galaxies.\n\n\n\n\\subsection{Detection of the cluster galaxies}\n\nBefore adopting an approach to detecting the emission from cluster\ngalaxies, we must first have some notion as to how bright we might\nexpect the emission to appear in this deep HRI image.  Previous X-ray\nobservations have shown that the X-ray luminosities of E and S0\ngalaxies in the 0.2-3.5 keV energy band range from $\\sim10^{39}$ to\n$\\sim 10^{42} \\ {\\rm erg \\ s^{-1}}$ (Kim, Fabbiano \\& Trinchieri\n1992a, b; Forman et al. 1985).  These limits at the distance of\nAbell~2634 correspond to fluxes of $5 \\times 10^{-16}$ to $5 \\times\n10^{-13} \\ {\\rm erg \\ s^{-1} \\ cm^{-2}}$.  We have used the PIMMS\nsoftware to convert these limits to count rates for the {\\it ROSAT}\nHRI detector.  The emission from the galaxies was modeled by a\nRaymond-Smith plasma (Raymond \\& Smith 1977) with a temperature\n$kT=0.862$ keV and a metal abundance of 25\\% solar; these quantities\nare consistent with the values previously found from observations of\nearly-type galaxies (Kim et al.  1992a; Matsushita et al. 1994; Awaki\net al. 1994). The absorption by the galactic hydrogen was also taken\ninto account by using the column density given by Stark et al. (1992)\nfor the direction of Abell~2634 ($N_{\\rm H}= 4.94 \\times 10^{20} \\\n{\\rm cm^{-2}}$).  These calculations predict that the $62.5\\,{\\rm\nksec}$ HRI observation of this cluster should yield somewhere between\n$\\sim1$ and $\\sim1200$ counts from each galaxy.  Motivated by this\nprediction of a respectable, but not huge, number of counts per\ngalaxy, we set out to detect emission associated with cluster\ngalaxies.\n\nWe are trying to detect this fairly modest amount of emission against\nthe bright background of the ICM emission.  We therefore seek to\nimprove the statistics by stacking together the X-ray images in the\nvicinity of the 40 E and S0 galaxies marked in Fig.~1.\nFig. 2. presents a contour plot of the combined image, which covers a\nregion of 1 arcmin radius around the stacked galaxies.  The centre of\nthe plot coincides with the optical centres of the individual\ngalaxies. Clearly, there appears to be X-ray emission associated with\nthe cluster galaxies, and it is centered at their optical positions.\nThis coincidence provides us with some confidence that the X-ray and\noptical frames are correctly registered.  We have also constructed a\ncomposite brightness profile for the 40 galaxies by adding the\nunsmoothed counts detected in concentric annuli centered on each\ngalaxy.  The width of each annulus in this profile was set at 6 arcsec\nand the local background, as measured in an annulus between 1.0 and\n2.0 arcmin around each galaxy, was subtracted.  The resulting profile\nis presented in Fig. 3.  Once again, the excess of emission in the\nvicinity of the cluster galaxies is apparent.\n\nIn order to assess the significance of this detection, we generated\n100 sets of simulated data from randomly selected points on the image.\nThe diffuse emission from the ICM varies systematically with radius,\nand so we might expect the probability that a galaxy is coincidentally\naligned with a clump in the ICM emission to vary systematically with\nradius.  Further, the sensitivity of the HRI varies with radius, and\nso the detectability of the emission from a single galaxy will vary\nwith radius as well.  We therefore constructed the simulated data sets\nby extracting counts from the HRI image at the same radii as the true\ngalaxy locations, but at randomized azimuthal angles.  The mean\nprofile and the RMS fluctuations amongst the simulated data sets are\nshown in Fig.~3.  As might be expected, the average number of counts\nin these random data sets is zero; the larger RMS error bars at small\nradii reflect the smaller sizes of these annuli.  From a $\\chi^{2}$\ncomparison between the observed galaxy profile and the simulated\nprofile, we can conclude that there is less than 0.1\\% probability\nthat the apparent peak in the galaxy emission is produced by chance.\nThus, the detection of emission from the galaxies in Abell~2634 is\nhighly statistically significant.\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{contour.ps}\n \\caption{Contour plot of the combined image of all the early-type\ngalaxies that belong to Abell~2634. The pixel size of the image is 2\narcsec and it has been smoothed with a Gaussian kernel of 2\npixels. The center of the plot coincides with the optical centres of\nthe galaxies. The contour lines are from 20 to 100 per cent the peak\nvalue and are spaced linearly in intervals of 5 per cent.}\n\\end{center}\n\\label{contour}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{40gal_simul.ps}\n \\caption{The combined surface brightness profile of all the\nearly-type galaxies (filled squares) normalized to one galaxy. Open squares\nrepresent the average profile from  the simulations.} \n\\end{center}\n\\label{earlysimul}\n\\end{figure}\n\n\\subsection{Origin of the X-ray emission}\n\nAs mentioned in the introduction, early-type galaxies have been found\nto retain large amounts of hot gas, which extends far beyond the\noptical limits of the galaxies. X-ray binaries also contribute to the\ntotal emission, and they become more dominant in X-ray faint galaxies.\nWe might also expect some of the emission to originate from faint\nactive galactic nuclei (AGNs) in the cores of these galaxies.\nAlthough none of the galaxies in our sample has been reported as an\nactive galaxy, there is increasing dynamical evidence that the vast\nmajority of elliptical galaxies contain central massive black holes\n(van der Marel et al. 1997; Kormendy et al.  1996a, 1996b; for a\nreview Kormendy \\& Richstone 1995), and so we might expect some\ncontribution from low-level activity in such systems.  We therefore\nnow see what constraints the observed X-ray properties of the galaxies\nin Abell~2634 can place on the origins of the emission.\n\n\n\\subsubsection{The extent of the X-ray emission}\n\nOne diagnostic of the origins of the X-ray emission is the measurement\nof its spatial extent.  AGN emission should be unresolved by the HRI,\nwhile emission from X-ray binaries should be spread over a similar\nspatial scale as the optical emission, and halos of hot gas should be\nstill more extended.\n\n\\begin{figure*}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.45\\hsize\n \\epsffile{cDplotno2.ps}\n \\epsfxsize 0.45\\hsize\n \\epsffile{s3plotno2.ps}\n\\end{center}\n\\begin{center}\n \\leavevmode \n \\epsfxsize 0.45\\hsize\n \\epsffile{s4plotno2.ps}\n \\epsfxsize 0.45\\hsize\n \\epsffile{s8plotno2.ps}\n\\end{center}\n\\caption{Surface brightness distribution of the bright sources in the\nHRI image. (s1, s3, s4, and s8). Source 1 is the central cD\ngalaxy. The profile is fitted by the appropriate HRI PSF for the\ndistance of the point source from the centre of the image (dash line)\nand a Gaussian (solid line). The calculated width ($\\sigma$) of the\nbest fit Gaussian is also given.}\n\\label{sources}\n\\end{figure*}\n \nIn order to assess the spatial extent of the X-ray emission, we need\nto characterize the PSF in this HRI observation.  The point sources\ndetected in these data are more extended than the model PSF for the\nHRI detector given by Briel et al. (1996) as is seen in Fig.~4, where\nfour of the sources are fitted by this model PSF (dash line).  This\ndiscrepancy can be attributed to residual errors in the reconstruction\nof {\\it ROSAT}'s attitude, which broaden the PSF in long integrations.\nWe have therefore empirically determined the PSF that is appropriate\nfor this observation by fitting the profiles of the point sources with\na Gaussian PSF model (Fig.~4, solid line). Only sources 1, 3, 4, 8 are\nused for the determination of the width of the Gaussian.  Source 6 is\nvery elongated and can not be represented by a symmetrical\nfunction. The mean dispersion of the best-fit model was found to be $(4.1\n\\pm 0.1)$ arcsec.  All of the point sources detected in this image\nhave widths consistent with this value, and so there is no evidence\nthat the PSF varies with radius.  We therefore adopt this PSF for the\nemission from all the galaxies in the observation.\n\n\nFigure~5 shows the comparison between the adopted PSF and the emission\nfrom the cluster galaxies.  The emission appears to be more extended\nthan the PSF; fitting the data to the PSF yields a $\\chi^{2}$ value of\n14.2 with 9 degrees of freedom, which is marginally consistent with\nthe emission being unresolved.  We can obtain a better fit by modeling\nthe radial profile of the emission using a Gaussian, which we convolve\nwith the PSF to model the observed profile.  Fitting this model to the\nobservations, we find that the intrinsic width of the X-ray emission\nis $4.3^{+2.2}_{-2.8}$ arcsec. The best-fit model is also shown in\nFig.~5.\n\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{40gal_gauss_2psfs.ps}\n\\caption{The combined surface brightness distribution of the\n40 early-type galaxies, normalized to one galaxy. The profile is fitted\nby the measured HRI PSF (dashed line) and the spatially-extended model \n(solid line).}\n\\end{center}\n\\label{earlyprof}\n\\end{figure}\n\nThe radius of the X-ray halos of early-type galaxies with optical\nluminosities comparable to those in this cluster has been shown to be\n$\\sim 20$ -- 60 kpc (e.g.~Fabbiano et al. 1992), with the lower values\ncharacterizing optically fainter galaxies. At the distance of\nAbell~2634 these values correspond to $\\sim 20$ -- 60 arcsec, much\nlarger than the upper limit of $\\sim$ 7 arcsec we found for the extent\nof the galactic X-ray emission.  Thus, the X-ray emission from the\ncluster galaxies, although apparently extended, clearly does not\noriginate from the large halos of hot gas found around comparable\ngalaxies in poorer environments.\n\nOne possible explanation for the spatial extent of the emission from\nthese galaxies is that it could arise from errors in the adopted\npositions for the galaxies.  Such errors would broaden the\ndistribution of X-rays when the data from different galaxies are\nco-added even if the individual sources are unresolved.  However, the\nzero-point of the X-ray reference frame is well tied-down by\nthe detected point sources in the field.  Further, the optical\nlocations of the galaxies come from CCD photometry with positional\nerrors of less than an arcsecond.  It therefore cannot explain the\n$\\sim 4$ arcsec extent of the observed X-ray emission.\n\nWe therefore now turn to the extent of the X-ray emission that we\nmight expect from X-ray binaries.  Nearly half of the early-type\ngalaxies that we use for our analysis have been imaged in the I-band\nby Scodeggio, Giovanelli \\& Haynes (1997).  They have fitted the\noptical galaxy profile with a de Vaucouleur law, and found a mean\nvalue for their effective radii of $\\sim$8 arcsec, with only 4\ngalaxies smaller than 3 arcsec and another 3 larger than 13 arcsec.\nThese values are directly comparable to the spatial extent of the\nX-ray emission derived above.  Thus, it would appear that the\nobservations are consistent with what we would expect if the X-ray\nemission from the galaxies in Abell~2634 originates from X-ray\nbinaries in these systems, although we have not ruled out the\npossibility that some fraction of the emission comes from AGN.\n\n\\subsubsection{The luminosity of the X-ray emission}\n\nA further test of the origins of the X-ray emission in the cluster\ngalaxies comes from its luminosity.  It has been found that the blue\nluminosities of galaxies correlates with their X-ray luminosities,\nwith the optically brighter galaxies being more luminous in X-rays\n(e.g.~Forman et al. 1985; Fabbiano et al.  1992).  This correlation\nfor the early-type galaxies in the Virgo cluster is presented in\nFig. 6. The optical and X-ray luminosities of these galaxies are taken\nfrom Fabbiano et al. (1992).  The line in this plot divides the\n$L_{\\rm B} - L_{\\rm X}$ plane into two distinct galaxy types (Fabbiano\n\\& Schweizer 1995).  In addition to the differences in the ratio of\nX-ray-to-optical luminosities, galaxies in these two regions have been\nshown to possess different spectral properties.  The spectra of the\nX-ray bright galaxies [group (I)] are well fitted by Raymond-Smith\nmodels of 1 keV temperature, and it is believed that these galaxies\nretain large amounts of hot ISM.  In the spectra of the X-ray faint\ngalaxies of group (II), on the other hand, a hard component is\npresent; X-ray binaries are believed to be the major source of the\nX-rays in these galaxies.\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{fabb.ps}\n\\caption{X-ray luminosity versus blue luminosity for the early-type\ngalaxies. The line delineates the boundary between the locations of\ngalaxies where the hot ISM makes a significant contribution to the\ntotal emission [region (I)], and the locations of galaxies where the\nentire emission can be ascribed to X-ray binaries [region (II)].  The\nlocations in this plane of galaxies that belong to the Virgo cluster\nare marked by filled circles.  The average properties of galaxies in\nAbell~2634 lying in\ndifferent optical luminosity ranges are indicated by crosses.}\n\\end{center} \n\\label{LxLB}\n\\end{figure}\n\n\nIn order to see where the galaxies of Abell~2634 lie in this plot, we\nmust calculate their optical and X-ray luminosities.  Butcher \\&\nOemler (1985) have measured {\\it J} and {\\it F} optical magnitudes for\na large number of galaxies in Abell~2634.  We have converted these\nmagnitudes to the blue band by applying the colour relations provided\nby Oemler (1974) and Butcher \\& Oemler (1985), correcting for galactic\nextinction, and using the appropriate K-correction.  We find that the\nabsolute blue magnitude of the galaxies in the HRI image lie in the\nrange from $-18.5$ to $-21.4$.  We have divided these galaxies into\nthree groups according to their optical luminosities: group A\n$(0.4-2.0) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with 17 galaxies; group B\n$(2.1-3.7) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with 8 galaxies; and group\nC $(3.8-5.5) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with only 2 galaxies.\n\nThe X-ray luminosity of each group was obtained by repeating the\nanalysis of \\S2.1 using just the galaxies in each sub-sample.  Using\nthe PIMMS software we converted the observed count rate from the HRI\nimage to X-ray luminosity in the energy range 0.2-3.5 keV.  The thermal\nmodel used for the conversion is the same that was used by Fabbiano et\nal. (1992) to derive the plot shown in Fig. 6, and is discussed in\n\\S2.1.   \n\nThe resulting values for optical and X-ray luminosities in each\nsub-sample are shown in Fig.~6.  The horizontal error bars represent\nthe width of each optical luminosity bin and the vertical ones show\nthe errors in the measured X-ray luminosities.  This plot shows that\nthe galaxies in our sample follow the established correlation: the\noptically-brighter galaxies are also more luminous in the X-rays.  The\nexistence of this correlation also implies that the detected X-ray\nflux from the galaxies in Abell~2634 is not dominated by a few bright\ngalaxies, but that the optically-fainter galaxies also contribute to\nthe detected X-ray emission.\n\nThe galaxies in Abell~2634 probe the fainter end of the $L_{\\rm B}$ --\n$L_{\\rm X}$ relation as covered by Virgo galaxies.  It should be borne\nin mind that there is a bias in the Virgo data which means that the\ntwo data sets in Fig.~6 are not strictly comparable.  At the lower\nflux levels, a large number of Virgo galaxies have not been detected\nin X-rays, and so this plot preferentially picks out any X-ray-bright\nVirgo galaxies.  For the Abell~2634 data, on the other hand, the\nco-addition of data from all the galaxies in a complete sample means\nthat the data points represent a true average flux.  However, it is\nclear that the X-ray fluxes from galaxies in these two clusters are\ncomparable.\n\nThe similarity between the X-ray properties of galaxies in these two\nclusters is of particular interest because their environments differ\nsignificantly.  The galaxies from the Virgo cluster shown in Fig.~6\nlie in a region between 360 kpc and 2 Mpc from the centre of the\ncluster.  Recent {\\it ROSAT} PSPC observations have shown that the\nnumber density of the hot ICM of this cluster drops from $3 \\times\n10^{-4}$ to $3 \\times 10^{-5} \\ {\\rm cm^{-3}}$ in this region\n(Nulsen \\& B\\\"{o}hringer 1995).  The galaxies from Abell~2634 that\nhave gone into this plot lie in the inner 0.8 Mpc of Abell~2634, and\nin this region the number density of the ICM varies between $1\n\\times 10^{-3}$ and $2 \\times 10^{-4} \\ {\\rm cm^{-3}}$ (Sakelliou \\&\nMerrifield 1997). Thus, the galaxies in the current analysis come from\na region in which the intracluster gas density is, on average, an order of\nmagnitude higher that surrounding the Virgo cluster galaxies.\n\nThe location of the galaxies in region (II) of Fig.~6 adds weight to\nthe tentative conclusion of the previous section that the X-ray\nemission from these galaxies can be explained by their X-ray binary\npopulations, since any significant ISM contribution would place them in\nregion (I).  Similarly, the low X-ray fluxes of these galaxies leaves\nlittle room for a significant contribution from weak AGN.  If the\nX-ray binary populations are comparable to those assumed by Fabbiano\n\\& Schweizer (1995) in calculating the dividing line in Fig.~6, then\nessentially all the X-ray emission from these galaxies can be\nattributed to the X-ray binaries.  Thus, any average AGN emission\nbrighter than a few times $10^{40} \\ {\\rm erg \\ s^{-1}}$ can be\nexcluded, as such emission would also move the galaxies into region\n(I) of the $L_{\\rm B}$ -- $L_{\\rm X}$ plane.\n\n\n\\subsection{Spiral galaxies}\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{spiral_sim.ps}\n \\caption{The combined surface brightness profile of the spiral \ngalaxies that lie in the field of view of the HRI (filled squares)\nnormalized to one galaxy. Open squares \nrepresent the average profile from the simulations.} \n\\end{center}\n\\label{spiral}\n\\end{figure}\n\nHaving discussed the X-ray properties of the early-type galaxies in\nAbell~2634 at some length, we now turn briefly to the properties of\nthe spiral galaxies in the cluster.  Abell~2634 is a reasonably rich\nsystem, and we therefore do not expect to find many spiral galaxies\nwithin it.  Indeed, only 7 of the 62 galaxies whose redshifts place\nthem at the distance of Abell~2634, and which lie within the field of\nthe HRI, have been classified as spirals.  The statistics are\ncorrespondingly poor when the X-ray emission around these galaxies is\nco-added: the combined profile is shown in Fig.~7, together with the\nresults from the control simulations (see \\S 2.1 for details).  It is\nclear from this figure that the spirals have not been detected in this\nobservation, and a $\\chi^2$ fit confirms this impression.\n\nThe failure to detect these galaxies is not surprising.  Not only are\nthere relatively few of them, but their X-ray luminosities are lower\nthan those of early-type galaxies.  In the {\\it Einstein} energy band\n(0.2 - 3.5 keV), their luminosities have been found to lie in the\nrange $\\sim 10^{38}$ to $\\sim 10^{41} \\ {\\rm erg \\ s^{-1}}$ (Fabbiano\n1989).  Modeling this emission using a Raymond-Smith model with a\nhigher temperature than for the early-type galaxies, as appropriate\nfor spiral galaxies (Kim et al. 1992a), we find that the expected\ncount rate for these galaxies is a factor of $\\sim 40$ lower than for\nthe ellipticals in the cluster.  It is therefore unsurprising that we\nfail to detect the small number of spiral galaxies present in the\ncluster.  \n\n\n\n\\section{Discussion}\n\nIn this paper, we have detected the X-ray emission from the normal\nelliptical galaxies in Abell~2634.  The limited spatial extent of this\nemission coupled with its low luminosity is consistent with it\noriginating from normal X-ray binaries in the galaxies' stellar\npopulations.   These galaxies do not seem to have the extended\nhot ISM found around galaxies that reside in poorer cluster\nenvironments.  We therefore now discuss whether this difference can be\nunderstood in terms of the physical processes outlined in the\nintroduction.\n\nIntuitively, the simplest explanation for the absence of an extensive\nhalo around a cluster galaxy is that it has been removed by ram\npressure stripping as the galaxy travels through the ICM.  A simple\ncriterion for the efficiency of this process can be obtained by\ncomparing the gravitational force that holds the gas within the\ngalaxy to the force due to the ram pressure, which tries to remove it\n(Gunn \\& Gott 1972).\n\nThe gravitational force is given by:\n\\begin{equation}\nF_{\\rm GR} \\sim G \\ \\frac{M_{\\rm gal} \\ M_{\\rm gas}}{R_{\\rm gal}^{2}}\n\\end{equation}\nwhere $M_{\\rm gal}$ is the total mass of the galaxy, $M_{\\rm gas}$ is\nthe mass of the X-ray emitting gas, and $R_{\\rm gal}$ is the radius of\nthe galaxy's X-ray halo.  For typical values for the masses of the\ngalaxy and the gas of $10^{12} \\ M_{\\sun}$ (Forman et al.\\ 1985) and\n$5 \\times 10^{9} \\ M_{\\sun}$ (e.g.~Canizares et al.\\ 1987)\nrespectively, and a mean value for $R_{\\rm gal}$ of $40 \\ {\\rm kpc}$\n(Canizares et al. 1986), which is a representative value for galaxies\nof the same optical luminosity as the galaxies in Abell~2634,\nequation~(1) implies that $F_{\\rm GR} \\sim 1 \\times 10^{30} \\ {\\rm\nN}$.\n \nThe force due to ram pressure is described by:\n\\begin{equation}\nF_{\\rm RP}=\\rho_{\\rm ICM} \\ v_{\\rm gal}^{2} \\ \\pi R_{\\rm gal}^{2} = \\mu\n\\ m_{\\rm p} \\ n\\ v_{\\rm gal}^{2} \\ \\pi R_{\\rm gal}^{2}\n\\end{equation}\nwhere $\\rho_{\\rm ICM}$ is the density of the ICM, $\\mu$ is the mean\nmolecular weight, $m_{\\rm p}$ is the proton mass, $n$ is the number\ndensity of the ICM, and $v_{\\rm gal}$ is the galaxy velocity.  From\nthe velocity dispersion profile of Abell~2634 presented by den Hartog\n\\& Katgert (1996) we find that the velocity dispersion, $\\sigma_{\\rm\nv}$, in the inner 15 arcmin of this system is $ \\approx 710 \\ {\\rm km\n\\ s^{-1}}$. Assuming an isotropic velocity field, the characteristic\nthree-dimensional velocity of each galaxy is hence $v_{\\rm gal}=\\surd\n3 \\ \\sigma_{\\rm v} \\approx 1230 \\ {\\rm km \\ s^{-1}}$.  The number\ndensity of the ICM in the same inner region has been derived from\nrecent {\\it ROSAT} PSPC data (Schindler \\& Prieto 1997) and this HRI\nobservation (Sakelliou \\& Merrifield 1997), and is found to vary from\n$1 \\times 10^{-3} \\ {\\rm cm^{-3}}$ down to $2 \\times 10^{-4} \\ {\\rm\ncm^{-3}}$, consistent with previous {\\it Einstein} observations (Jones\n\\& Forman 1984; Eilek et al. 1984).  Inserting these values into\nequation~(2), we find $F_{\\rm RP} \\sim (1 - 10) \\times 10^{30} \\ {\\rm\nN}$.\n\nThus, the ram pressure force exerted on the galaxies in Abell~2634 is\nfound to be larger than the force of gravity, and so we might expect\nram pressure stripping to be an effective mechanism for removing the\nISM from these galaxies.  In poorer environments, the density of the\nICM is likely to be at least a factor of ten lower, and the velocities\nof galaxies will be a factor of $\\sim 3$ smaller.  We might therefore\nexpect $F_{\\rm RP}$ to be a factor of $\\sim 100$ lower in poor\nenvironments.  Since such a change would make $F_{\\rm RP} < F_{\\rm\nGR}$, it is not surprising that galaxies in poor environments manage\nto retain their extensive halos.\n\nThe absence of extensive X-ray halos around the galaxies in Abell~2634\nimplies that ram pressure stripping dominates the processes of\naccretion and stellar mass loss which can replenish the ISM.  By\ncarrying out similar deep X-ray observations of clusters spanning a\nwide range of ICM properties, it will be interesting to discover more\nprecisely what sets of physical conditions can lead to the efficient\nISM stripping that we have witnessed in Abell~2634.\n\n \n\n\\section*{ACKNOWLEDGEMENTS} \n \nWe are indebted to the referee, Alastair Edge, for a very insightful\nreport on the first incarnation of this paper.  We thank Jason Pinkney\nfor providing us with the positions and redshifts of the galaxies and\nRob Olling for helpful discussions.  This research has made use of the\nNASA\/IPAC Extragalactic Database (NED) which is operated by the Jet\nPropulsion Laboratory, California Institute of Technology, under\ncontract with the National Aeronautics and Space Administration. Much\nof the analysis was performed using {\\sc iraf}, which is distributed\nby NOAO, using computing resources provided by STARLINK.  MRM is\nsupported by a PPARC Advanced Fellowship (B\/94\/AF\/1840).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent     \n\n\n In Si based semiconductor technology Sb is considered an important \ndopant for its role in the development of field effect transistors \nand infrared detectors \\cite{temp}. Ion implantation is a useful \ntechnique for fabricating such devices as it produces buried layers \nwith well defined interfaces, expanding possibility of designing \nnovel structures. The increased density in VLSI circuits also makes \nthe technological applications of the ion implantation , especially \nin MeV energy range, increasingly important. MeV implantation however \ncan also produce severe modifications in the material depending on \nthe nature and the energy of the impinging ion, and the implantation \ndose \\cite{tam1}.\nExtensive usage of ion implantation in device fabrication and the \ncontinued miniaturization of device structures has brought the \nissue of surface modifications, via ion implantations, to the \nforefront. However, the factors responsible for such modifications \nand the surface morphology after ion implantation, have received \nlittle attention \\cite{car}.\n\n Atomic force microscope (AFM) is a very effective tool for \nexamining surface modifications and surface structures. However,\nthere are very few studies in literature that have investigated \nthe morphological changes of the ion implanted surfaces by AFM. \nFurthermore, most of these surface studies are performed after keV \nimplantations \\cite{cou,pia,wan}, or at low fluences for individual \ncascade studies \\cite{wil}. Surface modifications after high energy,\n100 MeV, ion irradiation have also been investigated \\cite{sin}. \nHowever, the role of MeV ion implantation \non the surface topography remains poorly understood. In the present \nstudy we have made detailed investigation on Si(100) surfaces after \n1.5~MeV Sb implantation. The technique of AFM has been applied to \nunderstand the modification in roughness and morphology of silicon \nsurfaces upon ion implantation. We also investigate here the formation\n of nano-sized defects zones on Si(100) surfaces after MeV implantations.\n The results of shape transition in these nanostructures, from being\n elliptical at low fluences to becoming deformed circular at high fluences,\n will also be discussed here. Experimental procedure are mentioned in section~2\n and the results will be discussed in section~3. Conclusion are presented in section~4.\n \n\\section {Experimental}\n\\noindent     \n\nA mirror polished (100)-oriented Si single crystal (p-type) \nwafer was used in the\npresent study. The samples were implanted at room temperature\nwith a scanned beam of 1.5 MeV Sb$^{2+}$ ions at various fluences\nranging from  1$\\times 10^{11}$ to 5$\\times10^{14} ions\/cm^2$.\n The implantations were performed with\nthe samples oriented 7$^o$ off-normal to the incident beam to\navoid channeling effects. \n\nAFM Nanoscope E and Nanoscope III from Veeco were used to image\nthe implanted silicon sample surfaces. Images have been acquired\nin contact and tapping modes. Images ranging from 0.2 to 10 $\\mu{m}$ square \nwere obtained.\n\n\n\\section {Results and Discussion}\n\nFigure~1 shows \nthe $1 \\times 1\\mu{m^2}$ and $200 \\times 200 {nm^2}$ \n3D-images of the virgin silicon surface. It is observed that the\n virgin Si surface is smooth. 1.5 MeV implantation was carried out at \n various fluences and several $1 \\times 1\\mu{m^2}$ and $200 \\times 200 {nm^2}$ \nimages were taken at all the fluences. The  $1 \\times 1\\mu{m^2}$ images \nwere utilized for measuring the rms surface roughness of Si(100) surfaces\n after implantation.\nThe average roughness at each fluence is plotted \nin Figure~2. The rms roughness for a virgin Si(100) surface, \nmeasured to be 0.234~nm, is also marked. It can be clearly \nseen from Fig.~2 that the rms surface roughness exhibits \nthree prominent behaviors as a function of fluence. For \nlow fluences, up to $1\\times10^{13} ions\/cm^{2}$, the \nroughness is small and does not increase much compared to \nthe virgin surface roughness. Beyond this fluence an enhanced \nsurface roughness, increasing at a much steeper rate is observed. \nThis trend continues up to the fluence of \n$1\\times10^{14} ions\/cm^{2}$ where a high roughness of 0.296~nm \nis measured. A saturation in surface roughness with a slight \ndecrease in the roughness is observed beyond this fluence. \nThe decrease in surface roughness, at \n$5\\times10^{14} ions\/cm^{2}$, seems reasonable in view of high \nlevel of amorphicity at this dose \\cite{sdey2}, as beyond a certain \nhigh amorphicity, further higher levels of amorphization should \ntend to make the surface more homogeneous. A similar decrease \nin surface roughness with increasing fluence, beyond a critical \nfluence, has been observed for keV implantations of P and As \nin amorphous films \\cite{edr}.\n\n\n Our earlier RBS\/C and Raman scattering results \n\\cite{sdey2,sdey3} show that Si lattice disorder also displays\n3 similar behaviours as a function of ion fluence. Initially a low lattice \ndamage, due to simple defects, is seen upto the fluence of \n$1\\times10^{13}ions\/cm^{2}$. The disorder becomes larger with the\nonset of crystalline\/amorphous (c\/a) transition in Si-bulk \nat $1\\times10^{13}ions\/cm^{2}$. Finally the disorder \nsaturates with the Si-lattice as well as Si-surface becoming \namorphised at $5\\times10^{15}ions\/cm^{2}$.\nThe roughness on the Si\nsurface will be determined by several roughening and smoothening\nprocess that undergo on an ion implanted surface. Nuclear Energy loss\neffects are also crucial. In addition lattice disorder and the\nassociated stress will also be important in the evolution of the\nion implanted surface.\n\n\n High resolution $200\\times200nm^{2}$ \nimages of the Si-surfaces were acquired for all the fluences\nand are shown in Figure~3 for two representative Sb fluences\nof 1$\\times10^{13}$ and 5$\\times10^{14}ions\/cm^2$. \nThe images of Si surface acquired upto the  fluence of \n1$\\times10^{12}ions\/cm^2$ (not shown) are similar to the \nvirgin surface (of Fig.~1b) and their surface roughness \nis also similar (Fig.~2). However, after a \nfluence of 1$\\times10^{13}ions\/cm^2$, several nanostructures\ncan be seen on the Si-surface (see Fig.~3a). Fig.~4a is same \nas Fig.~3a and shows the approximate outlines of some of the \nnanostructures. The nanostructures represent the damage due to \nion implantation and are roughly of elliptical \nshape. For a quantitative analysis of \nthese nanostructures, the two axial lengths \nwere measured and the mean lengths of the minor and the major \naxes are found to be 11.6 and 23.0~nm respectively. The mean lengths \nof the two axes and the mean areas of the surface features \nare tabulated in Table~1 for various incident ion fluences. \n\nFor a Sb fluence of 5$\\times10^{13}ions\/cm^2$, although the \nsilicon surface is again found to be decorated with the \nelliptical nanostructures, the features have expanded\nalong both the axial directions (with the mean lengths \nof axes being 14.5 and 26.1~nm respectively). The average \narea of the nanostructures at this stage is calculated to be \n$297~nm^2$, which is about 41\\% higher than that at \n1$\\times10^{13}ions\/cm^2$. For a fluence of \n1$\\times10^{14}ions\/cm^2$, the area of these  nanostructures\nfurther inflates to $325\\pm31~nm^2$. Although the length of \nthe minor axis has not changed much at this fluence compared \nto that at 5$\\times10^{13}ions\/cm^2$, the major axis is elongated \nand has an average value of 31.6~nm.  Upto this stage the \neccentricity of the elliptical structures, for all fluences \nis found to be $\\sim$ 0.85$\\pm$ 0.4. The eccentricity of the \nelliptical structures, at each fluence is listed in Table~ 1.  \nInterestingly, after a fluence 5$\\times10^{14}ions\/cm^2$, the \nsurface structures undergo a shape transition with the \nnanostructures having axial lengths of $30.1\\pm4.4~nm$ and \n$30.7\\pm2.4~nm$, respectively (see Fig.~3b). Fig.~4b is same \nas Fig.~3b and shows the approximate outlines of some of the \nnanostructures. The nanostructures have become much bigger in \nsize and appear somewhat of circular shape. The nanostructures \nare not fully circular and have eccentricity $\\sim$ 0.19$\\pm$ 0.05 \n(the eccentricity for circle=0). However, the eccentricity of these \nnanostructures is much reduced compared to those at lower fluences \n(where eccentricity $\\sim$ 0.85$\\pm$ 0.4). Hence, we refer to these \nnanostructures as {\\it approximately circular}. An explosion in \nsize  ($\\sim$ 120\\%) of these features compared to that at \n1$\\times10^{14}ions\/cm^2$ suggests a tremendous modification in \nsurface morphology at this stage. Our results are in contrast to \nthe keV implantation study of Sb in Si where, for doses lower \nthan 1$\\times10^{14}ions\/cm^2$, no change in the surface topography \nwas observed \\cite{cou}.\n\nThe random arrival of ions on the surface constitute the stochastic\nsurface roughening. Surface diffusion, viscous flow, and surface sputtering\netc. contribute towards the smoothening of the surface \\cite{eklund}.\nThe mechanism for the formation of surface damage is also postulated as a\nresults of cascade collision due to nuclear energy loss ($S_n$). In the \npresent study also, $S_n$ seems to be the dominating factor in the creation of\nthe nanostructures after Sb implantation. In addition several factor like\nc\/a transition in Si lattice, the strain in the surface and in bulk, defect \nand disorder in the medium etc. may also responsible for the \nstructure formation at the surface. For Sb implantation in Si we observe \nthe formation of nanostructure at Si(100) surface only after the \nfluence of 1$\\times10^{13}ions\/cm^2$. These nanostructures inflate in size \nwith increasing fluence. The size inflation may also be related to the increased \ndisorder \\cite{sdey2,sdey3} in the Si lattice. The shape transition of \nnanostructures, from being elliptical at lower fluence to deformed  circular at \n5$\\times10^{14} ions\/cm^2$ , may be caused by the increase in the density\nof the electronic excitations. Our earlier studies \\cite{sdey2,sdey3} \nshow that the amorphization of Si-surface at this stage also leads to stress\nrelaxations on the ion implanted surface.\n\n\n\\section{Summary and conclusions}\n\\noindent\n\nIn the present study we have investigated  the modifications in \nthe morphology of the Si(100) surfaces after 1.5~MeV Sb implantation. \nWe observe the presence of nano-sized defect zones on the Si surfaces \nfor the Sb fluences of 1$\\times10^{13} ions\/cm^2$ and higher.\nThese nanostructures are elliptical in shape and their size \nincrease with fluence. We observe an abrupt\nincrease in size of nanostructures accompanied by a shape\ntransition after the fluence of  5$\\times10^{14}ions\/cm^2$.  \nThe nanostructures become approximately circular at this stage. \nWe have also investigated the modifications in the surface\nroughness of the ion implanted Si surfaces and find that\nsurface roughness demonstrates 3 different stages as a function \nof fluence. \n\n\\section{Acknowledgments}\n\\noindent\n\nThis work is partly supported by ONR grant no. N00014-97-1-0991.\nWe would like to thank A.M. Srivastava for very useful comments \nand suggestions. We would also like to thank Puhup Manas\nfor his help with the figures.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nVarious types of oscillating stars reside in binary systems and   therefore a precise determination of their system and stellar parameters can be performed. In particular the combined photometric-spectroscopic analysis of eclipsing binaries (EBs hereafter) leads to the determination of absolute masses and radii of the components in a direct way \\citep[e.g.][]{2013A&A...557A..79L, 2013A&A...556A.138F}. Thanks to spectral disentangling techniques \\citep[e.g.][]{1994A&A...281..286S, 1995A&AS..114..393H, 2001LNP...573..269I, 2019A&A...623A..31S}, faint components giving rise to a contribution of a few per cent can be detected in high-resolution spectra \\citep[e.g.][]{2013A&A...557A..79L, 2014MNRAS.438.3093T, 2014MNRAS.443.3068B}. Such techniques ensure the determination of precise (up to one per cent) model-independent dynamical masses of stars, which can further be confronted with asteroseismic values provided at least one of the binary components pulsates. This also means that the results can be used to calibrate the asteroseismic mass determination that becomes more and more important, in particular for planet hosting stars \\citep[e.g.][]{2007ApJ...670L..37H, 2008JPhCS.118a2016H, 2012A&A...543A..98H}.\n\nAlgol-type systems (Algols, hereafter) are semi-detached, interacting EBs consisting of a main-sequence star of spectral type B-A (primary, hereafter) and an evolved F-K type companion (secondary, hereafter). As a consequence of the above-mentioned facts, studying pulsating Algols from complementary spectroscopic and photometric data provides a valuable test of stellar evolutionary models. The group of oscillating eclipsing Algol stars (oEAs, hereafter) \\citep{2002ASPC..259...96M} consists of eclipsing Algols with mass transfer where the mass-accreting primary shows $\\delta$\\,Sct-like oscillations. These stars are extraordinary objects for asteroseismic studies because they allow us to investigate short-term dynamical stellar evolution during mass-transfer episodes, most probably caused by the magnetic activity cycle of the less massive secondary. Basic principles of the interaction between the magnetic cycle of the cool secondary, the occurrence of rapid mass-transfer episodes, the dynamical behaviour of the system, and the excitation of different non-radial pulsation (NRP hereafter) modes of the mass-gaining primary can be studied in great detail.  \\citet{2018MNRAS.475.4745M} showed for the first time that mass transfer and accretion influences amplitudes and frequencies of NRP modes of the oEA star RZ\\,Cas, where amplitude changes are caused by the sensitivity of the mode selection mechanism to conditions in the outer envelope \\citep[e.g.][]{1998A&A...333..141P}  and frequency variations by the acceleration of the outer layers by mass and angular momentum transfer.\n\nThe details of angular momentum exchange in Algols determining their evolution are still not completely understood. This is in particular valid for systems like the so-called R\\,CMa stars  \\citep[e.g.][]{2011MNRAS.418.1764B, 2013A&A...557A..79L, 2018A&A...615A.131L} that include companions of extremely low masses. But also for the oEA star RZ\\,Cas, calculations showed that its actual configuration cannot be explained when assuming a purely conservative mass-transfer scenario in the past \\citep{2008A&A...486..919M}. In several Algols, such as RZ\\,Cas \\citep{2008A&A...480..247L} or TW Dra \\citep{2008A&A...489..321Z, 2010AJ....139.1327T}, orbital period variations were observed that can be attributed to periods of rapid mass exchange. The gas stream hits the equatorial zones of the atmosphere of the pulsating star, transfers an essential amount of angular momentum, and forces the acceleration of its outermost surface layers, thus causing strong differential rotation. While rotation alters the frequencies of the individual axisymmetric modes \\citep[e.g.][]{1981ApJ...244..299S, 2008ApJ...679.1499L}, it also lifts the degeneracy in frequency for the non-axisymmetric modes as observed for some $\\beta$\\,Cep variables \\citep[e.g.][]{2004A&A...415..241A, 2005MNRAS.360..619J}. This means that NRP modes are sensitive to the acceleration of the surface layers and that it is possible to probe the acceleration via the rotational splitting effect, as suggested by \\citet{2018MNRAS.475.4745M} for RZ\\,Cas. The study of the corresponding frequency shifts, together with the direct measurement of changes in the projected equatorial velocity ($v\\sin{i}$, hereafter) of the primary from its spectral line profiles leads to an estimation of the amount of matter transferred to the primary. The results can be compared to those obtained from 3D hydrodynamical calculations, assuming different rates of mass transfer \\citep[e.g.][]{2007ASPC..370..194M}, and finally explain the kind of mass and angular momentum transfer of the Algols.\n\nOne further advantage of oEAs is that the so-called spatial filtration or eclipse mapping effect, occurring as a result of the obscuration of parts of the stellar disc of the oscillating primary by the secondary during eclipses, simplifies the identification of the pulsation modes. The effect was predicted and found in photometric \\citep[e.g.][]{2003ASPC..292..369G, 2004A&A...419.1015M, 2005ApJ...634..602R} and spectroscopic \\citep{2018A&A...615A.131L} observations. The dynamic eclipse mapping method was introduced by \\citet{2011MNRAS.416.1601B} with the aim of NRP mode identification by reconstructing the surface intensity patterns on EBs. With modern methods such as the least-squares deconvolution (LSD) technique \\citep{1997MNRAS.291..658D}  and the pixel-by-pixel method of the FAMIAS program \\citep{2008CoAst.157..387Z}, high-$l$-degree NRP modes can also be detected and identified from the signal-to-noise ratio (S\/N hereafter) enhanced line profiles; a unique identification however is difficult \\citep[e.g.][]{2009CoAst.159...45L}. \n\nRZ\\,Cas (spectral type A3\\,V\\,+\\,K0\\,IV) is a short-period ($P$\\,=\\,1.1953\\,d) Algol and one of the best studied oEA stars.  A partial eclipse is observed during primary minimum \\citep{1994AJ....107.1141N}. The primary was found by \\citet{1998IBVS.4581....1O, 2001AJ....122..418O} to exhibit short-period light variability with a dominant oscillation mode of a frequency of 64.2\\,c\\,d$^{-1}$. This finding was later confirmed from dedicated multi-site photometric campaigns by both \\citet{2003ASPC..292..113M} and \\citet{2004MNRAS.347.1317R}. The latter authors obtained simultaneous Str\\\"omgren uvby light curves of RZ\\,Cas. These authors present a detailed photometric analysis for both binarity and pulsation, deriving WD \\citep{1971ApJ...166..605W} solutions in the four mentioned passbands as well as absolute parameters of both components, and confirming the pulsational behaviour of the primary component as found by \\citet{1998IBVS.4581....1O}. We use the radii and flux ratios between the components derived by  \\citet{2004MNRAS.347.1317R} as a reference in our spectroscopic investigation. A comprehensive overview on the observations and analysis of RZ\\,Cas can be found in \\citet{2018MNRAS.475.4745M}. \n\nStarting our  spectroscopic investigation of RZ\\,Cas in 2001, we were the first to detect rapid oscillations in its spectra \\citep{2004A&A...413..293L}, and later on in spectra taken in 2006 \\citep{2008A&A...480..247L, 2009A&A...504..991T}. From the different amplitudes of the Rossiter-McLaughlin effect \\citep[][RME hereafter]{1924ApJ....60...15R, 1924ApJ....60...22M} observed during the primary eclipses in different seasons and the modelling of line profiles over the full orbital cycle using the Shellspec07\\_inverse program, we deduced that RZ\\,Cas was in an active phase of mass transfer in 2001, whereas in 2006 it was in a quiet state. To model the surface intensity distribution of the secondary of RZ\\,Cas, we had to include a large cool spot facing the primary, presumably originating from a cooling mechanism by the enthalpy transport via the inner Lagrangian point as suggested by \\citet{1994PASJ...46..613U}. Comparing the rapid oscillations found in the radial velocities (RVs hereafter) from 2001 and 2006 also with those derived from the light curves of RZ\\,Cas taken over many years \\citep[see][]{2018MNRAS.475.4745M}, we found that the NRP pattern of RZ\\,Cas changed from season to season. Different NRP modes have been excited with different amplitudes in different years and also frequency variations of single modes were observed.  \\citet{2018MNRAS.475.4745M} suggested that these frequency variations could be caused by a temporary acceleration of the outer layers of the primary owing to angular momentum exchange by mass-transfer effects.  \n\nAfter observing RZ\\,Cas in 2001 and 2006, we took new time series of high-resolution spectra in 2008 and 2009. The fact that we found a typical timescale of about nine years from the behaviour of the pulsation amplitudes but also from light-curve analysis \\citep[see][]{2018MNRAS.475.4745M} forced us to start a spectroscopic monitoring of the star covering the years 2013 to 2017. We now investigate the complete data set with the aim to use the spectra and  observed variations in RVs and line profile variations (LPV hereafter) to deduce stellar and system parameters, check for timely variable NRP pulsation patterns, and try to correlate all observed variations with the occurrence of quiet and active phases of the Algol system.  Observations are described in Sect.\\,\\ref{Sect2}. The spectra taken with the HERMES spectrograph are used for a detailed analysis with the aim to derive precise atmospheric parameters for both components of RZ\\,Cas (Sect.\\,\\ref{Sect3}). The extraction of RVs and calculation of mean, high-S\/N line profiles with the newly developed LSDbinary program is described in Sect.\\,\\ref{Sect4}. Using different methods, we measure the projected equatorial rotation velocity of the primary (Sect.\\,\\ref{Sect5}). In Sect.\\,\\ref{Sect6} we use the O-C values collected from literature to compute the orbital period variations of RZ\\,Cas over the last decades. We check for non-Keplerian effects in the orbital RV curves in Sect.\\,\\ref{Sect7.2} and try to model them using the PHOEBE \\citep{2005ApJ...628..426P} program. The results are discussed in Sect.\\,\\ref{Sect8} followed by concluding remarks in Sect.\\,\\ref{Sect9}.\n\nOur investigation of NRP is based on high-frequency oscillations in the RVs and in LPV. Applied methods and results will be described in a forthcoming article (Paper\\,II) and discussed together with the results presented in this work.\n\n\\begin{table} \\centering\n \\tabcolsep 1.8mm\n \\caption{Journal of observations listing the instrument, its spectral resolving power, year of observation, and mean Julian date. The last four columns give the number of spectra, total time span of observations in days, number of observed nights, and number of groups of observations.}\\label{Tab01}\n \\begin{tabular}{lclcrrrr}\n \\toprule\n Source & $R$     & Year & mean JD     & $s$ & $t$ & $n$ & $g$\\\\\n \\midrule\n TCES   & 32\\,000 & 2001     & 2\\,452\\,190 &  962 &   15 &13&1\\\\\n TCES   & 32\\,000 & 2006     & 2\\,453\\,800 &  517 &  150 &7 &3\\\\\n TCES   & 32\\,000 & 2008     & 2\\,454\\,717 &   94 &   27 &5 &2\\\\\n HERMES & 85\\,000 & 2009     & 2\\,455\\,156 &  228 &    5 &5 &1\\\\\n TCES   & 32\\,000 & 2013     & 2\\,456\\,600 &  835 &  157 &21&3\\\\\n TCES   & 32\\,000 & 2014     & 2\\,456\\,938 &  696 &   13 &8 &2\\\\ \n TCES   & 58\\,000 & 2015     & 2\\,457\\,300 &  998 &  152 &31&4\\\\\n TCES   & 58\\,000 & 2016     & 2\\,457\\,647 &  586 &   14 &10&1\\\\\n TCES   & 58\\,000 & 2017     & 2\\,458\\,097 & ~~43 &    4 &3 &1\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\section{Observations}\\label{Sect2}\n\nSpectra were taken over a total time span of 16 years with the TCES spectrograph at the 2 m Alfred Jensch Telescope of the Th\\\"uringer Landessternwarte (TLS) Tautenburg and over one year with the HERMES spectrograph \\citep{2011A&A...526A..69R} at the 1.25 m Mercator Telescope on La Palma. The TCES instrument is an echelle spectrograph in Coude focus and covers the wavelength range 4450-7550\\,\\AA. A 2015 upgrade resulted in improved efficiency so that this spectrograph could be used with higher spectral resolution (narrower entrance slit of one instead of two arcsec width). Table\\,\\ref{Tab01} gives the journal of observations. \n\nThe TCES spectra were reduced using standard MIDAS packages for Echelle spectrum reduction and HERMES spectra were reduced using the HERMES spectrum reduction pipeline. We used our own routines for the normalisation of the spectra to the local continuum. Instrumental shifts  were corrected using an additional calibration based on a large number of telluric O$_2$ absorption lines.\n\n\\section{Spectrum analysis}\\label{Sect3}\n\n\\subsection{Spectra}\n\nThe HERMES spectra taken in  2009 comprise the  H$_\\epsilon$ to H$_\\alpha$ range, whereas the TLS spectra only include H$_\\beta$ and H$_\\alpha$. Moreover, the HERMES spectra provide the higher spectral resolution. That is why we decided to use them for spectrum analysis. We were able to decompose the spectra into the spectra of the components using the KOREL program \\citep{1995A&AS..114..393H}. Analysing the decomposed spectra, we faced problems with the continuum normalisation. As a result of the very faint contribution of the secondary, small deviations in the continuum of the composite spectrum from the true continuum leads to large deviations in the continuum of the decomposed spectrum of the secondary. A first normalisation was applied during spectrum reduction by fitting higher polynomials to nodes assumed to represent the local continuum in each of the extracted echelle orders. This approach is not accurate enough when the contribution of the secondary is as faint as in the present case. Corrections to the local continuum have to be applied during spectrum analysis by a comparison with the continua of synthetic spectra. The spectra are then renormalised by multiplying them with the local ratio of the continua of the synthetic spectrum to that of the reduced spectrum. The spectrum decomposition with KOREL, on the other hand, is based on Fourier transform and introduces undulations in the continuum that are additive and have to be subtracted. This introduces an ambiguity with the multiplicative corrections described before that we could not solve to reach the required accuracy. Instead, we used the average of nine composite HERMES spectra of RZ\\,Cas taken at maximum separation of the components and having about the same RV. In this case, we have to apply multiplicative continuum corrections only, for which we used spline functions. Because the signal of the secondary gets fainter for smaller wavelengths and because of the occurrence of telluric lines in the red part of the spectra we limited the spectral range to 4000-5550\\,\\AA, including the three Balmer lines H$_\\beta$, H$_\\gamma$, and H$_\\delta$. \n\n\\subsection{Methods}\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{Fig01.png}\n \\caption{Continuum flux ratios between the components vs. wavelength. The dotted lines are calculated from synthetic continuum spectra with $T_{\\rm eff}$$_1$ of 8700\\,K and $T_{\\rm eff}$$_2$ of 3800\\,K to 5000\\,K. The best agreement with the flux ratio from photometry (red line) is obtained for $T_{\\rm eff}$$_2$\\,=\\,4400\\,K (black solid line). The upper green line represents $T_{\\rm eff}$$_1$\\,=\\,9000\\,K, $T_{\\rm eff}$$_2$\\,=\\,4500\\,K, the lower green line for $T_{\\rm eff}$$_1$\\,=\\,8400\\,K, $T_{\\rm eff}$$_2$\\,=\\,4300\\,K.}\n \\label{Fig01}\n\\end{figure}\n\n\\begin{table*}\n \\tabcolsep 4mm\n \\caption{Atmospheric parameters of primary and secondary of RZ\\,Cas derived with multiple methods.}\\label{Tab02}\n \\begin{tabular}{lllllll}\n \\toprule\n           & \\multicolumn{2}{c}{Method a)} & \\multicolumn{2}{c}{Method b)} & \\multicolumn{2}{c}{Method c)}\\\\\n\\midrule\n$T_{\\rm eff}$\\ (K)   &  ~8650$\\pm$60                 & ~4860$_{-150}^{+190}$    & ~8643$\\pm$57              & ~4474$\\pm$83              & ~8635$\\pm$49     & ~4800$_{-120}^{+130}$         \\vspace{1mm}\\\\\n$\\log{g}$\\ (dex) & ~~4.41$\\pm$0.09           & ~~3.7 fix                        & ~~4.42$\\pm$0.07                & ~~3.7 fix                    & ~~4.41$\\pm$0.06  & ~~3.7 fix                        \\vspace{1mm}\\\\\n$v_{\\rm turb}$\\ (km\\,s$^{-1}$)& ~~3.60$_{-0.15}^{+0.32}$  & ~~1.42$\\pm$0.80              & ~~~3.61$_{-0.13}^{+0.17}$ & ~~1.03$_{-0.96}^{+0.63}$ & ~~3.59$\\pm$0.13  & ~~1.83$\\pm$0.75                 \\vspace{1mm}\\\\  \n$[$Fe\/H$]$ & $-$0.43$_{-0.06}^{+0.01}$ & $-$0.50$\\pm$0.20             & $-$0.42$\\pm$0.03              & $-$0.49$\\pm$0.20      & $-$0.43$\\pm$0.02 & $-$0.38$\\pm$0.18              \\vspace{1mm}\\\\\n$[$C\/H$]$  & $-$0.82$_{-0.20}^{+0.14}$ & $-$0.53$_{-0.36}^{+0.27}$& $-$0.80$_{-0.18}^{+0.13}$ & $-$0.63$_{-0.42}^{+0.29}$& $-$0.82$_{-0.18}^{+0.13}$ & $-$0.24$_{-0.46}^{+0.18}$\\vspace{1mm}\\\\\n$v\\sin{i}$\\ (km\\,s$^{-1}$)& 65.40$\\pm$0.95                & ~~~84.5$_{-7.9}^{+9.3}$  & 65.65$\\pm$0.93            & ~~~92.3$_{-9}^{+12}$  & 65.74$\\pm$0.88   & ~~~81.5$_{-7.6}^{+8.7}$   \\vspace{1mm}\\\\\n$F_2\/F_1$              & \\multicolumn{2}{c}{free}                             & \\multicolumn{2}{c}{free}                          & \\multicolumn{2}{c}{taken from photometry}                \\\\\n$(R_2\/R_1)^{\\rm calc}$ & \\multicolumn{2}{c}{0.926$\\pm$0.008}      & \\multicolumn{2}{c}{~~~~~~~1.17 fixed}    & \\multicolumn{2}{c}{0.870}                           \\\\\n$(R_2\/R_1)^{\\rm adj}$  & \\multicolumn{2}{c}{0.780}                & \\multicolumn{2}{c}{1.065}                            & \\multicolumn{2}{c}{0.870}                             \\\\\nrms (line depth)       & \\multicolumn{2}{c}{0.004499}             & \\multicolumn{2}{c}{0.004696}                         & \\multicolumn{2}{c}{0.004525}                  \\\\ \n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure*}[hbt!]\n \\includegraphics[width=\\linewidth]{Fig02.png}\n \\caption{Results of spectrum analysis using method a) (cf. Sect.\\,\\ref{Sect3.3}, Table\\,\\ref{Tab02}), shown for the H$_{\\delta}$-H$_{\\gamma}$ region (left) and the region around the Mg\\,I\\,b triplet (right). First row: Observed, continuum adjusted composite spectrum (black), best-fitting synthetic spectrum (red), and shifted difference spectrum (green). The inverse of the applied continuum correction is shown in blue. Second row: The same for the secondary. Third row: Flux ratio of the secondary to primary from photometry (black) from spectrum analysis based on $R_2\/R_1$=0.926 (red) and assuming the adjusted ratio of 0.78 (green).}\n \\label{Fig02}\n \\vspace{4mm}\n \\includegraphics[width=\\linewidth]{Fig03.png}\n \\caption{As Fig.\\,\\ref{Fig02}, but using method b). Third row: Flux ratio from photometry (black) and from spectrum analysis based on $R_2\/R_1$=1.17 (red) and on the calculated ratio of 1.065 (green).}\n \\label{Fig03}\n \\vspace{4mm}\n \\includegraphics[width=\\linewidth]{Fig04.png}\n \\caption{As Fig.\\,\\ref{Fig02} but using method c). Third row: Flux ratio from photometry (black) from spectrum analysis based on the photometric flux ratio assuming $R_2\/R_1=1.17$ (red) and on the adjusted ratio of $R_2\/R_1=0.87$ (green). Bottom row: Ratio of the photometric flux ratio to the flux ratio obtained from spectrum analysis based on $R_2\/R_1=1.17$.}\n \\label{Fig04}\n\\end{figure*}\n\nWe used the GSSP program \\citep{2015A&A...581A.129T} to derive the atmospheric parameters of the components of RZ\\,Cas. It is based on the spectrum synthesis method and performs a grid search in stellar parameters. Synthetic spectra were computed with SynthV \\citep{1996ASPC..108..198T}, based on a library of atmosphere models computed with LLmodels \\citep{2004A&A...428..993S} for the hot primary and on MARCS models \\citep{2008A&A...486..951G} for the cool secondary. Atomic data were taken from the VALD database \\citep{2000BaltA...9..590K}.\n\nBesides the atmospheric parameters, GSSP also has to solve for the a priori unknown flux ratio of the components. In the following, we use the continuum flux ratio of the secondary (component\\,2) to  primary (component\\,1) for comparison.  It is interpolated from the $uvby$ luminosities provided by \\citet{2004MNRAS.347.1317R} and shown by the red line in Fig.\\,\\ref{Fig01}. To get an impression of the influence of the $T_{\\rm eff}$\\ of primary and secondary on the resulting flux ratios with wavelength, we computed synthetic continuum spectra for different $T_{\\rm eff}$\\ of the components, assuming $\\log{g}$$_2$\\,=\\,3.7  \\citep{2004MNRAS.347.1317R}, and $\\log{g}$$_1$\\,=\\,4.4 and [Fe\/H]=$-$0.42 as derived in Sect.\\,\\ref{Sect3.3}. Synthetic flux ratios were computed from the ratio of the continuum spectra assuming a radii ratio of $R_2\/R_1$\\,=\\,1.17, also taken from \\cite{2004MNRAS.347.1317R}. The best representative theoretical curve is shown by the black line in Fig.\\,\\ref{Fig01} and corresponds to $T_{\\rm eff1}$\\,=\\,8700\\,K, $T_{\\rm eff2}$\\,=\\,4400\\,K. From the green lines, we see that a variation of $T_{\\rm eff}$$_2$ by 100\\,K requires a variation of $T_{\\rm eff}$$_1$ by 300\\,K to approximately fit the photometric flux ratio.\n\nWe applied three different methods (labelled a, b, and c in what follows) to determine the atmospheric parameters together with the radii ratio and continuum flux ratio of the secondary to primary.\n\na) We use the standard method of the GSSP program that takes the wavelength dependence of the flux ratio from the ratio of the synthetic continuum spectra calculated with SynthV and scales it with the square of the radii ratio. The latter is obtained from comparing the observed with the synthetic normalised spectra on a grid of atmospheric parameters. To obtain the atmospheric parameters together with the radii ratio, we minimise\n\\begin{equation}\n \\chi^2 = \\sum_\\lambda\\left(\\frac{\\displaystyle o(\\lambda)-s(\\lambda)}{\\displaystyle\\sigma}\\right)^2,\n \\label{GSSP1}\n\\end{equation}\n where $o(\\lambda)$ and $s(\\lambda)$ are the observed and synthetic composite spectra, respectively, and $\\sigma$ is the estimated mean error of $o(\\lambda)$. The synthetic spectrum is computed from\n\\begin{equation}\n s(\\lambda) = \\frac{\\displaystyle s_1(\\lambda,v_1)+V_F(\\lambda)\\,s_2(\\lambda,v_2)}{\\displaystyle 1+V_F(\\lambda)},\n \\label{GSSP2}\n\\end{equation}\nwhere $s_1$ and $s_2$ are the synthetic spectra of the components shifted for their RVs $v_1, v_2$ (all spectra in line depths), and the flux ratio\n\\begin{equation}\n V_F(\\lambda) = \\left(\\frac{\\displaystyle R_2}{\\displaystyle R_1}\\right)^2\\frac{\\displaystyle C_2(\\lambda)}{\\displaystyle C_1(\\lambda)}\\label{GSSP3}    \n\\end{equation}\n is the product of the squared radii ratio and the ratio of the continuum fluxes $C_2$ and $C_1$ per unit surface, determined with SynthV.\n\nb) We use Equations\\,\\ref{GSSP1} to \\ref{GSSP3} as before, but fix the radii ratio to 1.17 as obtained from the uvby photometry.\n\nc) We replace the flux ratio $V_F$ computed so far from Eq.\\,\\ref{GSSP3} by the flux ratio obtained from the $uvby$ photometry. No synthetic spectra are needed in this case. The best-fitting radii ratio can then be determined from Eq.\\,\\ref{GSSP3} using a simple least-squares fit.\n\nBecause of the known degeneracy between various free parameters, we reduced their number. The spectrum of the secondary with its small contribution to total light suffers from low S\/N and we fixed its $\\log{g}$\\ to 3.7, as derived from light curve analysis \\citep{2004MNRAS.347.1317R}. Furthermore, we used the same values for its elemental abundances as derived for the primary, except for the iron and carbon abundances, which we determined separately. For the primary, we basically know the photometric value of $\\log{g}$\\,=\\,4.33(3) from \\citet{2004MNRAS.347.1317R}. Because of the good S\/N of the spectra and the fact that the Balmer lines shapes are very sensitive to $\\log{g}$\\ in this temperature range, we decided to use $\\log{g}$\\ of the primary as a free parameter to compare the results with the photometric results. \n\nThe GSSP allows us to iterate atmospheric parameters such as $T_{\\rm eff}$, $\\log{g}$, and microturbulent velocity $v_{\\rm turb}$\\ together with $v\\sin{i}$\\ and the surface abundance of one chemical element at the same time. We started, in this way, to optimise [Fe\/H] together with the other parameters, first for the primary, then for the secondary, and repeated until the change in parameters and $\\chi^2$ became marginal. In a next step, we fixed all parameters to the values obtained so far and optimised the elemental abundances of all other chemical elements for which significant contributions in the spectra could be found.\n\n\\subsection{Results}\\label{Sect3.3}\n\nTables\\,\\ref{Tab02} and \\ref{Tab03} list the results. $(R_2\/R_1)^{\\rm calc}$ in Table\\,\\ref{Tab02} is the radii ratio obtained from the analysis using methods a, b), or c). $(R_2\/R_1)^{\\rm adj}$ is the adjusted radii ratio that follows when we apply the least-squares fit based on Eq.\\,\\ref{GSSP3}, as used in method c) for the two other methods as well. This means that when keeping all other parameters obtained with the various methods, the radii ratio has to be changed from $(R_2\/R_1)^{\\rm calc}$ to $(R_2\/R_1)^{\\rm adj}$  to give the best agreement with the photometric flux ratio. For method c), both radii ratios are identical. The quantity $(R_2\/R_1)^{\\rm adj}$ is used in the following as a measure for the agreement of the spectroscopic with the photometric results. It should be equal to 1.17 in the optimum case.\n\nFigures\\,\\ref{Fig02} to  \\ref{Fig04} compare the results from the three methods. These figures show in the first row the observed spectrum together with the best-fitting synthetic composite spectrum. The second row compares the ``observed'' spectrum of the secondary with the best-fitting synthetic spectrum found for the secondary. The observed spectrum of the secondary was therefore computed by subtracting the best-fitting spectrum of the primary from the observed composite spectrum. These spectra are rescaled according to the obtained flux ratio and the observed spectrum of the secondary is correspondingly noisy. The bottom row compares the obtained flux ratio as a function of wavelength with that obtained from photometry. The differences seen by eye are marginal. \n\nFrom Table\\,\\ref{Tab02} and Figures\\,\\ref{Fig02} to \\ref{Fig04} we conclude as follows:\nFirst, there are only small differences in the atmospheric parameters and elemental abundances derived with the different methods for the primary.\nSecond, methods a) and c) give $T_{\\rm eff}$\\ of the secondary that agree with each other within the 1\\,$\\sigma$ error bars, but both are distinctly higher than the value expected from photometry.  The $T_{\\rm eff}$\\ derived from method b), on the other hand, is in agreement with that value.\nThird, the iron and carbon abundances of the secondary cannot be distinguished from those of the primary within the 1$\\sigma$ errors (Table\\,\\ref{Tab02}), but all three methods yield a distinctly lower [C\/H] of the primary component compared to the solar value. \nFourth, all three methods deliver flux ratios higher than the photometric ratio, as can be seen from the bottom panels in Figures\\,\\ref{Fig02} to \\ref{Fig04}.  Based on the atmospheric parameters derived with methods a) and c), RZ\\,Cas should have much smaller radii ratios (the $(R_2\/R_1)^{\\rm adj}$ in Table\\,\\ref{Tab02}) than computed. From the results of light curve analysis by \\citet{2004MNRAS.347.1317R}, however, we expect a radii ratio of 1.17 and can exclude that the radius of the secondary is larger than that of the primary. From method b), on the other hand, we obtain a value of $(R_2\/R_1)^{\\rm adj}$ that is larger than unity and not so far from 1.17.\n\n\\begin{table}\n \\tabcolsep 3.3mm\n \\caption{Elemental abundances. We list the solar values based on  \\citet{2009ARA&A..47..481A}, given as 12\\,+\\,log{(E\/H)}, and the abundances of the primary of RZ\\,Cas relative to the solar values, measured with the multiple methods.}\\label{Tab03}\n\\begin{tabular}{lcccccc}\n\\toprule\n & Solar & Method a) & Method b) & Method c) \\\\\n\\midrule\nC   &8.43 & $-0.82_{-0.20}^{+0.14}$ & $-0.80_{-0.18}^{+0.13}$ & $-0.82_{-0.18}^{+0.13}$\\vspace{1mm}\\\\\nO   &8.69 & $-0.11_{-0.34}^{+0.21}$ & $-0.22_{-0.48}^{+0.24}$ & $-0.17_{-0.39}^{+0.22}$\\vspace{1mm}\\\\\nMg  &7.60 & $-0.14_{-0.05}^{+0.05}$ & $-0.17_{-0.05}^{+0.05}$ & $-0.17_{-0.05}^{+0.05}$\\vspace{1mm}\\\\\nSi  &7.51 & $-0.20_{-0.14}^{+0.12}$ & $-0.18_{-0.14}^{+0.13}$ & $-0.19_{-0.14}^{+0.13}$\\vspace{1mm}\\\\\nCa  &6.34 & $-0.39_{-0.08}^{+0.07}$ & $-0.38_{-0.08}^{+0.08}$ & $-0.39_{-0.07}^{+0.07}$\\vspace{1mm}\\\\\nSc  &3.15 & $-0.23_{-0.09}^{+0.09}$ & $-0.21_{-0.09}^{+0.09}$ & $-0.24_{-0.09}^{+0.09}$\\vspace{1mm}\\\\\nTi  &4.95 & $-0.18_{-0.04}^{+0.04}$ & $-0.20_{-0.04}^{+0.04}$ & $-0.21_{-0.04}^{+0.04}$\\vspace{1mm}\\\\\nV   &3.93 & $-0.06_{-0.20}^{+0.15}$ & $-0.11_{-0.22}^{+0.16}$ & $-0.07_{-0.19}^{+0.15}$\\vspace{1mm}\\\\\nCr  &5.64 & $-0.35_{-0.07}^{+0.06}$ & $-0.36_{-0.06}^{+0.06}$ & $-0.36_{-0.06}^{+0.06}$\\vspace{1mm}\\\\\nMn  &5.43 & $-0.45_{-0.13}^{+0.11}$ & $-0.47_{-0.14}^{+0.12}$ & $-0.46_{-0.13}^{+0.11}$\\vspace{1mm}\\\\\nFe  &7.50 & $-0.43_{-0.06}^{+0.01}$ & $-0.42_{-0.04}^{+0.02}$ & $-0.43_{-0.02}^{+0.03}$\\vspace{1mm}\\\\\nNi  &6.22 & $-0.41_{-0.15}^{+0.12}$ & $-0.41_{-0.15}^{+0.12}$ & $-0.43_{-0.15}^{+0.12}$\\vspace{1mm}\\\\\nY   &2.21 & $-0.32_{-0.26}^{+0.19}$ & $-0.32_{-0.26}^{+0.20}$ & $-0.32_{-0.26}^{+0.19}$\\vspace{1mm}\\\\\nBa  &2.18 & $-0.45_{-0.18}^{+0.17}$ & $-0.42_{-0.18}^{+0.17}$ & \n$-0.45_{-0.17}^{+0.17}$\\\\            \n \\bottomrule                    \n \\end{tabular}                  \n\\end{table}\n\nThere is a large degeneracy between various parameters such as $T_{\\rm eff}$, $\\log{g}$, $v_{\\rm turb}$, [Fe\/H] (both components), [Mg\/H] (cool component), and radii ratio. This explains why we end up in both methods a) and c) with extraordinary small radii ratios on costs of higher $T_{\\rm eff}$\\  of the secondary (much too high when comparing with Fig.\\,\\ref{Fig01}). We assume that the number of degrees of freedom is too high when trying to optimise the radii ratio together with the other parameters; the signal from the faint companion in our composite spectra is simply too low for that. Thus we observe, although it does not give the smallest root mean square (rms) of the O-C residuals, that the most reliable results are obtained when fixing the flux ratio to the photometric ratio as we did in method b). \n\n\\section{Radial velocities and LSD profiles with LSDbinary}\\label{Sect4}\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\linewidth]{Fig05.png}\n \\caption{LSD profiles computed with LSDbinary from spectra taken during primary eclipse for the primary (left) and secondary (right) of RZ\\,Cas. Phase zero corresponds to Min\\,I.}\n \\label{Fig05}\n\\end{figure}\n\nThe classical method of LSD \\citep[][]{1997MNRAS.291..658D} is based on using one line mask as a template. In the result,  a strong deconvolved line profile is obtained for the star that best matches this template, whereas the contribution of the other component is more or less suppressed. \\citet{2013A&A...560A..37T} generalised the method so that it  can simultaneously compute an arbitrary number of LSD profiles from an arbitrary number of line masks. In this work, we used the LSDbinary program written by V. Tsymbal, which computes separated LSD profiles for the two stellar components using as templates two synthetic spectra that are based on two different atmosphere models. The program also delivers optimised values of the RVs of the components and their radii ratio. A first successful application of the program to the short-period Algol \\object{R CMa} is presented in \\cite{2018A&A...615A.131L}; this work also includes a short description of the program algorithms and shows the advantages of LSDbinary against TODCOR \\citep{1994ApJ...420..806Z} in the case of very small flux ratios between the components of binary stars.\n\nWe applied LSDbinary to the spectra of RZ\\,Cas to obtain the separated LSD profiles and RVs of its components. Figure\\,\\ref{Fig05} shows LSD profiles computed from RZ\\,Cas spectra taken during the primary eclipse in 2016 as an example. The RME can clearly be seen in the profiles of the primary, as well as the RV variation due to orbital motion in the profiles of both components. We will use the obtained RVs and LSD profiles for a detailed investigation of the stellar and system parameters of RZ\\,Cas in this first part and of the pulsations of its primary component in Paper\\,II.\n\n\\section{Rotation velocity of the primary}\\label{Sect5}\n\nWe applied three different methods to determine  $v\\sin{i}$\\ of the primary of RZ\\,Cas. In all cases, we only use spectra taken in out-of-eclipse phases.\n\n\\subsection{Fourier method}\\label{Sect5.1}\nFirst, we applied the Fourier method \\citep{1933MNRAS..93..478C, 1976PASP...88..809S, 2005oasp.book.....G} to the calculated LSD profiles. This method (DFT hereafter) is based on the determination of the rotation-broadening related zero points in the Fourier power spectra of the profiles. We assumed a linear limb darkening law with a limb darkening coefficient $\\beta$\\,=\\,1.5 (or $b$\\,=\\,0.6 with $b=\\beta\/(1+\\beta)$). Varying the value of $\\beta$ leads to slight systematic effects in $v\\sin{i}$, but changes were minor compared to the differences to the results from the two other methods described below. The selection of zero points was based on a 3$\\sigma$ clipping, comparing the $v\\sin{i}$\\ from single zero points with the mean values from all zero points.\n\n\\noindent In the result, we obtained $v\\sin{i}$\\ values strongly varying with orbital phase. Figure\\,\\ref{Fig06} shows an example.  We assume that this variation comes from Algol-typical effects such as an inhomogeneous circum-primary gas-density distribution and\/or surface structures caused by the influence of mass transfer and gas stream from the secondary. In a next step, we removed all the variations found in the line profiles with the pixel-by-pixel method of FAMIAS \\citep{2008CoAst.157..387Z} by subtracting all frequency contributions computed with the mentioned program from the profiles. This method will be explained in detail in Paper\\,II. For each season, we considered a certain pixel of all LSD profiles as part of one time series from which we subtracted the found contributions and did this pixel-by-pixel to build the undistorted profiles. We performed that for all seasons but 2008 for which we did not have enough data to apply the pixel-by-pixel method. The subtraction of the high-amplitude, low-frequency contributions found with FAMIAS (most of them are harmonics of the orbital frequency) are responsible for the cleaning, not the faint high-frequency oscillations due to pulsation.  The resulting $v\\sin{i}$\\ are shown in Fig.\\,\\ref{Fig06} in red. The distribution is now much flatter and we used it to calculate the $v\\sin{i}$\\ of the corresponding season as its arithmetic mean. Results are listed in the third column of Table\\,\\ref{Tab04}.\n\n\\subsection{Single spectral line} \nNext, we looked for stronger spectral lines of the primary in the composite spectra that are free of blends from the cool companion. We found only one line consisting of the Fe\\,II doublet 5316.61\/5316.78\\,\\AA\\ that fulfils that condition. For each season, we fitted the line profiles by synthetic spectra computed with the parameters listed in Table\\,\\ref{Tab02}, method b) to determine the best-fitting $v\\sin{i}$\\ and its error. This means that we were fixing all atmospheric parameters except for  $v\\sin{i}$\\ to the solution determined from the spectra observed in 2009 when the star was in a relatively quiet phase. However, this approach does not account for the effects in active phases of RZ\\,Cas such as the attenuation of light by circumstellar material as found for the year 2001 for example \\citep{2009A&A...504..991T}. Thus we used two more free parameters in our fit, correction factor $a$ counting for different line depths caused by the presumed effects, and factor $b$ to correct the continuum of the observed spectrum in the vicinity of the Fe\\,II line.  Both were determined from a least-squares fit\n\\begin{equation}\n    \\{1-[1-a\\,P_s(v\\sin{i})]-b\\,P_o\\}^2~\\longrightarrow~min.\n\\end{equation}\nwhere $P_s$ is the synthetic and $P_o$ the observed line profile. Finally, we built the mean $v\\sin{i}$\\ per season from the weighted mean of well-selected data points using  3$\\sigma$ clipping. Figure\\,\\ref{Fig07} shows two examples: one for 2001 in which the star was in an active phase, and one for 2014 in which the $v\\sin{i}$\\ shows a much smoother behaviour with orbital phase. The results are listed in the fourth column of Table\\,\\ref{Tab04}.\n\n\\begin{figure}\\centering\n \\includegraphics[angle=-90, width=\\linewidth]{Fig06.png}\n \\caption{Values of $v\\sin{i}$\\ obtained from DFT vs. out-of-eclipse phases measured from the LSD profiles observed in 2016 (black) and from the same profiles corrected for the LPV found with FAMIAS (red).}\n \\label{Fig06}\n\\end{figure}\n\n\\begin{figure}\\centering\n \\includegraphics[width=.82\\linewidth]{Fig07.png}\n \\caption{Values of $v\\sin{i}$\\ determined from the Fe\\,II 5317\\,\\AA\\ line vs. orbital phase, shown for 2001 and 2014. The mean values were built from the values indicated in green.}\n \\label{Fig07}\n\\end{figure}\n\n\\subsection{Using FAMIAS}\nIn Paper\\,II, we will \nuse the moment methods \\citep{1992A&A...266..294A} of the FAMIAS program for a mode identification of low-$l$ degree modes. For that, we cleaned the observed LSD profiles for all low-frequency distributions in the same way as described in Sect.\\,\\ref{Sect5.1}. One free parameter in applying the moment method is the $v\\sin{i}$\\ of the primary, which optimum value and 1$\\sigma$ error we obtained from the resulting $\\chi^2$-distribution. The values are listed in the last column of Table\\,\\ref{Tab04}. The number of observations in 2008 were not sufficient to apply this method.\n\n\\subsection{Comparison}\\label{Sect5.4}\nThe $v\\sin{i}$\\ determined with the three different methods are shown in Fig.\\,\\ref{Fig08}. The values obtained from DFT and FAMIAS agree in most cases within the 1$\\sigma$ error bars; there is a larger difference for 2006. A systematic offset can be observed for the $v\\sin{i}$\\ values obtained from the Fe\\,II line. This is in particular the case for 2001 when RZ\\,Cas was in an active phase. We assume that the offset comes from the cleaning procedures that we applied when using the other two methods. In these cases we removed the low-frequency contributions that we found with the pixel-by-pixel method of FAMIAS from the LSD profiles so that we removed the line broadening effects due to orbital-phase dependent dilution effects by circumstellar material in this way. \n\n\\begin{table}\n\\tabcolsep 1.4mm\n\\caption{Values of $v\\sin{i}$\\  obtained from the three methods in multiple years. JD is JD\\,2\\,450\\,000+. The last column gives the rotation-to-orbit synchronisation factor. Here and in the following, values in parentheses are the errors in units of the last digits.}\\label{Tab04}\n\\begin{tabular}{cccccc}\n\\toprule\nYear &  JD & $v\\sin{i}$$_{\\rm DFT}$ &$v\\sin{i}$$_{\\rm FeII}$ & $v\\sin{i}$$_{\\rm FAMIAS}$ & $F_1$ \\\\\n     &   &  (km\\,s$^{-1}$)        & (km\\,s$^{-1}$)          & (km\\,s$^{-1}$) & \\\\\n\\midrule\n2001 & 2192 & 68.9(1.2) & 74.1(2.4) & 69.47(39) & 0.978(15)\\\\\n2006 & 3768 & 64.33(74) & 67.0(1.6) & 65.95(28) & 0.921(12)\\\\\n2008 & 4717 & 63.39(27) & 63.6(1.0) &  --       & 0.896(11)\\\\\n2009 & 5156 & 63.91(45) & 64.49(59) & 62.99(25) & 0.897(11)\\\\\n2013 & 6635 & 65.16(87) & 66.9(1.3) & 64.43(35) & 0.916(13)\\\\\n2014 & 6936 & 64.33(61) & 65.3(1.1) & 64.94(26) & 0.914(12)\\\\\n2015 & 7278 & 64.31(67) & 66.11(91) & 64.08(23) & 0.908(12)\\\\\n2016 & 7646 & 63.62(65) & 64.6(1.0) & 64.87(29) & 0.909(12)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\\centering\n\\includegraphics[angle=-90, width=\\linewidth]{Fig08.png}\n\\caption{Values of $v\\sin{i}$\\  measured from a) DFT (black crosses), b) the Fe\\,II line (red squares), and c) FAMIAS (green plus signs). The solid and dotted lines show possible sinusoidal variations (see text).}\n\\label{Fig08}\n\\end{figure}\n\nFinally, we built a spline fit through the averaged values obtained from DFT and  FAMIAS and interpret this as the typical variation of $v\\sin{i}$\\ over the epoch of our observations.\nWe see that $v\\sin{i}$\\ was distinctly larger in 2001 compared to the other seasons. We assume that its value increased during the active phase of RZ\\,Cas near to 2001 because of the acceleration of the outer layers of  the primary by angular momentum transport via mass transfer from the cool component (see Sect.\\,\\ref{Sect8} for a more detailed discussion). A period search in the values averaged from DFT and FAMIAS gives a period of 9.6\\,yr, overlaid on a long-term trend (the solid line in Fig.\\,\\ref{Fig08}). A fit based on the 9.0\\,yr period as discussed in Sect.\\,\\ref{Sect7.2.3} is shown by the dotted line. The difference is marginal. The last column in Table\\,\\ref{Tab04} lists the rotation-to-orbit synchronisation factor of the primary, $F_1$, calculated from its radius, taken from \\citet{2004MNRAS.347.1317R} as 1.67\\,$\\pm$\\,0.02\\,R$_\\odot$, and the arithmetic mean of the $v\\sin{i}$\\ measured with DFT and FAMIAS. The result shows that the primary rotates sub-synchronously, only in 2001 it reaches almost synchronous rotation velocity.\n\n\\section{Orbital period changes}\\label{Sect6}\n\nIt is hard to search for orbital period changes in the range of a few seconds when the orbital RV curves are heavily distorted by non-Keplerian effects such as in the case of RZ\\,Cas  (see Fig.\\,\\ref{Fig13}). Table\\,\\ref{Tab05} lists the orbital periods and times of minimum ($T_{\\rm Min}$ hereafter) derived from the RVs in different seasons using the method of differential corrections \\citep{1910PAllO...1...33S, 1941PNAS...27..175S}. The 1$\\sigma$ errors of the period are of the order of two seconds or more, which is larger than the expected period changes.  Also the inclusion of effects such as Roche geometry of the components and spots on the stellar surfaces into the PHOEBE calculations (Sect.\\,\\ref{Sect7.2}) did not help us reach the desired accuracy in orbital period. The $T_{\\rm Min}$, on the other hand, could be very precisely determined.\n\n\\begin{table}\\centering\n\\tabcolsep 1.3mm\n\\caption{Orbital periods $P_{RV}$ and times of minimum $T_{\\rm RV}$ derived from the RVs from single seasons, and period changes $\\triangle P_{phot}$ and periods $P_{phot}$ derived from the photometric $T_{\\rm Min}$.}\n\\label{Tab05}\n\\begin{tabular}{cllrc}\n\\toprule\nYear  & \\multicolumn{1}{c}{$P_{RV}$}                  & \\multicolumn{1}{c}{$T_{\\rm RV}$} \n      & \\multicolumn{1}{c}{$\\triangle P_{phot}$} & \\multicolumn{1}{c}{$P_{phot}$}\\\\\n      & \\multicolumn{1}{c}{(d)}                    & \\multicolumn{1}{c}{BJD\\,2450000+}\n      & (sec)                                      & (d) \\\\\n\\midrule\n2001 &  1.19572(30)  & 2190.9954063(32) &    1.07 & 1.1952626\\\\\n2006 &  1.195248(20) & 3775.9076591(23) & $-$0.15 & 1.1952486\\\\\n2008 &  1.19525(26)  & 4717.7661978(51) & $-$0.03 & 1.1952500\\\\\n2009 &  1.1954(13)   & 5156.4217537(36) &    0.40 & 1.1952549\\\\\n2013 &  1.195307(65) & 6663.6333727(22) &    0.31 & 1.1952539\\\\\n2014 &  1.19533(28)  & 6936.1537115(17) &    0.45 & 1.1952554\\\\\n2015 &  1.195278(17) & 7279.1944163(12) &    0.70 & 1.1952583\\\\\n2016 &  1.19497(26)  & 7644.9432256(20) &    0.67 & 1.1952580\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=.9\\linewidth]{Fig09.png} \n\\caption{Radial velocities of the primary in 2006 (black) and 2001 (red) vs. orbital phase. Phase is calculated from $P$ and $T_{\\rm min}$ as derived from the RVs in 2006.}\n\\label{Fig09}\n\\end{figure}\n\nWhen we plot the RVs versus orbital phase where the latter is calculated from $T_{\\rm min}$ and $P$ of one single season, we see a clear phase shift in RV compared to other seasons (Fig.\\,\\ref{Fig09}). Since we do not have any information about the behaviour of the system in between, we cannot deduce period shifts from our RVs alone, however. To fix the problem, we used the photometric $T_{\\rm Min}$ from literature. We collected data from the O-C Gateway\\footnote{http:\/\/var2.astro.cz\/ocgate\/index.php?lang=en}, Bob Nelson's Data Base of O-C Values\\footnote{w.aavso.org\/bob-nelsons-o-c-files}, $T_{\\rm Min}$ kindly provided by J.\\,Kreiner\\footnote{https:\/\/www.as.up.krakow.pl\/o-c\/} \\citep[also see][]{2004AcA....54..207K}, and unpublished data from D.\\,Mkrtichian. Cross-checking for duplicates and rejecting all visual observations, we ended up with 605 $T_{\\rm Min}$ covering the time span from 1896 to 2019, to which we added the eight $T_{\\rm Min}$ derived from our spectra. We converted all dates given as HJD to BJD based on terrestial time TT. Then we computed the overall best fitting period from a least-squares fit $T_{\\rm Min}$ versus season number $E$, yielding $P_0$\\,=\\,1.195250392(61)\\,d and $T_0$\\,=\\,2\\,453\\,775.89453(79). The resulting values\n \\begin{equation}\n     O-C = T_E-T_0-P_0E\n \\end{equation}\nare shown in Fig.\\,\\ref{Fig10}. \n\nThere exist different approaches to determine the local period from an O-C diagram. The classic method is to fit segments by linear or parabolic functions to calculate constant periods or linearly changing periods per segment, respectively. But there are also approaches that assume a continuous change of the orbital period like that of \\citet{1994A&A...282..775K}, see \\citet{2001OAP....14...91R} for an overview on other methods. We applied a similar procedure as used in \\citet{2018MNRAS.475.4745M}, interpolating the O-C data to a grid of step width one in season $E$, using spline fits together with 3$\\sigma$-clipping and computing the period change from the local slope of the resulting fit. It is\n \\begin{equation}\\label{deltaP}\n     (O-C)_E-(O-C)_{E-1} = T_E-T_{E-1}-P_0\n \\end{equation}\nwhere $T_E$\\,$-$\\,$T_{E-1}$ is the local period and thus Eq.\\,\\ref{deltaP} describes the local difference $\\triangle P$\\,=\\,$P$\\,$-$\\,$P_0$. A necessary precondition for our method is that the observed data points are sufficiently dense so that the spline fit gives a reliable prediction of the behaviour between these points. For that reason, we only considered all $T_{\\rm Min}$ observed after BJD 2\\,437\\,000.\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=\\linewidth]{Fig10.png}\n\\caption{O-C values calculated from the corrected $T_{\\rm Min}$, full range in JD.}\n\\label{Fig10}\n\\end{figure}\n \n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig11.png}\n\\caption{O-C values and derived period changes. Top: O-C values (filled black circles) fitted by splines (red line). Values considered as outliers are shown by open circles and those derived from our spectra in green. Bottom: Period changes calculated from the local slope of the spline fit (see text). Open circles indicate the seasons of our spectroscopic observations.}\n\\label{Fig11}\n\\end{figure}\n\nFigure\\,\\ref{Fig11} shows in its top panel the O-C values together with the spline fit. The $T_{\\rm Min}$ obtained from our RVs ($T_{\\rm RV}$ in Table\\,\\ref{Tab05}) are shown in green and fit very well. The bottom panel shows the resulting period changes by the solid line, where we assumed a ``best-selected'' smoothness parameter for the underlying spline fit. To give an impression of the influence of the smoothness parameter, we also show (dotted line) the period changes resulting from a much larger parameter, leading to a  more relaxed spline fit. The positions of our spectroscopic observations are indicated. Obtained period changes and periods are listed in Table\\,\\ref{Tab05} as $\\triangle P_{LC}$ and $P_{LC}$. \n\n\\citet{2018MNRAS.475.4745M} derived typical timescales of 4.8, 6.1, and 9.2\\,yr from the O-C variations. We did a frequency search in the obtained period changes using the PERIOD04 program \\citep{2005CoAst.146...53L} and found the six periods listed in Table\\,\\ref{Tab06}. We do not assume that the observed variation can be described as strictly cyclic but consider the found periods as typical timescales that describe the behaviour in certain seasons. Figure\\,\\ref{Fig12} illustrates this. It shows the calculated period changes together with the best fit of the six periods and also single contributions from four of the six periods. The longest period of 52.7\\,yr describes a long-term trend in the data. The 6.3\\,yr period describes the variations before BJD 2\\,442\\,560 (segment A), and the 8.6\\,yr period the behaviour between 2\\,445\\,240 and 2\\,456\\,070 (segment B). The 14.5\\,yr period is responsible for an amplitude variation over a longer timescale. The remaining two periods of 6.9 yr and 10.8\\,yr cannot directly be linked to the variations in this way, but improve the fit by counting for their non-cyclicity. When doing a separate frequency search in segments A and B, we find the dominant periods as 5.9\\,yr and 8.4\\,yr, respectively. This shows that we can only estimate the order of the time scales underlying the orbital period variations but cannot use the errors obtained from frequency search as a measure for the accuracy of the obtained values.\n\n\\begin{figure}\\centering\n\\includegraphics[angle=-90, width=\\linewidth]{Fig12.png}\n\\caption{Calculated period changes (solid line) fitted by six frequencies (dashed line). The dotted lines show (from top to bottom) the contributions of the 52.7, 6.3, 8.6, and 14.5\\,yr periods.}\n\\label{Fig12}\n\\end{figure}\n\n\\begin{table}\n\\tabcolsep 2.65mm\n\\caption{Timescales and amplitudes of orbital period variation.}\n\\label{Tab06}\n\\begin{tabular}{lrrrrrr}\n\\toprule\n$P$ (y)  & 6.32 & 6.94 & 8.60 & 10.78 & 14.55 & 52.70\\\\\n$A$ (sec)& 0.34 & 0.37 & 0.37 &  0.24 &  0.26 &  0.45\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}\\centering\n\\includegraphics[width=0.76\\textwidth]{Fig13.png}\n\\caption{Radial velocity residuals. Left: the Rossiter-McLaughlin effect in the RVs of the primary in different years (red) after subtracting the best-fitting Keplerian orbits. For comparison, the RVs from 2006 are shown in black. In 2013, the RVs are divided into BJD\\,<\\,2\\,456\\,600 (2013a, red) and BJD\\,>\\,2\\,456\\,600 (2013b, green), and in 2015 into BJD\\,<\\,2\\,457\\,250 (2015a, red) and BJD\\,>\\,2\\,457\\,250 (2015b, green). Right: The same for the RVs of the secondary, shown over a larger range in orbital phase. Phase zero corresponds to Min\\,I in each case.}\n\\label{Fig13}\n\\end{figure*}\n\n\\section{Radial velocity variations with orbital phase}\n\\subsection{Keplerian approach}\\label{Sect7.1}\n\nFigure\\,\\ref{Fig13} shows the RVs folded with the orbital period after subtracting the best-fitting Keplerian orbits computed with the method of differential corrections as mentioned in Sect.\\,\\ref{Sect6}. We use the observations from 2006 when RZ\\,Cas was in a quiet state for comparison, shown by black dots. The deviations from a straight line (pure Keplerian motion) of the RVs of the secondary seen in 2006 are due to its non-spherical shape and inhomogeneous surface intensity distribution as discussed in \\citet{2009A&A...504..991T}, which we investigate in detail in Sect.\\,\\ref{Sect7.2}. More or less strong deviations from the behaviour in 2006 can be seen in different years. \n\nLooking at the behaviour of the RVs of the secondary, we find the strongest deviations in 2001 where we see large variations with orbital phase. Almost no differences to 2006 are observed in 2009, 2015, and 2016, whereas moderate differences occur in 2013 and 2014. Interpreting the strength of the deviations as activity indicator, we conclude that we observed RZ\\,Cas in or just after a mass-transfer episode in 2001 and in a quiet state in 2006 (as already stated in \\citet{2008A&A...480..247L}, and \\citet{2009A&A...504..991T}), and that this quiet phase continued until 2009, followed by a slightly more active phase around 2013 and 2014, and falling back into a quiet state in 2015 and 2016.\n\nLooking at the behaviour of the RVs of the primary, in particular at the amplitude and shape of the RME, we see a different picture. The amplitude of the RME in 2001 is much larger and its shape is strongly asymmetric. In all the other years, however, we see almost no difference to 2006, except for three nights of observation covering the ingress of the eclipses: one in 2009, one in 2013, and one in 2015. \n\n\\subsection{Analysis with PHOEBE}\\label{Sect7.2}\n\nRZ Cas is a semi-detached Algol-type binary system and its cool component fills its critical Roche lobe. Tidal distortions occur and lead to non-spherical shapes. According to \\citet{2013MNRAS.431.2024S}, non-negligible effects on RVs occur for $a<20\\,(R_1+R_2)$, that is when the separation of the components is smaller than 20 times the sum of their radii. Thus, the a priori assumption of Keplerian orbit, which considers the mass centre of the stellar disc coinciding with its intensity centre, is no longer valid. As already mentioned in previous sections, RZ Cas is undergoing stages of active mass transfer. A hot region around the equatorial belt has long been known \\citep{1982ApJ...259..702O}. Moreover, the recent comprehensive tomographic study of \\citet{2014ApJ...795..160R} revealed clear indicators of mass stream activity in several short period Algols, including RZ Cas. \\citet{1994PASJ...46..613U} found that the secondary of RZ\\,Cas can be modelled only when assuming an unusually large value of the gravity darkening exponent of 0.53 and inferred that dark spots are present on the front and back sides of the secondary with respect to the primary. These authors supposed that quasi-radial flow in the sub-adiabatic stellar envelope from the deep interior is the cause of darkening. \\citet{2008ysc..conf...33T, 2009A&A...504..991T} confirmed the finding of the two spots, resuming the interpretation. \\citet{2018MNRAS.481.5660D} and \\citet{2018A&A...611A..69B}, on the other hand, give an alternative explanation, showing for the short-period Algols \\object{$\\delta$~Lib} and \\object{$\\lambda$~Tau} that the mass stream can produce a light scattering cloud in front of the surface of the Roche lobe filling secondary facing the primary.\n\n\\subsubsection{Method}\n\nTo account for all these effects, we need to use sophisticated models to simulate RV changes during the whole orbital motion via integrating both components surface intensities. For this purpose, we used the well-known Wilson-Devinney code \\citep{1971ApJ...166..605W, 2005Ap&SS.296..121V} through the PHOEBE interface \\citep{2005ApJ...628..426P}. The WD code consists of light and\/or RV curve synthesiser (LC) and parameter optimiser (DC) for fitting purpose. As  we describe later, our attempts to optimise the multi-parameter fit failed with DC, which is expected for such complex configurations. Thus we used LC via our Markov Chain Monte Carlo (MCMC hereafter) optimiser written in Phoebe-scripter extension, to both optimise the parameters and explore the parameter space \\citep[see][]{2018MNRAS.481.5660D}.\n\nIn MCMC runs, we start from a random initial parameter set and add accepted parameters to the Markov chain. We need to run a lot of chains (or \"walkers\") to avoid the issue of one Markov chain sticking in a local $\\chi^2$ valley. That is why it is advisable to start the simulation with at least two times the number of walkers of the number of parameters to be optimised \\citep[see][]{2013PASP..125..306F}. For our last sets of simulation, we set the number of walkers > 20 and the lowest iteration numbers are dynamically increased to fulfil the condition by \\citet{2013PASP..125..306F} that the iteration number should be at least ten times the auto-correlation time (i.e. typically 500,000 iterations per season). \n\n\\subsubsection{Application}\n\nSpots are described in WD by two coordinates, co-latitude $\\delta$ and longitude $\\lambda$, and two physical parameters, temperature ratio compared to the normal surface $T$ and radius $R$. In our initial run, we started to model the RVs of the cool component by taking one spot (Spot1) on the cool secondary facing the primary into account to mimic a scattering cloud or diffusive material between the components. All of our models converged to spot position $\\delta\\approx 90^\\circ$, $\\lambda\\approx 0^\\circ$, that is the region that faces the primary. However, we found a strong correlation between spot size and temperature ratio of the form of $R^2T^4$\\,=\\,constant, balancing a lower temperature ratio by a more extended spot. Thus, we could only fit the \"strength\" or \"contrast\" of the spot with respect to the stellar surface, fixing the temperature ratio to a reasonable lower limit of 0.76. As mentioned at the beginning of Sect.\\,\\ref{Sect7.2}, \\citet{1994PASJ...46..613U} suggested dark spots at the front and back sides of the cool secondary towards the primary caused by mass transfer. We therefore implemented a second spot (Spot2) on the opposite side of the surface of the secondary to check if this offers further improvement.\n\nTo check if we can detect a variation of the filling factor between active and inactive phases of RZ\\,Cas, we used the \"unconstrained mode \" of WD and the surface potential of the secondary as a free parameter. In all of our runs, however, the filling factor of the cool component converged to unity. So for the final analysis, we fixed it to unity and used the \"semi-detached mode\". Eccentricity was set to zero and the orbital inclination to 82$^\\circ$. The temperature ratio was fixed to 0.76 for both spots on the secondary and, as already mentioned, the position of Spot1 to $\\delta_1$\\,=\\,90$^\\circ$, $\\lambda_1$\\,=\\,0$^\\circ$. The location of Spot2 converged to $\\delta_2$\\,=\\,90$^\\circ$ without showing larger scatter, so we fixed $\\delta_2$ as well.\n\n\\subsubsection{Results}\\label{Sect7.2.3}\n\nFree parameters were the synchronisation parameter for the primary, $F_1$, the radii of Spot1 and Spot2 on the secondary, $R_1^{sec}$ and $R_2^{sec}$, the longitude of Spot2, $\\lambda_2^{sec}$, and the systemic velocity $V_\\gamma$. The RV semi-amplitudes were allowed to vary within the error bars derived in previous attempts and led to the values as listed in Table\\,\\ref{Tab07}. \n\n\\begin{table}\\centering\n\\tabcolsep 4.1mm\n\\caption{Basic absolute values derived with PHOEBE.} \n    \\label{Tab07}\n    \\begin{tabular}{llll}\n    \\toprule  \n    $q$            & 0.350782(47) & $M_1$ (M$_\\sun$) & 1.9507(54)\\\\\n    $a$ (R$_\\sun$) & 6.5464(54)   & $M_2$ (M$_\\sun$) & 0.6843(13) \\\\\n    \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\tabcolsep 1.45mm\n\\caption{Synchronisation factor of the primary, radii of both spots on the secondary, longitude of Spot2 on the secondary, and mean scatter of the RV residuals.}\\label{Tab08}\n\\begin{tabular}{lccccc}\n\\toprule\nSeason & $F_1$ & $R_1^{sec}$ & $R_2^{sec}$ & $\\lambda_2^{sec}$ & rms\\\\\n       &       & \\small (deg)& \\small (deg)& \\small (deg)      & \\small (km\\,s$^{-1}$)\\\\\n\\midrule\n2001  & 1.217(20) & $23.09_{-1.40}^{+0.80}$  & $22.8_{-2.2}^{+2.1}$ & $136.7_{-4.1}^{+3.9}$& 5.31\\vspace{1mm}\\\\\n2006  & 0.924(13) & $41.04_{-0.92}^{+0.92}$  & $10.6_{-2.5}^{+1.9}$ & $150  _{-20 }^{+26 }$& 2.47\\vspace{1mm}\\\\\n2009  & 1.006(11) & $36.55_{-0.74}^{+0.74}$  & $10.8_{-1.7}^{+1.4}$ & $244.4_{-6.1}^{+6.6}$& 1.62\\vspace{1mm}\\\\ \n2013a & 0.897(08) & $28.69_{-1.08}^{+0.64}$  & $24.0_{-1.0}^{+0.9}$ & $180.0_{-2.4}^{+2.4}$& 3.46\\vspace{1mm}\\\\\n2013b & 0.991(14) & $28.01_{-1.09}^{+0.99}$  & $23.7_{-1.2}^{+1.2}$ & $218.4_{-2.9}^{+3.1}$& 3.60\\vspace{1mm}\\\\  \n2014  & 0.812(12) & $33.16_{-0.84}^{+0.96}$  & $21.3_{-1.2}^{+1.2}$ & $209.4_{-3.1}^{+3.3}$& 2.92\\vspace{1mm}\\\\  \n2015a & 1.057(13) & $42.51_{-0.72}^{+0.66}$  & $19.2_{-1.9}^{+1.8}$ & $194.4_{-3.2}^{+3.6}$& 2.71\\vspace{1mm}\\\\  \n2015b & 0.782(11) & $41.53_{-0.54}^{+0.69}$  & $14.3_{-1.0}^{+0.9}$ & $180.0_{-3.0}^{+3.2}$& 2.46\\vspace{1mm}\\\\\n2016  & 0.826(10) & $42.71_{-0.73}^{+0.63}$  & $16.2_{-1.1}^{+1.1}$ & $124.5_{-3.3}^{+3.0}$& 2.26\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\\centering\n\\includegraphics[width=\\linewidth]{Fig14.png}\n\\caption{Residuals of the RVs of the secondary after subtracting the best-fitting PHOEBE solutions without including spots (black), including one spot (red), and including two spots (green) on the secondary, obtained (from top to bottom) for seasons 2013a, 2015b, and 2016.}\\label{Fig14}\n\\end{figure}\n\n\\begin{table}\\centering\n\\caption{Absolute values of positive and negative RV deviations due to RME.}\n\\begin{tabular}{lccc}\n    \\toprule\nSeason& JD & $K_p$ (km\\,s$^{-1}$)& $K_n$ (km\\,s$^{-1}$)\\\\\n    \\midrule\n2001  & 2\\,452\\,192 & 26.5 & 32.4\\\\\n2006  & 2\\,453\\,768 & 19.9 & 22.4\\\\\n2009  & 2\\,455\\,156 & 24.7 & 21.0\\\\\n2013a & 2\\,456\\,584 & 22.6 & 21.7\\\\\n2013b & 2\\,456\\,672 & 26.4 & 20.0\\\\\n2014  & 2\\,456\\,936 & 19.3 & 19.7\\\\ \n2015a & 2\\,457\\,238 & 24.6 & 19.8\\\\\n2015b & 2\\,457\\,293 & 17.5 & 19.8\\\\\n2016  & 2\\,457\\,646 & 19.1 & 19.0\\\\\n\\bottomrule\n    \\end{tabular}\n    \\label{Tab09}\n\\end{table}\n\nIn Fig.\\,\\ref{FigA1}, we show the corner plot for season 2009. This shows that MCMC not only gives the most optimal parameter set, it also provides an error estimation on each parameter and possible correlations between the parameters. Finally obtained values are listed in Table\\,\\ref{Tab08}, in which we also included the mean scatter calculated from the RV residuals after subtracting the PHOEBE solutions obtained for each season. These residuals are shown in Fig.\\,\\ref{FigA2}. \n\nFigure\\,\\ref{Fig14} shows the influence of the inclusion of spots on the secondary into the model. We selected three seasons as examples. Including spots on the secondary did not effect the modelled RVs of the primary and vice versa. Thus, we only show the residuals of the RVs of the secondary. The black dots corresponds to the best-fitting solutions without spots, that is when only the non-spherical shapes of the components (in particular of the secondary) are taken into account. The RVs show a systematic deviation around Min\\,II that is distinctly reduced when including Spot1 that faces the primary, resulting in the red dots. The RVs show smaller, non-systematic deviations around Min\\,I, which is clearly present in the data from 2013a, small in 2016, and almost not visible in 2015b. Adding the second spot on the opposite side reduces this scatter, as shown by the green dots, but in some cases the reduction is marginal.\n\nOne explanation for the asymmetry observed in the RME in the years 2009, 2013b, and 2015a (see Fig.\\,\\ref{Fig13}) could be the impact of a hot spot on the primary on its RVs. For comparison, we therefore additionally included a hot spot on the primary for\nthese seasons, taking three more free parameters (temperature ratio, radius, and longitude) into account. The fit did not improve the solutions, however. \n\nTable\\,\\ref{Tab09} lists absolute values of positive and negative amplitudes of the RME (maximum deviations in RV during ingress and egress, respectively) obtained after subtracting the best-fitting PHOEBE solution for $F_1$\\,=\\,0, that is for a non-rotating primary, from the observed RVs. Panels h) and i) in Fig.\\,\\ref{Fig15} show that the positive amplitudes reveal a similar behaviour as $F_1$ or $R_2$, whereas the negative amplitudes show a systematically decreasing trend with time (see next Section for a discussion). \n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig15.png}\n\\caption{Time variations of different parameters (see text). The solid lines are calculated from the best-fitting periods and long-term trends. The dotted lines show possible sinusoidal variations with a period of 9\\,yr plus a long-term trend, except for panel f), which shows an 18 y variation. Values from 2013b and 2015a were considered as outliers.}\\label{Fig15}\n\\end{figure}\n\n\\begin{table}\\centering\n\\caption{Timescales in years determined from the seasonal variations of different parameters.}\n\\begin{tabular}{ccccccccc}\n\\toprule\nd$P$ & $v\\sin{i}$ & $F_1$ & $R_1^{sec}$ & $R_2^{sec}$ & $\\lambda_2^{sec}$ & rms & $K_p$ & $K_n$\\\\\n\\midrule\n8.6 & 9.6 & 8.4 & 9.4 & 8.8 & 16.7 & 9.3 & 8.5 & 9.6\\\\\n\\bottomrule\n\\end{tabular}\\label{Tab10}\n\\end{table}\n\nFigure\\,\\ref{Fig15} shows the seasonal variations of all investigated parameters. In panel a) we added the period change derived from the O-C values (cf. Sect.\\,\\ref{Sect6}), in panel b) the $v\\sin{i}$\\ values from the fit determined in  Sect.\\,\\ref{Sect5.4}, and in panel g) the mean scatter of the RV residuals after subtracting the best-fitting PHOEBE solutions from the input data (see Fig.\\,\\ref{FigA2}). All variations can be described by a sinusoid plus a long-term trend. Because the seasonal sampling was not sufficient to determine a second frequency describing the long-term trend, we fixed it to $10^{-6}$\\,c\\,d$^{-1}$\\ and determined only amplitudes and phases. Table\\,\\ref{Tab10} lists the timescales derived from the best-fitting sinusoids.\n\nThe best-fitting curves are shown in Fig.\\,\\ref{Fig15} by solid lines. We see that $F_1$, $R_2$, rms, and $K_p$ vary almost in phase, whereas $R_1$ varies in anti-phase. Moreover, $v\\sin{i}$\\ and $K_p$ show almost identical shapes of variation. Searching for a possible common period that explains the variations of all parameters, we found a period of 9 years that best fits, together with the mentioned long-term trend, the variations of all parameters except for $\\lambda_2^{sec}$ that can be fitted by a period of 18\\,years, twice the period of 9\\,years. For d$P$ we had to add a third (optimised) period of 6.3\\,yr. The resulting fits are shown in Fig.\\,\\ref{Fig15} by  dotted lines. In all cases, the quality is comparable to that obtained from the optimised values listed in Table\\,\\ref{Tab10}.\n\n\\section{Discussion}\\label{Sect8}\n\nOur spectral analysis yields atmospheric parameters of the components that are in agreement with the results of light curve analysis by \\citet{2004MNRAS.347.1317R}. For the first time, we determine the elemental surface abundances. For both components, we find [Fe\/H] close to $-0.4$, as for Ca, Cr, Mn, and Ni of the primary, whereas O, Mg, Si, Sc, Ti, and V show relative abundances between $-0.2$ and $-0.1$. For the carbon abundance of the primary, we find [C\/H]\\,=\\,$-0.80^{+0.13}_{-0.18}$, which is remarkably different from the other elements. We think that the difference is significant. First, the considered spectral range shows a sufficiently large number of C\\,I lines of the primary (SynthV computes line depths >5\\% for 32 rotationally unbroadened C\\,I lines). Second, differences between the fits with [C\/H] of $-0.4$ and $-0.8$ can clearly be seen by eye. Figure\\,\\ref{Fig16} shows this for the strongest carbon lines in the H$\\beta$ region of the spectrum of the primary (observed composite spectrum after subtracting the synthetic spectrum of the secondary). In the spectrum of the cool secondary, carbon is mainly present in form of the CH molecule bands. Compared to the primary, the signal is very weak and we can only say that [C\/H] is below solar abundance, between $-0.3$ and $-1.0$ within the 1$\\sigma$ error bars. \n\n\\citet{2008A&A...486..919M} tried to model the binary evolution of RZ\\,Cas and found consistent solutions only for an initial mass ratio of $q$\\,$\\approx$\\,3, which is about the inverse of the actual ratio. From the large initial mass of the donor in that case we can assume that the primary has switched during its evolution from pp-chain to CNO-cycle hydrogen burning, resulting in depleted carbon and enhanced nitrogen abundances in its core \\citep[e.g.][]{2010A&A...517A..38P}. From the reversal of mass ratio we conclude that the donor was stripped by mass loss down to its core so that the gainer accreted  CNO-cycled, carbon-deficient material in the late phase of its evolution. This material has then mixed with the surface layers of the gainer, leading to the observed carbon abundance. In that case, the surface carbon abundance of the gainer cannot be lower than that of the donor star, however. Multiple authors have used the sketched scenario to explain the surface abundance anomalies observed in several other Algol-type stars; these include \\citet{2012MNRAS.419.1472I} for \\object{GT~Cep}, \\object{AU~Mon}, and \\object{TU~Mon}, \\citet{2014MNRAS.444.3118K} for \\object{u~Her}, and \\citet{2018MNRAS.481.5660D} for $\\delta$\\,Lib.\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=\\linewidth]{Fig16.png}\n\\caption{Fit of the observed spectrum of the primary (black) in the H$_\\beta$\\,region by synthetic spectra with [C\/H] of 0.0 (blue), $-0.4$ (green), and $-0.8$ (red).}\n\\label{Fig16}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=.5\\linewidth]{Fig17a.png}\n\\includegraphics[width=.5\\linewidth]{Fig17b.png}\n\\caption{Logarithmic plots of gas density distributions (in $10^{11}$\\,cm$^{-3}$) obtained from 3D hydrodynamic simulations  of the RZ\\,Cas system (Nazarenko \\& Mkrtichian, priv. comm.)  for  mass-transfer rates of 1\\,$\\times$\\,$10^{-9}$\\,M$_\\odot$y$^{-1}$ (left) and 6\\,$\\times$\\,$10^{-8}$\\,M$_\\odot$y$^{-1}$ (right). The viewing angles at first ($\\phi$\\,=\\,$-0.1$) and last ($\\phi$\\,=\\,$+0.1$) contact are indicated.}\n\\label{Fig17}\n\\end{figure*}\n\nOur further investigation was based on the separated LSD profiles and RVs of the two components of RZ\\,Cas computed with LSDbinary, and on the times of minimum (O-C values) taken from literature. The change in orbital period computed from the slope of the O-C diagram cannot be characterised by one single timescale, as already found by \\citet{2018MNRAS.475.4745M}. These authors derived timescales of 4.8, 6.1, and 9.2 years. We observe from our slightly extended data set different timescales of variation in different seasons, such as 6.3\\,yr around JD\\,2\\,440\\,000, and 8.6\\,yr from 2\\,450\\,000 to 2\\,455\\,000 (which is about the time span of our spectroscopic observations), overlaid by longer periods. From our previous analysis using the Shellspec07\\_inverse code \\citep{2009A&A...504..991T}, we know that RZ\\,Cas was in an active state of mass transfer in 2001 and in a quiet state in 2006. Comparison of the RV residuals after subtracting a pure Keplerian orbit with those from 2006 gave us initial hints pointing to further activity periods (Fig.\\,\\ref{Fig13}). From the RVs of the primary we see that the RME was distinctly enhanced only in 2001. A comparison of the RV residuals of the secondary with those from 2006 shows the largest deviation in 2001, weaker deviations in 2013 and 2014, and almost no deviations in 2009, 2015, and 2016. The amount of scatter found in the RV residuals after subtracting the best-fitting PHOEBE solutions is strongly correlated with this finding. Thus, we conclude that no further mass-transfer episode as strong as in or close to 2001 occurred in RZ\\,Cas.\n\nThe modelling of the observed RVs of both components with MCMC-PHOEBE gave the best results when adding two dark spots on the surface of the cool companion: Spot1 facing the primary, Spot2 on the opposite side.  Spot1 was already found from the RVs from 2001 and 2006 by \\citet{2009A&A...504..991T}, whereas \\citet{1994PASJ...46..613U} included two spots into their model, explaining the observed  anomalous gravity darkening by a cooling mechanism by enthalpy transport due to mass outflow that leads to a reverse process of gravitational contraction. \n\nWe monitored both spots for the first time over decades.\nWe find that Spot1 always exactly points towards the primary, with radii (a synonym for strength or contrast as mentioned in before) between 23$^\\circ$ and 43$^\\circ$. Spot2, on the other hand, is much weaker, shows a variation in its position, and induces only a second order improvement in some of the observed seasons (cf. Fig.\\,\\ref{Fig14}). The main findings are that the strengths of the two spots vary in anti-phase and that Spot2 shows different positions in longitude, varying around the longitude of the L2 point of 180$^\\circ$.\n\nThe fact that the strength of Spot1 is largest when the star is between 2006 and 2009 in a quiet phase (Fig.\\,\\ref{Fig15}) speaks against the cooling by mass-outflow mechanism. Instead, we argue that the variations of the strengths of both spots can be explained by magnetic activity of the cool companion, assuming an activity cycle of 9 years, based on an 18-year cycle of magnetic field change, including a reversal of the magnetic poles. The 9-year cycle can be found in the variations of all investigated parameters except for the longitude of Spot2, as we showed in the last section, and was also found by \\citet{2018MNRAS.475.4745M}. Spot2 shows a migration in longitude, returning after an 18-year cycle to its position from 2001 (cf. Fig.\\,\\ref{Fig15}). We assume that we observe similar surface structures on the cool secondary of RZ\\,Cas such as for cool, rapidly rotating RS\\,CVn binaries or single rapidly rotating variables of FK Com and BY Dra type. Long-living active regions were observed on opposite sides of these stars. The spots are of different intensity and also show switching of activity from one spot to the other on timescales of years or decades, which is known as the flip-flop effect \\citep[see][and references herein]{2018A&A...613A...7Y}.\n\nWe believe that there is a  direct analogy with sunspot activity. Sunspots show a saucer-shaped depression in the photosphere caused by the Lorentz force of the strong magnetic field within the spot, the so-called Wilson depression \\citep{1774RSPT...64....1W}. This means that inside the spot the level of $\\tau$\\,=\\,1 is located below the level of the photosphere outside the spot. For the Sun, the geometric depth of the depression is of the order of 600\\,km \\citep[e.g.][]{1972SoPh...26...52G, 2018A&A...619A..42L}. For the Roche-lobe filling donor, the existence of such atmospheric depression close to the L1 point means that L1 is \"fed\" by atmospheric layers of lower density and thus the mass transfer is the more suppressed the deeper the depression is. The local magnetic field strength controls the strength (depression) of Spot1 and the height of atmospheric layers feeding the L1 point and in this way the mass-transfer rate. RZ\\,Cas showed, in perfect agreement with the drafted scenario, high mass-transfer rate in 2001 (and possibly in 2011, when observations are missing) when the Spot1 size was around 20 degrees and low activity state in 2006-2007 and in 2015-2016 when the spot size was about 40 degrees.\n\nThe explanation for the asymmetry and different amplitudes observed in the RME in the years 2001, 2009, 2013b, and 2015a (see Fig.\\,\\ref{Fig13}) could be the impact of the combined effect of acceleration of the photospheric layers in 2001 caused by the high mass-transfer rate and by screening the surface of the primary by the dense gas stream \\citep{2008A&A...480..247L}. The amplitudes of the RME in RZ\\,Cas in general show very different behaviour during ingress ($K_p$) and egress ($K_n$) of primary eclipse. The variation of $K_p$ is similar to those of $R_2$ and rms, whereas the shape of the $K_n$ variation perfectly matches that of $v\\sin{i}$\\ (see Fig.\\,\\ref{Fig15}). We assume that $K_n$ is related to the true $v\\sin{i}$\\ seen outside the eclipses and $K_p$ is strongly influenced by mass-transfer effects. This can be explained from Fig.\\,\\ref{Fig17}, showing the gas density distribution around RZ\\,Cas for two different mass-transfer rates, calculated from 3D-hydrodynamic simulations by Nazarenko \\& Mkrtichian (priv. comm.). It can be seen that the equatorial zone of the surface of the primary is masked during ingress (at orbital phases $\\phi$\\,=\\,$-0.1..0$) by the optical thick gas stream from the secondary, whereas we directly see the complete disc of the primary during egress ($\\phi$\\,=\\,$0..0.1$), only very slightly hampered by optical thin circumbinary matter. In consequence, $K_p$ is affected by the seasonal varying density of the gas stream (or mass-transfer rate) and $K_n$ is correlated with the surface rotation velocity of the primary. All the variability seen in RME in the years from 2006 on is mainly related to the variations of $K_p$, forced by changes in gas-stream density and variable screening and attenuation effects, while the rotation speed was nearly constant. \n\nThe synchronisation factor derived with PHOEBE is mainly based on the shape of the RME and this shape is strongly influenced by timely varying Algol-typical effects (see paragraph below). Our MCMC simulation did not count for the observed asymmetry in the RME either (i.e. the fact that the amplitudes $K_p$ and $K_n$ are different from each other) and so the synchronisation factor $F_1$ has to be considered as some kind of mean value. The shape of its time variation resembles the mean of the shapes of the $K_p$ and $K_n$ variations (cf. Fig.\\,\\ref{Fig15}). The results obtained for the synchronisation factor in Sect.\\,\\ref{Sect5.4} from radius and $v\\sin{i}$\\ of the primary, on the other hand, give sub-synchronous rotation of the primary. Its outer layers are only accelerated to almost synchronous rotation during the active phase\nin 2001. The finding of sub-synchronous rotation is surprising. It was suggested by \\citet{2014A&A...570A..25D} to occur during the rapid phase of mass exchange in Algols, when the donor star is spun down on a timescale shorter than the tidal synchronisation timescale and material leaving the inner-Lagrangian point is accreted back onto the donor, enhancing orbital shrinkage. But these authors state that once the mass-transfer rate decelerates and convection develops in the surface layers, tides should be effective enough to re-synchronise the primary. To our knowledge, sub-synchronous rotation was found so far in only one short-period Algol, namely TV\\,Cas, by \\cite{1992A&A...257..199K}, in which the authors found synchronisation factors of 0.85 for the primary and 0.65 for the secondary component.\n\nThe RV residuals after subtracting the solutions obtained with our MCMC simulations (Fig.\\,\\ref{FigA2}) finally show that our model distinctly reduces the scatter compared to the residuals obtained from pure Keplerian motion. On the other hand, we can still recognise many of the signs of activity that we discussed in connection with Fig.\\,\\ref{Fig13}. We assume that  these still unexplained features are due to the  distribution and density of circumbinary matter along the line of sight in different orbital phases, varying between different seasons according to the varying activity of the star. Finally, we can conclude from Fig.\\,\\ref{FigA2} in the same way as from Fig.\\,\\ref{Fig13} that RZ\\,Cas showed an extraordinary phase of activity around 2001 followed by a quiet phase in 2006 to 2009, slightly enhanced activity in 2013\/2014, followed by a quiet phase again. The calculated fits on the variation of the radii of the two spots and the RME amplitude $K_p$ based on the nine-year cycle, on the other hand, show extrema of the same amplitude such as in 2001 for the time around 2010-2011. Therefore, because of missing data in this period, we cannot exclude the possibility that a second mass-transfer episode of comparable strength like in 2001 occurred close to 2010-2011.\n\nThe orbital period was at maximum shortly after 2001, dropped down steeply until 2006, and was then more or less continuously rising. From Eq.\\,8 in \\citet{1973A&A....27..249B}, assuming conservative mass transfer and conservation of orbital angular momentum, we obtain a mass-transfer rate of 1.5\\,$\\times\\,10^{-6}$\\,M$_\\odot$\\,yr$^{-1}$. This is a typical value observed for Algol-type stars \\citep[e.g.][]{1976IAUS...73..283H} and also agrees \\citet{1976AcA....26..239H}, who calculate, from an O-C analysis, a mean mass-transfer rate of RZ\\,Cas of 1.0\\,$\\times\\,10^{-6}$\\,M$_\\odot$\\,yr$^{-1}$, averaged over several episodes of period change. As mentioned in the introduction, evolutionary scenarios for RZ\\,Cas have to count for mass loss from the system \\citep{2008A&A...486..919M}, which may be the case for Algols in earlier stages of their evolution in general \\citep[e.g.][]{2006MNRAS.373..435I, 2013A&A...557A..40D, 2015A&A...577A..55D}. Thus, conservatism is not necessarily justified and the assumption of mass conservation can only lead to a raw estimation of the amount of mass transferred during the active phase of RZ\\,Cas in 2001.\n\nWe assume that the enhanced value of $v\\sin{i}$\\ in 2001 points to an acceleration of the outer layers of the primary of RZ\\,Cas due to the mass-transfer episode occurring close to that year. This then dropped down by about 5\\,km\\,s$^{-1}$\\ and it could be that its slight increase after 2009 is correlated with a slightly increased activity as indicated by the variation of $F_1$ and rms.\n\n\\section{Conclusions}\\label{Sect9}\n\nIn this first of three articles related to spectroscopic long-term monitoring of RZ\\,Cas, we investigated high-resolution spectra with respect to stellar and system parameters based on the RVs and LSD profiles of its components calculated with the LSDbinary program. The main goal was to search for further episodes of enhanced mass transfer occurring after 2001 and for a general timescale of variations possibly caused by the magnetic cycle of the cool companion.\n\nFrom spectrum analysis we determined precise atmospheric parameters, among them low metal surface abundances, in particular [Fe\/H] of  $-0.42$ and [C\/H] of $-0.80$ for the primary and [Fe\/H] of the same order for the secondary. The carbon deficiency observed for the primary gives evidence that the outer layers of the cool secondary have been stripped in the fast mass-transfer phase down to its core so that CNO-cycled material was transferred to the outer layers of the primary in later stages of evolution. The derived $T_{\\rm eff}$\\ of the components and the $\\log{g}$\\ of the primary agree within the 1$\\sigma$ error bars with the results from LC analysis by \\citet{2004MNRAS.347.1317R}. From the RV analysis with PHOEBE, we derived very precise masses and separation of the components of $M_1$\\,=\\,1.951(5)\\,M$_\\odot$, $M_2$\\,=\\,0.684(1)\\,M$_\\odot$, and $a$\\,=\\,6.546(5)\\,R$_\\odot$.\n\nFrom several of the investigated parameters that show seasonal variations, such as orbital period, $v\\sin{i}$, strength of the spots on the secondary, synchronisation factor calculated with PHOEBE, and rms of the RV residuals after subtracting the PHOEBE solutions, we deduce a common time scale of the order of 9 years. The variation of the orbital period is complex and can be described in detail only when adding further periods. We conclude that we see the effects of a 9-year magnetic activity cycle of the cool companion of RZ\\,Cas, caused by an 18-year dynamo cycle that includes a reversal of polarity. This conclusion is strongly supported by the behaviour of the two dark spots on the surface of the secondary that show the flip-flop effect in their strengths and one  spot that shows an 18-year periodicity in longitudinal migration.\n\nFrom the variations of orbital period and $v\\sin{i}$\\ around 2001, we conclude that the determined $v\\sin{i}$\\ is the projected equatorial rotation velocity of the outer layers of the primary accelerated by transferred matter and does not stand for the rotation velocity of the star as a whole. In all other seasons, the measured $v\\sin{i}$\\ point to a sub-synchronous rotation of the gainer. At the present stage, we cannot give a physical explanation for this new and interesting finding, however.\n\nBased on our available data, we conclude that RZ\\,Cas was undergoing an episode of high mass transfer in 2001, in a quiet phase in 2006 and 2009, followed by a slightly more active phase in 2013 and 2014, and again in a quiet phase in 2015 and 2016. Because we did not observe the star in 2010 and 2011, we cannot exclude that a second episode of high mass transfer occurred in these years, which would agree with the derived magnetic activity cycle of nine years.\n\nThe results of our investigation of high-frequency oscillations of the primary of RZ\\,Cas will be presented in Part II of this article. In a third article, we will investigate the accretion-induced variability of He\\,I lines detected in the spectra.\n\n\\begin{acknowledgements}\nHL and FP acknowledge support from DFG grant LE 1102\/3-1. VT acknowledges support from RFBR grant No. 15-52-12371. AD is financially supported by the Croatian Science Foundation through grant IP 2014-09-8656 and Erciyes University Scientific Research Projects Coordination Unit under grant number MAP-2020-9749. The research leading to these results has (partially) received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), from the KU~Leuven Research Council (grant C16\/18\/005: PARADISE), from the Research Foundation Flanders (FWO) under grant agreement G0H5416N (ERC Runner Up Project), as well as from the BELgian federal Science Policy Office (BELSPO) through PRODEX grant PLATO. Results are partly based on observations obtained with the HERMES spectrograph, which is supported by the Research Foundation - Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Gen\\`eve, Switzerland and the Th\\\"uringer Landessternwarte Tautenburg, Germany.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nRecently much attention has been given to connections between\ncriticality and self-organized criticality (SOC) and evolutionary\nphenomenon, particularly punctuated equilibrium, and to\nconnections between SOC and synchronization. \nWe describe a model \nwhich we hope draws some connection between these 2 ideas.\nSOC has been proposed to describe out of equilibrium systems\nthat are critical, that self-organize into a scale invariant\ncritical state without tuning of a control\nparameter and show fractal time series.\\cite{rf:52} \nEvolution SOC type models \\cite{rf:60,rf:68,rf:45}\nhave been proposed to explain \npunctuated equilibrium. \\cite{rf:66,rf:67}\nPunctuated equilibrium is the phenomenon observed\nin the fossil record where long periods of stasis are interrupted \nby sudden bursts of evolutionary change. \nKaufman and Johnsen \\cite{rf:64} have modeled co-evolution,\nwhere agents live on \na coupled fitness landscape and walk around by\nrandom mutation, only \nmoves to higher fitnesses are allowed.\nOnce at a local maximum the walk stops until moves by another\nagent deform the lanscape so the agent is no longer at a maximum.\nKauffman \\cite{rf:64} has linked this to SOC. Bak,\nSneppen and Flyvbjerg \\cite{rf:60,rf:68} have taken a similar\napproach. They define a species as \na barrier to increasing fitness and choose the\nleast fit, then randomly change its barrier and the barriers of\nother agents. The\nsystem evolves to a critical state with a self-organized\nfitness threshold.\nSOC has also been linked to periodic\nbehaviour.\\cite{rf:50,rf:56,rf:58,rf:70} \nA.Corral et al and Bottani \\cite{rf:56,rf:50}, say there is a close relationship between\nSOC and synchronization. SOC\nappears when a system is perturbed which\notherwise should synchronise totally or partially. \nThe perturbation may be open boundary conditions rather than\nperiodic\\cite{rf:50,rf:70} or it may be\nrandomness present in initial conditions which is preserved by\nthe dynamic, or it may be the addition of noise.\nThe model we study appears to link these 2 ideas with the\nemergence of avalanches of partial synchronization on all\nscales. However all these models are real space models whereas\nours is a mean-field model with no spatial dimension. Our model \nis entirely determistic, the critical state is produced by\ncertain initial conditions, indeed other initial conditions\nproduce completely periodic states. \nOur model also is not strictly speaking an SOC model since\ncritical behaviour only occurs for a certain range of the\nparameter and then only for certain initial conditions. A\ncomplete analysis of the \nintial conditions is outside the scope of this paper.\n\nOur model was originally\nmotivated as a model of the behaviour of speculating traders in a \nfinancial market in the spirit of co-evolution.\nRecent results have shown stock price time\nseries to be fractal with Hurst exponent different from 0.5,\n\\cite{rf:72} \nand with positive lyapunov exponent.\n\\cite{rf:33,rf:34,rf:39}\nScrambling daily returns changes the Hurst exponent back to 0.5.  \nLarge crashes have been supposed\nto be due to exogenous shocks, where information enters the\nmarket randomly.\nHowever large crashes interspersed with periods of slow\ngrowth are \nstrongly reminiscent of punctuated equilibrium. Indeed\nMandlebrot has noted large changes of cotton prices occur in\noscillatory groups \nand the movement in tranquil periods is smoother than\npredicted.\\cite{rf:77} \nScaling behavior has been noted in a financial \nindex and in the size of companies.\\cite{rf:100,rf:101}  Stanley et\nal have noted `scaling laws used to describe complex systems\ncomprised of many interacting inanimate particles (as in many\nphysical systems) may be usefully extended to describe complex\nsystems comprised of many interacting animate subsystems (as in\neconomics).'\n\nVarious models have been proposed to explain\nmarket movements.\\cite{rf:36,rf:38,rf:6}\nSato and Takayasu have proposed a threshold type\nmodel.\\cite{rf:73,rf:73a}\nSince critical states can produce avalanches on all scales,\nwithout the need for \nexogenous shocks, we believe critical type dynamics\nare present in financial market dynamics. \n\n\\section{Model}\n\\noindent\nWe hope to model co-evolutionary phenomenon where the micro-level\nitself defines the macro-level but is also slaved to the\nmacro-level. This is very evident in speculative financial market dynamics\nwhere a collection of individuals (micro) trade therby creating a price time\nseries (macro), but determine their trading behaviour by reference to\nthis same price series and other macro variables.  \nWe desired to make a model in analogy to this phenomenon. \n\nThis is a highly stylized toy model of a stock-market.There are \n$N$ agents which are represented by spins $s_{i}(t)$, where\n$s_{i}(t)=1$, means the agent $i$ owns the stock and\n$s_{i}(t)=-1$ means doesn't own the stock at time t. Each agent also has an \nabsolute fitness $F_{i}(t)$ and a relative fitnes\n$f_{i}(t)=F_{i}(t)-F(t)$ where the mean-fitness\n$F(t)=\\frac{1}{N}\\sum_{i=1}^{N}F_{i}(t)$. \nWe\nbelieve speculative traders are part of 2 crowds, bulls and bears, and our\nmacrovariable is `groupthink' $G(t)$, \ndefined by, \n\\begin{equation}\nG(t)=\\Delta P(t)=\\frac{1}{N}\\sum_{i=1}^{N}s_{i}(t)\n\\end{equation}\nThe dynamic is:\n\\begin{equation}\n\\Delta s_{i}(t)=s_{i}(t+1)-s_{i}(t)=\\left\\{\\begin{array}{ll}\n-2s_{i}(t) & f_{i}(t)\\leq 0\\\\\n0 & f_{i}(t)>0\n\\end{array}\n\\right.\n\\end{equation}\n\\begin{equation}\n\\Delta F_{i}(t)=F_{i}(t+1)-F_{i}(t)=-\\frac{1}{2}\\Delta s_{i}(t)G(t)+\\frac{1}{2}|\\Delta s_{i}(t)|c\n\\end{equation}\n\nThe dynamic is synchronous and deterministic. First $G(t)$ and\n$F(t)$ are calculated then all agents are updated according to\n(2) and (3). The price $P(t)$ is defined by (1) and\n$P(0)=0$. Initially $s_{i}(0)$ are chosen randomly with\nprobablity 1\/2 and $F_{i}(0)$ are chosen randomly from the\ninterval [-1,1].\n\n$G(t)$ measures the bullishness or bearishness of the crowd.\nAlthough different to ours Callan and Shapiro mention groupthink\nin Theory of Social \nImitation\\cite{rf:75} and Vaga's\\cite{rf:76,rf:72} Coherent\nMarket Hypothesis explicitly includes a variable called\n{\\it groupthink}. \n\nWe believe speculative agents determine their spin state\ndependent on whether they \nbelieve the market will move in their favour in the future. \nTherefore \nour agents have an absolute fitness $F_{i}(t)$ which measures their\nperception of whether they are in a good position with respect\nto the future. If $F_{i}(t)$ is relatively high their state will\nbe stable and if $F_{i}(t)$ is relatively low they will want to\nchange their current state. Many ways to define $F_{i}(t)$ are\npossible. In this model we define it by \nanalogy to Plummer. Agents\nconsider the market to be `overbought' or `oversold'. \nIn our simplistic model this is measured by $G(t)$. An agent is fit if it is in the minority\ngroup.  According to Plummer when most agents \nare in one position then there must be less buying into this\nposition (because there are only a finite amount of agents) and\ntherefore the market will eventually correct itself (change\ndirection) because \nits growth will not be sustainable. It is always profitable then \nto be in the minority group before a correction. At a correction \nthe dominant crowd breaks, the macro position dissolves, the market\nmay crash, and \nsubsequently bull and bear crowds will begin to reform. \nIn fact at these times the agents may trade in\n2 macro-clusters or chaotically with the market attaining high\nvolatility which persists for some time.  \nThis type of\ntrader has \nbeen called a {\\it sheep trader} \\cite{rf:36,rf:38}, in\ncontrast to fundamentalists speculators and\nnon-speculators. \nTherefore in this model an agents fitness is increased if it\nchanges from the \nmajority group to the minority group, with the increase\nproportional to the size of the majority. The opposite is\napplied if it changes the other way. \nIf an agent doesn't change its state then it's absolute fitness\n$F_{i}(t)$ is not\nchanged, regardless of whether $G(t)$ changes.\nAn agent also has a relative fitness $f_{i}(t)$. The $f_{i}(t)$ are the\nbehaviour controlling variables in this model. They may change\nin 2 ways. Firstly an agent $\\alpha$ may change its state $s_{\\alpha}(t)$ thereby\ndirectly changing $F_{\\alpha}(t)$ and $f_{\\alpha}(t)$. This is similar to\na single adaptive\nmove on a fitness landscape by an individual optimizing\nagent. Secondly co-evolution may occur. Here an agent's\nrelative fitness $f_{\\beta}(t)$ may change due to changes in the\nother agents fitnesses $F_{i}(t)$ changing $F(t)$ while\n$F_{\\beta}(t)$ remains constant.\n\nTo model evolution then we follow natural selection by analogy\nand mutate unfit agents and leave fit agents unchanged (although \ntheir relative fitnesses may change). \nas in Kaufman,\\cite{rf:64} and Bak et al\\cite{rf:60,rf:68}.\nMutation is considered to\nbe a state change and this changes an agent's fitness according\nto (3). In this model since\ntheir are only 2 possible states this means we simply flip\nstate. (In a more extensive model this would correspond to\nchanges to an\nownership portfolio vector). To decide which spins flip we could\ncompare pairs of \nfitnesses and change the least fit. \nThat is we could choose 2 agents $\\alpha$ and $\\beta$ and let\nthem compete so that\n$F_{\\alpha}>F_{\\beta}$ then we say\n$s_{\\alpha}(t+1)=s_{\\alpha}(t)$ and\n$s_{\\beta}(t+1)=-s_{\\beta}(t)$. \nHowever in this paper we simply\ntake a mean-fitness approach. That is all agents $i$ whose\nfitnesses $F_{i}(t)$ fulfill $F_{i}(t)\\leq F(t)$ ie\n$f_{i}(t)\\leq 0$\nflip there spins and their fitnesses change according to (3).\nAll other agent's states and absolute fitnesses $F_{i}(t)$ do\nnot change although their relative fitnesses $f_{i}(t)$ of\ncourse do.  \nTherefore fit agents which could be considered to be at a local\nmaximum do not change their states until the mean-fitness $F(t)$ has\nbecome equal to their fitnesses $F_{i}(t)$.\n\nThis means the fitnesses are all internally defined emergent\nproperties as in co-evolution. Of course if there is no overall crowd\npolarisation then changing state does not change fitness.\n\nTherefore the fitness update rule (3) can be seen as the\nadaptive walk part and this is the reason why we do not simply\nset  $F_{i}(t)=-s_{i}(t)G(t)$ or $\\Delta F_{i}(t)=-s_{i}(t)G(t)$\ncontinuously for all agents. We \nhope agents will take time to walk out of unfit states and that\nfit maxima will be created which persist for some time. Our\nabsolute fitness is therefore cumulative and is only changed for unfit\nagents. More realistically we could\nthink of agents imperfectly sampling the market ie $G(t)$ at a series of\ntimes to determine their current absolute fitness. In\nfact we see that the concept of relative fitness and absolute\nfitness are very similar to the concept of `bounded\nrationality.' An agents rationality is bounded because he only\nmakes local adaptive moves and can percieve only his absolute\nfitness but not the overall mean-fitness or his relative fitness\n\n\nThis fitness of the position is natural in the sense that it can be seen as \na kind of potential for future profit, usually termed `utility'\nin economics. The fitter an agent is\nthe less likely it will want to change its state, the more\nstable it is, because it believes the market to be oversold in\nits favour.\n\nSince $G(t)$ will on average be $0$,\nin addition in equation (3) we include a very small control\nparameter $c$ which controls the driving rate. We add this to \nall fitnesses below the mean so that unfit agents on\naverage will be come fitter and interact with the fit\nagents. This is a general characteristic of evolutionary systems \nthat single entity moves on fitness landscape should be on\naverage uphill. \n\nOur market price $P(t)$ is defined by $\\Delta P(t)=G(t)$, ie\nprice increases while more people own the share than don't own\nit, and the price is theoretically unbounded as it should\nbe. Positive groupthink means positive increase and\nvice-versa. This is similar to the way prices are usually\ndefined by $\\Delta p(t)\\propto Z(t)$, where $Z(t)$ \nis the excess demand for something. \nThis model does not included a fixed amount of\nshares. Indeed any trader may independently buy or sell a share\nwithout the notion of swapping. This reflects the fact that this \nis a model of only speculative behaviour, and part of a much\nlarger of pool of shares. However a more realistic model should\ninclude a fixed amount of shares.\n\nThis model is intended to be a suggestive\nillustration rather than a realistic stock-market. \n\n\\section{Results}\n\nShown in Fig.1a is a time series for the fitnesses $F_{i}(t)$\nfor an $N=80$ system for $c=0.01$. Punctuated equilibrium behaviour is\nclearly visible, with periods of relative stasis interspersed\nwith sudden jumps.\nAlthough not shown the mean-fitness time series $F(t)$\nshows changes on all scales similar to a devils staircase. Also\nshown in Fig.1b is the corresponding daily returns time-series\n$\\Delta P(t)=G(t)$,\nthis also shows calm periods and sudden bursts of high\nvolatility. Infact this behaviour is a kind of intermittent\npartial synchronization. Shown in Fig.2 is the same time\nseries but with a small portion magnified. Macroscopic \nsynchronization can be seen. Partial synchronizations show various\ndifferent periods and complexities, and persist for various\nlengths of time.\nClustering allows\nsynchronized spins to trade in phase with groupthink $G(t)$ thereby\nrapidly increasing there fitnesses, or out of phase thereby\nbecoming less fit, this is the origin of the sudden large\nchanges in fitness. \nAlso at these times the fitness deviation suddenly\nincreases,(not shown). Between periods of large-scale partial\nsynchronization with high volatility, periods \nof calm are characterized by a small even number of spins\nflipping in anti-phase, they therefore increase their $F_{i}$\nonly slowly due to the driving parameter c, the returns $G(t)$\nremaining roughly \nconstant during these periods with the fitness deviation\ndecreasing. (Of course anti-phase flipping with no average\nincrease in fitness is prevented in a real market by a\nfixed transaction cost. A more realistic model must include this.) \nWhen the mean-fitness $F(t)$ which is usually increasing\ncrosses some non-flipping $F_{i}$, this spin flips and may cause the\n$F(t)$ to cross some more $F_{i}$, possibly starting an \navalanche. This only happens when the total fitness deviation is \nsmall.\n\nShown in Fig.3 are two price $P(t)$ time series for\n$c=0.013$. Their fractal slightly repetetive pattern is highly\nreminiscent of real financial time series.\n\nSince this model is deterministic, completely periodic\nstates are also possible. Shown in Fig.4 is the average of the\nquantity $<R(t)>$\nwhere $R(t)=\\sum_{i=1}^{N}|\\Delta s_{i}(t)|$\nis the amount of spins which flip at any time and $<\\ldots >$\ndenotes time averaging. In Fig.4a are time series for $c=0.0113$ \nwhile Fig.4b is $c=0.01$. \n\nFig.4a shows one time series finds\nthe 2 cluster periodic state, where 2 groups alternately topple. \nHere 59 time series are included in the non-periodic state. If\nthis is a\ntransient it is super long even for the moderate size \n$N=200$. Fig.4b shows at more regular $c$ values the system\ncan find periodic states with larger amounts of\nclusters. Roughly half of the 60 series investigated become\nperiodic by $t=2.5\\times 10^{7}$.\nShown in Fig.5a is $<R(t)>$ plotted against $c$. \nInfact to construct this plot we\ndiscarded $3\\times 10^{7}$ time steps and then averaged over the next\n$20,000$, each point represents a different initial condition\nand there are $8$ for each cost $c=0.001x+0.000138$, $x$ is an integer.\nFor small cost\ncritical type behaviour is evident \nwith a sudden phase transition at $c=0$. This is of course\nbecause at negative $c$ less fit spins continuously flip and\nnever interact with spins at greater than mean fitness. In fact\nthe system divides into a frozen solid fit component and an\nunfit gaseous type component for negative $c$. The size of the\nfrozen component \ndepends on the initial conditions as can be seen from the points \nat negative cost. For larger positive cost an upper branch of\nperiodic attractors at  \n$<R(t)>=100$, half the system size, is evident, the\nsystem has settled into 2 alternately toppling clusters which\ninterleave the mean-fitness $F(t)$. The lower branch is\ncharacterised by the punctuated equilibrium \nstate shown in Fig.1.\nFig.5b shows the time average $<S(t)>$ of an entropy type\nquantity of the fitness\ndistribution $S(t)$ given by,\n$S(t)=-\\sum_{i=1}^{N}\\frac{|f_{i}(t)|}{f(t)}ln\\frac{|f_{i}(t)|}{f(t)}$\nwhere $f_{i}(t)$ is the fitness deviation and \n$f(t)=\\sum_{i=1}^{N}|f_{i}(t)|$. The averaging is the same as\nfor $<R(t)>$. \nFor this $N=200$ system the maximum $S=ln200\\approx\n5.3$ and the periodic points at positive and negative cost are\nvery near this. \nThe punctuated equilibrium state which exists near\nthe transition is more ordered at lower entropy. \n\nThis is our first evidence of critical behaviour for small $c$. \nSecond evidence is obtained by \nlooking at the distribution of avalanches. In the punctuated\nequilibrium state the system finds a state characterised by\nfluctuations on all scales. Shown in Fig.6 is the\ndistribution of $P(R)$ against $R$ where $P(R)$ is the\nprobability of an avalanche of size $R$. These are distributions \nof avalanches for 1 time series for 3 different system\nsizes. They are not ensembles of time series, this distribution\nis independent of the initial conditions and any non-periodic\ntime series contains all avalanche sizes. \n\nThe time series\nwere of length $T=16,000,000$, near the transition point at\n$c=0.0113$. \nThe distribution shows scale invariance,$P(R)\\sim R^{\\alpha }$ up\nto about \nhalf the system size. At half the system size there is a peak\nwhere the system almost finds the periodic attractor and spends\nmore time in these states.\nAfter this \nthe distribution continues  \nto the cutoff near the system size. The scaling exponent $\\alpha \n$ taken\nfrom the $N=3000$ distribution is $\\alpha =-1.085\\pm 0.002$.\nAlso shown in Fig.7 is the distribution of magnitude of\nchanges in mean \nfitness $\\Delta F(t)=|F(t+1)-F(t)|$, the steps in the devils\nstaircase. \nThe time series are\nthe same as in Fig.6 for 2 different system sizes, there is\nno ensemble averaging. In fact the two distributions for\n$N=1500,923$ are almost identical, if we were to superpose them,\nonly one could be seen. This is also true for other system\nsizes. Peaks appear at $\\Delta F\n\\approx 0.12,0.5,0.75$. Between the peaks we see scaling\nregimes. Here we see at least 2\nscaling regimes, $P(\\Delta F)\\sim \\Delta F^{\\beta}$ where for\n$\\Delta F \\le 0.1$,  $\\beta =-1.25 \\pm 0.003$ and for $0.1\\le\n\\Delta F \\le 0.4$, $\\beta =-1.39 \\pm 0.02$. Possibly there is\nanother scaling regime for $0.55\\le \\Delta F \\le 0.7$.\n\\section{Conclusion}\n\\noindent\nThis model illustrates an interesting relation between critical\nphenomenon and punctuated equilibrium on the one hand, and\nbetween partial synchronization and punctuated equilibrium on\nthe other hand. The system synchronizes for certain cost $c$ and\ncertain intial conditions, otherwise it shows critical\nbehaviour, similar to the SOC models cited in the introduction. We\nbelieve this deserves further  \ninvestigation. We also find an interesting phase transition. \n\nSome typical behaviour of \nmoney markets is present here, especially the periods of low\nvolatility, where the price is relatively stable and the fitness \ngrows slowly while the fitness deviation decreases slowly, \ninterrupted by shorter periods of persistent high volatility and macroscopic\noscilations which are observed in real time series. We wonder if \nlike in earthquake dynamics, which are often modelled by SOC\ndynamics, a large crash in a real financial market is preceded\nby some smaller self-reinforcing oscilatory pre-shock, as is\nseen in our dynamics here. Also the time price time series is\nhighly suggestive of real time series, with \nformations similar to `double tops' and `rebounds' described in\nquantitative analysis, produced by the \nnear periodic macro behaviour which can appear. The\nslightly repetetive self-similarity reminds us of financial time series.    \n\nMany possible models of financial market dynamics can be\nplausibly suggested, included many exhibiting threshold\ndynamics, since data concerning the micro behaviour of\nindividual traders is not available.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nSo far, the LHC did not observe any particles beyond those of the\nStandard Model (SM). Complementary to direct high energy searches at\nthe LHC, there is a continuous effort in indirect searches for new\nphysics (NP). In this respect, a promising approach is the search for\nprocesses which are absent -- or extremely suppressed -- in the SM\nsuch as lepton flavour violation (LFV) which is forbidden in the SM in\nthe limit of vanishing neutrino masses. The experimental sensitivity\nfor rare LFV processes such as $\\ell\\rightarrow \\ell^{\\prime}\\gamma$,\n$\\mu \\to e$ conversion in nuclei and $\\ell\\rightarrow \\ell^{\\prime}\n\\mu^+\\mu^-$ or $\\ell\\rightarrow \\ell^{\\prime} e^+e^-$ will improve\nsignificantly in the near future, probing scales well beyond those\naccessible at foreseeable colliders.  Furthermore, the discovery of\nthe 125 \\ensuremath{\\,\\mbox{GeV}}{} Higgs boson $h$ \\cite{Aad:2012tfa,Chatrchyan:2012ufa}\nhas triggered an enormous experimental effort in measuring its\nproperties, including studies of its LFV decays. The most recent\nexperimental limits {on the LFV processes} are given in\nTable~\\ref{tab:leplim} {in Sec.~\\ref{sec:pheno}}.\n\nMany studies of LFV processes within the MSSM (and possible extensions\nof it) exist (see e.g. Refs.\\cite{Borzumati:1986qx, Casas:2001sr,\n  Masina:2002mv, Brignole:2004ah, Paradisi:2005fk, Fukuyama:2005bh,\n  Paradisi:2005tk, Dedes:2006ni, Dedes:2007ef, Antusch:2007dj,\n  Arganda:2008jj, Ilakovac:2009jf, Calibbi:2009ja,\n  Altmannshofer:2009ne, Calibbi:2006nq, Calibbi:2009wk, Hisano:2009ae,\n  Girrbach:2009uy, Biggio:2010me, Esteves:2010ff, Ilakovac:2012sh,\n  Arana-Catania:2013ggc, Goto:2014vga, Abada:2014kba, Vicente:2015cka,\n  Bonilla:2016fqd, Lindner:2016bgg} and Ref.~\\cite{Calibbi:2017uvl}\nfor a recent review).  In this article we revisit this subject in the\nlight of the new calculational methods which have been recently\ndeveloped~\\cite{Dedes:2015twa, Rosiek:2015jua}. These methods allow\nfor a systematic expansion of the amplitudes of the LFV processes in\nterms of mass insertions (MI), i.e. in terms of off-diagonal elements\nof the mass matrices.  We show that a transparent qualitative\nbehaviour of the amplitudes of the LFV processes is obtained by\nexpanding them not only in the flavour-violating off-diagonal terms in\nthe sfermion mass matrices but also in the flavour conserving but\nchirality violating entries related to the tri-linear $A$-terms as\nwell as in the off-diagonal terms of the gaugino and higgsino mass\nmatrices. This procedure is useful because in the MI approximation we\nwork directly with the parameters of the Lagrangian and can therefore\neasily put experimental bounds on them.  We compare the results of the\ncalculations performed in the mass eigenbasis (i.e. using a numerical\ndiagonalization of the slepton mass matrices) with those obtained at\nleading non-vanishing order of the MI approximation, in different\nregions of the supersymmetric parameter space and considering various\ndecoupling limits.  Of course, the MI\napproximation~\\cite{Gabbiani:1996hi, Misiak:1997ei} has already been\nexplored for many years as a very useful tool in flavour physics.\nHowever, a detailed comparison between the full calculation and the MI\napproximation is still lacking, partly because a fully systematic\ndiscussion of the MI approximation~\\cite{Dedes:2015twa} to any order\nand the technical tools facilitating it~\\cite{Rosiek:2015jua} have not\nbeen available until recently.\n\nConcerning the phenomenology, we summarise and update the bounds on\nthe flavour violating SUSY parameters, show their complementarity and\nexamine the impact of the anticipated increase in the experimental\nsensitivity.  We investigate in detail the decay $h \\to \\mu \\tau$\nshowing the results in various decoupling limits and analyse the role\nof the so-called non-holomorphic $A$-terms~\\cite{deWit:1996kc,\n  Matone:1996bj, Bellisai:1997ck, Dine:1997nq, ArkaniHamed:1998wc,\n  GonzalezRey:1998gz, Buchbinder:1998qd, Martin:1999hc} , which are\nusually neglected in literature.\nWe also avoid simplifying assumptions on the sparticle spectrum and\nassume neither degeneracies nor hierarchies among the supersymmetric\nparticles.\n\nThis article is structured as follows: in Sec.~\\ref{sec:efflag} we\nestablish our conventions and present the results for the 2-point,\n3-point, and 4-point functions related to flavour violating charged\nlepton interactions in the mass eigenbasis, i.e. expressed in terms of\nrotation matrices and physical masses.  Sec.~\\ref{sec:lfv} contains\nthe formula for the decay rates of the processes under investigation.\nIn Sec.~\\ref{sec:memi} we discuss the MI expansion and summarise\nimportant properties of the decoupling limits $M_{\\rm SUSY}\\to\\infty$\nand $M_A\\to \\infty$.  In Sec.~\\ref{sec:pheno} we present the numerical\nbounds on LFV parameters obtained from current experimental\nmeasurements and discuss the dependence of the results on the SUSY\nspectrum.  We also discuss the correlations between the radiative\ndecays and the 3-body decays of charged lepton as well as the\nnon-decoupling effects in LFV neutral Higgs decays.  Finally we\nconclude in Sec.~\\ref{sec:conclusion}.  All required Feynman rules\nused in our calculations are collected in appendix~\\ref{app:lagr}. The\ndefinitions of loop integrals can be found in appendix~\\ref{app:loop}.\nIn appendix~\\ref{app:ddiff} we explain the notation for the ``divided\ndifferences'' of the loop functions used in the expanded form of the\namplitudes. The expression for the 4-lepton box diagrams and for the\nMI-expanded expression of the amplitudes are given in the\nappendices~\\ref{app:fullbox} and ~\\ref{app:miexp}, respectively.\n\n\n\\section{Effective LFV interactions}\n\\label{sec:efflag}\n\nIn this Section we collect the analytical formula in the mass\neigenbasis for flavour violating interactions generated at the\none-loop level\\footnote{Note that these expressions are not valid in\n  the flavour conserving case where additional terms should be\n  included and renormalization is required.}.  We use the notation and\nconventions for the MSSM as given in Ref.~\\cite{Rosiek:1989rs,\n  Rosiek:1995kg}\\footnote{The conventions of~\\cite{Rosiek:1989rs,\n    Rosiek:1995kg} are very similar to the later introduced and now\n  widely accepted SLHA2~\\cite{Allanach:2008qq} notation, up to the\n  minor differences summarised in the Appendix~\\ref{app:lagr}.}.\n\n\nIn our analysis, we include the so-called non-holomorphic trilinear\nsoft SUSY breaking terms:\n\\begin{eqnarray}\nL_{nh} = \\sum_{I,J=1}^3\\sum_{i=1}^2 \\left(A_l^{'IJ} H_i^{2\\star} L_i^I\nR^J+ A_d^{'IJ} H_i^{2\\star} Q_i^I D^J + A_u^{'IJ} H_i^{1\\star} Q_i^I\nU^J + \\mathrm{H.c.} \\right)\\,,\n\\end{eqnarray}\nwhich couple up(down)-sfermions to the down(up)-type Higgs doublets.\nHere, {as throughout the rest of the paper}, capital letters\n$I,J=1,2,3$ denote flavour indices and the small letters $i=1,2$ are\n$SU(2)_L$ indices.\n\n\n\\subsection{$\\gamma-\\ell-\\ell^\\prime$ interactions}\n\nWe define the effective Lagrangian for flavour violating couplings of\nleptons to on-shell photons as\n\\begin{eqnarray}\nL_{\\ell \\gamma }^{} = - e\\sum\\limits_{I,J} {\\left(\n  {F_{\\gamma}^{JI}{{\\bar \\ell }^J}{\\sigma _{\\mu \\nu }}{P_L}{\\ell ^I} +\n    F_{\\gamma}^{IJ*}{{\\bar \\ell }^J}{\\sigma _{\\mu \\nu }}{P_R}{\\ell\n      ^I}} \\right)} {F^{\\mu \\nu }}\\,,\n\\label{eq:lgdef}\n\\end{eqnarray}\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{pl_llg-eps-converted-to.pdf}\n\\caption{\\small One-loop supersymmetric contributions to the LF\n  violating effective lepton-photon interaction (mirror-reflected\n  self-energy diagram not shown).\\label{fig:llg}}\n\\end{center}\n\\end{figure}\n\nThe SM contribution to $F_{\\gamma}^{JI}$ is suppressed by powers of\n$m_{\\nu}^2\/M_W^2$ and thus completely negligible. In the mass\neigenbasis the supersymmetric contributions to $F_{\\gamma}^{JI}$ come\nfrom the diagrams displayed in Fig.~\\ref{fig:llg}. Let us decompose\n$F_{\\gamma}$ in the following way\n\\begin{eqnarray}\nF_{\\gamma}^{JI} = F_{\\gamma A}^{JI} - m_J F_{\\gamma L B}^{JI} - m_I\nF_{\\gamma R B}^{JI}\\,,\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n(4\\pi)^2 F_{\\gamma A}^{JI} &=& \\sum_{K=1}^3 \\sum_{n=1}^2\nV_{\\ell\\tilde\\nu C, R}^{JKn*} V_{\\ell\\tilde\\nu C, L}^{IKn} \\; m_{C_n}\nC_{11}(m_{C_n},m_{\\tilde\\nu_K}) \\nonumber\\\\\n&-&\\frac{1}{2} \\sum_{k=1}^6\\sum_{n=1}^4 V_{\\ell \\tilde L N,R}^{Jkn*}\nV_{\\ell \\tilde L N,L}^{Ikn} \\; m_{N_n} C_{12}(m_{\\tilde\n  L_k},m_{N_n})\\,,\\nonumber\\\\[3mm]\n(4\\pi)^2 F_{\\gamma L B}^{JI} &=& - \\sum_{K=1}^3 \\sum_{n=1}^2\nV_{\\ell\\tilde\\nu C, L}^{JKn*} V_{\\ell\\tilde\\nu C, L}^{IKn} \\;\nC_{23}(m_{C_n},m_{\\tilde\\nu_K}) \\nonumber\\\\\n&+& \\frac{1}{2} \\sum_{k=1}^6\\sum_{n=1}^4 V_{\\ell \\tilde L N,L}^{Jkn*}\nV_{\\ell \\tilde L N,L}^{Ikn} C_{23}(m_{\\tilde L_k},m_{N_n})\\,.\n\\label{eq:fgamab}\n\\end{eqnarray}\nHere, $V$ abbreviates the tree-level lepton-slepton-neutrino and\nlepton-sneutrino-chargino vertices, i.e. the subscripts of $V$ stand\nfor the interacting particles and the chirality of the lepton\ninvolved. The super-scripts refer to the lepton or slepton flavour as\nwell as to the chargino and neutralino involved.  The specific form of\nthe chargino and neutralino vertices $V_{L(R)}$ is defined in\nAppendix~\\ref{app:lagr} and the 3-point loop functions $C_{ij}$ are\ngiven in Appendix~\\ref{app:loop}. $F_{\\gamma A}\\, (F_{\\gamma L B}$)\ndenotes the parts of the amplitude which is (not) proportional to the\nmasses of fermions exchanged in the loop. $F_{\\gamma R B}$ can be\nobtained from $F_{\\gamma L B}$ by exchanging $L\\leftrightarrow R$ on\nthe RHS of~\\eq{eq:fgamab}.\n\nGauge invariance requires that LFV (axial) vectorial photon couplings\nvanish for on-shell external particles. However, off-shell photon\ncontributions are necessary to calculate three body decays of charged\nleptons. The vectorial part of the amplitude for the $\\gamma\\ell\\ell'$\nvertex can be written as\n\\begin{eqnarray}\ni A_{\\gamma}^{JI\\;\\mu} = ie q^2 \\bar u_J (p_J) \\left(\\Gamma_{\\gamma\n  L}^{JI} P_L + \\Gamma_{\\gamma R}^{JI} P_R \\right) \\gamma^{\\mu}\nu_I(p_I)\\;,\n\\label{eq:phvect}\n\\end{eqnarray}\nwhere $q=p_I-p_J$ and $\\Gamma_{\\gamma L}^{JI}$ is at the leading order\nin $p^2\/M_{\\rm SUSY}^2$ momentum independent and reads\n\\begin{eqnarray}\n\\Gamma_{\\gamma L}^{JI} &=& \\sum_{K=1}^3 \\sum_{n=1}^2 V_{\\ell\\tilde\\nu C,\n  L}^{JKn*} V_{\\ell\\tilde\\nu C, L}^{IKn} \\;\nC_{01}(m_{C_n},m_{\\tilde\\nu_K}) \\nonumber\\\\ &-& \\sum_{k=1}^6\\sum_{n=1}^4\nV_{\\ell \\tilde L N,L}^{Jkn*} V_{\\ell \\tilde L N,L}^{Ikn} \\;\nC_{02}(m_{N_n},m_{\\tilde L_k})\\,.\n\\label{eq:llgv}\n\\end{eqnarray}\n$\\Gamma_{\\gamma R}^{JI}$ can be obtained by replacing\n$L\\leftrightarrow R$. Again, the loop functions $C_{01},C_{02}$ are\ndefined in Appendix~\\ref{app:loop}.\n\nFinally, one should note that for heavy MSSM spectrum the 2-loop\nBarr-Zee diagrams~\\cite{Barr:1990vd} involving the non-decoupling LFV\nHiggs interactions (see Sec.~\\ref{sec:hllndec}) are important and have\nto be included~\\cite{Chang:1993kw,Hisano:2006mj, Jung:2013hka,\n  Abe:2013qla, Ilisie:2015tra,Crivellin:2015hha}.\n\n\n\\subsection{$Z-\\ell-\\ell^\\prime$ interactions}\n\nIn order to calculate the three body decays of charged leptons as to\nbe considered in Sec.~\\ref{sec:llll} it is sufficient to calculate the\neffective $Z-\\ell-\\ell^\\prime$ interactions in the limit of vanishing\nexternal momenta. The Wilson coefficients of the effective Lagrangian\nfor the $Z$ coupling to charged leptons are generated at one-loop\nlevel by the diagrams shown in Fig.~\\ref{fig:zll} and can be written\nas\n\\begin{eqnarray}\nL_{\\ell Z}^{JI} = \\left(F_{ZL}^{JI}\\bar\\ell^J\\gamma_\\mu P_L \\ell^I +\nF_{Z R}^{JI}\\bar\\ell^J\\gamma_\\mu P_R \\ell^I\\right) Z^\\mu\\,,\n\\label{eq:lzdef}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nF_{ZL}^{JI} & = & \\Gamma_{ZL}^{JI} - \\frac{e(1 - 2 s_W^2)}{2 s_W\n  c_W}\\; \\Sigma_{VL}^{JI}(0)\\,,\\nonumber\\\\\nF_{ZR}^{JI} & = & \\Gamma_{ZR}^{JI} + \\frac{e s_W}{c_W}\\;\n\\Sigma_{VR}^{JI}(0)\\,.\n\\end{eqnarray}\nHere, $\\Gamma_{ZL(R)}$ denote the contribution originating from the\none-particle irreducible (1PI) vertex diagram and $\\Sigma_{VL(R)}$ is\nthe left-(right-)handed part of the lepton self-energy defined as\n\\begin{eqnarray}\n\\Sigma^{JI}(p^2) = \\Sigma_{VL}^{JI}(p^2)\\, \\slashed{p}\\,\nP_L+\\Sigma_{VR}^{JI}(p^2) \\,\\slashed{p}\\, P_R+\\Sigma_{mL}^{JI}(p^2)\n\\,P_L+\\Sigma_{mR}^{JI}(p^2) \\, P_R\\;.\n\\label{eq:sigdef}\n\\end{eqnarray}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{pl_zll-eps-converted-to.pdf}\n\\caption{\\small One-loop supersymmetric contributions to the LFV\n  effective lepton-$Z^0$ interaction (the mirror-reflected self-energy\n  diagram not shown).\\label{fig:zll}}\n\\end{center}\n\\end{figure}\n\nContrary to the left- and right-handed magnetic photon-lepton\ncouplings, which change chirality, the $Z\\bar\\ell^I\\ell^J$ coupling is\nchirality conserving. Therefore, the Wilson coefficients of the\nleft-handed and right-handed couplings are not related to each other\nbut rather satisfy $F_{Z L(R)}^{IJ} = F_{Z L(R)}^{JI*}$.  In the mass\neigenbasis the vectorial part of the lepton self-energy and the 1PI\ntriangle diagrams are given by (see Appendix~\\ref{app:lagr} for\ndefinitions of vertices $V$)\n\\begin{eqnarray}\n(4\\pi)^2 \\Sigma_{VL}^{JI}(p^2) &=& \\sum_{i=1}^2\\sum_{K=1}^3\nV_{\\ell\\tilde\\nu C,L}^{IKi} V_{\\ell\\tilde\\nu C,L}^{JKi\\,*}\nB_1(p,m_{\\tilde\\nu_K},m_{C_i})\\nonumber\\\\ &+& \\sum_{i=1}^4\\sum_{j=1}^6 V_{\\ell\n  \\tilde L N,L}^{Iji} V_{\\ell \\tilde L N,L}^{Jji\\,*}\nB_1(p,m_{L_j},m_{N_i})\\,,\n\\label{eq:sigmaV}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n(4\\pi)^2 \\Gamma_{ZL}^{JI} &=& \\frac{1}{2} \\sum_{i,j=1}^2\\sum_{K=1}^3\nV_{\\ell\\tilde\\nu C,L}^{IKi} V_{\\ell\\tilde\\nu C,L}^{JKj*} \\, \\left(\nV_{CCZ,L}^{ij} C_2(m_{\\tilde\\nu_K},m_{C_i},m_{C_j})\\right.\\nonumber\\\\\n&-&\\left.  2 V_{CCZ,R}^{ij} m_{C_i} m_{C_j}\nC_0(m_{\\tilde\\nu_K},m_{C_i},m_{C_j}) \\right)\\nonumber\\\\\n&+&\\frac{e}{4 s_W c_W} \\sum_{i=1}^2\\sum_{K=1}^3 V_{\\ell\\tilde\\nu\n  C,L}^{IKi} V_{\\ell\\tilde\\nu C,L}^{JKi*} \\,\nC_2(m_{\\tilde\\nu_K},m_{\\tilde\\nu_K},m_{C_i}) \\nonumber\\\\\n&+& \\frac{1}{2} \\sum_{j=1}^6\\sum_{i,k=1}^4 V_{\\ell \\tilde L N,L}^{Iji}\nV_{\\ell \\tilde L N,L}^{Jjk\\,*}\\,\\left(V_{NNZ,L}^{ik}\nC_2(m_{L_j},m_{N_i},m_{N_k}) \\right. \\nonumber\\\\\n&-&\\left. 2 V_{NNZ,R}^{ik} m_{N_i} m_{N_k}\nC_0(m_{L_j},m_{N_i},m_{N_k})\\right)\\nonumber\\\\\n&-& \\frac{1}{2} \\sum_{j,k=1}^6\\sum_{i=1}^4 V_{\\ell \\tilde L N,L}^{Iji}\nV_{\\ell \\tilde L N,L}^{Jki\\,*}\\,V_{LLZ}^{jk}\nC_2(m_{L_j},m_{L_k},m_{N_i})\\,,\n\\label{eq:VZ}\n\\end{eqnarray}\nat vanishing external momenta with obvious replacements\n$L\\leftrightarrow R$ for $\\Sigma_{VR}^{JI},\\, \\Gamma_{ZR}^{JI}$.\n\n\n\\subsection{LFV Higgs interactions}\n\nTo compactify the notation, we denote the CP-even Higgs boson decays\nby $H_0^K\\to \\bar\\ell^I \\ell^J$, where, following again the notation\nof~\\cite{Rosiek:1989rs,Rosiek:1995kg}, $H\\equiv H_0^1, h\\equiv\nH_0^2$. As usual, we denote CP-odd neutral Higgs boson by $A_0$.\n\nIn order to study $h\\to\\ell\\ell^\\prime$ decays precisely, we keep the\nterms depending on the external Higgs mass. Therefore, we assume the\nfollowing effective action governing the LFV Higgs-lepton interaction:\n\\begin{eqnarray}\nA_{H{\\rm eff}}^{\\ell} &=& \\bar \\ell^J(k_J) (F_{h\\ell}^{JIK}(k_J,k_I)\nP_L + F_{h\\ell}^{IJK*}(k_J,k_I) P_R )\\ell^I(k_I) H_0^K(k_I - k_J)\\nonumber\\\\\n& +& \\bar \\ell^J(k_J) (F_{A\\ell}^{JI}(k_J,k_I) P_L +\nF_{A\\ell}^{IJ*}(k_J,k_I) P_R )\\ell^I(k_I) A_0(k_I - k_J) \\,.\n\\end{eqnarray}\nIn addition, to calculate the $\\mu\\to e$ conversion rate one needs to\ninclude the effective Higgs-quark couplings. For this purpose, one can\nset all external momenta to zero and consider the effective Lagrangian\n\\begin{eqnarray}\nL_{H{\\rm eff}}^q &=& \\bar u^J (F_{hu}^{JIK} P_L + F_{hu}^{IJK*} P_R\n)u^I H_0^K + \\bar d^J (F_{hd}^{JIK} P_L + F_{hd}^{IJK*} P_R ) d^I\nH_0^K\\,.\n\\label{eq:lhdef}\n\\end{eqnarray}\nHowever, in this article we consider only the lepton sector and\ntherefore do not give the explicit forms of Higgs quark couplings. The\nrelevant 1-loop expressions in the same notation as used in the\ncurrent paper are given in Ref.~\\cite{Buras:2002vd} and the formulae\nthat take into account also non-decoupling chirally enhanced\ncorrections and 2-loop QCD corrections in the general MSSM can be\nfound in Refs.~\\cite{Crivellin:2010er, Crivellin:2011jt,\n  Crivellin:2012zz}\\footnote{Earlier accounts on chiral resummation\n  can be found in Refs.~\\cite{Hall:1993gn, Carena:1994bv,\n    Carena:1999py, Bobeth:2001sq, Babu:1999hn, Isidori:2001fv,\n    Dedes:2002er,Hofer:2009xb,Noth:2010jy}}.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{pl_hll-eps-converted-to.pdf}\n\\caption{\\small Slepton-neutralino diagrams contributing to the\n  $H_0^K\\to \\ell^I \\bar \\ell^J$ {and $A_0\\to \\ell^I \\bar \\ell^J$}\n  decays in the MSSM (the mirror-reflected self-energy diagram is\n  omitted).\\label{fig:diag}}\n\\end{center}\n\\end{figure}\n\nAt the 1-loop level there are eight diagrams contributing to the\neffective lepton Yukawa couplings.  The ones with slepton and\nneutralino exchange are displayed in Fig.~\\ref{fig:diag}, while\ndiagrams with the chargino exchange can be obtained by the obvious\nreplacements $N\\to C, L\\to\\tilde\\nu$.\n\nThe expressions for $F_h$ are obtained from 1PI triangle diagrams and\nthe scalar part of lepton self-energies (see \\eq{eq:sigdef}) while the\nchirality conserving parts of the self-energies are absorbed by a\nfield rotation required to go to the physical basis with a diagonal\nlepton mass matrix. Therefore,\n\\begin{eqnarray}\nF_h^{JIK}(k_J,k_I) &=& \\Gamma_{h}^{JIK}(k_J,k_I) -\n\\frac{Z_R^{1K}}{v_1} \\,\\Sigma_{mL}^{JI}(0)\\;,\\nonumber\\\\\nF_A^{JI}(k_J,k_I) &=& \\Gamma_{A}^{JI}(k_J,k_I) - \\frac{i\n  \\sin\\beta}{v_1} \\,\\Sigma_{mL}^{JI}(0)\\;,\n\\label{eq:yukeff}\n\\end{eqnarray}\nwhere the $Z_R$ denotes the CP-even Higgs mixing matrix (see\nAppendix~\\ref{app:lagr}) and the scalar self-energy contributions are\nevaluated at zero momentum transfer and given by:\n\\begin{eqnarray}\n(4\\pi)^2 \\Sigma_{mL}^{JI}(0) & = & \\sum_{i=1}^2\\sum_{L=1}^3 m_{C_i}\nV_{\\ell\\tilde\\nu C,L}^{ILi} V_{\\ell\\tilde\\nu C,R}^{JLi\\,*}\n\\,\\,B_0\\,(0,m_{\\tilde\\nu_L},m_{C_i})\\nonumber\\\\\n&+& \\sum_{i=1}^4\\sum_{j=1}^6 m_{N_i} V_{\\ell \\tilde L N,L}^{Iji}\nV_{\\ell \\tilde L N,R}^{Jji\\,*}\\,\\,B_0\\,(0,m_{L_j},m_{N_i})\n\\label{eq:sigmaLR}\n\\end{eqnarray}\nThe neutralino-slepton contributions to the 1PI vertex diagrams can be\nwritten as (the symbols in square brackets denote common arguments of\nthe 3-point functions)\\footnote{As we shall see later using MI\n  expanded formulae (see Appendix~\\ref{app:lhmi}), due to strong\n  cancellations the leading order terms in Eqs.~(\\ref{eq:sigmaLR},\n  \\ref{eq:hnn}) are suppressed by the ratios of $m_\\ell\/M_W$ or\n  $A'_l\/M_{SUSY}$. Additional terms linear in $m_\\ell\/M_W$, not\n  included in~\\eq{eq:hnn}, appear in 1PI vertex diagrams when external\n  lepton masses are not neglected. We calculated such terms and proved\n  explicitly that after performing the MI expansion they are\n  suppressed by additional powers of $v^2\/M_{SUSY}^2$ and therefore,\n  {\\em a posteriori}, negligible. Thus, we do not display such terms\n  in~\\eq{eq:hnn}.}  {\n\\begin{eqnarray}\n(4\\pi)^2 \\Gamma_h^{JIK}(k_J,k_I) &=& - \\sum_{n=1}^4 \\sum_{l,m=1}^6\nV_{\\ell\\tilde L N,L}^{Jmn*} V_{\\ell\\tilde L N,L}^{Iln} V_{H\\tilde L\n  \\tilde L}^{Klm} m_{N_n} C_0 [k_J,k_I-k_J,m_{N_n},m_{\\tilde\n    L_m},m_{\\tilde L_l}]\\nonumber\\\\\n&& \\hspace{-38mm} - \\sum_{l,n=1}^4 \\sum_{m=1}^6 V_{\\ell\\tilde L\n  N,R}^{Jnm*} V_{\\ell\\tilde LN,L}^{Inl} (V_{NHN,R}^{lKm} C_2 +\nV_{NHN,L}^{lKm} m_{N_l} m_{N_m} C_0 )[k_J,k_I-k_J,m_{\\tilde\n    L_n},m_{N_m},m_{N_l}]\\;,\\nonumber\\\\\n(4\\pi)^2 \\Gamma_A^{JI}(k_J,k_I) &=& - \\sum_{n=1}^4 \\sum_{l,m=1}^6\nV_{\\ell\\tilde L N,L}^{Jmn*} V_{\\ell\\tilde L N,L}^{Iln} V_{A\\tilde L\n  \\tilde L}^{1lm} m_{N_n} C_0 [k_J,k_I-k_J,m_{N_n},m_{\\tilde\n    L_m},m_{\\tilde L_l}]\\nonumber\\\\\n&& \\hspace{-38mm} - \\sum_{l,n=1}^4 \\sum_{m=1}^6 V_{\\ell\\tilde L\n  N,R}^{Jnm*} V_{\\ell\\tilde LN,L}^{Inl} (V_{NAN,R}^{l1m} C_2 +\nV_{NAN,L}^{l1m} m_{N_l} m_{N_m} C_0 )[k_J,k_I-k_J,m_{\\tilde\n    L_n},m_{N_m},m_{N_l}]\\;,\\nonumber\\\\\n\\label{eq:hnn} \n\\end{eqnarray}\n}\nwhile the chargino-sneutrino triangle diagram is obtained by replacing\n$\\tilde L\\to\\tilde\\nu, N\\to C$ and adjusting the summation limits\nappropriately in vertex factors $V_{\\ldots}^{\\ldots}$ (see\nAppendix~\\ref{app:lagr}).\n\n\\subsection{Box contributions}\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{pl_box-eps-converted-to.pdf}\n\\caption{\\small Box diagrams with external charged leptons or\n  quarks\\label{fig:box}}\n\\end{center}\n\\end{figure}\n\n4-fermion interactions are also generated by box diagrams. The\ncorresponding conventions for incoming and outgoing particles are\nshown in Fig.~\\ref{fig:box}. We calculate all box diagrams in the\napproximation of vanishing external momenta.  The effective Lagrangian\nfor the 4-lepton interactions involves the quadrilinear operators\n\\begin{eqnarray}\nO_{VXY}^{JIKL} & = & (\\bar\\ell^{J}\\gamma^{\\mu}P_X \\ell^I) \\times\n(\\bar\\ell^K\\gamma_{\\mu}P_Y \\ell^L)\\,,\\nonumber\\\\\nO_{SXY}^{JIKL} & = & (\\bar\\ell^{J}P_X \\ell^I) \\times (\\bar\\ell^K P_Y\n\\ell^L)\\,,\\nonumber\\\\\nO_{TX}^{JIKL} & = & (\\bar\\ell^{J}\\sigma^{\\mu\\nu}\\ell^I) \\times\n(\\bar\\ell^K\\sigma_{\\mu\\nu}P_X \\ell^L)\\,,\n\\label{eq:lboxdec}\n\\end{eqnarray}\nwhere $X,Y$ stands for the chirality $L$ or $R$\\footnote{Recall that\n  $(\\bar\\ell^{J}\\sigma^{\\mu\\nu}P_L\\ell^I) \\times\n  (\\bar\\ell^K\\sigma_{\\mu\\nu}P_R \\ell^L)=0$.}.  The Wilson coefficients\nof these operators are calculated from the box diagrams in\nFig.~\\ref{fig:box} and are denoted by $B_{NXY}^{JIKL}$ with $N=V$,$S$,\nor $B_{TX}^{JIKL}$.\n\nThe operator basis in \\eq{eq:lboxdec} is redundant. First, we note\nthat\n\\begin{eqnarray}\nO_{NXY}^{JIKL} &=& O_{NYX}^{KLJI}\\quad\\mathrm{for}~N=V,S,\\nonumber\\\\\nO_{TX}^{JIKL} &=& O_{TX}^{KLJI}.\n\\label{eq:flip}\n\\end{eqnarray}\nSecond, there are Fierz relations among different operators:\n\\begin{eqnarray}\nO_{VXX}^{JIKL} &=& O_{VXX}^{KIJL}, \\nonumber\\\\\nO_{VXY}^{JIKL} &=& -\\, 2\\, O_{SXY}^{KIJL} \\quad \\mathrm{for}~X\\neq Y,\n\\nonumber\\\\\nO_{TX}^{JIKL} &=& \\frac{1}{2} O_{TX}^{KIJL} -\\, 6\\, O_{SXX}^{KIJL},\\nonumber\\\\\nO_{SXX} ^{JIKL} &=& -\\frac{1}{2} O_{SXX}^{KIJL} - \\frac{1}{8}\nO_{TX}^{KIJL}.\n\\label{eq:fierz}\n\\end{eqnarray}\nFurthermore, we have \n\\begin{eqnarray}\nO_{VXY}^{JIKL\\,\\dagger} &=& O_{VXY}^{IJLK}, \\qquad\\qquad\nO_{SLL}^{JIKL\\,\\dagger} \\;=\\; O_{SRR}^{IJLK}, \\nonumber\\\\\nO_{SLR}^{JIKL\\,\\dagger} &=& O_{SRL}^{IJLK}, \\qquad\\qquad\nO_{TL}^{JIKL\\,\\dagger} \\;=\\; O_{TR}^{JILK}.\n\\label{eq:herm}\n\\end{eqnarray}\nEqs.~(\\ref{eq:flip}) to (\\ref{eq:herm}) must be taken into account\nwhen deriving the effective Lagrangian.\n\n\n\\subsubsection{Leptonic operators with $\\mathbf{J\\neq K}$ and\n  $\\mathbf{I\\neq L}$\\label{sec:jnotk}}\n\n\nThe case with both $J\\neq K$ and $I\\neq L$ covers the decays\n$\\tau^{\\mp} \\to \\mu^{\\mp} e^{\\mp}\\ell^{\\pm} $ with $\\ell=e$ or $\\mu$,\nbut does not appear in $\\mu^{\\mp}$ decays.  We can therefore specify\nto $I=3$ for the effective Lagrangian.  Furthermore, we can choose\neither $(J,K)=(1,2)$ or $(J,K)=(2,1)$ without the need to sum over\nboth cases: The Fierz identities in \\eq{eq:fierz} permit to bring all\noperators into the form $(\\overline{e}\\ldots\\tau) \\times (\\overline{\\mu}\\ldots\n\\ell)$ (corresponding to the case $(J,K)=(1,2)$) or into an\nalternative form with $e$ interchanged with $\\mu$. Thus we have\n\\begin{eqnarray}\nL_{4\\ell}^{J3KL} &=&\n\\sum_{L=1,2} \\left[ \\sum_{\\stackrel{N=V,S}{X,Y=L,R}} B_{NXY}^{J3KL}\n  O_{NXY}^{J3KL} \\, +\\!  \\sum_{X=L,R} B_{TX}^{J3KL} O_{TX}^{J3KL}\n  \\right]\n\\;+\\; \\mbox{h.c.}\\nonumber\\\\ &&\\qquad\\qquad\\qquad \\qquad\\mbox{with }J\\neq\nK\\mbox{ and }J,K,L \\leq 2,\n \\label{eq:lboxbasis}\n\\end{eqnarray}\nas the four-lepton interaction in the Lagrangian.  Note that the\n``$+$h.c.'' piece of $L_{4\\ell}^{JK}$ describes $\\tau^+$ decays.\n\nThe Wilson coefficients $B_{NXY}^{J3KL}$ and $ B_{TX}^{J3KL}$ in\n\\eq{eq:lboxbasis} are simply identical to the results of the sum of\nall contributing box diagrams to the decay amplitude.  The latter is\ngiven in \\eq{eq:lllla0} with the coefficients of the spinor structure\nin the right column of Tab.~\\ref{tab:lbox}. The relation to the\nanalytic expressions in Eqs.~(\\ref{eq:lboxa}) to (\\ref{eq:lboxd}) is\n\\begin{eqnarray}\nB_{NXY}^{JIKL} &=& B_{A\\, NXY}^{JIKL} + B_{B\\, NXY}^{JIKL} + \n  B_{C\\, NXY}^{JIKL} + B_{D\\, NXY}^{JIKL},\\qquad\\mbox{for }N=V,S \n         \\label{eq:boxsum}\n\\end{eqnarray}\nand an analogous expression for $B_{TX}^{JIKL}$. \n\n\\subsubsection{Leptonic operators with $\\mathbf{J= K}$ and $\\mathbf{I\\neq L}$}\n\nThe case $J=K$ occurs for the decays $\\mu^\\pm \\to e^\\pm e^\\pm e^\\mp$\nand $\\tau^\\pm \\to \\ell^\\pm \\ell^\\pm \\ell^{\\mp\\prime}$ with\n$\\ell,\\ell^\\prime=e,\\mu$. Thanks to the Fierz identities in\n\\eq{eq:fierz} we may restrict the operator basis to\n\\begin{eqnarray}\nO_{VXX}^{JIJL},\\qquad\\quad \nO_{VXY}^{JIJL}\\;=\\; -2 O_{SXY}^{JIJL},&& \\qquad \nO_{SXX}^{JIJL}\\;=\\; -\\frac{1}{12} O_{TX}^{JIJL},\\nonumber\\\\\n&& \n\\qquad \\quad\\mbox{with }X,Y=L,R\\mbox{ and }X\\neq Y.\n\\label{eq:jkbasis}\n\\end{eqnarray}\nThe four-lepton piece of the effective Lagrangian for the decay\n$\\ell^{I\\mp} \\to \\ell^{J\\mp} \\ell^{J\\mp} \\ell^{L\\pm} $ reads:\n\\begin{eqnarray}\nL_{4\\ell}^{JIJL} &=& \\sum_{L=1,2} \\left[ \\sum_{X,Y=L,R}\n  \\widetilde{C}_{VXY}^{JIJL} O_{VXY}^{JIJL} \\, +\\! \\sum_{X=L,R}\n  \\widetilde{C}_{SXX}^{JIJL} O_{SXX}^{JIJL} \\right] \\;+\\;\n\\mbox{h.c.}\\nonumber\\\\ &&\\qquad\\qquad\\qquad \\qquad\\mbox{with }L,J <I.\n\\label{eq:lboxbasis2}\n\\end{eqnarray}\nFor the matching calculation it is useful to quote the tree-level\nmatrix elements of the operators:\n\\begin{eqnarray}\n\\bra{l^{J-}(p_J,s_J) l^{J-}(p_J^\\prime,s_J^\\prime) l^{L+}(p_L,s_J) }\nO_{VXX}^{JIJL} \\ket{l^{I-}(p_I,s_I) } \\hspace{-7cm} \\nonumber\\\\\n&=& \\phantom{\\; - \\;} [\\bar u(p_J,s_J) \\gamma_{\\mu} P_X u(p_I,s_I) ]\n  [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma^{\\mu} P_X v(p_L,s_L)] \\nonumber\\\\\n&& \\; - \\; [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma_{\\mu} P_X u(p_I,s_I)\n  ] [\\bar u(p_J,s_J) \\gamma^{\\mu} P_X v(p_L,s_L)] \\nonumber\\\\\n\\qquad &=& \\;\\; 2 \\, [\\bar u(p_J,s_J) \\gamma_{\\mu} P_X u(p_I,s_I) ]\n       [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma^{\\mu} P_X v(p_L,s_L)]\n       \\nonumber\\\\[2mm]\n\\bra{l^{J-}(p_J,s_J) l^{J-}(p_J^\\prime,s_J^\\prime) l^{L+}(p_L,s_J) }\nO_{VXY}^{JIJL} \\ket{l^{I-}(p_I,s_I) } \\hspace{-7cm} \\nonumber\\\\\n&=& \\phantom{\\; - \\;} [\\bar u(p_J,s_J) \\gamma_{\\mu} P_X u(p_I,s_I) ]\n  [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma^{\\mu} P_Y v(p_L,s_L)] \\nonumber\\\\\n&& \\; - \\; [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma_{\\mu} P_X u(p_I,s_I)\n  ] [\\bar u(p_J,s_J) \\gamma^{\\mu} P_Y v(p_L,s_L)] \\nonumber\\\\\n\\qquad &=& \\phantom{\\; - \\; 2\\,} [\\bar u(p_J,s_J) \\gamma_{\\mu} P_X\n  u(p_I,s_I) ] [\\bar u(p_J^\\prime,s_J^\\prime) \\gamma^{\\mu} P_Y\n  v(p_L,s_L)] \\nonumber\\\\\n&& \\; - \\; 2\\, [\\bar u(p_J,s_J) P_X u(p_I,s_I) ] [\\bar\n     u(p_J^\\prime,s_J^\\prime) P_Y v(p_L,s_L)],\\qquad\\mbox{for } X\\neq\n   Y, \\nonumber\\\\[2mm]\n\\bra{l^{J-}(p_J,s_J) l^{J-}(p_J^\\prime,s_J^\\prime) l^{L+}(p_L,s_J) } \n  O_{SXX}^{JIJL} \\ket{l^{I-}(p_I,s_I) } \\hspace{-7cm} \\nonumber\\\\\n&=& \\phantom{\\; - \\;} [\\bar u(p_J,s_J) P_X u(p_I,s_I) ] [\\bar\n    u(p_J^\\prime,s_J^\\prime) P_X v(p_L,s_L)] \\nonumber\\\\\n&& \\; - \\; [\\bar u(p_J^\\prime,s_J^\\prime) P_X u(p_I,s_I) ] [\\bar\n    u(p_J,s_J) P_X v(p_L,s_L)] \\nonumber\\\\\n\\qquad &=& \\phantom{\\;-\\;} \\frac12 [\\bar u(p_J,s_J) P_X u(p_I,s_I) ]\n       [\\bar u(p_J^\\prime,s_J^\\prime) P_X v(p_L,s_L)] \\nonumber\\\\\n&& \\; - \\; \\frac18\\, [\\bar u(p_J,s_J) \\sigma_{\\mu\\nu} P_X u(p_I,s_I) ]\n   [\\bar u(p_J^\\prime,s_J^\\prime) \\sigma^{\\mu\\nu} P_X\n     v(p_L,s_L)] \\label{eq:melt}\n\\end{eqnarray}\nHere we have used the Fierz transform to group the spinors into the\ncanonical order $[\\bar u(p_J,...) ... u(p_I,...)][ \\bar\n  u(p_J^\\prime,...) ... v(p_L,...)]$. This allows us to use the same\nformula for spin-summed squared matrix elements as in the case of\n$J\\neq K$ of Sec.~\\ref{sec:jnotk}.\n\nTo quote the Wilson coefficients $\\widetilde{C}_{NXY}^{JIJL}$, $N=V,S$\nin terms of the box diagrams $B_{NXY}^{JIJL}$ in \\eq{eq:boxsum} we\nmust compare the results of the MSSM decay amplitude in\n\\eq{eq:lllla00} with the matrix elements in \\eq{eq:melt} and read off\ncoefficients of the various Dirac structures. The result is\n\\begin{eqnarray}\n\\widetilde{C}_{VXX}^{JIJL} &=& \\frac{1}{2} B_{VXX}^{JIJL},\\nonumber\\\\\n\\widetilde{C}_{VXY}^{JIJL} & =& B_{VXY}^{JIJL} \\qquad \\mbox{for }X\\neq\nY, \\nonumber\\\\\n\\widetilde{C}_{SXX}^{JIJL} & =& 2\\, B_{SXX}^{JIJL}.\n\\label{eq:wc}\n\\end{eqnarray}\nThe Fierz identities further imply the equalities \n\\begin{eqnarray}\nB_{SXY}^{JIJL} & = & -2 B_{VXY}^{JIJL} \\qquad \\mbox{for }X\\neq Y,\\nonumber\\\\\nB_{TX}^{JIJL} & =& -\\frac14 B_{SXX}^{JIJL} \\; .\\label{eq:feq}   \n\\end{eqnarray}\n\n\n\\subsubsection{Leptonic operators with $\\mathbf{J= K}$ and $\\mathbf{I=L}$}\n\nThese operators do not appear in lepton decays, but trigger\nmuonium-antimuonium transitions and describe muon or tau pair\nproduction in $e^-$--$e^-$ collisions at energies far below\n$M_{SUSY}$. Their Wilson coefficients are tiny in the MSSM.\n\n\\subsubsection{Operators with two leptons and two quarks}\n\nThe analogous Lagrangian for the 2-lepton--2-quark interactions reads\n\\begin{eqnarray}\nL_{2\\ell2q}^{IJKL} = \\sum_{N,X,Y} B^{IJKL}_{q\\,NXY}O^{JIKL}_{q\\,NXY}\n\\label{eq:qboxbasis}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nO_{q\\, VXY}^{IJKL} & = & (\\bar\\ell_{I}\\gamma^{\\mu}P_X \\ell_J) \\times\n(\\bar q_K\\gamma_{\\mu}P_Y q_L)\\,,\\nonumber\\\\\nO_{q\\,SXY}^{IJKL} & = & (\\bar\\ell_{I}P_X \\ell_J)\\times (\\bar q_L P_Y\nq_K)\\,,\\nonumber\\\\\nO_{q\\,TX}^{IJKL} & = & (\\bar\\ell_{I}\\sigma^{\\mu\\nu}\\ell_J) \\times\n(\\bar q_K \\sigma_{\\mu\\nu}P_X q_L)\\,.\n\\label{eq:qboxdec}\n\\end{eqnarray}\nAgain, we consider only purely leptonic contributions here in detail\nbut do not give explicit expressions for the 2-lepton--2-quark box\ndiagrams.  The relevant expressions in the mass eigenbasis can be\nfound using formulae of Appendix~\\ref{app:fullbox} and inserting\nproper quark vertices from Refs.~\\cite{Rosiek:1989rs, Rosiek:1995kg}\ninto these.\n\n\n\n\\section{Observables}\n\\label{sec:lfv}\n\nIn this Section we collect the formulae for the LFV observables in\nterms of the effective interactions defined in Sec.~\\ref{sec:efflag}.\nAll the processes listed here will be included in the future version\nof the SUSY\\_FLAVOR numerical library calculating an extensive set of\nflavour and CP-violating observables both in the quark and leptonic\nsectors~\\cite{Rosiek:2010ug, Crivellin:2012jv, Rosiek:2014sia}.\n\n\\subsection{Radiative lepton decays: $\\ell^I\\to\\ell^J\\gamma$}\n\nThe branching ratios for the radiative lepton decays\n$\\ell^I\\to\\ell^J\\gamma$ are given by\n\\begin{eqnarray}\n\\text{Br}(\\ell^I\\to\\ell^J\\gamma) &=&\n\\frac{48\\pi^2 e^2 }{m_I^2 G_F^2} \\left(|F_{\\gamma}^{JI}|^2 +\n|F_{\\gamma}^{IJ}|^2\\right) \\text{Br}(\\ell^I\\to e\\nu\\nu)\\,.\n\\end{eqnarray}\nHere we used $\\Gamma(\\ell^I\\to e\\nu\\nu)\\approx G_F^2 m_I^5\/(192 \\pi^3)$\nfor the tree-level leptonic decay width and the factors\n$\\text{Br}(\\mu\\to e\\nu\\nu)\\approx 1$,\n$\\text{Br}(\\tau\\to e\\nu\\nu) = 0.1785 \\pm 0.0005$~\\cite{Agashe:2014kda} are\nintroduced to account for the hadronic decay modes of the $\\tau$ lepton.\n\nEven though in our numerical analyses we restrict ourselves to LFV\nprocesses, we remind the reader that the expressions for the anomalous\nmagnetic moments and electric dipole moments of the charged leptons\ncan be also calculated in term of the quantities defined\nin~\\eq{eq:fgamab} and read:\n\\begin{eqnarray}\n\\Delta a_I &=& - 4 m_I \\text{\\,Re\\,}\\left[ F_{\\gamma A}^{II} - m_I\n  \\left(F_{\\gamma L B}^{II} + F_{\\gamma R B}^{II}\\right)\\right]\\,,\\\\\nd_l^I &=& - 2 e \\text{\\,Im\\,} F_{\\gamma A}^{II}\n\\end{eqnarray}\n\n\\subsection{$h(H)\\to \\bar\\ell^{I}\\ell^{J}$ decays}\n\\label{sec:hll}\n\nThe decay branching ratios for the CP-even and CP-odd Higgs bosons\nread:\n\\begin{eqnarray}\nBr(H_0^K\\to \\ell^{I+} \\ell^{J-}) & =&\n\\frac{m_{H_0^K}}{16\\pi\\Gamma_{H_0^K}} \\left( \\left|F_h^{IJK}\\right|^2\n+ \\left|F_h^{JIK}\\right|^2\\right)\\nonumber\\\\\nBr(A_0\\to \\ell^{I+} \\ell^{J-})& =& \\frac{m_{A}}{16\\pi\\Gamma_{A}}\n\\left( \\left|F_A^{IJ}\\right|^2 + \\left|F_A^{JI}\\right|^2\\right)\n\\label{eq:brhll}\n\\end{eqnarray}\nwith $F_h^{IJK}, F_A^{IJ}$ defined in~\\eq{eq:yukeff}. Note that\nsumming over lepton charges in the final state, $\\ell^{I+} \\ell^{J-}$\nand $\\ell^{J+} \\ell^{I-}$, would produce an additional factor of 2.\n\n\n\\subsection{$\\ell^I\\to \\ell^J \\ell^K\\bar \\ell^L$ decays}\n\\label{sec:llll}\n\nThe LFV decays of charged lepton into three lighter ones can be\ndivided into 3 classes, depending on the flavours in the final state:\n\\begin{itemize}\n\\item[(A)] $\\ell\\to\\ell^\\prime\\ell^\\prime\\ell^\\prime$: Three leptons\n  of the same flavour, i.e. $\\mu^\\pm \\to e^\\pm e^+ e^-$, $\\tau^\\pm \\to\n  e^\\pm e^+ e^-$ and $\\tau^\\pm \\to \\mu^\\pm \\mu^+ \\mu^-$, with a pair\n  of opposite charged leptons.\n\\item[(B)] $\\ell^\\pm \\to \\ell^{\\prime\\pm} \\ell^{\\prime\\prime+}\n  \\ell^{\\prime\\prime-}$: Three distinguishable leptons with\n  $\\ell^{\\prime}$ carrying the same charge as $\\ell$, i.e. $\\tau^\\pm\n  \\to e^\\pm \\mu^+ \\mu^-$ and $\\tau^\\pm \\to \\mu^\\pm e^+ e^-$.\n\\item[(C)] $\\ell^\\pm \\to \\ell^{\\prime\\mp} \\ell^{\\prime\\prime+}\n  \\ell^{\\prime\\prime-}$: Three distinguishable leptons with\n  $\\ell^{\\prime}$ carrying the opposite charge as $\\ell$,\n  i.e. $\\tau^\\pm \\to e^\\mp \\mu^\\pm \\mu^\\pm$ and $\\tau^\\pm \\to \\mu^\\mp\n  e^\\pm e^\\pm$.\n\\end{itemize}\n\nClass (C), representing a $\\Delta L=2$ processes, is tiny within the\nMSSM: it could only be generated at 1-loop level by box diagrams\nsuppressed by double flavour changes, or at the 2-loop level by double\npenguin diagrams involving two LFV vertices. Therefore, we will not\nconsider these process in our numerical analysis.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{pl_llll-eps-converted-to.pdf}\n\\end{center}\n\\caption{\\small Diagrams contributing to $\\ell^I\\to \\ell^J\\ell^K\\bar\n  \\ell^L$ decay. I): 1PI irreducible box diagrams; II): penguin\n  diagrams with $V=Z,\\gamma,h,H$ or $A$. For $K=J$ crossed diagrams\n  must be also included.\n\\label{fig:llll}}\n\\end{figure}\n\n\\noindent In order to calculate $\\text{Br}(\\ell^I\\to \\ell^J\\ell^K\\bar\n\\ell^L)$ we decompose the corresponding amplitude $A$ as\n\\begin{eqnarray}\nA = A_0 + A_\\gamma\\,, \\label{eq:llllampl}\n\\end{eqnarray}\nThe relevant diagrams are displayed in Fig.~\\ref{fig:llll}.  $A_0$\ncontains contributions from 4-lepton box diagrams and from penguin\ndiagrams (including vector-like off-shell photon couplings,\nsee~\\eq{eq:phvect}) which in the limit of vanishing external momenta\ncan be represented as the 4-fermion contact interactions.  $A_\\gamma$\nis the on-shell photon contribution originating from the magnetic\noperator (see~\\eq{eq:lgdef}) which has to be treated separately with\nmore care as the photon propagator becomes singular in the limit of\nvanishing external momenta.\n\nWe further decompose $A_0$ for the two cases (A) and (B) according to\nits Lorentz structure:\n\\begin{eqnarray}\nA_0^{(A)} &=& \\sum_{Q=V,S,T} C^{(A)}_{Q,\\,XY} [\\bar u(p_J)\n  \\Gamma_Q^\\prime P_X u(p_I)] [\\bar u(p_J^\\prime) \\Gamma_Q P_Y\n  v(p_L)]] \\;,\\label{eq:lllla00} \\\\\nA_0^{(B)} &=& \\sum_{Q=V,S,T} C^{(B)}_{Q,\\,XY} [\\bar u(p_J)\n  \\Gamma_Q^\\prime P_X u(p_I)][\\bar u(p_K) \\Gamma_Q P_Y v(p_L)]] \\;.\n  \\label{eq:lllla0} \n\\end{eqnarray}\nwith $X,Y=L,R$. Note that the amplitude $A_0^{(A)}$ in general\ncontains a second term which is obtained from the one given in\n\\eq{eq:lllla00} by replacing $(p_J\\leftrightarrow\np_J^\\prime)$. However, one can use Fierz identities to reduce it to\nthe structure given in \\eq{eq:lllla00}. The basis of Dirac\nquadrilinears $\\Gamma_Q$ is the same as the one used to decompose\n4-lepton box diagrams in~\\eq{eq:lboxdec}:\n\\begin{eqnarray}\n\\Gamma_S=1\\,, \\qquad \\Gamma_V = \\gamma^\\mu\\,, \\qquad \\Gamma_T =\n\\sigma_{\\mu\\nu}\\,,\n\\label{eq:qnbasis}\n\\end{eqnarray}\nand $\\Gamma_Q^\\prime$ is obtained from $\\Gamma_Q$ by lowering the\nLorentz indices.\n\nThe amplitudes originating from on-shell photon exchange are given by\n\\begin{eqnarray}\nA_\\gamma^{(A)} & = &\\! \\frac{e}{(p_I-p_J)^2} [\\bar u(p_J)\n  i\\sigma^{\\mu\\nu} (C_{\\gamma L} P_L + C_{\\gamma R} P_R) (p_I -\n  p_J)_\\nu u(p_I)] [\\bar u(p_J^\\prime) \\gamma_\\mu v(p_L)] \\nonumber\\\\\n&&- (p_J\\leftrightarrow p_J^\\prime) \\nonumber\\\\\nA_\\gamma^{(B)} & = &\\! \\frac{e}{(p_I-p_J)^2}[ \\bar u(p_J)\n  i\\sigma^{\\mu\\nu} (C_{\\gamma L} P_L + C_{\\gamma R} P_R) (p_I -\n  p_J)_\\nu u(p_I)][\\bar u(p_K) \\gamma_\\mu v(p_L)].\n\\label{eq:llllag}\n\\end{eqnarray}\n\n\\begin{table}[tb]\n  \\begin{center}\n    {\\small\n      \\begin{tabular}{|lp{-0mm}|l|l|}\n        \\hline \n        \\hspace{-32mm} && Decay (A) & Decay (B)  \\\\\n        \\hline\n       \n                      \n        $C_{VLL}$ && $ B_{VLL}^{JIJJ} -\\frac{e(1- 2 s_W^2)}{s_W c_W M_Z^2}\n                     F_{ZL}^{JI} + 2 e^2 V_{\\gamma L}^{JI} $ & \\scalebox{0.9}{$B_{VLL}^{JIKK} -\\frac{e(1-\n                                                               2 s_W^2)}{2s_W c_W M_Z^2} F_{ZL}^{JI}+ e^2 V_{\\gamma L}^{JI} $}\\\\[2mm]\n                                                          \n        $C_{VRR}$ && $ B_{VRR}^{JIJJ} +\\frac{2 e s_W}{c_W M_Z^2} F_{ZR}^{JI}+\n                     2 e^2 V_{\\gamma R}^{JI} $ & $B_{VRR}^{JIKK} + \\frac{e s_W}{c_W\n                                                 M_Z^2} F_{ZR}^{JI}+ e^2 V_{\\gamma R}^{JI} $\\\\[2mm]\n                                              \n        $C_{VLR}$ && \\scalebox{0.9}{$B_{VLR}^{JIJJ} +\\frac{e s_W}{c_W M_Z^2} F_{ZL}^{JI} +\n                     e^2 V_{\\gamma L}^{JI} + {\\frac{1}{2}Y_l^J (V_H^{IJ*} - V_A^{IJ*})}\n                     $} & $B_{VLR}^{JIKK} +\\frac{e s_W}{c_W M_Z^2} F_{ZL}^{JI}+ e^2\n                          V_{\\gamma L}^{JI} $ \\\\[2mm]\n                         \n        $C_{VRL}$ && \\scalebox{0.9}{$B_{VRL}^{JIJJ} -\\frac{e(1- 2 s_W^2)}{2 s_W c_W M_Z^2}\n                     F_{ZR}^{JI} + e^2 V_{\\gamma R}^{JI} + {\\frac{1}{2}Y_l^J (V_H^{JI} -\n                     V_A^{JI})} $} & \\scalebox{0.9}{$B_{VRL}^{JIKK} -\\frac{e(1- 2 s_W^2)}{2s_W c_W\n                                     M_Z^2} F_{ZR}^{JI} + e^2 V_{\\gamma R}^{JI}$} \\\\[2mm]\n                                \n        $C_{SLL}$ && $B_{SLL}^{JIJJ} + {\\frac{3}{2}Y_l^J (V_H^{JI} +\n                     V_A^{JI})}$ & $B_{SLL}^{JIKK} + {Y_l^K (V_H^{JI} +\n                                   V_A^{JI})}$\\\\[2mm]\n                                \n        $C_{SRR}$ && $B_{SRR}^{JIJJ} + {\\frac{3}{2} Y_l^J (V_H^{IJ*} +\n                     V_A^{IJ*})}$ & $B_{VRR}^{JIKK} + { Y_l^K (V_H^{IJ*} +\n                                    V_A^{IJ*})}$\\\\[2mm]\n                                 \n        $C_{SLR}$ && \\scalebox{0.9}{$-2 B_{VLR}^{JIJJ} -\\frac{2 e s_W}{c_W M_Z^2}\n                     F_{ZL}^{JI}- 2 e^2 V_{\\gamma L}^{JI} + {Y_l^J (V_H^{JI}\n                     - V_A^{JI})}$} \n                                    & $B_{SLR}^{JIKK} + {Y_l^K (V_H^{JI} -\n                                      V_A^{JI})}$\\\\[3mm]\n                                   \n        $C_{SRL}$ && \\scalebox{0.9}{$ - 2 B_{SRL}^{JIJJ} +\\frac{e(1- 2 s_W^2)}{s_W c_W M_Z^2}\n                     F_{ZR}^{JI} - 2 e^2 V_{\\gamma R}^{JI} + {Y_l^J (V_H^{IJ*} -\n                     V_A^{IJ*})}$} \n                                    & $B_{SRL}^{JIKK} + {Y_l^K (V_H^{IJ*} -\n                                      V_A^{IJ*})}$\\\\[2mm]\n                                   \n        $C_{TL}$ && $-\\frac{1}{4}B_{SLL}^{JIJJ} + {\\frac{1}{8}Y_l^J\n                    (V_H^{JI} + V_A^{JI})}$ & $B_{TL}^{JIKK}$\\\\[2mm]\n        $C_{TR}$ && $-\\frac{1}{4}B_{SRR}^{JIJJ} + \\frac{1}{8} Y_l^J\n                    (V_H^{IJ*} + V_A^{IJ*}) $ & $B_{TR}^{JIKK}$\\\\[2mm] $C_{\\gamma L}$\n                       && $-2 e F_\\gamma^{JI}$ & $-2 e F_\\gamma^{JI}$\\\\[2mm] $C_{\\gamma R}$\n                       && $- 2 e F_\\gamma^{IJ*}$ & $-2 e F_\\gamma^{IJ*}$\\\\[2mm]\n        \\hline\n      \\end{tabular}\n    }\n  \\end{center}\n  \\caption{\\small Coefficients $C_N, C_\\gamma$ of~\\eq{eq:lllla0} and\n    \\eq{eq:llllag} for decay types (A) and (B). $B_{QXY}$,$B_{TX}$\n    denote the irreducible box diagram contributions\n    (see~\\eq{eq:lboxbasis}), the terms with $F_Z$ stem from the $Z$\n    penguin Lagrangian (\\eq{eq:lzdef}), $V_\\gamma$ is the sum of the\n    vector-like photon contributions (\\eq{eq:phvect}), Higgs\n    contributions are defined in~\\eq{eq:vhadef} and the coefficients\n    $F_\\gamma$ of the magnetic operator are defined\n    in~\\eq{eq:lgdef}. \\label{tab:lbox}}\n\\end{table}\n\n\nThe full form of the coefficients $C_N^{(A,B)},\\;C_\\gamma$ is\ndisplayed in Table~\\ref{tab:lbox}, where we compactified the\nexpressions by using the following abbreviations for the Higgs penguin\ncontributions\\footnote{Note that we define lepton Yukawa coupling\n  appearing in Table~\\ref{tab:lbox} to be negative, $Y_l^I =\n  -\\sqrt{2}m_l^I\/v_1$}:\n\\begin{eqnarray}\nV_H^{JI} = \\sum_{N=1}^2 \\frac{Z_R^{1N}}{m_{H_0^N}^2} F_h^{JIN}\\;,\n\\qquad\\qquad\nV_{A}^{JI} = \\frac{i\\sin\\beta}{m_{A_0}^2} F_A^{JI}\\;.\n\\label{eq:vhadef}\n\\end{eqnarray}\nNote that in~\\eq{eq:lllla0} and~\\eq{eq:llllag} we do not explicitly\ndisplay flavour indices, but they are specified in\nTable~\\ref{tab:lbox}.\n\nNeglecting the lighter lepton masses whenever possible, the expression\nfor the branching ratios can be written down as (for comparison\nsee~\\cite{Ilakovac:2012sh}):\n\\begin{eqnarray}\n\\mathrm{Br}(\\ell^I\\to \\ell^J \\ell^K \\bar \\ell^L) &=& \\frac{N_c\n  \\text{Br}(\\ell^I\\to e\\nu\\nu)}{32 G_F^2} \\left(4 \\left(|C_{VLL}|^2 +\n|C_{VRR}|^2 + |C_{VLR}|^2 + |C_{VRL}|^2\\right)\\right.\\nonumber\\\\\n&+&|C_{SLL}|^2 + |C_{SRR}|^2+ |C_{SLR}|^2+ |C_{SRL}|^2 \\nonumber\\\\\n&+& \\left.   48 \\left(|C_{TL}|^2 + |C_{TR}|^2\\right) + X_\\gamma\\right)\n\\label{eq:br4l}\n\\end{eqnarray}\nwhere $N_c=1\/2$ if two of the final state leptons are identical\n(decays (A)), $N_c=1$ for decays (B) and $X_\\gamma$ denotes the\ncontribution to matrix element from the photon penguin $A_\\gamma$,\nincluding also its interference with the $A_0$ part of the amplitude\n($m$ denotes the mass of the heaviest final state lepton)\n\\begin{eqnarray}\nX_\\gamma^{(A)} & =& -\\frac{16e}{m_{\\ell^I}}\\mathrm{Re}\\left[(2\n  C_{VLL} + C_{VLR} - \\frac{1}{2} C_{SLR} )\\; C_{\\gamma R}^\\star +\n  ( 2 C_{VRR} + C_{VRL} - \\frac{1}{2} C_{SRL})\\; C_{\\gamma\n    L}^\\star \\right] \\nonumber\\\\\n&& + \\frac{64 e^2}{m_{\\ell^I}^2} \\left(\\log\\frac{m_{\\ell^I}^2}{m^2} -\n\\frac{11}{4} \\right)(|C_{\\gamma L}|^2 + |C_{\\gamma R}|^2) \\nonumber\\\\\nX_\\gamma^{(B)} & =& -\\frac{16e}{m_{\\ell^I}}\\mathrm{Re}\\left[ \\left(C_{VLL} +\n  C_{VLR}\\right) C_{\\gamma R}^\\star + \\left(C_{VRR} + C_{VRL}\\right)\n  C_{\\gamma L}^\\star \\right]\\nonumber\\\\\n&& +\\frac{32 e^2}{m_{\\ell^I}^2} \\left(\\log\\frac{m_{\\ell^I}^2}{m^2} -3\\right)\n(|C_{\\gamma L}|^2 + |C_{\\gamma R}|^2) \\;.\n\\label{eq:brg}\n\\end{eqnarray}\n\n\\subsection{$\\mu\\to e$ conversion in Nuclei}\n\\label{sec:mueconv}\n\nThe full 1-loop expressions for the $\\mu\\to e$ conversion in Nuclei\ndepend on both the squark and slepton SUSY breaking terms. Thus, in\nprinciple the resulting upper bounds on the slepton mass insertions\ndepend to some extent on the squark masses. Therefore, we do not\ninclude $\\mu\\to e$ conversion in nuclei in our numerical\nanalysis\\footnote{Recent discussion of interplay between the bounds on\n  MI's in the slepton and squark sectors can be found in\n  Ref.~\\cite{Ellis:2016yje}.}.  However, for completeness we collect\nhere the complete set of formulae required to calculate the rate of\nthis process.\n\n$\\mu\\to e$ conversion in nuclei is produced by the dipole, the vector,\nand the scalar operators already at the tree\nlevel~\\cite{Kitano:2002mt}.  Following the discussion of\nRef.~\\cite{Crivellin:2017rmk} we use the effective Lagrangian\n\\begin{eqnarray}\nL_{\\mu\\to e} &=& \\sum_{N,X,Y} C_{q_Iq_I}^{N\\;XY}O^{q_Iq_I}_{N\\;XY} +\nC^{gg}_X O^{gg}_X\n\\end{eqnarray}\nwhere $N=V,S$ and $X,Y=L,R$ with the operators defined as\n\\begin{eqnarray}\nO^{q_Iq_I}_{V\\;XY} &=& \\left( {\\bar e{\\gamma ^\\mu }{P_X}\\mu }\n\\right)\\left( {\\bar q_I{\\gamma _\\mu }{P_Y}q_I} \\right)\\nonumber\\\\\nO^{q_Iq_I}_{S\\;XY} &=& \\left( {\\bar e{P_X}\\mu } \\right)\\left( {\\bar\n  q_I{P_Y}q_I} \\right)\\nonumber\\\\\nO^{gg}_X &=& {\\alpha _s}{\\mkern 1mu} {m_\\mu }{G_F}\\left( {\\bar\n  e{P_X}\\mu } \\right)G_{\\mu \\nu }^aG_a^{\\mu \\nu }\n\\end{eqnarray}\nUsing the notation introduced in previous Sections, the corresponding\nWilson coefficients can be expressed as\n\\begin{eqnarray}\nC_{V\\;XL}^{{d_I}{d_I}} &=& C_{d\\ell {\\kern 1pt} VXL}^{12II} -\n\\frac{1}{{m_Z^2}}\\frac{e}{{2{s_W}{c_W}}}\\left( {1 - \\frac{2}{3}s_W^2}\n\\right)F_{ZX}^{12} - \\frac{1}{3}{e^2}V_{\\gamma X}^{JI}\\nonumber\\\\\nC_{V\\;XR}^{{d_I}{d_I}} &=& C_{d\\ell {\\kern 1pt} VXR}^{12II} +\n\\frac{1}{{m_Z^2}}\\frac{e}{{3{s_W}{c_W}}}s_W^2F_{ZX}^{12} -\n\\frac{1}{3}{e^2}V_{\\gamma X}^{JI}\\nonumber\\\\\nC_{V\\;XL}^{{u_I}{u_I}} &=& C_{u\\ell {\\kern 1pt} VXL}^{12II} +\n\\frac{1}{{m_Z^2}}\\frac{e}{{2{s_W}{c_W}}}\\left( {1 - \\frac{4}{3}s_W^2}\n\\right)F_{ZX}^{12} + \\frac{2}{3}{e^2}V_{\\gamma X}^{JI}\\nonumber\\\\\nC_{V\\;XR}^{{u_I}{u_I}} &=& C_{u\\ell {\\kern 1pt} VXR}^{12II} -\n\\frac{1}{{m_Z^2}}\\frac{e}{{{s_W}{c_W}}}\\frac{2}{3}s_W^2F_{ZX}^{12} +\n\\frac{2}{3}{e^2}V_{\\gamma X}^{JI}\\nonumber\\\\\nC_{S\\;LX}^{{d_I}{d_I}} &=& C_{d\\ell {\\kern 1pt} SLX}^{12II} +\n\\frac{1}{{{{\\left( {m_0^K} \\right)}^2}}}F_h^{12K}F_{hd}^{IIK}\\nonumber\\\\\nC_{S\\;LX}^{{u_I}{u_I}} &=& C_{u\\ell {\\kern 1pt} SLX}^{12II} +\n\\frac{1}{{{{\\left( {m_0^K} \\right)}^2}}}F_h^{12K}F_{hu}^{IIK}\\nonumber\\\\\nC_{S\\;RX}^{{d_I}{d_I}} &=& C_{d\\ell {\\kern 1pt} SRX}^{12II} +\n\\frac{1}{{{{\\left( {m_0^K} \\right)}^2}}}F_h^{21K*}F_{hd}^{IIK}\\nonumber\\\\\nC_{S\\;RX}^{{u_I}{u_I}} &=& C_{u\\ell {\\kern 1pt} SRX}^{12II} +\n\\frac{1}{{{{\\left( {m_0^K} \\right)}^2}}}F_h^{21K*}F_{hu}^{IIK}\n\\end{eqnarray}\n\nFor this process, a Lagrangian involving only quark, lepton and photon\nfields is not sufficient. Instead, an effective Lagrangian at the\nnucleon level containing proton and neutron fields is required. It can\nbe obtained in two steps. First, heavy quarks are integrated out. This\nresults in a redefinition of the Wilson coefficient of the gluonic\noperator~\\cite{Shifman:1978zn}\n\\begin{eqnarray}\nC^{gg}_L &\\to& \\tilde{C}^{gg}_L = C^{gg}_L - \\frac{1}{12 \\pi}\n\\sum_{q=c,b} \\frac{C^{qq}_{S\\;LL} + C^{qq}_{S\\;LR}}{G_F\\, m_\\mu m_q}\n\\label{anomaly}\n\\end{eqnarray}\nwith an analogous equation for $C_{gg}^R$. Second, the resulting\nLagrangian is matched at the scale of $\\mu_n=1$~GeV to an effective\nLagrangian at the nucleon level.  Following~\\cite{Cirigliano:2009bz}\nthe transition rate $\\Gamma_{\\mu\\to e}^N = \\Gamma(\\mu^- N \\to e^- N)$\ncan then be written as\n\\begin{eqnarray}\n\\Gamma_{\\mu\\to e}^N &=& \\frac{m_{\\mu}^{5}}{4} \\left| -e\\, C^{D}_{L} \\;\nF_\\gamma^{12}\/m_\\mu + 4\\left( G_F m_\\mu m_p \\tilde{C}^{(p)}_{SL}\nS^{(p)}_N + \\tilde{C}^{(p)}_{VR} \\; V^{(p)}_N + (p \\to n) \\right)\n\\right|^2\\nonumber\\\\\n&+& (L\\leftrightarrow R),\n\\label{Gconv}\n\\end{eqnarray}\nwhere $p$ and $n$ denote the proton and the neutron, respectively. The\neffective couplings in Eq.~(\\ref{Gconv}) can be expressed in terms of\nour Wilson coefficients as\n\\begin{eqnarray}\n\\tilde{C}^{(p\/n)}_{VR} &=& \\sum_{q=u,d,s} \n\\left(C^{qq}_{V\\;RL} + C^{qq}_{V\\;RR}\\right) \\; f^{(q)}_{Vp\/n} \\, , \n\\label{tildeCVR} \\\\\n\\tilde{C}^{(p\/n)}_{SL} &=& \\sum_{q=u,d,s}\n\\frac{\\left(C^{qq}_{S\\;LL}+C^{qq}_{S\\;LR}\\right)}{m_\\mu m_q G_F} \\;\nf^{(q)}_{Sp\/n} \\ + \\ \\tilde{C}^{gg}_L \\, f_{Gp\/n}\n\\label{tildeCSL}\n\\end{eqnarray}\nwith analogous relations for $L\\leftrightarrow R$. The Wilson\ncoefficients in Eqs.~(\\ref{tildeCVR}) and (\\ref{tildeCSL}) are to be\nevaluated at the scale $\\mu_n$.\n\nThe nucleon form factors for vector operators are fixed by\nvector-current conservation, i.e. $f_{Vp}^{(u)}=2$, $f_{Vn}^{(u)}=1$,\n$f_{Vp}^{(d)}=1$, $f_{Vn}^{(d)}=2$, $f_{Vp}^{(s)}=0$,\n$f_{Vn}^{(s)}=0$. Hence, the sum in Eq.~(\\ref{tildeCVR}) is in fact\nonly over $q=u ,d$. The calculation of the scalar form factors is more\ninvolved.  The values of the up- and down-quark scalar couplings\n$f_{S p\/n}^{(u\/d)}$ (based on the two-flavour chiral perturbation\ntheory framework of~\\cite{Crivellin:2013ipa}) can be found in\nRefs.~\\cite{Crivellin:2014cta, Hoferichter:2015dsa}, while the values\nof the $s$-quark scalar couplings $f_{S p\/n}^{(s)}$ can be borrowed\nfrom a lattice calculation~\\cite{Junnarkar:2013ac}\\footnote{For\n  earlier determinations of the pion-nucleon sigma terms\n  see~\\cite{Alarcon:2011zs,Alarcon:2012nr}}. In summary, one has\n\\begin{align}\nf_{Sp}^{(u)}&=(20.8\\pm 1.5) \\times 10^{-3},\\qquad\nf_{Sn}^{(u)}=(18.9\\pm 1.4) \\times 10^{-3}, \\nonumber\n\\\\\nf_{Sp}^{(d)}&=(41.1\\pm 2.8) \\times 10^{-3},\\qquad\nf_{Sn}^{(d)}=(45.1\\pm 2.7) \\times 10^{-3}, \\nonumber\n\\\\ f_{Sp}^{(s)}&=f_{Sn}^{(s)}=(53\\pm 27)\\times 10^{-3}.\n\\end{align}\nThe form factor for the gluonic operator can be obtained from a sum\nrule.  In our normalisation\n\\begin{align}\nf_{Gp\/n}&=-\\frac{8\\pi}{9}\\Big(1-\\sum_{q=u,d,s} f^{(q)}_{Sp\/n}\\Big) \\,.\n\\end{align}\nThe quantities $D_N$, $S^{(p\/n)}_N$, and $V^{(p\/n)}_N$ in\nEq.~(\\ref{Gconv}) are related to the overlap\nintegrals~\\cite{Czarnecki:1998iz} between the lepton wave functions\nand the nucleon densities.  They depend on the nature of the target\n$N$. Their numerical values can be found in Ref.~\\cite{Kitano:2002mt}:\n\\begin{align}\nD_{\\rm Au} &= 0.189, &S^{(p)}_{\\rm Au} &=0.0614, &V^{(p)}_{\\rm\n  Au}&=0.0974, &S^{(n)}_{\\rm Au}&=0.0918, &V^{(n)}_{\\rm Au}&=0.146;\n\\nonumber\\\\\nD_{\\rm Al} &= 0.0362, &S^{(p)}_{\\rm Al}&=0.0155, &V^{(p)}_{\\rm\n  Al}&=0.0161, &S^{(n)}_{\\rm Al}&=0.0167, &V^{(n)}_{\\rm Al}&=0.0173;\n\\end{align}\nfor gold and aluminium, respectively.\n\nFinally, the branching ratio is defined as the transition rate, (see\n\\eq{Gconv}), divided by the capture rate, the latter given in\nRef.~\\cite{Suzuki:1987jf}:\n\\begin{align}\n\\Gamma^{\\rm capt}_{{\\rm Au}}&=8.7\\times 10^{-15}~{\\rm MeV},&\n\\Gamma^{\\rm capt}_{{\\rm Al}}&=4.6\\times 10^{-16}~{\\rm MeV}\\, .\n\\end{align}\n\n\n\\section{Mass eigenstates vs. mass insertions calculations}\n\\label{sec:memi}\n\nFor each process, we have given the exact one-loop expressions\ncalculated in the mass eigenbasis (ME). These formulae are compact and\nwell suited for numerical computations, however, do not allow for an\neasy understanding of the qualitative behaviour of the LFV amplitudes\nfor various choices of the MSSM parameters.  Therefore, in this\nSection we expand the Wilson coefficients in terms of the ``mass\ninsertions'', defined as the off-diagonal elements (both flavour\nviolating and flavour conserving) of the mass matrices. Such an\nexpansion allows us to:\n\\begin{itemize}\n\\item Recover the direct analytical dependence of the results on the\n  MSSM Lagrangian parameters.\n\\item Prove analytically the expected decoupling features of the\n  amplitudes in the limit of a heavy SUSY spectrum. In the case of\n  Higgs boson decays, we also identify explicitly the terms decoupling\n  only with the heavy CP-odd Higgs mass $M_A$ (which also determines\n  the heavy CP even and the charged Higgs masses).  The decoupling\n  properties serve also as an important cross-check of the correctness\n  of our calculations.\n\\item Test the dependence of the results on the pattern of the MSSM\n  spectrum and the size of the mass splitting between SUSY particles.\n\\item Better understand the possible cancellations between various\n  types of contributions and correlations between different LFV\n  processes.\n\\end{itemize}\n\nThe mass insertion expansion in flavour off-diagonal terms has been\nused for a long time in numerous articles on the subject. However,\noften various simplifying assumptions have been made, i.e. some terms\nhave been neglected or a simplified pattern of the slepton spectrum\nwas considered.  This is understandable as a consistent MI expansion\nof the amplitudes for the LFV processes in the MSSM, mediated by the\nvirtual chargino and neutralino exchanges, is technically challenging.\nThe standard approach used in literature is to calculate\ndiagrammatically the LFV amplitudes with the ``mass insertions''\ntreated as the new interaction vertices. We follow the common practice\nand normalise such slepton mass insertions to dimensionless\n``$\\Delta$-parameters''\\footnote{We assume that trilinear $A_l$,\n  $A'_l$ terms scale linearly with the slepton mass scale.}:\n\\begin{eqnarray}\n\\Delta_{LL}^{IJ}\n=\\frac{(M_{LL}^2)^{IJ}}{\\sqrt{(M_{LL}^2)^{II}(M_{LL}^2)^{JJ}}}\\,,\n&\\qquad&\n\\Delta_{RR}^{IJ}\n=\\frac{(M_{RR}^2)^{IJ}}{\\sqrt{(M_{RR}^2)^{II}(M_{RR}^2)^{JJ}}}\\,,\\nonumber\\\\\n\\Delta_{LR}^{IJ} =\n\\frac{A_l^{IJ}}{\\left((M_{LL}^2)^{II}(M_{RR}^2)^{JJ}\\right)^{1\/4}}\\,,\n&\\qquad&\n\\Delta_{LR}^{'IJ} =\n\\frac{A_l^{'IJ}}{\\left((M_{LL}^2)^{II}(M_{RR}^2)^{JJ}\\right)^{1\/4}}\\,,\n\\label{eq:midef}\n\\end{eqnarray}\nwhere $M_{LL}^2, M_{RR}^2, A_l, A_l'$ are the slepton soft mass\nmatrices and trilinear terms. \n\nAs lepton flavour violation is already strongly constrained\nexperimentally, it is sufficient to expand the amplitudes up to the\nfirst order in flavour-violating $\\Delta$'s. For instance, the\n{effective} vertices listed in Sec.~\\ref{sec:lfv} take the\nschematic form:\n\\begin{eqnarray}\nF^{IJ} &=& \\frac{1}{(4\\pi)^2}\\left(F_{LL}^{IJ} \\Delta_{LL}^{IJ} +\nF_{RR}^{IJ} \\Delta_{RR}^{JI} \\right. \\nonumber\\\\\n&+& \\left. F_{ALR}^{IJ} \\Delta_{LR}^{JI} + F_{BLR}^{IJ}\n\\Delta_{LR}^{IJ*} + F_{ALR}^{'IJ} \\Delta_{LR}^{'JI} + F_{BLR}^{'IJ}\n\\Delta_{LR}^{'IJ*}\\right)\\;.\n\\label{eq:miform}\n\\end{eqnarray}\nThe MSSM contributions to $F_{LL}, \\ldots, F_{BLR}'$ can be classified\naccording to their decoupling behaviour, distinguishing the following\ntypes ($M$ denotes the average SUSY mass scale):\n\\begin{enumerate}\n \\item Effects related to the diagonal trilinear slepton soft terms or\n   to the off-diagonal elements of supersymmetric fermion mass\n   matrices, decoupling as $v^2\/M^2$.\n \\item Effects related to the external momenta of the (on-shell) Higgs\n   or $Z^0$ bosons, decoupling as $M_h^2\/M^2$ or $M_Z^2\/M^2$ (we\n     did not include the $M_Z$ dependence as it is not necessary for\n     the considered processes).\n \\item Non-decoupling effects related to the 2HDM structure of the\n   MSSM. Such contributions are constant in the limit of a heavy SUSY\n   scale $M$ but, in case of the SM-like Higgs boson $h$, decouple\n   with the CP-odd Higgs mass like $v^2\/M_{A}^2$ (the effective\n   couplings of heavier $H,A$ bosons do not exhibit such a\n   suppression). They are proportional either to the lepton Yukawa\n   couplings or to the non-holomorphic $A'_l$ terms.\n\\end{enumerate}\n\nThe structure of the box diagrams is more complicated as they carry 4\nflavour indices. Their MI expansion is given in\nAppendix~\\ref{app:lboxmi}.  All box diagram contributions decouple at\nleast as $v^2\/M^2$.\n\nCalculating consistently the quantities $F_{LL}, \\ldots, F_{BLR}'$ to\nthe order $v^2\/M^2$ is not trivial for chargino and neutralino\ncontributions.  If the MI expansion is used only for the sfermion mass\nmatrices but the calculations for the supersymmetric fermions are done\nin the mass eigenbasis, the direct dependence on the Lagrangian\nparameters is hidden and the decoupling properties of the amplitude\ncannot be seen directly. However, one can also treat the off-diagonal\nentries of the chargino and neutralino mass matrices as ``mass\ninsertions''.  With such an approach, the final result is expressed\nexplicitly in terms of Lagrangian parameters, but the computations can\nget very complicated.  At the order $v^2\/M^2$ one needs to include\ndiagrams with all combinations of two fermionic mass insertions (each\nproviding one power of $v\/M_1$, $v\/M_2$ or $v\/\\mu$) or flavour\ndiagonal slepton terms originating from trilinear $A$-terms (providing\npowers of $v A_l\/M^2$, $v A'_l\/M^2$). Thus, to obtain an expansion of\nthe $F$'s in~\\eq{eq:miform}, one needs to formally go to the 3rd order\nof MI expansion, adding all diagrams with up to two flavour conserving\nand one flavour violating mass insertion. Therefore, the number of\ndiagrams grows quickly with the order of the expansion and such a\nmethod is tedious and prone to calculational mistakes.\n\nIn our paper, we employ a recently developed technique using a purely\nalgebraic MI expansion of the ME amplitudes listed in\nSec.~\\ref{sec:lfv}, without the need for direct diagrammatic MI\ncalculations (``FET theorem'')~\\cite{Dedes:2015twa}, automatised in\nthe specialised {\\tt MassToMI} {\\em Mathematica}\npackage~\\cite{Rosiek:2015jua, Rosiek:2017cwn}.  The use of this\npackage and full automation of the calculations allows us to perform\nthe required 3rd order MI expansion for a completely general SUSY mass\nspectrum, without making any simplifying assumptions. Such a result\nwould be very difficult to obtain diagrammatically, as in the\nintermediate steps of the calculations (before accounting for the\ncancellations and simplifications between various contributions) the\nexpressions may contain up to tens of thousand terms, even if the\nfinal results collected in Appendix~\\ref{app:miexp} are again\nrelatively compact. In detail:\n\\begin{itemize}\n\\item We perform the expansion always up to the lowest non-vanishing\n  order in the slepton LFV terms, taking into account the possible\n  cancellations.  Compared to previous analyses, we consider the\n  non-holomorphic trilinear soft terms as well.\n\\item In the MI expanded expressions we include all terms decreasing\n  with the SUSY mass scale as $v^2\/M_{SUSY}^2$ (or slower), where\n  $M_{SUSY}$ denotes any of the relevant mass parameters in the MSSM\n  Lagrangian (apart from the soft Higgs mass terms): diagonal soft\n  slepton masses, gaugino masses $M_1,M_2$ or the $\\mu$ parameter.\n\\item\nWe do not assume degeneracy or any specific hierarchy for the\nsleptons, sneutrinos or supersymmetric fermion masses.\n\\item In calculating the LFV Higgs decays we keep the leading terms in\n  the external Higgs boson mass ($m_{h(H)}^2\/M_{SUSY}^2$).\n\\end{itemize}\nThe full set of the expanded expressions in the MI approximation for\nthe photon, $Z^0$ and CP-even Higgs leptonic penguins and for the\n4-lepton box diagrams is collected in Appendix~\\ref{app:miexp}.\n\n\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\begin{tabular}{ccc}\n    \\includegraphics[width=0.307\\textwidth]{cfll-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{cfrr-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{cflr-eps-converted-to.pdf}\\\\[3mm]\n    \\includegraphics[width=0.307\\textwidth]{czll-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{czrr-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{czlr-eps-converted-to.pdf}\\\\[3mm]\n    \\includegraphics[width=0.307\\textwidth]{chll-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{chrr-eps-converted-to.pdf}&\n    \\includegraphics[width=0.307\\textwidth]{chlr-eps-converted-to.pdf}\\\\[3mm]\n \\end{tabular}\n \\caption{\\small Accuracy of MI expansion for the penguin\n   amplitudes. The curves show the ratio defined in~\\eq{eq:miacc}.\n   Red and blue lines: $\\Delta F$ for left and right couplings\n   assuming a spectrum of~\\eq{eq:specdef} for both the MI and ME\n   expressions. Brown and green lines: $\\Delta F$ (again for left and\n   right couplings, respectively) assuming spectrum~(\\ref{eq:specdef})\n   for ME expressions but an universal degenerate sfermion mass in MI\n   expressions.  This assumption is inconsistent with non-zero\n   off-diagonal elements of the mass matrices, which imply\n   non-degenerate mass eigenstates. The plots show that the associated\n   error can be numerically sizeable.  The average SUSY mass scale $M$\n   (assumed to be equal to $M_2 = m_{\\tilde\\mu_L} = m_{\\tilde\\mu_R}$)\n   is shown on the horizontal axis. \\label{fig:acc}}\n\\end{center}\n\\end{figure}\n\n\n\n\nWe illustrate the accuracy of the derived MI formulae in\nFig.~\\ref{fig:acc}.  The plots show the ratio of the MI expanded\ncouplings over the ones obtained in the mass eigenbasis with exact\ndiagonalization. For this purpose, we start from the following setup\nwhere all mass parameters are given in GeV:\n\\begin{eqnarray}\n\\begin{array}{lp{1cm}lp{1cm}l} \n\\tan\\beta=5 && m_{\\tilde \\mu_L} = 300 && A_{\\mu\\mu} = A'_{\\mu\\mu} =\n  0.1 \\sqrt{m_{\\tilde \\mu_L} m_{\\tilde \\mu_R}}\\\\\n\\mu = 200+100i && m_{\\tilde \\tau_L} = 330\\\\\nM_1 = 150 && m_{\\tilde \\mu_R} = 300 && A_{\\tau\\tau} =\n  A'_{\\tau\\tau} = 0.1 \\sqrt{m_{\\tilde \\tau_L} m_{\\tilde \\tau_R}}\\\\\nM_2 = 300 && m_{\\tilde \\tau_R} = 350 \\\\\n\\end{array}\n\\label{eq:specdef}\n\\end{eqnarray}\nNext, to see the decoupling effects we scale this spectrum uniformly\nup to slepton masses of 2 TeV.  For each of the six penguin Wilson\ncoefficients describing the transition between 2nd and 3rd generation,\n$F_{\\gamma L(R)}^{23}$ (\\eq{eq:lgdef}), $F_{Z L(R)}^{23}$\n(\\eq{eq:lzdef}) and $F_{hL}^{23}\\equiv F_h^{232}$, $F_{hR}^{23}\\equiv\nF_h^{322}$ (\\eq{eq:lhdef}) we plot the quantity\n\\begin{eqnarray}\n\\Delta F = \\left|\\frac{F_\\mathrm{MI}}{F_\\mathrm{ME}}\\right| -1\\,,\n\\label{eq:miacc}\n\\end{eqnarray}\nas a function of the average slepton mass. The accuracy of left-handed\n(right-handed) Wilson coefficients is illustrated with red(blue)\nlines.  As can be seen from Fig.~\\ref{fig:acc}, the accuracy of MI\nexpanded amplitudes is very good even for light SUSY particles and for\n$M_{SUSY}>500$ GeV always better than 95\\%.\n\nMany analyses published to date for simplicity did not include the\ncomplete set of the contributions scaling like $v\/M$ order and\/or\nassumed a partially or fully degenerate SUSY spectrum.  This procedure\nis inconsistent with non-zero off-diagonal elements of mass matrices,\nbecause the latter enforce unequal eigenvalues of the corresponding\nmass matrix.  To illustrate the numerical effects arising from the\nincorrect neglect of SUSY mass splitting we plot the ratio of our\nexpressions in the MI approximation for penguin Wilson coefficients\ncalculated for degenerate slepton masses (equal to 300 GeV rescaled by\na common factor; other parameters as in~\\eq{eq:specdef}) and the exact\nmass eigenbasis formulae (calculated with non-degenerate sfermion\nspectrum of~\\eq{eq:specdef}) in Fig.~\\ref{fig:acc}.  The accuracy of\nleft-handed (right-handed) MI expanded Wilson coefficients with\ndegenerate slepton spectrum is shown in green(brown). In this case\ndiscrepancy is much larger, of the order of 10\\%-40\\%, and does not\ndisappear when increasing the total SUSY scale.\n\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\begin{tabular}{cc}\n    \\includegraphics[width=0.48\\textwidth]{gllscan.pdf}&\n    \\includegraphics[width=0.48\\textwidth]{hllscan.pdf}\\\\[3mm]\n    \\includegraphics[width=0.48\\textwidth]{glrscan.pdf}&\n    \\includegraphics[width=0.48\\textwidth]{hlrscan.pdf}\\\\[3mm]\n \\end{tabular}\n \\caption{\\small Accuracy of MI expansion for the $\\tau\\to\\mu\\gamma$\n   and $h\\to\\tau\\mu$ decay rates. Points show the ratio\n   $\\text{Br}(\\tau\\to\\mu\\gamma)^{MI}\/\\text{Br}(\\tau\\to\\mu\\gamma)^{ME}$ (lower and\n   upper left panel) and $\\text{Br}(h\\to\\tau\\mu)^{MI}\/\\text{Br}(h\\to\\tau\\mu)^{ME}$\n   (lower and upper right panel) as a function of LL and LR mass\n   insertion for $\\tan\\beta=5$ and random choice of other model\n   parameters (see~\\eq{eq:random}). \\label{fig:scan}}\n\\end{center}\n\\end{figure}\n\n\nSome papers on the LFV in the MSSM, like e.g.\nRefs.~\\cite{Paradisi:2005fk, Arganda:2015uca}, deal with general SUSY\nspectra. In order to compare the accuracy of the MI approximation\nderived in our analysis with previous works, we plotted in\nFig.~\\ref{fig:scan} the ratios of $Br(\\tau\\to\\mu \\gamma)$ and\n$Br(h\\to\\tau\\mu)$ calculated using the exact (ME) and MI-expanded\nformulae scanning over randomly chosen MSSM mass spectra.  In\nparticular, in Fig.~\\ref{fig:scan} we assume $\\tan\\beta=5$,\n$\\alpha-\\beta = -\\pi\/2-\\pi\/100$, $m_h=125$ GeV, diagonal $A$ terms\nwhich are proportional to the lepton Yukawa couplings ($A_l^{II} =\nA_l^{'II} = Y_l^I ((M_{LL}^2)_{II} (M_{RR}^2)_{JJ})^{1\/4}$) and we\nvary the mass parameters randomly and independently in the following\nranges (all values are given in GeV and we set $M_1= M_2\/2$):\n\\begin{eqnarray}\nM_A &\\in& (200,500) \\qquad\\qquad \\mu, M_2,\nm_{\\tilde\\tau_L},m_{\\tilde\\mu_L},m_{\\tilde\\tau_R},m_{\\tilde\\tau_R}\n\\in\\ (500,1000),.\n\\label{eq:random}\n\\end{eqnarray}\nAs can be seen from upper left panel of Fig.~\\ref{fig:scan}, even for\n$\\delta_{LL}^{32}=0.5$ the accuracy of our MI expansion is better than\nabout 15\\%. This can be compared with the corresponding right panel of\nFig.~8 in ref.~\\cite{Paradisi:2005fk} - there the difference between\nMI and ME calculation for the same value of $\\delta_{LL}^{32}=0.5$ is\n20\\%-70\\%, also the spread of points around the parabolic shape\narising from neglected $(\\delta_{LL}^{32})^2$ terms is much larger,\n50\\% against a maximum of 10\\% in our approach. It is worth noting\nthat the agreement in the lower left plot of Fig.~\\ref{fig:scan} is\nalmost perfect everywhere, which could be attributed to the fact that\nterms of higher order in $\\delta_{LR}$ are suppressed by additional\n$v\/M$ powers (with $A$ terms scaling linearly with $M$, like we\nchoose) and thus small even for large $\\delta_{LR}$ values.\n\nAccuracy of our MI expansion for $Br(h\\to\\tau\\mu)$ shown in the right\nupper panel of Fig.~\\ref{fig:scan} is worse, up to 20\\%, because we\ninclude only non-decoupling $LL$ terms in our formulae, which is not a\nfully satisfying approximation for a SUSY scale in the range of\n500-1000 GeV.  Concerning the expansion in $\\delta_{LR}^{32}$ (where\nonly decoupling terms contribute), we include them consistently and\nthe accuracy is much better.  This can be compared with\nRef.~\\cite{Arganda:2015uca}, considering the same process. The\nagreement for $\\delta_{LL}^{32}=0.5$ in upper left panel of Fig.~6\n(``general scenario'') in~\\cite{Arganda:2015uca} is better than ours,\nas they consistently include all $LL$ terms, not just non-decoupling\nones.  However, for $\\delta_{LR}^{32}$ (lower left panel of Fig.~6 in\n~\\cite{Arganda:2015uca}) numerical accuracy of our formulae seems to\nbe similar or even better. In general, no significant deviations\nshould be expected here, as for this process our approach and the\nanalysis of Ref.~\\cite{Arganda:2015uca} are equivalent up to the\nchosen calculational technique (FET vs. diagrammatic MI calculation)\nand, eventually, the selection of the included or neglected\ncontributions.\n\n\n\\section{Phenomenological analysis}\n\\label{sec:pheno}\n\n\\subsection{Generic bounds on LFV parameters}\n\\label{sec:genmi}\n\n\nAs outlined in the introduction, flavour violation in the charged\nlepton sector is strongly constrained experimentally. In\nTable~\\ref{tab:leplim} we collect the current and expected future\nexperimental bounds on the processes discussed so far.\n\n\\begin{table}[tb]\n\\begin{center}  \n  \\begin{tabular}{|l|r|c|r|c|}\n    \\hline\n & Experimental upper bound & CL & Future sensitivity & CL \\\\\n\\hline\n$ \\tau \\to e \\gamma $ & $ 3.3 \\times 10^{-8} $ \\cite{Aubert:2009ag} &\n90\\% & $ 10^{-9} $ \\cite{Hayasaka:2013dsa,Cavoto:2017kub} & 90\\% \\\\\n$ \\tau \\to \\mu \\gamma $ & $ 4.4 \\times 10^{-8} $\n\\cite{Aubert:2009ag,Hayasaka:2007vc} & 90\\% & $ 10^{-9} $\n\\cite{Hayasaka:2013dsa} & 90\\% \\\\\n$ \\mu \\to e \\gamma $ & $ 5.7 \\times 10^{-13} $ \\cite{Adam:2013mnn} &\n90\\% & $ 6 \\times 10^{-14} $ \\cite{Baldini:2018nnn} & 90\\% \\\\\n\\hline\n$ Z \\to \\mu e $ & $ 7.5 \\times 10^{-7} $ \\cite{Aad:2014bca} & 95\\% & &\n\\\\\n$ Z \\to \\mu \\tau $ & $ 1.2 \\times 10^{-5} $ \\cite{Abreu:1996mj} & 95\\%\n& & \\\\\n$ Z \\to \\tau e $ & $ 9.8 \\times 10^{-6} $ \\cite{Abreu:1996mj}& 95\\% &\n& \\\\\n\\hline\n$ \\mu \\to e^- e^+ e^- $ & $ 1.0 \\times 10^{-12} $\n\\cite{Bellgardt:1987du} & 90\\% & $ 10^{-16} $\n\\cite{Blondel:2013ia,Berger:2014vba} & 90\\% \\\\\n$ \\tau \\to e^- e^+ e^- $ & $ 2.7 \\times 10^{-8} $\n\\cite{Hayasaka:2010np} & 90\\% & & \\\\\n$ \\tau \\to \\mu^- \\mu^+ \\mu^- $ & $ 2.1 \\times 10^{-8}\n$\\cite{Hayasaka:2010np} & 90\\% & & \\\\\n$ \\tau \\to e^- \\mu^+ \\mu^- $ & $ 2.7 \\times 10^{-8} $\n\\cite{Hayasaka:2010np} & 90\\% & & \\\\\n$ \\tau \\to e^+ \\mu^- \\mu^- $ & $ 1.7 \\times 10^{-8} $\n\\cite{Hayasaka:2010np} & 90\\% & &\\\\\n$ \\tau \\to \\mu^- e^+ e^- $ & $ 1.8 \\times 10^{-8} $\n\\cite{Hayasaka:2010np} & 90\\% & & \\\\\n$ \\tau \\to \\mu^+ e^- e^- $ & $ 1.5 \\times 10^{-8} $\n\\cite{Hayasaka:2010np} & 90\\% & & \\\\\n\\hline\n$ h \\to e \\tau $ & $ 6.1 \\times 10^{-3} $ \\cite{CMS:2017onh} & 90\\%\n&&\\\\\n$ h \\to \\mu \\tau $ & $ 2.5 \\times 10^{-3} $ \\cite{CMS:2017onh} & 90\\%&\n& \\\\\n$ h \\to \\mu e $ & $ 3.6 \\times 10^{-4} $ \\cite{CMS:2015udp} & 90\\% &\n&\\\\\n\\hline\n$(\\mu\\to e)_{Au}$ & $7.0 \\times 10^{-13}$ \\cite{Bertl:2006up} & 90\\%\n&& \\\\\n$(\\mu\\to e)_{Al}$ & & & $10^{-16}$ \\cite{Abrams:2012er} & 90\\% \\\\\n\\hline\n\\end{tabular}\n\\caption{\\small Upper bounds on LFV decays of charged leptons. $h$\n  denotes the SM-like Higgs boson\\label{tab:leplim}}\n\\end{center}\n\\end{table}\n\n\nAssuming the absence of fine-tuned cancellations between different\nflavour violating parameters, the order of magnitude of the bounds on\na given flavour violating entry $\\Delta$ can be obtained by assuming\nthat it is the only source of flavour violation. At the lowest order\nin the MI expansion, any LFV observable $X$ scales like $\\Delta^2$:\n\\begin{eqnarray}\nX \\approx f(m_1,\\ldots, m_n) |\\Delta|^2\\,,\n\\end{eqnarray}\nwhere $f$ is a known (non-negative) function of diagonal mass\nparameters - for any given process it can be extracted from the\nexpanded expressions listed in Appendix~\\ref{app:miexp}.  Thus, the\nexperimental bound on $\\Delta$ from a given measurement can be written\nas:\n\\begin{eqnarray}\n|\\Delta| \\leq \\sqrt{\\frac{X^{\\rm exp}}{f(m_1,\\ldots, m_n)}}\n\\sqrt{\\frac{X^{\\rm future}}{X^{\\rm exp}}} \\equiv \\Delta(m_1,\\ldots, m_n)\n\\sqrt{\\frac{X^{\\rm future}}{X^{\\rm exp}}}\n\\end{eqnarray}\nwhere by $X^{\\rm exp}$ we denote one of the current experimental\nbounds listed in Sec.~\\ref{sec:genmi} and $X^{\\rm future}$ is the\nexpected future sensitivity.\n\nTo estimate the order of magnitude of the bounds on all types of mass\ninsertions, we assume a common mass scale $M$ for all flavour diagonal\nSUSY parameters:\n\\begin{eqnarray}\n&& m_{\\tilde e_{LI}} = m_{\\tilde e_{RI}} = M_1 = M_2 = \\mu = M_A = M\\,,\\nonumber\\\\\n&& A_{\\ell}^{II} = A_{\\ell}^{'II} = Y_{\\ell}^I\\; M\\,.\n\\label{eq:mpatt}\n\\end{eqnarray}\nCurrently, the strongest bounds on the dimensionless LFV parameters\n$\\Delta$ defined in~\\eq{eq:miform} originate from the radiative lepton\ndecays $\\ell\\to\\ell'\\gamma$. We list such bounds for the parameter\nsetup defined in~\\eq{eq:mpatt} and for the SUSY scale of $M=400$ GeV\nin Table~\\ref{tab:llgbounds}.\n\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|l|l|c|c|c|c|c|c|}\n\\hline\nProcess & $(I,J)$ & $\\Delta_{LL}^{IJ}$ & $\\Delta_{RR}^{IJ}$ &\n$\\Delta_{LR}^{IJ}$ & $\\Delta_{RL}^{IJ}$ & $\\Delta_{LR}^{'IJ}$ &\n$\\Delta_{RL}^{'IJ}$ \\\\\n\\hline\n$\\tan\\beta=2$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ \\mu\\to e \\gamma$ & $(2,1)$ & $ 8.4 \\cdot 10^{-4}$ & $ 5.0 \\cdot\n10^{-3}$ & $ 8.4 \\cdot 10^{-6}$ & $ 8.3 \\cdot 10^{-6}$ & $ 4.1 \\cdot\n10^{-6}$ & $ 4.1 \\cdot 10^{-6}$ \\\\\n$\\tau \\to \\mu \\gamma$ & $(3,2)$ &$ 5.3 \\cdot 10^{-1}$ & \n${\\cal O}(1)$ & $ 9.1 \\cdot 10^{-2}$ & $ 9.1 \\cdot 10^{-2}$ & $ 4.5\n\\cdot 10^{-2}$ & $ 4.5 \\cdot 10^{-2}$ \\\\\n$ \\tau \\to e \\gamma$ & $(3,1)$ &$ 4.6 \\cdot 10^{-1}$ &\n${\\cal O}(1)$ & $ 7.8 \\cdot 10^{-2}$ & $ 7.8 \\cdot 10^{-2}$ & $ 3.9\n\\cdot 10^{-2}$ & $ 3.8 \\cdot 10^{-2}$ \\\\\n\\hline\n$\\tan\\beta=20$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ \\mu\\to e \\gamma$ & $(2,1)$ & $ 1.0 \\cdot 10^{-4}$ & $ 4.5 \\cdot\n10^{-4}$ & $ 7.5 \\cdot 10^{-5}$ & $ 7.4 \\cdot 10^{-5}$ & $ 3.7 \\cdot\n10^{-6}$ & $ 3.7 \\cdot 10^{-6}$ \\\\\n$ \\tau \\to \\mu \\gamma$ & $(3,2)$ & $ 6.5 \\cdot 10^{-2}$ & $ 2.9 \\cdot\n10^{-1}$ & $ 8.2 \\cdot 10^{-1}$ & $ 8.2 \\cdot 10^{-1}$ & $ 4.0 \\cdot\n10^{-2}$ & $ 4.0 \\cdot 10^{-2}$ \\\\\n$ \\tau \\to e \\gamma$ & $(3,1)$ & $ 5.7 \\cdot 10^{-2}$ & $ 2.5 \\cdot\n10^{-1}$ & $ 7.0 \\cdot 10^{-1}$ & $ 7.0 \\cdot 10^{-1}$ & $ 3.4 \\cdot\n10^{-2}$ & $ 3.4 \\cdot 10^{-2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\small Upper bounds on the LFV parameters $\\Delta$ from\n  radiative charged lepton decays for the MSSM spectrum defined\n  in~\\eq{eq:mpatt} and a SUSY scale of $M=400$ GeV.  All bounds scale\n  (i.e. weaken) like $M^2$.\n\\label{tab:llgbounds}}\n\\end{table}\n\n\nThe 3-body decays of charged lepton lead to bounds which are\napproximately one order of magnitude weaker. In\nTable~\\ref{tab:l3bounds} we display the relative strength of such\nbounds comparing them to the ones obtained from the radiative lepton\ndecays, i.e. the ratios of bounds from radiative decays over the ones\nfrom 3-body decays. Such ratios remain constant with increasing $M$ up\nto the scale where the non-decoupling Higgs penguin contributions\nstart to contribute. However, such effects occurs for $M\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$} 30$ TeV\nfor $\\tau^\\pm\\to\\mu^\\pm\\mu^\\pm\\mu^\\mp$ and\n$\\tau^\\pm\\to e^\\pm\\mu^\\pm\\mu^\\mp$ decays and for even higher $M$ for\nthe decays with electron pair in the final state. For such a large $M$\nthe branching ratios for all 3-body decays are, anyway, below the\ncurrent experimental sensitivities even for\n${\\cal O}(\\Delta^{IJ})\\sim 1$.\n\nWe do not display the bounds from LFV violating $Z^0$ decays as they\nare much weaker (3 to 8 orders of magnitude depending on which\nparameter $\\Delta$ is chosen). This can be attributed to the large $Z$\nboson width -- for comparable $\\Gamma(Z\\to \\ell\\ell')$ and\n$\\Gamma(\\ell\\to\\ell'\\gamma)$ partial decay widths the difference in\ntotal widths leads to $\\text{Br}(\\ell\\to\\ell'\\gamma)\\gg \\text{Br}(Z\\to\n\\ell\\ell')$. Thus, bounds from $\\text{Br}(Z\\to \\ell\\ell')$ are not\ncompetitive (nor they will be in the foreseeable future) compared to\nthese from other observables.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|l|l|c|c|c|c|c|c|}\n\\hline\nProcess & $(I,J)$ & $\\Delta_{LL}^{IJ}$ & $\\Delta_{RR}^{IJ}$ &\n$\\Delta_{LR}^{IJ}$ & $\\Delta_{RL}^{IJ}$ & $\\Delta_{LR}^{'IJ}$ &\n$\\Delta_{RL}^{'IJ}$ \\\\\n\\hline\n$\\tan\\beta=2$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ \\mu \\to eee$ & $(2,1)$ & $ 1.7 \\cdot 10^{+1}$ & $ 1.5 \\cdot\n10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ & $ 1.6\n\\cdot 10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to \\mu\\mu\\mu$ & $(3,2) $ & $ 1.5 \\cdot 10^{+1}$ & $ 1.2 \\cdot\n10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ & $ 1.4\n\\cdot 10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to \\mu e^+ e^-$ & $(3,2) $ & $ 1.3 \\cdot 10^{+1}$ & $ 1.1\n\\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ & $\n1.2 \\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to eee$ & $(3,1) $ & $ 8.6 \\cdot 10^{+0}$ & $ 8.2 \\cdot\n10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ & $ 8.5\n\\cdot 10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ \\\\\n$\\tau \\to e \\mu^+ \\mu^-$ & $(3,1)$ & $ 6.9 \\cdot 10^{+0}$ & $ 6.7\n\\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ & $\n6.8 \\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ \\\\\n\\hline\n$\\tan\\beta=20$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ \\mu \\to eee$ & $(2,1)$ & $ 1.6 \\cdot 10^{+1}$ & $ 1.6 \\cdot\n10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ & $ 1.6\n\\cdot 10^{+1}$ & $ 1.6 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to \\mu\\mu\\mu$ & $(3,2) $ & $ 1.4 \\cdot 10^{+1}$ & $ 1.4 \\cdot\n10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ & $ 1.4\n\\cdot 10^{+1}$ & $ 1.4 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to \\mu e^+ e^-$ & $(3,2) $ & $ 1.3 \\cdot 10^{+1}$ & $ 1.2\n\\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ & $\n1.2 \\cdot 10^{+1}$ & $ 1.2 \\cdot 10^{+1}$ \\\\\n$ \\tau \\to eee$ & $(3,1) $ & $ 8.5 \\cdot 10^{+0}$ & $ 8.5 \\cdot\n10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ & $ 8.5\n\\cdot 10^{+0}$ & $ 8.5 \\cdot 10^{+0}$ \\\\\n$\\tau \\to e \\mu^+ \\mu^-$ & $(3,1)$ & $ 6.8 \\cdot 10^{+0}$ & $ 6.8\n\\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ & $\n6.8 \\cdot 10^{+0}$ & $ 6.8 \\cdot 10^{+0}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\small Ratios of upper bounds on the LFV parameters $\\Delta$\n  from the searches for 3-body and radiative decays of charged\n  leptons. The MSSM spectrum is defined in~\\eq{eq:mpatt}.\n\\label{tab:l3bounds}}\n\\end{table}\n\n\nAs can be seen in Table~\\ref{tab:hbounds}, the bounds on $\\Delta$\nparameters from LFV flavour Higgs boson decay searches are much weaker\nthan those from the radiative charged lepton decays. However, in the\nHiggs sector some effects proportional to lepton Yukawa couplings or\nto the non-holomorphic terms are non-decoupling and are not weakened\nby increasing $M$ like other contributions, for fixed Higgs sector\nparameters.  In Table~\\ref{tab:hbounds} we assume\n\\begin{eqnarray}\n\\alpha - \\beta = - \\frac{\\pi}{2} -\\gamma \\,,\n\\label{eq:alval}\n\\end{eqnarray}\nwith $\\gamma = \\pi\/100$. Using the tree-level relations of the MSSM\nHiggs sector in the limit of $\\tan\\beta>1$ and small values of\n$\\gamma$ one has\n\\begin{eqnarray}\nM_A = M_Z \\sqrt{\\frac{\\sin 2(\\alpha+\\beta)}{\\sin 2(\\alpha-\\beta)}}\n\\approx M_Z \\sqrt{\\frac{-\\sin 4\\beta}{2\\gamma}}\\;,\n\\end{eqnarray}\nthis corresponds to $M_A\\sim 350$ GeV for $\\tan\\beta=2$ and $M_A\\sim\n190$ GeV for $\\tan\\beta=20$ (the exact value including loop\ncorrections may vary, depending on the squark parameters which we do\nnot specify here).\n\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|l|l|c|c|c|c|c|c|}\n\\hline\nProcess & $(I,J)$ & $\\Delta_{LL}^{IJ}$ & $\\Delta_{RR}^{IJ}$ &\n$\\Delta_{LR}^{IJ}$ & $\\Delta_{RL}^{IJ}$ & $\\Delta_{LR}^{'IJ}$ &\n$\\Delta_{RL}^{'IJ}$ \\\\\n\\hline\n$\\tan\\beta=2$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ h\\to \\mu e$ & $(2,1)$ & $ 1.8 \\cdot 10^{+7}$ & $ 1.7 \\cdot 10^{+6}$\n& $ 2.6 \\cdot 10^{+7}$ & $ 2.6 \\cdot 10^{+7}$ & $ 1.0 \\cdot 10^{+7}$ &\n$ 1.0 \\cdot 10^{+7}$ \\\\\n$ h\\to \\tau \\mu$ & $(3,2) $ & $ 4.4 \\cdot 10^{+3}$ & $ 3.8 \\cdot\n10^{+2}$ & $ 6.3 \\cdot 10^{+3}$ & $ 6.3 \\cdot 10^{+3}$ & $ 2.5 \\cdot\n10^{+3}$ & $ 2.5 \\cdot 10^{+3}$ \\\\\n$ h\\to \\tau e$ & $(3,1) $ & $ 8.0 \\cdot 10^{+3}$ & $ 7.5 \\cdot\n10^{+2}$ & $ 1.1 \\cdot 10^{+4}$ & $ 1.1 \\cdot 10^{+4}$ & $ 4.9 \\cdot\n10^{+3}$ & $ 4.9 \\cdot 10^{+3}$ \\\\\n\\hline\n$\\tan\\beta=20$ & \\multicolumn{7}{|l|}{}\\\\\n\\hline\n$ h\\to \\mu e$ & $(2,1)$ & $ 3.9 \\cdot 10^{+6}$ & $ 8.3 \\cdot 10^{+6}$\n& $ 5.1 \\cdot 10^{+7}$ & $ 5.1 \\cdot 10^{+7}$ & $ 1.2 \\cdot 10^{+6}$ &\n$ 1.2 \\cdot 10^{+6}$ \\\\\n$ h\\to \\tau \\mu$ & $(3,2) $ & $ 9.5 \\cdot 10^{+2}$ & $ 1.9 \\cdot\n10^{+3}$ & $ 1.3 \\cdot 10^{+4}$ & $ 1.2 \\cdot 10^{+4}$ & $ 2.9 \\cdot\n10^{+2}$ & $ 2.9 \\cdot 10^{+2}$ \\\\\n$ h\\to \\tau e$ & $(3,1) $ & $ 1.7 \\cdot 10^{+3}$ & $ 3.5 \\cdot\n10^{+3}$ & $ 2.3 \\cdot 10^{+4}$ & $ 2.2 \\cdot 10^{+4}$ & $ 5.3 \\cdot\n10^{+2}$ & $ 5.4 \\cdot 10^{+2}$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\small Ratios of upper bounds on the LFV $\\Delta$ parameters\n  from leptonic Higgs boson decays and from radiative decays of\n  charged leptons. The MSSM spectrum is defined in~\\eq{eq:mpatt} (with\n  the exception of setting $A_l^{\\prime II}=0$) and a SUSY scale of\n  $M=400$ GeV. The ratios for $\\Delta_{LL}^{IJ}$, $\\Delta_{RR}^{IJ}$,\n  $\\Delta_{LR}^{'IJ}$, $\\Delta_{RL}^{'IJ}$ decrease with $M^2$,\n  assuming fixed masses and mixing angles in the Higgs sector.\n\\label{tab:hbounds}}\n\\end{table}\n\nThe bounds on $\\Delta_{LL}^{IJ}, \\Delta_{RR}^{IJ}, \\Delta_{LR}^{'IJ},\n\\Delta_{RL}^{'IJ}$ from the leptonic Higgs boson decay would decouple\nonly if also $M_A$ is scaled up simultaneously with SUSY particle\nmasses (thus assuring that the Higgs decay rates do not violate the\nAppelquist-Carrazone theorem~\\cite{Appelquist:1974tg}).  This\ninteresting feature is discussed in more details in\nSec.~\\ref{sec:hllndec}.\n\n\n\\subsection{Dependence on the mass splitting}\n\\label{sec:msplit}\n\nThe formulae derived in the previous Sections allow to analyse how the\nbounds on LFV mass insertions depend on the splitting between\ndifferent SUSY masses. However, any process involving transition\nbetween the generations $I$ and $J$ depends in general, even at lowest\norder in the flavour violating MI's, on many mass parameters: $\\mu$,\ngaugino masses $M_1, M_2$, left and right diagonal slepton soft masses\n$m_{\\tilde e_{LI}}, m_{\\tilde e_{LJ}}, m_{\\tilde e_{RI}}, m_{\\tilde\n  e_{RJ}}$, and for the Higgs decays also on $M_A$ or on $\\alpha$\nangle. To simplify the discussion, we only take into account the\nbounds from $\\ell\\to\\ell^\\prime \\gamma$ decays, which are currently\nmost constraining.\n\n\nIn Fig.~\\ref{fig:split_lr} we illustrate the dependence of the upper\nbounds on the $\\Delta$ parameters originating from $\\mu\\to e \\gamma$\non the mass splitting between left and right-handed sleptons for\n\\begin{center}\n\\begin{tabular}{cc}\n$\\tan\\beta=2$\\,, & $\\mu=M_1=M_2\\equiv M = 800$ GeV\\\\[1mm] $m_{\\tilde\n    e_L} =m_{\\tilde \\mu_L} = m_L$\\,, & $m_{\\tilde e_R} = m_{\\tilde\n    \\mu_R} = m_R$\\,,\\\\[1mm]\n$A_{\\ell}^{ee} = A_{\\ell}^{'ee} = Y_e\\; \\sqrt{m_L m_R}$\\,,\n  &$A_{\\ell}^{\\mu\\mu} = A_{\\ell}^{'\\mu\\mu} = Y_\\mu\\; \\sqrt{m_L\n    m_R}$\\,.  \\\\\n\\end{tabular}\n\\end{center}\nWe have chosen here an average SUSY mass scale of $M=800$~GeV, higher\nthan $M=400$~GeV used in Tables~\\ref{tab:llgbounds}-\\ref{tab:hbounds},\nto avoid the experimental bounds on slepton masses even in the case of\na large splitting between the left and right-handed masses.\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\begin{tabular}{cc}\n      $\\Delta_{LL}^{12}$ & $\\Delta_{RR}^{12}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{split_lr_ll.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{split_lr_rr.pdf}\\\\\n      $\\Delta_{LR}^{12},\\Delta_{LR}^{21}$ & $\\Delta_{LR}^{\\prime\n        12},\\Delta_{LR}^{\\prime 21}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{split_lr_lr.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{split_lr_lrp.pdf}\\\\\n\\end{tabular}\n\\caption{\\small Upper bounds on the LFV parameters from $\\mu\\to e\n  \\gamma$ for $\\tan\\beta=2$ and $M=800$ GeV as a function of the\n  splitting between the masses of gaugino and sleptons of different\n  chiralities. The normalised slepton masses $x_{L(R)}=m_{L(R)}\/M$ are\n  plotted on the axes.\\label{fig:split_lr}}\n  \\end{center}\n\\end{figure}\n\n\nThe features of plots in Fig.~\\ref{fig:split_lr} can be understood\nusing the expanded expressions for effective photon couplings\ncollected in Appendix~\\ref{app:llgmi}. As an example, let us consider\nthe interesting cancellation between different contributions in the\ncase of $\\Delta_{RR}^{12}$ (right upper panel of\nFig.~\\ref{fig:split_lr}).  For our parameter setup, the coefficient\n$X_{\\gamma N2}^{e\\mu}$ multiplying the RR parameter\n(see~\\eq{eq:llgmineut1}) can be reduced to the form\n\\begin{eqnarray}\nX_{\\gamma N2}^{e\\mu} = \\frac{v_1 Y_\\mu}{M^2}\\; f(x_L,x_R)\\,,\n\\end{eqnarray} \nwhere $f(x_L,x_R)$ is a known, although complicated, dimensionless,\nrational and logarithmic function of mass ratios whose analytical form\ncan be obtained using~\\eq{eq:llgmineut1}, the loop integrals collected\nin Appendix~\\ref{app:loop} and the definitions of divided differences\nfrom Appendix~\\ref{app:ddiff}. The properties of this function can be\nexamined analytically and numerically.  One finds\n\\begin{itemize}\n\\item For $x_R$ in the wide range $0.1-4$ the function $f$ vanishes\n  for $x_L\\sim 0.45$ (the exact value depends only weakly on\n  $x_R$). As a result, the bounds on $\\Delta_{RR}^{12}$ disappear\n  completely for $m_L \\sim 0.45 M$.\n\\item For large values of $x_R\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$} 5$ the position where the function\n  $f$ becomes zero shifts towards bigger values of $x_L$. In addition,\n  in this limit $f$ is suppressed by an overall factor $1\/x_R$, thus\n  the bounds on $\\Delta_{RR}^{12}$ become weaker for a larger values\n  of $x_R$.\n\\item For large values of $x_L$ the function $f$ depends on $x_R$\n  only. Therefore, the contour lines become horizontal.\n\\item For small values of $x_L$ the function $f$ behaves like\n  $1\/x_L$. Thus, the bounds on $\\Delta_{RR}^{12}$ become stronger.\n\\end{itemize}\nA similar analysis can be done for the bounds on\n$\\Delta_{LL}^{12}$. However, the coefficient multiplying\n$\\Delta_{LL}^{12}$ contains contributions from both chargino and\nneutralino loops and does not vanish for any mass pattern. Therefore,\nthere is no cancellation area in the upper left panel of\nFig.~\\ref{fig:split_lr}. In this case, the bound on $\\Delta_{LL}^{12}$\nis strongest for $m_L\\sim M$ and $m_R\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle<}{\\sim}\\;$} M$. For the case in which\nthe left slepton masses are much lighter or much heavier than the\nmasses of the SUSY fermion, the bounds become weaker.\n\nBounds on LR parameters, both holomorphic and non-holomorphic, are\ntypically 1-2 orders of magnitude stronger than for LL and RR ones.\nIn this case, the coefficient $X_{\\gamma N1}^{e\\mu}$ multiplying the\nLR terms has a much simpler functional form. Therefore, it never\nvanishes and in addition is explicitly symmetric (as follows from the\nproperties of divided differences) under the exchange of slepton mass\narguments, as visible in both lower panels of Fig.~\\ref{fig:split_lr}.\nFurthermore, one can see that bounds on LR parameters are strongest\nfor $m_L,m_R \\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle<}{\\sim}\\;$} M$ and become weaker when the slepton masses are\nmuch heavier than the chargino and neutralino masses. More\nquantitatively, $X_{\\gamma N1}^{e\\mu}$ is proportional to the divided\ndifference of the function $C_{12}$, which for $x\\equiv x_L = x_R$\n(corresponding to the diagonal of lower plots in\nFig.~\\ref{fig:split_lr}) has the simple asymptotic behaviour\n\\begin{eqnarray}\nC_{12}(\\{m_L,m_R\\},M)=\\left\\{\n\\begin{array}{lp{1cm}l}\n-\\frac{5}{2M^2} && x \\ll 1 \\\\[2mm]\n\\phantom{-}\\frac{1}{2M^2x^2} && x \\gg 1 \\\\\n\\end{array}\n\\right.\\;.\n\\label{eq:c12lr}\n\\end{eqnarray}\nFrom the form of~\\eq{eq:c12lr} it is immediately visible that the\nbounds become constant for small $x$ and fall like $1\/x^2$ for large\n$x$, as illustrated in the plots.\n\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\begin{tabular}{cc}\n      $\\Delta_{LL}^{12}$ & $\\Delta_{RR}^{12}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{split_ij_ll.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{split_ij_rr.pdf}\\\\\n    \\end{tabular}\n    \\caption{\\small Upper bounds on the flavour violating LL and RR\n      parameters using the current experimental limit on $\\text{Br}(\\mu\\to e\n      \\gamma)$ for $\\tan\\beta=2$ and $M=800$ GeV as a function of\n      splitting between the masses of gaugino and sleptons of various\n      flavours. The normalised selectron and smuon masses, $x_{e(\\mu)}\n      = m_{\\tilde e(\\tilde\\mu)}\/M$ are plotted on the axes.\n      \\label{fig:split_ij}}\n\\end{center}\n\\end{figure}\n\n\nUsing the formulae collected in Appendix~\\ref{app:miexp}, a similar\ndiscussion can be, if necessary, performed to explain the features or\ncancellation areas of other plots presented in this Section. However,\nas the general analytical formulae in the MI approximation are rather\ncomplicated, we illustrate here other scenarios with numerical plots\nonly.\n\nFig.~\\ref{fig:split_ij} shows similar bounds assuming identical\nleft-and right-handed slepton masses which however differ among the\ngenerations, so that we choose\n\\begin{center}\n\\begin{tabular}{cc}\n$\\tan\\beta=2$\\,, & $\\mu=M_1=M_2\\equiv M = 800$ GeV\\,,\\\\[1mm]\n$m_{\\tilde e_L} =m_{\\tilde e_R} = m_{\\tilde e}$ & $m_{\\tilde \\mu_L} =\n  m_{\\tilde \\mu_R} = m_{\\tilde \\mu}$\\,,\\\\[1mm]\n$A_{\\ell}^{ee} = A_{\\ell}^{'ee} = Y_e\\; m_{\\tilde e} $\\,,\n  &$A_{\\ell}^{\\mu\\mu} = A_{\\ell}^{'\\mu\\mu} = Y_\\mu\\; m_{\\tilde\n    \\mu}$\\,, \\\\\n\\end{tabular}\n\\end{center}\nand plot the results in terms of $x_e = m_{\\tilde e}\/M$ and $x_\\mu =\nm_{\\tilde \\mu}\/M$. Again, a cancellation only exists for the bounds on\n$\\Delta_{RR}^{12}$, for an almost constant ratio $m_{\\tilde \\mu}\\sim\n2.5 M$. In this case, the bounds on $\\Delta_{LL}^{12}$ are strongest\nfor small splitting between slepton and SUSY fermion masses, while the\nbounds on $\\Delta_{RR}^{12}$ are, apart from the cancellation region,\nstronger for $m_{\\tilde\\mu}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle<}{\\sim}\\;$} M$.  It is obvious from the form of\n$X_{\\gamma N1}^{e\\mu}$ in~\\eq{eq:llgmineut1} that the bounds on the LR\nparameters, both holomorphic and non-holomorphic, have an identical\nbehaviour as in the case of the $m_L-m_R$ splitting plotted in\nFig.~\\ref{fig:split_lr}, with the replacements $x_L \\leftrightarrow\nx_e$, $x_R \\leftrightarrow x_\\mu$.\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\begin{tabular}{cc}\n      $\\Delta_{LL}^{12}$ & $\\Delta_{RR}^{12}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{split_mg_ll.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{split_mg_rr.pdf}\\\\\n      $\\Delta_{LR}^{12},\\Delta_{LR}^{21}$ & $\\Delta_{LR}^{\\prime\n        12},\\Delta_{LR}^{\\prime 21}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{split_mg_lr.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{split_mg_lrp.pdf}\\\\\n\\end{tabular}\n    \\caption{\\small Upper bounds on the LFV parameters using the\n      current experimental limit on the $\\text{Br}(\\mu\\to e \\gamma)$ for\n      degenerate slepton masses $M=800$ GeV as a function of mass\n      splitting between the gaugino and the $\\mu$ related parameters,\n      $x_\\mu = \\mu\/M$, $x_2 = M_1\/M = M_2\/M$.\n      \\label{fig:split_mu}}\n  \\end{center}\n\\end{figure}\n\nFinally in Fig.~\\ref{fig:split_mu} we assume an identical mass of\n$m=400$ GeV for all sleptons but vary $M_1=M_2$ and $\\mu$. The results\nare displayed as a function of $x_2 = M_2\/M$, $x_\\mu = \\mu\/M$ (we do\nnot plot small values of $|\\mu|<100$~GeV which are excluded by the\ndirect searches for charginos and neutralinos). The structure of\ncancellation areas is more complicated, but again the ``blind spots'',\nwhere the bounds on MI's disappear, exist only for $\\Delta_{RR}^{12}$.\nAs expected from the form of $X_{\\gamma N1}^{e\\mu}$\nin~\\eq{eq:llgmineut1}, the bounds on $\\Delta_{LR}^{12}$,\n$\\Delta_{LR}^{'12}$ are at leading order independent of the $\\mu$\nparameter.  They are also correlated with the bounds displayed in\nlower plots of Fig.~\\ref{fig:split_lr}, as for a fixed slepton mass\nand varied $M_2$ the coefficient $X_{\\gamma N1}^{e\\mu}$ is now\nproportional to\n\\begin{eqnarray}\nC_{12}(\\{m,m\\},M_2)=\\left\\{\n\\begin{array}{lp{1cm}l}\n\\phantom{-}\\frac{1}{2m^2} && x \\ll 1 \\\\[2mm]\n-\\frac{5}{2m^2 x_2^2} && x \\gg 1 \\\\[2mm]\n\\end{array}\n\\right.\n\\label{eq:c12lra}\n\\end{eqnarray}\nso that again the bounds saturate for small $x_2$ and fall like\n$1\/x_2^2$ in the opposite limit.\n\n\n\nSimilar plots constraining $13$ and $23$ mass insertions have almost\nidentical shape; the bounds are just rescaled by constant factors. The\nbounds on $\\Delta_{LL}^{13}$ and $\\Delta_{RR}^{13}$\n($\\Delta_{LL}^{23}$ and $\\Delta_{RR}^{23}$) are approximately 550\n(650) times weaker than the bounds on $\\Delta_{LL}^{12}$ and\n$\\Delta_{RR}^{12}$, respectively.  The bounds on\n$\\Delta_{LR}^{13(31)}$ and $\\Delta_{LR}^{\\prime 13(31)}$\n($\\Delta_{LR}^{23(32)}$ and $\\Delta_{LR}^{\\prime 23(32)}$) are\nrespectively 9000 (11000) times weaker than the bounds on\n$\\Delta_{LR}^{12(21)}$ and $\\Delta_{LR}^{\\prime 12(21)}$.\n\n\n\n\\subsection{Correlations between LFV processes}\n\\label{sec:corr}\n\nThe correlations between various leptonic decays, in particular\nradiative and 3-body charged lepton decays, are important for\ndesigning new experiments searching for the LFV phenomena. In the\nphoton penguin domination scenario the ratio of decay rates for both\nprocesses is given by the simple formula:\n\\begin{eqnarray}\n\\frac{\\text{Br}(\\ell\\to 3\\ell')}{\\text{Br}(\\ell\\to\\ell'\\gamma)} \\approx\n\\frac{\\alpha_{em}}{3\\pi}\\left(\\log\\frac{m_\\ell^2}{m_{\\ell'}^2} -\n\\frac{11}{4}\\right)\\,.\n\\label{eq:phdomin}\n\\end{eqnarray}\nIn this case the decision which measurement is more promising depends\npurely on experimental accuracy achievable for each of them.  However,\nother type of contributions, like $Z$-penguin and box diagrams, can\nmodify the ratio~(\\ref{eq:phdomin}). Such contributions may be\nparticularly important for a ``blind spot'' scenario, like the\nweakened limit on $\\Delta_{RR}$ for some ratios of slepton and gaugino\nmasses.\n\nIn Fig.~\\ref{fig:correl} we plot the quantity $R_{\\ell\\ell'}$ defined as\n\\begin{eqnarray}\nR_{\\ell\\ell'} =\n\\frac{\\alpha_{em}}{3\\pi}\\left(\\log\\frac{m_\\ell^2}{m_{\\ell'}^2} -\n\\frac{11}{4}\\right)\\frac{\\text{Br}(\\ell\\to\\ell'\\gamma)}{\\text{Br}(\\ell\\to 3\\ell')}\\,,\n\\label{eq:rll}\n\\end{eqnarray}\nas a function of the SUSY mass splittings, in the same scenarios as\ndescribed in Fig.~\\ref{fig:split_lr} and Fig.~\\ref{fig:split_ij}. We\nassume non-vanishing $\\Delta_{LL}^{12}$ and $\\Delta_{RR}^{12}$ terms.\nFor LR terms, both holomorphic and non-holomorphic, a photon\npenguin dominated scenario is always realised and $R_{\\ell\\ell'}$ is\nvery close to 1.\n\n\\begin{figure}[tb!]\n  \\begin{center}\n    \\begin{tabular}{cc}\n      $\\Delta_{LL}^{12}$ & $\\Delta_{RR}^{12}$ \\\\[3mm]\n      \\includegraphics[width=0.47\\textwidth]{rat_lr_ll.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{rat_lr_rr.pdf}\\\\\n      \\includegraphics[width=0.47\\textwidth]{rat_ij_ll.pdf}&\n      \\includegraphics[width=0.47\\textwidth]{rat_ij_rr.pdf}\\\\\n\\end{tabular}\n\\caption{\\small The ratio $R_{\\ell\\ell'}$ as a function of the mass\n  splitting between left and right slepton masses (upper row) and\n  between the selectron and smuon masses (lower row) for $M=800$ GeV\n  as a function of the normalised slepton masses $x_{L(R)}=m_{L(R)}\/M$\n  and $x_{\\tilde e(\\tilde\\mu)} = m_{\\tilde e(\\tilde\\mu)}\/M$.\n  \\label{fig:correl}}\n  \\end{center}\n\\end{figure}\n\nAs one can see from Fig.~\\ref{fig:correl}, radiative and 3-body decays\nare almost always closely correlated, with $R_{\\ell\\ell'}$ differing\nfrom 1 by a few \\% at most.  Exceptions are only possible for\nparameter combinations for which $\\text{Br}(\\ell\\to\\ell'\\gamma)$ becomes\nsmall due to cancellations or some other type of suppression, like in\nscenarios with large mass splitting (compare Figs.~\\ref{fig:split_lr}\nand \\ref{fig:split_ij}). Simultaneously, $\\text{Br}(\\ell\\to 3\\ell')$ is\ngiven by the more complicated expression~(\\ref{eq:br4l}), which in the\nlimit of small photon penguin contribution becomes the sum of positive\nterms and cannot vanish. Thus, although both decays are usually\nstrongly correlated and only relative experimental sensitivities\ndecide which of them has better chances to discover generic LFV\neffects mediated by the slepton sector, for some particular ranges of\nMSSM parameter searches for 3-body charged lepton decays are a safer\nchoice, allowing to avoid blind spots appearing for such setups due to\nthe suppression of $\\ell\\to\\ell'\\gamma$ decay rates.\n\n\\subsection{Non-decoupling effects in LFV Higgs decays}\n\\label{sec:hllndec}\n\nLFV Higgs decays in the SM are absent at the tree level and strongly\nsuppressed also at the loop level. Examining LFV Higgs boson decays\nwithin the MSSM is very interesting because, contrary to the other\nprocesses discussed in this paper, some contributions to the Higgs\ndecay amplitudes proportional to the lepton Yukawa couplings or to the\nnon-holomorphic trilinear slepton soft terms do not decouple in the\nlimit of heavy SUSY masses and can be potentially large.\n\nAs can be seen from Tables~\\ref{tab:hbounds}, for an average SUSY mass\nscale of $M=400$ GeV and the parameter setup of~\\eq{eq:mpatt} the\nupper bounds on the flavour violating parameters from Higgs decays are\nmuch weaker than from the other processes.  However, the bounds from\nHiggs decays on the $\\Delta_{LL}^{IJ}$, $\\Delta_{RR}^{IJ}$ and on the\nnon-holomorphic LR terms $\\Delta_{LR}^{'IJ}$ do not scale like\n$1\/M^2$.  Thus, comparing the limits on $\\Delta_{LR}^{'13}$ and\n$\\Delta_{LR}^{'23}$ entries from $h\\to \\ell\\ell'$ and $\\ell\\to\\ell'\n\\gamma$ decays one can check that e.g. for $\\tan\\beta=20$ and\n\\begin{eqnarray}\nM_{SUSY}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$} \\frac{1.5}{\\sqrt{|\\cos(\\alpha-\\beta)|}}~\\mathrm{TeV}\\,,\n\\end{eqnarray}\nthe latter are becoming weaker. For $\\Delta_{LR}^{'12}$ the same\noccurs at much higher scale\n\\begin{eqnarray} M_{SUSY}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$}\n\\frac{220}{\\sqrt{|\\cos(\\alpha-\\beta)|}}~\\mathrm{TeV}\\,.\n\\end{eqnarray}\nThe bounds on $\\Delta_{LL}^{IJ}$, $\\Delta_{RR}^{IJ}$ are obtained\nassuming that the flavour diagonal $A'_l$ terms vanish, so that all\nnon-decoupling LL and RR contributions are proportional to the Yukawa\ncouplings (see~\\eq{eq:ly}). In this case the Higgs decays become most\nconstraining for slightly higher SUSY scales, again for $\\tan\\beta=20$\nand $\\alpha$ angle of~\\eq{eq:alval} bounds on $\\Delta_{LL}^{IJ}$ and\n$\\Delta_{RR}^{IJ}$ from Higgs decays become stronger than the bounds\nfrom $\\ell\\to\\ell' \\gamma$ decays for $M_{SUSY}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$}\n2\/\\sqrt{|\\cos(\\alpha-\\beta)|}$ TeV for $\\tau\\mu$ transitions,\n$M_{SUSY}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$} 3\/\\sqrt{|\\cos(\\alpha-\\beta)|}$ TeV for $\\tau e$\ntransitions and $M_{SUSY}\\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle>}{\\sim}\\;$} 200\/\\sqrt{|\\cos(\\alpha-\\beta)|}$ TeV\nfor $\\mu e$ transitions.\n\nThe Higgs decays in supersymmetric extensions of the SM have been\nalready studied e.g. in~\\cite{Arganda:2004bz, Azatov:2012wq,\n  Petersson:2012nv, Bartl:2014bka, Arana-Catania:2013xma,\n  Aloni:2015wvn, Arganda:2015uca, Barenboim:2015fya, Gomez:2017dhl}.\nIn this Section we analyse within the general MSSM the decays of the\nlighter CP-even Higgs boson $h$.  The mass eigenstates formulae for\nthe MSSM contributions to the effective leptonic Yukawa couplings of\n$h$ are given in Eqs.~(\\ref{eq:yukeff} -- \\ref{eq:hnn}) while the\nrelevant MI expressions are collected in Appendix~\\ref{app:lhmi}.  The\npotentially largest contributions to $h\\to \\ell\\ell'$ decays come from\nthe effects non decoupling in the limit of large SUSY masses and\nproportional to non-holomorphic trilinear terms (see~\\eq{eq:lnh}).\nAssuming that flavour violating $A_l^{'IJ}$ terms are the only source\nof LFV and using Eqs.~(\\ref{eq:brhll}), (\\ref{eq:lnh}) and\n(\\ref{eq:zrdef}), one can write\n\\begin{eqnarray}\nBr(h\\to \\ell^I \\bar\\ell^J) &\\approx& \\frac{e^4 M_h}{8192 \\pi^5 c_W^4\n  \\Gamma_h} \\frac{\\cos^2(\\alpha-\\beta)}{\\cos^2\\beta} \\left(\ng(x_{\\tilde e_{LI}}, x_{\\tilde e_{RJ}})^2\n\\left|\\Delta_{LR}^{'IJ}\\right|^2 \\right. \\nonumber\\\\\n&+& \\left. g(x_{\\tilde e_{LJ}}, x_{\\tilde e_{RI}})^2\n\\left|\\Delta_{LR}^{'JI}\\right|^2 \\right)\\,,\n\\end{eqnarray}\nwhere $\\alpha,\\beta$ are the mixing angles in the Higgs sector (see\nAppendix~\\ref{app:lagr}), the dimensionless mass ratios are\n\\begin{eqnarray}\nx_{\\tilde e_{L(R)I}} = \\frac{m_{\\tilde e_{L(R)I}}}{|M_1|}\\,,\n\\end{eqnarray}\nand we defined\n\\begin{eqnarray}\ng(x,y) = -\\sqrt{x y}\\; C_0(x,y,1) \\;.\n\\label{eq:gratdef}\n\\end{eqnarray}\n\n\\begin{figure}[tbp]\n  \\begin{center}\n    \\includegraphics[width=0.5\\textwidth]{ccrat-eps-converted-to.pdf}\n\\end{center}\n\\caption{\\small Dependence of function $g(x_L, x_R)$\n  of~\\eq{eq:gratdef} on the splitting between the slepton and bino\n  masses.\\label{fig:cprat}}\n\\end{figure}\n\nAs can be seen from of Fig.~\\ref{fig:cprat}, for reasonable mass\nsplittings $g(x,y)\\sim{\\cal O}(1)$ and, inserting the numerical values\nof known quantities, one has\n\\begin{eqnarray}\nBr(h\\to \\ell^I \\bar\\ell^J) \\sim 2\\cdot 10^{-4} \\;\n\\frac{\\cos^2(\\alpha-\\beta)}{\\cos^2\\beta}\\left\n|\\Delta_{LR}^{'IJ(JI)}\\right|^2\\,.\n\\label{eq:hlllim}\n\\end{eqnarray}\nEven if for large SUSY mass scale $\\Delta_{LR}^{'IJ}$ insertions are\nnot constrained experimentally by other LFV measurements, $Br(h\\to\n\\ell^I \\bar\\ell^J)$ cannot be arbitrarily large in the MSSM because\n$\\Delta_{LR}^{'IJ}$ are constrained to ${\\cal O}(1)$ by the vacuum\nstability conditions and the requirement of the absence of charge and\ncolour breaking (CCB) minima of the scalar potential (see\ne.g. discussion in~\\cite{Dedes:2014asa}).\n\nThe Higgs mixing angle $\\alpha$ is subject to strong radiative\ncorrections from the squark sector and thus from the point of view of\npure leptonic sector can be treated as a free parameter.  However, the\nallowed values of the Higgs mixing angles $\\alpha, \\beta$ are limited\nby the existing experimental constraints (see e.g. Fig.~6 in Appendix\nB of ref.~\\cite{Aloni:2015wvn}), thus also the overall pre-factor\n$\\frac{\\cos^2(\\alpha-\\beta)}{\\cos^2\\beta}$ in~\\eq{eq:hlllim} can be at\nmost ${\\cal O}(1)$. Summarising, the maximal $Br(h\\to \\ell^I\n\\bar\\ell^J)$ which can be generated with the non-holomorphic trilinear\nterms is ${\\cal O}(10^{-4})$, not much below the current experimental\nsensitivities collected in Table~\\ref{tab:leplim} (including of\ndecoupling contributions does not change this conclusion even for a\nlight SUSY spectrum~\\cite{Arana-Catania:2013xma, Aloni:2015wvn}).\nFurther searches may therefore find the effects of non-holomorphic\ntrilinear terms or provide stricter bounds on them.\n\nSimilar analysis could be done for non-decoupling contributions\nproportional to $\\Delta_{LL}^{IJ}$ and $\\Delta_{RR}^{IJ}$ parameters.\nHowever, in this case non-decoupling terms are proportional also\neither to the diagonal $A_l'$ soft terms or to lepton Yukawa\ncouplings, so the formulae become complicated and a more involved\nnumerical analysis is required. Terms proportional to\n$\\Delta_{LL}^{IJ}$ and $\\Delta_{RR}^{IJ}$ multiplied by diagonal\n$A_l'$ terms can generate similar LFV Higgs decay rates as the flavour\noff-diagonal non-holomorphic $A_l^\\prime$-terms. However, assuming\nthat all non-holomorphic terms vanish, and including only the Yukawa\nsuppressed contributions one has a much stricter bound $Br(h\\to \\ell^I\n\\bar\\ell^J) \\lower.7ex\\hbox{$\\;\\stackrel{\\textstyle<}{\\sim}\\;$} 10^{-4} (Y_l^I)^2$ in the MSSM.\n\nFor a complete phenomenological analysis of LFV Higgs decays in the\nMSSM one would need to go beyond the one-loop analysis of this\narticle.  First, one would need to perform the matching of the MSSM on\nthe 2HDM with generic Yukawa couplings including the resummation of\nthe higher order chirally enhanced effects (see for\nexample~\\cite{Crivellin:2010er, Crivellin:2011jt, Crivellin:2012zz}).\nThen, one has to calculate the loop effects for flavour observables\nwithin this generic 2HDM~\\cite{Crivellin:2013wna}.\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusion}\n\nNew precision data in the lepton flavour sector are expected to come\nin the foreseeable future.  In the search for beyond the SM effects,\nthey will require precision and efficient calculations in various BSM\nmodels. In this article lepton flavour violating processes within MSSM\nhave been calculated using the Flavour Expansion Theorem, a recently\ndeveloped new technique of a purely algebraic mass-insertion expansion\nof the amplitudes~\\cite{Dedes:2015twa}. Both flavour-violating\noff-diagonal terms and flavour-conserving mass-insertions are\nconsidered.  The expansion in the flavour conserving off-diagonal mass\nterms leads to a transparent qualitative understanding of the\ncoefficients in front of the flavour violating mass insertions\n(see~\\eq{eq:miform}) in various decoupling limits. Most flavour\nviolating one-loop amplitudes decouple as $v^2\/M^2$ where $M$ is one\nof the soft SUSY breaking mass parameters. The exception are the Higgs\nflavour violating decays where the amplitudes decouple as $v^2\/M^2_A$.\nWe find that our full MI approximation, both in flavour violating and\nflavour conserving off-diagonal mass terms is an excellent\napproximation to the calculations in the mass eigenstates basis for a\nvery broad pattern of supersymmetric spectra, in particular for highly\nnon-degenerate spectra.  This is useful because in the MI\napproximation we work directly with the Lagrangian parameters and can\nconstrain them with experimental limits.\n\nOn the physics side, the considered processes are:\n$\\ell \\to \\ell' \\gamma$, $\\ell \\to 3 \\ell'$, $\\ell \\to 2\\ell'\\ell''$,\n$h \\to \\ell\\ell'$ as well as $\\mu \\to e$ conversion in nuclei.  The\nbounds on the flavour changing parameters of the MSSM have been\nupdated and their sensitivity to the forthcoming experimental results\nin different channels has been discussed.  We have emphasised that,\ngiven the foreseen experimental progress, precision measurements of\ndifferent processes have very different potential for the discovery of\nsupersymmetric effects. The radiative and leptonic muon decays are\nlikely to remain the most important source of information on\nsupersymmetric LFV. The leptonic decays play a complementary role to\nthe radiative ones in eliminating some \"blind spots\" of weakly\nconstrained by the latter LFV mass insertions.  This is illustrated in\nSections~\\ref{sec:msplit} and~\\ref{sec:corr}. Our complete analytical\nMI expansion facilitates the investigation of the LFV processes when\nthe SUSY spectra are non-degenerate and finding such \"blind spots\"\nwith suppressed branching ratios and regions of correlations between\nvarious processes. This is illustrated in Sections~\\ref{sec:msplit}\nand~\\ref{sec:corr}. The LFV Higgs decays are discussed in some detail.\nFor the supersymmetric spectrum of order of 1 TeV, the current\nexperimental limits on the LFV Higgs decays give several orders of\nmagnitude weaker bounds on lepton violating MI than the radiative\nlepton decays.  However, for the superpartner masses of several TeV\nHiggs decays provide stronger bounds than the latter because the\nbounds from Higgs decays do not scale with superpartner masses.  We\nhave also analysed the role of the so-called non-holomorphic $A$-terms\nin the flavour-violating Higgs boson decays, which can give branching\nratios not much below the present experimental sensitivity.\n\n\n\n\n\n\\section*{Acknowledgements}\n\nZF, WM, SP and JR are supported in part by the National Science\nCentre, Poland, under research grants\nDEC-2015\/19\/B\/ST2\/02848,\nDEC-2015\/18\/M\/ST2\/00054\nand DEC-2014\/15\/B\/ST2\/02157.  \nU.N. is supported by BMBF under grant no.~05H15VKKB1. The work of\nA.C. is supported by an Ambizione Grant of the Swiss National Science\nFoundation (PZ00P2\\_154834). JR would also like to thank CERN for\nhospitality during his visits there. \n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nThis paper studies the following question: given an $n$-variate polynomial $f(X)$, over a ring $R$\\footnote{From now on, $R$ always denotes an integral domain, i.e. a commutative ring with a unit, which is also a domain, and $\\mathbb{F}$ a field ($\\mathbb{Q},\\mathbb{R}$ and $\\mathbb{C}$ are, as usual, the rational, real and complex fields, respectively).}, how difficult is the task of finding a shift $\\mathbf{b}\\in R^n$ such that $f(X +\\mathbf{b})$ has fewer monomials than $f$. \n\nBefore proceeding we would like to discuss the issue of representation of polynomials. There are several natural settings -- representation as vector of coefficients or as arithmetic circuits -- and two different models -- the white-box and black-box models. The most obvious representation is the \\emph{dense representation} in which $n$-variate polynomials of degree $d$ are represented as a vectors of coefficients of length ${n+d\\choose d}$. In this setting we assume that the vector is given as input to the algorithm. A more concise representation is the \\emph{sparse representation} in which a polynomial is represented as a list of pairs of exponent vectors and coefficients. In the black-box setting we only assume that the algorithm has black-box access to the polynomial  (though the important parameters such as number of variables and degree are known to the algorithm). I.e., the algorithm is restricted to asking the polynomial for its values on different inputs. \nAnother natural model is representing polynomials as arithmetic circuits. That is, the algorithm will get as input an arithmetic circuit computing the polynomial. In the white-box setting the algorithm is explicitly given the circuit so it has access to the graph of computation etc. In the black-box model the algorithm only has black-box access to the circuit (though the important parameters such as size, depth, number of variables etc. are known to the algorithm). \n\n\nOne of the most important questions in the area of Algebraic Complexity Theory is the problem of checking if two polynomials are equivalent under affine transformations. In generality this problem is also called the polynomial projection problem. Ignoring issues of representations the problem is the following.\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large Polynomial Projection ($\\mathsf{PolyProj}_\\mathbb{F}$):}\n    \\vspace{0.1cm}\\\\\n    \\hline\n    \\vspace{0.1cm}\n    Given two polynomials $f\\in \\mathbb{F}[y_1, \\ldots, y_m]$ and $g\\in \\mathbb{F}[x_1,\\ldots, x_n]$, over a field $\\mathbb{F}$, output an $m\\times n$ matrix $A$ and a vector $\\mathbf{b}\\in \\mathbb{F}^{m}$ such that $ g(x_1, \\ldots, x_n) = f(A\\cdot \\begin{bmatrix}x_1 & x_2 &\\ldots & x_n\\end{bmatrix}^T+\\mathbf{b})$ if such a pair exists, or output ``FAIL'' otherwise.\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\n\nFor example, the holy grail of algebraic complexity, Valiant's Extended Hypothesis  is an instance of the polynomial projection problem. Recall that the hypothesis says that the permanent of an $n\\times n$ matrix cannot be represented as a polynomial projection of determinant of any $m\\times m$ matrix, for any  $m$ that is polynomial in $n$~\\cite{v79}. Kayal~\\cite{Kay12} showed that the problem of polynomial projection is $\\mathsf{NP}$-hard in general. However, for specific instances of the polynomial  $g$, under the requirement that the matrix $A$ has full rank (or that it is random), Kayal~\\cite{Kay12} gave efficient randomized algorithms in the black-box model (i.e. assuming only black-box access to $f$). \n\n\n\n\nSince studying polynomial equivalence under such projections is $\\mathsf{NP}$-hard in general, the following \\emph{simpler} question was considered.\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large Polynomial Equivalence under Shifts ($\\mathsf{ShiftEquiv}_\\mathbb{F}$):}\n    \\vspace{0.1cm}\\\\\n    \\hline\n    \\vspace{0.1cm}\n    Given two polynomials $f,g\\in \\mathbb{F}[x_1, \\ldots, x_n]$ output a vector $(b_1, \\ldots, b_n)\\in \\mathbb{F}^{n}$ such that $ g(x_1, \\ldots, x_n) = f(x_1+b_1, \\ldots, x_n+b_n)$ if such a vector exists, or output ``FAIL'' otherwise.\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\nTo the best of our knowledge the notion of studying polynomial equivalence under shifts first appeared in \\cite{GK93} and it was formally addressed by Grigoriev in \\cite{G97}. For polynomials of degree $d$ over $n$ variables, Grigoriev~\\cite{G97} gave a deterministic algorithm over fields of zero characteristic, a randomized algorithm over prime residue fields, and a quantum algorithm over fields of characteristic $2$, all of which run in time polynomial in the dense representation. That is, the running time is polynomial in ${n+d \\choose d}$. If the degree of the polynomial grows as a function of the number of variables or vice versa,  the algorithms presented by Grigoriev require \\emph{exponential} time in the number of variables, even if the polynomial can be represented by a small arithmetic circuit or if it has polynomially many monomials. It is a natural question to ask if the complexity of the algorithms can be brought down when the input to the algorithm is provided in some succinct representation -- for example, as an arithmetic circuit. In such a setting, Dvir, Oliveira and Shpilka~\\cite{DOS14} showed that given just a black box access to the polynomials $f$ and $g$ on $n$ variables, and given a bound on the degree $d$ and circuit size $s$, there is a randomized algorithm that runs in time $\\operatorname{poly}(n,d,s)$ and solves the polynomial equivalence under shifts problem. \nThe randomness in their algorithm only stems from polynomial identity testing  (PIT), which is a sub-routine of their algorithm, and hence \nequivalence under shifts in this setting can be derandomized if and only if PIT can be derandomized (clearly PIT is a special case of equivalence under shifts when $g$ is the zero polynomial).  \n\n\nA polynomial $f$ is said to be $t$-sparse if the number of monomials with non-zero coefficients in $f$ is at most $t$.\nIn the literature, an  $n$ variate polynomial is generally said to be sparse if the  number of monomials in it with non-zero coefficients is at most $\\operatorname{poly}(n)$. Equivalently, a sparse polynomial is a polynomial that can be computed by a depth two $\\Sigma\\Pi$ arithmetic circuit with a polynomial bound on the top fan-in. \nSparse polynomials are extremely well studied because of their simplicity and as a result many efficient algorithmic results are known for them~\\cite{BT88, KLW90, BT91, CDGK91, GKS90, GK91, GK93, LS95, SY11}. \n\nA variant of the polynomial projection problem asks if a given polynomial is equivalent to a sparse polynomial under affine transformations. This can be seen as a variant of the classical Minimum Circuit Size Problem (MCSP) where given the truth table of a function we wish to find the minimal circuit computing it. In this case the circuit we are seeking is a very structured $\\Sigma\\Pi\\Sigma$ circuit that is obtained by composing a $\\Sigma\\Pi$ circuit with an affine transformation. As this set of polynomials is dense inside the class $\\Sigma\\Pi\\Sigma$ it is an interesting family to study (see \\cite{MediniS21}).  Grigoriev and Karpinski~\\cite{GK93} were the first to consider this variant of the polynomial projection problem. Specifically, they studied the following problem (in the dense representation model) -- given a polynomial $P(X)$, over the rationals, and a parameter $t$ output a matrix $A$ and a vector $\\mathbf{b}$, if they exist, such that the polynomial $P(A\\cdot X+\\mathbf{b})$ has at most $t$ monomials. They gave an algorithm whose complexity is $O(M\\cdot d^{n^4})$ where $M$ is a bound on the size of coefficients of the input polynomial. Lakshman and Saunders~\\cite{LS95} considered the problem of testing the equivalence of univariate polynomials (over $\\mathbb{Q}$) to $t$-sparse polynomials under just shifts instead of affine linear transformations. They provided sufficient conditions for uniqueness and rationality of a $t$-sparsifying shift. Grigoriev and Lakshman~\\cite{GL00} extended these criterion to multivariate polynomials. They also gave algorithms for polynomials with finitely many sparsifying shifts\\footnote{Over $\\mathbb{Z}, \\mathbb{Q}, \\mathbb{R}, \\mathbb{C}$ etc., it may happen that there are infinitely many $t$-sparsifying shifts for a given polynomial. Grigoriev and Lakshman~\\cite{GL00} give algorithms for polynomials that are guaranteed to have finitely many $t$-sparsifying shifts.} that run in deterministic time $(dt)^{O(n)}$ and randomized time $t^{O(n)}$. In the past two decades, these exponential time algorithms could not be improved and this is a major motivation behind our study of hardness of this problem. We state the following more general problem to allow polynomials over rings. \n\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large Sparsification of Polynomials via Shifts ($\\mathsf{SparseShift}_R$):}\n    \\vspace{0.1cm}\\\\\n    \\hline\n    \\vspace{0.1cm}\n    Given a polynomial $f\\in R[x_1, \\ldots, x_n]$,  decide if there exists a vector $(a_1, \\ldots, a_n)\\in R^n$ such that $f(x_1+a_1, \\ldots, x_n+a_n)$ has strictly fewer monomials with non-zero coefficients than $f(x_1, \\ldots, x_n)$, or output ``FAIL'' if no such vector exists.\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\nIn this paper we show that the problem $\\mathsf{SparseShift}_R$ is at least as hard as as checking if a given system of polynomial equations over $R[x_1,\\ldots, x_n]$ has a solution (Hilbert's Nullstellensatz).\n\n\\noindent\n\\emph{Hilbert's Nullstellensatz:} Given a system $S$ of polynomial equations $S=\\{f_1=0, \\ldots, f_r=0\\}$ over the polynomial ring $R[x_1, \\ldots, x_n]$, we say that the system is satisfiable if there exists an assignment $\\mathbf{a}\\in R^n$ to the variables that simultaneously satisfies all  equations in $S$. This problem has a great significance in Algebraic Geometry and has other important applications in diverse areas. We state a slightly restricted version of Hilbert's Nullstellensatz problem that asks for a common solution in a specific domain (the general version asks for a solution in the algebraic closure). This definition is similar to the definition in the Blum, Shub and Smale model of computation \\cite{BSS98}.\n\n\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large Hilbert's Nullstellensatz over a ring $R$ ($\\mathsf{HN}_R$):}\n    \\vspace{0.1cm}\\\\\n    \\hline\n    \\vspace{0.1cm}\n    Given a system of polynomial equations $\\{f_1=0, \\ldots, f_r=0\\}$ over $R[x_1, \\ldots, x_n]$, decide whether there exist a vector $(a_1, \\ldots, a_n)\\in R^n$ such that for all $i\\in[r]$, $f_i(a_1, \\ldots, a_n) = 0$, or output ``FAIL'' if no such vector exists.\\\\\n    \\hline\n  \\end{tabular}\n\\end{center}\n\n\nWith this background, we shall now state our  first main result that gives a reduction from Hilbert's Nullstellensatz problem to polynomial sparsification.\n\\begin{theorem}\\label{thm:HN-hard}\n  Let $R$ be an integral domain,  which is not a field. Then $\\mathsf{SparseShift}_R$ is $\\mathsf{HN}_R$-hard, in any of the white-box representations.\n\\end{theorem}\n\nIf $R$ is arbitrary, then the polynomials could have coefficients with arbitrary bit complexity. Thus, it is important for us to also specify the model of computation over which this problem is being considered. In the Turing machine model, assuming that $f_1,\\ldots,f_r\\in \\mathbb{C}[x_1,\\ldots, x_n]$ have integral coefficients, Koiran~\\cite{Koi96} showed (by assuming that the Generalized Riemann Hypothesis is true) that $\\mathsf{HN}_\\mathbb{C}$ can be solved in the second level of polynomial hierarchy. Without the GRH assumption, the only known upper bound for $\\mathsf{HN}_\\mathbb{C}$ is  $\\mathsf{PSPACE}$. \nFor $R=\\mathbb{Z}$, Matiyasevich~\\cite{Mat70} showed that this problem is undecidable (also see \\cite{Dav73}). Putting these together with \\cref{thm:HN-hard} we get the following  consequence. \n\n\\begin{corollary}\\label{cor:HN-Sparse}\n$\\mathsf{SparseShift}_\\mathbb{Z}$ is undecidable.\n\\end{corollary}\n\nIt is important to note that under sparse or dense representations, $\\mathsf{SparseShift}_{R}$ is in $\\mathsf{NP}_{{R}}$. That is, given $\\mathbf{a}$, we can efficiently verify if it is a sparsifying shift for a polynomial $P(X)$ using at most polynomially many algebraic operations using sparse polynomial interpolation, and sparse polynomial identity testing\\footnote{Given a polynomial $P(X)$ (of sparsity $t$) in its sparse representation, it is easy to see that we have access to evaluations of the polynomial $P(X+\\mathbf{a})$ as well. Let $Q(X)$ be the $(t-1)$-sparse polynomial obtained by polynomial interpolation (using \\cite{BT88} which uses at most polynomially many algebraic operations) using these evaluations. If $\\mathbf{a}$ were a sparsifying shift, then sparsity of $P(X+\\mathbf{a})$ would strictly be smaller than $t$ and in that case, $Q(X)$ would in fact be equal to $P(X+\\mathbf{a})$. This equivalence can be checked using polynomial identity testing of sparse polynomials (using \\cite{KS01} which also uses at most polynomially many algebraic operations).}. Thus for any integral domain $R$ (which is not a field) such that $\\mathsf{HN}_R$ is $\\mathsf{NP}_R$-complete, over the Blum-Shub-Smale model of computation~\\cite{BSS98}, we get the following corollary from the aforementioned statements and \\cref{thm:HN-hard}.\n\n\\begin{corollary}\\label{cor:bss-npc}\n Let $R$ be an integral domain (but not a field) such that $\\mathsf{HN}_R$ is $\\mathsf{NP}_R$-complete over the Blum-Shub-Smale model of computation. Then $\\mathsf{SparseShift}_{R}$ is also $\\mathsf{NP}_{{R}}$-complete.\n\\end{corollary}\nIn particular, we get that $\\mathsf{SparseShift}_{\\mathbb{Z}}$ is also $\\mathsf{NP}_{{\\mathbb{Z}}}$-complete.\\\\\n \nThese results, to some extent, shed a light on why this problem in general has been evading the efforts to provide efficient algorithms.\n\nNote that our problem $\\mathsf{SparseShift}_R$ can  also be viewed as a gap decision problem -- given a polynomial $P(X)$ of sparsity $t$, is there a vector $\\mathbf{a}$ such that $P(X+\\mathbf{a})$ has at most $t-1$ monomials. Let us formally define a more general gap version of $\\mathsf{SparseShift}_R$.\n\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large $\\alpha$-gap-$\\mathsf{SparseShift}_R$:}\\vspace{0.1cm}\\\\\n    \\hline\n    \\vspace{0.1cm}\n    Let $\\alpha>1$ be a parameter. Given a polynomial $P\\in R[X]$ and a parameter $t$,\n    \\begin{itemize}\n    \\item output YES if there exists a vector $\\mathbf{a}$ such that $P(X+\\mathbf{a})$ has at most $t$ monomials, and\n    \\item output NO if for all vectors $\\mathbf{a}$, $P(X+\\mathbf{a})$ has at least $\\alpha t$ monomials.\n    \\end{itemize}\\\\\n   \n    \\hline\n  \\end{tabular}\n\\end{center}\nUsing  gap amplification we reduce $\\mathsf{SparseShift}_R$ to $N^\\beta$-gap-$\\mathsf{SparseShift}_R$ for all functions $\\beta \\in o(1)$. We thus get our second main result.\n\n\\begin{theorem}\\label{thm:gap-hardnessZ}\nFor every function $\\beta:\\mathbb{N}\\to\\mathbb{R}_+$ such that $\\beta\\in o(1)$, $N^\\beta$-gap-$\\mathsf{SparseShift}_\\mathbb{Z}$ is undecidable (where $N$ is the input length).\n\n\\end{theorem}\n\n\nIn \\cref{thm:gap-hardnessZ}, we used the undecidability of $\\mathsf{HN}_\\mathbb{Z}$ to infer the undecidability of $N^\\beta$-gap-$\\mathsf{SparseShift}_\\mathbb{\\mathbb{Z}}$. However, we do not have such  results for  rings  $R\\neq \\mathbb{Z}$ (over Turing machine model). Furthermore, $HN_\\mathbb{Q}$ is not known to be undecidable and, as mentioned above, over $\\mathbb{C}$ it is decidable as well as over finite fields. Thus for $R=\\mathbb{F}_p, \\mathbb{Q}, \\mathbb{R}$ or $\\mathbb{Z}_q$, we present a different reduction of gap problems -- from $(1-\\varepsilon, \\delta)$-gap-$\\text{Max-3Lin}_R$ to $\\alpha$-gap-$\\mathsf{SparseShift}_R$ and infer  $\\mathsf{NP}$-hardness results for $\\alpha$-gap-$\\mathsf{SparseShift}_\\mathbb{R}$.\n\n\\begin{center}\n  \\begin{tabular}{|m{15cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    {\\large $(1-\\varepsilon,\\delta)$-gap-$\\text{Max-3Lin}_R$:}\\vspace{0.1cm} \\\\\n    \\hline\n    \\vspace{0.1cm}\n    Given a system of linear equations $\\{L_1=0, \\ldots, L_m=0\\}$ over $R[x_1, \\ldots, x_n]$ each of which depends on exactly $3$ variables,\n    \\begin{itemize}\n    \\item output YES if at least $(1-\\varepsilon)$ fraction of equations can be simultaneously satisfied, and\n    \\item output NO if at most $\\delta$ fraction of equations can be simultaneously satisfied.\n    \\end{itemize}\\\\\n   \n    \\hline\n  \\end{tabular}\n\\end{center}\n\nWe say that it is $\\mathsf{NP}$-hard to $(1-\\varepsilon, \\delta)$-approximate $\\text{Max-3Lin}_R$ \nif the decision problem $(1-\\varepsilon,\\delta)$-gap-$\\text{Max-3Lin}_R$ is $\\mathsf{NP}$-hard. Using this notion, we summarize non-exhaustively some known $\\mathsf{NP}$-hardness results for $(1-\\varepsilon, \\delta)$-approximating $\\text{Max-3Lin}_R$.\n\n\\begin{center}\n  \\begin{tabular}{|m{4.5cm}|m{3cm}|m{7cm}|}\n    \\hline\n    \\vspace{0.1cm}\n    Result & Ring $R$ & $\\mathsf{NP}$-Hardness for\\\\\n    \\hline\n    H{\\aa}stad~\\cite{has01} & $\\mathbb{F}_p$ & $\\forall~\\varepsilon>0$, $(1-\\varepsilon, \\frac{1+\\varepsilon}{p})$-approximation\\\\\n    \\hline\n    H{\\aa}stad~\\cite{has01} & $\\mathbb{Z}_q$ for $q\\in\\mathbb{N}$ & $\\forall~\\varepsilon, \\delta>0$, $(1-\\varepsilon, \\frac{1}{q}+\\delta)$-approximation\\\\\n    \\hline\n    Feldman, Gopalan, Khot and Ponnuswami~\\cite{FGKP06} & $\\mathbb{Q}$ & $\\forall~\\varepsilon>0$, $(1-\\varepsilon, \\varepsilon)$-approximation\\\\\n    \\hline\n    Gurswami and Raghavendra~\\cite{GR06, GR07} & $\\mathbb{Q}, \\mathbb{R}$& $\\forall~\\varepsilon, \\delta>0$, $(1-\\varepsilon, \\delta)$-approximation\\\\\n    \\hline\n  \\end{tabular}\n  \\captionof{table}{Non-exhaustive list of known $\\mathsf{NP}$-hardness results for approximating $\\text{Max-3Lin}_R$}\n  \\label{table:max3lin-hardness}\n\\end{center}\n\nThus, a gap reduction from $(1-\\varepsilon',\\delta')$-gap-$\\text{Max-3Lin}_R$ (where $\\varepsilon'$ and $\\delta'$ are as in the last column of \\cref{table:max3lin-hardness}) to $\\alpha$-gap-$\\mathsf{SparseShift}_R$ (for $\\alpha = \\alpha(\\varepsilon', \\delta', R)$) implies  hardness of $\\alpha$-gap-$\\mathsf{SparseShift}_R$ and using amplification we get our third main result. We first state it in the sparse representation model and then in the arithmetic circuit model.\n\n\n\\begin{theorem}[Sparse representation]\\label{thm:gap-sparse-input}\n    For $R=\\mathbb{F}_p, \\mathbb{Q}, \\mathbb{R}$ or $\\mathbb{Z}_q$ and for every $\\beta>1$  the $\\beta$-gap-$\\mathsf{SparseShift}_R$ problem is NP-hard. Furthermore, there exists a constant $\\alpha>1$ such that for every $d=O(1)$ the $\\alpha^d$-gap-$\\mathsf{SparseShift}_R$  problem is  NP-hard when given polynomials of degree at most $d$ as input. \n\\end{theorem} \n\n\nThe theorem is stated for the sparse representation model but as the polynomials under consideration  have many non-zero terms it can also be stated without any modification in the dense representation model. We next state the theorem in the arithmetic circuit model.\n\n\\begin{theorem}[Arithmetic circuit representation]\\label{thm:gap-circuit-input}\n  There exists a constant $\\alpha>1$ such that the following holds for $R=\\mathbb{F}_p, \\mathbb{Q}, \\mathbb{R}$ or $\\mathbb{Z}_q$. For every $d$  the $\\alpha^d$-gap problem is NP-hard when given polynomials of degree at most $d$ as input. Furthermore, our hard instances have circuit size $(nd+1)$.\n\\end{theorem} \n\nObserve that  if we take e.g. $d=n^c$ in the theorem above then the input size is $N=n^{c+1}$ and the gap is $\\exp(N^{1-1\/c})$.\n\n\n\\section{Preliminaries}\nWe use $[n]$ to refer to the set $\\inbrace{1,2,\\ldots, n}$.  We use capital letters $A$ and $C$ to represent matrices, capital letters $U,S$ and $T$ to represent systems of equations, and capital letters $X,Y$ and $Z$ to represent sets of variables. We reserve letters $x,y$ and $z$ with, or without subscripts, to represent variables. We use bold letters $\\mathbf{a}, \\mathbf{b}, \\ldots$ to indicate vectors and non-bold letters (apart from $x,y$ and $z$) with, or without subscripts, $e, b_i, a_j, A_{i,j}, C_{k, \\ell}, \\ldots$ to indicate scalars.\n\nLet $S$ be a system of polynomial equations $\\{f_i=0\\}_{i=1}^r$, where $\\deg(f_i)=d_i$. We use $\\mathrm{Vars}(f)$ to denote the variable support of the polynomial $f$, and for a system $S$ of polynomial equations we use $\\mathrm{Vars}(S)$ to denote the union of $\\mathrm{Vars}(f_i)$ for all equations $f_i=0$ in $S$. We denote with $L$ an upper bound on the bit-complexity of the coefficients of the polynomials in the system.\\footnote{When the underlying ring is an abstract ring one has to define this complexity, but for the usual rings and fields such as $\\mathbb{Z},\\mathbb{Q},\\mathbb{R},\\mathbb{C},\\mathbb{F}_q$ this is the natural definition. In the BSS model this complexity is called height and is indeed only defined for these natural domains \\cite{BSS98}.} The total degree of $S$ is\n$d=\\sum_i d_i$. \n\nIn this paper we shall consider two types of representations of polynomials (and hence of polynomial equations). The representation that is typically studied in the context of polynomial equations is the so called ``sparse\nrepresentation''. In this representation polynomials are given as a set of pairs consisting of exponent vectors together with the coefficients of the corresponding monomials. E.g. the polynomial $2x^2z-y\\in\\mathbb{F}[x,y,z,]$ is represented as $\\{((2,0,1),2),(0,1,0),-1)\\}$. This is called the sparse representation as  we do not charge for monomials whose coefficients are equal to $0$. In particular the size of the representation of a degree $d$ polynomial can be much smaller than ${n+d \\choose d}$. For a system of polynomial equations $S=\\{f_i=0\\}_{i=1}^{r}$, the complexity of $S$, or its size, is defined to be the total bit size of the sparse representations of the polynomials $\\{f_i\\}_{i\\in[r]}$. We note that this is always upper bounded by $\\sum_{i=1}^{r} {n+d_i \\choose d_i}\\cdot L$.\n\nThe second type of representation that we consider is when the polynomials $f_i$ are given as the outputs of arithmetic circuits.\\footnote{Arithmetic circuits are directed acyclic graphs whose leaf nodes are labeled by variables or constants from the underlying field, and every non-leaf node is labeled either by a $+$ or $\\times$. The fan-in of multiplication gates is $2$ while the fan-in of addition gates is unbounded. Every node computes a polynomial by operating on its inputs with the operation given by its label. The computation flows from the leaves to the output node (see \\cite{SY10, sapt} for more details).} In this paper we only consider the white-box version of this representation, i.e., when the computation graph of the circuit is explicitly given to the algorithm. In this case the complexity (or size) of the system $S$ is the total size of the input circuits times the maximal bit complexity of  coefficients in the circuits.\n\n\nAs we shall later see (\\cref{lem:sparserep-to-equations} and \\cref{lem:circuit-to-equations}), given a system of equations, either via arithmetic circuits or in the sparse representation model, one can easily construct an equivalent system $T$ of polynomial equations of degree $2$ and roughly of the same complexity, such that the system $S$ has a solution if and only if the system $T$ does. Hence, these two different representations have the same computational power. However, this reduction is not gap-preserving so we will have to give separate arguments for the gap problems.\n\n\n\\section{Reduction from $\\mathsf{HN}_R$ to $\\mathsf{SparseShift}_R$}\\label{sec:HN-hard}\n\nIn this section, we shall first show that given a system $S$ of $r$ many polynomial equations over $R[X]$, we can algorithmically construct a system $T$ of polynomial equations over $R[X''$] such that $X\\subseteq X''$; each polynomial in $T$ is of degree at most $2$; and if $\\mathbf{a}\\in R^{\\abs{X}}$ is a solution for the system $S$ then there exists an extension $\\mathbf{a}'$ of $\\mathbf{a}$ such that $\\mathbf{a}'$ is a solution for the system $T$. And vice versa, from a solution to $T$ we deduce a solution to $S$. From $T$ we shall then construct a polynomial $P_S\\in R[X'',W]$ such that $S$ has a solution if and only if the polynomial $P_S$ can be sparsified. \n\n\\subsection{Reduction to a system of polynomial equations of degree at most $2$}\\label{subsec:reduction-degree2}\n\n\\subsubsection{Case when input is provided in sparse representation}\nLet  $S = \\{f_1=0, \\ldots, f_r=0\\}$ be our input system of polynomial equations such that each $f_i$ is provided in sparse representation. \nLet $s_i$ denote the number of monomials with non-zero coefficients in the polynomial $f_i$. \nLet $Y$ and $Z$ be new disjoint sets of variables, that are disjoint from $X$ such that\n\\begin{align*}\n   Y = \\{y^{(i)}_{j,k}\\mid i\\in [r], j\\in [s_i]~\\text{and}~ k\\geq 1 \\}~\\text{and}~ Z = \\{z^{(i)}_{j}\\mid i\\in [r], j\\in [s_i]\\}.\n\\end{align*}\n\nLet the variables in $Y$ have a lexicographic ordering based on the indices $i,j$ and $k$, variables in $Z$ have a lexicographic ordering based on the indices $i$ and $j$, and the variables in $X$ have some arbitrary ordering. Across the sets $X,Y$ and $Z$, let the ordering be $Z\\succ Y \\succ X$. Given $S = \\{f_1=0, \\ldots, f_r=0\\}$, we construct an extended system $T$ of polynomial equations over the variables $X\\sqcup Y\\sqcup Z$ such that each polynomial equation is of degree at most $2$ and is such that there is a solution $\\mathbf{a}\\in R^{\\abs{X}}$  for the system $S$ if and only if there exists an extension $\\mathbf{a}'$ of $\\mathbf{a}$ such that $\\mathbf{a}'$ is a solution for the system $T$. This is a well known reduction (see, e.g., Lemma 6 in Chapter 2 of \\cite{BSS98}) but for completeness we repeat it here.\n\n\n\\cref{alg:extendSystem} and \\cref{alg:reduceMonomial} describe the construction of the extended system of equations. What the algorithms do is, roughly, for any monomial $m=x_{i_1}\\cdot x_{i_2} \\cdot x_{i_3}\\cdot\\ldots\\cdot x_{i_j}$ of degree greater than $2$,  introduce a new variable, say $y$, replace $m$ with the the monomial $y\\cdot x_{i_3}\\cdot\\ldots\\cdot x_{i_j}$ and introduce a new equation $y-x_{i_1}\\cdot x_{i_2}=0$ and for any monomial $m'$, of degree $1$, introduce a new variable $z$ and  a new equation $z-m' = 0$. Finally, an affine linear equation of the form $\\sum c_i z_i+c_0$ is added to account for the fact that the original sum of monomials has to be zero. In particular, at the termination of the algorithm, the system $T$ consists of  constant-free quadratic binomial\\footnote{We use the phrase constant-free quadratic binomial equation to refer to an equation with two non-constant monomials of degree  at most $2$.} equations and affine linear polynomial equations. It is also clear that there exists $\\mathbf{a}'$ that satisfies $T$ if and only if there is  $\\mathbf{a}$ that satisfies $S$. \n\n\\begin{algorithm}[H]\n  \\caption{ConstructExtendedSystem-SparseRepresentation($S$)}\\label{alg:extendSystem}\n  \\SetAlgoLined\n  \\KwResult{Given a system $S= \\{f_1=0, \\ldots, f_r=0\\}$ of polynomial equations provided in sparse representation, we construct a system $T$ of polynomial equations over an extended set of variables such that each polynomial in $T$ has degree at most 2 and if $\\mathbf{a}$ solves $S$ then there exists an extension $\\mathbf{a}'$ of $\\mathbf{a}$ such that $\\mathbf{a}'$ solves $T$ and vice-versa.}\n \n  $T\\gets \\emptyset$\\;\n  \n  \\For{$i\\in [r]$}{\n    $f' \\gets 0$\\;\n    Let $s_i$ be the sparsity of $f_i$\\;\n    \\For{$j\\in[s_i]$}{\n      Let $c_{i,j}$ be the coefficient of $j$'th monomial $m_{i,j}$ in $f_i$\\;\n      \\eIf{$\\deg(m_{i,j})\\geq 1$}{\n\t$U \\leftarrow \\text{ReduceMonomial($m_{i,j}$, $i$, $j$)}$\\;\n\t$T\\gets T\\cup U$\\;\n\t$f'\\gets f'+ c_{i,j}\\cdot z^{(i)}_{j}$\\;\n      }\n      {\n\t$f'\\gets f'+ c_{i,j}$\\;\n      }\n    }\n    $T\\gets T\\cup \\{f'=0\\}$\\;\\label{line:f'}\n  }\n  \\Return $T$\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n  \\caption{ReduceMonomial($m$, $i$, $j$)}\\label{alg:reduceMonomial}\n  \\SetAlgoLined\n  \\KwResult{Given the $j$'th monomial (of degree at least $1$) of the $i$'th polynomial, $m_{i,j}$, generate a collection of polynomial equations $U$ from it.}\n \n  $U\\gets \\emptyset$\\;\n  $k = 1$\\;\n  \\While{$\\deg(m) \\geq 2$}{\n    Let $u,v$ be the renaming of the two trailing variables under the ordering of $X$ and $Y$ variables as described above\\;\n    \n    $U\\gets U\\cup \\{y^{(i)}_{j, k}- u\\cdot v = 0\\}$\\;\n    $m\\gets \\frac{m}{u\\cdot v}\\cdot y^{(i)}_{j, k}$\\;\n    $k \\gets k+1$\\;\n  }\n  $U\\gets U\\cup \\{z^{(i)}_{j}- m = 0\\}$\\;\\label{line:linear-2}\n  \n  \\Return $U$\n\\end{algorithm}\n\n\\newpage\n\n\\begin{lemma}\\label{lem:sparserep-to-equations}\n  Let $S = \\{f_i(X) = 0\\}_{i=1}^r$ be a system of polynomial equations over the polynomial ring $R[X]$ such that for each $i\\in[r]$, $f_i(X)$ is a polynomial of degree $d_i$ and sparsity $s_i$, and the bit complexity of each coefficient is at most $L$. Then, \\cref{alg:extendSystem} runs in time $ \\operatorname{poly}(\\abs{X}, r, \\max_i\\{d_i\\}, \\max_i\\{s_i\\},L)$ and returns a set $T$ of polynomial equations $\\{g_j(X,Y,Z) = 0\\}_{j=1}^t$ over the polynomial ring $R[X, Y, Z]$ such that \n  \\begin{itemize}\n  \\item $t = \\operatorname{poly}(\\abs{X}, r, \\max_i\\{d_i\\}, \\max_i\\{s_i\\},L)$.\n  \n  \\item $\\abs{X\\sqcup Y\\sqcup Z} = \\operatorname{poly}(\\abs{X}, t, \\max_i\\{d_i\\}, \\max_i\\{s_i\\})$.\n\n  \\item The bit-complexity of the coefficients of the polynomial equations in $T$ is also at most $L$. \n  \\item Each polynomial equation $g_j(X,Y,Z) = 0$ in $T$ is either a quadratic binomial polynomial equation or an affine linear polynomial equation.\n  \\item $S$ has a solution $\\mathbf{a}\\in R^{\\abs{X}}$ if and only $T$ has a solution $\\mathbf{a}'\\in R^{\\abs{X\\sqcup Y \\sqcup Z}}$.\n  \\end{itemize} \n\\end{lemma}\n\nThe lemma is very easy to verify and so we only give a brief proof of the last claim.\n\n\\begin{proof}\nNote that by the aforementioned ordering of variables, in all quadratic binomial equations included into the set $T$ that have the form $u-v\\cdot w=0$, we have that $u$ is of the form $y^{(i)}_{j,k}$, and $v$ and $w$ could be of the form $y^{(i)}_{j,k'}$ or $x_{i'}$, and  the term $u$ is leading with respect to the terms $v$ and $w$. Further, all the quadratic equations in the set $T$ can be assumed to have some sort of a \\emph{topological order}.  Thus the values of all the $Y$ variables that appear in the variable support can be inductively inferred by just setting the $X$ variables. That is, if we want to satisfy all such equations, then the value of the term $u$ can be inferred from the value of terms $v$ and $w$ for every invocation of $u$, $v$ and $w$. \n\nFurther note that some of the linear polynomial equations in $T$ take the form $u'-v' =0$ (from \\cref{line:linear-2} of \\cref{alg:reduceMonomial}) where $u'$ is of the form $z^{(i)}_{j}$, and $v'$ could be of the form $y^{(i)}_{j,k}$ or $x_{i'}$. Similar to the case above, the value of the term $u'$ can be inferred from the value of the term $v'$ for every invocation of $u'$ and $v'$, and the value of $v'$ is already fixed as the values of all the $X$ and $Y$ variables that appear in the variable support were set in the aforementioned discussion. Observe that the rest of the linear polynomial equations in $T$ (from \\cref{line:f'}  of \\cref{alg:extendSystem}) correspond to the polynomial equations in $S$, and setting of $Z$ variables in the variable support of $T$, by the above procedure, satisfies all the linear polynomial equations in $T$. \n\nGiven an assignment $\\mathbf{a}$ to $X$ let $\\mathbf{a}'$ be its unique extension to the variable set $X\\sqcup Y\\sqcup Z$ according to the process described above.\nThe argument above shows that $\\mathbf{a}$ is a solution to the system $S$ if and only if $\\mathbf{a}'$ is a solution to the system $T$.\n\\end{proof}\n\n\\subsubsection{Case when input is provided in white-box circuit form}\n\n\n\nLet $S = \\{f_i=0\\}_{i=1}^r$ be our input system of polynomial equations such that each $f_i$ (for $i\\in[r]$) is provided as an arithmetic circuit $\\Phi_i$ of size $s_i$. Without loss of generality, for all $i\\in [r]$, we can assume that every product gate in $\\Phi_i$ has a fan-in of $2$.\n\nLet $Y  = \\inbrace{y^{(i)}_{j}\\mid i\\in[r]~\\text{and}~j\\in[s_i]}$ be a new set of variables disjoint from $X$.\nFor $i\\in [r]$, let $g^{(i)}_1, \\ldots, g^{(i)}_{s_i}$ be a topologically sorted enumeration of all nodes in circuit $\\Phi_i$. For all $j\\in[s_i]$, let node $g^{(i)}_{j}$ be labelled by the variable $y^{(i)}_{j}$. Corresponding to each node in $\\Phi_i$, we shall now define a polynomial equation of degree at most $2$ over the variable sets $X$ and $Y$. \n\n\n\\begin{algorithm}[!t]\n  \\caption{ConstructExtendedSystem-CircuitRepresentation(S)}\\label{alg:extendSystem-circuit}\n  \\SetAlgoLined\n  \\KwResult{Given a system $S= \\{f_1=0, \\ldots, f_r=0\\}$ of polynomial equations provided in white-box circuit representation, we construct a system $T$ of polynomial equations over an extended set of variables such that each polynomial in $T$ has degree of at most 2 and if $\\mathbf{a}$ solves $S$ then there exists an extension $\\mathbf{a}'$ of $\\mathbf{a}$ such that $\\mathbf{a}'$ solves $T$ and vice-versa.}\n  $T\\gets \\emptyset$\\;\n  \n  \\For{$i\\in [r]$}{ \n    Let $\\Phi_i$ be the circuit computing $f_i$, and $s_i = \\abs{\\Phi_i}$\\;\n    \\For{$j\\in[s_i]$}{\n        \\eIf{$g_j^{(i)}$ is a leaf node in $\\Phi_i$}{\n        Let $u$ be a variable or a constant labeling the input node in $\\Phi_i$\\;\n        $T \\gets T \\cup \\{y^{(i)}_j - u =0\\}$\\;\n      }{\n      \\eIf{$g^{(i)}_{j}$ is a product node in $\\Phi_i$}{\n        Let $u$ and $v$ be the renaming of the labels of the children of $g^{(i)}_{j}$\\;\n         $T \\gets T \\cup \\{y^{(i)}_j -  u\\cdot v = 0\\}$\\;\n      }{\n        $g^{(i)}_{j}$ is a sum node in $\\Phi_i$\\;\n        Let $u_1, \\ldots, u_k$ be the renaming of the labels of the children of $g^{(i)}_{j}$\\;\n        $T \\gets T \\cup \\{y^{(i)}_j - \\sum_{i=1}^k  u_i= 0\\}$\\;\n      }\n    }\n  }\n  }\n  \\Return $T$\\;\n\\end{algorithm}\n\n\\cref{alg:extendSystem-circuit} describes the construction of the extended system of equations. For each input node $g^{(i)}_{j}$, labeled by  $u$ (where  $u$ is either a variable $x_{k}$ or a constant $c$) in $\\Phi_i$, the algorithm adds the polynomial equation  $y^{(i)}_j - u= 0$ to the system $T$. For each product node $g^{(i)}_{j}$ with children labelled $u$ and $v$ (where $u,v$ could be of the form $y^{(i)}_{j'}$ for some $j>j'$ or $x_{k'}$), the algorithm introduces a new polynomial equation $y^{(i)}_j - u\\cdot v = 0$ to the system $T$, and for each sum node $g^{(i)}_{j}$ with children labelled $u_1, \\ldots, u_k$ (where $u_1, \\ldots, u_k$ could be of the form $y^{(i)}_{j'}$ for some $j>j'$ or $x_{k'}$), it introduces a new polynomial equation $y_j^{(i)} - \\sum_{i=1}^k  u_i = 0$. At the termination of the algorithm, the system $T$ consists of  either constant-free quadratic binomial equations or affine linear polynomial equations. It is also clear that there exists $\\mathbf{a}'$ that satisfies $T$ if and only if there is  $\\mathbf{a}$ that satisfies $S$. As before this is easy to see: by following the flow of computation in an arithmetic circuit from leaves to the root, we can infer the values of $y^{(i)}_j$ for all $i\\in[t]$ and $j\\in [s_i]$.\n\n\\begin{lemma}\\label{lem:circuit-to-equations}\n  Let $S = \\{f_i(X) = 0\\}_{i=1}^r$ be a system of polynomial equations over the polynomial ring $R[X]$ such that for each $i\\in[r]$, the polynomial $f_i(X)$ is provided as an arithmetic circuit $\\Phi_i$ of size $s_i$. Then, when given this as input, \\cref{alg:extendSystem-circuit} runs in time $\\operatorname{poly}(\\abs{X}, r, \\max_i\\{s_i\\})$ and returns a system $T$ of polynomial equations $\\{g_j(X,Y) = 0\\}_{j=1}^t$ over the polynomial ring $R[X, Y]$ such that \n  \\begin{itemize}\n\n  \\item $t = \\operatorname{poly}(\\abs{X}, r, \\max_i\\{s_i\\})$.\n    \\item $\\abs{Y} \\leq r\\cdot\\max_i\\{s_i\\}$.\n  \\item Each polynomial equation $g_j (X,Y) = 0$ in $T$ is either a quadratic binomial polynomial equation or an affine linear polynomial equation.\n  \\item $T$ has a solution in $R^{\\abs{X\\sqcup Y}}$ if and only $S$ has a solution in $R^{\\abs{X}}$.\n  \\end{itemize} \n\\end{lemma}\n\n\n\\subsection{Construction of  $P_S$}\\label{subsec:poly-defn}\n\nGiven a system $S$ of polynomial equations over a set of variables $X$, in \\cref{subsec:reduction-degree2} we constructed the system $T$ of polynomial equations of degree at most $2$ over the set of variables $X, Y$ and $Z$,\\footnote{In case the polynomials in the system $S$ of polynomial equations are provided as circuits, $Z = \\emptyset$.} such that $\\abs{X\\sqcup Y\\sqcup Z}$ is at most polynomial in the input size. Without loss of generality, let the variables in $X\\sqcup Y \\sqcup Z$ be renamed as the variable set $X' = \\inbrace{x_1,\\ldots, x_N}$ where $N = \\abs{X\\sqcup Y\\sqcup Z}$. \n\nLet $\\{g_1(X')=0, \\ldots, g_t(X')=0\\}$ be the enumeration of polynomial equations in $T$. \nWithout loss of generality, we can assume that the number of equations with a non-zero constant term is equal to $1$. Otherwise, given a system $T$ of polynomial equations, with $t'>1$ many of these polynomial equations with non-zero constant terms, we shall construct a new system $T'$ such that the number of polynomial equations in $T'$ that have non-zero constant terms is exactly equal to $1$, and a solution of $T$ is a solution of $T'$ and vice versa. Without loss of generality assume that $\\{g_1(X')=0, \\ldots, g_{t'}(X') =0\\}$ are the polynomial equations in $T$ with non-zero constant terms. By Lemmata~\\ref{lem:sparserep-to-equations} and~\\ref{lem:circuit-to-equations} it follows that $\\{g_1(X')=0,\\ldots,g_{t'}(X')=0\\}$ are affine linear equations.\nDenote the free term in $g_1(X'),\\ldots,g_{t'}(X')$ with  $c_1, \\ldots, c_{t'}$, respectively. We obtain $T'$ from $T$ by just updating each of the polynomials $g_i(X')$ (for $2\\leq i\\leq {t'}$) as follows. \n\\begin{align*}\n    g_i(X') \\gets c_1\\cdot g_i(X') - c_i\\cdot g_1(X')\\,.\n\\end{align*}\nThe rest of the polynomials from $T$ are directly added to $T'$. It is easy to see that any solution to the system $T$ is also a solution to the system $T'$ and vice-versa (as $R$ is a domain). Furthermore, the only equation in $T'$ with a non-zero constant term is an affine linear equation.\n\nConditioned on the aforementioned discussion, we shall assume that all polynomial equations in $T$ other than $g_1(X') = 0$, have no constant terms. For a new variable $x_0$, let $X'' = X' \\sqcup \\{x_0\\}$ and thus $\\abs{X''} = N+1$. Let $W=\\inbrace{w_1,\\ldots, w_{t}}$ be a new set of variables disjoint from $X''$. Let $\\gamma$ be an element in $R$ without a multiplicative inverse (recall that in \\Cref{thm:HN-hard} we assume that $R$ is not a field). We shall now define our polynomial $P_S$ in the polynomial ring $R[X'',W]$ as follows\n\\begin{equation}\\label{eqn:poly-defn}\n  P_S(X'', W) = \\underbrace{ w_1\\cdot g_1(X')}_{I} + \\underbrace{\\inparen{\\sum_{i =2}^t w_i\\cdot\\inparen{ \\gamma\\cdot g_i(X') + \\sum_{k=0}^Nx_k}}}_{II}\\,. \n\\end{equation}\nObserve that $\\deg(P_s)\\leq 3$.\n\n\\begin{remark}\\label{rem:poly-complexity}\n  The sparsity of the polynomial $P_S(X'',W)$, $\\sigma$, is equal to the sum of sparsities of polynomials in each of its summands, and it is equal to $(t-1)\\cdot (N+1)+ \\sum_{i=1}^t s'_i $ where $s'_i$ is the sparsity of the polynomial $g_i(X')$. On the other hand, $P_S(X'',W)$ can also be represented as a depth four arithmetic circuit, with at most $3t$ non-leaf nodes.\n\\end{remark}\n\nWe shall now show that it is sufficient to consider shifts with a certain structure for $P_S$. Further we shall show that a solution to the system $S$ of polynomial equations exists if and only if there exists a vector $\\mathbf{b}$ such that $P_S(X''+\\mathbf{b},W)$ has fewer monomials than $P_S(X'', W)$.\n\n\\begin{lemma}\n  \\label{lem:shift-structure}\n  Let $\\mathbf{a} = \\inbrace{a_1, \\ldots, a_N} \\in R^{N}$ be a solution to the system $T$ of polynomial equations. Let $\\mathbf{b} = \\inbrace{b_0, \\ldots, b_N}, \\mathbf{b}' =\\inbrace{b'_0, \\ldots, b'_N} \\in R^{N+1}$ and $\\mathbf{a}' =\\inbrace{a'_0, \\ldots, a'_N}$ be such that\n  \\begin{itemize}\n  \\item $b_i = b_i'$ for all $i\\in [N]$,\n  \\item $b'_0 \\neq b_0$ and $b_0 = - \\sum_{i\\in [N]} b_i$,\n  \\item $a_i' = a_i$ for all $i\\in [N]$, and\n  \\item $a'_0 = - \\sum_{i\\in [N]} a_i$.\n  \\end{itemize}\n  Let $\\mathbf{b}''$ and $\\mathbf{c}$ be any vectors in $R^{N+1}$ and $R^{t}$ respectively. Then,\n  \\begin{enumerate}\n  \\item The sparsity of $P_S(X''+\\mathbf{b}'', W+\\mathbf{c})$ is at least that of $P_S(X''+\\mathbf{b}'', W)$, \\label{item:1}\n  \\item The sparsity of $P_S(X''+\\mathbf{b}', W)$ is at least that of $P_S(X''+\\mathbf{b}, W)$.\n  \\label{item:2}\n  \\item The sparsity of $P_S(X'', W)$ is $1$ more than that of $P_S(X''+\\mathbf{a}',W)$.\\label{item:3}\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n  Given the structure of the polynomial $P_S(X'',W)$, proof of \\Cref{item:1} follows directly from the fact that the polynomial $P_S(X'',W)$ is linear in the $W$ variables and thus all terms of $P_S(X''+\\mathbf{b}'', W)$ also appear in $P_S(X''+\\mathbf{b}'', W+\\mathbf{c})$. Further, the difference  $P_S(X''+\\mathbf{b}'', W+\\mathbf{c}) - P_S(X''+\\mathbf{b}'', W)$ does not depend on any $W$ variable.\n\n  From their definition, the vectors $\\mathbf{b}$ and $\\mathbf{b}'$ are identical when projected down to their last $N$ coordinates and these exactly correspond to shifts of variables in $X'$. We shall use $\\mathbf{b}|_{X'}$ to denote this projection. In particular, for all $i\\in[t]$,\n  $g_i(X'+\\mathbf{b}|_{X'}) = g_i(X'+\\mathbf{b}'|_{X'})$.\n  It is easy to see that the polynomial $\\sum_{i=0}^N x_i$ is invariant under shift by $\\mathbf{b}$ (as $\\sum_{i=0}^N b_i = 0$) but not under shift by $\\mathbf{b}'$. Putting both of these facts together we can now say that $P_S(X''+\\mathbf{b}', W)$ contains all the terms that are contained in $P_S(X''+\\mathbf{b}, W)$, and it additionally contains a non-trivial linear polynomial in the $W$ variables. This proves \\Cref{item:2} of the lemma.\n\n  Towards proving \\Cref{item:3} of the lemma, we claim that under a shift by $\\mathbf{a}'$, as defined in the statement of the lemma, sparsity of part $II$ in \\cref{eqn:poly-defn} does not change, and sparsity of part $I$ definitely decreases. \n\n  All polynomial equations $\\{g_i(X') = 0\\mid 2 \\leq i\\leq t\\}$, are constant free and can either be quadratic binomial polynomial equations of the form $(x_{p} -  x_q\\cdot x_e) = 0$ (for some $p,q,e\\in [N]$) or homogeneous linear polynomial equations of the form $\\sum_{j=1}^k c_{i_j} x_{i_j} = 0$ (for some $i_1, \\ldots, i_k\\in [N]$ and scalars $c_{i_j}$). \n  \n\n  \\emph{When the equation is a quadratic binomial polynomial equation:} Since $\\mathbf{a} = \\mathbf{a}'|_{X'}$ solves $T$ we get that $a'_p- a'_q\\cdot a'_e = 0 $ and using this fact we can show that for each summand of this kind in part II, the sparsity does not change.\n  \\begin{align*}\n    &\\gamma\\cdot ((x_{p}+ a'_p) - (x_q+a'_q)\\cdot (x_e+a'_e)) + \\sum_{i=0}^N(x_{i}+a'_i)\\\\ \n    &= \\gamma\\cdot ((x_{p} -  x_q\\cdot x_e) -  (a'_q x_e +a'_e x_q)) + \\sum_{i=0}^N x_{i} \\qquad \\text{(Since $a'_p- a'_q\\cdot a'_e = 0 $ and $\\sum_{i=0}^N a'_i = 0$)}\\\\\n    &= \\gamma\\cdot(x_{p} -   x_q\\cdot x_e) + \\inparen{\\sum_{i\\in [N]\\setminus \\{q,e\\}}x_{i}} + (1-\\gamma\\cdot   a'_q)\\cdot x_e+ (1-\\gamma\\cdot    a'_e)\\cdot x_q.\n  \\end{align*}\n  Since $\\gamma$ has no multiplicative inverse, neither $(1-\\gamma\\cdot    a'_q)$ nor $(1-\\gamma\\cdot   a'_e)$ can be equal to $0$.\n  \n  \\emph{When the polynomial equation is a homogeneous linear polynomial equation:}  Since $\\mathbf{a} = \\mathbf{a}'|_{X'}$ solves $T$  we get that $\\sum_{j=1}^k c_{i_j}a'_{i_j} = 0$ and thus the sparsity remains invariant for such summands in part $II$.\n\n\nFinally consider the non-homogeneous linear polynomial equation $g_1(X')=0$. Without loss of generality, let $g_1(X') = c_1 x_1+\\ldots+ c_kx_k+c$. Note that $\\sum_{i=1}^k c_i a'_i+c =0$ as $\\mathbf{a} = \\mathbf{a}'|_{X'}$ solves $T$ and thus sparsity reduces by $1$ under shift by such a vector $\\mathbf{a}'$:\n$$\\sum_{i=1}^k c_i (x_i+a'_i)+c=\\sum_i c_i x_i+\\left(\\sum_{i=1}^k c_ia'_i+c\\right)=\\sum_{i=1}^k c_i x_i \\;.$$\n  \n\n  By putting together the analysis for all the summands we get that the polynomial $P_S(X''+\\mathbf{a}', W)$ has one monomial less than $P_S(X'',W)$. \n\\end{proof}\n\n\n\\begin{lemma}\n  \\label{lem:shift-implies-solution}\n  Let $\\mathbf{b} = (b_0, b_1, \\ldots, b_N) \\in R^{N+1}$ such that $b_0 = -\\sum_{i=1}^N b_i$, be a shift that sparsifies the polynomial $P_S(X'',W)$ by at least one monomial. Let $\\mathbf{a}\\in R^N$ be the projection of vector $\\mathbf{b}$ to its last $N$ coordinates. Then $\\mathbf{a}$ solves $T$.\n\\end{lemma}\n\\begin{proof}\n\n   The proof of \\cref{lem:shift-structure} shows that given the structure of the shift $\\mathbf{b}$, a reduction in sparsity can only come from $g_1(X')$. That is, the sparsity of polynomials $g_i(X')$ for $i\\geq 2$, can only increase upon a shift.\n  \n  For the sake of contradiction, let us assume that there exists a polynomial equation in $T$ that is not satisfied by $\\mathbf{a}$. \n  If for some $i\\geq 2$, $g_i(X') = 0$ is a polynomial equation that is not satisfied by $\\mathbf{a}$, then this contributes an increase of $1$ to sparsity of the polynomial $P_S(X'',W)$ upon the shift by $\\mathbf{b}$ (by adding a term of the form $c\\cdot w_i$). Else if $g_1(X') =0$ is not satisfied by $\\mathbf{a}$, then there is no contribution to reduction in sparsity from part $I$. This is due to the fact that the term of the form $c\\cdot w_1$ vanishes upon a shift by $\\mathbf{b}$ if and only if $\\mathbf{a}$ solves $g_1(X') =0$. In either of these cases, the sparsity of $P_s(X''+\\mathbf{b}, W)$ is not strictly less than that of $P_S(X'',W)$. This contradicts our assumption that $\\mathbf{b}$ sparsifies $P_S(X'',W)$ by at least one monomial.\n\\end{proof}\n\nBy putting together Lemmata \\ref{lem:sparserep-to-equations}, \\ref{lem:circuit-to-equations}, \\ref{lem:shift-structure}, and \\ref{lem:shift-implies-solution}, we get the following formal statement.\n\n\\begin{theorem}[$\\mathsf{HN}_R$ reduces to $\\mathsf{SparseShift}_R$]\\label{thm:reduction}\n  Given a system $S$ of polynomial equations over the polynomial ring $R[X]$, there exists a polynomial $P_S(X'', W)\\in R[X'', W]$ (where $X\\subseteq X''$) such that the system $S$ is solvable if and only if there exists a shift that sparsifies the polynomial $P_S$ by a monomial. Furthermore, the size of the polynomial instance $P_S$ is polynomially related to the input size of the system $S$ of polynomial equations. This holds true in both the sparse-representation and circuit-representation.\n\\end{theorem}\n\n\nWe thus get that if $\\mathsf{SparseShift}_R$ can be solved efficiently (in general) then $\\mathsf{HN}_R$ can also be solved efficiently. In other words, $\\mathsf{SparseShift}_R$ is at least as hard as $\\mathsf{HN}_R$. This completes the proof of \\cref{thm:HN-hard}. Putting \\cref{thm:HN-hard} together with the fact that $\\mathsf{HN}_\\mathbb{Z}$ is undecidable (due to \\cite{Mat70}). \n\n\n\\section{Undecidability of $\\beta$-gap-$\\mathsf{SparseShift}_\\mathbb{Z}$ problem}\\label{sec:approx-hard-Z}\n\nNote that $\\mathsf{SparseShift}_R$ can be rephrased as the following gap problem -- given a polynomial of sparsity $\\sigma$, decide if there is a shift that sparsifies the polynomial to at most $\\sigma-1$ monomials, or there is no shift that sparsifies the polynomial below $\\sigma$ monomials. We shall now show a reduction from this ($\\mathsf{SparseShift}_R$ problem) to $\\beta$-gap-$\\mathsf{SparseShift}_R$ for any $\\beta>1$.\n\nLet the sets $X''$ and $W$ be as defined in the construction of the polynomial $P_S$ in \\Cref{subsec:poly-defn}. Let $d$ be a parameter that we shall soon fix. Let $X^{(1)}, \\ldots, X^{(d)}$ and $W^{(1)}, \\ldots, W^{(d)}$ be $d$ many disjoint copies of variable sets $X''$ and $W$ respectively. Let $X_d = \\sqcup_{k=1}^d X^{(i)}$ and $W_d = \\sqcup_{k=1}^d W^{(i)}$. For the sake of brevity, let us use the following notation: Let $Y = X''\\sqcup W$. For all $k\\in[d]$, let $Y^{(k)} = X^{(k)}\\sqcup W^{(k)}$ and $\\abs{Y^{(k)}} = N'$. Let $Y_d = \\sqcup_{k=1}^d Y^{(i)}$, so that $|Y_d|=N'd$.\nLet the polynomial $Q_d(Y_d)$ be defined as follows.\n\\begin{align}\\label{eqn:amplification}\n    Q_d(Y_d) = \\prod_{k=1}^d P_S(Y^{(k)}).\n\\end{align}\nObserve that the sparsity of $Q_d(Y_d)$ is given by the product of sparsities of $d$ many instances of $P_S(X'',W)$. \n\n\\begin{lemma} \\label{lem:gap-amplification}\n  Let $P_S(X'',W)$ and $Q_d(Y_d)$ be the polynomials as defined above. Let $\\sigma$ be equal to the sparsity of the polynomial $P_S$. Then,\n  \\begin{enumerate}\n        \\item $\\deg(Q_d)\\leq 3d$.\n      \\item $Q_d(Y_d)$ has sparsity equal to $\\sigma^d$.\n      \\item If $P_S$ has a depth four circuit of size $s\\leq 3t+N'$ (recall Remark~\\ref{rem:poly-complexity}) then $Q_d$ has a depth five circuit of size $sd+1$.\n      \\item There is a vector $\\mathbf{a}\\in R^{N'}$ such that $P_S(Y+\\mathbf{a})$ has at most $\\sigma-1$ monomials if and only if there exists a vector $\\mathbf{a}_d \\in R^{N'\\cdot d}$ such that $Q_d(Y_d+\\mathbf{a}_d)$ has at most $(\\sigma-1)^d$ monomials.\n      \\item For all vectors $\\mathbf{a}\\in R^{N'}$, $P_S(Y+\\mathbf{a})$ has at least $\\sigma$ monomials if and only if for all vectors $\\mathbf{a}_d \\in R^{N'\\cdot d}$  $Q_d(Y_d+\\mathbf{a}_d)$ has at least $\\sigma^d$ monomials.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  The claim regarding the degree of $Q_d$ follows immediately from the fact that $\\deg(P_S)\\leq 3$.\n  Given that $P_S(X'',W)$ has a sparsity of $\\sigma$ and since $Q_d(Y_d)$ is defined to be a product of $d$ distinct copies of $P_S(X'',W)$, sparsity of $Q_d(Y_d)$ is equal to $\\sigma^d$. Similarly, if $P_S$ can be computed by  a circuit of size $s$, then there is a depth five circuit of size $(sd+1)$ that computes the polynomial $Q_d(Y_d)$ -- its output node is a product node into which $d$ copies of circuits of $P_S(X'',W)$ feed into.\n  \n  If there is a vector $\\mathbf{a}\\in R^{N'}$ such that $P_S(Y+\\mathbf{a})$ has at most $\\sigma-1$ monomials then by taking $\\mathbf{a}_d$ to be the concatenation of $\\mathbf{a}$, $d$ many times, we get that $Q_d(Y_d+\\mathbf{a}_d)$ has at most $(\\sigma-1)^d$ monomials. If there is $\\mathbf{a}_d \\in R^{N'\\cdot d}$ such that $Q_d(Y_d+\\mathbf{a}_d)$ has at most $(\\sigma-1)^d$ monomials then it cannot happen that there is no $\\mathbf{a}\\in R^{N'}$ such that $P_S(Y+\\mathbf{a})$ has at most $\\sigma-1$ monomials. \n  \n  If for all vectors $\\mathbf{a}\\in R^{N'}$, $P_S(Y+\\mathbf{a})$ has at least $\\sigma$ monomials, then $Q_d(Y_d+\\mathbf{a}_d)$ must have at least $\\sigma^d$ monomials for all $\\mathbf{a}_d\\in R^{N'\\cdot d}$.\n  On the other hand if for all vectors $\\mathbf{a}_d \\in R^{N'\\cdot d}$, $Q_d(Y_d+\\mathbf{a}_d)$ has at least $\\sigma^d$ monomials,  (for the sake of contradiction) let us suppose that there is a vector $\\mathbf{a}'\\in R^{N'}$ such that $P_S(Y+\\mathbf{a}')$ has at most $\\sigma-1$ monomials. As before (due to the product structure of $Q_d$) we get that there is a corresponding vector $\\mathbf{a}'_d\\in R^{N'\\cdot d}$ such that $Q_d(Y_d+\\mathbf{a}'_d)$ has at most $ (\\sigma-1)^d$ monomials. This contradicts our assumption. Thus, for all $\\mathbf{a}\\in R^{N'}$, $P_S(Y+\\mathbf{a})$ has at least $\\sigma$ monomials.\n\\end{proof}\n\n \n\\begin{theorem}\\label{thm:HN-to-gapZ}\n    Let $R$ be an integral domain but not a field.\n  Given a system $S$ of polynomial equations over the polynomial ring in $n$ variables $R[X]$, for any function $\\beta:\\mathbb{N}\\to\\mathbb{R}_+$ such that $\\beta\\in o(1)$, there exist $d = d(S, \\beta)\\in \\mathbb{Z}_{>0}$ and a polynomial $Q_d(Y_d)$, in $N'd$ variables, of degree at most $3d$, such that the system $S$ is solvable if and only if the $M^\\beta$-gap-$\\mathsf{SparseShift}_R$ problem for $Q_d(Y_d)$ is solvable, where $M$ is the representation length of $Q_d(Y_d)$ in the sparse representation.\\footnote{Recall that in sparse representation polynomials are given as a set of pairs consisting of exponent vectors together with the coefficient of the corresponding monomial. Thus, $M\\approx (\\#\\text{ monomials in } Q_d) \\times ((N'd)\\times O(\\log d)) \\times O(d \\cdot b)$, where $b$ is the maximal bit complexity of a coefficient in $P_S(X'', W)$.} \n\\end{theorem}\n\n\\begin{proof}\n  Given a system $S$ of polynomial equations, we can construct the polynomial $P_S(X'',W)$ (as defined in \\cref{eqn:poly-defn}). Let $\\sigma$ be the sparsity of $P_S(X'', W)$. Recall from \\cref{thm:reduction} that system $S$ has a solution if and only if there exists a shift that sparsifies the polynomial $P_S(X'',W)$ by a monomial. \n  \n    Recall that $Q_d(Y_d)$ has $M'=\\sigma^d$ many monomials. Let $\\alpha = \\frac{\\sigma}{\\sigma-1}$. \n    By putting together \\cref{thm:reduction} and \\cref{lem:gap-amplification}, we get that the system $S$ of polynomial equations is solvable if and only if the $\\alpha^d$-gap problem for $Q_d(Y_d)$ is solvable. Calculating we get that $\\alpha^d= \\left(\\frac{\\sigma}{\\sigma-1}\\right)^d \\approx e^{d\/(\\sigma-1)}=M'^{\\frac{1}{(\\sigma-1)\\log \\sigma}}$. Picking $d$ large enough so that $M'\\geq\\sqrt{M}$ and  $\\beta(M)<\\frac{1}{2(\\sigma-1)\\log \\sigma}$ the claim follows.\n  \\end{proof}  \n\nPutting \\cref{thm:HN-to-gapZ} together with the fact that $\\mathsf{HN}_\\mathbb{Z}$ is undecidable (due to \\cite{Mat70}) we get \\cref{thm:gap-hardnessZ}. \n\n\\section{Hardness of $\\beta$-gap-$\\mathsf{SparseShift}_R$ problem for $R=\\mathbb{F}_p, \\mathbb{Q}, \\mathbb{R}$, or $\\mathbb{Z}_q$}\\label{sec:approx-hard-K}\n\nIn this section we prove Theorems~\\ref{thm:gap-sparse-input} and~\\ref{thm:gap-circuit-input} by giving a reduction from  $\\text{Max-3Lin}_R$  to the $\\alpha$-gap-$\\mathsf{SparseShift}_R$ problem, for different domains $R=\\mathbb{F}_p, \\mathbb{Q}, \\mathbb{R}$ or $\\mathbb{Z}_q$. Observe that we now do not require that our ring $R$ is not a field. \n\nLet $X = \\{x_1, \\ldots, x_n\\}$. Let the given system $S$ of linear equations be $\\{L_1(X)=0, \\ldots, L_m(X)=0\\}$, where $L_i(X)\\in R[X]$, and each equation $L_i(X)=0$ depends on exactly $3$ variables. Let the given system of equations be expressed together as $A\\cdot X + \\mathbf{b} = \\mathbf{0}$ such that for all $i\\in[m]$, $A_i\\cdot X + b_i = 0$ is the $i$'th linear equation $L_i(X)=0$, where $A_i$ is the $i$'th row of the matrix $A$. Note that there are exactly three non-zero entries in each row of $A$. Let $w = \\max\\{2n, 2m\\}$. Let $C$ be a $w\\times w$ matrix such that\n\\begin{align*}\n  \\text{for all $1\\leq i,j \\leq w$}, \\quad C_{i,j} =\n  \\begin{cases}\n    A_{i,j-w+n} & \\text{if $i \\leq m$ and $j\\geq w-n+1$;}\\\\\n    0 & \\text{otherwise.}\n  \\end{cases}\n\\end{align*}\nIn other words, $A$ is the top right block of $C$ and the rest of $C$ is zeros. \nLet $\\mathbf{e} = (e_1, \\ldots, e_w)\\in R^{w}$ be such that $e_i = b_i$ for all $1\\leq i\\leq m$ and $e_i=0$ otherwise. Let $e_0$ be some constant. Let $Y=\\{y_1, \\ldots, y_w\\}$ be a new set of variables disjoint from $X$. Let the polynomial $Q_S(Y)\\in R[Y]$ be defined as follows.\n\\begin{align*}\n  Q_S(Y) = \\sum_{i,j\\in[w]}C_{i,j}\\cdot y_iy_j + \\sum_{i\\in [w]}e_i \\cdot y_i + e_0\\,.\n\\end{align*}\nNote that there are at most $3m$ many non-zero entries in $C$, and there are at most $m$ many non-constant linear terms. Thus the sparsity of this polynomial is at most $4m+1$. \n\nFor some vector $\\mathbf{a} = (a_1, \\ldots, a_w) \\in R^{w}$ let us  examine the structure of the polynomial $Q_S(Y+\\mathbf{a})$.\n\n\\begin{align*}\n  &Q_S(Y+\\mathbf{a})\n  = \\sum_{i,j\\in[w]}C_{i,j}\\cdot(y_i+a_i)(y_j+a_j) + \\sum_{i\\in [w]}e_i\\cdot (y_i+a_i) + e_0\\\\\n  &= \\sum_{i,j\\in[w]}C_{i,j}\\cdot(y_iy_j +a_iy_j+a_jy_i+a_ia_j) + \\sum_{i\\in [w]}e_i\\cdot (y_i+a_i) + e_0\\\\\n  &= \\sum_{i,j\\in[w]}C_{i,j}\\cdot(y_iy_j+a_ia_j) +\\sum_{i,j\\in[w]}a_i\\cdot y_j\\cdot C_{i,j}+ \\sum_{i,j\\in[w]}a_j\\cdot y_i\\cdot C_{i,j} + \\sum_{i\\in [w]}e_i\\cdot (y_i+a_i) + e_0\\\\\n  &= \\sum_{i,j\\in[w]}C_{i,j}\\cdot y_iy_j + \\sum_{i\\in [w]}y_i\\cdot(e_i + \\sum_{j\\in[w]}a_{j}\\cdot(C_{i,j}+C_{j,i})) + \\sum_{i,j\\in[w]}C_{i,j}\\cdot a_ia_j + \\sum_{i\\in[w]}a_i\\cdot e_i + e_0\\,.\n\\end{align*}\n\nObserve that the quadratic part of the polynomial $Q_S(Y)$ remains unperturbed under the shift but the affine linear part of it could get perturbed. We shall now show that every non-zero coefficient in the linear part corresponds to a linear equation in $S$.\n\\begin{lemma}\\label{lem:equivalence-approx}\n  Let $\\mathbf{a} = (a_1, \\ldots, a_w)\\in R^w$. Let $\\mathbf{a}' \\in R^n$ be the projection of $\\mathbf{a}$ down to its last $n$ elements, that is, for all $j\\in [n]$, $a'_j = a_{j+w-n}$. Then, for all $i\\in [m]$, the coefficient of $y_i$ in $Q_S(Y+\\mathbf{a})$ is zero if and only if $\\mathbf{a}'$ satisfies the $i$'th linear equation $L_i(X)=0$. Moreover, for all $i>m$, the coefficient of $y_i$ is zero.\n\\end{lemma}\n\\begin{proof}\n  For all $i\\in [m]$, the coefficient of $y_i$ in the polynomial $Q_S(Y+\\mathbf{a})$ is equal to $$e_i + \\sum_{j\\in[w]}a_{j}(C_{i,j}+C_{j,i})\\,.$$ Note that for this regime of $i\\in[m]$, $e_i = b_i$, $C_{j,i} = 0$, and $C_{i,j} = A_{i,j-w+n}$ for $j$ in $[w-n+1, w]$ and zero otherwise. Thus, the coefficient of such a $y_i$ reduces as follows.\n  \\begin{align*}\n   \\text{coef. of}~y_i~\\text{in}~Q_S(Y+\\mathbf{a}) =  e_i + \\sum_{j\\in[w]}a_{j}(C_{i,j}+C_{j,i}) = b_i + \\sum_{j'=1}^na_{w-n+j'}A_{i,j'} = b_i + \\sum_{j'=1}^na'_{j'}A_{i,j'}.\n  \\end{align*}\n  This is exactly the value obtained by evaluating the $i$'th linear polynomial $L_i$ at $\\mathbf{a}'$. Thus, we get that coefficient of $y_i$ (for $i\\in[m]$) is zero if and only if $\\mathbf{a}'$ satisfies the $i$'th linear equation, i.e.,\\ $L_i(\\mathbf{a}')=0$.\n\n  For all $i>m$, $e_i=0$ and $C_{i,j} =0$. Further, $\\mathbf{a}_j = 0$ for all $j\\leq w-n$. Hence,\n  \\begin{align*}\n    e_i + \\sum_{j\\in[w]}a_{j}(C_{i,j}+C_{j,i}) = \\sum_{j\\in[w]}a_{j}\\cdot C_{j,i} = \\sum_{j=w-n+1}^w a_{j}\\cdot C_{j,i} =0\\,.\n  \\end{align*}\n  The last equality in the math block above is due to the fact that the entries $C_{j,i}$ are equal to zero for $j\\geq w-n+1$ and $i\\geq m+1$ regardless of whether $n\\geq m$ or $m\\geq n$, from the construction of the matrix $C$. Thus the coefficients of the terms $y_i$ for all $i>m$ are zero.\n\\end{proof}\n\nUsing this correspondence, we can show the following reduction.\n\\begin{lemma}\\label{lem:gap-reduction}\n  Let $\\mathbf{a} = (a_1, \\ldots, a_w)\\in R^w$. Let $\\mathbf{a}' \\in R^n$ be the projection of $\\mathbf{a}$ down to its last $n$ elements, that is, for all $j\\in [n]$, $a'_j = a_{j+w-n}$. Then\n  \\begin{enumerate}\n  \\item $\\mathbf{a}'$ satisfies at most $\\delta$ fraction of equations in $S$ if and only if $Q_S(Y+\\mathbf{a})$ has at least $(4-\\delta)m$ non-constant monomials, and\n  \\item $\\mathbf{a}'$ satisfies at least $1-\\varepsilon$ fraction of equations in $S$ if and only if $Q_S(Y+\\mathbf{a})$ has at most $(3+\\varepsilon)m+1$ monomials.\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  From the aforementioned discussion, the sparsity of the polynomial $Q_S(Y+\\mathbf{a})$ is decided by the coefficients of the linear terms. Further, \\cref{lem:equivalence-approx} characterizes that the coefficient of a linear term is zero if and only if the \\emph{corresponding} linear polynomial equation is \\emph{satisfied}. Thus at most $\\delta$ fraction of equations in $S$ are satisfied if and only if at most $\\delta$ fraction of coefficients of linear terms are equal to zero. In other words, if at least $(1-\\delta)$ fraction of coefficients of linear terms are non-zero. The constant term $e_0$ could get cancelled out, that is, $\\sum_{i,j\\in[w]}C_{i,j}\\cdot a_ia_j + \\sum_{i\\in[w]}a_i\\cdot e_i+e_0$ could be zero.  \n  Thus, $\\mathbf{a}'$ satisfies at most $\\delta$ fraction of equations in $S$ if and only if $Q_S(Y+\\mathbf{a})$ has at least $3m+(1-\\delta)m = (4-\\delta)m$ non-constant monomials.\n  \n  Similarly at least $1-\\varepsilon$ fraction of equations in $S$ are satisfied if and only if at least $1-\\varepsilon$ fraction of coefficients of linear terms are equal to zero. In other words, if at most $\\varepsilon$ fraction of coefficients of linear terms are non-zero. Thus, $\\mathbf{a}'$ satisfies at least $1-\\varepsilon$ fraction of equations in $S$ if and only if $Q_S(Y+\\mathbf{a})$ has at most $(3+\\varepsilon)m+1$ monomials.\n\\end{proof}\n\nWe first prove a more restricted version of Theorems~\\ref{thm:gap-sparse-input} and~\\ref{thm:gap-circuit-input} that shows hardness of $\\alpha$-approximate $\\mathsf{SparseShift}_R$ for some small $\\alpha$. Then we shall amplify this hardness for any $\\beta >1$.\n\\begin{theorem}\\label{thm:small-gap}\n  Let $R = \\mathbb{F}_p, \n  \\mathbb{R}, \\mathbb{Q}$ or $\\mathbb{Z}_q$. For all $\\varepsilon, \\delta >0$ as given in the last column of  \\cref{table:max3lin-hardness}, there exists an $\\alpha = \\alpha(\\varepsilon, \\delta, R)$ such that it is $\\mathsf{NP}$-hard to $\\alpha$-approximate $\\mathsf{SparseShift}_R$, in either the sparse, dense or arithmetic circuit representation.\n\\end{theorem}\n\\begin{proof}\n  \n  We first note that as the input is a degree $2$ polynomial, all three representations are polynomially equivalent.\n  \n  Suppose for a regime of values of $\\varepsilon', \\delta' >0$ we are guaranteed the following. Given an instance of $\\text{Max-3Lin}_R$, it is $\\mathsf{NP}$-hard to distinguish the following cases -- if there is a assignment that satisfies at least $(1-\\varepsilon')$ fraction of linear equations or for all assignments at most $\\delta'$ fraction of linear equations are satisfied. Putting this together with \\cref{lem:gap-reduction}, we get that it is $\\mathsf{NP}$-hard to distinguish if there is a vector $\\mathbf{a}$ such that the polynomial $Q_S(Y+\\mathbf{a})$ has at most $t = 3m+\\varepsilon' m+1 $ monomials or if for all $\\mathbf{a}$, the polynomial $Q_S(Y+\\mathbf{a})$ has at least $\\alpha t = (4-\\delta')m$ non-constant monomials. Thus, we get that it is $\\mathsf{NP}$-hard to $\\alpha$-approximate $\\mathsf{SparseShift}_R$ where $\\alpha = \\alpha(\\varepsilon', \\delta')$ is obtained as follows.\n  \\begin{align*}\n    \\alpha = \\frac{4-\\delta'}{3+\\varepsilon' +\\frac{1}{m}} = \\frac{4}{3} - \\frac{4\\varepsilon' +3\\delta'+o(1)}{9+3\\varepsilon'+o(1)}\\,.\n  \\end{align*}\n  Each row of \\cref{table:max3lin-hardness} gives us a guarantee of the form that we assumed at the beginning of this proof. Thus by iterating through the rows of \\cref{table:max3lin-hardness}, we get our parameter $\\alpha = \\alpha(\\varepsilon, \\delta, R)$ for various settings of $R$.  This completes the proof.\n\\end{proof}\n\nLet $d$ be a parameter that we shall soon fix. Let $Y^{(1)}, \\ldots, Y^{(d)}$ be $d$ many disjoint copies of the variable set $Y = \\{y_1, \\ldots, y_w\\}$. Let $Y_d = \\sqcup_{k=1}^d Y^{(i)}$. Let the polynomial $F_{n,d}(Y_d)$ be defined as follows.\n\\begin{align}\\label{eqn:amplification-lin}\n    F_{n,d}(Y_d) = \\prod_{k=1}^d Q_S(Y^{(k)}).\n\\end{align}\nObserve that the sparsity of $F_{n,d}(Y_d)$ is given by the product of sparsities of $d$ many instances of $Q_S(Y)$. Further, if the polynomial $Q_S(Y)$ is computed by a circuit of size $s$ then the polynomial $F_{n,d}(Y_d)$ has a circuit of size at most $sd+1$.\n\n\\begin{lemma} \\label{lem:gap-amplification-large}\n  Let $S$ be a system of linear equations, and $F_{n,d}(Y_d)$ be the polynomial as defined above. \n  \\begin{enumerate}\n  \\item All vectors $\\mathbf{a}\\in R^n$ satisfy at most $\\delta$ fraction of equations in $S$ if and only if  all vectors $\\mathbf{b}_d\\in R^{wd}$ are such that $F_{n,d}(Y_d+\\mathbf{b}_d)$ has at least $((4-\\delta)m)^d$ non-constant monomials.\\label{item:4-d}\n  \\item There exists a vector $\\mathbf{a}\\in R^n$ such that it satisfies at least $1-\\varepsilon$ fraction of equations in $S$ if and only if there exists a vector $\\mathbf{b}_d\\in R^{wd}$ such that $F_{n,d}(Y_d+\\mathbf{b}_d)$ has at most $((3+\\varepsilon)m+1)^d$ monomials.\\label{item:3+e}\n  \\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFrom the product structure of $F_{n,d}(Y_d)$, we get that for  all vectors $\\mathbf{b}_d\\in R^{wd}$, the polynomial $F_{n,d}(Y_d+\\mathbf{b}_d)$ has at least $((4-\\delta)m)^d$ non-constant monomials if and only if for all vectors $\\mathbf{a}\\in R^n$, the polynomial $Q_S(Y+\\mathbf{a})$ has at least $(4-\\delta)m$ non-constant monomials. For the sake of contradiction, let us suppose that there is a vector $\\mathbf{b}'_d\\in R^{wd}$ such that $F_{n,d}(Y_d+\\mathbf{b}'_d)$ has at most $((4-\\delta)m)^d-1$ monomials. Because of the product structure of $F_{n,d}(Y_d)$, it must the case that there is a copy of $Q_S(Y)$, say $Q_S(Y^{(i)})$ such that $Q_S(Y^{(i)}+\\mathbf{b}'_d|_{Y^{(i)}})$ has at most $(4-\\delta)m-1$ non-constant monomials which contradicts our assumption. The other direction also follows trivially from the product structure. From \\cref{lem:gap-reduction}, we get that $\\mathbf{a}'$ satisfies at most $\\delta$ fraction of equations in $S$ if and only if $Q_S(Y+\\mathbf{a})$ has at least $(4-\\delta)m$ non-constant monomials. By putting both of these together, we get \\cref{item:4-d}.\n\nBy invoking \\cref{lem:gap-reduction} again, we get that  there is a vector $\\mathbf{a}\\in R^n$ such that it satisfies at least $1-\\varepsilon$ fraction of equations in $S$ if and only if there is a vector $\\mathbf{b}$ such that $Q_S(Y+\\mathbf{b})$ has at most $(3+\\varepsilon)m+1$  monomials. By taking $\\mathbf{b}_d$ to be the concatenation of $\\mathbf{b}$, $d$ many times, we get that $F_{n,d}(Y_d+\\mathbf{b}_d)$ has at most $((3+\\varepsilon)m+1)^d$ monomials. On the other hand, if there exists a vector $\\mathbf{b}_d\\in R^{wd}$ such that $F_{n,d}(Y_d+\\mathbf{b}_d)$ has at most $((3+\\varepsilon)m+1)^d$ monomials then because of the product structure of $F_{n,d}(Y_d)$, $Q_S(Y+\\mathbf{a})$ has at most $(3+\\varepsilon)m+1$  monomials where $\\mathbf{a} = \\mathbf{b}_d|_{Y^{(1)}}$. This completes the proof of \\cref{item:3+e}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorems~\\ref{thm:gap-sparse-input} and~\\ref{thm:gap-circuit-input}]\nGiven any $\\beta>1$ and $\\alpha$ as given by \\cref{thm:small-gap}, let $d = \\log_{\\alpha}{\\beta}$. Thus, $\\beta = \\alpha^d$. Let $F_{n,d}(Y_d)$ be the polynomial as defined in \\cref{eqn:amplification-lin}. From \\cref{lem:gap-amplification-large}, we get that $\\alpha$-gap-$\\mathsf{SparseShift}_R$ gap reduces to $\\alpha^d$-gap-$\\mathsf{SparseShift}_R$.\n  \n  To prove Theorem~\\ref{thm:gap-sparse-input} we note that if $Q_S(Y)$ is provided in sparse representation (recall that sparsity of $Q_S(Y)$ , denoted by $t$, is at most $4m+1$) and if $\\alpha^d$-gap-$\\mathsf{SparseShift}_R$ problem for $F_{n,d}(Y_d)$ can be solved efficiently in time $N^{O(1)}$ (where $N=t^d$ is the sparsity of the polynomial $F_{n,d}(Y_d)$) then $\\alpha$-gap-$\\mathsf{SparseShift}_R$ $Q_S(Y)$ can be solved in time $t^{O(d)}$. Thus, as long as $d=O(1)$ the gap reduction runs in polynomial time.\n  \n  Similarly, to prove  Theorem~\\ref{thm:gap-circuit-input} we note that if $Q_S(Y)$ is provided as a circuit of size $s$ and if $\\alpha^d$-gap-$\\mathsf{SparseShift}_R$ problem for $F_{n,d}(Y_d)$ can be solved efficiently in time $N^{O(1)}$ (where $N=sd+1$ is the input size of the instance provided as a circuit) then $\\alpha$-gap-$\\mathsf{SparseShift}_R$ problem $Q_S(Y)$ can be solved in time $\\inparen{sd}^{O(1)}$. As long as $d$ is at most a polynomial in $s$, the gap reduction runs in polynomial time.\n\\end{proof}\n\n\\bibliographystyle{alphaurl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent The most common version of Hilbert's fifth problem asks whether every locally euclidean topological group is a Lie group.  More precisely, if $G$ is a locally euclidean topological group, does there always exist a $C^\\omega$ structure on $G$ such that the group operations become $C^\\omega$?  In 1952, Gleason, Montgomery, and Zippin answered this question in the affirmative; see ~\\cite{G} and \\cite{MZ1}.  (``$C^\\omega$\" means ``real analytic,\" and by a $C^\\omega$ structure on a topological space $X$ we mean a real analytic manifold, everywhere of the same finite dimension, whose underlying topological space is $X$.)  \n\n\\\n\nHowever, Lie groups, as first conceived by Sophus Lie, were not actually groups but rather ``local groups\" where one could only multiply elements that were sufficiently near the identity.  (A precise definition of ``local group\" is given in the next section.)   It is thus natural to pose a local version of Hilbert's fifth problem (henceforth referred to as the Local H5):  Is every locally euclidean local group locally isomorphic to a Lie group?  In ~\\cite{J}, Jacoby claims to have solved the Local H5 affirmatively.  However, as pointed out by Plaut in ~\\cite{P}, Jacoby fails to recognize a subtlety in the way the associative law holds in local groups, as we will now explain.\n\n\\\n\nIn our local groups, if the products $xy, yz, x(yz),$ and $(xy)z$ are all defined, then $x(yz)=(xy)z$.  This condition is called ``local associativity\" by Olver in \\cite{O}.  A much stronger condition for a local group to satisfy is ``global associativity\" in which, given any finite sequence of elements from the local group and two different ways of introducing parentheses in the sequence, if both products thus formed exist, then these two products are in fact equal.  According to \\cite{O}, ``Jacoby's Theorem 8, which is stated without proof, essentially makes the false claim that local associativity implies global associativity.\"   Contrary to this ``Theorem 8\", ~\\cite{O} has examples of connected local Lie groups that are not globally associative.  Since, according to \\cite{O}, ``the rest of Jacoby's paper relies heavily on his incorrect Theorem 8\", this error invalidates his proof.  Olver also indicates that one could probably rework Jacoby's paper to extract the truth of the Local H5.  He suggests that this would be a worthwhile endeavor, as, for example, the solution of Hilbert's fifth problem for cancellative semigroups on manifolds heavily relies on the truth of the Local H5; see ~\\cite{B} and ~\\cite{Ho}.\n\n\\\n\nRather than reworking Jacoby's paper, we have decided to mimic the nonstandard treatment of Hilbert's fifth problem given by Hirschfeld \\cite{H}.  Whereas nonstandard methods simplified the solution of Hilbert's fifth problem, such methods are even more natural in dealing with the Local H5.  The nonstandard proof of Hilbert's fifth problem involves ``infinitesimals\"--elements in the nonstandard extension of the group infinitely close to the identity; therefore, much of \\cite{H} goes through in the local setting, since the infinitesimals form an actual group.  These infinitesimals are used to generate local one-parameter subgroups, as in \\cite{S} and \\cite{H}.\n\n\\\n\nWhile our solution of the Local H5 is along the lines of the proof given by Hirschfeld, many of the arguments have grown substantially in length due to the care needed in working with local groups.  We assume familiarity with elementary nonstandard analysis; otherwise, consult ~\\cite{D} or ~\\cite{He}.  We should also mention that McGaffey \\cite{ Mc} has a completely different\nnonstandard approach to Hilbert's fifth problem for local groups, using the connection\nbetween local Lie groups and finite-dimensional real Lie algebras to reduce to a\nproblem of approximating locally euclidean local groups in a\ncertain metric by local Lie groups.  This approach has not yet led to a solution.\n\n\\\n\nI would like to thank Lou van den Dries for many helpful suggestions. \n\n\\\n\n\\section{Preliminaries}\n\n\\\n\n\\noindent  In this section, we define the notion of local group and state precisely the Local H5 (in two forms).  We also introduce the fundamental objects in the entire story, the local 1-parameter subgroups of $G$.  After defining the local versions of familiar constructions in group theory, e.g. morphisms between local groups, sublocal groups, and local quotient groups, we prove some elementary properties of local groups that are used heavily throughout the paper.  We end this section by fixing some notation concerning the nonstandard setting.\n\n\\\n\n\\noindent Throughout, $m$ and $n$ range over $\\mathbb{N}:=\\{0,1,2,\\ldots\\}$.\n\n\\newpage\n\n\\noindent \\textbf{Statement of the Local H5}\n\n\\\n\n\\begin{df} A \\textbf{local group} is a tuple $(G,1,\\iota,p)$ where $G$ is a hausdorff topological space with a distinguished element $1\\in G$,  and $\\iota:\\t \\rightarrow G$ (the inversion map) and $p:\\o\\rightarrow G$ (the product map) are continuous functions with open $\\t \\subseteq G$ and open $\\o\\subseteq G\\times G$, such that $1\\in \\t$, $\\{1\\}\\times G \\subseteq \\o$, $G \\times \\{1\\} \\subseteq \\o$, and for all $x,y,z\\in G$:\n\n\\begin{enumerate}\n\\item $p(1,x)=p(x,1)=x$;\n\\item if $x\\in \\t$, then $(x,\\iota(x))\\in \\o$, $(\\iota(x),x)\\in \\o$ and $$p(x,\\iota(x))=p(\\iota(x),x)=1;$$\n\\item if $(x,y),(y,z)\\in \\o$ and $(p(x,y),z),(x,p(y,z))\\in \\o$, then \n\\begin{equation}\np(p(x,y),z)=p(x,p(y,z)).\\tag{A}\n\\end{equation}\n\\end{enumerate}\n\\end{df}\n\n\\\n\n\\noindent  In the rest of this paper $(G,1,\\iota,p)$ is a local group; for simplicity we denote it just by $G$. For the sake of readability, we write $x^{-1}$ instead of $\\iota(x)$, and $xy$ or $x\\cdot y$ instead of $p(x,y)$.  Note that $\\iota(1)=1$ and if $x,y\\in \\t$, $(x,y)\\in \\o$, and $xy=1$, then $y=x^{-1}$ and $x=y^{-1}$.\n\n\\\n\n\\begin{df}\n\n\\\n\n\\begin{enumerate}\n\\item Let $U$ be an open neighborhood of $1$ in $G$.  Then the \\textbf{restriction of $G$ to $U$} is the local group $G|U:=(U,1,\\iota|\\t_U,p|\\o_U)$, where $$\\t_U:=\\t \\cap U \\cap \\iota^{-1}(U) \\text{ and }\\o_U:=\\o \\cap (U\\times U)\\cap p^{-1}(U).$$\n\\item $G$ is \\textbf{locally euclidean} if there is an open neighborhood of $1$ homeomorphic to an open subset of $\\r^n$ for some $n$.\n\\item $G$ is a \\textbf{local Lie group} if $G$ admits a $C^\\omega$ structure such that the maps $\\iota$ and $p$ are $C^\\omega$.\n\\item $G$ is \\textbf{globalizable} if there is a topological group $H$ and an open neighborhood $U$ of $1_H$ in $H$ such that $G=H|U$.\n\\end{enumerate}\n\\end{df}\n\n\\\n\n\\noindent By a \\textit{restriction of $G$} we mean a local group $G|U$ where $U$ is an open neighborhood of $1$ in $G$.\n\n\\\n\n\\noindent \\textbf{Local H5-First Form}:  If $G$ is a locally euclidean local group, then some restriction of $G$ is a local Lie group.\n\n\\\n\n\\noindent \\textbf{Local H5-Second Form}:  If $G$ is a locally euclidean local group, then some restriction of $G$ is globalizable.\n\n\\\n\n\\noindent The equivalence of the two forms of the Local H5 is seen as follows.  It is known that if $G$ is a local Lie group, then some restriction of $G$ is equal to some restriction of a Lie group.  Thus the first form implies the second form.  Conversely, by the Montgomery-Zippin-Gleason solution to the original H5, if $G|U$ is globalizable, the global group containing $G|U$ will be locally euclidean and there will be a $C^\\omega$ structure on it making it a Lie group.  Then $G|U$ will be a local Lie group.\n\n\\\n\n\\begin{df}\nA \\textbf{subgroup} of $G$ is a subset $H\\subseteq G$ such that $1\\in H$, $H\\subseteq \\t$, $H\\times H\\subseteq \\o$ and for all $x,y\\in H$, $x^{-1}\\in H$ and $xy\\in H$.\n\\end{df}\n\n\\\n\n\\noindent Note that then $H$ is an actual group with product map $p|H\\times H$.  We will say $G$ has \\textbf{no small subgroups}, abbreviated $G$ is $\\operatorname{NSS}$, if there is a neighborhood $\\mathcal{U}$ of $1$ such that there are no subgroups $H$ of $G$ with $H\\not= \\{1\\}$ and $H\\subseteq \\mathcal{U}.$  In analogy with the solution to Hilbert's fifth problem in the global case, we will first show that every locally compact NSS local group has a restriction which is a local Lie group and then we will prove that every locally euclidean local group is NSS.\n\n\\bigskip\n\n\\noindent \\textbf{The Sets $\\u_n$}\n\n\\\n\n\\noindent As noted in the introduction, one obstacle in the local theory is the potential lack of generalized associativity.  The next definition captures the informal idea of being able to unambiguously multiply a finite sequence of elements.  \n\n\\\n\n\\begin{df}\nLet $a_1,\\ldots,a_n,b\\in G$ with $n\\geq 1$.  We define the notion $(a_1,\\ldots,a_n)$ \\textbf{represents} $b$, denoted $(a_1,\\ldots,a_n)\\rightarrow b$, by induction on $n$ as follows:\n\n\\begin{itemize}\n\\item $(a_1)\\rightarrow b$ iff $a_1=b$;\n\\item $(a_1,\\ldots,a_{n+1})\\rightarrow b$ iff for every $i\\in \\{1,\\ldots,n\\}$, there exists $b_i',b_i''\\in G$ such that $(a_1,\\ldots,a_i)\\rightarrow b_i'$, $(a_{i+1},\\ldots,a_{n+1})\\rightarrow b_i''$, $(b_i',b_i'')\\in \\o$ and $b_i'\\cdot b_i''=b$.\n\\end{itemize}\nBy convention, we say that $(a_1\\ldots a_n)$ represents $1$ when $n=0$.\n\\end{df}\n\n\\noindent We will say $a_1\\cdots a_n$ is \\textbf{defined} if there is $b\\in G$ such that $(a_1,\\ldots,a_n)$ represents $b$; in this case we will write $a_1\\cdots a_n$ for this (necessarily unique) $b$.  If $a_1\\cdots a_n$ is defined, then, informally speaking, all possible $n$-fold products are defined and equal.  If $a_1\\cdots a_n$ is defined and $a_i=a$ for all $i\\in \\{1,\\ldots,n\\}$, we say that $a^n$ is defined and denote the unique product by $a^n$.  In particular, $a^0$ is always defined and equals $1$.  \n\n\\\n\n\\noindent From now on, put $A^{\\times n}:=\\underbrace{A\\times \\cdots \\times A}_{n \\text{ times}}$.  Also, call $W\\subseteq G$ \\textbf{symmetric} if $W\\subseteq \\t$ and $W=\\iota(W)$.  Note that $U\\cap \\iota^{-1}(U)$ is a symmetric open neighborhood of $1$ in $G$ for every open neighborhood $U$ of $1$ contained in $\\t$. \n\n\\\n\n\\noindent We now prove a lemma which says that we can multiply any number of elements unambiguously provided that the elements are sufficiently close to the identity.  \n\n\\\n\n\\begin{lemma}\nThere are open symmetric neighborhoods $\\u_n$ of $1$ for $n>0$ such that $\\u_{n+1}\\subseteq \\u_n$ and for all $(a_1,\\ldots,a_n)\\in \\u_n^{\\times n}$, $a_1\\cdots a_n$ is defined.\n\\end{lemma}\n\n\\begin{proof}\nWe let $\\u_1$ be any open symmetric neighborhood of $1$ and choose $\\u_2$ to be a symmetric open neighborhood of $1$ such that $\\u_2 \\subseteq \\u_1$ and $\\u_2 \\times \\u_2\\subseteq \\o$.  \n\n\\\n\n\\noindent Assume inductively that $n\\geq 2$ and that for $m=1,\\ldots,n$, \n\\begin{itemize}\n\\item $\\u_m$ is a symmetric open neighborhood of $1$;\n\\item $\\u_{m+1}\\subseteq \\u_m$ if $m<n$;\n\\item for all $(a_1,\\ldots,a_m)\\in \\u_m^{\\times m}$, $a_1\\cdots a_m$ is defined;\n\\item the map $\\varphi_m:\\u_m^{\\times m}\\rightarrow G$ defined by $\\varphi_m(a_1,\\ldots,a_m)=a_1\\cdots a_m$ is continuous.\n\\end{itemize} \n\n\\noindent Let $\\u_{n+1}$ be a symmetric open neighborhood of $1$ such that $\\u_{n+1}\\subseteq \\u_n$ and $\\u_{n+1}^{\\times n}\\subseteq \\varphi_n^{-1}(\\u_2)$.  We show that this choice of $\\u_{n+1}$ works.  Let $(a_1,\\ldots,a_{n+1})\\in \\u_{n+1}^{\\times (n+1)}$.  We must show that $(a_1,\\ldots,a_{n+1})$ represents $a_1\\cdot(a_2\\cdots a_{n+1})$.  However, this is immediate by the inductive assumptions, using (A) several times.\n\\end{proof}\n\n\\\n\n\\noindent From now on, the sets $\\u_n$ will be as in the previous lemma.  Also, if $A\\subseteq \\u_n$, put $A^n:=\\{a_1\\cdots a_n \\ | \\ (a_1,\\ldots,a_n)\\in A^{\\times n}\\}$.\n\n\\\n\n\\noindent \\textbf{Local 1-parameter subgroups}\n\n\\\n\n\\begin{df}\nA \\textbf{local $1$-parameter subgroup} of $G$, henceforth abbreviated \\textbf{local $1$-ps} of $G$, is a continuous map $X:(-r,r)\\rightarrow G$, for some $r\\in (0,\\infty]$, such that \n\\begin{enumerate}\n\\item $\\operatorname{image}(X)\\subseteq \\t$, and \n\\item if $r_1,r_2,r_1+r_2\\in (-r,r)$, then $(X(r_1),X(r_2))\\in \\o$ and $$X(r_1+r_2)=X(r_1)\\cdot X(r_2).$$\n\\end{enumerate}\n\\end{df}\n\n\\\n\n\\noindent Suppose $X:(-r,r)\\rightarrow G$ is a local 1-ps of $G$ and $s\\in (-r,r)$.  It is easy to verify, by induction on $n$, that if $ns\\in (-r,r)$, then $X(s)^n$ is defined and $X(ns)=X(s)^n$.\n\n\\\n\n\\noindent We now define the local analog of the space which played a key role in the solution of Hilbert's fifth problem in the global setting.  \n\n\\\n\n\\begin{df}\nLet $X,Y$ be local 1-parameter subgroups of $G$.  We say that $X$ is \\textbf{equivalent to} $Y$ if there is $r\\in \\r^{>0}$ such that $$r\\in \\operatorname{domain}(X)\\cap \\operatorname{domain}(Y) \\text{ and }X|(-r,r)=Y|(-r,r).$$  We let $[X]$ denote the equivalence class of $X$ with respect to this equivalence relation.  We also let $L(G):=\\{[X] \\ | \\ X \\text{ is a local }1\\text{-ps of } G\\}$.  \n\\end{df}\n\n\\\n\n\\noindent In other words, $L(G)$ is the set of germs at $0$ of local 1-parameter subgroups of $G$.  We often write $\\mathbb{X}$ for an element of $L(G)$ and write $X\\in \\mathbb{X}$ to indicate that the local $1$-ps $X$ is a representative of the class $\\mathbb{X}$.  We have a \\textbf{scalar multiplication map} $(s,\\mathbb{X})\\mapsto s\\cdot \\mathbb{X}:\\r \\times L(G)\\rightarrow L(G)$, where $s \\cdot \\mathbb{X}$ is defined as follows.  Let $X\\in \\mathbb{X}$ with $X:(-r,r)\\rightarrow G$.  If $s=0$, then $0\\cdot \\mathbb{X}=\\O$, where $O:\\r \\rightarrow G$ is defined by $O(t)=1$ for all $t\\in \\r$ and $\\O=[O]$.  Else, define $s X:(\\frac{-r}{|s |},\\frac{r}{|s|})\\rightarrow G$ by $(s X)(t)=X(s t)$.  Then $sX$ is a local 1-ps of $G$, and we set $s \\cdot \\mathbb{X}=[s X]$.  It is easy to verify that for all $\\mathbb{X}\\in L(G)$ and $s,s' \\in \\r$, $1\\cdot \\mathbb{X}=\\mathbb{X}$ and $s \\cdot (s' \\cdot \\mathbb{X})=(ss')\\cdot \\mathbb{X}$.\n\n\\\n\n\\noindent Suppose $X_1$ and $X_2$ are local $1$-parameter subgroups of $G$ with $[X_1]=[X_2]$.  Suppose also that $t\\in \\operatorname{domain}(X_1)\\cap \\operatorname{domain}(X_2)$.  Then $X_1(t)=X_2(t)$.  To see this, let $r\\in \\r^{>0}$ be such that $X_1|(-r,r)=X_2|(-r,r)$ and choose $n>0$ such that $\\frac{t}{n}\\in (-r,r)$.  Then $X_1(t)=(X_1(\\frac{t}{n}))^n=(X_2(\\frac{t}{n}))^n=X_2(t)$.  Hence, it makes sense to define, for $\\mathbb{X}\\in L(G)$, \n$$\\operatorname{domain}(\\mathbb{X}):=\\bigcup_{X\\in \\mathbb{X}}\\operatorname{domain}(X)$$ and for $t\\in \\operatorname{domain}(\\mathbb{X})$, we will write $\\mathbb{X}(t)$ to denote $X(t)$ for any $X\\in \\mathbb{X}$ with $t\\in \\operatorname{domain}(X)$.  \n\n\\\n\n\n\\noindent \\textbf{The Category of Local Groups}\n\n\\\n\n\\noindent In this subsection, we define the analogs of a few ordinary group theoretic notions in the local group setting.  We will not need most of this material until Section 8.  The presentation given here borrows from ~\\cite{MZ} and ~\\cite{P}.\n\n\\\n\n\\begin{df}\nSuppose $G=(G,1,\\iota, p)$ and $ G'=(G',1',\\iota',p')$ are local groups with $\\operatorname{domain}(\\iota)=\\Lambda$, $\\operatorname{domain}(p)=\\o$, $\\operatorname{domain}(\\iota')=\\Lambda'$ and $\\operatorname{domain}(p')=\\o'$.  A \\textbf{morphism} from $G$ to $G'$ is a continuous function $f:G\\rightarrow G'$ such that:\n\\begin{enumerate}\n\\item $f(1)=1'$, $f(\\Lambda)\\subseteq \\Lambda'$ and $(f\\times f)(\\o)\\subseteq \\o'$,\n\\item $f(\\iota(x))=\\iota'(f(x))$ for $x\\in \\Lambda$, and\n\\item $f(p(x,y))=p'(f(x),f(y))$ for $(x,y)\\in \\o$.\n\\end{enumerate}\n\\end{df}\n\n\\\n\n\\noindent Suppose $U_1$ and $U_2$ are open neighborhoods of $1$ in $G$ and suppose that $f_1:G|U_1\\rightarrow G'$ and $f_2:G|U_2\\rightarrow G'$ are both morphisms.  We say that $f_1$ and $f_2$ are \\textbf{equivalent} if there is an open neighborhood $U_3$ of $1$ in $G$ such that $U_3\\subseteq U_1\\cap U_2$ and $f_1|U_3=f_2|U_3$.  This defines an equivalence relation on the set of morphisms $f:G|U \\rightarrow G'$, where $U$ ranges over all open neighborhoods of $1$ in $G$.  We will call the equivalence classes \\textbf{local morphisms} from $G$ to $G'$.  To indicate that $f$ is a local morphism from $G$ to $G'$, we write ``$f:G\\rightarrow G'$ is a local morphism,\" even though $f$ is not a function from $G$ to $G'$, but rather an equivalence class of certain partial functions.\n\n\\\n\n\\noindent Suppose $G,G',G''$ are local groups and $f:G \\rightarrow G'$ and $g:G' \\rightarrow G''$ are local morphisms.  Choose a representative of $f$ whose image lies in the domain of a representative of $g$.  Then these representatives can be composed and the composition is a morphism.  We denote by $g\\circ f$ the equivalence class of this composition.  The equivalence class of the identity function on $G$ is a local morphism and will be denoted by $\\operatorname{id}_G$.  We thus have the category \\textbf{LocGrp} whose objects are the local groups and whose arrows are the local morphisms with the composition defined above.  A \\textbf{local isomorphism} is an isomorphism in the category \\textbf{LocGrp}.  Hence, if $f:G \\rightarrow G'$ is a local isomorphism, then there is a representative $f:G|U \\rightarrow G'|U'$, where $U$ and $U'$ are open neighborhoods of $1$ and $1'$ in $G$ and $G'$ respectively, $f:U\\rightarrow U'$ is a homeomorphism, and $f:G|U\\rightarrow G'|U'$ and $f^{-1}:G'|U'\\rightarrow G|U$ are morphisms.  If there exists a local isomorphism between the local groups $G$ and $G'$, then we say that $G$ and $G'$ are \\textbf{locally isomorphic}.  \n\n\\\n\n\\begin{ex}\nOne can consider the additive group $\\r$ as a local group and $(-r,r)$ for $r\\in (0,\\infty]$ as a restriction of this local group, so that a local 1-ps $(-r,r)\\rightarrow G$ is a morphism of local groups.  In this way, the elements of $L(G)$ are local morphisms from $\\r$ to $G$.  Moreover, given any local morphism $f:G \\rightarrow G'$, one gets a map of sets $L(f):L(G)\\rightarrow L(G')$ given by $L(f)(\\mathbb{X})=f\\circ \\mathbb{X}$.  It can easily be checked that this yields a functor $L:\\textbf{LocGrp}\\rightarrow \\textbf{Sets}$.\n\\end{ex}\n\n\\\n\n\\begin{df} \nA \\textbf{sublocal group} of $G$ is a set $H\\subseteq G$ containing $1$ for which there exists an open neighborhood $V$ of $1$ in $G$ such that\n\\begin{enumerate}\n\\item $H\\subseteq V$ and $H$ is closed in $V$;\n\\item if $x\\in H\\cap \\t$ and $x^{-1}\\in V$, then $x^{-1}\\in H$;\n\\item if $(x,y)\\in (H\\times H)\\cap \\o$ and $xy\\in V$, then $xy\\in H$.\n\\end{enumerate}\nWith $H$ and $V$ as above, we call $H$ a sublocal group of $G$ with \\textbf{associated neighborhood} $V$.  A \\textbf{normal sublocal group} of $G$ is a sublocal group $H$ of $G$ with an associated neighborhood $V$ such that $V$ is symmetric and \n\\begin{enumerate}\n\\item[(4)] if $y\\in V$ and $x\\in H$ are such that $yxy^{-1}$ is defined and $yxy^{-1}\\in V$, then $yxy^{-1}\\in H$.\n\\end{enumerate}\nFor such $H$ and $V$, we also say that $H$ is a normal sublocal group of $G$ with associated \\textbf{normalizing} neighborhood $V$.\n\\end{df}\n\n\\\n\n\\noindent One checks easily that if $H$ is a sublocal group of $G$ with associated\nneighborhood $V$, then\n$$(H,1,\\iota|H\\cap \\Lambda \\cap \\iota^{-1}(V), p|(H\\times H) \\cap \\Omega \\cap\np^{-1}(V))$$ is a local group, to be denoted by $H$ for simplicity, and that\nthen the inclusion $H \\hookrightarrow G$ is a morphism.  Sublocal groups $H$ and $H'$ of $G$ are \\textbf{equivalent} if there is an open neighborhood $U$ of $1$ in $G$ such that $H\\cap U=H'\\cap U$.\n\n\\\n\n\\\n\n\\begin{ex}\nSuppose $f:G\\rightarrow G'$ is a morphism and $\\t=G$.  Then $\\ker(f):=f^{-1}(\\{1\\})$ is a normal sublocal group of $G$ with associated normalizing neighborhood $G$.\n\\end{ex}\n\n\\\n\n\\noindent The following lemma has a routine verification.\n\n\\\n\n\\begin{lemma}\nSuppose $H$ is a normal sublocal group of $G$ with associated normalizing neighborhood $V$.  Suppose $U\\subseteq H$ is open in $H$ and symmetric.  Let $U'\\subseteq V$ be a symmetric open neighborhood of $1$ in $G$ such that $U=H\\cap U'$. Then $H|U$ is a normal sublocal group of $G$ with associated normalizing neighborhood $U'$. \n\\end{lemma}\n\n\\\n\n\\noindent We now investigate quotients in this category.  For the rest of this subsection, assume that $H$ is a normal sublocal group of $G$ with associated normalizing neighborhood $V$.\n\n\\\n\n\\begin{lemma}\\label{L:localcoset}\nLet $W$ be a symmetric open neighborhood of $1$ in $G$ such that $W\\subseteq \\u_{6}$ and $W^{6}\\subseteq V$.  Then\n\\begin{enumerate}\n\\item The binary relation $E_H$ on $W$ defined by $$E_H(x,y)\\text{ if and only if }x^{-1}y\\in H$$ is an equivalence relation on $W$.\n\\item For $x\\in W$, let $xH:=\\{xh \\ | \\ h\\in H \\text{ and }(x,h)\\in \\o\\}$.  Then for $x,y\\in W$, $E_H(x,y)$ holds if and only if $(xH)\\cap W = (yH)\\cap W$.  In other words, if $E_H(x)$ denotes the equivalence class of $x$, then $E_H(x)=(xH)\\cap W$.  We call the equivalence classes \\textbf{local cosets} of $H$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFor (1), first note that the reflexivity of $E_H$ is trivial.  Now suppose $E_H(x, y)$ holds, i.e. that $x^{-1}y\\in H$.  Since $(x^{-1}y)^{-1}\\in V$, we have that $(x^{-1}y)^{-1}\\in H$.  It can easily be verified that $(x^{-1}y)^{-1}=y^{-1}x$, so $E_H(y,x)$ holds.  Finally, suppose $E_H(x, y)$ and $E_H(y, z)$ hold.  It follows from $(x^{-1}y)(y^{-1}z)\\in V$ that $(x^{-1}y)(y^{-1}z)\\in H$.  However, since $x,y,z\\in \\u_{4}$, we know that $(x^{-1}y)(y^{-1}z)=x^{-1}z$, so we have transitivity and (1) is proven.\n\n\\\n\n\\noindent For (2), first suppose that $E_H(x, y)$ holds.  Let $w\\in (yH)\\cap W$.  Let $h\\in H$ be such that $w=yh$.  Then $h=y^{-1}w\\in W^2$.  Since $(x^{-1}y)h\\in V$, we have $(x^{-1}y)h\\in H$.  But $(x^{-1}y)h=x^{-1}w$, so $w\\in (xH)\\cap W$.  By symmetry, $(xH)\\cap W = (yH)\\cap W$.  For the converse, suppose $(xH)\\cap W = (yH)\\cap W$.  Since $y\\in (yH)\\cap W$, we have $y\\in (xH)\\cap W$.  Let $h\\in H$ be such that $y=xh$.  Then since $xh\\in W$, we know that $x^{-1}(xh)$ is defined.  We thus have, by (A), $x^{-1}y=h\\in H$ and so $E_H(x, y)$ holds.   \n\\end{proof}\n\n\\\n\n\\noindent Let $\\pi_{H,W}:W \\rightarrow W\/E_H$ be the canonical map.  Give $W\/E_H$ the quotient topology.  Then $\\pi_{H,W}$ becomes an open continuous map.  \n\n\\\n\n\\noindent Let $\\iota_{H,W}:W\/E_H\\rightarrow W\/E_H$ be defined by $\\iota_{H,W}(E_H(x))=E_H(x^{-1})$.  Let $\\o_{H,W}:=(\\pi_{H,W} \\times \\pi_{H,W})((W\\times W)\\cap p^{-1}(W))$.  It is easy to check that one can define a map $p_{H,W}:\\o_{H,W}\\rightarrow W\/E_H$ by $p_{H,W}(E_H(x),E_H(y))=E_H(xy)$, where $x$ and $y$ are representatives of their respective local cosets chosen so that $xy\\in W$.  The following lemma is easy to verify.      \n\n\\\n\n\\begin{lemma}\nWith the notations as above, $$(G\/H)_W:=(W\/E_H,E_H(1),\\iota_{H,W},p_{H,W})$$ is a local group and $\\pi_{H,W}:G|W \\rightarrow (G\/H)_W$ is a morphism.      \n\\end{lemma}\n\n\\\n\n\\noindent In the construction of quotients, we have made some choices.  We would expect that, locally, we have the same local group.  The following lemma expresses this.\n\n\\\n\n\\begin{lemma}\nSuppose that $H'$ is also a normal sublocal group of $G$ with associated normalizing neighborhood $V'$ such that $H$ is equivalent to $H'$.  Let $W'$ be a symmetric open neighborhood of $1$ in $G$ used to construct $(G\/H')_{W'}$.  Then $(G\/H)_W$ and $(G\/H')_{W'}$ are locally isomorphic.\n\\end{lemma}\n\n\\begin{proof}\nLet $U$ be an open neighborhood of $1$ in $G$ such that $H\\cap U=H'\\cap U$.  Choose open $U_1$ containing $1$ such that $U_1^2 \\subseteq U$.  Let $U_2:=W\\cap W'\\cap U_1$.  Consider the map $\\psi:\\pi_{H,W}(U_2)\\rightarrow W'\/E_{H'}$ given by $\\psi(E_H(x))=E_{H'}(x)$ for $x\\in U_2$.  One can verify that $\\psi:(G\/H)_W|\\pi_{H,W}(U_2)\\rightarrow (G\/H')_{W'}$ induces the desired local isomorphism.\n\\end{proof}\n\n\\\n\n\\noindent From now on, we write $G\/H$ to denote $(G\/H)_W$, where $W$ is any open neighborhood of $1$ as in Lemma \\ref{L:localcoset}.  We will also write $\\pi:G\\rightarrow G\/H$ to denote the local morphism induced by $\\pi_{H,W}:G|W\\rightarrow (G\/H)_W$.  Since all of the various quotients defined in this way are locally isomorphic, fixing one's attention on one particular local coset space is no loss of generality for our purposes.\n\n\\\n\n\\\n\n\\noindent \\textbf{Further Properties of Local Groups}\n\n\\\n\n\\noindent In this subsection, we first show that we can place two further assumptions on our local groups which lead to no loss of generality in the solution of the Local H5.  Under these new assumptions, we derive several properties of the notion ``$a_1\\cdots a_n$ is defined.\" \n\n\\\n\n\\noindent Let $U:=\\t \\cap \\iota^{-1}(\\t)$, an open neighborhood of $1$ in $G$.  Note that if $g\\in U$, then $\\iota(g)\\in \\t$ and $\\iota(\\iota(g))=g\\in \\t$, implying that $\\t_U=U$.\n\n\\\n\n\\noindent \\textbf{From this point on, we assume that $\\t=G$ for every local group $G$ in this paper.}  By the preceding remarks, every local group has a restriction satisfying this condition and so this is no loss of generality for our purpose.  Note that, with these assumptions, if $(x,y)\\in \\o$ and $xy=1$, then $x=y^{-1}$ and $y=x^{-1}$.  In particular, $(x^{-1})^{-1}=1$ for all $x\\in G$, and $G$ is symmetric. \n\n\\\n\n\n\n\\\n\n\\noindent The following lemma is obvious for topological groups, but requires some care in the local group setting.\n\n\\\n\n\\begin{lemma}\\label{L:homog}(Homogeneity)\n\\begin{enumerate}\n\\item For any $g\\in G$, there are open neighborhoods $V$ and $W$ of $1$ and $g$ respectively such that $\\{g\\}\\times V\\subseteq \\o$, $gV\\subseteq W$, $\\{g^{-1}\\}\\times W\\subseteq \\o$, $g^{-1}W\\subseteq V$, and the maps $$v\\mapsto gv:V\\rightarrow W \\text{ and }w\\mapsto g^{-1}w:W\\rightarrow V$$ are each others inverses (and hence homeomorphisms).\n\\item$G$ is locally compact if and only if there is a compact neighborhood of $1$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nClearly (1) implies (2).  For any $g\\in G$, define $$\\o_g:=\\{h\\in G \\ | \\ (g,h)\\in \\o\\}.$$  Then $\\o_g$ is an open subset of $G$ and $L_g:\\o_g\\rightarrow G$, defined by $L_g(h)=gh$, is continuous.  Let $V:=(L_g)^{-1}(\\o_{g^{-1}})$.  Then $V$ is open and $1\\in V$.  Let $W= L_g(V)\\subseteq \\o_{g^{-1}}.$  Then $W$ is open since $W=L_{g^{-1}}^{-1}(V)$.  From our choices we can check that $L_g|V$ and $L_{g^{-1}}|W$ are inverses of each other.  \n\\end{proof}\n\n\\\n\n\n\\\n\n\\begin{cor}\nLet $U= \\u_3$.  Then for any $g,h\\in U$ such that $(g,h)\\in \\o_U$, one has $(h^{-1},g^{-1})\\in \\o_U$ and $(gh)^{-1}=h^{-1}g^{-1}$.\n\\end{cor}\n\n\\begin{proof}\nSuppose $g,h\\in U$ with $(g,h)\\in \\o_U$.  Then we have $h^{-1},g^{-1},gh\\in U$.  Hence $h^{-1}\\cdot g^{-1}\\cdot (gh)$ is defined and $$h^{-1}\\cdot g^{-1}\\cdot (gh)=h^{-1}\\cdot (g^{-1}\\cdot(gh))=h^{-1}h=1$$ by (A).  But $h^{-1}\\cdot g^{-1}\\cdot (gh)=(h^{-1}g^{-1})\\cdot (gh)$, so $h^{-1}g^{-1}=(gh)^{-1}$.  Also, since $h^{-1}g^{-1}=(gh)^{-1}\\in U$, we have $(h^{-1},g^{-1})\\in \\o_U$.    \n\\end{proof}\n\n\\\n\n\\noindent \\textbf{From now on, we make the following further assumption on our local group} $G$:  \n$$\\text{if } (g,h)\\in \\o \\text{, then }(h^{-1},g^{-1})\\in \\o \\text{ and }(gh)^{-1}=h^{-1}g^{-1}.$$ By the preceding corollary, every local group has a restriction satisfying this assumption and so this is no loss of generality for our purpose.\n\n\\\n\n\\begin{lemma}\\label{L:EZFacts}Let $a,a_1,\\ldots,a_n\\in G$.  Then \n\n\\begin{enumerate}\n\n\\item If $a_1\\cdots a_n$ is defined and $1\\leq i\\leq j \\leq n$, then $a_i\\cdots a_j$ is defined.  In particular, if $a^n$ is defined and $m\\leq n$, then $a^m$ is defined.  \n\\item If $a^m$ is defined and $i,j\\in \\{0,\\ldots,m\\}$ are such that $i+j=m$, then $(a^i,a^j)\\in \\o$ and $a^i \\cdot a^j = a^m$.\n\\item If  $a^n$ is defined and, for all $i,j\\in\\{1,\\ldots,n\\}$ with $i+j=n+1$, one has $(a^i,a^j)\\in \\o$, then $a^{n+1}$ is defined.  More generally, if $a_1\\cdots a_n$ is defined, $a_i \\cdots a_{n+1}$  is defined for all $i\\in\\{2,\\ldots,n\\}$ and $$(a_1\\cdots a_i,a_{i+1}\\cdots a_{n+1})\\in \\o \\text{ for all }i\\in\\{1,\\ldots ,n\\},$$ then $a_1 \\cdots a_{n+1}$ is defined. \n\\item If $a^n$ is defined, then $(a^{-1})^n$ is defined and $(a^{-1})^n=(a^n)^{-1}$.  (In this case, we denote $(a^{-1})^n$ by $a^{-n}$.)  More generally, if $a_1\\cdots a_n$ is defined, then $a_n^{-1}\\cdots a_1^{-1}$ is defined and $(a_1\\cdots a_n)^{-1}=a_n^{-1}\\cdots a_1^{-1}$. \n\\item If $k,l\\in \\mathbb{Z}$, $l\\not=0$, and $a^{k\\cdot l}$ is defined, then $a^k$ is defined, $(a^k)^l$ is defined and $(a^k)^l=a^{k\\cdot l}$.   \n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe proofs of (1) and (2) are immediate from the definitions.  (3) follows from repeated uses of (A).  We prove the first assertion of (4) by induction on $n$.  The cases $n=1$ and $n=2$ are immediate from our assumptions on local groups.  For the induction step, suppose $a^{n+1}$ is defined and $i,j\\in \\{1,\\ldots,n\\}$ with $i+j=n+1$. Then since $(a^j,a^i)\\in \\o$, by induction we have $((a^{-1})^i,(a^{-1})^j)\\in \\o$.  Hence, by (3) of the lemma, we know that $(a^{-1})^{n+1}$ is defined.  Also, $$(a^{-1})^{n+1}= (a^{-1})^n\\cdot a^{-1}=(a^n)^{-1}\\cdot a^{-1}=(a\\cdot a^n)^{-1}=(a^{n+1})^{-1}.$$  The second assertion of (4) is proved in the same manner.  \n\n\\\n\n\\noindent We now prove (5).  It is easy to see that it is enough to check the assertion for $k,l\\in \\mathbb{N}$, $l\\not= 0$.  Since $k\\leq k\\cdot l$, we have $a^k$ is defined by (1).  We prove the rest by induction on $l$.  This is clear for $l=1$.  Suppose the assertion is true for all $i\\leq l$ and suppose $a^{k\\cdot (l+1)}$ is defined  To see that $(a^k)^{l+1}$ is defined, we must check that $((a^k)^i,(a^k)^j)\\in \\o$ for $i,j\\in \\{1,\\ldots,l\\}$ such that $i+j=l+1$.  This is the case by (2), since for such $i,j$, we have $k\\cdot i + k\\cdot j=k\\cdot(l+1)$ and by induction $(a^k)^i=a^{k\\cdot i}$ and $(a^k)^j=a^{k\\cdot j}$.  Now that we know that $(a^k)^{l+1}$ is defined, we must have, by induction, that $$(a^k)^{l+1}=(a^k)^l\\cdot a^k=a^{k\\cdot l}\\cdot a^k=a^{k\\cdot l+k}.$$\n\\end{proof}\n\n\\\n\n\\noindent By (4) of the previous lemma, we can now speak of $a^k$ being defined for any $k\\in \\mathbb{Z}$, where $a^k$ being defined for $k<0$ means that $a^{-k}$ is defined. \n\n\\\n\n\\begin{cor}\\label{C:EZCorollary}\nSuppose $i,j\\in \\mathbb{Z}$ and $i\\cdot j <0$.  If $a^i$ and $a^j$ are defined and $(a^i,a^j)\\in \\o$, then $a^{i+j}$ is defined and $a^i \\cdot a^j=a^{i+j}$.\n\\end{cor}\n\n\\begin{proof}\n\\noindent The fact that $a^{i+j}$ is defined is clear from the preceding lemma.  Next note that the result is trivial if  $i=-j$.  We only prove the case that $i>0$ and $j<0$ and $i>|j|$, the other cases being similar.  By the preceding lemma, we have $a^i=a^{i+j}\\cdot a^{-j}$, whence $a^i \\cdot a^j=a^{i+j}$ by (A).\n\\end{proof}\n\n\\\n\n\\bigskip \\noindent \\textbf{The Nonstandard Setting}\n\n\\\n\n\\noindent We assume familiarity with this setting;\nsee \\cite{D} and \\cite{He} for details. Here we just fix notations and terminology.\nTo each relevant ``basic'' set $S$ corresponds functorially\na set $S^*\\supseteq S$,\nthe {\\em nonstandard extension\\\/} of $S$. In particular, $\\mathbb{N}, \\r, G$\nextend\nto $\\mathbb{N}^*, \\r^*, G^*$,\nrespectively. Also, any (relevant) relation $R$ and function\n$F$ on these basic sets extends functorially to a relation $R^*$ and\nfunction\n$F^*$ on the corresponding nonstandard extensions of these basic sets.\nFor example, the linear ordering $<$ on $\\mathbb{N}$ extends to a\nlinear ordering $<^*$ on $\\mathbb{N}^*$, and\nthe local group operation $p: \\o \\to G$ of $G$ extends to\nan operation $p^*: \\o^* \\to G^*$. For the sake of\nreadability\nwe only use a star in denoting the nonstandard extension of a basic set,\nbut drop the star when indicating the nonstandard extension\nof a relation or function on these basic sets. For example,\nwhen $x,y\\in \\r^*$ we write $x+y$ and $x<y$ rather than $x+^*y$ and $x<^*\ny$.\n\n\\\n\n\\noindent Given an ambient hausdorff space $S$ and $s\\in S$, the {\\em monad\\\/}\nof $s$, notation: $\\boldsymbol{\\mu}(s)$, is by definition the intersection\nof all $U^*\\subseteq S^*$ with $U$ a neighborhood of $s$ in $S$;\nthe elements of $\\boldsymbol{\\mu}(s)$ are the points\nof the nonstandard space $S^*$ that are {\\em infinitely close\\\/} to $s$.\nThe points of $S^*$ that are infinitely close to some $s\\in S$ are called\n{\\em nearstandard}, and $S_{\\operatorname{ns}}^*$ is the set of nearstandard\npoints\nof $S^*$:\n$$    S_{\\operatorname{ns}}^* = \\bigcup_{s\\in S} \\boldsymbol{\\mu}(s).$$\nSince $S$ is hausdorff, $\\boldsymbol{\\mu}(s)\\cap \\boldsymbol{\\mu}(s')=\\emptyset$ for distinct $s,s'\\in S$.  Thus, for $s^*\\in S^*_{\\operatorname{ns}}$, we can define the \\textit{standard part} of $s^*$, notation: $\\operatorname{st}(s^*)$, to be the unique $s\\in S$ such that $s^*\\in \\boldsymbol{\\mu}(s)$.  For $s_1,s_2 \\in S^*_{\\operatorname{ns}}$, we put $s_1\\sim s_2$ iff $\\operatorname{st}(s_1)=\\operatorname{st}(s_2)$.\n\n\\  \n\n\\noindent We let $\\boldsymbol{\\mu}:=\\boldsymbol{\\mu}(1)$ be the monad of $1$ in $G^*$; note that $\\boldsymbol{\\mu}$ is a group with product map $(x,y)\\mapsto xy:=p(x,y)$.  \n\n\\\n\n\\noindent From now on, we let $\\nu,\\sigma,\\tau,\\eta,N$ range over $\\mathbb{N}^*$ and $i$ and $j$ range over $\\mathbb{Z}^*$, but \\textbf{$m$ and $n$ will always denote elements of $\\mathbb{N}$}.  We use the Landau notation as follows:  $i=\\operatorname{o}(\\nu)$ means $|i|<\\frac{\\nu}{n}$ for all $n>0$ and $i=\\operatorname{O}(\\nu)$ means $|i|<n\\nu$ for some $n>0$.  \n\n\\\n\n\\noindent In our nonstandard arguments, we will assume as much saturation as needed.  The following lemma uses routine nonstandard methods and is left for the reader to verify.\n\n\\\n\n\\begin{lemma}\n$G$ is $\\operatorname{NSS}$ if and only if there are no internal subgroups of $\\boldsymbol{\\mu}$ other than $\\{1\\}$. \n\\end{lemma}\n\n\\\n\n\\noindent We also consider the internal versions of the notions and results that appear earlier in this section.  In particular, if $(a_1,\\ldots,a_\\nu)$ is an internal sequence in $G^*$, then it makes sense to say that $(a_1,\\ldots,a_\\nu)$  (internally) represents $b$, where $b\\in G^*$.  We say that $a_1\\cdots a_\\nu$ is defined if there exists $b\\in G^*$ such that $(a_1,\\ldots,a_\\nu)$ represents $b$.  Likewise, we have the notion ``$a^i$ is defined\" for $a\\in G^*, i\\in \\mathbb{Z}^*$.\n\n\\\n\n\\noindent The proof of the following lemma is trivial.\n\n\\\n\n\\begin{lemma}\n\n\\\n\n\\begin{enumerate}\n\\item Suppose $a,b\\in G$ and $a'\\in \\boldsymbol{\\mu}(a)$ and $b'\\in \\boldsymbol{\\mu}(b)$.\nIf $(a,b)\\in \\o$, then $(a',b')\\in \\o^*$, $a'\\cdot b' \\in G^*_{\\operatorname{ns}}$, and $\\operatorname{st}(a'\\cdot b')=a\\cdot b$.\n\\item For any $a\\in G^*_{\\operatorname{ns}}$ and $b\\in \\boldsymbol{\\mu}$, $(a,b),(b,a)\\in \\o^*$, $a\\cdot b, b\\cdot a\\in G^*_{\\operatorname{ns}}$,  and $\\operatorname{st}(a\\cdot b)=\\operatorname{st}(b\\cdot a)=\\operatorname{st}(a)$.\n\\item For any $a,b\\in G^*_{\\operatorname{ns}}$, if $(a,b^{-1})\\in \\o^*$ and $ab^{-1}\\in \\boldsymbol{\\mu}$, then $a\\sim b$.\n\\item For any $a\\in G$, $a'\\in \\boldsymbol{\\mu}(a)$, and any $n$, if $a^n$ is defined, then $(a')^n$ is defined and $(a')^n\\in \\boldsymbol{\\mu}(a^n)$. \n\\end{enumerate}\n\\end{lemma}\n\n\\\n  \n\\noindent A standard consequence of the last item of the preceding lemma is that the partial function $p_n:G\\rightharpoonup G$, defined by $p_n(a)=a^n$ if $a^n$ is defined, has an open domain and is continuous.  Note that $\\u_n \\subseteq \\operatorname{domain}(p_n)$.  \n\n\n\\\n\n\\begin{lemma}\\label{L:Overflow}\nSuppose $U$ is a neighborhood of $1$ in $G$ and $a\\in \\boldsymbol{\\mu}$.  Then there is a $\\nu> \\mathbb{N}$ such that $a^\\sigma$ is defined and $a^\\sigma \\in U^*$ for all $\\sigma \\in \\{1,\\ldots,\\nu\\}$.  \n\\end{lemma}\n\n\\begin{proof}\nLet $X=\\{\\sigma \\in \\mathbb{N}^* \\ | \\ a^\\sigma \\text{ is defined and } a^\\sigma \\in U^* \\}$.  Then $X$ is an internal subset of $\\mathbb{N}^*$ which contains $\\mathbb{N}$ since $\\boldsymbol{\\mu}$ is a subgroup of $G^*$.  Hence, by overflow, there is a $\\nu> \\mathbb{N}$ such that $\\{0,1,\\ldots,\\nu\\}\\subseteq X$.  \n\\end{proof} \n\n\\\n\n\\section{Connectedness and Purity}\n\n\n\\noindent In this section, we describe the different ways powers of infinitesimals can grow and its impact on the existence of connected subgroups of $G$.  We also show how to build local 1-parameter subgroups from ``pure\" infinitesimals.  \n\n\\\n\n\\noindent \\textbf{In the rest of this paper, we assume that $G$ is locally compact}.  In particular, $G$ is regular as a topological space.  Also, $a$ and $b$ will denote elements of $G^*$.\n \n\\\n\n\\noindent For $\\nu > \\mathbb{N}$, we set:\n$$G(\\nu):=\\{a\\in \\boldsymbol{\\mu} \\ | \\ a^i \\text{ is defined and }a^i\\in \\boldsymbol{\\mu} \\text{ for all }i=\\operatorname{o}(\\nu)\\},$$\n$$G^{\\operatorname{o}}(\\nu):=\\{a\\in \\boldsymbol{\\mu} \\ | \\ a^i \\text{ is defined and }a^i\\in \\boldsymbol{\\mu} \\text{ for all }i=\\operatorname{O}(\\nu)\\}.$$\n\\noindent Note that $G^{\\operatorname{o}}(\\nu)\\subseteq G(\\nu)$ and both sets are symmetric. \n\n\\\n\n\\begin{lemma}\\label{L:lilo}\nLet $a\\in \\boldsymbol{\\mu}$ and $\\nu>\\mathbb{N}$.  Then the following are equivalent:\n\\begin{enumerate}\n\\item $a^i$ is defined and $a^i\\in \\boldsymbol{\\mu}$ for all $i\\in \\{1,\\ldots,\\nu\\}$;\n\\item $a\\in G^{\\operatorname{o}}(\\nu)$;\n\\item there is $\\tau \\in \\{1,\\ldots,\\nu\\}$ such that $\\nu =\\operatorname{O}(\\tau)$ and $a^i$ is defined and $a^i\\in \\boldsymbol{\\mu}$ for all $i\\in \\{1,\\ldots,\\tau\\}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1) $\\Rightarrow$ (2):  Fix $\\sigma$ such that $\\sigma=\\operatorname{O}(\\nu)$.  We claim that if $a^\\sigma$ is defined, then $a^\\sigma\\in \\boldsymbol{\\mu}$.  To see this, write $\\sigma =n\\nu + \\eta$ for some $n$ and $\\eta < \\nu$.  By Lemma \\ref{L:EZFacts}, $$a^\\sigma =(a^\\nu)^n\\cdot a^\\eta \\in \\boldsymbol{\\mu} \\cdot \\boldsymbol{\\mu} \\subseteq \\boldsymbol{\\mu}.$$  We now prove by internal induction  that $a^i$ is defined for all $i\\in \\{1,\\ldots,\\sigma\\}$.  The assertion is true for $i=1$ and inductively suppose that $a^i$ is defined and $i+1\\leq \\sigma$.  By Lemma \\ref{L:EZFacts}, in order to show that $a^{i+1}$ is defined, it suffices to show that $(a^k,a^l)\\in \\o^*$ for all $k,l\\in \\{1,\\ldots,i\\}$ with $k+l=i+1$.  However, $(a^k,a^l)\\in \\boldsymbol{\\mu} \\times \\boldsymbol{\\mu} \\subseteq \\o^*$, finishing the induction.  \n\n\\\n\n\\noindent Clearly (2)$\\Rightarrow$ (1) and (3), so it remains to prove that (3) $\\Rightarrow (2)$.  So assume that there is $\\tau \\in \\{1,\\ldots,\\nu\\}$ as in (3).  By the proof of (1)$\\Rightarrow (2)$, we see that $a\\in G^{\\operatorname{o}}(\\tau)$.  Since $G^{\\operatorname{o}}(\\tau)\\subseteq G^{\\operatorname{o}}(\\nu)$, we are finished.   \n\\end{proof}\n\n\\\n\n\\begin{df}\nWe say that $a\\in \\boldsymbol{\\mu}$ is \\textbf{degenerate} if, for all $i$, $a^i$ is defined and $a^i\\in \\boldsymbol{\\mu}$.  \n\\end{df}\n\n\\\n\n\\noindent It is easy to see that $G$ is $\\operatorname{NSS}$ if and only if $\\boldsymbol{\\mu}$ has no nondegenerate elements other than $1$.\n\n\\\n\n\\begin{lemma}\nLet $a_1,\\ldots,a_\\nu$ be an internal sequence of elements of $G^*$ with $\\nu > 0$ such that for all $i\\in \\{1,\\ldots,\\nu\\}$ we have $a_i\\in \\boldsymbol{\\mu}$, $a_1\\cdots a_i$ is defined and $a_1\\cdots a_i\\in G^*_{\\operatorname{ns}}$.  Then the set $$S:=\\{\\operatorname{st}(a_1\\cdots a_i):1\\leq i \\leq \\nu\\}\\subseteq G$$ is compact and connected (and contains 1).\n\\end{lemma}\n\n\\\n\n\\noindent The proof is as in the global case, but we include it for completeness.\n \n\\begin{proof}\nSuppose for every compact $K\\subseteq G$, there is $i\\in \\{1,\\ldots, \\nu\\}$ such that $a_1\\cdots a_i \\notin K^*$.  Then, by saturation, there is $i\\in \\{1,\\ldots, \\nu\\}$ such that $a_1\\cdots a_i \\notin K^*$ for all compact $K\\subseteq G$, contradicting local compactness.  Hence there is compact $K\\subseteq G$ with $a_1\\cdots a_i\\in K^*$ for all $i\\in \\{1,\\ldots, \\nu\\}$, so $S\\subseteq K$.  It is well-known that $S$ must be closed, being the standard part of an internal nearstandard set.  Hence $S$ is  compact.\n\n\\\n\n\\noindent Suppose $S$ is not connected.  Then we have disjoint open subsets $U$ and $V$ of $G$ such that $S\\subseteq U\\cup V$ and $S\\cap U \\not= \\emptyset$, $S\\cap V \\not= \\emptyset$.  Assume $1\\in U$, so $a_1 \\in U^*$.  Choose $i\\in \\{1,\\ldots, \\nu\\}$ minimal such that $a_1\\cdots a_i\\in V^*$.  Then $i\\geq 2$ and $a_1\\cdots a_{i-1}\\in U^*$.  Now $a:=\\operatorname{st}(a_1\\cdots a_{i-1})=\\operatorname{st}(a_1\\cdots a_i)\\in S$.  If $a\\in U$, then $a_1\\cdots a_{i-1}\\in U^*$ and if $a\\in V$, then $a_1\\cdots a_i \\in V^*$; either option yields a contradiction.\n\\end{proof}\n\n\\\n\n\\noindent Until further notice, $U$ denotes a compact symmetric neighborhood of $1$ such that $U\\subseteq \\u_2$.\n\n\\\n\n\\begin{df}\\label{D:ordelement}\nLet $a\\in G^*$.  If, for all $i$, $a^i$ is defined and $a^i\\in U^*$, define ord$_U(a)=\\infty$.  Else, define ord$_U(a)=\\nu$ if, for all $i$ with $|i|\\leq \\nu$, $a^i$ is defined, $a^i \\in U^*$, and $\\nu$ is the largest element of $\\mathbb{N}^*$ for which this happens.\n\\end{df}\n\n\\\n\n\\noindent It is easy to see that ord$_U(a)=0$ iff $a\\notin U^*$.  By Lemma \\ref{L:Overflow}, we can see that ord$_U(a)>\\mathbb{N}$ if $a\\in \\boldsymbol{\\mu}$.  Note that if $a\\in \\boldsymbol{\\mu}$ and ord$_U(a)=\\nu \\in \\mathbb{N}^*$, then $a^{\\nu + 1}$ is defined.  To see this, it is enough to check that $(a^i,a^j)\\in \\o^*$ for all $i, j \\in \\{1,\\ldots,\\nu\\}$ such that $i+j=\\nu +1$.  However, this is true since $(a^i,a^j)\\in U^* \\times U^* \\subseteq \\u_2^*\\times \\u_2^* \\subseteq \\o^*$.  Hence, if ord$_U(a)=\\nu$, then this is not because $a^{\\nu+1}$ is undefined but rather for the more important reason that $a^{\\nu+1}$ is defined but not in $U^*$.  We have thus captured the correct adaptation of the global notion to our local setting.\n\n\\\n\n\\begin{lemma}\\label{L:connsubgroup}\nSuppose $a\\in \\boldsymbol{\\mu}$ and $a^i \\notin \\boldsymbol{\\mu}$ for some $i=\\operatorname{o}(\\operatorname{ord}_U(a))$.  Then $U$ contains a nontrivial connected subgroup of $G$. \n\\end{lemma}\n\n\\begin{proof}\nBy the previous lemma, the set $$G_U(a):=\\{\\operatorname{st}(a^i) \\ | \\ |i|=\\operatorname{o}(\\operatorname{ord}_U(a))\\}$$ is a union of connected subsets of $U$, each containing $1$, hence is connected.   It is also a subgroup of $G$ by Lemma \\ref{L:EZFacts} and Corollary \\ref{C:EZCorollary}.\n\\end{proof}\n\n\\\n\n\\begin{df}\nWe say that $a\\in \\boldsymbol{\\mu}$ is \\textbf{$U$-pure}\\footnote{We have decided to replace the use of ``regular\" in \\cite{H} with ``pure\" as ``regular\" already has a different meaning for topological spaces.} if it is nondegenerate and $a\\in G(\\tau)$, where $\\tau:=\\operatorname{ord}_U(a)$.  We say that $a\\in \\boldsymbol{\\mu}$ is \\textbf{pure} if it is $V$-pure for some compact symmetric neighborhood $V$ of $1$ such that $V\\subseteq \\u_2$.\n\\end{df}\n\n\\\n\n\\noindent The last lemma now reads that if $U$ contains no non-trivial connected subgroup of $G$, then every $a\\in \\boldsymbol{\\mu}$ which is nondegenerate is $U$-pure.\n\n\\\n\n\\begin{lemma}\\label{L:pure}\nLet $a\\in \\boldsymbol{\\mu}$.  Then $a$ is pure iff there is $\\nu > \\mathbb{N}$ such that $a^\\nu$ is defined, $a^\\nu \\notin \\boldsymbol{\\mu}$ and $a\\in G(\\nu)$.\n\\end{lemma}\n\n\\begin{proof}\nFirst suppose that $a$ is $V$-pure.  Let $\\nu = \\operatorname{ord}_V(a)$.  Then certainly $a^\\nu$ is defined.  Now since $a^{\\nu + 1}$ is defined, $a^\\nu \\notin \\boldsymbol{\\mu}$, else $a^{\\nu+1}=a^\\nu \\cdot a \\in \\boldsymbol{\\mu}$, contradicting  $a^{\\nu+1}\\notin V^*$.  However, $a^i\\in \\boldsymbol{\\mu}$ for $i=\\operatorname{o}(\\nu)$ by definition of $V$-purity.  Conversely, suppose one has $\\nu>\\mathbb{N}$ such that $a^\\nu$ is defined, $a^\\nu \\notin \\boldsymbol{\\mu}$ and $a\\in G(\\nu)$.  Since $a^\\nu \\notin \\boldsymbol{\\mu}$, there is a compact symmetric neighborhood $V$ of $1$ such that $V\\subseteq \\u_2$ and such that $a^\\nu \\notin V^*$.  Let $\\tau:=\\operatorname{ord}_V(a)$.  Then $\\tau < \\nu$, implying that $a\\in G(\\tau)$ and thus $a$ is $V$-pure.         \n\\end{proof}\n\n\\\n\n\\noindent Let $Q$ range over internal subsets of $G^*$ such that $Q\\subseteq \\boldsymbol{\\mu}$, $1\\in Q$, and $Q$ is symmetric.  If $\\nu$ is such that for every internal sequence $a_1,\\ldots,a_\\nu$ from $Q$, $a_1\\cdots a_\\nu$ is defined, then we define $Q^\\nu$ to be the internal subset of $G^*$ consisting of all $a_1\\cdots a_\\nu$ for $a_1,\\ldots,a_\\nu$ an internal sequence from $Q$.  In this situation, we say that $Q^\\nu$ is defined.  \n\n\\\n\n\\begin{df}\nIf for all $\\nu$, $Q^\\nu$ is defined and $Q^\\nu \\subseteq \\boldsymbol{\\mu}$, then we say that $Q$ is \\textbf{degenerate}.   If $Q^\\nu$ is defined and $Q^\\nu\\subseteq U^*$ for all $\\nu$, define ord$_U(Q)=\\infty$.  Else, define ord$_U(Q)=\\nu$ if $Q^\\nu$ is defined, $Q^\\nu \\subseteq U^*$, and $\\nu$ is the largest element of $\\mathbb{N}^*$ for which this happens.\n\\end{df}\n\n\\\n\n\\noindent Using an overflow argument similar to that in Lemma \\ref{L:Overflow}, we have that ord$_U(Q)>\\mathbb{N}$.  One can also check that if $\\operatorname{ord}_U(Q)\\in \\mathbb{N}^*$, then $Q^{\\operatorname{ord}_U(Q)+1}$ is defined.\n\n\\\n\n\\begin{lemma}\\label{L:singsubgroup}\nSuppose $Q^\\nu \\nsubseteq \\boldsymbol{\\mu}$ for some $\\nu=\\operatorname{o}(\\operatorname{ord}_U(Q))$.  Then $U$ contains a non-trivial connected subgroup of $G$.\n\\end{lemma}\n\n\\begin{proof}\nAs in the proof of Lemma \\ref{L:connsubgroup}, the set $$G_U(Q)=\\{\\operatorname{st}(a) \\ | \\ a\\in Q^\\nu \\text{ for some }\\nu=\\operatorname{o}(\\operatorname{ord}_U(Q))\\}$$ is a union of connected subsets of $U$, each containing $1$, and thus itself is a connected subset of $U$.  To see that $G_U(Q)$ is a group, suppose that $a\\in Q^\\nu$ and $b\\in Q^\\eta$ for $\\nu,\\eta=\\operatorname{o}(\\operatorname{ord}_U(Q))$.  Then $\\nu+\\eta=\\operatorname{o}(\\operatorname{ord}_U(Q))$ and so the concatenation of the sequences representing $a$ and $b$ represents an element of $G^*$, which must necessarily be $a\\cdot b$.  Hence $a\\cdot b\\in Q^{\\nu+\\eta}$ and $\\operatorname{st}(a)\\cdot \\operatorname{st}(b)=\\operatorname{st}(a\\cdot b)\\in G_U(Q)$.  That $G_U(Q)$ is closed under inverses is an easy consequence of Lemma \\ref{L:EZFacts}.\n\\end{proof}\n  \n\\\n\n\\begin{df}\nWe say that $Q$ is \\textbf{$U$-pure} if $Q$ is nondegenerate and if $Q^\\nu\\subseteq \\boldsymbol{\\mu}$ for all $\\nu=\\operatorname{o}(\\operatorname{ord}_U(Q))$.  We say that $Q$ is \\textbf{pure} if it is $V$-pure for some compact symmetric neighborhood $V$ of $1$ in $G$ such that $V\\subseteq \\u_2$.\n\\end{df}  \n  \n\\\n\n\\noindent The previous lemma now reads that if $U$ contains no nontrivial connected subgroups of $G$, then every nondegenerate $Q$ as above is $U$-pure.\n\n\\\n\n\\begin{lemma}\n$Q$ is pure iff there is some $\\tau > \\mathbb{N}$ such that $Q^\\tau$ is defined, $Q^\\tau \\nsubseteq \\boldsymbol{\\mu}$ and $Q^\\nu\\subseteq \\boldsymbol{\\mu}$ for all $\\nu=\\operatorname{o}(\\tau)$.  \n\\end{lemma}\n\n\\begin{proof}\nFirst suppose $Q$ is $V$-pure.  Let $\\tau=\\operatorname{ord}_V(Q)$.  Then $\\tau > \\mathbb{N}$, $Q^\\tau$ is defined, $Q^\\tau \\nsubseteq \\boldsymbol{\\mu}$ (otherwise $Q^{\\tau +1}\\subseteq \\boldsymbol{\\mu}\\subseteq V^*$, contradicting the definition of $\\tau$), and $Q^\\nu \\subseteq \\boldsymbol{\\mu}$ for $\\nu=\\operatorname{o}(\\tau)$ by definition of being $V$-pure.  Conversely, suppose we have $\\tau>\\mathbb{N}$ such that $Q^\\tau$ is defined, $Q^\\tau \\nsubseteq \\boldsymbol{\\mu}$, and $Q^\\nu\\subseteq \\boldsymbol{\\mu}$ for $\\nu=\\operatorname{o}(\\tau)$.  Since $Q^\\tau \\nsubseteq \\boldsymbol{\\mu}$, we can find a compact symmetric neighborhood $V$ of $1$ in $G$ such that $V\\subseteq \\u_2$ and $Q^\\tau \\nsubseteq V$.  Let $\\eta:=\\operatorname{ord}_V(Q)$.  Then $\\eta < \\tau$, implying that $Q^\\nu \\subseteq \\boldsymbol{\\mu}$ for all $\\nu =\\operatorname{o}(\\eta)$, that is $Q$ is $V$-pure.\n\\end{proof}\n\n\\\n\n\\begin{df}\n$G$ \\textbf{has no small connected subgroups}, henceforth abbreviated $G$ is $\\operatorname{NSCS}$, if there is a neighborhood of $1$ in $G$ that contains no connected subgroup of $G$ other than $\\{1\\}$.  We say that $G$ is \\textbf{pure} if every nondegenerate $Q$ as above is pure.  \n\\end{df}\n\n\\\n\n\\begin{cor}\nIf $G$ is $\\operatorname{NSCS}$, then $G$ is pure.\n\\end{cor}\n\n\\\n\n\\noindent \\textbf{Pure infinitesimals and local 1-parameter subgroups}\n\n\\\n\n\\noindent  Suppose $a\\in \\boldsymbol{\\mu}$ is $U$-pure and $0<\\nu=\\operatorname{O}(\\tau)$, where $\\tau:=\\operatorname{ord}_U(a)$.  Let $$\\Sigma_{\\nu,a,U}:=\\{r\\in \\r^{>0} \\ | \\ a^{[r\\nu]} \\text{ is defined and } a^i\\in U^* \\text{ if }|i|\\leq [r\\nu]\\}.$$  Since $\\nu=\\operatorname{O}(\\tau)$, $\\Sigma_{\\nu,a,U} \\not= \\emptyset$.  Let $r_{\\nu,a,U}=\\sup \\Sigma_{\\nu,a,U}$, where we let $r_{\\nu,a,U}=\\infty$ if $\\Sigma_{\\nu,a,U}=\\r^{>0}$.  Suppose $s<r_{\\nu,a,U}$.  Then $s\\in \\Sigma_{\\nu,a,U}$ and $a^{[s\\nu]}\\in U^*$.  Since $a^{[s\\nu]+1}$ is defined (see comment after Definition \\ref{D:ordelement}; all that was used there was that $U\\times U\\subseteq \\o$), we have that $a^{[s\\nu]+1}=a^{[s\\nu]}\\cdot a\\in G^*_{\\operatorname{ns}}$.  Hence we have that $a^{[s\\nu]}$ is defined and nearstandard for $s\\in (-r_{\\nu,a,U},0)$ as well since, for such $s$, we have that $[s\\nu]=-[(-s)\\nu]$ or $-[(-s)\\nu]-1$.\n\n\\\n\n\\begin{lemma}\\label{L:regelement}\nThe map $X:(-r_{\\nu,a,U},r_{\\nu,a,U})\\rightarrow G$ given by $X(s):=\\operatorname{st}(a^{[s\\nu]})$ is a local 1-ps of $G$.\n\\end{lemma}\n\n\\begin{proof}\nThroughout this proof, we let $r:=r_{\\nu,a,U}$.  We first show that $X$ satisfies the additivity condition in the definition of a local 1-ps of $G$.  Note that the preceding arguments show that $\\operatorname{image}(X)\\subseteq \\u_2$.  Suppose first that $s_1,s_2,s_1+s_2\\in (0,r)$.  Since $[s_1\\nu]+[s_2\\nu]=[(s_1+s_2)\\nu]$ or $[(s_1+s_2)\\nu]-1$, we have that $a^{[s_1\\nu]+[s_2\\nu]}$ is defined and $\\operatorname{st}(a^{[(s_1+s_2)\\nu]})=\\operatorname{st}(a^{[s_1\\nu]+[s_2\\nu]})$.  Hence $X(s_1+s_2)=X(s_1)\\cdot X(s_2)$.  The case that $s_1,s_2,s_1+s_2\\in (-r,r)$ with $s_1\\cdot s_2 <0$ works similarly, using Corollary \\ref{C:EZCorollary}.  Using the fact that $X(-s)=X(s)^{-1}$, we get the additivity property for $s_1,s_2\\in (-r,0)$ for which $s_1+s_2\\in (-r,0)$.\n\n\\\n\n\\noindent It remains to show that $X$ is continuous.  By Lemma \\ref{L:homog}, it is enough to show that $X$ is continuous at $0$.  Let $V$ be a compact neighborhood of $1$ in $G$.  If $i=\\operatorname{o}(\\nu)$, then $a^i$ is defined and $a^i\\in \\boldsymbol{\\mu} \\subseteq V^*$.  By saturation, we have $n$ such that $\\frac{1}{n}<r$ and such that if $|\\frac{i}{\\nu}|<\\frac{1}{n}$, then $a^i$ is defined and $a^i \\in V^*$.  Hence $X(s)\\in V$ for $|s|<\\frac{1}{n}$.\n\n\\end{proof}\n\n\\\n \n\\noindent We end this section with an extension lemma which will be used later in the paper.  From now on, $I:=[-1,1]\\subseteq \\r$.\n\n\\\n\n\\begin{lemma}\\label{L:extension}\nSuppose $X:I\\rightarrow G$ is a continuous function such that for all $r,s\\in I$, if $r+s\\in I$, then $(X(r),X(s))\\in \\o$ and $X(r+s)=X(r)\\cdot X(s)$.  Further suppose that $X(I)\\subseteq \\u_4$.  Then there exists $\\epsilon \\in \\r^{>0}$ and a local 1-parameter subgroup $\\overline{X}:(-1-\\epsilon,1+\\epsilon)\\rightarrow G$ of $G$ such that $\\overline{X}|I=X$.  \n\\end{lemma} \n\n\\begin{proof}\nFix $\\epsilon\\in (0,\\frac{1}{2})$.  We claim that $\\overline{X}:(-1-\\epsilon,1+\\epsilon)\\rightarrow G$ defined by $\\overline{X}|I=X$ and, for $0<\\delta < \\epsilon$, $\\overline{X}(1+\\delta)=X(1)\\cdot X(\\delta)$ and $\\overline{X}(-1-\\delta)=\\overline{X}(1+\\delta)^{-1}$ satisfies the conclusion of the lemma.  By Lemma \\ref{L:homog}, it is enough to show that $(\\overline{X}(r),\\overline{X}(s))\\in \\o$ and $\\overline{X}(r)\\overline{X}(s)=\\overline{X}(r+s)$ for all $r,s\\in (-1-\\epsilon,1+\\epsilon)$ for which $r+s\\in (-1-\\epsilon,1+\\epsilon)$.  We first note that $\\overline{X}(r),\\overline{X}(s)\\in \\u_4^2$ so that $(\\overline{X}(r),\\overline{X}(s))\\in \\o$ is immediate.  The additivity property of $\\overline{X}$ splits up into several cases.  \n\n\n\\noindent \\textbf{Case 1a:}  $r,s,r+s\\in [0,1]$.  This case is trivial.\n\n\\noindent \\textbf{Case 1b:} $r,s\\in [0,1+\\epsilon)$, $r+s\\in (1,1+\\epsilon)$.  Then\n$$\\overline{X}(r+s)=X(1)X(r+s-1)=X(1)X(r-1)X(s)=\\overline{X}(r)\\overline{X}(s).$$  \n\n\\noindent \\textbf{Case 2a:}  $r,s,r+s\\in [-1,0]$.  This case is trivial.\n\n\\noindent \\textbf{Case 2b:}  $r,s\\in (-1-\\epsilon,0]$, $r+s\\in (-1-\\epsilon,-1)$.  Then\n$$\\overline{X}(r+s)=\\overline{X}(-s-r)^{-1}=(\\overline{X}(-s)\\overline{X}(-r))^{-1}=\\overline{X}(r)\\overline{X}(s).$$\n\n\\noindent \\textbf{Case 3a:}  $r\\in [0,1]$, $s\\in [-1,0]$.  This case is trivial.\n\n\\noindent \\textbf{Case 3b:}  $r\\in [0,1]$, $s\\in (-1-\\epsilon,-1)$.  Then\n\\begin{align}\n\\overline{X}(r+s)&=\\overline{X}(-r-s)^{-1}\\notag \\\\ \\notag\n\t\t\t &=(X(1)X(-r-s-1))^{-1}\\\\ \\notag\n\t\t\t &=X(r+s+1)X(-1)\\\\ \\notag\n\t\t\t &=X(r)X(s+1)X(-1)\\\\ \\notag\n\t\t\t &=X(r)(X(1)X(-s-1))^{-1}\\\\ \\notag\n\t\t\t &=\\overline{X}(r)\\overline{X}(s).\\notag\n\\end{align}      \n\n\\noindent \\textbf{Case 3c:}  $r\\in (1,1+\\epsilon)$, $s\\in [-1,0]$.  Then by Case 3b, we have\n$$\\overline{X}(r+s)=\\overline{X}(-s-r)^{-1}=(\\overline{X}(-s)\\overline{X}(-r))^{-1}=\\overline{X}(r)\\overline{X}(s).$$\n\n\\noindent \\textbf{Case 3d:}  $r\\in (1,1+\\epsilon)$, $s\\in (-1-\\epsilon,-1)$.  For this case, we will need the observation that for any $\\delta \\in (-\\epsilon,\\epsilon)$, we have $X(1)X(\\delta)=X(\\delta)X(1)$.  We shall only prove this for $\\delta\\geq 0$, the proof being similar for $\\delta<0$.  The following chain of equalities verifies the observation:\n\n\\begin{align}\nX(1)X(\\delta)&=X(1-\\delta)X(\\delta)X(\\delta)\\notag \\\\ \\notag\n\t\t    &=X(1-\\delta)X(2\\delta)\\\\ \\notag\n\t\t    &=X(\\delta)X(1-2\\delta)X(2\\delta)\\notag \\\\\n\t\t    &=X(\\delta)X(1).\\notag\n\\end{align}\n\nNow, we have\n\\begin{align}\n\\overline{X}(r+s)&=X(r+s)X(1)X(-1) \\notag \\\\ \\notag\n\t                    &=X(1)X(r+s)X(-1) \\\\ \\notag\n\t                    &=X(1)X(r-1)X(s+1)X(-1)\\\\ \\notag\n\t                    &=\\overline{X}(r)\\overline{X}(s).\\notag\n\\end{align}\n\n\\noindent \\textbf{Case 4:}  $r\\in (-1-\\epsilon,0]$, $s\\in [0,1+\\epsilon)$.  This case is treated exactly like Case 3.\n\\end{proof}\n\n\\\n\n\\section{Consequences of $\\operatorname{NSS}$} \n\n\\\n\n\\noindent In this section we assume $G$ is $\\operatorname{NSS}$.  \n\n\\\n\n\\noindent  \\textbf{Special Neighborhoods}\n\n\\\n\n\\noindent The next lemma is from \\cite{K}, and we repeat its proof, with a more detailed\njustification of some steps.\n\n\\\n\n\\begin{lemma}\\label{L:specneigh}\nThere is a neighborhood $V$ of $1$ such that $V \\subseteq \\u_2$ and for all $x,y\\in V$, if $x^2=y^2$, then $x=y$.  \n\\end{lemma} \n\n\\begin{proof}\nChoose a compact symmetric neighborhood $U$ of $1$ in $G$ with $U\\subseteq \\u_6$ such that $U$ contains no subgroups of $G$ other than $\\{1\\}$.  Choose a symmetric open neighborhood $W$ of $1$ in $G$ such that $W^5\\subseteq U$.  Since $U$ is compact, we can find a symmetric open neighborhood $V$ of $1$ in $G$ such that $V^2\\subseteq W$ and $gVg^{-1}\\subseteq W$ for all $g\\in U$.  We claim that this $V$ fulfills the desired condition.  \n\n\\\n\n\\noindent Suppose $x,y\\in V$ are such that $x^2=y^2$.  Let $a:=x^{-1}y\\in V^2\\subseteq W \\subseteq U$.  Since $V\\subseteq \\u_5$, we have $$axa=(x^{-1}y)x(x^{-1}y)=x^{-1}y^2=x^{-1}x^2=x.$$\n\n\\\n\n\\noindent Claim:  For all $n$, $a^n$ is defined, $a^n\\in U$ and $a^n=xa^{-n}x^{-1}$.\n\n\\\n\n\\noindent We prove the claim by induction.  The case $n=1$ follows from the fact that $axa=x$.  For the inductive step, suppose the assertion holds for all $m\\in \\{1,\\ldots,n\\}$.  In order to show that $a^{n+1}$ is defined, it suffices to show that $(a^i,a^j)\\in \\o$ for all $i,j\\in \\{1,\\ldots,n\\}$ such that $i+j=n+1$.  However, by the inductive hypothesis, $(a^i,a^j)\\in U \\times U \\subseteq \\o$, proving that $a^{n+1}$ is defined.  Since $U\\subseteq \\u_6$, we have $$a^{n+1}=a^n\\cdot a=(xa^{-n}x^{-1})(xa^{-1}x^{-1})=xa^{-n-1}x^{-1}.$$  It remains to show that $a^{n+1}\\in U$.  First suppose that $n+1$ is even, say $n+1=2m$.  Then \n$$a^{n+1}=a^m\\cdot a^m=xa^{-m}x^{-1}a^m\\in xW\\subseteq W^2\\subseteq U.$$  Now if $n+1$ is odd, then $$a^{n+1}=a^n \\cdot a\\in W^2\\cdot W^2 \\subseteq U.$$  \n\n\\\n\n\\noindent This finishes the proof of the claim.  The claim yields a subgroup $a^\\mathbb{Z}$ of $G$ with $a^\\mathbb{Z}\\subseteq U$.  Thus $a=1$ and so $x=y$.\n\n\n\\end{proof}\n\n\\\n\n\\begin{df}\nA \\textbf{special neighborhood of $G$} is a compact symmetric neighborhood $\\u$ of $1$ in $G$ such that $\\u\\subseteq \\u_2$, $\\u$ contains no nontrivial subgroup of $G$, and for all $x,y\\in \\u$, if $x^2=y^2$, then $x=y$.\n\\end{df}\n\n\\\n\n\\noindent By the last lemma, we know that $G$ has a special neighborhood.  In the rest of this section, we fix a special neighborhood $\\u$ of $G$.  Then every $a\\in \\boldsymbol{\\mu} \\setminus \\{1\\}$ is $\\u$-pure.  For $a\\in G^*$, we set ord$(a):=\\operatorname{ord}_\\mathcal{U}(a).$  \n\n\\\n\n\\begin{lemma}\nLet $a\\in G^*$.  Then $\\operatorname{ord}(a)$ is infinite iff $a\\in \\boldsymbol{\\mu}$.\n\\end{lemma}\n\n\\begin{proof}\nWe remarked earlier that if $a\\in \\boldsymbol{\\mu}$, then ord$(a)$ is infinite.  Now suppose ord$(a)$ is infinite.  Then, for all $k\\in \\mathbb{Z}$, $a^k$ is defined and $a^k\\in \\u^*$.  It is easy to prove by induction that $\\operatorname{st}(a)^k$ is defined and $\\operatorname{st}(a)^k \\in \\u$ for all $k\\in \\mathbb{Z}$.  This implies that $\\operatorname{st}(a)=1$ since $\\u$ is a special neighborhood.\n\\end{proof}\n\n\\\n\n\\begin{lemma}\\label{L:sigparam}\nSuppose $\\sigma > \\mathbb{N}$ and $a\\in G(\\sigma)$.  Then:\n\\begin{enumerate}\n\\item if $a\\not= 1$, then $a$ is $\\u$-pure and $\\sigma=\\operatorname{O}(\\operatorname{ord}(a))$;\n\\item if $i=\\operatorname{o}(\\sigma)$, then $a^i$ is defined and $a^i \\in \\boldsymbol{\\mu}$;\n\\item Let $\\Sigma_a:=\\Sigma_{\\sigma,a,\\u}$ and $r_a:=r_{\\sigma,a,\\u}$.  Let $X_a:(-r_a,r_a)\\rightarrow G$ be defined by $X(s):=\\operatorname{st}(a^{[s\\sigma]})$.  Then $X_a$ is a local $1$-ps of $G$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nIt has already been remarked why $a$ is $\\u$-pure.  If $\\operatorname{ord}(a)=\\operatorname{o}(\\sigma)$, then $a^{\\operatorname{ord}(a)}\\in \\boldsymbol{\\mu}$, a contradiction.  Hence, $\\sigma =\\operatorname{O}(\\operatorname{ord}(a))$.  By (1), if $i=\\operatorname{o}(\\sigma)$, then $i=\\operatorname{o}(\\operatorname{ord}(a))$, whence $a^i$ is defined and $a^i\\in \\boldsymbol{\\mu}$ since $a$ is $\\u$-pure.  This proves (2).  (3) follows immediately from Lemma \\ref{L:regelement}.\n\\end{proof}\n\n\\\n\n\\begin{cor}\nSuppose $G$ is not discrete.  Then $L(G)\\not= \\{\\O\\}$.\n\\end{cor}\n\n\\begin{proof}\nTake $a\\in \\boldsymbol{\\mu}\\setminus \\{1\\}$.  Set $\\sigma:=\\operatorname{ord}(a)$.  Since $a\\in \\boldsymbol{\\mu}$, we must have $\\sigma > \\mathbb{N}$.  Since $a\\in G(\\sigma)$, we have $[X_a]\\in L(G)$, where $X_a$ is as defined in Lemma \\ref{L:sigparam}.  We need to show that $[X_a]\\not= \\O$.  It suffices to show that for every $n$ such that $\\frac{1}{n}\\in \\Sigma_a$, $a^{[\\frac{1}{n}\\sigma]}\\notin \\boldsymbol{\\mu}$.  Let $t:=[\\frac{1}{n}\\sigma]$.  Then $t=\\frac{1}{n}\\sigma-\\epsilon$, with $\\epsilon \\in [0,1)^*$.  Towards a contradiction, suppose that $a^t\\in \\boldsymbol{\\mu}$.  Then since $nt\\leq \\sigma$, $a^{nt}$ is defined and $a^{nt}=(a^t)^n\\in \\boldsymbol{\\mu}$ by Lemma \\ref{L:EZFacts}.  Also $n\\epsilon \\in \\mathbb{N}^*$ and $n\\epsilon < n$, whence $a^{n\\epsilon}\\in\\boldsymbol{\\mu}$.  But then $a^\\sigma=a^{nt+n\\epsilon}=a^{nt}\\cdot a^{n\\epsilon}\\in \\boldsymbol{\\mu}$, a contradiction.\n\\end{proof}\n\n\\\n\n\\begin{lemma}\\label{L:Gsigma properties}Let $a\\in G(\\sigma)\\setminus \\{1\\}$.  Then:\n\\begin{enumerate}\n\\item $a^{-1}\\in G(\\sigma)$ and $[X_{a^{-1}}]=(-1)\\cdot [X_a]$;\n\\item $b\\in \\boldsymbol{\\mu} \\Rightarrow bab^{-1}\\in G(\\sigma)$ and $[X_{bab^{-1}}]=[X_a]$;\n\\item $[X_a]=\\O \\Leftrightarrow a\\in G^{\\operatorname{o}}(\\sigma)$;\n\\item $L(G)=\\{ [X_a] \\ | \\ a\\in G(\\sigma)\\}$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\n(1) is straightforward.  As to (2), let $b\\in \\boldsymbol{\\mu}$ and let $\\tau:=\\operatorname{ord}(a)$.  We know that $\\sigma = \\operatorname{O}(\\tau)$.  One can show by internal induction on $\\eta$ that $(bab^{-1})^\\eta$ is defined and equal to $ba^\\eta b^{-1}$ for all $\\eta \\leq \\tau$.  Thus, for $i=\\operatorname{o}(\\sigma)$, we have $(bab^{-1})^i \\in \\boldsymbol{\\mu}$ and hence $bab^{-1}\\in G(\\sigma)$.  Again, for $r\\in \\Sigma_a$, we have that $(bab^{-1})^{[r\\sigma]}$ is defined and equals $ba^{[r\\sigma]}b^{-1}$, whence $[X_a]=[X_{bab^{-1}}]$.  \n\n\\\n\n\\noindent The proof of (3) is immediate from the definitions and Lemma \\ref{L:lilo}.  Now suppose $\\mathbb{X}\\in L(G)$ and $X\\in \\mathbb{X}$.  Suppose $\\operatorname{domain}(X)=(-r,r)$ and consider the nonstandard extension $X:(-r,r)^*\\rightarrow G^*$ of $X$.  Let $a=X(\\frac{1}{\\sigma})\\in \\boldsymbol{\\mu}$.  We claim that $\\mathbb{X}=[X_a]$.  To see this, let $s<\\min(r,r_a)$ and let $\\epsilon$ be an infinitesimal element of $\\r^*$ such that $s=\\frac{[s\\sigma]}{\\sigma}+\\epsilon$.  Then:\n\n\\begin{align}\nX(s)&=\\operatorname{st}(X(\\frac{[s\\sigma]}{\\sigma}))\\notag \\\\\n       &=\\operatorname{st}(X(\\frac{1}{\\sigma})^{[s\\sigma]})\\notag \\\\\n       &=\\operatorname{st}(a^{[s\\sigma]})\\notag \\\\\n       &=X_a(s) \\notag \\\\ \\notag\n\\end{align}\nHence we have shown that $X$ and $X_a$ agree on $(-r,r)\\cap (-r_a,r_a)$.  This proves (4).\n\n\\end{proof}\n\n\\noindent \\textbf{The Local Exponential Map}\n\n\\\n\n\n\\noindent Consider the following sets:\n\n$$\\k:=\\{\\mathbb{X} \\in L(G) \\ | I\\subseteq \\operatorname{domain}(\\mathbb{X}) \\text{ and }\\mathbb{X}(I)\\subseteq \\u\\}$$ and \n\n$$K:=\\{\\mathbb{X}(1) \\ | \\ \\mathbb{X}\\in \\k\\}.$$   \n\n\\\n\n\\begin{lemma}\nFor every $\\mathbb{X} \\in L(G)$, there is $s \\in (0,1)$ such that $s \\cdot \\mathbb{X} \\in \\k$. \n\\end{lemma}\n\n\\begin{proof}\nLet $X\\in \\mathbb{X}$ and suppose $X:(-r,r)\\rightarrow G$.  If $r\\leq 1$, pick any $0<s_1 <r$ and then $s_1 \\cdot X:(-\\frac{r}{s_1},\\frac{r}{s_1})\\rightarrow G$ will have $I$ contained in its domain.  If $r>1$, let $s_1=1$.  By continuity of $s_1 \\cdot X$, there is $0<s _2 < 1$ such that $(s_2 \\cdot (s_1 \\cdot X))(I)\\subseteq \\u$.  So for $s:=s_1\\cdot s_2$, $s \\cdot \\mathbb{X} \\in \\k$.\n\\end{proof}\n\n\\\n\n\\begin{lemma}\nThe map $\\mathbb{X} \\rightarrow \\mathbb{X}(1):\\k \\rightarrow K$ is bijective.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathbb{X}_1,\\mathbb{X}_2$ be in $\\k$ with representatives $X_1,X_2$ containing $I$ in their domains.  Suppose $X_1(1)=X_2(1)$.  Then $(X_1(\\frac{1}{2}))^2=(X_2(\\frac{1}{2}))^2$, implying that $X_1(\\frac{1}{2})=X_2(\\frac{1}{2})$ since $\\u$ is a special neighborhood.  Inductively, we have $X_1(\\frac{1}{2^n})=X_2(\\frac{1}{2^n})$ for all $n$, and thus $X_1(\\frac{k}{2^n})=X_2(\\frac{k}{2^n})$ for all $k\\in \\mathbb{Z}$ such that $|\\frac{k}{2^n}|\\leq 1$.  By density, we have $[X_1]=[X_2]$.\n\\end{proof}\n\n\\\n\n\\noindent From now on, for $\\mathbb{X} \\in \\k$, we let $E(\\mathbb{X})=\\mathbb{X}(1)$.  So the last lemma now reads $E:\\k \\rightarrow K$ is a bijection.  We will have much more to say about this \\textit{local exponential map} later and it will be a crucial tool in our solution of the Local H5.\n\n\\\n\n\\noindent \\textbf{A Countable Neighborhood Basis of the Identity}\n\n\\\n\n\\noindent For $Q$ as earlier, we put $\\operatorname{ord}(Q):=\\operatorname{ord}_\\u(Q)$.  Extend this to the standard setting as follows:  for symmetric $P\\subseteq G$ with $1\\in P$, let $\\operatorname{ord}(P)$ be the largest $n$ such that $P^n$ is defined and $P^n \\subseteq \\u$ if there is such an $n$ and set $\\operatorname{ord}(P):=\\infty$ if $P^n$ is defined and $P^n\\subseteq \\u$ for all $n$.\n\n\\\n\n\\noindent We set $V_n:=\\{x\\in G \\ |\\  \\operatorname{ord} (x) \\geq n\\}$.  Notice that for all $n$, $$p_1^{-1}(\\u)\\cap \\ldots \\cap p_n^{-1}(\\u)\\subseteq V_n \\subseteq \\u_n.$$\n\n\\\n\n\\begin{lemma}\n$(V_n:n\\geq 1)$ is a decreasing sequence of compact symmetric neighborhoods of $1$ in $G$, $\\operatorname{ord}(V_n)\\rightarrow \\infty$ as $n\\rightarrow \\infty$, and $\\{V_n:n\\geq 1\\}$ is a countable neighborhood basis of $1$ in $G$.\n\\end{lemma}\n\n\\begin{proof}\nFor $\\sigma > \\mathbb{N}$, consider the internal set $$V_\\sigma:=\\{g\\in G^* \\ | \\operatorname{ord}(g)\\geq \\sigma\\}.$$  Note that $V_\\sigma \\subseteq \\boldsymbol{\\mu}$, so given any neighborhood $U$ of $1$ in $G$, we have $x_1\\cdots x_m$ is defined and in $U^*$ for all $m$ and $x_1,\\ldots,x_m\\in V_\\sigma$.  It follows that for any neighborhood  $U$ of $1$ in $G$ and any $m$ we have $(V_n)^m$ is defined and contained in $U$ for all sufficiently large $n$.\n\\end{proof}\n\n\\\n\n\\section{Local Gleason-Yamabe Lemmas}\n\n\\noindent The trickiest and most technical part of the original solution of Hilbert's fifth problem concerned several ingenious lemmas proved by Gleason and Yamabe.  These lemmas really drive much of the proof and obtaining local versions of them is a crucial part of our current project.  Fortunately, it is not too difficult to obtain local versions of these lemmas, although carrying out some of the details can become laborious.\n\n\\\n\n\\noindent Throughout this section, we work with the following assumptions.  We suppose $\\u \\subseteq \\mathcal{W}$ are compact symmetric neighborhoods of $1$ such that $\\mathcal{W} \\subseteq \\u_M$ and $\\u^P \\subseteq \\mathcal{W}$ for fixed positive integers $P\\leq M$ large enough for all of the arguments below to make sense.  (A careful analysis of the arguments below could give precise values for $M$ and $P$, but this endeavor became quite tedious.)  We let $Q\\subseteq \\u$ be symmetric such that $1\\in Q$, $N:=\\operatorname{ord}_\\u(Q)\\not=\\infty$, and $Q^{N+1}$ is defined.  We first define the map $\\Delta=\\Delta_Q:G\\rightarrow [0,1]$ by \n\\begin{itemize}\n\\item $\\Delta(1)=0$;\n\\item $\\Delta(x)=\\frac{i}{N+1}$ if $x\\in Q^i \\setminus Q^{i-1}$, $1\\leq i \\leq N$;\n\\item $\\Delta(x)=1$ if $x\\notin Q^N$.\n\\end{itemize}\n\n\\noindent Then for all $x\\in G$, \n\n\\begin{itemize}\n\\item $\\Delta(x)=1$ if $x\\notin \\u$;\n\\item $|\\Delta(ax)-\\Delta(x)|\\leq \\frac{1}{N}$ for all $a\\in Q$ such that $(a,x)\\in \\o$.\n\\end{itemize}\n\n\\\n\n\\noindent To smooth out $\\Delta$, fix a continuous function $\\tau:G\\rightarrow [0,1]$ such that \n$$\\tau(1)=1, \\qquad \\tau(x)=0 \\text{ for all }x\\in G\\setminus \\u.$$\n\n\\\n\n\\noindent It is important later that $\\tau$ depends only on $\\u$, not on $Q$. \n\n\\\n\n\\noindent Next define $\\theta=\\theta_Q:G\\rightarrow [0,1]$ by setting \n\n\n\\[\n\t\\theta(x)=\n\t\\begin{cases}\n\t\\sup_{y\\in \\u}(1-\\Delta(y))\\tau(y^{-1}x)\t&\\text{if $x\\in \\mathcal{W}$}\\\\\n\t0 \t&\\text{if $x\\in G\\setminus \\mathcal{W}$}\n\t\\end{cases}\n\\]\n\n\\\n\n\\begin{lemma} The following properties hold for the above defined functions.\n\\begin{enumerate}\n\\item $\\theta$ is continuous and $\\theta(x)=0$ if $x\\notin \\u^2$;\n\\item $0\\leq \\tau \\leq \\theta \\leq 1$;\n\\item $|\\theta(ax)-\\theta(x)|\\leq \\frac{1}{N}$ for $a\\in Q$ if $(a,x)\\in \\o$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nIt is clear that $\\theta(x)=0$ if $x\\notin \\u^2$.  To see that $\\theta$ is continuous, let $x\\in G$ and $a\\in \\boldsymbol{\\mu}$.  We must show that $\\theta(ax)-\\theta(x)$ is infinitesimal.  First note that if $x\\notin \\mathcal{W}$, then $ax\\notin \\mathcal{W}^*$; in this case, $\\theta(ax)=\\theta(x)=0$.  So assume $x\\in \\mathcal{W}$.  If $ax\\notin \\mathcal{W}$, then $x\\notin \\operatorname{interior}(\\mathcal{W})$.  Since $\\u^2 \\subseteq \\operatorname{interior}(\\mathcal{W})$, we conclude that $x\\notin \\u^2$ and so $\\theta(x)=\\theta(ax)=0$.  Now suppose $ax\\in \\mathcal{W}^*$, so we can use the first clause in the definition of $\\theta$ for $\\theta(x)$ and $\\theta(ax)$.  In this case, $\\theta(ax)-\\theta(x)$ is infinitesimal by the continuity of $\\tau$.  $(2)$ is also clear from the definition.  \n\n\\\n\n\\noindent We now prove (3).  Let $a\\in Q$ and suppose $(a,x)\\in \\o$.  If $x\\notin \\mathcal{W}$, then $ax\\notin \\u^2$, else $x\\in \\u^3 \\subseteq \\mathcal{W}$, a contradiction.  Hence $\\theta(ax)=\\theta(x)=0$ in this case.  Now if $x\\in \\mathcal{W} \\setminus \\u^3$, then $ax\\notin \\u^2$, and once again $\\theta(ax)=\\theta(x)=0$.  Hence we can suppose $x\\in \\u^3$ so that $ax\\in \\u^4\\subseteq \\mathcal{W}$ and so we can use the first clause in the definition of $\\theta$ for $\\theta(x)$ and $\\theta(ax)$.  Let $y\\in \\u$.  Then $(a^{-1},y)\\in \\o$ and \n$$|(1-\\Delta(a^{-1}y))-(1-\\Delta(y))|\\leq \\frac{1}{N}.$$  Since $y^{-1}ax$ is defined, we have $y^{-1}ax=(a^{-1}y)^{-1}x$, so \n$$|(1-\\Delta(y))\\tau(y^{-1}ax)-(1-\\Delta(a^{-1}y)\\tau((a^{-1}y)^{-1}x))|\\leq \\frac{1}{N}.$$  (3) now follows from noting that the sets $\\{(1-\\Delta(y))\\tau(y^{-1}x) \\ | \\ y \\in \\u\\}$ and $\\{(1-\\Delta(a^{-1}y))\\tau((a^{-1}y)^{-1}x) \\ | \\ y \\in \\u, a^{-1}y\\in \\u\\}$ have the same supremum.\n\\end{proof}\n\n\n\\\n\n\\noindent Let $C:=\\{f:G\\rightarrow \\r \\ | \\ f \\text{ is continuous and }\\operatorname{supp}(f)\\subseteq \\mathcal{W}^2\\}.$  Then $C$ is a real vector space and we equip it with the norm given by \n$$\\|f\\|:=\\sup \\{|f(x)| \\ | \\ x\\in G\\}.$$\n\n\\noindent If $f\\in C$ is such that $\\operatorname{supp}(f)\\subseteq \\mathcal{W}$ and $a\\in \\mathcal{W}$, then we can define the new function $a\\cdot f$ by the rule\n\n\\[\n\t(a\\cdot f)(x)=\n\t\\begin{cases}\n\tf(a^{-1}x)\t\t&\\text{if $x\\in \\mathcal{W}^2$}\\\\\n\t0\t\t\t&\\text{otherwise}.\n\t\\end{cases}\n\\]\n\n\\noindent It is easy to see that $a\\cdot f\\in C$ and that if $f,g\\in C$ both have supports contained in $\\mathcal{W}$, then $a\\cdot (f+g)=a\\cdot f+a\\cdot g$.  We will often drop the $\\cdot$ and just write $af$ instead of $a\\cdot f$.  This should not be confused with the function $rf$ given by $(rf)(x)=rf(x)$ for $r\\in \\r$, which is defined for any $f:G\\rightarrow \\r$.  \n\n\\\n\n\n\\begin{lemma}\\label{L:triangle}\nSuppose $a,b\\in \\mathcal{W}$ and $f\\in C$ is such that $\\operatorname{supp}(f)\\subseteq \\mathcal{W}$ and $\\operatorname{supp}(bf)\\subseteq \\mathcal{W}$.  Then:\n\\begin{enumerate}\n\\item[(i)] $\\|af\\|=\\|f\\|$;\n\\item[(ii)] $a(bf)=(ab)f$ and $$\\|(ab)f -f\\|\\leq \\|af -f\\|+\\|bf-f\\|.$$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nTo see (i), note that \n\\begin{align}\n\\|af\\|&=\\sup \\{|f(a^{-1}x)| \\ | \\ x\\in \\mathcal{W}^2\\}\\notag \\\\\n       &=\\sup \\{|f(a^{-1}x)| \\ | \\ x\\in \\mathcal{W}^2, \\ a^{-1}x\\in \\mathcal{W}\\}\\notag \\\\\n       &=\\sup \\{|f(y)| \\ | \\ y\\in \\mathcal{W}\\}\\notag \\\\\n       &=\\|f\\| \\notag \\\\ \\notag\n\\end{align}\n\n\\noindent We now prove (ii).  Suppose $x\\notin \\mathcal{W}^2$.  In this case, $((ab)f)(x)=0$ and since $a^{-1}x\\notin \\mathcal{W}$ and $\\operatorname{supp}(bf)\\subseteq \\mathcal{W}$, we also have $(a(bf))(x)=0$.  Now suppose $x\\in \\mathcal{W}^2$.  Then $((ab)f)(x)=f((ab)^{-1}x)=f((b^{-1}a^{-1})x)$ and $(a(bf))(x)=(bf)(a^{-1}x)$.  If $a^{-1}x\\notin \\mathcal{W}^2$, then $b^{-1}a^{-1}x\\notin \\mathcal{W}$;  in this case, we have $((ab)f)(x)=0$ and $(a(bf)(x)=0$.  If $a^{-1}x\\in \\mathcal{W}^2$, then $(a(bf))(x)=(bf)(a^{-1}x)=f(b^{-1}(a^{-1}x))=((ab)f)(x)$.  We now finish the proof of (ii) as in the global case, that is\n\n\\begin{align}\n\\|(ab)f-f\\|&=\\|a(bf)-af+af-f\\| \\notag \\\\\n              &\\leq \\|a(bf)-af\\| + \\|af-f\\| \\notag \\\\\n              &\\leq \\|a(bf-f)\\|+\\|af-f\\| \\notag \\\\\n              &=\\|bf-f\\|+\\|af-f\\| \\notag \\\\ \\notag\n\\end{align}\n\\end{proof}\n\n\\\n\n\\noindent Since $\\operatorname{supp}(\\theta)\\subseteq \\u^2\\subset \\mathcal{W}$, we can consider the function $a\\theta\\in C$ for any $a\\in \\mathcal{W}$.  We will often need the following equicontinuity result.\n\\begin{enumerate}\n\\item[(4)]For each $\\epsilon \\in \\r^{>0}$, there is a symmetric neighborhood $V_{\\epsilon}$ of $1$ in $G$, independent of $Q$, such that $V_{\\epsilon}\\subseteq \\u$ and $\\|a\\theta -\\theta \\|\\leq \\epsilon$ for all $a\\in V_{\\epsilon}$.\n\\end{enumerate}\n\n\\noindent To see this, let $\\epsilon \\in \\r^{>0}$ and note by the uniform continuity of $\\tau$, we have a neighborhood $U$ of $1$ in $G$ such that $|\\tau(g)-\\tau(h)|<\\epsilon$ for all $g,h\\in \\u$ such that $gh^{-1}\\in U$.  Now take a symmetric neighborhood $V_{\\epsilon}$ of $1$ in $G$ such that $V_{\\epsilon}\\subseteq \\u$ and $y^{-1}ay\\in U$ for all $(a,y)\\in \\u \\times V_{\\epsilon}$.  The claim is that this choice for $V_{\\epsilon}$ works.  Fix $a\\in V_\\epsilon$.  We must compute $|(a\\theta)(x)-\\theta(x)|$ for various $x$.  Let us first take care of some of the trivial computations.  If $x\\notin \\mathcal{W}^2$, then $(a\\theta)(x)=\\theta(x)=0$.  Next suppose that $x\\in \\mathcal{W}^2 \\setminus \\mathcal{W}$, so $\\theta(x)=0$.  If $a^{-1}x\\notin \\mathcal{W}$, then $(a\\theta)(x)=0$.  If $a^{-1}x\\in \\mathcal{W}$, then $a^{-1}x\\notin \\u^2$, else $x\\in \\u^3\\subseteq \\mathcal{W}$, a contradiction.  So in this case also, $(a\\theta)(x)=0$.  Now if $x\\in \\mathcal{W} \\setminus \\u^3$, then $a^{-1}x\\notin \\u^2$, so once again $(a\\theta)(x)=\\theta(x)=0$.  Thus the only real case of interest is when $x\\in \\u^3$ and hence $a^{-1}x\\in \\u^4\\subseteq \\mathcal{W}$.  In this case, we have $|\\tau(y^{-1}a^{-1}x)-\\tau(y^{-1}x)|<\\epsilon$ for any $y\\in \\u$.  We now have\n\n\\begin{align}\n|(a\\theta)(x)-\\theta(x)|&=\\big|\\sup_{y\\in \\u} (1-\\Delta(y))\\tau(y^{-1}a^{-1}x)-\\sup_{y\\in \\u}(1-\\Delta(y))\\tau(y^{-1}x)\\big| \\notag \\\\\n\t\t\t\t&\\leq \\sup_{y\\in \\u}(1-\\Delta(y))|\\tau(y^{-1}a^{-1}x)-\\tau(y^{-1}x)| \\notag \\\\\n\t\t\t\t&\\leq\\epsilon. \\notag \\\\ \\notag\n\\end{align}\n\n\\noindent This finishes the proof of (4).\n\n\\\n\n\\noindent In ~\\cite{MZ}, Montgomery and Zippin construct a linear functional $$f\\mapsto \\int f:C \\rightarrow \\r$$ with the following properties:\n\n\\begin{enumerate}\n\\item[(i)] For any $f\\in C$, one has $|\\int f|\\leq \\int |f|$;\n\\item[(ii)] (Left invariance) Suppose $f\\in C$ satisfies $\\operatorname{supp}(f)\\subseteq \\mathcal{W}$.  Let $a\\in \\mathcal{W}$.  Then $\\int af=\\int f$.\n\\end{enumerate}\n\n\\noindent The above defined functional will be referred to as the \\textbf{local Haar integral} on $G$.  As is the case for topological groups, the local Haar integral is unique up to multiplication by a positive real constant.   By the Riesz Representation Theorem, for each local Haar integral as above, we get a Borel measure $\\mu$ on a certain $\\sigma$-algebra of subsets of $\\mathcal{W}^2$.  We will refer to the measure corresponding to a local Haar integral as a \\textbf{local Haar measure} on $G$.  If a given integral induces the measure $\\mu$, then we will denote the integral of a function $f$ by $\\int f d \\mu$, or if we wish to specify the variable, by $\\int f(x)d\\mu(x)$.  By the left invariance of the integral, the measure $\\mu$ is left-invariant in the sense that for any measurable subset $V\\subseteq \\mathcal{W}$ and any $a\\in \\mathcal{W}$, we have $\\mu(aV)=\\mu(V)$.\n\n\\\n\n\\noindent With these remarks behind us, we can now proceed with our discussion.  Fix a continuous function $\\tau_1:G\\rightarrow [0,1]$ such that \n$$\\tau_1(x)=1 \\text{ for all }x\\in \\u^2, \\qquad \\tau_1(x)=0 \\text{ for all }x\\in G\\setminus \\u^3.$$\n\n\\noindent It is important later that $\\tau_1$ depends only on $\\u$, not on $Q$.\n\n\\\n\n\\noindent Note that $0\\leq \\theta \\leq \\tau_1$.  Take the unique local Haar measure $\\mu$ such that   $\\int \\tau_1(x)d\\mu(x)=1$.  Then in particular we have \n\n\\\n\n\\begin{enumerate}\n\\item[(5)] $0\\leq \\int \\theta(x)d\\mu(x) \\leq 1.$\n\\end{enumerate}\n\n\\\n\n\\noindent By (4) above, we have an open neighborhood $V\\subseteq \\u$ of $1$ in $G$, independent of $Q$, such that $\\theta^2(x)\\geq \\frac{1}{2}$ on $V$.  We now introduce the function $$\\phi=\\phi_Q:G\\rightarrow \\r$$ defined by \n\n\\[\n\t\\phi(x)=\n\t\\begin{cases}\n\t\\int \\theta(xu)\\theta(u)d\\mu(u)\t&\\text{if $x\\in \\mathcal{W}$}\\\\\n\t0 \t&\\text{if $x\\in G\\setminus \\mathcal{W}$}\n\t\\end{cases}\n\\]\n\n\\noindent It should be clear that $\\phi$ is continuous.  The following are other useful properties of $\\phi$:\n\n\\begin{enumerate}\n\\item[(6)]$\\operatorname{supp}(\\phi)\\subseteq \\u^4\\subseteq \\mathcal{W}$;\n\\item[(7)] $\\phi(1)\\geq \\frac{\\mu(V)}{2}$;\n\\item[(8)] if $a\\in Q$, then $\\|a\\phi - \\phi\\| \\leq \\frac{1}{N}$;\n\\item[(9)] $\\|a\\phi - \\phi\\|\\leq \\|a\\theta -\\theta\\|$ for all $a\\in \\mathcal{W}$.\n\\end{enumerate}\n\n\\noindent In verifying properties (8) and (9), one needs to perform the case distinctions that we have done earlier.  As before, though, there is only one case where the first clause of the definition applies for both $a\\phi$ and $\\phi$ and in this case the results easily follow from the properties of the local Haar integral and the earlier derived properties (1)-(5).  We use that $\\u^6\\subseteq \\mathcal{W}$ in showing that there is only one genuine calculation to perform.\n\n\\\n\n\\noindent We should also point out that, by (6), $\\operatorname{supp}(a\\phi-\\phi)\\subseteq \\u^5\\subseteq \\mathcal{W}$ for all $a\\in \\mathcal{W}$, and so for all $b\\in\\mathcal{W}$, we can consider the function $b(a\\phi-\\phi)$.  We are finally in the position to prove the local versions of the Gleason-Yamabe Lemmas.\n\n\\begin{lemma}\\label{L:gleason1}\nLet $c,\\epsilon \\in \\r^{>0}$.  Then there is a neighborhood $U=U_{c,\\epsilon}\\subseteq \\u$ of $1$ in $G$, independent of $Q$, such that for all $a\\in Q$, $b\\in U$, and $m\\leq cN$,\n$$\\|b\\cdot m(a\\phi-\\phi)-m(a\\phi-\\phi)\\|\\leq \\epsilon, \\qquad \\|m(a\\phi-\\phi)\\|\\leq c.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $a\\in Q$, $b\\in \\u$.  Note that $\\operatorname{supp}(b\\cdot m(a\\phi-\\phi)-m(a\\phi-\\phi))\\subseteq \\u^6$.  So let $x\\in \\u^6$ and set $y:=b^{-1}x$.  Then\n\\begin{align*}  (a\\phi - \\phi)(x)\\ &=\\ \\int [\\theta(a^{-1}xu)-\\theta(xu)]\\theta(u)\\ d\\mu(u)\\\\ \nb(a\\phi-\\phi)(x)=(a\\phi - \\phi)(y)\\ &=\\ \\int [\\theta(a^{-1}yu)-\\theta(yu)]\\theta(u)\\ d\\mu(u).\n\\end{align*}\n\n\\noindent Assuming $P$ is large enough so that $y^{-1}x\\in \\mathcal{W}$, we can use left-invariance of the integral to replace $u$ by $x^{-1}yu$ in the function of $u$ integrated in the first identity.  So,\n$$(a\\phi-\\phi)(x)=\\int [\\theta(a^{-1}yu)-\\theta(yu)]\\theta(x^{-1}yu)d\\mu(u).$$\n\n\\noindent Taking differences gives \n$$[b\\cdot(a\\phi - \\phi)-(a\\phi-\\phi)](x)=\\int [(a\\theta-\\theta)(yu)][(\\theta-y^{-1}x\\theta)(u)]d\\mu(u).$$\n\n\\noindent By (4), we can take the neighborhood $U_{c,\\epsilon}\\subseteq \\u$ of $1$ in $G$ so small that for all $b\\in U_{c,\\epsilon}$ and $x\\in\\u^6$ we have $y^{-1}x\\in \\u$ and \n$$\\|\\theta-y^{-1}x\\theta\\|<\\frac{\\epsilon}{c\\mu(\\u^3)}.$$\n\n\n\\noindent It is straightforward to see that this choice of $U_{c,\\epsilon}$ works.\n\\end{proof}\n\n\\\n\n\\begin{lemma}\\label{L:gleason2}\nWith $c,\\epsilon \\in \\r^{>0}$, let $U=U_{c,\\epsilon}$ be as in the previous lemma and let $a\\in Q$ and $m,n$ be such that $m\\leq cN$, $n>0$, $a^n$ is defined, and $a^i\\in U$ for $i\\in\\{0,\\ldots,n\\}$.  Then\n$$\\|(\\frac{m}{n})(a^n\\phi - \\phi)-m(a\\phi-\\phi)\\|\\leq \\epsilon.$$\n\\end{lemma}\n\n\\begin{proof}\nIt is routine to check that $m(a^n\\phi - \\phi)=\\sum_{i=0}^{n-1} a^im(a\\phi-\\phi)$.  Hence we have \n$$m(a^n\\phi -\\phi)-nm(a\\phi-\\phi)=\\sum_{i=0}^{n-1}(a^im(a\\phi-\\phi)-m(a\\phi - \\phi)).$$ \n\n\\noindent By the previous lemma, we have for $i=0,\\ldots,n-1$,\n$$\\|a^im(a\\phi-\\phi)-m(a\\phi-\\phi)\\|\\leq \\epsilon,$$ which gives the desired result by summing and dividing by $n$.\n\\end{proof}\n\n\\\n\n\\noindent Now suppose that $Q$ is a symmetric internal subset of $\\boldsymbol{\\mu}$ with $1\\in Q$ such that $N:=\\operatorname{ord}_\\u(Q)\\in \\mathbb{N}^*$.  Note that $N>\\mathbb{N}$.  Then everything we have done so far in this section transfers to the nonstandard setting and yields internally continuous functions $\\theta:G^*\\rightarrow [0,1]^*$ and $\\phi:G^*\\rightarrow \\r^*$ satisfying the internal versions of (1)-(9) and Lemmas \\ref{L:gleason1} and \\ref{L:gleason2}.  We now have the following lemma.\n\n\\begin{lemma}\\label{L:gleason3}\nSuppose $a\\in Q$, $\\nu=\\operatorname{O}(N)$, $a^\\nu$ is defined and $a^\\sigma \\in \\boldsymbol{\\mu}$ for all $\\sigma \\leq \\nu$.  Then $\\|\\nu(a\\phi-\\phi)\\|$ is infinitesimal.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\epsilon \\in \\r^{>0}$.  Choose $c\\in \\r^{>0}$ such that $\\nu \\leq cN$.  Then by Lemma \\ref{L:gleason2}, with $m=n=\\nu$, we have $$\\|(a^\\nu\\phi -\\phi)-\\nu(a\\phi-\\phi)\\|\\leq \\epsilon.$$\n\n\\noindent Also $\\|a^\\nu \\phi-\\phi\\|\\leq \\|a^\\nu \\theta -\\theta\\|<\\epsilon$ by (9) and (4), so $\\|\\nu(a\\phi-\\phi)\\|\\leq 2\\epsilon.$\n\\end{proof}\n\n\\\n\n\\section{Consequences of the Gleason-Yamabe Lemmas} \n\n\\noindent  In this section, we bear the fruits of our labors from the previous section.  We derive some consequences of the Gleason-Yamabe lemmas which allow us to put a group structure on $L(G)$.  From this group structure on $L(G)$, we obtain the important corollary that in an $\\operatorname{NSS}$ local group, the image of the local exponential map is a neighborhood of $1$, which later will allow us to conclude that $\\operatorname{NSS}$ local groups are locally euclidean.\n\n\n\\\n\n\\noindent \\textbf{Group structure on $L(G)$}\n\n\\\n\n\\begin{lemma}\\label{L:prodmonad}\nSuppose $\\nu\\in \\mathbb{N}^*\\setminus \\mathbb{N}$ and $a_1,\\ldots,a_\\nu$ is a hyperfinite sequence such that $a_i\\in G^{\\operatorname{o}}(\\nu)$ for all $i\\in \\{1,\\ldots,\\nu\\}$.  Let $Q:=\\{1,a_1,\\ldots,a_\\nu,a_1^{-1},\\ldots,a_\\nu^{-1}\\}$.  Then $Q^\\nu$ is defined and $Q^\\nu \\subseteq \\boldsymbol{\\mu}$.\n\\end{lemma}\n\n\\begin{proof}\nWe first show that if $Q^\\nu$ is defined, then $Q^\\nu \\subseteq \\boldsymbol{\\mu}$.  Suppose, towards a contradiction, that $Q^\\nu \\nsubseteq \\boldsymbol{\\mu}$.  Take a compact symmetric neighborhood $U$ of $1$ in $G$ such that $Q^{\\nu+1}\\nsubseteq U^*$, so $\\operatorname{ord}_U(Q)\\leq \\nu$.  Fix $\\mathcal{W}$ as in the setting of the Gleason-Yamabe lemmas and choose $U$ small enough so that $U\\subseteq \\mathcal{W}$ and $U^P\\subseteq \\mathcal{W}$.  By decreasing $\\nu$ if necessary, and $Q$ accordingly, we arrange that $\\operatorname{ord}_U(Q)=\\nu$.\n\n\nConsider first the special case that $Q^i\\subseteq \\boldsymbol{\\mu}$ for all $i=\\operatorname{o}(\\nu)$.  Take $b\\in Q^\\nu$ such that $\\operatorname{st}(b)\\not=1$.  Choose $\\u$ as in the setting of the Gleason-Yamabe lemmas so that $\\u\\subseteq U$ and $\\operatorname{st}(b)\\notin \\u^4$.  Let $\\eta:=\\operatorname{ord}_{\\u}(Q)$ and note that $\\nu = \\operatorname{O}(\\eta)$, else we would have that $Q^{\\eta}\\subseteq \\boldsymbol{\\mu}$, a contradiction.  By the Transfer Principle, the previous section yields the internally continuous function $\\phi=\\phi_Q\\colon G^*\\rightarrow \\r^*$ satisfying the internal versions of all of the properties and lemmas from that section.  In particular, $\\phi(b^{-1})=0$ and $\\phi(1)$ is not infinitesimal.  Hence $\\|b\\phi - \\phi\\|$ is not infinitesimal.  Take a hyperfinite sequence $b_1,\\ldots,b_\\nu$ from $Q$ such that $b=b_1\\cdots b_\\nu$.  Then by the Transfer Principle and Lemma \\ref{L:triangle}, we have $$\\|b\\phi -\\phi\\|\\leq \\sum_{i=1}^\\nu \\|b_i\\phi -\\phi\\|$$ \n\\noindent and the right hand side of the inequality is infinitesimal by Lemma \\ref{L:gleason3}, yielding a contradiction.\n\n\nNow assume that $Q^i\\nsubseteq \\boldsymbol{\\mu}$ for some $i=\\operatorname{o}(\\nu)$.  Then the compact subgroup $G_U(Q)$ of $G$ as defined in Lemma \\ref{L:singsubgroup} is nontrivial (and contained in $U$).  By a standard consequence of the Peter-Weyl Theorem (see \\cite{MZ}, page 100), we can find a proper closed normal subgroup $K$ of $G_U(Q)$ and a compact symmetric neighborhood $V\\subseteq U$ of $1$ in $G$ such that $K\\subseteq \\operatorname{interior}(V)$ and every subgroup of $G_U(Q)$ contained in $V$ is contained in $K$.  Put $\\mu:=\\operatorname{ord}_V(Q)$, so $\\mu\\in \\mathbb{N}^*\\setminus \\mathbb{N}$ and $\\mu\\leq \\nu$.  Since the compact subgroup $G_V(Q)$ is contained in $V$, we must have $G_V(Q)\\subseteq K$.  Take $b\\in Q^\\mu$ such that $\\operatorname{st}(b)\\notin \\operatorname{interior}(V)$.  Then $\\operatorname{st}(b)\\notin K$, so we can take a compact symmetric neighborhood $\\u$ of $1$ as in the setting of the Gleason-Yamabe lemmas such that $K\\subseteq \\operatorname{interior}(\\u)$, $\\u\\subseteq V$, and $\\operatorname{st}(b)\\notin \\u^4$.  Put $\\eta:=\\operatorname{ord}_{\\u}(Q)$.  If $\\eta=\\operatorname{o}(\\mu)$, then $\\operatorname{st}(Q^\\eta)\\subseteq G_V(Q)\\subseteq K$, contradicting the fact that $\\operatorname{st}(Q^\\eta)\\nsubseteq \\operatorname{interior}(\\u)$.  Hence $\\mu=\\operatorname{O}(\\eta)$.  The rest of the proof now proceeds as in the special case considered earlier, with $\\nu$ replaced by $\\mu$, and $b_1,\\ldots,b_\\nu$ by an internal sequence $b_1,\\ldots,b_\\mu$ in $Q$ such that $b=b_1\\cdots b_\\mu$.\n\n\nWe now prove that $Q^\\nu$ is defined.  To do this, we first fix an internal set $E\\subseteq \\boldsymbol{\\mu}$.\n\n\n\\noindent \\textbf{Claim:}  Let $c_1,\\ldots,c_\\eta$ be an internal sequence such that $c_i^j$ is defined and $c_i^j\\in E$ for all $i,j\\in \\{1,\\ldots,\\eta\\}$.  Let  $R=\\{1,c_1,\\ldots,c_\\eta,c_1^{-1},\\ldots,c_{\\eta}^{-1}\\}$.  Then $R^{\\eta}$ is defined. \n\n\nWe prove this claim by internal induction on $\\eta$.  The claim is clearly true for $\\eta=1$ and we now suppose inductively that it holds for a given $\\eta$.  Suppose $c_1,\\ldots,c_{\\eta+1}$ is an internal sequence such that $c_i^j$ is defined and $c_i^j \\in E$ for all $i,j\\in \\{1,\\ldots,\\eta+1\\}$.  Let $R=\\{1,c_1,\\ldots,c_{\\eta+1},c_1^{-1},\\ldots,c_{\\eta+1}^{-1}\\}$.  Fix $d_1,\\ldots,d_{\\eta+1}\\in R$.  By the induction hypothesis and the first part of the proof, $d_1\\cdots d_\\eta$ is defined and infinitesimal; similarly, $d_1\\cdots d_i$ and $d_i\\cdots d_{\\eta+1}$ are defined and infinitesimal for all $i\\in \\{2,\\ldots,\\eta+1\\}$.  By Lemma \\ref{L:EZFacts}, $d_1\\cdots d_{\\eta+1}$ is defined.  Since $d_1,\\ldots,d_{\\eta+1}$ from $R$ were arbitrary, we have that $R^{\\eta+1}$ is defined.  This finishes the proof of the claim.\n\n\nNow given an internal sequence $a_1,\\ldots,a_\\nu$ as in the statement of the lemma, the claim implies that $Q^\\nu$ is defined by taking $E$ to be the set of all $a_i^j$ for $i,j\\in \\{1,\\ldots,\\nu\\}$.\n\\end{proof}\n\n\\\n\n\\begin{nrmk}\nIn the global version of this lemma appearing in \\cite{H}, it was required that $G$ was pure.  Van den Dries was able to remove this assumption and in the proof of the previous lemma we have adapted his method to the local setting. \n\\end{nrmk}\n\n\\\n\n\\begin{lemma}\\label{L:Gsigmagroup}\nSuppose $U$ is a compact symmetric neighborhood of $1$ in $G$ with $U\\subseteq \\u_2$. Let $\\nu > \\mathbb{N}$ be such that for all $i\\in\\{1,\\ldots,\\nu\\}$, $a^i$ and $b^i$ are defined and $a^i \\in U^*$, $b^i \\in \\boldsymbol{\\mu}$.  Then for all $i\\in\\{1,\\ldots,\\nu\\}$, we have that $(ab)^i$ is defined and $(ab)^i\\sim a^i$. \n\\end{lemma}\n\n\\begin{proof}\nSuppose $i\\in\\{1,\\ldots,\\nu\\}$.  Then $(a^i,b)\\in \\o^*$ and $a^ib\\sim a^i$.  Since $(\\operatorname{st}(a^i),\\operatorname{st}(a^{-i}))\\in \\o$, we have $(a^ib,a^{-i})\\in \\o^*$.  Similarly, $(a^i,ba^{-i})\\in \\o^*$.  Thus we can define the element $b_i:=a^iba^{-i}$.  For similar reasons, for any $i,j\\in\\{1,\\ldots,\\nu\\}$, we can define the element $a^ib^ja^{-i}$.  Note that $b_i\\in \\boldsymbol{\\mu}$ for all $i\\in\\{1,\\ldots,\\nu\\}$.  \n\n\\\n\n\\noindent Claim 1:  For all $i,j\\in\\{1,\\ldots,\\nu\\}$, $b_i^j$ is defined and $b_i^j=a^ib^ja^{-i}$.\n\n\\\n\n\\noindent We prove Claim 1 by internal induction on $j$.  The case $j=1$ is obvious.  Suppose the assertion is true for all $j'\\in \\{1,\\ldots,j\\}$ and further suppose $j+1\\leq \\nu$.  We first show that $b_i^{j+1}$ is defined.  Let $k,l\\in\\{1,\\ldots,j\\}$ be such that $k+l=j+1$.  By the induction hypothesis, we know that $b_i^k$ is defined and $b_i^k=a^ib^ka^{-i}$.  Since $b^k\\in \\boldsymbol{\\mu}$, we have $b_i^k \\in \\boldsymbol{\\mu}$.  Similarly, we have $b_i^l$ is defined and $b_i^l\\in \\boldsymbol{\\mu}$; thus $(b_i^k,b_i^l)\\in \\o^*$.  So by Lemma \\ref{L:EZFacts}, we have that $b_i^{j+1}$ is defined.  But now, using that $b_i^j \\in \\boldsymbol{\\mu}$, we have \n\n\\begin{align}\nb_i^{j+1}&=b_i^j \\cdot b_i \\notag \\\\\n\t      &=(a^ib^ja^{-i})(a^iba^{-i}) \\notag \\\\\n\t      &=((a^ib^ja^{-i})\\cdot (a^i))\\cdot (ba^{-i}) \\notag \\\\\n\t      &=(a^ib^j)\\cdot (ba^{-i}) \\notag \\\\\n\t      &=((a^ib^j)\\cdot b)a^{-i}\\notag \\\\\n\t      &=a^ib^{j+1}a^{-i}.\\notag \\\\ \\notag\n\\end{align}\n\n\\noindent This proves Claim 1.  Note that we have also shown that $b_i^j\\in \\boldsymbol{\\mu}$ for all $i,j\\in\\{1,\\ldots,\\nu\\}$.  Then by Lemma \\ref{L:prodmonad}, $b_1\\cdots b_i$ is defined and $b_1\\cdots b_i\\in \\boldsymbol{\\mu}$ for all $i\\in\\{1,\\ldots,\\nu\\}$.\n\n\n\\\n\n\\noindent The following claim finishes the proof of the lemma.\n\n\\\n\n\\noindent Claim 2:  For all $i\\in\\{1,\\ldots,\\nu\\}$, $(ab)^i$ is defined and $(ab)^i=(b_1\\cdots b_i)a^i$.\n\n\\\n\n\\noindent We prove Claim 2 by internal induction on $i$.  The assertion is clearly true for $i=1$ and now suppose it holds for all $j\\in \\mathbb{N}^*$ with $j\\leq i$.  Suppose $i+1\\leq \\nu$.  To prove that $(ab)^{i+1}$ is defined, it is enough to check that $((ab)^k,(ab)^l)\\in \\o^*$ for all $k,l\\in\\{1,\\ldots,i\\}$ with $k+l=i+1$.  By the induction hypothesis, $(ab)^k=(b_1\\cdots b_k)a^k\\sim a^k$ and $(ab)^l=(b_1\\cdots b_l)a^l\\sim a^l$ and the desired result now follows from the fact that $(a^k,a^l)\\in U^* \\times U^*$.  Next note that $a^{i+1}b=(a^{i+1}\\cdot b)\\cdot(a^{-i-1}a^{i+1})=(a^{i+1}ba^{-i-1})a^{i+1}=b_{i+1}a^{i+1}$.  Hence, by the induction hypothesis, we have\n\\begin{align}\n(ab)^{i+1}&=((ab)^i)\\cdot(ab)\\notag \\\\\n                 &=((b_1\\cdots b_i)a^i)\\cdot(ab)\\notag \\\\\n                 &=(b_1\\cdots b_i)\\cdot (a^i(ab))\\notag \\\\\n                 &=(b_1\\cdots b_i)(a^{i+1}b)\\notag \\\\\n                 &=(b_1\\cdots b_i)(b_{i+1}a^{i+1})\\notag \\\\\n                 &=((b_1\\cdots b_i)b_{i+1})a^{i+1}\\notag \\\\\n                 &=(b_1\\cdots b_{i+1})a^{i+1}.\\notag \\\\ \\notag\n\\end{align}\n\n\\end{proof}\n\n\\\n\n\\begin{lemma}\\label{L:commutator}\nLet $\\nu > \\mathbb{N}$ and $a\\in G(\\nu)$ be such that $a^\\nu$ is defined and $a^i\\in G^*_{\\operatorname{ns}}$ for all $i\\in \\{1,\\ldots,\\nu\\}$.  Suppose also that $b\\in \\boldsymbol{\\mu}$ is such that $b^\\nu$ is defined and $b^i\\in G^*_{\\operatorname{ns}}$ and $a^i \\sim b^i$ for all $i\\in\\{1,\\ldots,\\nu\\}$.  Then $a^{-1}b\\in G^{\\operatorname{o}}(\\nu)$.\n\\end{lemma}\n\n\\begin{proof}\nWe first make a series of reductions.  Note that by Lemma \\ref{L:Gsigmagroup}, we have $a^{-1}b\\in G(\\nu)$.  Let $V$ be a compact symmetric neighborhood of $1$ in $G$ such that $V\\subseteq \\u_2$.  Then since $\\nu=\\operatorname{O}(\\operatorname{ord}_V(a^{-1}b))$, by replacing $\\nu$ with $\\operatorname{ord}_V(a^{-1}b)$ if $\\nu > \\operatorname{ord}_V(a^{-1}b)$, we may as well assume that $(a^{-1}b)^i$ is defined and $(a^{-1}b)^i\\in G^*_{\\operatorname{ns}}$ for all $i\\in \\{1,\\ldots,\\nu\\}$.  Let $Q=\\{1,a,a^{-1},b,b^{-1}\\}$.  Suppose that $Q^\\nu$ is not defined.  Let $\\eta\\in \\mathbb{N}^*\\setminus \\mathbb{N}$ be the largest element of $\\mathbb{N}^*$ such that $Q^\\eta$ is defined.  If $\\eta=\\operatorname{o}(\\nu)$, then by Lemma \\ref{L:prodmonad}, $Q^i\\subseteq \\boldsymbol{\\mu}$ for all $i\\in \\{1,\\ldots,\\eta\\}$.  But then, by Lemma \\ref{L:EZFacts}, $Q^{\\eta+1}$ is defined, a contradiction.  Thus $\\nu=O(\\eta)$.  So, by replacing $\\nu$ by $\\eta$ if necessary, we can assume $Q^\\nu$ is defined.  If $a\\in G^{\\operatorname{o}}(\\nu)$, then by Lemma \\ref{L:Gsigmagroup}, $(a^{-1}b)^i$ is defined and $(a^{-1}b)^i\\sim a^{-i}\\in \\boldsymbol{\\mu}$ for all $i\\in \\{1,\\ldots,\\nu\\}$, yielding the desired conclusion.  So we may as well assume that $a\\notin G^{\\operatorname{o}}(\\nu)$ and by replacing $\\nu$ by a smaller element of its archimedean class, we may as well assume that $a^\\nu \\notin \\boldsymbol{\\mu}$.  \n\n\\\n\n\\noindent Now suppose, towards a contradiction, that there is $j\\in\\{1,\\ldots,\\nu\\}$ such that $(a^{-1}b)^j \\notin \\boldsymbol{\\mu}$.  Note we must have $\\nu =\\operatorname{O}(j)$.  Choose compact symmetric neighborhoods $\\u$ and $\\mathcal{W}$ of $1$ in $G$ as in the previous section so that $$a^\\nu \\in \\mathcal{W}^* \\setminus \\u^*\\text{ and }(a^{-1}b)^j\\in \\mathcal{W}^* \\setminus (\\u^*)^4.$$  Let $\\sigma = \\operatorname{ord}_\\u(Q)$.  Then $\\nu = \\operatorname{O}(\\sigma)$ by Lemma \\ref{L:prodmonad}.  Let $\\phi=\\phi_Q:G^* \\rightarrow \\r^*$ be the internally continuous function constructed in the previous section.  Then we know that $\\phi((a^{-1}b)^j)=0$ and $\\epsilon :=\\phi(1)>0$ and is not infinitesimal.  Also, note\n\n\\begin{align}\n\\epsilon &\\leq \\|(b^{-1}a)^j\\phi - \\phi \\| \\notag \\\\\n               &\\leq j\\|(b^{-1}a)\\phi - \\phi \\| \\notag \\\\\n               &=j\\|a\\phi - b\\phi \\| \\notag \\\\\n               &=\\|j(a\\phi-\\phi)-j(b\\phi -\\phi)\\|. \\notag \\\\ \\notag\n\\end{align}           \n\n\\noindent We will obtain a contradiction by showing that $\\|j(a\\phi - \\phi) - j(b\\phi -\\phi)\\|<\\epsilon.$  By Lemma \\ref{L:gleason2}, there is a compact symmetric neighborhood $U\\subseteq \\u$ of $1$ in $G$ such that for all $k>0$ in $\\mathbb{N}^*$, if $a^i \\in U^*$ and $b^i \\in U^*$ for all $i\\in\\{1,\\ldots,k\\}$, then \n$$\\|(\\frac{j}{k})(a^k\\phi - \\phi) - j(a\\phi - \\phi)\\|< \\frac{\\epsilon}{3},$$\n$$\\|(\\frac{j}{k})(b^k\\phi - \\phi) - j(b\\phi - \\phi)\\|< \\frac{\\epsilon}{3}.$$\n\n\\noindent Let $k=\\min\\{\\operatorname{ord}_U(a),\\operatorname{ord}_U(b)\\}$.  Then the above equalities hold for this $k$.  Note that $\\nu=\\operatorname{O}(k)$ and hence $j=O(k)$.  Since $k< \\nu$, we have $a^k \\sim b^k$ and so $$\\|(\\frac{j}{k})(a^k\\phi-\\phi)-(\\frac{j}{k})(b^k\\phi-\\phi)\\|=\\frac{j}{k}\\|a^k\\phi -b^k\\phi\\|\\sim 0.$$ \n\\noindent Combining this fact with the previous inequalities yields $$\\|j(a\\phi - \\phi) - j(b\\phi -\\phi)\\|<\\epsilon,$$ giving the desired contradiction.\n\\end{proof}\n\n\\\n\n\\begin{thm}\nSuppose $\\sigma > \\mathbb{N}$.  Then\n\\begin{enumerate}\n\\item $G(\\sigma)$ and $G^{\\operatorname{o}}(\\sigma)$ are normal subgroups of $\\boldsymbol{\\mu}$;\n\\item if $a\\in G(\\sigma)$ and $b\\in \\boldsymbol{\\mu}$, then $aba^{-1}b^{-1}\\in G^{\\operatorname{o}}(\\sigma)$;\n\\item $G(\\sigma)\/G^{\\operatorname{o}}(\\sigma)$ is abelian.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nWe have already remarked that $G(\\sigma)$ and $G^{\\operatorname{o}}(\\sigma)$ are symmetric and that $G(\\sigma)$ is closed under multiplication.  That $G(\\sigma)$ is a normal subgroup of $\\boldsymbol{\\mu}$ is part of the content of Lemma \\ref{L:Gsigma properties}, (2).  Lemma \\ref{L:Gsigmagroup} implies that $G^{\\operatorname{o}}(\\sigma)$ is a group.  Now suppose $a\\in G^{\\operatorname{o}}(\\sigma)$ and $b\\in \\boldsymbol{\\mu}$.  We need $bab^{-1}\\in G^{\\operatorname{o}}(\\sigma)$.  One can show by internal induction on $\\eta \\leq \\sigma$ that $(bab^{-1})^\\eta$ is defined and equal to $ba^\\eta b^{-1}$.  In particular, this shows that $bab^{-1}\\in G^{\\operatorname{o}}(\\sigma)$.  This finishes the proof of (1).\n\n\\\n\n\\noindent Let $a\\in G(\\sigma)$ and $b\\in \\boldsymbol{\\mu}$.  We need to show that $aba^{-1}b^{-1}\\in G^{\\operatorname{o}}(\\sigma)$.  Note that $ba^{-1}b^{-1} \\in G(\\sigma)$ by (1).  Let $U\\subseteq \\u_2$ be a compact symmetric neighborhood of $1$.  Choose $\\tau \\in \\{1,\\ldots,\\sigma\\}$ such that $\\sigma=\\operatorname{O}(\\tau)$, $a^\\tau$ and $(ba^{-1}b^{-1})^\\tau$ are defined, and $a^i\\in U^*$ for $i\\in \\{1,\\ldots,\\tau\\}$.  One can prove by internal induction that for $i\\in\\{1,\\ldots,\\tau\\}$, $(ba^{-1}b^{-1})^i=ba^{-i}b^{-1}$ and hence $a^{-i}\\sim (ba^{-1}b^{-1})^i$.  By Lemma \\ref{L:commutator}, $aba^{-1}b^{-1}\\in G^{\\operatorname{o}}(\\tau)\\subseteq G^{\\operatorname{o}}(\\sigma)$.  This finishes (2) and (2) obviously implies (3).\n\\end{proof}\n\n\\\n\n\\noindent Observe that if $\\mathbb{X}\\in L(G)$ and $\\sigma>\\mathbb{N}$, then $\\mathbb{X}(\\frac{1}{\\sigma})\\in G(\\sigma)$.  To see this, suppose that $i=\\operatorname{o}(\\sigma)$.  Then $(\\mathbb{X}(\\frac{1}{\\sigma}))^i=\\mathbb{X}(\\frac{i}{\\sigma})\\in \\boldsymbol{\\mu}$.\n\n\\\n\n\\begin{thm}\\label{T:addition}\nThe map $S:L(G)\\rightarrow G(\\sigma)\/G^{\\operatorname{o}}(\\sigma)$ defined by $$S(\\mathbb{X})=\\mathbb{X}(\\frac{1}{\\sigma})G^{\\operatorname{o}}(\\sigma)$$ is a bijection.  \n\\end{thm}\n\n\\begin{proof}\nSuppose $\\mathbb{X},\\mathbb{Y}\\in L(G)$ and $S(\\mathbb{X})=S(\\mathbb{Y})$.  Letting  $a:=\\mathbb{X}(\\frac{1}{\\sigma})$ and $b:=\\mathbb{Y}(\\frac{1}{\\sigma})$, we have that $a^{-1}b\\in G^{\\operatorname{o}}(\\sigma)$.    Let $U$ be a compact symmetric neighborhood of $1$ in $G$ with $U\\subseteq \\u_2$.  Let $\\tau:=\\min\\{\\operatorname{ord}_U(a),\\sigma\\}$.  By Lemma \\ref{L:Gsigmagroup}, we have $(a\\cdot (a^{-1}b))^i$ is defined, nearstandard and infinitely close to $a^i$ for all $i\\in\\{1,\\ldots,\\tau\\}$, i.e. $a^i \\sim b^i$ for all $i\\in\\{1,\\ldots,\\tau\\}$.  Then  for all $i\\in\\{1,\\ldots,\\tau\\}$ such that $\\frac{i}{\\sigma}\\in \\operatorname{domain}(\\mathbb{X})\\cap \\operatorname{domain}(\\mathbb{Y})$, we have $\\mathbb{X}(\\frac{i}{\\sigma})=\\mathbb{Y}(\\frac{i}{\\sigma})$.  Since $\\sigma=\\operatorname{O}(\\tau)$, this implies that $\\mathbb{X}=\\mathbb{Y}$.  Hence $S$ is 1-1.\n\n\\\n\n\\noindent Now suppose $b\\in G(\\sigma) \\setminus G^{\\operatorname{o}}(\\sigma)$.  Consider the local 1-ps $X_b$ of $G$ defined by $X_b(t)=\\operatorname{st}(b^{[t\\sigma]})$ on its domain; see Lemma \\ref{L:regelement}.  Let $b_1=X_b(\\frac{1}{\\sigma})\\in \\boldsymbol{\\mu}$.  Choose $\\tau\\in\\{1,\\ldots,\\sigma\\}$ such that $\\sigma = \\operatorname{O}(\\tau)$ and $\\frac{\\tau}{\\sigma}\\in \\operatorname{domain}(X_b)$.  Note then that $b_1^\\tau$ is defined.  For any $i\\in\\{1,\\ldots,\\tau\\}$, $b^i$ is defined and $b^i \\sim b_1^i$.  By Lemma \\ref{L:commutator}, $b^{-1}b_1\\in G^{\\operatorname{o}}(\\tau)\\subseteq G^{\\operatorname{o}}(\\sigma)$ and hence $bG^{\\operatorname{o}}(\\sigma)=b_1G^{\\operatorname{o}}(\\sigma).$    Hence $S$ is onto.\n\\end{proof}\n\n\\\n\n\\noindent We use Theorem \\ref{T:addition} to equip $L(G)$ with an abelian group operation $+_\\sigma$.  More explicitly, for $\\mathbb{X},\\mathbb{Y} \\in L(G)$, we define $\\mathbb{X}+_\\sigma \\mathbb{Y}:=S^{-1}(S(\\mathbb{X})\\cdot S(\\mathbb{Y}))$.  By the proof of Theorem \\ref{T:addition}, we see that if $t\\in \\operatorname{domain}(\\mathbb{X}+_\\sigma \\mathbb{Y})$ and $\\nu$ is such that $\\frac{\\nu}{\\sigma}\\sim t$, then \n$$(\\mathbb{X}+_\\sigma \\mathbb{Y})(t)\\sim [\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma})]^\\nu.$$   \n\n\\\n\n\\noindent  In ~\\cite{H}, a proof is given, in the global setting, that this group operation is independent of $\\sigma$.  A local version of that proof is probably possible, but this independence also follows from the Local H5, which we prove without using that  $+_\\sigma$ is independent of $\\sigma$.  In light of this independence of $\\sigma$, we write $\\mathbb{X}+\\mathbb{Y}$ instead of $\\mathbb{X}+_\\sigma \\mathbb{Y}$.\n\n\\\n\n\\noindent \\textbf{Existence of Square Roots}  \n\n\\\n\n\\noindent In the rest of this section, we assume that $G$ is $\\operatorname{NSS}$ and that $\\u$ is a special neighborhood of $1$ in $G$.  We now lead up to the proof of the crucial fact that the image of the local exponential map is a neighborhood of $1$ in $G$.  In particular, we will see that there is an open neighborhood of $1$ in $G$ which is \\textit{ruled by local 1-parameter subgroups} in the sense that every element in this open neighborhood lies on some local 1-ps of $G$.  In the rest of this section, we do not follow \\cite{H}, but rather \\cite{vdD}, which contains the following series of lemmas in the global setting.\n\n\\\n\n\\begin{lemma}\nSuppose $\\sigma > \\mathbb{N}$ and $a,b\\in G(\\sigma)$.  Then $[X_a]+[X_b]=[X_{ab}]$.\n\\end{lemma}\n\n\\begin{proof}\nFrom Theorem \\ref{T:addition}, we have \n\\begin{align}\nS([X_a]+[X_b])&=S([X_a])\\cdot S([X_b]) \\notag \\\\\n                          &=(aG^{\\operatorname{o}}(\\sigma))(bG^{\\operatorname{o}}(\\sigma)) \\notag \\\\\n                          &=(ab)G^{\\operatorname{o}}(\\sigma) \\notag \\\\\n                          &=S([X_{ab}]). \\notag \\\\ \\notag\n\\end{align}\n\\noindent The result now follows since $S$ is injective.\n\\end{proof}\n\n \\\n  \n \\noindent The global version of the next lemma is in Singer \\cite{S}, but the proof there is unclear. \n \\begin{lemma}\\label{L:Singer}\n Suppose $\\sigma > \\mathbb{N}$ and $a\\in G(\\sigma)$.  Then $a=bc^2$ with $b\\in G^{\\operatorname{o}}(\\sigma)$ and $c\\in G(\\sigma)$.  \n \\end{lemma}\n \n \\begin{proof}\nLet $c:=X_a(\\frac{1}{2\\sigma})\\in \\boldsymbol{\\mu}$.  Then $c^{-2}=X_a(-\\frac{1}{\\sigma})=X_{a^{-1}}(\\frac{1}{\\sigma})$, which implies that $[X_{c^{-2}}]=[X_{a^{-1}}]$.  From this, we conclude that $$[X_{ac^{-2}}]=[X_{a}]+[X_{c^{-2}}]=[X_1]=\\O.$$  Let $b:=ac^{-2}$.  Then since $[X_b]=\\O$, we have $b\\in G^{\\operatorname{o}}(\\sigma)$.         \n \\end{proof}\n \n \\\n \n \\begin{lemma}\\label{L:q}\n There exists $q\\in \\mathbb{Q}^{>0}$ such that for all $a,b,c\\in \\boldsymbol{\\mu}$, if $a=bc^2$ and $\\operatorname{ord}(b)\\geq \\operatorname{ord}(a)$, then $\\operatorname{ord}(c)\\geq q\\cdot \\operatorname{ord}(a)$.\n \\end{lemma}\n \n \\begin{proof}\n Suppose no such $q$ exists.  Then for all $n\\geq 1$, there are $a_n,b_n,c_n\\in \\boldsymbol{\\mu}$ such that $a_n=b_nc_n^2$ with ord$(b_n)\\geq \\operatorname{ord}(a_n)$ and $n \\operatorname{ord}(c_n) < \\operatorname{ord}(a_n)$.  By saturation, we have $a,b,c\\in \\boldsymbol{\\mu}$ such that $a=bc^2$, ord$(b)\\geq \\operatorname{ord}(a)$ and $n \\operatorname{ord}(c)<\\operatorname{ord}(a)$ for all $n$.  Let $\\sigma:=\\operatorname{ord}(c)>\\mathbb{N}$.  Then $$\\O=[X_a]=[X_{bc^2}]=[X_b]+[X_{c^2}]=\\O+2[X_c],$$ implying that $[X_c]=\\O$, an obvious contradiction.  \n \\end{proof}\n \n \\\n \n \\noindent Fix $q$ as in the above lemma.  For $a,b\\in \\boldsymbol{\\mu}$, we let $[a,b]:=aba^{-1}b^{-1}$, the commutator of $a$ and $b$.  We will frequently use the identity $ab=[a,b]ba$.\n \n \\\n \n \\begin{lemma}\n Let $a\\in \\boldsymbol{\\mu}$.  Then for all $\\nu \\geq 1$, there are $b_\\nu,c_\\nu \\in G^*$ such that $b_{\\nu}\\cdot c_\\nu \\cdot c_\\nu$ is defined, $a=b_\\nu c_\\nu^2$, $\\operatorname{ord}(b_\\nu) \\geq \\nu \\operatorname{ord}(a)$ and $\\operatorname{ord}(c_\\nu)\\geq q \\operatorname{ord}(a)$. \n \\end{lemma}\n \n \\begin{proof}\nNote that the assertion is trivial if $a=1$ and so we suppose $a\\not=1$.  Next note that if the assertion is true for a given $\\nu$, then we have $b_\\nu, c_\\nu \\in\\boldsymbol{\\mu}$ as they will have infinite order.  By Lemmas \\ref{L:Singer} and \\ref{L:q}, the assertion is true for some $\\nu>\\mathbb{N}$.  It is clearly then true for every $\\nu'\\in\\{1,\\ldots,\\nu\\}$.  Hence, if we can prove the assertion for $\\nu + 1$, then we will be done by internal induction.  If $b_\\nu=1$, then the assumed factorization witnesses the truth of the assertion for all $\\nu' $.  We thus assume $b_\\nu\\not=1$.  By Lemma \\ref{L:Singer}, there are $b,c\\in \\boldsymbol{\\mu}$ and $\\tau_1>\\mathbb{N}$ such that $b_\\nu=bc^2$ with $\\operatorname{ord}(b)\\geq \\tau_1 \\operatorname{ord}(b_\\nu)$.  Note that then $\\operatorname{ord}(b)\\geq \\tau_1 \\nu \\operatorname{ord}(a)$.  By Lemma \\ref{L:q}, we also know that $\\operatorname{ord}(c)\\geq q \\operatorname{ord}(b_\\nu)\\geq q \\nu \\operatorname{ord}(a)$.  Since $\\boldsymbol{\\mu}$ is a group, we have \n\\begin{align}\na=bc^2c_\\nu^2&=bccc_\\nu c_\\nu \\notag \\\\\n                           &=bc[c,c_\\nu]c_\\nu cc_\\nu \\notag \\\\\n                           &=b[c,[c,c_\\nu]][c,c_\\nu]cc_\\nu c c_\\nu \\notag \\\\\n                           &=b[c,[c,c_\\nu]][c,c_\\nu](cc_\\nu)^2 \\notag \\\\\n                           &=b_{\\nu+1}(c_{\\nu+1})^2, \\notag \\\\ \\notag\n\\end{align}\n   \n\\noindent where $b_{\\nu+1}:=b[c,[c,c_\\nu]][c,c_\\nu]$ and $c_{\\nu+1}:=cc_\\nu$.  Now note that there is $\\tau_2 >\\mathbb{N}$ such that \n$$\\operatorname{ord}([c,c_\\nu])\\geq \\tau_2 \\operatorname{ord}(c)\\geq \\tau_2 q\\nu \\operatorname{ord}(a).$$  Also there is $\\tau_3> \\mathbb{N}$ such that $$\\operatorname{ord}([c,[c,c_\\nu]])\\geq \\tau_3 \\tau_2 \\operatorname{ord}(c) \\geq \\tau_3 \\tau_2 q\\nu \\operatorname{ord}(a).$$  In combination with the facts that $\\operatorname{ord}(b)\\geq \\tau_1 \\nu \\operatorname{ord}(a)$ and $G(\\tau \\operatorname{ord}(a))$ is a group, where $\\tau:=\\min(\\tau_1,\\tau_2)$, we have that $\\operatorname{ord}(b_{\\nu+1})\\geq (\\nu+1)\\operatorname{ord}(a)$.  Finally, Lemma \\ref{L:q} yields that $\\operatorname{ord}(c_{\\nu+1})\\geq q\\operatorname{ord}(a)$ and so we are finished.       \n \\end{proof}\n \n \\\n \n \\begin{lemma}\\label{L:sqrt}\n For every $a\\in \\boldsymbol{\\mu}$, there is $b\\in \\boldsymbol{\\mu}$ with $a=b^2$.\n \\end{lemma}\n \n \\begin{proof}\n Fix $\\nu> \\mathbb{N}$.  Then by the previous lemma, the internal set of all $\\eta\\in \\mathbb{N}$ such that for each $x\\in V_\\eta$ there are $y,z\\in \\u^*$ such that $y \\cdot z \\cdot z$ is defined, $x=yz^2$, $\\operatorname{ord}(y)\\geq \\nu \\operatorname{ord}(x)$ and $\\operatorname{ord}(z)\\geq q\\operatorname{ord}(x)$ includes all infinite $\\eta$.  By underspill, there is some $n>0$ in this set.  Now let $x\\in V_n$.  Then there are $y,z\\in \\u^*$ such that $y\\cdot z \\cdot z$ is defined, $x=yz^2$ and $\\operatorname{ord}(y)\\geq \\nu \\operatorname{ord}(x)$.  Since $\\operatorname{ord}(y)$ is infinite, $y\\in \\boldsymbol{\\mu}$.  Hence $x=\\operatorname{st}(z)^2$, i.e. every element of $V_n$ has a square root in $\\u$.  By transfer, this implies that every $a\\in \\boldsymbol{\\mu}$ has a square root $b\\in \\u^*$ and $b\\in \\boldsymbol{\\mu}$, else the group $\\{1,\\operatorname{st}(b)\\}$ is contained in $\\u$, a contradiction.           \n \\end{proof}\n \n \\\n \n \\begin{lemma}\\label{L:ordinf}\n Suppose $a\\in \\boldsymbol{\\mu}$, $b\\in G^*$ and $\\nu>0$ are such that $\\operatorname{ord}(b)\\geq \\nu$ and $a=b^\\nu$.  Then for all $i\\in\\{1,\\ldots,\\nu\\}$, we have $b^i\\in \\boldsymbol{\\mu}$.\n \\end{lemma}\n \n \\begin{proof} \nWe begin the proof with a claim.\n\n\\\n\n\\noindent Claim:  For all $n$, $b^{n\\nu}$ is defined.\n\n\\\n\n\\noindent We prove this claim by induction on $n$, the case $n=1$ being true by assumption.  Assume inductively that $b^{n\\nu}$ is defined.  We claim that $$b^j\\in \\u_2^* \\text{ for all }j\\in \\{1,\\ldots,n\\nu\\}.\\quad (\\dagger)$$  To see this, write $j=n'\\nu +j'$, where $n'\\in \\mathbb{N}$ and $j'\\in \\{1,\\ldots,\\nu-1\\}$.  Then $$b^j=(b^\\nu)^{n'}\\cdot b^{j'}\\sim b^{j'}\\in \\u^*.$$  In order to show that $b^{n\\nu + \\nu}$ is defined, we prove by internal induction on $i$ that $b^{n\\nu+i}$ is defined and $b^{n\\nu+i}\\in \\u_2^*$ for all $i\\in \\{0,\\ldots,\\nu\\}$.  The case $i=0$ is true since $b^{n\\nu}=a^n\\in\\boldsymbol{\\mu}$.  Suppose, inductively, that it holds for a given $i$ and that $i+1\\leq \\nu$.  Let $k,l\\in \\{1,\\ldots,n\\nu+i\\}$ be such that $k+l=n\\nu +i+1$.  Then by $(\\dagger)$ and the inductive assumption, $(b^k,b^i)\\in \\u_2^*\\times \\u_2^*\\subseteq \\o^*$.  Hence $b^{n\\nu+i+1}$ is defined.  But then $b^{i+1}\\in \\u^*$ and $b^{n\\nu+i+1}\\sim b^{i+1}$.  Hence $b^{n\\nu+i+1}\\in \\u_2^*$.  This proves the claim.\n\n\\\n\n\\noindent Now suppose, towards a contradiction, that there is $i\\in\\{1,\\ldots,\\nu-1\\}$ such that $b^i\\notin \\boldsymbol{\\mu}$. Let $x:=\\operatorname{st}(b^i)\\in \\u\\setminus \\{1\\}$.  From the claim, we see that if $x^n$ is defined for a given $n$, then $x^n \\in \\u$.  From this, one can show by induction that for all $n$, $x^n$ is defined and $x^n\\in \\u$.  Hence $x^\\mathbb{Z} \\subseteq \\u$, a contradiction.    \n \\end{proof}\n \n\\\n \n \\begin{lemma}\n Given $a\\in \\boldsymbol{\\mu}$ and $\\nu$, there is $b\\in \\boldsymbol{\\mu}$ such that $\\operatorname{ord}(b)\\geq 2^\\nu$ and $a=b^{2^\\nu}$.\n \\end{lemma}\n \n \\begin{proof}\n We prove this by internal induction on $\\nu$, noting that the assertion is internal since $b\\in \\boldsymbol{\\mu}$ can be replaced by $b\\in G^*$ by the previous lemma.  The assertion is true for $\\nu=1$ by Lemma \\ref{L:sqrt}.  Now suppose the assertion is true for $\\nu$, i.e. that $a=b^{2^\\nu}$ with $\\operatorname{ord}(b)\\geq 2^\\nu$.  Since $b\\in \\boldsymbol{\\mu}$, there is $c\\in \\boldsymbol{\\mu}$ with $b=c^2$ by Lemma \\ref{L:sqrt}.  \n \n\\\n\n\\noindent Claim:  For all $i\\in\\{1,\\ldots,2^\\nu\\}$, $c^{2i}$ is defined and $(c^2)^i=c^{2i}$.\n\n\\\n\n\\noindent We prove the claim by internal induction.  It is obviously true for $i=1$ and suppose it is true for a given $i\\leq 2^\\nu$.  Suppose $i+1\\leq 2^\\nu$.  We first show $c^{2i+1}$ is defined.  By induction, we know that $c^{2i}$ is defined, and so by Lemma \\ref{L:EZFacts}, it is enough to show $(c^k,c^l)\\in \\o^*$ for $k,l\\in\\{1,\\ldots,2i\\}$ with $k+l=2i+1$.  If $k$ is even, then $$c^k=(c^2)^{\\frac{k}{2}}=b^{\\frac{k}{2}}\\in \\boldsymbol{\\mu}$$ by Lemma \\ref{L:ordinf} since $\\frac{k}{2}\\leq i\\leq 2^{\\nu}$.  Similarly, if $k$ is odd, then $$c^k=c\\cdot (c^2)^{\\frac{k-1}{2}}=cb^{\\frac{k-1}{2}}\\in \\boldsymbol{\\mu}.$$  Thus, for $k,l\\in\\{1,\\ldots,2i\\}$, one has $(c^k,c^l)\\in \\boldsymbol{\\mu} \\times \\boldsymbol{\\mu} \\subseteq \\o^*$.  Now one can repeat the same argument using that $c^{2i+1}$ is defined to conclude that $c^{2i+2}$ is defined.  Now that we know that $c^{2i+2}$ is defined, we have, by induction, that $(c^2)^{i+1}=(c^2)^i\\cdot c^2=c^{2i}\\cdot c^2=c^{2i+2}$.  \n \n \\\n \n \\noindent By the claim, we have that $a=(c^2)^{2^\\nu}=c^{2^{\\nu+1}}$.  It remains to show that ord$(c)\\geq 2^{\\nu+1}$.  Let $i\\in\\{1,\\ldots,2^{\\nu+1}\\}$.  Then if $i$ is even, we have $c^i=b^{\\frac{i}{2}}\\in \\boldsymbol{\\mu}$  by Lemma \\ref{L:ordinf} since $\\frac{i}{2}\\leq 2^\\nu \\leq \\operatorname{ord}(b)$.  If $i$ is odd, then $c^i=c\\cdot c^{\\frac{i-1}{2}}\\in \\boldsymbol{\\mu}$ for the same reason.  Thus $c^i\\in \\boldsymbol{\\mu}$ for all $i\\in\\{1,\\ldots,2^{\\nu+1}\\}$ and so ord$(c)\\geq 2^{\\nu+1}$.\n \\end{proof}\n  \n \\\n \n \\begin{thm}\\label{T:kneigh}\n $K$ is a neighborhood of $1$.\n \\end{thm}\n\n\\begin{proof}\nFix $\\nu> \\mathbb{N}$.  Then the internal set of all $\\eta$ such that for each $x\\in V_\\eta$ there is $y\\in \\u^*$ such that $\\operatorname{ord}(y)\\geq 2^\\nu$ and $x=y^{2^\\nu}$ contains all infinite $\\eta$.  So by underspill, there is some $n>0$ in this set.  Now given $x\\in V_n$, there is $a\\in \\u^*$ such that $\\operatorname{ord}(a)\\geq 2^\\nu$ and $x=a^{2^\\nu}$.  Let $\\sigma:=2^\\nu$.  Then for $t\\in I$, $X_a(t)=\\operatorname{st}(a^{[t\\sigma]})\\in \\u$ and $X_a(1)=x$.  Thus $[X_a]\\in \\k$ and $x=[X_a](1)\\in K$.  We have shown $V_n\\subseteq K$ and this completes the proof. \n\\end{proof}\n \n\\ \n  \n\\section{The Space $L(G)$}\n\n\\noindent Throughout this section, we assume that our (locally compact) $G$ is $\\operatorname{NSS}$.  We fix a special neighborhood $\\u$ of $1$ in $G$.  We will equip $L(G)$ with a topology in such a way that it becomes a locally compact finite dimensional real topological vector space, where the operations of scalar multiplication and vector addition are as previously defined.  The section culminates with the important fact that locally compact $\\operatorname{NSS}$ local groups are locally euclidean.\n\n\\\n\n\\noindent \\textbf{The Compact-Open Topology}\n\n\\\n\n\\noindent We first recall that if $X,Y$ are hausdorff spaces, we can turn $\\operatorname{C}(X,Y)$, the set of continuous maps from $X$ to $Y$, into a topological space by equipping it with the compact-open topology.  This topology has as a subbasis the sets $B(C,U):=\\{f\\in \\operatorname{C}(X,Y) \\ | \\ f(C)\\subseteq U\\}$, where $C$ ranges over all compact subsets of $X$ and $U$ ranges over all open subsets of $Y$.  In the solution of the H5, one equipped the space $\\operatorname{C}(\\r,G)$ with the compact-open topology and then gave the space of 1-parameter subgroups the corresponding subspace topology.\n\n\\\n\n\\noindent  Analogously, we will define a topology on $L(G)$ by declaring the subbasic open set to be the sets $B_{C,U}:=\\{ \\mathbb{X} \\in L(G) \\ | \\ C\\subseteq \\operatorname{domain}(\\mathbb{X}) \\text{ and }\\mathbb{X}(C)\\subseteq U\\}$, where $C$ ranges over all compact subsets of $(-2,2)$ and $U$ ranges over all open subsets of $G$.  We next describe the monad structure of $L(G)$ with respect to this topology. \n\n\\\n\n\\begin{lemma}\\label{L:monadLG}\nLet $\\mathbb{X}\\in L(G)$ and $\\mathbb{Y} \\in L(G)^*$.  Then $\\mathbb{Y} \\in \\boldsymbol{\\mu}(\\mathbb{X})$ if and only if $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{Y})$ and for every $t\\in \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)$ and every $t'\\in \\boldsymbol{\\mu}(t)$, $\\mathbb{Y}(t')\\in \\boldsymbol{\\mu}(\\mathbb{X}(t))$.\n\\end{lemma}\n\n\\begin{proof}\nFirst suppose that $\\mathbb{Y} \\in \\boldsymbol{\\mu}(\\mathbb{X})$.  Since $\\mathbb{Y} \\in B_{C,G}^*$ for every compact $C\\subseteq \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)$, we have $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{Y})$.  Now suppose $t\\in \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)$ and $U$ is an open neighborhood of $\\mathbb{X}(t)$ in $G$.  Let $C$ be a compact neighborhood of $t$ with $C\\subseteq \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)$ and such that $\\mathbb{X}(C)\\subseteq U$.  Since $\\mathbb{Y} \\in B_{C,U}^*$, we have $\\mathbb{Y}(C^*)\\subseteq U^*$ and so in particular $\\mathbb{Y}(t')\\in U^*$.  Thus $\\mathbb{Y}(t')\\in \\boldsymbol{\\mu}(\\mathbb{X}(t))$.  \n\n\\\n\n\\noindent For the converse, suppose $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{Y})$ and for all $t\\in \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)$ and $t'\\in \\boldsymbol{\\mu}(t)$, we have $\\mathbb{Y}(t')\\in \\boldsymbol{\\mu}(\\mathbb{X}(t))$.  Suppose $\\mathbb{X}\\in B_{C,U}$.  Then for all $t\\in C^*$, we have $\\mathbb{Y}(t)\\in \\boldsymbol{\\mu}(\\mathbb{X}(\\operatorname{st}(t)))\\subseteq U^*$.  This implies that $\\mathbb{Y}\\in B_{C,U}^*$ and thus $\\mathbb{Y}\\in \\boldsymbol{\\mu}(\\mathbb{X})$.\n\\end{proof}\n\n\\\n\n\\noindent In the rest of the paper, we assume the special neighborhood $\\u$ of an NSS local group is chosen so small that $\\u\\subseteq \\u_6$.\n\n\\\n\n\\begin{lemma}\\label{L:calkcompact}\n$\\k$ is a compact neighborhood of $\\O$ in $L(G)$.\n\\end{lemma}\n\n\\begin{proof}\nTo see that $\\k$ is a neighborhood of $\\O$, choose an open $W\\subseteq \\u$ with $1\\in W$.  Let $\\k_W:=\\{\\mathbb{X} \\in L(G) \\ | \\ I\\subseteq \\operatorname{domain}(\\mathbb{X}) \\text{ and }\\mathbb{X}(I)\\subseteq W\\}\\subseteq \\k$.  Clearly $\\O\\in \\k_W$ and $\\k_W$ is an open set in $L(G)$ by the definition of the topology.  \n\n\\\n\n\\noindent We now prove that $\\k$ is compact.  Let $\\mathbb{Y} \\in \\k^*$.  It suffices to find $\\mathbb{X} \\in L(G)$ such that $\\mathbb{Y}\\in \\boldsymbol{\\mu}(\\mathbb{X})$.  Choose $Y\\in \\mathbb{Y}$ with $I^*\\subseteq \\operatorname{domain}(Y)$.  Now define $X:I\\to G$ by $X(t)=\\operatorname{st}(Y(t))$.  By Lemma \\ref{L:extension}, we may extend the domain of $X$ to ensure that it is a local 1-ps.  We claim that $\\mathbb{X}:=[X]$ is such that $\\mathbb{Y}\\in \\boldsymbol{\\mu}(\\mathbb{X})$.  We first note that $(-2,2)^* \\subseteq \\operatorname{domain}(\\mathbb{Y})$.  To see this, suppose $t\\in (1,2)^*$.  Then $(Y(t-1),Y(1))\\in \\u^*\\times \\u^*\\subseteq \\o^*$, so one can define $Y(t):=Y(t-1)Y(1)$.  One defines $Y(t)$ for $t\\in (-2,-1)^*$ in a similar fashion.  An easy check verifies that $Y:(-2,2)^*\\to G^*$ is an internal local 1-parameter subgroup.  Furthermore, if $t\\in \\operatorname{domain}(\\mathbb{X})\\cap (1,2)$ and $t'\\in \\boldsymbol{\\mu}(t)$, we have $$\\mathbb{Y}(t')=\\mathbb{Y}(t'-1)\\mathbb{Y}(1)\\in \\boldsymbol{\\mu}(X(t-1)X(1))=\\boldsymbol{\\mu}(X(t)).$$ A similar argument deals with the case that $t\\in \\operatorname{domain}(\\mathbb{X})\\cap (-2,-1)$, finishing the proof of the lemma. \n\\end{proof}\n\n\\\n\n\\begin{cor}\\label{C:Ehomeo} \n$E:\\k \\rightarrow K$ is a homeomorphism.  \n\\end{cor}\n\n\\begin{proof}\nBy Lemma \\ref{L:monadLG}, it is clear that $E$ is continuous.  The corollary is now immediate from the fact that $E$ is a continuous bijection and $\\k$ is compact.\n\\end{proof}\n\n\\\n\n\\noindent Suppose $\\mathbb{X} \\in L(G)$ and $s\\in (0,2)$ is such that $[-s,s]\\subseteq \\operatorname{domain}(\\mathbb{X})$ and $\\mathbb{X}([-s,s])\\subseteq \\u_2$.  Let $U\\subseteq \\u_2$ be a neighborhood of $1$.  Consider the set $$N_\\mathbb{X}(s,U):=\\{\\mathbb{Y} \\in L(G) \\ | \\ \\mathbb{Y}(t)\\in \\mathbb{X}(t)U \\text{ for all }t\\in [-s,s]\\}.$$\n\n\\\n\n\\begin{lemma}\nFor each neighborhood $U$ of $1$ in $G$ with $U\\subseteq \\u_2$, $N_\\mathbb{X}(s,U)$ is a neighborhood of $\\mathbb{X}$ in $L(G)$ and the collection \n$$\\{N_\\mathbb{X}(s,U) \\ | \\ U \\text{ is a neighborhood of }1 \\text{ in }G \\text{ and }U\\subseteq \\u_2\\}$$ is a neighborhood base of $\\mathbb{X}$ in $L(G)$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\mathbb{Y} \\in \\boldsymbol{\\mu}(\\mathbb{X})$.  Since $[-s,s]\\subseteq \\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{Y})$, we have $[-s,s]^*\\subseteq \\operatorname{domain}(\\mathbb{Y})$.  Now fix $t\\in [-s,s]^*$ and $U\\subseteq \\u_2$ a neighborhood of $1$ in $G$.  Then $\\mathbb{Y}(t),\\mathbb{X}(t)\\in \\boldsymbol{\\mu}(\\mathbb{X}(\\operatorname{st}(t))$, whence $\\mathbb{X}(t)^{-1}\\mathbb{Y}(t)\\in \\boldsymbol{\\mu}$.  Thus $\\mathbb{Y}(t)\\in \\mathbb{X}(t)U^*$ and hence $\\mathbb{Y} \\in N_\\mathbb{X}(s,U)^*$.  This implies that $\\boldsymbol{\\mu}(\\mathbb{X})\\subseteq N_\\mathbb{X}(s,U)^*$, whence $N_\\mathbb{X}(s,U)$ is a neighborhood of $\\mathbb{X}$.\n\n\\\n\n\\noindent Now suppose that $\\mathbb{Y} \\in N_\\mathbb{X}(s,U)^*$ for every neighborhood $U$ of $1$ in $G$ with $U\\subseteq \\u_2$.  We must show $\\mathbb{Y}\\in \\boldsymbol{\\mu}(\\mathbb{X})$.  We first need to check that $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{Y})$.  Say $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)=(-r,r)$.  Choose $n>0$ such that $\\frac{1}{n}(-r,r)\\subseteq [-s,s]$.  Since $\\mathbb{Y} \\in N_\\mathbb{X}(s,\\u_{2n})^*$, we see that the domain of $\\mathbb{Y}$ contains $(-r,r)$ since the rule $\\mathbb{Y}(t)=\\mathbb{Y}(\\frac{t}{n})^n$ for $t\\in (-r,r)$ defines an internal local 1-parameter subgroup.  Now fix $t\\in (-r,r)$ and $t'\\in \\boldsymbol{\\mu}(t)$.  We must show that $\\mathbb{Y}(t')\\in \\boldsymbol{\\mu}(\\mathbb{X}(t))$.  Let $U$ be a neighborhood of $\\mathbb{X}(t)$.  We will show that $\\mathbb{Y}(t')\\in U^*$.  Since $\\mathbb{X}(\\frac{t}{n})\\in \\operatorname{domain}(p_n)$, there is an open neighborhood $W_1$ of $\\mathbb{X}(\\frac{t}{n})$ such that $W_1\\subseteq \\operatorname{domain}(p_n)$ and $p_n(W_1)\\subseteq U$.  By local compactness, we can choose a compact neighborhood $W$ of $1$ in $G$ with $W\\subseteq \\u_2$ such that $\\{\\mathbb{X}(\\frac{t}{n})\\}\\times W \\subseteq \\o$ and $\\mathbb{X}(\\frac{t}{n})W\\subseteq W_1$.  Since $\\mathbb{Y} \\in N_\\mathbb{X}(s,W)^*$, we know that $\\mathbb{Y}(\\frac{t'}{n})\\in \\mathbb{X}(\\frac{t'}{n})W^*\\subseteq W_1^*$.  Indeed, suppose $a\\in W^*$.  Then $\\mathbb{X}(\\frac{t'}{n})a=\\mathbb{X}(\\frac{t}{n})\\cdot (\\mathbb{X}(\\frac{t-t'}{n})a)$ and $\\mathbb{X}(\\frac{t-t'}{n})a\\sim a$, whence $\\operatorname{st}(\\mathbb{X}(\\frac{t-t'}{n})a)\\in W$.  Since $W_1$ is open, we see that $\\mathbb{X}(\\frac{t'}{n})a\\in W_1^*$.  We now see that $\\mathbb{Y}(\\frac{t'}{n})\\in \\operatorname{domain}(p_n)^*$ and $\\mathbb{Y}(t')=\\mathbb{Y}(\\frac{t'}{n})^n \\in U^*$.\n\\end{proof}\n\n\\\n\n\\begin{thm}\\label{T:L(G)vs}\n$L(G)$ is a locally compact, finite dimensional real topological vector space.\n\\end{thm}\n\n\\begin{proof}\nIf we can prove that $L(G)$ is a topological vector space, then since it is locally compact by Lemma \\ref{L:calkcompact}, it is finite-dimensional by a theorem of Riesz.  We first prove that scalar multiplication is continuous.  Let $r \\in \\r\\setminus \\{0\\}$ and let $f_r:L(G)\\rightarrow L(G)$ be defined by $f_r(\\mathbb{X})=r \\cdot \\mathbb{X}$.  Suppose $r'\\in \\boldsymbol{\\mu}(r)$ and $\\mathbb{X}'\\in \\boldsymbol{\\mu}(\\mathbb{X})$.  We must show that $r'\\mathbb{X}'\\in \\boldsymbol{\\mu}(r\\mathbb{X})$.  Since $\\operatorname{domain}(\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(\\mathbb{X}')$, it is easy to see that $\\operatorname{domain}(r\\mathbb{X})\\cap (-2,2)\\subseteq \\operatorname{domain}(r'\\mathbb{X}')$.  Furthermore, for $t\\in \\operatorname{domain}(r\\mathbb{X})\\cap (-2,2)$ and $t' \\in \\boldsymbol{\\mu}(t)$, one has $$(r'\\mathbb{X}')(t')=\\mathbb{X}'(r't')\\in \\boldsymbol{\\mu}(\\mathbb{X}(rt))=\\boldsymbol{\\mu}((r\\mathbb{X})(t)),$$ whence $r'\\mathbb{X}'\\in \\boldsymbol{\\mu}(r\\mathbb{X})$.  \n\n\\\n\n\\noindent We next prove that $+$ is continuous.  We first show that $+$ is continuous at $(\\O,\\O)$.  To do this, it suffices to show that for any subbasic open set $B_{C,U}$ of $L(G)$ containing $\\O$ (so that $1\\in U$), there is a subbasic open set $B_{C,W}$ of $L(G)$ containing $\\O$ such that $B_{C,W}+B_{C,W}\\subseteq B_{C,U}$.  Let $U_1$ be a compact neighborhood of $1$ such that $U_1\\subseteq U$.  Let $Z \\subseteq \\boldsymbol{\\mu}$ be an internally open set.  Then by Lemma \\ref{L:Gsigmagroup}, we know that for every $a,b \\in Z$ and for every $\\sigma$ such that for all $i\\in\\{1,\\ldots,\\sigma\\}$, $a^i$ and $b^i$ are defined and $a^i,b^i\\in Z$, we have $(ab)^\\sigma$ is defined and $(ab)^\\sigma \\in U_1^*$.  Hence, by transfer, there is an open set $W$ such that for all $a,b \\in W$ and all $n$ with $a^i$ and $b^i$ defined and $a^i,b^i \\in W$ for all $i\\in\\{1,\\ldots,n\\}$, we have $(ab)^n$ defined and $(ab)^n\\in U_1$.  We claim that this is the desired $W$.  Suppose $\\mathbb{X},\\mathbb{Y} \\in B_{C,W}$.  Let $t\\in C$ and suppose $\\nu$ is such that $\\frac{\\nu}{\\sigma}\\sim t$.  Then $(\\mathbb{X}+\\mathbb{Y})(t)\\sim [\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma})]^\\nu$.  Let $a:=\\mathbb{X}(\\frac{1}{\\sigma})$ and $b:=\\mathbb{Y}(\\frac{1}{\\sigma})$.  By assumption, for all $i\\in\\{1,\\ldots,\\nu\\}$, $a^i$ and $b^i$ are defined and $a^i,b^i \\in W^*$.  Thus, $(ab)^\\nu \\in U_1^*$, implying that  $(\\mathbb{X}+\\mathbb{Y})(t)\\in U_1\\subseteq U$ and so $\\mathbb{X}+\\mathbb{Y} \\in B_{C,U}$.\n\n\\\n\n\\noindent  In order to finish the proof that $+$ is continuous, it is enough to show that, for any $\\mathbb{Y} \\in L(G)$, the map $\\mathbb{X} \\mapsto \\mathbb{X}+\\mathbb{Y}:L(G)\\rightarrow L(G)$ is continuous at $\\O$ and the map $\\mathbb{X} \\mapsto \\mathbb{X}-\\mathbb{Y}:L(G)\\rightarrow L(G)$ is continuous at $\\mathbb{Y}$.  Let $W\\subseteq L(G)$ be a neighborhood of $\\mathbb{Y}$ in $L(G)$.  Fix $s\\in \\operatorname{domain}(\\mathbb{Y})\\cap (-2,2)$ such that $\\mathbb{Y}([-s,s])\\subseteq \\u_2$.  Fix a compact neighborhood $U$ of $1$ in $G$ with $U\\subseteq \\u_2$ so that $N_\\mathbb{Y}(s,U)\\subseteq W$.  Suppose $X\\in \\boldsymbol{\\mu}(\\O)\\subseteq L(G)^*$.  By Lemma 6.2, for all $i\\leq [s\\sigma]$ we have $(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^i$ is defined and infinitely close to $\\mathbb{Y}(\\frac{i}{\\sigma})$.  In particular, this means that for every $\\mathbb{X} \\in \\boldsymbol{\\mu}(\\O)$, whenever $i\\leq [s\\sigma]$, we have $((\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^i$ is defined and \n$$(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^i\\in Y(\\frac{i}{\\sigma})U^*.$$  By overspill, this gives a neighborhood $V$ of $\\O$ in $L(G)$ such that for all $\\mathbb{X} \\in V$ we have $((\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^i$ is defined and $$(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^i\\in Y(\\frac{i}{\\sigma})U^*$$ whenever $i\\leq [s\\sigma]$.  It now follows that for $\\mathbb{X} \\in V$, we have $$t\\in \\operatorname{domain}(\\mathbb{X}+\\mathbb{Y}) \\text{ and }(\\mathbb{X}+\\mathbb{Y})(t)\\in \\mathbb{Y}(t)U$$ for all $t\\in [-s,s]$, whence $\\mathbb{X}+\\mathbb{Y} \\in N_\\mathbb{Y}(s,U)\\subseteq W$.  This proves the continuity of $\\mathbb{X} \\mapsto \\mathbb{X}+\\mathbb{Y}$ at $\\O$.\n\n\\\n\n\\noindent The continuity of $\\mathbb{X}\\mapsto \\mathbb{X}-\\mathbb{Y}$ at $\\mathbb{Y}$ is proved in a similar fashion.  Let $W\\subseteq L(G)$ be a neighborhood of $\\O$.  Fix $s\\in \\operatorname{domain}(\\mathbb{Y})\\cap (-2,2)$.  Choose a compact neighborhood $U$ of $1$ in $G$ with $U\\subseteq \\u_2$ so that $N_\\O(s,U)\\subseteq W$.  Suppose $\\mathbb{X}\\in \\boldsymbol{\\mu}(\\mathbb{Y})\\subseteq L(G)^*$.  By Lemma 6.3, $(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(-\\frac{1}{\\sigma}))^i$ is defined and infinitesimal whenever $i\\leq [s\\sigma]$.  In particular, \n$$(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(-\\frac{1}{\\sigma}))^i\\in U^*$$ whenever $i\\leq [s\\sigma]$.  By overspill, this gives a neighborhood $V$ of $\\mathbb{Y}$ in $L(G)$ such that for all $\\mathbb{X} \\in V$, we have $(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(-\\frac{1}{\\sigma}))^i$ is defined and $$(\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(-\\frac{1}{\\sigma}))^i\\in U^*$$ whenever $i\\leq [s\\sigma]$.  It now follows that for $\\mathbb{X} \\in V$, we have $$t\\in \\operatorname{domain}(\\mathbb{X}-\\mathbb{Y}) \\text{ and }(\\mathbb{X}-\\mathbb{Y})(t)\\in U$$ for all $t\\in [-s,s]$, whence $\\mathbb{X}-\\mathbb{Y} \\in N_\\O(s,U)\\subseteq W$.     \n\n\\\n\n\\noindent We now need to prove that $L(G)$ is a vector space.  There is only one vector space axiom which is not trivial to verify, namely that $r \\cdot (\\mathbb{X}+\\mathbb{Y})=r \\cdot \\mathbb{X} + r \\cdot \\mathbb{Y}$.  We first consider the case that $r \\in \\mathbb{Z}$.  Let $a:=\\mathbb{X}(\\frac{1}{\\sigma})$ and $b:=\\mathbb{Y}(\\frac{1}{\\sigma})$.  So $S(r \\cdot \\mathbb{X} + r \\cdot \\mathbb{Y})=(a ^ r b^r) G^{\\operatorname{o}}(\\sigma)$.  Meanwhile, $S(r \\cdot (\\mathbb{X}+\\mathbb{Y}))=((ab)^r)G^{\\operatorname{o}}(\\sigma)$.  But $G(\\sigma)\/G^{\\operatorname{o}}(\\sigma)$ is abelian, so $$S(r \\cdot (\\mathbb{X}+\\mathbb{Y}))=((ab)^r)G^{\\operatorname{o}}(\\sigma)=(a ^ r b^r) G^{\\operatorname{o}}(\\sigma)=S(r \\cdot \\mathbb{X} + r \\cdot \\mathbb{Y}).$$  We have thus proven this axiom for the case $r \\in \\mathbb{Z}$.\n\n\\\n\n\\noindent Once again suppose $r \\in \\mathbb{Z}$, $r\\not=0$.  We now know that $r \\cdot (\\frac{1}{r}\\mathbb{X} + \\frac{1}{r}Y)=\\mathbb{X} + \\mathbb{Y}$, i.e. that $\\frac{1}{r}\\cdot (\\mathbb{X}+\\mathbb{Y})=\\frac{1}{r}\\mathbb{X}+\\frac{1}{r}\\mathbb{Y}$.  So for $r \\in \\mathbb{Q}$, we have $r \\cdot (\\mathbb{X}+\\mathbb{Y})=r \\cdot \\mathbb{X} + r \\cdot \\mathbb{Y}$.  We finally conclude that the axiom holds for all $r \\in \\r$ from the fact that the operations of addition and scalar multiplication are continuous.             \n\\end{proof}\n\n\\\n\n\\begin{rmk}\nSuppose $H$ and $H'$ are locally compact pure local groups.  Then the proof of Theorem \\ref{T:L(G)vs} shows that $L(H)$ is a real topological vector space.  The $\\operatorname{NSS}$ assumption was only used to obtain local compactness, and hence finite-dimensionality.  Moreover, given any local morphism $f:H\\rightarrow H'$, one can verify that the map $L(f):L(H)\\rightarrow L(H')$ is $\\r$-linear and continuous.\n\\end{rmk}\n\n\\\n\n\\begin{cor}\n$G$ is locally euclidean.\n\\end{cor}\n\n\\begin{proof}\nSince $L(G)$ is a finite dimensional topological vector space over $\\r$, we have an isomorphism $L(G)\\cong \\r^n$ of real vector spaces for some $n$, and any such isomorphism must be a homeomorphism.  Hence, by Lemma \\ref{L:calkcompact}, $\\k$ contains an open neighborhood $U$ of $1$ which is homeomorphic to an open subset of $\\r^n$.   Shrink $U$ if necessary so that $E(U)\\subseteq W$, where $W$ is an open neighborhood of $1$ in $G$ contained in $K$; such a $W$ exists by Theorem \\ref{T:kneigh}.  Then by Corollary \\ref{C:Ehomeo}, $E(U)$ is an open neighborhood of $1$ in $G$ homeomorphic to an open subset of $\\r^n$.\n\\end{proof}\n\n\\\n\n\\section{Local H5 for $\\operatorname{NSS}$ local groups}\n\n\\\n\n\\noindent The goal of this section is to prove the Local H5 for $\\operatorname{NSS}$ local groups using the local version of the Adjoint Representation Theorem.  \n\n\\\n\n\\noindent In this section we assume $G$ is $\\operatorname{NSS}$.  Recall that our special neighborhood has been chosen so that $\\u\\subseteq \\u_6$.\n\n\n\\\n\n\\noindent \\textbf{Local Adjoint Representation Theorem}\n\n\\\n\n\\noindent Fix $g\\in \\u_6$.  Let $\\mathbb{X} \\in L(G)$ and $X\\in \\mathbb{X}$.  Choose  $r\\in \\operatorname{domain}(X)\\cap \\r^{>0}$ such that $X((-r,r))\\subseteq \\u_6$.  Define the function $gXg^{-1}:(-r,r)\\rightarrow G$ by the rule $(gXg^{-1})(t)=gX(t)g^{-1}$.  We claim that $gXg^{-1}$ is a local 1-ps of $G$.  Certainly $gXg^{-1}$ is continuous.  Suppose $s,t,s+t\\in (-r,r)$.  Then since $g,g^{-1},X(t),X(s)\\in \\u_6$, we have \n\n\\begin{align}\n(gXg^{-1})(t+s)&=g(X(t)X(s))g^{-1} \\notag \\\\ \\notag\n                          &=(gX(t)g^{-1})(gX(s)g^{-1})\\\\ \\notag\n                          &=(gXg^{-1})(t)\\cdot (gXg^{-1})(s)\\\\ \\notag \\\\ \\notag\n\\end{align}\nThus $gXg^{-1}$ is a local 1-ps of $G$ and $\\operatorname{Ad}_g(\\mathbb{X}):=g\\mathbb{X} g^{-1}:=[gXg^{-1}]\\in L(G)$ is easily seen to be independent of the choice of $X\\in \\mathbb{X}$ and $r\\in \\operatorname{domain}(X)$ as chosen above.    \n\n\\\n\n\\begin{lemma}\nSuppose $\\sigma > \\mathbb{N}$, $a\\in G(\\sigma)$ and $g\\in \\u_6$.  Then  \n\\begin{enumerate}\n\\item $gag^{-1}\\in G(\\sigma)$;\n\\item $\\operatorname{Ad}_g([X_a])=[X_{gag^{-1}}]$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $\\tau:=\\operatorname{ord}(a)$.  Note that $\\sigma=\\operatorname{O}(\\tau)$.  \n\n\\\n\n\\noindent Claim:  If $i\\in \\{1,\\ldots,\\tau\\}$, then $(gag^{-1})^i$ is defined and $(gag^{-1})^i=ga^ig^{-1}$.\n\n\\\n\n\\noindent We prove this by internal induction on $i$.  This is certainly true for $i=1$ and we now assume that it holds for a given $i$ and that $i+1\\leq \\tau$.  Since we have assumed that $(gag^{-1})^i$ is defined, to show $(gag^{-1})^{i+1}$ is defined, it is enough to show, by Lemma \\ref{L:EZFacts}, that $((gag^{-1})^k,(gag^{-1})^l)\\in \\o^*$ for all $k,l\\in \\{1,\\ldots,i\\}$ such that $k+l=i+1$.  However, by induction, $(gag^{-1})^k=ga^kg^{-1}$ and $(gag^{-1})^l=ga^lg^{-1}$, and since $g,a^k,a^l\\in \\u_6^*$, we have $(ga^kg^{-1},ga^lg^{-1})\\in \\o^*$.  We thus know that $(gag^{-1})^{i+1}$ is defined.  Now $(gag^{-1})^{i+1}=(gag^{-1})^i(gag^{-1})=(ga^ig^{-1})(gag^{-1})=ga^{i+1}g^{-1}$ since $g,a^i,a,g^{-1}\\in \\u_6^*$.\n\n\\\n\n\\noindent By the claim, if $i=\\operatorname{o}(\\sigma)$, then $(gag^{-1})^i$ is defined and $(gag^{-1})^i=ga^ig^{-1}\\in \\boldsymbol{\\mu}$ since $a\\in G(\\sigma)$.  This proves (1).\n\n\\\n\n\\noindent  We now must prove (2).  Fix $r\\in \\r^{>0}$ such that $[r\\sigma]\\leq \\tau$, $X_a((-r,r))\\subseteq \\u_6$ and $gX_a((-r,r))g^{-1}\\subseteq \\u$.  Then for $t\\in (-r,r)$, we have \n\\begin{align}\n(\\operatorname{Ad}_g([X_a])(t))&=gX_a(t)g^{-1} \\notag \\\\ \\notag\n                          &=g(\\operatorname{st}(a^{[t\\sigma]}))g^{-1}\\\\ \\notag\n                          &=\\operatorname{st}(ga^{[t\\sigma]}g^{-1})\\\\ \\notag\n                          &=\\operatorname{st}((gag^{-1})^{[t\\sigma]})\\\\ \\notag\n                          &=X_{gag^{-1}}(t),\\\\ \\notag \\\\ \\notag\n\\end{align}\n\nwhich finishes the proof.\n\\end{proof}\n\n\\\n\n\\noindent The proof of the following lemma is routine.\n\n\\\n\n\\begin{lemma}\nFor $g\\in \\u_6$, $\\operatorname{Ad}_g:L(G)\\rightarrow L(G)$ is a vector space automorphism with inverse $\\operatorname{Ad}_{g^{-1}}$.\n\\end{lemma}\n\n\\\n\n\\noindent Let $\\operatorname{Aut}(L(G))$ denote the group of vector space automorphisms of $L(G)$.  Suppose $\\operatorname{dim}_\\r(L(G))=n$.  Take an $\\r$-linear isomorphism $L(G) \\cong \\r^n$; it induces a group isomorphism\n$\\operatorname{Aut}(L(G)) \\cong \\text{GL}_n(\\r) \\subseteq \\r^{n^2}$, and we take the\ntopology on $\\operatorname{Aut}(L(G))$ that makes this group isomorphism a homeomorphism.\n(This topology does not depend on the choice of $\\r$-linear isomorphism\n$L(G) \\cong \\r^n$.)  For $T\\in \\operatorname{Aut}(L(G))^*$, we see that $T$ is nearstandard if $T(\\mathbb{X})\\in L(G)^*_{\\operatorname{ns}}$ for all $\\mathbb{X}\\in L(G)$.  For $T,T'\\in \\operatorname{Aut}(L(G))^*_{\\operatorname{ns}}$, we see that $T\\sim T'$ if and only if $T(\\mathbb{X})\\sim T'(\\mathbb{X})$ for all $\\mathbb{X}\\in L(G)$.\n\n\\\n\n\\begin{thm}(Local Adjoint Representation Theorem) We have a morphism $\\operatorname{Ad}:G|\\u_6\\rightarrow \\operatorname{Aut}(L(G))$ of local groups given by $\\operatorname{Ad}(g):=\\operatorname{Ad}_g$.\n\\end{thm}\n\n\\begin{proof}\nSuppose $g,h,gh\\in \\u_6$ and $[X_a]\\in L(G)$.  Then\n\\begin{align}\n\\operatorname{Ad}_{gh}([X_a])&=[X_{(gh)a(gh)^{-1}}] \\notag \\\\ \\notag \n                        &=[X_{g(hah^{-1})g^{-1}}] \\\\ \\notag\n                        &=\\operatorname{Ad}_g(\\operatorname{Ad}_h([X_a])). \\notag \\\\ \\notag\n\\end{align}\nThus $\\operatorname{Ad}(gh)=\\operatorname{Ad}(g)\\circ \\operatorname{Ad}(h)$.  All that remains to prove is that $\\operatorname{Ad}$ is continuous.  To do this, it suffices to prove that $\\operatorname{Ad}$ is continuous at $1$.  Suppose $a\\in \\boldsymbol{\\mu}$ and $\\mathbb{X} \\in L(G)$.\n \n\\\n\n\\noindent Claim:  $\\operatorname{domain}(\\mathbb{X})=\\operatorname{domain}(\\operatorname{Ad}_a(\\mathbb{X}))\\cap \\r$.\n\n\\\n\n\\noindent In order to verify the claim, suppose $(-r,r)\\subseteq \\operatorname{domain}(\\mathbb{X})$.  We must verify that if $s,t,s+t\\in (-r,r)$, then $(a\\mathbb{X}(s)a^{-1},a\\mathbb{X}(t)a^{-1})\\in \\o^*$ and $(a\\mathbb{X}(s)a^{-1})(a\\mathbb{X}(t)a^{-1})=a\\mathbb{X}(s+t)a^{-1}$.  From the fact that $(\\mathbb{X}(s),\\mathbb{X}(t))\\in \\o$ and $a\\in \\boldsymbol{\\mu}$, we have that $(a\\mathbb{X}(s)a^{-1},a\\mathbb{X}(t)a^{-1})\\in \\o^*$ and we have that $(a\\mathbb{X}(s)a^{-1})(a\\mathbb{X}(t)a^{-1})=a\\mathbb{X}(s+t)a^{-1}$ from the usual calculations involving associativity when working with infinitesimals and nearstandard elements.\n\n\\  \n\n\\noindent Now notice that if $t\\in \\operatorname{domain}(\\mathbb{X})$, then $(\\operatorname{Ad}_a(\\mathbb{X}))(t)=a\\mathbb{X}(t)a^{-1}\\sim \\mathbb{X}(t)$.  So by Lemma \\ref{L:monadLG}, $\\operatorname{Ad}_a(\\mathbb{X})\\sim \\mathbb{X}$.  By the nonstandard characterization of the topology on $\\operatorname{Aut}(L(G))$, we have $\\operatorname{Ad}_a\\sim \\operatorname{id}_{L(G)}$, and hence $\\operatorname{Ad}$ is continuous.  \n\\end{proof}\n\n\n\\\n\n\\noindent \\textbf{Some Facts About Local Lie Groups}\n\n\\\n\n\\noindent In the global setting, the Adjoint Representation Theorem and some elementary Lie group theory imply that locally compact $\\operatorname{NSS}$ groups are Lie groups.  In the local group setting, we have to work a little harder.  We need a few ways of obtaining local Lie groups and the results that follow serve this purpose.  \n\n\\\n\n\\begin{df}\nA local group $G$ is \\textbf{abelian} if there is a neighborhood $U$ of $1$ in $G$ such that $U\\subseteq \\u_2$ and $ab=ba$ for all $a,b\\in U$.\n\\end{df}\n\n\\\n\n\\begin{thm}\\label{T:abelian}\nSuppose $G$ is abelian.  Then $G$ is locally isomorphic to a Lie group.\n\\end{thm}\n\n\\begin{proof}\nSuppose we have chosen our special neighborhood $\\u$ so small that elements of $\\u$ commute with each other.  Recall that we are still assuming that $\\u \\subseteq \\u_6$.\n\n\\\n\n\\noindent Claim:  Suppose $a,b\\in \\u$ are such that $(ab)^n,a^n$ and $b^n$ are all defined and $a^i,b^i\\in \\u$ for all $i\\in \\{1,\\ldots,n\\}$.  Then $(ab)^n=a^nb^n$.\n\n\\\n\n\\noindent We prove the claim by induction on $n$.  The case $n=1$ is trivial.  Now suppose $(ab)^{n+1}$ is defined as are $a^{n+1}$ and $b^{n+1}$.  Also suppose $a^i$ and $b^i$ are in $\\u$ for all $i\\in \\{1,\\ldots,n+1\\}$.  Then $$(ab)^{n+1}=(ab)^n\\cdot (ab)=(a^nb^n)\\cdot ab=a^nb^nab=a^{n+1}b^{n+1}.$$\n\n\\noindent This proves the claim.\n\n\\\n\n\\noindent Choose a symmetric open neighborhood $\\mathcal{V}$ of $\\O$ in $L(G)$ such that $\\mathcal{V}\\subseteq \\k$ and $\\mathbb{X}+\\mathbb{Y}\\in \\k$ for all $\\mathbb{X},\\mathbb{Y} \\in \\mathcal{V}$.  By the transfer of the claim, if $\\mathbb{X},\\mathbb{Y}\\in \\mathcal{V}$, then \n\n\\begin{align}\n(\\mathbb{X}+\\mathbb{Y})(1)&=\\operatorname{st}((\\mathbb{X}(\\frac{1}{\\sigma})\\mathbb{Y}(\\frac{1}{\\sigma}))^\\sigma) \\notag \\\\ \\notag\n                &=\\operatorname{st}(\\mathbb{X}(\\frac{1}{\\sigma})^\\sigma \\mathbb{Y}(\\frac{1}{\\sigma})^\\sigma) \\notag \\\\\n                &=\\mathbb{X}(1)\\mathbb{Y}(1). \\notag \\\\ \\notag\n\\end{align}\n\n\\noindent After possibly shrinking $\\mathcal{V}$, we can choose a symmetric open neighborhood $V$ of $1$ in $G$ such that $E(\\mathcal{V})=V$.  This witnesses that the equivalence class of the local exponential map is a local isomorphism from $L(G)$ to $G$.  Since $L(G)$ is a Lie group, we are done. \n\\end{proof}\n\n\\\n\n\\noindent To prove the next lemma, we need the following well-known theorem.\n\n\\\n\n\\begin{thm}(von Neumann, ~\\cite{MZ}, pg. 82)\nIf $H$ is a hausdorff topological group which admits an injective continuous homomorphism into $\\operatorname{GL}_n(\\r)$ for some $n$, then $H$ is a Lie group.\n\\end{thm}\n\n\\\n\n\\begin{lemma}\\label{L:embedding}\nSuppose $f:G\\rightarrow \\operatorname{GL}_n(\\r)$ is an injective morphism of local groups.  Then $G$ is a local Lie group.\n\\end{lemma}\n\n\\begin{proof}\nLet $G'=f(G)\\subseteq \\text{GL}_n(\\r)$.  Let $H$ be the subgroup of $\\text{GL}_n(\\r)$ generated by $G'$.  Let $\\mathcal{F}'$ be the filter of neighborhoods of $1$ in $G$ and $\\mathcal{F}''$ be the image of $\\mathcal{F}'$ under $f$.  Finally, let $\\mathcal{F}$ be the filter in $H$ generated by $\\mathcal{F}''$.  It is routine to verify that $\\mathcal{F}$ satisfies the properties needed to make it a neighborhood filter at $1$ for a topology on $H$ which makes $H$ a topological group.  By the definition of this topology, the inclusion map $H\\hookrightarrow \\text{GL}_n(\\r)$ is continuous.  By von-Neumann's theorem, $H$ is a Lie group.  The result now follows from the fact that $G$ is homeomorphic with $f(G)$ when $f(G)$ is given the induced topology from $H$.\n\\end{proof}\n\n\\\n\n\\noindent We use the following theorem of Kuranishi \\cite{K} to finish our proof of the $\\operatorname{NSS}$ case.  \n\n\\begin{thm}\nLet $G$ be a locally compact local group and let $H$ be a normal sublocal group of $G$.  Consider the local coset space $(G\/H)_W$ as in Lemma \\ref{L:localcoset}.  Suppose \n\\begin{enumerate}\n\\item $H$ is an abelian local Lie group;\n\\item $(G\/H)_W$ is a local Lie group;\n\\item there is a set $M\\subseteq W$ containing $1$ and $W'\\subseteq W$, an open neighborhood of $1$ in $G$, such that for every $(zH)\\cap W\\in \\pi(W')$, there is exactly one $a\\in M$ such that $a\\in (zH)\\cap W$; moreover, the map $\\pi(W')\\rightarrow M$ that assigns to each element of $\\pi(W')$ the corresponding $a$ in $M$ is continuous.\n\\end{enumerate}\nThen $G|W$ is a local Lie group.\n\\end{thm}\n\n\\\n\n\\noindent \\textbf{Proof of the $\\operatorname{NSS}$ Case}\n\n\\\n\n\\noindent We can now finish the proof that the Local H5 holds for $\\operatorname{NSS}$ local groups.  Let $G':=G|\\u_6$.  Recall the morphism of local groups $\\operatorname{Ad}:G' \\rightarrow \\operatorname{Aut}(L(G))$.   Then $H:=\\ker(\\operatorname{Ad})$ is a normal sublocal group of $G'$.  Since $H$ is also a locally compact $\\operatorname{NSS}$ local group, we know that $H$ must have an open (in $H$) neighborhood $V$ of $1$ ruled by local 1-parameter subgroups of $H$, which are also local 1-parameter subgroups of $G$.  We claim $gh=hg$ for all $g,h\\in V$, implying that $H$ is abelian.  Note that if $\\u'\\subseteq \\u$ is a special neighborhood for $H$ and if $\\mathbb{X}\\in L(H)$ is such that $I\\subseteq \\operatorname{domain}(\\mathbb{X})$ and $\\mathbb{X}(I)\\subseteq \\u'$, then $I\\subseteq \\operatorname{domain}(g\\mathbb{X} g^{-1})$.  So for $g,h\\in V$, writing $h=\\mathbb{X}(1)$ for some $\\mathbb{X}\\in L(H)$, we have $ghg^{-1}=h$.\n\n\\\n\n\\noindent   By Theorem \\ref{T:abelian}, we have that a restriction of $H$ is a local Lie group.  Let us abuse notation and denote this restriction, which is an equivalent sublocal group of $G'$, by $H$.  Note that if we let $G'':=(G'\/H)_W$ be some local coset space, then the adjoint representation induces an injective morphism $G''\\rightarrow \\operatorname{Aut}(L(G))$.  Thus $G''$ is a local Lie group by Lemma \\ref{L:embedding}.  In order to finish the proof, we need to show that condition (3) of Kuranishi's theorem is satisfied.  \n\n\\\n\n\\noindent Since $G''$ is a local Lie group, we can introduce \\textit{canonical coordinates of the second kind}.  More precisely, we can find an open neighborhood $\\u'$ of $1$ in $G''$ and a basis $\\mathbb{X}_1',\\ldots,\\mathbb{X}_r'$ of $L(G'')$ such that every element of $\\u'$ is of the form $\\mathbb{X}_1'(s_1)\\cdots \\mathbb{X}_r'(s_r)$ for a \\textit{unique} tuple $(s_1,\\ldots,s_r)\\in [-\\beta',\\beta']^r$.  Without loss of generality, we can suppose the closure of $\\u'$ is a special neighborhood of $G''$.  Choose a special neighborhood $\\u$ of $G'$ such that $\\pi(\\u)\\subseteq \\u'$.  Let $Z\\subseteq \\u$ be an open neighborhood of $1$ ruled by local 1-parameter subgroups of $G'$.  Fix $s_0\\in (0,\\beta')$ such that $\\mathbb{X}_1'(s_0),\\ldots,\\mathbb{X}_r'(s_0)\\in \\pi(Z)$.  Choose $x_i\\in Z$ such that $\\pi(x_i)=\\mathbb{X}_i'(s_0)$.  Let $\\mathbb{X}_i\\in L(G')$ be such that $\\mathbb{X}_i(s_0)=x_i$.  \n\n\\\n\n\\noindent Let $\\beta < s_0$ be so that $\\mathbb{X}_i(s)\\in \\u_{2r}$ if $|s|\\leq \\beta$ and $\\mathbb{X}_1(s_1)\\cdots \\mathbb{X}_r(s_r)\\in W$ if $|s_i|\\leq \\beta$ for all $i\\in \\{1,\\ldots,r\\}$.  A uniqueness of root argument yields that $\\pi(\\mathbb{X}_i(s))=\\mathbb{X}_i'(s)$ for $|s|\\leq \\beta$.  Set $$M:=\\{\\mathbb{X}_1(s_1)\\cdots \\mathbb{X}_r(s_r) \\ | \\ |s_i|\\leq \\beta \\text{ for all }i=1,\\ldots,r\\}$$ and let $W'\\subseteq W$ be an open neighborhood of $1$ in $G$ contained in the image of the map $$(s_1,\\ldots,s_r)\\mapsto \\mathbb{X}_1(s_1)\\cdots \\mathbb{X}_r(s_r):[-\\beta,\\beta]^r\\rightarrow G.$$  One can check that the choices of $M$ and $W'$ fulfill condition (3) of Kuranishi's theorem.\n\n\\begin{flushright}\n $\\Box$\n\\end{flushright}\n\\\n\n\\section{Locally Euclidean local groups are $\\operatorname{NSS}$}  \n\n\\\n\n\\noindent In order to finish the proof of the Local H5, we must now prove that every locally euclidean local group is $\\operatorname{NSS}$.  This proceeds in two stages:  locally euclidean local groups are $\\operatorname{NSCS}$ and locally connected $\\operatorname{NSCS}$ local groups are $\\operatorname{NSS}$.\n\n\\\n\n\\noindent \\textbf{Locally euclidean local groups are $\\operatorname{NSCS}$} \n\n\\\n\n\\noindent We will need the following facts.\n\n\\begin{thm}\\label{T:compOPS}(\\cite{MZ}, pg. 105)\nEvery compact connected nontrivial hausdorff topological group has a nontrivial 1-parameter subgroup.\n\\end{thm}\n\n\\\n\n\\begin{lemma}\\label{L:OPSfacts}\nLet $H$ be a topological group and $X:\\r \\rightarrow H$ a 1-parameter subgroup.  \n\\begin{enumerate}\n\\item If $H_1$ is a closed subgroup of $H$, then either $X(\\r)\\subseteq H_1$ or there is a neighborhood $D$ of $0$ in $\\r$ such that $X(D)\\cap H_1=\\{1\\}$.  \n\\item If $X$ is nontrivial, then there is a neighborhood $D$ of $0$ in $\\r$ on which $X$ is injective.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThis is immediate from the well-known fact that a closed subgroup of the additive group of $\\r$ different from $\\{0\\}$ and $\\r$ is of the form $\\mathbb{Z} r$ with $r\\in \\r^{>0}$.\n\\end{proof}\n\n\\\n\n\\noindent The following lemma is the local group version of Theorem 5.1 in ~\\cite{H}.  \n\n\\ \n\n\\begin{lemma}\\label{L:neighandgroup}\nSuppose $V$ is a neighborhood of $1$ in $G$.  Then $V$ contains a compact subgroup $H$ of $G$ and a neighborhood $W$ of $1$ in $G$ such that every subgroup of $G$ contained in $W$ is contained in $H$.\n\\end{lemma}\n\n\\begin{proof}\nLet $W$ be an internal neighborhood of $1$ in $G^*$ such that $W\\subseteq \\boldsymbol{\\mu}$.  Note that if $E_1,\\ldots,E_\\nu$ is an internal sequence of internal subgroups of $G^*$ contained in $W$ and $a_1,\\ldots,a_\\nu$ is an internal sequence such that $a_i \\in E_i$ for all $i\\in \\{1,\\ldots,\\nu\\}$, then $a_1\\cdots a_\\nu$ is defined by Lemma \\ref{L:prodmonad}.  We let $S$ be the set of all products $a_1\\cdots a_\\nu$, where $E_1,\\ldots,E_\\nu$ is an internal sequence of internal subgroups of $G^*$ contained in $W$ and $a_1,\\ldots,a_\\nu$ is an internal sequence such that $a_i\\in E_i$ for all $i\\in \\{1,\\ldots,\\nu\\}$.  By Lemma \\ref{L:prodmonad}, $S$ is an internal subgroup of $G^*$, $S\\subseteq \\boldsymbol{\\mu}$, and every internal subgroup of $G^*$ contained in $W$ is contained in $S$.  Furthermore, if $H$ is the internal closure of $S$ in $G^*$, then $H$ is an internally compact internal subgroup of $G^*$ containing all of the subgroups of $W$ and $H\\subseteq V^*$.  The desired result follows by transfer.\n\\end{proof}\n\n\\\n\n\\begin{df}\nFollowing Kaplansky \\cite{Ka}, we call a topological space \\textbf{feebly finite-dimensional} if, for some $n$, it does not contain a homeomorphic copy of $[0,1]^n$.  \n\\end{df}\n\n\\\n\n\\noindent Clearly locally euclidean local groups are feebly finite-dimensional.\n\n\\\n\n\n\n\n\n\\begin{lemma}\\label{L:eucNSCS}\nIf $G$ is feebly finite-dimensional, then $G$ is $\\operatorname{NSCS}$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $G$ is feebly finite-dimensional but, towards a contradiction, that $G$ is not $\\operatorname{NSCS}$.  We claim that for every compact symmetric neighborhood $U$ of $1$ in $G$ contained in $\\u_2$ and for every $n$, there is a compact subgroup of $G$ contained in $U$ which contains a homeomorphic copy of $[0,1]^n$.  Assume this holds for a given $n$ and let $U$ be given.  Lemma \\ref{L:neighandgroup} implies that there is a compact symmetric neighborhood $V$ of $1$ in $G$ containing $1$ and a compact subgroup $H\\subseteq U$ that contains every subgroup of $G$ contained in $V$.  Since, by assumption, $V$ contains a nontrivial connected compact subgroup of $G$, we have a nontrivial 1-parameter subgroup $X$ of $H$.  By shrinking $V$ if necessary, we can suppose $X(\\r)\\nsubseteq V$.  By assumption, we have a compact subgroup $G(V)\\subseteq V$ of $G$ and a homeomorphism $Y:[0,1]^n \\rightarrow Y([0,1]^n)\\subseteq G(V)$.  After replacing $X$ by $rX$ for suitable $r\\in (0,1)$, we can assume $X([0,1])\\subseteq V$, $X$ is injective on $[0,1]$, and $X([0,1])\\cap G(V)=\\{1\\}$.  Since $H$ is a group, we can define $Z:[0,1]\\times [0,1]^n \\rightarrow H$ by $Z(s,t)=X(s)Y(t)$, which is clearly continuous.  Suppose $Z(s,t)=Z(s',t')$ with $s\\geq s'$.  Then we have $X(s-s')=Y(t')Y(t)^{-1}\\in X([0,1])\\cap G(V)=\\{1\\}$.  Since $X$ is injective on $[0,1]$, we have $s=s'$, and since $Y$ is injective, we have $t=t'$.  It follows that $Z$ is injective, and thus a homeomorphism onto its image.  \n\\end{proof}\n\n\\\n\n\\noindent \\textbf{Locally connected $\\operatorname{NSCS}$ local groups are $\\operatorname{NSS}$}\n\n\\\n\n\\noindent The next lemma is the local analogue of Lemma 5.3 and Corollary 5.4 in \\cite{H}.  The proof there has a gap, which can be filled by changing the hypotheses as in our lemma.\n\n\\ \n\n\\begin{lemma}\\label{L:lift}\nSuppose $H$ is a normal sublocal group of $G$ which is totally disconnected.  Let $\\pi:G\\rightarrow G\/H$ be the canonical projection.  Then $L(\\pi)$ is surjective.  \n\\end{lemma}\n\n\\begin{proof}\nLet $G':=(G\/H)_W$, where $W$ is as in Lemma \\ref{L:localcoset}, and let $\\mathbb{Y} \\in L(G')$.  We seek $\\mathbb{X} \\in L(G)$ so that $\\pi \\circ \\mathbb{X} =\\mathbb{Y}$.  If $\\mathbb{Y}$ is trivial, this is obvious.  Thus we may assume, without loss of generality, that $1\\in \\operatorname{domain}(\\mathbb{Y})$ and $\\mathbb{Y}(1)\\not=1_{G'}$.  Fix $\\nu > \\mathbb{N}$ and let $h:=\\mathbb{Y}(\\frac{1}{\\nu})\\in \\boldsymbol{\\mu}(1_{G'})$.  Take a compact symmetric neighborhood $V$ of $1_{G'}$ in $G'$ such that $\\mathbb{Y}(1)\\notin V$.  Take a compact symmetric neighborhood $U$ of $1$ in $G$ with $U\\subseteq W$ such that $\\pi(U)\\subseteq V$.  Since $\\pi$ is an open map, we have $\\boldsymbol{\\mu}(1_{G'})\\subseteq \\pi(\\boldsymbol{\\mu})$.  Choose $a\\in \\boldsymbol{\\mu}$ with $\\pi(a)=h$.  Let $\\sigma:=\\operatorname{ord}_U(a)$.  If $\\nu \\leq \\sigma$, then $\\pi(a^\\nu)\\in V^*$, contradicting the fact that $\\pi(a^\\nu)=h^\\nu=1_{G'}\\notin V$.  Thus we must have $\\sigma < \\nu$.  If $i=\\operatorname{o}(\\sigma)$, then $\\pi(\\operatorname{st}(a^i))=\\operatorname{st}(h^i)=1_{G'}$, so $G_U(a)=\\{\\operatorname{st}(a^i) \\ | \\ i=\\operatorname{o}(\\sigma)\\}$ is a connected subgroup of $G$ contained in $H$.  Since $H$ is totally disconnected, we must have $G_U(a)=\\{1_G\\}$, i.e. $a\\in G(\\sigma)$.  Since $a\\notin G^{\\operatorname{o}}(\\sigma)$, we have $[X_a]\\not= \\O$.  Suppose $\\sigma =\\operatorname{o}(\\nu)$ and let $t\\in \\operatorname{domain}(\\pi \\circ [X_a])$.  Then $\\pi([X_a](t))=\\operatorname{st}(h^{[t \\sigma]})=1$, whence $[X_a]\\in L(H)$.  Since $H$ is totally disconnected, $L(H)$ is trivial and hence $[X_a]=\\O$, a contradiction.  Thus we have $\\sigma=(r+\\epsilon)\\nu$ for some $r\\in \\r^{>0}$ and infinitesimal $\\epsilon \\in \\r^*$.  Thus $\\pi \\circ [X_a]=r\\mathbb{Y}$ and $\\mathbb{X}:=\\frac{1}{r}[X_a]$ is the desired lift of $\\mathbb{Y}$. \n\\end{proof}\n\n\\\n\n\\noindent We need one fact from the global setting that we include here for completeness.  It is taken from ~\\cite{H}.\n\n\\\n\n\\begin{lemma}\\label{L:hirsch}\nSuppose $G$ is a pure topological group such that there are no nontrivial 1-parameter subgroups $X:\\r \\rightarrow G$.  Then $G$ has a neighborhood base at $1$ of open subgroups of $G$.  In particular, $G$ is totally disconnected.\n\\end{lemma}\n\n\\\n\n\\noindent Since locally euclidean local groups are locally connected, the next theorem, in combination with Lemma \\ref{L:eucNSCS}, finishes the proof of the Local H5.\n\n\\\n\n\\begin{thm}\\label{T:eucNSS}\nIf $G$ is locally connected and $\\operatorname{NSCS}$, then $G$ is $\\operatorname{NSS}$.\n\\end{thm}\n\n\\begin{proof}\nLet $V$ be a compact symmetric neighborhood of $1$ in $G$ such that $V$ contains no nontrivial connected subgroups.  Choose an open neighborhood $W$ of $1$ in $G$ with $W\\subseteq V$ and a compact subgroup $H_1$ of $G$ with $H_1\\subseteq V$ with the property that every subgroup of $G$ contained in $W$ is contained in $H_1$; these are guaranteed to exist by Lemma \\ref{L:neighandgroup}.  Without loss of generality, we can assume $W\\subseteq \\u_6$.  We now observe that if $a\\in \\boldsymbol{\\mu}$ is degenerate, then $a\\in H_1^*$, for the internal subgroup internally generated by $a$ is contained in $\\boldsymbol{\\mu}\\subseteq W^*$.  Since $G$ is pure, if $a\\in \\boldsymbol{\\mu}$ is nondegenerate, then it is pure; for such an $a$, if $a\\in H_1^*$, then $a^\\nu\\in H_1^*\\subseteq V^*$ for all $\\nu$, contradicting the fact that some power of $a$ lives outside of $V$.  Hence, nondegenerate infinitesimals live outside of $H_1^*$.  \n\n\\\n\n\\noindent Since $H_1\\subseteq V$, $H_1$ admits no nontrivial 1-parameter subgroups, and hence $H_1$ must be totally disconnected by Lemma \\ref{L:hirsch}.  Choose an open (in $H_1$) subgroup $H$ of $H_1$ contained in $H_1\\cap W$.  Write $H=H_1\\cap W_1$ with $W_1$ an open neighborhood of $1$ in $G$ and $W_1\\subseteq W$.  Since $H$ is an open subgroup of $H_1$, it is also closed in $H_1$, and thus $H$ is a compact subset of $G$.  This implies that the set $U:=\\{a\\in W_1 \\ | \\ aHa^{-1}\\subseteq W_1\\}$ is open.  Fix $a\\in U$.  Since $aHa^{-1}$ is a subgroup of $G$ contained in $W_1\\subseteq W$, we have $aHa^{-1}\\subseteq H_1$.  Thus $aHa^{-1}\\subseteq H_1\\cap W_1=H$ implying that $H$, considered as a sublocal group of $G$, is normal.  Replacing $U$ by $U\\cap U^{-1}$, we can take $U$ as the associated normalizing neighborhood for $H$.\n\n\\\n\n\\noindent Note that $H\\subseteq \\u_6$ and $H^6\\subseteq U$.  Find symmetric open neighborhoods $W_2$ and $W_3$ of $1$ in $G$ so that $H\\subseteq W_3 \\subseteq \\overline{W_3} \\subseteq W_2$ and so that $W_2 \\subseteq \\u_6$ and $W_2^6 \\subseteq U$; this is possible by the regularity of $G$ as a topological space and the fact that $H$ is compact.  Let $G':=(G\/H)_{W_2}$.  Suppose $a\\in \\boldsymbol{\\mu}$.  If $a$ is degenerate, then we have that $a\\in H_1^* \\cap W_1^* = H^*$, whence $\\pi(a)=1_{G'}$.  Now suppose that $a$ is nondegenerate.  Let $\\tau:=\\operatorname{ord}_{\\overline{W_3}}(a)$.  Then $a^\\tau \\in (\\overline{W_3})^*$, but $a^{\\tau+1}\\in W_2^* \\setminus (\\overline{W_3})^*$.  Choose an open neighborhood $U'$ of $1$ in $G$ so that $U'H\\subseteq W_3$.  Suppose $\\pi(a^{\\tau+1})=\\pi(x)$ for some $x\\in (U')^*$.  Then $a^{\\tau+1}=xh$ for some $h\\in H^*$, whence $a^{\\tau+1}\\in W_3^*$, a contradiction.  Hence $\\pi(a)^{\\tau+1}\\notin \\pi((U')^*)$.  Meanwhile, for $i=\\operatorname{o}(\\tau)$, we have $\\pi(a)^i\\in \\boldsymbol{\\mu}(G')$.  Thus $\\pi(a)$ is pure in $G'$ and we have shown that $G\/H$ has no degenerate infinitesimals other than $1_{G'}$; in other words, we have shown that $G'$ is $\\operatorname{NSS}$.\n\n\\\n\n\\noindent  Since $G$ and $G'$ are pure, $L(\\pi):L(G)\\rightarrow L(G')$ is a continuous $\\r$-linear map.  Since $H$ is totally disconnected, $L(\\pi)$ is surjective by Lemma \\ref{L:lift}.  Note that if $\\mathbb{X} \\in \\ker(L(\\pi))$, then $\\mathbb{X}$ can be considered a local 1-ps of $H$.  Since $H$ is totally disconnected, $L(H)$ is trivial, and hence $\\mathbb{X}=\\O$.  Since $L(G')$ is finite-dimensional, we can conclude that the map $L(\\pi)$ is an isomorphism of real topological vector spaces.\n\n\\\n\n\\noindent Now take a special neighborhood $\\u'$ of $G'$ and let $E':\\k'\\rightarrow K'$ denote the local exponential map for $G'$.  Take a connected neighborhood $\\u$ of $1$ in $G$ such that $\\u\\subseteq W_2$ and $\\pi(\\u)\\subseteq K'$.  Let $x\\in \\u$.  Since $E'$ is a bijection, there is a unique $\\mathbb{Y} \\in \\k'$ such that $\\pi(x)=E(\\mathbb{Y})$.  Since $L(\\pi)$ is a bijection, there is a unique $\\mathbb{X}\\in L(G)$ such that $\\pi\\circ \\mathbb{X}=\\mathbb{Y}$.  Thus we can write $x=\\mathbb{X}(1)x(H)$ where $x(H)\\in H$.  One can easily verify that the map which assigns to each $x\\in \\u$ the above $\\mathbb{Y} \\in L(G')$ is continuous.  From this and the fact that $L(\\pi)$ is a homeomorphism, we see that the map $x\\mapsto x(H):\\u \\rightarrow H$ is continuous.  Since $\\u$ is connected, $H$ is totally disconnected, and $1(H)=1$, it follows that $x(H)=1$ for all $x\\in \\u$.  Now by the injectivity of $L(\\pi)$ and $E'$, we have that $\\pi|\\u$ is injective, implying that $G$ cannot have any subgroups other than $\\{1\\}$ contained in $\\u$.  We thus conclude that $G$ is $\\operatorname{NSS}$.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMonitoring human activities in complex environments are finding an increasing number of applications (e.g.,~\\cite{chan2008review, corchado2008intelligent, Hwang2004}). Our current investigation is driven by the application of automated surveillance in a hospital, specifically, critical care units that house the sickest and most fragile patients. In 2012, the Institute of Medicine--the health arm of the National Academies of Sciences--released  their landmark report~\\cite{smith2013best} on developing digital infrastructures that enable rapid learning health systems; one of their key postulates is the need for improvement technologies for measuring the care environment. Currently, simple measures such as whether the patient has moved in the last 24 hours, or whether the patient has gone unattended for several hours require manual observation by a nurse, which is highly impractical to scale.\nThus, this paper tackles fully-automated multi-person tracking, a first step needed towards automated surveillance in such complex indoor environments.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{examples.pdf}\n\\caption{Example images show detection results using full-body (Red) and head (Green) detectors trained by deformable part models~\\cite{felzenszwalb2010object} (Better view in color). The head detector fails when (a) the two persons' heads get close to each other or (d) the head is far away from the camera and close to the wall (which makes the depth cue not discriminative). On the other hand, the full-body detector becomes less effective when two persons are close to each other as shown in (b) and (c). Moreover, human pose can change significantly, e.g., the patient goes from standing in (a) and (b) to sitting in (c) and lying in (d). (Due to privacy problem, color images captured in a real ICU room are blurred for de-identifying.)}\n\\vspace{-10pt}\n\\label{figExampleImgs}\n\\end{figure}\n\n\nPerson detection in images is an active research area~\\cite{azizpour2012object, chen2014detect, dalal2005histograms, felzenszwalb2010object, xia2011human, zeng2010robust}. For example, a widely used detector is the one by Dalal and Triggs \\cite{dalal2005histograms} that uses the histogram of oriented gradients to train a full-body detector. To tackle pose variations of articulated objects like humans, several works have introduced part-based models~\\cite{azizpour2012object, chen2014detect, felzenszwalb2010object}. Deformable part models (DPMs) \\cite{felzenszwalb2010object}, for example, train a discriminative detector with latent variables to jointly reason over individual parts and full-body locations. However, these do not take full advantage of the temporal dependence between successive images in video data.\n\nTo leverage temporal cues, visual tracking based approaches update the target location in a given frame conditioned on tracking results in the previous frame. Many appearance based tracking methods~\\cite{babenko2009visual, grabner2006line, hare2011struck, lan2014multi, li2013survey, luonline, wu2013online, yao2013part, zhang2012real} have been developed to tackle challenges due to changes in illumination, occlusion and scale variations. Nevertheless, these methods assume a hand annotated region of interest is available at the onset, rendering these methods inapplicable in the fully automated setting.\n\nTo avoid manual annotation of the target for tracker initialization, tracking-by-detection techniques~\\cite{andriluka2008people, andriyenko2010globally, andriyenko2012discrete, berclaz2011multiple, breitenstein2011online, chen2014multi, choi2013general, milan2013continuous, shu2012part, wen2014multiple, yang2012online} employ a single per-frame detector with temporal consistency constraints for tracking. In settings where occlusion is of concern, part-based models have been augmented with visibility variables corresponding to each part for occlusion reasoning~\\cite{shu2012part, yang2012online}. Although these methods achieve convincing results for fully automated tracking, reliance on a single detector may be suboptimal as shown in Fig.~\\ref{figExampleImgs}. To make use of the complementary information from multiple detectors, Choi \\etal~\\cite{choi2013general} propose to combine a set of detectors in a general multi-person tracking framework. However, their approach assumes that the spatial configuration between the detectors are fixed. Specifically, for a given detector output (e.g., the bounding box from a full-body detector),\ntheir approach specifies a mask for where the other detector output (e.g., the bounding box from a face\ndetector) should lies. This does not scale to domains where individuals show significant pose variations (see examples in Fig.~\\ref{figExampleImgs}).\n\nTo overcome these limitations, we propose to combine multiple detectors and model the deformable spatial relationship between them. More specifically, each detector seeks to detect a region of interest (e.g., full-body or head). On a single frame, a detector may yield one or more bounding boxes. Since bounding boxes from different detectors may belong to the same person, we first group the detection outputs to reduce the number of possible persons. Then, the deformable spatial relationship between different regions of interest are modeled by a least-square reconstruction constraint. At each frame, the inferred locations for persons are obtained by optimizing an objective function comprising of terms that penalize inferred person and individual detector locations that a) are distant from detector outputs, b) deviate from expected spatial structure, c) differ significantly in consecutive frames, and d) contain overlap in region-of-interest assignment for multiple persons (e.g. the inferred full-body locations for the two persons are at the same location). The proposed optimization objective is non-differentiable; we approximate each of these above-mentioned terms using a differentiable function, and develop a block gradient descent algorithm which provides an online estimate of the inferred person locations in the given video segment. To summarize, this paper makes the following key contributions:\n\n\n$\\bullet$ We develop a novel method for multi-person tracking by combining multiple detectors for a set of different regions of interest within a tracking-by-detection framework. Our method exploits the complementarity\nof cues from multiple detectors, that is, targets missed under one detector may be detected by another detector. Since the proposed method combines detectors in the level of detection outputs (bounding boxes), it is flexible and can incorporate new state-of-the-art detectors as they become available (e.g., \\cite{azizpour2012object, chen2014detect, dalal2005histograms, felzenszwalb2010object, xia2011human, zeng2010robust}). Moreover, these detectors can be geared towards different modalities (e.g., depth versus RGB versus RGBD).\n\n$\\bullet$ Our approach models the deformable spatial relationship between multiple detectors for multi-person tracking. We show that by doing so the tracking performance improves significantly for\ntracking articulated objects like humans in domains with significant pose variations.\n\n$\\bullet$ Empirically, we show that multi-person tracking performance can be significantly improved by combining multiple detectors on a novel and challenging dataset obtained from an Intensive Care Unit (ICU).\n\n\n\\vspace{-2pt}\n\\section{Related Work}\n\nExisting multi-target tracking methods can be divided into two categories: offline and online approaches.\n\nThe offline approach primarily used optimization based formulations in which the most likely trajectory is optimized over the entire video segment. In prior works, both discrete~\\cite{andriyenko2010globally, berclaz2011multiple} and continuous~\\cite{andriyenko2012discrete, milan2013continuous} formulations have been used for identifying latent person trajectories. Notably, in recent work by Milan~\\etal~\\cite{milan2013continuous}, they use temporal and appearance consistency based constraints coupled with an occlusion model and show state-of-the-art performance for multi-person tracking. While these methods process data in batch mode, they cannot be deployed to real-time system.\n\nOnline multi-target tracking methods~\\cite{breitenstein2011online, liu2012automatic, okuma2004boosted} widely use particle filter based framework to associate the tracker with detection outputs. To make use of multiple complementary cues, Choi \\etal~\\cite{choi2013general} propose to combine multiple detectors to estimate the likelihood of the target observation.\nThey perform score-level fusion to combine the multiple detectors. To estimate\nthese scores, for a given detector output (e.g., the bounding box from a full-body detector), they specify a mask for where the other detector output (e.g., the bounding box from a face\ndetector) should lie. Scores are computed based on the overlap of a detector\noutput with this mask. In domains with significant pose variations, the relative positions of the different regions of interest (e.g., head versus the full-body) may vary greatly. Thus, we develop a more general-purpose framework\nthat does not require making this restrictive assumption.\n\n\n\n\\vspace{-2pt}\n\\section{Proposed Method}\n\nOur method builds on the tracking-by-detection framework in~\\cite{milan2013continuous}. Their approach uses a single detector to do individual image level detection. They formulate an energy functional comprising spatial consistency and temporal consistency terms related to appearance and position. We extend this work to the multiple detector setting. In order to tackle pose variations, we model the relationship between detectors within a single frame using a deformable spatial model. Furthermore, our approach tackles tracking in an online setting.\n\n\nWe begin by giving an overview of our proposed method. We use a collection of existing detection methods to train $K$ detectors; each detector is geared towards detecting a region of interest (e.g., full-body or head). These detectors need not be semantically distinct, that is, one may choose to use more than one detector for a region of interest. We assume that ground truth bounding boxes corresponding to the regions of interest for each of the detectors are available in the training data. Let the number of regions of interest be $L$. We maintain $M^t$, the number of persons in any given frame $t$ as an unknown. Since bounding boxes from the $K$ detectors may come from the same person, we first group the detection outputs into $N^t$ groups as elaborated in Section~\\ref{secGroup}. In each group, there is at least one bounding box to represent all the detection evidence for one person. Deformable spatial relationship is modeled by minimizing the least-square errors between the expected location and projected one as introduced in Section~\\ref{secDeform}. The first frame uses an initialization step described in Section~\\ref{secInit} to infer the number of persons and their inferred bounding box locations. Thereafter, to determine the detector bounding box locations at time $t$, we proceed using three steps. First, a preprocessing step is used to determine for each person in the previous frame, whether they exist in the current frame. Then, an optimization objective is formulated for optimizing the bounding box locations at time $t$ conditioned on the inferred bounding box locations at time $t-1$ and the detector outputs at time $t$. Finally, a postprocessing step is used to update the appearance model and assign identities for each of the inferred bounding box locations. These steps are discussed in Section~\\ref{secUpdate}.\n\nNotations: We denote output bounding boxes from the $k$-th trained detector in the $t$-th frame as $D^t_k(1), \\cdots, D^t_k(N^t_k)$ and corresponding detection scores normalized by a exponential function as $w^t_k(1), \\cdots, w^t_k(N^t_k)$, where $N^t_k$ is the number of detected bounding boxes. Let the number of persons in a frame be $M^t$. For the $m$-th person, the corresponding bounding boxes from $L$ regions of interest are defined by $X^t_1(m), \\cdots, X^t_L(m)$. Let $X^t$ to refer to the collection of inferred locations of regions of interest for all persons in the frame. In this paper, each bounding box is represented by coordinates of the upper-left and lower-right corners.\n\n\\vspace{-2pt}\n\\subsection{Grouping Bounding Boxes from Multiple Detectors}\\label{secGroup}\n\n\nLet us consider two bounding boxes $D^t_k(n)$ and $D^t_{k'}(n')$ from any two detectors $k$ and $k'$, respectively. We calculate the probability that these two bounding boxes are from the same person as,\n\\begin{equation}\\label{eqProbAll}\n\\begin{split}\n    p = a p_{\\textrm{over}} + (1-a) p_{\\textrm{depth}}\n\\end{split}\n\\end{equation}\nwhere $a$ is a positive weight, $p_{\\textrm{over}}$ and $p_{\\textrm{depth}}$ measure the overlapping ratio and depth similarity between two bounding boxes, respectively. These two probability scores are\n\\begin{equation}\\label{eqProbOver}\n\\begin{split}\n    p_{\\textrm{over}} = \\frac{\\left|D^t_k(n) \\cap D^t_{k'}(n')\\right|}{\\min \\left( |D^t_k(n)|, |D^t_{k'}(n')| \\right)}\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{eqProbDepth}\n\\begin{split}\n    & p_{\\textrm{depth}} = \\frac{1}{2} e^{\\frac{-(v_k^t(n)-v_{k'}^t(n'))^2}{2 \\sigma_k^t(n)^2}} + \\frac{1}{2} e^{\\frac{-(v_k^t(n)-v_{k'}^t(n'))^2}{2 v_{k'}^t(n')^2}}\n\\end{split}\n\\end{equation}\nwhere $v$ and $\\sigma$ denote the mean and standard deviation of the depth values inside bounding box, respectively.\n\nIf two bounding boxes are overlapped with each other, e.g., the head is contained in the full-body, such probability should be large. Thus, we define the probability using the first term~\\eqref{eqProbOver} in~\\eqref{eqProbAll}. However, as shown in Fig.~\\ref{figGroup}, it is possible that $D^t_k(n)$ (e.g., the full-body bounding box on the left) does not overlap with any bounding boxes from detector $k'$ (e.g., head). This motivates another term for the grouping probability. Since regions of interest from the same person should have similar depth values, the depth similarity scores can help to better group the bounding boxes. Thus, the second term is defined by~\\eqref{eqProbDepth} based on the depth information.\n\n\\begin{figure}\n\\centering\n\\subfigure[Color image]{\\label{figGroupImg1}\\includegraphics[width=0.4\\linewidth]{example6.pdf}} \\hspace{4pt}\n\\subfigure[Depth image]{\\label{figGroupImg2}\\includegraphics[width=0.4\\linewidth]{example7.pdf}}\n\\caption{Example images to demonstrate the motivation for bounding boxes grouping (better view in color)}\n\\vspace{-10pt}\n\\label{figGroup}\n\\end{figure}\n\n\n\nTo group the bounding boxes from any two detectors $k$ and $k'$, we compute the grouping probability for each pair $n$ and $n'$ by~\\eqref{eqProbAll} to get a similarity matrix $P^t_{kk'}$. The largest element in $P^t_{kk'}$ is selected iteratively. Since one bounding box from detector $k$ is corresponding to only one bounding box from detector $k'$, the elements in the $n$-th row of $P^t_{kk'}$ are removed for the selected bounding box pair $(n,n')$. Similarly, we remove the $n'$-th column of $P^t_{kk'}$. If the selected elements are lager than a threshold $\\tau$, the bounding pair $(n,n')$ is considered to come from the same person.\n\nTo extend the algorithm to any number of detectors $K$, we first generate groups of bounding boxes $B^{t}_2(1), \\cdots, B^t_2(N^{t}_2)$ for two detectors by the grouping procedures mentioned above. Similarly, $B^{t}_2(1), \\cdots, B^t_2(N^{t}_2)$ and bounding boxes from another detector can be grouped to obtain $B^{t}_3(1), \\cdots, B^t_3(N^{t}_3)$. After $K-1$ iterations, we get $N^t$ groups of bounding boxes $B^{t}(1), \\cdots, B^t(N^{t})$ to represent $N^t$ possible persons.\n\nWith the groups of bounding boxes, we assume that the detector outputs $D^t_k(n)$ are more likely to be spatially contiguous with true positions for those regions of interest in the image. In other words, e.g. the head is more likely to be located near the candidate locations suggested by the head detector output. Thus, the cost function for this detection matching constraint is devised as\n\\begin{equation}\\label{eqDetCost}\n\\begin{split}\n    & E_{det}(X^{t},M^{t}) = \\\\\n    & \\sum_{n'=1}^{N^t} \\min_{1\\leq m \\leq M^t} \\sum_{D^t_k(n) \\in B^t(n')} w^t_k(n) \\left\\|D^t_k(n) - X^t_{\\iota(k)}(m)\\right\\|^2\n\\end{split}\n\\end{equation}\nwhere $\\iota$ maps the $k$-th detector to the $l$-th region of interest. If the detection confidence $w^t_k(n)$ is large, we assume greater likelihood of having a tracked bounding box $X^t_k(m)$ close to $D^t_k(n)$. Therefore, we use $w^t_k(n)$ to define the weighted penalty for each detected bounding box $D^t_k(n)$ in \\eqref{eqDetCost}. Note that, as presented in \\eqref{eqDetCost}, this term is not differentiable. To ensure differentiability of the cost function, we approximate the minimum operation by\n\\begin{equation}\\label{eqMinApprox}\n\\begin{split}\n    \\min_{1\\leq m \\leq M} \\{z_m\\} & = \\lim_{\\alpha\\rightarrow-\\infty} S_\\alpha(\\{z_m\\})  \\\\\n    S_\\alpha(\\{z_m\\}) & = \\frac{\\sum_{m=1}^{M} z_m e^{\\alpha z_m}}{\\sum_{m=1}^{M} e^{\\alpha z_m}}\n\\end{split}\n\\end{equation}\nWith the approximation by \\eqref{eqMinApprox}, the cost function becomes\n\\begin{equation}\\label{eqDetCostApprox}\n\\begin{split}\n    & E_{det}(X^{t},M^{t}) = \\\\\n    & \\sum_{n'=1}^{N^t} S_\\alpha \\left( \\left\\{ \\sum_{D^t_k(n) \\in B^t(n')} w^t_k(n) \\left\\|D^t_k(n) - X^t_{\\iota(k)}(m)\\right\\|^2 \\right\\} \\right)\n\\end{split}\n\\end{equation}\n\n\\subsection{Modeling Deformable Spatial Relationship}\\label{secDeform}\nIn addition to matching candidate detections, we wish to take advantage of the expected spatial structure between regions of interest for any given articulated pose variations. For example, when standing, we expect the location of the head bounding box to be in the top region of the full-body bounding box. Moreover, for two detectors that seek to detect the same region of interest (e.g., if we employed two different head detectors), the expected bounding box coordinates would be expected to be in the same location. We model these deformable spatial correlations using a function that penalizes deviations from expected configurations.\n\nLet $A_{ll'}^c$ be a projection matrix which learns the spatial relationship between the bounding box locations of two regions of interest $l$ and $l'$ for a given pose $c$. Specifically, given training data, for every pose $c \\in \\{1, \\cdots, C\\}$, it learns a mapping from the expected location $X^{t}_l(m)$ from the $l$-th region to the location $X^{t}_{l'}(m)$ from the $l'$-th one. Now, for any two proposed bounding boxes $X^{t}_{l'}(m)$ and $X^{t}_{l}(m)$ at frame $t$ for individual $m$, the deviation from the expected spatial configuration is quantified as the error between the expected location of the bounding box for the second region of interest (e.g., head) conditioned on the location of the first region of interest (e.g., full-body). The total cost is computed by summing for each of the $M^t$ individuals, the minimum cost for each of the $C$ subcategories, i.e.,\n\\begin{equation}\\label{eqSpaCost}\n\\begin{split}\n    E_{spa}(X^{t}, M^{t}) = \\sum_{m=1}^{M^t} \\min_{1\\leq c \\leq C} \\sum_{l \\neq l'}\\|A_{ll'}^c X^{t}_l(m) - X^t_{l'}(m)\\|^2\n\\end{split}\n\\end{equation}\nAgain, due to the minimization, this term is not differentiable. Similar to the relaxation we employ for the detection matching constraint, we approximate the minimization operation by \\eqref{eqMinApprox}. The resulting cost function \\eqref{eqSpaCost} becomes\n\\begin{equation}\\label{eqSpaCostApprox}\n\\begin{split}\n    & E_{spa}(X^{t}, M^{t}) \\\\\n    & \\quad = \\sum_{m=1}^{M^t} S_\\alpha \\left( \\left\\{ \\sum_{l \\neq l'}\\|A_{ll'}^c X^{t}_l(m) - X^{t}_{l'}(m)\\|^2 \\right\\} \\right)\n\\end{split}\n\\end{equation}\n\nTo obtain the projection matrices $A_{ll'}^1, \\cdots, A_{ll'}^C$, we employ the spatial features in~\\cite{chen2014detect} to separate the normalized training data into $C$ subcategories by \\emph{k}-means clustering. Then, $A_{ll'}^c$ is trained by solving the regularized regression problem with the training data in the $c$-th subcategory.\n\n\\subsection{Optimized Tracker Initialization}\\label{secInit}\nTo determine the optimal tracker initialization at time $t=1$, we aggregate the detection matching~\\eqref{eqDetCostApprox} and deformable spatial~\\eqref{eqSpaCostApprox} cost functions with two additional constraints (akin to~\\cite{milan2013continuous} for single detector) considering the mutual exclusion and regularization for multiple persons, i.e.,\n\\begin{equation}\\label{eqExcCost}\n\\begin{split}\n    E_{exc}(X^{t}, M^{t}) = \\sum_{l=1}^L \\sum_{m \\neq m'}  \\frac{1}{\\|X^t_l(m) - X^t_l(m')\\|^2}\n\\end{split}\n\\end{equation}\n\\begin{equation}\\label{eqRegCost}\n\\begin{split}\n    E_{reg}(X^{t}) = M^t\n\\end{split}\n\\end{equation}\nBy linearly weighting the cost functions \\eqref{eqDetCostApprox} \\eqref{eqSpaCostApprox} \\eqref{eqExcCost} \\eqref{eqRegCost}, the joint optimization is given by\n\\begin{equation}\\label{eqOptSpatial}\n\\begin{split}\n    \\min_{X^{t}, M^{t}} \\lambda_{det} E_{det} + \\lambda_{spa} E_{spa} + \\lambda_{exc} E_{exc} + \\lambda_{reg} E_{reg}\n\\end{split}\n\\end{equation}\nWe solve \\eqref{eqOptSpatial} incrementally for increasing values of $M^t$. Given candidate initializations, \\eqref{eqOptSpatial} can be optimized using gradient descent. Since \\eqref{eqOptSpatial} is not convex, clever initialization can both improve the computational efficiency and quality of the solution.\n\nWe can leverage the deformable spatial model to yield useful initializations. For each of the detected bounding boxes $D^t_k(n)$ for the $k$-th detector, for the $c$-th subcategory, using the projection matrices $A_{\\iota(k)1}^c, \\cdots, A_{\\iota(k)L}^c$, we predict the expected locations for each of the bounding boxes of other regions of interest as\n\\begin{equation}\\label{eqConfigEst}\n\\begin{split}\n     \\hat{X}^{tc}_1(n^t_k), \\cdots, \\hat{X}^{tc}_L(n^t_k) = (A_{\\iota(k)1}^c, \\cdots, A_{\\iota(k)L}^c) D^t_k(n)\n\\end{split}\n\\end{equation}\nThese predictions yield $C \\times \\sum_{k=1}^K N^t_k$ candidate sets of bounding boxes for persons in the image, where each set corresponds to all the regions of interest for pose $c$. This approach of predicting bounding box locations for the other detectors conditioned on the location for a given detector is useful when the original detector fails to identify any candidate locations. For example, as shown earlier in Fig.~\\ref{figExampleImgs}(a)(d), while the head detector cannot successfully detect the head of some persons, prediction using a full-body detector may successfully localize the head bounding box. For $M^t=1$, we select the candidate set of bounding boxs that minimizes \\eqref{eqOptSpatial}. As $M^t$ is increased, we retain the previous initializations and augment the set by selecting the next best candidate bounding box locations using the same criterion.\n\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithm}[t]\\small\n\\caption{Initialization at the first frame $t=1$}\n\\label{alg:Init}\n\\begin{algorithmic}[1]\n\\REQUIRE Detected bounding boxes $D^t_1(1), \\cdots, D^t_K(N^t_K)$ and corresponding weights $w^t_1(1), \\cdots, w^t_K(N^t_K)$;\n\\STATE For each $D^t_k(n)$ generate $C$ sets of bounding boxes by \\eqref{eqConfigEst} for $C$ types of deformations;\n\\STATE Set the objective cost as infinite;\n\\STATE Let $M^t=1$;\n\\STATE Select a set $\\hat{X}^{tc}_1(n^t_k), \\cdots, \\hat{X}^{tc}_L(n^t_k)$ that minimizes \\eqref{eqOptSpatial} as the initialized solution;\n\\STATE Employ gradient descend to obtain the optimal solution $X^t_1(1), \\cdots, X^t_L(M^t)$ for fixed $M^t$;\n\\IF{objective cost decreases}\n    \\STATE Save $X^t_1(1), \\cdots, X^t_L(M^t)$ as the optimal solution;\n    \\STATE Let $M^t = M^t + 1$;\n    \\STATE Add $\\hat{X}^{tc}_1(n^t_k), \\cdots, \\hat{X}^{tc}_L(n^t_k)$ to the previous initialization sets by minimizing \\eqref{eqOptSpatial};\n    \\STATE Go to step 5;\n\\ELSE\n    \\RETURN the saved optimal solution;\n\\ENDIF\n\\ENSURE Optimal bounding boxes $X^t_1(1), \\cdots, X^t_L(M^t)$.\n\\end{algorithmic}\n\\end{algorithm}\n\nIn summary, the algorithmic procedures for tracker initialization at the first frame $t=1$ is given in Algorithm~\\ref{alg:Init}.\n\n\\subsection{Online Tracking Update}\\label{secUpdate}\nIn the $t$-th frame, we are not only given the detection outputs $D^t_1(1), \\cdots, D^t_K(N^t_K)$ and corresponding weights $w^t_1(1), \\cdots, w^t_K(N^t_K)$ but also the tracked bounding boxes $X^{t-1}_1(1), \\cdots, X^{t-1}_L(M^{t-1})$ in the previous frame.\n\n\\textbf{Preprocessing:} Before introducing the cost functions to ensure the temporal consistence, the inferred bounding boxes in the previous frame are preprocessed to remove the persons who are likely to be out of the camera in the current frame. The velocity $v_l^{t-1}(m)$ of the position of the $l$-th region of interest for person $m$ at time $t-1$ is calculated by\n\\begin{equation}\\label{eqVel}\n\\begin{split}\n    v_l^{t-1}(m) = X^{t-1}_l(m) - X^{t-2}_l(m)\n\\end{split}\n\\end{equation}\nWith the velocity $v_l^{t-1}(m)$, the current bounding box position $\\tilde{X}^t_l(m)$ can be estimated by\n\\begin{equation}\\label{eqPred}\n\\begin{split}\n    \\tilde{X}^t_l(m) = X^{t-1}_l(m) + v_l^{t-1}(m)\n\\end{split}\n\\end{equation}\nIf the estimated position $\\tilde{X}^t_l(m)$ is close to the image boundary and there is no detected bounding boxes around $\\tilde{X}^t_l(m)$ for $l = 1, \\cdots, L$, person $m$ is thought to be out of the camera in the current frame. The temporal consistence is only valid for the $M^{t-1}_*$ persons who are still in the view of the camera at time $t$.\n\n\n\n\\textbf{Optimization:} For trajectory consistence, we define the cost function as\n\\begin{equation}\n\\begin{split}\n    E_{tra}(X^t) = \\sum_{l=1}^L \\sum_{m=1}^{M^{t-1}_*} \\left\\|v_l^{t}(m) - v_l^{t-1}(m)\\right\\|^2\n\\end{split}\n\\end{equation}\nwhere velocity is defined as in \\eqref{eqVel}. In addition to motion based consistency, we also impose appearance based consistency. This implements the natural assumption that a person between two consecutive frames will likely look similar. Let $\\psi$ be a normalized appearance feature extraction function. If the appearances of two bounding boxes $X^{t-1}_l(m)$ and $X^{t}_l(m)$ are similar, the inner product of their appearance feature vectors is large. Thus, we define the cost function for the appearance consistence term as\n\\begin{equation}\n\\begin{split}\n    E_{app}(X^t) = - \\sum_{l=1}^L \\sum_{m=1}^{M^{t-1}_*} \\psi\\left(X^{t-1}_l(m)\\right) \\cdot \\psi\\left(X^{t}_l(m)\\right)\n\\end{split}\n\\end{equation}\n\nTo ensure differentiability, we follow~\\cite{milan2013continuous} to define the feature extraction function $\\psi$ as\n\\begin{equation}\n\\begin{split}\n    \\psi_b\\left(X^{t}_l(m)\\right) & = \\sum_{\\textbf{\\emph{p}}} H_b(\\textbf{\\emph{p}}) \\mathcal{N}\\left(\\textbf{\\emph{p}}; \\mu_l^t(m), \\Sigma_l^t(m)\\right)\\\\\n\\end{split}\n\\end{equation}\nwhere $\\psi_b$ is the $b$-th element of the feature vector $\\psi$, $\\mathcal{N}(.)$ is a Gaussian distribution with the mean and covariance matrix of $\\mu_l^t(m)$ and $\\Sigma_l^t(m)$, respectively. $H_b(\\textbf{\\emph{p}})$ is a binning function of the $b$-th bin for the image.\n\nSince the upper-left and lower-right corners are used to represent the inferred bounding boxes $X^{t}_l(m)$, the mean $\\mu_l^t(m)$ is computed by\n\\begin{equation}\n\\begin{split}\n    \\mu_l^t(m) & = \\left(\n                     \\begin{array}{c}\n                       \\emph{\\textbf{a}}_1 \\cdot X^{t}_l(m) \\\\\n                       \\emph{\\textbf{a}}_2 \\cdot X^{t}_l(m) \\\\\n                     \\end{array}\n                   \\right) \\\\\n\\end{split}\n\\end{equation}\nwhere $\\cdot$ denotes the inner product, $\\emph{\\textbf{a}}_1$ and $\\emph{\\textbf{a}}_2$ are vectors for mean computation, i.e.  $\\emph{\\textbf{a}}_1 = (\\frac{1}{2}, 0, \\frac{1}{2}, 0)^T, \\emph{\\textbf{a}}_2 = (0, \\frac{1}{2}, 0, \\frac{1}{2})^T$. On the other hand, we assume that the covariance matrix $\\Sigma_l^t(m)$ is diagonal. Under normal distribution, 95.45\\% of the values lie within two standard deviations of the mean. Thus, we define the standard deviations as half of size of the bounding box so that 95.45\\% of energy concentrates with the bounding box, i.e.\n\\begin{equation}\n\\begin{split}\n    \\Sigma_l^t(m) & = \\left(\n                      \\begin{array}{cc}\n                        \\frac{\\left(\\textbf{\\emph{b}}_1 \\cdot X^{t}_l(m)\\right)^2}{2^2} & 0 \\\\\n                        0 & \\frac{\\left(\\textbf{\\emph{b}}_2 \\cdot X^{t}_l(m)\\right)^2}{2^2} \\\\\n                      \\end{array}\n                    \\right)\n\\end{split}\n\\end{equation}\nwhere $\\emph{\\textbf{b}}_1 = (-1, 0, 1, 0)^T$ and $\\emph{\\textbf{b}}_2 = (0, -1, 0, 1)^T$\n\n\nCombining the trajectory and appearance consistence constraints with the cost functions derived in Section \\ref{secInit}, the tracked bounding boxes are updated by solving the following optimization problem,\n\\begin{equation}\\label{eqOptSpaTem}\n\\begin{split}\n    \\min_{X^t, M^t} & \\lambda_{det} E_{det} + \\lambda_{spa} E_{spa} + \\lambda_{exc} E_{exc} \\\\\n    & \\qquad + \\lambda_{tra} E_{tra} + \\lambda_{app} E_{app} + \\lambda_{reg} E_{reg}\n\\end{split}\n\\end{equation}\nThis optimization problem can be solved incrementally by increasing values of $M^t$ similar to procedure used in Section~\\ref{secInit}. Differently, we have at least $M^{t-1}_*$ persons appear in the frame $t$, so the initial value of $M^t$ is set as $M^{t-1}_*$. The bounding box locations of these $M^{t-1}_*$ persons are initialized as in \\eqref{eqPred}. As $M^t$ is increased, we retain the previous initializations and augment the set by selecting the next best candidate bounding box location akin to the procedure used in Section~\\ref{secInit}. More specifically, as before, we obtain the set of candidate bounding boxes as in \\eqref{eqConfigEst}. We then initialize using the set that minimizes \\eqref{eqOptSpaTem}.\n\n\n\n\\renewcommand{\\algorithmicrequire}{\\textbf{Input:}}\n\\renewcommand{\\algorithmicensure}{\\textbf{Output:}}\n\\begin{algorithm}[t]\n\\caption{Temporal updating at time $t$}\n\\label{alg:Update}\n\\begin{algorithmic}[1]\\small\n\\REQUIRE Detected bounding boxes $D^t_1(1), \\cdots, D^t_K(N^t_K)$, corresponding weights as $w^t_1(1), \\cdots, w^t_K(N^t_K)$, previous tracking results $X^{t-1}_1(1), \\cdots, X^{t-1}_L(M^{t-1})$ and appearance models of all the previous targets $\\phi^{t-1}_1(1), \\cdots, \\phi^{t-1}_L(M^{t-1}_{all})$;\n\\STATE For each $D^t_k(n)$ generate $C$ sets of bounding boxes by \\eqref{eqConfigEst} for $C$ types of deformations;\n\\STATE Preprocess $X^{t-1}_1(m), \\cdots, X^{t-1}_L(m)$;\n\\STATE Set the objective cost as infinite;\n\\STATE Let $M^t = M^{t-1}_*$;\n\\STATE Initialize the solution as $\\tilde{X}^t_1(1), \\cdots, \\tilde{X}^t_L(M^{t-1}_*)$;\n\\STATE Employ gradient descend to obtain the optimal solution $X^t_1(1), \\cdots, X^t_L(M^t)$ for fixed $M^t$;\n\\IF{objective cost decreases}\n    \\STATE Save $X^t_1(1), \\cdots, X^t_L(M^t)$ as the optimal solution;\n    \\STATE Let $M^t = M^t + 1$;\n    \\STATE Add $\\hat{X}^{1c}_1(n^1_k), \\cdots, \\hat{X}^{1c}_L(n^1_k)$ to the previous initialization sets by minimizing \\eqref{eqOptSpaTem};\n    \\STATE Go to step 6;\n\\ENDIF\n\\STATE Assign identity to current targets $m$ ($1 \\leq m \\leq M^{t-1}_*$) as the same as the previous ones;\n\\STATE Calculate target matching confidence $\\eta(p')$ for $p' > M^{t-1}_*$ by~\\eqref{eqMatchCond};\n\\IF{$\\eta(p') < \\delta$}\n    \\STATE Assign a new identity to the current person $p'$;\n\\ELSE\n    \\STATE Assign the identity of the current person $p'$ as $p$ that achieves $\\eta(p')$;\n\\ENDIF\n\\STATE Update appearance models by~\\eqref{eqAppUp};\n\\ENSURE Optimal bounding boxes $X^t_1(1), \\cdots, X^t_L(M^t)$, updated appearance models $\\phi^{t}_1(1), \\cdots, \\phi^{t}_L(M^{t}_{all})$  and target identities.\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\textbf{Postprocessing:} After locating the optimal positions of persons in the current frame, identities of them must be assigned. For the persons which appear in the previous frame, their identities are the same as that in the previous frame. For a person $p'$ detected in this frame that does not have a corresponding detection in the previous frame, we use the appearance model to determine whether this detection corresponds to an individual previously seen, or a new individual. Towards this, we maintain and incrementally update $\\Phi^{t-1}$, the appearance model for all individuals previously seen. Assume there are $M^{t-1}_{all}$ total number of individuals observed by time $t-1$. For the $p$-th individual and the $l$-th region of interest, the appearance at $t-1$ is denoted by $\\phi^{t-1}_l(p)$. We determine the maximum similarity score $\\eta(p')$ between the appearance of $p'$ and all individuals in $\\Phi^{t-1}$ as\n\\begin{equation}\\label{eqMatchCond}\n\\begin{split}\n    \\eta(p') = \\max_{1 \\leq p \\leq M^{t-1}_{all}} \\max_{1 \\leq l \\leq l} \\phi^{t-1}_l(p) \\cdot \\psi\\left(X^{t}_l(p')\\right)\n\\end{split}\n\\end{equation}\nIf $\\eta(p')$ is larger than a positive threshold $\\delta$, the identity is assigned to $p$ that achieves $\\eta(p')$. Otherwise, we add $p'$ to $\\Phi^{t}$. Additionally, we update appearance models in $\\Phi^{t}$ for individuals that appear in frame $t$ as follows:\n\\begin{equation}\\label{eqAppUp}\n\\begin{split}\n    \\phi^{t}_l(p') = \\frac{F^{t-1}(p') \\phi^{t-1}_l(p') + \\psi\\left(X^{t}_l(p')\\right)}{F^{t-1}(p') + 1}\n\\end{split}\n\\end{equation}\nwhere $F^{t-1}(p')$ is the number of frames containing the $p'$-th person. The complete procedure for the online updating step is detailed in Algorithm~\\ref{alg:Update}.\n\n\n\\begin{table*}[htbp]\n  \\centering\\small\n    \\begin{tabular}{|r|r||r|r||r|r|r||r|r|}\n    \\hline\n    Tracker & full-body Detector & MOTA  $\\uparrow$ & MOTP $\\uparrow$ &  FP $\\downarrow$  &  FN $\\downarrow$  & IDs $\\downarrow$ & Recall  $\\uparrow$ & Precision $\\uparrow$ \\\\\n    \\hline\n    Choi~\\etal~\\cite{choi2013general} & Pre-Trained   & -18.88 & 66.98& 32.10 & 78.68 & {8.10} & 21.32 & 38.63 \\\\\n    \\hline\n    Choi~\\etal~\\cite{choi2013general} & ICU 1-Component DPM & -45.12 & 56.54& 57.36 & 76.82 & 10.94 & 23.18 & 27.06 \\\\\n    Milan~\\etal~\\cite{milan2013continuous} & ICU 1-Component DPM & -38.61 & 62.79& 55.76 & 56.92 & 25.94 & 43.08 & 43.39 \\\\\n    Ours & ICU 1-Component DPM & 13.50 & 67.04& 18.11 & 48.57 & 19.82 & 51.43 & 74.13 \\\\\n    \\hline\n    Milan~\\etal~\\cite{milan2013continuous} & ICU 3-Component DPM & -15.21 & 68.03& 45.17 & 37.40 & 32.64 & 62.60 & 61.25 \\\\\n    Ours & ICU 3-Component DPM & \\textbf{29.14} & \\textbf{68.64}& {12.77} & {36.43} & 21.66 & {63.57} & {83.06} \\\\\n    \\hline\n    \\end{tabular}%\n  \\caption{Results (\\%) on ICU data ($\\uparrow$ and $\\downarrow$ mean higher and lower scores for better performance, respectively)}\n  \\vspace{-4pt}\n  \\label{tabQuanICU}%\n\\end{table*}%\n\n\\begin{table*}[htbp]\n  \\centering\\small\n    \\begin{tabular}{|r|r||r|r||r|r|r||r|r|}\n    \\hline\n    Tracker & full-body Detector & MOTA  $\\uparrow$ & MOTP $\\uparrow$ &  FP $\\downarrow$  &  FN $\\downarrow$  & IDs $\\downarrow$ & Recall  $\\uparrow$ & Precision $\\uparrow$ \\\\\n    \\hline\n    Choi~\\etal~\\cite{choi2013general} & Pre-Trained  & 20.20 & 57.62& 20.90 & 57.62 & 1.28 & 42.38 & 67.67 \\\\\n    \\hline\n    Choi~\\etal~\\cite{choi2013general} & ICU 1-Component DPM & 16.28 & 59.83& {6.72}  & 76.43 & {0.56} & 23.57 & {80.21} \\\\\n    Milan~\\etal~\\cite{milan2013continuous} & ICU 1-Component DPM & 21.68 & \\textbf{74.61} & 34.10 & {40.27} & 3.95  & {59.73} & 64.37 \\\\\n    Ours & ICU 1-Component DPM & \\textbf{26.91} & 70.37 & 13.91 & 57.07 & 2.10  & 42.93 & 74.97 \\\\\n    \\hline\n    \\end{tabular}%\n  \\caption{Results (\\%) on RGBD Pedestrian Dataset ($\\uparrow$ and $\\downarrow$ mean higher and lower scores for better performance, respectively)}\n  \\vspace{-6pt}\n  \\label{tabQuanSpinello}%\n\\end{table*}%\n\n\\section{Experiments}\n\nIn this section, we evaluate the proposed method using RGBD data collected in a live Intensive Care Unit (ICU) as well as a publicly available pedestrian dataset. We report both quantitative and qualitative results in Section~\\ref{secExpQuant} and Section~\\ref{secExpQuali}, respectively.\n\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.92\\linewidth]{compareICU.pdf}\n\\caption{Example tracking results on the ICU data, top row: Choi~\\etal~\\cite{choi2013general}, middle: Milan~\\etal~\\cite{milan2013continuous}, bottom: ours (better view in color)}\n\\vspace{-10pt}\n\\label{figVisualICU}\n\\end{figure*}\n\n\n\n\\textbf{Datasets:} In our work, we are primarily motivated by person tracking in complex, structured environments such as an ICU. We report results on data that are collected in a live ICU environment using consumer RGBD cameras at a resolution of 480$\\times$640 at an adaptive sampling rate of 30 fps or 1 fps depending on the degree of motion in the room. The dataset contains 36 annotated sequences, each containing 1000 images, capturing 3 different patients. Head and full-body bounding boxes with a sub-sampled rate of 1\/10 from the original sequences are annotated\\footnote{The detection outputs, ground truth annotations and depth images can be released to public for evaluation.}. The current study is approved by the Institutional Review Board.\n\nWe also evaluate our method on a public available dataset. While there are many video datasets \\cite{andriluka2010monocular, ferryman2010pets2010, Ferryman2009pets2009, song2013tracking, wu2013online} in the tracking literature, these datasets only provide single object annotations \\cite{song2013tracking, wu2013online} or focus on RGB tracking \\cite{andriluka2010monocular, ferryman2010pets2010, Ferryman2009pets2009}. For RGBD based multi-target tracking, We use a RGBD pedestrian\\footnote{This dataset contains mostly upright walking and standing persons. In spite of lack of significant pose variations as in the ICU data, we still evaluate our method on this dataset, since no other public data can be found.} dataset \\cite{spinello2011people} which contains 3 RGBD sequences from three vertically mounted RGBD sensors. There are over 1000 images in each sequence.\n\n\n\\textbf{Experimental Setup:} For preprocessing, background subtraction is simply performed by subtracting the largest depth values over time for each depth image. Detected bounding boxes containing more than 60\\% background pixels are removed. HOG features are extracted from both color and depth images. In order to avoid overfitting of the appearance model to any single patient, we use images of 10 sequences from the two patients to train full-body and head detectors by 3-component deformable part models (DPMs)~\\cite{felzenszwalb2010object}. For the ICU data, sequences\nfrom the third patient are used for testing. For the pedestrian dataset, we select 1 component (corresponding to standing) in the trained 3-component DPM as the detector, since it contains mostly upright persons. All the 3 sequences are used for testing.\n\n\n\nThe widely used CLEAR MOT metrics~\\cite{keni2008evaluating} are employed for quantitative evaluation. The multiple object tracking precision (MOTP) measures the total error in estimating precise target positions compared with the ground truth. The multiple object tracking accuracy (MOTA) is computed by 100\\% minus three types of errors, false positive rate (FP), missed detection rate (MD) and identity switch rate (IDs).\n\nFor parameter selection, we empirically normalize the 6 function costs in~\\eqref{eqOptSpaTem} to ensure equal impotentness and set the parameters as 1. The number of deformable configurations $C$ is set as 4 for 4 types of pose variations. We assume equal weight for the overlapping ratio and depth information to define the grouping probability in~\\eqref{eqProbAll} and set $a$ as 0.5. The probability thresholds $\\tau, \\delta$ are set as 0.5.\n\nOur method is compared with two state-of-the-art tracking-by-detection algorithms by Choi~\\etal~\\cite{choi2013general} and Milan~\\etal~\\cite{milan2013continuous}. We use the publicly available implementations, i.e., Choi~\\etal\\footnote{http:\/\/cvgl.stanford.edu\/data2\/pr2dataset\/} and Milan~\\etal\\footnote{http:\/\/www.milanton.de\/contracking\/}, and default parameters in their codes, for our experiments\\footnote{It should be noticed that we used the RGB version in Choi~\\etal~\\cite{choi2013general}, since the RGBD version is not available}. Since Choi~\\etal~\\cite{choi2013general} combine multiple detectors using a fixed spatial configuration in score level, multiple component DPM is not applicable. Thus, 1-component DPM is used for their method on the ICU data. In addition, a pretrained detector is encoded in \\cite{choi2013general}. We also report the results using this detector for their method.\n\n\\vspace{-2pt}\n\\subsection{Quantitative Results}\\label{secExpQuant}\nThe averaged overall performance measurements MOTA and MOTP as well as individual measurements on the ICU data are recorded in Table~\\ref{tabQuanICU}. From Table~\\ref{tabQuanICU}, we can see that our method achieves the highest MOTA and MOTP by combining multiple detectors for different body parts (full-body and head) and modeling the deformable spatial relationship between them. Since person pose changes significantly on the ICU data, the results with an ICU 3-component DPM (multiple components corresponding to multiple poses) are better than those with an ICU 1-component DPM. When using the same full-body detector (i.e., ICU 1-Component DPM), our method outperforms the two state-of-the-art algorithms by MOTA, MOTP, and individual measures except IDs. The MOTA of our method with a 1-component DPM is even higher than that of Milan \\etal with a more discriminative 3-component DPM. This shows that our method can outperform existing single-strong-detector-based algorithms by combining multiple weaker detectors.\n\nThe results on the public RGBD pedestrian dataset are shown in Table~\\ref{tabQuanSpinello}. Though our method cannot achieve significant improvements like those for the ICU data, the MOTA of our method is higher than others and the MOTP is comparable with the highest one by Milan \\etal. While these results indicate that single-detector-based algorithm could achieve comparable results for multi-target tracking without extensive pose variations, our method can still achieve good performance by combining multiple detectors under this environment.\n\n\n\\vspace{-2pt}\n\\subsection{Qualitative Results}\\label{secExpQuali}\n\nExample tracking results from one sequence out of 26 on the ICU data and out of 3 on the pedestrian dataset are shown in Fig.~\\ref{figVisualICU} and Fig.~\\ref{figVisualSpinello}, respectively. From Figs.~\\ref{figVisualICU}(a)(b)(c), we can see that our method can track all the persons who are standing or lying. This shows that our method is more robust to significant pose variations by modeling the deformable spatial relationship between detectors. On the other hand, as shown in Figs.~\\ref{figVisualICU}(d)(e) and Figs.~\\ref{figVisualSpinello}(a)(b), our method also works better by combining multiple part-based (head) detector when part of the person is occluded or out of the boundary of the camera.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.96\\linewidth]{compareSpinello.pdf}\n\\caption{Example tracking results on RGBD pedestrian dataset}\n\\vspace{-10pt}\n\\label{figVisualSpinello}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn this paper, we propose an approach to combining multiple detectors and model the deformable spatial relationship between them within a tracking-by-detection framework. Detection outputs are grouped to reduce the number of false positives by the location and depth information. The deformable spatial correlations are modeled by minimizing the least-square errors between the expected positions and the projected ones. Person locations are inferred by minimizing an objective function including detection matching, spatial correlation, mutual exclusion, temporal consistence and regularization constraints. Experimental results on data from a live intensive care unit show that the proposed method significantly improves the multi-target tracking performance over state-of-the-art methods.\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{ieee}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSpontaneous symmetry breaking plays a key role in our current\nunderstanding of particle physics and is expected to have been a major\nfactor in determining the physical properties of the early\nuniverse \\cite{Vilenkin}. In cosmology, two aspects of symmetry\nbreaking are of great\ninterest: it typically happens far from thermal equilibrium and\nit is inherently nonlinear. In the context of the\nelectroweak phase transition, for example, an initially thermalized\nstate is tossed out of equilibrium as the Higgs evolves to acquire a\nnonzero expectation value. In inflation, a nonthermal state\nthermalizes to reheat the universe with an explosive energy transfer\nfrom the inflaton to other field modes. It is thus of great interest\nto study the dynamics of symmetry breaking in an expanding\nbackground numerically in order to isolate key features that may\nescape analytical techniques.\n\nHere, we report results on 3d simulations for two\nsituations: a single, self-interacting scalar field with a double-well\npotential, and an SU(2) non-Abelian Higgs model. In\nRefs. \\cite{graham_cos,farhi_cos} results have been obtained for the\ncase of a single scalar in 1d. It was shown that long-lived,\ntime-dependent field configurations known as oscillons\n\\cite{bogol,gleiser,copeland} emerged spontaneously and contributed an\namazing 50\\% of the total energy density. These initial results\ntriggered the present study in the context of more realistic\nmodels. There are two broad classes of scalar field\noscillons that have been studied in the literature, small and\nlarge-amplitude. Small-amplitude oscillons do not probe the \nhighly nonlinear domain of the potential, and typically have large\nspatial widths \\cite{fodor,farhi_cos,amin}. Their small amplitude\nmakes it possible to study them using linearization\ntechniques. Large-amplitude oscillons are harder to investigate\nanalytically \\cite{dosc,sicilia,hertzberg}. Simulations of scalar\nmodels in static 2d and 3d backgrounds \\cite{oscnuc} and expanding 1d\nbackgrounds \\cite{graham_cos,farhi_cos}\nindicate that mostly large-amplitude oscillons are excited during\nsymmetry breaking. As we show next, this is also the case for an\nexpanding 3d spacetime. The situation is different for SU(2)\nmodels, as we explain below.\n\nThis paper is organized as follows: in the next section, we introduce\nthe scalar field model in an expanding universe and discuss its\nlattice implementation. We report our results for a double well\npotential, showing that oscillons contribute about 1.2\\% of the energy\ndensity. In section III we show that, contrary to naive expectation,\nfor certain values of the expansion rate oscillons may have their\nlifetimes enhanced. We explain this result analytically by making use\nof parametric resonance. In section IV we introduce the SU(2)\nnon-Abelian Higgs model and discuss its lattice implementation in an\nexpanding universe. In section V we discuss the results for this\nmodel. In particular, we show that, as in the case of a real scalar\nfield, oscillons contribute a nontrivial percentage of the total energy\ndensity. Furthermore, our results indicate that the cosmological\nexpansion seems to favor the formation of oscillons for a wider range\nof parameters as compared to the static case, where oscillons were\nfound only in a 2:1 mass ratio for the Higgs and gauge boson. In\nsection VI, we briefly discuss possible application of oscillons in\ncosmology, which we hope to explore in forthcoming work, and\nconclude with a summary of our results.\n\n\\section{Scalar Field Model}\nWe consider a scalar field $\\Phi({\\bf x},t)$ propagating in\n$(3+1)$-dimensional de Sitter spacetime with Hubble constant $H = \\dot\na\/a$ and a double-well potential $\nV(\\Phi)=(\\lambda\/4)[\\Phi^2-\\mu^2\/\\lambda]^2$. Using $\\hbar=c=k_B=1$\nand defining dimensionless variables $\\phi =\n\\Phi(\\mu\/\\sqrt{\\lambda})^{-1}$ and $\\tilde{x}^{\\nu} = \\mu\nx^{\\nu}~(\\nu=0,1,2,3)$, the equation of motion satisfied by $\\phi$ is\n\\begin{equation}\n\\ddot{\\phi}+3 \\frac{\\dot{a}(t)}{a(t)}\\dot{\\phi}=\n\\frac{\\nabla^2\\phi}{a(t)^2}+\\phi-\\phi^3,\n\\label{Eq:EOM3D}\n\\end{equation}\nwhere overdot and $\\nabla$ denote derivatives with respect to\ndimensionless time $\\tilde{x}^0$ and space $\\tilde{x}^i$.\nThe expansion rate becomes $H = \\mu \\tilde{H}$, where\n$\\tilde{H}\\equiv d\\ln(a)\/d\\tilde{x}^0$.\n\nOur initial conditions simulate quasi-thermal states of the free\nmassive scalar field. The parameters that control the distribution of\nthe lattice modes are the temperature $T$ and the mass of the field\n$m=\\mu\\sqrt{2}$. The simulation space consists of a cube with comoving\nsize $L$ and volume $V=L^3$ discretized on a regular lattice with\nspacing $\\Delta x^i=\\Delta r~(i=1,2,3)$. We apply periodic\nboundary conditions and label the free field's normal modes by\n$\\mathbf{k}=(2\\pi \\mathbf{n}_i\/L)$, where $\\mathbf{n}=(n_x,n_y,n_z)$\nand the $n_i$ are integers $n_i=-N\/2+1\\ldots N\/2$. \nHere $N=L\/\\Delta r$ is the\nnumber of lattice points per side. Each free mode is\ndescribed by a harmonic oscillator with frequency\n$\\omega_k^2=(2\\sin(k\\Delta r\/2)\/\\Delta r)^2+m^2$,\nwhere $k=|\\mathbf{k}|$. The initial conditions for the field $\\phi$\nare then given by \n\\begin{eqnarray}\n \\phi(\\mathbf{r},t=0)&=&\\frac{1}{\\sqrt{V}}\n\\sum_\\mathbf{k}\\sqrt{\\frac{\\hbar}{2\\omega_k}}\\left[\\alpha_ke^{i\n\\mathbf{k}\\cdot\\mathbf{r}}+\n\\alpha_k^*e^{-i \\mathbf{k}\\cdot\\mathbf{r}}\\right],\\nonumber\\\\\n\\dot{\\phi}(\\mathbf{r},t=0)&=&\\frac{1}{\\sqrt{V}}\n\\sum_\\mathbf{k}\\sqrt{\\frac{\\hbar\\omega_k}{2}}\n\\left[\\alpha_ke^{i \\mathbf{k}\\cdot\\mathbf{r}}-\n\\alpha_k^*e^{-i \\mathbf{k}\\cdot\\mathbf{r}}\\right],\n\\label{Eq:InitialCond}\n\\end{eqnarray}\nwhere $\\alpha_k$ is a random complex variable with phase distributed\nuniformly on $[0,2\\pi)$ and magnitude drawn from a Gaussian\ndistribution such that\n$\\langle|\\alpha_k|^2\\rangle=[\\coth(\\hbar\\omega_k\/2T)-1]\/2$.\nThis is the amplitude distribution for a quantum harmonic oscillator\n\\cite{Landau} with the zero-point motion subtracted. On average, modes\nwith $\\hbar\\omega_k\\lesssim T$ get assigned energy $T$, in agreement\nwith equipartition, while the energy per mode goes\nrapidly to zero for $\\hbar\\omega_k\\gtrsim T$. We thus need a\nlattice fine enough to resolve the high $k$ modes that\nare excited at high temperatures. Using a value of $\\Delta r_0$ that\nis at least 10 times smaller than the wavelength of the mode\nsatisfying $\\hbar\\omega_k\\sim T$ is enough to provide a good continuum limit.\n\nWe discretize the equation of motion using second-order space\nderivatives with lattice spacing $\\Delta r$ in all directions. We then\nstep forward in time using a fourth-order R\\\"{u}nge-Kutta method. \nBy the Courant condition, we need to keep $\\Delta t<a(t)\\Delta r$ at\nall times. We impose a maximum physical lattice spacing $\\Delta\nr_{\\rm max}$ that is fine enough to resolve field configurations at\nphysical sizes that we expect for oscillons. When $a(t) \\Delta r\n\\ge \\Delta r_{\\rm max}$, we refine the lattice by bringing the lattice\nspacing back to $\\Delta r_{\\rm max}\/2$ and inserting points by polynomial\ninterpolation. We pick $\\Delta r_{\\rm max}$ and $\\Delta t$ small\nenough so that any further reduction does not significantly affect the\nfinal configuration of a run. All our simulations maintain energy\nconservation to a part in $10^3$ or better.\n\nWe evolve the field $\\phi(\\mathbf{x},t)$ in a box with $256^3$ lattice\npoints and $\\Delta r_0=0.05 \\mu^{-1}$. We keep $\\Delta t=0.01\\mu^{-1}$\nconstant throughout the simulation. As the lattice spacing increases\nto $\\Delta r_{\\rm max}=0.5\\mu^{-1}$, we insert points in the lattice,\nbringing the spacing down to $\\Delta r_{\\rm max}\/2$. We vary the\nvalues of the expansion factor $H$ and the initial temperature $T$\nand evolve the field until it cools down to $T\/a(t)=0.3\\mu$. After the\nuniverse has expanded and cooled, we observed persistent localized\nstructures as peaks in the energy density. We show a typical sequence\nof snapshots in  Fig.~\\ref{Fig:Snapshots}.\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.4]{fig1}\n\\caption{Sequence of time snapshots of the energy density. Time\nincreases from left to right and top to bottom at times\n$t\\mu={0,50,100,150,200,250}$, $H=0.01\\mu$ and $T=6.0\\mu$. All\nsnapshots show the energy density $u$\nwith an isosurface at $u=0.2\\mu^4$.}\n\\label{Fig:Snapshots}\n\\end{figure}\n\nBy isolating these peaks\nindividually, we find that they all share the typical signatures of\nspherically-symmetric oscillon configurations: their centers\noscillate with the typical oscillon frequency, as shown in the inset of\nFig.~\\ref{Fig:Energy_Fraction}, and their energies coincide with\nthe plateau energies found in detailed oscillon studies\n\\cite{gleiser,copeland,sicilia}.\nThe energy of a configuration is calculated by integrating the field's\ntotal energy around its peak using a radius $r=10\\mu^{-1}$. We\nconsistently found $E_{\\rm osc} \\simeq 45\\mu\/\\lambda$. We then\nmeasured the fraction of energy in oscillons (Fig.\\\n\\ref{Fig:Energy_Fraction}) and the number of oscillons nucleated as a\nfunction of temperature, which scales simply as $N_{\\rm osc} \\propto\nV=L^3 \\propto T^3$.\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.4]{fig2}\n\\caption{Fraction of energy in oscillons as a function of\ntemperature in units of $\\mu$. Here $\\Delta r_0=0.05 \\mu^{-1}$,\n$\\Delta r_{\\rm max}=0.5 \\mu^{-1}$, and $H=0.01 \\mu$. Error bars denote\nensemble averages over 10 runs. The inset shows the near-harmonic\noscillations of the oscillon core. A simulation of a typical run can\nbe viewed at \\href{http:\/\/www.youtube.com\/watch?v=_0co05XkNMY}\n{http:\/\/www.youtube.com\/watch?v=\\_0co05XkNMY}\n(web link on electronic version). }\n\\label{Fig:Energy_Fraction}\n\\end{figure}\n\nAlthough we quote results for $H=0.01\\mu$, we have performed simulations \nfor a slower expansion rate of $H=0.005 \\mu$\nobtaining similar qualitative behavior:  for a wide range of initial\ntemperatures, $\\rho_{\\rm osc}\/\\rho_{\\rm tot} =\\Omega_{\\rm osc} \\sim\n1.2\\%$. Smaller values of $H$ require impractical computation time,\nbut we don't expect any qualitative changes. We note that since the\nsimulations end with fairly large values of $\\Delta r=0.5\\mu^{-1}$, our\nresults are lower bounds on $\\rho_{\\rm osc}$. Of course, scalar field\noscillons are not stable in 3d and will decay after $\\tau_{\\rm osc} \\sim\n10^4\\mu^{-1}$. Nevertheless, during their lifetime, they may be\nresponsible for several important effects, as we discuss in section VI. We\nalso note that oscillons are prevented from forming if the horizon\nsize $1\/H$ is of the order of the oscillon size,\n$R_{\\rm osc} \\sim 4\\mu^{-1}$. For\n$H\\gtrsim0.1\\mu$, which fortunately is not very realistic,\nlarge-amplitude fluctuations are flattened out before the stabilizing\neffect of nonlinearities can kick in. In other words, for oscillons to\nbe cosmologically viable, we must have\n$R_{\\rm osc}\/\\lambda_H = \\tilde{R}_{\\rm osc}\\tilde{H} \\ll 1$,\nwhere $\\lambda_H = H^{-1}$ is the horizon length. This condition is\neasily satisfied for physics below the Planck scale.\n\n\\section{Lifetime Enhancement}\nHaving established that oscillons emerge dynamically in an expanding\nbackground, we need to examine how the expansion affects their\nlifetime. For numerical efficiency, we exploit the spherical symmetry\nof the final oscillon configuration and\nreduce our system to an effectively 1d problem by letting\n$\\nabla^2\\phi\\rightarrow\\partial^2\\phi\/\\partial\nr^2+(2\/r)\\partial\\phi\/\\partial r$ in Eq.~\\ref{Eq:EOM3D}. \nWe find oscillons by setting the initial field configuration to be\nGaussian, $ \\phi(r,0)=2\\exp(-r^2\/R_0^2)-1$, with\nboundary conditions $\\phi(r\\rightarrow\\infty,t)=-1\\textrm{,\n}\\phi'(0,t)=0\\textrm{, and } \\dot{\\phi}(r,0)=0$\n\\cite{gleiser,copeland}. In the absence of expansion, Gaussians with\n$2.4\\lesssim R_0 \\mu\\lesssim 4.5$ settle into long-lived oscillon\nconfigurations.\n\nWe follow the same procedure as in 3d so that as soon as the lattice\nspacing becomes $\\Delta r_{\\rm max}=0.1\\mu^{-1}$, \n$L_{\\rm max}\\gtrsim 2\/H$, we insert \npoints via polynomial interpolation, and bring the lattice spacing\nback to $\\Delta r=0.05\\mu^{-1}$. We then truncate the box to\n$L\\gtrsim 1\/H$, which can't affect the oscillon at $r=0$. We always use\na box of initial size $L_0=1\/H+50\\mu^{-1}$ in natural units, and we have\nverified that any run with $L_0\\gtrsim 1\/H$ gives identical results. In\nFig.~\\ref{Fig:Enhancement} we show the effects of expansion for a\nsample of initial configurations. There is a clear symmetry about\n$R_0=2.86\\mu^{-1}$, the longest-lived oscillon in the absence of expansion:\nradii to both sides of $R_0=2.86\\mu^{-1}$ experience an increase in lifetime\nfor a range of $H$, with the increase being more pronounced for shorter\nlifetimes. The longest-lived oscillon, in turn, doesn't experience any\nnoticeable enhancement. The inset of\nFig.~\\ref{Fig:Enhancement} shows the maximum fractional\nincrease in lifetime $(\\tau_{\\rm max} - \\tau_0)\/\\tau_0$ as a function\nof initial radius $R_0$. The lifetime enhancement follows an approximate\nscaling law around $R_0=2.86\\mu^{-1}$, $\\tau_{\\rm osc}\\mu \n\\sim |R_0\\mu - 2.86|^{0.05}$.\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.4]{fig3}\n\\caption{Lifetime, in units of $\\mu^{-1}$, for oscillons formed from\nGaussians with $2.4\\leq R\\mu\\leq4.4$, as a function of expansion\nrate $H$ in units of $\\mu$. The inset shows the maximal fractional\nincrease in oscillon lifetime for different radii at $H_{\\rm max}$.\n}\n\\label{Fig:Enhancement}\n\\end{figure}\n\nTo understand the origin of the lifetime enhancement caused by\nthe expansion, we decompose the field as\n$\\phi(\\textbf{x},t)=\\phi_{av}(t)+\\delta\\phi(\\textbf{x},t)$, where\n$\\phi_{av}$ is the volume averaged field. Linearizing\nEq.~\\ref{Eq:EOM3D} with respect to $\\delta\\phi(\\textbf{x},t)$ and\ntaking the Fourier transform, we obtain (for $k>0$)\n\n\\begin{equation}\n\\delta\\ddot{\\phi}(k,t) + 3H\\delta\\dot{\\phi}(k,t)+\n\\left(\\frac{k^2}{a(t)^2} + V''(\\phi_{av}(t))\\right)\\delta\\phi(k,t)=0.\n\\label{Eq:LinearEq}\n\\end{equation}\nOnce the Gaussian has settled into the oscillon stage,\n$V''(\\phi_{av}(t))$ can be approximated by\n$V''(\\phi_{av}(t))=\\Phi_0\\cos(\\omega t)+C$, where $\\Phi_0$ and $C$\nvary very slowly during an oscillon's lifetime and depend on the value\nof the initial radius $R_0$ and the Hubble constant $H$. Here $\\omega<m$ is\nthe oscillon's frequency of oscillation. \nIntroducing new variables $\\omega t=2z-\\pi$, and\n$\\delta\\phi=\\exp(-3Hz\/\\omega)\\chi$, Eq.~\\ref{Eq:LinearEq} becomes\n\\begin{equation}\n \\chi''+\\left[A_k-2q\\cos2z\\right]\\chi=0,\n\\label{Eq:Mathieu}\n\\end{equation}\nwhere $A_k=\\frac{1}{\\omega^2}\\left[4k^2\/a^2+4C-9H^2\\right]$,\n$q=2\\Phi_0\/\\omega^2$, and prime denotes differentiation with respect\nto the new variable $z$. Eq.~\\ref{Eq:Mathieu} is the Mathieu equation,\nwhich is known to exhibit parametric resonance when $A_k\\simeq l^2$,\n$l=1,2,\\ldots$\\cite{mathieu_inflation}. Thus, particular combinations\nof values of $C$, $H$ and $\\omega$ can lead to exponential\namplification in the oscillations of $\\chi$, and consequently\n$\\delta\\phi(k,t)$, at certain modes $k\/a$. Because $C$ is a positive\nnumber, the $A_k\\simeq1$ resonance window occurs for real values of\n$k\/a$ only for very large values of $H$, which destabilize the oscillon\nbefore it can lead to resonance. The $A_k\\simeq 9$ and higher windows\nlead to resonances that are too weak to overcome the damping due to\nthe expansion. The $A_k\\simeq 4$ window, however, can lead to\nparametric amplification of the dominant oscillon wavevectors for the\nvalues of $H$ that generate the observed lifetime enhancement depicted\nin Fig.~\\ref{Fig:Enhancement}.\n\nThe $A_k\\simeq4$ resonance window leads to exponential amplification\nto the oscillations in $\\chi\\propto e^{\\xi z}$, where \n$\\xi \\simeq\\sqrt{5}q^2\/48$ \\cite{mathieu_functions}.\nFor small values of $H$, the amplification overcomes the damping due\nto expansion. For higher values of $H$, the damping overcomes the\namplification, and the oscillon decays. An example is shown in\nFig. \\ref{Fig:Resonance}.\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.4]{fig4}\n\\caption{Oscillations in $\\delta\\phi(k,t)$ for an oscillon with\n$R_0=2.5\\mu^{-1}$ and expansion rate $H=0.006\\mu$, at times \n$3200\\leq t \\mu\\leq 3700$. Here\nwe plot a mode which starts at $k\/a\\simeq 1.4\\mu$ and gets redshifted to\n$k\/a\\simeq 0.1\\mu$. As $k\/a$ gets smaller, the mode enters the\nresonance window, and its Fourier component gets amplified and then\nredshifted away. In this case, $\\omega\\simeq 1.4 \\mu$ ($<m=\\sqrt{2}\n\\mu$) and $C=1.8 \\mu^2$, with $A_k=4$ for $k\/a\\simeq0.4 \\mu$, which is\nthe dominant wave vector of this oscillon ($R\\simeq 2.9\\mu^{-1})$.}\n\\label{Fig:Resonance}\n\\end{figure}\n\n\\section{SU(2) Model: Implementation}\nBecause the SU(2) model is considerably more expensive numerically to\nsimulate, the range of experiments we can carry out is\nlimited. However, these experiments show very similar behavior\nto the scalar model. We begin from the Lagrangian density in the\nabsence of expansion,\n\\begin{equation}\n{\\cal L} =  -\\frac{1}{4} {\\bm F}_{\\mu \\nu} \\cdot {\\bm F}^{\\mu \\nu}\n+ (D_\\mu \\Phi)^\\dagger (D^\\mu \\Phi)\n- \\lambda(|\\Phi|^2 - \\mu^2)^2 \\,,\n\\end{equation}\nwhere the boldface vector notation refers to isovectors. Here \nthe Higgs field $\\Phi$ is an SU(2) doublet, and the SU(2) field\nstrength and covariant derivatives are\n\\begin{eqnarray}\n\\bm{F}_{\\mu \\nu} &=& \\partial_\\mu {\\bm W}_\\nu - \\partial_\\nu \\bm{W}_\\mu\n- g{\\bm W}_\\mu\\times {\\bm W}_\\nu \\,, \\\\\nD_\\mu \\Phi &=& \\left(\\partial_\\mu \n+ i \\frac{g}{2} \\bm{\\tau} \\cdot \\bm{W}_\\mu\\right)\\Phi \\,, \\\\\nD^\\mu \\bm{F}_{\\mu \\nu} &=& \\partial^\\mu \\bm{F}_{\\mu \\nu}\n- g \\bm{W}^\\mu \\times \\bm{F}_{\\mu\\nu} \\,,\n\\end{eqnarray}\nwhere $\\bm{\\tau}$ represents the weak isospin Pauli matrices.\nWe obtain the equations of motion\n\\begin{equation}\nD_\\mu \\bm{F}^{\\mu\\nu} = \\bm{J}^\\nu \\,, \\qquad\nD^\\mu D_\\mu \\Phi = 2\\lambda(\\mu^2 - |\\Phi|^2) \\Phi\\,,\n\\end{equation}\nwhere the gauge current is\n$\\bm{J}_\\nu = g {\\, \\rm Im \\,} (D_\\nu \\Phi)^\\dagger \\bm{\\tau} \\Phi$\nand we work in the gauge\n$\\bm{W}_0 = \\bm{0}$. With this choice, the\ncovariant time derivatives become ordinary derivatives and we can\napply a Hamiltonian formalism. The $\\bm{W}_j$ fields have mass\n$m_W=g \\mu\/\\sqrt{2}$, and the Higgs field has mass \n$m_H=2 \\mu \\sqrt{\\lambda}$.\n\nTo include the effects of expansion, we again work in comoving\ncoordinates with a scale factor $a(t)$. We now have the action\n\\begin{equation}\nS = \\int d^3 \\bm{r} \\, a(t)^3 \\left[\n\\frac{1}{2} \\sum_{j=x,y,z}\n\\left(\\bm{E}_j\\cdot \\bm{E}_j -\n\\bm{B}_j\\cdot \\bm{B}_j \\right)\n+ \\dot \\Phi^\\dagger \\dot \\Phi\n- \\frac{1}{a(t)^2} \\sum_{j=x,y,z} (\\partial_j \\Phi^\\dagger)(\n\\partial_j \\Phi) - \\lambda(|\\Phi|^2 - \\mu^2)^2\n\\right] \\,,\n\\end{equation}\nwhere\n\\begin{equation}\n\\bm{E}_j = \\dot{\\bm{W}}_j \\hbox{\\quad and \\quad}\n\\bm{B}_j =  -\\frac{1}{2} \\sum_{j',j''=x,y,z}\n\\epsilon_{jj'j''} \\left(\\frac{1}{a(t)} \\partial_{j'} \\bm{W}_{j''}\n-g \\bm{W}_{j'} \\bm{W}_{j''}\\right) \\,.\n\\end{equation}\nHere dot indicates time\nderivative and Latin indices run over space dimensions.\n\nFor numerical computation we put the theory on a lattice,\nfollowing the conventions and techniques used in \\cite{SU2osc3d2}. \nThe field variables are the values of the $\\Phi^p$ field at the\nlattice sites $p$ and the spacelike Wilson lines\n\\begin{equation}\nU_j^p(t) = e^{i g \\bm{W}_j^p(t) \\cdot\\bm{\\tau} a(t) \\Delta x\/2}\n\\label{eq:Wilson}\n\\end{equation}\nemanating from lattice site $p$ in the spacelike direction $j$.\nSince the lattice equations are second order, we will find each field\nat the next time slice based on the previous two. We let $t$ be the\ntime for the current set of lattice points and spacelike links and\ndefine $t_+ = t+\\Delta t$  and $t_- = t-\\Delta t$ to be \nthe subsequent and previous times respectively. We also take $t_{+\/2}\n= t+\\Delta t\/2$ and $t_{-\/2} = t-\\Delta t\/2$ to be the times in\nbetween, which will be the times at which we evaluate the timelike links.\n\nWe define the Wilson line for the link emanating from\nlattice site $p$ in the negative $j^{\\rm th}$ direction to be the\nadjoint of the corresponding Wilson line emanating in the positive\ndirection from the neighboring site, $U_{-j}^p(t) = \nU_j^{p-j}(t)^\\dagger$,\nwhere the notation $p \\pm j$ indicates the adjacent lattice site to\n$p$, displaced from $p$ in direction $\\pm j$. At the edges of the\nlattice we use periodic boundary conditions. We define the elements\nof the field strength tensor, which are centered on the timelike and\nspacelike plaquettes of the lattice,\n\\begin{equation}\n\\bm{\\tau}\\cdot\n\\bm{E}_j^p(t_{+\/2}) = \\frac{2}{i g a(t_{+\/2}) \\Delta x\\Delta t}\n\\log U_j^p(t_+)U_j^p(t)^\\dagger \\hbox{\\quad and \\quad}\n\\bm{\\tau}\\cdot\n\\bm{B}_j^p(t) = \\frac{i}{g (a(t) \\Delta x)^2}\n\\sum_{j',j''=x,y, z}\n\\epsilon_{j j' j''} U^p_{\\square(j',j'')}(t) \\,,\n\\end{equation}\nwhere\n$U^p_{\\square(j,j')}(t) = U_j^p(t)\nU_{j'}^{p+j}(t) U_{-j}^{p+j+j'}(t)\nU_{-j'}^{p+j'}(t)$\nand  we have defined the logarithm of \na $2\\times 2$ matrix in the form of Eq.~\\ref{eq:Wilson} as\n\\begin{equation}\n\\log U_j^p(t) = \\frac{ig a(t) \\Delta x}{2} \\bm{W}_j^p(t)\\cdot\\bm{\\tau} \\,.\n\\label{eq:log}\n\\end{equation}\n\nWe note that $\\log XY \\neq \\log X + \\log Y$ when the matrices\ndo not commute. The logarithms and exponentials needed to\nconvert between the group and the algebra can be computed efficiently\nusing\n\\begin{equation}\ne^{i \\theta \\bm{\\hat n} \\cdot \\bm{\\vec \\tau}} =\n\\cos \\theta + i \\bm{\\hat n}\\cdot \\bm{\\vec \\tau} \\sin \\theta =\n\\begin{pmatrix}\n\\cos \\theta + i \\bm{\\hat n}_z \\sin \\theta &\ni\\bm{\\hat n}_x \\sin \\theta + \\bm{\\hat n}_y \\sin \\theta \\cr\ni\\bm{\\hat n}_x \\sin \\theta - \\bm{\\hat n}_y \\sin \\theta &\n\\cos \\theta - i \\bm{\\hat n}_z \\sin \\theta\n\\end{pmatrix}\n\\,,\n\\label{logformula}\n\\end{equation}\nwhere $\\bm{\\hat n}$ is a unit vector and the link matrices have\n$\\bm{\\hat n} \\theta = \\bm{W}_j^p(t) g a(t) \\Delta x\/2$. For efficiency\nwe replace $\\sin\\theta \\rightarrow \\theta$ and $\\cos\\theta \\rightarrow\n\\sqrt{1- \\theta^2}$ when computing both the logarithm and the\ncorresponding exponential. This discretization then is equivalent\n(without expansion) to what is used in other numerical studies of\nelectroweak symmetry breaking dynamics\n\\cite{Shaposhnikov,RajantieSim,SmitSim,vanderMeulen}. \n\nWe find the equation of motion for the Higgs field\n\\begin{equation}\n\\Phi^p(t_+) = \n\\frac{1}{1+\\frac{3H\\Delta t}{2}}\n\\left[2 \\Phi^p(t) - \n\\left(1-\\frac{3H\\Delta t}{2}\\right)\\Phi^p(t_-) + \\Delta t^2 \\ddot\n\\Phi^p (t)\\right] \\,,\n\\end{equation}\nwhere $H=\\frac{\\dot a(t)}{a(t)}$ is the Hubble constant and\n\\begin{equation}\n\\ddot \\Phi^p (t) = \\sum_{j=\\pm x,\\pm y, \\pm z}\n\\frac{U_j^p(t) \\Phi^{p+j}(t) - \\Phi^p(t)}{a(t)^2 \\Delta x^2}\n + 2\\lambda\\left(\\mu^2 - |\\Phi^p(t)|^2\\right)\\Phi^p(t) \\,.\n\\end{equation}\nFor the gauge fields, we have\n\\begin{eqnarray}\nU_j^p(t_+) &=&\n\\left(\\exp\\left\\{\\log \\left(U_j^p(t) \nU_j^p(t_-)^{\\frac{H\\Delta t}{2} - 1}\\right)\n\\phantom{\\left(\\frac{\\log U^p_{\\square(j,-j')}(t)}{\\Delta x^2}\\right)}\n\\right. \\right. \\cr && \\left. \\left.\n-\\left[\\sum_{j'\\neq j}  \\left(\n\\frac{\\log U^p_{\\square(j,j')}(t) + \n\\log U^p_{\\square(j,-j')}(t)}{a(t)^2 \\Delta x^2} \\right)\n+ \\frac{ia(t) \\Delta x}{2} g \\bm{J}_j^p (t)\\cdot \\bm{\\tau}\n\\right]\\Delta t^2 \\right\\}\nU_j^p(t)\\right)^{\\frac{1}{\\frac{H\\Delta t}{2} + 1}}\\,,\n\\label{eq:gauge}\n\\end{eqnarray}\nwhere the gauge current is\n\\begin{equation}\n\\bm{J}_j^p(t) = g {\\rm \\, Im \\,} \n\\frac{\\Phi^p(t)^\\dagger  \\bm{\\tau} U_{j}^p(t)\n\\Phi^{p+j}(t)}{a(t) \\Delta x}\n\\end{equation}\nand the logarithm in Eq.~\\ref{eq:log} is used to compute the\nexponents in Eq.~\\ref{eq:gauge}.\n\nAssuming it is obeyed by the initial conditions, time evolution\npreserves the Gauss's Law constraint,\n\\begin{eqnarray}\n\\sum_{j=x, y, z}\n\\frac{\\bm{E}_j^p(t_{+\/2}) + \\bm{E}_{-j}^p(t_{+\/2})} {a(t_{+\/2}) \\Delta x}\n=  \\bm{J}_0^p(t_{+\/2})\\,,\n\\label{eq:Gauss}\n\\end{eqnarray}\nwhere the charge density is given by\n\\begin{equation}\n\\bm{J}_0(t_{+\/2}) = g {\\rm \\, Im \\,} \\left(\\frac{\\Phi^{p}(t_+) -\n\\Phi^{p}(t)} {\\Delta t} \\right)^\\dagger \\bm{\\tau}\n\\Phi^{p}(t) \\,.\n\\end{equation}\nEnergy is not conserved because in the expanding background we have\n$dU = -p dV$, where $p$ is the pressure. In the lattice model we then\nhave\n\\begin{equation}\n\\frac{dU}{dt} = - (a(t) \\Delta x)^3 H \\sum_p\n\\left[\n\\frac{1}{2} \\sum_{j=x,y,z}\\left(\\bm{E}_j^p \\cdot \\bm{E}_j^p +\n\\bm{B}_j^p \\cdot \\bm{B}_j^p  \\right)\n+ 3 |\\dot \\Phi|^2 - \\sum_{j=x,y,z}\n\\left|\\frac{U_j^p \\Phi^{p+j} - \\Phi^p}{a(t) \\Delta x}\\right|^2\n- 3 \\lambda(|\\Phi^p|^2 - \\mu^2)^2\n\\right]\\,.\n\\end{equation}\nWe have verified that both the Gauss's Law and energy constraints are\nwell obeyed throughout our simulation.\n\nWe set initial conditions for the first two time-slices, which we\ndenote as $t_0$ and $t_1$, \nby occupying the modes of all the components of the Higgs and gauge\nfields at temperature $T$, as in the case of a single scalar field.\nThen we modify these initial conditions to make them obey Gauss's Law,\nby the following steps:\n\\begin{itemize}\n\\item\nFirst, since we have periodic boundary conditions, the total charge\nshould be zero. To enforce this constraint, we shift the $\\Phi^p$\nfield on both of the first two time slices $t_0$ and $t_1$ by the same\nconstant,\n\\begin{equation}\n\\Phi^p \\to \\Phi^p -\n\\frac{i}{g\\big|\\dot{\\bar{\\Phi}}\\big|^2}\n\\left(\\bar{\\bm{J}}_0 \\cdot \\bm{\\tau} \\right) \n\\dot{\\bar{\\Phi}} \\,,\n\\end{equation}\nwhere $\\bar{\\bm{J}}_0$ and $\\dot{\\bar{\\Phi}}$\nare the average values of $\\bm{J}_0$ and\n$\\dot \\Phi$ over the lattice at time $t_{1\/2}$, respectively.\n\n\\item\nNext, we fix the longitudinal component of the gauge fields, as\ndescribed in Ref.~\\cite{RajantieSim}. We take a discrete Fourier\ntransform of the initial charge $\\bm{J}_0^p(t_{1\/2})$\nand gauge field $\\bm{J}_j^p(t_{1\/2})$ to obtain\n$\\widetilde{\\bm{J}}_0^{\\vec k}(t_{1\/2})$ \nand $\\widetilde{\\bm{W}}_j^{\\vec k}(t_{1\/2})$ for the initial time step,\nwhere $\\vec k$ labels the Fourier transformed lattice. We\nthen modify the initial time derivative of $\\bm{W}_j^p$ (by changing\nits value on one of the first time slices but not the other) by sending\n\\begin{equation}\n\\dot{\\widetilde{\\bm{W}}}_j^{\\vec k}(t_{1\/2})\n\\to \\dot{\\widetilde{\\bm{W}}}_j^{\\vec k}(t_{1\/2})\n- \\left(\\sum_{j'=x,y,z} \\vec k_{j'}\n\\dot{\\widetilde{\\bm{W}}}_{j'}^{\\vec k}(t_{1\/2})\n+ i \\widetilde{\\bm{J}}_0^{\\vec k}(t_{1\/2})\n\\right) \\frac{\\vec{k}_j}{|\\vec k|^2}\n\\end{equation}\nand then inverting the discrete Fourier transform to obtain the\nmodified gauge fields, which in turn give the modified Wilson loops.\nNote that we don't make any modification for $\\vec k = \\vec 0$, where\nthis transformation breaks down; that case was already handled by the\nprevious step.\n\n\\item\nFinally, while in an Abelian theory the subtraction of the\nlongitudinal component would be sufficient to implement Gauss's Law,\nfor a nonabelian theory the nonlinear term in the field strength makes\nthis agreement only approximate. As a result, we adjust the phase of\n$\\Phi^p(t_1)$,\n\\begin{equation}\n\\Phi^p(t_1) \\to \n\\frac{|\\Phi^p(t_1)|}{|\\Phi^p(t_0)|}\n{\\cal U}^p(t_{1\/2}) \\Phi^p(t_0)\n\\hbox{\\quad with \\quad}\n{\\cal U}^p(t_{1\/2}) = \\exp\\left[\n-\\sum_{j=x, y, z}\n\\frac{\\log U_j^p(t_1) U_j^p(t_0)^\\dagger\n + \\log U_{-j}^p(t_1) U_{-j}^p(t_0)^\\dagger }\n{g^2 (a(t_{1\/2}) \\Delta x)^2   |\\Phi^p(t_1)| |\\Phi^p(t_0)|\/2}\n\\right] ,\n\\end{equation}\nleaving $\\Phi^p_0(t_0)$ unchanged, in order to assure that Gauss's Law is\nsatisfied.\n\n\\end{itemize}\n\n\\section{SU(2) Model: Results}\nWe begin with a universe of size $L a(t=0)=4\/\\mu$, temperature $T=4 \\mu$,\nlattice spacing $a(t=0) \\Delta x = 1\/(224 \\mu)$, and use a time step\n$\\Delta t = 1\/(448 \\mu)$. We allow the universe to expand at a constant\nrate, with Hubble constant $H = \\mu(\\log 2)\/12 \\approx 0.06\\mu$, and expand\nthe universe by a factor of $224$, so that the final lattice spacing is\n$a(t_{\\rm final}) \\Delta x = 1\/\\mu$. We measure the fraction of\nenergy in oscillons by including those points whose energy density is\nfour times the average energy density. Under ordinary thermal\nexpansion this fraction would stay constant, and during the initial stages\nof the expansion it is identically zero. At the end of the expansion,\napproximately $4\\%$ of the energy is found in oscillons by this measure.\nThe energy density is shown in Fig.~\\ref{Fig:oscsu2density}, and the\nevolution of the energy over time is shown in Fig.~\\ref{Fig:oscsu2time}.\nCompared to the scalar model, oscillons in the SU(2) model have\nsmaller amplitude and larger spatial extent, requiring a larger\nsimulation volume. The simulation is thus considerably more expensive\nnumerically, especially given the cost of evolving a total of\nthirteen real degrees of freedom per lattice site instead of one.\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.49\\linewidth]{fig5}\n\\includegraphics[width=0.49\\linewidth]{fig6}\n\n\\includegraphics[width=0.5\\linewidth]{fig7}\n\\caption{Energy density in units of $\\mu^4$ at the end of the expansion \nfor $g=\\sqrt{2}$ with $\\lambda = 1.0$ (left panel) and $\\lambda = 1.1$\n(right panel). To implement the Standard Model coupling of $g_{\\rm\nSM}=0.624$ with the same mass ratio, this energy would be scaled up by\na factor of $(g\/g_{\\rm SM})^2$.}\n\\label{Fig:oscsu2density}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\includegraphics[width=0.49\\linewidth]{fig8a}\n\\includegraphics[width=0.49\\linewidth]{fig8b}\n\\caption{\nFraction of energy in oscillons (solid line) and total energy (dashed\nline) as functions of time. Energy is given in units of $\\mu$ and time\nin units of $1\/\\mu$, with $g=\\sqrt{2}$ and $\\lambda = 1.0$ (left\npanel) and $\\lambda = 1.1$ (right panel). To implement the Standard\nModel coupling of $g_{\\rm SM}=0.624$ with the same mass ratio, this\nenergy would be scaled up by a factor of $(g\/g_{\\rm SM})^2$.\n}\n\\label{Fig:oscsu2time}\n\\end{figure}\n\nWe note that similar results are seen both for\n$g=\\sqrt{2}$, $\\lambda=1$, in which the Higgs and gauge fields are in\nthe $2:1$ mass ratio found in \\cite{SU2osc3d1,SU2osc3d2}, and for\n$g=\\sqrt{2}$, $\\lambda = 1.1$, where the masses are not in this\nratio. Because of the high numerical costs associated with the\nexpanding background simulation, we are not able to track the \nstability of oscillons formed in this way over long time scales, but\nthese results suggest that the expansion may broaden the range\nof parameters for which oscillons are stable. Also, by allowing\noscillons to form from a thermal background rather than a fixed\nansatz, this simulation is capable of scanning a wider\nrange of configuration space (and our results clearly show that\noscillon configurations represent attractors in this space).  Work is\ncurrently underway to investigate these questions in greater detail.\n\n\\section{Possible Impact on Cosmology and Summary}\nOur results indicate that oscillon-like configurations emerge\ndynamically during spontaneous symmetry breaking in expanding\ncosmological backgrounds. Furthermore, they contribute a significant\nfraction of the total energy density. They are thus poised to play an\nessential role in the dynamics of the early universe, be it during\npost inflationary reheating or during symmetry-breaking phase\ntransitions.\n\nIn order to briefly address the impact oscillons may have on\ncosmology, it is best to consider different energy scales\nseparately. For the sake of illustration, we focus on the GUT and\nelectroweak scales, which differ by roughly 13 orders of\nmagnitude. Also, it is important to differentiate between real scalars\nand Abelian and non-Abelian Higgs models. Thus, before we start, it\nmay be useful to summarize what is known of oscillon lifetimes in\nthese models.\n\nAs we mentioned before, for real scalar fields in 3d, the oscillon\nlifetime--with or without  the enhancement from the expansion reported\nhere--is typically $\\tau_{\\rm osc}\\sim 10^{3-4}\\mu^{-1}$\n\\cite{gleiser,sicilia}. For models with gauge fields, the evidence at\nhand points to {\\it very} large lifetimes. Studies for Abelian-Higgs\nmodels in 2d have not seen oscillons decaying, and report lifetimes in\nexcess of $10^5\\mu^{-1}$ \\cite{U1osc2d}. Studies of Abelian-Higgs\nmodels in 3d obtained similar results: oscillons have been observed to\npersist for times $t \\gtrsim 7\\times 10^5\\mu^{-1}$ without decaying\n\\cite{U1osc3d}. For non-Abelian Higgs models, the situation is\nsimilar: the data at hand indicates that once formed, oscillons live\nfor extremely long times. Those in the gauged-SU(2) Higgs model have\nnot been observed to decay after $t \\gtrsim 5\\times 10^5\\mu^{-1}$\n\\cite{SU2osc3d1,SU2osc3d2}. Thus, although a more detailed study of\nAbelian and non-Abelian Higgs oscillons and their stability is clearly\nwarranted, results so far indicate that they may be extremely\nlong-lived, even perturbatively stable. Of course, the key question is\nwhether their lifetime can be longer than the cosmological time scale\nat their formation. If that's the case, they behave as stable, localized\ndefects.\n\nAt the GUT scale, it is clear that the oscillon lifetime is at\nleast of order of the cosmological time scale, $\\tilde{H}^{-1}\\sim\n(M_{\\rm Planck}\/\\mu)\\sim 10^{3-4}$. Thus, for all practical purposes\nat GUT scales oscillons behave as stable localized defects. As has\nbeen shown elsewhere, in the context of first-order phase transitions,\nlong-lived bubble-like configurations such as oscillons can either\nbecome a critical bubble or coalesce to become one. In both cases, the\ndecay of the false vacuum is greatly accelerated, changing from\nexponentially-suppressed to power-law \\cite{vacdecay}. It\nhas been suggested that oscillons may accelerate the decay of the\nfalse vacuum during inflation, potentially solving the bubble\ncoalescence problem of old inflation. This has been recently\nillustrated within the context of a modified hybrid inflation model\n\\cite{oscinf}.\n\nOscillons may also have a key impact during post-inflationary\nreheating. As coherent field configurations, they naturally delay the\napproach to equilibrium, acting as bottlenecks for equipartition\n\\cite{gleiserhowell}. As such, they may influence (decrease) the\nreheating temperature, a possibility we are currently\ninvestigating. An interesting open question is how these\nnonequilibrium results apply in the context of gauge models.\n\nMoving on to the electroweak scale, since $\\tilde{H}_{\\rm ew} \\sim\n10^{-16}$, we are on a realm which is very distant from our numerical\nrange of $\\tilde{H}\\sim 10^{-2}$. Still, we suggest that there are at\nleast two ways in which oscillons may play a role at these relatively\nlow energy scales. Both depend on their lifetime. If non-Abelian\noscillons live for $t\\sim 10^{16}\\mu^{-1}$, that is, if they are\nperturbatively stable, they will remain active at\ncosmologically-relevant time scales. As at the GUT scale discussed\nabove, they may speed up vacuum decay in the context of a first-order\ntransition (which is ruled out in the Standard Model but not in all of\nits extensions) or they may delay thermalization. If they persist for\neven longer, they may even be relevant to dark matter or baryogenesis.\n\nOn the hand, if they live for shorter times $10^{4}< t \\mu < 10^{16}$, their\npresence may still affect the dynamics of symmetry breaking. As is\nwell-known, most phase transitions are initiated due to the presence\nof inhomogeneities or ``seeds'' \\cite{Landau}. If oscillons are\npresent in sufficient quantities, they will modify the effective\npotential nonperturbatively, affecting the dynamics of the transition\n\\cite{HecklerGleiser}. Although much work remains to be done to\ninvestigate such nonperturbative effects in more detail, these\nmechanisms suggest that even relatively short-lived oscillons will\nhave important effects during cosmological symmetry breaking.\n\n\n\\noindent\n{\\it Acknowledgements:}\nMG and NS were partially supported by a\nNational Science Foundation grant PHY-0653341. NG was supported in\npart by the National Science Foundation (NSF) through grant\nPHY-0855426 and a Baccalaureate College Development grant from Vermont\nEPSCoR. Numerical simulations were carried out using TeraGrid\nresources at the National Center for Supercomputing Applications\n(NCSA), with support from NSF, and at the Vermont Advanced Computing\nCenter. Visualizations were created by David Bock (NCSA).\\\\\n\n\\noindent{\\it Note added in proof:} A recent work by Mustafa A. Amin\noffers further support to our hypothesis that oscillons will have\nimportant effects in an expanding universe \\cite{Amin}. We thank the\nauthor for sending his manuscript to us.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{introduction}\n\t\n\n\t\n\n\t\n\tThere is a close relationship between tropical geometry and the geometry of degenerations of algebraic varieties. This relationship takes a particularly suggestive form in much of the recent work that reinterprets mirror symmetry and enumerative geometry in terms of fibrations of complex varieties over affine manifolds. \\cite{KS1} \\cite{Gross-Siebert} \\cite{KS2} \\cite{Gross}. An interesting manifestation of these ideas appears in the recent work of B. Parker \\cite{Parker1} \\cite{Parker2} \\cite{Parker3}. Working in the paradigm of symplectic manifolds, with a view toward applications to pseudoholomorphic curve counting, Parker defines topological spaces called {\\em exploded fibrations}, which he uses to construct log Gromov-Witten invariants \\cite{ParkerLog1}, as required by the Gross-Siebert program. Roughly speaking, an exploded fibration is a topological space with structure semiring, which can be glued along boundary strata from pieces of the form $\\big(\\mathbb{C}^{\\times}\\big)^{n}\\times P$, where $P$ is an integral polytope in $\\mathbb{R}^{m}$, for some $m$. For further details, see \\cite[\\S4]{Parker1} and \\cite[\\S3]{Parker2}. An illustrative example of an exploded fibration is the {\\em exploded curve} given by the fibration $p:Y\\longrightarrow\\!\\!\\!\\!\\rightarrow B$ of topological spaces where $B\\subset\\mathbb{R}^{2}$ is the union of the subspaces\n\t$$\n\t\\scalebox{.95}{$\n\tB_{1}=\\big\\{(x,0)\\in\\mathbb{R}^{2}:x\\ge0\\big\\},\n\t\\ \\ \\ \\ \n\tB_{2}=\\big\\{(0,y)\\in\\mathbb{R}^{2}:y\\ge0\\big\\},\n\t\\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ \n\tB_{3}=\\big\\{(x,x)\\in\\mathbb{R}^{2}:x\\le0\\big\\},\n\t$}\n\t$$\nand where the space $Y$ is glued from pieces\n\t$$\n\tB_{1}\\times\\mathbb{C}^{\\times}-\\big((0,0),\\ \\!0\\big),\n\t\\ \\ \\ \\ \\ \\ \\ \n\tB_{2}\\times\\mathbb{C}^{\\times}-\\big((0,0),\\ \\!1\\big),\n\t\\ \\ \\ \\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ \\ \\ \\ \n\tB_{3}\\times\\mathbb{C}^{\\times}-\\big((0,0),\\ \\!\\infty\\big)\n\t$$\nalong an inclusion of $\\big\\{(0,0)\\big\\}\\times\\big(\\mathbb{P}^{1}_{\\mathbb{C}}-\\{0,1,\\infty\\}\\big)$ into each $B_{i}\\times\\mathbb{C}^{\\times}$, $i=1,2,3$.\n\t\\begin{figure}[!htb]\n\t$$\n\t\\includegraphics[scale=.25]{EXPLODED_001.pdf}\n\t$$\n\t\\caption{An example of an {\\em exploded curve} in the sense of B. Parker \\cite{Parker2}.}\n\t\\label{figure: exploded curve}\n\t\\end{figure}\n\n\tA tropical geometer recognizes the exploded curve above as a close analogue to the family of initial degenerations over all rational points in the tropicalization of an algebraic $K$-curve $X$. In more detail, let $K$ be a non-Archimedean field, which is to say that $K$ is complete with respect to a non-Archimedean absolute value $|-|:K^{\\times}\\longrightarrow\\mathbb{R}_{\\ge0}$. Assume that $K$ is algebraically closed, with value group $\\Gamma\\overset{{}_{\\mathrm{def}}}{=}-\\mathrm{log}_{\\ \\!}|K^{\\times}|\\subset\\mathbb{R}$. Consider an $n$-dimensional split algebraic torus $\\mathbb{T}$ with character lattice $M$ and cocharacter lattice $N=\\mathrm{Hom}_{\\mathbb{Z}}(M,\\mathbb{Z})$. Each vector $v\\in N_{\\mathbb{R}}\\overset{{}_{\\mathrm{def}}}{=}\\mathbb{R}\\tensor_{\\mathbb{Z}}N$ determines the {\\em tilted group ring} $R[U_{v}]$, the ring generated by all those monomials $a\\chi^{u}$ for which $\\langle u,v\\rangle-\\mathrm{log}_{\\ \\!}|a|\\geq0$. Its spectrum $T_{v}\\overset{{}_{\\mathrm{def}}}{=}\\mathrm{Spec}_{\\ \\!}R[U_{v}]$ is an $R$-scheme with generic fiber $T_{v,K}\\cong\\mathbb{T}$. When $v\\in N_{\\Gamma}\\overset{{}_{\\mathrm{def}}}{=}\\Gamma\\tensor_{\\mathbb{Z}}N$, this ring $R[U_{v}]$ is finitely generated, with special fiber of the same dimension as its generic fiber $\\mathbb{T}$. If $X$ is any $K$-scheme, realized as a closed subvariety of the $K$-torus $\\mathbb{T}$ via some closed embedding $\\textit{\\i}:\\xymatrix{X\\ar@{^{(}->}[r]&\\mathbb{T}}$, then the {\\em initial degeneration} of $X$ at $v$, denoted $\\mathrm{in}_{v}X$, is the special fiber $(\\overline{X})_{k}$ of the closure $\\overline{X}$ inside $T_{v}$.\n\t\n\tThe Fundamental Theorem of Tropical Geometry \\cite[Theorem 3.2.4]{IntroTrop} says that the tropicalization of $X$ is the closure, in $N_{\\mathbb{R}}$, of the set of those vectors $v\\in N_{\\Gamma}$ at which the initial degeneration $\\mathrm{in}_{v}X$ is nonempty:\n$$\n\\mathrm{Trop}(X,\\textit{\\i})\n\\ =\\ \n\\overline{\\big\\{v\\in N_{\\Gamma}\\ \\!:\\ \\!\\mathrm{in}_{v}X\\ne \\emptyset \\}}.\n$$\nThis means that we can retain more information about the subvariety $X\\subset\\mathbb{T}$ by remembering not just the points of the tropical variety $\\mathrm{Trop}(X,\\textit{\\i})$, but the points of all initial degenerations of $X$ at all vectors $v\\in N_{\\Gamma}$. Payne makes use of this set-theoretical union\n\t$$\n\t\\bigsqcup_{v\\in\\mathrm{Trop}(X)\\cap N_{\\Gamma}}\\!\\!\\!\\!\\!\\!\\!\\!|\\mathrm{in}_{v}X|\n\t$$\nin order to study the fibers of the tropicalization map $X(K)\\longrightarrow\\mathrm{Trop}(X)$ \\cite{Fiber} \\cite{Fiber2}. He refers to this set as the ``exploded tropicalization,\" taking inspiration from Parker's exploded fibrations. In the present paper, we keep track of initial degenerations at all points of $\\mathrm{Trop}(X)$, $\\Gamma$-rational or not, and define the {\\em exploded tropicalization} to be the union\n\t$$\n\t\\mathfrak{Trop}(X)\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\!\n\t\\bigsqcup_{v\\in\\mathrm{Trop}(X)}\\!\\!\\!\\!\\!\\!|\\mathrm{in}_{v}X|.\n\t$$\n\t\n\tOne major difference between Parker and Payne's exploded objects is the presence of both a topology {\\em and} a structure sheaf (of semirings) on Parker's exploded fibrations, whereas Payne's exploded tropicalizations are mere sets. The present paper is an exposition that complements the forthcoming research paper \\cite{FP}, joint with Sam Payne, in which we explain how to put both a topological structure and a structure sheaf (of topological rings) on the exploded tropicalization $\\mathfrak{Trop}(X)$ of any closed subvariety $X$ of a torus $\\mathbb{T}$ over a non-Archimedean field $K$. Our construction turns $\\mathfrak{Trop}(X)$ into a locally topologically ringed space. We denote this space $\\mathrm{Ad}(X)$, and refer to it as the {\\em adic tropicalization} of $X$ in $\\mathbb{T}$.\n\t\n\tThe locally topologically ringed space $\\mathrm{Ad}(X)$ interacts with Huber's adic spaces \\cite{HubGen} \\cite{Huber} in much the same way that tropical varieties interact with Berkovich spaces. In \\cite{P09}, Payne shows that the if $X$ is a quasi-projective variety over an algebraically closed non-Archimedean field $K$, then the topological space underlying the Berkovich analytification $X^{\\mathrm{an}}$ is the inverse limit of all the tropical varieties $\\mathrm{Trop}(X,\\textit{\\i})$, taken over the inverse system formed by all closed embeddings $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ into quasi-projective toric varieties $Y_{\\Sigma}$. In \\cite{FGP}, P. Gross, S. Payne, and the author give criterion extending this result to the case of certain non-quasi-projective varieties $X$ over an algebraically closed, non-trivially valued non-Archimedean field $K$. In particular, \\cite[Theorem 1.2]{FGP} $X$ says that if $X$ is a closed subvariety of a toric $K$-variety, then the topological space underlying $X^{\\mathrm{an}}$ is the inverse limit of tropical varieties $\\mathrm{Trop}(X,\\textit{\\i})$, taken over the inverse system formed by all closed embeddings $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ into arbitrary toric varieties $Y_{\\Sigma}$. One of the central results of the forthcoming paper \\cite{FP} is a theorem stating that the Huber analytification $X^{\\mathrm{ad}}$ of a closed subvariety $X$ of a proper toric variety is isomorphic, as a locally topologically ringed space, to the inverse limit of the adic tropicalizations of $X$ associated to all closed embeddings of $X$ into proper toric varieties. Section \\ref{Section-Main Theorem} of the present paper provides a brief exposition on this result. See Theorem \\ref{Huberlimit} and Proposition \\ref{Nice system, man!} below for precise statements.\n\t\n\tBecause the present paper is expository, it omits most proofs. The interested reader can find all the proofs in \\cite{FP}. The present paper makes up for its lack of proofs by providing concrete details and examples that do not appear in \\cite{FP}. In what follows, we focus more heavily on the basic geometry of adic spaces, which underlies our geometrization of $\\mathfrak{Trop}(X)$, and on concrete examples of the resulting topological spaces and their relationship to the metrized curve complexes of O. Amini and M. Baker \\cite{Amini-Baker}.\n\n\\vskip .3cm\n\n\\noindent \\textbf{Acknowledgments.}  The author would like to thank the Simons Foundation and the organizers of the Simons Symposium on non-Archimedean and Tropical Geometry for bringing together a fantastic group of researchers for an incredibly stimulating week of mathematics. Special thanks go to S. Payne, as this paper is a report on joint with him. The author thanks M. Baker, F. Baldassarri, W. Gubler, and D. Ranganathan for helpful remarks. Finally, the author thanks the reviewer for comments that greatly improved this paper.\n\nThe author is supported by NSF RTG grant DMS-0943832.\n\n\n\\vskip .75cm\n\n\\section{Preliminaries on Huber analytification}\\label{section: preliminaries on Huber analytification}\n\n\tIn the present \\S\\ref{section: preliminaries on Huber analytification}, we review the basics of Huber's theory of adic spaces and Huber's analytification functor, which produces an adic space from any $K$-scheme $X$. The original sources for this material are \\cite{HubGen} and \\cite{Huber}. We also strongly recommend T. Wedhorn's notes \\cite{Wed} for a rather accessible introduction.\n\t\n\tThe reader already well versed in the theory of adic spaces can skip ahead to \\S\\ref{Section-adic trop}.\n\t\n\\vskip .3cm\n\n\\noindent\n{\\bf Notation.}\nFix a {\\em non-Archimedean} field $K$, i.e., a field complete with respect to some non-trivial, non-Archimedean valuation $v:K\\longrightarrow\\mathbb{R}\\sqcup\\{\\infty\\}$.\nAssume that $K$ is algebraically closed. Let $R$ denote the ring of integers in $K$, let $\\mathfrak{m}$ denote the unique maximal ideal in $R$, and let $k=R\/\\mathfrak{m}$ denote the residue field. Choose a real number $0<\\varepsilon<1$ once and for all, and let\n$|-|_{v}:K\\longrightarrow\\mathbb{R}_{\\geq0}$\ndenote the norm $|a|_{v}=\\varepsilon^{\\mathrm{val}(a)}$ associated to $v$.\n\t\n\tRecall that the {\\em Tate algebra}, denoted $K\\langle T_{1},\\dots,T_{n}\\rangle$, is the subalgebra of $K[[T_{1},\\dots,T_{n}]]$ consisting of those formal power series $f=\\sum a_{i_{1},\\dots,i_{n}}T^{m_{i_{1}}}_{i_1}\\cdots T^{m_{i_{n}}}_{i_n}$ for which $|a_{i_{1},\\dots,i_{n}}|_{v}\\rightarrow0$ as $i_{1}+\\cdots+i_{n}\\rightarrow\\infty$. The Tate algebra comes with the {\\em Gauss\\ norm} $\\|-\\|_{\\mathrm{G}}:K\\langle T_{1},\\dots,T_{n}\\rangle\\longrightarrow\\mathbb{R}_{\\ge0}$, which takes $f\\mapsto\\|f\\|_{\\mathrm{G}}\\overset{{}_{\\mathrm{def}}}{=}\\mathrm{max}|a_{i_{1},\\dots,i_{n}}|_{v}$.\nThe Gauss norm induces a {\\em quotient norm} $\\|-\\|_{\\mathrm{q}}$ on any quotient $\\mathrm{pr}:\\xymatrix{K\\langle T_{1},\\dots,T_{n}\\rangle\\ar@{->>}[r]&K\\langle T_{1},\\dots,T_{n}\\rangle\/\\mathfrak{a}}$, defined according to\n$$\n\\|f\\|_{\\mathrm{q}}\n\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\\mathrm{inf}\\big\\{\\|g\\|_{\\mathrm{G}}\\ \\!:\\ \\!g\\in\\mathrm{pr}^{-1}(f)\\big\\}.\n$$\nThis quotient norm $\\|-\\|_{\\mathrm{q}}$ gives $K\\langle T_{1},\\dots,T_{n}\\rangle\/\\mathfrak{a}$ the structure of a commutative $K$-Bannach algebra. A $K$-{\\em affinoid algebra} is any commutative Banach algebra $A$ admitting at least one isomorphism of $K$-Banach algebras $A\\cong K\\langle T_{1},\\dots,T_{n}\\rangle\/\\mathfrak{a}$. We use ``$\\|-\\|_{\\mathrm{q}}$\" to denote the norm on an arbitrary affinoid algebra $A$, even when there is no explicit mention of the presentation of $A$ that gives rise to $\\|-\\|_{\\mathrm{q}}$.\n\n\\vskip .3cm\n\n\n\\subsection{Adic spectra and adic spaces}\n\tFor an arbitrary totally ordered abelian group $\\Gamma'$, written multiplicatively, let $\\{0\\}\\sqcup\\Gamma'$ denote the totally ordered abelian semigroup where $0<\\gamma$ for all $\\gamma\\in\\Gamma'$, and where $0\\cdot\\gamma=0$ for all $\\gamma\\in\\{0\\}\\sqcup\\Gamma'$. A {\\em continuous seminorm} $\\|-\\|_{x}:A\\longrightarrow\\{0\\}\\sqcup\\Gamma'$ is any homomorphism of multiplicative semigroups satisfying\n$$\n\\|0\\|_{x}=0,\\ \\ \\ \\ \\ \\ \\|1\\|_{x}=1,\\ \\ \\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ \\ \\ \\|a+b\\|_{x}\\leq\\mathrm{max}(\\|a\\|_{x},\\|b\\|_{x}),\n$$\nsuch that the set\n\t$\n\t\\{a\\in A:\\|a\\|_{x}<\\gamma\\}\n\t$\nis open for each $\\gamma\\in\\Gamma'$. Two continuous seminorms\n\t$$\n\t\\|-\\|_{x}:A\\longrightarrow\\{0\\}\\sqcup\\Gamma_{1}\n\t\\ \\ \\ \\ \\ \\ \\ \\mathrm{and}\\ \\ \\ \\ \\ \\ \\ \n\t\\|-\\|_{y}:A\\longrightarrow\\{0\\}\\sqcup\\Gamma_{2}\n\t$$\nare {\\em equivalent} if there exists an inclusion $\\alpha:\\{0\\}\\sqcup\\Gamma_{1}\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  \\{0\\}\\sqcup\\Gamma_{2}$ of ordered abelian semigroups such that $\\alpha{}_{{}^{\\ \\!\\circ}}\\|-\\|_{x}=\\|-\\|_{y}$.\n\nLet $A$ be a $K$-affinoid algebra. The norm $\\|-\\|_{\\mathrm{q}}$ on $A$ determines the {\\em power bounded subring} $A^{\\circ}\\overset{{}_{\\mathrm{def}}}{=}\\big\\{f\\in A:\\|f\\|_{\\mathrm{q}}\\leq1\\big\\}$. The {\\em adic spectrum} of the pair $(A,A^{\\circ})$, denoted $\\mathrm{Spa}(A,A^{\\circ})$, is the set of all equivalence classes of continuous seminorms $\\|-\\|_{x}:A\\longrightarrow\\{0\\}\\sqcup\\Gamma$ such that $\\|a\\|_{x}\\leq1$ for all $a\\in A^{\\circ}$. The topology on $\\mathrm{Spa}(A,A^{\\circ})$ is the topology generated by its {\\em rational subsets}, that is, the coarsest topology containing the images of all inclusions\n\t$$\n\t\\xymatrix{\\mathrm{Spa}\\Big(A\\big\\langle\\!\\frac{f_{1},\\dots,f_{m}}{g}\\!\\big\\rangle,\\ A\\big\\langle\\!\\frac{f_{1},\\dots,f_{m}}{g}\\!\\big\\rangle^{\\!\\circ\\ }\\Big)\\ \\ \n\\ar@{^{(}->}[r]\n&\n\\ \\ \\mathrm{Spa}(A,A^{\\circ})},\n\t$$\nwhere $A\\Big\\langle\\!\\frac{f_{1},\\dots,f_{m}}{g}\\!\\Big\\rangle$ is any rational $K$-algebra determined by a set of elements $\\{f_{1},\\dots,f_{m},g\\}\\subset A$ generating the unit ideal in $A$ (see \\cite[\\S2.2.2.(ii)]{Berk} and \\cite[\\S8.1]{Wed}).\n\nIf $A$ is an affinoid algebra, with adic spectrum $X=\\mathrm{Spa}(A,A^{\\circ})$, and if $U$ is a rational subset of $X$, let $\\mathscr{O}_{X}(U)$ denote a choice of rational affinoid algebra associated to $U$ as above.\nThese rational algebras induce a {\\em structure presheaf} $\\mathscr{O}_{X}$ on the adic spectrum $X$, which returns, at each open subset $U'\\subset X$, the topological ring $\\mathscr{O}_{X}(U')$ given by the inverse limit\n$$\n\\mathscr{O}_{X}(U')\n\\ \\ =\\ \\ \n\\varprojlim_{U\\subset U'} \\mathscr{O}_{X}(U)\n$$\ntaken over all rational subsets $U$ of $U'$ in $X$. The theorem \\cite[Theorem 2.2]{HubGen} implies that the resulting structure presheaf $\\mathscr{O}_{\\!X}$ is in fact a sheaf. Furthermore, we can form the stalk $\\mathscr{O}_{X,x}$ at each point $x$ in $\\mathrm{Spa}(A,A^{\\circ})$. This stalk is a local topological ring, and the seminorm $\\|-\\|_{x}$ induces a continuous seminorm on $\\mathscr{O}_{X,x}$ \\cite[Proposition 1.6]{HubGen}.\n\n\\subsubsection{{\\bf Adic spaces.}}\n\tLet $\\bold{LRS}$ donote the category of locally ringed spaces. By a {\\em locally topologically ringed space}, we mean a locally ringed space $(Y,\\mathscr{O}_{Y})$ such that $\\mathscr{O}_{Y}$ comes with the structure of a sheaf of topological rings. Let $\\bold{LTRS}$ denote the category of locally topologically ringed spaces.\n\n\tAn {\\em adic space} over $K$ is a locally topologically ringed space $(X,\\mathscr{O}_{X})$ that comes equipped with a continuous seminorm $\\|-\\|_{x}$ on the stalk $\\mathscr{O}_{X,x}$ at each point $x\\in X$, and which admits an atlas $\\{U_{i}\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  X\\}$ in $\\bold{LTRS}$, whose charts $U_{i}$ are adic spectra of $K$-affinoid algebras, such that the seminorm $\\|-\\|_{x}$ on each stalk $\\mathscr{O}_{X,x}=\\mathscr{O}_{U_{i},x}$ coincides with the seminorm described in the previous paragraph. Note that this means, in particular, that the structure presheaf $\\mathscr{O}_{X}$ on any adic space must be a sheaf. A {\\em morphism} of adic spaces is any morphism $\\varphi:(X,\\mathscr{O}_{X})\\longrightarrow(Y,\\mathscr{O}_{Y})$ of locally topologically ringed spaces such that, at each point $x\\in X$, the induced morphism\n$$\n\\varphi^{\\#}:\\mathscr{O}_{Y,\\varphi(x)}\\longrightarrow\\mathscr{O}_{X,x}\n$$\non stalks satisfies $\\|-\\|_{x}{}_{{}^{\\ \\!\\circ}}\\varphi^{\\#}=\\|-\\|_{\\varphi(x)}$. We let $\\bold{Adic}_{K}$ denote the category of adic spaces over $K$.\n\n\\vskip .3cm\n\n\\subsubsection{{\\bf The sheaf of power bounded sections.}} \nEvery adic space $(X,\\mathscr{O}_{X})$ comes with a second sheaf of topological rings, called the {\\em sheaf of power bounded sections}. This is the subsheaf $\\mathscr{O}^{\\ \\!\\circ}_{\\!X}\\subset\\mathscr{O}_{X}$ that, on each open subset $U\\subset X$, returns those sections $f\\in\\mathscr{O}_{X}(U)$ for which $\\|f\\|_{x}\\leq1$ at every point $x\\in U$.\n\n\tThe sheaf of power bounded sections is closely related to the theory of degenerations of $K$-varieties to the special fiber in $\\mathrm{Spec}_{\\ \\!}R$, and it will play an important role in our formulation of the statement that ``Huber analytification is the limit of all exploded tropicalizations\" (see Theorem \\ref{Huberlimit} below).\n\n\n\\vskip .75cm\n\n\n\\subsection{Huber analytification}\nFix a separated scheme $X$ of finite type over $K$. It is a space defined locally by the vanishing of algebraic functions, and has no analytic structure. The Berkovich analytification $\\an{X}$ provides one way to realize an analytic structure on $X$. There is also an adic space $\\ad{X}$ that we can associate to $X$, called the Huber analytification of $X$. Intuitively, the Huber analytification of $X$ is the pullback of $X$ to the category of adic spaces over $K$ along the morphism $\\mathrm{Spa}(K,R)\\longrightarrow\\Spec{K}$ of locally ringed spaces.\n\nTo make this precise, observe that there is a forgetful functor $\\mathrm{U}:\\bold{Adic}_{K}\\longrightarrow\\bold{LRS}$ that takes a given adic space $Y$ and forgets both the seminorms on its stalks $\\mathscr{O}_{Y,y}$ and the topological structure on the sheaf $\\mathscr{O}_{Y}$. Let $\\bold{LRS}_{K}$ denote the category of locally ringed spaces over $\\Spec{K}$. There is a canonical isomorphism\n\t\\begin{equation}\\label{equation: forgetful functor}\n\t\\mathrm{U}\\big(\\mathrm{Spa}(K,R)\\big)\\xrightarrow{\\ \\sim\\ }\\Spec{K}\n\t\\end{equation}\nof locally ringed spaces, which lets us interpret the forgetful functor $\\mathrm{U}$ as a functor of the form\n$$\n\\mathrm{U}:\\bold{Adic}_{K}\\longrightarrow\\bold{LRS}_{K}.\n$$\n\n\\vskip .3cm\n\n\\begin{definition}\\label{adification}\n{\\bf (Huber analytification)}. If $X$ is a separated scheme of finite type over $K$, then its {\\em Huber analytification}, denoted $\\ad{X}$, is any adic space over $\\mathrm{Spa}(K,R)$ that comes equipped with a morphism $\\mathrm{U}(\\ad{X})\\longrightarrow X$ of locally ringed spaces over $\\Spec{K}$ satisfying the following universal property:\n\n\\begin{itemize}\n\\item[{\\bf (H)}]\nIf $Y$ is any adic space over $K$ that comes equipped with a map\n$\\varphi:\\mathrm{U}(Y)\\longrightarrow X$ of locally ringed spaces over $\\Spec{K}$, then there exists a unique morphism $\\widetilde{\\varphi}:Y\\longrightarrow\\ad{X}$ of adic spaces over $K$ for which the diagram\n$$\n\\ \\ \\ \\ \\ \\ \n\\begin{xy}\n(0,0)*+{X}=\"1\";\n(-17,10)*+{\\mathrm{U}(Y)}=\"2\";\n(-17,-10)*+{\\mathrm{U}(\\ad{X})}=\"3\";\n{\\ar@{->}^{\\varphi} \"2\"; \"1\"};\n{\\ar@{->} \"3\"; \"1\"};\n{\\ar@{-->}_{\\mathrm{U}({\\widetilde{\\varphi}})\\ } \"2\"; \"3\"};\n\\end{xy}\n$$\ncommutes.\n\\end{itemize}\n\\end{definition}\n\n\\vskip .2cm\n\n\\begin{remark}\nThe universal property {\\bf (H)} gives the Huber analytification the structure of a functor\n\t$$\n\t\\ad{(-)}:\\bold{Sch}_{K}\\longrightarrow \\bold{Adic}_{K}.\n\t$$\n\t\n\tFor an explicit construction of the adic space $\\ad{X}$ as a fiber product of locally ringed spaces, see \\cite[\\S 8.7 and Definition 8.63]{Wed}. Most relevant to our purposes is the alternate description of $\\ad{X}$ as an inverse limit over admissible formal models of $\\an{X}$, as established by M. van der Put and P. Schneider in \\cite{Points}. To this end, we briefly review the theory of admissible formal models of $X$.\n\\end{remark}\n\n\\vskip .3cm\n\n\\subsubsection{{\\bf Admissible formal models.}}\nRecall that $\\mathfrak{m}$ denotes the maximal ideal in $R$, and that $k=R\/\\mathfrak{m}$ denotes our residue field. The theory of admissible formal models provides us with a tool for degenerating adic spaces over $K$ to algebraic varieties over $k$. These degenerations are realized as certain formal $R$-schemes. Not all formal $R$-schemes will do. For instance, the category of formal $R$-schemes includes $k$-schemes as a full subcategory. A $k$-scheme is an example of a formal $R$-scheme that contains no information whatsoever over the generic point of $\\mathrm{Spec}_{\\ \\!}R$. We want to excise these from our category of interest.\n\nWe say that a topological $R$-algebra $A$ is {\\em admissible} if there exists an isomorphism\n$$\nA\n\\ \\ \\cong\\ \\ \nR\\langle t_{1},\\dots,t_{n}\\rangle\\big\/\\mathfrak{a}\n$$\nof topological rings, where $R\\langle t_{1},\\dots,t_{n}\\rangle\\big\/\\mathfrak{a}$ has its $\\mathfrak{m}$-adic topology, such that:\n\\vskip .2cm\n\\begin{itemize}\n\\item[{\\bf (i)}]\nthe ideal $\\mathfrak{a}\\subset R\\langle t_{1},\\dots,t_{n}\\rangle$ is finitely generated;\n\\item[{\\bf (ii)}]\n\\vskip .2cm\nthe ring $R\\langle t_{1},\\dots,t_{n}\\rangle\\big\/\\mathfrak{a}$ is free of $\\mathfrak{m}$-torsion.\n\\end{itemize}\n\\vskip .2cm\nAn {\\em admissible formal $R$-scheme} is any formal scheme $\\mathfrak{X}$ over the formal spectrum $\\mathrm{Spf}_{\\ \\!\\!}R$ of $R$ (see \\cite[Chp. II, \\S 9]{Hart} for details) that admits an open covering\n$\\big\\{\\mathfrak{U}_{i}\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\mathfrak{X}\\big\\}$ by formal spectra $\\mathfrak{U}_{i}=\\mathrm{Spf}A_{i}$, such that each $A_{i}$ is an admissible $R$-algebra.\n\nEach admissible formal $R$-scheme $\\mathfrak{X}$ has an associated adic space $(\\ad{\\mathfrak{X}},\\mathscr{O}_{\\ad{\\mathfrak{X}}})$.\nTo construct $\\ad{\\mathfrak{X}}$, choose a formal affine cover\n$\\big\\{\\mathfrak{U}_{i}\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  \\mathfrak{X}\\big\\}$,\nsay with $\\mathfrak{U}_{i}=\\mathrm{Spf}\\ \\!A_{i}$, where $A_{i}$ is an admissible $R$-algbera. Then $K\\tensor_{R}A_{i}$ is a $K$-affinoid algebra, and we can define\n$$\n\\ad{\\mathfrak{U}}_{i}\n\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\\mathrm{Spa}\\big(K\\tensor_{R}A_{i}\\ \\!,\\ (K\\tensor_{R}A_{i})^{\\circ}\\big).\n$$\nThese adic spaces glue along the intersections\n$\\ad{(\\mathfrak{U}_{i}\\cap\\mathfrak{U}_{j})}=\\ad{\\mathfrak{U}}_{i}\\cap\\ad{\\mathfrak{U}}_{j}$ to produce an adic space over $K$ that we denote $\\ad{\\mathfrak{X}}$. The adic space that we obtain in this way is independent of our choice of covering $\\big\\{\\mathfrak{U}_{i}\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  \\mathfrak{X}\\big\\}$ (see \\cite[Proposition 4.1]{HubGen} for details).\n\n\\vskip .3cm\n\n\\begin{definition}\\label{admissible model of a scheme}\n{\\bf (Admissible formal models of a scheme).}\nLet $X$ be any scheme locally of finite type over $K$. An {\\em admissible formal model of} $X$ is a datum consisting of an admissible formal $R$-scheme $\\mathfrak{X}$ and an isomorphism $\\ad{\\mathfrak{X}}\\xrightarrow{\\ \\sim\\ }\\ad{X}$ of adic spaces.\n\nA {\\em morphism} of admissible formal models of $X$ is a morphism $\\mathfrak{X}_{1}\\longrightarrow\\mathfrak{X}_{2}$ whose induced morphism $\\mathfrak{X}_{1}^{\\mathrm{ad}}\\longrightarrow\\mathfrak{X}_{2}^{\\mathrm{ad}}$ of $K$-analytic spaces commutes with the isomorphisms $\\mathfrak{X}_{1}^{\\mathrm{ad}}\\xrightarrow{\\ \\sim\\ }\\ad{X}$ and $\\mathfrak{X}_{2}^{\\mathrm{ad}}\\xrightarrow{\\ \\sim\\ }\\ad{X}$. We let $\\bold{AFS}_{R}$ denote the category of admissible formal $R$-schemes, and we let $\\bold{AFS}_{R}^{X}$ denote the category of admissible formal models of $X$.\n\\end{definition}\n\n\\vskip .3cm\n\n\\subsubsection{{\\bf Specializations of adic spaces.}}\\label{specializations of adic spaces}\n\tThe adic space $\\ad{\\mathfrak{X}}$ associated to the formal $R$-scheme\n$\\mathfrak{X}$ comes with a {\\em specialization morphism}\n\\begin{equation}\\label{formal specialization}\n\\mathrm{sp}_{\\mathfrak{X}}:(\\ad{\\mathfrak{X}},\\mathscr{O}^{\\ \\!\\circ}_{\\!\\ad{\\mathfrak{X}}})\\longrightarrow(\\mathfrak{X},\\mathscr{O}_{\\mathfrak{X}})\n\\end{equation}\nof locally topologically ringed spaces over $\\mathrm{Spec}_{\\ \\!}K$. Recall that $\\mathscr{O}^{\\ \\!\\circ}_{\\!\\ad{\\mathfrak{X}}}$ is the sheaf of power bounded sections on $\\ad{\\mathfrak{X}}$. The fact that $\\mathrm{sp}_{\\mathfrak{X}}$ is a morphism of locally ringed spaces is one feature of adic spaces that Berkovich spaces do {\\em not} satisfy, as the reduction map $X^{\\mathrm{an}}\\longrightarrow\\mathfrak{X}_{k}$ for Berkovich spaces fails to be continuous.\n\n\tThe specialization morphism (\\ref{formal specialization}) satisfies the following universal property:\n\\vskip .3cm\n\\begin{itemize}\n\\item[{\\bf (H${}_{\\bold{f}}$)}]\nIf $Y$ is any adic space over $K$ that comes equipped with a morphism $\\varphi:(Y,\\mathscr{O}^{\\ \\!\\!\\circ}_{\\!Y})\\longrightarrow(\\mathfrak{X},\\mathscr{O}_{\\mathfrak{X}})$ of locally topologically ringed spaces over $\\mathrm{Spec}_{\\ \\!}K$, then there exists a unique morphism $\\widetilde{\\varphi}:(Y,\\mathscr{O}_{Y})\\longrightarrow(\\ad{\\mathfrak{X}},\\mathscr{O}_{\\ad{\\mathfrak{X}}})$ of adic spaces over $K$ for which the diagram\n$$\n\\ \\ \\ \\ \\ \\ \n\\begin{xy}\n(0,0)*+{(\\mathfrak{X},\\mathscr{O}_{\\mathfrak{X}})}=\"1\";\n(-25,10)*+{(Y,\\mathscr{O}^{\\ \\!\\!\\circ}_{\\!Y})}=\"2\";\n(-25,-10)*+{(\\ad{\\mathfrak{X}},\\mathscr{O}^{\\ \\!\\!\\circ}_{\\!\\ad{\\mathfrak{X}}})}=\"3\";\n{\\ar@{->}^{\\varphi} \"2\"; \"1\"};\n{\\ar@{->}_{\\mathrm{sp}_{\\mathfrak{X}}} \"3\"; \"1\"};\n{\\ar@{-->}_{\\widetilde{\\varphi}\\ } \"2\"; \"3\"};\n\\end{xy}\n$$\ncommutes.\n\\end{itemize}\n\\vskip .3cm\n\n\\begin{remark}\\label{remark: Gillam}\nThe universal property {\\bf (H${}_{\\bold{f}}$)} gives us a functor\n$\\ad{(-)}:\\bold{AFS}_{R}\\longrightarrow\\bold{Adic}_{K}$\ninto the category of adic spaces over $K$. If $X$ is a separated finite-type $K$-scheme, then the resulting specialization morphisms\n\\begin{equation}\\label{adic specialization}\n\\mathrm{sp}_{\\mathfrak{X}}:(\\ad{X},\\mathscr{O}^{\\ \\!\\circ}_{\\!\\ad{X}})\\longrightarrow(\\mathfrak{X},\\mathscr{O}_{\\mathfrak{X}}),\n\\end{equation}\ninduced by the specialization morphisms (\\ref{formal specialization}), commute with morphisms in $\\bold{AFS}_{R}^{X}$.\n\n\tW. D. Gillam proved in \\cite[Corollary 5]{Gillam} that $\\bold{LRS}$ contains all inverse limits. It is not difficult to show that one can compute any inverse limit $\\varprojlim_{}(Y_{i},\\mathscr{O}_{Y_{i}})$ in $\\bold{LTRS}$ by first computing the inverse limit $(Y,\\mathscr{O}_{Y})$ in $\\bold{LRS}$ of the underlying locally ringed spaces, and then equipping each ring $\\mathscr{O}_{Y}(U)$ with the finest topology for which each of the morphisms \n\t\\begin{equation}\\label{individual restrictions}\n\t\\mathscr{O}_{Y_{i}}(V_{i})\\longrightarrow\\mathscr{O}_{X}(U),\n\t\\end{equation}\nfor each open subset $V_{i}\\subset Y_{i}$ mapping to $U$ under $Y_{i}\\longrightarrow Y$,\nbecomes continuous.\n\nThis means, in particular, that the inverse limit of an inverse system of formal schemes remains a topologically ringed space on which stalks are local rings. Thus the morphisms (\\ref{adic specialization}) induce a morphism\n\\begin{equation}\\label{specialization}\n\\mathrm{sp}:\\ \\!(\\ad{X},\\mathscr{O}^{\\ \\!\\!\\circ}_{\\!\\ad{X}})\\ \\xrightarrow{\\ \\ \\ \\ \\ }\\varprojlim_{\\bold{AFS}_{R}^{X}}\\!(\\mathfrak{X},\\mathscr{O}_{\\mathfrak{X}})\n\\end{equation}\nof locally topologically ringed spaces.\n\\end{remark}\n\n\\vskip .3cm\n\n\\begin{proposition}\\label{limit}\nThe morphism (\\ref{specialization}) is an isomorphism of locally topologically ringed spaces.\n\\end{proposition}\n\\begin{proof}\nFor a sketch of the proof of Proposition \\ref{limit}, see \\cite[Theorem 2.22]{Sch}.\n\\end{proof}\n\n\\vskip .75cm\n\n\n\\section{The exploded tropicalization as a locally ringed space}\\label{Section-adic trop}\n\n\n\\subsection{The combinatorics of toric degenerations}\\label{subsection: The combinatorics of toric degenerations}\n\tIn the present \\S\\ref{subsection: The combinatorics of toric degenerations}, we briefly review the combinatorial and algebraic geometry of toric degenerations of toric varieties. We work exclusively over the toric $k$-variety $\\mathbb{A}^{1}_{k}$, where $k=R\/\\mathfrak{m}$ is our original residue field. The reader already familiar with the geometry of toric degenerations over $\\mathbb{A}^{1}_{k}$, as developed by G. Kempf, F. F. Knudsen, D. Mumford, and B. Saint-Donat in \\cite{KKMSD}, may proceed to \\S\\ref{subsection: algebraic Gubler models}, where we review the formalism of toric $R$-schemes introduced by W. Gubler in \\cite{Gub}. The theory of toric $R$-schemes is further developed by W. Gubler and A. Soto in \\cite{GS}.\n\n\tFix a split, $n$-dimensional algebraic torus $\\mathbb{T}$ over $k$, with character lattice $M$. We have a canonical isomorphism $\\mathbb{T}\\cong\\mathrm{Spec}_{\\ \\!}k[M]$. Let $N\\overset{{}_{\\mathrm{def}}}{=}\\mathrm{Hom}_{\\mathbb{Z}}(M,\\mathbb{Z})$ be the cocharacter lattice, and let\n\t$$\n\t\\langle-,-\\rangle\\ \\!:\\ M\\times N_{\\mathbb{R}}\\ \\longrightarrow\\ \\mathbb{R}\n\t$$\ndenote the canonical paring.\n\t\n\tIf $Y_{\\Sigma}$ is a proper toric variety over $K$, built from a complete $n$-dimensional fan $\\Sigma$ in $N_{\\mathbb{R}}$, then we can freely produce degenerations of $Y_{\\Sigma}$ by writing down certain $(n+1)$-dimensional fans extending $\\Sigma$. Specifically, consider the $(n-1)$-dimensional half-space $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, along with its projection to the second factor\n\t\\begin{equation}\\label{inset: projection to second factor}\n\t\\mathrm{pr}_{2}\\ \\!:\\ N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}\\ \\longrightarrow\\!\\!\\!\\!\\rightarrow\\ \\mathbb{R}_{\\ge0}.\n\t\\end{equation}\nIf $\\Delta$ is any fan in $N_{\\mathbb{R}}\\times\\mathbb{R}$ with support $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, such that\n\t$$\n\t\\Delta\\ \\cap\\ \\big(\\ \\!N_{\\mathbb{R}}\\times\\{0\\}\\ \\!\\big)\n\t\\ \\ \\cong\\ \\ \n\t\\Sigma,\n\t$$\nthen the projection (\\ref{inset: projection to second factor}) induces a morphism of fans\n\t\\begin{equation}\\label{inset: morphisms of fans}\n\t\\mathrm{pr}_{2}\\ \\!:\\ \\Delta\\ \\longrightarrow\\!\\!\\!\\!\\rightarrow\\ \\big\\{0,\\mathbb{R}_{\\ge0}\\big\\}\n\t\\ \\ \\ \\ \\ \\ \\ \\ \\mbox{with}\\ \\ \\ \\ \\ \\ \\ \\ \n\t\\mathrm{pr}_{2}^{-1}(0)\\ =\\ \\Sigma.\n\t\\end{equation}\nThis induces a torus-equivariant morphism of toric varieties\n\t\\begin{equation}\\label{toric fibration}\n\tf_{\\mathrm{pr}_{2}}\\ \\!:\\ Y_{\\Delta}\\ \\longrightarrow\\!\\!\\!\\!\\rightarrow\\ \\mathbb{A}^{1}=\\mathrm{Spec}_{\\ \\!}k[t].\n\t\\end{equation}\nSee Figure \\ref{toric degeneration} below for an example. Note that the fiber $\\mathrm{pr}^{-1}_{2}(1)=\\Delta\\cap\\big(N_{\\mathbb{R}}\\times\\{1\\}\\big)$ is not a fan but a polyhedral complex. We identify it with a polyhedral complex $C_{\\Delta}$ in $N_{\\mathbb{R}}$ whose support is all of $N_{\\mathbb{R}}$. See Figure \\ref{tropcone} below for an example.\n\n\\vskip .3cm\n\t\n\\begin{proposition}\\label{proposition: how toric degenerations work}\n\tThe map $f_{\\mathrm{pr}_{2}}:Y_{\\Delta}\\longrightarrow\\!\\!\\!\\!\\rightarrow\\mathbb{A}^{1}$ induced by the projection (\\ref{inset: morphisms of fans}) realizes the complex toric variety $Y_{\\Delta}$ as an $\\mathbb{A}^{1}$-model of the toric variety glued from $\\Sigma$. More precisely:\n\\begin{itemize}\n\t\\item[{\\bf (i)}]\\vskip .25cm\n\tThe generic fiber of $f_{\\mathrm{pr}_{2}}$ is the toric $k(t)$-variety $Y_{\\Sigma}$.\n\t\\item[{\\bf (ii)}]\\vskip .25cm\n\tThe special fiber of $f_{\\mathrm{pr}_{2}}$ at $t=0$ is a union of toric varieties $Y_{\\mathrm{star}(v)}$ indexed by the vertices of $v$ of $C_{\\Delta}$, glued along torus-invariant divisors according to the higher-dimensional face incidences appearing in $C_{\\Delta}$.\n\\end{itemize}\n\\vskip .25cm\n\\end{proposition}\n\\begin{proof}\nThe complex toric variety $Y_{\\Delta}$ is glued from the affine open toric varieties $U_{\\delta}\\overset{{}_{\\mathrm{def}}}{=}\\mathrm{Spec}_{\\ \\!}k[S_{\\delta}]$, where $\\delta$ runs over all cones in $\\Delta$. Because $\\Delta$ is a fan in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, the semigroup $S_{\\delta}$ associated to $\\delta$ takes the form\n\t\\begin{equation}\\label{inset: suggestive description}\n\tS_{\\delta}\n\t\\ \\overset{\\mathrm{def}}{=}\\ \n\t\\big\\{(u,n)\\in M\\!\\times\\!\\mathbb{Z}:\\langle u,v\\rangle+rn\\ge0\\ \\mbox{for all}\\ (v,r)\\in\\delta\\big\\}.\n\t\\end{equation}\nFix a monomial $a\\ \\!t^{n}\\chi^{u}\\in k[S_{\\delta}]$, where $a\\in k$ and $u\\in M$. If $(u,0)\\notin S_{\\delta\\cap(N_{\\mathbb{R}}\\times\\{0\\})}$, then there exists some $(v',0)\\in\\delta\\cap\\big(N_{\\mathbb{R}}\\times\\{0\\}\\big)$ such that $\\langle u,v'\\rangle<0$. Thus, for some choice of $r\\gg0$, we have $\\langle u,v+rv'\\rangle<0$. Because $\\delta$ is closed under translation by $\\delta\\cap\\big(N_{\\mathbb{R}}\\times\\{0\\}\\big)$, this implies that $u\\in S_{\\delta\\cap(N_{\\mathbb{R}}\\times\\{0\\})}$. This proves part (i) of the proposition.\n\t\n\tPart (ii) is a consequence of the orbit-cone correspondence for toric varieties \\cite[\\S3.2]{CLS}, upon observing that the special divisors in $Y_{\\Delta}$ are exactly those codimension-$1$ orbit closures in $Y_{\\Delta}$ that correspond to rays in $\\Delta$ {\\em not} lying in $N_{\\mathbb{R}}\\times\\{0\\}$.\n\\end{proof}\n\n\\vskip .3cm\n\n\\begin{example}\\label{example: toric degeneration}\n\tLet $M=\\mathbb{Z}^{2}$, and let $\\Sigma$ be the complete, $2$-dimensional fan in $N_{\\mathbb{R}}=\\mathbb{R}^{2}$ pictured at left in Figure \\ref{tropcone} below. Then $Y_{\\Sigma}=\\mathbb{P}^{2}_{\\mathbb{C}}$. Let $\\Delta$ be the $3$-dimensional, $\\mathbb{Z}$-admissible fan in $\\mathbb{R}^{2}\\times\\mathbb{R}_{\\ge0}$ pictured in Figure \\ref{toric degeneration} below.\n\t\\begin{figure}[!htb]\n\t$$\n\t\\begin{xy}\n\t(0,0)*+{\\ \\ \\ \\ \\ \\includegraphics[scale=.3]{PIC_002_001.pdf}\\ \\ \\ \\ \\ }=\"1\";\n\t(60,0)*+{\\ \\ \\ \\ \\ \\includegraphics[scale=.4]{PIC_002_000.pdf}\\ \\ \\ \\ \\ }=\"2\";\n\t(-24,0)*+{\\Delta};\n\t(1,-14)*+{\\mbox{{\\smaller $0$}}};\n\t(62,-16)*+{\\mbox{{\\smaller $0$}}};\n\t(69,0)*+{\\big\\{0,\\mathbb{R}_{\\ge0}\\big\\}};\n\t{\\ar@{->>}^{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathrm{pr}_{2}} \"1\"; \"2\"};\n\t\\end{xy}\n\t$$\n\t\\caption{A map of fans $\\mathrm{pr}_{2}:\\Delta\\longrightarrow\\!\\!\\!\\!\\rightarrow\\big\\{0,\\mathbb{R}_{\\ge0}\\big\\}$ inducing toric morphism of toric varieties $f_{\\mathrm{pr}_{2}}:Y_{\\Delta}\\longrightarrow\\!\\!\\!\\!\\rightarrow\\mathbb{A}^{1}$.}\n\t\\label{toric degeneration}\n\t\\end{figure}\nBecause $\\Delta$ satisfies $\\Delta\\cap\\big(\\mathbb{R}^{2}\\times\\{0\\}\\big)=\\Sigma$, it produces a complex toric variety $Y_{\\Sigma}$ that comes equipped with a torus-equivariant map $Y_{\\Sigma}\\longrightarrow\\!\\!\\!\\!\\rightarrow\\mathbb{A}^{1}$. The generic fiber of the family is $\\mathbb{P}^{2}$ over $k(t)$. The special fiber over $t=0$ is a union of four complex surfaces: a copy of $\\mathbb{P}^{2}$, a copy of $\\mathbb{P}^{1}\\times\\mathbb{P}^{1}$, and two Hirzebruch surfaces.\n\\begin{figure}[!htb]\n\t$$\n\t\\underset{\\mbox{The fan $\\Sigma$}}{\\includegraphics[scale=.325]{PIC_002_002.pdf}}\n\t\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n\t\\underset{\\mbox{The complex $\\Delta\\cap\\big(\\mathbb{R}^{2}\\times\\{1\\}\\big)$}}{\\includegraphics[scale=.325]{PIC_002_003.pdf}}\n\t$$\n\t\\caption{The fan $\\Sigma\\cong\\Delta\\cap\\big(\\mathbb{R}^{2}\\times\\{0\\}\\big)$ and the polyhedral complex $\\Delta\\cap\\big(\\mathbb{R}^{2}\\times\\{1\\}\\big)$.}\n\t\\label{tropcone}\n\t\\end{figure}\nThese four surfaces are in one-to-one correspondence with the vertices of  the polyhedral decomposition $\\Delta\\cap\\big(\\mathbb{R}^{2}\\times\\{1\\}\\big)$ pictured at left in Figure \\ref{tropcone}. More precisely, they are  the toric varieties described by the stars at each vertex, and they are glued to one another along the torus-invariant divisors corresponding to the edges connecting their corresponding vertices in $C_{\\Delta}$.\n\\end{example}\n\t\n\\vskip .3cm\n\n\n\\subsection{Algebraic Gubler models}\\label{subsection: algebraic Gubler models}\n\tThe fundamental observation underlying Gubler's approach to integral toric geometry, already anticipated by Kempf, Knudsen, Mumford, and Saint-Donat in \\cite{KKMSD}, is that a significant part of the geometry of toric degenerations discussed in the previous \\S\\ref{subsection: The combinatorics of toric degenerations} works just as well upon replacing the curve $\\mathbb{A}^{1}=\\mathrm{Spec}_{\\ \\!}k[t]$ with the spectrum of the ring of integers $R$ in any non-Archimedean field $K$. This requires that we replace the factor $\\mathbb{Z}$ in $M\\times\\mathbb{Z}$ with the value group $\\Gamma=\\mathrm{val}(K^{\\times})\\subset\\mathbb{R}$ of $K$. The fact that we can express each semigroup $S_{\\delta}$ appearing in the proof of Proposition \\ref{proposition: how toric degenerations work} in the form (\\ref{inset: suggestive description}) implies that we can express the corresponding semigroup rings as \n\t\\begin{equation}\\label{evocative expression}\n\tk[S_{\\delta}]\\ \\ =\\ \\ \\Big\\{\\sum_{u\\in M}a_{u}(t)\\chi^{u}\\in k[t,t^{-1}][M]\\ :\\ \\langle u,v\\rangle+r\\ \\!\\mathrm{ord}_{t}\\big(a_{u}(t)\\big)\\ge0\\ \\mbox{for all}\\ (v,r)\\in\\delta\\Big\\},\n\t\\end{equation}\nwhere the coefficients $a_{u}(t)$ are Laurent polynomials in $t$, and where\n\t$$\n\t\\mathrm{ord}_{t}\\ \\!:\\ k(t)^{\\times}\\ \\longrightarrow\\ \\mathbb{Z}\n\t$$\nis the valuation on $k(t)$ given by order-of-vanishing at $t=0$. In Gubler's formalism, one replaces the ring $k[t,t^{-1}]$ in (\\ref{evocative expression}) with a non-Archimedean field $K$. This produces $R$-algebras that glue to give an $R$-model of the toric $K$-variety $Y_{\\Sigma}$. The present \\S\\ref{subsection: algebraic Gubler models} is devoted to a review of Gubler's formalism, with slight upgrades to the setting of formal $R$-schemes. In the next \\S\\ref{subsection: formal Gubler models}, we use the resulting formal $R$-schemes to put a topology and locally ringed structure sheaf on each exploded tropicalization.\n\n\t\\vskip .3cm\n\n\\begin{definition}\nA $\\Gamma$-{\\em admissible cone} is any convex cone\n$\\delta\\subset N_{\\mathbb{R}}\\!\\times\\!\\mathbb{R}_{\\geq0}$ that can be written as an intersection of the form\n$$\n\\delta\n\\ \\ =\\ \\ \n\\bigcap_{i=1}^{n}\\big\\{(v,c)\\in N\\!\\times\\!\\mathbb{R}_{\\geq0}:\\langle u_{i},v\\rangle+\\gamma_{i}c\\geq0\\big\\}\n$$\nfor vectors\n$(u_{1},\\gamma_{1}),\\dots,(u_{n},\\gamma_{n})$ in $M\\!\\times\\!\\Gamma$. A $\\Gamma$-{\\em admissible fan} is any fan $\\Delta$ in $N_{\\mathbb{R}}\\!\\times\\!\\mathbb{R}_{\\geq0}$ whose every cone $\\delta$ is $\\Gamma$-admissible.\n\\end{definition}\n\t\n\t\\vskip .3cm\n\t\n\\begin{definition}\n\tIf $\\delta$ is a $\\Gamma$-admissible cone in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, its associated {\\em $\\delta$-tilted algebra} is the ring\n$$\nR[U_{\\delta}]\n\\ \\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \\ \n\\Big\\{\\ \\!\\sum_{u\\in M}a_{u}\\chi^{u}\\in K[M]\\ :\\langle u,v\\rangle+\\mathrm{val}(a_{u})\\cdot c\\geq0\\ \\mathrm{for\\ all\\ }(v,c)\\in \\delta\\ \\!\\Big\\}.\n$$\n\\end{definition}\n\n\t\\vskip .3cm\n\t\n\tBy \\cite[Proposition 11.3]{Gub}, every $\\delta$-titled algebra $R[U_{\\delta}]$ is in fact a finite-type, flat $R$-algebra. We let $U_{\\delta}$ denote the $R$-scheme $U_{\\delta}=\\Spec{R[U_{\\delta}]}$.\n\t\n\tEach inclusion $\\delta'\\hookrightarrow\\delta$ of $\\Gamma$-admissible cones in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ realizing $\\delta'$ as a face of $\\delta$ induces a functorial, open embedding of $R$-schemes\n\t\\begin{equation}\\label{inset: induced inclusions}\n\tU_{\\delta'}\\ \\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\!\\ U_{\\delta}.\n\t\\end{equation}\nIf $\\Delta$ is a $\\Gamma$-admissible fan in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, then we can glue the affine $R$-schemes $\\{U_{\\delta}\\}_{\\delta\\in\\Delta}$ along these induced open embeddings (\\ref{inset: induced inclusions}) to obtain a single $R$-scheme $Y_{\\Delta}$. The $R$-scheme $Y_{\\Delta}$ is locally of finite type over $R$. Gubler shows \\cite{Gub} that when the support of $\\Delta$ is $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, the scheme $Y_{\\Delta}$ is in fact proper over $\\mathrm{Spec}_{\\ \\!}R$, with generic fiber canonically isomorphic to the toric $K$-variety $Y_{\\Sigma}$ glued from the fan\n\t$$\n\t\\Sigma\\ \\overset{\\mathrm{def}}{=}\\ \\Delta\\ \\cap\\ \\big(\\ \\!N_{\\mathbb{R}}\\times\\{0\\}\\ \\!\\big),\n\t$$\nupon interpreting $\\Sigma$ as a fan in $N_{\\mathbb{R}}$. In short:\n\t\\begin{equation}\\label{inset: non-formal generic fiber}\n\t(Y_{\\Delta})_{K}\\ \\ \\cong\\ \\ Y_{\\Sigma}.\n\t\\end{equation}\nIdentifying $N_{\\mathbb{R}}\\times\\{0\\}$ with $N_{\\mathbb{R}}$, we refer to the intersection $\\Delta\\cap\\big(N_{\\mathbb{R}}\\times\\{0\\}\\big)$ as the {\\em recession fan of $\\Delta$ in $N_{\\mathbb{R}}$}, and denote it $\\mathrm{rec}(\\Delta)$. Just as in Proposition \\ref{proposition: how toric degenerations work}.(ii), the special fiber of $Y_{\\Delta}$ over the unique maximal ideal in $\\mathrm{Spec}_{\\ \\!}R$ is a union of toric $k$-varieties indexed by the vertices in the {\\em height-$1$ polyhedral complex}\n\t\\begin{equation}\\label{inset: height-1, non-Arch}\n\t\\Delta\\ \\cap\\ \\big(\\ \\!N_{\\mathbb{R}}\\times\\{1\\}\\ \\!\\big),\n\t\\end{equation}\nand glued along torus-invariant divisors according to the incidence profile of higher-dimensional polyhedra in this height-$1$ complex.\n\t\n\tWe refer to $Y_{\\Delta}$ as the ({\\em algebraic}) {\\em Gubler model} associated to the $\\Gamma$-admissible fan $\\Delta$.\n\t\n\t\\vskip .3cm\n\t\n\\begin{example}\n\tLet $K=k((t))$, so that $R=k[[t]]$ and $\\Gamma=\\mathbb{Z}$. Then the fan $\\Delta$ from Example \\ref{example: toric degeneration} above, being integral, is $\\Gamma$-admissible. If $\\mathbb{Y}_{\\Delta}$ denotes the toric $k$-variety that we constructed in Example \\ref{example: toric degeneration}, equipped with its torus-equivariant fibration $f_{\\mathrm{pr}_{2}}:\\mathbb{Y}_{\\Delta}\\longrightarrow\\!\\!\\!\\!\\rightarrow\\mathbb{A}^{1}$, then the algebraic Gubler model associated to $\\Delta$ is the fiber\n\t$$\n\tY_{\\Delta}\\ \\ \\cong\\ \\ \\mathrm{Spec}_{\\ \\!}k[[t]]\\underset{\\ \\mathbb{A}^{1}\\!}{\\times}\\mathbb{Y}_{\\Delta}\n\t$$\nover the $k[[t]]$-valued point $\\mathrm{Spec}_{\\ \\!}k[[t]]\\longrightarrow\\mathbb{A}^{1}$ supported at $t=0$.\n\\end{example}\n\t\n\t\\vskip .3cm\n\t\n\\begin{example}\n\tLet $K=\\mathbb{Q}_{p}$, so that $R=\\mathbb{Z}_{p}$ and $\\Gamma=\\mathbb{Z}$. Define $M=\\mathbb{Z}$ and $N=\\mathbb{Z}$, so that the standard pairing is just multiplication. Let $\\Delta$ be the $\\mathbb{Z}$-admissible fan in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}=\\mathbb{R}\\times\\mathbb{R}_{\\ge0}$ pictured at left in Figure \\ref{figure over Z_p} below.\n\t\\vskip -.25cm\n\t\\begin{figure}[!htb]\n\t$$\n\t\\begin{array}{ccc}\n\t\\begin{xy}\n\t(0,0)*+{\\ \\ \\ \\ \\ \\includegraphics[scale=.4]{PIC_003_001.pdf}\\ \\ \\ \\ \\ };\n\t(-25,0)*+{\\Delta};\n\t(0,-13.5)*+{\\mbox{{\\smaller $0$}}};\n\t\\end{xy}\n\t&\n\t\\ \\ \\ \\ \\ \\ \n\t\\mbox{{\\larger\\larger\\larger $\\longmapsto$}}\n\t\\ \\ \n\t&\n\t\\begin{xy}\n\t(-1,0)*+{\\ \\ \\ \\ \\ \\includegraphics[scale=.25]{PIC_003_002.pdf}\\ \\ \\ \\ \\ };\n\t(-9,-10.75)+*{\\mathrm{Spec}_{\\ \\!}\\mathbb{Z}_{p}};\n\t(3,-10.5)+*{(p)};\n\t(-5,3)+*{Y_{\\Delta}};\n\t(4,-3)+*{(Y_{\\Delta})_{{}_{\\mathbb{F}_{p}}}};\n\t\\end{xy}\n\t\\end{array}\n\t$$\n\t\\caption{A $\\mathbb{Z}$-admissible fan in $\\mathbb{R}\\times\\mathbb{R}_{\\ge0}$, and its associated $\\mathbb{Z}_{p}$-model of $\\mathbb{P}^{1}_{\\mathbb{Q}_{p}}$.}\n\t\\label{figure over Z_p}\n\t\\end{figure}\n\t\n\tGoing from left to right, the $2$-dimensional cones in $\\Delta$ have corresponding tilted algebras\n\t$$\n\t\\mathbb{Z}_{p}[t^{-1}],\n\t\\ \\ \\ \\ \\ \\ \\\n\t\\mathbb{Z}_{p}[t,pt^{-1}]\\ \\cong\\ \\mathbb{Z}_{p}[x,y]\\big\/(xy-p),\n\t\\ \\ \\ \\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ \\ \\ \n\t\\mathbb{Z}_{p}[p^{-1}t].\n\t$$\nThus $Y_{\\Delta}$ describes the projective line over $\\mathbb{Q}_{p}$ degenerating to a pair of projective lines over $\\mathbb{F}_{p}$ intersecting at a node.\n\\end{example}\n\t\n\t\\vskip .3cm\n\t\n\\subsection{Formal Gubler models}\\label{subsection: formal Gubler models}\\ \n\tIf $\\Delta$ is a $\\Gamma$-admissible fan in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, whose support is all of $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, then we let $\\mathfrak{Y}_{\\Delta}$ denote the formal $R$-scheme obtained as the completion of $Y_{\\Delta}$ along its special fiber,\n\t$$\n\t\\mathfrak{Y}_{\\Delta}\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\t\\widehat{\\ Y_{\\Delta}}.\n\t$$\nIf, at each cone $\\delta$ in $\\Delta$, we let $R[\\mathfrak{U}_{\\delta}]$ denote the formal completion\n\t$$\n\tR[\\mathfrak{U}_{\\delta}]\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\t\\widehat{R[U_{\\delta}]\\ \\!}\n\t$$\nalong the ideal generated by $\\mathfrak{m}$ in $R[U_{\\delta}]$, then $\\mathfrak{Y}_{\\Delta}$ is glued from the formal spectra\n\t$$\n\t\\mathfrak{U}_{\\delta}\\ =\\ \\mathrm{Spf}_{\\ \\!}R[\\mathfrak{U}_{\\delta}].\n\t$$\nBecause each $\\mathfrak{U}_{\\delta}$ is an admissible formal $R$-scheme, this implies that $\\mathfrak{Y}_{\\Delta}$ is itself an admissible formal $R$-scheme.\n\t\n\tBecause $\\mathfrak{Y}_{\\Delta}$ is a formal completion, the ``generic fiber\" of $\\mathfrak{Y}_{\\Delta}$ is no longer a $K$-scheme as in the previous \\S\\ref{subsection: algebraic Gubler models}. Instead, \\cite[\\S2.4]{Bosch} tells us that in the formal setting, the topological $K$-algebras\n\t$$\n\tK\\underset{R}{\\tensor} R[\\mathfrak{U}_{\\delta}],\n\t\\ \\ \\ \\mbox{for}\\ \\ \\ \n\t\\delta\\ \\ \\mbox{in}\\ \\ \\Delta,\n\t$$\nare $K$-affinoid algebras. Their adic spectra\n\t\\begin{equation}\\label{inset: Berkovich spectra}\n\t\\mathfrak{U}^{\\mathrm{ad}}_{\\delta}\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\t\\mathrm{Spa}\\big(\\ \\!K\\!\\otimes_{R}\\!R[\\mathfrak{U}_{\\delta}],\\ R[\\mathfrak{U}_{\\delta}]\\ \\!\\big)\n\t\\end{equation}\nglue to produce an {\\em adic generic fiber} $\\mathfrak{Y}^{\\mathrm{ad}}_{\\Delta}$ of the formal $R$-scheme $\\mathfrak{Y}_{\\Delta}$. In place of the isomorphism of $K$-varieties (\\ref{inset: non-formal generic fiber}), we have a canonical isomorphism of adic spaces\n\t$$\n\t\\mathfrak{Y}^{\\mathrm{ad}}_{\\Delta}\\ \\ \\cong\\ \\ Y^{\\mathrm{ad}}_{\\Sigma}.\n\t$$\nIn the language of Definition \\ref{admissible model of a scheme}, the formal $R$-scheme $\\mathfrak{Y}_{\\Delta}$ is an admissible formal model of the $K$-scheme $Y_{\\Sigma}$.\n\n\\vskip .3cm\n\n\n\\subsection{Adic tropicalization of a closed embedding}\\label{Adic tropicalization of a closed embedding}\\ \n\n\tFix a proper algebraic variety $X$ over $K$. Even if $X$ fails to be projective, we can still ask for closed embeddings into proper toric $K$-varieties. Let us suppose that $X$ is a proper $K$-variety admiting at least one closed embedding into a proper toric $K$-variety\n\t$$\n\t\\textit{\\i}\\ \\!:\\ X\\ \\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\!\\ Y_{\\Sigma},\n\t$$\nwhere $\\Sigma$ is a complete fan in $N_{\\mathbb{R}}$. Using the machinery of \\S\\ref{subsection: algebraic Gubler models}, we can produce a flat, proper, algebraic $R$-model $Y_{\\Delta}$ of $Y_{\\Sigma}$ by making a choice of $\\Gamma$-admissible fan $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ with support equal to $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$. Our variety $X$ sits in the generic fiber of this model, and Gubler shows \\cite{Gub} that the closure $\\overline{X}$ of $X$ in $Y_{\\Delta}$ is itself a flat, proper $R$-model of $X$. Let\n\t$$\n\t\\mathfrak{X}_{(\\Delta,\\textit{\\i})}\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\t\\widehat{\\ \\overline{X}\\ }\n\t$$\ndenote the formal completion of the algebraic $R$-model $\\overline{X}$ along its special fiber. This formal completion admits a canonical isomorphism\n\t$\n\t\\mathfrak{X}^{\\mathrm{ad}}_{(\\Delta,\\textit{\\i})}\\ \\ \\cong\\ \\ X^{\\mathrm{ad}}.\n\t$\n\n\t\\vskip .3cm\n\t\n\\begin{definition}\t\n\tThe {\\em category of fans recessed over $\\textit{\\i}$}, denoted $\\bold{REC}_{\\Sigma,\\textit{\\i}}$, is the category with:\n\t\\begin{changemargin}{2cm}{0cm} \n\t\\begin{itemize}\n\t\\item[{\\bf\\em objects:}] \\vskip .15cm\n\tAny locally finite $\\Gamma$-admissible fan $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ with recession fan $\\mathrm{rec}(\\Delta)=\\Sigma$;\n\t\\item[{\\bf\\em morphisms:}] \\vskip .15cm\n\tAny pair of locally finite $\\Gamma$-admissible fans $\\Delta'$ and $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ such that each cone $\\delta'$ in $\\Delta'$ is contained in at least one cone $\\delta$ in $\\Delta$.\n\t\\end{itemize}\n\t\\end{changemargin}\n\t\\vskip .15cm\nWe refer to a morphism $\\rho:\\Delta'\\longrightarrow\\Delta$ in $\\bold{REC}_{\\Sigma,\\textit{\\i}}$ as a {\\em refinement of $\\Delta$}.\n\\end{definition}\n\n\\vskip .3cm\n\t\n\tIf $\\rho:\\Delta'\\longrightarrow\\Delta$ is a refinement of $\\Delta$, then it induces a morphism $f_{\\rho}:Y_{\\Delta'}\\longrightarrow Y_{\\Delta}$ of $R$-models of $Y_{\\Sigma}$. This map of $R$-models itself restricts to a morphism\n\t$$\n\tf_{\\rho}\\ \\!:\\ \\mathfrak{X}_{(\\Delta',\\textit{\\i})}\\ \\longrightarrow\\ \\mathfrak{X}_{(\\Delta,\\textit{\\i})}\n\t$$\nof admissible formal $R$-models of $X$. In this way, the assignment $\\Delta\\mapsto\\mathfrak{X}_{(\\Delta,\\textit{\\i})}$ becomes a functor $\\bold{REC}_{\\Sigma,\\textit{\\i}}\\longrightarrow\\bold{AFS}_{R}^{X}$ that takes values in the category of admissible formal models of $\\an{X}$.\n\n\t\\vskip .3cm\n\n\\begin{definition}\\label{adictrop}\n{\\bf (Adic tropicalization).}\nThe {\\em adic tropicalization} of the closed embedding\n$\n\\xymatrix{\n\\textit{\\i}:X\n\\ar@{^{(}->}[r]\n&\nY_{\\Sigma}\n}\n$\nis the inverse limit\n$$\n\\big(\\ \\!\\mathrm{Ad}(X,\\textit{\\i})\\ \\!,\\ \\mathscr{O}_{\\!\\mathrm{Ad}(X,\\textit{\\i})}\\ \\!\\big)\n\\ \\ \\ \\overset{\\mathrm{def}}{=}\\ \n\\varprojlim_{\\ \\bold{REC}_{\\Sigma,\\textit{\\i}}}\\big(\\mathfrak{X}_{_{(\\Delta,\\textit{\\i})}},\\mathscr{O}_{\\mathfrak{X}_{_{(\\Delta,\\textit{\\i})}}}\\big)\n$$\nin the category of locally topologically ringed spaces.\n\\end{definition}\n\n\\vskip .3cm\n\n\tThe utility of Definition \\ref{adictrop} comes from the fact that, as an immediate consequence of Remark \\ref{remark: Gillam}, it gives $\\mathrm{Ad}(X,\\textit{\\i})$ the structure of a locally topologically ringed topological space. But Definition \\ref{adictrop} obscures the relationship between $\\mathrm{Ad}(X,\\textit{\\i})$ and the exploded tropicalization of $X$. To clarify this relationship, first consider the special case where $\\textit{\\i}:X'\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\!\\mathbb{T}'$ is a closed embedding into a torus, with $N'$ the cocharacter lattice of $\\mathbb{T}'$. In this case, the {\\em exploded tropicalization} is as described in \\S\\ref{introduction}: it is the union\n\t$$\n\t\\mathfrak{Trop}(X',\\textit{\\i}')\n\t\\ \\ \\overset{\\mathrm{def}}{=}\\ \\ \n\t\\bigsqcup_{v\\in N'_{\\mathbb{R}}}|\\mathrm{in}_{v}X'|.\n\t$$\nFor a closed embedding $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ into an arbitrary toric variety $Y_{\\Sigma}$, the {\\em extended exploded tropicalization}, denoted $\\mathfrak{Trop}(X,\\textit{\\i})$, is the set-theoretical union\n\t$$\n\t\\mathfrak{Trop}(X,\\textit{\\i})\\ \\ =\\ \\ \\bigsqcup_{\\sigma\\in\\Sigma}\\mathfrak{Trop}\\big(\\ \\!\\textit{\\i}^{-1}\\big(O(\\sigma)\\big),\\ \\!\\textit{\\i}\\ \\!\\big),\n\t$$\nwhere $\\textit{\\i}^{-1}\\big(O(\\sigma)\\big)$ is the part of $X$ that maps to the torus-orbit $O(\\sigma)\\cong\\mathbb{T}_{N\/\\sigma\\cap N}$ under $\\textit{\\i}$ (see \\cite[\\S3.2]{CLS}). In other words, the extended exploded tropicalization is the union of all exploded tropicalizations encoded in $\\textit{\\i}$ as we run over all locally closed torus-orbit strata in $Y_{\\Sigma}$.\n\t\n\tNote that each $\\Gamma$-rational vector $v$ in $N_{\\mathbb{R}}$ determines a $\\Gamma$-admissible ray $\\tau_{v}\\overset{{}_{\\mathrm{def}}}{=}\\mathbb{R}_{\\ge0}\\ \\!(v,1)$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, and vice versa, as depicted at left  in Figure \\ref{points and rays}. When $v$ is {\\em not} $\\Gamma$-rational, the ray $\\tau_{v}$ is no longer $\\Gamma$-admissible, but there are still many $\\Gamma$-admissible cones $\\delta$ containing $\\tau_{v}$, as depicted at right in Figure \\ref{points and rays}.\n\t\\begin{figure}[!htb]\n\t$$\n\t\\ \\ \\ \\ \\ \\ \\ \\ \\ \n\t\\scalebox{.85}{$\n\t\\begin{xy}\n\t(0,0)*+{\\includegraphics[scale=.35,clip=true,trim=85pt 172pt 150pt 200pt]{ConeD3.pdf}};\n\t(10,28)*+{(v,1)};\n\t(28,29)*+{\\ \\ \\ \\ \\ \\ \\ \\ \\tau_{v}\\ \\!\\overset{\\mathrm{def}}{=}\\ \\!\\mathbb{R}_{\\ge0}(v,1)};\n\t\\end{xy}\n\t\\ \n\t\\begin{xy}\n\t(0,0)*+{\\includegraphics[scale=.35,clip=true,trim=80pt 171pt 101pt 210pt]{ConeD2.pdf}};\n\t(17,28)*+{(v,1)};\n\t(42.5,2)*+{\\mbox{$\\Gamma$-admissible cone}};\n\t(42.5,-2.75)*+{\\delta\\mbox{ containing the}};\n\t(42.5,-7)*+{\\mbox{ray $\\tau_{v}=\\mathbb{R}_{\\ge0}(v,1)$}};\n\t\\end{xy}\n\t$}\n\t$$\n\t\\caption{The ray $\\tau_{v}$ spanned by a vector $(v,1)$ in $N_{\\mathbb{R}}\\times\\{1\\}$, at left, and a $\\Gamma$-admissible cone $\\delta$ containing $\\tau_{v}$ when $\\tau_{v}$ is not itself $\\Gamma$-rational.}\n\t\\label{points and rays}\n\t\\end{figure}\n\n\\vskip .3cm\n\n\\begin{proposition}\\label{exploded lemma 1}\nFor any vector $v$ in $N_{\\mathbb{R}}$, we have a natural isomorphism\n\t\\begin{equation}\\label{inset: inverse limit giving initial degeneration}\n\t\\mathrm{in}_{v}X\n\t\\ \\ \\cong\\ \\ \n\t\\varprojlim_{\\delta\\supset\\tau_{v}}\\big(\\mathfrak{X}_{\\delta}\\big)_{k},\n\t\\end{equation}\nwhere the limit runs over all $\\Gamma$-admissible cones $\\delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ that contain the ray $\\tau_{v}$. When $v$ is $\\Gamma$-rational, the above isomorphism (\\ref{inset: inverse limit giving initial degeneration}) simplifies to an isomorphism\n\t$$\n\t\\mathrm{in}_{v}X\\ \\ \\cong\\ \\ \\big(\\mathfrak{X}_{\\tau_{v}}\\big)_{k}.\n\t$$\n\\end{proposition}\n\n\\vskip .3cm\n\n\tOne uses Proposition \\ref{exploded lemma 1} to prove that the topological space underlying $\\mathrm{Ad}(X,\\textit{\\i})$ coincides with the extended exploded tropicalization $\\mathfrak{Trop}(X,\\textit{\\i})$:\n\n\\vskip .3cm\n\n\\begin{theorem}\\label{theorem: topology on exploded trop}\n\tGiven any closed embedding $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ of a $K$-variety $X$ into a proper toric $K$-variety $Y_{\\Sigma}$, there is a natural bijection \n\t$$\n\t\\mathrm{Ad}(X,\\textit{\\i})\\ \\xrightarrow{\\ \\sim\\ }\\ \\mathfrak{Trop}(X,\\textit{\\i})\n\t$$\nfrom the underlying set of the adic tropicalization to the exploded extended tropicalization.\n\\end{theorem}\n\n\\vskip .3cm\n\n\n\\subsection{Adic tropicalization and metrized complexes}\\ \n\t\n\tIn this section, we describe the adic tropicalization of a generic hyperplane in the toric variety $\\mathbb{P}^{2}$ in detail, and we make some remarks about the relationship between adic tropicalizations and the metrized complexes of O. Amini and M. Baker \\cite{Amini-Baker}.\n\n\\vskip .3cm\n\n\\subsubsection{{\\bf Tropical complexes and metrized complexes}}\n\nFix a $K$-variety $X$ and a closed embedding $\\textit{\\i}:X\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  Y_{\\Sigma}$ into a toric variety. Each complete fan $\\Delta$ in $N_{\\mathbb{R}}\\!\\times\\!\\mathbb{R}_{\\ge0}$, with recession fan $\\mathrm{rec}(\\Delta)=\\Sigma$, determines a formal $R$-scheme $\\mathfrak{X}_{\\Delta}$. As explained in \\S\\ref{specializations of adic spaces}, the associated adic space $\\mathfrak{X}^{\\mathrm{ad}}_{\\Delta}\\cong X^{\\mathrm{ad}}$ comes with a specialization morphism\n\t$$\n\t\\mathrm{sp}_{\\mathfrak{X}_{\\Delta}}\\ \\!:\\ \n\tX^{\\mathrm{ad}}\n\t\\ \\longrightarrow\\ \n\t\\mathfrak{X}_{\\Delta}.\n\t$$\nBecause the Berkovich analytification $X^{\\mathrm{an}}$ is the maximal Hausdorff quotient $h:X^{\\mathrm{ad}}\\longrightarrow\\!\\!\\!\\!\\rightarrow X^{\\mathrm{an}}$, we also have a map\n\t$$\n\tq\\ \\!:\\ X^{\\mathrm{ad}}\\ \\xrightarrow{\\ \\ h}\\!\\!\\!\\!\\rightarrow\\ X^{\\mathrm{an}}\\ \\xrightarrow{\\mathrm{trop}}\\ \\mathrm{Trop}(X,\\textit{\\i}).\n\t$$\nDefine the {\\em tropical complex} ({\\em of varieties}) {\\em associated to $\\Delta$}, denoted $\\mathfrak{CX}_{\\Delta}$, to be the image, inside the product of topological spaces $\\mathrm{Trop}(X,\\textit{\\i})\\times\\mathfrak{X}_{\\Delta}$, of the product map\n\t$$\n\tq\\times\\mathrm{sp}_{\\mathfrak{X}_{\\Delta}}\n\t\\ \\!:\\ \n\tX^{\\mathrm{ad}}\n\t\\ \\longrightarrow\\ \n\t\\mathrm{Trop}(X,\\textit{\\i})\\times\\mathfrak{X}_{\\Delta}.\n\t$$\n\t\n\\vskip .3cm\n\n\\begin{example}\\label{metrized complex example}\n{\\bf Metrized complex associated to a generic hyperplane in $\\pmb{\\mathbb{P}^{2}}$.}\nFor a concrete example, let $Y_{\\Sigma}=\\mathbb{P}^{2}$ with its standard toric structure, let $\\textit{\\i}:X\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y\\!$ be the closed embedding of the hyperplane $X=V(x+y+1)$ into $\\mathbb{P}^{2}$, and let $C$ be any polyhedral complex decomposing $N_{\\mathbb{R}}=\\mathbb{R}^{2}$ such that the intersection $C_{X}\\overset{{}_{\\mathrm{def}}}{=}C\\cap\\mathrm{Trop}(X,\\textit{\\i})$ is the one pictured at left in Figure \\ref{metrized complex} below. Then our tropical complex $\\mathfrak{CX}_{\\Delta_{C}}$ is the {\\em metrized complex} ({\\em of curves}), in the sense of Amini and Baker \\cite[\\S1.2]{Amini-Baker}, pictured at right in Figure \\ref{metrized complex}.\n\n\\begin{figure}[!htb]\n$$\n\\vcenter{\\hbox{\n\\begin{xy}\n(0,0)*+{\\includegraphics[scale=.75]{adictrop3.pdf}};\n(4,-15)*+{C_{X}\\overset{{}_{\\mathrm{def}}}{=}C\\cap\\mathrm{Trop}(X,\\textit{\\i})\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!};\n(0,28)*+{{\\color{white}.\\color{black}}};\n\\end{xy}\n}}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\raisebox{-14pt}{\\huge $\\xmapsto{\\ \\ \\ }$}\\ \\ \n\\vcenter{\\hbox{\n\\begin{xy}\n(0,-5)*+{\\includegraphics[scale=.75]{adictrop5.pdf}};\n(-3,-20)*+{\\mathfrak{CX}_{\\Delta_{C}}};\n\\end{xy}\n}}\n$$\n\\caption{A $\\Gamma$-rational polyhedral decomposition of $\\mathrm{Trop}(X)$ induced by a convex polyhedral decomposition $C$ of $N_{\\mathbb{R}}$, and its associated tropical complex $\\mathfrak{CX}_{\\Delta_{C}}$.}\\label{metrized complex}\n\\end{figure}\n\\end{example}\n\n\\vskip .3cm\n \t\n\\subsubsection{{\\bf The general 1-dimesional case and metrized complexes.}}\nConsider the case where our embedding $\\textit{\\i}:X\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  Y_{\\Sigma}$ is a closed embedding of a smooth curve $X$ into the toric variety $Y_{\\Sigma}$. Recall that $\\textit{\\i}$ is {\\em sch\\\"on} if the initial degeneration $X_{v}$ is smooth at every vector $v$ in the tropicalization of each torus orbit in $Y_{\\Sigma}$.\n\t\n\tWe claim if $Y_{\\Sigma}$ is proper and $\\textit{\\i}$ is sch\\\"on, then each tropical complex $\\mathfrak{CX}_{\\Delta}$ is a metrized complex (of curves) in the sense of Amini and Baker \\cite[\\S1.2]{Amini-Baker}. Indeed, Lemma \\ref{exploded lemma 1}, in conjunction with the sch\\\"on condition, implies that if $\\delta$ is a cone in $\\Delta$ whose intersection $\\delta\\cap\\big(\\mathrm{Trop}(X,\\textit{\\i})\\times\\{1\\}\\big)$ is a (necessarily $\\Gamma$-rational) point $(v,1)\\in\\mathrm{Trop}(X,\\textit{\\i})\\times\\{1\\}$, then the $k$-scheme underlying $\\mathfrak{X}_{\\delta}$ is the nonsingular curve $\\mathrm{in}_{v}X$. Any point in $X^{\\mathrm{ad}}$ that gets mapped to $v$ under $q:X^{\\mathrm{ad}}\\longrightarrow\\mathrm{Trop}(X,\\textit{\\i})$ goes to a point of this nonsingular curve $\\mathrm{in}_{v}X$ under the specialization map $\\mathrm{sp}_{\\mathfrak{X}_{\\Delta}}\\!:X^{\\mathrm{ad}}\\longrightarrow\\mathfrak{X}_{\\Delta}$. Similarly, Lemma \\ref{exploded lemma 1} and the sch\\\"on condition imply that if $\\delta\\cap\\big(\\mathrm{Trop}(X,\\textit{\\i})\\times\\{1\\}\\big)$ is an edge connecting points $(u,1)$ and $(v,1)$ in $\\mathrm{Trop}(X,\\textit{\\i})\\times\\{1\\}$, then the $k$-scheme underlying $\\mathfrak{X}_{\\delta}$ is a $k$-curve with two smooth components meeting at a point, one of the components containing $\\mathrm{in}_{u}X$ and the other containing $\\mathrm{in}_{v}X$. The points of $\\mathrm{X}^{\\mathrm{ad}}$ mapping to $\\mathfrak{X}_{\\delta}-\\big(\\mathrm{in}_{u}X\\cup\\mathrm{in}_{v}X\\big)$ under the specialization map $\\mathrm{sp}_{\\mathfrak{X}_{\\Delta}}$ are mapped to interior points of the edge corresponding to $\\delta$ inside $\\mathrm{Trop}(X,\\textit{\\i})$.\n\t\n\tIn this way, we can describe the tropical complex $\\mathfrak{CX}_{\\Delta}$ using exactly the kind of gluing datum that Amini and Baker use to define a metrized complex in \\cite[\\S1.2]{Amini-Baker}.\n\n\\vskip .3cm\n\n\\begin{example}\\label{2-dimensional tropical complex}\n{\\bf An example of a 2-dimensional tropical complex.}\nLet $X=Y_{\\Sigma}=\\mathbb{P}^{2}$, where $\\Sigma\\subset N_{\\mathbb{R}}=\\mathbb{R}^{2}$ is the standard fan describing $\\mathbb{P}^{2}$ as a toric variety. Let $C$ be the polyhedral decomposition of $N_{\\mathbb{R}}=\\mathbb{R}^{2}$ pictured in Figure \\ref{tropical complex} below. It determines a fan $\\Delta_{C}$ in $N_{\\mathbb{R}}\\!\\times\\!\\mathbb{R}_{\\ge0}$ with recession fan $\\mathrm{rec}(\\Delta_{C})=\\Sigma$. Using Lemma \\ref{exploded lemma 1}, we can build the tropical complex $\\mathfrak{CX}_{\\Delta_{C}}$ from the datum of the $k$-varieties underlying the formal models $\\mathfrak{X}_{\\delta_{P}}$ associated to  each polygon $P$ in $C$.\n\\begin{figure}[!htb]\n$$\n\\begin{xy}\n(0,0)*+{\\includegraphics[scale=.55,clip=true,trim=80pt 310pt 50pt 290pt]{adictrop6.pdf}};\n(1,12)*+{\\mathbb{P}^{2}_{\\!k}};\n(14,-3.5)*+{\\mathbb{P}^{2}_{\\!k}};\n(-18.5,-2)*+{\\mathbb{P}^{2}_{\\!k}};\n(-8,5)*+{\\rotatebox{40}{{\\smaller\\smaller $D_{2}\\!\\times\\!d_{2}$}}};\n(11,6)*+{\\rotatebox{-45}{{\\smaller\\smaller $\\!\\!D_{1}\\!\\times\\!d_{1}$}}};\n(-1,-5.5)*+{\\rotatebox{10}{{\\smaller\\smaller $\\!\\!D_{3}\\!\\times\\!d_{3}$}}};\n(9,24)*+{\\rotatebox{70}{{\\smaller\\smaller $\\!\\!E_{2}\\!\\times\\!e_{2}$}}};\n(42,-0)*+{\\rotatebox{0}{{\\smaller\\smaller $\\!\\!E_{1}\\!\\times\\!e_{1}$}}};\n(-51,-20)*+{\\rotatebox{25}{{\\smaller\\smaller $\\!\\!E_{3}\\!\\times\\!e_{3}$}}};\n(2,2)*+{\\mbox{{\\smaller\\smaller $P_{0}$}}};\n(25,8)*+{\\mbox{{\\smaller\\smaller $P_{1}$}}};\n(-22,10)*+{\\mbox{{\\smaller\\smaller $P_{2}$}}};\n(0,-14)*+{\\mbox{{\\smaller\\smaller $P_{3}$}}};\n\\end{xy}\n$$\n\\caption{The tropical complex $\\mathfrak{CX}_{\\Delta_{C}}$ in Example \\ref{2-dimensional tropical complex}.}\n\\label{tropical complex}\n\\end{figure}\n\nThe polyhedral complex $C$ consists of three vertices, edges $d_{i}$ and $e_{i}$ for $1\\le i\\le 3$, and 2-dimensional polygons $P_{j}$ for $0\\le j\\le 3$. The tropical complex $\\mathfrak{CX}_{\\Delta_{C}}$ is glued from a copy of $\\mathbb{P}^{1}_{k}$ at each vertex, a product $D_{i}\\!\\times\\!d_{i}$ with a $1$-dimensional $k$-variety $D_{i}$ associated to each edge $d_{i}$, $1\\le i\\le 3$, and similarly for edges $e_{i}$, and then the three $2$-dimensional polygons $P_{j}$, $1\\le j\\le 3$, understood as products $\\mathrm{Spec}_{\\ \\!}k\\times P_{j}$. The data telling us how to glue these topological spaces comes from the incidence relations between the irreducible components of the $k$-variety underlying the formal model $\\mathfrak{X}_{\\Delta_{C}}$.\n\\end{example}\n\n\\vskip .3cm\n\n\\begin{lemma}\\label{tropical complex description of adic trop}\nIf $\\textit{\\i}:X\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}\\!  Y_{\\Sigma}$ is a closed embedding into a proper toric variety, then there is a canonical bijection $\\varprojlim_{\\bold{REC}_{\\Sigma,\\textit{\\i}}}\\!\\!\\mathfrak{CX}_{\\Delta}\\ \\xrightarrow{\\ \\sim\\ }\\ \\mathrm{Ad}(X,\\textit{\\i})$.\n\\end{lemma}\n\\begin{proof}\nEach projection $\\mathrm{Trop}(X,\\textit{\\i})\\times\\mathfrak{X}_{\\Delta}\\longrightarrow\\!\\!\\!\\!\\rightarrow\\mathfrak{X}_{\\Delta}$ induces a map $\\mathfrak{CX}_{\\Delta}\\longrightarrow\\mathfrak{X}_{\\Delta}$, and these maps give rise to a map\n\t\\begin{equation}\\label{comparison map tropical complex to adic trop}\n\t\\varprojlim_{\\bold{REC}_{\\Sigma,\\textit{\\i}}}\\!\\!\\mathfrak{CX}_{\\Delta}\\ \\longrightarrow\\ \\mathrm{Ad}(X,\\textit{\\i}).\n\t\\end{equation}\n\t\n\tFix a compatible system of points $x=(x_{\\Delta})_{\\Delta\\in\\mathrm{Rec}_{\\Sigma,\\textit{\\i}}}$, where $x_{\\Delta}$ lies in the $k$-variety underlying $\\mathfrak{X}_{\\Delta}$. This system describes a point $x$ in $\\mathrm{Ad}(X,\\textit{\\i})$. Because $\\mathfrak{X}_{\\delta_{1}}\\cap\\mathfrak{X}_{\\delta_{2}}=\\mbox{\\O}$ in $\\mathfrak{X}_{\\Delta}$ whenever $\\delta_{1}\\cap\\delta_{2}=\\mbox{\\O}$ in $N_{\\mathbb{R}}\\!\\times\\!\\mathbb{R}_{\\ge0}$, we know that the compatible system $x$ has an associated point $v_{x}\\in\\mathrm{Trop}(X,\\textit{\\i})$, determined by the condition that $v_{x}$ lie in the polygon $P_{\\delta}$ associated to a cone $\\delta$ in $\\Delta$ whenever $x_{\\Delta}$ lies in $\\mathfrak{X}_{\\delta}$. Here $P_{\\delta}$ denotes the unique polygon in $N_{\\mathbb{R}}$ satisfying $P_{\\delta}\\!\\times\\!\\{1\\}=\\delta\\cap\\big(N_{\\mathbb{R}}\\!\\times\\!\\{1\\}\\big)$. The very definition of $v_{x}$ implies that if $x'$ is any point of $\\mathfrak{X}^{\\mathrm{ad}}$ mapping to $x$ under $\\mathrm{sp}_{\\mathfrak{X}_{\\Delta}}:X^{\\mathrm{ad}}\\longrightarrow\\mathfrak{X}_{\\Delta}$, then $x'$ maps to $v_{x}$ under the map $q:X^{\\mathrm{ad}}\\longrightarrow\\mathrm{Trop}(X,\\textit{\\i})$. This implies subjectivity of the comparison map (\\ref{comparison map tropical complex to adic trop}).\n\t\n\tTo see that (\\ref{comparison map tropical complex to adic trop}) is injective, note that the argument of the previous paragraph implies that points given by pairs $(x,u)$ and $(y,v)$ in $\\mathfrak{CX}_{\\Delta}$ must map to distinct points of $\\mathrm{Ad}(X,\\textit{\\i})$ if their second coordinates $u,v\\in\\mathrm{Trop}(X,\\textit{\\i})$ differ. On the other hand, if $u=v$, but the inverse systems $y=(y_{\\Delta})_{\\Delta\\in\\mathrm{Rec}_{\\Sigma,\\textit{\\i}}}$ and $x=(x_{\\Delta})_{\\Delta\\in\\mathrm{Rec}_{\\Sigma,\\textit{\\i}}}$ differ, then they map to distinct points of $\\mathrm{Ad}(X,\\textit{\\i})$.\n\\end{proof}\n\n\\vskip .3cm\n\n\\begin{example}\n{\\bf The adic tropicalization of a generic hyperplane in $\\pmb{\\mathbb{P}^{2}}$.}\nAs in Example \\ref{metrized complex example} above, let $\\textit{\\i}:X\\!\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y\\!$ be the closed embedding of the hyperplane $X=V(x+y+1)$ into $\\mathbb{P}^{2}$. Lemma \\ref{tropical complex description of adic trop} says that we can figure out how $\\mathrm{Ad}(X,\\textit{\\i})$ looks by undrestanding it as the object that results, in the inverse limit, when we bubble of a copy of $\\mathbb{P}^{1}_{k}$ at every $\\Gamma$-rational point in $\\mathrm{Trop}(X,\\textit{\\i})$. The resulting topological space appears in Figure \\ref{adic trop drawing} below.\n\n\\begin{figure}[!htb]\n$$\n\\includegraphics[scale=.625]{adictrop.pdf}\n$$\n\\caption{Adic tropicalization of the hyperplane $V(x+y+1)$ in $\\mathbb{P}^{2}$. Compare this to the exploded curve, in the sense of B. Parker \\cite{Parker2}, appearing in Figure \\ref{figure: exploded curve}.}\n\\label{adic trop drawing}\n\\end{figure}\n\\end{example}\n\n\\vskip .3cm\n\n\\subsubsection{{\\bf Tropical ``type 5\" points and inverse systems of $R$-models.}}\\label{tropical type five points}\nWe take a moment to highlight a phenomenon that occurs in adic tropicalizations, related to the presence of ``type 5\" points in the adic affine line $(\\mathbb{A}^{1})^{\\mathrm{ad}}$.\n\t\n\tConsider a polygonal interval $P_{1}=\\begin{tikzpicture}\\draw[black, thick] (0,0) -- (1,0);\\fill[black] (0,0) circle (.08);\\fill[black] (1,0) circle (.08);\\end{tikzpicture}$ inside $N_{\\mathbb{R}}=\\mathbb{R}$. The $k$-variety underlying the model $\\mathfrak{X}_{P_{1}}$ consists of two copies of $\\mathbb{A}^{1}_{k}$ intersecting at a single $k$-point $x_{1}$ as pictured in Figure \\ref{bubbling off} below. If we subdivide $P_{1}$ by adding a single $\\Gamma$-rational vertex in the interior of $P_{1}$, then the $k$-variety underlying our formal $R$-scheme picks up a $\\mathbb{P}^{1}_{\\!k}$ component where $x_{1}$ used to be.\n\t\\begin{figure}[!htb]\n\t$$\n\t\\scalebox{.95}{$\n\t\\begin{array}{rcl}\n\t\\begin{xy}\n\t(0,0)*+{\\begin{tikzpicture}\n\t\t\\draw[red, very thick] (0,0) -- (2,0);\n\t\t\\fill[red] (0,0) circle (.075);\n\t\t\\fill[red] (2,0) circle (.075);\n\t\\end{tikzpicture}};\n\t(0,3)*+{{\\color{red} P_{1}}};\n\t(-10,-3)*+{{\\color{red} v}};\n\t\\end{xy}\n\t&\n\t\\ \\ \\ \\ \\ \\ \\ \\mbox{\\larger\\larger $\\xmapsto{\\ \\ \\ }$}\\ \\ \\ \\ \\ \\ \n\t&\n\t\\begin{xy}\n\t(0,0)*+{\\includegraphics[scale=.75]{growing_P1s_1.pdf}};\n\t(0,8.5)*+{{\\color{red} \\overset{\\mbox{$\\mathfrak{X}_{\\!P_{1}}$}}{{\\color{pink}\\overbrace{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }}}}};\n\t(-8.5,-6)*+{{\\color{red} \\mathfrak{X}_{v}}};\n\t(-1,-4.5)*+{\\begin{tikzpicture}\n\t\t\\draw[black, thick,->] (0,.25) to [out=0,in=-130] (0,1);\n\t\\end{tikzpicture}};\n\t(-3,-9)*+{\\mbox{{\\smaller $x_{1}$}}};\n\t\\end{xy}\n\t\\\\[30pt]\n\t\\begin{xy}\n\t(0,0)*+{\\begin{tikzpicture}\n\t\t\\draw[red, very thick] (0,0) -- (1,0);\n\t\t\\draw[black, very thick] (1,0) -- (2,0);\n\t\t\\fill[red] (0,0) circle (.075);\n\t\t\\fill[black] (2,0) circle (.075);\n\t\t\\fill[red] (1,0) circle (.075);\n\t\\end{tikzpicture}};\n\t(-4.5,3)*+{{\\color{red} P_{2}}};\n\t(-10,-3)*+{{\\color{red} v}};\n\t\\end{xy}\n\t&\n\t\\ \\ \\ \\ \\ \\ \\ \\mbox{\\larger\\larger  $\\xmapsto{\\ \\ \\ }$}\\ \\ \\ \\ \\ \\ \n\t&\n\t\\begin{xy}\n\t(0,0)*+{\\includegraphics[scale=.75]{growing_P1s_2.pdf}};\n\t(-3,7.5)*+{\\rotatebox{-15}{\\color{red} $\\overset{\\mbox{$\\mathfrak{X}_{\\!P_{2}}$}}{{\\color{pink}\\overbrace{\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ }}}$}};\n\t(-11.5,-6)*+{{\\color{red} \\mathfrak{X}_{v}}};\n\t(-3.5,-4.5)*+{\\begin{tikzpicture}\n\t\t\\draw[black, thick,->] (0,.25) to [out=0,in=-130] (0,1);\n\t\\end{tikzpicture}};\n\t(-5.5,-9)*+{\\mbox{{\\smaller $x_{2}$}}};\n\t\\end{xy}\n\t\\\\[30pt]\n\t\\begin{xy}\n\t(0,0)*+{\\begin{tikzpicture}\n\t\t\\draw[red, very thick] (0,0) -- (.5,0);\n\t\t\\draw[black, very thick] (.5,0) -- (2,0);\n\t\t\\fill[red] (0,0) circle (.075);\n\t\t\\fill[black] (2,0) circle (.075);\n\t\t\\fill[black] (1,0) circle (.075);\n\t\t\\fill[red] (.5,0) circle (.075);\n\t\\end{tikzpicture}};\n\t(-7.25,3)*+{{\\color{red} P_{3}}};\n\t(-10,-3)*+{{\\color{red} v}};\n\t\\end{xy}\n\t&\n\t\\ \\ \\ \\ \\ \\ \\ \\mbox{\\larger\\larger  $\\xmapsto{\\ \\ \\ }$}\\ \\ \\ \\ \\ \\ \n\t&\n\t\\begin{xy}\n\t(0,0)*+{\\includegraphics[scale=.75]{growing_P1s_3.pdf}};\n\t(-5,7)*+{\\rotatebox{-25}{\\color{red} $\\overset{\\mbox{$\\mathfrak{X}_{\\!P_{3}}$}}{{\\color{pink}\\overbrace{\\ \\ \\ \\ \\ \\ \\ }}}$}};\n\t(-13.5,-6)*+{{\\color{red} \\mathfrak{X}_{v}}};\n\t(-6,-4.5)*+{\\begin{tikzpicture}\n\t\t\\draw[black, thick,->] (0,.25) to [out=0,in=-130] (0,1);\n\t\\end{tikzpicture}};\n\t(-8,-9)*+{\\mbox{{\\smaller $x_{3}$}}};\n\t\\end{xy}\n\t\\\\[30pt]\n\t\\mbox{{\\larger\\larger $\\vdots$}}\\ \\ \\ \\ \\ \\ \\ \\ \\ \\!\n\t&\n\t\\mbox{{\\larger\\larger $\\vdots$}}\n\t&\n\t\\ \\ \\ \\ \\ \\ \\!\\mbox{{\\larger\\larger $\\vdots$}}\n\t\\end{array}\n\t$}\n\t$$\n\t\\caption{The formal $R$-schemes associated to several decompositions of the interval $P_{1}$ inside $N_{\\mathbb{R}}=\\mathbb{R}$. For each decomposition, the $k$-variety underlying the $R$-scheme $\\mathfrak{X}_{v}$ associated to the left endpoint $v$ is a copy of $\\mathbb{A}^{1}_{k}\\!-\\!\\{0\\}$ that does not change from decomposition to decomposition. The nodal point $x_{i}$ that partially compactifies $\\mathbb{A}^{1}_{k}\\!-\\!\\{0\\}$ in each of the $R$-schemes ``stays fixed\" as further decompositions move every $\\mathbb{P}^{1}_{\\!k}$ component further and further from $\\mathfrak{X}_{v}$.}\n\t\\label{bubbling off}\n\t\\end{figure}\n\t\nClearly $\\mathfrak{X}_{v}$ sits in the inverse limit of $R$-models associated to all subdivisions of the interval, but the inverse limit also contains a point in the closure of $\\mathfrak{X}_{v}$, which does {\\em not} map to the component of any model associated to a segment {\\em not} containing $v$. The sequence of points $x_{i}$ in Figure \\ref{bubbling off} form an inverse system, and hence determine a single point $x$ in the adic tropicalization $\\mathrm{Ad}(\\mathbb{P}^{1})$.\n\n\\vskip .75cm\n\n\n\\section{The adic limit Theorem}\\label{Section-Main Theorem}\n\n\tIn this final \\S\\ref{Section-Main Theorem}, we explain that adic tropicalizations satisfy a form of ``analytification is the limit of all tropicalizations,\" namely, that one can recover the Huber analytification $X^{\\mathrm{ad}}$ of any closed subvariety of a proper toric $K$-variety, along with the sheaf of power bound sections $\\mathscr{O}^{\\circ}_{\\!X^{\\mathrm{ad}}}$ on $X^{\\mathrm{ad}}$, from an inverse limit of adic tropicalizations of $X$. The exact statement appears in Theorem \\ref{Huberlimit} and Proposition \\ref{Nice system, man!} below. We begin with a brief review of the corresponding theorem in the setting of Berkovich analytic spaces, as proved in \\cite{P09} and \\cite{FGP}.\n\t\n\n\\subsection{The limit theorem in the Berkovich setting}\\label{systems}\\ \n\nSince each multiplicative seminorm $K[U_{\\sigma}]\\longrightarrow\\mathbb{R}_{\\geq0}$ induces a homomorphism $S_{\\sigma}\\longrightarrow\\mathbb{R}\\sqcup\\{\\infty\\}$ of semigroups,\neach closed embedding\n$\n\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}\n$\ncomes with a continuous map of topological spaces\n$$\n\\pi_{\\textit{\\i}}:\\an{X}\\longrightarrow\\Trop{X}{\\textit{\\i}},\n$$\ncalled the {\\em tropicalization map} associated to the closed embedding $\\textit{\\i}$.\n\nIf $f_{\\phi}:Y_{\\Sigma}\\longrightarrow Y_{\\Sigma'}$ is a toric morphism, necessarily induced by morphisms $\\phi:\\Sigma\\longrightarrow\\Sigma'$ of fans, and if\n$\n\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}\n$\nand\n$\n\\textit{\\i}':X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma'}\n$\nare closed embeddings for which the diagram\n\\begin{equation}\\label{systemmorphism}\n\\begin{aligned}\n\\begin{xy}\n(0,0)*+{X}=\"1\";\n(15,7)*+{Y_{\\Sigma}}=\"2\";\n(15,-7)*+{Y_{\\Sigma'}}=\"3\";\n{\\ar@{^{(}->}^{\\textit{\\i}} \"1\"; \"2\"};\n{\\ar@{^{(}->}_{\\textit{\\i}'} \"1\"; \"3\"};\n{\\ar@{->}^{f_{\\phi}} \"2\"; \"3\"};\n\\end{xy}\n\\end{aligned}\n\\end{equation}\ncommutes, then the corresponding tropicalization maps $\\pi_{\\!\\textit{\\i}}$ and $\\pi_{\\textit{\\i}'}$ fit into a commutative diagram\n\\begin{equation}\\label{tropmorphism}\n\\begin{aligned}\n\\ \\ \\ \\ \\ \\ \n\\begin{xy}\n(0,0)*+{\\an{X}\\!\\!}=\"1\";\n(20,10)*+{\\Trop{X}{\\textit{\\i}}}=\"2\";\n(20,-10)*+{\\Trop{X}{\\textit{\\i}'}}=\"3\";\n{\\ar@{->}^{\\pi_{\\!\\textit{\\i}}} \"1\"; \"2\"};\n{\\ar@{->}_{\\pi_{\\!\\textit{\\i}'}} \"1\"; \"3\"};\n{\\ar@{->}^{\\mathrm{Trop}(f_{\\phi})} \"2\"; \"3\"};\n\\end{xy}\n\\end{aligned}\n\\end{equation}\n\nLet $\\mathcal{S}$ denote any small category whose set of objects consists of some family of closed embeddings $\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}$,\nand in which each homset $\\mathrm{Hom}_{\\mathcal{S}}(\\textit{\\i},\\textit{\\i}')$ consists of all commutative diagrams of the above form (\\ref{systemmorphism}). Then the system of tropicalization maps $\\pi_{\\textit{\\i}}$ associated objects and morphisms in $\\mathcal{S}$ induces a map\n\\begin{equation}\\label{Berkcompmap}\n\\pi\\ \\overset{\\mathrm{def}}{=}\\ \\varprojlim_{\\mathcal{S}}\\pi_{\\textit{\\i}}\\ \\!:\\ \\an{X}\\longrightarrow\\varprojlim_{\\mathcal{S}}\\Trop{X}{\\textit{\\i}}\n\\end{equation}\nWe call this map the {\\em Berkovich comparison map}.\n\nIn \\cite{P09}, S. Payne showed that if $X$ admits at least one closed embedding into a quasi-projective toric variety $Y_{\\Sigma}$, and if $\\mathcal{S}$ is the system of all closed embeddings into quasiprojective toric varieties, then the Berkovich comparison map (\\ref{Berkcompmap}) is a homeomorphism.\nIn \\cite{FGP}, P. Gross, S. Payne, and the present author extended this result to the case of any $K$-scheme admitting at least one closed embedding into a toric variety. This result is a consequence of the following Theorem \\ref{Berkovich}, which describes conditions under which an arbitrary system $\\mathcal{S}$ of closed toric embeddings gives rise to a homeomorphism (\\ref{Berkcompmap}):\n\n\\vskip .4cm\n\n\\begin{theorem}\\label{Berkovich}\n{\\bf \\cite[Theorem 1.1]{FGP}}\\ \\ \nIf the system $\\mathcal{S}$ satisfies conditions {\\bf (C1)} and {\\bf (C2)} below, then the Berkovich comparison map (\\ref{Berkcompmap}) is a homeomorphism:\n\\vskip .3cm\n\\begin{itemize}\n\\item[{\\bf (C1)}]\nIf $\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}$ and $\\textit{\\i}':X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma'}$ are closed embeddings in $\\mathcal{S}$, then their product\n$\n\\textit{\\i}\\times\\textit{\\i}':X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma\\times\\Sigma'}\n$\nis also in $\\mathcal{S}$;\n\\item[{\\bf (C2)}]\n\\vskip .2cm\nThere exists a finite affine open cover $\\big\\{\\!\\xymatrix{U_{i}\\ar@{^{(}->}[r]&X}\\!\\big\\}$ such that for each nonzero regular function $f\\in K[U_{i}]$, there is some closed embedding\n$\n\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}\n$\nin $\\mathcal{S}$ that realizes $U_{i}$ as the preimage of a torus-invariant open affine, and realizes $f$ as the pullback of a monomial.\n\\end{itemize}\n\\end{theorem}\n\n\\vskip .3cm\n\n\n\\subsection{Adic tropicalization morphisms}\\label{Adic tropicalization morphisms}\\ \n\nFix a $K$-variety $X$, and let \n$\n\\xymatrix{\n\\textit{\\i}:X\n\\ar@{^{(}->}[r]\n&\nY_{\\Sigma}\n}\n$\nbe a closed embedding into a proper toric variety $Y_{\\Sigma}$. As explained in \\S\\ref{subsection: formal Gubler models}, each $\\Gamma$-admissible fan $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$, satisfying\n\t\\begin{equation}\\label{inset: two conditions}\n\t\\mathrm{supp}(\\Delta)\\ =\\ N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}\n\t\\ \\ \\ \\ \\ \\ \\ \\ \\mbox{and}\\ \\ \\ \\ \\ \\ \\ \\ \n\t\\Delta\\cap\\big(N_{\\mathbb{R}}\\times\\{0\\}\\big)\n\t\\ \\ =\\ \\ \\Sigma\\times\\{0\\},\n\t\\end{equation}\ngives rise to an admissible formal model $\\mathfrak{X}_{(\\textit{\\i},\\Delta)}$ of $X$. By \\ref{specializations of adic spaces}, this model $\\mathfrak{X}_{(\\textit{\\i},\\Delta)}$ comes with a morphism of locally topologically ringed spaces\n\t\\begin{equation}\\label{inset: second appearance of specialization}\n\\mathrm{sp}_{\\mathfrak{X}_{(\\textit{\\i},\\Delta)}}:\\big(\\ad{X},\\mathscr{O}^{\\ \\!\\!\\circ}_{\\ad{X}}\\big)\\longrightarrow\\big(\\mathfrak{X}_{(\\textit{\\i},\\Delta)},\\mathscr{O}_{\\mathfrak{X}_{(\\textit{\\i},\\Delta)}}\\big).\n\t\\end{equation}\nOur very construction of the adic tropicalization $\\mathrm{Ad}(X,\\textit{\\i})$ as an inverse limit implies that as $\\Delta$ ranges over all $\\Gamma$-admissible fans in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ satisfying the twin conditions (\\ref{inset: two conditions}), the resulting specialization morphisms (\\ref{inset: second appearance of specialization}) give rise to a single morphism of locally topologically ringed spaces\n\t\\begin{equation}\\label{adic tropicalization morphism}\n\t\\varpi_{\\textit{\\i}}\n\t\\ \\ \\overset{\\mathrm{def}}{=}\n\t\\varprojlim_{\\ \\bold{REC}_{\\Sigma,\\textit{\\i}}}\\mathrm{sp}_{(\\textit{\\i},\\Delta)}\\ \\ :\\ \\ (\\ad{X},\\mathscr{O}^{\\ \\!\\!\\circ}_{\\ad{X}})\\longrightarrow\\big(\\mathrm{Ad}(X,\\textit{\\i}),\\ \\!\\mathscr{O}_{\\mathrm{Ad}(X,\\textit{\\i})}\\big).\n\t\\end{equation}\nThis morphism (\\ref{adic tropicalization morphism}) is the adic counterpart to the tropicalization map\n\t$$\n\t\\pi_{\\textit{\\i}}:\\an{X}\\longrightarrow\\mathrm{Trop}(X,\\textit{\\i}),\n\t$$\nand so we refer to (\\ref{adic tropicalization morphism}) as the {\\em adic tropicalization morphism} at $\\textit{\\i}$.\n\n\\begin{definition} A {\\em system of embeddings} for $X$ is any category $\\mathcal{S}$ such that $\\mathrm{ob}_{\\ \\!}\\mathcal{S}$ is some class of closed embeddings $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ into proper toric varieties, and where the set of morphisms between two such embeddings is the set of all commutative diagrams\n\t\\begin{equation}\\label{system 2}\n\t\\begin{aligned}\n\t\\begin{xy}\n\t(0,0)*+{X}=\"1\";\n\t(15,7)*+{Y_{\\Sigma}}=\"2\";\n\t(15,-7)*+{Y_{\\Sigma'}}=\"3\";\n\t{\\ar@{^{(}->}^{\\textit{\\i}} \"1\"; \"2\"};\n\t{\\ar@{^{(}->}_{\\textit{\\i}'} \"1\"; \"3\"};\n\t{\\ar@{->}^{f_{\\phi}} \"2\"; \"3\"};\n\t\\end{xy}\n\t\\end{aligned}\n\t\\end{equation}\nwhere $f_{\\phi}$ is any torus-equivariant morphism induced by a morphism $\\phi:\\Sigma\\longrightarrow\\Sigma'$ of fans.\n\\end{definition}\n\n\\begin{remark}\nIf $\\Sigma$ and $\\Sigma'$ are complete fans in $N_{\\mathbb{R}}$ and $N'_{\\mathbb{R}}$, then for each fan $\\Delta'$ in $N'_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ satisfying the twin conditions (\\ref{inset: two conditions}), we can refine any fan $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ to a fan $\\Delta_{+}$ such that the product map\n\t$$\n\tf_{\\phi}\\times\\mathrm{id}:N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}\\ \\longrightarrow\\  N'_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}\n\t$$\ninduces a morphism of fans $f_{\\phi}\\times\\mathrm{id}:\\Delta_{+}\\longrightarrow\\Delta'$. Thus if $\\mathcal{S}$ is a system of embeddings for $X$, then each morphism (\\ref{system 2}) induces a morphism of locally topologically ringed spaces\n\t\\begin{equation}\\label{inset: induced adic trop map}\n\t\\mathrm{Ad}(X,f_{\\phi}):\\mathrm{Ad}(X,\\textit{\\i})\\longrightarrow\\mathrm{Ad}(X,\\textit{\\i}').\n\t\\end{equation}\nThese morphisms (\\ref{inset: induced adic trop map}) give the assignment $\\big(\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}\\big)\\longmapsto\\mathrm{Ad}(X,\\textit{\\i})$ the structure of a functor\n\t$$\n\t\\mathrm{Ad}(X,-):\\mathcal{S}\\longrightarrow\\bold{LTRS}.\n\t$$\nThe inverse limit of the adic tropicalization morphisms (\\ref{adic tropicalization morphism}) over $\\mathcal{S}$ provides us with a morphism of locally topologically ringed spaces\n\t\\begin{equation}\\label{inset: comparison map}\n\t\\varprojlim_{\\textit{\\i}\\in\\mathcal{S}}\\varpi_{\\textit{\\i}}\\ \\!:\\ (\\ad{X},\\mathscr{O}^{\\ \\!\\!\\circ}_{\\ad{X}})\\ \\!\\xrightarrow{\\ \\ \\ \\ \\ }\\ \\!\\varprojlim_{\\textit{\\i}\\in\\mathcal{S}}\\big(\\mathrm{Ad}(X,\\textit{\\i}),\\ \\!\\mathscr{O}_{\\mathrm{Ad}(X,\\textit{\\i})}\\big),\n\t\\end{equation}\nan adic counterpart to the comparison map (\\ref{Berkcompmap}) that appears in the context of Berkovich analytic spaces.\n\\end{remark}\n\n\\vskip .3cm\n\n\\begin{theorem}\\label{Huberlimit}\nLet $X$ be a proper variety over an algebraically closed, non-trivially valued non-Arhimedean field $K$, and let $\\mathcal{S}$ be a system of embeddings for $X$. If $\\mathcal{S}$ satisfies conditions {\\bf (C0)}, {\\bf (C1)}, and {\\bf (C2)} below, then the adic comparison morphism (\\ref{inset: comparison map}) is an isomorphism of locally topologically ringed spaces.\n\\begin{itemize}\n\\item[{\\bf (C0)}]\\vskip .25cm\nIf $\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}$ is a closed embedding in $\\mathcal{S}$, with $\\mathbb{T}$ the dense torus in $Y_{\\Sigma}$, then any translation of $\\textit{\\i}$ by a $K$-point of $\\mathbb{T}$ is again in $\\mathcal{S}$;\n\\item[{\\bf (C1)}]\\vskip .25cm\nIf $\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}$ and $\\textit{\\i}':X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma'}$ are closed embeddings in $\\mathcal{S}$, then their product\n$\n\\textit{\\i}\\times\\textit{\\i}':X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma\\times\\Sigma'}\n$\nis also in $\\mathcal{S}$;\n\\item[{\\bf (C2)}]\\vskip .25cm\nThere exists a finite affine open cover $\\big\\{\\!\\xymatrix{U_{i}\\ar@{^{(}->}[r]&X}\\!\\big\\}$ such that for each nonzero regular function $f\\in K[U_{i}]$, there is a closed embedding\n$\n\\textit{\\i}:X\\xymatrix{{}\\ar@{^{(}->}[r]&{}}Y_{\\Sigma}\n$\nin $\\mathcal{S}$ that realizes $U_{i}$ as the preimage of a torus-invariant open affine, and realizes $f$ as the pullback of a monomial.\n\\end{itemize}\n\\end{theorem}\n\n\\vskip .3cm\n\n\\begin{corollary}\n{\\bf (Adic limit theorem on point sets).}\nIf $\\mathcal{S}$ satisfies conditions {\\bf (C0)} through {\\bf (C2)}, then the point set underlying the Huber analytification of $X$ is in natural bijection with the inverse limit of all extended exploded tropicalizations of $X$:\n\t$$\n\t|X^{\\mathrm{ad}}|\n\t\\ \\ \\cong\\ \\ \n\t\\varprojlim_{\\textit{\\i}\\in\\mathcal{S}}\\ \\mathfrak{Trop}(X,\\textit{\\i})\n\t$$\n\\end{corollary}\n\\begin{proof}\nThis follows immediately from Theorems \\ref{theorem: topology on exploded trop} and \\ref{Huberlimit}.\n\\end{proof}\n\n\\vskip .3cm\n\n\\begin{remark}\nOne proves Theorem \\ref{Huberlimit} by showing that the system of all Gubler models of $X$, for all closed embeddings $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ in proper toric embeddings, is cofinal in the system of all admissible formal $R$-models of $X$. The key technical step in the proof is Proposition \\ref{majorizing} below. As in the statement of Theorem \\ref{Huberlimit}, let $X$ be a proper $K$-variety over an algebraically closed, non-trivially valued non-Arhimedean field $K$, and let $\\mathcal{S}$ be a system of embeddings for $X$ satisfying conditions {\\bf (C0)},  {\\bf (C1)}, and  {\\bf (C2)} of Theorem \\ref{Huberlimit}.\n\\end{remark}\n\n\\vskip .3cm\n\n\\begin{proposition}\\label{majorizing}\nEvery admissible formal $R$-model $\\mathfrak{X}$ of $X$ is dominated by a formal Gubler model. More precisely, if $\\mathfrak{X}$ is any admissible formal $R$-model of $X$, then there exists\n\t\\begin{itemize}\n\t\\item[{\\bf (i)}]\\vskip .25cm\n\ta closed embedding $\\textit{\\i}:X\\!\\xymatrix{{}\\!\\ar@{^{(}->}[r]&\\!{}}\\! Y_{\\Sigma}$ into the proper toric variety associated to some complete fan $\\Sigma$ in some $\\mathbb{R}$-vector space $N_{\\mathbb{R}}$,\n\t\\item[{\\bf (i)}]\\vskip .25cm\n\ta $\\Gamma$-admissible fan $\\Delta$ in $N_{\\mathbb{R}}\\times\\mathbb{R}_{\\ge0}$ satisfying the twin conditions (\\ref{inset: two conditions}),\n\t\\end{itemize}\n\t\\vskip .25cm\nsuch that the formal Gubler model $\\mathfrak{X}_{(\\textit{\\i},\\Delta)}$ associated to the pair $(\\textit{\\i},\\Delta)$ admits a proper morphism\n\t$$\n\t\\mathfrak{X}_{(\\textit{\\i},\\Delta)}\n\t\\ \\longrightarrow\\!\\!\\!\\!\\rightarrow\\ \n\t\\mathfrak{X}\n\t$$\nof formal $R$-models of $X$.\n\\end{proposition}\n\n\\vskip .3cm\n\nTo close, we give several examples of systems of embeddings that satisfy the hypotheses of Theorem \\ref{Huberlimit}, so that the comparison morphism (\\ref{inset: comparison map}) becomes an isomorphism.\n\n\\vskip .3cm\n\n\\begin{proposition}\\label{Nice system, man!}\nLet $X$ be a $K$-variety satisfying the two-point condition {\\bf (A2)} appearing in \\S\\ref{Adic tropicalization of a closed embedding} above. Then in each of the following situations, the system $\\mathcal{S}$ satisfies conditions {\\bf (C0)}, {\\bf (C1)}, and {\\bf (C2)} of Theorem \\ref{Huberlimit}:\n\\vskip .3cm\n\\begin{itemize}\n\\item[{\\bf (i)}]\n$X$ proper, $\\mathcal{S}$ the category of all closed embeddings of $X$ into proper toric varieties;\n\\vskip .2cm\n\\item[{\\bf (ii)}]\n$X$ projective, $\\mathcal{S}$ the category of all closed embeddings of $X$ into projective toric varieties;\n\\vskip .2cm\n\\item[{\\bf (iii)}]\n$X$ smooth and proper, $\\mathcal{S}$ the category of all closed embeddings of $X$ into smooth, proper toric varieties.\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nCase {\\bf (ii)} follows from \\cite[Lemma 4.3]{P09}. Cases {\\bf (i)} and {\\bf (iii)} follow from the construction described in the proof of \\cite[Theorem 4.2]{FGP}, using the fact that we can construct the toric embedding with the requisite property in each of these cases, as explained in \\cite[Theorem A]{Wlod}.\n\\end{proof}\n\n\\vskip .75cm\n\n\n\n\n\n\n\\bibliographystyle{acm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main result}\n\n\n\\subsection{Introduction}\nThroughout the paper $K$ is an algebraically closed field. By an algebra we mean a finite-dimensional algebra over $K$ and by a module a finite-dimensional left module.\n\nGiven an algebra $A$ and a nonnegative integer $d$, one defines the variety $\\mod (A, d)$, called the variety of $d$-dimensional $A$-modules, which consists of the $K$-algebra homomorphisms $A \\to \\mathbb{M}_d(K)$ and parameterizes $A$-modules of dimension $d$. Different aspects of its geometry have been objects of intensive studies during the last decades (see~\\cites{BobinskiRiedtmannSkowronski, Bongartz1998, Geiss, Zwara} for some references and reviews of results). In particular, one may ask, for a given algebra $A$, which properties the varieties $\\mod (A, d)$ have for all dimensions $d$. For example, Bongartz has proved in~\\cite{Bongartz1991} that the varieties $\\mod (A, d)$ are smooth for all $d$ if and only if $A$ is hereditary (i.e., $\\gldim A \\leq 1$).\n\nIn~\\cite{BobinskiSchroer2017a} we initiated the study of geometrically irreducible algebras, i.e., the algebras such that for each $d$ the connected components of $\\mod (A, d)$ are irreducible. In particular, we proved that the only geometrically irreducible algebras of finite global dimension are the hereditary ones. On the other hand, the truncated polynomial algebras are the only geometrically irreducible local algebras. Note that an algebra $A$ is Morita equivalent to a local algebra if and only if there is exactly one (up to isomorphism) simple $A$-module. In this paper we concentrate on algebras with exactly two simple modules.\n\n\\subsection{Main result}\nIn order to formulate the main result of the paper we need to introduce some families of algebras. For $h \\geq 0$, let $Q (h)$ be the quiver\n\\[\n\\xymatrix{\n0 \\ar@(ul,dl)_{\\varepsilon_0} & 1 \\ar@(ur,dr)^{\\varepsilon_1} \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l]\n}.\n\\]\nNote that $Q (h)$ is connected if and only if $h > 0$. For $n \\geq 1$ (and $h \\geq 1$), let\n\\[\n\\rho^{(n)} := \\sum_{i = 0}^n \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i.\n\\]\nFor $m_0, m_1 \\geq 2$ and $h, n \\geq 1$, let $A (h, m_0, m_1, n)$ be the path algebra of the quiver $Q (h)$ bounded by\n\\[\n\\varepsilon_0^{m_0}, \\; \\varepsilon_1^{m_1}, \\; \\rho^{(n)}.\n\\]\nSimilarly, for $m_0, m_1 \\geq 1$ and $h \\geq 0$, let $A' (h, m_0, m_1)$ be the path algebra of the quiver $Q (h)$ bounded by\n\\[\n\\varepsilon_0^{m_0}, \\; \\varepsilon_1^{m_1}.\n\\]\n\nThe following theorem is the first main result of the paper.\n\n\\begin{Thm}\\label{thm:intro3}\nAssume that $A$ is an algebra which has exactly two simples. If $A$ is geometrically irreducible, then $A$ is Morita equivalent to one of the following algebras:\n\\begin{enumerate}\n\n\\item\n$A (h, m, m, 1)$, for some $m \\geq 2$ and $h \\geq 1$, or\n\n\\item\n$A (h, m, m, m - 1)$, for some $m \\geq 2$ and $h \\geq 1$, or\n\n\\item\n$A' (h, m_0, m_1)$, for some $m_0, m_1 \\geq 1$ and $h \\geq 0$.\n\n\\end{enumerate}\n\\end{Thm}\n\nIt is easy to see that if $m_0, m_1 \\geq 1$ and $h \\geq 0$, then $A' (h, m_0, m_1)$ is geometrically irreducible (see Proposition~\\ref{prop Aprim}).\n\nNow let $A = A (h, m, m, 1)$, for $m \\geq 2$ and $h \\geq 1$. The category $\\mod(A)$ is related to the Birkhoff problem~\\cite{Birkhoff} and has been intensively studied (see for example~\\cites{KussinLenzingMeltzer, RingelSchmidmeier, Simson}). In~\\cite{BobinskiSchroer2017b} we prove that $A (h, m, m, 1)$ is geometrically irreducible.\n\nWe do not know if $A (h, m, m, m - 1)$ is geometrically irreducible, if $m \\geq 3$ and $h \\geq 1$.\n\n\\subsection{Minimal geometrically irreducible algebras without shortcuts}\nOur interest in two-vertex case and its significance follows from a conjecture formulated in~\\cite{BobinskiSchroer2017a} concerning a general form of geometrically irreducible algebras. We call an algebra $A$ a \\emph{gluing} of bound quiver algebras $K Q' \/ \\mathcal{I}'$ and $K Q'' \/ \\mathcal{I}''$ if $A$ is Morita equivalent to the algebra $K Q \/ \\mathcal{I}$, where\n\\begin{enumerate}\n\n\\item\n$Q_0 = Q'_0 \\cup Q''_0$ (the union is not necessarily disjoint);\n\n\\item\n$Q_1 = Q'_1 \\cup Q''_1$ and $Q'_1 \\cap Q''_1 = \\emptyset$;\n\n\\item\n$\\mathcal{I}$ is the ideal in $K Q$ generated by the union $\\mathcal{I}' \\cup \\mathcal{I}''$.\n\n\\end{enumerate}\nA connected algebra, which cannot be presented as the gluing of two algebras as above with $Q_1' \\neq \\emptyset \\neq Q_1''$, is called \\emph{minimal}. We conjecture that every geometrically irreducible minimal algebra has at most two simple modules. The second main result of the paper says this conjecture holds for a wide class of algebras. More precisely, an arrow $\\alpha$ in a quiver $Q$ is called a \\emph{shortcut}, if $\\alpha$ is not a loop and there is a path $\\alpha_1 \\cdots \\alpha_l$ of length at least~2 connecting the same vertices as $\\alpha$, such that none of the arrows $\\alpha_i$ is a loop. Then we have the following.\n\n\\begin{Thm}\\label{thm:conj}\nAssume that $A = K Q \/ \\mathcal{I}$ is a minimal geometrically irreducible algebra. If there are no shortcuts in $Q$, then $Q$ has at most two vertices, hence $A$ is Morita equivalent to one of the following algebras:\n\\begin{enumerate}\n\n\\item\n$K [X] \/ (X^m)$, for some $m \\geq 1$, or\n\n\\item\n$A (1, m, m, 1)$, for some $m \\geq 2$, or\n\n\\item\n$A (1, m, m, m - 1)$, for some $m \\geq 2$.\n\n\\end{enumerate}\n\\end{Thm}\n\nRecall we know that $K [X] \/ (X^m)$, for $m \\geq 1$, and $A (1, m, m, 1)$, for $m \\geq 2$, are geometrically irreducible, while it is open if $A (1, m, m, m - 1)$, for $m \\geq 3$, is geometrically irreducible.\n\n\\subsection{Acknowledgments}\nThe first author acknowledges the support of the National Science Center grant no.\\ 2015\/17\/B\/ST1\/01731. He would also like to thank the University of Bonn for its hospitality during his two one week visits in July 2016 and July 2017. The second author thanks the Nicolaus Copernicus University in Toru\\'n for one week of hospitality in October 2015, where this work was initiated. He thanks for a second week in Toru\\'n in March 2017, and he is also grateful to the SFB\/Transregio TR 45 for financial support.\n\n\n\\section{Schemes of representations}\n\n\n\\subsection{Quivers with relations} \\label{subsection quivers}\nBy a \\emph{quiver} we mean a quadruple $Q = (Q_0, Q_1, s, t)$, where $Q_0$ and $Q_1$ are finite sets of vertices and arrows, respectively, and $s, t \\colon Q_1 \\to Q_0$ are maps. We say that an arrow $\\alpha$ starts in $s (\\alpha)$ and ends in $t (\\alpha)$.\n\nA \\emph{path} of length $l \\geq 1$ in a quiver $Q$ is a sequence $\\sigma = (\\alpha_1, \\ldots, \\alpha_l)$ of arrows $\\alpha_1, \\ldots, \\alpha_l \\in Q_1$ such that $s (\\alpha_i) = t (\\alpha_{i + 1})$ for all $1 \\leq i \\leq l - 1$. We usually write $\\alpha_1 \\cdots \\alpha_l$ instead of $(\\alpha_1, \\ldots, \\alpha_l)$. Define $s (\\sigma) := s (\\alpha_l)$ and $t (\\sigma) := t (\\alpha_1)$. A path $\\sigma$ of positive length is called an \\emph{oriented cycle} if $s (\\sigma) = t (\\sigma)$. A \\emph{loop} is an oriented cycle of length one, i.e., an arrow $\\alpha$ such that $s (\\alpha) = t (\\alpha)$. Additionally to the paths of length $l \\geq 1$, there is, for each vertex $i \\in Q_0$, a path $e_i$ of length $0$ with $s (e_i) = t (e_i) = i$.\n\nBy $K Q$ we denote the path algebra of a quiver $Q$. It has all paths (including the length $0$ paths) as a $K$-basis, and the multiplication is defined via the composition of paths. Note that $K Q$ may be infinite-dimensional.\n\nBy a \\emph{relation} in a quiver $Q$ we mean a linear combination of paths of length at least~2, which have the same starting vertex and the same ending vertex. An ideal $\\mathcal{I}$ in $K Q$ is called \\emph{admissible} if the following hold:\n\\begin{itemize}\n\n\\item[(i)]\n$\\mathcal{I}$ is generated by a set of relations;\n\n\\item[(ii)]\nthere exists some $l \\geq 2$ such that all paths of length at least $l$ are contained in $\\mathcal{I}$.\n\n\\end{itemize}\nA pair $(Q, \\mathcal{I})$ consisting of a quiver $Q$ and an admissible ideal $\\mathcal{I}$ is called a \\emph{bound quiver}. If $(Q, \\mathcal{I})$ is a bound quiver, then we call $A := K Q \/ \\mathcal{I}$ a bound quiver algebra. One easily observes that $|Q_0|$ is the number of simple $A$-modules. If $i \\in Q_0$, the we put $A_i := e_i A e_i$.\n\nFor each algebra $A$ there exists a bound quiver algebra $B$ such that $A$ and $B$ are Morita equivalent. It follows from Bongartz~\\cite{Bongartz1991} that if algebras $A$ and $B$ are Morita equivalent, then $A$ is geometrically irreducible if and only if $B$ is geometrically irreducible. Consequently, from now on we assume that all considered algebras are bound quiver algebras.\n\n\\subsection{Schemes of representations}\nLet $A = K Q \/ \\mathcal{I}$ be a bound quiver algebra. A \\emph{representation} of $Q$ is a tuple $M = (M (i), M (\\alpha))$, where  for each vertex $i \\in Q_0$, $M (i)$ is a finite-dimensional $K$-vector space and, for each arrow $\\alpha \\in Q_1$, $M(\\alpha) \\colon M (s (\\alpha)) \\to M (t (\\alpha))$ is a $K$-linear map. The sequence $\\bdim M = (\\dim_K M (i))_{i \\in Q_0}$ is called the \\emph{dimension vector} of $M$. A \\emph{morphism} $f \\colon M \\to N$ of representations is a sequence $f = (f (i))$, where for each vertex $i \\in Q_0$, $f (i) \\colon M (i) \\to N (i)$ is a $K$-linear map. For a path $\\sigma = \\alpha_1 \\cdots \\alpha_l$ in $Q$ of positive length and a representation $M$ we define\n\\[\nM (\\sigma) := M(\\alpha_1) \\cdots M(\\alpha_l).\n\\]\nSimilarly, if $\\rho = \\sum_{i = 1}^k \\lambda_i \\sigma_i$ is linear combination of paths of positive lengths having the same starting vertex\nand the same ending vertex,\nwe set\n\\[\nM (\\rho) := \\sum_{i = 1}^k \\lambda_i M (\\sigma_i).\n\\]\nA representation $M$ of $Q$ is a \\emph{representation of $A$} (shortly an $A$-representa\\-tion) if $M (\\rho) = 0$ for all relations $\\rho \\in \\mathcal{I}$.\n\nFor a dimension vector $\\mathbf{d} = (d_i)_{i \\in Q_0} \\in \\mathbb{N}^{Q_0}$ let\n\\[\n\\rep (Q, \\mathbf{d}) :=  \\prod_{\\alpha \\in Q_1} \\Hom_K (K^{d_{s (\\alpha)}}, K^{d_{t (\\alpha)}})\n\\]\nbe the affine space of representations $M$ of $Q$ with $M(i) = K^{d_i}$ for each $i \\in Q_0$ and let\n\\begin{multline*}\n\\rep (A, \\mathbf{d}) := \\{ M = (M(\\alpha)) \\in \\rep(Q,\\mathbf{d}) \\mid\n\\\\\n\\text{$M (\\rho) = 0$ for all relations $\\rho \\in \\mathcal{I}$} \\}.\n\\end{multline*}\nThen $\\rep(A, \\mathbf{d})$ is an affine scheme, called the \\emph{scheme of $A${}-represen\\-tations} of dimension vector $\\mathbf{d}$. Using Bongartz~\\cite{Bongartz1991} again, we get that $A$ is geometrically irreducible if and only if $\\rep (A, \\mathbf{d})$ is irreducible for each dimension vector~$\\mathbf{d}$.\n\nThe group\n\\[\n\\GL (\\mathbf{d}) := \\prod_{i \\in Q_0} \\GL_{d_i} (K)\n\\]\nacts on $\\rep(A, \\mathbf{d})$ by conjugation. The $\\GL (\\mathbf{d})$-orbits are in bijection with the isomorphism classes of $A$-representations with dimension vector $\\mathbf{d}$. For a representation $M$ we denote the corresponding orbit by $\\mathcal{O}_M$. Then\n\\begin{equation} \\label{eq orbit}\n\\dim \\mathcal{O}_M = \\dim \\GL (\\mathbf{d}) - \\dim_K \\End_A (M).\n\\end{equation}\nAn orbit $\\mathcal{O}_M$ is called \\emph{maximal} if $\\mathcal{O}_M$ is not contained in the closure of another orbit.\n\n\n\\section{Preliminary observations}\n\n\n\\subsection{The possible quivers}\nWe recall some facts on geometrically irreducible algebras from~\\cite{BobinskiSchroer2017a}. If $A = K Q \/ \\mathcal{I}$ is a bound quiver algebra and $M$ is an $A$-representation, then $M$ is called \\emph{locally free} if, for each vertex $i \\in Q_0$, $M (i)$ is a free $A_i$-module (recall that $A_i := e_i A e_i$).\n\n\\begin{Prop} \\label{prop known}\nLet $A = K Q \/ \\mathcal{I}$ be a geometrically irreducible bound quiver algebra. Then the following hold:\n\\begin{enumerate}\n\n\\item\nEvery oriented cycle in $Q$ is a power of a loop.\n\n\\item\nFor each vertex $i \\in Q_0$, there is at most one loop $\\alpha$ with $s (\\alpha) = i = t (\\alpha)$.\n\n\\item\nProjective and injective $A$-representations are locally free.\n\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(1) is~\\cite{BobinskiSchroer2017a}*{Lemma~3.4}, (2) is~\\cite{BobinskiSchroer2017a}*{Lemma~3.2}, and (3) follows from~\\cite{BobinskiSchroer2017a}*{Lemma~4.1}.\n\\end{proof}\n\nFor $h \\geq 0$, let $Q' (h)$, $Q'' (h)$ and $Q''' (h)$ be the following quivers respectively:\n\\begin{gather*}\n\\\\\n\\xymatrix{\n0 \\ar@(ul,dl)_{\\varepsilon_0} & 1 \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l]\n}, \\qquad\n\\xymatrix{\n0 & 1 \\ar@(ur,dr)^{\\varepsilon_1} \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l]\n}, \\qquad\n\\xymatrix{\n0 & 1 \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l]\n}.\n\\end{gather*}\nWe have the following consequence of Proposition~\\ref{prop known}.\n\n\\begin{Cor} \\label{coro quiver}\nIf $A = K Q \/ \\mathcal{I}$ is a geometrically irreducible bound quiver algebra with exactly two simples, then $Q$ is one of the quivers $Q (h)$, $Q' (h)$, $Q'' (h)$ and $Q''' (h)$, $h \\geq 0$. In particular, $A \\cong A' (h, m_0, m_1) \/ \\mathcal{J}$, for some $m_0, m_1 \\geq 1$ and an ideal $\\mathcal{J}$.\n\\end{Cor}\n\n\\begin{proof}\nThe former part follows immediately from Proposition~\\ref{prop known}~(1) and~(2). For the latter observe, that if $Q = Q' (h)$, then we take $m_1 = 1$. If $Q = Q'' (h)$, then $m_0 = 1$, while for $Q = Q''' (h)$, $m_0 = 1 = m_1$ (and $\\mathcal{J} = 0)$. Finally, if $\\varepsilon_0$ (resp. $\\varepsilon_1$) is present, then there exists $m_0 \\geq 2$ (resp. $m_1 \\geq 2$) such that $\\varepsilon_0^{m_0} \\in \\mathcal{I}$ (resp. $\\varepsilon_1^{m_1} \\in \\mathcal{I}$), since $\\mathcal{I}$ is admissible.\n\\end{proof}\n\nThe following observation shows that $A$ is geometrically irreducible, if $\\mathcal{J} = 0$.\n\n\\begin{Prop} \\label{prop Aprim}\nIf $A \\cong A' (h, m_0, m_1)$, for $m_0, m_1 \\geq 1$ and $h \\geq 0$, then $A$ is geometrically irreducible.\n\\end{Prop}\n\n\\begin{proof}\nObserve that in the situation of the proposition, for each dimension vector $\\mathbf{d} = (d_0, d_1)$, we have\n\\[\n\\rep (A, \\mathbf{d}) \\cong \\rep (A_0, d_0) \\times \\rep (B, \\mathbf{d}) \\times \\rep (A_1, d_1),\n\\]\nwhere $B$ is the path algebra of the quiver $Q''' (h)$. Since $B$ is hereditary, while $A_0 = K [X] \/ (X^{m_0})$ and $A_1 = K [X] \/ (X^{m_1})$, we know that $B$, $A_0$ and $A_1$ are geometrically irreducible (see for example~\\cite{Gerstenhaber} for the claim in case of $A_0$ and $A_1$), hence the claim follows.\n\\end{proof}\n\nIn Section~\\ref{sect proof one} we study the case, when $\\mathcal{J} \\neq 0$.\n\n\\subsection{Representation theory of the truncated polynomial algebras}\nWe need to recall well-known facts from the representation theory of the truncated polynomial algebras.\n\nLet $\\Lambda = K [X] \/ (X^m)$, $m \\geq 1$. It is well-known that, for given $d \\geq 1$, the isomorphism classes of $d$-dimensional $\\Lambda$-modules are parameterized by the partitions of $d$ with parts at most $m$ (we denote the set of such partitions by $\\mathcal{P}_m (d)$): a partition $\\mathbf{p} = (p_1, \\ldots, p_l) \\in \\mathcal{P}_m (d)$ corresponds to a $\\Lambda$-module $M_\\mathbf{p}$ with the action of $X$ given by the matrix\n\\[\nJ_\\mathbf{p} := \\begin{bmatrix}\nJ_{p_1} & 0 & \\cdots & 0\n\\\\\n0 & J_{p_2} & \\ddots & \\vdots\n\\\\\n\\vdots & \\ddots & \\ddots & 0\n\\\\\n0 & \\cdots & 0 & J_{p_l}\n\\end{bmatrix},\n\\]\nwhere $J_p$ denotes the nilpotent Jordan matrix of size $p$ (with ones over the diagonal). Obviously, in the situation above we have\n\\[\nM_\\mathbf{p} = M_{(p_1)} \\oplus \\cdots \\oplus M_{(p_l)}.\n\\]\nIf $1 \\leq p, q \\leq m$, then\n\\begin{equation} \\label{eq Hom}\n\\dim_K \\Hom_\\Lambda (M_{(p)}, M_{(q)}) = \\min \\{ p, q \\}.\n\\end{equation}\nA partition $\\mathbf{p} \\in \\mathcal{P}_m (d)$ is called \\emph{maximal} if $\\mathbf{p} = (m, \\ldots, m, r)$, for some $1 \\leq r \\leq m$. A partition $\\mathbf{p}$ is maximal if and only if the orbit $\\mathcal{O}_\\mathbf{p} := \\mathcal{O}_{M_\\mathbf{p}}$ is maximal in $\\rep (\\Lambda, d)$. Note that~\\eqref{eq Hom} together with~\\eqref{eq orbit} gives a method for calculating $\\dim \\mathcal{O}_\\mathbf{p}$ for $\\mathbf{p} \\in \\mathcal{P}_m (d)$. As a consequence we get the following well-known formula, which we need later.\n\n\\begin{Lem} \\label{Lem orbits}\nIf $p \\geq 2$, then\n\\[\n\\dim \\mathcal{O}_{(p)} - \\dim \\mathcal{O}_{(p - 1, 1)} = 2.\n\\]\n\\end{Lem}\n\n\\begin{proof}\nWe have $\\dim_K \\End_\\Lambda (M_{(p)}) = p$ and\n\\[\n\\dim_K \\End_\\Lambda (M_{(p - 1, 1)}) = (p - 1) + 1 + 1 + 1 = p + 2,\n\\]\nhence the formula follows.\n\\end{proof}\n\n\\subsection{Stratification} \\label{sub strat}\nThe main tool which we use in the proof of Theorem~\\ref{thm:intro3} is the stratification by partitions, which we introduce now.\n\nLet $A = A' (h, m_0, m_1) \/ \\mathcal{J}$, for $m_0, m_1 \\geq 1$, $h \\geq 0$, and an ideal $\\mathcal{J}$. Since $A' (h, m_0, m_1)$ is a monomial bound quiver algebra (i.e., the defining ideal in $Q (h)$ can be generated paths), we may identify $A' (h, m_0, m_1)$ with the subspace of $K Q (h)$ spanned by paths which do not contain subpaths $\\varepsilon_0^{m_0}$ and $\\varepsilon_1^{m_1}$ (we stress that $A' (h, m_0, m_1)$ is not a subalgebra of $K Q (h)$). Consequently, if we assume that $m_0$, $m_1$ and $h$ are minimal, then $\\mathcal{J}$ consists of relations $\\rho$ such that $s (\\rho) = 1$ and $t(\\rho) = 0$.\n\nObserve that $A_0$ and $A_1$ are truncated polynomial algebras and if $M$ is an $A$-representation, then $M (0)$ and $M (1)$ together with the maps $M (\\varepsilon_0)$ and $M (\\varepsilon_1)$, respectively, are $A_0$- and $A_1$-modules, respectively. By abuse of notation we denote these modules by $M (0)$ and $M (1)$. Consequently, if $\\mathbf{d} = (d_0, d_1)$ is a dimension vector, $\\mathbf{p} \\in \\mathcal{P}_{m_0} (d_0)$ and $\\mathbf{q} \\in \\mathcal{P}_{m_1} (d_1)$, then we may define\n\\[\n\\mathcal{S}_{\\mathbf{p}, \\mathbf{q}} := \\{ M \\in \\rep (A, \\mathbf{d}) \\mid \\text{$M (0) \\cong M_\\mathbf{p}$ and $M (1) \\cong M_\\mathbf{q}$} \\}.\n\\]\nIf $\\rho$ is a relation in $Q (h)$, $M' \\in \\Hom_K(K^{d_0}, K^{d_0})$, $M'' \\in \\Hom_K(K^{d_1}, K^{d_1})$, and $M_1, \\ldots, M_h \\in \\Hom_K(K^{d_1}, K^{d_0})$, then we put\n\\[\n\\rho (M', M'', M_1, \\ldots, M_h) := M (\\rho),\n\\]\nwhere $M \\in \\rep (K Q (h), \\mathbf{d})$ is defined by\n\\begin{align*}\nM(\\alpha) :=\n\\begin{cases}\nM' & \\text{if $\\alpha = \\varepsilon_0$},\n\\\\\nM'' & \\text{if $\\alpha = \\varepsilon_1$},\n\\\\\nM_i & \\text{if $\\alpha = \\alpha_i$, $i = 1, \\ldots, h$}.\n\\end{cases}\n\\end{align*}\n\nThe following observations will be important.\n\n\\begin{Prop} \\label{prop strat}\nLet $A$, $\\mathbf{d}$, $\\mathbf{p}$ and $\\mathbf{q}$ be as above.\n\\begin{enumerate}\n\n\\item \\label{item strat1}\n$\\mathcal{S}_{\\mathbf{p}, \\mathbf{q}}$ is an irreducible locally closed subset of $\\rep (A, \\mathbf{d})$ of dimension\n\\[\n\\dim \\mathcal{O}_\\mathbf{p} + \\dim \\mathcal{O}_\\mathbf{q} + \\dim \\mathbb{A}_{\\mathbf{p}, \\mathbf{q}},\n\\]\nwhere\n\\begin{multline*}\n\\mathbb{A}_{\\mathbf{p}, \\mathbf{q}} := \\{ (M_1, \\ldots, M_h) \\in (\\Hom_K (K^{d_1}, K^{d_0}))^h \\mid\n\\\\\n\\text{$\\rho (J_\\mathbf{p}, J_\\mathbf{q}, M_1, \\ldots, M_h) = 0$ for each $\\rho \\in \\mathcal{J}$} \\}.\n\\end{multline*}\n\n\\item \\label{item strat2}\nIf $\\mathbf{p}$ and $\\mathbf{q}$ are maximal, then $\\overline{\\mathcal{S}}_{\\mathbf{p}, \\mathbf{q}}$ is an irreducible component of $\\rep (A, \\mathbf{d})$.\n\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\nSee~\\cite{BobinskiSchroer2017a}*{Subsection~5.3}.\n\\end{proof}\n\nLet $\\mathbf{d} = (d_0,d_1)$ be a dimension vector.\nIf $\\mathbf{p}_0 \\in \\mathcal{P}_{m_0} (d_0)$ and $\\mathbf{q}_0 \\in \\mathcal{P}_{m_1} (d_1)$ are maximal, then we say that $(\\mathbf{p}_0, \\mathbf{q}_0)$ is a \\emph{maximal pair of partitions} for $\\mathbf{d}$ and put $\\mathcal{S}_{\\max} := \\mathcal{S}_{\\mathbf{p}_0, \\mathbf{q}_0}$. Proposition~\\ref{prop strat}~\\eqref{item strat2} implies that in order to prove that $\\rep (A, \\mathbf{d})$ is reducible, it is sufficient to find $\\mathbf{p}_1 \\in \\mathcal{P}_{m_0} (d_0)$ and $\\mathbf{q}_1 \\in \\mathcal{P}_{m_1} (d_1)$ such that $(\\mathbf{p}_1, \\mathbf{q}_1)$ is not maximal and\n\\[\n\\dim \\mathcal{S}_{\\mathbf{p}_1, \\mathbf{q}_1} \\geq \\dim \\mathcal{S}_{\\max}.\n\\]\nFor $\\mathbf{p} \\in \\mathcal{P}_{m_0} (d_0)$ and $\\mathbf{q} \\in \\mathcal{P}_{m_1} (d_1)$, let $c_{\\mathbf{p}, \\mathbf{q}}$ be the codimension of $\\mathbb{A}_{\\mathbf{p}, \\mathbf{q}}$ inside $(\\Hom_K (K^{d_1}, K^{d_0}))^h$. Using Proposition~\\ref{prop strat}~\\eqref{item strat1} we get\n\\begin{multline} \\label{eq diff}\n\\dim \\mathcal{S}_{\\mathbf{p}_1, \\mathbf{q}_1} - \\dim \\mathcal{S}_{\\max} = (c_{\\mathbf{p}_0, \\mathbf{q}_0} - c_{\\mathbf{p}_1, \\mathbf{q}_1})\n\\\\\n+ (\\dim \\mathcal{O}_{\\mathbf{p}_1} - \\dim \\mathcal{O}_{\\mathbf{p}_0}) + (\\dim \\mathcal{O}_{\\mathbf{q}_1} - \\dim \\mathcal{O}_{\\mathbf{q}_0}).\n\\end{multline}\nIn all the situations we apply this argument we will have (up to duality) that $\\mathbf{q}_1 = \\mathbf{q}_0$, $\\mathbf{p}_0 = (p)$ and $\\mathbf{p}_1 = (p - 1, 1)$ with $p \\geq 2$. If this is the case, then Lemma~\\ref{Lem orbits} implies that $\\dim \\mathcal{S}_{\\mathbf{p}_1, \\mathbf{q}_1} \\geq \\dim \\mathcal{S}_{\\max}$ if and only if $c_{\\mathbf{p}_0, \\mathbf{q}_0} - c_{\\mathbf{p}_1, \\mathbf{q}_1} \\geq 2$. We formulate this result.\n\n\\begin{Cor} \\label{cor irr}\nLet $(\\mathbf{p}, \\mathbf{q})$ be a maximal pair of partitions for a dimension vector $\\mathbf{d}$, such that $\\mathbf{p} = (p)$ for some $p \\geq 2$. If\n\\[\nc_{\\mathbf{p}, \\mathbf{q}} \\geq c_{(p - 1, 1), \\mathbf{q}} + 2,\n\\]\nthen $\\rep (A, \\mathbf{d})$ is reducible. \\qed\n\\end{Cor}\n\nWe explain now how to calculate $c_{\\mathbf{p}, \\mathbf{q}}$. Observe that $c_{\\mathbf{p}, \\mathbf{q}}$ is the rank of the system of linear equations determined by the conditions\n\\[\n\\rho_i (J_\\mathbf{p}, J_\\mathbf{q}, M_1, \\ldots, M_h) = 0, \\qquad 1 \\leq i \\leq k,\n\\]\nwith $(\\rho_1, \\ldots, \\rho_k) = \\mathcal{J}$ and $M_1$, \\ldots, $M_h$ are matrices of indeterminates. If $\\rho$ is a relation, then we denote by $c_{\\mathbf{p}, \\mathbf{q}} (\\rho)$ the rank of the system of linear equations determined by the condition\n\\[\n\\rho (J_\\mathbf{p}, J_\\mathbf{q}, M_1, \\ldots, M_h) = 0.\n\\]\nAs a first observation we get the following, which is obvious in the light of the preceding remarks.\n\n\\begin{Lem} \\label{lem c part}\nAssume $\\mathcal{I} = (\\rho_1, \\ldots, \\rho_k)$ and there exists a partition $I_1$, \\ldots, $I_k$ of $\\{ 1, \\ldots, h \\}$ such that, for each $1 \\leq i \\leq k$,\n\\[\n\\rho_i = \\sum_{j \\in I_i} \\sum_{\\substack{0 \\leq a < m_0 \\\\ 0 \\leq b < m_1 \\\\ a + b \\geq 1}} \\lambda^{i, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b,\n\\]\ni.e., different relations $\\rho_i$ involve different arrows $\\alpha_j$. Then\n\\[\nc_{\\mathbf{p}, \\mathbf{q}} = \\sum_{i = 1}^k c_{\\mathbf{p}, \\mathbf{q}} (\\rho_i).\n\\]\n\\end{Lem}\n\nObserve that $c_{\\mathbf{p}, \\mathbf{q}}$ behaves additively, i.e., if $\\mathbf{p} = (p_1, \\ldots, p_l)$ and $\\mathbf{q} = (q_1, \\ldots, q_k)$, then\n\\[\nc_{\\mathbf{p}, \\mathbf{q}} = \\sum_{i = 1}^l \\sum_{j = 1}^k c_{(p_i), (q_j)}.\n\\]\nConsequently it is sufficient to study partitions consisting of only one part. We have the following rules, where by a linear combination of arrows we mean a linear combination of arrows in $Q (h)$ starting at $1$ and ending at $0$.\n\n\\begin{Prop} \\label{prop c formulas}\nLet $1 \\leq p \\leq m_0$, $1 \\leq q \\leq m_1$, and $\\alpha$ be a nonzero linear combinations of arrows.\n\\begin{enumerate}\n\n\\item \\label{item c formula2}\nIf $l \\leq p$, then\n\\[\nc_{(p), (1)} (\\varepsilon_0^l \\alpha) = p - l.\n\\]\n\n\\item \\label{item c formula3}\nIf $m_1 \\geq 2$ and $1 \\leq l < p$, then\n\\[\nc_{(p), (2)} (\\varepsilon_0^l \\alpha + \\varepsilon_0^{l - 1} \\alpha \\varepsilon_1) = 2 (p - l).\n\\]\n\n\\item \\label{item c formula4}\nIf $q \\leq p$, then\n\\[\nc_{(p), (q)} (\\varepsilon_0 \\alpha + \\alpha \\varepsilon_1) = q (p - 1).\n\\]\n\n\\item\nIf $q \\leq p$ and $\\beta$ is a linear combination of arrows such that $\\alpha$ and $\\beta$ are linearly independent, then\n\\[\nc_{(p), (q)} (\\varepsilon_0 \\alpha + \\beta \\varepsilon_1) = p q - 1.\n\\]\n\n\\item\nIf $2 \\leq q \\leq p$ and $\\beta$ is a linear combination of arrows such that $\\alpha$ and $\\beta$ are linearly independent, then\n\\[\nc_{(p), (q)} (\\varepsilon_0 \\alpha + \\alpha \\varepsilon_1 + \\beta \\varepsilon_1^{q - 1}) = q (p - 1) + 1.\n\\]\n\n\\item\nIf $l \\leq p, q$, then\n\\[\nc_{(p), (q)} \\Bigl( \\sum_{i = q - l}^{q - 1} \\varepsilon_0^{p + q - l - i - 1} \\alpha \\varepsilon_1^i \\Bigr) = l.\n\\]\n\n\\item\nIf $p, q \\geq 2$, $\\gamma$ is a linear combination of arrows and $\\lambda \\in K$, then\n\\[\nc_{(p), (q)} (\\varepsilon_0^{p - 1} \\alpha \\varepsilon_1^{q - 2} + \\lambda \\varepsilon_0^{p - 2} \\alpha \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma \\varepsilon_1^{q - 1}) = 2.\n\\]\n\n\\item\nIf $p, q \\geq 2$ and $\\beta$ and $\\gamma$ are linear combinations of arrows such that $\\alpha$ and $\\beta$ are linearly independent, then\n\\[\nc_{(p), (q)} (\\varepsilon_0^{p - 1} \\alpha \\varepsilon_1^{q - 2} + \\varepsilon_0^{p - 2} \\beta \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma \\varepsilon_1^{q - 1}) = 3.\n\\]\n\n\\item\nIf $p, q \\geq 3$, $\\gamma'$ and $\\gamma''$ are linear combinations of arrows and $\\lambda \\neq 1$, then\n\\[\nc_{(p), (q)} (\\varepsilon_0^{p - 1} \\alpha \\varepsilon_1^{q - 3} + \\varepsilon_0^{p - 2} \\alpha \\varepsilon_1^{q - 2} + \\lambda \\varepsilon_0^{p - 3} \\alpha \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 2} \\gamma' \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma'' \\varepsilon_1^{q - 1}) = 4.\n\\]\n\n\n\\item\nIf $3 \\leq l \\leq p, q$ and $\\beta$ and $\\gamma$ are linear combinations of arrows such that $\\alpha$ and $\\beta$ are linearly independent, then\n\\[\nc_{(p), (q)} \\Bigl( \\sum_{i = q - l}^{q - 1} \\varepsilon_0^{p + q - l - i - 1} \\alpha \\varepsilon_1^i  + \\varepsilon_0^{p - 2} \\beta \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma \\varepsilon_1^{q - 1} \\Bigr) = l + 1.\n\\]\n\n\\item\nIf $4 \\leq l \\leq p, q$, $\\gamma'$ and $\\gamma''$ are linear combinations of arrows and $\\lambda \\neq 0$, then\n\\[\nc_{(p), (q)} \\Bigl( \\sum_{i = q - l}^{q - 1} \\varepsilon_0^{p + q - l - i - 1} \\alpha \\varepsilon_1^i  + \\lambda \\varepsilon_0^{p - 3} \\alpha \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 2} \\gamma' \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma'' \\varepsilon_1^{q - 1} \\Bigr) = l + 1.\n\\]\n\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\nWe only prove the last formula, which seems to be the most complicated one. The remaining ones are proved similarly. If $M_1, \\ldots, M_h \\in \\Hom_K (K^q, K^p)$, then we write $M (\\alpha)$ in a matrix form\n\\[\nM (\\alpha) =\n\\begin{bmatrix}\nM_{p, 1} & \\cdots & M_{p, q}\n\\\\\n\\vdots & & \\vdots\n\\\\\nM_{1, 1} & \\cdots & M_{1, q}\n\\end{bmatrix}\n\\]\nwith unusual indexing. Similarly, we write\n\\[\nM (\\gamma') =\n\\begin{bmatrix}\nM_{p, 1}' & \\cdots & M_{p, q}'\n\\\\\n\\vdots & & \\vdots\n\\\\\nM_{1, 1}' & \\cdots & M_{1, q}'\n\\end{bmatrix}\n\\qquad \\text{and} \\qquad\nM (\\gamma'') =\n\\begin{bmatrix}\nM_{p, 1}'' & \\cdots & M_{p, q}''\n\\\\\n\\vdots & & \\vdots\n\\\\\nM_{1, 1}'' & \\cdots & M_{1, q}''\n\\end{bmatrix}.\n\\]\nIf $\\rho := \\sum_{i = q - l}^{q - 1} \\varepsilon_0^{p + q - l - i - 1} \\alpha \\varepsilon_1^i  + \\lambda \\varepsilon_0^{p - 3} \\alpha \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 2} \\gamma' \\varepsilon_1^{q - 1} + \\varepsilon_0^{p - 1} \\gamma'' \\varepsilon_1^{q - 1}$, then\n\\[\n\\rho (J_{(p)}, J_{(q)}, M_1, \\ldots, M_h) =\n\\begin{bmatrix}\n0 & \\cdots & 0 & x_2 & \\cdots & x_{l - 2} & x_{l - 1} & x_l & x_{l + 1}\n\\\\\n0 & \\cdots & \\cdots & 0 & x_2 & \\cdots & x_{l - 2} & x_{l - 1} & x_l'\n\\\\\n0 & \\cdots & \\cdots & \\cdots & 0 & x_2 & \\cdots & x_{l - 2} & x_{l - 1}'\n\\\\\n0 & \\cdots & \\cdots & \\cdots & \\cdots & 0 & x_2 & \\cdots & x_{l - 2}\n\\\\\n\\vdots & & & & & & \\ddots & \\ddots & \\vdots\n\\\\\n\\vdots & & & & & & & \\ddots & x_2\n\\\\\n\\vdots & & & & & & & & 0\n\\\\\n\\vdots & & & & & & & & \\vdots\n\\\\\n0 & \\cdots & \\cdots & \\cdots & \\cdots & \\cdots & \\cdots & \\cdots & 0\n\\end{bmatrix},\n\\]\nwhere\n\\begin{gather*}\nx_i := \\sum_{j = 1}^{l - 1} M_{j, l - j}, \\qquad 2 \\leq i \\leq l,\n\\\\\nx_{l - 1}' := x_{l - 1} + \\lambda x_2, \\qquad x_l' := x_l + \\lambda M_{2, 1} + M_{1, 1}'\n\\\\\n\\intertext{and}\nx_{l + 1} := x_l + \\lambda M_{3, 1} + M_{2, 1}' + M_{3, 1}''.\n\\end{gather*}\nNow one sees that $x_2$, \\ldots, $x_{l + 1}$ and $x_l'$ are linearly independent.\n\\end{proof}\n\n\\begin{Rem}\nThe formulas~\\eqref{item c formula2}, \\eqref{item c formula3} and~\\eqref{item c formula4} can be unified and generalized to the following formula: if $l < p$ and  $q \\leq p$, then\n\\[\nc_{(p), (q)} \\Bigl( \\sum_{i = 0}^{\\min \\{ l, q - 1 \\}} \\varepsilon_0^{l - i} \\alpha \\varepsilon_1^i \\Bigr) = q (p - l).\n\\]\nHowever in this case the analysis is more complicated than the one presented above. Moreover, we do not need this statement. As a result we decided not to include this claim in Proposition~\\ref{prop c formulas}.\n\\end{Rem}\n\n\n\\section{Proof of Theorem~\\ref{thm:intro3}} \\label{sect proof one}\n\n\n\\subsection{Notation} \\label{subsect not}\nWe present some notation we use throughout this section. Let $h \\geq 1$ and $m_0, m_1 \\geq 1$ be such that $m_0 + m_1 \\geq 3$. Put $Q := Q (h)$ and $B := A' (h, m_0, m_1)$. Fix relations $\\rho_1$, \\ldots, $\\rho_k$ in $B$ such that, for each $1 \\leq i \\leq k$, $s (\\rho_i) = 1$ and $t (\\rho_i) = 0$. Let $\\mathcal{J}$ be the ideal generated $\\rho_1$, \\ldots, $\\rho_k$, and let $A := B \/ \\mathcal{J}$. Observe that\n\\[\n\\mathcal{J} = \\sum_{i = 1}^k B_0 \\rho_i B_1,\n\\]\nand $A_0 = B_0$ and $A_1 = B_1$ (we again remind that $A_i := e_i A e_i$ and $B_i := e_i B e_i$, for $i = 0, 1$). We assume that $k$ is minimal in the sense that, if $\\rho_1'$, \\ldots, $\\rho_l'$ are relations such that\n\\[\n\\sum_{j = 1}^l B_0 \\rho_j' B_1 = \\mathcal{J},\n\\]\nthen $l \\geq k$. For each $1 \\leq i \\leq k$, write\n\\[\n\\rho_i = \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 0 \\leq b < m_1 \\\\ a + b \\geq 1}} \\lambda^{i, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b,\n\\]\nfor some $\\lambda^{i, j}_{a, b} \\in K$.\n\nOur aim is to prove that if $\\mathcal{J} \\neq 0$ and $A$ is geometrically irreducible, then we may assume $m_0 = m_1$, $k = 1$, and either $\\rho_1 = \\rho^{(1)}$ or $\\rho_1 = \\rho^{(m_0 - 1)}$, which will finish the proof of Theorem~\\ref{thm:intro3}.\n\n\\subsection{Each relation contains monomials of degree $(a, 0)$ and $(0, b)$}\nBefore we can proceed, we need some module theoretic considerations.\n\n\\begin{Lem}\\label{lem:a0prep}\nLet $\\Lambda$ be a $K$-algebra, and let $N_1 \\subseteq N \\subseteq M$ be $\\Lambda$-modules such that the following hold:\n\\begin{enumerate}\n\n\\item\n$N$ is a direct summand of $M$;\n\n\\item\n$N_1 + \\rad (M) = N + \\rad (M)$.\n\n\\end{enumerate}\nThen $N_1 = N$.\n\\end{Lem}\n\n\\begin{proof}\nWe have $M = N \\oplus N'$ for some submodule $N'$ of $M$. It follows that $\\rad (M) = \\rad (N) \\oplus \\rad (N')$, and therefore $N + \\rad (M) = N \\oplus \\rad (N')$ and $N_1 + \\rad (M) = (N_1 + \\rad (N)) \\oplus \\rad (N')$. Since $N_1 + \\rad (M) = N + \\rad (M)$, $N = N_1 + \\rad (N)$. This implies $N_1 = N$ by the Nakayama Lemma.\n\\end{proof}\n\nWe prove the following.\n\n\\begin{Lem} \\label{lem:a0}\nIf $A$ is geometrically irreducible, then for each $1 \\leq i \\leq k$, there exist $a$, $j_0$ and $b$, $j_1$ such that $\\lambda^{i, j_0}_{a, 0} \\neq 0$ and $\\lambda^{i, j_1}_{0, b} \\neq 0$.\n\\end{Lem}\n\n\\begin{proof}\nLet $\\Lambda := B_0$, $M := e_0 B e_1$ and $N := \\mathcal{J}$. Note that $N$ is a $\\Lambda$-submodule of $M$. Moreover, $N' := M \/ N = e_0 A e_1$. Since $A$ is geometrically irreducible, we know from Proposition~\\ref{prop known}~(3) that $N'$ is a free $\\Lambda$-module. It follows that $N$ is a direct summand of $M$.\n\nLet\n\\begin{gather*}\nI := \\{ 1 \\leq i \\leq k \\mid \\text{there exist $b$ and $j$ such that $\\lambda^{i, j}_{0, b} \\neq 0$} \\}\n\\\\\n\\intertext{and}\nN_1 := \\sum_{i \\in I} B_0 \\rho_i B_1.\n\\end{gather*}\nWe claim that\n\\[\nN_1 + \\rad_\\Lambda (M) = N + \\rad_\\Lambda (M).\n\\]\nObviously, we have $N_1 + \\rad_\\Lambda (M) \\subseteq N + \\rad_\\Lambda (M)$. For the converse inclusion, observe that $\\rho_i \\in \\rad_\\Lambda (M)$, for each $i \\notin I$. Consequently, $\\rho_i \\in N_1 + \\rad_\\Lambda (M)$, for each $1 \\leq i \\leq k$, hence $N \\subseteq N_1 + \\rad_\\Lambda (M)$. Obviously, $\\rad_\\Lambda (M) \\subseteq N_1 + \\rad_\\Lambda (M)$, hence the equality follows.\n\nNow we apply Lemma~\\ref{lem:a0prep} and get $N_1 = N$. Then minimality of $k$ implies that $I = \\{ 1, \\ldots, k \\}$, i.e., for each $1 \\leq i \\leq k$ there exist $b$ and $j$ such that $\\lambda^{i, j}_{0, b} \\neq 0$. Dually we prove that for each $1 \\leq i \\leq k$ there exist $a$ and $j$ such that $\\lambda^{i, j}_{a, 0} \\neq 0$.\n\\end{proof}\n\n\\subsection{Reduction to one relation}\nThe aim of this section is to prove the following result.\n\n\\begin{Lem}\\label{lem:1relation}\nIf $A$ is geometrically irreducible, then $k = 1$.\n\\end{Lem}\n\n\\begin{proof}\nAssume $k > 1$. Let $a_1$ be the minimal $a$ such that $\\lambda^{i, j}_{a, 0} \\neq 0$ for some $i, j \\geq 1$. Lemma~\\ref{lem:a0} ensures that such an $a_1$ exists. Without loss of generality we assume that $i = 1 = j$.\n\nThus $\\rho_1$ is of the form\n\\[\n\\rho_1 = \\varepsilon_0^{a_1} \\sum_{j = 1}^h \\sum_{a_1 \\leq a < m_0} \\lambda^{1, j}_{a, 0} \\varepsilon_0^{a - a_1} \\alpha_j + \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 1 \\leq b < m_1}} \\lambda^{1, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b.\n\\]\nRescaling by\n\\[\n\\alpha_1 \\leftarrow \\sum_{j = 1}^h \\sum_{a_1 \\leq a < m_0} \\lambda^{1, j}_{a, 0} \\varepsilon_0^{a - a_1} \\alpha_j\n\\]\nwe can assume that $\\rho_1$ is of the form\n\\[\n\\rho_1 = \\varepsilon_0^{a_1} \\alpha_1  + \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 1 \\leq b < m_1}} \\lambda^{1, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b\n\\]\n(in the paper we use the following notation: $\\alpha \\leftarrow \\omega$ means that $\\omega$ becomes the arrow $\\alpha$, while the remaining arrows do not change; note that the coefficients $\\lambda_{a, b}^{1, j}$ may change). Moreover, for $2 \\leq i \\le k$, we can now assume that $\\lambda_{a, 0}^{i, 1} = 0$ for any $a$. (This can be achieved by subtracting appropriate linear combinations of $\\varepsilon_0^{a - a_1} \\rho_1$ from $\\rho_i$.) On the other hand, if $\\lambda_{a, 0}^{i, j} \\neq 0$, for some $i, j \\geq 2$ and $a$, then $a \\geq a_1$.\n\nNext, let $a_2$ be the minimal $a$ such that $\\lambda^{i, j}_{a,0} \\neq 0$ for some $i, j \\geq 2$. Again existence of $a_2$ (in particular, that $h \\geq 2$) follows from Lemma~\\ref{lem:a0}. Observe that $a_2 \\geq a_1$. Without loss of generality we assume that $i = 2 = j$. After the arrow transformation\n\\[\n\\alpha_2 \\leftarrow \\sum_{j = 2}^h \\sum_{a_0 \\leq a < m_0} \\lambda^{2, j}_{a,0} \\varepsilon_0^{a - a_2} \\alpha_j\n\\]\nwe can obtain that $\\rho_1$ and $\\rho_2$ are of the form\n\\begin{gather*}\n\\rho_1 = \\varepsilon_0^{a_1} \\alpha_1  + \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 1 \\leq b < m_1}} \\lambda^{1, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b.\n\\\\\n\\intertext{and}\n\\rho_2 = \\varepsilon_0^{a_2} \\alpha_1  + \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 1 \\leq b < m_1}} \\lambda^{2, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b.\n\\end{gather*}\nMoreover, for $3 \\leq i \\le k$, we can again assume that $\\lambda_{a, 0}^{i, 1} = 0 = \\lambda_{a, 0}^{i, 2}$ for any $a$, while if $\\lambda_{a, 0}^{i, j} \\neq 0$, for some $i, j \\geq 3$ and $a$, then $a \\geq a_2$.\n\nContinuing inductively in this way, we can assume that, for each $1 \\leq i \\leq k$,\n\\[\n\\rho_i = \\varepsilon_0^{a_i} \\alpha_i + \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 1 \\leq b < m_1}} \\lambda^{i, j}_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b,\n\\]\nwith $0 < a_1 \\leq a_2 \\leq \\cdots \\leq a_k < m_0$. In particular, this means $k \\leq h$.\n\nLet $\\mathbf{d} = (a_2 + 1, 1)$. We show that $\\rep (A, \\mathbf{d})$ is reducible. Observe that $\\rep (A, \\mathbf{d})$ is described by the relations\n\\[\n\\varepsilon_0^{a_1} \\alpha_1, \\; \\varepsilon_0^{a_2} \\alpha_2, \\; \\cdots, \\; \\varepsilon_0^{a_k} \\alpha_k\n\\]\n(obviously, if $i > 2$ and $a_i > a_2$, then the relation $\\varepsilon_0^{a_i} \\alpha_i$ is redundant). The maximal partitions in $\\mathcal{P}_{m_0} (a_2 + 1)$ and $\\mathcal{P}_{m_1} (1)$ are $\\mathbf{p}_0 := (a_2 + 1)$ and $\\mathbf{q}_0 := (1)$, respectively. Using Lemma~\\ref{lem c part} we get\n\\[\nc_{\\mathbf{p}_0, \\mathbf{q}_0} = \\sum_{i = 1}^k c_{\\mathbf{p}_0, \\mathbf{q}_0} (\\varepsilon_0^{a_i} \\alpha_i).\n\\]\nApplying Proposition~\\ref{prop c formulas}~\\eqref{item c formula2} for $p = a_2 + 1$, $q = 1$ and $l = a_1$ ($l = a_2$, respectively), we get\n\\[\nc_{\\mathbf{p}_0, \\mathbf{q}_0} (\\varepsilon_0^{a_1} \\alpha_1) = a_2 + 1 - a_1 \\qquad \\text{and} \\qquad c_{\\mathbf{p}_0, \\mathbf{q}_0} (\\varepsilon_0^{a_2} \\alpha_2) = 1,\n\\]\nhence\n\\[\nc_{\\mathbf{p}_0, \\mathbf{q}_0} \\geq a_2 + 2 - a_1.\n\\]\nSimilarly, if we put $\\mathbf{p}_1 := (a_2, 1)$ and $\\mathbf{q}_1 := \\mathbf{q}_0$, then $\\mathcal{S}_{\\mathbf{p}_1, \\mathbf{q}_1}$ is described by the relation $\\varepsilon_0^{a_1} \\alpha_1$, hence we get\n\\[\nc_{\\mathbf{p}_1, \\mathbf{q}_1} = c_{\\mathbf{p}_1, \\mathbf{q}_1} (\\varepsilon_0^{a_1} \\alpha_1) = c_{(a_2), (1)} (\\varepsilon_0^{a_1} \\alpha_1) = a_2 - a_1,\n\\]\nwhich contradicts irreducibility according to Corollary~\\ref{cor irr}.\n\\end{proof}\n\n\\subsection{The one relation case}\nOur aim in this subsection is to prove the following result.\n\n\\begin{Prop} \\label{prop onerel}\nIf $\\mathcal{J} \\neq 0$ and $A$ is geometrically irreducible, then\n\\[\nA \\cong A (h, m_0, m_0, 1) \\qquad \\text{or} \\qquad A \\cong A (h, m_0, m_0, m_0 - 1).\n\\]\nIn other words, $m_0 = m_1$, $k = 1$, and we may assume either $\\rho_1 = \\rho^{(1)}$ or $\\rho_1 = \\rho^{(m_0 - 1)}$.\n\\end{Prop}\n\nThe proof of Proposition~\\ref{prop onerel} will be done in steps. We already know $k = 1$. Hence we may simplify our notation and put $\\rho := \\rho_1$.\nWithout loss of generality we may assume $m_0 \\geq m_1$. Then in particular $m_0 \\geq 2$ (recall that we assume $m_0 + m_1 \\geq 3$). Throughout the proof we use freely Corollary~\\ref{cor irr} and Proposition~\\ref{prop c formulas} without explicitly referring to them.\n\n\\begin{step} \\label{step1}\nThere exists $0 < n < m_0$ such that we may assume that, for each $0 \\leq p \\leq \\min \\{ n, m_1 - 1 \\}$,\n\\[\n\\rho = \\sum_{i = 0}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\omega \\varepsilon_1^{p + 1},\n\\]\nfor some $\\omega$. Furthermore, if $n > 1$, then $m_0 = n + 1$.\n\nIn particular, if $m_1 \\leq n + 1$, then we get\n\\[\n\\rho = \\sum_{i = 0}^{m_1 - 1} \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i.\n\\]\nOn the other hand, if $m_1 > n + 1$, then\n\\[\n\\rho = \\sum_{i = 0}^n \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\omega \\varepsilon_1^{n + 1}.\n\\]\n\\end{step}\n\n\\begin{proof}[Proof of Step~\\ref{step1}]\nThe proof will be inductive with respect to $p$.\n\n\\underline{\\textit{The base case}}\n\nWrite\n\\[\n\\rho = \\sum_{j = 1}^h \\sum_{\\substack{0 \\leq a < m_0 \\\\ 0 \\leq b < m_1 \\\\ a + b \\geq 1}} \\lambda^j_{a, b} \\varepsilon_0^a \\alpha_j \\varepsilon_1^b.\n\\]\nLet\n\\[\nn := \\min \\{ a \\mid \\text{$\\lambda_{a, 0}^j \\neq 0$ for some $j$} \\}.\n\\]\nNote that $n$ is well-defined (by Lemma~\\ref{lem:a0}) and $0 < n < m_0$. We may assume $\\lambda_{n, 0}^1 \\neq 0$ and make the substitution\n\\[\n\\alpha_1 \\leftarrow \\sum_{a, j} \\lambda_{a, 0}^j \\varepsilon_0^{a - n} \\alpha_j.\n\\]\nAs a result we get $\\rho$ of the form\n\\[\n\\varepsilon_0^n \\alpha_1 + \\omega \\varepsilon_1,\n\\]\nfor some $\\omega$, which shows the claim for $p = 0$. Note that this finishes the proof of Step~\\ref{step1} if $m_1 = 1$.\n\n\\underline{\\textit{The inductive step}}\n\nLet $0 < p \\leq \\min \\{ n, m_1 - 1 \\}$ and assume\n\\[\n\\rho = \\sum_{i = 0}^{p - 1} \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\omega \\varepsilon_1^p.\n\\]\nWrite\n\\[\n\\omega = \\sum_{a, j} \\lambda_a^j \\varepsilon_0^a \\alpha_j + \\omega' \\varepsilon_1,\n\\]\nand put\n\\[\n\\omega_q := \\sum_j \\lambda_q^j \\alpha_j \\qquad (q = 0, \\ldots, n - 1)\n\\]\nand\n\\[\n\\omega_n := \\sum_{a \\geq n, j} \\lambda_a^j \\varepsilon_0^{a - n} \\alpha_j.\n\\]\nIn particular,\n\\[\n\\rho = \\sum_{i = 0}^{p - 1} \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\sum_{q = 0}^n \\varepsilon_0^q \\omega_q \\varepsilon_1^p + \\omega' \\varepsilon_1^{p + 1}.\n\\]\n\n$1^\\circ$ ($q = n$). We show we may assume $\\omega_n = 0$.\n\nWe substitute\n\\[\n\\alpha_1 \\leftarrow \\alpha_1 + \\omega_n \\varepsilon_1^p,\n\\]\nand get\n\\[\n\\rho = \\sum_{i = 0}^{p - 1} \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\sum_{q = 0}^{n - 1} \\varepsilon_0^q \\omega_j \\varepsilon_1^p + \\omega'' \\varepsilon_1^{p + 1},\n\\]\nfor some $\\omega''$.\n\n$2^\\circ$ ($0 \\leq q \\leq n - p - 1$): We show that\n\\[\n\\omega_0 = \\cdots = \\omega_{n - p - 1} = 0\n\\]\n(there is nothing to prove if $p = n$). If this is not the case, let\n\\[\nl := \\min \\{ q \\in \\{ 0, \\ldots, n - p - 1 \\} \\mid \\omega_q \\neq 0 \\}.\n\\]\nTake the dimension vector $(l + 2, p + 1)$. Observe that $l + 2 \\leq n - p + 1$. Consequently, the variety is described by the relation\n\\[\n\\varepsilon_0^l \\omega_l \\varepsilon_0^p + \\varepsilon_0^{l + 1} \\omega_{l + 1} \\varepsilon_0^p.\n\\]\nThe maximal pair of partitions is $((l + 2), (p + 1))$ and $c_{(l + 2), (p + 1)} = 2$. On the other hand, we get no conditions for the stratum $\\mathcal{S}_{(l + 2), (p, 1)}$, hence $c_{(l + 2), (p, 1)} = 0$, which contradicts irreducibility.\n\nThus we get\n\\[\n\\rho = \\sum_{i = 0}^{p - 1} \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\sum_{q = n - p}^{n - 1} \\varepsilon_0^q \\omega_q \\varepsilon_1^p + \\omega'' \\varepsilon_1^{p + 1}.\n\\]\n\n$3^\\circ$ ($q = n - p$): We first show that $\\omega_{n - p} = \\lambda_{n - p}^1 \\alpha_1$ and $\\lambda_{n - p}^1 \\neq 0$.\n\nAssume first $\\omega_{n - p} = 0$ and consider the dimension vector $(n - p + 2, p + 1)$. Then the variety is described by the relation\n\\[\n\\varepsilon_0^{n - p + 1} \\alpha_1 \\varepsilon_1^{p - 1} + \\varepsilon_0^{n - p + 1} \\omega_{n - p + 1} \\varepsilon_1^p.\n\\]\nThe maximal pair of partitions is $((n - p + 2), (p + 1))$ and $c_{(n - p + 2), (p + 1)} = 2$. On the other there are no conditions for the stratum $\\mathcal{S}_{(n - p + 1, 1), (p + 1)}$, hence $c_{(n - p + 1, 1), (p + 1)} = 0$, which contradicts irreducibility.\n\nNext assume there exists $j \\neq 1$ such that $\\lambda_{n - p}^j \\neq 0$ and again consider the dimension vector $(n - p + 2, p + 1)$. In this case the variety is described by the relation\n\\[\n\\varepsilon_0^{n - p + 1} \\alpha_1 \\varepsilon_1^{p - 1} + \\varepsilon_0^{n - p} \\omega_{n - p} \\varepsilon_1^p + \\varepsilon_0^{n - p + 1} \\omega_{n - p + 1} \\varepsilon_1^p.\n\\]\nThe maximal pair of partitions is $((n - p + 2), (p + 1))$ and $c_{(n - p + 2), (p + 1)} = 3$ (note that $\\alpha_1$ and $\\omega_{n - p}$ are linearly independent). Let $(\\mathbf{p}, \\mathbf{q}) := ((n - p + 1, 1), (p + 1))$, if $n > p$, and $(\\mathbf{p}, \\mathbf{q}) := ((n - p + 2), (p, 1))$, if $n = p > 1$. Then $\\mathcal{S}_{\\mathbf{p}, \\mathbf{q}}$ is described by the condition $\\varepsilon_0^{n - p} \\omega_{n - p} \\varepsilon_1^p$ in the former case and by the condition $\\varepsilon_0^{n - p + 1} \\alpha_1 \\varepsilon_1^{p - 1}$ in the latter case. In both cases we get $c_{\\mathbf{p}, \\mathbf{q}} = 1$, which contradicts irreducibility.\n\nThus we only need to deal with the case $n = p = 1$. In this case we consider the dimension vector $(2 m_0, 2)$. Then the variety is described by the relation\n$\\varepsilon_0 \\alpha + \\omega_0 \\varepsilon_1$. The maximal pair of partitions is $((m_0, m_0), (2))$ and $c_{(m_0, m_0), (2)} = 2 (2 m_0 - 1) = 4 m_0 - 2$. On the other hand $c_{((m_0, m_0), (1, 1))} = 4 (m_0 - 1) = 4 m_0 - 4$, since $\\mathcal{S}_{(m_0, m_0), (1, 1)}$ is described by the condition $\\varepsilon_0 \\alpha$, which contradicts irreducibility.\n\nNow we show we may assume $\\lambda_{n - p}^1 = 1$. If $p = 1$, then we substitute\n\\[\n\\varepsilon_1 \\leftarrow \\lambda_{n - 1}^1 \\varepsilon_1.\n\\]\n\n\\underline{\\textit{Intermezzo}}\n\nObserve we have already proved our inductive claim for $p = 1$. In particular, this finishes the proof of Step~\\ref{step1} if $n = 1$. We additionally show that if $n > 1$, then $m_0 = n + 1$, which we use in the rest of the proof of Step~\\ref{step1}. Indeed, if this is not the case, then consider the dimension vector $(n + 2, 2)$. The variety is described by the relation $\\varepsilon_0^n \\alpha_1 + \\varepsilon_0^{n - 1} \\alpha_1 \\varepsilon_1$. The maximal pair of partitions is $((n + 2), (2))$ and $c_{((n + 2), (2))} = 4$. Moreover, $c_{(n + 1, 1), (2)} = 2$, which contradicts irreducibility.\n\n\\underline{\\textit{Inductive step (continuation of $3^\\circ$)}}\n\nAssume $p > 1$ and $\\lambda_{n - p}^1 \\neq 1$. If either $p < n$ or $p > 2$, then we consider the dimension vector $(n - p + 3, p + 1)$. The variety is described by the relation\n\\begin{multline*}\n\\varepsilon_0^{n - p + 2} \\alpha_1 \\varepsilon_1^{p - 2} + \\varepsilon_0^{n - p + 1} \\alpha_1 \\varepsilon_1^{p - 1} + \\lambda_{n - p}^1 \\varepsilon_0^{n - p} \\alpha_1 \\varepsilon_1^p\n\\\\\n+ \\varepsilon_0^{n - p + 1} \\omega_{n - p + 1} \\varepsilon_1^p + \\varepsilon_0^{n - p + 2} \\omega_{n - p + 2} \\varepsilon_1^p\n\\end{multline*}\n(note that $1^\\circ$ implies $\\omega_{n - p + 2} = 0$ if $p = 2$). The maximal pair of partitions is $((n - p + 3), (p + 1))$ and $c_{(n - p + 3), (p + 1)} = 4$. Let $(\\mathbf{p}, \\mathbf{q}) := ((n - p + 2, 1), (p + 1))$, if $p < n$, and $(\\mathbf{p}, \\mathbf{q}) := ((n - p + 3), (p, 1))$, if $n = p > 2$. Then $c_{\\mathbf{p}, \\mathbf{q}} = 2$, which contradicts irreducibility.\n\nFinally, assume $p = n = 2$ and $\\lambda_{n - p}^1 \\neq 1$. We consider the dimension vector $(2 m_0, p + 1) = (6, 3)$ (recall we proved $m_0 = n + 1$, provided $n > 1$). The variety is described by the relation\n\\[\n\\varepsilon_0^2 \\alpha_1 + \\varepsilon_0 \\alpha_1 \\varepsilon_1 + \\lambda_0^1 \\alpha_1 \\varepsilon_1^2 + \\varepsilon_0 \\omega_1 \\varepsilon_1^2\n\\]\nand the maximal pair of partitions is $((3, 3), (3))$. Then $c_{(3, 3), (3)} = 2 \\cdot 4 = 8$ and $c_{(3, 3), (2, 1)} = 2 \\cdot (2 + 1) = 6$, hence we again get a contradiction.\n\nThus we get\n\\[\n\\rho = \\sum_{i = 0}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\sum_{q = n - p + 1}^{n - 1} \\varepsilon_0^q \\omega_q \\varepsilon_1^p + \\omega'' \\varepsilon_1^{p + 1}.\n\\]\n\n$4^\\circ$ ($n - p + 1 \\leq q \\leq n - 2$): We prove that\n\\[\n\\omega_{n - p + 1} = \\cdots = \\omega_{n - 2} = 0\n\\]\n(obviously, there is nothing to prove if $p \\leq 2$). Assume this is not the case and put\n\\[\nl := \\min \\{ q \\in \\{ n - p + 1, \\ldots, n - 2 \\} \\mid \\omega_q \\neq 0 \\}.\n\\]\nIf there exists $j \\neq 1$ such that $\\lambda_l^j \\neq 0$, then we consider the dimension vector $(l + 2, p + 1)$. The variety is described by the relation\n\\[\n\\sum_{i = n - l - 1}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\varepsilon_0^l \\omega_l \\varepsilon_1^p + \\varepsilon_0^{l + 1} \\omega_{l + 1} \\varepsilon_1^p\n\\]\nand the maximal pair of partitions is $((l + 2), (p + 1))$. Note that $c_{(l + 2), (p + 1)} = p + l + 3 - n$, while $c_{(l + 2), (p, 1)} = p + l + 1 - n$, which contradicts geometric irreducibility of $A$. Thus $\\omega_l = \\lambda_l^1 \\alpha_1$.\n\nIf $\\lambda_l^1 \\neq 0$ and $l < n - 2$, then we consider the dimension vector $(l + 3, p + 1)$. The variety is described by the relation\n\\[\n\\sum_{i = n - l - 2}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\lambda_l^1 \\varepsilon_0^l \\alpha_1 \\varepsilon_1^p + \\varepsilon_0^{l + 1} \\omega_{l + 1} \\varepsilon_1^p + \\varepsilon_0^{l + 2} \\omega_{l + 2} \\varepsilon_1^p\n\\]\nand the maximal pair of partitions is $((l + 3), (p + 1))$. We have $c_{(l + 3), (p + 1)} = p + l + 4 - n$ and $c_{(l + 3), (p, 1)} = p + l + 2 - n$, which contradicts irreducibility.\n\nFinally, if $\\lambda_l^1 \\neq 0$ and $l = n - 2$, then we consider the dimension vector $(2 m_0, p + 1) = (2 n + 2, p + 1)$. In this case the variety is described by the relation\n\\[\n\\sum_{i = 0}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\lambda_l^1 \\varepsilon_0^{n - 2} \\alpha_1 \\varepsilon_1^p + \\varepsilon_0^{n - 1} \\omega_{n - 1} \\varepsilon_1^p\n\\]\nand the maximal pair of partitions is $((n + 1, n + 1), (p + 1))$. We have $c_{(n + 1, n + 1), (p + 1)} = 2 (p + 2) = 2 p + 4$ and $c_{(n + 1, n + 1), (p, 1)} = 2 (p + 1) = 2 p + 2$, hence the claim follows.\n\nThus if $p \\geq 2$, we get\n\\[\n\\rho = \\sum_{i = 0}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\varepsilon_0^{n - 1} \\omega_{n - 1} \\varepsilon_1^p + \\omega'' \\varepsilon_1^{p + 1}.\n\\]\n\n$5^\\circ$ ($q = n - 1$): If $p \\geq 2$, we may assume $\\omega_{n - 1} = 0$.\n\nFirst assume there exists $j \\neq 1$ such that $\\lambda_{n - 1}^j \\neq 0$ and consider the dimension vector $(2 m_0, p + 1) = (2 n + 2, p + 1)$. The variety is described be the relation\n\\[\n\\sum_{i = 0}^p \\varepsilon_0^{n - i} \\alpha_1 \\varepsilon_1^i + \\varepsilon_0^{n - 1} \\omega_{n - 1} \\varepsilon_1^p\n\\]\nand the maximal pair of partitions is $((n + 1, n + 1), (p + 1))$. We get $c_{(n + 1, n + 1), (p + 1)} = 2 (p + 2) = 2 p + 4$ and $c_{(n + 1, n + 1), (p, 1)} = 2 (p + 1) = 2 p + 2$. Thus irreducibility implies $\\omega_{n - 1} = \\lambda_{n - 1}^1 \\alpha_1$ and  we obtain the claim by substituting\n\\[\n\\varepsilon_1 \\leftarrow \\varepsilon_1 + \\lambda_{n - 1}^1 \\varepsilon_1^p.\n\\]\n(we use here that $p \\geq 2$).\n\\end{proof}\n\n\\begin{step} \\label{step2}\nIf $n > 1$, then $m_1 = n + 1$.\n\\end{step}\n\nNote that Step~\\ref{step2} finishes the proof if $n > 1$.\n\n\\begin{proof}[Proof of Step~\\ref{step2}]\nWe know $m_1 \\leq m_0 = n + 1$. If $m_1 < m_0$, then consider the dimension vector $(m_0 - m_1 + 1, 2 m_1)$. Then the variety is described by the relation\n\\[\n\\varepsilon_0^{m_0 - m_1} \\alpha_1 \\varepsilon_1^{m_1 - 1}\n\\]\nand the maximal pair of partitions is $((m_0 - m_1 + 1), (m_1, m_1))$. Since $c_{(m_0 - m_1 + 1), (m_1, m_1)} = 2 \\cdot 1 = 2$ and $c_{(m_0 - m_1, 1), (m_1, m_1)} = 0$, we get a contradiction to irreducibility.\n\\end{proof}\n\n\\begin{step} \\label{step3}\nIf $n = 1$, then we may assume that, for each $2 \\leq p \\leq m_1$,\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega \\varepsilon_1^p,\n\\]\nfor some $\\omega$. In particular, for $p = m_1$, we get\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1.\n\\]\n\\end{step}\n\n\\begin{proof}[Proof of Step~\\ref{step3}]\nThe proof will be inductive with respect to $p$.\n\n\\underline{\\textit{The base case}}\n\nIf $p = 2$, then the claim follows from Step~\\ref{step1}.\n\n\\underline{\\textit{The inductive step}}\n\nAssume $3 \\leq p \\leq m_1$ and\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega \\varepsilon_1^{p - 1},\n\\]\nfor some $\\omega$. Write\n\\[\n\\omega = \\sum_{a, j} \\lambda_a^j \\varepsilon_0^a \\alpha_j \\varepsilon_1 + \\omega' \\varepsilon_1,\n\\]\nand put\n\\[\n\\omega_0 := \\sum_j \\lambda_0^j \\alpha_j \\qquad \\text{and} \\qquad \\omega_1 := \\sum_{a \\geq 1, j} \\lambda_a^j \\varepsilon_0^{a - 1} \\alpha_j,\n\\]\nthus\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega_0 \\varepsilon_1^{p - 1} + \\varepsilon_0 \\omega_1 \\varepsilon_1^{p - 1} + \\omega' \\varepsilon_1^p.\n\\]\n\n$1^\\circ$ ($q = 1$). If we substitute\n\\[\n\\alpha_1 \\leftarrow \\alpha_1 + \\omega_1 \\varepsilon_1^{p - 1},\n\\]\nthen we get\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega_0 \\varepsilon_1^{p - 1} + \\omega'' \\varepsilon_1^p,\n\\]\nfor some $\\omega''$.\n\n$2^\\circ$ ($q = 0$). Assume first there exists $j \\neq 1$ such that $\\lambda_a^j \\neq 0$. If we consider the dimension vector $(2 m_0, p)$, then the variety is described by the relation\n\\[\n\\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega_0 \\varepsilon_1^{p - 1}\n\\]\nand the maximal pair of partitions is $((m_0, m_0), (p))$. We have $c_{(m_0, m_0), (p)} = 2 (p (m_0 - 1) + 1) = 2 p m_0 - 2 p + 2$ (note that $p \\leq m_1 \\leq m_0$), while $c_{(m_0, m_0), (p - 1, 1)} = 2 [(p - 1) (m_0 - 1) + (m_0 - 1)] = 2 p m_0 - 2 p$, which contradicts irreducibility.\n\nThus $\\omega_0 = \\lambda_0^1 \\alpha_1$. Now we substitute\n\\[\n\\varepsilon_1 \\leftarrow \\varepsilon_1 + \\lambda_0^1 \\varepsilon_1^{p - 1}\n\\]\nand we get\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1 + \\omega''' \\varepsilon_1^p,\n\\]\nfor some $\\omega'''$.\n\\end{proof}\n\n\\begin{step} \\label{step4}\nWe have $m_0 = m_1$.\n\\end{step}\n\n\\begin{proof}\nWe know the claim holds if $n > 1$, hence assume $n = 1$. In particular,\n\\[\n\\rho = \\varepsilon_0 \\alpha_1 + \\alpha_1 \\varepsilon_1.\n\\]\nIf $m_1 < m_0$, then consider the dimension vector $(m_0, 2 m_1)$. Then the maximal pair of partitions is $((m_0), (m_1, m_1))$ and $c_{(m_0), (m_1, m_1)} = 2 m_1 (m_0 - 1) = 2 m_0 m_1 - 2 m_1$. On the other hand $c_{(m_0 - 1, 1), (m_1, m_1)} = 2 m_1 (m_0 - 2) + 2 (m_1 - 1) = 2 m_0 m_1 - 2 m_1 - 2$. This contradiction finishes the proof of Step~\\ref{step4} and Proposition~\\ref{prop onerel}.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:intro3}}\nWe finish now Section~\\ref{sect proof one} explaining how the above imply Theorem~\\ref{thm:intro3}. Let $A$ be a geometrically irreducible algebra with exactly two simple modules. Corollary~\\ref{coro quiver} implies $A$ is Morita equivalent to $A' (h,m_0, m_1) \/ \\mathcal{J}$, for some $m_0, m_1 \\geq 1$, $h \\geq 0$, and an ideal $\\mathcal{J}$. We may also assume $m_0$, $m_1$ and $h$ are minimal. If $\\mathcal{J} = 0$, we are done, thus assume $\\mathcal{J} \\neq 0$. In this case $m_0 + m_1 \\geq 3$ and $h \\geq 1$, thus we are in setup of Section~\\ref{sect proof one}, hence Proposition~\\ref{prop onerel} implies the claim. \\qed\n\n\n\\section{Proof of Theorem~\\ref{thm:conj}} \\label{sect proof two}\n\n\n\\subsection{Degree of a relation}\nLet $Q$ be a quiver. For a path $\\sigma = \\alpha_1 \\cdots \\alpha_l$ we define its degree $\\deg \\sigma$ to be the number of indices $i$ such that $\\alpha_i$ is not a loop. If $\\rho = \\sum_{i = 1}^k \\lambda_i \\sigma_i$ with $\\lambda_1, \\ldots, \\lambda_k \\neq 0$ and $\\sigma_1$, \\ldots, $\\sigma_k$ pairwise different, then the degree $\\deg \\rho$ of $\\rho$ is the minimum of $\\deg \\sigma_i$, $i = 1, \\ldots, k$. Recall that by a shortcut we mean an arrow $\\alpha$ such that there exists a path $\\sigma$ of degree at least $2$ such that $s (\\sigma) = s (\\alpha)$ and $t (\\sigma) = t (\\alpha)$.\n\n\\begin{Lem} \\label{lem:degree}\nLet $A = K Q \/ \\mathcal{I}$ be a geometrically irreducible bound quiver algebra. If there are no shortcuts in $Q$, then $\\mathcal{I}$ is generated by relations of degree at most $1$.\n\\end{Lem}\n\n\\begin{proof}\nLet $R_0$ and $R_1$ be the sets of relations of degree $0$ and $1$ in $\\mathcal{I}$, respectively. Our aim is to show that $\\mathcal{I} \\subseteq \\mathcal{J} :=(R_0 \\cup R_1)$. Fix a relation $\\rho = \\sum_{i = 1}^k \\lambda_i \\sigma_i$ in $\\mathcal{I}$ and assume $\\deg \\rho \\geq 2$. For a path $\\sigma = \\alpha_1 \\cdots \\alpha_n$ let $\\supp (\\sigma) := \\{ t (\\sigma), s (\\alpha_1), \\ldots, s (\\alpha_n) \\} \\subseteq Q_0$. It is sufficient to show that $\\rho_I \\in \\mathcal{J}$, for each $I \\subseteq Q_0$, where\n\\[\n\\rho_I := \\sum_{i : \\supp (\\sigma_i) = I} \\lambda_i \\sigma_i.\n\\]\nIn other words, we may assume $\\supp (\\sigma_1) = \\cdots = \\supp (\\sigma_k) =: I$.\n\nLet $B := A \/ A e A$, where $e := \\sum_{i \\not \\in I} e_i$. Since there are nor shortcuts neither oriented cycles of positive degree in $Q$ (see Proposition~\\ref{prop known} for the latter claim), $B = K Q' \/ \\mathcal{I}'$, where $Q'$ is a subquiver of the quiver\n\\[\n\\xymatrix{\n0 \\ar@(ul,ur)^{\\varepsilon_0}\n&\n1 \\ar@(ul,ur)^{\\varepsilon_1} \\ar@<-1ex>[l]_{\\alpha_{11}} \\ar@<1ex>^{\\alpha_{h_11}}_{\\cdots}[l]\n&\n2 \\ar@(ul,ur)^{\\varepsilon_2} \\ar@<-1ex>[l]_{\\alpha_{12}} \\ar@<1ex>^{\\alpha_{h_22}}_{\\cdots}[l]\n&\n\\cdots \\ar@<-1ex>[l]_{\\alpha_{13}} \\ar@<1ex>^{\\alpha_{t3}}_{\\cdots}[l]\n&\nn \\ar@(ul,ur)^{\\varepsilon_n} \\ar@<-1ex>[l]_{\\alpha_{1n}} \\ar@<1ex>^{\\alpha_{h_nn}}_{\\cdots}[l]\n}.\n\\]\nObviously $\\rho \\in \\mathcal{I}'$. Since $B$ is geometrically irreducible, we know from \\cite{BobinskiSchroer2017a}*{Proposition~5.4} that $\\mathcal{I}'$ is generated by relations at most $1$. If $\\rho' \\in \\mathcal{I}'$, then there exist $\\omega_1, \\omega_2 \\in K Q$ such that $\\rho' + \\omega_1 e \\omega_2 \\in \\mathcal{I}$. If additionally we assume $\\deg \\rho' \\leq 1$, then $\\omega_1 e \\omega_2 = 0$, hence $\\rho' \\in \\mathcal{I}$. Indeed, if this is not the case, then $\\deg (\\omega_1 e \\omega_2) \\geq 2$, which is a contradiction, since there are neither shortcuts nor oriented cycles of positive degree in $Q$. The above implies $\\mathcal{I}' \\subseteq (R_0 \\cup R_1) = \\mathcal{J}$. In particular, $\\rho \\in \\mathcal{J}$, which finishes the proof.\n\\end{proof}\n\n\\subsection{No overlapping relations}\nRoughly speaking, the following lemma says that relations of degree $1$ cannot overlap.\n\n\\begin{Lem} \\label{lem:nooverlap}\nLet $A = K Q \/ \\mathcal{I}$. Assume $Q$ has three vertices and there are no shortcuts in $Q$. If $A$ is geometrically irreducible, then there exist relations $\\rho_1$, \\ldots, $\\rho_n$ of degree $0$ and a relation $\\rho$ of degree $1$ such that\n\\[\n\\mathcal{I} = (\\rho_1, \\ldots, \\rho_n, \\rho).\n\\]\n\\end{Lem}\n\n\\begin{proof}\nSince $A$ is geometrically irreducible and $Q$ contains no shortcuts, Proposition~\\ref{prop known} implies that up to duality either $Q$ is a subquiver of\n\\[\nQ' : \\xymatrix{\n0 \\ar@(ul,ur)^{\\varepsilon_0} & 1 \\ar@(ul,ur)^{\\varepsilon_1} \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l] & 2 \\ar@<1ex>[l]^{\\beta_l} \\ar@<-1ex>_{\\beta_1}^{\\cdots}[l] \\ar@(ul,ur)^{\\varepsilon_2}}\n\\]\nor a subquiver of\n\\[\nQ'' : \\xymatrix{\n0 \\ar@(ul,ur)^{\\varepsilon_0} & 1 \\ar@(ul,ur)^{\\varepsilon_1} \\ar@<-1ex>[l]_{\\alpha_1} \\ar@<1ex>^{\\alpha_h}_{\\cdots}[l]  \\ar@<1ex>[r]^{\\beta_1} \\ar@<-1ex>_{\\beta_l}^{\\cdots}[r] & 2 \\ar@(ul,ur)^{\\varepsilon_2}}.\n\\]\nWe concentrate on the former case.\n\nFirst observe, that \\cite{BobinskiSchroer2017a}*{Proposition~5.4} (or Lemma~\\ref{lem:degree}) implies that $\\mathcal{I}$ is generated by relations of degree at most $1$. Moreover, it follows from Theorem~\\ref{thm:intro3} that we may assume there is at most one relation $\\rho$ such that $s (\\rho) = 1$ and $t (\\rho) = 0$, and if there is such a relation, then $\\varepsilon_0$ and $\\varepsilon_1$ belong to $Q$. We have a similar claim for the vertices $1$ and $2$. Consequently, using Theorem~\\ref{thm:intro3} again we may assume $Q = Q'$ and $\\mathcal{I}$ is generated by relations $\\varepsilon_0^m$, $\\varepsilon_1^m$, $\\varepsilon_2^m$, $\\sum_{i = 0}^{n_1} \\varepsilon_0^i \\alpha_1 \\varepsilon_1^{n_1 - i}$ and $\\sum_{j = 0}^{n_2} (\\varepsilon_1')^j \\beta_1 \\varepsilon_2^{n_2 - i}$, for some $0 < n_1, n_2 < m$, where $\\varepsilon_1' = \\lambda_1 \\varepsilon_1 + \\cdots + \\lambda_{m - 1} \\varepsilon_1^{m - 1}$, for some $\\lambda_1, \\ldots, \\lambda_{m - 1} \\in K$, with $\\lambda_1 \\neq 0$. Finally, by symmetry we may assume $n_1 \\leq n_2$.\n\nWe explain the above claim more precisely. First, we apply Theorem~\\ref{thm:intro3} to the algebra $(e_1 + e_2) A (e_1 + e_2)$. In this way we get relations $\\varepsilon_1^m$, $\\varepsilon_2^m$ and $\\sum_{j = 0}^{n_2} \\varepsilon_1^j \\beta_1 \\varepsilon_2^{n_2 - i}$. Next, we apply Theorem~\\ref{thm:intro3} to the algebra $(e_0 + e_1) A (e_0 + e_1)$ and get relations $\\varepsilon_0^m$, $\\varepsilon_1^m$ (note that $m$ does not change) and $\\sum_{j = 0}^{n_1} \\varepsilon_1^j \\beta_1 \\varepsilon_2^{n_2 - i}$. However, in this step we may modify $\\varepsilon_1$ and that is why we need to introduce $\\varepsilon_1'$.\n\nNow consider the dimension vector $\\mathbf{d} = (1, n_2 + 1, 1)$. Then the variety $\\rep (A, \\mathbf{d})$ is described by the conditions $\\alpha_1 \\varepsilon_1^{n_1}$ and $\\varepsilon_1^{n_2} \\beta_1$. Let $\\mathcal{U}$ be the subset of $\\rep (A, \\mathbf{d})$ consisting of $M$ such that $M (\\varepsilon_1)$ is of rank (at least) $n_2$ (equivalently, $M (\\varepsilon_1)$ is similar to $J_{n_2 + 1}$). Then $\\mathcal{U}$ is an (irreducible) open subset of $\\rep (A, \\mathbf{d})$. Moreover, by arguments analogous to those presented in\nSubsection~\\ref{sub strat}, one calculates that\n\\[\n\\dim \\mathcal{U} = n_2^2 + n_2 + h n_2 + k n_2 - n_1 - 2.\n\\]\nOn the other hand, if $\\mathcal{V}$ be the subset of $\\rep (A, \\mathbf{d})$ consisting of $M$ such that $M (\\varepsilon_1)$ is similar to $J_{(n_2, 1)}$, then again\n\\[\n\\dim \\mathcal{V} = n_2^2 + n_2 + h n_2 + k n_2 - n_1 - 2.\n\\]\nConsequently, $\\mathcal{V}$ cannot be contained in the closure of $\\mathcal{U}$, which contradicts irreducibility and finishes the proof in this case.\n\nThe case, when $Q$ is a subquiver of $Q''$ is treated in the same way. The only difference is that we do not need to refer to~\\cite{BobinskiSchroer2017a}*{Proposition~5.4} in this case.\n\\end{proof}\n\n\\subsection{Proof of Theorem~\\ref{thm:conj}}\nWe present now the proof of Theorem~\\ref{thm:conj}. Let $A = K Q \/ \\mathcal{I}$ be a minimal geometrically irreducible bound quiver algebra and assume there are no shortcuts in $Q$. By definition minimality of $A$ implies that $Q$ is connected. In particular, if there are no arrows in $Q$, then $Q$ has exactly one vertex.\n\nWe know from Lemma~\\ref{lem:degree} that $\\mathcal{I}$ is generated by relations of degree at most $1$. Fix sets $R_0$ and $R_1$ of relations of degree $0$ and $1$, respectively, such that $R_0 \\cup R_1$ generates $\\mathcal{I}$. We also assume that the pair $(R_0, R_1)$ is minimal in the sense, that $|R_1|$ is minimal possible.\n\nAssume first there exists an arrow $\\alpha$ in $Q$, which is not a loop, such that there is no $\\rho \\in R_1$ with $s (\\rho) = s (\\alpha)$ and $t (\\rho) = t (\\alpha)$. Then minimality of $A$ implies that $A$ is the path algebra of the quiver\n\\[\n\\xymatrix{\ni & j \\ar[l]_{\\alpha}\n},\n\\]\nwhere $i := t (\\alpha)$ and $j := s (\\alpha)$, hence in particular $Q$ has two vertices. Indeed, if this is not the case, then we put $Q' := (\\{ i, j \\}, \\{ \\alpha \\})$, $\\mathcal{I}' := 0$, $Q'' := (Q_0, Q_1 \\setminus \\{ \\alpha \\})$, and $\\mathcal{I}'' := \\mathcal{I} \\cap K Q''$. Then obviously, $Q_0 = Q_0' \\cup Q_1''$, $Q_1 = Q_1' \\cup Q_1''$, $Q_1' \\cap Q_1'' = \\emptyset$, and $Q_1' \\neq \\emptyset \\neq Q_1''$ (the last claim uses connectivity again). Moreover, our assumption on $\\alpha$ implies $R_0 \\cup R_1 \\subseteq \\mathcal{I}''$, hence $\\mathcal{I}''$ generates $\\mathcal{I}$ in $K Q$. Altogether the above contradicts minimality of $A$.\n\nThus we may assume $Q_1 \\neq \\emptyset$ and for each arrow $\\alpha$ in $Q$ there exists $\\rho \\in R_1$ with $s (\\rho) = s (\\alpha)$ and $t (\\rho) = t (\\alpha)$. Fix $\\alpha \\in Q_1$ and $\\rho_1 \\in R_1$ such that $s (\\rho_1) = s (\\alpha)$ and $t (\\rho_1) = t (\\alpha)$. If $|Q_0| > 2$, then connectivity of $Q$ implies that there exists $\\beta \\in Q_1$ such that $\\{ s (\\beta), t (\\beta) \\} \\cap \\{ s (\\alpha), t (\\alpha) \\}$ has exactly one element. Fix $\\rho_2 \\in R_1$ such that $s (\\rho_2) = s (\\beta)$ and $t (\\rho_2) = t (\\beta)$. Put $I := \\{ s (\\alpha), t (\\alpha) \\} \\cup \\{ s (\\beta), t (\\beta) \\}$ (note that $|I| = 3$) and $B := A \/ A e A$, where $e := \\sum_{i \\not \\in I} e_i$. Then $B = K Q' \/ \\mathcal{I}'$, where $Q'$ is the full subquiver of $Q$ with the vertex set $I$. Note that $\\rho_1, \\rho_2 \\in \\mathcal{I}'$, as there are no shortcuts in $Q$. Since $B$ is geometrically irreducible, Lemma~\\ref{lem:nooverlap} implies there exists relations $\\rho_1'$, \\ldots, $\\rho_n'$ of degree $0$ and a relation $\\rho$ of degree $1$ such that $\\mathcal{I}' = (\\rho_1', \\ldots, \\rho_n', \\rho)$. Since there are neither shortcuts nor oriented cycles of positive degree in $Q$, $\\rho_1', \\ldots, \\rho_n', \\rho \\in \\mathcal{I}$ (compare the proof of Lemma~\\ref{lem:degree}). Consequently, if we put\n\\[\nR_0' := R_0 \\cup \\{ \\rho_1', \\ldots, \\rho_n' \\} \\qquad \\text{and} \\qquad R_1' := (R_1 \\setminus \\{ \\rho_1, \\rho_2 \\}) \\cup \\{ \\rho \\},\n\\]\nthen $\\mathcal{I}$ is generated by $R_0' \\cup R_1'$. Since $|R_1'| < |R_1|$, this contradicts minimality of the pair $(R_0, R_1)$, hence $|Q_0| \\leq 2$.\n\nThe list of possible forms of minimal geometrically irreducible algebras without shortcuts is now an immediate consequence of Theorem~\\ref{thm:intro3}. \\qed\n\n\\bibsection\n\n\\begin{biblist}\n\n\\bib{Birkhoff}{article}{\n   author={Birkhoff, G.},\n   title={Subgroups of abelian groups},\n   journal={Proc. London Math. Soc. (2)},\n   volume={38},\n   date={1935},\n   pages={385--401},\n}\n\n\\bib{BobinskiRiedtmannSkowronski}{collection.article}{\n   author={Bobi\\'nski, G.},\n   author={Riedtmann, C.},\n   author={Skowro\\'nski, A.},\n   title={Semi-invariants of quivers and their zero sets},\n   book={\n      title={Trends in Representation Theory of Algebras and Related Topics},\n      series={EMS Ser. Congr. Rep.},\n      publisher={Eur. Math. Soc., Z\\\"urich},\n   },\n   date={2008},\n   pages={49--99},\n}\n\n\\bib{BobinskiSchroer2017a}{article}{\n   author={Bobi\\'nski, G.},\n   author={Schr\\\"oer, J.},\n   title={Algebras with irreducible module varieties I},\n   eprint={arXiv:1709.05841},\n}\n\n\\bib{BobinskiSchroer2017b}{article}{\n   author={Bobi\\'nski, G.},\n   author={Schr\\\"oer, J.},\n   title={Algebras with irreducible module varieties III: Birkhoff varieties},\n   date={2017},\n   status={in preparation},\n}\n\n\\bib{Bongartz1991}{article}{\n   author={Bongartz, K.},\n   title={A geometric version of the Morita equivalence},\n   journal={J. Algebra},\n   volume={139},\n   date={1991},\n   number={1},\n   pages={159--171},\n}\n\n\\bib{Bongartz1998}{collection.article}{\n   author={Bongartz, K.},\n   title={Some geometric aspects of representation theory},\n   book={\n      title={Algebras and Modules, I},\n      series={CMS Conf. Proc.},\n      volume={23},\n      publisher={Amer. Math. Soc., Providence, RI},\n   },\n   date={1998},\n   pages={1--27},\n}\n\n\n\\bib{Geiss}{collection.article}{\n   author={Gei\\ss , Ch.},\n   title={Geometric methods in representation theory of finite-dimensional algebras},\n   book={\n      title={Representation theory of algebras and related topics},\n      series={CMS Conf. Proc.},\n      volume={19},\n      publisher={Amer. Math. Soc., Providence, RI},\n   },\n   date={1996},\n   pages={53--63},\n}\n\n\\bib{Gerstenhaber}{article}{\n   author={Gerstenhaber, M.},\n   title={On nilalgebras and linear varieties of nilpotent matrices. III},\n   journal={Ann. of Math. (2)},\n   volume={70},\n   date={1959},\n   pages={167--205},\n}\n\n\\bib{KussinLenzingMeltzer}{article}{\n   author={Kussin, D.},\n   author={Lenzing, H.},\n   author={Meltzer, H.},\n   title={Nilpotent operators and weighted projective lines},\n   journal={J. Reine Angew. Math.},\n   volume={685},\n   date={2013},\n   pages={33--71},\n}\n\n\n\\bib{RingelSchmidmeier}{article}{\n   author={Ringel, C. M.},\n   author={Schmidmeier, M.},\n   title={Invariant subspaces of nilpotent linear operators. I},\n   journal={J. Reine Angew. Math.},\n   volume={614},\n   date={2008},\n   pages={1--52},\n}\n\n\\bib{Simson}{article}{\n   author={Simson, D.},\n   title={Representation types of the category of subprojective representations of a finite poset over $K[t]\/(t^m)$ and a solution of a Birkhoff type problem},\n   journal={J. Algebra},\n   volume={311},\n   date={2007},\n   number={1},\n   pages={1--30},\n}\n\n\\bib{Zwara}{collection.article}{\n   author={Zwara, G.},\n   title={Singularities of orbit closures in module varieties},\n   book={\n      title={Representations of Algebras and Related Topics},\n      series={EMS Ser. Congr. Rep.},\n      publisher={Eur. Math. Soc., Z\\\"urich},\n   },\n   date={2011},\n   pages={661--725},\n}\n\t\t\n\n\\end{biblist}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThough the standard model (SM) has been verified to be correct times\nby times, many new physics beyond standard model are proposed to\nsolve both experimental and aesthetical problems,  such as neutrino\nmasses, $\\mu$ anomalous magnetic movement problem or hierarchy\nproblem, etc.  Many new models introduce  vector like particles (VLP) \\cite{VLP-rev}\nwhose right handed and left handed components transform in the same\nway under the weak SU(2)$\\times$U(1) gauge group. The extension is acceptable\nbecause the anomalies generated by the VLPs cancel automatically, \nand vector quarks can be heavy naturally. VLPs also arise\nin some grand unification theories. For example, \nin order to explain the little hierarchy problem between\nthe traditional GUT scale and string scale, a testable flipped $SU(5)\\times U(1)_X$\n model are proposed in Ref. \\cite{Jiang:2009zza}\nin which the TeV-scale VLPs were introduced~\\cite{Jiang:2006hf}.\nSuch kind of models can be constructed from the free fermionic string\nconstructions at the Kac-Moody level one~\\cite{Antoniadis:1988tt, Lopez:1992kg}\nand from the local F-theory model~\\cite{Beasley:2008dc, Jiang:2009zza}.\n\nHowever when we do the flavor physics with doublet VLPs in these models~\\cite{Li:2012xz, Li:2015nya},\na problem always appears when we are  dealing with the mass spectrum of quarks and leptons.\nLet us start with the SM in which all fermion masses come from the\nYukawa couplings. After the spontaneously gauge symmetry breaking, we\ncan get two separate mass matrices $M_U,~M_D$ for the up and down\ntype fermions. The mass eigen states are obtained  after the\ndiagonalization\n\\begin{eqnarray}\n  \\label{eq:1}\nZ_U^\\dagger M_U U_U=M_U^D,~~ Z_D^\\dagger M_D U_D=M_D^D,\n\\end{eqnarray}\n where\n$M_U^D={\\rm diag.}[m_u,m_c,m_t]$, $M_D^D={\\rm diag.}[m_d,m_s,m_b]$. \nThe physical measurable parameters are  $m_i$ and the so called CKM matrix\n\\begin{equation}\n  \\label{eq:defckm}\n  V_{\\rm CKM} = U_U^\\dagger U_D.\n\\end{equation}\nSince $M_U,~M_D$ come from separate Yukawa couplings, we can\nalways set one of the matrices diagonal, for example $M_U$,\nand  use the CKM matrix to get the Yukawa couplings\n\\begin{eqnarray}\n  \\label{eq:11}\nZ_D\\left(\n  \\begin{array}{ccc}\n    m_d & 0 & 0\\\\\n    0 & m_s & 0\\\\\n    0 & 0 & m_b\n  \\end{array}\n\\right)V_{\\rm CKM}^\\dagger=\n\\left(\n  \\begin{array}{ccc}\n    Y^D_{11}v & Y^D_{12}v  & Y^D_{13}v\\\\\n    Y^D_{21}v & Y^D_{22}v & Y^D_{23}v\\\\\n    Y^D_{31}v & Y^D_{32}v & Y^D_{33}v\n  \\end{array}\n\\right)\n\\end{eqnarray}\nfor the calculation in flavor physics. Note that $v$ is the vacuum\nexpectation value (VEV) of the Higgs, and $Z_D$ is a random unitary\nmatrix.\n\nSuch a trick can not be used in case of the participation of a vector\ndoublet, namely $Q$ with gauge charge $\\bf 3,~2,~\\frac{1}{6}$ and\n$\\bar Q$ with gauge charge $\\bf \\bar 3,~2,~-\\frac{1}{6}$, resulting\nbilinear term in the lagrangian $$M^VQ\\cdot\\bar Q.$$ It is\nclear that in the model, there are the same input parameters in the\nmatrices $M_U,~M_D$\n\\begin{eqnarray}\n  \\label{eq:mud}\n  M_U &=& \\left(  \\begin{array}{cccc}\n    Y^U_{11}v & Y^U_{12}v  & Y^U_{13}v & \\cdots\\\\\n    Y^U_{21}v & Y^U_{22}v & Y^U_{23}v & \\cdots\\\\\n    Y^U_{31}v & Y^U_{32}v & Y^U_{33}v & \\cdots\\\\\n    M^V_{41} & M^V_{42} & M^V_{43} &  \\cdots\n  \\end{array}\\right),~\n  M_D = \\left(  \\begin{array}{cccc}\n    Y^D_{11}v & Y^D_{12}v  & Y^D_{13}v & \\cdots\\\\\n    Y^D_{21}v & Y^D_{22}v & Y^D_{23}v & \\cdots\\\\\n    Y^D_{31}v & Y^D_{32}v & Y^D_{33}v & \\cdots\\\\\n    -M^V_{41} & -M^V_{42} & -M^V_{43} &  \\cdots\n  \\end{array}\\right).\n\\end{eqnarray}\nThe mass matrices for up and down type quarks are related to each\nother. Therefore,  we can not set one of the matrices diagonal and\nthe CKM matrix can not be got easily. The shooting method is always\nused to treat such an obstacle.  Random $M_U$ and $M_D$\nare generated to meet the requirements after diagonlization: the\nmass of eigen state and the measurements of elements of CKM matrix.\nHowever this is too much time consuming, and precise solution for\ndiagonalization is almost unavailable.  Although this is just  a\nnumerical problem, when one treats the VLP contributions to\nthe flavor physics seriously, diagonalization of quark matrices will\nbe the first and important step.\n\nIn this paper, we will first propose a general method to solve the\nobstacle in models with vector like quark doublets.  As its\napplication, we will study rare B decay $B\\to X_s\\gamma$ in the\nSM with one vector like quark doublet. The paper is\norganized as follows. We show the detail of the trick in\nSection~2.  The simple application to $B\\to X_s\\gamma$ process,\nincluding quark mass spectrum, Feynman rules\nand the Wilson coefficients, as well as the numerical analysis for\ncalculation of $B\\to X_s\\gamma$ is shown in Section~3.\nA summary is given in Section~4.\n\n\\section{The Trick of diagonalization of vector quark doublet}\\label{sec2}\nFirstly,  we address the problem clearly  on how to deal with the\ndiagonalization of $N\\times N$ matrix $M_U$ and $M_D$:\n\\begin{eqnarray}\n  \\label{eq:mud2}\nZ_U^\\dagger M_U U_U=M_U^D ,~ Z_D^\\dagger M_D U_D=M_D^D\n\\end{eqnarray}\nin which $M_U^D, M_D^D$ are the diagonal mass matrices for up and\ndown type quark, respectively. Note that $N$ should be greater than 3\nand the first three elements in the matices should be the three generations of quark\nmultiplates in the SM,  other elments with $N>3$ are the new multiplates introduced in\nnew physics beyond the SM. Then we have\n\\begin{eqnarray}\n  \\label{eq:mvud}\n  M_U = \\left(  \\begin{array}{ccccc}\n    Y^U_{11}v & Y^U_{12}v  & Y^U_{13}v &\\cdots & M_{U1N}\\\\\n    Y^U_{21}v & Y^U_{22}v  & Y^U_{23}v &\\cdots & M_{U2N}\\\\\n    Y^U_{31}v & Y^U_{32}v  & Y^U_{33}v &\\cdots & M_{U3N}\\\\\n    \\cdots & \\cdots  & \\cdots  &\\cdots & \\cdots \\\\\n    M^V_{N1} & M^V_{N2} & M^V_{N3}   & \\cdots  & M_{UNN}\n  \\end{array}\\right),~\n  M_D = \\left(  \\begin{array}{ccccc}\n    Y^D_{11}v & Y^D_{12}v  & Y^D_{13}v &\\cdots & M_{D1N}\\\\\n    Y^D_{21}v & Y^D_{22}v  & Y^D_{23}v &\\cdots & M_{D2N}\\\\\n    Y^D_{31}v & Y^D_{32}v  & Y^D_{33}v &\\cdots & M_{D3N}\\\\\n    \\cdots & \\cdots  & \\cdots  &\\cdots & \\cdots \\\\\n    -M^V_{N1} & -M^V_{N2} & -M^V_{N3}   & \\cdots  & M_{DNN}\n  \\end{array}\\right).\n\\end{eqnarray}\nThe last line of the two matrices has the same parameters except the\nlast elements.\n\nConsidering  that there are some same parameters in $M_U$ and $M_D$,\n we find that a very simple way is to add two matrices in Eq.\n(\\ref{eq:mvud}) \n\\begin{eqnarray}\n  M_U+M_D\n \n=\\left(Z_U M^D_U U_{\\rm CKMN}+Z_D M^D_D \\right) U^\\dagger_D.\n  \\label{eqvud}\n\\end{eqnarray}\nThe left side of the equation is\n\\begin{eqnarray}\n  \\label{eq:pvud}\n  M_U +M_D= \\left(  \\begin{array}{ccccc}\n    Y^U_{11}v+Y^D_{11}v & Y^U_{12}v+Y^D_{12}v  & Y^U_{13}v+Y^D_{13}v &\\cdots & M_{U1N}+M_{D1N}\\\\\n    Y^U_{21}v+Y^D_{21}v & Y^U_{22}v+Y^D_{22}v  & Y^U_{23}v+Y^D_{23}v &\\cdots & M_{U2N}+M_{D2N}\\\\\n    Y^U_{31}v+Y^D_{31}v & Y^U_{32}v+Y^D_{32}v  & Y^U_{33}v+Y^D_{33}v &\\cdots & M_{U3N}+M_{D3N}\\\\\n    \\cdots & \\cdots  & \\cdots  &\\cdots & \\cdots \\\\\n    0 & 0 & 0   & \\cdots  & M_{UNN}+M_{DNN}\n  \\end{array}\\right).\n\\end{eqnarray}\nObviously, the mass inputs from bilinear terms vanish. We can denote the matrix in the form as\n\\begin{eqnarray}\nM_U+M_D =M_{UD}=  \\left(  \\begin{array}{cc}\n    {\\bf M_{A}} & {\\bf M_{B}}  \\\\\n    {\\bf M_{0}}& M_C\n  \\end{array}\\right),\n\\end{eqnarray}\nin which ${\\bf M_{A}}$, ${\\bf M_{B}}$, ${\\bf M_{0}}$ are  $(N-1)\\times(N-1)$,\n $(N-1)\\times1$ and  $1\\times(N-1)$  matrices correspondingly.\n\nTo prepare for the diagonalization, we chose the diagonal mass matrix elements of\nquarks $(m_u,m_c,m_t,\\cdots m_X)$, $(m_d,m_s,m_b,\\cdots, m_Y)$\nand a matrix $U_{\\rm CKMN}$, which are determined partly by\nexperimental measurements as input parameters\n\\begin{eqnarray}\n  \\label{eq:ukmall}\n  U_{\\rm CKMN} &=& U^\\dagger_U U_D  =  \\left(  \\begin{array}{cc}\n    \\left(U_{\\rm CKM}\\right)_{3\\times 3}  &   \\cdots  \\\\\n  \\cdots   & U_{NN}\n  \\end{array}\\right)\n=  \\left(  \\begin{array}{cc}\n    \\left(  \\begin{array}{ccc}\n    U_{ud} & U_{us} & U_{ub}  \\\\\n    U_{cd} & U_{cs} & U_{cb}  \\\\\n    U_{td} & U_{ts} & U_{tb}\n  \\end{array}\\right)  &   \\cdots  \\\\\n  \\cdots   & U_{NN}\n  \\end{array}\\right).\n  \\label{CKM4}\n\\end{eqnarray}\nNote that above $Z_U,~Z_D,~U_U,~U_D$ are unitary matrices,\nbut $\\left(U_{\\rm CKM}\\right)_{3\\times 3}$ is not an ordinary CKM\nmatrix $V_{\\rm CKM}$ which is non-unitary in this case. Detailed dicussion\nwill be shown in the following section.\n\nWhat we need to do for the next is to generate a unitary matrix $U_D$. In\nthe similar way we denote $U_D$ as\n\\begin{eqnarray}\nU_D =  \\left(  \\begin{array}{cc}\n    {\\bf U_{DA}} & {\\bf U_{DB}}  \\\\\n    {\\bf U_{D0}}& U_{DNN}\n  \\end{array}\\right).\n\\end{eqnarray}\nBoth sides of Eq. (\\ref{eqvud}) times the matrix $U_D$, we can get\n\\begin{eqnarray}\nM_{UD}U_D &=&  \\left(  \\begin{array}{cc}\n    {\\bf M_{A}}{\\bf U_{DA}}+{\\bf M_{B}}{\\bf U_{D0}} & {\\bf M_{A}}{\\bf U_{DB}} + {\\bf M_{B}}U_{DNN} \\\\\n   M_C{\\bf U_{D0}}  & M_C U_{DNN}\n  \\end{array}\\right)\\nn\\\\\n&=& \\left(Z_U M^D_U U_{\\rm CKMN}+Z_D M^D_D \\right).\n\\end{eqnarray}\nFrom above equation, we can get the last line of $U_D$ simply by\ninputting $M^D_U,~M^D_D, ~U_{\\rm CKMN}$ and random $Z_U$, $Z_D$:\n\\begin{eqnarray}\n\\left(Z_U M^D_U U_{\\rm CKMN}+Z_D M^D_D \\right)_{\\mbox{last line}}&=&\n \\left(  \\begin{array}{cc}    M_C{\\bf U_{D0}}  & M_C U_{DNN}\n  \\end{array}\\right)\\nn\\\\\n&=&  M_C {\\bf U_{D}}_N  ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\bf U_{D}}_N&=& \\left(  \\begin{array}{cccc}  U_{DN1} &  U_{DN2}& \\cdots  & U_{DNN}\n  \\end{array}\\right)\n\\end{eqnarray} is a unit vector in  $N$ dimension.\n\nNext we use the unit vector to generate total $U_D$. Since \n${\\bf M_{A}}$  and ${\\bf M_{B}}$ are random matrix, $U_D$ can be random too.\nThe unit vector  ${\\bf U_{D}}_{N-1}$ of $U_D$\ncan be determined as\n\\begin{eqnarray}\n{\\bf U_{D}}_{N-1}&=& \\left(\n\\begin{array}{ccccc}  -\\frac{U_{DN2}^\\ast}{\\sqrt{|U_{DN1}|^2+|U_{DN2}|^2}} &\n\\frac{U_{DN1}^\\ast}{\\sqrt{|U_{DN1}|^2+|U_{DN2}|^2}} & 0 &\\cdots\n& 0\n\\end{array}\\right).\n\\end{eqnarray}\nIt is clear that the vector is orthogonal to ${\\bf U_{D}}_N$ and\nnormalized to 1. Then we use the first three elements of ${\\bf\nU_{D}}_N$ and ${\\bf U_{D}}_{N-1}$ to generate ${\\bf U_{D}}_{N-2}$:\nNormalize the algebraic complements of first line of the $3\\times 3$\nmatrix. Step by step, we can finally get (${\\bf U_{D}}_1$, ${\\bf\nU_{D}}_2$, $\\cdots$, ${\\bf U_{D}}_{N-1}$) and form a\nspecial $U_D^S$\n\\begin{eqnarray}\n  \\label{eq:mvud2}\n  U_D^S &=&\\left(\n    \\begin{array}{c}\n      {\\bf U_{D}}_1\\\\\n      \\cdots\\\\\n      {\\bf U_{D}}_{N-2}\\\\\n      {\\bf U_{D}}_{N-1}\\\\\n      {\\bf U_{D}}_N\n    \\end{array}\n\\right) =\\left(  \\begin{array}{ccccc}\n    U_{D11}   & U_{D12}    & U_{D13}   &\\cdots & U_{D1N}  \\\\\n    \\cdots & \\cdots  & \\cdots &\\cdots & 0\\\\\n    U_{D(N-2)1} & U_{D(N-2)2} & U_{D(N-2)3}   &\\cdots & 0\\\\\n    U_{D(N-1)1} & U_{D(N-1)2} & 0   & \\cdots  & 0 \\\\\n    U_{DN1} & U_{DN2} & U_{DN3}   & \\cdots  & U_{DNN}\n  \\end{array}\\right).\n\\end{eqnarray}\n\nFrom above steps, we can see that (${\\bf U_{D}}_1$, ${\\bf U_{D}}_2$,\n$\\cdots$, ${\\bf U_{D}}_{N-1}$) can be rotated into  any other\northogonal $N-1$ vectors to construct random matrix ${\\bf M_{A}}$\nand ${\\bf M_{B}}$,  only ${\\bf U_{D}}_N$ must be kept unchanged.\nTherefore,  a general unitary matrix can be realized by timesing a\nunitary $N\\times N$ matrix $U_R$,\n\\begin{eqnarray}\n  U_D = U_R U^S_D =\\left(\\begin{array}{cc}\n    {\\bf U_R}_{N-1}   & \\bf 0 \\\\\n    \\bf 0 & 1\n  \\end{array}\\right) U^S_D\n\\end{eqnarray}\nin which ${\\bf U_R}_{N-1}$ is a $(N-1)\\times (N-1)$ unitary matrix. We finish\nthe work by\n\\begin{eqnarray}\n  U^\\dagger_U &=& U_{\\rm CKMN}U^\\dagger_D\\\\\n  M_U  &=& Z_U M^D_U U^\\dagger_U\\\\\n  M_D  &=& Z_D M^D_D U^\\dagger_D\n \\end{eqnarray}\n\nAt this stage, we would like to summarize our method here\n\\begin{itemize}\n\\item Step 1: Chose ($m_u,m_c,m_t,\\cdots, m_X,m_d,m_s,m_b,\\cdots, m_Y$) and $U_{\\rm CKMN}$\n and  generate random unitary matrices $Z_U$ and $Z_D$  as the inputs for the model;\n\n\\item Step 2:  Determine the last line of matrix $Z_U M^D_U U_{\\rm CKMN}+Z_D M^D_D$ as\n\\begin{eqnarray}\n M_C\\left(  \\begin{array}{cccc}  U_{DN1} &  U_{DN2}& \\cdots  & U_{DNN}\n  \\end{array}\\right)\n\\end{eqnarray}\nand normalize it into  a unit vector ${\\bf U_D}_N$.\n\\item Step 3: Use the unit vector ${\\bf U_D}_N$ to generate other $N-1$ unitary vectors\n(${\\bf U_{D}}_1$, ${\\bf U_{D}}_2$, $\\cdots$, ${\\bf U_{D}}_{N-1}$),\nand form a special $U^S_D$\n\\begin{eqnarray}\n  \\label{eq:mvud3}\n  U_D^S &=&\\left( \n    \\begin{array}{ccccc}\n      {\\bf U_{D}}_1 &  \\cdots &  {\\bf U_{D}}_{N-2} & {\\bf U_{D}}_{N-1} & {\\bf U_{D}}_N\n    \\end{array}\n\\right)^T.\n\\end{eqnarray}\n\\item Step 4: Generate a $N-1$ unitary matrix ${\\bf U_R}_{N-1}$ to form a unitary matrix $U_R$\nwhich is\n\\begin{eqnarray}\n  U_R =\\left(\\begin{array}{cc}\n     {\\bf U_R}_{N-1}  & \\bf 0 \\\\\n    \\bf 0 & 1\n  \\end{array}\\right),\n\\end{eqnarray}\nthen, a general $U_D$ is obtained  by\n \\begin{eqnarray}\n  U_D = U_R U^S_D.\n\\end{eqnarray}\n\\item Step 5: Use these equations\n\\begin{eqnarray}\n  U^\\dagger_U &=& U_{\\rm CKMN}U^\\dagger_D,\\nonumber\\\\\n  M_U  &=& Z_U M^D_U U^\\dagger_U,\\nonumber\\\\\n  M_D  &=& Z_D M^D_D U^\\dagger_D,\n \\end{eqnarray}\nto get the inputs for the flavor physics.\n\\end{itemize}\n\nWe can see that by this trick we can skip the inputs of the bilinear\nmass terms $M^V_{Ni}$. In physical analysis, the mass of eigen states\n$m_{X,~Y}$ in the VLP models are inputs freely. $Z_{U}$\nand $Z_{D}$ can be generated randomly, $U_{U}$ and $U_D$ can also be\nscanned the most generally if we vary $U_R$ randomly. Thus the\nmethod can do the most general scan in the parameter space of mass\nmatrices in the models with VLPs for the numerical studies,  which\nwill be shown in the following section.\n\n\\section{$B\\to X_s\\gamma$ process in extension of the SM with one vector like quark doublet}\n\\label{sec3}\n\\subsection{The standard  model with vector like quarks}\n\\begin{table}\n  \\centering\n\\caption{A simple extension of the standard  model with one vector\nlike quarks doublet}\\label{tab1}\n  \\begin{tabular}{cc}\n  \\begin{tabular}{c|c}\n      & $\\bf SU(3),~SU(2),~U(1)$\\\\\n\\hline\n$ Q=\\left(\\begin{array}{c}\n U\\\\\n D\\end{array}\\right)_L$ & $\\bf 3,~2,~\\frac{1}{6}$\\\\\n $u_{R}$  &  $\\bf 3,~1,~\\frac{2}{3}$ \\\\\n $d_{R}$  &  $\\bf 3,~1,~-\\frac{1}{3}$\n  \\end{tabular}\n &   \\begin{tabular}{c|c}\n      & $\\bf SU(3),~SU(2),~U(1)$\\\\\n\\hline\n$ V_{Q}=\\left(\\begin{array}{c}\n\\bar{V}_{d}\\\\\n\\bar{V}_{u}\\end{array}\\right)_{R}$ & $\\bf \\bar{3},~2,~-\\frac{1}{6}$\\\\\n $\\bar{V}_{uL}$  &  $\\bf \\bar{3},~1,~-\\frac{2}{3}$ \\\\\n $\\bar{V}_{dL}$  &  $\\bf \\bar{3},~1,~\\frac{1}{3}$\n  \\end{tabular}\n  \\end{tabular}\n\\end{table}\nAs an application of  the method,  in this section we study the VLP\ncontribution to $B\\to X_s\\gamma$  in  a very simple VLP extension of\nSM for the demonstration. In the Tab. \\ref{tab1}, we list the gauge\nsymmetry of the matter multiplates in which the first two queues\nshow the quarks in the SM and the last two queues show the VLPs\nwith the anti-gauge symmetry. Note that we ignore partners of the last two queues\nwhose gauge symmetry are exactly the same as the first two queues of the SM.\nAs talked in the introduction, these VLPs can be heavy naturally.\nSince gauge symmetry of Higgs $H=(h^+,~h^0)^{\\rm T}$ is ($\\bf\n1,~2,~1\/2$), the lagrangian for two quarks  of the model is written as:\n\\begin{eqnarray}\n\\mathcal{L}=&&Y_{d}\\bar{Q}Hd_{R}+Y_{u}\\bar{Q}\\cdot\\bar{H}u_{R}+\nY_{Vu}\\bar{V_{Q}}H\\bar{V}_{uL}+Y_{Vd}\\bar{V_{Q}}\\cdot\\bar{H}\\bar{V}_{dL}\\nn\\\\\n    &&+M_{Q}V_{q}\\cdot Q+M_{u}\\bar{V}_{uL}u_{R}+M_{d}\\bar{V}_{dL}d_{R}+h.c.,\n\\end{eqnarray}\nin which $A\\cdot B=\\epsilon^{ij}A_i B_j$. The first line of the\nlagrangian is Yukawa terms, the second line is the bilinear terms.\nNote that $Y_u$, $Y_d$ are $3\\times 3$ matrix, without the\nbilinear terms, the model will be almost the same as the  fourth\ngeneration standard model (SM4).\n\nAfter the electro-weak symmetry breaking, we can get the mass matrices of\nup and down quarks in the basis of $(u,~c,~t,~V_u)$ and $(d,~s,~b,~V_d)$:\n\\begin{eqnarray}\nM_U= \\left(\\begin{array}{cccc}\nY_{u}^{11}v & Y_{u}^{12}v & Y_{u}^{13}v & M_{u}^{1}\\\\\nY_{u}^{21}v & Y_{u}^{22}v & Y_{u}^{23}v & M_{u}^{2}\\\\\nY_{u}^{31}v & Y_{u}^{32}v & Y_{u}^{33}v & M_{u}^{3}\\\\\n-M_{Q}^{1} & -M_{Q}^{2} & -M_{Q}^{3} & Y_{V_u}v\\end{array}\\right),~\n M_D=\\left(\\begin{array}{cccc}\nY_{d}^{11}v & Y_{d}^{12}v & Y_{d}^{13}v & M_{d}^{1}\\\\\nY_{d}^{21}v & Y_{d}^{22}v & Y_{d}^{23}v & M_{d}^{2}\\\\\nY_{d}^{31}v & Y_{d}^{32}v & Y_{d}^{33}v & M_{d}^{3}\\\\\nM_{Q}^{1} & M_{Q}^{2} & M_{Q}^{3} & Y_{V_d}v\\end{array}\\right),\n\\end{eqnarray}\nwhere $v$ is the VEV for $H$.\nThe first three elements of last line of the\nmatrices have the same parameter, making the scan of the parameter\nspace very difficult. These two matrices can be diagonalized by\nunitary matrices $U$ and $Z$,\n\\begin{eqnarray}\nZ_u^\\dagger M_UU_u={\\rm diag.}[m_{u},m_{c},m_{t},m_{X}],\\nn\\\\\nZ_d^\\dagger M_DU_d={\\rm diag.}[m_{d},m_{s},m_{b},m_{Y}].\n\\label{massdiag}\n \\end{eqnarray}\nProduct of the two matrices is denoted as\n\\begin{equation}\n  \\label{eq:ukm}\nU_{\\rm CKM4}=U_u^\\dagger U_d,\n\\end{equation}\nwhich is unitary $4\\times 4$ matrix. We stress that the trick we introduced in the above\nsection seems to just give us a numerical tool for quark masses and\nsome quark mixing matrices, but it is important in studying\nthe flavor physics in such models.\n\nFor studying VLP contributions to $B\\to X_s\\gamma$, we now present the\nFeynman rules for the interaction of  $\\bar{u}_ld_j\\chi^+, ~\\chi=W,~G$\n and $\\bar{d}_ld_jZ$ in the Feynman gauge which read:\n\\begin{eqnarray}\n&& {\\rm i}\\frac{g}{\\sqrt{2}}\\gamma^\\mu \\left[g^\\chi_L(i,j)P_L\n+g^W_R(i,j)P_R\\right],~~(\\chi=W,Z),\\label{gud}\\\\\n&& {\\rm i}\\frac{g}{\\sqrt{2}m_W}\\left[g^\\chi_L(i,j)P_L\n+g^\\chi_R(i,j)P_R\\right]~ (\\chi=G) \\label{hud}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\ng^W_L(i,j) &=& \\sum_{m=1}^3U_u^{*mi} U_d^{m,j},\\ \\\n  \\ g^W_R(i,j) =Z_u^{*4 i}Z_d^{4j}, \\label{ud_w}\\\\\ng^G_L(i,j) &=& \\sum_{k,m=1}^3\nY_{u}^{km}vZ_u^{*ki} U_d^{mj} + Y_{Vd}vZ_u^{*4i} U_d^{4j},\\label{ud_yl}\\\\\ng^G_R(i,j) &=& -\\sum_{k,m=1}^3 Y_{d}^{\\ast mk}vZ_d^{*k j} U_u^{mi} -\nY_{Vu}^\\ast vZ_d^{*4j} U_d^{4i}. \\label{ud_yr}\\\\\ng^Z_L(i,j) &=&\n-\\frac{1}{\\sqrt{2}\\cos\\theta_{W}}\\left[\\left(1-\\frac{2}{3}\\sin^{2}\n\\theta_{W}\\right)\\delta^{ij}-U_{d}^{*4i}U_{d}^{4j}\\right],\\label{ud_z1}\\\\\n g^Z_R(i,j) &=&  -\\frac{1}{\\sqrt{2}\\cos\\theta_{W}}\\left[-\\frac{2}{3}\\sin^{2}\n \\theta_{W}\\delta^{ij}+Z_{d}^{*4i}Z_{d}^{4j}\\right] .\\label{ud_z2}\n\\end{eqnarray}\nNote that $U(1)_{EM}$ interaction is not changed by the VLPs, thus \nthe vertices of photon and quarks are still the same as those in the SM.\nFrom above mass matrices and Feynman rules,  we can see that the  model\nhas two points to be explored:\n\\begin{itemize}\n\\item The CKM matrix is got from the $W^+\\bar{u}_id_j$ vertex in Eq. (\\ref{ud_w}) \n  \\begin{equation}\n    \\label{eq:vkm}\n  V^{ij}_{\\rm CKM4} = \\sum_{m=1}^3U_u^{*mi} U_d^{mj}\n   =U_{\\rm CKM4}^{ij}-U_u^{*4i} U_d^{4j}.\n  \\end{equation}\nwhich is non-unitary for that the indexes $i,j$ range form 1 to 4,\nbut the summation of index $m$ is from 1 to 3. \n$V^{ij}_{\\rm CKM4}$ is also a $4\\times 4$ matrix of which the upper left\nelements ($i,j\\ne4$)  are physical measurable value of CKM matrix $V$ as in the SM.\nThis is the key difference between VLP models and the SM4. Nevertheless,\nthe loop-level flavor change neutral current (FCNC) will be changed by the Yukawa interactions, \nthen the prediction of process $B\\to X_s\\gamma$  may be changed\nsignificantly.\n\\item The last terms in Eqs.(\\ref{ud_w})-(\\ref{ud_z2}), which we call the ``tail terms'',\nviolate the gauge universality of fermions and  cause tree-level FCNC processes\ninduced by the processes such as  $b \\to s\\ell^+\\ell^-$,  \nthen the  constraints on the parameter space need to be explored.\n\\end{itemize}\n\n\\subsection{Enhancement in $b\\to s$ transition}\n\\begin{figure}[hbtp]\n\\begin{center}\n\\scalebox{0.7}{\\epsfig{file=fig1.ps}} \n\\caption{Leading order Feynman diagram of $B\\to X_s\\gamma$ process.}\n\\label{fig1}\n\\end{center}\n\\end{figure}\nIn this subsection we focus attention on VLP contributions to\nrare B decay $B\\to X_s\\gamma$.\nThe starting point for rare B decays is the determination of  the\nlow-energy effective Hamiltonian obtained by integrating out the\nheavy degrees of freedom in the theory. For $b \\to s $ transition,\nthis can be written as\n\\begin{equation}\n{\\cal H}_{\\rm eff} = - \\frac{G_F}{\\sqrt{2}} V_{tb} V_{ts}^\\ast\n\\sum_{i=1}^{10} [C_i(\\mu) O_i(\\mu)+C^{'}_i(\\mu) O^{'}_i(\\mu)]~,~\\,\n\\label{eq:HeffBXsgamma}\n\\end{equation}\nwhere the effective operators $O_i$  are same as those in the SM\ndefined in Ref.~\\cite{BLOSM}. The chirality-flipped operators $O'_i$\nare obtained from $O_i$ by the replacement $\\gamma_5\\to -\\gamma_5$\nin quark current\\cite{Li:2012xz}. We calculate the Wilson\ncoefficient $C_7$ at matching scale $m_W$. The leading order \nFeynman diagrams are shown in FiG. \\ref{fig1} and  $C_7$ reads\n  \\begin{eqnarray}\n  C_{7}(m_W)&=&\\frac{1}{V_{tb}V_{ts}^{\\ast}}\\sum_{i=1}^{4}\\biggl[ g_{L}^{W\\ast}(i,2)g_{L}^{W}(i,3)A(x_i)\n + \\frac{g_{L}^{G\\ast}(i,2)g_{L}^{G}(i,3)}{m_{u_i}^2} x_i B(x_i)\\nn\\\\\n& &  + \\frac{g_{L}^{G\\ast}(i,2)g_{R}^{G}(i,3)}{m_{u_i}m_b} x_i\nC(x_i)  + \\frac{g_{L}^{W\\ast}(i,2)g_{R}^{G}(i,3)}{m_b}D(x_i)\\nn\\\\\n& & + \\frac{m_{u_i}}{m_b}g_{L}^{W\\ast}(i,2)g_{R}^{W}(i,3)E(x_i)\n    + \\frac{g_{L}^{G\\ast}(i,2)g_{R}^{W}(i,3)}{m_b}D(x_i)\\biggl] \\label{fc7}\n  \\end{eqnarray}\nwhere $x_i=m_{u_i}^2\/m_W^2$ and the loop function\n$A(x),~B(x),~C(x),~D(x),~E(x)$ are listed in the appendix.\nThe first two lines are the similar contribution as in the SM, while\nthe last lines are the terms come form the tail terms.  Note\nthat the contribution of right diagram in the second line of FIG. \\ref{fig1}\nis zero in the SM. The terms with $1\/m_b$ in above equation\nis extracted to compose the operator  ${\\cal O}_7$.\nThere are two differences in the calculation of $B\\to X_s\\gamma$ processes\ncompared with the SM. One is the tail terms of gauge or Yukawa interactions, another one\nis the new type of Yukawa interactions listed in Eqs. (\\ref{ud_yl}, \\ref{ud_yr})\nwhich can not be written into the simple form in the SM such as\n      \\begin{equation}\n        \\label{eq:smf}\n        g_L^{G,\\rm SM}(i,2)=m_{u_i}V_{is},~g_R^{G,\\rm SM}(i,3)=-m_{b}V_{ib}.\n      \\end{equation}\n\n\\begin{table}[htb]\n     \\caption[]{The CKM matrix elements constrained by the tree-level B decays.}\n     \\label{tab:c1B}\n     \\begin{center}\n       \\begin{tabular}{c||c|c}\n        \\hline\n        & absolute value   & direct measurement from \\\\ \\hline\n        $V_{ud}$ & $0.97425 \\pm 0.00022$  & nuclear beta decay \\\\\\hline\n        $V_{us}$ & $0.2252  \\pm 0.0009$   & semi-leptonic K-decay\\\\\\hline\n        $V_{ub}$ & $0.00415 \\pm 0.00049$  & semi-leptonic B-decay\\\\\\hline\n        $V_{cd}$ & $0.230   \\pm 0.011$    & semi-leptonic D-decay\\\\\\hline\n        $V_{cs}$ & $1.006   \\pm 0.023$    & (semi-)leptonic D-decay\\\\\\hline\n        $V_{cb}$ & $0.0409  \\pm 0.0011$   & semi-leptonic B-decay\\\\\\hline\n        $V_{tb}$ & $0.89  \\pm 0.07$       & (single) top-production\\\\\\hline\n       \\end{tabular}\n     \\end{center}\n \\end{table}\n\nIn the model with three generation quarks, the CKM matrix unitarity\nis already used in the calculations of the loop-level FCNC induced\nrare B decays. For consistency, in numerical analysis the\nconstraints on CKM matrix element are not from\nprocesses occurred at loop level, such as rare B decays, but from\ntree-level processes shown in Table~\\ref{tab:c1B}~\\cite{Beringer:1900zz, Eberhardt:2010bm}. \nSince there are no tree-level measurements of $V_{td}$, $V_{ts}$ now,\nwe use above inputs and the unitarity to get $3\\times3$ unitary matrix at first.\nThe method is that we scan  $(V_{ud}, V_{us}, V_{ub})$ randomly\n(keeping $|V_{ud}|^2+ |V_{us}|^2+ |V_{ub}|^2=1$ ) in range listed in Table~\\ref{tab:c1B},\nthen we define two parameters  $\\alpha,~\\beta$ and solve them by the equations\n\\begin{eqnarray}\n&&V_{ud}^\\ast (V_{cd}+\\alpha) + V_{us}^\\ast(V_{cs}+\\beta) + V_{ub}^\\ast V_{cb} =0,\\nn\\\\\n&&\\left|V_{cd}+\\alpha\\right|^2+  \\left|V_{cs}+\\beta\\right|^2+  \\left|V_{cb}\\right|^2  =1.\n\\end{eqnarray}\n$(V_{td}, V_{ts}, V_{tb})$ are got by the unitarity relation with \n$(V_{ud}, V_{us}, V_{ub})$ and $(V_{cd}, V_{cs}, V_{cb})$.\nAfter that we times the $3\\times3$ unitary matrix with three matrices\n\\begin{eqnarray}\n\\left(\n\\begin{array}{cccc}   {\\bf 1} &\\cdots &\\cdots &\\cdots \\\\\n\\cdots & \\cos \\theta_{4i}  & \\cdots  & \\sin \\theta_{4i} \\\\\n\\cdots & \\cdots  & {\\bf 1}  & \\cdots\\\\\n\\cdots & -\\sin \\theta_{4i}  & \\cdots  & \\cos \\theta_{4i}\n\\end{array}\\right)\n\\end{eqnarray} in which $i=1,~2,~3$ and $\\max(|\\theta_{41}|,~|\\theta_{42}|,~|\\theta_{43}|)<0.01\\pi$,\nto generate a $4\\times4$ unitary matrix $U_{\\rm CKM4}$.\n$V_{\\rm CKM4}$ are got by the Eq. (\\ref{eq:vkm}). All the corresponding \nelements should satisfy the experiment bound list in Table~\\ref{tab:c1B}\nand  $V_{td}$, $V_{ts}$ ($|V_{ts}|\\simeq 0.04$ which is consistent with the fitting results in \nRef. \\cite{Beringer:1900zz}) can be got too.\nWith this inputs in hand, the first task is to check  the scale of the\nmass parameter of model, such as $M_Q$, $m_X, m_Y$. From the\n$Z\\bar{b}b$ vertexes in Eq. (\\ref{ud_z1}) and Eq. (\\ref{ud_z2}),\nwe can see that in order to keep\ngauge universality of quarks, the tail terms in the Feynman rules\nmust be much smaller than the SM like terms, namely\n$|Z^{4i}_{u,d}|^2_{i=1,2,3},~|U^{4i}_{u,d}|^2_{i=1,2,3}\\ll\n\\sin^2\\theta_W$. Thus in the numerical studies we require\n\\begin{figure}[hbtp]\n\\begin{center}\n\\scalebox{0.4}{\\epsfig{file=fig2.ps}} \\caption{$M_V$ versus $M_Y$ under constraints\n$|Z^{4i}_{u,d}|^2_{i=1,2,3},~|U^{4i}_{u,d}|^2_{i=1,2,3}<\n10^{-4}.$}\\label{fig2}\n\\end{center}\n\\end{figure}\n\\begin{equation}\n  \\label{eq:vud2}\n  |Z^{4i}_{u,d}|^2_{i=1,2,3},~|U^{4i}_{u,d}|^2_{i=1,2,3}< 10^{-4}.\n\\end{equation}\nNote that though these elements are greater than $\\lambda^3$ \n(parameter in the Wolfenstein parameterization \\cite{Wolfenstein:1983yz}),\nthey are much smaller than the product of $V_{\\rm CKM3}^\\dagger V_{\\rm CKM3}$\n(almost equals $\\bf 1$), thus the requirements are suitable for indicating\nthe contraints from the deviation from unitarity.\n\nSince the scanning in the parameter space is freely, we set $ m_X = 1172{\\rm\nGeV}$ (mass of top quark plus 1000 GeV) and  scan $m_Y$ in the range of $(4.2, 1004) \\rm GeV$\n(mass of bottom quark plus 1000 GeV),\nand $Z_{u,d},~U_{u,d}$ randomly (ignoring the CP phases).  $M_V$ is defined by\n\\begin{equation}\n  \\label{eq:mv}\n  M_V = \\max(|M_Q^1|,~|M_Q^2|,~|M_Q^3|).\n\\end{equation} \nThe result for $M_V$ versus $M_Y$ is shown in the FIG. \\ref{fig2}\nwhich checks the mass input of vector doublet.  We can see that $M_V$\nincreases as  $m_Y$ growing up. However $M_V$ is much smaller than\n$m_X$ and $m_Y$. Small mixings lead to  parameter $M_Q$ which\ndetermine the mixing between SM quarks and vector like quarks are\nalso suppressed.  This is in agreement with that the deviation from\nunitarity is suppressed by the ratio $m\/m_{X,Y}$ where $m$ denotes\ngenerically the standard quark masses, which is  a typical result of VLP models.\n\\cite{Langacker:1988ur,delAguila:1982fs,delAguila:1987nn,Cheng:1991rr,Botella:2012ju}\n\n\\begin{figure}[hbtp]\n\\begin{center}\n\\scalebox{0.4}{\\epsfig{file=fig3-1.ps}}\n\\scalebox{0.4}{\\epsfig{file=fig3-2.ps}}\n\\caption{$K_1$, (red $\\triangle$) $K_2$ (green $\\Box$) versus $M_V$ and enhancement of $|C_7 (m_W)|$\ncase of $|Z^{4i}_{u,d}|^2_{i=1,2,3},~|U^{4i}_{u,d}|^2_{i=1,2,3}< 10^{-4}$ (color online).}\\label{fig3}\n\\end{center}\n\\end{figure}\nThe second task  is to check the VLP contribution to $B\\to X_s\\gamma$. \nWe find the Wilson coefficient of FCNC operator $O_7$ is\nnot so suppressed as the mixing.  The new contributions, from\nthe terms of the last line in Eq.(\\ref{fc7}) are  suppressed\nby the mixing, whereas the terms of first line are almost \nthe same as the SM. The enhancement comes mainly from the \n$g_R^G(4,3)\/m_b$ terms which are got from the\nGoldstone loop in $b\\to s$ transition. (The right diagram in the first line \nand diagrams in the second line of the Fig. \\ref{fig1}) In order to show the\nenhancement clearly, we define two factors\n\\begin{eqnarray}\n  \\label{eq:kfactor}\n K_1 &=& \\frac{g_R^{G}(4,3)g_L^{G}(4,2)^\\ast}{m_Xm_b V_{tb}V_{ts}^\\ast}\n      =  \\frac{g_R^{GVb}g_L^{GVs\\ast}}{m_Xm_b V_{tb}V_{ts}^\\ast}\\,, \\\\\n K_2 &=& \\frac{U^{43}U^{42\\ast}}{V_{tb}V_{ts}^\\ast}\n      =  \\frac{U^{Vb}U^{Vs\\ast}}{V_{tb}V_{ts}^\\ast}\\,,\n\\end{eqnarray}\nin which $K_2$ denotes the deviation from the unitarity of $3\\times 3$ CKM matrix,\nwhile $K_1$ shows the enhancement of the contribution from vector like particles.\n$K_1$ is in fact got from the coefficient of first term in the second line of analytical\nexpression of $C_7(m_W)$ in Eq. (\\ref{fc7}) when $i=4$. It will be changed into exactly \n$K_2$ in case of the SM4. Note that other terms with $g_R^G(4,3)\/m_b$ can \ngive enhancement too, we chose factor $K_1$ for a typical demonstration  since\nit seems that it will be suppressed by $m_X$. Results are shown in the\nFIG \\ref{fig3} in which the left panel shows $K_1$ and $K_2$ versus\n$M_V$ while the right panel shows $|C_7(m_W)|$ versus $K_1$. From the left\npanel, we can see that though $K_2$ increase as $M_V$ increases,  it\nis still much smaller than $V_{tb}V_{ts}^\\ast$, implying that\ndeviation of unitarity are negligible. However the factor $K_1$ can\nbe enhanced up to order ${\\cal O}(1)$  by the increase of $M_V$. \nFrom the right panel, we can see that $K_1$ enhances $C_7$ up to a \nvalue much larger that the result of the SM. The reason for the \nenhancement mainly comes from the new type of Yukawa couplings. \nCombining of Eqs. (\\ref{massdiag},~\\ref{ud_yl},~\\ref{ud_yr}), one\ncan get similar form compared with the SM4\n\\begin{eqnarray}\n \\label{eq:comsm}\n  \\frac{g^G_L(4,2)}{m_{X}} &=& U_{CKM4}^{42} \\label{eqglvs}\n              +\\frac{1}{m_{X}}\\left[\\sum_{m=1}^3\\left(M_{Q}^{m}U_{d}^{m2}Z_{u}^{\\ast44}\n-M_{u}^{m}U_{d}^{42}Z_{u}^{\\ast m4}\\right)\n+(Y_{Vd}-Y_{Vu})U_{d}^{42}Z_{u}^{\\ast44}v\\right],\\\\\n\\frac{g^G_R(4,3)}{m_{b}} &=&-U_{CKM4}^{\\ast 43} \\label{eqgrvb}\n            -\\frac{1}{m_{b}} \\left[\\sum_{m=1}^3\n\\left(M_{Q}^{\\ast m}U_{u}^{\\ast m4}Z_{d}^{43}+M_d^{\\ast m}U_{u}^{\\ast44}Z_{d}^{m3}\\right)\n+(Y^*_{Vd}-Y_{Vu}^\\ast)U_{u}^{\\ast44}Z_{d}^{43}v\\right].\n\\end{eqnarray}\nSince $m_X\\simeq Y_{Vu}v$, $Z_u^{44},~U_u^{44}\\simeq 1$, one can easily obtain that\n\\begin{equation}\n  \\label{eq:enfac1}\n  \\frac{g^G_L(4,2)}{m_{X}}\\sim  V_{\\rm CKM4}^{42},\n\\end{equation}\nthe suppression of $Z_{d}^{43}$ (order of $m\/m_{X,Y}$) in Eq. (\\ref{eqgrvb}) are enhanced by\nterms with factor such as $\\frac{Y_{Vu}v}{m_b}$, etc., resulting\n\\begin{equation}\n  \\label{eq:ehnfac2}\n  \\frac{g^G_R(4,3)}{m_{b}}\\gg V_{\\rm CKM4}^{43}.\n\\end{equation}\nThus the term $V_{4b}V_{4s}^\\ast$ satisfying  the unitary constraint\n\\begin{equation}\n  \\label{eq:enh4}\n  V_{ub}V_{us}^\\ast+V_{cb}V_{cs}^\\ast+V_{tb}V_{ts}^\\ast\n  +V_{4b}V_{4s}^\\ast=0\n\\end{equation}\nis enhanced greatly by heavy VLPs, then the factor leads the\nenhancement to $C_7$. This is different from those in the SM4 \nin which the contribution from the fourth generation can be neglected.\n\n\n\\begin{figure}[hbtp]\n\\begin{center}\n\\scalebox{0.4}{\\epsfig{file=fig4.ps}}\n\\caption{$B\\to X_s \\gamma$ prediction in random scan.}\\label{fig4}\n\\end{center}\n\\end{figure}\nIn the numerical scan, we vary $Z_{u,d}$ and $U_{u,d}$ randomly, keeping the constraints of\n$|V^{4i}_{u,d}|^2_{i=1,2,3},~|U^{4i}_{u,d}|^2_{i=1,2,3}$, \nscan $m_X$ and $m_Y$ in the range of  $(1, 2000) \\rm GeV$.\nApart from the CKM limits, we use the $B\\to X_s \\gamma$ process \nto constrain parameter space. The branching ratio of $B\\to X_s\\gamma$ \nis normalized by the process $B\\rightarrow X_{c}e\\bar{\\nu_{e}}$: \n \\begin{equation}\n  {\\rm Br}(B\\rightarrow X_{s}\\gamma)={\\rm Br}^{\\rm ex} (B\\rightarrow X_{c}e\\bar{\\nu_{e}})\n  \\frac{|V_{ts}^{\\ast}V_{tb}|^{2}}{|V_{cb}|^{2}}\\frac{6\\alpha}\n  {\\pi f(z)}[|C^{\\rm eff}_{7}(\\mu_b)|^{2}+|C^{\\prime,\\rm eff}_{7}(\\mu_b)|^{2}].\\label{bsg}\n \\end{equation}\nHere $z=\\frac{m_c}{m_b}$, and $f(z)=1-8z^2+8z^6-z^8-24z^4\\ln z$ is\nthe phase-space factor in the semi-leptonic B decay. \nThe method of running of the operators from $m_W$ scale to $\\mu_b$ scale can be found in\nRef. \\cite{Li:2012xz}. We use the following bounds on the calculation \\cite{Beringer:1900zz}\n     \\begin{eqnarray}\n     &&   {\\rm Br}^{\\rm ex}(b\\to ce\\overline{\\nu}_{e}) = (10.72 \\pm 0.13) \\times 10^{-2},\\\\\n     &&   {\\rm Br}^{\\rm ex}(B\\to X_s \\gamma)= (3.55 \\pm 0.24 \\pm 0.09) \\times 10^{-4}.\n     \\end{eqnarray}\nThe numerical results show that the $C^{\\prime,\\rm eff}_{7}(\\mu_b)$ is much  smaller\nthan $C^{\\rm eff}_{7}(\\mu_b)$, therefore we do not present the formula of\n$C^{\\prime,\\rm eff}_{7}(m_W)$ here.\n\nThe branching ratio as a function of $m_V$ is  shown in FIG. \\ref{fig4}, \nfrom which we can see that ${\\rm Br}({ B}\\to\nX_s \\gamma)$ can be enhanced much greater than the experiment bound.\nThen the measurements of FCNC process can give a stringent constraint on the\nvector like quark model, especially when the masses of vector\nquark are much greater than the electro-weak scale. \nA few remarks should be addressed:\n\\begin{itemize}\n\\item There is one point of view on the unitarity of the CKM matrix which \nis that the $3\\times 3 $ ordinary quark mixing matrix is  \nregarded as nearly unitary, deviation from unitarity is suppressed by \nheavy particle in the new physics beyond the SM. In other word, \none admits that the extended CKM matrix elements exist, they\napproach to zero while mass scale of the new physics approaches to infinity.\nAll the new  physical effects should decouple from the flavor sector and \nwhat should be checked is that if $3\\times 3$ unitariry is consistent\nin all kinds of flavor processes.\n\\item Another point of view is that, as in the SM case, \nthe $3\\times 3 $ ordinary quark mixing matrix elements are only\nextracted by experiments in the measurements \nof tree and loop level precesses.  The unitarity \nshould be checked, experiment measurements on the elements of matrix can be used\nas the constraints to the new physics beyond the SM.\nIn the numerical analysis, the elements of CKM matrix\nare regarded as inputs. Thus what should be done\nis to scan the parameter space generally under these constraints,\nno prejudice should be imposed.\nThen the enhancement effect in $B\\to X_s\\gamma$ will be more clear. \n\\end{itemize}\n\n\\begin{figure}[hbtp]\n\\begin{center}\n\\scalebox{0.4}{\\epsfig{file=fig5-1.ps}}\n\\scalebox{0.4}{\\epsfig{file=fig5-2.ps}}\n\\caption{Enhancement factor and deviation from unitarity versus $m_X$,\nred $\\triangle$ are excluded by the bound of $B\\to X_s \\gamma$ measurement which the\ngreen $\\Box$ are the survived points (color online).}\\label{fig5}\n\\end{center}\n\\end{figure}\nLarge parameter space is excluded by the measured branching\nfraction of $B\\to X_s \\gamma$ as shown in FIG. \\ref{fig4}.\nThe  enhancement effect of the VLPs can be seen in  FIG. \\ref{fig5}\nin which the left panel shows the enhancement factor $K_1$ \nversus $m_X$ while the right panel shows\n$K_2$ versus $m_X$. From the right panel we can see that deviation\nfrom unitarity are very small and almost irrelevant with $m_X$ since\nwe are doing a general scan of $Z_{u,d}$ and $U_{u,d}$. However as\nwe see from the left panel, as $m_X$ increases up, ${\\rm\nBr}({B}\\to X_s \\gamma)$ measurement will constrain the\nenhancement factor and then constrain the input parameter of $m_X$. In\nall, the enhancement can be summarized as that when mass\nof vector like particle increases up, it will increase the mass\nparameter $m_V$ thus give an enhancement factor under very small\ndeviation from  unitarity. This should be a special point when we\ndo the study on the vector like quark models.\n\n\\section{Summary}\\label{sec4}\nIn the model with  vector doublets, there exist bilinear terms in \nthe lagrangian, making the general scan of the Yukawa coupling very difficult.\nIn this paper, we show a trick to deal with the scan. Our scan method are\nexactly and the more efficient. We use the trick to study a very simple extension\nof the SM with  vector like quarks. We studied  one of the most important\nrare B decay $B\\to X_s \\gamma$ process in which we found that even the deviations\nfrom the unitarity of quark mixing matrix are small, \nthe enhancement to rare B decay from VLPs are still significant. \nThe enhanced effect is an important feature in the vector like particle model. \nIn this work we just show the scan method, the key point of the\nenhancement and how stringent constraints on the parameter space\nfrom $B\\to X_s \\gamma$ measurements. What should be done includes models like extension of the SM with VLPs,\ntwo higgs doublets models \\cite{Grinstein:1990tj} or supersymmetry models \\cite{Altmannshofer:2009ne}.\nSuch effect should be checked in all kinds of rare decays such\nas inclusive process $b\\to s \\ell^+\\ell^-$ and exclusive processes $B_s \\to \\mu^+\\mu^-$,\n$B_s \\to \\ell^+\\ell^- \\gamma$ and $B\\bar B$ mixing {\\it et. al.}\nThe detailed studies on the parameter space including other rare B decays\nand new models will appear in our future work.\n\n\\begin{acknowledgments}\nThis work was supported by the\nNatural Science Foundation of China under grant numbers 11375001\nand by talents foundation of education department of Beijing.\n\\end{acknowledgments}\n\\section*{Appendix}\n\\begin{itemize}\n\\item\nThe loop functions for calculating the Wilson coefficients at\nthe matching scale are the following\n\\end {itemize}\n\\begin{eqnarray}\nA(x)&=&\\frac{55-170x+127x^{2}}{36(1-x)^{3}}+\\frac{4x-17x^{2}+15x^{3}}{6(1-x)^{4}}\\ln x,\\nn\\\\\nB(x)&=&\\frac{-7+5x+8x^{2}}{36(1-x)^{3}}+\\frac{-2x+3x^{2}}{6(1-x)^{4}}\\ln x,\\nn\\\\\nC(x)&=& \\frac{3-5x}{6(1-x)^{2}}+\\frac{2-3x}{3(1-x)^{3}}\\ln x,\\nn\\\\\nD(x)&=&\\frac{3x-1}{4(1-x)^{2}}+\\frac{x^{2}}{2(1-x)^{3}}\\ln x,\\nn\\\\\nE(x)&=& \\frac{-17+19x}{6(1-x)^{2}}+\\frac{-8x+9x^{2}}{3(1-x)^{3}}\\ln x.\\nn\n\\end{eqnarray}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSemantic image segmentation is an important tool for deriving information from an image and plays a crucial role in many applications, facilitating higher-level image analysis \\cite{song2019vision, zahoor2017fast, ulmas2020segmentation}. The goal of any image segmentation algorithm is to partition an image into meaningful sub-regions. Edge-based segmentation algorithms do this by identifying boundaries, or edges between those regions. Human visual systems are able to interpret and recognise scenes and objects just based on the basic structure of their outlines \\cite{walther2011simple}, making edge-based approaches useful in mimicking some of the mechanisms of our own vision. Image segmentation is an active area of research since it is generally an ill-posed problem formulation. Consequently, much use is made of domain-specific implementations as no single segmentation algorithm suits all types of images.\n\nThis technique provides a tunable, robust, semi-automatic and probabilistic approach to edge-based image segmentation in images. The proposed methodology can be seen as a recursive Bayesian scheme, modelling the edge of interest (EoI) as a Gaussian process in a regression setting. Using a suitable edge map and tunable initialisation, the model iteratively searches for edge pixels, sequentially building an observation set to update and converge the posterior predictive distribution toward the EoI. This technique utilises the predictive variance of this distribution to search regions where no edge pixels have been observed and models uncertainty in regions with pixels already observed. This uncertainty quantification aids the tracing of edges which are degraded by noise and occlusions. There is no prior training required which allows the edge tracing technique to be applied to various different research fields. Moreover, the internal (hyper)parameters of the model allows the user to tune the fitting procedure to the EoI. For example, through specification of observation noise and choice of kernel, both of which have a strong impact on the fitting procedure.\n\n\\subsection{Related Work}\\label{review}\nMany early approaches to image segmentation used edge detection methods based on zero-crossings, discrete derivative operators or histogram thresholding \\cite{prewitt1970object, duda1970experiments, roberts1963machine, rosenfeld1969picture}. A variety of these early approaches to image segmentation can be found in Rosenfeld's work \\cite{rosenfeld1976digital}. The problem of edge detection has been well studied and with methods such as Canny \\cite{canny1986computational}, it can be straightforward to obtain some form of edge map, or image gradient. However, there is a distinct difference between identifying the edges in an image and linking them to form a meaningful contour, with direct knowledge of the pixel coordinates. Szeliski \\cite{szeliski2010computer} notes that this latter task is far more useful than the former and is more challenging due to the potential for extracting a poor quality edge map. This paper explicitly looks at solving this edge tracing task, with an edge map already obtained.\n\nAt present, the most popular frameworks employed to perform image segmentation are based on convolutional neural network models. These can produce highly accurate semantic or instance segmentation of images or videos \\cite{ren2015faster, ronneberger2015u, milletari2016v}. For literature reviews on deep learning techniques for image segmentation see Wu et al. \\cite{wu2019review} and for medical image segmentation see Hesamian et al. \\cite{hesamian2019deep}. Such methods require large amounts of high quality, labelled training data to perform well on a given type of image problem and do not model the edge structure of an image directly. Moreover, Renard et al. \\cite{renard2020variability} discuss the lack of interpretability and concern regarding the reproducibility of such methods in the medical image domain. Consequently, for many interesting applications it is still useful to work with edge-based segmentation models which are able to segment an image without training data and can provide a functional form of a given edge within an image.\n\nTwo common types of models suited for tracing individual edge structures are those which build a graph structure connecting the pixels within an image and those which adapt a contour to an edge structure using parametric energy-minimising splines. The latter are known as active contour models, originally introduced by Kass et al. \\cite{kass1988snakes}. For extensive graph-based literature reviews for image segmentation see Peng et al. \\cite{peng2013survey} and in medical image segmentation see Chen and Pan \\cite{chen2018survey}. For detailed literature reviews on active contour models in image segmentation see Jaiswal and Sarode \\cite{jaiswal2017review} and for medical image segmentation see Hemalatha et al. \\cite{hemalatha2018active}.\n\nThe use of Bayesian frameworks in computer vision and image processing is popular due to such models being able to feature non-parametric learning and uncertainty quantification. Some image processing applications involve denoising \\cite{liu2007using}, object categorisation \\cite{kapoor2010gaussian}, detection of pneumonia in chest X-rays \\cite{frank2020gaussian} or super-resolution of images \\cite{he2011single}. Relevance vector machines \\cite{tipping2001sparse} are related to Gaussian processes in its Bayesian formulation but has flexibility on kernel selection using localised basis functions and can induce sparsity due to its hyperparameter optimisation framework. Although this technique has been used in the computer vision community for classification \\cite{dong2012accelerating, wei2005relevance}, there is a general consensus that sparsity and kernel flexibility comes at a cost of unnatural variance behaviour in the predictive distributions, unlike in Gaussian processes \\cite{quinonero2004learning, rasmussen2005healing, martino2021joint, bishop2006pattern, rasmussen2003gaussian}, which can lead to undesirable results in modelling. We will not pursue relevance vector machines in this paper because the proposed edge tracing methodology relies heavily on the predictive variance for image exploration. Relevance vector machines do not have the capacity to trace well because of this unnatural variance behaviour.\n\nThere has not been an extensive amount of work on applying Gaussian processes to image segmentation. Freytag et al. \\cite{freytag2012efficient} perform semantic segmentation by using histogram intersection kernels for fast and exact Gaussian process classification. Tu and Zhu \\cite{tu2002image} proposed a Bayesian statistical framework to image segmentation by learning object appearance models for different region types, placing prior distributions over region size, region number and boundary smoothness. Simek and Barnard \\cite{simek2015gaussian} used a two-dimensional Gaussian process regression model to segment Arabidopsis leaves by modelling the blade and petiole as two random functions, joining them at their boundaries using a smoothing constraint. Our approach is in a similar vein to Simek and Barnard, but there is no trained prior based on a priori information. \n\n\\section{Gaussian Process Regression}\\label{gpr}\nIn Bayesian machine learning, a Gaussian process regressor (GPR) is viewed as a multivariate Gaussian distribution over function values. They are primarily used to model complex functions and quantify uncertainty in model fitting to continuous, real-valued data. The GPR is able to treat the learning process incrementally by updating beliefs in regions where observations can be incorporated. This belief, or uncertainty, is induced by the prior predictive distribution and influenced by subsequent observations which give rise to the posterior predictive distribution (PPD). The GPR models the underlying function $f(x)$ as a Gaussian process using noisy observations $y$ such that\n\\begin{equation}\\label{eq:gpr_assume_model}\n    y = f(x) + \\epsilon,\n\\end{equation}\nwith $f(x)$ fully specified by its mean function $\\mu(x)$ and kernel $k(x, x')$ defined by\n\\begin{align}\n    \\mu(x) &= \\mathbb{E}\\left[f(x)\\right];\\\\\n    k(x, x') &= \\mathbb{E}\\left[(f(x)-\\mu(x))(f(x')-\\mu(x'))\\right];\\\\\n    f(x) &\\sim \\mathcal{GP}\\big(\\mu(x), k(x, x')\\big)\\label{eq:gp_defn}.\n\\end{align}\nFor continuity with the remainder of this paper, we have assumed a one-dimensional domain and range, $x, x', f(x) \\in \\mathbb{R}$, as the proposed methodology works with a 2D image, utilising the real-valued intervals on the horizontal and vertical axes as the domain and range of the GPR. The identically, independently distributed Gaussian noise $\\epsilon \\sim \\mathcal{N}\\big(0, \\sigma_y^2\\big)$ is characterised by the observation noise variance $\\sigma_y^2$ which governs the uncertainty that the model assumes in the observation set. Without any loss of generality we can assume a trivial mean function, but it is the choice of kernel which has the largest impact on model fitting. The kernel is used to build the covariance matrix of random function values, which depend on their input observations. \n\nIn this work we are only concerned with isotropic, stationary kernels which are translation invariant, monotonic and dependent on the distance between input locations, $r_{ij} = | x_i - x_j |$. The kernel formulation imposes differentiability constraints while its hyperparameters dictate amplitude and sinuosity of the random curves which can be sampled from the PPD. It is common for these kernels to have at least two hyperparameters, the signal variance, $\\sigma_f^2$ and lengthscale, $\\ell$. $\\sigma_f^2$ dictates the average amplitude, or deviation, of the random curves from the mean while $\\ell$ roughly describes the typical distance between turning points, effectively controlling their sinuosity.\n\nThe squared exponential and Mat\\'{e}rn kernels are the isotropic, stationary kernels of interest in this paper. These kernels are scaled by a Gaussian kernel allowing them to compute covariance between function values monotonically with respect to the distance between their corresponding input locations, $k = k(r_{ij};\\hspace{2pt} \\sigma_f^2, \\ell)$. Although the squared exponential kernel is commonly used for drawing infinitely differentiable random curves, the strong assumption on continuity is one of its pitfalls when modelling real-life physical processes, as these tend to be more irregular. \n\nConsequently, the Mat\\'{e}rn class of kernels have more flexibility in the degree of differentiability and are preferred for modelling processes which are less smooth. They are enumerated by their unique hyperparameter $\\nu$, dictating the degree of differentiability of random curves they can generate. These kernels have been observed by Rasmussen and Williams \\cite{rasmussen2003gaussian} to be particularly useful when $\\nu = 3\/2$ or $\\nu=5\/2$. These values yield first-order and second-order differentiable curves from the PPD, respectively. This makes this kernel an ideal choice for modelling edges in real-life image segmentation problems. More information on these kernels, their theory and formulation can be found in Rasmussen and Williams' book on Gaussian processes for machine learning \\cite{rasmussen2003gaussian}.\n\n\\subsection{Posterior Predictive Distribution}\nAfter selection of a valid kernel $k$ and its kernel hyperparameters, we can construct the prior predictive distribution, from which we can draw random prior curves, $\\bm{f}^*$ evaluated along some domain $X^*$ such that\n\\begin{equation}\n    \\bm{f}^* \\sim \\mathcal{N}\\big(\\bm{0}, K_{**}\\big),\n\\end{equation}\nwith $K_{**}=K\\left(X^*, X^*\\right) = \\big[k(x_i, x_j)\\big]_{ij}$. However, in model fitting, we are more interested in using an observation set that has the form of equation \\eqref{eq:gpr_assume_model} to model an underlying function or process. These noisy observations, $\\mathcal{D} = \\big\\{X, \\bm{y}\\big\\}$ are integrated into the GPR by forming the joint distribution between them and the function values we seek but are unknown, $\\bm{f}^*$. The covariance matrix of this distribution is a block matrix comprising of the covariance pairs between the known target values, $\\bm{y}$ and the unknown function values, $\\bm{f}^*$ using their corresponding input locations, $X$ and $X^*$, respectively. This joint distribution is defined by\n\\begin{equation}\\label{eq:jointdist}\n    \\begin{bmatrix} \\bm{y} \\\\ \\bm{f}^*\\end{bmatrix} \\sim \\mathcal{N}\\bigg(\\bm{0}, \\begin{bmatrix} K + \\sigma_y^2\\mathbb{I} & K_* \\\\ K_*^{\\trans} & K_{**}\\end{bmatrix}\\bigg),\n\\end{equation}\nwhere $K_*=K(X, X^*)=K(X^*, X)^{\\text{T}}$. Here, we encode the observation noise into the covariance matrix between observations, $\\text{Cov}[\\bm{y}] = K(X, X) + \\sigma_y^2\\mathbb{I} = K +\\sigma_y^2\\mathbb{I}$. This observation noise promotes numerical stability and quantifies the freedom posterior curves have in not intersecting observations exactly. Collating all kernel hyperparameters and observation noise into $\\bm{\\theta}$, these are parameters which can be fixed during model fitting or optimised through maximising the log marginal likelihood of the observation set,\n\\begin{align}\n    \\bm{\\theta} &= \\argmin_{\\bm{\\theta}}\\Big\\{\\text{log}\\hspace{3pt} p(\\bm{y} \\hspace{3pt} | \\hspace{3pt} X, \\bm{\\theta})\\Big\\};\\label{eq:optim_theta}\\\\ \\label{eq:marginal_likelihood}\n    \\text{log} \\hspace{3pt} p(\\bm{y} \\hspace{3pt} | \\hspace{3pt} X, \\bm{\\theta}) &= -\\frac{1}{2}\\bm{y}^{\\trans}A^{-1}\\bm{y} - \\frac{1}{2}\\text{log}\\hspace{1pt}\\big|A\\big| - \\frac{m}{2}\\text{log}\\hspace{1pt}2\\pi,\n\\end{align}\nwith $m$ the sample size and $A = K+\\sigma_y^2\\mathbb{I}$. Once $\\mathcal{D}$ and $\\bm{\\theta}$ are known we can restrict the joint distribution in equation \\eqref{eq:jointdist} to only describe functions which accommodate the observations in $\\mathcal{D}$, inducing the PPD. This is done by conditioning on this joint distribution using $\\mathcal{D}$, $\\bm{\\theta}$ and the input locations of unknown function values $X^*$,\n\n\\begin{align}\\label{eq:gp_posterior}\n    \\bm{f}^* \\hspace{3pt}\\big|\\hspace{3pt} \\mathcal{D} &\\sim \\mathcal{N}\\big(\\mathbb{E}\\left[\\bm{f}^*\\hspace{3pt} | \\hspace{3pt} X, \\bm{y}\\right], \\text{Cov}\\left[\\bm{f}^* \\hspace{3pt} | \\hspace{3pt} X\\right]\\big);\\\\\\label{eq:gp_updated_mean}\n    \\mathbb{E}\\left[\\bm{f}^* \\hspace{3pt} | \\hspace{3pt} X, \\bm{y}\\right] &= K_*^{\\text{T}}\\left[K+ \\sigma^2_y\\mathbb{I}\\right]^{-1}\\bm{y};\\\\ \\label{eq:gp_updated_cov}\n    \\text{Cov}\\left[\\bm{f}^* \\hspace{3pt} | \\hspace{3pt} X\\right] &= K_{**} - K_*^{\\text{T}}\\left[K+ \\sigma^2_y\\mathbb{I}\\right]^{-1}K_*,\n\\end{align}\nallowing us to sample posterior curves in a similar fashion to the prior predictive distribution. For the equations above and for the remainder of this paper, we implicitly assume conditioning on $\\bm{\\theta}$ and test input locations $X^*$ for readability. The mathematical details of constructing this PPD is shown in appendix A.2 of Rasmussen and Williams' book \\cite{rasmussen2003gaussian}. \n\n\\section{Methodology}\\label{method}\nThe proposed procedure of tracing an EoI in an image of size $M \\times N$ is as follows. In the first instance, a suitable choice of kernel $k$ and hyperparameters $\\bm{\\theta}$ are selected and fixed to model the EoI (more on this in section \\ref{discussion}). An initial PPD, $\\bm{f}^{(1)} | \\mathcal{D}^{(0)}$, is induced through fitting an estimate for the edge endpoints, $\\mathcal{D}^{(0)} = \\big\\{X^{(0)}, \\bm{y}^{(0)}\\big\\}$. In each iteration $n$, $L$ posterior curves are sampled using the previous iteration's PPD, $\\bm{f}^{(n)} | \\mathcal{D}^{(n-1)}$, predicted at inputs along the image's horizontal axis, $X^* = [0, N-1] \\subset \\mathbb{Z}$. These curves are scored using the pre-computed image gradient $G$ and a proportion of the highest scoring posterior curves $\\epsilon L$ are used to decide on the next set of pixels that will be fitted to the model's observation set $\\mathcal{D}^{(n)}$ in the next iteration.\n\nThe individual elements of these optimal posterior curves are used to estimate a two-dimensional weighted smoothing kernel $\\phi^{(n)}$ which represents a frequency distribution of where these optimal curves pass through the image. Combining this density function with the image gradient, pixels are scored and only the higher scoring pixels are chosen, according to an adaptive threshold $T$, to form the next observation set, $\\mathcal{D}^{(n+1)} = \\big\\{X^{(n+1)}, \\bm{y}^{(n+1)}\\big\\}$. Each consecutive iteration refits the Gaussian process with these new observations, updating the belief in the PPD on where the EoI lies. In the final iteration, with enough observations to reconstruct the EoI, the kernel hyperparameters $\\bm{\\theta}$ are optimised according to equation \\eqref{eq:optim_theta} and the posterior predictive mean, $\\mathbb{E}\\left[\\bm{f}^{(n)} \\hspace{3pt} | \\hspace{3pt} \\mathcal{D}^{(n)}\\right]$, evaluated at $X^*$, and a 95\\% credible band is outputted to the user. While in sections \\ref{apps} and \\ref{discussion} we discuss circumventions for edges which are non-injective along the image width, for definiteness and simplicity we will assume the EoI can be modelled as an injective function and that its horizontal pixel length spans the width of the image, $N$. \n\n\\subsection{Initialisation}\nAt the beginning of the procedure, following a similar formulation to equation \\eqref{eq:gp_defn}, our prior distribution comes from a zero mean Gaussian process with kernel $k$ such that \n\\begin{equation}\n    f^{(0)}(x) \\sim \\mathcal{GP}\\big(0, k(x, x' ; \\bm{\\theta})\\big),\n\\end{equation}\ninducing a prior predictive distribution where a vector of random function values $\\bm{f}^{(0)}$ can be sampled using\n\\begin{equation}\\label{eq:init_prior}\n    \\bm{f}^{(0)} \\sim \\mathcal{N}\\big(\\bm{0}, K_{**}\\big).\n\\end{equation}\nTo induce the initial PPD requires an estimate for the edge endpoints, $\\mathcal{D}^{(0)} = \\big\\{X^{(0)}, \\bm{y}^{(0)}\\big\\}$. These user-specified pixel coordinates act as domain knowledge and are assumed to be within the vicinity of the true edge endpoints. As we may not know the true edge endpoints, these are fitted and imputed along with the remainder of the interior edge pixels. This is done by specifying the variance of the observation noise at these edge endpoints as $\\sigma_y^2$. In practice, the estimate for the true edge endpoints could be known and the algorithm has the flexibility of assuming a non-uniform noise model by setting the observation noise variance for these pixels as a very low value for numerical stability. This initialisation step tells the algorithm which edge the user is interested in tracing. For the remainder of this section, we will assume a uniform noise model. Fitting $\\mathcal{D}^{(0)}$ induces the initial PPD, $\\bm{f}^{(1)} | \\mathcal{D}^{(0)}$, from which we can draw samples evaluated at $X^*$.\n\nThe objective is to now impute the missing pixels between the edge endpoints. The predictive variance of the PPD increases in regions far away from the edge endpoints, allowing the posterior curves to evolve freely across the image, according to the properties imposed by the selected kernel. In fact, the signal variance $\\sigma_f^2$ dictates the maximum size of this predictive variance. With enough distance from the edge endpoints, the initial PPD will have a 95\\% credible interval at maximum of $\\overline{\\bm{y}}^{(0)} \\pm 2\\sigma_f$, where $\\overline{\\bm{y}}^{(0)}$ is the mean of the edge endpoint pixel heights. This predictive variance is something we take advantage of throughout the fitting procedure so that posterior curves explore around the EoI and propose possible locations for it. Figure \\ref{fig:init_post_dist} shows an initial PPD for an EoI, shown in bold red. The remainder of this section describes the methodology using an arbitrary iteration $n \\geq 0$.\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.75\\textwidth]{_FIGURES\/Init_Prior_PD.pdf}\n    \\caption{200 posterior curves drawn from an initial PPD. The grey shaded region represents the 95\\% credible region. The squared exponential was chosen as the kernel with $\\sigma^2_f = 75^2$, $\\ell = 20$ and observation noise variance $\\sigma^2_y = 1^2$.}\n    \\label{fig:init_post_dist}\n\\end{figure}\n\n\\subsection{Curve Scoring}\\label{subsec:curve_scoring}\nIn each iteration, $L$ posterior curves $\\bm{f}^{(n)}_l$, $l=1, \\dots, L$ are sampled from the PPD, evaluated at $X^*$. These curves represent a sample of the model's belief in where the EoI lies, given the current observation set. To select which pixel coordinates to add to the observation set, these curves must be scored by how well they fit the EoI. This is done using the pre-computed and user-defined image gradient $G$. The score for a curve is computed as the cumulative value of the gradient response along this curve per unit pixel length, \n\\begin{equation}\n    I(\\bm{f}^{(n)}_l) = \\int_{\\bm{f}^{(n)}_l} G\\big(\\bm{r}(s)\\big) \\hspace{2pt} ds \\bigg\/ \\int_{\\bm{f}^{(n)}_l} ds.\\label{eq:LI_1}\n\\end{equation}\n\nThis score is analogous to computing the area under the posterior curve along the interpolated image gradient surface, divided by the posterior curve's arc length. We use Simpson's rule and finite difference rules to estimate the line integral and arc length. The advantage of scaling curve scores by arc length is that curves will be penalised for making longer excursions. This can be helpful in the presence of noise in the image gradient. The cost function for scoring a posterior curve is the inverse of this gradient score, $C(\\bm{f}^{(n)}_l) = \\big(I(\\bm{f}^{(n)}_l)\\big)^{-1}$. This is so that higher scoring curves are minimally costly and the minimisation of this cost becomes part of the optimisation scheme to model the edge. \n\n\\subsection{Pixel Scoring}\nAn integer proportion $\\epsilon L$, $\\epsilon \\in (0, 1]$, of posterior curves which have the lowest costs are selected as the most optimal posterior curves. These optimal curves provide a best guess at where the current PPD believes the EoI lies and their coordinates together map out which regions have been visited the most. Given these curves lie in the vicinity of the EoI, collating their coordinates to construct a weighted smoothing kernel of where they explore in the image will provide an empirical approximation for where the PPD believes the EoI is. This weighted, frequency density function is defined by  \n\n\\begin{equation}\\label{eq:freq_kde}\n    \\phi^{(n)}(\\bm{z}) = \\frac{1}{\\sum_{l, i}w(\\bm{z}_{li})}\\sum_{l, i}w(\\bm{z}_{li})K_{\\mathbb{I}}(\\bm{z} - \\bm{z}_{li}),\n\\end{equation}\nwhere $\\bm{z}, \\bm{z}_{li} \\in [0, N-1] \\times [0, M-1] \\subset \\mathbb{R}^2$. $\\bm{z}_{li}$ corresponds to the $i^{\\textrm{th}}$ coordinate of the $l^{\\textrm{th}}$ posterior curve, $\\bm{f}^{(n)}_l$. The weighting for $\\bm{z}_{li}$ scores the point according to how optimal the curve $\\bm{f}^{(n)}_l$ is. This weighting will result in a higher density for regions where the most optimal curves lie in the image. The weight function $w$ is defined by\n\\begin{equation}\\label{eq:kde_weight}\n    w(\\bm{z}_{li}) = \\frac{I\\big(\\bm{f}^{(n)}_l\\big)}{\\sum^{\\epsilon L}_{l=1}I\\big(\\bm{f}^{(n)}_l\\big)}, \\hspace{2pt} \\forall i.\n\\end{equation}\n\n$K_{\\mathbb{I}}(\\bm{z}-\\bm{z}_{li})$ is the symmetric, multivariate kernel of the density function. Here we use the isotropic, two-dimensional Gaussian kernel with the identity matrix as its two-dimensional lengthscale,\n\\begin{equation}\\label{eq:gauss_kernel}\n    K_{\\mathbb{I}}(\\bm{z} - \\bm{z}_{li}) = (2\\pi)^{-1}\\exp\\left(-\\frac{1}{2}\\big(\\bm{z} - \\bm{z}_{li}\\big)^{\\text{T}}\\big(\\bm{z} - \\bm{z}_{li}\\big)\\right).\n\\end{equation}\n\nThe isotropic and unitary lengthscale imposes no dominant orientation in the Gaussian kernel centred at each real-valued point $\\bm{z}_{li}$. Moreover, the unitary property ensures each point's kernel contributes the majority of its density to only the 4 nearest pixel coordinates in the discretised space. \n\nFigure \\ref{fig:kde_plot} illustrates the process during the fitting procedure for a test case whose initial PPD is shown in figure \\ref{fig:init_post_dist}. Subfigure \\ref{fig:kde_plot}(a) shows a noisy and occluded sinusoidal edge. The true edge and edge predictions using the proposed methodology and two common edge tracing techniques are superimposed. A 95\\% credible band is shown in magenta around the proposed edge prediction. Subfigure \\ref{fig:kde_plot}(c) shows the $\\epsilon L$ optimal posterior curves from the iteration's PPD with a 95\\% credible interval represented by the grey shaded region. Note that almost all of the first two periods of the EoI are well mapped by the posterior curves, with a reduced 95\\% credible interval in the region of dense observations. Conversely, the posterior curves are more variable in the latter two periods due to a lack of observations. These two remarks can be observed in subfigure \\ref{fig:kde_plot}(d), where we see a significantly higher density covering the former two periods of the true edge in contrast to the latter.\n\n\\begin{figure}[!t]\n    \\centering\n    \\setlength{\\unitlength}{1cm}\n    \\begin{picture}(18,13.6)\n        \\put(0.0,0.0){\\includegraphics[width=\\textwidth]{_FIGURES\/GPET_Exemplar_Plot.pdf}}\n        \\put(0.35, 13.15){\\textbf{\\color{white}(a)}}\n        \\put(8.675, 13.15){\\textbf{\\color{white}(b)}}\n        \\put(0.35, 6.2){\\textbf{\\color{white}(c)}}\n        \\put(8.675, 6.2){\\textbf{\\color{white}(d)}}\n    \\end{picture}\n    \\caption{(a) Noisy and occluded test case. (b) Image gradient. (c) $\\epsilon L$ optimal posterior curves from a PPD fitted with an incomplete observation set. (d) Weighted frequency density function using optimal posterior curves from (c).}\n    \\label{fig:kde_plot}\n\\end{figure}\n\nIndividual pixel coordinates, $\\bm{p} = (x, y)$, can then be scored by combining their density, $\\bm{\\phi}^{(n)}(\\bm{p})$ and gradient response, $G(\\bm{p})$. This score, $s(\\bm{p}) \\in [0, 1]$, is defined by\n\\begin{equation}\\label{eq:score_eqn}\n    s(\\bm{p}) = \\frac{1}{3}\\left(\\phi^{(n)}(\\bm{p}) \\cdot G(\\bm{p}) + \\phi^{(n)}(\\bm{p}) + G(\\bm{p})\\right).\n\\end{equation}\n\n\\subsection{Accept-Discard Scheme}\nThe primary advantage of using a score function which includes the iteration-dependent frequency density $\\phi^{(n)}$ is that densities are updated in each consecutive iteration using the new PPD. This means that pixels fitted during previous iterations are re-scored using an updated density allowing a more accurate representation of how probable those pixels are of belonging to the EoI, given the new set of optimal posterior curves. This allows the possibility of older, incorrectly fitted pixels to be discarded from the model if their score drops sufficiently in later iterations. An adaptive, user-specified value $T \\in [0, 1]$ is chosen to threshold these pixels such that\n\\begin{equation}\\label{eq:thresh_set}\n   P^{(n)}_T = \\Big\\{\\bm{p} \\hspace{3pt} \\Big| \\hspace{3pt} s(\\bm{p}) \\geq T\\Big\\}.\n\\end{equation}\n\nThe thresholded pixels are divided into $\\ceil{\\nicefrac{N}{\\Delta x}}$ linearly spaced sub-intervals, $S_i = \\left[(i-1)\\Delta x, \\hspace{3pt} i\\Delta x\\right]$, $i = 1, \\dots, \\ceil{\\nicefrac{N}{\\Delta x}}$. The length of each sub-interval is a user-specified parameter, $\\Delta x$. As a continuous function is fitted, injectivity is enforced by using non-max suppression, selecting the highest scoring pixel per sub-interval, yielding\n\\begin{equation}\\label{eq:accepted_set} \n    P^{(n)}_A = \\Big\\{\\displaystyle\\argmax_{\\bm{p}_j}\\big\\{s(\\bm{p}_j)\\big\\}\\hspace{3pt} \\Big| \\hspace{3pt} x_j \\in S_i\\Big\\}_{i = 1}^{\\ceil{\\nicefrac{N}{\\Delta x}}}.\n\\end{equation}\n\nCoordinates are binned because it is unnecessary to fit a GPR with points in very close proximity in the domain, unless the underlying edge is very complex. Because $\\phi^{(n)}$ changes with each iteration, each sub-interval's highest scoring pixel coordinate may change dependant on the score it obtains compared to other possible pixels in its neighbourhood. The observation set for the next iteration $\\mathcal{D}^{(n)}$ is formed by re-scoring the previous iteration's observation set according to the new set of optimal posterior curves, $\\mathcal{D}_*^{(n-1)}$, and combining this with $P^{(n)}_A$ through binning and non-max suppression, yielding \n\\begin{equation}\\label{eq:concat_prev_new_set}\n    \\mathcal{D}^{(n)} = \\textrm{suppression}\\left(\\textrm{binning}\\big(P^{(n)}_A \\cup \\mathcal{D}_*^{(n-1)}\\big)\\right).\n\\end{equation}\n\n\\subsection{Refitting \\& Convergence}\nThe observations in $\\mathcal{D}^{(n)}$ represent the most probable pixel coordinates which form part of the EoI for the $n^{\\text{th}}$ iteration. An improved PPD can be formed using this new set of pixel coordinates, updating the model's belief in where the EoI lies,\n\\begin{align}\\label{eq:iter_posterior}\n    \\bm{f}^{(n+1)} \\hspace{3pt}\\big|\\hspace{3pt}\\mathcal{D}^{(n)} &\\sim \\mathcal{N}\\big(\\bm{\\mu}^{(n+1)}, \\bm{\\sigma}^{(n+1)}\\big);\\\\ \\label{eq:iter_predictive_mean}\n    \\bm{\\mu}^{(n+1)} &= K_{*n}\\left[K_{nn}+ \\sigma^2_y\\mathbb{I} \\right]^{-1}\\bm{y}^{(n)};\\\\\\label{eq:iter_predictive_cov} \n    \\bm{\\sigma}^{(n+1)} &= K_{**} - K_{*n}\\left[K_{nn}+ \\sigma^2_y\\mathbb{I}\\right]^{-1}K_{n*},\n\\end{align}\nwhere $K_{nn} = K(X^{(n)}, X^{(n)})$ and similarly for $K_{n*}$ and $K_{*n}$. The posterior curves $\\bm{f}^{(n+1)}$ we can sample are now more accurate in mapping regions of the EoI than in previous iterations. The user-specified observation noise variance $\\sigma^2_y$ is used to perturb the posterior curves from passing through the observation points exactly, allowing posterior curves to continue searching for pixels in the image that are in the vicinity of previously fitted pixels. This permits the possibility of accepting better edge pixels in each sub-interval in future iterations.\n\nThe algorithm reaches termination once all sub-intervals have selected a high-scoring pixel, i.e. $|\\mathcal{D}^{(n)}| = \\ceil{\\nicefrac{N}{\\Delta x}}$. The GPR is retrained once more using $\\mathcal{D}^{(n)}$, this time optimising $\\bm{\\theta}$ by maximising the log marginal likelihood according to equation \\eqref{eq:optim_theta}, yielding $\\hat{\\bm{\\theta}}$ and $\\hat{\\sigma}^2_y$. This maximisation scheme chooses kernel hyperparameters and noise variance which make $\\mathcal{D}^{(n)}$ seem probable. This optimisation is only carried out once the final observation set $\\mathcal{D}^{(n)}$ has enough pixels to reconstruct the EoI. This optimisation drives $\\sigma_y^2$ toward 0 because there are a sufficient enough edge pixels in $\\mathcal{D}^{(n)}$ to interpolate through the observations exactly. Note via equation \\eqref{eq:gp_updated_mean} that the smaller $\\sigma^2_y$ is to 0, the closer the posterior predictive mean is to all observations. It is this optimisation scheme which converges the final PPD to the EoI, using the optimised posterior predictive mean,\n\\begin{equation}\\label{eq:optimal_predict_mean}\n    \\hat{\\bm{f}}^{(n+1)} \\hspace{3pt}\\big|\\hspace{3pt}  \\mathcal{D}^{(n)}, \\hat{\\bm{\\theta}} = K_{*n}\\left[K_{nn}+ \\hat{\\sigma}^2_y\\mathbb{I}\\right]^{-1}\\bm{y}^{(n)},\n\\end{equation}\nas the proposed edge. This, as well as a 95\\% credible band, evaluated at $X^*$, is outputted to the user. Algorithm \\ref{alg:gpet} outlines the pseudocode of the proposed edge tracing algorithm.\n\n\\begin{algorithm}[t!]\\footnotesize\n    \\caption{Edge Tracing using Gaussian Process Regression}\n    \\textbf{Inputs}: Kernel $k$ and its hyperparameters $\\bm{\\theta}$, including observation noise variance $\\sigma_y^2$, image gradient $G$, edge endpoints $\\mathcal{D}^{(0)}$, number of posterior curves $L$ and proportion to keep as optimal $\\epsilon$, score threshold $T$ and sub-interval length $\\Delta x$. \\\\\n    \\textbf{Output}: Optimal posterior predictive mean with a 95\\% credible band, evaluated at $X^*$.\n    \\begin{algorithmic}[1]\n        \\State Induce prior predictive distribution, $f^{(0)}(x) \\sim \\mathcal{GP}\\big(0, k(x, x' ; \\bm{\\theta})\\big)$;\n        \\State Fit edge endpoints in $\\mathcal{D}^{(0)}$ to the GPR and induce the initial PPD, $\\bm{f}^{(1)} | \\mathcal{D}^{(0)}$, corrupting edge endpoints with observation noise $\\sigma^2_y$;\n        \\While{$|\\mathcal{D}^{(n)}| \\neq \\ceil{\\nicefrac{N}{\\Delta x}}$}\n            \\State \\multiline{Draw $L$ posterior curves from the current PPD;}\n            \\State \\multiline{Compute cost of each posterior curve and select $\\epsilon L$ curves which have the least cost;}\n            \\State \\multiline{Estimate frequency density function $\\phi^{(n)}$ using optimal posterior curves, weighted by their curve score;}\n            \\State \\multiline{Score and threshold pixel coordinates, yielding $P^{(n)}_T$;}\n            \\State \\multiline{Bin and use non-max suppression to obtain $P^{(n)}_A$;}\n            \\State \\multiline{Combine $\\mathcal{D}^{(n-1)}$ and $P^{(n)}_A$ to obtain $\\mathcal{D}^{(n)}$;}\n            \\State {While $|\\mathcal{D}^{(n)}| \\leq |\\mathcal{D}^{(n-1)}|$, reduce $T$ and repeat steps 7--9;}\n            \\State \\multiline{Retrain the GPR to induce an updated PPD, $\\bm{f}^{(n+1)} | \\mathcal{D}^{(n)}$, corrupting $\\mathcal{D}^{(n)}$ with observation noise $\\sigma^2_y$.}\n        \\EndWhile\n    \\State \\multiline{Optimise hyperparameters $\\bm{\\theta}$ and retrain GPR using final set of observations $\\mathcal{D}^{(n)}$ with optimised hyperparameters $\\hat{\\bm{\\theta}}$;}\n    \\State \\textbf{return} optimised posterior predictive mean, $\\hat{\\bm{f}}^{(n+1)} | \\mathcal{D}^{(n)}, \\hat{\\bm{\\theta}}$ evaluated at $X^*$, alongside a 95\\% credible band.\n    \\end{algorithmic}\n    \\label{alg:gpet}\n\\end{algorithm}\n\n\\section{Applications}\\label{apps}\nIn order to illustrate the effectiveness of the proposed methodology, we will compare its performance with two commonly used segmentation algorithms in medical and satellite image applications. These two segmentation algorithms are Dijkstra's shortest path algorithm \\cite{dijkstra1959note} and the active contour, energy minimising snake algorithm \\cite{kass1988snakes}, both of which are available from the scikit-image Python library \\cite{Walt2014skimage}. Table \\ref{tab:results_table} summarises the performance between the proposed methodology and the two popular edge tracing techniques using the Jaccard score \\cite{jaccard1912distribution} for comparing intersection-over-union of two binary segmentation masks. Execution time is also measured but with no expectation that the proposed algorithm will be faster given the scikit-image implementation of Dijkstra's algorithm is heavily optimised using Cython \\cite{Walt2014skimage}, unlike the proposed and active contour approaches. Due to the Mat\\'{e}rn kernel having better capacity for modelling real-life processes, all edges traced using our approach in this section utilise the Mat\\'{e}rn kernel with $\\nu = 2.5$. Note, for ease of reading, the 95\\% credible intervals are not shown in the applications of this section.\n\n\\subsection{Choroid Segmentation in OCT Imaging}\\label{subsec:OCT}\nOptical coherence tomography (OCT) imaging acquires micro-resolution, three-dimensional maps of biological tissue using low-coherence light. This technique is primarily used in ophthalmology to image the retinal and choroidal structures at the back of the eye. They are typically used to help clinicians deduce pathologies which affect or can be observed in the eye such as  glaucoma \\cite{lederer2003analysis}, chronic kidney disease \\cite{balmforth2016chorioretinal} and cardiovascular disease \\cite{farrah2020eye}. Because the layer boundaries in the retina and choroid are predominantly horizontal, this is a perfect application for the proposed methodology. The lower choroid boundary, the choroid--scleral interface, is particularly challenging to trace because of vessel shadowing and speckle noise. The latter is the result of the signal weakening the further it penetrates through the eye.\n\n\\begin{figure}[!t]\n    \\centering\n    \\setlength{\\unitlength}{1cm}\n    \\begin{picture}(20, 7.65)\n        \\put(0.0,0.0){\\includegraphics[width=\\textwidth]{_FIGURES\/OCT_Prop_Traces.pdf}}\n        \\put(0.175, 7.2){\\textbf{\\color{white}(a)}}\n        \\put(4.3, 7.2){\\textbf{\\color{white}(b)}}\n        \\put(8.375, 7.2){\\textbf{\\color{white}(c)}}\n        \\put(12.475, 7.2){\\textbf{\\color{white}(d)}}\n        \\put(0.175, 3.45){\\textbf{\\color{white}(e)}}\n        \\put(4.3, 3.45){\\textbf{\\color{white}(f)}}\n        \\put(8.375, 3.45){\\textbf{\\color{white}(g)}}\n        \\put(12.475, 3.45){\\textbf{\\color{white}(h)}}\n    \\end{picture}\n    \\caption{(a--d) Four cross-sectional OCT images. (b, d) Points propagated from traces in (e, g) in blue (more on this in section \\ref{subsec:propagate}). (e--h) Choroid segmentation of the three automated edge tracing techniques with manual labelling.}\n    \\label{fig:OCT_traces}\n\\end{figure}\nFigure \\ref{fig:OCT_traces} shows four scans from two exemplar OCT image sequences. Subfigures \\ref{fig:OCT_traces}(a, b) and \\ref{fig:OCT_traces}(c, d) are consecutive pairs of the same sequence. The bottom row shows the result of segmenting the choroidal region using the three automated approaches with manual segmentations provided by a trained, clinical ophthalmologist. The two OCT image sequences highlight the extremes in image quality of OCT cross-sectional scans. While subfigures \\ref{fig:OCT_traces}(e, f) show a clearer choroid region successfully segmented by all three approaches, subfigures \\ref{fig:OCT_traces}(g, h) illustrate the proposed algorithms superiority in dealing with low signal strength. This is due to utilising its Gaussian process back-end to interpolate through missing and noisy data. Note that the manual segmentation was unable to complete the lower choroid-scleral boundary in subfigures \\ref{fig:OCT_traces}(g, h).\n\n\\subsection{Optic Disc Segmentation in Fundus Imaging} \\label{subsec:fundus}\nThe optic disc is an approximately ellipsoidal retinal structure defining the outer rim of the optic nerve head (ONH). This is the the part of the eye which allows retina vessels to carry visual information to the brain. The ONH is usually shown to be a brighter region on the fundus than anywhere else. Accurate segmentation of the optic disc along the optic nerve head's circumference is the first step toward diagnosis of glaucoma \\cite{maccormick2019accurate}. Since the proposed methodology must enforce injectivity when modelling an EoI, the optic disc cannot immediately be traced. However, since it is a closed curve, we can utilise polar coordinate transformation so that the EoI becomes injective.  \n\nFigure \\ref{fig:fundus_traces} shows five fundus scans acquired from the RIM-ONE open database of healthy and glaucomatous images \\cite{fumero2011rim} and the scikit-image Python data module \\cite{Walt2014skimage}. Here, each EoI is occluded at several points as retinal vessels enter the disc. The proposed methodology consistently interpolates around the retinal vessels while the other automated approaches fall victim either to the inner retinal vessels or extraneous tissue surrounding the disc, as seen in subfigures \\ref{fig:fundus_traces}(h, j) or \\ref{fig:fundus_traces}(f, g, i), respectively. In particular, Dijkstra's algorithm fails due to its greedy one-shot approach, which always prefers the path of maximal image gradient response without taking into account the holistic geometric properties of the EoI.\n\n\\begin{figure*}[!t]\n    \\centering\n    \\setlength{\\unitlength}{1cm}\n    \\begin{picture}(20, 8.75)\n        \\put(0.0,0.0){\\includegraphics[width=\\textwidth]{_FIGURES\/Fundus_Traces.pdf}}\n        \\put(0.175, 8.3){\\textbf{\\color{white}(a)}}\n        \\put(3.45, 8.3){\\textbf{\\color{white}(b)}}\n        \\put(6.75, 8.3){\\textbf{\\color{white}(c)}}\n        \\put(10.05, 8.3){\\textbf{\\color{white}(d)}}\n        \\put(13.3, 8.3){\\textbf{\\color{white}(e)}}\n        \\put(0.175, 4){\\textbf{\\color{white}(f)}}\n        \\put(3.45, 4){\\textbf{\\color{white}(g)}}\n        \\put(6.75, 4){\\textbf{\\color{white}(h)}}\n        \\put(10.05, 4){\\textbf{\\color{white}(i)}}\n        \\put(13.3, 4){\\textbf{\\color{white}(j)}}\n    \\end{picture}\n    \\caption{(a--e) Five fundus images showing the optic disc and retinal vessels. The first two are glaucomatous and the remaining three are healthy. (f--j) Optic disc segmentations of the three automated edge tracing approaches with manual labelling.}\n    \\label{fig:fundus_traces}\n\\end{figure*}\n\n\\subsection{Satellite Image Segmentation}\\label{subsec:satellite}\nAnother important application of image processing is satellite imaging. This can help locate and map land and water disasters such as landslides \\cite{yu2020landslide} or mass oil spills \\cite{yekeen2020novel}. Figure \\ref{fig:sat_traces} shows a satellite image of part of the coast of California, acquired from the European Space Agency's Copernicus program \\cite{copernicusSatelliteHub}. Traces from all three automated segmentation approaches and manual labelling are shown in the bottom subfigure. The satellite image was chosen because the EoI is predominantly injective and geometrically complex, which can pose problems of short-circuiting. Short-circuiting is where a segmentation algorithm prefers a cheaper or smoother path to follow rather than the true edge, which may contain sharp corners for example. \n\nThe proposed methodology and Dijkstra's algorithm trace the coast successfully with minor issues of short-circuiting toward the right-hand side of the image, shown in the blue dotted square. Their main region of disagreement can be seen in the blue dashed square, where the coast becomes temporarily multi-valued which Dijkstra's algorithm can successfully trace, unlike the proposed methodology which short-circuits. However, the active contour model performs the most poorly, particularly in the region highlighted by the blue filled square. One of the difficulties of the active contour model is in knowing how to tune its parameters correctly, particularly the initial contour which is crucial and can be difficult to define a priori. \n\n\\begin{figure}[!t]\n    \\centering\n    \\setlength{\\unitlength}{1cm}\n    \\begin{picture}(20, 5)\n        \\put(0.0,0.0){\\includegraphics[width=\\textwidth]{_FIGURES\/Satellite_Traces_H.pdf}}\n        \\put(0.1, 4.35){\\textbf{\\color{white}(a)}}\n        \\put(8.45, 4.35){\\textbf{\\color{white}(b)}}\n    \\end{picture}\n    \\caption{(a) Satellite image showing the coast of California. (b) Coast segmentations of the three automated edge tracing techniques with manual labelling. Coloured squares represent discrepancies between traces.}\n    \\label{fig:sat_traces}\n\\end{figure}\n\n\\subsection{Image Sequence Segmentation}\\label{subsec:propagate}\nThe GPR at the beginning of the algorithm need not only be initialised with an estimate for the edge endpoints. Instead, the algorithm can be initialised with an evenly spaced set of pixels spanning the EoI. A use case for this feature is taking advantage of domain knowledge in image sequences, such as frames of a video or tomographic data. By tracing an EoI in one frame of the sequence, we can propagate a subset of these edge pixels into the model when tracing the same edge in consecutive images. This is under the assumption the edge changes incrementally, maintaining most of the same geometric and structural properties, between consecutive images. This allows the algorithm to start with a PPD which samples curves in the vicinity of the new edge, reducing the algorithms exploration space, quickening convergence along the new edge compared with initialising using only the edge endpoints.\n\nTo refer back to subsection \\ref{subsec:OCT}, subfigures \\ref{fig:OCT_traces}(b, d) show two cross-sectional scans next in the sequence from scans in \\ref{fig:OCT_traces}(a, c). Because OCT data represents a volumetric scan of biological tissue, there is necessarily some consistency and similarity in the geometric properties of the choroid and retinal interfaces between consecutive scans. Subfigures \\ref{fig:OCT_traces}(b, d) show blue, crossed points extracted from the traces of the proposed methodology in subfigures \\ref{fig:OCT_traces}(e, g), respectively. Using these pixels as initialisation, the proposed methodology successfully segments the choroid in subfigures \\ref{fig:OCT_traces}(f, h). Table \\ref{tab:results_table} summarises the performance across all three automated segmentation approaches compared to manual segmentations.\n\n\\begin{table}[!t]\n    \\centering\n    \\begin{tabular}{|c|c|l|l|l|}\n        \\hline\n        \\multicolumn{2}{|c|}{\\multirow{2}{*}{\\textbf{Application}}} & \\multicolumn{3}{c|}{\\textbf{Method}} \\\\ \\cline{3-5} \n        \\multicolumn{2}{|c|}{} & \\multicolumn{1}{c|}{Proposed} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Active Contour \\\\ \\cite{kass1988snakes}\\end{tabular}} & \\multicolumn{1}{c|}{\\begin{tabular}[c]{@{}c@{}}Dijkstra \\\\ \\cite{dijkstra1959note}\\end{tabular}} \\\\ \\hline\n        \n        \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}\\textbf{OCT} \\\\ (Fig. \\ref{fig:OCT_traces})\\end{tabular}} & $J$ (\\%) & $\\mathbf{91.3}$ ($\\pm$ 6.0) & 86.5 ($\\pm$ 5.5) & 85.2 ($\\pm$ 5.6) \\\\ \\cline{2-5} \n         & Time (s) & 8.2 ($\\pm$ 11.5) & 0.9 ($\\pm$ 1.5) & $\\mathbf{0.023}$ ($\\pm$ 0.01) \\\\ \\hline\n         \n        \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}\\textbf{Fundus} \\\\ (Fig. \\ref{fig:fundus_traces})\\end{tabular}} & $J$ (\\%) & $\\mathbf{91.8}$ ($\\pm$ 4.1) & 82.8 ($\\pm$ 2.4) & 89.9 ($\\pm$ 3.0) \\\\ \\cline{2-5} \n         & Time (s) & 28.1 ($\\pm$ 11.7) & 32.3 ($\\pm$ 37.0) & $\\mathbf{0.109}$ ($\\pm$ 0.1) \\\\ \\hline\n         \n        \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}\\textbf{Satellite} \\\\ (Fig. \\ref{fig:sat_traces})\\end{tabular}} & $J$ (\\%) & 99.7 & 98.1 & \\textbf{99.9} \\\\ \\cline{2-5} \n         & Time (s) & 100.5 & 21.0 & \\textbf{0.05} \\\\ \\hline \n         \n        \\multirow{2}{*}{\\begin{tabular}[c]{@{}c@{}}\\textbf{Sinusoid} \\\\ (Fig. \\ref{fig:kde_plot})\\end{tabular}} & $J$ (\\%) & \\textbf{99.6} & 93.1 & 77.0 \\\\ \\cline{2-5} \n         & Time (s) & 38.8 & 13.5 & \\textbf{0.1} \\\\ \\hline\n    \\end{tabular}\n    \\caption{Execution time and accuracy of all three automated edge-based segmentation techniques. Accuracy was measured using the Jaccard score, $J$.}\n    \\label{tab:results_table}\n\\end{table}\n\n\\section{Discussion}\\label{discussion}\nThe results in section \\ref{apps} highlight the benefits of the proposed methodology. Primarily, the novel technique is able to successfully trace discontinuous edges by interpolating between high gradient responses in a way that ensures a coherent overall trace for a given edge. This is, in part, due to converting the discrete optimisation problem to a continuous one using Gaussian process regression. This is advantageous for tracing edges that are relatively smooth, as having knowledge of some pixel coordinates in the image allows the GPR to sample curves which make appropriate assumptions on the path of the edge in neighbourhoods of such coordinates. Moreover, it can quantify uncertainty in regions of discontinuity and efficiently search it for adequate pixel coordinates which conform to the properties of the user-specified kernel, and also to the properties of the EoI when pixel coordinates surrounding these discontinuities are known. This allows the model to intelligently bridge gaps in otherwise discontinuous edges which may be degraded by noise or occluded by other artefacts. \n\nOur novel approach uses the image gradient to provide local edge information of potential edge pixels and combines this with global, structural information from the frequency distribution of optimal posterior curves. This is a direct consequence of modelling the EoI using Gaussian process regression and utilising its predictive variance. In each new iteration, the most optimal posterior curves are used to recalculate the frequency distribution. This frequency distribution shows higher density in regions where the most optimal posterior curves have visited. This allows the accept-discard scheme in each iteration to score previously and newly accepted pixel coordinates, dynamically refining the observation set of pixel coordinates depending on this frequency distribution. The balance between the image gradient and frequency distribution in equation \\eqref{eq:score_eqn} is key to deciding on the next set of pixels to fit to the GPR. Placing too much weighting on either image gradient or frequency distribution can have undesired effects. In the former, the model could be perturbed into accepting high-gradient, noisy non-edge pixels, while in the latter, local information is ignored in favour of selecting pixels which have been visited most, regardless of image gradient response. \n\nTo allow the GPR to continue exploring the space throughout the fitting procedure, the kernel hyperparameters and observation noise are not optimised in each iteration, contrary to typical application of Gaussian process regression. In our recursive Bayesian scheme, it is not the hyperparameters we wish to update but the PPD, utilising its corresponding predictive variance and posterior curves given a refined, but incomplete, observation set. Performing hyperparameter optimisation with an incomplete observation set, which necessarily have regions with missing data, will select kernel hyperparameters according to the current observations fitted to the GPR. Only posterior curves which fit the incomplete observation set will be sampled from then on, reducing the predictive variance in regions with missing data. This will limit the flexibility the PPD has in imputing the empty regions in later iterations. A disadvantage here is that an incorrect choice of kernel and hyperparameters could lead to a sub-optimal edge prediction due to the incorrect function properties encoded at the start of the algorithm. \n\nThroughout we have assumed that the edge's dominant direction is along the image width. We have already shown the use of polar image transformation to deal with simple, closed and elliptical edges. For edges whose dominant direction is not horizontal, a simple solution relies on being able to draw a line between the edge endpoints and checking that the edge does not appear multi-valued along that axis. Provided this is the case, then simple image rotation will convert the edge's dominant direction to be horizontal. Moreover, edges whose dominant direction changes in different regions of the image can be split into sub-edges and each can be traced individually, ensuring each sub-edge is injective along the image width. However, this introduces the extra computational overhead of estimating the edge endpoints for each sub-edge. We intend to address the issue of more complex and multi-valued edges in future work on this problem.\n\n\\subsection{Sensitivity Analysis}\\label{subsec:sens_anal} \nIn order to assess the robustness of the proposed methodology and its user-defined parameters, a comprehensive sensitivity analysis was conducted. This analysis used the test case in figure \\ref{fig:kde_plot} and the Jaccard score to measure how well the proposed methodology deals with perturbations in the user-defined input parameters. An edge prediction using an initial, pre-defined choice of parameters was compared with edge predictions resulting from perturbations of individual parameters, keeping all others fixed. Figure \\ref{fig:sensitivity_analysis} shows the results for assessing the robustness of eight model hyperparameters; score threshold $T$, number of posterior curves $L$, proportion of curves to keep $\\epsilon$, sub-interval length $\\Delta x$, noise variance $\\sigma^2_y$, signal variance $\\sigma_f^2$, lengthscale $\\ell$ and frequency distribution lengthscale $\\ell_\\phi$. For each of the eight hyperparameters, 100 values were chosen uniformly at random within a range of relative differences of over 5 times their initial value, resulting in 800 different experiments. The relative differences were chosen based on realistic perturbations from pre-defined, initial values. This meant excluding relative differences which were impossible, such as $T > 1$, or unnecessary, such as $L > 2000$ as it clearly adds no value or $\\ell > 65$ as there are already diminishing returns in Jaccard score for $\\ell > 50$ in figure \\ref{fig:sensitivity_analysis}.\n\\begin{figure*}[!t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{_FIGURES\/Sensitivity_Analysis.pdf}\n    \\caption{Sensitivity analysis using test case shown in figure \\ref{fig:kde_plot}. Heatmap annotated with average Jaccard scores for individual parameter perturbations, $p = p_0 + \\delta$, grouped according to relative change in parameter value from an initial choice, $p_0$.}\n    \\label{fig:sensitivity_analysis}\n\\end{figure*}\n\nThe sensitivity analysis provides an empirical guide for selecting parameters. The most robust parameters are the number of posterior curves $L$, keep ratio $\\epsilon$ and noise variance $\\sigma^2_y$, as there is negligible impact on edge prediction for almost all perturbations. $L$ should not be arbitrarily large as it has computational overhead of GPR sampling and costing of posterior curves. A value of $L$ between 100 and 1000 will suffice. $\\epsilon$ has little impact on edge prediction due to the weighting scheme in forming the frequency density function. A very small value is undesirable as the model is forced to rely on a small subset of posterior curves to choose pixels with. $\\sigma^2_y$, encoding the confidence in the models' pixels, $\\mathcal{D}^{(n)}$, should be selected between 0 and 2. The lower the value of $\\sigma^2_y$ implies a lower predictive variance in regions near observations in the PPD. Too large a value implies the model lacks confidence during pixel selection which can reduce accuracy.\n\nOther parameters which are slightly more sensitive are the score threshold $T$, sub-interval length $\\Delta x$ and lengthscale of the kernel used in constructing the frequency distribution $\\ell_\\phi$. A small score threshold or large sub-interval length quickens convergence since the margin for pixel acceptance is low or the number of observations in $\\mathcal{D}^{(n)}$ need not be large, respectively. However, this comes at a cost of accuracy. Because the score threshold is adaptive between iterations, setting $T = 1$ will initialise the search by only accepting pixels with a high gradient response and frequency density. $\\Delta x$ defines how many pixels are used to reconstruct the EoI and the choice of $\\Delta x$ depends on its complexity. For practical purposes, $\\Delta x = 5$ can reliably reconstruct edges of varying complexity. $\\ell_\\phi$ was chosen to mimic the lengthscale of a pixel, and for values $\\ell_\\phi \\geq 3$ the frequency density loses precision in where density is contributed. This causes density to spread to non-edge regions of the image leading to undesired effects, such as short-circuiting or acceptance of noisy, non-edge pixels.\n\nBased on the results in figure \\ref{fig:sensitivity_analysis}, the signal variance $\\sigma_f^2$ and kernel lengthscale $\\ell$ are the most sensitive user-defined parameters. $\\sigma_f^2$ dictates the predictive variance of the PPD. This value should be chosen large enough to contain the EoI in the 95\\% credible interval of the initial PPD, as seen in figure \\ref{fig:init_post_dist} using $\\sigma^2_f = 75^2$. The larger the kernel lengthscale $\\ell$ is, the less sinuous the posterior curves are. This prevents the model from reconstructing the complexity of the EoI, explaining why we see the Jaccard score reduce for $\\ell > 40$. The kernel lengthscale should never be set higher than the minimum distance between turning points in the EoI. Moreover, a smaller lengthscale results in highly sinuous posterior curves which slows convergence and begins to affect accuracy.\n\n\\section{Conclusion}\\label{concl}\nIn this paper, we have proposed a novel technique for edge tracing using Gaussian process regression. The algorithm follows a recursive Bayesian scheme, intelligently exploring the image for edge pixels using a well informed PPD. The algorithm combines local edge information and global structural information during the pixel scoring procedure, with the heuristics of the selection and convergence procedures based on Bayesian methods. Our algorithm is able to traverse and interpolate discontinuous edges through its ability to model uncertainty in gaps of the domain of the observation set, searching for the most sensible pixels to bridge said gap. This technique provides the user with a set of robust and simple parameters to define at initialisation, controlling aspects of convergence and model complexity.\n\nMoreover, the algorithm can be applied to numerous different image problems, a number of which we have illustrated and we have made favourable comparisons with two commonly used edge tracing algorithms. At present, the novel technique is unable to trace edges which are multi-valued along the predominant edge direction since it assumes injectivity. Many edges of interest have this property, particularly in medical imaging applications where we expect this algorithm to play a useful role. The proposed methodology can also be used efficiently for edge tracing in image sequences by utilising its ability to propagate a priori information from the edge in a previous frame of the sequence to new frames, limiting the search region to the vicinity of the new edge and speeding convergence.\n\n\\section*{Acknowledgements}\nThis work was supported by the Medical Research Council [grant number MR\/N013166\/1], as part of the Precision Medicine DTP with the University of Edinburgh. The authors would like to acknowledge Dr. Ian MacCormick, a clinical ophthalmologist at the University of Edinburgh for providing the manual ocular segmentations shown in section \\ref{apps}.\n\n\\bibliographystyle{unsrt}  \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe order in which customers\/items are served\/processed  is a common problem in queueing theory (see \\cite{Gro}). \nSee Down et al. (\\cite{Dow}) for queues with abandonment, and Raviv and Leshem for queues with deadlines. \nFor scheduling, see Conway et al. (\\cite{Con}) for discussion of RANDOM (Random), SPT (shortest processing time), \nMWKR (most work remaining), LWKR (least work remaining) orders of service.  \nNahmias (\\cite{Nah}) discussed perishable inventory including issues of out-dating. Survival analysis is a standard topic (\\cite{Hos})\nin actuarial modeling. \n\n  \nIn June and July of 2018, a group of thirteen persons (twelve boys and their coach) on a soccer team were trapped in flooded caves in Thailand. Their rescue attracted world attention.\nOne statement that initially appeared in the media stated that rescuers chose to take the strongest boys first. This was argued to give the best chance of survival.\nLater statements said that the weaker boys actually were removed first. In the end, all thirteen people were rescued, but tragically, one of the rescuers did not survive. \nIn this paper, we consider different models and criteria for which removing the stronger persons first may or may not be the best strategy.\nThe models to be presented do not fit precisely into standard scheduling or queueing or perishable inventory or survival analysis type models, and the objective here is different than in other settings. \n\n\\section{Details }\n\nIn the actual Thailand situation, the people were rescued in batches (4+4+5). For our analysis, we will change the setting so that we have $n$ items, and they are processed one at a time. The items are labeled $1,2,\\dots,n$ in the order of processing that is chosen (which can be changed as one chooses). Assume that at time 0, the $n$ items have probabilities of success $P_0(i), i=1,2,\\dots,n$. (For the Thailand situation, we have $n=13$). \nLet $P_0=(P_0(1),\\dots,P_0(n))$. \n\nTwo possible measures of success are (a) the expected number of successfully processed items (expected number of rescued people) (b) the probability that all items are successfully processed (probability that all people are rescued).\n\nFirst we assume that the initial probabilities of success do not change over time. \nLet $X_i=1$  if item $i$ is successfully processed and $X_i=0$ if item $i$ is not successfully processed.  Let $Y=X_1+\\dots +X_{n}$ be the total number of successes among the $n$ items.  \nThus we are dealing with a generalization of the binomial distribution with differing $p$'s. This topic has been studied extensively (see \\cite{Bol}, \\cite{Dar}, \\cite{Hoe}, \\cite{Sam}).  \nFor a generalized binomial distribution with $n$ independent trials with success probabilities $(p_1,\\dots,p_n)$, let $X_i=$ the success probability of item $i$, and let $Y=$ total number of successes. Then the probability of exactly $k$ successes in the $n$ trials is the coefficient of $z^k$ in the expansion of $\\prod_{i=1}^n  (1-p_i+p_i z)$. \n\n\\begin{Theorem}\nFor the generalized binomial model, both $E(Y)$ and \\\\\n$P(\\text{all successes})$ are constant regardless of the order in which items occur. \n\\end{Theorem}\n\\begin{proof}\nNote that $E(X_i)=0p_i+1(1-p_i)=p_i$. So \\\\\n\\begin{equation*}\nE(Y)=\\sum_{i=1}^n E(X_i)=\\sum_{i=1}^n p_i \\text{ and } P(\\text{all successes})=\\prod_{i=1}^n p_i.\n\\end{equation*}\nSince the expressions for $E(Y)$ and $P(\\text{all successes})$ do not change when we change the order of the $p_i$, the result follows. \n\\end{proof}\nIn the context of the rescue problem, the order of rescuing the 13 people does not matter as far as the expected number of successes or the probability of complete success is concerned, if the probabilities of success do not change over time. \n\nIt is more reasonable, however, to assume that as time goes by, the probabilities change  from the original vector $P_0$. Let  $0=t_1<t_2<\\dots<t_n$ be the times of start of processing for the $i$-th ($i=1,2,\\dots, n$) item. Then $T_i=t_{i+1}-t_i$ ($i=1,\\dots,n$) is the interval between individual processing time starts. \nThe probability of success of processing the $i$-th item is originally $P_0(i)$ but becomes $P_1(t_i)$ (to be specified) at the time that the $i$-th item is processed because of the delay to begin processing. Let $P_1=(P_1(t_1), P_1(t_2),\\dots, P_1(t_n))$. So\n\\begin{align*}\n&E(\\text{number of successes})=\\sum_{i=1}^n E(X_i)=\\sum_{i=1}^n P_1(t_i) \\\\\n&P(\\text{all successes})=\\prod_{i=1}^n P_1(t_i).\n\\end{align*}\n\n\\subsection{Additive Model}\nAssume that the interval between the processing start time of consecutive items is a constant value $T$. Then the ordered processing start times for the $n$ items are $(0, T, 2T, \\dots, (n-1)T)=(t_1,t_2,\\dots,t_{n})$ so $t_i=(i-1)T$. We assume an additive model for the values $P_1(t_i)$. Use the notation $x^+=max(x,0)$. Specifically, we assume $P_1(t_i)=\\max\\{P_0(i)+f(t_i),0\\} \\equiv (P_0(i)+f(t_i))^+$, where the function $f$ is chosen to take negative values ($f(t)<0$), and we assume $f'(t)\\leq 0$. The rationale for this assumption is that the probability of success decreases for every item over time.  \n\n\\begin{Theorem}\nIf $P_0(i)+f(t_i) >0$, $\\forall i$, then \n\\begin{align}\nE(\\# successes)&= \\sum_{i=1}^{n} P_0(i) +\\sum_{i=1}^{n} f((i-1)T) \\\\\nP(\\text{all successes}) &= \\prod_{i=1}^{n} (P_0(i)+f((i-1)T))\n\\end{align}\n\\end{Theorem}\n\\begin{proof} \n\\begin{align}\nE(\\# successes) &= \\sum_{i=1}^{n} P_1(t_i) \\notag \\\\\n&= \\sum_{i=1}^{n} (P_0(i)+f(t_i))^+= \\sum_{i=1}^{n} (P_0(i)+f(t_i)) \\notag\\\\\n&= \\sum_{i=1}^{n} P_0(i) +\\sum_{i=1}^{n} f((i-1)T) \\notag\\\\\nP(\\text{all successes}) &= \\prod_{i=1}^{n} P(\\text{i-th item is success})= \\prod_{i=1}^{n} P_1(i)\\notag\\\\\n&= \\prod_{i=1}^{n} (P_0(i)+f((i-1)T))\\notag\n\\end{align}\n\\end{proof}\n\\begin{Corollary} The expected  number of successes is independent of the order of service. \n\\end{Corollary}\n\\begin{proof}\n\\begin{equation*}\nE(\\# successes) = \\sum_{i=1}^{n} P_0(i) +\\sum_{i=1}^n f((i-1)T). \n\\end{equation*}\nThis expression does not depend on the order of service. \\end{proof}\n\nThus, in terms of expected number of successes, the order of the service does not matter. \nHowever the expression $P(\\text{all successes})$ does depend on the order of service. Recall the arithmetic\/geometric mean inequality. \n``If numbers $x_1,\\dots,x_n$ are not all the same, the geometric mean of these numbers is less than their arithmetic mean.''(\\cite{Kor}) Equality occurs only when the $x_i$ are all equal. So, with $\\sum_{i=1}^{n} P_0(i)+f((i-1)T)$ fixed, the product of $n$ terms would be largest when terms are closest to each other.\nIn terms of the Thailand rescue, the probability that all 13 people survive would be the maximized if the weakest person among the remaining is saved first. But the expected number of people successfully rescued stays the same regardless of the order chosen.  Thus, we would likely choose to rescue the weaker people first in order to satisfy objective (b). \n\nAs a simple example with $n=4$, take $P_0=(.8,.9,.7,.7)$  and $f(x)=-.1x\/T$ so $\\{f(t_i)\\}=\\{f((i-1)T)\\}=\\{-.1(i-1)\\}=\\{0,-.1,-.2,-.3\\}$. Thus $P_1=(P_1(t_1), P_1(t_2),P_1(t_3),P_1(t_4))=(.8-0,.9-.1,.7-.2,.7-.3)=(.8,.8,.5,.4)$.\\\\\nSo  $E(\\# successes)=  \\sum_{i=1}^{n} P_1(t_i)=.8+.8+.5+.4 = 2.5$ and \\\\\n$P(\\text{all successes}) =  \\prod_{i=1}^{n} P_1(t_i)=.8(.8)(.5)(.4) = .128$. \\\\\nIf we change the order  of $P_0$ to (.7,.7,.8,.9) but leave $f$ in its current form, we obtain $E(\\# successes) = 2.5$ and $P(\\text{all successes})=.1512$.\\\\\nIf we change the order of $P_0$ to (.9,.8,.7,.7) but leave $f$ in its current form, we obtain    $E(\\# successes) = 2.5$ and $P(\\text{all successes})=.126$. We see that using an increasing order in $P_0$ gives the optimal (highest) product. \n\nNext we consider the case when there exists an ordering such that for some $i$, we have $P_0(i)+f(t_i) < 0$.\n\\begin{Theorem} If for some $i$ we have $P_0(i)+f(t_i) < 0$, then\n\\begin{align}\nE(\\# successes) &= \\sum_{i=1}^{n} P_1(t_i)= \\sum_{i=1}^{n} (P_0(i)+f(t_i))^+ \\\\\nP(all successes) &=  \\prod_{i=1}^{n} P_1(t_i)= \\prod_{i=1}^{n} (P_0(i)+f(t_i))^+ = 0\n\\end{align}\n\\end{Theorem}\n\nWe change our earlier example and choose $P_0=(.8,.9,.1,.2)$, but leave $f$ as before with $\\{f(t_i)\\}=\\{f((i-1)T)\\}=\\{-.1(i-1)\\}=\\{0,-.1,-.2,-.3\\}$.\nThen  $\\{P_1(i)\\}=(.8-0,.9-.1,.1-.2,.2-.3)=(.8,.8,(-.1)^+,(-.1)^+.4)= (.8,.8,0,0)$\\\\\nSo  $E(\\# successes)=  \\sum_{i=1}^{n} P_1(t_i)=.8+.8+0+0 = 1.6$ and \\\\\n$P(\\text{all successes}) =  \\prod_{i=1}^{n} P_1(t_i)=.8(.8)(0)(0) = 0$. \\\\\nIf we change the order  of $P_0$ to (.1,.2,.8,.9) but leave $f$ in its current form, we obtain $E(\\# successes) = .1+.1+.6+.6=1.4$ and $P(\\text{all successes})=.1(.1)(.6)(.6)= .0036$.\\\\\nIf we change the order of $P_0$ to (.9,.8,.2,.1) but leave $f$ in its current form, we obtain  $E(\\# successes) = .9+.7+0+0=1.6$ and $P(\\text{all successes})=.9(.7)(0)(0)= 0$.\\\\\nWe see that we maximize $E(\\# successes)$ by choosing $P_0$ to have values in decreasing order but we maximize $P(\\text{all successes})$ by choosing $P_0$ components to be in increasing order. \nIn terms of the Thailand rescue, if for some $i$ we have $P_0(i)+f(t_i) < 0$, then we achieve a higher expected number of people rescued, if we rescue the stronger people first. This requires us to have a good estimate of the initial probabilities. However, we have a larger value of $P(\\text{all successes})$ if we rescue the weaker people first. \n\nWe can illustrate this as follows. We create a $13\\times 13$ matrix. Each column represents a person. The rows represent time and the $(i,j)$ entry represents the probability of success if person $j$ is rescued on the $i$th step (at time $t_i$). For row 1, we generate 13 random values  uniformly on (.5,1) to represent the probabilities $P_0(j)$of success  for person $j$ at time 0. The people are labeled so that each row is in increasing order, with the weaker person (lowest success probability) is in column 1 and the strongest person is in column 13. To get the later rows, we use the function $f$, described earlier. This time, we choose $f(x)=-0.06x\/T$. Then the $(2,j)$ element in row 2 is given by \\\\\n$(P_0(j)-.06(2-1))^+$, and the (3,j) element of row 3  consists of entry\\\\ $(P_0(j)-.06(3-1))^+$, and so on.   \nThe probabilities are presented in a matrix shown in Figure \\ref{color_matrix}, where the square at $i$-th row and $j$-th column corresponds to the $j$-th person's probability of survival if he is saved at the $i$-th stage. The bigger the square is and the darker its color, the bigger the probability.\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[scale=0.5]{colormatrix.pdf}\n\\caption{Prob that customer $j$ survives if served at stage $i$}\n\\label{color_matrix}\n\\end{figure}\n\nSee Figure \\ref{color_matrix}. We must choose one entry from each row and each column to indicate the order that the rescue attempts are made. If we always choose the weakest person to be taken earlier, then we are using the diagonal of the matrix from upper left to lower right. \nIf we always choose the strongest person to be taken earlier, then we are using the diagonal from lower left to upper right. The product (and hence $P(\\text{all survive})$) is maximized if the colors are closer to each other. It seems clear that the product of all entries for the case of stronger first strategy will give a value nearer to zero than the weaker first strategy because of the almost white values in the lower\nleft corner. So we maximize $P(\\text{all survive})$ by rescuing the weaker people first. However, it is not so clear as to the optimal order of rescue in terms of the expected values since we must sum the values rather than multiply them. \n\n\\section {Evaluating the model}\n\nOur matrix in Figure \\ref{color_matrix} was obtained by choosing the probabilities of success i.i.d. from $Unif(.5,1)$ and that these probabilities drop by $.06$  to a minimum of 0\n(after each of the first 12 stages, so $f(x)=-.06x\/T$. For such a model, define \\\\\n$\\pi_s = P(\\text{all successes with strongest first}) $ and \\\\$\\pi_w = P(\\text{all successes with weakest first})$.\\\\\n It is possible that the values that we generate by simulation will result in \n$\\pi_s=0$ or $\\pi_s>0$ (similarly for $\\pi_w$). As an evaluation of the model, we next compute $P(\\pi_s>0)$ and $P(\\pi_w>0)$, if we were to simulate the values. \n\nFirst consider $P(\\pi_s>0)$.  For our generator, let $P_0$ be the vector of initial probabilities generated. Let $P_1=(P_1(t_1),\\dots,P_1(t_n))$ be the vector of success probabilities for the items $(1,\\dots,n)$ at the processing start time for those items. Then $\\pi_s=0$  iff $\\exists k, s.t.\\, P_1(t_k)=0$. As earlier, take $t_i=(i-1)T$.  \n\\begin{align}\nP(\\pi_s >0) &=P(\\min \\{P_1(t_k)\\} >0) = P(\\min \\{P_0(k)\\} +f(t_{13})>0)\\notag\\\\\n&= P(\\min \\{P_0(k)\\} +f(12T))>0)= P(\\min \\{P_0(k)\\} -.72>0) \n\\notag \\\\\n&=\\prod\\limits_{k=1}^{13}P(P_0(k) > .72)=(\\frac{1-.72}{.5})^{13}=.000533 \n\\end{align}\n\nIf the weak ones were saved first, then the initial cases (all with $P_0(i)>.5$) would all have positive probabilities and the only cases with potentially zero probabilities generated would be\n the largest four values which must be larger than .06*.9, .06*10, .06*11,.6*12 respectively. The probability of such values being generated can be found frm the joint distribution of the largest four order statistics of values generated on Unif(.5,1). \n If we call these order statistics $t,u,v,w$ (from smaller to larger), then the joint distribution of the largest four order statistics is $g(t,u,v,w) = \\dfrac{13!}{9!1!1!1!1!}(2t-1)^9 2^4.$ \n This would be computed as \n\n \n\\begin{align*}\n P(\\pi_w >0) &= P(\\bigcap_{k=1}^{13} \\{ P_0(k) > -f(T(k-1))\\}  \\\\\n&=\\int_{.72}^1\\int_{.66}^w\\int_{.6}^v\\int_{.54}^u  g(t,u,v,w) dt\\, du\\, dv\\, dw=.9999677\n\\end{align*}\n \n\nThus, for the model that we are using to generate our values, if we are most concerned with $P(\\text{all successes})$, then it is best to processes highest probability of success items first (i.e. save the stronger boys first). \n\n\n\n\\section {Mutiplicative Model}\n\nIn the previous analysis, we assumed that we had an initial probability vector and that the probabilities decreased over time in an additive manner.\nHowver, it might be more reasonable to assume that the decrease in probabilities over time is multiplicative rather than additive. \ni.e. We begin wth $P_0=(P_0(1),\\dots,P_0(n)$ as the probabilities of success for $n$ items listed in order of processing. \nLet $p\\in (0,1)$ be the multiplciative factor resulting in the updated probability. After the first item is processed, the probabilites of the remaining $n-1$ items become $(P_0(2)*p,\\dots, P_0(n)*p)$, etc.\nAt the time of processing, the vector of probabilites of success is \\\\\n$P_1=(P_1(1),P_1(2),\\dots,P_1(n))=(P_0(1),P_0(2)*p,\\dots,,P_0*p^n)$.\n\n\\begin{Theorem} For initial success vector  $P_0=(P_0(1),\\dots,P_0(n))$and the multiplicative factor $p$, \n\\begin{align}\nE(\\# successes)&=\\sum_{i=1}^n P_0(i)p^i\\\\\nP(\\text{all successes})&=\\prod_{i=1}^n P_0(i)p^{(n-1)n\/2}.\n\\end{align}\n\\end{Theorem}\n\\begin{proof}\n\\begin{align*}\nE(\\# successes)&=\\sum_{i=1}^n P_1(i)=\\sum_{i=1}^n P_0(i)p^i\\\\\nP(\\text{all successes})&=\\prod{i=1}^n P_1(i)=\\prod_{i=1}^n P_0(i)P^i=\\prod_{i=1}^n P_0(i)p^{(n-1)n\/2}.\n\\end{align*}\\end{proof}\n\n\n\n\\begin{Corollary}\nIn the multiplicative model, $P(\\text{all successes})$ is independent of the order of processing.\n\\end{Corollary}\n\n\\begin{Corollary}\nIn the multiplicative model, $E(\\# successes)$ is maximized if $P_0$ is sorted so that $P_0(1)\\geq \\dots \\geq P_0(n)$\n\\end{Corollary}\n\\begin{proof} The result can be shown by induction. \\end{proof}\n\nIn the case of the cave rescue, the multiplicative model indicates that the order of rescue does not affect\n$P(\\text{all successes}$, but that in order to maximize the number of people saved, the stronger people should be rescued first.\n\n\\section{Conclusion}\nIn a rescue situation, one needs to clarify what the goal is. Generally, the logical goal would be to maximize the expcted number of successes.\nIn the additive model, the order does not matter, unless some of the items have their probabilities drop too much by the delay, in which case \nthe higher order items should be processed first. In the multiplicative model, the preferred order is to rescue the higher success probability items first.\nThis conflicts with our intuition, and it also contradicts our sense of fairness. A seemingly less important goal of maximizing the probability that all items are\nsuccessfully processed, results in the opposite preferred ordering.  \n\n\\noindent\n{\\bf Acknowledgements.} \nWe acknowledge funding and support  from MITACS Global Internship program, University of Windsor, Central South University, CSC Scholarship. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe relative phase between the strong and the electromagnetic amplitudes of the\ncharmonium decays is a basic parameter in understanding decay dynamics. Studies\nhave been carried out for many $\\jpsi$ two-body decay modes:\n$1^-0^-$~\\cite{dm2exp,mk3exp}, $0^-0^-$~\\cite{a00,lopez,a11},\n$1^-1^-$~\\cite{a11} and $N\\overline{N}$~\\cite{ann}. These analyses reveal that\nthere exists a relative orthogonal phase between the strong and the\nelectromagnetic amplitudes in $\\jpsi$ decays~\\cite{dm2exp, mk3exp, a00, lopez,\na11, ann, suzuki}. As to $\\psp$, there is also a theoretical argument which\nfavors the $\\pm90^\\circ$ phase~\\cite{gerard}. Experimentally, some\nanalyses~\\cite{wymppdk,wymphase,besklks1} based on limited $\\VP$ and $\\PP$ data\nindicate that the large phase is compatible with the data. Moreover, some\nefforts have been made to extend the phase study to $\\pspp$ decay\nphenomenologically~\\cite{Wang03b,wangp06} and experimentally~\\cite{psppklks}.\n\nThe great merit of the phase study lies in that it can provide the valuable\nclue for the relation between the strong and the electromagnetic interactions.\nNow with the upgraded accelerator and detector, BEPCII\/BESIII, on May 2009, the\nhigh luminosity of $3\\times 10^{32}$cm$^{-2}$s$^{-1}$ had achieved, which is\nthe highest luminosity in $\\tau$-charm energy region which ever existed. The\n106 M $\\psi^{\\prime}$ and 226 M $J\/\\psi$ events have been collected, even more\ncolossal data are to be collected in the forthcoming years, which gives a great\nopportunity to determine the phase between the strong and the electromagnetic\namplitudes with unprecedented statistical precision.\n\nA favorable way to measure the phase is through the scan experiment\nwhich is the most model-independent approach. However, even with high\nluminosity accelerator, the exclusive scan experiment of charmonium decay is\nfairly difficult due to low statistics at each energy point. Therefore, the\noptimization study for the data taking strategy is of great importance in order\nto obtain the most accurate results with the limited luminosity (equivalently\nwithin the limited data taking time).\n\nWithout losing generality, we focus on the mode of $\\psp$ decays to two\npseudoscalars. Because, as will be shown in the next section, this decay mode\ncan accommodate a comparatively simple parametrization form which is of great\nbenefit to extract the relative phase. To get the optimized data taking scheme,\nwe resort to the sampling simulation technique which is successfully used in\nthe study of the data taking strategy for a high precision $\\tau$ mass\nmeasurement~\\cite{taustat, taustat2}. For such kind of method, great many times\nof fits should be done, where a large number of calculations need to be\nperformed to get the theoretically expected observed cross section.\nUnfortunately, two nested integrations in this calculation take too\nlong time to make the actual optimization procedure impractical.\n\nIn this paper, we devote to the simplification of calculation for the\nobserved cross section of $\\psp$ decaying to pseudoscalar pair. Some\nreasonable assumptions lead us to obtain the analytic expression for the\nInitial State Radiative (ISR) corrected cross section. That is to say, we\ntransform the two-fold integral into a one-fold integral which speeds up the\ncalculation by one hundred times.\n\n\\section{Observed cross section}\n\nThe process of $\\psp$ decays to Pseudo-scalar and Pseudo-scalar (PP) final\nstate can be parameterized by merely two amplitudes~\\cite{a11, haber}, that is\n\n\\begin{equation}\n\t\\begin{array}{lcl}\n\t\tA_{\\pp} &=& A_{EM}~~, \\\\ A_{\\kk} &=& A_{EM} +\n\t\tA_{S},\\\\ A_{\\kskl} &=& A_{S}~~,\n\t\\end{array}\n\t\\label{em}\n\\end{equation}\nwhere $A_{EM}$ denotes the electromagnetic amplitude and $A_{S}$ the SU(3)\nbreaking strong amplitude. Here, the G-parity violating channel $\\pp$ is\nthrough the electromagnetic process~(the contribution from the\nisospin-violating part of QCD is expected to be small~\\cite{chernyak82} and is\nneglected), $\\kskl$ through the SU(3) breaking strong process, and $\\kk$\nthrough both. For $\\EE$ experiment, the actual amplitudes must include the\ncontribution of continuum which features the electromagnetic\nprocess~\\cite{Wang03,wymppdk,wymphase}:\n\n\\begin{equation}\n\t\\begin{array}{lcl}\n\t\tA_{\\pp} &=& A^c_{EM}+ A_{EM}~~, \\\\ A_{\\kk} &=&\n\t\tA^c_{EM}+ A_{EM} + A_{S},\\\\ A_{\\kskl} &=& A_{S}~~,\n\t\\end{array}\n\t\\label{emc}\n\\end{equation}\nwhere $ A^c_{EM}$ is the amplitude of the continuum contribution. Besides the\ncommon part, $A^c_{EM}$, $A_{EM}$ and $A_{S}$ can be expressed explicitly as\n\n\\begin{equation}\n\t\\begin{array}{lcl}\n\t\tA^c_{EM} &\\propto&{\\displaystyle\n\t\t\\frac{1}{s}}~~,{\\rule[-3.5mm]{0mm}{7mm}} \\\\ A_{EM}\n\t\t&\\propto&{\\displaystyle \\frac{1}{s} B(s)}~~, \\\\ A_{S}\n\t\t&\\propto&{\\displaystyle {\\cal C} e^{i \\phi} \\cdot \\frac{1}{s} B(s)\n\t\t}~~,\n\t\\end{array}\n\t\\label{ecem}\n\\end{equation}\nwhere the real parameters $\\phi$ and ${\\cal C}$ are the relative phase and the\nrelative strength between the strong and the electromagnetic amplitudes, and\n$B(s)$ is defined as~\\cite{wymppdk}\n\n\\begin{equation}\n\tB(s)=\\frac{3\\sqrt{s}\\Gamma_{ee}\/\\alpha}{s-M^2_{\\psp}+iM_{\\psp}\\Gamma_t}~~.\n\t\\label{bexpr}\n\\end{equation}\nHere $\\sqrt{s}$ is the center of mass energy, $\\alpha$ is the QED fine\nstructure constant; $M_{\\psp}$ and $\\Gamma_t$ are the mass and the total width\nof $\\psp$; $\\Gamma_{ee}$ is the partial width to $\\EE$.\n\nThe Born order cross sections for the three channels read\n\n\\begin{equation}\n\t\\begin{array}{ll}\n\t\t\\sigma_{Born}^{\\pp}(s) &= {\\displaystyle\n\t\t\\frac{4\\pi\\alpha^2}{s^{3\/2}}\\left[1+2\\Re B(s)+|B(s)|^2\\right] } \\\\ &\n\t\t\\times {\\displaystyle \\left| {\\cal F}_{\\pp} (s)\\right|^2{\\cal P}_{\\pp}\n\t\t(s)} ~,\n\t\\end{array}\n\t\\label{bnpp}\n\\end{equation}\n\n\\begin{equation}\n\t\\begin{array}{ll}\n\t\t\\sigma_{Born}^{\\kk}(s)&={\\displaystyle\n\t\t\\frac{4\\pi\\alpha^2}{s^{3\/2}}\\left[1+2\\Re ({C}_{\\phi} B(s)) +\\left|{\n\t\tC}_{\\phi} B(s)\\right|^2 \\right]} \\\\ & \\times {\\displaystyle\n\t\t\\left|{\\cal F}_{\\kk} (s) \\right|^2{\\cal P}_{\\kk} (s) }~,\n\t\\end{array}\n\t\\label{bnkk}\n\\end{equation}\n\n\\begin{equation}\n\t\\sigma_{Born}^{\\kskl}(s)=\n\t\\frac{4\\pi\\alpha^2}{s^{3\/2}} {\\cal C}^2 |B(s)|^2 |{\\cal F}_{\\kskl}\n\t(s)|^2{\\cal P}_{\\kskl} (s) ,\n\t\\label{bnkskl}\n\\end{equation}\nwhere $C_{\\phi} = 1 + {\\cal C}e^{i\\phi}$; ${\\cal F}_{f.s.}(s)=f_{f.s.}\/s$, with\n$f_{f.s.}$ being an energy independent constant, and $f.s.=\\pp,\\kk,\\kskl$;\n${\\cal P}_{f.s.}(s) = 2q^3_{f.s.}\/3s$, with $ q^2_{f.s.} = E^2_{f.s.} -\nm^2_{f.s.} = s\/4 - m^2_{f.s.}$.\n\nIt is obvious that in Eq.~\\eref{bnkk}, if ${ C}_{\\phi} =1$,\n$\\sigma^{Born}_{\\kk}(s)$ is identical to $\\sigma^{Born}_{\\pp}(s)$ while if ${\nC}_{\\phi} ={\\cal C}e^{i\\phi}$, $\\sigma^{Born}_{\\kk}(s)$ is identical to\n$\\sigma^{Born}_{\\kskl}(s)$. From the mathematical point of view, the cross\nsection expression of $\\sigma^{Born}_{\\kk}(s)$ is more general with the\nexpressions of $\\sigma^{Born}_{\\pp}(s)$ and $\\sigma^{Born}_{\\kskl}(s)$ as its\nspecial cases.  Therefore, in the following study, only tackled is the formula\nfor $\\kk$ final state and $f.s.$ is simply donated as $K$.\n\nIn $\\EE$ collision, the Born order cross section is modified by the ISR in the\nway~\\cite{Kuraev85}\n\n\\begin{equation}\n\t\\sigma_{r.c.} (s)=\\int_{0}^{X_f} dx F(x,s)\n\t\\frac{\\sigma_{Born}(s(1-x))}{|1-\\Pi (s(1-x))|^2},\n\t\\label{eq_isr}\n\\end{equation}\nwhere $X_f=1-s'\/s$. $F(x,s)$ has been calculated to an accuracy of\n0.1\\%~\\cite{Kuraev85,Altarelli,Berends} and $\\Pi(s)$ is the vacuum polarization\nfactor. In the upper limit of the integration, $\\sqrt{s'}$ is the\nexperimentally required minimum invariant mass of the final particles. In this\nwork, $X_f=0.15$ is used which corresponds to invariant mass cut of 3.4~GeV\/$c^2$.\n\nBy convention, $\\Gamma_{ee}$ has the QED vacuum polarization in its\ndefinition~\\cite{Tsai,Alexander}. Here it is natural to extend this convention\nto the partial widths of other pure electromagnetic decays, that is\n\n\\begin{equation}\n\t\\Gamma_{K} = 2\n\t\\tilde{\\Gamma}_{ee}\\left(\\frac{q_{K}}{M_{\\psp}}\\right)^3 \\left|{\\cal\n\tF} (M^2_{\\psp}) \\right|^2~,\n\t\\label{eq_defgf}\n\\end{equation}\nwhere\n\n$$\n\\tilde{\\Gamma}_{ee} \\equiv\n\\frac{\\Gamma_{ee}}{|1 - \\Pi (m^2_{\\psp})|^2}~,\n$$\nwith vacuum polarization effect included.\n\nThe $\\EE$ colliders have finite energy resolution which is much wider than the\nintrinsic width of $\\psp$. Such energy resolution is usually a Gaussian\ndistribution~\\cite{bk:Lee,bk:Wille}:\n\n$$\nG(W,W^{\\prime})=\\frac{1}{\\sqrt{2 \\pi} \\Delta} e^{\n-\\frac{(W-W^{\\prime})^2}{2 {\\Delta}^2} },\n$$\nwhere $W=\\sqrt{s}$ and $\\Delta$, a function of the energy, is the standard\ndeviation of the Gaussian distribution. The experimentally observed cross\nsection is the radiative corrected cross section folded with the energy\nresolution function\n\n\\begin{equation}\n\t\\sigma_{obs} (W)=\\int \\limits_{0}^{\\infty}\n\tdW^{\\prime} \\sigma_{r.c.} (W^{\\prime}) G(W^{\\prime},W).\n\t\\label{eqegsprd}\n\\end{equation}\n\nFor briefness, the variables $\\tilde{\\Gamma}_{ee}$, $M_{\\psp}$, and\n$\\Gamma_{t}$ are respectively written as ${\\Gamma}_{ee}$, $M$, and $\\Gamma$\nhereafter.\n\n\\section{Simplification of ISR correction}\n\nIn this section, we focus on the simplification of ISR correction of the\nobserved cross section. In the energy region we concerned (3.67 GeV\/$c^2$ --\n3.71 GeV\/$c^2$), the vacuum polarization factor could be concerned as constant\nand absorbed into $\\tilde \\Gamma_{ee}$ as in Eq.~\\eref{eq_defgf}. So we could begin\nwith this expression:\n\n\\begin{equation}\n\t\\sigma_{r.c.}(s) = \\int_0^{X_f} dx F(x,s) \\sigma_{Born}(s(1-x))~.\n\t\\label{eqisr}\n\\end{equation}\n$F(x,s)$ is the structure function, which can\nbe expressed as follows:\n\n\\begin{equation}\n\t\\begin{split}\n\t\tF(x,s) = x^{t-1}\\cdot B_1(t) + x^t \\cdot B_2(t) + \\\\\n\t\tx^{t+1}\\cdot B_3(t) + O(x^{t+1}t^2)\n\t\\end{split}~,\n\t\\label{eqfxs}\n\\end{equation}\nwhere\n\n\\begin{equation}\n\t\\begin{array}{ll}\n\t\tB_1(t) & ={\\displaystyle t\\cdot \\left[ 1 +\n\t\t\\frac{\\alpha}{\\pi}(\\frac{\\pi^2}{3} - \\frac{1}{2}) + \\frac{3}{4}t +\n\t\tt^2\\left(\\frac{9}{32} - \\frac{\\pi^2}{12} \\right) \\right] }~,\\\\\n\t\tB_2(t) &= {\\displaystyle-t-\\frac{t^2}{4} }~,\\\\\n\t\tB_3(t) &= {\\displaystyle \\frac{t}{2} - \\frac{3}{8}t^2}~,\n\t\\end{array}\n\t\\label{expbs}\n\\end{equation}\nwith\n\n\\[\n\\quad t = \\frac{2\\alpha}{\\pi} (\\textrm{ln}\\frac{s}{m_e^2} -1)~.\n\\]\n\nBased on Eq.~\\eref{bnkk}, the whole expression of the observed cross section is\nsubdivided into three terms: the continuum, the resonance, and the interference\nterms. The simplification of each term will be discussed separately.\n\n\\subsection{Continuum term}\n\\label{sbxn_cterm}\n\nIn the light of Eq.~\\eref{bnkk}, the Born order expression for the continuum is\nwritten explicitly as\n\n\\begin{equation}\n\t\\sigma_{Born}^C = \\frac{8\\pi\\alpha^2\n\tf_K^2}{3} \\cdot \\frac{(s\/4-m_K^2)^{3\/2}}{s^{9\/2}}~.\n\t\\label{eqbornc}\n\\end{equation}\n\nIn the above equation, the most crucial part is the factor\n\n\\[\nl_{9\/2}(s)=\\frac{(s\/4-m_K^2)^{3\/2}}{s^{9\/2}}~.\n\\]\n\n\\begin{center}\n\t\\includegraphics[angle=-90,width=8cm]{factor.eps}\n\t\\figcaption{\\label{figfactor}Variations of factor $l_{\\beta}(s)$ against\n\tcenter-of-mass energy ($\\sqrt{s}$) in the vicinity of $\\psp$\n\tresonance peak for $\\beta=9\/2,~4,~7\/2$. }\n\\end{center}\n\nFor the study of charmonium physics, $s$ is much greater than $m^2_K$,\ntherefore the factor $l_{9\/2}(s)$ variates almost linearly in the vicinity of\n$\\psp$ peak as shown in Fig.~\\ref{figfactor}. With this observation, it is\nnatural to approximate the factor $l_{9\/2}(s)$ with a linear function, viz.\n\n\\[\n\\bar{l}_{9\/2}(s) \\approx \\lambda_{9\/2} \\cdot s + \\zeta_{9\/2}~.\n\\]\n\nAs a matter of fact, the similar factors appear in the resonance and\ninterference terms as well. So generally, we define\n\\begin{equation}\n\tl_{\\beta}(s)=\\frac{(s\/4-m_K^2)^{3\/2}}{s^{\\beta}}~,\n\\end{equation}\nand utilizing the approximation\n\n\\begin{equation}\n\t\\bar{l}_{\\beta}(s) \\approx \\lambda_{\\beta} \\cdot s +\n\t\\zeta_{\\beta}~. \\label{eqlnrfac}\n\\end{equation}\nHere the coefficients $\\lambda_{\\beta}$ and $\\zeta_{\\beta}$ can be determined\nanalytically, the details are degraded into the appendix~\\footnote{The\ndetermination of linear coefficients $\\lambda_{\\beta}$ and $\\zeta_{\\beta}$ is\nsimilar to that of linear regression, where the optimization is used. However,\nfor linear regression, a linear function is used to fit a set of separated data\nwhile for our problem, a linear function is used to approximate another\nnon-linear function. Such an idea of linearization is referred to as the\ngeneralized linear regression.}.\n\nWith the linearization of the factor $l_{9\/2}(s)$, the $x$-concerned ISR\nintegral for the continuum term has actually the form\n\n\\begin{equation}\n\t\\rho_0=\\int_0^{X_f}x^{\\mu} dx~,\n\t\\label{rho0}\n\\end{equation}\nwhich can be integrated easily. So the ISR corrected cross section of the\ncontinuum is expressed analytically as follows\n\n\\begin{equation}\n\t\\sigma_{r.c.}^C =\n\t\\frac{8\\pi\\alpha^2 f_k^2}{3} \\cdot [ (\\lambda_{9\/2} \\cdot s +\n\t\\zeta_{9\/2} ) \\cdot H_0(s) - \\lambda_{9\/2} \\cdot s \\cdot H_1(s)]~,\n\t\\label{xcteqisrc}\n\\end{equation}\nwith\n\n\\[\nH_{\\mu}(s) \\equiv \\int_0^{X_f}x^{\\mu}F(x,s)dx =\n\\sum\\limits_{\\nu=1}^{3} \\frac{X_f^{t+\\mu+\\nu-1}}{t+\\mu + \\nu - 1} \\cdot\nB_{\\nu}(t)~.\n\\]\n\n\\subsection{Resonance term}\n\nIn the light of Eq.~\\eref{bnkk}, the Born order expression for the resonance is\nwritten explicitly as\n\n\\begin{equation}\n\t\\sigma_{Born}^R = \\frac{8\\pi\\alpha^2 f_k^2}{3} \\cdot \\frac{A_1}\n\t{(s-M^2)^2+M^2\\Gamma^2} \\frac{(s\/4-m_K^2)^{3\/2}} {s^{7\/2}}~,\n\t\\label{eqbornr}\n\\end{equation}\nwhere\n\n\\[\nA_1 = 9\\Gamma_{ee}^2\/\\alpha^2 \\cdot (1+{\\cal C}^2+2{\\cal C}\\cos\\phi)~.\n\\]\n\nAs far as the factor\n\n\\[\nl_{7\/2}(s)=\\frac{(s\/4-m_K^2)^{3\/2}}{s^{7\/2}}~\n\\]\nis concerned, the similar approximation as the previous section is adopted,\nviz.\n\n\\[\n\\bar{l}_{7\/2}(s) \\approx \\lambda_{7\/2} \\cdot s + \\zeta_{7\/2}~.\n\\]\n\nThe $x$-concerned ISR integral for the resonance term then reads\n\n\\begin{equation}\n\t\\rho(s,t) = \\int_0^{X_f}\\frac{x^{t-1}dx}{(s(1-x)-M^2)^2 +\n\tM^2\\Gamma^2},\n\t\\label{rhost}\n\\end{equation}\nwhich can be integrated analytically~\\cite{wangp90,cahn87}\n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\rho(s,t)= \\frac{1}{ts^2} \\cdot a^{t-2}\\frac{\\pi t\n\t\t\\sin[\\theta(1-t)]} {\\sin\\theta \\sin\\pi t} + \\frac{1}{s^2} \\cdot\n\t\t\\left[ \\frac{1}{t-2}\\cdot X_f^{t-2} + \\right. \\\\\n\t\t\\frac{2(s-M^2)}{(t-3)s}\\cdot X_f^{t-3}\n\t\t\\left. + \\frac{3(s-M^2)^2 - M^2\\Gamma^2}{(t-4)s^2}\\cdot X_f^{t-4}\n\t\t\\right]~,\n\t\\end{aligned}\n\t\\label{eqrhost}\n\\end{equation}\nwhere\n\n\\[\na^2 = \\left(1-\\frac{M^2}{s}\\right)^2 + \\frac{M^2\\Gamma^2}{s^2}\n\\quad (a>0), \\quad \\cos\\theta = \\frac{1}{a} \\cdot \\left(\\frac{M^2}{s}\n-1 \\right)~.\n\\]\n\nWith the expression of $\\rho(s,t)$, the ISR corrected cross section of the\nresonance is re-casted as\n\n\\begin{equation}\n\t\\sigma_{r.c.}^R =\n\t\\frac{8\\pi\\alpha^2 f_K^2}{3} \\cdot A_1 \\cdot [ (\\lambda_{7\/2} \\cdot s\n\t+ \\zeta_{7\/2} ) \\cdot G_0(s) - \\lambda_{7\/2} \\cdot s \\cdot G_1(s)]~,\n\t\\label{xcteqisrr}\n\\end{equation}\nwith\n\\begin{equation}\n\t\\begin{array}{rcl}\n\t\tG_{\\mu}(s) & = &\n\t\t\\int_0^{X_f}\\frac{x^{\\mu}\\cdot F(x,s)dx}{(s(1-x)-M^2)^2 + M^2\\Gamma^2}\\\\\n\t\t&=& \\sum\\limits_{\\nu=1}^{3} \\rho(s,t+\\mu+(\\nu-1))\\cdot\n\t\tB_{\\nu}(t)~.\n\t\\end{array}\n\t\\label{intgmu}\n\\end{equation}\n\n\\subsection{Interference term}\n\nThe Born order expression for the interference can be acquired readily from\nEq.~\\eref{bnkk}. However, for clearness the expression of the interference is\nfurther divided into two sub-terms as follows\n\n\\begin{equation}\n\t\\sigma_{Born}^{I_1} = \\frac{8\\pi\\alpha^2 f_K^2}{3} \\cdot\n\t\\frac{A_2\\cdot(s-M^2)}{(s-M^2)^2+M^2\\Gamma^2} \\cdot\n\t\\frac{(s\/4-m_k^2)^{3\/2}}{s^{4}}~,\n\t\\label{eqborni1}\n\\end{equation}\nand\n\n\\begin{equation}\n\t\\sigma_{Born}^{I_2} = \\frac{8\\pi\\alpha^2 f_k^2}{3} \\cdot \\frac{A_3}\n\t{(s-M^2)^2+M^2\\Gamma^2} \\cdot \\frac{(s\/4-m_k^2)^{3\/2}}{s^{4}}~,\n\t\\label{eqborni2}\n\\end{equation}\nwhere\n\n\\[\nA_2 = 6\\left(\\Gamma_{ee}\/\\alpha\\right) \\cdot (1+{\\cal C}\\cos\\phi)~,\n\\]\n\n\\[\nA_3 = 6\\left(\\Gamma_{ee}\/\\alpha\\right) \\cdot {\\cal C} M\\Gamma\\sin\\phi~.\n\\]\n\nThe simplification strategy is the same as those used for the continuum and\nresonance. First, the factor\n\n\\[\nl_{4}(s)=\\frac{(s\/4-m_k^2)^{3\/2}} {s^{4}}~\n\\]\nis approximated as\n\n\\[\n\\bar{l}_{4}(s) \\approx \\lambda_{4} \\cdot s + \\zeta_{4}~;\n\\]\nsecond, the $x$-concerned ISR integrals for the interference terms have the\nforms as those in Eqs.~\\eref{rho0} and \\eref{rhost}, which can be integrated\nout directly or by Formula \\eref{eqrhost}. Finally, the ISR corrected cross\nsection of the interference is obtained\n\n\\begin{equation}\n\t\\begin{split}\n\t\t\\sigma_{r.c.}^{I_1} = \\frac{8\\pi\\alpha^2 f_k^2}{3} \\cdot A_2\n\t\t\\cdot \\{(\\lambda_{4} \\cdot s + \\zeta_{4})(s-M^2) \\cdot G_0 \\\\\n\t\t- [2\\lambda_{4} \\cdot s^2 +\n\t\t(\\zeta_{4} - \\lambda_{4} M^2)s ] \\cdot\n\t\tG_1(s)+ \\lambda_{4} s^2 \\cdot G_2(s) \\}~,\n\t\\end{split}\n\t\\label{xcteqisri1}\n\\end{equation}\n\n\\begin{equation}\n\t\\sigma_{r.c.}^{I_2} = \\frac{8\\pi\\alpha^2 f_k^2}{3} \\cdot A_3 \\cdot\n\t[(\\lambda_{4} s + \\zeta_{4}) \\cdot G_0(s) - \\lambda_{4} s \\cdot\n\tG_1(s)] ~,\n\t\\label{xcteqisri2}\n\\end{equation}\nwhere $G_{\\mu}(s)$ is given by Formula \\eref{intgmu}.\n\nIn summary, the ISR corrected cross section formula is\n\\begin{equation}\n\t\\sigma_{r.c.} (s) = \\sigma_{r.c.}^{C} (s)+ \\sigma_{r.c.}^{R} (s)+\n\t\\sigma_{r.c.}^{I_1}(s) + \\sigma_{r.c.}^{I_2} (s)~,\n\t\\label{eqofsumsim}\n\\end{equation}\nwith expressions of the cross section for each term given in\nEqs.~\\eref{xcteqisrc}, ~\\eref{xcteqisrr}, ~\\eref{xcteqisri1}, and\n~\\eref{xcteqisri2}, respectively.\n\n\\section{Possible simplification for energy spread integral}\n\nAs indicated in Eq.~\\eref{eqegsprd}, the experimentally observed cross section\nis the $\\sigma_{r.c.}$ convoluted $G(W^{\\prime},W)$, which might be simplified\nfurther. Two methods, the Taylor Expansion (TE) method and the Fast Fourier\nTransformation (FFT) method, have been considered for such a simplification.\n\nFor the TE method, we begin from Eq.~\\eref{eqegsprd}, and Taylor expand the\n$\\sigma_{r.c.}$ at W, viz.\n\n$$\n\\sigma_{r.c.}(W^{\\prime}) = \\sum\\limits_{n=0}^{\\infty}\n\\frac{\\sigma_{r.c.}^{(n)}(W)}{n!} \\cdot (W^{\\prime}-W)^{n}~,\n$$\nwhere $\\sigma_{r.c.}^{(n)}(W)$ denotes the $n$-th derivative of function\n$\\sigma_{r.c.}$ at value $W$.  Replace the Taylor expansion of $\\sigma_{r.c.}$\ninto Eq.~\\eref{eqegsprd}, the integral to be calculated has the following form\n\n$$\n\\int \\limits_{-\\infty}^{\\infty} x^{n} e^{-x^2} dx~,\n$$\nwhich can be precalculated. However, in order to achieve a reasonable\nprecision, we need to calculate hundreds, or even thousands of terms in Taylor\nexpansion. This means that the fairly high order derivatives of $\\sigma_{r.c.}$\nhave to be calculated,and too much time is consumed, which is not acceptable.\n\nAs to FFT method\\footnote{http:\/\/en.wikipedia.org\/wiki\/Fast\\_Fourier\\_transform}, \nwe could easily find that the observed cross\nsection $\\sigma_{obs}(W)$ is a convolution of the radiative corrected cross\nsection and a gauss function. Considering the Convolution Theorem in Fourier\nTransformation\n\n$$\n\\mathfrak{F}(g\\bigotimes h) = \\mathfrak{F}(g) \\cdot \\mathfrak{F}(h)~,\n$$\nwhere $\\mathfrak{F}$ represents Fourier Transformation, $\\bigotimes$ represents\nconvolution. To calculate convolution efficiently, we use Fast Fourier\nTransformation. First, $\\sigma_{r.c.}$ and $G$ should be sampled in energy\nregion. After that, we get two series of numbers. Then DFT (Discrete Fourier\nTransformation) should be performed on both series, and the resulting series\nshould be multiplied to generate one final series. Finally IDFT (Inverse\nDiscrete Fourier Transformation) should be performed on this series and what we\nget is the distribution of $\\sigma_{obs}$ in energy region on which\n$\\sigma_{r.c.}$ and $G$ are sampled. This process is very fast, and we could\nget the result on the whole energy region at the same time rather than\ncalculating the integral one by one. To get accurate result, the sample number\nshould be very large (512 or 1024), which means a large number of cross\nsections should be calculated. In real energy scan, the number of data taking\npoints is usually not large (less than 20). The total integration time in a\nsmall number of energy points is less than the time cost by sampling a large\nnumber of cross sections and perform DFT and IDFT on it. So this method does\nnot fit our purpose.\n\n\\section{Investigation of simplified formula}\n\n\\subsection{Precision}\n\nThe accurate observe cross section ($\\sigma_{obs}$) is calculated by\nEq.~\\eref{eqegsprd} while the simplification one (denoted by $\\sigma^s_{obs}$)\nis also calculated by Eq.~\\eref{eqegsprd} but with ($\\sigma_{r.c.}$) replaced\nby the expression \\eref{eqofsumsim}. The relative error of two observed cross\nsections is defined as\n\n\\begin{equation}\n\tR_{\\sigma} =\\frac{\\sigma^s_{obs}-\\sigma_{obs}}{\\sigma_{obs}}~.\n\t\\label{eqrer}\n\\end{equation}\n\nIn the calculation of the observed cross section, all parameters of resonances\nare taken from PDG08~\\cite{pdg08}, $\\Delta=1.3$ MeV is used. Two real\nundetermined parameters are the relative phase ($\\phi$) and the relative\nstrength (${\\cal C}$) between the strong and the electromagnetic amplitudes.\nThe dependences of $R_{\\sigma}$ on $\\phi$ and ${\\cal C}$ are shown in\nFigs.~\\ref{erang} and~\\ref{erstg} respectively.\n\n\\begin{center}\n\t\\includegraphics[angle=-90,width=8cm]{erang.eps}\n\t\\figcaption{\\label{erang}Variations of $R_{\\sigma}$ against $\\sqrt{s}$\n\tin the vicinity of $\\psp$ resonance peak for $\\phi=0^{\\circ}$,\n\t$90^{\\circ}$, $180^{\\circ}$, and $270^{\\circ}$. In the calculation\n\tof the observed cross section, ${\\cal C}$ is fixed at 2.5.}\n\\end{center}\n\n\\begin{center}\n\t\\includegraphics[angle=-90,width=8cm]{erstg.eps}\n\t\\figcaption{\\label{erstg}Variations of $R_{\\sigma}$ against $\\sqrt{s}$\n\tin the vicinity of $\\psp$ resonance peak for ${\\cal C}=$1, 5, and\n\t10. In the calculation of the observed cross section, $\\phi$ is\n\tfixed at $90^{\\circ}$.}\n\\end{center}\n\nThe variations of $R_{\\sigma}$ against the center-of-mass energy ($\\sqrt{s}$)in\nthe vicinity of $\\psp$ resonance peak for $\\phi=0^{\\circ}$, $90^{\\circ}$,\n$180^{\\circ}$, and $270^{\\circ}$ are displayed in Fig.~\\ref{erang}, according\nto which we notice that firstly, the absolute value of $R_{\\sigma}$ is less\nthan one percent in the energy region we concerned; secondly, the difference\nbetween two cross sections fades away at the resonance peak; thirdly, the\ndifferences in off-resonance region are larger than that in on-resonance\nregion.  The similar dependence of $R_{\\sigma}$ on ${\\cal C}$ can be seen from\nFig.~\\ref{erstg}, where displayed are the variations of $R_{\\sigma}$ against\n$\\sqrt{s}$ in the vicinity of $\\psp$ resonance peak for ${\\cal C}=$1, 5, and\n10. It is obvious that the difference due to the variation of ${\\cal C}$ is\neven smaller, which is at the level of a few per mille.\n\n\\subsection{Computation time}\nThe symbol $T^s$ ($T^0$) denotes the computation time when $\\sigma^s_{obs}$\n($\\sigma_{obs}$) is used for the cross section calculation. The comparison of $T^s$\n(denoted by the solid line) and $T^0$ (denoted by the dotted line) at both resonance\nand off-resonance regions are shown in Fig.~\\ref{figtimecp}.\n\n\\begin{center}\n\t\\includegraphics[angle=-90,width=8cm]{time.eps}\n\t\\figcaption{\\label{figtimecp}Comparison of $T^s$ and $T^0$ at both\n\tresonance and off-resonance regions.}\n\\end{center}\n\nFrom comparison it can be seen that about one-hundred-time reduction of\ncomputation time is achieved by our simplification algorithm.  Although only\none-fold integral is simplified by analytic expression, the computation time is\nless than 0.1 second for each energy point which is fast enough for our scan\nsimulation study.\n\n\\subsection{Application}\n\nAs we mentioned in the introduction, the speed of calculation of the observed\ncross section is the crucial issue of data taking optimization study of the\nscan experiment. Without reasonably simplified formula, it will be a too long\ntime to perform the optimization fit, and the detailed scan optimization is\nimpractical.\n\nBesides the application in scan optimization, simplified cross section formulas\ncan also used for the uncertainty study~\\cite{moxh01} and correlation\nstudy~\\cite{moxhpsdctrep}. Since for both of these studies, the\nsampling-and-fitting method is also adopted, the fast computation of cross\nsection is needed as well.\n\n\\section{Summary}\n\nThe complete expressions for $\\psp \\rightarrow PP$ decays are presented, \nincluding the relative phase between the strong and the electromagnetic\namplitudes. After linearizing one non-linear kinematic factor, the integrand\nwith the initial state radiation is integrated analytically. Such a\nsimplification of two-fold integral into a one-fold integral reduces the total\ncomputing time by about one hundred times.\n\nThe possible approaches for simplification of energy spread integral are also\ndiscussed.\n\nThe simplified formulas of the observed cross sections obtained in this\npaper provide a practical tool for the further optimization study of the\nscan data taking, which is of great importance for the study of the relative\nphase between the strong and the electromagnetic amplitudes.\n\n\n\\end{multicols}\n\\vspace{10mm}\n\\begin{multicols}{2}\n\n\\subsection*{Appendices A}\n\n\\begin{small}\n\nAs we have noted in Subsection 3.1, the factor\n\n\\begin{subequations}\n\t\\renewcommand{\\theequation}{A\\arabic{equation}}\n\n\t\\[\n\tl_{9\/2}(s)=\\frac{(s\/4-m_K^2)^{3\/2}}{s^{9\/2}}~\n\t\\]\nvaried almost linearly in the vicinity of $\\psp$ peak, and its variation\nagainst $s$ is shown in Fig.~\\ref{figfactor}. Therefore, for the factor\n\n\\begin{equation}\nl_{\\beta}(s)=\\frac{(s\/4-m_K^2)^{3\/2}}{s^{\\beta}}~,\n\\end{equation}\na linear function (it refers to Eq.~\\eref{eqlnrfac}),\n\n\\begin{equation}\n\\bar{l}_{\\beta}(s) \\approx \\lambda_{\\beta} \\cdot s + \\zeta_{\\beta}~\n\\end{equation}\nis utilized to approximate it in the vicinity of resonance peak. The\ncoefficients $\\lambda_{\\beta}$ and $\\zeta_{\\beta}$ are determined by the\ngeneralized linear regression method. As the first step, we define the\nintegration\n\n\\begin{equation}\nI=\\int_{s1}^{s2}ds[(\\lambda_{\\beta} \\cdot s +\n\\zeta_{\\beta})-\\frac{(s\/4-m_K^2)^{3\/2}}{s^{\\beta}}]^2~.\n\\end{equation}\n\nThe needed values of coefficients $\\lambda_{\\beta}$ and $\\zeta_{\\beta}$ are\nobtained by the minimization of the integration $I$, that is\n\n\\begin{equation}\n\\frac{\\partial{I}}{\\partial{\\lambda_{\\beta}}}=0 ~~ \\text{and}~~\n\\frac{\\partial I}{\\partial \\zeta_{\\beta}} =0~.\n\\end{equation}\n\nFrom the above requirements, we acquire a set of linear equations of\n$\\lambda_{\\beta}$ and $\\zeta_{\\beta}$, solve it, we obtain\n\n\\begin{equation}\n\\lambda_{\\beta}=\\frac{\\delta_1C_1-\\delta_2C_2}{\\delta_1\\delta_3 -\n\\delta_2^2}~~ \\text{and} ~~\\zeta_{\\beta} =\n\\frac{\\delta_3C_2-\\delta_2C_1}{\\delta_1\\delta_3 - \\delta_2^2} ~,\n\\end{equation}\nwhere\n\n\\[\n\\delta_i = \\int_{s_1}^{s_2} s^{i-1}ds = \\frac{s_2^i-s_1^i}{i}~,\n\\]\n\n\\[\nC_1 = \\int_{s_1}^{s_2}ds\\frac{(s\/4-m_K^2)^{3\/2}}{s^{\\beta-1}} =\n\\frac{1}{8}D(\\beta-1)~,\n\\]\n\n\\[\nC_2 = \\int_{s_1}^{s_2}ds\\frac{(s\/4-m_K^2)^{3\/2}}{s^{\\beta}} =\n\\frac{1}{8}D(\\beta)~.\n\\]\n\nBoth $C_1$ and $C_2$ contain integral\n\n\\begin{equation}\nD(\\beta)=\\int_{s_1}^{s_2} dx \\frac{(x-u)^{3\/2}}{x^{\\beta}},\n\\end{equation}\nwhere\n\\[\n\\beta= 2,\\frac{5}{2},3,\\frac{7}{2},4,\\frac{9}{2}\n\\]\n\\[\nu=4m_K^2~.\n\\]\nFor different $\\beta$, we can calculate the integral analytically~\\footnote{The\nfollowing integrals are obtained by using Mathematica and checked by hands.}.\nFor $\\beta=2$,\n\n\\begin{equation}\nD(2)=\\left[\\sqrt{x-u}\\left(\\frac{u}{x}+2\\right)-3\\sqrt{u} \\textrm{tan}^{-1}\n\\left(\\frac{\\sqrt{x-u}}{\\sqrt{u}}\\right)\\right]\\bigg|_{s_1}^{s_2}~;\n\\end{equation}\nFor $\\beta={\\displaystyle \\frac{5}{2}}$,\n\n\\begin{equation}\nD\\left(\\frac{5}{2}\\right)=\\left[2\\textrm{log}\n\\left(2\\left(\\sqrt{x-u}+\\sqrt{x}\\right)\\right) +\n\\frac{2}{3}\\left(\\frac{u}{x^{\\frac{3}{2}}}-\n\\frac{4}{\\sqrt{x}}\\right)\\sqrt{x-u}\\right]\\bigg|_{s_1}^{s_2}~;\n\\end{equation}\nFor $\\beta=3$,\n\n\\begin{equation}\nD(3)=\\left[\\frac{3}{4\\sqrt{u}}\\textrm{tan}^{-1}\n\\left(\\frac{\\sqrt{x-u}}{\\sqrt{u}}\\right) + \\frac{1}{4}\\left(\\frac{2u}{x^2}-\n\\frac{5}{x}\\right)\\sqrt{x-u}\\right]\\bigg|_{s_1}^{s_2}~;\n\\end{equation}\nFor $\\beta={\\displaystyle \\frac{7}{2}}$,\n\n\\begin{equation}\nD\\left(\\frac{7}{2}\\right)=\\frac{2(x-u)^{\\frac{5}{2}}}\n{5ux^{\\frac{5}{2}}}\\bigg|_{s_1}^{s_2}~;\n\\end{equation}\nFor $\\beta=4$,\n\n\\begin{equation}\nD(4)=\\left[\\frac{\\textrm{tan}^{-1}\\left(\\frac{\\sqrt{x-u}}\n{\\sqrt{u}}\\right)}{8u^{\\frac{3}{2}}} + \\sqrt{x-u}\\left(\\frac{u}{3x^3} +\n\\frac{1}{8ux} - \\frac{7}{12x^2}\\right)\\right]\\bigg|_{s_1}^{s_2}~;\n\\end{equation}\nFor $\\beta={\\displaystyle \\frac{9}{2}}$,\n\n\\begin{equation}\nD\\left(\\frac{9}{2}\\right)=\\frac{2(x-u)^{\\frac{5}{2}}(5u+2x)}\n{35u^2x^{\\frac{7}{2}}}\\bigg|_{s_1}^{s_2}~.\n\\end{equation}\n\nIt could be easily checked that for the coefficients $\\lambda_{\\beta}$ and\n$\\zeta_{\\beta}$ we obtain,\n\\begin{equation}\n\\frac{\\partial^2 I}{\\partial \\lambda_{\\beta}^2} =\n2\\int_{s_1}^{s_2}s^2\\textrm{d}s = \\frac{2}{3} \\left(s_2^3-s_1^3\\right)>0~,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial^2 I}{\\partial \\zeta_{\\beta}^2} = 2\\int_{s_1}^{s_2}\\textrm{d}s =\n2(s_2 - s_1)>0~.\n\\end{equation}\nThis means what we get is the minimum of $I$, not maximum.\n\nThe relative error between the linearized formula and the original formula is\ndefined as follows:\n\\begin{equation}\nR_{l}=\\frac{|\\bar{l}_{\\beta}-l_{\\beta}|}{l_{\\beta}}~.\n\\end{equation}\nWhen $\\beta=9\/2,~4,~7\/2$, the variations of $R_{l}$ against the center-of-mass\nenergy ($\\sqrt{s}$) are shown in Fig.~\\ref{betaerror}.\n\n\\end{subequations}\n\\begin{center}\n\t\\includegraphics[angle=-90,width=8cm]{error.eps}\n\t\\figcaption{\\label{betaerror}The variations of $R_{l}$ against\n\t$\\sqrt{s}$ for $\\beta=9\/2,~4,~7\/2$.}\n\\end{center}\n\n\\end{small}\n\\end{multicols}\n\n\\vspace{-1mm}\n\\centerline{\\rule{80mm}{0.1pt}}\n\\vspace{2mm}\n\n\\begin{multicols}{2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe study of ``\\emph{infinite--dimensional diffusions}\" arising from the scaling limit of determinantal point processes attracted much attention in the last years, see for instance  \\cite{OxRMBook}, chapters 6,10,11,38 and references therein. One of the most popular model has been introduced by Dyson in \\cite{Dy:BrownianMotions}, where a dynamical version of the probability distributions arising in random matrix theory is defined. The idea is to study $N$ Brownian particles moving on the real line, conditioned not to intersect. When the starting and ending points are fixed and $N$ goes to infinity, the particles sweep out a certain region in space-time, whose shape depends on the number and relative position of the starting and ending points. Interesting infinite dimensional diffusions arise when studying the behavior of the particles near the boundary of this region. Near the points where this boundary is a smooth curve, the  fluctuations of the particles are described by the well known Airy process \\cite{PrSp,JoDetProc,Johansson3}. Near a cusp singularity we are lead to the study of the Pearcey process \\cite{TracyWidomPearcey,OkounkovReshetikhin}. The most recently studied case is the one of the so--called tacnode singularity, appearing when the boundary looks (locally) as two circles touching at one point. \nThe first expression for the kernel of the tacnode process has been written in \\cite{AFvMTacnode}, as a scaling limit of a model of random walks, continuous in time. Few months later, in \\cite{DelKuiZhaTacnode}, the authors found a different expression (for the one--time case) in terms of a $4\\times 4$ Riemann--Hilbert problem, this time starting from non--intersecting Brownian motions. Another formula for the multi--time case, again different from the previous ones, has been found by Johansson in \\cite{JohanssonTacnode}. There, for the first time, the resolvent of the Airy kernel appeared in the formula of the tacnode kernel (this is quite an important point for our present work, as it will be apparent later). In \\cite{AJvMTacnode}, finally, the authors analyzed the same process as arising from random tilings instead of Dyson Brownian motions: in this paper it has been proven that all the different formulations above are indeed equivalent, thus performing a significative step in the direction of universality for the tacnode process. A similar result has been obtained by Delvaux in \\cite{DelvauxEqualitiesTacnodes}. A more general formulation of this process in the ``asymmetric'' case has been studied in \\cite{FerrariVetoTacnode}.\n\nWhile the kernels describing the Airy and Pearcey processes can be expressed as simple double integrals of exponential functions (both in the one and multi--time case), studying the tacnode process presents an additional complication, since its kernel is highly transcendental. Indeed, it is written in terms of the resolvent of the Airy operator. In particular, given the known representations of the tacnode kernel,  the study of the associated (joint) gap probability (i.e. the Fredholm determinant associated to the kernel) appears as quite complicated.\n\nThe purpose of this short investigation is to show how to express the gap probability of the tacnode process in terms of Fredholm determinants associated to {\\em explicit} kernels, no more complicated than the standard Airy one. Explicitly the key theoretical result is Theorem \\ref{main} that expresses the gap probability of the tacnode process as a ratio of determinants (notation to be defined therein)\n\\begin{eqnarray}\n {\\rm Prob} \\left\\{ \\mathcal T_\\sigma(\\tau_i) \\not\\in E^{(i)},\\ \\ i=1,\\dots, r\\right\\} = \n\\frac {\\det \\left[ \\mathrm{Id} - \\mathbb H_E\\right]       } {  \\det \\left[\\mathrm{Id}_{[\\widetilde\\sigma,\\infty)} - \\pi_{\\widetilde \\sigma} K_{\\mathrm{Ai}} \\pi_{\\widetilde \\sigma} \\right]}\n\\label{13}\n{\\rm e}\nwhere $\\mathcal T_\\sigma(\\tau)$ denotes the tacnode point field at time $\\tau$, the operator $\\mathbb H$ acts on $$L^2\\left({\\bf R}_+\\sqcup {\\bf R}_0\\bigsqcup_{j=1}^r {\\bf R}_{\\tau_j} \\right)\\simeq\nL^2({\\bf R}_+)\\oplus L^2({\\bf R}_0) \\bigoplus_{j=1}^r L^2({\\bf R}_{\\tau_j})$$ and the subscript $_E$ means that it is restricted to the following subspace \n$$ L^2({\\bf R}_+)\\oplus L^2([\\widetilde \\sigma,\\infty)) \\oplus L^2\\left(E^{(1)}\\sqcup \\dots \\sqcup E^{(r)}\\right).$$\nIn the formula above, the subscripts on the various copies of ${\\bf R}$ are simply to distinguish them from each other and remind that some are associated to the times $\\tau_1,\\dots, \\tau_r$ and  each $E^{(j)},\\ j=1,\\dots, r$, is a finite union of bounded intervals.\nThe kernel $\\mathbb H$ is described explicitly in Theorem \\ref{main}, but here it suffices to say that it involves at most the standard Airy functions. In the one--time case it is a $3 \\times 3$ kernel acting on $L^2({\\mathbb R}_+)\\oplus L^2({\\mathbb R})\\oplus L^2({\\mathbb R})$ and reading as follows:\n$$\n\\mathbb H(x,y) =  \\left[\n\\begin{array} {c|c|c}\n  0\n& \n -\\mathrm{Ai}(x+y) \n&\n\\!\\!  \\mathrm{Ai}^{(-\\tau)} (x\\sqrt[3]{2}+\\sigma-y  ) \\!\\!\\!\n \\\\[10pt]\n \\hline &&\\\\ \n - \\mathrm{Ai}  (x+y ) \n&\n 0\n &\n   \\mathrm{Ai}^{(-\\tau)} (x\\sqrt[3]{2} + y-\\sigma ) \n \\\\[10pt]\n \\hline &&\\\\\n \\mathrm{Ai}^{(\\tau)} (\\sigma-x+y \\sqrt[3]{2}  )\\!\\!\n &\n\\!  \\mathrm{Ai}^{(\\tau)} (x-\\sigma+y\\sqrt[3]{2} ) \\!\\!\n  &\n   0\n\\end{array}\\right]\n$$\nwhere $\\mathrm{Ai}^{(\\tau)}(x) = 2^\\frac 1  6{\\rm e}^{ \\tau x+\\frac 2 3\\tau ^3 }\\mathrm{Ai} (x+\\tau^2)$ and $\\tau,\\sigma$ are parameters of the tacnode process, respectively the time and the ``pressure'' (or overlap, see also below for details).\nAs an application of our result, we compute numerically, following the method of \\cite{Bornemann}, the gap probability associated to the tacnode process, and its relationship with the Airy and Pearcey processes (a more detailed list of our results is given below). Theorem \\ref{main} can also be used as a starting point to identify the tacnode gap probability with the tau function of a given Riemann-Hilbert problem, in the spirit of \\cite{BertolaCafasso1,BertolaCafasso3}. A work in this direction is currently in preparation \\cite{BCG1}. \n\nBefore entering into the details of our results, let us spend few words about Fredholm determinants and their numerical evaluations.\nLet $(X,{\\rm d}\\mu)$ be a (sigma-finite) measure space. Given an integral operator $\\mathcal J$  on $L^2(X,{\\rm d} \\mu)$ (belonging to some trace ideal \\cite{Si}) the computation of the (regularized) Fredholm-Carleman determinant is an essentially transcendental problem.\nIf the kernel $\\mathbb J$ of the integral operator $\\mathcal J$ is known and sufficiently regular, then the numerical evaluation of the determinant can be approached by a suitable approximation scheme which involves projecting onto a suitable finite dimensional subspace. In this way the Fredholm determinant can be approximated by a finite determinant.\n\n\nThe numerical evaluation of Fredholm determinants of operators on ${\\bf R}$ (or disjoint unions of several copies thereof) has been recently shown to pose little obstacle \\cite{Bornemann}; the gist of the beautiful idea is basically the simplest approach of discretization, paired with skillful use of numerical estimates associated with quadrature rules.\n\nThe effectiveness of the method is vastly improved if  the kernel is {\\em explicit} in terms of functions that are already implemented or easily so.   The paramount example is  the Airy kernel and the corresponding Tracy-Widom \\cite{TracyWidomLevel} distribution\n\\begin{eqnarray}\nK_{\\mathrm{Ai}}(x,y):= \\frac {\\mathrm{Ai}(x) \\mathrm{Ai}'(y)-\\mathrm{Ai}'(x)\\mathrm{Ai}(y)}{x-y}\n\\ \\ F_2(s):= \\det \\left[\\mathrm{Id}_{[s,\\infty)} - \\pi_s K_{\\mathrm{Ai}} \\pi_s\\right]\\ ,\\\\ \\ \\ \\pi_s:L^2({\\bf R})\\to L^2([s,\\infty).\\label{TWd}\n\\end{eqnarray}\nAs we said before, the kernel of the tacnode process is, however, highly transcendental; it is precisely with the results contained in Theorem \\ref{main} that we make it directly amenable to the methods in \\cite{Bornemann}.\n\nThe following sections are organized as follows:\n\n\\begin{itemize}\n\t\\item In the second section, as a sort of warm-up, we show how to use the theoretical results in \\cite{BertolaCafasso1} to compute (numerically) the gap probability of the Pearcey process. As an application, we give numerical confirmations of the degeneration of the Pearcey gap probability to a couple of Tracy--Widom distribution (see \\cite{BertolaCafasso1} and, for a similar result, \\cite{ACvM2}).\n\t\\item In the third section, we start with our main theoretical result, i.e. the expression of the tacnode gap probability in terms of the ratio of two Fredholm determinants with simpler kernels. This allows us to show numerically, in the two subsections, how the tacnode gap probability degenerate, in different regimes, to the Pearcey and the Airy one. The degeneration tacnode-to-Pearcey has been already proven in \\cite{GeudensZhangTacPearcey} and another proof will be given in \\cite{BCG1}, together with the  tacnode-to-Airy one, which has not been proven yet (to the best of our knowledge).\n\t\\item In the appendix we prove that, given a determinantal point process and a given subset of the state space, the point process conditioned not to have particles on that given subset is also determinantal, and we explicitly write its kernel (see also \\cite{TaoWigner} for the case of point processes with a finite number of particles, and \\cite{Lyons} for the discrete case). In the third section we will relate (formally) this result to our representation of the tacnode gap probability as a ratio of two Fredholm determinants.\n\n\\end{itemize}\n\n\\section{Numerical evaluation of the Pearcey gap probabilities}\nSuppose that $I= [a_1,a_2]\\cup \\dots \\cup[a_{2N-1}, a_{2N}]$ and denote with $\\mathcal{P}$ the Pearcey process \\cite{TracyWidomPearcey,OkounkovReshetikhin}; \nin \\cite{BertolaCafasso1} it was proved that\n\\begin{eqnarray}\n\\mathbb P\\left\\{ \\mathcal P(\\tau) \\not\\in I \\right\\} = \\det \\left[\\mathbf 1- \\widetilde K_{\\vec a}\\right]\n\\label{BC1}\n{\\rm e}\nwhere the Fredholm determinant on the right hand side is on the space $L^2({\\gamma}_L\\cup i{\\bf R} \\cup {\\gamma}_R)$ and the operator $\\widetilde K$ has kernel $\\widetilde K(\\lambda,\\mu)$ given by  (here $\\chi_{_X}$ denotes the indicator function of a subset $X\\subset {\\mathbb C}$)\n\\begin{eqnarray}\\label{PearceyKernel}\n\\widetilde K_{\\vec a}(\\lambda,\\mu) = \\frac 1{2i\\pi} \\frac { {\\rm e}^{\\frac {\\Theta_0(\\lambda)-\\Theta_0(\\mu)}2} \\chi_{_{i{\\bf R} }}(\\mu)\\chi_{_{\\gamma_R\\cup \\gamma_L}}(\\lambda)  \n- \\sum_{j=1}^{2N}  (-)^j {\\rm e}^{\\frac {\\Theta_{0}(\\mu) -\\Theta_0(\\lambda)}2  + a_j \\lambda-a_j\\mu } \\chi_{_{\\gamma_R\\cup \\gamma_L}}(\\mu)\\chi_{_{i{\\bf R}}}(\\lambda)}{\\lambda - \\mu}\\\\\n\\Theta_0(\\lambda):= \\frac {\\lambda^4}4 - \\tau \\frac {\\lambda^2}2\n\\end{eqnarray}\nThe contours $\\gamma_{R}$ is a contour in the right  halfplane that comes from infinity along $\\arg \\lambda= \\frac \\pi 4$ and returns to infinity along $\\arg \\lambda = -\\frac \\pi 4$ while $\\gamma_L = -\\gamma_R$. Our aim is to use the equivalent representation of the Percey gap probability given in \\eqref{PearceyKernel} for numerical computations. \nFor the case of multi-time Pearcey gap probabilities we refer to \\cite{BertolaCafasso3}\\footnote{We remark that in the cited reference the formul\\ae\\ (3.11, 3.12) should have an additional overall denominator $2i\\pi$ and in the last line of formula (3.12) the alternating sign $(-1)^\\ell $ should be $(-1)^{\\ell +1}$. Similarly, the formul\\ae\\ (2.12, 2.13, 2.14) should all have an additional overall factor $\\frac 1{2i\\pi}$. }.\nIn order to apply Bornemann's procedure \\cite{Bornemann} we prefer to have kernels acting on the interval $[0,1]$, hence we have chosen two hyperbolas parametrized  by \n\\begin{eqnarray}\n\\phi_R(s) = \\frac 12 \\left[\\frac {s}{1-s} + \\frac {1-s}s \\right] -  \\frac i2 \\left[\\frac {s}{1-s} - \\frac {1-s}s \\right]:(0,1)\\to \\gamma_R\\\\\n\\phi_L(s) = -\\frac 12 \\left[\\frac {s}{1-s} + \\frac {1-s}s \\right] +  \\frac i2 \\left[\\frac {s}{1-s} - \\frac {1-s}s \\right]:(0,1)\\to \\gamma_L\\\\\n\\phi_0(s) = \\tan \\left(\\frac{2\\pi s-\\pi}2\\right):(0,1)\\to i{\\bf R}\n\\end{eqnarray}\nand, pulling back the kernel $\\widetilde K$ with the aid of the maps $\\phi_{L,R,0}$, we obtained a kernel on $(0,1)^3$ (it can be extended to the closure by defining it as zero). Then the numerical evaluation of the resulting kernel can be carried out as in \\cite{Bornemann}. We only point out that the Pearcey kernel involves special functions  (Pearcey integrals) that are {\\em not} implemented in any library that we could find. Thus it seems that the equivalent representation \\eqref{BC1} is convenient not only from the theoretical point of view to analyze the transition to the Airy process as in \\cite{BertolaCafasso1}, but also from the point of view of effective numerical implementation.\n\\subsection{From the Pearcey to the Airy process}\nLet us consider, for instance, the simplest gap probabilities for the Pearcey and Airy processes:\n\\begin{eqnarray}\nF_{\\mathrm{P}}([a_p,b_p]; \\tau) = \\mathbb P\\left\\{\\mathcal P(\\tau)\\not \\in [a _p, b_p]\\right\\}\\ ,\\qquad\nF_{2}([\\sigma,\\infty)) = \\mathbb P\\left\\{\\mathcal A\\not \\in [\\sigma,\\infty) \\right\\}\n\\end{eqnarray}\n\nIn \\cite{BertolaCafasso1} it was shown by using the Deift--Zhou nonlinear steepest descent method that, in particular\n\\begin{eqnarray}\nF_{\\mathrm P}\\left(\\left[ -2\\left(\\frac \\tau 3\\right)^\\frac 3 2 + (3\\tau)^\\frac 1  6 \\rho, 2\\left(\\frac \\tau 3\\right)^\\frac 3 2 - (3\\tau)^\\frac 1  6 \\sigma,   \\right]; \\tau \\right) \\mathop{\\longrightarrow}_{\\tau \\to\\infty} F_{2}([\\sigma,\\infty)) F_{2}([\\rho,\\infty))\n\\label{PtoAi}\n{\\rm e}\n(the statement of \\cite{BertolaCafasso1} was for an arbitrary finite union of bounded intervals, but here we restrict to the simplest nontrivial example). As an indirect confirmation that the numerical approach is sound, we verify this limit numerically plotting the graph of the ``\\emph{relative difference}'' $ 1 - \\frac {F_P([a_P, b_P];\\tau)}{F_2(\\sigma)F_2(\\rho)}$ with the parameters as in \\eqref{PtoAi} and $(\\rho,\\sigma)\\in [-3,1]\\times [-3,1]$. The two graphs in Figure 1 correspond to the case $\\tau = 3$ and $\\tau = 5.314$. In the second case, already for a relatively small value of $\\tau$, the relative difference is below $15\\%$ (in the range of space variables we considered).\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{Pearcey-to-Airy-tau=3.pdf}\n\\includegraphics[width=0.5\\textwidth]{Pearcey-to-Airy-tau=5_314.pdf}\n\\caption{\\footnotesize The relative difference $ 1 - \\frac {F_{\\mathrm P}([a_P, b_P];\\tau)}{F_2(\\sigma)F_2(\\rho)}$ with the parameters as in \\eqref{PtoAi}, $(\\rho,\\sigma)\\in [-3,1]\\times [-3,1]$ and $\\tau = 3$ (left) or $\\tau = 5.314$ (right).}\n\\end{figure}\n\\FloatBarrier\n\n\\section{Gap probabilities of the tacnode process as ratio of determinants}\n\nWe start recalling one of the three equivalent formulations of the tacnode kernel obtained in \\cite{AJvMTacnode}:\n\n\\begin{defn}[\\cite{AJvMTacnode} formula (19)]\n\\label{defKtac}\nThe kernel of the  {\\bf extended tacnode process with overlap $\\sigma$} is defined as\n \\begin{eqnarray}\n  {\\mathbb K}^{\\rm tac} (\\tau_1, \\xi_1;\\tau_2,\\xi_2 )  \n= K_{\\mathrm{Ai}}^{(\\tau_1,-\\tau_2)}(\\sigma -\\xi_1,\\sigma-\\xi_2)-\\mathrm{Id}_{\\tau_1>\\tau_2}\np(\\tau_1-\\tau_2;\\xi_1,\\xi_2)+\n\\nonumber\\\\+\n \\int_{\\tilde\\sigma}^{\\infty}\\left((\\mathbf 1- K_{\\mathrm{Ai}}\n )_{\\tilde\\sigma}^{-1}\n \\mathcal A_{\\xi_1-\\sigma}^{ \\tau_1}\\right)(u)\n\\mathcal A_{\\xi_2-\\sigma}^{ -\\tau_2 }(u)du.\n\\label{E127}\n\\end{eqnarray}\nwhere $\\widetilde \\sigma = 2^\\frac 23 \\sigma$ and  $K_{\\mathrm{Ai}}$ denotes the usual Airy kernel and\\footnote{For our convenience we slightly modified the definition of $\\mathrm{Ai}^{(\\tau)}$ multiplying it by the coefficient $2^{\\frac{1}6}$. Nevertheless formulas \\eqref{E127}, \\eqref{E5} and \\eqref{E6} are changed accordingly so that the kernel is truly the same as in \\cite{AJvMTacnode}. }\n\\begin{eqnarray}\np(\\Delta \\tau;\\xi_1,\\xi_2) &\\&:=\\frac{e^{-\\frac{(\\xi_1-\\xi_2)^2}{4\\Delta \\tau }}}{\\sqrt{4\\pi \\Delta \\tau }} \\\\\n\\mathrm{Ai}^{(\\tau)}(x)&\\&:=\n\\frac{2^\\frac 1  6 }{2\\pi i}\\int _{\\gamma_R }  dz~e^{ z^3\/3+z^2\\tau- zx }=\n2^\\frac 1  6e^{ \\tau x+\\frac 2 3\\tau ^3 }\\mathrm{Ai} (x+\\tau^2),\n\\\\\nK_{ \\mathrm{Ai} }^{(\\tau_1,-\\tau_2)}(x,y)&\\&:=\\int_0^{\\infty}\\!\\!\n\\mathrm{Ai}^{(\\tau_1)}(x\\!+\\!u \\sqrt[3]{2})  \\mathrm{Ai}^{(-\\tau_2)}(y\\!+\\!u \\sqrt[3]{2})du, \\label{E5}\n\\\\\n{\\cal A}^{\\tau}_\\xi(u) \n&\\& := \\mathrm{Ai}^{(\\tau)} ( \\xi+2^{1\/3} u)- \\int_0^\\infty \\mathrm{Ai}^{(\\tau)} (-\\xi+2^{1\/3} v )\n\\mathrm{Ai} (u+ v )dv  ,  \\label{E6}\n\\end{eqnarray}\n\\end{defn} \n\nGiven $r$ times $\\tau_1<\\tau_2<\\dots <\\tau_r$ we associate to each a copy of ${\\bf R}$ or, equivalently we consider the Cartesian product ${\\bf R}\\times \\{\\tau_1,\\dots, \\tau_r\\}$ which is set-theoretically isomorphic to ${\\bf R}^r$. Another convenient way of thinking about this set is as the {\\em disjoint union} of ${\\bf R}$ with itself, $r$ times. We will use the notation ${\\bf R}_{\\tau_j}$ to refer to the $j$-th copy (which should be thought of as a copy of ${\\bf R}$ ``at time $\\tau_j$''). \nIn each copy ${\\bf R}_{\\tau_j}$ we consider a Borel bounded subset $E^{(j)}$; although it is much less general, the reader should imagine that each $E^{(j)}$ is a finite union of bounded intervals as follows\n\\begin{eqnarray}\nE^{(j)} = [a_1^{(j)}, b_1^{(j)}] \\cup \\dots \\cup  [a_{\\ell_j} ^{(j)}, b_{\\ell_j} ^{(j)}]\\ ,\\ \\ a_1^{(j)}<b_1^{(j)}<a_2^{(j)}<b_2^{(j)}<\\dots<a_{\\ell_j}^{(j)}<b_{\\ell_j}^{(j)}.\n{\\rm e}\nWe denote by $\\mathcal T_\\sigma(\\tau)$ the tacnode field at time $\\tau$; the {\\em joint gap probability} of the multi-set $E:=E^{(1)} \\sqcup \\dots \\sqcup E^{(r)}$ is the probability that there are no points of the tacnode process in $E^{(j)}$ at time $\\tau_j$, for all $j=1,\\dots, r$: \n\\begin{eqnarray}\nF_{\\mathrm{tac}}(E, \\vec \\tau;\\sigma) := {\\rm Prob} \\left\\{\n\\mathcal T_\\sigma(\\tau_j) \\not \\in E^{(j)}\\ ,\\ \\ j=1,\\dots,r\n\\right\\}.\n\\end{eqnarray}\nThen the general theory of determinantal random point processes states that \n\\begin{eqnarray}\nF_{\\mathrm{tac}}(E, \\vec \\tau;\\sigma) =  \\det \\left( \\mathbf 1 - \\mathbb K^{\\rm tac}\\bigg|_E\\right)\\label{gaptac}\n\\end{eqnarray}\nwhere the operator $\\mathbb K^{\\rm tac}$ is intended as the operator on $L^2({\\bf R}_{\\tau_1} \\sqcup \\dots \\sqcup {\\bf R}_{\\tau_r})$ with kernel as in Definition \\ref{defKtac}.\nThe main goal is to express the determinant \\eqref{gaptac} as the {\\em ratio} of two simpler Fredholm determinants. The following two theorems, nevertheless, will be stated for the (more general) {\\em generating functions} of the occupation numbers (see \\cite{Sosh:RandomPointFields} for details); if $E^{(i)}$ is a disjoint union of intervals $E^{(j)} = \\bigsqcup_{\\alpha=1}^{\\ell_j}  E^{(j)}_{\\alpha}$, this latter is defined by\n\\begin{eqnarray}\nF_{\\rm tac}(E, \\vec \\tau;\\sigma, \\vec z):= \\left\\langle \\prod_{j=1}^r \\prod_{\\alpha=1}^{\\ell_j}   (z^{(j)}_{\\alpha})^{\\sharp_{E^{(j)}_\\alpha}}\\right\\rangle=\n\\det \\left[\n\\mathbf 1 - \n\\left(\\sum_{j=1}^r \\sum_{\\alpha=1}^{\\ell_j} (1-z^{(j)}_\\alpha) \\Pi_{E^{(j)}_{\\alpha}} \\right) \\mathbb K^{\\rm tac} \\vec \\Pi \n\\right]\\label{tacgen}\n{\\rm e}\nwhere $\\vec \\Pi = \\sum_{j=1}^r \\sum_{\\alpha=1}^{\\ell_j} \\Pi_{E^{(j)}_{\\alpha}}$ is the projector on the multi-interval $\\vec E$ and the generating function is a series on the variables $z_{\\alpha}^{(j)}$. The gap probability is nothing but the value of the generating function at $z^{(j)}_{\\alpha}=0$.\nWe shall use the notations \n\\begin{eqnarray}\n\\Pi^{(j)}_z:=  \\sum_{\\alpha=1}^{\\ell_j} (1-z^{(j)}_\\alpha) \\Pi_{E^{(j)}_{\\alpha}}\\ ,\\ \\ \\ \n\\vec \\Pi_z:= \\sum_{j=1}^r\\Pi^{(j)}_z.\n{\\rm e}\n\n\\begin{theorem}\n\\label{P1} \nThe generating function \\eqref{tacgen} admits the equivalent representation\n\\begin{eqnarray}\nF_{\\rm tac}(E, \\tau;\\sigma,\\vec z) =  F_2(\\widetilde \\sigma)^{-1}\\!\\det \\left( \\mathbf 1  - \\left[ \\begin{array}{c|ccc}\n \\pi K_\\mathrm{Ai}\\pi & -\\pi \\mathfrak A_{-\\tau_1}^T\\Pi^{(1)} & \\dots &  -\\pi \\mathfrak A_{-\\tau_r}^T\\Pi^{(r)} \\\\\n \\hline\n -\\Pi^{(1)}_z \\mathfrak A_{\\tau_1} \\pi &  \n \\\\\n \\vdots &&\\vec \\Pi_z\\left( \\mathbb K_0 - \\mathbb G\\right)\\vec \\Pi&\\\\\n  -\\Pi^{(r)}_z \\mathfrak A_{\\tau_r} \\pi & \n\\end{array}\\right]\\right) \\nonumber\\\\\n\\label{explode0}\n\\end{eqnarray}\nwhere $F_2(\\widetilde \\sigma)$ is the Tracy--Widom distribution \\eqref{TWd}\nand \\\\ $\\mathbb K_0, \\ \\mathbb G: L^2({\\bf R}_{\\tau_1}  \\sqcup \\dots \\sqcup {\\bf R}_{\\tau_r} ) \\to L^2({\\bf R}_{\\tau_1}  \\sqcup \\dots \\sqcup {\\bf R}_{\\tau_r} ) $ are the operators with kernels\n\\begin{eqnarray}\n(\\mathbb K_0)_{ij}(\\xi_1,\\xi_2) =K_\\mathrm{Ai}^{(\\tau_i, -\\tau_j)}(\\sigma-\\xi_1, \\sigma-\\xi_2)\\ ,\\qquad \n\\mathbb G_{ij}(\\xi_1,\\xi_2)  = \\mathbf 1_{\\tau_1>\\tau_2}\n\\frac{e^{-\\frac{(\\xi_1-\\xi_2)^2}{4(\\tau_1-\\tau_2) }}}{\\sqrt{4\\pi (\\tau_1-\\tau_2) }}\n\\end{eqnarray}\nwhile $\\mathfrak A_{\\vec \\tau}: L^2({\\bf R})\\to L^2({\\bf R}_{\\tau_1}  \\sqcup \\dots \\sqcup {\\bf R}_{\\tau_r} )  $ has kernel \n\\begin{eqnarray}\n[\\mathfrak A_{\\vec \\tau}(x,u)]_{j} =  \\mathrm{Ai}^{(\\tau_j)} ( x-\\sigma+2^{1\/3} u)- \\int_0^\\infty \\mathrm{Ai}^{(\\tau_j)} (\\sigma-x+2^{1\/3} v )\n\\mathrm{Ai} (v+u )dv.\n{\\rm e}\n\\end{theorem}\n{\\bf Proof}.\nWith the notations introduced above, the tacnode kernel defines an operator on\\\\ $L^2({\\bf R}_{\\tau_1}  \\sqcup \\dots \\sqcup {\\bf R}_{\\tau_r} ) $ that equals to $[\\mathbb K_0 - \\mathbb G]_{ij}+ \\mathfrak A_{\\tau_i}   (\\mathbf 1 -\\pi  K_\\mathrm{Ai} \\pi )^{-1} \\mathfrak A_{-\\tau_j}^T $.\\\\\nThe identity is based on the following operator identity (all being trace-class perturbations of the identity)\n\\begin{eqnarray}\n&& \\det \\left(\\mathbf 1  - \\left[ \\begin{array}{c|c}\n \\pi K_\\mathrm{Ai} \\pi & -\\pi \\mathfrak A_{-\\vec \\tau}^T\\vec \\Pi\\\\\n \\hline\n -\\vec \\Pi_z \\mathfrak A_{\\vec \\tau} \\pi &\\vec \\Pi_z\\left( \\mathbb K_0 - \\mathbb G\\right)\\vec \\Pi\n\\end{array}\\right]\\right) =\\cr\n&&=\n \\det\\left[\\begin{array}{c|c}\n \\mathbf 1 -\\pi K_\\mathrm{Ai} \\pi  & 0 \\\\ \n \\hline \n 0&\\mathbf 1 \\end{array}\n \\right] \\det  \\left[\n\\begin{array}{c|c}\n\\mathbf 1 & 0\\\\\n\\hline\n\\vec \\Pi_z \\mathfrak A_{\\vec \\tau }\\pi & \\mathbf 1\n\\end{array}\n\\right]\n\\det \\left[ \\begin{array}{c|c}\n \\mathbf 1 &(\\mathbf 1 -\\pi  K_\\mathrm{Ai} \\pi )^{-1}\\mathfrak A_{-\\vec \\tau}^T\\vec \\Pi\\\\\n \\hline\n 0 & \\mathbf 1 - \\vec \\Pi_z\\left( \\mathbb K_0 - \\mathbb G + \\mathfrak A_{\\vec \\tau}  (\\mathbf 1 -\\pi  K_\\mathrm{Ai} \\pi )^{-1}\\mathfrak A_{-\\vec \\tau}^T\\right)\\vec \\Pi\n\\end{array}\\right] = \\cr\n&&=\\underbrace{\\det \\left(\\mathbf 1- \\pi K_\\mathrm{Ai} \\pi     \\right)}_{F_2(\\widetilde \\sigma)} \\det( \\mathbf 1 -\\vec  \\Pi_z {\\mathbb K}^{\\rm tac} \\vec \\Pi) \n\\end{eqnarray}\n {\\bf Q.E.D}\\par \\vskip 4pt\n\n\\begin{remark}\\rm[Tacnode gap probabilities as (formal) conditioned process]\nThe gap probabilities of the tacnode process are  the ratio of two Fredholm determinants: the denominator is the  Tracy-Widom distribution, i.e., the gap probability of the Airy process. It is enticing to interpret thus the ratio as a {\\em conditional probability}.  For this interpretation to hold the numerator of formula \\eqref{explode0} should be a gap probability of a determinantal process on the configuration space ${\\bf R}_0 \\sqcup \\left(\\bigsqcup _{j=1}^r {\\bf R}_{\\tau_j}\\right)$ with kernel as indicated in the formula itself. \n\nThen the tacnode would be the {\\bf conditioned process} where the conditioning is that there are no points in $[\\widetilde \\sigma,\\infty)\\subset {\\bf R}_0$. This type of of conditioned processes have been already analyzed in \\cite{Lyons} for the discrete case, and in \\cite{TaoWigner} for the continuous case with a finite number of particles. In Appendix \\ref{condproc} we treat the continuous case with an arbitrary number of particles and give an explicit expression for the kernel of the conditioned process. However, the correlation functions of this ``extended'' tacnode process fail the positivity test (we checked on some numerical example) and thus the interpretation is only formal.\n\\end{remark}\nThe second theorem is an equivalent representation where the operator in the  Fredholm determinant in the numerator of Theorem \\ref{P1} is further ``unraveled'' in terms of elementary terms.\n\\begin{theorem}\n\\label{main}\nWith the same notations as in Theorem \\ref{P1} we have  \n\\begin{eqnarray}\nF_{\\rm tac}(E, \\vec \\tau;\\sigma, \\vec z)= \n\\frac {\\det \\left[ \\mathbf 1 -\\widehat  \\Pi_z  \\mathbb H\\widehat \\Pi \\right]       }{\\det \\left[\\mathbf 1 -\\pi K_{\\mathrm{Ai}} \\pi \\right]}\n{\\rm e}\nwhere the Fredholm determinant is defined on the space $$\nL^2({\\bf R}_+ \\sqcup {\\bf R}_0 \\sqcup {\\bf R}_{\\tau_1} \\sqcup\\dots, \\sqcup {\\bf R}_{\\tau_r}) \\simeq \nL^2({\\bf R}_+)\\oplus L^2( {\\bf R}_0)\\oplus \\left({\\bf R}_{\\tau_1} \\sqcup\\dots \\sqcup {\\bf R}_{\\tau_r}\\right).$$ \nDenoting the $L^2({\\bf R}_+) $ with the index $_{-1}$, the operators that appear are the following ones \n\\begin{eqnarray}\n\\widehat \\Pi_z := \\mathbf 1_{-1} \\oplus \\pi \\oplus \\vec \\Pi_z\\ ,\\ \\ \\widehat \\Pi = \\widehat \\Pi_0 = \\mathrm{Id}_{-1} \\oplus \\pi \\oplus \\vec \\Pi\n\\end{eqnarray} \nand their kernel $\\mathbb H$ is expressed in terms of ordinary Airy functions as follows\n\\begin{eqnarray}\n \\left[\n\\begin{array} {c|c|c}\n  \\mathbb H_{-1,-1} \\equiv 0\n& \n \\mathbb H_{-1,0}(x,y) = -\\mathrm{Ai}(x+y) \n&\n\\!\\!  \\mathbb H_{-1,j}(x,y) =  \\mathrm{Ai}^{(-\\tau_j)} (x\\sqrt[3]{2}+\\sigma-y  ) \\!\\!\\!\n \\\\[10pt]\n \\hline &&\\\\ \n \\mathbb H_{0,-1}(x,y) =  - \\mathrm{Ai}  (x+y ) \n&\n \\mathbb H_{0,0}\\equiv 0\n &\n   \\mathbb H_{0,j}(x,y) = \\mathrm{Ai}^{(-\\tau_j)} (x\\sqrt[3]{2} + y-\\sigma ) \n \\\\[10pt]\n \\hline &&\\\\\n\\!\\!\\!  \\mathbb H_{i,-1}(x,y) = \\mathrm{Ai}^{(\\tau_i)} (\\sigma-x+y \\sqrt[3]{2}  )\\!\\!\n &\n\\!  \\mathbb H_{i,0}(x,y) = \\mathrm{Ai}^{(\\tau_i)} (x-\\sigma+y\\sqrt[3]{2} ) \\!\\!\n  &\n   \\mathbb H_{i,j}(x,y) = - p(\\tau_i - \\tau_j; x,y)\\chi_{i >j}\n\\end{array}\n\\right],\\nonumber\\end{eqnarray}\nwith $1\\leq i,j \\leq r$.\n\\end{theorem}\n\\bigskip\n\nThe expression of the Fredholm determinant in Theorem \\ref{main} is the most suitable for the numerical computations because each entry of the kernel is either a simple modification of the Airy function or exponential.  The gap probabilities are obtained simply by setting all the $z_\\alpha^{(j)}$'s to zero, or --which is the same-- restricting the operator $\\mathbb H $ on the set  ${\\bf R}_+ \\sqcup [\\widetilde \\sigma,\\infty) \\sqcup\\left(\\bigsqcup_{j=1}^R E^{(j)}\\right)$.\n\n{\\bf Proof of Theorem \\ref{main}.}\nFor brevity we shall introduce the operators  with the kernels below\n\\begin{eqnarray}\n&& \\mathcal C_\\tau  (\\xi,u):=2^\\frac 1 6  \\mathrm{Ai}^{(\\tau)} \\left(\\xi -\\sigma+ u\\sqrt[3]{2}\\right):L^2({\\bf R}_+)\\to L^2({\\bf R}_\\tau)\\\\\n&& \\mathcal C(u,v):= \\mathrm{Ai}(u+v) :L^2({\\bf R}_+) \\to L^2({\\bf R}) \n\\end{eqnarray}\nThe first observation is that both $K_\\mathrm{Ai}$ and $K_\\mathrm{Ai}^{(\\tau_1,-\\tau_2)}$ can be written as compositions; \n\\begin{eqnarray}\nK_\\mathrm{Ai} = \\mathcal C \\mathcal C^T\\ ,\\qquad K_{\\mathrm{Ai}}^{(\\tau_1,-\\tau_2)} = \\mathcal C_{\\tau_1}  \\mathcal C_{-\\tau_2}^T\\ .\n{\\rm e}\nIf we denote the parity operator around $\\sigma$ by $P:L^2({\\bf R})\\to L^2({\\bf R})$ $(Pf)(x) = f(2\\sigma-x)$ then \nthe tacnode kernel can be written as\n\\begin{eqnarray}\n\\mathbb K^{\\rm tac}_{ij} =\n P\\mathcal C_{\\tau_i} \\mathcal C_{-\\tau_j}^T P \n + \\mathfrak A_{\\tau_i} \\pi \\left(\\mathbf 1 - \\pi \\mathcal C \\mathcal C^T \\pi\\right)^{-1}\\pi \\mathfrak A_{-\\tau_j}^T - \\mathbb G_{ij}\n{\\rm e}\nwhere \n\\begin{eqnarray}\n[\\mathfrak U_\\tau]_j = \\mathcal C_{\\tau_j} - P\\mathcal C_{\\tau_j} \\mathcal C: L^2({\\bf R}_+)\\to L^2({\\bf R}_{\\tau_j})\n\\end{eqnarray}\nWe denote $\\mathcal C_{\\vec \\tau}: L^2({\\bf R}_+)\\to  L^2(\\bigsqcup_{j=1}^r {\\bf R}_{\\tau_j})\\simeq \\bigoplus_{j=1}^r L^2({\\bf R}_{\\tau_j})$ as the operator with components $\\mathcal C_{\\tau_j}$; \nthen the proof relies upon\nthe following identity of determinants\n\\begin{eqnarray}\n\\det \\left(\\mathbf 1  - \\left[ \\begin{array}{c|c}\n \\pi \\mathcal C \\mathcal C^T \\pi & -\\pi \\mathfrak A_{-\\vec \\tau}^T\\vec \\Pi\\\\\n \\hline\n -\\vec \\Pi_z \\mathfrak A_{\\vec \\tau} \\pi &\\vec \\Pi_z(\\mathbb  K_0 - \\mathbb G)\\vec \\Pi\n\\end{array}\\right]\\right) \n= \\det  \\left[\n\\begin{array}{c|c|c}\n \\mathbf 1 &\\mathcal C^T\\pi  &    - \\mathcal C_{-\\vec \\tau}^T P \\vec \\Pi \\\\\n \\hline\n \\pi \\mathcal C & \\mathbf 1 & -\\pi\\mathcal C_{-\\vec \\tau}^T \\vec \\Pi  \\\\\n\\hline\n -\\vec \\Pi_z P \\mathcal C_{\\vec \\tau}  &-\\vec \\Pi_z \\mathcal C_{\\vec \\tau} \\pi  &\\mathbf 1+ \\vec \\Pi_z \\mathbb G \\vec \\Pi \n\\end{array}\n\\right] \\label{Hc}\n\\end{eqnarray}\nThe identity is seen by \n\\begin{eqnarray}\n \\left[\n\\begin{array}{c|c|c}\n \\mathbf 1 &\\mathcal C^T\\pi  &    - \\mathcal C_{-\\vec \\tau}^T P \\vec \\Pi \\\\\n \\hline\n \\pi \\mathcal C & \\mathbf 1 & -\\pi \\mathcal C_{-\\vec \\tau}^T\\vec  \\Pi  \\\\\n\\hline\n -\\vec \\Pi_z P \\mathcal C_{\\vec \\tau}   &-\\vec \\Pi_z \\mathcal C_{\\vec \\tau}\\pi  & \\mathbf 1+ \\vec \\Pi_z \\mathbb G \\vec \\Pi \n\\end{array}\n\\right] \n \\left[\n\\begin{array}{c|c|c}\n \\mathbf 1 &-\\mathcal C^T\\pi  &     \\mathcal C_{-\\vec \\tau}^T P \\vec \\Pi \\\\\n \\hline\n0 & \\mathbf 1 & 0  \\\\\n\\hline\n0  &0 & \\mathbf 1 \n\\end{array}\n\\right] \n= \\cr\n \\left[\n\\begin{array}{c|c|c}\n \\mathbf 1 &0 &   0 \\\\\n \\hline\n \\pi \\mathcal C & \\mathbf 1 -\\pi \\mathcal C\\mathcal C^T\\pi & -\\mathcal C_{-\\vec \\tau}^T \\vec \\Pi + \\pi \\mathcal C\\mathcal C_{-\\vec \\tau }^TP\\vec \\Pi   \\\\\n\\hline\n -\\vec \\Pi_z P \\mathcal C_{\\vec \\tau}  &-\\vec \\Pi_z \\mathcal C_{\\vec \\tau} + \\vec \\Pi_z P \\mathcal C_{\\vec \\tau} \\mathcal C^T\\pi  &\\mathbf 1+ \\vec\\Pi_z \\mathbb G \\vec \\Pi - \\vec\\Pi_z P \\mathcal C_{\\vec \\tau} \\mathcal C_{-\\vec \\tau}^T P \\vec \\Pi \n\\end{array}\n\\right] \n\\end{eqnarray}\nThe operator $\\mathbb H$ is read off \\eqref{Hc}. \n {\\bf Q.E.D}\\par \\vskip 4pt\n\n\\subsection{From the Tacnode to the Pearcey  process}\n\nGiven its origins in the Dyson Brownian diffusion, and by the same idea that led to considering the limit of the Pearcey to the Airy process,  it is physically expected that the tacnode process should converge to the Pearcey process under a suitable rescaling sending the overlap $\\sigma$ to $-\\infty$, which amounts to ``push'' closer and closer  the two sets of particles touching on the tacnode point. \nA rigorous approach based on the representation of the gap-probabilities in terms of a Riemann--Hilbert problem \\cite{BCG1} suggests that this convergence occurs in the following asymptotic regime\n\\begin{eqnarray}\na_{tac} = \\frac {a_{P}} {(-8\\sigma)^\\frac 18}\\ ,\\qquad \\tau_{tac} = \\pm \\sqrt{\\frac {-\\sigma}2} + \\frac {\\tau_P}{(-2^7\\sigma)^{\\frac 1 4}}\\ ,\\ \\ \\ \\sigma\\to -\\infty;\\label{goodone}\n{\\rm e}\nin other words we have that\n\\begin{eqnarray}\n\t\\lim_{\\sigma\\rightarrow -\\infty} F_{\\rm tac}\\left(\\left[\\frac {a_{P}} {(-8\\sigma)^\\frac 18},\\frac {b_{P}} {(-8\\sigma)^\\frac 18}\\right],\\pm\\sqrt{\\frac {-\\sigma}2} + \\frac {\\tau_P}{(-2^7\\sigma)^{\\frac 1 4}},\\sigma\\right) = F_{\\rm P}\\Big(\\big[a_P,b_P\\big],\\tau_P\\Big). \\label{goodone2}\n{\\rm e}\nIndeed, modulo a multiplicative rescaling of the variables (but without changing the scale exponents) the formula above is proven in \\cite{GeudensZhangTacPearcey}, Theorem 2.3 (more correctly, \\eqref{goodone2} is a direct consequence of the quoted theorem, since the uniform convergence of the kernels on compact sets implies the convergence of the Fredholm determinants). In \\cite{BCG1} we will give another proof of \\eqref{goodone2}, based on the nonlinear steepest descent analysis of a \\emph{different} Riemann-Hilbert problem giving directly the gap probability of the process rather than its kernel.\\\\\nHere we show numerically the validity of \\eqref{goodone2} by computing the following two gap-probabilities:\n\\begin{eqnarray}\nF_{\\rm tac} ([a_{tac} ,b_{tac}]; \\tau_{tac}, \\sigma) := \\mathbb P_\\sigma\\left\\{\\mathcal T_\\sigma(\\tau_{tac}) \\not\\in[a_{tac} , b_{tac}] \\right\\}\\\\\n F_{\\rm P} ([a_{P}, b_P], \\tau_{tac}) :=  \\mathbb P\\left\\{\\mathcal P(\\tau_{P}) \\not\\in[-a_{P} , a_{P}] \\right\\}\n\\end{eqnarray}\nand verifying numerically the convergence. This has been done for $-\\sigma$ up to $9$, after which our\\footnote{None of the author specializes in numerical computations, and hence we have implemented the ideas of \\cite{Bornemann} in its simplest form. See Appendix.} numerical implementation of the Fredholm determinant becomes too unstable, see Fig. \\ref{TactoP1t}.\nWe have also verified in a numerical example the convergence of a two-time gap probability, see Fig. \\ref{TactoP2t}.\n\n\\begin{figure}\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-1-abs.pdf}\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-1-err.pdf}\n\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-3-abs.pdf} \n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-3-err.pdf}\n\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-9-abs.pdf} \n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{sigma-9-err.pdf}\n\\caption{\\footnotesize The graphs of $F_{\\rm tac}([-a_{tac},a_{tac}],\\tau_{tac};\\sigma)$ and $F_{\\rm P}([-a_P,a_P]; \\tau_{P})$  with the parameters related by \\eqref{goodone}.}\n\\label{TactoP1t}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{T2Psigma-1-abs.pdf}\n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{T2Psigma-4-abs.pdf} \n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{T2Psigma-7-abs.pdf} \n\\includegraphics[height=0.4\\textwidth,width=0.45\\textwidth]{T2Psigma-1-4-7-rel.pdf}\n\\caption{\\footnotesize The multi--time case. Here the graphs of $F_{\\rm tac}\\left(\\left\\{[-a_{tac},a_{tac}]; \\tau_{tac,1},[-a_{tac},a_{tac}]; \\tau_{tac,2}\\right\\};\\sigma\\right)$ and $F_{\\rm P}\\left(\\left\\{[-a_P,a_P]; \\tau_{P,1},[-a_P,a_P]; \\tau_{P,2}\\right\\}\\right)$ are shown with the parameters related by \\eqref{goodone} and $\\tau_{P,1}=0, \\tau_{P,2}=1$. The last graph is the plot of the relative discrepancy between $F_{\\rm P}$ and $F_{\\rm tac}$.}\n\\label{TactoP2t}\n\\end{figure}\n\\subsection{From the tacnode to the Airy process}\nThere are two types of regimes in which the gap probability for the tacnode process ``degenerates'' into the Airy one:\n\\begin{enumerate}\n\\item large separation $\\sigma\\to+\\infty$; physically it corresponds to pull far apart the two sets of Brownian particles touching on the tacnode point.\n\\item large time $\\tau \\to \\pm \\infty$; it corresponds to move far away from the singular point along the boundary of the space-time region swept out by the particles.\n\\end{enumerate}\nLet us for simplicity describe the one-interval and one-time case. Keeping the overlap $\\sigma$ fixed, we will have that\n\\begin{eqnarray}\n\t\\lim_{\\tau\\rightarrow\\infty}F_{\\rm tac}\\left(\\left[a-\\sigma-\\tau^2,b-\\sigma-\\tau^2\\right],\\tau,\\sigma\\right) = F_2\\Big(\\big[a,b\\big]\\Big)\n\\label{conAi1}{\\rm e}\nwhile, keeping $\\tau$ fixed, in a complete analogous way we obtain\n\\begin{eqnarray}\n\t\\lim_{\\sigma\\rightarrow\\infty}F_{\\rm tac}\\left(\\left[a-\\sigma-\\tau^2,b-\\sigma-\\tau^2\\right],\\tau,\\sigma\\right) = F_2\\Big(\\big[a,b\\big]\\Big).\n\\label{conAi2}{\\rm e}\nThis convergence is simple to see by inspecting directly the kernel of the (extended) tacnode process \\eqref{E127} because  the term involving the resolvent of the Airy kernel tends to zero, uniformly over compact sets of the translated spatial variables $x-\\sigma - \\tau^2$. Here we will not study the rate of convergence in formulas \\eqref{conAi1},\\eqref{conAi2}. We simply point out that, if we knew some equation for the tacnode gap probability, we could perform a similar analysis to the one in \\cite{ACvM2}. The extension to multi--time case (and multi--intervals), though more cumbersome, does not present any additional difficulty.\n\nA perhaps more interesting situation, less obvious from the formula \\eqref{E127} but equally natural from the ``physical'' setting, is the one in which the tacnode process degenerates into a \\emph{couple} of Tracy--Widom distributions, in analogy with the Pearcey-to-Airy transition \\eqref{PtoAi} proved in \\cite{BertolaCafasso1}. In this case, roughly speaking, half of the space variables (endpoints of the gaps) moves far away from the tacnode following the left branch of the boundary of the space--time region swept by the particles, and the other half goes in the opposite direction. The simplest instance of this degeneration is the case of one-time and one-interval $E = [a -\\sigma-\\tau^2, -b+\\sigma+\\tau^2]:$ \n\\begin{eqnarray}\n\t\\lim_{\\sigma\\rightarrow\\infty} F_{\\rm tac}\\Big(\\big[a-\\sigma-\\tau^2,-b+\\sigma+\\tau^2\\big],\\tau,\\sigma\\Big) = F_2([a,\\infty))F_2([b,\\infty)).\n\\label{112}{\\rm e}\nSimilarly, the limit \\eqref{112} holds also  for $\\tau\\to \\pm\\infty$  and $\\sigma$ fixed. The formula \\eqref{112}, as well as its generalization to the multi--interval case, will be proven in \\cite{BCG1}, numerically these regimes are illustrated in Figure \\ref{TactoAiry}.\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{TactoAiry.pdf}\n\\includegraphics[width=0.49\\textwidth]{TactoAiry2.pdf}\n\\caption{\\footnotesize The relative values $1 - \\frac {F_{\\rm tac}([ a_{tac}, b_{tac}]; \\tau_{tac}, \\sigma)}{F_2(a)F_2(b)} $ with $a_{tac} =  a-\\sigma-\\tau_{tac}^2,\\ b_{tac} = b + \\sigma + \\tau_{tac}^2$, plotted against $\\tau_{tac}$ (left) and $\\sigma$ (right), showing the convergence of the tacnode gap probability to the product of two Tracy-Widom distributions. Here, by the way of example,  $a = -0.3, b = 0.5$. The convergence as $\\sigma\\to\\infty$ is clearly exponential (note that the graph on the right is on a semilog coordinate system).}\n\\label{TactoAiry}\n\\end{figure}\n\n\\FloatBarrier\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\vspace{-.2cm}\n\nIn lattice simulations of QCD on a cubic volume ($V=L^3$) with\nperiodic boundary conditions on the fields, the components of\nhadronic momenta $p_i$ are quantized in integer multiples of $\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}$.\nFor currently available lattices this implies that the lowest non-zero\nmomentum is large, typically $500\\,\\mathrm{MeV}$ or so, and there are\nlarge gaps between neighbouring momenta. This limits the\nphenomenological reach of simulations, particularly for momentum\ndependent quantities such as the form-factors of weak semileptonic\ndecays of hadrons. In ref.~\\cite{Bedaque:2004kc} Bedaque proposed the\nuse of \\textit{twisted} boundary conditions\\footnote{See the\nreferences cited in~\\cite{Bedaque:2004kc} and~\\cite{deDivitiis:2004kq}\nfor earlier related ideas.} for the quark fields $\\psi$\n\\begin{equation}\n\\label{eq:twistdef}\n\\psi(x_i+L)=e^{i\\theta_i}\\,\\psi(x_i)\\,.\n\\end{equation}\nTwisted boundary conditions allow for simulations with arbitrary\ncomponents of hadronic momenta. For example, the momentum of a meson\ncomposed of a quark with flavour $1$ satisfying boundary conditions\nwith a twisting angle $\\vec{\\theta}_1 =\n(\\theta_{11},\\theta_{12},\\theta_{13})$ and an antiquark of flavour $2$\nwith angle $\\vec{\\theta}_2$\\, is\n\\begin{equation}\\label{eq:momspectrum}\n\\vec{p}=\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}\\,\\vec{n}-\\frac{\\vec{\\theta}_1- \\vec{\\theta}_2}{L}\\,,\n\\end{equation}\nwhere $\\vec{n}$ is a vector of integers.\n\nThe practical difficulty in using twisted boundary conditions in\nlattice simulations with dynamical quarks is that it requires the\ngeneration of a new set of gauge field configurations for every choice\nof twisting angle(s). In refs.~\\cite{Sachrajda:2004mi,Bedaque:2004ax}\nit was shown that for many physical quantities one can use\n\\textit{partially twisted} boundary conditions, i.e. impose twisted\nboundary conditions for the valence quarks but periodic boundary\nconditions for the sea quarks, thus eliminating the need for new\nsimulations for every choice of momentum and making the technique\npracticable. The physical quantities for which partially twisted\nboundary conditions can be applied include those with at most a single\nhadron in the initial and final states (and possibly even in\nintermediate states), for which the finite-volume effects decrease\nexponentially with the volume. For these processes the finite-volume\neffects depend on the twisting angle(s) but remain exponentially\nsmall.\n\nFor some processes with energies above a two-body threshold, such\nas $K\\to\\pi\\pi$ decays with the two-pions in an isospin zero\nstate, the finite-volume effects decrease only as powers of the\nvolume and must be subtracted for acceptable precision to be\nreached. We are not able to perform these subtractions if\npartially twisted boundary conditions are used. Here we will only\nconsider processes for which such a problem does not arise.\n\nIn this letter we confirm the theoretical results of\nref.~\\cite{Sachrajda:2004mi} in a numerical study of partially twisted\nboundary conditions for dynamical, non-perturbatively improved Wilson\nfermions. In particular we find that:\n\\begin{itemize}\n\\item The energies of $\\pi$ and $\\rho$-mesons (with masses below the\n  two-pion threshold) satisfy the expected dispersion relation\n  \\begin{equation}\n  \\label{eq:disprel}\n  E_{\\pi,\\,\\rho}^2 =\n    m_{\\pi,\\,\\rho}^2+\\left(\\vec{p}_{\\rm lat}-{\\vec\\theta_1 -\n    \\vec\\theta_2 \\over L}\\right)^2,\n  \\end{equation}\n  where $\\vec\\theta_1$ and $\\vec\\theta_2$ are the twisting angles of\n  the two valence quarks and $\\vec{p}_\\mathrm{lat}=(\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L})\\vec{n}$ is\n  the contribution to the meson's momentum introduced by the Fourier\n  transform of the correlation function.\n\n  This study extends the one in ref.~\\cite{deDivitiis:2004kq} where\n  the dispersion relation for pseudo scalar mesons with twisted\n  boundary conditions in the quenched approximation was found to be\n  consistent with expectations.\n\\item The values of the leptonic decay constants of $\\pi$ and $\\rho$\n  mesons and of the matrix element $\\langle 0|P|\\pi\\rangle$ of the\n  pseudo scalar density $P$ are independent of the twisting angles as\n  expected.\n\\end{itemize}\n\nA further reassuring result of our study is that twisted boundary\nconditions do not introduce additional noise in the data. As we\nincrease the meson's momentum by suitably varying the angles\n$\\vec\\theta_{1,2}$, the statistical errors on meson masses and matrix\nelements increase smoothly. However, when comparing results obtained\nwith twisted and periodic boundary conditions with similar momenta\n(i.e. momenta close to $\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}$ or $\\sqrt{2}(\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L})$) the errors are\nfound to be comparable.\n\nThe plan of the remainder of this letter is as follows. In the next\nsection we present the details of our computation, the parameters of\nthe simulation (including the choice of twisting angles) and a\ndescription of the analysis. We present our results in\nsec.~\\ref{sec:results} and conclusions in sec.~\\ref{sec:concs}.\n\n\\vspace{-.5cm}\n\\section{Details of the Simulation and Analysis}\n\\vspace{-.2cm}\n\nWe study meson observables on sets of gauge configurations which were\ngenerated with two degenerate flavours of sea quarks using\nnon-per\\-tur\\-ba\\-tively improved Wilson fermions and the plaquette\ngauge action on the torus with periodic boundary conditions\n($\\beta=5.2$, $a\\approx0.1\\,\\mathrm{fm}$, $c_{\\rm SW}=2.0171$,\n$(L\/a)^3\\times T\/a =16^3\\times 32$). We used the ensembles of field\nconfigurations which were studied in detail\nin~\\cite{Allton:2001sk,Allton:2004qq} and took over the suggested\nseparation of measurements by $40$ trajectories in our analysis. The\nsimulated quark masses are summarized in table~\\ref{tab_sim-params}. \nPropagators and correlators were calculated using the FermiQCD\nlibraries~\\cite{DiPierro:2000bd,DiPierro:2001yu,DiPierro:2003sz}. \nWe stress that the aim of the present study is to investigate the\nconsistency and effectiveness of using partially twisted boundary\nconditions at fixed values of the quark mass. We do not attempt to\nperform a chiral extrapolation.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccc}\n\\hline\\hline\\\\[-1.5ex]\n$\\kappa_\\mathrm{val}=\\kappa_\\mathrm{sea}$ & $m_\\pi\/m_\\rho$ &\n $N_\\mathrm{meas}$\\\\[0.5ex]\n\\hline\\\\[-2ex]\n0.13500 & 0.697(11) & 200\\\\\n0.13550 & 0.566(16) & 200\\\\[0.2ex]\n\\hline \\hline\n\\end{tabular}\\caption{Simulation parameters.}\\label{tab_sim-params}\n\\end{center}\n\\vspace{-.8cm}\n\\end{table}\n\nFor each flavour of valence quark we impose the boundary\nconditions in eq.~(\\ref{eq:twistdef}) for a variety of twisting\nangles $\\vec{\\theta}=(\\theta_1,\\theta_2,\\theta_3)$. When\nevaluating the corresponding propagators we make use of the change\nof quark field variables\n\\begin{equation}\n\\psi(x) = e^{i{\\vec{\\theta}\\cdot\\vec{x} \\over\nL}}\\tilde{\\psi}(x),\n\\end{equation}\nwhere $\\tilde{\\psi}(x)$ satisfies periodic boundary conditions. The\nphase factor cancels in all terms of the lattice fermion action except\nfor the spatial hopping terms which now become (for $i=1,2,3$)\n\\begin{equation}\n\\overline{\\tilde{\\psi}}(x)\n \\left[ e^{i{a\\theta_i\\over L}} U_i(x)(1-\\gamma_i)\\tilde{\\psi}(x+\\hat i)\n      + e^{-i{a\\theta_i\\over L}}U^\\dagger_i(x-\\hat i)\n           (1+\\gamma_i)\\tilde{\\psi}(x-\\hat i)\n \\right].\n\\end{equation}\nIn practice therefore, the partially twisted quark propagator can be\ncomputed by inverting the standard improved Wilson-Dirac operator in a\ngauge field background where the link variables \\{$U_i(x)$\\} have been\nreplaced by \\{$e^{i{a\\theta_i\\over L}}U_i(x)$\\}.\n\nThe physical observables which we study in this letter are the\nenergies and leptonic decay constants of the pseudo scalar and vector\nmesons and the matrix element of the pseudo scalar density. In order to\ndetermine these, we compute the following correlation\nfunctions:\n\\begin{eqnarray}\nC_{A_0 P}(t,\\vec{p}) &=& \\sum\\limits_{\\vec x}\n  e^{i\\vec p_\\mathrm{lat}\\cdot\\,\\vec x}\n  \\langle 0|A^I_0(\\vec x,t)P^\\dagger(0)|0\\rangle\\,,\\label{corrA0P}\\\\\nC_{PP}(t,\\vec{p}) &=& \\sum\\limits_{\\vec x}\n  e^{i\\vec p_\\mathrm{lat}\\cdot\\,\\vec x}\n  \\langle 0|P(\\vec x,t)P^\\dagger(0)|0\\rangle\\,,\\label{corrPP}\\\\\nC_{A_0 A_0}(t,\\vec{p}) &=& \\sum\\limits_{\\vec x}\n  e^{i\\vec p_\\mathrm{lat}\\cdot\\,\\vec x}\\langle 0|A_0^I(\\vec x,t)\n  (A_0^I(0))^\\dagger|0\\rangle\\,,\\label{corrA0A0}\\\\\nC_{V_i V_i}(t,\\vec{p}) &=& \\sum\\limits_{\\vec x}\n  e^{i\\vec p_\\mathrm{lat}\\cdot\\,\\vec x}\\langle 0|V_i^I(\\vec\n  x,t)(V_i^I(0))^\\dagger|0\\rangle \\quad\\mbox{(no sum on $i$)}\n  \\label{corrViVi}\\,,\n\\end{eqnarray}\nwhere $P(x)$ is the pseudo scalar density\n\\begin{equation}\\label{eq:pdef}\nP(x)=\\overline\\psi_2(x)\\gamma_5\\psi_1(x)\n\\end{equation}\nfor quarks of flavour $1$ and $2$ (with twisting angles\n$\\vec{\\theta}_1$ and $\\vec{\\theta}_2$), and $V^I_\\mu(x)$ and\n$A^I_\\mu(x)$ are the improved vector and axial-vector currents\n\\begin{eqnarray*}\nV^I_\\mu(x)&=&\\overline\\psi_2(x)\\gamma_\\mu\\psi_1(x)+ac_V(g_0)\\,\n   {1\\over2}(\\partial^\\ast_\\nu+\\partial_\\nu)\n   \\overline\\psi_2(x)\\sigma_{\\mu\\nu}\\psi_1(x)\\label{eq:videf}\\\\\nA^I_\\mu(x)&=&\\overline\\psi_2(x)\\gamma_\\mu\\gamma_5\\psi_1(x)+a\nc_A(g_0){1\\over2}(\\partial^\\ast_\\mu+\\partial_\\mu)P(x)\\,.\\label{eq:aidef}\n\\end{eqnarray*}\nHere, $\\partial_\\mu$ and $\\partial^\\ast_\\mu$ are the forward and\nbackward derivatives and $c_V(g_0)$ and $c_A(g_0)$ are improvement\ncoefficients which we take from~\\cite{Harada:2002jh}\nand~\\cite{DellaMorte:2005se} respectively. Since we are primarily\ninterested in the effects of twisted boundary conditions we do not\nattempt to compute the renormalization constants of $P$, $V_\\mu$\nand $A_\\mu$, nor do we implement improvement factors of the form\n$1+b(g_0)m_qa$, where $m_q$ is the mass of the quark.\nThe inclusion of these factors would of course be necessary if we were\nattempting to determine the physical leptonic decay constants.\nHowever, they are overall factors for each choice of quark mass and\nare independent of the twisting angles, while it is precisely the\ndependence on these angles which is the object of our study.\n\nThe momentum, $\\vec{p}$, of the meson is given by\n\\begin{equation}\n\\vec p = \\vec p_\\mathrm{lat} - {\\vec \\theta_1 - \\vec\n\\theta_2\\over L}\\,,\n\\end{equation}\nwhere $p_\\mathrm{lat}=(\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L})\\vec{n}$ and $\\vec{n}$ is a vector of\nintegers.\n\nAt large values of $t$ the time dependences of\n(\\ref{corrA0P})--(\\ref{corrViVi}) approach:\n\\begin{eqnarray}\nC_{A_0 P}(t,\\vec{p}\\,)\\hspace{-2mm}&\\to&\\hspace{-2mm}{1\\over\nE_{\\pi}}\\,Z_P\\,M_0(\\vec{p}\\,)\\,\ne^{- E_{\\pi}\\, T\/2} \\sinh((t-T\/2)E_{\\pi}),\\label{CAP}\\\\\nC_{PP}(t,\\vec{p}\\,)\\hspace{-2mm}&\\to&\\hspace{-2mm}{1\\over\nE_{\\pi}}\\,Z_P^2\\,\ne^{-E_{\\pi}\\,T\/2} \\cosh((t-T\/2)E_{\\pi})\\label{CPP},\\\\\nC_{A_0 A_0}(t,\\vec{p}\\,)\\hspace{-2mm}&\\to&\\hspace{-2mm}{1\\over\nE_{\\pi}}\\,M_0^2(\\vec{p}\\,)\\,\n e^{-E_{\\pi}\\,T\/2} \\cosh((t-T\/2)E_{\\pi})\\label{CAA},\\\\\nC_{V_i V_i}(t,\\vec{p}\\,)\\hspace{-2mm}&\\to&\\hspace{-2mm}{1\\over\nE_{\\rho}}\\,N^2_{i}(\\vec p\\,)\\,e^{-E_{\\rho}\\,T\/2}\n\\cosh((t-T\/2)E_{\\rho})\\label{CVV}\\;\\quad(i=1,2,3),\n\\end{eqnarray}\nwhere, for each choice of quark masses, we have denoted the lightest\npseudo scalar and vector mesons by $\\pi$ and $\\rho$ respectively and\n$E_{\\pi}$ and $E_{\\rho}$ are the corresponding energies which we\nexpect to satisfy the dispersion relations in eq.~(\\ref{eq:disprel}).\nThe notation for the matrix elements is as follows:\n\\begin{eqnarray}\nZ_P &=& \\langle 0|P(0)|\\pi(\\vec{p}\\,)\\rangle, \\label{Z_P}\\\\\nM_0(\\vec{p}\\,) &=&\n  \\langle 0\\,|\\,A_0(0)\\,|\\,\\pi(\\vec{p}\\,)\\rangle = f_\\pi E_\\pi,\\label{M_0}\\\\\nN^2_{i}(\\vec{p}\\,)&=& \\sum_{\\lambda} |\\langle\n  \\,0|\\,V_i(0)\\,|\\,\\rho(\\vec{p},\\lambda)\\rangle|^2 = f_\\rho^2\n  m_\\rho^2 \\left(1+{p_{i}^2\\over m_\\rho^2}\\right)\\label{N_ii}\\,\n\\end{eqnarray}\nwhere the index $\\lambda$ labels the $\\rho$-meson's polarization\nstate.\n\nIn this letter we study the validity of the dispersion relation in\neq.~(\\ref{eq:disprel}) and the independence of $f_\\pi$, $f_\\rho$ and\n$Z_P$ of the momentum. We evaluate the quark propagators for four\nvalues of the twisting angle $\\vec\\theta$:\n\\begin{equation}\\label{eq:thetas}\n\\vec{\\theta} = \\vec{0},\\  (2,0,0),\\ (0,\\pi,0)\\ \\textrm{and}\\\n(3,3,3)\\,.\n\\end{equation}\nFor each value of $\\kappa_\\mathrm{val}$, quark and antiquark\npropagators with all possible pairs $\\vec{\\theta}_1$ and\n$\\vec{\\theta}_2$ were combined to construct correlation functions\nfor mesons with a variety of momenta. Moreover we also combined\nthem with Fourier momenta $\\vec{p}_\\mathrm{lat} =\n(0,\\pm\\,\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L},0)$ to increase the range of momenta which can be\nreached. When presenting our results in the following section, we\ninclude for comparison results without twisting\n($\\vec\\theta_1=\\vec\\theta_2=0$), obtained by averaging over the\n$12$ equivalent momenta with\n$|\\vec{p}_{\\mathrm{lat}}|=\\sqrt{2}\\times \\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}$ and those obtained\nby averaging over the eight equivalent momenta with\n$|\\vec{p}_{\\mathrm{lat}}|=\\sqrt{3}\\times\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}$. Of course this\naveraging reduces the statistical errors and this should be borne\nin mind when comparing the errors at these untwisted momenta with\nthose at momenta with $\\vec\\theta_1-\\vec\\theta_2\\neq\\vec0$ for\nwhich such averaging is not possible.\n\nApplying the jackknife procedure to the data for the correlation\nfunctions in eqs.~(\\ref{corrA0P})--(\\ref{corrA0A0}), we have extracted\nall observables in the pseudo scalar channel from a combined\nnon-linear $\\chi^2$ fit to the functional form suggested by\n(\\ref{CAP})--(\\ref{CAA}), (\\ref{Z_P}) and (\\ref{M_0}). The fit-ranges\nwere chosen to yield compatible results under variation of the range\nby at least one unit in $t\/a$. \nWe applied the same procedure in the vector channel, combining the\ndata for the correlation function~(\\ref{corrViVi}) for $i=1,2$ and $3$\nin one fit using the expressions in eqs.~(\\ref{CVV}) and\n(\\ref{N_ii}).\n\n\\vspace{-.5cm}\n\\section{Results}\\label{sec:results}\n\\vspace{-.2cm}\n\nThe series of plots in figures~\\ref{figE} and~\\ref{fig_ZP} show\nour data as a function of $(\\vec p L)^2$ in the range $|\\vec p\nL|\\in[0,\\sqrt{3}\\times2\\pi]$. Fig.~\\ref{figE} contains the results\nfor the energies as a function of momentum and fig.~\\ref{fig_ZP}\nthose for the decay constants and $Z_P$. To ease orientation, the\npositions of the discrete Fourier momenta $|\\vec p_{\\rm lat}\nL|=0$, $2\\pi$, $\\sqrt{2}\\times2\\pi$ and $\\sqrt{3}\\times2\\pi$ are\nindicated by dashed vertical lines. We emphasize that it is only\nat these values of momenta that one can obtain results using\nperiodic boundary conditions. In fig.~\\ref{figEzoom} we zoom into\nthe region $|\\vec{p}\\,L|\\le2\\pi$ for the dispersion relations. In\nthis region we would expect lattice artefacts to be small and the\nuse of twisted boundary conditions to be particularly useful.\n\nIn each plot, the (blue) triangles correspond to points in which the\ncorrelation function was evaluated with $\\vec{p}_\\mathrm{lat}=\\vec 0$,\nbut with all possible pairs of $\\vec\\theta_1$ and $\\vec\\theta_2$ from\nthe set in (\\ref{eq:thetas}). The (red) diamonds and (green) squares\nrepresent the results obtained with\n$\\vec{p}_\\mathrm{lat}=(0,2\\pi\/L,0)$ and\n$\\vec{p}_\\mathrm{lat}=-(0,2\\pi\/L,0)$ respectively, combined with all\npossible pairs of $\\vec\\theta_1$ and $\\vec\\theta_2$. The four points\nwith $\\vec\\theta_1=\\vec\\theta_2=\\vec 0$ with\n$|\\vec{p}_\\mathrm{lat}|=0,\\ 2\\pi\/L,\\ \\sqrt{2}\\times2\\pi\/L$ and\n$\\sqrt{3}\\times2\\pi\/L$ are denoted by (black) circles.\n\n\\begin{figure}\n\\begin{center}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.70$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{aEthetasq}[b][b][1][0]{$(aE_{\\pi\/\\rho})^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/disprel_PS_V_heavy.eps}\n \\end{minipage}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.57$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{aEthetasq}[b][b][1][0]{$(aE_{\\pi\/\\rho})^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/disprel_PS_V_mediu.eps}\n \\end{minipage}\\\\[4ex]\n \\begin{minipage}{.495\\linewidth}\n  \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.70$}\n   \\psfrag{relativeerror}[b][b][1][0]{$\\delta_{E_{\\pi}}$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/errors_heavy.eps}\n \\end{minipage}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.57$}\n   \\psfrag{relativeerror}[b][b][1][0]{$\\delta_{E_{\\pi}}$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/errors_mediu.eps}\n \\end{minipage}\\\\[4ex]\n\\end{center}\n\\caption{The plots in the first line illustrate the results for the\n  dispersion relation for the $\\pi$ and the $\\rho$ (empty and full\n  symbols respectively) for the two choices of the quark mass. In the\n  second line we show the corresponding relative error as a function\n  of the momentum.} \\label{figE}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.70$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{aEthetasq}[b][b][1][0]{$(aE_{\\pi\/\\rho})^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/disprel_PS_V_heavy_zoom.eps}\n \\end{minipage}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.57$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{aEthetasq}[b][b][1][0]{$(aE_{\\pi\/\\rho})^2$}\n  \\epsfig{scale=.275,angle=-90,file=plots\/disprel_PS_V_mediu_zoom.eps}\n \\end{minipage}\\\\[4ex]\n\\end{center}\n\\caption{Magnified view of the dispersion relation of fig.~\\ref{figE}\n  in the interval $|\\vec{p}_{\\rm lat}|\\in [0,2\\pi]$.}\n\\label{figEzoom}\n\\end{figure}\n\n\\begin{table}\n\\begin{small}\n\\begin{center}\n\\begin{tabular}{l@{\\hspace{0.5em}}cccccc}\n\\hline\n\\hline\\\\[-1.5ex]\n  &\\multicolumn{2}{c}{$\\kappa = 0.13500$}\n  &\\multicolumn{2}{c}{$\\kappa = 0.13550$}\\\\[0.5ex]\n  &$\\pi$ &$\\rho$ &$\\pi$ &$\\rho$ \\\\[0.5ex]\n\\hline\\\\[-1.5ex]\n$\\chi^2\/\\mathrm{d.o.f}|_\\mathrm{(\\ref{expecteddisprel2})}$\n& 0.3& 1.0& 0.6& 1.7\\\\\n$\\chi^2\/\\mathrm{d.o.f}|_\\mathrm{(\\ref{expecteddisprel1})}$\n& 1.8& 2.5& 0.9& 2.1\\\\[0.5ex]\n$\\Delta^2$ from~(\\ref{expecteddisprel2}) &0.0040(1) &0.0042(2)\n&0.0040(1)\n&0.0048(4) \\\\[0.5ex]\n\\hline \\hline\n\\end{tabular}\n\\caption{$\\chi^2$\/d.o.f. for the lattice data with respect the\n  expectations eqs.~(\\ref{expecteddisprel2})\n  and~(\\ref{expecteddisprel1}) with $\\Delta^2=a^2\/L^2=0.0039$ (first\n  two rows) and the results obtained from a fit\n  to~(\\ref{expecteddisprel2}) with $\\Delta^2$ left as a parameter of\n  the fit (third row).}\n\\label{tab_slopes}\n\\end{center}\n\\end{small}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.70$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{Fbare}[b][b][1][0]{$af_{\\pi\/\\rho}$}\n  \\epsfig{scale=.285,angle=-90,file=plots\/decay_constants_heavy.eps}\n \\end{minipage}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.57$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{Fbare}[b][b][1][0]{$af$}\n   \\psfrag{Fbare}[b][b][1][0]{$af_{\\pi\/\\rho}$}\n   \\epsfig{scale=.285,angle=-90,file=plots\/decay_constants_mediu.eps}\n \\end{minipage}\\\\[4ex]\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.70$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{ZP}[c][b][1][0]{$Z_P$}\n  \\epsfig{scale=.285,angle=-90,file=plots\/Z_P_heavy.eps}\n \\end{minipage}\n \\begin{minipage}{.495\\linewidth}\n   \\psfrag{kappa}[c][t][1][0]{$m_\\pi\/m_\\rho=0.57$}\n   \\psfrag{pLsq}[t][c][1][0]{$(\\vec{p}L)^2$}\n   \\psfrag{ZP}[c][b][1][0]{$Z_P$}\n  \\epsfig{scale=.285,angle=-90,file=plots\/Z_P_mediu.eps}\n \\end{minipage}\\\\[4ex]\n\\end{center}\n\\caption{The first line shows the results for the $\\pi$ and $\\rho$\n  decay constant (empty and full symbols respectively) and the second\n  line shows the matrix element~(\\ref{Z_P}) for the two choices of the\n  quark mass. In each plot the horizontal lines represent the central\n  value at $\\vec p_\\mathrm{lat} = \\vec\\theta_1 = \\vec\\theta_2 = 0$.}\n\\label{fig_ZP}\n\\end{figure}\n\nFor the discussion of our results it is convenient to rewrite the\ndispersion relation in eq.\\,(\\ref{eq:disprel}) in the form\n\\begin{equation}\n(aE_{\\pi\/\\rho})^2= (am_{\\pi\/\\rho})^2 + \\Delta^2(\\vec{p}\\, L)^2\n\\label{expecteddisprel2}\n\\end{equation}\nwhere $\\Delta^2=(a\/L)^2 = 0.0039$. The dispersion\nrelation~(\\ref{expecteddisprel2}) is displayed as the dashed line in\nthe plots of fig.~\\ref{figE}. In the first row of\ntable~\\ref{tab_slopes} we present the $\\chi^2\/$d.o.f of the comparison\nof our data to eq.~(\\ref{expecteddisprel2}) over the range\n$0\\le|\\vec{p}\\,|^2L^2\\le(2\\pi)^2$ using the values of the meson masses\nobtained from fits at zero momenta. In the third row of the table we\npresent the values of $\\Delta^2$ obtained by fitting the lattice data\nto the functional form in eq.~(\\ref{expecteddisprel2}) over the same\nrange in momentum, but allowing $\\Delta^2$ to be a parameter of the\nfit. \nWe note that our values for the ratios $m_\\pi\/m_\\rho$ agree with those\nfound earlier on the same configurations in~\\cite{Allton:2001sk}.\n\nAs the momentum of the meson grows so do the expected discretization\neffects in the dispersion relation. For example, a free scalar\nparticle with a wave-function $\\phi(x)$ satisfying the\n(Minkowski-Space) Klein-Gordon equation $(\\square+m^2)\\phi(x)=0$, with\na generic discretized second derivative defined by\n$\\partial^2f(x)\/\\partial x^2 = (f(x+a)+f(x-a)-2f(x))\/a^2$, satisfies\nthe following lattice dispersion relation:\n\\begin{equation}\n\\sinh^2\\bigg({aE_{\\pi\/\\rho}\\over2}\\bigg) =\n    \\sinh^2\\bigg({am\\over 2}\\bigg)\n  + \\sum_i\\sin^2\\bigg(\\Delta{p_{i}L\\over 2}\\bigg).\n  \\label{expecteddisprel1}\n\\end{equation}\nTo indicate the possible size of lattice artefacts we plot the\ndispersion relation of eq.~(\\ref{expecteddisprel1}) as the solid curve\nin fig.~\\ref{figE}. We stress, however, that in an interacting theory\nthe discretization errors will in general be different from those in\neq.~(\\ref{expecteddisprel1}). Indeed there seems to be no evidence\nfrom our data that eq.~(\\ref{expecteddisprel1}) is a particularly good\nrepresentation of the lattice artefacts (see the second row of\ntable~\\ref{tab_slopes}, where we show the $\\chi^2\/$d.o.f. from a\ncomparison of our data with~(\\ref{expecteddisprel1})). At small\nmomenta the solid curve merges of course with the dashed line\nrepresenting the continuum dispersion relation.\n\nWe conclude from the results for the dispersion relation plotted in\nfigs.~\\ref{figE} and~\\ref{figEzoom} that the use of partially twisted\nboundary conditions is beautifully consistent with expectations,\nparticularly at low momenta where the lattice artefacts are\nsmall\\footnote{The different data points at $|\\vec p L|=0$ and\n  $2\\pi$ correspond to $\\vec\\theta_1=\\vec\\theta_2$ but with different\n  choices of $\\vec\\theta_1$ and $\\vec\\theta_2$.}. An important further\nobservation is that there is no evidence in our data that the\nintroduction of twisted boundary conditions increases significantly\nthe statistical or systematic uncertainties. This is illustrated in\nthe second line of figure~\\ref{figE}, which shows the relative error\nin the pion energy\n\\begin{equation}\n\\delta_{E_{\\pi}}\\equiv{\\delta E_{\\pi}\\over E_{\\pi}}\n\\end{equation}\nas $|\\vec{p}\\, L|$ is varied. The error $\\delta E_{\\pi}$ is the\njackknife error, including the statistical error and the systematic\nuncertainty stemming from the improvement constants in the improved\nquark currents. The plot shows that the errors increase smoothly as\nthe momentum increases, and that no appreciable additional noise is\nintroduced by partial twisting\n\\footnote{For $\\kappa=0.13550$ at $|\\vec pL|=\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L}$ we observe a fluctuation in the effective mass from one of the gauge configurations and with our choice of the position of the source (this fluctuation has also been observed by our colleagues in the UKQCD collaboration \\cite{pricomMcNeile}). The fluctuation is particularly noticeable when twisting both quarks by  $\\vec\\theta =\n(3,3,3)$ and this is the reason for the larger jackknife error at this particular momentum and twist combination (see figure~\\ref{figE}).}.\nWe observe the same behaviour for all analyzed quantities.\n\n\nIn fig.~\\ref{fig_ZP} we plot our results for the decay constants\n$f_\\pi$ and $f_\\rho$ and for $Z_P$.\nThe values for $af_\\pi$ agree\nwith the ones obtained in \\cite{Allton:2004qq} at \n$|\\vec{p}\\, L|=0$. Again we see that the \nresults are completely consistent with theoretical expectations, being\nindependent of the twisting angles and Fourier momenta.\n\n\n\n\\section{Conclusions}\\label{sec:concs}\n\\vspace{-.2cm}\n\nWe have investigated the use of partially twisted boundary conditions\nin evaluating the energies of pseudo scalar and vector mesons and their\nleptonic decay constants. The results are very encouraging; it does\nappear that the method allows the evaluation of physical quantities\nwith any momentum. Moreover the use of these boundary conditions does\nnot appear to increase the errors in any appreciable way. It will be\nimportant to monitor whether this continues to be true as the quark\nmasses are decreased. Once the quark masses are such that two-pion\nintermediate states contribute significantly to the $\\rho$-meson's\ncorrelation function the finite-volume effects will no longer fall\nexponentially with the volume, but only as powers. For the pion\nobservables studied in this paper this is not the case.\n\nPartially twisted boundary conditions will be particularly useful for\nevaluating momentum-dependent physical quantities. One important\napplication is to the determination of the form-factors of\nsemileptonic weak decays of heavy ($D$ and $B$) mesons to light\nmesons. For these processes, with conventional periodic boundary\nconditions, the initial and final state hadrons are restricted to have\nmomenta $(\\mathchoice{{2\\pi\\over L}}{2\\pi\/L}{2\\pi\/L}{2\\pi\/L})\\vec{n}$ where $\\vec{n}$ is a vector of integers. In\norder to avoid lattice artefacts the possible values of $|\\vec{n}|$\nare frequently limited to $0$, $1$, and perhaps $\\sqrt{2}$. Thus, for\nany particular choice of quark masses, the number of values of the\nmomentum transfer, $q^2$, or the light meson energy, $E$, is also very\nlimited. Moreover, chiral extrapolations are conveniently performed at\nfixed $q^2$~\\cite{Bowler:1999xn,Abada:2000ty,Bowler:2004zb} or fixed\n$E$~\\cite{Aoki:2001rd,Aubin:2004ej,Shigemitsu:2004ft,Okamoto:2004xg}\n(and heavy quark extrapolations at fixed $E$), but $q^2$ and $E$ vary\nwith both the momentum and quark masses. Ans\\\"atze for the form\nfactors, such as the Becirevic-Kaidalov~\\cite{Becirevic:1999kt} model,\nare used to interpolate and extrapolate simulation data to sets of\ncommon $q^2$ or $E$ values before the extrapolations are performed.\nUsing twisted boundary conditions would enable the form factors to be\nevaluated directly at these common values, removing the need for the\nintermediate form-factor fit.\n\nFor some other physical quantities, such as the moments of hadronic\ndeep inelastic structure functions or light-cone distribution\namplitudes, it may be helpful to use twisted boundary conditions even\nthough it is not strictly necessary. The corresponding matrix elements\nare proportional to factors of $p_i$, where $\\vec{p}$ is the momentum\nof the hadron, so that the correlation functions must be computed with\n$\\vec{p}\\neq 0$. The use of twisted boundary conditions allows\n$|\\vec{p}\\,|$ to be decreased and hence the lattice artefacts to be\nreduced. Moreover by varying $\\vec{p}$ one can verify that the leading\ntwist component has been extracted correctly.\n\nFollowing the successful conclusion of this exploratory numerical\nstudy of the implementation of partially twisted boundary conditions\nwe now look forward to applying them in lattice computations of a wide\nvariety of phenomenologically important quantities.\n\n\n{\\bf Acknowledgements}\nWe warmly thank Daragh Byrne, Steve Downing and Craig McNeile for\ntheir support with QCDgrid and Massimo di Pierro for help using FermiQCD. \nWe thank the Iridis parallel\ncomputing team at the University of Southampton, in particular Oz\nParchment and Ivan Wolton, for their assistance. We also acknowledge\nAlan Irving, Craig McNeile and Chris Michael for correspondence on the\ngauge field configurations. \nThis work was\nsupported by PPARC grants PPA\/G\/S\/2002\/00467 and\nPPA\/G\/O\/2002\/00468.\n\n\\bibliographystyle{elsevier}\n\\vspace{-.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is well known that the positive part  of the integral form of quantum enveloping algebras of finite type was realized as a Ringel--Hall algebra (see \\cite{R90,R932}).  Using a beautiful geometric construction of $q$-Schur algebras, the entire quantum $\\frak{gl}_n$ over the rational function field $\\mathbb Q(v)$ (with $v$ being an indeterminant) was realized by A. A. Beilinson, G. Lusztig and R. MacPherson (BLM) in \\cite{BLM}.\n\n\nLet $U(n)$ be the Lusztig ${\\mathcal Z}$-form of quantum $\\frak{gl}_n$, where ${\\mathcal Z}=\\mathbb Z[v,v^{-1}]$. We will give BLM realization of $U(n)$ in this paper. More precisely, We will construct a certain ${\\mathcal Z}$-submodule of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$, denoted by ${\\mathcal V}(n)$, where ${\\boldsymbol{\\mathcal S}}(n,r)$ is the $q$-Schur algebra over $\\mathbb Q(v)$. We will show that ${\\mathcal V}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ and prove in \\ref{realization} that ${\\mathcal V}(n)$ is isomorphic to $U(n)$ as a ${\\mathcal Z}$-algebra. Similarly, we may construct the affine version of ${\\mathcal V}(n)$, denoted by ${\\mathcal V}_{\\!\\vartriangle\\!}(n)$, which is a certain ${\\mathcal Z}$-submodule of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$, where ${\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$ is the affine $q$-Schur algebra over $\\mathbb Q(v)$. We conjecture that ${\\mathcal V}_{\\!\\vartriangle\\!}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$. If this conjecture is true, then ${\\mathcal V}_{\\!\\vartriangle\\!}(n)$ is isomorphic to\nthe ${\\mathcal Z}$-module $\\widetilde{{\\mathfrak D}}_{\\!\\vartriangle\\!}(n)$ defined in \\cite[(3.8.1.1)]{DDF}.\n\nLet $k$ be a field containing\nan $l$-th primitive root $\\varepsilon$ of $1$ with $l\\geq 1$ odd.\nSpecializing $v$ to $\\varepsilon$, $k$ will be viewed as a ${\\mathcal Z}$-module.\nLet $U_k(n)=U(n)\\otimes_{\\mathcal Z} k$ and $\\ol{U_k(n)}=U_k(n)\/\\langle K_i^l-1\\mid 1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n-1\\rangle.$\nWe will prove that the algebra $\\ol{U_k(n)}$ can be realized as a $k$-subalgebra of\n$\\prod_{r\\geq 0}{\\mathcal S}_k(n,r)$, where ${\\mathcal S}_k(n,r)$ is the $q$-Schur algebra over $k$.\n\nWe organize this paper as follows. We recall some results of quantum $\\frak{gl}_n$ and $q$-Schur algebras in \\S2. We will establish some useful\nmultiplication formulas for $q$-Schur algebras\nin \\ref{formula 1} and \\ref{formula 2}. A certain ${\\mathcal Z}$-submodule of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$, denoted by ${\\mathcal V}(n)$, will be constructed in \\S4. We will use \\ref{formula 1} and \\ref{formula 2} to prove that ${\\mathcal V}(n)$ is BLM realization of $U(n)$. Furthermore, we will give realization of $\\ol{U_k(n)}$ in \\ref{realization of barUnk}.\n\n\nThroughout this paper, let ${\\mathcal Z}=\\mathbb Z[v,v^{-1}]$,  where $v$ is an\nindeterminate, and let $\\mathbb Q(v)$ be the fraction field of ${\\mathcal Z}$.\nFor $i\\in\\mathbb Z$ let $[i]=\\frac{v^i-v^{-i}}{v-v^{-1}}$ and $[\\![i]\\!]=\\frac{v^{2i}-1}{v^2-1}$.\nFor integers $N,t$ with $t\\geq 0$, let\n\\begin{equation*}\n\\left[{N\\atop t}\\right]=\\frac{[N][N-1]\\cdots[N-t+1]}{[t]^!}\\in{\\mathcal Z},\\quad\n\\left[\\!\\!\\left[{N\\atop t}\\right]\\!\\!\\right]=\\frac{[\\![N]\\!][\\![N-1]\\!]\\cdots[\\![N-t+1]\\!]}{[\\![t]\\!]^!}\\in{\\mathcal Z}\n\\end{equation*}\nwhere $[t]^{!}=[1][2]\\cdots[t]$ and $[\\![t]\\!]^{!}=[\\![1]\\!][\\![2]\\!]\\cdots[\\![t]\\!]$.\nFor $\\mu\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$ let $\\big[{\\mu\\atop{\\lambda}}\\big]=\\big[{\\mu_1\\atop{\\lambda}_1}\\big]\\cdots\\big[{\\mu_n\\atop{\\lambda}_n}\\big].$\n\n\\section{The quantum $\\frak{gl}_n$ and the $q$-Schur algebra}\n\nThe below definition of quantum $\\frak{gl}_n$ is a slightly\nmodified version of Jimbo \\cite{Ji}; see \\cite{GL,Ta}.\n\\begin{Def}\\label{definition of U(infty)}\nThe quantum enveloping algebra of ${\\frak {gl}}_n$ is the $\\mathbb Q(v)$-algebra ${\\mathbf U}(n)$\npresented by generators\n$$E_i,\\ F_i\\quad(1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n-1),\\  K_j,\\  K_j^{-1}\\quad(1\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant n)$$\nand relations\n\n$(a)\\ K_{i}K_{j}=K_{j}K_{i},\\ K_{i}K_{i}^{-1}=1;$\n\n$(b)\\ K_{i}E_j=v^{\\delta_{i,j}-\\delta_{i,j+1}} E_jK_{i};$\n\n$(c)\\ K_{i}F_j=v^{\\delta_{i,j+1}-\\delta_{i,j}} F_jK_i;$\n\n$(d)\\ E_iE_j=E_jE_i,\\ F_iF_j=F_jF_i\\ when\\ |i-j|>1;$\n\n$(e)\\ E_iF_j-F_jE_i=\\delta_{i,j}\\frac {\\widetilde\nK_{i}-\\widetilde K_{i}^{-1}}{v-v^{-1}},\\ where \\\n\\widetilde K_i =K_{i}K_{i+1}^{-1};$\n\n$(f)\\\nE_i^2E_j-(v+v^{-1})E_iE_jE_i+E_jE_i^2=0\\\n when\\ |i-j|=1;$\n\n$(g)\\\nF_i^2F_j-(v+v^{-1})F_iF_jF_i+F_jF_i^2=0\\\n when\\ |i-j|=1.$\n\\end{Def}\n\n\n\nFollowing \\cite{Lu90}, let $U(n)$ be the ${\\mathcal Z}$-subalgebra of ${\\mathbf U}(n) $\ngenerated by all $E_i^{(m)}$, $F_i^{(m)}$, $K_i^{\\pm 1}$ and\n$\\big[ {K_i;0 \\atop t} \\big]$, where for $m,t\\in\\mathbb N$,\n$$E_i^{(m)}=\\frac{E_i^m}{[m]^!},\\,\\,F_i^{(m)}=\\frac{F_i^m}{[m]^!},\\text{ and }\n\\bigg[ {K_i;0 \\atop t} \\bigg] =\n\\prod_{s=1}^t \\frac\n{K_iv^{-s+1}-K_i^{-1}v^{s-1}}{v^s-v^{-s}}.$$\n\n\n\nLet $\\Th(n)$ be the set of all $n\\times n$ matrices over\n$\\mathbb N$. Let $\\Th^\\pm(n)$ be the set of all $A\\in\\Th(n)$ whose diagonal entries are zero. Let $\\Th^+(n)$ (resp., $\\Th^-(n)$) be the subset of $\\Th(n)$ consisting of those matrices $A$ with $a_{i,j}=0$ for all $i>j$ (resp., $i<j$). For $A\\in\\Th^\\pm(n)$, write $A=A^++A^-$ with\n$A^+\\in\\Th^+(n)$ and $A^-\\in\\Th^-(n)$.\nFor $A\\in\\Th^\\pm(n)$ let\n\\begin{equation}\\label{order}\nE^{(A^+)}=\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant h<j\\leqslant}\\def\\geq{\\geqslant n}E_h^{(a_{i,j})}\\ \\text{and}\n\\ F^{(A^-)}=\\prod_{1\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant h<i\\leqslant}\\def\\geq{\\geqslant\nn}F_h^{(a_{i,j})}\\end{equation}\n The orders in which the products $E^{(A^+)}$ and $F^{(A^-)}$ are\ntaken are defined as follows. Put\n$$M_j=M_j(A^+)=E_{j-1}^{(a_{j-1,j})}(E_{j-2}^{(a_{j-2,j})}E_{j-1}^{(a_{j-2,j})})\n\\cdots(E_{1}^{(a_{1,j})}E_{2}^{(a_{1,j})}\\cdots\nE_{j-1}^{(a_{1,j})}).$$\n Similarly, put\n$$M_j'=(F_{j-1}^{(a_{j,1})}\\cdots\nF_{2}^{(a_{j,1})}F_{1}^{(a_{j,1})})\n\\cdots(F_{j-1}^{(a_{j,j-2})}F_{j-2}^{(a_{j,j-2})})\nF_{j-1}^{(a_{j,j-1})}.$$\nThen $E^{(A^+)}=M_nM_{n-1}\\cdots M_2$ and\n$F^{(A^-)}=M_2'M_3'\\cdots M_n'$.\nAccording to \\cite[4.5]{Lu90} and \\cite[7.8]{Lu901} we have the following result.\n\\begin{Prop}\\label{basis for Un}\nThe set $$\\{E^{(A^+)}\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}K_i^{\\delta_i}\\left[{K_i;0\\atop{\\lambda}_i}\\right] F^{(A^-)}\\mid A\\in\\Th^\\pm(n),\\,\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\,\\forall i\\}$$ forms a ${\\mathcal Z}$-basis of $U(n)$.\n\\end{Prop}\n\n\nSchur algebras are certain important finite-dimensional algebras. It is used to link representation of  general linear groups and symmetric groups. $q$-Schur algebras are quantum deformation of Schur algebras, which is defined by certain endomorphism algebras arising from Hecke algebras of type $A$. We now follow \\cite{DJ89,DJ91} to recall the definition of $q$-Schur algebras as follows.\nLet ${\\frak S}_r$ be the symmetric group on $r$ letters. The symmetric group ${\\frak S}_r$ is generated by the set $\\{s_i:=(i,i+1)\\mid 1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant r-1\\}$.\nThe  Hecke algebra ${{\\mathcal H}(r)}$ associated with ${\\frak S}_r$  is  the ${\\mathcal Z}$-algebra generated by\n$T_i$ ($1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant r-1$),\nwith the following relations:\n$$(T_i+1)(T_i-q)=0,\\;\\;\nT_iT_{i+1}T_i=T_{i+1}T_iT_{i+1},\\;\\;T_iT_j=T_jT_i\\;(|i-j|>1).\n$$\nwhere $q=v^2$. If $w=s_{i_1}s_{i_2}\\cdots s_{i_m}$ is reduced let $T_w=T_{i_1}T_{i_2}\\cdots T_{i_m}$. Then the set $\\{T_w\\mid w\\in{\\frak S}_r\\}$ forms a ${\\mathcal Z}$-basis for ${{\\mathcal H}(r)}$.\nLet $\\Lambda(n,r)=\\{{\\lambda}\\in\\mathbb N^{n}\\mid\\sigma({\\lambda}):=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}{\\lambda}_i=r\\}$.\n For ${\\lambda}\\in\\Lambda(n,r)$, let ${\\frak S}_{{\\lambda}}$ be the\nYoung subgroup of ${\\frak S}_r$ and let $x_{{\\lambda}}=\\sum_{w\\in\\frak\nS_{{\\lambda}}}T_w$. Let ${\\boldsymbol{\\mathcal H}}(r)={{\\mathcal H}(r)}\\otimes_{\\mathcal Z}\\mathbb Q(v)$. The endomorphism algebras\n$${\\mathcal S}(n,r):=\\operatorname{End}_{{{\\mathcal H}(r)}}\\bigg(\n\\bigoplus_{{\\lambda}\\in\\Lambda(n,r)}x_{{\\lambda}}{{\\mathcal H}(r)}\\bigg),\\quad {\\boldsymbol{\\mathcal S}}(n,r):=\\operatorname{End}_{{\\boldsymbol{\\mathcal H}}(r)}\\bigg(\n\\bigoplus_{{\\lambda}\\in\\Lambda(n,r)}x_{{\\lambda}}{\\boldsymbol{\\mathcal H}}(r)\\bigg)$$ are called\n$q$-Schur algebras.\nFor ${\\lambda},\\mu\\in\\Lambda(n,r)$ let $\\mathscr D_{{\\lambda},\\mu}$ be the set of\ndistinguished double $(\\frak S_{\\lambda},\\frak S_\\mu)$-coset\nrepresentatives.  For\n${\\lambda},\\mu\\in\\Lambda(\\eta,r)$, $d\\in\\mathscr D_{{\\lambda},\\mu}$, define\n$\\phi_{{\\lambda}\\mu}^d\\in{\\mathcal S}(n,r)$ by\n\\begin{equation*}\n\\phi_{{\\lambda}\\mu}^d(x_\\nu h)=\\delta_{\\mu,\\nu}\\sum_{x\\in{\\frak S}_{\\lambda}\nd{\\frak S}_\\mu}T_xh.\n\\end{equation*} According to \\cite[1.4]{DJ91}, the set $\\{\\phi_{{\\lambda}\\mu}^d\\mid {\\lambda},\\mu\\in\\Lambda(n,r),\\,d\\in\\mathscr D_{{\\lambda},\\mu}\\}$ forms a ${\\mathcal Z}$-basis for ${\\mathcal S}(n,r)$.\n\nLet\n$\\Thnr=\\big\\{A\\in\\Th(n)\\,\\big|\\,\\sigma(A):=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i,j\\leqslant}\\def\\geq{\\geqslant n}a_{i,j}=r\\big\\}.$\nThe basis for ${\\mathcal S}(n,r)$ can also be indexed by the set $\\Thnr$, which we now describe.\nFor $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$, and ${\\lambda}\\in\\Lambda(n,r)$ let\n\\begin{equation*}\nR_{i}^{{\\lambda}}=\\bigg\\{\\sum_{1\\leqslant}\\def\\geq{\\geqslant t\\leqslant}\\def\\geq{\\geqslant i-1}{\\lambda}_t+1,\\sum_{1\\leqslant}\\def\\geq{\\geqslant t\\leqslant}\\def\\geq{\\geqslant i-1}{\\lambda}_t+2,\\ldots,\\sum_{1\\leqslant}\\def\\geq{\\geqslant t\\leqslant}\\def\\geq{\\geqslant i-1}{\\lambda}_t+{\\lambda}_i\\bigg\\},\n\\end{equation*}\nAccording to \\cite[1.3.10]{JK}, there is\na bijective map\n\\begin{equation*}\n\\jmath:\\{({\\lambda}, d,\\mu)\\mid\nd\\in\\mathscr D_{{\\lambda},\\mu},{\\lambda},\\mu\\in\\Lambda(n,r)\\}\\longrightarrow\\Thnr\n\\end{equation*}\nsending $({\\lambda}, d,\\mu)$ to $A=(a_{k,l})$, where $a_{k,l}=|R_k^{\\lambda}\\cap dR_l^\\mu|$ for all $k,l\\in\\mathbb Z$.\nIf ${\\lambda},\\mu\\in\\Lambda(n,r)$ and $d\\in\\mathscr D_{{\\lambda},\\mu}$ are such that $A=\n\\jmath({\\lambda},d,\\mu)$, let\n\\begin{equation*}\n[A]=v^{-d_A}\\phi_{{\\lambda},\\mu}^d,\\quad\\text{ where } \\quad\nd_{A}=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n\\atop i\\geq k,j<l}a_{i,j}a_{k,l}.\n\\end{equation*}\nThen the set $\\{[A]\\mid A\\in\\Thnr\\}$ forms a ${\\mathcal Z}$-basis for ${\\mathcal S}(n,r)$.\n\nThe geometric definition of $q$-Schur algebra was given in \\cite[1.2]{BLM}.\nIt is proved in \\cite[A.1]{Du92} that the two definitions of $q$-Schur algebras are equivalent.\nAccording to  \\cite[1.2,1.3]{BLM}, for $\\lambda\\in\\Lambda(n,r)$ and $A\\in\\Thnr$, we have\n\\begin{equation}\\label{[diag(la)][A]}\n\\begin{aligned}\n\\ [\\operatorname{diag}({\\lambda})][A]=\n\\begin{cases}[A] & \\text{if}\\ \\lambda=ro(A)\\\\\n0 & \\text{otherwise;}\n\\end{cases}\n\\end{aligned} \\quad \\text{and}\\\n\\begin{aligned}\n\\ [A][\\operatorname{diag}({\\lambda})]=\n\\begin{cases}[A] & \\text{if}\\ \\lambda=co(A)\\\\\n0 & \\text{otherwise,}\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nwhere $\\text{\\rm ro}(A)=(\\sum_ja_{1,j},\\cdots,\\sum_ja_{n,j})$ and\n$\\text{\\rm co}(A)=(\\sum_ia_{i,1},\\cdots,\\sum_ia_{i,n})$ are the sequences of\nrow and column sums of $A$.\n\n\nThe algebra ${\\mathbf U}(n)$ and the $q$-Schur algebra ${\\boldsymbol{\\mathcal S}}(n,r)$ are\nrelated by an algebra epimorphism $\\zeta_r$ which we now\ndescribe. For $A\\in\\Th^\\pm(n)$, $\\delta\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$\nlet\n\\begin{equation*}\n\\begin{split}\nA(\\delta,{\\lambda},r)&=\\sum_{\\mu\\in\\Lambda(n,r-\\sigma(A))}v^{\\mu\\centerdot\\delta}\n\\left[{\\mu\\atop{\\lambda}}\\right][A+\\operatorname{diag}(\\mu)]\\in{\\boldsymbol{\\mathcal S}}(n,r);\\\\\nA(\\delta,r)&=\\sum_{\\mu\\in\\Lambda(n,r-\\sigma(A))}v^{\\mu\\centerdot\\delta}\n[A+\\operatorname{diag}(\\mu)]\\in{\\boldsymbol{\\mathcal S}}(n,r),\n\\end{split}\n\\end{equation*}\nwhere $\\mu\\centerdot\\delta=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}\\mu_i\\delta_i$.\nFurthermore we set\n\\begin{equation*}\n\\begin{split}\nA(\\delta,{\\lambda})&=(A(\\delta,{\\lambda},r))_{r\\geq 0}\\in\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r);\\\\\nA(\\delta)&=(A(\\delta,r))_{r\\geq 0}\\in\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r).\n\\end{split}\n\\end{equation*}\nThen by definition we have $A(\\delta)=A(\\delta,{\\mathbf{0}})$, where\n${\\mathbf{0}}=(0,\\cdots,0)\\in\\mathbb N^{n}$.\nFor $1\\leqslant}\\def\\geq{\\geqslant\ni,j\\leqslant}\\def\\geq{\\geqslant n$, let $E_{i,j}\\in\\Th(n)$ be the matrix $(a_{k,l})$ with\n$a_{k,l}=\\delta_{i,k}\\delta_{j,l}$.\nAccording to \\cite{BLM}, there is\nan algebra epimorphism $$\\zeta_r:{\\mathbf U}(n)\n\\twoheadrightarrow{\\boldsymbol{\\mathcal S}}(n,r)$$ satisfying\n$\\zeta_r(E_h)=E_{h,h+1}(\\mathbf 0,r)$, $\\zeta_r(K_1^{j_1}K_2^{j_2}\\cdots\nK_n^{j_n})=0(\\mathbf j,r)$ and $\\zeta_r(F_h)=E_{h+1,h}(\\mathbf\n0,r)$, for $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$ and ${\\mathbf{j}}\\in\\mathbb Z^{n}$.\n\nWe conclude this section by recalling an important triangular relation in $q$-Schur algebras.\nFor $A=(a_{s,t})\\in\\Th(n)$ and $i<j$, let\n$\\sigma_{i,j}(A)=\\sum_{s\\leqslant}\\def\\geq{\\geqslant i;t\\geq j}a_{s,t}$ and\n$\\sigma_{j,i}(A)=\\sum_{s\\leqslant}\\def\\geq{\\geqslant i;t\\geq j}a_{t,s}$. Define $A'\\preccurlyeq A$ iff\n$\\sigma_{i,j}(A')\\leqslant}\\def\\geq{\\geqslant\\sigma_{i,j}(A)$ and $\\sigma_{j,i}(A')\\leqslant}\\def\\geq{\\geqslant\\sigma_{j,i}(A)$\nfor all $1\\leqslant}\\def\\geq{\\geqslant i<j\\leqslant}\\def\\geq{\\geqslant n$. Put $A'\\prec A$ if $A'\\preccurlyeq A$ and, for some\npair $(i,j)$ with $i\\not=j$, $\\sigma_{i,j}(A')<\\sigma_{i,j}(A)$.\nAccording to \\cite[5.3 and 5.4(c)]{BLM}, we have the following result.\n\\begin{Prop}\\label{tri}\nFor $A\\in\\Th^\\pm(n)$, we have\n$$\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant h<j\\leqslant}\\def\\geq{\\geqslant n}(a_{i,j}E_{h,h+1})({\\mathbf{0}})\n\\cdot\\prod_{1\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant h<i\\leqslant}\\def\\geq{\\geqslant\n n}(a_{i,j}E_{h+1,h})({\\mathbf{0}})=A({\\mathbf{0}})+f$$\nwhere the ordering of the products in the left hand side of the above equation is the same as in \\eqref{order} and\n$f$ is the $\\mathbb Q(v)$-linear combination of $B({\\mathbf{j}})$ with $B\\in\\Th^\\pm(n)$, $B\\prec A$ and ${\\mathbf{j}}\\in\\mathbb Z^{n}$.\n\\end{Prop}\n\n\\section{The multiplication formulas for $q$-Schur algebras}\n\nWe will derive certain useful multiplication formulas for $q$-Schur algebras in \\ref{formula 1} and \\ref{formula 2}.\n\nWe need  some preparation before proving \\ref{formula 1} and \\ref{formula 2}.\nLet $\\bar\\ :{\\mathcal Z}\\rightarrow{\\mathcal Z}$ be the ring homomorphism defined by $\\bar v=v^{-1}$.\nThe following impotent multiplication formulas for  $q$-Schur algebras was proved in \\cite[3.4]{BLM}.\n\\begin{Prop}\\label{BLM formulas}\nLet $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$, $A\\in\\Thnr$ and ${\\lambda}=\\text{\\rm ro}(A)$.   Let $B_m=\\operatorname{diag}({\\lambda})+m E_{h,h+1}-m E_{h+1,h+1}$ and\n$C_m=\\operatorname{diag}({\\lambda})-m E_{h,h}+m E_{h+1,h}.$ Then in ${\\mathcal S}(n,r)$\n\n$(1)\\ \\displaystyle\n[B_m]\\cdot[A]=\\sum_{{\\mathbf{t}}\\in\\Lambda(n,m)\\atop \\forall\nu\\in\\mathbb Z,t_u\\leqslant}\\def\\geq{\\geqslant\na_{h+1,u}}v^{\\beta({\\mathbf{t}},A)}\\prod_{u\\in\\mathbb Z}\\overline{\\left[\\!\\!\\left[{a_{h,u}+t_u\\atop\nt_u}\\right]\\!\\!\\right]} \\biggl[\nA+\\sum_{u\\in\\mathbb Z}t_u(E^\\vartriangle_{h,u}-E^\\vartriangle_{h+1,u})\\biggr];$\\\\\nfor all $0\\leqslant}\\def\\ge{\\geqslant m\\leqslant}\\def\\ge{\\geqslant{\\lambda}_{h+1}$, where $\\beta({\\mathbf{t}},A)=\\sum_{j\\geq\nu}a_{h,j}t_u-\\sum_{j>u}a_{h+1,j}t_u+\\sum_{u<u'}t_ut_{u'}$.\n\n$(2)\\ \\displaystyle\n[C_m]\\cdot[A]=\\sum_{{\\mathbf{t}}\\in\\Lambda(n,m)\\atop \\forall\nu\\in\\mathbb Z,t_u\\leqslant}\\def\\geq{\\geqslant\na_{h,u}}v^{{\\gamma}({\\mathbf{t}},A)}\\prod_{u\\in\\mathbb Z}\\overline{\\left[\\!\\!\\left[{a_{h+1,u}+t_u\\atop\nt_u}\\right]\\!\\!\\right]} \\biggl[\nA-\\sum_{u\\in\\mathbb Z}t_u(E^\\vartriangle_{h,u}-E^\\vartriangle_{h+1,u})\\biggr],$\\\\\nfor all $0\\leqslant}\\def\\ge{\\geqslant m\\leqslant}\\def\\ge{\\geqslant{\\lambda}_{h}$, where ${\\gamma}({\\mathbf{t}},A)=\\sum_{j\\leqslant}\\def\\geq{\\geqslant\nu}a_{h+1,j}t_u-\\sum_{j<u}a_{h,j}t_u+\\sum_{u<u'}t_ut_{u'}$.\n\\end{Prop}\n\n\n\nWe also need the following formulas for Gaussian binomial coefficient (see \\cite{Liu}).\n\\begin{Lem}\\label{binomial}\nFor $m,n\\in\\mathbb Z$, $a,b\\in\\mathbb N$ we have\n\n$(1)$ $\\big[\\!\\!\\big[{n\\atop a}\\big]\\!\\!\\big]=\\sum\\limits_{0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant a}v^{2(m-j)(a-j)}\n\\big[\\!\\!\\big[{m\\atop j}\\big]\\!\\!\\big]\\big[\\!\\!\\big[{n-m\\atop a-j}\\big]\\!\\!\\big];$\n\n$(2)$\n$\\big[\\!\\!\\big[{m\\atop a}\\big]\\!\\!\\big] \\big[\\!\\!\\big[{m\\atop b}\\big]\\!\\!\\big]=\\sum\\limits_{0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant \\min\\{a,b\\}}\nv^{2(b-c)(a-c)}\\big[\\!\\!\\big[{m\\atop a+b-c}\\big]\\!\\!\\big] \\big[\\!\\!\\big[{a+b-c\\atop\nc,\\,a-c,\\,b-c}\\big]\\!\\!\\big],$\nwhere $\\big[\\!\\!\\big[{a+b-c\\atop\nc,\\,a-c,\\,b-c}\\big]\\!\\!\\big]=\\frac{[\\![a+b-c]\\!]^!}{[\\![c]\\!]^!\n[\\![a-c]\\!]^![\\![b-c]\\!]^!}.$\n\\end{Lem}\n\nLet $\\leqslant}\\def\\geq{\\geqslant$ be the partial order on $\\mathbb N^{n}$ defined by setting, for ${\\lambda},\\mu\\in\\mathbb N^{n}$,\n${\\lambda}\\leqslant}\\def\\geq{\\geqslant\\mu$ if and only if ${\\lambda}_i\\leqslant}\\def\\geq{\\geqslant\\mu_i$ for $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$.\nFor ${\\lambda},\\alpha,\\beta,{\\gamma}\\in\\mathbb N^{n}$ with ${\\lambda}=\\alpha+\\beta+{\\gamma}$ let $$\\left[{{\\lambda}\\atop\\alpha,\\,\\beta,\\,{\\gamma}}\n\\right]=\\prod\\limits_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}\\frac{[{\\lambda}_i]^!}{[\\alpha_i]^![\\beta_i]^![{\\gamma}_i]^!}.$$\nThe above lemma immediately yields the following corollary.\n\n\n\n\n\n\\begin{Coro}\\label{mul gauss binomial}\nFor ${\\lambda},\\mu\\in\\mathbb N^{n}$ and $\\alpha,\\beta\\in\\mathbb Z^{n}$ we have\n\n$(1)$\n$\\big[{\\alpha+\\beta\\atop{\\lambda}}\\big]=\\sum\\limits_{\\mu\\in\\mathbb N^{n},\\,\\mu\\leqslant}\\def\\geq{\\geqslant{\\lambda}}\nv^{\\alpha\\centerdot({\\lambda}-\\mu)-\\mu\\centerdot\\beta}\n\\big[{\\alpha\\atop\\mu}\\big]\\big[{\\beta\\atop{\\lambda}-\\mu}\\big];$\n\n$(2)$\n$\\big[{\\alpha\\atop{\\lambda}}\\big]\\big[{\\alpha\\atop\\mu}\\big]=\n\\sum\\limits_{{\\gamma}\\in\\mathbb N^{n}\\atop{\\gamma}\\leqslant}\\def\\geq{\\geqslant{\\lambda},\\,{\\gamma}\\leqslant}\\def\\geq{\\geqslant\\mu}v^{{\\lambda}\\centerdot\\mu\n-\\alpha\\centerdot{\\gamma}}\\big[{{\\lambda}+\\mu-{\\gamma}\\atop{\\gamma},\\,{\\lambda}-{\\gamma},\\,\\mu-{\\gamma}}\n\\big]\\big[{\\alpha\\atop{\\lambda}+\\mu-{\\gamma}}\\big]$.\n\\end{Coro}\n\nWe now use \\ref{BLM formulas} and \\ref{mul gauss binomial} to prove \\ref{formula 1} and \\ref{formula 2}.\n\n\\begin{Lem}\\label{formula 1}\nFor $A\\in\\Th^\\pm(n)$, ${\\lambda},\\mu\\in\\mathbb N^{n}$ and $\\delta,{\\gamma}\\in\\mathbb Z^{n}$\nwe have\n$$0({\\gamma},\\mu)A(\\delta,{\\lambda})=\\sum_{\\nu\\in\\mathbb N^{n},\\,\\nu\\leqslant}\\def\\geq{\\geqslant\\mu}a_\\nu A({\\gamma}+\\delta-\\nu,{\\lambda}+\\mu-\\nu),$$\nwhere $0$ stands for the zero matrix and\n$$a_\\nu=\\sum_{{\\mathbf{j}}\\in\\mathbb N^{n}\\atop\\nu-{\\lambda}\\leqslant}\\def\\geq{\\geqslant{\\mathbf{j}}\\leqslant}\\def\\geq{\\geqslant\\nu}\nv^{\\text{\\rm ro}(A)\\centerdot({\\gamma}+\\mu-{\\mathbf{j}})+{\\lambda}\\centerdot(\\mu-{\\mathbf{j}})}\n\\left[{\\text{\\rm ro}(A)\\atop{\\mathbf{j}}}\\right]\\left[{{\\lambda}+\\mu-\\nu\\atop\\nu-{\\mathbf{j}},\\,{\\lambda}-\\nu+{\\mathbf{j}},\\,\n\\mu-\\nu}\\right].$$\n\\end{Lem}\n\\begin{proof}\nAccording to \\eqref{[diag(la)][A]} we have\n\\begin{equation*}\n\\begin{split}\n0({\\gamma},\\mu,r)A(\\delta,{\\lambda},r)&=\\sum_{\\alpha\\in\\Lambda(n,r-\\sigma(A))}v^{(\\text{\\rm ro}(A)+\\alpha)\n\\centerdot{\\gamma}+\\alpha\\centerdot\\delta}\\left[{\\text{\\rm ro}(A)+\\alpha\\atop\\mu}\\right]\n\\left[{\\alpha\\atop{\\lambda}}\\right][A+\\operatorname{diag}(\\alpha)].\n\\end{split}\n\\end{equation*}\nFurthermore by \\ref{mul gauss binomial} we have\n\\begin{equation*}\n\\begin{split}\n\\left[{\\text{\\rm ro}(A)+\\alpha\\atop\\mu}\\right]\n\\left[{\\alpha\\atop{\\lambda}}\\right]&=\n\\sum_{{\\mathbf{j}}\\in\\mathbb N^{n},\\,\n{\\mathbf{j}}\\leqslant}\\def\\geq{\\geqslant\\mu}v^{\\text{\\rm ro}(A)\\centerdot(\\mu-{\\mathbf{j}})-\\alpha\\centerdot{\\mathbf{j}}}\n\\left[{\\text{\\rm ro}(A)\\atop{\\mathbf{j}}}\\right]\n\\left[{\\alpha\\atop\\mu-{\\mathbf{j}}}\\right]\\left[{\\alpha\\atop{\\lambda}}\\right]\\\\\n&=\\sum_{{\\mathbf{j}},\\beta\\in\\mathbb N^{n},\\,\n{\\mathbf{j}}\\leqslant}\\def\\geq{\\geqslant\\mu\\atop\\beta\\leqslant}\\def\\geq{\\geqslant{\\lambda},\\,\\beta\\leqslant}\\def\\geq{\\geqslant\\mu-{\\mathbf{j}}}\nv^{(\\text{\\rm ro}(A)+{\\lambda})\\centerdot(\\mu-{\\mathbf{j}})-\\alpha\\centerdot({\\mathbf{j}}+\\beta)}\n\\left[{\\text{\\rm ro}(A)\\atop{\\mathbf{j}}}\\right]\n\\left[{\\alpha\\atop{\\lambda}+\\mu-{\\mathbf{j}}-\\beta}\\right]\n\\\\&\\qquad\\qquad\\times\\left[{{\\lambda}+\\mu-{\\mathbf{j}}-\\beta\\atop\n\\beta,\\,{\\lambda}-\\beta,\\,\\mu-{\\mathbf{j}}-\\beta}\\right].\n\\end{split}\n\\end{equation*}\nThus we conclude that\n\\begin{equation*}\n\\begin{split}\n0({\\gamma},\\mu,r)A(\\delta,{\\lambda},r)&=\\sum_{{\\mathbf{j}},\\beta\\in\\mathbb N^{n},\\,\n{\\mathbf{j}}\\leqslant}\\def\\geq{\\geqslant\\mu\\atop\\beta\\leqslant}\\def\\geq{\\geqslant{\\lambda},\\,\\beta\\leqslant}\\def\\geq{\\geqslant\\mu-{\\mathbf{j}}}\nv^{\\text{\\rm ro}(A)\\centerdot({\\gamma}+\\mu-{\\mathbf{j}})+{\\lambda}\\centerdot(\\mu-{\\mathbf{j}})}\n\\left[{\\text{\\rm ro}(A)\\atop{\\mathbf{j}}}\\right]\n\\left[{{\\lambda}+\\mu-{\\mathbf{j}}-\\beta\\atop\n\\beta,\\,{\\lambda}-\\beta,\\,\\mu-{\\mathbf{j}}-\\beta}\\right]\\\\\n&\\qquad\\qquad\\qquad\\times A({\\gamma}+\\delta-{\\mathbf{j}}-\\beta,{\\lambda}+\\mu-{\\mathbf{j}}-\\beta,r)\\\\\n&=\\sum_{\\nu\\in\\mathbb N^{n},\\,\\nu\\leqslant}\\def\\geq{\\geqslant\\mu}a_\\nu A({\\gamma}+\\delta-\\nu,{\\lambda}+\\mu-\\nu,r).\n\\end{split}\n\\end{equation*}\nThe assertion follows.\n\\end{proof}\n\n\n\nFor simplicity, we set\n$A(\\delta,{\\lambda},r)= 0$ and $A(\\delta,{\\lambda})=0$ if $a_{i,j}< 0$ for some $i\\not=j$  for $A\\in M_n(\\mathbb Z)$.\n\n\\begin{Lem}\\label{formula 2}\nLet $A\\in\\Th^\\pm(n)$, $\\delta\\in\\mathbb Z^{n}$, ${\\lambda}\\in\\mathbb N^{n}$, $m\\in\\mathbb N$ and $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$.\n\n$(1)$ For ${\\mathbf{t}}\\in\\Lambda(n,m)$, $0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h$, $0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}$, and $0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_h,j\\}$, we set\n\\begin{equation*}\n\\begin{split}\n\\alpha^{\\mathbf{t}}_{j,c,k}&=\n\\bigg(\\sum_{h>u}t_u+{\\lambda}_h-j-c\\bigg)\\boldsymbol{e}_h+\\bigg({\\lambda}_{h+1}-k-\\sum_{h+1>u}t_u\n\\bigg)\n\\boldsymbol{e}_{h+1},\\\\\n\\beta^{\\mathbf{t}}_{j,c,k}&=(t_h+j-c-{\\lambda}_h)\\boldsymbol{e}_h+(k-{\\lambda}_{h+1})\\boldsymbol{e}_{h+1}\n\\end{split}\n\\end{equation*}\nand $$f^{\\mathbf{t}}_{j,c,k}=v^{g^{\\mathbf{t}}_{j,k}}\\prod_{u\\not=h}\\overline{\\left[\\!\\!\\left[{a_{h,u}+t_u\\atop t_u}\\right]\\!\\!\\right]}\n\\left[{-t_h\\atop{\\lambda}_h-j}\\right]\\left[{t_h+j-c\\atop c,\\,t_h-c,\\,j-c}\\right]\n\\left[{t_{h+1}\\atop{\\lambda}_{h+1}-k}\\right]$$\nwhere $g^{\\mathbf{t}}_{j,k}=\\sum_{j\\geq u,\\,j\\not=h}a_{h,j}t_u-\\sum_{j>u,\\,\nj\\not=h+1}a_{h+1,j}t_u+\\sum_{u'\\not=h,h+1,\\,u<u'}t_ut_{u'}-t_h\\delta_h\n+t_{h+1}\\delta_{h+1}+2jt_h-kt_{h+1}$.\nThen we have\n\\begin{equation*}\n\\begin{split}\n&\\qquad(mE_{h,h+1})({\\mathbf{0}})A(\\delta,{\\lambda})\\\\\n&=\\sum_{\\tiny{\\begin{array}{c}{\\mathbf{t}}\\in\\Lambda(n,m)\\\\\n0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h,\\,0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}\\\\\n0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_h,j\\}\\end{array}}}\nf^{\\mathbf{t}}_{j,c,k}\n\\bigg(A+\\sum_{u\\not=h}t_uE_{h,u}-\\sum_{u\\not=h+1}t_uE_{h+1,u}\\bigg)\n(\\delta+\\alpha^{\\mathbf{t}}_{j,c,k},{\\lambda}+\\beta^{\\mathbf{t}}_{j,c,k}).\n\\end{split}\n\\end{equation*}\n\n$(2)$ For ${\\mathbf{t}}\\in\\Lambda(n,m)$, $0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}$, $0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h}$, and $0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_{h+1},j\\}$, we set\n\n\\begin{equation*}\n\\begin{split}\n\\widetilde\\alpha^{\\mathbf{t}}_{j,c,k}&=\n\\bigg(\\sum_{h+1<u}t_u+{\\lambda}_{h+1}-j-c\\bigg)\n\\boldsymbol{e}_{h+1}+\\bigg({\\lambda}_{h}-k-\\sum_{h<u}t_u\n\\bigg)\n\\boldsymbol{e}_{h},\\\\\n\\widetilde\\beta^{\\mathbf{t}}_{j,c,k}&=(t_{h+1}+j-c-{\\lambda}_{h+1})\n\\boldsymbol{e}_{h+1}+(k-{\\lambda}_{h})\\boldsymbol{e}_{h}\n\\end{split}\n\\end{equation*}\nand\n$$\\widetilde f^{\\mathbf{t}}_{j,c,k}=v^{\\widetilde g^{\\mathbf{t}}_{j,k}}\n\\prod_{u\\not=h+1}\\overline{\\left[\\!\\!\\left[{a_{h+1,u}+t_u\\atop t_u}\\right]\\!\\!\\right]}\n\\left[{-t_{h+1}\\atop{\\lambda}_{h+1}-j}\\right]\\left[{t_{h+1}+j-c\\atop c,\\,t_{h+1}-c,\\,j-c}\\right]\n\\left[{t_{h}\\atop{\\lambda}_{h}-k}\\right]$$\nwhere $\\widetilde g^{\\mathbf{t}}_{j,k}=\\sum_{j\\leqslant}\\def\\geq{\\geqslant u,\\,j\\not=h+1}a_{h+1,j}t_u-\\sum_{j<u,\\,\nj\\not=h}a_{h,j}t_u+\\sum_{u\\not=h,h+1,\\,u<u'}t_ut_{u'}+t_h\\delta_h\n-t_{h+1}\\delta_{h+1}+2jt_{h+1}-kt_{h}$.\nThen we have\n\\begin{equation*}\n\\begin{split}\n&\\qquad(mE_{h+1,h})({\\mathbf{0}})A(\\delta,{\\lambda})\\\\\n&=\\sum_{\\tiny{\\begin{array}{c}{\\mathbf{t}}\\in\\Lambda(n,m)\\\\\n 0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1},\\,0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h}\\\\\n0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_{h+1},j\\}\\end{array}}}\n\\widetilde f^{\\mathbf{t}}_{j,c,k}\n\\bigg(A-\\sum_{u\\not=h}t_uE_{h,u}+\\sum_{u\\not=h+1}t_uE_{h+1,u}\\bigg)\n(\\delta+\\widetilde\\alpha^{\\mathbf{t}}_{j,c,k},{\\lambda}+\\widetilde\\beta^{\\mathbf{t}}_{j,c,k}).\n\\end{split}\n\\end{equation*}\n\\end{Lem}\n\\begin{proof}\nFor simplicity, for $A\\in M_n(\\mathbb Z)$  with $\\sigma(A)=r$, we set $[A] = 0\\in{\\boldsymbol{\\mathcal S}}(n,r)$ if $a_{i,j}<0$  for some $i,j$.\nAccording to \\ref{BLM formulas} we have\n\\begin{equation*}\n\\begin{split}\n&\\qquad(mE_{h,h+1})({\\mathbf{0}},r)A(\\delta,{\\lambda},r)\\\\\n&=\\sum_{\\alpha\\in\\Lambda(n,r-\\sigma(A))}\nv^{\\alpha\\centerdot\\delta}\\left[{\\alpha\\atop{\\lambda}}\\right][mE_{h,h+1}+\\operatorname{diag}(\\text{\\rm ro}(A)+\\alpha\n-m\\boldsymbol{e}_{h+1})]\\cdot[A+\\operatorname{diag}(\\alpha)]\\\\\n&=\\sum_{\\alpha\\in\\Lambda(n,r-\\sigma(A))\\atop{\\mathbf{t}}\\in\\Lambda(n,m)}\nv^{\\beta({\\mathbf{t}},A+\\operatorname{diag}(\\alpha))+\\alpha\\centerdot\\delta-t_h\\alpha_h}\n\\prod_{u\\not=h}\\overline{\\left[\\!\\!\\left[{a_{h,u}+t_u\\atop t_u}\\right]\\!\\!\\right]}\n\\left[{\\alpha_{h}+t_h\\atop t_h}\\right]\\left[{\\alpha\\atop{\\lambda}}\\right]\\\\\n&\\qquad\\qquad\\times\\bigg[A+\\sum_{u\\not=h}t_uE_{h,u}\n-\\sum_{u\\not=h+1}t_uE_{h+1,u}+\n\\operatorname{diag}(\\alpha+t_h\\boldsymbol{e}_h-t_{h+1}\\boldsymbol{e}_{h+1})\\bigg].\n\\end{split}\n\\end{equation*}\nLet $\\nu=\\alpha+t_h\\boldsymbol{e}_h-t_{h+1}\\boldsymbol{e}_{h+1}$. By \\ref{mul gauss binomial} we have\n\\begin{equation*}\n\\begin{split}\n\\left[{\\nu_h-t_h\\atop{\\lambda}_h}\\right]\n\\left[{\\nu_h\\atop t_h}\\right]&=\\sum_{0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h}v^{\\nu_h({\\lambda}_h-j)+jt_h}\\left[{-t_h\\atop{\\lambda}_h-j}\\right]\n\\bigg(\\left[{\\nu_h\\atop t_h}\\right]\\left[{\\nu_h\\atop j}\\right]\\bigg)\\\\\n&=\\sum_{0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h\\atop 0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\\{t_h,j\\}}\nv^{\\nu_h({\\lambda}_h-j-c)+2jt_h}\\left[{-t_h\\atop{\\lambda}_h-j}\\right]\\left[\n{\\nu_h\\atop t_h+j-c}\\right]\\left[{t_h+j-c\\atop c,\\,t_h-c,\\,j-c}\\right]\n\\end{split}\n\\end{equation*}\nand\n$\\big[{\\nu_{h+1}+t_{h+1}\\atop{\\lambda}_{h+1}}\\big]\n=\\sum_{0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}}v^{\\nu_{h+1}({\\lambda}_{h+1}-k)-kt_{h+1}}\n\\big[{\\nu_{h+1}\\atop k}\\big]\\big[{t_{h+1}\\atop{\\lambda}_{h+1}-k}\\big].$\nThis implies that\n\\begin{equation*}\n\\begin{split}\n\\left[{\\alpha_h+t_h\\atop t_h}\\right]\\left[{\\alpha\\atop{\\lambda}}\\right]&=\n\\prod_{s\\not=h,h+1}\\left[{\\nu_s\\atop{\\lambda}_s}\\right]\\sum_{0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1},\\,0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h\\atop 0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\\{t_h,j\\}}\nv^{x^{\\nu,{\\mathbf{t}}}_{j,c,k}}\\left[\n{-t_h\\atop{\\lambda}_h-j}\\right]\\left[{t_h+j-c\\atop c,\\,t_h-c,\\,j-c}\\right]\\\\\n&\\qquad\\qquad\\times\n\\left[{t_{h+1}\\atop{\\lambda}_{h+1}-k}\\right]\\left[{\\nu_h\\atop t_h+j-c}\\right]\n\\left[{\\nu_{h+1}\\atop k}\\right].\n\\end{split}\n\\end{equation*}\nwhere $x^{\\nu,{\\mathbf{t}}}_{j,c,k}\n=\\nu_h({\\lambda}_h-j-c)+\\nu_{h+1}({\\lambda}_{h+1}-k)+2jt_h-kt_{h+1}$.\nThus\n\\begin{equation*}\n\\begin{split}\n&\\qquad(mE_{h,h+1})({\\mathbf{0}},r)A(\\delta,{\\lambda},r)\\\\\n&=\\sum_{\\tiny{\\begin{array}{c}{\\mathbf{t}}\\in\\Lambda(n,m)\\\\\n0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h,\\,0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}\\\\\n0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_h,j\\}\\end{array}}}\\prod_{u\\not=h}\\overline{\\left[\\!\\!\\left[{a_{h,u}+t_u\n\\atop t_u}\\right]\\!\\!\\right]}\n\\left[{-t_h\\atop{\\lambda}_h-j}\\right]\\left[{t_h+j-c\\atop c,\\,t_h-c,\\,j-c}\\right]\n\\left[{t_{h+1}\\atop{\\lambda}_{h+1}-k}\\right]\\\\\n&\\qquad\\qquad\\times \\sum_{\\nu\\in\\Lambda(n,r-\\sigma(A)+t_h-t_{h+1})}v^{y^\\nu_{j,c,k}}\n\\prod_{s\\not=h,h+1}\\left[{\\nu_s\\atop{\\lambda}_s}\\right] \\left[{\\nu_h\\atop\nt_h+j-c}\\right]\\left[{\\nu_{h+1}\\atop k}\\right]\\\\\n&\\qquad\\qquad\\qquad\\times\n\\bigg[A+\\sum_{u\\not=h}t_uE_{h,u}\n-\\sum_{u\\not=h+1}t_uE_{h+1,u}+\n\\operatorname{diag}(\\nu)\\bigg]\\\\\n&=\\sum_{\\tiny{\\begin{array}{c}{\\mathbf{t}}\\in\\Lambda(n,m)\\\\\n0\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant{\\lambda}_h,\\,0\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant{\\lambda}_{h+1}\\\\\n0\\leqslant}\\def\\geq{\\geqslant c\\leqslant}\\def\\geq{\\geqslant\\min\n\\{t_h,j\\}\\end{array}}}\nf^{\\mathbf{t}}_{j,c,k}\n\\bigg(A+\\sum_{u\\not=h}t_uE_{h,u}-\\sum_{u\\not=h+1}t_uE_{h+1,u}\\bigg)\n(\\delta+\\alpha^{\\mathbf{t}}_{j,c,k},{\\lambda}+\\beta^{\\mathbf{t}}_{j,c,k}).\n\\end{split}\n\\end{equation*}\nwhere\n$y^{\\nu,{\\mathbf{t}}}_{j,c,k}=\\beta({\\mathbf{t}},A+\\operatorname{diag}(\\alpha))+\\alpha\\centerdot\\delta-t_h\\alpha_h\n+x^{\\nu,{\\mathbf{t}}}_{j,c,k}=g^{\\mathbf{t}}_{j,k}+\\nu\\centerdot(\\delta+\\alpha^{\\mathbf{t}}_{j,c,k})$.\nThe assertion (1) follows. The assertion (2) can be proved in a way  similar to the proof of (1).\n\\end{proof}\n\n\\section{Realization of $U(n)$ and $\\ol{U_k(n)}$}\n\n\nWe shall denote by ${\\mathcal V}(n)$  the ${\\mathcal Z}$-submodule of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ spanned by\n$\\{A(\\delta,{\\lambda})\\mid A\\in\\Th^\\pm(n),\\,\\delta\\in\\mathbb Z^{n},\\,{\\lambda}\\in\\mathbb N^{n}\\}.$\nLet ${\\mathcal V}^0(n)$ be the ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ generated by $0(\\pm\\boldsymbol{e}_i)$ and $0(0,t\\boldsymbol{e}_i)$ for $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$ and $t\\in\\mathbb N$,\nwhere\n$\\boldsymbol{e}_i=(0,\\cdots,0,\\underset i1,0\\cdots,0)\\in\\mathbb N^n.$\n\\begin{Lem}\\label{basis of zero part}\nThe set $\\{0(\\delta,{\\lambda})\\mid\n\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\}$\nforms a ${\\mathcal Z}$-basis for ${\\mathcal V}^0(n)$.\n\\end{Lem}\n\\begin{proof}\nLet $\\boldsymbol{{\\mathcal V}}^0(n)$ be the $\\mathbb Q(v)$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ generated by $0(\\pm\\boldsymbol{e}_i)$ for $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$. Since the set $\\{0({\\mathbf{j}})\\mid{\\mathbf{j}}\\in\\mathbb Z^{n}\\}$ forms a $\\mathbb Q(v)$-basis for $\\boldsymbol{{\\mathcal V}}^0(n)$ we conclude that\n$\\boldsymbol{{\\mathcal V}}^0(n)$ is isomorphic to ${\\mathbf U}^0(n)$, where ${\\mathbf U}^0(n)$ is the $\\mathbb Q(v)$-subalgebra of ${\\mathbf U}(n)$ generated by $K_i^{\\pm 1}$ for $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$.\nNow the assertion follows from \\cite[4.5]{Lu90}.\n\\end{proof}\nWe now describe several ${\\mathcal Z}$-bases for ${\\mathcal V}(n)$ as follows.\n\\begin{Lem}\\label{basis for Vn}\nEach of the following set forms a ${\\mathcal Z}$-basis for ${\\mathcal V}(n):$\n\n$(1)$ $\\frak B_1=\\{0(\\delta,{\\lambda})A({\\mathbf{0}})\\mid A\\in\\Th^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\};$\n\n$(2)$ $\\frak B_2=\\{A({\\mathbf{0}})0(\\delta,{\\lambda})\\mid A\\in\\Th^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\};$\n\n$(3)$ $\\frak B_3=\\{A(\\delta,{\\lambda})\\mid A\\in\\Th^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\}.$\n\\end{Lem}\n\\begin{proof}\nAccording to \\ref{formula 1} we have\n$$0(\\delta,{\\lambda})A({\\mathbf{0}})=v^{\\text{\\rm ro}(A)\\centerdot(\\delta+{\\lambda})}A(\\delta,{\\lambda})+\n\\sum_{{\\mathbf{j}}\\in\\mathbb N^{n},\\,{\\mathbf{0}}<{\\mathbf{j}}\\leqslant}\\def\\geq{\\geqslant{\\lambda}}\nv^{\\text{\\rm ro}(A)\\centerdot(\\delta+{\\lambda}-{\\mathbf{j}})}\\left[{\\text{\\rm ro}(A)\\atop{\\mathbf{j}}}\\right]\nA(\\delta-{\\mathbf{j}},{\\lambda}-{\\mathbf{j}}).$$\nIt follows that ${\\mathcal V}(n)$ is spanned by\n$\\{0(\\delta,{\\lambda})A({\\mathbf{0}})\\mid A\\in\\Th^\\pm(n),\\,\n\\delta\\in\\mathbb Z^{n},\\,{\\lambda}\\in\\mathbb N^{n}\\}$.\nThus by \\ref{basis of zero part} we have\n${\\mathcal V}(n)=\\operatorname{span}_{\\mathcal Z}{\\frak B}_1$.\nSince the set $\\{0({\\mathbf{j}})A({\\mathbf{0}})\\mid A\\in\\Th^\\pm(n),\\,{\\mathbf{j}}\\in\\mathbb Z^{n}\\}$ is linearly independent, by \\ref{basis of zero part} we conclude that\nthe set ${\\frak B}_1$ is linearly independent. Hence the set ${\\frak B}_1$ forms a ${\\mathcal Z}$-basis for ${\\mathcal V}(n)$. Similarly, the set ${\\frak B}_2$ forms a ${\\mathcal Z}$-basis for ${\\mathcal V}(n)$. It remains to prove that the set\n${\\frak B}_3$ forms a ${\\mathcal Z}$-basis for ${\\mathcal V}(n)$. For ${\\lambda}\\in\\mathbb N^{n}$ and $\\mu,\\delta\\in\\mathbb Z^{n}$\nwe have\n$$v^{\\delta_i\\mu_i}\\left[{\\mu_i\\atop{\\lambda}_i}\\right]\n=v^{{\\lambda}_i}(v^{{\\lambda}_i+1}-v^{-{\\lambda}_i-1})v^{(\\delta_i-1)\\mu_i}\\left[{\\mu_i\\atop\n{\\lambda}_i+1}\\right]+v^{2{\\lambda}_i+(\\delta_i-2)\\mu_i}\\left[{\\mu_i\\atop{\\lambda}_i}\\right].$$\nIt follows that\n\\begin{equation*}\n\\begin{split}\nA(\\delta,{\\lambda})&=v^{{\\lambda}_i}(v^{{\\lambda}_i+1}-v^{-{\\lambda}_i-1})A(\\delta-\\boldsymbol{e}_i,{\\lambda}+\\boldsymbol{e}_i)\n+v^{2{\\lambda}_i}A(\\delta-2\\boldsymbol{e}_i,{\\lambda})\\\\\n&=-v^{-{\\lambda}_i}(v^{{\\lambda}_i+1}-v^{-{\\lambda}_i-1})A(\\delta+\\boldsymbol{e}_i,{\\lambda}+\\boldsymbol{e}_i)\n+v^{-2{\\lambda}_i}A(\\delta+2\\boldsymbol{e}_i,{\\lambda})\n\\end{split}\n\\end{equation*}\nfor $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$, ${\\lambda}\\in\\mathbb N^{n}$ and $\\delta\\in\\mathbb Z^{n}$.\nThis shows that\n${\\mathcal V}(n)$ is spanned by ${\\frak B}_3$. Assume $$\\sum_{A\\in\\Th^\\pm(n),\\,{\\lambda},\\delta\\in\\mathbb N^{n}\\atop\\delta_i\\in\\{0,1\\},\\forall i}f_{A,\\delta,{\\lambda}}A(\\delta,{\\lambda})=0$$\nwhere $f_{A,\\delta,{\\lambda}}\\in\\mathbb Q(v)$.  Then\n$$\\sum_{A\\in\\Th^\\pm(n)\\atop\\mu\\in\\Lambda(n,r-\\sigma(A))}\\bigg\n(\\sum_{{\\lambda},\\delta\\in\\mathbb N^{n}\\atop\\delta_i\\in\\{0,1\\},\\forall i}f_{A,\\delta,{\\lambda}}\nv^{\\mu\\centerdot\\delta}\\left[{\\mu\\atop{\\lambda}}\\right]\\bigg)[A+\\operatorname{diag}(\\mu)]\n=\\sum_{A\\in\\Th^\\pm(n),\\,{\\lambda},\\delta\\in\\mathbb N^{n}\\atop\\delta_i\\in\\{0,1\\},\\forall i}f_{A,\\delta,{\\lambda}}A(\\delta,{\\lambda},r)=0.$$\nThis implies that $$\\sum_{{\\lambda},\\delta\\in\\mathbb N^{n}\\atop\\delta_i\\in\\{0,1\\},\\forall i}f_{A,\\delta,{\\lambda}}\nv^{\\mu\\centerdot\\delta}\\left[{\\mu\\atop{\\lambda}}\\right]=0$$ for $A\\in\\Th^\\pm(n)$ and $\\mu\\in\\mathbb N^{n}$.\nIt follows that\n$$\\sum_{{\\lambda},\\delta\\in\\mathbb N^{n}\\atop\\delta_i\\in\\{0,1\\},\\forall i}f_{A,\\delta,{\\lambda}}0(\\delta,{\\lambda},r)=\n0$$\nfor $r\\geq 0$ and $A\\in\\Th^\\pm(n)$. Thus by \\ref{basis of zero part} we conclude that\n$f_{A,\\delta,{\\lambda}}=0$ for all $A,\\delta,{\\lambda}$. This shows that\nthe set ${\\frak B}_3$ is linearly independent and hence the set ${\\frak B}_3$ forms\na ${\\mathcal Z}$-basis for ${\\mathcal V}(n)$.\n\\end{proof}\n\nWe now use \\ref{formula 1} and \\ref{formula 2} to prove that ${\\mathcal V}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$.\n\n\\begin{Prop}\\label{Z algebra}\n${\\mathcal V}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$. Furthermore\nthe elements $(mE_{h,h+1})({\\mathbf{0}})$, $(mE_{h+1,h})({\\mathbf{0}})$ and\n$0(\\delta,{\\lambda})$ $($for $m\\in\\mathbb N$, $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$, $\\delta\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n})$\ngenerate ${\\mathcal V}(n)$ as a ${\\mathcal Z}$-algebra.\n\\end{Prop}\n\\begin{proof}\nLet ${\\mathcal V}(n)_1$ be the ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ generated by\n$(mE_{h,h+1})({\\mathbf{0}})$, $(mE_{h+1,h})({\\mathbf{0}})$ and\n$0(\\delta,{\\lambda})$ for $m\\in\\mathbb N$, $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$, $\\delta\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$.\nFrom \\ref{formula 1} and \\ref{formula 2} we see that\n\\begin{equation}\\label{zeta(U(n))}\n{\\mathcal V}(n)_1\\subseteq{\\mathcal V}(n)_1{\\mathcal V}(n)\\subseteq{\\mathcal V}(n).\n\\end{equation}\nSo by \\ref{basis for Vn} it is enough to prove $A({\\mathbf{0}})0(\\delta,{\\lambda})\\in{\\mathcal V}(n)_1$ for $A\\in\\Th^\\pm(n)$, $\\delta,{\\lambda}\\in\\mathbb N^{n}$ with $\\delta_i\\in\\{0,1\\}$ ($1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$).\nWe shall prove this by induction on $|\\!|A|\\!|$, where\n$$|\\!|A|\\!|=\\sum_{r<s}\\frac{(s-r)(s-r+1)}{2}a_{rs}+\\sum_{r>s}\\frac{(r-s)(r-s+1)}{2}a_{rs}\\in\\mathbb N.$$\nIf $|\\!|A|\\!| = 0$, then $A({\\mathbf{0}})0(\\delta,{\\lambda})=0(\\delta,{\\lambda})\\in{\\mathcal V}(n)_1$.\nNow we assume that $\\ddet A>0$ and our statement is true\nfor $A'$ with $\\ddet {A'}<\\ddet A$.\nAccording to \\ref{tri}, for $A\\in\\Th^\\pm(n)$, we have\n$$\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant h<j\\leqslant}\\def\\geq{\\geqslant n}(a_{i,j}E_{h,h+1})({\\mathbf{0}})\n\\cdot\\prod_{1\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant h<i\\leqslant}\\def\\geq{\\geqslant\n n}(a_{i,j}E_{h+1,h})({\\mathbf{0}})=A({\\mathbf{0}})+f$$\nwhere $f$ is the $\\mathbb Q(v)$-linear combination of $B({\\mathbf{0}})0({\\mathbf{j}})$ with $B\\in\\Th^\\pm(n)$, $B\\prec A$ and ${\\mathbf{j}}\\in\\mathbb Z^{n}$. It follows that\n\\begin{equation}\\label{tri1}\n\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant h<j\\leqslant}\\def\\geq{\\geqslant n}(a_{i,j}E_{h,h+1})({\\mathbf{0}})\n\\cdot\\prod_{1\\leqslant}\\def\\geq{\\geqslant j\\leqslant}\\def\\geq{\\geqslant h<i\\leqslant}\\def\\geq{\\geqslant\n n}(a_{i,j}E_{h+1,h})({\\mathbf{0}})\\cdot 0(\\delta,{\\lambda})=A({\\mathbf{0}})0(\\delta,{\\lambda})+g\n\\end{equation}\nfor $\\delta,{\\lambda}\\in\\mathbb N^{n}$ with $\\delta_i\\in\\{0,1\\}$ ($1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$), where $g=f\\cdot 0(\\delta,{\\lambda})$. From \\eqref{zeta(U(n))}, \\ref{basis of zero part} and \\ref{basis for Vn} we see that $g$ must be a ${\\mathcal Z}$-linear combination of $B({\\mathbf{0}})0({\\gamma},\\mu)$ with $B\\in\\Th^\\pm(n)$, $B\\prec A$, ${\\gamma},\\mu\\in\\mathbb N^{n}$ and ${\\gamma}_i\\in\\{0,1\\}$ for $1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n$.\nNote that if $B\\in\\Th^\\pm(n)$ satisfy $B\\prec A$, then $|\\!|B|\\!| < |\\!|A|\\!|$ (see the proof of \\cite[4.2]{BLM}). Thus by induction we conclude that $g\\in{\\mathcal V}(n)_1$ and hence $A({\\mathbf{0}})0(\\delta,{\\lambda})\\in{\\mathcal V}(n)_1$. The assertion follows.\n\\end{proof}\n\n\\begin{Thm}\\label{realization}\nThere is a ${\\mathcal Z}$-algebra isomorphism\n$\\zeta:U(n)\\rightarrow{\\mathcal V}(n)$ satisfying\n$E_h^{(m)}\\mapsto(mE_{h,h+1})({\\mathbf{0}})$, $F_h^{(m)}\\mapsto(mE_{h+1,h})({\\mathbf{0}})$ and\n$\\prod_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}K_i^{\\delta_i}\\left[{K_i;0\\atop {\\lambda}_i}\\right]\\mapsto 0(\\delta,{\\lambda})$\nfor $m\\in\\mathbb N$, $1\\leqslant}\\def\\geq{\\geqslant h\\leqslant}\\def\\geq{\\geqslant n-1$, $\\delta\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$.\n\\end{Thm}\n\\begin{proof}\nThe maps $\\zeta_r:{\\mathbf{U}}(n)\\rightarrow{\\boldsymbol{\\mathcal S}}(n,r)$ induce an algebra homomorphism\n\\begin{equation*}\n\\zeta:{\\mathbf U}(n)\\rightarrow\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)\n\\end{equation*}\n satisfying $\\zeta(x)=(\\zeta_r(x))_{r\\geq 0}$ for $x\\in{\\mathbf U}(n)$.\nFrom \\ref{Z algebra} we see that $\\zeta(U(n))={\\mathcal V}(n)$. Furthermore by \\ref{basis for Un}, \\ref{basis for Vn} and \\eqref{tri1}, we conclude that $\\zeta$ is injective.\n\\end{proof}\n\\begin{Rem}\n(1) Let $\\boldsymbol{{\\mathcal V}}(n)$ be the $\\mathbb Q(v)$-subspace of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ spanned by $\\{A(\\delta)\\mid A\\in\\Th^\\pm(n),\\,\\delta\\in\\mathbb Z^{n}\\}.$ Then $\\boldsymbol{{\\mathcal V}}(n)\\cong{\\mathcal V}(n)\\otimes_{\\mathcal Z}\\mathbb Q(v)$. According to \\ref{Z algebra} and \\ref{realization} we conclude that $\\boldsymbol{{\\mathcal V}}(n)$ is a $\\mathbb Q(v)$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$ and ${\\mathbf U}(n)\\cong\\boldsymbol{{\\mathcal V}}(n)$.\n\n(2) Note that for $A\\in\\Th^\\pm(n)$ and ${\\lambda}\\in\\Lambda(n,r-\\sigma(A))$ we have $A({\\mathbf{0}},{\\lambda},r)=[A+\\operatorname{diag}({\\lambda})]$. Thus from \\ref{realization} we see that $\\zeta_r(U(n))={\\mathcal S}(n,r)$, which has been proved in \\cite[3.4]{Du95}.\n\\end{Rem}\n\nWe now use $q$-Schur algebras over $k$ to realize quantum $\\frak{gl}_n$ over $k$, where $k$ is a field containing\nan $l$-th primitive root $\\varepsilon$ of $1$ with $l\\geq 1$ odd.\nSpecializing $v$ to $\\varepsilon$, $k$ will be viewed as a ${\\mathcal Z}$-module. For $\\mu\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$ we shall denote the image of $\\big[{\\mu\\atop{\\lambda}}\\big]$ in $k$ by $\\big[{\\mu\\atop{\\lambda}}\\big]_\\varepsilon$.\nLet $U_k(n)=U(n)\\otimes_{\\mathcal Z} k$ and ${\\mathcal S}_k(n,r)={\\mathcal S}(n,r)\\otimes_{\\mathcal Z} k$.\nBy restriction, the map $\\zeta_{r}:{\\mathbf U}(n)\\rightarrow{\\boldsymbol{\\mathcal S}}(n,r)$ induces\nan algebra homomorphism $\\zeta_r:U(n)\\rightarrow{\\mathcal S}(n,r)$. By tensoring with the field $k$,\nwe get an algebra homomorphism $$\\zeta_{r,k}:=\\zeta_r\\otimes id:U_k(n)\\rightarrow{\\mathcal S}_k(n,r).$$\nLet $$\\ol{U_k(n)}=U_k(n)\/\\langle K_i^l-1\\mid 1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n-1\\rangle.$$\nSince $\\zeta_{r,k}(K_i^l)=1$, $\\zeta_{r,k}$ induces an algebra homomorphism\n$$\\bar\\zeta_{r,k}:\\ol{U_k(n)}\\rightarrow{\\mathcal S}_k(n,r)$$\nsatisfying $\\bar\\zeta_{r,k}(\\bar x)=\\zeta_{r,k}(x)$ for $x\\inU_k(n)$.\nThe maps $\\bar\\zeta_{r,k}$ induce an algebra homomorphism\n$$\\bar\\zeta_k:=\\prod_{r\\geq 0}\\bar\\zeta_{r,k}:\\ol{U_k(n)}\\rightarrow\\prod_{r\\geq 0}{\\mathcal S}_k(n,r)$$\nsatisfying $\\bar\\zeta_k(x)=(\\bar\\zeta_{r,k}(\\bar x))_{r\\geq 0}$ for $\\bar x\\in\\ol{U_k(n)}$.\nFor $A\\in\\Thnr$ we let\n$[A]_\\varepsilon=[A]\\otimes 1\\in{\\mathcal S}_k(n,r)$. Similarly, for $A\\in\\Th^\\pm(n)$, $\\delta\\in\\mathbb Z^{n}$ and ${\\lambda}\\in\\mathbb N^{n}$,  let $A(\\delta,{\\lambda},r)_\\varepsilon=A(\\delta,{\\lambda},r)\\otimes 1\\in{\\mathcal S}_k(n,r)$, $A(\\delta,{\\lambda})_\\varepsilon=(A(\\delta,{\\lambda},r)_\\varepsilon)_{r\\geq 0}\\in\\prod_{r\\geq 0}{\\mathcal S}_k(n,r)$\nand $A(\\delta)_\\varepsilon=A(\\delta,{\\mathbf{0}})_\\varepsilon$. From \\ref{basis for Vn} and \\ref{realization}\nwe see that\n$$\\bar\\zeta_k(\\ol{U_k(n)})=\\operatorname{span}_k\\{A(\\delta,{\\lambda})_\\varepsilon\\mid A\\in\\Th^\\pm(n),\\,\\delta,{\\lambda}\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\}.$$\n\n\n\n\\begin{Thm}\\label{realization of barUnk}\nThe algebra homomorphism $\\bar\\zeta_k$ is injective. Furthermore, the\nset $$\\frak B_k:=\\{A({\\mathbf{0}})_\\varepsilon 0(-{\\lambda},{\\lambda})_\\varepsilon\\mid\nA\\in\\Th^\\pm(n),\\,{\\lambda}\\in\\mathbb N^{n}\\}$$ forms a $k$-basis of $\\bar\\zeta_k(\\ol{U_k(n)})$.\n\\end{Thm}\n\\begin{proof}\nWe will identify $U(n)$ with ${\\mathcal V}(n)$ via the map $\\zeta$ defined in \\ref{realization}. From \\ref{basis for Vn} and \\cite[6.4(b)]{Lu90}, we see that\nthe set $$\\{A({\\mathbf{0}})0(l\\delta)0(-{\\lambda},{\\lambda})\\otimes 1\\mid A\\in\\Th^\\pm(n),\\,{\\lambda},\\delta\\in\\mathbb N^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\}$$\nforms a $k$-basis for $U_k(n)$. It follows that $\\bar\\zeta_k(\\ol{U_k(n)})$ is spanned by the set $\\frak B_k$. Thus it is enough to prove that the set $\\frak B_k$ is linearly independent.\n\nAssume $$\\sum_{A\\in\\Th^\\pm(n),\\,{\\lambda}\\in\\mathbb N^{n}}f_{A,{\\lambda}}A({\\mathbf{0}})_\\vep0(-{\\lambda},{\\lambda})_\\varepsilon=0$$\nwhere $f_{A,{\\lambda}}\\in k$. Then for any $r\\geq 0$\n\\begin{equation*}\n\\sum_{A\\in\\Th^\\pm(n)\\atop\\mu\\in\\Lambda(n,r-\\sigma(A))}\\bigg(\\sum_{{\\lambda}\\in\\mathbb N^{n}}f_{A,{\\lambda}}\n\\varepsilon^{-{\\lambda}\\centerdot\n(\\mu+\\text{\\rm co}(A))}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}}\\right]_\\varepsilon\\bigg)[A+\\operatorname{diag}(\\mu)]_\\varepsilon\n=0.\n\\end{equation*}\nIt follows that\nfor any $A\\in\\Th^\\pm(n)$, $\\mu\\in\\Lambda(n,r-\\sigma(A))$ with $r\\geq\\sigma(A)$, we have\n\\begin{equation}\\label{sum}\n\\sum_{{\\lambda}\\in\\mathbb N^{n}}f_{A,{\\lambda}}\n\\varepsilon^{-{\\lambda}\\centerdot\n(\\mu+\\text{\\rm co}(A))}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}}\\right]_\\varepsilon=0.\n\\end{equation}\nWe claim that for $A\\in\\Th^\\pm(n)$ and $\\mu,\\alpha\\in\\mathbb N^{n}$ we have\n\\begin{equation}\\label{sum1}\n\\sum_{{\\lambda}\\in\\mathbb N^{n},\\,{\\lambda}\\geq\\alpha}f_{A,{\\lambda}}\n\\varepsilon^{-{\\lambda}\\centerdot\n(\\mu+\\text{\\rm co}(A))}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}-\\alpha}\\right]_\\varepsilon=0.\n\\end{equation}\nWe apply induction on $\\sigma(\\alpha)$. For $A\\in\\Th^\\pm(n)$ and $\\mu,\\alpha\\in\\mathbb N^{n}$, we denote\n$$g_{A,\\alpha,\\mu}=\\sum_{{\\lambda}\\in\\mathbb N^{n},\\,{\\lambda}\\geq\\alpha}f_{A,{\\lambda}}\n\\varepsilon^{-{\\lambda}\\centerdot\n(\\mu+\\text{\\rm co}(A))}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}-\\alpha}\\right]_\\varepsilon.$$\n If $\\sigma(\\alpha)=0$ then the claim follows from\n\\eqref{sum}. Now we assume $\\sigma(\\alpha)>0$. There exist $\\beta\\in\\mathbb N^{n}$ such that\n$\\alpha=\\beta+\\boldsymbol{e}_i$.\nAccording to \\ref{mul gauss binomial}(1), for ${\\lambda}\\in\\mathbb N^{n}$ with ${\\lambda}\\geq\\beta$ we have\n$$\\left[{\\mu+\\boldsymbol{e}_i+\\text{\\rm co}(A)\\atop{\\lambda}-\\beta}\\right]_\\varepsilon=\\varepsilon^{{\\lambda}_i-\\beta_i}\\left[{\\mu+\\text{\\rm co}(A)\\atop\n{\\lambda}-\\beta}\\right]_\\varepsilon+\n\\varepsilon^{{\\lambda}_i-\\beta_i-1-\\mu_i-\\sum_{1\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant n}a_{k,i}}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}-\\beta-\\boldsymbol{e}_i}\\right]_\\varepsilon.$$\nThus by the induction hypothesis we conclude that\n\\begin{equation*}\n0=g_{A,\\beta,\\mu+\\boldsymbol{e}_i}=\\varepsilon^{-\\beta_i}g_{A,\\beta,\\mu}+\\varepsilon^{-\\beta_i-1-\\sum_{1\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant n}a_{k,i}-\\mu_i}g_{A,\\alpha,\\mu}=\\varepsilon^{-\\beta_i-1-\\sum_{1\\leqslant}\\def\\geq{\\geqslant k\\leqslant}\\def\\geq{\\geqslant n}a_{k,i}-\\mu_i}g_{A,\\alpha,\\mu}.\n\\end{equation*}\nfor $A\\in\\Th^\\pm(n)$ and $\\mu\\in\\mathbb N^{n}$.\nIt follows that $g_{A,\\alpha,\\mu}=0$ for $A\\in\\Th^\\pm(n)$ and $\\mu\\in\\mathbb N^{n}$, proving \\eqref{sum1}.\n\nLet ${\\mathcal X}=\\{{\\lambda}\\in\\mathbb N^{n}\\mid f_{A,{\\lambda}}\\not=0\\text{ for some }A\\in\\Th^\\pm(n)\\}$. If ${\\mathcal X}\\not=\\emptyset$, we may\nchoose a maximal element $\\nu$ in ${\\mathcal X}$ with respect to $\\leqslant}\\def\\geq{\\geqslant$.\nThen by \\eqref{sum1} we have\n\\begin{equation*}\nf_{A,\\nu}=\n\\varepsilon^{\\nu\\centerdot\n(\\mu+\\text{\\rm co}(A))}\n\\sum_{{\\lambda}\\in\\mathbb N^{n},\\,{\\lambda}\\geq\\nu}f_{A,{\\lambda}}\n\\varepsilon^{-{\\lambda}\\centerdot\n(\\mu+\\text{\\rm co}(A))}\\left[{\\mu+\\text{\\rm co}(A)\\atop{\\lambda}-\\nu}\\right]_\\varepsilon=0.\n\\end{equation*}\nfor $A\\in\\Th^\\pm(n)$. This is a\ncontradiction. Thus $f_{A,{\\lambda}}=0$ for all $A\\in\\Th^\\pm(n)$ and ${\\lambda}\\in\\mathbb N^{n}$.\nThe assertion follows.\n\\end{proof}\n\\begin{Rem}\n(1) Let ${\\mathcal U}(\\frak{gl}_n)$ be the universal enveloping algebra of $\\frak{gl}_n$ and let ${\\mathcal U}_\\mathbb Z(\\frak{gl}_n)$ be the Kostant $\\mathbb Z$-form of ${\\mathcal U}(\\frak{gl}_n)$. Let ${\\mathcal S}_\\mbq(n,r)={\\mathcal S}(n,r)\\otimes_{\\mathcal Z}\\mathbb Q$, $U_\\mbz(n)=U(n)\\otimes_{\\mathcal Z}\\mathbb Z$, where $\\mathbb Z$ and $\\mathbb Q$ are regarded as ${\\mathcal Z}$-modules by specializing $v$ to $1$.\nLet ${\\mathcal W}_\\mathbb Z(n)$ be the $\\mathbb Z$-submodule of $\\prod_{r\\geq 0}{\\mathcal S}_\\mbq(n,r)$ spanned by the set\n$\\{A({\\mathbf{0}},{\\lambda})_1\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,{\\lambda}\\in\\mathbb N^{n}\\}$.\nAccording to \\cite[6.7(c)]{Lu90}, \\ref{realization} and \\ref{realization of barUnk} we conclude that ${\\mathcal W}_\\mathbb Z(n)$ is a $\\mathbb Z$ algebra and ${\\mathcal U}_\\mathbb Z(\\frak{gl}_n)\\congU_\\mbz(n)\/\\langle K_i-1\\mid 1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n\\rangle\\cong{\\mathcal W}_\\mathbb Z(n).$\n\n\n(2) Assume $\\varepsilon=1\\in k$. Then $l=1$ and ${\\mathcal S}_k(n,r)$ is the Schur algebra over $k$.\nLet ${\\mathcal W}_k(n)$ be the $k$-subspace of $\\prod_{r\\geq 0}{\\mathcal S}_k(n,r)$ spanned by the set\n$\\{A({\\mathbf{0}},{\\lambda})_1\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,{\\lambda}\\in\\mathbb N^{n}\\}.$\nFrom \\cite[6.7(c)]{Lu90} and  \\ref{realization of barUnk} we see that ${\\mathcal W}_k(n)$ is a $k$-algebra and ${\\mathcal U}_\\mathbb Z(\\frak{gl}_n)\\otimes_\\mathbb Z k\\cong\\ol{U_k(n)}\\cong{\\mathcal W}_k(n).$\n\\end{Rem}\n\n\nWe end this paper with a conjecture on affine $q$-Schur algebras.\nLet\n $\\Theta_\\vtg(n)$  be the set of all $\\mathbb Z\\times\\mathbb Z$ matrices\n$A=(a_{i,j})_{i,j\\in\\mathbb Z}$ with $a_{i,j}\\in\\mathbb N$ such that\n\\begin{itemize}\n\\item[(a)]$a_{i,j}=a_{i+n,j+n}$ for $i,j\\in\\mathbb Z$, and \\item[(b)] for\nevery $i\\in\\mathbb Z$, both sets $\\{j\\in\\mathbb Z\\mid a_{i,j}\\not=0\\}$  and\n$\\{j\\in\\mathbb Z\\mid a_{j,i} \\not= 0\\}$ are finite.\n\\end{itemize}\nLet\n$\\mathbb Z_\\vtg^{n}=\\{({\\lambda}_i)_{i\\in\\mathbb Z}\\mid\n{\\lambda}_i\\in\\mathbb Z,\\,{\\lambda}_i={\\lambda}_{i-n}\\ \\text{for}\\ i\\in\\mathbb Z\\}$ and $\\mathbb N_\\vtg^{n}=\\{({\\lambda}_i)_{i\\in\\mathbb Z}\\in \\mathbb Z_\\vtg^{n}\\mid {\\lambda}_i\\ge0\\}$.\nFor $r\\in\\mathbb N$ let $\\Theta_\\vtg(n,r)=\\{A\\in\\Theta_\\vtg(n)\\mid\\sigma(A)=r\\}$ and\n$\\Lambda_\\vtg(n,r)=\\{{\\lambda}\\in\\mathbb N_\\vtg^{n}\\mid\\sigma({\\lambda})=r\\}$\nwhere\n$\\sigma({\\lambda})=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}{\\lambda}_i$ and $\\sigma(A)=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n,\\,\nj\\in\\mathbb Z}a_{i,j}$. For ${\\lambda}\\in\\Lambda_\\vtg(n,r)$, let\n$\\operatorname{diag}({\\lambda})=(\\delta_{i,j}{\\lambda}_i)_{i,j\\in\\mathbb Z}\\in\\Theta_\\vtg(n,r)$.\n\nLet ${\\mathcal S}_{\\vtg}(n,r)$ be the affine $q$-Schur algebra over ${\\mathcal Z}$. It has a normalized ${\\mathcal Z}$-basis\n$\\{[A]\\mid A\\in\\Theta_\\vtg(n,r)\\}$ (see \\cite[1.9]{Lu99}). We put ${\\boldsymbol{\\mathcal S}}_\\vtg(n,r)={\\mathcal S}_{\\vtg}(n,r)\\otimes_{\\mathcal Z}\\mathbb Q(v).$\n\n\nLet\n$\\Theta_\\vtg^\\pm(n)=\\{A\\in\\Theta_\\vtg(n)\\mid a_{i,i}\n=0\\text{ for all $i$}\\}.$\nFor $A\\in\\Theta_\\vtg^\\pm(n)$, $\\delta\\in\\mathbb Z_\\vtg^{n}$ and ${\\lambda}\\in\\mathbb N_\\vtg^{n}$\nlet\n\\begin{equation*}\n\\begin{split}\nA(\\delta,{\\lambda},r)&=\\sum_{\\mu\\in\\Lambda_\\vtg(n,r-\\sigma(A))}v^{\\mu\\centerdot\\delta}\n\\left[{\\mu\\atop{\\lambda}}\\right][A+\\operatorname{diag}(\\mu)]\\in{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)\\\\\nA(\\delta,{\\lambda})&=(A(\\delta,{\\lambda},r))_{r\\geq 0}\\in\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)\n\\end{split}\n\\end{equation*}\nwhere $\\mu\\centerdot\\delta=\\sum_{1\\leqslant}\\def\\geq{\\geqslant i\\leqslant}\\def\\geq{\\geqslant n}\\mu_i\\delta_i$ and $\\left[{\\mu\\atop{\\lambda}}\\right]=\\big[{\\mu_1\\atop{\\lambda}_1}\\big]\\cdots\\big[{\\mu_n\\atop{\\lambda}_n}\\big]$.\nLet $A(\\delta)=A(\\delta,{\\mathbf{0}})$, where\n${\\mathbf{0}}=(\\cdots,0,\\cdots,0,\\cdots)\\in\\mathbb N_\\vtg^{n}$.\n\n\nWe shall denote by ${\\mathcal V}_{\\!\\vartriangle\\!}(n)$  the ${\\mathcal Z}$-submodule of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$ spanned by\n$\\{A(\\delta,{\\lambda})\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,\\delta\\in\\mathbb Z_\\vtg^{n},\\,{\\lambda}\\in\\mathbb N_\\vtg^{n}\\}.$\n\n\n\\begin{Lem}\\label{basis for afVn}\nEach of the following set forms a ${\\mathcal Z}$-basis for ${\\mathcal V}_{\\!\\vartriangle\\!}(n):$\n\n$(1)$ $\\{0(\\delta,{\\lambda})A({\\mathbf{0}})\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N_\\vtg^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\};$\n\n$(2)$ $\\{A({\\mathbf{0}})0(\\delta,{\\lambda})\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N_\\vtg^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\};$\n\n$(3)$ $\\{A(\\delta,{\\lambda})\\mid A\\in\\Theta_\\vtg^\\pm(n),\\,\n\\delta,{\\lambda}\\in\\mathbb N_\\vtg^{n},\\,\\delta_i\\in\\{0,1\\},\\forall i\\}.$\n\\end{Lem}\n\\begin{proof}\nThe assertion can be proved in a way similar to the proof of \\ref{basis for Vn}.\n\\end{proof}\n\nAccording to \\ref{Z algebra}, ${\\mathcal V}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}(n,r)$. Thus, it is natural to formulate the following conjecture.\n\\begin{Conj}\\label{conj}\n${\\mathcal V}_{\\!\\vartriangle\\!}(n)$ is a ${\\mathcal Z}$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$.\n\\end{Conj}\n\\begin{Rems}\n(1) According to \\cite{Fu}, Conjecture \\ref{conj} is true in the classical ($v = 1$) case.\n\n(2)\nLet $\\boldsymbol{{\\mathcal V}}_{\\!\\vartriangle\\!}(n)$ be the $\\mathbb Q(v)$-subspace of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$ spanned by all\n$A(\\delta)$ for $A\\in\\Theta_\\vtg^\\pm(n)$ and $\\delta\\in\\mathbb Z^{n}$.\nIt is conjectured in \\cite[5.5(2)]{DF09} that $\\boldsymbol{{\\mathcal V}}_{\\!\\vartriangle\\!}(n)$ is a $\\mathbb Q(v)$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$.\nFrom \\ref{basis of zero part} and \\ref{basis for afVn}, we see that $\\boldsymbol{{\\mathcal V}}_{\\!\\vartriangle\\!}(n)\\cong{\\mathcal V}_{\\!\\vartriangle\\!}(n)\\otimes\\mathbb Q(v)$. Thus if Conjecture \\ref{conj} is true, then we conclude that\n$\\boldsymbol{{\\mathcal V}}_{\\!\\vartriangle\\!}(n)$ is a $\\mathbb Q(v)$-subalgebra of $\\prod_{r\\geq 0}{\\boldsymbol{\\mathcal S}}_\\vtg(n,r)$.\n\n(3) If Conjecture \\ref{conj} is true, then by \\cite[3.7.3]{DDF} we conclude that the conjecture formulated in \\cite[3.8.6]{DDF} is true and ${\\mathcal V}_{\\!\\vartriangle\\!}(n)$ is isomorphic to $\\widetilde{{\\mathfrak D}}_{\\!\\vartriangle\\!}(n)$, where $\\widetilde{{\\mathfrak D}}_{\\!\\vartriangle\\!}(n)$ is a certain ${\\mathcal Z}$-module defined in \\cite[(3.8.1.1)]{DDF}.\n\\end{Rems}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nA single highly charged ion (HCI) in a Penning trap is a unique system to investigate various facets of fundamental physics. Two parameters of a HCI are of particular interest in this investigation: the mass of a HCI and the magnetic moments of its atomic electrons. The Penning trap is a tool that enables a high-precision determination of the HCI parameters via measuring under well-controlled conditions the characteristic frequencies of the HCI motion confined to a very small volume in a Penning trap.\\\\\nThe mass of a HCI is an integral characteristic that reflects the masses of its constituents and their interactions. One gets access to the energetics of various nuclear reactions via measuring the masses of the involved nuclides and particles and hence gets a handle on their properties. One prominent example are experiments on the determination of the neutrino mass and the search for keV sterile neutrinos. These experiments \\cite{ECHo,HOLMeS,SterileNeutrino} can greatly benefit from an independent determination of the $Q$-values of certain beta-processes (the mass differences of the parent and daughter nuclides of the processes). Another example is a model-independent test of the equality of the speed of light and the limiting speed of massive matter which is considered a direct test of special relativity \\cite{Gree1991}. This test is based on the investigation of the energy balance of the non-resonant capture of a cold neutron by certain nuclides \\cite{Jent2018}. The crucial parameter to be known precisely in this investigation is a mass difference of the nuclides participating in the reaction.\\\\\nHCIs are key players in the development of new-generation ion clocks \\cite{KozlovRMP2018}. They can possess meta-stable electron configurations that exhibit just a very weak sensitivity to external electromagnetic perturbations and have configuration energies of a few ten to hundred eV. These two factors can facilitate a creation of a very stable ion clock for, e.g., probing the time variation of fundamental constants \\cite{safronova2018search}. Thus, a search for such meta-stable electron configurations is of great importance and can be accomplished by a  direct measurement of the mass differences of a nuclide of interest in the same charge state but in different electron configurations \\cite{Rima2020}.\\\\\nHighly charged heavy ions with just a few electrons in the atomic shell are probably at the moment the most suitable objects to test the validity of Quantum Electrodynamics (QED) in a regime of strong electromagnetic fields. For instance, the innermost electron in heavy nuclides like lead or uranium is exposed to a Coulomb field of the nucleus of approximately 10$^{16}$ V\/cm, the value which is hardly achievable in laboratories by other means. In such a strong Coulomb field QED effects manifest themselves in the binding energies and magnetic moments ($g$-factors) of the considered electrons on a level measurable by Penning-trap techniques.\\\\ \n\n\n\n\n\n\n\\section{Basics of Penning traps and ion cooling}\n\n\\subsection{Ion motion in a Penning trap}\n\nThe Penning trap  is a superposition of two static fields - a strong uniform magnetic field $B$ and a weak quadrupolar electrostatic field created by two electrodes shaped as infinite hyperboloids of revolution. The magnetic field confines the motion of a charged particle (ion) with mass $m$ and charge $q$ to a plane perpendicular to the magnetic field lines (radial plane) whereas the quadrupolar electrostatic  potential well constrains the ion motion along the magnetic field lines. In a Penning trap an ion performs three independent motions - cyclotron, magnetron and axial motions - with frequencies $\\nu_{+}, \\nu_{-}$ and $\\nu_{z}$, respectively  (see Figure~\\ref{trap_motion}). \n\\begin{figure*}[h!]\n\\includegraphics[width=1\\textwidth]{trap_motion.png}\n\\centering\n\\caption{(left) The ion motions in a Penning trap: (1) axial motion - a periodic oscillation with frequency $\\omega_z=2\\pi\\nu_z$ in the $z$-direction in the trap electrostatic potential, (2) cyclotron and magnetron motions with frequencies $\\omega_+=2\\pi\\nu_+$ and $\\omega_-=2\\pi\\nu_-$, respectively, in the $xy$-plane perpendicular to the magnetic field lines. (right) Cylindrical Penning trap - a set of usually five ring electrodes of equal diameter that allows a creation of sufficiently quadratic electrostatic potential in the volume of ion confinement.}\n\\label{trap_motion}\n\\end{figure*}\nThe cyclotron motion is a circular motion of an ion around the magnetic field lines. The axial motion is a harmonic oscillation of an ion along the magnetic field lines in the quadrupolar electrostatic potential well. The magnetron motion is a slow circular drift of an ion in the radial plane caused by the cross product of the magnetic and electrostatic fields. The frequency hierarchy is $\\nu_{c}>\\nu_{+}\\gg\\nu_{z}\\gg\\nu_{-}$ for typical trap parameters and ion masses.\\\\ \nThe recipe for determination of the ion mass is given by the invariance theorem \\cite{Brown1982}, which is applicable not only to an ideal Penning trap, but also to existing Penning-trap systems with small imperfections in the alignment of the electric and magnetic fields and with a non-vanishing ellipticity of the trap electrodes\n\\begin{equation}\n\\frac{1}{2\\pi} \\frac{q}{m} B = \\nu_{c}=\\sqrt{\\nu^{2}_{+}+\\nu^{2}_{-}+\\nu^{2}_{z}},\n\\label{eq:Invariance}\n\\end{equation}\nwhere $\\nu_{c}$ is the  frequency which the ion cyclotron motion would have in the pure magnetic field $B$. \\\\\nHigh-precision Penning-trap mass measurements on short-lived nuclides employ a simpler relation of the cyclotron frequency to the trap eigenfrequencies \n\\begin{equation}\n\\nu_{c}=\\nu_{+}+\\nu_{-}.\n\\label{eq:SideBand}\n\\end{equation}\nIt allows a determination of the mass ratio of two ions with close masses with a relative uncertainty down to at least few parts in $10^{10}$ for all existing on-line Penning-trap facilities \\cite{Gab2009}.\\\\\n\nBesides Penning-trap mass spectrometers there is a class of ultra-precise Penning-trap experiments devoted to the determination of the $g$-factor of charged particles.\nA charged particle with mass $m_s$, charge $q_s$ and a non-zero spin performs in a magnetic field $B$ a spin precession with Larmor frequency \n\\begin{equation}\n\\nu_L = \\frac{1}{2\\pi} \\frac{g}{2} \\frac{q_s}{m_s} B.\n\\label{eq:Larmor}\n\\end{equation}\nThus, in order to determine the $g$-factor, it is necessary to measure the Larmor frequency and the magnetic field strength $B$. For example, in the case of a $g$-factor of an electron bound in a highly charged ion with known mass $m$ and charge $q$, the magnetic field strength is determined by measuring the cyclotron frequency $\\nu_c$ of the ion. Such a combined measurement yields for the $g$-factor of a bound electron the following expression:\n\\begin{equation}\ng = 2\\frac{\\nu_L}{\\nu_c} \\frac{m_s}{m} \\frac{q}{q_s}.\n\\label{eq:g_factor}\n\\end{equation}\n\nA real Penning trap always differs from the ideal case. This leads to the fact that the measured eigenfrequency values in general exhibit certain non-vanishing offsets from the ideal Penning-trap values. There is quite a number of effects in the real Penning trap that can cause such frequency offsets. The most important ones are a misalignment of the trap and magnetic field axes, an ellipticity of the trap, a non-harmonicity of the electrostatic trap potential, and an inhomogeneity of the magnetic field. Also the image-charge effect and the relativistic dependence of the ion cyclotron frequency on its kinetic energy becomes non-negligible in the case of ultra-precise frequency-ratio measurements on highly charged ions.\\\\ \n\n\\subsection{Cooling of the ion motions in a Penning trap}\n\nA prerequisite for high-precision measurements of the ion cyclotron frequency is a confinement of the ion motions in the Penning trap to a small volume. This is achieved by various \"ion cooling\" techniques. The term \"ion cooling\" means in this context a reduction of the ion motion amplitudes. There are four distinctively different ion cooling techniques that are employed in Penning traps: (1) mass-selective buffer gas cooling \\cite{Sav1991}, (2) electron cooling \\cite{Rol1989,Ber2004}, (3) resistive cooling \\cite{Win1975} and (4) laser cooling \\cite{Lar1986}. \\\\\nThe mass-selective buffer gas cooling technique is a method of choice in online Penning-trap experiments \\cite{SHIPTRAP,TRIGATRAP,JYFLTRAP,ISOLTRAP,LEBIT,TITAN,APT}.  Short-lived nuclides of interest are synthesized in nuclear reactions in the form of high-energy highly charged ions, are slowed down in a gas-filled stopping chamber to thermal energy reducing their charge state to 1+, 2+ and rarely 3+, are converted in a gas-filled linear Paul trap from a continuous beam of low charged ions into ion bunches and finally are delivered to a Penning-trap mass spectrometer, which is usually a tandem of two Penning traps: cooler trap and measurement trap. The mass-selective buffer gas cooling technique is applied in the cooler trap, which is filled with helium gas at a typical pressure of a few 10$^{-5}$ mbar and a room temperature of about 300 K. The cyclotron and axial ion motions can be considered stable oscillators. Thus, in the cooler trap they come into thermal equilibrium with helium atoms. The amplitude of the magnetron motion being metastable increases due to collisions with helium atoms. The \"cooling\" of the magnetron motion is achieved by transferring its action to the cyclotron motion by means of a $\\pi$-pulse at the sideband frequency $\\nu_+$+$\\nu_-$. As a result, the \"cooled\" low charged ions in the cooler trap have cyclotron and magnetron amplitudes of just about 0.1 mm, whereas the amplitude of the axial motion does not exceed 1 mm. The virtue of this cooling technique is that it allows for fast cooling (sub-100 ms) of trap motions of low charged ions at room temperature to sub-mm amplitudes. The largest disadvantage of the technique is that it is not applicable to highly charged ions.\\\\\nOne Penning-trap facility, namely TITAN \\cite{TITAN}, stands out from the group of online Penning-trap mass spectrometers in that it employs highly charged ions to measure their masses. Since a \"cooling\" of highly charged ions in neutral gas is not possible, an electron cooling technique is proposed instead \\cite{Rol1989,Ber2004}. The TITAN's cooler trap contains a plasma of electrons that are injected into the cooler trap from an electron gun. Highly charged ions trapped in the cooler trap are sympathetically cooled by the co-trapped electron plasma to a common energy without bonding with the electrons.\\\\\nIn Penning traps that aim at a measurement of the ion cyclotron and Larmor frequencies with a relative uncertainty well below $10^{-10}$ a resistive cooling technique is a working-horse method of decreasing the ion trap-motion amplitudes to a $\\mu m$ scale \\cite{Win1975}. The resistive cooling is realized by connecting to one of the trap electrodes a superconducting NbTi resonator with a very high quality factor. The inductance of the resonator and the capacitance of the trap form a resonant circuit which resonance frequency is usually tuned to that of the ion's axial or cyclotron motion. The resonator has a certain impact on the ion motion. From one side, the voltage induced on the electrodes by the ion motion creates across the trap an electric field that tends to decrease the amplitude of the ion motion. From the other side, the thermal noise of the resonator tends to excite the ion motion. As a result, the ion motion finally comes into a thermal equilibrium with the resonator. Since the resonator and the trap are usually kept at a temperature of liquid helium, the amplitude of the thermalized ion motion does not exceed a few $\\mu m$.\\\\\nFor some applications temperatures below the 4K environment are required. This is the case in the very strongly inhomogeneous magnetic bottle fields required for the detection of the spin state  via the continuous Stern-Gerlach effect in $g$-factor experiments (see chapter \\ref{eq:g_factor}). Here, especially the residual cyclotron energy causes sizeable shifts of the axial frequency and can consequentially cause an unacceptable frequency instability if not kept constant carefully. As the heating rate due to residual noise on the electrodes scales approximately linearly with the total cyclotron energy \\cite{bohman2018sympathetic} it is advantageous or even mandatory to cool the motion to very low temperatures. Similarly, for high-precision laser spectroscopy, the linear Doppler shift resulting from the axial motion is the dominant source of line broadening if the ion is not cooled into or close to the Lamb-Dicke regime where individual motional sidebands can be resolved.\\\\\nDoppler laser cooling of trapped alkali-like ions such as Be$^+$ and Mg$^+$ can reach temperatures of about 0.5~mK or even lower. Since the ion is trapped, a single laser beam (for an almost closed transition) is sufficient to cool e.g. the axial motion. Radiofrequency coupling of the motional sidebands can then cool the other modes as well. Generally however, HCI do not feature fast optical transitions suitable for direct laser cooling. Consequentially, these ions of interest have to be cooled sympathetically via interaction with directly cooled ions. This interaction can take place either via Coulomb interaction for co-trapped ions, or via the induced image charges in shared electrodes. The latter technique has the important advantage that the motional frequencies of the HCI in that case are not significantly modified. This however comes at the cost of a much weaker coupling strength and consequentially slower cooling rate, which can only partially be counteracted by using not a single but a cloud of typically some hundred Be$^+$ ions. Currently, advanced techniques for the efficient coupling of the two species are in development at several experiments. Once successful, this will pave the way to ultra-high precision experiments on various HCI using laser and microwave spectroscopy as well as mass spectrometry.      \n\n\n\\subsection{Measurement of the free cyclotron and Larmor frequencies}\n\nThere is a considerable difference in methods employed at different classes of Penning traps for a measurement of the free cyclotron frequency. Each class is characterized with certain distinct features, which are listed in Table~\\ref{tab:Features}.\n\\begin{table} [h!]\n\\centering\n\\caption{A list of features which distinctly characterize each class of high-precision Penning traps.}\n \\label{tab:Features}\n\\begin{tabular*}{1\\textwidth}{@{\\extracolsep{\\fill}}cc}\n\\hline\n\\hline\non-line facilities for experiments & off-line setups for experiments\\\\\n on short-lived nuclides &  on long-lived or stable nuclides \\\\\n\\hline\n\\rowcolor{Gray}\noperated at room temperature & operated at 4.2 K \\\\\ntypically low charged ions used & highly charged ions used \\\\\n\\rowcolor{Gray}\nions with half-lives as short as a few ms & very long-lived or stable ions \\\\\nvery low on-line production & off-line production of ions\\\\\nrates of nuclides &  in large quantity \\\\\n\\rowcolor{Gray}\nuncertainty in mass determination &  uncertainty in mass-ratio determination \\\\\n\\rowcolor{Gray}\nis a few 100 eV\/c$^2$ up to a few keV\/c$^2$ &  is below a few eV\/c$^2$ \\\\\nfrequency-measurement technique: &  frequency-measurement technique: \\\\\nToF-ICR, PI-ICR; destructive &  FT-ICR; non-destructive \\\\\n\\hline\n\\hline\n\\end{tabular*}\n\\end{table}\n\nAt online facilities two cyclotron-frequency techniques are in use: the Time-of-Flight Ion-Cyclotron-Resonance (ToF-ICR) \\cite{ToF1980,ToF1995,ToF1992,ToF2007_1,ToF2007_2,ToF2007_3} and Phase-Image Ion-Cyclotron-Resonance (PI-ICR) \\cite{PIICR_1,PIICR_2} techniques. The ToF-ICR has been the method of choice until now due to its relative simplicity and sensitivity to a single ion in the trap at room temperature. Since this method is destructive, at least a few hundred ions are required to reach a sufficiently low uncertainty in mass determination.  The recently developed novel Phase-Image Ion-Cyclotron-Resonance (PI-ICR) technique is approximately by a factor of 25 faster than the ToF-ICR method and offers by a factor of 40 higher resolving power. It has a perspective to replace the ToF-ICR technique in mass measurements on very short-lived nuclides and low lying nuclear isomeric states \\cite{Manea2020}.\\\\\nIn cryogenic off-line setups a set of frequency-measurement techniques united under a common name \"the Fast-Fourier Ion-Cyclotron-Resonance (FT-ICR) technique\" is employed \\cite{FTICR_1,FTICR_2,FTICR_3,FTICR_4,FTICR_5}. It is based on a measurement of the frequency of a current (image current) induced by the ion trap motion in a superconducting resonant tank which is attached to one of the trap electrodes. This technique allows frequency measurements on a single ion with an extraordinary low uncertainty. The price to pay for this is a high complexity of the system, which requires a cooling of the trap as well as the associated frequency-measurement electronics to the temperature of liquid helium.\\\\ \nIn experiments on the determination of the $g$-factor one has to measure besides the free cyclotron frequency also the Larmor frequency of the spin precession. Since the spin precession is not associated with a charge oscillation, the techniques based on the image-current frequency measurement are not applicable. Instead, a technique that make use of the continuous Stern-Gerlach method is employed \\cite{dehmelt1986continuous,Stur2019}. It is based on the detection of the spin flip between two spin states of a charged particle of interest, which is placed in a Penning trap with strong quadratic magnetic inhomogeneity.\\\\      \n\n\\section{Production and properties of highly charged ions}\nWhile the Penning trap supports precision spectroscopy with arbitrary charged atomic and molecular ions, especially for fundamental physics applications a high charge state $Z$ can be a decisive advantage. This can be either simply due to the larger charge-to-mass ratio, which leads to a larger cyclotron frequency and correspondingly improved measurement precision, or due to the special atomic structure of such ions. When removing most of the electrons of an atom, the resulting HCI has a drastically simpler electron shell, which allows for high-precision atomic physics calculations. Moreover, the remaining electrons are then bound much stronger to the nucleus, where they experience the strongest electromagnetic fields accessible today in the laboratory in stable systems - up to $10^{16}$~V\/cm electric and 20,000~T magnetic fields. Accordingly, HCI are close to ideal candidates for tests of the fundamental laws of physics, the Standard Model, in extreme conditions. Furthermore, the presence of the strong field decreases the susceptibility to external influences such as stray electric and magnetic fields, again enabling even higher precision.\nThe strong fields also govern the scaling behaviour of the energy levels in the HCI. The expectation value of the electron radius moves closer to the nucleus with $1\/Z$, and correspondingly the electronic transitions of the gross structure scale roughly proportional to $Z^2$, which quickly shifts them into the x-ray and even gamma regime. On the other hand, the fine- and hyperfine structure transitions, which are typically in the low microwave range of frequencies for atoms, scale by $Z^4$ and $Z^3$, respectively, which makes them accessible to precision laser spectroscopy studies in a suitable range of $Z$. Only the Zeeman splittings, which originate from an externally applied magnetic field, remain in the millimeter regime (for typical 2-7 Tesla magnetic fields) throughout the complete range of $Z$. This makes these transitions and the associated $g$-factor especially versatile for fundamental physics tests.\nOn the other hand, the high $Z$ of course also increases the ionisation potential and correspondingly imposes significant difficulty for the production and storage of HCI. In the extreme cases of hydrogenlike Pb$^{81+}$ or Bi$^{82+}$ the energy required to remove the 1s electrons is higher than 100 keV. Such energies require specialized large-scale installations as there exist only a few world-wide. One possibility is to accelerate singly charged ions to GeV energies and then to strip the other electrons in a metallic foil. This procedure, as it is implemented at the experimental storage ring (ESR) at the GSI Helmholtzzentrum f\u00fcr Schwerionenforschung in Darmstadt, Germany \\cite{franzke1987heavy}, results in HCI produced in flight. To make these ions accessible for Penning-trap experiments they have to be cooled and decelerated almost to rest. This elaborate procedure is currently developed at the HITRAP facility at GSI \\cite{kluge2008hitrap}. \nThe alternative to the ion-accelerator based method is to use a high-energy electron beam impinging on the ions. As the energy transfer in electron-electron collisions is drastically more efficient than in the ion-electron case, here, energies in the 200~keV range suffice. This principle is used in electron-beam ion traps (EBITs). Here, the ions are confined in a strong magnetic field by means of cylindrical electrostatic electrodes as well as the space-charge of the electron beam which extends axially along the magnetic field lines. The trapped ions are then successively charge-bred to higher charge states, limited only by the available electron energy and recombination with rest-gas atoms from the imperfect vacuum. \nTechnically, however, it is challenging to supply strong electron beams in the order of Amp\\`eres with the required $>$100~keV energies. Currently, only very few machines world-wide operate in this regime - one of those is the Heidelberg Super-EBIT which is currently upgraded to enable the highest energies. The produced ion cloud consists of a distribution of charge-states at a typically quite high temperature owing to the collisions with the electron beam. However, unlike in the accelerator case, the energy distribution is typically only in the several eV range, allowing a direct capturing in a cryogenic Penning trap without preceding cooling steps, albeit with finite capturing efficiency. At the ALPHATRAP setup, ions are extracted from the HD-EBIT, selected by q\/m and injected into the cryogenic trap with typically a few percent efficiency, absolutely enough considering the experiment requires only a single ion. In hermetically sealed cryogenic traps the vacuum can be so good that recombinations are basically eliminated, allowing for storage times in the order of months even for HCI. In this case it becomes feasible to keep a number of the ions of interest in a dedicated \"reservoir\" trap \\cite{Stur2019}. This way, if an ion gets lost in the measurement or setup procedure, a new one can be extracted from the reservoir without the need to run the EBIT again, which drastically increases the productivity.       \n\n\n\\section{Precision Penning-trap mass spectrometry}\n\\label{ch:MassSpec}\nHigh-precision atomic and nuclear mass data have broad applications in numerous fields of research including nuclear and nuclear astrophysics, neutrino physics, metrology, tests of fundamental interactions and their symmetries as well as physics beyond the Standard Model. In the following some exciting examples will be discussed in more details.\n\n\\subsection{Nuclear masses for test of special relativity}\nSpecial relativity is one of the deepest principles in our understanding of modern physics. Its prominent role has motivated many tests of its foundations and predictions. These tests of special relativity can be divided in several categories according to which assumption they investigate: the isotropy of space, the independence of the speed of light from the velocity of the laboratory, the time dilation effect, and the equality of two distinct \"speeds of light\" \u2013 the velocity of propagation of an electromagnetic wave in vacuum, c, and the limiting velocity of a massive particle, cm. Only the latter does not assume the existence of a preferred frame of reference and hence can be considered a direct test of special relativity. \nIn 1991, G.L. Greene et al. proposed to test this equality, i.e., $c = c_m$ as special relativity predicts, by measuring the wavelength of a photon emitted in a transition where a mass $\\Delta m$ is converted into electromagnetic radiation \\cite{Gree1991}. From energy conservation it follows that $\\Delta mc_m^2 = hc\/\\lambda$, where $\\lambda$ is the photon wavelength and $h$ is the Planck constant. One considers the non-resonant capture of a cold  neutron in the reaction $^AX(n,\\gamma)^{A+1}X$: from energy conservation one obtains $[M(^AX) \u2013 M(^{A+1}X) + M(n)] c_m^2 = hc\/\\lambda_{A+1}$, where $\\lambda_{A+1}$ is the wave length of a $\\gamma$-ray emitted by the nuclide $^{A+1}X$ after a neutron capture by the nuclide $^AX$. The mass difference of two nuclides in the reaction can be measured very precisely with Penning-trap mass spectrometry, whereas the right tool to measure the wavelength is crystal Bragg spectroscopy \\cite{Jent2018}.\nIt took more than ten years until such a test was carried out by combining very accurate measurements of atomic-mass ratios of $^{29}$Si\/$^{28}$Si and $^{33}$S\/$^{32}$S with the MIT-trap \\cite{Corn1989} with a relative uncertainty of about $10^{-11}$, and of $\\gamma$-ray wavelengths in the GAMS4 experiment \\cite{Kess2001} with fractional accuracy of about $10^{-7}$. The test yielded a result of $1 - c_m\/c = -1.4 (4.4)\\cdot10^{-7}$, indicating that the velocity of propagation of an electromagnetic wave in vacuum and the limiting velocity of a massive particle are equal to a level of at least $4\\cdot10^{-7}$ \\cite{Rain2005}. To our knowledge, this is the most precise direct test of the famous equation yet described. \nThe sensitivity of about $4\\cdot10^{-7}$ on the equality of two velocities achieved in the MIT-trap and GAMS4 experiments is limited by the inability of the GAMS4 Bragg spectrometer to measure the wavelength of a $\\gamma$-ray with a fractional uncertainty better than $10^{-7}$. An order of magnitude improved precision in the $\\gamma$-ray measurement and thus in the test of special relativity seems to be possible with the upgraded GAMS6 spectrometer \\cite{G\u00fcnt2017}. However, a suitable nuclide must be selected, which should fulfil the following requirements: (1) it must have a relatively long lifetime on the order of at least several weeks to provide precise mass values, (2) it must have a large neutron capture cross section, (3) a mechanical form or chemical compound must be available, which can resist high temperatures inside the reactor, (4) the branching rate of a single cascade must be sufficiently high and (5) the $\\gamma$-decay scheme of the daughter nuclide in the reaction must be relatively simple. According to these requirements, the pair $^{35}$Cl\/$^{36}$Cl is one of the most promising in case the mass ratio gets measured with the required uncertainty of about $10^{-11}$. Presently, this mass ratio is known with a fractional uncertainty of only about $10^{-9}$ \\cite{Audi2017}. We aim to improve the mass uncertainties of $^{35,36}$Cl by two orders of magnitude. The measurement will be carried out with PENTATRAP Penning-trap mass spectrometer \\cite{Repp2012} using the laser Tip-EBIT technique for the production of highly charged Cl ions \\cite{Schw2019}. \n\n\n\\subsection{Nuclear masses for fifth force search}\nRecently, it has been suggested that ultrahigh-precision determinations of isotope shifts based on high-resolution laser spectroscopy can be used to detect interesting new particle candidates \\cite{Bere2018}. To that end the linearity of King plot isotope shift comparisons gets tested (see Figure~\\ref{fifth_force}) and provides bounds on the existence of new light force mediators which can improve existing bounds even by orders of magnitude \\cite{Frug2017}.  \n\nThe optical isotope shift between two isotopes $A$ and $A'$ can be written as:\n\\begin{equation}\n\\nu_i^{AA'} = K_i\\mu_{AA'}+F_i\\delta<r^{2}>_{AA'},\n\\label{eq:IS}\n\\end{equation}\nwith \n\\begin{equation}\n\\mu_{AA'}=\\frac{1}{m_A}-\\frac{1}{m_{A'}},\n\\label{eq:Massterm}\n\\end{equation}\nwhere the masses of isotopes $A$ and $A'$ are denoted as $m_A$ and $m_{A'}$, respectively. The first term in Eq.~(\\ref{eq:IS}) expresses the mass shift (normal as well as specific mass shift), the second term the leading-order contribution to the field shift, which scales with the difference of mean squares of nuclear charge radii between isotopes $A$ and $A'$. In order not to be limited by the masses of the involved isotopes, high-precision Penning-trap mass measurements on the isotopic chains of interest, e.g. Ca, Sr, and Yb \\cite{Solaro2020,Counts2020}, needs to be performed with relative mass uncertainties of $\\delta m\/m=1\\cdot10^{-11}$, as recently demonstrated by PENTATRAP \\cite{Rischka2020}.\n\\begin{figure*}[h!]\n\\includegraphics[width=0.5\\textwidth]{fifth_force.png}\n\\centering\n\\caption{Illustration of linearity (orange) and nonlinearity (red) in the King plot of the isotope shifts in transition 1 versus transition 2.}\n\\label{fifth_force}\n\\end{figure*}\n\n\\subsection{Nuclear masses for neutrino physics studies}\n\nA precise knowledge of the $Q$-values of a variety of nuclear single and double beta processes is essential for addressing such open issues in neutrino physics as (1) the neutrino mass value, (2) the neutrino type and (3) the existence of keV-sterile neutrinos.\\\\\n\\indent From several established approaches for the determination of the neutrino mass the investigation of single beta processes ($\\beta^-$ decay and electron capture (EC)) is the most convenient and moreover the only model-independent method to determine the mass of the electron neutrino - one of the neutrino flavor states. Several experiments currently struggle to reach a sub eV\/c$^2$ uncertainty in the determination of the (anti)neutrino mass from the analysis of the $\\beta^-$ decay of tritium \\cite{KATRIN,Project8} and the EC in $^{163}$Ho \\cite{ECHo,HOLMeS}. The most advanced experiment on the determination of the anti-neutrino mass from the tritium $\\beta^-$ decay - the Karlsruhe Tritium Neutrino experiment KATRIN - has recently obtained a value of 1.1 eV\/c$^2$ (90\\% confidence level) for an upper limit on the absolute mass scale of neutrinos \\cite{KATRIN2}. Presently, this is the most stringent limit on the electron neutrino mass ever reached. The KATRIN project can benefit from a very precise direct and independent measurement of the $Q$-value of tritium $\\beta^-$ decay. It can help to assess the systematic uncertainty in the neutrino-mass determination more reliably if the $Q$-value be measured with an uncertainty better than a few meV\/c$^2$. The FSU-trap has recently advanced towards this by having determined the $Q$-value with an uncertainty of 70 meV via measuring the mass difference of tritium and $^3$He \\cite{FSU-trap}. Another suitable beta process for a precise determination of the neutrino mass - the electron capture in $^{163}$Ho - underlies two experiments, ECHo \\cite{ECHo} and HOLMES \\cite{HOLMeS}. The neutrino mass is determined here from the analysis by means of cryogenic microcalorimetry electron de-excitation processes which follow the electron capture. Two Penning-trap experiments, PENTATRAP \\cite{Repp2012} and CHIP-trap \\cite{CHIP-trap}, have on their measurement list the determination of the $Q$-value of the EC in $^{163}$Ho with an uncertainty of approximately 1 eV. This will facilitate the determination of the neutrino mass on a sub eV\/c$^2$ precision level.\\\\\n\\indent It is not clear yet whether the neutrino and antineutrino are different (Dirac) or identical (Majorana) particles. Presently, the most suitable way to determine the neutrino type is the investigation of extremely rare double beta processes. Nuclides which can undergo these processes are in fact virtually stable. They can emit two electrons (2$\\beta^-$ decay) or can capture two atomic orbital electrons (2EC) either with or without an emission of two neutrinos. Of interest is only the neutrinoless mode (0$\\nu$2$\\beta^-$ decay and 0$\\nu$2EC), since it  can occur only if neutrinos are Majorana particles. Until now all large-scale experiments have been devoted to the search for 0$\\nu$2$\\beta^-$ decay \\cite{Double_Beta}, since in general, it is by far more probable than 0$\\nu$2EC. In these experiments one searches in the energy-sum spectrum of two emitted electrons for a peak at an energy which corresponds to the $Q$-value of the investigated 2$\\beta^-$ decay. The $Q$-value needs to be known with an uncertainty of approximately one keV in order (1) to localize the searched peak in the two-electron spectrum and (2) to compute with sufficient precision the nuclear matrix element of the process. All most prominent $2\\beta^-$ decay  transitions have been already addressed by various high-precision Penning-trap facilities by determining their $Q$-values  with sufficient accuracy \\cite{136Xe,76Ge,130Te,100Mo,82Se,116Cd,48Ca,150Nd,96Zr}. \nThe other neutrinoless double beta process, 0$\\nu$2EC, is in general so improbable that it is considered unobservable. Nevertheless, the probability of certain 0$\\nu$2EC transitions can be resonantly enhanced to such extent that it can become comparable to that of $2\\beta^-$ decay  transitions. The degree of the resonant enhancement of a certain 0$\\nu$2EC transition is solely defined by the difference of the binding energy of two captured electrons and the  $Q$-value of the transition.\nThree 0$\\nu$2EC transitions between the ground states and 12 0$\\nu$2EC transitions from excited states to the ground state  were considered to possess a certain degree of resonant enhancement. Their $Q$-values have been determined with high-precision Penning traps with uncertainties ranging from a few 100 eV down to a few 10 eV \\cite{2EC2012}. This extensive experimental $Q$-value measurement campaign has found two at least partially resonantly enhanced 0$\\nu$2EC-transitions, $^{152}$Gd$\\rightarrow^{152}$Sm \\cite{Ga152} and $^{156}$Dy$\\rightarrow^{156}$Gd \\cite{Dy156}. Their expected half-lives normalized to the effective Majorana neutrino mass value of 1 eV exceed 10$^{26}$ years rendering it impossible to observe the 0$\\nu$2EC-decay of these nuclides by means of present experimental technique. \\\\  \n\\indent Along with three active neutrinos one predicts the existence of sterile neutrinos - presumably Warm Dark Matter particles - with the mass ranging from 1 to 100 keV \\cite{SterileNeutrino}. Although sterile neutrinos do not undergo weak interaction, they mix with active neutrinos with certain mixing angles and thus reveal their existence in beta processes which are accompanied by an emission of neutrinos. One of the most suitable beta processes for searching for keV-sterile neutrinos is electron capture. The ECHo project on the determination of the neutrino mass plans to extend their measurement program to searching for keV-sterile neutrinos by analyzing the cryogenic microcalorimetric atomic de-excitation spectrum of the EC in $^{163}$Ho \\cite{SterileNeutrino}. If keV-sterile neutrinos exist, they manifest themselves as a small kink in the spectrum at the energy that corresponds to the difference of the $Q$-value of the process and the mass of the sterile neutrino (see Figure~\\ref{kink}).\n\\begin{figure*}[h!]\n\\includegraphics[width=1\\textwidth]{kink.png}\n\\centering\n\\caption{(a) Comparison between the expected calorimetrically measured $^{163}$Ho spectrum in the case of no sterile neutrino (black dashed line) and in the case of a heavy neutrino mass $m_4$ = 2 keV with a mixing of $U^2_{e4}$ = 0.5. (b) A magnification of (a) in the region of the kink. The plots are taken from \\cite{SterileNeutrino}.}\n\\label{kink}\n\\end{figure*}\nThe EC in $^{163}$Ho is sensitive to sterile neutrinos with mass smaller than 2.5 keV. The employment in the ECHo or similar experiment other EC transitions with larger $Q$-values will allow for a search for keV-sterile neutrinos with masses up to 100 keV. The list of the most suitable EC-transitions for the search for keV-sterile neutrinos includes, for instance, the EC in $^{157}$Tb, $^{163}$Ho, $^{179}$Ta, $^{193}$Pt, $^{205}$Pb \\cite{SterileNeutrino}. The precision in the determination of the position of the kink and hence the sterile neutrino mass is directly related to the uncertainty of the determination of the $Q$-value. Thus, an independent and direct determination of the $Q$-values of the above-mentioned EC-transitions is required on a level of at most a few eV-uncertainty and falls into the scope of the PENTATRAP project.\\\\\n\n\n\\subsection{Determination of electron binding energies and test of QED}\n\nThe validity of quantum electrodynamics (QED), so far the best confirmed theory, is persistently questioned in theory and experiments on various atomic systems.\nOf particular interest are hydrogen-like ions of heavy nuclides (Pb$^{81+}$, U$^{91+}$). These ions are simple systems of a single electron in a strong Coulomb field (10$^{16}$ V\/cm) of the ion nucleus. Thus, these systems give a unique opportunity to test QED in the strongest electromagnetic fields available in experiments.\nUnlike light-ion systems QED calculations of hydrogen-like heavy ions have to be performed non-perturbatively in $\\alpha Z$ regime. This imposes serious difficulties on these calculations requiring experimental investigations of such systems. One of such investigations is a measurement of the binding energy of the electron in hydrogen-like heavy ions, which allows for the determination of the Lamb shift - the difference between the exact binding energy and the point-nucleus Dirac binding energy.\nThe Lamb shift in H-like uranium is presently known with an uncertainty of 4.6 eV \\cite{LambShift}, thus providing a test of QED in the strong Coulomb field on a 2$\\%$ level.\nThe PENTATRAP experiment plans to substantially reduce the uncertainty in the electron binding energy determination by measuring the mass difference of hydrogen-like and bare heavy nuclides with a sub-eV\/c$^2$ uncertainty.\\\\\n\n\n\\section{Precision magnetic moment measurements of bound electrons}\nThe extraordinary control over the motion of single trapped ions in Penning traps, combined with novel cooling techniques, allows achieving impressive measurement precision of the magnetic moment of bound electrons in highly charged ions with numerous applications, which will be discussed in the following.\n\n\\subsection{Stringent test of bound-state quantum electrodynamics}\nThe determination of the $g$-factor of the electron bound in hydrogen like ions allows for stringent tests of bound-state quantum electrodynamics. The result of the experiment on $^{28}$Si$^{13+}$ for the $g$-factor of the single bound electron is $g_{exp} = 1.995\\,348\\,958\\,7\\,(5)(3)(8)$ \\cite{Stur2011}. The numbers in parenthesis correspond respectively to the systematical and statistical uncertainties and the error of the electron mass at that time. The result agrees well with the theoretical value $g_{theo}= 1.995\\,348\\,958\\,0\\,(17)$, which includes QED contributions up to the two-loop level of the order of $(Z\\alpha)^2$ and $(Z\\alpha)^4$ and represents the so far most stringent test of bound-state QED calculations in strong external fields (see Figure~\\ref{GFactor_Contrib}).\n\\begin{figure*}[h!]\n\\includegraphics[width=0.7\\textwidth]{GFactor_Contrib.png}\n\\centering\n\\caption{QED (blue and green), nuclear size (black) and recoil (orange) contributions to the $g$-factor bound in hydrogen-like ions for $Z=6$ (carbon) up to $Z=92$ (uranium) \\cite{Debierre2020}. The figure is adapted from \\cite{PhysRevLett.108.063005} and with results from \\cite{PhysRevLett.120.043203,PhysRevLett.89.081802,PhysRevA.99.012505}.}\n\\label{GFactor_Contrib}\n\\label{GFactor_Contrib}\n\\end{figure*}\n\nSimilar experiments with comparable precision have been obtained on H-like $^{12}$C$^{5+}$ \\cite{Haef2000} and $^{16}$O$^{7+}$ \\cite{Verd2004}. Also experiments using lithium like ions $^{40}$Ca$^{17+}$ and $^{48}$Ca$^{17+}$ \\cite{Koeh2016}, and $^{28}$Si$^{11+}$ \\cite{Wagn2013, Glazov2019} have been performed. Here the interaction of the additional 2 bound electrons modify slightly the value of the $g$-factor. The comparisons between theory and experiment test calculations of the inter-electronic interaction. Most recently the $g$-factor of the boron-like ion $^{40}$Ar$^{13+}$ has been determined with high precision \\cite{Arap2019} at the newly commissioned ALPHATRAP experiment at the Max-Planck-Institute for Nuclear Physics in Heidelberg, Germany \\cite{Stur2019}. The experimental result distinguishes between disagreeing predictions for the contribution of the electron-electron interaction.\n\nThe ALPHATRAP experiment opens the door for ultra-high precision $g$-factor experiments in the highly interesting high-$Z$ regime, and would allow the testing of the most stringently bound-state QED predictions and the achievement of a high sensitivity to physics beyond the Standard Model. The nuclide $^{208}$Pb seems to be the candidate of choice here since it is a heavy ion ($Z = 82$), its nuclear structure is best understood due to its doubly magic character (closed neutron and proton shell), and the high charge state of up to 81+ of these ions provides an about 3 orders of magnitude stronger field compared to silicon, the so far heaviest hydrogenlike system studied in a closed Penning trap. \n\n\\subsection{Measurement of fundamental constants}\nThe ability to measure the magnetic moment in a large variety of elements and charge states on the other hand allows selecting systems optimised for specific goals. Since the calculation of the $g$-factor, or specifically the magnetic moment of the ion depends on a number of fundamental constants it becomes possible to determine values of those constants by measuring the magnetic moment in a suitable ion or combination of ions and comparing it to the theory value. Of course, in order to do so, it is necessary to trust in the validity of the QED calculation. This, however, is feasible e.g. in the low-$Z$ regime, as we have already confirmed QED in the stronger fields of the medium-$Z$ regime. Furthermore, in the low-$Z$ regime the higher order (in $(Z\\alpha)$) contributions of QED are still relatively small and under control. In the past, we have demonstrated this principle by determining the atomic mass of the electron from a measurement of the magnetic moment of hydrogenlike $^{12}$C$^{5+}$ with an unprecedented precision of about 30~ppt. As the Larmor frequency of the bound electron is (anti-) proportional to the electron mass (via the Bohr magneton), but the cyclotron frequency of the ion depends on the ion mass, a $g$-factor measurement relates the electron mass to the ion mass, which can in the case of $^{12}$C$^{5+}$ easily be related to the atomic mass unit $u$. The measurement at the time had been limited by the measurement precision, dominantly by the magnetic field stability. Improvements in this respect make it feasible to revisit this measurement and improve the literature value, which is especially appealing in the light of currently ongoing efforts to achieve a comparable precision with laser spectroscopy of molecular HD$^+$ ions \\cite{alighanbari2020precise,patra2017proton}. \nThe developed techniques can of course also be applied to direct cyclotron frequency mass spectrometry as discussed in chapter \\ref{ch:MassSpec}. For fundamental atomic physics applications the lightest ions, such as the proton, deuteron, tritium and $^3$He as well as the neutron are of special interest. Measurements of these ions however suffer from the large energy-dependent systematic shifts originating from their small mass. This requires a specially optimised, extremely harmonic trap, which also allows detecting the small signals induced by these ions and exquisite control on the radiofrequency excitations and the resulting radii. At the Johannes Gutenberg-University in Mainz we have set up such a system, LIONTRAP \\cite{heisse2019high} , which has been recently able to perform previously unprecendented measurements of these light ions  and to unveil important discrepancies in the tabulated literature values \\cite{heisse2017high}.     \nAnother interesting opportunity is a determination of the electromagnetic finestructure constant $\\alpha_\\textrm{em}$, denoted here simply as $\\alpha$. This quantity directly enters the value of $g$ for bound electrons, on the one hand via the QED series solution, but also already on the relativistic level. Since the electron is bound to the nucleus it has to satisfy the Dirac equation for the hydrogenlike atom. The value for $g$ determined from that solution was first calculated by Breit \\cite{breit1928magnetic} and depends on $\\alpha_\\textrm{em}$. Especially for high-$Z$ systems the influence is sizeable and overcomes the impact of the QED contributions. Correspondingly, a determination of $\\alpha$ from the bound electron $g$-factor is mostly independent of alternative determinations from the free electron $g$-2 \\cite{hanneke2008new} or recoil spectroscopy of atoms \\cite{parker2018measurement}, allowing for stringent tests of QED across large variety of physical systems. Especially the $g$-factor of high-$Z$ systems is strongly influenced by the nuclear structure, particularly the nuclear charge radius. On the one hand, this can be exploited in order to determine values for the charge radii with competitive precision as we have shown in \\cite{sturm2011g,sturm2013g}, on the other hand the uncertainty of the tabulated nuclear properties poses a limitation on the achievable precision of an $\\alpha$ determination in the high-$Z$ regime.\nOne proposed solution \\cite{shabaev2006g} is to use specifically weighted differences of $g$-factors of HCI in different charge states, typically hydrogen- and boronlike. As a 2p-state the boronlike ions have very little overlap with the nucleus, while the 1s state has significantly more. By taking a weighted difference it is thus possible to cancel the small nuclear influence on the boronlike ion using a small admixture of the hydrogenlike $g$-factor. Currently, this procedure is hindered by the challenging theoretical calculation of the 5 electron system to the required precision, but work in this direction is currently actively ongoing. An alternative approach appears when considering low-$Z$ ions. Here, the theoretical precision is markedly higher and the influence of the nuclear properties is considerably smaller, however, also the dependence on $\\alpha$ is reduced to an extend that currently the experimentally achievable precision would be insufficient for a competitive determination of  $\\alpha_\\textrm{em}$. However, using advanced techniques that are currently under development it might be possible in the foreseeable future to determine differences of $g$-factors of two ions with significantly higher precision, rendering the low-$Z$ regime highly interesting.           \n\n\\subsection{Measurement of specific $g$-factor differences}\nGenerally, as the leading order contributions to the bound electron $g$-factor have been tested already at least in medium-strong fields, it becomes more interesting to investigate also higher-order contributions. Especially in the low-$Z$ regime these are however typically significantly smaller, either due to a strong reduction by factors of  $\\bigl( \\frac{\\alpha}{\\pi}\\bigr)^n$ and $(Z\\alpha)^n$, or by their dependence on nuclear properties. To specifically investigate those contributions, it is advantageous to look at differences of similar systems, where most of the contributions drop out, while the contribution of interest remains. In addition to the above-mentioned determination of $\\alpha$, an obvious case is the isotopic effect on the $g$-factor. This is a small contribution (as it depends on the nucleus), but can be observed very clearly by looking at the difference of the $g$-factors of two isotopes in the same charge state. Here, virtually all contributions of the (almost identical) electron shell drop out, while the nuclear contributions, which are typically in the $10^{-8}$ range for medium heavy systems, become clearly visible and enable a unique way to probe the validity of strong field QED beyond the Furry picture, where the nucleus is considered to be immotile \\cite{Koeh2016}.    \nOn the experimental side it is very unfavourable to measure the $g$-factors of both isotopes independently and then to take the difference of the two values, as this requires high precision for both the measurement and the masses of the ions. This precision for the $g$-factor is technically limited by the resolution of the magnetic field achievable by the cyclotron frequency measurement. \nA drastically better result can be achieved if it is possible to use the (much higher) Larmor frequency of a second ion as co-magnetometer. Normally, unavoidable fast magnetic field fluctuations lead to decoherence of the individual Larmor precessions on a time scale of milliseconds. If the two ions however see exactly the same magnetic field, the difference of the Larmor frequency, which is proportional to the sought-after difference of the $g$-factors, stays coherent for much longer times. This is possible by keeping both ions simultaneously in one trap, either in an axial Coulomb crystal, where both ions are (laser-) cooled to stable equilibrium positions along the trap axis, or on a common magnetron orbit, where both ions circulate around the trap axis (see figure \\ref{dgbeating}). \n\\begin{figure*}[h!]\n\\includegraphics[width=1\\textwidth]{dgbeating.pdf}\n\\centering\n\\caption{When two ions are stored on a common magnetron orbit they perform a deterministic motion and experience on average almost exactly the same magnetic field. This way, when observing the Larmor precession (using a Ramsey-type interrogation), while the individual (fast) Larmor precessions of the two ions rapidly become incoherent due to magnetic field fluctuations, their beating stays coherent for significantly longer times, provided the two Larmor frequencies are similar. This allows to directly extract the difference of the two similar $g$-factors $\\Delta g=g_1-g_2$ with drastically improved resolution compared to individual measurements. }\n\\label{dgbeating}\n\\end{figure*}\nThe techniques required to first combine the ions on a common orbit as well as to separate them after the measurement are currently developed. Once the viability of this technique has been shown, a leap in the achievable precision for small contributions to $g$ can be expected, which will consequently lead to intriguing new measurement opportunities. \n\n\\section{Precision laser spectroscopy on highly charged ions}\nMany of the most interesting transitions for fundamental atomic physics are at least electric dipole forbidden, as their long lifetimes enable reaching highest precision. However, this long lifetime also goes along with a very low fluorescent yield, rendering standard laser spectroscopy in many interesting cases impossible. Here, the continuous Stern-Gerlach effect (CSGE) offers a unique and versatile way  to  determine and prepare the quantum state of an ion, independent of the lifetime of the respective transition. To this end, the current sub-state of the ion is determined by driving a magnetic Zeeman transition in the ground state. A successful transition can be clearly seen via a change of the axial frequency of the ion, without the need to observe the actual transition. The change of the axial frequency, along with the knowledge of the millimeter-wave frequency used to drive the transition, allows unambiguously identifying the current sub-state. Consequently, if a probe excitation using a laser or any other radiation source successfully drives a transition this can be detected by re-determining the current state (see figure \\ref{Ar13structure} for an example). \n\\begin{figure*}[h!]\n\\includegraphics[width=1\\textwidth]{Ar13structure.pdf}\n\\centering\n\\caption{The finestructure of the ground state of boronlike $^{40}$Ar$^{13+}$. By using the continuous Stern-Gerlach effect and millimeter wave excitations at the ground-state Larmor frequency (green arrow), it is possible to prepare the ion deterministically in e.g. the $\\left|\\downarrow\\right>$ state. Then, a laser close to the transition frequency to the 2p$_{3\/2}$(m$_j$=+1\/2) state can lead (with about 66\\% probability) to a decay to the $\\left|\\uparrow\\right>$ state, which afterwards is virtually a dark state due to the detuning by the strong Zeeman shift. The successful transition can then be detected by determining the magnetic substate (now $\\left|\\uparrow\\right>$) with the CSGE. This method of detecting the result of the transition rather than the transition itself thus allows a rapid search for an unknown transition frequency by chirping a large range of frequencies (as indicated in the figure).}\n\\label{Ar13structure}\n\\end{figure*}\n\nA first proof-of-principle of this technique has recently been performed by measuring the finestructure transition in boronlike $^{40}$Ar$^{13+}$, an E1-forbidden transition with about 10~ms lifetime.  Here, the ground-state is a 2p$_{1\/2}$ doublet and a successful transition, probed by a 441~nm laser, effectively leads to pumping into the other magnetic sublevel of the doublet. As this method is sensitive to the result (the state change) of the transition rather than the transition itself, it is extremely powerful in locating unknown transition frequencies of narrow transitions. To this end, after preparing the ion in a known sub-state, the laser is chirped over a certain range of frequencies. Afterwards, a change of the sub-state indicates a successful transition within the chirp range, allowing to effectively narrow the search range until the transition is located. \nThis new method, combined with the extraordinarily long storage times enabled in the cryogenic vacuum and ultra-low temperatures resulting from sympathetic cooling of the ions of interest will allow in the future to study many systems that have been previously inaccessible to precision laser spectroscopy, such as the hyperfine structure in heavy HCI, e.g. $^{209}$Bi$^{82+}$ \\cite{ullmann2017high}, or the ro-vibrational structure of the hydrogen molecular ions, H$_2^+$ and HD$^+$ \\cite{alighanbari2020precise,parker2018measurement}, or even its antimatter counterpart $\\bar{\\textrm{H}}_2^-$ \\cite{myers2018c}, once this can be produced in antimatter facilities. \n\n\n\\section{Acknowledgments}\nThis project was funded by the European Research Council\n(ERC) under the European Union's Horizon 2020 research and\ninnovation programme under Grant Agreement No. 832848-FunI.\nFurthermore, we acknowledge funding and support by the Max-\nPlanck Society, and funding from the German\nResearch Foundation (DFG) Research UNIT FOR \n2202 under Project No. DU1334\/1-2 and from the Collaborative \nResearch Centre \"SFB 1225 (ISOQUANT).\"\n\n\\printbibliography\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{S:intro}\nThe hierarchy between electroweak and Planck scales can be\naddressed in extra dimensional models. Among these, the model proposed\nby Randall and Sundrum (RS) assumes warp geometry of the\nspace-time in 5 dimensions \\cite{RS}. The fifth dimension has Planck\nscale length $r_c$ and is compactified on the space $S^1\/Z_2$.\nTwo 3-branes are supported on either side of this fifth\ndimension. The exponential suppression\nalong the fifth dimension naturally suppresses Planck scale\nquantities of one 3-brane into electroweak scale on the second\n3-brane, which is identified as TeV-brane and can be interpreted\nas our universe. In the original RS model the standard model\nfields are assumed to lie on the TeV-brane while only gravity propagates in\nthe bulk. Later works have explored the phenomenology of bulk standard model fields\nin warped geometry model \\cite{GW,DHR,Pom,GN,CHNOY}. It was however shown that\nthe models with bulk gauge and Higgs fields where the spontaneous symmetry breaking\ntakes place in the bulk, encounters serious problems. \nThe two main problems are:\\\\\n\n(i) The non-Abelian gauge fields acquire masses through the Higgs vacuum\nexpectation value (vev) generated through spontaneous symmetry breaking \nin the bulk. This vev being a bulk parameter has a magnitude of the order\nof the Planck scale and therefore lends a very large bulk mass \n$\\sim$ Planck mass to the gauge boson in the bulk. As a result the lowest lying masses\nin the KK tower of the gauge boson \non the visible brane becomes $\\sim$ TeV which fails to comply with the $W$ and\n$Z$ boson masses $\\leq 100$ GeV. If one \ntries to reduce it by adjusting the bulk parameters then that would jeopardize\nthe unique feature of Planck scale to TeV scale warping,\n{\\it i.e.} the resolution of gauge hierarchy problem which was the original\nmotivation of such a warped geometry model.\n\n(ii) For an Abelian gauge boson with zero bulk mass, the massless Kaluza-Klein (KK)\nmode on the TeV-brane corresponds to\nphoton. However, the first excited state in the KK tower has an unacceptably\nlarge coupling with fermions.  This puts a stringent bound on the mass of this\nstate such that the model may survive the direct search bound at the Fermilab \nTevatron as well as  precision electroweak constraints. However, the mass of this\nfirst KK mode turns out to be much lower than the above bound. Once again it is\nimpossible either to reduce the coupling or to increase the mass \nby adjusting the bulk parameters without disturbing the resolution of the gauge hierarchy\/fine tuning problem.\\\\\n\nWe refer our readers to \\cite{DHR, CHNOY,DMS,higgs-others2,ssg1,cust} where both\nthese problems have been discussed in details. \nRecently, in a generalized 5-dimensional RS model with non-flat visible brane, by adjusting\nthe brane cosmological constant the problem coming from the\nprecision electroweak tests have been averted and also the bulk\nHiggs problem has been resolved \\cite{DMS}. \nThe  brane cosmological constant however was found to be negative implying that the visible brane in such a case is an \nanti-de Sitter 3-brane.\n \nIn the present work\nwe address these problems from a different viewpoint {\\it i.e.} in the backdrop of a 6-dimensional doubly\nwarped model with flat 3-branes \nwhich is an extension of the original RS model to more than one extra dimensions\\cite{CS}. In this 6-dimensional\nmodel two extra spatial coordinates are compactified such that\nthe space-time manifold is $[M^{(1,3)}\\times S^1\/Z_2]\n\\times S^1\/Z_2$. Compared to the RS model, here the two\nextra dimensions, denoted by angular coordinates $y,z$,\nare doubly warped. Four 4-branes are located at the orbifolded\npoints: $y=0,\\pi$, $z=0,\\pi$. The intersection of any two\n4-branes gives a 3-brane. The 3-brane located at $(y,z)=(\\pi,0)$\nis identified with our universe. Analogous to the RS setup,\nthe mass scale suppression can be felt along both the coordinates\n$y,z$. We can choose the moduli of these coordinates, say\n$R_y$ and $r_z$, such that TeV scale masses can be\ngenerated on the visible brane located at $(y,z)=(\\pi,0)$. Since\nthere is an extra freedom through an additional modulus in this model compared to the 5-dimensional RS model,\nwe explore if the above problems relating to bulk Higgs can be solved\nin the 6-dimensional doubly warped model by adjusting the\nmoduli $R_y,r_z$ suitably.\n\nWe organize our paper as follows.\nIn the following  section we explain some essential features of\nthe 6-dimensional doubly warped model. In Sec. 3 we\ndescribe the KK mode analysis of \ngauge bosons and fermions in 6-dimensional bulk and the corresponding modes on the visible 3-brane. In\nSec. 4 we present our results and argue that the precision\nelectroweak tests put no  additional constraints on this\nmodel. We conclude in Sec. 5.\n\n\\section{The 6-dimensional doubly warped model}\n\\label{S:mod}\n\nAs explained previously, the 6-dimensional doubly warped\nmodel has space-time of six dimensions and the extra two\nspatial dimensions are orbifolded by $Z_2$ symmetry \\cite{CS}. The\nmanifold under consideration is $[M^{(1,3)}\\times S^1\/Z_2]\n\\times S^1\/Z_2$ with four non-compact dimensions denoted\nby $x^\\mu$, $\\mu=0,\\cdots,3$. Since we are interested in\ndoubly warped model, the metric in this model can be chosen\nas\n\\begin{equation}\nds^2 = b^2(z)[a^2(y)\\eta_{\\mu\\nu}dx^\\mu dx^\\nu +R_y^2dy^2]+r_z^2dz^2\n\\label{E:metric}\n\\end{equation}\nAs explained before the angular coordinates $y,z$ represent the\nextra spatial dimensions with moduli $R_y,r_z$, respectively. The\nMinkowski matrix in the usual 4-dimensions has the form $\\eta_{\\mu\\nu} = {\\rm diag}\n(-1,1,1,1)$. The functions $a(y),b(z)$ give warp factors in\nthe $y$ and $z$ directions, respectively. The total bulk-brane action\nof this model has a form \\cite{CS}\n\\begin{eqnarray}\nS &=& S_6+S_5\n\\nonumber \\\\\nS_6 &=& \\int d^4xdydz\\sqrt{-g_6}(R_6-\\Lambda), \\quad\n\\nonumber \\\\\nS_5 &=& \\int d^4xdydz[V_1\\delta(y)+V_2\\delta(y-\\pi)]\n     +\\int d^4xdydz[V_3\\delta(z)+V_4\\delta(z-\\pi)]\n\\end{eqnarray}\nHere, $V_{1,2}$ and $V_{3,4}$ are brane tensions of the branes located at\n$y=0,\\pi$ and $z=0,\\pi$, respectively. $\\Lambda$ is the cosmological\nconstant in 6-dimensions. The 3-branes are located at the intersection points of the four 4-branes.\n\nAfter solving Einstein's\nequations the solutions to the warp functions\nof the metric as given in eq. (\\ref{E:metric}) have a form \\cite{CS}\n\\begin{eqnarray}\na(y) &=& \\exp(-c|y|), \\quad b(z) = \\frac{\\cosh(kz)}{\\cosh(k\\pi)}\n\\nonumber \\\\\nc&\\equiv & \\frac{R_yk}{r_z\\cosh(k\\pi)},\\quad k\\equiv r_z\\sqrt{\n\\frac{-\\Lambda}{10M_P^4}}\n\\label{E:sol}\n\\end{eqnarray}\nHere, $M_P$ is the Planck scale.\nThe warp factors $a(y)$ and $b(z)$ give\nlargest suppression from $y=0, z=\\pi$ brane to $y=\\pi z=0$ brane. For this reason\nwe can interpret the 3-brane formed out of the intersection of\n4-branes at $y=\\pi$ and $z=0$ as our standard model brane. The suppression\nfactor $f$ on the standard model brane can be written as\n\\begin{equation}\nf =\\frac{\\exp(-c\\pi)}{\\cosh(k\\pi)}\n\\label{E:supp}\n\\end{equation}\nThe desired suppression of $10^{-16}$ on the standard model brane\ncan be obtained for different combinations of the parameters $c$ and $k$. However, from\nthe relation for $c$ in eq. (\\ref{E:sol}) it can be noticed that\nin order not to have large hierarchy in the moduli $R_y$ and $r_z$, either of $c$\nor $k$ must be large where as other is small, e.g. $c\\sim$ 11.4 and $k\\sim$ 0.1.\nThis implies that the warping along $y$ is large whereas that along $z$ is small.\nIt has been argued that this feature may offer an explanation of the small mass hierarchy\namong the standard model fermions\\cite{CS}.  \n\nThe 6-dimensional model which has been described here is thus viable\nin explaining the hierarchy between Planck scale and the electroweak\nscale without introducing large hierarchy between the moduli\n$R_y$ and $r_z$. In this model KK modes of bulk scalar fields have\nbeen studied \\cite{KMS1}. Bulk fermion fields have also been\nstudied in this model with a possibility of localizing\nthem on a 4-brane \\cite{KMS2}. However the possibility of bulk gauge and Higgs field in this\nmodel has not been explored yet. In the following section we derive the KK modes of the gauge field, fermion fields  and the corresponding\ncouplings to estimate the viability of this model in respect to the problems discussed earlier. \nWe reiterate that our aim is to explore if we can put the Higgs in the bulk of such 6-dimensional\nmultiply warped model without invoking any contradiction with the precision electroweak test \n\\cite{DHR,CHNOY} as was encountered in the 5-dimensional RS model.\n\\section{Gauge bosons and fermions in the bulk}\n\nIn this section we explain the KK decomposition and eigenvalue\nequations of KK gauge bosons\nand KK fermions which arise from the respective bulk fields after\nintegrating over the two extra dimensions of the model.\n\n\\subsection{KK modes of the gauge bosons}\nFor simplicity, we consider a U(1) gauge theory, but our derivation\ngiven below is applicable to non-Abelian theory as well. In a realistic\nmodel the gauge fields can acquire non-zero masses due to spontaneous\nsymmetry breaking. In our model, Higgs mechanism can take place in the\nbulk of the 6-dimensions and the vev of the Higgs\nfield will be of the order of Planck scale. The vev of the Higgs field\ncontributes to generate the bulk mass for the gauge field. Hence, after spontaneous\nsymmetry breaking the invariant action can be written as\n\\begin{equation}\nS_G = \\int d^4xdydz\\sqrt{-G}\\left(-\\frac{1}{4}G^{MK}G^{NL}F_{KL}F_{MN}\n-\\frac{1}{2}M^2G^{MK}A_MA_K\\right),\n\\label{E:SG}\n\\end{equation}\nwhere, $M$ is the bulk mass $\\sim M_P$ and $G={\\rm det}(G_{AB})$ is the determinant\nof the metric $G_{AB}$ which is given in eq. (\\ref{E:metric}).\n$F_{KL}=\\partial_K A_L-\\partial_LA_K$ is the gauge field strength. Exploiting the\ngauge symmetry we can choose the gauge where\n$A_4=A_5=0$. The KK decomposition of the gauge field can be taken as\n\\begin{equation}\nA_\\mu = \\sum_{n,p}A^{(n,p)}_\\mu(x)\\xi_n(y)\\chi_p(z)\/\\sqrt{R_yr_z}.\n\\label{E:KKgau}\n\\end{equation}\nThe KK fields in the 4-dimensions $A_\\mu^{(n,p)}$ carry two indices\n$n,p$ due to the two additional dimensions of the model. The functions\n$\\xi_n(y)$ and $\\chi_p(z)$ give KK wave functions in the $y$ and $z$ directions,\nrespectively. Substituting the above KK decomposition in eq. (\\ref{E:SG})\nand integrating over the $y$ and $z$ coordinates, we demand\nthat the resulting action in the 4-dimensions must have a form\n\\begin{equation}\n\\sum_{n,p}-\\frac{1}{4}F_{\\mu\\nu}^{(n,p)}F^{(n,p)\\mu\\nu}-\\frac{1}{2}\nm_{n,p}^2A_{\\mu}^{(n,p)}A^{(n,p)\\mu},\n\\end{equation}\nwhere $m_{n,p}$ is the mass of the KK field $A_\\mu^{(n,p)}$. This can be achieved provided the KK wave functions \nsatisfy the following orthonormality condition.\n\\begin{equation}\n\\int dy~\\xi_n(y)\\xi_{n^\\prime}(y) = \\delta_{nn^\\prime},\n\\quad\n\\int dz~b(z)\\chi_p(z)\\chi_{p^\\prime}(z) = \\delta_{pp^\\prime}.\n\\label{E:gnorm}\n\\end{equation}\nMoreover, in addition to the above normalization conditions the following\neigenvalue equations for the $\\xi_n$ and $\\chi_p$ must also be satisfied:\n\\begin{eqnarray}\n\\frac{1}{R_y^2}\\partial_y(a^2\\partial_y\\xi_n)-m_p^2a^2\\xi_n &=& -m_{n,p}^2\\xi_n,\n\\nonumber \\\\\n\\frac{1}{r_z^2}\\partial_z(b^3\\partial_z\\chi_p)-M^2b^3\\chi_p &=& -m_{p}^2\\chi_p.\n\\label{E:KKyz}\n\\end{eqnarray}\nHere, $m_p$ is a mass parameter which is determined by solving the equation\nfor $\\chi_p(z)$, and the value of $m_p$ determines the KK mass $m_{n,p}$ through\nthe eigenvalue equation for $\\xi_n$, as given above.\n\nThe second of the eq. (\\ref{E:KKyz}) can be solved by approximating\n$b(z)\\sim\\exp[-k(\\pi-z)]=\\exp[-k\\tilde{z}]$. By writing\n$\\tilde{\\chi}_p(z)=\\exp(-3k\\tilde{z}\/2)\\chi_p(z)$ the eigenvalue\nequation for $\\tilde{\\chi}_p$ takes the form,\n\\begin{equation}\nz_p^2\\frac{d^2\\tilde{\\chi}_p}{dz_p^2}+z_p\\frac{d\\tilde{\\chi}_p}{dz_p}\n+(z_p^2-\\nu_p^2)\\tilde{\\chi}_p=0,\n\\end{equation}\nwhere $z_p=\\frac{m_p}{k^\\prime}\\exp(k\\tilde{z})$ and $\\nu_p^2=\\frac{9}{4}\n+\\left(\\frac{M}{k^\\prime}\\right)^2$. Here, $k^\\prime=k\/r_z$.  The\nsolutions to the above equation are Bessel functions of order $\\nu_p$,\nand we can write\n\\begin{equation}\n\\chi_p(z)=\\frac{1}{N_p}\\exp(\\frac{3}{2}k\\tilde{z})\n\\left[J_{\\nu_p}(z_p)+b_pY_{\\nu_p}(z_p)\\right],\n\\label{E:Zeigen}\n\\end{equation}\nwhere, $N_P$ and $b_p$ are some constants.\nBy demanding that the function $\\chi_p(z)$ be continuous at the\norbifold fixed points $z=0,\\pi$ we get the following approximate\nsolution which determines the spectrum for $m_p$.\n\\begin{equation}\n3J_{\\nu_p}(x_{\\nu_p})+x_{\\nu_p}(J_{\\nu_p - 1}(x_{\\nu_p}) - J_{\\nu_p + 1}(x_{\\nu_p})) = 0,\n\\label{E:Zeq}\n\\end{equation}\nwhere $x_{\\nu_p}=\\frac{m_p}{k^\\prime}\\exp(k\\pi)$. After solving\nfor $m_p$ using the above equation, we can compute the KK mass $m_{n,p}$\nby solving the first of eq. (\\ref{E:KKyz}). By writing $\\tilde{\\xi}_n\n=\\exp(-c|y|)\\xi_n$, the eigenvalue equation for $\\tilde{\\xi}_n(y)$ becomes,\n\\begin{equation}\ny_n^2\\frac{d^2\\tilde{\\xi}_n}{dy_n^2}+y_n\\frac{d\\tilde{\\xi}_n}{dy_n}\n+(y_n^2-\\nu_n^2)\\tilde{\\xi}_n=0,\n\\end{equation}\nwhere $y_n=\\frac{m_{n,p}}{k^\\prime}\\exp(c|y|)\\cosh(k\\pi)$ and\n$\\nu_n^2=1+\\left(\\frac{m_p}{k^\\prime}\\right)^2\\cosh^2(k\\pi)$. The solution\nfor $\\xi_n(y)$ can be written in terms of Bessel function of order\n$\\nu_n$ multiplied by growing exponential factor as\n\\begin{equation}\n\\xi_n(y)=\\frac{1}{N_n}\\exp(c|y|)\n\\left[J_{\\nu_n}(y_n)+b_nY_{\\nu_n}(z_n)\\right],\n\\label{E:Yeigen}\n\\end{equation}\nwhere, $N_n$ and $b_n$ are some constants. Again, by demanding that\nthe function $\\xi_n(y)$ be continuous at the orbifold fixed points\n$y=0,\\pi$ the following equation determines the KK\nmass $m_{n,p}$.\n\\begin{equation}\nJ_{\\nu_n}(x_{\\nu_n})+x_{\\nu_n}(J_{\\nu_n - 1}(x_{\\nu_n}) - J_{\\nu_n + 1}(x_{\\nu_n}))\/2 = 0,\n\\label{E:Yeq}\n\\end{equation}\nwhere,\n\\begin{equation}\nx_{\\nu_n}=\\frac{m_{n,p}}{k^\\prime}\\exp(c\\pi)\\cosh(k\\pi).\n\\label{E:xn}\n\\end{equation}\n\nThe actual KK mass of a gauge field is found by first solving the\neq. (\\ref{E:Zeq}) for $m_p$ and then solving the eq. (\\ref{E:Yeq}), which\nis described in the previous paragraph. The wave function of these\nKK gauge fields is product of wave functions given in eqs. (\\ref{E:Zeigen})\nand (\\ref{E:Yeigen}). \\\\\nIn the above analysis if we put the bulk gauge boson mass $M = 0$, we easily\nobtain the various KK mode solutions and the corresponding masses. In this case\nthe lowest lying KK mode is massless which corresponds to the standard model photon.\n\nA nice feature of the KK gauge fields in the\n6-dimensional doubly warped model is that their wave functions\ncan be decomposed into product of functions in the two extra dimensions,\na feature which may not be evident for the bulk fermion fields which\nis the subject of the next subsection.\n\n\\subsection{KK modes of the fermions}\n\nThe invariant action for a bulk fermion field $\\Psi$ in 6-dimensions is\n\\cite{DHR,GN,CHNOY}\n\\begin{equation}\nS_f = \\int d^4xdydz\\sqrt{-G}\\left\\{E^A_a\\left[\\frac{i}{2}\\left(\\bar{\\Psi}\\Gamma^a\n\\partial_A\\Psi - \\partial_A\\bar{\\Psi}\\Gamma^a\\Psi\\right)+\\frac{\\omega_{bcA}}{8}\n\\bar{\\Psi}\\{\\Gamma^a,\\sigma^{bc}\\}\\Psi\\right] - M_f\\bar{\\Psi}\\Psi\\right\\}\n\\label{E:Sf}\n\\end{equation}\nHere, the capital letter $A$ denotes index in the curved space\nand the lower case letters $a,b,c$ denote indices in the\ntangent space.\n$\\omega_{bcA}$ is spin connection and $E^A_a$ is inverse vielbein.\n$M_f$ is the bulk mass and $\\sigma^{bc}=\\frac{i}{2}[\\Gamma^b,\\Gamma^c]$.\nThe Dirac matrices $\\Gamma^a$ in 6-dimensions would be 8$\\times$8, and\nthey can be taken as \\cite{BDP}:\n\\begin{equation}\n\\Gamma^\\mu = \\gamma^\\mu\\otimes\\sigma^0,\\quad \\Gamma^4=i\\gamma_5\\otimes\\sigma^1,\n\\quad \\Gamma^5=i\\gamma_5\\otimes\\sigma^2\n\\end{equation}\nHere, $\\gamma^\\mu$ are the Dirac matrices in 4-dimensions and\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$. $\\sigma^i$, $i=1,2,3$, are the\nPauli matrices and $\\sigma^0$ is the 2$\\times$2 unit matrix.\nThe chirality in 6-dimensions is defined by the matrix $\\bar{\\Gamma}=\\Gamma^0\n\\Gamma^1\\Gamma^2\\Gamma^3\\Gamma^4\\Gamma^5$ as $\\bar{\\Gamma}\\Psi_\\pm = \\pm\\Psi_\\pm$.\nThe chiral fermions in 6-dimensions have both left- and right-handed chirality\nof 4-dimensions, which can be projected by the operators $P_{L,R} =\n(1\\mp i\\Gamma^0\\Gamma^1\\Gamma^2\\Gamma^3)\/2$.\n\nAs explained in Sec. \\ref{S:intro}, we are interested in estimating the gauge coupling\nof standard model fermions to the KK gauge bosons. We take the bulk mass\nof the fermions $M_f$ to be zero, since the masses of standard model fermions\nare much below the Planck scale. The term that is associated with the spin\nconnection in eq. (\\ref{E:Sf}) would give no contribution, since the metric\nin eq. (\\ref{E:metric}) is diagonal. Hence, in our particular case\nof interest we expand the first term of eq. (\\ref{E:Sf}),\nwhich has the following form:\n\\begin{eqnarray}\nS_f= &\\int d^4xdydz& \\left\\{ b^4a^3R_yr_z i\\left(\n\\bar{\\Psi}_{+L}\\Gamma^\\mu\\partial_\\mu\\Psi_{+L}+\n\\bar{\\Psi}_{+R}\\Gamma^\\mu\\partial_\\mu\\Psi_{+R}+\n\\bar{\\Psi}_{-L}\\Gamma^\\mu\\partial_\\mu\\Psi_{-L}+\n\\bar{\\Psi}_{-R}\\Gamma^\\mu\\partial_\\mu\\Psi_{-R}\\right) \\right.\n\\nonumber \\\\\n&& \\left.\n+\\left[\\bar{\\Psi}_{+L}\\left(\\Gamma^4 D_y+\\Gamma^5 D_z\\right)\\Psi_{+R}\n+\\bar{\\Psi}_{+R}\\left(\\Gamma^4 D_y+\\Gamma^5 D_z\\right)\\Psi_{+L}\n\\right. \\right.\n\\nonumber \\\\\n&& \\left. \\left.\n+\\bar{\\Psi}_{-L}\\left(\\Gamma^4 D_y+\\Gamma^5 D_z\\right)\\Psi_{-R}\n+\\bar{\\Psi}_{-R}\\left(\\Gamma^4 D_y+\\Gamma^5 D_z\\right)\\Psi_{-L}\n\\right]\\right\\},\n\\label{E:Sfin}\n\\end{eqnarray}\nwhere the differential operators are defined as:\n$ D_y = \\frac{i}{2}b^4r_z(a^4\\partial_y\n+\\partial_ya^4)$ and $ D_z = \\frac{i}{2}a^4R_y(b^5\\partial_z\n+\\partial_zb^5)$.\nIn the subscript of the fields $\\Psi$ the $\\pm$ indicates the chirality\nin 6-dimensions and the $L,R$ stands for the left- and right-handed chirality\nof the 4-dimensions. Terms in line 2 and 3 of the above equation give\neffective masses for the KK modes in the 4-dimensions. These terms indicate\nthat in general we cannot decompose the wave functions into $y$ and $z$ parts separately,\nlike what we have done for the KK wave functions of the gauge bosons as\ndescribed previously in eq. (\\ref{E:KKgau}). Hence, for the bulk fermions the KK decomposition\ncan be taken as\n\\begin{eqnarray}\n\\Psi_{+L,-R}(x^\\mu,y,z)&=&\\frac{1}{\\sqrt{R_yr_z}}\\sum_{j,k}\\psi^{(j,k)}_{+L,-R}(x^\\mu)\nf^{(j,k)}_{+L,-R}(y,z)\\otimes\\left(\\begin{array}{c}1\\\\0\\end{array}\\right),\n\\nonumber \\\\\n\\Psi_{-L,+R}(x^\\mu,y,z)&=&\\frac{1}{\\sqrt{R_yr_z}}\\sum_{j,k}\\psi^{(j,k)}_{-L,+R}(x^\\mu)\nf^{(j,k)}_{-L,+R}(y,z)\\otimes\\left(\\begin{array}{c}0\\\\1\\end{array}\\right).\n\\label{E:KKfer}\n\\end{eqnarray}\nIn the above equation various fields of the form $\\psi^{(j,k)}(x^\\mu)$ are the KK fields\nliving in the 4-dimensions and $f$'s are the KK wave functions depending on both\n$y$ and $z$ coordinates. Substituting the above KK decomposition into\neq. (\\ref{E:Sfin}) and integrating over the $y$ and $z$ we can get the action\nof the form\n\\begin{eqnarray}\nS_f = &\\int d^4x& \\sum_{j,k}\\bar{\\psi}^{(j,k)}_{+L}i\\gamma^\\mu\\partial\\psi^{(j,k)}_{+L}\n+\\bar{\\psi}^{(j,k)}_{+R}i\\gamma^\\mu\\partial\\psi^{(j,k)}_{+R}\n+\\bar{\\psi}^{(j,k)}_{-L}i\\gamma^\\mu\\partial\\psi^{(j,k)}_{-L}\n+\\bar{\\psi}^{(j,k)}_{-R}i\\gamma^\\mu\\partial\\psi^{(j,k)}_{-R}\n\\nonumber \\\\\n&&\n-M_{j,k}(\\bar{\\psi}^{(j.k)}_{+L}\\psi^{(j,k)}_{+R}+\\bar{\\psi}^{(j.k)}_{+R}\\psi^{(j,k)}_{+L}\n+\\bar{\\psi}^{(j.k)}_{-L}\\psi^{(j,k)}_{-R}+\\bar{\\psi}^{(j.k)}_{-R}\\psi^{(j,k)}_{-L}),\n\\end{eqnarray}\nprovided the following normalization and the eigenvalue equations\nfor the KK wave functions are satisfied:\n\\begin{equation}\n\\int dydz b^4(z)a^3(y)\\left(f^{(j,k)}_{+R,+L,-R,-L}(y,z)\\right)^*\nf^{(j^\\prime,k^\\prime)}_{+R,+L,-R,-L}(y,z) =\n\\delta^{j,j^\\prime}\\delta^{k,k^\\prime},\n\\label{E:fnorm}\n\\end{equation}\n\\begin{eqnarray}\n(i{\\cal D}_y+{\\cal D}_z)f^{(j,k)}_{+R}(y,z)&=&-M_{j,k}f^{(j,k)}_{+L}(y,z),\n\\nonumber \\\\\n(-i{\\cal D}_y+{\\cal D}_z)f^{(j,k)}_{+L}(y,z)&=&-M_{j,k}f^{(j,k)}_{+R}(y,z),\n\\nonumber \\\\\n(i{\\cal D}_y+{\\cal D}_z)f^{(j,k)}_{-L}(y,z)&=&M_{j,k}f^{(j,k)}_{-R}(y,z),\n\\nonumber \\\\\n(-i{\\cal D}_y+{\\cal D}_z)f^{(j,k)}_{-R}(y,z)&=&M_{j,k}f^{(j,k)}_{-L}(y,z),\n\\label{E:feigen}\n\\end{eqnarray}\nwhere the differential operators are: ${\\cal D}_y=\\frac{i}{2R_y}(4\\partial_ya\n+2a\\partial_y)$ and ${\\cal D}_z=\\frac{i}{2r_z}a(5\\partial_zb\n+2b\\partial_z)$.\nHere, $M_{j,k}$ is the mass of the KK fermion $\\psi^{(j,k)}$.\n\nAs explained previously, we are interested in standard model fermion\ncoupling with the KK modes of gauge field. The zero mode of the KK\nfermions are identified with the standard model fermions. The wave\nfunction for these fields can be solved from eq. (\\ref{E:feigen})\nby putting $M_{j,k}=0$. Here, we show the solution for the wave\nfunction $f_{+R}^{(0,0)}(y,z)$ and the solutions for other chiral\nfermions can be analogously worked out. The eigenvalue equation\nwe are interested in is\n\\begin{equation}\n(i{\\cal D}_y+{\\cal D}_z)f_{+R}^{(0,0)}(y,z)=0,\n\\end{equation}\nwhere the differential operators ${\\cal D}_y$ and ${\\cal D}_z$ are\ndefined below eq. (\\ref{E:feigen}). For the zero-mode case\nwe can write the function $f_{+R}^{(0,0)}$ as a product of\n$y$ and $z$ parts, say $f_{+R}^{(0,0)}(y,z)=f_y(y)f_z(z)$.\nThis simplification happens only for the zero-mode case, since\nthe factor $a(y)$ in the operators ${\\cal D}_{y,z}$ can be taken\nout and the right-hand side of the above equation is zero.\nSubstituting this form of $f_{+R}^{(0,0)}(y,z)$ in the above equation we get\n\\begin{equation}\n-\\frac{1}{R_y}\\frac{(-4c+2\\partial_y)f_y}{f_y}\n+i\\frac{1}{r_z}\\frac{(5\\partial_zb+2b\\partial_z)f_z}{f_z} = 0\n\\end{equation}\nSince the functional dependence on $y$ and $z$ are completely\nseparated out, we can solve for $f_y$ and $f_z$ by taking\n$\\frac{(-4c+2\\partial_y)f_y}{f_y}=c_1$, where $c_1$ is a separation\nconstant. In terms of $c_1$ the functional dependences of\n$f_y$ and $f_z$ are given below\n\\begin{equation}\nf_y(y)=\\exp\\left(\\frac{1}{2}(c_1+4c)y\\right),\\quad\nf_z(z)=\\frac{\\exp\\left(\\frac{-ic_1r_z}{kR_y}\\tan^{-1}(\\tanh(kz\/2))\\cosh(k\\pi)\\right)}\n{\\cosh^{5\/2}(kz)}\n\\label{E:SMwf}\n\\end{equation}\nThe value of $c_1$ can be worked out in terms of $c$ and $k$\nby normalizing the wave\nfunction $f_{+R}^{(0,0)}(y,z)$ using eq. (\\ref{E:fnorm}).\n\n\\section{Bulk phenomenology}\n\nIn the previous section we have given a description of the\nKK modes of the gauge bosons and fermions in the bulk of a\n6-dimensional doubly warped model. Now using the mode\nexpansion for the bulk fields, we can calculate the gauge\ncouplings of standard model fermions with the KK modes of the\ngauge bosons.\n\nWe now address the two problems mentioned in the beginning.\nRecall that in 5-dimensional RS model it is found\nthat for non-zero bulk mass for non-Abelian gauge field the lowest lying\nmode has mass much higher than $100$ GeV {\\it i.e.} the\nmasses for $W$ and $Z$ bosons. Also,\nfor the massless gauge boson the gauge coupling with the first exited KK gauge boson\nis larger than one and hence the standard model fermions are\nstrongly coupled \\cite{DHR,CHNOY}. Due to this a stringent\nlower bound of $\\sim$ 10 TeV\non the mass of the first exited KK boson arose because of\nthe precision electroweak tests.\n\nIn this section we repeat\nthis exercise in the 6-dimensional doubly warped model, and\nwill show that due to the presence of an additional modulus we can tune\nthe lowest KK mode mass for non-Abelian gauge field\nnear $100$ GeV although the spontaneous symmetry breaking takes place in the bulk\nwith a bulk Higgs field with vev $\\sim$ Planck scale. \nFurthermore, for a gauge boson with a zero bulk mass the coupling to mass ratio\nof the first excited KK mode can survive the precision \nelectroweak test without putting any  additional restriction on the model.\n\nThe action between the bulk fermions and gauge bosons\ncan be written as \\cite{DHR,CHNOY}\n\\begin{equation}\nS_{\\rm int} = \\int d^4xdydz\\sqrt{-G} g_{6d}\\bar{\\Psi}(x^\\mu,y,z)i\\Gamma^aE^A_aA_A(x^\\mu,y,z),\n\\Psi(x^\\mu,y,z)\n\\end{equation}\nwhere $g_{6d}$ is the gauge coupling in the 6-dimensions as has been discussed in Sec. 3.  \nSubstituting\nthe KK decomposition for the gauge and fermion fields in the above equation and also\nreminding that we are working in the gauge choice where $A_4=A_5=0$,\nwe get the gauge coupling in the 4-dimensions as\n\\begin{equation}\ng^{(j,k)(n,p)}_{+R} = \\int dydz~g_0\\pi b^4a^3\\left(f_{+R}^{(j,k)}(y,z)\\right)^*\nf_{+R}^{(j,k)}(y,z)\\xi_n(y)\\chi_p(z),\n\\label{E:coup}\n\\end{equation}\nwhere $g_0=g_{6d}\/\\sqrt{\\pi R_y\\pi r_z}$ is the effective 4-dimensional\ngauge coupling. In the above equation we have given gauge coupling for\nthe KK fermion $\\psi_{+R}^{(j,k)}$ with the KK gauge field $A_\\mu^{(n,p)}$.\nSimilarly, the gauge couplings with the other KK fermions can be easily\nobtained by replacing the $+$ with $-$ and $R$ with $L$ accordingly\nin the above equation. The KK wave functions: $f$'s, $\\xi$ and $\\chi$,\nin the above equation should be the normalized wave functions as given\nby eqs. (\\ref{E:gnorm}) and (\\ref{E:fnorm}). However, in our particular\ncase of interest, where we are interested in precision electroweak tests,\nwe compute gauge couplings of the standard model fermions with the\nKK gauge fields. Hence, the wave functions for the fermions are of\nthe form in eq. (\\ref{E:SMwf}) and the corresponding functions for\nthe KK gauge fields are given in eqs. (\\ref{E:Zeigen}) and (\\ref{E:Yeigen}).\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{||c|c|c||c|c|c||} \\hline\n\\multicolumn{3}{||c||}{$\\frac{M}{k^\\prime}=0.5$}\n& \\multicolumn{3}{|c||}{$\\frac{M}{k^\\prime}=1.0$}\\\\\\hline\n$m_{1,2}$ = 143.78 & $m_{1,3}$ = 194.15 & $m_{1,4}$ = 244.52 &\n$m_{1,2}$ = 148.33 & $m_{1,3}$ = 199.21 & $m_{1,4}$ = 249.58 \\\\\n$\\tilde{g}^{1,2}$ = 0.0015 & $\\tilde{g}^{1,3}$ = 0.0028 &\n$\\tilde{g}^{1,4}$ = 0.0009 &\n$\\tilde{g}^{1,2}$ = 0.0006 & $\\tilde{g}^{1,3}$ = 0.0027 &\n$\\tilde{g}^{1,4}$ = 0.0011\\\\\\hline\n$m_{2,1}$ = 180.23 & $m_{2,2}$ = 237.94 & $m_{2,3}$ = 295.40 &\n$m_{2,1}$ = 184.53 & $m_{2,2}$ = 243.25 & $m_{2,3}$ = 300.97 \\\\\n$\\tilde{g}^{2,1}$ = 0.0127 & $\\tilde{g}^{2,2}$ = 0.0012 &\n$\\tilde{g}^{2,3}$ = 0.0023 &\n$\\tilde{g}^{2,1}$ = 0.0124 & $\\tilde{g}^{2,2}$ = 0.0005 &\n$\\tilde{g}^{2,3}$ = 0.0022 \\\\\\hline\n$m_{3,1}$ = 261.73 & $m_{3,2}$ = 322.73 & $m_{3,3}$ = 383.48 &\n$m_{3,1}$ = 266.29 & $m_{3,2}$ = 328.30 & $m_{3,3}$ = 389.56 \\\\\n$\\tilde{g}^{3,1}$ = 0.0106 & $\\tilde{g}^{3,2}$ = 0.0010 &\n$\\tilde{g}^{3,3}$ = 0.0019 &\n$\\tilde{g}^{3,1}$ = 0.0104 & $\\tilde{g}^{3,2}$ = 0.0004 &\n$\\tilde{g}^{3,3}$ = 0.0019 \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The gauge couplings $g^{(0,0)(i,j)}$ of the standard model fermions\nare given in the form $\\tilde{g}^{i,j}=\\frac{g^{(0,0)(i,j)}}{g_0}$, where\n$g_0$ is the effective 4-dimensional gauge coupling. The masses of the KK\ngauge bosons $m_{i,j}$ are given in GeV units.\n$M$ is the bulk gauge boson mass and $k^\\prime = k\/r_z$.\nThe non-zero values for $\\frac{M}{k^\\prime}$ are indicated in\nthe table. $\\frac{1}{r_z}= 7\\times 10^{17}$ GeV,\n$k$ = 0.25 and $c$ = 11.52. The lowest lying mode $m_{1,1}$, which corresponds to $W$ or $Z$ boson, is not included in the table.}\n\\label{T:t1}\n\\end{table}\nNow, in order to compute the gauge couplings the unknown parameters\nthat need to be fixed are $k$, $c$, $r_z$ and the bulk mass of the gauge fields\n$M$. The non-zero value for the bulk gauge mass $M$ is around the Planck scale.\nWe can determine the remaining parameters by making the following demands:\n(a) the lowest non-zero mass of the KK tower of the bulk gauge boson should be\nidentified with either $W$ or $Z$ boson mass, (b) the suppression $f$ of\neq. (\\ref{E:supp}) should be $\\sim 10^{-16}$ and (c) the hierarchy\nbetween the moduli $R_y$ and $r_z$ should not be too large. The expression for the\nKK gauge boson mass is given in eq. (\\ref{E:xn}). For the lowest non-zero\nKK gauge boson mass which is identifie as $m_{1,1}$ the root $x_{\\nu_n}$ would be ${\\cal O}(1)$. The factor $\\exp(c\\pi)\n\\cosh(k\\pi)$ in this equation, which is the inverse of $f$,\nshould be $\\sim 10^{16}$. By demanding that the lowest\nKK mode $m_{1,1}$ has mass of $\\sim 100$ GeV, from eq. (\\ref{E:xn}) we can naively\nestimate that $k^\\prime\\sim 10^{17}$ GeV. Since we would argue that\n$k\\sim 0.1$, a consistent value for the scale $r_z$ is\n$\\frac{1}{r_z}=7\\times 10^{17}$ GeV, which is about 14 times smaller than\nthe Planck scale. The parameters $k$ and $c$ can\nbe determined from the fact that we should not get large hierarchy between\nthe moduli $R_y$ and $r_z$ and also we should get the desired suppression\nof $f\\sim 10^{-16}$ on the standard model brane. We have estimated that for\n$k$ = 0.25 and $c$ = 11.52, the ratio between the moduli is $\\frac{R_y}{r_z}$ = 61,\nwhich is not unacceptably large, and also the suppression $f$\ncame out to be $1.45\\times 10^{-16}$. For these particular values of\n$k$, $c$ and $1\/r_z$ we have given the gauge\ncouplings and the corresponding masses of the excited KK gauge fields\nin Table \\ref{T:t1}. In this table the gauge couplings $g^{(i,j)}$,\nwhere $i,j$ are integers, of the standard\nmodel fermion are given as a fraction of the 4-dimensional coupling\n$g_0$. In the case of $\\frac{M}{k^\\prime}$ = 0.5 or = 1.0,\nwe have found that the lowest non-zero mode has mass of about 95 GeV.\nHence this mode can be identified with the $W$ or $Z$ gauge boson. Since\nwe have got the right amount of $W,Z$ boson masses for the above\ndescribed values of $k,c,\\frac{1}{r_z}$, we use the same set of\nvalues to get the gauge couplings and KK gauge boson masses in the\ncase where the bulk mass $M$ is zero. In this case we have found\nthat the lowest mode $m_{0,0}$ has zero mass which can be identified with\nthe photon state. The non-zero KK masses of the photon field and\ntheir corresponding gauge coupling values are given in Table \\ref{T:t2}.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{||c|c|c||} \\hline\n\\multicolumn{3}{||c||}{$\\frac{M}{k^\\prime}=0$}\\\\ \\hline\n$m_{1,1}$ = 93.15 & $m_{1,2}$ = 142.0 & $m_{1,3}$ = 192.38 \\\\\n$\\tilde{g}^{1,1}$ = 0.0168 & $\\tilde{g}^{1,2}$ = 0.0019 &\n$\\tilde{g}^{1,3}$ = 0.0028 \\\\\\hline\n$m_{2,1}$ = 178.45 & $m_{2,2}$ = 235.91 & $m_{2,3}$ = 293.12 \\\\\n$\\tilde{g}^{2,1}$ = 0.0128 & $\\tilde{g}^{2,2}$ = 0.0015 &\n$\\tilde{g}^{2,3}$ = 0.0023 \\\\\\hline\n$m_{3,1}$ = 259.96 & $m_{3,2}$ = 320.71 & $m_{3,3}$ = 381.21 \\\\\n$\\tilde{g}^{3,1}$ = 0.0107 & $\\tilde{g}^{3,2}$ = 0.0012 &\n$\\tilde{g}^{3,3}$ = 0.0020 \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The gauge couplings $g^{(0,0)(i,j)}$ of the standard model fermions\nare given in the form $\\tilde{g}^{i,j}=\\frac{g^{(0,0)(i,j)}}{g_0}$, where\n$g_0$ is the effective 4-dimensional gauge coupling. The masses of the KK\ngauge bosons $m_{i,j}$ are given in GeV units.\n$M$ is the bulk gauge boson mass, which is taken to be zero and $k^\\prime = k\/r_z$.\nThe values of $k$, $c$ and $r_z$ in this case are same as that in Table \\ref{T:t1}. The lowest\nlying mode $m_{0,0}$, which corresponds to photon, is not included in the table.}\n\\label{T:t2}\n\\end{table}\n\nAs stated in Sec. \\ref{S:intro} that the 5-dimensional RS model\nsuffers from the precision electroweak tests due to the fact that\nthe first excited KK gague boson has coupling larger than one with\nthe standard model field. To parameterize the precision electroweak\nconstraints in extra dimensional models the following quantity\nhas been defined \\cite{DHR,RW}.\n\\begin{equation}\nV=\\sum_n\\left(\\frac{g_n}{g_0}\\frac{m_W}{M_n}\\right)^2.\n\\label{E:peV}\n\\end{equation}\nHere, $m_W$ is the mass of the $W$ gauge boson and $M_n$ is\nthe higher KK gauge boson mass.\nThe summation on $n$ in the above equation is over all the higher\nKK gauge masses $M_n$ with corresponding gauge couplings $g_n$.\nIn our 6-dimensional model the index $n$ would be replaced by\na pair of integers and we should sum over all non-zero higher order modes.\nIt has been shown that by fitting to the precision electroweak\nobservables the quantity $V$ should satisfy the condition:\n$V <$ 0.0013 at 95$\\%$ confidence level \\cite{DHR}. It can be easily checked\nthat this bound can be respected by the gauge couplings and\nthe KK masses of the tables \\ref{T:t1} and \\ref{T:t2}.\nFrom both these tables, we can notice that the\ngauge couplings are decreasing with increasing the KK gauge masses\nfor a particular value of $\\frac{M}{k^\\prime}$. Hence, in the summation\nof eq. (\\ref{E:peV}) only the first few higher KK modes are relevant.\nWe have checked that for $\\frac{M}{k^\\prime}$ = 0.5, $V$ has come out\nto be about $5\\times 10^{-5}$. In the case of photon where bulk mass\nis zero the value of $V$ is found out to be about 0.000257. From these\nresults we can conclude that precision electroweak tests can be\nsatisfied in the 6-dimensional doubly warped model without\nintroducing too much hierarchy in the moduli $R_y$ and $r_z$.\n\nAt the Tevatron the higher KK gauge bosons have been searched\nin the channel $P\\bar{P}\\to X\\to e^+e^-$ and a\nlimit of $M_T>$ 700 GeV on the heavy vector gauge boson ($X$) mass has\nbeen put-in \\cite{Teva}.\nIn our case the gauge couplings of the higher KK modes have\nbeen reduced from the 4-dimensional coupling $g_0$ by some\nfactors which are given in tables \\ref{T:t1} and \\ref{T:t2}.\nHence, in our case, the Tevatron bounds for the non-zero KK mode\nmasses should be greater than 700$\\times\\tilde{g}^{i,j}$ GeV.\nAs an example, the KK mode of 93.15 GeV mass of the Table \\ref{T:t2}\nhas the gauge coupling ratio of 0.0168. Hence, the lower bound\nfrom the Tevatron on this KK mode mass would be about 12 GeV,\nwhich is much lower than our calculated value of 93.15 GeV.\nLike wise, from each column of the tables \\ref{T:t1} and \\ref{T:t2} it\ncan be easily seen that the above mentioned Tevatron bounds\ncan be satisfied. So the 6-dimensional warped model is not only\nfree from the precision electroweak constraints but also from\nthe Tevatron limits.\n\n\\section{Conclusions}\nThe extra dimensional phenomenological models in a warped geometry encounters\nproblems in putting the Higgs and the gauge\nfields in the bulk. It was shown that it is impossible to\nconstruct proper $W$ and $Z$ boson masses on the brane from the KK modes of a\nnon-Abelian bulk gauge field through spontaneous symmetry breaking\nin the bulk. Also proper coupling and masses for the first KK excitation of\na massless bulk gauge field consistent with\nelectroweak precision test as well as Fermilab Tevatron mass bound is hard to obtain\nwithout changing the bulk parameter of the theory\nfrom their desired values. In this work we have shown that it is possible to resolve both\nthese problems in a multiply warped geometry model\nwhere there are more than one modulus. Considering a 6-dimensional model we have shown that\nby setting one of the modulus approximately \ntwo orders smaller than the Planck scale, we can have the the mass for the lowest lying mode of the\nbulk gauge field ( with bulk mass $\\sim M_P$,\nacquired through a spontaneous symmetry breaking in the bulk) on the TeV-brane \nto be of the order of $100$ GeV which therefore may be identified with the $W$, $Z$ boson mass.\nMoreover, such a choice for the moduli which does\nnot contradict the main spirit of the RS model lowers the coupling of the first\nKK mode excitation of a massless bulk gauge field so that\nit can escape the electroweak precision test. We have determined the KK mode masses\nas well as their couplings for different choices of\nthe parameter of the theory namely the ratio of the bulk mass and the bulk\ncosmological constant. In the entire analysis the value of the\nwarp factor is maintained at $10^{-16}$ so that the resolution of the gauge hierarchy problem,\nthe main objective of these models can be achieved. These findings can be easily extended to models with even larger\nnumber of warped extra dimensions \\cite{cs}. One would then arrive at similar conclusions with a lesser hierarchy among different\nmoduli. We can therefore conclude that\na consistent description of bulk Higgs and gauge field with spontaneous symmetry breaking\nin the bulk can be obtained in a warped geometry model\nif the RS model in 5-dimensions is generalized to six or higher dimensions with more than one moduli. The phenomenology of these models \ntherefore becomes an interesting area of study for the forthcoming collider experiments.        \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMultifunctional material systems unite different, e.g., electric and magnetic, functionalities in either a single phase or a heterostructure. They thus are of great fundamental and technological interest. Given that the electric and magnetic properties furthermore are coupled, it becomes possible to control either the magnetization by the application of an electric field or the electric polarization via a magnetic field alone. An electric field control of magnetization is particularly appealing, as it removes the need to generate magnetic fields of sufficient strength for magnetization switching on small length scales -- thus enabling novel concepts for high density magnetic data storage applications.\n\\begin{figure}\n\\centering\n \n  \\includegraphics[]{img\/figure1a_to_1e}\\\\\n  \\caption{(a) Schematic illustration of the ferromagnetic thin film\/piezoelectric actuator hybrid\\xspace. (b), (d) The application of a voltage $\\ensuremath{V_\\mathrm{p}}\\xspace\\neq\\unit{0}{\\volt}$ to the actuator results in a deformation of the actuator and the affixed ferromagnetic film. The relaxed actuator at $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{0}{\\volt}$ is shown by dotted contours. (c), (e) Schematic free energy contours in the film plane.  The magnetic easy axis (e.a.) shown by the thick dashed line is oriented parallel to the compressive strain and can thus be rotated by 90\\degree by changing the polarity of $\\ensuremath{V_\\mathrm{p}}\\xspace$.}\\label{fig:panel1}\n\\end{figure}\nThus, techniques to control the magnetization by means of electric fields or currents have recently been vigorously investigated and several schemes for an electric control of magnetization have been reported. These include the spin-torque effect~\\cite{Slonczewski:1996, berger:1996, tsoi:1998, zhang:2002} in spin-valves and magnetic tunnel junctions and the direct electric field control of magnetization in intrinsically multiferroic materials~\\cite{spaldin:multiferroics, Eerenstein:2006, Fiebig2005,ramesh:2007,Lottermoser2004, gajek:2007, Chu2008, Zhao:2006} or in ferromagnetic\/ferroelectric heterostructures~\\cite{Stolichnov:2008, binek:2005}. A third, very attractive approach for the electric field control of magnetization takes advantage of the elastic channel, i.e., magnetostrictive coupling~\\cite{bootsmann:2005, wan:2006, Doerr:2006, Goennenwein:2008, brandlmaier:2008, Bihler:2008, overby:2008, jungwirth:2008, wang:2005}.\nWe here show that magnetoelasticity -- that is the effect of lattice strain on magnetic anisotropy -- in a polycrystalline nickel film\/piezoelectric actuator hybrid structure allows to switch the magnetic easy axis in the Ni film by 90\\degree at room temperature by simply changing the polarity of the voltage applied to the actuator.\nThis can be used to achieve either an irreversible magnetization orientation control of 180\\degree or a reversible control close to 90\\degree, depending on whether the magnetization orientation is prepared in a local or global minimum of the free energy via a magnetic field sweep prior to the electric control experiment.\nWe use ferromagnetic resonance (FMR) to quantify the effect of an electric field on magnetic anisotropy and superconducting quantum interference device (SQUID) magnetometry to directly record the evolution of magnetization as a function of the applied electric field. These experiments demonstrate that the strain-mediated electric field control of magnetization indeed is a viable technique in technologically relevant ferromagnetic thin films at room temperature.\n\\section{Sample preparation and experimental techniques}\\label{sec:samples}\nWe fabricated ferromagnetic thin film\/piezoelectric actuator structures by depositing nickel (Ni) thin films onto piezoelectric Pb(Zr$_x$Ti$_{1-x}$)O$_3$-based actuators~\\cite{manual:piezo}. Nickel was chosen as the ferromagnetic constituent as it is a prototype 3d itinerant ferromagnet with a Curie temperature $T_\\mathrm{c}=\\unit{627}{\\kelvin}$ well above room temperature~\\cite{kittel:ssp}, a high bulk saturation magnetization $M_\\mathrm{s}=\\unit{411}{\\kilo\\ampere\\per\\meter}$~\\cite{danan:1968} and sizeable volume magnetostriction~\\cite{chikazumi:ferromagnetism, lee:1955} $\\overline{\\lambda}=\\frac{2}{5}\\lambda_{100}+\\frac{3}{5}\\lambda_{111}=-32.9\\times10^{-6}$ with $\\lambda_{100}$ and $\\lambda_{111}$ being the single crystal saturation magnetostriction for a magnetic field applied along a crystalline $<100>$ or $<111>$ axis, respectively.\nThe actuators exhibit a hysteretic mechanical stroke of up to $1.3\\times10^{-3}$ along their dominant elongation axis [cf. Fig.~\\ref{fig:panel1}(a)] if voltages $\\unit{-30}{\\volt}\\leq \\ensuremath{V_\\mathrm{p}}\\xspace \\leq\\unit{+150}{\\volt}$ are applied.\nPrior to the deposition of the Ni film, the actuators were mechanically polished to a size of $x\\times y\\times z=\\unit{3\\times2.6\\times2}{\\milli\\meter\\cubed}$ [cf. Fig.~\\ref{fig:panel1}(a)] to accommodate the size of the sample to the restrictions imposed by the ferromagnetic resonance setup. We then used electron beam evaporation at a base pressure of \\unit{2.0\\times10^{-8} }{\\milli\\bbar} to deposit a \\unit{70}{\\nano\\meter} thick Ni film onto an area of \\unit{5}{\\milli\\meter\\squared} on the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-\\ensuremath{\\mathbf{\\hat{y}}}\\xspace face of the  actuators. To prevent oxidation of the Ni film, a \\unit{10}{\\nano\\meter} thick Au film was deposited in-situ on top of the Ni layer. The multifunctional hybrid obtained after Ni deposition is sketched schematically in Fig.~\\ref{fig:panel1}(a) together with the definition of the angles that describe the orientation of the magnetization $\\mathbf{M}=(M,\\Theta,\\Phi)$ and the external magnetic field $\\mathbf{H}=(H,\\theta,\\phi)$ in the sample-affixed coordinate system.\n\nTo determine the static magnetic response of the ferromagnetic thin film\/piezoelectric actuator hybrid\\xspace  we employ superconducting quantum interference device (SQUID) magnetometry. The Quantum Design MPMS-XL-7 SQUID magnetometer is sensitive to the projection $m=\\mathbf{m}\\mathbf{\\hat{H}}$ of the total magnetic moment $\\mathbf{m}$ onto the unit vector $\\mathbf{\\hat{H}}=\\frac{\\mathbf{H}}{H}$. We corrected $m$ for the paramagnetic contribution of the actuator and used the Ni film volume $V=\\unit{3.5\\times10^{-13}}{\\meter\\cubed}$ to calculate the respective projection $M=m\/V$ of the magnetization onto $\\mathbf{\\hat{H}}$. All magnetometry data shown in the following were recorded at a temperature $T=\\unit{300}{\\kelvin}$.\n\nThe magnetic anisotropy of the ferromagnetic thin film\/piezoelectric actuator hybrid\\xspace was measured  by ferromagnetic resonance (FMR) at room temperature. We use a Bruker ESP 300 spin resonance spectrometer with a TE$_{102}$ cavity operating at a constant microwave frequency $\\nu_\\mathrm{MW}=\\unit{9.3}{\\giga\\hertz}$. The sample can be rotated in the FMR setup with respect to the external dc magnetic field, so that either $\\theta$ or $\\phi$ [cf. Fig.~\\ref{fig:panel1}(a)] can be adjusted at will. To allow for lock-in detection we use magnetic field modulation at a modulation frequency of \\unit{100}{\\kilo\\hertz} with a modulation amplitude of $\\mu_0 H_\\mathrm{mod}=\\unit{3.2}{\\milli\\tesla}$.\n\n\\section{Phenomenology of strain-induced magnetic anisotropy}\\label{sec:theory}\n\nThe piezoelectric actuator deforms upon the application of a voltage $\\ensuremath{V_\\mathrm{p}}\\xspace \\neq \\unit{0}{\\volt}$. Due to its elasticity, an elongation (contraction) along one cartesian direction is always accompanied by a contraction (elongation) in the two orthogonal directions. Therefore, for $\\ensuremath{V_\\mathrm{p}}\\xspace>\\unit{0}{\\volt}$, the actuator expands along its dominant elongation axis \\ensuremath{\\mathbf{\\hat{y}}}\\xspace and contracts along the two orthogonal directions \\ensuremath{\\mathbf{\\hat{x}}}\\xspace and \\ensuremath{\\mathbf{\\hat{z}}}\\xspace. The Ni film affixed to the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-\\ensuremath{\\mathbf{\\hat{y}}}\\xspace face of the actuator is hence strained tensilely along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace and compressively along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace for $\\ensuremath{V_\\mathrm{p}}\\xspace>\\unit{0}{\\volt}$ [cf. Fig.~\\ref{fig:panel1}(b)]. For $\\ensuremath{V_\\mathrm{p}}\\xspace<\\unit{0}{\\volt}$ the actuator contracts along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace and thus expands along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace and \\ensuremath{\\mathbf{\\hat{z}}}\\xspace  and the Ni film thus exhibits a compressive strain along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace and a tensile strain along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace [cf. Fig.~\\ref{fig:panel1}(d)].\n\nTo describe the impact of this lattice strain on the Ni magnetization orientation we use a magnetic free energy density approach. The free energy $F_\\mathrm{tot}$ is a measure for the angular dependence of the magnetic hardness, with maxima in $F_\\mathrm{tot}$ corresponding to magnetically hard directions and minima to magnetically easy directions. In equilibrium, the magnetization always resides in a local minimum of $F_\\mathrm{tot}$. Contrary to single-crystalline films, the evaporated Ni films are polycrystalline and thus show no net crystalline magnetic anisotropy which may compete with strain-induced anisotropies and hereby reduce the achievable magnetization orientation effect~\\cite{brandlmaier:2008}. $F_\\mathrm{tot}$ is thus given by\n\\begin{equation}\\label{eq:Ftot}\n  F_\\mathrm{tot}=F_\\mathrm{stat}+F_\\mathrm{demag}+F_\\mathrm{magel}\\;.\n\\end{equation}\nThe first term $F_\\mathrm{stat} = -\\mu_0 M H(\\sin\\Theta \\sin\\Phi \\sin\\theta \\sin\\phi +\\cos\\Theta \\cos\\theta+\\sin\\Theta \\cos\\Phi \\sin\\theta \\cos\\phi)$ in Eq.~\\eref{eq:Ftot} is the Zeemann term and  describes the influence of an external magnetic field $\\mathbf{H}$ on the orientation of $\\mathbf{M}$. The uniaxial demagnetization term $F_\\mathrm{demag}=\\frac{\\mu_0}{2} M^2 \\sin^2\\Theta\\cos^2\\Phi$ is the anisotropy caused by the thin-film shape of the sample~\\cite{Morrish2001}. The last contribution to Eq.~\\eref{eq:Ftot}\n\\begin{eqnarray}\\label{eq:Fmagel}\n    F_\\mathrm{magel}&=&\\frac{3}{2} \\overline{\\lambda} \\left(c^\\mathrm{Ni}_{12}-c^\\mathrm{Ni}_{11}\\right)\n  \\bigl[\\varepsilon_1(\\sin^2\\Theta\\sin^2\\Phi-1\/3) \\\\\n   &&+\\varepsilon_2(\\cos^2\\Theta-1\/3)+\\varepsilon_3(\\sin^2\\Theta\\cos^2\\Phi-1\/3) \\bigr]\\nonumber\n\\end{eqnarray}\ndescribes the influence of the lattice strains on the magnetic anisotropy~\\cite{chikazumi:ferromagnetism}. The strains along the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-, \\ensuremath{\\mathbf{\\hat{y}}}\\xspace- and \\ensuremath{\\mathbf{\\hat{z}}}\\xspace-axis are denoted in Voigt (matrix) notation~\\cite{nye:1985} as $\\varepsilon_1, \\varepsilon_2$ and $\\varepsilon_3$, respectively. Furthermore, $c^\\mathrm{Ni}_{11}=\\unit{2.5\\times10^{11}}{\\newton\\per\\meter^2}$ and $c^\\mathrm{Ni}_{12}=\\unit{1.6\\times10^{11}}{\\newton\\per\\meter^2}$ are the elastic moduli of Ni~\\cite{lee:1955}. The effects of shear strains ($\\varepsilon_i, i \\in \\{4,5,6\\}$) average out in our polycrystalline film and thus are neglected.\n\nUsing $\\sin^2\\Theta\\sin^2\\Phi+\\cos^2\\Theta+\\sin^2\\Theta\\cos^2\\Phi=1$ and omitting isotropic terms, Eq.~\\eref{eq:Fmagel} can be rewritten as\n\\begin{equation}\\label{eq:Fmagel_final}\n    F_\\mathrm{magel}=K_\\mathrm{u,magel,y} \\cos^2\\Theta + K_\\mathrm{u,magel,z} \\sin^2\\Theta \\cos^2\\Phi\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{eq:Kmagel}\n  K_\\mathrm{u,magel,y} &=& \\frac{3}{2} \\overline{\\lambda}(c^\\mathrm{Ni}_{12}-c^\\mathrm{Ni}_{11})(\\varepsilon_2-\\varepsilon_1) \\\\\n \\nonumber K_\\mathrm{u,magel,z} &=& \\frac{3}{2} \\overline{\\lambda} (c^\\mathrm{Ni}_{12}-c^\\mathrm{Ni}_{11})(\\varepsilon_3-\\varepsilon_1)\\;.\n\\end{eqnarray}\n\nDue to the elasticity of the actuator and the Ni film, the strains $\\varepsilon_i$ are not independent of each other. The strains $\\varepsilon_1$ and $\\varepsilon_2$ in the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-\\ensuremath{\\mathbf{\\hat{y}}}\\xspace plane are linked by the Poisson ratio $\\nu=0.45$ of the actuator~\\cite{manual:piezo} according to\n\\begin{equation}\\label{eq:poisson_ip}\n    \\varepsilon_1=-\\nu \\varepsilon_2 \\;.\n\\end{equation}\nFurthermore, due to the elasticity of the Ni film, the out-of-plane strain $\\varepsilon_3$ can be expressed as~\\cite{brandlmaier:2008}\n\\begin{equation}\\label{eq:strain_oop}\n    \\varepsilon_3=-\\frac{c^\\mathrm{Ni}_{12}}{c^\\mathrm{Ni}_{11}}(\\varepsilon_1+\\varepsilon_2) \\;.\n\\end{equation}\nAssuming a linear expansion of the actuator with lateral dimension $L$ parallel to its dominant elongation axis, we can calculate the strain in the Ni film parallel to the actuator's dominant elongation axis as\n\\begin{equation}\\label{eq:strain_voltage}\n    \\varepsilon_2=\\frac{\\delta L}{L}\\frac{\\ensuremath{V_\\mathrm{p}}\\xspace}{\\unit{180}{\\volt}}\\;,\n\\end{equation}\nwhere $\\delta L \/L=1.3\\times10^{-3}$  is the nominal full actuator stroke for the full voltage swing $\\unit{-30}{\\volt}\\leq \\ensuremath{V_\\mathrm{p}}\\xspace \\leq\\unit{+150}{\\volt}$.\nWith Eqs.~\\eref{eq:poisson_ip}, \\eref{eq:strain_oop} and~\\eref{eq:strain_voltage} all strains in the Ni film can thus directly be calculated for any given voltage \\ensuremath{V_\\mathrm{p}}\\xspace. This allows to determine the equilibrium orientation of the magnetization as a function of \\ensuremath{V_\\mathrm{p}}\\xspace by minimizing the total magnetoelastic free energy density $F_\\mathrm{tot}$ [Eq.~\\eref{eq:Ftot}].\n\nDue to the negative magnetostriction ($\\overline{\\lambda}<0$) of Ni and $c^\\mathrm{Ni}_{11}>c^\\mathrm{Ni}_{12}$, the in-plane easy axis of $F_\\mathrm{tot}$ is oriented orthogonal to tensile and parallel to compressive strains in the absence of external magnetic fields. Due to the strong uniaxial out-of-plane anisotropy caused by $F_\\mathrm{demag}$, the in-plane easy axis is the global easy axis. For $\\ensuremath{V_\\mathrm{p}}\\xspace>\\unit{0}{\\volt}$,  the Ni film exhibits a tensile strain along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace ($\\varepsilon_2>0$) and a compressive strain along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace  ($\\varepsilon_1<0$). The easy axis of $F_\\mathrm{tot}$ is thus oriented along the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-direction [cf. Fig.~\\ref{fig:panel1}(c)]. Accordingly, for $\\ensuremath{V_\\mathrm{p}}\\xspace<\\unit{0}{\\volt}$, the easy axis of $F_\\mathrm{tot}$ is oriented parallel to the compressive strain along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace and orthogonal to the tensile strain along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace [cf. Fig.~\\ref{fig:panel1}(e)]. We thus expect a 90\\degree rotation of the in-plane easy axis of $F_\\mathrm{tot}$ upon changing the polarity of \\ensuremath{V_\\mathrm{p}}\\xspace.\nNote that in the absence of external magnetic fields there are always two energetically equivalent antiparallel orientations of $\\mathbf{M}$ along the magnetic easy axis. However, for simplicity, we only show one of the two resulting possible in-plane orientations of $\\mathbf{M}$ in Figs.~\\ref{fig:panel1}(b) and~\\ref{fig:panel1}(d).\nNevertheless these panels show that it should be possible to change the orientation of $\\mathbf{M}$ from $\\mathbf{M} || \\ensuremath{\\mathbf{\\hat{x}}}\\xspace$ to $\\mathbf{M} || \\ensuremath{\\mathbf{\\hat{y}}}\\xspace$ via the application of appropriate voltages $\\ensuremath{V_\\mathrm{p}}\\xspace$ to the actuator.\n\nThe total magnetic free energy density $F_\\mathrm{tot}$ in Eq.~\\eref{eq:Ftot} is experimentally accessible by ferromagnetic resonance measurements. The FMR equations of motion~\\cite{smit:1954, smit:1955, suhl:1955}\n\\begin{eqnarray}\\label{eq:FMR_motion}\n\t \\left(\\frac{\\omega}{\\gamma}\\right)^2&=&\\frac{1}{M_\\mathrm{s}^2\\sin^2\\Theta} \\bigl.\\Bigl(\\left(\\partial_{\\Phi}^2F_\\mathrm{tot}\\right)\\left(\\partial_\\Theta^2 F_\\mathrm{tot}\\right)-\\left(\\partial_{\\Phi}\\partial_\\Theta F_\\mathrm{tot}\\right)^2\\Bigr)\\bigr|_{\\Theta_0,\\Phi_0}\\\\\n\\nonumber &&\\mathrm{and}~\\bigl.\\partial_\\Theta F_\\mathrm{tot}\\bigr|_{\\Theta=\\Theta_0} = \\bigl.\\partial_{\\Phi} F_\\mathrm{tot}\\bigr|_{\\Phi={\\Phi}_0}=0\n    \\end{eqnarray}\nlink the experimentally determined ferromagnetic resonance field $\\mu_0H_\\mathrm{res}$ to the magnetic free energy density $F_\\mathrm{tot}$. In Eq.~\\eref{eq:FMR_motion}, $\\gamma$ is the gyromagnetic ratio and $\\omega=2\\pi\\nu_\\mathrm{MW}$ with the microwave frequency $\\nu_\\mathrm{MW}$.\nThe effect of damping is neglected in Eq.~\\eref{eq:FMR_motion}, as damping only affects the lineshape which will not be discussed in this article. We note that the FMR resonance field $\\mu_0H_\\mathrm{res}$  measured in experiment can be considered as a direct indicator of the relative magnetic hardness along the external dc magnetic field direction. From Eqs.~\\eref{eq:Ftot} and~\\eref{eq:FMR_motion} one finds that smaller resonance fields correspond to magnetically easier directions and larger resonance fields to magnetically harder directions.\n\\section{Ferromagnetic resonance}\\label{sec:fmr}\n\\begin{figure}\n \n  \\includegraphics[width=\\textwidth]{img\/figure2a_2b}\\\\\n  \\caption{(a) FMR spectra recorded with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace at different voltages \\ensuremath{V_\\mathrm{p}}\\xspace. An increasing resonance field $\\mu_0 H_\\mathrm{res}$ (solid squares) is observed for increasing $\\ensuremath{V_\\mathrm{p}}\\xspace$ while the lineshape is not significantly altered. (b) The dependence of $\\mu_0 H_\\mathrm{res}$ on \\ensuremath{V_\\mathrm{p}}\\xspace is qualitatively different for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace.  Full symbols correspond to increasing $\\ensuremath{V_\\mathrm{p}}\\xspace$ and open symbols to decreasing $\\ensuremath{V_\\mathrm{p}}\\xspace$. The solid lines represent the resonance fields calculated from magnetoelastic theory (cf. Sec.~\\ref{sec:theory}) yielding very good agreement with the measurement. For \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace, increasing \\ensuremath{V_\\mathrm{p}}\\xspace increases $\\mu_0 H_\\mathrm{res}$ and thus the magnetic hardness of this direction, while for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace, increasing \\ensuremath{V_\\mathrm{p}}\\xspace decreases $\\mu_0 H_\\mathrm{res}$ and thus the magnetic hardness of this direction. The hysteresis of $\\mu_0 H_\\mathrm{res}$ is due to the hysteretic stress-strain curve of the actuator.}\\label{fig:panel2}\n\\end{figure}\n\\begin{figure}\n \n  \\includegraphics[width=\\textwidth]{img\/figure3a_3b}\\\\\n  \\caption{(a) The symbols show the FMR resonance field $\\mu_0 H_\\mathrm{res}(\\theta)$ obtained for a constant actuator voltage $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ (open triangles) and $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+20}{\\volt}$ (solid circles) as a function of the orientation $\\theta$ of $\\mathbf{H}$ in the sample plane. A uniaxial (180\\degree periodic) anisotropy of $\\mu_0 H_\\mathrm{res}(\\theta)$ is observed for both voltages. However, the easy axis is rotated by 90\\degree as $\\ensuremath{V_\\mathrm{p}}\\xspace$ is changed from \\unit{+20}{\\volt} to \\unit{-30}{\\volt}. The full lines show the resonance fields simulated using the anisotropy constants from Eq.~\\eref{eq:constants}. (b) Corresponding experiments with $\\mathbf{H}$ rotated in the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace plane (from within the Ni film plane to out-of-plane). Regardless of \\ensuremath{V_\\mathrm{p}}\\xspace, a strong uniaxial anisotropy with the hard axis perpendicular to the film plane is observed. The inset shows that the resonance fields for $\\mathbf{H}$ in the film plane ($\\phi=90\\degree$ and $\\phi=270\\degree$) are still shifted as a function of \\ensuremath{V_\\mathrm{p}}\\xspace in accordance to the data shown in (a) for $\\theta=90\\degree$ and $\\theta=270\\degree$.}\\label{fig:panel3}\n\\end{figure}\nIn this Section, we quantitatively determine the magnetic anisotropy of our ferromagnetic thin film\/piezoelectric actuator hybrid\\xspace structure using FMR measurements. Figure~\\ref{fig:panel2}(a) shows four FMR spectra recorded at room temperature with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and a voltage $\\ensuremath{V_\\mathrm{p}}\\xspace \\in \\{\\unit{-30}{\\volt},\\unit{0}{\\volt},\\unit{30}{\\volt},\\unit{90}{\\volt}\\}$ applied to the actuator, respectively. Each spectrum shows one strong ferromagnetic resonance which -- due to the magnetic field modulation and lock-in detection -- has a lineshape corresponding to the first derivative of a Lorentzian line~\\cite{Poole1996}. The resonance field $\\mu_0H_\\mathrm{res}$ of a given FMR line is determined as the arithmetic mean of its maximum and minimum and depicted by the full squares in Fig.~\\ref{fig:panel2}(a). The figure shows that, with the external magnetic field \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace, $\\mu_0H_\\mathrm{res}$ is shifted to higher magnetic fields for increasing $\\ensuremath{V_\\mathrm{p}}\\xspace$, while the lineshape is not significantly changed. According to Eqs.~\\eref{eq:Ftot} and~\\eref{eq:FMR_motion}, this implies that the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace direction becomes increasingly harder as $\\ensuremath{V_\\mathrm{p}}\\xspace$ is increased.\n\nTo determine the evolution of $\\mu_0 H_\\mathrm{res}$ with \\ensuremath{V_\\mathrm{p}}\\xspace in more detail, we recorded FMR spectra similar to those shown in Fig.~\\ref{fig:panel2}(a) for \\ensuremath{V_\\mathrm{p}}\\xspace increasing from $\\unit{-30}{\\volt}$ to $\\unit{+90}{\\volt}$ (upsweep) and decreasing back to $\\unit{-30}{\\volt}$ (downsweep) in steps of $\\Delta\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{10}{\\volt}$. For \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace, we obtain the resonance fields shown by the squares in Fig.~\\ref{fig:panel2}(b). Here, full squares depict the upsweep and open squares the downsweep of \\ensuremath{V_\\mathrm{p}}\\xspace. As discussed in the context of Fig.~\\ref{fig:panel2}(a), Fig.~\\ref{fig:panel2}(b) shows that the FMR resonance field for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace increases with increasing \\ensuremath{V_\\mathrm{p}}\\xspace and decreases with decreasing \\ensuremath{V_\\mathrm{p}}\\xspace. The small hysteresis between up- and downsweep is due to the hysteretic expansion of the actuator~\\cite{manual:piezo}. Carrying out the same series of FMR measurements with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace yields the resonance fields shown by the triangles in Fig.~\\ref{fig:panel2}(b). For this magnetic field orientation, $\\mu_0 H_\\mathrm{res}$ decreases for increasing \\ensuremath{V_\\mathrm{p}}\\xspace and vice versa. In terms of magnetic anisotropy we thus can conclude that the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace direction becomes easier for increasing \\ensuremath{V_\\mathrm{p}}\\xspace while at the same time the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace direction simultaneously becomes harder.\n\nFor a more quantitative discussion we have also plotted the behavior expected from Eqs.~\\eref{eq:Ftot} and~\\eref{eq:FMR_motion} as solid lines in Fig.~\\ref{fig:panel2}(b). These lines show the resonance fields obtained by assuming a linear, non-hysteretic voltage-strain relation [cf. Eq.~\\eref{eq:strain_voltage}] and solving Eq.~\\eref{eq:FMR_motion} with $F_\\mathrm{tot}$ from Eq.~\\eref{eq:Ftot}. We use a saturation magnetization $M_\\mathrm{s}=\\unit{370}{\\kilo\\ampere\\per\\meter}$ as determined by SQUID measurements and a $g$-factor of 2.165~\\cite{meyer:1961}.\nConsidering the fact that the actuator expansion saturates at high voltages and the data in Fig.~\\ref{fig:panel2}(b) only show a minor loop of the full actuator swing, the simulation is in full agreement to the experimental results. This demonstrates that for $\\ensuremath{V_\\mathrm{p}}\\xspace<\\unit{0}{\\volt}$ the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace-direction is magnetically easier than the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-direction, while for $\\ensuremath{V_\\mathrm{p}}\\xspace>\\unit{0}{\\volt}$ the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-direction is easier than the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace-direction. Moreover, at $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{0}{\\volt}$, both measurement and simulation yield resonance fields of $\\mu_0H_\\mathrm{res}\\approx\\unit{150}{\\milli\\tesla}$ for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace as well as \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace. Thus, at this voltage, both orientations \\ensuremath{\\mathbf{\\hat{x}}}\\xspace and \\ensuremath{\\mathbf{\\hat{y}}}\\xspace are equally easy in our sample.\n\nTo quantitatively determine the full magnetic anisotropy as a function of $\\ensuremath{V_\\mathrm{p}}\\xspace$, we recorded FMR traces at constant \\ensuremath{V_\\mathrm{p}}\\xspace for several different $\\mathbf{H}$ orientations. The FMR resonance fields thus determined in experiment are shown in Fig.~\\ref{fig:panel3} together with simulations according to Eqs.~\\eref{eq:Ftot} and~\\eref{eq:FMR_motion}. The open triangles in Fig.~\\ref{fig:panel3} represent the resonance fields obtained for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and the full circles those obtained for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+20}{\\volt}$. Note that in the polar plots in Fig.~\\ref{fig:panel3}, the distance of $\\mu_0 H_\\mathrm{res}$ from the coordinate origin is an indicator of the magnetic hardness, with easy directions corresponding to small distances and hard directions to large distances.\n\nIf $\\mathbf{H}$ is rotated in the film plane [cf. Fig.~\\ref{fig:panel3}(a), $\\phi=90\\degree$], the obtained $\\mu_0H_\\mathrm{res}(\\theta)$ shows minima at $\\theta=0\\degree$ and $\\theta=180\\degree$ for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and at $\\theta=90\\degree$ and $\\theta=270\\degree$ for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+20}{\\volt}$, respectively. Thus, a clear 180\\degree periodicity of the resonance fields and hence a uniaxial magnetic anisotropy is observed for both \\ensuremath{V_\\mathrm{p}}\\xspace.  As the orientations $\\theta$ corresponding to minima of $\\mu_0H_\\mathrm{res}$ for one voltage coincide with maxima for the other voltage, we conclude that the direction of the easy axis is rotated by 90\\degree if \\ensuremath{V_\\mathrm{p}}\\xspace is changed from $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ to $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+20}{\\volt}$. This is exactly the behaviour expected according to Figs.~\\ref{fig:panel1}(c) and~\\ref{fig:panel1}(e).\n\nIf $\\mathbf{H}$ is rotated from within the film plane ($\\phi=90\\degree$, $\\theta=90\\degree$) to perpendicular to the film plane ($\\phi=0\\degree$, $\\theta=90\\degree$), we obtain the resonance fields shown in Fig.~\\ref{fig:panel3}(b). In this case, we observe a strong uniaxial anisotropy with a hard axis perpendicular to the film plane regardless of \\ensuremath{V_\\mathrm{p}}\\xspace, stemming from the \\ensuremath{V_\\mathrm{p}}\\xspace-independent contribution $F_\\mathrm{demag}$ to $F_\\mathrm{tot}$. The inset in Fig.~\\ref{fig:panel3}(b) shows that for $\\mathbf{H}$ in the film plane, the resonance fields for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+20}{\\volt}$ are shifted by approximately \\unit{10}{\\milli\\tesla} in full accordance to the data in Fig.~\\ref{fig:panel3}(a).\n\nAs discussed in Section~\\ref{sec:theory}, the \\ensuremath{V_\\mathrm{p}}\\xspace dependence of the resonance field is given by a \\ensuremath{V_\\mathrm{p}}\\xspace dependence of $F_\\mathrm{magel}$. We thus evaluate the measurement data shown in Figs.~\\ref{fig:panel3}(a,b) using an iterative fitting procedure of $K_\\mathrm{u,magel,y}$ in Eq.~\\eref{eq:Ftot} to fulfill Eq.~\\eref{eq:FMR_motion}. We obtain\n\\begin{eqnarray}\\label{eq:constants}\n  K_\\mathrm{u,magel,y}(\\unit{-30}{\\volt})\/M_\\mathrm{s} &=& \\unit{-3.4}{\\milli\\tesla} \\\\\n  \\nonumber K_\\mathrm{u,magel,y}(\\unit{+20}{\\volt})\/M_\\mathrm{s} &=& \\unit{4.4}{\\milli\\tesla}\\;.\n\\end{eqnarray}\nThe resonance fields calculated using these anisotropy fields as well as Eqs.~\\eref{eq:FMR_motion} and~\\eref{eq:Ftot} are depicted by the lines in Fig.~\\ref{fig:panel3}. The good agreement between simulation and experiment shows that $F_\\mathrm{tot}$ given by Eq.~\\eref{eq:Ftot} is sufficient to describe the magnetic anisotropy of the Ni film.\n\nIn summary, the FMR experiments conclusively demonstrate that it is possible to invert the in-plane magnetic anisotropy of our hybrid structure, i.e. to invert the sign of $ K_\\mathrm{u,magel,y}$ \\textit{solely by changing \\ensuremath{V_\\mathrm{p}}\\xspace.} As the FMR experiment furthermore allows to quantitatively determine all contributions to the free energy, Eq.~\\eref{eq:Ftot}, the orientation of the magnetization vector $\\mathbf{M}$ in our sample can be calculated a priori for arbitrary $\\mathbf{H}$ and \\ensuremath{V_\\mathrm{p}}\\xspace. However, it is not possible to directly measure the magnetization orientation as a function of \\ensuremath{V_\\mathrm{p}}\\xspace with FMR. To demonstrate that the piezo-voltage control of magnetic anisotropy indeed allows for a voltage control of $\\mathbf{M}$, we now turn to magnetometry.\n\n\n\\section{Magnetometry}\\label{sec:squid}\n\\begin{figure}\n \n  \\centering\n  \\includegraphics[]{img\/figure4a_4b}\\\\\n  \\caption{(a) $M(H)$-loops recorded with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace show a higher remanent magnetization for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ than for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$, thus the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace-axis is magnetically easier for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ than for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$. (b) The \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-axis is magnetically easier for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ than for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$.}\\label{fig:panel4}\n\\end{figure}\n\\begin{figure}\n\\centering\n \n  \\includegraphics[]{img\/figure5a_to_5c}\\\\\n  \\caption{SQUID $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$-loops show the projection $M$ of the magnetization $\\mathbf{M}$ onto the direction of the applied magnetic field $\\mathbf{H}$ as a function of $\\ensuremath{V_\\mathrm{p}}\\xspace$. The symbols represent the experimental data and the lines show the simulation of $M$ resulting from a minimization of $F_\\mathrm{tot}$ [Eq.~\\eref{eq:Ftot}], with $\\varepsilon_2$ determined using a strain gauge to explicitly take into account the actuator hysteresis. (a) For \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace with $\\mu_0 H=\\unit{-3}{\\milli\\tesla}$, $\\mathbf{M}$ exhibits an \\textit{irreversible} rotation from A to B followed by a \\textit{reversible} rotation from B to C. (b), (c) For \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace with $\\mu_0 H=\\unit{-5}{\\milli\\tesla}$, $\\mathbf{M}$ exhibits a \\textit{reversible} rotation from A to B and back to A.}\\label{fig:panel5}\n\\end{figure}\n\\begin{figure}\n\\centering\n \n  \\includegraphics[]{img\/figure6a_to_6c}\\\\\n  \\caption{Calculated free energy contours in the film plane at the points of the $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$-loop depicted by capital letters in Fig.~\\ref{fig:panel5} (solid lines: $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$, dotted lines: $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$). To clarify the lifting of the free energy degeneracy by the magnetic field, an angle of 10\\degree between the magnetic field and the $\\ensuremath{\\mathbf{\\hat{x}}}\\xspace$ or $\\ensuremath{\\mathbf{\\hat{y}}}\\xspace$ axis was assumed. The open downward-oriented arrows depict the orientation of $\\mathbf{H}$ during the field preparation at \\unit{7}{\\tesla} and the closed downward-oriented arrows depict the orientation of $\\mathbf{H}$ during the actual measurement. Capital letters indicate the equilibrium $\\mathbf{M}$ orientation at the corresponding positions in Fig.~\\ref{fig:panel5}.  (a) Subsequent to the field preparation at \\unit{-30}{\\volt}, $\\mathbf{M}$ resides in a \\textit{local} minimum of $F_\\mathrm{tot}$  (point A) at $\\mu_0H=\\unit{-3}{\\milli\\tesla}$ and rotates to the \\textit{global} minimum of $F_\\mathrm{tot}$ at (point B) as \\ensuremath{V_\\mathrm{p}}\\xspace is increased to $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$. Sweeping the voltage from \\unit{+120}{\\volt} to \\unit{-30}{\\volt} now results in $\\mathbf{M}$ following the \\textit{global} minimum of $F_\\mathrm{tot}$. (b), (c)  For $\\mu_0H=\\unit{-5}{\\milli\\tesla}$, $\\mathbf{M}$ follows the \\textit{global} minimum of $F_\\mathrm{tot}$ as \\ensuremath{V_\\mathrm{p}}\\xspace is changed.}\\label{fig:panel6}\n\\end{figure}\nIn this Section, we show that it is possible to not only change the magnetic anisotropy but to deliberately \\textit{irreversibly} and\/or \\textit{reversibly} rotate $\\mathbf{M}$ simply by changing \\ensuremath{V_\\mathrm{p}}\\xspace. To this end, we need to employ an experimental technique that is directly sensitive to the orientation of $\\mathbf{M}$ rather than to magnetic anisotropies. Here we used SQUID magnetometry to record the projection $m$ of the total magnetic moment $\\mathbf{m}$ onto the direction of the external magnetic field $\\mathbf{H}$.\n\nIn a first series of experiments we recorded $m$ as a function of the external magnetic field magnitude $\\mu_0H$ at fixed orientations of $\\mathbf{H}$ and fixed voltages \\ensuremath{V_\\mathrm{p}}\\xspace at $T=\\unit{300}{\\kelvin}$. Figure~\\ref{fig:panel4}(a) shows $M=m\/V$ measured with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace as a function of the external magnetic field strength at constant voltage $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ (full circles) and $\\ensuremath{V_\\mathrm{p}}\\xspace={-30}{\\volt}$ (open triangles). The Ni film shows a rectangular $M(H)$ loop for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$, while for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ the remanent magnetization is lowered by a factor of approximately three. According to, e.g., Morrish~\\cite{Morrish2001}, the rectangular loop for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ indicates a magnetically easy axis, while the smooth, s-shaped loop for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ indicates a magnetically harder axis. Thus Fig.~\\ref{fig:panel4}(a) shows that the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace-direction is magnetically \\textit{easier} for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and \\textit{harder} for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ -- as expected from the FMR experiments. Changing the orientation of $\\mathbf{H}$ to \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace yields the $M(H)$-loops shown in Fig.~\\ref{fig:panel4}(b). Following the same line of argument we can conclude that the \\ensuremath{\\mathbf{\\hat{x}}}\\xspace-direction is magnetically \\textit{easier} for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$ and \\textit{harder} for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$. Altogether these results show that we observe an in-plane anisotropy, the easy axis of which is parallel to \\ensuremath{\\mathbf{\\hat{y}}}\\xspace for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and parallel to \\ensuremath{\\mathbf{\\hat{x}}}\\xspace for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+30}{\\volt}$. These observations are fully consistent with the FMR results, and corroborate the simple model shown in Fig.~\\ref{fig:panel1}.\n\nBefore discussing further experimental magnetometry results, we note that the free energy minima shown in Fig.~\\ref{fig:panel1}(c,e) as well as the in-plane resonance fields in Fig.~\\ref{fig:panel3}(a) are degenerate by 180\\degree. This degeneracy may lead to demagnetization due to domain formation if the polarity of $\\ensuremath{V_\\mathrm{p}}\\xspace$ is repeatedly inverted as there are always two energetically equally favorable but opposite directions of $\\mathbf{M}$. Thus, to achieve a reversible control of $\\mathbf{M}$ orientation, the degeneracy of $F_\\mathrm{tot}$ needs to be lifted. This can be achieved if a unidirectional anisotropy is superimposed on the uniaxial anisotropy. Regarding $F_\\mathrm{tot}$ in Eq.~\\eref{eq:Ftot}, only $F_\\mathrm{stat}$ possesses the desirable unidirectional anisotropy. Hence, a small but finite external magnetic field $\\mathbf{H}\\neq0$ can be used to lift the 180\\degree degeneracy. This approach works for all $\\mathbf{H}$ orientations, except for $\\mathbf{H}$ exactly parallel \\ensuremath{\\mathbf{\\hat{x}}}\\xspace or \\ensuremath{\\mathbf{\\hat{y}}}\\xspace. For \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace (with $H>0$), the easy axis parallel to \\ensuremath{\\mathbf{\\hat{x}}}\\xspace in Fig.~\\ref{fig:panel1}(c) is replaced by an easier positive \\ensuremath{\\mathbf{\\hat{x}}}\\xspace direction and a harder negative \\ensuremath{\\mathbf{\\hat{x}}}\\xspace direction, thus the degeneracy is lifted for $\\ensuremath{V_\\mathrm{p}}\\xspace>0$. However, for $\\ensuremath{V_\\mathrm{p}}\\xspace<0$ [cf. Fig.~\\ref{fig:panel1}(e)] a magnetic field \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace is orthogonal to the easy axis which thus remains degenerate. The same consideration holds for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace where we expect degenerate free energy minima for $\\ensuremath{V_\\mathrm{p}}\\xspace>0$ and a preferred $\\mathbf{M}$ orientation for $\\ensuremath{V_\\mathrm{p}}\\xspace<0$. However, in experiment it is essentially impossible to orient $\\mathbf{H}$ \\textit{exactly} along \\ensuremath{\\mathbf{\\hat{x}}}\\xspace or \\ensuremath{\\mathbf{\\hat{y}}}\\xspace; any small misorientation between $\\mathbf{H}$ and \\ensuremath{\\mathbf{\\hat{x}}}\\xspace or \\ensuremath{\\mathbf{\\hat{y}}}\\xspace is sufficient to lift the degeneracy.\n\nWe now turn to the experimental measurement of the voltage control of $\\mathbf{M}$ orientation. To show that $\\mathbf{M}$ can be rotated by varying $\\ensuremath{V_\\mathrm{p}}\\xspace$ alone, we change $\\ensuremath{V_\\mathrm{p}}\\xspace$ at constant external magnetic bias field $\\mathbf{H}$. As SQUID magnetometry is limited to recording the projection of $\\mathbf{M}$ on the direction of the external magnetic field $\\mathbf{H}$, it is important to choose appropriate $\\mathbf{H}$ orientations in the experiments. As evident from Figs.~\\ref{fig:panel1} and \\ref{fig:panel4}, \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace and \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace are the most interesting orientations of the external magnetic field. In view of the circumstance discussed in the previous paragraph, we applied $\\mathbf{H}$ to within 1\\degree of \\ensuremath{\\mathbf{\\hat{x}}}\\xspace or \\ensuremath{\\mathbf{\\hat{y}}}\\xspace, respectively. We still refer to these orientations of $\\mathbf{H}$ as \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace and \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace for simplicity, but take into account a misalignment of 1\\degree in the calculations. As the experimental results and the corresponding simulations will show, this slight misalignment is sufficient to lift the degeneracy in the free energy regardless of $\\ensuremath{V_\\mathrm{p}}\\xspace$.\n\nRecording $M$ as a function of $\\ensuremath{V_\\mathrm{p}}\\xspace$ for two complete voltage cycles $\\unit{-30}{\\volt}\\leq \\ensuremath{V_\\mathrm{p}}\\xspace \\leq \\unit{+120}{\\volt}$ with \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and $\\mu_0H=\\unit{-3}{\\milli\\tesla}$ yields the data points shown in Fig.~\\ref{fig:panel5}(a). Since \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace, $M$ is the projection of $\\mathbf{M}$ to the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace direction in the experiment. Prior to the first voltage sweep starting at point A, the voltage was set to $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and the sample was magnetized to a single domain state by applying $\\mu_0 H=\\unit{+7}{\\tesla}$ along \\ensuremath{\\mathbf{\\hat{y}}}\\xspace which is the easy axis for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$. The magnetic field was then swept to $\\mu_0 H=\\unit{-3}{\\milli\\tesla}$ which is close to but still below the coercive field of the rectangular loops [cf. Fig.~\\ref{fig:panel4}] and the acquisition of the $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ data was started. The fact that $M$ is positive at first (starting from point A), while the field is applied along the negative \\ensuremath{\\mathbf{\\hat{y}}}\\xspace direction, shows that $\\mathbf{M}$ and $\\mathbf{H}$ are essentially antiparallel at first. Upon increasing \\ensuremath{V_\\mathrm{p}}\\xspace in steps of $\\unit{+5}{\\volt}$ to $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$ (point B), $M$ vanishes, which indicates an orthogonal orientation of $\\mathbf{M}$ and $\\mathbf{H}$. Upon reducing $\\ensuremath{V_\\mathrm{p}}\\xspace$ to its initial value of \\unit{-30}{\\volt}, $M$ becomes negative (point C), evidencing a parallel orientation of $\\mathbf{M}$ and $\\mathbf{H}$. The \\ensuremath{V_\\mathrm{p}}\\xspace cycle was then repeated once more, with $M$ now reversibly varying between its negative value at point C and zero at B.  Hence the evolution of $M$ is qualitatively different in the first and in the second \\ensuremath{V_\\mathrm{p}}\\xspace cycle, with an \\textit{irreversible} $\\mathbf{M}$ rotation in the first cycle and a \\textit{reversible} $\\mathbf{M}$ rotation in the second cycle. This behaviour is expected from the magnetic free energy [cf. Eq.~\\eref{eq:Ftot}] as the preparation (point A) yields $\\mathbf{M}$ in a metastable state ($\\mathbf{M}$ antiparallel $\\mathbf{H}$).\n\nAfter a renewed preparation with $\\mu_0 H=\\unit{+7}{\\tesla}$ at $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$, we repeated the experiment with the external magnetic field \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace but with a slightly larger magnitude $\\mu_0 H=\\unit{-5}{\\milli\\tesla}$ and again recorded $M$ for a complete voltage cycle. The magnetic field magnitude of $\\mu_0 H=\\unit{-5}{\\milli\\tesla}$ was chosen as it exceeds the coercive field (cf. Fig.~\\ref{fig:panel4}) while keeping the influence of the Zeemann term in Eq.~\\eref{eq:Ftot} on the total magnetic anisotropy comparable to the influence of the strain-induced anisotropies [cf. Eq.~\\eref{eq:constants}]. The experimental data are shown by the symbols in Fig.~\\ref{fig:panel5}(b). Here a parallel orientation of $\\mathbf{M}$ and $\\mathbf{H}$ is already observed at point A ($\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$) and $\\mathbf{M}$ rotates reversibly by approximately 70\\degree towards point B ($\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$) and back when \\ensuremath{V_\\mathrm{p}}\\xspace is reduced to $\\unit{-30}{\\volt}$ again (point A).\n\nTo complete the picture, we repeated the $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ experiment again, but now applied \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace. The magnetic preparation with $\\mu_0H=\\unit{7}{\\tesla}$ was now performed at $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$ to ensure that the preparation field was again applied along an easy axis. The data  recorded subsequently at $\\mu_0H=\\unit{-5}{\\milli\\tesla}$  [cf. Fig.~\\ref{fig:panel5}(c)] show a reversible rotation of $\\mathbf{M}$ by approximately 60\\degree between points A, B ($\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$) and back to A.\n\nTo quantitatively simulate the evolution of $M$ with \\ensuremath{V_\\mathrm{p}}\\xspace depicted in Fig.~\\ref{fig:panel5}, we again took advantage of the fact that the free energy of the Ni film is known from the FMR experiments. To account for the hysteresis of the actuator expansion which is responsible for hysteresis in the $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ loops in Fig.~\\ref{fig:panel5}, we hereby used $\\varepsilon_2$ as measured using a Vishay general purpose strain gauge in the voltage range of $\\unit{-30}{\\volt}\\leq\\ensuremath{V_\\mathrm{p}}\\xspace\\leq\\unit{+120}{\\volt}$. The $\\varepsilon_2$ data thus obtained are in accordance with the actuator data sheet.\nUsing $\\varepsilon_2(\\ensuremath{V_\\mathrm{p}}\\xspace)$ as well as the material constants given above we obtained in-plane free energy contours for each voltage \\ensuremath{V_\\mathrm{p}}\\xspace. Figure~\\ref{fig:panel6} exemplarily shows such free energy contours at the voltages depicted by the capital letters in Fig.~\\ref{fig:panel5}. To clearly visualize the effect of $\\mathbf{H}$ misalignment with respect to the sample coordinate system on $F_\\mathrm{tot}$, the plots in Fig.~\\ref{fig:panel6} were calculated assuming a misalignment in the in-plane orientation of the external magnetic field $\\theta$ of 10\\degree, so that $\\theta=10\\degree$  for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and $\\theta=100\\degree$ for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace. Figure~\\ref{fig:panel6} clearly shows that under these conditions the local minima of $F_\\mathrm{tot}$ are non-degenerate for $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{-30}{\\volt}$ and $\\ensuremath{V_\\mathrm{p}}\\xspace=\\unit{+120}{\\volt}$.\n\nTo determine the orientation of $\\mathbf{M}$, we traced the minimum of the total free energy $F_\\mathrm{tot}$ as a function of \\ensuremath{V_\\mathrm{p}}\\xspace. This was done by minimizing Eq.~\\eref{eq:Ftot} with respect to the in-plane orientation $\\Theta$ of $\\mathbf{M}$, whilst assuming that $\\Phi=90\\degree$ due to the strong shape anisotropy.\n\nTo simulate the experimental $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$-loops [cf. Fig.~\\ref{fig:panel5}], we assume a more realistic misalignment of 1\\degree in $\\theta$ and numerically minimize $F_\\mathrm{tot}(\\Theta)$ as a function of $\\ensuremath{V_\\mathrm{p}}\\xspace$. For this, we set the initial value of $\\Theta$ antiparallel to the external magnetic field for $\\mu_0H=\\unit{-3}{\\milli\\tesla}$ and parallel to the external magnetic field for $\\mu_0H=\\unit{-5}{\\milli\\tesla}$. Minimizing $F_\\mathrm{tot}(\\Theta)$ determines the $\\mathbf{M}$ orientation which we project onto the \\ensuremath{\\mathbf{\\hat{y}}}\\xspace or \\ensuremath{\\mathbf{\\hat{x}}}\\xspace axis to yield $M$. For this, the magnitude of $\\mathbf{M}$ is chosen to give a good fit to the experimental data at points A and is assumed to remain constant independent of \\ensuremath{V_\\mathrm{p}}\\xspace.\n\nThe resulting simulations of $M$ are shown by the solid lines in Fig.~\\ref{fig:panel5}. The simulation yields the experimentally observed $\\textit{irreversible}$ $\\mathbf{M}$ rotation during the first voltage sweep in Fig.~\\ref{fig:panel5}(a) as well as the $\\textit{reversible}$ $\\mathbf{M}$ rotations in the second voltage sweep in Fig.~\\ref{fig:panel5}(a) and in Figs.~\\ref{fig:panel5}(b,c). The simulated total swing of $M$ is in excellent agreement with the experimental results for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{y}}}\\xspace and in good agreement for \\ensuremath{\\mathbf{H} || \\mathbf{\\hat{x}}}\\xspace. The fact that the experimental results exhibit a more rounded shape than the simulation is attributed to domain formation during the magnetization reorientation, which is neglected in the simulation.\n\nTaken together, the free energy density of Eq.~\\eref{eq:Ftot} with the contributions quantitatively determined from FMR and our simple simulation of $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ yield excellent agreement with experiment. In particular, we would like to emphasize that for the experiment and the simulation the application of \\ensuremath{V_\\mathrm{p}}\\xspace leads to a \\textit{rotation} of $\\mathbf{M}$ and not to a decay into domains. This is evident from the fact that in Figs.~\\ref{fig:panel5}(a) and \\ref{fig:panel5}(b), a large projection of $\\mathbf{M}$ onto \\ensuremath{\\mathbf{\\hat{y}}}\\xspace is accompanied by a small projection onto \\ensuremath{\\mathbf{\\hat{x}}}\\xspace and vice versa. Combining FMR and SQUID magnetometry, we thus have unambiguously and quantitatively demonstrated that $\\mathbf{M}$ can be rotated reversibly by about 70\\degree at room temperature solely via the application of appropriate voltages \\ensuremath{V_\\mathrm{p}}\\xspace.\n\n\\section{Summary}\nThe FMR measurements show that the in-plane anisotropy of Ni\/piezoactor hybrids can be inverted if the polarity of the voltage applied to the actuator is changed.  The magnetometry results corroborate this result and furthermore show that it is possible to irreversibly or reversibly rotate $\\mathbf{M}$ solely by changing \\ensuremath{V_\\mathrm{p}}\\xspace. For an \\textit{irreversible} $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ rotation, an appropriate preparation of $\\mathbf{M}$ using a magnetic field sweep is necessary -- i.e. $\\mathbf{M}$ must be aligned in a local free energy minimum. It then can be rotated out of this minimum by changing \\ensuremath{V_\\mathrm{p}}\\xspace and the corresponding $\\mathbf{M}$ orientation change can amount to up to 180\\degree.\nHowever, this voltage control of $\\mathbf{M}$ is irreversible in the sense that $\\mathbf{M}$ can not be brought back into the original orientation by changing \\ensuremath{V_\\mathrm{p}}\\xspace alone. Rather, a second appropriate magnetic field sweep is required to align $\\mathbf{M}$ into its original orientation.\nIn contrast, the \\textit{reversible} $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ reorientation requires a preparation of $\\mathbf{M}$ only once. In this case, $\\mathbf{M}$ is oriented along a global free energy minimum, and can be rotated within up to 70\\degree (90\\degree in the ideal case) at will by applying an appropriate \\ensuremath{V_\\mathrm{p}}\\xspace. The $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$-loops simulated by minimizing the single-domain free energy [Eq.~\\eref{eq:Ftot}] are in excellent agreement with experiment, showing that the $\\mathbf{M}$ orientation at vanishing magnetic field strengths can still be accurately calculated from the free energy determined from FMR. Finally, we would like to point out that the hysteretic expansion\/contraction of the actuator visible as a hysteresis in $M(\\ensuremath{V_\\mathrm{p}}\\xspace)$ [cf. Fig.~\\ref{fig:panel5}] in particular also leads to distinctly different $M(\\ensuremath{V_\\mathrm{p}}\\xspace=0)$, depending on the \\ensuremath{V_\\mathrm{p}}\\xspace history. Thus, our data also demonstrate that a remanent $\\mathbf{M}$ control is possible in our ferromagnetic thin film\/piezoelectric actuator hybrids.\n\n\\ack\nThe work at the Walter Schottky Institut was supported by the Deutsche Forschungsgemeinschaft (DFG) via SFB 631. The work at the Walther-Meissner-Institut was supported by the DFG via SPP 1157 (Project No.~GR 1132\/13), DFG Project No.~GO 944\/3-1 and the German Excellence Initiative via the \"Nanosystems Initiative Munich (NIM)\".\n\n\\section*{References}\n\\bibliographystyle{prsty2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCameras are becoming increasingly ubiquitous and pervasive. Millions of surveillance cameras are recording people's everyday behavior at public places, and people are using wearable cameras designed for lifelogging (e.g., GoPro and Narrative Clips) to obtain large collections of egocentric videos. Furthermore, robots at public places are equipped with multiple cameras for their operation and interaction.\n\n\n\nSimultaneously, such abundance of cameras is also causing a big societal challenge: privacy protection from unwanted video recordings. We want a camera system (e.g., a robot) to recognize important events and assist human daily life by understanding its videos, but we also want to ensure that it is not intruding the user's or others' privacy. This leads to two contradicting objectives. More specifically, we want to (1) prevent the camera system from obtaining detailed visual data that may contain private information (e.g., faces), desirably at the hardware-level. Simultaneously, we want to (2) make the system capture as much detailed information as possible from its video, so that it understands surrounding objects and ongoing events for surveillance, lifelogging, and intelligent services.\n\nThere have been previous studies corresponding to such societal needs. Templeman et al. \\shortcite{templeman14} studied scene recognition from images captured with wearable cameras, detecting locations where the privacy needs to be protected (e.g., restrooms). This will allow the device to be automatically turned off at sensitive locations. One may also argue that limiting the device to only process\/transfer feature information (e.g., HOG and CNN) instead of visual data will make it protect privacy. However, recent studies on feature ``visualizations'' \\cite{vondrick15} showed that it actually is possible to recover a good amount of visual information (i.e., images and videos) from the feature data. Furthermore, all these methods described above rely on software-level processing of original high-resolution videos (which may already contain privacy sensitive data), and there is a possibility of these original videos being snatched by cyber attacks.\n\nA more fundamental solution toward the construction of a privacy-preserving vision system is the use of \\emph{anonymized videos}. Typical examples of anonymized videos are videos made to have extreme low resolution (e.g., 16x12) by using low resolution (LR) camera hardware, or based on image operations like Gaussian blurring and superpixel clustering \\cite{butler15}. Instead of obtaining high-resolution videos and trying to process them, this direction simply limits itself to only obtain anonymized videos. The idea is that, if we are able to \\textbf{develop reliable computer vision approaches that only utilize such anonymized videos}, we will be able to do the recognition while preserving privacy. Such a concept may even allow cameras that can intelligently select their resolution; it will use high-resolution cameras only when it is necessary (e.g., emergency), determined based on extreme low-resolution video analysis.\n\nThere were previous attempts under such paradigm: \\cite{dai15}. This conventional approach was to resize the original training videos to fit the target resolution, making training videos to visually look similar to the testing videos. However, there is an intrinsic problem: because of natural limitations of what a pixel can capture in a LR video, features being extracted from LR videos change a lot depending on sub-pixel camera viewpoint changes even when they contain the exact same object\/human: Figure \\ref{fig:isr-motivation}. This makes the decision boundary learning unstable.\n\n\n\n\n\n\n\n\n\n\n\nIn this paper, we introduce the novel concept of \\emph{inverse super resolution} to overcome such limitations. Super resolution reconstructs a single high-resolution image from a set of low-resolution images. Inverse super resolution is the reverse of this process: we learn to generate a set of informative LR images from one HR image. Although it is natural to assume that the system obtains only one low-resolution `testing' video for privacy protection, in most cases, the system has an access to a rich set of high-resolution `training' videos publicly available (e.g., YouTube). The motivation behind inverse super resolution is that, if it really is true that a set of low-resolution images contains comparable amount of information to a high-resolution image, then we can also generate a set of LR training images from a HR image so that the amount of training information is maintained. Our approach learns the optimal set of LR transformations to make such generation possible, and uses the generated LR videos to obtain LR decision boundaries (Figure \\ref{fig:isr}).\n\n\n\n\n\n\n\n\\section{Related works}\n\nHuman activity recognition is a computer vision area with a great amount of attention \\cite{aggarwal11}. There not only have been studies to recognize activities from YouTube-style videos \\cite{google15} or surveillance videos, but also first-person videos from wearable cameras \\cite{kitani11,ramanan12,lee12,li13,poleg14} and robots \\cite{ryoo13}. However, they only focused on developing features\/methods for more reliable recognition, without any privacy aspect consideration.\n\n\n\n\n\nOn the other hand, as mentioned in the introduction, there are research works whose goal is to specifically address privacy concerns regarding unwanted video taking. Templeman et al. \\shortcite{templeman14} designed a method to automatically detect locations where the cameras should be turned off. \\cite{tran16} was similar. Dai et al. \\shortcite{dai15} studied human activity recognition from extreme low resolution videos, different from conventional activity recognition literature that were focusing mostly on methods for images and videos with sufficient resolutions. Although their work only focused on recognition from 3rd-person videos captured with static cameras, they showed the potential that computer vision features and classifiers can also work with very low resolution videos. However, they followed the `conventional paradigm' described in Figure \\ref{fig:isr} (a), simply resizing original training videos to make them low resolution. \\cite{shokri15} studied privacy protection for convolutional neural networks (CNNs), but they consider the privacy protection only at the training phase and not at the testing phase, unlike ours.\n\n\n\n\n\n\\begin{figure}\n\t\\centering\n\t\\resizebox{1.0\\linewidth}{!}{\n\t  \\includegraphics{images\/ISR_concept2.pdf}\n\t}\n\t\\caption{A figure comparing (a) the conventional learning framework for low-resolution videos vs. (b) our learning framework using the proposed inverse super resolution.}\n\t\\label{fig:isr}\n\\end{figure}\n\n\n\\section{Inverse super resolution}\n\\label{sec:isr}\n\n\\emph{Inverse super resolution} (ISR) is the concept of generating a set of low-resolution training images from a single high-resolution image, by `learning' different image transforms optimized for the recognition task. Such transforms may include sub-pixel translation, scaling, rotation, and other affine transforms emulating possible camera motion.\n\nOur ISR targets the realistic scenario where the system is prohibited from obtaining high-resolution videos in the \\emph{testing} phase due to privacy protection but has an access to a rich set of high-resolution training videos publicly available (e.g., YouTube). Instead of trying to enhance the resolution of the testing video (which is not possible with our scale factor x20), our approach is to make the system learn to benefit from high-resolution training videos by imposing multiple different sub-pixel transformations. This enables us to better estimate the decision boundary in the low-resolution feature space, as illustrated in Figure \\ref{fig:isr}.\n\n\n\n\n\n\n\n\n\n\nFrom the super resolution perspective, this is a different way of using the super resolution formulation, whose assumption is that multiple LR images may contain a comparable amount of information to a single HR image. It is called `inverse' super resolution since it follows the super resolution formulation, while the input and the output is the reverse of the original super resolution.\n\n\n\n\n\n\\subsection{Inverse super resolution formulation}\nThe goal of the super resolution process is to take a series of low resolution images $Y_k$, and generate a high resolution output image $X$ \\cite{huang10}. This is done by considering the sequence of images $Y_k$ to be different views of the high resolution image $X$, subject to camera motion, lens blurring, down-sampling, and noise. Each of these effects is modeled as a linear operator, and the sequence of low resolution images can be written as a linear function of the original high resolution image: \n\\begin{equation}\nY_k = D_k H_k F_k X + V_k, ~~k = 1 \\ldots n\n\\end{equation}\nwhere $F_k$ is the motion transformation, $H_k$ models the blurring effects, $D_k$ is the down-sampling operator, and $V_k$ is the noise term. $X$ and $Y_k$ are both images flattened into 1D vectors. In the original super resolution problem, none of these operators are known exactly, resulting in an ill-posed, ill-conditioned, and most likely rank deficient problem. Super resolution research focuses on estimating these operators and the value of $X$ by adding additional constraints to the problem, such as smoothness requirements.\n\n\nOur inverse super resolution formulation can be described as its inverse problem. We want to generate multiple (i.e., $n$) low resolution images (i.e., $Y_k$) for each high resolution training image $X$, by applying the optimal set of transforms $F_k$, $H_k$, and $D_k$. We can simplify our formulation by removing the noise term $V_k$ and the lens blur term $H_k$, since there is little reason to further contaminate the resulting low-resolution images and confuse the classifiers in our case:\n\\begin{equation}\n\\label{eq:isr}\nY_k = D_k F_k X, ~~k = 1 \\ldots n.\n\\end{equation}\nIn principle, $F_k$, the camera motion transformation, can be any affine transformation. We particularly consider combinations of shifting, scaling, and rotation transforms as our $F_k$. We use the standard average downsampling as our $D_k$.\n\nThe main technical challenge here is that we need to learn the set of different motion transforms $S = \\{F_k\\}^n_{k=1}$ from a very large pool, which is expected to maximize the recognition performance when applied to generate the training set for classifiers. Such learning should be done based on training data and should be dependent on the features and the classifiers being used, solving the optimization problem in the feature space.\n\n\nOnce $S$ is learned, the inverse super resolution allows generation of multiple low resolution images $Y_k$ from a single high resolution image $X$ by following Equation \\ref{eq:isr}. Low resolution `videos' can be generated in a similar fashion to the case of images. We simply apply the same $D_k$ and $F_k$ for each frame of the video $X$, and concatenate the results to get the new video $Y_k$.\n\n\n\n\n\n\n\\subsection{Recognition with inverse super resolution}\n\nGiven a set of transforms $S = \\{F_k\\}^n_{k=1}$, the recognition framework with our inverse super resolution is as follows: For each of original high resolution training video $X_i$, we apply Equation \\ref{eq:isr} to generate $n$ number of $Y_{ik} = D_k F_k X_i$. Let us denote the ground truth label (i.e., activity class) of the video $X_i$ as $y_i$. Also, we describe the feature vector of the video $Y_{ik}$ more simply as $x_{ik}$. The features extracted from all LR videos generated using inverse super resolution become training data. That is, the training set $T_n$ can be described as $T(S) = \\cup_i\\{ \\langle x_{ik}, y_i \\rangle \\}_{k=0}^n$ where $n$ is the number of LR training samples to be generated per original video. $x_{i0}$ is the original sample resized to LR as is.\n\nBased on $T(S)$, a classification function $f(x)$ with the parameters $\\theta$ is learned. The proposed approach can cope with any types of classifiers in principle. In the case of SVMs with the non-linear kernels we use in our experiments,\n\\begin{equation}\n\t\\begin{aligned}\n\t\tf_{\\theta}(x) = \\sum_j \\alpha_j y_j K(x, x_j) + b\n\t\\end{aligned}\n\\end{equation}\nwhere $\\alpha_j$ and $b$ are SVM parameters, $x_j$ are support vectors, and $K$ is the kernel function being used.\n\n\n\\section{Transformation learning}\n\\label{sec:learning}\n\nIn the previous section, we presented a new framework that takes advantage of LR training videos generated from HR videos assuming a `given' set of transforms. In this section, we present methods to `learn' the optimal set of such motion transforms $S = \\{F_k\\}^n_{k=1}$ based on video data. Such $S$ learned from video data is expected to perform superior to transforms randomly selected or uniformly selected, which we further confirm in our experiments.\n\n\\subsection{Method 1 - decision boundary matching}\n\nHere, we present a Markov chain Monte Carlo (MCMC)-based search approach to find the optimal set of transforms providing the ideal activity classification decision boundaries. The main idea is that, if we have an infinite number of transforms $F_k$ generating LR training samples, we would be able to learn the best low-resolution classifiers for the problem. Let us denote such ideal decision boundary as $f_{\\theta^*}$. By trying to minimize the distance between $f_{\\theta^*}$ and the decision boundary that can be learned with our transforms, we want to find a set of transformations $S^*$:\n\\begin{equation}\n    \\label{eq:boundary-sim}\n\t\\begin{aligned}\n\t\tS^* &= \\argmin_{S} \\left| f_{\\theta^*} - f_{\\theta(S)} \\right|\\\\\n\t\t    & \\approx \\argmin_{S} \\sum_{x \\in A} \\left| f_{\\theta^*}(x) - f_{\\theta(S)}(x) \\right|\\\\\n\t\t\\textrm{s.t.}~~ & |S^*| = n\n\t\\end{aligned}\n\\end{equation}\nwhere $f_{\\theta(S)}(x)$ is a classification function (i.e., a decision boundary) learned from the training set $T(S)$ (i.e., LR videos generated using transforms in $S$). $A$ is a validation set with activity videos, being used to measure the empirical similarity between two classification functions.\n\nIn our implementation, we further approximate the above equation, since learning $f_{\\theta^*}(x)$ conceptually requires an infinite (or very large) number of transform filters $F_k$. That is, we assume $f_{\\theta^*}(x) \\approx f_{\\theta(S_L)}(x)$ where $S_{L}$ is a set with a large number of transforms. We also use $S_{L}$ as the `pool' of transforms we consider: $S \\subset S_L$.\n\n\nWe take advantage of a MCMC sampling method with Metropolis-Hastings algorithm, where each MCMC action is adding or removing a particular motion transform filter $F_k$ to\/from the current set $S^{t}$. The transition probability $a$ is defined as \n\\begin{equation}\n\t\\begin{aligned}\n\t\ta = \\frac{\\pi(S') \\cdot q(S', S^{t})}{\\pi(S^{t}) \\cdot q(S^{t}, S')}\n\t\\end{aligned}\n\\end{equation}\nwhere the target distribution $\\pi(S)$ is computed by\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\pi(S) \\propto e^{-\\sum_{x \\in A} \\left| f_{\\theta^*}(x) - f_{\\theta(S)}(x) \\right|},\n\t\\end{aligned}\n\\end{equation}\nwhich is based on the $\\argmin$ term of Equation \\ref{eq:boundary-sim}. We model the proposal density $q(S', S^{(t)})$ with a Gaussian distribution $|S'| \\sim N(n, \\sigma^2)$ where $n$ is the number of inverse super resolution samples we are targeting (i.e., the number of filters). The proposal $S'$ is accepted with the transition probability $a$, and it becomes $S^{t+1}$ once accepted.\n\nUsing the above MCMC formulation, our approach goes through multiple iterations from $S^0 = \\{ \\}$ to $S^m$ where $m$ is the number of maximum iterations we consider. Based on the sampled $S^0, \\ldots, S^m$, the one with the maximum $\\pi(S)$ value is finally chosen as our transforms: $S^* = \\argmax_{S^t} \\pi(S^t)$ with the condition $|S| \\leq n$.\n\n\n\n\\subsection{Method 2 - maximum entropy}\n\nIn this subsection, we propose an alternative methodology to learn the optimal set of transform filters $S^*$. Although the above methodology of directly comparing the classification functions provides us a good solution for the problem, a fair number of MCMC sampling iterations is needed for a reliable solution. It also requires a separate validation set $A$, which often means that the system is required to split the provided training set into the real training set and the validation set. This makes the transformation set learning itself to use less training data in practice.\n\nHere, we present another approach of using the \\emph{entropy} measure. Entropy is an information-theoretic measure that represents the amount of information needed, and it is often used to measure uncertainty (or information gain) in machine learning \\cite{settles2010active}. Our idea is to learn the set $S^*$ by iteratively finding transform filters $F_{1 \\cdots n}$ that will provide us the maximum amount of information gain when applied to the (original HR) training videos we have.\n\nWe formulate the problem similar to the active learning problem. At each iteration, we select $F_k$ that will generate new LR samples with the most uncertainty (i.e., maximum entropy) measured based on the current classifier trained with the current set of transforms: $f_{\\theta(S^t)}$. Adding such samples to the training set makes the new classifier to have the most information gain. That is, we iteratively update our set as $S^{t+1} = S^t \\cup \\{F_*^{t}\\}$ where\n\\begin{equation}\n\t\\begin{aligned}\n\t\tF^{t}_* &= \\argmax_k \\sum_i H(D_kF_kX_i)\\\\\n\t\t    &= \\argmax_k - \\sum_i \\sum_j P_{\\theta(S^t)}(y_j | D_kF_kX_i)\\\\\n\t\t    &\\relphantom{= \\argmax - \\sum \\sum}\\log  P_{\\theta(S^t)}(y_j | D_kF_kX_i).\n\t\\end{aligned}\n\\end{equation}\nHere, $X_i$ is each video in the training set, and $P_{\\theta(S^t)}$ is the probability computed from the classifier $f_{\\theta(S^t)}$. We are essentially searching for the filter that will provide the largest amount of information gain when added to the current transformation set $S^t$. More specifically, we sum the entropy $H$ (i.e., uncertainty) of all low resolution training videos that can be generated with the filter $F_k$: $H(D_kF_kX_i)$.\n\nThe approach iteratively adds one transform $F^t_*$ at every iteration $t$, which is the greedy strategy based on the entropy measure, until it reaches the $n$th round: $S^* = S^n$. Notice that such entropy can be measured with any videos with\/without ground truth labels. This makes the proposed approach suitable for the unsupervised (transform) learning scenarios as well.\n\n\n\\section{Experiments}\n\nWe confirm the effectiveness of inverse super resolution using low resolution version (16x12 and 32x24) of three different datasets (HMDB, DogCentric, and JPL-Interaction).\n\n\\subsection{Resized datasets}\n\nWe selected three public datasets and resized them to obtain low resolution (e.g., 16x12) videos.\n\nHMDB dataset \\cite{kuehne11} is a dataset popularly used for video classification. It is composed of videos with 51 different action classes, containing $\\sim$7000 videos. The videos include short clips, mostly from movies, obtained from YouTube. DogCentric dataset \\cite{ryoo14dog} and JPL-Interaction dataset \\cite{ryoo13} are the first-person video datasets taken with wearable\/robot cameras. They are smaller scale datasets, having $\\sim$200 videos and $\\sim$10 activity classes. DogCentric activity dataset is composed of first-person videos taken from a wearable camera mounted on top of a dog interacting with humans and surroundings. The dataset contains a significant amount of ego-motion, and it serves as a benchmark to test whether an approach is able to capture information in activities while enduring\/capturing strong camera ego-motion. JPL-Interaction dataset contains human-robot activity videos taken from a robot's point-of-view.\n\n\n\n\n\nWe emphasize once more that we made all these videos in the datasets have significantly lower resolution (Figure \\ref{fig:videos}), which is a much more challenging setting compared to the original datasets with their full resolution. Our main video resolution setting was 16x12, and we also tested the resolution of 32x24. For the resizing, we used the approach of averaging pixels in the original high-resolution videos that fall within LR pixel boundaries. A video cropping was used for the videos with non-4:3 aspect ratio.\n\n\\begin{figure}\n\t\\centering\n\t\\resizebox{0.95\\linewidth}{!}{\n\t  \\includegraphics{images\/low_resolution_videos.pdf}\n\t}\n\t\\caption{The original resolution videos (top) and their 16x12 resized videos (bottom) from the three datasets used. We can confirm that the videos are properly anonymized (i.e., we cannot distinguish human faces) by resizing them to extreme low resolution, but activity recognition from them is becoming more challenging due to the loss of details.}\n\t\\label{fig:videos}\n\\end{figure}\n\n\n\\subsection{Implementation}\n\n{\\flushleft\\textbf{Feature descriptors\/representation:} We extracted three different types of popular video features and tested our inverse super resolution with each of them and their combinations. The three features are (i) histogram of oriented gradients (HOG), (ii) histogram of optical flows (HOF), and (iii) convolutional neural network (CNN) features. These feature descriptors were extracted from each frame of the video. In order to make the CNN handle our low-resolution frames, we newly designed and utilized a 3-layer network with dense convolution and minimal pooling, illustrated in Figure \\ref{fig:LR_cnn}. Next, we use Pooled Time Series (PoT) feature representation \\cite{ryoo15} with temporal pyramids of level 1 or 4 on top of these four descriptors.}\n\n\n\n\n\n\n\n\n\n\n{\\flushleft\\textbf{Classifier:} Standard SVM classifiers with three different kernels were used: a linear kernel and two non-linear multi-channel kernels ($\\chi^2$ and the histogram-intersection kernels).}\n\n\n\n{\\flushleft\\textbf{Baselines:} The conventional approach for low resolution activity recognition is to simply resize original training videos to fit the target resolution (Figure \\ref{fig:isr} (a)). We use this as our baseline, while making it utilize the identical features and representation. The parameters were tuned for each system individually. In addition, we implemented the \\emph{data augmentation} (DA) approach similar to \\cite{karpathy14} that randomly selects LR transformations to increase the number of training samples. We added random rotation transformations to the data augmentation as well (DA+rotation), and also tested the uniform transformation selection strategy.}\n\n\n\\begin{figure}\n\t\\centering\n\t\\resizebox{0.8\\linewidth}{!}{\n\t  \\includegraphics{images\/LR-cnn.pdf}\n\t}\n\t\\caption{CNN architecture for extracting 256-D features from very low resolution images.}\n\t\\label{fig:LR_cnn}\n\\end{figure}\n\n\n\n\\subsection{Evaluation}\n\nWe conducted experiments with the videos downsampled to 16x12 (or 32x24) as described above. We followed the standard evaluation setting for each of the datasets. In the HMDB experiment, we used the provided 3 training\/testing splits and performed the 51-class classification. In the experiments with the DogCentric dataset, multiple rounds of random half-half training\/test splits were used to measure the accuracy. In the case of JPL-Interaction dataset with robot videos, 12-fold leave-one-set-out cross validation was used.\n\n\n\\begin{figure*}\n\t\\centering\n\t\\resizebox{1.0\\linewidth}{!}{\n\t  \\includegraphics{images\/dogcentric_results.pdf}\n\t}\n\t\\caption{Experimental results with different features on 16x12 DogCentric dataset. X-axis shows the number of LR samples obtained using ISR or data augmentation (i.e., $n$), and Y-axis is the classification accuracy (\\%). The blue horizontal line in each plot shows the activity classification performance without ISR. ISR shows a superior performance in all cases.}\n\t\\label{fig:exp-dog}\n\\end{figure*}\n\n\\subsubsection{16x12 DogCentric dataset}\n\\label{subsubsec:dog}\n\n\n\\begin{table}\n    \\small\n\t\\caption{Performances (\\%) of different methods tested with 16x12 DogCentric dataset, using three different kernels. $n=16$ and PoT level 1 was used with all features. Standard deviations were $\\sim0.3$, and the behaviors were very consistent.}\n\t\\label{table:kernels}\n\n\t\\small\n\t\\center\n\t\\setlength\\extrarowheight{0pt}\n\n\t\t\\begin{tabular}\t{c|c|c|c}\n\t\t\t\\hline \t & ~~Linear~~ & ~~~~$\\chi^2$~~~~ & Histogram \\tabularnewline\n\t\t\t\\hline \tBaseline (PoT) & 58.47\t& 63.33\t& 58.98  \\tabularnewline\n\t\t\t        DA & 60.86\t& 63.36\t& 62.08  \\tabularnewline\n\t\t\t        DA + rotation & 60.94\t& 64.17\t& 62.85  \\tabularnewline\n\t\t\t        Uniform & 60.56\t& 63.95\t& 62.29  \\tabularnewline\n\t\t\t\\hline\n\t\t\t\t\tISR-method1  & 61.73\t& 64.85\t& 63.35  \\tabularnewline\n\t\t\t\t\tISR-method2 & \\textbf{61.96}\t& \\textbf{64.91}\t& \\textbf{63.61} \\tabularnewline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\n\\end{table}\n\nIn this experiment, we used 4 types of features, 3 different types of kernels (i.e., linear, $\\chi^2$, and histogram-intersection), and 6 different settings for the number of LR samples generated per original video (i.e., $n$). For each of the settings, five different types of sample generation methods were tested: data augmentation, data augmentation with rotations, uniform sampling, and our ISR transform learning methods (method1 and method2).\n\n\n\n\n\n\n\n\n\\begin{table}\n\t\\caption{16x12 DogCentric dataset result comparison: Notice that \\cite{wang13} was not able to extract any trajectories from 16x12 videos. For the PoT and our ISR, we are reporting the result with $\\chi^2$ kernel with PoT level 4.}\n\t\\label{table:dogcentric}\n\n\t\\small\n\t\\center\n\t\\setlength\\extrarowheight{0pt}\n\n\t\t\\begin{tabular}\t{c|c|c}\n\t\t\t\\hline \tApproach & Resolution &Accuracy \\tabularnewline\n\t\t\t\\hline \tIwashita et al. \\shortcite{ryoo14dog} & 320x240\t& \t60.5 \\%  \\tabularnewline\n\t\t\t        Wang and Schmid \\shortcite{wang13} & 320x240\t& \t67.6 \\%  \\tabularnewline\n\t\t\t        PoT (Ryoo et al. 2015) & 320x240\t& \t73.0 \\%  \\tabularnewline\n\t\t\t\\hline\n\t\t\t        Iwashita et al. \\shortcite{ryoo14dog} & 16x12\t& \t46.2 \\%  \\tabularnewline\n\t\t\t        Wang and Schmid \\shortcite{wang13} & 16x12\t& \t10.0 \\%  \\tabularnewline\n\t\t\t\t\tPoT (HOG + HOF + CNN)   & 16x12 & \t64.6 \\%  \\tabularnewline\n\t\t\t\t\t\\textbf{ISR} ($n = 16$)  & 16x12 &  67.4 \\% \\tabularnewline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\n\\end{table}\n\n\n\\begin{table*}\n\t\\caption{HMDB performances with and without our ISR. Classification accuracies (\\%) on 16x12 and 32x24 are reported. We show means and standard deviations of repeated experiments. The best performance per feature is indicated with bold.}\n\t\\label{table:exp-hmdb}\n\n\t\\centering\n\t\\resizebox{1.0\\linewidth}{!}{%\n\t  \\includegraphics{images\/hmdb_results2.pdf}\n\t}\n\\end{table*}\n\n\nFigure \\ref{fig:exp-dog} shows the results of our ISR tested with multiple different features while using a linear kernel, and Table \\ref{table:kernels} shows the results with three different kernels. We are able to observe that our method1 always performs superior to conventional approaches including data augmentation or uniform transform selection, very reliably. Our method2 performance was less consistent due to that it only uses information gain instead of taking advantage of sample ground truth labels when learning transforms.\nNevertheless, method2 showed an overall performance meaningfully superior to the other approaches (e.g., method2 - 62.0 vs. data augmentation - 60.9 with a linear kernel).\n\nTable \\ref{table:dogcentric} compares our approach with other state-of-the-art approaches. The best performance report on the DogCentric dataset is 73\\% with 320x240 videos using the PoT feature representation \\cite{ryoo15}, but this method only obtains the accuracy of 64.6\\% with 16x12 anonymized videos. Our inverse super resolution is further improving it to 67.4\\% while using the same features, representation, and classifier.\n\n\n\n\\subsubsection{Tiny-HMDB dataset: 16x12 and 32x24}\n\n\n\n\nTable \\ref{table:exp-hmdb} shows the results with our ISR-method2. The result clearly suggests that inverse super resolution improves low resolution video recognition performances in all cases. Even with a small number of additional ISR samples (e.g., $n = 2$), the performance improved by a meaningful amount compared to the same classifier without inverse super resolution. Naturally, the classification accuracies with 32x24 videos were higher than those with 16x12. CNN performances were similar in both 16x12 and 32x24, since our CNN takes 16x12 resized videos as inputs.\n\nNotice that even with 16x12 downsampled videos, where visual information is lost and the use of trajectory-based features are prohibited, our methods were able to obtain performance superior to several previous methods such as the standard HOF\/HOG classifier (20.0\\% \\cite{kuehne11}) and ActionBank (26.9\\% \\cite{corso12}). That is, although we are extracting features from 16x12 videos where a person is sometimes as small as a few pixels, it is performing superior to certain methods using the original HR videos (i.e., videos larger than 320x240). The approach \\cite{wang13} obtaining state-of-the-art performance of 57.2\\% with 320x240 HR videos got the performance of $\\sim$2\\% in LR videos, since no trajectories could be extracted from 16x12 or 32x24.\n\n\n\n\n\n\\subsubsection{16x12 JPL-Interaction dataset}\n\nWe also conducted experiments with the segmented version of JPL-Interaction dataset \\cite{ryoo13}, containing robot-centric videos. Figure \\ref{fig:videos} (c) shows examples of its 16x12 version, where we can observe that human faces are anonymized.\n\nTable \\ref{table:jpl-interaction} shows the results. We are able to confirm once more that the proposed concept of inverse resolution benefits low resolution video recognition. Surprisingly, probably due to the fact that each activity in the dataset shows very distinct appearance\/motion different from the others, we were able to obtain the activity classification accuracy comparable to the state-of-the-arts while only using 16x12 extreme low resolution videos.\n\n\n\n\n\n\n\\section{Conclusion}\n\nWe present an \\emph{inverse super resolution} method for improving classification performance on extreme low resolution video. We experimentally confirm its effectiveness using three different public datasets. The overall recognition was particularly successful with first-person video datasets, where capturing ego-motion is the most important. Our approach is also computationally efficient in practice, requiring learning iterations linear in the number of ISR samples when using our method 2. In contrast, to achieve similar performance with traditional data augmentation, an order of magnitude more examples are needed (e.g., $n$=$16$ vs. $n$=$175$).\n\n\\begin{table}\n    \\small\n\t\\caption{Recognition performances of our approach tested with 16x12 resized JPL-Interaction dataset. Although our result is based on extremely low resolution 16x12 videos, it obtained comparable performance to the other methods \\cite{ryoo13,wang13,narayan14} tested using much higher resolution 320x240 videos.}\n\t\\label{table:jpl-interaction}\n\n\t\\small\n\t\\center\n\t\\setlength\\extrarowheight{0pt}\n\n\t\t\\begin{tabular}\t{c|c|c}\n\t\t\t\\hline \tApproach & Resolution &Accuracy \\tabularnewline\n\t\t\t\\hline \tRyoo and Matthies \\shortcite{ryoo13} & 320x240\t& \t89.6 \\%  \\tabularnewline\n\t\t\t        Wang and Schmid \\shortcite{wang13} & 320x240\t& \t96.1 \\%  \\tabularnewline\n\t\t\t        Narayan et al. \\shortcite{narayan14} & 320x240\t& \t96.7 \\%  \\tabularnewline\n\t\t\t\\hline\n\t\t\t        Ryoo and Matthies \\shortcite{ryoo13} & 16x12\t& \t74.5 \\%  \\tabularnewline\n\t\t\t\t\tPoT   & 16x12 & \t92.9 \\%  \\tabularnewline\n\t\t\t\t\tOurs (PoT + \\textbf{ISR})  & 16x12 &  96.4 \\% \\tabularnewline\n\t\t\t\\hline\n\t\t\\end{tabular}\n\n\\end{table}\n\n\n\n\n\n\\section{Discussions}\n\nOne natural question is whether the resolution in our testing videos is small enough to prevent the human\/machine recognition of faces (i.e., whether our videos are really privacy-preserving videos).\n\n\\begin{figure}\n\t\\centering\n\t\\resizebox{1.0\\linewidth}{!}{\n\t  \\includegraphics{images\/HR_recovery_examples.pdf}\n\t}\n\t\\caption{Example resolution enhancement attempts using \\cite{kim16sr}. We can observe that face details are not being recovered properly, even after the x4 scale enhancement. This is particularly so with our 16x12 videos.}\n\t\\label{fig:hr-recovery}\n\\end{figure}\n\nThe state-of-the-art low resolution face recognition (i.e., face identification) algorithm using convolutional neural networks \\cite{lrface16} obtained around 50$\\sim$60\\% accuracy with 16x16 human face images. This 50$\\sim$60\\% classification accuracy is based on the dataset with 180 subjects, and the performance is expected to go even lower in real-world environments with more subjects to be considered for the classification. On the other hand, in our extreme low resolution videos (i.e., 16x12 videos), the resolution of human face is at most 5x7. In often cases, the face resolution was as small as 2x2 or even 1x1. This suggests that reliable face recognition from our extreme low resolution videos will be difficult for both machines and humans. Furthermore, there is a user study \\cite{butler15} reporting that anonymizing videos in a way similar to what we have done significantly lowers human's privacy sensitivity.\n\nAnother relevant question is whether enhancing the resolution of the testing video (i.e., recovering high resolution faces from LR images) is possible with our extreme LR videos. In order to confirm that such recovery is not possible due to the information loss, we applied the state-of-the-art deep learning-based recovery approach \\cite{kim16sr} to the video frames. Figure \\ref{fig:hr-recovery} illustrates the results. Notice that these images are based on the scale factor of x4. Any attempt with the higher scale factor gave us worse results. We observe that this deep learning-based resolution enhancement is not recovering the face details, particularly in 16x12 videos. The algorithm sharpens the edges compared to bicubic interpolation, but fails to recover actual details.\n\n\n\n\n\n\\renewcommand{\\thefootnote}{\\fnsymbol{footnote}}\n\n\\section*{Acknowledgement}\n\nRyoo and Yang's research in this work was conducted as a part of EgoVid Inc.'s research activity on privacy-preserving computer vision. Ryoo and Yang are the corresponding authors.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\small\n\\bibliographystyle{aaai}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{intro}\n\nIt is well known that prime factorization is a computationally difficult problem and the security of the RSA-type classical cryptographic systems  derive from this difficulty \\cite{rivest1978method}. Although, various RSA-type schemes for public key cryptography are still in use, the confidence in the\nsecurity provided by RSA-type cryptosystem has been considerably weakened since the pioneering work of Shor \\cite{shor1999polynomial,shor1995scheme,shor2000simple,shor1994algorithms}. Specifically, in \\cite{shor1999polynomial}, Shor showed that the factorization of bi-primes can be performed in polynomial time if a scalable quantum computer can be built. In other words, building of a scalable quantum computer would imply the end of RSA-type classical cryptosystem and a set of other classical cryptosystems, too. This fundamental importance of the factorisation problem and the benefit\nof implementing it using a quantum computer have led to a set of schemes for prime factorization, mainly experimental \\cite{Vandersypen2001Experimental,lu2007demonstration,lanyon2007experimental,politi2009shor,matthews2009compiled}. Initial implementations were based on nuclear magnetic resonance (NMR) and mostly followed Shor's original algorithm. For example,  15 was factorised using an NMR-based 7-qubit  quantum computer \\cite{Vandersypen2001Experimental}. Interestingly, largest number that was factored using Shor's algorithm until 2012 was 21 and a 10 qubit quantum computer was used for the purpose \\cite{lucero2012computing}. Due to the fact that considerably large quantum registers are required in Shor's original algorithm, now it's not usually used in its original form \\cite{Vandersypen2001Experimental}. Though, Shor's algorithm, in principle, guarantees  factorization of a bi-prime number in polynomial time, the requirement of very large quantum registers restricted its applicability in factorising relatively large bi-primes. This fact motivated researchers to look for alternative approaches. One such approach is to use a hybrid scheme, variants of this approach are reported in Refs. \\cite{peng2008quantum, pal2019hybrid, li2017high}. These variants are quite close to each other and each of them require relatively less quantum resources than that required in Shor's original approach. Here, it would be apt to note that hybrid schemes refer to those schemes, where part of the factorisation task is done classically to reduce the requirement of quantum resources which are costly. \n\nIn 2008, Peng et al. \\cite{peng2008quantum}, devised an algorithm utilizing adiabatic quantum computing, on the basis of the work of Farhi et al. \\cite{farhi2001quantum} and demonstrated factorization of 21 using 3-qubits. Furthermore, in 2012, Xu et al., have improved the scheme by solving some equations mathematically. They have further demonstrated the beauty and power of the factorisation algorithms of this class by factorizing 143 using a 4-qubit NMR quantum processor\\cite{xu2012quantum}. Two years later, Dattani and Bryans established that Xu et al., had actually factored 3599, 11663, and 56153, but could not recognize that. In the work of Dattani and Bryans, classical resources were used for partially simplifying a set of bit-wise factoring equations which allowed them to reduce the quantum overhead for a set of numbers \\cite{dattani2014quantum}. In  2019, Pal et al.,  have demonstrated a hybrid  scheme for factorization of 551 using 3-qubit system  resources  \\cite{pal2019hybrid}. \nThe progress has been continuing and very recently (in 2017),\nfactorization of 35 was demonstrated using a single solid spin system under ambient conditions by Xu et al., \\cite{xu2017experimental}. Furthermore, a set of relatively large numbers have recently been experimentally factorized. Specifically, combining the concepts of quantum annealing and computational algebraic geometry, a new approach for quantum factorization is developed by Dridi and Alghassi \\cite{dridi2017prime} and the same has been successfully used to factorize all bi-primes up to 200099 using the D-Wave 2X processor \\cite{anschuetz2019variational} and the experimental factorization of 291311 is performed by Li et al., \\cite{li2017high} using a hybrid adiabatic quantum algorithm.\n\nThe importance of the factorization problem and the facts that (i) quantum factorization has yet been performed using only a few potential candidates for the scalable quantum computer and (ii) hybrid schemes have potential to factorize large bi-primes using small and noisy quantum computers available today,  have motivated us to modify the hybrid scheme given in \\cite{pal2019hybrid,xu2012quantum} to obtain a new scheme which can be implemented in another experimental platform, namely IBM Quantum processor. Specifically, the algorithm proposed here is designed for factorization of bi-primes using a Josephson-qubit based quantum computer \\cite{IBMQE,devitt2016performing}.\nInterested readers may find a detailed user guide on how to use this computer at \\cite{IBMQE},\nand a lucid description of the working principle of a Josephson-qubit based quantum computer in Ref. \\cite{steffen2011quantum}. This quantum computer was placed in the cloud in 2016. It immediately drew considerable attention of the quantum information processing community, and several quantum information tasks have already been performed using this quantum computer. Specifically, in the\ndomain of quantum communication, properties\nof different quantum channels that can be used for quantum communication have been studied experimentally \\cite{wei2018efficient} and experimental realizations of a\nquantum analogue of a bank cheque \\cite{behera2017experimental}\nthat is claimed to work in a banking system having a quantum network,\nand solving set of linear equations \\cite{doronin2020solving}\nand two-qubit quantum states using optimal resources \\cite{sisodia2017design}, have been reported; in the field of quantum foundation, violation\nof multi-party Mermin inequality has been realized for 3, 4, and 5\nparties \\cite{alsina2016experimental}; an information\ntheoretic version of uncertainty and measurement reversibility has\nalso been implemented \\cite{berta2016entropic}; in \nthe area of quantum computation, a comparison of two architectures\nusing demonstration of an algorithm has been performed \\cite{linke2017experimental},\na quantum permutation algorithm \\cite{yalccinkaya2017optimization},\nand a quantum eigensolver method based experimental search for ground state energy  energy for molecules of increasing  size  up  to   $\\rm{BeH_2}$   \\cite{kandala2017hardware} have been implemented recently. Further, a non-abelian braiding of surface code defects \\cite{wootton2017demonstrating} and a compressed simulation of the transverse field one-dimensional Ising interaction (realized as a four-qubit Ising chain that utilizes only two qubits) \\cite{hebenstreit2017compressed}\nhave also been demonstrated. Clearly, the IBM quantum computer\nhas already been used for the experimental realizations of various\nquantum information processing tasks. However, to the best of our\nknowledge, IBM quantum computer has not yet been used to realize Shor's algorithm or hybrid algorithms for factorization. This paper aims to address that gap.\n\nThis paper is organized as follows. \nSec. \\ref{intro} sets motivation behind choosing factorization problem, followed by detailing the gradual development of hybrid (i.e., combined classical and quantum) strategies to obtain efficient solution. In Sec. \\ref{theory}, we revisit the general scheme for hybrid factorization. In Sec. \\ref{fact35}, we elaborate on a specific case-  factorization of 35 using hybrid scheme of factorization, to be precise, we construct bit-wise equations and bit-wise matrix, optimize them to calculate unknown constants, and obtain relation between bit variables, formulate corresponding problem Hamiltonian using the procedure given in Sec. \\ref{theory}. In Sec. \\ref{exp}, we illustrate experimental implementation of the quantum part of the hybrid scheme for factorizing 35 using IBM architecture. For the purpose, we use Adiabatic evolution for ground state search of the problem Hamiltonian constructed in the previous section, which is the desired solution. In Sec. \\ref{results}, we show the experimental results, which reveals the unknown qubit state and hence one factor of the composite number, and ultimately We conclude in Sec. \\ref{concl}.\n\n\\section{Theory} \\label{theory}\n\nLet's consider a $b_{n}$-bit number $N=\\sum_{j=0}^{b_{n}-1}2^{j}n_{j}$. Although, there may be various sets of factors of a composite number of bit length $b_{n}$ with maximum number of possibilities equals $\\lceil \\frac{b_{n}}{2} \\rceil$. Here, $\\lceil x \\rceil$ corresponds to a  ceiling function. Acquainted with the requirement of cryptosystems, here we consider  $N$ as a distinct bi-prime. Let us assume that the two factors of $N$ are $P=\\sum_{k=0}^{b_{p}-1}2^{k}p_{k}$ and $Q=\\sum_{l=0}^{b_{q}-1}2^{l}q_{l}$ with bit length $b_{p}$ and $b_{q}$ respectively, such that, either $b_n=b_p+b_q$ or $b_{n}=b_{p}+b_{q}-1$. From the definition of prime factors, $p_{0}=q_{0}=1$  and $p_{b_{p}-1}$ = $q_{b_{q}-1}$=1. Thus, identity $N=PQ$ becomes,\n\\begin{equation}\n\\sum_{j=0}^{b_{n}-1}2^{j}n_{j} = \\sum_{k=0}^{b_{p}-1}2^{k}p_{k} \\sum_{l=0}^{b_{q}-1}2^{l}q_{l}. \\label{eq1}\n\\end{equation}\nAs follows from the above equation either $b_{n}=b_{p}+b_{q}$ or $b_{n}= b_{p}+b_{q}-1$. The hybrid scheme for factorization can be divided into the following steps.\n\\begin{itemize}\n\\item Formulating multiplication table for $P$ and $Q$.\n\\item Constructing bit-wise equations from multiplication table.\n\\item Simplifying bit-wise equations using classical computation.\n\\item Constructing bit-wise Hamiltonian and hence problem Hamiltonian.\n\\item Obtaining unitaries corresponding to adiabatic evolution starting from given Hamiltonian to the problem Hamiltonian.\n\\item Decomposition of a given unitary using gates available in IBM Clifford library.\n\\item Obtaining solution of problem Hamiltonian using adiabatic quantum computation.\n\\end{itemize}\n\nIn the following, we illustrate the hybrid scheme for factorization with an example of $N=35$. We would also report experimentally obtained factors using this scheme. Experimental implementation of quantum part of the scheme, i.e., constrained minimization using adiabatic evolution has been done in 5-qubit IBM quantum processor involving superconducting qubits. \n\\subsection{Simplification of bit-wise constraint equation}\nFor the purpose of bit-wise comparison of coefficients of both sides, we need to rewrite the above equation by introducing new index $c=k+l$. In terms of this new index $c$ modified equation becomes  \n\\begin{equation}\n\\sum_{j=0}^{b_{n}-1}2^{j}n_{j} = \\sum_{c=0}^{b_{p}+b_{q}-2}2^{c} \\sum_{l=c_{min}}^{c_{max}}p_{c-l}q_{l}.\n\\label{rewritten_m}\n\\end{equation}\nHere, $c_{min}= max(0,c-lp-1)$ and $c_{max}= min(c,lq-1)$. Further, term $p_{c-l}q_{l}$ can be broken as $p_{c-l}q_{l}+C_{c-1,c}=s_{c,l}+2C_{c,c+1}$. Here, $C_{c-1,c}$ is the carry from $(c-1)^{th}$ column to $c^{th}$ column and $C_{c,c+1}$ is the carry from $(c)^{th}$ column to $(c+1)^{th}$ column. Such a decomposition allows us an easy understanding of the construction of the multiplication table. In order to get simplified bit-wise factoring equations, without any loss of generality, we now add all carries in the given column. So the bit-wise factoring equation takes following form\n\\begin{equation}\n\\sum_{l=c_{min}}^{c_{max}}p_{c-l}q_{l}+C_{c}-2C_{c+1}= n_{j}.\n\\label{bitequation}\n\\end{equation}\nHere, cumulative carry $C_{c}= \\sum_{c_{min}}^{c_{max}}C_{c-1,c}$ and $n_{j}$ is the bit value of number $N$ for the $j^{th}$ order (power) of the base value in the binary system. \n\\subsection{Constraint optimization using classical resources} \\label{classicaloptimization}\n\nConsider the following equation,\n\\begin{equation}\n    \\sum_{j=0}^{b_{n}-1}2^{j}n_{j} = \\sum_{c=0}^{b_p+b_q-2} 2^{c}n_{c}+2C_{c+1}-C_{c} \n\\label{lastequation1}\n\\end{equation}\nand equation\n\\begin{equation}\n \\sum_{j=0}^{b_{n}-1}2^{j}n_{j}= \\sum_{c=0}^{b_p+b_q-2} 2^{c}n_{c}+ C_{b_{p}+b_{q}-1}\n \\label{lastequation2}\n\\end{equation}\nobtained after putting the value of $$\\sum_{l=c_{min}}^{c_{max}}p_{c-l}q_{l}$$ from Eq.(\\ref{bitequation}) into Eq.(\\ref{rewritten_m}). Writing Eq. (\\ref{lastequation2}) in such a way allows us to calculate values of carry $C_{b_{p}+b_{q}-1}$. A direct comparison of the L$.$H $.$S$.$ $\\mathrm{with}$ R$.$H$.$ S$.$ reveals the values of the carry $ C_{b_{p}+b_{q}-1} $. Also, as there is no incoming carry to the first column, we set $C_{0}=0$. Moreover, bit-wise equation for $c=0$ and $c=b_{p}+b_{q}-2$, i.e., $1+C_{0}=1-2C_{1}$ and $C_{b_{p}+b_{q}-2}+1=n_{b_{p}+b_{q}-2}+2C_{b_{p}+b_{q}-1}$ give $C_{1}$ and $C_{b_{p}+b_{q}-2}$. Next we obtain the following equality by rewriting the bit-wise equation. \n\\begin{equation}\n    \\mathrm{max}\\lfloor C_{c+1} \\rfloor = \\lfloor \\mathrm{max}(\\frac{1}{2}\\sum_{l=C_{min}}^{C_{max}} p_{c-l}q_{l}+\\frac{1}{2}C_{i})-\\frac{n_{i}}{2}  \\rfloor.\n\\end{equation}\nThis equality can be used to calculate the upper bounds over $C_{j+1}$, and this upper bound can be iteratively used to obtain upper bound on next $C$ value. The bit equation obtained under these  constrain, further allows us to simplify complete set of bit equations with minimum number of independent parameters.  \nFor the two cases, namely, Case A $\\colon$ when $b_n=b_p+b_q$ and Case B $\\colon$ when $b_n=b_p+b_q-1$ the values of the carry are $1$ and $0$ respectively.\n\n\\subsection{Construction of problem Hamiltonian} \\label{probHam}\nIn 2008, Peng et al. developed a framework for solving factorization problem using quantum adiabatic algorithm. For the purpose, they formulated the factorization problem as a minimization problem by constructing a function $f(p,q)=(N-pq)$. The solution of this equation should reveal the values of classical variables $p$ and $q$. They  further suggested that any corresponding quantum approach to the minimization problem must involve finding the ground state of the Hamiltonian, which can be considered to be of the form $H=f(p,q)\\ket{p,q}\\bra{p,q}$, where $f(p,q)$ is the ground state eigenvalue and $\\ket{p,q}$ is the corresponding product state of states $\\ket{p}$ and $\\ket{q}$. The problem Hamiltonian for the factorization problem takes the form $H=(N\\operatorname{I_{2^{n}}}-PQ)^2$, where $\\operatorname{P}=\\sum_{i}2^iA_{i}$ $\\operatorname{Q}=\\sum_{i}2^iA_{i}$ and $\\operatorname{A_{i}}= \\frac{\\operatorname{I}-\\operatorname{\\sigma_{iz}}}{2}$ with eigenvalues 0 and 1 for eigenstates $\\ket{0}$ and $\\ket{1}$, respectively. Furthermore, Xu et al. \\cite{xu2012quantum} have used another approach to construct a Hamiltonian which uses relatively less quantum resources than that used in Ref. \\cite{peng2008quantum} but still uses more resources than used by Pal et al. in Ref. \\cite{pal2019hybrid}. In this article, in order to construct the problem Hamiltonian we have used the same approach as was used by Xu et al. in Ref. \\cite{xu2012quantum} i.e., to begin with we have transformed the classical bit variable $p_{i}$ and $q_{i}$ into operators such that $p_{i} \\rightarrow A_{i}$ and $q_{i} \\rightarrow A_{i+b_{k}-2}$.  \n\n \\subsection{Quantum adiabatic algorithm for ground state search}\nQuantum adiabatic algorithm states that, during the evolution of a quantum system under a slowly varying (as per adiabaticity condition \\cite{farhi2000quantum}) time dependent Hamiltonian $H(t)$, system remains in the same eigenstate in which the system is prepared initially \\cite{mesiah1961quantum}. Given a problem, adiabatic quantum computation typically involves encoding the solution to the problem in the ground-state of the  final Hamiltonian. A suitable initial Hamiltonian $H_{i}(0)$ is chosen for which ground-state can be prepared easily and evolved to the final Hamiltonian $H_{f}(T)$. \n\nThen the Hamiltonian of the system is slowly varied such that the system stays in the ground state of the instantaneous Hamiltonian. The instantaneous Hamiltonian can be designed as an interpolation (linear or nonlinear) between the initial Hamiltonian and the final Hamiltonian \\cite{farhi2000quantum}. For the linear interpolation parameter $s=\\frac{t}{T}$, such that $0 \\leq s \\leq 1$, where $t$ is the evolution time and $T= \\vert \\frac{max(\\frac{dH(s)}{ds})}{\\epsilon \\Delta^2\/\\hbar} \\vert$, is the total evolution time. The adiabatic theorem guarantees that system will evolve to the ground state of the final Hamiltonian with probability $1-\\epsilon^2$, and the transformed Hamiltonian would become\n\\begin{equation}\n\\operatorname{H(s)} = (1-s)\\operatorname{H_i} + s\\operatorname{H_f}. \n\\end{equation}\nConsidering the Hamiltonian as piecewise constant Hamiltonian with M pieces the time ($t$) can be rewritten as $t=\\frac{m}{M}T$, where $0 \\le m \\le M$ and the Hamiltonian for the $m^{th}$ piece is\n\\begin{equation}\nH_{m} = (1-\\frac{m}{M})H_{i} + (\\frac{m}{M})H_{f}.\n\\end{equation}\nThe unitary evolution $U_{m} = \\exp{(-\\iota H_{m}\\delta t)}$ governed by the corresponding Hamiltonian $H_{m}$, where $t$ is the duration of $m^{th}$ piece of evolution. Unitary operation for the total evolution is $U=\\prod_{m=1}^{M} U_{m}$.  \n\n\\section{Factorization of 35 using IBM's 5-qubit quantum processor} \\label{fact35}\nIn order to demonstrate the method, we take the example of 35. As mentioned in Sec. \\ref{theory} there are two possible cases for the choice of the bit-length of the bi-prime factors, we start with the case $b_n=b_p+b_q$, thus $b_{p}=3$ and $b_{q}=3$. Although, one can start with any of the two cases and in case of obtaining an inconsistent solution in the first case, consistent solution will be guaranteed in the other case. We start by obtaining the multiplication table (see Tab. \\ref{bitwiseMultiplication})  for the composite number $N=35$. \n\n\\begin{table}[b]\n\\begin{tabular} { | c |c| c| c| c | c | c | c | c | c | c | c | c |} \n\\hline\n \\rm{ \\textbf {j}} $\\rightarrow$&5&4&3&2&1&0\\\\\n\\rm{\\textbf{l}} $\\downarrow$&&&&&&\\\\\n\\hline\n&&&&1&$p_{1}$&1\\\\ \n\n\n&&&&1&$q_{1}$&1\\\\\n\\hline\n0&&&&1 & $p_{1}$&1\\\\\n\\hline\n1&&& $q_{1}$&$p_{1}q_{1}$&$q_{1}$&\\\\\n\\hline\n2&&$1$ & $p_{1}$ &  $1$&&\\\\\n\\hline\nCarry&$c_{4,5}$ & $c_{3,4}$&$c_{2,3}$&$c_{1,2}$&&\\\\\n& & $c_{2,4}$&&&&\\\\\n\\hline\nCumulative Carry &&&&&&\\\\\n$C_{c}$&$C_{5}$&$C_{4}$&$C_{3}$&$C_{2}$&$C_{1}$&$C_{0}$\\\\\n\\hline\n$n_{j}$&1&0&0&0&1&1\\\\\n\n\\hline\n\\end{tabular}\n\\caption{Bit-wise multiplication table for 35. Columns in the table correspond to parameter $c$ introduced to get simplified bit-wise equations (see Eq. (\\ref{lastequation2})) while rows correspond to $l$ values (see Eq. (\\ref{lastequation1})). Bit values $n_{j}$ for $j=0\\colon5$ are provided in the last row of the table.}\n\\label{bitwiseMultiplication}\n\\end{table}\n\nThe bit-wise equations obtained from the  multiplication table (Table \\ref{bitwiseMultiplication}) are$\\colon$\n\\begin{align*}\n 1+C_{0}=1,\\\\\n    p_{1}+q_{1}+C_{1}-1=2C_{2},\\\\\n    1+p_{1}q_{1}+1+C_{2}-0=2C_{3},\\\\\n    p_{1}+q_{1}+C_{3}-0=2C_{4},\\\\\n    1+C_{4}-0=2C_{5},\\\\\n    C_{5}=1.\\\\    \n\\end{align*}\nWe then optimize above set of equations using the constrain optimum condition given in Sec. \\ref{classicaloptimization}. The carries thus obtained are $C_{0}=0, C_{1}=0, C_{2}=0,C_{3}=1, C_{4}=1, C_{5}=1$. This leaves us with only one bit equation, i.e., $p_1+q_{1}=1$. We then construct the operators corresponding to bit values $p_{1}$ and $q_{1}$ as discussed in \\ref{probHam}. Thus, the operators corresponding to the bit values $p_{1}$ and $q_{1}$ are $P=Q=\\sum_{i}A_{i}=A_{1}$ and for $A_{1}= \\frac{\\operatorname{I}-\\operatorname{\\sigma_{1z}}}{2}$. Now the problem Hamiltonian becomes \n\n\\begin{align*}\nH_{p}= (P_{1}+Q_{1}-1)^2 \\\\       \n     = (\\frac{\\operatorname{I}-\\operatorname{\\sigma_{z}}}{2}+\\frac{\\operatorname{I}-\\operatorname{\\sigma_{z}}}{2}-\\operatorname{I})^2\\\\\n     = {\\operatorname{\\sigma_{z}}}^2\\\\\n     = {\\operatorname{I}}.    \n\\end{align*}\n\nWe now use quantum adiabatic evolution for finding the ground state of the final Hamiltonian $H_{p}$ starting from the ground state of the easily initializable Hamiltonian in the IBM's $QX_{4}$ processor, i.e., $H_{i}=J \\sigma_{z}$, in the units of $\\hbar$ and $J \\approx 2\\pi\\times10^{6}$ rad\/sec. The ground state of the initial Hamiltonian $H_{i}$ is $\\ket{-}=\\frac{\\ket{0}-\\ket{1}}{2}$ and ground state of the final Hamiltonian $H_{f}$ is degenerate and are $\\ket{0}$ and $\\ket{1}$ with eigenvalues 1. We decided to adiabatically evolve the Hamiltonian in 8 steps. During each step the Hamiltonian can be considered as piecewise constant and the corresponding unitary operators can be obtained using the formula $U_{m}=\\exp{(-\\iota H_{m} \\Delta t)}$, where $H_{m}$ is the Hamiltonian in the $m^{th}$ step and $\\Delta t$ is the duration of the step. \n\nWe first optimize the number of steps to check the adiabaticity condition being satisfied by the given set of Hamiltonians i.e., we check if the ground state of the initial Hamiltonian reaches the ground state of the final Hamiltonian without anticrossing as  shown in the Fig. \\ref{fig:Adb}. \n\n\\begin{figure}[h]\n\\includegraphics[width=8cm,trim=5cm 3cm 7cm 3cm]{fig4}\n\\caption{\\label{fig:Adb}(Color online) Simulation shows no anticrossing between two states of the qubit q[0] of the IBM's $QX_{4}$ quantum processor during adiabatic evolution for time chosen $T=10 \\mu s$ in 8 steps. The Hamiltonian used are $H_{i}= J \\sigma_{z}$ and $H_{f}= J\\operatorname{I}$.}\n\\end{figure}\n\\subsection{Decomposition of Unitaries}\nWe now decompose the unitaries obtained in Sec. \\ref{fact35} for each of the 8 steps into the single-qubit Clifford+T gates. In actual implementation of Adiabatic evolution of the system from the ground state of the initial Hamiltonian to the ground state of the final Hamiltonian. The actual decomposition for a general unitary is shown in Eq. (\\ref{nine}) and the exact values of $\\theta$s for each step are given in Tab. \\ref{theta}.  In this stage, we need to be specific to the quantum processor to be used, as the available gate library and the ease with which different gates can be realized in a particular implementation\/architecture are different. Here, we are interested in implementing  the proposed scheme using an IBM QX4 processor, which restricts us to decompose the unitaries in terms of the available Clifford gate library of IBM Quantum Experience (QE). To be specific, to implement our scheme in IBM's QX4, any general unitary which we require to implement as part of our scheme (circuit), has to be decomposed in terms of the Clifford+T gates. In general, a single-qubit unitary obtained with the chosen initial and obtained final Hamiltonians are of the form, \n$U = \\left(\n\\begin{array}{cc}\n    a & 0 \\\\\n    0 & b\n\\end{array}\\right)$.\nEach element in the given unitary is a complex number. Thus, $ a= r_{1}\\exp(\\iota \\theta_{1}) $ and,  $ b= r_{2}\\exp(\\iota \\theta_{2})$ hence corresponding to each unitary we have two matrices corresponding to r and $\\theta$ values i.e., $R = \\left(  \\begin{array}{cc}r_{1} & 0 \\\\ 0 & r_{2} \\end{array} \\right)$ and $ \\Theta = \\left(  \\begin{array}{cc} \\exp(\\iota \\theta_{1}) & 0 \\\\ 0 & \\exp(\\iota \\theta_{2}) \\end{array} \\right)$ such that \n\\begin{equation}\n\\operatorname{U}= \\operatorname{R} \\cdot \\operatorname{\\Theta}.\n\\end{equation}\nThe values of $\\theta$ are given in Tab. \\ref{theta}. For the case of 35, $R$ matrices for all unitaries are identity, so $\\operatorname{U}= \\operatorname{\\Theta}$.\nThe IBM Clifford+T gate library consists of following single-qubit unitaries:\\\\\n(i) The Pauli gates: $\\operatorname{I}$, $\\operatorname{X}$, $\\operatorname{Y}$, and $\\operatorname{Z}$\\\\\n(ii) General gates: $\\operatorname{U_{1}\\left(\\theta \\right)}$, $\\operatorname{U_{2}}(\\phi,\\lambda)$, and $\\operatorname{U_{3}}(\\theta,\\phi,\\lambda)$.\\\\\n(iii) Phase Gates: $\\operatorname{S} (\\operatorname{S^{\\dagger}})$, $\\operatorname{T} (\\operatorname{T^{\\dagger}})$, and\\\\\n(iv) Other Gates: $\\operatorname{H}$ (Hadamard) \\\\\nIn what follows, we will use phase gate and Pauli $X$-gate to construct unitaries to be implemented in our experiments. The decomposition of $\\Theta$ in Clifford+T gate library can be obtained as\n\n\\begin{widetext}\n\\begin{eqnarray}\n\\operatorname{\\Theta} &=&\\left({\\begin{array}{cc} 1 & 0\\\\ 0 & \\exp(\\iota \\theta_{2})\\end{array}}\\right)  \\left({\\begin{array}{cc}  0 & 1\\\\ 1 & 0\\end{array}}\\right)\\left({\\begin{array}{cc} 1 & 0\\\\ 0 & \\exp(\\iota \\theta_{1})\\end{array}}\\right) \\left({\\begin{array}{cc}  0 & 1\\\\ 1 & 0\\end{array}}\\right)\\\\   \\nonumber\n&=& \\operatorname{U_{1}}\\left(\\theta_{2}\\right) \\cdot \\operatorname{X} \\cdot \\operatorname{U_{1}}\\left(\\theta_{1}\\right) \\cdot \\operatorname{X}.\n\\label{nine}\n\\end{eqnarray}\n\\end{widetext}\nCombining $R$ and $\\Theta$ matrices, the unitary U$_{m}$ for m$^{th}$ step can be written as\n \\begin{equation}\nU_{m} = \\operatorname{U_{1}^{m}}\\left(\\theta_{1}^{m}\\right) \\cdot \\operatorname{X} \\cdot \\operatorname{U_{1}^{m}}\\left(\\theta_{2}^{m}\\right) \\cdot \\operatorname{X}, \n\\label{um}\n \\end{equation}\nwhere, $\\theta_{1}^{m}$ and $\\theta_{2}^{m}$ are the angles in the m$^{th}$ unitary.\nThus, the total unitary for quantum part of the hybrid factorization scheme is $\\operatorname{U}= \\prod_{m=8}^{m=1} U_{m}$.\nHere, it is important to mention that gates in the unitary have been applied in the right to left order. The values for $\\theta_{1}^{m}$ and $\\theta_{2}^{m}$ for different steps are provided in Tab. \\ref{theta}.\n\n\\begin{center}\n\\begin{table}[b]\n\\begin{tabular}{|m{.5cm}|m{.8cm}|m{.8cm}|m{1.5cm}|m{1.5cm}|}\n\\hline\nm & $r_{1}$ & $r_{2}$ & $\\theta_{1}^{m}$ & $\\theta_{2}^{m}$\\\\\n\\hline\n1 & 1 & 1 & -1.2500 & 0.9375\\\\\n\\hline\n2 &  & 1 & -1.2500 & 0.6250 \\\\\n\\hline\n3 & 1 & 1 &-1.2500 & 0.3125\\\\\n\\hline\n4 & 1 & 1 & -1.2500 & 0 \\\\\n \\hline\n5 & 1 & 1 & -1.2500 & -0.3125\\\\\n \\hline\n 6& 1 & 1 & -1.2500 & -0.6250\\\\\n \\hline\n 7 & 1 & 1 & -1.2500 & -0.9375 \\\\\n \\hline\n 8 & 1 & 1 & -1.2500 & -1.2500\\\\\n  \\hline\n\\end{tabular}\n\\caption{The r and $\\theta$ values for unitaries in each step of adiabatic evolution.} \\label{theta}\n\\end{table}\n\\end{center}\n\n\\begin{figure}[h]\n\\includegraphics[width=5cm,trim=9cm 3cm 10cm 1cm]{fig3.pdf}\n\\caption{(Color online) Schematic procedure of complete scheme for hybrid factorization. Top trace shows the steps of the scheme. Second trace shows circuit for obtaining ground state of the final Hamiltonian $H_{f}$ starting from $H_{i}$ through adiabatic evolution. The trace below that shows quantum circuit for implementing quantum adiabatic evolution part on IBM's QX4 processor for factorization of 35, by initializing the qubit system to the ground state i.e., $\\ket{1}$ of the initial Hamiltonian $J \\sigma_{z}$ and the lowest trace shows the gate decomposition in IBM's gate library of $k^{th}$ unitary $U_{k}$. }\n\\label{schematic}\n\\end{figure}\n\n\\section{Experiments} \\label{exp}\n This experiment has been performed on an open access 5-qubit quantum processor placed on cloud by IBM corporation. In particular, we have used the architecture of IBM's QX4 (IBM Q 5 Tenerife)\\cite{IBMQE}, which consists of superconducting Transmon qubits \\cite{malkoc2013quantum}. A schematic diagram of this architecture and description of the architecture can be found in \\cite{sisodia2017design,sisodia2017experimental} and references therein. The basic gate library used for the single-qubit gates are $\\operatorname{H}$, Pauli operators $\\operatorname{X}$, $\\operatorname{Y}$, $\\operatorname{Z}$, parametric gates $\\operatorname{U_{1}}$, $\\operatorname{U_{2}}$, and $\\operatorname{U_{3}}$. The operator $\\operatorname{U_{1}}$ depends on single parameter $\\theta$, operator $\\operatorname{U_{2}}$ depends on two parameters $\\theta, \\phi$, and operator $\\operatorname{U_{3}}$ depends on three parameters $\\theta, \\phi, \\lambda$. We chose qubit q[0] for implementation of quantum adiabatic evolution of ground state search. We initialize the system to the ground state of the initial Hamiltonian by applying the $\\operatorname{X}$ gate ($\\sigma_{x}$) to the qubit q[0]. To evolve this state adiabatically we apply a set of eight unitaries, in their decomposed form as shown in the previous section. In order to extract the probabilities of the final state we perform quantum state tomography after each step. The directly measured observable in IBM processor are $\\ket{0}\\bra{0}$ and $\\ket{1}\\bra{1}$ which allow us to calculate $\\expv{Z}$. This is sufficient to reveal the probabilities $p_{0}$ and $p_{1}$. In order to measure real and imaginary part of the coherence term, we have used the method described in our earlier work \\cite{shukla2020complete} i.e., we have applied, $H$ gate followed by $\\operatorname{Z}$-measurement to reveal real part and $S^{\\dagger}H$ followed by $\\operatorname{Z}$-measurement for measuring its imaginary part.\n\n\\section{Results} \\label{results}\nThe QST results, are shown in Fig. \\ref{fig:results1} and Fig. \\ref{fig:results2}. To measure the final state we have the corresponding measurement operator in the Clifford library. The ground state of the final Hamiltonian provides us the solution to our problem, in our case, we would obtain probability of  $p_{0}$ and $p_{1}$ for the states $\\ket{0}$ and $\\ket{1}$, respectively, after performing the experiment 8192 times (i.e., the maximum number of runs that one can select from the interface provided by IBM QE). The tomography results reveal that after the full adiabatic evolution of the system in 8 steps, system is in the ground state of the final Hamiltonian. Since the Hamiltonian at this point is degenerate, consequently, solution of the problem Hamiltonian are any of the two states $\\ket{0}$ and $\\ket{1}$. If we consider $\\ket{0}$ as the solution, then the corresponding classical bit value would be $p_{1}=0$ leading to the first factor of the composite number 35 in the binary system as $1 p_{1} 1 =101$ and consequently, in decimal system as $P=\\sum_{k} 2^{k} p_{k}=5$. The conjugate bit value by using the identity $p_{1}+q_{1}=1$ is $q_{1}=1$ and the corresponding prime factor in the binary system is $1 q_{1} 1=111$, and consequently the number, in the decimal system would be $Q=\\sum_{l} 2^{l} q_{l}= 7$. The two factors can also be obtained in the same way if we consider $\\ket{1}$ as the solution of the ground state search of the adiabatic evolution. \n\n\\begin{figure}[h]\n\\includegraphics[width=7cm,trim=5cm 9cm 5cm 8cm]{fig1.pdf}\n\\caption{\\label{fig:results1} (Color online)  Figure shows density matrices in rows, with the columns representing the real and imaginary parts of the density matrix. Top row indicates the initial state, subsequent rows represent state at even instances of applying unitaries i.e., 2, 4, 6, and 8 times, while going down.   Ticks 1, 2, 3, 4 correspond to the elements $\\ket{0}\\bra{0}$, $\\ket{0}\\bra{1}$, $\\ket{1}\\bra{0}$, $\\ket{1}\\bra{1}$.}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=9cm,trim=3cm 8cm 3.5cm 8cm]{fig2.pdf}\n\\caption{\\label{fig:results2} (Color online) Figure shows density matrices in rows, with the columns representing the real and imaginary parts of the density matrix. Rows represent state at odd instances of applying unitaries to the initial state i.e., 1, 3, 5, and 7 times, while going down. Ticks 1, 2, 3, 4 correspond to the elements $\\ket{0}\\bra{0}$, $\\ket{0}\\bra{1}$, $\\ket{1}\\bra{0}$, $\\ket{1}\\bra{1}$. The result reveals the system to be in the ground state of the final Hamiltonian.}\n\\end{figure}\n\n \n\n\\section{Conclusions} \\label{concl}\nAs discussed above, factorisation of bi-prime numbers are important for hacking of RSA type cryptographic schemes and various related problems. However, factorisation of some bi-primes are not as difficult as that of the others. Specifically, for an even bi-prime, we already know that one factor is 2 and it's trivial to find the other factor. Further, there are excellent algorithms for finding square root, so factorisation of square bi-primes are easy. What we are left with is odd and square-free bi-primes which are used in cryptography. Here, we report a quantum-classical hybrid scheme for factorization of such (i.e., odd and square-free) bi-prime numbers.  The scheme proposed here is hybrid in nature, implying that the scheme utilizes both classical optimization techniques and adiabatic quantum optimization techniques. The advantage of such hybrid schemes underlies in the fact that they require less quantum resources (which are fragile and costly at the moment) in comparison with the purely quantum schemes designed for the same purpose. For, example, it's already understood that the extremely large quantum registers required in Shor's original protocol are not required in the hybrid schemes proposed later on. The same is true for our scheme as well as the schemes \\cite{pal2019hybrid,xu2012quantum}  which have been extended here. Thus, in short the proposed scheme has the capability to factorise relatively large odd and square-free bi-primes using a small amount of quantum resources. To illustrate this, we have explicitly performed factorisation of 35 (which is an odd and square free bi-prime) using the smallest quantum computer available on the cloud (i.e., a five qubit quantum processor called IBM's QX4).  The quantum processor used here is known to be noisy, but here we have correctly obtained the prime factors of 35  with some small signatures of noise depicting the strength of the algorithm. We conclude the paper with a hope that with the availability of larger quantum processors, larger bi-primes will be factored using this algorithm and it will be found useful in the future development of the hybrid algorithm designing. \n\n\\section{Acknowledgements}\nAuthors thank to Defense Research And Development Organization (DRDO), India for the support provided through the project number ERIP\/ER\/1403163\/M\/01\/1603. Abhishek Shukla thanks to Applied Physics department, The Hebrew University of Jerusalem for the support.\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAutonomous driving requires the vehicles to efficiently sense and understand the surrounding 3D world in real time. Currently, most of the autonomous vehicles (Google, Uber, Ford, Baidu, etc) are equipped with high-end 3D scanning systems such as Velodyne which could provide accurate 3D measurements that are critical to the autonomous vehicles' decision making and planning. However these high-end 3D scanning systems such as Velodyne LIDAR sensors are quite expensive with the cost comparable to the cost of the whole vehicle, which may hinder their admittance to the global consumer market.\n\nTo effectively reduce the cost in surrounding 3D world sensing, a natural and cost efficient alternative would be using passive sensors such as monocular cameras or stereo cameras, which could not only provide 3D measurements (by using structure-from-motion (SfM) or simultaneous localization and mapping (SLAM)) but also semantic information (not available from the point clouds). However, the computer vision based system is either not robust (different weather conditions could result in dramatically different vision measurements) or inaccurate (compared with LiDAR based 3D scanning). \n\nIn this paper, we propose to investigate effective fusion to integrate sparse 3D point clouds (low cost) and high-resolution colour images, such that generating dense 3D point clouds comparable to high-end 3D scanning systems. Specifically, the objective of this paper is to ``\\textbf{generate\/predict a dense depth map and the corresponding 3D point clouds from very sparse depth measurements with\/without colour images and colour-depth image datasets}''.  Under our framework, we use a low-resolution LIDAR and a high-resolution colour image to generate dense depth maps\/3D point clouds (Fig.~\\ref{fig:conceptual-illustration}). This can be understood as using the high-resolution colour image to augment the sparse LIDAR map and predict a dense depth map\/3D point clouds. Our proposed methods can not only provide accurate dense depth maps but also provide visually pleasant 3D point clouds, which are critical for autonomous driving in urban scenes. Specifically, we have presented a novel way in perceiving 3D surrounding environments, which owns low-cost compared with the high-end LIDAR sensors and high precision and efficiency compared with the colour camera only solutions. The proposed framework owns great potentials in developing compact and high-resolution LIDAR sensors, at very low cost, for domain-specific applications (e.g. ADAS, autonomous driving).\n\n\\begin{figure}[!htp]\n \\centering\n    \\includegraphics[width=1\\columnwidth]{Sparse-Dense.png} \n  \\caption[Problem definition.]{\\label{fig:sparse-dense-concept} Conceptual illustration of the problem of predicting dense 3D point clouds\/depth maps from sparse LiDar input, where the input is a collection of sparse 3D point clouds and desired output is a dense 3D point cloud\/depth map.}\n  \\label{fig:conceptual-illustration}\n\\end{figure}\n\nOur dense depth map prediction framework differs from general depth interpolation methods and depth super-resolution methods in the following aspects: 1) Irregular depth points pattern; 2) Very sparse depth measurements; and 3) Missing real obstacles or generating false obstacles in a certain range is unacceptable. The irregular depth points pattern creates a barrier in applying deep convolutional neural network (CNN) based methods as it requests irregular convolution pattern, which is still an unsolved problem in deep learning. On the other hand, even though many traditional methods \\cite{Li2012,ferstl2013b,YangAR2014,Barron2016,Li2016} could deal with irregular patterns, they suffer from the dependency on the strong correlation between colour images and depth maps, where boundaries that only exist in the colour images will mislead the methods and generate false obstacles on road. For those general smooth interpolation methods such as nearest neighbor, bilinear interpolation, bicubic interpolation and \\cite{Garcia20101167}, since they neglect the colour images in problem formation, they tend to generate over-smoothed results and miss real obstacles.\n\nIn this paper, targeting at handling the above difficulties with existing methods, we propose a dense depth prediction approach that only uses the boundaries in colour image (i.e., the superpixel over-segmentation). We resort to the piecewise planar model for general scene reconstruction, where the desired depth map\/point clouds are represented as a collection of 3D planar and the reconstruction problem is formulated as the optimization of planar parameters. The resultant optimization involves the unary term evaluated at the sparse depth measurements, the smoothness term across neighboring planes. Thirdly, as the urban driving scenarios are well structured, we propose a specifically designed model for urban driving scenario called ``cardboard world'' model, i.e., front-parallel orthogonal piecewise planar model, where each segment can only be assigned to either the road plane or a front-parallel object plane orthogonal to the road plane. We formulate the problem as a continuous CRF optimization problem and solve it through particle based method (MP-PBP) \\cite{Yamaguchi14}. Extensive experiments on the KITTI visual odometry dataset show that the proposed methods owns high resistance to false object boundaries and can generate useful 3D point clouds without missing obstacles.\n\n\\section{Related Work}\nGiven an incomplete depth map, depth completion task is to fill the missing depth values to get a dense depth map. A colour image that is captured with a camera is often used as the guided image. Since most range sensors can only achieve sparse or semi-dense depth maps, completing missing depth values can increase the quality of depth maps for better applicability. This task has drawn increasing attention recently due to the rising interest of autonomous driving, augmented\/virtual reality and robotics. Although this task can be partially addressed by traditional image in-painting techniques, extra knowledge such as depth smoothness, normal smoothness, colour guidance and \\emph{etc. }} \\def\\vs{\\emph{vs. } is yet to be utilized for achieving an accurate dense depth map. \n\nDepth completion includes three sub-tasks: depth inpainting, depth super resolution and depth reconstruction from sparse samples. Anisotropic diffusion \\cite{Perona1990} is a popular method for image inpainting, and it has been successfully adapted to the depth inpainting task in \\cite{Miao2012}. Energy Minimization based depth inpainting methods \\cite{Liu2013icip,Chongyu2015} take characteristics of depth maps as a regularization term in energy minimization and generate more plausible results. Exemplar-based Filling \\cite{Criminisi2004} works well in image inpainting but creates extra challenges in depth inpainting due to the lacking of textures in depth maps. Matrix Completion \\cite{Lu2014} assumes that a depth map lies in a low-dimensional subspace and can be approximated by a low-rank matrix. The main goal for depth super-resolution task is to increase the spatial resolution of a depth image to match a high resolution colour image. The low-resolution depth maps are completed in this case. By assuming depth discontinuities are often aligned with colour changes in the reference image, this task can be solved through Markov Random Fields (MRF) \\cite{DiebelT05,Andreasson07}. Such methods can be easily adapted to depth reconstruction from sparse samples task as they do not require the depth points to be regularly sampled \\cite{DolsonBPT10}. In order to intensively use the structural correlation between colour and depth pairs in colour guided depth super-resolution, non-local means (NLM) was introduced as a high-order term in regularization \\cite{Park2011}. \\cite{YangAR2014} swaps the Gaussian kernel in the standard NLM to a bilateral kernel to enhance the structural correlation in colour-depth pairs and proposed an adaptive colour-guided auto-regressive (AR) model that formulates the depth upsampling task as AR prediction error minimization, which owns a closed-form solution. A few approaches employ sparse signal representations for guided upsampling making use of the Wavelet domain \\cite{Hawe2011}, learned dictionaries \\cite{Li2012} or co-sparse analysis models \\cite{Gong2014}. \\cite{Barron2016} and \\cite{Li2016} leverage edge-aware image smoothing techniques and formulate it as a weighted least squares problem while \\cite{Li2016} also applied coarse-to-fine strategy to deal with the very sparse situation. \\cite{ku2018defense} proposed an efficient depth completion algorithm that achieves the state-of-the-art performance on KITTI depth completion benchmark \\cite{Uhrig2017THREEDV} that only uses image processing operations.\n\nWith the booming development of deep learning technique, deep networks were introduced to geometry learning tasks such as monocular depth estimation~\\cite{Zhong2018ECCV}, stereo matching~\\cite{zhong2017self,Zhong2018ECCV_rnn,zhong2020nipsstereo,zhong2020displacementinvariant}, optical flow~\\cite{Zhong_2019_CVPR,zhong2020nipsflow}, and achieved state-of-the-art performance on most benchmarks. By treating a depth map as a gray-scale image, image super-resolution networks \\cite{Dong2014} can be directly applied \\cite{Song2016}. Riegler \\emph{et~al.} \\cite{Riegler2016} proposed a deeper depth super-resolution network that has faster convergence and better performance. A natural barrier for applying deep methods to other depth completion tasks is the irregular depth pattern as standard convolution layers are designed for regular grid inputs. Sparsity Invariant CNNs \\cite{Uhrig2017THREEDV} addresses this problem and tried to handle the sparse and irregular inputs by introducing invalid masks in convolution layers. On the other hand, \\cite{jaritz2018sparse} claims that standard convolution layers can handle sparse inputs of various densities without any additional mask input. In \\cite{semantic2016}, it proves that leveraging semantic information can improve the depth completion performance and vice versa~\\cite{zhong20183d}. In the recent stereo-LiDAR fusion work~\\cite{Cheng_2019_CVPR}, they also apply the piecewise planner model as a soft constraint in their network loss functions.\n\n\\section{Approach}\nReal world driving scenarios generally consist of road, surrounding buildings, vehicles, pedestrians and etc., which can be well approximated with piece-wise planar model in 3D representation. By representing the traffic scenes with piece-wise planar model, we are able to exploit the structural information in the scenes and the number of variables can be greatly reduced (For each plane, we only need 3 parameters to represent.). In this way, the number of measurements of sparse depths could be greatly reduced, which enables us the ability to work with very sparse LIDAR measurements. Furthermore, the parametric model owns the ability to handle outlying LIDAR measurements. \n\n\\subsection{An overview of the proposed method}\nA high level description of our method is given as follows: assume the LIDAR sensor and the camera have been geometrically calibrated (both intrinsically and extrinsically).  Given the input of a single frame monocular image with corresponding sparse depth points, we first perform image over-segmentation to obtain fixed number of super-pixels using the SLIC algorithm \\cite{SLIC2012}. Then we conduct interpolation on the sparse depth points to generate an initial dense depth map by using the penalized least squares method \\cite{Garcia20101167}. This dense depth map is used to provide initial planar parameters for each segment. As shown in Fig.~\\ref{fig:input}, on average each superpixel region contains a single depth point measurement. Also note some superpixels do not contain any depth measurement.  After fitting the initial depth measurements inside each superpixel with a plane, we formulate a Conditional Random Field (CRF) to optimize all the plane parameters and recalculate the depth map. A flowchart of our approach is given in Fig.~\\ref{fig:processcrf}. \n\n\\begin{figure}[!htp]\n\\begin{center} \n \\subfigure{\n\\includegraphics[width=1\\columnwidth]{input3.png} }\n \\subfigure{\n   \\includegraphics[width=1\\columnwidth]{input.png} }\n \\caption[Inputs of our algorithm.]{\\label{fig:input} \\textbf{Inputs of our algorithm implementation.} Top: The SLIC over-segmentation of the input image. Bottom: the input of our algorithm: the super-pixel segmentation and sparse depth measurements. Note that as the depth measurements are very sparse, there are only a few depth measurements in each super-pixel and there are considerable super-pixels where no depth measurements are available.}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[!htp]\n\\centering\n\\begin{center}\n\\includegraphics[width=.65\\columnwidth]{process.png\n\\caption[Pipeline of our method.]{\\label{fig:processcrf}\\textbf{Pipeline of our method.} Given high resolution colour image and sparse depth measurements as input, our approach predicts dense depth map with the same resolution as the colour image.}\n\\end{center}\n\\end{figure}\n\n\\subsection{Mathematical Formulation}\nUnder our piece-wise planar model for the scenes, each segment is well approximated with a plane. In this way, the dense depth prediction problem is transformed to the optimization of the plane parameters. Furthermore, we assume that the boundaries in the depth maps are a subset of the boundaries in the colour images, which enables us the freedom to decide the real depth boundaries from the colour boundaries.\n\n\nSpecifically, let $\\mathcal{S}$ denote the set of superpixels and each superpixel $i\\in \\mathcal{S}$ is associated with a segment $\\mathcal{R}_i$ in the image plane and a random variable $\\mathbf{s}_i = \\mathbf{n}_i$, where $\\mathbf{n}_i \\in \\mathbb{R}^3$ describes the plane parameter in 3D. Our goal is to infer the 3D geometry of each superpixel $\\mathbf{s}_i$ given the sparse depth measurements $\\mathrm{d}_i(\\mathbf{x})$. We define the energy of the system to be the sum of a data term $\\varphi_i$ and a smoothness term $\\psi_{i,j}$,\n\\begin{equation}\nE(\\mathbf{s}) = \\sum_{i\\in \\mathcal{S}}\\varphi_i(\\mathbf{s}_i) + \\sum_{i\\sim j}\\psi_{i,j}(\\mathbf{s}_i,\\mathbf{s}_j),\n\\label{energy}\n\\end{equation}\nwhere $\\varphi$ denotes the data term while $\\psi$ denotes the smoothness term. $\\mathbf{s} = \\{\\mathbf{s}_i|i\\in \\mathcal{S}\\}$ and $i\\sim j$ denotes the set of adjacent superpixels in $\\mathcal{S}$.\n\n\\noindent\\textbf{Data Term:} The data term encourages the depth measurement to lie on the plane by penalizing the discrepancy between the measurement and the prediction. To better enforce this constraint, we choose the $\\ell_2$ norm to amplify the cost. Therefore, our data term can be written as:\n\\begin{equation}\n\\varphi_i(\\mathbf{s}_i) = \\theta_1 \\sum_{\\mathbf{x}\\in \\mathcal{X}}\\|\\widehat{\\mathrm{d}}(\\mathbf{s}_i,\\mathbf{x}) - \\mathrm{d}_i(\\mathbf{x})\\|_2^2,\n\\label{dataenergy}\n\\end{equation}\nwhere $\\mathcal{X}$ is the set of pixels that has depth measurements. $\\widehat{\\mathrm{d}}(\\mathbf{s}_i,\\mathbf{x}) = \\frac{-\\widetilde{\\mathrm{d}}_i}{\\mathbf{n}_i^T \\mathbf{K}^{-1}\\mathbf{x}}$ represents the estimated depth value on a pixel $x$, $\\widetilde{\\mathrm{d}}_i$ represents the distance between the plane and the origin, $\\mathbf{n}_i^T$ is the normal vector of the plane, $\\mathbf{K}$ is the camera intrinsic matrix and $\\mathbf{x} = (u,v,1)^T $ is the homogeneous representation of the pixel $x$. $\\mathrm{d}_i(x)$ is the depth measurement on pixel $x$. \n\n\\noindent\n\\textbf{Smoothness Term:}\nThe smoothness term encourages coherence of adjacent superpixels in terms of both depth and orientation. The depth coherence is defined by the difference between the depths of neighboring superpixels' boundaries while the orientation coherence is defined as the difference between the surface normal of neighbouring superpixels. Considering the discontinuity in neighboring superpixels due to scene structure, we use the truncated $\\ell_1$ norm to allow discontinuity in both depth and orientation. \n\n\nFollowing \\cite{Menze2015CVPR}, our smoothness potential $\\psi_{i,j}(\\mathbf{s}_i,\\mathbf{s}_j)$ can be decomposes as:\n\\begin{equation}\n\\psi_{i,j}(\\mathbf{s}_i,\\mathbf{s}_j) = \\theta_2 \\psi_{i,j}^{depth}(\\mathbf{n}_i,\\mathbf{n}_j) + \\theta_3 \\psi_{i,j}^{orient}(\\mathbf{n}_i,\\mathbf{n}_j),\n\\label{smenergy}\n\\end{equation}\nwith weights $\\theta$ and\n\\begin{equation*}\n\\psi_{i,j}^{depth}(\\mathbf{n}_i,\\mathbf{n}_j) = \\sum_{\\mathbf{p}\\in \\mathcal{B}_{i,j}}\\rho_{\\tau_1}(d(\\mathbf{n}_i,\\mathbf{p})-\\mathrm{d}(\\mathbf{n}_j,\\mathbf{p})),\n\\end{equation*}\n\\begin{equation*}\n\\psi_{i,j}^{orient}(\\mathbf{n}_i,\\mathbf{n}_j) = \\rho_{\\tau_2}(1-|\\mathbf{n}_i^T\\mathbf{n}_j|\/(|\\mathbf{n}_i||\\mathbf{n}_j|)),\n\\end{equation*}\nwhere $\\mathcal{B}_{i,j}$ is the set of shared boundary pixels between superpixel $i$ and superpixel $j$, and $\\rho$ is the robust $\\ell_1$ penalty function $\\rho_{\\tau}(x) = \\min(|x|,\\tau)$.\n\n\\subsection{CRF Optimization}\nThe optimization of the above continuous CRF defined in Eq.~\\eqref{energy} is generally NP-hard. In order to efficiently solve this optimization problem, we discretize the continuous variables and leverage particle convex belief propagation (PCBP) \\cite{PCBP2011}, an algorithm that is guaranteed to converge and gradually approach the optimum. It works in the following way: after initialization, for each random variable, particles are sampled around current states. Then these particles act as labels in discretized MRF\/CRF that can be solved by any MRF\/CRF solving methods such as multi-label graph cut, sequential tree-reweighted message passing (TRW-S) \\cite{Kolmogorov2006TRWS} and update the MAP estimation to current solution. The process is repeated for a fixed number of iterations or until convergence.\n\n\\begin{algorithm}[t!]\n\\caption{Solving Eq.~\\eqref{energy} via the PCBP}\n\\label{Algorithm 1}\n\\begin{algorithmic}\n\\REQUIRE ~~\\\\\nSuperpixels $\\mathbf{S}$ and sparse depth measurements $\\mathbf{d}$, number of particles $n_p$, number of iterations $n_i$, parameters $\\theta_1$, $\\theta_2$, $\\theta_3$, $\\tau_1$, $\\tau_2$, $\\rho$, $\\sigma$. \n \\\\ \\vspace{0.2cm}\n\\hspace{-0.3cm}{\\bf Initialize:} Initial plane parameters $\\mathbf{s}_i$ for each superpixel segment. \\\\ \\vspace{0.2cm}\n\\WHILE {iteration $< n_i$}\n\\STATE 1). Sample particles: the first particle for each superpixel is the result of previous iteration, the next $n_p\/2$ particles are randomly sampled around the state in the previous iteration and the remaining $n_p\/2-1$ particles are sampled from the neighboring superpixels' current states;\n\n\\STATE 2). Evaluate the data term \\eqref{dataenergy} and the smoothness term \\eqref{smenergy};\n\n\\STATE 3). Solve the resultant discrete problem with TRW-S and update plane parameters for each superpixel.\n\n\\ENDWHILE\n\\ENSURE ~Plane parameters $\\mathbf{S}$, recovered depth map $\\mathbf{D}$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\textbf{Implementation Details:} A 3D plane is defined by: \n\\begin{equation}\naX_i+bY_i+cZ_i+d = 0\n\\label{eq:abcdplane}\n\\end{equation}\nwhere $(a,b,c,d)$ are the plane parameters, $(X_i,Y_i,Z_i)$ is the 3D points coordinates that can be computed by\n\\begin{equation}\nX_i = (u_i - C_x)\\times Z_i\/f,\n\\end{equation}\n\\begin{equation}\nY_i = (v_i - C_y)\\times Z_i\/f,\n\\end{equation}\nwhere $Z_i = d_i$. $(u_i,v_i)$ is the point coordinate in the image plane, $(C_x,C_y)$ is the camera principal point offset and $f$ is the camera focal length. \n\nThus, the $i^{th}$ particle is defined as a $4\\times 1$ vector $(a_i,b_i,c_i,d_i)^T$. $a_i,b_i,c_i,d_i$ are independently generated through a normal distribution with standard deviation $\\sigma$ and mean $\\mu$, where $\\sigma$ is given by user setting and $\\mu$ is the state in the previous iteration.\n\nOur approach to solve Eq.~\\eqref{energy} is outlined in Algorithm 1, where parameters $\\theta_1$ $\\theta_2$ $\\theta_3$ $\\tau_1$ $\\tau_2$ have been already defined in previous equations, $\\rho$ is the decay rate for generating particles and $\\sigma$ is a $4\\times 1$ vector that contains the standard deviation for the 4 parameters of each particle.\n\n\\textbf{Particle Generation:} Instead using of the regular PCBP particle generation scheme, which generates particles only from the MCMC framework, we partially adopt the PMBP \\cite{PMBP2014} scheme by adding neighboring plane parameters into the particles, thus the candidate particles are a mixture of MCMC sampling around the previous states and the states from neighboring planes. The advantage of this modification is that it allows neighboring superpixels to fuse together thus decreasing the energy. For example, the road area is often segmented into several segments. By using the regular PCBP particle sampling strategy, each segment's parameters are independently generated by a normal distribution. Even though the smoothness term encourages the planes be close to each other, there will still exist small gap between them as the algorithm could not find exactly the same parameters from its candidate particles. However, in our modified version, the neighboring superpixels can share their parameters and therefore resolve the issue.\n\nIn each PCBP iteration, each superpixel has fixed number of particles, and we need to find the sub-optimal combination that has the lowest energy. This problem can be efficiently solved through tree-reweighted max-product message passing (TRW-S) algorithm. The processing time depends on the number of superpixels and the number of particles.\n\n\\subsection{Cardboard World Model: A More Constrained Model}\nIn the above section, we described our piece-wise planar model for solving the dense depth prediction problem, where each plane has three freedoms in 3D space. \n\nHere we notice that for real-world driving applications, man-made road scenes often have stronger structured information (i.e. stronger prior).  In particular, we realize that modeling a front-view road scene as a combination of ground-plane and many front-parallel obstacles planes will be convenient for the task of drivable free-space detection task. Next, we will show how to infer such a simplified 3D road scene model by using the same method of our CRF--conditional random field framework. \n\nBased on these observations, we propose our ``cardboard world'' model for representing the driving scenes. Under the ``cardboard world'' model, there only consist two kinds of planes: the road plane and the object plane. We assume there is only one road plane in the scene and all the other planes are object plane. These two kinds of planes are orthogonal to each other and object planes are front parallel. Fig.~\\ref{fig:cw} illustrates an example of our ``cardboard world'' model. \n\n\\begin{figure}[!htp]\n\\begin{center} \n\\includegraphics[width=1\\columnwidth]{cw001.png} \n \\caption[``Cardboard world'' model.]{\\label{fig:cw} \\textbf{An example of our ``cardboard world'' model:} Object planes are front parallel planes that orthogonal to the road plane.}\n \\end{center}\n\\end{figure}\n\nThere are three advantages in replacing the slanted plane model in the piecewise planar model with our proposed ``cardboard world'' model:\n\\begin{compactenum}[1)]\n\\item The recovered 3D point clouds are more visually pleasant and the location of each object is more accurate. In some case, when there are two depth points with very different depth values in a single superpixel, the slanted plane model will use a very slanted plane to fit these two points. As a result, the shape of this area will be largely distorted. On the other hand, in the ``cardboard world'' model, it will force the plane to become front parallel therefore maintain the object shape.\n\\item As a byproduct, this method provides a free-space for autonomous driving vehicles. We embed a binary labeling problem in our task that classifies superpixels into two clusters: road and object. Superpixels labeled as road belong to the free-space.\n\\item Processing time can be greatly reduced. By applying a front-parallel constraint and orthogonal constraint, the number of parameters to optimize have been greatly reduced thus decreasing the processing time.\n\\end{compactenum}\n\nA diagram of our cardboard model approach is shown in Fig.~\\ref{fig:process}.\n\\begin{figure}[!htp]\n\\centering\n\\begin{center}\n\\includegraphics[width=.65\\columnwidth]{process2.png\n\\caption[Process flow of our method.]{\\label{fig:process}\\textbf{Process flow of our approach.} Given high resolution colour image and sparse depth measurements as input, our approach predicts dense depth map with the same resolution as the colour image.}\n\\end{center}\n\\end{figure}\n\n\\textbf{Initialization:} Different from the piecewise planar based method, this new method requires an initial road plane estimation for initialization. To do so, we fit the road plane directly from sparse depth point using RANSAC. After getting the initial dense depth map using \\cite{Garcia20101167}, for each superpixel segment, we project all the depth points into 3D and calculate the distance between them and the road plane by summing the Euclidean distance of each point. If the distance is smaller than a given threshold $\\epsilon$, we label the superpixel as road plane and fit it with the initial road plane parameters. Otherwise, the superpixel is labeled as object and will be fitted with a front parallel plane with a depth of the mean depth of this superpixel.\n\nThe optimization problem remains the same as the previous one. However, since we need to enforce the orthogonal relationship as well as the front parallel constraint, given road plane parameter $(a_r,b_r,c_r,d_r)^T$, the normal of object planes are fixed as well and are equal to $(0, -\\frac{c_r}{b_r},1)^T$. Hence there is only one freedom in object planes, i.e., the unknown depth. Note that we naturally embed a binary labeling problem in optimization step since superpixels are divided into two groups: road plane group and object planes group. This labeling problem is jointly solved through PCBP process by adding road plane parameters into particle sets of every superpixels, so that every superpixel has a choice to joint the road plane when the fitting cost is low enough. Our approach to solving Eq.~\\eqref{energy} using our ``cardboard world'' model is outlined in Algorithm 2.\n\n\n\\begin{algorithm}[t]\n\\caption{Solving Eq.~\\eqref{energy} under ``cardboard world'' model via the PCBP}\n\\label{Algorithm 2}\n\\begin{algorithmic}\n\\REQUIRE ~~\\\\\nSuperpixels $\\mathbf{S}$ and sparse depth measurements $\\mathbf{d}$, number of particles $n_p$, number of iteration $n_i$, parameters $\\theta_1$, $\\theta_2$, $\\theta_3$, $\\tau_1$, $\\tau_2$, $\\rho$, $\\sigma$, $\\epsilon$. \n\\\\ \n\\hspace{-0.3cm}{\\bf Initialize:} Estimate road plane parameter $\\mathbf{s}_d$, initial dense depth map $\\mathrm{D}_0$  \n\\FORALL{$\\mathbf{S_i} \\in \\mathbf{S}$}\n    \\STATE Project every depth points into 3D;\n    \\STATE Calculate the sum of Euclidean distance between each 3D point the road plane $\\mathbf{s}_d$;    \\IF{The sum of Euclidean distance is less than a given threshold $\\epsilon$}\n      \\STATE Fit $\\mathbf{S}_i$ with $\\mathbf{n}_\\mathrm{d}$\n    \\ELSE\n      \\STATE Fit $\\mathbf{S}_i$ with front parallel plane and orthogonal to $\\mathbf{n_{\\mathrm{d}}}$\n    \\ENDIF\n\\ENDFOR\n\\WHILE {iteration $< n_i$}\n\\STATE 1). Sample particles: the first particle for each superpixel is the state in previous iteration, then randomly generate $n_p\/2-1$ particles and add $n_p\/2-1$ neighboring superpixels' parameters to the particle set and add the last particle with $\\mathbf{n_d}$;\n\\STATE 2). Calculate data energy through \\eqref{dataenergy} and smoothness energy through \\eqref{smenergy};\n\\STATE 3). Solve through TRWS and update current MAP estimation ${\\bf n}$ to the first particle;\\\\\n\\ENDWHILE\n\\ENSURE ~Plane parameters ${\\bf n}$, recovered depth map ${\\bf D}$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Experiments}\n\\subsection{Experimental setup}\nWe perform both quantitative and qualitative evaluations of our methods on the KITTI VO dataset. The KITTI VO dataset consists of 22 sequences 43,596 frames which includes various driving scenarios such as highway, city and country road. It provides stereo images and semi-dense $360^\\circ$ LIDAR points. In our experimental setting, we only use the left images and the sparse LIDAR points. Note, we recover the depth for the every 10th frames, thus 4,359 frames are recovered in total. \n\nWe perform evaluation of two subset of KITTI dataset: KITTI stereo and KITTI VO, which both consist of challenging and varied road scene imagery collected from a test vehicle. Ground truth depth maps are obtained from 64-line LIDAR data. The main difference between these two dataset is that the ground truth depth maps for the stereo dataset were manually corrected and interpolated based on the neighboring frames. Note we only used the lower half part of the images ($200\\times 1226$) as the upper half part generally include large part of sky and there is no depth measurements available. The input of our experiment was synthesized low-resolution LIDAR points which was downsampled by a factor of 6 in horizontal direction and 3 in vertical direction.\n\n\\noindent\n\\textbf{Piece-wise planar method setting:} As this method performs on superpixel level, we use SLIC \\cite{SLIC2012} segmentation method to provide segments and the number of super-pixels is manually set to 800. The number of particles is the set to 10. The total iteration of PCBP is set to 40 as in Fig.~\\ref{fig:energyDist} as the energies - both total and unary one - become flat since then.\n\n\\begin{figure}[t]\n\\begin{center} \n\n \\subfigure[Unary energy versus the number of iterations.]{\n\\includegraphics[width=0.45\\columnwidth]{unaryEnergyDistribution.png} }\n\n \\subfigure[Total energy versus the number of iterations.]{\n   \\includegraphics[width=0.45\\columnwidth]{EnergyDistribution.png} }\n \\caption[Energy distribution.]{\\label{fig:energyDist} This figure shows the energy with respect to the number of iterations in CRF based method.}\n \\end{center}\n\\end{figure}\n\n\n\\noindent\n\\textbf{``Cardboard world'' method setting:} We use SLIC \\cite{SLIC2012} segmentation method to provide segments and the number of super-pixels is manually set to 1200. In the TRW-s inference step, the number of particles is set to 10, and divided into 3 parts: 1 particle represents the road plane, 4 particles are sampled from its neighbors and 5 particles are newly generated from MCMC process. The number of iteration of PCBP is set to 20 as there is only 1 parameter that need to optimize in this case.\n\n\\subsection{Error Metrics}\nTo evaluate the performance of depth interpolation, we use the following three quantitative metrics:\n\\begin{compactenum}[1)]\n\\item \\textbf{Mean relative error} (MRE), which is defined as:\n\\begin{equation}\n\\mathrm{e}_{MRE} = \\frac{1}{N}\\sum_{i=1}^N\\frac{|\\mathrm{d}_i-\\widehat{\\mathrm{d}_i}|}{\\mathrm{d}_i},\n\\end{equation}\nwhere $\\mathrm{d}_i$ and $\\widehat{\\mathrm{d}_i}$ are the ground truth depth and inferred depth respectively. A lower $\\mathrm{MRE}$ indicates a better dense depth prediction performance.\n\n\\item \\textbf{Bad pixel ratio} (BPR) measures the percentage of erroneous positions in total, where a depth prediction result is determined as erroneous if the absolute depth prediction error is beyond a given threshold $\\mathrm{d}_{th}$. In our experiment, we set the bad pixel threshold as $\\mathrm{d}_{th} = 3$ meters in VO dataset. A lower bad pixel ratio indicates a better depth prediction results. \n\n\\item \\textbf{Mean absolute error} (MAE) is defined as:\n\\begin{equation}\n\\mathrm{e}_{MAE} = \\frac{1}{N}\\sum_{i=1}^N|\\mathrm{d}_i-\\widehat{\\mathrm{d}_i}|,\n\\end{equation}\nwhere $d_i$ and $\\widehat{\\mathrm{d}_i}$ are the ground truth depth and depth prediction respectively. A lower mean absolute error indicates a better dense depth prediction performance achieved. It also indicates the average depth estimation error in meters.\n\\end{compactenum}\nBad pixel ratio, mean relative error and mean absolute error measure different statistics of the dense depth prediction results, which jointly evaluate the prediction performance.\n\n\\subsection{Experiment Results}\nOur quantitative results are shown in Table~\\ref{kittivo}. The piece-wise planar method outperforms all the other methods. However, since our goal is to generate both useful and visual pleasant 3D point cloud, we also provide qualitative results as shown in Fig.~\\ref{fig:p3}. For better comparison, we compare our method with several other state-of-the-art depth super-resolution methods: bilateral solver\\cite{Barron2016}, as well as our colour-guided PCA based depth interpolation method. The initial depth map used for the bilateral solver \\cite{Barron2016} was generated by a general smooth interpolation method \\cite{Garcia20101167}.\n\n\\begin{table}[!htb]\n\\centering\n\\caption[Evaluation on the KITTI VO dataset.]{Evaluation on the KITTI VO dataset.}\n\\label{kittivo}\n\\tabcolsep=0.38cm\n\\begin{tabularx}{\\columnwidth}{c|c|c|c|c}\n\\hline\n        & Bilateral\\cite{Barron2016}  & colour PCA        & Piece-wise      & ``Cardboard\"\\\\ \\hline\nMRE(\\%)     & 7.36                        & 5.73             & \\textbf{4.87}   & 7.85 \\\\ \\hline\nBPR(\\%)     & 9.46                        & 7.51             & \\textbf{5.82}   & 8.57 \\\\ \\hline\nMAE(m)\t& 1.20                        & 1.04             & \\textbf{0.80}   & 1.29 \\\\ \\hline\n\\end{tabularx}\n\\end{table}\n\nAs we can observe, bilateral solver \\cite{Barron2016} provides over-smoothed results, large distortion can be observed in the areas with different colours. For example, in Fig.~\\ref{fig:p3}, when there are shadows on road, it tends to assign same depth to same colour areas when lacking information, therefore create stripes effects in 3D point clouds. However, in our PCA based colour guided method, with the help of high order smoothness term and the PCA bases as a global constrain, it shows some resistances to the misleading of false boundaries that introduced by the colour images. Also, it can recover the shape of cars. However, as it is a pixel-wise algorithm, it is hard to estimate every pixel with the right depth. Therefore, we can find there are holes on the road plane. This can be harmful for autonomous driving system as it creates many false road pits and\/or false obstacles.\n\nThe ``cardboard world'' algorithm, on the other hand, does not have these drawbacks. As we can see from all these results, none of the road plane was fooled by shadows or marks. For better illustration of our algorithm, in Fig.~\\ref{fig:pcVO1}, we provide the input and output of our algorithm and the road plane segmentation as well.  From top to bottom, each figure consists of the reference colour image, the input of our algorithm (LIDAR measurements and super-pixel segmentation), our recovered dense depth map and our road plane segmentation result correspondingly. \\emph{Note that the colour image is only used to generate the super-pixel segmentation and only sparse LIDAR measurements are used in the optimization.}\n\n\\begin{figure}[t]\n\\begin{center} \n \\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_0_00_L00.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_0_00_L01.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_0_00_L02.png} }\\vspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_300_00_L00.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_300_00_L01.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_300_00_L02.png} }\\vspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_620_00_L00.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_620_00_L01.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{00_620_00_L02.png} }\\vspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{02_190_00_L00.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{02_190_00_L01.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.3\\columnwidth]{02_190_00_L02.png} }\n\\caption[Comparison in coloured 3D point clouds.]{\\label{fig:p3} \\textbf{Comparison in coloured 3D point clouds.} Left column: results from \\cite{Barron2016}; Middle column: results from PCA based colour guide method; Right column: results from the ``cardboard world'' method.}\n \\end{center}\n\\end{figure}\n\n\\begin{figure}[!htp]\n\\begin{center} \n \\subfigure{\n\\includegraphics[width=0.48\\columnwidth]{Results1_00_0.png} } \\hspace{-.3cm}\n\\subfigure{\n\\includegraphics[width=0.48\\columnwidth]{Results1_00_250.png} }\n\\caption[Results of ``Cardboard world'' method.]{\\label{fig:pcVO1}\\textbf{Example results of ``Cardboard world'' method:} Top to bottom: reference colour frames, inputs of our method, recovered depth map and segmented free space (coloured with green).}\n \\end{center}\n \\vspace{-4mm}\n\\end{figure}\n\n\n\nAs we can see from the figures, all dominant road plane space (labeled by dark green) have been accurately extracted. In typical city road scenarios (i.e., Fig.~\\ref{fig:pcVO1}), our method can successfully extract cars that are parked on side road and a motorbike over 22 meters away from only 3 LIDAR points on it.\n\nTo better illustrate the advantage of our ``cardboard world'' model over the standard piece-wise planar model, we also provided quality comparison between them. As we can see from Fig.~\\ref{fig:cp1}, both methods successfully recover road plane with resistant to shadows. The quantitative results show that the CRF based method achieves better performance. However, in turns of the quality of 3D point cloud, the CRF method largely distorts the shape of cars as in (a) and (g) while the ``cardboard world'' method provides less distorted, much clean and visually look better results. \n\n\\begin{figure}[!htp]\n\\begin{center} \n\\subfigure[\\scriptsize MRE:4.9 BPR:6.1 MAE:0.846 ]{\n\\includegraphics[width=0.48\\columnwidth]{pc00_L00.png} } \\hspace{-.3cm}\n \\subfigure[\\scriptsize MRE:6.5 BPR:8.7 MAE:1.041]{\n\\includegraphics[width=0.48\\columnwidth]{pc00_L01.png} }\\vspace{-.3cm}\n\\subfigure[\\scriptsize MRE:5.2 BPR:4.2 MAE:0.583]{\n\\includegraphics[width=0.48\\columnwidth]{pc03_L00.png} } \\hspace{-.3cm}\n \\subfigure[\\scriptsize MRE:9.5 BPR:6.2 MAE:1.060]{\n\\includegraphics[width=0.48\\columnwidth]{pc03_L01.png} }\n \\caption[CRF based method \\vs ``Cardboard world'' method.]{\\label{fig:cp1} Comparison between CRF based method (left) and ``cardboard world'' method (right).}\n \\end{center}\n  \\vspace{-4mm}\n\\end{figure}\n\n\\subsection{Processing time}\nUnlike pixel-level colour guided methods where the processing time largely depends on the input image resolution, the processing time of our method depends on the number of superpixels, the number of particles and the number of parameters to optimize. The higher the numbers, the longer the processing time is. In piece-wise planar model based solution, when we use 800 superpixels, 10 particles for each superpixel, the running time is about 10 seconds on average. However, in the more constrained case, our ``cardboard world'' method with more superpixels only needs 1 seconds on average. \n\n\\section{Conclusion}\nThis paper aims at tackling the challenging task of predicting dense depth maps from very sparse measurements, we have proposed two different methods by exploits various local and global constraints inside the problem. The first method is based on the piecewise planar model of the scene, where dense depth prediction is reformulated as the optimization of planar parameters. The second method enforces strong regularization on the scene model to exploit the structural information in outdoor traffic scenes, which is more suitable for autonomous driving tasks. Unlike existing depth super-resolution methods that can be easily misled by marks or shadows on road, our methods inherently resist to these false guidance. Experimental results on the KITTI VO dataset show that our methods can efficiently recover dense depth map from less than $1000$ LIDAR points without losing important information for autonomous driving, i.e., obstacles on road, or creating false obstacles that may misleading self-driving vehicles. In future, we plan to exploit the temporal information in constraining the dense depth maps.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nLet $K$ be a commutative ring and $A$ an algebra over $K$ which we assume to be $K$-projective. The \\textit{Hochschild cohomology algebra of $A$} (\\textit{over} $K$) can be expressed as $\\operatorname{HH}\\nolimits^\\ast(A) = \\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,A)$, where $A^\\mathrm{ev} = A \\otimes_K A^\\mathrm{op}$ is the \\textit{enveloping algebra of $A$ over $K$}, over which $A$ naturally becomes a left module. The cup product $\\smallsmile$ on Yoneda extensions, see \\cite{CaEi56} and \\cite{Yo54}, turns $\\operatorname{HH}\\nolimits^\\ast(A)$ into a graded commutative algebra; that is, a $\\mathbb Z$-graded algebra such that $\\alpha \\smallsmile \\beta = (-1)^{mn} \\beta \\smallsmile \\alpha$ for homogeneous elements $\\alpha$ and $\\beta$ of degrees $\\abs{\\alpha} = m$ and $\\abs{\\beta} = n$. The multiplicative structure on $\\operatorname{HH}\\nolimits^\\ast(\\Lambda)$ has been used in \\cite{EHTSS04} and \\cite{SnSo04} (see also \\cite{Sn09} and \\cite{PSS14}) to develop a support variety theory for Artin algebras $\\Lambda$, analogous to the classical ones for group (see \\cite{Ev91}) and cocommutative Hopf algebras. \n\nHowever, the graded commutative product is just part of richer structure on $\\operatorname{HH}\\nolimits^\\ast(A)$, for, in 1963, Gerstenhaber discovered a graded Lie bracket,\n$$\n\\{-,-\\}_A \\colon \\operatorname{HH}\\nolimits^{m}(A) \\times \\operatorname{HH}\\nolimits^n(A) \\longrightarrow \\operatorname{HH}\\nolimits^{m+n-1}(A),\n$$\nlowering the degree by $1$ and a (divided) squaring map,\n$$\nsq_A \\colon \\operatorname{HH}\\nolimits^{2n}(A) \\longrightarrow \\operatorname{HH}\\nolimits^{4n-1}(A),\n$$\nwhich takes over the role of the map $\\alpha \\mapsto 2^{-1}\\{\\alpha,\\alpha\\}$ if $2 \\notin K^\\times$. The bracket $\\{-,-\\}_A$ is compatible with $\\smallsmile$ in that it acts on $\\operatorname{HH}\\nolimits^\\ast(A)$ through graded derivations, and a derived invariant as shown in \\cite{Ke04} (compare with \\cite{He14b} as well). In the initial article \\cite{Ge63}, see also \\cite{Ge68}, Gerstenhaber presented $\\{-,-\\}_A$ as a tool to study and classify deformations of $A$. However, by theorems such as the one of Hochschild-Kostant-Rosenberg (see \\cite{HKR62}), Gerstenhaber's bracket found its way into the realm of differential and Poisson geometry, and other branches of mathematics. See \\cite{Ko03}, \\cite{KoSo09} and \\cite{Xu94} for additional applications.\n\\medskip\n\nIt is, of course, a fundamental question whether, or when, an exact functor $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ gives rise to a graded homomorphism $\\operatorname{HH}\\nolimits^\\ast(A) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(B)$ taking the strict Gerstenhaber structure on the source to the one on the target. Significant progress in this re\\-gard was initiated in \\cite{NeRe96}, \\cite{Re86} and \\cite{Sch98} by taking homotopy theoretical properties of \\textit{extension categories} $\\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ into account, leading to a description of the bracket $\\{-,-\\}_A$ in terms of Yoneda extensions; see also \\cite{Bu11} for some adaptations of those methods in the setting of triangulated categories. In \\cite{He14b}, we formulated and investigated a generalisation of the construction presented in \\cite{Sch98}, leading to the insight, that many monoidal functors $(\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}), \\otimes_A, A) \\rightarrow (\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ give rise to very well-behaved maps between Hochschild cohomology algebras. This is a powerful tool that we will instrument at various stages of this article, as it will allow us to study the Lie bracket in Hochschild cohomology by means of homological algebra.\n\\medskip\n\nOur first main result involves \\textit{recollements of abelian categories}, or more specifically, \\textit{recollements of module categories} which are the analogue of split short exact sequences taken in the ``category'' of abelian categories. As such, a recollement $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ between abelian categories ${\\mathsf A}$, ${\\mathsf B}$ and ${\\mathsf C}$ thus comes with exact functors ${\\mathsf A} \\rightarrow {\\mathsf B} \\rightarrow {\\mathsf C}$, which admit ``nice'' left and right adjoints. Recollements of abelian categories first appeared in the famous article of Be{\\u\\i}linson--Bernstein--Deligne on perverse sheaves, see \\cite{BBD82}, wherein they were obtained from their triangulated analogues, i.e., recollements of triangulated categories. Since then, recollements have been studied by various authors, with different backgrounds; see e.g. \\cite{ClPaSc88} and \\cite{Ku94}. Note that frequently a recollement of abelian categories induces such of triangulated categories, by deriving the whole diagram.\n\nIf the categories ${\\mathsf A}$, ${\\mathsf B}$ and ${\\mathsf C}$ in $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ are module categories over $K$-algebras $A$, $B$ and $C$, with $K$-linear functors in between them, we call $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ a \\textit{$K$-linear recollement of module categories}, and denote it by $\\mathcal R(A,B,C)$. The theorem in this context predicts that -- under mild assumptions -- a recollement of module categories gives rise to a map between Hochschild cohomology algebras that preserve all of the rich structure that was mentioned above. It makes use of recent results by Psaroudakis--Vit{\\'o}ria; see \\cite{PsVi14}.\n\n\\begin{Thm1}[$=$ Proposition \\ref{prop:envrecoll} \\& Theorem \\ref{thm:mainthm}] Let $\\mathcal R = \\mathcal R(A,B,C)$ be a $K$-linear recollement of module categories. Assume that $B$ is projective as a $K$-module. Further, denote by $j$ the functor $\\operatorname{\\mathsf{Mod}}\\nolimits(B) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C)$ in $\\mathcal R$ and by $(-)^\\vee$ the functor $\\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(-,B)$. \n\n\\begin{enumerate}[\\rm(1)]\n\\item The functor $j^\\mathrm{ev}(-) = j(B) \\otimes_B (-) \\otimes_B j(B)^\\vee$ determines a $K$-linear recollement $\\mathcal R^\\mathrm{ev} = \\mathcal R(\\operatorname{Ker}\\nolimits(j^\\mathrm{ev}), \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev}))$. Moreover, the image of $B$ under $j^\\mathrm{ev} \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ is $K$-projective and canonically isomorphic to $C$ $($as a $C^\\mathrm{ev}$-module$)$. \n\\item Under the additional assumption that \n$$\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0\\,,\n$$\nthe graded $K$-algebra homomorphism\n$$\nj^\\mathrm{ev}_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\xrightarrow{\\ j^\\mathrm{ev} \\ } \\operatorname{Ext}\\nolimits^\\ast_{C^\\mathrm{ev}}(j^\\mathrm{ev}(B), j^\\mathrm{ev}(B)) \\xrightarrow{\\ \\sim \\ } \\operatorname{HH}\\nolimits^\\ast(C)\n$$\nis a homomorphism of strict Gerstenhaber algebras, i.e., the diagrams\n$$\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^m(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} \\ar[d]_-{j^\\mathrm{ev}_m \\times j^\\mathrm{ev}_n} & \\operatorname{HH}\\nolimits^{m+n-1}(B) \\ar[d]^-{j^\\mathrm{ev}_{m + n -1}} \\\\\n\\operatorname{HH}\\nolimits^{m}(C) \\times \\operatorname{HH}\\nolimits^n(C) \\ar[r]^-{\\{-,-\\}_C} & \\operatorname{HH}\\nolimits^{m+n-1}(C)\n}\n\\ \\quad\\quad \\\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^{2n}(B) \\ar[r]^{sq_B} \\ar[d]_-{j^\\mathrm{ev}_{2n}} & \\operatorname{HH}\\nolimits^{4n-1}(B) \\ar[d]^-{j^\\mathrm{ev}_{4n-1}}\\\\\n\\operatorname{HH}\\nolimits^{2n}(C) \\ar[r]^{sq_C} & \\operatorname{HH}\\nolimits^{4n-1}(C)\n}\n$$\ncommute.\n\\end{enumerate}\n\\end{Thm1}\n\nIf $\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0$, one may want to call the functor $j: \\operatorname{\\mathsf{Mod}}\\nolimits(B) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C)$, or the recollement $\\mathcal R(A,B,C)$ that it belongs to, \\textit{pseudoflat}, a terminology that has been established in \\cite{Bu98} for ring homomorphisms $R \\rightarrow S$ with $\\operatorname{Tor}\\nolimits^R_1(S,S) = 0$.\n\\medskip\n\nFrequently, the functor $\\operatorname{\\mathsf{Mod}}\\nolimits(A) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B)$ in a recollement $\\mathcal R(A,B,C)$ of module categories is given by the restriction along a \\textit{homological epimorphism} $\\pi\\colon B \\rightarrow A$, as introduced and studied by Geigle--Lenzing in \\cite{GeLe91}. Such a homological epimorphism $\\pi$ gives rise to a homological epimorphism $\\pi^\\mathrm{ev} \\colon B^\\mathrm{ev} \\rightarrow A^\\mathrm{ev}$, and the left adjoint to the associated restriction functor $\\pi^\\mathrm{ev}_\\star\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ has notable properties.\n\n\\begin{Thm2}[= Theorem \\ref{thm:homepi}]\nLet $\\pi\\colon B \\rightarrow A$ be a $K$-linear homological epimorphism between $K$-projective $K$-algebras. Then the functor\n$$\nA \\otimes_B (-) \\otimes_B A \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})\n$$\nmaps $B$ to $A$ and, moreover, induces a homomorphism\n$$\n\\pi_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) = \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B) \\longrightarrow \\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,A) = \\operatorname{HH}\\nolimits^\\ast(A)\n$$\nof strict Gerstenhaber algebras, that is, $\\pi_\\ast$ is a graded $K$-algebra homomorphism and the diagrams\n$$\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^m(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} \\ar[d]_-{\\pi_m \\times \\pi_n} & \\operatorname{HH}\\nolimits^{m+n-1}(B) \\ar[d]^-{\\pi_{m + n -1}} \\\\\n\\operatorname{HH}\\nolimits^{m}(A) \\times \\operatorname{HH}\\nolimits^n(A) \\ar[r]^-{\\{-,-\\}_A} & \\operatorname{HH}\\nolimits^{m+n-1}(A)\n}\n\\ \\quad\\quad \\\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^{2n}(B) \\ar[r]^{sq_B} \\ar[d]_-{\\pi_{2n}} & \\operatorname{HH}\\nolimits^{4n-1}(B) \\ar[d]^-{\\pi_{4n-1}}\\\\\n\\operatorname{HH}\\nolimits^{2n}(A) \\ar[r]^{sq_A} & \\operatorname{HH}\\nolimits^{4n-1}(A)\n}\n$$\ncommute.\n\\end{Thm2}\n\nAs the functor $A \\otimes_B (-) \\otimes_B A \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$ is, in general, not exact, the map $\\pi_\\ast$ will not be given by applying $A \\otimes_B (-) \\otimes_B A$ to extensions directly. However, one may pass to a suitable exact subcategory ${\\mathsf C}$ of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ on which $A \\otimes_B (-) \\otimes_B A$ is exact, and which is, in the sense of Paragraph \\ref{nn:extclosed}, entirely extension closed in that extension groups in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ may be computed by extensions in ${\\mathsf C}$. This is indeed some sort of a derived functor, as we will notice along the way.\n\\medskip\n\nIn the context of triangular algebras, one-point-extensions and stratifying ideals, various long exact sequences for Hochschild cohomology algebras have been established; see, for instance, \\cite{BeGu04, Bu03, Ci00, GoGo01, GrMaSn03, GrSo02, Ha14, Ha89, KoNa09, MiPl00}. The graded maps within those sequences are known to respect the multiplicative structure. Theorems 1 and 2 may be applied, in oder to realise that even more structure is preserved. We take care of a long exact sequence of Green--Solberg first; see \\cite{GrSo02}. Let $B$ be a $K$-algebra.\n\n\\begin{Thm3}[= Theorem \\ref{thm:greensol}]\nLet $e \\in B$ be an idempotent, $e' = 1 - e$ its complementary idempotent and $C = eBe$, $C' = e'Be'$. If\n$$\n\\operatorname{Tor}\\nolimits^{C}_i(Be,eB) = 0 = \\operatorname{Tor}\\nolimits^{C'}_i(Be',e'B) \\quad \\text{$($for all $i > 0)$},\n$$\nthen there is a long exact sequence\n$$\n\\xymatrix@C=14.5pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(B,B) \\ar[r]^-{g_n} & \\operatorname{Ext}\\nolimits^n_{C^\\mathrm{ev}}(C,C) \\oplus \\operatorname{Ext}\\nolimits^n_{(C')^\\mathrm{ev}}(C', C') \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(\\Omega^1_B,B) \\ar[r] & \\cdots\n}\n$$\ninduced from the canonical short exact sequence\n$$\n\\xymatrix@C=14pt{\n0 \\ar[r] & {\\Omega}^1_B \\ar[r] & \\displaystyle (Be \\otimes_{C} eB) \\oplus (Be' \\otimes_{C'} e'B) \\ar[r] & B \\ar[r] & 0\n}.\n$$\nIf, moreover, $B$ is $K$-projective, then the map $g_\\ast\\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(eBe) \\times \\operatorname{HH}\\nolimits^\\ast(e'Be')$ is a homomorphism of strict Gerstenhaber algebras.\n\\end{Thm3}\n\nNote that by specialising to $B = \\left(\\begin{smallmatrix} R & M\\\\ 0 & K\\end{smallmatrix}\\right)$ for a $K$-algebra $R$ and an $R$-module $M$, Theorem 3 recovers the long exact sequence for one-point-extensions introduced by Happel in \\cite{Ha89}. Thus the maps in Happel's sequence preserve the bracket as well. The exact sequences in the upcoming statement are due to Koenig--Nagase; see \\cite{KoNa09}. Recall that an ideal $I \\subseteq B$ is called \\textit{stratifying}, if there is an idempotent element $e \\in B$ with $I = BeB$, such that $\\operatorname{Tor}\\nolimits^{eBe}_i(Be,eB) = 0$ for $i > 0$ and the multiplication map $Be \\otimes_{eBe} eB \\rightarrow BeB = I$ is an isomorphism. In this situation, the canonical surjection $\\pi\\colon B \\rightarrow B\/I$ is a homological epimorphism (cf.\\,Lemma \\ref{lem:strattorvan}).\n\n\\begin{Thm4}[$\\subseteq$ Theorem \\ref{thm:koenignagase}]\nLet $I \\subseteq B$ be a \\emph{stratifying ideal}, and $A = B\/I$. Assume further that $B$ and $A$ are $K$-projective. Then the following hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item The short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$ gives rise to a long exact sequence\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(B,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{k_n} & \\operatorname{HH}\\nolimits^n(A) \\ar[r] & \\operatorname{Ext}\\nolimits^{n+1}_{B^\\mathrm{ev}}(B,I) \\ar[r] & \\cdots\n}\n$$\nin Hochschild cohomology. The map $k_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(A)$ is a homomorphism of strict Gerstenhaber algebras.\n\n\\item The short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$ gives rise to a long exact sequence\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^n(A,B) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{l_n} & \\operatorname{HH}\\nolimits^n(eBe) \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^{n+1}(A,B) \\ar[r] & \\cdots\n}\n$$\nin Hochschild cohomology. The map $l_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(eBe)$ is a homomorphism of strict Gerstenhaber algebras. \n\\end{enumerate}\n\\end{Thm4}\n\nRecall that for a subset $S \\subseteq \\operatorname{HH}\\nolimits^\\ast(A)$ of homogeneous elements, the (\\textit{weak}) \\textit{Gerstenhaber ideal} generated by $S$ is the smallest homogeneous ideal $G(S)$ of $\\operatorname{HH}\\nolimits^\\ast(A)$ that contains $S$ and which satisfies\n$$\n\\{ \\gamma, \\gamma' \\}_A \\in G(S) \\quad \\text{(for all $\\gamma, \\gamma' \\in G(S)$).}\n$$\nIf $\\mathrm{char}(K) = 2$ or $2 \\in K^\\times$, and if $S = N$ is the set of all homogeneous nilpotent elements, then the quotient (Gerstenhaber) algebra $\\operatorname{HH}\\nolimits^\\ast_G(A) = \\operatorname{HH}\\nolimits^\\ast(A) \/ G(N)$ is commutative. In light of Theorems 3 and 4, we end by asking the following question.\n\n\\begin{Quest}[= Question \\ref{conj:snashsol}]\nLet $A$ be a finite dimensional algebra over a field $K$. If $N$ denotes the set of all homogeneous nilpotent elements in $\\operatorname{HH}\\nolimits^\\ast(A)$, is the commutative $K$-algebra\n$$\n\\operatorname{HH}\\nolimits^\\ast_G(A) = \\operatorname{HH}\\nolimits^\\ast(A) \/ G(N)\n$$\nfinitely generated? More specifically, can we use (any of) the long exact sequences to deduce that for the one-point-extension $B = \\Gamma[M] = \\left( \\begin{smallmatrix} K (\\mathbb Z_2 \\times \\mathbb Z_2) & K\\mathbb Z_2\\\\ 0 & K \\end{smallmatrix} \\right)$ of $\\Gamma = K(\\mathbb Z_2 \\times \\mathbb Z_2)$ by $M = K\\mathbb Z_2$, the quotient algebra\n$$\n\\operatorname{HH}\\nolimits^\\ast_G(B) = \\operatorname{HH}\\nolimits^\\ast(B) \/ G(N)\n$$\nis finitely generated (if $\\mathrm{char}(K) = 2$)?\n\\end{Quest}\n\nNote that $B = \\Gamma[M]$ is the initial counterexample to the finite generation conjecture of Snashall--Solberg, see \\cite{SnSo04}, which was first mention in \\cite{Xu08}, and studied in further detail in \\cite{Sn09} thereafter. It remains to be checked whether, or when, the algebra $\\operatorname{HH}\\nolimits^\\ast_G(A)$ may be used to establish a fruitful support variety theory on a sufficiently broad class of $A$-modules. This question shall be studied in future work.\n\\medskip\n\nThis article is organised as follows. In Sections \\ref{sec:prereq} we will built up the necessary foundations on exact and monoidal categories. Afterwards, in Section \\ref{sec:gerstHH}, we will give a recap on the theory of Hochschild cohomology, and the definition of the Lie bracket associated with it. We are also going to briefly discuss Schwede's exact sequence interpretation of the Lie bracket. Section \\ref{sec:recoll} concerns itself with recollements of module categories and provides all the detail to state and prove Theorem 1. In Sections \\ref{sec:buchweitz} and \\ref{sec:greensol} we apply Theorem 1 to long exact sequences due to Buchweitz and Green--Solberg, thus covering a proof of Theorem 3. We make further observations in the context of one-point-extensions and Happel's long exact sequence. Section \\ref{sec:homepi} sets up the necessary peripherals on homological epimorphisms in oder to formulate and prove Theorem 2. The latter theorem, along with Theorem 1, will be used in Section \\ref{sec:koenagase} to improve results relating the long exact sequences due to Koenig--Nagase, i.e., to obtain Theorem 4. Finally, in Section \\ref{sec:snashsol}, we try to motivate how our results may be used to formulate a refined version of the finite generation conjecture of Snashall--Solberg.\n\\medskip\n\nThe present paper is a vastly extended version of a talk given during the \\textit{International Conference on Representations of Algebras 2014} in Sanya City, China. I would like to thank L.\\,Angeleri-H\\\"ugel, \\O.\\,Solberg and J.\\,Vit\\'oria for awakening my interest in the theory of recollements and homological epimorphisms through most intriguing presentations at various occasions throughout this year. Many thanks also go to C.\\,Psaroudakis for valuable discussions on the topic. Finally, I would like to thank the referees for their careful reading and numerous comments. The research that led to this paper was supported by the project ``Triangulated categories in algebra'' (Norwegian Research Council project NFR 221893).\n\n\n\\section{Prerequisites on exact and monoidal categories}\\label{sec:prereq}\n\n\\begin{nn}\nLet us recall the notions of exact and monoidal categories and structure preserving functors (exact and monoidal functors) between them. For further details on exact categories, we refer to \\cite{Ke90} and \\cite{Qu72}, whereas the textbooks \\cite{AgMa10} and \\cite{MaL98} provide background material on monoidal categories. In the following section, and in fact for the entire article, we fix a commutative ring $K$.\n\\end{nn}\n\n\\begin{nn}\nA full subcategory $\\mathsf U$ of an abelian category ${\\mathsf A}$ is \\textit{extension closed in ${\\mathsf A}$} if $0 \\rightarrow U'' \\rightarrow U \\rightarrow U' \\rightarrow 0$ is an exact sequence in ${\\mathsf A}$ with $U', U'' \\in {\\mathsf U}$, then also $U \\in \\mathsf U$. An \\textit{exact} $K$-category is a pair $({\\mathsf C}, i_{\\mathsf C})$ consisting of an additive $K$-category ${\\mathsf C}$ and a full and faithful embedding $i_{\\mathsf C}\\colon {\\mathsf C} \\rightarrow {\\mathsf A}_{\\mathsf C}$ into an abelian $K$-category ${\\mathsf A}_{\\mathsf C}$, such that the essential image\n$$\n\\Im(i_{\\mathsf C}) = i_{\\mathsf C} {\\mathsf C} = \\{ A \\in {\\mathsf A}_{\\mathsf C} \\mid i_{\\mathsf C}(C) \\cong A \\text{ for some $C \\in {\\mathsf C}$}\\}\n$$\nof $i_{\\mathsf C}$ is an extension closed subcategory of ${\\mathsf A}_{\\mathsf C}$. Exact categories admit a sensible notion of exact sequences. Let ${\\mathsf C} = ({\\mathsf C}, i_{\\mathsf C})$ be an exact $K$-category. A sequence $0 \\rightarrow C'' \\rightarrow C \\rightarrow C' \\rightarrow 0$ is an \\textit{admissible short exact sequence in ${\\mathsf C}$}, if its image $0 \\rightarrow i_{\\mathsf C}(C'') \\rightarrow i_{\\mathsf C}(C) \\rightarrow i_{\\mathsf C}(C') \\rightarrow 0$ under $i_{\\mathsf C}$ is exact in ${\\mathsf A}_{\\mathsf C}$. Given such an admissible short exact sequence $0 \\rightarrow C'' \\rightarrow C \\rightarrow C' \\rightarrow 0$, the morphism $C'' \\rightarrow C$ is called an \\textit{admissible monomorphism}, whereas $C \\rightarrow C'$ is an \\textit{admissible epimorphism}. The class of admissible short exact sequences is closed under taking isomorphisms and direct sums (in the category of chain complexes over ${\\mathsf C}$).\n\\end{nn}\n\n\\begin{nn}\nLet $({\\mathsf C},i_{\\mathsf C})$ be an exact $K$-category. We say that ${\\mathsf C}$ is\n\\begin{enumerate}[\\rm(1)]\n\\item \\textit{closed under kernels of epimorphisms} if the functor $i_{\\mathsf C} \\colon {\\mathsf C} \\rightarrow {\\mathsf A}_{\\mathsf C}$ detects admissible epimorphisms, that is, if\n$$\n\\text{$f \\in {\\mathsf C}$ is an admissible epimorphism} \\quad \\Longleftrightarrow \\quad \\text{$i_{\\mathsf C}(f)$ is an epimorphism in ${\\mathsf A}_{\\mathsf C}$}.\n$$\n\\item \\textit{closed under cokernels of monomorphisms} if the functor $i_{\\mathsf C} \\colon {\\mathsf C} \\rightarrow {\\mathsf A}_{\\mathsf C}$ detects admissible monomorphisms, that is, if\n$$\n\\text{$f \\in {\\mathsf C}$ is an admissible monomorphism} \\quad \\Longleftrightarrow \\quad \\text{$i_{\\mathsf C}(f)$ is a monomorphism in ${\\mathsf A}_{\\mathsf C}$}.\n$$\n\\end{enumerate}\n\\end{nn}\n\n\\begin{nn}\nEach exact $K$-category $({\\mathsf C}, i_{\\mathsf C})$ is closed under taking pushouts along admissible monomorphisms and pullbacks along admissible epimorphisms. Assume that $({\\mathsf D}, i_{\\mathsf D})$ is another exact $K$-category, and let $\\mathfrak X\\colon {\\mathsf C} \\rightarrow {\\mathsf D}$ be an \\textit{exact} $K$-linear functor, that is, it takes admissible short exact sequences in ${\\mathsf C}$ to admissible short exact sequences in ${\\mathsf D}$. Each such functor preserves pushouts along admissible monomorphisms and pullbacks along admissible epimorphisms.\n\\end{nn}\n\n\\begin{lem}\\label{lem:smallexactfunc}\nLet ${\\mathsf C}$ and ${\\mathsf D}$ be exact $K$-categories, with corresponding abelian categories ${\\mathsf A}_{\\mathsf C}$ and ${\\mathsf A}_{\\mathsf D}$. Let $\\mathfrak X \\colon {\\mathsf C} \\rightarrow {\\mathsf A}_{\\mathsf D}$ be an exact and $K$-linear functor. Then the full subcategory ${\\mathsf C}' \\subseteq {\\mathsf C}$ defined by \n$$\nX \\in {\\mathsf C} \\text{ belongs to ${\\mathsf C}'$} \\quad \\Longleftrightarrow \\quad \\mathfrak X(X) \\text{ belongs to $i_{\\mathsf D} {\\mathsf D}$}\n$$\n$($that is, ${\\mathsf C}' = \\mathfrak X^{-1}(i_{\\mathsf D} {\\mathsf D}))$ is an exact $K$-category $($along with the obvious inclusion ${\\mathsf C}' \\hookrightarrow {\\mathsf C} \\hookrightarrow {\\mathsf A}_{\\mathsf C})$ and $\\mathfrak X$ restricts to an exact functor $\\mathfrak X \\mathord{\\upharpoonright}_{{\\mathsf C}'}\\colon {\\mathsf C}' \\rightarrow i_{\\mathsf D} {\\mathsf D}$.\n\\end{lem}\n\n\\begin{proof}\nAdditivity and $K$-linearity are obvious. If $0 \\rightarrow i_{\\mathsf C} (X'') \\rightarrow X \\rightarrow i_{\\mathsf C} (X') \\rightarrow 0$ is an exact sequence in ${\\mathsf A}_{\\mathsf C}$ with $X', X'' \\in {\\mathsf C}'$, it follows that $X \\in i_{\\mathsf C} {\\mathsf C}$ by definition; say $X \\cong i_{\\mathsf C}(Y)$ for some object $Y \\in {\\mathsf C}$. We have thus an admissible short exact sequence $0 \\rightarrow X'' \\rightarrow Y \\rightarrow X' \\rightarrow 0$ in ${\\mathsf C}$, and hence, after applying $\\mathfrak X$ to it, such in ${\\mathsf D}$. As $\\mathfrak X(X')$ and $\\mathfrak X(X'')$ belong to $i_{\\mathsf D} {\\mathsf D}$, so must $\\mathfrak X(Y)$. Therefore $Y$ belongs to ${\\mathsf C}'$, as required.\n\\end{proof}\n\n\\begin{nn}\nFor an integer $n \\geqslant 1$, a sequence\n$$\nS \\quad \\equiv \\quad 0 \\longrightarrow C'' \\longrightarrow C_{n-1} \\longrightarrow \\cdots \\longrightarrow C_0 \\longrightarrow C' \\longrightarrow 0\n$$\nof morphisms in an exact $K$-category $({\\mathsf C}, i_{\\mathsf C})$ is called an \\textit{admissible $n$-extension} (\\textit{of $C'$ by $C''$}) in case $i_{\\mathsf C} S$ is exact in ${\\mathsf A}_{\\mathsf C}$, and $\\operatorname{Ker}\\nolimits(i_{\\mathsf C}(C_0 \\rightarrow C'))$ and $\\operatorname{Ker}\\nolimits(i_{\\mathsf C}(C_k \\rightarrow C_{k-1}))$ belong to $i_{\\mathsf C} {\\mathsf C}$ for all $k = 1, \\dots, n-1$. As for abelian categories, we let $\\operatorname{Ext}\\nolimits^n_{\\mathsf C}(C',C'')$ be the set of $n$-extensions of $C'$ by $C''$ modulo the equivalence relation generated by \\textit{morphisms} of admissible $n$-extensions.\n\\end{nn}\n\n\\begin{nn}\\label{nn:extclosed}\nLet ${\\mathsf C}$ be an additive subcategory of an abelian $K$-category ${\\mathsf A}$ and let $n \\geqslant 1$ be an integer. The category ${\\mathsf C}$ is \\textit{$n$-extension closed in ${\\mathsf A}$}, if the maps\n$$\n\\operatorname{Ext}\\nolimits^i_{\\mathsf C}(C,D) \\longrightarrow \\operatorname{Ext}\\nolimits^i_{\\mathsf A}(C,D) \\quad \\text{(for all $0 \\leqslant i \\leqslant n$)}\n$$\ninduced by the inclusion functor ${\\mathsf C} \\rightarrow {\\mathsf A}$ are isomorphisms for every pair $C,D$ of objects in ${\\mathsf C}$. If ${\\mathsf C}$ is $n$-extension closed for all $n \\in \\mathbb N$, then we say that ${\\mathsf C}$ is \\textit{entirely extension closed in ${\\mathsf A}$}. Note that ${\\mathsf C}$ being $1$-extension closed is equivalent to $({\\mathsf C}, {\\mathsf C} \\hookrightarrow {\\mathsf A})$ being an exact category. One has the following criterion as to when ${\\mathsf C}$ is entirely extension closed in ${\\mathsf A}$.\n\\end{nn}\n\n\\begin{prop}[see {\\cite[Prop.\\,2.4.6]{He14b}}]\\label{prop:entireextclo}\nAssume that ${\\mathsf C}$ is $1$-extension closed in ${\\mathsf A}$ and that ${\\mathsf A}$ has enough projective objects. If $\\operatorname{\\mathsf{Proj}}\\nolimits({\\mathsf A}) \\subseteq {\\mathsf C}$ and ${\\mathsf C}$ is, when viewed as an exact category, closed under kernels of epimorphisms, then ${\\mathsf C}$ is entirely extension closed in ${\\mathsf A}$.\n\\end{prop}\n\n\\begin{rem}\nIndependently, Coulembier--Mazorchuk and Psaroudakis introduced and studied entirely extension closed (abelian) subcategories in the context of the representation theory of algebraic groups and associative algebras (see \\cite{CoMa14a, CoMa14b} and \\cite{Ps14}). However, their terminology differs from ours. Coulembier--Mazorchuk refer to those subcategories as \\textit{extension full subcategories}, while Psaroudakis investigated \\textit{homological embeddings}, i.e., exact functors between abelian categories giving rise to isomorphisms on $\\operatorname{Ext}\\nolimits$-groups.\n\\end{rem}\n\nLet us turn to monoidal categories.\n\n\\begin{nn}\nRecall that a \\textit{monoidal category} is a 6-tuple $(\\mathsf C, \\otimes, \\mathbbm 1, \\alpha, \\lambda, \\varrho)$, where $\\mathsf {\\mathsf C}$ is a category, $\\otimes\\colon {\\mathsf C} \\times {\\mathsf C} \\rightarrow {\\mathsf C}$ is a functor, $\\mathbbm 1$ is an object in ${\\mathsf C}$, and\n\\begin{gather*}\n\\alpha\\colon - \\otimes (- \\otimes-) \\longrightarrow (- \\otimes -) \\otimes - \\ , \\\\\n\\lambda\\colon \\mathbbm 1 \\otimes - \\longrightarrow \\operatorname{Id}\\nolimits_{{\\mathsf C}} \\ , \\\\\n\\varrho\\colon - \\otimes \\mathbbm 1 \\longrightarrow \\operatorname{Id}\\nolimits_{{\\mathsf C}}\n\\end{gather*}\nare isomorphisms of functors such that, for all objects $W, X,Y,Z$ in ${\\mathsf C}$,\n$$\n\\xymatrix@C=40pt{\nW \\otimes (X \\otimes (Y \\otimes Z)) \\ar[r]^-{\\alpha_{W,X,Y \\otimes Z}} \\ar[d]_-{W \\otimes \\alpha_{X,Y,Z}} & (W\\otimes X) \\otimes (Y \\otimes Z) \\ar[r]^{\\alpha_{W \\otimes X, Y, Z}} & ((W \\otimes X) \\otimes Y) \\otimes Z \\ \\ \\\\\nW \\otimes ((X \\otimes Y) \\otimes Z) \\ar[rr]^-{\\alpha_{W,X \\otimes Y,Z}} & & (W \\otimes (X \\otimes Y)) \\otimes Z \\ar[u]_{\\alpha_{W,X,Y} \\otimes Z} \\ ,\n}\n$$\ncommutes and $(\\varrho_X \\otimes Y) \\circ \\alpha_{X,\\mathbbm 1,Y} = X \\otimes \\lambda_Y$. In this situation, $\\otimes$ is a \\textit{monoidal} (or \\textit{tensor}) \\textit{product functor for ${\\mathsf C}$} and $\\mathbbm 1$ is the \\textit{$($tensor$)$ unit} of $\\otimes$.\n\\end{nn}\n\\begin{rem}\\label{rem:opposite} Let $({\\mathsf C}, \\otimes, \\mathbbm 1, \\alpha, \\lambda, \\varrho)$ be a monoidal category.\n\\begin{enumerate}[\\rm(1)]\n\\item We will often suppress a huge part of the structure morphisms and simply write $({\\mathsf C}, \\otimes, \\mathbbm 1)$ instead of $({\\mathsf C}, \\otimes, \\mathbbm 1, \\alpha, \\lambda, \\varrho)$; if they are needed without priorly having been mentioned, we will refer to them as $\\alpha_{\\mathsf C}$, $\\lambda_{\\mathsf C}$ and $\\varrho_{\\mathsf C}$.\n\\item It follows from the axioms (cf.\\,\\cite[Prop.\\,1.1]{JoSt93}) that the following equations hold true for all $X, Y, Z \\in \\operatorname{Ob}\\nolimits {\\mathsf C}$:\n$$\n\\lambda_{\\mathbbm 1} = \\varrho_{\\mathbbm 1}, \\quad \\varrho_{X \\otimes Y} \\circ \\alpha_{X,Y,{\\mathbbm 1}} = X \\otimes \\varrho_Y, \\quad (\\lambda_X \\otimes Y) \\circ \\alpha_{{\\mathbbm 1}, X, Y} = \\lambda_{X \\otimes Y} \\, .\n$$\n\\item Note that if $(\\mathsf C, \\otimes, \\mathbbm 1, \\alpha, \\lambda, \\varrho)$ is a monoidal category, then so is $\\mathsf C^\\mathrm{op}$ together with the structure morphisms $\\alpha^{-1}$, $\\lambda^{-1}$ and $\\varrho^{-1}$.\n\\end{enumerate}\n\\end{rem}\n\\begin{nn}\nWe say that a monoidal category $({\\mathsf C}, \\otimes, \\mathbbm 1)$ is a \\textit{tensor $K$-category}, if ${\\mathsf C}$ is $K$-linear and the tensor product functor $\\otimes\\colon {\\mathsf C} \\times {\\mathsf C} \\rightarrow {\\mathsf C}$ is $K$-bilinear on morphisms, that is, it factors through the \\textit{tensor product category} ${\\mathsf C} \\otimes_K {\\mathsf C}$ which is defined as follows:\n\\begin{align*}\n\\operatorname{Ob}\\nolimits({\\mathsf C} \\otimes_K {\\mathsf C}) &{\\coloneqq} \\operatorname{Ob}\\nolimits({\\mathsf C} \\times {\\mathsf C}),\\\\\n\\operatorname{Hom}\\nolimits_{{\\mathsf C} \\otimes_K {\\mathsf C}}(\\underline{X}, \\underline{Y}) &{\\coloneqq} \\operatorname{Hom}\\nolimits_{\\mathsf C}(X_1,Y_1) \\otimes_K \\operatorname{Hom}\\nolimits_{\\mathsf C}(X_2, Y_2)\n\\end{align*}\nfor objects $\\underline{X} = (X_1, X_2)$ and $\\underline{Y} = (Y_1, Y_2)$ in ${\\mathsf C} \\times {\\mathsf C}$. \n\\end{nn}\n\\begin{nn}\\label{nn:monoclosure}\nThe triple $({\\mathsf C}, \\otimes, {\\mathbbm 1})$ is \\textit{weak exact monoidal $K$-category} if ${\\mathsf C}$ is an exact $K$-category and $({\\mathsf C}, \\otimes, {\\mathbbm 1})$ is a tensor $K$-category such that every object in ${\\mathsf C}$ is either \\textit{flat} (that is, the functors $X \\otimes - \\colon {\\mathsf C} \\rightarrow {\\mathsf C}$ are exact for all $X \\in \\operatorname{Ob}\\nolimits {\\mathsf C}$) or \\textit{coflat} (that is, the functors $- \\otimes X \\colon {\\mathsf C} \\rightarrow {\\mathsf C}$ are exact for all $X \\in \\operatorname{Ob}\\nolimits {\\mathsf C}$). A \\textit{strong exact monoidal category} is a weak exact monoidal $K$-category whose underlying exact category is closed under kernels of epimorphisms or under cokernels of monomorphisms. Note that these definitions are not quite the same as introduced in \\cite[Sec.\\,1.2]{He14b} and \\cite[Sec.\\,3.3]{He14b}, however weak\/strong exact monoidal categories in the above sense are weak\/strong exact monoidal categories in the sense of \\cite[Def.\\,1.2.6]{He14b}\/\\cite[Def.\\,3.3.3]{He14b}. For the remainder of this article, the term exact monoidal category will mostly stand for \\textit{strong} exact monoidal category, unless explicitly stated otherwise.\n\nLet ${\\mathsf U}$ be a full and extension closed subcategory of ${\\mathsf C}$ that contains ${\\mathbbm 1}$. Define $\\overline {\\mathsf U}$ to be the full subcategory of ${\\mathsf U}$ consisting of all objects $X \\in {\\mathsf U}$ such that $X \\otimes Y \\in {\\mathsf U}$ for all objects $Y \\in {\\mathsf U}$.\n\\end{nn}\n\\begin{lem}\\label{lem:monoclosure}\nKeeping the notations from above, the triple $(\\overline {\\mathsf U}, \\otimes, {\\mathbbm 1})$ is a full, additive, extension closed and monoidal subcategory of $({\\mathsf C},\\otimes, {\\mathbbm 1})$. It is closed under kernels of epimorphisms $($cokernels of monomorphisms$)$ if ${\\mathsf U}$ is closed under kernels of epimorphisms $($cokernels of monomorphisms$)$.\n\\end{lem}\n\\begin{proof}\nAdditivity follows from the additivity of $\\otimes$. If $0 \\rightarrow X'' \\rightarrow X \\rightarrow X' \\rightarrow 0$ is an admissible exact sequence, then so is\n$$\n0 \\longrightarrow X'' \\otimes Y \\longrightarrow X \\otimes Y \\longrightarrow X' \\otimes Y \\longrightarrow 0\n$$\nfor all $Y \\in {\\mathsf U}$. Therefore, if $X', X''$ are in $\\overline {\\mathsf U}$, it follows that $X \\otimes Y$ belongs tu ${\\mathsf U}$, thus, that $X$ belongs to $\\overline {\\mathsf U}$. Let $X', X'' \\in \\overline {\\mathsf U}$ and $Y \\in {\\mathsf U}$. Then, by the associativity isomorphisms,\n$$\n(X' \\otimes X'') \\otimes Y \\cong X' \\otimes (X'' \\otimes Y) \\ \\in \\ {\\mathsf U},\n$$\ni.e., $X' \\otimes X''$ is a member of $\\overline {\\mathsf U}$. Of course, ${\\mathbbm 1} \\in \\overline {\\mathsf U}$ by the unit isomorphisms.\n\\end{proof}\n\\begin{nn}\\label{def:monoidalfunc}\nWe are going to recall the definition of certain structure preserving functors between monoidal categories. Let $(\\mathsf C, \\otimes_{\\mathsf C}, \\mathbbm 1_{\\mathsf C})$ and $(\\mathsf D, \\otimes_{\\mathsf D}, \\mathbbm 1_{\\mathsf D})$ be monoidal categories. Let $\\mathfrak A\\colon \\mathsf C \\rightarrow \\mathsf D$ be a functor, and\n\\begin{align*}\n& \\phi_{X,Y}\\colon \\mathfrak A X \\otimes_{\\mathsf D} \\mathfrak A Y \\longrightarrow \\mathfrak A (X \\otimes_{\\mathsf C} Y),\\\\\n& \\psi_{X,Y}\\colon \\mathfrak A (X \\otimes_{\\mathsf C} Y) \\longrightarrow \\mathfrak A X \\otimes_{\\mathsf D} \\mathfrak A Y,\n\\end{align*}\nbe natural morphisms in ${\\mathsf D}$ (for $X,Y \\in \\operatorname{Ob}\\nolimits\\mathsf C$). Further, let $\\phi_0\\colon \\mathbbm 1_{\\mathsf D} \\rightarrow \\mathfrak A \\mathbbm 1_{\\mathsf C}$ and $\\psi_0\\colon \\mathfrak A \\mathbbm 1_{\\mathsf C} \\rightarrow \\mathbbm 1_{\\mathsf D}$ be morphisms in $\\mathsf D$.\n\nThe triple $(\\mathfrak A, \\phi, \\phi_0)$ is called an \\textit{almost strong monoidal functor} if $\\phi_0$ is invertible and the following diagrams commute for all $X,Y,Z \\in \\operatorname{Ob}\\nolimits \\mathsf C$.\n\n\\begin{equation*}\n\\xymatrix@C=40pt{\n\\mathfrak A X \\otimes_{\\mathsf D} (\\mathfrak A Y \\otimes_{\\mathsf D} \\mathfrak A Z) \\ar[r]^{\\mathfrak A X \\otimes_{\\mathsf D} \\phi_{Y,Z}} \\ar[d]_{\\alpha_{\\mathsf D}\\mathfrak A} & \\mathfrak A X\n\\otimes_{\\mathsf D} \\mathfrak A(Y \\otimes_{\\mathsf C} Z) \\ar[r]^{\\phi_{X,Y \\otimes_{\\mathsf C} Z}} & \\mathfrak A(X \\otimes_{\\mathsf C} (Y \\otimes_{\\mathsf C} Z)) \\ar[d]^{\\mathfrak A \\alpha_{\\mathsf C}}\\\\\n(\\mathfrak A X \\otimes_{\\mathsf D} \\mathfrak A Y) \\otimes_{\\mathsf D} \\mathfrak A Z \\ar[r]^-{\\phi_{X,Y} \\otimes_{\\mathsf D} \\mathfrak A Z} & \\mathfrak A (X \\otimes_{\\mathsf C} Y) \\otimes_{\\mathsf D} \\mathfrak A Z\n\\ar[r]^-{\\phi_{X \\otimes_{\\mathsf C} Y,Z}} & \\mathfrak A ((X \\otimes_{\\mathsf C} Y) \\otimes_{\\mathsf C} Z)\n}\n\\end{equation*}\n\\begin{equation*}\n\\xymatrix{\n\\mathbbm 1_{\\mathsf D} \\otimes_{\\mathsf C} \\mathfrak A X  \\ar[d]_{\\phi_0 \\otimes_{\\mathsf D} \\mathfrak A X} \\ar[r]^-{\\lambda_{{\\mathsf D}}\\mathfrak A} & \\mathfrak A X  \\\\\n\\mathfrak A \\mathbbm 1_{\\mathsf C} \\otimes_{\\mathsf D} \\mathfrak A X \\ar[r]^-{\\phi_{\\mathbbm 1_{\\mathsf C}, X}} & \\mathfrak A(\\mathbbm 1_{\\mathsf C} \\otimes_{\\mathsf C} X) \\ar[u]_{\\mathfrak A \\lambda_{{\\mathsf C}}}\n}\n\\quad\n\\xymatrix{\n\\mathfrak A X \\otimes_{\\mathsf D}  \\mathbbm 1_{\\mathsf D} \\ar[d]_{\\mathfrak A X \\otimes_{\\mathsf D} \\phi_0} \\ar[r]^-{\\varrho_{{\\mathsf D}}\\mathfrak A} & \\mathfrak A X  \\\\\n\\mathfrak A X \\otimes_{\\mathsf D} \\mathfrak A \\mathbbm 1_{\\mathsf C} \\ar[r]^-{\\phi_{X, \\mathbbm 1_{\\mathsf C}}} & \\mathfrak A(X \\otimes_{\\mathsf C} \\mathbbm 1_{\\mathsf C})\n\\ar[u]_{\\mathfrak A \\varrho_{{\\mathsf C}}}\n}\n\\end{equation*}\nThe triple $(\\mathfrak A, \\psi, \\psi_0)$ is called an \\textit{almost costrong monoidal functor} if $(\\mathfrak A^\\mathrm{op}, \\psi, \\psi_0)$ is an almost strong monoidal functor. The triple $(\\mathfrak A, \\phi, \\phi_0)$ is called a \\textit{strong monoidal functor} if it is an almost strong monoidal functor and $\\phi$ is invertible, whereas $(\\mathfrak A, \\psi, \\psi_0)$ is called a \\textit{costrong monoidal functor} if it is an almost costrong monoidal functor and $\\psi$ is invertible.\n\\end{nn}\n\nThe following two examples will reappear at later stages of this article.\n\n\\begin{exa}\nLet $A$ be a $K$-algebra which is projective when considered as a $K$-module and $A^\\mathrm{ev} = A \\otimes_K A^\\mathrm{op}$ the \\textit{enveloping algebra} of $A$ over $K$, with factorwise multiplication. Note that there are forgetful functors $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A)$ and $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{op})$. Let ${\\mathsf P}_\\lambda$ and ${\\mathsf P}_\\varrho(A)$ be the following full subcategories of $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$. A module $M \\in \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$ belongs to\n\\begin{enumerate}\n\\item ${\\mathsf P}_\\lambda(A)$ if, and only if, $M$ is projective as a left $A$-module.\n\\item ${\\mathsf P}_\\varrho(A)$ if, and only if, $M$ is projective as a right $A$-module.\n\\end{enumerate}\nBoth categories are non-empty, as they contain $A$ and $A \\otimes_K A$. As short exact sequences ending in a projective module split, ${\\mathsf P}_\\lambda(A)$ is closed under extensions and kernels of admissible epimorphisms. Summands and arbitrary direct sums of projectives are projective, so that ${\\mathsf P}_\\lambda(A)$ is closed under taking direct summands and arbitrary direct sums in $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$. It follows that ${\\mathsf P}_\\lambda(A)$ is entirely extension closed, due to Proposition \\ref{prop:entireextclo}. Moreover, if $M$ and $N$ belong to ${\\mathsf P}_\\lambda(A)$, then so does $M \\otimes_A N$, since $\\operatorname{Hom}\\nolimits_A(M \\otimes_A N, -) \\cong \\operatorname{Hom}\\nolimits_A(N, \\operatorname{Hom}\\nolimits_A(M,-))$ is an exact functor. Thus ${\\mathsf P}_\\lambda(A)$ is monoidal, and the functors $- \\otimes_A M \\colon {\\mathsf P}_\\lambda(A) \\rightarrow {\\mathsf P}_\\lambda(A)$ are exact for all $M \\in {\\mathsf P}_\\lambda(A)$, thus ${\\mathsf P}_\\lambda(A)$ is a strong exact monoidal subcategory of $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$. Likewise, the analogue statements hold true for ${\\mathsf P}_\\varrho(A)$. It follows that the category ${\\mathsf P}(A) = {\\mathsf P}_\\lambda(A) \\cap {\\mathsf P}_\\varrho(A)$ of $A^\\mathrm{ev}$-modules which are projective on either side is entirely extension closed and strong exact monoidal as well.\n\\end{exa}\n\n\\begin{exa}\nLet $A$ be a $K$-algebra which is flat as a $K$-module. Replacing the term $A$-projective (on the left or right) by $A$-flat in the previous example, we obtain three full subcategories ${\\mathsf F}_\\lambda(A)$, ${\\mathsf F}_\\varrho(A)$ and ${\\mathsf F}(A) = {\\mathsf F}_\\lambda(A) \\cap {\\mathsf F}_\\varrho(A)$ of $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$. They contain $A$ and $A \\otimes_K A$, as $A$ is $K$-flat. Moreover, by the Snake Lemma, they are closed under taking extensions and tensor products over $A$, thus are weak exact monoidal subcategories of $(\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}), \\otimes_A, A)$. However, neither the kernel of a map $M \\twoheadrightarrow N$ in ${\\mathsf F}(A)$ nor the cokernel of a map $M \\hookrightarrow N$ in ${\\mathsf F}(A)$ needs to be $A$-flat on any side. Therefore one cannot expect those subcategories to be entirely extension closed in general.\n\\end{exa}\n\n\n\n\\section{Gerstenhaber algebras and Hochschild cohomology}\\label{sec:gerstHH}\n\nFix a unital and associative $K$-algebra $A$. The symbol $\\otimes$ will always stand for $\\otimes_K$, i.e., the tensor product over the base ring $K$. Let $A^\\mathrm{ev} = A \\otimes A^\\mathrm{op}$ be the enveloping algebra of $A$ over $K$. It is a very well-known fact, that $A$-bimodules with symmetric $K$-action bijectively correspond to left modules over $A^\\mathrm{ev}$. In this section, we will recall the definition of Hochschild cohomology, and its (higher) structures, as introduced in \\cite{CaEi56}, \\cite{Ge63} and \\cite{Ho45}.\n\n\\subsection{Reminder on Hochschild cohomology}\nLet $M$ be a $A^\\mathrm{ev}$-module. The \\textit{Hochschild cocomplex} $\\mathbb C(A,M) = (C^\\ast(A,M), \\partial_M)$ with coefficients in $M$ is the cocomplex conentrated in non-negative degrees which is given by\n$$\nC^n(A,M) = \\operatorname{Hom}\\nolimits_K(A^{\\otimes n},M) \\quad \\text{(for $n \\geqslant 0$)}\n$$\nand $\\partial_M^n \\colon C^n(A,M) \\rightarrow C^{n+1}(A,M)$,\n\\begin{align*}\n\\partial_M^n(f)(a_1 \\otimes \\cdots a_{n+1}) = & a_1 f(a_2 \\otimes \\cdots \\otimes a_{n+1})\\\\ &+ \\sum_{i=1}^{n}(-1)^n f(a_1 \\otimes \\cdots \\otimes a_{i-1} \\otimes a_i a_{i+1} \\otimes a_{i+1} \\otimes \\cdots \\otimes a_{n+1})\\\\\n&+ (-1)^{n+1}f(a_1 \\otimes \\cdots \\otimes a_{n})a_{n+1}.\n\\end{align*}\nThe Hochschild cocomplex admits an exterior pairing, in that there is a map\n$$\n\\smallsmile \\colon C^m(A,M) \\times C^n(A,N) \\longrightarrow C^{m+n}(A,M \\otimes_A N)\n$$\nfor any pair $M, N \\in \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$. Namely,\n$$\n(f \\smallsmile g)(a_1 \\otimes \\cdots a_{m + n}) = f(a_1 \\otimes \\cdots \\otimes a_{m}) \\otimes g(a_{m+1} \\otimes \\cdots \\otimes a_{m+n}).\n$$\nIn fact, through $\\smallsmile$, and after identifying $A \\otimes_A A \\cong A$, $M \\otimes_A A \\cong M \\cong A \\otimes_A M$, the DG module $\\mathbb C(A,A)$ is a DG $K$-algebra, and $\\mathbb C(A,M)$ will be a left and a right DG module over it. In particular, the cohomology of $\\mathbb C(A,A)$ will be a graded $K$-algebra. We refer to the map $\\smallsmile$ on $\\mathbb C(A,A)$ (and also to the induced map in cohomology) as the \\textit{cup product}.\n\n\\begin{defn}\nFor an integer $n \\geqslant 0$, the \\textit{$n$-th Hochschild cohomology module} with coefficients in $M$ is given by\n$$\n\\operatorname{HH}\\nolimits^n(A,M) = H^n \\mathbb C(A,M).\n$$\nThe graded module\n$$\n\\operatorname{HH}\\nolimits^\\ast(A,M) = \\bigoplus_{n \\geqslant 0}\\operatorname{HH}\\nolimits^n(A,M)\n$$\nis the \\textit{Hochschild cohomology module} of $A$ with coefficients in $M$. We abbriviate $\\operatorname{HH}\\nolimits^n(A) = \\operatorname{HH}\\nolimits^n(A,A)$ for $n \\geqslant 0$ and call $\\operatorname{HH}\\nolimits^\\ast(A)$ the \\textit{Hochschild cohomology algebra} of $A$.\n\\end{defn}\n\n\\begin{nn}\nThere are two immediate observations.\n\\begin{enumerate}[\\rm(1)]\n\\item The module $\\operatorname{HH}\\nolimits^0(A,M)$ coincides with $M^A$, where\n$$\nM^A = \\{m \\in M \\mid am = ma \\text{ for all $a \\in A$}\\}.\n$$\nIn particular, $\\operatorname{HH}\\nolimits^0(A) = Z(A)$ is the center of $A$.\n\\item The module $\\operatorname{HH}\\nolimits^1(A,M)$ coincides with the module $\\operatorname{Out}\\nolimits_K(A,M)$ of \\textit{outer derivations} which is given by\n$$\n\\operatorname{Out}\\nolimits_K(A,M) = \\frac{\\operatorname{Der}\\nolimits_K(A,M)}{\\operatorname{Inn}\\nolimits_K(A,M)},\n$$\nwhere $\\operatorname{Der}\\nolimits_K(A,M)$ denotes the derivations of $M$, and $\\operatorname{Inn}\\nolimits_K(A,M)$ the submodule of inner derivations.\n\\end{enumerate}\n\\end{nn}\n\n\\begin{nn}\nThe \\textit{bar resolution} $\\mathbb BA = (B_\\ast, \\beta_\\ast)$ of $A$, is the following exact resolution of $A$ by $A^\\mathrm{ev}$-modules. To begin with, $B_n = A^{\\otimes(n+2)}$ is the $(n+2)$-fold tensor product of $A$ with itself (over $K$ and for $n \\geqslant 0$). The $A^\\mathrm{ev}$-linear map $\\beta_{n+1}\\colon B_{n+1} \\rightarrow B_n$,\n$$\n\\beta_{n+1}(a_0 \\otimes \\cdots \\otimes a_{n+2}) = \\sum_{i=0}^{n+1}(-1)^i a_0 \\otimes \\cdots \\otimes a_i a_{i+1} \\otimes \\cdots \\otimes a_{n+2},\n$$\nturns $\\mathbb BA = (B_\\ast, \\beta_\\ast)$ into a complex, which is acyclic in all degrees but in degree $0$, wherein its homology is isomorphic to $A$. The multiplication map $\\mu\\colon A \\otimes A \\rightarrow A$ provides a suitable augmentation $\\mathbb BA \\rightarrow A \\rightarrow 0$.\n\nNow, the adjunction isomorphism\n$$\n\\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(A^{\\otimes(n+2)}, M) \\longrightarrow \\operatorname{Hom}\\nolimits_K(A^{\\otimes n}, M), \\ \\varphi \\mapsto \\varphi(1 \\otimes - \\otimes \\cdots \\otimes - \\otimes 1)\n$$\nis compatible with the differentials $\\partial_M$ and $\\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(\\mathbb \\beta_\\ast, M)$, that is, the cocomplexes $\\mathbb C(A,M)$ and $\\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(\\mathbb BA, M)$ are isomorphic. Thus $\\operatorname{HH}\\nolimits^\\ast(A,M) \\cong H^\\ast \\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(\\mathbb BA, M)$ and, since $\\mathbb BA \\rightarrow A \\rightarrow 0$ is exact, there is a graded map\n$$\n\\chi_{M}^\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(A,M) \\longrightarrow \\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,M).\n$$\nIt is given by sending a cocycle $\\varphi \\in \\operatorname{Ker}\\nolimits \\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(\\beta_{n+1}, M)$ to the equivalence class defined by the lower sequence in the pushout diagram below.\n$$\n\\xymatrix@C=20pt{\n\\cdots \\ar[r]^-{\\beta_{n+1}} & A^{\\otimes (n + 2)} \\ar[r]^-{\\beta_{n}} \\ar[d]_-\\varphi & A^{\\otimes (n + 1)} \\ar[r]^-{\\beta_{n-1}} \\ar[d] & \\cdots \\ar[r]^-{\\beta_2} & A^{\\otimes 3} \\ar[r]^-{\\beta_1} \\ar@{=}[d] & A \\otimes A \\ar[r]^-{\\beta_0} \\ar@{=}[d] & A \\ar[r] \\ar@{=}[d] & 0\\\\\n0 \\ar[r] & M \\ar[r] & Q \\ar[r] & \\cdots \\ar[r]^-{\\beta_2} & A^{\\otimes 3} \\ar[r]^-{\\beta_1} & A \\otimes A \\ar[r]^-{\\beta_0} & A \\ar[r] & 0\n}\n$$\nThis map respects the graded ring structures if $M = A$, but will in general not be a bijection (as we will remark later, $\\operatorname{HH}\\nolimits^\\ast(A)$ is always \\textit{graded commutative}, whereas a sufficient criterion for $\\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,A)$ being graded commutative is $\\operatorname{Tor}\\nolimits_i^K(A,A) = 0$ for all $i > 0$; see \\cite[Sec.\\,2.2]{BuFl08} and \\cite{SnSo04}). However, one easily checks that $\\mathbb BA$ will be a resolution by projective $A^\\mathrm{ev}$-modules, if $A$ is projective over $K$. Thus the above map is going to be an ismorphism,\n$$\n\\chi^\\ast_M \\colon \\operatorname{HH}\\nolimits^\\ast(A,M) \\xrightarrow{\\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,M),\n$$\nif $A$ is $K$-projective.\n\\end{nn}\n\n\\subsection{The Gerstenhaber bracket in Hochschild cohomology} Let us first recall the definition of a Gerstenhaber algebra over $K$ (following \\cite{Ge63}, \\cite{GeSch86} and \\cite{Lei80}).\n\n\\begin{defn}\\label{def:galgebra}\nLet $G = \\bigoplus_{n \\in {\\mathbb Z}}{G^n}$ be a graded $K$-algebra. Further, let $\\{-,-\\}\\colon G \\times G \\rightarrow G$ be a $K$-bilinear map of degree $-1$ (that is, $\\{a,b\\} \\in G^{\\abs{a}+\\abs{b}-1}$ for all homogeneous $a,b \\in G$). The pair $(G,\\{-,-\\})$ is a \\textit{Gerstenhaber algebra} over $K$ if\n\\begin{enumerate}[\\rm(G1)]\n\\item $G$ is graded commutative, i.e., $ab = (-1)^{\\abs{a}\\abs{b}}ba$ for all homogeneous $a,b \\in G$;\n\\item $\\{a,b\\} = -(-1)^{(\\abs{a}-1)(\\abs{b}-1)}\\{b,a\\}$ for all homogeneous $a,b \\in G$;\n\\item $\\{a,a\\} = 0$ for all homogeneous $a \\in G$ of odd degree;\n\\item $\\{\\{a,a\\},a\\} = 0$ for all homogeneous $a \\in G$ of even degree;\n\\item\\label{def:galgebra:5} the graded Jacobi identity holds:\n$$\n\\{a,\\{b,c\\}\\} = \\{\\{a,b\\},c\\} + (-1)^{(\\abs{a}-1)(\\abs{b}-1)}\\{b,\\{a,c\\}\\}\n$$\nfor all homogeneous $a,b,c \\in G$;\n\\item\\label{def:galgebra:6} the graded Poisson identity holds:\n$$\n\\{a,bc\\} = \\{a,b\\}c + (-1)^{(\\abs{a}-1)\\abs{b}} b\\{a,c\\}\n$$\nfor all homogeneous $a,b,c \\in G$.\n\\end{enumerate}\nAssume that $(G,\\{-,-\\})$ is a Gerstenhaber algebra over $K$. We call $(G,\\{-,-\\})$ a \\textit{strict} Gerstenhaber algebra over $K$ if there is a map $sq\\colon G^{2\\ast} \\rightarrow G^{4\\ast-1}$ of degree $0$ such that\n\\begin{enumerate}[\\rm(G1)]\n\\setcounter{enumi}{6}\n\\item\\label{def:galgebra:7} $sq(ra) = r^2 sq(a)$ for all $r \\in K$ and all homogeneous $a \\in G^{2\\ast}$;\n\\item\\label{def:galgebra:8} $sq(a+b) = sq(a) + sq(b) + \\{a,b\\}$ for all homogeneous $a,b \\in G^{2\\ast}$;\n\\item $\\{a,sq(b)\\} = \\{\\{a,b\\},b\\}$ for all homogeneous $a,b \\in G^{2\\ast}$;\n\\item $sq(ab) = a^2sq(b) + sq(a)b^2 + a\\{a,b\\}b$ for all homogeneous $a,b \\in G^{2\\ast}$.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{nn}\nThe map $\\{-,-\\}$ is called a \\textit{Gerstenhaber bracket} for $G$, whereas $sq$ is a \\textit{$($devided$)$ squaring map} for the Gerstenhaber algebra $(G, \\{-,-\\})$. Note that any graded commutative $K$-algebra can be viewed as (strict) a Gerstenhaber algebra over $K$ with trivial bracket (and trivial squaring map).\n\\end{nn}\n\n\\begin{nn}\nThe axioms (G\\ref{def:galgebra:5}) and (G\\ref{def:galgebra:6}) may be read as follows: The graded Jacobi identity measures the failure of $\\{-,-\\}$ from being associative, whereas the graded Poisson identity translates to $\\{a,-\\}$ being a graded derivation of $G$ of degree $\\abs{a} - 1$ (for $a \\in G$ homogeneous). Also note that by (G\\ref{def:galgebra:7}) and (G\\ref{def:galgebra:8}), there is precisely one squaring map $sq$ for a Gerstenhaber algebra $(G,\\{-,-\\})$ if $2 \\in K^\\times$, in which case it is given by $sq(a) = 2^{-1}\\{a, a\\}$.\n\\end{nn}\n\n\\begin{nn}\\label{nn:Gbracket}\nLet $M$ be a fixed $A^\\mathrm{ev}$-module and $f \\in C^m(A,M)$, $g \\in C^n(A,A)$ for integers $m, n \\geqslant 0$. For $i=1, \\dots, m$, let $f \\bullet_i g \\in C^{m + n -1}(A,M)$\n\\begin{align*}\n&\\qquad(f \\bullet_i g)(a_1 \\otimes \\cdots \\otimes a_{m+n-1})\\\\ = f(a_1 \\otimes \\cdots & \\otimes a_{i-1} \\otimes g(a_i \\otimes \\cdots \\otimes a_{i+n-1}) \\otimes a_{i+n} \\otimes \\cdots \\otimes a_{m+n-1}).\n\\end{align*}\nDenote by $f \\bullet g \\in C^{m+n-1}(A,M)$ the alternating sum of the $f \\bullet_i g$:\n\\begin{align*}\nf \\bullet g = \\sum_{i=1}^m (-1)^{(i-1)(n-1)}f \\bullet_i g.\n\\end{align*}\nThe product $\\bullet$ is, in general, non-unital and highly non-associative. However, the external pairing $\\smallsmile$ and $\\bullet$ are related by the following fundamental formula:\n\\begin{equation}\\tag{$\\dagger$}\\label{eq:fundform}\n\\partial_M(f \\bullet g) + (-1)^n \\partial_M(f) \\bullet g = f \\bullet \\partial_A(g) + (-1)^{n}[g \\smallsmile f - (-1)^{mn}f \\smallsmile g].\n\\end{equation}\nSee \\cite[Thm.\\,3]{Ge63} for a proof.\n\\end{nn}\n\n\\begin{nn}\nThe fundamental formula (\\ref{eq:fundform}) yields three important insights.\n\\begin{enumerate}[\\rm(1)]\n\\item If $M = A$, and $f$ and $g$ are \\textit{cocycles} (i.e., $\\partial_A(f) = 0 = \\partial_A(g)$), then $f \\smallsmile g = (-1)^{mn} g \\smallsmile f$. It follows that $\\operatorname{HH}\\nolimits^\\ast(A)$ is a graded commutative $K$-algebra.\n\\item The map\n\\begin{align*}\n\\{-,-&\\} \\colon C^m(A,A) \\times C^n(A,A) \\longrightarrow C^{m+n-1}(A,A)\\\\\n&\\{f,g\\} = f \\bullet g - (-1)^{(m-1)(n-1)}g \\bullet f\n\\end{align*}\ninduces a well-defined map\n$$\n\\{-,-\\} = \\{-,-\\}_A \\colon \\operatorname{HH}\\nolimits^m(A) \\times \\operatorname{HH}\\nolimits^n(A) \\longrightarrow \\operatorname{HH}\\nolimits^{m+n-1}(A).\n$$\n\\item The map\n\\begin{align*}\nsq \\colon C^{2n}(A,A) \\longrightarrow C^{4n-1}(A,A), \\ sq(f) = f \\bullet f\n\\end{align*}\ninduces a well-defined map\n$$\nsq = sq_A \\colon \\operatorname{HH}\\nolimits^{2n}(A) \\longrightarrow \\operatorname{HH}\\nolimits^{4n-1}(A).\n$$\n\\end{enumerate}\nIt is the main observation of \\cite{Ge63} that the hereby obtained quadruple $(\\operatorname{HH}\\nolimits^\\ast(A), \\smallsmile, \\{-,-\\}, sq)$ is a strict Gerstenhaber algebra over $K$, in the sense of Definition \\ref{def:galgebra}.\n\\end{nn}\n\n\\subsection{Schwede's loop bracket} Let us briefly recall Schwede's exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology; we will give a rough outline, and refer to \\cite{He16a}, \\cite{He14b} and \\cite{Sch98} for details. Let $A$ be a $K$-algebra which is projective as a $K$-module. For an integer $n \\geqslant 1$, we let $\\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ be the \\textit{category of $n$-self extensions of $A$ over $A^\\mathrm{ev}$}. Its objects are $n$-extensions $S$ of the form $0 \\rightarrow A \\rightarrow E_{n-1} \\rightarrow \\cdots \\rightarrow E_0 \\rightarrow A \\rightarrow 0$ in $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$, whereas the morphisms are given by \\textit{morphisms of extensions}, i.e., morphisms $(f_n)_{n \\in \\mathbb Z} \\colon S \\rightarrow S'$ of complexes with $f_{-1} = f_n = \\operatorname{id}\\nolimits_A$. The $0$-th homotopy group $\\pi_0 \\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ of $\\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ coincides with $\\operatorname{Ext}\\nolimits^n_{A^\\mathrm{ev}}(A,A)$. Moreover, by a theorem of Retakh, see \\cite{Re86}, one has isomorphisms\n$$\n\\operatorname{Ext}\\nolimits^n_{A^\\mathrm{ev}}(A,A) = \\pi_0 \\mathcal Ext^{n-1}_{A^\\mathrm{ev}}(A,A) \\xrightarrow{\\, \\sim \\, } \\pi_1(\\mathcal Ext^n_{A^\\mathrm{ev}}(A,A), S)\n$$\nfor each base point $S \\in \\operatorname{Ob}\\nolimits \\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$. Given $\\alpha \\in \\operatorname{HH}\\nolimits^m(A)$ and $\\beta \\in \\operatorname{HH}\\nolimits^n(A)$, one may thus think of $\\{\\alpha, \\beta\\}_A$ as an element of $\\pi_1 \\mathcal Ext^{m+n}_{A^\\mathrm{ev}}(A,A)$ (at some basepoint $S'$), that is, a \\textit{loop} (see \\cite{Qu72}, \\cite{Sch98}) in the category $\\mathcal Ext^{m+n}_{A^\\mathrm{ev}}(A,A)$ based at $S'$. In fact, if $\\alpha$ and $\\beta$ are the equivalence classes of extensions $S = S(\\alpha)$ and $T = T(\\beta)$, the loop $\\Omega_A(S,T)$ that corresponds to $\\{\\alpha, \\beta\\}_A$ may be realised as\n$$\n\\xymatrix@!C=15pt@R=15pt{\n& S \\boxtimes_A T \\ar[dr] \\ar[dl] & \\\\\nS \\# T & & (-1)^{mn} T \\# S\\\\\n& (-1)^{mn} T \\boxtimes_A S \\ar[ur] \\ar[ul] &\n}\n$$\nby \\cite[Thm.\\,3.1]{Sch98}, with base point $S \\#T$, where $S \\# T$ denotes the Yoneda composite of $S$ and $T$. The $(m + n)$-extension $S \\boxtimes_A T$ is obtained by taking the tensor product of the truncated complexes $0  \\rightarrow A \\rightarrow E_{m-1} \\rightarrow \\cdots \\rightarrow E_0$ and  $0 \\rightarrow A \\rightarrow F_{n-1} \\rightarrow \\cdots \\rightarrow F_0$, denoted by $S^\\natural$ and $T^\\natural$ for short, followed by the canonical augmentation $E_0 \\otimes_A F_0 \\twoheadrightarrow A \\otimes_A A \\cong A$. Caution is advised, as the complex $S \\boxtimes_A T$ a priori does not need to be acyclic, which however may be bypassed by, for instance, considering the categories $\\mathcal Ext^n_{{\\mathsf P}(A)}(A,A)$ of admissible extensions over ${\\mathsf P}(A)$ instead of $\\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ (recall that $\\operatorname{Ext}\\nolimits^n_{{\\mathsf P}(A)}(A,A) \\cong \\operatorname{Ext}\\nolimits^n_{A^\\mathrm{ev}}(A,A)$). Schwede's construction also covers an interpretation of the squaring map; the loop corresponding to $sq_A(\\alpha)$ for some $\\alpha \\in \\operatorname{HH}\\nolimits^{2n}(A)$ is given by the northern hemisphere of the loop $\\Omega_A(S,S)$.\n\nNote that the loop above only covers the cases $m, n \\geqslant 1$; see \\cite{He16a} for the remaining cases $m \\geqslant 0$, $n = 0$.\n\n\\begin{lem}\\label{lem:loopcoinc}\nAssume that $S \\in \\mathcal Ext^m_{A^\\mathrm{ev}}(A,A)$ and $T \\in \\mathcal Ext^n_{A^\\mathrm{ev}}(A,A)$ such that $S \\boxtimes_A T$ is exact. Let $\\mathsf C$ be an exact monoidal subcategory of $(\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}, \\otimes_A, A)$ which is entirely extension closed. Assume further, that there are morphisms $S' \\rightarrow S$ and $T' \\rightarrow T$ of extensions with $S' \\in \\mathcal Ext^m_{{\\mathsf C}}(A,A)$ and $T' \\in \\mathcal Ext^m_{{\\mathsf C}}(A,A)$. Then there exists a morphism $v \\colon S' \\# T' \\rightarrow S \\# T$ and the conjugate of the loop $\\Omega_A(S,T)$ by $f$ is homotopically equivalent to $\\Omega_A(S',T')$.\n\\end{lem}\n\n\\begin{proof}\nOf course, the morphisms $S' \\rightarrow S$ and $T' \\rightarrow T$ may be combined to obtain a morphism $v \\colon S' \\# T' \\rightarrow S \\# T$ as claimed. They also yield a morphism $u \\colon S' \\boxtimes_A T' \\rightarrow S \\boxtimes_A T$. These morphisms render the square\n$$\n\\xymatrix@!C=35pt@R=15pt{\n& S' \\boxtimes_A T' \\ar[r]^-u \\ar[dl] & S \\boxtimes_A T \\ar[dl]\\\\\nS' \\# T' \\ar[r]^-{\\, v} & S \\# T &\n}\n$$\ncommutative, where $S' \\boxtimes_A T' \\rightarrow S' \\# T'$ is the northwestern face morphism in the loop $\\Omega_A(S',T')$. By similar arguments, we obtain commutative diagrams for the remaining face morphisms as well, whence the claim follows.\n\\end{proof}\n\n\\begin{thm}\\label{thm:gersthochcomp}\nLet $A$ and $B$ be $K$-algebras which are projective when viewed as $K$-modules. Let $\\mathsf C$ be an exact monoidal subcategory of $(\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}), \\otimes_A, A)$ which is entirely extension closed, and let $\\mathfrak X \\colon (\\mathsf C, \\otimes_A, A) \\rightarrow (\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ be an exact and almost strong $($or almost costrong$)$ monoidal functor such that one of the following conditions holds true for all objects $X \\in {\\mathsf C}$.\n\\begin{equation}\\label{eq:exactnesscond}\\tag{\\textbf E}\n\\begin{aligned}\n0 \\longrightarrow L \\longrightarrow &M \\longrightarrow N \\longrightarrow 0 \\ \\text{admissible exact in ${\\mathsf C}$}\\\\\n\\Longrightarrow \\quad &  0 \\longrightarrow \\mathfrak X(L) \\otimes_B \\mathfrak X(X) \\longrightarrow \\mathfrak X(M) \\otimes_B \\mathfrak X(X) \\longrightarrow \\mathfrak X(N) \\otimes_B \\mathfrak X(X) \\longrightarrow 0\\\\\n&\\text{exact in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$.}\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:exactnesscondprime}\\tag{\\textbf E'}\n\\begin{aligned}\n0 \\longrightarrow L \\longrightarrow &M \\longrightarrow N \\longrightarrow 0 \\ \\text{admissible exact in ${\\mathsf C}$}\\\\\n\\Longrightarrow \\quad &  0 \\longrightarrow \\mathfrak X(X) \\otimes_B \\mathfrak X(L) \\longrightarrow \\mathfrak X(X) \\otimes_B \\mathfrak X(M) \\longrightarrow \\mathfrak X(X) \\otimes_B \\mathfrak X(N) \\longrightarrow 0\\\\\n&\\text{exact in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$.}\n\\end{aligned}\n\\end{equation}\nThen the inclusion functor induces an isomophism $\\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf C}}(A,A) \\xrightarrow{\\sim } \\operatorname{HH}\\nolimits^\\ast(A)$ and the map\n$$\n\\mathfrak X_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(A) \\xrightarrow{\\, \\sim \\, } \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(A,A) \\xrightarrow{\\, \\ \\, } \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B) \\xrightarrow{\\, \\sim \\, } \\operatorname{HH}\\nolimits^\\ast(B)\n$$\nthat sends the equivalence class of an $n$-extension $0 \\rightarrow A \\rightarrow E_{n-1} \\rightarrow \\cdots \\rightarrow E_0 \\rightarrow A \\rightarrow 0$ to the equivalence class of\n$$0 \\longrightarrow B \\cong \\mathfrak X(A) \\longrightarrow \\mathfrak X (E_{n-1}) \\longrightarrow \\cdots \\longrightarrow \\mathfrak X (E_0) \\longrightarrow \\mathfrak X(A) \\cong B \\longrightarrow 0$$\nrenders the following diagrams commutative for all $m, n \\geqslant 0$.\n$$\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^m(A) \\times \\operatorname{HH}\\nolimits^n(A) \\ar[r]^-{\\{-,-\\}_A} \\ar[d]_-{\\mathfrak X_m \\times \\mathfrak X_n} & \\operatorname{HH}\\nolimits^{m+n-1}(A) \\ar[d]^-{\\mathfrak X_{m + n -1}} \\\\\n\\operatorname{HH}\\nolimits^{m}(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} & \\operatorname{HH}\\nolimits^{m+n-1}(B)\n}\n\\ \\quad\\quad \\\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^{2n}(A) \\ar[r]^{sq_A} \\ar[d]_-{\\mathfrak X_{2n}} & \\operatorname{HH}\\nolimits^{4n-1}(A) \\ar[d]^-{\\mathfrak X_{4n-1}}\\\\\n\\operatorname{HH}\\nolimits^{2n}(B) \\ar[r]^{sq_B} & \\operatorname{HH}\\nolimits^{4n-1}(B)\n}\n$$\nHere $\\{-,-\\}_A$ and $sq_A$ denote the Gerstenhaber bracket and the squaring map on $\\operatorname{HH}\\nolimits^\\ast(A)$, whereas $\\{-,-\\}_B$ and $sq_B$ denote the Gerstehaber bracket and the squaring map on $\\operatorname{HH}\\nolimits^\\ast(B)$.\n\\end{thm}\n\n\\begin{proof}\nThe theorem summarises some results relating the exact sequence interpretation of the Gerstenhaber bracket in Hochschild cohomology that we mentioned above, and its generalisation to exact monoidal categories, see \\cite{He14b}, which incorporates the statement that, under appropriate assumptions, the (generalised) bracket commutes with almost strong (or almost costrong) exact monoidal functors.\n\nTo be more precise, the theorem is obtained by puzzling together \\cite[Thm.\\,3.1]{Sch98}, \\cite[Lem.\\,5.3.3]{He14b}, \\cite[Thm.\\,5.2.12]{He14b} and \\cite[Thm.\\,5.1]{He16a}; note that the exactness conditions (\\ref{eq:exactnesscond}) and (\\ref{eq:exactnesscondprime}) ensure that $\\mathfrak X(S) \\boxtimes_A \\mathfrak X(T)$ is exact in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ for any choice of admissible extensions $S \\in \\mathcal Ext^m_{{\\mathsf C}}(A,A)$ and $T \\in \\mathcal Ext^n_{\\mathsf C}(A,A)$ by (the proof of) the statement \\cite[Lem.\\,3.3.4]{He14b}. Thus the homotopy class of the corresponding loop $\\Omega_B(\\mathfrak X(S),\\mathfrak X(T))$ coincides (up to conjugation) with the homotopy class of the loop for the corresponding admissible exact sequences $S'$ and $T'$, with morphisms $S' \\rightarrow \\mathfrak X(S)$, $T' \\rightarrow \\mathfrak X(T)$, taken in, for instance, ${\\mathsf P}(B)$, by Lemma \\ref{lem:loopcoinc}.\n\\end{proof}\n\n\\begin{rem}\nLet $\\mathfrak X \\colon (\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}), \\otimes_A, A) \\rightarrow (\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ be an exact and almost strong\/costrong monoidal functor. The exactness condition on $\\mathfrak X$ stated in the above theorem is automatically satisfied, if $\\mathfrak X$, for instance, restricts as\n$$\n\\mathfrak X \\colon ({\\mathsf P}(A), \\otimes_A, A) \\longrightarrow ({\\mathsf P}(B), \\otimes_B, B)\\,.\n$$\nSimilar situations will appear frequently hereinafter. \n\\end{rem}\n\n\n\\section{Recollements of module categories}\\label{sec:recoll}\n\n\\begin{nn}\nLet ${\\mathsf A}$, ${\\mathsf B}$ and ${\\mathsf C}$ be abelian categories. Recall that a diagram\n\\begin{equation}\\tag{$\\diamond$}\\label{eq:recollement}\n\\begin{aligned}\n\\xymatrix{\n\\\\\\mathcal R({\\mathsf A}, {\\mathsf B}, {\\mathsf C}) \\qquad \\equiv \\qquad\\\\\n}\n\\xymatrix@!C=50pt{\n\\\\ {\\mathsf A} \\ar[r]^-{i} & {\\mathsf B} \\ar[r]^-{j} \\ar@\/^2pc\/[l]^-{i_\\varrho} \\ar@\/_2pc\/[l]_-{i_\\lambda} & {\\mathsf C} \\ar@\/^2pc\/[l]^-{j_\\varrho} \\ar@\/_2pc\/[l]_-{j_\\lambda}\\\\\n}\n\\end{aligned}\n\\end{equation}\nof additive functors is called a \\textit{recollement} (\\textit{of abelian categories}) if $(i_\\lambda, i , i_\\varrho)$ and $(j_\\lambda, j , j_\\varrho)$ are adjoint triples, the functors $i$, $j_\\lambda$, $j_\\varrho$ are fully faithful, and $\\Im(i) = \\operatorname{Ker}\\nolimits(j)$. The functor $j : {\\mathsf B} \\rightarrow {\\mathsf C}$ is referred to as the \\textit{quotient functor} of $\\mathcal R({\\mathsf A}, {\\mathsf B}, {\\mathsf C})$. We call a recollement (\\ref{eq:recollement}) \\textit{$K$-linear} if all the occurring categories and functors are $K$-linear. It is well-known, see \\cite{FrPi04}, that in any recollement situation\n\\begin{enumerate}[\\rm(1)]\n\\item the functors $i\\colon {\\mathsf A} \\rightarrow {\\mathsf B}$ and $j\\colon {\\mathsf B} \\rightarrow {\\mathsf C}$ are exact;\n\\item the functor $j\\colon {\\mathsf B} \\rightarrow {\\mathsf C}$ is essentially surjective;\n\\item the units of the adjoint pairs $(i, i_\\varrho)$, $(j_\\lambda, j)$ and the counits of the adjoint pairs $(i_\\lambda, i)$ and $(j,j_\\varrho)$ are isomorphisms;\n\\item for each object $B \\in {\\mathsf B}$, there are objects $A, A' \\in {\\mathsf A}$ and exact sequences\n$$\n0 \\longrightarrow i(A) \\longrightarrow (j_\\lambda \\circ j)(B) \\longrightarrow B \\longrightarrow (i \\circ i_\\lambda)(B) \\longrightarrow 0 \\, ,\n$$\n$$\n0 \\longrightarrow (i \\circ i_\\varrho)(B) \\longrightarrow B \\longrightarrow (j_\\varrho \\circ j)(B) \\longrightarrow i(A') \\longrightarrow 0 \\,\\,\n$$\ninduced by the units and counits of the adjunctions.\n\\end{enumerate}\nIn particular, for all $X \\in \\operatorname{Ob}\\nolimits {\\mathsf A}$, the functor $j\\colon {\\mathsf B} \\rightarrow {\\mathsf C}$ gives rise to a homomorphism\n$$\nj_\\ast^{X} \\colon \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf B}(X,X) \\longrightarrow \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(j(X),j(X)), \\ j_\\ast(\\alpha) = [j(S)] \\quad \\text{(for $\\alpha = [S] \\in \\operatorname{Ext}\\nolimits^n_{\\mathsf B}(X,X)$)}\n$$\nof graded rings (for $j$ preserves pullbacks and pushouts). We make the following definition.\n\\end{nn}\n\n\\begin{defn}\nLet ${\\mathsf B}$ and ${\\mathsf C}$ be abelian categories and let $f_\\ast \\colon \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf B}(B,B) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(C,C)$ be a homomorphism of graded rings for some objects $B \\in \\operatorname{Ob}\\nolimits {\\mathsf B}$ and $C \\in \\operatorname{Ob}\\nolimits{\\mathsf C}$. \n\n\\begin{enumerate}[\\rm(1)] \n\\item Let $\\mathfrak X \\colon {\\mathsf B} \\rightarrow {\\mathsf C}$ be an exact functor. The map $f_\\ast$ is \\textit{induced by the functor} $\\mathfrak X : {\\mathsf B} \\rightarrow {\\mathsf C}$ if there is an isomorphism $C \\cong \\mathfrak X(B)$ in ${\\mathsf C}$ such that the diagram \n$$\n\\xymatrix@!C=20pt@!R=20pt{\n& \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf B}(B,B) \\ar[dr]^-{\\mathfrak X^B_\\ast} \\ar[dl]_-{f_\\ast} &\\\\\n\\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(C,C) \\ar[rr]^-\\cong & & \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(j(B),j(B))\n}\n$$\ncommutes.\n\\item Let $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ be a recollement of abelian categories. The map $f_\\ast$ is \\textit{induced by the recollement} $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ if $f_\\ast$ is induced by the quotient functor $j \\colon {\\mathsf B} \\rightarrow {\\mathsf C}$ inside $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$.\n\\end{enumerate}\n\\end{defn}\n\nNote that for any given recollement $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ of abelian categories, and any isomorphism $\\alpha \\colon C \\xrightarrow{\\sim} j(B)$ in ${\\mathsf C}$, the graded ring homomorphism\n$$\nj_\\ast = j_\\ast^{B,\\alpha} \\colon \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf B}(B,B) \\xrightarrow{\\ j_\\ast^B \\ } \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(j(B),j(B)) \\xrightarrow{ \\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(C,C)\n$$\nis (by definition) induced by the recollement $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$.\n\n\\begin{nn}\\label{nn:morrec}\nLet $\\mathcal R = \\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ and $\\mathcal R' = \\mathcal R({\\mathsf A}',{\\mathsf B}',{\\mathsf C}')$ be recollements of abelian categories. A triple $(F,G,H)$ of functors $F\\colon {\\mathsf A} \\rightarrow {\\mathsf A}'$, $G\\colon {\\mathsf B} \\rightarrow {\\mathsf B}'$ and $H\\colon {\\mathsf C} \\rightarrow {\\mathsf C}'$ is a \\textit{morphism of recollements}, if the following functors agree up to natural isomorphisms.\n\\begin{multicols}{2}\n\\begin{enumerate}[\\rm(1)]\n\\item\\label{nn:morrec:1} $i' \\circ F$ and $G \\circ i$;\n\\item\\label{nn:morrec:2} $F \\circ i'_\\lambda$ and $i_\\lambda \\circ G$;\n\\item\\label{nn:morrec:3} $F \\circ i'_\\varrho$ and $i_\\varrho \\circ G$;\n\\item\\label{nn:morrec:4} $j' \\circ G$ and $H \\circ j$;\n\\item\\label{nn:morrec:5} $G \\circ j'_\\lambda$ and $j_\\lambda \\circ H$;\n\\item\\label{nn:morrec:6} $G \\circ j'_\\varrho$ and $j_\\varrho \\circ G$.\n\\end{enumerate}\n\\end{multicols}\nA morphism $(F,G,H)\\colon \\mathcal R \\rightarrow \\mathcal R'$ is an \\textit{equivalence}, if $F$, $G$ and $H$ are equivalences of categories. Note that in this case, $j' \\circ G$ and $H \\circ j$ being naturally isomorphic already implies the same for the composites with the respective adjoint functors of $i$ and $i'$, and $j$ and $j'$.\n\nEvery pair $(G,H)$ of functors satisfying \\ref{nn:morrec}(\\ref{nn:morrec:4})--(\\ref{nn:morrec:6}) gives rise to a (up to natural isomorphism) unique functor $F\\colon {\\mathsf A} \\rightarrow {\\mathsf A}'$ such that $i' \\circ F$ and $G \\circ i$ are naturally isomorphic. The functor $F$ is an equivalence if $G$ and $H$ were and in that case, it also is compatible with the adjoint functors of $i'$ (in the sense of \\ref{nn:morrec}(\\ref{nn:morrec:1})--(\\ref{nn:morrec:3})). Thus naming an equivalence $(F,G,H) \\colon \\mathcal R \\rightarrow \\mathcal R'$ of recollements is the same as naming two equivalences $G\\colon {\\mathsf B} \\rightarrow {\\mathsf B}'$ and $H\\colon {\\mathsf C} \\rightarrow {\\mathsf C}'$ of categories such that $j' \\circ G$ and $H \\circ j$ are naturally isomorphic.\n\\end{nn}\n\n\\begin{defn}[see {\\cite[Def.\\,4.1]{PsVi14}}]\\label{defn:recequ}\nLet $\\mathcal R = \\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ and $\\mathcal R' = \\mathcal R({\\mathsf A}',{\\mathsf B}',{\\mathsf C}')$ be recollements of abelian categories. The recollements $\\mathcal R({\\mathsf A},{\\mathsf B},{\\mathsf C})$ and $\\mathcal R({\\mathsf A}',{\\mathsf B}',{\\mathsf C}')$ are \\textit{equivalent} in case there are equivalences $G \\colon {\\mathsf B} \\rightarrow {\\mathsf B}'$ and $H\\colon {\\mathsf C} \\rightarrow {\\mathsf C}'$ of categories such that the functors $j' \\circ G$ and $H \\circ j$ are naturally isomorphic. In other words, $\\mathcal R$ and $\\mathcal R'$ are equivalent if there is an equivalence $(F,G,H) \\colon \\mathcal R \\rightarrow \\mathcal R'$ of recollements.\n\\end{defn}\n\n\\begin{nn}\nA recollement $\\mathcal R({\\mathsf A},{\\mathsf B}, {\\mathsf C})$ is a \\textit{recollement of module categories}, if the categories ${\\mathsf A}$, ${\\mathsf B}$ and ${\\mathsf C}$ are module categories over some rings $A$, $B$ and $C$. In that case, we write $\\mathcal R(A,B,C)$ instead of $\\mathcal R(\\operatorname{\\mathsf{Mod}}\\nolimits (A), \\operatorname{\\mathsf{Mod}}\\nolimits (B), \\operatorname{\\mathsf{Mod}}\\nolimits(C))$. Recall from, for instance, \\cite{AnFu92} that an equivalence $(G,H) \\colon \\mathcal R(A,B,C) \\rightarrow \\mathcal R(A',B',C')$ between recollements of module categories is determined by progenerators $P$ for $B$ and $Q$ for $C$ such that $B' \\cong \\operatorname{End}\\nolimits_B(P)^\\mathrm{op}$ and $C' \\cong \\operatorname{End}\\nolimits_C(Q)^\\mathrm{op}$, with the functors $G$ and $H$ thus being equivalent to $\\operatorname{Hom}\\nolimits_B(P,-) \\cong \\operatorname{Hom}\\nolimits_B(P,B) \\otimes_B -$ and $\\operatorname{Hom}\\nolimits_C(Q,-) \\cong \\operatorname{Hom}\\nolimits_C(Q,C) \\otimes_C -$ respectively. A choice of quasi-inverse functors is given by $P \\otimes_{B'} -$ and $Q \\otimes_{C'} -$.\n\nAny ring $R$ along with any choice of an idempotent $e \\in R$ gives rise to a recollement of module categories, which we will refer to as the \\textit{standard recollement} associated to $(R,e)$.\n\n\\begin{exa}[Standard recollement]\nLet $R$ be a ring and let $e \\in R$ be idempotent. Let us put $\\overline R = R\/ReR$ and let $\\pi \\colon R \\rightarrow \\overline{R}$ be the natural projection. If we write $\\pi_\\star \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(\\overline R) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R)$ for the restriction along $\\pi$, then the diagram\n\\begin{equation*}\n\\begin{aligned}\n\\xymatrix{\\\\\\mathcal R(\\overline R, R, eRe) \\qquad \\equiv \\qquad\\\\}\n\\xymatrix@!C=50pt{\\\\ \\operatorname{\\mathsf{Mod}}\\nolimits(\\overline{R}) \\ar[r]^-{\\pi_\\star} & \\operatorname{\\mathsf{Mod}}\\nolimits(R) \\ar[r]^-{e(-)} \\ar@\/^2pc\/[l]^-{\\operatorname{Hom}\\nolimits_{R}(\\overline R,-)} \\ar@\/_2pc\/[l]_-{\\overline R \\otimes_{R} -} & \\operatorname{\\mathsf{Mod}}\\nolimits(eRe) \\ar@\/^2pc\/[l]^-{\\operatorname{Hom}\\nolimits_{eRe}(eR,-)} \\ar@\/_2pc\/[l]_-{Re \\otimes_{eRe} -}\\\\}\n\\end{aligned}\n\\end{equation*}\nof functors defines a recollement which we will denote by $\\mathcal R (R,e)$. This recollement will be $K$-linear, in case $R$ is an algebra over $K$.\n\\end{exa}\n\\end{nn}\n\n\\begin{thm}[see {\\cite[Thm.\\,5.3]{PsVi14}}]\\label{thm:recmodeq}\nLet $\\mathcal R(A,B,C)$ be a recollement of module categories. Then there is a ring $R$ and an idepotent $e \\in R$ such that $\\mathcal R(A,B,C)$ is equivalent to $\\mathcal R(R,e)$ in the sense of Definition $\\ref{defn:recequ}$.\n\\end{thm}\n\nLet $\\mathcal R(A,B,C)$ be as above. The theorem yields that if $B$ is a $K$-algebra, being $K$-projective, then so is $C$. This follows from the following two observations for a $K$-projective $K$-algebra $\\Gamma$.\n\n\\begin{enumerate}[\\rm(1)]\n\\item Let $P$ be a finitely generated projective module over $\\Gamma$. Then $\\operatorname{End}\\nolimits_\\Gamma(P)$ is $K$-projective. Indeed, if $P \\oplus P' \\cong \\Gamma^n$ for some $n \\geqslant 0$, then the $K$-module $\\operatorname{End}\\nolimits_\\Gamma(P)$ is a direct summand of $\\Gamma^{n^2}$.\n\\item Let $\\varepsilon \\in \\Gamma$ be an idempotent. The \\textit{Peirce decomposition} of $\\Gamma$ with respect to $\\varepsilon$ yields\n$$\n\\Gamma = \\varepsilon\\Gamma \\varepsilon \\oplus \\varepsilon\\Gamma (1-\\varepsilon) \\oplus (1-\\varepsilon)\\Gamma \\varepsilon \\oplus (1-\\varepsilon)\\Gamma (1-\\varepsilon)\\,,\n$$\nimplying that $\\varepsilon \\Gamma \\varepsilon$ is $K$-projective.\n\\end{enumerate}\n\nSince the functors $\\operatorname{\\mathsf{Mod}}\\nolimits(R\/ReR) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R)$, $e(-) = eR \\otimes_R (-) \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(eRe)$, as well as their respective left adjoints $(R\/ReR) \\otimes_R (-) \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R\/ReR)$, $Re \\otimes_{eRe}(-) \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(eRe) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R)$ are right exact and preserve arbitrary direct sums, the following consequence is imminent.\n\n\\begin{cor}\\label{cor:recolldirectsums}\nLet $\\mathcal R(A,B,C)$ be a recollement of module categories. Then the functors $i \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(A) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B)$ and $j\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C)$, along with their left adjoints $i_\\lambda$ and $j_\\lambda$ are right exact and preserve arbitrary direct sums.\\qed\n\\end{cor}\n\nLet $R$ and $S$ be $K$-algebras. Watt's theorem (see \\cite{Wa60}) tells that every right exact functor $\\mathfrak X \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(S)$ which commutes with arbitrary direct sums is of the form $M \\otimes_R -$ for some $S \\otimes R^\\mathrm{op}$-module $M$. In fact, $M \\cong \\mathfrak X(R)$ necessarily is implied. Note that $\\mathfrak X$ gives rise to a $K$-algebra homomorphism \n$$\nR^\\mathrm{op} \\cong \\operatorname{End}\\nolimits_R(R) \\longrightarrow \\operatorname{End}\\nolimits_S(\\mathfrak X(R)) \\subseteq \\operatorname{End}\\nolimits_K(\\mathfrak X(R))\\,,\n$$\nthat is, $\\mathfrak X(R)$ is $S$-$R$-bimodule. Theorem \\ref{thm:recmodeq} yields Watt's theorem for functors within a recollement.\n\n\\begin{cor}\nLet $\\mathcal R(A,B,C)$ be a recollement of module categories. Then $i(-) \\cong i(A) \\otimes_A -$, $j(-) \\cong j(B) \\otimes_B -$ and $i_\\lambda(-) \\cong i_\\lambda(B) \\otimes_B -$, $j_\\lambda(-) \\cong j_\\lambda(C) \\otimes_C -$.\n\\end{cor}\n\n\\begin{proof}\nAssume that $\\mathcal R(A,B,C)$ is equivalent to $\\mathcal R(R,e)$, and let $P$ and $Q$ be progenerators for $B$ and $C$ with $R \\cong \\operatorname{End}\\nolimits_B(P)^\\mathrm{op}$ and $eRe \\cong \\operatorname{End}\\nolimits_C(Q)^\\mathrm{op}$ giving rise to the defining equivalences $\\operatorname{\\mathsf{Mod}}\\nolimits(B) \\xrightarrow \\sim \\operatorname{\\mathsf{Mod}}\\nolimits(R)$ and $\\operatorname{\\mathsf{Mod}}\\nolimits(C) \\xrightarrow \\sim \\operatorname{\\mathsf{Mod}}\\nolimits(eRe)$. Then\n\\begin{align*}\nj(-) & \\cong Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P,-)\\\\\n& \\cong Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P, B) \\otimes_B -\\\\\n& \\cong j(B) \\otimes_B - \\, .\n\\end{align*}\nSimilarly, $i(-) \\cong i(A) \\otimes_A -$ follows, as well as the statements for the left adjoints.\n\\end{proof}\n\n\\begin{nn}\\label{nn:Rev}\nLet $R$ and $S$ be $K$-algebras and $\\mathfrak X \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(S)$ be an additive functor. By taking the dual of $\\mathfrak X(R)$ with respect to $R$ we get the $R$-$S$-bimodule $\\mathfrak X(R)^\\vee = \\operatorname{Hom}\\nolimits_{R^\\mathrm{op}}(\\mathfrak X(R),R)$. The functor\n$$\n\\mathfrak X^\\mathrm{ev} \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev}), \\ \\mathfrak X^\\mathrm{ev}(M) = \\mathfrak X(R) \\otimes_R M \\otimes_R \\mathfrak X(R)^\\vee\n$$\nwill be referred to as the \\textit{enveloping functor} of $\\mathfrak X$.\n\\end{nn}\n\n\\begin{prop}\\label{prop:envrecoll}\nLet $A$, $B$ and $C$ be $K$-algebras, and let $\\mathcal R(A,B,C)$ be a recollement which is equivalent to a standard recollement $\\mathcal R(R,e)$. Let $S = R \\otimes R^\\mathrm{op}$, $f$ be the idempotent $e \\otimes e^\\mathrm{op}$ in $S$ and $\\overline S = S\/SfS$. Then the functor $j^\\mathrm{ev}$ fits inside a recollement $\\mathcal R(\\overline S, B^\\mathrm{ev}, C^\\mathrm{ev})$ of module categories which is equivalent to the standard recollement $\\mathcal R(S, f)$.\n\\end{prop}\n\n\\begin{proof}\nLet $P$ be a progenerator for $B$ with $R \\cong \\operatorname{End}\\nolimits_{B}(P)^\\mathrm{op}$ and $Q$ be a progenerator for $C$ with $eRe \\cong \\operatorname{End}\\nolimits_C(Q)^\\mathrm{op}$. Let $G \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev})$ and $H \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits((eRe)^\\mathrm{ev})$ be the equivalences\n$$\nG(M) = \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_B(P,B), M), \\quad H(N) = \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(Q \\otimes \\operatorname{Hom}\\nolimits_C(Q,C), N) \\, .\n$$\nIt suffices to show that the functor $j^\\mathrm{ev}$ is equivalent to $H^{-1} \\circ e(-)e \\circ G$. Indeed, as\n\\begin{align*}\nG(M) & \\cong \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_B(P,B), B^\\mathrm{ev}) \\otimes_{B^\\mathrm{ev}} M\\\\\n& \\cong \\big(\\operatorname{Hom}\\nolimits_B(P,B) \\otimes \\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(\\operatorname{Hom}\\nolimits_B(P,B), B)\\big) \\otimes_{B^\\mathrm{ev}} M\\\\\n& \\cong \\operatorname{Hom}\\nolimits_B(P,B) \\otimes_B M \\otimes_B P \\, ,\n\\end{align*}\n\\begin{align*}\n\\operatorname{Hom}\\nolimits_{B}(Pe \\otimes_{eRe} \\operatorname{Hom}\\nolimits_C(Q,C),B) &\\cong \\operatorname{Hom}\\nolimits_{eRe}(\\operatorname{Hom}\\nolimits_C(Q,C), \\operatorname{Hom}\\nolimits_B(Pe,B))\\\\\n& \\cong Q \\otimes_{eRe} \\operatorname{Hom}\\nolimits_B(Pe,B)\\\\\n& \\cong Q \\otimes_{eRe} \\operatorname{Hom}\\nolimits_R(Re,\\operatorname{Hom}\\nolimits_B(P,B))\\\\\n& \\cong Q \\otimes_{eRe} e \\operatorname{Hom}\\nolimits_B(P,B)\\\\\n& \\cong j(B)\n\\end{align*}\nand $H^{-1}(N') \\cong Q \\otimes_{eRe} N' \\otimes_{eRe} \\operatorname{Hom}\\nolimits_C(Q,C)$, we may conclude\n\\begin{align*}\n(H^{-1} \\circ e (-)& e \\circ G)(M) \\\\\n& \\cong Q \\otimes_{eRe} e \\operatorname{Hom}\\nolimits_B(P,B) \\otimes_B M \\otimes_B P e \\otimes_{eRe} \\operatorname{Hom}\\nolimits_C(Q,C)\\\\\n& \\cong j(B) \\otimes_B M \\otimes_B j(B)^\\vee\n\\end{align*}\nto finish the proof.\n\\end{proof}\n\n\\begin{nn}\nKeep the notation from Proposition \\ref{prop:envrecoll}. Note that in general, the category $\\operatorname{\\mathsf{Mod}}\\nolimits(S\/SfS)$ will be far away from being equivalent to the category of modules over the enveloping algebra of $\\overline{R} = R \/ ReR$. There is, however, a canonical epimorphism\n$$\n\\pi \\colon  \\overline S = S\/SfS \\longrightarrow \\overline R \\otimes \\overline R^\\mathrm{op} \\cong \\left(\\frac{(1-e)R(1-e)}{(1-e)ReR(1-e)} \\right)^\\mathrm{ev}\n$$\nof algebras, where the righthand isomorphism is induced by\n$$\n\\overline R = \\frac{R}{ReR} \\xrightarrow{\\ \\sim \\ } \\frac{(1-e)R(1-e)}{(1-e)ReR(1-e)}, \\, (r + ReR) \\mapsto (1-e)r(1-e) + (1-e)ReR(1-e)\\,.\n$$\nThe epimorphism $\\pi$ fits inside the diagram\n\\begin{equation}\\label{eq:shortpi}\\tag{$\\dagger$}\n\\begin{aligned}\n\\xymatrix@R=14pt@C=14pt{\n& 0 \\ar[d] & 0 \\ar[d] & &\\\\\n& ReR \\otimes ReR \\ar@{=}[r] \\ar[d] & SfS \\ar[d] & &\\\\\n0 \\ar[r] & (ReR \\otimes R) + (R \\otimes ReR) \\ar[r] \\ar[d] & S \\ar[r]^-p \\ar[d] & \\displaystyle\\left(\\frac{(1-e)R(1-e)}{(1-e)ReR(1-e)} \\right)^\\mathrm{ev} \\ar[r] \\ar@{=}[d] & 0\\\\\n0 \\ar[r] & \\displaystyle\\frac{(ReR \\otimes R) + (R \\otimes ReR)}{ReR \\otimes ReR} \\ar[r] \\ar[d] & \\overline{S} \\ar[r]^-{\\pi} \\ar[d] & \\displaystyle\\left(\\frac{(1-e)R(1-e)}{(1-e)ReR(1-e)} \\right)^\\mathrm{ev} \\ar[r] & 0\\\\\n& 0 & 0 & &\n}\n\\end{aligned}\n\\end{equation}\nof $S$-modules -- which has exact rows provided that $R$ and $ReR$ are $K$-flat, and $\\operatorname{Tor}\\nolimits_1^K(\\overline{R}, \\overline{R}) = 0$. Indeed, the commutative diagramm\n$$\n\\xymatrix{\n& 0 \\ar[d] & 0 \\ar[d] & 0 \\ar[d] &\\\\\n0 \\ar[r] & ReR \\otimes ReR \\ar[dr]^-{i} \\ar@{ >->}[d] \\ar@{ >->}[r] & ReR \\otimes R^\\mathrm{op} \\ar@{ >->}[d] \\ar[r] & ReR \\otimes \\overline{R}^\\mathrm{op} \\ar[d] \\ar[r] & 0\\\\\n0 \\ar[r] & R \\otimes ReR \\ar[d] \\ar@{ >->}[r] & R \\otimes R^\\mathrm{op} \\ar@\/_3pc\/[ddd]_-{\\operatorname{coker}\\nolimits(i)} \\ar[d] \\ar[r] \\ar[dr]^-p  & R \\otimes \\overline{R}^\\mathrm{op} \\ar[d] \\ar[r] & 0\\\\\n0 \\ar[r] & \\overline{R} \\otimes ReR \\ar@{ >->}[r]|(.575){ \\ \\ } \\ar[d] & \\overline{R} \\otimes R^\\mathrm{op} \\ar[d] \\ar[r] & \\overline{R} \\otimes \\overline{R}^\\mathrm{op} \\ar[r] \\ar[d] & 0\\\\\n& 0 & 0 & 0 &\\\\\n& & S\/SfS \\ar@{-->}[uur]_-\\pi & &\n}\n$$\nhas exact rows and columns by the assumed $K$-flatness and $\\operatorname{Tor}\\nolimits$-vanishing, whence the kernel of $p$ is $(ReR \\otimes R) + (R \\otimes ReR)$ by \\cite[Sec.\\,II.6, Exer.\\,3]{Mac95}. Further, as $p \\circ i = 0$, the universal property of the cokernel map induces the (necessarily surjective) dashed arrow as indicated. The following lemma can be read off from (\\ref{eq:shortpi}) immediately.\n\\end{nn}\n\n\\begin{lem}\\label{lem:PIisoiff}\nAssume that $R$ and $ReR$ are $K$-flat, and that $\\operatorname{Tor}\\nolimits_1^K(\\overline{R}, \\overline{R}) = 0$. Then the $K$-algebra\n$$\n\\overline S = \\frac{R \\otimes R^\\mathrm{op}}{(R \\otimes R^\\mathrm{op})(e \\otimes e^\\mathrm{op})(R \\otimes R^\\mathrm{op})} = \\frac{R \\otimes R^\\mathrm{op}}{(ReR) \\otimes (R^\\mathrm{op} e^\\mathrm{op} R^\\mathrm{op})}\n$$\nis isomorphic to $\\overline R^\\mathrm{ev} = \\overline R \\otimes \\overline R^\\mathrm{op}$, through $\\pi$, if, and only if, $(ReR \\otimes R) + (R \\otimes ReR) = SfS$. \\qed\n\\end{lem}\n\n\\begin{nn}\nThe lemma tells us that in the case of finite dimensional algebras over a field $K$, $\\overline R \\otimes \\overline R^\\mathrm{op}$ being isomorphic to $\\overline S$ is equivalent to $ReR \\in \\{0,R\\}$, thus, that it is (almost) never the case.\n\nHowever, the map $\\pi$ is a split surjection of $\\overline{S}$-modules in many cases. For instance, if $K$ is an algebraically closed field and $R = KQ$ is the path algebra of the quiver\n$$\n\\xymatrix@R=0pt{\n\\bullet \\ar[r] & \\bullet \\ar[r] & \\cdots \\ar[r] & \\bullet\\\\\n1 & 2 & & n\n} \\, ,\n$$\ni.e., of the Dynkin graph $A_n$ with linear orientation, then the quiver of $R \\otimes R^\\mathrm{op} $ is given by $Q^\\mathrm{ev} = (A_n)^\\mathrm{ev}$\n$$\n\\overbrace{\n\\xymatrix{\n\\bullet \\ar[r]^-x \\ar[d]_-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[d]^-y \\ar[r]^-x & \\bullet \\ar[d]^-y\\\\\n\\bullet \\ar[r]^-x \\ar[d]_-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[d]^-y \\ar[r]^-x & \\bullet \\ar[d]^-y \\\\\n\\tvdots \\ar[d]_-y & \\tvdots \\ar[d]^-y & & \\tvdots \\ar[d]^-y & \\tvdots \\ar[d]^-y \\\\\n\\bullet \\ar[r]^-x \\ar[d]^-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[r]^-x \\ar[d]^-y & \\bullet \\ar[d]^-y\\\\\n\\bullet \\ar[r]^-x & \\bullet \\ar[r]^-x & \\cdots \\ar[r]^-x & \\bullet \\ar[r]^-x & \\bullet \n}}^\\text{$n^2$ vertices}\n$$\nbounded by the commutativity relations $xy = yx$. Considering the primitive idempotent $e = \\varepsilon_1$ that corresponds to the vertex $1$, the algebra $\\overline S$ comes from the quiver \n\\begin{equation}\\label{eq:quiverSbar}\n\\begin{aligned}\n\\overbrace{\n\\xymatrix{\n\\bullet \\ar[r]^-x \\ar[d]_-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[d]^-y \\ar[r]^-x & \\bullet \\ar[d]^-y\\\\\n\\bullet \\ar[r]^-x \\ar[d]_-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[d]^-y \\ar[r]^-x & \\bullet \\ar[d]^-y \\\\\n\\tvdots \\ar[d]_-y & \\tvdots \\ar[d]^-y & & \\tvdots \\ar[d]^-y & \\tvdots \\ar[d]^-y \\\\\n\\bullet \\ar[r]^-x \\ar[d]^-y & \\bullet \\ar[r]^-x \\ar[d]^-y & \\cdots \\ar[r]^-x & \\bullet \\ar[r]^-x \\ar[d]^-y & \\bullet\\\\\n\\bullet \\ar[r]^-x & \\bullet \\ar[r]^-x & \\cdots \\ar[r]^-x & \\bullet &\n\\save \"4,1\"+DL \\PATH ~={**@{.}}\n       '+<10.2pc,0pc> '+<0pc,9.85pc> '+<-10.5pc,0pc> '+<0pc,-9.85pc> '+<0.3pc,0pc>\n\\restore\n}}^\\text{$n^2 - 1$ vertices}\n\\end{aligned}\n\\end{equation}\nwith commutativity relations $xy = yx$, whereas $\\overline R \\otimes \\overline R^\\mathrm{op}$ is the path algebra of $(A_{n-1})^\\mathrm{ev}$, bounded by the commutativity relations. It is thus apparent that $\\pi$ splits, and a right inverse is given by identifying $\\overline R \\otimes \\overline R^\\mathrm{op}$ with the submodule of $\\overline S$ corresponding to the highlighted piece of the quiver (\\ref{eq:quiverSbar}).\n\\end{nn}\n\n\\begin{nn}\nAs $\\operatorname{Hom}\\nolimits_{\\overline{S}}(\\overline R \\otimes \\overline R^\\mathrm{op}, - )$ is right adjoint to the full and faithful restriction functor $\\operatorname{\\mathsf{Mod}}\\nolimits(\\overline R \\otimes \\overline R^\\mathrm{op}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(\\overline S)$ along $\\pi$, and $\\overline R \\otimes \\overline R^\\mathrm{op} \\cong \\operatorname{End}\\nolimits_{\\overline{S}}(\\overline R \\otimes \\overline R^\\mathrm{op})^\\mathrm{op}$, Morita theory yields the following statement.\n\\end{nn}\n\n\\begin{prop}\nAssume that $R$ and $ReR$ are $K$-flat, and that $\\operatorname{Tor}\\nolimits_1^K(\\overline{R}, \\overline{R}) = 0$. Then the restriction functor\n$$\n\\pi_\\star \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(\\overline R \\otimes \\overline R^\\mathrm{op}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(\\overline S)\n$$\nis an equivalence of categories if, and only if, the $\\overline S$-module epimorphism $\\pi$ splits and there is a surjective $\\overline S$-module homomorphism $(\\overline R \\otimes \\overline R^\\mathrm{op})^n \\rightarrow \\overline S$ for some integer $n \\geqslant 1$. \\qed\n\\end{prop}\n\nThe main theorem of this section is the following.\n\n\\begin{thm}\\label{thm:mainthm}\nLet $\\mathcal R = \\mathcal R(A,B,C)$ be a $K$-linear recollement of module categories. Assume that $B$ is projective as a $K$-module and that \n$$\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0 \\, .\n$$\nThe image of $B$ under $j^\\mathrm{ev}$ is canonically isomorphic to $C$ $($as a $C^\\mathrm{ev}$-module$)$ and the graded map\n$$\nj^\\mathrm{ev}_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\xrightarrow{\\ j^\\mathrm{ev} \\ } \\operatorname{Ext}\\nolimits^\\ast_{C^\\mathrm{ev}}(j^\\mathrm{ev}(B), j^\\mathrm{ev}(B)) \\xrightarrow{\\ \\sim \\ } \\operatorname{HH}\\nolimits^\\ast(C)\n$$\ninduced by $\\mathcal R^\\mathrm{ev}$ is a homomorphism of strict Gerstenhaber algebras.\n\\end{thm}\n\nThe proof of the theorem is spread over the next couple of statements. We begin by proving a fact that should be well-known, for which, however, we were not able to track down a reference.\n\n\n\\begin{lem}\\label{lem:Tor1equ}\nLet $\\mathcal R(A,B,C)$ be a recollement of module categories being equivalent to a standard recollement $\\mathcal R(R,e)$. Then\n\\begin{align*}\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee,j(L)) = 0& \\ \\ \\text{$($for all $L \\in {\\mathsf P}(B))$}\\\\ &\\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0\\\\ \n&\\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^{eRe}_1(Re,eR) = 0 \\\\\n&\\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^{eRe}_1(Me,eN) = 0 \\ \\ \\text{$($for all $M,N \\in {\\mathsf P}(R))$}\\, .\n\\end{align*}\n\\end{lem}\n\n\\begin{proof}\nIndeed, the first equivalence is due to\n$$\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee,j(F)) \\cong \\bigoplus_{i \\in I}\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee,j(B))\n$$\nfor each free $B$-module $F = \\bigoplus_{i \\in I}{B}$ (see Corollary \\ref{cor:recolldirectsums}). For the second equivalence, let $P$ and $Q$ be progenerators for $B$ and $C$ giving rise to the equivalence $\\mathcal R(A,B,C) \\xrightarrow \\sim \\mathcal R(R,e)$. We start by taking a projective resolution $\\mathbb P(eR) \\rightarrow eR \\rightarrow 0$ of $eR$ over $eRe$ and assume that $\\operatorname{Tor}\\nolimits^{eRe}_1(Re,eR) = 0$. The complex\n$$\nRe \\otimes_{eRe} \\mathbb P(eR) \\longrightarrow Re \\otimes_{eRe} eR \\longrightarrow 0\n$$\nconsists of projective $R$-modules in degrees $\\geqslant 0$ and, moreover, has vanishing homology in degrees $0$ and $1$ (as its $i$-th homology computes as $\\operatorname{Tor}\\nolimits^{eRe}_i(Re,eR)$). Thus we obtain an exact sequence\n$$\n0 \\longrightarrow Re \\otimes_{eRe} X \\longrightarrow Re \\otimes_{eRe} P_1 \\longrightarrow Re \\otimes_{eRe} P_0 \\longrightarrow Re \\otimes_{eRe} eR \\longrightarrow 0 \\,,\n$$\nwhere $P_0$ and $P_1$ are projective $eRe$-modules. The composition $j \\circ (P \\otimes_R -)$ of $j$ with the quasi-inverse functor $P \\otimes_R -$ of $\\operatorname{Hom}\\nolimits_B(P,-)$ takes that sequence to\n\\begin{equation}\\label{eq:sequence}\\tag{$\\diamond$}\n0 \\longrightarrow j(Pe \\otimes_{eRe} X) \\longrightarrow j(Pe \\otimes_{eRe} P_1) \\longrightarrow j(Pe \\otimes_{eRe} P_0) \\longrightarrow j(Pe \\otimes_{eRe} eR) \\longrightarrow 0 \\,.\n\\end{equation}\nAs $j \\circ (P \\otimes_R -)$ is naturally isomorphic to $(Q \\otimes_{eRe} -) \\circ e(-)$, the exact sequence above is (as a complex of $C$-modules) isomorphic to\n$$\n0 \\longrightarrow Q \\otimes_{eRe} X \\longrightarrow Q \\otimes_{eRe} P_1 \\longrightarrow Q \\otimes_{eRe} P_0 \\longrightarrow Q \\otimes_{eRe} eR \\longrightarrow 0 \\,.\n$$\nIn particular, $j(Pe \\otimes_{eRe} P_1)$ and $j(Pe \\otimes_{eRe} P_0)$ are projective $C$-modules. Let us write $\\overline{j}(-) = j(Pe \\otimes_{eRe} (-))$ for abbreviation. We show that the exact sequence (\\ref{eq:sequence}) remains exact after applying $j(B)^\\vee \\otimes_C -$, which will imply that $\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(Pe \\otimes_{eRe} eR)) = 0$. Indeed, as already mentioned above, we have natural $C$-module isomorphisms\n$$\n\\overline{j}(E) = j(Pe \\otimes_{eRe} E) \\cong Q \\otimes_{eRe} E \\quad \\text{(for all modules $E \\in \\operatorname{\\mathsf{Mod}}\\nolimits(eRe)$)}\n$$\nand the $B$-$C$-bimodule isomorphism\n\\begin{align*}\nj(B)^\\vee &\\cong \\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P,B), B)\\\\\n& \\cong \\operatorname{Hom}\\nolimits_{(eRe)^\\mathrm{op}}(Q, \\operatorname{Hom}\\nolimits_{R^\\mathrm{op}}(eR, \\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(\\operatorname{Hom}\\nolimits_B(P,B),B)))\\\\\n& \\cong \\operatorname{Hom}\\nolimits_{(eRe)^\\mathrm{op}}(Q, Pe)\\\\\n&\\cong Pe \\otimes_{eRe} \\operatorname{Hom}\\nolimits_{(eRe)^\\mathrm{op}}(Q,eRe) \\,.\n\\end{align*}\nThe map $\\operatorname{Hom}\\nolimits_{(eRe)^\\mathrm{op}}(Q,eRe) \\otimes_C Q \\xrightarrow{\\sim} \\operatorname{Hom}\\nolimits_{C}(Q,C) \\otimes_C Q \\xrightarrow{\\ \\, } eRe, \\, \\varphi \\otimes q \\mapsto \\varphi(q)$, see for instance \\cite[\\S 22]{AnFu92}, is an $eRe$-bimodule isomorphism which gives rise to an isomorphism of complexes,\n$$\n\\xymatrix@C=11.5pt{\n0 \\ar[r] & j(B)^\\vee \\otimes_C \\overline{j}(X) \\ar[r] \\ar[d] & j(B)^\\vee \\otimes_C \\overline{j}(eP_1) \\ar[r] \\ar[d] & j(B)^\\vee \\otimes_C \\overline{j}(eP_0) \\ar[r] \\ar[d] & j(B)^\\vee \\otimes_C \\overline{j}(eR) \\ar[r] \\ar[d] & 0\\, \\,\\\\\n0 \\ar[r] & Pe \\otimes_{eRe} X \\ar[r] & Pe \\otimes_{eRe} eP_1 \\ar[r] & Pe \\otimes_{eRe} eP_0 \\ar[r] & Pe \\otimes_{eRe} eR \\ar[r] & 0\\, ,\n}\n$$\nwhence the top complex is acyclic as claimed. It thus follows that, for some $n \\geqslant 1$,\n\\begin{align*}\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee,j(B)) &\\cong \\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P,B))\\\\ \n&\\subseteq \\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P,P^n))\\\\\n& \\cong \\bigoplus_{i = 1}^n \\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, Q \\otimes_{eRe} eR)\\\\\n& \\cong \\bigoplus_{i = 1}^n \\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(Pe \\otimes_{eRe} eR)) = 0\n\\end{align*}\nwhence one implication of the second equivalence follows. Similar arguments apply to deduce the implication\n$$\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0 \\quad \\Longrightarrow \\quad \\operatorname{Tor}\\nolimits^{eRe}_1(Re,eR) = 0 \\,.\n$$\nFinally, the last equivalence follows from\n$$\n\\operatorname{Tor}\\nolimits^{eRe}_1(F_1 e, eF_2) \\cong \\bigoplus_{i_1 \\in I_1}\\bigoplus_{i_2 \\in I_2}\\operatorname{Tor}\\nolimits^{eRe}_1(Re,eR)\n$$\nfor all free right $R$-modules $F_1 = \\bigoplus_{i_1 \\in I_1}{R}$ and all free left $R$-modules $F_2 = \\bigoplus_{i_2 \\in I_2}{R}$.\n\\end{proof}\n\n\\begin{cor}\nLet $\\mathcal R(A,B,C)$ and $\\mathcal R(A',B',C')$ be equivalent recollements of module categories. Then\n$$\n\\operatorname{Tor}\\nolimits^C_1(j(B)^\\vee, j(B)) = 0 \\quad \\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^{C'}_1(j'(B')^\\vee, j'(B')) = 0 \\,.\n$$\\qed\n\\end{cor}\n\nFor the next proposition only, we let $B$ be a $K$-algebra, $e \\in B$ be an idempotent, $C = eBe$ and $\\mathfrak A_e$ the functor $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$, $\\mathfrak A_e(M) = eMe$.\n\n\\begin{prop}\\label{prop:funcproper}\nThe following statements hold true for the functor $$\\mathfrak A_e = e(-)e \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev}).$$\n\\begin{enumerate}[\\rm(1)]\n\\item\\label{prop:funcproper:1} $\\mathfrak A_e$ is exact.\n\\item\\label{prop:funcproper:2} $\\mathfrak A_e$ is an almost costrong monoidal functor. It will be a strong monoidal functor, if the restricted multiplication map $\\mu_e \\colon Be \\otimes_{eBe} eB \\rightarrow B$ is an isomorphism.\n\\item\\label{prop:funcproper:3} The functor\n$$\nBe \\otimes_C (-) \\otimes_C eB \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})\n$$\nis a right invers of $\\mathfrak A_e$, that is, $\\mathfrak A_e (Be \\otimes_C (-) \\otimes_C eB) \\cong \\operatorname{Id}\\nolimits_{\\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})}$. More precisely, $(Be \\otimes_C (-) \\otimes_C eB, \\mathfrak A_e)$ forms an adjoint pair, whose unit is an isomorphism. For modules $M \\in \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ and $N \\in \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, unit and counit are given by\n$$\n\\eta_M \\colon M \\longrightarrow eBe \\otimes_C M \\otimes_C eBe, \\ m \\mapsto e \\otimes m \\otimes e,\n$$\n$$\n\\varepsilon_N\\colon Be \\otimes_C e N e \\otimes_C eB \\longrightarrow N, \\ be \\otimes ene \\otimes eb' \\mapsto (be) n (eb').\n$$\n\\item\\label{prop:funcproper:4} If $eB$ is projective as a left $C=eBe$-module $($$Be$ is projective as a right $C=eBe$-module$)$, and if $M$ is a $B^\\mathrm{ev}$-module which is projective as a left $B$-module $($as a right $B$-module$)$, then $\\mathfrak A_e(M)$ is projective as a left $C$-module $($as a right $C$-module$)$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nAs $e(-)e \\cong eB \\otimes_B (-) \\otimes_B Be$, the statement (\\ref{prop:funcproper:1}) is obvious. As for the second item, we have to name an isomorphism $t_0\\colon C = eBe \\rightarrow \\mathfrak A_e(B)$ as well as natural homomorphisms\n$$\nt_{M,N} \\colon \\mathfrak A_e(M) \\otimes_C \\mathfrak A_e(N) \\longrightarrow \\mathfrak A_e(M \\otimes_C N) \\quad \\text{(for all $M, N \\in \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$)}.\n$$\nTo begin with, notice that $\\mathfrak A_e(B) = eBe = C$. The multiplication map $\\mu_e \\colon Be \\otimes_{eBe} eB \\rightarrow B$ gives rise to the homomorphism $t'_{M,N} = eB \\otimes_B M \\otimes_B \\mu_e \\otimes_B N \\otimes_B Be$,\n$$\nt'_{M,N} \\colon eB \\otimes_B M \\otimes_B (Be \\otimes_{eBe} eB) \\otimes_B N \\otimes_B Be \\longrightarrow eB \\otimes_B M \\otimes_B B \\otimes_B N \\otimes_B Be \\, ,\n$$\nwhich leads to a homomorphism $t_{M,N}$ between\n\\begin{align*}\n\\mathfrak A_e(M) \\otimes_C \\mathfrak A_e(N) \n&= eMe \\otimes_{eBe} eNe\\\\ \n&= (eB \\otimes_B M \\otimes_B Be) \\otimes_{eBe} (eB \\otimes_B N \\otimes_B Be)\\\\\n&= eB \\otimes_B M \\otimes_B (Be \\otimes_{eBe} eB) \\otimes_B N \\otimes_B Be\n\\intertext{and}\neB \\otimes_B M \\otimes_B B \\otimes_B N \\otimes_B Be\n&\\cong eB \\otimes_B M \\otimes_B N \\otimes_B Be\\\\ \n&\\cong eM \\otimes_B Ne\\\\\n&= \\mathfrak A_e(M \\otimes_B N) \\, .\n\\end{align*}\nNaturality of the $t_{M,N}$ is easily established. Item (\\ref{prop:funcproper:3}) is either a straightforward verification, or a reminder of the fact that $\\mathfrak A_e$ actually belongs to the standard recollement $\\mathcal R(B^\\mathrm{ev}, e \\otimes e^\\mathrm{op})$. As for the last item, if $eB$ is $C$-projective, then so is $eM$, and thus\n$$\n\\operatorname{Hom}\\nolimits_{C}(eMe, -) \\cong \\operatorname{Hom}\\nolimits_{C}(eM \\otimes_B Be, -) \\cong \\operatorname{Hom}\\nolimits_B(Be, \\operatorname{Hom}\\nolimits_{C}(eM,-))\n$$\nis exact as required.\n\\end{proof}\n\n\\begin{prop}\\label{prop:moritamonoidal}\nLet $R$ be a $K$-algebra, $P$ be a progenerator for $R$ and let $S = \\operatorname{End}\\nolimits_R(P)^\\mathrm{op}$. Let $\\mathfrak A_P$ be the equivalence\n$$\n\\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_R(P,R), -) \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev})\n$$\nof categories. The following statements hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item $\\mathfrak A_P$ is exact.\n\\item\\label{prop:moritamonoidal:2} $\\mathfrak A_P$ is a strong monoidal functor.\n\\item\\label{prop:moritamonoidal:3} If $M$ is an $R^\\mathrm{ev}$-module that is projective as a left $R$-module $($as a right $R$-module$)$, then $\\mathfrak A_P(M)$ is projective as a left $S$-module $($as a right $S$-module$)$.\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nWe will show that $\\mathfrak A_P$ actually may be expressed as the composite of functors arising from certain idempotents, so that all statements will follow directly from Proposition \\ref{prop:funcproper}. The matrix\n$$\nB = \\left(\\begin{matrix}\nR & P\\\\\n\\operatorname{Hom}\\nolimits_R(P, R) & \\operatorname{End}\\nolimits_R(P)^\\mathrm{op}\n\\end{matrix}\\right) = \n\\left(\\begin{matrix}\nR & P\\\\\n\\operatorname{Hom}\\nolimits_R(P, R) & S\n\\end{matrix}\\right)\n$$\ncan be understood as an algebra through the following natural bimodule isomorphisms:\n$$\nP \\otimes_S \\operatorname{Hom}\\nolimits_R(P,R) \\longrightarrow R, \\ p \\otimes \\varphi \\mapsto \\varphi(p) \\,,\n$$\n$$\n\\operatorname{Hom}\\nolimits_R(P, R) \\otimes_R P \\longrightarrow S, \\ \\varphi \\otimes p \\mapsto (p' \\mapsto \\varphi(p')p) \\, .\n$$\nWe have two canonical idempotens in $B$, namely\n$$\ne_R = \\left(\\begin{matrix}\n1 & 0\\\\\n0 & 0\n\\end{matrix}\\right)\n\\quad \\text{and} \\quad\ne_S = \\left(\\begin{matrix}\n0 & 0\\\\\n0 & 1\n\\end{matrix}\\right) \\, .\n$$\nThe quotient algebras $\\overline{B}_R = B\/B e_R B$ and $\\overline{B}_S = B\/B e_S B$ both vanish, whence the functors\n$$\ne_\\Sigma (-) e_\\Sigma \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits((e_\\Sigma B e_\\Sigma)^\\mathrm{ev}) = \\operatorname{\\mathsf{Mod}}\\nolimits(\\Sigma^\\mathrm{ev}) \\quad \\text{(inside of $\\mathcal R(B, e_\\Sigma)$)}\n$$\nare equivalences for $\\Sigma \\in \\{R, S\\}$ by Lemma \\ref{lem:PIisoiff}. Thus, the right adjoint of $e_R(-)e_R$, given by\n$$\n\\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(e_R B \\otimes B e_R, -) = \\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(\\left(\\begin{smallmatrix}\nR & P\\\\\n0 & 0\n\\end{smallmatrix}\\right) \\otimes \\left(\\begin{smallmatrix}\nR & 0\\\\\n\\operatorname{Hom}\\nolimits_R(P,R) & 0\n\\end{smallmatrix}\\right), -)\n\\, ,\n$$\nis also an isomorphism and, as $e_R (-) e_R$ is strong monoidal, a strong monoidal functor. When composed with $e_S(-)e_S$, we obtain the functor\n\\begin{align*}\ne_S\\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(e_R B \\otimes B e_R, -)e_S &\\cong \\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}((e_R B e_S)\\otimes (e_S B e_R), -)\\\\ &= \\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_R(P,R), -)\\\\\n&= \\mathfrak A_P(-)\n\\end{align*}\nwhich is a strong monoidal equivalence $(\\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}), \\otimes_R, R) \\xrightarrow{\\sim} (\\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev}),\\otimes_S, S)$, as desired. It takes left projective modules to left projective modules by Proposition \\ref{prop:funcproper}.\n\\end{proof}\n\nWe can now prove our main theorem.\n\n\\begin{proof}[Proof of Theorem $\\ref{thm:mainthm}$]\nThe functor $j^\\mathrm{ev}\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ restricts to an exact and almost costrong monoidal functor\n$$\nj^\\mathrm{ev} \\colon ({\\mathsf P}(B), \\otimes_B, B) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})\n$$\nby Propositions \\ref{prop:funcproper} and \\ref{prop:moritamonoidal}. In order to apply Theorem \\ref{thm:gersthochcomp}, it suffices to show (since ${\\mathsf P}(B)$ is closed under kernels of epimorphisms) that if $0 \\rightarrow L \\rightarrow M \\rightarrow N \\rightarrow 0$ is an admissible exact sequence in ${\\mathsf P}(B)$, then\n$$\n0 \\longrightarrow j^\\mathrm{ev}(L) \\otimes_C j^\\mathrm{ev}(X) \\longrightarrow j^\\mathrm{ev}(M) \\otimes_C j^\\mathrm{ev}(X) \\longrightarrow j^\\mathrm{ev}(N) \\otimes_C j^\\mathrm{ev}(X) \\longrightarrow 0\n$$\nis exact for all modules $X \\in {\\mathsf P}(B)$. In view of Proposition \\ref{prop:moritamonoidal}(\\ref{prop:moritamonoidal:2})+(\\ref{prop:moritamonoidal:3}), this is equivalent to showing that if $0 \\rightarrow L \\rightarrow M \\rightarrow N \\rightarrow 0$ is an admissible exact sequence in ${\\mathsf P}(R)$, then\n$$\n0 \\longrightarrow eLe \\otimes_{eRe} eXe \\longrightarrow eMe \\otimes_{eRe} eXe \\longrightarrow eNe \\otimes_{eRe} eXe \\longrightarrow 0\n$$\nis exact for all modules $X \\in {\\mathsf P}(R)$. Indeed, one has the exact sequence\n$$\n\\operatorname{Tor}\\nolimits^{eRe}_1(Ne,eX) \\xrightarrow{\\ a \\ } Le \\otimes_{eRe} eX \\longrightarrow Me \\otimes_{eRe} eX \\longrightarrow Ne \\otimes_{eRe} eX \\longrightarrow 0\n$$\nwherein the map $a$ vanishes by Lemma \\ref{lem:Tor1equ}; thus, applying $e(-)e$ yields the exact sequence\n$$\n0 \\longrightarrow eLe \\otimes_{eRe} eXe \\longrightarrow eMe \\otimes_{eRe} eXe \\longrightarrow eNe \\otimes_{eRe} eXe \\longrightarrow 0\n$$\nas desired.\n\\end{proof}\n\nIn the following sections, we will illustrate the theorem's potential by a couple of applications. However, let us close this section by observing that a recollement of module categories $\\mathcal R(A,B,C)$ gives rise to very nicely behaved exact and monoidal functors if we assume that that $j(B)$ or $\\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(j(B),B)$ are flat over $C$.\n\n\\begin{nn}\\label{nn:exactsubcats}\nLet us define a subcategory ${{\\mathsf P}}_\\lambda(B;e) \\subseteq {\\mathsf F}_\\lambda(B) \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ by letting $M \\in {\\mathsf P}(B)$ belong to ${{\\mathsf P}}_\\lambda(B;e)$ if, and only if, $\\mathfrak A_e(M)$ belongs to ${\\mathsf F}_\\lambda(C)$. The subcategory ${{\\mathsf P}}_\\lambda(B;e)$ is non-empty, as $B$ belongs to it. Define further $\\overline{{\\mathsf P}}_\\lambda(B;e) \\subseteq {{\\mathsf P}}_\\lambda(B;e)$ to be the full subcategory that consists of all modules $M$ in ${{\\mathsf P}}_\\lambda(B;e)$ such that $M \\otimes_B N \\in {{\\mathsf P}}_\\lambda(B;e)$ for all $N \\in {\\mathsf P}_e(B)$; it is evident, that $B$ is a member of that category. By replacing ${\\mathsf F}_\\lambda(B)$ by ${\\mathsf F}_\\varrho(B)$ we obtain, in a similar way, a full subcategory $\\overline{{\\mathsf P}}_\\varrho(B;e)$ of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$. Note that ${{\\mathsf P}}_\\lambda(B;1) = \\overline{{\\mathsf P}}_\\lambda(B;1) = {\\mathsf P}_\\lambda(B)$ and ${{\\mathsf P}}_\\varrho(B;1) = \\overline{{\\mathsf P}}_\\varrho(B;1) = {\\mathsf P}_\\varrho(B)$. The following observation is immediate from Lemma \\ref{lem:monoclosure}.\n\\end{nn}\n\n\\begin{lem}\nThe subcategories $\\overline{{\\mathsf P}}_\\lambda(B;e)$ and $\\overline{{\\mathsf P}}_\\varrho(B;e)$ of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ are closed under taking arbitrary direct sums and direct summands. Furthermore, they are closed under extensions and under taking tensor products over $B$. Thus, $(\\overline{{\\mathsf P}}_\\lambda(B;e), \\otimes_B, B)$ and $(\\overline{{\\mathsf P}}_\\varrho(B;e), \\otimes_B, B)$ are exact monoidal subcategories of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$. \\qed\n\\end{lem}\n\n\\begin{lem}\\label{lem:projflatidemp}\nLet $\\mathcal R = \\mathcal R(A,B,C)$ be a recollement of module categories, and let $\\mathcal R(R,e)$ be a standard recollement that is equivalent to it. Then\n\\begin{enumerate}[\\rm(1)]\n\\item $j(B)$ is $C$-flat if, and only if, $eR$ is $eRe$-flat,\n\\item $j(B)^\\vee = \\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(j(B), B)$ is $C$-flat if, and only if, $Re$ is $eRe$-flat,\n\\item\\label{lem:projflatidemp:3} $j(B)$ is $C$-projective if, and only if, $eR$ is $eRe$-projective, and\n\\item $j(B)^\\vee$ is $C$-projective if, and only if, $Re$ is $eRe$-projective.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nLet $P$ and $Q$ be progenerators for $B$ and $C$ defining the equivalence $\\mathcal R(A,B,C) \\xrightarrow \\sim \\mathcal R(R,e)$. We only take care of the first item in detail, as the remaining items may be proven similarly. As $\\mathcal R$ and $\\mathcal R(R,e)$ are equivalent, there are progenerators $P$ for $B$ and $Q$ for $C$ with $R \\cong \\operatorname{End}\\nolimits_{B}(P)^\\mathrm{op}$ and $eRe \\cong \\operatorname{End}\\nolimits_C(Q)^\\mathrm{op}$ such that there is a natural isomorphism\n\\begin{align*}\nj(-) \\xrightarrow{\\, \\sim \\,} \\left( Q \\otimes_{eRe} (-) \\right)& \\circ e(-) \\circ \\operatorname{Hom}\\nolimits_B(P,-)\\\\ &= Q \\otimes_{eRe} e \\operatorname{Hom}\\nolimits_B(P,-) \\, .\n\\end{align*}\nIt thus follows that \n\\begin{equation}\\label{eq:isolem1}\n\\begin{aligned}\nj(B) &\\cong Q \\otimes_{eRe} e \\operatorname{Hom}\\nolimits_B(P,B)\\\\ &\\cong Q \\otimes_{eRe} eR \\otimes_R \\operatorname{Hom}\\nolimits_B(P,B)\\,.\n\\end{aligned}\n\\end{equation}\nAs the functors $Q \\otimes_{eRe} (-)$ and $\\operatorname{Hom}\\nolimits_B(P,-)$ are invertible, with quasi-invers functors $\\operatorname{Hom}\\nolimits_C(Q,-)$ and $P \\otimes_R (-)$, we also have\n\\begin{align*}\neR \\otimes_R (-) & \\cong \\operatorname{Hom}\\nolimits_C(Q, j(P \\otimes_R(-)) \\\\\n& \\cong \\operatorname{Hom}\\nolimits_C(Q,C) \\otimes_C j(P \\otimes_R (-)) \\,,\n\\end{align*}\nas $Q$ is finitely generated projective over $C$, and hence \n\\begin{equation}\\label{eq:isolem2}\neR \\cong \\operatorname{Hom}\\nolimits_C(Q,C) \\otimes_C j(P)\n\\end{equation}\nObserve that $- \\otimes_C j(B)$ is exact if, and only if, $- \\otimes_C j(P)$ is exact. The modules $Q \\in \\operatorname{\\mathsf{Mod}}\\nolimits(C)$, $\\operatorname{Hom}\\nolimits_B(P,B) \\in \\operatorname{\\mathsf{Mod}}\\nolimits(R)$, $P \\in \\operatorname{\\mathsf{Mod}}\\nolimits(B)$ and $\\operatorname{Hom}\\nolimits_C(Q,C) \\in \\operatorname{\\mathsf{Mod}}\\nolimits(eRe)$ are projective; thus, by the equations (\\ref{eq:isolem1}) and (\\ref{eq:isolem2}), it follows that $- \\otimes_C j(B)$ is exact if, and only if, $- \\otimes_{eRe} eR$ is exact.\n\\end{proof}\n\n\\begin{nn}\nLet $\\mathcal R = \\mathcal R(A,B,C)$ be a recollement of module categories, being equivalent to a standard recollement $\\mathcal R(R,e)$. Substituting $\\mathfrak A_e$ in Paragraph \\ref{nn:exactsubcats} by $j^\\mathrm{ev}$, we obtain full, exact monoidal subcategories $(\\overline{{\\mathsf P}}_\\lambda(\\mathcal R), \\otimes_B, B)$ and $(\\overline{{\\mathsf P}}_\\varrho(\\mathcal R), \\otimes_B, B)$ of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$. Note that $B$ indeed belongs to either of these categories, as $j^\\mathrm{ev}(B) \\cong C$.\n\\end{nn}\n\n\\begin{prop}\\label{prop:flatprojrecoll}\nLet $\\mathcal R = \\mathcal R(A,B,C)$ be a recollement of module categories.\n\\begin{enumerate}[\\rm(1)]\n\\item\\label{prop:flatprojrecoll:1} The functor $j^\\mathrm{ev}: \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ restricts to an exact and monoidal functor\n$$\n(\\overline{{\\mathsf P}}_\\lambda(\\mathcal R), \\otimes_B, B) \\longrightarrow (\\mathsf F_\\lambda(C), \\otimes_C, C)\n$$\nbetween exact monoidal subcategories. Moreover, the inclusion functor induces an isomorphism\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}_\\lambda(\\mathcal R)}(B,B) \\xrightarrow{\\,\\sim\\,} \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B) \n$$\nif $j(B)$ is $C$-projective.\n\n\\item The functor $j^\\mathrm{ev}\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B)^\\mathrm{ev} \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ restricts to an exact and monoidal functor\n$$\n(\\overline{{\\mathsf P}}_\\varrho(\\mathcal R), \\otimes_B, B) \\longrightarrow (\\mathsf F_\\varrho(C), \\otimes_C, C)\n$$\nbetween exact monoidal subcategories. Moreover, the inclusion functor induces an isomorphism\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}_\\varrho(\\mathcal R)}(B,B) \\xrightarrow{\\,\\sim\\,} \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B)\n$$\nif $j(B)^\\vee$ is $C$-projective.\n\\end{enumerate}\nFurthermore, if for $? \\in \\{\\lambda, \\varrho\\}$ the map $\\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}_?(\\mathcal R)}(B,B) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B)$ is an isomorphism, then the induced map\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B) \\xrightarrow{\\,\\sim\\,} \\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}_?(\\mathcal R)}(B,B) \\xrightarrow{\\, \\ \\,} \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf F_?(C)}(C,C) \\xrightarrow{\\,\\ \\,} \\operatorname{Ext}\\nolimits^\\ast_{C^\\mathrm{ev}}(C,C)\n$$\nagrees with the map $j^\\mathrm{ev}_\\ast$ as mentioned in the statement of Theorem $\\ref{thm:mainthm}$.\n\\end{prop}\n\n\\begin{proof}\nWe only prove item (\\ref{prop:flatprojrecoll:1}). The restriction of $j^\\mathrm{ev}$ to $(\\overline{{\\mathsf P}}_\\lambda(\\mathcal R), \\otimes_B, B)$ is of course exact, and also monoidal by Propositions \\ref{prop:funcproper} and \\ref{prop:moritamonoidal}. It is by definition that $j^\\mathrm{ev}$ maps $\\overline{{\\mathsf P}}_\\lambda(\\mathcal R)$ to ${\\mathsf F}_\\lambda(C)$. Now assume that $j(B)$ is $C$-projective. Replacing ${\\mathsf F}_\\lambda(eBe)$ by ${\\mathsf P}_\\lambda(C)$, and $\\mathfrak A_e$ by $j^\\mathrm{ev}$ in Paragraph \\ref{nn:exactsubcats}, we obtain extension closed subcategories $\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R) \\subseteq {{\\mathsf P}}^1_\\lambda(\\mathcal R) \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, the former being an exact monoidal subcategory of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$. Further, $\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R)$ is closed under kernels of epimorphisms, as ${\\mathsf P}_\\lambda(C)$ is, and contains $\\operatorname{\\mathsf{Proj}}\\nolimits(B^\\mathrm{ev})$. Indeed, the functor $j^\\mathrm{ev}$ sends $B \\otimes B$ to\n$$\nQ \\otimes_{eRe} \\left(e\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_B(P,B), B \\otimes B) e\\right) \\otimes_{eRe} \\operatorname{Hom}\\nolimits_C(Q,C) \\,.\n$$\nThe module $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(P \\otimes \\operatorname{Hom}\\nolimits_B(P,B), B^\\mathrm{ev})$ is a progenerator for $R^\\mathrm{ev}$ which consequently is a direct summand of $(R^\\mathrm{ev})^n$ for an integer $n \\geqslant 1$. Further $eR \\otimes Re$ is a projective left $eRe$-module by Lemma \\ref{lem:projflatidemp}(\\ref{lem:projflatidemp:3}), therefore so is\n$$\nQ \\otimes_{eRe} (eR \\otimes Re) \\otimes_{eRe} \\operatorname{Hom}\\nolimits_C(Q,C)\n$$\nover $C$; thus $j^\\mathrm{ev}(B \\otimes B)$ is a direct summand of a $C$-projective $C^\\mathrm{ev}$-module, hence itself $C$-projective. It follows that $\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R)$ is entirely extension closed by Proposition \\ref{prop:entireextclo}, whence the commutative diagram\n$$\n\\xymatrix@C=4pt{\n& \\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R)}(B,B) \\ar[dr]^-{\\sim} \\ar[dl]_-{\\sim} & \\\\\n\\operatorname{Ext}\\nolimits^\\ast_{\\overline{{\\mathsf P}}_\\lambda(\\mathcal R)}(B,B) \\ar[rr] & & \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,B)\n}\n$$\ninduced by the inclusion functors $\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R) \\hookrightarrow \\overline{{\\mathsf P}}_\\lambda(\\mathcal R)$ and $\\overline{{\\mathsf P}}^1_\\lambda(\\mathcal R) \\hookrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ yields the claim.\n\\end{proof}\n\n\n\n\\section{The long exact sequence of Buchweitz}\\label{sec:buchweitz}\n\n\\begin{nn}\nIn this section, we investigate a long exact sequence that has been introduced by Buchweitz in \\cite{Bu03}. By $B$ we will denote a $K$-algebra which is projective as a $K$-module. Further, we let $e \\in B$ be an idempotent element such that\n$$\n\\operatorname{Tor}\\nolimits^{eBe}_1(Be,eB) = 0\n$$\nand abbriviate $\\overline B = B \/ BeB$, $C = eBe$. Of course, $C$ is a $K$-projective $K$-algebra as well, whereas $\\overline B$ does not have to be. We like to point out that the above $\\operatorname{Tor}\\nolimits$-vanishing condition will only be relevant for the proof of Theorem \\ref{thm:chigersten} (as one easily believes in light of Theorem \\ref{thm:mainthm}), but not for the construction of the map it concerns itself with.\n\\end{nn}\n\n\\begin{nn}\\label{nn:buchweitz}\nAs usual, let $\\mathbb B B$ denote the bar resolution of $B$, and let $\\mathbb B C$ be the bar resolution for $C$. There is an injection\n$$\n\\tilde{\\mu}_e \\colon Be \\otimes_C \\mathbb B C \\otimes_C eB \\longrightarrow \\mathbb B B\n$$\nof complexes that lifts the multiplication map $\\mu_e \\colon Be \\otimes_C eB \\rightarrow B$. It is induced by the canonical isomorphisms\n$$\n\\xymatrix@C=16pt{\nBe \\otimes_C C^{\\otimes (n + 2)} \\otimes_C eB \\ar[r]^-{\\sim} & Be \\otimes (eBe)^{\\otimes n} \\otimes eB \\subseteq B^{\\otimes(n+2)}\n}\n$$\nfor $n \\geqslant 0$, which are compatible with the occurring differentials and therefore identify $Be \\otimes_C \\mathbb B C \\otimes_C eB$ with a subcomplex of $\\mathbb B B$. We put\n$$\n\\mathbb B(B\/C) = \\operatorname{Coker}\\nolimits(\\tilde{\\mu}_e \\colon Be \\otimes_C \\mathbb B C \\otimes_C eB \\rightarrow \\mathbb B B)\\,,\n$$\nand hence obtain a short exact sequence\n$$\n0 \\longrightarrow Be \\otimes_C \\mathbb B C \\otimes_C eB \\longrightarrow \\mathbb B B \\longrightarrow \\mathbb B(B\/C) \\longrightarrow 0\n$$\nof complexes. It is semi-split, in that in each degree, the $B^\\mathrm{ev}$-module homomorphisms are split. Indeed, $Be \\otimes (eBe)^{\\otimes n} \\otimes eB$ is an evident direct summand of \n$$\nB^{\\otimes (n+2)} = (Be \\oplus Be') \\otimes (eBe \\oplus eBe' \\oplus e'Be \\oplus e'Be')^{\\otimes n} \\otimes (eB \\oplus e'B)\\,,\n$$\nwhere $e' = 1 - e$. Since\n$$\n\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_C C^{\\otimes(n+2)} \\otimes_C eB, M) \\cong \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(C^{\\otimes(n+2)}, eMe) \\quad (\\text{for $n \\geqslant 0$})\n$$\nby adjunction, we obtain a long exact sequence in Hochschild cohomology:\n$$\n\\cdots \\longrightarrow \\operatorname{HH}\\nolimits^i(B\/C,M) \\longrightarrow \\operatorname{HH}\\nolimits^i(B,M) \\longrightarrow \\operatorname{HH}\\nolimits^i(C, eMe) \\longrightarrow \\operatorname{HH}\\nolimits^{i+1}(B\/C,M) \\longrightarrow \\cdots \\, ,\n$$\nwhere $\\operatorname{HH}\\nolimits^\\ast(B\/C,M)$ denotes the cohomology of $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(\\mathbb B(B\/C),M)$. If we specialise to $M = B$, we in particular get maps\n$$\n\\chi_{B\/C}^i = \\operatorname{HH}\\nolimits^i(\\mu_e,B) \\colon \\operatorname{HH}\\nolimits^{i}(B) \\longrightarrow \\operatorname{HH}\\nolimits^i(C) \\quad \\text{(for $i \\geqslant 0$).}\n$$\n\\end{nn}\n\n\\begin{thm}[see {\\cite[Thm.\\,4.7]{Bu03}}]\\label{thm:chiringhom}\nThe graded map $\\chi_{B\/C} = \\chi^\\ast_{B\/C} \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(C)$ is a homomorphism of graded $K$-algebras.\n\\end{thm}\n\nThe goal of the upcoming considerations is to show that the algebra homomorphism $\\chi^\\ast_{B\/C}$ also preserves the Gerstenhaber bracket and the squaring map.\n\n\\begin{prop}\\label{prop:Ieischi}\nLet $\\mathfrak A_e$ be the functor $e(-)e\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ as in Proposition $\\ref{prop:funcproper}$. The induced map $$\\mathfrak A_e^\\ast\\colon \\operatorname{HH}\\nolimits^\\ast(B) \\longrightarrow \\operatorname{HH}\\nolimits^\\ast(C)$$ coincides with the map $\\chi_{B\/C}$ occurring in the long exact sequence of Paragraph $\\ref{nn:buchweitz}$. In particular, $\\chi_{B\/C}$ is a homomorphism of graded $K$-algebras $($cf.\\,Theorem $\\ref{thm:chiringhom})$.\n\\end{prop}\n\n\\begin{proof}\nWe show a slightly stronger statement. Let $\\mathsf B \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ and ${\\mathsf C} \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$ be entirely extension closed subcategories, respectively containing $B$ and $C$, as well as their enveloping algebras, such that $\\mathfrak A_e \\mathsf B \\subseteq {\\mathsf C}$. Then the induced homomorphism\n$$\n\\mathfrak A_e^\\ast \\colon \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf B}}(B,B) \\longrightarrow \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(C,C)\n$$\ncoincides with $\\chi_{B\/C}$. To begin with, through the adjunction isomorphism of the adjoint pair $(Be \\otimes_C (-) \\otimes_C eB, \\mathfrak A_e)$, the morphism $\\tilde{\\mu}_e$ yields an inclusion\n\\begin{align*}\n\\tilde{\\mu}_e^\\sharp \\colon \\mathbb B C \\cong \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}&(C \\otimes C, \\mathfrak A_e(Be \\otimes_C \\mathbb B C \\otimes_C eB))\\\\\n&\\longrightarrow \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(C \\otimes C, \\mathfrak A_e\\mathbb B B)\\\\\n&\\xrightarrow{\\hspace*{1.7pt}\\sim\\hspace*{1.7pt}} \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes eB, \\mathbb B B) \\cong e (\\mathbb B B) e\n\\end{align*}\nof complexes, which can also be obtained from the fact that $C^{\\otimes (n+2)}$ is a direct summand of $e (B^{\\otimes(n+2)}) e = eB \\otimes B^{\\otimes n} \\otimes Be$. Observe that $\\tilde \\mu^\\natural_e$ agrees with the map\n$$\n\\xymatrix@C=20pt{\n\\mathbb B C \\ar[r]^-\\sim & eBe \\otimes_{eBe} \\mathbb B C \\otimes_{eBe} eBe \\ar[r]^-{e(\\tilde{\\mu}_e) e} & e (\\mathbb B B) e\n},\n$$\nthat is, the diagram\n$$\n\\xymatrix{\n\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(\\mathbb B B, B) \\ar[r]^-{- \\circ \\tilde{\\mu}_e} \\ar[d]_-{\\mathfrak A_e} & \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_C \\mathbb B C \\otimes_C eB, B) \\ar[d]^-{\\mathrm{adj}} \\\\\n\\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(e(\\mathbb B B) e, C) \\ar[r]^-{- \\circ \\tilde\\mu_e^\\sharp} & \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(\\mathbb B C, C)\n}\n$$\ncommutes, where $\\mathrm{adj}$ denotes the adjunction isomorphism from above, given by\n\\begin{align*}\n& \\ \\ \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_C \\mathbb B C \\otimes_C eB, B) \\longrightarrow \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(\\mathbb B C, C)\\\\\n& \\mathrm{adj}(\\varphi)(c_0 \\otimes \\cdots \\otimes c_{n+1}) = e\\varphi(e \\otimes c_o \\otimes \\cdots \\otimes c_{n+1} \\otimes e)e.\n\\end{align*}\nThus, the $i$-th component of the map $\\chi_{B\/C}$ is given by $\\chi_{B\/C}^i = H^i(- \\circ \\mu^\\sharp_e) \\circ H^i(\\mathfrak A_e(-))$.\n\nDue to the assumptions made, $\\mathbb B B$ is a complex in ${\\mathsf B}$, and similarly, $\\mathbb BC$ is a complex in ${\\mathsf C}$. Pick an $n$-extension\n$$\n\\xymatrix@C=20pt{\nS & \\equiv & 0 \\ar[r] & B \\ar[r]^-{d_n} & E_{n-1} \\ar[r]^-{d_{n-1}} & \\cdots \\ar[r]^-{d_1} & E_0 \\ar[r]^-{d_0} & B \\ar[r] & 0\n}\n$$\nin ${\\mathsf B}$ along with a comparison map\n$$\n\\xymatrix@C=16pt{\n& & \\cdots \\ar[r] & B^{\\otimes(n+2)} \\ar[d]_-{\\varphi_{n}} \\ar[r] & B^{\\otimes(n+1)} \\ar[d]_-{\\varphi_{n-1}} \\ar[r] & \\cdots \\ar[r] & B \\otimes B \\ar[r] \\ar[d]^-{\\varphi_0} & B \\ar[r] \\ar@{=}[d] & 0 \\\\\nS & \\equiv & 0 \\ar[r] & B \\ar[r] & E_{n-1} \\ar[r] & \\cdots \\ar[r] & E_0 \\ar[r] & B \\ar[r] & 0\n}\n$$\nThe map $\\mathfrak A_e^n$ sends $S$ to $eSe$,\n$$\n\\xymatrix@C=18pt{\n\\mathfrak A_e^n S & \\equiv & 0 \\ar[r] & C \\ar[r] & eE_{n-1}e \\ar[r] & \\cdots \\ar[r] & eE_0e \\ar[r] & C \\ar[r] & 0\\,,\n}\n$$\nwhereas $\\chi_{B\/C}^n$ maps it to the lower sequence in the pushout diagram below.\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & C^{\\otimes(n+2)} \\ar[d]_-{e\\varphi_{n}e} \\ar[r] & C^{\\otimes(n+1)} \\ar[d] \\ar[r] & \\cdots \\ar[r] & C \\otimes C \\ar[r] \\ar@{=}[d] & C \\ar[r] \\ar@{=}[d] & 0 \\\\\n0 \\ar[r] & C \\ar[r] & C \\oplus_{C^{\\otimes(n+2)}} C^{\\otimes(n+1)} \\ar[r] & \\cdots \\ar[r] & C \\otimes C \\ar[r] & C \\ar[r] & 0\n}\n$$\nTo make sense out of the map $e(\\varphi_n)e\\colon C^{\\otimes(n+2)} \\rightarrow C$, remember that $C^{\\otimes(n+2)} = (eBe)^{\\otimes(n+2)}$ is a direct summand of $e(B^{\\otimes(n+2)})e$ to which any map from $e(B^{\\otimes(n+2)})e$ may be restricted to; in fact, $(e \\varphi_n e) \\mathord{\\upharpoonright}_{C^{\\otimes(n+2)}}$ is simply $(e \\varphi_n e) \\circ \\mu_e^\\sharp$. However, the commutative diagram\n$$\n\\xymatrix@C=16pt{\n0 \\ar[r] & C \\ar[r] \\ar@{=}[d] & C \\oplus_{C^{\\otimes(n+2)}} C^{\\otimes(n+1)} \\ar[r] \\ar@{-->}[d] & C^{\\otimes n} \\ar[r] \\ar[d]_-{e\\varphi_{n-2}e} & \\cdots \\ar[r] & C \\otimes C \\ar[r] \\ar[d]^-{e\\varphi_{0}e} & C \\ar[r] \\ar@{=}[d] & 0 \\, \\,\\\\\n0 \\ar[r] & eBe \\ar[r] & eE_{n-1}e \\ar[r] & eE_{n-2}e \\ar[r] & \\cdots \\ar[r] & eE_0e \\ar[r] & eBe \\ar[r] & 0 \\, ,\n}\n$$\nwith the dashed arrow being induced by the map\n$$\nC \\oplus C^{\\otimes(n+1)} \\longrightarrow eE_{n-1}e, \\ (c, c_0 \\otimes \\cdots \\otimes c_n) \\mapsto e\\left(d_n(c) + \\varphi_{n-1}(c_0 \\otimes \\cdots \\otimes c_n)\\right)e\\,,\n$$\nnow shows that both sequences define the same equivalence class in $\\operatorname{Ext}\\nolimits^n_{C^\\mathrm{ev}}(C,C)$ (even in $\\operatorname{Ext}\\nolimits_{\\mathsf C}^\\ast(C,C)$), whence the claim follows.\n\\end{proof}\n\nThe proposition yields the direct consequence below.\n\n\\begin{cor}\nThe map $\\chi_{B\/C} \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(C)$ is bijective if $\\overline B = B \/ BeB = 0$.\\qed\n\\end{cor}\n\n\\begin{thm}\\label{thm:chigersten}\nThe graded map $\\chi_{B\/C} \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(C)$ is a homomorphism of strict Gerstenhaber algebras.\n\\end{thm}\n\n\\begin{proof}\nBy Proposition \\ref{prop:Ieischi}, the map $\\chi_{B\/C}$ is given (on extensions) by applying the functor $e(-)e = eB \\otimes_B (-) \\otimes_B Be \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev})$, that is, it is induced by the standard recollement $\\mathcal R(B^\\mathrm{ev},e \\otimes e^\\mathrm{op})$. The assertion now follows from Theorem \\ref{thm:mainthm}.\n\\end{proof}\n\nRecall the following homological invariant of a module (see \\cite[Def.\\,2.1]{Bu03}).\n\n\\begin{defn}\nLet $X$ be a module over a ring $R$. The \\textit{grade} of $X$ is given by\n$$\n\\operatorname{grade}\\nolimits_R X = \\inf\\{i \\geqslant 0 \\mid \\operatorname{Ext}\\nolimits^i_R(X,R) \\neq 0\\} \\quad \\in \\quad \\mathbb Z \\cup \\{\\infty\\}.\n$$\n\\end{defn}\n\nIt is obvious that $\\overline B = 0$ implies $\\operatorname{grade}\\nolimits_B \\overline B = \\infty$. However, the converse, in general, does not need to hold true. Several conjectures though predict a strong mutual dependance of $\\overline B = 0$ and $\\operatorname{grade}\\nolimits_B \\overline B = \\infty$ for certain pairs $(B,e)$. See \\cite[Sec.\\,3]{Bu03} for a very clear exposition of those conjectures, and their interplay. \n\nThe following result is \\cite[Thm.\\,5.5]{Bu03}. It tells us that the grade of $\\overline{B}$, to some extend, controls whether $\\chi_{B\/C}$ is an isomorphism or not. Indeed, its proof utilises the long exact sequence we briefly introduced in Paragraph \\ref{nn:buchweitz} and vanishing criteria for its somewhat mysterious members $\\operatorname{HH}\\nolimits^i(B\/C,B)$ (for $i \\geqslant 0$) in terms of $\\mathrm{grade}_B \\overline B$ (see \\cite[Sec.\\,5]{Bu03}).\n\n\\begin{thm}\\label{thm:grade}\nLet $\\operatorname{grade}\\nolimits_B \\overline B = g$ and $E = \\operatorname{Ext}\\nolimits^g_C(\\overline{B},B)$. Then the maps\n$$\n\\chi_{B\/C}^j \\colon \\operatorname{HH}\\nolimits^{j}(B) \\longrightarrow \\operatorname{HH}\\nolimits^j(C)\n$$\nare isomorphisms for $j \\leqslant g-2$, and there is an exact sequence\n$$\n0 \\longrightarrow \\operatorname{HH}\\nolimits^{g-1}(B) \\longrightarrow \\operatorname{HH}\\nolimits^{g-1}(C) \\longrightarrow \\operatorname{HH}\\nolimits^0(B,E) \\longrightarrow \\operatorname{HH}\\nolimits^g(B) \\longrightarrow \\operatorname{HH}\\nolimits^g(C).\n$$\\qed\n\\end{thm}\n\nLet $e' = 1 - e$ the complementary idempotent of $e$ in $B$, and $C' = e'Be'$. As a consequence of the Theorems \\ref{thm:chigersten} and \\ref{thm:grade}, we obtain the following result.\n\n\\begin{thm}\\label{thm:isogerstengrade}\nThe following hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item If $\\operatorname{grade}\\nolimits_B \\overline B \\geqslant 3$, then $\\chi_{B\/C}^1$ defines an isomorphism\n$$\n\\frac{\\operatorname{Der}\\nolimits_K(B)}{\\operatorname{Inn}\\nolimits_K(B)} = \\operatorname{HH}\\nolimits^1(B) \\longrightarrow \\operatorname{HH}\\nolimits^1(C) = \\frac{\\operatorname{Der}\\nolimits_K(C)}{\\operatorname{Inn}\\nolimits_K(C)}\n$$\nof Lie algebras. Thus, if $\\operatorname{Tor}\\nolimits^{C'}_1(Be',e'B) = 0$, the map $\\chi^1_{B\/C'} \\circ (\\chi^1_{B\/C})^{-1}$ defines a $($in general non-trivial$)$ Lie algebra homomorphism $\\operatorname{HH}\\nolimits^1(C) \\rightarrow \\operatorname{HH}\\nolimits^1(C')$.\n\\item If $\\operatorname{grade}\\nolimits_B \\overline B = \\infty$, then $\\chi^\\ast_{B\/C}$ defines an isomorphism $\\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(C)$ of strict Gerstenhaber algebras. Thus, there is a homomorphism $\\operatorname{HH}\\nolimits^\\ast(C) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(C')$ of strict Gerstenhaber algebras.\\qed\n\\end{enumerate}\n\\end{thm}\n\n\n\\section{The long exact sequence of Cibils, Green--Solberg and Michelena--Platzeck}\\label{sec:greensol}\n\n\\begin{nn}\nIn this section, we investigate long exact sequences independently studied and introduced by Cibils, Green--Solberg and Michelena--Platzeck in \\cite{Ci00}, \\cite{GrSo02} and \\cite[Sec.\\,2]{MiPl00}; see also \\cite{BeGu04}, \\cite{GeSch85} and \\cite{GoGo01}. The results we will present here are fairly similar in taste to the ones in Section \\ref{sec:buchweitz}, however with different outcome, and different applications. Let us fix a $K$-algebra $B$ which we assume to be projective over $K$. Let $e \\in B$ be an idempotent, and $e' = 1 - e$ be its complementary idempotent. As usual, we abbreviate $C = eBe$ and $C' = e'Be'$.\n\\end{nn}\n\n\\begin{lem}\\label{lem:isoidealtor}\nAssume that $\\operatorname{Tor}\\nolimits^{eBe}_i(Be,eB) = 0$ for $i > 0$. Then the functor\n$$\nBe \\otimes_{eBe} (-) \\otimes_{eBe} eB \\colon \\operatorname{\\mathsf{Mod}}\\nolimits((eBe)^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})\n$$\ngives rise to an isomorphism \n$$\n\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(Be \\otimes_{eBe} eB, B) \\xrightarrow{\\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{(eBe)^\\mathrm{ev}}(eBe,eBe) = \\operatorname{HH}\\nolimits^\\ast(eBe)\n$$\nof graded $K$-modules.\n\\end{lem}\n\n\\begin{proof}\nIndeed, if $\\mathbb PC \\rightarrow C \\rightarrow 0$ is a projective resolution of $C$ over $C^\\mathrm{ev}$, then $Be \\otimes_C \\mathbb PC \\otimes_C eB$ is a complex of projectives, since $e(-)e$ is exact and right adjoint to $Be \\otimes_C (-) \\otimes_C eB$. Moreover,\n$$\nH_i (\\mathbb PC \\otimes_C eB) = \\operatorname{Tor}\\nolimits_i^C(C, eB) = 0 \\quad \\text{(for all $i \\neq 0$)}\n$$\nfor $\\mathbb PC \\rightarrow C \\rightarrow 0$ in particular is a resolution of $C$ by flat right $C$-modules ($C \\otimes C$ is $C$-flat on both sides since $B$, and therefore $C$ are $K$-flat). Furthermore,\n$$\n(-) \\otimes_C ((C \\otimes C) \\otimes_C eB) \\cong (- \\otimes_C C) \\otimes (C \\otimes_C eB) \\cong (-) \\otimes eB\n$$\nas functors $\\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{op}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(K)$, whence $(C \\otimes C) \\otimes_C eB$ is a flat left $C$-module. It follows that $\\mathbb PC \\otimes_C eB$ resolves $eB$ by flat left $C$-modules. From\n$$\nH_i (Be \\otimes_C \\mathbb PC \\otimes_C eB) = \\operatorname{Tor}\\nolimits_i^C(Be, eB) = 0 \\quad \\text{(for all $i \\neq 0$)}\n$$\nwe deduce that $Be \\otimes_C \\mathbb PC \\otimes_C eB \\rightarrow Be \\otimes_C eB \\rightarrow 0$ is a $B^\\mathrm{ev}$-projective resolution of $Be \\otimes_C eB$. Thus,\n\\begin{align*}\n\\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(Be \\otimes_{eBe} eB,B) = H^n & \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_{eBe} \\mathbb P(eBe) \\otimes_{eBe} eB, B)\\\\ &\\xrightarrow{\\hspace*{1.7pt}\\sim\\hspace*{1.7pt}} H^n \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(\\mathbb P(eBe), C) = \\operatorname{Ext}\\nolimits^n_{(eBe)^\\mathrm{ev}}(eBe,eBe) \\, ,\n\\end{align*}\nwhere the isomorphism is by adjunction.\n\\end{proof}\n\n\\begin{nn}\\label{nn:fundexact}\nWe have the following fundamental short exact sequence of $B^\\mathrm{ev}$-modules.\n\\begin{equation}\\tag{$\\diamond$}\\label{eq:fundexactsequ}\n0 \\longrightarrow \\Omega^1_B \\longrightarrow (Be \\otimes_C eB) \\oplus (Be' \\otimes_{C'} e'B) \\longrightarrow B \\longrightarrow 0 \\, .\n\\end{equation}\nThe second map is induced by the multiplications $\\mu_e\\colon (Be \\otimes_C eB) \\rightarrow B$ and $\\mu_{e'}\\colon Be' \\otimes_{C'} e'B \\rightarrow B$, where surjectivity follows from\n$$\n\\mu_e(ce \\otimes e) + \\mu_{e'}(ce' \\otimes e') = c \\quad \\text{(for $c \\in B$)}.\n$$\nThe sequence (\\ref{eq:fundexactsequ}) gives rise to a long exact sequence (after applying $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(-,B)$) which, locally, looks as follows.\n$$\n\\cdots \\xrightarrow{\\ \\ } \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(B,B) \\xrightarrow{\\ \\ } \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}((Be \\otimes_C eB) \\oplus (Be' \\otimes_{C'} e'B),B) \\xrightarrow{\\ \\ } \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(\\Omega_B^1,B) \\xrightarrow{\\ \\ } \\cdots \\, .\n$$\nBy Lemma \\ref{lem:isoidealtor}, this long exact sequence reads as\n$$\n\\xymatrix@C=14.5pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(B,B) \\ar[r] & \\operatorname{Ext}\\nolimits^n_{C^\\mathrm{ev}}(C,C) \\oplus \\operatorname{Ext}\\nolimits^n_{(C')^\\mathrm{ev}}(C',C') \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(\\Omega^1_B,B) \\ar[r] & \\cdots \\, .\n}\n$$\nTo summarise, we have obtained a stronger statement than \\cite[Thm.\\,1.2]{GrSo02}, in the following sense.\n\\end{nn}\n\n\\begin{thm}\\label{thm:greensol}\nLet $e \\in B$ be an idempotent, $e' = 1 - e$ its complementary idempotent and $C = eBe$, $C' = e'Be'$. If\n$$\n\\operatorname{Tor}\\nolimits^{C}_i(Be,eB) = 0 = \\operatorname{Tor}\\nolimits^{C'}_i(Be',e'B) \\quad \\text{$($for all $i > 0)$},\n$$\nthen there is a long exact sequence\n$$\n\\xymatrix@C=14.5pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(B,B) \\ar[r]^-{g_n} & \\operatorname{Ext}\\nolimits^n_{C^\\mathrm{ev}}(C,C) \\oplus \\operatorname{Ext}\\nolimits^n_{(C')^\\mathrm{ev}}(C', C') \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(\\Omega^1_B,B) \\ar[r] & \\cdots\n}\n$$\ninduced from the canonical short exact sequence\n$$\n\\xymatrix@C=14pt{\n0 \\ar[r] & {\\Omega}^1_B \\ar[r] & \\displaystyle (Be \\otimes_{C} eB) \\oplus (Be' \\otimes_{C'} e'B) \\ar[r] & B \\ar[r] & 0\n}.\n$$\nIf, moreover, $B$ is $K$-projective, then the map $g_\\ast\\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(eBe) \\times \\operatorname{HH}\\nolimits^\\ast(e'Be')$ is a homomorphism of strict Gerstenhaber algebras.\n\\end{thm}\n\n\\begin{proof}\nIt suffices to prove that the component maps $g^C_\\ast$ and $g^{C'}_\\ast$ of $g_\\ast$ coincide with the maps $\\chi = \\chi_{B\/C}$ and $\\chi' = \\chi_{B\/C'}$ that we have studied earlier. We will only take care of $g_n^C = \\chi$ as the proof for $g_\\ast^{C'} = \\chi'$ is identical. Indeed, recall that by Proposition \\ref{prop:Ieischi}, the map $\\chi$ is given by applying the functor\n$$\ne(-)e \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(C^\\mathrm{ev}) \\, ,\n$$\ni.e., it is induced by the standard recollement $\\mathcal R(B^\\mathrm{ev},e \\otimes e^\\mathrm{op})$. We show that the same holds true for $g_\\ast$. To begin with, let $\\mathbb PC \\rightarrow C \\rightarrow 0$ be a $C^\\mathrm{ev}$-projective resolution of $C$. By the proof of Lemma \\ref{lem:isoidealtor}, $Be \\otimes_C \\mathbb PC \\otimes_C eB \\rightarrow Be \\otimes_{C} eB \\rightarrow 0$ is a projective resolution of $Be \\otimes_C eB$ over $B^\\mathrm{ev}$. Further, pick a lifting\n$$\n\\small{\\xymatrix@C=11pt{\nBe \\otimes_C \\mathbb PC \\otimes_C eB \\ar[d]_-{\\Phi} & \\cdots \\ar[r] & Be \\otimes_C P_n \\otimes_{C} eB \\ar[d]_-{\\varphi_n} \\ar[r] & \\cdots \\ar[r] & Be \\otimes_C P_0 \\otimes_{C} eB \\ar[r] \\ar[d]^-{\\varphi_0} & Be \\otimes_C eB \\ar[r] \\ar[d]^-{\\mu_e} & 0\\\\\n\\mathbb B B & \\cdots \\ar[r] & B^{\\otimes(n+2)} \\ar[r] & \\cdots \\ar[r] & B \\otimes B \\ar[r] & B \\ar[r] & 0\n}}\n$$\nof $\\mu_e \\colon Be \\otimes_C eB \\rightarrow B$. If now $\\alpha \\in H^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(\\mathbb B B, B)$ is represented by a cocycle $\\varphi\\colon B^{\\otimes(n+2)} \\rightarrow B$, the map $g_n^C$ will take $\\alpha$ to the equivalence class of the cocycle\n$$\n\\varphi \\circ \\varphi_n \\colon Be \\otimes_C P_n \\otimes_{C} eB \\longrightarrow B \n$$\nin $H^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_C \\mathbb PC \\otimes_C eB, B)$, which, in turn, will be taken to the map\n$$\n(e\\varphi e) \\circ (e \\varphi_n e) = e(\\varphi \\circ \\varphi_n)e \\colon P_n \\xrightarrow{\\, \\sim \\,} eBe \\otimes_C P_n \\otimes_{C} eBe \\xrightarrow{\\, \\ \\, } eBe = C\n$$\nby the canonical isomorphism $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(Be \\otimes_C \\mathbb PC \\otimes_C eB, B) \\cong \\operatorname{Hom}\\nolimits_{C^\\mathrm{ev}}(\\mathbb P C,C)$.\n\nOn the other hand, $\\varphi$ corresponds to the Yoneda extension $S \\in \\mathcal Ext^n_{B^\\mathrm{ev}}(B,B)$ given by the lower row in the pushout diagram\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & B^{\\otimes(n+2)} \\ar[d]_-{\\varphi} \\ar[r] & B^{\\otimes(n+1)} \\ar[r] \\ar[d] & \\cdots \\ar[r] & B \\otimes B \\ar@{=}[d] \\ar[r] & B \\ar@{=}[d] \\ar[r] & 0 \\, \\, \\\\\n0 \\ar[r] & B \\ar[r] & B \\oplus_{B^{\\otimes(n+2)}} B^{\\otimes(n+1)} \\ar[r] & \\cdots \\ar[r] & B \\otimes B \\ar[r] & B \\ar[r] & 0 \\, ,\n}\n$$\nwhereas $\\psi = e(\\varphi \\circ \\varphi_n) e$ corresponds to the lower sequence $T$ in \n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & P_n \\ar[d]_-{\\psi} \\ar[r] & P_{n-1} \\ar[r] \\ar[d] & \\cdots \\ar[r] & P_0 \\ar@{=}[d] \\ar[r] & C \\ar@{=}[d] \\ar[r] & 0 \\, \\, \\\\\n0 \\ar[r] & C \\ar[r] & C \\oplus_{P_n} P_{n-1} \\ar[r] & \\cdots \\ar[r] & P_0 \\ar[r] & C \\ar[r] & 0 \\, .\n}\n$$\nNow the map\n$$\nC \\oplus P_{n-1} \\longrightarrow C \\oplus e (B^{\\otimes(n+1)}) e, \\ (c, p) \\mapsto (c,e\\varphi_{n-1}(p)e)\n$$\ninduces a well-defined map\n$$\n\\tilde{\\varphi}_{n-1} \\colon C \\oplus_{P_n} P_{n-1} \\longrightarrow e \\left(B \\oplus_{B^{\\otimes(n+2)}} B^{\\otimes(n+1)}\\right)e \\,.\n$$\nTo conclude, observe that we obtain the commutative diagram\n$$\n\\xymatrix@C=15pt{\nT \\ar[d] & 0 \\ar[r] & C \\ar[r] \\ar@{=}[d] & C \\oplus_{P_n} P_{n-1} \\ar[r] \\ar[d]_-{\\tilde{\\varphi}_{n-1}} & \\cdots \\ar[r] & P_0 \\ar[d]^-{e\\varphi_0 e} \\ar[r] & C \\ar[r] \\ar@{=}[d] & 0\\,\\, \\\\\neSe & 0 \\ar[r] & eBe \\ar[r] & e\\left(B \\oplus_{B^{\\otimes(n+2)}} B^{\\otimes(n+1)}\\right)e \\ar[r] & \\cdots \\ar[r] & eB \\otimes Be \\ar[r] & eBe \\ar[r] & 0 \\, ,\n}\n$$\nthat is, a morphism $T \\rightarrow eSe$ of $n$-extensions over $C^\\mathrm{ev}$, which finishes the proof.\n\\end{proof}\n\n\\begin{exa}\\label{exa:matrixalg}\nLet $R$ and $S$ be $K$-algebras, and let $M$ be a $R \\otimes S^\\mathrm{op}$-module. Assume further, that $R$, $S$ and $M$ are projective as $K$-modules. Then the matrix algebra\n$$\nB = \\left(\\begin{matrix}\nR & M\\\\\n0 & S\n\\end{matrix}\n\\right)\n$$\nalong with the canonical idempotents\n$$\ne = \\left(\\begin{matrix}\n1 & 0\\\\\n0 & 0\n\\end{matrix}\\right) \\quad \\text{and} \\quad e' = \\left(\\begin{matrix}\n0 & 0\\\\\n0 & 1\n\\end{matrix}\\right)\n$$\nfulfils the requirements of Theorem \\ref{thm:greensol}, because of $\\operatorname{Tor}\\nolimits^{eBe}_i(Be,eB) = \\operatorname{Tor}\\nolimits^R_i(0,M) = 0$ and $\\operatorname{Tor}\\nolimits^{e'Be'}_i(Be',e'B) = \\operatorname{Tor}\\nolimits^S_i(M,0) = 0$ for all $i > 0$.\n\\end{exa}\n\n\\begin{cor}\\label{cor:tensorvanish}\nLet $B$ be a matrix algebra as above. Then there is a long exact sequence\n\\begin{align*}\n\\cdots \\longrightarrow \\operatorname{HH}\\nolimits^n(B) \\xrightarrow{\\, g_n \\, } \\operatorname{HH}\\nolimits^n(R) \\oplus \\operatorname{HH}\\nolimits^n(S) \\xrightarrow{\\, h_n \\, } \\operatorname{Ext}\\nolimits^{n}_{R \\otimes S^\\mathrm{op}}(M,M) \\longrightarrow \\cdots\n\\end{align*}\nin Hochschild cohomology. The graded map $g_\\ast\\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(R) \\times \\operatorname{HH}\\nolimits^\\ast(S)$ is a homomorphism of strict Gerstenhaber algebras.\n\\end{cor}\n\n\\begin{proof}\nUnder the given assumptions, the module $\\Omega^1_B$ evidently is isomorphic to $\\left(\\begin{smallmatrix}0 & M\\\\ 0 & 0 \\end{smallmatrix}\\right) = M$. Moreover, we have$$\n\\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(M,B) \\xrightarrow{\\, \\sim \\, } \\operatorname{Ext}\\nolimits^{n}_{R \\otimes S^\\mathrm{op}}(M,M)\n$$\nas any projective resolution $\\mathbb P M \\rightarrow M \\rightarrow 0$ over $R \\otimes S^\\mathrm{op}$ is already a $B^\\mathrm{ev}$-projective resolution $\\mathbb P M \\rightarrow M \\rightarrow 0$. Indeed, we have canonical isomorphisms\n$$\nBe \\otimes_R \\mathbb P M \\otimes_S e'B \\cong R \\otimes_R \\mathbb P M \\otimes_S S \\cong \\mathbb P(M)\n$$\nwhich yield the desired identification. Thus, by adjunction,\n$$\nH^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(\\mathbb P M, B) \\cong H^n \\operatorname{Hom}\\nolimits_{R \\otimes S^\\mathrm{op}}(\\mathbb PM, eB \\otimes_B B \\otimes_B Be') \\cong \\operatorname{Hom}\\nolimits_{R \\otimes S^\\mathrm{op}}(\\mathbb PM, M)\n$$\nand the assertion follows from Theorem $\\ref{thm:greensol}$.\n\\end{proof}\n\n\\begin{cor}\\label{cor:strexc}\nLet $B$ be a matrix algebra as above. \n\\begin{enumerate}[\\rm(1)]\n\\item If $d = \\operatorname{pd}\\nolimits_{R \\otimes S^\\mathrm{op}}(M)$ is finite, then $g_n \\colon \\operatorname{HH}\\nolimits^n(B) \\rightarrow \\operatorname{HH}\\nolimits^n(R) \\times \\operatorname{HH}\\nolimits^n(S)$ is an isomorphism for $n > d$.\n\n\\item If $M$ is \\emph{strongly exceptional} over $R \\otimes S^\\mathrm{op}$ $($i.e., $\\operatorname{Ext}\\nolimits^n_{R \\otimes S^\\mathrm{op}}(M,M) = 0$ for $n > 0$ and $\\operatorname{End}\\nolimits_{R \\otimes S^\\mathrm{op}}(M) = K)$, then there is the short exact sequence\n$$\n0 \\longrightarrow \\operatorname{HH}\\nolimits^0(B) \\xrightarrow{\\ g_0 \\, } \\operatorname{HH}\\nolimits^0(R) \\oplus \\operatorname{HH}\\nolimits^0(S) \\longrightarrow K \\longrightarrow 0\n$$\nand $g_{> 0} \\colon \\operatorname{HH}\\nolimits^{> 0}(B) \\rightarrow \\operatorname{HH}\\nolimits^{> 0}(R) \\times \\operatorname{HH}\\nolimits^{> 0}(S)$ is an isomorphism of non-unital strict Gerstenhaber algebras.\n\\end{enumerate}\\qed\n\\end{cor}\n\nThe special case $M = 0$ also yields an interesting consequence.\n\n\\begin{cor}\nLet $R$ and $S$ be $K$-algebras being projective as $K$-modules. Then\n$$\n\\operatorname{HH}\\nolimits^\\ast(R \\times S) \\cong \\operatorname{HH}\\nolimits^\\ast(R) \\times \\operatorname{HH}\\nolimits^\\ast(S)\n$$\nas strict Gerstenhaber algebras; that is, the strict Gerstenhaber structure on Hochschild cohomology commutes with finite products.\\qed\n\\end{cor}\n\nWe continue with an analysis of the map $\\operatorname{HH}\\nolimits^\\ast(R) \\times \\operatorname{HH}\\nolimits^\\ast(S) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{R \\otimes S^\\mathrm{op}}(M,M)$.\n\n\\begin{lem}\\label{lem:grsolfunctor}\nLet $B$ be a matrix algebra as in Example $\\ref{exa:matrixalg}$. The respective restrictions of the functors $- \\otimes_R M \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ and $M \\otimes_S -\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ to ${\\mathsf P}(R)$ and ${\\mathsf P}(S)$ are exact. Moreover, the following statements hold true.\n\\begin{enumerate}[\\rm(1)]\n\\item\\label{lem:grsolfunctor:1a} Assume that $M = eBe'$ is a projective left $R = eBe$-module. Then the functors $- \\otimes_R M \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ and $M \\otimes_S - \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ restrict to exact functors\n$$\n- \\otimes_R M \\colon \\mathsf P(R) \\longrightarrow \\mathsf P_R(R,S), \\quad M \\otimes_S - \\colon {\\mathsf P}(S) \\longrightarrow \\mathsf P_R(R,S)\n$$\nwhere $\\mathsf P_R(R,S) \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ denotes the full subcategory of those modules that are projective as left $R$-modules.\n\n\\item\\label{lem:grsolfunctor:1b} Assume that $M = eBe'$ is a projective right $S = e'Be'$-module. Then the functors $- \\otimes_R M \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ and $M \\otimes_S - \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(S^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ restrict to exact functors\n$$\n- \\otimes_R M \\colon \\mathsf P(R) \\longrightarrow \\mathsf P_S(R,S), \\quad M \\otimes_S - \\colon {\\mathsf P}(S) \\longrightarrow \\mathsf P_S(R,S)\n$$\nwhere $\\mathsf P_S(R,S) \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(R \\otimes S^\\mathrm{op})$ denotes the full subcategory of those modules that are projective as right $S$-modules.\n\n\\item\\label{lem:grsolfunctor:2} The induced maps\n$$\n\\operatorname{HH}\\nolimits^\\ast(R) \\cong \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf P}(R)}(R,R) \\xrightarrow{\\ \\ } \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf P_S(R,S)}(M, M) \\xrightarrow{\\, \\sim \\, } \\operatorname{Ext}\\nolimits^\\ast_{R \\otimes S^\\mathrm{op}}(M,M)\n$$\nand\n$$\n\\operatorname{HH}\\nolimits^\\ast(S) \\cong \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf P}(S)}(S,S) \\xrightarrow{\\ \\ } \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf P_R(R,S)}(M, M) \\xrightarrow{\\, \\sim \\, } \\operatorname{Ext}\\nolimits^\\ast_{R \\otimes S^\\mathrm{op}}(M,M)\n$$\nare the component maps in the long exact sequence of Corollary $\\ref{cor:tensorvanish}$ $($up to the sign $(-1)^{\\delta_{\\Sigma,S}}$, where $\\delta_{\\Sigma,S}$ is the Kronecker delta for $\\Sigma \\in \\{R, S\\})$. In particular, those $($signed$)$ component maps are homomorphisms of graded $K$-algebras.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nExactness of the restricted functors is automatic, as they are additive and the bimodules they are defined on are projective on either side. If $X$ is a module in $\\mathsf P(R)$, then\n$$\n\\operatorname{Hom}\\nolimits_{R}(X \\otimes_R M, -) \\cong \\operatorname{Hom}\\nolimits_{R}(M, \\operatorname{Hom}\\nolimits_{R}(X,-))\n$$\nis exact by assumption; thus $X \\otimes_R M \\in \\mathsf P_R(R,S)$. Further, for $Y \\in \\mathsf P(S)$,\n$$\n\\operatorname{Hom}\\nolimits_{R}(M \\otimes_S Y,-) \\cong \\operatorname{Hom}\\nolimits_{S^\\mathrm{op}}(Y, \\operatorname{Hom}\\nolimits_{R}(M,-)),\n$$\nso that $M \\otimes_S Y \\in {\\mathsf P}_R(R,S)$, whence (\\ref{lem:grsolfunctor:1a}) follows. Similar arguments apply to deduce (\\ref{lem:grsolfunctor:1b}). As for item (\\ref{lem:grsolfunctor:2}), notice that the bar resolution $\\mathbb B R \\rightarrow R \\rightarrow 0$ gives rise to a projective resolution $Be \\otimes_R \\mathbb B R \\otimes_R eB \\rightarrow Be \\otimes_R eB \\rightarrow 0$ over $R \\otimes S^\\mathrm{op}$ (indeed, as $M$ is projective as a right $S$-module, the functor $\\operatorname{Hom}\\nolimits_{R \\otimes S^\\mathrm{op}}((R \\otimes R) \\otimes_R M, -) \\cong \\operatorname{Hom}\\nolimits_R(R, \\operatorname{Hom}\\nolimits_{S^\\mathrm{op}}(M,-))$ is exact). The latter resolution may be expressed as\n$$\n\\xymatrix{\nBe \\otimes_R \\mathbb B R \\otimes_R eB \\ar[r] \\ar@{=}[d] & Be \\otimes_R eB \\ar[r] \\ar@{=}[d] & 0\\, \\,\\\\\n\\displaystyle {\\left(\\begin{matrix}\n\\mathbb B R & \\mathbb B R \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[r] & \\displaystyle {\\left(\\begin{matrix}\nR & M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[r] & 0 \\, .\n}\n$$\nThe adjunction isomorphism yields, for $n \\geqslant 0$,\n\\begin{align*}\nH^n \\operatorname{Hom}\\nolimits_{R^\\mathrm{ev}}(\\mathbb B R, R) \\xrightarrow{\\, \\sim \\, } H^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}&(Be \\otimes_R \\mathbb B R \\otimes_R eB, B)\\\\ &= H^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}\\left({\\left(\\begin{matrix}\n\\mathbb B R & \\mathbb B R \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)}, B\\right)\n\\end{align*}\nwhich sends the equivalence class of a cocylce $\\varphi \\colon R^{\\otimes(n + 2)} \\rightarrow R$ to the map $\\psi(\\varphi)$ given by\n$$\n\\psi(\\varphi){\\left(\\begin{matrix}\nr & r' \\otimes m\\\\\n0 & 0\n\\end{matrix}\\right)} = \\left(\\begin{matrix}\n\\varphi(r) & \\varphi(r')m\\\\\n0 & 0\n\\end{matrix}\\right) \\ \\in \\ B \\\n\\quad (\\text{for $r, r' \\in R^{\\otimes(n + 2)}$ and $m \\in M$}).\n$$\nThe cocycle $\\psi = \\psi(\\varphi)$ corresponds to the middle sequence in the diagram with exact rows below, whereas the rightmost sequence represents its image (under $h^R_n$) in $\\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(M,B)$.\n$$\n\\xymatrix@C=24pt{\n\\ar[d] \\tvdots & 0 \\ar[d] & 0 \\ar[d]\\\\\n{\\left(\\begin{matrix}\nR^{\\otimes(n+2)} & R^{\\otimes(n+2)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[r]^-{\\psi} \\ar[d] & B \\ar[d] & \\ar@{=}[l] \\ar[d] B\\\\\n{\\left(\\begin{matrix}\nR^{\\otimes(n+1)} & R^{\\otimes(n+1)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[r] \\ar[d] & P(\\psi) \\ar[d] & P(\\psi) \\ar[d] \\ar@{=}[l] \\\\\n\\tvdots \\ar[d] & \\tvdots \\ar[d] & \\tvdots \\ar[d] \\\\\n{\\left(\\begin{matrix}\nR \\otimes R & (R \\otimes R) \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar@{=}[r] \\ar[d] & {\\left(\\begin{matrix}\nR \\otimes R & (R \\otimes R) \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[d] & \\ar[l] Q(\\mathrm{inc}) \\ar[d]\\\\\n{\\left(\\begin{matrix}\nR & M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar@{=}[r] \\ar[d] & {\\left(\\begin{matrix}\nR & M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[d] & M \\ar@{ >->}[l]_-{\\mathrm{inc}} \\ar[d]\\\\\n0 & 0 & 0\n}\n$$\nHere $P(\\psi)$ denotes the pushout and $Q(\\mathrm{inc})$ denotes the pullback of the obvious diagrams. Observe that $Q = Q(\\mathrm{inc})$ is canonically isomorphic to $(R \\otimes R) \\otimes_R M \\cong R \\otimes M$. Further, if $\\beta_n \\colon R^{\\otimes(n+2)} \\rightarrow R^{\\otimes(n+1)}$ denotes the $n$-th differential in $\\mathbb B R$ and $\\tilde{\\beta}_n$ is the map\n$$\n{\\left(\\begin{matrix}\n\\beta_n & \\beta_n \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\colon {\\left(\\begin{matrix}\nR^{\\otimes(n+2)} & R^{\\otimes(n+2)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\longrightarrow {\\left(\\begin{matrix}\nR^{\\otimes(n+1)} & R^{\\otimes(n+1)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\, ,\n$$\nthe module $P(\\psi)$ is given by the cokernel in the exact sequence\n$$\n\\xymatrix@C=24pt{\n{\\left(\\begin{matrix}\nR^{\\otimes(n+2)} & R^{\\otimes(n+2)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar[r]^-{\n\\left(\n\\begin{smallmatrix}\n\\psi\\\\\n- \\tilde{\\beta}_n\n\\end{smallmatrix}\n\\right)\n} & B \\oplus {\\left(\\begin{matrix}\nR^{\\otimes(n+1)} & R^{\\otimes(n+1)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)} \\ar@{->>}[r] & \\operatorname{Coker}\\nolimits{\\left(\\begin{smallmatrix}\n\\psi\\\\\n- \\tilde{\\beta}_n\n\\end{smallmatrix}\\right)}\n}\n$$\nand therefore applying the exact functor $e(-)e' \\cong eB \\otimes_B (-) \\otimes_B Be'$ yields the exact sequence\n$$\n\\small{\n\\xymatrix@C=10pt{\ne{\\left(\\begin{matrix}\nR^{\\otimes(n+2)} & R^{\\otimes(n+2)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)}e' \\ar[r] \\ar@{=}[d] & eBe' \\oplus e{\\left(\\begin{matrix}\nR^{\\otimes(n+1)} & R^{\\otimes(n+1)} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)}e' \\ar@{->>}[r] \\ar@{=}[d] & e\\operatorname{Coker}\\nolimits{\\left(\\begin{smallmatrix}\n\\psi\\\\\n- \\tilde{\\beta}_n\n\\end{smallmatrix}\\right)}e' \\ar[d]^-\\cong\\\\\nR^{\\otimes(n+2)} \\otimes_R M \\ar[r]^-{\\left(\\begin{smallmatrix}\n\\varphi \\otimes_R M\\\\\n- {\\beta}_n \\otimes_R M\n\\end{smallmatrix}\\right)} & M \\oplus R^{\\otimes(n+1)} \\otimes_R M \\ar@{->>}[r] & \\operatorname{Coker}\\nolimits{\\left(\\begin{smallmatrix}\n\\varphi \\otimes_R M\\\\\n- {\\beta}_n \\otimes_R M\n\\end{smallmatrix}\\right)}\n}}\n$$\nAs ${\\left(\\begin{smallmatrix}\n\\varphi \\otimes_R M\\\\\n- {\\beta}_n \\otimes_R M\n\\end{smallmatrix}\\right)} = {\\left(\\begin{smallmatrix}\n\\varphi\\\\\n- {\\beta}_n\n\\end{smallmatrix}\\right) \\otimes_R M}$, it thus follows that $eP(\\psi)e'$ is the module $P(\\varphi) \\otimes_R M$, where $P(\\varphi)$ denotes $\\operatorname{Coker}\\nolimits{\\left(\\begin{smallmatrix}\n\\varphi\\\\\n- {\\beta}_n\n\\end{smallmatrix}\\right)}$, i.e., the pushout\n$$\n\\xymatrix{\nR^{\\otimes(n+2)} \\ar[r]^-{\\beta_n} \\ar[d]_-{\\varphi} & R^{\\otimes(n+1)} \\, \\, \\ar[d] \\\\\nR \\ar[r] & P(\\varphi) \\, .\n}\n$$\nThus, under the isomorphism $\\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(M,B) \\cong \\operatorname{Ext}\\nolimits^n_{R \\otimes S^\\mathrm{op}}(M,M)$, the sequence that corresponds to $\\psi$ is taken to\n$$\n\\small{\n\\xymatrix@C=10pt@R=20pt{\n0 \\ar[r] & eBe' \\ar@{=}[d] \\ar[r] & eP(\\psi)e' \\ar@{=}[d] \\ar[r] & e{\\left(\\begin{matrix}\nR^{\\otimes n} & R^{\\otimes n} \\otimes_R M\\\\\n0 & 0\n\\end{matrix}\\right)}e' \\ar@{=}[d] \\ar[r] & \\cdots \\ar[r] & e Q e' \\ar[r] \\ar@{=}[d] & eMe' \\ar@{=}[d] \\ar[r] & 0\\\\\n0 \\ar[r] & M \\ar[r] & P(\\varphi)\\otimes_R M \\ar[r] & R^{\\otimes n} \\otimes_R M \\ar[r] & \\cdots \\ar[r] & (R \\otimes R) \\otimes_R M \\ar[r] & M \\ar[r] & 0\n}}\n$$\nwhich is precisely the image under $- \\otimes_R M$ of the sequence that corresponds to $\\varphi$. The second assertion follows similarly.\n\\end{proof}\n\n\\begin{nn}\nAssume that $R = S$ and $M = R$; then ${\\mathsf P}_R(R,S) = {\\mathsf P}_\\lambda(R)$, and the functors\n$$\n- \\otimes_R R,\\, R \\otimes_R - \\colon {\\mathsf P}(R) = {\\mathsf P}_\\lambda(R) \\cap {\\mathsf P}_\\varrho(R) \\longrightarrow {\\mathsf P}_\\lambda(R)\n$$\nare isomorphic to the inclusion functor. In particular, these are very strong monoidal functors\n$$\n({\\mathsf P}(R), \\otimes_R, R) \\longrightarrow ({\\mathsf P}_\\lambda(R), \\otimes_R, R) \\, .\n$$\nThus Lemma \\ref{lem:grsolfunctor} implies, when combined with Theorem \\ref{thm:gersthochcomp} and Theorem \\ref{thm:greensol}, the following statement.\n\\end{nn}\n\n\\begin{cor}\\label{cor:tripleR}\nLet $R$ be a $K$-algebra which is projective as a $K$-module. Further, let $B$ be the matrix algebra\n$$\nB = \\left(\\begin{matrix}\nR & R\\\\\n0 & R\n\\end{matrix}\n\\right) \\, .\n$$\nThen the long exact sequence\n\\begin{align*}\n\\cdots \\longrightarrow \\operatorname{HH}\\nolimits^n(B) \\xrightarrow{\\, g_n \\, } \\operatorname{HH}\\nolimits^n(R) \\oplus \\operatorname{HH}\\nolimits^n(R) \\xrightarrow{\\, h_n \\, } \\operatorname{Ext}\\nolimits^{n}_{R \\otimes R^\\mathrm{op}}(R,R) = \\operatorname{HH}\\nolimits^n(R) \\longrightarrow \\cdots\n\\end{align*}\nas stated in Theorem $\\ref{thm:greensol}$, or Corollary $\\ref{cor:tensorvanish}$, decomposes into short exact sequences\n$$\n0 \\longrightarrow \\operatorname{HH}\\nolimits^n(B) \\xrightarrow{\\, g_n \\, } \\operatorname{HH}\\nolimits^n(R) \\oplus \\operatorname{HH}\\nolimits^n(R) \\xrightarrow{\\, h_n \\, } \\operatorname{HH}\\nolimits^n(R) \\longrightarrow 0\\,,\n$$\nwherein the graded map $g_\\ast \\colon \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(R) \\times \\operatorname{HH}\\nolimits^\\ast(R)$ and the component maps $\\operatorname{HH}\\nolimits^\\ast(R) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{R^\\mathrm{ev}}(R,R) \\cong \\operatorname{HH}\\nolimits^\\ast(R)$, $\\operatorname{HH}\\nolimits^\\ast(R) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{R^\\mathrm{ev}}(R,R) \\cong \\operatorname{HH}\\nolimits^\\ast(R)$ of $h_\\ast$ are homomorphisms of strict Gerstenhaber algebras.\\qed\n\\end{cor}\n\n\\subsection{The long exact sequence of Happel}\\label{subsec:happel}\nBy specialising the long exact sequence in Theorem \\ref{thm:greensol} to a matrix algebra $B = \\left(\\begin{smallmatrix} R & M\\\\ 0 & K \\end{smallmatrix}\\right)$, where $M$ is an $R$-module, and $B = R[M]$ is thus the \\textit{one-point-extension} of $R$ by $M$, we arrive at a long exact sequence\n$$\n\\xymatrix@C=24pt@R=6pt{\n0 \\ar[r] & \\operatorname{HH}\\nolimits^0(B) \\ar[r]^-{g_0} & \\operatorname{HH}\\nolimits^0(R) \\oplus K \\ar[r]^-{(h_0^R \\, h_0^K)} & \\operatorname{Hom}\\nolimits_R(M,M) \\ar[r] & \\operatorname{HH}\\nolimits^{1}(B) \\ar[r] & \\cdots\\\\\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{g_n} & \\operatorname{HH}\\nolimits^n(R) \\ar[r]^-{h_n^R} & \\operatorname{Ext}\\nolimits^{n}_R(M,M) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B) \\ar[r] & \\cdots\n}\n$$\nfirst discovered by Happel in \\cite{Ha89}, and picked up for further analysis by Green-Marcos-Snashall in \\cite{GrMaSn03}, resulting in the insight, that the maps $g_\\ast$ and $h_\\ast^R$ define homomorphisms of graded $K$-algebras. By the results established previously, we immediately deduce the following.\n\n\\begin{prop}\\label{prop:happel}\nLet $R$ be a $K$-algebra and $M$ be an $R$-module. Assume further, that $R$ and $M$ are projective as $K$-modules. Then the one-point-extension $B = R[M]$ of $R$ by $M$ fits inside a long exact sequence\n$$\n\\xymatrix@C=20pt{\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{g_n} & \\operatorname{HH}\\nolimits^n(R) \\ar[r]^-{h_n^R} & \\operatorname{Ext}\\nolimits^{n}_R(M,M) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B) \\ar[r] & \\cdots\n}\n$$\nsuch that the map $g_\\ast$ is a homomorphism of strict Gerstenhaber algebras, and $h^R_\\ast$ is a homomorphism of graded $K$-algebras.\\qed\n\\end{prop}\n\n\\begin{rem}\nHappel's sequence comes in handy, when examining \\textit{accessible algebras}, as it has been done in work by Lenzing and de la Pe{\\~n}a (see \\cite{LedlP08} and also \\cite{BaLe03}, \\cite{Gei02}). Recall that, for a $K$-algebra $B$, a $B$-module $E$ is called \\textit{exceptional} if $\\operatorname{End}\\nolimits_B(E) \\cong K$ and $\\operatorname{Ext}\\nolimits^1_B(E,E) = 0$. The $K$-algebra $B$ is called \\textit{accessible} if there is an \\textit{accessible tower}\n$$\nK = B_0 , B_1, \\dots, B_n = B \\,;\n$$\nthat is, a sequence of $K$-algebras as indicated above, such that $B_{t} \\cong B_{t-1}[E_{t-1}]$ is the one-point-extension of $B_{t-1}$ by some exceptional $(B_{t-1})$-module $E_{t-1}$ (for $1 \\leqslant t \\leqslant n$). Thus, the Gerstenhaber algebra structure on $\\operatorname{HH}\\nolimits^\\ast(B)$ may be understood via the same for the Hochschild cohomology of its \\textit{steps} $B_t$. For instance, we immediately deduce from Happel's sequence, and the exceptionality of the $E_t$, that there are short exact sequences\n$$\n0 \\longrightarrow Z(B_t) \\longrightarrow Z(B_{t-1}) \\oplus K\\longrightarrow K \\longrightarrow 0 \\quad \\text{(for $1 \\leqslant t \\leqslant n$)}\n$$\nwhich induce algebra isomorphisms $Z(B_t) \\cong Z(B_{t-1})$, and Lie algebra isomorphisms\n$$\n\\operatorname{HH}\\nolimits^1(B_{t}) = \\operatorname{Out}\\nolimits_K(B_t) \\xrightarrow{\\, \\sim \\, } \\operatorname{Out}\\nolimits_K(B_{t-1}) = \\operatorname{HH}\\nolimits^1(B_{t-1}) \\quad \\text{(for $1 \\leqslant t \\leqslant n$)}.\n$$\nCompare this with Corollary \\ref{cor:strexc}.\n\\end{rem}\n\n\\begin{nn}\nLet us specialise further. Let $R$ be a $K$-projective $K$-algebra and consider\n$$\nB = \\left(\\begin{matrix}\nR \\otimes R^\\mathrm{op} & R\\\\\n0 & K\n\\end{matrix}\\right)\\, ,\n$$\nthat is, the one-point-extension $B = U[R]$ of $U = R^\\mathrm{ev}$ by $R$. By Proposition \\ref{prop:happel}, we already know that the map $g_\\ast$ defines a homomorphism of strict Gerstenhaber algebras. In the following, we take the map $h_\\ast$ under further analysis.\n\\end{nn}\n\n\\begin{nn}\nTo this end, consider the full subcategory $\\overline{{\\mathsf P}}(U)$ of ${\\mathsf P}(U)$ consisting of all objects $X \\in {\\mathsf P}(U)$ such that $X \\otimes_U R$ belongs to ${\\mathsf P}(R)$. Of course, $U$ and $U \\otimes U$ belong to $\\overline{{\\mathsf P}}(U)$. Further, let $\\tilde{{\\mathsf P}}(U)$ be the full subcategory of $\\overline{{\\mathsf P}}(U)$ consisting of all objects $X \\in \\overline{{\\mathsf P}}(U)$ such that $X \\otimes_U Y$ belongs to $\\overline{{\\mathsf P}}(U)$ for all objects $Y \\in \\overline{{\\mathsf P}}(U)$. We will argue that $\\tilde{{\\mathsf P}}(U)$ defines an exact monoidal subcategory of $(\\operatorname{\\mathsf{Mod}}\\nolimits(U^\\mathrm{ev}), \\otimes_U, U)$.\n\nFirstly, $\\tilde{{\\mathsf P}}(U)$ is non-empty as $U$ and $U \\otimes U$ belong to $\\tilde{{\\mathsf P}}(U)$. Indeed, for $Y \\in \\overline{{\\mathsf P}}(U)$,\n$$\nU \\otimes_U Y \\cong Y \\ \\in \\ \\overline{{\\mathsf P}}(U)\n\\quad\n\\text{and}\n\\quad\n(U \\otimes U) \\otimes_U Y \\cong U \\otimes Y \\ \\in \\ \\overline{{\\mathsf P}}(U)\\,.\n$$\nThe subcategory $\\tilde{{\\mathsf P}}(U)$ is closed under arbitrary direct sums and summands (taken in $\\operatorname{\\mathsf{Mod}}\\nolimits(U^\\mathrm{ev})$). Moreover, $\\tilde{{\\mathsf P}}(U)$ is extension closed and closed under kernels of epimorphisms, as any admissible short exact sequence $0 \\rightarrow X'' \\rightarrow X \\rightarrow X' \\rightarrow 0$ in ${\\mathsf P}(U)$ with $X' \\in \\tilde{{\\mathsf P}}(U)$ gives rise to a split exact sequence\n$$\n0 \\longrightarrow (X'' \\otimes_U Y) \\otimes_U R \\longrightarrow (X \\otimes_U Y) \\otimes_U R \\longrightarrow (X' \\otimes_U Y) \\otimes_U R \\longrightarrow 0 \\quad \\text{(for $Y \\in \\overline{{\\mathsf P}}(U)$)}\n$$\nof left and right $R$-modules. It follows from Proposition \\ref{prop:entireextclo} that $\\tilde{{\\mathsf P}}(U)$ is entirely extension closed in $\\operatorname{\\mathsf{Mod}}\\nolimits(U^\\mathrm{ev})$. Finally, $\\tilde{{\\mathsf P}}(U)$ is closed under taking tensor products over $U$, as $(X \\otimes_U X') \\otimes_U Y \\cong X \\otimes_U (X' \\otimes_U Y) \\in \\overline{P}(U)$ for all $X, X' \\in \\tilde{P}(U)$ and $Y \\in \\overline{P}(U)$.\n\\end{nn}\n\n\\begin{lem}\nThe functor $- \\otimes_{U} R \\colon \\operatorname{\\mathsf{Mod}}\\nolimits(U^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(U) = \\operatorname{\\mathsf{Mod}}\\nolimits(R^\\mathrm{ev})$ restricts to an exact functor\n$$\n- \\otimes_{U} R \\colon (\\tilde{{\\mathsf P}}(U), \\otimes_U, U) \\longrightarrow ({\\mathsf P}(R), \\otimes_R, R),\n$$\ntaking $U$ to $R$. The induced map\n$$\n\\operatorname{HH}\\nolimits^\\ast(U) \\cong \\operatorname{Ext}\\nolimits^\\ast_{\\tilde{{\\mathsf P}}(U)}(U,U) \\longrightarrow \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf P}(R)}(R,R) \\cong \\operatorname{HH}\\nolimits^\\ast(R)\n$$\nis a split surjection $($in the category of graded $K$-algebras$)$ and agrees with $h_\\ast^U \\colon \\operatorname{HH}\\nolimits^\\ast(U) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(R)$.\n\\end{lem}\n\n\\begin{proof}\nBy definition of $\\tilde{{\\mathsf P}}(U)$ it is apparent, that $- \\otimes_U R$ restricts as claimed; since $\\tilde{{\\mathsf P}}(U) \\subseteq {\\mathsf P}(U)$, the restriction will also be exact. Moreover, we have the natural isomorphism\n$$\nR \\xrightarrow{\\ \\sim \\ } U \\otimes_U R, \\ r \\mapsto 1 \\otimes r \\, .\n$$\nThe exact functor\n$$\n- \\otimes_R U = - \\otimes_K R \\colon ({\\mathsf P}(R), \\otimes_R, R) \\longrightarrow (\\operatorname{\\mathsf{Mod}}\\nolimits(U^\\mathrm{ev}), \\otimes_U, U)\n$$\ngives rise to an algebra homomorphism $\\operatorname{HH}\\nolimits^\\ast(R) \\cong \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf P}(R)}(R,R) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_{U^\\mathrm{ev}}(U,U) \\cong \\operatorname{HH}\\nolimits^\\ast(U)$ which is right inverse to the one induced by $- \\otimes_U R$ (see \\cite[Thm.\\,6.3.12]{He14b}). Finally, that we retain the map $h_\\ast$ follows by a similar argument as in the proof of Lemma \\ref{lem:grsolfunctor}(\\ref{lem:grsolfunctor:2}).\n\\end{proof}\n\n\\begin{cor}\nIn the long exact sequence of Proposition $\\ref{prop:happel}$ applied to $B = U[R]$,\n$$\n\\xymatrix@C=20pt{\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{g_n} & \\operatorname{HH}\\nolimits^n(U) \\ar[r]^-{h_n} & \\operatorname{Ext}\\nolimits^{n}_U(R,R) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B) \\ar[r] & \\cdots \\, ,\n}\n$$\nthe graded map $g_\\ast\\colon \\operatorname{HH}\\nolimits^\\ast(U[R]) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(U) \\times K$ is a homomorphisms of strict Gerstenhaber algebras and the graded map $h_\\ast^U \\colon \\operatorname{HH}\\nolimits^\\ast(U) \\rightarrow \\operatorname{Ext}\\nolimits^\\ast_U(R,R) = \\operatorname{HH}\\nolimits^\\ast(R)$ is a homomorphism of graded $K$-algebras. Furthermore, the map $h_\\ast^U$ is surjective, so that the long exact sequence decomposes into short exact sequences\n$$\n0 \\longrightarrow Z(B) \\longrightarrow Z(U) \\oplus K \\longrightarrow Z(R) \\longrightarrow 0\n$$\nand\n$$\n0 \\longrightarrow \\operatorname{HH}\\nolimits^n(B) \\longrightarrow \\operatorname{HH}\\nolimits^n(U) \\longrightarrow \\operatorname{HH}\\nolimits^n(R) \\longrightarrow 0\n$$\nfor each integer $n \\geqslant 1$. \\qed\n\\end{cor}\n\n\n\\section{Homological epimorphisms}\\label{sec:homepi}\n\n\\begin{nn}\nThe notion of a homological epimorphism has been introduced by Geigle--Lenzing in \\cite{GeLe91}. Recall that a ring homomorphism $\\pi \\colon B \\rightarrow A$ is an epimorphism in the category of rings, if $f \\circ \\pi = g \\circ \\pi$ for some ring homomorphisms $f, g\\colon A \\rightarrow R$ implies that $f = g$. Note that such homomorphisms are in general far from being surjective. For example, the canonical embedding $\\mathbb Z \\rightarrow \\mathbb Q$ is an epimorphism in the above sense, but certainly not surjective. More generally, if $B$ is commutative and $\\Sigma \\subseteq B$ is a multiplicative subset, then the canonical map $B \\rightarrow B[\\Sigma^{-1}]$ is an epimorphism. However, any surjective ring homomorphism will, of course, be an epimorphism.\n\\end{nn}\n\n\\begin{nn}\\label{nn:equhomepi}\nBy \\cite[Prop.\\,1.1]{Si67}, a ring homomorphism $\\pi\\colon B \\rightarrow A$ is a ring epimorphism if, and only if, the multiplication map $A \\otimes_B A \\rightarrow A$ is an isomorphism, which is if, and only if, the restriction functor $\\pi_\\star = \\mathrm{Res}(\\pi)\\colon \\operatorname{\\mathsf{Mod}}\\nolimits(A) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B)$ is full and faithful.  A ring epimorphism $\\pi\\colon B \\rightarrow A$ is \\textit{homological} if $\\operatorname{Tor}\\nolimits_i^B(A,A) = 0$ for $i > 0$. This is equivalent to the functor\n$$\n\\mathbf D^b(\\pi_\\star) {\\colon} \\mathbf D^b(\\operatorname{\\mathsf{Mod}}\\nolimits(A)) \\longrightarrow \\mathbf D^b(\\operatorname{\\mathsf{Mod}}\\nolimits(B))\n$$\nbeing full and faithful, see \\cite[Thm.\\,4.4]{GeLe91}.\n\\end{nn}\n\nIn what follows, we will assume that $A$ and $B$ are $K$-algebras, and $\\pi {\\colon} B \\rightarrow A$ is a $K$-linear homological epimorphism. We will further assume that $A$ and $B$ are \\textit{projective} as $K$-modules.\n\n\\begin{lem}\\label{lem:piev}\nThe $K$-algebra homomorphism $\\pi^\\mathrm{ev} = \\pi \\otimes \\pi^\\mathrm{op} {\\colon} B^\\mathrm{ev} \\rightarrow A^\\mathrm{ev}$ is a homological epimorphism.\n\\end{lem}\n\n\\begin{proof}\nIndeed, one has\n$$\n\\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_i(A \\otimes A^\\mathrm{op}, A \\otimes A^\\mathrm{op}) \\cong \\operatorname{Tor}\\nolimits^B_i(A, A \\otimes A) = 0 \\quad \\text{(for $i > 0$)}\n$$\nby \\cite[Chap.\\,IX, Thm.\\,2.8]{CaEi56} and \\cite[Thm.\\,4.4]{GeLe91}. Furthermore, if $f, g {\\colon} A^\\mathrm{ev} \\rightarrow R$ are ring homomorphisms with $f \\circ \\pi^\\mathrm{ev} = g \\circ \\pi^\\mathrm{ev}$, they must fulfil $f(- \\otimes 1) = g(- \\otimes 1)$ and $f(1 \\otimes - ) = g(1 \\otimes -)$ as $\\pi$ and $\\pi^\\mathrm{op}$ are epimorphisms. Thus $f = g$.\n\\end{proof}\n\nWe state an immediate consequence of Lemma \\ref{lem:piev} and Paragraph \\ref{nn:equhomepi}.\n\n\\begin{cor}\nThe graded $K$-algebras $\\operatorname{HH}\\nolimits^\\ast(A)$ and $\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(A,A)$ are isomorphic. \\qed\n\\end{cor}\n\n\\begin{nn}\nThe restriction functor along $\\pi^\\mathrm{ev}$ has a left adjoint, given by\n$$\nA^\\mathrm{ev} \\otimes_{B^\\mathrm{ev}} (-) \\cong A \\otimes_B (-) \\otimes_B A.\n$$\nThe following theorem states, that it gives rise to a very well behaved map between the Hochschild cohomology algebras of $B$ and $A$.\n\\end{nn}\n\n\\begin{thm}\\label{thm:homepi}\nThe graded $K$-algebra map\n$$\nA \\otimes_B^{\\mathbf L} (-) \\otimes_B^{\\mathbf L} A {\\colon} \\operatorname{HH}\\nolimits^\\ast(B) = \\operatorname{Hom}\\nolimits_{\\mathbf D^b(B^\\mathrm{ev})}(B,B[\\ast]) \\longrightarrow \\operatorname{Hom}\\nolimits_{\\mathbf D^b(A^\\mathrm{ev})}(A,A[\\ast]) = \\operatorname{HH}\\nolimits^\\ast(A)\n$$\nis a homomorphism of strict Gerstenhaber algebras; that is, it renders the diagrams\n$$\n\\xymatrix@C=30pt{\n\\operatorname{HH}\\nolimits^m(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} \\ar[d] & \\operatorname{HH}\\nolimits^{m+n-1}(B) \\ar[d] \\\\ \n\\operatorname{HH}\\nolimits^m(A) \\times \\operatorname{HH}\\nolimits^n(A) \\ar[r]^-{\\{-,-\\}_A} & \\operatorname{HH}\\nolimits^{m+n-1}(A)\n}\\quad\\quad\n\\xymatrix{\n\\operatorname{HH}\\nolimits^{2m}(B) \\ar[r]^-{sq_B} \\ar[d] & \\operatorname{HH}\\nolimits^{4m-1}(B) \\ar[d] \\\\ \n\\operatorname{HH}\\nolimits^{2m}(A) \\ar[r]^-{sq_A} & \\operatorname{HH}\\nolimits^{4m-1}(A)\n}\n$$\ncommutative for all integers $m, n \\geqslant 0$.\n\\end{thm}\n\nThe proof requires several preliminary observations whose main goal is to construct an exact and entirely extension closed monoidal subcategory $\\mathsf C$ of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$, such that $(A \\otimes_B (-) \\otimes_B A)\\mathord{\\upharpoonright}_{{\\mathsf C}}$ fulfils the requirements of Theorem \\ref{thm:gersthochcomp}.\n\n\\begin{nn}\\label{nn:tinftycinfty}\nRecall that ${\\mathsf P}(B) = {\\mathsf P}_\\lambda(B) \\cap {\\mathsf P}_\\varrho(B)$ denotes the full, exact monoidal subcategory of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ of those $B^\\mathrm{ev}$-modules which are projective when considered as left and right $B$-modules. Let $\\mathsf T_\\infty(B)$ be the full subcategory of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ consisting of all $B^\\mathrm{ev}$-modules $M$ with\n$$\n\\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_i(A \\otimes A^\\mathrm{op}, M) = 0 \\quad (\\text{for all $i > 0$}).\n$$\nSo, in different words, $\\mathsf T_\\infty(B) = \\operatorname{Ker}\\nolimits \\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_{> 0}(A \\otimes A^\\mathrm{op},-)$. Further, we let ${\\mathsf P}_\\infty(B)$ be the full subcategory ${\\mathsf P}(B) \\cap \\mathsf T_\\infty(B)$ of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$. As one has $\\operatorname{Tor}\\nolimits^K_i(A,A) = 0 = \\operatorname{Tor}\\nolimits^{B^\\mathrm{op}}_i(A,X) = \\operatorname{Tor}\\nolimits^B_i(X,A)$ for all $X \\in {\\mathsf P}(B)$ and $i > 0$, we may apply \\cite[Chap.\\,IX, Thm.\\,2.8]{CaEi56} to conclude that\n$$\n\\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_i(A \\otimes A^\\mathrm{op}, X) \\cong \\operatorname{Tor}\\nolimits^B_i(A, X \\otimes_{B} A) \\quad \\text{(for $i > 0$)}.\n$$\nThus, for $X \\in {\\mathsf P}(B)$,\n$$\nX \\in {\\mathsf P}_\\infty(B) \\quad \\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^{B}_i(A,X \\otimes_B A) = 0 \\quad (\\text{for all $i > 0$}),\n$$\nand consequently ${\\mathsf P}_\\infty(B) = {\\mathsf P}(B) \\cap \\operatorname{Ker}\\nolimits \\operatorname{Tor}\\nolimits^B_{>0}(A, - \\otimes_B A)$.\n\\end{nn}\n\n\\begin{lem}\nThe subcategory ${\\mathsf P}_\\infty(B)$ is extension closed in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, and contains $B$ and $B \\otimes B$. Moreover, it is closed under taking direct summands and arbitrary direct sums in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$. In particular, the subcategory ${\\mathsf P}_\\infty(B)$ is entirely extension closed in the sense of Paragraph $\\ref{nn:extclosed} $. Lastly, the subcategory ${\\mathsf P}_\\infty(B)$ is, viewed as an exact category, closed under kernels of epimorphisms, that is, if $f{\\colon} M \\rightarrow N$ is a surjective map in ${\\mathsf P}_\\infty(B)$, then $\\operatorname{Ker}\\nolimits(f)$ belongs to ${\\mathsf P}_\\infty(B)$.\n\\end{lem}\n\n\\begin{proof}\nAs the subcategory ${\\mathsf P}(B)$ fulfils all of the properties stated above, it suffices to show that $\\mathsf T_\\infty(B)$ does as well. To begin with, $\\operatorname{Hom}\\nolimits_B(B \\otimes A, -) \\cong \\operatorname{Hom}\\nolimits_K(A,-)$ tells, that the module $(B \\otimes B) \\otimes_B A \\cong B \\otimes A$ is in particular $B$-flat, and thus\n$$\n\\operatorname{Tor}\\nolimits^B_i(A,B \\otimes_B A) \\cong \\operatorname{Tor}\\nolimits^B_i(A,A) = 0 = \\operatorname{Tor}\\nolimits^B_i(A,(B \\otimes B) \\otimes_B A) \\quad (\\text{for $i > 0$}).\n$$\nConsequently $B$ and $B \\otimes B$ belong to $\\mathsf{T}_\\infty(B)$. As $\\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_i(A \\otimes A^\\mathrm{op}, -)$ commutes with (arbitrary) direct sums, $\\mathsf{T}_\\infty(B)$ is closed under taking summands and sums. Next, the long exact homology sequence\n$$\n\\xymatrix@C=11.5pt@R=4pt{\n\\cdots \\ar[r] & \\operatorname{Tor}\\nolimits^{B^\\mathrm{ev}}_{i+1}(A^\\mathrm{ev}, N) \\ar[r] & \\operatorname{Tor}\\nolimits^{B^\\mathrm{ev}}_i(A^\\mathrm{ev}, L) \\ar[r] & \\operatorname{Tor}\\nolimits^{B^\\mathrm{ev}}_i(A^\\mathrm{ev},M) \\ar[r] & \\operatorname{Tor}\\nolimits^{B^\\mathrm{ev}}_i(A^\\mathrm{ev},N) \\ar[r] & \\cdots & \\\\\n\\cdots \\ar[r] & \\operatorname{Tor}\\nolimits^{B^\\mathrm{ev}}_1(A^\\mathrm{ev},N) \\ar[r] & A \\otimes_B L \\otimes_B A \\ar[r] &  A \\otimes_B M \\otimes_B A \\ar[r] & A \\otimes_B N \\otimes_B A \\ar[r] & 0\n}\n$$\nfor some short exact sequence $0 \\rightarrow L \\rightarrow M \\rightarrow N \\rightarrow 0$ in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, shows that if $L$ and $N$ (or $M$ and $N$) belong to $\\mathsf{T}_\\infty(B)$, then so does $M$ (or, respectively, $L$). Finally, we conclude that $\\mathsf T_\\infty(B)$ is entirely extension closed in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ by Proposition \\ref{prop:entireextclo}.\n\\end{proof}\n\n\\begin{lem}\\label{lem:funcexactres}\nThe functor $A \\otimes_B (-) \\otimes_B A {\\colon} \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$ gives rise to an almost costrong monoidal functor\n$$\nA \\otimes_B (-) \\otimes_B A {\\colon} (\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B) \\longrightarrow (\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}), \\otimes_A, A).\n$$\nMoreover, when restricted to ${\\mathsf P}_\\infty(B)$, it defines an exact functor ${\\mathsf P}_\\infty(B) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$.\n\\end{lem}\n\n\\begin{proof}\nLet $M$ and $N$ be arbitrary $B^\\mathrm{ev}$-modules. It is a straightforward verification that the natural homomorphisms\n$$\n\\psi_0 {\\colon} A \\otimes_B B \\otimes_B A \\xrightarrow{\\, \\sim \\, } A \\otimes_B A \\xrightarrow{\\, \\sim \\, } A, \\ a \\otimes b \\otimes a' \\mapsto aba'\n$$\nand\n\\begin{align*}\n\\psi_{M,N} {\\colon} A \\otimes_B (M \\otimes_B N) \\otimes_B A & \\xrightarrow{\\ \\sim \\, } A \\otimes_B (M \\otimes_B B \\otimes_B N) \\otimes_B A\\\\ & \\xrightarrow{\\text{can}} A \\otimes_B (M \\otimes_B A \\otimes_B N) \\otimes_B A\\\\ & \\xrightarrow{\\ \\sim \\, } (A \\otimes_B M \\otimes_B A) \\otimes_A (A \\otimes_B N \\otimes_B A)\n\\end{align*}\nturn $A \\otimes_B (-) \\otimes_B A$ into an almost costrong monoidal functor $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}, \\otimes_A, A)$. As for the exactness, each choice of $X \\in {\\mathsf P}_\\infty(B)$ and $i > 0$ satisfies the equality\n$$\n\\operatorname{Tor}\\nolimits^{B \\otimes B^\\mathrm{op}}_i(A \\otimes A^\\mathrm{op}, X) = 0 = \\operatorname{Tor}\\nolimits^B_i(A, X \\otimes_{B} A) \\quad \\text{(for $i > 0$)}\n$$\nby Paragraph \\ref{nn:tinftycinfty}. Therefore, the sequence\n$$\n0 \\longrightarrow A \\otimes_B L \\otimes_B A \\longrightarrow A \\otimes_B M \\otimes_B A \\longrightarrow A \\otimes_B N \\otimes_B A \\longrightarrow 0\n$$\nis exact for each admissible short exact sequence $0 \\rightarrow L \\rightarrow M \\rightarrow N \\rightarrow 0$ in ${\\mathsf P}_\\infty(B)$, whence the exactness of the restriction of $A \\otimes_B (-) \\otimes_B A$ to ${\\mathsf P}_\\infty(B)$ follows. \n\\end{proof}\n\n\\begin{prop}\\label{prop:exactmono}\nThere is an exact and entirely extension closed monoidal subcategory $({\\mathsf C}, \\otimes_B, B)$ of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ such that\n\\begin{enumerate}[\\rm(1)]\n\\item ${\\mathsf C}$ is a full and exact subcategory of ${\\mathsf P}_\\infty(B)$ which is closed under taking direct summands and arbitrary direct sums, and\n\\item\\label{prop:exactmono:2} the functor $A \\otimes_B (-) \\otimes_B A$ restricts to an exact and almost costrong monoidal functor\n$$\n\\mathfrak A = (A \\otimes_B (-) \\otimes_B A)\\mathord{\\upharpoonright}_{{\\mathsf C}} {\\colon} ({\\mathsf C}, \\otimes_B, B) \\longrightarrow ({\\mathsf P}(A), \\otimes_A, A),\n$$\nthat is, one has the following commutative diagram of almost costrong monoidal functors.\n$$\n\\xymatrix@C=55pt{\n{\\mathsf C} \\ar[r]^-{\\mathfrak A} \\ar@{ >->}[d]_-{\\mathrm{inc}} & {\\mathsf P}(A) \\ar@{ >->}[d]^-{\\mathrm{inc}} \\\\\n\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\ar[r]^-{A \\otimes_B (-) \\otimes_B A} & \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})\n}\n$$\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nObserve that as soon as we have named an appropriate category ${\\mathsf C} \\subseteq {\\mathsf P}_\\infty(B)$ with $A \\otimes_B {\\mathsf C} \\otimes_B A \\subseteq {\\mathsf P}(A)$, the exactness of $\\mathfrak A$, as well as the fact that it is almost costrong monoidal, will follow from Lemma \\ref{lem:funcexactres} immediately. In order to find ${\\mathsf C}$, we proceed in several steps.\n\n\\medskip\n\n\\textbf{Step 1}. Let ${\\mathsf C}_1$ be the full subcategory of ${\\mathsf P}_\\infty(B)$ defined as follows: An object $M \\in {\\mathsf P}_\\infty(B)$ belongs to ${\\mathsf C}_1$ if, and only if, $M \\otimes_B X$ belongs to ${\\mathsf P}_\\infty(B)$ for all $X \\in {\\mathsf P}_\\infty(B)$. Apperently, ${\\mathsf C}_1$ is extension closed in ${\\mathsf P}_\\infty(B)$. It is further closed under direct summands and arbitrary direct sums, as well as under kernels of epimorphisms. Furthermore, for any $X \\in {\\mathsf P}_\\infty(B)$, $B \\otimes_B X \\cong X$ so that $B \\in {\\mathsf C}_1$. Also, the module $(B \\otimes B) \\otimes_B X \\cong B \\otimes X$ is $B$-projective on either side. Moreover, $(-) \\otimes_B (B \\otimes B) \\otimes_B X \\otimes_B A \\cong (-) \\otimes_B (B \\otimes X) \\otimes_B A \\cong (-) \\otimes (X \\otimes_B A)$ is an exact functor, since $X \\otimes_B A$ is $K$-projective, and hence\n$$\n\\operatorname{Tor}\\nolimits^B_i(A, (B \\otimes B) \\otimes_B X \\otimes_B A) = 0 \\quad \\text{(for $i > 0$)}.\n$$\nIt follows that $B \\otimes B$ belongs to ${\\mathsf C}_1$. From Proposition \\ref{prop:entireextclo} we deduce that ${\\mathsf C}_1$ is entirely extension closed in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$. Finally, ${\\mathsf C}_1$ is a monoidal subcategory of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ as, for all $M, N \\in {\\mathsf C}_1$ and $X \\in {\\mathsf P}_\\infty(B)$,\n$$\n(M \\otimes_B N) \\otimes_B X \\cong M \\otimes_B (N \\otimes_B X) \\ \\in \\ {\\mathsf P}_\\infty(B) \\, .\n$$\n\\medskip\n\n\\textbf{Step 2.} Let ${\\mathsf C}_2$ be the full subcategory of ${\\mathsf C}_1$ which is defined as follows. An object $M \\in {\\mathsf C}_1$ belongs to ${\\mathsf C}_2$ if, and only if, $A \\otimes_B M \\otimes_B A$ belongs to ${\\mathsf P}(A)$. As $A \\otimes_B B \\otimes_B A \\cong A$ and $A \\otimes_B (B \\otimes B) \\otimes_B A \\cong A \\otimes A$, the modules $B$ and $B \\otimes B$ belong to ${\\mathsf C}_2$. Except for being monoidal, the subcategory ${\\mathsf C}_2 \\subseteq \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ has the same properties as ${\\mathsf C}_1$, that is, it is extension closed, it is closed under direct summands and arbitrary direct sums and it is closed under kernels of epimorphisms. Yet again, these properties combined with $B \\otimes B \\in {\\mathsf C}_2$ yield that ${\\mathsf C}_2$ is entirely extension closed. We therefore have enforced the existence of an exact functor $A \\otimes_B (-) \\otimes_B A {\\colon} {\\mathsf C}_1 \\rightarrow {\\mathsf P}(A)$.\n\\medskip\n\n\\textbf{Step 3.} In the third and conclusive step, we define a full subcategory ${\\mathsf C} = {\\mathsf C}_3$ of ${\\mathsf C}_2$ as follows. A module $M \\in {\\mathsf C}_2$ belongs to ${\\mathsf C}$ if, and only if, $M \\otimes_B X$ belongs to ${\\mathsf C}_2$ for all $X \\in {\\mathsf C}_2$. Evidently, $B$ has that property, thus $B \\in {\\mathsf C}$. Let $X \\in {\\mathsf C}_2$; by adjunction, $\\operatorname{Hom}\\nolimits_A(A \\otimes X \\otimes_B A, -) \\cong \\operatorname{Hom}\\nolimits_K(X \\otimes_B A, -)$ and thus $A \\otimes_B (B \\otimes B) \\otimes_B X \\otimes_B A \\cong A \\otimes X \\otimes_B A$ is projective as a left $A$-module. On the other hand, $\\operatorname{Hom}\\nolimits_{A^\\mathrm{op}}(A \\otimes X \\otimes_B A, -) \\cong \\operatorname{Hom}\\nolimits_{B^\\mathrm{op}}(A \\otimes X, -)$. Since $A \\otimes X$ is a projective right $B$-module, $A \\otimes X \\otimes_B A$ is projective as a right $A$-module. It thus follows that $B \\otimes B \\in {\\mathsf C}$.\n\nThe subcategory ${\\mathsf C}$ inherits all other properties that ${\\mathsf C}_2$ has, that is, it is extension closed, it is closed under direct summands and arbitrary direct sums and it is closed under kernels of epimorphisms. Naturally, it is entirely extension closed as well. Moreover, it is a monoidal subcategory of $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, as for all $M, N \\in {\\mathsf C}$ and $X \\in {\\mathsf C}_2$\n$$\nA \\otimes_B (M \\otimes_B N) \\otimes_B X \\otimes_B A \\cong A \\otimes_B M \\otimes_B (N \\otimes_B X) \\otimes_B A  \\ \\in \\ {\\mathsf P}(A) \\, .\n$$\nThus the proof of the proposition is established.\n\\end{proof}\n\n\\begin{rem}\nLet $\\mathfrak A {\\colon} \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$ be a $K$-linear right exact monoidal functor that commutes with arbitrary direct sums. Assuming that the left derived functors $L_i \\mathfrak A$ (for $i > 0$) vanish on $B$, one can construct an exact monoidal subcategory ${\\mathsf C}(\\mathfrak A) \\subseteq {\\mathsf P}(B)$ of $(\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}), \\otimes_B, B)$ so that $\\mathfrak A$ restricts to an exact functor $({\\mathsf C}(\\mathfrak A), \\otimes_B, B) \\rightarrow ({\\mathsf P}(A), \\otimes_A, A)$.\n\\end{rem}\n\n\\begin{rem}\nObserve that the categories ${\\mathsf C}_1$ and ${\\mathsf C} = {\\mathsf C}_3$ in the above proof agree with $\\overline{P}_\\infty(B)$ and $\\overline {\\mathsf C}_2$, as introduced in Paragraph \\ref{nn:monoclosure}, and thus are extension closed and monoidal by Lemma \\ref{lem:monoclosure}. Further, ${\\mathsf C}_2$ is an exact category that arises in the way as described in Lemma \\ref{lem:smallexactfunc}.\n\\end{rem}\n\n\\begin{lem}\\label{lem:barexact}\nIf $\\mathbb B B$ denotes the bar resolution for $B$, then the complex\n$$\n\\mathbb B B \\otimes_B A \\longrightarrow B \\otimes_B A \\cong A \\longrightarrow 0\n$$\nis a flat resolution of $A$ over $B$ and, moreover,\n$$\nA \\otimes_B \\mathbb B B \\otimes_B A \\longrightarrow A \\otimes_B A \\cong A \\longrightarrow 0\n$$\nis a projective resolution of $A$ over $A^\\mathrm{ev}$.\n\\end{lem}\n\n\\begin{proof}\nFirst of all, $A \\otimes_B (-) \\otimes_B A$ is left-adjoint to the restriction functor $\\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev}) \\rightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$ along the epimorphism $\\pi^\\mathrm{ev} {\\colon} B \\otimes B^\\mathrm{op} \\rightarrow A \\otimes A^\\mathrm{op}$ (cf.\\,Lemma \\ref{lem:piev}) and thus takes projective $B^\\mathrm{ev}$-modules to projective $A^\\mathrm{ev}$-modules. When considered as a sequence of right $B$-modules, the bar resolution $\\mathbb B B \\rightarrow B \\rightarrow 0$ splits in every degree (in that each degree is the direct sum of the kernel and the image of the bordering homomorphisms). It follows that $\\mathbb B B \\otimes_B A \\rightarrow B \\otimes_B A \\cong A \\rightarrow 0$ is an exact resolution of $A$ by left $B$-modules. Since\n$$\n(-) \\otimes_B B^{\\otimes n} \\otimes_B A \\cong (-) \\otimes B^{\\otimes(n-2)} \\otimes A \\quad \\text{(for all $n \\geqslant 2$)}\n$$\nthis is even a resolution by flat left $B$-modules. Thus,\n$$\nH_i(A \\otimes_B \\mathbb B B \\otimes_B A) = \\operatorname{Tor}\\nolimits^B_i(A,A) = 0 \\quad \\text{(for all $i > 0$)},\n$$\nand, consequently, $A \\otimes_B \\mathbb B B \\otimes_B A \\rightarrow A \\otimes_B A \\cong A \\rightarrow 0$ is a projective resolution of $A$ over $A^\\mathrm{ev}$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:mapsequal}\nLet ${\\mathsf C}$ be as in the statement of Proposition $\\ref{prop:exactmono}$. The maps\n$$\nA \\otimes_B^{\\mathbf L} (-) \\otimes_B^{\\mathbf L} A {\\colon} \\operatorname{Hom}\\nolimits_{\\mathbf D^-(B^\\mathrm{ev})}(B,B[\\ast]) \\longrightarrow \\operatorname{Hom}\\nolimits_{\\mathbf D^-(A^\\mathrm{ev})}(A,A[\\ast])\n$$\nand\n$$\n\\operatorname{HH}\\nolimits^\\ast(B) \\cong \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(B,B) \\xrightarrow{\\, \\mathfrak A_\\ast \\,} \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf P(A)}(A,A) \\cong \\operatorname{HH}\\nolimits^\\ast(A)\n$$\nidentify under the canonical isomorphisms between $\\operatorname{Hom}\\nolimits$ in the derived category and $\\operatorname{Ext}\\nolimits$ in the module category.\n\\end{lem}\n\n\\begin{proof}\nBy Lemma \\ref{lem:barexact}, $(A \\otimes_B^{\\mathbf L} (-) \\otimes_B^{\\mathbf L} A)(B)$ computes as $A \\otimes_B \\mathbb B B \\otimes_B A$ which is, of course, quasi-isomorphic to $A$. If $\\mathbb X = (X_\\ast, \\partial_\\ast)$ is a complex in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, we will denote by $\\mathbb X^\\natural$ the complex obtained from $\\mathbb X$ by replacing $X_n$ by $0$ for $n \\leqslant -1$; that is, $\\mathbb X^\\natural$ is the truncation of $\\mathbb X$ in negative degrees. Fix an integer $n \\geqslant 1$ and an admissible $n$-extension\n$$\n\\xymatrix@C=16pt{\nS & \\equiv & 0 \\ar[r] & B \\ar[r] & E_{n-1} \\ar[r] & \\cdots \\ar[r] & E_0 \\ar[r] & B \\ar[r] & 0\n}\n$$\nin ${\\mathsf C}$. Moreover, fix a comparison map\n$$\n\\xymatrix@C=16pt{\n(\\mathbb B B \\rightarrow B) \\ar[d]_-\\Phi & & \\cdots \\ar[r] & B^{\\otimes(n+2)} \\ar[r]^-{\\beta_n} \\ar[d]_-{\\varphi_{n}} & B^{\\otimes (n+1)} \\ar[d]_-{\\varphi_{n-1}} \\ar[r] & \\cdots \\ar[r] & B \\otimes B \\ar[d]^-{\\varphi_{0}} \\ar[r]^-{\\beta_0} & B \\ar[r] \\ar@{=}[d] & 0 \\,\\,\\\\\nS & & 0 \\ar[r] & B \\ar[r] & E_{n-1} \\ar[r] & \\cdots \\ar[r] & E_0 \\ar[r] & B \\ar[r] & 0 \\, .\n}\n$$\nThe map $\\Phi^\\natural {\\colon} \\mathbb B B \\rightarrow S^\\natural$ is a quasi-isomorphism. Recall from, for instance, \\cite{HiSt97} or \\cite{Wei94}, that the canonical isomorphism $\\operatorname{Ext}\\nolimits^n_{\\mathsf C}(B,B) \\xrightarrow{\\sim} \\operatorname{HH}\\nolimits^n(B) \\xrightarrow{\\sim} \\operatorname{Hom}\\nolimits_{\\mathbf D^-(B)}(B,B[n])$ (the first being due to the fact that ${\\mathsf C}$ is entirely extension closed) sends the equivalence class of $S$ to the equivalence class $f_S$ of the roof\n$$\n\\xymatrix@!C=18pt@R=22pt{\nB & \\cdots \\ar[r] & 0 \\ar[r] & 0 \\ar[r]^-{} & 0 \\ar[r]^-{} & \\cdots \\ar[r]^{} & B \\ar[r] & 0 \\ar[r] & \\cdots \\\\\nS^\\natural \\ar[u]^-{\\text{qis}} \\ar[d]  & \\cdots \\ar[r] & 0 \\ar[r] \\ar[u] \\ar[d] & B \\ar[r] \\ar@{=}[d] \\ar[u] & E_{n-1} \\ar[d] \\ar[u] \\ar[r] & \\cdots \\ar[r] & E_0 \\ar[u] \\ar[r] \\ar[d] & 0 \\ar[r] \\ar[d] \\ar[u] & \\cdots \\\\\nB[n] & \\cdots \\ar[r] & 0 \\ar[r] & B \\ar[r]^-{} & 0 \\ar[r]^-{} & \\cdots \\ar[r]^{} & 0 \\ar[r] & 0 \\ar[r] & \\cdots\n}\n$$\nin $\\operatorname{Hom}\\nolimits_{\\mathbf D^-(B)}(B,B[n])$. We claim that there is a morphism $\\Psi {\\colon} \\mathbb B B \\rightarrow \\mathbb B B[n]$ of chain complexes in $\\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev})$, fitting inside the commutative diagram below.\n\\begin{equation}\\label{eq:diagramqis}\n\\begin{aligned}\n\\xymatrix{\n\\mathbb B B \\ar[d]_-{\\text{qis}} & \\mathbb B B \\ar@{=}[l] \\ar[r]^-\\Psi \\ar[d]^-{\\text{qis}}_-{\\Phi^\\natural} & \\mathbb B B[n] \\ar[d]^-{\\text{qis}}\\\\\nB & S^\\natural \\ar[r] \\ar[l]_-{\\text{qis}} & B[n]\n}\n\\end{aligned}\n\\end{equation}\nIndeed, $\\varphi_n$ uniquely factors through $\\Im(\\beta_n)$ as $\\varphi_n \\circ \\beta_{n+1} = 0$. Further, there is some $\\overline{\\varphi}_n {\\colon} B^{\\otimes(n+2)} \\rightarrow B \\otimes B$ with $(B \\otimes B \\twoheadrightarrow B) \\circ \\overline{\\varphi}_n = \\varphi_n$ since $B^{\\otimes(n+2)}$ is a projective $B^\\mathrm{ev}$-module. A lifting\n$$\n\\xymatrix@C=16pt{\n\\cdots \\ar[r] & B^{\\otimes (n+4)} \\ar[r] \\ar[d]_-{\\psi_{n+2}} & B^{\\otimes (n+3)} \\ar[d]_-{\\psi_{n+1}} \\ar[r] & B^{\\otimes(n+2)} \\ar[r] \\ar[d]_-{\\psi_{n}}|-{=}^-{\\overline{\\varphi}_n} & \\Im(\\beta_n) \\ar[r] \\ar[d] & 0\\\\\n\\cdots \\ar[r] & B^{\\otimes 4} \\ar[r] & B^{\\otimes 3} \\ar[r] & B \\otimes B \\ar[r] & B \\ar[r] & 0\n}\n$$\nof $\\overline{\\varphi}_n$ gives rise to the desired morphism $\\Psi$. It thus follows that $A \\otimes_B^{\\mathbf L} f_S \\otimes_B^{\\mathbf L} A$ is represented by the roof $(\\operatorname{id}\\nolimits_{\\mathbb BB}, A \\otimes_B \\Psi \\otimes_B A)$ conjugated by the canonical quasi-isomorphism $\\mathbb B B \\rightarrow B$.\n\nAs the diagram (\\ref{eq:diagramqis}) lives in ${\\mathsf C}$, all the quasi-isomorphisms in it will remain so if we apply $A \\otimes_B (-) \\otimes_B A$ to it. After identifying $A$ with $A \\otimes_B B \\otimes_B A$ (which we are allowed to do as $\\pi$ is homological; see Paragraph \\ref{nn:equhomepi}), we arrive at the commutative diagram\n$$\n\\xymatrix@!C=60pt{\nA \\otimes_B \\mathbb B B \\otimes_B A \\ar[d]_-{\\text{qis}} & A \\otimes_B \\mathbb B B \\otimes_B A \\ar@{=}[d] \\ar[r] \\ar@{=}[l] & A \\otimes_B \\mathbb B B \\otimes_B A[n] \\ar[d]^-{\\text{qis}}\\\\\nA & A \\otimes_B \\mathbb B B \\otimes_B A \\ar[d]^-{\\text{qis}} \\ar[r]^-{\\text{qis}} \\ar[l]_-{\\text{qis}} & A[n]\\\\\n& (A \\otimes_B S \\otimes_B A)^\\natural \\ar[ur] \\ar[ul]^-{\\text{qis}} &\n}\n$$\nwhich tells us that $A \\otimes_B^{\\mathbf L} f_S \\otimes_B^{\\mathbf L} A$ is represented by the roof $A \\leftarrow (A \\otimes_B S \\otimes_B A)^\\natural \\rightarrow A[n]$. To summarise, we have shown that within the diagram\n$$\n\\xymatrix@C=18pt{\n\\operatorname{Hom}\\nolimits_{\\mathbf D^-(B^\\mathrm{ev})}(B,B[\\ast]) \\ar[rrr]^-{A \\otimes_B^{\\mathbf L} (-) \\otimes_B^{\\mathbf L} A} & & &  \\operatorname{Hom}\\nolimits_{\\mathbf D^-(A^\\mathrm{ev})}(A,A[\\ast])\\\\\n& \\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(B,B) \\ar@{-->}[ul]_-\\cong \\ar[dl]_-\\cong \\ar[r]^-{\\mathfrak U_\\ast} & \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf P}(A)}(A,A) \\ar[dr]^-\\cong \\ar@{-->}[ur]^-\\cong &\\\\\n\\operatorname{HH}\\nolimits^\\ast(B) \\ar[uu]^-\\cong \\ar@{-->}[rrr] & & & \\operatorname{HH}\\nolimits^\\ast(A) \\ar[uu]_-\\cong\n}\n$$\nthe sub-diagram given by the solid arrows commutes, whence the result follows.\n\\end{proof}\n\nAfter all preparations made, the proof of Theorem \\ref{thm:homepi} is now a single-liner. Recall that it asserted that the functor $A \\otimes_B^{\\mathbf L} (-) \\otimes_B^{\\mathbf L} A$ defines a homomorphism $\\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(A)$ of strict Gerstenhaber algebras.\n\n\\begin{proof}[Proof of Theorem $\\ref{thm:homepi}$]\nUse Proposition \\ref{prop:exactmono}(\\ref{prop:exactmono:2}) and Lemma \\ref{lem:mapsequal} with Theorem \\ref{thm:gersthochcomp}.\n\\end{proof}\n\nWe will apply the insights acquired in this and previous sections to long exact sequences by Koenig--Nagase.\n\n\n\\section{The long exact sequences of Koenig--Nagase}\\label{sec:koenagase}\n\n\\begin{nn}\nIn the case of \\textit{stratifying ideals}, the long exact Hochschild cohomology sequences stated in this paragraph are due to Koenig--Nagase; see \\cite{KoNa09}. In those sequences, we will identify certain homomorphisms as such of strict Gerstenhaber algebras. Proofs for them being multiplicative can, for $I$ being stratifying, already be found in \\cite{KoNa09}. In the following, we let $B$ be a $K$-algebra, $I \\subseteq B$ a two-sided ideal, $A = B\/I$ and $\\pi {\\colon} B \\rightarrow A$ the canonical surjection. We further assume that $B$ and $A$ are \\textit{projective} as $K$-modules. Observe that one has the following commutative diagrams with exact rows:\n\\begin{equation*}\n\\begin{aligned}\n\\xymatrix@C=18pt@R=18pt{\n0 \\ar[r] & \\operatorname{Tor}\\nolimits_1^B(I,A) \\ar[d]_-\\cong \\ar[r] & I \\otimes_B I \\ar[r] \\ar@{=}[d] & I \\otimes_B B \\ar[r] \\ar[d]^-\\cong & I \\otimes_B A \\ar[r] \\ar[d]^-\\cong & 0\\, \\ \\\\\n0 \\ar[r] & \\operatorname{Tor}\\nolimits_2^B(A,A) \\ar[r] & I \\otimes_B I \\ar[r]^-{\\mu_I} & I \\ar[r] & I\/I^2 \\ar[r] & 0 \\, ,\n}\n\\end{aligned}\n\\end{equation*}\nwhere $\\mu_I{\\colon} I \\otimes_B I \\rightarrow I$ denotes the (restricted) multiplication map, and\n\\begin{equation*}\n\\begin{aligned}\n\\xymatrix@C=18pt@R=18pt{\n0 \\ar[r] & \\operatorname{Tor}\\nolimits_1^B(A,A) \\ar@{=}[d] \\ar[r] & I \\otimes_B A \\ar[r]^-0 \\ar[d]_-\\cong & B \\otimes_B A \\ar[r] \\ar[d]^-\\cong & A \\otimes_B A \\ar[r] \\ar@{=}[d] & 0\\, \\ \\\\\n0 \\ar[r] & \\operatorname{Tor}\\nolimits_1^B(A,A) \\ar[r] & I\/I^2 \\ar[r] & A \\ar[r] & A \\otimes_B A \\ar[r] & 0 \\, .\n}\n\\end{aligned}\n\\end{equation*}\nThus, $\\operatorname{Tor}\\nolimits_1^B(A,A) \\cong I\/I^2$ and $A \\cong A \\otimes_B A$ as $A^\\mathrm{ev}$-modules. It follows that $\\operatorname{Tor}\\nolimits_1^B(A,A) = 0 = \\operatorname{Tor}\\nolimits_2^B(A,A)$ holds true if, and only if, the map $\\mu_I{\\colon} I \\otimes_B I \\rightarrow I$ is an isomorphism and $I = I^2$. See \\cite{APT92} for a thorough study of ideals with such properties, and \\cite{GeLe91} for some related observations. \n\\end{nn}\n\n\\begin{lem}\\label{lem:ideal}\nIf $\\operatorname{Tor}\\nolimits^B_i(A,A) = 0$ for $i > 0$, then the functor $$A \\otimes_B (-) \\otimes_B A {\\colon} \\operatorname{\\mathsf{Mod}}\\nolimits(B^\\mathrm{ev}) \\longrightarrow \\operatorname{\\mathsf{Mod}}\\nolimits(A^\\mathrm{ev})$$ gives rise to an isomorphism\n$$\n\\operatorname{HH}\\nolimits^\\ast(B,A) = \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(B,A) \\xrightarrow{\\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{A^\\mathrm{ev}}(A,A) = \\operatorname{HH}\\nolimits^\\ast(A)\n$$\nof graded modules over $\\operatorname{HH}\\nolimits^\\ast(A)$.\n\\end{lem}\n\n\\begin{proof}\nBy Lemma \\ref{lem:barexact}, $A \\otimes_B \\mathbb B B \\otimes_B A \\rightarrow A \\otimes_B A \\cong A \\rightarrow 0$ is a projective resolution of $A$ over $A^\\mathrm{ev}$. Therefore, for all $n \\geqslant 0$,\n\\begin{align*}\n\\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(B,A) & \\cong H^n \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(\\mathbb B B, A) && \\text{(by definiton)}\\\\\n&\\cong H^n \\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(A \\otimes_B \\mathbb B B \\otimes_B A, A) && \\text{(by adjunction)}\\\\\n&\\cong \\operatorname{Ext}\\nolimits^n_{A^\\mathrm{ev}}(A,A) && \\text{(by the remark above)},\n\\end{align*}\nwhence the claim follows.\n\\end{proof}\n\nThe following definition is taken from \\cite[Sec.\\,2.1]{ClPaSc96}.\n\n\\begin{defn}\nLet $\\Lambda$ be a ring. An idempotent element $e \\in \\Lambda$ is a \\textit{stratifying idempotent} if the multiplication map $\\Lambda e \\otimes_{e\\Lambda e} e\\Lambda \\rightarrow \\Lambda e \\Lambda$ is an isomorphism and $\\operatorname{Tor}\\nolimits^{e\\Lambda e}_i(\\Lambda e, e\\Lambda) = 0$ for $i > 0$. A twosided ideal $J \\subseteq \\Lambda$ is a \\textit{stratifying ideal} if there is a stratifying idempotent $e \\in \\Lambda$ that generates $J$, i.e., $J = \\Lambda e \\Lambda$.\n\\end{defn}\n\n\\begin{exas}\nLet $R$ and $S$ be $K$-algebras, and $M$ an $R \\otimes S^\\mathrm{op}$-module. \n\\begin{enumerate}[\\rm(1)]\n\\item Consider the matrix algebra\n$$\nB = \\left(\\begin{matrix}\nR & M\\\\\n0 & S\n\\end{matrix}\n\\right)\n$$\nalong with the canonical idempotents\n$$\ne = \\left(\\begin{matrix}\n1 & 0\\\\\n0 & 0\n\\end{matrix}\\right) \\quad \\text{and} \\quad e' = \\left(\\begin{matrix}\n0 & 0\\\\\n0 & 1\n\\end{matrix}\\right).\n$$\nClearly $I = BeB$ is isomorphic to $eB$, and $R = Be = eBe$. Hence $R \\otimes_{R} eB = Be \\otimes_{eBe} eB \\rightarrow BeB = eB$ is an isomorphism and\n$$\n\\operatorname{Tor}\\nolimits^{eBe}_i(Be, eB) = \\operatorname{Tor}\\nolimits^{eBe}_i(eBe, eB) = 0 \\quad \\text{(for $i > 0$)}.\n$$\nConsequently, $e$ is a stratifying idempotent in $B$, and $I = BeB$ is a stratifying ideal. By similar arguments, $e'$ is a stratifying idempotent as well, and thus, $I' = Be'B$ is stratifying.\n\\item The above example can be extended. Let $N$ be a $S \\otimes R^\\mathrm{op}$-module and $\\phi \\colon N \\otimes_R M \\rightarrow S$, $\\psi \\colon M \\otimes_S N \\rightarrow R$ be bimodule homomorphisms. These additional data give rise to a \\textit{generalised matrix algebra}:\n$$\nB = \\left(\\begin{matrix}\nR & M\\\\\nN & S\n\\end{matrix}\n\\right)\\,.\n$$\nBy \\cite[Prop.\\,4.1]{GaPs15}, the idempotent $e$ is stratifying in $B$ if, and only if, $\\phi$ is a monomorphism and $\\operatorname{Tor}\\nolimits^{eBe}_i(Be,eB) = 0$ for all $i > 0$. Similarly, $e'$ is stratifying in $B$ if, and only if, $\\psi$ is a monomorphism and $\\operatorname{Tor}\\nolimits^{e'Be'}_i(Be',e'B) = 0$ for all $i > 0$.\n\\end{enumerate}\n\\end{exas}\n\n\\begin{rem}\nA great deal of examples of algebras that are controlled by certain stratifying ideals is given by \\textit{quasi-hereditary algebras}, as studied in, for instance, \nthe very clear note \\cite{DlRi89b}. Let $\\Lambda$ be a ring and $\\mathrm{Rad}(\\Lambda)$ its Jacobsen radical. Recall that a twosided ideal $J$ of $\\Lambda$ is called a \\textit{heredity ideal}, if $J = \\Lambda e \\Lambda$ for some idepotent element $0 \\neq e \\in \\Lambda$, $J \\mathrm{Rad}(\\Lambda) J = 0$ and $J$ is a projective right $\\Lambda$-module. By \\cite[Appendix, Statement 6]{DlRi89b} every idempotent ideal is of the form $\\Lambda e \\Lambda$ for some idempotennt $e \\in \\Lambda$, if $\\Lambda$ is \\textit{semiprimary} (that is, $\\mathrm{Rad}(\\Lambda)$ is nilpotent and $\\Lambda \/ \\mathrm{Rad}(\\Lambda)$ is semisimple artinian; therefore, every finite dimensional algebra over a field is semiprimary), so that in this situation the first requirement translates to $J = J^2$.\n\nLet $J = \\Lambda e \\Lambda \\subseteq \\Lambda$ be a heredity ideal. It follows from \\cite[Appendix, Statement 7]{DlRi89b} that the multiplication map $\\Lambda e \\otimes_{e\\Lambda e} e\\Lambda \\rightarrow J$ defines a bimodule isomorphism. Moreover, as $J$ is a projective right $\\Lambda$-module,\n$$\n\\operatorname{Hom}\\nolimits_{(e\\Lambda e)^\\mathrm{op}}(\\Lambda e, (-)e) \\cong \\operatorname{Hom}\\nolimits_{\\Lambda^\\mathrm{op}}(\\Lambda e \\otimes_{e\\Lambda e} e\\Lambda, -) \\cong \\operatorname{Hom}\\nolimits_{\\Lambda^\\mathrm{op}}(J,-)\n$$\nis exact, thus $\\Lambda e$ is a projective right $e\\Lambda e$-module and, in particular, $J$ is a stratifying ideal in $\\Lambda$. The ring $\\Lambda$ is called \\textit{quasi-hereditary}, if there is a \\textit{heredity chain}\n$$\n0 = J_0 \\subsetneq J_1 \\subsetneq \\cdots \\subsetneq J_{n-1} \\subsetneq J_n = \\Lambda \\, ,\n$$\nthat is, a chain of ideals in $\\Lambda$ such that $J_t\/J_{t-1}$ is a heredity ideal in $\\Lambda\/J_{t-1}$ (for $1 \\leqslant t \\leqslant n$). We thus obtain canonical short exact sequences\n$$\n0 \\longrightarrow J_t\/J_{t-1} \\longrightarrow \\Lambda \/ J_{t-1} \\longrightarrow \\Lambda \/ J_t \\longrightarrow 0 \\quad \\text{(for $1 \\leqslant t \\leqslant n$)},\n$$\nwhere the leftmost term in particular is a stratifying ideal of the middle term. Every semiprimary ring of global dimension $2$ is quasi-hereditary (see \\cite[Thm.\\,2]{DlRi89b}). Moreover, any semiprimary algebra $R$ may be realised as $e \\Lambda e$ for some quasi-hereditary algebra $\\Lambda$ and some idempotent $e \\in \\Lambda$ (see \\cite{DlRi89a}). Over fields, the Brauer algebras $B(r,\\delta)$ and the partition algebras $P(r,\\delta)$ are quasi-hereditary for many choices of the parameters $r \\geqslant 1$ and $\\delta \\in K$ (see \\cite[Thm.\\,1.3 and Thm.\\,1.4]{KoXi99}).\n\\end{rem}\n\n\\begin{lem}\\label{lem:homepi}\nIf the ideal $I$ is a stratifying ideal, then $\\operatorname{Tor}\\nolimits^B_i(A,A) = 0$ for $i > 0$, that is, $\\pi{\\colon} B \\rightarrow A$ is a homological epimorphism.\n\\end{lem}\n\n\\begin{proof}\nObserve that for $i > 0$,\n$$\n\\operatorname{Tor}\\nolimits_i^B(I,A) \\cong \\operatorname{Tor}\\nolimits^B_i(Be \\otimes_{eBe} eB, A) \\cong \\operatorname{Tor}\\nolimits^{eBe}_i(Be, eB \\otimes_B A) \\cong \\operatorname{Tor}\\nolimits^{eBe}_i(Be, eA) = 0\n$$\nby \\cite[Chap.\\,IX, Thm.\\,2.8]{CaEi56} and, moreover,\n$$\n\\operatorname{Tor}\\nolimits^B_1(A,A) = I\/I^2 = 0, \\quad \\operatorname{Tor}\\nolimits_{i+1}^B(A,A) \\cong \\operatorname{Tor}\\nolimits_i^B(I,A)\n$$\nby the long exact homology sequence arsing from applying $- \\otimes_B A$ to the canonical short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:isoextstr}\nThe lemma implies that the restriction functor induces isomorphisms\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{A}(M,N) \\xrightarrow{\\, \\sim \\, } \\operatorname{Ext}\\nolimits^\\ast_B(M,N)\\, ,\n$$\nfor all $A$-modules $M$ and $N$, as $\\mathbf D^b(\\pi_\\star)$ is full and faithful by Paragraph \\ref{nn:equhomepi}. The following statement can also be found in \\cite{GeLe91}. We still give the (short) proof for completeness.\n\\end{rem}\n\n\\begin{lem}\\label{lem:strattorvan}\nIf the ideal $I$ is a stratifying ideal, then $\\operatorname{Tor}\\nolimits^B_i(A,M) = 0$ for each $A$-module $M$ and each integer $i > 0$.\n\\end{lem}\n\n\\begin{proof}\nIndeed, for all $i > 0$,\n\\begin{align*}\n0 & = \\operatorname{Ext}\\nolimits^i_A(A,\\operatorname{Hom}\\nolimits_{\\mathbb Z}(M, \\mathbb Q \/ \\mathbb Z))&& \\text{(as $A$ is $A$-pojective)}\\\\\n&\\cong \\operatorname{Ext}\\nolimits^i_B(A,\\operatorname{Hom}\\nolimits_{\\mathbb Z}(M, \\mathbb Q \/ \\mathbb Z)) && \\text{(by Remark \\ref{rem:isoextstr})}\\\\\n&\\cong \\operatorname{Hom}\\nolimits_{\\mathbb Z}(\\operatorname{Tor}\\nolimits^B_i(A,M), \\mathbb Q\/ \\mathbb Z) && \\text{(by \\cite[Chap.\\,VI, Prop.\\,5.1]{CaEi56})}.\n\\end{align*}\nAs $\\mathbb Q\/\\mathbb Z$ is an injective cogenerator for $\\mathbb Z$,\n$$\n\\operatorname{Hom}\\nolimits_{\\mathbb Z}(\\operatorname{Tor}\\nolimits^B_i(A,M), \\mathbb Q\/ \\mathbb Z) = 0 \\quad \\Longleftrightarrow \\quad \\operatorname{Tor}\\nolimits^B_i(A,M) = 0 \\quad \\text{(for $i > 0$)},\n$$\nthus the claim follows.\n\\end{proof}\n\nThe following observation is an immediate consequence of the above lemma, and Remark \\ref{rem:isoextstr}.\n\n\\begin{lem}[see {\\cite[Prop.\\,3.3]{KoNa09}}]\\label{lem:isostrat}\nAssume that the ideal $I$ is a stratifying ideal, and let $e \\in B$ be the stratifying idem\\-po\\-tent that generates $I$, i.e., $I = BeB$. If $M$ is a $B^\\mathrm{ev}$-module, then\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(I,M) \\xrightarrow{\\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{(eBe)^\\mathrm{ev}}(eBe, eMe) = \\operatorname{HH}\\nolimits^\\ast(eBe, eMe)\n$$\nas graded $K$-modules.\n\\end{lem}\n\n\\begin{proof}\nSince $I \\cong Be \\otimes_{eBe} eB$ as $B^\\mathrm{ev}$-modules, we conclude\n$$\n\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(I,M) \\xrightarrow{\\ \\sim \\ } \\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(Be \\otimes_{eBe} eB, M) \\xrightarrow{\\ \\sim \\ } \\operatorname{HH}\\nolimits^\\ast(eBe, eMe)\n$$\nby Lemma \\ref{lem:isoidealtor}.\n\\end{proof}\n\n\\begin{nn}\nBy applying $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(B,-)$ and $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(-,B)$ to the canonical short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$ we obtain long exact sequences\n$$\n\\xymatrix@C=16pt{\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{HH}\\nolimits^n(B,A) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B,I) \\ar[r] & \\cdots\n}\n$$\nand\n$$\n\\xymatrix@C=16pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(A,B) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(I,B) \\ar[r] & \\operatorname{Ext}\\nolimits^{n+1}_{B^\\mathrm{ev}}(A,B) \\ar[r] & \\cdots\n}\\, .\n$$\nThe following is our main observation in this section.\n\\end{nn}\n\n\\begin{thm}\\label{thm:koenignagase}\n\\begin{enumerate}[\\rm(1)]\n\\item\\label{thm:koenignagase:1} If $\\operatorname{Tor}\\nolimits_i^B(A,A) = 0$ for all $i > 0$, then the canonical short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$ gives rise to a long exact sequence\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{k_n} & \\operatorname{HH}\\nolimits^n(A) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B,I) \\ar[r] & \\cdots\n}\n$$\nin Hochschild cohomology. The map $k_\\ast {\\colon} \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(A)$ is a homomorphism of strict Gerstenhaber algebras, that is, $k_\\ast$ is a homomorphism of graded $K$-algebras and\n$$\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^m(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} \\ar[d]_-{k_m \\times k_n} & \\operatorname{HH}\\nolimits^{m+n-1}(B) \\ar[d]^-{k_{m + n -1}} \\\\\n\\operatorname{HH}\\nolimits^{m}(A) \\times \\operatorname{HH}\\nolimits^n(A) \\ar[r]^-{\\{-,-\\}_A} & \\operatorname{HH}\\nolimits^{m+n-1}(A)\n}\n\\ \\quad\\quad \\\n\\xymatrix@C=35pt{\n\\operatorname{HH}\\nolimits^{2n}(B) \\ar[r]^{sq_B} \\ar[d]_-{k_{2n}} & \\operatorname{HH}\\nolimits^{4n-1}(B) \\ar[d]^-{k_{4n-1}}\\\\\n\\operatorname{HH}\\nolimits^{2n}(A) \\ar[r]^{sq_A} & \\operatorname{HH}\\nolimits^{4n-1}(A)\n}\n$$\ncommute.\n\n\\item\\label{thm:koenignagase:2} If the ideal $I$ is a stratifying ideal, with corresponding stratifying idempotent $e \\in B$, then the short exact sequence $0 \\rightarrow I \\rightarrow B \\rightarrow A \\rightarrow 0$ gives rise to a long exact sequence\n$$\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^n(A,B) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{l_n} & \\operatorname{HH}\\nolimits^n(eBe) \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^{n+1}(A,B) \\ar[r] & \\cdots\n}\n$$\nin Hochschild cohomology. The map $l_\\ast {\\colon} \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(eBe)$ is a homomorphism of strict Gerstenhaber algebras, that is, $l_\\ast$ is a homomorphism of graded $K$-algebras and\n$$\n\\xymatrix@C=30pt{\n\\operatorname{HH}\\nolimits^m(B) \\times \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{\\{-,-\\}_B} \\ar[d]_-{l_m \\times l_n} & \\operatorname{HH}\\nolimits^{m+n-1}(B) \\ar[d]^-{l_{m + n -1}} \\\\\n\\operatorname{HH}\\nolimits^{m}(eBe) \\times \\operatorname{HH}\\nolimits^n(eBe) \\ar[r]^-{\\{-,-\\}_{eBe}} & \\operatorname{HH}\\nolimits^{m+n-1}(eBe)\n}\n\\quad\\quad\n\\xymatrix@C=30pt{\n\\operatorname{HH}\\nolimits^{2n}(B) \\ar[r]^{sq_B} \\ar[d]_-{l_{2n}} & \\operatorname{HH}\\nolimits^{4n-1}(B) \\ar[d]^-{l_{4n-1}}\\\\\n\\operatorname{HH}\\nolimits^{2n}(eBe) \\ar[r]^{sq_{eBe}} & \\operatorname{HH}\\nolimits^{4n-1}(eBe)\n}\n$$\ncommute. \n\\end{enumerate}\n\\end{thm}\n\nWe will turn ourselves to the proof after a remark and an immediate corollary of the theorem.\n\n\\begin{rem}\nLet $e \\in B$ be a stratifying idempotent, $I = BeB$ and $A = B\/I$. In \\cite{KoNa09}, the authors also construct a long exact sequence\n$$\n\\xymatrix@C=14pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(A,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{HH}\\nolimits^n(A) \\times \\operatorname{HH}\\nolimits^n(eBe) \\ar[r] & \\operatorname{Ext}\\nolimits^{n+1}_{B^\\mathrm{ev}}(A,I) \\ar[r] & \\cdots\n} \\, ,\n$$\nwhere the component maps of $\\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(A) \\times \\operatorname{HH}\\nolimits^\\ast(eBe)$ precisely are $k_\\ast$ and $l_\\ast$ as introduced in Theorem \\ref{thm:koenignagase}. Thus the map $\\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(A) \\times \\operatorname{HH}\\nolimits^\\ast(eBe)$ in that sequence is a homomorphism of strict Gerstenhaber algebras as well. The first part of the corollary below makes use of this additional long exact sequence.\n\\end{rem}\n\n\\begin{cor}\nAssume that the ideal $I$ is a stratifying ideal, with corresponding stratifying idempotent $e \\in B$.\n\\begin{enumerate}[\\rm(1)]\n\\item If $d = \\operatorname{id}\\nolimits_{B^\\mathrm{ev}}(I)$ is finite, then $k_n {\\colon} \\operatorname{HH}\\nolimits^n(B) \\rightarrow \\operatorname{HH}\\nolimits^n(A)$ is an isomorphism for $n > d$. Thus, $\\operatorname{HH}\\nolimits^{>d}(B) \\cong \\operatorname{HH}\\nolimits^{>d}(A)$ as non-unital strict Gerstenhaber algebras. In particular, $\\operatorname{HH}\\nolimits^{>d}(eBe) = 0$.\n\\item If $d' = \\operatorname{pd}\\nolimits_{B^\\mathrm{ev}}(A)$ is finite, then $l_n {\\colon} \\operatorname{HH}\\nolimits^n(B) \\rightarrow \\operatorname{HH}\\nolimits^n(eBe)$ is an isomorphism for $n > d'$. Thus, $\\operatorname{HH}\\nolimits^{>d'}(B) \\cong \\operatorname{HH}\\nolimits^{>d'}(eBe)$ as non-unital strict Gerstenhaber algebras.\n\\qed\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}[Proof of Theorem $\\ref{thm:koenignagase}$]\nLet us first prove item (\\ref{thm:koenignagase:2}). As $\\operatorname{Ext}\\nolimits^\\ast_{B^\\mathrm{ev}}(I,B) \\xrightarrow \\sim \\operatorname{HH}\\nolimits(eBe)$ by Lemma \\ref{lem:isostrat}, the existence of the exact sequence is imminent. By arguments being similar to those in the proof of Thereom \\ref{thm:greensol}, one shows that the map $l_\\ast$ is induced by the recollement $\\mathcal R(B,e)$, thus, the assertion follows from Theorem \\ref{thm:mainthm}. \n\nThanks to Lemma \\ref{lem:ideal}, we obtain a long exact sequence as claimed in (\\ref{thm:koenignagase:1}). Let us conclude that $k_\\ast$ is a homomorphism of strict Gerstenhaber algebras. In the following, we let $\\pi: B \\rightarrow A$ be the canonical surjection. By Proposition \\ref{prop:exactmono}, one has the following diagram of graded maps.\n$$\n\\xymatrix{\n\\operatorname{Ext}\\nolimits^\\ast_{\\mathsf C}(B,B) \\ar[d]_-\\cong \\ar[r]^-{\\mathfrak{A}_\\ast} & \\operatorname{Ext}\\nolimits^\\ast_{{\\mathsf C}(A)}(A,A) \\ar[d]^-{\\cong}\\\\\n\\operatorname{HH}\\nolimits^\\ast(B) \\ar[r]^-{k_\\ast} & \\operatorname{HH}\\nolimits^\\ast(A)\n}\n$$\nWe are going to prove that this square commutes -- and thus an application of Theorem \\ref{thm:homepi} yields the desired result, since the map $\\mathfrak{A}_\\ast$ coincides with the one discussed in there (see Lemma \\ref{lem:mapsequal}). Let $\\varphi{\\colon} B^{\\otimes(n+2)} \\rightarrow B$ be a cocycle representing an element $\\alpha \\in \\operatorname{HH}\\nolimits^n(B)$. The preimage of $\\alpha$ in $\\operatorname{Ext}\\nolimits^n_{\\mathsf C}(B,B)$, under the vertical isomorphism, is given by the equivalence class of the lower sequence in the following pushout diagram.\n$$\n\\xymatrix@C=18pt@R=18pt{\n\\cdots \\ar[r] & B^{\\otimes(n+2)} \\ar[r]^-{\\beta^B_n} \\ar[d]_-{\\varphi} & B^{\\otimes(n+1)} \\ar[r]^-{\\beta^B_{n-1}} \\ar[d] & \\cdots \\ar[r]^-{\\beta^B_1} & B \\otimes B \\ar[r]^-{\\beta^B_0} \\ar@{=}[d] & B \\ar[r] \\ar@{=}[d] & 0\\\\\n0 \\ar[r] & B \\ar[r] & B \\oplus_{B^{\\otimes(n+2)}} B^{\\otimes(n+1)} \\ar[r] & \\cdots \\ar[r] & B \\otimes B \\ar[r] & B \\ar[r] & 0\n}\n$$\nPut $P = B \\oplus_{B^{\\otimes(n+2)}} B^{\\otimes(n+1)}$. Identifying $A$ with $A \\otimes_B B \\otimes_B A$, the homomorphism $\\mathfrak A_n$ maps this equivalence class, by definition, to the equivalence class of the admissible $n$-extension\n\\begin{equation}\\label{eq:imageA}\\tag{$\\diamond$}\n0 \\longrightarrow A \\longrightarrow A \\otimes_B P \\otimes_B A \\longrightarrow \\cdots \\longrightarrow A \\otimes_B (B \\otimes B) \\otimes_B A \\longrightarrow A \\longrightarrow 0 \\, ,\n\\end{equation}\nwhich, thanks to the vertical isomorphism on the right (= the inclusion), sits inside $\\operatorname{HH}\\nolimits^n(A)$. We show that this class coincides with $k_n(\\alpha)$. Indeed, if $j_1 {\\colon} A \\otimes_B A \\otimes_B A \\xrightarrow{\\sim} A$ and $j_2{\\colon} A \\otimes_B B \\otimes_B A \\xrightarrow{\\sim} A$ denote the canonical isomorphisms, then $k_n$ maps $\\alpha$ to the equivalence class of the cocycle $\\tilde{\\varphi} = j_1 \\circ (A \\otimes_B (\\pi \\circ \\varphi) \\otimes_B A) = j_2 \\circ (A \\otimes_B \\varphi \\otimes_B A)$,\n$$\n\\tilde{\\varphi}{\\colon} A \\otimes B^{\\otimes n} \\otimes A \\longrightarrow A, \\ \\tilde{\\varphi}(a \\otimes b_1 \\otimes \\cdots \\otimes b_n \\otimes a') = a \\cdot (\\pi \\circ \\varphi)(1 \\otimes b_1 \\otimes \\cdots \\otimes b_n \\otimes 1) \\cdot a'\\, ,\n$$\nin $\\operatorname{Hom}\\nolimits_{A^\\mathrm{ev}}(A \\otimes_B \\mathbb B B \\otimes_B A, A)$. As $A \\otimes_B B^{\\otimes(m+2)} \\otimes_B A \\cong A \\otimes B^{\\otimes m} \\otimes A$ belongs to ${\\mathsf P}(A)$ for all $m \\geqslant 0$, a preimage of the equivalence class of $\\tilde{\\varphi}$ in $\\operatorname{Ext}\\nolimits^n_{{\\mathsf P}(A)}(A,A)$ is given by the equivalence class of the lower sequence in the pushout diagram\n$$\n\\xymatrix@C=18pt@R=18pt{\n\\cdots \\ar[r] & A \\otimes B^{\\otimes n} \\otimes A \\ar[r] \\ar[d]_-{\\tilde{\\varphi}} & A \\otimes B^{\\otimes(n-1)} \\otimes A \\ar[r] \\ar[d] & \\cdots \\ar[r] & A \\otimes A \\ar[r] \\ar@{=}[d] & A \\ar[r] \\ar@{=}[d] & 0\\\\\n0 \\ar[r] & A \\ar[r] & Q \\ar[r] & \\cdots \\ar[r] & A \\otimes A \\ar[r] & A \\ar[r] & 0\n}\n$$\nwhich, as $\\mathfrak A$ is exact and thus commutes with pushouts along admissible epimorphisms, coincides with the equivalence class of the sequence (\\ref{eq:imageA}).\n\\end{proof}\n\n\\subsection{The long exact sequence of Michelena--Platzeck} We review yet another long exact sequence which is also due to Michelena--Platzeck; see \\cite[Sec.\\,1]{MiPl00} for its original construction which, however, is given in a far more restrictive setting.\n\nIn what follows, we will consider upper triangular matrix algebras of the form\n$$\nB =\n\\left(\n\\begin{matrix}\nR & M\\\\\n0 & S\n\\end{matrix}\n\\right) \\,,\n$$\nwhere, as in the previous sections, $R$ and $S$ are algebras over the fixed commutative base ring $K$ and $M$ is an $R \\otimes S^\\mathrm{op}$-module such that $R$, $S$ and $M$ are projective as $K$-modules. The canonical idempotents $e$ and $e' = 1 - e$ in $B$, associated to $R$ and $S$ respectively, will once again play a key role.\n\n\\begin{nn}\nTo begin with, notice that\n$$\nBe' = Be'B = \\left(\n\\begin{matrix}\n0 & M\\\\\n0 & S\n\\end{matrix}\n\\right) \\quad \\text{and} \\quad\neB = BeB = \\left(\n\\begin{matrix}\nR & M\\\\\n0 & 0\n\\end{matrix}\n\\right)\\,.\n$$\nAs we have already observed, these are stratifying ideals in $B$. There are four algebra homomorphisms, relating $R$, $S$ and $B$, namely\n$$\nR \\longrightarrow B, \\ r \\mapsto \\left(\n\\begin{matrix}\nr & 0\\\\\n0 & 1_S\n\\end{matrix}\n\\right)\\, \\qquad B \\longrightarrow R, \\ \\left(\n\\begin{matrix}\nr & m\\\\\n0 & s\n\\end{matrix}\n\\right) \\mapsto r\n$$\nand\n$$\nS \\longrightarrow B, \\ s \\mapsto \\left(\n\\begin{matrix}\n1_R & 0\\\\\n0 & s\n\\end{matrix}\n\\right)\\, \\qquad B \\longrightarrow S, \\ \\left(\n\\begin{matrix}\nr & m\\\\\n0 & s\n\\end{matrix}\n\\right) \\mapsto s \\,.\n$$\nNote that, for instance, the right $B$-module structure of $Be'$ coincides with the one given by restricting the right $S$-module structure along $B \\rightarrow S$. We now deduce the following consequence of Theorem \\ref{thm:koenignagase}, generalising the statement \\cite[Thm.\\,1.14]{MiPl00}.\n\\end{nn}\n\n\\begin{cor}\\label{cor:mpla}\nPut $I = Be'B = Be'$. Then there is a long exact sequence\n$$\n\\xymatrix@C=18pt@R=6pt{\n0 \\ar[r] & \\mathrm{Ann}_S(M) \\cap Z(S) \\ar[r] & \\operatorname{HH}\\nolimits^0(B) \\ar[r]^-{m_0} & \\operatorname{HH}\\nolimits^0(R) \\ar[r] & \\operatorname{Ext}\\nolimits^{1}_{B^\\mathrm{op} \\otimes S}(S,I) \\ar[r] & \\cdots\\\\\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{op} \\otimes S}(S,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r]^-{m_n} & \\operatorname{HH}\\nolimits^n(R) \\ar[r] & \\operatorname{Ext}\\nolimits^{n + 1}_{B^\\mathrm{op} \\otimes S}(S,I) \\ar[r] & \\cdots\n}\n$$\nwherein the graded map $m_\\ast{\\colon} \\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(R)$ is a homomorphisms of strict Gerstenhaber algebras.\n\\end{cor}\n\n\\begin{proof}\nIndeed, by Theorem \\ref{thm:koenignagase}(\\ref{thm:koenignagase:1}), we are given a long exact sequence which is induced by the short exact sequence\n\t$$\n\t0 \\longrightarrow Be' = Be'B \\longrightarrow B \\longrightarrow B\/Be' = R \\longrightarrow 0\n\t$$\nof $B^\\mathrm{ev}$-modules, connecting $\\operatorname{HH}\\nolimits^\\ast(B)$ and $\\operatorname{HH}\\nolimits^\\ast(R)$ by a homomorphism of strict Gerstenhaber algebras. It thus remains to be shown, that $\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(B,I) \\cong \\mathrm{Ann}_S(M) \\cap Z(S)$ and $\\operatorname{Ext}\\nolimits^{n}_{B^\\mathrm{ev}}(B,I) \\cong \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{op} \\otimes S}(S,I)$ for all $n \\geqslant 1$. If $f \\in \\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(B,I)$, then $f$ is determined by the image\n\t$$\n\tf(1_B) = \\left(\\begin{matrix}\n0 & m\\\\\n0 & s\n\\end{matrix}\\right)\\,.\n\t$$\n\tAs\n\t$$\n\t\\left(\\begin{matrix}\n0 & 0\\\\\n0 & s's\n\\end{matrix}\\right) = \\left(\\begin{matrix}\n1 & 0\\\\\n0 & s'\n\\end{matrix}\\right) \\cdot f(1_B) = f(1_B) \\cdot \\left(\\begin{matrix}\n1 & 0\\\\\n0 & s'\n\\end{matrix}\\right) = \\left(\\begin{matrix}\n0 & ms'\\\\\n0 & ss'\n\\end{matrix}\\right)\n$$\nmust hold for all $s' \\in S$, it follows that $m = 0$ and $s \\in Z(S)$. \tFurther, the equation\n\t$$\n\t\\left(\\begin{matrix}\n0 & m's\\\\\n0 & 0\n\\end{matrix}\\right) = \\left(\\begin{matrix}\n0 & m'\\\\\n0 & 0\n\\end{matrix}\\right) \\cdot f(1_B) = f(1_B) \\cdot \\left(\\begin{matrix}\n0 & m'\\\\\n0 & 0\n\\end{matrix}\\right) = \\left(\\begin{matrix}\n0 & 0\\\\\n0 & 0\n\\end{matrix}\\right)\\,,\n$$\nwhich holds for all $m' \\in M$, implies that $s \\in \\mathrm{Ann}_S(M)$, so that \n\t$$\n\t\\operatorname{Hom}\\nolimits_{B^\\mathrm{ev}}(B,I) \\longrightarrow \\mathrm{Ann}_S(M) \\cap Z(S), \\ f \\mapsto f(1_B)\n\t$$\n\tis a $K$-linear isomorphism. As $\\operatorname{Tor}\\nolimits^B_i(B, S) = 0 = \\operatorname{Ext}\\nolimits^i_{S^\\mathrm{op}}(S, I)$ for all $i > 0$, we may apply \\cite[Chap.\\,IX, Thm.\\,2.8a]{CaEi56} to deduce\n\t\\begin{align*}\n\t\\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(B, \\operatorname{Hom}\\nolimits_{S^\\mathrm{op}}(S,I)) &\\cong \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{op} \\otimes S}(B \\otimes_B S, I)\\\\ &\\cong \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{op} \\otimes S}(S, I)\n\t\\end{align*}\n\tfor $n > 0$. To conclude, we observe that $I \\cong \\operatorname{Hom}\\nolimits_{S^\\mathrm{op}}(S,I)$ as $B \\otimes S^\\mathrm{op}$-modules, hence also as $B^\\mathrm{ev}$-modules.\n\\end{proof}\n\n\\begin{rem}\nNotice that by replacing $e$ by $e'$, and thus $R$ by $S$, we obtain a similar long exact sequence relating $\\operatorname{Ext}\\nolimits^\\ast_{B \\otimes R^\\mathrm{op}}(B,eB)$, $\\operatorname{HH}\\nolimits^\\ast(B)$ and $\\operatorname{HH}\\nolimits^\\ast(S)$ with the map $\\operatorname{HH}\\nolimits^\\ast(B) \\rightarrow \\operatorname{HH}\\nolimits^\\ast(S)$ being a homomorphism of strict Gerstenhaber algebras.\n\\end{rem}\n\n\\begin{nn}\nLet us close this section by summarising the observations made so far. Let $e \\in B$ be a stratifying idempotent, $e' = 1 - e$ and $I = BeB$. The standard recollement\n\\begin{equation*}\n\\begin{aligned}\n\\xymatrix{\\\\\\mathcal R(B,e) \\qquad \\equiv \\qquad\\\\}\n\\xymatrix@!C=50pt{\\\\ \\operatorname{\\mathsf{Mod}}\\nolimits(B\/I) \\ar[r]^-{\\mathrm{res}} & \\operatorname{\\mathsf{Mod}}\\nolimits(B) \\ar[r]^-{e(-)} \\ar@\/^2pc\/[l]^-{\\operatorname{Hom}\\nolimits_{B}(B\/I,-)} \\ar@\/_2pc\/[l]_-{(B\/I) \\otimes_{B} -} & \\operatorname{\\mathsf{Mod}}\\nolimits(eBe) \\ar@\/^2pc\/[l]^-{\\operatorname{Hom}\\nolimits_{eBe}(eB,-)} \\ar@\/_2pc\/[l]_-{Be \\otimes_{eBe} -}\\\\}\n\\end{aligned}\n\\end{equation*}\ngives rise to two homomorphisms of strict Gerstenhaber algebras:\n$$\n\\operatorname{HH}\\nolimits^\\ast(B) \\longrightarrow \\operatorname{HH}\\nolimits^\\ast(eBe), \\ \\quad \\ \\operatorname{HH}\\nolimits^\\ast(B) \\longrightarrow \\operatorname{HH}\\nolimits^\\ast(B\/I).\n$$\nThe first one is given by applying the functor $e(-)e$, whereas the latter comes from deriving $(B\/I) \\otimes_B (-) \\otimes_B (B\/I)$. These homomorphisms fit into several long exact sequences.\n\\begin{gather*}\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{HH}\\nolimits^n(B,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{HH}\\nolimits^n(A) \\ar[r] & \\operatorname{HH}\\nolimits^{n+1}(B,I) \\ar[r] & \\cdots\n}\\\\\n\\xymatrix@C=18pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^n(A,B) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{HH}\\nolimits^n(eBe) \\ar[r] & \\operatorname{Ext}\\nolimits_{B^\\mathrm{ev}}^{n+1}(A,B) \\ar[r] & \\cdots\n}\\\\\n\\xymatrix@C=14pt{\n\\cdots \\ar[r] & \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(A,I) \\ar[r] & \\operatorname{HH}\\nolimits^n(B) \\ar[r] & \\operatorname{HH}\\nolimits^n(A) \\oplus \\operatorname{HH}\\nolimits^n(eBe) \\ar[r] & \\operatorname{Ext}\\nolimits^{n+1}_{B^\\mathrm{ev}}(A,I) \\ar[r] & \\cdots\n}\n\\end{gather*}\nand, provided that $\\operatorname{Tor}\\nolimits^{e'Be'}_i(Be',e'B) = 0$ for $i > 0$,\n\\begin{align*}\n\\cdots \\xrightarrow{\\ \\ } \\operatorname{HH}\\nolimits^n(B) \\xrightarrow{\\ \\ } \\operatorname{HH}\\nolimits^n(eBe) \\oplus \\operatorname{HH}\\nolimits^n(e'Be') \\xrightarrow{\\ \\ }  \\operatorname{Ext}\\nolimits^n_{B^\\mathrm{ev}}(\\Omega_B^1, B) \\xrightarrow{\\ \\ } \\operatorname{HH}\\nolimits^{n+1}(B) \\xrightarrow{\\ \\ } \\cdots \\, .\n\\end{align*}\nAny upper or lower triangular matrix algebra, with canonical idempotents, is an example that gives rise to such a situation, and in that case, the first exact sequence simplifies to the one in Corollary \\ref{cor:mpla}.\n\\end{nn}\n\n\n\\section{The finite generation conjecture of Snashall--Solberg}\\label{sec:snashsol}\n\n\\begin{nn}\nThe Hochschild cohomology ring of an algebra is usually far away from being finitely generated. This changes, if one factors out the ideal generated by all homogeneous nilpotent elements. The resulting quotient still has very nice properties, to some extend even nicer ones than the ring we started with, since it will be a commutative ring (no signs involved!) if one works over a field. Also this quotient is finitely generated in many cases, but, however, not always, as it was proven by example in \\cite{Xu08}, and then even further generalised in \\cite{Sn09}, contrasting the initial conjecture by Snashall--Solberg; see \\cite{SnSo04}.\n\nHowever, it seems completely unknown and unstudied, whether or when the quotient by the (strict) Gerstenhaber ideal generated by the homogeneous nilpotent elements is finitely generated.\n\\end{nn}\n\n\\begin{nn}\nFor a $K$-algebra $B$ and a subset $S \\subseteq \\operatorname{HH}\\nolimits^\\ast(B)$ of homogeneous elements, we will denote by\n\\begin{enumerate}[\\rm(1)]\n\\item $I(S)$ the graded ideal of $\\operatorname{HH}\\nolimits^\\ast(B)$ generated by $S$.\n\t\\item $\\mathcal I(S)$ the graded Lie ideal of $\\operatorname{HH}\\nolimits^\\ast(B)$ generated by $S$.\n\t\\item $G(S)$ the \\textit{weak Gerstenhaber ideal} generated by $S$, that is, the smallest graded Lie subalgebra of $\\operatorname{HH}\\nolimits^\\ast(B)$ containing $S$ and being an ideal with respect to $\\cup$.\n\t\\item $\\mathcal G(S)$ the \\textit{Gerstenhaber ideal generated by $S$}, that is, the smallest subset of $\\operatorname{HH}\\nolimits^\\ast(B)$ containing $S$ and being both an ideal with respect to $\\cup$ and $\\{-,-\\}_B$.\n\\end{enumerate}\nClearly, we have $\\mathcal I(S) \\subseteq \\mathcal G(S)$ and $I(S) \\subseteq G(S) \\subseteq \\mathcal G(S)$, whereas there does not need to be any relation between $\\mathcal I(S)$ and $G(S)$. If $K$ is a field and $S$ is the set of homogeneous nilpotent elements, then $\\operatorname{HH}\\nolimits^\\ast(B)\/G(S)$ and $\\operatorname{HH}\\nolimits^\\ast(B)\/\\mathcal G(S)$ are \\textit{commutative} algebras (in the strict sense). In the following, we will review a couple of examples.\n\\end{nn}\n\n\\begin{nn}\\label{nn:elementab}\nFor the remainder of the section, and in fact of this article, assume that $K$ is a field of characteristic $p > 0$. Let $\\mathbb Z_ p = \\mathbb Z\/ p\\mathbb Z$ and $\\mathbb Z_p^r$ be the finite elementary abelian $p$-group of rank $r > 0$, with corresponding group algebra $\\Gamma = K \\mathbb Z_p^r$. Let $\\{-,-\\}$ be the Gerstenhaber bracket on $\\operatorname{HH}\\nolimits^\\ast(\\Gamma)$. By \\cite{LeZh13}, the Gerstenhaber algebra structure of $\\operatorname{HH}\\nolimits^\\ast(\\Gamma)$ is given as described in the following two paragraphs. \n\\end{nn}\n\n\\begin{nn}\nIf $p = 2$, then\n$$\n\\operatorname{HH}\\nolimits^\\ast(\\Gamma) \\cong \\frac{K[x_1, \\dots, x_r, y_1, \\dots, y_r]}{(x_1^2, \\dots, x_r^2)}\\,, \\quad \\abs{x_i} = 0, \\, \\abs{y_i} = 1\n$$\nand\n$$\n\\{x_i,x_j\\} = \\{y_i,y_j\\} = 0\\, , \\quad \\{x_i, y_j\\} = \\delta_{ij}\n$$\nfor all $i,j = 1, \\dots, r$. The ideal $I(N) \\subseteq \\operatorname{HH}\\nolimits^\\ast(\\Gamma)$ generated by the homogeneous nilpotent elements $N$ in $\\operatorname{HH}\\nolimits^\\ast(\\Gamma)$ is precisely the ideal $(x_1, \\dots, x_r)$, which also coincides with $G(N)$, whereas $\\mathcal I(N) = K \\oplus \\mathrm{Span}_K\\{x_iy_j^{2v} \\mid 1 \\leq i,j \\leq r, v \\in \\mathbb Z_{\\geq 0}\\}$ and $\\mathcal G(N) = \\operatorname{HH}\\nolimits^\\ast(\\Gamma)$ as $\\{x_i, y_i^n\\} = n y_i^{n-1}$ for $n \\geq 1$. It thus follows that\n$$\n\\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/I(N) = \\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/G(N) \\cong K[y_1, \\dots, y_r] \\quad \\text{and} \\quad \\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/\\mathcal G(N) \\cong 0 \\, .\n$$\nA squaring map for $\\{-,-\\}$ is given by $sq(\\alpha) = \\alpha^2$ (for $\\alpha$ of even degree). \n\\end{nn}\n\n\\begin{nn}\nIf $p \\neq 2$, one has\n$$\n\\operatorname{HH}\\nolimits^\\ast(\\Gamma) \\cong \\frac{K[x_1, \\dots, x_r, z_1, \\dots, z_r]}{(x_1^p, \\dots, x_r^p)} \\otimes \\Lambda(y_1, \\dots, y_r)\\,, \\quad \\abs{x_i} = 0, \\, \\abs{y_i} = 1, \\, \\abs{z_i} = 2\n$$\nand\n$$\n\\{x_i,x_j\\} = \\{y_i,y_j\\} = 0\\, , \\quad \\{x_i, y_j\\} = \\delta_{ij}\\,, \\quad \\{z_i,-\\} = 0\n$$\nfor all $i,j = 1, \\dots, r$. Denoting $\\{a, a+1, \\dots, b\\}$ by $[a,b]$ for $b-a \\in \\mathbb N$, we get $I(N) = (x_1, \\dots, x_r, y_1, \\dots, y_r)$, $\\mathcal I(N) = \\mathrm{Span}_K\\{x_i^u y_j^{v} \\mid i,j \\in [1,r], u \\in [0,p-1], v \\in [0,1]\\} \\oplus \\mathrm{Span}_K\\{x_i^u z_j^w \\mid i,j \\in [1,r], u \\in [0,p-1], w \\in \\mathbb N\\}$ and $G(N) = \\mathcal G(N) = \\operatorname{HH}\\nolimits^\\ast(\\Gamma)$, whence\n$$\n\\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/I(N) \\cong K[z_1, \\dots, z_r] \\quad \\text{and} \\quad \\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/G(N) = \\operatorname{HH}\\nolimits^\\ast(\\Gamma)\/\\mathcal G(N) \\cong 0 \\, .\n$$\nAs we assumed that $p \\neq 2$, \\textit{the} squaring map for $\\{-,-\\}$ is given by $sq(\\alpha) = 2^{-1} \\{\\alpha, \\alpha\\}$ (for $\\alpha$ of even degree). It sends the $x_i, y_i^2$ and $z_i$ to zero, but $x_i + y_i^{2n}$ to $n y_i^{2n-1} \\neq 0$ (for $p \\nmid n$).\n\\end{nn}\n\n\\begin{nn}\\label{nn:E_0}\nLet ${\\mathsf C}$ be a \\textit{finite EI-category}, that is, a category with $\\abs{\\operatorname{Mor}\\nolimits({\\mathsf C})} < \\infty$ and $\\operatorname{End}\\nolimits_{\\mathsf C}(X) = \\operatorname{Aut}\\nolimits_{\\mathsf C}(X)$ for all $X \\in \\operatorname{Ob}\\nolimits({\\mathsf C})$. Finite groups and finite acyclic quivers are examples of such categories. The \\textit{category algebra of ${\\mathsf C}$} (over $K$) has $K \\operatorname{Mor}\\nolimits({\\mathsf C})$ as underlying $K$-module, and the product $f \\cdot g$ of morphisms $f, g$ in ${\\mathsf C}$ is $f \\circ g$, whenever this makes sense, and $0$ otherwise. If ${\\mathsf C}$ is a finite group or a finite acylcic quiver, $K{\\mathsf C}$ agrees with the corresponding group or path algebra. For an example that is neither a group algebra nor a path algebra of a quiver, consider the finite EI-category ${\\mathsf E}_0$\n$$\n\\xymatrix@C=1pt{\n{\\ X\\ } \\ar@(u,l)_h \\ar@(l,d)_g \\ar@(d,r)_{gh} \\ar@(r,u)_{\\operatorname{id}\\nolimits_X} & & \\ar@<3pt>[rrrrrr]^-\\alpha\n\\ar@<-3pt>[rrrrrr]_-\\beta & &&&&& Y \\ar@(ur,dr)^{\\operatorname{id}\\nolimits_Y}\n}\n$$\nsubject to the relations\n\\begin{equation*}\ng^2 = \\operatorname{id}\\nolimits_X = h^2, \\ gh = hg, \\ \\alpha h= \\beta g = \\alpha, \\ \\alpha g = \\beta h = \\beta.\n\\end{equation*}\n\nLet $N \\subseteq \\operatorname{HH}\\nolimits^\\ast(K{\\mathsf E}_0)$ denote the set of all homogeneous nilpotent elements, and $I(N)$ the graded ideal generated by $N$. It is one of the main results of \\cite{Xu08}, that $\\operatorname{HH}\\nolimits^\\ast(K{\\mathsf E}_0) \/ I(N)$ is not finitely generated as a graded $K$-algebra if $K$ is a field of characteristic $2$. In this setting, one has the following obvious interpretation of the algebra $K{\\mathsf E}_0$.\n\\end{nn}\n\n\\begin{lem}\\label{lem:isoonepoint}\nAssume that $\\mathrm{char}(K) = 2$. The category algebra $K{\\mathsf E}_0$ is isomorphic to the one-point-extension $B = \\Gamma[M] = \\left( \\begin{smallmatrix} K (\\mathbb Z_2 \\times \\mathbb Z_2) & K\\mathbb Z_2\\\\ 0 & K \\end{smallmatrix} \\right)$ of $\\Gamma = K(\\mathbb Z_2 \\times \\mathbb Z_2)$ by $M = K\\mathbb Z_2$. \\qed\n\\end{lem}\n\nThe following questions shall occupy us in future work.\n\n\\begin{quest}\\label{conj:snashsol}\nLet $A$ be a finite dimensional algebra over a field $K$. If $N$ denotes the set of all homogeneous nilpotent elements in $\\operatorname{HH}\\nolimits^\\ast(A)$, is the commutative $K$-algebra\n$$\n\\operatorname{HH}\\nolimits^\\ast_G(A) = \\operatorname{HH}\\nolimits^\\ast(A) \/ G(N)\n$$\nfinitely generated? More specifically, can we use (any) of the long exact sequences discussed in this article, along with \\ref{nn:elementab}, to deduce that\n$$\n\\operatorname{HH}\\nolimits^\\ast_G(K{\\mathsf E}_0) = \\operatorname{HH}\\nolimits^\\ast(K{\\mathsf E}_0) \/ G(N)\n$$\nis finitely generated (where ${\\mathsf E}_0$ denotes Xu's EI-category reviewed above and $\\mathrm{char}(K) = 2$)? Especially the results obtained in Subsection \\ref{subsec:happel} should be of help.\n\\end{quest}\n\n\\begin{quest}\nUnder the assumptions of the above question, does the commutative algebra $\\operatorname{HH}\\nolimits^\\ast_G(A)$ possess enough structure to establish a sensible support variety theory for (special classes of) $A$-modules, in a similar way as described in \\cite{SnSo04}?\n\\end{quest}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe famous Sard's Theorem asserts that the set of critical values of a ${\\mathcal C}^\\infty$ map $f:\\mathbb{R}^n\\to\\mathbb{R}^k$ is of Lebesgue measure zero. \nLet us recall that the set of critical values of $f$ is the image under $f$ of all the elements at which $f$ fails to be a submersion. \nIn this article, we focus on semi-algebraic functions on real closed fields. \nThe notion of Lebesgue measure does not really make sense on a real closed field.\nHowever, the dimension of a semi-algebraic set can always be defined \nand it is well known that one can prove that the set of critical values of a mapping $f:R^n \\to R^k$ has dimension less than $k$ \\cite{bcr}, \nif $R$ stands for a real closed field. In this note, we establish a related result. \nWe introduce a more general notion of critical values and show a Sard type theorem for these critical values, \nestimating the size of the set of our generalized critical values. There is no nice geometric measure theory on real closed fields. \nWe therefore make use of the notion of $v$-thin sets introduced in \\cite{v}. Further features of these critical values will be developped in forthcoming articles.\n\n We then give an application of our Sard's theorem to the study of asymptotic critical values of real mappings $f:X \\to \\mathbb{R}^k$, $X\\subset \\mathbb{R}^n$. We shall show that the set of asymptotic critical values of a semi-algebraic mapping $f:X \\to \\mathbb{R}^k$ has dimension less than $k$.  A similar result was also obtained by K. Kurdyka, P. Orro and S. Simon \\cite{kos} who showed that the set of  generalized critical values at infinity for semi-algebraic mappings has dimension less than $k$. This result lead them to establish a generalization of Ehresmann's fibration theorem for nonproper mappings. This provided an effective way to ensure stability and yield that the set of bifurcation points of a semi-algebraic mapping is relatively small. \n \n\n\n\n\n\n\nBy ${R}$ we denote a real closed field. We refer to \\cite{bcr} for basic facts about semi-algebraic sets. We denote by $cl(X)$ and $int(X)$ the closure and interior of a set and set $\\delta X:=cl(X)\\setminus int(X)$ as well as $fr(X):=cl(X)\\setminus X$.  We will write $|x|$ for the Euclidean norm of $x\\in R^n$ and $d(\\cdot, \\cdot)$ for the Euclidean distance, while $B^n(x,r)$ will stand for the ball of radius $r$ centered at $x$ for this metric.\n\n\\section{$v$-thin sets}We introduce the notion of $v$-thin sets as it was defined in \\cite{v}.\n\n\n\n\\begin{dfn}\\label{z-thin-set}\nLet $X\\subset{R}^n$  be a semi-algebraic subset and let $z\\in{R}$ be positive.  A $k$-dimensional semi-algebraic subset\n$X$ of $R^n$ is called {\\bf $z$-thin} if for a generic orthogonal projection $\\pi:{R}^n\\to H^k$, where $H^k$ is a $k$-dimensional vector subspace of $R^n$, the image $\\pi(X)$ does not contain any open ball of radius $z$.\n\n\n\\end{dfn} \n\n\n\n\nLet $v\\subset {R}$ be a convex subgroup, i.e. a set such that for any $x,y\\in v$ and for any $t\\in [0,1]\\subset{R}$ we have $tx+(1-t)y\\in v$. \n\n\n For example, suppose that  the field $R$ is the field of real algebraic Puiseux series $k(0_+)$ (on a right-hand-side neighborhood of $0$ in $\\mathbb{R}$) endowed with the order that makes the indeterminate $T$ positive and smaller than any positive real number. Then, one can choose $v$ as the group of all the series satisfying $|x| \\leq N T^2$ for  any $N$ in  $\\mathbb{R}$. In the Hardy field of $ln-exp$ definable germs of one variable functions\n (in a right-hand side neighborhood of zero)\none may consider  the set of all the $L^p$ integrable germs of\nseries.\n\n\\begin{dfn}\\label{thin-set}\nWe say that a set $X$ is {\\bf $v$-thin} if it is $z$-thin for some positive $z\\in v$.  \n\nLet $j\\ge \\dim X $ be an integer.  For simplicity we say that $X$ is {\\bf$(j,v)$-thin} if  it is   $v$-thin or if  $\\dim X<j$ (with the convention $\\dim \\emptyset:=-1$). \n\n\\end{dfn}\n\n\n\nWe shall need the fact that a semi-algebraic family of $v$-thin sets is $v$-thin (Proposition \\ref{family}). \nThis requires two preliminary lemmas.  Given $z \\in R$ and $Y\\subset R^n$, we define {\\bf the $z$-neighborhood of $Y$} as the set $$\\mathcal{V}_z(Y):=\\{ x \\in R^n| d(x,Y)<z \\}.$$ \n\n\n\\begin{lem}\\label{lem_z_nghbd}\n If $Y\\subset R^n$ is a semi-algebraic subset of dimension less than $n$, then for any $z \\in v$, the set  $\\mathcal{V}_z(Y)$ is $v$-thin.\n\\end{lem}\n\\begin{proof}Let $Y$ be of dimension $<n$ and   $z \\in v$ be positive. We shall indeed show  that $\\mathcal{V}_z(Y)$ is $N z$-thin for some $N \\in \\mathbb{N}$.\nSince $\\dim Y< n$ we can decompose the set $Y$ into semi-algebraic sets $A_1\\cup\\dots\\cup A_k$ such that there exist some orthogonal automorphisms $\\varphi_i:R^n\\to R^n$, $i=1,\\dots , k$,  such that $\\Gamma_i:=\\varphi_i(A_i)$ is the graph of a semi-algebraic $C$-Lipschitz function $\\xi_i:B_i\\to \\mathbb{R}$, with $C\\in \\mathbb{N}$, $B_i \\subset \\mathbb{R}^{n-1}$ (see \\cite{f}).\n\n We will proceed by induction on $k$, the case $k=0$ being vacuous. For simplicity, we will assume $\\varphi_k=Id_{R^n}$ (we can work up to an orthogonal map).  Assume the result true for $(k-1)$, $k\\ge 1$, and take $M \\ge 4(C+1)$, $M\\in \\mathbb{R}$. \n\nLet $a \\in A_k$. We claim that there is $b\\in B^n(a,M^2 \\cdot z)$, such that  $B^n(b,M\\cdot z)\\cap  A_k=\\emptyset$.  Assume otherwise. It means that $d(b,A_k) \\le M z$, for any $b\\in B^n(a,M^2\\cdot z)$. Take $b\\in B^n(a,M^2 \\cdot z)$ such that $|b-a|\\ge \\frac{M^2}{2}z$ and  $\\pi(b)=\\pi(a)$, where $\\pi:R^n\\to R^{n-1}$ is the canonical projection. Then, as $A_k$ is the graph of a $C$-Lipschitz function, a straightforward computation gives (since $M\\ge \\frac{C+1}{4}$): \n $$|b-a|\\le (C+1)d(b,A_k)\\le (C+1)Mz\\le \\frac{M^2}{4}z,$$\na contradiction.\n\nBy induction on $k$, if we set $Y':= A_1\\cup \\dots \\cup A_{k-1}$ then $\\mathcal{V}_z(Y')$ must be $N z$-thin, for some $N\\in\\mathbb{N}$.   Hence, for any $b \\in \\mathcal{V}_z(Y')$, the ball $B^n(b,N\\cdot z)$ does not fit in $\\mathcal{V}_z(Y')$. Let $z':=2M^2 z$, where $M:=N+1$. We are going to show that $B^n(a,z')$ does not fit in $\\mathcal{V}_z(Y)$, which will yield the desired result. By the above paragraph (we may assume $N\\ge 4(C+1)$), we know that there is $b\\in B^n(a,M^2\\cdot z)$  such that $B^n(b,M\\cdot z)$ does not meet $A_k$. Hence, $B^n(b, N\\cdot z)$ does not meet  $\\mathcal{V}_z(A_k)$.  As (by choice of $N$) this ball is not included in $\\mathcal{V}_z(Y')$, it therefore cannot be included in $\\mathcal{V}_z(Y)=\\mathcal{V}_z(Y')\\cup \\mathcal{V}_z(A_k)$.  \nAs $B^n(b, N\\cdot z)\\subset B^n(a,z')$, we are done.\n\\end{proof}\n\n\nLet us recall that given two nonempty subsets $X$ and $Y$ of a semialgebraic subset of $ R^n$, the Hausdorff distance is defined as:\n$$d_\\mathcal{H}(X,Y):= \\max \\;(\\sup_{x \\in X} \\inf_{y \\in Y} d(x,y), \\, \\sup_{y\\in Y} \\inf_{x\\in X} d(x,y)).$$\nHere, we adopt the convention that $d_\\mathcal{H}(X,Y)$ is equal to infinity if one of these suprema is infinite.\n\n Given $X \\in R^{m+n}$ and $t \\in R^m$ we set\n $$X_t:=\\{x \\in R^n |(x,t)\\in  X\\}.$$\nAny family $(X_t)_{t \\in R^m}$ constructed in this way is then called {\\bf a semi-algebraic family of sets}.\n\n\\begin{lem}\\label{thin}\n Let $X\\subset{R}^n$ be a semi-algebraic  set of dimension $k$. The set $X$ is $v$-thin iff there exists a semi-algebraic set $Y\\subset R^n$ of dimension less than $k$ such that\n $d_{\\mathcal H}(X,Y)\\in v$.  Moreover, if $(X_t)_{t \\in R^m}$ is a semi-algebraic family of subsets of $R^n$, then the corresponding family  $(Y_t)_{t \\in R^m}$ can be chosen semi-algebraic as well.\n\\end{lem}\n\\begin{proof\n We start with the ``only if'' part. If  the set $X$ has nonempty interior, it is enough to take  $Y$ as the set $\\delta X$. Indeed, if  $d_{\\mathcal H}(X,Y)\\notin v$, it follows that there exists a ball of radius $z\\in v$ which fits in $X$, which implies that $X$ is not $v$-thin.  In the case where  $X$ is of empty interior, we can decompose  this set into a finite union of graphs of Lipschitz mappings and the result follows from the case $\\dim X=n$.  \n\n\nLet us establish the ``if'' part. By Lemma \\ref{lem_z_nghbd}, the projection of $X$ onto a generic $k$-dimensional hyperplane lies in a $z$-neighborhood of $Y$  for $z\\in v$, which, by the previous Lemma, is a $v$-thin set.  Hence, $X$ is $v$-thin.\n\n\n By construction of $Y$ and quantifier elimination, if $(X_t)_{t \\in R^m}$ is a semi-algebraic family of subsets of $R^n$, then the corresponding family  $(Y_t)_{t \\in R^m}$ is semi-algebraic as well (since $\\delta X_t$ is then a semi-algebraic family).\n\\end{proof}\n\nAs a matter of fact, we get the following proposition which will be useful to establish our Sard type theorem.\n\n\n\\begin{pro}\\label{family}\nLet $X\\subset R^{m+n}$ be a semi-algebraic set. If $X_t$ is  $v$-thin for all $t\\in R^m$ then $X$ is a $v$-thin set.\n\\end{pro}\n\\begin{proof}\nObserve that by Lemma \\ref{thin} for any $t \\in R^m$ we have a semi-algebraic set $Y_t$ of smaller dimension and $z\\in v$ such that $X_t\\subset \\mathcal{V}_z(Y_t)$. Clearly, the set $$Y=\\bigcup\\limits_{t\\in R^m} Y_t\\times\\{t\\}$$ has dimension less than $\\dim X$. As $X \\subset \\mathcal{V}_z(Y)$, by Lemma \\ref{thin}, this shows that the set $X$ is $v$-thin.    \n\\end{proof}\n\n\n\\section{Sard's theorem}\nLet us recall the definition of the Rabier function \\cite{r} which measures the distance of a linear operator $A$ to the set of singular operators: $$\\nu(A):=\\inf_{||\\varphi||=1}||A^\\ast\\varphi||.$$\n See \\cite{kos,r} for basic properties of this function. In the case where $A$ is an operator from $R^n$ to $R^k$ with $n\\ge k$, one can equivalently characterize $\\nu(A)$ by $$\\nu(A)=\\inf\\{|Av|| v \\perp \\ker A, |v|=1 \\}.$$\n\n We say that $z\\in R$ is an {\\bf infinitesimal} if $|z|$ is smaller than any positive rational number. \nWe will say that a subset of $R^n$ is {\\bf  bounded} if it is included in a ball $B^n(0,N)$ in $R^n$, with $N \\in \\mathbb{N}$.   The field $R$ being not necessarily Archimedean it is worthy of notice that we require $N$ to be an  integer.\n\n\n\\begin{dfn}\nLet $X\\subset{R}^n$ be a bounded manifold, $z\\in{R}$ be an infinitesimal,  and  \nlet $f:X\\to {{R}}^k$ be a ${\\mathcal C}^1$ semi-algebraic mapping. \nWe say that  $x\\in X$ is a {\\bf $z$-critical point of $f$}  if $\\nu(d_xf)$ is less than $z$.\nWe denote the set of all $z$-critical points by $c_z(f)$, i.e. $$c_z(f)=\\{x\\in X|\\, \\nu(d_xf)<z\\}.$$\n\\end{dfn}\n\nWe now come to the main theorem of this article.\n\n\\begin{thm}\\label{sard}Let $v$ be a convex subgroup of ${R}$  and  \nlet $f:X\\to {{R}}^k$ be a ${\\mathcal C}^1$ semi-algebraic function, $X$ bounded manifold. \nFor any  infinitesimal $z\\in v$ the image $f(c_z(f))$ is $(k,v)$-thin.\n\\end{thm}\n\\begin{proof}\n We shall argue by induction on $k$. \n We may assume that $f$ is everywhere a submersion, since, by the classical Sard's theorem \\cite{bcr}, the image of the singular points has dimension less than $k$. \n\n\nAssume first that $k=1$ and take a partition $\\Sigma$ of  $c_z(f)$ such that for every $S\\in \\Sigma$, every couple of points $x,y$ of $S$ can be joint by a piecewise ${\\mathcal C}^1$ arc $\\gamma:[0,1]\\to  S$ that has derivative bounded by $N \\in \\mathbb{N}$ (that such a partition exists follows from existence of $L$-regular cell decompositions \\cite{f}). Take two arbitrary points $x$ and $y$ in $S\\in \\Sigma$ as well as such an arc $\\gamma$. Let $\\phi(t):=f(\\gamma(t))$ and observe that $$|\\phi'(t)|=|<\\partial_{\\gamma(t)}f, \\gamma'(t)>|\\le Nz$$ which means in particular that $\\phi'(t)$ belongs to $v$.  \n  By the mean value theorem, there is $ t \\in [0, 1]$ such that $$\\phi(1)-\\phi(0) =\\phi'(t).$$\nAs $|\\phi'(t)|\\le  Nz$ for all $t \\in [0,1]$, we get that   $|f(x)-f(y)|\\le  N z$. This shows that $f(c_z(f))$ is $(1,v)$-thin.\n\n\n We now assume the result true for $(k-1)$, $k \\ge 2$.  Let $\\Gamma$ denote the graph of $f_{|c_z(f)}$.\n   Let $n$ be such that the set $X$ is included in $R^n$ and let $\\pi:R^{n+k}\\to R^k$ be the projection onto the  $k$ last coordinates. For each $x\\in \\Gamma$ let $w(x)$ be a unit vector such that \n\\begin{equation}|\\pi(w(x))|=\\sup\\{|\\pi(w)|\\,\\,|\\,\\, |w|=1\\,\\, \\mbox{\\rm and}\\,\\, w\\in T_x\\Gamma\\}.\\end{equation}\nBy semi-algebraic choice, the vector field $w$ can be required to be a semi-algebraic.  The vector $w(x)$ is normal to \n$$K_x:=\\ker \\pi_{|T_x\\Gamma}=T_x\\Gamma\\cap (R^{n}\\times\\{0_{R^k}\\}).$$\nLet $v(x)$ be  a semi-algebraic vector field such that \n\\begin{equation}\n|\\pi(v(x))|=\\min\\{|\\pi(v)|\\,\\,|\\,\\, |v|=1, v\\perp K_x, v\\in T_x\\Gamma\\}.\n\\end{equation}\nLet \n$$ b(x):=|\\pi(w(x))|.$$ Since $k\\ge 2$, the vector $v$  can be required to be ortogonal to $w$. \nAs $w$ is a semi-algebraic unit vector field, for any positive rational number $\\varepsilon$, there is a finite partition of $\\Gamma$, say $V_1,\\dots, V_p$, such that for any $i$, there is $f_i\\in S^{k-1}$ such that for each $x\\in V_i$ we have:\n$$\n|\\frac{\\pi(w(x))}{|\\pi(w(x))|}-f_i|<\\varepsilon.$$\nWe can work separately on every $V_i$  and up to a linear isometry of $R^k$. We will thus assume for simplicity that for all $x\\in \\Gamma$ we have \\begin{equation}\\label{1}|e_{k}-\\frac{\\pi(w(x))}{|\\pi(w(x))|}|<\\varepsilon ,\\end{equation} where $e_{k}$ is the last vector of canonical basis of $R^k$.  We do not specify $\\varepsilon$ as it simply needs to be a small positive constant in $\\Q$.\nLet also $$\\Delta_x:=e_{n+k}^{\\perp}\\cap T_x\\Gamma.$$\nWe claim that \n\\begin{equation}\\label{3}\nd(w(x),\\Delta_x)\\ge 1-\\varepsilon>0.\n\\end{equation} \nAssume otherwise i.e. $w(x)=w_1(x)+w_2(x)$ with $w_1(x)\\in\\Delta_x$ and $|w_2(x)|< 1-\\varepsilon $. \nBy definition of $w$ and $b$ we get\n\\begin{equation}\\label{2}\n|<e_{k},\\pi(w(x))>| \\overset{(\\ref{1})}{\\ge}  b(x) (1-\\varepsilon).\\end{equation}\nMoreover, $$|\\pi(w_2(x))|\\le b(x)|w_2(x)|< b(x)(1-\\varepsilon).$$\nAs $<e_{k},\\pi(w_1)>\\equiv 0$ we get \n$$|<e_{k},\\pi(w(x))>|=|<e_{k},\\pi(w_2(x))>|<b(x)(1-\\varepsilon),$$ contradicting (\\ref{2}).\n\nSince $v(x)$ is orthogonal to $w(x)$ we thus have by (\\ref{3}) for $\\varepsilon$ small enough:\n\\begin{equation}\\label{omega}\n d(v(x),\\Delta_x)\\le\\frac{1}{2}.\n\\end{equation}\n We can decompose the vector $v$ as follows:\n$$v=v_1+\\alpha w,\\; \\,\\, \\mbox{ with}\\, v_1\\in\\Delta_x, \\;\\alpha\\in\\mathbb{R}.$$\nAs $w$ and $v$ are normal to $K_x$, so is $v_1$.\nLet $\\pi_2:R^{k}\\to e_{k}^\\perp $  be the projection along the span $<\\pi(w)>$. Observe that it follows from (\\ref{1}) that $|\\pi_2| \\le 2$ (provided $\\varepsilon$ was chosen small enough). Note also that, by (\\ref{3}) and (\\ref{omega}), we have $\\alpha< \\frac{3}{4}$ (for $\\varepsilon$ small enough),  so that \\begin{equation}\\label{eq_v1}\n|v_1(x)|\\ge 1-\\frac{3}{4}= \\frac{1}{4}.\\end{equation}\n                                         \n\nHence, as $\\pi(v_1(x))\\in e_{k}^\\perp$ (since $v_1(x)\\in e_{n+k}^\\perp$) , we have for any $x\\in \\Gamma $:\n\\begin{equation}\\label{eq_pi_v_1}\n|\\pi(v_1(x))|=|\\pi_2(\\pi(v_1(x)))|=|\\pi_2(\\pi(v(x)))| \\le 2|\\pi(v(x))|\\le 2z. \n \\end{equation}\n\n\n\nIt means that for every $t \\in R$, every point $x\\in  \\Gamma_t$ is an $8z$-critical point of $\\pi_{|\\Gamma_t}$ (since $v_1(x)\\perp K_x$ and, by (\\ref{eq_v1}), $|v_1(x)|\\ge \\frac{1}{4}$). \nLet $$A_t:=\\Lambda( \\Gamma_t),$$ where $\\Lambda:R^n \\times R^k \\to R^n$ is the projection onto the first factor. Clearly, $$\\dim A_t=\\dim \\Gamma_t\\le l-1$$ (the set $\\Gamma_t$ cannot be $l$-dimensional for $f$ is a submersion). For $t\\in R$,  write $f(x,t)$ as $(g_t(x),t)\\in R^{k-1}\\times R$ for each   $x\\in A_t$.  This defines a mapping $g_t:A_t \\to R^{k-1}$ (observe that the graph of $g_t$ is $\\Gamma_t$).\n\n\n We claim that every point of $A_t$ is a $16z$-critical for  $g_t$, i.e. that  $c_{16z}(g_t)=A_t$. To see this, pick a point $\\xi \\in A_t$. As $x=(\\xi,g_t(\\xi))$ is  $8z$-critical point of $\\pi_{|\\Gamma_t}$, we know that there is a unit vector $\\omega\\in T_x \\Gamma_t$ such that $|\\pi(\\omega)| \\le 8z$. This implies that $$|\\Lambda(\\omega)|\\ge \\sqrt{1-(8 z)^2 }\\ge \\frac{1}{2}$$ (since $z$ is an infinitesimal). Hence, if we set $\\omega':= \\frac{\\Lambda(\\omega)}{|\\Lambda(\\omega)|}$ we get  $$|d_x g_t(\\omega')|\\le 2| \\pi(\\omega)|\\le 16 z,$$ \nas required. \n\n Consequently, by induction on $k$, the set  $g_t(A_t)$ is $v$-thin for every $t\\in R$.  As, by definition of $g$,  $g_t(A_t)=f(c_z(f))_t$, by Lemma \\ref{family}, this entails that the set $f(c_z(f))$ is $v$-thin.\n\\end{proof}\n\n\n\n\\section{Application to the field of the real numbers}\nLet $\\mathcal{S}(0,\\varepsilon)$ denote the ring of semi-algebraic functions on $(0,\\varepsilon)$ (in $\\mathbb{R}$), $\\varepsilon $ positive real number, \nand let   $k(0_+)$ be the ordered field defined by the direct limit \n$\\underset{\\longrightarrow}{lim}\\, \\mathcal{S}((0,\\varepsilon))$. \nIt is the set of semi-algebraic functions $f:(0,\\varepsilon)\\to \\mathbb{R}$, quotiented by the equivalence relation that identifies two semi-algebraic functions that coincide on a right-hand-side neighborhood of zero.\n Let then $v_{0_+}$ be the additive subgroup of $k(0_+)$ consisting of the germs at $0$ of functions of $\\mathcal{S}(0,\\varepsilon)$  tending to zero at $0$. It is nothing but the set of all the infinitesimals of $k(0_+)$.\n\n Let $(A_t)_{t \\in \\mathbb{R}}$ be a semi-algebraic family of subsets of $\\mathbb{R}^n$ and denote by $A_{0_+}$ the generic fiber of the family, that is to say the subset of $k(0_+)^n$ constituted by all the germs of map-germs $f\\in k(0_+)^n$ such that $f(t) \\in A_t$ for all $t$ positive and small enough (see \\cite{bcr} for more details on generic fibers). If $(g_t)_{t \\in \\mathbb{R}}$ is a semi-algebraic family of functions,   we then denote by $g_{0_+}$ the generic fiber of the family $g_t$, i.e. the mapping whose graph is the generic fiber of the family constituted by the graphs of the functions $g_t$, $t \\in \\mathbb{R}$.\n\n\n\\begin{lem}\\label{lem_familles}\n If  $A_{0_+}$ is $v_{0_+}$-thin then the set $$B:=cl( \\bigcup_{t>0} (A_t \\times \\{t\\}))\\cap (\\mathbb{R}^n \\times \\{0\\})$$ has dimension less than $\\dim A_t$, for $t >0$ small ($\\dim A_t$ is constant for $t>0$ small).  \n\\end{lem}\n\\begin{proof}\nIf $A_{0_+}$ is $v_{0_+}$-thin then by  Lemma \\ref{thin} there is a subset $Y_{0_+}$ of $k(0_+)^n$ such that $d_{\\mathcal{H}}(A_{0_+},Y_{0_+})$ is an infinitesimal and $\\dim Y_{0_+}<\\dim A_{0_+}$. It means that the set $Y_{0_+}$ is the generic fiber of a family $(Y_t)_{t \\in \\mathbb{R}}$ such that $d_{\\mathcal{H}}(A_{t},Y_{t})$ tends to zero as $t$ positive goes to zero. This shows that \n$$cl( \\bigcup_{t\\in \\mathbb{R}} (A_t \\times \\{t\\}))\\cap (\\mathbb{R}^n \\times \\{0\\})=cl( \\bigcup_{t\\in \\mathbb{R}} (Y_t \\times \\{t\\}))\\cap (\\mathbb{R}^n \\times \\{0\\}),$$\nwhich is of dimension less than $\\dim A_{0_+}$. \n\\end{proof}\n\n\n\n\n\\subsection{Asymptotic critical values.}\nLet $f:X \\to \\mathbb{R}^k$ be a semi-algebraic mapping, with $X$ (not necessarily bounded) manifold. \nRecall that the set of critical values of $f$ is\n$$K_0(f):=\\{y\\in \\mathbb{R}^k\\;|\\; \\exists x\\in f^{-1}(y)\\, , \\nu(d_x f)=0\\}.$$\n\\begin{dfn}\\label{dfn_critical_values}\nThe set \n$$K(f):=\\{y\\in\\mathbb{R}^k\\;|\\; \\exists  x_n \\in X\\, ,  f(x_n)\\to y\\, \\mbox{\\rm and}\\, (1+|x_n|)\\nu(d_{x_n}f)\\to 0\\}.$$\nis called the {\\bf set of generalized critical values}.\n\\end{dfn}\n\nThis set of course contains all the critical values of $f$ but also other elements that are sometimes called {\\bf asymptotic critical values}. In particular, $K(f)$ contains {\\bf the asymptotic critical values at infinity} studied in \\cite{kos} and defined as:\n$$K_{\\infty}(f):=\\{y\\in\\mathbb{R}^k\\;|\\; \\exists x_n \\in X\\, , |x_n|\\to+\\infty\\, , f(x_n)\\to y,\\, \\mbox{\\rm and}\\, |x_n|\\nu(d_{x_n}f)\\to 0\\}.$$\nWe also set $$K_1(f):=\\{y\\in\\mathbb{R}^k\\;|\\; \\exists x_n\\in X\\, , |x_n|\\to fr( X),  f(x_n)\\to y,\\, \\mbox{\\rm and}\\, \\nu(d_{x_n}f)\\to 0\\}.$$ \n\nThese are the asymptotical critical values at the points of $fr( X)$. Clearly, $$K(f)=K_0(f)\\cup K_1(f)\\cup K_\\infty(f).$$\n\nIt is proved in  \\cite{kos} that the set $K_\\infty(f)$ is of dimension less than $k$.  Theorem \\ref{sard} entails  a Sard Theorem for asymptotic critical values\n\n\\begin{thm}\\label{thm_kurdyka}\nLet $f:X \\to \\mathbb{R}^k$ be a ${\\mathcal C}^1$ semi-algebraic mapping, where $X$ is a ${\\mathcal C}^1$-submanifold of $\\mathbb{R}^n$. The set of asymptotic critical values has dimension less than $k$. \n\\end{thm}\n\\begin{proof}\nWe first establish that $K_\\infty(f)$ has dimension less than $k$.\nIt easily follows from Curve Selection Lemma that  we have $$K_\\infty(f)=\\cup_{i=1} ^\\infty C_i,$$ where $C_i$ is the semi-algebraic set defined by \n$$C_i:=\\{y \\in \\mathbb{R}^k|\\exists (x_l)_{l\\ge l_0}, x_l  \\in  X, |x_l|=l, \\lim f(x_l) = y,  \\nu(d_{x_l} f)\\le l^{-(1+1\/i)} \\} .$$\nBy the Baire Property, it suffices to establish that $C_i$ has empty interior for all $i$. For this purpose, let for $t \\in \\mathbb{R}$, $ Y_{t}:=\\{x \\in B^n(0,1)|\\frac{x}{t} \\in X\\}$ and, for $x\\in Y_t$,  $g_t(x):=f(\\frac{x}{t})$. \n Let also\n\n\n$$Z_{i,t} := \\{x \\in Y_t\\,|\\;\\nu (d_x g_t)\\le t^{1\/i} \\}.$$ Observe that $Z_{i,0_+}=c_{t^{\\frac{1}{i}}}(g_{0_+})$, so that, by Theorem \\ref{sard},  $g_{0_+}(Z_{i,0_+})$ is $(k,v_{0_+})$-thin. By Lemma \\ref{lem_familles}, this implies that the set\\begin{equation}\\label{eq_points_critiques}\ncl( \\bigcup_{t\\in \\mathbb{R}} (g_t(Z_{i,t}) \\times \\{t\\}))\\cap (\\mathbb{R}^k \\times \\{0\\})\\end{equation}\nhas dimension less than $k$. As this set contains $C_i$, this yields $\\dim C_i<k$, as required.\n\n It remains to show that $K_1(f)$ has dimension less than $k$. We shall make use of a similar argument. It again easily follows from Curve Selection Lemma  that  $$K_1(f)=\\cup_{i =1}^\\infty C_i',$$ where $C_i'$ is the semi-algebraic set defined by \n$$C_i':=\\{y \\in \\mathbb{R}^k|\\exists (x_l)_{l\\ge l_0} \\mbox{ such that } x_l\\in  X, d(x_l,fr( X))=\\frac{1}{l}, \\lim f(x_l) = y,  \\nu(d_{x_l} f)\\le l^{\\frac{-1}{i}} \\} .$$\nAgain, by the Baire Property, it suffices to establish that $C_i'$ has empty interior for all $i$.  For $t>0$, tet  $h_t$ be the restriction of $f$ to the set $Y'_{t}:=\\{x \\in X| \\frac{t}{2}<d(x,fr( X))< 2t\\}$. By Theorem \\ref{sard}, $h_{0_+}(c_{t^{\\frac{1}{i}}}(h_{0_+}))$ is $(k,v_{0_+})$-thin for all $i$.\n By Lemma \\ref{lem_familles} this establishes that $\\dim C'_i<k$  (with the same argument as for $C_i$ and $c_{t^{\\frac{1}{i}}}(g_{0_+})$, see (\\ref{eq_points_critiques})).\n\\end{proof}\n\n\n\n\\begin{rem}\\label{rem_fin}These theorems remain valid on an arbitrary o-minimal structure with minor or no modifications of the proofs. Namely, if the o-minimal structure is polynomially bounded then the same argument applies. If it is not assumed to be polynomially bounded, only the proof of Theorem \\ref{thm_kurdyka}  requires a slight modification.\n \\end{rem}\n\n\n  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $M^{n}$ be a (connected) submanifold in the $m$-dimensional Euclidean space $E^{m}$. Let $\\boldsymbol{x},\\boldsymbol{H}$ be the position vector\nfield and the mean curvature field of $M^{n}$ respectively. Denote by $\\Delta ^{I}$ the second Beltrami-Laplace operator corresponding to the first\nfundamental form $I$ of $M^{n}$. Then, it is well known that \\cite{R3}\n\\begin{equation}\n\\Delta ^{I}\\boldsymbol{x}=-n\\boldsymbol{H}.  \\notag\n\\end{equation}\n\nFrom this formula one can see that $M^{n}$ is a minimal submanifold if and only if all coordinate functions, restricted to $M^{n}$, are eigenfunctions\nof $\\Delta ^{I}$ with eigenvalue $\\lambda =0$. Moreover in \\cite{R15} T.Takahashi showed that a submanifold $M^{n}$ for which $\\Delta^{I}\\boldsymbol{x} =\\lambda\\boldsymbol{x}$, i.e. for which all coordinate functions are eigenfunctions of $\\Delta ^{I}$ with the same eigenvalue $\\lambda$, is either minimal in $E^{m}$ with eigenvalue $\\lambda=0$ or minimal in a hypersphere of $E^{m}$ with eigenvalue $\\lambda>0$. Although the class of finite type submanifolds\nin an arbitrary dimensional Euclidean spaces is very large, very little is known about surfaces of finite type in the Euclidean 3-space $E^{3}$.\nActually, so far, the only known surfaces of finite type corresponding to the first fundamental form in the Euclidean 3-space are the minimal\nsurfaces, the circular cylinders and the spheres. So in \\cite{R5} B.-Y. Chen mentions the following problem\n\n\\begin{problem}\n\\label{(1)}Determine all surfaces of finite Chen $I$-type in $E^{3}$.\n\\end{problem}\n\nIn order to provide an answer to the above problem, important families of surfaces were studied by different authors by proving that finite type ruled\nsurfaces, finite type quadrics, finite type tubes, finite type cyclides of Dupin and finite type spiral surfaces are surfaces of the only known\nexamples in $E^{3}$. However, for another classical families of surfaces, such as surfaces of revolution, translation surfaces as well as helicoidal\nsurfaces, the classification of its finite type surfaces is not known yet. For a more details, the reader can refer to \\cite{R6}.\n\nLater in \\cite{R11} O. Garay generalized T. Takahashi's condition studied surfaces in $E^{3}$ for which all coordinate functions $\\left(\nx_{1},x_{2},x_{3}\\right)$ of $\\boldsymbol{x}$ satisfy $\\Delta^{I}\\boldsymbol{x}_{i} = \\lambda_{i}x_{i}, i = 1,2,3$, not necessarily with the same\neigenvalue. Another generalization was studied in \\cite{R9} for which surfaces in $E^{3}$ satisfy the condition $\\Delta^{I}\\boldsymbol{x}= A\\boldsymbol{x} + B$ $(\\ddag)$ where $A \\in\\mathbb{Re}^{3\\times3} ;B \\in\\mathbb{Re}^{3\\times1}$. It was shown that a surface $S$ in $E^{3}$ satisfies $(\\ddag)$ if and only if it is an open part of a minimal surface, a sphere, or a circular cylinder. Surfaces satisfying $(\\ddag)$ are said to be of coordinate finite type.\n\nIn the thematic circle of the surfaces of finite type in the Euclidean space in $E^{3}$, S. Stamatakis and H. Al-Zoubi in \\cite{R13} restored attention\nto this theme by introducing the notion of surfaces of finite type corresponding to the second or third fundamental forms of $S$ in the\nfollowing way: A surface $S$ is said to be of finite type corresponding to the fundamental form $J$, or briefly of finite $J$-type, $J=II,III$, if the\nposition vector $\\boldsymbol{x}$ of $S$ can be written as a finite sum of nonconstant eigenvectors of the operator $\\Delta ^{J}$, that is if\n\\begin{equation}\n\\boldsymbol{x}=\\boldsymbol{x}_{0}+\\sum_{i=1}^{k}\\boldsymbol{x}_{i},\\ \\ \\ \\ \\\n\\ \\Delta ^{J}\\boldsymbol{x}_{i}=\\lambda _{i}\\boldsymbol{x}_{i},\\ \\ \\\ni=1,...,k,  \\label{r1}\n\\end{equation\nwhere $\\boldsymbol{x}_{0}$ is a fixed vector and $\\boldsymbol{x}_{1},...\n\\boldsymbol{x}_{k}$ are nonconstant maps such that $\\Delta ^{J}\\boldsymbol{\\\nx}_{i}=\\lambda _{i}\\boldsymbol{x}_{i},i=1,...,k$. If, in particular, all\neigenvalues $\\lambda _{1},\\lambda _{2},...,\\lambda _{k}$ are mutually\ndistinct, then $S$ is said to be of $J$-type $k$, otherwise $S$ is said to\nbe of infinite type. When $\\lambda _{i}=0$ for some \\emph{i} = 1,..., \\emph{\n}, then $S$ is said to be of null $J$-type $k$.\n\nIn general when $S$ is of finite type $k$, it follows from (\\ref{r1}) that there exist a monic polynomial, say $R(x)\\neq 0,$ such that $R(\\Delta ^{J})(\\boldsymbol{x}-\\boldsymbol{c})=\\mathbf{0}.$ If we suppose that $R(x)=x^{k}+\\sigma _{1}x^{k-1}+...+\\sigma _{k-1}x+\\sigma _{k},$ then coefficients $\\sigma _{i}$\\ are given by\n\n\\begin{eqnarray}\n\\sigma _{1} &=&-(\\lambda _{1}+\\lambda _{2}+...+\\lambda _{k}),  \\notag \\\\\n\\sigma _{2} &=&(\\lambda _{1}\\lambda _{2}+\\lambda _{1}\\lambda_{3}+...+\\lambda\n_{1}\\lambda _{k}+\\lambda _{2}\\lambda _{3}+...+\\lambda _{2}\\lambda\n_{k}+...+\\lambda _{k-1}\\lambda _{k}),  \\notag \\\\\n\\sigma _{3} &=&-(\\lambda _{1}\\lambda _{2}\\lambda _{3}+...+\\lambda\n_{k-2}\\lambda _{k-1}\\lambda _{k}),  \\notag \\\\\n&&.............................................  \\notag \\\\\n\\sigma _{k} &=&(-1)^{k}\\lambda _{1}\\lambda _{2}...\\lambda _{k}.  \\notag\n\\end{eqnarray}\n\nTherefore the position vector $\\boldsymbol{x}$ satisfies the following equation \\cite{R3}\n\n\\begin{equation*}\n(\\Delta ^{J})^{k}\\boldsymbol{x}+\\sigma _{1}(\\Delta ^{J})^{k-1}\\boldsymbol{x\n+...+\\sigma _{k}(\\boldsymbol{x}-\\boldsymbol{c})=\\boldsymbol{0}.\n\\end{equation*}\n\nIn this paper we will pay attention to surfaces of finite $III$-type. Firstly, we will establish a formula for $\\Delta ^{III}\\boldsymbol{x}$ by\nusing Cartan's method of the moving frame. Further, we continue our study by proving finite coordinate type surfaces for an important class of surfaces,\nnamely, translation surfaces in $E^{3}$.\n\n\\section{Preliminaries}\n\nLet $S$ be a (connected) surface in the Euclidean 3-space $E^{3}$, whose\nGaussian curvature $K$ never vanishes. Let $\\wp =\\{\\boldsymbol{\\varepsilon\n_{1}}(u,v),\\boldsymbol{\\varepsilon _{2}}(u,v),\\boldsymbol{\\varepsilon _{3}\n(u,v)\\}$ is a moving frame of $S$, $\\boldsymbol{\\varepsilon _{3}}\n\\boldsymbol{n}$ is the Gauss map of $S$ and $\\det (\\boldsymbol{\\varepsilon\n_{1}},\\boldsymbol{\\varepsilon _{2}},\\boldsymbol{\\varepsilon _{3}})=1$. Then\nit is well known that there are linear differential forms $\\omega\n_{1},\\omega _{2},\\omega _{31},\\omega _{32}$ and $\\omega _{12}$, such that\n\\cite{R10}\n\n\\begin{equation*}\nd\\boldsymbol{x}=\\omega _{1}\\boldsymbol{\\varepsilon _{1}}+\\omega _{2\n\\boldsymbol{\\varepsilon _{2}},\\ \\ \\ d\\boldsymbol{n}=\\omega _{31}\\boldsymbol\n\\varepsilon _{1}}+\\omega _{32}\\boldsymbol{\\varepsilon _{2}},\n\\end{equation*}\n\n\\begin{center}\n\\begin{equation*}\nd\\boldsymbol{\\varepsilon _{1}}=\\omega _{12}\\boldsymbol{\\varepsilon _{2}\n-\\omega _{31}\\boldsymbol{\\varepsilon _{3}},\\ \\ \\ d\\boldsymbol{\\varepsilon\n_{2}}=-\\omega _{12}\\boldsymbol{\\varepsilon _{1}}-\\omega _{32}\\boldsymbol\n\\varepsilon _{3}},\n\\end{equation*}\n\\end{center}\n\nand functions $a,b,c,q_{1},q_{2}$ of $u,v$ such that\n\n\\begin{equation*}\n\\omega _{31}=-a\\omega _{1}-b\\omega _{2},\\ \\ \\ \\omega _{32}=-b\\omega\n_{1}-c\\omega _{2},\\ \\ \\ \\omega _{12}=q_{1}\\omega _{1}+q_{2}\\omega _{2}.\n\\end{equation*}\n\nWe can choose the moving frame of $S$, such that the vectors $\\boldsymbol\n\\varepsilon_{1}},\\boldsymbol{\\varepsilon_{2}}$ are the principle directions\nof $S$. Then $a$, $c$ are the principle curvatures of $S$ and $b=0$, so the\ndifferential forms $\\omega_{1}$ and $\\omega_{2}$ become\n\n\\begin{equation}\n\\omega_{1} =-\\frac{1}{a}\\omega_{31}, \\ \\ \\ \\ \\omega_{2} =-\\frac{1}{c\n\\omega_{32}.  \\notag\n\\end{equation}\nThe Gauss and mean curvature are respectively\n\n\\begin{equation}\nK= ac, \\ \\ \\ \\ H=\\frac{a+c}{2}.  \\notag\n\\end{equation}\n\nLet $\\nabla _{1}f,\\nabla _{2}f$ be the derivatives of Pfaff of a function \nf(u,v)\\in C^{1}$ along the curves $\\omega _{2}=0,\\omega _{1}=0$\nrespectively. Then we have the following well known relations \\cite{R2}\n\n\\begin{eqnarray*}\n\\nabla _{1}\\boldsymbol{x} &=&\\boldsymbol{\\varepsilon _{1}},\\ \\ \\ \\nabla _{2\n\\boldsymbol{x}=\\boldsymbol{\\varepsilon _{2}}, \\\\\n\\nabla _{1}\\boldsymbol{\\varepsilon _{1}} &=&q_{1}\\boldsymbol{\\varepsilon _{2\n}+a\\boldsymbol{n},\\ \\ \\ \\nabla _{2}\\boldsymbol{\\varepsilon _{1}}=q_{2\n\\boldsymbol{\\varepsilon _{2}}, \\\\\n\\nabla _{1}\\boldsymbol{\\varepsilon _{2}} &=&-q_{1}\\boldsymbol{\\varepsilon\n_{1}},\\ \\ \\ \\ \\nabla _{2}\\boldsymbol{\\varepsilon _{2}}=-q_{2}\\boldsymbol\n\\varepsilon _{1}}+c\\boldsymbol{n}, \\\\\n\\nabla _{1}\\boldsymbol{n} &=&-a\\boldsymbol{\\varepsilon _{1}},\\ \\ \\ \\nabla\n_{2}\\boldsymbol{n}=-c\\boldsymbol{\\varepsilon _{2}},\n\\end{eqnarray*}\n\nWe denote by $\\widetilde{\\nabla}_{1}f$ and $\\widetilde{\\nabla}_{2}f$ the derivatives of Pfaff of $f$ along the curves $\\omega_{32}=0, \\omega_{31}=0$\nrespectively. One finds\n\n\\begin{equation}\n\\widetilde{\\nabla}_{1}f =-\\frac{1}{a}\\nabla_{1}f,\\ \\ \\ \\widetilde{\\nabla}_{2}f =-\\frac{1}{c}\\nabla_{2}f.  \\notag\n\\end{equation}\n\nIt follows that\n\n\\begin{equation}\n\\widetilde{\\nabla }_{1}\\boldsymbol{x}=-\\frac{1}{a}\\boldsymbol{\\varepsilon\n_{1}},\\ \\ \\ \\widetilde{\\nabla }_{2}\\boldsymbol{x}=-\\frac{1}{c}\\boldsymbol\n\\varepsilon _{2}},  \\label{r3}\n\\end{equation\n\\begin{equation}\n\\widetilde{\\nabla }_{1}\\boldsymbol{\\varepsilon _{1}}=p_{1}\\boldsymbol\n\\varepsilon _{2}}-\\boldsymbol{n},\\ \\ \\ \\widetilde{\\nabla }_{2}\\boldsymbol\n\\varepsilon _{1}}=p_{2}\\boldsymbol{\\varepsilon _{2}},  \\label{r4}\n\\end{equation\n\\begin{equation}\n\\widetilde{\\nabla }_{1}\\boldsymbol{\\varepsilon _{2}}=-p_{1}\\boldsymbol\n\\varepsilon _{1}},\\ \\ \\ \\ \\widetilde{\\nabla }_{2}\\boldsymbol{\\varepsilon _{2\n}=-p_{2}\\boldsymbol{\\varepsilon _{1}}-\\boldsymbol{n},  \\label{r5}\n\\end{equation\n\\begin{equation}\n\\widetilde{\\nabla }_{1}\\boldsymbol{n}=\\boldsymbol{\\varepsilon _{1}},\\ \\ \\\n\\widetilde{\\nabla }_{2}\\boldsymbol{n}=\\boldsymbol{\\varepsilon _{2}},  \\notag\n\\end{equation\nwhere $p_{1}=-\\frac{1}{a}q_{1},p_{2}=-\\frac{1}{c}q_{2}$ are the geodesic\ncurvatures of the spherical curves $\\omega _{32}=0$ and $\\omega _{31}=0$\nrespectively. The Mainardi-Codazzi equations have the following forms\n\n\\begin{equation}\n\\widetilde{\\nabla }_{1}\\frac{1}{c}=p_{2}\\Big(\\frac{1}{a}-\\frac{1}{c}\\Big),\\\n\\ \\ \\widetilde{\\nabla }_{2}\\frac{1}{a}=p_{1}\\Big(\\frac{1}{a}-\\frac{1}{c}\\Big\n.  \\label{r6}\n\\end{equation}\n\nLet $f$ be a sufficient differentiable function on $S$. Then the second differential parameter of Beltrami corresponding to the fundamental form \nIII $ of $S$ is defined by\n\\begin{equation}  \\label{777}\n\\Delta ^{III}f= -\\widetilde{\\nabla}_{1}\\widetilde{\\nabla}_{1}f-\\widetilde\n\\nabla}_{2}\\widetilde{\\nabla}_{2}f-p_{2}\\widetilde{\\nabla}_{1}f+p_{1\n\\widetilde{\\nabla}_{2}f.\n\\end{equation}\n\nApplying (\\ref{777}) to the position vector $\\boldsymbol{x}$, gives\n\n\\begin{equation}\n\\Delta ^{III}\\boldsymbol{x}= -\\widetilde{\\nabla}_{1}\\widetilde{\\nabla}_{1\n\\boldsymbol{x}-\\widetilde{\\nabla}_{2}\\widetilde{\\nabla}_{2}\\boldsymbol{x\n-p_{2}\\widetilde{\\nabla}_{1}\\boldsymbol{x}+p_{1}\\widetilde{\\nabla}_{2\n\\boldsymbol{x}.  \\notag\n\\end{equation}\n\nFrom (\\ref{r3}) we obtain\n\n\\begin{equation}\n\\Delta ^{III}\\boldsymbol{x}=\\widetilde{\\nabla }_{1}\\Big(\\frac{1}{a\n\\boldsymbol{\\varepsilon _{1}}\\Big)+\\widetilde{\\nabla }_{2}\\Big(\\frac{1}{c\n\\boldsymbol{\\varepsilon _{2}}\\Big)+\\frac{1}{a}p_{2}\\boldsymbol{\\varepsilon\n_{1}}-\\frac{1}{c}p_{1}\\boldsymbol{\\varepsilon _{2}}.  \\label{r8}\n\\end{equation}\n\nUsing (\\ref{r4}) and (\\ref{r5}), equation (\\ref{r8}) becomes\n\n\\begin{equation}\n\\Delta ^{III}\\boldsymbol{x}=\\Big(\\widetilde{\\nabla }_{1}\\frac{1}{a}\\Big\n\\boldsymbol{\\varepsilon _{1}}+\\Big(\\frac{1}{a}-\\frac{1}{c}\\Big)p_{2\n\\boldsymbol{\\varepsilon _{1}}+\\Big(\\widetilde{\\nabla }_{2}\\frac{1}{c}\\Big\n\\boldsymbol{\\varepsilon _{2}}+\\Big(\\frac{1}{a}-\\frac{1}{c}\\Big)p_{1\n\\boldsymbol{\\varepsilon _{2}}-\\Big(\\frac{1}{a}+\\frac{1}{c}\\Big)\\boldsymbol{n\n.  \\label{r9}\n\\end{equation}\n\nTaking into account the Mainardi-Codazzi equations (\\ref{r6}), so equation \n\\ref{r9}) reduces to\n\n\\begin{equation*}\n\\Delta ^{III}\\boldsymbol{x}=\\Big(\\widetilde{\\nabla }_{1}\\big(\\frac{1}{a}\n\\frac{1}{c}\\big)\\Big)\\boldsymbol{\\varepsilon _{1}}+\\Big(\\widetilde{\\nabla \n_{2}\\big(\\frac{1}{a}+\\frac{1}{c}\\big)\\Big)\\boldsymbol{\\varepsilon _{2}}-\\big\n\\frac{1}{a}+\\frac{1}{c}\\big)\\boldsymbol{n}\n\\end{equation*\nor equivalently, (see \\cite{R13})\n\n\\begin{equation}\n\\Delta ^{III}\\boldsymbol{x}=grad^{III}\\Big(\\frac{2H}{K}\\Big)-\\Big(\\frac{2H}{\n}\\Big)\\boldsymbol{n}.  \\label{r11}\n\\end{equation}\n\n\\begin{remark}\nS. Stamatakis and H. Al-zoubi proved in \\cite{R13} relation (\\ref{r11}) by\nusing tensors calculus.\n\\end{remark}\n\nFrom (\\ref{r11}) we obtain the following results which were proved in \\cit\n{R13}.\n\n\\begin{theorem}\n\\label{T1} A surface $S$ in $E^{3}$ is of 0-type 1 corresponding to the third fundamental form if and only if $S$ is minimal.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{T2} A surface $S$ in $E^{3}$ is of $III$-type 1 if and only if $S$ is part of a sphere.\n\\end{theorem}\n\n\\begin{corollary}\n\\label{C1.1} The Gauss map of every surface $S$ in $E^{3}$ is of $III$-type 1. The corresponding eigenvalue is $\\lambda=2$.\n\\end{corollary}\n\nUp to now, the only known surfaces of finite $III$-type in $E^{3}$ are parts of spheres, minimal surfaces and parallel surfaces to minimal ones\nwhich are of null $III$-type 2. So the following question seems to be interesting:\n\n\\begin{problem}\n\\label{p1}Other than the surfaces mentioned above, which surfaces in $E^{3}$ are of finite $III$-type?\n\\end{problem}\n\nAnother generalization of the above problem is to study surfaces in $E^{3}$ whose position vector $\\boldsymbol{x}$ satisfies\n\\begin{equation}\n\\Delta ^{III}\\boldsymbol{x}=\\Lambda \\boldsymbol{x,}  \\label{r12}\n\\end{equation\nwhere $\\Lambda \\in \\mathbb{Re}^{3\\times 3}$.\n\nFrom this point of view, we also pose the following problem\n\n\\begin{problem}\n\\label{p2}Classify all surfaces in $E^{3}$ whose position vector $\\boldsymbol{x}$ satisfies relation (\\ref{r12}).\n\\end{problem}\n\nConcerning this problem, in \\cite{R14} S. Stamatakis and H. Al-Zoubi studied the class of surfaces of revolution and they proved that:\n\n\\begin{theorem}\n\\label{TC} A surface of revolution $S$ in \n\\mathbb{R}\n^{3},$ satisfies (\\ref{r12}), if and only if S is a catenoid or a part of a\nsphere.\n\\end{theorem}\n\nRecently, the same authors in \\cite{R1} studied the class of ruled surfaces and the class of quadric surfaces. More precisely, they proved that\n\n\\begin{theorem}\n\\label{TA} The only \\textit{ruled surfaces in the 3-dimensional Euclidean space that satisfies (\\ref{r12}), are the helicoids.}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{TB} The only quadric surfaces in the 3-dimensional \\textit{Euclidean } space that satisfies (\\ref{r12}), are the spheres.\n\\end{theorem}\n\nThe present paper contributes to the solution of problem \\ref{p2}, for the class of translation surfaces in the Euclidean 3-space $E^{3}$, meanwhile\nproblem \\ref{p1} is still unsolved for this family of surfaces.\n\n\\section{Translation surfaces}\n\nLet $S$ : $M^{2}\\rightarrow E^{3}$ be a translation surface in the Euclidean 3-space. Then the position vector $\\boldsymbol{x}$ of $S$ is given by \\cite{R25}\n\n\\begin{equation}\nS:\\boldsymbol{x}(s,t)=\\left\\{ s,\\ t,\\ \\widetilde{f}(s)+\\widetilde{h\n(t)\\right\\} ,\\ \\ \\ \\ \\ \\ \\ \\ (s,t)\\in D\\subset M^{2},  \\label{1}\n\\end{equation\nwhere $\\widetilde{f}(s)$ and $\\widetilde{h}(t)$ are two sufficiently differentiable functions on $S$. We put\n\n\\begin{equation*}\nf:=\\frac{d\\widetilde{f}}{ds},\\ \\ \\ \\ h:=\\frac{d\\widetilde{h}}{dt}.\n\\end{equation*}\n\nIt is easily verified that the first and the second fundamental forms of $S$ are given by\n\\begin{align*}\nI& =(1+f^{2})ds^{2}+2fhdsdt+(1+h^{2})dt^{2}, \\\\\nII& =\\frac{f_{s}}{\\sqrt{\\mu }}ds^{2}+\\frac{h_{t}}{\\sqrt{\\mu }}dt^{2},\n\\end{align*\nwhere\n\n\\begin{equation*}\nf_{s}:=\\frac{df}{ds},\\ \\ \\ \\ h_{t}:=\\frac{dh}{dt}\n\\end{equation*\nand $\\mu :=\\det (g_{ij})=1+f^{2}+h^{2}.$\n\nThe Gauss and mean curvatures of $S,$ are respectively\n\n\\begin{equation*}\nK=\\frac{f_{s}h_{t}}{\\mu ^{2}},\n\\end{equation*}\n\n\\begin{equation}\n2H=\\frac{(1+f^{2})h_{t}+(1+h^{2})f_{s}}{\\mu \\sqrt{\\mu }}.  \\label{2}\n\\end{equation}\n\nSince $S$ does not contain parabolic points, so $f_{s}\\neq 0$ and $h_{t}\\neq0$ for each $(s,t)\\in D.$\n\nOur main result is the following\n\n\\begin{theorem}\n\\label{T2.1} The only\\textit{\\ translation surface in the 3-dimensional Euclidean space that satisfies (\\ref{r12}) is Scherk's surface.}\n\\end{theorem}\n\nOur discussion is local, which means that we show in fact that any open part of a translation surface satisfies (\\ref{r12}), if it is minimal.\n\nBefore we start proving our main result we prove the following\n\n\\begin{proposition}\n\\label{C2.1} A \\textit{minimal translation surface with parametric represantation (\\ref{1}) has the form \n\\begin{equation}\nS:\\boldsymbol{x}(s,t)=\\left\\{ s,\\ t,\\ c-\\frac{1}{a_{1}}\\ln \\left\\vert \\cos\n(a_{1}s+a_{2})\\right\\vert +\\frac{1}{a_{1}}\\ln \\left\\vert \\cos\n(-a_{1}t+b_{2})\\right\\vert \\right\\} ,  \\label{3}\n\\end{equation\n\\begin{equation*}\n(s,t)\\in D\\subset M^{2},\n\\end{equation*}\n\nwhere $a_{1},a_{2},b_{2}$ and $c$ are integration constants.\\footnote{This proposition is due to H. F. Scherk. For a more details, the reader is referred to \\cite{R17}.}\n\\end{proposition}\n\n\\begin{proof}\nFrom (\\ref{2}) we obtain that $S$ is minimal if and only if\n\n\\begin{equation}\n\\frac{1+f^{2}}{f_{s}}+\\frac{1+h^{2}}{h_{t}}=0.  \\label{e1}\n\\end{equation}\n\nSince the first factor of this equation depends only on the parameter $s$ and the second factor depends only on the parameter $t,$ (\\ref{e1}) implies\nthat $\\frac{1+f^{2}}{f_{s}}=a_{1}$ and $\\frac{1+h^{2}}{h_{t}}=b_{1},$ where $a_{1},$ $b_{1}$ are constants and $a_{1}+$ $b_{1}=0.$\n\nIntegrating each factor with respect to the corresponding parameter, we obtain\n\n\\begin{equation*}\nf(s)=\\tan (a_{1}s+a_{2}),\\ \\ \\ \\ \\ \\ h(t)=\\tan (b_{1}t+b_{2}).\n\\end{equation*}\n\nwhere $a_{2},$ $b_{2}$ are integration constants. Integrating again, it then follows that\n\n\\begin{equation*}\n\\widetilde{f}(s)=d_{1}-\\frac{1}{a_{1}}\\ln \\left\\vert \\cos\n(a_{1}s+a_{2})\\right\\vert ,\\ \\ \\ \\ \\widetilde{h}(t)=d_{2}-\\frac{1}{b_{1}}\\ln\n\\left\\vert \\cos (b_{1}t+b_{2})\\right\\vert .\n\\end{equation*}\n\nwhere $d_{1},$ $d_{2}$ are constants. Putting $b_{1}=-a_{1}$ and $d_{1}+d_{2}=c$. Thus we have\\textit{\\ }(\\ref{3}).\n\\end{proof}\n\n\n\\section{Proof of Theorem \\protect\\ref{T2.1}}\n\nAs in section 2, a parametric represantation of a translation surface $S$ in $E^{3}$ is\n\n\\begin{equation*}\nS:\\boldsymbol{x}(s,t)=\\left\\{ s,\\ t,\\ \\widetilde{f}(s)+\\widetilde{h\n(t)\\right\\} ,\\ \\ \\ \\ \\ \\ \\ \\ (s,t)\\in D\\subset M^{2},\n\\end{equation*\nwhere $\\widetilde{f}(s),\\widetilde{\\ h}(t)$ are two sufficiently differentiable functions on $S$. For convenience, we put\n\n\\begin{equation*}\nM(s)=\\frac{(1+f^{2})f_{ss}}{f_{s}^{3}}-\\frac{2f}{f_{s}},\\ \\ \\ \\ N(t)=\\frac\n(1+h^{2})h_{tt}}{h_{t}^{3}}-\\frac{2h}{h_{t}}.\n\\end{equation*}\n\nThen the Beltrami operator with respect to the third fundamental form, after a lengthy computation, can be expressed as follows \\cite{R0}\n\n\\begin{equation}\n\\Delta ^{III}=\\left( -\\frac{1+f^{2}}{f_{s}^{2}}\\frac{\\partial ^{2}}{\\partial\ns^{2}}-\\frac{1+h^{2}}{h_{t}^{2}}\\frac{\\partial ^{2}}{\\partial t^{2}}-\\frac\n2fh}{f_{s}h_{t}}\\frac{\\partial ^{2}}{\\partial s\\partial t}+M(s)\\frac\n\\partial }{\\partial s}+N(t)\\frac{\\partial }{\\partial t}\\right)\\mu .\n\\label{eq 4}\n\\end{equation}\n\nFor later use, we remark that\n\n\\begin{equation}\n-\\frac{d}{ds}\\left( \\frac{1+f^{2}}{f_{s}}\\right) =M(s)f_{s},  \\label{4}\n\\end{equation}\n\n\\begin{equation}\n-\\frac{d}{dt}\\left( \\frac{1+h^{2}}{h_{t}}\\right) =N(t)h_{t}.  \\label{5}\n\\end{equation}\n\nApplying (\\ref{eq 4}) on the coordinate functions $x_{i},i=1,2,3$ of the position vector $\\boldsymbol{x}$ of $S,$ we have\n\n\\begin{equation*}\n\\Delta ^{III}x_{1}=\\Delta ^{III}s=M(s)\\mu ,\n\\end{equation*}\n\n\\begin{equation*}\n\\Delta ^{III}x_{2}=\\Delta ^{III}t=N(t)\\mu ,\n\\end{equation*}\n\n\\begin{equation*}\n\\Delta ^{III}x_{3}=\\Delta ^{III}(\\widetilde{f}+\\widetilde{h})=\\left(\nM(s)f+N(t)h-\\frac{1+f^{2}}{f_{s}}-\\frac{1+h^{2}}{h_{t}}\\right) \\mu .\n\\end{equation*}\n\nWe denote by $\\lambda _{ij},i,j=1,2,3$ the entries of the matrix $\\Lambda .\\\n$Using the last three equations and condition (\\ref{r12}) one find\n\\begin{equation}\nM(s)\\mu =\\lambda _{11}s+\\lambda _{12}t+\\lambda _{13}(\\widetilde{f}\n\\widetilde{h}),  \\label{eq 5}\n\\end{equation}\n\n\\begin{equation}\nN(t)\\mu =\\lambda _{21}s+\\lambda _{22}t+\\lambda _{23}(\\widetilde{f}\n\\widetilde{h}),  \\label{eq 6}\n\\end{equation}\n\n\\begin{equation}\n\\left( M(s)f+N(t)h-\\frac{1+f^{2}}{f_{s}}-\\frac{1+h^{2}}{h_{t}}\\right) \\mu\n=\\lambda _{31}s+\\lambda _{32}t+\\lambda _{33}(\\widetilde{f}+\\widetilde{h}).\n\\label{eq 7}\n\\end{equation}\n\nDifferentiating (\\ref{eq 5}) with respect to $t,$ we find\n\n\\begin{equation*}\n2Mhh_{t}=\\lambda _{12}+\\lambda _{13}h.\n\\end{equation*}\n\nHence, $M(s)$ must be constant.\nSimilarly, if we take the derivative of (\\ref{eq 6}) with respect to $s$ we\nobtain that $N(t)$ must be constant. If we put $M(s)=c_{1}$ and $N(t)=c_{2}\n, where $c_{1}$ and $c_{2}$ are constants, then equations (\\ref{eq 5}), (\\re\n{eq 6}) and (\\ref{eq 7}) become\n\n\\begin{equation}\nc_{1}\\mu =\\lambda _{11}s+\\lambda _{12}t+\\lambda _{13}(\\widetilde{f}\n\\widetilde{h}),  \\label{eq 8}\n\\end{equation}\n\n\\begin{equation}\nc_{2}\\mu =\\lambda _{21}s+\\lambda _{22}t+\\lambda _{23}(\\widetilde{f}\n\\widetilde{h}),  \\label{eq 9}\n\\end{equation}\n\n\\begin{equation}\n\\left( c_{1}f+c_{2}h-\\frac{1+f^{2}}{f_{s}}-\\frac{1+h^{2}}{h_{t}}\\right) \\mu\n=\\lambda _{31}s+\\lambda _{32}t+\\lambda _{33}(\\widetilde{f}+\\widetilde{h}).\n\\label{eq 10}\n\\end{equation}\n\nDifferentiating (\\ref{eq 10}) with respect to $s,$ and taking into account \n\\ref{4}), we have\n\n\\begin{equation*}\n2c_{1}f_{s}\\mu +2\\left( c_{1}f+c_{2}h-\\frac{1+f^{2}}{f_{s}}-\\frac{1+h^{2}}\nh_{t}}\\right) ff_{s}=\\lambda _{31}+\\lambda _{33}f.\n\\end{equation*}\n\nAnother derivation with respect to $t,$ and taking into account (\\ref{5}),\ngives\n\n\\begin{equation*}\n4f_{s}h_{t}(c_{1}f+c_{2}h)=0.\n\\end{equation*}\n\nSince the first factor can not be zero, we conclude that $c_{1}f+c_{2}h=0,$\nand hence $c_{1}=c_{2}=0.$\n\nFrom (\\ref{eq 8}) and (\\ref{eq 9}) we find that $\\lambda\n_{ij}=0,i=1,2,j=1,2,3,$ and (\\ref{eq 10}) becomes\n\n\\begin{equation}\n-\\left( \\frac{1+f^{2}}{f_{s}}+\\frac{1+h^{2}}{h_{t}}\\right) \\mu =\\lambda\n_{31}s+\\lambda _{32}t+\\lambda _{33}(\\widetilde{f}+\\widetilde{h}).\n\\label{eq 11}\n\\end{equation}\n\nAlso, the system of equations (\\ref{4}) and (\\ref{5}) reduces to\n\n\\begin{equation}\n\\frac{d}{ds}\\left( \\frac{1+f^{2}}{f_{s}}\\right) =0,  \\label{eq 12}\n\\end{equation}\n\n\\begin{equation}\n\\frac{d}{dt}\\left( \\frac{1+h^{2}}{h_{t}}\\right) =0.  \\label{eq 13}\n\\end{equation}\n\nFrom (\\ref{eq 12}) and (\\ref{eq 13}) we obtain\n\n\\begin{equation}\n\\frac{1+f^{2}}{f_{s}}=\\frac{1}{a_{1}},  \\label{eq 14}\n\\end{equation}\n\n\\begin{equation}\n\\frac{1+h^{2}}{h_{t}}=\\frac{1}{b_{1}},  \\label{eq 15}\n\\end{equation\nwhere $\\frac{1}{a_{1}}$ and $\\frac{1}{b_{1}}$ are constants.\n\nConsequently, equation (\\ref{eq 11}) becomes\n\n\\begin{equation}\n-\\left( \\frac{1}{a_{1}}+\\frac{1}{b_{1}}\\right) \\mu =\\lambda _{31}s+\\lambda\n_{32}t+\\lambda _{33}(\\widetilde{f}+\\widetilde{h}).  \\label{e2}\n\\end{equation}\n\nBy taking the derivative of (\\ref{e2}) with respect to $s$, we find that\n\n\\begin{equation}\n-2\\left( \\frac{1}{a_{1}}+\\frac{1}{b_{1}}\\right) ff_{s}=\\lambda _{31}+\\lambda\n_{33}f,  \\label{eq 16}\n\\end{equation}\n\nAnother integration of (\\ref{eq 14}) and (\\ref{eq 15}) gives\n\n\\begin{equation*}\nf(s)=\\tan (a_{1}s+a_{2}),\\ \\ \\ \\ \\ \\ h(t)=\\tan (b_{1}t+b_{2}),\n\\end{equation*\nwhere $a_{2},$ $b_{2}$ are integration constants.\n\nInserting $f(s)=\\tan (a_{1}s+a_{2})$ and its derivation $f_{s}$ in (\\ref{eq 16}), we get\n\n\\begin{equation*}\n-2a_{1}\\left( \\frac{1}{a_{1}}+\\frac{1}{b_{1}}\\right) \\frac{\\sin\n(a_{1}s+a_{2})}{\\cos ^{3}(a_{1}s+a_{2})}=\\lambda _{31}+\\lambda _{33}\\frac\n\\sin (a_{1}s+a_{2})}{\\cos (a_{1}s+a_{2})}\n\\end{equation*\nor, equivalently\n\n\\begin{equation*}\n\\lambda _{31}\\cos ^{3}(a_{1}s+a_{2})+\\lambda _{33}\\sin (a_{1}s+a_{2})\\cos\n^{2}(a_{1}s+a_{2})\n\\end{equation*}\n\n\\begin{equation}\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ +2a_{1}\\left( \\frac{1}\na_{1}}+\\frac{1}{b_{1}}\\right) \\sin (a_{1}s+a_{2})=0.  \\label{eq 17}\n\\end{equation}\n\nNotice that $a_{1}\\neq 0,$ otherwise $f_{s}=0.$ Putting $z=\\cos\n(a_{1}s+a_{2}),$ (\\ref{eq 17}) becomes\n\n\\begin{equation*}\n\\lambda _{31}z^{3}+\\lambda _{33}z^{2}\\sqrt{1-z^{2}}+2a_{1}\\left( \\frac{1}\na_{1}}+\\frac{1}{b_{1}}\\right) \\sqrt{1-z^{2}}=0.\n\\end{equation*}\n\nSince the functions $z^{3}$ and $z^{j}\\sqrt{1-z^{2}}$, $0\\leq j\\leq 2$ are linearly independent, we have\n\n\\begin{equation*}\n\\lambda _{31}=\\lambda _{33}=0,\n\\end{equation*}\n\n\\begin{equation}\n\\frac{1}{a_{1}}+\\frac{1}{b_{1}}=0.  \\label{eq 18}\n\\end{equation}\n\nFinally, from (\\ref{e2}) and the last two equations we find that $\\lambda_{32}=0$. Also, from (\\ref{eq 18}) we get $a_{1}+$ $b_{1}=0.$ Integrating \nf(s)=\\tan (a_{1}s+a_{2})$ with respect to $s,$ $h(t)=\\tan (b_{1}t+b_{2})$ with respect to $t$ and using proposition \\ref{C2.1}, we find that the only\ntranslation surfaces that satisfies (\\ref{r12}), are the minimals with corresponding matrix $\\Lambda $ the null matrix. This ends the proof of our theorem.\n\n\\begin{acknowledgement}\nThe authors would like to express their thanks to the referee for his useful remarks.\n\\end{acknowledgement}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Summary and conclusions}\n\\label{sec:conclusions}\n\nThe vortex dynamics resulting from the interaction of an $AR=18$ rectangular orifice synthetic jet with a turbulent boundary layer was experimentally investigated by measuring the three dimensional three-component velocity field within the jet using stereoscopic particle image velocimetry. \nThree common orifice geometries and three practical actuation conditions were tested to explore the impact of factors of actuator design and operation that govern the jet flow physics. \nThe measurements revealed a variety of different vortex structures and enabled valuable connections to be made between the vortex dynamics and the jet's influence on the boundary layer.\n\nOne prominent effect of the cross-flow interaction was to diminish the circulation strength of the upstream side of the initial vortex ring while strengthening the downstream side. \nWhen the synthetic jet interacted strongly with the cross-flow, as was the case in the wall-normal spanwise-oriented orifice, the weak upstream side of the ring quickly decayed, and the downstream flow field was dominated by a train of vortices which rotated clockwise in the spanwise direction. \nWhile the initial vortex structure was connected to the wall through legs which branched off from the primary vortex, downstream a new topology developed in the form of hairpin vortices which were connected to each other via their legs. \nRotating the orifice $90^\\circ$ to align the long axis with the streamwise direction produced a more intact vortex ring which propagated rapidly away from the wall and axis-switched once before the self-induced deformations subsided. \nAlthough the circulation strength still varied around the circumference of the vortex rings from the streamwise-oriented orifice, evidenced by the presence of leg and bridge vortices, the differences were smaller than those seen in the spanwise-oriented jet at the same conditions. \n\nIt should be noted that hairpin vortices are not exclusive to spanwise-oriented jets, and that distorted vortex rings are not restricted to actuators with streamwise-oriented orifices. \nIt is possible to generate both types of structures from either orientation of the orifice. \nThe difference between these two orifices is the conditions at which the jet transitions from containing one type of vortex structure to the other. \nBecause jets from spanwise-oriented orifices interact strongly with the boundary layer, due to the large cross-section they present to the cross-flow, a larger blowing ratio is required to produce a structure which behaves like a vortex ring than the blowing ratio required for a jet issuing from a streamwise-oriented orifice. \nThis blowing ratio threshold likely depends also on the aspect ratio of the orifice, the thickness of the boundary layer, and whether the flow is laminar or turbulent. \nFor example, \\citet{Wang_Feng_JFM_2020} found that their $AR=3$ spanwise-oriented rectangular orifice synthetic jet produced axis-switching vortex rings in a laminar cross-flow at a blowing ratio of one. \nAdditionally, in the present study the centreline velocity field of the streamwise-oriented synthetic jet suggests that the vortices produced at a blowing ratio of 0.5 began to behave more like hairpin vortices and less like the modified vortex rings observed at a blowing ratio of one.\n\nAnother factor influencing the type of vortex structures produced by the synthetic jet was the pitch angle of the orifice. \nPitching a spanwise-oriented orifice downstream had the opposite effect on the circulation strength of the vortex ring as the cross-flow did. \nNamely, the upstream side of the vortex ring became stronger and the downstream side weaker. \nTherefore, at a blowing ratio of one the pitched orifice produced a train of counterclockwise-rotating arch-shaped vortices which remained connected to the wall as they advected downstream. \n\nThe effect that the jets had at a blowing ratio of one on the boundary layer varied significantly with the type of vortex structure formed. \nThe wall-normal jets induced mixing which transported high momentum fluid towards the wall, accelerating the fluid there, while low momentum fluid transported off the wall was deposited into a roughly circular wake higher in the cross-flow.\nAlternatively, the pitched jet accelerated a thick layer of fluid near the wall through direct momentum injection, and, although a layer of fluid higher in the cross-flow was decelerated slightly by the jet, there was no localized wake.\n\nFrom the perspective of delaying separation, the pitched jet is believed to be more likely to offer effective control at a blowing ratio of one than the wall-normal jets did because the layer of accelerated fluid it produced was thicker and the increase in streamwise velocity produced was larger than those in the other jets. \nInterestingly, when the actuator average blowing velocity was decreased, reducing blowing ratio to 0.5, the pitched jet became a net detriment on or sink for the cross-flow momentum while the wall-normal jets were still able to enhance the near-wall flow speed, albeit to a lesser degree than they did at the higher blowing ratio. \nThe differences in the relative performance of the different jets at a lower blowing ratio relates to the mechanism by which they alter the flow field. \nWhereas the wall-normal jets were still able to increase mixing in the turbulent boundary layer at the lower blowing ratio the pitched jet no longer had sufficient excess momentum relative to the cross-flow to be able to accelerate the near-wall fluid. \n\nThe performance of the wall-normal synthetic jets in the two orifice orientations tested was subject to a trade-off between the near-wall persistence of the periodic structures and the strength of their induced velocity.\nThe more complete vortex rings produced by the streamwise-oriented orifice induced a strong near-wall vertical velocity when the vortex ring first formed, but they also propagated away from the wall quickly, which diminished their influence downstream.\nConversely, the wall-normal velocity induction and rate of mixing was more modest in the spanwise-oriented synthetic jet, but the periodic structure which developed into the hairpin vortices remained relatively close to the wall, extending the domain of their near-wall influence.\nThe spanwise-oriented synthetic jet also enhanced the near-wall flow speed in the areas to the sides of the jet centreline more than the streamwise-oriented jet did.\nThe net result of the trade-offs between the two wall-normal orifice orientations was that the downstream difference in the near-wall flow speed was small, but a slight advantage was measured for a spanwise orifice orientation when the total footprint of the jet was considered instead of just the jet centreline.\n\nAn interesting discovery was made at the third condition tested where a blowing ratio of 0.5 was achieved by increasing the cross-flow speed, instead of by reducing the actuator average blowing velocity. \nIncreasing the cross-flow speed reduced the Strouhal number of the jet, i.e., it increased the spacing between the vortices in the boundary layer. \nWhile the relative rate of wall-normal fluid transport within the mean field dropped significantly from what it had been at a blowing ratio of one, the wall-normal jets exhibited particularly high unsteady mixing at the lower Strouhal number. \nThe unsteady mixing was driven by the alternating direction of vertical velocity induction from the spanwise-oriented portion of the vortices and increased when the vortices were spaced further apart, i.e., when they interacted with each other more weakly. \nFurthermore, when the cross-flow speed was doubled the clockwise-rotating vortices produced by the wall-normal jets had roughly double the circulation strength that they had at a blowing ratio of one, despite the fact that the driving conditions of the actuator were unchanged. \nThis phenomenon can be attributed to the fact that, at the reduced blowing ratio, the imbalance in the jet vorticity production due to the cross-flow interaction was enhanced such that it skewed further in favor of clockwise vorticity. \nThus the detrimental impact that reducing the blowing ratio had on the relative strength of induced velocity from the vortices in the jets was partially mitigated in the wall-normal jets --- which were dominated by clockwise-rotating structures --- and amplified in the pitched jet --- which was dominated by counterclockwise-rotating vortices.\n\nThe present experimental study sought to investigate the fundamental vortex dynamics of synthetic jet-boundary layer interactions, but the conditions and geometries tested were selected intentionally for their relevance to the application of active flow control. \nThe insight provided by this work can help connect particular vortex structures to their characteristic influence on the surrounding flow field, whether that is the height and footprint of the jet, the intensity of mean and unsteady mixing, or the location within the boundary layer where streamwise momentum of the flow is increased or diminished. \nFurthermore, the insight gained on how orifice pitch angle, skew angle and jet blowing ratio can be used to alter the types of vortices produced by the actuator can be a helpful guide for selectively generating synthetic jets that are composed of specific vortex structures. \nThe ability to create a jet dominated by clockwise-rotating or counterclockwise-rotating vortices versus vortex pairs with equal strength has significant practical implications. \nFinally, the results have underscored the importance of understanding the underlying physics because the dynamics within a synthetic jet may change considerably when critical parameters are varied.\n\n\\bigskip\n\n\\textbf{Acknowledgements}\n\nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship Programme under Grant No. (DGE 1144083).\n\n\\textbf{Declaration of Interests}\n\nThe authors report no conflict of interest.\n\\section{Experimental methods}\n\\label{sec:expmethods}\n\n\\subsection{The facility}\n\\label{sec:facility}\n\nThe SJBLI experiments were conducted in the low-speed wind tunnel within the Experimental Aerodynamics Laboratory at the University of Colorado Boulder. \nThis open-return facility is configured with a centrifugal blower at the wind tunnel inlet and is able to generate air speeds in the test section of up to 65~ms$^{-1}$. \nThe test section has a square cross-sectional area of 0.76~m $\\times$ 0.76~m and is 3.5~m long. \nA boundary layer plate with a development length of 3.4~m was installed 0.31~m off of the test section floor and spanned the full width of the test section. \nThe BL plate leading edge had an 8:1 modified super elliptical shape while the trailing edge was mounted with a flap which was adjusted to achieve a zero incidence at the leading edge~\\citep{Lin_Reed_Saric_StabTranTurb_1992}. \nTo trip the flow a 25.4~mm wide strip of 36 grit distributed sand grain roughness was installed across the BL plate at a distance of 0.19~m downstream from the leading edge. \nThis provided over 1000 trip strip heights of development length for the flow before it reached the actuator which was installed on the plate centreline 1.62~m downstream from the leading edge.\n\n\\subsection{The actuator}\n\\label{sec:actuator}\n\nTo study the interaction of synthetic jets with turbulent BLs a modular actuator was designed. \nInterchangeable plates containing the actuator orifice were fabricated using a stereolithography (SLA) printer. \nThe jet was driven by a 63.5~mm diameter piezoelectric disk actuator which was clamped against the orifice plate (figure~\\ref{fig:sjascheme}). \nThe resulting cylindrical cavity connected to the orifice was 58.4~mm in diameter and 3.18~mm tall. \nUnder the piezoelectric disk was a second cavity that was sealed with a ring shaped gasket against a cast acrylic disk. \nClear acrylic was used for the floor of the actuator under-cavity to provide optical access for laser vibrometer measurements of the piezoelectric disk. \nThe actuator assembly was secured into a circular cut-out in the BL plate using a retaining ring, which, when loose, allowed the actuator to be rotated $360^{\\circ}$. \n\nThe present experiment involved three common rectangular orifice configurations. \nIn all cases the orifice had a throat width, $h$, of 1~mm and an $AR$ of 18. \nWhat differentiates the three configurations are the orifice pitch angle ($\\alpha$, the angle between the centreline of the orifice channel and the cross flow surface) and skew angle ($\\beta$, the angle between the orifice major axis and the spanwise direction) as highlighted in figure~\\ref{fig:sjascheme}. \nUsing this nomenclature the three configurations are (1) $\\alpha=90^\\circ$, $\\beta=0^\\circ$, a spanwise-oriented orifice with the flow exiting normal to the cross-flow surface, (2) $\\alpha=90^\\circ$, $\\beta=90^\\circ$, a streamwise-oriented orifice with the flow exiting normal to the cross-flow surface, and (3) $\\alpha=45^\\circ$, $\\beta=0^\\circ$, a spanwise-oriented orifice with the flow pitched $45^\\circ$ relative to the cross-flow surface. \nOrifice geometries (1) and (2) are referred to as ``wall-normal orifices'' while (3) is called the ``pitched orifice''.\nThe origin of the coordinate system used in this study sits at the centroid of the orifice exit while the $x$, $y$ and $z$ directions align with the cross-flow, wall-normal and spanwise directions, respectively (figure~\\ref{fig:sjascheme}). \n\n\\subsection{Actuator performance}\n\\label{sec:actuatorperf}\n\nThe piezoelectric disk was driven with a sinusoidal voltage waveform with a constant amplitude and frequency. \nTo establish the SJA performance at different driving conditions, benchtop experiments were conducted prior to cross-flow testing. \nAn A.A. Lab Systems AN-1003 constant-temperature anemometer system was used to measure the synthetic jet velocity inside the orifice. \nThese measurements were made on the $\\alpha=90^\\circ$ orifice centreline with a Dantec miniature single-component hot-wire probe inserted to half the depth of the orifice. \nSimultaneously, the velocity of the underside of the piezoelectric disk was measured using a Polytec PDV 100 laser vibrometer. \nUsing these two instruments the performance of the actuator was mapped for a range of driving voltages and frequencies. \n\n\\begin{equation} \n\\label{eq:Uo}\nU_o = \\frac{1}{T} \\int_{0}^{T\/2} u_o (t) dt\n\\end{equation}\n\nThe average blowing velocity, $U_o$, of a synthetic jet is obtained by integrating the instantaneous velocity of the jet at the orifice throat $u_o(t)$ over the blowing phase of the cycle and averaged over the period of the full cycle, $T$ (\\ref{eq:Uo}). \nThe average outstroke velocity of the piezoelectric disk, $U_d$, can be computed by applying the same calculation to the instantaneous velocity of the disk centre point. \nA linear relationship was observed between the average blowing velocity and the average disk outstroke velocity and this relationship remained consistent even after disassembly and reassembly of the actuator. \nTherefore, during cross-flow testing the synthetic jet performance was set by matching the average disk outstroke velocity which corresponded to the desired average blowing velocity. \n\nMeasuring the average blowing velocity of the pitched $\\alpha=45^\\circ$ orifice jet was not attempted due to the geometric challenges of inserting the hot-wire probe into the pitched orifice and the fact that the jet velocity profile no longer had a top-hat shape. \nInstead, the volumetric flux out of the wall-normal and pitched orifices was matched by selecting actuator driving voltages which achieved the same average disk outstroke velocities. \nTherefore, the average blowing velocities reported for the pitched orifice cases are actually the values measured in the wall-normal orifice for the same volume of ejected fluid and driving frequency.\nDisassembly and reassembly of the actuator to install different orifice plates produced small variations in the actuator performance, as did the interaction of the jet with different speed cross-flows. \nTo address these sources of variability the actuator was calibrated in situ before every day of testing using the laser vibrometer. \nTo enable these measurements, the floor of the wind tunnel and underside of the BL plate were fabricated from clear cast acrylic. \nThe average disk outstroke velocity was mapped for a range of driving voltages at all planned cross-flow speeds and the driving voltage necessary to achieve the desired test points was then interpolated from the calibration data.\n\n\\subsection{SPIV measurements}\n\\label{sec:SPIV}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/experimental_setup_filled_b.eps}\n\\caption{Schematic of BL plate and SPIV measurement system.}\n\\label{fig:SPIVsetup}\n\\end{figure}\n\nStereoscopic particle image velocimetry (SPIV) was used to measure the three-component velocity field in the SJBLI experiments. \nThe LaVision SPIV system was comprised of two 2560$\\times$2160 pixel 16-bit dynamic range scientific CMOS cameras and a Quantel Evergreen 200 dual-pulsed 532~nm Nd:YAG laser. \nThe laser was mounted above the test section on two Velmex linear traverses in a $y-z$ configuration. \nA series of optical elements were used to turn and focus the laser beam generating a thin sheet in the $x-y$ plane (figure~\\ref{fig:SPIVsetup}).\n\nThe two SPIV cameras were mounted on one side of the test section with a three-axis ($x-y-z$) Velmex linear traverse system. \nOne camera was positioned upstream of the actuator looking downstream towards the jet while the other was positioned downstream looking upstream, such that their orientations mirrored each other about the $x\/h=40$ plane (figure~\\ref{fig:SPIVsetup}). \nThe $67^\\circ$ of separation between the viewing angles of the two cameras enabled all three velocity components to be computed during processing of the raw images. \nIdentical Nikon 105~mm 2.8D MicroNikkor lenses set to an aperture of f\/8.0 were used on both cameras. \nThe lenses were attached to the camera bodies by LaVision Scheimpflug mounts that allowed the full measurement plane to be brought into focus despite the off-axis arrangement of the cameras. \nThe resulting image scale factor was 24.5 pixels~mm$^{-1}$.\n\nTwo types of SPIV measurements were obtained within the SJBLI domain: time-averaged and phase-locked data. \nFor time-averaged measurements 500 instantaneous velocity fields were captured at a frequency which temporally aliased the actuation frequency. \nThe result of averaging these 500 instantaneous measurements was the approximate mean jet flow field. \nAlternatively, in phase-locked measurements 250 instantaneous velocity fields were captured at a fixed phase angle of the actuation cycle. \nAveraging these measurements revealed the influence of the periodic actuation and the coherent structures at a particular instance in the actuation cycle. \nOn the centreline of the jet eight evenly distributed actuator phase angles were obtained while off-centreline a subset of four actuator phases were measured.\n\nProcessing of the raw flow field images to obtain the 2D three-component vector field was conducted using the multi-pass stereoscopic cross-correlation algorithm in LaVision's DaVis 8.4.0 software. \nThe first pass was run with 64$\\times$64 pixel interrogation windows followed by two passes with 32$\\times$32 pixel integration windows. \nA Gaussian weighting function was applied to the interrogation windows in the final pass and a 75\\% overlap was used between the windows. \nThe results after processing and masking was a 366$\\times$253 vector velocity field with a vector resolution of 3.1 vectors per mm. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/exp_diagram_data_planes2_b.eps}\n\\caption{SPIV measurement plane locations relative to the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ orifice where solid green lines represent time-averaged data and dashed blue lines represent phase-locked data.}\n\\label{fig:SPIVplanes}\n\\end{figure}\n\nThe measured vector field after image alignment and cropping extended roughly $x=-19$ to 100~mm and $y=1$ to 83~mm, with the origin at the orifice centre. \nFor off-centreline measurements the laser and cameras were traversed the same distance in the spanwise ($z$) direction such that the laser sheet and lens focal planes remained coplanar. \nFigure~\\ref{fig:SPIVplanes} displays an example distribution of the time-average and phase-locked SPIV data planes in the jet interaction domain. \nThe time-averaged data was obtained on planes spaced every 1~mm in the $z$-direction from $z=-$12~mm to 12~mm. \nConversely, the phase-locked data was obtained every $\\Delta z$ = 0.5~mm over a reduced span of the interaction domain to increase the data resolution within the coherent structures. \n\nIn the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets phase-locked data was obtained in one half of the interaction domain spanning from centreline out to a distance of $z=$11.5~mm. \nA subset of three more widely spaced planes were obtained on the opposite side of centreline from the high resolution data for the purposes of confirming the location of the jet plane of symmetry. \nTo reconstruct the 3D flow field in these cases the data obtained in one half of the jet was mirrored onto the other side of the jet symmetry plane. \nIn the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet phase-locked data was obtained between $z=-$6~mm and 6~mm, therefore, no data mirroring was required for the 3D flow field reconstruction in this case.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{c c c c}\n& \\multicolumn{1}{c}{Baseline} & \\multicolumn{2}{c}{Actuation on} \\\\  [1.5ex]\n\\cline{2-4} \\\\  [-0.5ex]\n& Time-averaged & Time-averaged & Phase-locked \\\\\n\\hline\n$\\sigma_u^{rms}\/U_e$ & 0.19--0.28\\% & 0.26--0.45\\% & 0.35--0.58\\% \\\\ [1ex]\n$\\sigma_v^{rms}\/U_e$ & 0.09--0.15\\% & 0.14--0.48\\% & 0.18--0.55\\% \\\\ [1ex]\n$\\sigma_w^{rms}\/U_e$ & 0.13--0.21\\% & 0.18--0.39\\% & 0.25--0.60\\% \\\\ [1ex]\n\\end{tabular}\n\\caption{Uncertainty in SPIV measurements within the interaction domain for all cases and conditions.}\n\\label{tab:uncert}\n\\end{table}\n\nUncertainty in the SPIV velocity measurements was quantified using the correlation statistics method implemented within DaVis~\\citep{Wieneke_MeasSciTech_2015}. \nFor reporting purposes the root-mean-square (RMS) of the uncertainty field within the SJBLI domain was calculated. \nThe edge of this domain, $\\delta_{99}(x)$, was defined as the height above the wall where the streamwise flow speed first dropped below $0.99U_e$ when moving from the upper edge of the measurement domain towards the wall, where $U_e$ is the velocity of the flow at the edge of the viscous BL. \nRestricting the RMS calculation to the interaction domain eliminated the outer flow velocity data where uncertainty was particularly low. \nThe RMS values of the measurement uncertainty in the three-component directions are reported in table~\\ref{tab:uncert} as percentages of $U_e$. \nThe ranges represent the minimum and maximum RMS values observed in all of the averaged velocity fields obtained in the unactuated and actuated cases, inclusive of all measurement planes, phase angles, test conditions, and actuator configurations.\n\nA number of steps were taken to mitigate laser reflections in the SPIV images. \nThese included framing the SPIV images to cut-off the primary reflection on the BL plate and polishing build lines out of the SLA parts. \nAn acrylic paint mixed with Rhodamine 6G powder, which absorbs 532~nm light and re-emits it at a lower wavelength, was also applied to the SLA parts. \nThis coupled with band pass filters installed on the camera lenses helped reduce the intensity of laser reflections in the images. \nTwo smaller reflections off of the inner edge of the ring retaining the SLA part were addressed in cases where they appeared in the 3D flow field reconstructions by either reflecting valid data from the opposite side of centreline or replacing bad vectors via interpolation. \n\n\\subsection{Vortex identification and visualization}\n\\label{sec:vortexid}\n\nTwo methods were used to identify and visualize vortices in the SPIV data. \nTo identify vortices in 2D planar data on the jet centreline the $\\mathcal{Q}$\u2013criterion was used, where $\\mathcal{Q}$ is the second invariant of the velocity gradient tensor~\\citep{Hunt_Wray_CTR_1988}. \nUnder this criterion a vortex is defined as a region of the flow where rotation dominates strain. \nWithin a region of the $\\mathcal{Q}$-field identified as containing a vortex, the vortex centre was defined as the location of the extremum in the vorticity field, obtained by linear interpolation of the discrete flow field data.\n\nThe swirling strength criterion is used to visualize vortex structures in phase-locked 3D three-component velocity fields~\\citep{Zhou_Adrian_JFM_1999}. \nThis criterion examines the magnitude of the imaginary part of the velocity gradient tensor eigenvalues, denoted $\\lambda_{ci}$ in (\\ref{eq:lambdaci}). \nWhen $\\lambda_{ci}$ is greater than zero swirling motion is present in the flow. \nVortices may therefore be visualized by plotting isosurfaces at a chosen threshold of $\\lambda_{ci}$. \nThe particular thresholds used to generate the following figures were selected to highlight the structures of interest.\n\n\\begin{equation} \\label{eq:lambdaci}\n\\mathrm{\\bm{\\nabla}} \\bm{u} = [\\bm{\\nu_r} \\; \\bm{\\nu_{cr}} \\; \\bm{\\nu_{ci}}]\\begin{bmatrix} \\lambda_r&0&0\\\\\n0&\\lambda_{cr}&\\lambda_{ci}\\\\\n0&-\\lambda_{ci}&\\lambda_{cr}\\end{bmatrix}[\\bm{\\nu_r} \\; \\bm{\\nu_{cr}} \\; \\bm{\\nu_{ci}}]^T\n\\end{equation}\n\nDifferentiation of the velocity field tends to amplify the noise within the SPIV data. \nTo mitigate this effect the velocity gradient tensor used to compute $\\lambda_{ci}$ was obtained using a non-local differentiation technique.\nIn this method the spatial derivate at a point is computed using the adjacent velocity field within a spherical stencil 1~mm in radius.\nAdditional details on this method are provided in \\citet{Straccia_Farnsworth_JFM_2020}.\n\n\\subsection{Cross-flow conditions}\n\\label{sec:crossflowcond}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in]{figures\/HW_profile_U12_x0_c.eps}\n\\caption{Baseline turbulent BL profile at $x\/h=0$, $z\/h=0$ for $U_e$ = 12~ms$^{-1}$ (Hot-wire profile data, $\\bullet$; Composite profile fit, \\sampleline{}).}\n\\label{fig:BLprof}\n\\end{figure}\n\nThe baseline (no actuation) flow conditions and BL shape was characterized using a Dantec single element BL hot-wire probe which was traversed in the wall-normal direction. \nData was obtained at three streamwise locations ($x=-$0.1, 0, 0.1~m) for all cross-flow speeds. \nTo aid in assessing whether the measured BL profiles were consistent with a fully-developed zero pressure gradient turbulent BL the data was fit with the composite profile validated by \\citet{Chauhan_Monkewitz_FluidDynRes_2009}. \nThe fitting process also yielded estimates of the full BL thickness $\\delta$ (i.e., the point on the edge of the BL where $u(y)=U_e$) and confirmed that wall-normal distances ($y$) were correctly zeroed at the wall~\\citep{RodriguezLopez_Bruce_ExpFluids_2015}. \nFigure~\\ref{fig:BLprof} presents an example fit of the $U_e$ = 12~ms$^{-1}$ BL at $x\/h=0$. \nThe remarkably good match between the data and the composite fit confirm that a canonical fully developed turbulent BL has been achieved. \nThe fitting results were equally good at $U_e$ = 24~ms$^{-1}$. \n\n\\begin{table}\n\\setlength{\\tabcolsep}{6pt}\n\\caption{Conditions of the baseline cross-flow and boundary layer.}\n\\centering\n\\begin{tabular}{ccc}\n\\hline \\hline\n& $U_e$ = 12 ms$^{-1}$ & $U_e$ = 24 ms$^{-1}$  \\\\\n\\hline \\\\ [-2.0ex]\n$\\delta$ (mm) & 29.5 & 35.5 \\\\\n$H$ & 1.400 & 1.358 \\\\\n$Re_\\tau$ & 837 & 1810 \\\\\n$Re_\\theta$ & 1780 & 4634 \\\\\n$\\beta_{PG}$ & -0.021 & -0.030 \\\\\n$I_x$ & 0.27\\% & 0.40\\% \\\\\n\\hline \\hline\n\\end{tabular}\n\\label{tab:crossflowcond}\n\\end{table}\n\nThe conditions of the baseline BL and outer flow above the orifice ($x$ = 0~mm) are summarized in table~\\ref{tab:crossflowcond} for the two cross-flow speeds tested. \nThe BL thickness values, $\\delta$, were obtained as an output of the composite profile fitting process described earlier. \nNote that the BL at $U_e$ = 24~ms$^{-1}$ was slightly thicker than at $U_e$ = 12~ms$^{-1}$ because at $U_e$ = 24~ms$^{-1}$ the flow transitioned sooner and thus the BL grew at a faster rate for a longer streamwise distance. \nThe shape of the BL is characterized using the shape factor, $H$, which is defined as the ratio between the displacement thickness $\\delta^*$ and the momentum thickness $\\theta$ (\\ref{eq:H}).The shape factor values measured are consistent with a fully developed zero pressure gradient turbulent BL~\\citep{Schlichting_2017}. \nThe Reynolds number of the BL based on the friction velocity ($u_\\tau$), defined in \\ref{eq:Retau}, and the edge velocity ($U_e$), defined in \\ref{eq:Retheta}, are also reported in table~\\ref{tab:crossflowcond}. \nFinally, the freestream streamwise turbulence intensity ($I_x=\\langle u'u' \\rangle U_e^{-2}$), calculated from hot-wire anemometry data taken outside the BL ($y>2\\delta$), is included.\n\n\\begin{equation} \n\\label{eq:H}\nH = \\frac{\\delta^*}{\\theta} \n\\end{equation}\n\n\\begin{equation} \n\\label{eq:Retau}\nRe_\\tau = \\frac{u_\\tau \\delta}{\\nu} \n\\end{equation}\n\n\\begin{equation} \n\\label{eq:Retheta}\nRe_\\theta = \\frac{U_e \\theta}{\\nu} \n\\end{equation}\n\nThe pressure gradient in the wind tunnel test section was quantified using an array of twelve pressure taps running along the centreline of the BL plate. \nThe Clauser pressure gradient parameter ($\\beta_{PG}$) values computed using these data are listed in table~\\ref{tab:crossflowcond}~\\citep{Clauser_JAeroSci_1954}. \nThe values indicate that the favorable pressure gradient that was present was very mild, which was expected given the excellent match between the BL profile measurements and the canonical zero pressure gradient turbulent BL profile from \\citet{Chauhan_Monkewitz_FluidDynRes_2009}. \n\n\\subsection{Synthetic jet-boundary layer interaction conditions}\n\\label{sec:SJBLIcond}\n\nSeveral parameters are utilized in this paper to describe the interaction between the periodic jet and the cross-flow. \nThe relative strength of the jet was quantified using blowing ratio, $C_b$, which relates the average blowing velocity of the jet to the velocity of the cross-flow (\\ref{eq:Cb}). \nLow values of blowing ratio ($C_b<1$) describe cross-flow-dominated interactions while high values ($C_b>1$) correspond to jet-dominated flows. \nA common alternative to blowing ratio is momentum coefficient, $C_\\mu$, which is the ratio of jet momentum to the cross-flow momentum (\\ref{eq:Cmu}). \nAlthough the experiments were designed based on blowing ratio, momentum coefficient is also reported for convenience. \nTo characterize the effect of actuation frequency the Strouhal number, $St$, which relates the actuation frequency to the freestream convective frequency, is also reported (\\ref{eq:St}). \nAs $St$ is increased coherent structures become more tightly spaced in the streamwise direction and the mutual interaction between coherent structures from different actuator cycles becomes stronger.\n\n\\begin{equation} \n\\label{eq:Cb}\nC_b = \\frac{U_o}{U_e} \n\\end{equation}\n\n\\begin{equation} \n\\label{eq:Cmu}\nC_{\\mu} = \\frac{U^2_o h}{U^2_e \\delta} \n\\end{equation}\n\n\\begin{equation} \n\\label{eq:St}\nSt = \\frac{f \\delta}{U_o} \n\\end{equation}\n\n\\begin{table}\n\\setlength{\\tabcolsep}{6pt}\n\\caption{Synthetic jet-boundary layer interaction conditions tested.}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline \\hline\nCondition & $U_o$ (ms$^{-1})$ & $f$ (Hz) & $U_e$ (ms$^{-1}$) & $C_b$ & $C_\\mu$ & $St$ \\\\\n\\hline \\\\ [-2.0ex]\n1 & 12 & 540 & 12 & 1.0 & 0.0339 & 1.33\\\\\n2 & 6 & 540 & 12 & 0.5 & 0.0085 & 1.33\\\\\n3 & 12 & 540 & 24 & 0.5 & 0.0070 & 0.80\\\\\n\\hline \\hline\n\\end{tabular}\n\\label{tab:caseconds}\n\\end{table}\n\nThe three SJBLI conditions investigated in this paper are summarized in table~\\ref{tab:caseconds}. \nAt the $C_b=1$ condition SPIV data was obtained on an array of parallel planes throughout the interaction domain such that the 3D flow field could be reconstructed. \nFor the other two conditions where $C_b=0.5$ only centreline data are presented. \nThe actuation frequency was held constant for all three conditions such that the Strouhal number at $U_e$ = 24~ms$^{-1}$ is roughly half that at $U_e$ = 12~ms$^{-1}$. \nTherefore, the three conditions tested allow both the influence of blowing ratio (or momentum coefficient) and Strouhal number to be investigated independently.\n\\section{Introduction}\n\\label{sec:Intro}\n\nThe interaction of synthetic jets with wall-bounded cross-flows is a fascinating dynamic phenomenon in which both the jet and the boundary layer (BL) may be substantially modified. \nThe physical details of this interaction are an active area of research, especially with regard to the behaviour of the coherent structures. \nSuch studies are valuable not only due to their scientific merit, but also because they provide the critical engineering knowledge needed to more effectively apply synthetic jet-based flow control to practical problems. \nIn addition to exploring in depth the flow physics of synthetic jet\u2013boundary layer interactions (SJBLI), this paper will also consider how changes to the jet may influence the effectiveness of the actuation in controlling smooth wall flow separation.\n\nSynthetic jet actuators (SJA) are a type of active flow control device which adds momentum to a flow field without a net addition of mass. \nTo generate this zero-net-mass-flux jet the driver draws external fluid into a cavity during the suction phase and then expels that fluid back out of the orifice during the blowing phase. \nThe periodic expulsion of fluid by the actuator leads to the rapid formation of vortex rings which travel away from the orifice carrying the momentum introduced by the actuator outstroke~\\citep{Xia_Mohseni_PoF_2015}. \nA vortex ring is formed with every cycle of the actuator, sometimes at a rate of thousands per second, resulting in a train of vortex rings which constitute the early jet~\\citep{Smith_Glezer_PoF_1998}.\n\n\\subsection{Vortex dynamics in synthetic jets}\n\\label{sec:vortdynSJ}\n\nVortex dynamics plays an important role in establishing the characteristics of synthetic jets. \nNot only are the vortex rings the vehicles by which momentum is transported in the synthetic jet nearfield, but they also influence the jet's entrainment rate, shape, and momentum distribution. \nWhen the performance of equivalent steady and synthetic jets is compared, synthetic jets demonstrate a markedly higher entrainment rate, especially near the orifice~\\citep{Smith_Glezer_PoF_1998, Cater_Soria_JFM_2002, Smith_Swift_2003}. \nThis is attributed to the formation of the vortex rings which engulf surrounding fluid as the vortex sheet generated by the expelled fluid slug rolls up. \nVortex ring-driven entrainment is further enhanced when the jet is non-circular.\n\nVortex rings induce a velocity in the fluid around them, as described by the Biot-Savart law. \nThis effect extends to the fluid within the vortex core itself leading to what is known as a self-induced velocity. \nThe speed of the local self-induced velocity varies with the curvature of the vortex ring axis; specifically, high curvature regions experience higher local self-induction than low or zero curvature zones~\\citep{Arms_Hama_PoF_1965}. \nThus, non-circular vortex rings, which by definition have variable curvature along their axes, deform as they propagate. \nIn vortex rings with symmetric shapes these deformations lead to oscillatory changes in the ring's width known as axis-switching~\\citep{Viets_Sforza_PoF_1972, Dhanak_Bernardinis_JFM_1981}. \nThe oscillations in the vortex ring width are mirrored in the jet velocity field and can be quite significant in larger aspect ratio ($AR$: the ratio of orifice length to width) jets~\\citep{Van_Buren_Whalen_JFM_2014, Straccia_Farnsworth_PRF_2021}. \nThis growth and contraction of the vortex rings in the transverse direction is responsible for enhancing mixing between the jet and the surrounding fluid~\\citep{Ho_Gutmark_JFM_1987, Wang_Feng_ExpFluids_2018}.\n\nWhen a slug of fluid ejected from an orifice or nozzle issues into a cross-flow the resulting vortex structure may deviate significantly from the classic vortex ring shape. \nOne effect of the cross-flow is to weaken the upstream side of the cylindrical vortex sheet formed by the ejection of the fluid slug~\\citep{Lim_Lua_2008}. \n\\citet{Sau_Mahesh_2008} observed this type of modification to the vortex rings formed by a circular nozzle in a laminar BL in their direct numerical simulations. \nAt high blowing ratios (i.e., ratio of average nozzle exit velocity to the free-stream cross-flow velocity) roughly circular vortex rings formed, but as blowing ratio was reduced, no vortex rolled up on the upstream side of the nozzle, resulting instead in the formation of a hairpin vortex. \n\nTo understand the shift from symmetric vortex rings in quiescent conditions to hairpin vortices in a cross-flow, it is helpful to think about the interaction from the perspective of the vorticity field. \nThe vorticity generated on the upstream side of the jet is of the opposite sign to the vorticity in the viscous BL, and when fluids containing anti-parallel vorticity come into close proximity, segments of the counter rotating vortex lines are annihilated in the contact zone by viscous cross-diffusion~\\citep{Melander_Hussain_CTR_1988}. \nMeanwhile, at the edge of the contact zone the remaining sections of the vortex lines crosslink to form a new vortex topology. \nThe annihilation of vorticity in one part of a vortex causes the circulation in that segment to drop relative to the rest of the structure \\citep{Oshima_Izutsu_FDR_1988, Straccia_Farnsworth_JFM_2020}. \nSimultaneously, the downstream vortex may be strengthened by the cross-flow interaction because the ambient fluid entrained during the vortex roll-up carries vorticity of the same sign~\\citep{Cheng_Lou_PoF_2009}.\n\nThe difference in circulation strength between the upstream and downstream sides of a vortex ring formed in a cross-flow may also be explained from the perspective of the velocity field. \nWhen the cross-flow collides with the jet the pressure field at the orifice exit is modified such that fluid leaves the upstream edge at a slower speed, reducing vorticity production, and exits the downstream edge are a higher speed, increasing vorticity production \\citep{Sau_Mahesh_2008}. \nThese two explanations are both manifestations of the same effect, merely viewed from the perspective of the vorticity field versus the velocity and pressure fields.\n\nThis variability in the strength of the original vortex tube is made possible by branches in the tube that contain the `missing' circulation~\\citep{Melander_Hussain_CTR_1988, Kida_Takaoka_JFM_1991}. \nFor vortex rings formed in BLs, these branches look like `legs' which trail behind the main vortex and lead back towards the wall~\\citep{Jabbal_Zhong_2010}. \nThe legs are composed of BL vorticity which crosslinked to vorticity within the vortex during reconnection. \nAs the circulation strength of the upstream side of the vortex is reduced by vorticity reconnection the strength of the vortex tube branches must increase by an equivalent amount \\citep{Straccia_Farnsworth_JFM_2020}. \nFor very low blowing ratios, where no detectable upstream side of the vortex ring forms, the vortex tube branches become the hairpin legs and the unannihilated downstream portion of the vortex ring becomes the hairpin head.\n\n\\citet{Jabbal_Zhong_2010} employed dye-visualization to study how the vortex structures formed by a circular orifice synthetic jet issuing into a laminar BL changed with blowing ratio. \nAt low blowing ratios a train of hairpin vortices formed which remained relatively close to the wall. \nAs blowing ratio increased the structures took the form of stretched vortex rings with trailing legs. \nFinally, at the highest blowing ratios tilted vortex rings formed which penetrated deeper into the cross-flow than in the other two scenarios. \nThe vortex rings which formed at the higher blowing ratios rotated in the counter-clockwise (CCW) direction (cross-flow left-to-right) such that their direction of propagation was tilted into the oncoming flow. \nThe vortex ring rotation was driven by the variable propagation speed around the ring resulting from uneven circulation strength in the upstream versus the downstream side of the ring~\\citep{Lim_Lua_2008, Cheng_Lou_PoF_2009}.\n\nThe dynamics of vortex rings generated by rectangular orifice synthetic jets in a cross-flow differ in a few ways from those of circular jets. \nThe cross-flow interaction still weakens the upstream side of the vortex ring; however, for moderate or high $AR$ orifices which are oriented with their long axis aligned with the spanwise direction (i.e., spanwise-oriented orifices) the vortex pair on centreline tilts clockwise (CW) instead of CCW (flow left-to-right)~\\citep{Sahni_Wood_JFM_2011, Wang_Feng_JFM_2020, Elimelech_Vasile_2011}. \nThe direction of rotation of the vortex pair on centreline is CW because motion is dominated by velocity induction from the adjacent counter-rotating side of the vortex ring instead of by local self-induction. \nUnder the right conditions the upstream side of the vortex ring may even be rolled up and over the stronger downstream vortex segment and deposited near the wall~\\citep{Sahni_Wood_JFM_2011}. \nIf the upstream side of the vortex sheet is weakened sufficiently by the SJBLI, hairpin-like vortex structures may form in spanwise-oriented rectangular orifice synthetic jets~\\citep{Belanger_Zingg_AIAA_2020}. \nUnlike the hairpin vortices from circular orifices, however, these form after a rapid spanwise contraction of the initial vortex structure due to self-induction and the influence of the wall (image vortex effect). \nConversely, if conditions produce a sufficiently intact vortex ring a form of axis-switching occurs as the non-circular vortex rings propagate into the cross-flow~\\citep{Wang_Feng_JFM_2020}. \n\nIn both circular and rectangular orifice synthetic jets, increasing the blowing ratio increases the self-induced propagation speed of the vortex relative to the cross-flow speed and leads to an overall more complete vortex ring structure. \nThus, the higher the blowing ratio, the deeper the synthetic jet tends to penetrate into the cross-flow~\\citep{Berk_Hutchins_JFM_2018, Jabbal_Zhong_2010, Sahni_Wood_JFM_2011, Belanger_Zingg_AIAA_2020}. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/SJA_side_N_top_view_sketch.eps}\n\\caption{Side view cross-section and top view of SJA installed in BL plate.}\n\\label{fig:sjascheme}\n\\end{figure}\n\nDue to the lack of axisymmetry in rectangular orifices the orientation of the orifice major axis relative to the spanwise direction (i.e., the skew angle, $\\beta$) also influences the resulting vortex dynamics (figure~\\ref{fig:sjascheme}). \nAs a rectangular orifice's skew angle is modified the side of the vortex ring which is diminished as a result of the SJBLI changes, as does the location of the vortex legs~\\citep{Van_Buren_Leong_PoF_2016, Wang_Feng_JFM_2020}. \nJets from spanwise-oriented orifices are strongly influenced by the cross-flow because it is the long sides of the vortex ring which are modified. \nConversely, jets from streamwise-oriented orifices (i.e., orifices with the major axis aligned with the cross-flow direction) present a much smaller frontal area to the cross-flow so, in this case, the vortex rings are minimally modified by the SJBLI. \nThus vortex rings formed in the streamwise-orientation lift off more rapidly from the wall and penetrate deeper into the cross-flow than those formed in a spanwise orientation~\\citep{Van_Buren_Leong_PoF_2016, Smith_2002, Wang_Feng_JFM_2020}. \n\nThe interaction of a cross-flow with a synthetic jet issuing normal to the local surface produces a low momentum wake region in the lee of the jet. \n\\citet{Berk_Ganapath_JFM_2019} investigated three potential contributors to the formation of a wake: (1) momentum lost by the cross-flow in accelerating the jet, (2) upwash of low momentum fluid off the wall by the induction from the vortex ring on the jet centreline, and (3) velocity induction from the vortex ring which may be oriented counter to the cross-flow direction. \nBased on their experiments they concluded that, to varying degrees, all three factors contributed to the formation of the wake in different regions of the interaction domain. \nWhen blowing ratio is increased, the wake moves further from the wall and has a larger peak velocity deficit~\\citep{Berk_Hutchins_JFM_2018} which is due to the formation of stronger vortex rings that penetrated deeper into the cross-flow and induce higher velocities in the fluid around them.\n\nAlthough the mixing introduced by wall-normal synthetic jets contributes to the formation of a low-momentum wake it also has a salutary effect on the near-wall BL. \nSpecifically, the vortex rings or hairpin vortices induce a downward velocity around the outside of these structures, entraining high momentum fluid into the lower BL. \nThe result is a thinned BL surrounding the footprint of the jet and accelerated flow near the wall~\\citep{Smith_2002, Jabbal_Zhong_2010, Van_Buren_Leong_PoF_2016}. \nTherefore, wall-normal synthetic jets can be effective at enhancing the wall shear stress in a cross-flow despite reducing fluid momentum along the jet centreline further from the wall~\\citep{Jabbal_Zhong_2010}.\n\nThe SJBLI is not the only factor that influences the structure of the synthetic jet vortex rings. \nThe geometry of a nozzle or orifice can also lead to variable circulation and a branched topology of the vortices. \nFor example, this occurs when the flow channel through which the fluid is ejected (i.e., the orifice or throat of the jet) is pitched or angled away from the wall-normal direction, i.e., pitch angle $\\alpha \\neq 90^\\circ$ (figure~\\ref{fig:sjascheme}). \nPitched orifices produce vortex rings which have stronger circulation at the acute edge of the orifice than at the obtuse edge~\\citep{Zhong_Garcillan_AIAA_2004, Yehoshua_Seifert_2006, Li_Sahni_2014}. \nStudies involving piston-driven vortex ring formation from inclined nozzles found that a difference in the magnitude of the pressure gradient on the short and long sides of the flow channel leads to non-uniform vorticity production at the nozzle exit~\\citep{Webster_Longmire_1998,Le_Borazjani_2011}. \nAn alternate or potentially complementary explanation relates to the theory that orifices with pitched geometries tend to induce separation in the flow on one wall of the orifice channel resulting in a skewed velocity profile at the orifice exit~\\citep[p.57-58]{Zhong_Garcillan_AIAA_2004, Bray_Thesis_1998}.\n\nWhen an orifice is pitched in the downstream direction with respect to the cross-flow, the two modifying effects on the vortex circulation strength discussed above are in competition. \nOn one hand, the SJBLI will cause vortex rings to form with lower circulation strength on the upstream side of the ring than on the downstream section, due to the rotational sign of the vorticity in the BL. \nOn the other hand, pitching the orifice downstream has the opposite effect, which is the enhancement of the circulation strength of the upstream or acute edge of the orifice. \nTherefore, a balance exists which is governed by the orifice pitch angle and the relative characteristics of the cross-flow (i.e., blowing ratio and BL profile). \n\\citet{Yehoshua_Seifert_2006} experimentally observed that the circulation asymmetry in the vortex pair produced by a pitched high $AR$ orifice was reduced as the cross-flow velocity was increased (i.e., blowing ratio was reduced). \n\\citet{Li_Sahni_2014} found in their simulations of an infinite slot that under the right conditions the two effects essentially cancel each other out and a roughly symmetric vortex pair forms. \n\nThe structure and behaviour of the vortices in a synthetic jet are also influenced by the frequency at which they are formed by the actuator.\nAs actuation frequency is increased the circulation strength per vortex decreases~\\citep{Straccia_Farnsworth_JFM_2020}; therefore, the trajectories of synthetic jets formed at higher frequencies generally remain closer to the wall than those of jets formed at lower frequencies~\\citep{Berk_Hutchins_JFM_2018, Berk_Ganapath_JFM_2019}. \nBut while the circulation strength of an individual vortex decreases with increasing actuation frequency, the time-averaged (mean) circulation produced by a jet remains constant. \n\\citet{Berk_Ganapath_JFM_2019} argued that this explained the minimal change in the jet wake momentum deficit when they varied the actuation frequency. \nIncreasing actuation frequency also reduces the spacing between adjacent vortex structures within a BL. \nWhen the spacing is small enough, adjacent vortices will interact and may merge. \nThus the flow field can shift from one characterized by a train of discrete vortex rings or hairpin vortices into one of persistent streamwise vortices in the domain downstream of the actuator~\\citep{Sahni_Wood_JFM_2011, Van_Buren_Leong_PoF_2016, Belanger_Zingg_AIAA_2020}.\n\n\\subsection{Context for the present study}\n\\label{sec:sumpriorwork}\n\nThe combination of periodic suction and blowing, orifice skew, orifice pitch, and a wall-bounded cross-flow leads to complex vortex dynamics within the SJBLI zone and results in a range of vortex topologies. \nInvestigating the details of these varied and multifaceted flow fields is one objective of the present study. \nTo put the present work into context, table~\\ref{tab:priorlit} provides a partial list of prior studies which sought to characterize the SJBLI on a flat plate.\nThe table includes information on (1) the BL type (laminar or turbulent), (2) the orifice shape (circular, slot or rectangular) and $AR$, (3) whether three-dimensional, three-component (3D3C) velocity field data are presented, (4) whether instantaneous or phase-locked flow field representations are presented (as opposed to the mean flow field), and (5) the type of measurements including flow visualization (FV), hot-wire anemometry (HW), particle image velocimetry (PIV) or computational fluid dynamics (CFD). \nA tilde symbol in the `yes' or `no' columns indicates that a particular category is only partially met or that the data relevant to that category are limited.\n\nAlthough several dozen prior studies have sought better understanding of fundamental vortex dynamics in synthetic jets issuing into zero or near-zero pressure gradient cross-flows the configurations and conditions which may be most relevant to real-world flow control applications are less well studied (table~\\ref{tab:priorlit}). \nFor example, many of the prior vortex dynamics studies have focused on circular orifices. \nBy contrast, applied research studies investigating the use of synthetic jets to solve flow control problems (e.g., separation control) frequently employ rectangular orifices with an $AR$ of 18 or larger. \nBut the behaviour of vortices from circular versus finite span orifices can be quite different, as discussed earlier. \n\n\\begin{table}\n\\centering\n\\begin{tabular}{c c c c c c c}\nArticle & BL Type & Orifice Shape; $AR$ & 3D3C & Instant. Data & Measurement \\\\ [0.5ex]\n\\hline\n\\citet{Zhong_Garcillan_AeroJ_2005} & Lam & Circ; 1 & No & Yes & FV \\\\\n\\citet{Shuster_Pink_2005} & Lam & Circ; 1 & No & Yes & PIV \\\\\n\\citet{Jabbal_Zhong_2008} & Lam & Circ; 1 & No & Yes & FV \\\\\n\\citet{Jabbal_Zhong_2010} & Lam & Circ; 1 & No & Yes & FV, PIV \\\\\n\\citet{Xia_Mohseni_AIAAJ_2017} & Lam & Circ; 1 & No & $\\sim$ & HW, PIV \\\\\n\\citet{Zhou_Zhong_CompFluids_2009} & Lam & Circ; 1 & Yes & Yes & CFD \\\\\n\\citet{Mittal_Rampunggoon_AIAA_2001} & Lam & Slot; $\\infty$ & No & Yes & CFD \\\\\n\\citet{Li_Sahni_2014} & Lam & Slot; $\\infty$ & $\\sim$ & Yes & CFD \\\\\n\\citet{Yehoshua_Seifert_2006} & Lam & Rect; 135 & No & Yes & FV, HW, PIV \\\\\n\\citet{Van_Buren_Beyar_PoF_2016} & Lam & Rect; 6, 12, 18 & Yes & $\\sim$ & SPIV \\\\\n\\citet{Van_Buren_Leong_PoF_2016} & Lam & Rect; 18 & Yes & $\\sim$ & SPIV \\\\\n\\citet{Wang_Feng_JFM_2020} & Lam & Rect; 3 & Yes & Yes & Tomo PIV \\\\\n\\citet{Milanovic_Zaman_AIAAJ_2005} & Turb & Circ; 1 & No & No & HW \\\\\n\\citet{Garcillan_Liddle_AIAA_2006} & Turb & Circ; 1 & No & Yes & PIV \\\\\n\\citet{Ramasamy_Wilson_JAircraft_2010} & Turb & Circ; 1 & No & No & HW, PIV \\\\\n\\citet{Cui_Agarwal_AIAA_2003} & Turb & Slot; $\\infty$ & No & Yes & CFD \\\\\n\\citet{Smith_2002} & Turb & Rect; 45 & No & No & HW \\\\\n\\citet{Bridges_Smith_AIAAJ_2003} & Turb & Rect; 98 & No & No & HW \\\\\n\\citet{Milanovic_Zaman_2004} & Turb & Rect; 4, 6, 18 & No & No & HW \\\\\n\\citet{Berk_Ganapath_JFM_2019} & Turb & Rect; 13 & No & Yes & SPIV \\\\\n\\citet{Housley_PhDThesis_2020} & Turb & Rect; 18 & Yes & Yes & SPIV \\\\\n\\citet{Belanger_Zingg_AIAA_2020} & Turb & Rect; 13 & Yes & Yes & CFD \\\\ [1ex]\n\\end{tabular}\n\\caption{Summary of prior flat plate SJBLI studies.}\n\\label{tab:priorlit}\n\\end{table}\n\nMany SJBLI studies are also conducted in laminar BLs. \nHowever, separation control problems in engineering flows often involve high Reynolds numbers and turbulent BLs. \nThis distinction can be important because the distribution of vorticity in a BL may influence vorticity annihilation and the types of structures resulting from SJBLIs. \nFurthermore, vortices embedded in turbulent BLs distort the Reynolds stress field and, in turn, gradients in the Reynolds stress field accelerate diffusion of the embedded vortices~\\citep{Perkins_JFM_1970, Pauley_Eaton_1988}. \nTherefore, one cannot necessarily expect a particular vortex structure in a laminar BL to evolve in the same manner as it would in a turbulent one. \nAnother challenge with extending SJBLI results in laminar BLs to turbulent flow fields relates to flow transition. \nIn fundamental flow control studies the BL shape or wall shear stress is often used as an indicator of the potential effectiveness for separation control. \nHowever, the tendency of a large perturbation to force transition in laminar BLs makes it difficult to determine whether a particular actuator induces superior mixing or is just more effective at tripping the flow, a task easily achieved by simpler passive devices.\n\nOf the subset of experiments conducted to investigate the interaction between rectangular orifice synthetic jets and turbulent BLs many have only obtained time-averaged or mean data, typically using hot-wire anemometry. \nAlthough mean flow data may reveal the average effect of the actuation, it does not provide a definitive picture of the underlying vortex dynamics, i.e., the why. \nIn a similar vein, data revealing the instantaneous vortex structure on just the centreline plane provides some information about the dynamics but can miss critical 3D effects like spanwise contraction of the jet or the presence of hairpins legs that straddle the symmetry plane. \nTherefore, it is helpful when trying to understand the detailed vortex dynamics in SJBLIs to obtain both time-averaged and the phase-locked measurements and to do so for the three-component velocity field throughout the interaction domain. \n\n\\subsection{Outline of the present study}\n\\label{sec:outpresstudy}\n\nThe current study presents detailed experimental data revealing the vortex dynamics resulting from a SJBLI for an infrequently studied, but highly relevant, combination of actuator geometry and flow field conditions. \nFor its relevance to common separation control problems, the baseline flow field, upon which the actuator acts, is that of a fully developed turbulent BL. \nAdditionally, the actuator employs an $AR=18$ rectangular orifice which falls within the range of values often considered in applied studies. \nThree of the most common variations of the orifice orientation are explored including (1) a wall normal jet with the orifice major axis aligned with the spanwise direction, (2) a wall normal jet with the orifice major axis aligned with the streamwise direction, and (3) a jet pitched 45 degrees downstream with the orifice major axis aligned with the spanwise direction. \nTwo blowing ratios of 0.5 and 1.0 are considered, values which fall within the practical range for typical engineering problems. \nAdditionally, the influence of vortex spacing in the BL is also explored by varying the Strouhal number of the actuation. \nThe data needed to reconstruct both the mean influence of the SJBLI along with the development of 3D instantaneous structures was captured using particle image velocimetry measurements on an array of planes throughout the domain. \nFinally, in addition to exploring the flow physics in these various cases the resulting flow fields are also analyzed to infer their potential effectiveness in delaying separation.\n\nThe experimental results of this study are presented as follows. \nSection~\\ref{sec:expmethods} provides details on the actuator design, the conditions studied, and the experimental methods employed. \nSection~\\ref{sec:results1} looks in detail at the dynamics in the three jets individually, at a single actuation condition. \nSection~\\ref{sec:results2} compares the effects of the three jets on the flow field at that same condition. \nNext, the influences of different actuation conditions on the SJBLI are explored in \\S~\\ref{sec:results3}. \nFinally, the results are summarized and the conclusions are presented in \\S~\\ref{sec:conclusions}.\n\\section{SJBLI vortex dynamics at \\texorpdfstring{$C_b=1$}{TEXT}}\n\\label{sec:results1}\n\nThe following section looks in detail at the 3D reconstructed flow field data for each of the three jet geometries at $C_b=1$ to establish a foundational understanding of the jet structure and behaviour. \n\n\\subsection{The \\texorpdfstring{$\\alpha=90^\\circ$}{TEXT}, \\texorpdfstring{$\\beta=0^\\circ$}{TEXT} orifice SJA}\n\\label{sec:AR18Beta0}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta0_Uo12_Uinf12_vortex_isosurface_thresh0_5_rfilter3D_r2w1_annot_bot_cbar.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.5$ coloured by spanwise vorticity in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet at (a) $\\phi=90^\\circ$, (b) $\\phi=180^\\circ$, (c) $\\phi=270^\\circ$, and (d) $\\phi=360^\\circ$.}\n\\label{fig:AR18Beta0LambdaciQuad}\n\\end{figure}\n\nA synthetic jet issuing from a $\\alpha=90^\\circ$, $\\beta=0^\\circ$ orifice interacts strongly with cross-flows due to the large spanwise extent of the interaction zone and the $90^\\circ$ misalignment between the directions of the jet and the cross-flow. \nThis results in a large disturbance of the BL and a significant modification to the vortex structures.\n\n\\subsubsection{Vortex structure and development}\n\\label{sec:AR18Beta0vortex}\n\nFigure~\\ref{fig:AR18Beta0LambdaciQuad} presents phase-locked isosurfaces of $\\lambda_{ci}$ for the purpose of visualizing the coherent structures produced by a $\\alpha=90^\\circ$, $\\beta=0^\\circ$ synthetic jet. \nThe isosurfaces are coloured by spanwise vorticity ($\\omega_z$) such that red represents positive or counterclockwise rotation (i.e., rotating counter to the BL vorticity), blue represents negative or clockwise rotation (i.e., rotating with the BL vorticity), and white represents regions with zero spanwise vorticity. \nFor convenience, the different sections of a vortex ring will be distinguished by the direction of their rotation in the spanwise direction such that clockwise and counterclockwise rotating regions of the ring will be referred to as the CW and CCW side, respectively. \n\nThe actuator phase angle, $\\phi$, is quoted relative to the start of the blowing cycle ($\\phi=0^\\circ$), such that $90^\\circ$ is roughly the point of peak blowing, $180^\\circ$ is the start of the suction cycle, and $270^\\circ$ is roughly the point of peak suction. \nThe different generations of vortices are annotated at each phase angle in figure~\\ref{fig:AR18Beta0LambdaciQuad}, with G1 being the eldest vortex visible and G3 being the youngest one.\nFinally, the cross-flow direction is from bottom left to top right and the location of the orifice exit is indicated with a black outline.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/AR18_beta0_sketch7.eps}\n\\caption{Sketch of the vortices in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet with rotational direction indicated by blue arrows at (a) $\\phi=180^\\circ$, (b) $\\phi=270^\\circ$, (c) $\\phi=360^\\circ$, and (d) $\\phi=90^\\circ$.}\n\\label{fig:AR18Beta0Sketch}\n\\end{figure}\n\nObtaining a complete picture of the topology of the primary and secondary vortex structures and how they develop requires a detailed inspection of several parameters, from many viewpoints, and at numerous isosurface levels. \nDue to the infeasibility of presenting that quantity of data the authors have summarized their findings in a pictorial representation of the periodic structures (e.g., figure~\\ref{fig:AR18Beta0Sketch}). \nThese graphics are meant to conceptually relate our understanding of the flow field instead of faithfully reproducing a particular parameter at one specific contour level. \n\nThe periodic structure which forms during the blowing phase of the synthetic jet is a distorted vortex ring, visible just downstream from the orifice in figure~\\ref{fig:AR18Beta0LambdaciQuad}a (G3). \nThe upstream CCW rotating side of the vortex ring appears as a narrow discontinuous segment of the ring which has been diminished by vorticity cancelation with the counter-rotating BL vorticity. \nConversely, the downstream CW rotating side of the vortex ring is a large robust structure, bolstered by the co-rotating BL vorticity. \nNote that the relatively stronger velocity induction from the dominant CW side of the vortex ring has already begun to rotate the weaker CCW segment off the wall and downstream. \nThe balance of the circulation between the weak upstream side and the strong downstream side of the vortex ring is carried by short broad legs originating from a branch of the primary vortex tube at the spanwise ends of the vortex ring and terminating on their other end at the wall (figure~\\ref{fig:AR18Beta0Sketch}a).\n\nIn the next phase of the cycle, the CW side of the vortex ring begins to deform due to self-induction (G3 in figure~\\ref{fig:AR18Beta0LambdaciQuad}b). \nIt is helpful to imagine the vortex legs together with the CW vortex as forming half of an elongated vortex ring, with the other half represented by the image vortex due to the effect of the wall. \nThus the high curvature ends of this virtual vortex ring induce a velocity pointed in the upstream (negative $x$) direction which is stronger than the velocity induced locally on the roughly rectilinear mid-span portion of the ring. \nThis differential causes the ends of the virtual ring to bend upstream and towards centreline in a manner similar to an axis-switching vortex ring propagating through quiescent fluid in the $-x$ direction \\citep{Straccia_Farnsworth_PRF_2021}. \nConcurrently, the CCW side of the vortex ring continues to be rolled up and over the stronger CW vortex, and although only the curved ends of the structure are visible at this isosurface level, those segments are still connected in the spanwise direction by CCW vorticity (figure~\\ref{fig:AR18Beta0Sketch}b).\n\nThe structure of the G3 vortex in figure~\\ref{fig:AR18Beta0LambdaciQuad}b is actually more complicated than it appears at this $\\lambda_{ci}$ level. \nThe spanwise contraction of the virtual vortex ring induces compressive normal strain ($S_{zz}<0$) on centreline in the middle of the patch of CW vorticity associated with the vortex (figure~\\ref{fig:AR18Beta0Tight}a). \nThe resulting divergence in the $x-y$ plane causes a local drop in angular velocity and a spreading out of the surrounding spanwise vorticity due to the requirements of continuity.\nThis phenomenon presents as a broadening of the CW vortex cross-section in figure~\\ref{fig:AR18Beta0LambdaciQuad}b. \nThe net result is that a portion of the CW vorticity generated by the actuator outstroke splits off from the primary vortex and forms a secondary structure which remains close to the wall as it advects downstream (figure~\\ref{fig:AR18Beta0Sketch}).\nThe initial phase of this splitting process can be seen in the $\\lambda_{ci}=1.75$ isosurface in figure~\\ref{fig:AR18Beta0Tight}b.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta0_540Hz_U012_Uing12_263deg_vortex_closeup.eps}\n\\caption{Close up view of (a) coloured contours of spanwise vorticity overlaid with black contour curves of spanwise normal strain every 20\\% of $|S_{zz}|_{max}$ from -90\\% to 90\\% (\\sampleline{dashed}, $S_{zz}<0$; \\sampleline{}, $S_{zz}>0$) and the location of the vortex centre ($\\times$) on centreline and (b) isosurfaces of phased-locked $\\lambda_{ci}$ at levels of 0.5 (aqua) and 1.75 (green) in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=180^\\circ$.}\n\\label{fig:AR18Beta0Tight}\n\\end{figure}\n\nIn figure~\\ref{fig:AR18Beta0LambdaciQuad}c the virtual vortex ring has continued to contract in the spanwise direction which introduces a bend in the vortex axis on centreline (see G3). \nThe self-induced velocity of the CW rotating vortex is pointed upstream and away from the wall in the bend region and its magnitude increases as the bend becomes tighter (figure~\\ref{fig:AR18Beta0Sketch}c).\nThe progression of these deformations continue to follow those of the half-ring in the axis-switching analogy.\n\nMeanwhile, the CW vorticity which split off from the primary vortex forms a secondary vortex close to the wall (see I in figure~\\ref{fig:AR18Beta0LambdaciQuad}c). \nThis upstream pointing chevron structure continues to be visible in the next two phases as it slowely advects downstream. \nAt the ends of the chevron structure in figure~\\ref{fig:AR18Beta0LambdaciQuad}c the vortex tube turns $90^\\circ$ towards the wall-normal direction before turning again towards centreline to form the CCW branch of the original vortex ring (not visible at this $\\lambda_{ci}$ level). \nIn essence, this structure is like a secondary vortex ring which has partially separated from the upstream vortex and exists in the space near the wall between the different generations of the primary vortices (figure~\\ref{fig:AR18Beta0Sketch}c). \n\nIn the next few phases the primary structure continues to lift off the wall forming an arch-shaped vortex with a hairpin head and legs which terminate at the wall (see G2 in figure~\\ref{fig:AR18Beta0LambdaciQuad}c). \nHowever, the topology of the legs begins to change by G2 in figure~\\ref{fig:AR18Beta0LambdaciQuad}d. \nAt this phase a new streamwise branch in the vortex tube appears in the $\\lambda_{ci}$ isosurface which bridges the gap between the G2 and G3 vortices (see II in figure~\\ref{fig:AR18Beta0LambdaciQuad}d). \nThe SPIV data implies that this branch was created by a complex combination of tilting and stretching of the secondary vortex which sits in the interstitial space between the primary structures (figure~\\ref{fig:AR18Beta0Sketch}c, d).\nWhen the secondary structure comes close to the two primary structures, vorticity reconnection occurs, forming a new interconnected topology.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/AR18_beta0_540Hz_Uo12_Uinf12_3D3C_83deg_vortex_isosurface_train_thresh0_5_rfilter3D_r2w1_annot.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.25$ coloured by spanwise vorticity in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=360^\\circ$ showing the train of hairpin vortices.}\n\\label{fig:AR18Beta0Train}\n\\end{figure}\n\nInitially, the new streamwise branches are smaller and weaker than the legs of the arch-shaped vortex (see II in figure~\\ref{fig:AR18Beta0LambdaciQuad}d), but as the structures travel downstream the streamwise branches grow at the expense of the original legs (see II in figure~\\ref{fig:AR18Beta0LambdaciQuad}a) until they are the only set of legs visible (see II in figure~\\ref{fig:AR18Beta0LambdaciQuad}b). \nTherefore, the primary vortex disconnects from the wall between $x\/h=25$ and $x\/h=35$ and its new streamwise-orientated legs terminate instead on the leeward side of the upstream vortex ring. \nIn the subsequent phases this hairpin-like vortex continues to lift off the wall and eventually disappears from view at $\\lambda_{ci}=0.5$ in figure~\\ref{fig:AR18Beta0LambdaciQuad}.\n\nIn summation, the interaction of a $\\alpha=90^\\circ$, $\\beta=0^\\circ$ orifice jet with a cross-flow produces a train of vortices that look and behave much like hairpin vortex packets known to form naturally in turbulent BLs (figure~\\ref{fig:AR18Beta0Train}). \nThe role of natural hairpin vortices in driving sweeps and ejections that enhance wall-normal mixing makes these synthetic hairpin vortices a potentially desirable product of the SJBLI \\citep{Adrian_PoF_2007}.\n\n\\subsubsection{Modification to velocity field}\n\\label{sec:AR18Beta0velocity}\n\nThe influence of the flow control on the velocity field within the SJBLI domain is presented in figures \\ref{fig:AR18Beta0DeltaPavg} and \\ref{fig:AR18Beta0DeltaTavg}. \nPlotted are the normalized change in streamwise ($\\Delta u\/U_e$) and wall-normal velocities ($\\Delta v\/U_e$) which were obtained by subtracting the baseline velocity field (no-actuation) from the velocity field with actuation on. \nAccordingly, the red and blue isosurfaces enclose regions of the flow field where the flow velocity has increased or decreased, respectively, as a result of actuation. \nFigure~\\ref{fig:AR18Beta0DeltaPavg} presents the phase-locked velocity field at a single representative actuator phase angle with gray isosurfaces of $\\lambda_{ci}$ depicting the periodic coherent structures.\nThe $\\Delta u\/U_e$ and $\\Delta v\/U_e$ for the mean velocity field are also presented (figure~\\ref{fig:AR18Beta0DeltaTavg}).\nAdditionally, the location of the edge of the BL, $\\delta(x)$, in the baseline flow field is marked with a dashed line for reference. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta0_540Hz_Uo12_Uinf12_3D3C_173deg_UVdiff_cont_rfilter3D_r2w1_thres0_5_scf.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.5$ (grey) in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=90^\\circ$ with isosurfaces of (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue).}\n\\label{fig:AR18Beta0DeltaPavg}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta0_540Hz_Uo12_Uinf12_3D3C_Tavg_UVdiff_cont_scf.eps}\n\\caption{Isosurfaces of time-averaged (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue) in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet.}\n\\label{fig:AR18Beta0DeltaTavg}\n\\end{figure}\n\nThe hairpin vortices formed by the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA induce a velocity on centreline, below the head and between the legs, which is pointed upstream and away from the wall (figure~\\ref{fig:AR18Beta0Sketch}).\nThe result is that every generation of vortex in the phase-locked flow field encloses a region of strong upwash (figure~\\ref{fig:AR18Beta0DeltaPavg}a). \nThis upwash along centreline is balanced by downward motion of the fluid surrounding the hairpin vortex, as indicated by the regions enclosed by the blue isosurfaces. \nIn the mean flow field the average effect of the periodic structures advecting downstream is to lift low momentum fluid off the wall on centreline and entrain high momentum fluid towards the wall on both sides of the centreline (figure~\\ref{fig:AR18Beta0DeltaTavg}a). \nIn addition, a region of downward moving fluid which is generated during the suction cycle of the actuator appears on the upstream side of the orifice in the time-averaged flow field. \nSuction dominates blowing on average in this region of the mean flow field because the outward oriented momentum of the jet during the blowing cycle is turned downstream by the cross-flow.\n\nThe $\\Delta u\/U_e<0$ isosurfaces reveal that a low momentum wake with a roughly circular cross-section runs through the centre of the hairpin vortex heads (figure~\\ref{fig:AR18Beta0DeltaTavg}b). \nAn additional mound of low momentum fluid directly above the orifice forms during the blowing cycle of the actuator when fluid ejected from the orifice acts as an obstruction to the cross-flow. \nThe effect of this viscous blockage is visible in the region of decelerated streamwise velocity directly above and upstream of the orifice in the phase-locked data (figure~\\ref{fig:AR18Beta0DeltaPavg}b). \n\nDownstream of the orifice the velocity induced by the hairpin vortices on the surrounding fluid contributes to the development of a wake in two ways (figure~\\ref{fig:AR18Beta0DeltaTavg}b). \nFirst, the upward velocity induced on centreline lifts low momentum fluid off the wall and deposits it deeper into the cross-flow. \nSecond, the upstream pointing component of the induced velocity opposes the cross-flow at the centre of the hairpin heads, decelerating the fluid in the wake further (figure~\\ref{fig:AR18Beta0Sketch}).\nTherefore, the trajectory of the wake is inherently tied to the motion and changing shape of the hairpins vortices (figure~\\ref{fig:AR18Beta0DeltaPavg}b).\n\nOn the sides of each hairpin vortex is an accompanying patch of high streamwise velocity introduced by the entrainment of high momentum fluid towards the wall by the vortices (figure~\\ref{fig:AR18Beta0DeltaPavg}b). \nAs the vortices travel downstream the regions of elevated streamwise velocity spread in the spanwise direction towards centreline due to induction from the hairpin vortex legs. \nThus, the average effect of the actuation is to move low momentum fluid off the wall into a consolidated wake higher in the BL and to transport high momentum fluid down into a layer of accelerated flow close to the wall (figure~\\ref{fig:AR18Beta0DeltaTavg}b).\n\n\\subsection{The \\texorpdfstring{$\\alpha=90^\\circ$}{TEXT}, \\texorpdfstring{$\\beta=90^\\circ$}{TEXT} orifice SJA}\n\\label{sec:AR18Beta90}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta90_Uo12_Uinf12_vortex_isosurface_thresh0_75_rfilter3D_r2w1_annot_bot_cbar.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.75$ coloured by spanwise vorticity in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet at (a) $\\phi=90^\\circ$, (b) $\\phi=180^\\circ$, (c) $\\phi=270^\\circ$, and (d) $\\phi=360^\\circ$.}\n\\label{fig:AR18Beta90LambdaciQuad}\n\\end{figure}\n\nWith the actuator skewed $90^\\circ$ such that the long axis of the orifice is aligned with the streamwise direction the initial jet presents a relatively small frontal area to the cross-flow, resulting in the formation of a more intact vortex ring than in the previous case. \n\n\\subsubsection{Vortex structure and development}\n\\label{sec:AR18Beta90vortex}\n\nFigure~\\ref{fig:AR18Beta90LambdaciQuad} depicts the coherent structures at four phases of the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet. \nA pictorial representation of the vortex structures is presented in figure~\\ref{fig:AR18Beta90Sketch}.\nThe elongated vortex ring forming above the orifice in figure~\\ref{fig:AR18Beta90LambdaciQuad}a (G3) looks much as it would have if there had been no cross-flow.\nThe ends of the vortex ring have even started to lift away from the wall ahead of the ring centre due to the higher local self-induced velocity in those tightly curved segments of the ring. \nThere are some subtle indications, however, that the vortex ring has been affected by the cross-flow. \nFor one, the vortex ring has translated downstream a short distance. \nIn addition, the velocity gradient in the BL has altered the vortex ring topology slightly. \nIn particular, the upstream end of the vortex ring is slightly diminished in strength by vorticity cancelation with the BL, and from the upstream corners of the ring two short legs branch off carrying the balance of the circulation to the wall (figure~\\ref{fig:AR18Beta90Sketch}a).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/AR18_beta90_sketch7.eps}\n\\caption{Sketch of the vortices in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet with rotational direction indicated by blue arrows at (a) $\\phi=90^\\circ$, (b) $\\phi=180^\\circ$, (c) $\\phi=270^\\circ$, and (d) $\\phi=360^\\circ$.}\n\\label{fig:AR18Beta90Sketch}\n\\end{figure}\n\nThe legs become more evident at $\\phi=180^\\circ$ (G3 in figure~\\ref{fig:AR18Beta90LambdaciQuad}b). \nIt is also clear that the vortex ring is undergoing typical axis-switching deformation which has caused the ring to shrink in the $x$-direction and to grow slightly in the $z$-direction. \nIn addition, two more structures are now visible on the vortex ring. \nThe first is a CW rotating vortex running across the middle of the ring (see I in figure~\\ref{fig:AR18Beta90LambdaciQuad}b). \nThis structure is also present in figure~\\ref{fig:AR18Beta90LambdaciQuad}a but is not visible at this isosurface level, except for a small stub annotated I (see figure~\\ref{fig:AR18Beta90Sketch}a for the full structure).\nSimilar structures were detected in quiescent conditions by \\citet{Straccia_Farnsworth_JFM_2020} which they called `formation bridges'. \nLike the bridge vortices formed by vorticity reconnection during the collision of a vortex ring with itself after axis-switching, formation bridges occur when the two sides of a wide-cored vortex ring come in contact during the vortex's formation. \nSPIV data obtained by \\citet{Housley_PhDThesis_2020} in a similar experiment reveal that this initial vorticity reconnection event can be strong enough to cause the vortex ring to bifurcate into two smaller rings. \nWhat is curious on first look is the absence of a counter-rotating (CCW) formation bridge which should accompany the CW-rotating vortex. \nA closer look at the vorticity field on the jet centreline suggests that the reconnected CCW vorticity sits below the visible CW formation bridge in the central region of the vortex ring where the high induced strain prevents its detection by vortex identification schemes like $\\mathcal{Q}$-criterion and $\\lambda_{ci}$. \nThis proposed topology appears to be supported by figure 3.44 in \\citet{Housley_PhDThesis_2020}. \n\nThe other structure visible on the vortex ring in figure~\\ref{fig:AR18Beta90LambdaciQuad}b is a vortex running spanwise between the legs below the primary vortex ring (annotated II). \nSimilar ladder-rung-like structures between the legs were detected by \\citet{Wang_Feng_JFM_2020} on lower $AR$ vortex rings in a laminar BL. \nThe interlinking structure rotates in the same direction as the upstream side of the vortex ring and appears to develop after the vortex ring.\nIt is likely that this vortex forms from trailing vorticity generated later in the actuator stroke that is not incorporated into the primary vortex ring, possibly due to a modified form of vortex pinch-off \\citep{Gharib_Rambod_JFM_1998}. \nSuch secondary vortices have been shown to form in synthetic jets issuing into quiescent fluid under the right conditions \\citep{Straccia_Farnsworth_JFM_2020}. \nThe CCW rotating interlinking vortex is detectable, although not visible in figure~\\ref{fig:AR18Beta90LambdaciQuad}, for the next three phase-angles until it becomes too weak to distinguish from background noise by G2 at $\\phi=180^\\circ$. \nIn addition to increasing unsteady mixing below the primary vortex ring, the interlinking vortex also induces a downstream pointing velocity component near the wall (figure~\\ref{fig:AR18Beta90Sketch}b).\n\nAt $\\phi=270^\\circ$ the formation bridge is still stretched between the ever widening spanwise ends of the G3 vortex ring, although it is not visible in the figure~\\ref{fig:AR18Beta90LambdaciQuad}c isosurface (see figure~\\ref{fig:AR18Beta90Sketch}c for the full structure). \nThe vortex ring continues to advect downstream and propagate away from the wall while widening in the spanwise direction due to the self-induced axis-switching deformations (figure~\\ref{fig:AR18Beta90LambdaciQuad}d). \nWhen the subsequent vortex ring forms, the legs of the older G2 ring terminate slightly upstream of the end of the orifice. \nThe result is that the legs from the prior vortex ring end up becoming tied to the downstream end of the new ring (figure~\\ref{fig:AR18Beta90Sketch}a). \nInitially, this branch in the vortex tube is located towards the upper surface of the new vortex ring, but as the strong induced velocity around the ring acts, this connection point migrates to the underside of the upstream vortex ring.\n\nThe spanwise-oriented G2 vortex ring and its legs are visible downstream of the youngest ring in all four panels of figure~\\ref{fig:AR18Beta90LambdaciQuad}. \nThe vortex ring eventually extends past the edge of the measurement domain and then decays to a point where it is undetectable at the isosurface level set in figure~\\ref{fig:AR18Beta90LambdaciQuad}, starting with the weaker CCW rotating side first. \nThe vortex ring appears to roughly complete the axis-switching cycle by G2 in figure~\\ref{fig:AR18Beta90LambdaciQuad}b. \nAfter that point the vortex rings deform very little and instead the older structures advect along as a train of spanwise rollers loosely tied together by their legs.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta90_540Hz_Uo12_Uinf12_3D3C_172deg_UVdiff_cont_rfilter3D_r2w1_thres0_75_scf.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.75$ (grey) in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet at $\\phi=90^\\circ$ with isosurfaces of (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue).}\n\\label{fig:AR18Beta90DeltaPavg}\n\\end{figure}\n\n\\subsubsection{Modification to velocity field}\n\\label{sec:AR18Beta90velocity}\n\nThe effects that the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ SJA has on the velocity field are similar to those of the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA in a general sense, but the strengths and locations of particular features are quite different. \nAn especially strong upwash is induced at the centre of each vortex ring in this case because the collective action of the counter-rotating sides of the vortex ring results in an induced velocity vector that points predominantly in the wall-normal direction (figure~\\ref{fig:AR18Beta90DeltaPavg}a). \nThe vortex rotation also induces downward flow in the fluid around the circumference of the vortex ring (figure~\\ref{fig:AR18Beta90Sketch}a).\nIn the time-averaged field the $\\Delta v\/U_e=0.05$ isosurface enclosing the upward moving fluid on the jet centreline extends to the end of the measurement domain thanks to the strength and longevity of the vortex rings (figure~\\ref{fig:AR18Beta90DeltaTavg}a). \n \nA pronounced wake forms in the streamwise velocity field of the jet (figure~\\ref{fig:AR18Beta90DeltaPavg}b). \nUpwash of low momentum fluid from lower in the BL is again a factor in producing the wake. \nIn addition, the pair of legs induce a velocity that is pointed partially upstream, counter to the cross-flow direction, as does the underside of the dominant clockwise rotating side of the vortex ring (figure~\\ref{fig:AR18Beta90Sketch}a).\nThe wake which results from the combined upwash and velocity induction has a mean cross-sectional shape at the end of the measurement domain which is broader at the top (due to the spanwise orientation of the vortex ring) and narrower at the bottom (due to the small gap between the legs) (figure~\\ref{fig:AR18Beta90DeltaTavg}b). \nNotably absent from both the phase-locked and time-averaged flow fields is the region of low streamwise momentum fluid upstream of the orifice and near the wall which was seen in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet. \nThis is because the streamwise-oriented jet only minimally obstructs the cross-flow and therefore there is less viscous blockage introduced into the flow than there is by the spanwise-oriented jet.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18_beta90_540Hz_Uo12_Uinf12_3D3C_Tavg_UVdiff_cont_scf.eps}\n\\caption{Isosurfaces of time-averaged (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue) in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet.}\n\\label{fig:AR18Beta90DeltaTavg}\n\\end{figure}\n\nStreamwise velocity is enhanced above the CW rotating sides of the vortex ring, above the formation bridge, and on the outer sides of the legs (figures~\\ref{fig:AR18Beta90Sketch}a and \\ref{fig:AR18Beta90DeltaPavg}b). \nThis is the reason for the sheath of elevated streamwise velocity upstream and to the sides of the wake near the orifice in the mean flow (figure~\\ref{fig:AR18Beta90DeltaTavg}b). \nFurther downstream a streak of elevated streamwise velocity forms near the wall and extends to the end of the measurement domain (figure~\\ref{fig:AR18Beta90DeltaTavg}b). \nThis high velocity stream is formed by the entrainment of fluid from higher in the BL along either side of the jet which rushes in to fill the space left by the upwash on centreline. \n\n\\subsection{The \\texorpdfstring{$\\alpha=45^\\circ$}{TEXT}, \\texorpdfstring{$\\beta=0^\\circ$}{TEXT} orifice SJA}\n\\label{sec:AR18P45Beta0}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18P45_beta0_Uo12_Uinf12_vortex_isosurface_thresh0_5_rfilter3D_r2w1_annot_bot_cbar.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.5$ coloured by spanwise vorticity in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet at (a) $\\phi=90^\\circ$, (b) $\\phi=180^\\circ$, (c) $\\phi=270^\\circ$, and (d) $\\phi=360^\\circ$.}\n\\label{fig:AR18P45Beta0LambdaciQuad}\n\\end{figure}\n\nPitching a spanwise-oriented orifice $45^\\circ$ downstream reduces the intensity of the SJBLI and gives the momentum introduced by the jet a streamwise component. \nThe result is a dramatically different vortex structure and flow field, as compared with the wall-normal jets. \n\n\\subsubsection{Vortex structure and development}\n\\label{sec:AR18P45Beta0vortex}\n\nThe coherent structures produced by the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet are depicted in the $\\lambda_{ci}$ isosurfaces of figure~\\ref{fig:AR18P45Beta0LambdaciQuad} for four evenly spaced actuation phase angles. \nA pictorial representation of the vortex structures is presented in figure~\\ref{fig:AR18P45Beta0Sketch}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4.25in]{figures\/AR18P45_beta0_sketch7.eps}\n\\caption{Sketch of the vortices in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet with rotational direction indicated by blue arrows at (a) $\\phi=90^\\circ$, (b) $\\phi=180^\\circ$, (c) $\\phi=270^\\circ$, and (d) $\\phi=360^\\circ$.}\n\\label{fig:AR18P45Beta0Sketch}\n\\end{figure}\n\nThe G3 vortex ring forming near the orifice in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a is tilted downstream due to the pitching of the jet. \nPitching the jet downstream enhances the circulation strength in the upstream side of the vortex ring, offsetting the vorticity cancelation between the CCW rotating vortex and the CW rotating BL vorticity. \nSimultaneously, the downstream CW rotating side of the vortex ring is weakened in the pitched jet. \nAs a result, the flow field within the SJBLI domain is dominated by a CCW rotating vortex in this case. \nThe CW side of the new vortex ring is partially obscured by the CCW rotating vortex in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a but can be seen more clearly in figure~\\ref{fig:AR18P45Beta0Tight}. \nNote that there is a secondary patch of $\\lambda_{ci}$ upstream of the vortex ring in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a marking the location of additional CW vorticity from the actuator outstroke that was not incorporated into the primary vortex (see I in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{figures\/AR18P45_beta0_540Hz_Uo12_Uinf12_3D3C_150deg_vortex_isosurface_closeup_thresh0_75_rfilter3D_r2w1_scf.eps}\n\\caption{Close up view of isosurfaces of phased-locked $\\lambda_{ci}=0.75$ coloured by spanwise vorticity in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=90^\\circ$.}\n\\label{fig:AR18P45Beta0Tight}\n\\end{figure}\n\nAs the vortex ring propagates downstream the ends of the ring outrun the middle due to the higher local self-induction (G3 in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}b). \nMeanwhile, the trailing vorticity upstream of the vortex ring tilts into the streamwise direction and begins to coalesce into two pairs (four total) of counter-rotating streamwise vortices (see II and III in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}b and proposed topology in figure~\\ref{fig:AR18P45Beta0Sketch}b). \nThe inner vortex pair appears to form from tilted and consolidated vorticity in the trailing CCW rotating shear layer and is connected to the upstream side of the CCW rotating primary vortex. \nThe outer vortex pair may form out of the CW vorticity identified in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a, which can already be seen tilting into the streamwise direction at its ends. \nHow the outer vortex pair connects to the primary ring is difficult to precisely resolve in the present data, but the connection points appear to initially be near the ends of the CW rotating side of the ring. \n\nBy $\\phi=270^\\circ$ the CW-rotating side of the G3 vortex ring is greatly dissipated and is becoming difficult to detect in the $\\lambda_{ci}$ field (figure~\\ref{fig:AR18P45Beta0LambdaciQuad}c). \nThe rest of the vortex ring has continued to progress through the typical axis-switching deformations, and a highly curved head has started to form along centreline. \nUpstream of the vortex ring the outer pair of secondary streamwise vortices are clearly visible (annotation II) while the inner pair show up only as small knobs that protrude upstream from the CCW-rotating side of the primary vortex near the centreline (annotation III). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/AR18P45_beta0_540Hz_Uo12_Uinf12_3D3C_150deg_vortex_isosurface_train_thresh0_5_rfilter3D_r2w1_annot.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.3$ coloured by spanwise vorticity in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=90^\\circ$ showing the train of arch-shaped vortices.}\n\\label{fig:AR18P45Beta0Train}\n\\end{figure}\n\nIn figure~\\ref{fig:AR18P45Beta0LambdaciQuad}d the CW side of the G3 vortex is no longer detected and the primary structure now takes the form of an arch-shaped vortex with its apex leaned into the flow and its legs terminating at the wall. \nAt this time only a single set of streamwise secondary vortices remain. \nThe vortex tube topology at $\\phi=360^\\circ$ implies that the inner vortex pair have merged with their co-rotating partners from the outer vortex pair, at least at the surface of the primary vortex. \nVorticity annihilation across centreline may also play a role in the disappearance of the inner vortex pair closer to the wall. \nThe remaining streamwise secondary vortices (annotation IV) may appear to be extensions of the G3 primary vortex's legs in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}d but in fact the streamwise vorticity in the secondary structure is counter-rotating to that of the primary vortex. \nInstead, the secondary structures are branches off the primary structure and have a rotational direction which indicates that the head of the arch-shaped vortex has lower circulation strength than the legs do (figure~\\ref{fig:AR18P45Beta0Sketch}b). \nThis may explain why the apex of the arch-shaped vortex is the first to drop below the isosurface level for both vorticity magnitude and $\\lambda_{ci}$ (see, for example, the broken isosurface of the G2 vortex in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}d and the G1 vortex in figure~\\ref{fig:AR18P45Beta0LambdaciQuad}a).\n\nAt later phase angles the primary vortex continues to deform in such a way that the apex of the arch lifts up from the wall and rotates downstream relative to the legs due to its self-induced velocity (figure~\\ref{fig:AR18P45Beta0Train}). \nThe more upright arch-shaped vortex also becomes narrower in the spanwise direction. \nThis geometric progression is similar to the deformations of an axis switching vortex ring traveling roughly in the $x$-direction. \nCompared to the other two orifice geometries, it is noteworthy that the vortices in the present case remain tied to the wall via their legs for as long as they are detectable, instead of detaching from the wall. \nFurthermore, while the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA produced a train of hairpin-shaped vortices, the somewhat similar train of arch-shaped vortices in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet rotate in the opposite direction to the hairpin structures (compare figures \\ref{fig:AR18Beta0Train} and \\ref{fig:AR18P45Beta0Train}). \nThe secondary streamwise vortex pair advect downstream along the wall, located in the space between subsequent vortex generations (annotation IV). \nAs it travels the streamwise pair also converges in the spanwise direction, a behaviour that is to be expected based on its direction of rotation and the wall-induced image vortex effect (figure~\\ref{fig:AR18P45Beta0Sketch}a).\n\\citet{Housley_PhDThesis_2020} also detected a streamwise vortex pair in a SJBLI with a pitched orifice, and they speculated that this structure may have originated as the CW side of the vortex ring; however, they were unable to resolve that side of the vortex ring because of the gap between their lowest SPIV measurement and the wall.\n\n\\subsubsection{Modification to velocity field}\n\\label{sec:AR18P45Beta0velocity}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18P45_beta0_540Hz_Uo12_Uinf12_3D3C_150deg_UVdiff_cont_rfilter3D_r2w1_thres0_5_scf.eps}\n\\caption{Isosurfaces of phased-locked $\\lambda_{ci}=0.5$ (grey) in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet at $\\phi=90^\\circ$ with isosurfaces of (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue).}\n\\label{fig:AR18P45Beta0DeltaPavg}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_AR18P45_beta0_540Hz_Uo12_Uinf12_3D3C_Tavg_UVdiff_cont_scf.eps}\n\\caption{Isosurfaces of time-averaged (a) $\\Delta v\/U_e=0.05$ (red), $\\Delta v\/U_e=-0.05$ (blue) and (b) $\\Delta u\/U_e=0.05$ (red), $\\Delta u\/U_e=-0.10$ (blue) in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet.}\n\\label{fig:AR18P45Beta0DeltaTavg}\n\\end{figure}\n\nThe pitched orifice has a markedly different effect on the velocity field within the BL than the other two orifice geometries have. \nThe early vortex ring induces a strong upwash within the ring and downwash around the ring (figure~\\ref{fig:AR18P45Beta0DeltaPavg}a), similar to the early $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (figure~\\ref{fig:AR18Beta90DeltaPavg}a). \nThe difference is that by pitching the vortex ring downstream the induced velocity within the ring also has a significant streamwise component (figure~\\ref{fig:AR18P45Beta0DeltaPavg}b). \nWhen the arch-shaped vortex first forms and its apex leans upstream, upwash is induced downstream of the crook formed by the vortex while fluid is induced towards the wall on the upstream side of the structure (figures~\\ref{fig:AR18P45Beta0Sketch}a and \\ref{fig:AR18P45Beta0DeltaPavg}a). \nAs the arch-shaped vortex straightens out, velocity induction in the wall-normal direction steadily drops. \nInstead, the upright arch-shaped vortex induces a strong streamwise velocity under the arch (figure~\\ref{fig:AR18P45Beta0DeltaPavg}b). \n\nThe time-averaged isosurfaces of wall-normal velocity change are relatively small for this case (figure~\\ref{fig:AR18P45Beta0DeltaTavg}a). \nA patch of downward motion upstream of the orifice is the result of the actuator suction cycle while an additional region of upward moving fluid exists within the pitched jet itself. \nGenerally speaking, the velocity field does not indicate strong wall-normal mixing in a time-averaged sense. \nRather, the large region of enhanced streamwise momentum which exists downstream of the actuator is the result of direct momentum addition from the pitched orifice (figure~\\ref{fig:AR18P45Beta0DeltaTavg}b). \nFrom a vortex dynamics perspective, the arch-shaped vortices are the vehicles which carry the high streamwise momentum in the pitched synthetic jet. \nIn addition to producing a much thicker and faster moving layer of accelerated fluid near the wall than the other two orifices, the pitched orifice also does not generate a detectable wake in figure~\\ref{fig:AR18P45Beta0DeltaTavg}b. \nIt will be shown that a layer of fluid of reduced streamwise velocity does exist further from the wall, due to the upstream pointing velocity induction above the arch-shaped vortex (figure~\\ref{fig:AR18P45Beta0Sketch}a), but that the velocity reduction is small and the slower moving fluid is not consolidated into a well-defined wake.\n\n\\section{Comparison of the three SJAs at \\texorpdfstring{$C_b=1$}{TEXT}}\n\\label{sec:results2}\n\nHaving established the types of periodic structures formed by each of the jets and how they alter the flow field in \\S~\\ref{sec:results1}, this section focuses on comparing the flow fields of the three jets and assessing their relative potential to control separation.\n\n\\subsection{Trajectory and propagation speed of periodic structures}\n\\label{sec:vorttraj}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_orifices_Uo12_Uinf12_Tavg_centerline_vortex_trajectories_n_spacing2.eps}\n\\caption{Centerline vortex centre (a) locations and (b) streamwise spacing from phase-locked data for the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ ($\\circ$), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ ($\\vartriangle$), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ ($\\lozenge$) jets.}\n\\label{fig:AllOrfVortTraj}\n\\end{figure}\n\nThe distinctive periodic structures produced by the three actuator geometries travel through the BL at different rates and on different paths. \nThe spatial trajectories of the vortices for these cases are depicted in figure~\\ref{fig:AllOrfVortTraj}a. \nIndicated are the centres of the dominant vortex on the jet centreline, extracted from the eight phased-locked data sets. \nNote that the CW rotating vortex is the dominant structure in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet cases while the CCW rotating vortex is dominant in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet. \nThe discrete vortex location data was fitted with spline curves which are plotted as solid lines in figure~\\ref{fig:AllOrfVortTraj}a while a dotted line near the top of the figure indicates the height of the BL edge for the baseline case. \n\nThe initial trajectory of the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet vortex follows an S-curve path emblematic of axis-switching in moderate $AR$ vortex rings (figure~\\ref{fig:AllOrfVortTraj}a). \nThis motion is brought about by the self-induced deformations of the virtual vortex ring which is formed by the primary structure together with its wall image. \nThe vortex trajectory up to $x\/h=25$ closely matches that presented in figure 9i of \\citet{Straccia_Farnsworth_PRF_2021} for an $AR=19$ rectangular orifice synthetic jet issuing into quiescent fluid, albeit rotated by $90^\\circ$. \nAt $x\/h \\approx 27.5$ the vortex detaches from the wall and propagates downstream as a hairpin vortex on a roughly linear trajectory. \nThe arrangement of the hairpin vortex train produced by the SJBLI is reminiscent of the linear ramp structure observed in naturally occurring turbulent BL hairpin vortex packets \\citep{ Adrian_Meinhart_JFM_2000}. \nBy the end of the measurement domain the head of the hairpin vortices have penetrated to a height that is roughly half the thickness of the baseline BL. \n\nAxis-switching of the largely intact vortex ring in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet case also modified the initial trajectory of the vortices on the jet centreline (figure~\\ref{fig:AllOrfVortTraj}a). \nThe CW vortex moves almost straight up from the wall initially, because the rapid contraction in width of the vortex ring offsets the downstream transport of the ring by the cross-flow. \nOnce the self-induced deformations cease and the vortex ring assumes a spanwise orientation, the vortex follows an essentially linear path downstream of $x\/h \\approx 20$. \nThe upward motion of the vortex rings in the streamwise-oriented jet is more rapid than the motion of the vortices of the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet case, leading to their deeper penetration into the cross-flow. \nIn fact, by $x\/h=100$ the upper surface of the vortex rings have passed through the original edge of the BL as measured in the baseline case.\n\nThe motion of the vortices in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet is initially driven by velocity induction between the counter-rotating sides of the vortex ring, which is pointed downstream in the pitched jet (figure~\\ref{fig:AllOrfVortTraj}a). \nBy $x\/h=15$ the CW vortex is no longer detectable, and the motion of the CCW vortex is dictated from there on by the self-induced motion of the resulting arch-shaped vortex. \nAs the apex of the arch-shaped vortex rotates downstream relative to the legs, the arch straightens out and the CCW vortex travels away from the wall. \nThe CCW vortex thus reaches its furthest distance from the wall at $x\/h \\approx 45$, roughly where the vortex had deformed into an upright arch shape. \nDownstream of this point the CCW vortex penetrated no further into the cross-flow. \nOf the three cases the jet from the pitched orifice remains the closest to the wall, with the vortex penetrating to a height no deeper than a third of the thickness of the baseline BL.\n\nThe vortex centre location data plotted in figure~\\ref{fig:AllOrfVortTraj}a was also used to compute the spacing of the periodic structures in the BL. \nThe results are presented in figure~\\ref{fig:AllOrfVortTraj}b where $\\overline{x_{vc}}$ is the streamwise mid-point between two adjacent vortices of subsequent generations and $\\Delta x_{vc}$ is the streamwise distance between those vortices.\nThe parameter $\\Delta x_{vc}$ has been normalized by the thickness of the baseline BL. \nStrouhal number, as defined in \\ref{eq:St}, may be physically interpreted as the spatial frequency of the periodic structures within a cross-flow. \nTherefore, the ordinate of figure~\\ref{fig:AllOrfVortTraj}b is the reciprocal of $St$ and accordingly the dotted line in the figure indicates the value of $1\/St$, i.e., the vortex spacing that would be achieved if the periodic structures advected downstream at a speed of $U_e$. \n\nThe vortices in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet are spaced more closely than is implied by the $St$ value (figure~\\ref{fig:AllOrfVortTraj}b). \nOn the other hand, after initial self-induced deformations subside, the vortices in both the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets converge to a much larger vortex spacing, one that is only slightly smaller than the theoretical $St$ value. \nThe $St$ equation typically used in cross-flow studies makes the simplifying assumption that the periodic structures advect at the BL edge velocity, $U_e$, which is not physically accurate while the structures are embedded in the slower moving BL fluid. \nBut there are more factors that influence the vortex spacing than just the distance of the structures from the wall.\nThis is clear from the fact that the vortices in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets have very similar streamwise spacing despite being at very different heights in the BL (figure~\\ref{fig:AllOrfVortTraj}). \nAt this point it is helpful to look at the advective speed of the periodic structures in these jets.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_orifices_Uo12_Uinf12_Tavg_centerline_vortex_U_N_Udiff_speed.eps}\n\\caption{Centerline vortex (a) absolute streamwise velocity and (b) change in streamwise velocity for the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ ($\\circ$), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ ($\\vartriangle$), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ ($\\lozenge$) jets.}\n\\label{fig:AllOrfVortVel}\n\\end{figure}\n\nFigure~\\ref{fig:AllOrfVortVel}a is a plot of the streamwise advective velocity of the dominant vortices for the three cases. \nThese velocities were computed using the spline curves fit to the vortex centre locations (solid lines in figure~\\ref{fig:AllOrfVortTraj}a) and the known time increments separating the eight measured phases of the actuator. \nAdditionally, figure~\\ref{fig:AllOrfVortVel}b compares the vortex propagation speed to the speed of the unforced flow. \nThe parameter $\\Delta u_{vc}$ is the difference between the vortex advective velocity and the baseline BL flow speed at the height of the vortex centre. \nTherefore, when $\\Delta u_{vc}>0$ the vortex travels faster than a passive scalar would, on average, at that location in the baseline BL, and when $\\Delta u_{vc}<0$ the vortex travels slower than the average baseline flow.\n\nThe CW rotating vortex in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet initially has low streamwise velocity because the jet issues normal to the wall (figure~\\ref{fig:AllOrfVortVel}a). \nAfter formation, the vortex quickly accelerates in the downstream direction due to momentum transfer from the cross-flow. \nThe velocity of the coherent structure is also influenced by vortex induction. \nFor example, between $x\/h=5$ and $x\/h=8.5$ the streamwise velocity of the vortex actually exceeds the average baseline flow speed at that height in the BL (figure~\\ref{fig:AllOrfVortVel}b). \nDuring this period in the jet development the CCW vortex is rolled up and over the CW vortex such that the velocity induction between the counter rotating sides of the vortex ring is pointed downstream. \nThe streamwise velocity of the CW vortex then declines as the CCW vortex rolls back towards the wall and dissipates. \nThis deceleration continues as axis-switching deformations in the virtual vortex ring cause the CW rotating head of the structure to rotate off the wall and move upstream, relative to the legs.\n\nWhile the streamwise velocity of the CW vortex increases slowly between $x\/h=11$ and $x\/h=90$ it remains lower than the baseline flow speed for the duration of the measurement (figure~\\ref{fig:AllOrfVortVel}). \nThe advective speed of the hairpin vortices is slow because the hairpin head induces an upstream pointing velocity on itself. \nThe gradual convergence of the vortex streamwise velocity with the baseline flow speed comes about as the strength of self-induction drops due to dissipation of the vortex (figure~\\ref{fig:AllOrfVortVel}b). \nBy the end of the measurement domain the hairpin head travels at $0.8U_e$ which is similar to the speed at which naturally occurring hairpin vortex packets travel in turbulent BLs \\citep{Adrian_Meinhart_JFM_2000}.\n\nThe combination of a wall-normal pitch angle in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet, along with the self-induced deformations depicted in figure~\\ref{fig:AllOrfVortTraj}a, lead to low initial streamwise velocity in the CW rotating side of the vortex ring. \nOnce the axis switching deformations reorient the vortex ring major axis in the spanwise direction the streamwise component of the self-induced velocity drops significantly and the CW vortex is transported downstream by the cross-flow. \nThis motion in the later vortex ring is evident from the essentially zero value of $\\Delta u_{vc}\/U_e$ starting around $x\/h=28$. \nThe streamwise velocity of the vortex in the furthest downstream measurement is $0.97U_e$ due both to the propagation of the vortex ring to the edge of the BL (figure~\\ref{fig:AllOrfVortTraj}a), where the cross-flow speed is high, and to the passive nature of the ring's motion after the axis switching deformations are complete (figure~\\ref{fig:AllOrfVortVel}b).\n\nUnsurprisingly, the initial streamwise velocity of the CCW vortex is very high in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet because the vortex ring formed by the pitched orifice is oriented downstream and therefore induces a high streamwise velocity on itself (figure~\\ref{fig:AllOrfVortVel}b). \nThe induced velocity drops however as the core of the CW rotating vortex loses coherence. \nThe streamwise velocity of the CCW vortex reaches a minimum between $x\/h=15$ and $x\/h=20$, roughly where the CW vortex becomes undetectable. \nStreamwise velocity rises again starting around $x\/h=20$ as the CCW vortex deforms into an arch-shaped structure. \nThis arch-shaped vortex induces a downstream velocity on itself, and so these structures travel faster than the fluid in the baseline case (figure~\\ref{fig:AllOrfVortVel}b). \nAs the apex of the arch lifts off the wall, and rotates downstream relative to the legs the streamwise velocity continues to rise. \nBy $x\/h=70$ the CCW vortex is traveling at over $0.99U_e$ (figure~\\ref{fig:AllOrfVortVel}b) despite remaining relatively close to the wall (figure~\\ref{fig:AllOrfVortTraj}a). \n\nInterestingly, the dominant vortices in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets end up with roughly the same velocities downstream despite the periodic structures being very different from each other (figure~\\ref{fig:AllOrfVortVel}a). \nIn the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet the high streamwise velocity is due to rapid penetration of the vortex ring into the faster moving fluid near the edge of the BL while the arch-shaped vortices in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet achieved a high streamwise velocity due to a self-induced velocity that is oriented downstream. \nThe relatively slow streamwise velocity of the vortices in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet is in turn caused by only moderate penetration of the periodic structures into the BL and a self-induced velocity that was pointed upstream. \nThese factors explain the significant differences in streamwise spacing of the dominant vortices in the three jets observed in figure~\\ref{fig:AllOrfVortTraj}b.\n\n\\subsection{Velocity field near the wall}\n\\label{sec:wallU}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_all_orifices_Uo12_Uinf12_floorplane_Udiff_hy2.eps}\n\\caption{Contours of time-averaged streamwise velocity change near the wall ($y\/h=1$) for the (a) $\\alpha=90^\\circ$, $\\beta=0^\\circ$, (b) $\\alpha=90^\\circ$, $\\beta=90^\\circ$, and (c) $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets.}\n\\label{fig:AllOrfWallVel}\n\\end{figure}\n\nThe influence that the SJA has on the flow velocity near the wall is of particular interest because of the implication it has for wall shear stress and the susceptibility of the BL to separation. \nFigure~\\ref{fig:AllOrfWallVel} displays the normalized change in streamwise velocity for the three orifice geometries on a measurement plane at a height of $y\/h=1$ ($y\/\\delta=0.035$ or $y^+=29$). \nThe contour plots are from the time-averaged data sets, and the position of the orifice is indicated by a rectangular outline with flow traveling from left to right. \n\nA prominent feature of the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet in the near-wall plane is the region of reduced flow velocity downstream of the orifice (figure~\\ref{fig:AllOrfWallVel}a). \nInitially, the primary vortex is connected to the wall by the legs which induce an upstream pointing velocity that decelerates the fluid between them. \nAs the vortex contracts in the spanwise direction and the legs detach from the wall the region of decelerated flow shrinks in width. \nBetween $x\/h=25$ and $x\/h=35$ the hairpin vortex lifts off the wall, and the legs, which have reconnected in the streamwise direction, now induce high momentum fluid down around their sides and then inwards towards the jet centreline, increasing the velocity there. \nHowever, that redistribution of fluid is weakly opposed by the chevron-shaped secondary structures which rotate in the opposite direction as the hairpin legs (figure~\\ref{fig:AR18Beta0Sketch}a). \nThe net result is a persistent streak of slightly lower momentum fluid on centreline as compared with the fluid located to the sides of centreline (see arrow in figure~\\ref{fig:AllOrfWallVel}a). \nDespite the spanwise variability in the fluid momentum near the wall all of the fluid in that layer still travels faster than it did before actuation was applied (downstream of $x\/h \\approx 30$).\n\nIn the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet the streamwise velocity is higher than it was in the baseline flow almost everywhere in the near-wall plane (figure~\\ref{fig:AllOrfWallVel}b). \nA region of low velocity fluid does not exist downstream of the orifice in this case because the relatively weaker legs emanating from the vortex rings do not remain connected to the wall past the end of the orifice.\nDownstream of the orifice the region of highest streamwise velocity is found on centreline where the upwash from the vortex ring is strongest. \nRegions of slightly lower velocity on either side of centreline (see arrows in figure~\\ref{fig:AllOrfWallVel}b) sit just outboard of two streaks of secondary streamwise vorticity found very close to the wall. \nThat streamwise vorticity, which rotates counter to the streamwise vorticity in the vortex ring, may be generated by the viscous interaction of spanwise velocity induction from the primary vortex and the no-slip condition at the wall. \n\nThe effectiveness of the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJA in enhancing streamwise velocity near the wall is remarkable when compared with the performance of the wall-normal jets (figure~\\ref{fig:AllOrfWallVel}c). \nThe highest streamwise velocity occurs in the region enclosed by the arch-shaped vortices. \nThis region narrows downstream of the orifice as the arch-shaped vortices contract in width. \nHowever, even outside of the footprint of the primary vortices streamwise velocity is elevated as a result of the actuation, presumably due to mixing and diffusion of momentum.\n\n\\subsection{Mean mixing in a midfield plane}\n\\label{sec:meanmix}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5.25in]{figures\/CompFig_All_orifices_Uo12_Uinf12_omegax_cont_w_vectors_hy.eps}\n\\caption{Contours of time-averaged streamwise vorticity overlaid with velocity vectors on the $x\/h=25$ plane for the (a) $\\alpha=90^\\circ$, $\\beta=0^\\circ$, (b) $\\alpha=90^\\circ$, $\\beta=90^\\circ$, and (c) $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets.}\n\\label{fig:AllOrfMidfield}\n\\end{figure}\n\nThe advecting train of vortices downstream of the actuator lead to a rotational mean flow within the SJBLI domain, as revealed in the time-averaged streamwise vorticity contours of figure~\\ref{fig:AllOrfMidfield} on the $x\/h=25$ plane. \nOverlain on the contour plot are the in-plane time-average velocity vectors, which have a consistent scale factor applied across the three panels. \n\nIn all of the jets the advecting periodic structures establish counter-rotating patches of fluid in the mean field such that the common flow on centreline is pointed upward off the wall. \nIt is tempting to interpret the time-average vorticity field as representing ``vortices'' but in an unsteady flow field this interpretation is inaccurate and can easily be misleading. \nTo be rigorous these fields should be viewed as the result of the mean influence of the evolving and advecting vortices within the unsteady field. \nIn the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet the mean rotation is induced by the sides and legs of the hairpin vortices (figure~\\ref{fig:AllOrfMidfield}a). \nIn this jet the rotation is strong and proximate to the wall which makes the actuation particularly effective at entraining fast moving fluid into the lower BL. \n\nIn the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet the centres of the mean flow rotation, seen at a height of $y\/h \\approx 16$ in figure~\\ref{fig:AllOrfMidfield}b, align with the ends of the vortex ring, which by $x\/h=25$ has completed its axis switch and is now widest in the spanwise direction. \nThe influence of the legs and interlinking vortex structures is apparent closer to the wall between $y\/h=5$ and 10. \nIn the early vortex ring, while the long sections of the vortex are aligned in the streamwise direction and are relatively close together in the spanwise direction, the wall-normal velocity of the mean flow is very high. \nThis region of rapid mean upwash occurs directly above the orifice. \nBut as the vortex ring axis-switches and the ends of the ring --- which carry the streamwise vorticity --- move apart, mean upwash on centreline drops. \nThus by $x\/h=25$ the average wall-normal flow speed on centreline is higher in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet than in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet. \nThe rapid upward propagation of the vortex rings in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet also causes the induced flow redistribution to act further from the wall. \nThis lessening influence of the primary vortex on the fluid at lower $y\/h$ is mitigated somewhat by the trailing legs which remain closer to the wall.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_all_orifices_Uo12_Uinf12_exitplane_Udiff_hy.eps}\n\\caption{Contours of time-averaged streamwise velocity change at $x\/h \\approx 100$ for the (a) $\\alpha=90^\\circ$, $\\beta=0^\\circ$, (b) $\\alpha=90^\\circ$, $\\beta=90^\\circ$, and (c) $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets.}\n\\label{fig:AllOrfExitPlane}\n\\end{figure}\n\nThe mean redistribution of flow is generally weakest in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (figure~\\ref{fig:AllOrfMidfield}c). \nThis close to the orifice the head of the arch-shaped vortex is still leaned upstream, introducing a streamwise component to rotation in the mean flow field around $y\/h=7$. \nUnlike the wall-normal orifices, however, the mean rotation does not persist downstream because the straightening out of the arch-shaped vortex tilts the streamwise component of the vorticity into the wall-normal direction. \n\n\\subsection{Velocity field in the \\texorpdfstring{$x\/h=100$}{TEXT} plane}\n\\label{sec:exitplaneu}\n\nFigure~\\ref{fig:AllOrfExitPlane} displays the normalized change in streamwise velocity at the end plane of the measurement domain ($x\/h \\approx 100$) for all three orifice geometries tested. \nActuation enhanced the near-wall flow velocity in all cases, but the flow fields further from the wall differ significantly from each other. \nThe $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA produced a roughly circular wake which sat above a region of accelerated flow near the wall (figure~\\ref{fig:AllOrfExitPlane}a). \nThe $x\/h \\approx 100$ plane of the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet looks similar to the spanwise-oriented jet, albeit stretched in the $y$-direction (figure~\\ref{fig:AllOrfExitPlane}b). \nThe streamwise-oriented jet penetrates almost twice as deeply into the cross-flow as the spanwise-oriented one, and therefore the wake sits much further from the wall. \nA larger area of the BL has elevated streamwise velocity in the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet case, but the magnitude of the increase is generally lower than it is in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet. \nThe flow field of the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet is distinct from the other two jets in that a localized wake did not form (figure~\\ref{fig:AllOrfExitPlane}c). \nInstead, a diffuse region with a relatively small reduction in streamwise velocity exists above the jet. \nUnsurprisingly, the core region of accelerated fluid near the wall takes the shape of a semi-ellipse that mirrors the projection of the arch-shaped vortices onto the $x$-plane. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/CompFig_All_orifices_Uo12_Uinf12_Tavg_centerline_exit_xavg_U_N_Udiff_profile_lt.eps}\n\\caption{Centerline time-averaged (a) streamwise velocity and (b) change in streamwise velocity profiles at $x\/h \\approx 100$ for the baseline flow (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black!40!white,solid,line width = 0.5pt](0,0) -- (7mm,0);}}), $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{}), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (\\sampleline{dash pattern=on .475em off .3em on .475em off .3em}), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{dash pattern=on .55em off .2em on .05em off .2em}).}\n\\label{fig:AllOrfBLProf}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/CompFig_All_orifices_Uo12_Uinf12_Tavg_exit_plane_momentum_N_delta_momentum_flux_scf.eps}\n\\caption{Time-averaged (a) streamwise momentum and (b) change in streamwise momentum profiles at $x\/h \\approx 100$ for the baseline flow (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black!40!white,solid,line width = 0.5pt](0,0) -- (7mm,0);}}), $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{}), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (\\sampleline{dash pattern=on .475em off .3em on .475em off .3em}), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{dash pattern=on .55em off .2em on .05em off .2em}).}\n\\label{fig:AllOrfMomProf}\n\\end{figure}\n\nThe centreline velocity profile and the change in streamwise velocity are compared for the three orifice geometries at the end of the measurement domain in figure~\\ref{fig:AllOrfBLProf}. \nThe velocity profiles were obtained by averaging the SPIV data from the last 3~mm of the measurement domain in the streamwise direction up to $x\/h \\approx 100$. \nThe velocity profile of the pitched orifice exceeds the velocity of the wall-normal orifices at all $y$, in some places by over 10\\% of $U_e$ because the pitched orifice jet has both a shallower ``wake'' and induces higher flow speeds near the wall. \nConversely, the maximum velocity deficit in the wakes of the wall-normal jets were roughly the same despite being offset by $y\/\\delta=0.3$ in height relative to each other (figure~\\ref{fig:AllOrfBLProf}b).\nBetween the wake region and the region of enhanced velocity near the wall, a transitional layer exists in all three cases where $\\frac{\\partial u}{\\partial y}$ is roughly zero (figure~\\ref{fig:AllOrfBLProf}a). \n\nThe enhancement of the flow velocity near the wall by the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJA is not only much larger than it is in the other two cases, but the region of elevated velocity is also much thicker in the $y$-direction (figure~\\ref{fig:AllOrfBLProf}). \nIn terms of near-wall flow speed on centreline, the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet is a distant second. \nBut while the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet produced a larger velocity enhancement on centreline than the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet, away from the centreline the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet was more effective at increasing the near-wall flow speed.\n\nTo account for the effect of the actuation across the entire plane, the fluid momentum was integrated in the spanwise direction. \nThe resulting momentum profiles are compared in figure~\\ref{fig:AllOrfMomProf}, where $\\dot{M}_e$ is the momentum of the flow outside the BL and $\\Delta \\dot{M}_(y)$ is the difference in the integrated momentum between the actuated and baseline flows.\nAccounting for the flow field variability in the spanwise direction reduces the relative differences in performance between the pitched and wall-normal orifices. \nFurthermore, while the pitched jet still achieves higher flow momentum than the other two cases do at all $y\/\\delta$, the slope of the momentum profiles in figure~\\ref{fig:AllOrfMomProf}b suggest that closer to the wall the wall-normal jets may match or exceed the momentum of the pitched jet. \nWhat is clear from the measurement data is that on average the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet is more effective at enhancing the momentum of the near-wall BL than the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet is when the full measurement domain is considered.\n\n\\subsection{Unsteady velocity field on the jet centreline}\n\\label{sec:unsteadymix}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_all_orifices_Uo12_Uinf12_Re_stress_hy2.eps}\n\\caption{Time-averaged Reynolds stress contours on the jet centreline for the (a) $\\alpha=90^\\circ$, $\\beta=0^\\circ$, (b) $\\alpha=90^\\circ$, $\\beta=90^\\circ$, and (c) $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJAs.}\n\\label{fig:AllOrfReStress}\n\\end{figure}\n\nTo analyze the unsteady flow field within the SJBLI domain a Reynolds decomposition of the instantaneous velocity field was conducted. \nPer \\ref{eq:uprime} the velocity fluctuations ($u'$) were computed by subtracting the time-averaged velocity ($\\overline{u}$) from the instantaneous velocity. \nNote that the velocity fluctuation $u'$ is composed of a combination of the periodic and stochastic fluctuations. \nTo investigate unsteady mixing, Reynolds shear stress ($\\langle u'v' \\rangle$) contour plots on the centreline of the three jets are presented in figure~\\ref{fig:AllOrfReStress}. \nA particular Reynolds stress value can be assigned to one of four quadrants based on where a specific instantaneous velocity falls on a plot of $u'$ vs $v'$. \nNegative values of $\\langle u'v' \\rangle$ are associated with enhanced mixing within the BL corresponding to either ``ejections'' --- which move low momentum fluid away from the wall (quadrant two: $u'<0$, $v'>0$) --- or ``sweeps'' --- which entrain high momentum fluid towards the wall (quadrant four: $u'>0$, $v'<0$). \nConversely, positive values of $\\langle u'v' \\rangle$ enhance the natural gradient of momentum within a BL by either moving low momentum fluid towards the wall (quadrant three: $u'<0$, $v'<0$) or by moving high momentum fluid away from the wall (quadrant one: $u'>0$, $v'>0$).\n\n\\begin{equation} \n\\label{eq:uprime}\nu(\\bm{x},t)=\\overline{u}(\\bm{x})-u'(\\bm{x},t)\n\\end{equation}\n\nAll three jets exhibit similar features in the flow field around the orifice (figure~\\ref{fig:AllOrfReStress}). \nUpstream of the orifice a region of negative Reynolds shear stress is associated with the localized influence of the suction phase of the actuator which draws higher momentum fluid down towards the actuator orifice (Q4). \nAlternatively, the high velocity jet produced by the blowing cycle of the actuator which is bent downstream by the cross-flow is responsible for the region of positive Reynolds shear stress around the orifice's downstream corner (Q1). \nDownstream of the actuator orifice the unsteady flow fields of the pitched and wall-normal jets follow different trends.\n\nIn the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jets a streak of negative $\\langle u'v' \\rangle$ traces a trajectory which generally follows the path swept out by the primary vortex structures. \nIn this region of the flow field the action of the synthetic jet vortices enhances unsteady mixing within the BL. \nConversely, in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet the region of positive $\\langle u'v' \\rangle$ originating at the orifice extends into the far field. \nIn this jet the periodic ejections of high momentum fluid from the pitched jet present as Q1 events and unsteady mixing is considerably lower than it is in the other two jets.\n\nTo quantify the intensity of the velocity fluctuations induced by the different jets, the kinetic energy associated with the fluctuations was computed. \nHere the quantity $K(x)$ was obtained by integrating the local kinetic energy of the velocity fluctuations in the wall-normal ($y$) and spanwise ($z$) directions (\\ref{eq:Kfluc}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/All_orifices_Uo12_Uinf12_Tavg_3D3D_integrated_tke_streamwise_profile_lt.eps}\n\\caption{Streamwise variation in the time-averaged, integrated fluctuation kinetic energy for the baseline flow (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black!40!white,solid,line width = 0.5pt](0,0) -- (7mm,0);}}), $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{}), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (\\sampleline{dash pattern=on .475em off .3em on .475em off .3em}), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{dash pattern=on .55em off .2em on .05em off .2em}) cases.}\n\\label{fig:AllOrfKfluc}\n\\end{figure}\n\n\\begin{equation} \n\\label{eq:Kfluc}\nK(x)=\\iint \\frac{1}{2} \\Big( \\big \\langle u'^2 \\big \\rangle + \\big \\langle v'^2 \\big \\rangle + \\big \\langle w'^2 \\big \\rangle \\Big) dydz\n\\end{equation}\n\nThe results for the baseline flow field and the three jets are presented in figure~\\ref{fig:AllOrfKfluc}. \nIn all three jets the fluctuating kinetic energy peaks a short distance downstream from the orifice and then decreases downstream as the vortices dissipate.\nThe kinetic energy is markedly higher in the jets which issue normal to the wall than it is in the pitched jet. \nThe $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet in particular maintains a higher magnitude of $K(x)$ in the far field because, after the vortex ring axis-switches, the majority of the ring length is aligned in the spanwise direction. \nWhile streamwise vortices induce mean mixing in the flow field, as shown in \\S~\\ref{sec:meanmix}, the alternating sign of the wall-normal velocity induced on the upstream versus the downstream side of spanwise-oriented vortices leads to high unsteadiness as the vortices advect downstream. \nFinally, fluctuating kinetic energy remains relatively low in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet, which is to be expected since pitching the orifice aligns the jet closer to the direction of the cross-flow which in turn decreases the magnitude of the mean shear stress associated with the SJBLI.\n\n\\section{Influence of actuation conditions}\n\\label{sec:results3}\n\nThe results presented in \\S~\\ref{sec:results1} and \\ref{sec:results2} were all obtained at a single condition, $C_b=1$, $St=1.33$ ($U_o$ = 12~ms$^{-1}$, $U_e$ = 12~ms$^{-1}$). \nIn the following section the influence of blowing ratio and Strouhal number on the SJBLI in the centreline plane, for all three orifice geometries, is explored.\n\n\\subsection{Trajectory and propagation speed of periodic structures}\n\\label{sec:vorttraj2}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_conditions_AR18_beta0_Tavg_centerline_vortex_trajectories_N_spacings2.eps}\n\\caption{$\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet centreline vortex centre (a) locations and (b) streamwise spacing from phase-locked data for $C_b=1.0$, $St=1.33$ ($\\circ$), $C_b=0.5$, $St=1.33$ ($\\vartriangle$), and $C_b=0.5$, $St=0.8$ ($\\lozenge$).}\n\\label{fig:AllCondVortTraj}\n\\end{figure}\n\nThe influence of the actuation and cross-flow conditions on the dominant vortex trajectory and streamwise spacing is depicted in figure~\\ref{fig:AllCondVortTraj}a and figure~\\ref{fig:AllCondVortTraj}b, respectively, for the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet. \nAdditionally, the absolute and relative streamwise velocity of the dominant vortices is presented in figure~\\ref{fig:AllCondVortVel}a and figure~\\ref{fig:AllCondVortVel}b, respectively. \nThe data presentation format and parameter definitions associated with figure~\\ref{fig:AllCondVortTraj} and \\ref{fig:AllCondVortVel} follow the descriptions given for figure~\\ref{fig:AllOrfVortTraj} and \\ref{fig:AllOrfVortVel} in \\S~\\ref{sec:results2}. \nFurthermore, the results at the $C_b=1$, $St=1.33$ condition, which were previously presented in figure~\\ref{fig:AllOrfVortTraj} and \\ref{fig:AllOrfVortVel}, are repeated here for comparison.\n\nComparing the $C_b=0.5 $, $St=1.33$ and $C_b=1.0$, $St=1.33$ SJBLI conditions provides insight into the effect of reducing $C_b$ while holding $St$ constant. \nReducing the synthetic jet average blowing velocity significantly decreases the distance the jet penetrates into the cross-flow (figure~\\ref{fig:AllCondVortTraj}a) which is an expected outcome since the ratio of the jet wall-normal momentum to the cross-flow streamwise momentum has dropped. \n\nThe reduction in $C_b$ between $C_b=1.0$, $St=1.33$ and $C_b=0.5 $, $St=1.33$ was achieved by reducing the average blowing velocity of the actuator. \nIn quiescent conditions the circulation strength of the vortex ring scales linearly with $U_o^2\/f$, all else being equal \\citep{Straccia_Farnsworth_JFM_2020}. \nThus it would be reasonable to predict that a reduction in $U_o$ from 12~ms$^{-1}$ to 6~ms$^{-1}$ should result in a drop in circulation strength by a factor of four. \nHowever, predicting how a reduction of $U_o$ would influence the circulation strength of the upstream (CCW) or downstream (CW) side of the vortex ring formed in a cross-flow is more complicated.\nSpecifically, as $C_b$ drops the outstroke velocity profile at the orifice exit becomes more skewed, such that CW vorticity production is favored over CCW vorticity \\citep{Sau_Mahesh_2008}. \nThe net result of reducing $C_b$ from 1 to 0.5 while holding $St$ constant is that the CW-rotating hairpin vortices produced by the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA have lower circulation strength, albeit by less than a factor of four. \nCirculation strength calculations conducted by integrating the velocity field along circuits that enclosed the vortex in the centreline phased-locked data indicated that the CW vortices at $C_b=0.5 $, $St=1.33$ were between two to three times weaker than at $C_b=1.0$, $St=1.33$. \nThe lower CW vortex strength affected not just the jet penetration but also the rate of the self-induced deformations which is evident from the fact that the S-curve trajectory of the CW vortex is much shallower at the lower $C_b$ (figure~\\ref{fig:AllCondVortTraj}a).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_conditions_AR18_beta0_Tavg_centerline_vortex_U_N_Udiff_speed.eps}\n\\caption{$\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet centreline vortex (a) absolute streamwise velocity and (b) change in streamwise velocity for $C_b=1.0$, $St=1.33$ ($\\circ$), $C_b=0.5$, $St=1.33$ ($\\vartriangle$), and $C_b=0.5$, $St=0.8$ ($\\lozenge$).}\n\\label{fig:AllCondVortVel}\n\\end{figure}\n\nThe vortex spacing for the two $St=1.33$ cases are fairly similar despite moderate differences in penetration height and circulation strength (figure~\\ref{fig:AllCondVortTraj}b). \nWhile the $C_b=0.5$, $St=1.33$ hairpin vortices remain closer to the wall in slower moving fluid (figure~\\ref{fig:AllCondVortTraj}a) than the $C_b=1.0$, $St=1.33$ vortices, they also induce a smaller upstream velocity on themselves due to their lower circulation strength, i.e., their $\\Delta u_{vc}\/U_e$ is closer to zero (figure~\\ref{fig:AllCondVortVel}b).\nThe net result is that the $C_b=0.5$, $St=1.33$ vortices travel somewhat slower (figure~\\ref{fig:AllCondVortVel}a) and are spaced slightly closer together in the streamwise direction (figure~\\ref{fig:AllCondVortTraj}b) than the $C_b=1.0$, $St=1.33$ vortices, but the difference is small.\n\nThe third SJBLI condition tested was achieved by doubling the cross-flow speed to $U_e$ = 24~ms$^{-1}$, while holding $U_o$ = 12~ms$^{-1}$, which reduced $St$ to 0.8 and $C_b$ to 0.5. \nAlthough the $C_b=0.5$, $St=0.8$ jet did not penetrate as deeply into the cross-flow as the $C_b=1$, $St=1.33$ jet, the difference is fairly small compared with the low penetration of the $C_b=0.5$, $St=1.33$ jet (figure~\\ref{fig:AllCondVortTraj}a). \nThis finding may come as something of a surprise given that the actuator driving conditions ($U_o$ and $f$) are the same between the $C_b=0.5$, $St=0.8$ and $C_b=1$, $St=1.33$ cases, but the vortices in the $C_b=0.5$, $St=0.8$ case encounter a much faster moving cross-flow. \nWhile these two jets would produce vortex rings with identical circulation strength in quiescent conditions the difference in $C_b$ alters how circulation is distributed around the ring. \nThe CW side of the vortex ring, which eventually develops into the hairpin vortices, is stronger in the $C_b=0.5$, $St=0.8$ jet than in the $C_b=1$, $St=1.33$ jet due to enhanced CW vorticity production in the more highly skewed orifice exit velocity profile. \nThe fact that the $C_b=0.5$, $St=0.8$ jet penetration depth and spatial rate of self-induced deformations (i.e., the slope of vortex S-curve trajectory) is only slightly lower than that of the $C_b=1$, $St=1.33$ jet implies that the CW vortices have around twice the circulation strength (figure~\\ref{fig:AllCondVortTraj}a), a supposition supported by vortex circulation strength calculations on centreline.\n\nThe lower $St$ of the $C_b=0.5$, $St=0.8$ jet leads to a larger streamwise spacing between CW vortices as compared with the $St=1.33$ jets (figure~\\ref{fig:AllCondVortTraj}b). \nIn fact, the ratio of $\\Delta x_{vc}\/\\delta(x=0)$ between the $C_b=0.5$, $St=0.8$ and $C_b=1.0$, $St=1.33$ jets is roughly 1.62, only slightly lower than the ratio of their $St$, which is 1.66. \nThe normalized relative streamwise velocity of the vortices in the $C_b=0.5$, $St=0.8$ jet is also similar to the $C_b=1.0$, $St=1.33$ jet, albeit slightly closer to zero indicating a slightly weaker self-induced velocity relative to the cross-flow speed (figure~\\ref{fig:AllCondVortVel}b).\n\nIt is interesting that the hairpin vortices at all three jet conditions travel downstream at similar rates relative to $U_e$, after the initial self-induced deformations subside (figure~\\ref{fig:AllCondVortVel}a). \nThis convergence of propagation speeds is a testament to the competing effects that the vortex trajectory and self-induction have on the vortex velocity when the vortex circulation strength is varied in the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_conditions_AR18_beta90_AR18P45_beta0_Tavg_centerline_vortex_Udiff_speed.eps}\n\\caption{Change in vortex streamwise velocity in the (a) $\\alpha=90^\\circ$, $\\beta=90^\\circ$ and (b) $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jets for $C_b=1.0$, $St=1.33$ ($\\circ$), $C_b=0.5$, $St=1.33$ ($\\vartriangle$), and $C_b=0.5$, $St=0.8$ ($\\lozenge$).}\n\\label{fig:AllCondVortVel2}\n\\end{figure}\n\nThe jet trajectory from the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ SJA follows similar trends with changing $C_b$ and $St$ as the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA case does. \nNamely, penetration is highest at $C_b=1$, $St=1.33$, somewhat lower at $C_b=0.5$, $St=0.8$, and much lower at $C_b=0.5$, $St=1.33$. \nThe most notable differences between the jets formed by the wall-normal orifice geometries occur in the vortex spacing and velocity. \nWhile the $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet vortex rings at a $C_b$ of 0.5 initially undergo similar self-induced deformations as they did at $C_b=1$ early in the jet development, the now weaker CCW side of the vortex ring has a diminished influence farther downstream. \nThe behaviour of the vortex diverges further from that of an ideal vortex ring with constant circulation strength about its circumference and is increasingly dominated by its relatively stronger CW rotating side. \nThe resulting $C_b=0.5$ periodic structures behave more like the hairpin vortices described in \\S~\\ref{sec:AR18Beta0}, inducing an upstream pointing velocity on themselves and thus travel at a slower streamwise velocity than the baseline flow (i.e., $\\Delta u_{vc}\/U_e<0$)(figure~\\ref{fig:AllCondVortVel2}a). \nTherefore, the vortices have a tighter streamwise spacing at $C_b=0.5$, $St=1.33$ than they did at $C_b=1$, $St=1.33$.\n\nThe influence changing $C_b$ and $St$ has on the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet is quite different from the situation with the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet because their dominant vortices rotate in opposite directions. \nReducing $C_b$ to 0.5 in the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet caused its BL penetration to drop significantly, for both of the low $C_b$ cases. \nHowever, for this orifice, the $C_b=0.5$, $St=0.8$ jet travelled the shortest distance from the wall.\nAs described earlier, reducing $C_b$ causes the CW side of the vortex ring to become stronger and the CCW side to become weaker. \nUnlike with the other orifice geometries this change further weakens the dominant vortex instead of mitigating some of the reduction in the strength of the vortex relative to the cross-flow. \nTherefore, instead of traveling faster than the baseline flow, as the arch-shaped vortices did at the $C_b=1$, $St=1.33$ conditions, the vortices at $C_b=0.5$, $St=0.8$ traveled at roughly the same speed as the baseline flow ($\\Delta u_{vc}\/U_e=0$), and the vortices at $C_b=0.5$, $St=1.33$ traveled somewhat slower ($\\Delta u_{vc}\/U_e<0$)(figure~\\ref{fig:AllCondVortVel2}b).\nThe result is a tighter streamwise spacing between the periodic structures at $C_b=0.5$, $St=1.33$ than at $C_b=1.0$, $St=1.33$, despite the actuation frequency being the same at the two conditions.\n\n\\subsection{Velocity profile on centreline at \\texorpdfstring{$x\/h=100$}{TEXT}}\n\\label{sec:exituprof}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/CompFig_All_orifices_Uo6_Uinf12_Tavg_centerline_exit_xavg_U_N_Udiff_profile_lt.eps}\n\\caption{Centerline time-averaged (a) streamwise velocity and (b) change in streamwise velocity profiles at $x\/h \\approx 100$ for the baseline flow (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black!40!white,solid,line width = 0.5pt](0,0) -- (7mm,0);}}), $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{}), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (\\sampleline{dash pattern=on .475em off .3em on .475em off .3em}), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{dash pattern=on .55em off .2em on .05em off .2em}) at $C_b=0.5$, $St=1.33$.}\n\\label{fig:AllOrfBLProf2}\n\\end{figure}\n\nThe velocity profiles at $C_b=0.5$, $St=1.33$ and $C_b=0.5$, $St=0.8$ for all three orifices are presented in figures \\ref{fig:AllOrfBLProf2} and figure~\\ref{fig:AllOrfBLProf3}, respectively.\nTo compare the effect of actuation at all three SJBLI conditions tested figure~\\ref{fig:AllOrfBLProf} should be referenced in conjunction with figures \\ref{fig:AllOrfBLProf2} and figure~\\ref{fig:AllOrfBLProf3}.\n\nOne striking observation when comparing figures \\ref{fig:AllOrfBLProf}, \\ref{fig:AllOrfBLProf2}, and \\ref{fig:AllOrfBLProf3} is that the superiority of the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJA at $C_b=1.0$ does not extend to either of the $C_b=0.5$ conditions tested. \nIn fact, at $C_b=0.5$, $St=1.33$ the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJA actually decreased the centreline flow velocity everywhere within the BL (figure~\\ref{fig:AllOrfBLProf2}b). \nThe results are somewhat more favorable at $C_b=0.5$, $St=0.8$ where the $\\alpha=45^\\circ $, $\\beta=0^\\circ $ jet did enhance near-wall flow velocity, but the effect is much weaker than at $C_b=1.0$, $St=1.33$ and more localized (figure~\\ref{fig:AllOrfBLProf3}b). \n\nAt both conditions plotted in figures \\ref{fig:AllOrfBLProf2} and figure~\\ref{fig:AllOrfBLProf3} the $\\alpha=45^\\circ$, $\\beta=0^\\circ$ SJA was the least effective of the three actuators at enhancing the near-wall flow speed, highlighting an important difference between the flow control mechanisms of the pitched versus wall-normal orifices. \nWhile the wall-normal orifices modify the BL shape via mixing and flow redistribution within the BL the pitched orifice acts primarily through direct momentum addition. \nThus the effectiveness of a pitched orifice is highly dependent on the $C_b$. \nIf $C_b$ is set at a level such that the streamwise component of the outstroke is too low relative to the flow speed near the wall then the actuation may act as a net sink on the fluid momentum (see for example figure~\\ref{fig:AllOrfBLProf2}b). \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=4in]{figures\/CompFig_All_orifices_Uo12_Uinf24_Tavg_centerline_exit_xavg_U_N_Udiff_profile_lt.eps}\n\\caption{Centerline time-averaged (a) streamwise velocity and (b) change in streamwise velocity profiles at $x\/h \\approx 100$ for the baseline flow (\\protect\\raisebox{2pt}{\\tikz{\\draw[-,black!40!white,solid,line width = 0.5pt](0,0) -- (7mm,0);}}), $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{}), $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet (\\sampleline{dash pattern=on .475em off .3em on .475em off .3em}), and $\\alpha=45^\\circ$, $\\beta=0^\\circ$ jet (\\sampleline{dash pattern=on .55em off .2em on .05em off .2em}) at $C_b=0.5$, $St=0.8$.}\n\\label{fig:AllOrfBLProf3}\n\\end{figure}\n\nAt $C_b=0.5$, $St=1.33$ the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=90^\\circ$, $\\beta=90^\\circ$ SJAs had a similar influence on the centreline BL profile as they did at $C_b=1.0$, $St=1.33$, but the modifications are merely more muted (figure~\\ref{fig:AllOrfBLProf2}b). \nSpecifically, the wakes are shallower in depth and located closer to the wall while the velocity enhancement at the wall is weaker. \nThese changes are to be expected for a reduction in the momentum output by the actuator.\n\nAt $C_b=0.5$, $St=0.8$ the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ and $\\alpha=90^\\circ$, $\\beta=90^\\circ$ jet velocity profiles also contain shallow wakes which remain relatively close to the wall (figure~\\ref{fig:AllOrfBLProf3}b). \nWhat is interesting is that the near-wall flow velocity enhancement is still very robust in this case (figure~\\ref{fig:AllOrfBLProf3}b). \nIn fact, the enhancement of $\\Delta u\/U_e$ near the wall by the $\\alpha=90^\\circ$ SJAs at $C_b=0.5$, $St=1.33$ is similar in magnitude to what was observed at $C_b=1.0$, $St=1.33$, despite operating at a lower blowing ratio (compare figure~\\ref{fig:AllOrfBLProf}b to figure~\\ref{fig:AllOrfBLProf3}b). \nIn the following sections the analysis of the mean and fluctuating velocity fields on the jet centreline reveals some interesting differences between the influence of the actuation at $C_b=1.0$, $St=1.33$ and $C_b=0.5$, $St=0.8$.\n\n\\subsection{Average vertical mixing on the jet centreline}\n\\label{sec:deltavfield}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_conditions_AR18_beta0_Vdiff_hy2.eps}\n\\caption{Contours of time-averaged wall-normal velocity change on the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet centreline for (a) $C_b=1.0$, $St=1.33$, (b) $C_b=0.5$, $St=1.33$, and (c) $C_b=0.5$, $St=0.8$.}\n\\label{fig:AllCondvfield}\n\\end{figure}\n\nTo compare the mean flow redistribution induced by the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA at the three conditions tested, contours of $\\Delta v$ were plotted on the centreline plane in figure~\\ref{fig:AllCondvfield}. \nAlthough the magnitude of the induced wall-normal velocity on centreline varies between the cases the prominent flow field features are the same. \nJust upstream and downstream of the orifice a region of downward moving fluid (blue contours) is the manifestation of the localized influence of the suction cycle along with the downwards velocity induction around the early vortex ring.\nThe path of the jet out of the orifice during the blowing cycle introduces a streak of upward moving fluid through the middle of the blue contours. \nThis region of upward moving fluid in the mean velocity field extends downstream from the orifice, with peak upwash occurring between $x\/h=10$ and 15 before decreasing downstream. \n\nThe relative strength of the upwash at the two $C_b=0.5$ conditions was lower than at $C_b=1.0$, which is the expected outcome of reducing the blowing ratio (figure~\\ref{fig:AllCondvfield}). \nOf the two $C_b=0.5$ cases, the $C_b=0.5$, $St=0.8$ jet achieved a higher peak $\\Delta v\/U_e$ value than the $C_b=1.0$, $St=1.33$ jet did, although, it still had only half the peak $\\Delta v\/U_e$ value measured in the $C_b=1.0$, $St=1.33$ jet.\nWhile it is unsurprising that changing the actuation conditions from $C_b=1.0$, $St=1.33$ to $C_b=0.5$, $St=0.8$ led to a decrease in the relative strength of upwash on centreline, the implied reduction in mean flow redistribution within the BL is at odds with the noteworthy effectiveness of the $C_b=0.5$, $St=0.8$ jet in enhancing the near-wall flow velocity, as seen in figure~\\ref{fig:AllOrfBLProf3}b.\nTherefore, another factor must be at play.\n\n\\subsection{Reynolds stress field on the jet centreline}\n\\label{sec:Restressfield}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=5in]{figures\/CompFig_All_conditions_AR18_beta0_Re_stress_hy2.eps}\n\\caption{Time-averaged Reynolds stress contours on the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ jet centreline for (a) $C_b=1.0$, $St=1.33$, (b) $C_b=0.5$, $St=1.33$, and (c) $C_b=0.5$, $St=0.8$.}\n\\label{fig:AllCondReStress}\n\\end{figure}\n\nTo help understand the surprisingly good performance of the $\\alpha=90^\\circ$, $\\beta=0^\\circ$ SJA at $C_b=0.5$, $St=0.8$ a Reynolds decomposition, per \\ref{eq:uprime}, of the instantaneous velocity fields was conducted at the three driving conditions. \nThe resulting Reynolds stress fields on centreline are presented in figure~\\ref{fig:AllCondReStress}. \nNote that the data plotted in figure~\\ref{fig:AllCondReStress}a are the same as in figure~\\ref{fig:AllOrfReStress}a but with the contour levels now rescaled for easier comparison with the $C_b=0.5$ condition results. \nWhat is conspicuous in these results is the remarkably high Reynolds shear stress in the $C_b=0.5$, $St=0.8$ jet as compared with the $C_b=1.0$, $St=1.33$ condition, despite the blowing ratio being much lower. \nIn fact the peak value of $\\langle u'v' \\rangle$ at $C_b=0.5$, $St=0.8$ is 2.7 times higher in magnitude than measured at $C_b=1.0$, $St=1.33$. \n\nDifferences in vortex spacing may be one factor that contributes to the observed disparity in unsteady mixing.\nAt high $St$ subsequent generations of periodic structures are spaced more closely together, and the velocity induction from the spanwise portion of one vortex destructively interferes with the velocity induction of its neighboring structures. \nReducing $St$ increases the spacing between the structures which in turn reduces the mutual interaction between different generations of vortices and allows each vortex to induce stronger unsteady vertical mixing. \nFurthermore, the formation of relatively stronger CW rotating vortices at the lower blowing ratio --- as described in \\S~\\ref{sec:vorttraj2} --- also enables strong velocity induction by the hairpin vortices. \nThe phase-locked velocity fields (not presented here) support this interpretation. \nThus while the relative strength of the mean mixing was lower at $C_b=0.5$, $St=0.8$ as compared with $C_b=1.0$, $St=1.33$, the unsteady mixing was much higher, resulting in an unexpectedly strong enhancement of the near-wall flow speed at the lower $C_b$ and $St$. ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{The classical Heisenberg model}\n\nWe start with the definition of the classical Heisenberg  model.\nAs a classical Hamiltonian system, we need to introduce the phase space\nand its symplectic structure as well as the Hamiltonian. The classical\nvariables are the following spin variables $S(x)$:\n \\begin{eqnarray} \nS(x)=\\sum_{i=1}^3 S^i(x) \\sigma_i\\quad\n{\\rm with}\\quad \\sum_{i=1}^3 S^i(x)^2=s^2 \\nonumber\n \\end{eqnarray} \nwhere the $\\sigma_i$ are the Pauli matrices with $[\\sigma}\t\\def\\Sig{\\Sigma_i,\\sigma}\t\\def\\Sig{\\Sigma_j]=\n2i\\epsilon_{ijk}\\sigma}\t\\def\\Sig{\\Sigma_k$ and $tr(\\sigma}\t\\def\\Sig{\\Sigma_j\\sigma}\t\\def\\Sig{\\Sigma_k)=2\\delta}\t\t\\def\\De{\\Delta_{jk}$.\nHere and below, $s$ is a fixed real number.\nThe Poisson bracket are defined by\n \\begin{eqnarray} \n\\{ S^i(x),S^j(y)\\} = \\epsilon^{ijk} S^k(x)\\delta(x-y) \\label{Ki}\n \\end{eqnarray} \nThey are symplectic thanks to the constraint $\\sum_i S^i(x)^2 = s^2$.\nThe Hamiltonian is\n \\begin{eqnarray} \nH_1=-{1\\over 4}\\int_0^L dx~ tr(\\partial_x S \\partial_x S) \\label{Kii}\n \\end{eqnarray} \nThe equations of motion deduced from this Hamiltonian are:\n \\begin{eqnarray} \n\\partial_t S\n=-{\\textstyle{i\\over 2}} \\Bigr[ S ,\\partial_x^2 S \\Bigr]={\\textstyle{i\\over 2}}\n\\partial_x \\Bigr[\\ S_x ,S\\ \\Bigr]  \\label{EHv}\n \\end{eqnarray} \nNotice that these equations are the conservation laws for a $su(2)$-valued\ncurrent.\nAs is well known, the Heisenberg model is a completely integrable model.\nIts integrability relies on the fact that the equations (\\ref{EHv})\ncan be written as a zero curvature condition for an auxiliary linear\nproblem. Namely, consider the Lax connexion,\n \\begin{eqnarray} \nA_x& =& {i\\over \\lambda}S(x)  \\nonumber\\\\ \nA_t& =&- {{2is^2}\\over \\lambda^2} S(x) +\n{1\\over{2\\lambda}}[S(x),\\partial_x S(x)]\\label{EHvii}\n \\end{eqnarray} \nthen, the zero curvature condition, $\\[ \\partial_t + A_t , \\partial_x + A_x \\] = 0$,\nis equivalent to the equations of motion.\nNote that the Lax connexion (\\ref{EHvii}) is an element of the $su(2)$ loop\nalgebra ${\\widetilde {su(2)}}=su(2)\\otimes C\\[\\lambda}\t\\def\\La{\\Lambda,\\lambda}\t\\def\\La{\\Lambda^{-1}\\]$.\n\nAn important ingredient is the transfer matrix $T(x,\\lambda)$;\nit is defined as\n \\begin{eqnarray} \nT(x,\\lambda)&=&P \\exp\\left[-\\int_0^x A_x(y,\\lambda) dy\\right]\n\\label{EHviii}\n \\end{eqnarray} \nThe monodromy matrix $T(\\lambda)$ is simply $T(L,\\lambda)$.\n From its definition, $T(x,\\lambda)$ is analytic in $\\lambda$\nwith an essential singularity at $\\lambda =0$. From eq.(\\ref{EHviii}), we can\neasily find an expansion around $\\lambda =\\infty $:\n \\begin{eqnarray} \nT(x,\\lambda )= 1 -{\\textstyle{i\\over \\lambda}}\\int_0^x dy S(y)\n -{\\textstyle{1\\over \\lambda^2}}\\int_0^x dy S(y) \\int_0^y dz S(z) + \\cdots\n\\nonumber\n \\end{eqnarray} \nThis development in $1\/\\lambda}\t\\def\\La{\\Lambda$ has an infinite radius of convergence.\nWe will study it later in more details in relation with the\nnon-Abelian symmetries of the model.\n\nAs is well known, the importance of the monodromy matrix lies in\nthe fact that one can calculate the Poisson bracket of its  matrix\nelements. One finds \\cite{Sk79}:\n \\begin{eqnarray} \n\\Bigl\\{ T(\\lambda) \\stackrel{\\otimes}{,} T(\\mu)\\Bigr\\}={\\textstyle{1\\over 2}}\n\\Bigr[\\ r(\\lambda,\\mu)\\ ,\\ T(\\lambda)\\otimes T(\\mu)\\ \\Bigr] \\label{EHix}\n \\end{eqnarray} \nwhere\n \\begin{eqnarray} \nr(\\lambda ,\\mu)= \\inv{ \\lambda-\\mu}\\ {\\sum_i \\sigma_i \\otimes \\sigma_i}\n\\label{EHx}\n \\end{eqnarray} \n From this result, it follows that $tr(T(\\lambda))$ is a generating function\nfor quantities in involution.\n\nAssuming periodic boundary conditions, the monodromy matrix $T(\\lambda)$\ncan be written as follows:\n$T(\\lambda)= \\cos P_0(\\lambda)~{\\rm Id} +i \\sin P_0(\\lambda)~ M(\\lambda)$,\nwith $M(\\lambda)$ traceless.\nTherefore, the trace of the transfer matrix is:\n$tr(T(\\lambda ))=2 \\cos P_0(\\lambda)$, and\nwe can use $P_0(\\lambda)$ as a generating function\nfor the commuting conserved quantities. The local conserved charges\nare found by expanding not around $\\lambda=\\infty$ but around\n$\\lambda= 0$:\n \\begin{eqnarray} \nP_o(\\lambda) = -\\frac{sL}{\\lambda}\n+ \\sum_{n=0}^{\\infty} \\lambda^n\\ I_n \\nonumber\n \\end{eqnarray} \nThe quantities $I_n$ are integral of local densities.\nThe first two, $I_0$ and $I_1$, correspond to momentum and energy respectively.\n\n\\subsection{ Non-Abelian symmetries.}\n\nClearly, since the Hamiltonian is a $su(2)$ scalar, the equations\nof motion are $su(2)$ invariant. Therefore, for any element $v\\in su(2)$,\nthe transformation,\n \\begin{eqnarray} \n\\delta_v\\ S(x)\\ =\\ i\\Bigr[\\ v\\ ,\\ S(x)\\ \\Bigr] \\nonumber\n \\end{eqnarray} \nis a symmetry of the equations of motion.\nActually, the symmetry group is much bigger.\nFor any $v\\in\\ su(2)$ and any non-negative integer $n$,\nthe transformations $S(x)\\to\\ \\delta_v^nS(x)$  defined by:\n\\begin{eqnarray}\n\\delta^0_v\\ S(x)\\ &=&\\ i\\Bigr[\\ v\\ ,\\ S(x)\\ \\Bigr]  \\nonumber\\\\ \n\\delta^n_v\\ S(x)\\ &=&\\ i\\Bigr[\\ Z^n_v(x)\\ ,\\ S(x)\\ \\Bigr]\\label{dt4}\n\\end{eqnarray}\nwhere the functions $Z^k_v(x)$ are  recursively computed  by\n\\begin{eqnarray}\n\\partial_x Z^k_v(x) +i \\Bigr[ S(x),Z^{k-1}_v(x)\\Bigr] =0\\qquad;\n\\qquad Z_v^0 =v \\label{dt3}\n\\end{eqnarray}\nare symmetries of the equations of motion.\n\nMoreover, these transformations form a representation of\nthe loop algebra (more precisely of the sub-algebra\n$su(2)\\otimes C[\\lambda]$ of the $su(2)$ loop algebra); i.e. we have:\n \\begin{eqnarray} \n\\Bigr[\\ \\delta_v^n\\ , \\delta_w^m\\ \\Bigr] = \\ \\delta_{[v,w]}^{n+m}\n \\end{eqnarray} \nfor any non-negative intergers $n$ and $m$, and any $v,\\ w \\in su(2)$.\nIn other words, the symmetry group is the loop group\n(more precisely the sub-group of the loop group which consists\nof loops regular at zero).\n\nThis can be proved as follows.\nFor any $v\\in su(2)$, let us define the functions $Z^k_v(x)$ by:\n\\begin{eqnarray}\n(TvT^{-1})(x,\\lambda}\t\\def\\La{\\Lambda) =\\sum_{k=0}^\\infty \\lambda^{-k} Z^k_v(x) \\nonumber\n\\end{eqnarray}\nwhere we have used the $(\\frac{1}{\\lambda})$ expansion of the transfer matrix.\nThe differential equations satisfied by these functions are consequences\nof those satisfied by the transfer matrix $T(x,\\lambda}\t\\def\\La{\\Lambda)$. For any positive\ninteger $n$, we set:\n$\\Theta_v^n(x,\\lambda) =i\\sum_{k=0}^n \\lambda^{n-k} Z_v^k(x)$.\nConsider now the following gauge transformations acting on the Lax connexion:\n \\begin{eqnarray} \n\\delta_v^n A_x &=&- [A_x, \\Theta_v^n] - \\partial_x \\Theta_v^n \\nonumber \\\\\n\\delta_v^n A_t &=&- [A_t, \\Theta_v^n] - \\partial_t \\Theta_v^n \\nonumber\n \\end{eqnarray} \nBy construction, these transformations preserve the zero curvature\ncondition, since they are gauge transformations.\nTherefore, they will be symmetries if\nthe form of the components, $A_t$ and $A_x$, of the  Lax connexion is\npreserved.\nIt is a simple exercise to check this fact; e.g. for $A_x$  we have:\n\\begin{eqnarray}\n\\delta_v^n A_x &=& \\lambda^{-1} [ S, Z^n_v ] -\n i\\sum_{k=0}^{n-1} \\left\\{ \\partial_x Z_v^{k+1}\n +i [S, Z_v^k] \\right\\}\\lambda^{n-k-1} \\nonumber \\\\\n &=& \\lambda^{-1}  [ S, Z^n_v ] \\nonumber\n \\end{eqnarray}\nThe last sum vanishes by vitue of eq.(\\ref{dt3}) and we are left with\n$\\delta^n_v S(x) = i\\[ Z^n_v(x),S(x)\\]$.\nOne can check similarly that the form of $A_t$ is unchanged, and\nits variation is compatible with eq.(\\ref{dt4}).\nThis proves that eqs.(\\ref{dt4}) define symmetries of the equations\nof motion.\n\nTo these symmetries correspond an infinite number of conserved currents.\nIndeed, as we already remarked the equations of motion\nhave the form of a conservation law $\\partial_t J_t - \\partial_x J_x =0$\nwith $ J_t =S$ and $J_x={i\\over 2}[S_x,S]$.\nSince the transformations are symmetries,\ntransforming this local current produces new\ncurrents which form an infinite multiplet of currents:\n\\begin{eqnarray}\nJ_t^{n,v} &=& \\delta^n_v S = i[Z_v^n,S ] \\nonumber \\\\\nJ_x^{n,v} &=&- {\\textstyle{1\\over 2}}[ \\partial_x [Z_v^n,S ],S]\n-{\\textstyle{1\\over 2}} [S_x ,[Z_v^n,S]] \\label{Ecur}\n\\end{eqnarray}\nfor any $n\\geq 0$ and $v\\in su(2)$.\nNote that these currents are non-local. By construction they are conserved:\n$ \\partial_t J^{n,v}_t - \\partial_x J^{n,v}_x =0$.\nTo them correspond charges which are defined by:\n\\begin{eqnarray}\nQ^n_v =\\int_0^L J_t^{n,v} (x) dx  = Z_v^{n+1}(L) \\label{Echar}\n\\end{eqnarray}\nSince the currents are non local, the charges are not conserved. We have\n\\begin{eqnarray}\n{d\\over dt}Q_v^n = J_x^{n,v} (L) - J_x^{n,v} (0) \\nonumber\n\\end{eqnarray}\nIn the infinite volume limit ($L\\to\\infty$), with an appropriate choice\nof the boundary conditions the charges can eventually be conserved.\nEven if they are not conserved, they are nevertheless important because,\nas we will see, they are the generators of the non-Abelian  transformations.\n\n\\subsection{ The generators of the symmetries and the\nsemi-classical Yangians.}\n\nWe now discuss in which sense the non-local charges $Q_v^n$\nare the generators of the non-Abelian symmetries (\\ref{dt4}); i.e.\nin which sense the infinitesimal transformation laws (\\ref{dt4}) of\nthe spin variables are given by Poisson brackets between\nthe non-local charges and the dynamical variables $S(x)$.\nAs we will see, contrary to what happens for symplectic\ntransformations, the infinitesimal variations $\\delta_v^nS(x)$\nare not linearly generated by the charges.\n\nThe charges $Q_v^n$ take values in the Lie algebra $su(2)$.\nLet us introduce their components, $Q_{ij}^n$ and\n$Q_i^n$, in the basis of the Pauli matrices $\\sigma_i$:\n \\begin{eqnarray} \nQ^n_{ij} ={1 \\over 2}\\ tr\\({ Q_{\\sigma_i}^n\\ \\sigma_j}\\)\\quad\n{\\rm and}\\quad\nQ^n_i = \\sum_{j,k=0}^3 \\epsilon_{ijk} Q_{jk}^n \\label{CIv}\n \\end{eqnarray} \nIn particular,  for $n=0$ and $n=1$, we have the simple formula:\n \\begin{eqnarray} \nQ_i^0 &=& 4 \\int_0^L\\ dx\\ S^i(x)  \\nonumber\\\\ \nQ_i^1 &=& 4 \\int_0^L\\ dx \\int_0^x dy\\ \\epsilon^{ijk}\\  S^j(x) S^k(y) \\label{CIvi}\n \\end{eqnarray} \nNotice that $Q_i^0$ are local whereas $Q_i^1$ are not.\nThey generate the transformations $\\delta_i^0S(y)$ and\n$\\delta^1_iS(y)$ in the following way:\n\\begin{eqnarray}\n\\delta^0_{i} S(y)&=&{\\textstyle{1\\over 2}}  \\{ Q^0_i,S(y) \\}  \\nonumber\\\\ \n \\delta_i^1 S(y)&=&{\\textstyle{1\\over 2}} \\{Q_i^1 ,S(y) \\}\n-{\\textstyle{1\\over 8}} \\epsilon^{ijk} Q_j^0 \\{ Q_k^0 ,S(y) \\}\n\\label{EZv}\n\\end{eqnarray}\nThe first equation in (\\ref{EZv}) implies that the $su(2)$\nsymmetry $\\delta^0_iS(y)$ is a symplectic action as it should\nbe. The non-linearity in the second equation is the sign that these\ntransformations are not symplectic but is characteristic to\nLie-Poisson actions, a notion which generalize the notion of symplectic\nactions.\n\nThe non-linearity in eq. (\\ref{EZv}) has another echo. The Poisson algebra\nof the charges $Q_i^0$ and $Q^1_i$ is not the $su(2)$ loop algebra\nbut a deformation of it. Indeed recall that the $su(2)$ loop algebra\ncan be presented as the associative algebra generated by the\nelements $\\delta^0_i$ and $\\delta^1_i$ satisfying the following relations:\n \\begin{eqnarray} \n\\Bigr[ \\delta_i^0,\\delta_j^0\\Bigr] &=&  \\epsilon_{ijk} \\delta_k^0  \\nonumber\\\\ \n\\Bigr[ \\delta_i^0,\\delta_j^1\\Bigr] &=&  \\epsilon_{ijk} \\delta_k^1 \\label{CIx}\\\\\n\\[\\delta^1_i,\\[\\delta^1_j,\\delta^0_k\\]\\]\n&-&\\[\\delta^0_i,\\[\\delta^1_j,\\delta^1_k\\]\\] =0  \\nonumber\\\\ \n\\[\\[\\delta^1_i,\\delta^1_j\\],\\[\\delta^0_k,\\delta^1_l\\] \\]\n&+&\\[\\[\\delta^1_k,\\delta^1_l\\],\\[\\delta^0_i,\\delta^1_j\\]\\]=0 \\nonumber\n \\end{eqnarray} \n\nOn the other hand, given the explicit expressions of the charges\n$Q_i^0$ and $Q_i^1$ in terms of the spin variables, one can\ncompute their Poisson brackets and check that they satisfy the following\nrelations:\n \\begin{eqnarray} \n\\{ Q_i^0,Q_j^0\\} &=& 4 \\epsilon_{ijk} Q^0_k  \\nonumber\\\\ \n\\{ Q_i^0,Q_j^1\\} &=& 4 \\epsilon_{ijk} Q^1_k \\label{CIxi}\\\\\n\\{Q^1_i,\\{Q^1_j,Q^0_k\\}\\} -\\{Q^0_i,\\{Q^1_j,Q^1_k\\}\\} &=&\nA_{ijk}^{lmn} Q^0_l Q^0_m Q^0_n  \\nonumber\\\\ \n\\{\\{Q^1_i,Q^1_j\\},\\{Q^0_k,Q^1_l\\} \\}\n+\\{\\{Q^1_k,Q^1_l\\},\\{Q^0_i,Q^1_j\\}\\} &=&\n8( A_{ija}^{mnp}\\epsilon_{kla} + A^{mnp}_{kla} \\epsilon_{ija}) Q^0_m Q^0_n Q^1_p\n\\nonumber\n \\end{eqnarray} \nwith $A_{ijk}^{lmn}= {\\textstyle{2\\over\n3}}\\epsilon_{ila}\\epsilon_{jmb}\\epsilon_{knc}\\epsilon^{abc}$\n\nOne sees that these relations form a deformation of those defining\nthe $su(2)$ loop algebra. Therefore the Poisson algebra of the charges\nis a deformation of the $su(2)$ loop algebra.\nIt is called the semi-classical $su(2)$ Yangian.\nThere is no extra relation between the charges $Q^0_n$ and $Q^1_k$ since\nthere is no extra relation between the generators $\\delta}\t\t\\def\\De{\\Delta^0_n$ and $\\delta}\t\t\\def\\De{\\Delta^1_k$\nin the $su(2)$ loop algebra.\n\\par\n\nAll the non-local charges $Q_{ij}^n$ can be expressed as multiple\nPoisson brackets between the two first charges $Q_i^0$ and $Q_i^1$.\nTherefore, the Poisson algebra of the symmetries is generated by these two\ncharges.\n(This fact is more easily proved by introducing the\ntransfer matrix and its Poisson brackets as it will be done\nin the next section.)\n\nAlso, since the $su(2)$ loop algebra is generated only by\n$\\delta^0_i$ and $\\delta^1_i$, all the infinitesimal transformations\n$\\delta^n_iS(y)$ can be expressed non-linearly in terms of\nthe non-local charges $Q_i^0$ and $Q_i^1$, or alternatively,\n in terms of the charges $Q^n_{ij}$. A close expression is:\n\\begin{eqnarray}\n\\delta^n_{i} S(y)= {\\textstyle{1\\over 2}} \\{ Q^n_i , S(y) \\}\n +{\\textstyle{1\\over 2}}\\sum_{p=0}^{n-1}\\left[ Q^{n-p-1}-tr(Q^{n-p-1})Id\n\\right]_{ik} \\delta_k^p S(y)\n\\nonumber\n\\end{eqnarray}\nThis equation can be interpreted in two different ways: (i)\nthis is a system of equations allowing to express recursively $\\delta_v^n\nS(y)$ in terms of Poisson brackets; or (ii) it allows to express the\ntransformed spins $\\delta}\t\t\\def\\De{\\Delta_v^nS$ in a non-linear way in terms of the charges $Q_i$.\n\n\n\\subsection{The transfer matrix and an alternative presentation\nof the semi-classical Yangians.}\n\nWe now show that the knowledge of the charges is equivalent to the\ndata of the monodromy matrix $T$. Let us introduce a notation for\nthe matrix elements of the monodromy matrix:\n\\begin{eqnarray}\nT(\\lambda)=\\pmatrix{A(\\lambda )& B(\\lambda ) \\cr C(\\lambda ) & D( \\lambda ) }\n\\qquad;\\qquad Det\\ T(\\lambda) = AD-BC=1  \\nonumber\n\\end{eqnarray}\nRecall that from eq. (\\ref{CIv}), we have:\n \\begin{eqnarray} \nQ_{ij}(\\lambda) &=& \\delta_{ij}+ \\sum_{n=0}^\\infty \\lambda^{-n-1} Q^n_{ij} =\n{1 \\over 2} tr( T \\sigma_i T^{-1} \\sigma_j )  \\nonumber\\\\ \nQ_i(\\lambda) &=& \\sum_{n=0}^\\infty \\lambda^{-n-1} Q^n_i =\n\\epsilon_{ijk} Q_{jk}(\\lambda) \\nonumber\n \\end{eqnarray} \nTherefore, the quantities $Q_{ij}(\\lambda)$ and $Q_i(\\lambda)$ are quadratic\nfunctions of the matrix elements of $T(\\lambda)$.\nWe also point out that it is possible to express the charges $Q_{ij}^n$\nin terms of $Q_i^n$.\n\nBut, one can invert these relations and express $T(\\lambda)$ in terms\nof the generating functions $Q_i(\\lambda)$.\nThe relation between the transfer matrix and the\nnon-local charges $Q_i(\\lambda}\t\\def\\La{\\Lambda)$ is:\n\\begin{eqnarray}\nT(\\lambda)=\n{\\inv{2}}W(\\lambda)\\ Id -{\\frac{i}{2}} W^{-1}(\\lambda)\n\\sum_i Q_i(\\lambda) \\sigma_i\n\\nonumber\n\\end{eqnarray}\nwith $ W(\\lambda)= \\sqrt{ 2 + \\sqrt{4-\\vec{Q}^2(\\lambda)} }$.\n\\par\nSo, the charges $Q_i(\\lambda)$ contains the same amount of\ninformation as the monodromy matrix.\nIt is interesting in this context to examine more closely the relation\nbetween the $Q_i(\\lambda)$ and the matrix elements of $T(\\lambda}\t\\def\\La{\\Lambda)$.\nWe have\n\\begin{eqnarray}\nQ_+(\\lambda) &=& Q_1(\\lambda) +i Q_2(\\lambda) = 2i\nW(\\lambda)\\  C(\\lambda) \\label{nlc1} \\\\\nQ_-(\\lambda) &=& Q_1(\\lambda) -i Q_2(\\lambda) = 2i\nW(\\lambda)\\  B(\\lambda) \\label{nlc2}\n\\end{eqnarray}\nMoreover, the trace of the transfer matrix is:\n$tr~T(\\lambda)=W[\\vec{Q}^2(\\lambda)]$.\nTherefore, $\\vec{Q}^2(\\lambda)$ is also a generating\nfunction for commuting quantities. Its relation with the generating\nfunction $P_0(\\lambda}\t\\def\\La{\\Lambda)$ is:\n \\begin{eqnarray} \n\\vec{Q}^2(\\lambda}\t\\def\\La{\\Lambda)\\ =\\ 4\\sin^2\\Bigr( 2P_0(\\lambda}\t\\def\\La{\\Lambda)\\Bigr) \\nonumber\n \\end{eqnarray} \nHowever, expanding $\\vec{Q}^2(\\lambda}\t\\def\\La{\\Lambda)$ around $\\lambda}\t\\def\\La{\\Lambda=\\infty$ gives\nnon-local commuting quantities while expanding $P_0(\\lambda}\t\\def\\La{\\Lambda)$ around\n$\\lambda}\t\\def\\La{\\Lambda=0$ gives the local commuting quantities. The links between\nthem are hidden in the subtilities of the analytic properties\nof the monodromy matrix (these analytic properties should be encoded\ninto the representation theory of the Yangian).\n\\par\n\nSince, the transfer matrix encodes the same amount of information\nas the charges, the semi-classical $su(2)$ Yangian (\\ref{CIxi}) can\nalso be presented in terms of the transfer matrix:\nthe semi-classical $su(2)$ Yangians is the Poisson algebra generated by\nthe transfer matrix, $T(\\lambda)$, of unit determinant\nand with Poisson brackets:\n \\begin{eqnarray} \n\\Bigl\\{ T(\\lambda) \\stackrel{\\otimes}{,} T(\\mu)\\Bigr\\}={\\textstyle{1\\over 2}}\n\\Bigr[\\ r(\\lambda,\\mu)\\ ,\\ T(\\lambda)\\otimes T(\\mu)\\ \\Bigr] \\label{EHixb}\n \\end{eqnarray} \nThe $\\frac{1}{\\lambda}$ expansion of $T(\\lambda)$ is implicitely\nassumed in the definition. For the components $t^{(n)}_{ab}$ of $T(\\lambda)$,\ndefined by $T(\\lambda)_{ab}=\\delta_{ab}+\\sum_{n=0}^{\\infty}\nt^{(n)}_{ab}\\lambda^{-n-1}$, we get:\n \\begin{eqnarray} \n\\Bigl\\{  t^{(n)}_{ab},t^{(m)}_{cd}  \\Bigr\\}\n= \\delta_{cb}t^{(n+m)}_{ad}- \\delta_{ad} t^{(n+m)}_{cb}\n+\\sum_{p=0}^{n-1}\\Bigr(  t^{(m+p)}_{ad}t^{(n-1-p)}_{cb}\n- t^{(n-1-p)}_{ad}t^{(m+p)}_{cb} \\Bigr)\n\\label{CIxv}\n \\end{eqnarray} \nIn particular, this last relation shows that the Poisson algebra is\neffectively generated by the two first charges $Q_i^0$ and $Q_i^1$.\n\nFinally, let us describe how the monodromy matrix generates\nthe transformations (\\ref{dt4}).\nBy an explicit computation of the Poisson brackets between\nthe monodromy matrix and the spin variables, one can easily check\nthat the variation $\\delta_v^n S(y)$ of the spin variables are\ngiven by following formula:\n\\begin{eqnarray}\n\\delta_v^n S(y)=\\ i\\oint {{d\\lambda }\\over{2i\\pi}} \\lambda^n\\\ntr_1\\Bigr(\\ (vT^{-1}(\\lambda)\\otimes 1)\\Bigl\\{\nT(\\lambda)\\otimes 1 , 1\\otimes S(y) \\Bigr\\}\\ \\Bigr) \\label{CIxvi}\n\\end{eqnarray}\nHere $tr_1$ denotes the trace over the first space in the tensor product.\nFor $n=0$ or $1$, eq. (\\ref{CIxvi}) is equivalent to eq. (\\ref{EZv}).\nIt indicates that $T(\\lambda) $ is the generator of\nthe non-Abelian symmteries and characterizes the transformations\n$S(y)\\to \\delta^n_vS(y)$ as Lie-Poisson actions.\n\n\\subsection{Digression on Poisson-Lie groups.}\n\nIn this section we present a few basic facts about Lie-Poisson\nactions \\cite{Dr86,Se83}. Let $M$ be a sympletic manifold.\nWe denote by $\\{\\ ,\\ \\}_M$ the Poisson bracket in $M$.\n\nBefore describing Lie-Poisson actions,\nwe recall some well known facts about Hamiltonian actions.\nLet $H$ be a Lie group and $\\CH$ its Lie algebra. The action of\na one parameter subgroup $(h^t)$ of $H$ is said to be symplectic if for any\nfunctions $f_1$ and $f_2$ on $M$ ,\n\\begin{eqnarray}\n\\{ f_1(h^t.x), f_2(h^t.x)\\}_M\\ =\\ \\{f_1,f_2\\}_M(h^t.x) \\label{EIi}\n\\end{eqnarray}\nIntroducing the vector field $X$ on $M$ corresponding to the infinitesimal\naction, $\\delta}\t\t\\def\\De{\\Delta_X.f(x) = \\frac{d}{dt}f(h^t.x)|_{t=0}$, the condition (\\ref{EIi})\nbecomes:\n\\begin{eqnarray}\n\\{ \\delta}\t\t\\def\\De{\\Delta_X.f_1 , f_2 \\}_M\\ +\\ \\{ f_1 , \\delta}\t\t\\def\\De{\\Delta_X.f_2 \\}_M\\ =\\\n\\delta}\t\t\\def\\De{\\Delta_X.\\{f_1,f_2\\}_M \\nonumber\n\\end{eqnarray}\nWe have the standard property that the action of any one parameter subgroup\nof $H$ is locally Hamiltonian. This means that there\nexists a function $H_X$, locally defined on $M$, such that:\n\\begin{eqnarray}\n\\delta}\t\t\\def\\De{\\Delta_X.f\\ =\\ \\{H_X, f\\}_M \\label{EIiii}\n\\end{eqnarray}\nThe proof is standard.\nThe global existence of $H_X$ is another state of affair.\nThe Hamiltonians $H_X$ are used to define the moment map.\nThese properties generalize to Lie-Poisson actions.\n\nA Poisson- Lie group $H$ is a Lie group equipped with a Poisson\nstructure such that the multiplication in $H$ viewed as map\n$H\\times H \\to H$ is a Poisson mapping.\nLet us be more explicit. Any Poisson bracket $\\{,\\}_H$ on a Lie\ngroup $H$ is uniquely characterized by the data of a\n$\\CH \\otimes \\CH$-valued function: $h\\in H\\to \\eta(h)\\in\\CH \\otimes \\CH$.\nIndeed, introducing a basis $(e_a)$ of $\\CH$, the Poisson\nbracket  for any functions\n$f_1$ and $f_2$ on $H$ can be written as :\n\\begin{eqnarray}\n\\{f_1,f_2\\}_H(h)=\\sum_{a,b}\\ \\eta^{ab}(h)(\\nabla^R_af_1)(h)\n\t\t(\\nabla^R_bf_2)(h) \\label{Epi}\n\\end{eqnarray}\nwhere $\\eta(h)=\\sum_{a,b}\\eta^{ab}(h)e_a \\otimes  e_b$ and\n$\\nabla^R_a$ is the right-invariant vector field corresponding\nto the element $e_a\\in\\CH$:\n$\\nabla^R_a f(h)= { {d\\over{dt}}f(e^{te_a}h)}\\Big\\vert_{t=0}$.\nThe antisymmetry of the Poisson bracket (\\ref{Epi})\nrequires $\\eta_{12}=-\\eta_{21}$, and the Jacobi identity is equivalent\nto a quadratic relation for $\\eta$ which can be easily written down.\nThe Lie Poisson property of the Poisson brackets (\\ref{Epi}) is the\nrequirement that they transform covariantly under the multiplication\nin $H$; It requires that $\\eta(h)$ is a cocycle \\cite{Dr86}:\n$\\eta(hg) = \\eta(h) + Ad\\, h\\cdot \\eta(g)$.\n\nThe bracket $\\{,\\}_H$ can be used to define a Lie algebra\nstructure on $\\CH^*$ by\n$\\[ d_ef_1, d_ef_2 \\]_{\\CH^*}$ $= d_e\\{f_1,f_2\\}_H$,\nwith $d_ef\\ \\in\\CH^*$ the differential of the function $f$\non $H$ evaluated at the identity of $H$.\nIn a basis $(e^a)$ in $\\CH^*$, dual to\nthe basis $(e_a)$ in $\\CH$, the differential at the identity can\nwritten as $d_ef=\\sum_a e^a(\\nabla^L_af)(e)\\in\\CH^*$ where\n$\\nabla^L_a$ are the left-invariant vector fields on $H$, and\nthe Lie structure in $\\CH^*$ is:\n \\begin{eqnarray} \n\\[e^a,e^b\\]_{\\CH^*}\\ =\\ f^{ab}_c\\ e^c \\label{Epv}\n \\end{eqnarray} \nwhere the structure constants are $f^{ab}_c=(\\nabla^L_c\\eta^{ab})(e)$.\nThe Lie bracket eq.(\\ref{Epv}) satisfies the Jacobi identity thanks to the\nJacobi identity for the Poisson bracket in $H$.\nWe denote by $H^*$ the Lie group with Lie algebra $\\CH^*$.\n\nThe action of a Poisson-Lie group on a symplectic manilfold is a Lie-Poisson\naction if the Poisson brackets transform covariantly;\ni.e. if for any $h\\in H$ and any function $f_1$ and $f_2$ on $M$,\n\\begin{eqnarray}\n\\{f_1(h.x),f_2(h.x)\\}_{H\\times M}\\ =\\\n\\{f_1,f_2\\}_M(h.x) \\label{EIviii}\n\\end{eqnarray}\nThe Poisson structure on $H\\times M$ is the product Poisson structure.\n\nLet $X\\in\\CH$ and denote also by $\\delta}\t\t\\def\\De{\\Delta_X$ the vector field on $M$ corresponding\nto the infinitesimal transformation generated by $X$.\nIntroducing two dual basis of the Lie algebras $\\CH$ and $\\CH^*$,\n$e_a\\in\\CH$ and $e^a\\in\\CH^*$ with $<e^a,e_b>=\\delta^a_b$,\nwhere $<,>$ denote the pairing between $\\CH$ and $\\CH^*$,\neq. (\\ref{EIviii}) becomes:\n\\begin{eqnarray}\n\\{\\delta}\t\t\\def\\De{\\Delta_{e_a}.f_1,f_2\\}_M\\ +\\\n\\{f_1,\\delta}\t\t\\def\\De{\\Delta_{e_a}.f_2\\}_M\\ +\\ f_a^{bd} (\\delta}\t\t\\def\\De{\\Delta_{e_b}.f_1)(\\delta}\t\t\\def\\De{\\Delta_{e_d}.f_2)\\ =\\\n\\delta}\t\t\\def\\De{\\Delta_{e_a}.\\{f_1,f_2\\}_M \\label{EIxi}\n\\end{eqnarray}\nIt follows immediately from eq.(\\ref{EIxi}) that a Lie-Poisson action cannot\nbe Hamiltonian unless the algebra $\\CH^*$ is Abelian.\n However, in general, we have a non-Abelian analogue of\nthe Hamiltonian action eq~.~(\\ref{EIiii})~\\cite{Lu90}:\nThere exists a function\n$\\Ga$, locally defined on $M$ and taking values in the group $H^*$,\nsuch that for any function $f$ on $M$,\n\\begin{eqnarray}\n\\delta}\t\t\\def\\De{\\Delta_X.f\\ =\\ <\\ \\Ga^{-1}\\ \\{f,\\Ga\\}_M,X>\\quad,\\quad \\forall\\ X\\in\\CH\n\\label{EIxii}\n\\end{eqnarray}\nWe refer to $\\Ga$ as the non-Abelian Hamiltonian of the Lie-Poisson\naction.\nThe moment map ${\\cal P}}\t\\def\\CQ{{\\cal Q}}\t\\def\\CR{{\\cal R}$ for the Lie-Poisson action is the map\n${\\cal P}}\t\\def\\CQ{{\\cal Q}}\t\\def\\CR{{\\cal R}$ from $M$ to $H^*$ defined by $x\\longrightarrow \\Ga(x)$.\\par\nThe proof is the following. Introduce the Darboux coordinates $(q^i,p^i)$.\nLet $\\Om= e^a\\ \\Om_a$ be the $\\CH^*$-valued one-form\ndefined by $\\Om_a=e_a^{q^i}dp^i\\ -\\ e_a^{p^i}dq^i$ where\n$e_a^{q^i}$, $e^{p^i}_a$ are the components of the vector field $e_a$,\n$e_a=e_a^{q^i}\\partial_{q^i}+e_a^{p^i}\\partial_{p^i}$. Eq. (\\ref{EIxi}) is then\nequivalent to the following zero-curvature condition for $\\Om$:\n$d\\Om\\ + \\[\\Om,\\Om\\]_{\\CH^*}\\ =\\ 0 $.\nTherefore, locally on $M$, $\\Om=\\Ga^{-1}\\ d\\Ga$.\nThis proves eq.(\\ref{EIxii}).\nThe converse is true: an action generated by a non-Abelian\nHamiltonian as in eq.(\\ref{EIxii}) is Lie-Poisson since then we have:\n\\begin{eqnarray}\n\\delta}\t\t\\def\\De{\\Delta_X.\\{ f_1 ,f_2 \\}_M &-& \\{\\delta}\t\t\\def\\De{\\Delta_X.f_1 , f_2 \\}_M\n- \\{f_1 , \\delta}\t\t\\def\\De{\\Delta_X.f_2 \\}_M \\nonumber \\\\\n&=& <\\Bigr[ \\Gamma^{-1} \\{ f_1 ,\\Ga \\}_M ~,~\n\\Ga^{-1}\\{f_2 ,\\Ga \\}_M \\Bigr] _{\\CH^*} ,X > \\nonumber\n\\end{eqnarray}\n\nThe non-Abelian symmetries we descrided above thus provide an exemple\nof Lie-Poisson actions, since they are generated by a non-Abelian\nHamiltonian. They actually are a particular exemple\nof more general transformations called dressing transformations\nin solitons theories \\cite{ZaSh79,Se85}.\n\n\n\\section{The quantum Heisenberg chain}\n\nWe introduce the quantum Heisenberg chain as a quantization\nof the discrete analogue of the  classical Heisenberg model.\nSince, it is not more difficult to deal with the $su(p)$ algebra,\nwe generalize it from $su(2)$ to $su(p)$.\nSo let us assume that we have discretized the interval on which the\nmodel was defined into $N$ segments of identical length that we denote\nby $h$. The extremities of the segments form a chain of sites on which\nthe $su(p)$ spin variables leave. We denote the $su(p)$ spin variables at\nthe sites $j$ by $S_j^{ab}$ with $a,\\ b =1,\\cdots,p$. They satisfy the\ncommutation relations of the discretized $su(p)$ loop algebra:\n \\begin{eqnarray} \n\\Bigr[\\ S^{ab}_j\\ ,\\ S^{cd}_k\\ \\Bigr] = \\delta}\t\t\\def\\De{\\Delta_{jk}\n\\Bigr(\\ \\delta}\t\t\\def\\De{\\Delta^{cb}\\ S^{ad}_j - \\delta}\t\t\\def\\De{\\Delta^{ad}\\ S^{cb}_j\\ \\Bigr) \\label{CIIi}\n \\end{eqnarray} \nThis corresponds to the quantization of the Poisson brackets (\\ref{Ki}).\nApproximating the derivatives by finite differences,\nand using the constraints $S^2_k=s^2={\\rm const}$,\nthe discretized version of the Hamiltonian (\\ref{Kii}) becomes,\n(up to a constant term):\n \\begin{eqnarray} \nH = \\sum_{k=1}^{N}\\sum_{ab} S^{ab}_k S^{ba}_{k+1}\n\\label{CIIii}\n \\end{eqnarray} \nHere, we have assumed periodic boundary conditions.\nAs is well known, in order to preserve the integrability we have to\nchoose the  spin operators $S^{ab}_k$ to act on the fundamental vector\nrepresentation of $su(p)$. So on each sites there is a copy of\n$\\ifmmode\\Cmath\\else$\\Cmath$\\fi^p$ and the operator $S^{ab}_j$  which acts only the $j^{th}$ copies of\n$\\ifmmode\\Cmath\\else$\\Cmath$\\fi^p$ is represented by the canonical matrice $\\ket{a}\\bra{b}$.\n\nIn the discretized quantized model, the monodromy matrix, which is a\n$p \\times p$ matrix with operator entries, has matrix elements:\n \\begin{eqnarray} \nT_{ab}(\\lambda)&=&P \\exp\\left[-\\int A_x(y,\\lambda) dy\n \\right]_{ab}^{ {\\rm discretized} }  \\nonumber\\\\ \n &=&  \\sum_{a_1\\cdots a_{N-1}}\\\n\\Bigr[{1 + \\frac{h}{\\lambda} S_1}\\Bigr]^{a a_1}\n\\Bigr[{1 + \\frac{h}{\\lambda} S_2}\\Bigr]^{a_1 a_2}\\cdots\n\\Bigr[{1 + \\frac{h}{\\lambda} S_N}\\Bigr]^{a_{N-1} b}\n\\label{CIIIiii}\n \\end{eqnarray} \nAs for the classical theory, the\ntransfer matrix admits an $\\inv{\\lambda}$ expansion:\n \\begin{eqnarray} \nT_{ab}(\\lambda) = \\delta_{ab} + \\frac{h}{\\lambda}\\sum_k S^{ab}_k\n +\\frac{h^2}{\\lambda^2} \\sum_{j<k} \\sum_d S^{ad}_j S^{db}_k +\\cdots\n\\nonumber\n \\end{eqnarray} \n\nThe important properties of this monodromy matrix is that we\ncan compute the commutation relations between its matrix elements.\nThese relations can be gathered into the famous relations of the\nalgebraic Bethe anstaz, see e.g. \\cite{Fa82,Ga0}:\n \\begin{eqnarray} \nR(\\lambda-\\mu) (T(\\lambda)\\otimes 1) (1\\otimes T(\\mu))\n=( 1\\otimes T(\\mu))(T(\\lambda)\\otimes 1) R(\\lambda-\\mu)\n\\label{CIIiv}\n \\end{eqnarray} \nwhere $R(\\lambda)$ is Yang's solution of the Yang-Baxter equation:\n \\begin{eqnarray} \nR(\\lambda) = \\lambda - h\\ P \\nonumber \\end{eqnarray} \nwith $P$ the exchange operator $P(x\\otimes y)= y \\otimes x$.\nFor the matrix elements of the monodromy matrix this becomes:\n \\begin{eqnarray} \n(\\lambda-\\mu) \\Bigr[ T_{ab}(\\lambda), T_{cd}(\\mu)\\Bigr] =\nh \\Bigr( T_{ad}(\\lambda)T_{cb}(\\mu) - T_{ad}(\\mu) T_{cb}(\\lambda) \\Bigr)\n\\label{CIIva}\n \\end{eqnarray} \nThese relations are the quantum analogues of the Poisson brackets (\\ref{EHix}).\nAs in the classical theory, it implies that the trace of the\nmonodomy matrix, or more accurately the logarithm\nof the trace of the monodromy matrix,\n form a generating function of commuting quantities.\nIn the same as in the previous section, the\nlocal Hamiltonians are not found by expanding around $\\lambda=\\infty$\nbut around $\\lambda=0$. In particular, we recover the Hamiltonian\nby expanding the logarithm of the trace to first order:\n$ H \\propto \\partial_\\lambda \\log \\lambda^NT(\\lambda)\n\\Big\\vert_{\\lambda=0}$.\n\n\n\\subsection{Non-Abelian symmetries and the quantum Yangians.}\n\nThe Hamiltonian is clearly $su(p)$ invariant. But, in the same as\nfor the classical theories, the symmetry group is much bigger.\nThe naive discrete quantum analogue of the classical charges $Q^0$\nand $Q^1$ introduced  in eq. (\\ref{CIv}) is:\n \\begin{eqnarray} \nQ^0_{ab} &=& \\sum_k S^{ab}_k   \\nonumber\\\\ \nQ^1_{ab} &=& \\frac{h}{2} \\sum_{j<k} \\sum_d\n(S^{ad}_jS^{db}_k - S^{ad}_k S^{db}_j)\n\\label{CIIvi}\n \\end{eqnarray} \nThis naive ansatz turns out to be correct:\nthe charges $Q^0_{ab}$ commutes with the Hamiltonian and the\ncharges $Q^1_{ab}$ formally commutes for chains of infinite\nlength. For finite chain the commutation is broken by boundary\nterms. However, the corresponding currents, which are conserved on\nthe lattice (of any size), were constructed in \\cite{BF91}.\n\nThe charges (\\ref{CIIvi}) form a non-Abelian algebra, which is\nnot a Lie algebra.\nThey satisfy the following commutations relations:\n \\begin{eqnarray} \n\\Bigr[ Q^0_{ab},Q^0_{cd} \\Bigr] &=& \\delta}\t\t\\def\\De{\\Delta_{cb} Q^0_{ad} - \\delta}\t\t\\def\\De{\\Delta_{ad} Q^0_{cb} \\nonumber\\\\ \n\\Bigr[ Q^0_{ab},Q^1_{cd} \\Bigr] &=& \\delta}\t\t\\def\\De{\\Delta_{cb} Q^1_{ad} - \\delta}\t\t\\def\\De{\\Delta_{ad} Q^1_{cb}\n\\label{CIIix}\\\\\n\\Bigr[ Q^1_{ab},Q^1_{cd} \\Bigr] &=& \\delta}\t\t\\def\\De{\\Delta_{cb} Q^2_{ad} - \\delta}\t\t\\def\\De{\\Delta_{ad} Q^2_{cb}\n + \\frac{h^2}{4} Q^0_{ad}(\\sum_e Q^0_{ce}Q^0_{eb}) -\n \\frac{h^2}{4} (\\sum_e Q^0_{ae}Q^0_{ed})Q^0_{cb} \\nonumber\n \\end{eqnarray} \nHere $Q^2_{ab}$ is a new operator whose explicit expression is\nirrelevent for our discussion.\nThe remarkable fact is that the extra non-linear term in the last\nequation can be expressed only in terms of $Q^0_{ab}$. This last\nequation implies a relation involving only $Q^0_{ab}$\nand $Q^1_{ab}$:\n \\begin{eqnarray} \n\\Bigr[ Q^0_{ab},\\Bigr[ Q^1_{cd},Q^1_{ef}\\Bigr]\\BBR &-&\n\\Bigr[ Q^1_{ab},\\Bigr[ Q^0_{cd},Q^1_{ef}\\Bigr]\\BBR  \\nonumber\\\\ \n= \\frac{h^2}{4} \\sum_{p\\, q}\\Bigr(\n\\Bigr[ Q^0_{ab},\\Bigr[ Q^0_{cp}Q^0_{pd},Q^0_{eq}Q^0_{qf}\\Bigr]\\BBR &-&\n\\Bigr[ Q^0_{ap}Q^0_{pb},\\Bigr[ Q^0_{cd},Q^0_{eq}Q^0_{qf}\\Bigr]\\BBR \\Bigr)\n\\label{CIIIx}\n \\end{eqnarray} \nThe associative algebra generated by elements $Q^0_{ab}$ and\n$Q^1_{ab}$ satisfying the relations (\\ref{CIIix}) and (\\ref{CIIIx})\nis called the $su(p)$ Yangians \\cite{Dr86}. As it can be seen by comparing\nwith eq. (\\ref{CIx}), this algebra is a deformation of the $su(p)$\nloop algebra.\n\nThe $su(p)$ Yangian is not an Lie algebra but a Hopf algebra.\nIt is therefore equipped with a comultiplication $\\De$, which\nis a homomorphism from the algebra into the tensor product\nof two copies of the same algebra. For the $su(p)$ Yangians,\nthe comultiplication is given by:\n \\begin{eqnarray} \n\\De Q^0_{ab} &=& Q^0_{ab}\\otimes 1 + 1 \\otimes Q^0_{ab} \\label{Ecom}\\\\\n\\De Q^1_{ab} &=& Q^1_{ab}\\otimes 1 + 1 \\otimes Q^1_{ab}\n +\\frac{h}{2} \\sum_d\\Bigr( Q^0_{ad}\\otimes Q^0_{db} -\n Q^0_{db}\\otimes Q^0_{ad} \\Bigr) \\nonumber\n \\end{eqnarray} \nIt can be used to construct tensor products of representations.\nFor $h=0$ it reduces to the comultiplication of the $su(p)$\nloop algebra.\n\n\\subsection{The quantum transfer matrix and an alternative presentation\nof the Yangians.}\n\nThe charges $Q_{ab}^0$ and $Q^1_{ab}$ appears as the first terms in the\n$\\inv{\\lambda}$ expansion of the quantum monodromy matrix $T(\\lambda)$.\nTherefore, the $su(p)$ Yangian can also be presented in terms\nof $T(\\lambda)$, or more precisely, in terms of its components\nin an $\\inv{\\lambda}$ expansion:\n \\begin{eqnarray} \nT_{ab}(\\lambda) = \\delta}\t\t\\def\\De{\\Delta_{ab} + h \\sum_{n=0}^\\infty \\lambda^{-n-1} t^{(n)}_{ab}\n\\label{CIIxi}\n \\end{eqnarray} \nTherefore, the alternative presentation of\nthe $su(p)$ Yangian is as the associative algebra generated by the\nelements $t^{(n)}_{ab}$ with relations:\n \\begin{eqnarray} \n\\Bigr[  t^{(n)}_{ab},t^{(m)}_{cd}  \\Bigr]\n= \\delta_{cb}t^{(n+m)}_{ad}- \\delta_{ad} t^{(n+m)}_{cb}\n+ h\\sum_{p=0}^{n-1}\\Bigr(  t^{(m+p)}_{ad}t^{(n-1-p)}_{cb}\n- t^{(n-1-p)}_{ad}t^{(m+p)}_{cb} \\Bigr) \\label{CIIxii}\n \\end{eqnarray} \nThese relations are equivalent to :\n \\begin{eqnarray} \n\\Bigr[  t^{(0)}_{ab},t^{(m)}_{cd}  \\Bigr]\n&= &\\delta_{cb}t^{(m)}_{ad}- \\delta_{ad} t^{(m)}_{cb} \\label{CIIxiii}\\\\\n\\Bigr[  t^{(n+1)}_{ab},t^{(m)}_{cd}  \\Bigr]\n -\\Bigr[  t^{(n)}_{ab},t^{(m+1)}_{cd}  \\Bigr]\n&=& h\\Bigr(  t^{(m)}_{ad}t^{(n)}_{cb}\n- t^{(n)}_{ad}t^{(m)}_{cb} \\Bigr) \\nonumber\n \\end{eqnarray} \nwhich, in their turn, are equivalent to the fundamental commutation\nrelations (\\ref{CIIiv}).\nIn the quantum theory, the unit determinant constraint is modified\ninto the condition that the so-called quantum determinant of the\ntransfer matrix is one:\n \\begin{eqnarray} \nDet_q\\ T(\\lambda) \\equiv \\sum_{\\sigma}\t\\def\\Sig{\\Sigma {\\rm perm.}} \\epsilon(\\sigma)\nT_{\\sigma(p) p}(\\lambda) ..... T_{\\sigma(1) 1}(\\lambda+ph) = 1\n\\label{CIIxiv}\n \\end{eqnarray} \nThe sum is over the permutation of $p$ objects and $\\epsilon(\\sigma}\t\\def\\Sig{\\Sigma)$ is\nthe signature of the permutation $\\sigma$.\nThe quantum determinant commutes with all the components of the\nmonodromy matrix, so this constraint can be imposed in a consistent\nway.\n\nWith the quantum determinant constraint,\nthe $\\inv{\\lambda}$ expansion of the\nmonodromy matrix can be reconstructed from its\ntwo first components $t^{(0)}_{ab}$ and $t^{(1)}_{ab}$.\nFinally, the relation between these two components and\nthe quantum charges $Q^0_{ab}$ and $Q^1_{ab}$ is:\n \\begin{eqnarray} \nQ^0_{ab} &=& t^{(0)}_{ab}  \\nonumber\\\\ \nQ^1_{ab} &=& t^{(1)}_{ab} - \\frac{h}{2} \\sum_d t^{(0)}_{ad} t^{(0)}_{db}\n\\label{CIIxv}\n \\end{eqnarray} \nThis shows that, as for the classical theory, the knowledge of the\nfirst two non-local charges $Q^0_{ab}$ and $Q^1_{ab}$ is equivalent ot\nthe knowledge of the $\\inv{\\lambda}$ expansion of the monodromy matrix.\n\nThe comultiplication for the transfer matrix is given by:\n \\begin{eqnarray} \n\\De T_{ab}(\\lambda) = \\sum_d T_{ad}(\\lambda)\\otimes T_{db}(\\lambda)\n\\nonumber\n \\end{eqnarray} \nFor $Q^0_{ab}$ and $Q^1_{ab}$ it reduces to eq. (\\ref{Ecom}).\nThe adjoint action of the transfer matrix on an operator $\\Phi$ is:\n \\begin{eqnarray} \n\\Bigr[ {\\rm Adj.} T_{ab}(\\lambda)\\Bigr] \\Phi = \\sum_d T_{ad}(\\lambda) \\\n\\Phi \\ T_{db}^{-1}(\\lambda) \\nonumber\n \\end{eqnarray} \nwhere $T_{ab}^{-1}(\\lambda)$ is the matrix of operators characterized\nby $\\sum_dT_{ad}(\\lambda}\t\\def\\La{\\Lambda) T^{-1}_{db}(\\lambda}\t\\def\\La{\\Lambda)=\\delta}\t\t\\def\\De{\\Delta_{ab}$. This adjoint action\nis the quantum analogue of the semi-classical equation (\\ref{CIxvi}).\n\n\n\\subsection{Digression on the double Yangians.}\n\nThe Yangians are deformation of only half of the loop algebras,\ni.e. they are deformation of only the sub-algebra of loops\nregular at the origin. A quantum deformation of the complete\nloop algebra can be obtained by introducing the quantum\ndouble of the Yangian, following Drinfel'd \\cite{Dr86}.\nIn quantum field theory, the double Yangian was introduced\nin \\cite{LS91}. It can be presented in the following multiplicative\nform similar to eq. (\\ref{CIIiv}). Let $T^+_{ab}(\\lambda}\t\\def\\La{\\Lambda)$\nand $T^-_{ab}(\\lambda}\t\\def\\La{\\Lambda)$ be two $p\\times p$ matrices of operators\nwith the following expansion:\n \\begin{eqnarray} \nT^+_{ab}(\\lambda) &=& \\delta}\t\t\\def\\De{\\Delta_{ab} +\n h \\sum_{n=0}^{+\\infty} \\lambda^{-n-1} t^{(n)}_{ab} \\nonumber\\\\ \nT^-_{ab}(\\lambda) &=& h \\sum_{n=0}^{+\\infty} \\lambda^{n} t^{(-n-1)}_{ab}\n\\nonumber\n \\end{eqnarray} \nNote that these expansions correspond to expansions into\nseries regular at the origin or at infinity.\nThe $gl(p)$ double Yangian is the algebra generated by the\nelements $t^{(\\pm n)}_{ab}$ with relations:\n \\begin{eqnarray} \nR(\\lambda-\\mu) (T^\\pm(\\lambda)\\otimes 1) (1\\otimes T^\\pm(\\mu))\n&=&( 1\\otimes T^\\pm(\\mu))(T^\\pm(\\lambda)\\otimes 1) R(\\lambda-\\mu) \\nonumber\\\\ \nR(\\lambda-\\mu) (T^+(\\lambda)\\otimes 1) (1\\otimes T^-(\\mu))\n&=&( 1\\otimes T^-(\\mu))(T^+(\\lambda)\\otimes 1) R(\\lambda-\\mu)\n\\nonumber\n \\end{eqnarray} \nIf in a finite dimensional representation of the Yangian the transfer matrix\n$T^+_{ab}(\\lambda)$ is a polynomial of degree $N$ in $1\/\\lambda$, then\nthe double Yangian is represented in the same vector space by\n$T^-_{ab}(\\lambda}\t\\def\\La{\\Lambda)\\propto \\lambda}\t\\def\\La{\\Lambda^N T^+_{ab}(\\lambda}\t\\def\\La{\\Lambda)$.\n\n\\vskip 2cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nMarkov random fields (MRF) are frequently used as prior distributions\nin spatial statistics. A common situation is that we have an observed \nor latent field $x$ which we model as an MRF, $p(x|\\phi)$, conditioned \non a vector of model parameters $\\phi$. The most common situation in\nthe literature is to consider $\\phi$ as fixed, see for instance\nexamples in \\citet{Besag1986} and \\citet{moller2003}, but several\narticles have also considered a fully Bayesian approach by assigning a \nprior on $\\phi$. A fully Bayesian model is computationally \nsimplest when $x|\\phi$ is a Gaussian Markov random field (GMRF)\nand this case is therefore especially well developed. A flexible\nimplementation of the GMRF case is given in the integrated\nnested Laplace approximation (INLA) software, see\n\\citet{art129} and \\citet{art143}. The case when the components of\n$x$ are discrete variables is computationally much harder and therefore \nless developed in the literature. However, some articles have considered\nthe fully Bayesian approach also in this case, see in particular the \nearly \\citet{art109} and \\citet{art144} and the more recent \n\\citet{art120}, \\citet{art117}, \\citet{phd6}, \\citet{art145} and \\citet{tjelmeland2012}.\n\n\nMRFs is a very flexible class of models. Formally, any distribution is an MRF with \nrespect to a neighbourhood system where all nodes are neighbours of \neach other. For the MRF formulation to be of any help, however, reasonably small \nneighbourhoods must be adopted. The typical choice in the literature is \nto assume each node to have the nearest four, eight or 24 other nodes as \nneighbours. Moreover, in the model specification it is common to restrict \noneself to models that include interactions between pairs of nodes only.\nSuch pairwise interaction priors are just token priors, unable to \nspecify more spatial structure than that nodes close to each other\nshould tend to have the same value. In the literature it is often argued\nthat such token priors are sufficient in many applications, as the information that neighbour nodes should tend to have the \nsame value is the information lacking in the observed data. In \nparticular one typically gets much better results based on such a token prior\nthan by not including any spatial prior information in the analysis at all.\nThe main reason for resorting to pairwise interaction priors, in addition\nto the argument that these are good enough, is that the class of \nMRFs with higher-order interactions is so large that it becomes \ndifficult both to select a reasonable parametric form for the prior\nand to specify associated parameter values, not to mention the \nspecification of a hyper-prior if a fully Bayesian approach is \nadopted. However, \\citet{DescombesEtAl} and \\citet{TjelmelandBesag}\ndemonstrate that it is possible to \nspecify MRFs with higher-order interactions that are able to model \nmore spatial structure than a pairwise interaction MRF, \nand where the model parameters have a reasonable interpretation.\n\nIn this article we consider the fully Bayesian approach \nand for simplicity we limit the attention to the case \nwhere the components of $x$ are binary. Our focus is \non the specification of a prior distribution and on \nsimulation from the associated posterior distribution.\nWe define priors both on the parametric form of the MRF \nand on the parameter values. To the best of our knowledge\nthis is the first attempt on putting a prior on the \nparametric form of a discrete MRF. Other articles\nconsidering a fully Bayesian approach in such a setting,\nare using a fixed parametric model and put \na prior on the parameter values only. One should note that\nby assigning a prior to the parametric structure of the model, \nincluding the number of parameters, we get an automatic model \nchoice when simulating from the posterior distribution. \n\nTo be able to define a reasonable prior it is essential\nto adopt a model where the parameters have a natural \ninterpretation. In this article we consider two ways to\nparametrise the MRF. The first approach is inspired by \nthe so-called $u$-parameters commonly used in \nthe log-linear and graphical model literature \nfor contingency tables\n\\citep{Bishop1975,Forster1999,massam2009,overstall2012}. Here the \nparameters are interactions of different orders. To \nlimit the complexity of the model is easy by \nrestricting some of the parameters to be zero,\nbut we argue that the interpretation of the \nparameters is difficult. The second parametrisation \nwe consider is inspired by the MRF formulation in\n\\citet{TjelmelandBesag}. The parameters then represent \npotentials for configurations in maximal cliques, \nand we limit the model complexity by restricting\ndifferent configurations to have the same potential.\nIn \\citet{TjelmelandBesag} this grouping of \nconfigurations is done manually, whereas we assign a \nprior to the grouping so that it is done \nautomatically in the posterior simulation. Thereby we \ndo not need, for example, to specify apriori whether or not the field \nis isotropic. We argue that the interpretation of the\nconfiguration potentials is much easier than\nfor the interactions, and unless any particular\nprior information is available and suggest the opposite,\nit is natural to assume the configuration potentials\nto be on the same scale.\n\n\n\n\n\n\n\n\nTo explore the resulting posterior distribution we construct\na reversible jump MCMC (RJMCMC) algorithm \\citep{green1995}.\nTo run this algorithm we have to cope with the computationally \nintractable normalising constant of the MRF. In the literature\nseveral strategies for handling this have been proposed.\nWe adopt an approximation strategy for binary MRFs \nintroduced in \\citet{phd6}, where a partially ordered\nMarkov model (POMM), see \\citet{art119}, approximation to the MRF is defined.\nWe simply replace the MRF with the corresponding POMM approximation.\n\n\nThe article has the following organisation. In Section \\ref{sec:MRF}\nwe discuss the two parametrisations of binary MRFs, and in \nparticular we identify the maximal number of free parameters for a\nmodel with specified maximal cliques. In Section \\ref{sec:prior}\nwe define our prior for $\\phi$, and in Section \\ref{sec:sampling}\nwe discuss how to handle the computationally intractable \nnormalising constant and describe our RJMCMC algorithm for \nsimulating from the posterior distribution.\nIn Section \\ref{sec:results} we present results for one simulated data example \nand for one real data example. One additional simulated example is\ngiven in the supplementary materials. Finally, \nsome closing remarks are provided in Section \\ref{sec:discussion}.\n\n\n\\section{MRF}\n\\label{sec:MRF}\nIn this section we give a brief introduction to MRFs, see\n\\cite{cressie1993} and \\cite{moller2003} for more details, and\nin particular we focus on binary MRFs and the parametrisation in this\ncase. We close with one example of a binary MRF, the Ising model. This section provides\nthe theoretical background needed in order to understand the\nconstruction of our prior distribution in Section \\ref{sec:prior}, and\nthe RJMCMC algorithm given in Section \\ref{sec:sampling}.    \n\n\\subsection{Binary MRF}\n\\label{sec:binaryMRF}\n\nConsider a rectangular lattice of dimension $n\\times m$, and let the\nnodes be identified by $(i,j)$ where $i=0,...,n-1$ and $j=0,...,m-1$. To each node $(i,j)\\in\nS=\\{(i,j);i=0,...,n-1,j=0,...,m-1\\}$ we associate a binary variable\n$x_{i,j}\\in \\{0,1\\}$, and let $x=(x_{i,j};(i,j)\\in S)$ be the collection of these\nbinary variables. We let $x_A=(x_{i,j};(i,j)\\in A)$ denote the collection of \nthe binary variables with indices belonging to an index set\n$A\\subseteq S$, and let\n$x_{-(i,j)}=x_{S\\setminus \\{(i,j)\\}}$. Associating zero with black and one with white we may \nthereby say the $x$ specifies a colouring of the nodes. We let \n$\\mathcal{N}=\\{\\mathcal{N}_{0,0},...,\\mathcal{N}_{n-1,m-1}\\}$ be\na neighbourhood system on $S$, where $\\mathcal{N}_{i,j}\\subseteq S \\setminus \\{(i,j)\\}$ is \nthe set of neighbour\nnodes of node $(i,j)$. We assume symmetry in the neighbour sets, so if $(i,j)\\in \\mathcal{N}_{t,u}$,\nthen also $(t,u)\\in \\mathcal{N}_{i,j}$. Now, $x$ is a binary MRF if $p(x)>0$ for all\n$x$, and $p(x_{i,j}|x_{-(i,j)})$ fulfils the Markov property \n\\begin{equation}\np(x_{i,j}|x_{-(i,j)})=p(x_{i,j}|x_{\\mathcal{N}_{i,j}}) \\text{ for all}\n\\ (i,j)\\in S.\n\\end{equation}\nA clique is defined to be a set $\\lambda\\subseteq\nS$, where $(i,j)\\in \\mathcal{N}_{t,u}$ for all distinct pair of\nnodes $(i,j),(t,u)\\in \\lambda$,\nand we denote the set of all cliques by $\\mathcal{L}$. Note that by this\ndefinition sets containing only one node and the empty set are cliques. A\nmaximal clique is defined to be a clique that is not a subset of\nanother clique, and we denote the set of all maximal cliques by\n$\\mathcal{L}_m$. Moreover, for $\\lambda\\in \\mathcal{L}$ we let \n$\\mathcal{L}_m^\\lambda$ denote the set of all maximal \ncliques that contains $\\lambda$, i.e. $\\mathcal{L}_m^\\lambda=\\{\\Lambda \\in \\mathcal{L}_m;\\lambda \\subseteq\n\\Lambda\\}$.\nIn the following we use $\\Lambda$ and $\\Lambda^\\star$ to denote\nmaximal cliques, i.e. $\\Lambda,\\Lambda^* \\in \\mathcal{L}_m$, whereas we use $\\lambda$ and\n$\\lambda^\\star$ to denote cliques that do not need to be maximal,\ni.e. $\\lambda,\\lambda^\\star\\in \\mathcal{L}$. To denote an $x$ where $x_{i,j}=1$\nfor all $(i,j)\\in A$ for some $A\\subseteq S$ and $x_{i,j}=0$ otherwise,\nwe use $\\bold 1^A = \\{ x_{i,j}=I( (i,j)\\in A); (i,j)\\in S\\}$. Thereby a colouring\nof the nodes in a maximal clique $\\Lambda\\in\\mathcal{L}_m$ may be specified\nby $\\bold 1^\\lambda_\\Lambda$, where $\\lambda\\subseteq \\Lambda$\nspecifies the set of nodes in $\\Lambda$ that has the value one.\n\n\n\nAccording to the Hammersley-Clifford theorem\n\\citep{Clifford}, the most general form the distribution $p(x)$ of an\nMRF can take is \n\\begin{equation}\np(x)=Z\\exp\\{ U(x)\\} \\mbox{~~with~~} \\ U(x)=\\sum_{\\Lambda \\in \\mathcal{L}_m}\n    V_{\\Lambda}(x_{\\Lambda}),\n\\label{eq:cliqueRep}\n\\end{equation}\nwhere $Z$ is the\ncomputationally demanding normalising constant,\n$U(x)$ is frequently called the energy function, \nand $V_{\\Lambda}(x_{\\Lambda})$ is a potential function for\n$\\Lambda$. A naive parametrisation of $V_{\\Lambda}(x_{\\Lambda})$ is to\nintroduce one parameter for each\npossible $\\Lambda\\in \\mathcal{L}_m$ and $x_{\\Lambda}\\in\n\\{0,1\\}^{|\\Lambda|}$ by setting\n\\begin{equation}\\label{eq:phi}\nV_{\\Lambda}(\\bold 1^\\lambda_{\\Lambda})=\\phi^\\lambda_{\\Lambda}.\n\\end{equation} \nIt is a well known fact the $\\phi^\\lambda_\\Lambda$ parameters do not\nconstitute a unique representation of $U(x)$. Thereby, in the resulting parametric model $p(x)$ the $\\phi^\\lambda_\\Lambda$ parameters\nare not identifiable, meaning\nthat different choices for the $\\phi^\\lambda_\\Lambda$ parameters may give the same\nmodel $p(x)$. For example, adding the same value to all $\\phi^\\lambda_\\Lambda$ \nparameters will not change the model, \nas this will be compensated for by a corresponding change in the normalising \nconstant $Z$. If the set of maximal cliques $\\mathcal{L}_m$\nconsists of, for example, all $2\\times 2$ blocks of nodes a perhaps less \nobvious way to change the parameter values without changing\nthe model and neither the normalising constant, is to add an \narbitrary value to $\\phi^{\\{ (i,j)\\}}_\\Lambda$ for some\n$(i,j)\\in\\Lambda\\in\\mathcal{L}_m$, and to subtract the same \nvalue from $\\phi^{\\{(i,j)\\}}_{\\Lambda^\\star}$ for some \n$\\Lambda^\\star\\in\\mathcal{L}_m$, $\\Lambda^\\star\\neq \\Lambda$\nfor which $(i,j)\\in\\Lambda^\\star$.\n\n\nAn alternative way to represent an MRF is through a parametrisation of\nthe cliques. The energy function $U(x)$ is a pseudo-Boolean function and \nwhen it is given as in (\\ref{eq:cliqueRep}) \\cite{tjelmeland2012} show that \nit can be represented as\n\\begin{equation}\nU(x)=\\sum_{\\lambda \\in\n    \\mathcal{L}}\\beta^{\\lambda}\\prod_{(i,j)\\in\\lambda}x_{i,j},\n\\label{eq:DAGrep}\n\\end{equation} \nwhere $\\beta^{\\lambda}$ is referred to as the interaction parameter for\nclique $\\lambda$, which is said to be of $|\\lambda|$'th order. More\ndetails on pseudo-Boolean functions and their properties can be found\nin \\cite{Grabisch2000} and \\cite{Hammer1992}. Since\nthis representation consists of linearly independent functions of $x$,\nit is clear that the set of interaction parameters is a unique representation of $U(x)$. \nFurthermore, in the corresponding parametric model $p(x)$ the $\\beta^\\lambda$ \nparameters become\nidentifiable if fixing $\\beta^\\emptyset$ to zero (say). \nWe \nnote in passing that \\citet{Besag1974} uses the \nrepresentation in \\eqref{eq:DAGrep} in a proof for \nthe Hammersley--Clifford theorem.\n\nIn the following we define a set of constraints on the $\\phi^\\lambda_\\Lambda$\nparameters in \\eqref{eq:cliqueRep} and show that subject to these \nconstraints there is a one-to-one relation between the \n$\\phi^\\lambda_\\Lambda$ parameters and the\ninteraction parameters $\\beta^\\lambda$.\nThe constrained $\\phi^\\lambda_\\Lambda$ parameters thereby constitute an \nalternative unique representation of $U(x)$.\n\n\\begin{definition}\\label{def:constraints}\nThe constrained set of $\\phi^\\lambda_\\Lambda$ parameters are defined by \nrequiring that $\\phi^\\lambda_\\Lambda=\\phi^\\lambda_{\\Lambda^\\star}$ for all\n$\\Lambda,\\Lambda^\\star\\in\\mathcal{L}_m$, $\\lambda\\subseteq \\Lambda \\cap \\Lambda^\\star$.\nTo simplify the notation we then write $\\phi^\\lambda_\\Lambda=\\phi^\\lambda$.\n\\end{definition}\nTo understand the implication of the constraint one may again consider the \nsituation where the set of maximal cliques $\\mathcal{L}_m$ consists of all \n$2\\times 2$ blocks of nodes, and focus on the two overlapping maximal cliques\n$\\Lambda = \\{ (i,j-1),(i+1,j-1),(i,j),(i+1,j)\\}$ and \n$\\Lambda^\\star = \\{(i,j),(i+1,j),(i,j+1),(i+1,j+1)\\}$. For $\\lambda=\n\\{ (i,j),(i+1,j)\\}$ the constraint is that the potential $V_\\Lambda(x_\\Lambda)$ for the colouring $\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 0 & 1 \\end{array}$\nin $\\Lambda$ is the same as the potential $V_{\\Lambda^\\star}(x_{\\Lambda^\\star})$ for the\ncolouring\n$\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 0 \\\\ 1 & 0 \\end{array}$\nin $\\Lambda^\\star$. One should also note that the constraint implies that \n$\\phi^\\emptyset_\\Lambda$ is the same for all $\\Lambda\\in\\mathcal{L}_m$, so in \nthe $2\\times 2$ maximal cliques case the potential for the colouring \n$\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 0 & 0 \\end{array}$\nmust be the same for all maximal cliques.\n\n\n\n\n\\begin{theorem}\\label{th:onetoone}\nConsider an MRF and constrain the $\\phi$ parametrisation of the potential functions as described in\nDefinition \\ref{def:constraints}. Then there is a one-to-one relation between \n$\\{\\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$\nand $\\{\\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$. \n\\end{theorem}\nThe proof is given in the supplemental material, and the result is \nshown by establishing recursive equations\nshowing how to compute the $\\beta^\\lambda$'s from \nthe $\\phi^\\lambda$'s and vice versa.\n\nTo simplify the definition of a prior for the parameter\nvector of an MRF in the next section, we first limit\nthe attention to {\\em stationary} MRFs defined on \na rectangular  $n\\times m$ lattice, and to obtain \nstationarity we assume {\\em torus boundary conditions}.\nIn the following we\ndefine the concepts of stationarity and \ntorus boundary conditions and states two \ntheorems which identify what properties the\n$\\{\\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$ parameters\nand the $\\{ \\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$ \nparameters must have\nfor the MRF to be stationary.\n\\begin{definition}\\label{def:torus}\nIf, for a rectangular lattice\n$S = \\{ (i,j);i=0,\\ldots,n-1,\nj=0,\\ldots,m-1\\}$, the translation of a node\n$(i,j)\\in S$ with an amount $(t,u)\\in S$\nis defined to be\n\\begin{equation*}\n(i,j) \\oplus (t,u) = (i+t \\mod n, j+u \\mod m),\n\\end{equation*}\nwe say that the lattice has torus boundary conditions.\n\\end{definition}\nTo denote translation of a set of nodes $A\\subseteq S$ by an\namount $(t,u)\\in S$ we write\n$A \\oplus (t,u) = \\{ (i,j)\\oplus (t,u) ; (i,j)\\in A\\}$.\nWith this notation stationarity of an MRF defined\non a rectangular lattice with torus boundary conditions \ncan be defined as follows.\n\\begin{definition}\\label{def:stationarity}\nAn MRF $x$ defined on a rectangular lattice\n$S$ with torus boundary\nconditions is said to be stationary if and only if\n$p(\\bold 1^A) = p(\\bold 1^{A \\oplus (t,u)})$ for all\n$A\\subseteq S$ and $(t,u)\\in S$.\n\\end{definition}\nTo explore what restrictions the stationarity assumption\nputs on the $\\beta^\\lambda$ and $\\phi^\\lambda$ parameters\nwe assume the set of maximal cliques to consist of all \npossible translations of a given nonempty template set $\\Lambda_0\\subset S$, i.e.\n\\begin{equation}\\label{eq:Lm}\n\\mathcal{L}_m = \\{ \\Lambda_0 \\oplus (t,u); (t,u)\\in S\\}.\n\\end{equation}\nFor example,\nwith $\\Lambda_0 = \n\\{ (0,0),(0,1),(1,0),(1,1)\\}$ the set of maximal cliques\nwill consist of all $2\\times 2$ blocks of nodes. One \nshould note that with the\ntorus boundary assumption there is always $|\\mathcal{L}_m|=nm$ \nmaximal cliques.\n\\begin{theorem}\\label{th:betaStationarity}\nAn MRF $x$ defined on a rectangular lattice\n$S=\\{ (i,j);i=0,\\ldots,n-1,j=0,\\ldots,m-1\\}$ with \ntorus boundary conditions and $\\mathcal{L}_m$ given\nin \\eqref{eq:Lm} is stationary if and only \nif $\\beta^{\\lambda}=\\beta^{\\lambda\\oplus (t,u)}$ for all \n$\\lambda \\in \\mathcal{L}$, $(t,u)\\in S$. We then say\nthat $\\beta^\\lambda$ is translational invariant.\n\\end{theorem}\nThe proof is again given in the supplemental material. We proof the \nif part of the theorem by induction on $|\\lambda|$, and the only if part\nby direct manipulation with the expression for the energy function.\n\nTo better understand the effect of the theorem we can again consider the \n$2\\times 2$ maximal clique case, i.e. $\\mathcal{L}_m$ is \ngiven by \\eqref{eq:Lm} with $\\Lambda_0=\\{ (0,0),(0,1),(1,0),(1,1)\\}$. \nThe translational invariance means that all first-order interactions \n$\\{\\beta^{\\{ (i,j)\\}},(i,j)\\in S\\}$ must be equal and in the following we \ndenote their common value by $\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} & \\\\\\framebox(5.0,5.0){} & \\end{array}}$, \nwhere the idea is that the superscript represents any node \n$(i,j)\\in S$. Correspondingly we use $\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\\\ &\\end{array}}$, where the superscript\nrepresent any two horizontally first-order neighbours, to denote \nthe common value for all \n$\\{\\beta^{\\{ (0,0),(0,1)\\}\\oplus (t,u)},(t,u)\\in S\\}$.\nContinuing in this way we get, in addition to $\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} & \\\\\\framebox(5.0,5.0){} & \\end{array}}$, $\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\\\ &\\end{array}}$ \nand the constant term $\\beta^\\emptyset$,\nthe parameters\n$\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){}  & \\end{array}}$, $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\\\ &\\framebox(5.0,5.0){}\\end{array}}$, $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}}$, $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}}$, $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ & \\framebox(5.0,5.0){} \\end{array}}$, \n$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}$, $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}$ and $ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}$.\nWe collect the eleven parameter values necessary to represent $U(x)$\nin this stationary MRF case \ninto a vector which we denote by $\\beta$, i.e.\n\\begin{equation}\n\\beta = \\left(\n\\beta^\\emptyset,\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} & \\\\\\framebox(5.0,5.0){} & \\end{array}},\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\\\ &\\end{array}},\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){}  & \\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\\\ &\\framebox(5.0,5.0){}\\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}},\n \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ & \\framebox(5.0,5.0){} \\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}, \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}\\right).\n\\label{eq:beta2x2}\n\\end{equation}\n\nThe next theorem gives a similar result for the $\\phi^\\lambda$ \nparameters as Theorem \\ref{th:betaStationarity} did for the \ninteraction parameters $\\beta^\\lambda$.\n\\begin{theorem}\\label{th:phiStationarity}\nAn MRF $x$ defined on a rectangular lattice \n$S=\\{ (i,j);i=0,\\ldots,n-1,j=0,\\ldots,m-1\\}$ with torus boundary\nconditions and $\\mathcal{L}_m$ given in \n\\eqref{eq:Lm} is stationary if and only if\n$\\phi^{\\lambda}=\\phi^{\\lambda\\oplus (t,u)}$ for all \n$\\lambda \\in \\mathcal{L}$ and $(t,u)\\in S$. We then say\nthat $\\phi^\\lambda$ is translational invariant. \n\\end{theorem}\nThe proof is again given in the supplemental material. Given the result in \nTheorem \\ref{th:betaStationarity} it is \nsufficient to show that $\\phi^\\lambda$ is translational invariant\nif and only if $\\beta^\\lambda$ is translational invariant, and we show\nthis by induction on $|\\lambda|$.\n\nIt should be noted that the \ninterpretation of the $\\phi^\\lambda$ parameters is very different from the \ninterpretation of the $\\beta^\\lambda$ parameters. Whereas the $\\beta^\\lambda$ \nparameters relates to cliques $\\lambda$ of different sizes, all the \n$\\phi^\\lambda$'s represent the potential of a maximal clique $\\Lambda\\in\\mathcal{L}_m$,\nwhich are all of the same size. The effect of the above theorem is that we\nget groups of configurations in maximal cliques that must be assigned the same potential, hereafter referred to as configuration sets. \nWe let $\\cal C$ denote the set of these configuration sets. In the $2\\times 2$ maximal clique\ncase for example, we get\n\\begin{eqnarray} \\nonumber\n{\\cal C} &=& \\left\\{ \\left\\{ \\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 0 & 0 \\end{array}\\right]\\right\\},\n\\left\\{ \\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 0 \\\\ 0 & 0 \\end{array}\\right],\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 0 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 1 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 0 & 1 \\end{array}\\right]\\right\\},\n\\left\\{ \\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 0 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 1 & 1 \\end{array}\\right]\\right\\},\n\\left\\{ \\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 0 \\\\ 1 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 0 & 1 \\end{array}\\right]\\right\\},\n\\right.\\\\\n&& \\left. \n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 0 \\\\ 0 & 1 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 1 & 0 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 0 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 0 & 1 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 0 \\\\ 1 & 1 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 1 & 1 \\end{array}\\right]\\right\\},\n\\left\\{\\left[ \\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}\\right]\\right\\}\n\\right\\}\n\\label{eq:confsubset}\n\\end{eqnarray}\nWe denote these sets of configurations by $c^0$, $c^1$, $c^{11}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}}$,\n$c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}}$, $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}$ and $c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}$\nwhen listed in the same order as in \\eqref{eq:confsubset}, where the idea of the notation is that the \n$1$'s in the superscript can be placed anywhere inside a maximal clique and the remaining nodes takes the value of \nzero. One should note that a similar notation can be used in other sets of maximal cliques. In the $3\\times 3$\nmaximal clique case we have for example\n\\[\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}} = \\left\\{\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 1 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 0 & 1 & 1 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 0 \\end{array}\\right],\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 0 & 0 & 0 \\\\ 1 & 1 & 0 \\\\ 1 & 0 & 0 \\end{array}\\right],\n\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 0 & 0 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 1 & 0 \\end{array}\\right]\\right\\}.\n\\]\nAssociated to each member $c\\in {\\cal C}$ we thus have a corresponding parameter value $\\phi(c)$ \nwhich is the potential assigned\nto any maximal clique configuration in the set $c$. We use corresponding superscripts for the $\\phi$ parameters as we did\nfor the sets $c\\in{\\cal C}$. In the $2\\times 2$ maximal clique case we thereby get the parameter vector\n\\[\n\\phi = \\left (\\phi^0, \\phi^1, \\phi^{11}, \\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}}, \n\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}},\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}} \\right),\n\\]\nwhere for example $\\phi^1$ is the potential for the four maximal clique configurations in $c^1$.\n\n\n\n\n\n\n\n\nWe end this section with a discussion on how the above stationary MRF\ndefined with torus boundary condition \ncan be modified in the free boundary case. Using the same template maximal clique $\\Lambda_0$\nas before, the set of maximal cliques $\\mathcal{L}_m$ now has to be redefined relative to the torus boundary condition case.\nIn the free boundary case we let $\\mathcal{L}_m$ contain all translations of $\\Lambda_0$ that are \ncompletely inside our $n\\times m$ lattice, i.e.\n\\[\n\\mathcal{L}_m = \\{ \\Lambda_0 + (t,u); t=-n,\\ldots,n,u=-m,\\ldots,m, \\Lambda_0 + (t,u)\\subseteq S\\},\n\\]\nwhere $\\Lambda_0 + (t,u) = \\{ (i+t,j+u); (i,j)\\in \\Lambda_0\\}$.\nOne should note that for a free boundary MRF the translational invariance property \nof the $\\phi^\\lambda$ parameters identified in Theorem \\ref{th:phiStationarity} no \nlonger apply, and neither will such a model be stationary. However,\nthe extra free $\\phi$ parameters that may be introduced in the \nfree boundary case will only model properties sufficiently close to\na boundary of the lattice. Our strategy in the free boundary case \nis to keep the same $\\phi$ parameter vector as in the torus case,\nto adopt translational invariant potential functions $V_\\Lambda(\\bold 1_\\Lambda^\\lambda)=\\phi^\\lambda$\nfor all maximal cliques $\\Lambda\\in\\mathcal{L}_m$ just as in the torus case, but to add non-zero \npotential functions for some (non-maximal) cliques at the \nboundaries of the lattice. Our motivation for this is to reduce the\nboundary effect and, hopefully, to get a model which is less non-stationary. To define our \nnon-zero potential functions at the boundaries, imagine that our\n$n\\times m$ lattice is included in a much larger lattice and that \nthis extended lattice also has maximal cliques that are translations of $\\Lambda_0$. \nWe then\ninclude a non-zero potential function for every maximal clique\nin the extended lattice which is partly inside and partly outside \nour original $n\\times m$ lattice. In such a maximal clique in the extended lattice, let $\\lambda$ denote \nthe set of nodes that are inside our $n\\times m$ lattice, and let $\\lambda^\\star$\ndenote the set of nodes outside. As we have assumed\nthat the maximal clique is partly inside and partly outside \nour original $n\\times m$ lattice, $\\lambda$ and $\\lambda^\\star$ are both non-empty and \n$\\lambda\\cup\\lambda^\\star$ is clearly a maximal clique in the extended \nlattice. For the (non-maximal) clique $\\lambda$ we define \nthe potential function\n\\begin{equation}\\label{Uborder}\nV_\\lambda(x_\\lambda) = \\frac{1}{2^{|\\lambda^\\star|}}\\sum_{x_{\\lambda^\\star}}\nV_{\\lambda\\cup\\lambda^\\star}(x_{\\lambda\\cup\\lambda^\\star}),\n\\end{equation}\nwhere $V_{\\lambda\\cup\\lambda^\\star}(x_{\\lambda\\cup\\lambda^\\star})$\nis the same (translational invariant) potential function we are \nusing for maximal cliques inside our $n\\times m$ lattice. One can note \nthat (\\ref{Uborder}) corresponds to averaging over the \nvalues in the nodes outside our lattice, assuming them to \nbe independent, and to take the values $0$ or $1$ with \nprobability a half for each.\n\n\n\\subsection{Example: The Ising model}\n\\label{sec:Ising}\nThe Ising model \\citep{Besag1986} is given by \n\\begin{equation}\np(x)=Z\\exp \\left \\{- \\omega \\sum_{(i,j)\\sim (t,u)}I(x_{i,j}\\neq x_{t,u})\\right\\},\n\\label{eq:Ising}\n\\end{equation}\nwhere the sum is over all horizontally and vertically adjacent sites, \nand $\\omega$ is a parameter controlling the probability of\nadjacent sites having the same value. We use the Ising model as an\nexample also later in the paper, and in particular we fit an MRF with\n$2\\times 2$ maximal cliques to data simulated from this model. \nAssuming torus boundary conditions and using that for binary variables\nwe have $I(x_{i,j}\\neq x_{t,u})=x_{i,j}+x_{t,u}-2x_{i,j}x_{t,u}$, we\ncan rewrite \\eqref{eq:Ising} as \n\\begin{equation*}\np(x)=Z\\exp \\left \\{-4\\omega\\sum_{(i,j)\\in S}x_{i,j} +2\\omega\\sum_{(i,j)\\sim(t,u)}x_{i,j}x_{t,u} \\right\\}.\n\\end{equation*}\nThus, the $\\beta^\\emptyset$ can be given any value as this will be\ncompensated for by the normalising constant, whereas\n$\\beta^{\\{(i,j)\\}}=-4\\omega$, $\\beta^{\\{(i,j),(i,j)\\oplus(1,0)\\}}=2\\omega$ and\n$\\beta^{\\{(i,j),(i,j)\\oplus(0,1)\\}}=2\\omega$, and $\\beta^{\\lambda}=0$ for\nall other cliques $\\lambda$. The corresponding\n$\\phi^{\\lambda}$ parameters can then be found using the recursive\nequation (S2) identified in the proof of Theorem\n\\ref{th:onetoone}. Using the notation introduced above for the $2\\times 2$\nmaximal clique case this gives $\\phi^0=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}=\\eta$, $\\phi^1=\\phi^{11}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}=\n\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}=-\\omega+\\eta$ and $\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}=-2\\omega+\\eta$,\nwhere $\\eta$ is an arbitrary value originating from the\narbitrary value that can be assign to $\\beta^\\emptyset$.\n\n\\section{Prior specification}\n\\label{sec:prior}    \n\nIn this section we define a generic prior for the parameters \nof an MRF with maximal cliques specified as in \\eqref{eq:Lm}. \nThe first step in the specification is to choose what parametrisation of the MRF to consider. \nIn the previous section we discussed two\nparametrisations for the MRF, with parameter vectors \n$\\beta$ and $\\phi$, respectively. When choosing between the two\nparametrisations and defining the prior we primarily have the torus\nversion of the MRF in mind. However, as the free boundary version of \nthe model is using the same parameter vectors, the prior we end up with \ncan also be used in that case. It should be remembered that \nthe parametrisations using $\\beta$ and $\\phi$ are non-identifiable,\nbut that it is sufficient to add one restriction to make them \nidentifiable. The perhaps easiest way to do this is to \nrestrict one of the parameters to equal zero, but other alternatives also\nexist. We return to this issue below. \nThe dimension of the $\\beta$ and $\\phi$ parameter vectors grows rapidly with \nthe number of elements in the set $\\Lambda_0$ defining the set of maximal \ncliques. Table \\ref{tab:ktimesl} \n\\begin{figure*}[h]\\center\nTable \\ref{tab:ktimesl} approximately here.\n\\end{figure*}\ngives the number of parameters in the\nidentifiable models, which we in the following \ndenote by $N_{\\Lambda_0}$, when $\\Lambda_0$ is a $k\\times l$ block of nodes.\nWe see that the number of parameters grows rapidly with the size of $\\Lambda_0$.\nIt is therefore natural to \nlook for prior formulations which include the possibility for a reduced\nnumber of free parameters. For the $\\beta$ parametrisation the \nperhaps most natural strategy to do this is to assign positive prior \nprobability to the event that one or several of the interaction \nparameters are exactly zero.\nThe interpretation of the $\\phi$ parameters\nis different from the interpretation of the $\\beta$ parameters,\nand it is not natural to assign positive probability for elements\nof the $\\phi$ vector to be exactly zero. A more reasonable scheme here\nis instead to set a positive prior probability for the event that groups of \n$\\phi$ parameters have exactly the same value. \n\nIn the Bayesian contingency tables literature\nthe $\\beta$ parametrisation is popular, see for example\n\\citet{Forster1999}, \\citet{massam2009} and \\citet{overstall2012} and\nreferences therein, where the second article develops a conjugate prior \nfor this parametrisation. \nHowever, these results do not\ndirectly apply for an MRF where one restricts the potential functions\nto be translational invariant. More importantly, however, the various\n$\\beta$ parameters relates to cliques of different sizes and this makes \nthe interpretation of the parameters difficult. In \\citet{Forster1999} and\nin \\citet{overstall2012} effort is made in order to create a\nreasonable multinormal prior for the $\\beta$ parameters. In contrast, the $\\phi$ \nparameters all represent the potential of a configuration of a maximal clique, \nwhich is all of the same size. Unless particular prior information is available and suggests the opposite, \nit is therefore natural to assume that all $\\phi$ parameters are on the\nsame scale. A tempting option is therefore first to assign identical and independent normal\ndistributions to these parameters, and obtain identifiability by\nconstraining the sum of the parameters to be zero. Thereby the elements of $\\phi$ are exchangeable \\citep{diaconis1980}. Note that the $\\beta$ parameters\nbecome multinormal also in our case, see for instance (S3) in the\nsupplementary materials.  In the following we therefore focus on specifying a prior for $\\phi$. \nWe first introduce notation necessary to define the groups of \nconfiguration parameters $\\phi$ that should have the same value and \nthereafter discuss possibilities for how to define the prior.\n\nTo define groups of configuration set parameters that should have the \nsame value, let $C_1,\\ldots,C_r$ be a partition of the configuration sets in $\\mathcal{C}$ with $C_i\\neq \\emptyset$ for $i=1,\\ldots,r$. \nThus, $C_i\\cap C_j = \\emptyset$ for $i\\neq j$ and \n$C_1\\cup\\ldots\\cup C_r=\\mathcal{C}$. For each $i=1,\\ldots,r$ we thereby assume \n$\\phi(c)$ to be equal for all $c\\in C_i$, and we denote this common value by $\\varphi_i$.\nSetting $z=\\{ (C_i,\\varphi_i),i=1,\\ldots,r\\}$ we thus can write the resulting potential\nfunctions as\n\\begin{equation}\\label{eq:VFinal}\nV_\\Lambda(x_\\Lambda|z) = \\sum_{(C,\\varphi)\\in z} \\varphi I\\left(x_\\Lambda \\in \\bigcup_{c\\in C}  c\\right).\n\\end{equation}\nWe define \na prior on the $\\phi$ parameters by specifying a prior for $z$. An\nalternative to this construction would be to build up $\\{C_1,...,C_r\\}$ in a non-random fashion, \nconstraining the $\\phi$ parameters\naccording to properties like symmetry and rotational invariance. However, our goal is that such properties can be\ninferred from observed data. \n\nGiven all configuration sets, we want to\nassign positive probability to the event that groups of configuration\nsets have exactly the same parameter value. For instance, the\nthree groups in Section \\ref{sec:Ising} is an example of such a\ngrouping for a $2\\times 2$ maximal clique. Since we do not allow empty groups $C_i$, the maximum number of\ngroups one can get is $N_{\\Lambda_0}+1$.\nOur prior distribution for $z$ is on\nthe form\n\\begin{equation*}\np(z)=p(\\{C_1,....,C_r\\})p(\\{\\varphi_1,...,\\varphi_r\\}|r)\n\\end{equation*}\nwhere $p(\\{C_1,...,C_r\\})$ is\na prior for the grouping of the configuration sets, while $p(\\{\\varphi_1,...,\\varphi_r\\}|r)$\nis a prior for the group parameters given the number of groups $r$. Two possibilities\nfor $\\{C_1,...,C_r\\}$ immediately comes to\nmind. The first is to assume a uniform distribution on the groupings, i.e.\n\\begin{equation*}\np_1(\\{C_1,...,C_r\\})\\propto const,\n\\end{equation*}\nmeaning that each grouping is apriori equally likely. However for $p(r)$, the marginal probability of the number of\ngroups, this\nmeans that most of the probability is put on groupings with approximately\n$(N_{\\Lambda_0}+1)\/2$ groups. In fact the probability $p(r)$ becomes equal to\n\\begin{equation*}\np(r)=\\frac{g(N_{\\Lambda_0}+1,r)}{\\sum_{i=1}^{N_{\\Lambda_0}+1}g(N_{\\Lambda_0}+1,i)},\n\\end{equation*}\nwhere $g(N_{\\Lambda_0}+1,r)$ is the number of ways $N_{\\Lambda_0}+1$ configuration sets can be\norganised into $r$ unordered groups, remembering that no empty groups\nare allowed. The function $g(N_{\\Lambda_0}+1,r)$ can be written as\n\\begin{equation*}\ng(N+1,r)=\\frac{1}{r!}\\sum_{i=0}^r\\binom{r}{i}(-1)^{r-i}i^{N+1},\n\\end{equation*}\nand is known as the Stirling number of the second kind \\citep{Graham1988}. \nFor the $2\\times 2$ maximal clique this means for instance\nthat $p(r=1)=p(r=11)\\approx 10^{-6}$ while $p(r=5)=0.36$. An\nalternative for $p(\\{C_1,...,C_r\\})$ is to make the\ndistribution for the number of groups uniform. This can be done by defining the probability distribution\n\\begin{equation*}\np_2(\\{C_1,...,C_r\\})=\\frac{1}{(N_{\\Lambda_0}+1)g(N_{\\Lambda_0}+1,r)}.\n\\end{equation*}\nWith this prior a particular grouping with many or few groups will have a larger\nprobability than a particular grouping with approximately\n$(N_{\\Lambda_0}+1)\/2$ groups. In the $2\\times 2$ case for example, the\nprobability of the grouping where all\nconfiguration sets are assigned to the same group or the grouping\nwith 11 groups is $p(\\{C_1\\})=p(\\{C_1,...,C_{11}\\})=0.09$, while\nthe probability of a particular grouping with 5 groups is\n$p(\\{C_1,...,C_5\\})\\approx 10^{-7}$. Observe however, that with both\npriors we have that the groups are uniformly distributed when the number of groups is given.\nAs a compromise between the two prior\ndistributions we propose  \n\\begin{equation*}\np(\\{C_1,...,C_r\\})\\propto p_1(\\{C_1,...,C_r\\})^{1-\\gamma}p_2(\\{C_1,...,C_r\\})^{\\gamma},\n\\end{equation*}\nwhere $0 \\leq\\gamma\\leq 1$. \n\nAs also discussed above, to get an identifiable model we need to put one additional restriction\non the elements of $\\phi$, or alternatively on the $\\varphi_i$ parameters. As we want the \ndistribution $p(\\varphi_1,\\ldots,\\varphi_r|r)$ to be exchangeable we want the restriction also\nto be exchangeable in the $\\varphi_i$ parameters, and set\n\\begin{equation}\n\\sum_{(C,\\varphi)\\in z}\\varphi=0.\n\\label{eq:identifiability}\n\\end{equation}\nUnder this sum-to-zero restriction we assume the $\\varphi_i$ apriori \nto be independent normal with zero mean and with a common variance\n$\\sigma_{\\varphi}^2$. \nThis fully defines\nthe prior for $z$, except that we have not specified values for the\ntwo hyper-parameters $\\gamma$ and $\\sigma_{\\varphi}^2$.\n   \n\\section{Posterior sampling}\\label{sec:sampling}\nIn this section we first discuss different strategies proposed\nin the literature for how to handle the computationally intractable \nnormalising constant in discrete MRFs, and in particular\ndiscuss their applicability in our situation. Thereafter\nwe describe the RJMCMC algorithm we adopt for simulating\nfrom our posterior distribution.\n\n\n\n\n\\subsection{Handling of the normalising constant}\n\\label{sec:normconstant}\nDiscrete MRFs contain a computationally intractable \nnormalising constant and this makes the fully Bayesian approach\nproblematic. Three strategies have been proposed to \ncircumvent or solve this problem. The first alternative is to replace \nthe MRF likelihood with a computationally tractable \napproximation. The early \\citet{art109} use the pseudo-likelihood for \nthis, \\citet{art117} and \\citet{art145} adopt a reduced dependency \napproximation (RDA), and \\citet{phd6} and \\citet{tjelmeland2012}\nconstruct a POMM approximation by making use of theory for \npseudo-Boolean functions. The second strategy, used in \n\\citet{art144}, is to adopt an estimate of the normalisation constant obtained\nby some Markov chain Monte Carlo (MCMC) procedure prior to \nsimulating from the posterior, and the third alternative is to include an auxiliary variable sampled \nfrom the MRF $p(x|\\phi)$ in the posterior simulation algorithm. \n\\citet{art120} is the first article using the third approach, and \nthe exchange algorithm of \\citet{pro20} falls within the same\nclass. \\cite{CaimoFriel2011} and \\citet{Everitt2012} adopt an \napproximate version of this third approach, by replacing\nperfect sampling from $p(x|\\phi)$ with approximate sampling\nvia an MCMC algorithm.\n\n\n\nThe three approaches all have their advantages and disadvantages.\nFirst of all, only the third approach is without approximations\nin the sense that it defines an MCMC\nalgorithm with limiting distribution exactly equal to the \nposterior distribution of interest. However, for this approach \nto be feasible perfect sampling from $p(x|\\phi)$ must \nbe possible, and computationally reasonably efficient,\nfor all values of $\\phi$. The strategy used in the second \nclass requires in practice that the parameter vector\n$\\phi$ is low dimensional. The approximation \nstrategy does not have restrictions on the dimension of $\\phi$\nand perfect sampling from $p(x|\\phi)$ is not needed. In that \nsense this approach is more flexible, but of course the \napproximation quality may depend on the the parametric form \nof the MRF and the value of $\\phi$.\n\nIn principle \nany of the approaches discussed above may be used in our\nsituation, but the\ncomplexity of the parameter space makes the prior estimation of the \nnormalisation constant approach impractical. Moreover, the \naccuracy of the pseudo-likelihood approximation is known to be quite poor, and\nin simulation exercises we found that perfect sampling from $p(x|\\phi)$\nwas in practice infeasible for many of the\nhigher-order interaction models visited by our RJMCMC algorithm.\nThe approximate version in \\citet{CaimoFriel2011} is, however,\na viable alternative.\nWe are thereby left with the RDA approach, the POMM approximation,\nand the strategy proposed in \\citet{CaimoFriel2011}.\nIn our simulation examples we adopt the second of these, but the other two\ncould equally well have been used. In fact, in one of our simulation \nexamples we use also the strategy from \\citet{CaimoFriel2011} to check\nthe approximation quality obtained when replacing the MRF with the \nPOMM approximation.\n\n\n\\subsection{MCMC algorithm}\nAssume that an observed binary $n\\times m$ image is \navailable. We consider this image as a realisation from our MRF \nwith the free boundary conditions defined in Section \\ref{sec:MRF}. As\na prior for the MRF parameters we adopt the prior specified in \nSection \\ref{sec:prior}.\nThe focus in this section is then on how to sample from the \nresulting posterior distribution. One should note that in \nthis section we formulate the algorithm as if one can evaluate\nthe MRF likelihood, including the normalising constant. This is \nof course not feasible in practice, so when running the algorithm \nwe replace the MRF likelihood with the corresponding POMM \napproximation discussed above.\n\nLetting $x$ denote the observed\nimage, the posterior distribution we want to sample from is \ngiven by\n\\begin{equation*}\np(z|x) \\propto p(x|z) p(z),\n\\end{equation*}\nwhere $p(x|z)$ and $p(z)$ are the MRF defined by \\eqref{eq:VFinal}\nand the prior defined in Section\n\\ref{sec:prior}, respectively. \nTo simulate from this posterior we adopt a reversible \njump Markov chain Monte Carlo (RJMCMC) algorithm \\citep{green1995} with \nthree types of updates. The detailed proposal mechanisms are specified\nin the supplementary materials, here we just give a brief description\nof our proposal strategies.\n\n\nThe first proposal in our algorithm is simply first to propose a\nchange in an existing $\\varphi$ parameter by a random walk proposal\nwith variance $\\sigma^2$,\nand thereafter to subtract the same value from all $\\varphi$\nparameters to commit with the sum-to-zero constraint. In the second proposal we draw a pair of groups and propose to move\none configuration set from the first group to the second group,\nensuring that the two groups are still non-empty. In the last proposal\ntype, we propose a new state by either increasing or\ndecreasing the number of groups with one. When increasing the number of\ngroups by one we randomly choose a configuration set from a\nrandomly chosen group and propose this configuration set to be a new\ngroup. When proposing to reduce the number of parameters with\none, we randomly choose a group with only one configuration\nset and propose to merge this group into another group. In the\ntrans-dimensional proposals we ensure that the proposed parameters\ncommit with the sum-to-zero constrain by subtracting the same value\nfrom all $\\varphi$ parameters.\n\n\\section{Simulation examples}\n\\label{sec:results}\n\nIn this section we first present an example based on a simulated\ndata set from the Ising model, and thereafter present results for a data set of census \ncounts of red deer in the Grampians Region of north-east Scotland. In\naddition, another example based on simulated data is included in the\nsupplementary materials. In all the simulation experiments we use the prior distribution\nas defined in Section \\ref{sec:prior}. In this prior the values of the two \nhyper-parameters $\\sigma_\\varphi$ and $\\gamma$ must be specified.\nWe have fixed $\\sigma_\\varphi=10$ and tried $\\gamma=0$, $0.5$ and\n$1$. When discussing simulation results we first present results for \n$\\gamma=0.5$. As the likelihood function we use the MRF discussed in \nSection \\ref{sec:MRF} and we use $2\\times 2$ maximal cliques except in the last part of\nthe real data example where we also discuss results for $3\\times 3$ \nmaximal cliques. To cope with the computationally intractable normalising \nconstant of the MRF likelihoods, we adopt the approximation strategy of\n\\cite{tjelmeland2012}. The MRF is then approximated with a \npartially ordered Markov model (POMM), see \\citet{art119}, \nwhere the conditional distribution of one variable given all \nprevious variables is allowed to depend on maximally $\\nu$ \nprevious variables. We have tried different values for $\\nu$\nand found that in all our examples $\\nu=7$ is sufficient to obtain\nvery good approximations, so all the results presented here\nare based on this value of $\\nu$. To simulate from\nposterior distributions we use the reversible jump \nMCMC algorithm defined in Section \\ref{sec:sampling}. In our sampling algorithm we have\nan algorithmic tuning parameter $\\sigma^2$ as the variance in Gaussian\nproposals. Based on the results of\nsome preliminary runs we set $\\sigma=0.3$. One\niteration of our sampling algorithm is defined to be one proposal of each type. Lastly we note that parallel\ncomputing was used in order to reduce computational time, and the\ntechnique that is used is explained in the supplementary materials.\n\n \n\n\n\\subsection{The Ising model}\n\\label{sec:IsingResults}\n\nWe generated a realisation from the Ising model given in\nSection \\ref{sec:Ising} with $\\omega=0.4$ on a $100\\times 100$ lattice, \nconsider this as our observed data $x$ and simulate by the RJMCMC algorithm\nfrom the resulting posterior distribution. The $x$ was obtained using \nthe perfect sampler presented in \\cite{propp1996}. From\nthe calculations in Section \\ref{sec:Ising} we ideally want the correct\ngroups,\n$\\{c^0,c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$ $\\{c^1,c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$, and $\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}\\}$,\nto be visited frequently by our sampler. Note that due to our\nidentifiability restriction in\n\\eqref{eq:identifiability} the configuration set parameters\nshould be close to the values given in Section \\ref{sec:Ising} with\n$\\eta=\\omega$. We run our sampler for 20000\niterations and study the simulation\nresults after convergence. A small convergence study is included in\nthe supplementary materials for the other simulated data set. The acceptance rate for the parameter value proposals is 19\\%, whereas the acceptance\nrates for the other two types of proposals are both around 1\\%. The\nestimated distribution for the number of groups is 94\\%, 5\\% and 1\\%,\nfor 3, 4 and 5 groups respectively.  \n\nIn Figure \\ref{fig:groupingMatrixIsing} we have plotted the\nmatrix representing the estimated posterior probability of two\nconfiguration sets being\nassigned to the same group. \n\\begin{figure*}[h]\\center\nFigure \\ref{fig:groupingMatrixIsing} approximately here.\n\\end{figure*}\nAs we can see in this\nfigure, the configuration sets are separated into 3 groups, and these\ngroups correspond to the correct grouping shown in grey. About $94\\%$ of the realisations is assigned to\nthis particular grouping, and almost all other groupings that are simulated correspond to\ngroupings where the middle group is split in various ways, while some\nvery few are splits of the groups $\\{c^0,c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$ and $\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}\\}$. Every one\nof these alternative groupings\nhave an estimated posterior probability of less than 0.5\\%. \n\nOne informative way to look at the result of the simulation is to estimate the posterior\ndistribution for the interaction parameters $\\beta$. Histograms and\nestimated 95\\% credibility intervals for each of the parameters are\ngiven in Figure \\ref{fig:interactionIsing}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:interactionIsing} approximately here.\n\\end{figure*}\nAs we can see, all the\ntrue values of the interaction parameters are within the estimated\ncredibility intervals, however the modes of the distributions for the\npairwise horizontal and vertical second order interactions, see Figure\n\\ref{fig:Isingi2} and \\ref{fig:Isingi3}, seem to be somewhat lower than the\ncorrect value. \n\nTo study the properties of the MRF $p(\\cdot |z)$ when \n$z$ is a sample from the posterior $p(z|x)$ we take $5000$ samples\nfrom the MCMC run for $p(z|x)$ and generate for each of these\na corresponding realisation from the MRF $p(\\cdot |z)$. \nTo analyse these $5000$ images we use six statistics describing\nlocal properties of the images. The statistics used and resulting\ndensity estimates (solid) of the distribution of these statistics\nare shown in Figures \\ref{fig:compareSimIsing} (a)-(f).\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:compareSimIsing} approximately here.\n\\end{figure*}\nIn the same figures we also show \ndensity estimates of the same statistics when images are generated \nfrom the Ising model with the true parameter value (dashed),\nand when images are generated from the Ising model with \nparameter value $\\omega$ generated from the posterior distribution\ngiven our observed image $x$ (dotted). In this last case, a zero mean\nGaussian prior with standard deviation equal to ten is\nused for $\\omega$. In these figures we also see that the data we use for posterior sampling\n(black dots) of\n$z$ is a realisation from the Ising model with low values for \nthe number of equal horizontal and vertical\nadjacent sites, see Figure \\ref{fig:is2} and \\ref{fig:is3}, which\ncauses, as already observed above, our simulations of the second order\ninteractions between horizontal and vertical adjacent sites to be \nsomewhat lower than the true values.\nIn fact we can see that the simulations from the\nIsing model using posterior samples for the parameter value closely\nfollows that of our $2 \\times 2$ model. This means that the results\nfrom our model is\nas accurate as the result one gets when knowing that the true model is the\nIsing model without knowing the model parameter. \n\nTo evaluate the quality of the POMM approximation\nin this example, we also simulate from the posterior\ndistribution with the same RJMCMC algorithm using the \napproximate exchange algorithm of \\cite{pro20}, as discussed \nin Section \\ref{sec:normconstant}.\nWe compare in Figures \\ref{fig:compareSimIsing} (g) and (h) the \nresults using the POMM approximation (solid) to the results from \nthe approximate \nexchange algorithm (dashed) using two of our six statistics. We \nobserve that the differences are minimal for these two, and indeed we \nget as accurate results for the four other statistics as well. \nThat these two very different approximation strategies produces\nessentially the same results strongly indicate that both \nprocedures are very accurate.     \n\nAll the above results are for $\\gamma=0.5$, but as mentioned in the introduction of this section we also investigate\nthe results for $\\gamma=0$ and $1$. For $\\gamma=0$ the configuration sets\nare organised into 3 (66\\%), 4 (31\\%) or 5 (3\\%) groups, and for $\\gamma=1$ we get 3 (96\\%) or 4\n(4\\%) groups. From these numbers we see the effect of varying $\\gamma$. In particular when increasing $\\gamma$ from\n$0.5$ to $1.0$ the tendency to group\nmore configuration sets together becomes stronger for this data set.\n\n\n\n\\subsection{Red deer census count data}\n\\label{sec:redDeer}\nIn this section we analyse a data set of census counts of\nred deer in the Grampians Region of north-east Scotland. A full\ndescription of the data set is found in \\cite{augustin1996} and\n\\cite{buckland1993}. The data is obtained by dividing the region of\ninterest into $n=1277$ grid cells on a lattice and observing the presence or\nabsence of red deer in each cell. In our notation this is our observed\nimage $x$, but in this example we also have the four covariates\naltitude, mires, north coordinate and east coordinate available in each\ngrid cell. The\nbinary data $x$ and the two first covariates are shown in Figure\n\\ref{fig:redDeerData}. \n\\begin{figure*}[h]\\center\nFigure \\ref{fig:redDeerData} approximately here.\n\\end{figure*}\nWe denote the covariate $k$ at a location $(i,j)$ by\n$y_{i,j,k}, \\ j=1,2,3,4$, and model them into the likelihood function in\nthe following way \n\\begin{equation}\\label{eq:likelihoodDeer}\np(x|z,\\theta^C,y)=Z\\exp{\\left(\n    \\sum_{\\Lambda\\in \\mathcal{L}_m}V_\\Lambda(x_{\\Lambda}|z)+\\sum_{(i,j)\\in S}x_{i,j}\\sum_{k=1}^4\\theta^C_ky_{i,j,k}\\right )},\n\\end{equation}  \nwhere $\\theta^C=(\\theta^C_1,...,\\theta^C_4)$ are the parameters for\nthe covariates.\n\nWe put independent zero mean Gaussian prior distributions with standard\ndeviation equal to 10 on $\\theta^C_j$, $j=1,...,4$. In the sampling algorithm\nthese covariates are updated using random walk, i.e. we uniformly choose\none of the four covariates to update and propose a new value using a\nGaussian distribution with the old parameter value as\nthe mean and a standard deviation of $0.1$.\n\nWe ran our algorithm for 50000 iterations, and the acceptance rates\nfor the parameter random walk proposal is 42\\%, the group changing\nproposal is 33\\%, the trans-dimensional proposal is 5\\%, and the\ncovariate proposal is 48\\%. The posterior most probable grouping \nbecomes $\\{c^0\\}$,\n$\\{c^1,c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$ and $\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$ \nwith probability 33.2\\%. In total more\nthan 2500 different groupings\nare visited, and except for the posterior most probable grouping the posterior probabilities of all other groupings are less than 5\\%. The estimated\nposterior probability distribution for the number of groups becomes\n43\\% for 3 groups, 48\\% for 4 groups, 8\\% for 5 groups and 1\\% for\n6 groups. In particular, the realisations with four or more groups are\nmostly groupings where the set $\\{c^1,c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$  is split in various\nways. This can also be seen in Figure\n\\ref{fig:configurationMatrixDeer}, which shows\nthe estimated posterior probability of two configuration sets\nbeing assigned to the same group.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:configurationMatrixDeer} approximately here.\n\\end{figure*}\nThe grey blocks in this figure show the estimated posterior most probable grouping described\nabove. Next\nwe estimate the posterior density for the interaction parameters, see\nFigure \\ref{fig:interactionsDeer}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:interactionsDeer} approximately here.\n\\end{figure*}\nAs we can see, most of the higher\norder interaction parameters becomes significantly different from zero,\nsuggesting that a $2\\times 2$ clique system is needed for this\ndata set. \nFigure \\ref{fig:covariatesDeer} shows the estimated\nposterior density for the covariate parameters.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:covariatesDeer} approximately here.\n\\end{figure*}\nAs we can see from the\ncredibility intervals, all these parameters are significantly different\nfrom zero, which justifies the need to include them. \nSimulations of $p(x|z,\\theta^C,y)$ for\nthree randomly chosen posterior samples of $z$ and $\\theta^C$ are\nshown in Figure \\ref{fig:simulationsDeer}. \n\\begin{figure*}[h]\\center\nFigure \\ref{fig:simulationsDeer} approximately here.\n\\end{figure*}\nAs we can see the spatial\ndependency in these realisations looks similar to the data which\nindicates that the features of this data set are captured with this\nmodel. \n\n\nAs discussed above, the estimated marginal posterior densities \nfor the interaction parameters in Figure \\ref{fig:interactionsDeer}\nindicate that higher order interaction parameters are needed for this\ndata set. To investigate this further we also run a corresponding MCMC simulation\nwith a prior where the spatial interaction is as in the nearest neighbour\nautologistic model defined in \\citet{Besag1972}, whereas the covariates\nare included as in (\\ref{eq:likelihoodDeer}). This pairwise interaction\nprior has three interaction parameters, for first-order interactions\nand for horisontal and vertical second-order interactions, respectively, \nand apriori\nwe assume these three parameters to be independent and \nGaussian distributed with zero-mean\nand standard deviations equal to ten. To simulate these three\nparameters we randomly choose one and propose a zero mean Gaussian\nchange with standard deviation equal to 0.3 to the chosen parameter. For the $\\theta_j^C$ parameters\nwe adopt the same prior and proposals as before. For the pairwise interaction prior and our original prior\nin (\\ref{eq:likelihoodDeer}), Figure \\ref{fig:compareSimDeer} shows \n\\begin{figure*}[h]\\center\nFigure \\ref{fig:compareSimDeer} approximately here.\n\\end{figure*}\nestimates of the resulting\nmarginal posterior distributions for the same six statistics studied in \nour Ising simulation example. For several of the statistics we see that \nthere is a clear difference between the results for the two priors. \nThe differences for the higher-order interaction statistics are perhaps\nless surprising, but one should note that the distribution of the \nfirst-order statistic in \nFigure \\ref{fig:compareSimDeer}(a) also changes quite much when\nallowing higher order interactions. One should also note that \nour $2\\times 2$ model fits better to the statistics of the data, \nshown as black dots in the figures.\n\n\n\nReturning to the $2\\times 2$ prior, using\n$\\gamma=0$ in the prior for this data set gives the estimated posterior probability\ndistribution 24\\%, 63\\%, 11\\% and 2\\% for 3, 4, 5 and 6 groups\nrespectively, whereas for $\\gamma=1$ we obtain 60\\%, 35\\% and 5\\% for 3, 4\nand 5 groups respectively. Again we see that higher values of $\\gamma$\nresults in more realisations with fewer number of\ngroups. However, for all the three values of $\\gamma$ the estimated most probable\ngrouping is the same.\n\nWe end our discussion of this data set by mentioning that some results when assuming a clique size of $3\\times 3$ is\nincluded in the supplementary material of this paper. These results\nindicate that no more significant structure is introduced in the $3\\times\n3$ case for this data set. \n\n\n\\section{Closing remarks}\n\\label{sec:discussion}\nOur main focus in this paper is to design a generic prior distribution\nfor the parameters of an MRF. This is done by assuming a set of maximal \ncliques defined from a template maximal clique $\\Lambda_0$, \nbut as the number of free parameters grows quickly as a function of\nthe number of elements in $\\Lambda_0$ we construct our prior\ndistribution such that it gives a positive \nprobability for groups of parameters to have\nexactly the same value. In that way we reduce the effective number of\nparameters, still keeping the flexibility large cliques \nprovides. Proposal distributions that enable\nus to simulate from the resulting posterior distributed is also\npresented. However, to evaluate the likelihood we use a\npreviously defined approximation to MRFs \\citep{phd6}, and the trade\noff between accuracy and computational complexity limits in practice the size of the cliques that can be assumed. An\nalternative to approximations is perfect sampling \\citep{propp1996}, but this was in all\nour examples too computationally intensive. A third alternative would be\nto use an MCMC sample of $x$ instead of a perfect sample, as described\nin for instance \\citet{Everitt2012}. An issue with this approach is the need to set a burn in period for the sampler of $x$, where a\ntoo long burn in period would make the parameter sampler too\nintensive. Lastly, we illustrate the effect of our prior distribution and\nsampling algorithm on two examples. \n\nOur focus in this paper is on binary\nMRFs. It is however possible to generalise our framework to discrete MRFs, i.e. where $x_i\\in \\{0,1,...,K\\}$ for\n$K \\geq 2$. An identifiable parametrisation of a\ndiscrete MRF using clique potentials can with a small effort be\ndefined in a similar way to what is done in the binary case, and ones this parametrisation is established, the prior distribution\npresented in this paper can be used unchanged. The same apply\nto our sampling strategy.    \n\nWith our prior distribution the size of the maximal cliques, and\nthereby the number of configuration sets, act as a hyper parameter and must\nbe set prior to any sampling algorithm. One could imagine putting a\nprior also on $\\Lambda_0$, introducing the need to\nconstruct algorithms for trans-dimensional sampling also for $\\Lambda_0$. \nAnother way to avoid the need to set the number of\nconfiguration sets would be to construct a\nprior distribution for the $\\beta$ parameters. A natural choice would\nbe to construct a positive prior probability for these parameters to\nbe exactly zero, and in this way the significant interactions of an MRF can\nbe inferred from data. However, as discussed above, \nit is not clear to us how to design generic\nprior distributions for the values of these interaction parameters, as\nhigher order interactions intuitively would be different from lower\norder interaction. Also, grouping $\\beta$ parameters together in order\nto reduce the number of parameters would, for the same\nreason as above, make little sense. An ideal solution would be somehow\nto draw strength from both of the two\nparametrisations in order to assign a prior distribution to both the \nappearance of different\ncliques and the number of free parameters. This idea is currently work\nin progress. \n\n\\section*{Supporting Information}\n\nAdditional Supporting Information may be found in the online version\nof this article:\\\\\n\n{\\noindent \\bf Section S.1:} Proof of one-to-one relation\nbetween $\\phi$ and $\\beta$.\n\n{\\noindent \\bf Section S.2:} Proof of translational\ninvariance for $\\beta$.\n\n{\\noindent \\bf Section S.3:} Proof of translational invariance for $\\phi$.\n\n{\\noindent \\bf Section S.4:} Details for the MCMC sampling algorithm.\n\n{\\noindent \\bf Section S.5:} The independence model with\ncheck of convergence.\n\n{\\noindent \\bf Section S.6:} Reed deer data with $3 \\times 3$ maximal cliques.\n\n{\\noindent \\bf Section S.7:} Parallelisation of the\n  sampling algorithm.\n\n\n\n\\bibliographystyle{jasa}\n\n\\section{Proof of one-to-one relation between $\\phi$ and $\\beta$}\n\\label{sec:S1}\n\n\\begin{theorem}\nConsider an MRF and constrain the $\\phi$ parametrisation of the potential functions as described in\nDefinition 1. Then there is a one-to-one relation between \n$\\{\\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$\nand $\\{\\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$. \n\\end{theorem}\n\n\\begin{proof}We prove the theorem by\nestablishing recursive equations showing how to compute the $\\beta^\\lambda$'s from\nthe $\\phi^\\lambda$'s and vice versa.\n\nSetting $x=\\bold 1^\\lambda$ for some $\\lambda\\in\\mathcal{L}$ into the two \nrepresentations of $U(x)$ in (2) and (4), we get\n\\begin{equation*}\n\\sum_{\\Lambda\\in \\mathcal{L}_m}V_{\\Lambda}(\\bold\n1^{\\lambda}_{\\Lambda})=\\sum_{\\lambda^\\star\\in \\mathcal{L}}\\beta^{\\lambda^\\star}\n\\prod_{(i,j)\\in \\lambda^\\star} \\bold 1^{\\lambda}_{\\{(i,j)\\}}.\n\\end{equation*}\nUsing (3) and Definition 1 this gives\n\\begin{equation}\n\\sum_{\\Lambda\\in \\mathcal{L}_m}\\phi^{\\lambda\\cap\n  \\Lambda}=\\sum_{\\lambda^\\star\\subseteq \\lambda}\\beta^{\\lambda^\\star}.\n\\label{eq:equal}\n\\end{equation}\nSplitting the sum on the left hand side into one sum over \n$\\Lambda\\in\\mathcal{L}_m^\\lambda$ and one sum over\n$\\Lambda\\in\\mathcal{L}_m\\setminus\\mathcal{L}_m^\\lambda$,\nand using that $\\lambda\\cap \\Lambda=\\lambda$ for \n$\\Lambda\\in \\mathcal{L}_m^\\lambda$ we get\n\\begin{equation*}\n|\\mathcal{L}_m^{\\lambda}|\\phi^{\\lambda}+\\sum_{\\Lambda\\in \\mathcal{L}_m\\setminus \\mathcal{L}_m^\\lambda}\\phi^{\\lambda\\cap\n  \\Lambda}=\\sum_{\\lambda^\\star\\subseteq \\lambda}\\beta^{\\lambda^\\star}.\n\\end{equation*}\nSolving for $\\phi^\\lambda$ gives\n\\begin{equation}\n\\phi^{\\lambda}=\\frac{1}{|\\mathcal{L}_m^{\\lambda}|}\\left\n  [\\sum_{\\lambda^\\star\\subseteq\n    \\lambda}\\beta^{\\lambda^\\star}-\\sum_{\\Lambda\\in \\mathcal{L}_m\\setminus\n\\mathcal{L}_m^\\lambda}\\phi^{\\lambda\\cap \\Lambda}\\right].\n\\label{eq:claim}\n\\end{equation}\nClearly $|\\lambda\\cap\\Lambda| < | \\lambda|$ when\n$\\Lambda\\in \\mathcal{L}_m\\setminus \\mathcal{L}_m^\\lambda$,\nso \\eqref{eq:claim} implies that we can compute all \n$\\{ \\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$ recursively from \n$\\{ \\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$. First \nwe can compute $\\beta^\\lambda$ for $|\\lambda|=0$, i.e. \n$\\phi^\\lambda=\\phi^\\emptyset = \\beta^\\emptyset\/|\\mathcal{L}_m|$, then\nall $\\beta^\\lambda$ for which $|\\Lambda|=1$, thereafter\nall $\\beta^\\lambda$ for which $|\\Lambda|=2$ and so on until\n$\\beta^\\lambda$ has been computed for all $\\lambda\\in\\mathcal{L}$.\nThus, $\\{ \\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$ is uniquely\nspecified by $\\{ \\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$.\n\nSolving \\eqref{eq:equal} with respect to $\\beta^\\lambda$ we get\n\\begin{equation}\n\\beta^{\\lambda}=\\sum_{\\Lambda\\in \\mathcal{L}_m}\\phi^{\\lambda\n  \\cap \\Lambda}-\\sum_{\\lambda^\\star \\subset \\lambda}\\beta^{\\lambda^\\star},\n\\label{eq:claim2}\n\\end{equation}\nand noting that clearly $|\\lambda^\\star|<|\\lambda|$ when \n$\\lambda^\\star \\subset \\lambda$ we correspondingly  get \nthat $\\{ \\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$ can be \nrecursively computed from $\\{ \\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$.\nOne must first compute $\\beta^\\lambda$ for $|\\lambda|=0$, i.e.\n$\\beta^\\emptyset$, then all $\\beta^\\lambda$ for which \n$|\\lambda|=1$, thereafter all $\\beta^\\lambda$ for which \n$|\\lambda|=2$ and so on. Thereby $\\{ \\beta^\\lambda;\\lambda\\in\\mathcal{L}\\}$\nis also uniquely specified by $\\{ \\phi^\\lambda;\\lambda\\in\\mathcal{L}\\}$\nand the proof is complete.\n\\end{proof}\n\n\n\\section{Proof of translational invariance for $\\beta$}\n\\label{sec:S2}\n\\begin{theorem}\nAn MRF $x$ defined on a rectangular lattice\n$S=\\{ (i,j);i=0,\\ldots,n-1,j=0,\\ldots,m-1\\}$ with \ntorus boundary conditions and $\\mathcal{L}_m$ given\nin (5) is stationary if and only \nif $\\beta^{\\lambda}=\\beta^{\\lambda\\oplus (t,u)}$ for all \n$\\lambda \\in \\mathcal{L}$, $(t,u)\\in S$. We then say\nthat $\\beta^\\lambda$ is translational invariant.\n\\end{theorem}\n\n\\begin{proof}\nWe first prove the only if part of \nthe theorem by induction on $|\\lambda|$. Since $\\emptyset\n\\oplus (t,u)=\\emptyset$ we clearly have\n$\\beta^{\\emptyset}=\\beta^{\\emptyset \\oplus (t,u)}$ and thereby\n$\\beta^{\\lambda}=\\beta^{\\lambda \\oplus (t,u)}$ for $|\\lambda|=0$. \nNow assume all interaction parameters up to order $o$\nto be translational invariant, i.e. $\\beta^{\\lambda}=\\beta^{\\lambda \\oplus\n(t,u)}$ for $|\\lambda|\\leq o$. Now focusing on any $\\lambda^\\star\\in \\mathcal{L}$\nwith $|\\lambda^\\star|=o+1$, the assumed stationarity in particular gives \nthat we must have $p(\\bold 1^{\\lambda^\\star})=p(\\bold 1^{\\lambda^\\star\\oplus (t,u)})$.\nUsing (2) and (4) it follows that\n\\begin{equation*}\n\\beta^{\\lambda^\\star}+\\sum_{\\lambda \\subset \\lambda^\\star}\n\\beta^{\\lambda}=\\beta^{\\lambda^\\star\\oplus(t,u)}+\\sum_{\\lambda \\subset \\lambda^\\star\\oplus(t,u)}\\beta^{\\lambda}.\n\\end{equation*}\nRewriting the sum on the right hand side we get\n\\begin{equation*}\n\\beta^{\\lambda^\\star}+\\sum_{\\lambda \\subset \\lambda^\\star}\\beta^{\\lambda}=\\beta^{\\lambda^\\star\\oplus(t,u)}+\n\\sum_{\\lambda \\subset \\lambda^\\star}\\beta^{\\lambda\\oplus(t,u)},\n\\end{equation*}\nwhere the induction assumption gives that the two sums must be equal, and \nthereby $\\beta^{\\lambda^\\star}=\\beta^{\\lambda^\\star\\oplus(t,u)}$, which \ncompletes the only if part of the proof.\n\nTo prove the if part of the theorem we need to show that if the interaction parameters\nare translational invariant then $U(\\bold 1^A) = U( \\bold 1^{A\\oplus (t,u)})$ for any\n$A\\subseteq S$ and $(t,u)\\in S$. For $U(\\bold 1^A)$ we have\n\\begin{eqnarray*}\nU(\\bold 1^A) &=& \\sum_{\\lambda\\in\\mathcal{L}} \\beta^\\lambda \\prod_{(i,j)\\in\\lambda} \\bold 1^A_{i,j}\\\\\n&=& \\sum_{\\lambda\\in \\mathcal{L}} \\beta^{\\lambda\\oplus (t,u)} \\prod_{(i,j)\\in\\lambda\\oplus (t,u)}\n\\bold 1_{i,j}^A\\\\\n&=& \\sum_{\\lambda\\in\\mathcal{L}} \\beta^{\\lambda\\oplus (t,u)} \\prod_{(i,j)\\in \\lambda}\n\\bold 1_{i,j}^{A\\oplus (t,u)} \\\\\n&=& \\sum_{\\lambda\\in\\mathcal{L}} \\beta^{\\lambda} \\prod_{(i,j)\\in \\lambda}\n\\bold 1_{i,j}^{A\\oplus (t,u)} = U(\\bold 1^{A\\oplus (t,u)}),\n\\end{eqnarray*}\nwhere the first equality follows from (4), the second equality\nis true because $\\{ \\lambda\\oplus (t,u);\\lambda\\in \\mathcal{L}\\} =\n\\mathcal{L}$ for any $(t,u)\\in S$, the third equality follows \nfrom the identity $\\bold 1^A_{(i,j)\\oplus (t,u)} = \n\\bold 1_{i,j}^{A\\oplus (t,u)}$, and the fourth equality is using the assumed \ntranslational invariance of the interaction parameters. Thereby the proof\nis complete.\n\\end{proof}\n\n\n\\section{Proof of translational invariance for $\\phi$}\n\\label{sec:S3}\n\n\\begin{theorem}\nAn MRF $x$ defined on a rectangular lattice \n$S=\\{ (i,j);i=0,\\ldots,n-1,j=0,\\ldots,m-1\\}$ with torus boundary\nconditions and $\\mathcal{L}_m$ given in \n(5) is stationary if and only if\n$\\phi^{\\lambda}=\\phi^{\\lambda\\oplus (t,u)}$ for all \n$\\lambda \\in \\mathcal{L}$ and $(t,u)\\in S$. We then say\nthat $\\phi^\\lambda$ is translational invariant. \n\\end{theorem}\n\n\\begin{proof}\nGiven the result in Theorem 2 it is \nsufficient to show that $\\phi^\\lambda$ is translational invariant\nif and only if $\\beta^\\lambda$ is translational invariant. We first\nassume $\\beta^\\lambda$ to be translational invariant for all $\\lambda\\in\\mathcal{L}$ \nand need to show that then also $\\phi^\\lambda$ must be translational invariant for \nall $\\lambda\\in\\mathcal{L}$. Starting with \n\\eqref{eq:claim}, repeatedly using the specific form we are using for $\\mathcal{L}_m$ and the assumed\ntranslational invariance for $\\beta^\\lambda$, we get for any $\\lambda\\in\\mathcal{L}$, $(t,u)\\in S$ \n\\begin{eqnarray*}\n\\phi^{\\lambda\\oplus (t,u)} &=& \\frac{1}{\\left|\\mathcal{L}_m^{\\lambda\\oplus (t,u)}\\right|}\n\\left[ \\sum_{\\lambda^\\star\\subseteq \\lambda\\oplus (t,u)}\\beta^{\\lambda^\\star} -\n\\sum_{\\Lambda\\in\\mathcal{L}_m} \\phi^{(\\lambda\\oplus (t,u))\\cap\\Lambda}\n+ \\sum_{\\Lambda\\in\\mathcal{L}_m^{\\lambda\\oplus (t,u)}} \\phi^{(\\lambda\\oplus (t,u))\\cap \\Lambda}\\right] \\\\\n&=& \\frac{1}{|\\mathcal{L}_m^\\lambda|}\\left[ \\sum_{\\lambda^\\star\\subseteq \\lambda}\n\\beta^{\\lambda^\\star\\oplus (t,u)} - \n\\sum_{\\Lambda\\in\\mathcal{L}_m} \\phi^{(\\lambda\\oplus (t,u))\\cap (\\Lambda\\oplus (t,u))}\n+ \\sum_{\\Lambda\\in\\mathcal{L}_m^\\lambda} \\phi^{(\\lambda\\oplus (t,u))\\cap (\\Lambda\\oplus (t,u))}\\right]\\\\\n&=& \\frac{1}{|\\mathcal{L}_m^\\lambda|} \\left[ \\sum_{\\lambda^\\star\\subseteq \\lambda} \\beta^{\\lambda^\\star}\n- \\sum_{\\Lambda\\in\\mathcal{L}_m} \\phi^{(\\lambda\\cap \\Lambda)\\oplus (t,u)} +\n\\sum_{\\Lambda\\in\\mathcal{L}_m^\\lambda}\\phi^{(\\lambda\\cap\\Lambda) \\oplus (t,u)}\\right] \\\\\n&=& \\frac{1}{|\\mathcal{L}_m^\\lambda|}\\left[ \\sum_{\\lambda^\\star\\subseteq\\lambda} \\beta^{\\lambda^\\star} - \n\\sum_{\\Lambda\\in\\mathcal{L}_m\\setminus\\mathcal{L}_m^\\lambda} \\phi^{(\\lambda\\cap\\Lambda)\\oplus (t,u)}\\right].\n\\end{eqnarray*}\nFrom this we can use induction on $|\\lambda|$ to show that $\\phi^\\lambda$ is \ntranslational invariant for all $\\lambda\\in\\mathcal{L}$. Setting $\\lambda=\\emptyset$\nwe get $\\phi^{\\lambda\\oplus (t,u)} = (1\/|\\mathcal{L}_m^\\emptyset|)\\beta^\\emptyset$ which \nis clearly not a function of $(t,u)$. Then assuming $\\phi^{\\lambda\\oplus (t,u)}=\\phi^\\lambda$\nfor all $\\lambda\\in \\mathcal{L}$ with $|\\lambda| \\leq o$, considering the above \nrelation for a $\\lambda$ with $|\\lambda|=o+1$, and observing that \n$|\\lambda\\cap \\Lambda|\\leq o$ when $|\\lambda|=o+1$ and $\\Lambda\\in \\mathcal{L}_m\\setminus\n\\mathcal{L}_m^\\lambda$, we get that also $\\phi^{\\lambda\\oplus (t,u)}=\\phi^\\lambda$ for \n$|\\lambda|=o+1$, and the induction proof is complete.\n\nNext we assume $\\phi^\\lambda$ to be translational invariant for all $\\lambda\\in\\mathcal{L}$\nand need to show that then also $\\beta^\\lambda$ is translational invariant for all \n$\\lambda\\in\\mathcal{L}$. Starting with \\eqref{eq:claim2}, using the assumed translational \ninvariance of $\\beta^\\lambda$, \nand again repeatedly using the specific form of $\\mathcal{L}_m$, we get\n\\begin{eqnarray*}\n\\beta^{\\lambda\\oplus (t,u)} &=& \\sum_{\\Lambda\\in\\mathcal{L}_m}\\phi^{(\\lambda\\oplus (t,u))\\cap \\Lambda}\n-\\sum_{\\lambda^\\star\\subset \\lambda\\oplus (t,u)}\\beta^{\\lambda^\\star} \\\\\n&=& \\sum_{\\Lambda\\in\\mathcal{L}_m} \\phi^{(\\lambda\\oplus (t,u))\\cap (\\Lambda\\oplus (t,u))}\n- \\sum_{\\lambda^\\star \\subset \\lambda} \\beta^{\\lambda^\\star \\oplus (t,u)}\\\\\n&=&\\sum_{\\Lambda\\in\\mathcal{L}_m} \\phi^{(\\lambda\\cap \\Lambda)\\oplus (t,u))}\n- \\sum_{\\lambda^\\star \\subset \\lambda} \\beta^{\\lambda^\\star \\oplus (t,u)}.\n\\end{eqnarray*}\nUsing this we can easily use induction on $|\\lambda|$ to show that we must have \n$\\beta^{\\lambda\\oplus (t,u)}=\\beta^\\lambda$. The proof is thereby complete.\n\\end{proof}   \n\n\\section{Details for the MCMC sampling algorithm}\n\\label{sec:S4}\n\nIn this section we provide details of the proposal distributions that we use when sampling from the posterior distribution\n\\begin{equation*}\np(z|x) \\propto p(x|z) p(z),\n\\end{equation*}\nwhere $p(x|z)$ and $p(z)$ are the MRF\nand the prior given in the paper, respectively.\n\nTo simulate from this posterior distribution we adopt a reversible \njump Markov chain Monte Carlo (RJMCMC) algorithm with \nthree types of updates. The first update type uses a random walk proposal\nfor one of the $\\varphi$ parameters, the second proposes to move one \nconfiguration set to a new group, and the third proposes to \nchange the number of groups, $r$, in the partition of the \nconfiguration sets. In the following we describe the \nproposal mechanisms for each of the three update types. \nThe corresponding acceptance probabilities are given by standard formulas. It should be noted that only the \nlast type of proposal produces a change in the dimension of \nthe parameter space.\n   \n\\subsection{\\it Random walk proposal for parameter values}\n\nThe first proposal in our algorithm is simply to propose\na new value for an already existing parameter using a random walk proposal, but\ncorrecting for the fact that the parameters should sum to zero. More precisely, we first draw a change $\\varepsilon \\sim \\mbox{N}(0,\\sigma^2)$,\nwhere $\\sigma^2$ is an algorithmic tuning parameter. Second, we\nuniformly draw one element from the current state\n$z=\\{ (C_i,\\varphi_i),i=1,\\ldots,r\\}$, $(C_i,\\varphi_i)$ say, and\ndefine the potential new state as\n\\begin{equation*}\nz^* = \\left \\{ \\left (C_j,\\varphi_j - \\frac{1}{r}\\varepsilon \\right ),\nj=1,\\ldots,i-1,i+1,\\ldots,r \\right \\} \\cup \\left \\{\n\\left (C_i,\\varphi_i+\\varepsilon - \\frac{1}{r}\\varepsilon \\right )\n\\right \\}.\n\\end{equation*}\n\n\\subsection{\\it Proposing to change the group for one configuration set}\n\\label{sec:swappingProposal}\nLetting the current state be $z=\\{ (C_i,\\varphi_i),i=1,\\ldots,r\\}$,\nwe start this proposal by drawing a pair of groups, \n$C_i$ and $C_j$ say, where the first set $C_i$ is restricted to \ninclude at least two configuration sets. We draw $C_i$ and $C_j$ so \nthat the difference between the \ncorresponding parameter values, $\\varphi_i - \\varphi_j$, tend to be small.\nMore precisely, we draw $(i,j)$ from the joint distribution\n\\begin{eqnarray*}\nq(i,j)\\propto\n\\begin{cases}\n\\exp\\left(-(\\varphi_i-\\varphi_j)^2\\right ) & \\mbox{if $i\\neq\n  j$ and group $C_i$ contains at least two configuration sets,}\\\\\n0  & \\mbox{otherwise.}\n\\end{cases}\n\\end{eqnarray*}\nThereafter\nwe draw uniformly at random one of the configuration sets in $C_i$, \n$c$ say. Our potential new state is then obtained by moving $c$ from \n$C_i$ to $C_j$. Thus, our potential \nnew state becomes \n\\begin{equation*}\nz^* = \\left ( z \\setminus \n\\{ (C_i,\\varphi_i),(C_j,\\varphi_j)\\}\\right ) \n\\cup \\left \\{ (C_i\\setminus c,\\varphi_i ), (C_j\\cup\nc,\\varphi_j )\\right \\}. \n\\end{equation*}\n\n\\subsection{\\it Trans-dimensional proposals}\n\\label{sec:transDimProposal}\nLet again the current state be $z=\\{ (C_i,\\varphi_i),i=1,\\ldots,r\\}$. In\nthe following we describe how we propose a new state by either\nincreasing or reducing the number of groups, $r$, with one. There\nwill be a one-to-one transition in the proposal, meaning that the\nopposite proposal, going from the new state to the old state has a non-zero\nprobability. We make no attempt to jump between states where the\ndifference between the dimensions are larger than one. \n\nFirst we draw whether to increase or to decrease the number of\ngroups. If the number of groups are equal to the number of\nconfigurations sets, no proposal to increase the number of groups can\nbe made due to the fact that empty groups have zero prior\nprobability. In that case we propose to decrease the number\nof dimensions with probability 1. In our proposals we also make the\nrestriction that only groupings containing at least one group with\nonly one configuration set can be subject to a dimension reducing\nproposal. In a case where no such group exists, a proposal of\nincreasing the number of dimensions are made with probability 1. In a\ncase where both proposals are allowed we draw at random which to do with\nprobability 1\/2 for each. Note that at least one of the two proposals\nis always valid. \n\nWe now explain how to propose to increase the number of groups by one.\nWe start by drawing uniformly at random one of the groups with more \nthan one configuration set, $C_i$ say, which we want to split into two \nnew groups. Thereafter we draw uniformly at random one of the \nconfiguration sets in $C_i$, $c$ say, and form a new partition of \nthe configuration sets by extracting $c$ from $C_i$ and adding a \nnew group containing only $c$. Next we need to draw a parameter value for \nthe new group $\\{ c\\}$, and the parameter values for the other groups\nalso need to be modified for the proposal to conform with the requirement\nthat the sum of the (proposed) parameters should equal zero. \nWe do this by \nfirst drawing a change $\\varepsilon\\sim \\mbox{N}(0,\\sigma^2)$ in \nthe parameter value for $c$, where $\\sigma^2$ is the same tuning\nparameter as in the random walk proposal. We then define the potential new state as\n\\begin{eqnarray*}\nz^* &=& \\left \\{ \\left (C_j,\\varphi_j - \\frac{1}{r+1}(\\varphi_i +\n    \\varepsilon)\\right ),\nj=1,\\ldots,i-1,i+1,\\ldots,r\\right \\} \\cup \\\\\n&&\\left \\{\n\\left (C_i\\setminus c,\\varphi_i - \\frac{1}{r+1}(\\varphi_i + \\varepsilon)\\right),\n\\left(\\{ c\\},\\varphi_i + \\varepsilon - \\frac{1}{r+1}(\\varphi_i +\n  \\varepsilon)\\right)\\right \\}.\n\\end{eqnarray*}\n\nNext we explain the proposal we make when the dimension is to be decreased\nby one. Since we need a one-to-one transition in our proposals, we\nget certain restrictions for these proposals. In particular, the fact that\nonly groupings containing at least one group with only one\nconfiguration set are possible outcomes from a dimension increasing\nproposal dictates that dimension decreasing proposals only can be made\nfrom such groupings. Assume again our current model to be $z=\\{\n(C_i,\\varphi_i),i=1,\\ldots,r\\}$, where at least one group contains\nonly one configuration set. The strategy is to propose to merge one group consisting of only one \nconfiguration set into another group. As in Section \\ref{sec:swappingProposal}, we draw the two configuration\nsets to be merged so that the difference between the corresponding\nparameter values tend to be small. More precisely, we let the two \ngroups be $C_i$ and $C_j$ where $(i,j)$ is sampled according to the \njoint distribution\n\\begin{eqnarray*}\nq(i,j)\\propto\n\\begin{cases}\n\\exp\\left(-(\\varphi_i-\\varphi_j)^2\\right ) & \\mbox{if $i\\neq j$ and $C_i$ consists of \nonly one configuration set,}\\\\\n0  & \\mbox{otherwise.}\n\\end{cases}\n\\end{eqnarray*}\nNext we need to specify potential new parameter values. As\nthese must conform with how we generated potential new values in the split\nproposal, we have no freedom left in how to do this. The potential \nnew state must be\n\\begin{equation*}\nz^* = \\left \\{ \\left (C_k,\\varphi_k + \\frac{1}{r-1}\\varphi_i\\right\n ), k\\in \\{ 1,\\ldots,r\\} \\setminus \\{ i,j\\}\\right \\}\n\\cup \\left \\{ \\left (C_j\\cup C_i,\\varphi_j +\n    \\frac{1}{r-1}\\varphi_i\\right )\\right \\}.\n\\end{equation*}\nThe split and merge steps produce a change in the dimension of \nthe parameter space, so to calculate the acceptance probabilities \nfor such proposals we need corresponding Jacobi determinants.\nIt is straightforward to show that the Jacobi determinants for \nthe merge and split proposals become $\\frac{r}{r-1}$ and\n$\\frac{r}{r+1}$, respectively.\n\n\\section{The independence model with check of convergence}\n\\label{sec:S5}\n\nConsider a model where the variables are all independent of each other\nand $p(x_{i,j})=p^{x_{i,j}}(1-p)^{1-x_{i,j}}$ for each $(i,j)\\in S$ and where $p$ is the\nprobability of $x_{i,j}$ being equal to $1$. We get \n\\begin{equation}\\label{eq:independence}\np(x)=\\prod_{(i,j)\\in S}p^{x_{i,j}}(1-p)^{1-x_{i,j}}\\propto\\exp\\left(\\alpha \\sum_{(i,j)\\in S}x_{i,j}\\right),\n\\end{equation}\nwhere\n\\begin{equation*}\n\\alpha=\\ln \\left ( \\frac{p}{1-p}\\right ).\n\\end{equation*}\nIn this section we use the independence model as an example, and in particular we fit an MRF with $2 \\times 2$ maximal cliques to\ndata simulated from this model. Therefore it is helpful to know\nhow one\ncan represent the independence model using $2 \\times 2$ maximal\ncliques, and this can be done using the same strategy that was used\nfor the Ising model in Section 2.2 in the paper. We get $\\phi^0=\\eta$, $\\phi^1=\\frac{\\alpha}{4}+\\eta$, $\\phi^{11}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}}=\n\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}=\\frac{\\alpha}{2}+\\eta$, $\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}=\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}}=\n\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}=\\frac{3\\alpha}{4}+\\eta$ and $\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}=\\alpha+\\eta$, where \n$\\eta$ is an arbitrary value coming from the arbitrary value for $\\beta^\\emptyset$.\nIf $p=0.5$ we see that $\\alpha=0$ and all the configuration set\nparameters are equal.  \n\nWe generate a realisation from the independence model with $p=0.3$ on a $100\\times 100$\nlattice, consider this as our observed data $x$, and simulate by the\nMCMC algorithm defined in Section 4 from the\nresulting posterior distribution. Using the notation for the configuration\nsets in a $2\\times 2$ maximal clique and the results above, we ideally want our algorithm to produce realisations\nwith the groups $\\{c^0\\}$, $\\{c^1\\}$, $\\{c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}\\}$,\n$\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$ and\n$\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$. Note that due to our\nidentifiability restriction in\n(11) the configuration set parameters\nshould be close to the solution above with\n$\\eta=-\\alpha\/2$. To check convergence we investigated trace plots of\nvarious statistics, see Figure \\ref{fig:tracePlotsMarginal}, and the\nconclusion was that the algorithm converges very quickly. \n\\begin{figure*}[h]\\center\nFigure \\ref{fig:tracePlotsMarginal} approximately here.\n\\end{figure*}\nThe acceptance rate for the parameter value proposals\nis $24\\%$, whereas the acceptance rates for the other two types of \nproposals are both around $2\\%$. We run\nour sampling algorithm for 20000 iterations, and estimate the\nposterior probability of the number of groups. The configuration sets\nare organised into 4 (77\\%), 5\n(21\\%) or 6 (2\\%) groups, so for these data the grouping tends to be a little bit\ntoo strong compared to the correct number of groups. This can also be seen\nfrom the estimated posterior probability of two configuration sets being\nassigned to the same group, shown in Figure \\ref{fig:pairMatrixMarginal}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:pairMatrixMarginal} approximately here.\n\\end{figure*}\nThis figure suggests the four groups\n$\\{c^0\\},\\{c^1\\},\\{c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}\\}$, $\\{c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$\nwhich is also calculated to be the most probable\ngrouping estimated by counting the number of occurrences. In fact the\nposterior probability for this grouping is as high as $55 \\%$. In Figure \\ref{fig:pairMatrixMarginal} we also see how the most probable\ngrouping differ from the correct model grouping, shown in grey. The group $\\{ c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$ in the correct model is split in two\nin the most probable grouping, and the subsets \n$\\{ c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}\\}$ and \n$\\{ c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}\\}$ are inserted into the correct model groups \n$\\{ c^{11},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}},c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}}\\}$  \nand $\\{ c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\\}$, respectively.\n\nAs in the Ising model example in Section 5.1 we estimate the posterior distribution for the\ninteraction parameters, see Figure \\ref{fig:marginalInteractionParameters}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:marginalInteractionParameters} approximately here.\n\\end{figure*}\nAs we can\nsee, the true value of the interaction parameters are mostly within the\ncredibility intervals, but the tendency to group the configurations too\nmuch is in this case forcing some of the true values into a tail of\nthe marginal posterior distributions.  \n\nAs in the Ising example we compare the distribution of the same six\nstatistics from simulations from our $2\\times 2$ model with\nposterior samples of $z$, the independence model with correct parameter\nvalue, and the independence model with parameter value obtained by posterior\nsampling, see Figure \\ref{fig:compareSimIndependence}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:compareSimIndependence} approximately here.\n\\end{figure*}\nAs we can see, our model captures approximately the\ncorrect distribution of the chosen statistics. It is interesting to\nnote that for some statistics the realisations from the independence\nmodel with simulated $\\alpha$ values follows our model tightly\nwhereas for the other statistics it is close to the correct model.\n\nAlso for this data set we investigated the case where $\\gamma=0$ and\n$\\gamma=1$. For $\\gamma=0$ the configuration\nsets are organised into 4 (75\\%), 5\n(23\\%) or 6 (2\\%) groups, and for $\\gamma=1$ we get 4 (93\\%), 5 (6\\%),\n6 (1\\%) groups.  As expected we again see the tendency\ntowards stronger grouping when $\\gamma$ is increased. \n\nWe also did experiments were the value of $p$ was changed. If the\nvalue of $p$ is close to 0.5 the tendency to group the\nconfigurations too much becomes stronger. This makes perfectly sense,\nsince the correct grouping for $p=0.5$ is to put all\nconfiguration sets into only one group. In the other end, choosing $p$\ncloser to 0 or 1 gives a stronger tendency to group the configurations\naccording to the correct solution. This illustrates the fact that\nthe algorithm tries to find a good model for the data using as few\ngroups as possible, but as the difference between the true parameter values\nof the groups becomes larger the price to pay for choosing a model with fewer\nparameters increases. \n\n\n\n\n\\section{Red deer data with $3\\times 3$ maximal cliques}\n\\label{sec:S6}\n\nIn this section we present some results when assuming maximal clique size of\n$3\\times 3$ for the red deer data set presented in Section 5.2 in the\npaper. The main drawback\nwith our approach is computational time, which is very dependent on\nthe approximation parameter $\\nu$. One also needs to keep in mind that even data from simple models\nwill need many groups in the $3 \\times 3$ case to be modelled correctly. For instance, for the\nindependence model the 401 configuration sets would need to be separated\ninto 10 groups, while for the Ising model one would need 11 groups to\nget the correct model grouping. Similarly, the posterior most\nprobable grouping found for the $2 \\times 2$ case for the reed deer\nexample would need 38 groups to be modelled in the $3\\times 3$\ncase. Thus it is important not to assume larger maximal cliques than\nneeded. However for this data set it is possible to run the sampling algorithm\nwith $3\\times 3$ clique, even though this is computationally\nexpensive. \n\nTo get convergence we need a small generalisation to the\nproposal distribution for the trans-dimensional sampling step\npresented in Section \\ref{sec:transDimProposal}. In particular we\nallow for several configuration sets to be split out into a new group\nat a single proposal,  and correspondingly allow for the\npossibility of several configuration sets to be merge into another\ngroup in one single proposal. The estimated marginal\ndistribution of the number of groups is 1\\%, 65\\%, 33\\%, and 1\\%\nfor 29, 30, 31 and 32 groups respectively. Three realisations from\nthe likelihood for\nthree randomly chosen realisations of $z$ are shown in Figure\n\\ref{fig:3t3}(a), and comparing with the realisations for\nthe $2\\times 2$ case, see Figure 8 in the paper, it is hard\nto see any differences in the spatial structure of the realisations.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:3t3} approximately here.\n\\end{figure*}\n\nWe also investigated the distribution of three statistics for 5000\nrealisation from the likelihood of each of the two clique sizes, see Figure \\ref{fig:3t3}(b), and it appears to be\nlittle difference also here. These results indicate that $2\\times\n2$ maximal cliques might have sufficient complexity to explain this data set.          \n\n\\section{Parallelisation of the sampling algorithm}\n\\label{sec:S7}\n\nMost of the computing time for running our sampling algorithm is used\nto evaluate the likelihood in (2). In order to reduce\nthe running time we adopt a scheme that do multiple updates of the Markov chain by evaluating likelihoods in parallel.\n\nAssume we are in a state $z$ and propose to split\/merge into a new\nstate $z_1$. Now there are two possible outcomes for this\nproposal. Either we reject the proposal, which results in state $z$, or\nwe accept the proposal, which results in state $z_1$. Either way we\nalways propose a parameter update in the next step, and proposing this\nstep from both the two states $z$ and $z_1$ before evaluating the\nacceptance probability for the split\/merge step is possible. The\npossible outcomes for these three proposals are $z$, $z_1$, $z_2$\nand $z_{12}$, where $z$ is the outcome where neither the split\/merge\nproposal nor the following parameter proposal is accepted, $z_1$ is the\noutcome where the split\/merge proposal is accepted but not the following\nparameter proposal, $z_2$ is the outcome where the split\/merge\nproposal is not accepted but the parameter proposal is, and $z_{12}$ is the\noutcome where both the split\/merge proposal and the following parameter proposal are accepted. If we continue the argument we can do the same to\npropose updates where configurations are moved from one group to\nanother group, and in the red deer example we even include a\nlevel where updates of covariates are proposed. After making all\nproposals we evaluate the likelihoods for each possible state in\nparallel. The result is that we do need to evaluate too many\nlikelihoods, but if the number of CPUs that are available is larger\nthan or equal to the number of likelihoods we need to evaluate, a\ncomputational gain close to the number of levels is obtained. The\nupdating scheme is illustrated in Figure \\ref{fig:parallellScheme}.\n\\begin{figure*}[h]\\center\nFigure \\ref{fig:parallellScheme} approximately here.\n\\end{figure*}\n\n\n\n\\counterwithin{figure}{section}\n\\setcounter{section}{5}\n\\setcounter{figure}{0}\n\n\\begin{figure}\n  \\centering\n\\subfigure[][Trace plot for the number of groups.]{\\includegraphics[scale=0.65]{nrOfGroupsMarginal_gamma05}\\label{fig:g1}}\\\\\n\\subfigure[][Trace plot for $\\phi^0$(black), $\\phi^1$(dark grey) and\n$\\phi^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}$(light grey).]{\\includegraphics[scale=0.65]{tracePhi_gamma05}\\label{fig:g2}}\n\\caption{Independence model example: Trace plots for the first quarter\n  of the posterior simulation run. Solid curves are the result from a\n  simulation where the initial number of groups is 1, and dashed\n  curves are from a run with an initial value of 11 (maximal) number of groups.}\n\\label{fig:tracePlotsMarginal}\n\\end{figure} \n\n\\begin{figure}\n\\centering\n\\begin{equation*}\n  \\arraycolsep=4.5pt\\def0.6}\\begin{array}{cccccccccccc{0.6}\\begin{array}{cccccccccccc}\nc^0 &\\vv1.00&&&&&&&&&&\\\\\nc^1 &&\\vv1.00&&&&&&&&&\\\\\nc^{11}&&&\\vv1.00&\\vv 0.94&\\vv 0.88&\\vv 0.90&0.13&0.77&&&\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}} &&&\\vv 0.94&\\vv 1.00&\\vv 0.85&\\vv 0.93&0.12&0.75&&&\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}} &&&\\vv0.88&\\vv0.85&\\vv 1.00&\\vv0.85&0.19&0.83&&&\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}} &&&\\vv 0.9&\\vv 0.93&\\vv 0.85&\\vv 1.00&0.12&0.75&&&\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}}&&&0.13&0.12&0.19&0.12&\\vv 1.00&\\vv 0.28&\\vv0.78&\\vv0.78&0.74\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}}&&&0.77&0.75&0.83&0.75&\\vv 0.28&\\vv 1.00&\\vv\n0.12&\\vv0.12 &0.09\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}}&&&&&&&\\vv0.78&\\vv0.12&\\vv 1.00&\\vv 0.93&0.93\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}}&&&&&&&\\vv0.78&\\vv0.12&\\vv0.93 &\\vv 1.00&0.93\\\\\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}&&&&&&&0.74&0.09&0.93&0.93&\\vv 1.00\\\\\n&c^0 & c^1 & c^{11} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}} 1 \\\\ 1 \\end{array}} &\nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 &  \\\\  & 1 \\end{array}} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 &  \\end{array}} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 &  \\end{array}} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\  & 1 \\end{array}} & \nc^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & \\\\ 1 & 1 \\end{array}} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}}  & 1 \\\\ 1 & 1 \\end{array}} & c^{\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 1 \\end{array}}\n\\end{array}\n\\end{equation*}\n\\caption{\\label{fig:pairMatrixMarginal}Independence model example: Estimated posterior probabilities for two configuration\n    sets to be grouped together. The correct grouping is shown\n    in grey, and only probabilities larger than $5 \\%$ are given.}\n\\end{figure}\n\n\n\\begin{figure}\n  \\centering\n\\subfigure[][$\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} & \\\\\\framebox(5.0,5.0){} & \\end{array}}\\ (-1.08,-0.80)$.]{\\label{fig:b1}\\includegraphics[scale=0.5]{marginalBeta2}}\n\\subfigure[][$\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\\\ &\\end{array}}\\ (-0.05,0.22)$.]{\\label{fig:b2}\\includegraphics[scale=0.5]{marginalBeta3}}\\\\\n\\subfigure[][$\\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){}  & \\end{array}}\\ (-0.05,0.23)$.]{\\label{fig:b3}\\includegraphics[scale=0.5]{marginalBeta4}}\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}}\\framebox(5.0,5.0){} & \\\\ &\\framebox(5.0,5.0){}\\end{array}}\\ (-0.11,0.12)$.]{\\label{fig:b4}\\includegraphics[scale=0.5]{marginalBeta5}}\\\\\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}}\\ (-0.03,0.17)$.]{\\label{fig:b5}\\includegraphics[scale=0.5]{marginalBeta6}}\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\end{array}}\\ (-0.37,0.21)$.]{\\label{fig:b9}\\includegraphics[scale=0.5]{marginalBeta10}}\\\\\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ & \\framebox(5.0,5.0){} \\end{array}}\\ (-0.16,0.26)$.]{\\label{fig:b8}\\includegraphics[scale=0.5]{marginalBeta9}}\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){}& \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}\\ (-0.37,-0.05)$.]{\\label{fig:b7}\\includegraphics[scale=0.5]{marginalBeta8}}\\\\\n\\subfigure[][$ \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}\\ (-0.42,-0.06)$.]{\\label{fig:b6}\\includegraphics[scale=0.5]{marginalBeta7}}\n\\subfigure[][$  \\beta^{\\scriptsize\\arraycolsep=0.0pt\\def0.6}\\begin{array}{cccccccccccc{0.0}\\begin{array}{@{}l@{}l@{}} \\framebox(5.0,5.0){} &\\framebox(5.0,5.0){} \\\\ \\framebox(5.0,5.0){} & \\framebox(5.0,5.0){} \\end{array}}\\ (-0.05,0.69)$.]{\\label{fig:b10}\\includegraphics[scale=0.5]{marginalBeta11}}\n\\caption[Caption For LOF]%\n{\\label{fig:marginalInteractionParameters}Independence model example: Estimated marginal posterior distribution of the interaction parameters. True\n  values are shown with a black dot and estimated 95\\% credibility interval is given for\n  each parameter.}\n\\end{figure}\n\n\\begin{figure}\n  \\centering\n\\subfigure[][$g(x)=\\sum_ix_i$]{\\label{fig:i1}\\includegraphics[scale=0.5]{ind1}}\n\\subfigure[][$g(x)=\\sum_{i,j:\\text{vertical adjacent sites}}I(x_i=x_j)$]{\\label{fig:i2}\\includegraphics[scale=0.5]{ind2}}\\\\\n\\subfigure[][$g(x)=\\sum_{i,j:\\text{horizontal adjacent sites}}I(x_i=x_j)$]{\\label{fig:i3}\\includegraphics[scale=0.5]{ind3}}\n\\subfigure[][$g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left( x_{\\Lambda}=\n  {\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 0 \\\\ 0 & 0 \\end{array}\\right]}\\right)$]{\\label{fig:i4}\\includegraphics[scale=0.5]{ind4}}\\\\\n\\subfigure[][$g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left ( x_{\\Lambda}=\n  {\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 0 & 1 \\\\ 1 & 0 \\end{array}\\right]}\\right)$]{\\label{fig:i5}\\includegraphics[scale=0.5]{ind7}}\n\\subfigure[][$g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left ( x_{\\Lambda}=\n  {\\left[\\scriptsize \\def0.6}\\begin{array}{cccccccccccc{0.3}\\begin{array}{@{}c@{}c@{}} 1 & 1 \\\\ 1 & 0 \\end{array}\\right]}\\right)$]{\\label{fig:i6}\\includegraphics[scale=0.5]{ind15}}\n\\caption{\\label{fig:compareSimIndependence}Independence model example: Distribution of six statistics of realisations from our $2\\times 2$ model with\nposterior samples of $z$ (solid), the independence model with correct\nparameter value (dashed), and the independence model with posterior samples of the\nparameter value (dotted). The data evaluated with\neach statistic is shown with a black dot.}\n\\end{figure}\n\n\\counterwithin{figure}{section}\n\\setcounter{section}{6}\n\\setcounter{figure}{0}\n\n\\begin{figure}\n  \\centering\n\\subfigure[Three realisations from the likelihood for three random\nsamples of $z$ from the posterior distribution.]{\\label{fig:simulate_3t3}\\includegraphics[scale=0.5]{xSim_3t3}}\\\\\n\\subfigure[][Distribution of three functions of realisations from the\nlikelihood with $3\\times 3$ cliques (solid) and $2\\times 2$ cliques (dashed). The three functions are $g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left ( x_{\\Lambda}={\\left[\\scriptsize \\def\\arraystretch{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array}\\right ]}\\right )$ (left), $g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left ( x_{\\Lambda}=\n  {\\left[\\confBBBBBBBAA\\right ]}\\right )$ (middle), and $g(x)=\\sum_{\\Lambda\\in\\mathcal{L}_m}I\\left ( x_{\\Lambda}=\n  {\\left[\\scriptsize \\def\\arraystretch{0.2}\\begin{array}{@{}c@{}c@{}c@{}} 1 & 1 & 1 \\\\ 1 & 1 & 0 \\\\ 1 & 0 & 0 \\end{array}\\right ]}\\right )$ (right). ]{\\label{fig:gfunc_Deer_3t3}\\includegraphics[scale=0.5]{gfuncDeer}}\n\\caption{Red deer $3\\times 3$ example: Posterior results with $\\gamma=0.5$.}\n                \\label{fig:3t3}\n\\end{figure}\n\n\\counterwithin{figure}{section}\n\\setcounter{section}{7}\n\\setcounter{figure}{0}\n \\begin{figure}\n  \\centering\n\\centerline{\n\\xymatrix{\n  \\\\\n&&&&&&&*++[F]{z}\\ar[d]\\ar[dllll]\\\\\n&&&*++[F=]{z_1}\\ar[d]\\ar[dll]&&&&*++[F]{z}\\ar[d]\\ar[dll]\\\\\n&*++[F=]{z_{12}}\\ar[d]\\ar[dl]&&*++[F]{z_1}\\ar[d]\\ar[dl]&&*++[F=]{z_2}\\ar[d]\\ar[dl]&&*++[F]{z}\\ar[d]\\ar[dl]\\\\\n*++[F=]{z_{123}}&*++[F]{z_{12}}&*++[F=]{z_{13}}&*++[F]{z_1}&*++[F=]{z_{23}}&*++[F]{z_2}&*++[F=]{z_3}&*++[F]{z}\n  }}\n\\vspace{0.25cm}\n  \\caption{Proposal scheme for parallel likelihood evaluations. Starting in model $z$, proposals are made down the graph. Arrows pointing straight down represents rejection of proposal while arrow pointing down and left represent acceptance. Double squares are used to represents states where a new likelihood evaluation is needed.}\n  \\label{fig:parallellScheme}\n\\end{figure}\n\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe model $C^{*}$, introduced by Kennedy, Magidor and V\\\"{a}\\\"{a}n\\\"{a}nen\n in \\cite{IMEL}, is the model of sets constructible using the logic\n$\\LL(Q_{\\omega}^{\\mathrm{cf}})$ -- first order logic augmented with\nthe ``cofinality $\\omega$'' quantifier. As in the case of $L$\n-- the model of sets constructible using first order logic, this\nis a model of $\\zfc$, and one can phrase the formula ``$V=C^{*}$'',\ni.e. $\\forall x\\exists\\alpha(x\\in L'_{\\alpha})$ where $L'_{\\alpha}$\nis the $\\alpha$-th level in the construction of $C^{*}$. Unlike\n$L$, however, it is not always true that $C^{*}\\vDash V=C^{*}$,\nwhich is equivalent to the question whether $\\left(C^{*}\\right)^{C^{*}}=C^{*}$.\nThis is clearly the case if $V=L$, so the interesting question is\nwhether this can hold with $C^{*}\\ne L$. In section \\ref{sec:V=00003DC*}\nwe show that this is consistent relative to the consistency of $\\zfc$.\nNext we investigate the possibilities of $C^{*}\\nvDash V=C^{*}$.\nIn such a case, it makes sense to define recursively the iterated\n$C^{*}$s: \n\\begin{align*}\nC^{*0} & =V\\\\\nC^{*\\left(\\alpha+1\\right)} & =\\left(C^{*}\\right)^{C^{*\\alpha}}\\,\\text{ for any \\ensuremath{\\alpha}}\\\\\nC^{*\\alpha} & =\\bigcap_{\\beta<\\alpha}C^{*\\beta}\\,\\,\\text{ for limit \\ensuremath{\\alpha}}.\n\\end{align*}\n This type of construction was first investigated by McAloon \\cite{mcaloon2}\nregarding $\\mathrm{HOD}$, where he showed that it is equiconsistent\nwith $\\zfc$ that there is a strictly decreasing sequence of iterated\n$\\mathrm{HOD}$ of length $\\omega$, and the intersection of the sequence\ncan be either a model of $\\zfc$ or of $\\mathrm{ZF+\\neg AC}$. Harrington\nalso showed (in unpublished notes, cf. \\cite{ZADROZNY}) that the\nintersection might not even be a model of $\\mathrm{ZF}$. Jech \\cite{jech1975descending}\nshowed that it is possible to have a strictly decreasing sequence\nof iterated $\\mathrm{HOD}$ of any arbitrary ordinal length, and later\nZadro\\.{z}ny \\cite{zadrozny1981transfinite} improved this to get\nan $\\ord$ length sequence. In section \\ref{sec:Iterating C*} we\nshow that unlike the case of $\\mathrm{HOD}$, without large cardinals\nwe can only have finite decreasing sequences of iterated $C^{*}$,\nand that assuming the existence of a measurable cardinal is equivalent\nto the consistency of a strictly decreasing sequence of length $\\omega$.\n\n\\section{\\label{sec:V=00003DC*}Relative consistency of \\textquotedblleft$V=C^{*}\\protect\\ne L$\\textquotedblright}\n\nIn this section we follow the method of Zadro\\.{z}ny \\cite{ZADROZNY}\nto obtain the following result:\n\\begin{thm}\n\\label{thm:V=00003DC*}If $\\zfc$ is consistent then so is $\\zfc+V=C^{*}\\ne L\\,+$\n$2^{\\aleph_{0}}=\\aleph_{1}$.\n\\end{thm}\n\nThe idea\\footnote{I'd like to thank Kameryn Williams for his exposition of this and\nrelated results in his blog -- \\href{http:\/\/kamerynjw.net\/2019\/12\/04\/omegath-hod.html}{http:\/\/kamerynjw.net\/2019\/12\/04\/omegath-hod.html}} of Zadro\\.{z}ny's proofs, which are based on results of McAloon's\n\\cite{McAloon1,mcaloon2}, is to add a generic object (to make $V\\ne L$),\ncode it using some other generic object, then code the coding, and\nso on, iterating until we catch our tail. Our coding tool will be\nthe modified Namba forcing of \\cite[section 6]{IMEL}, which adds\na countable cofinal sequence to any element of some countable sequence\nof regular cardinals $>\\aleph_{1}$ (and only to them). Revised countable\nsupport iterations of this forcing preserves $\\omega_{1}$. In \\cite[theroem 6.7]{IMEL},\nthese tools are used to produce a model of $\\zfc+V=C^{*}\\ne L\\,+$\n$2^{\\aleph_{0}}=\\aleph_{2}$, but this requires an inaccessible cardinal\n(as proven there as well).\n\nThese two results covers all possibilities, since in \\cite[corollary to theroem 5.20]{IMEL},\nit is shown that the statement $V=C^{*}$ implies that $2^{\\aleph_{0}}\\in\\left\\{ \\aleph_{1},\\aleph_{2}\\right\\} $\nand for any $\\kappa>\\aleph_{0}$ $2^{\\kappa}=\\kappa^{+}$ .\n\nTo prove theorem \\ref{thm:V=00003DC*}, we begin with $V=L\\left[A_{0}\\right]$\nwhere $A_{0}$ is a countable set of ordinals such that $A_{0}$ does\nnot contain any of its limit points. Set $\\p_{0}=\\left\\{ 1\\right\\} $.\nInductively we assume that $\\p_{n}$ forces the existence of a countable\nset of ordinals $A_{n}$, $\\xi_{n}=\\sup A_{n}$ and we set $\\p_{n+1}=\\p_{n}*\\dot{\\q}_{n+1}$\nwhere $\\dot{\\q}_{n+1}$ is the modified Namba forcing to add a Namba\nsequence $E_{\\alpha}$ to each $\\aleph_{\\xi_{n}+\\alpha+2}^{L}$ such\nthat $\\alpha\\in A_{n}$. We can require that $E_{\\alpha}\\con\\left(\\aleph_{\\xi_{n}+\\alpha+1}^{L},\\aleph_{\\xi_{n}+\\alpha+2}^{L}\\right)$,\nso that $A_{n+1}:=\\bigcup\\left\\langle E_{\\alpha}\\mid\\alpha\\in A_{n}\\right\\rangle $\ndoes not contain any of its limit points. $\\p_{\\omega}$ is the full\nsupport (which is in our case also the revised countable support)\niteration. Let $G\\con\\p_{\\omega}$ generic, and denote $A=\\bigcup_{n<\\omega}A_{n}$.\nBy the properties of the modified Namba forcing, for any $\\gamma$,\n$V\\left[G\\right]\\vDash\\cof{\\aleph_{\\gamma+2}^{L}}=\\omega$ iff $\\gamma=\\xi_{n}+\\alpha$\nfor $\\alpha\\in A_{n}$. \n\\begin{rem}\n\\label{rem:codings}\n\\begin{enumerate}\n\\item \\label{rem:enu: suprema of codings}$\\xi_{n+1}=\\aleph_{\\xi_{n}\\cdot2}^{L}$\n, so inductively depends only $A_{0}$ and not on the generics. $\\xi_{\\omega}:=\\sup A$\nsatisfies $\\xi_{\\omega}=\\aleph_{\\xi_{\\omega}}^{L}$ and is of cofinality\n$\\omega$.\n\\item \\label{rem:enu:reconstruct An}From $\\otp(A_{0})$ and $A$ we can\ninductively reconstruct each $A_{n}$ -- $A_{0}$ is the first $\\otp(A_{0})$\nelements of $A$, and if we know $A_{n}$, then $A_{n+1}$ are the\nfirst $\\otp(A_{n})\\cdot\\omega$ elements of $A$ above $\\sup A_{n}$.\n\\item \\label{rem:enu:otp}Hence for each $n$, $\\otp(A_{n})=\\otp(A_{0})\\cdot\\omega^{n}$.\n\\item \\label{rem:enu:reconstruct E_alpha}If $\\alpha\\in A_{n}$, then $E_{\\alpha}=A\\cap\\left(\\aleph_{\\xi_{n}+\\alpha+1}^{L},\\aleph_{\\xi_{n}+\\alpha+2}^{L}\\right)$.\n\\end{enumerate}\n\\end{rem}\n\n\\begin{prop}\n$\\left(C^{*}\\right)^{V\\left[G\\right]}=L\\left[A\\right]=V\\left[G\\right]$\n\\begin{proof}\nBy the properties of the modified Namba forcing, at each stage of\nthe iteration the only cardinals of $L\\left[A_{0}\\right]$ receiving\ncofinality $\\omega$ are the ones in $A_{n}$. The whole iteration\nwill also add new $\\omega$ sequences to $\\sup A$, but this already\nhad cofinality $\\omega$ as we noted earlier. So $V[G]\\vDash\\cof{\\gamma}=\\omega$\niff either $V\\vDash\\cof{\\gamma}=\\omega$ or there is $n$ s.t. $V\\vDash\\cof{\\gamma}=\\aleph_{\\xi_{n}+\\alpha+2}^{L}$\nfor $\\alpha\\in A_{n}$. And on the other hand, if $\\alpha\\in A$ then\n$V\\left[G\\right]\\vDash\\cof{\\aleph_{\\xi_{n}+\\alpha+2}^{L}}=\\omega$\\,.\nSo \n\\[\nA=\\bigcup_{n<\\omega}\\left\\{ \\alpha\\in[\\xi_{n-1},\\xi_{n})\\mid V\\left[G\\right]\\vDash\\cof{\\aleph_{\\xi_{n}+\\alpha+2}^{L}}=\\omega\\right\\} \\in\\left(C^{*}\\right)^{V\\left[G\\right]}\n\\]\n(where $\\xi_{-1}=0$\\,) hence $L\\left[A\\right]\\con\\left(C^{*}\\right)^{V\\left[G\\right]}$.\n\nAs we noted, for every $\\alpha\\in A$, $E_{\\alpha}$ can be reconstructed\nfrom $A$ and $\\alpha$, so $\\left\\langle E_{\\alpha}\\mid\\alpha\\in A\\right\\rangle $\nis in $L\\left[A\\right]$. $G$ can be reconstructed from this sequence,\nhence $V\\left[G\\right]\\con L\\left[A\\right]\\con\\left(C^{*}\\right)^{V\\left[G\\right]}$,\nso the equality follows.\n\\end{proof}\n\\end{prop}\n\nThis finishes the proof of theorem \\ref{thm:V=00003DC*} since for\nany non-empty $A_{0}$ we'll get a model of ``$V=C^{*}\\ne L$'',\nand $2^{\\aleph_{0}}=\\aleph_{1}$ still holds.\n\nBefore moving to the next section we prove a useful lemma:\n\\begin{lem}\n\\label{lem:C*=00003DL=00005BE=00005D}Let $E=\\left\\{ \\alpha<\\omega_{2}^{V}\\mid\\cof{\\alpha}=\\omega\\right\\} $.\n\\begin{enumerate}\n\\item \\label{enu:no 0=000023}If $0^{\\sharp}$ does not exist, then $C^{*}=L\\left[E\\right]$.\n\\item \\label{enu:no L-mu}If there is no inner model with a measurable cardinal,\nthen $C^{*}=K\\left[E\\right]$ where $K$ is the Dodd-Jensen core model.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n1. Clearly $E\\in C^{*}$ so $L\\left[E\\right]\\con C^{*}$. Let $\\alpha\\in\\ord$.\nIf $\\cof{\\alpha}\\geq\\omega_{2}^{V}$ then also $L\\vDash\\cof{\\alpha}\\geq\\omega_{2}^{V}$\nso in particular $L\\vDash\\cof{\\alpha}>\\omega$. If $\\cof{\\alpha}<\\omega_{2}^{V}$,\nlet $A\\con\\alpha$ be cofinal, so $\\left|A\\right|\\leq\\aleph_{1}$.\nBy the covering theorem, there is $B\\in L$, $A\\con B\\con\\alpha$\ns.t. $\\left|B\\right|=\\aleph_{1}+\\left|A\\right|=\\aleph_{1}$. Let $\\bar{\\alpha}=\\otp(B)$,\nso $\\bar{\\alpha}<\\omega_{2}^{V}$ , and $\\cof{\\alpha}=\\cof{\\bar{\\alpha}}$,\nso $\\cof{\\alpha}=\\omega$ iff $\\bar{\\alpha}\\in E$. \n\nTo summarize, we get that for every $\\alpha$, in $L\\left[E\\right]$\nwe can determine whether $\\cof{\\alpha}=\\omega$ or not, so $C^{*}\\con L\\left[E\\right]$.\n\n2. The proof is exactly the same, noting that $K\\con C^{*}$ by \\cite[theroem 5.5]{IMEL},\nand that our assumption implies the covering theorem holds for $K$.\n\\end{proof}\n\n\n\\section{\\label{sec:Iterating C*}Iterating $C^{*}$ }\n\\begin{thm}\nIf $\\zfc$ is consistent then so is the existence of a model with\na decreasing $C^{*}$-sequence of any finite length.\n\\end{thm}\n\n\\begin{proof}\nGoing back to the proof of theorem \\ref{thm:V=00003DC*}, we note\nthat for any $n$, $\\left(C^{*}\\right)^{L\\left[\\bigcup_{k=0}^{n+1}A_{k}\\right]}=L\\left[\\bigcup_{k=0}^{n}A_{k}\\right]$:\n$A_{n}$ can be computed from $A_{n+1}$ using the cofinality-$\\omega$\nquantifier, which gives $\\supseteq$, and on the other hand, from\n$\\bigcup_{k=0}^{n}A_{k}$ we know exactly which ordinals will have\ncofinality $\\omega$ in $L\\left[\\bigcup_{k=0}^{n+1}A_{k}\\right]$,\nwhich gives $\\subseteq$. So by starting e.g. from $A_{0}=\\omega$,\n$L\\left[\\bigcup_{k=0}^{n}A_{k}\\right]$ has the decreasing $C^{*}$\nchain \n\\[\nL\\left[\\bigcup_{k=0}^{n}A_{k}\\right]=C^{*0}\\supsetneq C^{*1}\\supsetneq...\\supsetneq C^{*n}=L.\\qedhere\n\\]\n\\end{proof}\nWithout large cardinals this is best possible:\n\\begin{thm}\nIf there is no inner model with a measurable cardinal, then there\nis $k<\\omega$ such that $C^{*k}=C^{*(k+1)}$.\n\\end{thm}\n\n\\begin{proof}\nBy applying lemma \\ref{lem:C*=00003DL=00005BE=00005D}.\\ref{enu:no L-mu}\ninside each $C^{*n}$, for every $n$ we have $C^{*(n+1)}=K\\left[E_{n}\\right]$\nwhere \n\\[\nE_{n}=\\left\\{ \\alpha<\\omega_{2}^{C^{*n}}\\mid C^{*n}\\vDash\\cof{\\alpha}=\\omega\\right\\} .\n\\]\nThe sequence $\\left\\langle \\left(\\omega_{1}^{C^{*n}},\\omega_{2}^{C^{*n}}\\right)\\mid n<\\omega\\right\\rangle $\nis non-increasing in both coordinates, hence it stabilizes. Let $k$\nsuch that $(\\omega_{1}^{C^{*k}},\\omega_{2}^{C^{*k}})=(\\omega_{1}^{C^{*(k+1)}},\\omega_{2}^{C^{*(k+1)}})$,\nand we claim that $C^{*(k+1)}=C^{*(k+2)}$. To simplify notation we\nassume w.l.o.g $k=0$, i.e $\\omega_{i}^{C^{*}}=\\omega_{i}^{V}$ for\n$i=1,2$ (so we can omit the superscript) and we want to show that\n$\\left(C^{*}\\right)^{C*}=C^{*}$. We have:\n\\begin{align*}\nC^{*} & =K\\left[\\left\\{ \\alpha<\\omega_{2}\\mid V\\vDash\\cof{\\alpha}=\\omega\\right\\} \\right]=K\\left[E_{0}\\right]\\\\\n\\left(C^{*}\\right)^{C*} & =K\\left[\\left\\{ \\alpha<\\omega_{2}\\mid C^{*}\\vDash\\cof{\\alpha}=\\omega\\right\\} \\right]=K\\left[E_{1}\\right].\n\\end{align*}\n Clearly $E_{1}\\con E_{0}$. On the other hand, if $\\alpha\\in\\omega_{2}\\smin E_{1}$,\nthis means that $C^{*}\\vDash\\cof{\\alpha}=\\omega_{1}$, and since $\\omega_{1}^{C^{*}}=\\omega_{1}$,\nwe get that also $V\\vDash\\cof{\\alpha}=\\omega_{1}$, so $\\alpha\\in\\omega_{2}\\smin E_{0}$,\nthus $E_{1}=E_{0}$, and our claim is proved.\n\\end{proof}\nOur next goal is to show that this is precisely the consistency strength\nof a decreasing sequence:\n\\begin{thm}\n\\label{thm:decreasing}If there is an inner model with a measurable\ncardinal, then it is consistent that the sequence $\\left\\langle C^{*n}\\mid n<\\omega\\right\\rangle $\nis strictly decreasing. \n\\end{thm}\n\nWe work in $L^{\\mu}$ where $\\mu$ is a measure on $\\kappa$, and\ndenote by $M_{\\alpha}$ the $\\alpha$-th iterate of $L^{\\mu}$ by\n$\\mu$, $j_{\\alpha,\\beta}:M_{\\alpha}\\to M_{\\beta}$ the elementary\nembedding and $\\kappa_{\\alpha}=j_{0,\\alpha}(\\kappa)$.\n\nIn \\cite[theroem 5.16]{IMEL} the authors show that if $V=L^{\\mu}$\nthen $C^{*}=M_{\\omega^{2}}\\left[E\\right]$ where $E=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $.\nWe improve this by showing that $C^{*}$ is unchanged after adding\na Prikry sequence to $\\kappa$, and then investigate the $C^{*}$-chain\nof $L^{\\mu}$. First we prove two useful lemmas.\n\\begin{lem}\n\\label{lem:Prikry-generic}$E=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $\nis generic over $M_{\\omega^{2}}$ for the Prikry forcing on $\\kappa_{\\omega^{2}}$\ndefined from the ultrafilter $U^{(\\omega^{2})}\\in M_{\\omega^{2}}$.\n\\end{lem}\n\n\\begin{proof}\nWe use Mathias's characterization of Prikry forcing:\n\\begin{fact}[Mathias, cf. \\cite{mathias1973prikry}]\n Let $M$ be a transitive model of $\\zfc$, $U$ a normal ultrafilter\non $\\kappa$, then $S\\con\\kappa$ of order type $\\omega$ is generic\nover $M$ for the Prikry forcing defined from $U$ iff for any $X\\in U$,\n$S\\smin X$ is finite.\n\\end{fact}\n\nSo we need to show that for any $X\\in U^{(\\omega^{2})}$, $E\\smin X$\nis finite. The ultrafilter $U^{(\\omega^{2})}$ is defined by $X\\in U^{(\\omega^{2})}$\niff $\\exists\\alpha<\\kappa_{\\omega^{2}}$$\\left\\{ \\kappa_{\\beta}\\mid\\alpha\\leq\\beta<\\kappa_{\\omega^{2}}\\right\\} \\con X$.\nFor $X\\in U^{(\\omega^{2})}$, choose some $\\alpha<\\kappa_{\\omega^{2}}$\nsuch that $\\left\\{ \\kappa_{\\beta}\\mid\\alpha\\leq\\beta<\\kappa_{\\omega^{2}}\\right\\} \\con X$,\nthen $E\\smin\\alpha=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid\\omega\\cdot n<\\alpha\\right\\} $,\nand since $\\kappa_{\\omega^{2}}=\\sup\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $,\nthis set is finite. Hence $E=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $\nsatisfies the characterization.\n\\end{proof}\n\\begin{lem}\n\\label{lem:k-cof preserves}For any $\\beta<\\kappa$ and any $\\alpha$,\n$M_{\\beta}\\vDash\\cof{\\alpha}=\\kappa$ iff $V\\vDash\\cof{\\alpha}=\\kappa$.\n\\end{lem}\n\n\\begin{proof}\n $\\kappa$ is regular in $V$, thus it is regular in every $M_{\\beta}$\nwhich is an inner model of $V$. If $M_{\\beta}\\vDash\\cof{\\alpha}=\\kappa$,\nthen there is a cofinal $\\kappa$-sequence in $\\alpha$ (in both $M_{\\beta}$\nand $V$), and since $\\kappa$ is regular we get $V\\vDash\\cof{\\alpha}=\\kappa$.\nIf $V\\vDash\\cof{\\alpha}=\\kappa$, then the same argument rules out\n$M_{\\beta}\\vDash\\cof{\\alpha}<\\kappa$. So the only case left to rule\nout is $V\\vDash\\cof{\\alpha}=\\kappa$ $\\land$ $M_{\\beta}\\vDash\\cof{\\alpha}>\\kappa$.\nIf $\\beta=\\gamma+1$, then $M_{\\beta}$ is contained in $M_{\\gamma}$\nand closed under $\\kappa$-sequences in it, so they agree on cofinality\n$\\kappa$, and by induction we get that they agree with $V$ as well.\nSo assume $\\beta$ is limit and let $\\left\\langle \\alpha_{\\eta}\\mid\\eta<\\kappa\\right\\rangle $\nbe a cofinal sequence in $\\alpha$. by definition of the limit ultrapower,\neach $\\alpha_{\\eta}$ is of the form $j_{\\bar{\\beta},\\beta}(\\bar{\\alpha}_{\\eta})$\nfor some $\\bar{\\beta}<\\beta$. We can also assume that each such $\\bar{\\beta}$\nis large enough so that $\\alpha\\in\\rng(j_{\\bar{\\beta},\\beta})$. Since\n$\\beta<\\kappa$, there is some $\\bar{\\beta}$ fitting $\\kappa$ many\n$\\alpha_{\\eta}$s, so w.l.o.g we can assume $\\bar{\\beta}$ fits all\nof them. We can assume $\\bar{\\beta}>0$ so $\\kappa$ is a fixed point\nof $j_{\\bar{\\beta},\\beta}$. If $\\bar{\\alpha}=\\sup\\left\\{ \\bar{\\alpha}_{\\eta}\\mid\\eta<\\kappa\\right\\} $\\footnote{Note that we can't assume this sequence is in $M_{\\bar{\\beta}}$}\nthen, since $\\alpha=\\sup\\left\\{ j_{\\bar{\\beta},\\beta}(\\bar{\\alpha}_{\\eta})\\mid\\eta<\\kappa\\right\\} $\nand $\\alpha\\in\\rng(j_{\\bar{\\beta},\\beta})$, we must have that $\\alpha=j_{\\bar{\\beta},\\beta}(\\bar{\\alpha})$.\n$\\bar{\\alpha}$ is of cofinality $\\kappa$ in $V$, so by induction\nalso in $M_{\\bar{\\beta}}$, hence by elementarity $M_{\\beta}\\vDash\\cof{\\alpha}=\\kappa$.\n\\end{proof}\n\\begin{prop}\n\\label{prop:C* in Prikry ext}If $V=L^{\\mu}$ where $\\mu$ is a measure\non $\\kappa$, $G$ is generic for Prikry forcing on $\\kappa$, then\n$\\left(C^{*}\\right)^{V\\left[G\\right]}=\\left(C^{*}\\right)^{V}$.\n\\end{prop}\n\n\\begin{proof}\nAfter forcing with Prikry forcing, the only change of cofinalities\nis that $\\kappa$ becomes of cofinality $\\omega$. So $V\\left[G\\right]\\vDash\\cof{\\alpha}=\\omega$\niff $V\\vDash\\cof{\\alpha}\\in\\left\\{ \\omega,\\kappa\\right\\} $. We now\nfollow the proof of \\cite[theroem 5.16]{IMEL}. \n\nConsider $M_{\\omega^{2}}$, the $\\omega^{2}$ iterate of $V$, and\nlet $E=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $ and\nfix an ordinal $\\alpha$. As in the proof of \\cite[theroem 5.16]{IMEL},\n$V\\vDash\\cof{\\alpha}=\\omega$ iff ($M_{\\omega^{2}}\\left[E\\right]\\vDash\\cof{\\alpha}\\in\\left\\{ \\omega\\right\\} \\cup E\\cup\\left\\{ \\sup E\\right\\} $).\nRegarding cofinality $\\kappa$ -- by lemma \\ref{lem:k-cof preserves},\n$V\\vDash\\cof{\\alpha}=\\kappa$ iff $M_{\\omega^{2}}\\vDash\\cof{\\alpha}=\\kappa$.\nAs we noted, $M_{\\omega^{2}}=L^{\\nu}$ where $\\nu$ is a measure on\n$\\kappa_{\\omega^{2}}$ and by lemma \\ref{lem:Prikry-generic}, $E$\nis Prikry generic over it, hence, since cofinality $\\kappa$ is unaffected\nby Prikry forcing on $\\kappa_{\\omega^{2}}$, we get $V\\vDash\\cof{\\alpha}=\\kappa$\niff $M_{\\omega^{2}}\\left[E\\right]\\vDash\\cof{\\alpha}=\\kappa$. Putting\nthese facts together, in $M_{\\omega^{2}}\\left[E\\right]$ we can detect\nwhether $V\\vDash\\cof{\\alpha}\\in\\left\\{ \\omega,\\kappa\\right\\} $, so\nwe know whether $V\\left[G\\right]\\vDash\\cof{\\alpha}=\\omega$, hence\nwe can construct $\\left(C^{*}\\right)^{V\\left[G\\right]}$ inside $M_{\\omega^{2}}\\left[E\\right]$. \n\nThe other direction of the proof is almost the same as in \\cite[theroem 5.16]{IMEL}:\n$E$ is the set of ordinals in the interval $(\\kappa,\\kappa_{\\omega^{2}})$\nwhich have cofinality $\\omega$ in $V\\left[G\\right]$ and are regular\nin the core model\\footnote{Note that here we had to avoid $\\kappa$ which also satisfies this.},\nwhich is contained in any $C^{*}$, so $E\\in\\left(C^{*}\\right)^{V\\left[G\\right]}$,\nand from $E$ one can define $M_{\\omega^{2}}$, so $M_{\\omega^{2}}[E]\\con\\left(C^{*}\\right)^{V\\left[G\\right]}$.\n\\end{proof}\nNow we can analyze the $C^{*}$-chain of $V=L^{\\mu}$. To avoid confusion\nwe stick to the notation $C^{*\\alpha}$, starting from $C^{*0}=L^{\\mu}=M_{0}$,\nwith $M_{\\alpha}$ being the $\\alpha$th iterate of $L^{\\mu}$ and\n$\\kappa_{\\alpha}$ the $\\alpha$th image of the measurable cardinal.\nSo by \\cite[theroem 5.16]{IMEL} we have $C^{*1}=M_{\\omega^{2}}\\left[E_{1}\\right]$\nfor $E_{1}=\\left\\{ \\kappa_{\\omega\\cdot n}\\mid n<\\omega\\right\\} $.\nAs we noted earlier $M_{\\omega^{2}}$ is also of the form $L^{\\mu'}$\nfor the measurable $\\kappa_{\\omega^{2}}$, and by lemma \\ref{lem:Prikry-generic},\n$E_{1}$ is Prikry generic over it, so by proposition \\ref{prop:C* in Prikry ext}\n\\[\nC^{*2}=\\left(C^{*}\\right)^{M_{\\omega^{2}}[E]}=\\left(C^{*}\\right)^{M_{\\omega^{2}}}\n\\]\nwhich is again the $\\omega^{2}$-th iterate of $M_{\\omega^{2}}$,\ni.e. $M_{\\omega^{2}+\\omega^{2}}$, plus the corresponding sequence\n-- $E_{2}=\\left\\{ \\kappa_{\\omega^{2}+\\omega\\cdot n}\\mid n<\\omega\\right\\} $.\nSo $C^{*2}=M_{\\omega^{2}\\cdot2}\\left[E_{2}\\right]$. We can continue\ninductively, and get the following:\n\\begin{thm}\nIf $V=L^{\\mu}$ then for every $m\\geq1$ $C^{*m}=M_{\\omega^{2}\\cdot m}\\left[E_{m}\\right]$\nwhere $M_{\\alpha}$ is the $\\alpha$th iterate of $V$, $\\kappa_{\\alpha}$\nthe $\\alpha$th image of the measurable cardinal and $E_{m}=\\left\\{ \\kappa_{\\omega^{2}\\cdot(m-1)+\\omega\\cdot n}\\mid n<\\omega\\right\\} $. \n\\end{thm}\n\nThis concludes the proof of theorem \\ref{thm:decreasing}. To analyze\n$\\left(C^{*\\omega}\\right)^{L^{\\mu}}$, we will use the following theorem,\ndue to Bukovsk\\'{y} \\cite{bukovsky1973changing,bukovsky1977iterated}\nand Dehornoy \\cite{dehornoy1978iterated}:\n\\begin{fact}\n\\label{thm:Buk-Deho}If $\\kappa$ is measurable, $M_{\\alpha}$ is\nthe $\\alpha$-th iterate of $V$as by a normal ultrafilter on $\\kappa$\nand $\\kappa_{\\alpha}$ the $\\alpha$-th image of $\\kappa$, then for\nany limit ordinal $\\lambda$ exactly one of the following holds:\n\\begin{enumerate}\n\\item If $\\exists\\alpha<\\lambda$ s.t. $M_{\\alpha}\\vDash\\cof{\\lambda}>\\omega$\nthen $\\bigcap_{\\alpha<\\lambda}M_{\\alpha}=M_{\\lambda}$\n\\item If $\\lambda=\\alpha+\\omega$ for some $\\alpha$, then $\\left\\langle \\kappa_{\\alpha+n}\\mid n<\\omega\\right\\rangle $\nis Prikry generic over $M_{\\lambda}$ and $\\bigcap_{\\alpha<\\lambda}M_{\\alpha}=M_{\\lambda}\\left[\\left\\langle \\kappa_{\\alpha+n}\\mid n<\\omega\\right\\rangle \\right]$\n\\item Otherwise, $\\bigcap_{\\alpha<\\lambda}M_{\\alpha}$ is a quasi-generic\nextension of $M_{\\lambda}$, hence satisfies $\\mathrm{ZF}$, but doesn't\nsatisfy $\\mathrm{AC}$.\n\\end{enumerate}\n\\end{fact}\n\n\\begin{cor}\n$C^{*\\omega}=\\bigcap_{\\alpha<\\omega^{3}}M_{\\alpha}$ and it satisfies\n$\\mathrm{ZF}$ but not $\\mathrm{AC}$.\n\\end{cor}\n\n\\begin{proof}\nBy definition and our previous calculation, \n\\[\nC^{*\\omega}=\\bigcap_{m<\\omega}C^{*m}=\\bigcap_{1\\leq m<\\omega}M_{\\omega^{2}\\cdot m}\\left[E_{m}\\right]\n\\]\nand for each $m\\geq1$, $E_{m}\\in M_{\\omega^{2}\\cdot(m-1)}$ so \n\\[\n\\bigcap_{1\\leq m<\\omega}M_{\\omega^{2}\\cdot m}\\left[E_{m}\\right]=\\bigcap_{m<\\omega}M_{\\omega^{2}\\cdot m}=\\bigcap_{\\alpha<\\omega^{3}}M_{\\alpha}.\n\\]\n Since $\\omega^{3}$ is of cofinality $\\omega$ but not of the form\n$\\alpha+\\omega$, the conclusion follows from (3) of fact \\ref{thm:Buk-Deho}.\n\\end{proof}\n\n\\section{Conclusion and open questions}\n\nWe summarize what is now known in terms of equiconsistency:\n\\begin{enumerate}\n\\item $\\zfc$ is equiconsistent with $V=C^{*}\\ne L\\,+$ $2^{\\aleph_{0}}=\\aleph_{1}$.\n\\item Existence of an inaccessible cardinal is equiconsistent with $V=C^{*}+$\n$2^{\\aleph_{0}}=\\aleph_{2}$.\n\\item Existence of a measurable cardinal is equiconsistent with $\\forall n<\\omega(C^{*n}\\supsetneq C^{*(n+1)})$\nand $C^{*\\omega}\\vDash\\mathrm{ZF+\\neg AC}$.\n\\end{enumerate}\nCompared to the results regarding $\\mathrm{HOD}$, the following questions\nremain open:\n\\begin{question}\n\\begin{enumerate}\n\\item Is it possible, under any large cardinal hypothesis, that $\\forall n<\\omega$\n$C^{*n}\\supsetneq C^{*(n+1)}$ and $C^{*\\omega}\\vDash\\zfc$? More\ngenerally, for which ordinals $\\alpha$ can we get a decreasing $C^{*}$\nsequence of length $\\alpha$?\n\\item Is it possible, under any large cardinal hypothesis, that $\\forall n<\\omega$\n$C^{*n}\\supsetneq C^{*(n+1)}$ and $C^{*\\omega}\\nvDash\\mathrm{ZF}$?\n\\end{enumerate}\n\\end{question}\n\nA natural first attempt towards answering the first question would\nbe to try and work in a model with more measurable cardinals. However,\nit seems that it would require at least \\emph{measurably many} measurables:\nin a forthcoming paper \\cite{yaar2021short-seq}, we generalize \\cite[theroem 5.16]{IMEL}\nand our proposition \\ref{prop:C* in Prikry ext} and show the following:\n\\begin{thm}\nAssume $V=L\\left[\\mathcal{U}\\right]$ where $\\mathcal{U}=\\left\\langle U_{\\gamma}\\mid\\gamma<\\chi\\right\\rangle $\nis a sequence of measures on the increasing measurables $\\left\\langle \\kappa^{\\gamma}\\mid\\gamma<\\chi\\right\\rangle $\nwhere $\\chi<\\kappa^{0}$. Iterate $V$ according to $\\mathcal{U}$\nwhere each measurable is iterated $\\omega^{2}$ many times, to obtain\n$\\left\\langle M_{\\alpha}^{\\gamma}\\mid\\gamma<\\chi,\\alpha\\leq\\omega^{2}\\right\\rangle $,\nwith iteration points $\\left\\langle \\kappa_{\\alpha}^{\\gamma}\\mid\\gamma<\\chi,\\alpha\\leq\\omega^{2}\\right\\rangle $,\nand set $M^{\\chi}$ as the directed limit of this iteration. Let $G$\nbe generic over $V$ for the forcing adding a Prikry sequence to every\n$\\kappa^{\\gamma}$. Set for every $\\gamma<\\chi$ $E^{\\gamma}=\\left\\langle \\kappa_{\\omega\\cdot n}^{\\gamma}\\mid1\\leq n<\\omega\\right\\rangle $\nand $\\hat{E}^{\\gamma}=\\left\\langle \\kappa_{\\omega\\cdot n}^{\\gamma}\\mid0\\leq n<\\omega\\right\\rangle $\nthen \n\\begin{align*}\n\\left(C^{*}\\right)^{V} & =M^{\\chi}\\left[\\left\\langle E^{\\gamma}\\mid\\gamma<\\chi\\right\\rangle \\right]\\\\\n\\left(C^{*}\\right)^{V\\left[G\\right]} & =M^{\\chi}\\left[\\left\\langle \\hat{E}^{\\gamma}\\mid\\gamma<\\chi\\right\\rangle \\right].\n\\end{align*}\n\\end{thm}\n\nSo, if $V=L\\left[\\mathcal{U}\\right]$ as above, $C^{*}$ is of the\nform $L\\left[\\mathcal{U}_{1}\\right]\\left[G_{1}\\right]$ for some sequence\nof measures and a sequence of Prikry sequences on it's measures, and\nso $C^{*2}$ is again of that form, where we iterated the measures\nin $\\mathcal{U}$ $\\omega^{2}\\cdot2$ many times and add Prikry sequences.\nSo again, as we've done here, we'll get that $C^{*\\omega}$ is the\nintersection of the models $M_{\\omega^{2}\\cdot n}^{\\chi}$ where we\niterated each measure $\\omega^{2}\\cdot n$ times . This is due to\nthe facts that changing the order of iteration between the measures\ndoesn't change the final result, and that the Prikry sequences ``fall\nout'' during the intersection. Now, we don't have a complete analysis\nof intersections of iterations by more than one measure, but Dehornoy\nproves the following more general fact:\n\\begin{fact}[{\\cite[section 5.3 proposition 3]{dehornoy1978iterated}}]\n For every $\\alpha$ let $N_{\\alpha}$ be the $\\alpha$th iteration\nof $V$ by some measure. Assume $\\lambda$ is such that for every\n$\\alpha<\\lambda$, $N_{\\alpha}\\vDash\\mathrm{cf}(\\lambda)=\\omega$\nbut there is no $\\rho$ such that $\\lambda=\\rho+\\omega$. Then if\n$M$ is a transitive inner model of $\\zfc$ containing $\\bigcap_{\\alpha<\\lambda}N_{\\alpha}$,\nthen there is some $\\alpha<\\lambda$ such that $N_{\\alpha}\\con M$.\n\\end{fact}\n\nSo, if we take $N_{\\alpha}$ to be the iteration of $V$ by the first\nmeasure in $\\mathcal{U}$, we get that $C^{*\\omega}$ contains $\\bigcap_{\\alpha<\\omega^{3}}N_{\\alpha}$,\nbut doesn't contain any $N_{\\alpha}$ for $\\alpha<\\omega^{3}$, so\n$C^{*\\omega}$ cannot satisfy $\\mathrm{AC}$. Hence a different approach,\nor larger cardinals, would be required to answer this question. \n\nA different line of inquiry stems from the following fact:\n\\begin{fact}[\\cite{IMEL}]\n If there is a proper class of Woodin cardinals then the theory of\n$C^{*}$ is unchanged by forcing.\n\\end{fact}\n\nSo the question whether $C^{*}\\vDash V=C^{*}$ cannot be changed under\nforcing in the presence of class many Woodin cardinals. If the sequence\nof $C^{*\\alpha}$ is definable (perhaps up to some ordinal) then this\nwill also be in the theory of $C^{*}$ (note that on the face of it\neven the sequence up to $\\omega$ may not be definable). \n\\begin{question}\nWhat can be deduced on the sequence of iterated $C^{*}$ from a proper\nclass of Woodin cardinals?\n\\end{question}\n\n\\bibliographystyle{amsplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Introduction}\n\n\n\nThe nature and origin of cosmic dark matter is still unknown.  The standard Cold Dark Matter (``CDM'') model is actually a place-holder with certain assumed properties -- e.g. it behaves\nduring the structure formation era as a\nnonrelativistic, collisionless gas of particles that couple only weakly to each other and to other\ncomponents of the Universe, via gravity alone,\ncontributing to the expansion rate of the Universe as a pressure-free component whose mass density, conserved over time, is measured astronomically.  A further assumption is made \nthat the CDM particles are born so ``cold'' -- i.e. with such small peculiar motion -- that their free-streaming could only have washed out primordial density fluctuations on the tiniest\nof scales, many orders of magnitude below the scale that affects the formation of galaxies or even their internal structure. With these assumptions, it is possible to work out, within the\ncontext of the standard big bang model and inflationary paradigm, how structure formation proceeds from linear-perturbation initial conditions through nonlinear formation of galaxies, clusters and the cosmic web.  This model has successfully accounted for many observational properties of the Universe on large scales, from galaxy formation scales on up.\nIt has, however, been challenged by over-predicting\nthe level of small-scale structure inside galaxies\ncompared to that observed in a variety of systems (see, for instance, \\citealp{BBK17} and references, therein).  This may reflect\nour incomplete knowledge of the impact on galaxy\nformation of the baryonic component, coupled via\ngravity to CDM.  However, it has also led\nto a consideration of microscopic properties\nof CDM that go beyond the standard assumptions\ndescribed above, in ways that can affect\nsmall-scale structure without spoiling its success on larger scales.  \n\nThe best-studied origin story\nof CDM involves \nWeakly Interacting Massive Particles (``WIMPs'') with masses in the range $\\sim$ GeV - TeV, predicted\nby theories of supersymmetry, which form as big bang thermal relics whose abundance is set\nwhen their annihilation at a weak-interaction-like rate becomes slower than the Hubble expansion rate.\nThis so-called ``standard CDM'' -- more\nproperly referred to as ``WIMP-CDM'' -- \nis more specific than the CDM whose\ngeneric properties we identified above, with additional observable consequences, and, as such,\nis more tightly constrained.  \nIn particular, it predicts that the same annihilations might lead to their indirect detection from astronomical sources \n(e.g. by gamma-rays), while their \nscattering interactions with nucleons \nmight permit their direct detection.  So far,\nhowever, despite considerable effort,  \nthere are no confirmed detections, and the allowed\nrange of particle models and parameters that remain\nviable, consistent with that nondetection, has\nshrunk \\citep[see e.g.][]{Schumann19}.  This has added some urgency to the consideration of alternative models for\nthe origin and microscopic nature of CDM, even\napart from the possible conflicts it has of\nover-predicting small-scale structure on\ngalactic scales.  We emphasize this last point\nat the outset, because it is often missed when\nevaluating alternatives to WIMP-CDM, by\ndismissing the alternatives as soon as it\nappears they are merely ``CDM-like'' at\nall scales of interest, i.e. failing to\nprovide the ``Deus ex machina'' of\nreducing the apparent over-prediction of small-scale structure by CDM, while retaining\nall of its successes on larger scales. \nThat is, as long as the origin of CDM is\nunknown, any microscopic model\nthat is observationally indistinguishable\nfrom it is just as successful as any other, until it is disfavored\neither by further theoretical considerations, or by predictions of additional observables\nthat can distinguish it from the standard model.  \n\nOne such alternative model of great interest is Scalar Field Dark Matter (``SFDM'') comprised of ultralight ($m \\gtrsim 10^{-22}$ eV$\/c^2$) \nbosons, originally suggested because of its \nnovel structure-formation dynamics as a Bose-Einstein condensate and quantum superfluid, described by coupled Schr\\\"odinger-Poisson equations.\\footnote{The field of SFDM study has an extensive literature, well-beyond our ability to cite it here.  A few early references not otherwise cited in this paper include \\citet*{Sin94,LK96,MGU00,GU04,MVM09,SC17,DKR18}. Some reviews with additional references therein include \\citet*{SRM14,RDS14,Marsh16}.} \nIn the free-field (``fuzzy'') limit of SFDM, or fuzzy dark matter (``FDM''), the small-scale \nstructure which we described above for \n``generic'' CDM is suppressed below the \nde~Broglie wavelength, $\\lambda_\\text{deB}$, but resembles CDM on larger scales.  \nVirialized FDM haloes have been shown to have\nsolitonic cores of radius $\\sim\\lambda_\\text{deB}$,\nsurrounded by CDM-like envelopes (\\citealp*{SCB14, SNE16}; \\citealp{Mocz17}). When a strong enough repulsive self-interaction (SI)\nis also present, \nstructure can be inhibited below a second length scale, $\\lambda_\\text{SI}$, \nwith $\\lambda_\\text{SI}>\\lambda_\\text{deB}$ -- referred to as the Thomas-Fermi (``TF'') regime.\nThis TF regime of SFDM (henceforth, ``SFDM-TF''),\nassuming a quartic potential for the repulsive SI, has been referred to variously as ``BEC-CDM'' \\citep{RDS12,RDS14},\nSFDM \\citep*{Li14,Li17}, ``Repulsive Dark Matter'' \\cite{Goodman2000}, or more generally ``Fluid Dark Matter'' in \\cite{Peebles2000}.\\footnote{\nNote that SFDM-TF should not be confused with another CDM alternative known as self-interacting dark matter (SIDM). \nThe latter behaves as a collection of point-particles undergoing elastic scatterings, which\namounts to adding 2-body elastic collisions to the standard, collisionless gravitational dynamics of CDM, while the former is a BEC, best thought of as a classical field evolving under a continuous self-interaction potential (along with the gravitational potential).\nThe two models are phenomenologically distinct, as is evident by the r\\^ole SI plays in their respective non-relativistic fluid equations. In the SFDM-TF model (as discussed in \\S\\ref{subsection:fluidapprox}), SI enters as a potential or pressure term in the momentum equation (equation~\\ref{eq:1momLagrangian}), whereas in the SIDM model, SI enters as a heat conduction term in the energy equation \\citep*{BSI02,AS05}.}\n\nStructure formation in the TF regime differs significantly from FDM.  \nIn \\citet*{paper1} \\citepalias[henceforth][]{paper1}, we discussed the nonlinear dynamics and internal structure \nof haloes that form from gravitational instability in the TF regime.  Here we revisit this in the \ncontext of halo and large-scale structure formation from cosmological perturbations, \nincluding observational constraints. As this is a companion paper (``Paper II''), we \nwill sometimes refer the reader to \\citetalias{paper1} for details, \nfor the sake of brevity and to avoid\nredundancy.  \n\nIn \\citetalias{paper1}, we developed equations\nand tools to calculate the dynamical evolution of inhomogeneity\nin response to gravitational instability in this model, \nstarting from the nonlinear Schr\\\"{o}dinger equation (``NLSE'').\nA key development was to take\nfull account of both quantum pressure (associated with inhomogeneity on the tiny\nscale of the de~Broglie wavelength) and SI, based upon a\nreformulation of the NLSE, smoothed on scales larger than \nthe de~Broglie scale. In that case,\nthe NLSE reduced to the collisionless Boltzmann equation\nthat describes CDM, except with\nthe SI potential added to\nthe usual gravitational potential. \nFrom this, a fluid approximation \nfor SFDM-TF was derived, which\nadded a new pressure force for\nSI to equations we \npreviously derived for CDM.\nThis enabled us to describe the scalar field by the familiar \nhydrodynamical conservation\nequations for a $\\gamma=5\/3$ ideal gas, with an ideal gas pressure,\n$P_{\\!\\sigma}$, \nassociated with particle velocity dispersion (which accounts\nfor the large-scale effects of\nquantum pressure), and with a\nsecond pressure, $P_\\text{SI}$, which accounts for the\nrepulsive SI force. This\nfluid approximation allowed us to\nevolve SFDM-TF computationally\nin spherical-1D with the very \nhigh resolution and dynamic\nrange which are required\n(but not currently possible)\nfor full 3D simulations of SFDM-TF.\nWith this tool, we solved the Newtonian problem of virialized halo formation by the nonlinear collapse of a Jeans-unstable perturbation,\nfrom non-cosmological (i.e. static)\ninitial conditions, for the\nfirst time. These simulations in \n\\citetalias{paper1} showed that\nhaloes formed in this way have \ncore-envelope density profiles, with\nflattened cores of size $R_\\text{TF}$ -- \nthe radius of the ($n=1$)-polytrope that forms for a self-gravitating fluid in hydrostatic equilibrium when supported by SI pressure alone, referred to as the Thomas-Fermi radius --\nsurrounded by an envelope just like the one that\nformed when CDM dynamics \nwas simulated, instead, \nfrom the same initial\nconditions. This explained why, in the case of FDM, when haloes form in 3D simulations (which are possible\nfor FDM, where the de~Broglie wavelength is large enough to be \nfully resolvable, unlike SFDM-TF,\nfor which de~Broglie wavelengths are unresolvably small),\ndensity profiles are also CDM-like\noutside their solitonic cores (of size close to the de~Broglie wavelength).\n\nIn \\citetalias{paper1}, we showed\nhow SFDM-TF haloes with the right value of $R_\\text{TF}$ could resolve\nthe cusp-core and ``too-big-to-fail'' (TBTF) problems\nof CDM, without suffering\nfrom the problem identified for FDM\nby \\cite*{VBB19} in which the\nsolitonic cores of larger-mass FDM haloes were too dense to be consistent with the rotation curves\nof larger-mass galaxies, if the FDM\nparticle mass were tuned to \n$m\\simeq10^{-22}$~eV$\/c^2$ to match observations of smaller-mass galaxies.  We found\nthat $R_\\text{TF}\\gtrsim1$ kpc is\nrequired.  It was left to Paper II,\nhere, to determine if such a value\nof $R_\\text{TF}$ was consistent\nwith the cosmological formation of\nthose haloes.\n\nIn what follows, we study the cosmological context for structure formation in the SFDM-TF model, \nboth by simulating individual halo formation cosmologically,\nand by deriving the initial conditions for Gaussian-random\nprimordial fluctuations from\nlinear perturbation theory. \nIn \\S\\ref{sec:basicequations}, we\npresent the model and basic equations we use to study SFDM-TF\ncosmologically.  In \\S\\ref{sec:infall}, we\napply these equations to simulate\nindividual halo formation by spherical infall and collapse in a cosmologically expanding SFDM-TF universe.  The overall scenario for cosmic structure in the SFDM-TF model is developed in \\S\\ref{sec:LinPertsTransferFn}, including\nlinear perturbation theory to\nderive the transfer function and\npower spectrum, as well as their\napplication to statistical\nmeasures of structure formation like\nthe halo mass function.  We apply\nthe same approach to the FDM\nmodel, as well, for comparison.\nIn \\S\\ref{sec:results}, we present\nthe results of this linear perturbation theory and use \nobservational constraints on the\nFDM model, fully parameterized by\nits particle mass $m$,\nas a proxy to derive\nconstraints on the SFDM-TF model,\nparameterized by $R_\\text{TF}$.\nOur summary and conclusions are presented in \\S\\ref{sec:Conclusion}. \n\n\n\\section{The model and basic equations}\n\\label{sec:basicequations}\n\n\\subsection{The fully relativistic description}\n\\label{sec:fullyrelativistic}\n\nWe model SFDM as a classical, massive, complex\nscalar field $\\phi$ with a quartic, repulsive self-interaction (SI), which obeys the Klein-Gordon equation\n\\begin{equation}\n    \\Box \\phi + \\frac{m^2 c^2}{\\hbar^2}\\phi + \\frac{2 g}{\\hbar^2} |\\phi|^2 \\phi = 0\n    \\label{eq:KGE}\n\\end{equation}\nwhere $\\Box \\equiv \\nabla_\\mu \\nabla^\\mu$ is the d'Alembertian for a general 4D spacetime metric, $m$ is the boson mass, \nand $g$ is the SI coupling strength. This equation is coupled to the Einstein equations, with a metric tensor given \nby the Friedmann-Lemaitre-Robertson-Walker metric for a homogeneous and isotropic background universe, \nperturbed by primordial fluctuations.  The scalar field contributes terms to the stress-energy tensor in the Einstein equations, which contain the field and its derivatives.  \n\nAs we described in \n\\citet{Li14,Li17}, the evolution of the homogeneous background scalar field corresponds to that of a perfect fluid. However, the scalar field oscillates \nwith an angular frequency which eventually\nexceeds the Hubble expansion rate, when \n\\begin{equation}\n    \\omega \\simeq \\frac{mc^2}{\\hbar} \\sqrt{1 + \\frac{2g}{m^2 c^2}|\\phi|^2 } \\gg H\n\\end{equation}\n\\citep[see equation 21 in][]{Li14},\nwhere the square of the modulus of the field, $|\\phi|^2$, is, at late times, related to the SFDM mass density by  $\\rho_b=\\langle|\\phi|^2\\rangle=\\rho_b(a=0)a^{-3}$ (the subscript ``$b$'' refers to background), and the Hubble parameter $H(t) = (da\/dt)\/a$ with scale factor $a=a(t)$ depending on cosmic time $t$.  \nPrior to this rapid oscillation epoch, the \nfield evolution was shown to start from a kinetic-energy-dominated, stiff, \nrelativistic phase during the slow-oscillation period when $\\omega \\ll H$, but eventually to transition \nover time to a radiation-like, still relativistic phase during the rapid oscillation period, \nwhen the SI term in equation~(\\ref{eq:KGE}) dominates the quadratic mass term.\nFinally, at later times, when the mass term dominates, there is a transition to a nonrelativistic, matter-like phase.  More precisely, as a perfect fluid, the field can be characterized by an equation of state parameter $w$ defined as the ratio of pressure to energy density, which varies over time to reflect these phases of evolution.  Initially, $w=1$ (stiff phase), which must end prior to the epoch of big bang nucleosynthesis (BBN), in order not to spoil the agreement between predictions of the standard model involving primordial nucleosynthesis in a radiation-dominated (RD) universe with \nthe observed relic abundances of the light elements. Eventually, when the field oscillates much more rapidly than the Hubble expansion rate,\nwe can average the pressure $P$ and energy-density $\\varepsilon$ of the field over the oscillation period and take the ratio of these average values to give the average value of the equation\nof state parameter $w$, according to the following \\citep[which is equation 26 in][]{Li14}:\n\\begin{equation}\n    \\langle w \\rangle \\equiv \\frac{\\langle P \\rangle}{\\langle \\varepsilon \\rangle} = \\frac{1}{3} \\left( \\frac{1}{1 + \\frac{2m^2c^2}{3g \\langle |\\phi|^2 \\rangle}} \\right)\n    \\label{eq:averagew}\n\\end{equation}\nWhen the SI term dominates during this phase,\n$\\langle w \\rangle \\simeq 1\/3$ (radiation-like), but at late times, before the\nuniverse made its transition from radiation-\nto matter-dominated (MD), the mass term\nmust come to dominate and $\\langle w \\rangle \\simeq 0$ thereafter, rendering SFDM matter-like, as it should be in order to account for the MD era.\\footnote{This description above\nfrom \\citet{Li14,Li17}, of the evolution of the EOS\nof the homogeneous scalar field, from stiff,\nrelativistic ($w=1$) at early times, before\nthe rapid-oscillation phase, to radiation-like\n\\big($\\langle w \\rangle \\simeq 1\/3$\\big) when rapid oscillation begins and SI dominates, and finally\nto nonrelativistic, matter-like \n\\big($\\langle w \\rangle \\simeq 0$\\big),\nwas derived for a complex scalar\nfield.  However, we are only interested here\nin the rapid-oscillation phase, during which\nthe EOS is radiation-like when the SI term \ndominates over the mass term in the Lagrangian,\nthen transitions to matter-like when the\nlatter term dominates.  As such, \nwhat happens before the rapid-oscillation \nphase begins does not affect our calculations here.  \nIn that case, we do not expect our calculations\nto depend upon whether the field is complex or\nreal, since the description above of the EOS\nevolution during the rapid-oscillation phase\napplies equally well to the real and complex cases.}\n  \n\n\n\n\\subsection{Matter-like regime and the Newtonian approximation}\n\\label{sec:matterlike}\n\nIn this latter \\big($\\langle w \\rangle \\simeq 0$\\big) regime, in the limit of weak-field (i.e. Newtonian) gravity and even in the presence of inhomogeneities, as long as they are\non scales well-within the horizon (i.e. $\\ll c\/H$), we can adopt the Newtonian approximation in which gravity obeys the Poisson equation. The equation of motion of the scalar field in this nonrelativistic regime then simplifies to the nonlinear Schr\\\"odinger equation (NLSE) in an expanding background, by factorizing the field into a slowly-varying ``mean'' field $\\psi$ and an oscillatory component, as follows:\\footnote{\nIf $\\phi$ is a real field, then one should add to the right side of equation~(\\ref{eq:meanfield}) the complex conjugate, i.e. $\\phi = \\psi e^{-imc^2t\/\\hbar} + \\psi^* e^{imc^2t\/\\hbar}$, where $\\psi$ is still a complex field.}\n\\begin{equation}\n    \\phi = \\psi e^{-imc^2t\/\\hbar} \\quad \\quad \\quad |\\partial_t \\psi| \\ll \\left| \\frac{mc^2}{\\hbar} \\psi \\right|\n    \\label{eq:meanfield}\n\\end{equation}\nIn terms of the comoving coordinate $\\bm{x}$, defined with respect to the proper coordinate $\\bm{r}$ such that\n\\begin{align}\n    &\\bm{x} = \\bm{r}\/a \\\\\n    &\\nabla \\equiv \\nabla_{\\bm{x}} = a\\nabla_{\\bm{r}}\n\\end{align}\nand\n\\begin{equation}\n    \\partial_t \\equiv (\\partial_t)_{\\bm{x}} = (\\partial_t)_{\\bm{r}} + H \\bm{r} \\cdot \\nabla_{\\bm{r}}\n\\end{equation}\nwhere $(\\partial_t)_{\\bm{x}~(\\bm{r})}$ represents the partial time derivative at fixed comoving (proper) coordinate $\\bm{x}$ ~($\\bm{r}$),\nthe NLSE and Poisson equations are then\n\\begin{equation}\n    i\\hbar \\left(\\partial_t + \\frac{3}{2}H \\right)\\psi = \\left( -\\frac{\\hbar^2}{2ma^2} \\nabla^2 + \\frac{g}{m}|\\psi|^2 + m\\Phi \\right) \\psi\n\\end{equation}\nand\n\\begin{equation}\n    \\nabla^2 \\Phi = 4\\pi G a^2 (\\rho - \\rho_b)\n\\end{equation}\nwith\n\\begin{equation}\n    \\rho = |\\psi|^2\n\\end{equation}\nwhere $\\Phi$ is the peculiar Newtonian gravitational potential, $\\rho$ is the mass density of the scalar field, and $\\rho_b=\\rho_b(a=0)a^{-3}$ is the homogeneous background value.\\footnote{\nWe shall generally assume here, unless otherwise stated, that the nonrelativistic matter density is dominated by the scalar field dark matter density, so we do not need to keep separate track of other nonrelativistic mass components like the baryons.}\n\n\nThese equations were presented in \\citetalias{paper1}\nin the Newtonian limit in the\nabsence of cosmological expansion.  The difference\nhere is that we represent derivatives and spatial coordinates in the comoving frame,\nwhich introduces the homogeneous expansion term involving Hubble parameter $H(t)$,\nand the peculiar gravitational potential $\\Phi$.\nIf the scale factor were constant, $a(t) = 1$,\nwe would, of course, recover the equations of \\citetalias{paper1}.\n\nAs we discussed in detail in \\citetalias{paper1}, \nthere is a fully equivalent formulation of this\nnonrelativistic NLSE as quantum hydrodynamics\nequations (``QHD'').  In particular, the NLSE\ncan be transformed into a pair of hydrodynamical\nmass and momentum conservation equations via the Madelung transformation,\nin which the field is expressed in polar form as\n\\begin{equation}\n    \\psi = |\\psi|e^{iS}\n    \\label{eq:Madelung}\n\\end{equation}\nand the conservation of current density\nallows us to identify the peculiar bulk fluid velocity in the current flux density as\n\\begin{equation}\n    \\bm{v} = \\frac{\\hbar}{ma} \\nabla S\n\\end{equation}\nThis leads to\n\\begin{equation}\n    \\partial_t \\rho + 3H\\rho + \\frac{1}{a} \\nabla \\cdot (\\rho \\bm{v}) = 0\n\\end{equation}\nand\n\\begin{equation}\n    \\partial_t \\bm{v} + H\\bm{v} + \\frac{1}{a} (\\bm{v} \\cdot \\nabla) \\bm{v} + \\frac{1}{a} \\nabla (Q + V_\\text{SI} + \\Phi) = 0\n\\end{equation}\nwhere the ``quantum potential'' $Q$ and\nself-interaction potential $V_\\text{SI}$ are given\nby \n\\begin{equation}\n    Q = -\\frac{\\hbar^2}{2m^2 a^2} \\frac{\\nabla^2 \\sqrt{\\rho}}{\\sqrt{\\rho}}\n\\end{equation}\nand\n\\begin{equation}\n    V_\\text{SI} = \\frac{g \\rho}{m^2}\n\\end{equation}\nThese may also be written as pressure terms:\n\\begin{equation}\n    \\nabla Q = \\frac{1}{\\rho} \\nabla \\cdot \\bm{\\Pi}\n\\end{equation}\nwhere $\\bm{\\Pi}$ is the momentum flux density tensor, also known as the ``quantum pressure'' tensor, given by\n\\begin{equation}\n    \\Pi_{ij} = \\Big(\\frac{\\hbar}{2m a}\\Big)^2 \\Big( \\frac{1}{\\rho}\\frac{\\partial \\rho}{\\partial x_i}\\frac{\\partial \\rho}{\\partial x_j} - \\frac{\\partial^2 \\rho}{\\partial x_i \\partial x_j} \\Big)\n    \\label{eq:QPT}\n\\end{equation}\nand\n\\begin{equation}\n    \\nabla V_\\text{SI} = \\frac{1}{\\rho} \\nabla P_\\text{SI}\n    \\label{eq:SIpressure}\n\\end{equation}\nwhere $P_\\text{SI}$ is the self-interaction (SI) pressure, given by\n\\begin{equation}\\label{eq:SIpressure2}\n    P_\\text{SI} = \\frac{g\\rho^2}{2m^2}\n\\end{equation}\nIn the matter-like phase when $\\langle w \\rangle \\simeq 0$, the parameter $g\/m^2$ is related to the radius $R_\\text{TF}$ of the ($n=1$)-polytrope which results when gravity balances SI pressure in hydrostatic equilibrium, sometimes referred to as the Thomas-Fermi (TF) radius,\naccording to\n\\begin{equation}\n    R_\\text{TF} = \\pi\\sqrt{\\frac{g}{4 \\pi G m^2}} \\simeq 1.1 \\text{ kpc} \\left( \\frac{g\/(mc^2)^2}{2\\times 10^{-18} \\text{ eV}^{-1}\\text{cm}^3} \\right)^{1\/2}\n    \\label{eq:RTF}\n\\end{equation}\n\n\n\\subsection{The TF regime and the limit of small de~Broglie wavelength}\n\\label{sec:smalldeB}\n\nAs discussed in \\citetalias{paper1}, the same\nQHD equations can also by derived by\na reformulation of the NLSE in phase space,\ninvolving momentum moments of a quantum phase space distribution function.  In one version\nof that approach, involving a smoothed phase space\ndistribution, we showed that when the\nsmoothing scale is much larger than the de~Broglie\nwavelength $\\lambda_\\text{deB}$, the NLSE is\nequivalent to the Collisionless Boltzmann \nEquation (``CBE''), except with the gravitational\npotential term replaced by the sum of the\ngravitational and SI potentials.  This \nreduction of the NLSE to the CBE in the\nsmall-$\\lambda_\\text{deB}$ limit, for the free-field case \\textit{without} the modification to account for SI introduced in \\citetalias{paper1},\nwas previously discussed by \\citet{Mocz18}, in regards to the FDM model,\nfollowing earlier work by \\citet{WK93} and \\cite*{SKV89}.  Since the\ndescription of standard CDM as a gas of point-particles which are nonrelativistic, \nnoninteracting (except gravitationally), and\ncollisionless (i.e. the 2-body gravitational relaxation time is much longer than the\nlifetime of the system) is consistent with\nthe CBE, this explains why FDM behaves just like CDM, dynamically,\non scales much\nlarger than $\\lambda_\\text{deB}$,\ni.e. when viewed in this coarse-grained way that does not\nresolve substructure on scales smaller than\n$\\lambda_\\text{deB}$.  \nFor example, numerical simulations of \nthe NLSE in the FDM limit (no SI) show that,\nwhen haloes form gravitationally and virialize, \nthey have a core-envelope structure in which\na solitonic core of size $\\sim\\lambda_\\text{deB}$\n(with a centrally-flattened density profile \nlike the ground-state,\nstatic equilibrium solution of the 1D, spherical\nNLSE) is surrounded by an envelope\nwith a profile like that which would have \noccurred in standard CDM.  With strong enough SI\nto be in the TF regime, however, we showed\nin \\citetalias{paper1} that \nthe solitonic core would be replaced\nby a core,  much larger than \n$\\lambda_\\text{deB}$ (the latter is arbitrarily small\nin this TF regime), \nsupported instead by SI pressure, while\noutside this SI-supported core, \na CDM-like envelope would still be expected. \nHowever, to solve the latter problem in this TF regime, \nby the same kind of 3D simulations as were done for FDM, \nis much more difficult, since it would \nrequire fully resolving length scales well-below\n$\\lambda_\\text{deB}$ in order to produce the correct CDM-like behavior\noutside the core. In the TF regime, $\\lambda_\\text{deB}$ is\nmuch smaller than the size of the SI-smoothed halo\ncores, which are, themselves, much smaller than the \nhaloes, and even smaller relative to \nthe large-scale structure from which they formed, \nso the required resolution and dynamic range becomes prohibitively expensive.\nWe solved\nthis problem by finding an approximate fluid\ndescription for SFDM-TF, as described below. \n\n\n\\subsection{The fluid approximation for SFDM-TF}\n\\label{subsection:fluidapprox}\n\nIn \\citetalias{paper1}, we were tasked with calculating the dynamical formation of virialized haloes in SFDM-TF by evolving the scalar field\nfrom a nearly-uniform, linearly-perturbed, initial condition through many orders of magnitude of gravitational collapse, all the while\nkeeping track of the dominant dynamical effects involving quantum pressure,\nwhich result from inhomogeneities on the scale of the unresolvably-small $\\lambda_\\text{deB}$.\nThe fact discussed there (noted above\nin \\S\\ref{sec:smalldeB}), that, in\nthe small-$\\lambda_\\text{deB}$ limit, the NLSE\nfor SFDM-TF can be shown to reduce to\nthe same CBE as for CDM, except \nmodified by adding the SI potential to\nthat for gravity, suggests that,\nin principle, an SI-modified N-body method \nmight be developed to solve this problem,\nif we were willing to forego knowledge\nof the detailed substructure on the scale of\n$\\lambda_\\text{deB}$, while still \naccounting for the effects\nof quantum pressure on larger scales in\na coarse-grained way.\nHowever, such an N-body method would still require\nan extremely large dynamic range \nto simulate this in 3D, in order to\nresolve the small-scale suppression effects\nof SI pressure on scales below $R_\\text{TF}$, even in this limit in which \nthe (much smaller) de~Broglie scale is \nleft unresolved -- a high computational barrier.\nInstead, we avoided the complicated challenge\nof such a grand computational solution\nby closing the infinite hierarchy of the \nmomentum moments of the CBE by following the approach developed for standard CDM dynamics by \\citet{AS05}.\n\n\\citet{AS05} reduced the CBE equation for\nCDM to a set of equations \nidentical to the hydrodynamical \nequations of conservation of mass, momentum, and\nenergy for an ideal gas with ratio of specific\nheats $\\gamma = 5\/3$, in 1D, spherical symmetry, by assuming the velocity distribution is\nskewless and isotropic in the frame of bulk motion. As discussed there, these are assumptions that \napproximate the results of CDM N-body simulations \nreasonably well, for the virialized regions\nidentified as haloes.\\footnote{Outside of\nvirialized regions, infall is highly supersonic; \nthe velocity of bulk peculiar motions \ngreatly exceeds that associated with \nthe velocity dispersion due to random motions \nin that frame. Hence, these assumptions make\nno appreciable difference in the dynamics\noutside virialized regions.} \nIn the case of SFDM-TF, however,\nas we showed in \\citetalias{paper1}, we must \nreplace the gravitational potential\nin the CDM case of \\citet{AS05}\nby the sum of the gravitational and SI potentials.\nAs a result, a new kind of \npressure force appears in the momentum equation,\nin addition to the ideal gas pressure, namely \nthe SI pressure force, involving the gradient of\nthe SI pressure given above in equations~(\\ref{eq:SIpressure}) and (\\ref{eq:SIpressure2}).  \n\nThe resulting spherically-symmetric fluid equations for SFDM-TF are presented below in proper coordinates, in Eulerian form:\n\\begin{align}\n        \\text{[continuity]}& \\quad \\quad \\frac{\\partial \\rho}{\\partial t} + \\frac{1}{r^2}\\frac{\\partial (r^2 \\rho v)}{\\partial r} = 0 \\label{eq:0momSpherical}\\\\\n        \\text{[momentum]}& \\quad \\quad \\frac{\\partial v}{\\partial t} + v \\frac{\\partial v}{\\partial r} + \\frac{1}{\\rho}\\frac{\\partial}{\\partial r}(P_{\\!\\sigma}+P_\\text{SI}) + \\frac{\\partial \\Phi}{\\partial r} = 0 \\label{eq:1momSpherical} \\\\\n        \\text{[energy]}& \\quad \\quad \\frac{\\partial}{\\partial t}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) + v \\frac{\\partial}{\\partial r}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) + \\frac{P_{\\!\\sigma}}{\\rho r^2}\\frac{\\partial (r^2 v)}{\\partial r} = 0\n        \\label{eq:2momSpherical}\n\\end{align}\nwhere $P_{\\!\\sigma} = \\rho \\sigma^2$ is an isotropic pressure associated with the velocity dispersion $\\sigma$ obtained from velocity moments of the phase space distribution function. This\npressure, which behaves like the pressure\nof an ideal gas with ratio of specific heats $\\gamma = 5\/3$ and, hence, contributes \na pressure-gradient force to the \nmomentum equation, while changing\nits specific internal energy $u \\equiv (P_{\\!\\sigma}\/\\rho)\/(\\gamma -1)$ to account for\n$P_{\\!\\sigma} dV$-work, thereby accounts for the large-scale effects of the quantum pressure tensor ($\\bm{\\Pi}$), as \ndiscussed in \\citetalias{paper1}.\nIf we define the mass interior to radius $r$ as $M_r$, according to\n\\begin{equation}\n    M_r \\equiv M(\\leq r) = \\int_0^r 4\\pi (r')^2 \\rho~dr'\n    \\label{eq:interiormass}\n\\end{equation}\nwe can rewrite equations~(\\ref{eq:0momSpherical}) - (\\ref{eq:2momSpherical}) in Lagrangian form, \nthough still in proper coordinates, as follows:\n\\begin{align}\n        \\text{[continuity]}& \\quad \\quad \\frac{D \\rho}{D t} = -\\frac{\\rho}{r^2}\\frac{\\partial (r^2 v)}{\\partial r}  \\label{eq:0momLagrangian}\\\\\n        \\text{[momentum]}& \\quad \\quad \\frac{D v}{D t} = -4\\pi r^2 \\frac{\\partial}{\\partial M_r}(P_{\\!\\sigma}+P_\\text{SI}) - \\frac{G M_r}{r^2} \\label{eq:1momLagrangian} \\\\\n        \\text{[energy]}& \\quad \\quad \\frac{D}{D t}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) = -P_{\\!\\sigma}\\frac{D (1\/\\rho)}{D t} \\label{eq:2momLagrangian}\n\\end{align}\nwhere the Lagrangian, or total, time derivative is given by\n\\begin{equation}\n    \\frac{D}{Dt} \\equiv \\frac{\\partial}{\\partial t} + v \\frac{\\partial}{\\partial r}\n\\end{equation}\nIn \\citetalias{paper1}, we also stated the\nadiabatic shock jump conditions that \naccompany these differential conservation\nequations when there are discontinuities\npresent.  The reader is referred to \\citetalias{paper1} for those.  \n\n\\section{Halo formation by cosmological infall: spherical collapse and virialization}\n\\label{sec:infall}\n\n\n\\subsection{The non-cosmological case: nonlinear Jeans instability from a static initial condition}\n\\label{sec:staticICPaperI}\n\nIn \\citetalias{paper1}, we solved the equations above to follow the spherical collapse of a Jeans-unstable linear perturbation in a static background. We simulated the nonlinear\nevolution of a region containing a mass much larger than the Jeans mass, initially at rest, \nfrom a uniform background density, perturbed\nby a spherically-symmetric perturbation that is slightly concentrated toward the center.\nAs it was gravitationally unstable,\nthe mass collapsed, first at the center, \nuntil the rising SI pressure inside the central region of radius $\\simeq R_\\text{TF}$ halted the gravitational collapse of the innermost\nmass shells, and a strong accretion shock \nformed there to decelerate other \nincoming shells.  As more\nmass collapsed and would have, if entirely pressure-free, \nreached the origin in a finite time, with\nshells originating at successively larger radii\npredicted to reach the\norigin at successively later times, each\nshell, instead, encountered the shock at\na finite radius and was halted by it.\nOutside this accretion shock, \nthe infall was highly supersonic and the\nideal gas ``temperature'' (i.e. the velocity dispersion in momentum space, as \nmeasured in the frame of bulk radial motion) was\nvery small, so the ideal gas pressure associated with it was too small to halt the collapse. The\nSI pressure was also low there, so it, too, was\nunable to decelerate the infall.  When\na mass shell crossed the accretion shock, however,\nits ``temperature'' and ideal gas pressure jumped \nby orders of magnitude, \nwhile its density jumped only by a factor\nof four (i.e. the value for a strong, adiabatic\nshock jump in an ideal gas with a ratio of specific heats $\\gamma = 5\/3$).  As a result, \nthe post-shock gas was dominated by ideal gas pressure, high enough to bring the mass shell to\nan approximate rest in hydrostatic \nequilibrium (``HSE'').\nIn this post-shock HSE, ideal gas pressure balanced gravity everywhere inside the shock, except inside $\\sim R_\\text{TF}$, \nwhere it was SI pressure that provided the support, instead,\nand the ideal gas pressure was unimportant. \nOver time, more mass shells reached the\naccretion shock, which gradually moved outward,\nencompassing an ever-increasing mass.\nAs new mass shells passed through the shock and\nsteadily added their mass to that of the previously-shocked shells, they extended the\nradius of the post-shock region in hydrostatic equilibrium as the shock\nmoved outward. \nThe\nstructure interior to the post-shock region\ncorresponds to the ``virialized halo'' in\ncosmological simulations.   \n\nIn \\citetalias{paper1},\nwe compared the outcomes of\ntwo simulations from the same initial\nconditions, one that included the SI\npressure for SFDM-TF and one that did not, \nso the latter was entirely ``CDM-like'', \ntherefore. Since the initial conditions for\nboth of these simulations was that for a\nJeans-unstable density perturbation on a\nstatic homogeneous background, the\ncollapse that ensued left a post-shock\ndensity profile in the CDM-like case\n(i.e. with no SI) with a radial dependence\n$\\rho\\propto r^{-12\/7}$.  As we explained there,\nthis slope results when cold, infalling\ngas with a density profile with this\nslope -- the same as reported in earlier work on infall from instability in a spherical, Jeans-unstable gas cloud\n\\citep [e.g.][]{Penston69} -- encounters the strong accretion shock. The shock jump converts the kinetic energy of this cold, supersonic infall into\nideal-gas internal energy, with enough post-shock gas pressure to establish hydrostatic equilibrium.\nAccording to the strong, adiabatic shock jump conditions, the density must also increase,\njumping by a constant factor of 4 \n(for $\\gamma = 5\/3$).  As a result, the static\npost-shock density profile inside the shock \nmust have the same power-law profile as the pre-shock infall, outside the shock, namely, $\\rho\\propto r^{-12\/7}$.\nThe SFDM-TF solution was \nshown to be identical to\nthis CDM solution everywhere outside of the\nSI-pressure-supported core of radius equal \nto $\\sim R_\\text{TF}$.  Inside this radius, however,\nthe density profile was flattened, following\nthat of the $(n=1$)-polytropic profile that would result\nif SI pressure alone supported the scalar\nfield against its own self-gravity. Its \ncentral density was set by the condition that\nit conserved the mass that would have been\nthere inside that radius if the CDM-like\nprofile had continued onward to the origin,\nuninterrupted by SI pressure.  \n\nIn short, the\ngeneric structure of the haloes that formed\nas the nonlinear outcome of Jeans instability\nin SFDM-TF, therefore, could be described\nsimply as the core-envelope structure in which\na flattened, SI-polytropic core of \nradius $\\sim R_\\text{TF}$\nwas surrounded by the same CDM-like envelope\nit would have had if there were no SI pressure,\nand the mass inside the polytropic core radius was\nthe same as it would have been in that CDM-like\nprofile in the absence of SI.  As such, this\nalso served to compare the SFDM-TF profile with\nthe corresponding FDM (i.e. ``free-field'' limit) profile that would have\nresulted from the same initial conditions if $\\lambda_\\text{deB}$ is small, since the\nlatter was shown in \\citetalias{paper1} to be\nidentical to the CDM profile, when smoothed \non scales large compared with $\\lambda_\\text{deB}$.\n\nWe note that, although the description above\napplies to the entire post-shock region at\na given time, we can make our description\napply, as well, to cosmological haloes,\nwhich are customarily identified as the\n``virialized regions'' bounded by \na ``virial radius'' inside of which\nthe average mass density is equal \nto some fixed constant multiple of the \nunperturbed background density\nof the universe at that cosmic time.  In the Jeans-unstable collapse\ncalculations in \\citetalias{paper1} from static\ninitial conditions, there is not a cosmological\nbackground density to relate to the overdensity\ninside some radius within\nthe post-shock region, but we are still free\nto adopt an overdensity like the conventional\ndefinition, relative to the unperturbed\nbackground density, in order to determine\nthe radius (within the post-shock sphere) which we can \ncall the ``virial radius''.  \n\n\\subsection{The cosmological case: spherical infall in the expanding universe}\n\\label{sec:cosmologicalcase}\n\nThe description above is a guide to\nwhat we can expect below, when we re-consider the\nnonlinear Jeans instability in SFDM-TF here,\nto place it in the proper cosmological context of\nthe expanding universe in the MD era, i.e. after matter-radiation equality, $a > a_\\text{eq}$.  To do this, we will \nsolve the same equations, as written\nin proper coordinates, but now with cosmological initial and boundary conditions,\\footnote{\nAlthough our dynamical equations (\\ref{eq:0momSpherical}) - (\\ref{eq:2momSpherical}) \ndo not appear to be cosmological, for a spherically-symmetric distribution of \nmatter in the nonrelativistic and weak gravitational field limits, \nBirkhoff's Theorem implies that Newtonian gravity is sufficient for \ndescribing cosmological dynamics. Therefore, our cosmology is taken \ncare of by the Newtonian potential and the inclusion of \nHubble flow (and its perturbation)\nin the initial velocity.}\nto follow the spherical collapse of a linear perturbation in the expanding universe.\nThis is fully cosmological, for structure\nformation on scales that are well within the\nhorizon when we evolve them.  \nAs we shall see, the novel effect of cosmological\nexpansion is to allow mass shells that were\ninitially inside the characteristic radius\nof the effects of SI pressure to expand outward\nfor a time, carrying them beyond the reach of this repulsive force, and then to fall back again, \nto re-encounter it after turn-around and collapse.\nAn imprint of this initial encounter remains, \nhowever, and has important consequences for their\nfinal resting place.  \n\n\\subsubsection{CDM-like initial conditions: the infall rate that makes an NFW profile} \\label{sec:NFW}\n\nOur goal in this paper (``Paper II'')\nis to generalize the non-cosmological results in \\citetalias{paper1} to investigate the cosmological formation of SFDM-TF haloes.\nOur first approach in this direction follows\nthe tradition of previous study of structure formation in the free-field limit of the\nSFDM model (FDM),\nin which the first cosmological\nsimulations assumed scalar field dynamics\nbut initial conditions like those for the CDM model. In particular, we will simulate the formation of individual haloes from the same initial conditions for both CDM and SFDM-TF, in order to learn what difference results \nfrom scalar field dynamics with repulsive SI, \nas opposed to the collisionless dynamics of CDM. \n\nIt is useful to begin by summarizing what is\nknown about the CDM case and why it is that\nwe have confidence in the 1D, spherical \ncollapse calculations we propose to use to\nstudy the contrast between CDM and SFDM-TF\nhalo dynamics.\nIn \\cite*{AAS03}, \\citet{Shapiro04}, \\citet{AS05} and \\citet{Shapiro06}, we showed that the 1D, spherical infall model involving the same fluid \napproximation as used here and in \\citetalias{paper1}, derived by \\citet{AS05}  \nfrom the \ncollisionless Boltzmann equation, \ncan explain\nthe universal structure of haloes in N-body simulations of nonlinear structure formation from \nGaussian-random-noise density fluctuations in\nthe CDM universe. As a first step, \n\\citet{AS05} applied\nthis fluid approximation for CDM to derive\nan analytical similarity solution for halo formation from spherical infall. As described below, this\nproduced a density profile inside the accretion shock in this solution which was a\nsurprisingly good match to various \nempirical formulae developed to fit density profiles of CDM haloes in N-body simulations, e.g. the NFW profile \\cite*{NFW97}. \nHowever, while the agreement\nwas impressive at a given snapshot of time,\nthe N-body results showed that, \nif captured at different times in the mass assembly history (``MAH'') of the same halo,\nthe shape of its profile would evolve,\nwhile that of a self-similar profile does not.\nAs \\citet{AS05} pointed out, this reflected\nthe fact that infall rates\nof individual haloes derived by fitting the Lagrangian MAH of haloes in CDM N-body simulations \n\\textit{vary} over the lifetime of each halo --\nrapid early, then slowing down late -- \nwhile the rate in their self-similar\nsolution ($d \\ln M\/d \\ln a = 6$) is fixed, corresponding to the early, rapid-infall phase of the N-body results.  \nTo better match the \\textit{non-self-similar} shape evolution of the N-body haloes, in \\citet{AAS03} we replaced the self-similar initial perturbation\nin \\citet{AS05} by a \\textit{non-self-similar} spherically-symmetric perturbation profile calculated to make a time-varying\ninfall rate that matched the MAH of CDM\nN-body haloes.  Solving the 1D fluid\napproximation equations for CDM in this case\nrequired a numerical hydrodynamics\ncode.  The haloes that resulted were shown\nto be in excellent agreement with \nthe universal average behavior of the haloes in N-body simulations.  Since this earlier work,\nalso summarized in \\citet{Shapiro04} and\nupdated in \\citet{Shapiro06}, is central to\nour adoption of CDM-like initial conditions here\nto demonstrate the difference between SFDM and\nCDM dynamics on cosmological halo formation,\nwe describe it in more detail below.\n\n\\subsubsection{CDM haloes from self-similar spherical infall: $\\epsilon=1\/6$}\n\\label{sec:selfsimilar}\n\nIn an Einstein-de Sitter universe, a scale-free,\nspherically-symmetric linear perturbation \n$\\delta M\/M \\propto M^{-\\epsilon}$ leads to\nself-similar halo formation around the density\npeak at the center. According to the \ntheory of halo formation\nfrom peaks in the density field which result from Gaussian-random-noise initial density fluctuations,\nthe average density profile around such peaks can be related to the shape of the fluctuation power spectrum \\citep{HS85,Bardeen86}.  These density peaks in the linear regime are the progenitors of nonlinear structures like haloes.  It is useful to\ndescribe such peaks in terms of the RMS mass fluctuation, $\\sigma_\\textsc{m}$, of the overdensity field smoothed on some mass scale $M$. The average linear overdensity profiles around peaks of \noverdensity $\\nu\\sigma_\\textsc{m}$ are triaxial, in general, but approach spherical symmetry as $\\nu$ increases  (e.g. for $\\nu \\geq 3$), with cumulative overdensity $\\Delta_0$ inside radius $r$, given by $\\Delta_0 \\propto r^{-(n+3)}$, where $n$ is\nthe effective index of the (transferred) \npower spectrum $P(k)$ if approximated as a power-law $P(k)\\propto k^n$ at the wavenumber $k$ that corresponds to that halo mass.\nIn terms of fractional mass perturbation, \n$\\Delta_0 = \\delta M\/M \\propto  M^{-(n+3)\/3}$, \nso a power-law $P(k)\\propto k^n$ leads \nnaturally to a scale-free initial condition with $\\epsilon = (n+3)\/3$.  While the actual transferred\n$P(k)$ for CDM is not a power-law, it is\npossible to use the actual shape to derive an effective power-law index which varies with halo mass $M$ as a good approximation.  \\citet{AS05} calculated this $n_\\text{eff}$ for the current $\\Lambda$CDM universe, for the full range of halo masses from $10^3$ to $10^{15} \\text{ M}_\\odot$, showing that galactic haloes are well-approximated by $n = -2.5$, since $n\\cong-2.5\\pm0.2$ for $M$ in the range from\n$10^3 \\text{ M}_\\odot$ to $10^{11} \\text{ M}_\\odot$.\nFor haloes of galactic mass, formation by\nspherical infall from a self-similar initial perturbation with $\\epsilon=1\/6$ is therefore a reasonable\nfirst approximation (for the\nMD era, for which the universe\nis well-approximated as Einstein-de~Sitter). \n\nThe resulting self-similar density profile agreed with that for the NFW profile to within\n$10\\%$, at radii $r \\lesssim 0.6 R_{200}$, for a best-fitting concentration parameter of 3, including a density cusp at small radii with a logarithmic\nslope $\\approx{-1.27}$ \\big(for $4 \\times 10^{-3} < r\/R_{200} < 1.4 \\times 10^{-2}$\\big), where $R_{200}$\nis the radius within which the average density is\n200 times the cosmic mean density.   Furthermore,\nas we showed in \\citet{Shapiro06}, the analytical similarity solution yields a profile for the quantity $\\rho\/\\sigma_V^3$, sometimes referred to as the ``phase-space density'', which is \nbest-fit by a power-law $\\rho\/\\sigma_V^3\\propto r^{-1.91}$, in close agreement at all times with\nthe universal profile reported in the literature\nfor CDM N-body simulation results cited there. \nHowever, the\nradius of the accretion shock that bounds the virialized region in this solution is somewhat smaller than $R_{200}$, located instead at the\nradius which encloses an average overdensity of\n564 times this mean density, at $R_{564} \\simeq 0.6 R_{200}$.  This suggests that the rapid early phase of MAH in the N-body results, which deposits the\nmass of the density profile at inner radii\nthat remains there at later times, is well-represented by this self-similar infall solution, but the departure from self-similarity,\nespecially as related to the late\ntapering-off of the mass accretion is not.  \n\n\\subsubsection{CDM haloes from non-self-similar spherical infall}\n\\label{sec:nonselfsimilar}\n\nIf the infall rate was given, instead, by \nthe spherically-averaged, Lagrangian MAH \nof individual \nhaloes in CDM N-body simulations, however, then\nthe resulting haloes shared most of the\nempirical CDM N-body halo structural\nproperties \\textit{and} their time-evolution.\nIn particular, in \\citet{AAS03}, \nwe took as our initial\ncondition, a spherical density perturbation\nwhose radial dependence ensured that\nthe rate at which infalling mass reached the virial radius of the emerging halo followed\nthe empirical MAH derived by \\cite{Wechsler02}\nfrom N-body simulations of CDM.  With this\nsimple assumption, we found that the virialized\nhaloes which result have density profiles that\nare well-fit by the NFW density profile\n(at all radii $r\/R_\\text{vir}\\geq0.01$), and \nthe radius of the accretion shock is safely\noutside $R_{200}$. These\nhalo density profiles also \\textit{evolve} like\nCDM N-body haloes, with concentration\nparameter that grows with time just as it\ndoes for CDM N-body haloes.  In addition, \nas shown in \\citet{Shapiro06},\nthe resulting halo phase-space density profile, $\\rho\/\\sigma_V^3$, best-fit by a power-law\n$\\propto r^{-1.93}$, was found to be in\nremarkable agreement at all times with \nthe universal profile reported for CDM\nN-body haloes.  We concluded, therefore,\nthat the time-varying mass accretion rate,\nor equivalently, the initial perturbation\nprofile shape around the density peak that\nled to the formation of the halo, is the\ndominant influence on the internal structure\nof CDM haloes, which can be well-understood\nfrom this simple spherical infall model involving\nsmooth accretion and \nthe fluid approximation derived by \\citet{AS05}\nfrom the CBE.\n\n\\subsubsection{SFDM vs. CDM haloes from non-self-similar spherical infall}\n\\label{sec:SFDMnonselfsimilar}\n\nIn \\citetalias{paper1}, we showed that\nthe fluid approximation used above to model CDM halo\ndynamics also models SFDM dynamics\nin the free-field limit of FDM, as long as we\ndo not need to resolve substructure\non scales as small as the de~Broglie \nwavelength but still wish to account faithfully\nfor the influence of quantum pressure on\nlarger scales.\nThis, then, is a useful tool for the study\nof halo formation in the FDM regime, \nas long as we do not need to resolve the\nsolitonic core in the central region.  \nThe results described above, which\nexplain how the NFW profile\nand its related universal properties\nin the CDM model can arise as a simple consequence\nof spherical infall of smoothly-accreted matter, \nalso explain why 3D simulations\nof FDM haloes report NFW-like density profiles \n\\textit{outside} their solitonic cores. \n\nIt remains\nfor us to determine here, what happens to this\nresult when a repulsive SI is added to \nSFDM.  In \\citetalias{paper1},\nwe \\textit{generalized} the\nfluid approximation  of \\citet{AS05}\nto include the effects of SI pressure in the\nTF regime, which corresponds to\nthe regime in which the de~Broglie wavelength\ncan be assumed to be arbitrarily small,\njust right for application of the fluid\napproximation derived from the CBE.\nOur first approach to putting halo formation\nin the SFDM-TF model in a cosmological context\nwill be to adopt the spherical infall\nmodel described above for the CDM model and\nsolve the 1D, fluid conservation equations for \nSFDM-TF of \\citetalias{paper1} with cosmological initial conditions to determine\nwhat the dynamical effect of adding SI to the scalar\nfield is.   \n\nWe consider a spherically-symmetric, adiabatic\ndensity perturbation as our initial condition.  We \ntake our initial time to be when the \nscale factor $a_i = a_\\text{eq}$, and further\nassume that the universe remains matter-dominated, thereafter, at the critical density, corresponding\nto an Einstein-de Sitter (EdS) cosmology. \nThe initial linear density perturbation profile\ncan be described in terms of the mass perturbation $\\Delta_\\textsc{l}(M,a_i)$, where $M$ is\nthe mass inside a sphere of initial (perturbed) radius $r(M,a_i)$, related to the mass $M_b(M,a_i)$ that would have been inside a sphere of that radius in the unperturbed background universe, according to\n\\begin{equation}\n    \\Delta_\\textsc{l}(M,a_i) \\equiv \\frac{M-M_b(M, a_i)}{M_b(M,a_i)}\n    \\label{eq:DeltaLMass}\n\\end{equation}\nwhere $M_b(M,a_i) = (4\\pi\/3) r^3(M,a_i) \\rho_b(a_i)$, and $\\rho_b(a_i)$ is the background matter\ndensity at the initial redshift, given by \n$\\rho_b(a_i) = \\rho_\\text{crit}(a_i) \\equiv 3H^2(a_i)\/8\\pi G$, where the Hubble parameter\nevolves according to $H^2(a)=H^2_0a^{-3}$.\nWe adopt the value $H_0 = 67$ km\/s\/Mpc for the present-day Hubble parameter.\nThe initial radius of the Lagrangian\nmass shell which encloses that mass $M$ is then given by\n\\begin{equation}\n    r(M,a_i) = \\Bigg(\\frac{M}{\\frac{4\\pi}{3} \\big[1+\\Delta_\\textsc{l}(M,a_i)\\big] \\rho_b(a_i)} \\Bigg)^{1\/3}\n    \\label{eq:initialradius}\n\\end{equation}\nwhile its initial velocity corresponds to the \nperturbed Hubble flow for the growing mode of the \nperturbation with that linear overdensity, given by \\citep*[see, e.g.,][\\S5.1.1]{MVW10}\n\\begin{equation}\n    v(M,a_i) = H(a_i) r(M,a_i)\\big(1 -\\Delta_\\textsc{l}(M,a_i)\/3\\big)\n\\end{equation}\n\nThe quantity $\\Delta_\\textsc{l}(M,a_i)$ is also the\naverage fractional overdensity inside the sphere which encloses the mass $M$ at $a_i$.  This average overdensity is chosen to be a function of mass $M$\nsuch that, for an initial linear perturbation with this profile in a CDM universe, \nthe ensuing growth would lead to infall and\nnonlinear collapse that reproduces the average MAH of \nthe virialized haloes that form in CDM N-body simulations, as described above.\nThe density profiles of these N-body simulation haloes are well-fit by NFW profiles\\footnote{Note that in \\citetalias{paper1}, we denoted $A_\\text{NFW}$ as $\\delta_\\text{NFW}$. } \\citep{NFW97}:\n\\begin{align}\n    &\\rho = \\frac{\\rho_\\text{crit} A_\\text{NFW}}{(c_\\textsc{nfw}r\/R_{200})(1+c_\\textsc{nfw}r\/R_{200})^2} \\\\\n    &A_\\text{NFW} = \\frac{200}{3}\\frac{c_\\textsc{nfw}^3}{\\ln{(1+c_\\textsc{nfw})} - c_\\textsc{nfw}\/(1+c_\\textsc{nfw})}\n\\end{align}\nwhere $R_{200}$ is the radius within which the average density is 200$\\rho_\\text{crit}$, and $c_\\textsc{nfw}$ is the concentration parameter, defined by $c_\\textsc{nfw}\\equiv R_{200}\/R_s$, where $R_s$ is the ``scale radius'' at which the logarithmic slope of the density profile equals $-2$. \\cite{Wechsler02} found that the MAH of CDM haloes in N-body simulations followed a simple rule -- the logarithmic slope of the mass accretion is inversely proportional to the scale factor:\n\\begin{align}\n    \\frac{d\\ln{M_{200}}}{d\\ln{a}} &\\propto \\frac{1}{a} \\\\\n    \\Rightarrow M_{200}(a) &= M_\\infty\\exp{(-s a_f\/a)} \\label{eq:MAH}\n\\end{align}\nwhere $M_{200}$ is the mass within $R_{200}$, $M_\\infty$ is the asymptotic halo mass as $a\\rightarrow\\infty$, and $a_f$ is the ``formation scale factor'' at which the logarithmic slope equals $s$. By convention, $s=2$. In \\citet{AAS03} and \\cite{Shapiro04}, we found an initial perturbation profile for an EdS universe that is consistent with this MAH,\ngiven by\n\\begin{equation}\n    \\Delta_\\textsc{l}(M,a_i) = \\Delta_{\\textsc{l},i} \\ln{(M_\\infty\/M)}\n    \\label{eq:NFWPert}\n\\end{equation}\nwhere $\\Delta_{\\textsc{l},i} \\equiv \\Delta_\\textsc{l}\\big( M_{200}(s a_f \\!) , a_i \\big)$ is the initial average linear overdensity within the shell whose ballistic motion brings it to a radius which corresponds to\n$R_{200}$ at $a=s a_f$. \nSince $a_f$ corresponds roughly to\nthe ``formation time'' of the halo, we can linearly extrapolate this initial overdensity to $a_f$, when the (linearly-extrapolated) average overdensity should be roughly unity:\n\\begin{equation}\n    \\Delta_{\\textsc{l},f} = \\Delta_{\\textsc{l},i} \\frac{a_f}{a_i} \\approx 1 \\Longrightarrow \\Delta_{\\textsc{l},i} \\approx \\frac{a_i}{a_f}\n    \\label{eq:deli}\n\\end{equation}\nThe exact value we use in our code is $\\Delta_{\\textsc{l},i} = 0.8 a_i\/a_f$, which we obtained by calculating the linearly-extrapolated overdensity of a spherical top-hat in an EdS universe that has collapsed to a nonlinear overdensity of 200,\nat $a=s a_f$. We show our calculation in Appendix~\\ref{sec:deli}.\n\nAs described above in \\S\\ref{sec:nonselfsimilar}, \nthe result of adopting this\nlinear perturbation profile and simulating its\ngrowth and collapse in a CDM universe, using the\nfluid approximation of \\citet{AS05}, is to\nproduce a virialized region inside the accretion\nshock whose structure and evolution closely \nresembles that reported for CDM N-body haloes. \nFor example, not only are the halo profiles that result from these initial conditions well-fit by an NFW density profile at\nall times following the formation of that\naccretion shock, but the concentration parameter \nof that profile increases over time as the halo grows, according to  \n\\begin{equation}\n    c_\\textsc{nfw}=4.25 a\/a_f \\quad \\quad \\text{for  } a\\gtrsim a_f\n    \\label{eq:c200}\n\\end{equation}\n\\citep{Shapiro04,AAS03}. \nThis is close to the relationship reported by \\citet{Wechsler02} in their analysis of 3D N-body simulations: $c_\\textsc{nfw}=4.1a\/a_f$.\n\nIn practice, if we want to form an individual\nhalo with a given mass and concentration parameter\nat a given scale factor, we combine  equations~(\\ref{eq:MAH}) and (\\ref{eq:c200}) and\nsolve them for the free parameters of this NFW-producing initial condition, ($M_\\infty$, $a_f$), that will yield the desired halo.\nFor example, consider a $10^9 \\text{ M}_\\odot$ halo with a concentration parameter of 18 at the\npresent \\citep[the median value for this mass as per the mass-concentration relation found by][]{Klypin16}. Equation~(\\ref{eq:c200}) then gives $a_f \\simeq 0.24$ for this halo, in which case equation~(\\ref{eq:MAH}) gives $M_\\infty \\simeq 1.6 \\times 10^9 \\text{ M}_\\odot$.\nThe results of a simulation run using these values in the initial condition will be discussed below,\nin \\S\\ref{sec:CDMNFWsims}.\n\n\n\\subsubsection{Numerical Method}\n\\label{sec:numericalmethod}\n\nAll simulations of cosmological halo formation from spherical infall in this paper will use the \nsame 1D, Lagrangian hydrodynamics\nmethod and set-up as described in \\citetalias{paper1}.   We adopt the same\nnon-self-similar initial conditions\ndescribed above in \\S\\ref{sec:SFDMnonselfsimilar}\nand Appendix \\ref{sec:deli}, for both CDM\nand SFDM-TF, starting from $a = a_\\text{eq}$.  \nThis is designed \nto allow us to compare the dynamical\nconsequences of SFDM-TF and CDM directly,\nfor evolution from the same initial perturbation.\nUnless otherwise noted, each simulation uses\n2000 Lagrangian shells, logarithmically-spaced\nin mass.  \n\n\\subsection{Simulation results for cosmological halo formation from spherical infall and collapse}\n\\label{sec:simulationresults}\n\n\\subsubsection{CDM halo formation from NFW-producing infall rate}\n\\label{sec:CDMNFWsims}\n\nIn Fig.~\\ref{fig:NFW}, we show results of our 1D\nnumerical hydrodynamics simulation of the formation of a CDM halo, based upon the fluid approximation of \\citet{AS05} for CDM described above in \\S\\ref{subsection:fluidapprox} and \\citetalias{paper1}.  We adopted the non-self-similar cosmological perturbation described in \\S\\ref{sec:SFDMnonselfsimilar} and Appendix~\\ref{sec:deli} as our initial condition,\nstarting at $a = a_\\text{eq}$.  Our purpose here is\nto illustrate the fact that this fluid approximation\nfor CDM (\\textit{without} the SI pressure term\nwhich must be added to represent SFDM-TF) \nfaithfully reproduces the universal halo properties\nand their evolution over time of the NFW-profile\nhaloes reported for CDM N-body simulations \nwhen the initial perturbation is tuned to\nproduce a smooth accretion rate that matches\nthe MAH from \\citet{Wechsler02}. \nThe results in Fig.~\\ref{fig:NFW} confirm what we\npreviously reported in \\citet{Shapiro04}, \\citet{AAS03}, and \\citet{Shapiro06}, as they\nshould, since we are following the same \nprocedure.  Our purpose in the \\textit{next} \nsection will be to show what \\textit{difference} \nis made if these same CDM-like initial conditions\nare used to simulate the formation of an \\textit{SFDM-TF} halo, instead, when \nthe important dynamical effects\nof both quantum pressure and SI-pressure\nare properly taken into account.\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{NFW_fit.pdf}\n    \\caption{\\textbf{The fluid approximation results for halo formation from spherical infall and collapse in CDM vs. the NFW fit to N-body simulations}. Resulting density profile from our 1D hydrodynamics code without repulsive self-interaction ($g=0$) starting from NFW-producing initial conditions. The red dashed line shows the corresponding NFW profile with the same value of $R_{200}$ as the numerical profile, but with $c_\\textsc{nfw}$ chosen to fit the numerical profile in the domain $10^{-2} < r\/R_{200} < 1$. The profiles are plotted at 5 scale factors, $a\/a_f = \\{1, 2, 3, 4, 5\\}$, for which the best-fit NFW concentration parameters are $c_\\textsc{nfw} = \\{4.6, 8.5, 12.7, 17.0, 21.2\\}$, respectively. The curves for $a\/a_f = 1, 2, 3$, and 4 have been shifted downward by 4, 3, 2, and 1 order(s) of magnitude, respectively, for visual clarity. The radius $R_{200}$ and background density $\\rho_b$, used to normalize the radius on the x-axis and density on the y-axis, are evaluated at each time-slice, separately.\n    As expected, these results reproduce those presented previously in \\citet{AAS03}, \\citet{Shapiro04}, and \\citet{Shapiro06}.  }\n    \\label{fig:NFW}\n\\end{figure}\n\n\\subsubsection{SFDM-TF halo formation from  NFW-producing infall rate: (SFDM-TF dynamics) + (CDM-like initial conditions)}\n\\label{sec:TFNFWsims}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{NumericalProfiles_Rtf=1kpc.pdf}\n    \\includegraphics[width=\\columnwidth]{NumericalProfiles_Rtf=10pc.pdf}\n    \\caption{\\textbf{The fluid approximation simulation results for SFDM-TF halo formation from spherical infall and collapse of a cosmological perturbation.} \n    Each panel shows profiles of hydrodynamical variables at select times: \n    from top to bottom, (non-dimensionalized) \n    mass density, bulk velocity, ideal gas pressure (i.e. velocity-dispersion ``pressure''), \n    and ``temperature'' (i.e. velocity dispersion) \n    vs. radius, for two example SFDM-TF haloes, from the same NFW-producing initial conditions at $a = a_\\text{eq}$ as for the CDM simulations in Fig. \\ref{fig:NFW}, with\n    $\\{M_\\infty, a_f~(z_f), R_\\text{TF}\\} = \\{5.7\\times10^{14} \\text{ M}_\\odot, 0.87~(0.15), 1 \\text{ kpc}\\}$ (left) and $\\{1.6\\times10^{9} \\text{ M}_\\odot, 0.24~(3.2), 10 \\text{ pc}\\}$ (right). \n    The thick black lines show the profiles at the ``formation'' scale factor $a = a_f$. The thin grey lines show the profiles at earlier times in the simulation, labeled by $a\/a_f$.\n    Here, $\\rho_b$ is the time-dependent background density, \n    while $\\{M_{200}, R_{200}\\} \\simeq \\{7\\times10^{13} \\text{ M}_\\odot, 740 \\text{ kpc}\\}$ (left) and $\\{2\\times10^{8} \\text{ M}_\\odot, 3 \\text{ kpc}\\}$ (right) are the mass and radius of the halo at $a=a_f$. \n    The discontinuity near $R_{200}$ is the accretion shock, which sets the virial radius of the halo. \n    The thin red lines are profiles for a corresponding CDM run using the same initial conditions. The SFDM-TF density profile closely tracks the NFW-like CDM profile at large radii, but more closely resembles a power-law, $\\rho \\propto r^{-12\/7}$, \n    at smaller radii (blue lines), until, at still\n    smaller radii, $r \\lesssim R_\\text{TF}$, flattens to the shape of an SI-supported polytropic core (red long-dashed lines).\n    For SFDM-TF at early times, the centrally-peaked initial density profile was flattened, out to $\\sim R_\\text{TF}$, by an outward-propagating, SI-pressure-driven sound wave. Circles mark the shell that was located at $r=R_\\text{TF}$ at the time the sound wave reached that radius, while triangles mark the shell that was located at $r=R_\\text{TF}$ at the initial time $a_i$ -- these are indicators of where the profile deviates from CDM.\n    The long-dashed blue line in the $P_{\\!\\sigma}$ panel shows the profile for $P_\\text{SI} = g\\rho^2\/2m^2$ at $a=a_f$ with the same normalization.}\n    \\label{fig:NFWSI}\n\\end{figure*}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{EarlySmoothing_Rtf=10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{EarlySmoothing_Rtf=1kpc.pdf}\n    \\caption{\\textbf{The beginning of spherical collapse from NFW-producing initial conditions: early Jeans-filtering by SI-pressure}. Simulation results at several early time-slices, soon after the initial scale factor $a_i$, for two cases with the same\n    initial perturbation, with $M_\\infty=1.6\\times10^{9} \\text{ M}_\\odot$ and $a_f = 0.24$, but with\n    different SI strengths for SFDM-TF: \\textbf{(a) (top panel)} $R_\\text{TF} = 10$ pc,  and \\textbf{(b) (bottom panel)} $R_\\text{TF} = 1$ kpc. Corresponding CDM runs are shown for comparison (dashed lines). The density profiles are normalized by the time-dependent background density. The figures illustrate the initial SI sound wave that propagates outward and flattens the density profile out to $R_\\text{TF}$. In the case of $R_\\text{TF} = 1$ kpc, this sound wave passes over the entire profile (which encompasses the entire computational domain, extending to $M_\\infty$), and the perturbation is prevented from growing (i.e. it is Jeans-filtered) within the age of the universe. Circles mark the shell that was located at $r=R_\\text{TF}$ at the time the sound wave reached that radius, while triangles mark the shell that was located at $r=R_\\text{TF}$ at the initial time $a_i$.}\n    \\label{fig:SIsmoothing}\n\\end{figure}\n\nIn Figs.~\\ref{fig:NFWSI} and \\ref{fig:SIsmoothing}, we show illustrative\nresults of our 1D numerical hydrodynamics simulations of SFDM-TF halo formation from \nthe same initial conditions as used in the\nprevious section for the CDM halo formation\nshown in Fig.~\\ref{fig:NFW}.\nIn this case, however, our fluid equations are now\nthe modified ones for SFDM-TF, which\ninclude the dynamical\neffects of both SI pressure $P_\\text{SI}$ and\nquantum pressure, where the latter\nis accounted for by the ideal gas pressure $P_{\\!\\sigma}$ \n(i.e. CDM-like ``velocity-dispersion'' \npressure), which represents the dynamical \neffect on large scales that results from inhomogeneity on the tiny, unresolved scale of the \nde~Broglie wavelength. Two cases are shown\nin Fig.~\\ref{fig:NFWSI}, involving different\nSI strengths, parameterized by their \nadopted $R_\\text{TF}$-values, with initial perturbations chosen to make haloes of different mass at $a=a_f$ in each\ncase.  Profiles of hydrodynamical variables vs. proper\nradius are plotted at selected times, including \nmass density, bulk velocity, ideal gas pressure \n(i.e. velocity-dispersion ``pressure''), \nand ``temperature'' (i.e. velocity-dispersion), \nfor two example SFDM-TF haloes, formed\nfrom the same NFW-producing initial conditions \nat $a = a_\\text{eq}$ as for the \nCDM simulations in Fig. \\ref{fig:NFW}, with\n$\\{M_\\infty, a_f~(z_f), R_\\text{TF}\\} = \\{5.7\\times10^{14} \\text{ M}_\\odot, 0.87~(0.15), 1 \\text{ kpc}\\}$ (left) \nand $\\{1.6\\times10^{9} \\text{ M}_\\odot, 0.24~(3.2), 10 \\text{ pc}\\}$ (right).  For comparison, we also plot\nthe end results for CDM, from the same initial conditions. \n\nThe evolution is similar in both cases, with some elements common to the non-cosmological infall and\ncollapse we simulated in \\citetalias{paper1}, \nfrom static, Jeans-unstable initial conditions, while\nothers are uniquely cosmological, as follows:\n\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item Growth of the initial perturbation slows its\n    expansion relative to the unperturbed Hubble flow\n    far from the center, so the perturbed density \n    in units of the time-dependent mean background density increases with time, even as the latter drops.\n    \n    \n    \\item Close to the center, a sound wave driven by\n    the SI pressure gradient quickly \n    propagates outward until it reaches the physical\n    radius $R_\\text{TF}$, one sound-crossing-time after\n    the initial time at $a_i$, travelling \n    at the SI sound speed given by \n    \\begin{equation}\n        c^2_{s,\\text{SI}} = \\frac{\\partial P_\\text{SI}}{\\partial \\rho} = \\frac{g\\rho}{m^2}\n        \\label{eq:SI_soundspeed}\n    \\end {equation}\n    evaluated at a density roughly equal to the initial\n    background density. This sound wave\n    smooths out the density as it goes,\n    leaving it nearly uniform within this radius,\n    and remaining uniform as it expands, i.e. it\n    expands homologously. We will say more about this sound-wave-smoothed central region below.\n    \n    \\item Thereafter, all mass shells continue expanding until\n    -- at later times for shells that started at\n    larger radii -- they reach maximum expansion, halt \n    and collapse inward, supersonically.\n    \n    \\item As the density of these infalling shells\n    rises, adiabatic compression\n    increases their ideal gas pressure \n    more slowly \\big(since $P_{\\!\\sigma} \\propto \\rho^{5\/3}$\\big) than it does their SI pressure \\big(for which $P_\\text{SI} \\propto\\rho^2$\\big). The \n    latter eventually becomes large enough in the center\n    to halt the infall of the innermost\n    shells and provide their hydrostatic support\n    within a radius $\\simeq R_\\text{TF}$, with a flattened\n    profile like that of an ($n=1$)-polytrope. This\n    was also seen in the non-cosmological results of\n    \\citetalias{paper1}.\n    \n    \\item At this point, by $a\/a_f \\gtrsim 0.4$, \n    the incoming mass shell just outside this polytropic core\n    -- just about to slam into it -- is halted by the \n    formation of a strong accretion shock that converts\n    its kinetic energy of infall into ``thermal energy'' (i.e. energy due to dispersion).\n    This boosts its ideal gas pressure to be\n    comparable to the SI pressure within the core\n    and extends the hydrostatic inner profile of the\n    latter to larger radii, bounded by the shock, \n    supported outside the core by post-shock\n    $P_{\\!\\sigma}$, instead of $P_\\text{SI}$. \n    This, too, is similar to the non-cosmological case\n    in \\citetalias{paper1}.\n    \n    \\item By $a\/a_f = 1$, the accretion shock has moved \n    outward in mass, just beyond the radius $R_{200}$, \n    to encompass a virialized halo, within which the\n    ``temperature'' profile associated with velocity dispersion\n    is approximately isothermal, except inside the polytrope,\n    where it is cold, because it remained interior to the \n    mass shell at which the shock formed.  However, matter interior to that shock is everywhere hydrostatic, supported\n    inside the polytropic core by SI pressure and outside\n    it by post-shock ideal gas pressure.\n    \n    \\item The post-shock density profile, thereafter,\n    has three distinct regions in \n    radius:\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item a flattened, polytropic core inside $R_\\text{TF}$,\n        \\item an intermediate zone with power-law density profile, $\\rho \\propto r^{-12\/7}$, and \n        \\item an outer envelope which\n        follows the same NFW-profile as for CDM\n        from the same initial conditions, inside\n        the advancing accretion shock, which remains\n        just outside of the time-varying $R_{200}$.\n    \\end{enumerate}\n    Regions 7(a) and (b) were also present in the non-cosmological results in\n    \\citetalias{paper1}, while 7(c) is unique to the\n    cosmological case. \n    \n    \\item Outside the shock, the density follows\n    the pre-shock, ballistic infall profile of a cold, pressure-free gas, just like the CDM case.\n\\end{enumerate}\n\n\nAs described in \\S\\ref{sec:staticICPaperI},  \\citetalias{paper1} found a generic core-envelope\nstructure for the virialized objects formed\nby the nonlinear evolution of the Jeans \ninstability in SFDM-TF from non-cosmological \n(i.e. static) initial conditions.  As in\nthe cosmological case described above, an accretion\nshock bounded the virialized region. However,\nunlike the cosmological case above, the entire envelope\nbetween the polytropic core and the shock\nhad the same power-law density profile $\\rho \\propto r^{-12\/7}$ as reported above for the intermediate \nzone 7(b), and there was no NFW-like zone 7(c).\nThis reflected the fact that, in this \\textit{non}-cosmological\ncase, the Jeans-instability problem involved a nearly uniform initial density, which gravitational instability\nturned into an infall density profile \n(outside the accretion shock) with\nthat power-law.  The condition of hydrostatic\nequilibrium inside the post-shock region then combined with\nthe constant (shock-velocity-independent) factor\nof density jump at the shock to make the\nprofile of \\textit{post-shock} density the same\nas that of the \\textit{pre-shock} infall.  \nIndeed, for this reason, our simulation from\nthe same non-cosmological initial conditions but\nwith CDM dynamics, instead, \nproduced the exact same power-law\nprofile inside the shock, except with no SI-flattened \npolytropic core in the center.  In fact, the mass\ninside the core in the SFDM-TF case, with polytropic\nradius $R_\\text{TF}$, was the same as that \ninside that radius in the CDM-like power-law profile.\nSo, apparently, outside of $R_\\text{TF}$, where \nSFDM-TF and CDM dynamics was the same, it was the non-cosmological \\textit{initial} conditions\nof the SFDM-TF and CDM simulations that\nresulted in making their infall \nprofiles $\\rho \\propto r^{-12\/7}$, instead of\nthe cosmological infall profile derived from\nthe halo mass accretion rate in CDM N-body \nsimulations, which makes the post-shock profile NFW-like.  \n\nFrom that description, we might reasonably have expected\nhalo formation from\n\\textit{cosmological} initial conditions to be\nsimilar, with the same polytropic\ncore, surrounded again by a CDM-like envelope.\nIn particular, we might\nhave expected this CDM-like envelope\nto be \\textit{NFW-like}, as it is in our CDM simulation \n(without SI) for this same cosmological case.\nThen why did we find, as reported \nin item (7), that while this is true for \nregions (a) and (c), there is an intermediate zone (b) \nin which the density profile follows \nthe same power-law profile \\big($\\rho \\propto r^{-12\/7}$\\big)\nas in the \\textit{non-cosmological} simulations of \\citetalias{paper1}, instead?   \nThe answer to this question encapsulates\nthe fundamental distinction between static and\ncosmological Jeans instability in SFDM-TF and\nis at the heart of our cosmological story-line here\nin Paper II. \n\nWhat distinguishes a \\textit{cosmological} from a \\textit{non-cosmological} density perturbation is their different initial velocity perturbations.  \nWithin the Newtonian approximation, our \nspherically-symmetric Jeans instability problem\nis transformed from a non-cosmological one into a cosmological one entirely by the presence of perturbed Hubble expansion in the latter that replaces the static initial condition of the former.  This small change has\nprofound consequences, however, because the SI Jeans \nlength is fixed in physical coordinates by the constant\nvalue of $R_\\text{TF}$.  For CDM in an Einstein-de Sitter\nuniverse, a spherical perturbation that is overdense\nwill expand more slowly than unperturbed Hubble expansion, until it eventually\nreaches a maximum expansion radius and recollapses.\nThis will be true in SFDM-TF, as well, if it is overdense\nacross a region much larger than $R_\\text{TF}$.  \nBut in that case,\nmatter close to the center of the perturbation,\nwhich is initially expanding, will be affected by the\npresence of SI pressure, too, which responds to density\ninhomogeneity by driving a sound wave outward from\nthe center, as far as $R_\\text{TF}$, which smooths out the density perturbation as it goes.  \nThis is what we found in item (2) above.\nThe effect of this early SI-smoothing imprints the\nevolution to come for the mass shells inside the\nsphere smoothed-out by the sound wave, even though they\ncontinue to expand homologously, thereafter, \nwhich carries them far outside the physical radius \noriginally overtaken by the sound wave.\n\n\nIn Fig. \\ref{fig:SIsmoothing}, we illustrate this\nearly SI-sound-wave smoothing by\ntwo examples from the same NFW-producing\nCDM-like initial perturbation, with $M_\\infty=1.6\\times10^{9} \\text{ M}_\\odot$ and $a_f = 0.24$,\nbut with different SI strengths, by plotting density\nprofiles for a sequence of early time-slices of \nour SFDM-TF simulation, compared with\ntheir CDM counterparts. Let us focus first\non the top panel, Fig. \\ref{fig:SIsmoothing}(a),\nthe case for $R_\\text{TF} = 10$ pc. \nThe central density profile is quickly \nmade nearly uniform out to $R_\\text{TF}$, by a sound\nwave from the origin that reaches $R_\\text{TF}$ \nlong before that shell reaches maximum expansion.  Thereafter, this inner\nsphere evolves just like a top-hat perturbation.\nAs such, it continues to expand until it reaches maximum expansion, turns around, and recollapses.\nAt maximum expansion, the spherical region is \nmomentarily at rest, so it is at that\nmoment essentially the same as the initial perturbation\nof the \\textit{non-cosmological} problem in \\citetalias{paper1}!\nIts evolution, thereafter, must, therefore, be similar to\nwhat we found there.  It should yield \nthe same core-envelope structure inside the\nadvancing accretion shock as in \\citetalias{paper1}, with\npolytropic core of proper radius $R_\\text{TF}$\nand the same power-law density profile, $\\rho \\propto r^{-12\/7}$, outside the core, but only out through\nthe mass that was initially smoothed by the SI-pressure-driven sound wave.  As we can see in Fig. \\ref{fig:SIsmoothing}, the sound wave smooths out the density as it travels to $\\sim R_\\text{TF}$. Each time-slice in Fig. \\ref{fig:SIsmoothing}(a)\nis marked (by a filled circle) to identify the particular mass shell that was located at $R_\\text{TF}$ when the \nsound wave reached that radius.  The mass within that \nradius was basically the SI-Jeans mass of the background universe at that time, since the density \nperturbation was only linear then. \nThe time-varying radius so marked,\nwhich contains this fixed mass as it\nevolves, provides a rough estimate (i.e. lower-limit) \nfor the outer radius of the intermediate zone 7(b)\nwithin which we expect the final \ndensity profile to follow the power-law \n$\\rho \\propto r^{-12\/7}$, \nonce this mass is inside the shock.  We also\nmark this radius on the density\nprofiles in the top panels of Fig. \\ref{fig:NFWSI},\nfor all the time-slices, to confirm this.  \n\nThe transition from power-law to NFW-like post-shock\ndensity profiles inside the shock extends beyond the \nfilled circles, encompassing a somewhat larger mass.\n(The filled circles in \nFig. \\ref{fig:NFWSI} enclose masses of $5\\times10^{11} \\text{ M}_\\odot$ (left) and $5\\times10^5 \\text{ M}_\\odot$ (right), for $R_\\text{TF} = 1$ kpc and $10$ pc, respectively, as compared with their Jeans masses at the initial time of $2\\times10^{13} \\text{ M}_\\odot$ and $2\\times10^7 \\text{ M}_\\odot$.)\nWe can understand why and bound this transition from\nabove by a second estimate, as follows, which we \nmark with filled triangles on the density profiles\nin both Figs. \\ref{fig:NFWSI} and \\ref{fig:SIsmoothing}(a).\nAs we shall see, in \\S\\ref{sec:LinPertsTransferFn},\nthe cosmological SI Jeans mass at any scale factor \nis the mass inside a sphere of fixed physical radius \n$R_\\text{TF}$, containing the mean density at that time, \na mass which drops with scale factor as $a^{-3}$\n(see equation \\ref{eq:Jeansmass}).  As\na result, even the small, finite time for\nthe sound wave to reach $R_\\text{TF}$ is enough\ntime for the Jeans mass to drop significantly from its value at the initial scale factor $a_i$.\nTherefore, the effects of SI Jeans-filtering extend to\nencompass a mass which is between this \n(larger) \\textit{initial} Jeans mass \n(marked by filled triangles)\nand the (smaller) Jeans mass when the sound\nwave reached $R_\\text{TF}$ (marked by filled\ncircles). \n\nNow we are ready to focus on the second example of\nSI-smoothing, plotted in the bottom panel\nFig. \\ref{fig:SIsmoothing}(b). This case differs from\nthat in the top panel Fig. \\ref{fig:SIsmoothing}(a)\nonly by having a much stronger SI, corresponding to\n$R_\\text{TF} = 1$ kpc, instead of $10$ pc.  As with\nthe latter case, the initial SI sound wave propagates outward and flattens the density profile out to $R_\\text{TF}$. In the case of $R_\\text{TF} = 1$ kpc, \nhowever, this sound wave passes over the entire computational domain, which extends to $M_\\infty$ \n(i.e. the radial position of the outermost shell containing $M_\\infty$ is $\\ll R_\\text{TF}$), \nand the perturbation is, thereby, prevented from growing \nto form the collapsed virialized object it would\nhave formed in the CDM case.\n\nThis reveals the essential factor that distinguishes\nthe outcomes of non-cosmological vs. cosmological initial conditions for the formation of an SFDM-TF halo with \na given mass at a given epoch. The former yields a\nvirialized halo in the post-shock region, with a\npolytropic core of radius $R_\\text{TF}$ surrounded by\na CDM-like density profile that extends to the\naccretion shock radius at that epoch.  The mass encompassed\nby that shock grows with time just by adding new\nmass shells to the outside, at the\ngravitational free-fall times defined by the mean\nmass densities initially interior to each mass shell.\nThe mass in the core is fixed by\nextrapolating the CDM-like profile inward from $R_\\text{TF}$\nand integrating out to that radius.\nSI-smoothing in this case is confined to the region within\na fixed physical radius $R_\\text{TF}$, \nand to the mass which ends up inside this\nradius at the final epoch, while the dynamics outside this\nradius -- and the final halo structure there -- are CDM-like.\nFor the \\textit{cosmological} case, however, \nthe mass that is SI-smoothed is the much larger \namount that was inside that same sphere of physical radius $R_\\text{TF}$ at a time\nclose to the \\textit{initial} time, at the \nmean matter density of the universe at that time,\nbefore expansion carried it well beyond\n$R_\\text{TF}$, enroute to its turn-around radius.   \nSince SI-smoothing quickly flattens the initial perturbation\nout to $R_\\text{TF}$,\nthe turn-around and recollapse of any shell interior\nto the one which bounds the smoothed mass\nwill be delayed relative to their timing without\nSI-smoothing. That bounding sphere and every shell inside it\nwill expand, turn-around and recollapse like\na uniform, top-hat perturbation with the same mean density.\nSince SI-smoothing out to $R_\\text{TF}$\nmust conserve mass, the average overdensity inside\nthe smoothed sphere must be about\nthe same as in the unsmoothed perturbation, \nso the bounding sphere and its contents\nturn-around and recollapse just like the\nbounding sphere in the case without SI-smoothing.\nAs long as the latter is destined to collapse,\nthat means SI-smoothing does not \\textit{prevent}\nthe smoothed mass from collapsing, \nit only \\textit{delays} the collapse of interior\nshells to the time at which the bounding sphere, itself,\nwould have, in the absence of SI-smoothing. \n\n\nThis assumes, of course, that the average overdensity\ninside the bounding sphere in the initial\nperturbation was positive,\nas it would be for the self-similar perturbations described above in \\S\\ref{sec:selfsimilar}. \nThe CDM-like perturbation that leads to an NFW-like\npost-shock profile, however, is a \\textit{non}-self-similar one,\nin which the overdensity is ``compensated'' by an underdensity at large radii, so that\nthe average overdensity\ninside the sphere which encompasses mass $M_{\\infty}$\nis zero.   Spherical shells outside that mass will,\ntherefore, remain unperturbed and cannot collapse.\nIn that case, if the mass affected by \nSI-smoothing exceeds $M_{\\infty}$, as it does\nin Fig. \\ref{fig:SIsmoothing}(b), then the \nbounding sphere described above is actually \n\\textit{prevented} from collapsing, altogether.  \nFor those cases that \\textit{do} collapse, however,\nour simulations above tell us that\nthe virialized object that forms inside the accretion shock\nwill have an inner portion that \nresembles that in \\citetalias{paper1} from  \\textit{non-cosmological} (static) initial conditions,\nencompassing the initial SI-smoothed mass, \nwith the core-envelope structure reported there\n\\big(polytropic core of radius $R_\\text{TF}$, \nsurrounded by a power-law density profile, $\\rho \\propto r^{-12\/7}$\\big), but beyond the radius \ncontaining this initial SI-smoothed mass, \nit will transition to the NFW-like profile. \n\nThis description above\nassumes we started from CDM-like initial conditions.\nWe made this assumption as the first step in\nunderstanding the fully nonlinear\ndynamics of SFDM-TF halo formation.  \nThat was a reasonable first step for a range\nof perturbation mass scales, in view of our\ndiscussion in \\citetalias{paper1} of the fact\nthat, at scales beyond $R_\\text{TF}$, the dynamics\nof SFDM-TF is CDM-like.  As such, we were able to\nconfirm that, beyond the inner mass smoothed by SI,\nSFDM-TF halo profiles simulated with nonlinear\ndynamics will follow the same NFW-like profile\nexpected for the CDM model.  However, we\nalso learned that this SI-smoothed inner\nmass can be as large as the mass inside\nthe fixed radius $R_\\text{TF}$ at early\ntimes, when the universe was much denser,\nand that there is an intermediate zone of\ndensity profile outside the polytropic core\nof radius $R_\\text{TF}$ in the final object, which\ndiffers from the NFW profile.  Such a feature\nmight be a signature of SFDM-TF that distinguishes\nit observationally from CDM.  However, it remains to\nbe seen from 3D-cosmological SFDM-TF\nsimulations, \nfrom initial conditions \nthat are self-consistently based upon\nlinear perturbation theory for SFDM-TF,\nif this intermediate zone will survive, or else\nrelax to the NFW-like profile over time.\n\n{ In addition, such 3D simulations are required to determine what MAH results for halo growth from SFDM initial conditions and how it differs from that for CDM.\nThe results we\npresent here by assuming the smoothed accretion rate derived from N-body simulations of the CDM model are likely to be robust on the smallest and largest scales with respect to replacing the initial conditions with the proper MAH of SFDM. The shells at large radii are likely to follow CDM-like trajectories even with correct initial conditions, while the shells at small radii are quickly smoothed by SI Jeans filtering and so should also follow the correct trajectories.}\n\n{ As we shall see below, from linear perturbation analysis, SFDM initial conditions differ from those of CDM by suppressing power on small scales, due to Jeans filtering.   Halos can form, however, on all mass scales, even those below that filter scale, although with lower abundance for those smaller masses. Filtering may smooth the initial density perturbation responsible for building a halo by spherical infall, out to a larger radius than in the case with CDM-like initial conditions, resulting in a more nearly uniform collapse.  In that case, we might expect the nonlinear collapse to produce a post-shock density profile outside the polytropic core, which follows the power-law $\\rho \\propto r^{-12\/7}$, over a larger range of radii than in our results above.  } \n\nIn the meantime,  we need linear \nperturbation theory to determine the statistical \nmeasures of structure formation, e.g. how many haloes of a given mass are likely\nto form by a given epoch, in order to constrain\nthe model parameter $R_\\text{TF}$ against observations.  \nWe consider that in the remainder of this\npaper.\n\n\n\n\\section{Linear perturbation theory and the transfer function for SFDM-TF}\n\\label{sec:LinPertsTransferFn}\n\n\\subsection{Qualitative overview}\n\\label{sec:LinPertOverview}\n\nThe growth of density perturbations in a universe with complex SFDM in the \npresence of a repulsive SI differs from both that in the \nCDM model and that in the Fuzzy Dark Matter (FDM) limit of SFDM.    \nIn all models, the assumed, primordial initial condition involves \nsmall-amplitude, 3D, Gaussian-random density fluctuations with zero mean, \ncharacterized by a scale-invariant\\footnote{More generally, \n$P(k,t_i) \\propto k^{n_s}$ where $n_s$ is the primordial spectral index of scalar perturbations, and the value we adopt here is\n$n_s = 1$, the Harrison-Peebles-Zeldovich spectrum, which is consistent with current observational constraints.} power spectrum  \n\\begin{equation}\n    P(k,t_i) = |\\delta_k(t_i)|^2 \\propto k\n\\end{equation}\nwhere the linear overdensity $\\delta_k (t)$ of a Fourier mode of \nwavenumber $k$ at a cosmic time $t$ is related to its \namplitude $\\delta_k (t_i)$ at initial time $t_i$, according to\n\\begin{equation}\\label{eq:dP}\n    \\delta_k(t) = \\mathcal{T}_k(t,t_i)\\delta_k(t_i)\n\\end{equation}\nin terms of the transfer function $\\mathcal{T}_k(t,t_i)$.\nDifferences in the structure formation histories of these models arise while \nthey are still in their linear amplitude phase, as characterized by different \ntransfer functions for each model.\nThey have one very important thing in common, however.  \nWhen fluctuation modes of different proper wavelength $\\lambda$ (or, equivalently, comoving $k=2\\pi a \/ \\lambda$) \nfirst re-enter the horizon \\big(i.e. near the time\n$t=t_\\textsc{h}$ at which their proper wavelength equals the Hubble diameter, \ni.e. $\\lambda(t) = 2c\/H(t)$, in which case\ncomoving $k = k_\\text{H} \\equiv \\frac{\\pi}{c}a_\\textsc{h}H(a_\\textsc{h})$\\big), they do so with the same, wavelength-independent, fractional overdensity, $\\delta_k(t_\\textsc{h}) \\ll 1$, for all models. \nAlthough modes of different $k$ enter the horizon at different times \n$t_\\textsc{h}$ (later for smaller $k$), and it is even possible that \nsome modes of the same $k$ in the different models enter the horizon at different times, \nthe value of $\\delta_k(t_\\textsc{h})$ in all cases is the same for all values of $k$ and \nall values of $t_\\textsc{h}$. \n\nThe value of $t_\\textsc{h}$ for a given wavenumber depends only on the time-dependence \nof the Hubble parameter, so $t_\\textsc{h}(k)$ is the same in all models only if $H(t)$ \nis the same at all times in different models.  We expect this to be the case for \nCDM and SFDM models, in general, in our observed Universe, for modes that re-enter the \nhorizon during the radiation-dominated (RD) and matter-dominated (MD) eras of interest to us here, since all models are required \nby observational constraints to share a common expansion history from the time the wavenumbers\nassociated with galaxy formation began to enter the horizon during the RD era through the present.\nThis condition is not met for all of the models at all times. For modes that enter before \nthe RD era -- e.g. for complex SFDM, with or without SI, there is an early stiff phase ($w=1$), \nprior to the RD era, in which the scalar field dominates.  \nHence, even though the Hubble parameter during the RD and MD eras, for example, is basically the same at a given scale factor for all models, it differs during the SFDM \\textit{stiff} phase at earlier times, in that case, when much \nsmaller scales entered the horizon. In fact,\neven \\textit{during} the RD epoch, but well before the epoch of matter-radiation equality, \n$a_\\text{eq}$ (for both complex and real field\ncases), \nthe Hubble parameter can be slightly enhanced relative to its value in CDM and FDM, \nby a small boost to the total energy density contributed by SFDM in the presence of SI,\nwhen its EOS is radiation-like \\big($\\langle w \\rangle=1\/3$\\big), between its stiff- and matter-like phases of evolution, \nbut this small enhancement is gone well before $a_\\text{eq}$.\\footnote{When we\ndescribe the EOS parameter $w$ of the \nscalar field during its rapid-oscillation phase, \nin which the value averaged over oscillation\nperiods, $\\langle w \\rangle$, evolves\nfrom $\\langle w \\rangle = 1\/3$ to \n$\\langle w \\rangle = 0$, we will, henceforth,\ngenerally just write\n$w$ to refer to this averaged quantity and\nomit the brackets.}\n                           \nWhat happens to that amplitude next, after a mode is already inside the horizon, depends upon the microscopic nature of the dark matter and other components that contribute to the total mass-energy of the Universe. It also depends on whether the background Universe is RD or MD during the phase where the galaxy- and galaxy-cluster-scale fluctuations are still linear.\n\nIn the standard CDM model, dark matter is a nonrelativistic gas of cold, collisionless particles, so the ratio $w$ of its pressure to its energy-density is well-approximated by $w = 0$.  In that case, its fluctuation growth rate after horizon entry depends upon whether the Universe is in the RD or MD era.  For the scales of interest for structure formation, spanning the range of galaxy- and cluster-mass objects, horizon entry is during the RD era. At that time, \nradiation dominates the gravitational potential fluctuations that drive perturbation growth in CDM, but fluctuations\nin the radiation component do not amplify during this phase, because a relativistic fluid with $w = 1\/3$ has a speed of sound close to $c\/\\sqrt{3}$,  and density inhomogeneities are thus smoothed out by pressure disturbances that can travel a distance comparable to the Hubble radius within the age of the universe at that time. As a result, gravitational fluctuation growth in the CDM component is driven only by its own fluctuation as a subdominant contribution to the total energy-density. This yields only a relatively small amount of growth between horizon entry at scale factor $a_\\textsc{h}$, and matter-radiation equality at $a_\\text{eq}$, which is a logarithmic function of those scale factors. \n\nWe adopt cosmological parameters from \\citet{planck18} as given in Table~\\ref{tab:params}.\n\\begin{table}\n    \\centering\n    \\begin{tabular}{c|c}\n        \\hline\\hline\n        parameter & value \\\\\n        \\hline\n        $h$ & 0.67 \\\\\n        $\\Omega_\\text{dm} h^2$ & 0.12 \\\\\n        $\\Omega_m h^2$ & 0.14 \\\\\n        $\\Omega_\\Lambda h^2$ & 0.31 \\\\\n        $\\Omega_\\text{rad} h^2$ & $4\\times10^{-5}$ \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{\\textbf{Cosmological parameters} from \\citet{planck18}, which we adopt for our linear perturbation analysis in \\S\\ref{sec:LinPertsTransferFn}.}\n    \\label{tab:params}\n\\end{table}\nThe scale factor of matter-radiation equality is given by\n\\begin{equation}\n    a_\\text{eq} \\simeq 2.9 \\times 10^{-4}\n\\end{equation}\nFor a comoving wavelength $\\lambda_0$ that enters the Hubble volume during the RD epoch,\n\\begin{equation}\n    a_\\textsc{h} \\simeq 10^{-6} \\left( \\frac{\\lambda_0}{\\text{1 Mpc}} \\right)\n\\end{equation}\nAnd the unperturbed mass of nonrelativistic dark matter contained in a sphere of proper radius $\\lambda(t)\/2$, where $\\lambda(t)\/a(t) = \\lambda_0\/a_0$, is given at all times by\n\\begin{equation}\n    M = \\frac{4\\pi}{3} \\rho_{\\text{dm},0} \\left( \\frac{\\lambda_0}{2} \\right)^3 \\simeq 1.45 \\times 10^{11} \\text{ M}_\\odot \\big(\\Omega_\\text{dm} h^2\\big) \\left( \\frac{\\lambda_0}{\\text{1 Mpc}} \\right)^3\n\\end{equation}\nwhere the mean dark matter density at present is given by\n\\begin{equation}\n    \\rho_{\\text{dm},0} \\equiv \\Omega_\\text{dm}\\frac{3 H_0^2}{8\\pi G} \\simeq 2.8 \\times 10^{11}~\\frac{\\text{M}_\\odot}{\\text{Mpc}^{3}} \\big(\\Omega_\\text{dm} h^2\\big)\n\\end{equation}\nWe can also write the scale factor when a given mass fills the Hubble volume, according to\n\\begin{equation}\n    a_\\textsc{h} \\simeq 1.9 \\times 10^{-6} \\big(\\Omega_\\text{dm} h^2\\big)^{-1\/3} \\left( \\frac{M}{10^{12} \\text{ M}_\\odot} \\right)^{1\/3}\n\\end{equation}\nThe comoving wavelength that entered the horizon at $a_\\text{eq}$ is given by\n\\begin{equation}\n    \\lambda_\\text{0,eq} \\simeq 290 \\text{ Mpc}\n\\end{equation}\nand the mass\\footnote{If we had chosen to define the mass associated with a horizon-filling mode as the mass inside a sphere of comoving radius \n$\\lambda_0$, instead of  $\\lambda_0\/2$ as we did above, then the mass associated with the mode that enters at $a_\\text{eq}$ would be 8 times larger than this, or $3 \\times 10^{18} \\text{ M}_\\odot$.  \nAlternatively, if we had, instead, defined the mass associated with a mode as just $\\rho_{\\text{dm},0} \\lambda_0^3$,  rather than $4\\pi\/3  \\rho_{\\text{dm},0} (\\lambda_0\/2)^3$, as above, then the mass above, which is evaluated based upon the latter, should be multiplied by $6\/\\pi \\simeq 1.91$, instead, which would make $M_\\text{eq} \\simeq 8 \\times 10^{17} \\text{ M}_\\odot$.   These are just matters of definition, of course, as the horizon-filling wavelength and the unperturbed mass associated with a given wavelength mode are not quantities of precise meaning, but we should try to compare different mode-associated masses using the same choice of definition.} that filled the Hubble volume at $a_\\text{eq}$ is, therefore, given by\n\\begin{equation}\n    M_\\text{H,eq} \\simeq 3.5 \\times 10^{18} \\text{ M}_\\odot \\big(\\Omega_\\text{dm} h^2 \\big)\n\\end{equation}\nwhich, for $\\Omega_\\text{dm} h^2=0.12$, yields $M_\\text{H,eq} \\simeq 4 \\times 10^{17} \\text{ M}_\\odot$.\n\nFor CDM, modes of mass scale $M$ that enter the horizon during RD experience only logarithmic growth between $a_\\textsc{h}$ and $a_\\text{eq}$, given by (see equation~\\ref{eq:deltaCDM})\n\\begin{equation}\n    \\frac{\\delta_\\text{eq}}{\\delta_\\textsc{h}} \\approx 15\\ln \\left(\\frac{a_\\text{eq}}{a_\\textsc{h}}\\right)\n    \\label{logarithmicgrowth}\n\\end{equation}\nwhere $\\delta_\\text{eq} \\equiv \\delta_k (a_\\text{eq})$ and $\\delta_\\textsc{h} \\equiv \\delta_k (a_\\textsc{h})$.\nHenceforth, we will omit the subscript $k$ when referring to the Fourier mode overdensity $\\delta$.\nBy $a_\\text{eq}$, all modes of interest in forming galaxies and clusters had already entered the horizon, so they all experience the same factor of growth between $a_\\text{eq}$ and any later $a$ (i.e. for early times during the MD era, growth is the same as in an Einstein-de Sitter universe, in which case $\\delta \\propto a$).\n\nIt is convenient to describe the growth of density fluctuations in alternative dark matter models like SFDM in terms of their growth relative to that in CDM. \nFor a range of wavenumbers, the growth of density fluctuations in SFDM with SI in the \nTF regime, for modes of interest to us in galaxy and large-scale structure formation, is the same \nafter horizon crossing as for CDM. This is the case\nfor modes of small enough wavenumber, as we shall see. \n\nThere are two effects, however, that distinguish the two models.   \nCDM behaves as a pressure-free, matter-like gas (with $w=0$) at all times, both during the RD era when it is a subdominant component of the total mass-energy density of the universe and during the MD era when it dominates that energy-density.    \nSFDM differs in that regard.   \nSFDM with repulsive SI is only nonrelativistic and matter-like (with $w = 0$) at late times, after it transitions from being relativistic and  radiation-like (with $w  = 1\/3$).  \nAlthough this transition is not instantaneous, it is fast enough that we can characterize it as a sudden transition at scale factor $a_t$.   \nAs we discuss below, this transition scale factor $a_t$ is set by the value of the ratio $g\/m^2$.  \nAs pointed out by \\citet{Li14}, observations of the CMB anisotropy constrain the allowed values of this ratio by requiring that the EOS of the background universe must transition from RD to MD at close to the same value of  $a_\\text{eq}$ in both SFDM and CDM.   \nThis places an upper bound on the ratio, by requiring that $a_t \\leq a_\\text{eq}$.\nAs such, i.e. as long as $a_t \\leq a_\\text{eq}$, modes that enter the horizon before $a_t$, when SFDM-TF behaves like a radiation-like perfect fluid ($w=1\/3$),  \nwill not experience the logarithmic growth between $a_\\textsc{h}$ and $a_t$ that the same modes \nexperience in CDM.  In fact, these modes that enter \nthe horizon prior to $a_t$ should evolve\njust like those of the dominant radiation component, which oscillate\nlike sound waves of a relativistic gas\nwith sound speed $c\/\\sqrt{3}$, without changing their amplitude. If so, then these modes that entered the horizon before $a_t$ and before $a_\\text{eq}$\nwould be deprived of the interval of logarithmic\ngrowth which the same modes would have experienced\nafter horizon entry in CDM between $a_\\textsc{h}$ and\n$a_\\text{eq}$, according to equation (\\ref{logarithmicgrowth}).   \nIn the absence of CDM-like \ngrowth during this interval of scale factor, therefore, we might expect a net amount of growth for these modes over the entire RD era, \nincluding the time between $a_t$\nand $a_\\text{eq}$, equal to that which would result for CDM modes if we replaced $a_\\textsc{h}$ in their logarithmic growth factor in equation (\\ref{logarithmicgrowth}) by $a_t$.   \n\nThere is a second effect, however, which may also interfere with the growth of SFDM modes, if their proper wavelength is below the Jeans length that results from the back reaction of SI pressure forces that act to cancel their gravitational instability.    \nTo be more specific, we must estimate this Jeans length and its dependence on scale factor, and consider which of these two effects dominates, for which modes, and during which intervals of scale factor. \n \nLet us start by finding the transition scale factor $a_t$ and its dependence on SI strength $g$ and particle mass $m$. \nSuppose we identify the transition between $w = 1\/3$ and $w = 0$ in equation (\\ref{eq:averagew}), which goes through these values monotonically with increasing scale factor, by solving that equation for the scale factor at which $w=1\/9$,\nassuming that, for small $w$, it is sufficient to take the homogeneous background value of \n$\\langle|\\phi|^2\\rangle$ as given by\n$\\langle|\\phi|^2\\rangle\\simeq\\rho_\\text{dm}=\\rho_\\text{dm,0}a^{-3}$,\nwhich then yields  \n\\begin{equation}\n    a_{1\/9} = \\left(3\\rho_\\text{dm,0} \\frac{g}{m^2c^2}\\right)^{1\/3}\n\\end{equation}\nWe can then write the scale factor for any \n$w$\nas follows: \n\\begin{equation}\n    a_w = a_{1\/9}\\left (\\frac{1}{6w}-\\frac{1}{2}\\right)^{1\/3}\n\\end{equation}\nIn terms of $R_\\text{TF}$ in equation~(\\ref{eq:RTF}), we can write the transition scale factor as follows:\n\\begin{align}\n    a_{1\/9} &= \\left( \\frac{12G \\rho_\\text{dm,0} R_\\text{TF}^2}{\\pi c^2} \\right)^{1\/3} \\nonumber \\\\\n    &\\simeq 1.8\\times 10^{-5} \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{2\/3} \\left( \\frac{R_\\text{TF}}{\\text{1 kpc} } \\right)^{2\/3}\n\\end{align}\nIf we now take $a_t = a_{1\/9}$, then $a_t(R_\\text{TF})$ follows\n\\begin{equation}\n    a_t(R_\\text{TF}) \\propto R_\\text{TF}^{2\/3}\n    \\label{eq:atransitionRTF}\n\\end{equation}\nIn the radiation-dominated era, for which $H(a) \\propto a^{-2}$, the mass that fills the Hubble volume is\n\\begin{equation}\n    M_\\text{H,RD}(a) = \\frac{4\\pi}{3}\\rho_\\text{dm,0}a^{-3} \\Big(\\frac{c}{H(a)}\\Big)^3 \\propto a^3\n\\end{equation}\nwhich, evaluated at $a = a_t$ for a given value of $R_\\text{TF}$, is then\n\\begin{equation}\n    M_\\text{H,RD}(a_t(R_\\text{TF})) \\propto a_t^3(R_\\text{TF}) \\propto R_\\text{TF}^2\n\\end{equation}\nCombining this with the relationship above between $a_t$ and $R_\\text{TF}$, we can write the horizon-filling mass at the transition scale factor in terms of $R_\\text{TF}$ as\n\\begin{equation}\n    M_\\text{H,RD}(a_t(R_\\text{TF})) = M_\\text{H,eq}\\left(\\frac{R_\\text{TF}}{R_\\text{TF,eq}}\\right)^2\n\\end{equation}\nwhere $R_\\text{TF,eq}$ is the value that makes $a_t(R_\\text{TF,eq}) = a_\\text{eq}$.\n\nAs discussed in \\citetalias{paper1}, \nwhen $w\\simeq0$ \n(i.e. $a\\gtrsim a_t$), the Jeans length scale below which gravitational instability is opposed by the back-reaction of SI pressure is the proper wavelength\n$\\lambda_\\text{J,0} = 2 R_\\text{TF}$\n(where the subscript ``0'' refers to $w=0$ and\nthe subscript ``SI'' is removed for simplicity \nbut is implied).\nThe associated SFDM-TF Jeans mass scale is given by the dark matter mass inside a sphere of radius $R_\\text{TF}$, at any scale factor $a$, according to\n\\begin{align}\n    M_\\text{J,0}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} R_\\text{TF}^3 \\nonumber \\\\\n    &\\simeq 4.6\\times 10^{12} \\text{ M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left(\\frac{a}{10^{-3.5}}\\right)^{-3} \\left(\\frac{R_\\text{TF}}{\\text{1 kpc}}\\right)^3\n    \\label{eq:Jeansmass}\n\\end{align}\nFor future reference, it will be useful to define the scale factor $a_\\textsc{j,0}(M)$ at which the mass $M$ equals the Jeans mass, by inverting equation~(\\ref{eq:Jeansmass}) to write\n\\begin{align}\n    a_\\text{J,0}(M) &= R_\\text{TF} \\left( \\frac{ 4\\pi \\rho_\\text{dm,0} }{ 3M } \\right)^{1\/3} = \\frac{R_\\text{TF} k}{\\pi} \\nonumber \\\\\n    &\\simeq 5\\times10^{-4} \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/3}\\left( \\frac{M}{10^{12} \\text{ M}_\\odot} \\right)^{-1\/3} \\left(\\frac{R_\\text{TF}}{\\text{1 kpc}}\\right)\n    \\label{eq:aJeans0}\n\\end{align}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{JeansMass.pdf}\n    \\caption{\\textbf{Cosmic evolution of the SFDM-TF Jeans-filter scale: \\textit{The Incredible Shrinking Jeans Mass}.}  The Jeans mass $M_\\text{J}$ \\big(i.e. $M_\\text{J} = \\frac{4\\pi}{3} \\rho_\\text{dm,0} ( \\pi\/k_\\text{J})^3$\\big) associated with the critical wavenumber $k_\\text{J}$ for linear density fluctuation modes in SFDM-TF that marks the boundary between growth (for $M>M_\\text{J}$) and oscillation (for $M<M_\\text{J}$), vs. scale factor $a$, for \n    several illustrative values of the Thomas-Fermi radius $R_\\text{TF}$.  \n    Curve going from lower left to upper right is $M_\\text{H}(a)$, the mass that fills the horizon \n    (i.e. the Hubble volume, of radius $c\/H(a)$) at each scale factor.  The diagonal curves from upper left to lower right represent the curves of $M_\\text{J}(a)$ for different values of the $R_\\text{TF}$  (1000, 100, 10, and 1 pc, in order from top to bottom).  The dashed vertical lines and the discontinuities on each of these curves for different values of $R_\\text{TF}$ locates the value of the transition scale factor $a_t$ ([1.8, 8.4, 39, 180] $\\times 10^{-7}$, in order from left to right)\n    for the SFDM EOS, to the\n    left of which $w=1\/3$, while to the right $w=0$.}\n    \\label{fig:JeansMass}\n\\end{figure}\n\nWe are now in a position to compare this Jeans mass scale $M_\\text{J,0}$ given above as a function of scale factor, for a given value of  $R_\\text{TF}$, with the horizon-filling mass at each scale factor, to show the range of fluctuation mass-scales that are free of Jeans filtering from the moment those fluctuations re-enter the horizon during the RD era.   \nFor those fluctuation modes, CDM-like perturbation growth can be assumed to apply to the SFDM fluctuations, as well, as long as their horizon entry took place after SFDM transitions from a radiation-like to a matter-like EOS.   \nWe plot these quantities in Fig.~\\ref{fig:JeansMass} for 4 different values of $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc.    \nAs the figure indicates, all modes that enter the horizon super-Jeans (i.e. $M > M_\\text{J}$ when $M < M_\\text{H}$)  do so when the SFDM EOS was matter-like.   \nThat means that the Jeans mass calculated above\nfor the case when SFDM is matter-like is applicable, as we have assumed.  \nModes of smaller wavelength, for which horizon entry preceded the transition scale factor of the SFDM EOS, were relativistic and radiation-like upon entry, so their Jeans filtering during that phase should be calculated for a relativistic fluid, instead, because their effective sound speed is close to the speed of light and the Jeans scale is comparable to the horizon size, at which scale the Newtonian approximation breaks down. \n\nRoughly speaking, when $w = 1\/3$ \n(i.e. $a \\lesssim a_t$), \nthe Jeans length is equal to the distance a \nrelativistic sound wave can travel at the\nsound speed $c\/\\sqrt{3}$ in a Hubble time:\n\\begin{equation}\n    R_\\text{J,1\/3}(a) = \\frac{c}{H\\sqrt{3}}\n\\end{equation}\nThe mass enclosed by this radius\nis the same as the mass inside the usual\nacoustic horizon during the RD era, which is\njust smaller than the mass inside the Hubble\nvolume, according to\n\\begin{align}\n    M_\\text{J,1\/3}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} R_\\text{J,rad}^3 \\nonumber \\\\\n    &\\simeq 8.7\\times 10^{16} \\text{ M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left(\\frac{a}{10^{-3.5}}\\right)^{3}\n\\end{align} \n\\textit{independent of} $R_\\text{TF}$,\naside from the dependence of transition scale factor $a_t$ on it (see, for example, equation~\\ref{eq:atransitionRTF}).\nThis radiation-like Jeans mass defines the $a<a_t$ segment of each colored solid line\nin Fig.~\\ref{fig:JeansMass}.\n\nFor any given mode, we can describe its cosmological evolution by drawing a horizontal line across the box in Fig.~\\ref{fig:JeansMass}\nat that mass scale and following it in time\nfrom small scale factor to large, for a fixed\nchoice of SI strength, expressed by the\nvalue of $R_\\text{TF}$, as follows.  At early\ntimes such that $a<a_t$, the SFDM-TF is \nradiation-like ($w\\approx{1\/3}$), so its fluctuations behave during this phase\njust like those of fluctuations of the\nsame wavenumber $k$ in the dominant radiation component.  They are indistinguishable from the\nlatter, in fact \n(i.e. Lagrangian volume changes in the\nscalar field and radiation are the same for these\nmodes, during this phase of their evolution, so\nthe fluctuation in dark matter \\textit{mass} density $\\delta_\\text{SFDM}$ for these modes\nis simply related to \nthe energy-density fluctuation in the radiation\n$\\delta_\\text{rad}$ by $\\delta_\\text{SFDM}= (3\/4)\\delta_\\text{rad}$). The smaller the\nvalue of $R_\\text{TF}$ is, the earlier this\nradiation-like phase of EOS will end. \nHowever, there is always \na range of modes for which the\nmass is small enough (i.e. $k$ is large enough) to make horizon entry early enough that $a_\\textsc{h}< a_t$ for those modes, \nso they enter while the EOS of the scalar field is radiation-like. According to Fig.~\\ref{fig:JeansMass}, these modes are sub-Jeans ($M<M_\\text{J,1\/3}$) once they\nare within the horizon.  Radiation fluctuations that enter the horizon during this epoch\n(which is RD), all\nstart from the same $k$-independent amplitude when they are far outside the horizon, but\noscillate like sound waves with constant amplitude after entry. This also describes\nthe behavior of SFDM fluctuations of the \nsame wavenumber, therefore, until the\nend of their radiation-like phase at $a=a_t$.\nAt this time, the EOS of the\nscalar field transitions to matter-like ($w\\approx0$) and, thereafter, the character\nof the evolution departs from that of the\ndominant radiation component, as described \nbelow.  \n\nThere are three kinds of \nmodes to track, as we follow their evolution \\textit{after}\n$a=a_t$, when the EOS turns matter-like. There\nare the smaller-mass modes, described above, that entered the horizon prior to this time.  At the other extreme, there are  \nmodes that are so massive that they are matter-like and\n\\textit{super-Jeans} when they enter the horizon\n(i.e. $M>M_\\text{J,0}$ when $M<M_\\text{H}$).  \nIn between these two mass ranges, there are\nmodes which are matter-like and \\textit{sub-Jeans} upon horizon entry (i.e. $M<M_\\text{J,0}$ when $M<M_\\text{H}$).\nIn the first category, the\nhigh-$k$ modes that entered prior to this\nphase will still be sub-Jeans initially, after\n$a_t$, but with the Jeans mass now given by \n$M_\\text{J,0}$, rather than $M_\\text{J,1\/3}$. \nTheir fluctuations will transition\nfrom the radiation-like behavior described\nabove to a new, matter-like behavior, \nthereafter. For the range of modes that entered the horizon after $a=a_t$, however, \nthe EOS is matter-like at all times, thereafter.  Their behavior once within the horizon \ndepended upon whether they were above or below\nthe Jeans mass $M_\\text{J,0}$ \nat the time of their horizon entry.  \nWe expect perturbation modes that were super-Jeans upon horizon entry (i.e. modes with $M > M_\\text{J}$ when $M < M_\\text{H}$) during this phase\nto experience CDM-like growth, thereafter.  For the intermediate-mass modes that were sub-Jeans ($M<M_\\text{J}$) and\nmatter-like ($w\\approx0$ at $a>a_t$) when they were \\textit{superhorizon}\n($M>M_\\text{H}$), their evolution was also no different from that of CDM.  After their horizon entry, however, \nthese sub-Jeans modes oscillated like sound waves \nwith an amplitude that grew as $a^{1\/4}$, \nas described by \\citet{SC15}\n(see below).  This sub-Jeans oscillatory\nbehavior during the matter-like phase ($a>a_t$) \nof the scalar field, with amplitude growing as $a^{1\/4}$, was also shared by the higher-$k$ \nmodes that entered the horizon \\textit{before}\n$a=a_t$, although these latter modes were \nalready oscillating at \\textit{constant} \namplitude prior to $a_t$, just like fluctuations in the dominant radiation component at that time.\n\nThe latter sub-Jeans behavior during the matter-like\nphase of SFDM-TF contrasts with the\nbehavior of fluctuations in another matter component subject to Jeans-filtering by pressure forces, the baryons. For the baryons,\nsubhorizon baryonic fluctuations below the Jeans scale in the CDM universe also oscillated like sound waves, but with an amplitude that actually\n\\textit{decayed} during the MD era, prior to their decoupling from photons. \nHowever, this sub-Jeans behavior of the baryonic component in which the amplitude does not grow but even decays prior to photon decoupling has a much smaller impact on structure formation in the standard CDM model than the corresponding\nsub-Jeans behavior of SFDM-TF described above,\neven though the latter involves oscillations\nthat grow rather than decay.  The reason is \nthat the baryonic fluctuations are not required\nto ``carry the load'' as the driver of structure formation in the standard CDM model.  \nIn that model, subhorizon fluctuations in the \n\\textit{dark matter} were always able to grow.\nHence, in that case, even though sub-Jeans baryonic fluctuations could not, the baryons were \nlater able to catch up (after recombination) to those dark-matter fluctuations that had a head-start at growing during the RD era and \nbeyond and never damped.  On scales above\nthe baryonic Jeans-filter mass \n\\textit{after} recombination, the CDM fluctuations were never\nfiltered and so provided potential wells into which the baryons \nwere able to fall, even if their own density\nfluctuations were initially\nnegligible on these scales.  \nHere, by contrast, SFDM \\textit{is} the dark matter (\\textit{all} of it, in fact),\nand, even though its Jeans-filtering\nbears some superficial similarity to the \nJeans-filtering of baryonic fluctuations in CDM, \nthere is no other component on which it\ncan rely to drive its growth at later times.\nHence, it is not possible for fluctuations in SFDM on these smaller scales to ``re-animate'' later,\nas those of the baryons did for the CDM\nmodel, by being driven to catch-up to perturbations in some other component that were able to grow while those in the SFDM did not.    \n\nThat said, the Jeans mass in SFDM-TF decreases rapidly with time, so one might wonder whether this would free some fluctuation modes of their Jeans filtering after they transition from sub-Jeans to super-Jeans.  \nThe Jeans scale in the matter-like phase of\nSFDM corresponds to a fixed proper length, so it encompasses a smaller and smaller mass over time, scaling as it does with $a^{-3}$.\nAs such, whatever a mode's amplitude is when it reaches this super-Jeans epoch following its sub-Jeans phase, it will be able to grow, thereafter, at the matter-like growth rate.   \nIt remains to be determined below,\nif that will be a significant effect, or,  if not, if the growth of just those modes that were always super-Jeans from their horizon-entry epoch forward was sufficient on its own to form the full mass range of haloes at different redshifts required by astronomical observations. \n\n\n\\subsection{How the small-scale cut-off in the SFDM-TF transfer function depends on the polytrope radius}\n\\label{sec:cutoff}\n\nThe effect of Jeans-filtering described above\nis to introduce a small-scale cut-off in the \ntransfer function for SFDM-TF, while for larger\nscales, the CDM and SFDM-TF transfer functions \nare the same. We will derive this effect in detail\nby solving the perturbation equations below.\nFirst, however, we will anticipate the result of those calculations by estimating the \ncomoving wavenumber $k_\\text{cut}$ and \nassociated mass scale $M_\\text{cut}$ \nat which the cut-off begins in the resulting\ntransfer function, such that power is reduced \nrelative to that in CDM (for $k \\gtrsim k_\\text{cut}$, or equivalently, $M \\lesssim M_\\text{cut}$),\nas follows.  \nThis cut-off will occur roughly \nat the wavenumber of the first mode to enter\nthe horizon (i.e. the mode with the smallest $k$-value, corresponding to the largest mass \nscale) which is sub-Jeans when it enters.\nThis is the mode for which \n$a_\\textsc{j,0} = a_\\textsc{h}$, or equivalently, $R_\\text{TF} = c\/H(a_\\textsc{h})$:  \n\\begin{equation}\n    R_\\text{TF} = \\frac{c a_\\textsc{h}^2}{H_0 \\sqrt{\\Omega_\\text{rad}}} = \\frac{ \\pi^2 H_0 \\sqrt{\\Omega_\\text{rad}} }{ c k_\\text{cut,TF}^2  }\n\\end{equation}\nwhere $H(a_\\textsc{h})$ is evaluated assuming all modes of interest enter the horizon when the Universe is RD, and the last expression follows from our definition of $a_\\textsc{h}$: $ck\/\\pi = a_\\textsc{h} H(a_\\textsc{h})$. Solving for $k_\\text{cut}$ gives\n\\begin{align}\n    k_\\text{cut,TF} &= \\left(\\frac{ \\pi^2 H_0 \\sqrt{\\Omega_\\text{rad}} }{c R_\\text{TF}}\\right)^{1\/2} \\nonumber \\\\\n    &\\simeq 0.2~h~{\\rm Mpc}^{-1} \\left( \\frac{R_\\text{TF}}{\\rm 1~kpc} \\right)^{-1\/2} \\label{eq:kcut-TF}\n\\end{align}\n\\begin{align}\n    M_\\text{cut,TF} &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} \\left( \\frac{\\pi}{k_\\text{cut,TF}} \\right)^3 \\nonumber \\\\\n    &\\simeq 1.4\\times10^{15}~{\\rm M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left( \\frac{R_\\text{TF}}{\\rm 1~kpc} \\right)^{3\/2}\n\\end{align}\n\nIt will be useful to be able to compare the\ncut-off scale for SFDM-TF to that which occurs\nwhen SI is not present and the SFDM model is\nin the FDM limit, instead.  In the latter case, as\nwe shall show below, there is a different \nJeans mass which depends\nupon the particle mass $m$, rather than on\nthe ratio $g\/m^2$ involving SI, as it \ndoes for SFDM-TF.  For FDM, the EOS\nis always matter-like during the \ngalaxy-formation era (i.e. from horizon entry\nforward, for all modes that are relevant), \nand, just as for SFDM-TF and CDM, \ngalaxy-formation modes\nenter the horizon \nwhen the Universe is RD.  In that case,\nas we shall show in \\S\\ref{sec:LinPerts}, \nthe Jeans mass and its dependence \non scale factor are then\ngiven as follows:\n\\begin{equation}\n    a_\\textsc{j,f} = \\frac{\\hbar^2 k^4}{16\\pi G \\rho_\\text{dm,0} m^2}\n\\end{equation}\n\\begin{equation}\n    \\lambda_\\text{J,F} = \\left( \\frac{\\pi^3 \\hbar^2}{ G \\rho_\\text{dm,0} a^{-3} m^2} \\right)^{1\/4}\n\\end{equation}\n\\begin{align}\n    M_\\text{J,F}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} \\left( \\frac{\\lambda_\\text{J,F}}{2} \\right)^3 \\nonumber \\\\\n    &\\simeq 6.2 \\times 10^9 \\text{ M}_\\odot~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/4} \\left(\\frac{a}{10^{-3.5}}\\right)^{-3\/4} m_{22}^{-3\/2}\n\\end{align}\nwhere $m_{22} \\equiv m\/(10^{-22}~{\\rm eV}\/c^2)$. \nWe can then do a similar calculation for FDM\nas we did above for SFDM-TF, to estimate\nthe cut-off scale by setting \n$a_\\textsc{j,f} = a_\\textsc{h}$, instead:\n\\begin{equation}\n    \\frac{\\hbar^2 k_\\text{cut,F}^4}{16\\pi G \\rho_\\text{dm,0} m^2} = \\frac{c k_\\text{cut,F} a_\\textsc{h}^2}{ \\pi H_0 \\sqrt{\\Omega_\\text{rad}} } = \\frac{ \\pi H_0 \\sqrt{\\Omega_\\text{rad}} }{c k_\\text{cut,F}}\n\\end{equation}\n\\begin{align}\n    k_\\text{cut,F} &= \\left( \\frac{16 \\pi^2 G H_0 \\rho_\\text{dm,0} \\sqrt{\\Omega_\\text{rad}} m^2 }{c\\hbar^2} \\right)^{1\/5} \\nonumber \\\\\n    &\\simeq 3.9 ~h ~{\\rm Mpc}^{-1}~ \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/5} m_{22}^{2\/5} \\label{eq:kcut-FDM}\n\\end{align}\n\\begin{align}\n    M_\\text{cut,F} &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} \\left( \\frac{\\pi}{k_\\text{cut,F}} \\right)^3 \\nonumber \\\\\n    &\\simeq 2.3\\times10^{11}~{\\rm M}_\\odot~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{2\/5} m_{22}^{-6\/5}\n\\end{align}\n\nGiven this, we can equate equations~(\\ref{eq:kcut-TF}) and (\\ref{eq:kcut-FDM}) to find the equivalent FDM model that cuts off at the same scale as a given SFDM-TF model, in terms of $m$ and $R_\\text{TF}$. That is, the cut-off scale for a FDM model with a particle mass of $m_\\text{22,cut}$ is the same as that for a SFDM-TF model with a TF radius of $R_\\text{TF,cut}$ given by\n\\begin{equation}\n    R_\\text{TF,cut} \\simeq 3 \\text{ pc }~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{-2\/5} m_\\text{22,cut}^{-4\/5}\n    \\label{eq:RTFversusmFDM}\n\\end{equation}\n\nAccording to equation~(\\ref{eq:RTFversusmFDM}),\nthe FDM model with particle mass $m_{22} = 1$ \nresults in suppression of small-scale structure relative to the CDM model below a cut-off scale that is comparable to the\ncut-off scale where suppression begins in the SFDM-TF regime if $R_\\text{TF}\\approx3 \\text{ pc}$.  This is interesting, since\nthis fiducial value chosen for the FDM model is about the value required to flatten the density\nprofiles of present-day dwarf galaxies relative to the CDM model, by making solitonic cores of\nradius equal to the de~Broglie length there,\nof order $\\sim 1\\text{ kpc}$.  According to our results\nin \\citetalias{paper1} and here in \\S\\ref{sec:infall}, however, \nwhen a halo forms in SFDM-TF, it has a \ncore-envelope structure in which the envelope\nis CDM-like but the center flattens to\nthe profile of the $(n=1$)-polytropic SI\ncore, of radius $\\simeq R_\\text{TF}$.\nAs such, the value of $R_\\text{TF}~\\approx~3 \\text{ pc}$, which our estimate above tells us will\nproduce a cut-off of small-scale structure\nin the power-spectrum of density fluctuations\nrelative to that for CDM,\nat a similar wavenumber to\nthat for the FDM model when the latter \nhas $m_{22} = 1$, is very much smaller than\nthat $\\simeq 1\\text{ kpc}$ solitonic core\nsize for the corresponding FDM model.\n\nThis estimate will guide our expectations from the detailed calculation of linear perturbation growth in the two models below. However, while it would  \nbe tempting to think we can use this estimate of the correspondence of the two models, with their parameters so chosen to place their small-scale\ncutoffs at the same wavenumber, the actual\noutcome of structure formation in the two\nmodels depends upon the full spectrum of\nperturbation modes and their cosmic evolution, \ndown into the nonlinear era. \nA contrast between the two models \nmay arise because the TF Jeans mass drops so \nmuch more rapidly with scale factor than does\nthe FDM Jeans mass, for example, so \nscales that are Jeans-filtered in SFDM-TF at\nsome epoch may be able to grow again\nat later times, while there is no such\neffect in FDM.  Moreover, this effect of\nthe ``incredible shrinking Jeans mass'' for\nSFDM-TF means that, while some structures may be\nunderabundant relative to CDM as a result of\ntheir suppression in the linear perturbation stage -- an effect which may be similar to\nthat for the corresponding FDM model --\nthe internal structures of haloes\nthat result in the\ntwo models will differ considerably, if \nFDM solitonic cores in that model are much\nlarger than the polytropic cores in the \ncorresponding SFDM-TF haloes.   The true\ncomparison will require that we first derive \nthe transfer function for the full spectrum of modes,\nwhich we do in the next section,\nbefore we can compare and distinguish the actual impact on galaxy and large-scale structure formation of the two models.  \n\n\n\n\\subsection{Linear perturbations in SFDM-TF}\n\\label{sec:LinPerts}\n\nWe are now ready to calculate the growth of linear perturbations for all modes of interest to us in the formation of galaxies and large-scale structure, and from this, the transfer function\nfor the SFDM-TF model, for comparison with those \nof the CDM and FDM models.   We do this semi-analytically, by adopting several simplifying assumptions.  \nThe full treatment without approximation would require solving the relativistic \nKlein-Gordon-Einstein equations of motion for the scalar field, in a FLRW background universe, along with all other components of the standard cosmological model (i.e. radiation, baryons, neutrinos). This would be necessary\nto be able to consider the\nsmooth entry of modes into the horizon, in general,\nfrom well-outside to deep within the horizon.  \nThis full treatment would also need to incorporate the non-standard expansion history which results from the evolution of the SFDM EOS, as described by \\citet{Li14,Li17}, which depends upon the contribution of SFDM to the total energy density at each moment and, hence, depends on the value of particle parameters $m$ and $g\/m^2$.  In the most general case, the field can be either real or complex.\nIn both cases, the homogeneous field can be described as a perfect fluid, which makes possible\na fluid description for its evolution.  However,\nwhen treating perturbations in the complex case,\nthe perfect fluid description can break down in\nits stiff phase.   We will, therefore, \ngreatly facilitate the semi-analytical solution\nby the following approximations:\n\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item We will assume that SFDM model parameters are\n    constrained such that the structure formation era,\n    when all modes of interest to us enter the horizon\n    and evolve, is insignificantly affected by \n    the ``back-reaction'' of the SFDM and its EOS\n    evolution on the expansion history of the universe.\n    It turns out that this is a very good approximation, since the departures from standard expansion history during this structure formation era are related to the transitions which occur between\n    the EOS from matter-like ($w=0$) to \n    radiation-like ($w=1\/3$) and, thence,\n    either to cosmological-constant-like ($w=-1$) \n    or stiff ($w=1$), as one moves \n    backward in time.  The TF regime under consideration here ensures that the structure formation era is confined to the rapid oscillation phase of the scalar field, and the only transition\n    that takes place during this period is that between radiation-like and matter-like.  When that happens, as we said before, it must take place early enough that it does not perturb\n    the value of $a_\\text{eq}$ inferred from the measured CMB anisotropy.  As pointed out\n    in \\citet{Li14,Li17},\n    this means there is an upper bound to the value of the SI strength\n    in terms of $g\/m^2$, and, hence, of the value of $R_\\text{TF}$.\n    In particular, $g\/m^2c^4 \\leq 4 \\times 10^{-17} \\text{ eV}^{-1} \\text{ cm}^3$ is required\n    to satisfy $w(z_\\text{eq})\\leq0.001$, which\n    is equivalent to requiring that $R_\\text{TF} \\leq 5 \\text{ kpc}$. \n    When this bound is placed, there\n    is a corresponding limitation placed on the\n    contribution of the SFDM radiation-like energy density to the total during the RD era, which\n    means that SFDM does not appreciably alter the\n    expansion rate of the universe then.\n    \n    \\item We will assume that SFDM behaves as a perfect\n    fluid, so a fluid description is possible, with\n    pressure and energy density related accordingly, by\n    the EOS parameter in equation~(\\ref{eq:averagew}). \n    \n    \\item We will assume that SFDM is radiation-like\n    with $w = 1\/3$ at $a\\leq a_t$ and matter-like with\n    $w = 0$ at $a>a_t$, i.e. we assume an instantaneous transition. \n    We will take $a_t=a_{1\/9}$.\n    \n    \\item Once modes are well-within the horizon, we will treat them in the Newtonian approximation. \n\n\\end{enumerate}\n\nFor each mode, we shall calculate as our\nreference solution, that for the growth of dark matter perturbations in the CDM model.\nThe latter will be treated in the conventional way, as a nonrelativistic, collisionless, pressure-free gas (i.e. ``dust''). \nAn analytical solution is available for CDM perturbation modes when they are sub-horizon ($a \\gg a_\\textsc{h}$), given by \\citep[see, e.g.,][\\S6.3]{Mukhanov}:\n\\begin{align}\n    &\\frac{\\delta_\\text{CDM}(a)}{\\delta_\\text{H}} = 6 \\left[ \\ln \\left( \\frac{4 y }{ x } \\right) + \\gamma_\\textsc{e} - \\frac{7}{2} \\right] \\left(1 + \\frac{3 x}{2} \\right) \\nonumber\\\\\n    & \\quad \\quad \\quad \\quad \\quad- 6 \\left[ \\left(1 + \\frac{3 x}{2} \\right) \\ln \\left( \\frac{\\sqrt{1 + x} + 1}{\\sqrt{1 + x} - 1} \\right) - 3 \\sqrt{1 + x} \\right] \\label{eq:deltaCDM}\\\\\n    & x \\equiv \\frac{a}{a_\\text{eq}} \\\\\n    & y \\equiv \\frac{c k a}{H_0 \\sqrt{3\\Omega_\\text{rad}}}\n\\end{align}\nwhere $\\gamma_\\textsc{e} \\simeq 0.577$ is the Euler-Mascheroni constant, and $\\delta_\\text{H}$ is the asymptotic superhorizon value for the (mass) density contrast of a given mode.\nTo match this analytical subhorizon solution in equation~(\\ref{eq:deltaCDM}),\nvalid when a mode is deep within the horizon, to the solution of a more exact, fully-relativistic, numerical solution that follows the perturbation smoothly as it\nenters the horizon from far outside to deep within, we\ndefine an effective horizon-entry scale factor for each mode, $a_{\\textsc{h},\\text{eff}}(k)$,\nthat makes the two solutions agree\nwhen the modes are deep within\nthe horizon. This is equivalent\nto our adopting a thin-horizon approximation\nsuch that for \n$a < a_{\\textsc{h},\\text{eff}}$, we treat the perturbations as frozen at their superhorizon values $\\delta_\\text{H}$, while for $a > a_{\\textsc{h},\\text{eff}}$, the perturbations grow according to equation~(\\ref{eq:deltaCDM}).\nTo ensure that the density contrast is continuous across horizon-entry, then, we must choose $a_{\\textsc{h},\\text{eff}}$ such that, by equation~(\\ref{eq:deltaCDM}),  \n\\begin{equation}\n    \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{H} \\label{eq:aHeff}\n\\end{equation}\nfor each mode $k$. For SFDM,\nas long as the scalar field is matter-like at this value of $a_{\\textsc{h},\\text{eff}}$ \n(i.e. $a_{\\textsc{h},\\text{eff}}>a_t$), \nwe can also use this condition to set the horizon-entry value for an SFDM perturbation.\nThat is, the superhorizon and horizon-entry values for a dark matter perturbation should be independent of the dark matter model (i.e. only their subhorizon evolution should differ):\n\\begin{equation}\n    \\delta_\\text{SFDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{H}    \\label{eq:horentryamp}\n\\end{equation}\n(Henceforth, in \\S\\ref{sec:LinPerts}, the horizon-entry scale factor will refer to\nthis $a_{\\textsc{h},\\text{eff}}$, unless otherwise noted.  Elsewhere, we refer more generally to ``horizon-entry'' in terms of a mode filling the Hubble volume, for which\n$k = k_\\text{H} \\equiv \\frac{\\pi}{c}a_\\textsc{h}H(a_\\textsc{h})$.  The two definitions are quantitatively similar; we compare them in Appendix~\\ref{sec:aH}, Fig.~\\ref{fig:aHeff}.)\n\nFor perturbations in SFDM-TF, there are\ntwo possible phases of behavior to consider,\naccording to whether the scalar field\nis radiation-like or matter-like when a given\nmode is inside the horizon. \nModes that enter the horizon when the\nscalar field is \\textit{radiation-like} (i.e. with $a_{\\textsc{h},\\text{eff}} \\leq a_t$)  \nevolve just like those in the radiation component that dominates the total energy density.  \nSince we will only consider models for which the field transitions from radiation-like to matter-like at a scale factor $a_t < a_\\text{eq}$, the field will behave like radiation in a radiation-dominated universe for $a < a_t$, which has a known analytical solution, given by\n\\citep[see, e.g.,][\\S7.3]{Mukhanov}:\n\\begin{align}\n        \\frac{\\delta_\\text{rad}(a)}{\\delta_\\text{rad}(0)} = -3 \\left[ \\left( \\frac{2-y^2}{y^2} \\right) (\\text{sinc }y - \\cos y) - \\text{sinc } y \\right] \\label{eq:deltarad}\n\\end{align}\nwhere $\\text{sinc } y \\equiv \\sin y\/y$.\nAccording to this solution, such modes oscillate\nlike sound waves in a relativistic fluid\nwith sound speed $c\/\\sqrt{3}$, with constant\namplitude.  This analytical solution is fully-relativistic and\nproperly follows the perturbations from far\noutside the horizon through horizon entry, so\nthe time at which oscillations begin as a\nfunction of wavenumber is not arbitrary, but\nreflects the synchronization of horizon\nentry for all modes\\footnote{Our only\napproximation in this case is that, in practice,\nwe assign $a_t$ as the scale factor at which the average EOS parameter $w$ for the scalar field equals $1\/9$, and assume that SFDM instantaneously transitions from radiation-like to matter-like at $a_t$.} of a\ngiven $k$.   \n\nFor modes that are inside the horizon when\nthe scalar field is \\textit{matter-like}, \ninstead \\big(i.e. whenever $a > \\max[a_{\\textsc{h},\\text{eff}},a_t]$\\big), \neach mode evolves thereafter according to the following differential equation \\citep{SM11,Chavanis12}:\n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{ \\hbar^2 k^4}{4 m^2 a^4} + \\frac{\\bar{c}_{s,\\text{SI}}^2 k^2}{a^2} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpertgen}\n\\end{equation}\nwhere overdots denote differentiation with respect to cosmic time $t$, and $\\bar{c}_{s,\\text{SI}}$ is the sound speed of the background associated with SI pressure, given by:\n\\begin{equation}\n    \\bar{c}^2_{s,\\text{SI}} = \\frac{g}{m^2} \\rho_\\text{dm,0} a^{-3}\n    \\label{eq:SI_soundspeed_bg}\n\\end{equation}\nThe first term inside the parentheses in equation (\\ref{eq:linpertgen}) is the quantum pressure term, while the second is the effect of SI pressure.  The third term\nis responsible for gravitational growth.\nThe negative sign\nin front of that third term means that the first two, both positive-definite, \nact to oppose gravitational growth. The sum of all three terms inside the parentheses can be zero for a certain value of wavenumber, which we\nshall call the generalized Jeans wavenumber, \n$k_\\text{J}$.  Modes with wavenumber $k<k_\\text{J}$ are able to grow, while those with\n$k>k_\\text{J}$ oscillate \nlike sound waves for which\nthe evolution of their amplitude depends upon\nwhich of those first two terms dominates. \nWe note that the scale factor dependences of\nthose two first terms are different, so,\nin principle, if one term dominates at some\ntime, the other may dominate at other times. \n\n\nIn the TF regime, the\nregime of primary interest to us here, we can\ntake the limit in which ${\\hbar k^2 \/ H m \\to 0}$\nin equation~(\\ref{eq:linpertgen}) above\nand drop the first term inside the\nparentheses. This means that, in this\nlinear regime, SI dominates and\nquantum pressure is unimportant.  This contrasts\nthe linear perturbation regime with the nonlinear\nregime in which inhomogeneities even on the scale of the de~Broglie wavelength (and their contribution to quantum pressure) cannot be neglected, since they are responsible for transporting momentum and, thereby, affecting larger scales,\neven in a spatially-averaged description, as we discussed in \\citetalias{paper1}.  \nIn this linear regime, then, subhorizon\nperturbations in SFDM-TF \nduring its matter-like phase obey the following\nequation:\n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{\\bar{c}_{s,\\text{SI}}^2 k^2}{a^2} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpert}\n\\end{equation}\nThis equation is the origin of our expression\nfor the proper TF Jeans wavelength $\\lambda_\\text{J,0}$ in\n\\S\\ref{sec:LinPertOverview}, which we obtain by solving for the comoving Jeans wavenumber ($k_\\text{J,0} = 2\\pi a\/\\lambda_\\text{J,0}$)\nthat makes the terms inside the parentheses in equation~(\\ref{eq:linpert}) sum to zero:\n\\begin{equation}\n    k_\\text{J,0} = a\\sqrt{\\frac{4\\pi G m^2}{g}} = \\frac{\\pi a}{R_\\text{TF}}\n\\end{equation}\nAccording to equations~(\\ref{eq:linpertgen}) and (\\ref{eq:SI_soundspeed_bg}),\nif we are in the TF regime at the end\nof the evolution period of interest,  \nthen we were always in the TF regime at all earlier times, as well,\nso equation~(\\ref{eq:linpert}) remains valid\nthroughout this phase of matter-like EOS.  \n\nIn practice, we solve equation~(\\ref{eq:linpert}) for $\\delta \\equiv \\delta_\\text{SFDM}(a)\/\\delta_\\text{H}$.\nThe remaining uncertainty is then about what initial condition to impose on the solution. There are two regimes:\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent]\n    \\item $a_{\\textsc{h},\\text{eff}} > a_t$ -- modes enter the horizon when the field is matter-like:\n    \n    In this case, since all dark matter models should behave the same for superhorizon modes, we use the CDM horizon entry amplitude and slope\n    \\begin{align}\n        &\\delta_\\text{SFDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) \\label{eq:delICmatter}\\\\\n        &\\delta_\\text{SFDM}'(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}'(a_{\\textsc{h},\\text{eff}}) \\label{eq:delprimeICmatter}\n    \\end{align}\n    or, equivalently,\n    \\begin{align}\n        &\\delta(a_{\\textsc{h},\\text{eff}}) = 1 \\label{eq:delnormICmatter}\\\\\n        &\\delta'(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}'(a_{\\textsc{h},\\text{eff}})\/\\delta_\\text{H} \\label{eq:delprimenormICmatter}\n    \\end{align}\n    \n    \n    \\item $a_{\\textsc{h},\\text{eff}} \\leq a_t$ -- modes enter the horizon when the field is radiation-like:\n    \n    In this case, since SFDM should behave like radiation prior to $a_t$, we use the (normalized) amplitude and slope of the radiation solution at $a_t$\n    \\begin{align}\n        &\\delta(a_t) = \\delta_\\text{rad}(a_t)\/\\delta_\\text{rad}(0) \\\\\n        &\\delta'(a_t) = \\delta_\\text{rad}'(a_t)\/\\delta_\\text{rad}(0)\n    \\end{align}\n\\end{enumerate}\n\n\n\\subsection{The transfer function: SFDM-TF vs. CDM}\n\\label{Transferfunction}\n\nIt is customary to describe the growth of density fluctuations in alternative dark matter models to CDM in terms of a \\textit{normalized} transfer function, defined as follows:\n\n\\begin{align}\n    T_\\text{SFDM}^2(k,a) &\\equiv \n    \\frac{\\mathcal{T}^2_{\\text{SFDM},k}(a,a_i)}{\\mathcal{T}^2_{\\text{CDM},k}(a,a_i)} =\n    \\frac{P_\\text{SFDM}(k,a)}{P_\\text{CDM}(k,a)} \\nonumber\\\\\n    &= \\left|\\frac{\\delta_\\text{SFDM}(k,a)}{\\delta_\\text{CDM}(k,a)}\\right|^2\n\\end{align}\n(see also equation \\ref{eq:dP}),\nwhere $P(k,a)$ is the linear-fluctuation power spectrum of the dark matter for each model, as labeled. \n\n\n\n\\subsection{Comparison with FDM}\n\nAs a proof of principle, we can also compute the \ntransfer function for the FDM model, \nusing a procedure similar to that outlined above for SFDM-TF, and compare our results to those of the publicly-available code \\textsc{axioncamb}\\footnote{https:\/\/github.com\/dgrin1\/axionCAMB}\nby \\cite{axionCAMB}.\nThis code is presumably more accurate since it\nsolves the fully-relativistic perturbation \nequations numerically, from superhorizon through  \nsubhorizon regimes for each $k$, including their coupling\nto fluctuations in other components of energy-density, as\nwell. To derive the transfer \nfunction for FDM, \nwe drop the SI-term in equation~(\\ref{eq:linpertgen}):  \n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{ \\hbar^2 k^4}{4 m^2 a^4} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpertFDM}\n\\end{equation}\nThis equation for self-gravitating perturbations \nin the free-field (no-SI) case \nwas also discussed elsewhere, e.g. \\cite*{KMZ}.\nOnce again, we obtain the Jeans wavelength and wavenumber by setting the terms in parentheses equal to zero:\n\\begin{equation}\n    k_\\text{J,F} = \\left( \\frac{16\\pi G \\rho_\\text{dm,0} m^2 a}{\\hbar^2} \\right)^{1\/4}\n\\end{equation}\n\nThere are three noteworthy distinctions between the FDM and SFDM-TF models with regards to the solutions of their linear perturbation equations:\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n\\item when an FDM mode is sub-Jeans, it oscillates with a constant amplitude, whereas a sub-Jeans SFDM-TF mode oscillates with an amplitude that grows like $a^{1\/4}$;\n\\item the FDM Jeans mass shrinks much more slowly over time \nthan that for SFDM-TF \\big($M_\\text{J,F} \\propto a^{-3\/4}$ vs. $M_\\text{J,0} \\propto a^{-3}$\\big),\nso sub-Jeans modes of SFDM-TF can become super-Jeans\nsooner and with more time left to grow like CDM; and\n\\item in the FDM limit, the radiation-like phase is absent,\nso the initial conditions are set by equations~(\\ref{eq:delICmatter}) - (\\ref{eq:delprimenormICmatter}) for all modes.\n\\end{enumerate}\nThe first two of these distinctions explain why FDM has a sharper cut-off \nin the transfer function \nthan SFDM-TF does.\\footnote{The sharp cut-off in the \ntransfer function for FDM was also discussed by \\cite*{Hu2000},\nan important early contribution to this subject. \nThey reported an approximate fitting formula for the solution for different wavenumbers \n$k$ at a given scale factor, parameterized in terms of the ratio of that wavenumber to the Jeans wavenumber at that scale factor, $k\/k_\\text{J}$, which differs somewhat from the solutions presented here, which are numerical solutions of equation~(\\ref{eq:linpertFDM}). \nWhile they do not give the linear perturbation equations they solved numerically, \nthey say they treated FDM as a perfect fluid \nwith an effective sound speed, which, in its \nrapid-oscillation phase of matter-like EOS, \nis given by\n$c_{s,\\textsc{f}}^2=\\hbar^2 k^2\/\\big(4 m^2 a^2\\big)$. If so, then the dependence of the\nJeans wavenumber on scale factor should agree with ours derived by setting the bracket in our \nequation~(\\ref{eq:linpertFDM}) to\nzero, namely $k_\\text{J}\\propto a^{1\/4}$, in \\textit{both} the RD and MD eras.  \nWhile agreeing with this for the MD era, however, \\cite{Hu2000} states \nthat $k_\\text{J}$ is independent of $a$ during the important RD era, which is not correct.\n}\n\n\\subsection{The Halo Mass Function}\n\\label{sec:HMF}\n\nThe linear transfer function for the\nSFDM-TF model computed above can be used to\ncalculate the halo mass function (``HMF'') -- defined as the comoving number density of haloes \nper unit logarithm of halo mass $M$, $\\frac{dn}{d\\ln M}$.  There is an extensive literature on\nthe various choices for analytical and empirical\nprescriptions for computing the HMF from the\ntransfer function so as to predict the results of\ncosmological N-body simulation and their analysis\nby different halo-finding algorithms \\citep[see, e.g.,][and references therein]{Knebe13,Diemer18}. As our purpose here is to compare and contrast the HMF of\nthe SFDM-TF model with those of CDM and FDM, and\nthere are not yet N-body results for the former\nwith which to perform this comparison (and hardly\nany for FDM, either), we will adopt the\nmost straightforward prescription for calculating\nthe HMFs semi-analytically \nfor all three models, from their respective\nlinear power spectra $P(k,a)$, \nusing the standard Press-Schechter (``PS'') formalism \\citep{PS74}. \nThe latter is based upon the Gaussian statistics of the initial density perturbations, smoothed on a given mass scale, and the ansatz that a given local overdensity in the smoothed, linearly-perturbed density field will collapse and virialize at the time predicted for a spherical top-hat perturbation of that initial overdensity to reach infinite density.   A more\nrigorous underpinning for this remarkably successful ansatz was subsequently provided by the Excursion Set Theory of \\citet{Bond91}, including derivation of the PS formula for the HMF for the case of ``sharp $k$-space filtering'', about which we will say more below.\n\nThe PS HMF depends, not only on the underlying\npower spectrum of the initial, Gaussian-random density fluctuations, \nbut also on the shape of the function adopted to\nsmooth the density field.  The purpose of this\nsmoothing function is to squelch \nhigh-frequency modes that correspond to fluctuations of wavelength smaller than that which encompasses a given halo mass scale $M$, on average.  The mass variance $\\sigma_\\textsc{m}^2(M,a)$ of the linear overdensity field smoothed in coordinate space with a spherical window function \n$W(\\bm{x},R)$ -- where $R$ is the comoving radius of a sphere in the unperturbed\nbackground universe which encompasses the mass $M$, $R = \\big[M\/\\big(\\frac{4\\pi}{3} \\rho_\\text{dm,0}\\big)\\big]^{1\/3}$ -- is related to\nthe power spectrum $P(k,a)$ and the Fourier transform of this window function, $\\mathcal{W}(k,R)$, as\nfollows:\n\\begin{equation}\n    \\sigma_\\textsc{m}^2(M,a) = \\frac{1}{2\\pi^2} \\int_0^\\infty P(k,a) \\mathcal{W}(k,R) k^2 dk\n\\end{equation}\nThe window function is usually taken to be a top-hat filter, expressed in Fourier space as \n\\begin{equation}\n    \\mathcal{W}_\\text{top-hat}(k,R) = 3\\frac{\\sin(kR) - kR \\cos(kR)}{(kR)^3}\n\\end{equation}\nHowever, for models with sharp cut-offs in the power spectrum such as FDM, a sharp-$k$ filter is preferred in order to have the cut-off reflected in the HMF \\citep{KO20}:\n\\begin{equation}\n    \\mathcal{W}_{\\text{sharp-}k}(k,R) =\n    \\begin{cases}\n        1 & k < \\alpha\/R \\\\\n        0 & k > \\alpha\/R\n    \\end{cases}\n\\end{equation}\nwhere $\\alpha$ is a free parameter which we set to 1.9 in order to make the resulting HMF agree with that computed using the top-hat filter at large mass.\nThe PS HMF is then given by\n\\begin{equation}\n    \\frac{dn}{d\\ln M} = - \\sqrt{\\frac{2}{\\pi}} \\frac{\\rho_\\text{dm,0}}{M} \\frac{\\delta_c}{\\sigma_\\textsc{m}} e^{-\\delta_c^2\/2\\sigma_\\textsc{m}^2} \\frac{d\\ln \\sigma_\\textsc{m}}{d\\ln M}\n    \\label{eq:PSHMF}\n\\end{equation}\nwhere $\\delta_c = 1.686$ is the critical linear overdensity for halo formation derived from the spherical top-hat collapse model.\\footnote{\n\\citet{KO20} used $\\alpha=2.5$ following the work of \\citet{Benson13}, who found that value to be a good fit to warm dark matter simulations, and re-scaled $\\delta_c$ by a factor of 1.195 in order to produce a sharp-$k$ filtered HMF that was consistent with the top-hat filtered HMF at large mass. In view of the fact that we are here comparing the HMF's of CDM, SFDM-TF and FDM, these choices would not be appropriate for all three models, especially in view of the strong differences we find between SFDM-TF and FDM. However, we also find that adopting these choices would not affect our general conclusions.}\n\n\\subsection{Results of linear perturbation theory: SFDM-TF vs. CDM vs. FDM}\n\\label{sec:results}\n\n\\subsubsection{Perturbation growth for a single mode}\n\\label{sec:pertresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{LinPertSoln.pdf}\n    \\includegraphics[width=\\columnwidth]{LinPertSoln_FDM.pdf}\n    \\caption{\\textbf{Linear perturbation growth after horizon entry, for a single Fourier mode.} Solutions of \n    the differential equations for the evolution of linear-perturbation fractional overdensity $|\\delta\/\\delta_\\text{H}|$ vs. scale factor\n    $a$ are plotted for a single Fourier mode,\n    with wavenumber $k$ associated with a mass\n    $M=10^{9} \\text{ M}_\\odot$ \n    \\big(where $M = \\frac{4\\pi}{3} \\rho_\\text{dm,0}(\\pi\/k)^3$ and $k \\simeq 16$ Mpc$^{-1}$\\big) in the SFDM-TF (top panel) and FDM (bottom panel) models, for\n    two different cases for each. \n    The purple and green SFDM-TF lines correspond to $R_\\text{TF} = 1$~pc and $R_\\text{TF} = 100$~pc, respectively. The light and dark grey FDM lines correspond to $m_{22} = 0.8$ and $m_{22} = 29$, respectively. The black line is the CDM solution. Dots of the same color as the curves on which they reside mark the Jeans scale factors, $a_\\textsc{j,0}$ (SFDM-TF) and $a_\\textsc{j,\\textsc{f}}$ (FDM). The blue dashed line in the top panel shows the SFDM-TF solution for $R_\\text{TF} = 100$ pc when the scalar field is radiation-like (i.e. $w=1\/3$) (given by equation~\\ref{eq:deltarad}), and the blue star \n    marks the transition scale factor $a_t$ for this case, after which SFDM-TF is matter-like (green solid line). The red dashed lines in both panels marks radiation-matter equality.}\n    \\label{fig:delta_ex}\n\\end{figure}\n\nIn Fig. \\ref{fig:delta_ex}, we plot solutions \nof the differential equations described in \\S\\ref{sec:LinPerts} for the fractional linear overdensity $|\\delta\/\\delta_\\text{H}|$ vs. scale factor $a$, \nfor the subhorizon evolution of \na single, representative Fourier mode $k$. \nWe compare the analytical\nsolution for this mode in the CDM model\nwith our solutions for \nSFDM-TF and FDM, for illustrative cases with different parameters for the two kinds of \nSFDM models. The wavenumber $k$ for our\nrepresentative mode is that associated with a mass $M=10^{9} \\text{ M}_\\odot$ \\big(where $M = (4\\pi\/3) \\rho_\\text{dm,0}(\\pi\/k)^3$\\big). \nFor SFDM-TF, we present two cases, with \n$R_\\text{TF} = 1$ and $100$ pc, respectively. For FDM, we present two cases\nas well, with values of the particle mass\n$m_{22} = 0.8$ and $29$, respectively.\n\nThe perturbation evolution for this mode in \npressure-free CDM plotted in\nFig. \\ref{fig:delta_ex} shows the familiar\ntransition from RD-era logarithmic growth\nto the MD-era growth rate ($\\delta \\propto a$). \nThis transition for CDM is gradual, but\nis centered around $a = a_\\text{eq}$. By contrast,\nthe perturbation evolution for both the SFDM-TF \nand FDM solutions plotted in Fig. \\ref{fig:delta_ex}\nshows the effects of pressure and, hence,\nJeans filtering, at early times. \nThe evolution changes when this mode passes from \nsub-Jeans ($k>k_\\text{J}$) to \nsuper-Jeans ($k<k_\\text{J}$) at \n$a = a_\\textsc{j}$, where the values of $a_\\textsc{j}$ (marked by a dot on each curve) depend upon the model parameters and differ\nfor each case.\nWhen a mode of either SFDM-TF or FDM\nis sub-Jeans, its fractional overdensity oscillates\nlike a sound wave.  As such, \nthe periods of oscillation in the two models are inversely proportional to their effective sound speeds. For SFDM-TF, this sound speed increases with $R_\\text{TF}$ when the EOS is matter-like, but is a constant, independent of $R_\\text{TF}$, \nwhen the EOS is radiation-like.  The scale\nfactor $a_t$ at which this transition\nfrom radiation-like to matter-like EOS occurs\nfor SFDM-TF also depends on $R_\\text{TF}$,\nwith $a_t$ increasing with $R_\\text{TF}$.   \nFor FDM, which is always matter-like during\nthe era we study here, the sound speed decreases\nwith increasing $m_{22}$. These points allow us\nto understand the behavior of the different\ncurves in Fig. \\ref{fig:delta_ex}, as follows.\n\nConsider, first, the curves for\n$m_{22} = 0.8$ and $R_\\text{TF} = 100$ pc, respectively.  For these values of $R_\\text{TF}$ and $m_{22}$, \nthe periods of sub-Jeans oscillation are smaller than a Hubble time during that phase of evolution. \nIn that case, their fractional overdensities oscillate rapidly enough to \npass from positive to negative and back again\na number of times, so the\ncurves of their absolute value displayed in Fig. \\ref{fig:delta_ex} appear like trajectories\nof a bouncing ball.  For FDM, \nthe sub-Jeans oscillations have a constant amplitude, whereas, when the\nsub-Jeans SFDM-TF mode oscillates, its amplitude\nis initially constant for a time (the blue-dashed portion), during its radiation-like phase,\nbut then grows like $a^{1\/4}$ after it transitions\nat $a=a_t$ (as marked on the curve) \nto a matter-like EOS. In both cases, the oscillations cease near $a\\approx a_\\textsc{j}$, and perturbation\ngrowth rates become CDM-like for $a\\gg a_\\textsc{j}$.\n\n\nIn the case of the other two \ncurves in Fig. \\ref{fig:delta_ex},\nthat for SFDM-TF with $R_\\text{TF}= 1$ pc\nand FDM with $m_{22} = 29$, the \noscillatory behavior during their sub-Jeans\nphase is apparently absent, but this reflects\nthe fact that the change of model parameters \nhas pushed their sound speeds to be smaller \nand their oscillation ``periods'' to be\nlarger than a Hubble time.  In that case,\nthe background universe is expanding too fast for\nthe density to complete a single oscillation.\nThese cases therefore follow the CDM trajectory fairly closely, albeit with their growth slightly hampered due to the brief initial sub-Jeans phase.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction.pdf}\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction_rescaled.pdf}\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction_FDM.pdf}\n    \\caption{\\textbf{Linear transfer functions.} \\textbf{(a) (top panel)} Normalized transfer functions $T_\\text{SFDM}^2(k,a)$ for SFDM-TF \n    vs. comoving wavenumber \n    $k$, for $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc (left to right). For comparison, we plot FDM transfer functions $T_\\text{FDM}^2(k,a)$\n    for $m_{22}= \\{0.8, 29\\}$, generated by \\textsc{axioncamb}.\n    Dots locate values of $k_\\text{cut}$ derived in \\S\\ref{sec:cutoff}\n    to estimate the cut-off wavenumber above which high-frequency modes are suppressed relative to CDM. \n    \\textbf{(b) (middle panel)} SFDM-TF transfer functions in (a), \n    but with $k$-axis rescaled by dependence of $k_\\text{cut}$ on $R_\\text{TF}$, \n    so that locations on the horizontal axis \n    at which $k = k_\\text{cut}$ are the same\n    for all $R_\\text{TF}$. \n    SFDM-TF curves for larger $R_\\text{TF}$ start to deviate from CDM more sharply than for smaller \n    $R_\\text{TF}$, so $T^2_\\text{SFDM}(k_\\text{cut})$ is closer to 1 for smaller $R_\\text{TF}$.\n    \\textbf{(c) (bottom panel)} \n    FDM transfer functions   \n    in (a) (light and dark red lines),\n    generated by \\textsc{axioncamb}, are \n    compared to our solutions of subhorizon\n    linear perturbation growth for\n    FDM (dark and light blue), \n    i.e. same procedure we used for\n    SFDM-TF, but using the FDM linear perturbation equation (\\ref{eq:linpertFDM}), instead.}\n    \\label{fig:transfer}\n\\end{figure}\n\n\\subsubsection{Normalized transfer functions}\n\\label{sec:transferfunctionresults}\n\n\n\nIn Figs. \\ref{fig:transfer}(a) and (b), \nour \nsolutions for linear perturbation growth for different $k$-modes are used to plot the \nnormalized transfer functions \n$T_\\text{SFDM}^2(k,a)$ for SFDM-TF vs. $k$, for several illustrative cases, with $R_\\text{TF} = 1000, 100, 10, \\text{and}~1$ pc (left to right). \nFor comparison, \nwe also plot FDM transfer functions\n$T_\\text{FDM}^2(k,a)$, for the same two masses, $m_{22}=0.8$ and $29$, as in Fig. \\ref{fig:delta_ex}\nabove.   In Fig. \\ref{fig:transfer}(a),\ntransfer functions for FDM were generated by \\textsc{axioncamb}, while in \nFig. \\ref{fig:transfer}(c), we compare those\nwith our solutions\nfor FDM, to show their close agreement.\nThis agreement \nserves to validate our\nsolutions for both FDM and SFDM-TF, \nsince the former were derived\nby the same procedure as for our SFDM-TF \nsolutions.  Dots locate values of $k_\\text{cut}$ derived in \\S\\ref{sec:cutoff} to estimate the cut-off wavenumber above which high-frequency modes are suppressed relative to CDM.\n\nIn Fig. \\ref{fig:transfer}(b) (the middle panel),\nwe replot the SFDM-TF transfer functions in \nFig. \\ref{fig:transfer}(a) (the top panel) \nwith the $k$-axis rescaled by the dependence \nof $k_\\text{cut}$ on $R_\\text{TF}$, \nso that locations on the horizontal axis \nat which $k = k_\\text{cut}$ are the same\nfor all $R_\\text{TF}$. This shows that the smaller\nthe value of $R_\\text{TF}$ is, the more gradual\nis the decline relative to CDM at wavenumbers\nwhere the SFDM-TF transfer function begins to\ndecline relative to CDM at high $k$.\nIn addition, since curves for larger $R_\\text{TF}$ start \nto deviate from CDM more sharply than for smaller \n$R_\\text{TF}$, $T^2_\\text{SFDM}(k_\\text{cut})$ \nis closer to 1 for smaller $R_\\text{TF}$.\n{We can also see from this figure that the amplitudes of the oscillations at high wavenumber are smaller for larger $R_\\text{TF}$. \nThis may be due, in part, to the fact that, for larger $R_\\text{TF}$, modes with $k \\gg k_\\text{cut}$ have more of their sub-Jeans evolution occur during the matter-dominated era ($a > a_\\text{eq}$).\nSince CDM modes grow faster during this era (linearly with scale factor, as opposed to logarithmically for $a < a_\\text{eq}$), the $k \\gg k_\\text{cut}$ SFDM-TF modes for larger $R_\\text{TF}$ cases ``miss out'' on more of this comparatively rapid, linear growth phase than the corresponding modes of the smaller $R_\\text{TF}$ cases, and so are more suppressed relative to CDM. }\n\n\nAccording to our analytical estimates in \\S\\ref{sec:LinPertOverview}, it is possible\nto choose values of $R_\\text{TF}$ for SFDM-TF\nand $m_{22}$ for FDM so as to facilitate their direct comparison,\nby selecting values that make the\nsmall-scale cut-offs in their respective\ntransfer functions occur at roughly the same \nwavenumber (see equation \\ref{eq:RTFversusmFDM}).  In that case, for example, our solution for \nFDM with $m_{22} = 0.8$ can be compared with that\nfor SFDM-TF with $R_\\text{TF} \\simeq 10$ pc.\nIn this way, observational constraints on\nFDM can also serve as a rough proxy for constraints on \nSFDM-TF, since both models have perturbation growth \nwhich is CDM-like at small $k$ but suppressed \nat high $k$, i.e. values of $k$ above \nthe cut-off. \n\nThis estimator for the $R_\\text{TF}$-values that make\nthe transfer function cut-offs for SFDM-TF and \nFDM for a given $m_{22}$ ``align'' in wavenumber by\nsolving equation (\\ref{eq:RTFversusmFDM})\nshould be used with caution, however. \nThe \\textit{shapes} of their transfer functions below the cut-off wavenumber differ somewhat. \nAs a result, if different observables measure the\namount of structure on different scales, \ni.e. at different $k$, then the correspondence between \nthe $m_{22}$- and $R_\\text{TF}$-values that make\nthe two model predictions agree may depend upon which\nobservable is used to constrain them.  \nAs an example,\nthe two models can have very\ndifferent half-power wavenumbers, $k_{1\/2}$\nin their transfer functions,\neven when their $k_\\text{cut}$-values are identical.  \n\nThe two illustrative values of\nFDM particle mass we chose for Figs. \\ref{fig:delta_ex} and \\ref{fig:transfer}\nwill appear again in later figures, too,\nin which we compare the SFDM-TF and FDM models \nfurther.  These values roughly\nspan the current range of constraints in the\nFDM literature, placed by various observational probes,  namely, $m_{22}=0.8$ \n(\\citealp[e.g. fiducial case of ][]{VBB19} and \\citetalias{paper1}) and $m_{22}=29$\n\\citep[e.g. lower bound from][]{Nadler21}.\nFor these values of $m_{22}$, de~Broglie wavelengths \ninside a present-day dwarf galaxy of mass $10^{10}\\text{ M}_\\odot$ with\nvelocity dispersion $\\sigma_\\text{v} \\approx 30$ km s$^{-1}$, for \nexample, are $\\approx 4$ kpc and $0.1$ kpc, respectively, so haloes of this mass that\nformed in FDM would have solitonic cores of these sizes.\nIf we chose $R_\\text{TF}$-values of this size, \nso as to make the SI-polytropic cores inside \nSFDM-TF haloes as large as this, then\nthe $k_\\text{cut}$-values for SFDM-TF will be\nmuch smaller than the $k_\\text{cut}$-values for\nthese FDM models.  The transfer function for\nSFDM-TF in that case would begin to cut-off at a much higher\nmass scale than FDM, so the amount of structure formed  \non such small mass scales in SFDM-TF would be well below\nthat in the corresponding, core-size-matched FDM model. \nOn the other hand, if we align the $k_\\text{cut}$-values \nof the two models, instead, which Fig. \\ref{fig:transfer} tells us requires $R_\\text{TF} \\lesssim 1$ pc, then much\n\\textit{more} small-scale power is retained below the\ncut-off in SFDM-TF than in FDM, as will be even\nclearer from the results for their power spectra\nand HMFs in the sections below.  Hence,\nlower limits like those above on the \nparticle masses for FDM, derived to keep FDM from \\textit{underproducing} structure on small scales, \nwhen used to infer upper limits on\n$R_\\text{TF}$ based upon $k_\\text{cut}$-matching \nFDM, might then be overly restrictive in limiting $R_\\text{TF}$. \n\n\n\n\n\n\\subsubsection{Power spectra and RMS mass fluctuations}\n\\label{sec:powerspecsigmaresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{PowerSpectrum.pdf}\n    \\includegraphics[width=\\columnwidth]{SigmaM.pdf}\n    \\includegraphics[width=\\columnwidth]{dlogSigma-dlogM.pdf}\n    \\caption{\\textbf{Power spectrum and RMS mass fluctuation at $\\bm{z=0}$}. \\textbf{(a) (top panel)} Present-day matter power spectra $P(k)$, \\textbf{(b) (middle panel)} RMS mass fluctuation $\\sigma_\\textsc{m}(M)$, and \\textbf{(c) (bottom panel)} the logarithmic derivative of $\\sigma_\\textsc{m}$,  for CDM (black), FDM (grey), and SFDM-TF (colored) models, as labeled. The CDM and FDM curves are generated using \\textsc{axioncamb}. The SFDM-TF curves are generated by multiplying the transfer functions of Fig.~\\ref{fig:transfer} (obtained through our linear perturbation calculation) by the CDM power spectrum from \\textsc{axioncamb}. In the middle and bottom panels, solid lines represent top-hat filtering, while dashed lines represent sharp-$k$ filtering. The logarithmic derivative is much steeper for sharp-$k$ filtering than for top-hat filtering for FDM, while for SFDM-TF, top-hat filtering captures the envelope of the rapidly-oscillating sharp-$k$-filtered result,\n    so they are essentially the same.}\n    \\label{fig:PandSigma}\n\\end{figure}\n\nIn \\S\\ref{sec:transferfunctionresults} above,\nwe presented the normalized transfer function\nresults for SFDM-TF and FDM, which are,\nby definition, expressed in units of the \ntransfer function for CDM.  In Fig.\n\\ref{fig:PandSigma}(a), we also present the power\nspectra $P(k)$ vs. $k$ for each model, separately,\nby multiplying those normalized transfer functions\nby the CDM transfer function generated by the\npublicly-available \\textsc{camb} code.\nWe present these power spectra, for several\nillustrative values of $R_\\text{TF}$ for\nSFDM-TF and $m_{22}$ for FDM, respectively,\nas labeled, linearly-extrapolated to $z=0$.\nThis takes proper account of the effect of\nthe smooth transition of our observed\nbackground Universe at late times, from MD\nto $\\Lambda$-dominated (``$\\Lambda$D''), \nsince all modes of interest to us have,\nby then, long-since\ntransitioned to pressure-free CDM-like \nperturbation growth according to linear theory.\n\nWith these power spectra, we can also compute\nthe RMS mass fluctuations $\\sigma_\\textsc{m}(M)$\nand their logarithmic derivatives, \nas plotted vs. $M$ \nin Figs.~\\ref{fig:PandSigma}(b) and (c).\nThe curves of $\\sigma_\\textsc{m}(M)$ for SFDM-TF\nand FDM follow that for CDM at high mass, \nincreasing towards smaller $M$, \nbut they flatten to a \ncontant value in the mass range\nassociated with wavenumbers of modes which are\nsuppressed by Jeans-filtering relative to the\ngrowth of CDM perturbations of the same\nwavenumber.  From the plots of $\\sigma_\\textsc{m}(M)$, alone, it is difficult\nto appreciate the differences between the SFDM-TF\nand FDM results, \neven when overplotting them so as to compare them \ndirectly, by choosing model parameters\nfor SFDM-TF and FDM which give them similar\nvalues of $k_\\text{cut}$.\nIn particular, it is difficult to see from the\ncurves in Fig.~\\ref{fig:PandSigma}(b), the\nimportant distinction between the behaviors of\nthe two models when switching between \ntop-hat and sharp-$k$ filtering.  The difference\nis more evident, however, when viewed by\nplotting the logarithmic derivatives of $\\sigma_\\textsc{m}(M)$, instead, as\nshown in Fig. \\ref{fig:PandSigma}(c).  \nFor FDM, this logarithmic derivative \nat values of $M$ associated with \nwavenumbers above the cut-off in\nthe transfer function\napproaches zero (from below) much more rapidly as $M$ decreases, for sharp-$k$ filtering than for top-hat filtering. We will see just how\nsignificant this difference is below, \nwhen we apply these results to calculate the\nHMF for FDM.  For SFDM-TF, however, the logarithmic\nderivative shown in Fig. \\ref{fig:PandSigma}(c) for top-hat filtering captures the envelope of the rapidly-oscillating behavior when a sharp-$k$-filter is adopted, instead, so, when averaged over these oscillations, the results for the\ntwo filter functions are essentially the same.\n\n\n\n\\subsubsection{Halo mass functions}\n\\label{sec:HMFresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{HMF.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio.pdf}\n    \\caption{\\textbf{Halo mass function at $\\bm{z=0}$.}\n    \\textbf{(a) (top panel)} HMFs, \n    $dn\/d\\ln M$, are \n    plotted vs. halo mass $M$ for SFDM-TF, CDM\n    and FDM. SFDM-TF curves are for different $R_\\text{TF}$-values,  $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc (colored curves from lowest to highest). CDM HMF (black curve) is computed from the transfer function generated by \\textsc{camb}. FDM HMFs (light and dark grey curves) are for different particle masses  $m_{22}=\\{0.8, 29\\}$,\n    respectively, as computed from the transfer functions generated by \\textsc{axioncamb}. \\textbf{(b) (bottom panel)} Same as (a), but normalized by the CDM HMF.}\n    \\label{fig:HMF}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{HMF_evo_10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio_evo_10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio_evo_mfid.pdf}\n    \\caption{\\textbf{Evolution of the halo mass function.} Same as Fig.~\\ref{fig:HMF}, except for different redshifts as labeled, and the SFDM-TF curves are all for $R_\\text{TF} = 10$~pc, while the FDM curves are all for $m_{22}=0.8$. \n    \\textbf{(a) (top panel)} Same as Fig.~\\ref{fig:HMF}(a) (but note the expanded $y$-axis scale), except for different redshifts and a single SFDM-TF case with $R_\\text{TF} = 10$~pc.\n    CDM curves are dashed for readability. \\textbf{(b) (middle panel)} Same as\n    Fig.~\\ref{fig:HMF}(b), except for different redshifts and a single SFDM-TF case with $R_\\text{TF} = 10$~pc.\n    \\textbf{(c) (bottom panel)} Same as (b), except for a FDM case with $m_{22} = 0.8$.}\n    \\label{fig:HMFevo}\n\\end{figure}\n\nIn Fig. \\ref{fig:HMF}, we plot HMFs\nat $z=0$, computed from the RMS mass fluctuations and their logarithmic\nderivatives plotted in Fig. \\ref{fig:PandSigma},\nfor SFDM-TF (for $R_\\text{TF} = 1000, 100, 10$ and $1$ pc), FDM (for $m_{22}=0.8$ and 29), \nand CDM.  As expected, all three models produce\nthe same HMF at high mass, well above the scales\naffected by Jeans-filtering. Since \nJeans-filtering of small-scale structure produces\ncut-offs at high wavenumber in the power spectra\nand transfer functions of both FDM and SFDM-TF, \ncut-offs appear in both their HMFs at small halo\nmass $M$, as well.  \nHowever, as Fig. \\ref{fig:HMF} makes\nclear, the shape and depth of this HMF cut-off\nfor SFDM-TF are dramatically different\nfrom that for FDM.  In particular, the cut-off\nfor SFDM-TF is much weaker and more gradual\nthan that for FDM.   This will have strong\nconsequences for the comparison between any\nobservables for the two models that reflect\na turn-down in their abundance of small-mass\ngalactic haloes relative to that in the \nstandard CDM model.\n\nThe origin of this difference can be understood\nfrom the curves for $\\sigma_\\textsc{m}(M)$ and\nits logarithmic derivative for the two models,\nin Fig. \\ref{fig:PandSigma}(b) and (c). As we\nnoted above, the sharp-$k$-filtered result\nfor FDM is much more sharply cut-off at small\nmass than is the top-hat-filtered result.\nThis is why sharp-$k$-filtered results are \nstrongly preferred for this model \\citep{KO20}.\nThis is generally true for other models with\na similar cut-off of power on\nsmall scales, e.g. warm dark matter (``WDM''),\nas well.  For this reason, we used\nthe sharp-$k$-filtered results to compute the\nHMF for FDM in Fig.~\\ref{fig:HMF}.\nFor SFDM-TF, however, the choice of \nfilter does not make a difference in how\nsharply $\\sigma_\\textsc{m}(M)$ cuts off.\nUsing the top-hat filter for SFDM-TF merely\nsmooths over high-frequency \noscillations in the logarithmic derivative\nof the sharp-$k$-filtered\n$\\sigma_\\textsc{m}(M)$, but still\ntraces their envelope very well, so the \nsharpness of the cut-off is the same for the two filters.  \\textit{Regardless} of which filter is\nused, that cut-off is much more gradual\nand less severe for the SFDM-TF model than for FDM.\nWe chose to plot the SFDM-TF HMF using the\ntop-hat filter, rather than the sharp-$k$-filter,\nto avoid giving false prominence to the\ncorresponding oscillations that appear\nin the HMF in this case, if the latter were used,\ninstead. Their prominence and phases are \nartificially related to our thin-horizon\nand thin-EOS-transition approximations.  \nHowever, our conclusion that the overall\ncut-off for the SFDM-TF case is much shallower\nand more gradual than for FDM is robust, \nas it is the same for both filters.   \n\nIn view of the importance of this difference in\nthe $z=0$ HMFs for SFDM-TF and FDM, \nwe have explored the\ntime-dependence of this difference in Fig. \\ref{fig:HMFevo}.  By comparing\nthe HMFs for the two models at\nredshifts ranging from\n$z=0$ to $z=10$, it is apparent\nthat the small-mass cut-off\nfor FDM, relative to that for\nCDM, is much sharper and much \ndeeper than that for SFDM-TF, \nat \\textit{all} redshifts in this range.   \nMoreover, there is a strong trend in the\nHMF for SFDM-TF at masses below the cut-off,\nfor it to recover over time, gradually rising\ntoward that for CDM.  There is no comparable\nrecovery for the more\nsharply-reduced HMF in the FDM case, \nsufficient to fill in\nthe missing halo abundance at small mass\nrelative to that of CDM.\n\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{l|c c c c c}\n        \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad & \\multicolumn{2}{c}{$10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$} & \\quad\\quad & \\multicolumn{2}{c}{$10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$} \\\\\n        \\hline\n        \\hline\n         &  $n_\\textsc{lmh}$ \\big(Mpc$^{-3}$\\big) & $\\rho_\\textsc{lmh}$ \\big($10^9$ M$_\\odot$ Mpc$^{-3}$\\big) & & $n_\\textsc{lmh}$ \\big(Mpc$^{-3}$\\big) & $\\rho_\\textsc{lmh}$ \\big($10^9$ M$_\\odot$ Mpc$^{-3}$\\big)\\\\\n         \\hline\n        $R_\\text{TF} = 10$ pc & 29    & 3.2 & & 1.1   & 3.0\\\\\n        $m_{22} = 0.8$        & 0.086 & 2.6 & & 0.086 & 2.6\\\\ \\\\\n        $R_\\text{TF} = 1$ pc  & 61    & 7.1 & & 4.2   & 6.7\\\\\n        $m_{22} = 29$         & 7.6   & 7.0 & & 6.0   & 6.9\\\\\n        \\hline\n        & $n_\\textsc{lmh}\/n_{\\textsc{lmh},\\textsc{cdm}}$ & $\\rho_\\textsc{lmh}\/\\rho_{\\textsc{lmh},\\textsc{cdm}}$ & & $n_\\textsc{lmh}\/n_{\\textsc{lmh},\\textsc{cdm}}$ & $\\rho_\\textsc{lmh}\/\\rho_{\\textsc{lmh},\\textsc{cdm}}$\\\\\n        \\hline\n        $R_\\text{TF} = 10$ pc & 0.07   & 0.32 & & 0.14  & 0.39\\\\\n        $m_{22} = 0.8$        & 0.0002 & 0.26 & & 0.011 & 0.33\\\\ \\\\\n        $R_\\text{TF} = 1$ pc  & 0.14   & 0.71 & & 0.55  & 0.87\\\\\n        $m_{22} = 29$         & 0.017  & 0.70 & & 0.79  & 0.90\\\\\n        \\hline\n    \\end{tabular}\n    \\caption{\\textbf{Integrated halo number density and mass density of low-mass haloes} (``LMH'')  in the range $10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$ (left 2 columns) and $10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$ (right 2 columns) at $z=0$ for SFDM-TF and FDM, including their ratios to the CDM quantities, for\n    illustrative values of $R_\\text{TF}$ and $m_{22}$.}\n    \\label{tab:halocounts}\n\\end{table*}\n\nFor some purposes, when comparing predictions of\n``cut-off models'' like these to CDM and to\nobservations, the integrated HMFs over some \nlow-mass halo (``LMH'') mass \nrange affected by the cut-off is \nalso of interest.  In\nTable \\ref{tab:halocounts}, we show\nthe integrated halo number and\nmass densities, given by\n\\begin{equation}\n n_\\textsc{lmh} = \\int \\frac{dn}{d \\ln M} d\\ln M = \\int \\frac{1}{M}\\frac{dn}{d \\ln M} dM\n \\label{eq:nlmh}\n\\end{equation}\nand \n\\begin{equation}\n \\rho_\\textsc{lmh} = \\int \\frac{dn}{d \\ln M} d M\n \\label{eq:rholmh}\n\\end{equation}\nevaluated at $z=0$, for halo masses in the ranges $10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$ and $10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$,\nfor two illustrative sets of\npairings of model parameters\nfor SFDM-TF vs. FDM, $R_\\text{TF}=10$ pc vs.\n$m_{22}=0.8$, and \n$R_\\text{TF}=1$ pc vs.\n$m_{22}=29$, respectively,  \nplotted in\nprevious figures, including their results relative\nto CDM. When the minimum halo mass\nis $10^8\\text{ M}_\\odot$, \nSFDM-TF with $R_\\text{TF}=1$ pc\nhas a halo number comparable to\nthat of FDM with $m_{22}=29$, and within a factor of 2 of that for CDM, while the masses in all three\nmodels are very close. For $R_\\text{TF}=10$ pc, however, SFDM-TF has many more haloes\nthan does FDM with $m_{22}=0.8$,\nby an order of magnitude, and \n$14\\%$ of the CDM number, while\nthe masses in both\nSFDM-TF and FDM are comparable\nand $\\approx{1\/3}$ of that for CDM. If the minimum halo mass is lowered to $10^6\\text{ M}_\\odot$, \nhowever, SFDM-TF retains\nmany more haloes than FDM for\n\\textit{both} example cases --\nby an order of magnitude for\n$R_\\text{TF}=1$ pc vs. $m_{22}=29$,\nand by more than a factor of 300 for\n$R_\\text{TF}=10$ pc vs. \n$m_{22}=0.8$ --\nand somewhat more halo mass \nin this low-mass range than the corresponding FDM models, despite the fact that \nthe SFDM-TF cases have slightly smaller cut-off mass scales ($M_\\text{cut}$) than the corresponding FDM cases. \nIn fact, the $R_\\text{TF} = 10$ pc case even retains more haloes (by number) than \nthe $m_{22} = 29$ case. \nThese examples make it clear that one must be careful\nwhen treating lower limits on $m_{22}$ for FDM as\na proxy for providing upper limits on $R_\\text{TF}$\nfor SFDM-TF.  If the former is \nderived to ensure that FDM has a high enough \nHMF at low mass, it may be overly restrictive if\nused to place an upper limit on the SFDM-TF $R_\\text{TF}$\nsimply by solving for the\nvalue that makes their cut-off wavenumbers the same.  \n\n\nThis gradual cut-off of the HMF is a defining feature of SFDM-TF and distinguishes it from any other DM alternative with a cut-off in its power spectrum that reduces small-scale power relative to CDM as sharply as does FDM.\nIts origin is rooted in the ``incredible shrinking Jeans mass'' of SFDM-TF, which leads to a ``recovery'' of small-scale perturbations that\nwere initially \nretarded below the cut-off when the Jeans scale was large, then\nfree to grow like CDM after exceeding the Jeans scale, with plenty of time left to close the gap with CDM, at least partially. As a\nresult, even haloes of mass \nwell below $M_\\text{cut}$ in the transfer function can still be present in significant numbers\nat late times.  \n\n\nIn principle, this suggests that SFDM-TF might have an advantage over FDM, when it comes to resolving the problem CDM has of overpredicting dwarf satellites compared with observations of the Milky Way\nand Local Group.  \nIn particular, if FDM uses its\ncut-off to suppress the abundance\nof dwarf satellites so as to\nreconcile with observations\nat the higher-mass end (i.e.\nat the scale of the so-called\n``too big to fail'' dwarfs -- henceforth, ``TBTF''), it \ntends to eliminate lower-mass\nhaloes (e.g. the ultra-faint dwarfs)\ntoo drastically.  Since the\nmore gradual cut-off in the HMF of SFDM-TF might avoid this difficulty, we consider this further, as follows. \n\nPredicted dwarf satellite abundances\ncan be expressed in terms either of\nthe number of haloes per interval\nof halo mass or their number per\ninterval of halo circular velocity\n(as measured at some radius), which decline with increasing halo mass\nand velocity, respectively.\nThe observational problem for CDM\nis usually expressed in terms of the latter, more observable property, involving\nvelocity, since it can be determined from a rotation curve, observed\nonly part-way out in total mass.  There are, in principle, two ways\nto modify the predicted velocity distribution function\nin the TBTF mass range\nso as to reduce the overabundance\nin CDM N-body simulations to \nmake them agree with the observed\ndistribution: \n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item reduce the halo number per unit mass without \n    altering their internal densities relative to CDM haloes, \n    or \n    \\item reduce the internal densities of the haloes \n    without changing their number,\n    so haloes of a given mass appear at\n    lower velocity but match the\n    lower abundance of CDM haloes\n    of higher velocity.\n\\end{enumerate}\n(TBTF haloes are so named because\nthey are too massive to be made\nundetectable by suppressing their ability to form stars, which would otherwise be a third option for avoiding their apparent overabundance in CDM.)  \n\nFDM has\ntrouble with both options, \nas follows.  If $m_{22}\\approx1$,\nTBTF haloes (e.g. with $M\\approx{10^{10}\\text{ M}_\\odot}$) have their internal densities lowered by the presence of solitonic cores as large as their de~Broglie wavelengths of a few kpc or more.  However, while\nthis helps to solve the TBTF problem by option (2),\nit also reduces the \nabundance in the HMF, including a catastrophic reduction at lower-mass where ultra-faint dwarfs are detected. \nIf, on the other hand, FDM tries\nto avoid the latter problem by choosing a large enough $m_{22}$,\n$m_{22} \\geq 29$ \\citep{Nadler21},\nthen neither option (1) nor (2)\nare available to \nsolve the TBTF problem. \nWhile the value of $m_{22}=29$ \nis just large enough to make sure\nthe HMF in the range of ultrafaint dwarfs is not over-reduced, the\ncut-off is so sharp that\nthe number of haloes is then\nnot significantly reduced in the\nTBTF mass range, either, so\noption (1) fails.\nAnd, unfortunately, the \nsolitonic cores of the TBTF\nmass range are smaller \nin that case, since their de~Broglie wavelengths are then \n$\\lesssim0.1$ kpc,\nwhich is too small to flatten the central density profiles enough to lower the rotation-curve velocities to make option (2) possible. \nSuch small cores are also too small for FDM to solve the cusp-core\nproblem of CDM.  For that problem,\nit turns out, even the smaller value of $m_{22}\\approx1$ is not a good\nsolution, since the solitonic\ncores of galaxies even \\textit{more} massive than TBTF galaxies \n(e.g. $M\\gtrsim{10^{11}\\text{ M}_\\odot}$) are then\ntoo dense, even denser than in CDM \\citep{VBB19}.\n\nFor SFDM-TF, on the other\nhand, option (2) requires $R_\\text{TF}\\gtrsim1$~kpc, in\norder to make large enough\npolytropic cores to lower the densities of TBTF haloes sufficiently, which would also\nhelp to resolve the cusp-core problem \\citepalias{paper1}.\nHowever, for halo formation\nfrom cosmological\ninitial conditions, \nour results \nin Fig. \\ref{fig:HMF} show that,\nif $R_\\text{TF}$ is this large,\nthe cut-off mass in the HMF is\nso large that halo abundances at\nall galactic halo masses\nare substantially reduced\nrelative to CDM. Hence,\neven though it is\na much more gradual cut-off \nthan FDM's, it still over-reduces \nthe dwarf satellite abundance\nrelative to CDM.  So, option (2)\nis not available to SFDM-TF, either.\nAccording to the results \nin Table \\ref{tab:halocounts}, in fact, if $m_{22}\\geq29$ is required for FDM to avoid\nunder-predicting the observed\nabundance of ultrafaint dwarfs, then SFDM-TF probably requires $R_\\text{TF} \\lesssim 1$ pc.\nAccording to\nthe HMF in Fig. \\ref{fig:HMF} for\n$R_\\text{TF}=1$ pc, however, the halo abundance at the TBTF mass scale is then just the same\nas for CDM, so option (1) is\nunavailable just as it was\nfor FDM with $m_{22}=29$.  This\nis true despite the fact that\nthe FDM HMF cut-off just below the assumed ultrafaint dwarf mass\nrange of $\\gtrsim{10^{8}\\text{ M}_\\odot}$ is much sharper \nthan that for SFDM-TF \nwith $R_\\text{TF}=1$ pc.\nThat property does\ngive SFDM-TF one advantage\nover FDM, however, since it\nenables SFDM-TF to contribute a population of even smaller-mass subhalos which may help to account for perturbers of stellar streams observed in the Milky Way, which\nFDM may not \\citep[e.g.][]{Banik21}.\n\n\n\n\nWhile this may suggest that SFDM-TF \nis not much better at resolving the TBTF problem than FDM, in the face\nof the recently tightened constraint on the latter imposed by  \\citet{Nadler21} based on the abundance of ultrafaint dwarfs in the Milky Way and Local Group, there are caveats to this conclusion.\nFirstly, our discussion is so far based\nonly upon the comparison of\nthe global HMFs of SFDM-TF, FDM\nand CDM, while\ndwarf satellites are subhalos\nwithin these larger halos.\nA conditional HMF which\npredicts the abundances of subhalos\nof a given mass within a halo of\na larger mass at a given epoch\nmight make the comparison between\nFDM and SFDM-TF different from\nthat based upon comparing\ntheir global HMFs. \nSecondly, if the masses\nof ultrafaint dwarfs, which are\nuncertain, are determined to be below $10^{8}\\text{ M}_\\odot$, then\nthe more gradual\ncut-off of SFDM-TF gives it a much\nbetter chance of explaining TBTF than FDM,\nby allowing a \nlarger value of $R_\\text{TF}$\n(e.g. \n$R_\\text{TF}\\gtrsim{10}$ pc) without then fatally\nunderproducing the ultrafaint dwarfs. This is apparent \nif we compare the results for the two \nhalo mass ranges in Table \\ref{tab:halocounts}:\nin the range $10^6 \\leq M\/\\text{M}_\\odot \\leq 10^{11}$, the $R_\\text{TF} = 10$ pc SFDM-TF model\ncontains \\textit{more} haloes than the $m_{22} = 29$ FDM limit, while this is not true in the range \n$10^8 \\leq M\/\\text{M}_\\odot \\leq 10^{11}$.\nThirdly, comparisons of the\nobservations with these predicted\nHMFs also says nothing about the impact\nof dynamical processes inside the parent\nhalo which can alter the subhalo HMFs\nthere over time, e.g. by tidal\ndisruption, which may also be\ndifferent for SFDM-TF and FDM.\nUltimately, 3D-cosmological simulations with high resolution\nand dynamic range, from initial\nconditions that led to the formation\nof the Local Group and its\nsurroundings, may be required to\nsettle the matter. \n\nFor values\nof $R_\\text{TF}\\ll1$ pc or $m_{22}\\gg29$, the HMFs and\ninternal halo structure of both models approach that of CDM,\nfor most observables.  In that\ncase, the same solutions\nto the small-scale structure \nproblems of CDM that depend upon\nmodifications of halo structure\nand abundance at the low-mass\nend by baryonic effects, are\nalso available to solve the \nsame problems for SFDM-TF or FDM.  However, it may still be possible to\ndistinguish the three models at\neven smaller halo mass, e.g. in the \nminihalo mass range, below $\\approx{10^{8}\\text{ M}_\\odot}$.\nAccording to the results for $z=0$ in Table\n\\ref{tab:halocounts}, the more\ngradual cut-off of the HMF for SFDM-TF, relative to FDM, means\nthat, for $R_\\text{TF}=1$ pc, there is still a significant population of minihalos for SFDM-TF, albeit several times reduced from that for CDM, but\nan order of magnitude more than\nfor FDM with $m_{22} = 29$.\n\n\n\n\\section{Summary and Conclusions}\n\\label{sec:Conclusion}\n\n\nIn this paper and its companion \\citetalias{paper1}, we considered the gravitational dynamics\nand quantum hydrodynamics of an alternative form of cosmic dark matter to standard CDM -- \nSFDM-TF -- a scalar field comprised\nof ultralight bosons with a \nrepulsive self-interaction (SI)\nstrong enough to make the\nlength scale, which characterizes\nthat SI, larger than the de~Broglie\nwavelength (i.e. the Thomas-Fermi (TF)\nregime). \nHere, in Paper II, we studied structure formation in a universe with this kind of dark matter in a cosmological context, including the dynamics of both nonlinear collapse leading to halo formation and\nthe linear perturbations which\nform the initial conditions for\nstructure formation in this model.\nFor the former, we applied the tools developed in \\citetalias{paper1}\nto simulate cosmological infall and collapse to form individual virialized haloes.  As the first\nstep in understanding how cosmological initial and boundary\nconditions affect the outcome,\nwe replaced the non-cosmological initial conditions of \\citetalias{paper1} with CDM-like\nlinear initial conditions.\nThe latter were derived in order to make spherical infall occur at the\nrate which matches the empirical mass-accretion history (MAH)\nof haloes derived from\nN-body simulations of CDM.  Our previous application of this method\nfor fixing the initial perturbation for CDM simulations showed that\nCDM haloes formed in this way \nhave NFW-like mass and phase-space density profiles, including the time-dependence of the NFW \nparameters of an individual halo as it grows, in close agreement with\nCDM N-body simulations\n\\citep{AAS03,Shapiro04,Shapiro06}.  So, this provided an\nexcellent test-bed for determining the impact on this problem of replacing CDM with SFDM-TF in the cosmological context of the expanding universe.\n\nThe SFDM-TF haloes that resulted\n(see \\S\\ref{sec:simulationresults})\nshared some properties of those in \\citetalias{paper1}, namely, polytropic cores within\nradius $R_\\text{TF}$ and outer envelopes that matched those \nof a simulation with CDM dynamics from the same initial conditions,\nalthough in this case, the\nouter-envelope profiles were NFW-like, as expected.\nIn \\citetalias{paper1}, the envelopes were power-law\nprofiles, $\\rho \\propto r^{-12\/7}$,\neverywhere outside the cores,\njust as for CDM simulations\nfrom the same \\textit{non}-cosmological (i.e. static) initial conditions.\nIn the cosmological case here, \nhowever, between the core\nand the NFW-like outer envelope\nwas an intermediate zone with the same power-law profile, $\\rho \\propto r^{-12\/7}$, as \n\\citetalias{paper1} found\nfor its entire envelope outside\nthe core. The result\nof \\citetalias{paper1} that haloes\nform with polytropic cores surrounded by CDM-like envelopes is still an accurate description of the cosmological case, too, because the infall and collapse that\nleads to accretion shocks and virialized regions inside\nthe shock are essentially CDM-like\ndynamics outside the polytropic cores in both cases.  In other words, the term \\textit{CDM-like} characterizes the structure outside the cores in both cases, just from different initial conditions.  The reason the\ncosmological SFDM-TF \noutcome exhibits\n\\textit{both} kinds of CDM-like\nenvelope structure in the same\nhalo -- a power-law like the CDM simulation from non-cosmological initial conditions, surrounded \nby the NFW-like profile of CDM\nsimulated in the cosmological case -- is explained by the early\nsmoothing effect that SI has on\nthose  \ninitial conditions\nwhile they are still linear. \nAn SI-pressure-driven sound wave was found to smooth-out the mass within a proper radius $R_\\text{TF}$\nof the center, in\na time short compared to the time for that mass to expand  cosmologically, turn around and recollapse.  Once smoothed,\nthat mass continues expanding, but thereafter evolves like a top-hat perturbation, at the smoothed-out overdensity.\nWhen this mass collapses, the\noutcome for it is just like\nthat in \\citetalias{paper1}, while\nthe mass outside it evolves just\nas it would for those CDM-like initial conditions with no smoothing effect.\n\nThis has profound consequences for\nhalo formation from cosmological perturbations in SFDM-TF.\nIf SI is too strong, i.e. $R_\\text{TF}$ is too large, the\nSI-smoothing sound wave can\nsmooth-out a mass as large as that\nof the halo destined to form at some epoch in CDM from the same initial conditions,\npreventing its formation,\naltogether, or at least delaying\nit significantly.  In \\citetalias{paper1}, we showed\nhow SFDM-TF haloes with the right value of $R_\\text{TF}$ could resolve\nthe cusp-core and TBTF problems \nof CDM, without suffering\nfrom the problem identified for FDM\nby \\citet{VBB19} in which the\nsolitonic cores of larger-mass FDM haloes were too dense to be consistent with the rotation curves\nof more massive galaxies, if FDM\nparticle mass were tuned to \n$m_{22}\\simeq1$ to solve\nthe other problems.  We found\nthat $R_\\text{TF}\\gtrsim1$ kpc is\nrequired.  It was left to Paper II,\nhere, to determine if such a value\nof $R_\\text{TF}$ was consistent\nwith the cosmological formation of\nthose haloes.  Our cosmological simulations here indicate\nthat the SI-smoothed mass \nat early times is large \nenough to suppress\nthe formation of such dwarf galaxies if $R_\\text{TF}\\gtrsim1$ kpc, if\nwe start from CDM initial conditions.  At the other extreme,\nif $R_\\text{TF}\\lesssim1$ pc, then\nthe internal structure of haloes\nthat form from such CDM-like \ninitial conditions would closely resemble those of CDM, over most\nof the observable range of radii.\n\n\nThis still left open \nthe question of what\nrealistic cosmological initial conditions are for SFDM-TF halo formation, as distinct from those\nfor CDM.  To answer that requires\nlinear perturbation theory, \nfrom which the statistical likelihood of finding haloes of different masses at different epochs, can be determined. That\nwas the subject of \\S\\ref{sec:LinPertsTransferFn}.\nThere, we derived the transfer function for SFDM-TF, normalized\nby that for CDM, and, for comparison, following the same\nprocedure, did the same for\nFDM, to contrast the three models. With the transfer function, we were able to calculate statistical measures of structure formation for the SFDM-TF model, for the first time, including its halo mass function (HMF).  Just as\nthe FDM model is fully parameterized\nby its particle mass $m$, so\nSFDM-TF is parameterized entirely\nby its SI strength parameter,\n$g\/m^2$, which is directly related\nto the radius $R_\\text{TF}$ of the\n($n=1$)-polytropic sphere in hydrostatic equilibrium in which \ngravity is balanced by SI\npressure. Like FDM, SFDM-TF is\na model with a cut-off of power\non small scales, reflected as a\ncut-off at high wavenumber in\nthe transfer function.  As a\nresult, we used these findings\nto make a first cut at estimating\nthe range of $R_\\text{TF}$-values\nwhich are consistent with \nastronomical observations, by\nusing constraints on the FDM\nmodel derived elsewhere, \nexpressed as constraints on its $m_{22}$, as a proxy.\n\nAccording to our linear perturbation theory presented in \\S\\ref{sec:LinPertsTransferFn},\nthe Fourier mode at which SFDM-TF cuts off relative to CDM has a comoving mass associated with it that equalled the time-varying Jeans mass at the moment that mode entered the horizon (i.e. filled the Hubble volume).  With a Jeans length equal to $R_\\text{TF}$, fixed in proper coordinates, the comoving\nJeans mass during this phase decreased rapidly with scale factor, as $M_\\text{J,0}\\propto a^{-3}$. \nModes of smaller wavenumber entered the horizon later, with larger associated masses, \\textit{above} the Jeans mass, \nso they always grew like CDM. \nModes of higher wavenumber entered earlier, however, when their associated masses were \\textit{below} the Jeans mass and, so, were initially retarded relative to CDM, but, as soon as that rapidly shrinking Jeans mass reached their mass scale, they, too, were free, thereafter, to grow like CDM.\nThe cosmological story of structure formation in SFDM-TF is, therefore,\nvery much a tale of \\textit{``the incredible shrinking Jeans mass''}, which is the key to understanding how it differs\nfrom other cut-off models \nlike FDM \\big(for which the Jeans\nmass only drops weakly, as\n$M_\\text{J,F}\\propto a^{-3\/4}$\\big)\nwhich do not share this unique property.\nAnother factor that distinguishes\nSFDM-TF from FDM, in particular,\nis that, even when SFDM-TF\nmodes are \\textit{sub-Jeans}, \nthey still grow, albeit modestly, as $a^{1\/4}$ (when the field is\nmatter-like), while for FDM, there is \\textit{no} sub-Jeans growth. \n\nThis explains why, when we \napplied the same treatment to FDM to identify its cut-off wavenumber (as a function of $m_{22}$), and solved for the $R_\\text{TF}$-values required to align the cut-off wavenumbers of the two models, SFDM-TF haloes had SI-polytropic halo cores\n(of radius $R_\\text{TF}$) much\nsmaller than the solitonic cores of\nFDM haloes of the same mass.  \nFor example,\nFDM with $m_{22}\\simeq 1$, for \nwhich a $10^{10}\\text{ M}_\\odot$ halo has a solitonic core size\n(of order the de~Broglie wavelength) of $\\simeq 4$ kpc, cuts off at close\nto the same wavenumber as SFDM-TF\nwith $R_\\text{TF}\\simeq3$ pc, \nwhich gives a halo of that mass a polytropic core  $\\approx$\\,1000 times smaller.  This is\nconsistent with our simulations in \\S\\ref{sec:simulationresults},\nwhich showed that halo formation\nin SFDM-TF from a CDM-like perturbation experiences early SI-smoothing that, in effect, ``Jeans-filters'' a mass closer to the \\textit{initial} value\nof the Jeans mass than to its \nsmaller value after the collapse is\nfinished.  As a result, although\nSFDM-TF haloes with\nflattened cores as large\nas $R_\\text{TF}\\!\\gtrsim\\!1$ kpc,\nformed by Jeans \ninstability and collapse from \na static initial condition,\nwere shown in \\citetalias{paper1}\nto be capable of resolving the \ncusp-core problem of CDM while \\textit{simultaneously} solving its\nTBTF problem, as well,\nthis unified explanation \nis disfavored, \nonce formation is placed in the context of\ncosmological perturbation growth.\n\nObservational constraints on FDM\ncan be used as a proxy for those\non SFDM-TF by solving for the matching $m_{22}$ and $R_\\text{TF}$ values that make their respective transfer functions cut off at the same wavenumber, e.g. as estimated analytically by equation (\\ref{eq:RTFversusmFDM}).  Care\nshould be taken to \navoid FDM constraints that are based upon phenomenology that might distinguish it from SFDM-TF because of their different dynamics.  \nSince lowering \n$m_{22}$ increases the de~Broglie\nwavelength and extends the mass range suppressed by FDM to larger scales, many of its observational constraints are expressed as lower limits on $m_{22}$ required to avoid over-suppressing structure.   \nThere is\na long list of attempts to\nestimate or limit $m_{22}$, with results roughly spanning\nthe range from 1 to 30 \n(\\citealp[as summarized, e.g., in][]{AWP}). Some are based\nupon the comparison of the abundance of haloes or subhaloes with the\npredicted HMF for FDM.  \\citet{Nadler21}, for example,\ncompare the predicted\nHMF of subhaloes to the observed\ndwarf galaxies in the Local\nGroup and find that $m_{22}\\gtrsim{29}$ is required,\ncorresponding to $\\lambda_\\text{deB} \\lesssim 0.5$~kpc. Others\nare based upon more complicated,\nnonlinear dynamical phenomena. \nFor example,\nthe requirements of not dynamically over-heating the Milky Way disk or its stellar streams were\nused to place bounds of $m_{22} \\gtrsim 0.6-1.5$ \\citep{AL18, Church19}. Still\nothers are based upon structure on larger scales, at\nepochs when it is only\nquasi-linear, but where one must model effects on the baryonic component, too, such as\nfluctuations in the matter-density\nfield inferred from the\npower spectrum of measured fluctuations in the Lyman-$\\alpha$\nforest. Bounds inferred from the Lyman-$\\alpha$ forest were the \nfirst to pose the ``catch-22 problem'' for FDM, in fact, \nfinding $m_{22} \\gtrsim 20-30$ \\citep{sc22,sc23}, now\nconsistent with the recent\nlower limits\nquoted above from the abundance\nof dwarf satellites in the Local\nGroup. \n\nIf the observational \nconstraints on FDM we use as our proxy for those on SFDM-TF\nare taken to be \n$1 \\lesssim m_{22} \\lesssim 30$,\nthen the range of $R_\\text{TF}$-values for which\nthe two models have the same cut-off wavenumber corresponds\nroughly to \n$3 \\text{ pc} \\gtrsim R_\\text{TF} \\gtrsim 0.2$ pc.  Of all the methods for\nconstraining the FDM model we \nmight use, as a more \\textit{specific} proxy than this\nto constrain SFDM-TF, those comparing\ntheir HMFs are,\nto first approximation, less affected by additional differences in their dynamics\nthan others.  For this reason,\nwe focused most of our additional\ncomparison here on the results of\ntheir HMF predictions, as discussed\nin \\S\\ref{sec:HMFresults}.\nWe showed there that, for the same\ncut-off wavenumbers, the SFDM-TF \nHMF is much more gradually cut off\nthan that of FDM, so it retains\nmany more haloes at the low-mass\nend.  As a result, if we\nwere, instead, to solve for the\n$R_\\text{TF}$-value that yields the\nsame halo number as FDM\nfor some value of $m_{22}$, the\ncorresponding $R_\\text{TF}$-value\nis generally larger than the\none that makes the cut-off wavenumbers match, described\nabove.  For example, if we\nintegrate over their HMFs\nto count haloes above \n$10^8\\text{ M}_\\odot$, \nSFDM-TF with $R_\\text{TF}=1$ pc\nhas a halo number comparable to\nthat of FDM with $m_{22}=29$, and within a factor of 2 of that for CDM, while the integrated collapsed\nfractions in this halo mass range\nin all three models are very close.\nIn that case, we should\nmultiply the $R_\\text{TF}$-value\nabove that matches the cut-off wavenumber for this $m_{22}$ by\na factor of 5, from 0.2 to 1 pc,\nto use the HMF FDM constraint as a proxy for SFDM-TF. Moreover, \nif $R_\\text{TF}=10$ pc, \nSFDM-TF has many more haloes\nthan does FDM with $m_{22}=0.8$,\nby an order of magnitude, and\nstill has $14\\%$ of the CDM number.\nIn that case, we should refine\nthe first proxy range \nestimated above, upward,\nat least to $10 \\text{ pc} \\gtrsim R_\\text{TF} \\gtrsim 1$ pc.  If we compare\nthe integrated halo counts \nin the two models down to even smaller halo mass, the corresponding $R_\\text{TF}$-values shift upward\neven further. If the minimum halo mass is lowered to $10^6\\text{ M}_\\odot$, for example,\nthen SFDM-TF with $R_\\text{TF}=1$ pc has an order of magnitude more haloes than does FDM with $m_{22}=29$, while \nif $R_\\text{TF}=10$ pc, it has \nmore by a factor of 300 than FDM\nwith $m_{22}=0.8$. \nThis indicates that one must be\ncareful not to take the simple\nalignment of the cut-off wavenumbers of SFDM-TF and FDM as a basis\nfor using the observational\nconstraints on $m_{22}$ to solve\nfor the corresponding constraints\non $R_\\text{TF}$ -- that will generally be overly restrictive,\nwith upper limits on $R_\\text{TF}$\nwhich are too low. It further\nsuggests that SFDM-TF may\nhave some advantages \nover FDM since, even when\ntheir model parameters are tuned\nso as to \\textit{match} HMFs \nat larger\nmass scales, SFDM-TF may still have\nmore smaller-mass haloes.  For\nexample, when FDM and SFDM-TF\nare both limited so as to avoid\nover-suppressing the\ndwarfs in the Local Group, \nSFDM-TF may still have enough even-smaller-mass subhalos to account for the perturbers of stellar streams observed in the Milky Way, even if FDM does not \\citep[e.g.][]{Banik21}. \n\nTo test SFDM-TF against observations more directly in future work, \nit will be necessary to revisit\nthe full list of phenomenological\nconstraints that have been applied\nto FDM and other models that\nsuppress small-scale structure\nas a built-in feature (e.g. warm dark matter),\nwith SFDM-TF initial conditions and dynamics, instead.  This will require us to apply the cosmological initial conditions derived here for SFDM-TF in fully 3D simulations of galaxy and large-scale structure formation, which are under development. This includes the coupling of dark matter and baryonic components, as well as pure dark matter simulation.  \nFor ``cut-off models'' like these,\nthe nature of structure formation\non scales at or below the cut-off\nis different for different types\nof dark matter, even when they\nstart from transfer functions that\nhave similar cut-off scales,\nas our results here demonstrate\nby comparing the HMFs for SFDM-TF and FDM.   Additional effects,\nlike fragmentation of \npancakes and filaments, for\nexample, can also introduce novel small-scale structures, unfamiliar from the hierarchical clustering paradigm of CDM \\citep[e.g.][]{Mocz19,Valinia97}.  The small-$R_\\text{TF}$-regime favored \nthus far by our results presented above, for which \nthe polytropic core sizes are\nsubgalactic (i.e. sub-kpc), may\npresent other novel features\nwhich are detectable, as well,\nworth exploring further.   As discussed in \\citet{Padilla20}, for example, such cores, \nif dense enough, may\nbe subject to general relativistic\ninstability leading to collapse to\nform supermassive black holes or\ntheir seeds.  \n\n\n{\nIn this paper, we considered the r\\^ole of SI in structure formation in the SFDM model, in the TF regime.   We also compared the results in this regime with the other limiting case, of FDM, in which SI is absent.   As we showed in \\S\\ref{sec:LinPerts}, as long as the particle mass and interaction strength place the scalar field in this regime at late times (e.g. the present), it is in the TF regime at earlier times, as well. In the future, we will consider the possibility of a transitional regime between these two limits.\n}\n \n\n\\section*{Acknowledgements}\n\nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1610403. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. \nT.R.-D. is supported by the Austrian Science Fund FWF through an Elise Richter fellowship, Grant No. V 656-N28. \nSimulations presented here were conducted on the Texas Advanced Computing Center's Stampede2 supercomputer under accounts asoz-630 and NSF XSEDE account TG-AST090005.  \nThe authors gratefully acknowledge  \n\\textit{The Incredible Shrinking Man }(1957)\n~[Film], directed by Jack Arnold, Universal-International Pictures Co., Inc., as the inspiration for our subtitle.\n \n\n\n\\section*{Data Availability Statement}\n\nThe data underlying this article are available in the article and in its online supplementary material.\n\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\\label{sec:Introduction}\n\n\n\nThe nature and origin of cosmic dark matter is still unknown.  The standard Cold Dark Matter (``CDM'') model is actually a place-holder with certain assumed properties -- e.g. it behaves\nduring the structure formation era as a\nnonrelativistic, collisionless gas of particles that couple only weakly to each other and to other\ncomponents of the Universe, via gravity alone,\ncontributing to the expansion rate of the Universe as a pressure-free component whose mass density, conserved over time, is measured astronomically.  A further assumption is made \nthat the CDM particles are born so ``cold'' -- i.e. with such small peculiar motion -- that their free-streaming could only have washed out primordial density fluctuations on the tiniest\nof scales, many orders of magnitude below the scale that affects the formation of galaxies or even their internal structure. With these assumptions, it is possible to work out, within the\ncontext of the standard big bang model and inflationary paradigm, how structure formation proceeds from linear-perturbation initial conditions through nonlinear formation of galaxies, clusters and the cosmic web.  This model has successfully accounted for many observational properties of the Universe on large scales, from galaxy formation scales on up.\nIt has, however, been challenged by over-predicting\nthe level of small-scale structure inside galaxies\ncompared to that observed in a variety of systems (see, for instance, \\citealp{BBK17} and references, therein).  This may reflect\nour incomplete knowledge of the impact on galaxy\nformation of the baryonic component, coupled via\ngravity to CDM.  However, it has also led\nto a consideration of microscopic properties\nof CDM that go beyond the standard assumptions\ndescribed above, in ways that can affect\nsmall-scale structure without spoiling its success on larger scales.  \n\nThe best-studied origin story\nof CDM involves \nWeakly Interacting Massive Particles (``WIMPs'') with masses in the range $\\sim$ GeV - TeV, predicted\nby theories of supersymmetry, which form as big bang thermal relics whose abundance is set\nwhen their annihilation at a weak-interaction-like rate becomes slower than the Hubble expansion rate.\nThis so-called ``standard CDM'' -- more\nproperly referred to as ``WIMP-CDM'' -- \nis more specific than the CDM whose\ngeneric properties we identified above, with additional observable consequences, and, as such,\nis more tightly constrained.  \nIn particular, it predicts that the same annihilations might lead to their indirect detection from astronomical sources \n(e.g. by gamma-rays), while their \nscattering interactions with nucleons \nmight permit their direct detection.  So far,\nhowever, despite considerable effort,  \nthere are no confirmed detections, and the allowed\nrange of particle models and parameters that remain\nviable, consistent with that nondetection, has\nshrunk \\citep[see e.g.][]{Schumann19}.  This has added some urgency to the consideration of alternative models for\nthe origin and microscopic nature of CDM, even\napart from the possible conflicts it has of\nover-predicting small-scale structure on\ngalactic scales.  We emphasize this last point\nat the outset, because it is often missed when\nevaluating alternatives to WIMP-CDM, by\ndismissing the alternatives as soon as it\nappears they are merely ``CDM-like'' at\nall scales of interest, i.e. failing to\nprovide the ``Deus ex machina'' of\nreducing the apparent over-prediction of small-scale structure by CDM, while retaining\nall of its successes on larger scales. \nThat is, as long as the origin of CDM is\nunknown, any microscopic model\nthat is observationally indistinguishable\nfrom it is just as successful as any other, until it is disfavored\neither by further theoretical considerations, or by predictions of additional observables\nthat can distinguish it from the standard model.  \n\nOne such alternative model of great interest is Scalar Field Dark Matter (``SFDM'') comprised of ultralight ($m \\gtrsim 10^{-22}$ eV$\/c^2$) \nbosons, originally suggested because of its \nnovel structure-formation dynamics as a Bose-Einstein condensate and quantum superfluid, described by coupled Schr\\\"odinger-Poisson equations.\\footnote{The field of SFDM study has an extensive literature, well-beyond our ability to cite it here.  A few early references not otherwise cited in this paper include \\citet*{Sin94,LK96,MGU00,GU04,MVM09,SC17,DKR18}. Some reviews with additional references therein include \\citet*{SRM14,RDS14,Marsh16}.} \nIn the free-field (``fuzzy'') limit of SFDM, or fuzzy dark matter (``FDM''), the small-scale \nstructure which we described above for \n``generic'' CDM is suppressed below the \nde~Broglie wavelength, $\\lambda_\\text{deB}$, but resembles CDM on larger scales.  \nVirialized FDM haloes have been shown to have\nsolitonic cores of radius $\\sim\\lambda_\\text{deB}$,\nsurrounded by CDM-like envelopes (\\citealp*{SCB14, SNE16}; \\citealp{Mocz17}). When a strong enough repulsive self-interaction (SI)\nis also present, \nstructure can be inhibited below a second length scale, $\\lambda_\\text{SI}$, \nwith $\\lambda_\\text{SI}>\\lambda_\\text{deB}$ -- referred to as the Thomas-Fermi (``TF'') regime.\nThis TF regime of SFDM (henceforth, ``SFDM-TF''),\nassuming a quartic potential for the repulsive SI, has been referred to variously as ``BEC-CDM'' \\citep{RDS12,RDS14},\nSFDM \\citep*{Li14,Li17}, ``Repulsive Dark Matter'' \\cite{Goodman2000}, or more generally ``Fluid Dark Matter'' in \\cite{Peebles2000}.\\footnote{\nNote that SFDM-TF should not be confused with another CDM alternative known as self-interacting dark matter (SIDM). \nThe latter behaves as a collection of point-particles undergoing elastic scatterings, which\namounts to adding 2-body elastic collisions to the standard, collisionless gravitational dynamics of CDM, while the former is a BEC, best thought of as a classical field evolving under a continuous self-interaction potential (along with the gravitational potential).\nThe two models are phenomenologically distinct, as is evident by the r\\^ole SI plays in their respective non-relativistic fluid equations. In the SFDM-TF model (as discussed in \\S\\ref{subsection:fluidapprox}), SI enters as a potential or pressure term in the momentum equation (equation~\\ref{eq:1momLagrangian}), whereas in the SIDM model, SI enters as a heat conduction term in the energy equation \\citep*{BSI02,AS05}.}\n\nStructure formation in the TF regime differs significantly from FDM.  \nIn \\citet*{paper1} \\citepalias[henceforth][]{paper1}, we discussed the nonlinear dynamics and internal structure \nof haloes that form from gravitational instability in the TF regime.  Here we revisit this in the \ncontext of halo and large-scale structure formation from cosmological perturbations, \nincluding observational constraints. As this is a companion paper (``Paper II''), we \nwill sometimes refer the reader to \\citetalias{paper1} for details, \nfor the sake of brevity and to avoid\nredundancy.  \n\nIn \\citetalias{paper1}, we developed equations\nand tools to calculate the dynamical evolution of inhomogeneity\nin response to gravitational instability in this model, \nstarting from the nonlinear Schr\\\"{o}dinger equation (``NLSE'').\nA key development was to take\nfull account of both quantum pressure (associated with inhomogeneity on the tiny\nscale of the de~Broglie wavelength) and SI, based upon a\nreformulation of the NLSE, smoothed on scales larger than \nthe de~Broglie scale. In that case,\nthe NLSE reduced to the collisionless Boltzmann equation\nthat describes CDM, except with\nthe SI potential added to\nthe usual gravitational potential. \nFrom this, a fluid approximation \nfor SFDM-TF was derived, which\nadded a new pressure force for\nSI to equations we \npreviously derived for CDM.\nThis enabled us to describe the scalar field by the familiar \nhydrodynamical conservation\nequations for a $\\gamma=5\/3$ ideal gas, with an ideal gas pressure,\n$P_{\\!\\sigma}$, \nassociated with particle velocity dispersion (which accounts\nfor the large-scale effects of\nquantum pressure), and with a\nsecond pressure, $P_\\text{SI}$, which accounts for the\nrepulsive SI force. This\nfluid approximation allowed us to\nevolve SFDM-TF computationally\nin spherical-1D with the very \nhigh resolution and dynamic\nrange which are required\n(but not currently possible)\nfor full 3D simulations of SFDM-TF.\nWith this tool, we solved the Newtonian problem of virialized halo formation by the nonlinear collapse of a Jeans-unstable perturbation,\nfrom non-cosmological (i.e. static)\ninitial conditions, for the\nfirst time. These simulations in \n\\citetalias{paper1} showed that\nhaloes formed in this way have \ncore-envelope density profiles, with\nflattened cores of size $R_\\text{TF}$ -- \nthe radius of the ($n=1$)-polytrope that forms for a self-gravitating fluid in hydrostatic equilibrium when supported by SI pressure alone, referred to as the Thomas-Fermi radius --\nsurrounded by an envelope just like the one that\nformed when CDM dynamics \nwas simulated, instead, \nfrom the same initial\nconditions. This explained why, in the case of FDM, when haloes form in 3D simulations (which are possible\nfor FDM, where the de~Broglie wavelength is large enough to be \nfully resolvable, unlike SFDM-TF,\nfor which de~Broglie wavelengths are unresolvably small),\ndensity profiles are also CDM-like\noutside their solitonic cores (of size close to the de~Broglie wavelength).\n\nIn \\citetalias{paper1}, we showed\nhow SFDM-TF haloes with the right value of $R_\\text{TF}$ could resolve\nthe cusp-core and ``too-big-to-fail'' (TBTF) problems\nof CDM, without suffering\nfrom the problem identified for FDM\nby \\cite*{VBB19} in which the\nsolitonic cores of larger-mass FDM haloes were too dense to be consistent with the rotation curves\nof larger-mass galaxies, if the FDM\nparticle mass were tuned to \n$m\\simeq10^{-22}$~eV$\/c^2$ to match observations of smaller-mass galaxies.  We found\nthat $R_\\text{TF}\\gtrsim1$ kpc is\nrequired.  It was left to Paper II,\nhere, to determine if such a value\nof $R_\\text{TF}$ was consistent\nwith the cosmological formation of\nthose haloes.\n\nIn what follows, we study the cosmological context for structure formation in the SFDM-TF model, \nboth by simulating individual halo formation cosmologically,\nand by deriving the initial conditions for Gaussian-random\nprimordial fluctuations from\nlinear perturbation theory. \nIn \\S\\ref{sec:basicequations}, we\npresent the model and basic equations we use to study SFDM-TF\ncosmologically.  In \\S\\ref{sec:infall}, we\napply these equations to simulate\nindividual halo formation by spherical infall and collapse in a cosmologically expanding SFDM-TF universe.  The overall scenario for cosmic structure in the SFDM-TF model is developed in \\S\\ref{sec:LinPertsTransferFn}, including\nlinear perturbation theory to\nderive the transfer function and\npower spectrum, as well as their\napplication to statistical\nmeasures of structure formation like\nthe halo mass function.  We apply\nthe same approach to the FDM\nmodel, as well, for comparison.\nIn \\S\\ref{sec:results}, we present\nthe results of this linear perturbation theory and use \nobservational constraints on the\nFDM model, fully parameterized by\nits particle mass $m$,\nas a proxy to derive\nconstraints on the SFDM-TF model,\nparameterized by $R_\\text{TF}$.\nOur summary and conclusions are presented in \\S\\ref{sec:Conclusion}. \n\n\n\\section{The model and basic equations}\n\\label{sec:basicequations}\n\n\\subsection{The fully relativistic description}\n\\label{sec:fullyrelativistic}\n\nWe model SFDM as a classical, massive, complex\nscalar field $\\phi$ with a quartic, repulsive self-interaction (SI), which obeys the Klein-Gordon equation\n\\begin{equation}\n    \\Box \\phi + \\frac{m^2 c^2}{\\hbar^2}\\phi + \\frac{2 g}{\\hbar^2} |\\phi|^2 \\phi = 0\n    \\label{eq:KGE}\n\\end{equation}\nwhere $\\Box \\equiv \\nabla_\\mu \\nabla^\\mu$ is the d'Alembertian for a general 4D spacetime metric, $m$ is the boson mass, \nand $g$ is the SI coupling strength. This equation is coupled to the Einstein equations, with a metric tensor given \nby the Friedmann-Lemaitre-Robertson-Walker metric for a homogeneous and isotropic background universe, \nperturbed by primordial fluctuations.  The scalar field contributes terms to the stress-energy tensor in the Einstein equations, which contain the field and its derivatives.  \n\nAs we described in \n\\citet{Li14,Li17}, the evolution of the homogeneous background scalar field corresponds to that of a perfect fluid. However, the scalar field oscillates \nwith an angular frequency which eventually\nexceeds the Hubble expansion rate, when \n\\begin{equation}\n    \\omega \\simeq \\frac{mc^2}{\\hbar} \\sqrt{1 + \\frac{2g}{m^2 c^2}|\\phi|^2 } \\gg H\n\\end{equation}\n\\citep[see equation 21 in][]{Li14},\nwhere the square of the modulus of the field, $|\\phi|^2$, is, at late times, related to the SFDM mass density by  $\\rho_b=\\langle|\\phi|^2\\rangle=\\rho_b(a=0)a^{-3}$ (the subscript ``$b$'' refers to background), and the Hubble parameter $H(t) = (da\/dt)\/a$ with scale factor $a=a(t)$ depending on cosmic time $t$.  \nPrior to this rapid oscillation epoch, the \nfield evolution was shown to start from a kinetic-energy-dominated, stiff, \nrelativistic phase during the slow-oscillation period when $\\omega \\ll H$, but eventually to transition \nover time to a radiation-like, still relativistic phase during the rapid oscillation period, \nwhen the SI term in equation~(\\ref{eq:KGE}) dominates the quadratic mass term.\nFinally, at later times, when the mass term dominates, there is a transition to a nonrelativistic, matter-like phase.  More precisely, as a perfect fluid, the field can be characterized by an equation of state parameter $w$ defined as the ratio of pressure to energy density, which varies over time to reflect these phases of evolution.  Initially, $w=1$ (stiff phase), which must end prior to the epoch of big bang nucleosynthesis (BBN), in order not to spoil the agreement between predictions of the standard model involving primordial nucleosynthesis in a radiation-dominated (RD) universe with \nthe observed relic abundances of the light elements. Eventually, when the field oscillates much more rapidly than the Hubble expansion rate,\nwe can average the pressure $P$ and energy-density $\\varepsilon$ of the field over the oscillation period and take the ratio of these average values to give the average value of the equation\nof state parameter $w$, according to the following \\citep[which is equation 26 in][]{Li14}:\n\\begin{equation}\n    \\langle w \\rangle \\equiv \\frac{\\langle P \\rangle}{\\langle \\varepsilon \\rangle} = \\frac{1}{3} \\left( \\frac{1}{1 + \\frac{2m^2c^2}{3g \\langle |\\phi|^2 \\rangle}} \\right)\n    \\label{eq:averagew}\n\\end{equation}\nWhen the SI term dominates during this phase,\n$\\langle w \\rangle \\simeq 1\/3$ (radiation-like), but at late times, before the\nuniverse made its transition from radiation-\nto matter-dominated (MD), the mass term\nmust come to dominate and $\\langle w \\rangle \\simeq 0$ thereafter, rendering SFDM matter-like, as it should be in order to account for the MD era.\\footnote{This description above\nfrom \\citet{Li14,Li17}, of the evolution of the EOS\nof the homogeneous scalar field, from stiff,\nrelativistic ($w=1$) at early times, before\nthe rapid-oscillation phase, to radiation-like\n\\big($\\langle w \\rangle \\simeq 1\/3$\\big) when rapid oscillation begins and SI dominates, and finally\nto nonrelativistic, matter-like \n\\big($\\langle w \\rangle \\simeq 0$\\big),\nwas derived for a complex scalar\nfield.  However, we are only interested here\nin the rapid-oscillation phase, during which\nthe EOS is radiation-like when the SI term \ndominates over the mass term in the Lagrangian,\nthen transitions to matter-like when the\nlatter term dominates.  As such, \nwhat happens before the rapid-oscillation \nphase begins does not affect our calculations here.  \nIn that case, we do not expect our calculations\nto depend upon whether the field is complex or\nreal, since the description above of the EOS\nevolution during the rapid-oscillation phase\napplies equally well to the real and complex cases.}\n  \n\n\n\n\\subsection{Matter-like regime and the Newtonian approximation}\n\\label{sec:matterlike}\n\nIn this latter \\big($\\langle w \\rangle \\simeq 0$\\big) regime, in the limit of weak-field (i.e. Newtonian) gravity and even in the presence of inhomogeneities, as long as they are\non scales well-within the horizon (i.e. $\\ll c\/H$), we can adopt the Newtonian approximation in which gravity obeys the Poisson equation. The equation of motion of the scalar field in this nonrelativistic regime then simplifies to the nonlinear Schr\\\"odinger equation (NLSE) in an expanding background, by factorizing the field into a slowly-varying ``mean'' field $\\psi$ and an oscillatory component, as follows:\\footnote{\nIf $\\phi$ is a real field, then one should add to the right side of equation~(\\ref{eq:meanfield}) the complex conjugate, i.e. $\\phi = \\psi e^{-imc^2t\/\\hbar} + \\psi^* e^{imc^2t\/\\hbar}$, where $\\psi$ is still a complex field.}\n\\begin{equation}\n    \\phi = \\psi e^{-imc^2t\/\\hbar} \\quad \\quad \\quad |\\partial_t \\psi| \\ll \\left| \\frac{mc^2}{\\hbar} \\psi \\right|\n    \\label{eq:meanfield}\n\\end{equation}\nIn terms of the comoving coordinate $\\bm{x}$, defined with respect to the proper coordinate $\\bm{r}$ such that\n\\begin{align}\n    &\\bm{x} = \\bm{r}\/a \\\\\n    &\\nabla \\equiv \\nabla_{\\bm{x}} = a\\nabla_{\\bm{r}}\n\\end{align}\nand\n\\begin{equation}\n    \\partial_t \\equiv (\\partial_t)_{\\bm{x}} = (\\partial_t)_{\\bm{r}} + H \\bm{r} \\cdot \\nabla_{\\bm{r}}\n\\end{equation}\nwhere $(\\partial_t)_{\\bm{x}~(\\bm{r})}$ represents the partial time derivative at fixed comoving (proper) coordinate $\\bm{x}$ ~($\\bm{r}$),\nthe NLSE and Poisson equations are then\n\\begin{equation}\n    i\\hbar \\left(\\partial_t + \\frac{3}{2}H \\right)\\psi = \\left( -\\frac{\\hbar^2}{2ma^2} \\nabla^2 + \\frac{g}{m}|\\psi|^2 + m\\Phi \\right) \\psi\n\\end{equation}\nand\n\\begin{equation}\n    \\nabla^2 \\Phi = 4\\pi G a^2 (\\rho - \\rho_b)\n\\end{equation}\nwith\n\\begin{equation}\n    \\rho = |\\psi|^2\n\\end{equation}\nwhere $\\Phi$ is the peculiar Newtonian gravitational potential, $\\rho$ is the mass density of the scalar field, and $\\rho_b=\\rho_b(a=0)a^{-3}$ is the homogeneous background value.\\footnote{\nWe shall generally assume here, unless otherwise stated, that the nonrelativistic matter density is dominated by the scalar field dark matter density, so we do not need to keep separate track of other nonrelativistic mass components like the baryons.}\n\n\nThese equations were presented in \\citetalias{paper1}\nin the Newtonian limit in the\nabsence of cosmological expansion.  The difference\nhere is that we represent derivatives and spatial coordinates in the comoving frame,\nwhich introduces the homogeneous expansion term involving Hubble parameter $H(t)$,\nand the peculiar gravitational potential $\\Phi$.\nIf the scale factor were constant, $a(t) = 1$,\nwe would, of course, recover the equations of \\citetalias{paper1}.\n\nAs we discussed in detail in \\citetalias{paper1}, \nthere is a fully equivalent formulation of this\nnonrelativistic NLSE as quantum hydrodynamics\nequations (``QHD'').  In particular, the NLSE\ncan be transformed into a pair of hydrodynamical\nmass and momentum conservation equations via the Madelung transformation,\nin which the field is expressed in polar form as\n\\begin{equation}\n    \\psi = |\\psi|e^{iS}\n    \\label{eq:Madelung}\n\\end{equation}\nand the conservation of current density\nallows us to identify the peculiar bulk fluid velocity in the current flux density as\n\\begin{equation}\n    \\bm{v} = \\frac{\\hbar}{ma} \\nabla S\n\\end{equation}\nThis leads to\n\\begin{equation}\n    \\partial_t \\rho + 3H\\rho + \\frac{1}{a} \\nabla \\cdot (\\rho \\bm{v}) = 0\n\\end{equation}\nand\n\\begin{equation}\n    \\partial_t \\bm{v} + H\\bm{v} + \\frac{1}{a} (\\bm{v} \\cdot \\nabla) \\bm{v} + \\frac{1}{a} \\nabla (Q + V_\\text{SI} + \\Phi) = 0\n\\end{equation}\nwhere the ``quantum potential'' $Q$ and\nself-interaction potential $V_\\text{SI}$ are given\nby \n\\begin{equation}\n    Q = -\\frac{\\hbar^2}{2m^2 a^2} \\frac{\\nabla^2 \\sqrt{\\rho}}{\\sqrt{\\rho}}\n\\end{equation}\nand\n\\begin{equation}\n    V_\\text{SI} = \\frac{g \\rho}{m^2}\n\\end{equation}\nThese may also be written as pressure terms:\n\\begin{equation}\n    \\nabla Q = \\frac{1}{\\rho} \\nabla \\cdot \\bm{\\Pi}\n\\end{equation}\nwhere $\\bm{\\Pi}$ is the momentum flux density tensor, also known as the ``quantum pressure'' tensor, given by\n\\begin{equation}\n    \\Pi_{ij} = \\Big(\\frac{\\hbar}{2m a}\\Big)^2 \\Big( \\frac{1}{\\rho}\\frac{\\partial \\rho}{\\partial x_i}\\frac{\\partial \\rho}{\\partial x_j} - \\frac{\\partial^2 \\rho}{\\partial x_i \\partial x_j} \\Big)\n    \\label{eq:QPT}\n\\end{equation}\nand\n\\begin{equation}\n    \\nabla V_\\text{SI} = \\frac{1}{\\rho} \\nabla P_\\text{SI}\n    \\label{eq:SIpressure}\n\\end{equation}\nwhere $P_\\text{SI}$ is the self-interaction (SI) pressure, given by\n\\begin{equation}\\label{eq:SIpressure2}\n    P_\\text{SI} = \\frac{g\\rho^2}{2m^2}\n\\end{equation}\nIn the matter-like phase when $\\langle w \\rangle \\simeq 0$, the parameter $g\/m^2$ is related to the radius $R_\\text{TF}$ of the ($n=1$)-polytrope which results when gravity balances SI pressure in hydrostatic equilibrium, sometimes referred to as the Thomas-Fermi (TF) radius,\naccording to\n\\begin{equation}\n    R_\\text{TF} = \\pi\\sqrt{\\frac{g}{4 \\pi G m^2}} \\simeq 1.1 \\text{ kpc} \\left( \\frac{g\/(mc^2)^2}{2\\times 10^{-18} \\text{ eV}^{-1}\\text{cm}^3} \\right)^{1\/2}\n    \\label{eq:RTF}\n\\end{equation}\n\n\n\\subsection{The TF regime and the limit of small de~Broglie wavelength}\n\\label{sec:smalldeB}\n\nAs discussed in \\citetalias{paper1}, the same\nQHD equations can also by derived by\na reformulation of the NLSE in phase space,\ninvolving momentum moments of a quantum phase space distribution function.  In one version\nof that approach, involving a smoothed phase space\ndistribution, we showed that when the\nsmoothing scale is much larger than the de~Broglie\nwavelength $\\lambda_\\text{deB}$, the NLSE is\nequivalent to the Collisionless Boltzmann \nEquation (``CBE''), except with the gravitational\npotential term replaced by the sum of the\ngravitational and SI potentials.  This \nreduction of the NLSE to the CBE in the\nsmall-$\\lambda_\\text{deB}$ limit, for the free-field case \\textit{without} the modification to account for SI introduced in \\citetalias{paper1},\nwas previously discussed by \\citet{Mocz18}, in regards to the FDM model,\nfollowing earlier work by \\citet{WK93} and \\cite*{SKV89}.  Since the\ndescription of standard CDM as a gas of point-particles which are nonrelativistic, \nnoninteracting (except gravitationally), and\ncollisionless (i.e. the 2-body gravitational relaxation time is much longer than the\nlifetime of the system) is consistent with\nthe CBE, this explains why FDM behaves just like CDM, dynamically,\non scales much\nlarger than $\\lambda_\\text{deB}$,\ni.e. when viewed in this coarse-grained way that does not\nresolve substructure on scales smaller than\n$\\lambda_\\text{deB}$.  \nFor example, numerical simulations of \nthe NLSE in the FDM limit (no SI) show that,\nwhen haloes form gravitationally and virialize, \nthey have a core-envelope structure in which\na solitonic core of size $\\sim\\lambda_\\text{deB}$\n(with a centrally-flattened density profile \nlike the ground-state,\nstatic equilibrium solution of the 1D, spherical\nNLSE) is surrounded by an envelope\nwith a profile like that which would have \noccurred in standard CDM.  With strong enough SI\nto be in the TF regime, however, we showed\nin \\citetalias{paper1} that \nthe solitonic core would be replaced\nby a core,  much larger than \n$\\lambda_\\text{deB}$ (the latter is arbitrarily small\nin this TF regime), \nsupported instead by SI pressure, while\noutside this SI-supported core, \na CDM-like envelope would still be expected. \nHowever, to solve the latter problem in this TF regime, \nby the same kind of 3D simulations as were done for FDM, \nis much more difficult, since it would \nrequire fully resolving length scales well-below\n$\\lambda_\\text{deB}$ in order to produce the correct CDM-like behavior\noutside the core. In the TF regime, $\\lambda_\\text{deB}$ is\nmuch smaller than the size of the SI-smoothed halo\ncores, which are, themselves, much smaller than the \nhaloes, and even smaller relative to \nthe large-scale structure from which they formed, \nso the required resolution and dynamic range becomes prohibitively expensive.\nWe solved\nthis problem by finding an approximate fluid\ndescription for SFDM-TF, as described below. \n\n\n\\subsection{The fluid approximation for SFDM-TF}\n\\label{subsection:fluidapprox}\n\nIn \\citetalias{paper1}, we were tasked with calculating the dynamical formation of virialized haloes in SFDM-TF by evolving the scalar field\nfrom a nearly-uniform, linearly-perturbed, initial condition through many orders of magnitude of gravitational collapse, all the while\nkeeping track of the dominant dynamical effects involving quantum pressure,\nwhich result from inhomogeneities on the scale of the unresolvably-small $\\lambda_\\text{deB}$.\nThe fact discussed there (noted above\nin \\S\\ref{sec:smalldeB}), that, in\nthe small-$\\lambda_\\text{deB}$ limit, the NLSE\nfor SFDM-TF can be shown to reduce to\nthe same CBE as for CDM, except \nmodified by adding the SI potential to\nthat for gravity, suggests that,\nin principle, an SI-modified N-body method \nmight be developed to solve this problem,\nif we were willing to forego knowledge\nof the detailed substructure on the scale of\n$\\lambda_\\text{deB}$, while still \naccounting for the effects\nof quantum pressure on larger scales in\na coarse-grained way.\nHowever, such an N-body method would still require\nan extremely large dynamic range \nto simulate this in 3D, in order to\nresolve the small-scale suppression effects\nof SI pressure on scales below $R_\\text{TF}$, even in this limit in which \nthe (much smaller) de~Broglie scale is \nleft unresolved -- a high computational barrier.\nInstead, we avoided the complicated challenge\nof such a grand computational solution\nby closing the infinite hierarchy of the \nmomentum moments of the CBE by following the approach developed for standard CDM dynamics by \\citet{AS05}.\n\n\\citet{AS05} reduced the CBE equation for\nCDM to a set of equations \nidentical to the hydrodynamical \nequations of conservation of mass, momentum, and\nenergy for an ideal gas with ratio of specific\nheats $\\gamma = 5\/3$, in 1D, spherical symmetry, by assuming the velocity distribution is\nskewless and isotropic in the frame of bulk motion. As discussed there, these are assumptions that \napproximate the results of CDM N-body simulations \nreasonably well, for the virialized regions\nidentified as haloes.\\footnote{Outside of\nvirialized regions, infall is highly supersonic; \nthe velocity of bulk peculiar motions \ngreatly exceeds that associated with \nthe velocity dispersion due to random motions \nin that frame. Hence, these assumptions make\nno appreciable difference in the dynamics\noutside virialized regions.} \nIn the case of SFDM-TF, however,\nas we showed in \\citetalias{paper1}, we must \nreplace the gravitational potential\nin the CDM case of \\citet{AS05}\nby the sum of the gravitational and SI potentials.\nAs a result, a new kind of \npressure force appears in the momentum equation,\nin addition to the ideal gas pressure, namely \nthe SI pressure force, involving the gradient of\nthe SI pressure given above in equations~(\\ref{eq:SIpressure}) and (\\ref{eq:SIpressure2}).  \n\nThe resulting spherically-symmetric fluid equations for SFDM-TF are presented below in proper coordinates, in Eulerian form:\n\\begin{align}\n        \\text{[continuity]}& \\quad \\quad \\frac{\\partial \\rho}{\\partial t} + \\frac{1}{r^2}\\frac{\\partial (r^2 \\rho v)}{\\partial r} = 0 \\label{eq:0momSpherical}\\\\\n        \\text{[momentum]}& \\quad \\quad \\frac{\\partial v}{\\partial t} + v \\frac{\\partial v}{\\partial r} + \\frac{1}{\\rho}\\frac{\\partial}{\\partial r}(P_{\\!\\sigma}+P_\\text{SI}) + \\frac{\\partial \\Phi}{\\partial r} = 0 \\label{eq:1momSpherical} \\\\\n        \\text{[energy]}& \\quad \\quad \\frac{\\partial}{\\partial t}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) + v \\frac{\\partial}{\\partial r}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) + \\frac{P_{\\!\\sigma}}{\\rho r^2}\\frac{\\partial (r^2 v)}{\\partial r} = 0\n        \\label{eq:2momSpherical}\n\\end{align}\nwhere $P_{\\!\\sigma} = \\rho \\sigma^2$ is an isotropic pressure associated with the velocity dispersion $\\sigma$ obtained from velocity moments of the phase space distribution function. This\npressure, which behaves like the pressure\nof an ideal gas with ratio of specific heats $\\gamma = 5\/3$ and, hence, contributes \na pressure-gradient force to the \nmomentum equation, while changing\nits specific internal energy $u \\equiv (P_{\\!\\sigma}\/\\rho)\/(\\gamma -1)$ to account for\n$P_{\\!\\sigma} dV$-work, thereby accounts for the large-scale effects of the quantum pressure tensor ($\\bm{\\Pi}$), as \ndiscussed in \\citetalias{paper1}.\nIf we define the mass interior to radius $r$ as $M_r$, according to\n\\begin{equation}\n    M_r \\equiv M(\\leq r) = \\int_0^r 4\\pi (r')^2 \\rho~dr'\n    \\label{eq:interiormass}\n\\end{equation}\nwe can rewrite equations~(\\ref{eq:0momSpherical}) - (\\ref{eq:2momSpherical}) in Lagrangian form, \nthough still in proper coordinates, as follows:\n\\begin{align}\n        \\text{[continuity]}& \\quad \\quad \\frac{D \\rho}{D t} = -\\frac{\\rho}{r^2}\\frac{\\partial (r^2 v)}{\\partial r}  \\label{eq:0momLagrangian}\\\\\n        \\text{[momentum]}& \\quad \\quad \\frac{D v}{D t} = -4\\pi r^2 \\frac{\\partial}{\\partial M_r}(P_{\\!\\sigma}+P_\\text{SI}) - \\frac{G M_r}{r^2} \\label{eq:1momLagrangian} \\\\\n        \\text{[energy]}& \\quad \\quad \\frac{D}{D t}\\Big(\\frac{3P_{\\!\\sigma}}{2\\rho}\\Big) = -P_{\\!\\sigma}\\frac{D (1\/\\rho)}{D t} \\label{eq:2momLagrangian}\n\\end{align}\nwhere the Lagrangian, or total, time derivative is given by\n\\begin{equation}\n    \\frac{D}{Dt} \\equiv \\frac{\\partial}{\\partial t} + v \\frac{\\partial}{\\partial r}\n\\end{equation}\nIn \\citetalias{paper1}, we also stated the\nadiabatic shock jump conditions that \naccompany these differential conservation\nequations when there are discontinuities\npresent.  The reader is referred to \\citetalias{paper1} for those.  \n\n\\section{Halo formation by cosmological infall: spherical collapse and virialization}\n\\label{sec:infall}\n\n\n\\subsection{The non-cosmological case: nonlinear Jeans instability from a static initial condition}\n\\label{sec:staticICPaperI}\n\nIn \\citetalias{paper1}, we solved the equations above to follow the spherical collapse of a Jeans-unstable linear perturbation in a static background. We simulated the nonlinear\nevolution of a region containing a mass much larger than the Jeans mass, initially at rest, \nfrom a uniform background density, perturbed\nby a spherically-symmetric perturbation that is slightly concentrated toward the center.\nAs it was gravitationally unstable,\nthe mass collapsed, first at the center, \nuntil the rising SI pressure inside the central region of radius $\\simeq R_\\text{TF}$ halted the gravitational collapse of the innermost\nmass shells, and a strong accretion shock \nformed there to decelerate other \nincoming shells.  As more\nmass collapsed and would have, if entirely pressure-free, \nreached the origin in a finite time, with\nshells originating at successively larger radii\npredicted to reach the\norigin at successively later times, each\nshell, instead, encountered the shock at\na finite radius and was halted by it.\nOutside this accretion shock, \nthe infall was highly supersonic and the\nideal gas ``temperature'' (i.e. the velocity dispersion in momentum space, as \nmeasured in the frame of bulk radial motion) was\nvery small, so the ideal gas pressure associated with it was too small to halt the collapse. The\nSI pressure was also low there, so it, too, was\nunable to decelerate the infall.  When\na mass shell crossed the accretion shock, however,\nits ``temperature'' and ideal gas pressure jumped \nby orders of magnitude, \nwhile its density jumped only by a factor\nof four (i.e. the value for a strong, adiabatic\nshock jump in an ideal gas with a ratio of specific heats $\\gamma = 5\/3$).  As a result, \nthe post-shock gas was dominated by ideal gas pressure, high enough to bring the mass shell to\nan approximate rest in hydrostatic \nequilibrium (``HSE'').\nIn this post-shock HSE, ideal gas pressure balanced gravity everywhere inside the shock, except inside $\\sim R_\\text{TF}$, \nwhere it was SI pressure that provided the support, instead,\nand the ideal gas pressure was unimportant. \nOver time, more mass shells reached the\naccretion shock, which gradually moved outward,\nencompassing an ever-increasing mass.\nAs new mass shells passed through the shock and\nsteadily added their mass to that of the previously-shocked shells, they extended the\nradius of the post-shock region in hydrostatic equilibrium as the shock\nmoved outward. \nThe\nstructure interior to the post-shock region\ncorresponds to the ``virialized halo'' in\ncosmological simulations.   \n\nIn \\citetalias{paper1},\nwe compared the outcomes of\ntwo simulations from the same initial\nconditions, one that included the SI\npressure for SFDM-TF and one that did not, \nso the latter was entirely ``CDM-like'', \ntherefore. Since the initial conditions for\nboth of these simulations was that for a\nJeans-unstable density perturbation on a\nstatic homogeneous background, the\ncollapse that ensued left a post-shock\ndensity profile in the CDM-like case\n(i.e. with no SI) with a radial dependence\n$\\rho\\propto r^{-12\/7}$.  As we explained there,\nthis slope results when cold, infalling\ngas with a density profile with this\nslope -- the same as reported in earlier work on infall from instability in a spherical, Jeans-unstable gas cloud\n\\citep [e.g.][]{Penston69} -- encounters the strong accretion shock. The shock jump converts the kinetic energy of this cold, supersonic infall into\nideal-gas internal energy, with enough post-shock gas pressure to establish hydrostatic equilibrium.\nAccording to the strong, adiabatic shock jump conditions, the density must also increase,\njumping by a constant factor of 4 \n(for $\\gamma = 5\/3$).  As a result, the static\npost-shock density profile inside the shock \nmust have the same power-law profile as the pre-shock infall, outside the shock, namely, $\\rho\\propto r^{-12\/7}$.\nThe SFDM-TF solution was \nshown to be identical to\nthis CDM solution everywhere outside of the\nSI-pressure-supported core of radius equal \nto $\\sim R_\\text{TF}$.  Inside this radius, however,\nthe density profile was flattened, following\nthat of the $(n=1$)-polytropic profile that would result\nif SI pressure alone supported the scalar\nfield against its own self-gravity. Its \ncentral density was set by the condition that\nit conserved the mass that would have been\nthere inside that radius if the CDM-like\nprofile had continued onward to the origin,\nuninterrupted by SI pressure.  \n\nIn short, the\ngeneric structure of the haloes that formed\nas the nonlinear outcome of Jeans instability\nin SFDM-TF, therefore, could be described\nsimply as the core-envelope structure in which\na flattened, SI-polytropic core of \nradius $\\sim R_\\text{TF}$\nwas surrounded by the same CDM-like envelope\nit would have had if there were no SI pressure,\nand the mass inside the polytropic core radius was\nthe same as it would have been in that CDM-like\nprofile in the absence of SI.  As such, this\nalso served to compare the SFDM-TF profile with\nthe corresponding FDM (i.e. ``free-field'' limit) profile that would have\nresulted from the same initial conditions if $\\lambda_\\text{deB}$ is small, since the\nlatter was shown in \\citetalias{paper1} to be\nidentical to the CDM profile, when smoothed \non scales large compared with $\\lambda_\\text{deB}$.\n\nWe note that, although the description above\napplies to the entire post-shock region at\na given time, we can make our description\napply, as well, to cosmological haloes,\nwhich are customarily identified as the\n``virialized regions'' bounded by \na ``virial radius'' inside of which\nthe average mass density is equal \nto some fixed constant multiple of the \nunperturbed background density\nof the universe at that cosmic time.  In the Jeans-unstable collapse\ncalculations in \\citetalias{paper1} from static\ninitial conditions, there is not a cosmological\nbackground density to relate to the overdensity\ninside some radius within\nthe post-shock region, but we are still free\nto adopt an overdensity like the conventional\ndefinition, relative to the unperturbed\nbackground density, in order to determine\nthe radius (within the post-shock sphere) which we can \ncall the ``virial radius''.  \n\n\\subsection{The cosmological case: spherical infall in the expanding universe}\n\\label{sec:cosmologicalcase}\n\nThe description above is a guide to\nwhat we can expect below, when we re-consider the\nnonlinear Jeans instability in SFDM-TF here,\nto place it in the proper cosmological context of\nthe expanding universe in the MD era, i.e. after matter-radiation equality, $a > a_\\text{eq}$.  To do this, we will \nsolve the same equations, as written\nin proper coordinates, but now with cosmological initial and boundary conditions,\\footnote{\nAlthough our dynamical equations (\\ref{eq:0momSpherical}) - (\\ref{eq:2momSpherical}) \ndo not appear to be cosmological, for a spherically-symmetric distribution of \nmatter in the nonrelativistic and weak gravitational field limits, \nBirkhoff's Theorem implies that Newtonian gravity is sufficient for \ndescribing cosmological dynamics. Therefore, our cosmology is taken \ncare of by the Newtonian potential and the inclusion of \nHubble flow (and its perturbation)\nin the initial velocity.}\nto follow the spherical collapse of a linear perturbation in the expanding universe.\nThis is fully cosmological, for structure\nformation on scales that are well within the\nhorizon when we evolve them.  \nAs we shall see, the novel effect of cosmological\nexpansion is to allow mass shells that were\ninitially inside the characteristic radius\nof the effects of SI pressure to expand outward\nfor a time, carrying them beyond the reach of this repulsive force, and then to fall back again, \nto re-encounter it after turn-around and collapse.\nAn imprint of this initial encounter remains, \nhowever, and has important consequences for their\nfinal resting place.  \n\n\\subsubsection{CDM-like initial conditions: the infall rate that makes an NFW profile} \\label{sec:NFW}\n\nOur goal in this paper (``Paper II'')\nis to generalize the non-cosmological results in \\citetalias{paper1} to investigate the cosmological formation of SFDM-TF haloes.\nOur first approach in this direction follows\nthe tradition of previous study of structure formation in the free-field limit of the\nSFDM model (FDM),\nin which the first cosmological\nsimulations assumed scalar field dynamics\nbut initial conditions like those for the CDM model. In particular, we will simulate the formation of individual haloes from the same initial conditions for both CDM and SFDM-TF, in order to learn what difference results \nfrom scalar field dynamics with repulsive SI, \nas opposed to the collisionless dynamics of CDM. \n\nIt is useful to begin by summarizing what is\nknown about the CDM case and why it is that\nwe have confidence in the 1D, spherical \ncollapse calculations we propose to use to\nstudy the contrast between CDM and SFDM-TF\nhalo dynamics.\nIn \\cite*{AAS03}, \\citet{Shapiro04}, \\citet{AS05} and \\citet{Shapiro06}, we showed that the 1D, spherical infall model involving the same fluid \napproximation as used here and in \\citetalias{paper1}, derived by \\citet{AS05}  \nfrom the \ncollisionless Boltzmann equation, \ncan explain\nthe universal structure of haloes in N-body simulations of nonlinear structure formation from \nGaussian-random-noise density fluctuations in\nthe CDM universe. As a first step, \n\\citet{AS05} applied\nthis fluid approximation for CDM to derive\nan analytical similarity solution for halo formation from spherical infall. As described below, this\nproduced a density profile inside the accretion shock in this solution which was a\nsurprisingly good match to various \nempirical formulae developed to fit density profiles of CDM haloes in N-body simulations, e.g. the NFW profile \\cite*{NFW97}. \nHowever, while the agreement\nwas impressive at a given snapshot of time,\nthe N-body results showed that, \nif captured at different times in the mass assembly history (``MAH'') of the same halo,\nthe shape of its profile would evolve,\nwhile that of a self-similar profile does not.\nAs \\citet{AS05} pointed out, this reflected\nthe fact that infall rates\nof individual haloes derived by fitting the Lagrangian MAH of haloes in CDM N-body simulations \n\\textit{vary} over the lifetime of each halo --\nrapid early, then slowing down late -- \nwhile the rate in their self-similar\nsolution ($d \\ln M\/d \\ln a = 6$) is fixed, corresponding to the early, rapid-infall phase of the N-body results.  \nTo better match the \\textit{non-self-similar} shape evolution of the N-body haloes, in \\citet{AAS03} we replaced the self-similar initial perturbation\nin \\citet{AS05} by a \\textit{non-self-similar} spherically-symmetric perturbation profile calculated to make a time-varying\ninfall rate that matched the MAH of CDM\nN-body haloes.  Solving the 1D fluid\napproximation equations for CDM in this case\nrequired a numerical hydrodynamics\ncode.  The haloes that resulted were shown\nto be in excellent agreement with \nthe universal average behavior of the haloes in N-body simulations.  Since this earlier work,\nalso summarized in \\citet{Shapiro04} and\nupdated in \\citet{Shapiro06}, is central to\nour adoption of CDM-like initial conditions here\nto demonstrate the difference between SFDM and\nCDM dynamics on cosmological halo formation,\nwe describe it in more detail below.\n\n\\subsubsection{CDM haloes from self-similar spherical infall: $\\epsilon=1\/6$}\n\\label{sec:selfsimilar}\n\nIn an Einstein-de Sitter universe, a scale-free,\nspherically-symmetric linear perturbation \n$\\delta M\/M \\propto M^{-\\epsilon}$ leads to\nself-similar halo formation around the density\npeak at the center. According to the \ntheory of halo formation\nfrom peaks in the density field which result from Gaussian-random-noise initial density fluctuations,\nthe average density profile around such peaks can be related to the shape of the fluctuation power spectrum \\citep{HS85,Bardeen86}.  These density peaks in the linear regime are the progenitors of nonlinear structures like haloes.  It is useful to\ndescribe such peaks in terms of the RMS mass fluctuation, $\\sigma_\\textsc{m}$, of the overdensity field smoothed on some mass scale $M$. The average linear overdensity profiles around peaks of \noverdensity $\\nu\\sigma_\\textsc{m}$ are triaxial, in general, but approach spherical symmetry as $\\nu$ increases  (e.g. for $\\nu \\geq 3$), with cumulative overdensity $\\Delta_0$ inside radius $r$, given by $\\Delta_0 \\propto r^{-(n+3)}$, where $n$ is\nthe effective index of the (transferred) \npower spectrum $P(k)$ if approximated as a power-law $P(k)\\propto k^n$ at the wavenumber $k$ that corresponds to that halo mass.\nIn terms of fractional mass perturbation, \n$\\Delta_0 = \\delta M\/M \\propto  M^{-(n+3)\/3}$, \nso a power-law $P(k)\\propto k^n$ leads \nnaturally to a scale-free initial condition with $\\epsilon = (n+3)\/3$.  While the actual transferred\n$P(k)$ for CDM is not a power-law, it is\npossible to use the actual shape to derive an effective power-law index which varies with halo mass $M$ as a good approximation.  \\citet{AS05} calculated this $n_\\text{eff}$ for the current $\\Lambda$CDM universe, for the full range of halo masses from $10^3$ to $10^{15} \\text{ M}_\\odot$, showing that galactic haloes are well-approximated by $n = -2.5$, since $n\\cong-2.5\\pm0.2$ for $M$ in the range from\n$10^3 \\text{ M}_\\odot$ to $10^{11} \\text{ M}_\\odot$.\nFor haloes of galactic mass, formation by\nspherical infall from a self-similar initial perturbation with $\\epsilon=1\/6$ is therefore a reasonable\nfirst approximation (for the\nMD era, for which the universe\nis well-approximated as Einstein-de~Sitter). \n\nThe resulting self-similar density profile agreed with that for the NFW profile to within\n$10\\%$, at radii $r \\lesssim 0.6 R_{200}$, for a best-fitting concentration parameter of 3, including a density cusp at small radii with a logarithmic\nslope $\\approx{-1.27}$ \\big(for $4 \\times 10^{-3} < r\/R_{200} < 1.4 \\times 10^{-2}$\\big), where $R_{200}$\nis the radius within which the average density is\n200 times the cosmic mean density.   Furthermore,\nas we showed in \\citet{Shapiro06}, the analytical similarity solution yields a profile for the quantity $\\rho\/\\sigma_V^3$, sometimes referred to as the ``phase-space density'', which is \nbest-fit by a power-law $\\rho\/\\sigma_V^3\\propto r^{-1.91}$, in close agreement at all times with\nthe universal profile reported in the literature\nfor CDM N-body simulation results cited there. \nHowever, the\nradius of the accretion shock that bounds the virialized region in this solution is somewhat smaller than $R_{200}$, located instead at the\nradius which encloses an average overdensity of\n564 times this mean density, at $R_{564} \\simeq 0.6 R_{200}$.  This suggests that the rapid early phase of MAH in the N-body results, which deposits the\nmass of the density profile at inner radii\nthat remains there at later times, is well-represented by this self-similar infall solution, but the departure from self-similarity,\nespecially as related to the late\ntapering-off of the mass accretion is not.  \n\n\\subsubsection{CDM haloes from non-self-similar spherical infall}\n\\label{sec:nonselfsimilar}\n\nIf the infall rate was given, instead, by \nthe spherically-averaged, Lagrangian MAH \nof individual \nhaloes in CDM N-body simulations, however, then\nthe resulting haloes shared most of the\nempirical CDM N-body halo structural\nproperties \\textit{and} their time-evolution.\nIn particular, in \\citet{AAS03}, \nwe took as our initial\ncondition, a spherical density perturbation\nwhose radial dependence ensured that\nthe rate at which infalling mass reached the virial radius of the emerging halo followed\nthe empirical MAH derived by \\cite{Wechsler02}\nfrom N-body simulations of CDM.  With this\nsimple assumption, we found that the virialized\nhaloes which result have density profiles that\nare well-fit by the NFW density profile\n(at all radii $r\/R_\\text{vir}\\geq0.01$), and \nthe radius of the accretion shock is safely\noutside $R_{200}$. These\nhalo density profiles also \\textit{evolve} like\nCDM N-body haloes, with concentration\nparameter that grows with time just as it\ndoes for CDM N-body haloes.  In addition, \nas shown in \\citet{Shapiro06},\nthe resulting halo phase-space density profile, $\\rho\/\\sigma_V^3$, best-fit by a power-law\n$\\propto r^{-1.93}$, was found to be in\nremarkable agreement at all times with \nthe universal profile reported for CDM\nN-body haloes.  We concluded, therefore,\nthat the time-varying mass accretion rate,\nor equivalently, the initial perturbation\nprofile shape around the density peak that\nled to the formation of the halo, is the\ndominant influence on the internal structure\nof CDM haloes, which can be well-understood\nfrom this simple spherical infall model involving\nsmooth accretion and \nthe fluid approximation derived by \\citet{AS05}\nfrom the CBE.\n\n\\subsubsection{SFDM vs. CDM haloes from non-self-similar spherical infall}\n\\label{sec:SFDMnonselfsimilar}\n\nIn \\citetalias{paper1}, we showed that\nthe fluid approximation used above to model CDM halo\ndynamics also models SFDM dynamics\nin the free-field limit of FDM, as long as we\ndo not need to resolve substructure\non scales as small as the de~Broglie \nwavelength but still wish to account faithfully\nfor the influence of quantum pressure on\nlarger scales.\nThis, then, is a useful tool for the study\nof halo formation in the FDM regime, \nas long as we do not need to resolve the\nsolitonic core in the central region.  \nThe results described above, which\nexplain how the NFW profile\nand its related universal properties\nin the CDM model can arise as a simple consequence\nof spherical infall of smoothly-accreted matter, \nalso explain why 3D simulations\nof FDM haloes report NFW-like density profiles \n\\textit{outside} their solitonic cores. \n\nIt remains\nfor us to determine here, what happens to this\nresult when a repulsive SI is added to \nSFDM.  In \\citetalias{paper1},\nwe \\textit{generalized} the\nfluid approximation  of \\citet{AS05}\nto include the effects of SI pressure in the\nTF regime, which corresponds to\nthe regime in which the de~Broglie wavelength\ncan be assumed to be arbitrarily small,\njust right for application of the fluid\napproximation derived from the CBE.\nOur first approach to putting halo formation\nin the SFDM-TF model in a cosmological context\nwill be to adopt the spherical infall\nmodel described above for the CDM model and\nsolve the 1D, fluid conservation equations for \nSFDM-TF of \\citetalias{paper1} with cosmological initial conditions to determine\nwhat the dynamical effect of adding SI to the scalar\nfield is.   \n\nWe consider a spherically-symmetric, adiabatic\ndensity perturbation as our initial condition.  We \ntake our initial time to be when the \nscale factor $a_i = a_\\text{eq}$, and further\nassume that the universe remains matter-dominated, thereafter, at the critical density, corresponding\nto an Einstein-de Sitter (EdS) cosmology. \nThe initial linear density perturbation profile\ncan be described in terms of the mass perturbation $\\Delta_\\textsc{l}(M,a_i)$, where $M$ is\nthe mass inside a sphere of initial (perturbed) radius $r(M,a_i)$, related to the mass $M_b(M,a_i)$ that would have been inside a sphere of that radius in the unperturbed background universe, according to\n\\begin{equation}\n    \\Delta_\\textsc{l}(M,a_i) \\equiv \\frac{M-M_b(M, a_i)}{M_b(M,a_i)}\n    \\label{eq:DeltaLMass}\n\\end{equation}\nwhere $M_b(M,a_i) = (4\\pi\/3) r^3(M,a_i) \\rho_b(a_i)$, and $\\rho_b(a_i)$ is the background matter\ndensity at the initial redshift, given by \n$\\rho_b(a_i) = \\rho_\\text{crit}(a_i) \\equiv 3H^2(a_i)\/8\\pi G$, where the Hubble parameter\nevolves according to $H^2(a)=H^2_0a^{-3}$.\nWe adopt the value $H_0 = 67$ km\/s\/Mpc for the present-day Hubble parameter.\nThe initial radius of the Lagrangian\nmass shell which encloses that mass $M$ is then given by\n\\begin{equation}\n    r(M,a_i) = \\Bigg(\\frac{M}{\\frac{4\\pi}{3} \\big[1+\\Delta_\\textsc{l}(M,a_i)\\big] \\rho_b(a_i)} \\Bigg)^{1\/3}\n    \\label{eq:initialradius}\n\\end{equation}\nwhile its initial velocity corresponds to the \nperturbed Hubble flow for the growing mode of the \nperturbation with that linear overdensity, given by \\citep*[see, e.g.,][\\S5.1.1]{MVW10}\n\\begin{equation}\n    v(M,a_i) = H(a_i) r(M,a_i)\\big(1 -\\Delta_\\textsc{l}(M,a_i)\/3\\big)\n\\end{equation}\n\nThe quantity $\\Delta_\\textsc{l}(M,a_i)$ is also the\naverage fractional overdensity inside the sphere which encloses the mass $M$ at $a_i$.  This average overdensity is chosen to be a function of mass $M$\nsuch that, for an initial linear perturbation with this profile in a CDM universe, \nthe ensuing growth would lead to infall and\nnonlinear collapse that reproduces the average MAH of \nthe virialized haloes that form in CDM N-body simulations, as described above.\nThe density profiles of these N-body simulation haloes are well-fit by NFW profiles\\footnote{Note that in \\citetalias{paper1}, we denoted $A_\\text{NFW}$ as $\\delta_\\text{NFW}$. } \\citep{NFW97}:\n\\begin{align}\n    &\\rho = \\frac{\\rho_\\text{crit} A_\\text{NFW}}{(c_\\textsc{nfw}r\/R_{200})(1+c_\\textsc{nfw}r\/R_{200})^2} \\\\\n    &A_\\text{NFW} = \\frac{200}{3}\\frac{c_\\textsc{nfw}^3}{\\ln{(1+c_\\textsc{nfw})} - c_\\textsc{nfw}\/(1+c_\\textsc{nfw})}\n\\end{align}\nwhere $R_{200}$ is the radius within which the average density is 200$\\rho_\\text{crit}$, and $c_\\textsc{nfw}$ is the concentration parameter, defined by $c_\\textsc{nfw}\\equiv R_{200}\/R_s$, where $R_s$ is the ``scale radius'' at which the logarithmic slope of the density profile equals $-2$. \\cite{Wechsler02} found that the MAH of CDM haloes in N-body simulations followed a simple rule -- the logarithmic slope of the mass accretion is inversely proportional to the scale factor:\n\\begin{align}\n    \\frac{d\\ln{M_{200}}}{d\\ln{a}} &\\propto \\frac{1}{a} \\\\\n    \\Rightarrow M_{200}(a) &= M_\\infty\\exp{(-s a_f\/a)} \\label{eq:MAH}\n\\end{align}\nwhere $M_{200}$ is the mass within $R_{200}$, $M_\\infty$ is the asymptotic halo mass as $a\\rightarrow\\infty$, and $a_f$ is the ``formation scale factor'' at which the logarithmic slope equals $s$. By convention, $s=2$. In \\citet{AAS03} and \\cite{Shapiro04}, we found an initial perturbation profile for an EdS universe that is consistent with this MAH,\ngiven by\n\\begin{equation}\n    \\Delta_\\textsc{l}(M,a_i) = \\Delta_{\\textsc{l},i} \\ln{(M_\\infty\/M)}\n    \\label{eq:NFWPert}\n\\end{equation}\nwhere $\\Delta_{\\textsc{l},i} \\equiv \\Delta_\\textsc{l}\\big( M_{200}(s a_f \\!) , a_i \\big)$ is the initial average linear overdensity within the shell whose ballistic motion brings it to a radius which corresponds to\n$R_{200}$ at $a=s a_f$. \nSince $a_f$ corresponds roughly to\nthe ``formation time'' of the halo, we can linearly extrapolate this initial overdensity to $a_f$, when the (linearly-extrapolated) average overdensity should be roughly unity:\n\\begin{equation}\n    \\Delta_{\\textsc{l},f} = \\Delta_{\\textsc{l},i} \\frac{a_f}{a_i} \\approx 1 \\Longrightarrow \\Delta_{\\textsc{l},i} \\approx \\frac{a_i}{a_f}\n    \\label{eq:deli}\n\\end{equation}\nThe exact value we use in our code is $\\Delta_{\\textsc{l},i} = 0.8 a_i\/a_f$, which we obtained by calculating the linearly-extrapolated overdensity of a spherical top-hat in an EdS universe that has collapsed to a nonlinear overdensity of 200,\nat $a=s a_f$. We show our calculation in Appendix~\\ref{sec:deli}.\n\nAs described above in \\S\\ref{sec:nonselfsimilar}, \nthe result of adopting this\nlinear perturbation profile and simulating its\ngrowth and collapse in a CDM universe, using the\nfluid approximation of \\citet{AS05}, is to\nproduce a virialized region inside the accretion\nshock whose structure and evolution closely \nresembles that reported for CDM N-body haloes. \nFor example, not only are the halo profiles that result from these initial conditions well-fit by an NFW density profile at\nall times following the formation of that\naccretion shock, but the concentration parameter \nof that profile increases over time as the halo grows, according to  \n\\begin{equation}\n    c_\\textsc{nfw}=4.25 a\/a_f \\quad \\quad \\text{for  } a\\gtrsim a_f\n    \\label{eq:c200}\n\\end{equation}\n\\citep{Shapiro04,AAS03}. \nThis is close to the relationship reported by \\citet{Wechsler02} in their analysis of 3D N-body simulations: $c_\\textsc{nfw}=4.1a\/a_f$.\n\nIn practice, if we want to form an individual\nhalo with a given mass and concentration parameter\nat a given scale factor, we combine  equations~(\\ref{eq:MAH}) and (\\ref{eq:c200}) and\nsolve them for the free parameters of this NFW-producing initial condition, ($M_\\infty$, $a_f$), that will yield the desired halo.\nFor example, consider a $10^9 \\text{ M}_\\odot$ halo with a concentration parameter of 18 at the\npresent \\citep[the median value for this mass as per the mass-concentration relation found by][]{Klypin16}. Equation~(\\ref{eq:c200}) then gives $a_f \\simeq 0.24$ for this halo, in which case equation~(\\ref{eq:MAH}) gives $M_\\infty \\simeq 1.6 \\times 10^9 \\text{ M}_\\odot$.\nThe results of a simulation run using these values in the initial condition will be discussed below,\nin \\S\\ref{sec:CDMNFWsims}.\n\n\n\\subsubsection{Numerical Method}\n\\label{sec:numericalmethod}\n\nAll simulations of cosmological halo formation from spherical infall in this paper will use the \nsame 1D, Lagrangian hydrodynamics\nmethod and set-up as described in \\citetalias{paper1}.   We adopt the same\nnon-self-similar initial conditions\ndescribed above in \\S\\ref{sec:SFDMnonselfsimilar}\nand Appendix \\ref{sec:deli}, for both CDM\nand SFDM-TF, starting from $a = a_\\text{eq}$.  \nThis is designed \nto allow us to compare the dynamical\nconsequences of SFDM-TF and CDM directly,\nfor evolution from the same initial perturbation.\nUnless otherwise noted, each simulation uses\n2000 Lagrangian shells, logarithmically-spaced\nin mass.  \n\n\\subsection{Simulation results for cosmological halo formation from spherical infall and collapse}\n\\label{sec:simulationresults}\n\n\\subsubsection{CDM halo formation from NFW-producing infall rate}\n\\label{sec:CDMNFWsims}\n\nIn Fig.~\\ref{fig:NFW}, we show results of our 1D\nnumerical hydrodynamics simulation of the formation of a CDM halo, based upon the fluid approximation of \\citet{AS05} for CDM described above in \\S\\ref{subsection:fluidapprox} and \\citetalias{paper1}.  We adopted the non-self-similar cosmological perturbation described in \\S\\ref{sec:SFDMnonselfsimilar} and Appendix~\\ref{sec:deli} as our initial condition,\nstarting at $a = a_\\text{eq}$.  Our purpose here is\nto illustrate the fact that this fluid approximation\nfor CDM (\\textit{without} the SI pressure term\nwhich must be added to represent SFDM-TF) \nfaithfully reproduces the universal halo properties\nand their evolution over time of the NFW-profile\nhaloes reported for CDM N-body simulations \nwhen the initial perturbation is tuned to\nproduce a smooth accretion rate that matches\nthe MAH from \\citet{Wechsler02}. \nThe results in Fig.~\\ref{fig:NFW} confirm what we\npreviously reported in \\citet{Shapiro04}, \\citet{AAS03}, and \\citet{Shapiro06}, as they\nshould, since we are following the same \nprocedure.  Our purpose in the \\textit{next} \nsection will be to show what \\textit{difference} \nis made if these same CDM-like initial conditions\nare used to simulate the formation of an \\textit{SFDM-TF} halo, instead, when \nthe important dynamical effects\nof both quantum pressure and SI-pressure\nare properly taken into account.\n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{NFW_fit.pdf}\n    \\caption{\\textbf{The fluid approximation results for halo formation from spherical infall and collapse in CDM vs. the NFW fit to N-body simulations}. Resulting density profile from our 1D hydrodynamics code without repulsive self-interaction ($g=0$) starting from NFW-producing initial conditions. The red dashed line shows the corresponding NFW profile with the same value of $R_{200}$ as the numerical profile, but with $c_\\textsc{nfw}$ chosen to fit the numerical profile in the domain $10^{-2} < r\/R_{200} < 1$. The profiles are plotted at 5 scale factors, $a\/a_f = \\{1, 2, 3, 4, 5\\}$, for which the best-fit NFW concentration parameters are $c_\\textsc{nfw} = \\{4.6, 8.5, 12.7, 17.0, 21.2\\}$, respectively. The curves for $a\/a_f = 1, 2, 3$, and 4 have been shifted downward by 4, 3, 2, and 1 order(s) of magnitude, respectively, for visual clarity. The radius $R_{200}$ and background density $\\rho_b$, used to normalize the radius on the x-axis and density on the y-axis, are evaluated at each time-slice, separately.\n    As expected, these results reproduce those presented previously in \\citet{AAS03}, \\citet{Shapiro04}, and \\citet{Shapiro06}.  }\n    \\label{fig:NFW}\n\\end{figure}\n\n\\subsubsection{SFDM-TF halo formation from  NFW-producing infall rate: (SFDM-TF dynamics) + (CDM-like initial conditions)}\n\\label{sec:TFNFWsims}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{NumericalProfiles_Rtf=1kpc.pdf}\n    \\includegraphics[width=\\columnwidth]{NumericalProfiles_Rtf=10pc.pdf}\n    \\caption{\\textbf{The fluid approximation simulation results for SFDM-TF halo formation from spherical infall and collapse of a cosmological perturbation.} \n    Each panel shows profiles of hydrodynamical variables at select times: \n    from top to bottom, (non-dimensionalized) \n    mass density, bulk velocity, ideal gas pressure (i.e. velocity-dispersion ``pressure''), \n    and ``temperature'' (i.e. velocity dispersion) \n    vs. radius, for two example SFDM-TF haloes, from the same NFW-producing initial conditions at $a = a_\\text{eq}$ as for the CDM simulations in Fig. \\ref{fig:NFW}, with\n    $\\{M_\\infty, a_f~(z_f), R_\\text{TF}\\} = \\{5.7\\times10^{14} \\text{ M}_\\odot, 0.87~(0.15), 1 \\text{ kpc}\\}$ (left) and $\\{1.6\\times10^{9} \\text{ M}_\\odot, 0.24~(3.2), 10 \\text{ pc}\\}$ (right). \n    The thick black lines show the profiles at the ``formation'' scale factor $a = a_f$. The thin grey lines show the profiles at earlier times in the simulation, labeled by $a\/a_f$.\n    Here, $\\rho_b$ is the time-dependent background density, \n    while $\\{M_{200}, R_{200}\\} \\simeq \\{7\\times10^{13} \\text{ M}_\\odot, 740 \\text{ kpc}\\}$ (left) and $\\{2\\times10^{8} \\text{ M}_\\odot, 3 \\text{ kpc}\\}$ (right) are the mass and radius of the halo at $a=a_f$. \n    The discontinuity near $R_{200}$ is the accretion shock, which sets the virial radius of the halo. \n    The thin red lines are profiles for a corresponding CDM run using the same initial conditions. The SFDM-TF density profile closely tracks the NFW-like CDM profile at large radii, but more closely resembles a power-law, $\\rho \\propto r^{-12\/7}$, \n    at smaller radii (blue lines), until, at still\n    smaller radii, $r \\lesssim R_\\text{TF}$, flattens to the shape of an SI-supported polytropic core (red long-dashed lines).\n    For SFDM-TF at early times, the centrally-peaked initial density profile was flattened, out to $\\sim R_\\text{TF}$, by an outward-propagating, SI-pressure-driven sound wave. Circles mark the shell that was located at $r=R_\\text{TF}$ at the time the sound wave reached that radius, while triangles mark the shell that was located at $r=R_\\text{TF}$ at the initial time $a_i$ -- these are indicators of where the profile deviates from CDM.\n    The long-dashed blue line in the $P_{\\!\\sigma}$ panel shows the profile for $P_\\text{SI} = g\\rho^2\/2m^2$ at $a=a_f$ with the same normalization.}\n    \\label{fig:NFWSI}\n\\end{figure*}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{EarlySmoothing_Rtf=10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{EarlySmoothing_Rtf=1kpc.pdf}\n    \\caption{\\textbf{The beginning of spherical collapse from NFW-producing initial conditions: early Jeans-filtering by SI-pressure}. Simulation results at several early time-slices, soon after the initial scale factor $a_i$, for two cases with the same\n    initial perturbation, with $M_\\infty=1.6\\times10^{9} \\text{ M}_\\odot$ and $a_f = 0.24$, but with\n    different SI strengths for SFDM-TF: \\textbf{(a) (top panel)} $R_\\text{TF} = 10$ pc,  and \\textbf{(b) (bottom panel)} $R_\\text{TF} = 1$ kpc. Corresponding CDM runs are shown for comparison (dashed lines). The density profiles are normalized by the time-dependent background density. The figures illustrate the initial SI sound wave that propagates outward and flattens the density profile out to $R_\\text{TF}$. In the case of $R_\\text{TF} = 1$ kpc, this sound wave passes over the entire profile (which encompasses the entire computational domain, extending to $M_\\infty$), and the perturbation is prevented from growing (i.e. it is Jeans-filtered) within the age of the universe. Circles mark the shell that was located at $r=R_\\text{TF}$ at the time the sound wave reached that radius, while triangles mark the shell that was located at $r=R_\\text{TF}$ at the initial time $a_i$.}\n    \\label{fig:SIsmoothing}\n\\end{figure}\n\nIn Figs.~\\ref{fig:NFWSI} and \\ref{fig:SIsmoothing}, we show illustrative\nresults of our 1D numerical hydrodynamics simulations of SFDM-TF halo formation from \nthe same initial conditions as used in the\nprevious section for the CDM halo formation\nshown in Fig.~\\ref{fig:NFW}.\nIn this case, however, our fluid equations are now\nthe modified ones for SFDM-TF, which\ninclude the dynamical\neffects of both SI pressure $P_\\text{SI}$ and\nquantum pressure, where the latter\nis accounted for by the ideal gas pressure $P_{\\!\\sigma}$ \n(i.e. CDM-like ``velocity-dispersion'' \npressure), which represents the dynamical \neffect on large scales that results from inhomogeneity on the tiny, unresolved scale of the \nde~Broglie wavelength. Two cases are shown\nin Fig.~\\ref{fig:NFWSI}, involving different\nSI strengths, parameterized by their \nadopted $R_\\text{TF}$-values, with initial perturbations chosen to make haloes of different mass at $a=a_f$ in each\ncase.  Profiles of hydrodynamical variables vs. proper\nradius are plotted at selected times, including \nmass density, bulk velocity, ideal gas pressure \n(i.e. velocity-dispersion ``pressure''), \nand ``temperature'' (i.e. velocity-dispersion), \nfor two example SFDM-TF haloes, formed\nfrom the same NFW-producing initial conditions \nat $a = a_\\text{eq}$ as for the \nCDM simulations in Fig. \\ref{fig:NFW}, with\n$\\{M_\\infty, a_f~(z_f), R_\\text{TF}\\} = \\{5.7\\times10^{14} \\text{ M}_\\odot, 0.87~(0.15), 1 \\text{ kpc}\\}$ (left) \nand $\\{1.6\\times10^{9} \\text{ M}_\\odot, 0.24~(3.2), 10 \\text{ pc}\\}$ (right).  For comparison, we also plot\nthe end results for CDM, from the same initial conditions. \n\nThe evolution is similar in both cases, with some elements common to the non-cosmological infall and\ncollapse we simulated in \\citetalias{paper1}, \nfrom static, Jeans-unstable initial conditions, while\nothers are uniquely cosmological, as follows:\n\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item Growth of the initial perturbation slows its\n    expansion relative to the unperturbed Hubble flow\n    far from the center, so the perturbed density \n    in units of the time-dependent mean background density increases with time, even as the latter drops.\n    \n    \n    \\item Close to the center, a sound wave driven by\n    the SI pressure gradient quickly \n    propagates outward until it reaches the physical\n    radius $R_\\text{TF}$, one sound-crossing-time after\n    the initial time at $a_i$, travelling \n    at the SI sound speed given by \n    \\begin{equation}\n        c^2_{s,\\text{SI}} = \\frac{\\partial P_\\text{SI}}{\\partial \\rho} = \\frac{g\\rho}{m^2}\n        \\label{eq:SI_soundspeed}\n    \\end {equation}\n    evaluated at a density roughly equal to the initial\n    background density. This sound wave\n    smooths out the density as it goes,\n    leaving it nearly uniform within this radius,\n    and remaining uniform as it expands, i.e. it\n    expands homologously. We will say more about this sound-wave-smoothed central region below.\n    \n    \\item Thereafter, all mass shells continue expanding until\n    -- at later times for shells that started at\n    larger radii -- they reach maximum expansion, halt \n    and collapse inward, supersonically.\n    \n    \\item As the density of these infalling shells\n    rises, adiabatic compression\n    increases their ideal gas pressure \n    more slowly \\big(since $P_{\\!\\sigma} \\propto \\rho^{5\/3}$\\big) than it does their SI pressure \\big(for which $P_\\text{SI} \\propto\\rho^2$\\big). The \n    latter eventually becomes large enough in the center\n    to halt the infall of the innermost\n    shells and provide their hydrostatic support\n    within a radius $\\simeq R_\\text{TF}$, with a flattened\n    profile like that of an ($n=1$)-polytrope. This\n    was also seen in the non-cosmological results of\n    \\citetalias{paper1}.\n    \n    \\item At this point, by $a\/a_f \\gtrsim 0.4$, \n    the incoming mass shell just outside this polytropic core\n    -- just about to slam into it -- is halted by the \n    formation of a strong accretion shock that converts\n    its kinetic energy of infall into ``thermal energy'' (i.e. energy due to dispersion).\n    This boosts its ideal gas pressure to be\n    comparable to the SI pressure within the core\n    and extends the hydrostatic inner profile of the\n    latter to larger radii, bounded by the shock, \n    supported outside the core by post-shock\n    $P_{\\!\\sigma}$, instead of $P_\\text{SI}$. \n    This, too, is similar to the non-cosmological case\n    in \\citetalias{paper1}.\n    \n    \\item By $a\/a_f = 1$, the accretion shock has moved \n    outward in mass, just beyond the radius $R_{200}$, \n    to encompass a virialized halo, within which the\n    ``temperature'' profile associated with velocity dispersion\n    is approximately isothermal, except inside the polytrope,\n    where it is cold, because it remained interior to the \n    mass shell at which the shock formed.  However, matter interior to that shock is everywhere hydrostatic, supported\n    inside the polytropic core by SI pressure and outside\n    it by post-shock ideal gas pressure.\n    \n    \\item The post-shock density profile, thereafter,\n    has three distinct regions in \n    radius:\n    \\begin{enumerate}[label=(\\alph*)]\n        \\item a flattened, polytropic core inside $R_\\text{TF}$,\n        \\item an intermediate zone with power-law density profile, $\\rho \\propto r^{-12\/7}$, and \n        \\item an outer envelope which\n        follows the same NFW-profile as for CDM\n        from the same initial conditions, inside\n        the advancing accretion shock, which remains\n        just outside of the time-varying $R_{200}$.\n    \\end{enumerate}\n    Regions 7(a) and (b) were also present in the non-cosmological results in\n    \\citetalias{paper1}, while 7(c) is unique to the\n    cosmological case. \n    \n    \\item Outside the shock, the density follows\n    the pre-shock, ballistic infall profile of a cold, pressure-free gas, just like the CDM case.\n\\end{enumerate}\n\n\nAs described in \\S\\ref{sec:staticICPaperI},  \\citetalias{paper1} found a generic core-envelope\nstructure for the virialized objects formed\nby the nonlinear evolution of the Jeans \ninstability in SFDM-TF from non-cosmological \n(i.e. static) initial conditions.  As in\nthe cosmological case described above, an accretion\nshock bounded the virialized region. However,\nunlike the cosmological case above, the entire envelope\nbetween the polytropic core and the shock\nhad the same power-law density profile $\\rho \\propto r^{-12\/7}$ as reported above for the intermediate \nzone 7(b), and there was no NFW-like zone 7(c).\nThis reflected the fact that, in this \\textit{non}-cosmological\ncase, the Jeans-instability problem involved a nearly uniform initial density, which gravitational instability\nturned into an infall density profile \n(outside the accretion shock) with\nthat power-law.  The condition of hydrostatic\nequilibrium inside the post-shock region then combined with\nthe constant (shock-velocity-independent) factor\nof density jump at the shock to make the\nprofile of \\textit{post-shock} density the same\nas that of the \\textit{pre-shock} infall.  \nIndeed, for this reason, our simulation from\nthe same non-cosmological initial conditions but\nwith CDM dynamics, instead, \nproduced the exact same power-law\nprofile inside the shock, except with no SI-flattened \npolytropic core in the center.  In fact, the mass\ninside the core in the SFDM-TF case, with polytropic\nradius $R_\\text{TF}$, was the same as that \ninside that radius in the CDM-like power-law profile.\nSo, apparently, outside of $R_\\text{TF}$, where \nSFDM-TF and CDM dynamics was the same, it was the non-cosmological \\textit{initial} conditions\nof the SFDM-TF and CDM simulations that\nresulted in making their infall \nprofiles $\\rho \\propto r^{-12\/7}$, instead of\nthe cosmological infall profile derived from\nthe halo mass accretion rate in CDM N-body \nsimulations, which makes the post-shock profile NFW-like.  \n\nFrom that description, we might reasonably have expected\nhalo formation from\n\\textit{cosmological} initial conditions to be\nsimilar, with the same polytropic\ncore, surrounded again by a CDM-like envelope.\nIn particular, we might\nhave expected this CDM-like envelope\nto be \\textit{NFW-like}, as it is in our CDM simulation \n(without SI) for this same cosmological case.\nThen why did we find, as reported \nin item (7), that while this is true for \nregions (a) and (c), there is an intermediate zone (b) \nin which the density profile follows \nthe same power-law profile \\big($\\rho \\propto r^{-12\/7}$\\big)\nas in the \\textit{non-cosmological} simulations of \\citetalias{paper1}, instead?   \nThe answer to this question encapsulates\nthe fundamental distinction between static and\ncosmological Jeans instability in SFDM-TF and\nis at the heart of our cosmological story-line here\nin Paper II. \n\nWhat distinguishes a \\textit{cosmological} from a \\textit{non-cosmological} density perturbation is their different initial velocity perturbations.  \nWithin the Newtonian approximation, our \nspherically-symmetric Jeans instability problem\nis transformed from a non-cosmological one into a cosmological one entirely by the presence of perturbed Hubble expansion in the latter that replaces the static initial condition of the former.  This small change has\nprofound consequences, however, because the SI Jeans \nlength is fixed in physical coordinates by the constant\nvalue of $R_\\text{TF}$.  For CDM in an Einstein-de Sitter\nuniverse, a spherical perturbation that is overdense\nwill expand more slowly than unperturbed Hubble expansion, until it eventually\nreaches a maximum expansion radius and recollapses.\nThis will be true in SFDM-TF, as well, if it is overdense\nacross a region much larger than $R_\\text{TF}$.  \nBut in that case,\nmatter close to the center of the perturbation,\nwhich is initially expanding, will be affected by the\npresence of SI pressure, too, which responds to density\ninhomogeneity by driving a sound wave outward from\nthe center, as far as $R_\\text{TF}$, which smooths out the density perturbation as it goes.  \nThis is what we found in item (2) above.\nThe effect of this early SI-smoothing imprints the\nevolution to come for the mass shells inside the\nsphere smoothed-out by the sound wave, even though they\ncontinue to expand homologously, thereafter, \nwhich carries them far outside the physical radius \noriginally overtaken by the sound wave.\n\n\nIn Fig. \\ref{fig:SIsmoothing}, we illustrate this\nearly SI-sound-wave smoothing by\ntwo examples from the same NFW-producing\nCDM-like initial perturbation, with $M_\\infty=1.6\\times10^{9} \\text{ M}_\\odot$ and $a_f = 0.24$,\nbut with different SI strengths, by plotting density\nprofiles for a sequence of early time-slices of \nour SFDM-TF simulation, compared with\ntheir CDM counterparts. Let us focus first\non the top panel, Fig. \\ref{fig:SIsmoothing}(a),\nthe case for $R_\\text{TF} = 10$ pc. \nThe central density profile is quickly \nmade nearly uniform out to $R_\\text{TF}$, by a sound\nwave from the origin that reaches $R_\\text{TF}$ \nlong before that shell reaches maximum expansion.  Thereafter, this inner\nsphere evolves just like a top-hat perturbation.\nAs such, it continues to expand until it reaches maximum expansion, turns around, and recollapses.\nAt maximum expansion, the spherical region is \nmomentarily at rest, so it is at that\nmoment essentially the same as the initial perturbation\nof the \\textit{non-cosmological} problem in \\citetalias{paper1}!\nIts evolution, thereafter, must, therefore, be similar to\nwhat we found there.  It should yield \nthe same core-envelope structure inside the\nadvancing accretion shock as in \\citetalias{paper1}, with\npolytropic core of proper radius $R_\\text{TF}$\nand the same power-law density profile, $\\rho \\propto r^{-12\/7}$, outside the core, but only out through\nthe mass that was initially smoothed by the SI-pressure-driven sound wave.  As we can see in Fig. \\ref{fig:SIsmoothing}, the sound wave smooths out the density as it travels to $\\sim R_\\text{TF}$. Each time-slice in Fig. \\ref{fig:SIsmoothing}(a)\nis marked (by a filled circle) to identify the particular mass shell that was located at $R_\\text{TF}$ when the \nsound wave reached that radius.  The mass within that \nradius was basically the SI-Jeans mass of the background universe at that time, since the density \nperturbation was only linear then. \nThe time-varying radius so marked,\nwhich contains this fixed mass as it\nevolves, provides a rough estimate (i.e. lower-limit) \nfor the outer radius of the intermediate zone 7(b)\nwithin which we expect the final \ndensity profile to follow the power-law \n$\\rho \\propto r^{-12\/7}$, \nonce this mass is inside the shock.  We also\nmark this radius on the density\nprofiles in the top panels of Fig. \\ref{fig:NFWSI},\nfor all the time-slices, to confirm this.  \n\nThe transition from power-law to NFW-like post-shock\ndensity profiles inside the shock extends beyond the \nfilled circles, encompassing a somewhat larger mass.\n(The filled circles in \nFig. \\ref{fig:NFWSI} enclose masses of $5\\times10^{11} \\text{ M}_\\odot$ (left) and $5\\times10^5 \\text{ M}_\\odot$ (right), for $R_\\text{TF} = 1$ kpc and $10$ pc, respectively, as compared with their Jeans masses at the initial time of $2\\times10^{13} \\text{ M}_\\odot$ and $2\\times10^7 \\text{ M}_\\odot$.)\nWe can understand why and bound this transition from\nabove by a second estimate, as follows, which we \nmark with filled triangles on the density profiles\nin both Figs. \\ref{fig:NFWSI} and \\ref{fig:SIsmoothing}(a).\nAs we shall see, in \\S\\ref{sec:LinPertsTransferFn},\nthe cosmological SI Jeans mass at any scale factor \nis the mass inside a sphere of fixed physical radius \n$R_\\text{TF}$, containing the mean density at that time, \na mass which drops with scale factor as $a^{-3}$\n(see equation \\ref{eq:Jeansmass}).  As\na result, even the small, finite time for\nthe sound wave to reach $R_\\text{TF}$ is enough\ntime for the Jeans mass to drop significantly from its value at the initial scale factor $a_i$.\nTherefore, the effects of SI Jeans-filtering extend to\nencompass a mass which is between this \n(larger) \\textit{initial} Jeans mass \n(marked by filled triangles)\nand the (smaller) Jeans mass when the sound\nwave reached $R_\\text{TF}$ (marked by filled\ncircles). \n\nNow we are ready to focus on the second example of\nSI-smoothing, plotted in the bottom panel\nFig. \\ref{fig:SIsmoothing}(b). This case differs from\nthat in the top panel Fig. \\ref{fig:SIsmoothing}(a)\nonly by having a much stronger SI, corresponding to\n$R_\\text{TF} = 1$ kpc, instead of $10$ pc.  As with\nthe latter case, the initial SI sound wave propagates outward and flattens the density profile out to $R_\\text{TF}$. In the case of $R_\\text{TF} = 1$ kpc, \nhowever, this sound wave passes over the entire computational domain, which extends to $M_\\infty$ \n(i.e. the radial position of the outermost shell containing $M_\\infty$ is $\\ll R_\\text{TF}$), \nand the perturbation is, thereby, prevented from growing \nto form the collapsed virialized object it would\nhave formed in the CDM case.\n\nThis reveals the essential factor that distinguishes\nthe outcomes of non-cosmological vs. cosmological initial conditions for the formation of an SFDM-TF halo with \na given mass at a given epoch. The former yields a\nvirialized halo in the post-shock region, with a\npolytropic core of radius $R_\\text{TF}$ surrounded by\na CDM-like density profile that extends to the\naccretion shock radius at that epoch.  The mass encompassed\nby that shock grows with time just by adding new\nmass shells to the outside, at the\ngravitational free-fall times defined by the mean\nmass densities initially interior to each mass shell.\nThe mass in the core is fixed by\nextrapolating the CDM-like profile inward from $R_\\text{TF}$\nand integrating out to that radius.\nSI-smoothing in this case is confined to the region within\na fixed physical radius $R_\\text{TF}$, \nand to the mass which ends up inside this\nradius at the final epoch, while the dynamics outside this\nradius -- and the final halo structure there -- are CDM-like.\nFor the \\textit{cosmological} case, however, \nthe mass that is SI-smoothed is the much larger \namount that was inside that same sphere of physical radius $R_\\text{TF}$ at a time\nclose to the \\textit{initial} time, at the \nmean matter density of the universe at that time,\nbefore expansion carried it well beyond\n$R_\\text{TF}$, enroute to its turn-around radius.   \nSince SI-smoothing quickly flattens the initial perturbation\nout to $R_\\text{TF}$,\nthe turn-around and recollapse of any shell interior\nto the one which bounds the smoothed mass\nwill be delayed relative to their timing without\nSI-smoothing. That bounding sphere and every shell inside it\nwill expand, turn-around and recollapse like\na uniform, top-hat perturbation with the same mean density.\nSince SI-smoothing out to $R_\\text{TF}$\nmust conserve mass, the average overdensity inside\nthe smoothed sphere must be about\nthe same as in the unsmoothed perturbation, \nso the bounding sphere and its contents\nturn-around and recollapse just like the\nbounding sphere in the case without SI-smoothing.\nAs long as the latter is destined to collapse,\nthat means SI-smoothing does not \\textit{prevent}\nthe smoothed mass from collapsing, \nit only \\textit{delays} the collapse of interior\nshells to the time at which the bounding sphere, itself,\nwould have, in the absence of SI-smoothing. \n\n\nThis assumes, of course, that the average overdensity\ninside the bounding sphere in the initial\nperturbation was positive,\nas it would be for the self-similar perturbations described above in \\S\\ref{sec:selfsimilar}. \nThe CDM-like perturbation that leads to an NFW-like\npost-shock profile, however, is a \\textit{non}-self-similar one,\nin which the overdensity is ``compensated'' by an underdensity at large radii, so that\nthe average overdensity\ninside the sphere which encompasses mass $M_{\\infty}$\nis zero.   Spherical shells outside that mass will,\ntherefore, remain unperturbed and cannot collapse.\nIn that case, if the mass affected by \nSI-smoothing exceeds $M_{\\infty}$, as it does\nin Fig. \\ref{fig:SIsmoothing}(b), then the \nbounding sphere described above is actually \n\\textit{prevented} from collapsing, altogether.  \nFor those cases that \\textit{do} collapse, however,\nour simulations above tell us that\nthe virialized object that forms inside the accretion shock\nwill have an inner portion that \nresembles that in \\citetalias{paper1} from  \\textit{non-cosmological} (static) initial conditions,\nencompassing the initial SI-smoothed mass, \nwith the core-envelope structure reported there\n\\big(polytropic core of radius $R_\\text{TF}$, \nsurrounded by a power-law density profile, $\\rho \\propto r^{-12\/7}$\\big), but beyond the radius \ncontaining this initial SI-smoothed mass, \nit will transition to the NFW-like profile. \n\nThis description above\nassumes we started from CDM-like initial conditions.\nWe made this assumption as the first step in\nunderstanding the fully nonlinear\ndynamics of SFDM-TF halo formation.  \nThat was a reasonable first step for a range\nof perturbation mass scales, in view of our\ndiscussion in \\citetalias{paper1} of the fact\nthat, at scales beyond $R_\\text{TF}$, the dynamics\nof SFDM-TF is CDM-like.  As such, we were able to\nconfirm that, beyond the inner mass smoothed by SI,\nSFDM-TF halo profiles simulated with nonlinear\ndynamics will follow the same NFW-like profile\nexpected for the CDM model.  However, we\nalso learned that this SI-smoothed inner\nmass can be as large as the mass inside\nthe fixed radius $R_\\text{TF}$ at early\ntimes, when the universe was much denser,\nand that there is an intermediate zone of\ndensity profile outside the polytropic core\nof radius $R_\\text{TF}$ in the final object, which\ndiffers from the NFW profile.  Such a feature\nmight be a signature of SFDM-TF that distinguishes\nit observationally from CDM.  However, it remains to\nbe seen from 3D-cosmological SFDM-TF\nsimulations, \nfrom initial conditions \nthat are self-consistently based upon\nlinear perturbation theory for SFDM-TF,\nif this intermediate zone will survive, or else\nrelax to the NFW-like profile over time.\n\n{ In addition, such 3D simulations are required to determine what MAH results for halo growth from SFDM initial conditions and how it differs from that for CDM.\nThe results we\npresent here by assuming the smoothed accretion rate derived from N-body simulations of the CDM model are likely to be robust on the smallest and largest scales with respect to replacing the initial conditions with the proper MAH of SFDM. The shells at large radii are likely to follow CDM-like trajectories even with correct initial conditions, while the shells at small radii are quickly smoothed by SI Jeans filtering and so should also follow the correct trajectories.}\n\n{ As we shall see below, from linear perturbation analysis, SFDM initial conditions differ from those of CDM by suppressing power on small scales, due to Jeans filtering.   Halos can form, however, on all mass scales, even those below that filter scale, although with lower abundance for those smaller masses. Filtering may smooth the initial density perturbation responsible for building a halo by spherical infall, out to a larger radius than in the case with CDM-like initial conditions, resulting in a more nearly uniform collapse.  In that case, we might expect the nonlinear collapse to produce a post-shock density profile outside the polytropic core, which follows the power-law $\\rho \\propto r^{-12\/7}$, over a larger range of radii than in our results above.  } \n\nIn the meantime,  we need linear \nperturbation theory to determine the statistical \nmeasures of structure formation, e.g. how many haloes of a given mass are likely\nto form by a given epoch, in order to constrain\nthe model parameter $R_\\text{TF}$ against observations.  \nWe consider that in the remainder of this\npaper.\n\n\n\n\\section{Linear perturbation theory and the transfer function for SFDM-TF}\n\\label{sec:LinPertsTransferFn}\n\n\\subsection{Qualitative overview}\n\\label{sec:LinPertOverview}\n\nThe growth of density perturbations in a universe with complex SFDM in the \npresence of a repulsive SI differs from both that in the \nCDM model and that in the Fuzzy Dark Matter (FDM) limit of SFDM.    \nIn all models, the assumed, primordial initial condition involves \nsmall-amplitude, 3D, Gaussian-random density fluctuations with zero mean, \ncharacterized by a scale-invariant\\footnote{More generally, \n$P(k,t_i) \\propto k^{n_s}$ where $n_s$ is the primordial spectral index of scalar perturbations, and the value we adopt here is\n$n_s = 1$, the Harrison-Peebles-Zeldovich spectrum, which is consistent with current observational constraints.} power spectrum  \n\\begin{equation}\n    P(k,t_i) = |\\delta_k(t_i)|^2 \\propto k\n\\end{equation}\nwhere the linear overdensity $\\delta_k (t)$ of a Fourier mode of \nwavenumber $k$ at a cosmic time $t$ is related to its \namplitude $\\delta_k (t_i)$ at initial time $t_i$, according to\n\\begin{equation}\\label{eq:dP}\n    \\delta_k(t) = \\mathcal{T}_k(t,t_i)\\delta_k(t_i)\n\\end{equation}\nin terms of the transfer function $\\mathcal{T}_k(t,t_i)$.\nDifferences in the structure formation histories of these models arise while \nthey are still in their linear amplitude phase, as characterized by different \ntransfer functions for each model.\nThey have one very important thing in common, however.  \nWhen fluctuation modes of different proper wavelength $\\lambda$ (or, equivalently, comoving $k=2\\pi a \/ \\lambda$) \nfirst re-enter the horizon \\big(i.e. near the time\n$t=t_\\textsc{h}$ at which their proper wavelength equals the Hubble diameter, \ni.e. $\\lambda(t) = 2c\/H(t)$, in which case\ncomoving $k = k_\\text{H} \\equiv \\frac{\\pi}{c}a_\\textsc{h}H(a_\\textsc{h})$\\big), they do so with the same, wavelength-independent, fractional overdensity, $\\delta_k(t_\\textsc{h}) \\ll 1$, for all models. \nAlthough modes of different $k$ enter the horizon at different times \n$t_\\textsc{h}$ (later for smaller $k$), and it is even possible that \nsome modes of the same $k$ in the different models enter the horizon at different times, \nthe value of $\\delta_k(t_\\textsc{h})$ in all cases is the same for all values of $k$ and \nall values of $t_\\textsc{h}$. \n\nThe value of $t_\\textsc{h}$ for a given wavenumber depends only on the time-dependence \nof the Hubble parameter, so $t_\\textsc{h}(k)$ is the same in all models only if $H(t)$ \nis the same at all times in different models.  We expect this to be the case for \nCDM and SFDM models, in general, in our observed Universe, for modes that re-enter the \nhorizon during the radiation-dominated (RD) and matter-dominated (MD) eras of interest to us here, since all models are required \nby observational constraints to share a common expansion history from the time the wavenumbers\nassociated with galaxy formation began to enter the horizon during the RD era through the present.\nThis condition is not met for all of the models at all times. For modes that enter before \nthe RD era -- e.g. for complex SFDM, with or without SI, there is an early stiff phase ($w=1$), \nprior to the RD era, in which the scalar field dominates.  \nHence, even though the Hubble parameter during the RD and MD eras, for example, is basically the same at a given scale factor for all models, it differs during the SFDM \\textit{stiff} phase at earlier times, in that case, when much \nsmaller scales entered the horizon. In fact,\neven \\textit{during} the RD epoch, but well before the epoch of matter-radiation equality, \n$a_\\text{eq}$ (for both complex and real field\ncases), \nthe Hubble parameter can be slightly enhanced relative to its value in CDM and FDM, \nby a small boost to the total energy density contributed by SFDM in the presence of SI,\nwhen its EOS is radiation-like \\big($\\langle w \\rangle=1\/3$\\big), between its stiff- and matter-like phases of evolution, \nbut this small enhancement is gone well before $a_\\text{eq}$.\\footnote{When we\ndescribe the EOS parameter $w$ of the \nscalar field during its rapid-oscillation phase, \nin which the value averaged over oscillation\nperiods, $\\langle w \\rangle$, evolves\nfrom $\\langle w \\rangle = 1\/3$ to \n$\\langle w \\rangle = 0$, we will, henceforth,\ngenerally just write\n$w$ to refer to this averaged quantity and\nomit the brackets.}\n                           \nWhat happens to that amplitude next, after a mode is already inside the horizon, depends upon the microscopic nature of the dark matter and other components that contribute to the total mass-energy of the Universe. It also depends on whether the background Universe is RD or MD during the phase where the galaxy- and galaxy-cluster-scale fluctuations are still linear.\n\nIn the standard CDM model, dark matter is a nonrelativistic gas of cold, collisionless particles, so the ratio $w$ of its pressure to its energy-density is well-approximated by $w = 0$.  In that case, its fluctuation growth rate after horizon entry depends upon whether the Universe is in the RD or MD era.  For the scales of interest for structure formation, spanning the range of galaxy- and cluster-mass objects, horizon entry is during the RD era. At that time, \nradiation dominates the gravitational potential fluctuations that drive perturbation growth in CDM, but fluctuations\nin the radiation component do not amplify during this phase, because a relativistic fluid with $w = 1\/3$ has a speed of sound close to $c\/\\sqrt{3}$,  and density inhomogeneities are thus smoothed out by pressure disturbances that can travel a distance comparable to the Hubble radius within the age of the universe at that time. As a result, gravitational fluctuation growth in the CDM component is driven only by its own fluctuation as a subdominant contribution to the total energy-density. This yields only a relatively small amount of growth between horizon entry at scale factor $a_\\textsc{h}$, and matter-radiation equality at $a_\\text{eq}$, which is a logarithmic function of those scale factors. \n\nWe adopt cosmological parameters from \\citet{planck18} as given in Table~\\ref{tab:params}.\n\\begin{table}\n    \\centering\n    \\begin{tabular}{c|c}\n        \\hline\\hline\n        parameter & value \\\\\n        \\hline\n        $h$ & 0.67 \\\\\n        $\\Omega_\\text{dm} h^2$ & 0.12 \\\\\n        $\\Omega_m h^2$ & 0.14 \\\\\n        $\\Omega_\\Lambda h^2$ & 0.31 \\\\\n        $\\Omega_\\text{rad} h^2$ & $4\\times10^{-5}$ \\\\\n        \\hline\n    \\end{tabular}\n    \\caption{\\textbf{Cosmological parameters} from \\citet{planck18}, which we adopt for our linear perturbation analysis in \\S\\ref{sec:LinPertsTransferFn}.}\n    \\label{tab:params}\n\\end{table}\nThe scale factor of matter-radiation equality is given by\n\\begin{equation}\n    a_\\text{eq} \\simeq 2.9 \\times 10^{-4}\n\\end{equation}\nFor a comoving wavelength $\\lambda_0$ that enters the Hubble volume during the RD epoch,\n\\begin{equation}\n    a_\\textsc{h} \\simeq 10^{-6} \\left( \\frac{\\lambda_0}{\\text{1 Mpc}} \\right)\n\\end{equation}\nAnd the unperturbed mass of nonrelativistic dark matter contained in a sphere of proper radius $\\lambda(t)\/2$, where $\\lambda(t)\/a(t) = \\lambda_0\/a_0$, is given at all times by\n\\begin{equation}\n    M = \\frac{4\\pi}{3} \\rho_{\\text{dm},0} \\left( \\frac{\\lambda_0}{2} \\right)^3 \\simeq 1.45 \\times 10^{11} \\text{ M}_\\odot \\big(\\Omega_\\text{dm} h^2\\big) \\left( \\frac{\\lambda_0}{\\text{1 Mpc}} \\right)^3\n\\end{equation}\nwhere the mean dark matter density at present is given by\n\\begin{equation}\n    \\rho_{\\text{dm},0} \\equiv \\Omega_\\text{dm}\\frac{3 H_0^2}{8\\pi G} \\simeq 2.8 \\times 10^{11}~\\frac{\\text{M}_\\odot}{\\text{Mpc}^{3}} \\big(\\Omega_\\text{dm} h^2\\big)\n\\end{equation}\nWe can also write the scale factor when a given mass fills the Hubble volume, according to\n\\begin{equation}\n    a_\\textsc{h} \\simeq 1.9 \\times 10^{-6} \\big(\\Omega_\\text{dm} h^2\\big)^{-1\/3} \\left( \\frac{M}{10^{12} \\text{ M}_\\odot} \\right)^{1\/3}\n\\end{equation}\nThe comoving wavelength that entered the horizon at $a_\\text{eq}$ is given by\n\\begin{equation}\n    \\lambda_\\text{0,eq} \\simeq 290 \\text{ Mpc}\n\\end{equation}\nand the mass\\footnote{If we had chosen to define the mass associated with a horizon-filling mode as the mass inside a sphere of comoving radius \n$\\lambda_0$, instead of  $\\lambda_0\/2$ as we did above, then the mass associated with the mode that enters at $a_\\text{eq}$ would be 8 times larger than this, or $3 \\times 10^{18} \\text{ M}_\\odot$.  \nAlternatively, if we had, instead, defined the mass associated with a mode as just $\\rho_{\\text{dm},0} \\lambda_0^3$,  rather than $4\\pi\/3  \\rho_{\\text{dm},0} (\\lambda_0\/2)^3$, as above, then the mass above, which is evaluated based upon the latter, should be multiplied by $6\/\\pi \\simeq 1.91$, instead, which would make $M_\\text{eq} \\simeq 8 \\times 10^{17} \\text{ M}_\\odot$.   These are just matters of definition, of course, as the horizon-filling wavelength and the unperturbed mass associated with a given wavelength mode are not quantities of precise meaning, but we should try to compare different mode-associated masses using the same choice of definition.} that filled the Hubble volume at $a_\\text{eq}$ is, therefore, given by\n\\begin{equation}\n    M_\\text{H,eq} \\simeq 3.5 \\times 10^{18} \\text{ M}_\\odot \\big(\\Omega_\\text{dm} h^2 \\big)\n\\end{equation}\nwhich, for $\\Omega_\\text{dm} h^2=0.12$, yields $M_\\text{H,eq} \\simeq 4 \\times 10^{17} \\text{ M}_\\odot$.\n\nFor CDM, modes of mass scale $M$ that enter the horizon during RD experience only logarithmic growth between $a_\\textsc{h}$ and $a_\\text{eq}$, given by (see equation~\\ref{eq:deltaCDM})\n\\begin{equation}\n    \\frac{\\delta_\\text{eq}}{\\delta_\\textsc{h}} \\approx 15\\ln \\left(\\frac{a_\\text{eq}}{a_\\textsc{h}}\\right)\n    \\label{logarithmicgrowth}\n\\end{equation}\nwhere $\\delta_\\text{eq} \\equiv \\delta_k (a_\\text{eq})$ and $\\delta_\\textsc{h} \\equiv \\delta_k (a_\\textsc{h})$.\nHenceforth, we will omit the subscript $k$ when referring to the Fourier mode overdensity $\\delta$.\nBy $a_\\text{eq}$, all modes of interest in forming galaxies and clusters had already entered the horizon, so they all experience the same factor of growth between $a_\\text{eq}$ and any later $a$ (i.e. for early times during the MD era, growth is the same as in an Einstein-de Sitter universe, in which case $\\delta \\propto a$).\n\nIt is convenient to describe the growth of density fluctuations in alternative dark matter models like SFDM in terms of their growth relative to that in CDM. \nFor a range of wavenumbers, the growth of density fluctuations in SFDM with SI in the \nTF regime, for modes of interest to us in galaxy and large-scale structure formation, is the same \nafter horizon crossing as for CDM. This is the case\nfor modes of small enough wavenumber, as we shall see. \n\nThere are two effects, however, that distinguish the two models.   \nCDM behaves as a pressure-free, matter-like gas (with $w=0$) at all times, both during the RD era when it is a subdominant component of the total mass-energy density of the universe and during the MD era when it dominates that energy-density.    \nSFDM differs in that regard.   \nSFDM with repulsive SI is only nonrelativistic and matter-like (with $w = 0$) at late times, after it transitions from being relativistic and  radiation-like (with $w  = 1\/3$).  \nAlthough this transition is not instantaneous, it is fast enough that we can characterize it as a sudden transition at scale factor $a_t$.   \nAs we discuss below, this transition scale factor $a_t$ is set by the value of the ratio $g\/m^2$.  \nAs pointed out by \\citet{Li14}, observations of the CMB anisotropy constrain the allowed values of this ratio by requiring that the EOS of the background universe must transition from RD to MD at close to the same value of  $a_\\text{eq}$ in both SFDM and CDM.   \nThis places an upper bound on the ratio, by requiring that $a_t \\leq a_\\text{eq}$.\nAs such, i.e. as long as $a_t \\leq a_\\text{eq}$, modes that enter the horizon before $a_t$, when SFDM-TF behaves like a radiation-like perfect fluid ($w=1\/3$),  \nwill not experience the logarithmic growth between $a_\\textsc{h}$ and $a_t$ that the same modes \nexperience in CDM.  In fact, these modes that enter \nthe horizon prior to $a_t$ should evolve\njust like those of the dominant radiation component, which oscillate\nlike sound waves of a relativistic gas\nwith sound speed $c\/\\sqrt{3}$, without changing their amplitude. If so, then these modes that entered the horizon before $a_t$ and before $a_\\text{eq}$\nwould be deprived of the interval of logarithmic\ngrowth which the same modes would have experienced\nafter horizon entry in CDM between $a_\\textsc{h}$ and\n$a_\\text{eq}$, according to equation (\\ref{logarithmicgrowth}).   \nIn the absence of CDM-like \ngrowth during this interval of scale factor, therefore, we might expect a net amount of growth for these modes over the entire RD era, \nincluding the time between $a_t$\nand $a_\\text{eq}$, equal to that which would result for CDM modes if we replaced $a_\\textsc{h}$ in their logarithmic growth factor in equation (\\ref{logarithmicgrowth}) by $a_t$.   \n\nThere is a second effect, however, which may also interfere with the growth of SFDM modes, if their proper wavelength is below the Jeans length that results from the back reaction of SI pressure forces that act to cancel their gravitational instability.    \nTo be more specific, we must estimate this Jeans length and its dependence on scale factor, and consider which of these two effects dominates, for which modes, and during which intervals of scale factor. \n \nLet us start by finding the transition scale factor $a_t$ and its dependence on SI strength $g$ and particle mass $m$. \nSuppose we identify the transition between $w = 1\/3$ and $w = 0$ in equation (\\ref{eq:averagew}), which goes through these values monotonically with increasing scale factor, by solving that equation for the scale factor at which $w=1\/9$,\nassuming that, for small $w$, it is sufficient to take the homogeneous background value of \n$\\langle|\\phi|^2\\rangle$ as given by\n$\\langle|\\phi|^2\\rangle\\simeq\\rho_\\text{dm}=\\rho_\\text{dm,0}a^{-3}$,\nwhich then yields  \n\\begin{equation}\n    a_{1\/9} = \\left(3\\rho_\\text{dm,0} \\frac{g}{m^2c^2}\\right)^{1\/3}\n\\end{equation}\nWe can then write the scale factor for any \n$w$\nas follows: \n\\begin{equation}\n    a_w = a_{1\/9}\\left (\\frac{1}{6w}-\\frac{1}{2}\\right)^{1\/3}\n\\end{equation}\nIn terms of $R_\\text{TF}$ in equation~(\\ref{eq:RTF}), we can write the transition scale factor as follows:\n\\begin{align}\n    a_{1\/9} &= \\left( \\frac{12G \\rho_\\text{dm,0} R_\\text{TF}^2}{\\pi c^2} \\right)^{1\/3} \\nonumber \\\\\n    &\\simeq 1.8\\times 10^{-5} \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{2\/3} \\left( \\frac{R_\\text{TF}}{\\text{1 kpc} } \\right)^{2\/3}\n\\end{align}\nIf we now take $a_t = a_{1\/9}$, then $a_t(R_\\text{TF})$ follows\n\\begin{equation}\n    a_t(R_\\text{TF}) \\propto R_\\text{TF}^{2\/3}\n    \\label{eq:atransitionRTF}\n\\end{equation}\nIn the radiation-dominated era, for which $H(a) \\propto a^{-2}$, the mass that fills the Hubble volume is\n\\begin{equation}\n    M_\\text{H,RD}(a) = \\frac{4\\pi}{3}\\rho_\\text{dm,0}a^{-3} \\Big(\\frac{c}{H(a)}\\Big)^3 \\propto a^3\n\\end{equation}\nwhich, evaluated at $a = a_t$ for a given value of $R_\\text{TF}$, is then\n\\begin{equation}\n    M_\\text{H,RD}(a_t(R_\\text{TF})) \\propto a_t^3(R_\\text{TF}) \\propto R_\\text{TF}^2\n\\end{equation}\nCombining this with the relationship above between $a_t$ and $R_\\text{TF}$, we can write the horizon-filling mass at the transition scale factor in terms of $R_\\text{TF}$ as\n\\begin{equation}\n    M_\\text{H,RD}(a_t(R_\\text{TF})) = M_\\text{H,eq}\\left(\\frac{R_\\text{TF}}{R_\\text{TF,eq}}\\right)^2\n\\end{equation}\nwhere $R_\\text{TF,eq}$ is the value that makes $a_t(R_\\text{TF,eq}) = a_\\text{eq}$.\n\nAs discussed in \\citetalias{paper1}, \nwhen $w\\simeq0$ \n(i.e. $a\\gtrsim a_t$), the Jeans length scale below which gravitational instability is opposed by the back-reaction of SI pressure is the proper wavelength\n$\\lambda_\\text{J,0} = 2 R_\\text{TF}$\n(where the subscript ``0'' refers to $w=0$ and\nthe subscript ``SI'' is removed for simplicity \nbut is implied).\nThe associated SFDM-TF Jeans mass scale is given by the dark matter mass inside a sphere of radius $R_\\text{TF}$, at any scale factor $a$, according to\n\\begin{align}\n    M_\\text{J,0}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} R_\\text{TF}^3 \\nonumber \\\\\n    &\\simeq 4.6\\times 10^{12} \\text{ M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left(\\frac{a}{10^{-3.5}}\\right)^{-3} \\left(\\frac{R_\\text{TF}}{\\text{1 kpc}}\\right)^3\n    \\label{eq:Jeansmass}\n\\end{align}\nFor future reference, it will be useful to define the scale factor $a_\\textsc{j,0}(M)$ at which the mass $M$ equals the Jeans mass, by inverting equation~(\\ref{eq:Jeansmass}) to write\n\\begin{align}\n    a_\\text{J,0}(M) &= R_\\text{TF} \\left( \\frac{ 4\\pi \\rho_\\text{dm,0} }{ 3M } \\right)^{1\/3} = \\frac{R_\\text{TF} k}{\\pi} \\nonumber \\\\\n    &\\simeq 5\\times10^{-4} \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/3}\\left( \\frac{M}{10^{12} \\text{ M}_\\odot} \\right)^{-1\/3} \\left(\\frac{R_\\text{TF}}{\\text{1 kpc}}\\right)\n    \\label{eq:aJeans0}\n\\end{align}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{JeansMass.pdf}\n    \\caption{\\textbf{Cosmic evolution of the SFDM-TF Jeans-filter scale: \\textit{The Incredible Shrinking Jeans Mass}.}  The Jeans mass $M_\\text{J}$ \\big(i.e. $M_\\text{J} = \\frac{4\\pi}{3} \\rho_\\text{dm,0} ( \\pi\/k_\\text{J})^3$\\big) associated with the critical wavenumber $k_\\text{J}$ for linear density fluctuation modes in SFDM-TF that marks the boundary between growth (for $M>M_\\text{J}$) and oscillation (for $M<M_\\text{J}$), vs. scale factor $a$, for \n    several illustrative values of the Thomas-Fermi radius $R_\\text{TF}$.  \n    Curve going from lower left to upper right is $M_\\text{H}(a)$, the mass that fills the horizon \n    (i.e. the Hubble volume, of radius $c\/H(a)$) at each scale factor.  The diagonal curves from upper left to lower right represent the curves of $M_\\text{J}(a)$ for different values of the $R_\\text{TF}$  (1000, 100, 10, and 1 pc, in order from top to bottom).  The dashed vertical lines and the discontinuities on each of these curves for different values of $R_\\text{TF}$ locates the value of the transition scale factor $a_t$ ([1.8, 8.4, 39, 180] $\\times 10^{-7}$, in order from left to right)\n    for the SFDM EOS, to the\n    left of which $w=1\/3$, while to the right $w=0$.}\n    \\label{fig:JeansMass}\n\\end{figure}\n\nWe are now in a position to compare this Jeans mass scale $M_\\text{J,0}$ given above as a function of scale factor, for a given value of  $R_\\text{TF}$, with the horizon-filling mass at each scale factor, to show the range of fluctuation mass-scales that are free of Jeans filtering from the moment those fluctuations re-enter the horizon during the RD era.   \nFor those fluctuation modes, CDM-like perturbation growth can be assumed to apply to the SFDM fluctuations, as well, as long as their horizon entry took place after SFDM transitions from a radiation-like to a matter-like EOS.   \nWe plot these quantities in Fig.~\\ref{fig:JeansMass} for 4 different values of $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc.    \nAs the figure indicates, all modes that enter the horizon super-Jeans (i.e. $M > M_\\text{J}$ when $M < M_\\text{H}$)  do so when the SFDM EOS was matter-like.   \nThat means that the Jeans mass calculated above\nfor the case when SFDM is matter-like is applicable, as we have assumed.  \nModes of smaller wavelength, for which horizon entry preceded the transition scale factor of the SFDM EOS, were relativistic and radiation-like upon entry, so their Jeans filtering during that phase should be calculated for a relativistic fluid, instead, because their effective sound speed is close to the speed of light and the Jeans scale is comparable to the horizon size, at which scale the Newtonian approximation breaks down. \n\nRoughly speaking, when $w = 1\/3$ \n(i.e. $a \\lesssim a_t$), \nthe Jeans length is equal to the distance a \nrelativistic sound wave can travel at the\nsound speed $c\/\\sqrt{3}$ in a Hubble time:\n\\begin{equation}\n    R_\\text{J,1\/3}(a) = \\frac{c}{H\\sqrt{3}}\n\\end{equation}\nThe mass enclosed by this radius\nis the same as the mass inside the usual\nacoustic horizon during the RD era, which is\njust smaller than the mass inside the Hubble\nvolume, according to\n\\begin{align}\n    M_\\text{J,1\/3}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} R_\\text{J,rad}^3 \\nonumber \\\\\n    &\\simeq 8.7\\times 10^{16} \\text{ M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left(\\frac{a}{10^{-3.5}}\\right)^{3}\n\\end{align} \n\\textit{independent of} $R_\\text{TF}$,\naside from the dependence of transition scale factor $a_t$ on it (see, for example, equation~\\ref{eq:atransitionRTF}).\nThis radiation-like Jeans mass defines the $a<a_t$ segment of each colored solid line\nin Fig.~\\ref{fig:JeansMass}.\n\nFor any given mode, we can describe its cosmological evolution by drawing a horizontal line across the box in Fig.~\\ref{fig:JeansMass}\nat that mass scale and following it in time\nfrom small scale factor to large, for a fixed\nchoice of SI strength, expressed by the\nvalue of $R_\\text{TF}$, as follows.  At early\ntimes such that $a<a_t$, the SFDM-TF is \nradiation-like ($w\\approx{1\/3}$), so its fluctuations behave during this phase\njust like those of fluctuations of the\nsame wavenumber $k$ in the dominant radiation component.  They are indistinguishable from the\nlatter, in fact \n(i.e. Lagrangian volume changes in the\nscalar field and radiation are the same for these\nmodes, during this phase of their evolution, so\nthe fluctuation in dark matter \\textit{mass} density $\\delta_\\text{SFDM}$ for these modes\nis simply related to \nthe energy-density fluctuation in the radiation\n$\\delta_\\text{rad}$ by $\\delta_\\text{SFDM}= (3\/4)\\delta_\\text{rad}$). The smaller the\nvalue of $R_\\text{TF}$ is, the earlier this\nradiation-like phase of EOS will end. \nHowever, there is always \na range of modes for which the\nmass is small enough (i.e. $k$ is large enough) to make horizon entry early enough that $a_\\textsc{h}< a_t$ for those modes, \nso they enter while the EOS of the scalar field is radiation-like. According to Fig.~\\ref{fig:JeansMass}, these modes are sub-Jeans ($M<M_\\text{J,1\/3}$) once they\nare within the horizon.  Radiation fluctuations that enter the horizon during this epoch\n(which is RD), all\nstart from the same $k$-independent amplitude when they are far outside the horizon, but\noscillate like sound waves with constant amplitude after entry. This also describes\nthe behavior of SFDM fluctuations of the \nsame wavenumber, therefore, until the\nend of their radiation-like phase at $a=a_t$.\nAt this time, the EOS of the\nscalar field transitions to matter-like ($w\\approx0$) and, thereafter, the character\nof the evolution departs from that of the\ndominant radiation component, as described \nbelow.  \n\nThere are three kinds of \nmodes to track, as we follow their evolution \\textit{after}\n$a=a_t$, when the EOS turns matter-like. There\nare the smaller-mass modes, described above, that entered the horizon prior to this time.  At the other extreme, there are  \nmodes that are so massive that they are matter-like and\n\\textit{super-Jeans} when they enter the horizon\n(i.e. $M>M_\\text{J,0}$ when $M<M_\\text{H}$).  \nIn between these two mass ranges, there are\nmodes which are matter-like and \\textit{sub-Jeans} upon horizon entry (i.e. $M<M_\\text{J,0}$ when $M<M_\\text{H}$).\nIn the first category, the\nhigh-$k$ modes that entered prior to this\nphase will still be sub-Jeans initially, after\n$a_t$, but with the Jeans mass now given by \n$M_\\text{J,0}$, rather than $M_\\text{J,1\/3}$. \nTheir fluctuations will transition\nfrom the radiation-like behavior described\nabove to a new, matter-like behavior, \nthereafter. For the range of modes that entered the horizon after $a=a_t$, however, \nthe EOS is matter-like at all times, thereafter.  Their behavior once within the horizon \ndepended upon whether they were above or below\nthe Jeans mass $M_\\text{J,0}$ \nat the time of their horizon entry.  \nWe expect perturbation modes that were super-Jeans upon horizon entry (i.e. modes with $M > M_\\text{J}$ when $M < M_\\text{H}$) during this phase\nto experience CDM-like growth, thereafter.  For the intermediate-mass modes that were sub-Jeans ($M<M_\\text{J}$) and\nmatter-like ($w\\approx0$ at $a>a_t$) when they were \\textit{superhorizon}\n($M>M_\\text{H}$), their evolution was also no different from that of CDM.  After their horizon entry, however, \nthese sub-Jeans modes oscillated like sound waves \nwith an amplitude that grew as $a^{1\/4}$, \nas described by \\citet{SC15}\n(see below).  This sub-Jeans oscillatory\nbehavior during the matter-like phase ($a>a_t$) \nof the scalar field, with amplitude growing as $a^{1\/4}$, was also shared by the higher-$k$ \nmodes that entered the horizon \\textit{before}\n$a=a_t$, although these latter modes were \nalready oscillating at \\textit{constant} \namplitude prior to $a_t$, just like fluctuations in the dominant radiation component at that time.\n\nThe latter sub-Jeans behavior during the matter-like\nphase of SFDM-TF contrasts with the\nbehavior of fluctuations in another matter component subject to Jeans-filtering by pressure forces, the baryons. For the baryons,\nsubhorizon baryonic fluctuations below the Jeans scale in the CDM universe also oscillated like sound waves, but with an amplitude that actually\n\\textit{decayed} during the MD era, prior to their decoupling from photons. \nHowever, this sub-Jeans behavior of the baryonic component in which the amplitude does not grow but even decays prior to photon decoupling has a much smaller impact on structure formation in the standard CDM model than the corresponding\nsub-Jeans behavior of SFDM-TF described above,\neven though the latter involves oscillations\nthat grow rather than decay.  The reason is \nthat the baryonic fluctuations are not required\nto ``carry the load'' as the driver of structure formation in the standard CDM model.  \nIn that model, subhorizon fluctuations in the \n\\textit{dark matter} were always able to grow.\nHence, in that case, even though sub-Jeans baryonic fluctuations could not, the baryons were \nlater able to catch up (after recombination) to those dark-matter fluctuations that had a head-start at growing during the RD era and \nbeyond and never damped.  On scales above\nthe baryonic Jeans-filter mass \n\\textit{after} recombination, the CDM fluctuations were never\nfiltered and so provided potential wells into which the baryons \nwere able to fall, even if their own density\nfluctuations were initially\nnegligible on these scales.  \nHere, by contrast, SFDM \\textit{is} the dark matter (\\textit{all} of it, in fact),\nand, even though its Jeans-filtering\nbears some superficial similarity to the \nJeans-filtering of baryonic fluctuations in CDM, \nthere is no other component on which it\ncan rely to drive its growth at later times.\nHence, it is not possible for fluctuations in SFDM on these smaller scales to ``re-animate'' later,\nas those of the baryons did for the CDM\nmodel, by being driven to catch-up to perturbations in some other component that were able to grow while those in the SFDM did not.    \n\nThat said, the Jeans mass in SFDM-TF decreases rapidly with time, so one might wonder whether this would free some fluctuation modes of their Jeans filtering after they transition from sub-Jeans to super-Jeans.  \nThe Jeans scale in the matter-like phase of\nSFDM corresponds to a fixed proper length, so it encompasses a smaller and smaller mass over time, scaling as it does with $a^{-3}$.\nAs such, whatever a mode's amplitude is when it reaches this super-Jeans epoch following its sub-Jeans phase, it will be able to grow, thereafter, at the matter-like growth rate.   \nIt remains to be determined below,\nif that will be a significant effect, or,  if not, if the growth of just those modes that were always super-Jeans from their horizon-entry epoch forward was sufficient on its own to form the full mass range of haloes at different redshifts required by astronomical observations. \n\n\n\\subsection{How the small-scale cut-off in the SFDM-TF transfer function depends on the polytrope radius}\n\\label{sec:cutoff}\n\nThe effect of Jeans-filtering described above\nis to introduce a small-scale cut-off in the \ntransfer function for SFDM-TF, while for larger\nscales, the CDM and SFDM-TF transfer functions \nare the same. We will derive this effect in detail\nby solving the perturbation equations below.\nFirst, however, we will anticipate the result of those calculations by estimating the \ncomoving wavenumber $k_\\text{cut}$ and \nassociated mass scale $M_\\text{cut}$ \nat which the cut-off begins in the resulting\ntransfer function, such that power is reduced \nrelative to that in CDM (for $k \\gtrsim k_\\text{cut}$, or equivalently, $M \\lesssim M_\\text{cut}$),\nas follows.  \nThis cut-off will occur roughly \nat the wavenumber of the first mode to enter\nthe horizon (i.e. the mode with the smallest $k$-value, corresponding to the largest mass \nscale) which is sub-Jeans when it enters.\nThis is the mode for which \n$a_\\textsc{j,0} = a_\\textsc{h}$, or equivalently, $R_\\text{TF} = c\/H(a_\\textsc{h})$:  \n\\begin{equation}\n    R_\\text{TF} = \\frac{c a_\\textsc{h}^2}{H_0 \\sqrt{\\Omega_\\text{rad}}} = \\frac{ \\pi^2 H_0 \\sqrt{\\Omega_\\text{rad}} }{ c k_\\text{cut,TF}^2  }\n\\end{equation}\nwhere $H(a_\\textsc{h})$ is evaluated assuming all modes of interest enter the horizon when the Universe is RD, and the last expression follows from our definition of $a_\\textsc{h}$: $ck\/\\pi = a_\\textsc{h} H(a_\\textsc{h})$. Solving for $k_\\text{cut}$ gives\n\\begin{align}\n    k_\\text{cut,TF} &= \\left(\\frac{ \\pi^2 H_0 \\sqrt{\\Omega_\\text{rad}} }{c R_\\text{TF}}\\right)^{1\/2} \\nonumber \\\\\n    &\\simeq 0.2~h~{\\rm Mpc}^{-1} \\left( \\frac{R_\\text{TF}}{\\rm 1~kpc} \\right)^{-1\/2} \\label{eq:kcut-TF}\n\\end{align}\n\\begin{align}\n    M_\\text{cut,TF} &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} \\left( \\frac{\\pi}{k_\\text{cut,TF}} \\right)^3 \\nonumber \\\\\n    &\\simeq 1.4\\times10^{15}~{\\rm M}_\\odot \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right) \\left( \\frac{R_\\text{TF}}{\\rm 1~kpc} \\right)^{3\/2}\n\\end{align}\n\nIt will be useful to be able to compare the\ncut-off scale for SFDM-TF to that which occurs\nwhen SI is not present and the SFDM model is\nin the FDM limit, instead.  In the latter case, as\nwe shall show below, there is a different \nJeans mass which depends\nupon the particle mass $m$, rather than on\nthe ratio $g\/m^2$ involving SI, as it \ndoes for SFDM-TF.  For FDM, the EOS\nis always matter-like during the \ngalaxy-formation era (i.e. from horizon entry\nforward, for all modes that are relevant), \nand, just as for SFDM-TF and CDM, \ngalaxy-formation modes\nenter the horizon \nwhen the Universe is RD.  In that case,\nas we shall show in \\S\\ref{sec:LinPerts}, \nthe Jeans mass and its dependence \non scale factor are then\ngiven as follows:\n\\begin{equation}\n    a_\\textsc{j,f} = \\frac{\\hbar^2 k^4}{16\\pi G \\rho_\\text{dm,0} m^2}\n\\end{equation}\n\\begin{equation}\n    \\lambda_\\text{J,F} = \\left( \\frac{\\pi^3 \\hbar^2}{ G \\rho_\\text{dm,0} a^{-3} m^2} \\right)^{1\/4}\n\\end{equation}\n\\begin{align}\n    M_\\text{J,F}(a) &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} a^{-3} \\left( \\frac{\\lambda_\\text{J,F}}{2} \\right)^3 \\nonumber \\\\\n    &\\simeq 6.2 \\times 10^9 \\text{ M}_\\odot~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/4} \\left(\\frac{a}{10^{-3.5}}\\right)^{-3\/4} m_{22}^{-3\/2}\n\\end{align}\nwhere $m_{22} \\equiv m\/(10^{-22}~{\\rm eV}\/c^2)$. \nWe can then do a similar calculation for FDM\nas we did above for SFDM-TF, to estimate\nthe cut-off scale by setting \n$a_\\textsc{j,f} = a_\\textsc{h}$, instead:\n\\begin{equation}\n    \\frac{\\hbar^2 k_\\text{cut,F}^4}{16\\pi G \\rho_\\text{dm,0} m^2} = \\frac{c k_\\text{cut,F} a_\\textsc{h}^2}{ \\pi H_0 \\sqrt{\\Omega_\\text{rad}} } = \\frac{ \\pi H_0 \\sqrt{\\Omega_\\text{rad}} }{c k_\\text{cut,F}}\n\\end{equation}\n\\begin{align}\n    k_\\text{cut,F} &= \\left( \\frac{16 \\pi^2 G H_0 \\rho_\\text{dm,0} \\sqrt{\\Omega_\\text{rad}} m^2 }{c\\hbar^2} \\right)^{1\/5} \\nonumber \\\\\n    &\\simeq 3.9 ~h ~{\\rm Mpc}^{-1}~ \\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{1\/5} m_{22}^{2\/5} \\label{eq:kcut-FDM}\n\\end{align}\n\\begin{align}\n    M_\\text{cut,F} &= \\frac{4\\pi}{3} \\rho_\\text{dm,0} \\left( \\frac{\\pi}{k_\\text{cut,F}} \\right)^3 \\nonumber \\\\\n    &\\simeq 2.3\\times10^{11}~{\\rm M}_\\odot~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{2\/5} m_{22}^{-6\/5}\n\\end{align}\n\nGiven this, we can equate equations~(\\ref{eq:kcut-TF}) and (\\ref{eq:kcut-FDM}) to find the equivalent FDM model that cuts off at the same scale as a given SFDM-TF model, in terms of $m$ and $R_\\text{TF}$. That is, the cut-off scale for a FDM model with a particle mass of $m_\\text{22,cut}$ is the same as that for a SFDM-TF model with a TF radius of $R_\\text{TF,cut}$ given by\n\\begin{equation}\n    R_\\text{TF,cut} \\simeq 3 \\text{ pc }~\\left( \\frac{\\Omega_\\text{dm} h^2}{0.12} \\right)^{-2\/5} m_\\text{22,cut}^{-4\/5}\n    \\label{eq:RTFversusmFDM}\n\\end{equation}\n\nAccording to equation~(\\ref{eq:RTFversusmFDM}),\nthe FDM model with particle mass $m_{22} = 1$ \nresults in suppression of small-scale structure relative to the CDM model below a cut-off scale that is comparable to the\ncut-off scale where suppression begins in the SFDM-TF regime if $R_\\text{TF}\\approx3 \\text{ pc}$.  This is interesting, since\nthis fiducial value chosen for the FDM model is about the value required to flatten the density\nprofiles of present-day dwarf galaxies relative to the CDM model, by making solitonic cores of\nradius equal to the de~Broglie length there,\nof order $\\sim 1\\text{ kpc}$.  According to our results\nin \\citetalias{paper1} and here in \\S\\ref{sec:infall}, however, \nwhen a halo forms in SFDM-TF, it has a \ncore-envelope structure in which the envelope\nis CDM-like but the center flattens to\nthe profile of the $(n=1$)-polytropic SI\ncore, of radius $\\simeq R_\\text{TF}$.\nAs such, the value of $R_\\text{TF}~\\approx~3 \\text{ pc}$, which our estimate above tells us will\nproduce a cut-off of small-scale structure\nin the power-spectrum of density fluctuations\nrelative to that for CDM,\nat a similar wavenumber to\nthat for the FDM model when the latter \nhas $m_{22} = 1$, is very much smaller than\nthat $\\simeq 1\\text{ kpc}$ solitonic core\nsize for the corresponding FDM model.\n\nThis estimate will guide our expectations from the detailed calculation of linear perturbation growth in the two models below. However, while it would  \nbe tempting to think we can use this estimate of the correspondence of the two models, with their parameters so chosen to place their small-scale\ncutoffs at the same wavenumber, the actual\noutcome of structure formation in the two\nmodels depends upon the full spectrum of\nperturbation modes and their cosmic evolution, \ndown into the nonlinear era. \nA contrast between the two models \nmay arise because the TF Jeans mass drops so \nmuch more rapidly with scale factor than does\nthe FDM Jeans mass, for example, so \nscales that are Jeans-filtered in SFDM-TF at\nsome epoch may be able to grow again\nat later times, while there is no such\neffect in FDM.  Moreover, this effect of\nthe ``incredible shrinking Jeans mass'' for\nSFDM-TF means that, while some structures may be\nunderabundant relative to CDM as a result of\ntheir suppression in the linear perturbation stage -- an effect which may be similar to\nthat for the corresponding FDM model --\nthe internal structures of haloes\nthat result in the\ntwo models will differ considerably, if \nFDM solitonic cores in that model are much\nlarger than the polytropic cores in the \ncorresponding SFDM-TF haloes.   The true\ncomparison will require that we first derive \nthe transfer function for the full spectrum of modes,\nwhich we do in the next section,\nbefore we can compare and distinguish the actual impact on galaxy and large-scale structure formation of the two models.  \n\n\n\n\\subsection{Linear perturbations in SFDM-TF}\n\\label{sec:LinPerts}\n\nWe are now ready to calculate the growth of linear perturbations for all modes of interest to us in the formation of galaxies and large-scale structure, and from this, the transfer function\nfor the SFDM-TF model, for comparison with those \nof the CDM and FDM models.   We do this semi-analytically, by adopting several simplifying assumptions.  \nThe full treatment without approximation would require solving the relativistic \nKlein-Gordon-Einstein equations of motion for the scalar field, in a FLRW background universe, along with all other components of the standard cosmological model (i.e. radiation, baryons, neutrinos). This would be necessary\nto be able to consider the\nsmooth entry of modes into the horizon, in general,\nfrom well-outside to deep within the horizon.  \nThis full treatment would also need to incorporate the non-standard expansion history which results from the evolution of the SFDM EOS, as described by \\citet{Li14,Li17}, which depends upon the contribution of SFDM to the total energy density at each moment and, hence, depends on the value of particle parameters $m$ and $g\/m^2$.  In the most general case, the field can be either real or complex.\nIn both cases, the homogeneous field can be described as a perfect fluid, which makes possible\na fluid description for its evolution.  However,\nwhen treating perturbations in the complex case,\nthe perfect fluid description can break down in\nits stiff phase.   We will, therefore, \ngreatly facilitate the semi-analytical solution\nby the following approximations:\n\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item We will assume that SFDM model parameters are\n    constrained such that the structure formation era,\n    when all modes of interest to us enter the horizon\n    and evolve, is insignificantly affected by \n    the ``back-reaction'' of the SFDM and its EOS\n    evolution on the expansion history of the universe.\n    It turns out that this is a very good approximation, since the departures from standard expansion history during this structure formation era are related to the transitions which occur between\n    the EOS from matter-like ($w=0$) to \n    radiation-like ($w=1\/3$) and, thence,\n    either to cosmological-constant-like ($w=-1$) \n    or stiff ($w=1$), as one moves \n    backward in time.  The TF regime under consideration here ensures that the structure formation era is confined to the rapid oscillation phase of the scalar field, and the only transition\n    that takes place during this period is that between radiation-like and matter-like.  When that happens, as we said before, it must take place early enough that it does not perturb\n    the value of $a_\\text{eq}$ inferred from the measured CMB anisotropy.  As pointed out\n    in \\citet{Li14,Li17},\n    this means there is an upper bound to the value of the SI strength\n    in terms of $g\/m^2$, and, hence, of the value of $R_\\text{TF}$.\n    In particular, $g\/m^2c^4 \\leq 4 \\times 10^{-17} \\text{ eV}^{-1} \\text{ cm}^3$ is required\n    to satisfy $w(z_\\text{eq})\\leq0.001$, which\n    is equivalent to requiring that $R_\\text{TF} \\leq 5 \\text{ kpc}$. \n    When this bound is placed, there\n    is a corresponding limitation placed on the\n    contribution of the SFDM radiation-like energy density to the total during the RD era, which\n    means that SFDM does not appreciably alter the\n    expansion rate of the universe then.\n    \n    \\item We will assume that SFDM behaves as a perfect\n    fluid, so a fluid description is possible, with\n    pressure and energy density related accordingly, by\n    the EOS parameter in equation~(\\ref{eq:averagew}). \n    \n    \\item We will assume that SFDM is radiation-like\n    with $w = 1\/3$ at $a\\leq a_t$ and matter-like with\n    $w = 0$ at $a>a_t$, i.e. we assume an instantaneous transition. \n    We will take $a_t=a_{1\/9}$.\n    \n    \\item Once modes are well-within the horizon, we will treat them in the Newtonian approximation. \n\n\\end{enumerate}\n\nFor each mode, we shall calculate as our\nreference solution, that for the growth of dark matter perturbations in the CDM model.\nThe latter will be treated in the conventional way, as a nonrelativistic, collisionless, pressure-free gas (i.e. ``dust''). \nAn analytical solution is available for CDM perturbation modes when they are sub-horizon ($a \\gg a_\\textsc{h}$), given by \\citep[see, e.g.,][\\S6.3]{Mukhanov}:\n\\begin{align}\n    &\\frac{\\delta_\\text{CDM}(a)}{\\delta_\\text{H}} = 6 \\left[ \\ln \\left( \\frac{4 y }{ x } \\right) + \\gamma_\\textsc{e} - \\frac{7}{2} \\right] \\left(1 + \\frac{3 x}{2} \\right) \\nonumber\\\\\n    & \\quad \\quad \\quad \\quad \\quad- 6 \\left[ \\left(1 + \\frac{3 x}{2} \\right) \\ln \\left( \\frac{\\sqrt{1 + x} + 1}{\\sqrt{1 + x} - 1} \\right) - 3 \\sqrt{1 + x} \\right] \\label{eq:deltaCDM}\\\\\n    & x \\equiv \\frac{a}{a_\\text{eq}} \\\\\n    & y \\equiv \\frac{c k a}{H_0 \\sqrt{3\\Omega_\\text{rad}}}\n\\end{align}\nwhere $\\gamma_\\textsc{e} \\simeq 0.577$ is the Euler-Mascheroni constant, and $\\delta_\\text{H}$ is the asymptotic superhorizon value for the (mass) density contrast of a given mode.\nTo match this analytical subhorizon solution in equation~(\\ref{eq:deltaCDM}),\nvalid when a mode is deep within the horizon, to the solution of a more exact, fully-relativistic, numerical solution that follows the perturbation smoothly as it\nenters the horizon from far outside to deep within, we\ndefine an effective horizon-entry scale factor for each mode, $a_{\\textsc{h},\\text{eff}}(k)$,\nthat makes the two solutions agree\nwhen the modes are deep within\nthe horizon. This is equivalent\nto our adopting a thin-horizon approximation\nsuch that for \n$a < a_{\\textsc{h},\\text{eff}}$, we treat the perturbations as frozen at their superhorizon values $\\delta_\\text{H}$, while for $a > a_{\\textsc{h},\\text{eff}}$, the perturbations grow according to equation~(\\ref{eq:deltaCDM}).\nTo ensure that the density contrast is continuous across horizon-entry, then, we must choose $a_{\\textsc{h},\\text{eff}}$ such that, by equation~(\\ref{eq:deltaCDM}),  \n\\begin{equation}\n    \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{H} \\label{eq:aHeff}\n\\end{equation}\nfor each mode $k$. For SFDM,\nas long as the scalar field is matter-like at this value of $a_{\\textsc{h},\\text{eff}}$ \n(i.e. $a_{\\textsc{h},\\text{eff}}>a_t$), \nwe can also use this condition to set the horizon-entry value for an SFDM perturbation.\nThat is, the superhorizon and horizon-entry values for a dark matter perturbation should be independent of the dark matter model (i.e. only their subhorizon evolution should differ):\n\\begin{equation}\n    \\delta_\\text{SFDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{H}    \\label{eq:horentryamp}\n\\end{equation}\n(Henceforth, in \\S\\ref{sec:LinPerts}, the horizon-entry scale factor will refer to\nthis $a_{\\textsc{h},\\text{eff}}$, unless otherwise noted.  Elsewhere, we refer more generally to ``horizon-entry'' in terms of a mode filling the Hubble volume, for which\n$k = k_\\text{H} \\equiv \\frac{\\pi}{c}a_\\textsc{h}H(a_\\textsc{h})$.  The two definitions are quantitatively similar; we compare them in Appendix~\\ref{sec:aH}, Fig.~\\ref{fig:aHeff}.)\n\nFor perturbations in SFDM-TF, there are\ntwo possible phases of behavior to consider,\naccording to whether the scalar field\nis radiation-like or matter-like when a given\nmode is inside the horizon. \nModes that enter the horizon when the\nscalar field is \\textit{radiation-like} (i.e. with $a_{\\textsc{h},\\text{eff}} \\leq a_t$)  \nevolve just like those in the radiation component that dominates the total energy density.  \nSince we will only consider models for which the field transitions from radiation-like to matter-like at a scale factor $a_t < a_\\text{eq}$, the field will behave like radiation in a radiation-dominated universe for $a < a_t$, which has a known analytical solution, given by\n\\citep[see, e.g.,][\\S7.3]{Mukhanov}:\n\\begin{align}\n        \\frac{\\delta_\\text{rad}(a)}{\\delta_\\text{rad}(0)} = -3 \\left[ \\left( \\frac{2-y^2}{y^2} \\right) (\\text{sinc }y - \\cos y) - \\text{sinc } y \\right] \\label{eq:deltarad}\n\\end{align}\nwhere $\\text{sinc } y \\equiv \\sin y\/y$.\nAccording to this solution, such modes oscillate\nlike sound waves in a relativistic fluid\nwith sound speed $c\/\\sqrt{3}$, with constant\namplitude.  This analytical solution is fully-relativistic and\nproperly follows the perturbations from far\noutside the horizon through horizon entry, so\nthe time at which oscillations begin as a\nfunction of wavenumber is not arbitrary, but\nreflects the synchronization of horizon\nentry for all modes\\footnote{Our only\napproximation in this case is that, in practice,\nwe assign $a_t$ as the scale factor at which the average EOS parameter $w$ for the scalar field equals $1\/9$, and assume that SFDM instantaneously transitions from radiation-like to matter-like at $a_t$.} of a\ngiven $k$.   \n\nFor modes that are inside the horizon when\nthe scalar field is \\textit{matter-like}, \ninstead \\big(i.e. whenever $a > \\max[a_{\\textsc{h},\\text{eff}},a_t]$\\big), \neach mode evolves thereafter according to the following differential equation \\citep{SM11,Chavanis12}:\n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{ \\hbar^2 k^4}{4 m^2 a^4} + \\frac{\\bar{c}_{s,\\text{SI}}^2 k^2}{a^2} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpertgen}\n\\end{equation}\nwhere overdots denote differentiation with respect to cosmic time $t$, and $\\bar{c}_{s,\\text{SI}}$ is the sound speed of the background associated with SI pressure, given by:\n\\begin{equation}\n    \\bar{c}^2_{s,\\text{SI}} = \\frac{g}{m^2} \\rho_\\text{dm,0} a^{-3}\n    \\label{eq:SI_soundspeed_bg}\n\\end{equation}\nThe first term inside the parentheses in equation (\\ref{eq:linpertgen}) is the quantum pressure term, while the second is the effect of SI pressure.  The third term\nis responsible for gravitational growth.\nThe negative sign\nin front of that third term means that the first two, both positive-definite, \nact to oppose gravitational growth. The sum of all three terms inside the parentheses can be zero for a certain value of wavenumber, which we\nshall call the generalized Jeans wavenumber, \n$k_\\text{J}$.  Modes with wavenumber $k<k_\\text{J}$ are able to grow, while those with\n$k>k_\\text{J}$ oscillate \nlike sound waves for which\nthe evolution of their amplitude depends upon\nwhich of those first two terms dominates. \nWe note that the scale factor dependences of\nthose two first terms are different, so,\nin principle, if one term dominates at some\ntime, the other may dominate at other times. \n\n\nIn the TF regime, the\nregime of primary interest to us here, we can\ntake the limit in which ${\\hbar k^2 \/ H m \\to 0}$\nin equation~(\\ref{eq:linpertgen}) above\nand drop the first term inside the\nparentheses. This means that, in this\nlinear regime, SI dominates and\nquantum pressure is unimportant.  This contrasts\nthe linear perturbation regime with the nonlinear\nregime in which inhomogeneities even on the scale of the de~Broglie wavelength (and their contribution to quantum pressure) cannot be neglected, since they are responsible for transporting momentum and, thereby, affecting larger scales,\neven in a spatially-averaged description, as we discussed in \\citetalias{paper1}.  \nIn this linear regime, then, subhorizon\nperturbations in SFDM-TF \nduring its matter-like phase obey the following\nequation:\n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{\\bar{c}_{s,\\text{SI}}^2 k^2}{a^2} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpert}\n\\end{equation}\nThis equation is the origin of our expression\nfor the proper TF Jeans wavelength $\\lambda_\\text{J,0}$ in\n\\S\\ref{sec:LinPertOverview}, which we obtain by solving for the comoving Jeans wavenumber ($k_\\text{J,0} = 2\\pi a\/\\lambda_\\text{J,0}$)\nthat makes the terms inside the parentheses in equation~(\\ref{eq:linpert}) sum to zero:\n\\begin{equation}\n    k_\\text{J,0} = a\\sqrt{\\frac{4\\pi G m^2}{g}} = \\frac{\\pi a}{R_\\text{TF}}\n\\end{equation}\nAccording to equations~(\\ref{eq:linpertgen}) and (\\ref{eq:SI_soundspeed_bg}),\nif we are in the TF regime at the end\nof the evolution period of interest,  \nthen we were always in the TF regime at all earlier times, as well,\nso equation~(\\ref{eq:linpert}) remains valid\nthroughout this phase of matter-like EOS.  \n\nIn practice, we solve equation~(\\ref{eq:linpert}) for $\\delta \\equiv \\delta_\\text{SFDM}(a)\/\\delta_\\text{H}$.\nThe remaining uncertainty is then about what initial condition to impose on the solution. There are two regimes:\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent]\n    \\item $a_{\\textsc{h},\\text{eff}} > a_t$ -- modes enter the horizon when the field is matter-like:\n    \n    In this case, since all dark matter models should behave the same for superhorizon modes, we use the CDM horizon entry amplitude and slope\n    \\begin{align}\n        &\\delta_\\text{SFDM}(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}(a_{\\textsc{h},\\text{eff}}) \\label{eq:delICmatter}\\\\\n        &\\delta_\\text{SFDM}'(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}'(a_{\\textsc{h},\\text{eff}}) \\label{eq:delprimeICmatter}\n    \\end{align}\n    or, equivalently,\n    \\begin{align}\n        &\\delta(a_{\\textsc{h},\\text{eff}}) = 1 \\label{eq:delnormICmatter}\\\\\n        &\\delta'(a_{\\textsc{h},\\text{eff}}) = \\delta_\\text{CDM}'(a_{\\textsc{h},\\text{eff}})\/\\delta_\\text{H} \\label{eq:delprimenormICmatter}\n    \\end{align}\n    \n    \n    \\item $a_{\\textsc{h},\\text{eff}} \\leq a_t$ -- modes enter the horizon when the field is radiation-like:\n    \n    In this case, since SFDM should behave like radiation prior to $a_t$, we use the (normalized) amplitude and slope of the radiation solution at $a_t$\n    \\begin{align}\n        &\\delta(a_t) = \\delta_\\text{rad}(a_t)\/\\delta_\\text{rad}(0) \\\\\n        &\\delta'(a_t) = \\delta_\\text{rad}'(a_t)\/\\delta_\\text{rad}(0)\n    \\end{align}\n\\end{enumerate}\n\n\n\\subsection{The transfer function: SFDM-TF vs. CDM}\n\\label{Transferfunction}\n\nIt is customary to describe the growth of density fluctuations in alternative dark matter models to CDM in terms of a \\textit{normalized} transfer function, defined as follows:\n\n\\begin{align}\n    T_\\text{SFDM}^2(k,a) &\\equiv \n    \\frac{\\mathcal{T}^2_{\\text{SFDM},k}(a,a_i)}{\\mathcal{T}^2_{\\text{CDM},k}(a,a_i)} =\n    \\frac{P_\\text{SFDM}(k,a)}{P_\\text{CDM}(k,a)} \\nonumber\\\\\n    &= \\left|\\frac{\\delta_\\text{SFDM}(k,a)}{\\delta_\\text{CDM}(k,a)}\\right|^2\n\\end{align}\n(see also equation \\ref{eq:dP}),\nwhere $P(k,a)$ is the linear-fluctuation power spectrum of the dark matter for each model, as labeled. \n\n\n\n\\subsection{Comparison with FDM}\n\nAs a proof of principle, we can also compute the \ntransfer function for the FDM model, \nusing a procedure similar to that outlined above for SFDM-TF, and compare our results to those of the publicly-available code \\textsc{axioncamb}\\footnote{https:\/\/github.com\/dgrin1\/axionCAMB}\nby \\cite{axionCAMB}.\nThis code is presumably more accurate since it\nsolves the fully-relativistic perturbation \nequations numerically, from superhorizon through  \nsubhorizon regimes for each $k$, including their coupling\nto fluctuations in other components of energy-density, as\nwell. To derive the transfer \nfunction for FDM, \nwe drop the SI-term in equation~(\\ref{eq:linpertgen}):  \n\\begin{equation}\n    \\Ddot{\\delta} + 2H\\Dot{\\delta} + \\left( \\frac{ \\hbar^2 k^4}{4 m^2 a^4} - 4\\pi G \\rho_\\text{dm,0} a^{-3} \\right) \\delta = 0\n    \\label{eq:linpertFDM}\n\\end{equation}\nThis equation for self-gravitating perturbations \nin the free-field (no-SI) case \nwas also discussed elsewhere, e.g. \\cite*{KMZ}.\nOnce again, we obtain the Jeans wavelength and wavenumber by setting the terms in parentheses equal to zero:\n\\begin{equation}\n    k_\\text{J,F} = \\left( \\frac{16\\pi G \\rho_\\text{dm,0} m^2 a}{\\hbar^2} \\right)^{1\/4}\n\\end{equation}\n\nThere are three noteworthy distinctions between the FDM and SFDM-TF models with regards to the solutions of their linear perturbation equations:\n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n\\item when an FDM mode is sub-Jeans, it oscillates with a constant amplitude, whereas a sub-Jeans SFDM-TF mode oscillates with an amplitude that grows like $a^{1\/4}$;\n\\item the FDM Jeans mass shrinks much more slowly over time \nthan that for SFDM-TF \\big($M_\\text{J,F} \\propto a^{-3\/4}$ vs. $M_\\text{J,0} \\propto a^{-3}$\\big),\nso sub-Jeans modes of SFDM-TF can become super-Jeans\nsooner and with more time left to grow like CDM; and\n\\item in the FDM limit, the radiation-like phase is absent,\nso the initial conditions are set by equations~(\\ref{eq:delICmatter}) - (\\ref{eq:delprimenormICmatter}) for all modes.\n\\end{enumerate}\nThe first two of these distinctions explain why FDM has a sharper cut-off \nin the transfer function \nthan SFDM-TF does.\\footnote{The sharp cut-off in the \ntransfer function for FDM was also discussed by \\cite*{Hu2000},\nan important early contribution to this subject. \nThey reported an approximate fitting formula for the solution for different wavenumbers \n$k$ at a given scale factor, parameterized in terms of the ratio of that wavenumber to the Jeans wavenumber at that scale factor, $k\/k_\\text{J}$, which differs somewhat from the solutions presented here, which are numerical solutions of equation~(\\ref{eq:linpertFDM}). \nWhile they do not give the linear perturbation equations they solved numerically, \nthey say they treated FDM as a perfect fluid \nwith an effective sound speed, which, in its \nrapid-oscillation phase of matter-like EOS, \nis given by\n$c_{s,\\textsc{f}}^2=\\hbar^2 k^2\/\\big(4 m^2 a^2\\big)$. If so, then the dependence of the\nJeans wavenumber on scale factor should agree with ours derived by setting the bracket in our \nequation~(\\ref{eq:linpertFDM}) to\nzero, namely $k_\\text{J}\\propto a^{1\/4}$, in \\textit{both} the RD and MD eras.  \nWhile agreeing with this for the MD era, however, \\cite{Hu2000} states \nthat $k_\\text{J}$ is independent of $a$ during the important RD era, which is not correct.\n}\n\n\\subsection{The Halo Mass Function}\n\\label{sec:HMF}\n\nThe linear transfer function for the\nSFDM-TF model computed above can be used to\ncalculate the halo mass function (``HMF'') -- defined as the comoving number density of haloes \nper unit logarithm of halo mass $M$, $\\frac{dn}{d\\ln M}$.  There is an extensive literature on\nthe various choices for analytical and empirical\nprescriptions for computing the HMF from the\ntransfer function so as to predict the results of\ncosmological N-body simulation and their analysis\nby different halo-finding algorithms \\citep[see, e.g.,][and references therein]{Knebe13,Diemer18}. As our purpose here is to compare and contrast the HMF of\nthe SFDM-TF model with those of CDM and FDM, and\nthere are not yet N-body results for the former\nwith which to perform this comparison (and hardly\nany for FDM, either), we will adopt the\nmost straightforward prescription for calculating\nthe HMFs semi-analytically \nfor all three models, from their respective\nlinear power spectra $P(k,a)$, \nusing the standard Press-Schechter (``PS'') formalism \\citep{PS74}. \nThe latter is based upon the Gaussian statistics of the initial density perturbations, smoothed on a given mass scale, and the ansatz that a given local overdensity in the smoothed, linearly-perturbed density field will collapse and virialize at the time predicted for a spherical top-hat perturbation of that initial overdensity to reach infinite density.   A more\nrigorous underpinning for this remarkably successful ansatz was subsequently provided by the Excursion Set Theory of \\citet{Bond91}, including derivation of the PS formula for the HMF for the case of ``sharp $k$-space filtering'', about which we will say more below.\n\nThe PS HMF depends, not only on the underlying\npower spectrum of the initial, Gaussian-random density fluctuations, \nbut also on the shape of the function adopted to\nsmooth the density field.  The purpose of this\nsmoothing function is to squelch \nhigh-frequency modes that correspond to fluctuations of wavelength smaller than that which encompasses a given halo mass scale $M$, on average.  The mass variance $\\sigma_\\textsc{m}^2(M,a)$ of the linear overdensity field smoothed in coordinate space with a spherical window function \n$W(\\bm{x},R)$ -- where $R$ is the comoving radius of a sphere in the unperturbed\nbackground universe which encompasses the mass $M$, $R = \\big[M\/\\big(\\frac{4\\pi}{3} \\rho_\\text{dm,0}\\big)\\big]^{1\/3}$ -- is related to\nthe power spectrum $P(k,a)$ and the Fourier transform of this window function, $\\mathcal{W}(k,R)$, as\nfollows:\n\\begin{equation}\n    \\sigma_\\textsc{m}^2(M,a) = \\frac{1}{2\\pi^2} \\int_0^\\infty P(k,a) \\mathcal{W}(k,R) k^2 dk\n\\end{equation}\nThe window function is usually taken to be a top-hat filter, expressed in Fourier space as \n\\begin{equation}\n    \\mathcal{W}_\\text{top-hat}(k,R) = 3\\frac{\\sin(kR) - kR \\cos(kR)}{(kR)^3}\n\\end{equation}\nHowever, for models with sharp cut-offs in the power spectrum such as FDM, a sharp-$k$ filter is preferred in order to have the cut-off reflected in the HMF \\citep{KO20}:\n\\begin{equation}\n    \\mathcal{W}_{\\text{sharp-}k}(k,R) =\n    \\begin{cases}\n        1 & k < \\alpha\/R \\\\\n        0 & k > \\alpha\/R\n    \\end{cases}\n\\end{equation}\nwhere $\\alpha$ is a free parameter which we set to 1.9 in order to make the resulting HMF agree with that computed using the top-hat filter at large mass.\nThe PS HMF is then given by\n\\begin{equation}\n    \\frac{dn}{d\\ln M} = - \\sqrt{\\frac{2}{\\pi}} \\frac{\\rho_\\text{dm,0}}{M} \\frac{\\delta_c}{\\sigma_\\textsc{m}} e^{-\\delta_c^2\/2\\sigma_\\textsc{m}^2} \\frac{d\\ln \\sigma_\\textsc{m}}{d\\ln M}\n    \\label{eq:PSHMF}\n\\end{equation}\nwhere $\\delta_c = 1.686$ is the critical linear overdensity for halo formation derived from the spherical top-hat collapse model.\\footnote{\n\\citet{KO20} used $\\alpha=2.5$ following the work of \\citet{Benson13}, who found that value to be a good fit to warm dark matter simulations, and re-scaled $\\delta_c$ by a factor of 1.195 in order to produce a sharp-$k$ filtered HMF that was consistent with the top-hat filtered HMF at large mass. In view of the fact that we are here comparing the HMF's of CDM, SFDM-TF and FDM, these choices would not be appropriate for all three models, especially in view of the strong differences we find between SFDM-TF and FDM. However, we also find that adopting these choices would not affect our general conclusions.}\n\n\\subsection{Results of linear perturbation theory: SFDM-TF vs. CDM vs. FDM}\n\\label{sec:results}\n\n\\subsubsection{Perturbation growth for a single mode}\n\\label{sec:pertresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{LinPertSoln.pdf}\n    \\includegraphics[width=\\columnwidth]{LinPertSoln_FDM.pdf}\n    \\caption{\\textbf{Linear perturbation growth after horizon entry, for a single Fourier mode.} Solutions of \n    the differential equations for the evolution of linear-perturbation fractional overdensity $|\\delta\/\\delta_\\text{H}|$ vs. scale factor\n    $a$ are plotted for a single Fourier mode,\n    with wavenumber $k$ associated with a mass\n    $M=10^{9} \\text{ M}_\\odot$ \n    \\big(where $M = \\frac{4\\pi}{3} \\rho_\\text{dm,0}(\\pi\/k)^3$ and $k \\simeq 16$ Mpc$^{-1}$\\big) in the SFDM-TF (top panel) and FDM (bottom panel) models, for\n    two different cases for each. \n    The purple and green SFDM-TF lines correspond to $R_\\text{TF} = 1$~pc and $R_\\text{TF} = 100$~pc, respectively. The light and dark grey FDM lines correspond to $m_{22} = 0.8$ and $m_{22} = 29$, respectively. The black line is the CDM solution. Dots of the same color as the curves on which they reside mark the Jeans scale factors, $a_\\textsc{j,0}$ (SFDM-TF) and $a_\\textsc{j,\\textsc{f}}$ (FDM). The blue dashed line in the top panel shows the SFDM-TF solution for $R_\\text{TF} = 100$ pc when the scalar field is radiation-like (i.e. $w=1\/3$) (given by equation~\\ref{eq:deltarad}), and the blue star \n    marks the transition scale factor $a_t$ for this case, after which SFDM-TF is matter-like (green solid line). The red dashed lines in both panels marks radiation-matter equality.}\n    \\label{fig:delta_ex}\n\\end{figure}\n\nIn Fig. \\ref{fig:delta_ex}, we plot solutions \nof the differential equations described in \\S\\ref{sec:LinPerts} for the fractional linear overdensity $|\\delta\/\\delta_\\text{H}|$ vs. scale factor $a$, \nfor the subhorizon evolution of \na single, representative Fourier mode $k$. \nWe compare the analytical\nsolution for this mode in the CDM model\nwith our solutions for \nSFDM-TF and FDM, for illustrative cases with different parameters for the two kinds of \nSFDM models. The wavenumber $k$ for our\nrepresentative mode is that associated with a mass $M=10^{9} \\text{ M}_\\odot$ \\big(where $M = (4\\pi\/3) \\rho_\\text{dm,0}(\\pi\/k)^3$\\big). \nFor SFDM-TF, we present two cases, with \n$R_\\text{TF} = 1$ and $100$ pc, respectively. For FDM, we present two cases\nas well, with values of the particle mass\n$m_{22} = 0.8$ and $29$, respectively.\n\nThe perturbation evolution for this mode in \npressure-free CDM plotted in\nFig. \\ref{fig:delta_ex} shows the familiar\ntransition from RD-era logarithmic growth\nto the MD-era growth rate ($\\delta \\propto a$). \nThis transition for CDM is gradual, but\nis centered around $a = a_\\text{eq}$. By contrast,\nthe perturbation evolution for both the SFDM-TF \nand FDM solutions plotted in Fig. \\ref{fig:delta_ex}\nshows the effects of pressure and, hence,\nJeans filtering, at early times. \nThe evolution changes when this mode passes from \nsub-Jeans ($k>k_\\text{J}$) to \nsuper-Jeans ($k<k_\\text{J}$) at \n$a = a_\\textsc{j}$, where the values of $a_\\textsc{j}$ (marked by a dot on each curve) depend upon the model parameters and differ\nfor each case.\nWhen a mode of either SFDM-TF or FDM\nis sub-Jeans, its fractional overdensity oscillates\nlike a sound wave.  As such, \nthe periods of oscillation in the two models are inversely proportional to their effective sound speeds. For SFDM-TF, this sound speed increases with $R_\\text{TF}$ when the EOS is matter-like, but is a constant, independent of $R_\\text{TF}$, \nwhen the EOS is radiation-like.  The scale\nfactor $a_t$ at which this transition\nfrom radiation-like to matter-like EOS occurs\nfor SFDM-TF also depends on $R_\\text{TF}$,\nwith $a_t$ increasing with $R_\\text{TF}$.   \nFor FDM, which is always matter-like during\nthe era we study here, the sound speed decreases\nwith increasing $m_{22}$. These points allow us\nto understand the behavior of the different\ncurves in Fig. \\ref{fig:delta_ex}, as follows.\n\nConsider, first, the curves for\n$m_{22} = 0.8$ and $R_\\text{TF} = 100$ pc, respectively.  For these values of $R_\\text{TF}$ and $m_{22}$, \nthe periods of sub-Jeans oscillation are smaller than a Hubble time during that phase of evolution. \nIn that case, their fractional overdensities oscillate rapidly enough to \npass from positive to negative and back again\na number of times, so the\ncurves of their absolute value displayed in Fig. \\ref{fig:delta_ex} appear like trajectories\nof a bouncing ball.  For FDM, \nthe sub-Jeans oscillations have a constant amplitude, whereas, when the\nsub-Jeans SFDM-TF mode oscillates, its amplitude\nis initially constant for a time (the blue-dashed portion), during its radiation-like phase,\nbut then grows like $a^{1\/4}$ after it transitions\nat $a=a_t$ (as marked on the curve) \nto a matter-like EOS. In both cases, the oscillations cease near $a\\approx a_\\textsc{j}$, and perturbation\ngrowth rates become CDM-like for $a\\gg a_\\textsc{j}$.\n\n\nIn the case of the other two \ncurves in Fig. \\ref{fig:delta_ex},\nthat for SFDM-TF with $R_\\text{TF}= 1$ pc\nand FDM with $m_{22} = 29$, the \noscillatory behavior during their sub-Jeans\nphase is apparently absent, but this reflects\nthe fact that the change of model parameters \nhas pushed their sound speeds to be smaller \nand their oscillation ``periods'' to be\nlarger than a Hubble time.  In that case,\nthe background universe is expanding too fast for\nthe density to complete a single oscillation.\nThese cases therefore follow the CDM trajectory fairly closely, albeit with their growth slightly hampered due to the brief initial sub-Jeans phase.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction.pdf}\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction_rescaled.pdf}\n    \\includegraphics[width=.95\\columnwidth]{TransferFunction_FDM.pdf}\n    \\caption{\\textbf{Linear transfer functions.} \\textbf{(a) (top panel)} Normalized transfer functions $T_\\text{SFDM}^2(k,a)$ for SFDM-TF \n    vs. comoving wavenumber \n    $k$, for $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc (left to right). For comparison, we plot FDM transfer functions $T_\\text{FDM}^2(k,a)$\n    for $m_{22}= \\{0.8, 29\\}$, generated by \\textsc{axioncamb}.\n    Dots locate values of $k_\\text{cut}$ derived in \\S\\ref{sec:cutoff}\n    to estimate the cut-off wavenumber above which high-frequency modes are suppressed relative to CDM. \n    \\textbf{(b) (middle panel)} SFDM-TF transfer functions in (a), \n    but with $k$-axis rescaled by dependence of $k_\\text{cut}$ on $R_\\text{TF}$, \n    so that locations on the horizontal axis \n    at which $k = k_\\text{cut}$ are the same\n    for all $R_\\text{TF}$. \n    SFDM-TF curves for larger $R_\\text{TF}$ start to deviate from CDM more sharply than for smaller \n    $R_\\text{TF}$, so $T^2_\\text{SFDM}(k_\\text{cut})$ is closer to 1 for smaller $R_\\text{TF}$.\n    \\textbf{(c) (bottom panel)} \n    FDM transfer functions   \n    in (a) (light and dark red lines),\n    generated by \\textsc{axioncamb}, are \n    compared to our solutions of subhorizon\n    linear perturbation growth for\n    FDM (dark and light blue), \n    i.e. same procedure we used for\n    SFDM-TF, but using the FDM linear perturbation equation (\\ref{eq:linpertFDM}), instead.}\n    \\label{fig:transfer}\n\\end{figure}\n\n\\subsubsection{Normalized transfer functions}\n\\label{sec:transferfunctionresults}\n\n\n\nIn Figs. \\ref{fig:transfer}(a) and (b), \nour \nsolutions for linear perturbation growth for different $k$-modes are used to plot the \nnormalized transfer functions \n$T_\\text{SFDM}^2(k,a)$ for SFDM-TF vs. $k$, for several illustrative cases, with $R_\\text{TF} = 1000, 100, 10, \\text{and}~1$ pc (left to right). \nFor comparison, \nwe also plot FDM transfer functions\n$T_\\text{FDM}^2(k,a)$, for the same two masses, $m_{22}=0.8$ and $29$, as in Fig. \\ref{fig:delta_ex}\nabove.   In Fig. \\ref{fig:transfer}(a),\ntransfer functions for FDM were generated by \\textsc{axioncamb}, while in \nFig. \\ref{fig:transfer}(c), we compare those\nwith our solutions\nfor FDM, to show their close agreement.\nThis agreement \nserves to validate our\nsolutions for both FDM and SFDM-TF, \nsince the former were derived\nby the same procedure as for our SFDM-TF \nsolutions.  Dots locate values of $k_\\text{cut}$ derived in \\S\\ref{sec:cutoff} to estimate the cut-off wavenumber above which high-frequency modes are suppressed relative to CDM.\n\nIn Fig. \\ref{fig:transfer}(b) (the middle panel),\nwe replot the SFDM-TF transfer functions in \nFig. \\ref{fig:transfer}(a) (the top panel) \nwith the $k$-axis rescaled by the dependence \nof $k_\\text{cut}$ on $R_\\text{TF}$, \nso that locations on the horizontal axis \nat which $k = k_\\text{cut}$ are the same\nfor all $R_\\text{TF}$. This shows that the smaller\nthe value of $R_\\text{TF}$ is, the more gradual\nis the decline relative to CDM at wavenumbers\nwhere the SFDM-TF transfer function begins to\ndecline relative to CDM at high $k$.\nIn addition, since curves for larger $R_\\text{TF}$ start \nto deviate from CDM more sharply than for smaller \n$R_\\text{TF}$, $T^2_\\text{SFDM}(k_\\text{cut})$ \nis closer to 1 for smaller $R_\\text{TF}$.\n{We can also see from this figure that the amplitudes of the oscillations at high wavenumber are smaller for larger $R_\\text{TF}$. \nThis may be due, in part, to the fact that, for larger $R_\\text{TF}$, modes with $k \\gg k_\\text{cut}$ have more of their sub-Jeans evolution occur during the matter-dominated era ($a > a_\\text{eq}$).\nSince CDM modes grow faster during this era (linearly with scale factor, as opposed to logarithmically for $a < a_\\text{eq}$), the $k \\gg k_\\text{cut}$ SFDM-TF modes for larger $R_\\text{TF}$ cases ``miss out'' on more of this comparatively rapid, linear growth phase than the corresponding modes of the smaller $R_\\text{TF}$ cases, and so are more suppressed relative to CDM. }\n\n\nAccording to our analytical estimates in \\S\\ref{sec:LinPertOverview}, it is possible\nto choose values of $R_\\text{TF}$ for SFDM-TF\nand $m_{22}$ for FDM so as to facilitate their direct comparison,\nby selecting values that make the\nsmall-scale cut-offs in their respective\ntransfer functions occur at roughly the same \nwavenumber (see equation \\ref{eq:RTFversusmFDM}).  In that case, for example, our solution for \nFDM with $m_{22} = 0.8$ can be compared with that\nfor SFDM-TF with $R_\\text{TF} \\simeq 10$ pc.\nIn this way, observational constraints on\nFDM can also serve as a rough proxy for constraints on \nSFDM-TF, since both models have perturbation growth \nwhich is CDM-like at small $k$ but suppressed \nat high $k$, i.e. values of $k$ above \nthe cut-off. \n\nThis estimator for the $R_\\text{TF}$-values that make\nthe transfer function cut-offs for SFDM-TF and \nFDM for a given $m_{22}$ ``align'' in wavenumber by\nsolving equation (\\ref{eq:RTFversusmFDM})\nshould be used with caution, however. \nThe \\textit{shapes} of their transfer functions below the cut-off wavenumber differ somewhat. \nAs a result, if different observables measure the\namount of structure on different scales, \ni.e. at different $k$, then the correspondence between \nthe $m_{22}$- and $R_\\text{TF}$-values that make\nthe two model predictions agree may depend upon which\nobservable is used to constrain them.  \nAs an example,\nthe two models can have very\ndifferent half-power wavenumbers, $k_{1\/2}$\nin their transfer functions,\neven when their $k_\\text{cut}$-values are identical.  \n\nThe two illustrative values of\nFDM particle mass we chose for Figs. \\ref{fig:delta_ex} and \\ref{fig:transfer}\nwill appear again in later figures, too,\nin which we compare the SFDM-TF and FDM models \nfurther.  These values roughly\nspan the current range of constraints in the\nFDM literature, placed by various observational probes,  namely, $m_{22}=0.8$ \n(\\citealp[e.g. fiducial case of ][]{VBB19} and \\citetalias{paper1}) and $m_{22}=29$\n\\citep[e.g. lower bound from][]{Nadler21}.\nFor these values of $m_{22}$, de~Broglie wavelengths \ninside a present-day dwarf galaxy of mass $10^{10}\\text{ M}_\\odot$ with\nvelocity dispersion $\\sigma_\\text{v} \\approx 30$ km s$^{-1}$, for \nexample, are $\\approx 4$ kpc and $0.1$ kpc, respectively, so haloes of this mass that\nformed in FDM would have solitonic cores of these sizes.\nIf we chose $R_\\text{TF}$-values of this size, \nso as to make the SI-polytropic cores inside \nSFDM-TF haloes as large as this, then\nthe $k_\\text{cut}$-values for SFDM-TF will be\nmuch smaller than the $k_\\text{cut}$-values for\nthese FDM models.  The transfer function for\nSFDM-TF in that case would begin to cut-off at a much higher\nmass scale than FDM, so the amount of structure formed  \non such small mass scales in SFDM-TF would be well below\nthat in the corresponding, core-size-matched FDM model. \nOn the other hand, if we align the $k_\\text{cut}$-values \nof the two models, instead, which Fig. \\ref{fig:transfer} tells us requires $R_\\text{TF} \\lesssim 1$ pc, then much\n\\textit{more} small-scale power is retained below the\ncut-off in SFDM-TF than in FDM, as will be even\nclearer from the results for their power spectra\nand HMFs in the sections below.  Hence,\nlower limits like those above on the \nparticle masses for FDM, derived to keep FDM from \\textit{underproducing} structure on small scales, \nwhen used to infer upper limits on\n$R_\\text{TF}$ based upon $k_\\text{cut}$-matching \nFDM, might then be overly restrictive in limiting $R_\\text{TF}$. \n\n\n\n\n\n\\subsubsection{Power spectra and RMS mass fluctuations}\n\\label{sec:powerspecsigmaresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{PowerSpectrum.pdf}\n    \\includegraphics[width=\\columnwidth]{SigmaM.pdf}\n    \\includegraphics[width=\\columnwidth]{dlogSigma-dlogM.pdf}\n    \\caption{\\textbf{Power spectrum and RMS mass fluctuation at $\\bm{z=0}$}. \\textbf{(a) (top panel)} Present-day matter power spectra $P(k)$, \\textbf{(b) (middle panel)} RMS mass fluctuation $\\sigma_\\textsc{m}(M)$, and \\textbf{(c) (bottom panel)} the logarithmic derivative of $\\sigma_\\textsc{m}$,  for CDM (black), FDM (grey), and SFDM-TF (colored) models, as labeled. The CDM and FDM curves are generated using \\textsc{axioncamb}. The SFDM-TF curves are generated by multiplying the transfer functions of Fig.~\\ref{fig:transfer} (obtained through our linear perturbation calculation) by the CDM power spectrum from \\textsc{axioncamb}. In the middle and bottom panels, solid lines represent top-hat filtering, while dashed lines represent sharp-$k$ filtering. The logarithmic derivative is much steeper for sharp-$k$ filtering than for top-hat filtering for FDM, while for SFDM-TF, top-hat filtering captures the envelope of the rapidly-oscillating sharp-$k$-filtered result,\n    so they are essentially the same.}\n    \\label{fig:PandSigma}\n\\end{figure}\n\nIn \\S\\ref{sec:transferfunctionresults} above,\nwe presented the normalized transfer function\nresults for SFDM-TF and FDM, which are,\nby definition, expressed in units of the \ntransfer function for CDM.  In Fig.\n\\ref{fig:PandSigma}(a), we also present the power\nspectra $P(k)$ vs. $k$ for each model, separately,\nby multiplying those normalized transfer functions\nby the CDM transfer function generated by the\npublicly-available \\textsc{camb} code.\nWe present these power spectra, for several\nillustrative values of $R_\\text{TF}$ for\nSFDM-TF and $m_{22}$ for FDM, respectively,\nas labeled, linearly-extrapolated to $z=0$.\nThis takes proper account of the effect of\nthe smooth transition of our observed\nbackground Universe at late times, from MD\nto $\\Lambda$-dominated (``$\\Lambda$D''), \nsince all modes of interest to us have,\nby then, long-since\ntransitioned to pressure-free CDM-like \nperturbation growth according to linear theory.\n\nWith these power spectra, we can also compute\nthe RMS mass fluctuations $\\sigma_\\textsc{m}(M)$\nand their logarithmic derivatives, \nas plotted vs. $M$ \nin Figs.~\\ref{fig:PandSigma}(b) and (c).\nThe curves of $\\sigma_\\textsc{m}(M)$ for SFDM-TF\nand FDM follow that for CDM at high mass, \nincreasing towards smaller $M$, \nbut they flatten to a \ncontant value in the mass range\nassociated with wavenumbers of modes which are\nsuppressed by Jeans-filtering relative to the\ngrowth of CDM perturbations of the same\nwavenumber.  From the plots of $\\sigma_\\textsc{m}(M)$, alone, it is difficult\nto appreciate the differences between the SFDM-TF\nand FDM results, \neven when overplotting them so as to compare them \ndirectly, by choosing model parameters\nfor SFDM-TF and FDM which give them similar\nvalues of $k_\\text{cut}$.\nIn particular, it is difficult to see from the\ncurves in Fig.~\\ref{fig:PandSigma}(b), the\nimportant distinction between the behaviors of\nthe two models when switching between \ntop-hat and sharp-$k$ filtering.  The difference\nis more evident, however, when viewed by\nplotting the logarithmic derivatives of $\\sigma_\\textsc{m}(M)$, instead, as\nshown in Fig. \\ref{fig:PandSigma}(c).  \nFor FDM, this logarithmic derivative \nat values of $M$ associated with \nwavenumbers above the cut-off in\nthe transfer function\napproaches zero (from below) much more rapidly as $M$ decreases, for sharp-$k$ filtering than for top-hat filtering. We will see just how\nsignificant this difference is below, \nwhen we apply these results to calculate the\nHMF for FDM.  For SFDM-TF, however, the logarithmic\nderivative shown in Fig. \\ref{fig:PandSigma}(c) for top-hat filtering captures the envelope of the rapidly-oscillating behavior when a sharp-$k$-filter is adopted, instead, so, when averaged over these oscillations, the results for the\ntwo filter functions are essentially the same.\n\n\n\n\\subsubsection{Halo mass functions}\n\\label{sec:HMFresults}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{HMF.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio.pdf}\n    \\caption{\\textbf{Halo mass function at $\\bm{z=0}$.}\n    \\textbf{(a) (top panel)} HMFs, \n    $dn\/d\\ln M$, are \n    plotted vs. halo mass $M$ for SFDM-TF, CDM\n    and FDM. SFDM-TF curves are for different $R_\\text{TF}$-values,  $R_\\text{TF} = \\{1000, 100, 10, 1\\}$ pc (colored curves from lowest to highest). CDM HMF (black curve) is computed from the transfer function generated by \\textsc{camb}. FDM HMFs (light and dark grey curves) are for different particle masses  $m_{22}=\\{0.8, 29\\}$,\n    respectively, as computed from the transfer functions generated by \\textsc{axioncamb}. \\textbf{(b) (bottom panel)} Same as (a), but normalized by the CDM HMF.}\n    \\label{fig:HMF}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{HMF_evo_10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio_evo_10pc.pdf}\n    \\includegraphics[width=\\columnwidth]{HMF_CDMratio_evo_mfid.pdf}\n    \\caption{\\textbf{Evolution of the halo mass function.} Same as Fig.~\\ref{fig:HMF}, except for different redshifts as labeled, and the SFDM-TF curves are all for $R_\\text{TF} = 10$~pc, while the FDM curves are all for $m_{22}=0.8$. \n    \\textbf{(a) (top panel)} Same as Fig.~\\ref{fig:HMF}(a) (but note the expanded $y$-axis scale), except for different redshifts and a single SFDM-TF case with $R_\\text{TF} = 10$~pc.\n    CDM curves are dashed for readability. \\textbf{(b) (middle panel)} Same as\n    Fig.~\\ref{fig:HMF}(b), except for different redshifts and a single SFDM-TF case with $R_\\text{TF} = 10$~pc.\n    \\textbf{(c) (bottom panel)} Same as (b), except for a FDM case with $m_{22} = 0.8$.}\n    \\label{fig:HMFevo}\n\\end{figure}\n\nIn Fig. \\ref{fig:HMF}, we plot HMFs\nat $z=0$, computed from the RMS mass fluctuations and their logarithmic\nderivatives plotted in Fig. \\ref{fig:PandSigma},\nfor SFDM-TF (for $R_\\text{TF} = 1000, 100, 10$ and $1$ pc), FDM (for $m_{22}=0.8$ and 29), \nand CDM.  As expected, all three models produce\nthe same HMF at high mass, well above the scales\naffected by Jeans-filtering. Since \nJeans-filtering of small-scale structure produces\ncut-offs at high wavenumber in the power spectra\nand transfer functions of both FDM and SFDM-TF, \ncut-offs appear in both their HMFs at small halo\nmass $M$, as well.  \nHowever, as Fig. \\ref{fig:HMF} makes\nclear, the shape and depth of this HMF cut-off\nfor SFDM-TF are dramatically different\nfrom that for FDM.  In particular, the cut-off\nfor SFDM-TF is much weaker and more gradual\nthan that for FDM.   This will have strong\nconsequences for the comparison between any\nobservables for the two models that reflect\na turn-down in their abundance of small-mass\ngalactic haloes relative to that in the \nstandard CDM model.\n\nThe origin of this difference can be understood\nfrom the curves for $\\sigma_\\textsc{m}(M)$ and\nits logarithmic derivative for the two models,\nin Fig. \\ref{fig:PandSigma}(b) and (c). As we\nnoted above, the sharp-$k$-filtered result\nfor FDM is much more sharply cut-off at small\nmass than is the top-hat-filtered result.\nThis is why sharp-$k$-filtered results are \nstrongly preferred for this model \\citep{KO20}.\nThis is generally true for other models with\na similar cut-off of power on\nsmall scales, e.g. warm dark matter (``WDM''),\nas well.  For this reason, we used\nthe sharp-$k$-filtered results to compute the\nHMF for FDM in Fig.~\\ref{fig:HMF}.\nFor SFDM-TF, however, the choice of \nfilter does not make a difference in how\nsharply $\\sigma_\\textsc{m}(M)$ cuts off.\nUsing the top-hat filter for SFDM-TF merely\nsmooths over high-frequency \noscillations in the logarithmic derivative\nof the sharp-$k$-filtered\n$\\sigma_\\textsc{m}(M)$, but still\ntraces their envelope very well, so the \nsharpness of the cut-off is the same for the two filters.  \\textit{Regardless} of which filter is\nused, that cut-off is much more gradual\nand less severe for the SFDM-TF model than for FDM.\nWe chose to plot the SFDM-TF HMF using the\ntop-hat filter, rather than the sharp-$k$-filter,\nto avoid giving false prominence to the\ncorresponding oscillations that appear\nin the HMF in this case, if the latter were used,\ninstead. Their prominence and phases are \nartificially related to our thin-horizon\nand thin-EOS-transition approximations.  \nHowever, our conclusion that the overall\ncut-off for the SFDM-TF case is much shallower\nand more gradual than for FDM is robust, \nas it is the same for both filters.   \n\nIn view of the importance of this difference in\nthe $z=0$ HMFs for SFDM-TF and FDM, \nwe have explored the\ntime-dependence of this difference in Fig. \\ref{fig:HMFevo}.  By comparing\nthe HMFs for the two models at\nredshifts ranging from\n$z=0$ to $z=10$, it is apparent\nthat the small-mass cut-off\nfor FDM, relative to that for\nCDM, is much sharper and much \ndeeper than that for SFDM-TF, \nat \\textit{all} redshifts in this range.   \nMoreover, there is a strong trend in the\nHMF for SFDM-TF at masses below the cut-off,\nfor it to recover over time, gradually rising\ntoward that for CDM.  There is no comparable\nrecovery for the more\nsharply-reduced HMF in the FDM case, \nsufficient to fill in\nthe missing halo abundance at small mass\nrelative to that of CDM.\n\n\\begin{table*}\n    \\centering\n    \\begin{tabular}{l|c c c c c}\n        \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad & \\multicolumn{2}{c}{$10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$} & \\quad\\quad & \\multicolumn{2}{c}{$10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$} \\\\\n        \\hline\n        \\hline\n         &  $n_\\textsc{lmh}$ \\big(Mpc$^{-3}$\\big) & $\\rho_\\textsc{lmh}$ \\big($10^9$ M$_\\odot$ Mpc$^{-3}$\\big) & & $n_\\textsc{lmh}$ \\big(Mpc$^{-3}$\\big) & $\\rho_\\textsc{lmh}$ \\big($10^9$ M$_\\odot$ Mpc$^{-3}$\\big)\\\\\n         \\hline\n        $R_\\text{TF} = 10$ pc & 29    & 3.2 & & 1.1   & 3.0\\\\\n        $m_{22} = 0.8$        & 0.086 & 2.6 & & 0.086 & 2.6\\\\ \\\\\n        $R_\\text{TF} = 1$ pc  & 61    & 7.1 & & 4.2   & 6.7\\\\\n        $m_{22} = 29$         & 7.6   & 7.0 & & 6.0   & 6.9\\\\\n        \\hline\n        & $n_\\textsc{lmh}\/n_{\\textsc{lmh},\\textsc{cdm}}$ & $\\rho_\\textsc{lmh}\/\\rho_{\\textsc{lmh},\\textsc{cdm}}$ & & $n_\\textsc{lmh}\/n_{\\textsc{lmh},\\textsc{cdm}}$ & $\\rho_\\textsc{lmh}\/\\rho_{\\textsc{lmh},\\textsc{cdm}}$\\\\\n        \\hline\n        $R_\\text{TF} = 10$ pc & 0.07   & 0.32 & & 0.14  & 0.39\\\\\n        $m_{22} = 0.8$        & 0.0002 & 0.26 & & 0.011 & 0.33\\\\ \\\\\n        $R_\\text{TF} = 1$ pc  & 0.14   & 0.71 & & 0.55  & 0.87\\\\\n        $m_{22} = 29$         & 0.017  & 0.70 & & 0.79  & 0.90\\\\\n        \\hline\n    \\end{tabular}\n    \\caption{\\textbf{Integrated halo number density and mass density of low-mass haloes} (``LMH'')  in the range $10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$ (left 2 columns) and $10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$ (right 2 columns) at $z=0$ for SFDM-TF and FDM, including their ratios to the CDM quantities, for\n    illustrative values of $R_\\text{TF}$ and $m_{22}$.}\n    \\label{tab:halocounts}\n\\end{table*}\n\nFor some purposes, when comparing predictions of\n``cut-off models'' like these to CDM and to\nobservations, the integrated HMFs over some \nlow-mass halo (``LMH'') mass \nrange affected by the cut-off is \nalso of interest.  In\nTable \\ref{tab:halocounts}, we show\nthe integrated halo number and\nmass densities, given by\n\\begin{equation}\n n_\\textsc{lmh} = \\int \\frac{dn}{d \\ln M} d\\ln M = \\int \\frac{1}{M}\\frac{dn}{d \\ln M} dM\n \\label{eq:nlmh}\n\\end{equation}\nand \n\\begin{equation}\n \\rho_\\textsc{lmh} = \\int \\frac{dn}{d \\ln M} d M\n \\label{eq:rholmh}\n\\end{equation}\nevaluated at $z=0$, for halo masses in the ranges $10^6 \\le M\/\\text{M}_\\odot \\le 10^{11}$ and $10^8 \\le M\/\\text{M}_\\odot \\le 10^{11}$,\nfor two illustrative sets of\npairings of model parameters\nfor SFDM-TF vs. FDM, $R_\\text{TF}=10$ pc vs.\n$m_{22}=0.8$, and \n$R_\\text{TF}=1$ pc vs.\n$m_{22}=29$, respectively,  \nplotted in\nprevious figures, including their results relative\nto CDM. When the minimum halo mass\nis $10^8\\text{ M}_\\odot$, \nSFDM-TF with $R_\\text{TF}=1$ pc\nhas a halo number comparable to\nthat of FDM with $m_{22}=29$, and within a factor of 2 of that for CDM, while the masses in all three\nmodels are very close. For $R_\\text{TF}=10$ pc, however, SFDM-TF has many more haloes\nthan does FDM with $m_{22}=0.8$,\nby an order of magnitude, and \n$14\\%$ of the CDM number, while\nthe masses in both\nSFDM-TF and FDM are comparable\nand $\\approx{1\/3}$ of that for CDM. If the minimum halo mass is lowered to $10^6\\text{ M}_\\odot$, \nhowever, SFDM-TF retains\nmany more haloes than FDM for\n\\textit{both} example cases --\nby an order of magnitude for\n$R_\\text{TF}=1$ pc vs. $m_{22}=29$,\nand by more than a factor of 300 for\n$R_\\text{TF}=10$ pc vs. \n$m_{22}=0.8$ --\nand somewhat more halo mass \nin this low-mass range than the corresponding FDM models, despite the fact that \nthe SFDM-TF cases have slightly smaller cut-off mass scales ($M_\\text{cut}$) than the corresponding FDM cases. \nIn fact, the $R_\\text{TF} = 10$ pc case even retains more haloes (by number) than \nthe $m_{22} = 29$ case. \nThese examples make it clear that one must be careful\nwhen treating lower limits on $m_{22}$ for FDM as\na proxy for providing upper limits on $R_\\text{TF}$\nfor SFDM-TF.  If the former is \nderived to ensure that FDM has a high enough \nHMF at low mass, it may be overly restrictive if\nused to place an upper limit on the SFDM-TF $R_\\text{TF}$\nsimply by solving for the\nvalue that makes their cut-off wavenumbers the same.  \n\n\nThis gradual cut-off of the HMF is a defining feature of SFDM-TF and distinguishes it from any other DM alternative with a cut-off in its power spectrum that reduces small-scale power relative to CDM as sharply as does FDM.\nIts origin is rooted in the ``incredible shrinking Jeans mass'' of SFDM-TF, which leads to a ``recovery'' of small-scale perturbations that\nwere initially \nretarded below the cut-off when the Jeans scale was large, then\nfree to grow like CDM after exceeding the Jeans scale, with plenty of time left to close the gap with CDM, at least partially. As a\nresult, even haloes of mass \nwell below $M_\\text{cut}$ in the transfer function can still be present in significant numbers\nat late times.  \n\n\nIn principle, this suggests that SFDM-TF might have an advantage over FDM, when it comes to resolving the problem CDM has of overpredicting dwarf satellites compared with observations of the Milky Way\nand Local Group.  \nIn particular, if FDM uses its\ncut-off to suppress the abundance\nof dwarf satellites so as to\nreconcile with observations\nat the higher-mass end (i.e.\nat the scale of the so-called\n``too big to fail'' dwarfs -- henceforth, ``TBTF''), it \ntends to eliminate lower-mass\nhaloes (e.g. the ultra-faint dwarfs)\ntoo drastically.  Since the\nmore gradual cut-off in the HMF of SFDM-TF might avoid this difficulty, we consider this further, as follows. \n\nPredicted dwarf satellite abundances\ncan be expressed in terms either of\nthe number of haloes per interval\nof halo mass or their number per\ninterval of halo circular velocity\n(as measured at some radius), which decline with increasing halo mass\nand velocity, respectively.\nThe observational problem for CDM\nis usually expressed in terms of the latter, more observable property, involving\nvelocity, since it can be determined from a rotation curve, observed\nonly part-way out in total mass.  There are, in principle, two ways\nto modify the predicted velocity distribution function\nin the TBTF mass range\nso as to reduce the overabundance\nin CDM N-body simulations to \nmake them agree with the observed\ndistribution: \n\\begin{enumerate}[labelwidth=2em, leftmargin=2em, listparindent=\\parindent, label=(\\arabic*)]\n    \\item reduce the halo number per unit mass without \n    altering their internal densities relative to CDM haloes, \n    or \n    \\item reduce the internal densities of the haloes \n    without changing their number,\n    so haloes of a given mass appear at\n    lower velocity but match the\n    lower abundance of CDM haloes\n    of higher velocity.\n\\end{enumerate}\n(TBTF haloes are so named because\nthey are too massive to be made\nundetectable by suppressing their ability to form stars, which would otherwise be a third option for avoiding their apparent overabundance in CDM.)  \n\nFDM has\ntrouble with both options, \nas follows.  If $m_{22}\\approx1$,\nTBTF haloes (e.g. with $M\\approx{10^{10}\\text{ M}_\\odot}$) have their internal densities lowered by the presence of solitonic cores as large as their de~Broglie wavelengths of a few kpc or more.  However, while\nthis helps to solve the TBTF problem by option (2),\nit also reduces the \nabundance in the HMF, including a catastrophic reduction at lower-mass where ultra-faint dwarfs are detected. \nIf, on the other hand, FDM tries\nto avoid the latter problem by choosing a large enough $m_{22}$,\n$m_{22} \\geq 29$ \\citep{Nadler21},\nthen neither option (1) nor (2)\nare available to \nsolve the TBTF problem. \nWhile the value of $m_{22}=29$ \nis just large enough to make sure\nthe HMF in the range of ultrafaint dwarfs is not over-reduced, the\ncut-off is so sharp that\nthe number of haloes is then\nnot significantly reduced in the\nTBTF mass range, either, so\noption (1) fails.\nAnd, unfortunately, the \nsolitonic cores of the TBTF\nmass range are smaller \nin that case, since their de~Broglie wavelengths are then \n$\\lesssim0.1$ kpc,\nwhich is too small to flatten the central density profiles enough to lower the rotation-curve velocities to make option (2) possible. \nSuch small cores are also too small for FDM to solve the cusp-core\nproblem of CDM.  For that problem,\nit turns out, even the smaller value of $m_{22}\\approx1$ is not a good\nsolution, since the solitonic\ncores of galaxies even \\textit{more} massive than TBTF galaxies \n(e.g. $M\\gtrsim{10^{11}\\text{ M}_\\odot}$) are then\ntoo dense, even denser than in CDM \\citep{VBB19}.\n\nFor SFDM-TF, on the other\nhand, option (2) requires $R_\\text{TF}\\gtrsim1$~kpc, in\norder to make large enough\npolytropic cores to lower the densities of TBTF haloes sufficiently, which would also\nhelp to resolve the cusp-core problem \\citepalias{paper1}.\nHowever, for halo formation\nfrom cosmological\ninitial conditions, \nour results \nin Fig. \\ref{fig:HMF} show that,\nif $R_\\text{TF}$ is this large,\nthe cut-off mass in the HMF is\nso large that halo abundances at\nall galactic halo masses\nare substantially reduced\nrelative to CDM. Hence,\neven though it is\na much more gradual cut-off \nthan FDM's, it still over-reduces \nthe dwarf satellite abundance\nrelative to CDM.  So, option (2)\nis not available to SFDM-TF, either.\nAccording to the results \nin Table \\ref{tab:halocounts}, in fact, if $m_{22}\\geq29$ is required for FDM to avoid\nunder-predicting the observed\nabundance of ultrafaint dwarfs, then SFDM-TF probably requires $R_\\text{TF} \\lesssim 1$ pc.\nAccording to\nthe HMF in Fig. \\ref{fig:HMF} for\n$R_\\text{TF}=1$ pc, however, the halo abundance at the TBTF mass scale is then just the same\nas for CDM, so option (1) is\nunavailable just as it was\nfor FDM with $m_{22}=29$.  This\nis true despite the fact that\nthe FDM HMF cut-off just below the assumed ultrafaint dwarf mass\nrange of $\\gtrsim{10^{8}\\text{ M}_\\odot}$ is much sharper \nthan that for SFDM-TF \nwith $R_\\text{TF}=1$ pc.\nThat property does\ngive SFDM-TF one advantage\nover FDM, however, since it\nenables SFDM-TF to contribute a population of even smaller-mass subhalos which may help to account for perturbers of stellar streams observed in the Milky Way, which\nFDM may not \\citep[e.g.][]{Banik21}.\n\n\n\n\nWhile this may suggest that SFDM-TF \nis not much better at resolving the TBTF problem than FDM, in the face\nof the recently tightened constraint on the latter imposed by  \\citet{Nadler21} based on the abundance of ultrafaint dwarfs in the Milky Way and Local Group, there are caveats to this conclusion.\nFirstly, our discussion is so far based\nonly upon the comparison of\nthe global HMFs of SFDM-TF, FDM\nand CDM, while\ndwarf satellites are subhalos\nwithin these larger halos.\nA conditional HMF which\npredicts the abundances of subhalos\nof a given mass within a halo of\na larger mass at a given epoch\nmight make the comparison between\nFDM and SFDM-TF different from\nthat based upon comparing\ntheir global HMFs. \nSecondly, if the masses\nof ultrafaint dwarfs, which are\nuncertain, are determined to be below $10^{8}\\text{ M}_\\odot$, then\nthe more gradual\ncut-off of SFDM-TF gives it a much\nbetter chance of explaining TBTF than FDM,\nby allowing a \nlarger value of $R_\\text{TF}$\n(e.g. \n$R_\\text{TF}\\gtrsim{10}$ pc) without then fatally\nunderproducing the ultrafaint dwarfs. This is apparent \nif we compare the results for the two \nhalo mass ranges in Table \\ref{tab:halocounts}:\nin the range $10^6 \\leq M\/\\text{M}_\\odot \\leq 10^{11}$, the $R_\\text{TF} = 10$ pc SFDM-TF model\ncontains \\textit{more} haloes than the $m_{22} = 29$ FDM limit, while this is not true in the range \n$10^8 \\leq M\/\\text{M}_\\odot \\leq 10^{11}$.\nThirdly, comparisons of the\nobservations with these predicted\nHMFs also says nothing about the impact\nof dynamical processes inside the parent\nhalo which can alter the subhalo HMFs\nthere over time, e.g. by tidal\ndisruption, which may also be\ndifferent for SFDM-TF and FDM.\nUltimately, 3D-cosmological simulations with high resolution\nand dynamic range, from initial\nconditions that led to the formation\nof the Local Group and its\nsurroundings, may be required to\nsettle the matter. \n\nFor values\nof $R_\\text{TF}\\ll1$ pc or $m_{22}\\gg29$, the HMFs and\ninternal halo structure of both models approach that of CDM,\nfor most observables.  In that\ncase, the same solutions\nto the small-scale structure \nproblems of CDM that depend upon\nmodifications of halo structure\nand abundance at the low-mass\nend by baryonic effects, are\nalso available to solve the \nsame problems for SFDM-TF or FDM.  However, it may still be possible to\ndistinguish the three models at\neven smaller halo mass, e.g. in the \nminihalo mass range, below $\\approx{10^{8}\\text{ M}_\\odot}$.\nAccording to the results for $z=0$ in Table\n\\ref{tab:halocounts}, the more\ngradual cut-off of the HMF for SFDM-TF, relative to FDM, means\nthat, for $R_\\text{TF}=1$ pc, there is still a significant population of minihalos for SFDM-TF, albeit several times reduced from that for CDM, but\nan order of magnitude more than\nfor FDM with $m_{22} = 29$.\n\n\n\n\\section{Summary and Conclusions}\n\\label{sec:Conclusion}\n\n\nIn this paper and its companion \\citetalias{paper1}, we considered the gravitational dynamics\nand quantum hydrodynamics of an alternative form of cosmic dark matter to standard CDM -- \nSFDM-TF -- a scalar field comprised\nof ultralight bosons with a \nrepulsive self-interaction (SI)\nstrong enough to make the\nlength scale, which characterizes\nthat SI, larger than the de~Broglie\nwavelength (i.e. the Thomas-Fermi (TF)\nregime). \nHere, in Paper II, we studied structure formation in a universe with this kind of dark matter in a cosmological context, including the dynamics of both nonlinear collapse leading to halo formation and\nthe linear perturbations which\nform the initial conditions for\nstructure formation in this model.\nFor the former, we applied the tools developed in \\citetalias{paper1}\nto simulate cosmological infall and collapse to form individual virialized haloes.  As the first\nstep in understanding how cosmological initial and boundary\nconditions affect the outcome,\nwe replaced the non-cosmological initial conditions of \\citetalias{paper1} with CDM-like\nlinear initial conditions.\nThe latter were derived in order to make spherical infall occur at the\nrate which matches the empirical mass-accretion history (MAH)\nof haloes derived from\nN-body simulations of CDM.  Our previous application of this method\nfor fixing the initial perturbation for CDM simulations showed that\nCDM haloes formed in this way \nhave NFW-like mass and phase-space density profiles, including the time-dependence of the NFW \nparameters of an individual halo as it grows, in close agreement with\nCDM N-body simulations\n\\citep{AAS03,Shapiro04,Shapiro06}.  So, this provided an\nexcellent test-bed for determining the impact on this problem of replacing CDM with SFDM-TF in the cosmological context of the expanding universe.\n\nThe SFDM-TF haloes that resulted\n(see \\S\\ref{sec:simulationresults})\nshared some properties of those in \\citetalias{paper1}, namely, polytropic cores within\nradius $R_\\text{TF}$ and outer envelopes that matched those \nof a simulation with CDM dynamics from the same initial conditions,\nalthough in this case, the\nouter-envelope profiles were NFW-like, as expected.\nIn \\citetalias{paper1}, the envelopes were power-law\nprofiles, $\\rho \\propto r^{-12\/7}$,\neverywhere outside the cores,\njust as for CDM simulations\nfrom the same \\textit{non}-cosmological (i.e. static) initial conditions.\nIn the cosmological case here, \nhowever, between the core\nand the NFW-like outer envelope\nwas an intermediate zone with the same power-law profile, $\\rho \\propto r^{-12\/7}$, as \n\\citetalias{paper1} found\nfor its entire envelope outside\nthe core. The result\nof \\citetalias{paper1} that haloes\nform with polytropic cores surrounded by CDM-like envelopes is still an accurate description of the cosmological case, too, because the infall and collapse that\nleads to accretion shocks and virialized regions inside\nthe shock are essentially CDM-like\ndynamics outside the polytropic cores in both cases.  In other words, the term \\textit{CDM-like} characterizes the structure outside the cores in both cases, just from different initial conditions.  The reason the\ncosmological SFDM-TF \noutcome exhibits\n\\textit{both} kinds of CDM-like\nenvelope structure in the same\nhalo -- a power-law like the CDM simulation from non-cosmological initial conditions, surrounded \nby the NFW-like profile of CDM\nsimulated in the cosmological case -- is explained by the early\nsmoothing effect that SI has on\nthose  \ninitial conditions\nwhile they are still linear. \nAn SI-pressure-driven sound wave was found to smooth-out the mass within a proper radius $R_\\text{TF}$\nof the center, in\na time short compared to the time for that mass to expand  cosmologically, turn around and recollapse.  Once smoothed,\nthat mass continues expanding, but thereafter evolves like a top-hat perturbation, at the smoothed-out overdensity.\nWhen this mass collapses, the\noutcome for it is just like\nthat in \\citetalias{paper1}, while\nthe mass outside it evolves just\nas it would for those CDM-like initial conditions with no smoothing effect.\n\nThis has profound consequences for\nhalo formation from cosmological perturbations in SFDM-TF.\nIf SI is too strong, i.e. $R_\\text{TF}$ is too large, the\nSI-smoothing sound wave can\nsmooth-out a mass as large as that\nof the halo destined to form at some epoch in CDM from the same initial conditions,\npreventing its formation,\naltogether, or at least delaying\nit significantly.  In \\citetalias{paper1}, we showed\nhow SFDM-TF haloes with the right value of $R_\\text{TF}$ could resolve\nthe cusp-core and TBTF problems \nof CDM, without suffering\nfrom the problem identified for FDM\nby \\citet{VBB19} in which the\nsolitonic cores of larger-mass FDM haloes were too dense to be consistent with the rotation curves\nof more massive galaxies, if FDM\nparticle mass were tuned to \n$m_{22}\\simeq1$ to solve\nthe other problems.  We found\nthat $R_\\text{TF}\\gtrsim1$ kpc is\nrequired.  It was left to Paper II,\nhere, to determine if such a value\nof $R_\\text{TF}$ was consistent\nwith the cosmological formation of\nthose haloes.  Our cosmological simulations here indicate\nthat the SI-smoothed mass \nat early times is large \nenough to suppress\nthe formation of such dwarf galaxies if $R_\\text{TF}\\gtrsim1$ kpc, if\nwe start from CDM initial conditions.  At the other extreme,\nif $R_\\text{TF}\\lesssim1$ pc, then\nthe internal structure of haloes\nthat form from such CDM-like \ninitial conditions would closely resemble those of CDM, over most\nof the observable range of radii.\n\n\nThis still left open \nthe question of what\nrealistic cosmological initial conditions are for SFDM-TF halo formation, as distinct from those\nfor CDM.  To answer that requires\nlinear perturbation theory, \nfrom which the statistical likelihood of finding haloes of different masses at different epochs, can be determined. That\nwas the subject of \\S\\ref{sec:LinPertsTransferFn}.\nThere, we derived the transfer function for SFDM-TF, normalized\nby that for CDM, and, for comparison, following the same\nprocedure, did the same for\nFDM, to contrast the three models. With the transfer function, we were able to calculate statistical measures of structure formation for the SFDM-TF model, for the first time, including its halo mass function (HMF).  Just as\nthe FDM model is fully parameterized\nby its particle mass $m$, so\nSFDM-TF is parameterized entirely\nby its SI strength parameter,\n$g\/m^2$, which is directly related\nto the radius $R_\\text{TF}$ of the\n($n=1$)-polytropic sphere in hydrostatic equilibrium in which \ngravity is balanced by SI\npressure. Like FDM, SFDM-TF is\na model with a cut-off of power\non small scales, reflected as a\ncut-off at high wavenumber in\nthe transfer function.  As a\nresult, we used these findings\nto make a first cut at estimating\nthe range of $R_\\text{TF}$-values\nwhich are consistent with \nastronomical observations, by\nusing constraints on the FDM\nmodel derived elsewhere, \nexpressed as constraints on its $m_{22}$, as a proxy.\n\nAccording to our linear perturbation theory presented in \\S\\ref{sec:LinPertsTransferFn},\nthe Fourier mode at which SFDM-TF cuts off relative to CDM has a comoving mass associated with it that equalled the time-varying Jeans mass at the moment that mode entered the horizon (i.e. filled the Hubble volume).  With a Jeans length equal to $R_\\text{TF}$, fixed in proper coordinates, the comoving\nJeans mass during this phase decreased rapidly with scale factor, as $M_\\text{J,0}\\propto a^{-3}$. \nModes of smaller wavenumber entered the horizon later, with larger associated masses, \\textit{above} the Jeans mass, \nso they always grew like CDM. \nModes of higher wavenumber entered earlier, however, when their associated masses were \\textit{below} the Jeans mass and, so, were initially retarded relative to CDM, but, as soon as that rapidly shrinking Jeans mass reached their mass scale, they, too, were free, thereafter, to grow like CDM.\nThe cosmological story of structure formation in SFDM-TF is, therefore,\nvery much a tale of \\textit{``the incredible shrinking Jeans mass''}, which is the key to understanding how it differs\nfrom other cut-off models \nlike FDM \\big(for which the Jeans\nmass only drops weakly, as\n$M_\\text{J,F}\\propto a^{-3\/4}$\\big)\nwhich do not share this unique property.\nAnother factor that distinguishes\nSFDM-TF from FDM, in particular,\nis that, even when SFDM-TF\nmodes are \\textit{sub-Jeans}, \nthey still grow, albeit modestly, as $a^{1\/4}$ (when the field is\nmatter-like), while for FDM, there is \\textit{no} sub-Jeans growth. \n\nThis explains why, when we \napplied the same treatment to FDM to identify its cut-off wavenumber (as a function of $m_{22}$), and solved for the $R_\\text{TF}$-values required to align the cut-off wavenumbers of the two models, SFDM-TF haloes had SI-polytropic halo cores\n(of radius $R_\\text{TF}$) much\nsmaller than the solitonic cores of\nFDM haloes of the same mass.  \nFor example,\nFDM with $m_{22}\\simeq 1$, for \nwhich a $10^{10}\\text{ M}_\\odot$ halo has a solitonic core size\n(of order the de~Broglie wavelength) of $\\simeq 4$ kpc, cuts off at close\nto the same wavenumber as SFDM-TF\nwith $R_\\text{TF}\\simeq3$ pc, \nwhich gives a halo of that mass a polytropic core  $\\approx$\\,1000 times smaller.  This is\nconsistent with our simulations in \\S\\ref{sec:simulationresults},\nwhich showed that halo formation\nin SFDM-TF from a CDM-like perturbation experiences early SI-smoothing that, in effect, ``Jeans-filters'' a mass closer to the \\textit{initial} value\nof the Jeans mass than to its \nsmaller value after the collapse is\nfinished.  As a result, although\nSFDM-TF haloes with\nflattened cores as large\nas $R_\\text{TF}\\!\\gtrsim\\!1$ kpc,\nformed by Jeans \ninstability and collapse from \na static initial condition,\nwere shown in \\citetalias{paper1}\nto be capable of resolving the \ncusp-core problem of CDM while \\textit{simultaneously} solving its\nTBTF problem, as well,\nthis unified explanation \nis disfavored, \nonce formation is placed in the context of\ncosmological perturbation growth.\n\nObservational constraints on FDM\ncan be used as a proxy for those\non SFDM-TF by solving for the matching $m_{22}$ and $R_\\text{TF}$ values that make their respective transfer functions cut off at the same wavenumber, e.g. as estimated analytically by equation (\\ref{eq:RTFversusmFDM}).  Care\nshould be taken to \navoid FDM constraints that are based upon phenomenology that might distinguish it from SFDM-TF because of their different dynamics.  \nSince lowering \n$m_{22}$ increases the de~Broglie\nwavelength and extends the mass range suppressed by FDM to larger scales, many of its observational constraints are expressed as lower limits on $m_{22}$ required to avoid over-suppressing structure.   \nThere is\na long list of attempts to\nestimate or limit $m_{22}$, with results roughly spanning\nthe range from 1 to 30 \n(\\citealp[as summarized, e.g., in][]{AWP}). Some are based\nupon the comparison of the abundance of haloes or subhaloes with the\npredicted HMF for FDM.  \\citet{Nadler21}, for example,\ncompare the predicted\nHMF of subhaloes to the observed\ndwarf galaxies in the Local\nGroup and find that $m_{22}\\gtrsim{29}$ is required,\ncorresponding to $\\lambda_\\text{deB} \\lesssim 0.5$~kpc. Others\nare based upon more complicated,\nnonlinear dynamical phenomena. \nFor example,\nthe requirements of not dynamically over-heating the Milky Way disk or its stellar streams were\nused to place bounds of $m_{22} \\gtrsim 0.6-1.5$ \\citep{AL18, Church19}. Still\nothers are based upon structure on larger scales, at\nepochs when it is only\nquasi-linear, but where one must model effects on the baryonic component, too, such as\nfluctuations in the matter-density\nfield inferred from the\npower spectrum of measured fluctuations in the Lyman-$\\alpha$\nforest. Bounds inferred from the Lyman-$\\alpha$ forest were the \nfirst to pose the ``catch-22 problem'' for FDM, in fact, \nfinding $m_{22} \\gtrsim 20-30$ \\citep{sc22,sc23}, now\nconsistent with the recent\nlower limits\nquoted above from the abundance\nof dwarf satellites in the Local\nGroup. \n\nIf the observational \nconstraints on FDM we use as our proxy for those on SFDM-TF\nare taken to be \n$1 \\lesssim m_{22} \\lesssim 30$,\nthen the range of $R_\\text{TF}$-values for which\nthe two models have the same cut-off wavenumber corresponds\nroughly to \n$3 \\text{ pc} \\gtrsim R_\\text{TF} \\gtrsim 0.2$ pc.  Of all the methods for\nconstraining the FDM model we \nmight use, as a more \\textit{specific} proxy than this\nto constrain SFDM-TF, those comparing\ntheir HMFs are,\nto first approximation, less affected by additional differences in their dynamics\nthan others.  For this reason,\nwe focused most of our additional\ncomparison here on the results of\ntheir HMF predictions, as discussed\nin \\S\\ref{sec:HMFresults}.\nWe showed there that, for the same\ncut-off wavenumbers, the SFDM-TF \nHMF is much more gradually cut off\nthan that of FDM, so it retains\nmany more haloes at the low-mass\nend.  As a result, if we\nwere, instead, to solve for the\n$R_\\text{TF}$-value that yields the\nsame halo number as FDM\nfor some value of $m_{22}$, the\ncorresponding $R_\\text{TF}$-value\nis generally larger than the\none that makes the cut-off wavenumbers match, described\nabove.  For example, if we\nintegrate over their HMFs\nto count haloes above \n$10^8\\text{ M}_\\odot$, \nSFDM-TF with $R_\\text{TF}=1$ pc\nhas a halo number comparable to\nthat of FDM with $m_{22}=29$, and within a factor of 2 of that for CDM, while the integrated collapsed\nfractions in this halo mass range\nin all three models are very close.\nIn that case, we should\nmultiply the $R_\\text{TF}$-value\nabove that matches the cut-off wavenumber for this $m_{22}$ by\na factor of 5, from 0.2 to 1 pc,\nto use the HMF FDM constraint as a proxy for SFDM-TF. Moreover, \nif $R_\\text{TF}=10$ pc, \nSFDM-TF has many more haloes\nthan does FDM with $m_{22}=0.8$,\nby an order of magnitude, and\nstill has $14\\%$ of the CDM number.\nIn that case, we should refine\nthe first proxy range \nestimated above, upward,\nat least to $10 \\text{ pc} \\gtrsim R_\\text{TF} \\gtrsim 1$ pc.  If we compare\nthe integrated halo counts \nin the two models down to even smaller halo mass, the corresponding $R_\\text{TF}$-values shift upward\neven further. If the minimum halo mass is lowered to $10^6\\text{ M}_\\odot$, for example,\nthen SFDM-TF with $R_\\text{TF}=1$ pc has an order of magnitude more haloes than does FDM with $m_{22}=29$, while \nif $R_\\text{TF}=10$ pc, it has \nmore by a factor of 300 than FDM\nwith $m_{22}=0.8$. \nThis indicates that one must be\ncareful not to take the simple\nalignment of the cut-off wavenumbers of SFDM-TF and FDM as a basis\nfor using the observational\nconstraints on $m_{22}$ to solve\nfor the corresponding constraints\non $R_\\text{TF}$ -- that will generally be overly restrictive,\nwith upper limits on $R_\\text{TF}$\nwhich are too low. It further\nsuggests that SFDM-TF may\nhave some advantages \nover FDM since, even when\ntheir model parameters are tuned\nso as to \\textit{match} HMFs \nat larger\nmass scales, SFDM-TF may still have\nmore smaller-mass haloes.  For\nexample, when FDM and SFDM-TF\nare both limited so as to avoid\nover-suppressing the\ndwarfs in the Local Group, \nSFDM-TF may still have enough even-smaller-mass subhalos to account for the perturbers of stellar streams observed in the Milky Way, even if FDM does not \\citep[e.g.][]{Banik21}. \n\nTo test SFDM-TF against observations more directly in future work, \nit will be necessary to revisit\nthe full list of phenomenological\nconstraints that have been applied\nto FDM and other models that\nsuppress small-scale structure\nas a built-in feature (e.g. warm dark matter),\nwith SFDM-TF initial conditions and dynamics, instead.  This will require us to apply the cosmological initial conditions derived here for SFDM-TF in fully 3D simulations of galaxy and large-scale structure formation, which are under development. This includes the coupling of dark matter and baryonic components, as well as pure dark matter simulation.  \nFor ``cut-off models'' like these,\nthe nature of structure formation\non scales at or below the cut-off\nis different for different types\nof dark matter, even when they\nstart from transfer functions that\nhave similar cut-off scales,\nas our results here demonstrate\nby comparing the HMFs for SFDM-TF and FDM.   Additional effects,\nlike fragmentation of \npancakes and filaments, for\nexample, can also introduce novel small-scale structures, unfamiliar from the hierarchical clustering paradigm of CDM \\citep[e.g.][]{Mocz19,Valinia97}.  The small-$R_\\text{TF}$-regime favored \nthus far by our results presented above, for which \nthe polytropic core sizes are\nsubgalactic (i.e. sub-kpc), may\npresent other novel features\nwhich are detectable, as well,\nworth exploring further.   As discussed in \\citet{Padilla20}, for example, such cores, \nif dense enough, may\nbe subject to general relativistic\ninstability leading to collapse to\nform supermassive black holes or\ntheir seeds.  \n\n\n{\nIn this paper, we considered the r\\^ole of SI in structure formation in the SFDM model, in the TF regime.   We also compared the results in this regime with the other limiting case, of FDM, in which SI is absent.   As we showed in \\S\\ref{sec:LinPerts}, as long as the particle mass and interaction strength place the scalar field in this regime at late times (e.g. the present), it is in the TF regime at earlier times, as well. In the future, we will consider the possibility of a transitional regime between these two limits.\n}\n \n\n\\section*{Acknowledgements}\n\nThis material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1610403. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. \nT.R.-D. is supported by the Austrian Science Fund FWF through an Elise Richter fellowship, Grant No. V 656-N28. \nSimulations presented here were conducted on the Texas Advanced Computing Center's Stampede2 supercomputer under accounts asoz-630 and NSF XSEDE account TG-AST090005.  \nThe authors gratefully acknowledge  \n\\textit{The Incredible Shrinking Man }(1957)\n~[Film], directed by Jack Arnold, Universal-International Pictures Co., Inc., as the inspiration for our subtitle.\n \n\n\n\\section*{Data Availability Statement}\n\nThe data underlying this article are available in the article and in its online supplementary material.\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\columnwidth]{growthSequence_bubbletraj.pdf}\n\\caption{(A) A series of transmission electron microscopy images of a bubble growing under high, non-uniform confinement witinh a liquid cell, FIG.  S1 and movie S1. The bubble formation occurred while imaging an aqueous solution of gold nanorods that contained a trace amount of the surfactant cetrimonium bromide (CTAB) with TEM at 300 keV, beam current $\\sim$1\u221210 nA, and beam radius $\\sim$2 $\\mu$m. The bubble grows anisotropically and in the last frame detaches from the nucleation size (dashed white circle), the growth of this bubble is compared to theory in FIG. \\ref{fig:compareshapes}. (B) Normalized intensity (contours) with bubble trajectories (data points), nucleation site (dashed ellipse), and approximate location of bubble breakup (solid line).}\\label{fig:growthsequence}\n\\end{figure}\n\nThe motion and shape of droplets and bubbles in confined geometries have attracted considerable attention in the scientific community due to their importance in industrial processes, multi-phase flows through porous media and recently, microfluidic devices. Typically at the macroscale, pressure driven flow or buoyancy drive motion of bubbles and fluid-fluid interfaces. G. I. Taylor and Saffman \\cite{Saffman1958,Taylor1959} and Tanveer and Saffman \\cite{Tanveer1987} studied the migration of droplets in Hele-Shaw cells. Matched asymptotic methods have subsequently been used to elucidate the geometry of gas bubbles in cylindrical tubes \\cite{Bretherton1961} and in Hele-Shaw cells when the continuous phase either perfectly wets the surface \\cite{Park1984} or when contact line dynamics are at play \\cite{Weinstein1990}.\n\nIn the absence of applied pressure gradients and when drops are small such that the Bond number is much less than unity, gradients in substrate elasticity \\cite{Style2012,Style2013}, chemistry \\cite{Chaudhury1992, Darhuber2005}, temperature, and electric field, among others, can spontaneously drive the motion of contact lines and interfaces. Similarly, geometric gradients can induce capillary forces and promote transport of drops\/bubbles confined in tapered capillaries \\cite{Renvoise2009}  and residing on wires with varying cross-sections  \\cite{Hanumanthu2006, Lorenceau2004}. To minimize their surface energy, wetting drops seek confinement while non-wetting drops and bubbles avoid it. The dynamics of bubbles and droplets in tapered Hele-Shaw devices have been modeled and compared to experiment \\cite{Jenson2014,  Metz2009, Reyssat2014}. In these cases a dynamically created film wets the substrate. Furthermore, the mass of the bubble or drop is fixed and so the process is purely relaxational. Droplets and bubbles begin in a non-equilibrium configuration and transport to their equilibrium positions, where they can exist as spheres (for completely wetting substrates) or spherical sections (for partially wetting substrates) that satisfy the equilibrium contact angle \\cite{Langbein2002}. Laplace pressure gradients drive the disperse phase and hydrodynamics either in the bulk or at the contact line provide dissipation, setting the time scale for the process.\n\nIn recent years, there has been a considerable interest in surface-bound nanobubbles and their stability  \\cite{Alheshibri2016, Ball2012, Craig2011, Fang2016}.  Understanding such bubbles has the potential to improve processes such as acoustic surface scrubbing \\cite{Liu2008, Liu2009, Yang2011}  and designing surface textures that optimize onset of nucleate boiling to enhance heat transfer \\cite{Dong2014,Fazeli2015,Chu2012}. Free nano-bubbles can also be used as contrast agents in ultra-sonic imaging \\cite{Cai2015} and so developing processes which reliably produce nanobubles could have applications in medical imaging as well. Due to the extremely high Laplace pressure of free bubbles, mass transfer-driven dissolution can dominate dynamics when the bubbles are below their critical radius. As the radius of curvature increases $R^{-1}$, the internal pressure $P$ and the equilibrium surface concentration $C_\\text{s}$ rise; the latter in accordance with Henry's law,  $C_\\text{s}\\propto P\\propto R^{-1}$.  This positive feedback results in rapid dissolution; a bubble 100nm in radius will dissolve in $\\sim$100$\\mu$s \\cite{Epstein1950, Ljunggren1997}. In contrast to the above prediction, surface-bound nanobubbles have been observed to persist for many hours \\cite{Zhang2008, Seddon2011, Sun2016}. Researchers have proposed various mechanisms for this anomalous behavior, including a perpetual dynamic equilibrium \\cite{Petsev2013,Seddon2011}, the stabilizing effect of organic contaminants \\cite{Zhang2012}, and contact line pinning \\cite{Weijs2013,Lohse2015,Maheshwari2016}. The constraint of a pinned contact line introduces a negative feedback by forcing the radius of curvature to decrease as the mass of the bubble decreases. As the surface concentration of dissolved gases decreases and approaches that of the bulk concentration, mass transfer slows.\n\n\nInspired by this simple feedback mechanism between the geometry of a bubble and its \\emph{growth}, we reaxmine the transport of bubbles in tapered Hele-Shaw cells when mass transfer and contact line dynamics dominate the system behavior. We develop our model around experimental observations of nanobubbles $R\\sim 10-100$nm growing and migrating in tapered Hele-Shaw cells with plate gaps $\\sim$10-100s of nanometers, FIG. \\ref{fig:growthsequence}\\cite{Grogan2014}. The bubbles grow anomalously slow when compared to classical mass transfer limited growth theories \\cite{Epstein1950} and despite their small size and velocities (the capillary numbers are in the range $10^{-9}-10^{-7}$ and Bond numbers $10^{-11}-10^{-9}$) are rarely spherical in shape. To explain both observations, we posit that the contact line is slow to relax compared to the rate of mass transfer and dominates bubble shape, growth rate, and translation in a regime where the gas-liquid interface is always at mechanical equilibrium (uniform Laplace pressure). We utilize the Blake-Haynes (BH) mechanism to model contact line movement. When a portion of the contact line is immobilized, a teardrop shaped bubbles result, similar to those observed in our experiments.\n\n\\section{Bubble Growth and Migration Model}\nOur goal is to model contact line evolution of a bubble growing in a supersaturated solution confined between two plates diverging at an $2\\Phi$.  FIG. \\ref{fig:schematic}A  depicts top view of the bubble. The bubble is symmetric with respect to the $x$-axis. The inner contour (solid line) is the projection of the bubble's contact line with the confining plates onto the $xy$-plane  and is defined with the polar coordinate $\\rho\\left(\\psi,t\\right)$, where $\\psi$ is the azimuthal angle and $t$ is time. The intersection of the bubble's surface with the $xy$-plane is shown as the outer, dashed contour. FIG. \\ref{fig:schematic}B depicts the cross-section of the bubble confined in the tapered conduit. The two plates are distance $2h\\left(\\mathbf{x},t\\right)$ apart. We restrict our analysis to cases when $R\\left(t \\right) \\ll \\min\\limits_{\\psi}\\rho\\left(\\psi,t\\right)$.  In the limit of zero capillary and Bond numbers, the pressure inside the bubble is nearly uniform and dominated by the smallest radius of curvature $R$. Hence, we take $R$ to be $\\psi$-independent. $\\theta\\left(\\psi\\right)$ is the dynamic contact angle between the continuous phase (liquid) and the plates. The contact angle may vary as a function of position.\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{schematic_3.pdf}\n\\caption{: (A) Schematic of the geometry of our problem with global perspectives of a bubble shape (top) as seen from the top and a cross section of the interface as viewed from the local co\u00f6rdinate system (\\emph{i}); the unit vectors of the latter are defined by the local tangent and normal vectors of the bubble's interface at the $xy$-plane; both the slope $\\varphi$ and contact angle $\\theta$ are local properties and depend on the polar angle $\\psi$.  (B) Top and cross-section (through the plane \\emph{ii}) view of the initial condition used for the model, a sphere with a circular contact line that when projected onto the $xy$-plane is an ellipse with major and minor radii $\\{a,a^*\\}$.}\\label{fig:schematic}\n\\end{figure}\n\nThe projection of a contact line point on the $x-y$ plane is given by the two-dimensional vector\n\\begin{equation}\n{\\bf{x}}\\left( {\\psi ,t} \\right) = \\left\\{ {\\rho \\left( {\\psi ,t} \\right)\\cos \\psi  + X\\left( t \\right),\\rho \\left( {\\psi ,t} \\right)\\sin \\psi } \\right\\}.\n\\label{eq:contactline}\n\\end{equation}\nIn the above, $X$ is the distance of the origin of $\\rho$ from the center of the initial $\\left(t=0\\right)$ bubble geometry, which we will later define to be a sphere (eqn. \\ref{eq:RIC}). $2h_0$ is the height of the conduit at $X=0$.\n\nWe define local planar coordinates aligned with the normal $\\hat{\\mathbf{n}}$ and tangent $\\hat{\\mathbf{t}}$ to the projection of the contact line on the $xy$-plane. Using FIG. \\ref{fig:schematic}\\emph{i} as a guide, we relate the radius of curvature $R$ to the slope of the wedge, contact angle, and channel height at the contact line in the plane that is both normal to the contact line at position $\\mathbf{x}$ and the $xy$- plane. \n\\begin{equation}\nh = R\\cos \\left( {\\theta \\left( \\psi  \\right) - \\varphi \\left( \\psi  \\right)} \\right).\n\\label{eq:localheight1}\n\\end{equation}\nAdditionally,\n\\begin{equation}\nh = {h_0} + x\\tan \\Phi  = {h_0} + \\left( {X + \\rho \\cos \\psi } \\right)\\tan \\Phi.\n\\label{eq:localheight2}\n\\end{equation}\nEquating eqn. \\ref{eq:localheight1} and \\ref{eq:localheight2} and solving for the contact angle $\\theta$ gives\n\\begin{equation}\n\\theta  = {\\cos ^{ - 1}}\\left[ {\\frac{{{h_0} + \\left( {X + \\rho \\cos \\psi } \\right)\\tan \\Phi }}{R}} \\right] + \\varphi.\n\\label{eq:localcontactanlge}\n\\end{equation}\nThe local wedge angle $\\varphi\\left(\\theta\\right)$ relates to the global or maximum wedge angle $\\Phi$ through\n\\begin{equation}\n\\tan \\varphi  = {\\bf{\\hat n}} \\cdot {{\\bf{\\hat e}}_x}\\tan \\Phi \n\\label{eq:localslopeA}\n\\end{equation}\nWhen  ${\\left. {{\\bf{\\hat n}}} \\right|_{\\psi  = 0}} = {{\\bf{\\hat e}}_x}$ eqn. \\ref{eq:localslopeA} yields ${\\left. \\varphi  \\right|_{\\psi  = 0}} = \\Phi $; when ${\\left. {{\\bf{\\hat n}}} \\right|_{\\psi  = \\pi \/2}} = {{\\bf{\\hat e}}_y}$ , ${\\left. \\varphi  \\right|_{\\psi  = \\pi \/2}} = 0$.  In radial coordinates, the normal vector to the contact line curve \n\\begin{equation}\n{\\bf{\\hat n}} = \\left\\{ {\\begin{array}{*{20}{c}}\n{\\rho \\cos \\psi  + \\frac{{\\partial \\rho }}{{\\partial \\psi }}\\sin \\psi }\\\\\n{\\rho \\sin \\psi  - \\frac{{\\partial \\rho }}{{\\partial \\psi }}\\cos \\psi }\n\\end{array}} \\right\\}\\mathcal{N}^{-1},\n\\end{equation}\nwhere\n$\\mathcal{N}= \\sqrt{{\\rho ^2} + {{\\left( {{\\partial \\rho }\/{{\\partial \\psi }}} \\right)}^2}}$, which follows from ${\\bf{\\hat n}} = {{\\frac{{d{\\bf{\\hat t}}}}{{d\\psi }}} \\mathord{\\left\/\n {\\vphantom {{\\frac{{d{\\bf{\\hat t}}}}{{d\\psi }}} {\\left| {\\frac{{d{\\bf{\\hat t}}}}{{d\\psi }}} \\right|}}} \\right.\n \\kern-\\nulldelimiterspace} {\\left| {\\frac{{d{\\bf{\\hat t}}}}{{d\\psi }}} \\right|}}$, where ${\\bf{\\hat t}} = {{\\frac{{d{\\bf{x}}}}{{d\\psi }}} \\mathord{\\left\/\n {\\vphantom {{\\frac{{d{\\bf{x}}}}{{d\\psi }}} {\\left| {\\frac{{d{\\bf{x}}}}{{d\\psi }}} \\right|}}} \\right.\n \\kern-\\nulldelimiterspace} {\\left| {\\frac{{d{\\bf{x}}}}{{d\\psi }}} \\right|}}$ is the unit tangent vector. The full equation for the local inclination angle\n \\begin{equation}\n \\varphi  = \\tan^{-1}\\left[ {\\tan \\Phi} \\left( {\\rho \\cos \\psi  + \\frac{{\\partial \\rho }}{{\\partial \\psi }}\\sin \\psi } \\right) \\mathcal{N}^{-1} \\right].\n \\label{eq:localslopeB}\n \\end{equation}\n The full expression for the local contact angle is therefore given by substituting eqn. \\ref{eq:localslopeB} into eqn. \\ref{eq:localcontactanlge}. The projection of the linearized Blake-Haynes (BH) velocity \\cite{Blake1969, Blake2006} for contact line motion onto the $x-y$ plane gives the normal velocity of the interface\n\\begin{equation}\n\\frac{{\\partial {\\mathbf{x}}}}{{\\partial t}} \\cdot {\\bf{\\hat n}} = {U_0}\\cos \\varphi \\left( {\\cos \\theta  - \\cos {\\theta _0}} \\right),\n\\end{equation}\nwhere\n\\begin{equation}\n\\frac{{\\partial {\\mathbf{x}}}}{{\\partial t}} = \\left\\{ {\\frac{{dX}}{{dt}} + \\frac{{\\partial \\rho }}{{\\partial t}}\\cos \\psi ,\\frac{{\\partial \\rho }}{{\\partial t}}\\sin \\psi } \\right\\}.\n\\label{eq:BH}\n\\end{equation}\n$U_0=\\gamma\/\\eta_{\\text{CL}}$ is the contact line's velocity scale defined in terms of the contact line viscosity $\\eta_{\\text{CL}}$ and the surface tension  of the gas-liquid interface $\\gamma$. We assume that the continuous phase (water) only partially wets the solid substrate (silicon nitride) ($\\theta_0>0$). Previously, the BH equation has been successfully applied to model the adsorption dynamics of colloids at interfaces \\cite{Kaz2012}. Since the observed interfacial velocities in our experiments are well below the threshold required to support a dynamically-formed liquid film \\cite{Brochard-Wyart2001}, we assume that the gaseous phase contacts the solid surfaces at all times. In the supplement, we compare bulk and contact line hydrodynamic dissipation with that of BH wetting\/de-wetting and determine that the latter dominates the system. Although the disjoining pressure may influence the details of the interface geometry and play a role in wetting and de-wetting dynamics when the gap between the plates is small, this effect is not accounted for by the coarse-grain BH model.\n\nSubstititung eqns. \\ref{eq:localcontactanlge} and \\ref{eq:localslopeB} into eqn.  \\ref{eq:BH} and linearizing for $\\Phi\\ll1$ and $\\partial\\rho\/\\partial\\psi\\ll1$ about $\\Phi=0$ and $\\partial\\rho\/\\partial\\psi=0$, we obtain\n\\begin{multline}\n\\frac{{\\partial \\rho }}{{\\partial t}} + \\frac{{dX}}{{dt}}\\left( {\\frac{{\\sin \\psi }}{\\rho }\\frac{{\\partial \\rho }}{{\\partial \\psi }} + \\cos \\psi } \\right) + \\mathcal{O}\\left(\\left( {{{\\frac{{\\partial\\rho }}{{\\partial\\psi }}}}} \\right)^2\\right) =\\\\\nU_0\\left\\{\\frac{h_0}{R}+\\Phi\\left[\\frac{X}{R}+\\cos\\psi\\left(\\frac{\\rho}{R}-H\\right)-\\frac{\\sin\\psi H}{\\rho}\\frac{\\partial\\rho}{\\partial\\psi}\\right.\\right.\\\\\n\\left.\\left.+\\mathcal{O}\\left(\\left(\\frac{\\partial\\rho}{\\partial\\psi}\\right)^2\\right)\\right]-\\cos\\theta_0+\\mathcal{O}\\left(\\Phi^2\\right)\\right\\}\n\\label{eq:BHunpinned}\n\\end{multline}\nwhere $H=\\sqrt{1-\\left(h_0\/R\\right)^2}$.\n\nIn our experiments, we observed bubbles with an advancing front and a pinned aft, FIG. \\ref{fig:growthsequence}. To accommodate such situations, we modified the BH theory, introducing the weighing pre-factor   $\\left(1+\\cos\\psi\\right)\/2$ such that eqn. \\ref{eq:BHunpinned} becomes\n\n\\begin{multline}\n\\frac{{\\partial \\rho }}{{\\partial t}} + \\frac{{dX}}{{dt}}\\left( {\\frac{{\\sin \\psi }}{\\rho }\\frac{{\\partial \\rho }}{{\\partial \\psi }} + \\cos \\psi } \\right) =\\\\\nU_0\\frac{\\left(1+\\cos\\psi\\right)}{2}\\left\\{\\frac{h_0}{R}+\\Phi\\left[\\frac{X}{R}+\\cos\\psi\\left(\\frac{\\rho}{R}-H\\right)\\right.\\right.\\\\\n\\left.\\left.-\\frac{\\sin\\psi H}{\\rho}\\frac{\\partial\\rho}{\\partial\\psi}\\right]-\\cos\\theta_0\\right\\}\n\\label{eq:BHpinned}\n\\end{multline}\nThe above weighting function smoothly reduces the contact line velocity from its maximum value at $\\psi=0$ to zero at $\\psi=\\pi$ .\n\nIn defining our coordinate system, we introduced the variable $X$, denoting the position of the bubble's center, without providing a rigorous definition. We do so now by selecting $X\\left(t\\right)$ such that $\\rho\\left(0,t\\right)=\\rho\\left(\\pi,t\\right)$ at all times. Equivalently, $\\left.\\partial\\rho\/\\partial t \\right|_{\\psi=\\pi}=\\left.\\partial\\rho\/\\partial t \\right|_{\\psi=0}$  with $\\rho\\left(0,0\\right)=\\rho\\left(\\pi,0\\right)$. Evaluating eqn. \\ref{eq:BHunpinned} for $\\psi=0$ and $\\psi=\\pi$, subtracting the results, and incorporating $\\left(\\partial\\rho\/\\partial t\\right)|_{\\psi=\\pi}=\\left(\\partial\\rho\/\\partial t\\right)|_{\\psi=0}$  we obtain the evolution equation for $X$ for the unpinned case\n\n\\begin{equation}\n2R\\left(\\frac{dX}{dt}+ {U_0}\\Phi H \\right)=\n{U_0}\\Phi \\left( {\\rho \\left( {0,t} \\right) + \\rho \\left( {\\pi ,t} \\right)} \\right)\n\\label{eq:Xunpinned}\n\\end{equation}\nFollowing the same procedure but with eqn. \\ref{eq:BHpinned} gives the evolution for $X$ for the pinned case \n\n\\begin{multline}\nR \\left(2\\frac{dX}{dt}+U_0 \\Phi H \\right)=\\\\\nU_0\\left\\{\\left[h_0+\\Phi\\left(X+\\rho\\left(0,t\\right)\\right)\\right]-R\\cos\\theta_0\\right\\}\n\\label{eq:Xpinned}\n\\end{multline}\n\nThe evolution of the unpinned contactline is governed by eqns. \\ref{eq:BHunpinned} and \\ref{eq:Xunpinned}; the pinned contactline is governed by \\ref{eq:BHpinned} and \\ref{eq:Xpinned}.  To complete our model, we still need to address the dependence of $R$ on time.  For example, in an isobaric process, one would expect $\\dot{R}=0$. In the supplement, we consider a Gedanken experiment, wherein the bubble's volume is controlled in a specific way $V\\propto t^3$.  Although perhaps not practical, this case is instructive as it admits geometrically self-similar growth in the asymptotic limit $t\\rightarrow\\infty$, which is helpful for understanding contact line dynamics when the growth rate is independent of $R$. Here, however, we are interested in mass transfer-driven growth, which we address below.\n\nWe begin by writing an expression for the volume of the bubble; it can be divided into two contributions, a nearly cylindrical portion\n\\begin{equation}\n{V_\\text{I}} =\n\\int\\limits_0^{2\\pi } {{\\rho ^2}\\left( {{h_0} + \\rho \\cos \\psi {\\mathop{\\rm tan}\\nolimits} \\Phi } \\right){\\rm{d}}\\psi }\n\\end{equation}\nand the portion that bulges out from the contact line \n\\begin{equation}\n{V_{\\text{II}}} = \\int\\limits_0^{2\\pi } {\\frac{{{R^2}}}{2}\\left( \\begin{array}{l}\n\\pi  - 2\\theta + 2\\varphi\\\\\n - \\sin \\left( {2\\theta - 2\\varphi } \\right)\n\\end{array} \\right){\\cal N}{\\rm{d}}\\psi }\n\\label{eq:volII}\n\\end{equation}\n\nIn eqn. \\ref{eq:volII}, we assumed $R\/\\rho\\ll1$  to simplify the integrand. The integrals can be further simplified by applying, as before, the linearization, $\\Phi\\ll 1$ and $\\partial\\rho \/\\partial \\psi \\ll 1$, to yield\n\\begin{multline}\nV=V_{\\text{I}}+V_{\\text{II}}=\\\\\n\\int\\limits_0^{2\\pi } {\\frac{1}{2}\\rho \\left\\{ {2{h_0}\\rho  + {R^2}\\left[ {\\pi  - 2{{{\\mathop{\\rm cos}\\nolimits} }^{ - 1}}\\left( {\\frac{{{h_0}}}{R}} \\right) - \\sin \\left( {2{{{\\mathop{\\rm sin}\\nolimits} }^{ - 1}}\\left( {\\frac{{{h_0}}}{R}} \\right)} \\right)} \\right]} \\right\\}}\\\\\n+ \\Phi \\left[ \\frac{2}{3}{\\rho ^3}\\cos \\psi  - h_0^2\\frac{{\\partial \\rho }}{{\\partial \\psi }}\\sin \\psi  + {\\rho ^2}\\left( {\\frac{{2h_0^2\\cos \\psi }}{{RH}} + X} \\right)\\right.\\\\\n\\left.+ \\rho h_0^2\\left( {\\cos \\psi  + \\frac{{2X}}{{RH}}} \\right) \\right]\n+ \\mathcal{O}\\left( {{\\Phi ^2}} \\right)\\mathcal{O}\\left( {{{\\frac{{\\partial \\rho }}{{\\partial \\psi }}}^2}} \\right){\\rm{d}}\\psi\n\\label{eq:volume}\n\\end{multline}\n\nNext, we derive the expression for the effective surface area of a bubble available for mass transport. While the bubble is not perfectly cylindrical, the radius of the outward bulge is small compared to the overall radius of the contact line. We therefore expect the concentration field around the bubble to be essentially similar to that around a cylindrical bubble. Hence, the surface area \n\\begin{equation}\nS\\sim\\int\\limits_0^{2\\pi } {\\int\\limits_{ - h\\left( \\rho  \\right)}^{h\\left( \\rho  \\right)} \\mathcal{N} {\\rm{d}}z{\\rm{d}}\\psi }.\n\\end{equation}\nIntegration in the $z$-direction gives\n\\begin{equation}\nS = 2\\int\\limits_0^{2\\pi } {\\left( {{h_0} + \\left( {X + \\rho \\cos \\psi } \\right){\\mathop{\\rm tan}\\nolimits} \\Phi } \\right)\\mathcal{N} } {\\rm{d}}\\psi\n\\end{equation}\nand linearizing yields\n\\begin{equation}\nS=2\\int\\limits_0^{2\\pi } {\\rho \\left( {{h_0} + \\Phi \\left( {X + \\rho \\cos \\psi } \\right)} \\right) + \\mathcal{O}\\left( {{\\Phi ^2}} \\right)\\mathcal{O}{{\\left( {\\frac{{\\partial \\rho }}{{\\partial \\psi }}} \\right)}^2}{\\rm{d}}\\psi }.\n\\label{eq:area}\n\\end{equation}\n\nNext, we estimate the quasi-static total mass flux $\\dot{n}$  into a bubble growing in a supersaturated solution with fixed far field concentration $C_{\\infty}$ (at $\\rho=\\rho_{\\infty}$) by solving the steady diffusion equation in cylindrical coordinates. We assume that the bubble's contact line is nearly circular with radius $\\rho\\sim\\rho_0$ (where $\\rho_0$ is the leading order coefficient describing the shape of the contact line and will be described in eqn. \\ref{eq:expansion}). The gas concentration next to the bubble's surface $C_s=C\\left(\\rho=\\rho_0\\right)$ is given by a combination of Laplace's equation and Henry's law $C_s=K\\left(P_{\\infty}+\\gamma\/R\\right)$, where $K$ is the Henry constant. In the above, we again took advantage of $R\/\\rho\\ll1$ by assuming that only the confinement radius $R$ contributes to the Laplace pressure. For simplicity, we assume that the state of the gas inside the bubble is described with the ideal gas equation of state $PV=nBT$, where $P$ is the pressure, $B$ is the universal gas constant, and $T$ is the absolute temperature. Taking the time derivative of the state equation and using eqns. \\ref{eq:volume} and \\ref{eq:area} gives the differential equation that couples geometry and pressure (radius of curvature)\n\\begin{multline}\n\\left( {{P_\\infty } + \\frac{\\gamma }{R}} \\right)\\dot V - \\gamma V\\frac{{\\dot R}}{{{R^2}}} =\\\\\n S\\frac{{D\\left[ {{C_\\infty } - K\\left( {{P_\\infty } + \\gamma \/R} \\right)} \\right]}}{{\\ln \\left( {{\\rho _0}\/{\\rho _\\infty }} \\right)}}BT\n \\label{eq:state}\n\\end{multline}\n\nWe define supersaturation $\\alpha$ or the excess dissolved gas concentration relative to the surface concentration of the initial bubble\n\\begin{equation}\n\\alpha  = {\\log _{10}}\\left(C_{\\infty}\/C_0  - 1 \\right),\n\\label{eq:alpha}\n\\end{equation}\nwhere ${C_0} = K\\left( {{P_\\infty } + \\gamma \/R\\left( {t = 0} \\right)} \\right)$. When supersaturation is large, the system is driven at a rate that is incommensurate with the BH velocity and the contact angle goes to zero somewhere along the contact line. To avoid such a situation, we restrict our analysis to small and moderate supersaturations. \n\nFor concreteness, we consider here circumstances when the initial geometry of the bubble is a spherical section with equilibrium contact angle along the entire contact line. While a spherical bubble does not satisfy our assumption $R\/\\rho\\ll1$, it is the \\emph{only} geometry that creates a uniform contact angle along the entire contact line. Any other geometry would introduce additional dynamics at early times, which we wish to avoid. While this initial state is somewhat artificial, we find that, in practice, the condition $R\/\\rho\\ll1$ is reached quickly. FIG. \\ref{fig:schematic}B depicts the geometry of the initial state of our bubble. Based on simple geometric considerations, we have the following relationship between the initial radius and the height of the channel at the center of the sphere \n\\begin{equation}\nR_0=R\\left(t=0\\right)=h_0 \\frac{\\cos\\Phi}{\\cos\\theta_0},\n\\label{eq:RIC}\n\\end{equation}\nwhere $h_0$ is used in the definition of our coordinate system eqn. \\ref{eq:localheight2}. The initial contact line shape is a circle with radius $R_0\\sin\\theta_0$. The projection of this circle onto the $x-y$ plane is an ellipse with major and minor radii $\\left\\{a,a^* \\right\\}$ given by\n\\begin{equation}\n\\left\\{ {\\begin{array}{*{20}{c}}\na\\\\\na^*\n\\end{array}} \\right\\} = \\left\\{ {\\begin{array}{*{20}{c}}\n{{{R}_0}\\sin {\\theta _0}}\\\\\n{{{R}_0}\\sin {\\theta _0}\\cos \\Phi }\n\\end{array}} \\right\\} = \\left\\{ {\\begin{array}{*{20}{c}}\n{{h_0}\\tan {\\theta _0}\\cos \\Phi }\\\\\n{{h_0}\\tan {\\theta _0}{{\\cos }^2}\\Phi }\n\\end{array}} \\right\\}\n\\end{equation}\nIn our polar coordinate system centered about $X$, the contact line is described by\n\\begin{equation}\n\\rho \\left(\\psi, {t = 0} \\right) = \\frac{{{h_0}\\tan {\\theta _0}{{\\cos }^2}\\Phi }}{{\\sqrt {{{\\cos }^2}\\psi  + {{\\cos }^2}\\Phi {{\\sin }^2}\\psi } }}\n\\label{eq:rhoIC}\n\\end{equation}\nwhere the initial center of the bubble\n\\begin{equation}\nX\\left( {t = 0} \\right) =  - {h_0}\\sin \\Phi \\cos \\Phi.\n\\label{eq:XIC}\n\\end{equation}\nAs illustrated by FIG. \\ref{fig:schematic}B, $X\\left(t=0\\right)$ is negative because the center of the initial contact line is to the left of the center of the initial sphere, which serves as the origin of the global coordinate system. To summarize, the problem statement consists of a partial differential equation (the BH relationship for contact line velocity for unpinned eqn. \\ref{eq:BHunpinned} and pinned cases eqn. \\ref{eq:BHpinned} ), equation of state eqn. \\ref{eq:state}, evolution equation for the bubble's center (unpinned eqn. \\ref{eq:Xunpinned} and pinned eqn. \\ref{eq:Xpinned}), and initial conditions (eqns. \\ref{eq:rhoIC} and \\ref{eq:XIC}). \n\nTo simplify the numerical solution of this hybrid system of equations, we assume that $\\rho$ is a continuous function of $\\psi$ and use a spectral decomposition of the contact line's position\n\\begin{equation}\n\\rho \\left( {\\psi ,t} \\right) = \\sum\\limits_{n = 0}^N {{\\rho _n}\\left( t \\right)\\cos \\left( {n\\psi } \\right)}.\n\\label{eq:expansion}\n\\end{equation}\nOnly cosine terms are used in the above expansion because we restrict our analysis to bubbles symmetric with respect to the principal axis of the wedge  $\\psi=0,\\pi$). We substitute eqn. \\ref{eq:expansion} into the BH equation (eqns. \\ref{eq:BHunpinned} or \\ref{eq:BHpinned})  and require that it is satisfied in the sense of the weighted residuals\n\\begin{equation}\n\\int\\limits_0^{2\\pi } {\\cos \\left( {n\\psi } \\right)\\left( {\\text{BH}_{\\text{LHS}} - \\text{BH}_{\\text{RHS}}} \\right){\\rm{d}}} \\psi  = 0,{\\rm{   }}n = 0,1,2 \\cdots.\n\\end{equation}\nWe show terms for the case $N=2$ in the electronic supplement. The above decomposition has the virtue of not only reducing the system to a set of non-linear ordinary differential equations, but also readily incorporating integral (eqn. \\ref{eq:state}) and evolution equations (eqn. \\ref{eq:Xunpinned} or eqn. \\ref{eq:Xpinned})  in the model. We find rapid convergence and very few modes are needed when no-pinning is used (\\emph{i.e.} when bubbles are free to translate, the contact line is nearly circular at all times). In the presence of pinning, the model with $N=$5 functions well for moderate time and is able to model bubble growth to the point they detached in the experiment. For long times, higher frequency modes begin to dominate, which invalidates assumptions made in our model. While we only use cosine terms, the problem could be generalized to include sine terms if, for example, initial conditions with broken symmetry were of interest or if one wished to include additional forces acting transverse to the wedge axis; otherwise, there is no physical reason for information to flow into the odd functions given the problem's symmetry. \n\n\n\n\\section{Experimental Method and Observations}\n\nWe imaged bubble nucleation, growth, and migration in our custom-made liquid cell (the nanoaquarium) \\cite{Grogan2010,Grogan:2011dj, Grogan2014} with a transmission electron microscope (TEM). The liquid cell consists of a thin (nominally 200 nm) liquid layer confined and hermetically sealed between two very thin (50nm) silicon nitride membranes (100 $\\mu$m x 100 $\\mu$m). See FIG.  S1 in the electronic supplement for a schematic depiction of our liquid cell. Additional details on the structure of the liquid cell and its fabrication are available in Grogan and Bau \\cite{Grogan2010}. The liquid is sealed from the vacuum of the electron microscope and the entire assembly is thin enough to be transparent to electrons.  In our experiments, the liquid cell is filled with a solution of water with trace amount of the surfactant cetrimonium bromide (CTAB). The liquid cell is imaged at 300kV using a beam of diameter 4 $\\mu$m and current 1nA. FIG. \\ref{fig:growthsequence} (movie S1) shows a series of bubbles nucleating from a location that is most likely a defect such as a pit on the interior surface of one of the liquid cell windows. These bubbles form because of radiolysis of water by the electron beam that generates gaseous species. The primary species are hydrogen and oxygen \\cite{Schneider2014}, their production induces bubble nucleation and sustains bubble growth. Liquid cell electron microscopy is a new that can provide nanometer resolution of aqueous samples \\cite{Ross2015}. In terms of imaging fluid dynamics phenomena, the technique is still evolving: the liquid layer geometry can not be defined accurately by the microscopist and the nucleation sites for bubbles are determined by random defects in the silicon nitride windows. This restricts our observations to somewhat uncontrolled experiments and precludes quantitative comparison between our theoretical predictions and data.  Nevertheless, the liquid cell provides sufficient information to allow us to qualitatively compare our theoretical predictions with experimental data. TEM liquid cell microscopy has been used previously to image condensation \\cite{Bhattacharya:2014cr}, motion of droplets \\cite{Mirsaidov2012a}, dewetting phenomena \\cite{Mirsaidov2012b} and bubble formation by heating \\cite{White2012} however, the physics behind these phenomena have not been modeled extensively and appear distinct from the phenomena discussed here.\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{growthstages_modified_2.pdf}\n\\caption{(A-C) A series of schematics (1st column), bright field electron micrographs (2nd column) and intensity profiles taken alone the midline of the bubble indicated on the images (3rd column) of a bubbles growing in under confinement. The bubble in (C) is pinned to a nucleation site.}\\label{fig:growthstages}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=3in]{naqrm_collapse_nano.pdf}\n\\caption{(A) and (B), respectively, show a microscopy image and corresponding intensity profiles ( \\emph{i} is taken along the centerline of the bubble, while \\emph{ii} is taken parallel to it in a nearby region of the device ). Since the transmission of (\\emph{ii}) is flat and similar to that of the bubble (\\emph{i}) we conclude that the membranes have contacted, along the line labled in (A) and (B).}\n\\label{fig:collapsenano}\n\\end{figure}\n\nFIG. \\ref{fig:growthstages}A-C shows bright field electron microscope images of bubbles growing under confinement in our liquid cell. Three bubbles at different stages in their growth are shown along with a schematic side view (first column) that interprets the intensity profiles (third column). The scatter of electrons by the irradiated medium and the darkness of the image scale with the integrated atomic number density along the beam's path, or in our case, the local thickness of water. Darker (lower intensity) sections of the image therefore indicate a thicker liquid layer. The flat intensity profile of the bubble FIG. \\ref{fig:growthstages}C indicates that the bubble has contacted both membranes, while the ``shaded'' bubbles in (A) and (B) indicate that electrons encounter both liquid and vapor along their path and that the bubbles have therefore not contacted both surfaces.\n\nWhen we observe nucleation, the bubbles appear to emanate periodically from a presumed impurity in the silicon nitride film. FIG. \\ref{fig:growthsequence} shows snapshots of bright field electron microscope micrographs of a bubble growing under high confinement (movie S1). We compare our theory to bubbles like these. Nucleated bubbles are first observed when they are $\\sim80$ nm in diameter and depart when they are $\\sim250$nm in size. The bubbles depart by breaking into two bubbles: a smaller bubble that remains attached to the nucleation site and a larger bubble that continues to translate. The process repeats with a frequency of 0.3 Hz.  The reproducibility of the process from one bubble to the next suggests quasi-static far field supersaturation \\cite{Grogan2014}. Indeed, theory suggests that the radiolysis process is self-regulating and that the concentration of radiolysis products achieve steady state \\cite{Schneider2014}. Since we cannot measure the dissolved gas concentration, we use it as a fitting parameter in our model, $\\alpha$ eqn. \\ref{eq:alpha}. \n\nThe experimental data suggests that locally, bubbles always migrate in the same direction towards lower confinement, FIG. \\ref{fig:growthsequence}B.  In support of this, we show a bubble at a different nucleation site FIG. \\ref{fig:collapsenano}A; FIG. \\ref{fig:collapsenano}B depicts two intensity profiles along parallel lines (\\emph{i}) and (\\emph{ii}), shown as arrows in FIG. \\ref{fig:collapsenano}A.  Profile (\\emph{i}) includes the bubble and profile \\emph{ii} goes alongside the bubble.  The recorded intensity declines as we go from left to right along the line (\\emph{ii}), suggesting that the thickness of the liquid layer increases from left to right. Furthermore, the similar transmitted intensity of the bubble compared to the region from which it emanates indicate a very thin liquid layer and possibly membrane contact. During loading of the liquid cell, capillary forces can pull the silicon nitride membranes into close proximity. We show such collapsed membranes in the supplement at lower magnification using optical microscopy (FIG.  S2). FIG. \\ref{fig:collapsenano} and FIG.  S3 show that bubbles grow and migrate in the tapered conduit in the direction of diverging plates.  This is the situation for which we built our model in Section II above.  Like FIG. \\ref{fig:growthstages}C, the detected intensity along the bubble (\\emph{i}) is nearly uniform and the transition from the bubble to the bulk liquid is relatively rapid compared to the size of the bubble's plateau, one can infer that the bubble is highly confined, justifying the assumption made in the theory section $R\/\\rho \\ll 1$. \n\nUsing simple image processing algorithms, we automate measurements of bubble features such as centroid position, area, and contact line shape and stitch together bubble trajectories using established techniques \\cite{Schneider:2016hm,Blair}. In the Section V, we compare experimental observations with theoretical predictions.\n\n\\section{Model Results and Discussion}\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{theory_2x2_2.pdf}\n\\caption{Model predictions for (A) bubble size, (B) velocity and (C) radius of curvature and (D) aspect ratio for two different slopes and contact line viscosities:  $\\xi=10^2$ (gray) and $\\xi=10^4$(black), and slopes $\\Phi=10^{-4}$ (dotted) and $\\Phi=10^{-3.5}$ (solid) . In all cases $h_0=100$ nm and $\\alpha=-6$. \\label{fig:theory}}\n\\end{figure}\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{contactangleevolution.pdf}\n\\caption{(A) Contact angle $\\theta\\left(\\psi,t\\right)$ relative to the equilibrium contact angle $\\theta_0$ as a function of polar angle $\\psi$ at a few different stages in bubble growth. (B) $\\theta_0-\\theta$ at the leading (solid) and trailing (dashed) edges of the bubble as a function of time; for $t>0.3$ s dewetting occurs at the leading edge and wetting occurs at the trailing edge. Inset (\\emph{i}) shows short dynamics in which, the bubble expands in all directions. $h_0=100$nm, $\\alpha=-6$, $\\xi=10^4$, and $\\Phi=10^{-4}$. \\label{fig:contactangle}}\n\\end{figure}\n\nIn all our theoretical predictions, we assume that the gaseous species is hydrogen and that the liquid cell is at room temperature ( $T=298$ K) and pressure ( $P=0.1$ MPa) \\cite{Grogan2014}. We assume that as bubbles grow they do not significantly increase the pressure of the device since their volume is miniscule to that of the device.  We use  $\\theta_0=30^{\\circ}$ \\cite{Arkles2006}, $K_{\\text{H}_2}=7.74\\times10^{-6}$ mol\/Pa $\\text{m}^3$ \\cite{Sander2015},  $D_{\\text{H}_2}$ $\\text{m}^2$\/s \\cite{Cussler2009},  $\\eta_{\\text{CL}}=k_{\\text{B}}T\/\\left(\\kappa_0\\lambda^3\\right)=\\xi\\eta_0$ Pa s  \\cite{Petrov1997,Blake2006}, bulk viscosity $\\eta_0=8.9\\times 10^{-4}$ Pa s, and surface tension of the vapor-liquid interface and $\\gamma=40$ mN\/m. We use surface tension of the gas-liquid interface lower than that of pure water to account for the presence of trace amounts of the CTAB surfactant in our experiment \\cite{Berg2010}. The negative value for $\\alpha$ corresponds to a far field gas concentration that is only slightly above the equilibrium concentration at the surface of the initial bubble.  In the expression for the contact line viscosity $\\lambda$ is the lattice spacing, $k_\\text{B}$ is the Boltzman constant, and  $\\kappa_0$ is the hopping frequency. We explore a range of  dimensionless contact line viscosities ($\\xi=\\eta_{\\text{CL}}\/\\eta_{0}=10^2-10^6$) because literature values vary greatly ($\\lambda\\sim10^{-10}-10^{-9}$ m and  $\\kappa_0\\sim10^3-10^9$ $\\text{s}^{-1}$  \\cite{Blake2006}), are not known for our system, and may be altered by the presence of surfactants \\cite{Petrov1997}. The local slope of the device is also not precisely known, but estimated to be in the range $\\Phi=10^{-3}-10^{-2}$ as further discussed in the  electronic supplement. \n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{Plots3DPinnedUnpinnedExpCompare_4.pdf}\n\\caption{Comparison of measured (points) and predicted average bubble size $\\rho_0$ as a function of time for several bubbles emerging from the same nucleation site (colors indicate different bubbles nucleating from the same site). Model results for unpinned (black) and pinned (gray) are shown along with the evolution of the radius of curvature $R$ (dashed). The open circle denotes the end of the validity of the pinned model. \\label{fig:comparemetrics}}\n\\end{figure}\n\nBefore comparing to experiment, we examine the impact of some of the key parameters in the model: the slope of the channel $\\Phi$ and contact line viscosity $\\xi$ in FIG. \\ref{fig:theory}. We fix initial channel height $h_0=100$nm and supersaturation $\\alpha=-4$. FIG. \\ref{fig:theory} depicts (A) the bubble size $\\rho_0$, (B) the velocity of the bubble's center of mass $\\dot{X}$, (C) the radius of curvature R, and (D) the ratio of curvatures  $R\/\\left(\\rho_0+R\\left(1-\\sin\\theta_0\\right)\\right)$ as functions of time when $\\xi=10^2$ (gray line) and $10^4$ (black line) and when $\\Phi=10^{-3.5}$ (solid line) and $10^{-4}$ (dashed line). The effect of the contact line viscosity is intuitive. The greater the dissipation at the contact line, the more the contact angle must be pushed out of equilibrium to achieve the same velocity leading to slower growth. Interestingly, changing the slope of the taper also changes the time at which the bubble will appreciably depart from its initial size. To understand why this is the case we consider the short time dynamics more closely.\n\nInitially, the bubble grows slowly because it takes time for the contact line to move. Contact-line motion is needed to enable the Laplace pressure and the equilibrium concentration of the dissolved gas next to the bubble's surface to decrease. At short times, before the contact line moves, the only way to accommodate mass is by \\emph{decreasing} $R$, which bulges the bubble further beyond the contact line, increasing Laplace pressure, slowing mass transfer. Thus at short times, contact line resistance introduces a negative feedback mechanism on bubble growth. This dynamic is apparent when examining the contact angle distribution at short times ($t<0.3$ s) in FIG. \\ref{fig:contactangle}, the local contact angle $\\theta\\left(\\psi,t\\right)$ is for all $\\psi$ less than the equilibrium contact angle $\\theta_0$ indicating that the bubble has displaced fluid in all directions. \n\nAccording to BH theory, such a contact angle distribution will causes the contact line to advance into the liquid (de-wet) and give way to the growing bubble. While this occurs at all points on the contact line initially, the contact angle distribution eventually breaks symmetry giving rising to a faster velocity at the leading edge, FIG. \\ref{fig:contactangle}. As the interface at the leading edge moves $h$ increases, allowing $R$ to increase, the equilibrium concentration of the dissolved gas to decrease, and the mass transport to increase. In other words, the increasing channel height provides a positive feedback mechanism that accelerates the bubble's growth once the contact line begins moving. The larger the opening slope or smaller the contact line viscosity, the quicker the switch between negative and positive feedback occurs. As the bubble's geometry evolves, the contact angle distribution switches from being non-uniformly below the equilibrium contact angle $\\theta_0$ (corresponding to de-wetting) to a portion of the rear contact line being greater than the equilibrium value. In other words, once the bubble begins translating down the conduit, de-wetting occurs at the leading edge ($\\psi=0$), and wetting occurs at the trailing edge ($\\psi=\\pi$), FIG. \\ref{fig:contactangle} tracks the evolution of the contact angle at these two locations.  The moment of the sign change of the trailing contact line is made clear in the inset of FIG. \\ref{fig:contactangle}B\\emph{i} (dashed lined).\n\nInterestingly, while the velocity $\\dot{X}$ initially increases (FIG. \\ref{fig:theory}B), it achieves a maximum velocity and then slowly declines as $t\\rightarrow\\infty$. This velocity scaling is correlated with the aspect ratio $\\varepsilon=R\/\\left(\\rho_0+R\\left(1-\\sin\\theta_0\\right) \\right)$ tending towards unity, indicating that the bubble's geometry is becoming more spherical over time. Since this must also means that the contact angle distribution is becoming more uniform (tending towards $\\theta\\left(\\psi\\right)\\rightarrow \\theta_0$), we consider the limit of fast contact line relaxation $\\eta_{\\text{CL}}=0$ in the supplement. In this case, the  contact angle is always uniform and at equilibrium; the bubble is therefore a spherical section that grows self-similarly and translates down the conduit at a velocity required to satisfy geometric constraints. In the supplement we show that $\\dot{X}\\propto t^{-1\/2}$, just like FIG. \\ref{fig:theory}B for large times. For the parameters used, our theory therefore predicts that confinement and contact line dynamics become less important once bubbles are on the order of $\\sim$ 100 $\\mu$m - 1 mm in extent.\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{PseudoSpectral_PlotProfilePinnedCompare_9.pdf}\n\\caption{: (Top) Experimental observations of a bubble growing from a nucleation site (dashed white lines). (Bottom) Pinned (black lines, eqn. \\ref{eq:BHpinned}) and unpinned (dashed lines, eqn. \\ref{eq:BHunpinned}) model predictions compared to contact line from top row (thick gray lines).\\label{fig:compareshapes}}\n\\end{figure*}\n\n\\section{Comparison with Experimental Observations}\n\nFIG. \\ref{fig:comparemetrics} compares our model predictions, with (gray) and without (black) pinning (modeled with the weighted eqn. \\ref{eq:BHpinned} that smoothly interpolate between a completely pinned contact line at $\\psi=\\pi$ and full mobility at $\\psi=0$), to experimental observations of bubbles that are pinned to their nucleation site (symbols). While we know the bubbles to be pinned, for comparison we show predictions made by both pinned and unpinned versions of the model. For both models,  $\\Phi=10^{-3}$, $h_0=80$nm, $\\alpha=-2.9$, and $\\xi=5.4$. FIG. \\ref{fig:comparemetrics}A depicts  $\\rho_0$ (solid line) and $R$ (dashed line) as functions of time. When $t<10$s, we find no significant difference in the leading order growth rates predicted in the absence and presence of pinning. This is because the feedback of higher order geometric terms  $\\rho_{1,2,\\cdots}$ on $R$ is weak when the bubble nearly circular. The open circle in FIG. \\ref{fig:comparemetrics} indicates the end of theoretical predictions for the pinned case; while the model fails at finite time, we note that in the experiment the bubble breaks apart prior to this point. In this paper, we do not consider instabilities that may lead to detachment. Before breakup, there is good qualitative agreement between the model and theory. In FIG. \\ref{fig:compareshapes}, we more closely compare the predicted geometry of the contact line to observations of the same data set.  We see that the theoretical predictions are in good agreement with the observations away from the pinning\/nucleation site (dashed circle).\n\nFinally, in FIG. \\ref{fig:aspectratio} we examine the relationship between the size $\\rho_0$ and the \\emph{projected} aspect ratio $\\mathscr{E}$ of the bubble (not to be confused the aspect ratio $\\varepsilon$ plotted in FIG. \\ref{fig:comparemetrics}D)\n\\begin{equation}\n\\mathscr{E}=\\left(\\rho_0-\\rho_2+\\rho_4 \\right)\/\\left(\\rho_0+\\rho_2+\\rho_4\\right)\n\\label{eq:aspectratio}\n\\end{equation}\nwhere $\\rho_{2,4}$ are shape corrections to $\\rho_0$ (for $N=$5).  We find that the faster growing bubbles (those with higher superstation) become elongated at smaller size. Our model therefore predicts that contact line dynamics and the tapered channel together introduce a growth rate-dependent geometric evolution, which would otherwise be absent in the zero capillary and Bond number regime we consider.  Since bubble breakup likely depends on the extent to which the bubbles elongate, we posit that bubble creation frequency and departure size will depend on super-saturation. Our current single-curvature model does not allow us model the breakup mechanism. Before breakup, a neck forms which invalidates assumptions of our model; however, controlled experiments on droplet breakup in tapered conduits in the quasi-static regime strongly implicate a geometric instability\\cite{Dangla2013}.\n\n\\begin{figure}[h]\n\\includegraphics[width=\\columnwidth]{PsuedoSpectral_AspectRatioParametricExperimentCompare_6.pdf}\n\\caption{Model predictions (lines) of bubble size $\\rho_0$ as a function of aspect ratio $\\mathscr{E}$ (eqn. \\ref{eq:aspectratio}) for three different supersaturation values compared to experiment (points, colors indicate different bubbles nucleating from the same site). Arrows show the progression of time and circle highlights the point at which the bubbles break away from their nucleation sites ($t\\sim$1-2 s).}\n\\label{fig:aspectratio}\n\\end{figure}\n\n\\section{Conclusion}\n\n\n\nWe imaged and analyzed the growth and motion of sub-micron bubbles in a supersaturated liquid confined between two narrowly separated, rigid, diverging plates in the asymptotic limit of zero Capillary and Bond numbers. To enable imaging sub-micrometer-size bubbles, we used the experimental technique liquid cell electron microscopy to  provide spatial and temporal resolution that is not readily available by any other types of microscopy. The motion of macroscopic bubbles is often driven by capillary and buoyancy forces, both of which are negligible at the length scales considered here.  We hypothesize that bubble motion and growth are rate-limited by contact line dynamics. To predict contact line motion, bubble growth, and interface geometry, we use the Blake-Haynes mechanism to describe contact line motion as a function of contact angle.  Variations in contact angle result from gas mass flow into the bubble under supersaturated conditions. Our theoretical predictions agree qualitatively with experimental observations.  At early stages, bubbles grow slowly as contact line-mediated curvature mitigates the positive feedback mechanism that would otherwise enhance mass transfer into the bubble and drive rapid growth. This is somewhat similar to one of the more recent mechanisms that has been proposed to explain the longevity of surface-bound nano-bubbles; the pinning of the contact line controls curvature \\cite{Weijs2013,Lohse2015a}. At longer times, our model predicts that the growth rate of our bubbles accelerates due to positive feedback: decreased radius of curvature reduces the equilibrium gas concentration next to the bubbles' surfaces, enhancing mass transport, and decreased confinement. \n\nOur observations and model predictions have implications for various processes where bubbles nucleate and grow within surface defects, fissures, or tailored nanostructures such as are used for boiling and hydrolysis. Clearly, one could exploit geometry to clear bubbles; this much has been explored for microgravity applications \\cite{Jenson2014},  albeit relying on driving forces that are negligible at the nanoscale as in our experiments.\n\nPerhaps more intriguing is our model's implication that bubble geometry depends on growth rate. This is somewhat unexpected given the overdamped regime but arises as the result of an additional time scale in the problem, the contact line relaxation. Growth-rate dependence is particularly apparent when aft contact line pinning is included in the model. Our prediction that the aspect ratio of growing teardrop bubbles can be controlled by controlling the supersaturation level is a novel mechanism that could be exploited in device design.\n\nThis paper has examined a fluid mechanical problem with the emerging experimental method of liquid cell electron microscopy.  We demonstrate that liquid cell electron microscopy with its few nanometer and video-rate resolution can provide meaningful data that would be difficult, if not impossible, to obtain by other means. Although our liquid cell  imaging provided qualitative information to support our theoretical predictions, the method is still at its infancy  in terms of yielding quantitative data. Some calibration tools exist, for example the use of electron energy loss spectroscopy to measure liquid thickness \\cite{Jungjohann2012}, but the accuracy is not sufficient for these types of experiments.   We hope that in the future, liquid cells will be better equipped with diagnostic tools such as means to measure sufficiently accurately the distance between the confining plates and the conditions (pressure, temperature, chemical composition) of the liquid inside the liquid cell.  Such developments would allow for better controlled experiments and will enable quantitative comparisons between theoretical predictions and experimental data.  \n\n\n\\section{Acknowledgements}\nThe research was supported, in part, by the National Science Foundation CBET 1066573 and the Nano\/Bio Interface Center through the National Science Foundation NSEC DMR08-32802. All movies were recorded by Joseph. M. Grogan at the IBM T. J. Watson Research Center.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe setting of this paper will be the $d$-dimensional torus, $\\T^d$. We may equivalently think of the problem in the setting of $\\R^d$ with periodic initial data. Throughout the paper we will consider only real valued functions. We consider the Cauchy problem for the dissipative equation \n\\begin{equation}\n\\left\\{ \\begin{array}{l}\n\\theta_t = (u \\cdot \\nabla) \\theta - (-\\Delta)^{\\alpha\/2} \\theta\\\\\n\\theta(x,0)= \\theta_0(x)\\\\\n\\end{array}\\right., \\label{SQG}\n\\end{equation} \nwhere $u = R\\theta$, $R$ is a certain divergence free operator, and $(-\\Delta)^{\\alpha\/2}$ is the fractional Laplacian. In the case of the surface quasi-geostrophic equation (SQG for brevity), $d=2$ and $u = (-R_2\\theta, R_1\\theta)$, where the $R_j$s are the standard Riesz transforms. These operators are defined on a suitably smooth class of functions by multiplication on the Fourier side. For $n \\in \\mathbb{Z}^d$, if \n$$\\widehat{\\theta}(n) = \\frac{1}{(2\\pi)^d}\\int_{\\T^d} \\theta(x) e^{-i n \\cdot x}\\,dx $$\nis the $n^{th}$ Fourier coefficient of $\\theta$, then for $n\\neq 0$\n$$ \\widehat{(R_j\\theta)}(n) = i\\frac{n_j}{|n|} \\widehat{\\theta}(n) \\quad \\mbox{and} \\quad [(-\\Delta)^{\\alpha\/2}\\theta]\\,\\,\\widehat{}\\,\\,(n) = |n|^{\\alpha} \\widehat{\\theta}(n),$$\nand $\\widehat{(R_j\\theta)}(0) = [(-\\Delta)^{\\alpha\/2}\\theta]\\,\\,\\widehat{}\\,\\,(0)= 0.$ \n\nThe parameter $\\al$ ranges between $0$ and $2$. The case when $\\al \\in (1,2]$ is referred to as the subcritical case. In the subcritical case, the global well-posedness has been established in the case of smooth initial data (See \\cite{CCW} and the references therein). The critical case, $\\al=1$, has been the source of much study in recent years. In \\cite{CCW}, Constantin, Cordoba, and Wu proved that if the $L^{\\infty}$ norm on the initial data is small enough, then there is a global regular solution. Later, Kiselev, Nazarov, and Volberg introduced the modulus of continuity method in \\cite{KNV}. This method was used to prove the global well posedness of the critical SQG for smooth periodic initial data by finding a priori bounds on $\\nrm{\\nabla \\theta}_{\\infty}$. In the supercritical case, $\\al <1$, many open questions remain. \n\nIn \\cite{CV}, Caffarelli and Vasseur used De Giorgi iteration to show that a uniform bound in $BMO$ of the velocity term in the drift diffusion equation implies that certain weak solutions are locally H\\\"{o}lder continuous.  In \\cite{KN}, Kiselev and Nazarov showed that, in the case of the critical surface quasi-geostrophic equation, a uniform bound in $BMO$ on a smooth velocity leads to a certain degree of H\\\"{o}lder continuity. In this way they were able to give yet another proof of the existence of global smooth solutions to the critical surface quasi-geostrophic equation. Their approach relies on passing the evolution onto a special class of functions which is ``dual\" to the class of  H\\\"{o}lder continuous functions. The reason for studying the H\\\"{o}lder continuity of solutions to the SQG can be seen from results of Constantin and Wu \\cite{CW}. They showed that if you have a uniform bound on the $C^{1- \\al + \\delta}$ ($\\delta >0$) norm of a certain weak solution to the SQG on a time interval, then in fact you have a smooth classical solution on that interval. Recently in \\cite{S}, Silvestre proved that if the dissipative power is slightly smaller than $1\/2$, namely the power of the Laplacian is $\\frac{1-\\epsilon}{2}$ for small $\\epsilon$, then certain weak solutions become H\\\"{o}lder continuous after a certain time. The proof employs De Giorgi-type estimates to show that on a parabolic cylinder the oscillation of a certain continuation of the solution is not more than a fraction of the oscillation of the continuation of the solution on a twice larger parabolic cylinder under the assumption that the $L^{2d\/\\al}$ norm of the velocity is uniformly bounded. \n\nBefore stating the main result, we define  \n$$ \\nrm{f}_{k,p} = \\nrm{\\nabla^k f}_{p},$$\nwhich for mean zero functions can be shown to be equivalent to the standard Sobolev norm by the Poincar\\'{e} inequality. Motivated by the smooth class constructed in the work of Nazarov and Kiselev and the work of Silvestre, we will prove the following theorem:\n\n\\begin{thm} Suppose that $R$ is a divergence free vector-valued operator that, for every $k\\geq 0$ and every $1<p<\\infty$, satisifes $\\nrm{Rf - Rg}_{k,p} \\leq C(k,p) \\nrm{f-g}_{k,p}$ for some constants $C(k,p)$, and, for every $\\epsilon >0$ satisifes $\\nrm{\\nabla (R f)}_{\\infty} \\leq C(\\epsilon) \\nrm{\\nabla f}_{C^{\\epsilon}}$ for some constant $C(\\epsilon)$. There is a time $T = T(\\al,\\nrm{\\theta_0}_{\\infty})$ such that if $\\theta \\in C^{\\infty}(\\T^d \\times [0,T])$ is a solution to the Cauchy problem\n$$\\left\\{ \\begin{array}{l}\n\\theta_t = (R \\theta \\cdot \\nabla) \\theta - (-\\Delta)^{\\alpha\/2} \\theta\\\\\n\\theta(x,0)= \\theta_0(x)\\\\\n\\end{array}\\right.,\n$$\nthen $\\theta$ extends to a solution in $C^{\\infty}(\\T^d \\times [0,\\infty)).$\n\\end{thm}\n\nA consequence of this theorem is \n\n\\begin{thm}[Eventual Regularization for the Supercritical SQG] There is a time $T = T(\\al,\\nrm{\\theta_0}_{\\infty})$ such that if $\\theta \\in C^{\\infty}(\\T^2 \\times [0,T])$ is a solution to\n$$\n\\left\\{ \\begin{array}{l}\n\\theta_t = (R^{\\perp}\\theta \\cdot \\nabla) \\theta - (-\\Delta)^{\\alpha\/2} \\theta\\\\\n\\theta(x,0)= \\theta_0(x)\\\\\n\\end{array}\\right.,\n$$\nthen $\\theta$ extends to a solution in  $C^{\\infty}(\\T^2 \\times [0,\\infty)).$ \n\\end{thm}\n\nClassical results about Riesz transforms imply $R^{\\perp}$ satisfies the conditions in Theorem $1$ (See \\cite{St}). Both Theorems tell us that for any value of $\\al$ in the supercritical range, if we have a solution that is smooth up to a certain time, then it remains smooth forever. \n\n\n\n\\section{Dualizing the Problem}\n\nWe now define a variant of the class introduced in \\cite{KN}. Let $A > 1$ be a parameter to be fixed later. \n\\begin{defn} We will say that a smooth function $\\psi$ defined on $\\T^d$ is in $\\mathcal{U}(r)$ if \n\\begin{equation}\n\\int_{\\T^d} |\\psi(x)|^p \\,dx \\leq A r^{-(p-1)d} \\label{size}\n\\end{equation}\nand\n\\begin{equation}\n\\sup\\Big\\{ \\Big| \\int_{\\T^d} f(x)\\psi(x)\\,dx \\Big| : f \\in C^{\\infty} \\cap \\text{Lip}(1)\\Big\\} \\leq r \\label{smooth}\n\\end{equation}\n\n\\end{defn}\nIn this definition we have used the notation $Lip(M)$ to denote the class of all functions $f$ such that $|f(x) -f(y)| \\leq M|x-y|$ for all $x,y\\in \\T^d.$\nSince all constant functions are Lipschitz, if $(\\ref{smooth})$ holds, then the function $\\psi$ must have mean zero. Also notice that if a function $\\psi$ satisfies $|\\!|\\psi|\\!|_1 \\leq 1$ and $|\\!|\\psi|\\!|_{\\infty} \\leq A^{1\/(p-1)} r^{-d}$ (as were the conditions in \\cite{KN}), then interpolation shows that $\\psi$ satisfies $(\\ref{size})$. If $\\f$ is supported in $B_r$ (the ball of radius $r$ centered at the origin in $\\T^d$), has mean zero and $\\nrm{\\f}_{p} \\leq r^{-d\/q}$, then $\\f \\in \\mathcal{U}(r)$. In what follows we will write $f \\in B\\,\\mathcal{U}(r)$ if $f\/B \\in \\mathcal{U}(r).$ \n\nRecall that a function $g$ is H\\\"{o}lder continuous with exponent $\\beta \\in (0,1)$ if $|g(x)-g(y)| \\leq C |x-y|^{\\beta}$ for some constant $C$ and all $x,y \\in \\T^d.$ We will denote the class of  H\\\"{o}lder continuous functions with exponent $\\beta$ on $\\T^d$ by $C^{\\beta}(\\T^d)$ . Paley-Littlewood projections can be used to characterize $C^{\\beta}(\\T^d)$ as follows: we let $\\omega$ be a smooth compactly supported function on $\\mathbb{R}^d$ that is identically $1$ when $|x| \\leq 1$, radially decreasing, and vanishing for $|x|\\geq 2$. Define $\\f(x) = \\omega(x)- \\omega(2x)$ and $\\f_j(x) = \\f(x\/2^j).$ For an integrable function $f$ on $\\T^d$, we define for any non-negative integer $j$\n$$ \\Delta_j f (x) =  \\sum_{n \\in \\mathbb{Z}^d} \\f_j (n) \\widehat{f}(n) e^{in\\cdot x}.$$\nThe operators $\\Delta_j$ are essentially smooth projections on the frequency scale $2^{j}$.  Recall that a bounded function $g$ on $\\T^d$ is $C^{\\beta}(\\T^d)$ if and only if for every $j \\geq 0$,\n\\begin{equation}\n\\nrm{\\Delta_j g}_{\\infty} \\leq M 2^{-\\beta j} ,\\label{CB}\n\\end{equation}\nwith some $M>0$. See \\cite{K} for a proof. \nWrite $\\ip{f,g}$ to denote $\\int_{\\T^d} f(x)g(x)\\,dx.$ If we have control over \n$$|\\ip{g,\\mathcal{U}(r)}| := \\sup \\big\\{|\\ip{g,\\psi}|: \\psi \\in \\mathcal{U}(r) \\big\\},$$ \nthen we get control over the H\\\"{o}lder $C^{\\beta}$ seminorm of $g$. More precisely, we have the following \n\\begin{lem} Suppose that a function $g$ on $\\T^d$ has the property that \n$$ \\Big| \\int_{\\T^d} g(x) \\psi(x) \\,dx \\Big| \\leq r^{\\beta},$$\nfor all $\\psi \\in \\mathcal{U}(r)$ and $0<r\\leq 1$. Then $g \\in C^{\\beta}(\\T^d)$ and \n$$ \\nrm{g}_{C^{\\beta}} := \\sup_{x\\neq y} \\frac{|g(x) - g(y)|}{|x-y|^{\\beta}} \\leq C(\\beta).$$\n\\end{lem}\n\n\\proof Let $\\f_j$ be the function defined above and let $\\mathcal{F}^{-1}$ denote the inverse Fourier transform. Since $\\f$ is smooth, compactly supported, and vanishes at the origin, $\\mathcal{F}^{-1}{\\f}$ is a Schwartz function with mean zero. We can therefore find a constant $C$ such that $\\int_{\\R^d} |\\mathcal{F}^{-1}{\\f}(x)| \\,dx \\leq C, \\int_{\\R^d}|x|\\cdot|\\mathcal{F}^{-1}{\\f}(x)| \\,dx \\leq C$, and $|\\mathcal{F}^{-1}\\f(y)| \\leq C (1+|y|^2)^{-d}$ for all $y \\in \\R^d$. Scaling these inequalities gives for all $j\\geq 0$,\n\\begin{enumerate}\n\\item[(i)] $\\int_{\\R^d} |\\mathcal{F}^{-1}{\\f_j}(x)| \\,dx \\leq C$\n\\item[(ii)] $\\int_{\\R^d}|x|\\cdot|\\mathcal{F}^{-1}{\\f_j}(x)| \\,dx \\leq C2^{-j}$\n\\item[(iii)] $|\\mathcal{F}^{-1}\\f_j(y)| \\leq C 2^{jd}(1+|2^jy|^2)^{-d}$ for all $y \\in \\R^d$\n\\end{enumerate}\n\nNow define for $x \\in \\T^d,$  \n$$\\Phi_j(x) = c \\sum_{n \\in \\mathbb{Z}^d} (\\mathcal{F}^{-1} \\f_j) (x+2\\pi n),$$\nfor some constant $c$ to be chosen later.  We claim that if we choose $c$ sufficiently small independently of $j$, then $\\Phi_j \\in \\mathcal{U}(2^{-j}).$ Inequality (iii) implies that $\\nrm{\\Phi_j}_{\\infty} \\leq cC' 2^{jd}$ and inequality (i) implies $\\nrm{\\Phi_j}_{1} \\leq cC'$ for some constant $C'>C$. Interpolation shows the norm condition, $(\\ref{size})$, is satisfied provided $c$ is small. Inequality (ii) and mean zero property of $\\mathcal{F}^{-1}\\f_j$ show that\n$$ \\Big| \\int_{\\R^d} f(x) \\mathcal{F}^{-1} \\f_j (x) \\,dx \\Big| \\leq C 2^{-j},$$\nfor any smooth $2\\pi$-periodic function $f$ in $Lip(1)$ on $\\R^d$. This implies that the smooth condition, $(\\ref{smooth})$, is satisfied, provided $c$ is small enough. \nSince $\\Delta_j$ is a convolution with $\\Phi_j$ on the space side, we have for any $y \\in \\T^d$,\n$$ \\int_{\\T^d} g(x) \\Phi_j(x-y)\\,dx = c \\Delta_j g(y).$$\nSince the class $\\mathcal{U}(r)$ is invariant under translations $\\Phi_j(\\cdot-y) \\in \\mathcal{U}(2^{-j})$ for any $y\\in \\T^d$, which implies the left hand side of the above equality is not more than $2^{-j\\beta}$ by assumption. Therefore, we have $\\nrm{\\Delta_j g}_{\\infty} \\leq c^{-1} 2^{-j\\beta}$ for all $j\\geq 0$. From $(\\ref{CB})$ we conclude that $g \\in C^{\\beta}(\\T^d)$. \\qed \n\\vspace{.1in}\n\nIf we wish to prove that the solution to the SQG at time $t$ is in $C^{\\beta}$, we must estimate $|\\ip{\\theta(\\cdot, t), \\mathcal{U}(r)}|$. We do so by determining how the class $\\mathcal{U}(r)$ evolves under the backward equation. Let $u = R\\theta$ and suppose that $\\psi$ is a solution to the equation\n\\begin{equation}\n\\psi_s = (u \\cdot \\nabla) \\psi + (-\\Delta)^{\\alpha\/2} \\psi \\label{URSQG}\n\\end{equation}\nLater we will impose some ``future condition\" that at some moment of time, the solution to $(\\ref{URSQG})$ is in the class $\\mathcal{U}(r)$. Consider the pairing function defined for $\\tau>0$,\n$$P(\\tau)=\\ip{\\theta(\\cdot, \\tau), \\psi(\\cdot, \\tau)} = \\int_{\\T^d} \\theta(x, \\tau) \\psi(x,\\tau)\\,dx$$ \nEquations $(\\ref{SQG})$ and $(\\ref{URSQG})$ together with the fact that $u$ is a divergence free vector field imply that $P$ has zero derivative and is therefore constant:\n\\begin{equation}\n\\int_{\\T^d} \\theta(x, t-s) \\psi(x,t-s)\\,dx = \\int_{\\T^d} \\theta(x,t) \\psi(x,t)\\,dx, \\label{duality}\n\\end{equation}\nfor any pair of times $t\\geq s\\geq 0.$ \nOur next objective will be to find functions $F$ and $G$ such that if $\\psi$ is a solution to $(\\ref{URSQG})$ and $\\psi(\\cdot,t) \\in \\mathcal{U}(r)$ for some fixed time $t$, then $\\psi(\\cdot, t-s) \\in F(s,r)\\mathcal{U}(G(s,r))$ for $s$ sufficiently small. This is what is meant by dualizing the problem: $(\\ref{duality})$ allows us to move the dynamics from a solution to $(\\ref{SQG})$ onto a solution of $(\\ref{URSQG})$. Now we can determine the H\\\"{o}lder regularity of the solution to $(\\ref{SQG})$ by determining how $(\\ref{URSQG})$ alters the class $\\mathcal{U}(r)$.  \n\n\\section{Evolution of the Class $\\mathcal{U}(r)$}\n\nThe heart of the proof of Theorem $1$ is the following\n\n\\begin{lem} (Class Evolution) Given $\\al,\\beta \\in (0,1)$ and $p>1$ such that $\\al +\\beta -d\/q >1$ where $q$ is the conjugate exponent to $p$, there are parameters $\\delta, r_0>0$ with the following property: if $0< r \\leq r_0$, $s \\leq r^{\\al}$, and $|\\ip{\\theta(\\cdot, \\tau),\\mathcal{U}(R)}| \\leq R^{\\beta}$ for all $R\\geq r e^{\\delta} $ and $\\tau \\in [t-s,t]$, then every solution $\\psi$ to $(\\ref{URSQG})$ with $\\psi(\\cdot,t) \\in \\mathcal{U}(r)$ satisfies\n\\begin{equation}\n\\psi(\\cdot, t-s) \\in \\exp(-\\delta s r^{-\\al})\\mathcal{U}\\big(r\\exp(\\delta \\beta^{-1} sr^{-\\al})\\big).\n\\label{evolve}\n\\end{equation}\n\\end{lem}\n\n\\proof Let $\\widetilde{\\chi}$ be a smooth radially decreasing function on $\\R^d$ supported in $\\{x: |x| \\leq 1\\}$ and has mean $1$. Then define $\\widetilde{\\chi}_r(x) = r^{-d} \\widetilde{\\chi}(x\/r)$, so that $ \\widetilde{\\chi}_r$ has mean $1$ for every $r>0$ and $\\nrm{\\nabla \\widetilde{\\chi}_r}_{\\infty} \\leq Cr^{-d-1}$. For $r\\in(0,1]$, the function $\\widetilde{\\chi}_r$ is supported in $(-\\pi,\\pi)^d$. Using the identification of $\\T^d$ with $[-\\pi,\\pi]^d$, we identify $\\widetilde{\\chi}_r$ with a function $\\chi_r$ on $\\T^d$. \n\nSuppose now that $f = f(x,\\tau)$ is a solution to the smooth forward evolution:\n\\begin{equation}\n\\left\\{ \\begin{array}{l}\nf_{\\tau}= (u_{r} \\cdot \\nabla) f - (-\\Delta)^{\\alpha\/2} f\\\\\nf(x,t-s) = f_0(x)\n\\end{array}\\right., \\label{LIPSQG}\n\\end{equation}\nwhere $u_{r} = R(\\theta \\ast \\chi_r)$ and $R$ is the divergence free operator in the statement of Theorem $1$.\nConsider the pairing function $\\Pi$ defined in the interval $[t-s,t]$ by\n$$\\Pi(\\tau) = \\ip{f(\\cdot,\\tau), \\psi(\\cdot,\\tau)} = \\int_{\\T^d} f(x,\\tau) \\psi(x,\\tau)\\,dx,$$\nwhere $\\psi$ is a solution to $(\\ref{URSQG})$. Differentiating $\\Pi$ and using that $u_{r}$ and $u$ are divergence free, we see that \n$$\\Pi'(\\tau) = \\int_{\\T^d} [(u_{r}(x,\\tau) - u(x,\\tau))\\cdot \\nabla f(x,\\tau)] \\psi(x,\\tau)\\,dx.$$\nIntegrating the above expression we have\n\\begin{equation}\n\\int_{\\T^d} \\psi(x,t-s) f_0(x) \\,dx = \\int_{\\T^d} \\psi(x,t) f(x,t)\\,dx + \\int_{t-s}^{t} \\int_{\\T^d} [(u-u_{r})(x,\\tau)\\cdot \\nabla f(x,\\tau)] \\psi(x,\\tau) \\,dx\\,d\\tau. \\label{lipduality}\n\\end{equation}\nCall the absolute value of the first integral on the right hand side of the above equation the \\emph{smooth part}, which we denote by $I$, and the absolute value of the second integral the \\emph{rough part}, which we denote by $II$. Then we have  \n$$ \\Big|\\int_{\\T^d} \\psi(x,t-s) f_0(x) \\,dx \\Big| \\leq I + II.$$\nIn what follows we will estimate the quantities $I$ and $II$. \n\n\n\\section{Modulus of Continuity Redux: The Smooth Part}\n\nUsing a modulus of continuity argument inspired by \\cite{KNV}, we will determine the size of the smooth part.\n\\begin{lem}(Lipschitz Evolution)\nLet $v$ be a smooth divergence free vector field such that $v(x,\\tau) \\in Lip(M)$ for all $\\tau \\in [t-s,t].$ Suppose that $f$ is a solution to the system \n$$\\left\\{ \\begin{array}{l}\nf_{\\tau} = (v \\cdot \\nabla) f - (-\\Delta)^{\\alpha\/2} f\\\\\nf(x,t-s) = f_0(x)\n\\end{array}\\right.,\n$$ \nwhere $f_0$ is a smooth function in $Lip(1)$. Then $f(x,t-k) \\in Lip(\\exp(M(s-k))),$ for all $k\\in[0,s].$ \n\\end{lem} \n\n\n\\proof Fix $\\epsilon > 0$ and consider \n$$ \\kappa = \\sup \\big\\{k\\in[0,s] : \\exists x,y \\in \\T^d \\,\\,\\text{such that}\\,\\, |f(x,t-k)-f(y,t-k)| \\geq e^{M(s-k)}(|x-y| + \\epsilon)\\big\\}.$$\nThe global regularity theory for an equation of the form $(\\ref{LIPSQG})$ with smooth velocity implies that $f$ is smooth for all times. In what follows we will omit the absolute value signs around the quanitity $f(x,\\tau)-f(y,\\tau)$ as we may always make it is non-negative by exchanging $x$ and $y$ if necessary. Suppose $\\kappa \\geq 0$. \nFirst, we notice that $\\kappa \\neq s$. Indeed, if it were, then there would be sequences of points $x_n,y_n \\in \\T^d$ and $k_n \\rightarrow s$ such that \n$$ f(x_n,t-k_n) - f(y_n, t-k_n) \\geq \\exp(M(s-k_n))(|x_n-y_n|+\\epsilon).$$\nThe compactness of $\\T^{2d}$ implies there are points $x$ and $y$ such that\n$$ f(x,t-s) - f(y,t-s) \\geq  |x-y|+\\epsilon,$$\nwhich contradicts the assumption on $f_0$. \nIt follows that for $\\kappa < k < s$ and all $x,y \\in \\T^d$ we have \n$$f(x,t-k)-f(y,t-k) < \\exp(M(s-k))(|x-y| + \\epsilon).$$\nPassing to the limit as $k$ tends to $\\kappa$ in the previous inequality, the continuity of $f$ implies that for all $x,y \\in \\T^d$,\n$$f(x,t-\\kappa) - f(y,t-\\kappa) \\leq \\exp(M(s-\\kappa))(|x-y| + \\epsilon).$$ \nUsing the same compactness argument as above, we see that there are points $x,y \\in \\T^d$ such that\n\\begin{equation}\nf(x,t-\\kappa) - f(y,t-\\kappa) = \\exp(M(s-\\kappa))(|x-y| + \\epsilon).\\label{lipequal}\n\\end{equation}\nWe now claim that there is a $k > \\kappa$ such that at these points $x,y \\in \\T^d$ we have \n\\begin{equation}\nf(x,t-k) - f(y,t-k) \\geq \\exp(M(s-k)(|x-y| + \\epsilon).\\label{liplarge}\n\\end{equation}\nTo this end, we will now compute \n\\begin{equation}\n\\partial_{k} (f(x, t-k) - f(y, t-k))\\Big|_{k= \\kappa}. \\label{derivative}\n\\end{equation}\n\nThe velocity term is the derivative of $f$ in the direction of $v$; more precisely, the chain rule gives $(v \\cdot \\nabla f)(x) = \\frac{d}{dh} f(x + h v(x)) |_{h=0}.$ At the breaking points $x$ and $y$ we have\n$$ f(x+ hv(x),t-\\kappa) - f(y + hv(y),t-\\kappa) \\leq \\exp(M(s-\\kappa))[(|x-y|  + h |v(x) - v(y)|) + \\epsilon].$$  \nSubtracting $\\exp(M(s-\\kappa))(|x-y|+\\epsilon)$ from both sides, dividing by $h$, and passing to the limit gives\n$$  (v \\cdot \\nabla f)(x,t-\\kappa) - (v \\cdot \\nabla f)(y,t-\\kappa) \\leq \\exp(M(s-\\kappa)) M |x-y|.$$ \n\nThe next contribution to $(\\ref{derivative})$ comes from the dissipative term. Consider the pure dissipative equation \n$$\\left\\{ \\begin{array}{l}\ng_{\\tau} = - (-\\Delta)^{\\alpha\/2} g\\\\\ng(\\cdot,t-\\kappa) = f(\\cdot,t-\\kappa)\n\\end{array}\\right..\n$$ \nThe solution to this equation is $g(z,\\tau) = f(\\cdot,t-\\kappa) \\ast \\Phi(z,\\tau)$, where $\\widehat{\\Phi}(\\xi,\\tau) = \\exp(-|\\xi|^{\\al} \\tau)$. The estimates on $f$ at time $t-\\kappa$ imply \n$$g(z_1,t-\\kappa) - g(z_2,t-\\kappa) \\leq \\exp(M(s-\\kappa))(|z_1-z_2| + \\epsilon),$$\nfor all $z_1,z_2 \\in \\T^d$. Since the solutions to the purely dissipative equation perserve the modulus of continuity, $g(x,\\tau) - g(y,\\tau) \\leq \\exp(M(s-\\kappa))(|x-y|+\\epsilon)$ for all $\\tau \\geq t-\\kappa$.  The contribution of the dissipative part to $(\\ref{derivative})$ is exactly the same as $\\partial_{\\tau}(g(x,\\tau) -g(y,\\tau))|_{\\tau =t-\\kappa}$. Since $g(x,t-\\kappa) - g(y,t-\\kappa) = \\exp(M(s-\\kappa))(|x-y|+\\epsilon)$ and $g(x,\\tau) - g(y,\\tau) \\leq \\exp(M(s-\\kappa))(|x-y|+\\epsilon)$ for all $\\tau \\geq t-\\kappa$, the function $g(x,\\tau) - g(y,\\tau)$ has a local maximum at $\\tau = t-\\kappa$. It follows that $\\partial_{\\tau}(g(x,\\tau) -g(y,\\tau))|_{\\tau =t-\\kappa} \\leq 0$ and \n$$ [- (-\\Delta)^{\\al\/2} f] (x,t-\\kappa) - [- (-\\Delta)^{\\al\/2} f] (y,t-\\kappa) \\leq 0.$$ \n\n\nCombining the estimates for the velocity and the dissipation we see that\n\\begin{equation} \n\\partial_{k} (f(x,t-k) - f(y,t-k)) \\Big|_{k=\\kappa} \\geq - \\exp(M(s-\\kappa))M|x-y| \\label{lipest}\n\\end{equation}\nThe $k$ derivative of the growth condition, $\\exp(M(s-k))(|x-y|+ \\epsilon)$, at the point $\\kappa$ is $-M\\exp(M(s-\\kappa))(|x-y|+ \\epsilon)$, which is strictly smaller than the right hand side of $(\\ref{lipest})$. It follows that for $k$ slightly larger than $\\kappa$ we have $(\\ref{liplarge})$, which contradicts the choice of $\\kappa$. From this we conclude\n$$\\big\\{k\\in[0,s] : \\exists x,y \\in \\T^d \\,\\,\\text{such that}\\,\\, |f(x,t-k)-f(y,t-k)| \\geq e^{M(s-k)}(|x-y|+\\epsilon) \\big\\}$$\nis empty for every $\\epsilon>0$ and the lemma follows. \\qed \n\\vspace{.1in}\n\nWe now wish to apply the previous lemma to the solution $f$ of $(\\ref{LIPSQG})$. Since $\\theta_r=\\theta \\ast \\chi_r$ is a smooth function, the velocity term $u_{r} = R (\\theta_r)$ is a $C^1$ divergence free vector field. In order to find a uniform bound on the Lipschitz constant of $u_{r}$, we must first estimate the H\\\"{o}lder norm of $\\nabla\\theta_r$. To this end, we notice that the choice of $\\chi_r$ implies that  $\\nabla \\chi_r$ is a mean zero vector-valued function supported in the set $B_r$ and has $L^{\\infty}$ norm at most $Cr^{-d-1}$. It follows that $r \\nabla \\chi_r \\in C\\mathcal{U}(2r)$ (meaning each component is in $C\\mathcal{U}(2r)$). By the assumption of the Class Evolution Lemma, after the time $t-s$ the solution pairs well against $\\mathcal{U}(R)$ for $R \\geq re^{\\delta}$; therefore provided $\\delta<1\/2$, we see that for any $\\tau \\in[t-s,t]$ we have  $\\nrm{\\nabla \\theta_r(\\cdot,\\tau)}_{\\infty} = \\nrm{\\theta(\\cdot,\\tau) \\ast \\nabla \\chi_r}_{\\infty} \\leq C|\\ip{\\theta(\\cdot, \\tau), r^{-1}\\mathcal{U}(2r)}| \\leq C r^{\\beta -1}.$ Similarly, $\\nrm{\\nabla(\\theta(\\cdot,\\tau) \\ast \\nabla \\chi_r)}_{\\infty} \\leq C r^{\\beta -2}.$  Let $\\epsilon >0$. Interpolation implies that the $C^{\\epsilon}$ norm of $\\nabla \\theta_r$ is no more than $Cr^{\\beta -1 - \\epsilon}$. The norm assumption on $R$ implies $\\nrm{\\nabla R \\theta_r(\\cdot, \\tau)}_{\\infty} \\leq C r^{\\beta-1-\\epsilon}$. Thus, $R \\theta_r(\\cdot, \\tau) \\in Lip(Cr^{\\beta -1 -\\epsilon})$ for any $\\tau \\in [t-s,t]$. The Lipschitz Evolution Lemma implies $f(\\cdot,t) \\in Lip (\\exp(Csr^{\\beta-1-\\epsilon}).$ Since $\\psi(\\cdot, t) \\in \\mathcal{U}(r)$ by assumption, we have the following estimate for the smooth part:\n\\begin{equation}\nI \\leq \\Big| \\int_{\\T^d} \\psi(x,t) f(x,t)\\,dx \\Big| \\leq r\\exp(Csr^{\\beta-1-\\epsilon}).\\label{S}\n\\end{equation} \n\n\\section{Mean Zero Duality: The Rough Part}\n\nIn this section, we will estimate the rough part of the evolution. Recall that the rough part was the expression\n$$ II = \\Big|  \\int_{t-s}^{t} \\int_{\\T^d} \\big((u-u_{r})(x,\\tau)\\cdot \\nabla f(x,\\tau)\\big) \\psi(x,\\tau) \\,dx\\,d\\tau \\Big|.$$ \nTrivially estimating $II$ by H\\\"{o}lder's inequality yields\n\\begin{equation}\nII \\leq s \\sup_{\\tau \\in [t-s,t]} \\big(\\nrm{(u-u_{r})(\\cdot,\\tau)}_{q}  \\nrm{\\psi(\\cdot,\\tau)}_{p} \\nrm{\\nabla f(\\cdot,\\tau)}_{\\infty}\\big).\\label{rough1}\n\\end{equation}\nThe maximum principle implies $\\nrm{\\psi(\\cdot,\\tau)}_{p} \\leq \\nrm{\\psi(\\cdot,t)}_{p} \\leq A^{1\/p} r^{-d\/q}$ for $\\tau \\in [t-s,t]$. The Lipschitz Evolution Lemma implies that $\\nrm{\\nabla f(\\cdot,\\tau)}_{\\infty}$ is not more than $\\exp(Csr^{\\beta-1-\\epsilon})$. Since $R$ is Lipchitz in the $L^q$ norm,  $\\nrm{(u-u_{r})(\\cdot,\\tau)}_{q} \\leq C_q \\nrm{(\\theta -\\theta_r)(\\cdot,\\tau)}_{q},$  so it suffices to bound $\\nrm{(\\theta -\\theta_r)(\\cdot,\\tau)}_{q}.$ \n\nRecall that $\\chi_r$ was chosen to have mean $1$, so for any constant $c$ \n\\begin{equation}\n\\nrm{(\\theta-\\theta_r)(\\cdot, \\tau) \\chi \\ci{B_{r}}}_q \\leq \\nrm{(\\theta-c)(\\cdot, \\tau) \\chi \\ci{B_{r}}}_q + \\nrm{(c-\\theta)_r(\\cdot, \\tau) \\chi \\ci{B_{r}}}_q \\leq 2 \\nrm{(\\theta-c)(\\cdot, \\tau) \\chi \\ci{B_{3r}}}_q. \\label{uncertain}\n\\end{equation}\nWe claim that for some choice of $c$ the above expression is not more than $Cr^{\\beta +d\/q}.$ We will prove this using the smoothness on larger scales along with the following \n \n\\begin{lem}[Mean Zero Duality] For any $\\rho >0,$ there is a constant $c$ such that\n\\begin{equation} \n\\Big(\\int_{B_{\\rho}} |\\theta-c|^{q}\\,dx\\Big)^{1\/q} \\leq \\sup \\{\\rho^{d\/q} |\\ip{\\theta,\\psi}| : \\psi \\in \\mathcal{U}(\\rho)\\}.\\label{meanzeroduality}\n\\end{equation} \n\\end{lem}\n\n\\proof Choose $c$ so that  $\\text{sgn}(\\theta-c)|\\theta-c|^{q-1}$ has mean zero on $B_{\\rho}$ and define $\\lambda = \\rho^{-d\/q} \\nrm{(\\theta-c)\\chi \\ci{B_{\\rho}}}_{q}^{-q\/p}.$ With these choices $\\psi = \\lambda\\, \\text{sgn}(\\theta-c)|\\theta-c|^{q-1} \\chi \\ci{B_{\\rho}}$ is a mean zero function supported in $B_{\\rho}$. A direct computation shows $\\nrm{\\psi}_p^p \\leq  \\rho^{-(p-1)d} \\leq A \\rho^{-(p-1)d}$ since $A >1$. As mentioned previously, the mean zero, support, and norm properties of $\\psi$ imply the Lipshitz pairing condition. We know choose a sequence of smooth functions $\\psi_j \\in \\mathcal{U}(\\rho)$ which converge to $\\psi$ in $L^p$ norm. Since $\\rho^{d\/q} |\\ip{\\theta,\\psi_j}| \\rightarrow \\rho^{d\/q} |\\ip{\\theta,\\psi}|$ and the latter expression is left hand side of $(\\ref{meanzeroduality})$, the Lemma follows. \\qed \n \n\\vspace{.1in}\n\nApplying the lemma to the left hand side of $(\\ref{uncertain})$ with $\\rho = 3r$ for any $\\tau \\in [t-s,t]$ gives\n$$ \\Big(\\int_{B_r} |(\\theta- \\theta_r)(x,\\tau)|^q \\,dx\\Big)^{1\/q} \\leq C(3r)^{d\/q} |\\ip{\\theta, \\mathcal{U}(3r)}| \\leq C r^{d\/q+\\beta}.$$\nSince $\\T^d$ can be covered by a constant multiple of $r^{-d}$ balls of radius $r$, adding the $q^{th}$ powers of the left hand sides of the above inequalities for all these balls yields \n\\begin{equation}\n\\sup_{\\tau \\in [t-s,t]} \\nrm{(\\theta - \\theta_r)(\\cdot, \\tau)}_q = \\sup_{\\tau \\in [t-s,t]} \\Big(\\int_{\\T^d} |(\\theta- \\theta_r)(x, \\tau)|^q \\,dx\\Big)^{1\/q} \\leq Cr^{-d\/q} r^{d\/q+\\beta} \\leq C  r^{\\beta}. \\label{roughlq}\n\\end{equation}\nThe above estimates, $(\\ref{rough1})$, and $(\\ref{roughlq})$ imply\n \n\\begin{equation}\nII\\leq C_q C A^{1\/p} r^{\\beta -d\/q}s \\exp(Csr^{\\beta-1-\\epsilon})\\label{R}\n\\end{equation}\n\nAdding the contributions of the smooth part $(\\ref{S})$ and the rough part $(\\ref{R})$ and choosing $\\epsilon < d\/q$, we have for all $\\tau \\in [t-s,t]$ and for some $C'_q >0,$\n\\begin{equation}\n\\sup_{f_0 \\in Lip(1)}\\Big|\\int_{\\T^d} \\psi(x,\\tau) f_0(x) \\,dx \\Big| \\leq r\\exp\\big(C'_q A^{1\/p} sr^{\\beta-1-d\/q}\\big). \\label{anylipevolve}\n\\end{equation}\nIn particular, we have \n\\begin{equation}\n\\sup_{f_0 \\in Lip(1)}\\Big|\\int_{\\T^d} \\psi(x,t-s) f_0(x) \\,dx \\Big| \\leq r\\exp\\big(C'_q A^{1\/p} sr^{\\beta-1-d\/q}\\big). \\label{lipevolve}\n\\end{equation}\nIt follows that for $r\\leq r_0$, $(\\ref{lipevolve})$ is stronger than what we need for $(\\ref{evolve})$ provided \n\\begin{equation}\nC'_q A^{1\/p} r_0^{\\beta-d\/q -1 +\\al} \\leq \\delta (\\beta^{-1} -1).\\label{r0}\n\\end{equation}\n\\section{The Decay of the $L^p$ Norm}\n\nIn this part, we show the way it decays of the $L^p$ norm on scale $r$ is stronger than what we need for $(\\ref{evolve})$. Computing the derivative of the $p^{th}$ power of the $L^p$ norm of $\\psi(\\cdot,\\tau)$ gives\n\\begin{equation}\n\\frac{d}{d\\tau} \\int_{\\T^d} |\\psi(x,\\tau)|^p \\,dx = p \\int_{\\T^d} \\Psi (x,\\tau) (-\\Delta)^{\\al\/2} \\psi(x,\\tau) \\,dx, \\label{diff}\n\\end{equation}\nwhere $\\Psi(x,\\tau) = |\\psi(x,\\tau)|^{p-2} \\psi(x,\\tau)$ (here we used the fact that the velocity was divergence free). We also have the well-known formula \n\\begin{equation}\n(-\\Delta)^{\\al\/2} \\psi(x,\\tau) = C_{\\al} \\sum_{n \\in \\mathbb{Z}^d} \\text{p.v.} \\int_{\\T^d} \\frac{\\psi(x,\\tau) -\\psi(y,\\tau)}{|x-y-n|^{\\al +d}}\\,dy. \\label{laplace}\n\\end{equation}\nSee \\cite{CC} for a proof of $(\\ref{laplace})$. If we plug $(\\ref{laplace})$ into $(\\ref{diff})$ and symmetrize, we see that the derivative of the $p^{th}$ power of the $L^p$ norm is \n\\begin{equation}\n\\frac{p}{2}C_{\\al} \\sum_{n \\in \\mathbb{Z}^d} \\lim_{l \\rightarrow 0} \\int_{D_l} \\frac{ (\\Psi(x,\\tau) -\\Psi(y,\\tau))(\\psi(x,\\tau) -\\psi(y,\\tau))}{|x-y-n|^{\\al +d}}\\,dy\\,dx,\\label{purediff}\n\\end{equation}\nwhere $D_l = \\{(x,y)\\in \\T^d \\times \\T^d: |x-y| \\geq l \\}$. Notice that the integrand in $(\\ref{purediff})$ is non-negative. If $r\\leq r_)$, $s\\leq r^{\\al}$, and $\\delta \\beta^{-1} < \\log 2$, $(\\ref{anylipevolve})$ implies that for the smooth function $\\eta(\\cdot, \\tau) = \\Psi(\\cdot, \\tau) \\ast \\chi_r$ \n\\begin{equation}\n\\int_{\\T^d} \\psi(x,\\tau) \\eta(x,\\tau) \\,dx < 2r\\nrm{\\nabla \\eta}_{\\infty} = 2 r \\nrm{\\Psi(\\cdot,\\tau) \\ast \\nabla \\chi_r}_{\\infty} \\leq 2r \\nrm{\\Psi(\\cdot,\\tau)}_q\\nrm{\\nabla \\chi_r}_p. \\label{smallev}\n\\end{equation}\nThe choice of $\\chi_r$ implies that $2r \\nrm{\\nabla \\chi_r}_p \\leq C r^{-d} r^{d\/p} = C r^{-d\/q},$ for some constant $C = C(\\chi)$.\nNotice that $\\nrm{\\Psi(\\cdot, \\tau)}_q = \\nrm{\\psi(\\cdot,\\tau)}_p^{p-1}.$ \n\nWe may assume $\\nrm{\\psi(\\cdot, t-s)}_p \\geq A^{1\/p} \\frac{r^{-d\/q}}{2}$ (provided $\\delta(1 + d(\\beta q)^{-1}) \\leq \\log 2$), otherwise the evolution would already be satisfied. The maximum principle implies $\\nrm{\\psi(\\cdot,\\tau)}_p \\leq \\nrm{\\psi(\\cdot,t)}_p$ for any $\\tau \\in [t-s,s]$, so substituting the above inequalities into $(\\ref{smallev})$ yields \n\\begin{equation}\n\\int_{\\T^d} \\int_{\\T^d} \\psi(x,\\tau) \\Psi(y,\\tau)\\chi_r(x-y) \\,dy\\,dx \\leq  C r^{-d\/q}\\nrm{\\psi(\\cdot,\\tau)}_p^{p-1}  \\leq  2 CA^{-1\/p}\\nrm{\\psi(\\cdot,\\tau)}_p^p . \\label{minorant}\n\\end{equation}\nThe same inequality holds if $x$ and $y$ are interchanged. Let $I(x,y,\\tau)$ be the numerator of the integrand in $(\\ref{purediff})$, then $I(x,y,\\tau) \\geq 0$ by the above comments. Notice that $|x-y|^{-\\al -d} \\geq c' r^{-\\al} \\chi_r(x-y)$, for all $x,y \\in \\T^d$ with some constant $c'$ depending only on $\\chi$. Therefore, the kernel in $(\\ref{purediff})$ dominates $c' r^{-\\al} \\chi_r(x-y).$ Leaving only the central cell contribution ($n=0$) in $(\\ref{purediff})$ and scaling $\\Lambda$ by  $C_{\\al} c' r^{-\\al} \\frac{p}{2}$ gives   \n\n$$\\frac{d}{d\\tau} \\int_{\\T^d} |\\psi(x,\\tau)|^p \\,dx \\geq C_{\\al} \\frac{p}{2} \\int_{\\T^d} \\int_{\\T^d} I(x,y,\\tau)|x-y|^{-\\al -d}\\,dx\\,dy \\geq  $$\n$$C_{\\al} c' \\frac{p}{2} r^{-\\al} \\int_{\\T^d}\\int_{\\T^d}  I(x,y,\\tau)  \\chi_r(x-y) \\,dx\\,dy $$\n$I(x,y,\\tau)$ is a sum of four terms: two of the form $\\Psi(x,\\tau) \\psi(x, \\tau) = |\\psi(x,\\tau)|^p$ and two of the form $-\\Psi(x,\\tau) \\psi(y,\\tau).$ Since $\\chi_r$ has mean $1$, the former terms contribute $C_{\\al} c' p r^{-\\al} \\nrm{\\psi(\\cdot,\\tau)}_p^{p}.$ \nThe latter terms contribute no less than $-2C C_{\\al} c' A^{-1\/p}  p r^{-\\al} \\nrm{\\psi(\\cdot,\\tau)}_p^{p}$ by $(\\ref{minorant})$ . Therefore, we have the lower bound\n\\begin{equation}\n\\frac{d}{d\\tau} \\int_{\\T^d} |\\psi(x,\\tau)|^p \\,dx \\geq C_{\\al}c'p(1-2CA^{-1\/p})r^{-\\al} \\nrm{\\psi(\\cdot,\\tau)}_p^p.\\label{normbound}\n\\end{equation} \nProvided we choose $p$ first, we can choose $A$ large enough so that \n\\begin{equation}\n1-2CA^{-1\/p} > 1\/2\\label{A}\n\\end{equation} and integrate the inequality $(\\ref{normbound})$ to get\n\\begin{equation}\n\\int_{\\T^d} |\\psi(x,t-s)|^p \\,dx \\leq A r^{-(p-1)d}\\exp(-cpsr^{-\\al}), \\label{normevolve}\n\\end{equation}\nwith $c  = c(\\chi,\\al,p) = C_{\\al} c'\/2$. It follows that for $r\\leq r_0$ and $s \\leq r^{\\al},$ $(\\ref{normevolve})$ is stronger than what we need for $(\\ref{evolve})$ provided \n\\begin{equation}\n\\delta \\leq \\text{min}\\{\\beta \\log 2,(1 + d(\\beta q)^{-1})^{-1}\\log 2, (1 + d(\\beta q)^{-1})^{-1} c\\} \\label{delta}\n\\end{equation} \nThis proves the Class Evolution Lemma provided $\\delta$ and $r_0$ are small and $A$ is large. \n\\qed\n\n\\section{The Proof of Theorem 1}\n\nIf we are given a solution $\\theta$ to $(\\ref{SQG})$ with initial data whose mean is $\\bar{\\theta}_0 \\neq 0$, we define $\\widetilde{\\theta}(x,t) = \\theta(x,t) - \\bar{\\theta}_0$ and an operator $\\widetilde{R}(\\f) = R(\\f + \\bar{\\theta}_0)$. The modified operator $\\widetilde{R}$ still satisfies the assumptions of Theorem $1$. Furthermore, $\\widetilde{\\theta}$ is mean zero and $\\theta$ solves $(\\ref{SQG})$ if and only if $\\widetilde{\\theta}$ solves the equation\n$$ \\widetilde{\\theta}_t = (\\widetilde{R}\\widetilde{\\theta}\\cdot \\nabla)\\widetilde{\\theta} - (-\\Delta)^{\\al\/2}\\widetilde{\\theta}.$$ \nIf we can show there is a time $T$ such that every $\\widetilde{\\theta} \\in C^{\\infty}(\\T^d \\times [0,T])$ can be extended to a function in $C^{\\infty}(\\T^d \\times [0,\\infty))$, then the same conclusion holds for $\\theta$. It follows that we may assume that $\\theta_0$ has mean zero. \n\nLet $\\al <1$, choose $\\beta >1-\\al$, and then choose $p>1$ so that $\\beta + \\al -d\/q >1$ and $q = 2^n$ for some positive integer $n$. Now we select the parameters from the Class Evolution Lemma. Choose $A$ large enough so that $(\\ref{A})$ is true. Now, we choose $\\delta$ small enough so that $(\\ref{delta})$ is true. Finally, we choose $r_0$ sufficiently small so $(\\ref{r0})$ is true.  \n\nSince the initial data $\\theta_0$ has mean zero, the maximum principle implies that the $L^{q}$ norm of $\\theta$ decays exponentially. More precisely, $\\nrm{\\psi(\\cdot,\\tau)}_q \\leq C(\\nrm{\\theta_0}_{\\infty}) \\exp(-\\tau\/q)$. Indeed, the proof of Lemma $2.4$ in \\cite{CC} implies for $\\theta_0$ with mean zero,\n$$ \\frac{d}{d\\tau} \\nrm{\\theta(\\cdot,\\tau)}_q^q = -q\\int_{\\T^d} |\\theta(x,\\tau)|^{q-2} \\theta(x,\\tau) (-\\Delta)^{\\frac{\\al}{2}} \\theta(x,\\tau)\\,dx \\leq -\\int_{\\T^d} |(-\\Delta)^{\\frac{\\al}{4}} \\theta^{\\frac{q}{2}}(x,\\tau)|^2\\,dx.$$\nSince $\\widehat{\\theta}(\\cdot,\\tau) (0) = 0$, by passing to the Fourier side we see \n$$ \\frac{d}{d\\tau} \\nrm{\\theta(\\cdot,\\tau)}_q^q \\leq - \\int_{\\T^d} |\\theta^{q\/2}(x,\\tau)|^2 \\,dx = - \\nrm{\\theta(\\cdot,\\tau)}_q^q.$$\nThis implies $\\nrm{\\theta(\\cdot,\\tau)}_q^q \\leq \\nrm{\\theta_0}_q^q \\exp(-\\tau) \\leq C \\nrm{\\theta_0}_{\\infty}^q \\exp(-\\tau)$. \nIt follows that there is a time $T_0$ (depending on $\\nrm{\\theta_0}_{\\infty}$) such that$|\\ip{\\theta(\\cdot,\\tau), \\mathcal{U}(r)}| \\leq r^{\\beta}$ if $\\tau \\geq T_0$ and $r_0 \\leq r \\leq 1$. Define \n$$ T_k = T_0 + \\beta r_0^{\\al} \\sum_{j=0}^{k-1} e^{-\\delta \\al j}.$$ \nWe now claim that if $t\\geq T_k$, then $|\\ip{\\theta(\\cdot,t), \\mathcal{U}(r)}| \\leq r^{\\beta}$ for $ r\\geq r_0 e^{-\\delta k}$. This is certainly true for $k=0$ by definition of $T_0$. Suppose that the claim is true for some $k$. Let $r \\in [r_0 e^{-\\delta(k+1)},r_0 e^{-\\delta k})$ and $s = \\beta r^{\\al}$.\nSuppose that $t\\geq T_{k+1}$ and $\\psi(\\cdot,t) \\in \\mathcal{U}(r)$. Notice that $re^{\\delta} \\geq r_0 e^{-\\delta k}$ and $t-s \\geq T_k$. The Class Evolution Lemma and $(\\ref{duality})$ imply  \n$$ \\Big|\\int_{\\T^d} \\theta(x,t) \\psi(x,t)\\,dx \\Big| = \\Big|\\int_{\\T^d} \\theta(x,t-s) \\psi(x,t-s)\\,dx \\Big| \\leq e^{-\\delta\\beta}(r  e^{\\delta})^{\\beta} \\leq r^{\\beta}.$$\nIt follows that the claim is true for all $k\\geq 0$. Passing to the limit, we see that $\\theta$ pairs well against any $\\mathcal{U}(r)$ after the moment\n$$\nT = T(\\al, \\nrm{\\theta_0}_{\\infty}) = \\lim_{k\\rightarrow \\infty} T_k = T_0 + \\frac{\\beta r_0^{\\al}}{1-\\exp(-\\delta \\al)}.\n$$\nFor any time $t \\geq T$, we have\n\\begin{equation} \n\\Big|\\int_{\\T^d} \\theta(x,t) \\f(x)\\, dx \\Big| \\leq  r^{\\beta},\\label{endgame}\n\\end{equation}\nfor all $\\f \\in \\mathcal{U}(r)$ and all $0< r \\leq 1$. It follows that past the moment $T(\\al, \\nrm{\\theta_0}_{\\infty})$, the solution is H\\\"{o}lder continuous with exponent $\\beta$ with H\\\"{o}lder norm uniformly bounded. \n\nWe have now shown that there is a time after which we have a uniform bound on the $C^{\\beta}$ norm on the solution for $\\beta$ as close to $1$ as we wish. A generalization of the argument of Constantin, Cordoba, and Wu (\\cite{CCW}) shows that this is sufficient to conclude that the solution is smooth past this moment.  \n\n\n\\section{Concluding Remarks}\n\nThe method presented above can be used to prove that a viscosity weak solution of $(\\ref{SQG})$ eventually becomes smooth and therefore eventually becomes a classical solution. The terminology and following definitions come from \\cite{CC}. Given $\\theta_0 \\in C^{\\infty}(\\T^d),$ a viscosity solution to the $(\\ref{SQG})$ is a weak limit (in $L^2$) of solutions to \n\\begin{equation}\n\\left\\{ \\begin{array}{l}\n\\theta_t = (R \\theta \\cdot \\nabla) \\theta - (-\\Delta)^{\\alpha\/2} \\theta + \\epsilon \\Delta \\theta\\\\\n\\theta(x,0)= \\theta_0(x)\\\\\n\\end{array}\\right., \\label{WEAKSQG}\n\\end{equation}\nas $\\epsilon \\rightarrow 0$. The pertubation by the Laplacian and smooth initial initial data guarantee a solution, $\\theta^{\\epsilon}$, to $(\\ref{WEAKSQG})$ is smooth for all times. The extra dissipative term doesn't affect the above estimates which allows us to conclude $\\nrm{\\theta^{\\epsilon}(\\cdot,t)}_{C^{\\delta}}\\leq C$ with $\\delta > 1-\\al$ for all $t \\geq  T(\\al, \\nrm{\\theta_0}_{\\infty})$ uniformly in $\\epsilon$. The results from \\cite{CCW} now give the desired regularity. In particular, the above argument gives another proof of the main result in \\cite{S}.  \n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{.1in}\n\\textbf{Acknowledgment.} The author is grateful to Fedor Nazarov for his guidance in the completion of this project. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nBooks by Fern Michaels\n\nA Family Affair\n\nForget Me Not\n\nThe Blossom Sisters\n\nBalancing Act\n\nTuesday's Child\n\nBetrayal\n\nSouthern Comfort\n\nTo Taste the Wine\n\nSins of the Flesh\n\nSins of Omission\n\nReturn to Sender\n\nMr. and Miss Anonymous\n\nUp Close and Personal\n\nFool Me Once\n\nPicture Perfect\n\nAbout Face\n\nThe Future Scrolls\n\nKentucky Sunrise\n\nKentucky Heat\n\nKentucky Rich\n\nPlain Jane\n\nCharming Lily\n\nWhat You Wish For\n\nThe Guest List\n\nListen to Your Heart\n\nCelebration\n\nYesterday\n\nFinders Keepers\n\nAnnie's Rainbow\n\nSara's Song\n\nVegas Sunrise\n\nVegas Heat\n\nVegas Rich\n\nWhitefire\n\nWish List\n\nDear Emily\n\nChristmas at Timberwoods\n\nThe Sisterhood Novels:\n\nKiss and Tell\n\nBlindsided\n\nGotcha!\n\nHome Free\n\nD\u00e9j\u00e0 Vu\n\nCross Roads\n\nGame Over\n\nDeadly Deals\n\nVanishing Act\n\nRazor Sharp\n\nUnder the Radar\n\nFinal Justice\n\nCollateral Damage\n\nFast Track\n\nHokus Pokus\n\nHide and Seek\n\nFree Fall\n\nLethal Justice\n\nSweet Revenge\n\nThe Jury\n\nVendetta\n\nPayback\n\nWeekend Warriors\n\nThe Godmothers Series:\n\nClassified\n\nBreaking News\n\nDeadline\n\nLate Edition\n\nExclusive\n\nThe Scoop\n\nE-Book Exclusives:\n\nSerendipity\n\nCaptive Innocence\n\nCaptive Embraces\n\nCaptive Passions\n\nCaptive Secrets\n\nCaptive Splendors\n\nCinders to Satin\n\nFor All Their Lives\n\nFancy Dancer\n\nTexas Heat\n\nTexas Rich\n\nTexas Fury\n\nTexas Sunrise\n\nAnthologies:\n\nSecret Santa\n\nA Winter Wonderland\n\nI'll Be Home for Christmas\n\nMaking Spirits Bright\n\nHoliday Magic\n\nSnow Angels\n\nSilver Bells\n\nComfort and Joy\n\nSugar and Spice\n\nLet it Snow\n\nA Gift of Joy\n\nFive Golden Rings\n\nDeck the Halls\n\nJingle All the Way\nFERN MICHAELS\n\nPicture Perfect\n\nZEBRA BOOKS\n\nKensington Publishing Corp.   \nwww.kensingtonbooks.com\n\nAll copyrighted material within is Attributor Protected.\nTable of Contents\n\nBooks by Fern Michaels  \nTitle Page  \nChapter 1  \nChapter 2  \nChapter 3  \nChapter 4  \nChapter 5  \nChapter 6  \nChapter 7  \nChapter 8  \nChapter 9  \nChapter 10  \nChapter 11  \nChapter 12  \nChapter 13  \nChapter 14  \nChapter 15  \nCopyright Page\n\nChapter 1\n\nDavey Taylor didn't like the shine of the street lamp that cut through the darkness and played against the filmy curtains in his bedroom. The lamp created shadows that danced on the wall, menacing his toy chest and his favorite stuffed animals on the shelf above. Each night Davey would move his ragged, beloved Panda Bear from the shelf and place it where the shadows couldn't touch it.\n\nRight now the lights in his room were all lit and the shadows were held at bay. If Davey moved back the curtain, he could even see his own reflection in the glass. But later, after Mom turned off the lights, those dark invaders would enter his room. His mother said he was too old for a night-light.\n\nStraightening his room before he went to bed, as he had been taught, Davey pursed his mouth as he studied his dog calendar.\n\n\"Today is Sunday; yesterday was Saturday,\" he told the dog sitting quietly near his feet. \"I'm supposed to change my pajamas on Saturdays, Tuesdays, and Fridays.\" His brow knit into worried lines. \"I can't remember if I changed them last night or not, Duffy.\"\n\nThe Yorkshire terrier squirmed, as if uncertain of the tone of Davey's voice.\n\n\"See, I make an X on the days I change my PJs. There's no X for yesterday.\" Davey looked down at his dog, who tipped her head to one side. Shaking his head over his forgetfulness, he walked over to his chair and flopped down.\n\n\"Changing my pajamas,\" he said, with seven-and-a-half-year-old authority, \"is one of those 'almost' things. You know, Duff, like I can almost reach the top cupboards. I can almost tell the time. I can almost walk to school by myself. Everything is 'almost.' I can't wait to grow up so I can be most.\"\n\nThe tan-and-black dog woofed in agreement.\n\nDavey swiveled his bright blue eyes to the clothes tree. There were no colorful pajamas on the peg. A cherry-red windbreaker and a yellow slicker with matching hood were the only garments hanging there. Davey ran his stubby, little-boy fingers through his thick, flaxen hair, a sign that he was relieved. His breath exploded in a loud whoosh. He must have changed his PJs the night before after all, and put them under his pillow, otherwise they'd be hanging from the peg. As if sensing her master's relief, Duffy yipped happily.\n\n\"See these, Duff? They're my first pair of Reeboks!\" Davey said proudly. \"And I almost got them dirty today. I'm wearing them tomorrow with my new red jacket when we leave with Aunt Lorrie to go camping. Mom says you can't go camping with dirty shoes, Duff.\"\n\nDuffy rolled over on the meadow of green carpet, taking Davey's excitement as a sign that it was time to play. Instead, Davey leaned over to pull up his pants leg. Duffy watched as first one strap and then another was loosened. She growled deep in her throat when the brace fell against the side of the desk. Crawling on her belly, she stretched herself to her entire two feet in length to show her disapproval.\n\nDavey stood erect. He could walk without the brace; he just wasn't supposed to be ram . . . rambunctious. He liked that word even though he wasn't exactly certain what it meant.\n\nFinding the PJs under his pillow, he stripped down and pulled the top over his head, then completed the job with the long-legged bottoms. Jumping onto the bed, he settled himself down with his new book, Elliott the Lovesick Swan. The long hand on his 101 Dalmatians watch told him it was almost time for his parents to come in and say good night. The new watch, a gift from Aunt Lorrie, was special. The only time he took it off was when he had a shower.\n\n\"Oh, man, I forgot to brush my teeth!\" With only a few minutes until bedtime, he didn't want to waste them brushing his teeth. Davey threw back the covers and marched to the bathroom. He turned on the water, put his toothbrush under the flow and wet it. His eyes danced merrily as he purposely splashed a little water onto the marble vanity top. A giggle erupted as he gave the toothpaste tube a quick squeeze in the middle, then set it down. He scurried back to bed where Duffy watched him with droopy eyes.\n\n\"I made it look like I brushed my teeth, Duff, but I really didn't.\"\n\nPicking up his picture book, he flipped through the pages. He wasn't interested in Elliott tonight. If only he could talk to his friend Digger on the CB radio. But \"if only\" was like \"almost.\"\n\n\"Time for lights out, buddy,\" his father said, opening Davey's door all the way.\n\nDavey looked up to see his mom and dad standing in the doorway. \"I know, Dad. See, the big hand is almost on the six. Do I call it six eight or eight six?\"\n\nThere was a trace of annoyance in Sara Taylor's voice when she answered for her husband. \"No, Davey. You call it eight thirty or half past eight. The little hand tells you the hour and the big hand tells you the minutes.\" She refused to call the hands on his Dalmatian watch \"paws,\" as her sister Lorrie had suggested. The boy would learn to tell time properly. \"We went over all this on Saturday afternoon. I can see where we'll have to practice extensively when you get back from your camping trip.\"\n\nDavey was undaunted by her displeasure. \"I don't think I can fall asleep tonight. I can't wait for tomorrow. Gee, this is almost better than Christmas,\" he said, his voice bubbling with excitement. Almost. It was going to be better than Christmas, he just knew it.\n\nAndrew Taylor walked over to the bed and grinned as he bent to kiss his son good night. \"I think you're absolutely right. Do you have all your gear ready?\"\n\nDavey nodded. \"I've had it ready for a whole week. Are you going to miss me, Dad?\"\n\n\"Of course we're going to miss you,\" Sara replied instead. \"By the time you get back from your trip, we'll be back from ours. We'll all be together again in just a few days. Did you brush your teeth, Davey?\"\n\nDavey squirmed. \"Go see the toothbrush,\" he answered, avoiding the lie. He looked at his parents, noticing how close they were to each other. They were always like that, he thought. And Mom always knew what Dad was thinking or was going to say. He had heard the phrase \"matched pair\" and that was how he thought of his parents. A pair. Like a pair of socks or shoes. They matched. Wanting to be part of a pair, Davey drew Duffy close.\n\nWordlessly, Sara Taylor pushed the terrier off the bed. \"I think I will check that toothbrush,\" she said, then leaned over to kiss him good night. \"Did you find your PJs under the pillow?\"\n\n\"Yep. See?\" He lifted the pillow for her inspection.\n\n\"Davey, we do not say 'yep.' It's a slang term and I don't want you to use it.\"\n\nHer voice was firm and Davey made a note to try and remember. Mom's voice was always firm. He liked Dad's voice better because there was usually a smile in it. But he liked Aunt Lorrie's voice best of all because there was usually a secret waiting to be told. Hers was a tickly, fun kind of voice. You couldn't fool Aunt Lorrie. She would have known about the toothbrush right away.\n\nDavey felt guilty for liking Aunt Lorrie's voice more than his mom's. Impulsively, he reached out, hugging her around the neck. The stretching pulled back his pajama sleeves from his arms.\n\nSara Taylor's cinnamon-brown eyes fell on the needle marks dotting her son's arms. But her movements, when she extricated herself, were icily controlled. There was no hugging pressure on her part, no smile in her eyes, when she firmly pushed him back onto his nest of pillows. \"Good night, Davey. Sleep well.\"\n\n\"G'night, Mom. G'night, Dad,\" the boy said quietly. He felt funny inside, as if he'd done something wrong. He lay very still until the door closed behind them.\n\nSeconds later, he scooted to the bottom of the bed. Duffy lay stretched out on a small carpet bearing her name.\n\n\"C'mon, Duff. You can get up here now.\" The little dog was on the bed in one leap, her tail wagging furiously. \"I've got that funny feeling again, Duff. As if we did something wrong.\"\n\nChubby hands cupped the terrier's face in a firm grip. Bright blue eyes stared unblinkingly into Duffy's melting brown ones. \"We didn't do anything wrong today, did we?\" Duffy wriggled, trying to get free to snuggle in the warmth of the blankets.\n\nDavey stared into the dim corners of the room, trying not to look at the light that filtered through his curtains. Why did his stomach feel so funny after Mom and Dad said good night? All those times in the hospital, his stomach had felt bad, too. The tubes going into his veins, his sore puffy knees making him want to cry. But he hadn't cried. Instead he'd gripped the pillows and clenched his teeth so hard he'd been afraid they'd crack into pieces. \"Don't cry, Davey. Only babies cry,\" his mother had cautioned. \"You must be brave and not do anything to upset your father.\" He had felt sick whenever Mom said that to him, her eyes willing him not to cry.\n\nHe remembered the day the tall doctor told him he was going to get a different kind of treatment\u2014a blood transfusion through the jugular vein. Davey had steeled himself not to cry in front of his dad. Instead, he'd grinned and waved to his dad as they wheeled him to the special room for blood transfusions.\n\nPills and shots, shots and pills for days afterward, and Davey had taken it all, like the brave little man his mother had told him he must be for his father's sake. He'd been carefully instructed from infancy that Daddy's wants and needs came first. Even when the pain in his joints was so bad he couldn't walk, still he hadn't cried. Davey's eyes had searched his father's each time he visited. There was always acceptance in his father's eyes, an acceptance that was totally ignorant of the price Davey was paying just so that his dad could laugh and smile when he came to visit.\n\nDavey had hoped his mom would be proud of him, but if she was she hadn't said so. He didn't understand. He'd done as he'd been told. He'd been brave. Behaved like a grown-up. With a child's sure instinct, he'd recognized that he was a trial to his parents, less than perfect, a disappointment.\n\nOnce when his aunt had visited him, she'd commented on the shadows under his eyes and asked if he was in pain. He'd been hesitant to say yes, but Aunt Lorrie had persisted until he admitted it.\n\n\"Why didn't you say something to your mom when she was in here?\"\n\n\"I . . . I wanted to but\u2014\"\n\nBefore Davey could finish his answer, his mother and father had entered the room and stopped him from continuing. That night, after visiting hours, Lorrie came back. Wordlessly, she lowered the bars of the youth bed and sat on the edge. There, in the dark, she took him in her arms and held him.\n\n\"It's okay to feel tired and sore, Davey,\" she told him, her voice as soft and sweet as the darkness. \"It's all right. You can cry if you want to. No one will hear. I know it hurts,\" she crooned, reaching out to share the weight of his misery, acknowledging Davey's pain, accepting it.\n\nSilently Davey clung to her, taking from her the courage to continue with his charade and face the ordeal. At last he slept, his body weak with exhaustion. But he hadn't cried. Not then, nor the last time either. But knowing that it would be okay to cry lightened his burden.\n\nNow the worst was behind him; he was home, and there were just the daily shots of antigen. He had done what his mother wanted; he had been brave. He hadn't cried. He hadn't upset his dad.\n\nNow, sitting with Duffy in the darkness of his room, trying to avoid the light coming through the curtains, Davey felt that tightness in his middle, again the alarm that said he'd done something wrong. That his mother didn't approve.\n\nIn a flash he was off the bed and across the room, dodging the light. He created a windmill of motion as he pulled his toys from the toy chest and sent them sailing across the bedroom. \"See, Duff. Here it is,\" he whispered triumphantly, grabbing onto his stuffed giraffe.\n\nBack on the bed, with Duffy crouched between his legs, Davey held the stuffed giraffe up for inspection. \"You see, Duff, how shiny Jethroe's eyes are?\" The giraffe's bright, shoe-button eyes stared back at him. \"Look, Duff,\" the little boy commanded, \"Jethroe's eyes never change, no matter how I move him. I don't like this giraffe!\" he cried suddenly, and his lower lip trembled as he stared at the toy. \"You know why I don't like that old thing, Duff? I'll tell you. It's . . . it's 'cause I feel like Jethroe sometimes. All wobbly and tired. Aunt Lorrie says it's okay to feel that way sometimes. But Mom doesn't.\" Seven-and-a-half-year-old wisdom rose to the fore. \"If I cry and act like Jethroe, Dad will get upset. Mom doesn't want to see Dad upset.\"\n\nDuffy snuggled deeper into the covers. \"Aunt Lorrie knows I hurt sometimes. She knows I feel like Jethroe. She says it's okay to feel like that because those trips to the hospital for blood tests take all the . . . energy out of me. Energy, Duff. Like what's in the batteries that make my RC car go. The blood tests take all my energy.\"\n\nDavey pitched the giraffe across the room. The backward motion of his hand nearly toppled the picture of his parents that rested on his nightstand. \"Whew, that was close,\" he sighed as he grappled with the slippery frame. Even in the near darkness it seemed he could see the photograph of his smiling parents. Holding the frame carefully by the edges, he turned the picture to the light that came through the curtains. His gaze intent, he brought the faces closer, then held them at arm's length again. Gingerly, he replaced the picture on the nightstand.\n\nHis whisper was fierce, almost savage, as he pulled up the covers. \"I like Aunt Lorrie best! Mom and Dad really only like each other.\"\n\nA soft whine and much wiggling and the little dog was safely tucked against the pillow next to Davey. \"You know, Duff, when I get all my energy back, I'm going to . . .\"\n\nHe was asleep before he could complete his thought.\n\nIn the corridor leading to their bedroom, Sara linked her arm through her husband's and squeezed. \"I want to talk to you about something, Andrew.\"\n\nAndrew smiled around the pipe clenched in his teeth. \"I'm all yours, as soon as the door closes.\" He turned and leered suggestively at his wife.\n\nSara laughed, tossing her blond head. \"That, too.\"\n\n\"Why don't we go down into the den and have a nightcap? It's early and we're all packed and ready to go.\"\n\n\"Mr. Sanders is downstairs. I hardly think an FBI agent, even one as nice as Stuart Sanders, is conducive to a relaxed drink and lovemaking. Why don't you,\" she said, dropping her voice to a whisper, \"turn on the gas log, shower, and wait for me? I'll go down and lock up and bring our wine up here. We haven't made love in front of the fire for ages. It's time,\" she purred.\n\nSara always had a better idea, or so it seemed to Andrew, as he returned her grin. \"Hurry,\" was all he could say. God, how he loved and desired her. He would never cease to be amazed that she returned his feelings. A man could search his life through for the right woman and never find her, but he'd found Sara and she was perfect. She fulfilled his every need. There seemed no amount of energy and caring that Sara would not put forth for his happiness. She had even interrupted her career as an English literature professor to bear him a son. At the time, she had been thirty-nine years old. He knew it had been no small concession on her part to make their union even more perfect.\n\nDesire, hot and potent, coursed through him as he turned the key to light the fire. Sara would return in exactly the amount of time it would take him to shower, dry off and put on the bathrobe she'd bought him for his birthday.\n\nSara descended the long, circular staircase. Halfway down she called softly, \"It's all right, Mr. Sanders. I'm just coming down to lock up and get a drink for my husband and myself.\"\n\nStuart Sanders waited at the bottom of the steps. His appraising, businesslike gaze took in the woman's cool blond beauty and her regal bearing. He could appreciate her neutral tone of voice. He wasn't a servant, or even a family friend; he was an acquaintance and Mrs. Taylor addressed him as such. It was acceptable.\n\n\"I'll stay with you, Mrs. Taylor, until you go back upstairs.\"\n\nSara recognized the order behind the words. \"Of course, Mr. Sanders.\"\n\nStuart followed her from one end of the house to the other as she checked the locks and turned off the lights. Even though he had locked up himself, she'd explained that the nightly ritual helped her to sleep better. He waited in the doorway of the den while she retrieved a couple of glasses and a bottle of wine from the built-in bar fridge. They weren't just glasses, he told himself, they were antique wine goblets and the wine was one of those fifty-dollar-a-bottle varieties.\n\nHe felt no envy as he surveyed the expensively appointed room. The whole house reflected Sara Taylor's conservative style and exacting taste. It was totally unlike his own place, where the furnishings\u2014bought one at a time\u2014never seemed to match. The clink of the crystal echoed through the room as Sara prepared to go back upstairs. There was nothing personal in Stuart's gaze at her. She looked like a sophisticated movie actress in her ivory satin robe and slippers. Too thin for his tastes. He liked a little more flesh on his women. Besides that, he'd never cared much for blondes; Sara's smooth delicate complexion lacked the vibrant flush he preferred.\n\nSara's sister, on the other hand, Lorrie\u2014now there was a woman. She was just the opposite of Sara in coloring and temperament. On top of that she was unattached. He'd liked her the moment he'd met her.\n\n\"Would you get the lights for me, Mr. Sanders?\"\n\n\"Sure. Can I help you carry any of that?\" Stuart offered.\n\n\"It's quite all right, I can manage. I like doing things for my husband. It's all part of being a good wife.\" She smiled at him, her widening lips and soft tone belied by her expressionless eyes.\n\nStuart Sanders returned to his position in front of the television screen. He didn't like Sara Taylor. She was cold. Icy. At first he'd thought she was only that way with him, but then he'd realized she acted like that with everybody, and worse with her sister.\n\nSibling rivalry, he thought. Maybe there was something in their past that had come between them. Whatever the reason, it was none of his business. His business was to protect the Taylors, not get involved in their lives.\n\nMaybe after the trial, when he wasn't on assignment, he could ask Lorrie out.\n\n\"You're something, honey,\" Andrew said, taking the wine bottle from her. \"Right on schedule. I just this minute stepped from the shower.\"\n\nSara laughed, a warm rich sound that sent tingles up Andrew's spine. He loved to watch her when she laughed. The mirth began around her mouth and ended in her eyes, and he knew it was for him alone. Wanting to savor the moment, he poured the wine slowly while Sara settled herself against a mound of cushions in front of the fire. He handed her a glass and sat down beside her. \"A toast. How about to\u2014\"\n\n\"Our happiness,\" Sara said, extending her glass. Her eyes were glowing, full of desire as she met Andrew's gaze.\n\nIt was Andrew who looked away first. \"Tell me, what did you want to talk about?\"\n\nSara placed her goblet on the raised hearth. \"I've been thinking that Lorrie is spending too much time with Davey. What do you think?\"\n\nAndrew's mind raced back in time. He frowned. \"You may be right. We can't allow her to intrude into our lives and Davey's affections. I wish you'd mentioned it sooner, Sara. How long has this been troubling you?\"\n\n\"A while. I wasn't certain I should say anything. Not until I saw the way Davey looked at that ridiculous Dalmatian watch, and noticed the way he's beginning to use slang words. Lorrie's responsible for that, I think. After this camping trip, we should have a talk with her. And,\" she held up a warning hand, \"we have to be prepared for some hysterics.\"\n\nSara brushed the hair back from Andrew's forehead. Her touch was cool, confident and soothing. Beneath her fingers, his brow wrinkled in a frown at the thought of the inevitable confrontation with his sister-in-law. He knew that she loved Davey almost to a fault. That was the problem: Sara found fault with that love. How like Sara to put Davey's welfare above her love for her sister\u2014her only living relative. Andrew was glad Sara would deal with the unpleasantness herself. She would handle it just the way she handled every situation he found disturbing or distasteful. He trusted her judgment\u2014she always did the right thing at the right time. Still, Andrew really liked Lorrie and he knew Davey loved her. An unsettling sensation grew in the pit of his stomach. \"We must think of Davey first . . .\" he began, half-developed contradictions forming in his mind. He had never been any good at personal relationships. He was really only comfortable with the undeniable truths of the laws of physics and higher calculus that he taught at Montclair College. And, of course, with Sara.\n\n\"Yes,\" Sara smiled warmly, \"Davey must come first.\"\n\n\"The little guy is really excited about the camping trip. I think it will be a good experience for him. Since Lorrie is a doctor, we can leave for Florida without worrying about him. I meant to go up to his room this afternoon and set up his train tracks for him, but I got involved with something else and never got around to it. I'll have some free time when all this trial business is over, I'll be able to do it then.\" Reaching for Sara's hand, he asked, \"Want to sit in with the grand old master of locomotives when he does his thing?\"\n\n\"I'd love to,\" Sara assured him, pleased that Andrew always included her in his plans. \"I was thinking of taking some time off myself, a day or so at least, and taking Davey to the apple orchard. We could watch them bake pies and buy some to bring home. Davey does love apple pie.\"\n\nAndrew frowned. \"I thought you were going to take him a couple of weeks ago. Didn't you?\"\n\nSara laughed ruefully. \"Unfortunately, no. Something came up and I couldn't make it.\"\n\n\"Was he disappointed?\"\n\n\"No, not that I could see.\" Sara sipped at her drink, eyeing her husband over the rim of the glass.\n\n\"Okay. Next thing we have to talk about is our trip tomorrow. Nervous?\"\n\n\"No,\" she answered flatly.\n\n\"I wish we didn't have to go through with this. I hate the whole thing. And I never liked the FBI's decision to place Sanders and his partner in this house. You know, Sara, I've been thinking. You don't really have to go with me. I'm the one who has to testify, and I don't want you to be upset.\"\n\n\"I'm going and that's final. I wouldn't dream of letting you go off without me. We belong together. That's the way it's always been. Where you go, I go. Final.\"\n\nAndrew ran his fingers through his thatch of dark hair salted with gray. Sara smiled, knowing the gesture signified relief. \"I don't like the fact that our names are being splashed all over the papers. And calling me a hostile witness . . .\"\n\n\"Andrew, I don't pay any attention to nonsense like that. The media is the media. Period. You know how they like to latch on to what they think is a story. Everything is going to work out, so I don't want you losing any sleep over this. Promise me. After tomorrow, or the next day at most, this whole ordeal will be over.\"\n\nAndrew drank his wine. \"I never thought they'd link me with this business, Sara. Not after we took the precaution of moving out of Miami and coming here to New Jersey.\"\n\n\"I know all that, darling. I thought we'd escape this dreadful mess too, but it hasn't worked out that way. Don't blame yourself, Andrew.\"\n\n\"Jason Forbes was a good student, Sara. Bright. Lots of potential. And now he's dead. Maybe if I'd come between them there in the university library . . .\"\n\n\"It wouldn't have made any difference,\" she assured him. \"Kids make drug deals every day\u2014in libraries, in classrooms, even in churches. You just happened to witness a buy. You couldn't have known it would end up in murder.\"\n\n\"But that doesn't excuse the fact that I didn't go to the police the minute I heard about the murder. Now, because I didn't, and because Jason told his roommate that I'd witnessed his buy, it looks as though I was trying to cover up something.\"\n\n\"Well, we both know you weren't doing anything of the kind. Mr. Sanders says that the only reason you're called a hostile witness is because you didn't come forward voluntarily but had to be subpoenaed. Once you testify, the State will have its case wrapped up, and we can go back to our normal lives. And Mr. Sanders and his partner can go home and leave us alone.\"\n\n\"I should have stepped forward voluntarily, Sara. I should have reported the threats I'd overheard as soon as the body was discovered.\"\n\n\"Hush, darling, you'll only upset yourself.\" Sara cradled Andrew's head against her soft bosom. \"You were only trying to protect Davey and me, and we love you for it. Even the FBI recognizes that our lives are endangered, otherwise they wouldn't have put us under twenty-four-hour guard. I love you, Andrew Taylor, with all my heart for all my life.\"\n\nAndrew's pulses pounded as Sara's face swam before his eyes. It never failed to happen when Sara prompted their lovemaking with those words. God, how he loved her. He knew his life would be meaningless without her. They shared their lives, careers, and interests; theirs was a coming-together, a blending, a loving. His hand slipped beneath the soft velour of her robe, touching her breast. Through the years he had learned the special phrases and words that heightened her response and brought her to life beneath his touch. He told her how he loved her, how they fitted one another like hand and glove. How perfect she made his life, how perfect she was, her beauty, her womanliness. How complete they were, one with the other, inseparable. And Sara responded, listening, prompting his words with touches, kisses, and murmurs.\n\nHer eyes became liquid, her mouth ripe and open for him, accepting his kiss, his tongue. He loved her like this, soft and yielding. His pulses quickened, his senses sharpened as he waited, knowing she would slip out from under him and turn, leaning over him, assuming her usual dominant position.\n\nHer thighs were lean and hard-muscled as they closed around his body, the heated, warm center of her pressed against his belly, rubbing, pleasuring. He submitted himself to her mastery without any inclination to assert a masculine role, trusting her implicitly, always trusting himself to her.\n\nWineglass in hand, Sara watched his reaction as she tipped the rim, allowing the sweet liquid to trickle down his chest, pooling on his belly. The chilled wine, her hot tongue. She felt his hands stroking and pressing her head, heard him groan with pleasure. \"Your mouth, Sara, your beautiful mouth . . .\"\n\nContact between their bodies was wet, slick, so warm. Artfully, she lowered herself onto him and felt him fill her body. She felt she was dissolving, melting. He seemed to become a part of herself. The muscles in her pelvis became rigid; she could feel her womb contract. It was as though she were birthing him.\n\nAt the moment of climax she brought her hard-tipped breast to his lips, encouraging him to suckle. And while she held his head, feeling the life spurt into her, she crooned, \"Sara's baby, Sara's sweet, perfect baby.\"\n\nBlue light glaring from the television washed the faded colors of the sparsely furnished room. Hands gripping the arms of the chair, he sat with his booted feet planted solidly on the floor, his bulky torso leaning slightly forward, poised as though he were about to spring up. The images on the screen flickered. He stared at them, unblinking, but didn't see any of the action, didn't hear the blaring sound. Chill, wet patches on his back betrayed his anxiety. Perspiration broke out above his sullen mouth and on his scalp beneath his dark, military-short hair. Cudge Balog was waiting, listening for the dull thud of hooves deep inside his head. Cutting hooves which dug into his brain matter, tearing and gouging at it. It would start slowly, with only a hint of the weight and power to come.\n\nHe had been watching TV, his thoughts on Lenny Lombardi, who, Cudge knew, would soon be pounding on the door, demanding repayment of the borrowed fifty dollars. There was a crap game in the neighborhood tonight and Lenny would want to sit in well heeled. Little bastard. He didn't need the fifty. Lombardi got more than he could spend from his drug-dealing racket. It was only pot, none of the big stuff, because he didn't want trouble with the syndicate. Still, Lombardi made more in a week than Balog would see in a month of breaking his ass on construction jobs.\n\nCudge's short, thick fingers dug into the threadbare fabric covering the chair. Pressure crowded the back of his brain, driving his squarish head into his neck as his powerful shoulders hunched to bear the weight. Soon, he knew, the hooves would pound through his skull\u2014an unleashed power, irrevocable and ruthless. A dark, hulking shape would break loose from that area of his mind where he kept it penned, under control. Thinking about Lombardi had opened the gate.\n\nAs far back as Cudge could remember, the hoofed beast had lived inside his head. As a kid, he'd thought of it as a huge prehistoric monster with a long, arching neck and rows of jagged, fierce teeth. But then, at a summer camp for underprivileged city children, he'd seen a bull. He'd known then, he'd recognized the thick hulking body, the menacing drift of weight. Black, with dagger-sharp horns and fiery snorts of breath. He feared it but, in doing so, he feared a part of himself. When he was provoked and lost control of the gate, it was there\u2014lurking, skulking, ready to burst forth from the recesses of his brain. A pounding, all-powerful force, hooves striking, horns slashing, searching for escape. Finding none, it would stampede wildly, smashing his reserve, pulverizing his restraint, compelling and dominating him until he became it.\n\nSome said it was temper. Cudge knew better. It was the bull.\n\nBrenda Kopec\u2014or Elva St. John as she preferred to be called\u2014sat on the lumpy daybed, her back against the wall. Her attention was riveted on the man in front of the television. She watched his profile with feral alertness, knowing he was a firecracker about to go off.\n\nThe instant Cudge had turned on the TV, she'd immediately lowered the volume of her small cassette player and jammed the earphones onto her head. Elva knew the words to Elvis Presley's \"Blue Suede Shoes\" by heart, but she wanted to hear the song from beginning to end. As her foot tapped to the rhythm, the scowl on Balog's face deepened. Elva knew he wasn't really watching the TV. She'd known that from the minute he had turned it on. He was thinking about that little rat-faced Lenny Lombardi. Cudge was mad and getting madder by the minute.\n\nAs though feeling her eyes on him, Balog turned and glared at her. His square, snub-nosed face registered contempt. Veins swelled in his short, thick neck. With a speed that belied his bulk, he tore the earphones from her head; when she grappled for them, he struck her. Hard.\n\nElva brought up her arms defensively. \"Why'd you do that?\" she whined. If she cried, Cudge would hit her again.\n\n\" 'Cause you're breathin'. Shut that damn thing off and sit still. I'm trying to watch TV.\"\n\n\"No, you're not. Anyway, you've seen that one before. It's the one where\u2014\" Instantly, she was sorry she'd opened her mouth. Cudge sent her another look which made her cower and slip off the end of the daybed.\n\nHe stood and loomed over her. \"How many times you heard that dumb song, Brenda? Oh, 'scuse me,\" he sneered at her, \"you wanna be called Elva now. In honor of Elvis Presley. Well, he's dead and you're nothin' but a dummy. Say it, Elva\u2014you ain't nothin' but a dummy.\"\n\nElva swallowed hard. The side of her head smarted from the blow. She knew better than to argue with Cudge. \"So, okay, I'm a dummy.\"\n\n\"You always get hit because you never know when to shut up.\" His words were accusing, placing the blame for his actions on her. \"Now, shut up, if you know what's good for you. Already I missed the first part of the show.\"\n\nRighting herself, cautious to stay out of his reach, Elva put the cassette player in the paper shopping bag on the floor beside her, where she kept all her meager possessions. If Cudge decided they were moving on, he wouldn't give her five minutes to get her gear together. Wishing she were invisible, she settled herself again on the worn daybed. She wanted to cry. She wanted to run. But she never would. Cudge scared her sometimes, but the outside world scared her more. At least Cudge took care of her. Sometimes he wasn't so bad, she told herself. Once he'd bought her a purple scarf, and he often took her to the movies. Every Elvis cassette she owned had come from Cudge. So why did she take such pleasure in goading him the way she did? Even when he was raging at her, even when he hit her, there was a small part of her that took abject pleasure in it. Not that she was a pervert, or an S&M freak, or anything like that. No, it was more that she was little and helpless, so it felt good to be able to get a rise out of a hulk like Cudge. It gave her a kind of power, knowing she could set him off anytime she wanted. It made her stronger than him, in some strange way. But Cudge was right\u2014she was a dummy. Someone smarter would know how to get a rise out of Cudge and aim it at somebody else. Whenever she set him off, she was bound to get the brunt of it.\n\nSuddenly she felt contrite with tenderness for Cudge. He had his own problems to deal with. And he wasn't so bad, not really. So what if this dumpy room wasn't the Ritz? People like her and Cudge would never make the Ritz. They'd be lucky if they ever saw the inside of a Holiday Inn. She risked a quick, sidelong glance at Cudge to see if he really was watching TV. If he was, she could lean back and relax. She stared at the screen, fearful that any movement would alert Cudge that she was restless or scared. Her toothache was coming back and she wanted to massage her cheek but she was afraid to move.\n\n\"Someday I'm gonna get one of those portable satellite dishes so's I can see some really sexy shows,\" Cudge said during a commercial.\n\nElva shrugged.\n\n\"Why ain't you sayin' anything?\" he demanded irritably.\n\n\"You told me to shut up, that's why. I'm a dummy, remember?\"\n\n\"That's your trouble, you never know when to shut your mouth. Here,\" he said, fishing in his pocket for money, \"go get us a pizza, and I want the change. And listen\u2014\"\n\n\"I know, I know\u2014I should tell that guy to put on extra cheese and not charge me for it.\"\n\nCudge laughed. \"You really think that old man has the hots for you, don't you? Well, he don't. And if he did, he knows better than to mess with you. Thirty minutes, Elva, and you better be back here handing me my first slice. Don't lose the change!\" He laughed again, his flat blue eyes narrowing.\n\nFeeling like a trapped rat, Elva scuttled away. If she ran, she might make it there and back again in thirty minutes. Tony might be nice and give her somebody else's pizza when she told him it was for Cudge. Tony would do that for her, maybe.\n\nHer skinny body bent into the wind, she hurried along the deserted streets of Newark's Ironbound section. The tap of her high heels echoed hollowly off the sleeping, brick-fronted tenements. She was wary, jumping at imagined shadows, at the prowlings of a conspiracy of cats lurking in an alley. Her worn navy parka was warm but it hung loosely on her slight frame. She pulled it higher, burrowing her chin into it against the late October cold.\n\nJust ahead, less than a block away, she saw the dim red halo outlining the storefront of Tony's Pizzeria. She broke into a run, eager to be near the warmth of Tony's ovens and out of the menacing darkness. For an instant she panicked. Pushing her hand deep into the pocket of her parka, she searched for the ten-dollar bill Cudge had given her to pay for the pizza. Torn tissues and gum wrappers tumbled out, were caught in the wind and fell onto the sidewalk. Biting her lower lip, she prayed silently that the ten would magically appear. The last time Cudge had sent her out to buy something, she'd stupidly lost the money and had to go back to face his rage. She gave an audible sigh of relief when her skinny, twitching fingers found the bill. Holding tightly to the money, as though fearful some unseen force might pluck it away, she made a dash for the pizzeria.\n\nThe glass-paned door was steamed up, dripping moisture from the heat of the ovens meeting the cold outside. Throwing her weight against it, she entered into the light and warmth of the restaurant. The jukebox was playing a popular song and Tony, behind the counter, was singing along in his broken English.\n\n\"Hey, Elva! Whatcha doin' out so late? Don't y'know li'l girls should be in bed by now? I'm just closin' up. Business, she's bad tonight. Every Monday, it's the same.\" His white apron was stained with tomato sauce and the bright overhead lights accentuated the stubble on his jowly face.\n\n\"I ain't so little,\" she protested shyly. \"I told you, I was eighteen last month.\"\n\n\"Elva, you always gonna be a li'l girl. It make no difference how old you gonna get.\" He smiled at her, showing a space between his front teeth.\n\nElva liked Tony. He was always friendly and he seemed to know instinctively how scared she was of everything and everyone. \"Cudge wants a pizza.\"\n\n\"So? He wants a pizza. I'm just closing up.\" Tony saw the dread in her dark eyes. \"Why you wait so long? It's late. I've got a family waitin' for me,\" he complained, leaning over the counter. \"Hey, how's your eye? It's not so nice what he does to you, that guy. Why you wanna stay with him?\" His finger touched her cheek just below her left eye where, only last week, she had been black and blue from another of Cudge's beatings. \"Poor little thing,\" Tony commiserated. \"You oughta leave that son of a bitch.\" He stared at her, pity in his eyes. \"Sure, Elva, for you, anything. What kinda pizza you want?\"\n\nCudge heard the door slam as Elva ran out. He really had to hand it to her\u2014when she wanted she could really get that skinny ass of hers moving.\n\nHe wished he had a beer. The dull thudding in his head was getting louder; a beer might help. It was a piss-poor world when a man couldn't have a beer. Elva always had her Kool-Aid in the fridge. His sullen mouth turned down. He was starting to hate Elva almost as much as he hated that sticky-sweet, artificial drink. It was getting to be time to rip the rug out from under old Elva. Time to move on and he liked to travel light.\n\nCudge let his eyes drift back to the blurry picture on the TV. It was an old rerun. Hutch was saying something to Starsky. Now that Starsky was a real man. Starsky, if you hoot with the owls all night, you won't be able to soar with the eagles in the morning. Cudge rolled Hutch's words around in his head then said them aloud. He liked the sound and the meaning. He repeated the sentence four times, till he was sure he'd remember it. It was just the kind of thing a guy would say to his best buddy.\n\nA knock sounded and the door opened. \"Cudge, you in here?\"\n\nLenny! The thudding in his brain matched the beating of his pulses. He knew it! It'd been a sure bet that as soon as Lenny'd heard about the floating crap game, he'd come looking for that fifty. Some best buddy Lenny was. Lenny Lombardi would pick the gold from a dead man's teeth; he didn't deserve the words Cudge had just heard on the TV. He was a jerk. The whole world was full of jerks.\n\nThe muscles in Cudge's neck went into a spasm. He feigned a smile, showing his teeth. \"C'mon in, Len. Wanna drink? Elva's got some Kool-Aid in the fridge.\" He liked the stupid look on Lenny's face.\n\n\"Nah. I didn't come for Kool-Aid. I saw that Olive Oyl old lady of yours runnin' down the street. What did you do? Threaten to beat her again?\" Lenny loved to torment Cudge about his uncontrollable temper.\n\n\"What's it to you?\" Cudge drawled menacingly.\n\n\"Nothin'. I come for the bread you owe me. There's a hot crap game and I want to sit in.\" Lenny sauntered around the room, hands jammed into his pockets. \"Cough it up, I'm in a hurry.\"\n\nCudge's fist tightened. The lone ten-dollar bill in his pants pocket seemed to be burning his leg. He didn't need this cocky little dude with his pointed shoes giving him grief. \"I ain't got it.\"\n\nLenny's pinched face flattened. He worked his tongue between the space in his front teeth, making a hissing noise that set Cudge's nerves on edge. \"You told me that three weeks ago. Your time ran out, now pay up.\"\n\nCudge laughed, an obscene sound. \"I told you I ain't got it. Gimme another week. Christ, Lenny, we been friends for a long time. You gonna blow it all for a lousy fifty bucks?\" He watched Lenny keenly.\n\nLenny looked nervously over his shoulder before turning back to Cudge. It was a habit Cudge found irritating. Always looking away and then back again, diverting his attention, making him look over Lenny's shoulder himself, making him half expect to see someone there.\n\n\"Looks like I'm gonna have to take it in trade, old buddy.\"\n\nCudge's mouth tightened. Both hands balled into fists. \"Yeah? How?\"\n\n\"By taking that camper sittin' down at the curb, that's how. And your truck goes with it. Give me the keys. When you come up with the bread, you get it all back. Simple.\"\n\n\"You ain't taking my rig so get that idea right out of your head. You want collateral, take Elva's cassette player and tapes.\"\n\n\"Hey, man, I don't want your junk. Just give me the keys to your wheels. I gotta get going if I wanna sit in on the game.\"\n\nCudge's mind raced. The hooves pounded in his brain. Without his truck he'd be sunk, unable to get to the construction sites where he could pick up some money, even though he had to work his balls off just to keep body and soul together. He had to think of something. Think fast. Before the thundering hooves blotted out all reason. Lenny was a sneak, a real bastard, when it came to money. He had to get rid of him somehow.\n\n\"Don't even think about pulling a fast one, Balog. I know you got money. You think I'm stupid or somethin'? Your old lady was going into Tony's, probably for a pizza and some beer. If you can eat, you can pay your debts.\"\n\nCudge got to his feet, Elva's tape player in hand. He had no plan as he stared at Lenny Lombardi. He could almost hear the creak of the gate that kept his rage penned in the back of his head. His shoulders hunched from the weight pressing against the top of his spinal column. \"I ain't got it. If you can't take my word for it, you ain't my friend.\"\n\n\"Friends don't welch on loans,\" Lenny told him. At the look in Balog's eyes, he edged back.\n\nCudge laughed, an unpleasant sound. Lenny backed up another step, lurching into the kitchen table. His eyes seemed to measure the distance to the door. \"Okay, okay. Forget the wheels. I'll give you another week to come up with the scratch. Look, I gotta go now,\" he bleated as he put the table between himself and Cudge.\n\nSuddenly the beast was loose. It took off at a gallop, snorting fire. Pressure moved from the back of Cudge's head to a point at the center of his skull. Instinct told him that if he frightened Lenny enough the fifty bucks would be called even, and he could forget about ever paying it back. He took a deliberate step in Lenny's direction, hefting the cassette player in his beefy hand.\n\nIt was the sheer terror on Lenny's face more than his words that provoked Balog. \"You're crazy, man! Crazy!\"\n\nHavoc broke loose in Cudge's brain. He became the beast, sensing his prey, moving in for the kill. Blood surged into his face; his skull throbbed and pounded. Fiery breaths scorched his thoughts; dagger horns gouged and ripped.\n\nLenny stood speechless, his eyes round with fear. Urine pooled around his shoes. A sound erupted from his throat\u2014a sick, choking sound. He made a run for the door but Balog was there ahead of him, blocking the way.\n\nCudge snorted; saliva glistened on his chin. Lenny froze. Only his eyes moved as the cassette player lifted and crashed down on top of his skull.\n\n\"Take my wheels, will you?\" Cudge raged, slamming the cassette player again and again into Lenny's head. \"You ain't my friend. Now get your ass out of here before I throw you down four flights of stairs.\"\n\nLenny lay with his face pressed against the floor. Cudge stood over him, seeing only the back of his friend's head. \"Get up! Move, you little turd!\"\n\nHe prodded the still form with his boot, was surprised when there was no movement. He squeezed his eyes shut against the sudden stab of pain in his temples. When he reopened them he noticed the widening pool of blood on the floor.\n\nCautiously Cudge crouched to the ground, the cassette player still clutched in his hand. He turned Lenny faceup, thinking how light he felt, how his still form offered little resistance. The wide, staring eyes panicked him and the cassette player fell from his hand.\n\nJesus. He didn't need anyone to tell him that Lenny was dead. The jerk was dead! Jesus. Oh, Jesus. He had killed his best friend!\n\nAs Tony punched down the yeasty dough and stretched it over the shiny pan, he watched her. As always, his heart went out to her. She was still a kid. Other girls, by the time they were eighteen, were more woman than child. But not Elva. She would always remain a child, a frightened, winsome, confused child. Too bad she had to meet up with that animal, Balog. A nice guy could be the salvation of a timid kid like Elva, but in the hands of the hulk she was damned. Pity. She wasn't a bad-looking girl. Too skinny, of course, and a little pinched-looking, and her eyes were always on the edge of panic, but she was pretty in a shy sort of way. With a little fixing she could be really pretty. A haircut and a little meat on her bones would make a world of difference. And something, Tony thought, or someone, to take that haunted look from her eyes.\n\nAs he scattered mozzarella cheese on the pizza, Tony found a chunk and handed it to Elva, noticing her severely bitten fingernails. She took the cheese from him with a shy smile and nibbled at it. He pushed the prepared pizza into the oven and went back to cleaning the counter. \"Elva\u2014what kind of name is that? Ol' Tony never hear it before you come here.\"\n\n\"It's a name I just like,\" she answered between nibbles.\n\n\"So, it's not your name?\"\n\n\"It is now. My name used to be Brenda Kopec,\" she said, putting the last morsel into her mouth.\n\n\"Brenda! That's a nice name. Soft, like you. So, how come you change it? My own two daughters, they want names like Brandy and Tiffany. What's wrong with Maria and Theresa anyhow? I'm never gonna understand them. So, tell Tony, how come you changed your name?\"\n\n\"I call myself Elva after Elvis Presley. I heard somewhere that Elva was the girl's name for Elvis.\"\n\n\"Elvis, huh? He dead long time now, you know.\"\n\n\"Gone but not forgotten. As far as I'm concerned, he's still the king and I love him!\" she said with rare emotion.\n\nTony glanced up, struck by the sadness in her voice. It held the same note he had heard in his wife's voice whenever she mentioned their own dead son.\n\n\"I've read all the books written about him, seen all his old movies, and I've got all his songs on tape. He was a gentleman, Tony. A real gentleman. And generous.\" She pulled at her dull brown hair, her fingers working in agitation.\n\n\"You like your fellas generous? So what are you doing with that cheap son of a bitch, Balog?\"\n\n\"He ain't so bad. Sometimes I think he's scared inside, just like me. Only he don't show it like I do.\"\n\nTony shrugged. There was no accounting for these American girls. He only prayed that his own daughters wouldn't end up with anybody like Cudge Balog. If Elva was right about Balog being scared of something, Tony couldn't imagine what it might be. He'd seen guys that Balog had worked over and he knew what the man's fists could do to a face. It was only a matter of time before he killed someone, and Tony hoped that it wouldn't be Elva. She was a good kid, even if she was a little stupid. Maybe if she weren't so scared all the time she wouldn't be so dumb.\n\nElva hurried back to the apartment, balancing the hot pizza carefully so the gooey cheese wouldn't run to one side. She wondered how long she'd been gone. It seemed like a long time, and Cudge would get mad if she kept him waiting. Suddenly, she couldn't remember if she'd picked up the change from Tony's counter. Cudge was a real stickler when it came to money. She stopped in front of a tenement and propped her leg on the stoop, balancing the pizza on her knee. The heat penetrated the cardboard box and stung her leg as she frantically dug through her pockets, looking for the change. Her panic began to turn to hysteria when she couldn't find it. She thought of going back to Tony's to see if she'd left it on the counter and glanced back along the street. The red light over Tony's door had gone out. What should she do? Maybe she could catch up with him at his car . . . Just then her fingers touched cold metal and relief flooded through her. She'd found it; she hadn't been stupid after all. For safekeeping, Elva dumped the coins into her bra, then gripped the pizza box again and hurried back to Cudge.\n\nShe smiled in the darkness. Everything had gone right for a change. Cudge wouldn't have anything to yell about.\n\nWhen Elva turned down Courtland Street, she recognized the familiar outline of Cudge's Chevy pickup truck and the flat square shape of the pop-up camper hitched to its rear. They rarely went camping, but just a few days ago Cudge had talked about taking a weekend in the country. Like so many things Cudge talked about, Elva never expected to see it come to anything.\n\nShe loped up the front stoop of their building and into the dimly lit hallway. Urine and stale cooking odors came to her nostrils. Just as her foot was on the first step leading upstairs, the door to the landlady's apartment swung open.\n\n\"Oh, it's you. I thought maybe it was you he was knocking around up there.\" Mrs. Fortunati's thin gray hair fell over her eyes and she brushed it away with an impatient gesture of her work-worn hands. \"You'd better get your ass up there and see what's going on. I was thinking about calling the cops.\"\n\nElva gulped at the sinking feeling in the pit of her stomach. The night was ruined; Cudge had done it again. Now it wouldn't matter that she had bought the pizza and brought home the right change and had done everything just exactly right. Cudge was going to be nasty and find something, anything, to be mad about anyway.\n\n\"Well, what are you waiting for? Get up there! From the sound of it he was tearing the place apart.\" She moved to the banister and watched Elva go up the stairs as she issued her last warnings. \"I'm telling you now, there better not be any trouble or out you go! The both of you! Him in particular!\"\n\nElva waited outside the door, dreading going in. For all Mrs. Fortunati's ravings, it was quiet now. Only the cries of the baby from up in 4B broke the silence.\n\nShe fumbled with the doorknob, balancing the pizza box on her knee. The door opened a mere three inches. Cudge had latched the chain hook. Puzzled, Elva opened her mouth to call him, then winced. The temperamental tooth with its rice grain of decay was going to ache all night.\n\n\"Cudge,\" she whimpered, \"open the door, will you?\"\n\n\"Elva?\" It was a hoarse whisper from the other side of the door.\n\nSomething was wrong. Cudge never whispered. He yelled and put his fists through walls, but he never whispered. \"Yeah, it's me,\" she responded. \"What's the matter? Why are you whispering?\"\n\nThe door was forced shut, jamming against a corner of the pizza box, and she heard him fumbling with the chain latch. Then it swung open again and he grabbed hold of her arm and pulled her into the apartment. The bare lightbulb over the kitchen table swung back and forth, creating wild shadows and rhythmic patterns of light.\n\n\"Get in here, dummy. Where the hell were you?\" He was angry but he was still whispering, and the annoyance on his face was mingled with something else. Something dreadful she had never seen there before. Now it wouldn't matter that she had done everything exactly right. Nothing would matter except that Cudge was mad and, one way or another, she would pay for it.\n\n\"I . . . I went for the pizza like you told me. I even got the change.\"\n\n\"Shut up. I gotta think!\"\n\nElva shrank back, still clutching the pizza box. Something was wrong, awfully wrong. What? She'd never seen Cudge like this, so quiet and scared. He moved away from her and sank down on the edge of the daybed, his head in his hands. The TV was still on but the sound had been turned off. She watched him, not daring to turn her eyes away.\n\nThen suddenly, like an uncoiled spring, he jumped to his feet and punched the wall, his lips drawn back over his teeth in a frightening grimace.\n\n\"Stupid little shit! He never should've tried to bust my hump. He should've known I didn't have fifty bucks to pay him back.\" His fist pounded the wall again, punctuating his words. \"Thought he'd take my truck and rig. Thought I was stupid or something. He should've known!\"\n\nElva pressed against the wall, eyes wide with terror. In all the time she had lived with Cudge she'd never seen him like this. Cudge was scared. Scared shitless.\n\n\"Don't look at me that way!\" He turned on her, slamming his fist into the cardboard pizza box, knocking it to the floor.\n\n\"You ruined it.\" Automatically she bent down to pick up the box but Cudge hoisted her to her feet.\n\n\"What the hell are you messing with that for?\"\n\n\"I . . . I just wanted to clean it up.\"\n\nHe shook her, almost making her teeth chatter. \"Oh, yeah? Well, see what you can do about cleaning that up!\" He turned her around so she came face-to-face with Lenny Lombardi. Lenny was lying on the floor, his face barely recognizable. If it hadn't been for his familiar trench coat and slick dark hair, she wouldn't have known him.\n\nElva knelt down beside him, her hands extended in a gesture of helplessness. Lenny wasn't breathing!\n\nHer mouth opened but before the sound could rip from her throat, Cudge had his beefy hand clasped over her lips, covering her nose, cutting off her air. Waiting for her to be quiet, he hissed a warning not to scream.\n\nShe stared up at him over his hand, her eyes wild and panicky, then shook her head violently, fighting for breath.\n\n\"Will you shut up?\" Cudge growled. \" 'Cause if you don't, you'll get some of the same.\"\n\nThe cords in Elva's neck threatened to burst; she was feeling dizzy and sparks were shooting off inside her head. Frantically, she nodded.\n\nCudge waited a long moment before removing his hand. For an instant, she believed he never would, that he would hold her there forever and ever. Her feet kicked out, touching the soft, unyielding body wedged against the wall. Sickened, she ceased her struggles.\n\n\"Now, shut up. One sound out of you and you'll look just like him!\" Cudge warned in that creepy whisper, a scared look narrowing his eyes.\n\nFull of revulsion, Elva made her way to the daybed, away from Lenny\u2014from what used to be Lenny. She clamped her hands over her mouth to stay the questions. Unable to control herself any longer, she began to tremble as the words tumbled out.\n\n\"Why? Why'd you kill him? He was your friend! My God! You killed him!\"\n\nCudge raised his hand, threatening her. \"I told you to shut up! I don't wanna hear your mouth! Shut up!\"\n\nElva was beyond the point of hysteria, she was verging on dementia. \"God! You killed him! You killed Lenny! Your best friend! God!\"\n\n\"If you don't shut up so I can think, you're gonna get what he got!\" Cudge knocked the lamp beside her onto the floor. \"One more word, Elva, one more word and you're gonna get it! You stupid broad! I gotta think!\"\n\n\"But the police! What are you gonna do? They'll find out!\"\n\n\"Quit your babbling, I gotta think!\"\n\nShe shuddered with horror. Cudge had killed Lenny Lombardi and he would do the same to her if she didn't keep quiet. Everyone always said Cudge would kill somebody someday and Elva had silently agreed with them, never realizing how his potential for violence fascinated her. But Lenny was his friend.\n\nCudge paced the floor, his hands constantly kneading his skull in exasperation. While he paced, he kept up a constant monologue, muttering curses at Lenny, whining complaints and praying to God for a solution.\n\nBit by bit the quarrel between Lenny and Cudge became clear to Elva as she stole quick looks at the body that lay stuffed between the table and the wall.\n\n\"We have to get out of here,\" Cudge said, intensity sharpening his blunt features. \"And we have to get him out of here, too, before anybody starts wondering where he got to.\"\n\nElva looked up, puzzled.\n\n\"You're an accessory, you know,\" he informed her. \"If I hang, you're gonna hang, too!\"\n\n\"Me? I didn't do anything! I just came in here and found . . . him.\"\n\n\"It don't matter, baby,\" Cudge told her, his voice showing concern. He knew how easy Elva was to handle\u2014stupid, dumb broad. All he had to do was make her think he cared for her and she came crawling, willing to do anything he demanded. \"Look, baby. According to the law, you should have run out of here and gone straight to the cops. You didn't, so that means you're aiding and abetting. That makes you an accessory and what I get, you get too! Understand?\"\n\nElva really didn't understand, but she knew that Cudge was smart when it came to the law and he sounded as though he knew what he was talking about. If he said she was an accessory, then she must be one. He'd been busted by the cops enough times to know what he was saying. \"But what can we do? Where can we go?\"\n\nCudge smiled to himself. Poor, stupid Elva. \"Look, baby, we've got to get out of here and we have to take him with us. I figure we can stuff him into the camper and take off somewhere and bury him.\"\n\nA tear trickled down her cheek. \"Poor Lenny.\"\n\n\"What about 'poor Cudge'? What about me? That stupid jerk tried to rip me off\u2014he got what was coming to him! And now I ain't got no best friend.\" He knew that would bring Elva around. There was nothing that could swing Elva around like cheering for the underdog. Just make her feel sorry for you and you could lead her around by the nose.\n\n\"Cudge, I didn't mean anything like that.\" She went over to put her arms around him. \"Sure I know how hard this must be on you and all. Lenny was your friend and I know you didn't mean to . . . to hurt him.\"\n\n\"That's right, honey. I never mean to hurt nobody. I just don't know what comes over me sometimes. Hey, I'm sorry I hit you before. Real sorry. Sometimes I don't know my own strength. But don't back out on me, baby. I need you. More now than ever.\"\n\nElva's heart went out to him. Poor Cudge, he just couldn't help himself. Any more than her father had been able to control his temper. Hadn't Mama always forgiven him? Hadn't Mama known that she was Daddy's very own salvation here on earth? Daddy had known it too. He always called Mama his own angel. Deciding she couldn't do any less for Cudge, Elva squeezed him hard. \"Just tell me what you want me to do. I'll do anything to help you, Cudge, you know that.\"\n\n\"Good girl.\" He answered her hug with a kiss on the cheek. \"Only don't go getting the idea you're doing it for me. It's for you too, baby. Christ, what would I do if they ever took you away from me because you're an accessory?\"\n\n\"Don't worry, Cudge,\" she said soothingly. \"Nobody will ever take me away from you.\"\n\nCudge Balog smiled and began formulating his plans for moving the remains of Lenny Lombardi into the camper.\nChapter 2\n\nListening to Sara's slow, regular breathing, Andrew knew she had fallen asleep. She had climbed into bed beside him after their lovemaking in front of the fire, placed her head on his shoulder and settled down. Now, as he lay beside her, he thought about their trip to Florida the next morning.\n\nNo need to worry about packing; Sara had seen to it, and much better than he could have done himself. They would be escorted by federal agents to the airport; and there was no concern about tickets or reservations; the government had seen to everything. They would simply board the plane and, once in Miami, go right to the courthouse. Their hotel accommodation was being kept secret even from them, so there was no possibility of a leak.\n\nNo matter how often Sara tried to reassure him, Andrew still felt uneasy for not coming forward to testify of his own volition. Nothing should have kept him from going to the authorities as soon as he'd learned that Jason Forbes's body had been discovered behind an all-night supermarket. Forbes had been only twenty, a promising student in Andrew's second-year physics class. While Andrew hadn't known the young man outside class, he had found him to be affable and to have an above average aptitude for higher mathematics.\n\nTomorrow Andrew would be asked to review his acquaintance with Forbes on the witness stand. There was little he could say beyond an impersonal recital of Forbes's class attendance and scholastic record. The prosecuting attorney wouldn't be looking for a personal history, Andrew reminded himself. He would want to know the details of the last time Andrew had seen Forbes alive.\n\nIt had been in May. The university library had been dim and cool, especially in the stacks where he was doing research in preparation for the coming week's classes. It had been quiet, so quiet one could almost hear the proverbial pin drop. With a scholar's contentment in the musty, hushed atmosphere, Andrew had gathered up the heavy physics texts. This was a little-used area and he had expected to spend the entire time alone at a table in the far alcove. He had been so immersed in his work that he hadn't even been aware of any noise until he heard angry voices. Curious, he had stopped to listen.\n\nThere were two voices: one with the unmistakable tenor of youth; the other harsher, older, more authoritative. They were arguing in hushed tones but their words were clear and distinct. The older voice was accusing the other of \"holding out . . . starting your own business . . .\"\n\n\"No!\" the younger man had protested. \"That was all I picked up. Honest!\" He continued to protest the accusation, his tone becoming more nervous, fearful and wheedling. Impelled by curiosity, and a vague recognition of the younger voice, Andrew had quietly closed his book and moved to the archway of the alcove.\n\nHe'd recognized Forbes immediately. His accuser was a man in his late forties, of heavy build, wearing a Hawaiian-print shirt. Andrew had never seen the man before but there was something menacing about him and the way he was leaning toward Forbes. He was a dangerous man, Andrew thought, one he wouldn't want to deal with himself. He wondered if he would have Forbes's courage in standing up to him.\n\n\"We know you're lying, kid. I've been told to tell you that you'd better get the rest of the stuff to us by ten o'clock tonight . . . or else.\" The man had jabbed his index finger into Forbes's shoulder for emphasis. \"We don't like kids who hold out on us. We know you picked up ten kilos, so how come you only delivered eight? At street prices that would make you a nice little bundle, wouldn't it, kid? Think about it, you've been playing around with the big boys and you'd better come across.\"\n\nForbes's complexion had turned pasty white beneath his Florida tan, and he'd choked out his words with difficulty. \"I don't have it, I tell you. You've got this all wrong.\"\n\n\"You've got it all right, kid. We know you do. Word got back to us about that little sale you made yesterday. Ten o'clock, kid, the usual place. Be there with the stuff. Oh, and don't try running home to mama. We've got our connections in St. Louis, too.\" With the speed of lightning, the man had punched Forbes in the stomach, the blow sending the young man gasping to his knees. Then he'd turned and strolled away, his heels clicking along the aisles of books.\n\nAndrew had hurried over to Forbes to see if he was hurt. As he stepped forward, Forbes staggered to his feet, rubbing his abdomen. A low whistle escaped him, and Andrew thought he heard him mutter a curse.\n\n\"Are you all right?\"\n\nThe young man stood frozen, staring at Andrew.\n\nAndrew had taken a step toward the student, but the forbidding expression on Forbes's face stopped him. Jamming his hands in his jeans pockets, Forbes walked out.\n\nFor the rest of the day, the scene he had witnessed replayed in Andrew's mind. That Forbes was involved with drugs was obvious. Florida was the U.S. entry point used by many drug smugglers; from there the drugs were distributed to major cities. There was also no doubt that Forbes had gotten hold of two kilos of whatever\u2014cocaine, heroin or marijuana\u2014and had kept it. The only uncertainty was whether or not he would return it. If he had made a sale the day before, did that mean he no longer had the drugs? What would happen to him if that were the case? The question had occupied Andrew throughout dinner, until Sara had complained. Finally, he had confided in her.\n\nThe following day, Forbes's body had been found behind the supermarket. Was that the \"usual place\" the older man had spoken of? The campus was thrown into turmoil by the murder, and an investigation was launched. Detectives and policemen scoured the campus for information. No one approached Andrew to question him; yet he was becoming increasingly uneasy. Two weeks passed and still he wrestled with the question of whether or not to come forward with his information. Then more information had come to light. Forbes had been found to be involved in drug trafficking, along with his roommate, Franklin Pell. Still, nothing in the newspaper reports indicated that the police were any closer to finding Forbes's killer. Andrew knew that the police rarely revealed all information to the press, yet every day he scanned the papers, finding out more about Forbes after his death than he had ever known about him in life. It became an obsession.\n\nIn the midst of it all, Andrew and Sara had had to respond to job offers made to both of them earlier that year by Montclair College in New Jersey. They had talked it over and agreed they would be fools to leave the \"Sunshine State\" for the long, cold Jersey winters. But now the opportunity to leave Florida and escape even the remotest involvement in the Forbes case appealed to them. They would leave as soon as classes ended in order to set up housekeeping and prepare for the fall semester.\n\nBy the middle of July, the Taylors had moved into an old Victorian house on the outskirts of Montclair. Andrew seemed to be his normal self again, and Sara was occupied with setting up housekeeping and redecorating. Even Davey seemed to adjust to the move with little difficulty; in fact, he thrived on the additional attention provided by Sara's sister, Lorrie, who lived nearby. September came, and with it a structured schedule, marshaled by Sara, of classes, study, chores, and school for Davey. Then the murder in Florida had caught up with them\u2014Andrew and Sara were picked up during their classes by federal agents. Franklin Pell had informed the police about the circumstances leading to his roommate's death, revealing that Forbes had told him that Professor Taylor had witnessed the confrontation in the library. Pell's testimony had leaked to the newspapers, but before the news could be picked up by the wire services, the government hastily arranged for the Taylors to be picked up and questioned.\n\nThe drug ring Forbes had been involved with was one of the biggest in the country. Andrew's testimony was vital in linking the syndicate to Forbes's death, and hence to other crimes. The government had been waiting for such a link\u2014it would be instrumental in breaking up a wide circle of corruption which it was determined to destroy. Without Andrew's testimony, the connection between Forbes and the syndicate would be weak, and if the government knew this, so did the killers. Even Franklin Pell had no direct contact with the syndicate. His only contact had been Forbes himself.\n\nWith awesome speed and thoroughness, the government put the Taylor family under twenty-four-hour guard. The syndicate's main objective would be to prevent Andrew from appearing in court\u2014an end most likely to be achieved the same way they had dealt with Forbes.\n\nSara had been wonderful throughout, dealing with the government interlopers with the same efficiency with which she ran the house and her career. Davey liked the men who followed him to school and stood outside his classroom even though he didn't understand exactly why they were there. Sara had deemed it unnecessary to tell him too much. It was more important, she said, that Davey saw his parents coping with the unusual happenings in their household and going about their lives as normally as possible.\n\nAndrew hated the news media most of all. They referred to his caution in coming forward with the information that could \"crack the most infamous drug ring operating in the United States\" as though he had deliberately held back. While he didn't have any illusions about being a hero, Andrew despised their use of the word \"caution.\" He knew it was a non-libelous euphemism for \"coward.\"\n\nSara turned over onto her side, sleeping deeply now, her blond hair fanned over the pillow. The sweet, round shape of her haunches pressed against Andrew and he smoothed his hand over her hip, following the line to the slim valley of her waist. He envied Sara her peace, wanted it for himself, wished he could find sleep instead of lying there thinking.\n\nA soft cough from Davey's room drew his attention. He smiled as he thought of his son. Andrew didn't share Sara's reservations about Davey's growing dependence on Lorrie, but he didn't disagree either. Sara, as his mother, was much closer to Davey and his needs than he was; he would rely on her instincts.\n\nAndrew and Sara had been married nearly fifteen years before Davey was born. They had been resigned to never having a child when the miracle was announced. Methodical thinkers both, they spent the time prior to Davey's birth discussing and rediscussing their ideas of child rearing. Happily, they'd found they agreed on almost every point. At the time, they had been teaching in a small college in upstate New York, their professional lives neatly blending with their home life. It had been an idyllic time, filled with scholastic achievements and music and love. And even though both were just past forty, they were certain that their long-awaited child could only enhance their lives.\n\nShortly after Davey's birth they had learned he was a hemophiliac. When he was nine months old, Sara had discovered a swelling near the base of his spine. When the doctor entered the examination room, the first thing he'd asked was, \"How long has he been this color?\" Andrew remembered how upset Sara had been, feeling she had been remiss, but neither one of them had noticed a change in the baby's complexion. Thinking about what had happened next still caused Andrew to break out in a sweat. After a preliminary blood test the doctor had informed them: \"This baby is dying. Get him to the hospital, fast! Your son is a hemophiliac and his condition is critical.\"\n\nThat immediate crisis had passed, but it had launched a whole new lifestyle for the Taylors, one predicated on preventing Davey suffering even the slightest injury. Simple things would send them rushing to the hospital emergency room\u2014bumps, bruises, cutting baby teeth, Davey biting his tongue when he fell taking his first steps.\n\nThose had been bitter days and Sara, in particular, had agonized over the situation. Hemophilia was a blood disorder passed from mother to son. Having no brothers or uncles on her maternal side, Sara had been completely ignorant of the fact that she carried the gene. She was burdened with a guilt that could never be overcome. It was her fault that Davey was imperfect. She had immediately planned a campaign to protect her son in every way possible. The baby's crib and play areas were padded. Expensive special shoes with rubber soles were purchased so Davey wouldn't slip. Occasionally braces were necessary to assure that his limbs grew straight. Each little episode bordered on catastrophe. In addition to dealing with the harrowing medical problems of hemophilia, the Taylors had to live with anxiety and uncertainty every day. And still accidents would occur. Because of Davey's young age, his tiny veins sometimes could not accommodate transfusion equipment, and he would have to be strapped to a hospital bed for hours.\n\nSara's strength of will kept the family on as even a keel as was possible. She could ease Andrew's mind and offer hope when there was none, holding out for the day when Davey could be put on an antigen program. Before researchers had succeeded in isolating the two anti-hemophilic factors\u2014Factor VIII and Factor IX\u2014patients with bleeding had to lie in bed while bottles of plasma containing clotting factors were dripped into their veins. Then pharmaceutical companies had developed a way to freeze and concentrate the factors, making it possible for patients to self-administer the drug with daily shots. Unfortunately, Davey had developed antibodies against Factor VIII. It took more and more of the concentrate to block his bleeding, and Sara and Andrew feared that he would die from his next injury or spontaneous bleed. So when he was accepted into the antigen program researching suppression of antibodies, his parents were relieved. As long as Davey received an uninterrupted daily dosage of antigen, he would be able to live a normal life.\n\nThat word \"uninterrupted\" still made Andrew uneasy. In this life of uncertainties, was that really possible? It had to be. At this point, Davey's hemophilia was controlled. But, by the very nature of the antigen, it was possible that Davey's own body defenses could reject the drug. Then it would be useless, and the only treatment that would be effective would entail hours strapped to transfusion devices. Sara and Andrew had been instructed how vitally important it was that Davey received the antigen at the same time every single day. Even one day without the injection could set up a chemical reaction in his body, whereby the drug would be rejected and rendered useless forever. There would be no turning back. Once a week, Davey's antigen level had to be checked; it was imperative that it be kept at a level which coincided with his growth.\n\nSara's sister, Lorrie, was a doctor; she realized the problem. So well, in fact, that she had made a decision not to have children, knowing the odds were against her since she, too, carried the hemophiliac gene. Lorrie had happily anticipated getting married and having a family until Davey's condition was discovered. After that she'd devoted herself to her pediatric practice.\n\nSara threw her arm over her husband's chest and Andrew nuzzled against her, aware of the fresh scent of her shampoo which lingered in her hair. Trying to set his worries aside, he settled into an uneasy sleep.\n\nCudge sat beside Elva on the daybed. He was certain he had used the right words to settle her down and make her help him. She'd come up with an idea for disposing of Lenny's body; though he'd never admit it, it was more than he could have done.\n\n\"Question is, how do we get him downstairs to the camper?\" Cudge touched Elva's cheek, brushing back a strand of hair. \"We can't just drag him down the stairs and pretend he's drunk. Not with his face all . . .\"\n\nElva's eyes scanned the room and came to rest on the three paper bags that held the laundry. A small bottle of fabric softener and a box of detergent were beside the bags. Frowning, she saw the rusty ironing board with its scorched, dirty gray cover leaning against the wall. She forced her eyes to go to Lenny's body and then back to the ironing board. \"I got an idea, Cudge.\"\n\n\"Yeah? What?\" He knew it was going to be something stupid but he'd better hear her out and make her feel important. This wasn't the time to fight with her and get her screaming loud enough to wake the dead.\n\n\"Like you said, we can't just take him down the stairs and pretend he's drunk or sick or something. That nosy old Mrs. Fortunati is awake and she already told me she was thinking of calling the cops because of the noise coming from our apartment. That must have been when you were hitting Lenny and\u2014\"\n\nCudge's hair almost stood on end; the black look in his eyes stopped Elva's flow of words. He clenched his hands to keep them from shaking. \"What did that old snoop say she heard, Elva?\" It was a monumental effort to keep his voice steady.\n\n\"Nothing. She just said it sounded like you were tearing the place apart.\" Elva saw him relax. That wasn't anything new with the landlady. She was always saying she was going to call the cops, and she even did once because the baby in 4B was sick and cried all night long. \"Anyway, I was thinking we could put Lenny on the ironing board, pile the dirty clothes on top of him and pretend we're going to the laundromat. Then you can put him in the camper and drive him away somewhere.\"\n\nCudge's eyes widened. \"Sometimes you ain't so stupid after all. It just might work. Hey, you look like you're gonna be sick. If you gotta puke, do it in the toilet.\"\n\n\"I ain't gonna puke. I just don't like seeing dead people. It reminds me of . . . They look like chalk, and they don't move no more. Not ever.\"\n\nThe regret in her voice was lost on Cudge who was busy putting the ironing board next to the body to measure it. \"He'll fit. Come on, don't just sit there, give me a hand.\"\n\n\"Cudge, I . . . I can't touch him,\" Elva cried.\n\n\"Move it, Elva!\"\n\nElva shuffled over to the body and bent down to grasp the legs while Cudge took hold of it under the armpits. Lenny landed with a thump on the rickety board. Elva backed away, her hands going to her mouth to stifle a retch.\n\n\"We'll tie him on it. I don't want him rolling off in front of Mrs. Fortunati's apartment door.\"\n\n\"I don't like that Mrs. Fortunati. She's got little beady eyes that see everything.\"\n\nCudge ignored Elva and began to heap dirty clothes over Lenny's body. Beads of perspiration dotted his upper lip as he leaned back against the table. \"Okay, that was the first step. Now we gotta clean out this place and make tracks. Get your junk together.\"\n\n\"I got it all in a paper bag. The rest is on top of your friend. Now what?\" Elva asked, moving as far away as possible from the ironing board.\n\n\"Get everything together. We ain't coming back here ever again. Take all the food and then clean up this mess.\" He gestured to the pool of blood under the table. \"Make damn sure you do a good job. First thing I'll take down to the camper is the TV.\"\n\n\"Cudge, I hate blood. It makes me sick. I can't do it!\"\n\n\"You're going to do it and you'll do it now, before I punch a hole in that thing you call a head. Move it!\"\n\n\"I always get the shitty jobs,\" Elva protested as she kicked at a filthy dish towel. With the toe of her shoe she picked up the rag and dropped it into a supermarket grocery bag. The towel was so threadbare it barely soaked up any of the blood. Not wanting to be alone with Lenny in the kitchen, she raced to the bathroom and waited till Cudge came back into the apartment. A roll of toilet paper in her hand, she walked back to the dingy kitchen. She unrolled the sheets and wiped the mess up with her foot. Satisfied that the blood had been wiped up, she poured a glass of water on the floor and repeated her actions. It was kinda sad, she thought, one roll of toilet paper was all it took to wipe up a man's life.\n\n\"You got everything?\" Cudge asked belligerently.\n\nElva was tossing food from the refrigerator into a paper bag. \"Should I take the eggs?\"\n\nHe rolled his eyes. \"Yes, you should take the eggs,\" he mimicked. \"Take everything. Come on, we ain't got all night. We'll take him down first, but I want to make sure the coast is clear. I left the back of the camper open\u2014all we have to do is stuff him in.\"\n\nElva gritted her teeth before picking up her end of the ironing board. \"Wait! We have to put the detergent and softener on top to make it look real.\"\n\n\"Christ, Elva, we ain't really going to the laundry. Leave the damn stuff.\"\n\nElva was not to be deterred. The detergent and fabric softener were plopped on top of Lenny's stomach. Halting abruptly in mid stride, Elva's voice was a high-pitched stuttering squeak. \"You can't, you just can't . . . We have to spray him with something.\"\n\nCudge's fists were white-knuckled tight. \"Why?\"\n\nElva gulped. \"Be-because he'll smell. Dead bodies smell. They start to . . . to rot or something. I'm telling you what to do\u2014I didn't say I knew what to use,\" she blurted. Her toothache was pounding away like a trip hammer.\n\nCudge stared at Elva. His voice was almost patient. \"I ain't exactly planning on carrying Lenny around for very long. I don't think he'll have a chance to smell.\"\n\n\"Soon as he gets stiff, he'll smell.\"\n\nCudge hated the certainty in Elva's voice. \"We don't have anything around to spray him with. Come on, grab your end.\"\n\n\"What about . . . what about the mothballs in the bottom of the sink? That's enough to kill any kind of smell. You could stick some in Lenny's coat pockets.\"\n\nIt was evident to Elva that Cudge was going to go along with her idea by the way his gaze shifted to the bottom of the sink. She darted between the table and the body. Her skinny arm was trembling so badly that Cudge jerked the container of mothballs from her hand. \"This better work, you dizball.\"\n\nElva backed away till she was standing in the dingy living room. Cudge sneezed four times in rapid succession as he stuffed the white pellets into Lenny's pockets. \"Okay, he's preserved now. You got any more crazy ideas, now is the time to spit 'em out. I ain't planning on touching him again. Let's go. Get back over here\u2014you think I can do this myself?\"\n\nElva advanced one step then backed up two. \"I can't, Cudge, I just can't do it,\" she whined.\n\n\"Listen, you're the one who came up with this whole idea. Now grab hold of the damn thing and let's get him down to the camper.\"\n\nShe stared at Cudge. As usual he was right. The ironing board and the mothballs had been her idea. It never occurred to her that if Cudge hadn't lost his temper and killed Lenny, she wouldn't be standing here now, ready to cart a dead body down to the street.\n\n\"Grab hold, and God help you if you let him slip,\" Cudge snarled.\n\nElva shivered as she picked up the narrow end of the board. Her hold secure, she stopped again, the wide end of the board jamming Cudge in the small of his back.\n\n\"What now?\"\n\n\"The detergent and bottle of softener,\" she explained. With a mighty effort, she reached to the top of the sink.\n\nAll the way out of the apartment and up to the top of the stairs, Lenny's body jounced with each step they took. Elva tried to look anywhere but at the ironing board. Halfway down the stairs she spotted the feathery trail of soap powder spilling onto the steps. Maybe if she didn't say anything, Cudge wouldn't notice. She repeated the thought over and over again, thinking about the trailing powder, remembering the fairy tale about the children who left a trail of bread crumbs in the forest so they could find their way back home. Elva St. John had had a home once, but something told her she never would again.\n\nA sound like rain pelting on a roof made Cudge stop at the fourth step from the bottom. Elva's eyes popped open as she toppled over the ironing board. Cudge lost his hold and the load slid down the remaining steps. Bile rose in Elva's throat and soured her mouth. \"Oh, my God, quick, pick him up! Somebody might hear! Pick him up!\"\n\nCudge moved swiftly to right the board and rearrange the laundry on top of its burden. \"I guess you know that funny noise was those damn mothballs falling out of his pockets. Run around here and get that front door open.\"\n\nElva obeyed, never letting her eyes rest on the bundled board. Cudge grimaced with effort as he dragged the board forward, holding the door open with his shoulder. \"Get around and grab the other end. You and your bright ideas.\"\n\nHuffing at her end of the board, Elva whispered, \"I ain't even sure they'll work in his pockets anyway. Maybe you should've stuffed them in his ears.\" She could almost see the stubble stand on end on the back of Cudge's neck.\n\n\"Either you shut up, Elva, or you're gonna be in the back with Lenny, and it's you who's gonna get mothballs in the ears. Dumb shit!\"\n\nHis voice was tight and choked. Elva smiled to herself. She could risk being smug. With her own eyes she'd seen Cudge gag while he was stuffing the pellets into Lenny's pockets. It was nice to know that Cudge was afraid. \"I'm just trying to help,\" she whined. \"I keep tellin' you, I ain't smart. I just know things.\"\n\n\"Right the first time,\" Balog grunted, bearing almost the full weight of the board and Lenny down the front steps and out to the curb. \"Okay, now when I say 'shove' you push your end in. You got that?\"\n\nElva let her breath out in a sob the minute the camper was closed and the top half lowered. She could imagine poor Lenny squashed in there, in the darkness. \"Cudge?\" she said hesitantly.\n\n\"Jeez, what do I have to do to make you shut up? What?\" His tone was hushed, little more than a whisper instead of his usual yelling. Elva liked that; it was the one good thing that had happened all night.\n\n\"Should we go back and pick up the mothballs?\"\n\n\"I'm going back in to get the food and make sure you didn't leave anything. You wait right here and don't go getting any ideas about taking off. You go when I say you go. Got that? And quit your worrying about the mothballs.\"\n\nIt was spooky sometimes the way Cudge could read her mind. No matter what he said, she wasn't getting into the cab of the pickup until he was right there beside her. She didn't know what hurt her more\u2014her shoulders from carrying Lenny down two flights of stairs, or her tooth.\n\nThe streetlight seemed friendly as Elva leaned against the pole. Her thoughts, however, were gray. She'd been this close to death before, only this time it was easier. She didn't really care about Lenny Lombardi, not the way she had about little BJ. She couldn't cry for Lenny\u2014she had spent all her tears on her little brother. And she didn't feel guilty either. She hadn't even been there when Cudge killed Lenny. Not like with BJ when, if she hadn't been so scared for herself, she might have been able to do something to save him.\n\nThe worst behind him, Cudge looked around the apartment and felt nothing except relief. There were no visible signs that anything had happened in the kitchen. He walked through the apartment, taking his time to see that nothing had been left behind. If\u2014and it was a big if\u2014they had to come back for any reason, they could just walk in the door. The rent had been paid for the next month; there was no reason why anyone should come nosing around.\n\nCudge made his way down the dim stairway. The mothballs littered the filthy stair treads and, unconsciously, he counted them. When he reached the fourth step from the bottom, he stopped and inched his way closer to the greasy wall. It hit him then. He was a murderer. He felt like a land mine ready to explode. A grin spread across his face, easing the tension between his shoulders. He was the only one who knew he was a murderer\u2014he and Elva. Nine mothballs. Wiseass Elva would have to go. It would be his secret then; no one would ever know. One balled fist smacked into the other.\n\nHe was still grinning when he loped down the steps to the sidewalk. The camper looked just fine, and Elva was leaning against the telephone pole. \"Get in, Elva, and don't you so much as blubber. Not even once, hear me?\"\n\nElva's toothache was getting worse. If only she could fall asleep and wake up when the pain was gone. If there was someone who could promise her that, she might give up one of her Elvis tapes.\n\n\"I'm gonna drive around for a while till I decide what to do with Lenny. You keep your eye peeled for cops, or anything that might get us into trouble.\"\n\n\"Like what?\" Elva whispered. \"I want to know so I don't get us into trouble.\"\n\n\"Like some goddamn jerk trying to mug us as we drive around. I have to think, so watch with both your eyes. I know that trick you use sometimes, when you close one eye and stare with the other. Maybe you think it's sexy, but you look like a ghoul. Now, start looking.\"\n\nCudge drove carefully, his eyes alert. While he was searching the apartment, he'd toyed with the idea of burying Lenny in the Watchung Mountains. It was as good a place as any. There wouldn't be too many people on the steep mountain roads. The trees were thick and dark around Bernardsville, and he had more than enough gas to get him there and back. Back to what? He couldn't think that far ahead.\n\nAs he drove along, his thoughts were confused. It seemed like he was always in trouble of some kind. When he was a kid, he had been in one scrape after another, but he'd always managed to save his hide at the last moment. So far, the cops had nothing on him except a few drunk driving charges. He had been deprived. He was being deprived now. Pity was drowning out all reason. Pity for himself, pity for the circumstances that controlled his life. When he was a kid, he'd never had anything, except the clothes on his back and what he could steal. Now he was a man, but still he had nothing.\nChapter 3\n\nLorrie Ryan maneuvered the small motor home around a bend. She enjoyed driving the RV, found it relaxing.\n\nShe had been looking forward to spending time with Davey for weeks now. She'd gotten Dr. Petti to cover for her. He was a good pediatrician and his small patients liked him, to say nothing of their mothers. Tall, dark, and handsome Douglas Petti. Too bad he was already spoken for. Lorrie was at an age where most of the men she met were already spoken for, or divorced, or losers.\n\nHer thoughts turned back to the camping trip ahead of her. She'd made a dozen cupcakes for her and Davey to share. Not the best diet in the world, but Davey did so love them and this trip was special. If he wanted, she would serve them for breakfast, lunch, and dinner. Davey had a sweet tooth, just like her. She knew she'd better not let Sara know that cupcakes were on the menu for all of Davey's meals. Sara followed a strict regimen in the kitchen, even down to how many times Davey chewed his food.\n\nSara was okay, Lorrie told herself, she was just a little overorganized. Even as a child, Lorrie could remember her older sister saying \"A place for everything; everything in its place.\" Today, that included Andrew and Davey, she thought.\n\nSara was Davey's mother and of course she wanted what was best for her son. Lucky Sara, she'd got the best material to work with. Davey was the most remarkable, resilient child Lorrie had ever known. Lorrie looked ahead, anticipating a left-hand turn several blocks on. Andrew was the one who surprised her, she admitted to herself. She'd always thought of him as the absentminded professor, oblivious to everything except mathematical theories\u2014a little sloppy, disorganized, terribly forgetful. The fact that he didn't balk at Sara's \"organization\" surprised her. Andrew was a good guy, a dear really, even if he was just a little too comfortable and secure. That was Sara's fault. Lorrie didn't think it would even occur to Andrew to rock the boat, to make a fuss. He seemed to think that if he was happy, everyone else must be too.\n\nUnfortunately, that didn't hold for his son. Davey had been over-protected, over-organized and raised strictly by the book. Lorrie remembered how Sara hadn't been able to keep a nursemaid while she was teaching at the college. No one was able to adhere to her stringent schedules. Time to get the baby up, time to feed him, time for a bath\u2014Christ! The woman was a living automaton. Lorrie wanted to blame it on the fact that they came from a big family and that every time their poor mother even tried to set a time for dinner, she had a mutiny on her hands, but the fact was, Sara was different. Lorrie knew that Sara didn't approve of her sister's influence over her son. Andrew was a great guy, but he wouldn't stand in Sara's way if she suddenly decided to put an end to Lorrie's relationship with Davey.\n\nLorrie didn't understand Sara. Never had, and probably never would. The fifteen-year gap in their ages had caused a gulf of understanding that couldn't be bridged\u2014until Davey, that was. Davey had brought them together. His medical condition had forced Sara to turn to Lorrie for help and answers.\n\nCarefully executing the left-hand turn, Lorrie glanced in her side mirror. All this business with the drug racket in Florida had put a crimp in Sara's style. Especially since it was plastered all over the news. It was difficult to picture Andrew as the lead witness. A hostile witness, since he had to be subpoenaed to come forward. It was going to be a rough few days for him, she was sure. He wouldn't like being cross-examined on the witness stand. Lester Weinberg was representing the syndicate, and he wasn't exactly a pussycat lawyer. When he had Andrew on the stand, he would do whatever he could to rip apart Andrew's testimony. There was a lot at stake\u2014more than just some thug threatening a kid entangled in the rackets. That thug was a known connection with the syndicate, and if the government could pin the murder on him, it would lead to other important convictions.\n\n\"Poor Andrew,\" Lorrie whispered aloud. \"And poor Sara. She must be fit to be tied.\" She wondered how Sara was coping with all those FBI agents underfoot; Sara who didn't even like live-in help. During a recent phone conversation, Andrew had told her that Sara had had one of those agents removed from the case. The man had removed his jacket in front of Davey, exposing his shoulder holster and gun. Lorrie couldn't say she blamed Sara for getting angry, but she could have cut the man some slack and just asked him to keep his jacket on. But that was Sara\u2014unbending, unyielding, unforgiving.\n\nLorrie would have liked to make excuses for her sister\u2014after all, Sara only wanted what was best for Davey. But she went too far. She demanded perfection from people. And, if they didn't live up to her standards, she cut them down ruthlessly, without conscience, all in the name of what was best for her son. And for Andrew, too, now that Lorrie thought about it.\n\nLorrie had a good idea of where she stood in Sara's eyes and had learned to accept it. One of her friends, a staff psychologist, had suggested that Sara might be jealous of anyone besides herself having a place in Davey's or Andrew's life. He'd warned her to be careful, not to let herself get hurt, no matter how much Davey meant to her.\n\nTears welled in Lorrie's blue eyes as she thought about the love she felt for Davey. She didn't want to steal Davey away from Sara; she only wanted to have a part in his life, to be his favorite aunt, even if she was his only aunt. For all that Lorrie felt sorry for Andrew and Sara and all they were going through, she was glad for the chance to spend some time alone with Davey. She would love him and pamper him as he had never been loved and pampered before. And maybe, just maybe, after all this was over and done, Sara would consent to letting her take him someplace again.\n\nDavey kept vigil at the front windows, waiting for the first sight of his aunt's motor home coming down the street. It turned the corner, a white and green RV with a wide expanse of tinted windows, and gold decals that proclaimed it was \"King of the Road.\" \"She's here! She's here!\" he shouted excitedly, hurrying into the dining room with Duffy chasing at his heels.\n\nAndrew and Sara Taylor sat at the table with their morning coffee. \"Well, get over to the door and help Mr. Sanders let her in, Davey,\" Andrew said, smiling as he put down his mug. \"Mr. Sanders knows she's coming.\"\n\nDavey raced to the front door, almost running headlong into Stuart Sanders who was at his usual post in the foyer.\n\n\"Hey, little buddy, what's your hurry?\" Sanders asked, well aware of Davey's excitement about his first camping trip.\n\n\"Aunt Lorrie is here to take me camping and to the zoo! I'm going to sleep in the RV and cook outside and everything!\" Davey's bright blue eyes shone, and a heightening flush colored his little-boy cheeks.\n\n\"So I hear,\" Sanders said jovially, keeping a professionally watchful eye on the RV as it pulled into the Taylors' wide driveway.\n\n\"Let her in, Mr. Sanders, she's taking me camping!\" Davey pleaded.\n\nSanders had his keys ready to insert into the double locks as soon as Dr. Ryan reached the door. As a matter of routine, he had already unsnapped the flap on his shoulder holster although, as per his instructions, it was done in such a way that Davey didn't notice.\n\nLorrie knew her every movement was being watched as she climbed the steps to her sister's front door. What a way to live! Constantly monitored and watched, unable even to open the front door without a security check. Hearing Davey's voice on the other side of the door, she didn't bother to knock.\n\n\"Hurry up, Mr. Sanders,\" Davey was urging.\n\nThe moment the barrier was gone, the little boy threw himself into his aunt's arms, hugging her tightly. \"Can we go now? Huh? Can we?\"\n\nLorrie laughed as she knelt to hug her nephew. \"Yep, in just a few minutes. Are you all packed and ready to go?\"\n\n\"I've been ready since before it got light.\" Davey sniffed the air. \"What's that smell?\"\n\n\"Smell? What smell?\" Lorrie asked, sniffing the air as Davey had done. \"I don't smell anything.\"\n\n\"I do,\" Stuart Sanders said. \"I believe it's your perfume, Dr. Ryan. Chanel Number Five, isn't it?\"\n\nLorrie released Davey and stood. \"Why yes, as a matter of fact it is, Mr. Sanders,\" she said. \"Does your wife wear it?\" she asked, fishing.\n\n\"I'm not married.\"\n\n\"Oh, I see . . .\"\n\n\"I worked a case once that involved a perfume manufacturer,\" Sanders said, as if to explain.\n\nLorrie nodded. In spite of his stern expression, Stuart Sanders was a very good-looking man. She'd noticed him the last time she'd come to see her sister and, though he hadn't spoken more than a few polite words, she'd found herself discreetly watching him, wondering about him.\n\n\"Where are your mom and dad, Davey?\" Lorrie asked, looking around.\n\n\"In the dining room. Here comes Mom now,\" Davey said quietly, moving back. Duffy scampered between his legs, knocking him slightly off balance.\n\n\"Whoa there, young man.\" Mr. Sanders grabbed his shoulder and held firm until Davey was steady on his feet.\n\nSara Taylor's eyes held a guarded look as she walked toward her sister and the FBI agent.\n\n\"You're right on schedule,\" she said, offering her cheek for Lorrie's kiss. \"Come and sit down for a few minutes.\"\n\n\"But Mom . . .\" Davey whined with impatience.\n\nSara rested her hand on top of his head. \"Your aunt and I have a few things to discuss before you go, Davey. You go on up to your room and get your bag.\"\n\n\"Okay, but hurry!\"\n\nLorrie followed her sister back to the dining room.\n\n\"Lorrie! You look like a real camper, decked out in Wranglers and all.\" Andrew stood up and gave his sister-in-law a quick kiss.\n\n\"I am a real camper, Andrew. I've even got my American Trails card.\" Lorrie sat down and looked toward the stairs. \"Davey's really excited, isn't he?\"\n\n\"That he is!\" Sara answered. \"I only hope he's this happy at bedtime. He's never spent the night away from us before, except for hospital stays.\"\n\nLorrie reached across the table and patted her sister's hand. \"Now, don't go worrying. I'll take good care of him, and if he has any problem tonight he can just crawl in the bunk with me.\"\n\nSara pulled her hand away. \"I don't want you taking Davey into bed with you, Lorrie,\" she said flatly. \"Andrew and I have never done that and, frankly, we find it unacceptable.\"\n\n\"Hey, okay,\" Lorrie agreed, trying to make it sound like it was no big deal. \"If that's the way you want it, I don't have a problem.\"\n\nAndrew leaned toward his wife. \"It might be rough on him, Sara, being away from home for the first time. What harm\u2014\"\n\n\"No, Andrew,\" Sara interrupted. \"I won't hear of it. Once you start something like that . . .\" She shook her head.\n\nLorrie smiled. \"I understand, sis. If you say no, then it's no. I really don't anticipate any problem anyway. He'll probably be so worn out, he'll go to sleep right after supper.\" Lorrie hated to ingratiate herself with her sister this way, but she would do anything to prevent Sara from coming between her and Davey. A familiar sadness crept up on her as she thought of the children she would never have because of the gene she carried.\n\nAs if reading her thoughts, Sara moved away from the table, returning with a small box which she set before her sister. \"Here's Davey's medication. Keep it refrigerated. His usual time is at noon. That makes it easy to remember.\" A hint of a frown drew her brows together.\n\n\"Noon,\" Lorrie repeated. \"I won't forget.\"\n\n\"I know I shouldn't worry, but I do,\" Sara said. \"Davey's been doing so well, and his doctors say it's mainly due to the strict regimen we adhere to. Last month he had another test for antibodies and he's still very low. As far as we're concerned, that's great. He could possibly go on using the AHG for a long time to come.\"\n\nLorrie had made a point of keeping up on the latest research. Approximately eight percent of all hemophiliacs would develop an antibody with specificity against Factor VIII, making it less effective in stopping the bleeding. AHG was more or less a replacement for the factor that Davey's own body didn't manufacture. By adhering to a strict daily time schedule, it was more likely that the antibody buildup could be avoided. In Davey's case, it seemed to be working and, with any luck, he would be able to live an almost normal life, free of the spontaneous bleeding beneath the skin and into the joints that resulted from what was, for most kids, minor trauma\u2014a spill from his bicycle or a skinned knee. Without AHG, the antibody levels would shoot up, especially if there was a disproportionate amount of stress present. That's when all systems were go, as far as the body was concerned\u2014the adrenaline flow increased and the heart pumped faster. AHG was touted as a miracle treatment, as long as the antibodies didn't develop to cancel out the new, refined Factor VIII.\n\n\"Well, you don't have to worry about Davey this weekend. I'll take good care of him.\" Wanting to change the subject, Lorrie pulled a brochure out of her pocket. \"I've got some really great plans for the next two days,\" she said, sliding the brochures toward her sister. \"I thought we'd spend most of today at the Philadelphia Zoo, then come back into Jersey and spend the night at my favorite campground. It has all the modern conveniences,\" she added. \"Then tomorrow morning, I'll take him to New York City. I've got tickets to the laser show at the planetarium. Then back to the campground for tomorrow night, and home Wednesday afternoon. What do you think?\"\n\n\"Sounds great!\" Andrew interjected before Sara could find fault with the plans. He knew she would worry about Davey becoming over-tired and over-excited.\n\n\"Yes, I . . . I suppose that will be all right.\"\n\n\"Of course it is, honey,\" Andrew said. \"Davey will love it. Only one word of warning, though,\" he added, looking at Lorrie. \"Don't tell Davey about the planetarium until tomorrow or he won't sleep through the night. Ever since he saw the advertisement for the laser show on TV, he's been begging me to take him and, well, with everything that's been going on around here lately, I haven't been able to take him anywhere.\" Andrew smiled and shook his head. \"All your plans sound like so much fun I feel like playing hooky and coming with you.\"\n\n\"Oh, no you don't,\" Lorrie scolded her brother-in-law playfully. \"This is my chance to have Davey all to myself and you're not invited. I intend to treat that kid as if he was my own. Better.\"\n\nSara straightened her spine. \"I suggest you remember that Davey's our child, not yours, Lorrie,\" Sara said, avoiding Andrew's warning glance. She knew he didn't think this was the right time to tell Lorrie they intended to reduce her time with Davey.\n\nLorrie paled, her sister's cruel words cutting her to the quick.\n\nAt a sound from the doorway everyone turned. Davey was standing there, Duffy clutched in his arms. The little boy's face looked crestfallen.\n\n\"What's the matter, son? Not having second thoughts, are you?\" Andrew teased.\n\nDavey sniffed. \"It's Duffy. She doesn't want me to go. Look at her eyes, she's been crying.\"\n\n\"Duffy will be fine in the kennel,\" Sara told her son.\n\n\"I don't want her to stay all alone in a kennel. I want her to go with me!\" He glanced shyly at his aunt, feeling like a crybaby yet unable to help himself.\n\nLorrie smiled warmly. \"Come here, Davey.\" Slowly Davey walked over to her, Duffy still held tightly in his arms.\n\n\"That's one sorry-looking dog you've got there. I think you're right. She doesn't look to me as though she'd last a day without you. We'd better take her with us.\"\n\nImmediately Davey brightened. \"Really? Did you hear that, Duff? Aunt Lorrie said you can come with us!\" He set her down. \"I'll go get her dog food,\" he said, turning to leave. \"C'mon, Duff. We need your leash and your dish, too.\" Davey scampered off to the kitchen, leaving his aunt to bear the weight of his parents' remonstrances.\n\n\"Lorrie, are you sure you want to do that?\" Andrew asked, looking doubtful.\n\n\"Where will Duffy stay while you're at the planetarium and the zoo?\" Sara questioned, not liking a last-minute change to the plans she had carefully made.\n\n\"In the motor home. She'll be fine. I should have thought of it before.\"\n\nDavey ran through the dining room carrying Duffy's red plastic dish, a metal-link leash and a bag of dry dog food. He headed for the front door with Duffy scrambling after him, yipping at the sight of her leash. \"I'm going to tell Mr. Sanders that Duffy's going with us!\"\n\nDavey dropped the bag of dog food and the rest of Duffy's possessions next to the front door. \"Duffy's coming too!\" he told Stuart Sanders.\n\n\"Is that right? Good for Duffy! Hey, Davey, come over here. I've got something for you.\" Sanders withdrew something from his pocket. \"It's a penlight, just like the Boy Scouts use.\" He flipped the switch and displayed the light against the palm of his hand.\n\n\"Gee! Thanks, Mr. Sanders.\"\n\n\"And see, it has a chain attached to it so you can hook it through your belt loop, like this.\" Sanders attached the penlight to Davey's narrow belt loop and dropped it into the boy's pocket. \"For your very first camping trip.\"\n\nDavey's smile was the agent's reward. Shyly the youngster reached out to shake hands, just the way his mother had taught him.\n\nLorrie called Davey back to the dining room and held him on her lap, the brochure spread out on the table in front of them. She pointed to the pictures of the campground where they would be spending the night.\n\nDavey pored over the pictures, then looked up. \"Mom, you said I could call Digger on the CB to say goodbye. Can I tell him about my camping trip, Mom, can I? I want to tell him that we're staying at a campground right near Wild Adventure Park, and that at night you can hear the lions roar.\"\n\n\"Yes, Davey, you can call him but don't be long.\"\n\nAfter Davey had left the room, Andrew turned to Lorrie. \"That CB was a godsend. Best thing you could have given him. He loves it.\"\n\nLorrie smiled, pleased that Andrew was pleased. \"Well, I knew Davey would be cooped up in the house for a while because of the trial and all. I'd remembered one of my patients, a little boy with severe physical limitations, talking about a Junior CB Club and how much fun he had talking to other kids.\"\n\n\"Davey would be on it all the time if we let him,\" Andrew said. \"But Sara and I explained that operating a citizen's band radio is a responsibility, and I'm proud to say that Davey doesn't abuse it. One half hour each afternoon, that's all. After all this is over, we promised Davey he could meet this 'Digger' in person.\"\n\n\"Digger? That's a funny handle. I've heard some of them are outrageous. What's Davey's?\" Lorrie asked.\n\n\"Panda Bear,\" Sara answered, long white tapered fingers shredding the paper napkin near her cup. \"It's the only one he came up with that Andrew and I would agree to. You should have heard some of the others.\"\n\n\"I'm so glad you approve of Davey belonging to the Junior CB Club, Sara. I wasn't sure you would.\"\n\nSara faced her sister. \"I don't approve of some of the 'lingo' that the CBers use. It goes against my grain. Whatever happened to the Queen's English?\"\n\nElva and Cudge sat stiffly on the front seat of the Chevy pickup truck, fully aware of Lenny Lombardi's body behind them on the floor of the pop-up camper.\n\nCudge kept a careful eye out for traffic signals and speed limits. This was definitely not the time to attract the attention of some cop who might demand an inspection of the camper. Cudge damned himself for painting the truck with wild colors and applying decals that just barely came within the limits of the law. Cops were always stopping vans and pickups, looking for drugs, and the wild decorations on the old Chevy just begged for police to get nosy, if only on principle.\n\nThe CB on the dashboard emitted its usual static, picking up a trucker here and there. He intended to keep the CB going all the while they were on the road, regardless of Elva's whining protests that she wanted to listen to her Elvis cassettes. If there were speed traps or any cops in the vicinity, he wanted to know while he could still do something about it.\n\nThey headed west out of Newark, traveling city streets. Elva had never seen Cudge drive so carefully. She knew better than to complain about the static coming from the CB that was giving her a roaring headache. This was no time for Cudge to lose his cool.\n\nAbout nine o'clock they were going through the town of Montclair. As they entered the town, the transmission on the CB became clearer.\n\n\". . . that sounds great, Panda Bear. When will you get back? Over.\" A click.\n\n\"We'll be back Wednesday, Digger. How long will you stay in the hospital to get your legs fixed? Over.\" A click.\n\n\"Don't know, Panda Bear. Where are you camping? Over.\" A click. The second young voice sounded louder, clearer, coming through the speaker.\n\n\"Down at a campground near the Wild Adventure Amusement Park. Aunt Lorrie says you can hear the lions roar at night. Over.\"\n\n\"Too bad the park isn't open yet,\" Digger said. \"I know you'd like the Ferris wheel they have there. Over.\" Click.\n\n\"That's what Aunt Lorrie says is so great. When the park is open, it's so crowded you can never get in; this time of year there won't be hardly anybody there. It'll be like being out all alone in the woods. Over.\"\n\n\"Have a great time, Panda Bear. Wish I could go with you rather than to the hospital. Over.\"\n\n\"Digger? I hope they fix your legs this time. Over.\"\n\n\"Yeah, me too. Gotta go now, Panda Bear. Have a great time for me. Out.\"\n\n\"I will, Digger. I know I will! Out.\"\n\nThe transmission ended and the static sounded again.\n\n\"Get that map out of the glove compartment, Elva,\" Cudge instructed, keeping his eye on the road. \"Find out where that Wild Adventure place is.\"\n\nFederal Bureau of Investigation Special Agent Stuart Sanders stood outside Davey's door. He could hear the excited voices on Davey's CB. He still had some time. He would wait till the conversation was over before he said goodbye.\n\nThe agent looked into Davey's immaculate room, so different from his nephews' bedrooms. Where were the personal touches? Everything was expensive, from the porcelain clowns on the shelf to the leather-bound editions of Dickens. An interior decorator might find it in perfect taste, but to Sanders it seemed like a model room in a department store. Where were the bits and pieces of games that were standard in a kid's room? Where were the stubby crayons in coffee cans? Or the slipper without a mate? Where were the comics and storybooks that took a kid on trips to fantasy land? He knew if he looked under the bed he would find nothing but a clean, vacuumed carpet. There would be no forgotten toys, no dirty pajamas, no pennies. Just a clean floor. As he watched the Yorkshire terrier nip at Davey's pant leg he thought, at least Duffy was real. Davey was real.\n\nIt was impossible, but the bed was already made. It annoyed him. His sister's kids never made their beds till five minutes before their mother came in from the office, and then all they did was throw the comforter up over the pillow. There was no life in this room. Sanders had a sudden urge to scoop up the dog and the kid and take them to his sister's house with the noise and disorder that four boys made. Nancy would take one look at Davey and cradle him in her arms and croon some indistinguishable words that only kids understood. Nancy called it \"mother magic.\" Disgust washed through him. This room, this kid, could use a double dose of Nancy.\n\nThe minute Davey broke his connection Duffy barked to let Stuart know it was all right to enter the room. Walking in, he held out his hand. \"I came to say goodbye, Davey.\"\n\nManfully, Davey extended his hand. \"It was nice of you to come up to see me and Duffy, sir.\"\n\nStuart Sanders squatted down till he was at eye level with the little boy. \"Listen, tiger, I hope you have a whale of a good time on your camping trip. Take good care of this bundle of dynamite,\" he said, motioning to the dog.\n\n\"I will, sir. And, sir, thanks again for the flashlight. I have it in my jacket pocket so I can use it whenever I want.\"\n\nStuart stared into Davey's bright eyes. Despite the room, the kid would be all right. Stuart felt it in his gut. His eyes dropped to the needle marks on Davey's arms.\n\nDavey grinned when he noticed Stuart staring. Playfully, he punched the agent on the arm. \"I almost don't need them anymore.\"\n\nSuddenly, Stuart didn't want to leave. He had never in his life believed in premonitions or anything supernatural, but he was experiencing something now. And the dog, the dog was staring up at him with what Stuart would later describe as questioning eyes. \"Davey, I'm going to give you a card that has my cell phone number on it. If you ever want to talk to me, or if you think I might be able to help you sometime\u2014you know, if you ever join the Scouts or something like that\u2014you call me. It doesn't make any difference what time it is, day or night. If I don't answer, you just leave a message and I'll get right back to you.\"\n\n\"Sure, Mr. Sanders,\" Davey said, taking the business card and sticking it in his hip pocket. \"Maybe you should give me two, one to take with me and one to leave here in my desk.\"\n\nStuart got to his feet. He didn't know why but it was important for him to make the kid promise. \"Okay, but first I want you to promise me that you'll call me if you need me.\" Deliberately, he made his tone light.\n\nDuffy barked sharply and Davey's tone was solemn. \"I promise, Mr. Sanders.\"\n\n\"Listen, tiger, you got any money on you?\"\n\nDavey shook his head. \"Mom and Dad said I don't need any money.\"\n\nSanders winced slightly. \"I'm sure they're right, but when I was a kid, my dad always gave me some pocket change to carry around in case I needed to make a phone call and all I could find was a pay phone. I think you should have some. Here, take these three quarters.\"\n\nDutifully, Davey accepted the three quarters and stuffed them down into his pocket with the business card. The jangle sounded nice.\n\n\"As long as you have money in your pocket, you're always just a phone call away from having someone to talk to. I have to go, tiger. You and Duffy take good care of each other, okay?\" He choked a little as he tousled Davey's hair.\n\n\"Got everything, sport? Ready to go? Your mom and dad have a plane to catch to Florida and we can't hold them up.\" Although Davey was aware that his parents were flying to Florida, Lorrie knew he wasn't aware of the reason. He had been told it was on school business.\n\n\"Everything's ready. Right, Mom?\"\n\n\"You'd better be on your way, son,\" Andrew said, then turned to his sister-in-law. \"His gear is on the front porch. Sanders took it out a while ago.\"\n\nSara, issuing last-minute instructions, was still talking when she pecked Davey on the cheek. Her hand lingered a moment on his shoulder before she stepped back.\n\nAndrew Taylor clapped Davey gently on the back and patted his head. \"I want you to be a good boy for your aunt, and don't give her any trouble.\"\n\n\"And keep Duffy out of trouble,\" Sara instructed. \"We'll see you in a few days,\" she added quietly.\n\nLorrie knew she had been dismissed. It was Sara's inimitable style. She didn't just say hello or goodbye like other people. Not Sara. She welcomed you for the moment and then, when some invisible clock in her head reached the appropriate time, she dismissed you. As she ushered Davey out the door, Lorrie wondered if Sara ever dismissed Andrew that way, and what he thought of it if she did. Or if he was even aware of it. Sara was her sister and she loved her dearly, but she had to acknowledge she was different.\n\n\"Doctor Ryan,\" Stuart Sanders said, following Lorrie out the door and down the steps. \"If I could have a word with you, please.\"\n\nLorrie turned around. \"Sure, just let me open the door to the motor home so Davey can get in.\"\n\n\"Of course. I'll just wait right here,\" he said, standing back.\n\nWhile Lorrie unlocked the motor home door, Davey turned to see if his parents were watching him from the long panes beside the front door. Not seeing them, he scanned the living room windows. There was no shadow behind the lace curtains.\n\n\"C'mon, Duff,\" he said, turning toward the motor home. Aunt Lorrie stood beside the open door, waiting to give him a hand up the steps. \"I can do it myself,\" he said, wanting to show her that he was self-sufficient.\n\n\"Okay then, but how about if I give Duffy a boost?\" Lorrie's eyes met Stuart Sanders's amused gaze. At Davey's nod of agreement, Lorrie lifted Duffy and put her inside. \"You two look around while I talk to Mr. Sanders, all right?\"\n\n\"All right, but hurry!\" Davey replied, then disappeared into the motor home behind Duffy.\n\nStuart Sanders rolled his shoulders and straightened his tie in preparation for his talk with Lorrie Ryan. It was funny, where criminals were concerned, he had always been aggressive and sure of himself, but with women\u2014especially unattached, attractive women\u2014he was a different man entirely, awkward and never quite knowing the right thing to say or do. \"I only need a minute,\" he said, reaching into his coat pocket as Lorrie came toward him. \"I wanted to give you my card. It has my cell phone number on it. If you need me . . . for anything, I want you to call.\"\n\nLorrie stared at him, worry etching lines into her forehead. \"You don't think\u2014\"\n\nSanders put his hand on her arm, stopping her from completing her sentence. \"Let's not borrow trouble by speculating on things that may never happen. I just want you to know that I'm only a phone call away, and that means help is only a phone call away. Understand?\"\n\nShe nodded. \"That's very considerate of you, Mr. Sanders.\"\n\n\"Stuart. Call me Stuart.\"\n\n\"Stuart, then.\" She took the card from his hand, looked it over, then slid it down into her right front pocket.\n\n\"You do have a cell phone, don't you?\"\n\n\"Yes, it's in the motor home. Do you want the number?\"\n\nHe took a pen and a little spiral notebook out of his inside pocket and jotted down the number. \"I gave Davey a card, too,\" he told her. \"He does know how to use a telephone, doesn't he?\"\n\n\"Of course, he does. Along with a CB and a computer,\" she said proudly.\n\nHe smiled. \"Davey's a lucky boy to have an aunt who loves him so much.\"\n\nLorrie's eyes started to tear up. \"No, Stuart,\" she said, shaking her head, \"I'm the lucky one to have a nephew like Davey. He's one in a million.\"\n\n\"I know. So are you.\"\n\n\"Aunt Lorrie!\"\n\nLorrie turned toward the sound of Davey's voice. \"Coming, Davey.\"\n\n\"You'd better hurry,\" Sanders said.\n\n\"Yes, I . . .\" Something\u2014Lorrie wasn't exactly sure what\u2014had just passed between them, something that left her feeling a little lightheaded. \"Will you be here when Davey and I come back from our big adventure?\"\n\n\"I'm not sure. It depends on what happens with your brother-in-law.\"\n\n\"Aunt Lorrie!\"\n\n\"Goodbye, Stuart. And thank you for your concern.\"\n\n\"You're welcome, and have a good trip.\"\n\nLorrie climbed up into the motor home and closed the door behind her. \"So, what do you think, Davey? This is going to be your home for the next couple of days. Like it?\"\n\nDavey looked around, his eyes wide with excitement. \"It's a house! A real house! On wheels!\"\n\nLorrie smiled. She was proud of the recreation vehicle and had selected each accessory with great deliberation. The thick, forest-green carpeting added softness to the hard-surfaced, utilitarian appliances. The first thing Davey asked her to show him was his bed. Laughing, Lorrie pointed to a bunk over the driver and passenger seats.\n\n\"Where are you going to sleep?\" Davey questioned, looking around for a big king-sized bed like his parents used.\n\n\"In the back,\" Lorrie said, sliding back the bedroom door. \"You see, just like home.\"\n\n\"Wow. It is. It really is.\"\n\n\"Okay, now. Sit down and I'll buckle you in.\" As Lorrie secured the safety belt across Davey's chest and lap, she said, \"I want you to know this expedition is in no way a Mickey Mouse production. This is a first-class, grade-A, super-colossal expedition to the Philadelphia Zoo and other points of interest. Any comments?\"\n\n\"Nope,\" Davey responded with mock solemnity.\n\n\"First things first. Let's go through our check list. Your medicine. What time do you get your shot?\"\n\n\"You know. Every day at noon.\"\n\n\"Right. I'll set the alarm on my watch for twelve noon just in case we're having so much fun we forget what time it is. Dog food! Did you remember to bring dog food?\"\n\n\"Yep. It's under the table. Duffy eats twice a day.\"\n\n\"Okay, then. I guess that's it.\"\n\nDavey chattered excitedly as the RV ate up the miles on the crowded highway. He was having a good time just sitting in his seat, talking to Aunt Lorrie about his CB buddies and the movies he'd seen. He thought about his mom and dad when he saw an airplane and wondered if it was their plane. He waved, just in case it was and they were looking out the window. Were they thinking about him the way he was thinking about them? He listened with half an ear to Lorrie talking about the Philadelphia Zoo. He had never been to a zoo before. His CB buddy, Digger, had told him it was humungous, whatever that meant. It must be good, he reasoned, because his buddy got all excited just telling him about it. He'd said it was almost as good as being there just telling Davey so he would know what to expect. A giggle erupted in his throat. Panda Bear was going to the zoo!\n\nA large white dome with red lettering caught his eye as the RV slowed to merge with the traffic. \"What's that, Aunt Lorrie?\"\n\n\"The dome? It says 'Cherry Hill.' That's the name of a town near Philadelphia. It's a kind of welcome for travelers, it lets them know that Cherry Hill is the next turnoff.\"\n\nDavey shrugged. His attention was diverted when a battered pickup truck painted with bright designs pulled up beside the motor home. He stared at it and saw it was pulling a big square box on wheels. The glare on the window made it impossible to see who was sitting in the front. It didn't really matter who it was. He would never see them again once Aunt Lorrie turned off the road for the zoo. On closer inspection he saw that the pickup truck was pulling a pop-up trailer. Davey watched the truck as it moved still closer to the RV. A man and a woman were sitting in the front. He stared at the woman; she looked scared. His eyes widened when he saw the man reach out and slap her across the face. He swallowed hard as he clutched Duffy.\n\n\"Another five minutes and we'll be getting off the turnpike. Are you excited, Davey?\" Lorrie asked heartily.\n\n\"Uh-huh,\" Davey mumbled as he slid a piece of gum into his mouth. The woman in the pickup wasn't like Aunt Lorrie or his mother. She was more like Millicent, the babysitter who used to watch him when his parents went out. Only she wasn't pretty like Millicent. And Millicent would never look scared like that, or cry like the woman in the truck. Millicent said only babies cried, and that you had to be tough to survive. When he'd repeated what Millicent had said, his mother had got him a new babysitter, Mrs. Goodeve.\n\nDavey's golden eyebrows drew together as he stared pointedly at the girl in the truck. She gazed back at him through the glass before wiping at her eyes with the back of her hand.\n\n\"Get ready, Davey, here we go, exit four. Time to get off and make tracks for the zoo. I vote to see the monkey cage first. What about you, Davey, what do you want to see first?\"\n\nDavey turned, trying to get a last look at the pickup and the girl with the white face.\n\n\"The elephants,\" he said, distracted. Why had the man hit the girl? What could she have done? She had just been looking out the window.\n\nCudge Balog caught a look at the RV beside him. Sitting in the passenger seat there was a blond kid holding a dumb-looking pooch. Wet nose prints slopped up the side window.\n\nCudge was driving as carefully as he could. No way did he want a trooper pulling him over. Damn, he wasn't even going to sneeze for fear his foot would jam down on the accelerator. Slow and easy was the way to do it. No wiseass trooper with polished sunglasses was going to chew him out for anything, real or imagined.\n\nCudge's hands tightened on the steering wheel. A bead of sweat dotted his upper lip. \"Why ain't you saying anything, Elva? It ain't like you to sit so quiet. You up to something or what? If you got any ideas at all about jumping out or taking off, forget it.\" His voice was mean and low.\n\n\"You told me to sit here and keep my mouth shut. Make up your mind. Either you want me to talk or you want me to keep quiet.\" God, now why had she said that? She was so in tune with Cudge's voice that she knew exactly what his next move and statement would be. You did what Cudge said when Cudge said it, and you didn't ask questions. You didn't volunteer anything with Cudge either. What was wrong with her? Why was she acting this way? Her stomach churned and she felt bile rise in her throat. She was afraid to stay with him and afraid to leave him. Fear of walking the streets, with no place to go at the end of the day, was worse than living with Cudge. Everyone needed someone, something\u2014a place that was their own at the end of the day when darkness fell. She remembered only too well the darkness of the pantry where her father had locked her to punish her. She hated the dark and the creatures that came out in the dark. In a way, Cudge was like one of the rats back in the apartment; his eyes were just as beady, his lips as thin, his ears as pointed.\n\nCudge ignored her; this was no time to let Elva get under his skin. He had to concentrate. \"I want you to keep your eyes peeled for anything suspicious, like a trooper in an unmarked car, that kind of thing. Keep what wits you have sharp. If I get pulled over for any reason, you just sit there, deaf, dumb and blind. Don't open that mouth of yours. You got that, Elva?\"\n\n\"Yeah, I'm watching. How am I supposed to spot a trooper in an unmarked car? They don't wear their trooper hats in plain cars. I might pick the wrong person and then you'll get mad,\" she whined.\n\n\"You can always tell a trooper because they wear those fancy polished sunglasses. I'm obeying all the traffic rules, so I think we're safe, but that's usually when something goes wrong. There's no way I could make a run for it in this old buggy. If we get caught, it's jail for both of us. You're an accessory and don't you ever forget it.\"\n\n\"I won't,\" Elva said. She had to think about what she was going to do when they got to the Wild Adventure campground. She knew she couldn't stay with Cudge after this. Enough was enough. Even her fear of the dark and the creatures that prowled in the night weren't as bad as winding up as dead as Lenny, and that's what would happen if she stayed with Cudge. She couldn't let him know what she was planning. The thought of leaving him was so daring, so alien to her, that she broke into a cold sweat. Fear\u2014it always came down to fear. If Cudge made her help dig the grave for Lenny, she would bawl. And if he made the grave wider than it needed to be for Lenny, she would die on the spot\u2014he wouldn't have to kill her.\n\n\"Damn you, Elva, didn't you see that trooper? What the hell is wrong with you? Look, in the next lane, that's a trooper or my name ain't Cudge Balog. Pay attention. I ain't gonna tell you again.\"\n\n\"He don't look like a trooper to me,\" Elva said defensively. \"There's two kids in the backseat. Troopers don't ride with kids. I'm watching the best I can.\"\n\n\"Okay, okay. He could still be a trooper. Just because he had kids doesn't mean a thing. Always go by the sunglasses. They try and trick honest drivers. I'm gonna move up now and get in the right lane. Keep your eyes peeled on the road and don't screw up, Elva.\"\n\n\"If you're moving to the right lane, what should I be looking for? There ain't no traffic to the right of the right lane.\" Her gaze shifted from Cudge's profile to her window. BJ! The kid in the RV looked just like her little brother BJ. Another bummer. There was really something wrong with her. BJ was dead. She stared at him for a moment before she turned back to Cudge.\n\nCudge clenched his teeth. One long arm reached out and yanked at Elva's shoulder. Before she knew what was happening, she felt the hard sting of his hand against her face. She blinked as scalding tears burned her eyes. She turned away before Cudge could see the result of his handiwork. The little boy in the RV, his dog clutched to him, was staring at her as she wiped the tears with the back of her hand. He looked scared, the way BJ used to look scared.\n\n\"You bawl one more time and you've bought it, Elva. We'll get off this pike and get something to eat. We have to kill time. I don't want to show up at that campground too early.\"\n\nElva said nothing, wishing she knew who the little boy with the dog was. He looked just like BJ\u2014same color hair, same bright blue eyes, same scared look. Only this kid had a dog to love. Poor little BJ had only had her, and what good had that done him? When it counted, she hadn't helped him at all. The little kid in the RV was alive and BJ was dead. And here she was with Cudge. How soon would it be before she joined BJ, wherever he was? Fear of the unknown or fear of Cudge\u2014it was six of one and half dozen of the other.\n\nIt was mid afternoon when Lorrie and Davey left the reptile house. Lorrie took long gulps of fresh air to get over the creepy feeling the snakes had given her. Duffy frolicked at their feet, evidently glad to be outdoors even if she was confined to a leash.\n\nLorrie glanced at her watch. \"I think we've had enough zoo for one day. What do you say we head for the campground and set up camp?\"\n\n\"I didn't like the smell in that snake house, did you, Aunt Lorrie?\"\n\n\"No. It was awful,\" she said with a shiver. \"Are you tired, Davey?\" she asked, noticing how slowly he was walking.\n\n\"A little. But that sure was fun.\" His voice was full of awe. \"I think Duffy is more tired than I am. What are you cooking for dinner, Aunt Lorrie?\"\n\n\"Hamburgers, baked beans, and cupcakes for dessert. How does that sound?\" Lorrie asked, looking down at the little boy.\n\n\"Great. I love hamburgers with lots and lots of ketchup.\"\n\n\"Climb up then, and let's head for camp,\" Lorrie said as she unlocked the RV. \"Next stop, Wild Adventure!\"\n\n\"Duffy and me, we'll sit here at the table, okay?\"\n\nLorrie belted him in then headed for the driver's seat. She followed the yellow exit arrows through the parking lot and, just before pulling out onto the highway, turned to check on Davey. He was sitting up, sound asleep, Duffy lying across his lap. She hoped she hadn't overdone it. That brace was pretty heavy to lug around. Poor little guy. He was exhausted. She wondered if maybe she shouldn't have left sooner.\n\nLorrie shook away the bad thoughts and concentrated on the fun they'd had. She was acting like a mother hen, overprotective of her chick. It was just that she was so crazy about him she wanted to do everything right. She would hate to see Sara's wrath if something went wrong.\n\nShe turned off the highway into the campground and followed the signs to the office. The place looked pretty deserted and she wondered if maybe she shouldn't find another campground, somewhere more of a year-round variety. On the other hand, the lack of people meant that she and Davey would have more of an opportunity to spend time together, without distractions.\n\n\"We're here,\" she called behind her as she turned the key in the ignition. In an instant, Davey went from being sound asleep to wide awake. Before Lorrie could get out of the driver's seat, he had his seat belt unbuckled. \"You wait here while I get us signed in and find out where our campsite is. Did you ever see such glorious colors?\" she asked as she craned her neck to look at the giant trees with their spirals of autumn leaves.\n\nA few minutes later, Lorrie climbed back into the RV with a map and a card to hang from the rearview mirror. \"I took a site at the far side of the pond.\" The manager had told her she would be near an elderly couple from Massachusetts with an Air Stream trailer, and a New Jersey couple with a pop-up trailer.\n\nThe RV slid into a deep rut and bounced out. Davey struggled to hang onto Duffy. \"Aunt Lorrie, I have to go to the bathroom.\"\n\nLorrie grinned. \"Hang on till we're parked, Davey. I don't want you walking around in the motor home while I'm driving.\"\n\n\"I can't wait to get out and see everything. Right, Duffy?\" Exactly on cue, the terrier woofed.\n\nLorrie passed a garishly decorated pickup truck with a pop-up trailer parked in a grove of trees and knew she was heading in the right direction. \"Our site must be just around the bend and to the right,\" she said, more for her own benefit than Davey's.\n\nDavey turned to look out the long side window. He rubbed at his eyes and stared at the pickup. It looked like the one he'd seen on the turnpike. His thoughts were diverted when Lorrie deftly maneuvered the RV into the assigned slot. As soon as they stopped, he unbuckled his seat belt and headed for the bathroom.\n\nStepping down out of the RV, Lorrie stretched luxuriously. Davey joined her seconds later, zipping his jeans. \"Is it okay if Duffy and I take a walk?\"\n\nTousling his blond head, Lorrie was once again taken with her nephew's grammar. Sara's influence, no doubt. Most of the kids who came into Lorrie's office would have said, \"Duffy and me.\" \"Sure,\" she said, \"but first let's set down the rules. Look at your watch. The big hand should be on the nine and the little hand almost on the five. What time is that?\"\n\n\"It's fifteen minutes before five o'clock,\" Davey said proudly.\n\n\"Right. Now, when the big hand is on the three, I want you back here. And you must stay within earshot. Do you know what that means?\"\n\n\"That means you have to hear me if I call, or I have to hear you. Right?\"\n\n\"Right again. Now, Davey, what time do you have to be back here?\" she questioned, just to make sure he understood.\n\n\"Fifteen minutes after five,\" he said.\n\n\"Don't let Duffy wander off, and if she does, call her, don't chase after her or you could end up getting lost.\"\n\n\"Aw, you shouldn't worry, Aunt Lorrie, Mr. Sanders gave me a real Boy Scout flashlight and money for a phone call. See!\" He rummaged in his jeans pocket for his two gifts.\n\n\"And I'm impressed.\" Not for the world would she tell Davey there were no phones attached to the trees. And as far as the little penlight went, he'd be lucky if he could see his hand in the dark with it. \"I think you can make it. Remember, big hand on the three.\"\n\nHe started to leave then winced.\n\n\"What's the matter?\" Lorrie asked, concern drawing her brows together as she noticed the pained expression on the little boy's face.\n\n\"It's . . . my leg brace\u2014I must have it strapped too tight or something. Dad knows how to fix it.\" A shadow slipped over Davey's face, and Lorrie knew he was suddenly anxious about being separated from his parents for the first time.\n\n\"Let's have a look. Maybe I can help.\"\n\nDavey lifted his pant leg. Hunkering down to inspect the brace, Lorrie grimaced when she saw how the leather straps were cutting into the fragile flesh of his calf. \"No wonder. Your sock slipped down. How long has this been irritating you, Davey?\" From the abrasion on his leg and the look on his face, Lorrie knew it must have been bothering him for some time. \"Oh honey, this was hurting you at the zoo, wasn't it? And you didn't say anything. Why?\"\n\n\"Mom and Dad told me not to cause any trouble.\"\n\n\"Davey, you're the best kind of trouble I could ever have. I love you, don't you know that? There's no way you could be any trouble to me. Now, sit up here on the step so I can fix that brace. I'll pull up your sock and loosen the strap a notch.\" As she worked, Lorrie noticed the new shoes that had never had an opportunity to get dirty. It wasn't fair, she told herself. Davey was too overprotected, too housebound. Instinct told her it wasn't the hemophilia that made Andrew and Sara restrict Davey to the house. A child who played in his room all day was a lot less trouble than one who was in and out, bringing in dirt onto Sara's spotlessly clean carpet.\n\n\"How's that?\"\n\nDavey tested the brace and smiled. \"Way better. Do you like my new shoes? Mom says I can't wear them all the time. Is that because Reeboks are special?\"\n\nLorrie thought a moment. \"Well yes, Davey, in a way they are. They're for playing and running. You should consider yourself very lucky. Some children who wear a leg brace must wear heavy, high-top shoes. But your brace is only to give strength to your leg. See how it's made? This strap slips under your foot so you can wear it with any shoe.\"\n\nDavey nodded. \"I saw a boy in the doctor's office and his braces went over his knees. He had to push these clamps so he could sit down. But I don't know why he had to wear those braces. And I don't know why I have to wear mine.\"\n\n\"Didn't the doctor tell you that it's only for a short time? When you were a very little boy, before you started your shots, you injured your knee and it bled. You don't have to worry about that anymore\u2014the bleeding, I mean. The brace keeps your shin bone in line with your knee joint.\" She ran her fingers down the length of Davey's leg to show him what she meant.\n\n\"Oh,\" the boy murmured, pondering her explanation. \"But I can walk real good, and I can run.\" Suddenly he wrapped his arms around Lorrie's neck, giving her a tight hug. \"I love you, Aunt Lorrie. You're a foxy lady.\"\n\n\"And just where did you hear that term, young man?\"\n\n\"In the Junior CB Club. But don't tell Mom what I said. She doesn't like it when I learn new words.\"\n\n\"I promise. It's our secret. I'm your secret foxy lady.\"\n\n\"Hey, Aunt Lorrie, that could be your handle on the CB. I'm Panda Bear and you're Foxy Lady!\" Delighted with himself, Davey trotted off, Duffy at his heels.\n\nLorrie watched them go. There was a slight unevenness to Davey's step because of the brace, and his red nylon windbreaker blended with the autumn colors in the woods. Her eyes focused on the blond head and she wanted to reach out and touch him again. Duffy's short legs stirred the thick layer of leaves on the ground as she followed her master. With the same sense of loss that she always felt whenever she parted from the little boy, Lorrie turned to go into the motor home and begin dinner.\n\nShe was stirring some brown sugar into the beans when a mental picture of Stuart Sanders came to mind. It had been a long time since she'd felt an attraction toward a man, and never a man like Stuart Sanders\u2014big, tall, a rugged sort of guy. She supposed he had to be rugged to be an FBI agent, chasing after all those criminals.\n\n\"Too much TV,\" she said aloud, then laughed at herself. Too much TV was right. After a long day at work, all she wanted to do was sit down in front of the TV and be entertained. The hospital shows were always good for a few laughs. The truth of it was, the long hours were of her own doing. A workaholic, her peers called her. The last time she'd had a date was . . . She was horrified to realize she couldn't remember her last date.\n\nPrior to Davey's birth she'd dated a lot, and had thoughts of marriage and a family. But once Davey had been diagnosed as a hemophiliac, things had changed. She had changed. A family was no longer an option, at least not until medical science gave prospective parents the option of choosing the sex of their children. Until then, she would remain childless. But she needn't have given up men, she realized. It had just happened. She wasn't sure exactly why.\n\nNow Stuart Sanders had reignited a fire within her, a fire she'd thought was dead. And if she wasn't missing her guess, he was equally attracted to her. She hoped she wasn't missing her guess. It would be nice to get to know him better.\n\n\"Stuart.\" She tested his name on her tongue. It had a nice sound. \"Stuart Sanders.\"\n\nDavey and Duffy kicked their way through the fallen leaves until they reached the pond. The water was still and a deep, mossy green, and the red and orange of the sunset broke through the surrounding trees to create a golden path on the pond's surface.\n\n\"I saw this on TV, Duff,\" Davey said. \"Now watch. You pick up a stone and skip it across the water.\" Duffy sat patiently on her haunches, watching pebble after pebble hit the water with a splash, then sink.\n\n\"Guess it's another 'almost,' Duff. I can almost do it, can't I?\" Losing interest, Davey picked up a twig and poked at the dry pine needles and soft leaves. He was surprised to see the snowy shoelaces on his new shoes were now a dirty brown. He laughed delightedly; now they looked used, like Aunt Lorrie's.\n\nPushing back the sleeve of his windbreaker, Davey took note of his watch. The big hand was on the one. He had two more numbers to go. \"C'mon, Duff, let's see what's over on the other side of the pond.\"\n\nJohn and Sophie Koval stood outside their silver trailer, watching the newcomers' motor home bump down the road. Sophie wiped her hands on her apron, relieved that there were going to be other campers besides themselves and those . . . those hippies! Camping certainly wasn't turning out to be the lark that John and the brochures had promised. \"Meet new friends, see the country, get back to nature.\" The \"new friends\" had turned out to be large families who couldn't afford to get away from home any other way, and instead of the nice, sociable bridge games to liven the evenings that Sophie had envisioned, she'd had to endure the sounds of children squabbling, their parents yelling at them, and the sight of endless lines of laundry hanging between the trees. The smell of that disinfectant they used got everywhere, even into the dirt. In every campsite it was the same, and now even the shiny new inside of their trailer smelled of it. At night it filled her nose and seemed to parch her throat. Whoever it was who said there was no place like home must have gone camping.\n\nNow that summer was over and the northeastern states were well into fall, John and Sophie felt lonely and apart from everyone and everything. Most of the campgrounds were desolate, like this one, stopping places only. They would rest overnight then, in the dew-heavy mornings, they would break camp and ride on, sometimes for hours at a time, pushing the speedometer and the clock to arrive at their next destination before dark. As the days grew shorter, camping was becoming more of an ordeal, just like the very thing John had promised they would leave behind.\n\nSophie longed for the ease and comfort of her home back in Massachusetts, where she could spend the afternoons watching her soap operas on TV instead of growing stiff from long hours in the car, watching for obscure turnoffs to their next destination. John's retirement was becoming a trial and a punishment.\n\n\"Why couldn't we have gotten one of those nice buses, John? Then I could stay right in the back and prepare lunch or dinner and still be able to talk to you.\"\n\n\"Watch your soap operas, you mean. I've already told you, Sophie, those things guzzle gas. We've got to watch our pennies now. Social Security doesn't bring much and you know it.\" He watched her from behind his wire-rimmed glasses. The annoyance that he felt whenever she complained about their new lifestyle made him chomp down on his pipe. For over thirty years he'd made a nice home for her, given her an easy life, while he went to work every day at the mill and dreamed of the day he could retire. He'd given up most of his dreams and all of his energies for Sophie's comfort, and he wasn't going to let her make him feel guilty now.\n\n\"They looked like nice people, didn't you think?\" she asked, changing the subject, almost able to read his mind after all those years. \"Maybe we could go over and meet them later.\"\n\nJohn's eyes narrowed. \"No, don't think so. Be getting dark soon and you've got supper to fix. Imagine they do too, what with setting up camp and all.\"\n\nSophie frowned. She didn't like this new John, and she certainly didn't like their new lifestyle. She didn't know who or what she could trust anymore. Gone were all the comfortable, familiar things from over the years. Instead, in their place, everything was stiff and new. She comforted herself with the knowledge that she had at least refused to allow John to sell their home. She'd just wait for him to tire of this vagabond existence, and then she could go back to her nice electric range and a refrigerator that could hold a week's supply of food. She couldn't get used to shopping every other day at strange supermarkets where she couldn't seem to find the simplest item.\n\nWorst of all were the doubts she had about John. It had been fine back in Massachusetts when he drove her to the neighborhood supermarket and even to the shopping mall. But out here, on the open road, John had become a man she didn't know, a stranger. A . . . a Seattle cowboy! His driving had become aggressive instead of defensive. He was constantly cursing under his breath about this one cutting him off, or that one driving with his brights on. She was worried he would do something stupid, like flip someone off. Nowadays, instead of flipping you back, they just shot you! Road rage\u2014that was another part of this new life that frightened her.\n\nLooking at her husband, she saw him staring off in the direction of the camp where those hippies were parked. She knew they were hippies because of the way their pickup truck was painted. \"You're just as glad as I am that we're not alone here with those hippies, aren't you?\" More than a question, her words were a challenge.\n\n\"Sophie, they don't call them hippies anymore. And I don't see what the problem is. We're bound to meet people from different walks of life. You've got to learn to live and let live.\" He'd never admit it, but he was glad there were other campers besides themselves. Elderly people were easy prey for some types. \"Go on in and get dinner, Sophie. We've got a long drive ahead of us in the morning, and I want to get started early.\" He refolded a road map and traced the lines with his fingers. \"The manager of this camp told me about a real nice place down near Virginia Beach.\"\n\nCudge watched Elva set up the barbecue grill. \"Why are you using that? Why don't you use the stove?\"\n\nElva continued to pour briquettes into the grill. \" 'Cause we don't have any more propane, that's why.\" She lifted her head and glared at him.\n\nIgnoring her unspoken accusation, Cudge leaned against the camper. \"What's for supper?\"\n\n\"Eggs, corned beef hash, and Kool-Aid.\"\n\n\"Shit.\"\n\n\"Well, if you'd stopped at the store the way I wanted we could be having hamburgers or something. Eggs is all we got. And beans. You want beans?\"\n\n\"Shit. That's what it is. Shit. I could go for a steak, rare and juicy, and maybe a baked potato.\"\n\n\"Yeah, well, couldn't we all,\" she snipped.\n\n\"Don't we have anything besides Kool-Aid? A beer maybe?\"\n\n\"If we did, you'd have drunk it before now. All we have is Kool-Aid.\"\n\n\"Hey, I don't like your tone. Don't get smart with me.\" His voice was suddenly menacing. Elva hung her head a little lower and went on trying to start the fire.\n\n\"Be careful with that charcoal lighter. I don't need you setting fire to yourself and having to go to the hospital. We still got some work to do.\" He gestured to the closed pop-up with his head.\n\nAfter cranking up the top section that served as the roof, Cudge opened the front and back canvas wings that expanded into bunks. \"We have to do this right or else somebody might ask questions. Just keep the door closed so nobody can see inside,\" he warned.\n\n\"I need the eggs and stuff,\" Elva said. \"And I don't want to go in there, so you'll have to get them.\"\n\n\"What's the matter? Little Elva afraid that Lenny'll jump up and bite her ass?\" Cudge mocked.\n\n\"So what if I'm afraid? I don't like dead people. You killed him, so you go in there and get the food.\"\n\n\"Don't get fresh, Elva. I'm warning you. I've had more than a man can take today, and I've been real patient with you.\"\n\n\"It's not my fault that we were stopped by the state trooper. You're the one who had the truck painted that way.\"\n\n\"Shut up, Elva.\"\n\nElva looked at Cudge, saw the way his lips were curled in a snarl, saw his eyes boring into her and the way his neck seemed to disappear into his shoulders. Prickles of apprehension rose on her arms; she could feel them through the scratchy woolen poncho she had crocheted for herself. \"You gonna get me the stuff?\" she squeaked.\n\n\"Yeah, I'll get it. But you better not turn chicken on me tonight when we haul his ass out of here. I took a walk around before and I found the ideal spot. It's down in that shallow gully.\" He pointed to a spot near the pond on the far side of the campground.\n\n\"How are we going to get him there? I can't carry him that far.\"\n\n\"Well, you're going to. Just make up your mind. What do you think I feed you for? Just remember, you're an accessory, Elva, and I don't want to hear any of your shit. You just do what I tell you, understand?\" His fingers closed over her arm like a vise and Elva wriggled away.\n\n\"Lemme go! I understand. Just lemme go!\"\n\n\"I don't want you screwin' up the way you did with that trooper.\"\n\n\"That wasn't my fault! He was on your side and you should've seen him in the side-view mirror. It wasn't like he was in an unmarked car or anything like that.\"\n\n\"So you say. And I say you should've been watching! That was a close call, too close.\"\n\nThey had been on the turnpike after riding the back roads around Brick Township. It was too early in the day to appear at the campgrounds without provoking questions. Elva had her nose buried in the map, looking for the exit, and Cudge was tuned to the CB, listening to the truckers' conversations over the airwaves. Usually he could count on truckers to report a \"Smokey\" in the area, but not this time. The next thing Cudge knew, the bright blue light of a trooper's car was flashing and he was being instructed to pull over.\n\n\"Not a word, Elva,\" he said before getting out of the pickup.\n\n\"I'll need to see your driver's license and your registration, please,\" the trooper said, the afternoon sun reflecting off his mirrored sunglasses. Like most troopers he was tall, taller than Cudge, but he didn't have Cudge's bulk. Sizing him up, Cudge decided he wouldn't have the least bit of trouble in pounding the man to the ground. Then he saw the trooper's side holster and the blue-black grip of his pistol.\n\n\"Where are you heading?\"\n\nCudge was so preoccupied with half-formed plans of escape he hardly heard the question. \"Huh? Oh, we're camping. Heading north.\"\n\n\"Sir, your license says you're from Newark. That is north.\"\n\n\"Uh, we have been camping, in Maryland. We're heading home.\"\n\nThe trooper leaned forward to peer through the dirty window. The passenger was a young woman, at least ten years younger than her companion. \"Would you open the door, sir,\" he asked Cudge.\n\n\"What for? There ain't nothing in there that shouldn't be,\" Cudge bristled. He didn't want the trooper talking to Elva, and he didn't like his condescending attitude. The trooper called him \"sir\" but it could have been \"shit\" from the way he said it.\n\n\"Open the door, sir.\"\n\n\"Okay, okay.\" Cudge pulled the door open. The trooper looked in at Elva, who was sitting with her legs up underneath her on the seat. His trained eye observed the girl's long, dull brown hair, her painfully thin body, and the worn clothes. He also noticed the panic in her eyes. \"Everything all right, miss?\" He spoke from over Cudge's shoulder, some instinct telling him he didn't want this man behind him where he couldn't see him.\n\nElva nodded.\n\n\"Are you Mrs. . . .\" he glanced at Cudge's license, \"Balog?\"\n\nElva shook her head.\n\n\"How long have you known Mr. Balog?\" He deliberately asked a question that would require a verbal answer. Something was wrong here\u2014he could almost smell it. His mind clicked back to his last radio transmission, calculating how long it would take for assistance to arrive.\n\nElva remained silent, her eyes searching Cudge's.\n\n\"Miss, how long have you known Mr. Balog?\" the uniformed trooper persisted.\n\nA long pause ensued. Elva's eyes were locked with Cudge's. The scent of trouble grew stronger in the trooper's nostrils.\n\n\"Jesus Christ! Will you tell him, Elva? What's wrong with you anyway?\" He was shouting, knowing that he was alarming the trooper, yet unable to control himself. The beast in his brain pawed, the hooves biting into tissue, alerting him. Struggling for control, Cudge lowered his voice. \"Elva, for crissakes!\" He thought of Lenny hidden in the pop-up, saw the blue flash of the dome light on the trooper's car, and felt the threat of the trooper's pistol.\n\n\"Miss, would you like to get out of the truck, please?\"\n\nElva shook her head, her face whitening.\n\n\"Elva, for crissakes, do what he tells you.\"\n\n\"Would you please assist the lady from the cab, Mr. Balog?\" Something was amiss here. The girl's silence, the panic in her eyes, the man's agitation. Even the pickup truck looked like trouble\u2014electric colors, slogans, and decals that were just short of obscene. Only the pop-up trailer was still a light beige, devoid of decorations except for a few four-letter words fingered in the grime. In fact, the pop-up was in pretty good condition compared to the old Chevy pickup pulling it. \"I'd like to see your registration for the trailer.\"\n\nCudge's eyes widened, his hand shook as he pulled Elva from the truck. What in the hell did the cop want to see the registration to the trailer for? His glance darted from the trooper to the trailer and back again. \"Damn you, Elva, this is all your fault. Why wouldn't you answer him like he wanted?\"\n\n\"You told me to shut my mouth and not to say a word,\" she whined.\n\n\"Shut up,\" he hissed. \"If he wants a look inside, you better be in this truck when I take off or else I'll leave you on the side of the road.\"\n\n\"Cudge, I didn't\u2014\"\n\n\"Shut up!\"\n\n\"Any problem over there?\" the trooper inquired.\n\n\"No, no, nothing's wrong. Right, Elva?\"\n\nElva nodded.\n\n\"Crissakes! Tell him, willya?\" Cudge saw the trooper's mouth tighten and he could sense his eyes narrowing behind those glasses.\n\n\"Your trailer registration, please.\"\n\nDigging in his wallet again, Cudge presented his registration.\n\n\"What have you got in the back?\" the face behind the glasses asked, checking the information on the card Cudge had handed him.\n\n\"Whatever goes into a camper\u2014a toilet, sink, beds.\" The trooper was starting to piss him off. He'd like to punch his fist right into those glasses and push them through the back of his head. The silent pawing in his head became the restless shifting of weight, leaning against the gate, wanting to be let out.\n\n\"Let's take a look.\"\n\n\"Christ, you know how much trouble it is to open one of those things? To say nothing of putting it together again. I've been thinking of getting a regular trailer, one of those Air Streams or something. This kind is a real pain in the ass.\"\n\n\"Let's take a look, buddy. Now!\"\n\nGone was the \"Mr. Balog.\" No more \"sir.\" Now it was \"buddy.\" Cudge's spine stiffened and a light spray of sweat broke out on his brow. He moved to the rear of the trailer; his mind raced but no solution came to him. This was all Elva's fault. Damn Elva.\n\n\"Mind telling me what I did wrong? I wasn't speeding or nothing. I was just cruising along. This is harassment.\"\n\n\"Police brutality!\" Elva prompted. \"Tell him, Cudge, you want to talk to your lawyer!\"\n\nBalog's eyes rolled back in his head. If she said one more word, he was going to let her have it.\n\n\"Finally found your tongue, miss? What's this about police brutality?\"\n\n\"You have no right to bother us. We weren't doing anything. You can't just sneak up on people like that. Cudge told me to watch out for cops, and I was, but you came up on the wrong side.\"\n\n\"Shut up, Elva, for crissakes, shut up!\"\n\n\"While you're cranking open your camper, I'll call in your license,\" the trooper told him in a somber tone that made Cudge wonder whether or not he'd heard what Elva had said about watching out for the law. \"Get going!\" There it was, the authority, the suspicion that made Cudge's skin prickle.\n\nThe trooper moved toward his patrol car, senses alert for danger. It wasn't unheard of for a trooper to be shot down just asking a guy for his registration. Damn those politicians for cutting back on two-man teams. If he ever needed a partner, it was now. There was something fishy going on here, and he didn't like this Balog. And when that mouse he was traveling with decided finally to open her mouth, it was to tell him that she'd been on the look out for the law. Was she trying to tell him she was being kidnapped? What was in the camper? Drugs, contraband cigarettes? What? Not only was he going to call in Balog's license, he was going to ask for assistance.\n\nTurning his back on the couple, he took the few steps to the patrol car. He could feel Balog's eyes on him, piercing and angry. For a second back there, when he'd asked what was in the camper, Balog's anger had become almost tangible, thick and viscous. For that one second he had felt as though all his air had been cut off and he was trying to breathe through a vacuum.\n\nChrist! He was getting paranoid, jumping at shadows. Still, that feeling had been real\u2014a warning. Fleetingly, he thought of how many times his wife had begged him to change jobs. His brother-in-law had even offered him a job managing one of his used-car lots. Maybe it was time to consider it. Scared troopers didn't do their jobs properly; and the ones who did often ended up with half their head blown away.\n\nHe wondered again how far away the nearest assistance was.\n\nThe patrol car was still flashing its blue lights when a loud squawk sounded through the open window. \"Car 169, Car 169, proceed to mile 43 southbound. Emergency vehicle needed. Car 169, collision at mile 43, New Jersey Turnpike. Assistance en route, do you read?\"\n\nThe trooper looked at Cudge and Elva and then back at the patrol car. He hadn't realized how damp his shirt had become or how eager he was to get away before he found out what Balog was hiding in the camper. Thrusting the registration and license back into Cudge's hand, he hurried to his car and took off, lights flashing, siren blaring.\n\n\"Get in the truck, Elva. Move!\"\n\n\"Cudge,\" she whined. \"Cudge . . .\"\n\nCudge blinked and Elva's whining brought him back to the present. \"Are you going to get me the eggs? This fire's ready. And don't forget the black iron frypan.\"\n\nFor a long moment Cudge looked at her, the menace and power in his gaze so tangible that she backed away. \"Cudge, what's the matter? Cudge, don't look at me that way, I don't like it.\"\n\nWordlessly, Balog stepped inside. If he had to look at her for one minute longer, he would kill her.\n\nDuffy scampered ahead of Davey, intent on chasing a gray squirrel. Davey shook his head with exasperation. The squirrel would run up a tree, and Duffy would have led him on a merry chase for nothing. Nearby a cook fire was burning; he could identify the smell from when Dad barbecued steaks in the backyard. As expected, the squirrel escaped up a tree, leaving Duffy barking near the base.\n\n\"C'mon, Duffy, you can't climb trees and that squirrel won't come down until we're gone. C'mon!\"\n\nReluctantly, Duffy obeyed, coming to heel near her master. Davey looked at his watch. The big hand was on the two. That gave him one more number to get back to Aunt Lorrie. He frowned. Running through the woods with Duffy was so much fun, maybe Aunt Lorrie wouldn't be too upset if he got back when the finger was almost off the three. That would make it almost one more number. No, he'd promised to be back on time and he would be.\n\nThe trail led away from the pond to the road. If he turned to the right and followed it around, he would come back to their campsite. Breaking through the growth at the side of the gravel road, Davey took another glance at his watch. He wasn't certain how far it was back to the site, and he thought he would have to hurry. Anyway, his legs were tired, and even though Aunt Lorrie had fixed his sock and adjusted the strap, his calf was still sore.\n\n\"C'mon, Duffy! I'll race you back to the RV!\"\n\nDuffy woofed, stubby tail wagging and ears lifted, watching Davey for his next move. When she saw that Davey wanted to run, she obliged happily, scooting ahead to lead the way.\n\nJust around the road's bend, Elva and Cudge were eating when their attention was caught by Duffy's yips and Davey calling to her. They watched the little dog and the boy run headlong into their campsite. Elva recognized Davey immediately. He was the little boy who looked so much like BJ, the one that she had seen in the RV on the turnpike.\n\n\"That's a cute little dog you've got there,\" Elva called to Davey, bringing him up short. Davey's pride in Duffy was evident in his shy smile. He liked hearing that Duffy was smart and cute.\n\n\"Her name's Duffy,\" he told Elva, realizing that she was the woman from the pickup who looked like Millicent, his old babysitter. Remembering the man who had hit her, Davey turned to see Cudge sitting on a camping stool, his elbows on his knees, and an empty dish in his hands. Davey didn't like this man who had such a mean face and eyes that could eat you up.\n\n\"Where's your campsite, kid?\" Cudge asked. It was a piece of information worth having. If their campsite was too close to where he intended to bury Lenny's body, then he'd have to make other plans while it was still light out.\n\n\"Over there, on the other side of the pond,\" Davey told him, fidgeting beneath Balog's appraising stare. Duffy's exploring nose was put to the ground and she followed it to the tip of the man's boots. For an instant Davey thought the man was reaching down to pet Duffy; for some reason he couldn't explain, he didn't want the man to touch his dog. \"Duffy, come here,\" he ordered.\n\nHe wrinkled his nose; something smelled strange over here near the man's camper, a bit like the way his snow jacket smelled before his mother hung it on the line to air. It was the little white balls in the pocket that made it smell that way\u2014mothballs, Mom called them. Curious, he dropped his eyes to the ground, expecting to see some sign of the candy-like balls. He decided he liked the smell on the other side of the pond better.\n\n\"Hey, kid, how many brothers and sisters you got camping with you? Not too many, I hope. I don't like all kinds of noise at night when I'm trying to sleep.\"\n\n\"I'm the only . . . I don't have any brothers or sisters. I'm camping with my Aunt Lorrie.\"\n\n\"I don't want to scare you or anything, but you shouldn't go walking through the woods after dark. I hear there's bears and wild cats that'll eat that dog of yours for supper.\" A satisfied smirk lifted the side of Balog's mouth as he saw Davey gulp.\n\n\"Aw, why d'ya have to scare the kid that way? There ain't no bears in these woods!\" Elva raised her voice and moved forward, her half-eaten eggs sliding around on her dish.\n\n\"Shut up, will you? That's right, kid\u2014there's no bears, only tigers, and there's nothing they like better for supper than a nice juicy little dog like that one.\" Balog looked around for Duffy. His eyes darted to the open door of the pop-up\u2014in the shadows he could just make out the small dog nosing around the blanket-wrapped corpse. In two long steps he was at the camper and a second later Duffy came flying out.\n\nDuffy grappled for her footing, barking in pain. To Davey's wide-eyed horror, Cudge leapt down and grabbed the dog by the scruff of the neck, shaking her helpless body. \"Damn little bitch! I'll teach you to go snoopin' around where you don't belong!\"\n\n\"Cudge! Stop it! Cudge!\" Elva screamed. \"For crissakes, leave the dog alone! Put her down, Cudge! Cudge!\" She grabbed hold of Cudge's arm.\n\nWith a mutter of disgust, Cudge threw the dog to the ground, ready to plant a boot in her side if she came after him.\n\n\"Get your dog, little boy, and get away from here!\" Elva shouted.\n\nSparked to action by Elva's commands, Davey moved forward and scooped Duffy into his arms.\n\n\"Get out of here!\" Cudge thundered. \"Don't come snooping around here no more, you hear? And if I see that dog again, I'll kill her! Understand?\"\n\nDavey didn't move. His mouth dropped open, his wide eyes stared at the man. He couldn't think, couldn't breathe.\n\n\"You brazen little son of a bitch!\" Cudge made a motion toward the boy, who was clutching his panting dog.\n\n\"No, Cudge, don't. You can't!\" Elva knew Davey wasn't being stubborn and brazen, he was just scared stiff. Just like BJ. Just like herself. \"Don't hurt the kid, okay?\" she pleaded with Cudge. \"He's scared, that's all. Just scared.\"\n\nCudge shook Elva's hand from his arm, the movement tossing her backward to the ground. The behemoth in his brain was charging the gate now; he had to get control and force it back. His heart pounded with the effort; his knees became rubbery, incapable of holding his weight. Warning signals were going off inside his head, telling him that if he wasn't careful, he'd have trouble with the kid's family.\n\nJaw jutting forward, fists clenched, Balog leaned over, placing himself at eye level with Davey. \"Get out of here,\" he growled, \"and if you want that dog to stay alive, keep her away from me!\"\n\nSuddenly, charged with a rush of energy that enabled him to move his feet, Davey turned and ran. Though the brace on his leg hampered free movement, and Duffy's weight was heavy in his arms, he still found the strength to run. His Reeboks pounded the dirt road, sending up spirals of dust. His laces snapped and twisted together in agitation, his socks slipped downward, wrinkling at the ankles. The nylon windbreaker whispered, louder and louder, urging him to run. Faster! Faster!\n\nDown the road, through the trees, Aunt Lorrie's RV stood sentinel, waiting for Davey's return. He could see it now, all lit from within, a welcome refuge. He put Duffy down and watched her carefully, checking her for injuries. He walked on and Duffy followed\u2014her movements seemed normal. She was all right.\n\nDavey bent down and gathered Duffy close to him. Her cold wet nose brushed against his flushed cheek. One thought only raced through Davey's head\u2014he must never let Duffy go near that bad man again.\nChapter 4\n\nLorrie was concerned about Davey's behavior. He hadn't said more than a few words since he'd returned. She sat watching him as he ate his hamburger. She could hear his shoes rubbing against each other restlessly beneath the picnic table\u2014when kids did that, something was wrong. She restrained herself from lifting the colorful picnic cloth and peeking to see if she was right. She sipped her coffee, never taking her eyes from Davey.\n\nDavey reached for his glass of chocolate milk and swigged it down. \"I'm finished, Aunt Lorrie. That was good. I even liked the hard stuff on the hamburgers.\"\n\nLorrie threw back her head and laughed. \"Tell it like it is, Davey. Your aunt flubbed up. That hard stuff is called burnt crust. I promise you the cupcakes will be much better.\"\n\nDavey grimaced slightly. \"Can I save mine till later?\" He wanted to go inside the camper and lie down and think about the woman who looked like Millicent. He waited dutifully to be excused from the table. The moment his aunt nodded, he went inside the RV and sat down at the table. If he thought about the woman, he would have to think about the bad man.\n\n\"He was mean, Duff,\" he whispered to the little dog. \"And bad. We have a lot of things to think about, Duff, so you be quiet or Aunt Lorrie will come in and take my temperature.\" Duffy settled herself on Davey's lap.\n\nAll manner of jumbled thoughts raced through the little boy's mind as he absently reached for his new Goosebumps book. His voice, when he spoke next, was controlled yet squeaky. \"You weren't around, Duff, when Puffy got put in the ground. She was killed by a bad person who was going too fast in his car. Then they gave me you. You were supposed to be a boy dog, but Aunt Lorrie got all mixed up and gave you a name before she figured out that you were a girl dog. I don't want that bad man to get you, Duff. I don't want you to get killed. That's why we aren't going back there again. Now, you have to listen to me when we wake up in the morning. You have to stay with me.\" He wagged a finger under Duffy's nose. \"We're going to stay on this side of the pond so we don't see him!\" For an answer, Duffy snuggled closer to the little boy. \"I was scared and so were you. I could tell.\"\n\nIdly, Davey flipped a page of the book then played with the sharp corner until it was limp. When he saw how he'd curled the edge of the paper, he spat on his finger and tried to smooth it out. He pursed his lips then quickly turned three pages before he risked a look at the door. Aunt Lorrie wouldn't really care if he curled the pages. Books were supposed to be fun\u2014she had even given him some of her own from when she was a little girl that had crayon marks and spilled jam on the pages.\n\n\"Now I've forgotten what I was thinking about.\" Then Davey swallowed hard as the specter of Cudge's face appeared before him again. He didn't want to think about him, or the woman who looked like Millicent anymore. \"C'mon, Duff, let's go get our cupcakes, and maybe Aunt Lorrie will let me talk on the CB.\"\n\nHe opened the door and looked out. \"Aunt Lorrie, can I have the cupcake now, and can Duffy have one, too?\"\n\n\"You bet.\" Lorrie smiled happily as she untied her red checkered apron. What a relief. Davey was fine. He'd just been tired from his long tramp through the woods. Maybe it wouldn't hurt to take his temperature, just to be on the safe side. Immediately she quashed the idea; she wasn't going to be a mother hen. \"So tell me, big guy, where did you and Duffy go and what did you see?\" Lorrie asked, peeling back the cupcake wrapper.\n\nDavey settled himself on the picnic bench. He couldn't lie, especially to Aunt Lorrie. \"We walked around the pond and we visited the people in the camper. You can't see their truck from here, but it's there. I guess that's all we did. Duffy doesn't like it over there.\"\n\n\"I see,\" Lorrie said as she watched Duffy devour her treat.\n\nDavey wiped his hands on his blue jeans. \"Aunt Lorrie, will you let me talk on the CB before I go to bed?\"\n\n\"Sure I will, Davey. Tell you what, why don't you get ready for bed, and when you're clean, you can talk for a while.\"\n\nLorrie's gaze followed the little boy inside the RV. Something was wrong, she just knew it. He'd been distracted throughout dinner. Was he homesick? It was perfectly reasonable that he should miss Sara and Andrew. After all, this was his first time away from them. He was doing remarkably well, all things considered.\n\nCudge's voice was hardly more than a whisper. \"It's almost ten o'clock, Elva. Time to take Lenny out. I've been watching that RV on the other side of the pond and their lights have been out for more than an hour. Get that shovel out from under the bunk. I can't wait to get rid of old Lenny and split.\"\n\nElva was snuggled down in a blanket near the campfire. She sat up. \"Are we leaving here tonight?\"\n\n\"Nah! I told you before, we can't make any false moves. Everything has to look natural. C'mon, give me a hand.\"\n\nThis was the moment she had been dreading\u2014the moment when she would have to go near Lenny. \"How are you gonna get him to that place you picked out this afternoon?\" Her dun-colored eyes drifted away from Cudge towards the fire, the light. She didn't want to hear Cudge say she had to help carry him. If he told her she had to touch Lenny, she'd be sick right here and now.\n\n\"Same way we got him out of the apartment\u2014on the ironing board. He's still tied on it, ain't he?\"\n\n\"Cudge, I don't want to do it! It's dark and I don't like dead people.\" Her whining grated on Cudge's already tightly strung nerves.\n\n\"You'll do what I tell you. Now, get your ass over here and grab your end of the ironing board.\" He cursed viciously as he dragged the narrow end of the board over the aluminum step so Elva could grasp it. \"Now, pull it out and I'll grab the other end.\" He checked his pocket for the flashlight.\n\nBalog led the way through the light underbrush near the road and into the darkness under the trees. \"Just keep up and make sure you hold up your end.\"\n\n\"It's too heavy.\"\n\n\"You made it down the stairs at the apartment all right. You've got the light end, so shut up and quit your whining.\"\n\n\"Maybe people get heavier when they're dead. Maybe that's what they mean when they say a 'dead weight.' I don't think\u2014\"\n\n\"That's right, you don't think! I've had enough of your bright ideas. Christ! The smell of those mothballs is making me sick.\"\n\nThe harvest moon shone through the trees and, once their eyes had adjusted to the night, they were able to see surprisingly well. \"See, it ain't so dark. What's to be scared of? We'll be through before you know it.\"\n\nElva trudged along behind Cudge, struggling with her end of the ironing board, wishing she could put it down even for a minute. The aluminum frame was biting into her fingers. If only she could stop and change her grip.\n\n\"Okay, here it is,\" Cudge whispered, huffing with the exertion. \"Drop your end and I'll slide him the rest of the way down. Take this flashlight and turn it on so I can see what I'm doing. And for crissakes, keep it pointed at the ground.\"\n\nElva took the flashlight and turned it on, pointing it at the ground in front of Cudge. She trembled violently, making the light skitter in all directions.\n\nAfter sliding down the embankment to the shallow gully, Cudge asked for the shovel that had traveled along on Lenny's stomach. Careful not to touch the blanket, or to think about the body beneath it, Elva gingerly lifted the shovel and handed it to Cudge.\n\nCudge felt better now that he was at Lenny's last resting place. It wasn't bad, he told himself. Lots of trees, quiet, shade in the summer\u2014Lenny could have done worse.\n\n\"Are we gonna bury him with my ironing board?\" Elva's voice squeaked.\n\n\"Dumb broad. No! How'd you like to spend the rest of eternity strapped to an ironing board? Point that light over this way.\" Clearing away the underbrush, Cudge put the spade into the loamy earth and forced it deep with his foot. Shovelful by shovelful, the rich, black dirt piled up beside him as he dug deeper, calculating the size of the hole into which Lenny would fit.\n\n\"Your turn, Elva, and I don't want to hear any complaints. Just dig while I take a breather.\"\n\nCudge's venomous growl left no room for argument. It wasn't long before the perspiration was dripping down Elva's face and between her shoulders. She didn't want to be a grave digger. How deep was deep enough? Her arms ached from the unfamiliar exertion. Now she had to stand in the hole to excavate the heavy earth. The woods surrounding them seemed unnaturally quiet. \"Quiet as a grave,\" her mother used to say, and now Elva understood it.\n\n\"Move over, Elva. I want this done and over with. You operate somewhere between slow and stop, and we ain't got all night.\" The grave was almost four feet deep\u2014Cudge decided that was deep enough. \"Get over there and untie him from that board. Then you can help me roll him into the hole.\"\n\n\"I . . . I can't,\" she gulped. \"Don't make me, I can't touch him!\"\n\n\"When I tell you to do something, I mean for you to do it. He'll roll easy; don't think about it.\"\n\nHis no-nonsense tone propelled Elva toward the body. Shakily, she untied the knot that held Lenny to the board, trying not to touch the body or feel the yield of flesh beneath the blanket. One good thing, she thought. Cudge was right about Lenny rolling easy. The incline of the gully aided her efforts as she prodded and pushed toward the grave. She nearly gagged from each dull thud the body made as it rolled over. \"Watch out, Cudge, here he comes,\" she said in a quivering voice. The loud thwack of Lenny's body falling into the grave made Elva squeal with fright. An awful sound. The last sound poor Lenny would ever make.\n\n\"Right. He's down. Now we shovel the dirt back on top of him.\"\n\n\"No! Wait! He's . . . he's lying on his face.\" The flashlight beam revealed Elva to be right.\n\n\"So what, he'll be facing the way he's going.\"\n\n\"No, we can't, you have to turn him over. He was your best friend, you can't leave him like that.\"\n\nRevulsion filled Balog at the thought of touching Lenny. What if the blanket slipped and Lenny was looking up at him? Christ! No man should be expected to see his best friend before he threw the dirt over him. Yet Elva was right. Besides, if he left Lenny face down, Elva would never stop whining or let him forget it. He was tempted to tell Elva to turn Lenny over, but the grave was shallow and narrow; it would take a fair amount of strength to position the body within its confines. He shuddered. \"Here, hold the light and get ready with the shovel. And whatever you do, don't shine it on his face. I don't want to look at him.\"\n\nTaking the flashlight, Elva trained it into the grave. Cudge slid in near Lenny's feet and grasped the body near the shoulders to turn it. Some demon of perversity made Elva flash the light on Lenny's face just as Cudge turned him over.\n\n\"God!\" Scrambling as though his life depended on it, Balog climbed out of the hole, his stomach heaving, his lower intestines loosening. The underbrush holding his legs seemed to be Lenny's hands pulling him back into the grave. He propelled himself forward into the darkness, where he retched and felt himself almost foul his pants.\n\n\"Cudge, Cudge, you all right? I didn't mean for the light . . .\"\n\n\"So help me, Elva, what the hell's wrong with you anyway ?\" Cudge spat, clearing his mouth, knowing the sour taste of his own bile. \"Get that shovel and start covering him over, and if you stop, even for a minute, you're going in there with him. You got me?\"\n\nElva grabbed the shovel, pushing the dirt into the hole to cover Lenny as fast as she could. She'd seen Lenny's face, and it was something she never wanted to see again. It was worse than what had happened to BJ.\n\nTwenty minutes later, they were finished. Cudge played the flashlight around the area, scanning the gully till he was satisfied everything looked normal. At the last minute he added a fallen branch, heavy with red-gold leaves, to the new grave.\n\n\"Can we go now, Cudge? I hate this place, I really hate it.\" In the chill night air, Elva shivered all the way back to the campsite.\n\n\"There's no point in hanging around here,\" Cudge told her in the light of their campfire. He brushed the dirt from his jeans. \"I'm going to take a ride and find a bar. I need a few beers. You're staying here so nothing looks suspicious.\"\n\n\"You want me to stay in that pop-up after . . . Cudge, a dead body was in there and it stinks.\"\n\n\"The stink is your own fault. Just where else do you think you're gonna sleep tonight? It's cold out here and we've only got thin blankets. What's the matter, Elva\u2014afraid of spooks coming back to haunt you?\" he taunted.\n\n\"I'm not as scared as you,\" she shot back. \"I ain't the one who almost shit his pants.\"\n\nWithout another word, Cudge jumped into the pickup and drove down the road. The taillights became fainter and fainter until they were just tiny red eyes staring at her through the darkness. Now she was all alone in the darkness, with Lenny's body only yards away in the gully. It was like being left in a graveyard. Death was all around\u2014she could feel it and smell it. It was cold, so cold. If only she wasn't afraid to go into the camper, but Lenny's body had been there for all those hours.\n\nA gust of wind swayed the trees, knocking a dead branch to the ground near her feet. Elva screamed and covered her face with her bony hands. Then she crawled up the steps into the pop-up.\nChapter 5\n\nMorning arrived fresh and clear. The sunlight streaming through the hardwood trees sent a kaleidoscope of color through the motor home windows. Davey climbed down out of bed, yawned and looked around for Duffy.\n\n\"I let her out to go to the bathroom,\" Lorrie said, hugging him. Davey yawned again. \"You are a sleepyhead this morning, aren't you?\" Lorrie kissed his forehead, discreetly checking for fever. It was cool. He was fine. And she was a worrier. \"After you get dressed, I thought we'd go fishing for an hour or so, then we'll break camp and head for our next destination.\"\n\n\"Where are we going?\"\n\n\"It's a surprise.\" She hunkered down to his level. \"But you're gonna love it.\" She brushed his light, silky hair back from his forehead. It would need cutting soon.\n\n\"I am?\"\n\n\"Yep. Now get a move on. Time's a'wastin',\" she said, patting him on the backside. As soon as she stood up, she heard Duffy barking outside the door. \"C'mon, girl. Time for you to eat, too,\" she said, opening the door. The little dog bounded up the steps into the motor home and greeted Davey as if she hadn't seen him in a week.\n\nBreakfast consisted of cereal, a banana, chocolate milk and a cupcake. Afterwards, Davey and Lorrie got their fishing gear and headed toward the pond.\n\n\"They don't seem to be biting,\" Lorrie said morosely after only fifteen minutes.\n\n\"How do you know there are any fish in the pond?\"\n\n\"I don't, but it looks like a good place for fish, doesn't it? Besides, Davey my boy, there's more to fishing than meets the eye. It's not always important to bring home the dinner, if you know what I mean.\"\n\nDavey looked up at his aunt, his head tilted to one side, trying to understand.\n\n\"Look, it's this way. Fishing is about getting out into the fresh air and sunshine. You sit around, watch the water, listen to the birds, and commune with nature. Understand?\"\n\n\"I guess. If you say so.\"\n\n\"Davey, you have the instincts of a true sportsman!\"\n\nIt was nice when Aunt Lorrie smiled at him the way she was now. Her eyes got crinkly and it made Davey feel warm inside.\n\n\"Tell you what. We've got a little time to kill before we take off, so why don't you take Duffy for a short walk and I'll stay here and fish. I'll prop your pole up and whistle if there's a tug on your line. Then you can come back and reel in your fish. How's that?\"\n\n\"Aunt Lorrie, maybe if you didn't whistle, the fish would bite.\"\n\n\"Davey! I have to whistle to let the fish know where to find the bait.\" She watched with amusement as Davey pondered her answer. When the boy shook his head from side to side, Lorrie burst into laughter. It was clear Davey had his own views on fishing.\n\nElva watched Cudge sleep. Even though she had pretended to be asleep when he'd returned to the camper in the early morning hours, she hadn't been able to relax. He had come back drunk; within seconds he'd fallen across the bunk, his loud snores bouncing off the canvas walls of the pop-up. She had been even more afraid with him there than she had been alone. She'd seen him like this before, many times. When Cudge got drunk, he got even meaner.\n\nInside the pop-up everything smelled of mold and mildew, sour liquor, and mothballs. Bright sunlight shone through the nylon mesh windows, which were backed by clear plastic to keep out the wind. The plastic warped the sun's rays, making lacy patterns on the filthy floor. Elva scraped at the floor with the toe of her shoe. The grime was embedded so deeply that her rubber sole left no mark. That annoyed her still more.\n\nElva needed to use the bathroom. There was a pot stashed in the camper, but she didn't want to use that. There were showers and bathrooms at the end of the road, but she didn't want to have to walk that far. She decided to wait.\n\nCudge mumbled in his sleep and thrashed about on the narrow bunk. Elva held her breath. Was he going to wake up or would he sleep some more? His long arm hung disjointedly; from where she sat she could read the numerals on his watch. Ten minutes past eight.\n\nMost people were up by eight o'clock. The woman and boy in the RV across the pond were probably up and eating their breakfast. Maybe they were finished with breakfast by now\u2014they were probably arranging their day, the kitchen all tidied up, both of them dressed in neat, clean clothes. They must have money, Elva decided. You had to have money to drive an RV that only got six or eight miles to the gallon, and dress your kid in new clothes to go camping!\n\nSlowly Elva stretched one cramped leg from beneath her. She rubbed her aching calf before bringing her other leg out straight in front of her, careful not to make a sound. She took turns massaging each leg and, when she was certain she could stand, she slowly inched to her knees, holding onto the handle of the small refrigerator door for support.\n\nA hand gripped her from behind and she heard Cudge's voice. \"Wherever you thought you were going, Elva, forget it. We have something we have to do this morning, and then we'll get out of here.\"\n\n\"I wasn't going nowhere. And what do we have to do this morning?\" She was surprised to see him so lucid and alert. Something told her that he had been awake all this time, thinking. And when Cudge Balog started thinking, it didn't bode well for her.\n\nBalog swung his legs over the side of the bunk. His watchful eyes never left Elva's face. \"We have to dig Lenny up.\"\n\nElva gasped, the wind knocked from her. The thought of going down into that gully, even in bright daylight, started her shaking. \"Why?\"\n\n\" 'Cause we don't have any money, that's why. Lenny was on his way to a crap game, wasn't he? Well, he wouldn't go unless he was well heeled. I blew what money I had last night in the tavern, or else I was rolled in the can. Don't open your yap, Elva; I'm in no mood for you.\"\n\nOh, God! She couldn't look into that hole again and see Lenny's eyes staring up at her. It was a sin to open a grave, wasn't it? She couldn't do it; but one look at Cudge's face told her she would.\n\nStanding, he loomed over her. \"You're the one with the bright ideas, so how come you never thought about taking the money out of Lenny's pockets? Why do I have to think of everything? Every cent we've got is buried in that hole with what's left of Lenny Lombardi.\"\n\nCudge was looking at her strangely. And his voice, while hardly pleasant, was different somehow. He wasn't yelling. Normally, he'd be yelling at her. Something about him was making Elva's flesh crawl.\n\n\"Go outside and take a look around. Where's that Kool-Aid you made last night? Christ, I feel as though my mouth's lined with cotton.\"\n\nElva handed him the plastic pitcher of grape Kool-Aid. \"We can't go down there in the daytime. Somebody might see.\"\n\n\"So what if they see? All they'll see is us taking a little walk. What's wrong with that? One of us will stand guard while the other digs Lenny up. We need that money.\"\n\nAll Elva heard was the word \"we.\" \"We\" meant both of them, which made them a couple. She liked it that Cudge was thinking of them in those terms. It made her feel special, like she meant something to him.\n\n\"Go outside and see if anybody's moving around out there. We don't want anybody to come up on us when we have Lenny laid bare to the daylight.\" He laughed, a horrible sound that made the bile rise in Elva's throat. She nodded and closed the door behind her.\n\nShading her eyes as she peered off into the distance, she couldn't see anyone or anything. The campground was thick with trees and undergrowth, making it impossible to see very far in any direction. The campsites were all set away from the road, with a modicum of privacy. Off in the direction of the pond and the gully there was nothing to be seen. With only two other camping parties besides themselves, Elva really hadn't expected to see anyone.\n\nStepping out beyond the perimeter of their campsite into the bushes, she dropped her jeans to her ankles. She'd better go now, otherwise she'd probably pee in her pants when Cudge opened Lenny's grave. Cudge's bellow made her hurry, pulling her jeans up quickly.\n\n\"Took you long enough,\" he grumbled as she stepped back into the clearing. \"Get over here and give me a hand with the camper. I'm going to close up the sides so's all we have to do when we're finished is throw the shovels in and crank the top down.\"\n\nObediently, Elva helped him fold in the canvas eaves and put away the barbecue grill and utensils they had used the day before. She knew they should have been put in before the canvas eaves were folded, but she was afraid to argue with Cudge this morning. In went the ice chest, the nearly empty bag of briquettes, and the dirty frying pan. Used paper plates and plastic cups littered the campsite and she began to clear the area.\n\n\"Leave it! We paid enough to stay here; let them clean it up. I ain't no garbage picker. Grab the shovel and let's go.\"\n\nFollowing him through the woods, Elva dragged her feet. The thought of what they were about to do revolted her. It was a sin, she knew it.\n\nAt the edge of the pond Cudge stopped, coming up short, waving her back with his hand. Across the pond, a woman sat fishing. \"Dumb broad, she's not going to catch anything there,\" Balog whispered. \"I don't like it.\"\n\n\"Me neither. Cudge, let's forget it. Huh? It's a sin to dig up a body. I know it is.\"\n\n\"Shut up. It don't look like she's going to walk this way, and even if she did, it'd take her a while to go around the pond to the gully. By that time, we'll have old Lenny dug up and planted again. This time, permanently.\"\n\nElva swallowed hard as she followed Cudge.\n\nDavey tossed a stick high in the air, watching to see where it would land. \"Go get it, Duff. Bring it back.\" The little dog scampered off to do his bidding. Davey followed her into the grove of trees that circled the pond.\n\nAgain and again he threw the stick and followed Duffy. Once or twice he turned around and looked back; his Aunt Lorrie was growing smaller and smaller as he walked away. Now he couldn't see her at all. Looking at his watch, he realized he'd walked for three numbers, and his eyes widened as he looked around him. He was on the other side of the pond now, the side where the mean man was camping.\n\nHis first thought was of Duffy. He could see her, carrying the stick back. There was a funny smell in the air. The smell like his snow jacket. \"Good girl, Duffy. Here, give me the stick. Time to go back and see if we've caught any fish. I don't like it over here; I don't like the way it smells.\"\n\nDuffy cocked her head to one side, growling deep in her throat. Someone was there. Davey looked around, trying to detect the direction the sound came from. He thought he heard a voice. He held his arms out to Duffy who needed no second urging. \"Shhh,\" Davey whispered close to the dog's ear. \"It's that mean man. Be real quiet, Duff.\"\n\nThe little boy moved away from where he thought the sounds were coming; crouching low. Afraid for his dog, he went deeper into the woods, hiding in the low growth of scrubby pines.\n\nHe heard the sounds again, and he knew he had chosen the wrong direction. They were there, right in front of him. He could look between the green branches and see them. \"Quiet, Duff. Quiet. That man said he'd kill you if he saw you again. We have to be still, and hide and wait till they go away.\" Davey wished he and Duffy hadn't walked so far, hadn't gone out of earshot of Aunt Lorrie.\n\nThe girl who looked like his babysitter kept wiping at her eyes. She looked scared. The man was digging with a short-handled shovel, and it crossed Davey's mind that he was digging for buried treasure. Davey wanted to know what was in the ditch; he wanted to see what buried treasure looked like. He dropped to his knees to get a better look and accidentally let go of Duffy.\n\n\"I hit something,\" Cudge said. \"God, he's buried deeper than I thought.\"\n\nElva forced herself to look down into the open grave. She couldn't speak; her tongue was stuck to the roof of her mouth.\n\nDavey looked at the open hole. A man's shoe. A man's leg. A blanket. What was the man doing in the hole, and why did the mean man have to dig out the dirt?\n\nThe truth dawned on Davey. After his cat had been run over by a car, his dad had dug a hole in the backyard and buried him. His dad wrapped the cat in a clean towel, not like the dirty blanket the man was wrapped in. When you were dead and didn't breathe anymore, they put you in a hole and planted flowers. But there weren't any flowers here. He felt a vague disappointment that he wasn't going to see a buried treasure of gold coins and jewels. Just a dead man.\n\n\"Get in there and go through Lenny's pockets, Elva.\" Cudge's voice was cool and controlled. He had purposely kept a thick layer of soil over Lenny's face, knowing he didn't have the stomach to see it in broad daylight. Besides, something was eating at him. Elva kept saying it was a sin to open a grave. What the hell? It was a sin to kill!\n\n\"What? I can't . . . no, Cudge, I won't do that. I'll puke.\" Elva's face was a sickly green.\n\n\"Do what I tell you and quit your yapping. We need that money, Elva. I did all the digging.\"\n\nHis last remark stopped her. Illogically, she saw the practicality in what he wanted her to do. And he had done all the digging. Hesitantly, she jumped down and braced her legs on either side of Lenny's, trying hard not to retch.\n\nCudge saw her hesitate. \"C'mon, Elva, we ain't got all day. Just get the wallet. Try the back right-hand pocket first.\"\n\nElva's hand made contact with the blanket, but it might have been the slime of a giant slug, the way it made her recoil.\n\n\"Do it, damn it! Get it and hand it up!\"\n\nLenny was heavy and cold. She could feel how cold he was, right through the blanket, even through his clothes. She managed to lift him slightly and get her hand underneath him, feeling for the rectangular bulge that would be his wallet. Closing her eyes, she found it and pulled it free. Cudge reached down and grabbed it away from her.\n\n\"Good girl, now try the other pockets. He must have had a good-sized roll.\" While Elva struggled with her revulsion and Lenny's stiff body, Cudge opened the wallet and pulled out two twenties and a single. \"Find it, Elva. This can't be all he had. Lenny liked to go into a game well heeled.\" Putting the bills into his own pockets, he dropped the wallet onto Lenny's chest.\n\n\"I think I got it, Cudge,\" Elva told him, her spirits lifting because he had called her a good girl. \"It's a wad, all right.\"\n\n\"Hand it over.\" Must be a couple of hundred dollars, he told himself, satisfied. Ol' Lenny was worth a lot more dead than alive. \"Try those other pockets.\"\n\n\"I already did and there was nothing. Cudge, I don't want to touch him anymore. And jeez, this blanket stinks. I think we used too many mothballs.\"\n\n\"Search him, Elva, do like I tell you.\" His hands closed over the shovel handle, eyes intent, watching. He wouldn't like hitting Elva and rolling her into Lenny's grave, but he had to do it. She was stupid, and stupid was dangerous.\n\nElva looked up to plead with him, to tell him that it made her sick to touch Lenny. She had to convince him that there was nothing else in the earth-damp pockets. Her mouth opened but no sound came out. Cudge was staring down at her, shovel held high over his head. Doom crowded her and she knew she would soon be lying next to Lenny. \"Cudge! No, don't!\"\n\nA sharp yipping followed by an angry bark broke the stillness.\n\n\"What the hell?\" Cudge shouted, leaping back from the open grave. The shovel flew from his hands and clattered to the ground. \"I thought I told you to have a look around, Elva! Now you've blown it!\" He grabbed her hand and yanked her out of the grave. \"It was that kid's mutt. Where the hell did it go?\"\n\n\"I don't know! I don't know!\" she wailed.\n\n\"Get him! Find him! I'll fill in the hole!\" Cudge's words were cut off by a streak of black fur charging from the bushes.\n\nDavey was stunned. He stood transfixed by the sight of Duffy running toward the mean man and the crying girl. Growls and snarls burst from Duffy as she tried unsuccessfully to chew at the man's leg. Davey wanted to run to get his dog, but the girl was running toward him. For one instant he looked directly into her eyes.\n\n\"Run, kid! Run as fast as you can,\" she ordered in a loud whisper. \"If he gets his hands on you, he'll kill you and you'll end up like Lenny. Run, damn you! Run!\" That's what she had said to BJ, but BJ couldn't run. BJ had just stared at her, willing her with his great saucer eyes to help him. But she hadn't. Couldn't. This kid wasn't moving either. Panicking, she reached for him, pushing, shoving. \"Now, damn it! Run! Get away from here!\"\n\nDavey eyes went from the girl to the man. He had to get Duffy before the man did. Davey charged past the girl, calling Duffy's name. Quicker than mercury, the boy caught up with his dog, grabbed her and ran as fast as he could. Duffy was still growling, erupting into agitated barks as Davey climbed the ridge of the gully, heading for safety, for Aunt Lorrie.\n\nEncumbered by his leg brace, Davey slowed. Duffy fell from his arms, tumbling into the leaves. Half dragging his leg behind him, Davey scooted beneath low overhanging branches, in and out of the brush, Duffy loping behind him. Behind him, he could hear something tearing through the woods, breaking branches, snapping twigs, pounding the soft earth.\n\nDavey had to hide; he had to keep Duffy safe. The dog ran ahead of him, ears erect, tail held high. Davey followed her through the brush into the bright sunshine, toward a winding road. They had to hide. He'll kill you, the girl had said.\n\nOn and on Davey ran, shoelaces snapping.\n\nThe truck and pop-up sprang into Davey's line of vision. The camper was half-closed; the sides were folded in, but the door stood open.\n\nDuffy picked up speed, heading away from the pop-up, toward Aunt Lorrie's RV. Davey was breathless, nearly exhausted. He knew the man was close and could probably see him now that he was out in the open. The decision was made. He ran up the aluminum step into the camper and slammed the door shut behind him. He had to hide where the man couldn't see him. Duffy was smart; she would find her way back to Aunt Lorrie.\n\nCovering his mouth with his hand to quiet his ragged breathing, he crawled to the back of the camper, down between some empty boxes and the bunk. He waited, listening. There was no sound, no scratching from outside to tell him that Duffy was out there. Just silence.\n\nA loud bellow ripped close to Davey, making him jump. \"That goddamn kid got away and it's your fault, Elva. You dumb broad! Once that kid tells his folks what he's seen, it's curtains. The manager has our license number and knows who we are. That means jail, Elva. We'll be locked up for the rest of our lives.\"\n\nDavey held his breath, expecting the man to walk into the camper any minute. Instead, a rusty creaking sound filled the pop-up, making it vibrate. The scraping of metal against metal pierced Davey's ears, making his teeth hurt. Daring a glance upward, he saw the camper's roof lowering, coming down to squash him like a bug.\nChapter 6\n\nElva climbed into the truck, her face white with terror. She had every reason to be terrified of Cudge, but being left behind was somehow more frightening than what he might do to her. \"You were going to kill me back there!\" she hissed as she lifted herself onto the seat beside him.\n\nHis face was set in lines of panic and it gave her some small satisfaction to see him this way. Big, bad Cudge Balog, scared of what a little kid could do to him! \"Don't try and lie to me, I seen it in your face.\" She wanted to hear him say she was wrong, that she was crazy and imagined things. She needed to believe that she was safe with Cudge Balog.\n\nHe turned the ignition key, foot pressing on the gas pedal. The engine cranked and almost caught before winding down. Again, he pressed the gas, twisting the ignition key viciously as he pumped the pedal, willing the engine to turn over, dreading the thought that he might have flooded it. He was sweating. Elva looked at him, desperately wanting him to defend his actions back at Lenny's grave. \"I ain't going with you!\"\n\nBefore she could think twice she was out of the truck and running across the dusty road, heading for the cover of the trees. Bullet-swift, Cudge was out of the cab and racing after her.\n\nResolution died in Elva even before she felt his hands on her shoulders. \"This is all your fault, Elva! You're not running out on me now. Get it through your head. Get in the truck and don't open your mouth unless I tell you. In a couple of hours we're gonna be wanted for murder because of that kid and his mutt. Murder, Elva! And it's all your fault.\"\n\nCudge wasn't sure which way to go\u2014north or south? Maybe he'd stand a better chance if he ditched the camper. No, he'd worked too hard to get it, and he wasn't going to part with it. Why did that mangy mutt have to show up? Of all the bad luck! And that kid\u2014what had happened to him? If only he could have gotten his hands on him. \"By now that kid is spilling his guts to his aunt. Hear me, Elva? That kid is blabbing and his aunt is going to the cops. We got another hour of freedom and then . . . pow!\" His arm shot out toward Elva but she ducked and managed to miss it.\n\nElva huddled against the door, unable to move, fearing that if she did, Cudge would try to hit her again. If the cops got her, she would be locked up. If she made a move, Cudge would kill her. Why couldn't she win, just once? At least the kid had gotten away. If it hadn't been for her, he would be dead and his parents would be crying over his body, trying to figure out what had happened. She was a heroine of sorts. She had saved the kid and got the good-looking guy. Only Cudge wasn't a good-looking guy and she didn't want him. Still, the kid had got away, thanks to her, and she felt good about it. She wished she could bless herself and maybe go to confession. She could do it in her mind. Cudge wouldn't have to know she was praying and confessing. In the name of the Father and the Son and the Holy Ghost, amen. Bless me, Father, for I have sinned. It has been many years since my last confession, since . . . since BJ. Father help me, somebody help me, she cried inwardly.\n\nThere was no priest behind a screen in a small confessional. She was in a dirty pickup with a murderer. What good was pretending to go to confession? She needed a real priest to give her penance. The kid was safe; that's what was important. If there was a God up there somewhere, then He would know she had saved the little boy.\n\n\"Get the map out, Elva, and make it snappy. You know where we are; find some back roads and give me directions. We'll head south and maybe our chances will be better once we hit Delaware and Maryland. We'll ditch the pop-up as soon as we can. I'll smear the license plates with mud and maybe we can hole up in some other campground. For now, it's the only thing I can think of. Don't even say you're sorry, because I don't want to hear your sniveling. I'm dumping you, Elva, first chance I get. You ain't nothin' but trouble.\"\n\nElva clenched her teeth and then bit her tongue. Dump me, my ass, she thought bitterly. Kill me is more like it. Still, she was glad she'd helped the kid. She would do it again in a heartbeat. She felt suddenly defiant as she flipped the map over. Maybe she was stupid like Cudge said and, then again, maybe she wasn't. \"If you take Route 535 south for a while, you can either pick up 33, or at that point look for some other back roads. There ain't too much on this map, or if there is, I can't see it, the print is too small. This map must be twenty years old. The amusement park ain't even on it.\"\n\n\"Do you see any campgrounds listed?\" Cudge asked.\n\n\"No.\"\n\n\"Then get out the camp guide and find one. Do I have to think for you, too?\"\n\nElva dug under the seat and pulled out a tattered loose-leaf book. With nimble fingers she found the page she wanted. \"There's two KOA camps open and the others are closed for the year. This is October.\"\n\n\"Shit!\"\n\nDavey was wedged in between the bunks on either side of the pop-up. It hurt when he took a deep breath and something was pounding inside his chest. If only Duffy were here to hug. He sniffled, wishing he had a tissue to blow his nose. The dark didn't scare him, only the smell of mothballs was making him sick.\n\nMotion, rocking\u2014the camper was moving! The tires were bouncing over the road; the bad man was taking him away. Davey realized that the man didn't know he was trapped inside the camper. If he just stayed very quiet and waited, he would have his chance to get away. Wait, instinct told him. Wait.\n\nThe rocking wasn't so bad now. It didn't seem as though the wheels of the camper were bounding over holes and ruts. No, it seemed almost smooth, like when he rode his bicycle off the back lawn and onto the paved drive. It was a road\u2014a highway, Davey thought, making the connection.\n\nHis legs hurt. The leather strap from the brace was cutting into his good knee, making it throb like a drum. He wanted to cry, but instead he bit his lip and tried to work the strap free of his good leg. If he could just catch the metal brace against the side of the refrigerator, he might be able to push with both hands and roll free. Time and again he tried and failed. A sob caught in his throat. If only Duffy were here. Again he tried hooking the side of the brace that curved around his shoe against the greasy refrigerator, and this time he was successful. The metal brace clanked against the refrigerator with a loud bang. Would the man and woman hear? What would they do to him? Would she help him get away again? Somehow he knew the man wouldn't let that happen.\n\nIt was cold in the pop-up; a draft was coming up from the crack where the sides and floor of the camper met.\n\nHow long was he going to have to hide in here? He had to go to the bathroom. He wished he could see what time it was. When he didn't get back at the time Aunt Lorrie had said, she would start looking for him.\n\nHe had to get out of here and back to Aunt Lorrie. He sighed deeply. How nice it would be if she suddenly appeared from out of nowhere and took him in her arms and held him close. He would sniff and sniff until he couldn't sniff anymore. Then he would hug her as tight as she hugged him.\n\nManeuvering a little, he tried to relieve the pressure in his abdomen, but it wouldn't go away. A look of horror crossed his face when he felt a warm trickle. He'd tried, but hadn't been able to hold it. Tears stung his eyes as the wetness seeped into his clothes. Only babies wet their pants.\n\nLorrie clapped her hands in delight when Davey's fishing pole bobbed up and down in the water. She looked around, calling his name, wanting him to be the one to reel in the fish.\n\nA quick glance at her watch told her it was just after ten. Davey should have been back by now. \"Davey! Davey!\" she called. When he didn't immediately appear, she reeled the fish in herself, then removed the hook and set it free. \"Oh, well. Maybe next time,\" she said to herself as she gathered up their fishing gear. Lorrie shielded her eyes against the bright sun. \"Just where is that boy?\"\n\nAs if in answer to her question, Duffy came scampering toward her. \"Hey, there, Duff, where's your master? He was supposed to be back here five minutes ago.\" Duffy stood at Lorrie's feet, wagging her tail. Fully expecting to see Davey at any moment, Lorrie laughed and headed back toward the motor home. Maybe Davey was waiting for her there.\n\nHe wasn't. She set her gear down, put her hand over her eyes and scanned the campground for a sign of her nephew. \"Davey! Davey, it's time to leave,\" she yelled, her voice rising.\n\nShe looked down at Duffy. Davey never left Duffy alone, and Duffy never left Davey alone. Something was wrong. She could feel it. Oh, God, she had to find him. What if something happened to him?\n\nShe felt herself begin to panic, but couldn't help it. She raced around the campground, shouting Davey's name over and over. Eventually she noticed that the little dog was right on her heels. \"Some watchdog you are. Where is he, Duffy? Find Davey,\" she pleaded. \"Go on, girl, find Davey.\"\n\nDuffy raced ahead, her short legs whirling the dry leaves in her wake like spindrift. Faster and faster the little dog raced, Lorrie right behind her. Gasping for breath, Lorrie skidded to a stop when Duffy pulled up short, wildly barking.\n\nLorrie walked around, peering into the brush and beneath the low-spreading evergreens. Nothing. Over and over she called Davey's name, her voice becoming hoarser with each agitated call. Davey was nowhere that she could see. She looked down at Duffy. She was staring at something but Lorrie couldn't see what it was. \"What is it, girl? Is it Davey?\"\n\nDuffy woofed with excitement and ran around in circles. Lorrie knew that dogs had sharper eyes than people, but for the life of her she couldn't see anything. She scratched her nose. It must be the smell of mothballs that was making Duffy act strangely. \"I don't see him, Duff,\" she said, scooping the dog up into her arms. Her heart raced as she headed back to the motor home, wondering what to do. Practically every day the news carried a report of a child being abducted, sexually assaulted and killed. Every day! This was the nineties for God's sake. Things were different from when she'd grown up. If only she'd thought about that before she'd let Davey go off exploring.\n\nOh, God, she groaned. What was she going to tell Sara? \"I lost your son.\" Where was Davey?\n\nStop it! she told herself. You're jumping to conclusions. In all likelihood, Davey had just forgotten the time. After all, it was his first camping trip and there were lots of things to see and do, things he'd never seen or done before. She would wait a little longer. Eventually, he would look at his watch and come running. If he wasn't back in a half hour, she would take action.\n\nLorrie let her thoughts jump ahead, creating scenarios she prayed wouldn't happen. First, she supposed she should visit the other campsites and ask if anyone had seen Davey, then she should alert the campground managers and enlist their help in a search. If they didn't turn anything up, then she should call the police. Once it got dark, she would call Sara and Andrew. Or should she wait till morning? No, they had a right to know as soon as possible. After all, Davey was their son. Their son, not hers. Never hers again after this. Christ, she'd be lucky if Sara ever let her set eyes on the kid again. Maybe on his eighteenth birthday, Sara would take pity on Lorrie and let her see her nephew come of age. Her thoughts were getting more ridiculous by the second. Goddamn it, where could the kid be? Why had Duffy come back by herself? Did Davey send her back for help or had the little dog just tired of the walk and wandered back on her own?\n\nShe waited.\n\nAfter precisely thirty minutes, she headed out to visit the other campers. They'd gone. Both couples. The smell of mothballs made her realize this was the same spot she and Duffy had stood at earlier. She hadn't noticed before, but the site was littered with trash. She remembered the pickup truck, the way it was painted, and wasn't at all surprised.\n\nLorrie took off at a run for the manager's office and told them her problem. They grabbed their jackets, locked the door and headed out to search for Davey.\n\n\"Don't worry, ma'am. We'll find him. Kids wander off all the time but they always turn up,\" the manager assured her.\n\nLorrie nodded, thankful for his assurance, but not altogether convinced it would turn out that way this time.\n\n\"Now, tell us what he looks like and what he's wearing.\"\n\nDavey sneezed then immediately clapped his hands over his mouth. He sneezed again as dry, gritty dust blew up at him through the crack between the floor and sides of the pop-up. Angrily, he pounded small fists against the side of the refrigerator and tried to kick out at the cardboard boxes near the toes of his shoes. It was so dark inside the pop-up, he couldn't even see his shoes. He'd be willing to bet they were dirty. Maybe even ruined. As soon as he found Aunt Lorrie he would throw them away. Would three quarters buy new Reeboks? Three quarters for two shoes sounded right to him. He sneezed again, then again. He was hungry and he wanted a drink. He wanted out of this dark, smelly place. He lashed out with his foot at the carton filled with Cudge Balog's barbells. His foot shot back as quick as a rattler. Gently, he nursed his aching foot by holding it in both hands. \"I want out of here!\" he shouted. \"Let me out of here!\" The only response was a jolting thump as the pop-up hit a bump; another spiral of dust came in through the crack.\n\nSuddenly, his head jerked up and he strained to hear. Was the camper slowing down? His tongue worked frantically in his mouth as he tried to wet his lips. He needed more spit if he was going to shout so someone could hear him. The woman would let him out when she heard him call. But how was he going to know where the bad man was? What would he do if it was the man who raised the top? The woman would try to help him, he was sure of it. The camper was slowing down. He would be quiet and wait.\n\nCudge eased up on the throttle. \"Keep your eyes peeled for a gas station, Elva. This ain't the time to run out of gas. You listening to me?\"\n\n\"Yeah, I hear you. I don't see anything but grass and trees. I'm hungry,\" she whined.\n\n\"Ain't we all. My advice to you is to suck in your gut because it might be a long time before we eat.\"\n\n\"Can't we stop and get something from the pop-up? How long would it take?\" Elva persisted. \"I can't remember the last time I had something to eat. I'm really hungry, Cudge. If I don't eat, I'm gonna be sick. I can feel it in my stomach.\"\n\n\"Jesus Christ! You don't hear too good, do you? After that dumb stunt you pulled this morning, you don't deserve to eat. You had the kid, Elva. You actually had him in your hands, and what do you do? You let him get away. That kid is spilling his goddamn guts to the police right now, and all you can think of is your stomach.\"\n\nElva slouched back against the seat. Cudge was probably right. Scared as he was, he was right. Maybe, when they stopped for gas, she could crank open the top and get something. How long could it take? A minute, two or three at the most. It took that long for the tank to be filled. She decided she would risk it. She needed her strength to run if she found the opportunity. Her stomach seemed to settle down with her decision.\n\nShe wondered where the little boy was right now. Was he talking to the police like Cudge had said? She had tried to help. Cudge would never understand; he was too concerned with not being blamed for Lenny's murder. Her stomach heaved as she remembered the body in the open grave. Poor Lenny, he didn't even have a coffin. The worms and bugs would eat through the blanket real quick. Her stomach heaved, then eased as she swallowed hard.\n\nElva risked a quick glance at Cudge. She really wasn't hungry. She might not be the smartest person in the world, but right now, this minute, if somebody offered her a Big Mac, she wouldn't be able to swallow past the fear in her throat. Cudge had tried to kill her back there and he would try again. She must never forget that, never pretend to herself she hadn't seen that look on his face when he wanted to put her in that hole with Lenny. It was okay to pretend sometimes, when reality hurt too much, but this time pretending could get her killed.\n\n\"Dammit, Elva!\" Cudge spat. \"I can't depend on you to do anything. See that gas station? That's where you get gas. I thought I told you to keep your eyes open.\" Elva shrugged. When you were going to die, what did it matter if you saw a gas station or not? Cudge's foot moved from the gas pedal; he swung it to the right and brought it down with all his force on Elva's ankle bone. She yelped in pain as she jerked her foot away. \"Next time you do what I tell you.\" Cudge laughed at the expression on Elva's face. \"Now, try and act normal.\"\n\nThe pickup truck bounced over and through deep ruts as Cudge maneuvered around the entrance ramp to the gas station. A homemade sign with big red letters said shocks were a specialty of the station. \"Rip-offs,\" Cudge muttered, \"they probably dug the damn holes themselves.\"\n\nThe truck pulled alongside the pump. It was so old it looked like something out of a Presley movie, none of the fancy digital stuff that was on the gas pumps in the cities. No, this one was a real antique. For that matter, so was the station itself.\n\n\"You sit tight, I gotta take a leak first. Jesus, this place don't even look like a gas station.\" Elva looked around for some sign of life. The place looked empty. Disobeying Cudge's orders, she opened the door and got out. Limping, Elva walked to the opposite side of the pumps. Maybe she should pump the gas herself to save time. She was just about to lift the nozzle from the rack when she heard music and saw a needle-thin youth coming out of the garage, carrying a boombox. \"What'll it be?\"\n\nElva almost laughed. She couldn't remember when the last time was that she'd seen a gas station attendant. Usually, there was just some guy sitting in a glass box taking money.\n\n\"Fill it up with unleaded, please,\" she said in a loud voice. She moved away from the pump toward the pop-up.\n\n\"Let me out of here,\" she heard someone say. She looked around. The voice came again, louder. It was coming from inside the pop-up. It was the kid. Oh, God. Elva's brain felt like cold, wet spaghetti as her eyes went to the door marked MEN.\n\n\"Hey!\" the muffled voice called again. \"Let me out!\"\n\nElva slouched against the side of the pop-up. \"Is that you, little boy?\" She waited, hardly daring to breathe as the attendant danced around on one foot, watching the nozzle with unseeing eyes.\n\nDavey's eyes closed in relief. She'd heard him. Where was the man? They must have stopped at a gas station. He could smell the fumes and \"fill it up with unleaded\" was what his dad always said when he stopped to get gas. \"Please let me out,\" he yelled excitedly.\n\n\"You say something to me?\" the boy asked, turning down the volume on the boombox. \"You want the water and oil checked? Hey, are you all right? You look kinda sick.\"\n\nElva's eyes remained glued to the restroom door. \"Sick? No, I'm not sick. The water and oil are okay.\" As if he cared whether or not she was sick. He was just being polite. She felt sorry for him; he kept scratching at his acne. For want of anything better to say, she blurted, \"You got a problem or what? How come you bounce around like that on one foot and then the other?\"\n\nThe boy held up his boombox. \"I got music in my soul. My boss, he don't understand. He likes rock'n'roll. You sure you're okay?\"\n\n\"Yeah, I'm okay. I like rock'n'roll too. Especially Elvis Presley. I have every single one of his tapes but I usually listen to them with earphones,\" Elva confided.\n\n\"Whatcha doin' way out here?\" the boy asked.\n\nElva clamped her mouth shut. What business was it of his what she was doing out here\u2014wherever here was. She'd better play it cool. It would be just like Cudge to get jealous.\n\n\"Let me out, please let me out!\" came the muffled plea to Elva's left.\n\n\"Shhh,\" she whispered into the crack. \"Be quiet. I have to think. If Cudge hears . . .\"\n\n\"What did you say?\" the pimply faced youth asked as he slammed the gas pump back onto the rack.\n\n\"Nothing,\" Elva replied. Her eyes flew to the old-fashioned numbers on the gas pump. Twenty-four dollars ninety. Cudge would have a fit. Let him. Right now she had enough problems. Where was he anyway?\n\n\"I heard you say something and I saw your lips move. My ma used to talk to herself before they took her away. You better watch it. She said she didn't talk to herself either.\"\n\nElva's heart fluttered. What if this kid said something when Cudge got back? What if he said she was talking to herself? Cudge wasn't dumb. \"Yeah, you're right, I wasn't actually talking, I was kind of singing. I miss playing my tapes. I was just saying the words to a song to myself. That's what you saw me doing. I ain't like your mother, believe me, I ain't.\"\n\nThe boy looked skeptical as he held out his hand for the money.\n\n\"You have to wait a few minutes till my . . . till Cudge . . . Here he comes.\" She didn't know if she was sorry or relieved. \"Hey, why don't you turn your set up a little so I can hear that song? I haven't heard anything but the CB for two days.\" Anything to drown out the feeble, muffled pleas of the little boy.\n\n\"You got it!\" Never taking his eyes from Elva, the attendant turned up the volume. \"And now for all you Metallica fans, here's their latest . . .\" The disc jockey bellowed so loud Elva clamped her hands over her ears.\n\n\"Shut that goddamned thing off,\" Cudge shouted.\n\nThe gas station attendant's eyes widened. Then his eyes locked with Elva's. Defiantly, he turned the volume up even louder. Blaring music ricocheted around the pumps.\n\n\"I thought I told you to shut that thing off!\" Cudge bellowed.\n\n\"That's what you told me all right, but this is my turf, buddy, and I don't give a damn what you say. Pay up and get that junker of yours out of here.\"\n\nCudge balled his hands into hard fists. Who the hell did the kid think he was with all those ugly pimples on his face? He was just about to raise his fist when he saw the boy looking at the license plate. \"Okay, okay, play your damn radio. I got a headache and that's why I asked you to turn it off, but never mind. Here, keep the change.\"\n\n\"Big spender, a whole quarter,\" the kid smirked.\n\n\"It's twenty-three cents more than what your service was worth!\" Cudge shot back.\n\nElva stared at the boy. As defiant as he was, she knew he would have helped her if she'd asked. Why hadn't she asked? Why had she just stood there and done nothing? Now it was too late. Her window of opportunity had closed with Cudge's return. The kid stared back at her, pity in his eyes as she climbed into the cab of the pickup truck.\n\n\"You're a loser, Elva. I saw the way you was sucking up to that kid. Well, let me tell you something. I saw that bastard look at our license plates. He's going to remember us. You in particular.\"\n\n\"No, he isn't,\" Elva said defensively. God, what was she going to do about the little boy? She had to get him out before he suffocated. Reason, crystal clear, seemed to come back to her. He couldn't suffocate with the cracks in the outer shell of the pop-up. She knew there had to be hundreds because of the way road dust filtered inside and stuck to everything. The boy might be stiff and sore but he wouldn't die, not like BJ. Not if she could help it anyway. She saw herself letting the little boy out of the pop-up, spiriting him away and taking him back to his family. They would call her a hero and give her a reward and everyone would live happily ever after. Everyone but Cudge.\n\nShe was cold, almost as cold as she imagined Lenny must be. Cudge was perspiring. Served him right, she thought viciously, as she rolled up her window.\n\nThe pickup came to life beneath Cudge's hands. Slowly, it moved past the attendant, who made a show of turning down the volume of his boombox.\n\n\"Let me out of here,\" Davey yelled. \"Open the door and let me out!\"\n\nThe attendant's eyes widened in question. He followed the trailer for a few feet, his head cocked, then stood watching as it pulled out onto the highway.\n\nThe station owner walked out to where the kid was standing. \"Anything wrong?\" he asked.\n\n\"I thought I heard . . .\"\n\n\"What?\" the owner prompted.\n\nThe boy shrugged. \"Ah, it must have been my imagination. It sounded like there was someone in that pop-up. A kid, but . . . Nah! That ain't possible.\"\n\n\"Why not?\" the man asked, staring after the pickup.\n\n\" 'Cause there ain't no room in there once those things get folded up. If there was somebody in there, they'd be smashed. Must have been my imagination.\"\n\n\"It was probably one of those new radio commercials,\" the owner said, pointing to the kid's boombox. \"Back to work,\" he said, then headed toward the grease pit.\n\n\"I think I'll write down their license plate anyway. You never know.\"\n\nDavey knew they were on the move again. \"Let me out of here!\" he screamed. \"Open the door! Let me out!\" Why hadn't the girl opened the door? She'd heard him. He knew she'd heard him because she'd answered him. So why?\n\nDavey thought for a moment. Maybe she was scared the man would see her letting him out. Maybe she was afraid he would punish her if he saw her. Davey scrunched himself even tighter into the space between the refrigerator and the hard, wooden bunk. It was dark and smelly, but it was better than having the man catch him. The girl would let him out as soon as she could. She might even take him back to Aunt Lorrie. His eyes drooped wearily\u2014he was so tired, and he didn't feel well.\nChapter 7\n\nStuart Sanders watched the people in the anteroom with clinical detachment. It was a hell of a job, all things considered. When you came all the way from New Jersey to Florida you expected to get a little sun and fun, not to sit inside a courthouse waiting for the prize witness to be recalled to the stand.\n\nThe waiting was hard on the Taylors, too. Annoyance was clearly written on both their faces, yet Sanders would have staked a week's salary that he was the only one aware of it. There they sat, chatting in companionable comfort, occasionally puffing on a cigarette.\n\nAs he watched them, he realized it was difficult to imagine Sara Taylor without her husband, just as it was difficult to picture Andrew without Sara at his side. He tried, unsuccessfully, to complete the family picture by imagining Davey between them. His eyes narrowed as he watched a stocking-clad leg move just a fraction closer to the trouser leg. No, there was no place for a little boy. Sanders saw an intimate smile play around Andrew's mouth as he looked up from his magazine, acknowledging the pressure of Sara's leg. There was no trace of a smile on her face as she lowered her gaze to her own magazine.\n\nSanders had seen their silent communication when Andrew Taylor had been questioned on the stand by the prosecutor, Roman DeLuca. The preliminary questions had focused on Andrew's university position, and were designed to show the jury that he was a solid, upstanding citizen, testifying at personal cost and possible jeopardy to himself and his family.\n\nThen the prosecuting attorney's questions shifted to Andrew's relationship with Jason Forbes. It had become clear to Sanders that, whenever a question was put to Taylor, he sought out his wife, who was sitting in the first row, just behind the table where Roman DeLuca's assistants were taking notes.\n\nAndrew Taylor's poise and confidence were unshakable, even under the hostile stare of the accused killer, and his confidence was reinforced by small, almost imperceptible nods of Sara's head. Her eyes never left her husband and, from the proud set of her shoulders and the slight smile around her lips, Sanders realized that she was giving Andrew her approval.\n\nSanders turned his attention to DeLuca, saw him look back and forth between Andrew and Sara, much as he had done himself. DeLuca was sharp, almost as sharp as him. Sanders studied DeLuca's expression and knew that he, too, had seen the silent signals between husband and wife. Sanders wondered how unshakable Andrew would be in his testimony if Sara weren't in the courtroom.\n\nHe hoped the trial wouldn't continue much longer. He'd always considered himself a patient man, but he decided now that he liked his company a little more jovial. He disliked the hushed whispers and dusty rooms, the smell of furniture polish and dry, cracked leather. Most of all he disliked the stale air and the austerity of the countless courthouses in which he had spent half of his life. And, to be honest, he disliked the Taylors. The thought surprised him\u2014he'd never thought himself capable of such intense dislike. It bordered on hate. Everyone had faults, he supposed, and God knew, he was far from perfect himself. That was it! The Taylors were just too perfect to suit his tastes.\n\nAt that moment he would have given his soul to be with Lorrie Ryan and Davey, visiting the zoo, laughing at the animals' antics, eating popcorn and drinking sodas. Or had the trip to the zoo been planned for yesterday? He couldn't remember.\n\nNow, where had all that come from? he asked himself. He glanced at the clock above the judge's head. In the weeks he'd stayed with the Taylors, he'd grown especially fond of Davey. Maybe because he felt sorry for the little kid. In the Taylor family, Davey was the odd man out, so to speak. Wanted, yet a bother. A flawed intruder. A wrinkle in the fabric.\n\nSanders had never had any kids of his own; his one marriage, ten years ago to a career-minded book editor, hadn't produced any children. His wife hadn't wanted them, and at the time, neither had he, though now he couldn't remember the reason why.\n\nSpending time in Davey's company had brought out feelings in him that he hadn't known existed\u2014feelings about home and family.\n\nLorrie was also an intruder. Where her feelings went, she wore her heart on her sleeve, loving Davey as if he were her own. Men weren't supposed to see or understand the subtleties that went on between women, but any fool could see that Sara disliked her sister and especially disliked Lorrie's love for Davey.\n\nIt wasn't right, he thought. Any kid who had gone through what Davey had gone through deserved a lot more love and nurturing than Sara offered. Davey should have been Lorrie's kid, not Sara's. Sara only loved Andrew and Andrew . . . Sanders wasn't sure what was up with Andrew; he seemed deaf and dumb to what Sara was doing. Why they'd had a child at all was beyond him, yet he was sure Davey hadn't been a mistake. The Taylors didn't make mistakes.\n\nSanders wished he had a cigarette, but he'd emptied his pack of Marlboros a half hour ago. He'd have to go out into the hall to see if he could bum one off one of the court clerks.\n\n\"Mrs. Taylor, I'm going outside for a smoke. If you need me, I'll be right outside the door.\" That was another thing that angered him\u2014why did he always defer to her?\n\nSara nodded, not bothering to glance up from her magazine.\n\nOutside the courtroom, Sanders spotted a clerk standing next to a potted palm, smoking a cigarette. The man nodded at Sanders's request but was slow to get his pack out of his pocket. As Sanders waited, he imagined he could taste the acrid, hot smoke on his tongue.\n\n\"Thanks,\" he told the clerk as he put the filter tip between his lips. A lighter flashed and he drew deeply, the spiraling smoke making his eyes narrow. A vibration against his waist told him he was being beeped. He pulled the cell phone out of its belt case, opened it up and pushed the button to retrieve his voice message.\n\n\"This is Lorrie Ryan. I need to speak with you. It's an emergency,\" the voice said.\n\nSanders recalled the premonition that had prompted him to give Davey and Lorrie his business card.\n\nHe pushed a second button to see the phone number, then pushed it again to dial it automatically. \"Lorrie, this is Stuart Sanders. What's wrong?\"\n\n\"Oh, Stuart. It's Davey. He's gone. I've looked everywhere and can't find him.\"\n\n\"What happened, Lorrie?\"\n\n\"I\u2014Oh, God.\" Her voice was an anguished cry.\n\n\"You have to calm yourself, otherwise I can't help you,\" he said in his FBI voice.\n\nShe sniffled. He could tell that she was trying to regroup. \"Right after breakfast, we went fishing,\" she began. \"There's this cute little pond . . . We weren't catching anything and Davey got bored so I suggested he go for a short walk with Duffy. He knew we would be leaving soon . . .\"\n\n\"Lorrie?\"\n\n\"Duffy came back without Davey.\"\n\n\"Christ,\" Sanders swore beneath his breath. \"How long has he been gone?\"\n\n\"Duffy never leaves Davey's side,\" Lorrie said, not answering his question. \"This is all my fault. I should have been more careful. I shouldn't have let him out of my sight, not even for a minute. If anything's happened to him, I'll never forgive myself.\"\n\n\"Let's not borrow trouble, Lorrie. You don't know that anything has happened to him. How long has he been gone?\" he asked again.\n\n\"Since about nine thirty this morning.\"\n\n\"For God's sake, Lorrie. It's almost four. What took you so long to call me?\"\n\n\"What? I . . . we . . . the managers and I\u2014we've been searching the campground and\u2014\"\n\n\"Who else knows he's missing?\"\n\n\"The police. I called them when we couldn't find him.\"\n\n\"I wish you'd called me first, Lorrie.\" Sanders glanced at his watch. Three fifty. Another five minutes and the court would adjourn for the day.\n\n\"Called you first? But I thought it was more important to call the police and get them started looking for him. I called you because . . . because I need you to tell Sara and Andrew.\"\n\nSanders's sharp gray eyes traveled the length of the corridor, coming to rest on the nattily dressed young lawyer who was assistant to the defense. In the old days, he would have been called a mouthpiece. Not for the first time Sanders wondered why a promising young attorney would align himself with known syndicate bosses. It occurred to him that Davey might have been kidnapped by a branch of the syndicate and was being used as a pawn to discourage Andrew Taylor from testifying.\n\n\"I'll tell them.\" He pulled a notepad out of his pocket. \"Tell me exactly where you are, Lorrie.\"\n\nLorrie quietly gave Sanders the name and location of the campground. \"Stuart . . . this couldn't have anything to do with . . . What I mean is\u2014you don't think Davey's disappearance has anything to do with Andrew's testifying in this case, do you?\"\n\nSanders hesitated before answering. \"Anything's possible, and the thought has occurred to me.\" He was instantly sorry he'd admitted his suspicions. \"I want you to sit tight. Leave your cell phone on so I can get back to you.\" He swallowed the lump in his throat. \"Lorrie?\" He wished he could reach out and touch her, comfort her.\n\n\"What?\"\n\n\"Everything's going to turn out all right. I promise. You'll get Davey back unharmed.\"\n\n\"I pray to God you're right, Stuart. I can't tell you how much I love that little boy.\"\n\n\"You don't have to. I already know.\" He paused. \"I'll be back in touch with you in a couple of hours.\" He pushed the \"end\" button and stood there, staring into space.\n\nDavey had been kidnapped. It was exactly what the FBI had been guarding against; the reason he'd been moved into the Taylors' house. What kind of scum would snatch a kid? The kind of scum Andrew Taylor was testifying against, he answered himself. They were everywhere. And kidnapping was only the tip of the iceberg.\n\nHis steps were heavy as he made his way down the hall. In a husky whisper he repeated the phone conversation to his associate, Jake Matthews, giving him orders to report to headquarters. \"I'll be waiting here, so make it snappy. I don't want to tell the Taylors until I get an okay from upstairs. There's a good possibility the boy is just lost.\"\n\nMatthews nodded and took off at a run. He was back ten minutes later. \"Something is brewing around here, Sanders,\" the younger man told him. \"I get the feeling that everybody is watching and waiting. What's your hunch? Do you think that friends of the syndicate have snatched the Taylor kid?\"\n\nBefore he replied, Stuart Sanders took a last drag on his cigarette, watching the conversation outside the courtroom door between Lester Weinberg, counsel for the defense, and his young assistant. \"Matthews, it's enough to know that the bad guys probably already know the kid is gone. Our aim, right now, is to keep the Taylors from knowing until we get the go-ahead. You'd better get word to Roman DeLuca. As state prosecutor, he's got to know that his case might fly out the window if Taylor refuses to jeopardize his son by testifying.\"\n\nLorrie's white-knuckled fingers dug into the flesh of her upper arms. In spite of Stuart's words, she couldn't find the comfort she so desperately needed. All she could do was wait. She kept going over her phone conversation with Stuart. Was it possible that Davey had been kidnapped by someone affiliated with the syndicate? If Davey had been kidnapped, then it was reasonable to assume that he was safe somewhere. But if he'd just wandered off\u2014what had happened to him? Where was he? More often than not, kidnapped children were found dead. It was easier to think that Davey was lost and that, before too long, he'd come wandering out of the woods, safe and sound.\n\nLorrie watched the police prowl the woods, their dogs sniffing and straining at their leashes. The local police had called in the state troopers for extra manpower. It was late in the day, and Lorrie had been told that if the boy wasn't found by morning, the police would alert the media and widen the search. The blood had rushed to Lorrie's head at the thought of Sara seeing her son's picture on national news.\n\nWhere was Davey? What could have happened to him?\n\nDavey Taylor was as close to a flesh-and-blood son as she was ever going to get. He was part of her, no matter what Sara or Andrew said to the contrary. She loved him as much as Sara did\u2014maybe more. A sob rose in her throat and tears spilled down her cheeks.\n\nLeaning against the door frame, looking out, Lorrie thought about Stuart Sanders. She wondered what difference it would have made to call him first. He was in Florida. How could he help find Davey from there?\n\nIt was nearing dusk when Lorrie saw a tall, lanky police officer come into the clearing. Suddenly uniformed figures were everywhere. \"What? What is it?\" Lorrie yelled.\n\nThe lanky officer backed off a step when he saw Dr. Ryan running toward him. \"Stay here, ma'am. This isn't for you to see. No,\" he answered her unasked question, \"we didn't find the boy. This is something else. Stay here,\" he repeated firmly.\n\n\"No,\" Lorrie said just as firmly. \"I want to see what it is. Whatever it is, will it help you find Davey?\"\n\n\"I don't know,\" the officer replied.\n\nIn spite of his orders, Lorrie followed him through a stand of trees to what looked like a hole . . . or a grave.\n\n\"Put some muscle into it, Delaney. If there was a six-pack in that hole you'd have it out by now!\" a local policeman with a noticeable beer paunch bellowed.\n\n\"Whatever's down here isn't going to come up any faster with you yelling, Jackowsky. And who the hell appointed you my superior? You want speed, grab a shovel and do it yourself. If not, shut up and let me do it my way.\" Delaney hefted his shovel and gently prodded into the soft dirt.\n\nStanding at the edge of the hole, Lorrie saw the outline of a body. An adult's body, not a child's. She breathed a huge sigh of relief.\n\n\"Simpson, get in there with Delaney and help him,\" Jackowsky ordered. \"We don't want any marks on the body made by our department.\"\n\nSimpson sneezed. \"Jesus! What's that smell?\"\n\n\"It's mothballs!\" Delaney lifted up a shovelful of dirt containing several of the white balls. \"Suppose this is somebody's idea of how to preserve a body.\"\n\nJackowsky didn't laugh.\n\n\"Okay, so it's not funny. What we got here is a white male, twenty-five to twenty-eight years old. Hey, his eyes are open!\"\n\n\"Cover his face,\" Jackowsky said, offering a used handkerchief. \"This ain't no funeral parlor, you know,\" he grumbled. \"Lift him out while I call this in.\"\n\nMinutes later, Jackowsky returned. \"The chief is on the way with the coroner, and if you're one of those guys that's gonna puke, do it somewhere else. What's that you got, Simpson?\"\n\n\"Looks like the guy's wallet. Hell, I picked it up and now my prints are on it.\"\n\n\"Then throw it back in.\" Jackowsky sighed as if to say how tired he was of dealing with rookies like Simpson and Delaney.\n\nLorrie whispered a prayer of thanks. It wasn't Davey. Thank God, it wasn't Davey.\n\n\"Now, ma'am,\" Jackowsky said. \"You'd best be getting back to your RV and taking a pill or something. Or you won't be able to sleep tonight.\"\n\n\"Officer, I appreciate your concern, but I'm a doctor. I've seen dead bodies before. Who is he?\"\n\n\"I don't know, but we intend to find out. Ma'am, why don't you go back to the camp and we'll get back to you as soon as we clear this up.\"\n\nLorrie looked around and counted a total of seven policemen and troopers. \"I know this is going to sound ungrateful, but there are seven of you here. Why do you need seven people to dig up one body? Why aren't at least four or five of these men out looking for my nephew? I'm sorry that the man is dead, but nothing can be done for him. Davey is just a little boy, and he needs all the help he can get, and you aren't doing anything.\"\n\nLorrie felt herself bordering on hysteria. She decided to leave the gravesite before she said something she was sure to regret. Behind her, the seven men were silent.\n\nLorrie shivered. As soon as she got back to camp, she would make herself a cup of coffee\u2014no, a pot of coffee\u2014and build a campfire. A big campfire, one that could be seen from a long distance. It would warm the searchers as they came back with their reports, and it would serve as a beacon for Davey, a way for him to find his way back through the dark of night.\n\nToday had been the kind of day you prayed would never happen, the kind of day that always seemed to happen to other people. Tomorrow had to be better. But, unless Davey was found, tomorrow would be worse, and every day after that would be worse than the one before. And then there was Sara. Oh, God, Sara. Sara would arrive and . . . She didn't want to think about it.\n\nLorrie imagined her sister receiving the news. Of course, she would take the next plane out of Miami. Lorrie shivered again, not with cold but with fear of Sara and how she would react\u2014or rather, retaliate. Because that's what Sara would do. Lorrie felt guilty thinking that of her sister, but she knew her well enough after all these years. It would be Sara against Lorrie and the police; Sara would stay on the sidelines, yet somehow she would manage to control everything and everyone. There would be no closeness, no weeping together, no hoping together. Sara never shared her emotions with anyone but Andrew. She would be the judge and the jury, and Lorrie would never get an acquittal. Sara would find her guilty and judge her accordingly.\n\nLorrie turned her thoughts to building the fire. She piled up the kindling and set more rocks around the firepit. When everything was ready, she set a match to the construction and watched the flames spread through it.\n\nShe wished Stuart Sanders would call again and reassure her, like he had earlier. This was one of those times when she really needed a good man to lean on. Someone she could share her fears with. Someone who would hold her while she cried her eyes out. When this was all over, and Davey was safe and sound, she was going to think seriously about pursuing a relationship with the man.\n\nLorrie looked up to see three patrol cars coming her way. Their lights were flashing but their sirens were silent. Must be the coroner and his entourage, she thought. A short while later, Officer Delaney came over to speak to her. \"We still have men combing the woods.\"\n\nLorrie nodded.\n\n\"The coroner says that the man we found in the grave died of massive head injuries. Either he hit something or something hit him. We're pretty certain it was the latter. As you know, there was a wallet in the grave. The description on his driver's license seems to indicate that the deceased is Leonard Lombardi of Newark, New Jersey. Do you know him?\"\n\n\"No, I don't. It was murder, wasn't it?\"\n\n\"It's looking that way. Right now, there's an all-points bulletin out for the two other campers who were here last night. According to the manager here, one was an older, retired couple. The other couple, they were a lot younger, real lowlifes from what the manager's wife says. Seems they were camped in the vicinity of the grave. Both couples left early this morning. We're checking it all out.\"\n\n\"What about Davey?\" Lorrie asked anxiously. It seemed to her that the excitement of finding the body was taking precedence over the search for her nephew.\n\nDelaney became defensive. \"We're working on that, Dr. Ryan. There's always the possibility that your nephew's disappearance is linked with the dead man. I'm not saying that's the case, ma'am, only that it's a possibility.\"\n\nLorrie's voice was low and controlled when she questioned the young officer. \"Did the manager say if he knew where the other couples were heading? Campers, as a rule, usually make inquiries about where their next stop is going to be.\"\n\n\"The older couple, the Kovals, said they were going to Virginia Beach. They should be setting up camp there around about now. We have a call in to the local police and we're waiting for them to get back to us. The young couple is another story. They paid in advance and didn't stop on their way out. The guy didn't even have a hookup, so we don't know when he left or where he was heading.\" Delaney looked up as someone boomed his name. \"Hold it, looks like something just came in. Stay here and I'll be right back, ma'am.\"\n\n\"I'll wait, officer,\" Lorrie said quietly. \"I won't get in your way.\"\n\nShe dug a trail in the powdery dirt at her feet, fighting to keep her emotions in check. Delaney returned shortly.\n\n\"What did you find out?\" Lorrie asked, hoping, praying.\n\n\"The young couple I was telling you about\u2014well, the pickup is registered to an Edmund Balog of Newark. Yesterday he was stopped on the turnpike by a state police officer. Seems there was some uneasiness on the part of the trooper. He was checking Balog's license when an emergency call came in. There was an accident further down on the pike and he had to cover it. We have a call in to the officer now. He'll be going off duty soon and will call in. I'm afraid that's all I have to report for now. Why don't you go back to your RV? I give you my word that as soon as I hear something, I'll report to you. The other teams will be checking in, and the new crew will be coming on duty. We may need you later. What do you say?\" he asked hopefully.\n\n\"Did you get an address for the couple in Newark?\" Lorrie asked.\n\nOfficer Delaney looked pained. \"Yes, ma'am, and right now there's a team of officers on the way. I don't know if you know anything about Newark, but the address is in the Ironbound section. Tough neighborhood, if you know what I mean.\"\n\n\"I know the area. I worked at a free clinic there for six months.\" Lorrie assumed the officer suspected Edmund Balog of kidnapping Davey or he wouldn't have brought him up. \"Officer, if this goes out to the media, you need to let them know about Davey's medical condition.\"\n\n\"When was he supposed to have his shot?\"\n\n\"Noon.\"\n\nDelaney looked at his watch. \"It's five-thirty now.\"\n\n\"I guess all I can do is pray that nothing happens to Davey to cause him to bleed. I don't have to tell you that, without the proper medical care, he could bleed to death in a very short time.\"\n\n\"We're doing everything we can, ma'am. I wish we could do more, but we can't. Things like this take time.\"\n\n\"Thank you, Officer Delaney. I appreciate you keeping me posted.\" Lorrie began to go back inside the motor home, then turned to the officer. \"I just made a fresh pot of coffee. Would you like a cup?\"\n\nDelaney nodded. \"Make it hot and strong. The manager's wife thinks coffee is colored water.\" He gave Lorrie a conspiratorial wink, then excused himself a moment and hurried back to the police car where Jackowsky was gesturing to him.\n\n\"She okay?\" Jackowsky nodded toward the motor home.\n\n\"As good as can be expected,\" Delaney replied. \"Shouldn't we have something on the kid by now? Too many hours have gone by for him to have simply wandered off.\"\n\nLorrie fixed herself and Officer Delaney a cup of coffee. She stood in the doorway of the van, waiting for him to return with whatever news he had just been given. Over and over she pleated the hem of her cotton shirt, her long, tapered fingers creasing the material, smoothing it out, and pleating it again.\n\n\"I used to love this time of year,\" she said when Delaney returned. \"Now I hate it. It's starting to get chilly. It was downright cold last night. Davey had only that windbreaker on; it just has a thin flannel lining. I wonder if he's hungry.\"\n\n\"Nothing much new to report,\" Delaney said as he took the cup of coffee from her hands.\n\n\"Whatever it is, please tell me.\"\n\n\"That older couple that was going to Virginia Beach\u2014they turned out to be duds. Zero. They didn't see your nephew. They had some thoughts about the young couple, but that's all they were. Mrs. Koval says she heard a lot of yelling and screaming, sounded like they were having a verbal altercation. Mr. Koval said he was sleeping, and his wife always hears yelling and screaming. He blames it on all the soap operas she used to watch and something about a fish tank. Sometimes this happens. People don't want to get involved and they mind their own business. The Kovals are a mind-your-own-business couple\u2014and those are the words of Detective First-Grade Harry Thatcher. They checked out fine. Sorry.\"\n\n\"Thanks for coming back to tell me, Officer Delaney. You'll let me know about any further developments, right?\"\n\n\"I'm going off duty shortly. One of the other men will check in with you. We've set up temporary offices next to the manager's quarters. If you need us, or if the boy comes back on his own, you'll know where to find us. Try not to worry.\"\n\nSanders's ulcer was beginning to act up, and he had the two-and-a-half-hour plane ride back to New Jersey to look forward to. And an airline dinner. He popped two Rolaids into his mouth, hoping to ward off what he knew would be an acute case of indigestion. The thought of sitting beside Sara Taylor for the entire trip played hell with his whole body chemistry. Perhaps she would sleep, or he would. Although the latter was out of the question\u2014he'd have to remain awake to play bodyguard. Ridiculous. It was like playing nursemaid to a barracuda in open water.\n\nHe wanted this case to be over and done with. He wanted Davey back home, safe and sound. He yearned to take Lorrie Ryan into his arms and tell her everything was going to be all right, but he didn't know if it would be. It had to be, dammit. It just had to be. Davey Taylor was one hell of a special little boy and Sanders couldn't bear the thought of anything bad happening to him.\n\nHis footsteps were silent in the thickly carpeted hallway which led to the Taylors' hotel suite. Could it have been only half an hour ago that he'd presented them with the facts about Davey, carefully omitting any mention of the possibility of foul play? He'd intentionally reinforced their suppositions that Davey was lost or had wandered off. Whatever it was that he'd expected, it hadn't happened. Sanders knew that he had wanted to be the one to tell them because he'd wanted to see Sara Taylor go to pieces. He should have known better, he told himself, and wondered when he'd become so vicious in his thinking. If it had been any other woman, he would have dreaded telling her that her son was missing. But not Sara Taylor. For once he wanted to see her rattled, confused, and desperate. Out of control.\n\nAndrew Taylor was the one Sanders pitied. The man kept running a frantic hand through his hair, his features drawn and pained. Sara had been the strong one, comforting her husband, telling him that she agreed with Sanders and was certain Davey had wandered off.\n\nFor a long moment, Sara's eyes had commanded Sanders's attention. Was he wrong, or had he seen an accusation in their depths, even while her hand tenderly stroked Andrew's arm? A thought came to him. Andrew Taylor's witnessing of that scene in the university library had upset the order of Sara's household. Things were beyond her control and she blamed Andrew.\n\nMatthews stood outside the Taylor suite, arms crossed over his chest. \"They wanted to be alone,\" he explained.\n\nSanders nodded and rapped on the door. Andrew opened it, the anguished lines of half an hour ago gone from his face. Sara had worked her magic with him once again. But how, Sanders wondered, did you get a man to forget that his son might be in grave danger?\n\n\"Mrs. Taylor, our plane leaves at six ten. A car will pick us up. The airport is only a few minutes from here. Can you be ready?\"\n\nSara's eyebrows shot up. \"Mr. Sanders, I won't be leaving with you after all. My husband and I talked it over, and I'm going to remain here with him.\"\n\nSanders was incredulous. Wild horses wouldn't have kept his sister away if one of her kids had been in trouble.\n\n\"And Mr. Sanders,\" Sara continued in her cool voice, \"tell my sister I'll be calling her shortly.\" The words held a threat; Sanders pitied Lorrie Ryan, and Davey too. Every child needed a loving, tender mother figure in his life. Somehow he just knew that if Sara had anything to say about it, and she would, Lorrie Ryan might as well forget she'd ever known and loved little Davey Taylor.\n\n\"Now, Mr. Sanders, I don't want you to waste your time trying to persuade me to go back with you. My mind is made up. I must be here with Andrew. We started this together, and we'll finish it together. What kind of wife would I be if I deserted my husband now when he needs me? You'd better hurry; you don't have much time and there's bound to be traffic at this hour. We wouldn't want to be responsible for you missing your plane. After all, it is your job.\"\n\nSanders turned to Andrew. \"Mr. Taylor, do you agree with your wife's decision to remain here?\" Make her go, dammit! Make her be a mother to that kid for once in her life. But Sanders did not give voice to his thoughts. He'd been warned that the Taylors weren't to be unnecessarily alarmed\u2014at least Andrew Taylor wasn't. He was the key witness in the trial, and his testimony was all important in the case against the multimillion-dollar drug ring. The State needed to prove a connection between the murdered man and the syndicate, and Andrew would provide the connection.\n\n\"Both you and my wife have assured me that Davey has just wandered off from the campsite. What would Sara's presence accomplish? Lorrie will do all she can; we know that. Right now, as I see it, the imminent danger to Davey is his missing his shot. Unfortunately, nothing can be done about that until he returns.\"\n\nSanders couldn't believe his ears. Taylor was talking as though Davey were a naughty child who had wandered away, and would return when he was good and ready! Sanders wanted to take Andrew and shake him and hear his stiff neck crack. Still, he couldn't go against departmental orders. Even a hint that Davey might be held by friends of the Miami syndicate could discourage Andrew's testimony.\n\nAs time went on, Sanders was becoming more and more skeptical of that possibility. Surely, if Davey had been kidnapped, the Taylors would have been apprised of that fact; word would have been gotten to them, either directly or through the Bureau itself.\n\n\"As my husband has told you, Mr. Sanders, we'll be staying here until Andrew takes the stand. Tomorrow afternoon, at the latest, and then we'll be back in New Jersey. By that time, Davey will have returned.\"\n\nSanders stiffened. What kind of mother was Sara Taylor? That poor little kid. Christ, he felt like smashing something. Smashing her! Something was on fire in his stomach\u2014his damn ulcer was acting up. At least he could sleep on the plane and hope for the best. Just because she gave birth to Davey didn't make her a mother. Schooling his face to impassivity, he said goodbye and walked away. He hoped he would never have to see either one of them again. Back in New Jersey he would be in the field, his responsibility to the Taylors over. From this moment on, his only contact with them would be through Lorrie Ryan.\n\nLeaving their hotel room, he sent up a silent prayer. Hang in there, little buddy. I'll find you somehow. He dismissed the lump in his throat. Must have something to do with the ulcer, he told himself.\n\nAndrew Taylor watched his wife through narrowed lids. Another day and they would be back home, safe in New Jersey. Now why had he used the word \"safe\"? Here they were, locked in a hotel room with an armed guard outside the door. How much safer could he be than he was at this moment? He still wasn't certain of Sara's decision to stay with him. Was it a mistake for Sanders to go back alone? Sara had made the decision to stay at his side; he'd known she would. Her argument that two heads were better than one convinced him. Sara was always right. Everything that could possibly be done for Davey was being done. Sanders would take charge. Davey was alone, lost in the woods, but Sanders would find him and take him home safely. Sanders was a man to depend on.\n\nAndrew's shoulders slumped as he recalled the look on the agent's face when Sara had told him she wasn't flying back with him. Her duty, she had said, was to stay with her husband. Sanders should have understood that, but Andrew knew he hadn't. At first the look had been one of disbelief, but that had given way to sour acceptance. His goodbye had been barely audible. Sara was right; she was always right. Davey had just lost his bearings and would turn up before dark.\n\nSara walked around the edge of the bed and lightly touched Andrew's shoulder. \"Would you like roast beef or chicken for dinner?\"\n\n\"What do you suggest?\"\n\n\"I think a broiled spring chicken sounds fine. With a garden salad and some fresh peas\u2014it's much too hot for a heavy dinner. Andrew, you aren't upset with me because I decided to stay on with you, are you?\"\n\n\"Of course not, Sara. Why would you even think such a thing? I know everything is being done to find Davey. The truth of the matter is, we would probably be in the way. Besides, he'll turn up before dark. I know I always did.\" He laughed. \"I used to get lost at least once a week and always managed to find my way back.\"\n\n\"And Davey is his father's son,\" Sara said briskly. \"After dinner I'm going to call Lorrie. I want to know how this happened. She always was a featherbrain. I'm so disappointed in her that I hardly know what to do. It just bears out, Andrew, what I told you before we left home. We must curtail her involvement with Davey.\"\n\nAny doubts Andrew might have had concerning Lorrie's attentions toward his son were wiped away with Sara's words. It was uncanny how she was always right. \"I quite agree, Sara.\"\n\n\"Police!\" Sara made the term sound obscene. \"It isn't bad enough that we're surrounded with guards and police twenty-four hours a day, but now we have to be subjected to more of them. What would you like to drink, Andrew? Why don't we order wine and we can have it for the rest of the evening. It will help us to unwind and relax.\"\n\nAll Andrew heard was the word \"relax.\" He had been looking forward to a double scotch, straight up. A couple of liters of wine while he went over his testimony would certainly help. Sara was one step ahead of him all the time. Scotch wasn't really a drink to sip in a hotel room. He hoped the wine-glasses had long stems.\n\n\"I think I have it all now. You don't want dessert, do you?\" Sara said, putting the hotel pen back in the desk drawer.\n\nIf he ordered dessert, he would still be eating after Sara was finished.\n\n\" No. \"\n\n\"I didn't think so.\" Sara's eyes twinkled down at him. Fondly, she stroked his hair. \"I'll give this to the guard and you can watch the news while we wait.\"\n\nJust as Sara was about to open the door a sharp knock sounded. She glanced at Andrew\u2014he was already engrossed in the early evening news. It was probably just the guard wondering why she was taking so long with the dinner menu. She smiled as she opened the door, the menu extended ready to hand over, then backed off a step, her eyes flashing between the bellboy and then to the guard.\n\n\"A message came in for you a few minutes ago, ma'am,\" the bellboy offered.\n\nSara looked at Jake Matthews. \"It's all right,\" he assured her. \"We've checked it out. It just came in. Is that your menu? Good, the boy can take it down when he leaves. I'll tip him, Mrs. Taylor. You go back inside now and lock the door.\"\n\nSara made a face the minute the door was shut\u2014as if she needed to be told to lock it. The message was probably from Lorrie and full of all kinds of apologies, or else to say Davey had returned to the campsite. She sat down next to Andrew and withdrew a slip of paper from the envelope. She read it once, then read it again. The message was brief: It's urgent that you call 943-0773 as soon as possible. There was no signature, just the time the message had been logged in.\n\nSara read it again. Was it a local number? It was certainly a Florida number or the operator would have included an area code. What could it mean? Should she show it to Andrew? He would worry. Roman DeLuca. Of course. It didn't matter that she detested the suave district attorney. She would wait till a commercial came on before telling Andrew.\n\nThe anchorman's voice droned on as he reported on the plight of the Albanian refugees. A sleek Mercedes flashed on the screen as a pitch came on for a local car dealer. \"Darling, look at me,\" Sara began. \"It seems we have a message. A bellboy gave it to me when I opened the door to give our dinner menu to the guard. Read it and tell me what you think.\"\n\nAndrew read the curt message. He shrugged. \"Did you show it to the guard?\"\n\n\"Of course not, Andrew. After all, we are entitled to some privacy. Perhaps it's something personal, although I can't think what it could be. It appears to be a local number, or at least a Florida number. At first I thought it might be a message from Lorrie, or even Mr. Sanders, but I don't think that's the case.\"\n\n\"Well, there's only one way to find out\u2014call the number and see what it is,\" Andrew suggested.\n\n\"I don't know if that's such a good idea, Andrew. Perhaps we should wait for Roman DeLuca to get here and show it to him. I really don't want to get involved in anything else right now. It could be some pervert, or a weird person who watches the news and does things like this. You know how some people get a thrill out of tormenting others. It could even be a threat against us.\"\n\n\"I don't want anyone threatening you, ever,\" Andrew said adamantly. \"We'll wait for Roman to get here and let him handle it.\"\n\nSara smiled. \"How gallant of you, Andrew. I think it's wise to wait, too.\"\n\nAndrew let out a long sigh. She agreed with him. He smiled at her in an intimate way and she responded in kind. Gently, she blew him a kiss. He smiled again as the caravan of Mercedes cars left the screen to be replaced again with the silver-haired, somber-faced news commentator.\n\nSanders settled himself in his aisle seat aboard the 747. He hoped the flight would take off on time and that they wouldn't have to wait in line on the runway. He hated waiting. It seemed all he ever did was wait. Wait for this. Wait for that. Waiting was something most people thought he did well because he never complained. But he didn't do it well. Not at all. His insides always felt as though they were on fire. He detested inactivity almost as much as he hated certain people. Once he hated someone, God Almighty couldn't get him to revise his opinion. And by the same token, when someone got close to his heart, that person stayed close forever. His sister said he loved with a vengeance. Maybe he did, he thought sheepishly.\n\nOut of habit, he reached for his pack of cigarettes, took one out, then remembered: no smoking. What the hell! The half dozen he would have smoked during the flight would be that many fewer nails in his coffin. He'd always thought that when he went down for the count it would be with the big C. He was already coughing and hacking in the morning. The thought of his own death didn't bother him. It was the death of other people that made him want to lash out.\n\nDavey Taylor. Was the kid suffering, wherever he was? Sanders leaned back and grasped the cross he always wore around his neck. He didn't like takeoffs any more than he liked landings. It used to be that as long as he could smoke and sleep in between, he didn't care. But things were different now. The silver bird climbed and climbed then leveled off. The \"No Smoking\" sign, a laughable relic of the last decade, went off. Maybe if he had a bourbon and water, he would be all right for the next two and a half hours. He opened his briefcase and pulled out the Polaroid he'd taken of Davey Taylor wearing his red windbreaker. A deep, paternal feeling for the little boy swelled in him. There was something about this kid, something that bothered him. Something he just couldn't put his finger on. He would have to chalk it up to a gut feeling, and twenty-three years with the Bureau.\n\nBack in Miami he'd felt that someone, or some force, was at work against the kid. Over the years he'd learned to trust his feelings, even if others didn't. When he'd told the chief about it, he'd listened, nodded and asked for some concrete evidence or substantiated facts. Sanders didn't have anything concrete\u2014just a feeling, a gut feeling. Both agreed it was possible that the little boy was being used as a pawn, but Sanders thought it more improbable than probable. So what was it then\u2014this feeling of his?\n\nAll of a sudden Sara Taylor's face appeared before him. She had a cool, patrician kind of beauty, he had to give her that. But that was all he would give her. She'd said her place was with her husband. How could she ignore her son like that? Was it possible she knew something Sanders didn't? Had she really bought into his bogus theory that the kid had just wandered off? Who knew what she was thinking behind that frozen mask she wore. Sanders might not know what made Sara Taylor tick, but there was one man who thought he did know\u2014Roman DeLuca. Sanders had watched the suave district attorney during his first meeting with the Taylors. DeLuca sized Sara up in five minutes flat, and immediately discounted her\u2014much to Sara's chagrin.\n\nSanders stared at the snapshot of Davey for a while longer. He couldn't shake his feelings. He kept thinking that he'd see something in the photo that would make it all clear. Some answer, some clue. But there was nothing. He laid the picture back on top of a yellow legal pad, groped for a paper clip and attached it to a wad of sheets. He didn't want to lose it. Satisfied that the picture was secure, he fished and fumbled inside the briefcase, looking for his favorite treat\u2014salted peanuts. He found two bags and laid them on the seat next to him.\nChapter 8\n\nCudge Balog wished he had three eyes\u2014one for watching the road, one for watching his back, and the third for watching Elva. He didn't like the way she was acting, nor did he care for the creepy way she kept looking at him. And, he thought grimly, she hadn't bawled at all when he'd kicked her ankle. That wasn't like Elva. Crying and whining were the things she did best, besides making Kool-Aid. Time to give her another jolt.\n\nHe took his eyes from the road for an instant and sneaked a look at her. She was nervous, of that he was certain, but it was annoying him that she was in control of it. \"I was thinkin', Elva. You know, in prison they give you anything you want to eat for your last meal. What would you order, Elva? This is as good a time as any to decide because you been belly-aching how hungry you are. What would you order?\"\n\nElva knew exactly what Cudge was trying to do. Unnerve her. Make her fall apart, and then, when she was hysterical, he would kill her. She couldn't let that happen. She had to help the little boy get out of the camper and get away. She couldn't let Cudge do anything to her until the boy was safe.\n\nCautiously, she turned in her seat and stared at Cudge for a long time. It was Cudge who was becoming unnerved now as he kept glancing at her then quickly looking back at the road. Her voice was light, almost airy, and seemed to come from far away. \"I think I'd order veal cutlets, because of what they cost. And shrimp. Big juicy ones with a real spicy sauce to start with. Baked potatoes, with lots of sour cream. And rolls. Hot ones with butter. And that dessert I always hear about, Cherries Jubilee; whatever it is, it sounds good. And beer. A really icy cold beer. Coffee, with rich cream, none of that fake stuff, to top it off. Maybe a peppermint Lifesaver to settle my stomach as I walk down death row. What are you gonna order, Cudge?\"\n\nCudge glared at Elva. She'd turned the tables on him. She made her question sound as though it was certain he'd have to order his last meal because he was definitely going to be convicted. \"You're dumb, Elva. You'd be so piss-ass scared you wouldn't be able to eat a thing. And if you did, you'd puke your guts out.\"\n\n\"Maybe you'd be too scared to eat, but I wouldn't.\" That should settle him down for the time being, she thought. She had to think about the little boy and how to get him out of the pop-up. She wouldn't mess up this time, not again. Poor little thing, he must be miserable, cooped up in there. I won't let anything happen to you this time, BJ, she crooned to herself. This time I'll make sure you're safe. Don't worry, BJ, I'm gonna take care of you. Somehow, I'll make him let you go.\n\nCudge didn't like Elva's wiseass answers. By now, she should have been a glob of putty, jabbering away at him to stop tormenting her. Stupid broad. Instead, she was sitting there in a dream world, smiling to herself. It was her smile that started to awaken his anger. Maybe she was thinking about Elvis Presley again. No, that wasn't the usual dumb smile she had when she was dreaming about Elvis. This was more ominous. Maybe she was plotting his death, just the way he was planning hers\u2014this very minute. Old Elva might be skinny as a pencil, but she was wiry and strong, like a bull terrier.\n\nHe hated Elva and everyone else. But mostly, he hated that little bastard who was finking on him to the cops. If he'd had a gun, he would have killed the two of them on the spot. People like Elva and that brat kid could ruin a person's life. He'd get out of this mess yet, and without Elva.\n\nHours passed as Cudge concentrated on his driving. For Elva, it was impossible to think clearly with Cudge being so controlled at the wheel. The road map in her lap was nothing more than a blur. Her neck and shoulders were stiff with the effort she was making to sit straight in the jouncing truck seat. Her ankle throbbed painfully and her tooth was aching. She was sure of just one thing: she had to find a way to open the pop-up and set BJ free. Even if she died doing it. Once in a while God and the fates allowed you a second chance, and this was hers. She couldn't flub up, not now. Maybe she should pray for a miracle. Or just pray. Did she remember how? Maybe there wasn't a God after all. If there was a God, what was He thinking of, to allow BJ to be penned in the camper and her to be stuck with Cudge Balog? God was as make-believe as Ali Baba and the Forty Thieves.\n\nRight now Cudge would have bet the remainder of Lenny's bankroll that Elva was up to something. He didn't like the ramrod stiffness of her back, nor the way she stared straight ahead. This silence was different from the times when she had her damned earphones on. \"There's a rest stop ahead at the next exit. I think we can get some eats. Elva, are you listening to me?\"\n\n\"I ain't hungry.\"\n\n\"What do you mean, you ain't hungry? You been busting me to stop and eat, and now you ain't hungry. Why ain't you hungry, Elva?\" he asked suspiciously.\n\n\"Because I ain't.\"\n\n\"Well, I am, and we're going to stop. We can get a carryout order. You go in and get it, and I'll wait for you. We need to fill up the water bottles anyway. Make up your mind, we're stopping.\"\n\nElva hoped her relief wasn't evident. If she had learned anything at Cudge Balog's knee, it was that to disagree would make him more determined. Just a little more playacting and maybe she could free BJ. \"I can't do it, Cudge. My foot hurts too bad, and I'll be limping. Someone will remember that I limped. You have to be careful and use your head now.\"\n\n\"I got a bad feeling about you, Elva, like you're planning something.\"\n\n\"Yeah, like what I'm gonna buy you for Christmas. What I said is the truth. If you want to be stupid, go ahead, but don't say I didn't warn you. I didn't ask you to kick me.\"\n\nCudge eased up on the gas pedal as he turned off at the exit sign. It was a Denny's.\n\n\"Where you gonna park this thing?\" Elva asked quietly. \"Maybe it would be best if you park right in front. That way no one will think it's at all suspicious. It's just a suggestion.\" Now, if she was half as smart as she thought she was, Cudge would park at the end of the parking lot, away from all the families with their little kids. The last slot would be perfect. She could open the pop-up, scoop up BJ and run like hell.\n\n\"Maybe that's what you would do, but it ain't what I'm gonna do. I'm parking at the end of the lot. There ain't enough cars around to lose ourselves in the crowd. We don't need to be noticed. You keep your yap shut, and do what I tell you when I tell you. You got that?\"\n\n\"Does that mean you're going in for the food? I told you I can't walk. Look at how swollen my ankle is. I think you broke a bone or something.\"\n\n\"You're walking, Elva, and I'm gonna be right behind you. And I don't want to hear any more of your crap about someone noticing you either. Ain't nobody gonna pay any attention to you.\"\n\n\"Oh, yeah, what about those two guys?\" Elva said, pointing toward the entrance.\n\nCudge's eyes followed her finger. Two troopers in tight gray pants and the inevitable polished sunglasses were just getting out of an unmarked car. His features tightened. One hand curled into a fist. The two men walked slowly through the entrance. There were booths alongside the long row of windows. Would the troopers sit at the counter or take a booth? Either way, he had to make some kind of move. If they sat down in one of the window booths, they had a clear view of the entire parking lot. Should he go in alone, as Elva had suggested, and take out the food, or should he drag Elva with him? Christ, what should he do? It was the smirk on Elva's face that forced his decision. She wouldn't dare try anything. \"You're coming with me. We'll eat inside. Act like we didn't do nothin' and everything will be all right.\"\n\nElva squared her shoulders. \"We didn't do anything; you did it.\" She felt more powerful at that moment than she had ever felt in her life. \"Ain't you afraid I'll spill my guts to those cops? I could, you know. Just remember that, Cudge. You should have killed me back at the campground when you had the chance.\"\n\nCudge's pent-up rage swelled and his eyes rolled back. Elva watched him, neither afraid nor relieved. Nothing mattered to her anymore, except freeing the little boy who looked like BJ.\n\n\"Okay, okay. You stay here and watch the rig. Make sure no one comes nosing around.\" Each word was slow and distinct, spoken with great effort. He had to close the gate now, before it was too late. A loud sigh escaped Cudge's lips. What he had to do was convince Elva he was going to take care of her.\n\n\"I been thinking, Elva, about what you just said, that I was gonna kill you. You're wrong. I wasn't gonna kill you. You ain't never done anything bad to me like Lenny did. Jesus, Elva, you're all I live for. Look what you've done for me so far. Without you, my ass would be in jail. I ain't never gonna forget that. I mean it, and as soon as we get out of this mess, I'm gonna bankroll this wad and take you to the Poconos, to that lodge where they have heart-shaped bathtubs. I was even thinking of marrying you. I don't expect you to believe me, but it's what I want and I think it'll work. What do you say, honey?\"\n\n\"I ain't as dumb as you think I am, Cudge. I know wives can't testify against their husbands. I ain't ready to get married, and if I never see a heart-shaped bathtub, I'll live. You can't snow me anymore, Cudge, so forget it. If you're gonna get the eats, get 'em. Them troopers must be wondering what you're doing just sitting here. Most people either go to the bathroom or the restaurant when they stop at a place like this. You're gonna look suspicious if you don't get moving.\"\n\nCudge knew she was right. If she was telling the truth and she couldn't walk, then he was safe. \"Okay, I'm going in. You get out and stretch your legs like everyone else. Hop around on one foot and stay put. Don't you have to go to the bathroom?\" he asked as he suddenly remembered that she hadn't gone when they'd stopped for gas.\n\n\"I don't have to go. I didn't drink any Kool-Aid today.\"\n\n\"You better be here when I get back, Elva.\"\n\nElva nodded. She would be here, but the kid wouldn't. There was no point in fooling herself\u2014there was no way she could walk, much less run, with her swollen ankle.\n\n\"While you're getting the food, I'll open the pop-up and get out the water bottles to save time.\"\n\n\"Like hell you will. I can just see you cranking it the wrong way and bam!\u2014one pop-up shot to hell. You know you can't do nothin' right. I'll do it when I get back. If you even think of trying to open my rig, your neck is gonna look like your foot. Now, you get out and we'll talk a minute and then I'll walk real slow into the restaurant, and you damn well better act like you know what the hell you're doing.\"\n\nCudge stretched luxuriously, as if he had all the time in the world. He worked first one leg and then the other. He hitched up his jeans and threw back his shoulders like a man who was bone tired from traveling all day. Walking to the back of the rig, he thumped on the pop-up. It was his, all his. The one thing in his whole stinking life that was his alone. No one was going to take it away from him, and no skinny bag of bones was going to crank it open and screw it up. \"You stay right here, Elva. I'm gonna be watching from the restaurant.\"\n\nDavey woke with a start. The trailer wasn't moving anymore. He squirmed around, trying to get comfortable. Why had they stopped? Was it nighttime? Where was the woman? Would she let him out? He was about to shout when he heard the man's voice. He had to be quiet and not make a sound. He had to act the way he did in the hospital and be brave. Be quiet and still. No matter how scared he was. One, two, three strikes and you're out! His dad often said that. This was his third time. If the woman didn't get him free, he was out. His dad had never said what happened when you were out. But Davey thought he knew. He was tired, almost too tired to care. He frowned in the darkness. It was a different kind of \"almost.\"\n\nAs soon as Cudge entered the restaurant, Elva made her way to the back of the rig. \"Little boy, can you hear me?\"\n\n\"I hear you. Let me out of here! I can't move. Please let me out.\"\n\n\"I'm gonna let you out soon as I can. Can you be quiet for just a little longer? I know you must be hungry, but I can't do anything about it right now. Cudge just went inside to get some food. I think he's going to open the pop-up, and you gotta be ready to run. Do you think you can run? Little boy, can you run?\"\n\nDavey's thoughts were jumbled. He knew he couldn't run. He was too tired and too weak. But he had to say yes or she wouldn't help him. \"Yes,\" he said loudly.\n\n\"Good. There's two cops here, and if you run inside the restaurant, they'll take you back home. But you have to be quick. Real quick. Are you okay? You ain't sick or anything are you, little boy?\"\n\nDavey thought a minute. \"I'm almost sick. I didn't get my shot today.\"\n\n\"What shot?\" Elva asked fearfully.\n\n\"I'm a hemophiliac.\"\n\n\"You mean you bleed?\" Elva asked in horror.\n\n\"I used to, but now I get shots. Please, can't you let me out before he comes back?\"\n\n\"You have to trust me, little boy. I'm gonna help you.\" If the kid didn't get his shots, he might die. She would be just as guilty as Cudge this time around. He was just a kid, a little boy like BJ. She had to do something. The hell with what Cudge said. What did she care if the cranks fell apart? Her hand was on the crank ready to turn when Cudge walked through the doorway. Behind him were the two troopers. Elva froze. \"Little boy, something's wrong. Be real quiet.\"\n\n\"Here we go, honey. Food at last,\" Cudge shouted with false gaiety. Elva stared at him. Didn't he see the troopers behind him?\n\nSuddenly a voice shouted. \"Sir, sir, the cashier wants to see you.\" Cudge ignored the voice. Elva gagged and her face drained of all color. As if in slow motion, she watched one of the troopers put a hand on Cudge's shoulder. \"Hey, mister, you deaf or somethin'? The cashier wants you back inside.\"\n\nCudge turned slowly and stared at the two troopers. His brain was swelling, every instinct prepared him to fight, to defend himself. The very sight of a uniform could do this to him, but to have the trooper speak to him\u2014touch him! A hoof cut into the soft tissue of his brain. A hulking, dark shape shouldered the restraining gate. Only a superhuman effort quelled the restlessness of the beast and held him behind the gate. \"Me?\" he asked stupidly.\n\n\"Yes, you. The cashier wants you back inside.\"\n\nThe trooper's partner smirked as Cudge turned and walked back into the restaurant. \"They're all alike, those campers. A couple of days on the road and they're in another world. Kind of airless around here, isn't it?\" His fingers worked at his collar.\n\n\"I thought it was me,\" the first trooper said quietly. \"You ever hear about the Santa Ana winds? They say they make people do crazy things, kinda like a full moon.\" He laughed sheepishly. \"Come on, let's get it in gear and hit the road. I got a date tonight that would set your hair on end. She's got the biggest knockers on the East Coast.\"\n\n\"Seeing is believing,\" the second trooper grinned. \"I don't take a cop's word for anything. Everyone knows you can't trust a cop.\"\n\n\"Yeah, right,\" the first trooper said with a grin as he climbed behind the wheel. The car came to life just as Cudge hit the parking lot. \"Would you believe they shortchanged me?\" he said, holding out his palm with thirty-seven cents in it for proof.\n\nThe trooper stared at the change for a full second. His eyes behind the polished glasses were cold. He nodded curtly and switched from park to reverse. Cudge stood back respectfully and watched till the yellow Plymouth left the parking lot.\n\n\"Close your mouth, Elva, and get in the truck.\"\n\n\"What about the water bottles? We ain't even got a drop. We need the water. You crank open the top and I'll get the bottles. You'll have to fill them though.\"\n\n\"Christ, you're stupid. Two cops just left here. We were eyeball to eyeball, and you want to hang around here to get water? We ain't opening that pop-up till we make camp. When the shit hits the fan, those guys are gonna remember me. Now, get your ass in that truck and let's move!\"\n\nAs Elva started to hobble back to the truck, a noisy family got out of a maroon station wagon close by. Seven children squealed and shouted as they romped about the parking lot, trying to catch a frisky, fuzzy-looking dog. Cudge and Elva were suddenly surrounded by yelling kids and a barking dog. \"C'mere, Bizzy. Good girl. C'mon, we got some popcorn for you.\"\n\n\"Jesus Christ!\" Cudge shouted to be heard above the noise. \"Get that damn dog out of here and get him out now! All of you get out of here. I want to start this rig up and I'm running late.\"\n\n\"Bizzy is just sniffing your pop-up, mister. You got something in there she likes. She's not hurting anything. See, she's just trying to get in to see what you got,\" a boy in tattered overalls grinned.\n\n\"Well, I ain't got nothin' in here for your dog, so get her out of here.\"\n\n\"We can't catch her,\" a little girl with pigtails complained. \"She's fast, mister. Her mother's name was Flash; that's why we picked her from the litter.\"\n\n\"I don't give a damn what her mother's name was. Get her out of here!\"\n\n\"Kids, kids, what's going on here? Where's the dog?\" The kids' mother approached, gathering her children into a close group like baby chicks. Her eyes flicked over them as if taking a habitual head count. Her work-worn hands pushed back frizzy hair from her forehead. \"Hey, Max, you better get your butt out here and settle this. These people want to get on their way and your kids are holding up the works. Max, you hear me?\"\n\n\"Yeah, yeah, I hear you. I been listening to you all the way from Milwaukee. What's wrong this time?\" a bear of a man demanded as he climbed from the dust-streaked station wagon.\n\n\"There isn't any problem,\" Cudge said patiently. \"We just want you to get your dog, Flash, out of here so we can be on our way.\"\n\n\"Name's Bizzy. Flash was her mother,\" the girl with the pigtails chirped.\n\n\"What you carrying in that pop-up, mister, that attracts our dog? She usually ain't interested in other people and their belongings. Kind of a mind-your-own-business mutt, if you know what I mean,\" the man said.\n\n\"Just get that dog out of here. We're running behind schedule now.\"\n\nMax LeRoy stared at Cudge in a way that said he didn't like what he saw. \"Bizzy, up,\" he commanded in a sharp, clear voice. The little dog leaped in midair to land in his arms. He nodded curtly to Cudge and, without so much as a word, the kids backed off, allowing Cudge and Elva to enter the pickup. It was Mrs. LeRoy who memorized the license plate, without realizing what she was doing.\n\nElva sighed heavily. She felt like an old newspaper that had been cut up to paper-train a new pup. How much more could she take? And the kid\u2014how much more could he take? What if he started to bleed back there, all alone, and then died on her? Please, God, don't let anything happen to the little boy. I don't care about me, just don't let anything happen to him. I know that I was thinking before that there wasn't a God, but you must be real or Cudge would have found him by now. I don't need a miracle, just a diversion of some kind. You can't let him die, he's just a little boy, hardly more than a baby? Elva choked back a tear. She knew God did let little boys die; he'd let BJ die.\n\n\"What are you doing, Elva? Tell me you ain't talking to yourself. Please tell me you ain't.\"\n\n\"Okay, I ain't talking to myself. I was praying.\"\n\nCudge's eyes widened. His head bobbed up and down. \"It figures,\" was all he said.\n\n\"Where we going and how much longer are we going to stay on the road?\" Elva demanded.\n\n\"Soon as I see a spot that looks like it's off the beaten track, I'll stop. You feeling frisky, Elva?\" he baited her. \"That why you wanna stop so quick?\"\n\nShe ignored him, her mind racing ahead. The kid had said he could run, so he must be all right. Little kids were always hungry. Whatever it was Cudge had bought, she would save her portion for the little boy to take with him.\n\nCudge was so full of it. First he tried to kill her, then he thought he could make it all better by calling her honey and telling her he'd take her to a lodge in the Poconos with a heart-shaped bathtub. As if that would make everything all right. The thought was too ludicrous even to warrant a smile. She was just beginning to realize how stupid Cudge really was. Now, when it was too late.\n\n\"Elva, get that map out and take a look-see. I just seen something that makes me nervous. Take a look over there and tell me what you see. Ain't that the same diner and ain't that the same gas station we seen when we started out? It is!\" he bellowed. \"There's that mom-and-pop camp store. Of all the friggin' luck!\" For the first time in his life Cudge felt raw gut fear. They were right back at the scene of the burial. The kid would have told his folks. There'd be cops everywhere. Fear swooped up into his throat and he gagged. What should he do? Head back to Newark? Turn around and drive all night? Camp? Where?\n\nElva felt herself go limp. This was her miracle. Think, she had to think. \"That's good, ain't it, Cudge? Who would ever think we'd come back here? We can even stop at the store and get the water and stuff.\" She hardly dared breathe. \"We don't have to camp in the Wild Adventure campground. We don't need no hookups. You could pull deep into the woods. A flashlight is all we need. What do you think, Cudge?\" she asked anxiously. \"But we better get off the road now, before it gets dark.\" The little boy will be able to find his way to the highway before the black night descends, she thought. \"Take a look at the sky, looks like rain to me. If we're going to stop, we better do it now.\" Childishly she crossed her fingers, waiting for his answer. Make it the right answer, she pleaded silently. You're almost free, little boy, she said over and over in her mind.\n\nCudge risked a glance at Elva, then made up his mind. She was probably right, but he wouldn't give her the satisfaction of telling her so. Without thinking, he signaled a right turn and was off the road.\n\nHe had done what she wanted; now she would let the little boy out. Elva's teeth clamped together in relief.\n\n\"You're right, it's gonna rain pretty soon.\" Cudge frowned. \"Elva, you notice anything, feel anything?\"\n\nNow what? \"What's wrong?\"\n\n\"Feels like we're limping, like maybe we're getting a flat. I'll have to check it out when we stop.\"\n\nThis couldn't be happening. All the tools were in the pop-up in a tool kit under the bunk. Cudge would crank it open and then it would be all over for her and the little boy. She tried for a light tone. \"You're just uptight, Cudge. I don't think there's anything wrong.\"\n\n\"We'll know soon enough.\" He drove a little way beyond the campground to a general store. \"I'll pull up around the back and take a look. Here's some money\u2014you hobble in and get a few things. No gabbin', Elva.\"\n\nThe moment Cudge braked the pickup, Elva had the door open and was outside in the damp air, pretending to check the tires on her side. She moved toward the pop-up. \"Little boy,\" she whispered, \"whatever you do, don't make a sound.\"\n\nCudge bent down to inspect the tires on his side. \"The rear tire's really low. Must have a slow leak. I'm gonna have to change it, but I think we can hold out till we make camp. Go get our stuff and make it snappy.\"\n\n\"Cudge, I gotta go to the bathroom real bad. You go in and get the stuff,\" she said, holding out the money.\n\n\"All right, but I ain't buying no Kool-Aid; goes against my grain to buy that crap.\" He watched her limp to the restroom.\n\nWhen Elva was done, she opened the door and looked around the tiny parking area. Thank God he hadn't cranked open the camper. \"Little boy, can you hear me? Say something. Are you all right? You ain't bleeding, are you? I been real worried.\"\n\n\"I'm okay. Let me out. When are you going to let me out?\" Davey pleaded.\n\n\"I was going to do it now, but Cudge is watching me. We're going to stop real soon because it looks like it might rain. You gotta be ready to run real fast. I'm sorry I couldn't let you out before, but he was watching all the time. I couldn't take the chance. You just have to hang tight for a little longer. I'll get you out, I promise.\"\n\nA promise. That was real, Davey thought. Something true that happened even when you thought it wouldn't. Like when Aunt Lorrie made a promise. \"Okay, I can wait.\"\n\n\"I ain't gonna be able to run with you, but you'll be able to make it by yourself. I got a flashlight you can use. It's the best I can do. Shhh, he's coming now.\"\n\n\"Who the hell you talking to, Elva? I seen your mouth going a mile a minute from inside.\"\n\nIf it had worked once, it would work again. \"I was singing to myself, like this,\" she said miming the words to an Elvis favorite. \"I miss not hearing my music.\"\n\n\"Get in. I got eggs, bacon, instant coffee, bottled water, beer and some cupcakes. They had a special on Coke so I got two six-packs. Here's a can for you,\" he said generously. \"I got us a little information. A quarter of a mile down the road there's a deserted quarry. What do you think?\"\n\nDeserted. Would the kid get lost, or hurt, trying to get away with only a flashlight to guide him? \"Yeah, sounds good to me,\" she said. She hoped it wasn't too far off the road or the kid wouldn't be able to make it to the campground. \"You don't want to be driving on dirt roads with that tire like that,\" she cautioned.\n\n\"I won't. Just till we're out of sight. Beats the shit out of me how we ended up back where we started. See those trees over there? That's the back of the Wild Adventure Park, and to the right of that is where old Lenny's planted. It's your fault, Elva. Somehow you screwed this all up. Everybody said that 1998 was a big year for assholes, but you carried over to '99.\"\n\nThere was no point in arguing with Cudge. The more he talked, the less time he had to think. She could listen to his harangue with one ear and still use her brain to figure something out for the little boy. God, where was it all going to end? Imagine returning to the same place they'd started out from. She knew in her gut that God was punishing her and Cudge. But the little boy shouldn't be punished. He hadn't done anything. Cudge wasn't going to hurt him, except over her dead body.\n\n\"Here we go\u2014hold on now.\" Deftly, Cudge maneuvered the truck around potholes as big as craters. \"Son-of-a-bitch!\" he exploded as he swerved to avoid one yawning hole only to hit another. His head hit the roof of the truck and Elva bounced almost as high. \"Goddammit! I think the tire blew.\" He shifted into neutral, opened the door and banged it shut. \"Shit!\"\n\n\"Did it blow?\" Elva asked fearfully.\n\n\"Damn right it did. I don't even know if I can get this rig out of here. Slide over, Elva, and gun it. I'll push from here.\"\n\nElva did as she was told. The rig jockeyed back and forth and then was free.\n\n\"Okay, we're stopping right over there. There's room to back the pop-up under the trees. If it rains, we'll have some protection. I gotta put the spare on now before it gets too dark.\"\n\nElva was out of the pickup like a whirlwind. \"Let me help you. Where's the spare, Cudge? And the toolbox? Just crank open the top and I'll get it for you.\"\n\n\"How come you're so helpful all of a sudden? You want to help, get me a beer and then shut up.\"\n\nIn her haste to pop open the beer can, Elva shook it and bubbles of foam shot up, soaking the front of her blouse. With shaking hands she held it out to Cudge, who grabbed it from her and consumed it in one swallow. \"Gimme another and don't shake it this time.\"\n\nHe upended the second can, gulped down its contents in several noisy swallows, then tossed it into the brush. Before Elva knew what was happening, he had the spare tire off its rack on the rear of the pop-up. \"If this ain't one hell of a mess,\" he said disgustedly. \"I didn't get the spare fixed the last time.\"\n\nElva felt overwhelming relief. Now he wouldn't open the pop-up.\n\n\"I'm gonna have to unhitch the rig and try to make it into the nearest town to get the tire fixed. What time is it, Elva?\"\n\nShe shrugged and looked at the sky. \"Must be after five, at least.\"\n\n\"I suppose they roll up the sidewalks early, so I better get moving. I want some supper when I get back. That greasy hamburger wasn't fit to eat. Help me unhook the camper.\" Elva's hands were shaking so badly, she was next to useless. \"Get out of here! You ain't no help at all. Okay, it's off. Now look, me and you, we gotta have a talk. I don't like the idea of leaving you here while I go into town, but I ain't got no other choice. This is my rig, see, and I gotta take care of it. It's all I got in the whole world\u2014except for you, Elva. I want you to promise that you'll be here cooking supper when I get back. I shouldn't be gone more than an hour, two at the most.\"\n\nShe'd promise anything as long as he left. \"Okay, Cudge. I'll be cooking supper when you get back. Where else can I go anyway? Don't worry, just get the tire fixed and get back here. You know I don't like the dark.\"\n\n\"First, I have to get my tool kit out the camper.\"\n\nA drowning fear engulfed Elva. She couldn't let him open the pop-up, not now. \"Cudge, if the garage man sees you with your own tool kit, he ain't gonna do nothin' for you. You know how those guys are. He's gonna think you couldn't do it yourself, and you're only going to him as a last resort because you botched up the job. You don't like people making a fool out of you. You don't need it.\"\n\nAn ominous roll of thunder helped Cudge with his decision. He threw the spare tire into the back of the pickup and climbed behind the wheel.\n\nElva's tongue was so thick she could only nod in farewell. Miracles\u2014they did really happen, and at the oddest times. Somehow she just knew that roll of thunder had come from God.\n\nShe limped down the road, watching till the pickup hit the highway, then hobbled back to the pop-up, grasping at low branches for support along the way.\n\n\"Little boy, I'm going to let you out now. Can you hear me?\" she gasped. Not waiting for a reply, she yanked viciously at the crank and lifted the top. The worn, khaki canvas unfolded like a flower. \"Where are you? It's so dark I can barely see you. Come on, little boy, I'll help you.\"\n\nDavey wriggled out from his makeshift nest and slid down to the cardboard cartons. Elva quickly shoved them aside and reached for the boy. How wonderful he felt in her arms. He was okay. \"You ain't bleeding, are you, little boy?\" she demanded as she gathered him to her.\n\nDavey let himself be hugged and petted. He liked it. He liked this girl who talked to him and told him not to be afraid. She was as scared as he was; he could tell by the way her arms were shaking, even if she was holding him tight. \"No, I'm not bleeding. I told you, I get shots so I don't bleed. Can you call my aunt now and tell her to come and get me?\" he asked hopefully.\n\n\"I ain't got no cell phone. All I can do is give you a flashlight and something to eat. We're real close to where your motor home was parked. You see those trees over there? That's where you were camped. And just over there is where Len\u2014I mean, that's where we were parked. I think that if you go through the trees, you stand a better chance of finding the motor home than going to the highway. But you gotta get going. Cudge will be coming back pretty soon and I don't want him to find you here. I know it's scary in the dark. I hate the dark and don't like to be by myself, but you're gonna have to be brave. Do you think you can make it?\"\n\nShe was hugging him so hard, Davey could barely breathe. With all his might, he pushed against her arms and gasped, \"I think I can.\" When she put him down he asked, \"What's your name? I have to know so I can tell my aunt how you helped me.\"\n\n\"Brenda. Brenda Kopec. C'mon now, we have to get you ready and on your way. Cudge will be back soon. Are you hungry?\"\n\n\"Starved,\" Davey responded, with a weariness in his voice that made Elva clasp him to her again, squeezing and crooning softly. BJ's little face floated before her eyes. An acrid smell wafted into her nostrils. Dried urine. Poor kid. She thought about all the times BJ had wet his pants because he was scared. And the beatings, the constant beatings. As if a thrashing could make a little kid stop wetting his pants. Poor little BJ.\n\nA quick glance at the sky told her the rain would arrive shortly. And then what? Could the kid find his way in the rain? It would be cold, and he'd get soaked in minutes. God, what if he started to bleed in the rain? \"Rain, rain, go away, come again some other day,\" she crooned.\n\n\"Here we go, you sit down and eat. You gotta be quick, BJ, and get away from here before Cudge comes back.\" Elva's eyes were glazed and watery as she watched the little boy wolf down the cupcakes Cudge had just bought. Starved. BJ was always starved. He could stuff himself one minute and be looking for something else five minutes later. A bottomless pit was what BJ was. A bottomless pit who wet his pants when he was scared.\n\n\"Can I have something to drink?\" Davey asked as he licked at his fingers.\n\n\"I saved my Coke for you. Drink it all, I'm not thirsty. It's kind of watery now because all the ice melted.\" Davey gulped and gulped until the watery Coke was all gone. Carefully, Elva pulled at the hem of her blouse and wiped at the cupcake crumbs around his mouth. \"There now, your mouth is clean and no one will know you were snacking.\"\n\nDavey sat quietly on the step of the pop-up, trusting the woman to help him. So what if she called him BJ? Grown-ups made mistakes sometimes. Not all grown-ups; he corrected the thought and sucked in his cheeks. His mother never made mistakes. Thunder roared overhead and he flinched. If you think a thing through to the end, you won't make a mistake. That's what his mother always said. Lightning danced across the sky, bathing Elva's face in eerie yellow light. Davey flinched again. Her eyes looked like Duffy's in the dark, all bright and shiny.\n\nHe inched his way to the end of the step and waited, his eyes on the trees. The woods were dark and gloomy, like on the cartoons at Halloween. He wondered what time it was. When the lightning flashed he saw on his watch that it was close to six o'clock. Davey frowned.\n\nElva saw the little boy check the time. Jesus. She didn't remember him being so smart, and where did he get a watch? A stolen watch would only get him another beating. BJ wouldn't steal, though. Or would he? She looked around. It was darker now; she didn't have time to worry about where he'd got the watch.\n\nDavey held out his hand. \"It's raining, Brenda, what should we do?\"\n\n\"Come on, get inside, and I'll light the lantern. I have to get you ready to go even if it is raining. I don't have an umbrella,\" she said fretfully. \"I don't have a single thing that will keep you dry. I don't want you to get sick, BJ. You know there's no money for doctors.\"\n\n\"Aunt Lorrie is a doctor,\" Davey said helpfully. \"If I can get to her, she'll take care of me.\"\n\n\"Sometimes you talk crazy, BJ. We don't have an Aunt Lorrie, just an Aunt Stella and an Aunt Helen. Don't you remember? There, the lantern is on. Isn't this cozy, BJ? Just me and you. I hate the dark as much as you hate the beatings. I think I'd rather have a beating than be locked in a dark closet with the rats and roaches.\"\n\nDavey wondered if Brenda was making up a story so he wouldn't think about Aunt Lorrie. But he didn't want to be entertained. He just wanted to be out of here. He had to leave Brenda and the security of the pop-up, but first he had to get the flashlight she'd promised him. The one Mr. Sanders had given him was in his pocket, but he didn't think the light would be big enough to find a way through the trees.\n\nRain pounded the canvas roof. \"It's raining hard, BJ. I don't think you should go out. Stay here with me. I hate the dark,\" Elva said, hugging her thin arms against her chest.\n\nDavey felt confused. He didn't understand. \"If I stay here with you, the man will catch me. I thought you wanted to help me get away.\"\n\nIt was Elva's turn to be confused. \"BJ,\" she said patiently, \"I'm supposed to take care of you, and I will. But you have to stay with me. I won't let anything happen to you. Not again. The last time it was all a mistake. I won't let that happen again.\"\n\n\"I want to go back to Aunt Lorrie. She's worried about me, I know she is. I want to go home.\" He had an idea. \"You can come with me, Brenda.\"\n\n\"What time is it?\" she demanded fearfully.\n\n\"The big hand is on the three and the little hand is on the six.\"\n\nElva screwed her pinched features into a frown. \"Quarter after six. God, Cudge will be back here any minute now. You have to get out of here fast.\"\n\nDavey didn't like the way Brenda's hands were shaking. He sighed. \"I know. You said to go through the trees, right?\"\n\n\"It's raining so hard. Let me think. The rain will slow Cudge down and he won't be able to see the holes in the road. Yes, through the trees. That way you won't pass him on the road. Will you be afraid to go through the woods?\"\n\nDavey thought about her question. The road and the bad man, or the woods and the rain and the dark. \"No, but I need that flashlight you promised me.\"\n\nA sob caught in Elva's throat. She couldn't let him go out alone in the dark and rain. God, she was tired, almost too tired to think. Where was Cudge? Probably in some bar. For a moment, the thought pleased her. If he was in a bar, he wouldn't be coming through the door any second now. The kid was safe, for the moment. If he were in a bar, he would start off with beer and switch to boilermakers, and from there anything that came in a bottle. He would be good for a couple of hours. She and BJ were safe, for now.\n\nA deep bellow of thunder sounded above them. Elva shivered and drew Davey into her arms for comfort. Streak after streak of lightning skittered overhead, lighting the inside of the camper. Elva swallowed hard. She feared thunder and lightning almost as much as she feared the dark.\n\n\"We're going to sit quietly for a little while so I can think. I have to think about what's best for you. Do you understand that, BJ?\"\n\nDavey nodded and relaxed into the crook of her bony arm. He felt almost safe. Almost.\nChapter 9\n\nThe sky was darkening, more from the threatening storm than from the oncoming night. The roads were almost deserted, even the main road that led back to the garage. The occasional car that passed was going in the opposite direction; the drivers intent on getting home for supper after a long day on the job.\n\nCudge stopped the pickup in the bay area of the garage. He was relieved to see a dim light shining through the door. Quickly he climbed out of the cab in search of the mechanic on duty.\n\n\"Hey, fellah,\" he said, trying for his most pleasant tone, \"I've got a couple of tires what needs fixin'\u2014could be I bent up the rim driving on a flat. I'm camped down the road and I left my wife there, havin' a fit because she's afraid of storms. I'll throw in a few extra bucks for you if you can fix it now.\"\n\n\"Ah, sure, why not? I was about to close up for the night, but a fellah can always use a few extra bucks. Keeps the wolves away from the door, if you know what I mean. It's gonna take me at least an hour.\" The tall, slim man wiped his grease-stained hands on the seat of his coveralls. \"Why don't you go next door and have a beer? They got nude dancers in there for the after-work crowd.\"\n\nCudge relaxed; the mechanic seemed more interested in the tire than in him. \"Sounds good to me, I could use a cold beer.\" Sensing that the man expected him to make some remark about the dancers, he added, \"I haven't seen a good set of tits in a long time.\"\n\n\"Ain't it the truth,\" the mechanic said, grinning. \"I thought my wife had a good pair when I married her. You should see them now\u2014two lemons and all nipple. Hell, it don't hurt to look, so I go in every so often and get my fill. Ain't nothin' wrong in lookin', I always say.\"\n\nCudge followed the man out to the pickup, where he retrieved the tire. Lowering his voice and looking over his shoulder, the man continued, \"There's one little gal in there that's a piece of work. I ain't never been with her, but I heard about her. Calls herself Candy Striper and she's got knockers on her that'll knock your eyes out. You can watch her dance and dream a little.\"\n\n\"Sounds like a waste of good beer to me,\" Cudge grunted. Everywhere you looked there were perverts. \"You said an hour, right?\"\n\n\"More or less. When I'm done, I'll throw the spare back in your truck. But you'll have to pay up now, 'cause I'll be leaving soon as I'm done with your job.\"\n\nCudge handed over the money, with some extra as promised. He walked across the gravel-topped parking area surrounding the bar. A weather-worn sign beneath the orange neon proclaimed BEER\u2014SNOOKIE'S. The thrum of music blared as someone opened the door to leave. Cudge had been in bars like this before; the place smelled of beer and cigarettes, and sawdust littered the floor. There was a crowd at the curving bar but he spotted an empty stool.\n\nResting his elbows on the bar, he ordered a draft. He glanced around at the people; the place was packed. He grinned, thinking of what the mechanic had said about his wife's breasts. Hell, anything was better than Elva's\u2014her hard little breasts reminded him of overgrown walnuts, and you had to look to find the nipple.\n\nTinny music pounded as two scantily costumed girls mounted a step-stool to get up to the beer-splattered bar. Cudge stared at their jiggling bodies but his thoughts were a million miles away. Well, a few, anyway. On Lenny. On Elva. On the blabber-mouthed kid. On how he was going to get out of this mess he'd gotten himself into. He knew he had to think of a plan. Naked bosoms and gyrating pelvises would have to wait for another day. It was getting late. Time for another beer and then he had to hit the road.\n\nThe music ended just as the bartender slid Cudge's beer down the length of the bar. Cudge took a long swallow and looked around. He hated dumps like this. Someday he was going to get all duded up and strut into a first-class cocktail lounge and drink champagne. He'd have a pearl-gray Lincoln Town Car, like undertakers drove, and a cigar clamped in his mouth. Cigars always added a touch of class, especially if they were Havana cigars. He'd go to Canada to get them if he had to. He shook his head at his fantasy. This bar was his speed, and he'd never move on. The lousy cops would be on his tail anytime now. No point in fooling anyone, least of all himself. He had to get rid of Elva before dawn and travel light. Even if it meant ditching the pop-up and coming back for it later. He was shaken from his thoughts by a sweat-soaked man beside him calling out, \"C'mon, Candy, show an old man how it's done.\"\n\n\"And after I turn you on, what am I going to do with you?\" the second girl from the end shouted over the music.\n\n\"The same thing you did last week, baby. I got thirty bucks if you're interested.\"\n\nThe girl laughed and swayed her hips and cupped her breasts, making them stand erect. Cudge's interest peaked. He nudged the guy sitting next to him. \"You got into her?\" he demanded.\n\nThe man stared at Cudge. \"She'll take on anything, mister, as long as you got thirty bucks. She locks in on thirty for some reason. Not a penny less. She likes two tens, a five, and five ones. Anything else is a problem for her.\" His round brown eyes stared at Cudge. \"She gives you your money's worth, too. You interested?\"\n\n\"Not if I have to pay for it,\" Cudge snorted. The day he'd pay a hooker would be the day he tied his cock in a knot.\n\n\"You're in the wrong place then, buddy. Candy don't hand out no freebies. She's putting her baby brother through medical school.\"\n\nMy ass she is, Cudge thought to himself.\n\n\"She likes her money up front too. You can't get around old Candy\u2014she's a businesswoman.\"\n\nCudge took a better look at the prancing woman on the bar. \"Old Candy\" was right, but for looks she wasn't half bad. Better than scrawny Elva, at any rate. He'd been to bed with a bag of bones for so long, he'd almost forgotten how nice it was to be cushioned between a pair of thighs that had some meat on them. Her belly was slightly rounded and fleshy; it jiggled as she danced. Nice to slap his own belly into, he thought, feeling a sensation of life in his loins.\n\nThirty bucks. That's what she wanted. A dollar for each year of her age. Still, she did have a nice smile. Friendly. There hadn't been too many friendly people in his life lately, God knew. And her legs were long. Real long. Long enough to wrap around a man when he was . . . Cudge laughed at himself then, a dry, humorless laugh that made the man beside him turn to stare. What did he care?\n\nCandy's bright smile took in everyone at the bar. Her skin was white, pale white, almost translucent under the blue-tinted lights. Again Cudge felt that stirring below his belt as she gyrated, her breasts bouncing. A long time ago, he'd seen one of the old-fashioned burlesque shows. The stripper had worn tassels on her breasts and could make them twirl in different directions at the same time. He bet Candy could do that; he'd love to watch her try.\n\n\"You got thirty bucks?\" he asked the sweating man next to him.\n\n\"Right here,\" the man said, slapping the bills on the bar.\n\n\"You a gambling man?\"\n\n\"In a manner of speaking.\"\n\n\"I got thirty bucks that says I can get Candy to leave here with me with no cash up front. A freebie. You want to cover the bet or not?\"\n\n\"You're on, man. I've known Candy a long time, and she don't give it away.\"\n\n\"Hey, Candy, come on over here,\" Cudge shouted.\n\nCandy Striper stared down the length of the bar. A new dude. Looked loaded. Been a while since she'd lain with someone she didn't know. What the hell. His thirty bucks was just as good as anybody's. She didn't like his eyes though. Pig eyes. And he had tattoos. She didn't like tattoos either. Candy weaved her way among the bar glasses till she was standing in front of the man. She squatted down so he could whisper in her ear.\n\nShe laughed delightedly. \"You putting me on?\"\n\nCudge shook his head. \"Tell my sweaty friend here that you're coming with me and you're givin' me a freebie.\"\n\n\"You heard the man,\" Candy told Cudge's neighbor, who was watching her with a shocked expression.\n\nCudge scooped up the money and put it in his shirt pocket. When you were on the run, thirty bucks could be the means to an end. \"C'mon, let's go.\"\n\n\"Not so fast. I got a job to do here.\"\n\n\"Yeah, well, forget it,\" he answered. \"I can't wait around all night.\"\n\n\"Listen, I got a break coming after the next set. Twenty minutes at the most and then I'm all yours. Have another beer and watch me dance while I warm things up for you.\"\n\nCudge almost told her to forget it, but the promise in her eyes stopped him. And what was waiting for him back at the pop-up? Elva? Skinny, scared Elva. She would be there when he got back, cowering inside the pop-up, cheeks streaked with tears, body shaking with every clap of the storm.\n\n\"Okay, do your thing, baby,\" he told her, assuming his most debonair manner. \"You ain't got nothin' I can't wait for.\"\n\nA half hour later they left the bar. Outside, in the wind and rain, he looked at Candy. \"How far do you live from here?\"\n\n\"Behind the bar. This dump used to be a motel and those little one-room cottages are still out back. Were you serious about Las Vegas? I never been outside this town, much less to Las Vegas. Do you really know show people there who can get me a job on the stage? When are we leaving?\"\n\n\"Right after you give me your freebie.\"\n\n\"Man, you can have all the freebies you want if you take me to Las Vegas. On the hour if you want. I gotta be honest with you though. I got this little problem. I don't mind giving if you don't mind getting.\"\n\n\"What the hell?\" Then it dawned on him what she'd said. He had a condom in his wallet that he'd been carrying around for just such an emergency. He wasn't \"getting\" anything he didn't bargain for. \"No problem.\"\n\n\"Where's your motor home? I like those things\u2014all the comforts of home. You don't look like a rich promoter.\"\n\n\"Now, did you ever see a rich promoter?\" Cudge demanded as he hustled her around the side of the bar. \"I parked it behind the garage. This looks like the kind of neighborhood that means trouble.\"\n\n\"You scared me there for a minute. I thought that pile of junk over there was yours. I wouldn't be caught dead riding in something like that.\" She motioned to his Chevy pickup, which was parked in front of the garage bay.\n\nCudge's eyes narrowed. \"Wouldn't you now? Probably belongs to some hard-working slob who's sitting in that bar watching you gals toss your tits around.\"\n\nConcentrating on keeping her balance on the gravel in her incredibly high heels, Candy led him toward the cottage nearest the road. A slat-ribbed dog near the door growled to show he resented their intrusion. Cudge watched the dog while Candy dug in her purse for the keys, clutching the edges of the sweater she had thrown over her shoulders to hide her nakedness.\n\n\"Don't let the dog bother you, honey. Just like the rest of us, he's only looking for a good meal.\" The door swung open and she reached inside for the light switch. \"'Course, things are different out in Las Vegas. Nobody, but nobody, goes hungry out there. Say, how long did you say it would take to get there?\"\n\n\"I didn't,\" Cudge answered, his voice harsh. He didn't like answering questions, especially ones that challenged his lies. Besides, if she was dumb enough to think she was good enough for Las Vegas, she deserved anything she got.\n\n\"Here it is, home sweet home. Should I pack first or do you want to ball? Don't make no difference to me.\"\n\n\"You got any alcohol around here?\"\n\n\"Wine. A whole gallon. You want some?\"\n\n\"Get it out, I feel like getting drunk. The guy sitting next to me at the bar said you'd take on anything for thirty bucks. Is that right?\"\n\n\"Thirty bucks is thirty bucks. I just close my eyes.\"\n\n\"How come you sell yourself so cheap?\"\n\n\"I really need the money. I'm putting my kid brother through medical school,\" Candy said, uncorking the jug of wine.\n\n\"You need a better story than that one.\"\n\n\"I'm not putting you on. My kid brother, Jackie, is going to be an orthopedic surgeon someday. If you get me a job on the line in Las Vegas, I can send him more money. He's having a real tough time. Do you know, just one medical book costs seventy-five dollars, sometimes more! Here,\" she handed Cudge a full glass of wine. \"I think I'll have some myself to get in the mood.\" She poured herself a glass, then took a sip. \"So, how do you like it, anyway?\"\n\n\"The wine? It's vinegary.\"\n\n\"No, not the wine. Sex. How do you like your sex? I don't want you to be afraid to tell me what you want.\"\n\n\"What\u2014are you kiddin'? Afraid? Me?\" Cudge laughed until his eyes started to water. When he'd settled down, he glanced around the room. This dump was no better than that furnished apartment he'd left in Newark. The iron bed in the corner looked like George Washington could have slept in it, and the sheets probably hadn't been changed since.\n\nThere was a Panasonic radio-CD player on the table. Candy's brightly polished fingernail pressed the \"on\" button and Kenny Rogers started to sing \"Lady.\" Candy casually shrugged her sweater off her shoulders. She always enjoyed watching the really cool-acting guys get excited when they got her alone. But something about the way this one was looking at her with those pig eyes of his worried her. Suddenly she realized that she'd been a gullible fool, and that he was nothing but a drifter with a silver tongue. He was no more a big promoter than she was Miss America. Still, a girl had to have hope, and working in a dump like Snookie's turned those hopes to needs.\n\n\"You were fibbing to me about Las Vegas, weren't you?\" she asked, hearing the quake in her voice. The guy's eyes widened, then narrowed, and his expression turned evil. Candy decided she definitely should have stuck with the men who frequented Snookie's. There wasn't one who would go out of his way to do anything for her, but none of them scared her. She had to get out of here, away from this man.\n\n\"What's the matter with you?\" Cudge asked. \"You always get jumpy when it's time to hit the sack?\" His voice was almost genial and Candy thought it must be a trick of the light making his pig eyes glitter and emphasizing the grim lines around his mouth.\n\nForcing herself to relax, she leaned over to undo the tiny straps on her shoes.\n\n\"No, leave them on. They make your legs look nice and long.\"\n\nNow this was the kind of talk she was used to hearing from her customers. So what if he was a little kinky? She had a steady who liked to cross-dress. And even if he didn't take her to Vegas, there was still the thirty bucks.\n\nStuart Sanders lost no time once the plane touched down. He ran the length of the seemingly endless concourse; having no patience for the escalator, he bounded down the steps two at a time and elbowed his way through the milling travelers in search of the car that would be waiting for him.\n\nMac Feeley was waiting behind the wheel, cigar clamped in his mouth. He reached over to open the door for Sanders. \"How goes it, big guy?\"\n\n\"You don't want to know,\" Sanders replied in way of greeting. \"Let's move it. Take the turnpike. Is this one of those souped-up jobs the motor pool hands out to speed demons like you?\"\n\nFeeley grinned. \"This little number is slick\u2014it starts out at ninety and works up to one eighty. You think you're flying. Five bucks says I get you there in thirty minutes.\"\n\n\"Is that with or without the siren?\" Sanders asked irritably.\n\n\"Either or, you name it.\"\n\n\"I'd hate like hell to get pulled over and lose time.\"\n\n\"No way. This car has official government plates with the right code numbers, and I guess you didn't see the State seal on the door. You look done in. You hear anything? How did things go?\"\n\n\"Nothing on my end. Mrs. Taylor elected to stay behind with her husband. She said she trusts me and the others to find her son.\"\n\nFeeley switched the cigar from one side of his mouth to the other. \"There's a lot to be said for motherly love.\" His voice was sour.\n\n\"What's new here? Any leads? How's the aunt handling things? What's the latest on the guy with the painted truck? How did the old couple check out?\"\n\n\"Dr. Ryan is worried but she's holding on. We tracked the older couple down in Virginia and asked a few questions, but it was a dead end. The local police think they have a few leads, but they aren't ones for sharing\u2014they think this is their baby all the way down. Our guys had to pull badges to remind them who's in charge\u2014what can I tell you? From here on out it's going to be legwork.\"\n\n\"What's your hunch, Feeley?\"\n\n\"I'd like to know more about the body they found. Coroner's making his report in the morning, but it's pretty certain the stiff was beaten to death. Nasty head wounds. Leonard Lombardi, twenty-eight or so, lived in Newark. Putting the pieces together, we believe he was already dead when he was brought to the campground. Best lead we've got is a guy named Edmund Balog, aka Cudge Balog. Everything is pointing in his direction, right down to a report from a trooper who stopped him on the road. Said he was real nervous about opening the pop-up rig he was dragging. Emergency called the trooper away so he never got a look inside.\"\n\n\"Any connection with the syndicate, you think?\"\n\n\"Haven't found any so far. And there's been no ransom demand. My thought is that the kid saw something he wasn't supposed to. Other than that . . .\" Feeley shrugged \". . . your guess is as good as mine.\"\n\nSanders nodded. \"I let this one get under my skin as far back as house duty in Montclair. I feel responsible, somehow. Did Dr. Ryan tell you Davey is a hemophiliac? He has to have a shot every day, regular as clockwork, to keep it under control.\"\n\n\"Yes. I know that if he misses his shots there's no telling if the drug will work for him anymore.\"\n\n\"Right. It's sort of an internal sabotage, his own body rejecting the drug that would keep him from bleeding to death. How many has he missed so far?\"\n\n\"According to Dr. Ryan, only one\u2014at twelve noon today. It's been over thirty hours since his last injection. Dr. Ryan has been on the phone with a specialist. No one knows how fast those antibodies in the kid's body will develop. Their best guess is that at forty-eight hours it starts getting critical. By the way, we have a code name for radio contact. Transmissions relating to this case are called in to Panda Bear.\"\n\nSanders laughed and shook his head. \"That's the handle Davey uses in the CB club he belongs to.\"\n\n\"Yeah, we know. Dr. Ryan told us.\" Feeley bit down hard on his cigar. Everyone knew Sanders's feelings about kids, especially his sister's. Kids were okay, in their place, Feeley thought, but he didn't want any of his own. Not now, anyway. As if he had a choice, he laughed wryly to himself. His first and only marriage had come and gone before he'd realized what was happening. He hadn't been a bad husband, but he hadn't been a good one either.\n\n\"I thought you said you were going to get us there in thirty minutes.\"\n\nFeeley looked at his watch. \"You want to quibble about one minute and ten seconds, go right ahead.\"\n\nAs soon as the car stopped, Sanders grunted and climbed out. Why spoil Feeley's day, or night? \"Who's in charge?\"\n\n\"You are, now that you're here. You're senior officer. The local police will bow and kiss your hand, if you play your cards right. We've a temporary office set up behind the manager's office.\"\n\nSanders grunted again as he made his way to the camp store's grimy office. The fresh pungent scent of pine was everywhere. The smell reminded him of the air freshener the cleaning crew used in his own office. The autumn leaves heralded a new season that would soon give way to sharp, cold winds and, he hoped, snow. He liked snow, didn't even mind driving in it. Hell, he liked life and everything it had to offer. He felt a chill and stopped mid stride. \"What do you think the chances are for a frost tonight?\"\n\nFeeley worked at the cigar in his mouth. He remembered Dr. Ryan saying that Davey was wearing a light windbreaker. A little kid could freeze, especially one in his condition. And the forecasts were predicting rain for tonight, possibly thunderstorms. \"Jesus, I'm no goddamn weatherman. Fifty\u2013fifty would be my guess.\"\n\nInside the storeroom, Feeley lounged against stacked cartons of cereal. He looked around at the boxes of merchandise that would eventually fill the camp-store shelves. Beans, Spam, instant coffee; he grimaced. His tastes ran to prime rib, baked potato, garden salad, fresh vegetables. Key lime or pecan pie for dessert. He looked around again to see what the storeroom held in the way of dessert. A meal wasn't a meal without dessert. He snorted as he removed the mangled cigar from his mouth. He should have known\u2014canned fruit cocktail. He trampled the ruined cigar underfoot and put a fresh one in his mouth. Sanders was shaking hands with the local police. Feeley dexterously bit off the end of the cigar. A butane lighter snapped to life, almost singeing his thick eyebrows. He paid it no mind. Black, evil-smelling smoke circled and spiraled around his head. If nothing else, it would stop the locals getting in his face and expounding their theories, which, in his eyes, weren't theories at all, but assumptions that any first-week rookie cop could make.\n\n\"You've got a dead body with positive identification,\" Sanders said. \"That's good. I have a missing kid who's a hemophiliac. It seems to me you've been doing a hell of a lot of work on a case that should be cut and dried by now. It beats the hell out of me how you haven't managed to pick up that psychedelic truck and pop-up.\"\n\nEyebrows raised, jaws clenched and lips thinned as Sanders drove home his point to the group of policemen. \"Good police instinct should have told you that the boy is mixed up in this somehow. Your all-points bulletin is worthless. And sitting around waiting is pointless. We have to get moving. Now!\" He looked at each in turn and wondered if he'd ever been so naive. \"Okay, listen up. I'm the chief and you're the Indians. You got it? If not, I have a badge saying that's the way it goes.\"\n\nFeeley blew another cloud of smoke in Sanders's direction. He grinned to himself. Old war horses could take charge quickly when they wanted to.\n\n\"Feeley,\" Sanders called over his shoulder, \"see that this picture is in every morning paper in the area. Start with the Asbury Park Press. Don't let them give you any crap about it being a color Polaroid shot either. If they even think about giving you trouble with the deadline, tell them you know the paper isn't put to bed till ten, and then there's a two-hour grace period. Call ahead if you want. Just get it done.\"\n\nSanders issued his remaining orders grimly. He believed that Davey was somewhere near the park. The moment the others cleared out, he spoke to Feeley across the makeshift table.\n\n\"If you're asking me what I think professionally, I have to say that I go along with the department. If you're asking me what I think off the record, I think the kid saw something and was picked up. He could be anywhere; he could be dead.\"\n\n\"What's he like\u2014the kid, I mean?\" Feeley asked curiously.\n\n\"He's got a lot of savvy, if you know what I mean. He's been through a rough time, and he's just now coming into his own. I know some adults who couldn't take the medical treatment this kid has been through. Do you think you could stand being transfused through your jugular vein? No? That kid has, many times. I'll tell you who he looks like. You ever see that airline commercial where the kid gets a pair of wings from the flight captain and then says, 'Oooh, thank you, Captain'? Well, he looks just like him. I'm sure there are thousands of kids who look like that, but Davey Taylor is special. Very special. I'm going over to the aunt's campsite now. Which way is it?\" Talking about Davey made Sanders's stomach churn.\n\n\"Take the main road and follow it to the fork, then bear left. It's the only RV in sight.\"\n\nJust as Sanders stepped outside, Lorrie Ryan walked up. Sanders saw a woman who was stripped-down, naked with hurt. Nobody deserved to go through what she was going through.\n\n\"Stuart\u2014I mean, Mr. Sanders\u2014when did you get here? Where's Sara? Do you have word of Davey, is that why you're here?\" Lorrie clasped her hands together and stared at him with wide, frightened eyes.\n\n\"Hey!\" he said, taking her firmly by the shoulder. \"Slow down.\" He tried to quiet her with his own calmness. \"There's no word yet. And your sister\u2014well, she elected to stay behind with Mr. Taylor.\" How bitter the words sounded.\n\n\"She what?\" Lorrie looked incredulous.\n\nSanders tried to make his voice sound neutral, feeling that his own judgment would only serve to aggravate the situation. Christ, maybe he was getting too old for fieldwork. A desk job, that was what he needed. He repeated his carefully chosen words. \"Mrs. Taylor elected to stay behind.\"\n\n\"Just what the hell does that mean?\" Lorrie burst out.\n\nSanders kept control of his voice. \"Mr. Taylor took the stand for thirty minutes today.\"\n\n\"What the hell does that have to do with Sara and Davey? She's the kid's mother! You did tell her what happened, didn't you?\"\n\nSanders looked straight into her eyes. \"Of course, I did, but . . .\"\n\nLorrie clutched at Sanders's forearm and spoke softly. \"I can't believe she would stay there. She has to be worried sick\u2014how could she stay behind? Is there something you aren't telling me, something I should know?\"\n\nSanders shook his head. He was truly at a loss for words to explain Sara's behavior. \"Your sister, Mrs. Taylor, she said she trusted me to handle the matter.\" He paused to let his words sink in. \"Both Mr. and Mrs. Taylor believed that Davey . . . that he's simply wandered off and will find his way back.\" Should he volunteer the rest or be quiet? If he was going by the book, he'd keep his mouth shut, but he hadn't worked by the book in a long time. \"Mr. Taylor was a very convincing witness today. It was unfortunate that the court adjourned at such an early hour or it could have been wrapped up and they both could have come.\"\n\nLorrie's eyes flashed with anger. \"So, Sara's little Andrew made a good showing for himself under Mommy's watchful eye, did he?\" She laughed with disgust. \"Of course, you know she didn't come back to Jersey with you because she's afraid he'll screw up without her there to protect him. And if he screws up\u2014well, let's just say she would feel that he had disgraced her. And God forbid that should happen. She's disgusting!\"\n\nSanders sighed deeply.\n\n\"Poor Davey. He always comes last.\" Lorrie flung herself against the agent's chest and clung to him.\n\nSanders wrapped his arms around her and rested his chin on the top of her head. He'd wanted to hold her like this since the first time he'd seen her, but he would have preferred their first embrace to be under more pleasant circumstances. \"Some people just have their priorities screwed up,\" he said, brushing her temple with his lips.\n\n\"Don't make up excuses for her. There is no excuse for her not coming here. None, and you know it, don't you?\" She leaned back in his arms and looked up at him, waiting for him to answer.\n\nIf he hadn't known before, Sanders knew now that he was too close to this case. His emotions were involved. For an FBI agent, that was a cardinal sin.\n\n\"Lorrie,\" he began, not sure what he was going to say. \"I . . .\"\n\nLorrie glared at him, challenging him to answer, to be truthful with her. \"Tell me I'm wrong, Stuart. Tell me there's a better reason for her to stay with Andrew than for her to come here.\"\n\n\"It's not for me to say, Lorrie.\"\n\nWith a pained expression, Lorrie glanced away. \"You're right. It isn't. I'm sorry. I shouldn't have tried to involve you in our family's matters.\"\n\nHe smiled down at her. \"I understand. You're upset.\"\n\nRegretfully, Lorrie eased herself out of his arms. \"Yes, I'm upset. But I'm sick and tired of Davey taking a backseat. He should be in the front seat all the time. Especially with his medical problems and\u2014\" She broke off, shaking her head.\n\n\"Why don't we walk back to your campsite? I could use a cup of coffee.\"\n\n\"Excuse me, sir.\" A young officer was heading towards Sanders.\n\n\"Yeah, what is it?\"\n\n\"Phone call for Dr. Ryan.\" The cop held up Lorrie's cell phone. \"You left it on the table outside your motor home. I thought you'd probably want someone to answer it,\" he said, handing it to her.\n\n\"Who is it?\" she asked.\n\n\"Mrs. Taylor. From Miami.\"\n\nSanders watched as Lorrie seemed to shrink into herself. His voice was protective when he spoke. \"Why don't you let me take the call? I can tell her you're trying to take a nap or . . .\"\n\nShe smiled up at him. \"Thanks, but that's the easy way. I'll talk to her.\"\n\nSanders watched her walk away. \"I don't envy her that conversation,\" he said to Feeley, who had walked up behind him. \"Sara Taylor is like a barracuda and Lorrie's a guppy in comparison.\"\n\n\"I don't know about Mrs. Taylor,\" Feeley said, \"but I think you're wrong about Dr. Ryan. She'll handle the call.\"\n\nSanders nodded morosely.\n\nLorrie walked slowly in the direction of the motor home. \"Is that you, Sara?\" she asked calmly. She refused to permit Sara to rattle her.\n\n\"Yes, this is Sara, Lorrie. How could you have allowed my son to wander off? I trusted you. Andrew and I just can't believe you allowed this to happen. It's unforgivable. Well, has he returned yet?\"\n\n\"No, Sara, he hasn't come back to camp yet. The police are still searching for him. Mr. Sanders arrived a short while ago. I don't think he's 'wandered off' as you put it. We all think it's more serious than a little boy lost in the woods. I'm not sure if you know it or not, but the police found a dead body near where Davey was playing. We're not sure if Davey saw or overheard something. We're doing everything we can. I'm sorry, Sara. I take full responsibility.\" Lorrie kept her voice firm and controlled.\n\n\"That's absurd. What could Davey possibly have to do with a dead body? You're grasping at straws, Lorrie, to cover your own ineffectiveness. There won't be a next time, I assure you. Andrew and I both feel Davey's wandered off. Andrew admitted that he used to do the same thing when he was a child. Like father, like son. I want you to call me the minute Davey gets back, no matter what time of the day or night it is.\" Her voice was frigid, bordering on open hostility. \"I suppose if there's anything we can be grateful for in this . . . this outing you insisted on, it's that Davey's had his shot.\"\n\nLorrie swallowed hard. Evidently Sara didn't know Davey had been missing since early morning. \"Davey missed his shot. He wandered off shortly after breakfast. I thought you knew,\" Lorrie said, gripping the phone tighter.\n\n\"Oh, for God's sake, Lorrie. What have you done?\" Lorrie's control snapped. \"Why aren't you here, Sara? Why didn't you come with Mr. Sanders? Your place is here where Davey\u2014\"\n\n\"How dare you tell me where my place is! I'm here seeing to something very important and I trusted my son to you. The only reason you want me there is so I can make everything right for you, the way I did when we were children. You never did anything right, Lorrie. I was always the one who had to pull your chestnuts out of the fire and get you out of one scrape after another. Not this time. You, and you alone, are responsible for my son. I don't want to hear another word from you until you call me and tell me Davey is safe. Do you understand me, Lorrie?\"\n\nLorrie felt goose bumps on her arms, not from the cold but from Sara's icy voice. Hang up, her mind shrieked. Don't pay attention to what Sara's saying. Sara was changing everything around, just like she always had. It was always Sara who began the arguments, not her. Yet she, Lorrie, had always ended up paying the piper. Ignore her. She's upset. Don't say another word.\n\n\"Lorrie, are you there? Answer me.\"\n\n\"I'm here, Sara. I was thinking about something.\" It took every ounce of willpower she possessed to square her shoulders. \"I always tried to like you because you were my sister, but I don't like you, Sara. I didn't like you when we were children, and I don't like you now that we're adults. I love Davey\u2014you know that. I won't bore you with what I've gone through today. There's no way you could possibly understand. Just because you're Davey's biological parent doesn't mean you love him more\u2014or even as much\u2014as I do. If you did, you'd be here. I don't have anything else to say, so I'm going to hang up. If I hear anything I'll let you know.\" Lorrie pressed the button to end the call. Damn Sara to hell.\n\n\"Are you all right?\" Stuart Sanders moved around in front of Lorrie. He'd been right behind her while she was talking to her sister, not eavesdropping, just waiting. He'd wanted to be there for her if Sara Taylor said something to send her over the edge.\n\nWhen Lorrie turned around, he saw that her eyes were glistening with tears. \"I'm okay,\" she said, her words belying her expression.\n\n\"Yeah, sure you are.\" He put his arm around her shoulders and led her toward the motor home.\nChapter 10\n\nThe phone was still in Sara's hand when Andrew admitted Roman DeLuca into the hotel suite. Her conversation with Lorrie had not been satisfying, and it rankled that Lorrie had hung up on her. But Sara schooled her face to impassivity before facing the prosecuting attorney. Everything about him annoyed her. He was as bogus as a three-dollar bill, and he capitalized on his movie-star good looks, right from his meticulously clipped gray hair to the tips of his manicured fingers. A snakeskin briefcase matched his shoes and belt, and his dove-gray suit and sparkling white shirt were custom-tailored to his slim body in an expensive salute to his vanity.\n\nAndrew had scoffed when she'd told him that she thought the impeccable attorney had his eye on the governor's seat. But Sara knew she was right, and Roman DeLuca seemed to read her thoughts. It didn't upset him, and somehow his reaction added to Sara's dislike.\n\n\"Mrs. Taylor,\" DeLuca said quietly. Sara nodded slightly in response. The phone receiver was still clutched in her hand. Was he wrong, or was there a glimmer of self-righteousness in her eyes? \"Were you about to make a phone call, Mrs. Taylor?\" he asked.\n\nSara glanced down at the phone. \"Oh, no, as a matter of fact, I've just completed one. To my sister in New Jersey.\" Damn, now why had she needed to explain?\n\nDeLuca noticed the stiffening of her shoulders; he had been right about the self-righteous glimmer. Sara Taylor must have just finished berating her sister, blaming her for the boy's disappearance. He wished he had arrived just a moment earlier to hear Sara's end of the conversation for himself. He would have liked to see her in action.\n\n\"Andrew, we must talk.\" DeLuca seated himself near the window, careful of the creases in his slacks. \"Don't be alarmed. I want to congratulate you on your fine performance on the stand this afternoon. There's only one point that's annoying me.\" He looked up at Andrew from beneath his dark, bushy brows. \"It's the way you look for approval from your wife before you answer a question.\" At Andrew's confused look, DeLuca continued. \"Obviously you didn't realize that every time I asked you a question about what you'd witnessed, or your acquaintance with Jason Forbes, you looked to your wife before you answered. You almost never took your eyes off her. I even attempted to block your line of vision but, when I did, you started to get rattled and acted unsure of yourself and your statements. Rather than make you appear a fool, I allowed the visual contact between yourself and Mrs. Taylor.\"\n\nAndrew was stunned. He knew he hadn't been a star witness, but he hadn't thought he'd done badly. And Sara had told him how wonderfully he'd performed on the witness stand.\n\n\"If the judge had let the defense attorney begin his cross-examination, you'd be on the scrap heap by now, my friend. The judge adjourned as a favor to me when he saw how perturbed I was by your attention to your wife.\"\n\n\"I'm sorry. I didn't realize . . .\"\n\n\"Let's just hope the jury didn't notice. If they did, it may jeopardize our case.\"\n\n\"I don't know what to say.\"\n\n\"Say you won't do it tomorrow. Because if you do, I'll be forced to ask Mrs. Taylor to leave the courtroom. Do you understand what I'm saying?\"\n\n\"Of course he understands, Mr. DeLuca. He's not a child,\" Sara snapped. \"It was your own inept questioning technique that rattled Andrew.\"\n\nRoman DeLuca's eyes narrowed to bits of chipped ice. \"You should know, my dear, what makes your husband tick. I say he behaved like a sophomore, and so does the judge. Instead of giving me his undivided attention, he gave it to you. Why don't you ask him why?\"\n\n\"I don't have to ask Andrew why. I was there. I saw what went on. You could have put Andrew on the stand earlier, but you didn't. Why? You deliberately waited till the end of the day before making your move. I want to know why, Mr. DeLuca,\" Sara said coldly. \"I want to know now.\"\n\nDeLuca smiled wryly. He didn't like Sara Taylor. Despite being sapling thin she had a ramrod strength. He possessed similar traits, but in Sara Taylor he detested them.\n\n\"Strategy, Mrs. Taylor. I don't exactly understand what you're accusing me of, and I don't believe you do either. I think you're just trying to turn the question I'm asking away from you.\" He gave a sly smile that he knew would irritate her. \"For the sake of argument, let's just put it down to your desire to protect your husband. I'm merely making the point that a witness who continually directs the jury's attention away from his testimony is of little or no use. I know that Mr. Taylor's looking to you is merely a loving habit, but it's one that could lose this case for the State, and I won't have it.\" His tone was smooth, and he waited for the Taylors to digest what he had said before continuing. \"Tomorrow is another day, Mr. Taylor . . . or didn't you say I should call you Andrew? Andrew, I'm here tonight to go over your testimony with you. I've studied your written deposition, and I think we'll work directly from that. If we can get it all together, you should be unwavering in the face of cross-examination.\" DeLuca smiled.\n\nHe's like a snake, Sara thought. She disliked DeLuca intensely, almost hated him. \"You're very confident for a man who was so angry a moment ago,\" she remarked.\n\nDeLuca's voice was urbane now, charming and suave. \"Mrs. Taylor, what you mistook for anger was concern for Andrew, and yourself. As they would say back in New York, this is my turf and I know every blade of grass. I always win, Mrs. Taylor, remember that.\"\n\nSara's eyes narrowed. It sounded like a threat, or was it a warning? How dare the man? How dare he!\n\nAndrew intervened. \"Sara, before we go over my testimony, why don't you show Mr. DeLuca the message that was delivered before dinner?\"\n\nSara gathered her indignation close about her and walked stiffly over to the desk. She picked up the square of white paper and held it out, eyes cold and hard.\n\nDeLuca scanned the message. \"Well, who was it and what did they want?\" he asked, listening closely for her answer.\n\n\"We didn't call the number. We waited to show it to you,\" Sara said. God, how she detested this phony movie-star lawyer.\n\n\"Let me understand something, Mrs. Taylor. You say this message was delivered before dinner, which, I assume, was between six and seven. Am I right?\" At Andrew's nod he continued. \"It's nine forty-five now,\" he said, teeth clenched. \"Didn't it occur to either of you that this might be a message concerning your son who is missing, possibly kidnapped?\" DeLuca glanced between the Taylors. Andrew looked horrified; Sara looked . . . Damn, nothing got to her. Nothing. He watched her carefully, almost admiring the way she took in his statement and managed to appear unruffled. Was it possible she didn't make the connection? He opened his mouth to drive it home. \"Mrs. Taylor, I feel it is my duty to tell you there's a possibility that someone associated with the drug syndicate we're prosecuting may have kidnapped your son.\" For one instant Sara appeared to be taken aback. But the heavy intake of breath belonged to Andrew, who sank down onto a chair. \"You must believe I care about your son,\" DeLuca added in his most sober tone.\n\nSara was the first to rally. \"Your concern is most appreciated, Mr. DeLuca, but neither Andrew nor I owe you any explanations concerning our son or our private life. Experts, Mr. DeLuca, are handling matters back in New Jersey,\" she said stiffly, not liking the look in Andrew's eyes. \"Andrew is here to do a job, and I'm here because I choose to be here. Do we understand each other? There's nothing in this case to suggest kidnapping. Davey has merely wandered off.\" Sara hadn't told Andrew about the corpse that had been discovered near where Davey had been playing. And she wasn't about to tell him now, not with DeLuca standing there ready to spring on her like a cat.\n\n\"But, of course, my dear. I understand everything you've done and said since our first meeting. I was merely thinking of you and your family. However, I must warn you\u2014if the media gets hold of this . . . Why, I can't be responsible. They tend to start off with the term 'unfit mother' and from there it's usually a quick slide to a 'philandering, alcoholic husband. ' I'm not saying any of it's true, or that I condone it\u2014I'm just pointing out the possibilities to you. If my information is correct, Mrs. Taylor, Mr. Sanders urged you to take the next plane home. I would second that. Your husband, believe it or not, is perfectly capable of speaking for himself.\"\n\nSara was visibly shaken but she recovered quickly. She grabbed the message out of DeLuca's hand, picked up the phone and dialed for an outside line. Seconds later she replaced the phone and turned to face the two men. Her voice, when she spoke, was neutral. \"It was a recording. Some insane person says he has our son and, if Andrew testifies, we'll never see Davey again. Obviously, this person is demented. Davey has just wandered off and is temporarily lost. If it was anything more serious, my sister would have told me. There are some people, Mr. DeLuca, who enjoy trying to make other people miserable. In spite of your seconding Mr. Sanders's suggestion, I'm standing by my decision to stay with Andrew. My son's escapade is in capable hands.\"\n\n\"Is that what you call it? An escapade?\" Roman DeLuca questioned, his tone rising with disbelief. What a cold bitch she was.\n\n\"In my opinion, Mr. DeLuca, this matter is closed. Andrew, what's your feeling?\"\n\nAndrew looked from his wife to the attorney. \"I quite agree, Sara. Why don't we get down to business, Mr. DeLuca? It's after ten now, and we have an early day in court tomorrow.\"\n\n\"Can I order something for either of you from room service?\" Sara asked as if nothing had happened. Both men shook their heads. \"Well, then, I think I'll leave you alone.\" Regally, she left the sitting room to go into the bedroom.\n\nIt was twelve forty-five when Andrew tapped on the door. \"Mr. DeLuca is leaving now, darling.\"\n\n\"One moment, dear,\" Sara said, retying the sash of her peach-colored dressing gown. She opened the door and studied her husband's appearance. \"Oh, Andrew, you look terrible. Take a shower, a nice hot one. It'll do you a world of good. Stand under the spray for at least ten minutes. I can see how tense you are. Relax, darling, this is almost over. By this time tomorrow it will be nothing more than a bad memory. You go along now and I'll see Mr. DeLuca out. I've laid fresh pajamas out for you in the bathroom. I'll even pour us a nightcap.\"\n\nSara moved past Andrew, her scent circling above and around her. Andrew sniffed, liking it. Was it new, or were his senses exceptionally keen tonight?\n\n\"Mr. DeLuca, I'll see you out,\" Sara said formally.\n\nThe attorney dropped all pretense. Lowering his voice, he said curtly, \"Mrs. Taylor, I want you on the morning plane back to New Jersey.\"\n\n\"I won't do that. My place is here with my husband,\" she answered tightly.\n\n\"Then let me put it to you another way. If you aren't on that morning plane, certain friends of mine will not be responsible for your son's safety. Now do we understand one another?\"\n\nSara's world was turning upside down. She knew DeLuca would not repeat himself, nor would he ever admit to saying what he had just told her. \"Certain friends\" he called them; he obviously meant the syndicate against whom Andrew was testifying.\n\n\"Why?\"\n\n\"It's not important for you to know why. Just be on that plane.\"\n\nIf he was saying what she thought he was, Roman DeLuca knew Davey's whereabouts, which did make it a kidnapping. Yet Stuart Sanders had led her to believe that Davey had wandered off and was having difficulty finding his way back. Or had she just accepted the first thing he'd said? She couldn't remember.\n\nPoor Andrew. If anything happened to Davey, Andrew would be devastated. She needed time to think.\n\n\"Can I have my office make your airline reservation, Mrs. Taylor?\"\n\n\"No, you can't. I'm staying. You wouldn't dare have my son harmed. You're too respectable for that, Mr. DeLuca. And I understand you're thinking of running for governor of Florida.\"\n\n\"It's crossed my mind,\" DeLuca replied. \"I meant what I said about your son. And remember something else\u2014I always win.\"\n\nA chill washed over Sara. Her tone, however, was just as menacing as his. \"So do I, Mr. DeLuca.\"\n\n\"There's a first time for everything. You're out of your depth here. This is my turf, remember?\" He smiled down at her. \"I'll have a reservation made just in case you change your mind. I believe the flight leaves at ten after ten.\"\n\n\"Why are you doing this?\" Sara hissed.\n\n\"Because I don't want you in the courtroom. Good night, Mrs. Taylor.\"\n\nSara Taylor stood for a full five minutes at the door after closing it on the attorney. She had caught a glimpse of Sanders's colleague in the hallway and, furious and frightened, had thought of calling him in to tell him about DeLuca's threat. But she knew he'd never believe her. Still, perhaps she should tell him about the phone message and the result of her call. Maybe he could trace it . . . But no, still no one would believe her. Who would seriously consider that the upstanding Roman DeLuca would subvert the very laws he had sworn to uphold? He hadn't achieved his current status by being careless. More compelling still was her own feeling that this was a personal confrontation between the attorney and herself. All she had to do to win was be smarter and faster than he was. As for going back to New Jersey, she would have to do that, too. She was going to have to buckle under to a hoodlum in a thousand-dollar suit!\n\n\"Sara? Where are you, darling?\"\n\n\"Right here, dear. I was just turning off the lights. I'll be there in a moment.\"\n\nSettled in the bed beside Andrew, Sara searched for the right words. She had to be very careful what she said.\n\n\"Andrew, there's something I want to talk to you about. I've been thinking that maybe I should go home after all. That I could be of more use there than here. There's a flight around ten tomorrow morning. What do you think?\"\n\n\"I think that's a good idea. Mr. DeLuca assured me I would finish with my testimony by noon so I'll take the one-fifteen flight. I'm pleased you're going, Sara. I think it's important that, when Davey finds his way back, you're there for him.\"\n\nNot for anything in the world would Sara let Andrew know that she was going back because of Roman DeLuca's implied threat. No, it hadn't been implied, it had been clearly stated and she had to deal with it. \"I'll go with you to the courthouse and see that you're settled in, then leave from there. I want to be sure that you're all right before I go.\"\n\n\"That's not necessary, Sara.\"\n\n\"Maybe it isn't, but it's what I want to do.\"\n\n\"Is there anything else on your mind, Sara? You haven't been yourself since DeLuca arrived. I know he upset you with all that eye-contact nonsense, but now that you're going home it's hardly important.\"\n\n\"You always see right through me, don't you, darling? Yes, something has been bothering me. It's about your testimony this afternoon. I can't be certain, but I think there's a small problem we've overlooked. Remember when we were in school and had to learn the answers to a long list of questions? I don't know about you, but I always had the answers down pat as long as the questions were asked in order. Once the questions were out of sequence, I failed miserably. I heard Mr. DeLuca going over your testimony with you. Everything was in sequence, and you had all the facts down perfectly. But what would happen if those questions were asked out of sequence?\"\n\n\"By whom?\" Andrew asked.\n\n\"Anyone. The attorney for the defense\u2014anyone. It could rattle you, Andrew. It could rattle anyone!\" she added, touching his shoulder lightly. \"Darling, I have every confidence that you'll be wonderful. But perhaps it wouldn't hurt to have our own rehearsal. I'll go over the questions with you, just like Mr. DeLuca did, only I'll mix them up so you'll be prepared for any eventuality. Darling, in an hour or so, you'll be letter perfect. Remember that old saying, 'Anything worth doing is worth doing well'? Andrew, you've put too much time and effort\u2014and, yes, sacrifice to yourself and our family\u2014not to be the best possible witness you can.\"\n\nSara waited, hardly daring to breathe, for his answer.\n\n\"Darling, I think you're right. You've hit the nail on the head. But why didn't Roman DeLuca warn me? Why didn't he think of it?\"\n\nSara shrugged. Because, dear Andrew, Roman DeLuca is part of the syndicate. He's a hoodlum and a crook, and with the help of \"certain friends\" he wants to buy himself the governorship. Roman DeLuca knows that without me in the courtroom he can make you appear a fool and your testimony worthless. If the State doesn't win a conviction for this murder, there will be no connection with the syndicate. And it will all seem as though it's your fault\u2014the absentminded professor. That's why I have to go back to New Jersey and rescue our son. But I'll have both, Andrew darling\u2014Davey and my pride in you. I'll have won!\n\n\"Sara? You didn't answer me. Why didn't DeLuca think of asking me the questions out of sequence?\"\n\nSara smiled warmly. \"Who knows why lawyers do the things they do? Your job is to get on that stand tomorrow and be a credible witness. Now, let's get down to work. I want you to make me proud. I want to pick up the evening paper and see that you've proven yourself to the court. I want no sly innuendoes about my husband being an absentminded professor. Now, darling, here's the first question.\"\n\nSara mercilessly drilled her husband, making him word-perfect, unshakable, no matter what tack her questioning took. Hour after hour she pounded away, refusing to hear Andrew's complaints of weariness. At last, his responses were clear, confident and, above all, honest. None of her grilling had removed his spontaneity. Roman DeLuca was going to be in for a big surprise when Andrew took the stand; he would do an excellent job. Her husband wasn't going to be intimidated by some glib attorney, not if she had anything to do with it. DeLuca's mistake had been to underestimate her. He hadn't realized how important Andrew was to her, or to what lengths she would go to be certain he appeared first-rate in the public eye. Sara smiled in the early morning light. It always paid to know one's adversary. A pity DeLuca hadn't applied that rule to her.\n\n\"Enough, darling, you've got it down pat. There's no possibility of becoming mixed up now. Truth is on your side, and you've shown me a confidence I hadn't realized you possessed. Each day, darling, I love you more. I'm so proud of you.\"\n\nAndrew returned Sara's smile as he basked in her praise. He did feel confident now, able to handle anything. Sleepily, he reached out to his wife.\n\n\"Darling, you still have time for a short nap,\" Sara said. \"I'm going to shower and pack for us. I'll lay your suit out so you can sleep till the last minute. I want you bright and relaxed when you enter the courtroom.\"\n\n\"I'll be so relieved when you get back to Jersey, Sara. If anyone can find Davey, it's you. You're so capable. I have every faith you'll know exactly what to do. All these hours I've been thinking about testifying, and I haven't given Davey much thought. I can't understand why we haven't heard anything. . .\"\n\n\"Hush, darling, everything will be fine. You'll see. I don't want you worrying about Davey, or anyone. You have your civic duty to perform and you'll make me proud. I know it. Now, close your eyes and don't think about anything or anyone.\"\n\n\"Hmmm,\" he answered, closing his eyes obediently and nodding off.\n\nPoor baby, he looked so tired, she thought. A nap would refresh him then, within a few hours, he'd rock DeLuca back on his heels and the attorney would be powerless to do a thing about it. She would have followed orders by returning to New Jersey, and Andrew would have testified in the service of justice, making himself a hero.\n\nRoman DeLuca was a loser. He had counted on Andrew making a poor showing, something he must have depended upon from the first because Andrew had been a hostile witness. When Andrew had made his deposition with the State Attorney's office in New Jersey, it had been too late for DeLuca to do anything other than base his case on Andrew's testimony. And it certainly would have looked strange if Andrew refused to testify because she or Davey was threatened. No, DeLuca had to go along with appearing to be on the side of the law, while secretly undermining Andrew's testimony. But he hadn't counted on Sara, which was why he was forcing her to return to New Jersey.\n\nBut Sara could handle Roman DeLuca. He was nothing more than a scab on a sore, and scabs could be pulled off. In a few hours, that's exactly what would happen. She leaned back on the pillows, folding her hands primly on her rising bosom. She would sleep now and shower later.\n\nPrecisely twenty minutes later the alarm on her watch beeped softly and Sara woke instantly, ready for the day ahead. She climbed carefully out of bed and began to move about the hotel suite, putting everything in order. Noticing that Andrew was huddled beneath the covers, she adjusted the thermostat in the room. She looked around to see if she had missed anything. Satisfied that everything was in order, she stepped out of her nightgown, folded it neatly, and placed it on top of her other clothes. She walked naked into the bathroom and stared at her reflection with clinical interest. For her age, she had a firm, supple body. High, round breasts, narrow waist, flat belly. Long, slender legs that, when flexed, were like steel springs, or so Andrew said. She smiled.\n\nTurning to adjust the showerhead and regulate the water, she wondered briefly what kind of women Roman DeLuca liked. Probably fleshy, big-bosomed women who wore heavy makeup. He had strong hands. Would those hands be gentle or savage on a woman's body? Her tense muscles relaxed as the steam spiraled upward in search of the vent. What kind of words would he whisper into a woman's ear? Her eyes narrowed. She could almost imagine. He would be a demanding lover, uncontrollable when driven by lust. An animal, savage in his intent, and even more savage as he pounded against his prey.\n\nSara shook her thoughts away as she stepped beneath the spray. She didn't want to think about that handsome, leonine head, the profile that could have been minted into an ancient Roman coin. The width of his shoulders, the slimness of his waist, the length of his thighs, the power of the man\u2014the challenge.\n\nThe needle-sharp spray hit her breasts, erecting her nipples. A sudden intake of breath brought with it a mouthful of water. She doused her head, struggling to control her traitorous thoughts.\n\nHe was there with her, watching her, his eyes touching her with their glinting desire. He was inches away from her, just outside the shower spray. She could feel the heat from his naked body, hotter than the shower, touching the most intimate parts of her with the flaring licks of a blast furnace. He watched her, face impassive, eyes devouring, as she soaped her breasts, lifting them, displaying them for him, knowing that the drumbeat of his desire was quickening to wildness.\n\nShe was mad with her own sense of power over this man who had become her enemy. She was returning his gaze brazenly, impudence lifting the corners of her mouth. If she wanted, she could reach out and touch him, sending him beyond the parameters of sanity. She was preening beneath his gaze, perpetuating his madness for her.\n\nA growing ache was voicing an appeal between her strong, supple thighs. Her skin felt pink, shining from an inner glow. Her breasts were heavy, hard, thrust high on her torso in a silent petition to be crushed beneath his hands.\n\nAs she was watching him, his hands dropped to his loins, holding himself. Taking her cue from her fantasy, her own fingers slipped between her legs; she was surprised by the heat that rivaled the steamy spray. She threw back her head, the water pelting the full length of her nakedness. As a low, throaty moan came from her throat, she recognized the need building in her. The misty steam was fast obliterating her fantasy image. Spasm after spasm racked her body. Her knees crumpled beneath her as she sank to the bottom of the tub, doubling over in a paroxysm of seemingly endless joy.\nChapter 11\n\nIt had been a long day. One of the longest Stuart Sanders could remember. How had he allowed himself to get so caught up in this case? A case should stay a case, and a kid a kid. But it wasn't like that this time.\n\nHis stomach rumbled ominously. He'd taken one look at the peppers and steak on the airline tray and pushed it away; he had no desire to tempt his ulcer. Lorrie led him to her motor home where, over a glass of milk and a turkey sandwich, Sanders had her tell him everything that had happened since breakfast.\n\n\"So Davey knew you'd be leaving soon to go to the laser show?\" he asked.\n\n\"Yes. He knew, and he was very excited. He promised to stay close.\"\n\n\"Do you consider him responsible\u2014I mean, as responsible as a kid his age can be?\"\n\n\"Davey is very responsible,\" she answered. \"He's had to be because of his medical condition.\"\n\n\"Right,\" Sanders said, nodding his head. He'd wanted to talk to Lorrie and get her take on the whole thing because she would have a different perspective from the cops.\n\n\"You think he was kidnapped, don't you?\" she asked.\n\n\"I don't know what to think, Lorrie. There are too many pieces to this puzzle, if you know what I mean. They can't all fit.\" He got out from behind the table and took his plate and glass to the sink. \"I'm going to go outside and walk around. Do I go left or right around the lake to find the grave?\"\n\n\"There's a path that sort of veers to the right. It's a couple of hundred feet away. You can't miss it\u2014there's a deep gully. You're not going to be able to see much and, from the sound of the wind out there, you might get blown away.\"\n\n\"Are you going to be up for a while longer?\" he asked.\n\nLorrie gave him a watery smile. \"You don't really think I'd go to bed while Davey's still missing, do you?\"\n\n\"No, but . . .\"\n\n\"I'm not Sara,\" Lorrie said, voicing his thoughts. \"I know she would think it advisable to get some sleep so she'd be refreshed and ready for anything, but I'm not so logical or so . . .\"\n\n\"Cold,\" Stuart inserted, without thinking. He shook his head. \"I'm sorry. I have no right to criticize your sister.\"\n\n\"You don't have to apologize. I agree with you completely. There's no better word to describe Sara, except maybe for . . . heartless. I don't understand her and, even if I did, there's no excuse for the way she treats her own son.\"\n\nLooking at Lorrie, knowing the guilt and pain she was suffering, made Stuart want to take her in his arms. As if reading his mind, she reached out and touched his cheek. \"I know if there's anyone who can find him, it's you.\"\n\n\"I'll do my best.\" He put his arms around her and pulled her close.\n\n\"That's all anyone can do,\" she whispered.\n\nRegretfully, Sanders left the motor home, the white beam of his flashlight guiding his way. He stopped as he neared the pond, sensing a presence. He turned and was about to reach for the gun under his armpit when he looked down into the pool of light and saw Duffy.\n\n\"How goes it, Duff? Kind of late for you, isn't it? You're supposed to bark when you see someone. Some watchdog,\" Sanders said as he dropped to his haunches to fondle the dog's ears. Duffy rolled over and let her paws go straight in the air, her signal that she wanted her belly scratched. Sanders obliged. \"What happened, Duff? Where's the kid? Where's Davey? Jesus, if only you could talk.\"\n\nAt the sound of Davey's name Duffy woofed softly. She waited expectantly for Sanders to get on with his brisk massage. \"You're one of those safe dogs that are meant to be good companions for little boys. I'm not blaming you, you understand. Hey, what's this?\" Sanders's fingers encountered a swelling in the dog's groin. Duffy yelped and rolled over. \"Let me see what that is.\" Obediently, the little dog came back to the crouching man. It was apparent that she trusted him as she started to nuzzle his ankle and whine. Sanders could feel the dog tremble as he gently ran his hands over her body. There weren't any other swollen areas but there was a small cut on her face. \"Did somebody hurt you, Duff? It's okay, girl. No one is going to hurt you anymore.\" He picked up the fuzzy bundle. \"We're going for a walk, Duffy. We're going to that grave and maybe you can tell me something in your own way.\"\n\nSanders unzipped his vest and held Duffy close to his chest, making sure the dog was comfortable. \"I can't take any chances on you getting lost. You've been hurt and you deserve a ride.\" Duffy yipped, then buried her head in the man's warm shirt.\n\nThe wind whipped through the trees, whistling and shrieking. A three-quarter moon slid from behind a cloud, lighting the way for Sanders.\n\nDuffy growled low in her throat. \"Gotcha, girl. We're close, is that it?\" Some familiar scent teased at Sanders's nostrils. It wasn't the leaves or the pine needles; it was something more commonplace, something he smelled only once in a while. He wrinkled his nose, trying to place the faint aroma. Christ, it was mothballs. Cautiously, Sanders approached the gully, the flashlight held straight in front of him. Duffy cowered against his chest, whimpering and whining. He hunkered down and peered into the gully.\n\nDuffy jumped out of his arms. At first the little dog's bark was fretful. She approached the yawning gully then backed away, growling ferociously. Gaining her courage against some unseen or remembered terror, she used her front paws to dig into the soft earth. Sanders lowered the flashlight, illuminating the spot where Duffy was digging. Nosing and pawing, she made high-pitched whining noises, finally breaking into a howl, stopping only to stare up at him, waiting.\n\nSanders sifted through the dirt, looking for whatever the dog wanted him to find. The small, round pellet felt rock hard as he fingered it. It was a mothball! No bigger than the gumballs that kids got out of candy machines. \"Good dog, good Duffy. I know this means something, but for the life of me I can't imagine what it is.\"\n\nSanders spent another half hour walking around, searching the area near the gully. He found nothing. Aside from the mothball in his pocket, it was a dry run, a bust. What had he hoped to find? Davey, of course. Always Davey.\n\nDuffy backed off a step and barked. She planted her feet firmly on the soft earth. Her ears were straight up when she barked again.\n\n\"You got something in mind?\"\n\nDuffy backed off.\n\n\"Okay, show me.\" Duffy was swift as she turned to follow Sanders's order, and he was hard-pressed to keep up with the terrier in the dim light of the flashlight. Too much high-calorie food and too many cigarettes. He was breathless when he stopped next to Duffy. In all the time he had spent guarding the Taylors, he had never seen the dog so agitated. He lifted the flashlight higher into the leaves of the low-branched trees. A covered garbage can and a picnic table told him they were at a campsite. The campsite, if he was understanding Duffy correctly.\n\n\"Good girl,\" Sanders said, squatting down. \"You've gotten me this far, now what? Where's Davey? What happened here?\" He didn't feel foolish talking to the dog. Duffy whined then howled. Sanders swallowed. \"I need more than that, Duff, if we're gonna find Davey.\" Duffy stared at Sanders, continuing to whimper and whine. The noise set the agent's nerves on edge. \"Look, I get it\u2014we're at a campsite. But what does it mean? Was Davey here? Were you here?\" Duffy cocked her head as though she understood every word the man was saying.\n\nThen she barked once, twice, a mean, chilling bark. Sanders watched as the dog dropped to her belly, shimmying toward him. When she was within five feet of him, she lay down then rolled over, belly up. Sanders reached out to touch Duffy, to touch the tender groin. Duffy let loose with a low growl and then leaped to her feet. Playfully, she pranced around Sanders, circling him Indian fashion.\n\n\"Son of a bitch!\" Sanders exclaimed as he scooped up the little dog. \"The bastard who was camped here is the one who hurt you. Kicked you, probably. Is he the one who has Davey? Show me, Duffy. Davey, where's Davey? What happened to Davey?\"\n\nSanders lowered the beam of his flashlight to the ground. He squatted down and was able to make out tire tracks in the hardpacked ground. The truck with the pop-up had camped in this spot. It wasn't a startling discovery\u2014any of the police officers could have told him if he'd asked. What was interesting was the way Duffy kept circling the back end of the campsite, where the pop-up had rested. To his mind, it suggested that Davey had been in the pop-up. He squinted in the yellowish light at the panting dog. \"Davey was in the pop-up. You got hurt here and made it back to the motor home. But Davey wasn't that lucky, was he?\"\n\nSanders kept walking around the back end of the campsite, trying to envisage what had happened to Davey Taylor. \"You got hurt and probably ran off to lick your wounds. I'm not saying I blame you but the question is, did Davey hide in the pop-up or did the couple grab him and keep him in the truck with them? I think he hid. What do you think, Duffy old girl?\"\n\nThe terrier whined in answer, nuzzling Sanders's leg. Satisfied that the agent was watching her, she lay down, her head on her paws. Sanders frowned. \"Are you tired, Duffy?\" The dog lay quietly, staring at him with wet brown eyes.\n\n\"I see, you're sticking with your story. That's good enough for me. C'mon, you deserve a ride back.\" He held out his arms to the little dog, allowing her into his shirt where she nestled her head on his chest.\n\nHe was walking back to Lorrie's RV when he saw a mothball laying on the ground. Thinking it was the mothball from the gravesite, that he'd somehow dropped it, he bent down and picked it up. \"I'm getting careless in my old\u2014\" He broke off when his hand dipped into his pocket and discovered the first mothball still there. \"Well, I'll be damned!\"\n\nMac Feeley was waiting for him outside the motor home, an evil-smelling cigar clamped between his teeth. Sanders wondered what the agent looked like without the cigar. Jesus, probably naked.\n\n\"Everything's taken care of\u2014the kid's picture will be in all the morning papers. Nothing's going on here. What say we get some shut-eye?\" Feeley's tone was hopeful but he was resigned to what he knew Sanders's answer would be.\n\n\"I have to call headquarters. But first I want to talk to Officer Ordway.\"\n\nFeeley grinned. \"The guy's a piss, Sanders. He's combed his hair four times in the half hour I've been standing here.\"\n\nSanders chuckled as he approached the young, blond policeman. \"Officer Ordway, isn't it?\" he asked. \"I'm Stuart Sanders, FBI. Were you around when they dug up Lombardi's body?\"\n\n\"Yeah, and I'm still around. Pulling double duty is no way to spend your life, if you know what I mean,\" the young man told him.\n\n\"Did you see the body? Was there anything unusual about it?\"\n\n\"Yeah, if you consider that bodies don't grow in the ground around these parts,\" came the flip reply.\n\nForcing himself to keep a cool head, Sanders persisted. \"Were you at the scene?\"\n\n\"I wasn't the one who dug him up, but it made the ones who did pretty sick. We don't have much crime down here. Drunks, petty thefts\u2014this is a small town. Ever since Wild Adventure was built, things have picked up a little. Hired two new policemen last year, and there's room in the budget for another one this year. My brother's got a bid in for the job.\"\n\nSanders listened with pretended interest. \"Sharp guy like you, you could go far. You look the type who keeps on his toes. Bet you've got it all over these locals. You should set your sights higher; maybe I can do something for you with the Bureau.\"\n\n\"No, thanks. If this is the kind of case you guys deal in every day of the week, I'll just sit back and collect a salary for traffic control. A few weekend family squabbles, a couple of buzzed-out teenagers, an occasional robbery\u2014that's my speed. Don't get dirty that way.\" He picked an imaginary speck of lint off his uniform jacket.\n\n\"I heard digging up that body was real dirty work,\" Sanders tried again.\n\n\"The worst. The stench was awful. Don't think I'll ever be able to smell that stuff again without remembering.\"\n\n\"What stuff?\" Sanders asked, getting to the point.\n\nOrdway stiffened as if realizing he'd spoken out of turn. \"Ah, you know\u2014the smell of death.\"\n\n\"Oh, sure,\" Sanders nodded. \"But I thought Lombardi had been dead only twenty hours or so. Bodies don't usually start to stink that soon.\"\n\nOrdway shrugged expansively. \"Yeah, well, twenty hours or not, that son of a bitch stunk.\" He turned back to the magazine he was reading, apparently determined not to say any more on the subject.\n\nSanders grinned wryly. He'd figured the local police would hold back. It was always the same story\u2014the search for glory. They wanted to be the ones to come up with the answers, and the only way to do that was to withhold information. It was a fact of life that Sanders understood, and he'd come up against it before. But what he had to do was make a definite connection between Balog and the dead man, and he could have what he needed right in his pocket. Lorrie had been there when they'd dug up the body. Maybe she could shed some light on things.\n\nLorrie was sitting at the table, staring out the window when Sanders opened the door to the RV. \"Did you find anything?\" The look on her face was hopeful.\n\n\"I'm not sure. You tell me.\" He pulled the two mothballs out of his pocket and set them on the table. \"I found one of these at the gravesite.\"\n\n\"There were dozens of them in the grave with the body,\" she confirmed, then glanced up at him. \"You say you found one of the mothballs at the grave. Where did you find the other one?\"\n\n\"At a campsite.\"\n\n\"Was there litter all around it?\"\n\nHe nodded.\n\n\"That's where the couple with the pop-up were parked.\"\n\n\"I thought so.\" Sanders squeezed Lorrie's hand and went back outside. The cops had an all points on Balog and his companion for questioning but, as of now, they were prime suspects in a murder inquiry, and possibly a kidnapping.\n\nWalking up to Feeley, Sanders said in a low tone, \"Keep pretty boy over there occupied. I want to call in and I don't want him listening to my end of the conversation. Two can play this game.\"\n\n\"We got nothing in common,\" Feeley grumbled.\n\n\"Then asphyxiate him with your damn cigar. But keep him occupied.\"\n\nThe Bureau chief picked up the phone on the first ring. \"Carmichael,\" he barked.\n\n\"Sanders here. Buzz, I want the okay to send Feeley into Newark. I've got a ninety-nine percent pure-gold lead I want him to check for me. We're just holding up the walls here, and the locals are withholding information. Feeley can be back by morning. I want him to check out the apartment that Balog was renting.\"\n\n\"You got it. No new leads on the kid, or is this part of it? You've only been there a few minutes and already you've come up with a snitch. How reliable is he?\"\n\nSanders's face was pained. He could hardly tell the chief that his \"snitch\" was a dog. \"Yeah, she's reliable,\" he answered without elaborating.\n\n\"Stu, I don't mind telling you this entire case is a pain in the ass. If I never see or hear from another local cop again, I'll die happy. Feeley tells me that Chief of Police Allen and his squad have been dubbed Allen's Assholes. True or false?\"\n\nSanders strained to see Officer Ordway through the thick smoke coming from Feeley's cigar. He was combing his hair. \"Sounds fairly accurate to me. I'll let you know my opinion tomorrow.\"\n\n\"You holding anything back on the kid, Stu?\"\n\n\"Only theories and personal opinions. You know they count for zilch as far as the Bureau goes.\"\n\n\"Tell me about it.\"\n\n\"I'm pretty sure the kid wasn't snatched by any syndicate. It doesn't fit. What I've got is a possible connection between this Balog character and the dead man. I say, if I find Lombardi's killer, I'll find the Taylor boy. I think the kid saw something he shouldn't have, got scared and ended up in Balog's pop-up camper.\"\n\n\"Your snitch drop that in your lap?\"\n\nThe pained look crossed Sanders's features again. \"In a manner of speaking. You could say she pointed it out to me. I promised her the biggest steak dinner of her life if it pays off.\"\n\n\"Is that on or off the expense account? You sly dog, I always knew you had it in you. A word of caution from one who's been burned\u2014that young stuff out there can be hard to handle. They don't mind givin' if you don't mind gettin'. You know what I'm talking about, don't you?\"\n\nSanders grinned. \"This little lady, I can handle. I'll check in midmorning. Have a good night, Buzz.\"\n\n\"You too, Stu.\"\n\nA harsh violet-colored light was coming through the plastic drapes when Cudge opened his eyes. For the briefest instant he couldn't remember where he was, or why. Then he realized the glare was the security light in the parking lot behind Snookie's.\n\nCandy Striper was in the bed next to him, turned on her side to face him. She wasn't pretty now with her makeup streaked and her mouth partially open. Why did he always end up with the dogs? Each time she exhaled, the odor of stale wine and cigarettes assaulted him. She was a pig, he decided. He'd known it last night, but he'd thought she may have something to offer. He only hoped what she was offering wasn't the AIDS virus. \"Lie down with dogs, get up with fleas\" as the old saying went.\n\nHis head felt heavy and he could hardly lift his arm to read his watch dial. Four forty. In the morning? It had to be. It was yesterday evening when he'd brought the tire to the gas station next door to Snookie's. He sat up quickly, holding on to the wall for support. Shit! That mechanic had said he'd throw the tire into the back of the truck when he'd finished with it. What if someone had stolen it? All those drunks coming out of Snookie's\u2014what if they stole his truck?\n\n\"Hey, what's the matter?\" Candy groaned. \"What time is it?\"\n\n\"I gotta get outta here,\" Cudge said, and rolled over her to get out of bed. The mattress sank beneath his weight and the bedsprings screeched in protest. Candy's body was lush and soft beneath him and he remembered flashes of what it had been like with her last night. Soft, warm, like he'd been plowing into a featherbed.\n\n\"You got an old lady someplace looking for you?\" she asked sleepily.\n\nElva. How could he have forgotten about Elva? What if she decided to skip out on him? Go to the cops herself? Pin the whole rap about Lenny on him. Then it wouldn't be just that little kid to worry about, there would be Elva, too. Cursing under his breath that he'd ever gone into Snookie's and got entangled with Candy, Cudge searched for his clothing. He never should have gotten sidetracked. He should have stayed to watch the mechanic fix the tire then gone right back to the camper, just the way he'd planned.\n\n\"Hey, babe, any more wine left in that bottle? My mouth's dry.\" Candy's voice was husky with sleep and hangover.\n\n\"Get it yourself,\" he snapped. \"I gotta get going. Where the hell're my boots?\"\n\n\"Find 'em yourself.\"\n\nHe bristled. \"Watch your mouth. Where's my boots?\"\n\n\"How should I know? They're your boots. What did you do with them?\"\n\n\"I don't even remember taking them off.\"\n\n\"You didn't. I did,\" Candy smiled with satisfaction. \"Hey, babe, why don't you crawl back in bed here? I got something to show you.\"\n\n\"Save it,\" he answered uncomfortably, not liking women to take the initiative. \"Didn't you get enough last night?\" He found his boots and something soft and rotten came off the bottom onto his hands. \"Shit!\"\n\n\"What's that?\"\n\n\"Dog shit,\" he moaned, looking for something to wipe his hand on, finally deciding on the bedcovers.\n\n\"Hey! Don't do that!\" Candy sat up suddenly to inspect the damage. \"Christ, ain't you got no manners? Don't wipe that stuff on the covers. One thing at least,\" she said, turning on the lamp for a closer examination, \"it's not dog do, it's apples. Even so, watch where you put it.\"\n\n\"Apples!\" Sure, now he remembered those rotting apples on the road where he'd pulled over in the pop-up. He must have stepped on them and gotten bits of the pulp wedged between the ridges of his soles. Apples. Elva. He had to get back!\n\nCandy settled back against the pillows in a languorous pose. \"What's your hurry, babe? It's not even morning. Come on back to bed. This time it's for free.\"\n\n\"It's always free,\" Cudge growled, tying one of his laces. \"Your kind always are.\"\n\n\"Like hell! That little parlor game we played is going to cost you exactly thirty bucks.\"\n\n\"Don't bug me, Candy, I ain't in the mood. Right now you're treading on thin ice. You think I'd pay to screw an old whore like you?\"\n\n\"Old whore! Listen, you'd have to pay anybody to jump in the sack with you. And double if they'd been there once already. You ain't exactly Casanova,\" she mocked. \"Tell me, you always have that much trouble gettin' it up or was it just the wine?\"\n\nCudge loomed over her, standing close to the bed, hands curling into fists. \"One more word and you're askin' for it. One more word. I don't want no more trouble, so just shut up!\" He seemed to be fighting himself as he stood over her, forcing himself to back down, to quell his sudden rage. \"Where's my shirt?\" Not waiting for an answer, he looked around the shabby room.\n\nWhen he moved away from her, Candy propped herself up on her elbow. \"You don't really have any connections in Vegas, do you?\"\n\n\"What do you think?\"\n\n\"I think you were puttin' me on, so the deal's off. I told you, I've got expenses and that wine we drank together last night really knocked me on my ass. Far as I'm concerned, last night was a lost cause, and I ain't got nothing to show for it. So, how's about that thirty dollars I helped you win from that guy?\"\n\n\"You gotta be kiddin'. That's my thirty bucks. I won it. You already got all you're gonna get.\"\n\nCandy sat up in bed, the blankets falling away to reveal her full, round breasts. \"The way I see it, you parked your ass in my bed and that'll cost you thirty dollars.\"\n\nBalog faced her in the yellow lamplight, his eyes cold and hard. The air between them seemed to become dense and still. His square, cropped head burrowed into his powerfully wide shoulders; his barrel-staved chest expanded with each deep intake of air. Candy heard a high, frightened whine and realized with horror it was her own.\n\nThe corral gate in Cudge's brain swung open with nerve-cracking velocity. The beast lunged forward to freedom, sharp-edged hooves cutting into his brain tissue. Blackness obscured his vision. Rage ripped through him; he was almost senseless. Razor-tipped horns gored his sanity and attacked again and again.\n\nCandy was helpless, defenseless as a rag doll in the jaws of a mad dog. She struggled against unconsciousness, fighting to stay alert, knowing it to be her only chance of survival. But when oblivion came, she surrendered gratefully.\n\nThe woman was still, no longer crawling across the floor to escape his onslaught. How long had she been like that? Still as death. Cudge backed away, stumbling over an overturned chair.\n\nIncredulous, he looked down at his hands, just beginning to feel the tingle that preceded pain. He'd broken his little finger; it was sticking out at a crazy angle. Gritting his teeth, he straightened it and heard a sharp crack before it settled into place. Thinking he should wrap it to keep it in place, he searched the room, seeing one of the black stockings Candy had worn during her dance on Snookie's bar. It was thin and long, suitable. Winding the nylon around his knuckles, he looked around for anything he might have left behind. Now he had to pick up his truck and make tracks.\n\nHe didn't want to think about what he'd done to Candy. He didn't want to think about anything. He'd have to keep his head, keep his temper under control. From now on, he promised himself, every move he made would be planned. He wouldn't let the gate swing open again.\n\nDavey woke slowly, trying to remember where he was. He opened his eyes to see Brenda still sleeping. Her arms were wrapped tightly around him. He had to go to the bathroom; it must be morning. He shouldn't have stayed, but Brenda had been so scared; he had felt her heart pounding when she held him. He didn't know what he had feared more: going off alone in the dark, or what the man would do if he found him.\n\nHe strained to see the hands on his watch. The big hand was on the seven and the little hand between the five and six. He felt confused, but knew he must leave the safety of the pop-up and try to find Aunt Lorrie. Through the mesh windows he could see daylight between the trees, streaking the sky a gunmetal gray. First, he had to go to the bathroom. He needed some food to take with him too, in case he got lost again. He had to be prepared. Should he wake Brenda and say goodbye? He should thank her for helping him.\n\nCarefully and quietly, Davey inched from Elva's arms and stood up. He held his breath when his foot touched his discarded leg brace alongside the small refrigerator. Should he put it on? Maybe Aunt Lorrie would feel better if he wore it. He struggled with the familiar straps, trying to be quiet. Wincing in discomfort when the padded straps touched his chafed ankle, he remembered to pull his sock up the way his aunt had shown him.\n\nHurry, hurry, he told himself. He looked at Brenda. He knew he should thank her but if he woke her up she might want to keep him with her the way she'd done the night before. Aunt Lorrie had told him once that sometimes the right thing to do was often the hardest. So it must be the right thing\u2014to leave without thanking her, or saying goodbye.\n\nThe sound of a car coming close stopped him.\n\n\"Brenda, Brenda, wake up. He's here. Wake up, please wake up,\" Davey coaxed.\n\nElva sat up. \"He's here, Brenda. I heard the truck.\"\n\n\"Oh, my God!\" Holding Davey's hand, she scrambled out of the camper. \"Run, BJ, run!\"\n\nDavey didn't hesitate. He was out of the camper and running as fast as his legs would carry him toward the trees. On and on he ran, the patch pockets of the red windbreaker ballooning. He fought for air as he staggered through the sodden leaves and twigs. It was good that they were wet and didn't make any noise.\n\nHe needed to take off the brace; it was slowing him down. Could he stop? Did he have time? He had to stop. He had to take the time or the man would catch him. He sat down on a matted pile of leaves. The straps were tight and still damp from yesterday when he'd wet himself. The brace was always easier to put on than take off. There, it was off. He couldn't take the time to appreciate how good it felt to have his leg free. He had to run. He had to run fast.\n\nCudge stood over Elva, his hands flexing as though working an invisible pair of grips. The long tendons in his forearms were hard and prominent under his skin. The meager light of early morning was behind him, throwing his silhouette into relief and hiding the brain-exploding rage on his blunt-featured face.\n\nElva backed away, shaking. She tried to tell herself that Cudge hadn't seen BJ escape, but she knew that was just wishful thinking. His posture was threatening, lethal. But BJ was safe.\n\n\"What were you doing here with that kid? How long have you been hiding him?\" Her father's voice boomed in her ears, bouncing off her fear. She'd always been afraid of him, ever since she was a little girl. She'd never understood why Mama had married him or had children with him. There was no kindness in the man, only a love for the cheap hooch he bought in town.\n\n\"Papa, don't. I didn't do nothing wrong. Honest, I didn't.\"\n\nShe could smell the familiar liquor, sweet yet rancid, each time he exhaled. He'd been drinking again. There wasn't food on the table, but the welfare check could buy hooch the same as it could buy food. Poor BJ. He was just a little boy and he needed milk, butter and eggs. Instead, all he got was thin potato soup made with dry skim milk. Poor BJ. All eyes, scared eyes, that watched the doorway waiting for Papa to come home. Praying he wouldn't. And Mama, grim mouth, hunched shoulders, red hands from too much laundry and trying to keep the place clean.\n\nPapa advanced on her, striking her in the thigh with his heavy boot. \"How long have you been hiding that little bastard? You tell me or I'll give it to you. I'll finish you for good!\"\n\nThe words were different, but they meant the same. Just like the night Papa had come home and heard BJ crying. BJ had never cried again. But Brenda\u2014she had never stopped crying.\n\nElva was back in the poverty-ridden shack in the Pennsylvania foothills. The smell was the same. Liquor, dirt, fear. And the smell of dry urine from where little BJ slept. But this was the second chance she had always prayed for. BJ would be all right; she'd seen to it by making him run away.\n\nShe could hear the low intake of her father's breath; it sounded like the growl of a beast. He moved toward her with sure, slow steps. She could sense his lust for the kill; she could almost taste it. An icy finger touched her shoulder, but she was too paralyzed to move, trapped in his gaze like a rabbit in the beam of a headlight. Somewhere in the back of her brain she heard her mother screaming: \"Don't touch my baby!\"\n\nHis hands were in her hair, yanking her to her feet. \"No! Leave her alone!\"\n\nWith all the force she could muster, she kicked, aiming between his legs. She felt him flinch and heard his sudden gasp. He flung her from him, sending her on to the floor. The pop-up rocked wildly on its moorings as the world tilted before her.\n\nPop-up? This wasn't Papa. Mama wasn't screaming, those were her own screams. BJ was dead, long dead. Cowering in a corner beside the refrigerator, Elva shielded her head with her arms. This was Cudge and he was just like Papa. Nasty, mean\u2014a killer. Before the cast-iron frying pan came crashing down on her, Elva had an insight that she'd never been quite able to put together before. She didn't have to wonder any more why she stayed with Cudge\u2014she knew. He was her punishment. She'd always known it would end like this. He was Papa all over again and, because he was, he gave her a certain security. Violence had been her birthright. And violence was Cudge. Through him, she would be able to pay for her sin of not saving BJ. She would die, just the way her little brother had died, and then her soul would be saved.\n\nHer arms came away from her head. She allowed his weapon to fall, allowed herself to succumb to the blows. And always there was little BJ's face, eyes watching, waiting for her.\n\nThe round, plastic disc on the zipper of his jacket bounced against his neck as Davey ran through the woods, his breathing harsh and ragged. Faster, his mind screamed, don't let him get you. You have to find Aunt Lorrie. He was getting tired. And then he heard the words, and his Reeboks picked up speed.\n\n\"I'll get you, you little bastard,\" the man yelled. \"You ain't getting away from me!\"\n\nThe trees were thinning so it was easier to run between them. Perspiration streamed down Davey's forehead as he staggered ahead. It was so hard to breathe. Maybe he could hide somewhere and wait till he felt better, just a little while. No, he couldn't stop. If he did, the man would find him and kill him. If only Duffy were here. If only Aunt Lorrie were here. But they weren't\u2014he was alone, and he had to find Duffy and Aunt Lorrie all by himself.\n\nSaliva formed in the corners of his mouth as he heard curses coming from his left. His gasping breaths were making too much noise; his hand over his mouth to seal in the harsh sounds, Davey floundered ahead. He was slowing down now and tears of frustration gathered in his eyes. He stopped and listened to the early morning stillness. He was at the edge of the woods; ahead was a row of young trees then an open field. Davey looked ahead then back behind him. If he went across the field, he would be out in the open. If he stayed where he was, the man would catch him. Hide, he thought. He had to hide for a little while and rest. Davey looked around. There was no hiding place among the saplings.\n\nQuickly he veered off his straight path, almost backtracking, but to the right. Hardly daring to breathe, he crouched down behind a wide tree and clamped one hand over his mouth to still his harsh gasps. His other hand fumbled with the zipper on his windbreaker. His eyes closed as he anticipated the sound the zipper would make as he pulled it down. It moved down but got stuck in the metal at the end. He needed two hands to make the jacket open. A second was all he needed, but he had to decide which was more important\u2014keeping his hand over his mouth or getting rid of the red jacket. He crouched lower against the tree trunk. He crossed his fingers then yanked at the plastic disc and struggled out of the jacket, careful to make no sound. What should he do with the jacket? Mom would be mad if he left it behind. She always said you had to value your things and take care of them. He looked at his Reeboks. He might get away with the ruined sneakers, but not the jacket as well. He had to take it with him. If only it wasn't red. He looked at it with disgust and saw that the inside was brown. Dirt brown. He pulled the sleeves inside out and put the jacket back on. Now he blended in with the forest colors. The man would have to have real good eyes to find him. He felt confident as he started out again, but a sharp noise close ahead made him drop to the ground.\n\n\"I know you're here somewhere, you little twerp. I seen your jacket. You ain't gonna get away from me this time. I took care of Elva, and now I'm gonna take care of you.\"\n\nDavey flinched. The voice was so close, almost at his head. The tone changed from threatening to cajoling. \"You know I'm gonna catch you, so you might as well come out from wherever you're hiding. You come out now, and I'll take you to your folks. I'm gonna count to three, and you come out, okay?\"\n\nDavey remained still. The man was lying. He was trying to trick him, to make him come out, and then . . .\n\n\"One, two, three. Okay, you had your chance!\"\n\nSounds of crashing and swearing fell on Davey's ears. How close the man must be. Davey was shaking with fear. He didn't dare lift his head or make any movement at all. It was a good thing he'd turned his jacket inside out. Gradually, the sounds grew quieter, as if the man were moving farther away, but his curses and threats could still be heard. Davey lifted his head and risked a look; there was no one staring back at him. A long sigh escaped him.\n\nHe struggled to his feet and looked at his watch. The little hand was on the seven and the big hand on the twelve. Seven o'clock, he told himself. Seven o'clock in the morning. If he were at home, Mom and Dad would be in the kitchen eating breakfast. He wished he'd had time to take one of the cupcakes out of the bag on the refrigerator before he left the pop-up. But he couldn't think about food now. He had to get out of the woods and find a road where there were cars.\n\nDavey trudged back through the woods. The man had gone in the other direction, so he didn't feel he had to run. His legs ached furiously, and the knee that needed the brace kept bending when he put his weight on it. His hands were scratched and bleeding, and he could feel where the side of his face had scraped into a tree. Touching it lightly, he brought droplets of blood away on his fingertips. When had he had his last shot? Was it still working? Would the bleeding stop? His ankles and knees felt tender. Was it from running, or was it like before he started the shots?\n\nPainfully tired and frightened, Davey hid in the brush under the trees. He would sit there and listen, watchful for the bad man. Lying on his belly, head cradled in his arms, he tried to stay awake.\nChapter 12\n\nSara Taylor stepped out of the bathroom after her morning shower, dry, powdered, cologned and naked. Slipping her toiletries into an Yves Saint Laurent cosmetic case, she placed it at the top of her suitcase. Andrew would bring both their suitcases home on his late afternoon flight. Sara hoped no unexpected problems would arise, and that Andrew's appearance in court would be completed by the afternoon. She could count on it, she thought, remembering how well she had drilled him on his testimony.\n\nShe must see about breakfast. For Andrew, hot cakes, sausages, one scrambled egg, and toast. For her, half a melon. A large pot of hot coffee for the two of them. She had to eat something to get her thoughts in order. She didn't want to think about what had happened in the shower ever again. She would put it out of her mind and never let it resurface.\n\nSara opened the door and handed the ever-present guard her breakfast menu. \"Would you please tell room service that Mr. Taylor and I would appreciate our breakfast now. We have to be in court by nine.\" Without waiting for a reply, she closed the door, then walked over to the bed and gently shook Andrew awake.\n\n\"Time to get up, darling. All you have to do is shower. Everything is laid out for you. Breakfast will be up shortly. I don't want to rush you, Andrew, but sometimes you have a tendency to dawdle. Today you have to hurry\u2014it's already seven o'clock.\" If Andrew noticed her annoyed tone, he didn't say anything. Instead, he dutifully got up, went into the bathroom and took his shower.\n\nSara stood outside the bathroom, listening to the sound of water splashing against the tiles. Her thoughts strayed back to DeLuca and her sexual fantasy. She shook her head to make them go away but they wouldn't. Maybe the television would distract her. The set came to life with a sexy, light-hearted comedy show. Sara frowned\u2014everything had sexual connotations of one sort or another. She changed the channel to an early-morning talk show. An author was describing her latest book about men's sexual fantasies. Damn. Again, she pushed the button on the remote. Cartoons. That was certainly safe.\n\nShe heard the shower being turned off. Andrew would be in the sitting room in another minute, and she could question him again, give him a short run-through to be sure he hadn't forgotten his lines. What was she going to do on the plane for two and a half hours? She would have all that time to think, to fantasize . . . Davey\u2014she would think about Davey, and Lorrie, and what she was going to say to her. How could Lorrie lose her son? How dare she allow him out of her sight? Yes, Davey and Lorrie would be safe ground.\n\nAndrew was fixing his tie when a knock sounded at the door. A waiter carried in a breakfast tray laden with plates and silverware. After he'd left, Sara lifted the stainless-steel covers. She shook out Andrew's napkin and handed it to him.\n\n\"Looks good. Aren't you eating, darling?\" he said.\n\n\"Just some melon. You know I don't like to eat before a plane trip. This is more than enough. That little round scoop of butter makes the hot cakes look so appealing, don't you think, Andrew?\"\n\nAndrew nodded as he chewed industriously. \"This breakfast is a surprise\u2014all my favorites!\"\n\n\"You deserve only the best, Andrew, and that's what I want for you.\" Sara knew her voice lacked conviction, but Andrew was so busy eating he wasn't paying any attention.\n\n\"Are we taking the bags to the courtroom or do you think I should come back for them?\" he asked minutes later, as he pushed his empty plate aside.\n\n\"You might want to freshen up before you leave for your plane. Leave them here or they'll just be a hindrance. The limo will be waiting, so there shouldn't be any problem. You might even want to shower. I heard just a short while ago that it's going to be eighty-nine degrees today.\"\n\n\"I'm glad we don't live here anymore, I can tell you that,\" Andrew volunteered as he pushed back his chair. \"I like the change of seasons at home. What about you, Sara? Do you ever miss Florida?\"\n\n\"No, darling, I don't. Perhaps though, we should start thinking about moving out of New Jersey. It's so . . . so crime-ridden. Connecticut would be nice; let's consider it in the spring. I'm ready when you are.\"\n\n\"Then let's go and get this over with, so I can be home with you and Davey for dinner this evening.\"\n\n\"We don't know for certain that Davey has been found. As of last night he was still lost, but I'm sure they must have found him by now. He did miss his shot though, and that's what worries me.\"\n\n\"I feel just as you do\u2014Davey will be waiting for both of us when we get back. Sara, instead of punishing him for wandering off, why don't you make a wonderful dinner with all his favorites? We'll make a party of it, and afterward have a serious talk with him. We don't want him to think we're angry with him, but he has got to realize what he's done. My father used to do that with me. The real punishment came from having to stand there with no defense for what I'd done, and my father understanding perfectly. My mother, too. They were never divided when it came to me. Whatever one said, the other agreed. Like we do with Davey.\"\n\n\"It sounds like a tremendous idea.\" That was something she could do on the plane ride. She could plan the menu and think about the talk she and Andrew would have with Davey.\n\n\"I, for one, won't be sorry to see the last of this hotel,\" Andrew said, closing the door behind him. \"You'd better give me the key, Sara.\"\n\n\"Darling, I really don't want to go and leave you here. I'd much rather be here with you, but our son . . .\"\n\n\"Sara,\" Andrew said, putting his arm around her shoulder, \"I don't want to hear another word. We agreed and the matter is settled. I'll be fine. I'm just sorry that I made such a botch of things yesterday. Today I'll do much better. The State will have no problems. Mr. DeLuca will be more than satisfied.\"\n\n\"I rather doubt that,\" Sara said, sotto voce.\n\n\"What was that, darling?\"\n\n\"Nothing. Just woolgathering. It wasn't important. We're fortunate that we left when we did. The traffic is building up. You're going to be the first witness, so it is imperative that you be on time.\"\n\nThirty-five minutes later the limousine pulled up at the curb outside the courthouse. Sara hated the pushing and shoving crowd of reporters shouting questions. As she walked beside Andrew up the stairs, she kept her gaze straight ahead, her step firm and sure. It wasn't till one of the security guards opened the door for Sara that she saw Roman DeLuca. She felt a flush creep up her neck.\n\n\"Good morning, Andrew, Mrs. Taylor,\" DeLuca said. Sara wondered why DeLuca called Andrew Andrew and her Mrs. Taylor. Andrew was whisked off by two court bailiffs, and Sara was left standing next to Roman DeLuca and one of the men from Sanders's unit, Michael Jonas. Stuart Sanders had introduced Jonas to Sara and Andrew, identifying him as an assistant supplied by the State of Florida. Sara wondered if he, too, was part of the syndicate.\n\n\"You don't strike me as the type to make foolish mistakes, Mrs. Taylor.\" DeLuca's eyes narrowed. \"Why didn't you take my advice? You'll have no one to blame but yourself.\"\n\nSara felt herself grow rigid. \"I don't make mistakes. I always take full responsibility for my actions. As for your advice. . .\"\n\n\"Spare me, my dear. You had your chance last night and you ignored it. We don't play games.\" DeLuca turned to his associate. Sara felt mesmerized as she watched him. His lips barely moved. \"Take care of the lady, Jonas.\"\n\n\"Yes, sir,\" came the clipped reply.\n\nIt happened so fast. One minute the three of them were standing alone, the next minute there was a swarm of reporters around them. Sara stepped aside, clearing the way for DeLuca to take center stage. Jonas moved at the same time, jostling against Sara. She knew, without seeing the gun, that the man was wearing a shoulder holster; she felt the hardness of the leather as she was thrown against him. The color drained from her face as she understood what DeLuca had meant by \"take care of her.\" He must have thought she wasn't leaving. It was her own fault for allowing him to think she wasn't giving in to his threats. The fact that she was on the verge of telling him meant nothing. She couldn't tell him now, while he was holding court to the press. She would leave now while all the commotion was going on. DeLuca would find out sooner or later that she was on the flight. As she turned to go she felt a touch on her arm.\n\n\"We'll leave together. Don't say anything, just move casually.\"\n\n\"Take your hands off me,\" Sara hissed.\n\nJonas's grip became more secure. \"Smile, pretty lady, so the reporters can snap your picture. I mean it.\"\n\nHe did, she could feel it. \"You wouldn't dare do anything here in full view of all these people. You aren't an associate of Stuart Sanders. You're one of DeLuca's thugs, not a government agent. I also know what you're wearing under that custom-tailored suit.\"\n\n\"Then you know I mean business. Keep smiling and walk slowly, straight out to the car.\"\n\n\"Why? Where are we going? That's assuming I go with you.\"\n\n\"For a little ride. You talk too much.\"\n\n\"I was just going to the airport.\" Sara didn't like the man's looks or the cold, controlled way he spoke, knowing that he had no intention of taking her to the airport. The shoulder holster said all there was to be said. She had to think quickly. Where was that seedy-looking reporter from the slick tabloid? He had offered five hundred dollars for her exclusive interview late yesterday afternoon. Her mind raced. What was his name? Peter? Percy. Percy Strang, that was it.\n\n\"Let's get on with it, Mrs. Taylor. You're wasting time. I'm not going to tell you again.\"\n\n\"I can see that,\" Sara smiled up at him. \"One moment and then I'll be glad to join you for coffee.\" She made her voice purposely high and shrill, knowing one or more of the reporters would look her way. She gave them all a dazzling smile then asked for Percy Strang. The reporter from the Miami Herald looked disgusted. Sara could almost read his mind\u2014the gossip tabloids always got the dirt because they paid for it. It always came down to the buck. An honest reporter had to hustle, and then he was lucky to get two inches of printable news.\n\n\"Yo! Here I am, Mrs. Taylor. Changed your mind, I see.\"\n\n\"Why not? I can spend five hundred dollars just as easily as the next person. Why don't we go for coffee?\" Sara asked.\n\n\"You got it. I'll even buy.\"\n\n\"How generous of you.\" She felt panic rise in her chest as the clamor around Roman DeLuca stilled; everyone was listening to the exchange between the wife of the State's star witness and the reporter from the Informer. \"When will you pay me?\" Sara demanded as an afterthought.\n\n\"As soon as these nimble fingers get everything down in black and white.\"\n\n\"A lady could hardly ask for more.\" Sara felt all eyes on her as she shook Jonas's hand off her arm. \"I'm so sorry, Mr. Jonas, to renege on our breakfast date. However, five hundred dollars is five hundred dollars. I'll see all of you later. Gentlemen.\" She nodded to the crowd and moved quickly to Percy Strang's side.\n\nSara linked her arm through Strang's, walking him down the corridor, out the door and down the steps to the street. The limousine was standing at the curb; the driver inside was reading the paper and smoking a cigarette. She dragged the reporter with her as she leaped into the backseat of the luxurious car. \"Take me to the airport, driver, and please hurry. You don't mind, do you, Mr. Strang? I mean about going to the airport. If we have time, we can have coffee. I thought it would be best if we talked on the way. This way you won't feel I'm trying to put you off. I don't have much time, and we'll have to stop the interview when I board.\"\n\nAt the reporter's disappointed look she went on hastily. \"Of course, you can always come to New Jersey, or call me long-distance. My son disappeared yesterday, and that's why I'm going home now. No one knows yet. It's your exclusive, Mr. Strang. However, I don't want to talk about that just yet. We'll get to my son later. For now, why don't you and I get to know each other? You tell me what it is you want to know, and I'll answer to the best of my ability.\"\n\n\"Are you saying your son's been kidnapped? Jesus, Mrs. Taylor, I don't know what to say!\"\n\n\"I knew you'd feel like that which is why I said we would talk about it later. I do have one rather small favor to ask of you though.\"\n\n\"For an exclusive like that, Mrs. Taylor, I would try to walk on water for you.\"\n\n\"Yes, well, that won't be necessary. I want you to stay next to me every minute till I board the plane. Don't leave my side. If you see . . . What I mean is, if you suspect . . . Just don't leave my side. Is it a deal, Mr. Strang?\"\n\n\"Of course it's a deal. This is the kind of stuff reporters only dream about. How am I ever going to thank you?\"\n\n\"By staying next to me and not letting me out of your sight.\"\n\n\"Is there a microphone in this limo? How do you give the driver instructions?\"\n\nSara felt faint. \"I don't know . . . I'm not sure.\"\n\nPercy Strang fiddled with the small grille above the ashtray. \"Is this a rented limo or did the District Attorney provide it for you and your husband?\"\n\n\"The district attorney?\" Oh, God. \"Mr. Strang, look outside. Is this the way to the airport?\"\n\n\"Beats me, Mrs. Taylor. I'm based out of Chicago\u2014I was sent here to cover the trial. We could be going to Cuba for all I know.\"\n\n\"Press that gadget and ask the driver how much further it is. Do it now!\" Her voice was shrill. Percy did as he was instructed.\n\n\"Another five minutes, ma'am,\" came the reply. \"The turnoff is just ahead. You can see the sign from here.\"\n\n\"Thank God,\" Sara sighed.\n\n\"Mrs. Taylor, I don't know why you're nervous, but I do think there's something you should know. When we pulled away from the curb back at the courthouse, a car pulled out right behind us. It's been following us ever since. Being a reporter, I'm trained to notice such things,\" Strang said importantly, hoping the elegant woman would be impressed.\n\n\"Oh, God,\" Sara moaned. Whoever was in the car wasn't following her to the airport to see if she got on the plane. If she knew nothing else, she knew that for a fact. She was going to be killed. She could taste her own death; a bitter sourness in her mouth.\n\n\"Do you read the Informer, Mrs. Taylor?\"\n\n\"All the time. My housekeeper buys it and leaves it in the kitchen,\" Sara said feverishly. She should have told Andrew. Left a message, or something. What lies would Roman DeLuca tell her husband? Poor Andrew, he expected to get home to a wife, a wonderful celebration dinner, and everyone living happily ever after. \"Shit!\" she said succinctly.\n\nPercy Strang frowned. He pursed his narrow lips in disapproval. Mrs. Taylor had slipped a notch in his esteem. Back at the courthouse she had seemed so elegant, so regal, like a princess, and she would certainly look smashing on the front page, right next to his story about the woman who had grown another kidney\u2014a genuine medical miracle.\n\nThe chauffeur leaned his head back and spoke. \"Which airline, ma'am?\"\n\n\"Airline?\" Sara repeated.\n\n\"Yes, ma'am. Where do you want to be dropped off?\"\n\nSara's mind raced. Eastern? Delta? Pan Am? She couldn't remember. They had come down on Eastern, but that was no reason to assume DeLuca had made her return reservation on Eastern. However, \"Eastern,\" she said firmly. The check-ins couldn't be that far apart.\n\nSara's mind raced as the limousine pulled up at the curb. Perhaps the driver would deliver a message to Andrew, or should she have the reporter deliver it? She couldn't make up her mind. If she did write a note to Andrew, she might place him in danger, and Andrew didn't function well in a crisis. He needed her and she was failing him. Should she write the note or not? The decision was taken out of her hands the moment she stepped out of the car. Just as the porter asked if she had any luggage, DeLuca's man, Jonas, leaped from his car.\n\n\"Mrs. Taylor, Mrs. Taylor, wait for me,\" Jonas shouted, a smile on his face.\n\nSara could hear Percy Strang calling to her to stop as she ran for the Eastern check-in counter, but she ignored him. Where were the police? The security guards? Miami had a crime rate to warrant an officer every twenty feet, but she couldn't see anyone to help her. How could Jonas kill her in a public space?\n\nIt had been so long since she'd prayed that Sara felt the words catch in her throat. She couldn't remember the simple prayers she had learned as a child. She had never been one to rely on the Almighty to help her, preferring to handle her own affairs in her own way. If she didn't see a uniform soon, she would have to go to the check-in and tell the reservation clerk. Tell him what? her mind shrieked.\n\n\"Mrs. Taylor, where are you going? Why are you so upset?\" Strang had caught up with her. \"Didn't you hear me? There's a man running after us who wants to talk to you. Perhaps he's got news of your son.\"\n\nSara turned. \"That man you are so concerned about, Mr. Strang, is trying to kill me. He may even kill you. That's why I'm trying to get away from him. Where is he?\"\n\nIntrigue, kidnapping, killing\u2014all the ingredients for a first-class story. Some journalist in the sky must be looking out for him, Percy thought happily. He almost tripped over his own feet as he tried to keep up with Sara. \"He's right behind us, and I think you should know he's gaining rapidly. Look, Mrs. Taylor, I'm no he-man type. Why don't you find a cop?\"\n\nSara picked up speed. \"I would if I could. Do you see one anywhere?\" she shot back.\n\n\"You could shout for help,\" Strang said.\n\n\"And then what? The man following us is connected with the syndicate. He works for the District Attorney, who is also connected with the syndicate. Where do you think that leaves me, Mr. Strang?\"\n\nSara wanted to scream with frustration. Where was she going anyway? Who was she fooling? If she did find a policeman, what was she going to say? Jonas would whip out his credentials and tell the officer that she was distraught. The officer would gladly hand her over to the District Attorney's right-hand man. She didn't stand a chance, and she knew it. She wasn't going to be allowed to get on any plane. Jonas could even say she was mentally unstable. Who was going to believe her against someone with his credentials? He had Madison Avenue, Ivy League, the military, and law enforcement on his side. And what did she have? She was just a highly strung, perspiring, middle-aged woman being trailed by the star reporter from the Informer. Once she was in Jonas's custody, that would be the end. The local police would probably give him an escort out of the terminal.\n\n\"Andrew, Andrew, I'm so sorry,\" she murmured. Frantically she looked around, trying to see some way of escaping the man trailing her. Maybe she would be better off outside; she could walk around the airport building endlessly and still not get away. She needed an exit. A diversion. It was getting difficult to breathe.\n\n\"I don't understand any of this,\" Strang bleated. \"We really have to stop and catch our breath. Mrs. Taylor, are you listening to me? If you sit down calmly to talk and think, we might come up with a solution to this . . . this predicament. What I'm saying is, this is a busy, well-policed airport. The man behind us isn't going to . . . Besides, isn't he one of the men assigned to guard you and Mr. Taylor? Mrs. Taylor, you aren't listening to a word I'm saying. You're overwrought; you must be mistaken.\"\n\n\"You're right, Mr. Strang, I'm not listening, and that's because you don't know what you're talking about. Back in the courthouse I purposely singled you out to get me away from the man following us. He is wearing a gun in a shoulder holster. As long as you walk directly behind me, as you're doing now, he won't dare shoot me. I'm frightened, and I don't mind if you know. You should be frightened, too, Mr. Strang. These people are evil; they will stop at nothing. They've already kidnapped my son. Now do you see why I have to get away from that man? He has no intention of letting me board any plane. I did something foolish this morning, and I'm going to pay for it with my life. But I'm not giving up without a fight, I can tell you that.\" Her voice was barely a whisper, hoarse and frightened.\n\n\"Come now, Mrs. Taylor, you're upset about your son, and the fact that your husband is a key witness in a murder case. It's understandable that you're building mountains out of molehills. What do you say to a cup of coffee?\"\n\nOh, God, she was going to die and he wanted a cup of coffee. She felt like throwing up. Then she saw the diversion she needed. A group of South American soccer players was advancing down the concourse with good-natured backslapping and hilarity. If she quickened her stride, she could reach the exit sign at about the same time the players did. And if Strang could manage to join the crowd of players, he could stall for time and allow her to get outside where she could run.\n\nThe words came in controlled gasps. \"Mr. Strang, will you help me? If you do this, I promise you the story of your life. Ten minutes, that's all I need.\"\n\nThe story of his life! The big scoop! They were the words every journalist dreams of. The big by-line. He would do it\u2014it would spread across pages one and two. The woman with three kidneys would have to wait for another issue. This was the big one, the one he had dreamed about ever since walking into the Informer office to apply for a job. They had asked him two questions only. Can you type? Can you spell? He had answered \"yes\" to both and they'd hired him on the spot. And now his big chance had come! Even if Sara Taylor was crazy, it was a great story.\n\nSara felt the color drain from her face as she closed the distance between the soccer players and herself. How lightheaded she felt. Maybe she could get out of this after all. The exit sign blurred. \"Now,\" she whispered hoarsely.\n\nShe thought Strang hadn't heard her, but suddenly he threw up his arms and walked smack into one of the approaching players. \"My God, you're a sight for sore eyes,\" he shouted as he wrapped his skinny arms around the player's neck. \"I want you to endorse some new soccer balls for me. You and your friends move right over here.\" He pushed the players together into a huddle under the exit sign.\n\nSara gasped\u2014she'd made it! She was through the door. The question was, what was on the other side? Where was she? Paying no heed to the signs which read \"Authorized Personnel Only,\" she raced along the concrete corridor and down a short flight of stairs. Red letters on the door ahead shouted \"No Admittance\" but Sara was beyond stopping now. She pushed open the door; behind her she could hear the pounding of footsteps. Jonas knew what she had done, and he was following her again.\n\nThere wasn't much time left. Still, she had to try. She felt a rush of air on her face\u2014she was outside. Her mind was racing, her thoughts incoherent. Maybe she should have taken her chances with the airport police. Maybe, maybe, maybe. But the end would have been the same, of that she was convinced. Only the location would have been different\u2014a lonely road with a bullet through her head, or worse, a blow to her head and the car set on fire. She knew she had to die. She had disobeyed Roman DeLuca, tried to outsmart him, and now it was too late! He'd never let her get away with it, especially if Andrew helped the State's case on the witness stand. She should have obeyed DeLuca's rules.\n\nSara ran wildly, legs pumping furiously, breath labored and painful. She was heedless of her path, aware only of open space around her and concrete beneath her feet.\n\nDimly, in the distance, she heard a voice on the public address system. \"Unauthorized persons on the runway! I repeat\u2014unauthorized persons on the runway! Clear the runway! I repeat\u2014clear the runway!\" Over and over the message was repeated, and each time the volume increased. Sara realized they were referring to her, and to Jonas. She saw running figures converging on her path\u2014help was on the way. She was within a hair's breadth of winning!\n\n\"Clear the runway! Will somebody clear the goddamn runway? A man and a woman are on runway six. Clear the runway! Aircraft approaching! Aircraft approaching!\" the PA shrilled.\n\nSara kept running, knowing her life depended on it. She risked a glance behind her. Jonas was still there, grim and determined. Safety was only yards away. There was a roar near her head, loud and piercing. She mustn't stop, mustn't think! She was almost home, a winner again.\n\nThe aircraft appeared out of nowhere. Out of the corner of her eye she could see a monstrous moving object but she had to get across\u2014she would.\n\nThe whining in her ears became louder. Then, a shadow pressed down on her. She was losing control of her movements in the turbulence. It felt as if the plane would flatten her onto the runway\u2014but of course it wouldn't land flat\u2014there were the wheels.\n\nBut it was the wheels that cut that thought short.\n\nSeconds later, Michael Jonas skidded to a stop next to the motionless form crumpled on the runway. His chest heaved with exertion, and his eyes were wide with horror. But he'd seen worse. He had known exactly what he was doing when he chased her onto the runway. DeLuca might protest, but he would be pleased. If there was one way to climb to the top, it was by obeying orders. DeLuca had said to take care of her, and he had. Aside from being windblown and out of breath, he was none the worse for wear. It was over.\n\nSanders trudged into the campground just as dawn broke. He felt more certain than ever that Davey Taylor was close by. Feeley wasn't back yet and he wondered who was going to replace the blond policeman behind the desk. A quick catnap was what he needed now, so he could make a fresh start when Feeley returned. Sanders nodded curtly to the young officer, who was slipping a long-handled comb back into his shirt pocket.\n\nFeeley shook him awake at eight forty-five and handed him a cup of coffee. \"You know something, Stu, I can't drink coffee out of a cup anymore,\" he joked as Sanders roused himself. \"If it doesn't come out of a Styrofoam container and have a lid, I can't get it down.\"\n\n\"Never mind the coffee. What did you find?\"\n\n\"The same thing the cops did\u2014traces of blood. That apartment was a hellhole\u2014I wouldn't let a dog live there. I rousted the landlady and, let me tell you, she was a piece of work. She was nipping on a bottle of beer at five in the morning. Said it beat coffee for a pick-me-up. She'd heard sounds of a fight and a lot of banging around\u2014same thing she told the cops. I did find a couple of mates to this,\" he said, holding out six mothballs. \"I found them on the steps. I looked all over the apartment but couldn't find any more. There was an empty box in a paper bag by the sink. No other trash. I hung around till the workforce crawled out of the woodwork. If there wasn't someone to hold up the corners at seven A.M., Newark would fall apart. No one saw a thing,\" he said disgustedly.\n\nAt Sanders's bleak look, he added hastily: \"I don't know if this is worth anything, but as I was getting into my car\u2014thanking God it was still there\u2014a little kid came up. He wanted a quarter, probably to play the numbers. The long and the short of it is, he was playing stickball outside the building when Balog and the girl came out carrying an ironing board which they loaded into the pop-up. The kid said Elva\u2014that's her name\u2014always carried the dirty clothes in a paper bag to the laundromat around the corner. He said he saw her do it lots of times. The kid was a regular little wiseass, said you don't take ironing boards to the laundromat because you can't iron there. He also said the clothes looked heavy, and that Balog was sweating. I gave him a few bucks.\" Feeley could tell Sanders was pleased.\n\n\"Heavy, huh? This kid say anything about talking to the cops?\"\n\n\"You gotta be putting me on. He's a street kid. Street kids don't talk to the cops unless it's to tell them to drop dead. Nah, he thought I was a relative.\"\n\nSanders put two Rolaids into his mouth. He'd give his right arm for a home-cooked meal of French toast, or pancakes, or scrambled eggs with lots of bacon on the side.\n\n\"They could have been carrying a body,\" Feeley said. \"You have to admit it was a clever idea, if that's the way they got him out. This Balog is our man, I'm sure of it. The locals must know it too, only they think they've got the jump on us. And who are we, anyway? Just some tired men looking for a lost kid.\"\n\nSanders nodded. \"It's adding up, that's for sure. The mothballs sort of frost it, if you know what I mean.\"\n\n\"Yeah,\" Feeley said, mangling the soggy end of his cigar. \"While you were playing Rip van Winkle, I was watching the bird on the phone desk. He kept looking over at me while he was carrying on this conversation. Call came in a little after eight. I asked him point blank if it was something we should know, and he told me it was a personal call. You want to check it out?\"\n\nSanders rubbed the stubble on his chin. \"Do you think it was a personal call?\"\n\nFeeley shrugged his shoulders.\n\n\"Then let's check it out.\"\n\nSanders got right to the point. \"Now, let's make sure we understand each other,\" he told the officer manning the phone. \"This is a bureau office. Any calls that come in pertain to our case. My partner said you received a call a short while ago. Let me see the log sheet.\"\n\nThe officer's face drained of color. \"It was a personal call. A pal of mine was going off duty and they put him through here to me.\"\n\n\"We can call your pal, but that would take time. Make it easy for all of us.\"\n\nThe pale face flushed. \"Well, you see, what I mean . . . My buddy and me know this hooker. She's okay, if you know what I mean,\" he added hastily, watching Sanders's face. \"Anyway, she got busted up during the night. Some dude from out of town fed her a line about making her a showgirl in Vegas, and she fell for it. Somebody found her and called the hospital in Point Pleasant. She's being operated on right now. My buddy went over after he finished his shift to see if he could get a line on how it happened.\"\n\n\"And?\"\n\n\"All he came up with was some guy driving a pickup which he left parked behind the garage. Gus, the guy who runs the garage, said this dude came in late yesterday to have a tire fixed. He steered him to the saloon where Candy dances. That's it.\"\n\n\"Did the guy from the garage say anything about the truck, like what color it was?\"\n\n\"No, but my buddy said it was one of those hippie rigs, all painted up and\u2014Oh, Jesus, that was the guy, right? Jesus!\"\n\n\"Which hospital?\" Sanders barked.\n\n\"Point Pleasant General. She's being operated on now for a busted spleen, fractured collarbone, and two cracked ribs. My buddy says her face will never be the same again.\"\n\nSanders's stomach turned sour. He chewed up two more Rolaids and turned to Feeley. \"Sack out for an hour or so. I'll check the police report. No point in hanging around the hospital\u2014we'll get to her doctor later on this morning.\"\n\n\"Go away, I'm asleep already.\" It was true, Sanders thought in amazement. Loud, gusty snores rippled around the room. The cigar hadn't moved.\n\nOutside the office, Sanders stopped to gather his thoughts. Balog was still in the area\u2014everything pointed to it. The beaten-up woman, the flat tire, the description of the pickup by the garage owner. But where was Davey? He would have staked his life on it that Davey was with Balog and his traveling companion. Sanders snapped his fingers. Right, there were two of them. One could have stayed with the boy while the other went into town. But it didn't fit. A man on the run with a kid in tow just didn't take the time to visit a hooker. It didn't fit.\n\nUnless . . . unless Davey was already dead. He'd seen too much; Balog knew it and had disposed of him. Still, it didn't feel right. What about the woman with Balog? Elva, Feeley had called her. Would Balog have left her behind to guard the kid while he took himself off for a little relaxation? Nah, no woman would stand for that. Okay, so this Elva didn't know Balog was paying a house call. He'd left her alone somewhere with the camper and Davey\u2014but still, where did Davey fit?\n\nPerhaps he'd been right earlier. Perhaps Davey had seen Balog burying Lombardi. Davey had been spotted; he'd run. Balog couldn't have caught him, or he'd have killed the boy right there at the campsite and dumped the body into the grave with Lombardi. Sanders snapped his fingers\u2014that was why the grave had been left open. Balog had chased off after Davey, and then panicked when he couldn't find the kid. He'd packed up camp and made a run for it. But where was Davey now? Sanders remembered Duffy's attempts to communicate with him. Perhaps Davey had hidden in the trailer, not realizing he was getting himself into worse trouble.\n\nThe thought that Davey had already spent a night with a probable killer, and was still nowhere to be found, made Sanders's blood run cold.\n\nTired and hungry, Davey made his way to what looked like a farm. The roof of a large barn caught his eye, along with the silver silo standing beside it. He almost laughed aloud at his discovery, but some inner voice warned him that he couldn't count on victory just yet. There was still the open field to cross, and then he had to find someone to call his aunt. Whoever lived near the barn might take him to the police, and they would let him wear a police cap and give him an ice cream. They always did that on television.\n\nCautiously, Davey walked to the edge of the forest and looked across the open field. He had to cross it to get to the barn. But what if the man saw him? Overhead, a squirrel scurried out onto a low-lying branch then dropped to one beneath it. Petrified by the sound, Davey dropped to his belly and dug his head into his folded arms. He waited, lying motionless, barely breathing, until the squirrel was a foot in front of him. Davey opened one eye and stared with relief into the shiny brown buttons that were the squirrel's eyes.\n\nCrawling on his belly, he inched his way out into the field. He wriggled and squirmed across the muddy terrain. Mom sure was going to be mad when she got a look at his clothes. Aunt Lorrie would never be able to get them clean. Davey felt part of the earth now, and for a few brief seconds he reveled in the sensation of his fingers clawing through the muddy ground, pulling him closer and closer to the red barn. By the time he reached the barn he was exhausted but exhilarated. He had done it; he had gotten away from the man. Now, if only he could find someone to call Aunt Lorrie, everything would be all right.\n\nDavey got to his knees and brushed his hands together to get them clean. Mud spattered every which way. He laughed delightedly. If Mom could see him now she would take a fit. Aunt Lorrie would just laugh and turn a hose on him. But he couldn't think about that now\u2014he had to find someone to help him. He looked over his shoulder to see if the man was anywhere near but the open field was empty.\n\nThe hems of his new jeans flapped at his ankles as Davey trudged around the side of the barn. There were turkeys there, daintily picking at the corn that littered the barnyard. Everything smelled sweet and clean. He wished he had a drink; even toothpaste would taste good in his mouth now. As he walked toward the turkeys, they started to gobble and scatter. A white-haired woman came out of the barn, holding a pitchfork.\n\n\"Well, well, what have we here?\" she said kindly.\n\n\"Lady, I need someone to help me call my aunt. Will you call her for me?\"\n\nThe woman laid the pitchfork on the ground. \"You're lost, is that it? Bet you wandered off from the amusement park. I thought it was closed for the winter.\"\n\n\"It is. I was camping there with my aunt, and I don't know how to get back.\"\n\n\"How old are you?\" she asked.\n\n\"I'll be eight after Christmas,\" Davey said proudly. \"I would 'preciate it if you called my aunt. I have to get a shot at noontime. Do you know what time it is now?\"\n\nThe old woman looked up at the sun. \"Pretty near one o'clock.\" She reached out and took his hand in hers. \"You come along with me and I'll try to clean you up a little. You need hosing down in the trough, but it's a mite too chilly for that. Wait on the back porch while I make the call. Would you like some cookies and milk? I make the best ginger cookies in these parts.\"\n\n\"I'd like that,\" Davey said agreeably. They walked side by side toward the farmhouse.\n\n\"Do you know the campsite number?\"\n\n\"It's close to the pond\u2014I don't know the number. But we have an RV and it's the only one in the campground.\"\n\n\"How did you get so muddy?\"\n\n\"I crawled on my belly across the field,\" Davey replied truthfully.\n\n\"You ask a dumb question and you get a dumb answer,\" the old woman laughed. \"I'll be back in two shakes,\" she said, leaving him on the porch.\n\nDavey sat down on the step. His leg was aching and he was tired. The cookies and milk were going to taste good.\n\nThe screen door banged shut as the old woman brought out his food. \"Here you go. What's your name so I know who I'm talking about when I call the campground?\"\n\n\"Davey Taylor, but my aunt's name is Lorrie Ryan.\"\n\nAll of a sudden the turkeys started gobbling again. \"Mercy me, what's that ruckus? Looks like we have another visitor. Land sakes, weeks and months go by and nary a soul stops by, and today we have two visitors.\"\n\nDavey laid the cookie he was about to eat back on the plate.\n\n\"There you are, you little rascal,\" Cudge Balog accused playfully as he climbed from the pickup. \"Thought you would give me a scare running off like that, did you? Excuse me, ma'am, this is my son, and he ran off on me this morning. You see, he didn't want to do his chores around the campground. I have a rule that each child does his share but this rascal likes to play. I'm sorry if he gave you any trouble, ma'am.\"\n\nFear gave Davey the impetus to get to his feet. He moved over to the woman and clung onto her dress. \"He's not my father. He's a mean, bad man and he's telling you a lie. Please, call my aunt and tell her to come and get me. He's not my father.\"\n\n\"Now, why are you upsetting this nice lady with your stories? Someday this boy's going to write books. I just know it,\" Cudge said airily. \"He does have some imagination.\"\n\n\"You're a bad man! You killed that man and tried to bury him. And you kicked my dog!\" Davey stared with imploring eyes at the old lady. She didn't believe him, he could tell.\n\n\"That's enough of that,\" Cudge said. \"Come along now and give this nice woman some peace and quiet.\"\n\n\"Now, just a minute,\" Elsie Parsons said sharply. \"This little boy don't look like no liar to me. What's it gonna hurt if I call the campground to see if his aunt is there? It's only going to take a few minutes.\"\n\n\"What's going to take a few minutes?\" a nasally voice inquired from inside the house. \"What's going on here?\"\n\n\"Sid, this here boy is Davey Taylor. This man claims to be the boy's father but the boy says he ain't. I found him by the barn this morning looking like this. He wants me to call the campground for his aunt. I think we should call the police and let them straighten it all out.\"\n\n\"Now, Ma, you don't want to go sticking your nose in someone else's business and get yourself in trouble,\" Sid said warily. \"If the man says he's the kid's father, he is. Who you gonna believe\u2014the guy or the kid? Kids lie all the time. He probably done something wrong and lit out.\" There was no way Sid wanted police around the place. First thing they'd be tramping all over and find his patch in the cornfield. Smoking pot was one thing, but growing it was something else.\n\nCudge grinned and slapped the youth on the shoulder. \"You're absolutely right. Davey wasn't in the mood to clean up the campsite this morning and just took off. He's a mighty big source of worry to his mother, I can tell you.\" He turned to Davey. \"Now you get your butt in that truck before I take a switch to you. Apologize to these nice people for the trouble you caused them, and we'll be on our way. Ma'am, I do want to thank you for taking care of my boy here.\" Cudge held out his hand to the old lady.\n\nElsie backed off one step and then another. What could she do? She'd seen the look in Sid's eyes at the mention of police. Good Lord, what had he done this time? Well, your own came first, and then you worried about someone else's kids. Davey seemed like a nice little boy, well-mannered and polite. The father\u2014if he was the father\u2014left something to be desired.\n\n\"You mean you aren't going to help me?\" Davey asked incredulously.\n\n\"No, she ain't gonna help you,\" Sid told him. \"You go on with your old man and stop bothering people and telling lies, or you're going to wake up some morning with a nose a mile long.\"\n\nDavey threw himself against the old woman's legs. \"He's not my father! He's not! He kills people! He kicked my dog.\" He could feel the lady tense as he tried to hold on to her. Just as Cudge reached for him, Davey dropped to his knees, crawled quickly around him and down the steps of the back porch.\n\nSid raced after him, caught him by the collar of the windbreaker and literally lifted the little boy off the ground. \"Where's your respect for your old man, kid? Now you get in that truck and act the way you're supposed to. I'll personally tan your hide if I hear another peep out of you.\"\n\nBefore Davey knew what was happening, he'd been thrown into the cab of the pickup. Sid's leering face staring at him through the passenger window made him want to cry, but only babies cried. He wiped at his eyes with his muddy sleeve, leaving streaks of dirt on his cheeks. He was never going to get away now. He was going to be dead. His eyes went to the CB unit on the dashboard and then to Cudge, who was stepping down from the front porch. Quickly Davey locked both doors. He had the CB speaker to his mouth before Cudge's feet had hit the ground. He switched to the emergency channel. \"Breaker, do you read? This is Panda Bear. Breaker! Breaker!\"\n\nCudge felt the heat of his anger the minute he saw Davey with the CB. It was all over if the kid knew what he was doing. His rage intensified as he raced to the pickup. Out of the corner of his eye, he saw Sid walk around the back of the house. The old woman had gone inside the moment Cudge had stepped off the porch. He was alone with Davey.\n\nDavey looked around wildly. \"Help, help me. This is Panda Bear. Breaker, Breaker, do you read? This is Panda Bear! Breaker, he's coming. Somebody answer me. Do you read?\" Frustration gripped him when he saw Cudge dig in his pocket for the keys. He had to get out before Cudge got in. If he stayed in the truck, the man would kill him. Whatever he was going to do, he had to do it alone, just like before. There was no one to help him. He had to think; above all, he couldn't cry. If he cried, he wouldn't be able to see. Swallowing hard, Davey tossed the speaker onto the cracked leather of the driver's seat and pulled at the lock on the cab door. The moment Cudge opened the driver's door to climb into the truck, Davey opened his side. He jumped to the ground and took off down the road, away from the farmhouse, back across the open field toward the woods. He knew that if he had the farm at his back, the amusement park would be ahead of him. That meant Aunt Lorrie would be close by. He had to run straight, and he couldn't stop for anything. Run, run, run.\n\nHis tattered shoelaces slapped the wet ground as his short legs pumped away. Danger was behind him; he could feel it, smell it. On and on he ran, his arms flailing the air as he fought for breath. Once he fell sprawling in the muddy field. He picked himself up and raced on, not looking behind him to see if Cudge was gaining on him. He knew Cudge must be close, but he couldn't run any faster. He made himself think about Aunt Lorrie waiting for him, her arms outstretched to hug him, mud and all. She would laugh and tell him a story about making mudpies, and then he would laugh too. He wanted to hear the story, he wanted her to hug him. If Cudge caught him, he would never see Aunt Lorrie again. His Reeboks picked up speed. Behind him, curses sailed through the air, but he ignored them. Careening wildly from right to left, Davey headed into the welcome darkness of the forest. He didn't stop. Faster. Run. Aunt Lorrie. A sob caught in his throat.\n\nA thicket of low underbrush caught his eye as he raced ahead. Without thinking, he dived down. Brambles and stickers scratched at his face as he burrowed deep into the undergrowth. Cudge was close; Davey could hear him now, feel the cold stream of danger getting closer and closer. He waited, his eyes squeezed shut. He didn't want to see. He had to be quiet and still.\n\nAt first he couldn't comprehend the sound. It was a roar, a deep, hard rumble that came from the belly; like when his dad threw back his head and had a good laugh. He wanted to lift his head to look around to see where the sound came from, to try to identify it. The man chasing him didn't make this kind of sound. This was something different. Davey tilted his head and listened as the rumbling was repeated. It was from the wildlife reserve. He must be close to the animals. That meant that on the other side was safety. His breathing eased; he wasn't afraid of the animals in their cages.\n\nCautiously Davey inched his way out of his nest. Everything looked so big as he lay on his stomach; even the scrubby bushes looked immense. He waited a moment, hoping the animals would roar again so he could tell in which direction he should go. If only he hadn't been so afraid he might be able to remember now. For the thousandth time he wished Duffy was with him.\n\nHe crawled backward on his belly from his hiding place and looked around again. Satisfied that the man wasn't anywhere around, he stood up. A wave of dizziness overcame him and he swayed, feeling sick to his stomach. Both hands grappled for the bushes; he needed all his willpower to steady himself. He couldn't get sick now. He shook his head several times; he was hungry and he really didn't feel well. How much farther did he have to go?\n\nThe old woman stood at the screen door, looking out at Cudge. Her son Sid watched him too, waiting to see if he was going to run after the boy, or jump into the truck and try to overtake him before he ran across the open field into the woods.\n\n\"Watcha lookin' at, old lady?\" Cudge bellowed. \"If you two had minded your own business, I would've had him!\"\n\nElsie Parsons stared at him a moment longer, then retreated into the house, closing the door firmly.\n\nSid was belligerent. \"Ma don't think that's really your kid.\"\n\nCudge turned on the teenager. \"Oh, yeah? And what do you think, punk? You should've kept out of it, and I would've had him.\"\n\nSid gathered his courage. He really didn't want any trouble with this man, but he felt compelled to stand his ground. \"I agree with Ma. I don't think that boy's your kid, either.\"\n\n\"I don't give a damn what either one of you think. That's my kid and I mean to get him and beat the tar out of him. What's on the other side of the woods?\" Cudge demanded, stepping closer to Sid.\n\n\"Wild Adventure Park. Only it's closed now.\"\n\n\"Where's the campground from here?\"\n\n\"Due north . . .\"\n\n\"I didn't ask you that, did I?\" Cudge bellowed in rage, reaching out and grasping Sid's shirt front. \"I asked you where.\"\n\nSid pointed across the field to the right, in the opposite direction from that the boy had taken. Cudge grinned; he knew Sid was afraid and it made him feel powerful.\n\n\"Stay out of my way.\" He shoved Sid aside. \"I eat punks like you for breakfast and spit them out before lunch. I knew a kid like you once. I took care of him, and I could do the same to you.\" He saw a defiance in Sid's eyes he didn't like. \"Don't think about calling anybody. A man has a right to his own kid, don't he? Besides, you may not like it if I told anybody that you've got a patch of marijuana growing out behind the house.\"\n\nSid was amazed. \"How do you know?\"\n\n\"I always know, punk. You reek of it. Besides, you just told me all I need to know. Even if you weren't growing the stuff, you'd be real unhappy if the cops came beating at your door and found your little stash.\"\n\nCudge jumped into the truck and fired the ignition. It wasn't going to be easy to find that kid, but he'd do it, just the way he'd tracked him here to the farmhouse. He'd drive back to the park and leave the truck in the cover of the trees where it wouldn't be spotted. Then he'd grab the kid and shut his mouth for good.\n\nCudge took his time driving back to the campsite. He didn't need a broken axle now, not when the kid was within his reach. \"It's either him or me. And it ain't gonna be me!\"\n\nThe turns in the road took his full concentration. The weather was blowing up and storm clouds were gathering. For a fleeting moment he thought about Elva lying in his pop-up. He was glad she was dead, and he didn't have to listen to her squawking that he should leave the kid alone. Elva had never been a survivor, not like him. If he could just get his hands on that kid, his problems would be over. \"It's always the little things that trip a guy up,\" he muttered aloud, \"little boy, little dog. Elva with her little bit of brains. Candy Striper for a little piece of ass. Lenny, for a little bit of money.\" He groaned. How had this all happened? This wasn't the way things were supposed to work out! When he was a kid, he had believed he'd be able to overcome the poverty, the filth, his ignorance. And all he'd done was carry it all with him.\n\nAs he drove he kept his eyes trained on the edge of the woods, expecting a mud-smeared figure to emerge at any moment. Christ, that kid was smart. Still, when he got his hands on the brat's neck, he was going to finish him for good.\n\nThe CB in the truck squawked. \"Breaker, Breaker. Do you read me, Panda Bear? Come in, Panda Bear!\"\n\nPanda Bear. Where had he heard that before? Then he remembered. He'd been cruising around in north Jersey, looking for a place to dump Lenny, and Panda Bear had come on the CB channel and talked about the campground. So that's your name, eh, kid? Cudge thought. Panda Bear.\n\n\"Breaker, Breaker, do you read? Come in, Panda Bear.\"\n\nCudge reached over and flicked off the CB. The sharp click made him think of how he would snap Panda Bear's neck when he caught him.\nChapter 13\n\nStuart Sanders checked in with his chief then set out in Feeley's motor-pool car. He couldn't explain why, but he knew Davey Taylor was close by, close enough to touch if he could just reach out in the right direction.\n\nUp and down the dirt roads he drove, but with no success. He braked hard and sat for a while, pondering his next move. His instincts told him to head for the Wild Adventure Park, the same route he had followed on foot last night. There was something out there, something he'd missed.\n\nIt was well after one o'clock when he returned to the camp office to clean up and get on with the day.\n\nLorrie was sitting on the steps of the motor home. She looked as beat as he felt.\n\n\"Hi,\" he said, when he got within speaking distance.\n\n\"Hi, yourself. Anything new to report?\"\n\nSanders shook his head. \"Not yet, but not to worry. Everything is going to be okay.\"\n\nLorrie tilted her head sideways and gave him a questioning look. \"How can you be so sure?\"\n\n\"It's just a feeling.\" He shrugged.\n\n\"A feeling,\" she repeated.\n\nHe hunkered down in front of her, his hands braced on his thighs. \"I've been in this business a long time, Lorrie, and after a while you start to rely on your gut instinct. And mine tells me Davey is alive and well.\"\n\nLorrie bowed her head. \"I hope you're right.\"\n\nHe could see the tears welling in her eyes and, without thinking, reached out to touch her face. \"When this is all over and Davey is safe and sound, you and I are going to get to know each other a little better.\"\n\nLorrie smiled at him through her tears. \"There's nothing I'd like better but . . .\"\n\n\"But what?\" he asked, worry creeping into his voice.\n\n\"I may not be alive after my sister gets through with me.\"\n\n\"Your sister . . .\" Sanders began, anger welling behind his words. \"I still can't believe she refused to come with me. What kind of mother\u2014\" He broke off, realizing he had spoken out of turn.\n\nLorrie leaned forward and put her arms around his neck. \"It's all right. I feel the same way you do.\"\n\nThe anger Sanders had been carrying around with him dissolved when he gazed into Lorrie Ryan's eyes. \"Lorrie, I . . .\" Whatever he'd been about to say was forgotten the moment Lorrie's mouth pressed against his. Slowly, he rose to his feet, drawing her up with him. Needs that he'd left too long unsatisfied rose to the surface. There were other feelings too\u2014new feelings he couldn't put a name to. Together they created a terrible hunger.\n\nLorrie broke the kiss and stepped back, a look of surprise on her face. \"What were you going to say?\"\n\nSanders shook his head. \"I have absolutely no idea.\"\n\n\"Oh.\"\n\n\"Yeah . . . oh.\"\n\nLorrie smiled sheepishly. \"Well, I guess I'd better go inside and fix something to eat.\" She started to turn away.\n\n\"And I guess I'd better head for the showers,\" Sanders said, moving backwards. \"I'll see you a little later.\"\n\nLorrie climbed the steps to the motor home and disappeared inside.\n\nSanders went to the camp showers, where he cleaned up quickly. Out of season the campgrounds offered cold water only, which seemed appropriate under the circumstances. A loud knock startled him. \"Come in,\" he shouted as he put away his shaving gear.\n\n\"Yeah, what is it?\" he demanded of the reflection that met his eyes in the mirror over the sink.\n\n\"Your man thought you would want to hear about this.\"\n\nIt was Officer Delaney. Sanders liked him on sight, from the top of his neat haircut to the tips of his polished shoes that were buffed to a high sheen.\n\n\"Feeley? I thought he was asleep. Something come in that sounds important?\" He noticed the alert look in the young officer's eyes.\n\n\"Yes, sir. He was asleep, but he sleeps like me, with an eye open. You have to do that when you're in law enforcement.\"\n\n\"Tell me about it,\" Sanders mumbled as they walked through the sodden leaves. \"That was some storm we had last evening.\"\n\n\"Good thing it was over early. As it was, the power company was out all night working. We get storms like that around here this time of year. I hear there's another blowing up.\" Delaney's tone was easy but his respect for Sanders was evident. \"There was a call from a shut-in who monitors police calls, Citizens Band, short-wave, you name it. He said he didn't know if it was important or not, but he heard a child calling for help on the emergency channel a little while ago. I have to be honest with you, this turkey calls in on a regular basis. He sees UFOs once a week, hears calls for help, and once he said he heard a gang rape going on in the back of an eighteen-wheeler.\"\n\n\"The kid have a handle?\" Sanders waited, hardly daring to breathe, for Delaney's answer.\n\n\"Yes, sir. Panda Bear.\"\n\n\"Jesus Christ!\" Sanders exploded as he broke into a run.\n\n\"Your man has the caller on hold,\" Delaney shouted. For a big man, Sanders moved fast as he raced ahead to the offices. Delaney looked after him, wondering if he would ever join the state police. That was the big-time. He'd get to wear a snappy uniform, a snap-brim hat and, of course, those polished sunglasses.\n\nDelaney took his work seriously; he was even willing to forgo marriage so he could devote his life to law enforcement. That was the supreme sacrifice. A talk with Sanders could be helpful. He hoped the caller had his marbles all in one bag this time around, and that the child on the emergency channel was the Taylor kid, and that he was okay. Dr. Ryan was a nice person, she deserved some good news. There was a chance that he would be the one to tell her. He would get pleasure out of that, seeing her eyes light up. Childishly, he crossed his fingers.\n\nHe nodded briefly to Feeley as he took his position again behind the desk. Sanders was just hanging up the phone. Delaney waited, not sure if he liked the look on Sanders's face. It was Sanders's case and the agent didn't have to confide in Delaney. Delaney crossed his fingers again.\n\n\"Feeley, you stick around and work the phone. Find out if anyone else heard the call. Let's see if we can't pinpoint it a little more accurately.\" Sanders looked at the scrawled note in his hand. \"The guy's name is Rob Benton. He lives right here in Jackson. Delaney, find out how many turkey farms there are around here. This Benton is certain he heard turkeys in the background.\" Sanders grimaced. \"He said he hates turkeys\u2014actually what he said was, he's afraid of them.\" He looked directly at Delaney, defying him to say the caller was a crackpot.\n\nDelaney's gaze was unblinking. \"You ever have a bunch of turkeys gang up on you? It's hairy\u2014I know what he's talking about. And yes, there are three or four turkey farms around here. As a matter of fact, there's one right next to the amusement park. The bottom end of the farm runs parallel with the wildlife reserve. An old lady owns the farm\u2014her son is on every cop's list from here to Forked River.\"\n\n\"Check it out, Delaney. I'll see you in a little while.\" Sanders wished he had a lucky horseshoe, or a rabbit's foot.\n\nAs he steered the high-powered car down the dirt road, he tried to calculate how long Davey had gone without a shot, but he gave up. Waiting for the lights to change at the main road, he spotted some newspapers in a vending machine on the corner. Leaving the door open and the engine running, he picked up a paper\u2014Davey's picture was on the front page. The traffic light changed and several horns beeped. He ran back to the car, tossing the paper on to the seat beside him.\n\n\"I know you're on the loose, kid,\" Sanders mumbled to himself. \"You can find your way back, I know you can. I got a steak dinner going on your getting back okay. You just hang tight. I'm going to find you.\"\n\nNinety minutes later Stuart Sanders was back at the same light, waiting for it to change again. Rob Benton's story was unshakable. The guy had heard exactly what he'd repeated on the phone. He also reported that he'd monitored the channel from that moment on, and there had been no more calls. Sanders believed him.\n\nDavey started off, looking back over his shoulder every few minutes. He still didn't feel well, but it wasn't as bad as it had been a few minutes ago. Something jingled in his pocket\u2014the three quarters Mr. Sanders had given him. They clinked companionably together. The flashlight should be in the zipper pocket on the sleeve of his windbreaker. He felt for it\u2014it was still there. If he could just keep away from that bad man, he would be okay.\n\nHe cocked his head and listened. The woods were silent, except for the rustling of the leaves overhead. From time to time a squirrel raced through the treetops. Davey grinned. The squirrels were getting ready for winter. Just like he was getting ready for whatever was going to happen to him next. So far he had missed two lunches, one dinner, and one breakfast. He ticked the meals off on his fingers. Four! He would tell Digger he had missed four meals and was still alive. When you didn't eat, you turned into skin and bones and died. Digger knew what he was talking about. He said it had almost happened to him on one of his trips to the hospital. They'd fed him with a tube because he was almost skin and bones. Davey's face puckered up as he tried to figure it all out.\n\nDavey walked slowly on through the woods for another half hour. His leg ached and he wished he had someone to talk to. It would feel so good to have Duffy scampering around his feet, even if he was too tired to play with her. Duffy would be able to smell Cudge if he got too close. She would bark to warn him in time to find a hiding place. Davey would have to tell Digger how good he was at finding hiding places. Digger would appreciate his low-flying dives into the brambles. Poor Digger, he hoped the doctors fixed his legs right this time.\n\nDavey stopped, every sense alert. It was quiet. There were no squirrels, no rabbits running through the brush. He was still safe. Then Davey noticed two things: a loud banging noise, and the way the woods were thinning out. He strained to identify the banging sound. He had heard it before back home, or had it been Aunt Lorrie's house? He grinned\u2014how could he have forgotten? He himself had helped make the sounds. Last year, Aunt Lorrie had let him bang in the nails for the tree house they'd built in the backyard. Aunt Lorrie had hurt her thumb and then quit to go make lemonade.\n\nHe walked slowly to the very edge of the trees, careful to shield himself by not stepping out into the bright sunshine. He dropped to his knees and then to his belly. It looked like the muddy field by the old lady's house, but different. He propped his elbows on the ground, letting his chin rest in the hollow of his cupped hands. It took a moment to make sense of what he was seeing\u2014all kinds of posts, gravel . . . It was a parking lot, he thought jubilantly. He hadn't realized it at first because there weren't any cars. Now he knew where he was\u2014the amusement park. If he was really smart, he might be able to find his way back to Aunt Lorrie without anyone's help.\n\nIn his mind, he had begun to think of Cudge as a wild animal\u2014a wild animal who was chasing him, wanting to eat him. He remembered how Cudge had sounded like an animal tearing through the woods, pounding the ground. Davey had heard its heavy panting and pictured a red-eyed beast with sharp horns and hoofed feet. A hard knot in his stomach seemed to squeeze out his breath, making him feel he was going to be sick.\n\nHe needed to find the man who was banging in nails. He had been successful once by crawling on his belly; he would do it again. If he stood up he would be an easy target. The ground was muddy, and a shard of broken glass winked at him from the left. He would have to be careful. When he reached the other side, where would he be? Shielding his eyes from the sun, he peered across the parking lot. He could see the tip of a roof, some trees and bushes, and a very high fence, the kind that had little holes in it. Dad had told him it was called chain-link. Fences had gates, he knew, like at home where the gate was always closed so Duffy couldn't get out of the yard. Sometimes Dad locked the gate when they went on vacation.\n\nDavey was fifty feet from the end of the parking lot when he heard the animal roar. \"Hey, kid!\"\n\nDavey's heart pounded in his chest. He lifted his head to look around and saw Cudge at the far end of the parking lot, pointing at him. Davey shot to his feet and scrambled to the fence. Frantically, he looked for a gate. He couldn't let Cudge get him. He tripped and sprawled in the coarse gravel, and he saw a small hole under the fence. Using his hands, he dug at the wet earth. He dug faster, sending the loosened earth flying this way and that, the way he'd seen Duffy do it. He heard Cudge yelling at him and knew the beast was close. Lying flat, he pushed his head through the opening, then wriggled one shoulder under, then the other. He winced as the points of the fence dug into his jacket and his back. He worked his way loose and pulled himself through to the other side.\n\n\"Get back here, you little bastard! You ain't getting away this time,\" Cudge yelled. He grabbed for Davey's leg.\n\nDavey shrieked in terror as Cudge pulled at him. He jerked his leg and felt his Reebok slide off his foot. He was through! Safe! Cudge only had his sneaker. Davey struggled to his feet to see Cudge's rage-filled face staring at him through the fence. Slowly, Davey backed away, then turned and ran.\n\nA roar ripped from Cudge's throat. \"I'll get you yet! You ain't getting away from me this time, you stupid kid. This place is all closed up. You won't be able to get out, and there's no one in there to help you.\"\n\nDavey knew he had to move fast to find the man with the hammer. Screaming for help, he ran toward the center of the park. Where was he? Why couldn't he hear the banging now? Maybe the man had finished his work and gone home. The thought was so terrible, he wanted to cry.\n\nCudge dropped the single Reebok as though it had burned his fingers. He'd been so close\u2014he'd almost had him. It was becoming impossible to think; there was a roaring noise in his head, or was it the thunder? Panting with rage, blind with frustration, he was unable to think what his next move should be.\n\nAn eight-foot-tall fence separated him from the kid; by the time he climbed it, the kid would be long gone. He twisted the small black-and-white leather shoe in his hands. Panda Bear\u2014it was a stupid CB handle if ever he'd heard one. Only the kid wasn't so stupid\u2014Cudge would give him that.\n\nCudge had been smart, too, when he was a kid, but he wondered how smart he was now. Here he was, chasing a little boy, when he should have been planning his own escape.\n\nStuart Sanders tapped his fingers on the steering wheel while he waited for the light to change. He'd been riding around for what seemed like hours, thinking, reviewing, but always on the lookout for Davey.\n\nA car full of teenagers passed him just as the amber light flicked to green. Surprised, he checked his watch; it was two-thirty already, and the high school was letting out. That was when he noticed a billboard with an arrow pointing to Wild Adventure Park. Maybe he could spot the turkey farm on the way. He allowed a van and a sports car to pass him before inching into the moving traffic.\n\nWithin minutes the main gates to the park were in sight, but it was the appearance of a secondary road, probably for employees, which caught his interest. After following a circuitous route for several miles, the road ended in a graveled parking area. The gates were chained shut.\n\nSanders turned off the engine and surveyed his surroundings, attempting to pinpoint his location. Last night he hadn't come this far through the woods. On the map, the distance between the park and the campground hadn't appeared so close. He knew he wouldn't have any difficulty gaining admittance to the park; all he had to do was flash his credentials. He could even climb the fence if he had a mind to.\n\nBack at the car he retrieved the binoculars he'd borrowed from Feeley. Training the sights on the horizon, he scanned the treetops, spotting the tall girders of the Ferris wheel. Rides, thrills, and adventure, all waiting till spring, when flocks of children would swarm over them, laughing and shrieking. He wondered if Davey Taylor had ever ridden on a Ferris wheel or a carousel.\n\nLighting a cigarette, Sanders turned to his left, and walked along the fence. He took a drag on the cigarette as he retraced his steps past the gate, along the perimeter of the parking lot. He would have missed it if he hadn't dropped the cigarette to grind it out with the heel of his shoe. There was a hole under the fence. Dropping to his heels, Sanders saw the small mud-caked Reebok lying close by. He hadn't realized he had been holding his breath until it exploded from his lungs, making him lightheaded. Both hands reached for the bedraggled leather shoe. The hole under the fence was just about big enough for the kid to belly through, and it was freshly dug.\n\n\"Good boy. You're almost there. Just a little longer and I'll find you, Davey.\"\n\nThe pain that had been gnawing at his stomach stilled. Davey was free, not in Balog's hands. At least not for the moment. Sanders gripped the shoe hard, as though trying to squeeze information out of it. Looking at it, thinking about Davey, the chewing in his stomach began again. Davey wasn't the kind of kid who was careless with his things. He would never have left his shoe behind unless something had prevented him from retrieving it. Or someone. Balog.\n\nDavey Taylor was on the run. His path took him across a park-like area littered with outbuildings that looked like the Quonset huts that came with his army and soldier set. All the buildings had wide doors like garages, and there were stacks and stacks of trash cans nestled inside one another. There was no safety to be found here, nowhere to hide. He ran on blindly, not stopping to take his bearings or note his surroundings. Past the utility buildings, alongside the storehouses and equipment garages he ran. His stockinged foot hurt from the pebbles on the ground and his sock was cold and wet from the puddles left by last night's rain.\n\nAhead of him was the visitors' area of the amusement park. The rides looked stark and alien against the vibrant golds and reds of the autumn leaves. To Davey's right was a tall, semicircular amphitheater with a blue dolphin pictured on the stark white concrete. Beyond this, the brick path widened into an expanse of cement where the desolate rides were located. Between two towers hung a huge pirate ship, painted bright red and suspended over a now-empty pool.\n\nGlancing up at the sky, Davey saw the tall pylons which supported the guide wires for the Sky Ride. There were lots of buildings dotting the area between the trees. Once, for an instant, he drew up short, staring at a candyland structure with peppermint sticks for columns, supporting a sugar-frosting roof and ice-cream-cone facade. He was reminded of how long it had been since he'd eaten, and how good a cupcake would taste right now. The chocolate browns, vanilla whites and shiny reds of the building fascinated him.\n\nThen he remembered his predicament and, with great effort, forced his weary legs on. The knee that usually had the support of the brace was sore and throbbing. Mustering his courage yet again, he ran onward, heading for a small building where he hoped he could hide.\n\nThe building had two doors. By stretching back, Davey could see the pictures on them: a lady and a man. He reached for the knob to the men's room then hastily withdrew his hand. The man would definitely look for him there. Without another second's hesitation, he opened the door to the women's bathroom. The heavy hinges stopped the door from closing immediately. Davey leaned against it to push it closed; now he felt almost safe. But then he noticed there was no lock on the doorknob. If he'd gotten in, the man could get in too. Frantically he looked around for something to block the door.\n\nHe could see sinks, toilets with doors, and a garbage can\u2014everything looked clean and forgotten, as though no one was coming back. There was nothing here, not even a sliver of soap on the sink. He decided not to wash his hands and dirty the sink, but he had to go to the bathroom. Six of the doors had silver coin slots, and were locked shut, but one door at the end stood open. He had only Mr. Sanders's three quarters\u2014he didn't want to spend them going to the bathroom. He looked in the open door as he unzipped his jeans. Carefully, he held up the seat while he urinated. He liked watching the steady stream as it hit the water. As he zipped up his pants a door slammed close by. Arrow swift, Davey had the door closed and locked behind him. He hopped up onto the seat and braced his hands against the door. He sucked in his breath.\n\nHe could tell Cudge was mad by the way the door banged against the tiled wall. Davey waited while footsteps sounded. Looking down from his perch, he could see heavy yellow boots, caked with mud, appear and disappear as the man stalked back and forth. With a growl, he moved toward the door. Davey waited for the sound of the door closing, but he didn't hear it. Did that mean the man hadn't pulled it closed, or did it mean he was still there, waiting to catch him? Davey wished he could hear the sound of the workman banging the nails. He was tired, and his arms ached, but he would wait a little longer. He couldn't get caught now, not when he was so close to Aunt Lorrie. He had to be more careful than ever.\n\nJust when he thought he couldn't stand another second of waiting, Davey heard the snick of the closing door. There were no more footsteps, no muttered curses. The man was gone. Davey gingerly pulled back one arm and then the other. Dropping to the floor, he reached out a quivering arm to flush the toilet, but then quickly withdrew it. The frothy bubbles would have to stay\u2014toilets made a lot of noise in places like this. It wasn't safe here; he had to leave. If the man came back, he would see the open door, and then he would know Davey had tricked him. The thought pleased Davey\u2014he really had tricked him.\n\nNow, when he walked out of the bathroom, he would go to his left, because if he went right he would end up back at the hole in the fence. He had to keep going in the opposite direction; he had to stay behind the man.\n\nAs Davey started out, he listened for sounds of hammering; his entire body was alert to any movement within his line of vision. He shivered, it was getting cold. Now that he was moving again, he realized he still didn't feel well. If only he could lie down and take a nap, but he couldn't. It was important to keep going, to find Aunt Lorrie. If he lay down and fell asleep, the man would find him.\n\nDavey gradually slowed as he trudged around the park, bewildered by the shadows the giant rides created. He knew he had to be quick to hide at the first sound that fell on his ears. He wished he could read better so he would know what all the signs meant. Why couldn't he find the man with the hammer? Why hadn't he seen anyone to ask for help? Then he saw Cudge, just ahead, stalking the area in front of a hamburger restaurant. The little round tables and chairs were painted to look like polka-dotted mushrooms and toadstools, and Cudge was bending down, peering underneath them. Davey crouched low. Sometimes, like now, he was glad he was small. When you were little, there wasn't so much of you to see. He maneuvered his way behind a big red trash can and watched Cudge work his way around the perimeter of the restaurant. His heart hammered in his chest and he felt as though the ocean was slapping at his ears.\n\nHe was getting colder and he was so tired. He stifled a yawn, never taking his eyes from Cudge's slouched form. He drew back suddenly. Cudge was standing upright now and looking around, deciding which way to go. Davey risked another quick look and saw him head for a low, white building with a red and black sign. He watched as Cudge opened the door and looked inside. He didn't go inside, just looked. Then he closed the door and moved on. As soon as Cudge was out of sight, Davey ran across the space and into the building. If Cudge had already looked in there, he probably wouldn't come back there again.\n\nOne wall was lined with lockers, the other with open-stall showers. To the left, in a room littered with cartons and boxes, Davey saw a desk with a push-button telephone sitting on it. Frantically, he dialed 911. He waited and waited but no one picked up the other end. He tried again but achieved the same result. Defeated, he sat down on the swivel chair, his face puckered with despair. Mom had told him to dial 911 if ever he had an emergency. But it didn't work. There wasn't anybody there.\n\nMaybe this telephone was broken. Maybe if he could find another telephone it would work. Outside, in the locker room, he saw a pay phone on the wall. He knew it was a pay phone because he'd been with his mom once when she'd used one to call his dad. For almost a full minute Davey stared at the phone. He couldn't reach it. Maybe if he dragged the bench over and stood on it . . . He had to try.\n\nPulling and tugging, he managed to drag the heavy wooden bench directly underneath the telephone. He climbed onto it, lifted the receiver and listened to the dial tone. He wasn't sure if he was supposed to put the money in before or after he dialed.\n\nAgain, he dialed 911 but, as before, no one answered. He tried it again, putting the money in first. No answer. When he'd gotten out his money, he'd felt Mr. Sanders's card in his pocket. He would call Mr. Sanders! Mr. Sanders would help him. He would know what to do.\n\nWith shaking hands Davey dialed the numbers on the card. A voice came on the line and told him to deposit fifty cents. He dropped two quarters into the slot and waited. Three, four, five rings.\n\n\"Hello, this is Stuart Sanders,\" a voice said. \"I'm away from the phone right now but you can page me at\u2014\"\n\n\"Hey you,\" a gruff voice shouted outside. \"What are you doing in here? The park is closed, mister. How did you get in?\"\n\n\"I climbed the fence, that's how!\" It was Cudge. \"My kid got in here by digging under the fence out by the parking lot. I had to climb over, because the hole he dug wasn't big enough for me. Are you sure you ain't seen him? He's about this high, has blond hair? I gotta find him before his mother takes a fit!\"\n\nDavey put the receiver back on the hook. He wanted to run outside and ask the man Cudge was talking to for help, but he was afraid to because of what had happened at the farmhouse.\n\n\"Mister,\" the other man continued, \"I haven't seen any kids around here. I've been working all day over by the roller coaster, dismantling the cars and getting them ready for next spring. There's no kid around here. You'd better be on your way before I call the police.\"\n\nSuddenly the man's voice changed, as though something had choked off his words. \"Now look, mister, if you want I'll go around and ask the shut-down crew if they've seen anything.\" He was speaking faster, higher pitched, as though he were scared. Davey knew that the man was afraid of Cudge too.\n\n\"I want to find my kid. I'm not leaving till I do!\"\n\n\"I know, I know. I've got kids of my own. Come on with me, we'll go around and ask the other guys. We'll be punching out for the day pretty soon. Maybe somebody's seen him.\"\n\nPanic-stricken, Davey dropped the phone and jumped down from his perch atop the bench. He had to get away.\n\nHe looked around\u2014there was no back door! Saliva dribbled down his chin; he was too frightened to swallow. Into the small storage area behind the locker room he ran, hoping there might be a door through there. He saw some double doors with a big red exit sign above them. Silently, he inched open one of the doors and peered out.\n\nThe sun was gone now and the sky looked dark. A strong breeze that smelled of rain was sending fallen leaves and paper spiraling along the ground. He could hear voices outside. His heart pounding, he inched the door back into place then ran. Faster and faster\u2014up the incline, past the miniature golf course, around the bend to the haunted house, down the rise to the old-fashioned carousel. Sobbing, gasping for breath, he ran blindly, not caring where he went as long as he was putting distance between Cudge and himself.\n\nWould Cudge go into the building? Would he notice that the bench had been pulled up to the phone? He'll catch me! Faster, always faster, Davey staggered onward. He had to keep running away. He had to make himself safe. Safe so Aunt Lorrie could find him.\nChapter 14\n\nThe police station was located behind the city hall, and it was here that Stuart Sanders found a distracted Chief Allen. The ensuing confrontation added further fuel to the fire in Sanders's digestive tract.\n\n\"What's this bullshit you're giving me about not having any available men? You have a fifteen-man force here\u2014or is that just for the taxpayers' benefit?\"\n\n\"I do have fifteen men, but two are out with the flu, one's on vacation, and one has a death in the family. That leaves me eleven\u2014count 'em, eleven!\" Allen glowered at Sanders.\n\n\"Call in for help if you need it,\" Sanders suggested.\n\n\"Don't need it. This is our baby and we'll handle it. Eleven men, two murders.\"\n\nSanders remembered the beaten-up prostitute. \"When did she die?\"\n\n\"Who?\"\n\n\"Who else\u2014the hooker your boys are so fond of.\" He restrained his rising temper. Discipline, he told himself. Discipline.\n\n\"No, Candy is holding her own. We're not stupid here, Sanders. We found traces of rotting apples on the floor and bedcovers in Candy's cottage. All her shoes were clean, so we knew it had to have been tracked in by whoever beat her up. Turns out we were right. A few of these dirt roads leading out from the orchards are littered with dropped apples that bounce off the trucks.\"\n\n\"Another little tidbit that didn't appear in the police report. Just like the mothballs,\" Sanders slipped in. He was rewarded by the surprised look on Allen's face. \"So, what did you find?\"\n\n\"A pop-up camper. The same one that Balog was pulling, according to the plates. And a woman. Beaten the same way Candy was. Only it's too late for this one.\"\n\nSanders clenched his teeth. \"Balog. That man is responsible for beating up two women and killing one, as well as the murder of the body you dug up in the campground. He's involved with the Taylor boy. Wherever Davey is, you'll find Balog close by. Now, are you going to make a move, or do I have to move you myself?\"\n\n\"Forget it, Sanders. This doesn't involve you. Whatever you do about the kid is fine with me, but stay out of my business. We'll handle the murders.\" Allen reached for his hat.\n\n\"Where are you going now?\"\n\n\"Out to the camper. The body hasn't been removed yet and I want to take a look for myself. Another thing\u2014there's a storm brewing, due to hit here in the next hour or so. A good rain will obliterate any leads so we've got to work fast.\"\n\n\"I'm coming with you,\" Sanders said. \"I'll follow in my own car.\"\n\nBouncing down the track behind Chief Allen, Sanders hit a pothole big enough to swallow the front end of the car. The pop-up was parked less than a mile from the main highway; Allen's men were already crawling all over it.\n\n\"He doesn't need a rainstorm to 'obliterate his leads,'\" Sanders swore under his breath, \"his men are doing it for him.\"\n\nHopping out of the car, Sanders moved quickly toward the trailer, elbowing through the uniformed men. The pop-up was rocking on its moorings with the extra weight of Allen's men. The interior was dark and filthy, littered with cartons, discarded cupcake wrappers and empty soft-drink cups. The clear vinyl windows were all in place, keeping the air within stale and close.\n\nSanders noticed an odor he couldn't immediately identify. Whatever it was, it was hours old, and smoking had weakened his sense of smell. He turned to one of the other men. \"What's that smell?\"\n\n\"Smell? Oh, yeah, it must be the mothballs. We found a few of them rolling around in here.\"\n\n\"No, it's something else. Ammonia?\"\n\nThe young policeman shrugged his shoulders. \"Urine maybe. Now that you mention it, it sort of reminds me of my kid brother's bedroom when he used to wet the bed.\"\n\nThat was it\u2014dry urine. Poor Davey, trapped in the camper, no way to get out, not even to go to the bathroom. Sanders could imagine the little boy's discomfort.\n\n\"There she is, Mr. Sanders.\" Allen drew his attention to the form beneath the blanket. \"Not a pretty sight, is she?\"\n\n\"Seen worse,\" Sanders told him, meaning it.\n\nThe girl was small and scrawny; she could only be seventeen or eighteen. The coroner would know for sure. The side of her face was battered, and one arm was twisted at an unnatural angle. From the way she was lying curled on the floor she had probably been trying to defend herself from being kicked. Sanders's eyes followed the line of her body, coming back again to the girl's hand, relaxed now, and open. Her nails were bitten down to the quick.\n\nThe man who had done this was an animal. And every instinct told him that Davey was marked as his next victim. Sanders went back to the campground to call in for additional men.\n\nMac Feeley knew better than to utter a single word when he saw Sanders stride into the camp office. He had that look that said he was at the end of his rope.\n\nSanders grabbed the phone and punched out the section chief's number. \"Buzz, there are some new developments here and I need help.\" Quickly, he reviewed the situation.\n\n\"Can do, Stu,\" his chief responded, \"but they won't get there till around five this afternoon. It's the best I can do. Take it easy\u2014we'll get him before it gets dark. If the kid's in the park and moving around under his own power, we're okay. Sounds like your theory was right. I want to meet that kid when this is all over.\"\n\n\"You and a lot of other people. Talk to you later.\"\n\nSanders turned to Feeley. \"A second body's turned up. Have Delaney . . . Forget it, he's being recalled. Where is he?\"\n\n\"He left just before you arrived. The body you're talking about was found on the other side of the highway in a pop-up camping trailer. According to Delaney, there was no positive identification, but it seems likely that the woman was traveling with Balog. I don't understand where this guy is, or what happened to his pickup. Beats the hell out of me why we haven't come up with something on that rig. From the description I've heard, it should be easy to spot.\"\n\n\"I need someone to stay here by the phone.\" Sanders chewed his lip. \"Hey, wait a minute\u2014I'm forgetting Dr. Ryan. Go get her, Feeley, and bring her back here. She can handle the phones. You camp out in Allen's office in case something comes in and he's not generous about passing it on. I'll go to the park myself. It's the best we can do till five, when Buzz's troops arrive. Doesn't this remind you of the time we were in Birmingham and only had three men, working around the clock for four days?\"\n\nFeeley's eyes were dreamy. \"There was this waitress that made the best damn goulash I ever ate. She had other talents too, but the goulash was her specialty.\" He shook himself back to the present. \"I'll call and see if I can get the park opened up. And if it comes down to Allen being the only one with authority, I suggest you storm the gates. Let Buzz take the heat.\"\n\nLorrie burst through the door, out of breath from running. Sanders quickly filled her in, ending by showing her the Reebok he'd found. Lorrie collapsed on to a chair and let her tears flow as she held the muddy shoe. \"Is it a positive or a negative sign?\" she asked eventually.\n\n\"I think it's safe to say it's positive. I'm going back out there now\u2014I'm convinced Davey's somewhere inside the amusement park. I'm taking Duffy with me. I've called in for more men but it'll be a while before they get here. If you could handle the phones\u2014\"\n\nLorrie stood up. \"No! I'm going with you.\" Sanders shook his head but Lorrie went on. \"I mean it, I'm going with you. You forget, I'm a doctor and Davey is a very sick little boy. The sooner I get to him, the better chance he'll have of coming through this.\"\n\nSanders knew she was right, but it was against everything he believed in to take her with him. If Balog was still in the park too, things could get dangerous.\n\n\"All right, but there are conditions,\" he said finally. \"You have to do as I say. No going off on your own. We stay together\u2014you hear me?\"\n\nLorrie smiled. \"I hear you.\" She started for the door. \"Let's go!\"\n\nDavey was tired of running. He leaned back against the rough bark of a tree in a sheltered grove overlooking the entrance to a giant flume ride. He knew he wasn't safe here, that he should keep going, but his knee hurt and he needed to rest a little while.\n\nHe was so tired. He wanted to just curl up somewhere and fall asleep. And he was hungry\u2014hungrier than he'd ever been before. Spaghetti would taste good now. He would suck the long strands through his teeth and not care if little drops of the sauce splattered all over his cheeks and shirt. Mom was always showing him how to roll it on the fork. His mouth started to water. Suddenly, a sound behind him made his heart leap in his chest. He relaxed again when he saw it was only a fat, gray squirrel searching through the trash cans.\n\nHe couldn't go much farther. He needed to find a place to hide so he could take a nap, but everything was locked up or out in the open. He needed somewhere enclosed. The sky was getting blacker by the second. A streak of lightning tore across the sky, making Davey's heart pound in his chest. A rumble of thunder echoed around him, frightening him still more.\n\nIt was almost dark when he thought of hiding beneath the carousel's platform. He dropped to his belly and crawled underneath the circular structure; there was more room than he'd expected. Torrents of rain beat against the colorful carousel, and the heavy thunder and lightning made a Fourth of July display.\n\nDavey curled into a ball. Embraced by the darkness, he was shielded from the warring elements overhead.\n\nThe storm was building as Lorrie and Sanders pulled up in the parking lot just outside the gates of Wild Adventure. Duffy sat between them, her nose keen for Davey's scent.\n\nLeaving the car, Lorrie matched Sanders's athletic stride. Both glanced at the ominous gathering of thunderheads, each dreading the rain that would delay their search for Davey.\n\n\"We don't have time to wait for someone to open the gates, Lorrie, so the best thing we can do is climb the fence. I'll give you a hand up and you go over first. Find a garbage can and toss it over the fence, and I'll follow you.\"\n\n\"Okay.\" Lorrie placed her foot into Sanders's cupped hands and grasped the fine links of fencing. They cut into her fingers. She was going to feel this tomorrow, she thought, as she dropped to the ground. The first roll of thunder sounded. She tried to ignore it\u2014to acknowledge it was to accept that Davey might be caught in the storm, unsheltered and afraid. Quickly she found a trash can and ran back to the fence with it.\n\n\"This storm is worse than I thought. I think you'd better come back to this side for now,\" Sanders said. \"We can sit in the car until it blows over.\"\n\nSeconds after Lorrie got back over the fence a river of rain came down out of the sky. They made a mad dash for the car.\n\n\"I guess you know we're stuck here. I can't even see to drive in this. Jesus, I haven't seen rain like this since I left the farm thirty years ago, and then only once. Later on the weather guys said it had been a hurricane.\"\n\nLorrie reluctantly resigned herself to waiting. \"You know something, Stuart, I'd give everything I own, everything I hope to own for the rest of my life, if I could see Davey safe and sound right now. I love that little boy more than you can possibly imagine.\"\n\n\"I'm sort of nuts about him myself,\" Sanders admitted. \"There's just something about him that makes you want to hold him and love him, and it doesn't have a damn thing to do with his medical problem. It's Davey. He's a very special kid.\"\n\n\"Do you get this involved in all your cases?\" Lorrie asked.\n\nSanders chuckled. \"No way.\"\n\nA brilliant flash of light ricocheted off the windshield. Sanders swallowed hard as he caught the look on Lorrie's face. \"Stuart, tell me that Duffy is in the backseat.\"\n\nHe would deserve the whiplash he got from the quick swivel he did. \"Oh, Jesus. She went under the hole in the fence where I found the sneaker. She must have picked up Davey's scent. When you got the trash can, did you see her?\"\n\n\"No, but I wasn't looking. I thought she'd stayed with you.\"\n\n\"If I were a crying man, I would bawl my head off about now,\" Sanders said.\n\n\"There must be something we can do. We can't just sit here. That dog weighs all of six pounds\u2014the wind will kill her if she doesn't find a safe spot to take shelter. She's not an outside dog, Stuart, she's a house dog. She won't even go outside to pee when it rains. The cook spreads a paper by the back door for her. She's frightened to death of thunder and lightning.\"\n\nSanders hated negatives of any kind. \"I know. I also know that you are an intelligent woman. No one with reasonable intelligence would even think about opening this car door. Believe me when I tell you I know what you're going through.\"\n\nLorrie's head fell back against the headrest. \"We wait, is that it?\"\n\n\"At least until it starts to let up. Come here,\" he said, putting his arm around her shoulders and pulling her toward him.\n\nIt was shortly after eight when a call came in over the radio from the campground manager, asking Sanders to return to the camp offices. Though it was still raining cats and dogs, Sanders was able to inch the car out of the parking lot and onto the road that led to the campground.\n\n\"I have good news, excellent news as a matter of fact,\" the manager said importantly. \"Your man Feeley called in just a few minutes after you left. He said that an elderly woman stopped by the police station shortly after he arrived; she was carrying a copy of the Asbury Park Press. She said Davey had been at her turkey ranch around noon.\"\n\n\"Oh, Stuart, he's all right!\" Lorrie threw her arms around Stuart's neck and hugged him.\n\n\"What did I tell you?\" He hugged her back. One of these days he wouldn't have to let her go.\n\nThe manager went on. \"Mr. Feeley says all the roads are flooded, especially the causeway, so your men might be delayed even longer, Mr. Sanders. They closed off the parkway an hour ago. He's been most kind, calling in every twenty minutes or so to keep me alerted. Lordy, I almost forgot. The woman from the turkey ranch said that a man who claimed to be Davey's father came after him.\"\n\n\"Balog?\"\n\nThe manager shrugged. \"Feeley didn't say. The woman told him she didn't think the man was the boy's father because he was too mean.\"\n\n\"Anything else?\" Sanders inquired. It would be easier to get blood out of a turnip than the full story out of the campground manager.\n\n\"Feeley said . . . I'm not sure, but I think he said that the woman's son put the boy into the man's truck.\" He shook his head. \"Well, anyway, the woman said the boy started talking on the CB. Then, when the man got into the truck, the kid jumped out and ran across the field toward the amusement park.\"\n\n\"That clinches the shut-in call,\" Sanders mumbled. He hated to ask but he had to, for his own peace of mind. \"Did Feeley say anything about how the boy looked?\"\n\n\"I asked that and apparently the woman said the boy was hungry but looked right as rain to her. Oh\u2014and he looked like he had been wallowing in the pig trough.\" The manager laughed, but sobered at once.\n\n\"Then he is in the amusement park. I knew it.\"\n\n\"What about Balog?\" Lorrie asked. \"Did he follow Davey?\"\n\nThe manager shrugged. \"I don't know. Feeley didn't say.\"\n\n\"If he'd followed him to the turkey ranch, he'd probably follow him to the amusement park,\" she surmised. \"How long can Davey elude him?\"\n\n\"I don't know, but so far Davey's given him a run for his money,\" Sanders said.\n\n\"If Balog does find Davey, do you think he'll kill him?\" Lorrie's voice was unsteady.\n\nSanders couldn't answer any other way. \"Yes.\"\n\nLorrie sucked in her breath. \"Are we going back out there now?\"\n\n\"As soon as I can find us some raincoats or slickers.\" Sanders's gaze shifted back to the manager. \"It's still coming down in torrents. When our back-up men arrive, you'll have to direct them to the park. You can do that, can't you?\"\n\n\"Of course I can. I'll stay right here by the phone and write everything down.\"\n\n\"Here we go\u2014best I could do.\" Sanders handed Lorrie a pair of knee-high green boots and a bright orange hooded slicker. \"I think I should warn you, if we fall into a pond or a lake we'll both go straight to the bottom. This stuff must weigh eighteen pounds.\"\n\nLorrie leaned over to kiss Sanders on the cheek then wrinkled her nose. He smelled like mildew and detergent. \"Come on, let's go find Davey,\" she said.\n\nSanders drove carefully through the heavy rain. No matter how bad the storm got, there was no way he was going back to the camp offices now. He knew Davey Taylor was in the amusement park and he was going to find him\u2014tonight.\n\n\"We aren't going over that fence again, are we?\" Lorrie asked.\n\n\"Hell, no. We're going to do it the way they do in the movies. We're going to shoot off the locks and walk right in.\"\n\n\"Good. I was wondering how we would make it over the fence in these Armani originals.\"\n\n\"That's what I like about you, Lorrie\u2014your keen wit and utter logic,\" Sanders laughed. He felt more confident now, sure they would find Davey soon. He wouldn't think about the cold rain and the dark. \"I should be back at the offices waiting for the others right now. And having you along with me is a no-no, right off the bat. We never involve civilians in anything that might be dangerous. I'm not going by the book is what I'm saying.\"\n\n\"To hell with the book,\" Lorrie said briskly.\n\n\"That's what I say. Okay, we're here. Let's hope the thunder muffles the shots.\"\n\nLorrie stood back, holding the flashlight while Sanders aimed the gun. Two shots and the lock was still intact. \"Just how the hell rusty can it be?\" he asked, not considering for a minute that he might have missed. He fired again and the lock fell off.\n\n\"Okay, here we go!\" Sanders swung open the wide gate.\n\nLorrie held the flashlight in front of her as she waded through six-inch-deep puddles. When was this rain going to let up? She shouted for Davey and Duffy till she was hoarse. The wind and rain just carried her cries back to her. It was eerie\u2014she didn't like it. Over and over she shouted, though she knew it was probably useless. Even if Duffy heard her calling, and barked in response, would she be able to hear the dog?\n\n\"I know you're out there, kid,\" Sanders mumbled. \"Give us a break, Davey. Do something\u2014anything would be a help at this point. Help me to help you.\" The rain was running down his arm, and he switched his flashlight from one hand to the other to shake the water from out of his sleeve. It was cold; the boy must be freezing.\n\nThe agent stopped to check their surroundings. To their immediate left was the roller coaster and the thrill rides; to their right was the Lehigh Valley Express train for little kids. Beyond the tracks, a small section of the park was devoted to the children's rides. None of the areas offered shelter of any kind. All he could see were the steel skeletons of framework. Should he go to the left or the right?\n\nAn hour later they were almost ready to give up. Just then the rain slackened, then stopped. In the dim, yellow-white glow of the flashlight, Sanders could at least see where he was going now. Just ahead were the carousel and the end of the children's area\u2014they had almost reached the wildlife preserve. He doubted Davey would go in there, not if he knew about the wild animals. Something teased at him, some comment his nephew had made last spring. One of the animals\u2014Delilah, the lioness\u2014had given birth to six cubs. That was it. Her mate, Samson, was extremely protective of Delilah and the cubs. God help anyone who strayed into the lions' den.\n\nEverywhere Sanders looked there were fallen branches and uprooted trees; anything that hadn't been nailed down had been scattered by the storm. Swinging the flashlight wide to get a better view, he was startled to see a monstrous oak lying across the Lehigh Valley Railroad. It would take a team of ten men to move it.\n\nHe didn't like the destruction he saw all around. \"Where are you, Davey?\" Maybe he was crazy; maybe Davey had never come back to the amusement park at all.\nChapter 15\n\nUpon awakening, the little boy's movements were slow, sluggish. At first he was unaware of the softness nestled close to his chest. But when he came fully awake, he wanted to howl with glee at the comforting feel of his little dog.\n\n\"Duffy!\" he cried. \"How did you find me? Where did you come from? Where's Aunt Lorrie? Good girl, you can lick my face all you want,\" Davey laughed happily. \"I'm so glad to see you!\"\n\nHe put his arms around his dog and hugged her, then suddenly remembered why he was hiding under the carousel. \"We have to be quiet, Duff. You can't bark and don't run away this time. It's good you're here, Duff. I don't know my way back to camp and you can show me the way. Boy, is everybody going to be surprised when we come back together. Just you and me, Duff. We're a team, just like Mom and Dad. I only wish Mr. Sanders was with you\u2014he'd know what to do. But he's in Florida with Mom and Dad.\"\n\nIt was good talking to Duffy. Davey felt reassured that soon he would be safe at home. \"You're all wet,\" he noticed suddenly, \"and so am I.\" But Duffy's companionship relieved some of his hunger and weariness\u2014as long as Duffy was with him, he could talk to her and it would make the hike back to camp easier.\n\nBrows puckered, he decided it must be very late. \"We'll just have to wake Aunt Lorrie up, right, Duff?\" He smiled to himself in the darkness, visualizing the expression on his aunt's face when he opened the door of the RV. Duffy whined low in her throat and continued with her furious licking.\n\n\"You're tickling me, Duff! We've got to get out of here. Do you think you can find your way in the dark?\" Davey grunted the question from the effort of inching out from under the carousel. He shivered in his wet clothes. Now that he was exposed to the wind, he could feel the cold all the way down to his stockinged foot.\n\n\"Be quiet, Duff, real quiet. I don't know where that man is now but I know he's here somewhere, waiting to catch me. I'm going to be right behind you. If he catches us this time, he's going to kill us both. I don't want to be dead, and I don't want you to be dead, so don't bark,\" he cautioned in a firm voice. Duffy trotted off with Davey following close behind.\n\nThe rain was coming down steadily and Davey was cold. Duffy must be cold too. The pavement dipped\u2014Duffy growled but it was too late. Davey's foot slipped and he found himself in water up to his chin. He thrashed about wildly as he struggled to reach higher ground. The rain continued to beat down in steady driving torrents. Carefully, not allowing himself to panic, he trod water till he was at the edge of the little pond. He groped for a handhold, only to find himself slipping back into the water. Duffy stood sentinel, barking loudly to offer encouragement. Her stubby tail swished furiously against her haunches as she crept up to the edge then backed off.\n\n\"It's no use, Duff,\" Davey called out. \"I can't get out of here\u2014there's nothing to hold on to. I need someone to pull me out. Quiet Duff, stop barking and go get Aunt Lorrie. Go on, girl, go get her. I'm not going to drown. I know how to float on my back. Go on, girl. Go!\" Purposely, with all his might, he forced his voice to be like his mother's when she gave an order. \"It's the best thing for me, Duff. Go get Aunt Lorrie.\"\n\nDavey watched Duffy run away and hoped she'd understood him. He rolled over in the water. It was better lying on his back than playing dead-man's float. He never wanted to play that game again, not now that he knew what being dead meant. Blackness engulfed him, closing off the world. He could feel fear closing in too, choking off his air. He stiffened, feeling himself going under, the weight of his clothing dragging him down. Tentatively, he straightened his legs, the one shoeless little foot stabbing out a speculative toe to touch bottom. There it was, not very far down, but it felt slick and slimy, unpleasant. Yet, by standing on tiptoe, his head and face were out of the water.\n\nHe could feel the tears pricking at his eyelids, but then he imagined what Mom would say at a time like this. \"David, crying solves nothing. You must use your brain and think. Tears are a sign of immaturity. You must reserve your tears for important things, such as grieving and weddings.\" He wasn't exactly sure what grieving was, and he only had a vague idea about weddings, but what he did know was that Mom wouldn't consider that the position he was in called for tears. He swallowed hard and wished for a light so he could see how big the pond was and find a way out. He had a light! Mr. Sanders had given him the penlight and it was still in the zippered pocket of his sleeve. He would have to unzip his jacket and take it off, or at least get his arm free of the clingy, nylon windbreaker.\n\nExperimentally, he shifted his arm, trying to take it out of the sleeve, working at the zipper and sleeve under water. He had to balance himself on his toes so he didn't go under. He yanked at the zipper and managed to get it down halfway, and then it stuck. His toes gave out and he slipped beneath the water. Gagging and spluttering, he rose to the surface and flailed out with his arms. His teeth began to chatter and he had to control them, because the more noise his teeth made the colder he felt. Flopping over on his back, the way he'd been taught, he let the rain drizzle down on him. It reminded him of a waterfall he had seen in one of his picture books. He wasn't afraid of the rain or the water. He just had to think of a safe way to get out of the pond.\n\nHe was concentrating on testing the bottom with his foot to find a shallow place to get a foothold, when he heard a muffled curse. A dim circle of light played near him. Cudge! If he went under the water, he could only stay under until he counted to four. He would make noise coming to the top and then he would have to take deep breaths. Cudge would be sure to hear him.\n\nWhen the light reached the edge of the little pond, Davey could make out the big man's work boots and wet trouser legs.\n\n\"You're really a pain in the ass, kid. You made me tramp this goddamn place in the rain for hours. I thought you might have gotten away while I went back to the truck to get the flashlight. Guess I was wrong. Get over here and I'll pull you out. C'mon, over to the edge. Move it!\"\n\nDavey hesitated. To put himself into the hands of the man was unthinkable.\n\n\"Better do what I say, kid. I got your dog. I'll hold her under the water and finish her off real fast. Move it!\"\n\nHe'd got Duffy?\n\n\"You hear me, kid? Get your ass over here so's I can grab you. You want to see your dog drown?\"\n\nHe had got Duffy! Davey obeyed unwillingly. Maybe it was better to be out of the water. Maybe he could get away from him again. As long as he didn't hurt Duffy. Please don't let him hurt Duffy!\n\nThe little dog snapped and snarled as Cudge reached down to lift Davey out of the pond. He hauled him out effortlessly. \"I thought I heard this mutt of yours. Even the rain couldn't muffle her barking.\" He set Duffy down on the ground and immediately she attacked his leg. \"Call her off, kid, or I'll kick her from here to kingdom come.\"\n\n\"Duffy! Duffy\u2014down girl. Down!\" The dog sat back on her haunches and tilted her head, deep growls rolling in her throat.\n\n\"Pick her up!\" the man ordered. \"Pick her up and carry her. I got someone I want you to meet. As a matter of fact, he's dying to meet the both of you.\"\n\n\"My aunt is going to find me,\" Davey said, feeling his lower lip quiver. \"She's going to come and get me and take me away from you. And Duffy, too!\"\n\n\"Well, ain't your aunt gonna be surprised when she gets here and you're not. C'mon, walk! My friend is waiting.\"\n\nDavey knew he didn't want to meet anyone this man knew. Still, he couldn't resist asking \"Who?\" in the hope that it might be Brenda. Maybe he was wrong and Brenda was okay; maybe the man hadn't hurt her.\n\n\"The sign says his name is Samson. A real live lion, like in the jungle. The kind that eats kids like you and doesn't even burp. Now shut up and walk!\"\n\nLorrie and Sanders came to a standstill in front of the Space Port. When the park was open it would be packed with kids plunking their quarters into video games like Space Invaders and Indy 500. Now it was deathly silent.\n\n\"He's here,\" Sanders said bitterly, \"I know it!\"\n\n\"What now?\" Lorrie asked, despair in her voice.\n\n\"I'm going to call in to see where our backups are. And I want my section chief to get in touch with somebody to organize turning on all the goddamned lights in this park. Even if Davey's not in the park, but somewhere nearby, the lights should attract him.\"\n\n\"Good idea. But wouldn't the storm have knocked the power out?\"\n\nSanders's face fell. \"Well, it's worth a try. Anyway, it's time to call in.\" He walked over to the closest building and sheltered under its eaves.\n\nLorrie waited nearby, her nerves shattered. As soon as Sanders had finished the call, she was at his side. \"Well?\"\n\n\"They're on their way.\"\n\n\"And the lights?\"\n\n\"I don't know. They're going to try, maybe use the generator.\"\n\n\"Let's go back down the main concourse,\" Lorrie suggested. Even though she was exhausted she just couldn't give up.\n\n\"Right,\" Sanders said grimly. It was worth another try.\n\nThey walked side by side, their flashlights fanning the ground, searching for clues that might lead them to Davey. Each kept their thoughts to themselves. It was beginning to look as though they were on the wrong track. Every lead was cold.\n\n\"Hey, watch out,\" Sanders cautioned. \"What's that?\" His light sought out the shiny object that had reflected the beam back at him. \"There!\"\n\nLorrie's eyes followed the beam of light. The disc was silver, shining and new. A quarter. She picked it up, holding it for Sanders to see, her manner almost reverent.\n\n\"Hot damn!\" Sanders slapped his thigh. \"That's one of the quarters I gave Davey. I know it is because of the nail polish on the edge!\"\n\nSuddenly their steps were lighter and their gloom lifted. \"Davey!\" Lorrie called, loud and clear, the rain drizzling into her mouth unnoticed. \"Davey!\"\n\nStrong fingers reached out and yanked at his collar, pulling him backward so violently that he almost lost his hold on Duffy. Davey hadn't realized how heavy the little dog was until he'd had to carry her for so long. They had left the main amusement park behind now and were circling the far side of the parking lot, heading for the trees again. The rain had almost stopped and the wind had died down. The air was cold and Davey was colder. His one shoe squished with each step he took, and he could feel a blister growing on the bottom of his foot.\n\nThey came to a cyclone fence like the one at the parking lot, only this one was higher and had spiked wire strung along the top. Davey could hardly see the top, even when Cudge held the flashlight high. Duffy was nosing into his neck and sometimes he could feel her shiver. She was cold too. He wrapped his arms protectively around her, warming her, trying to keep her from Cudge's notice. Somehow the responsibility of looking after Duffy gave Davey the courage to go on.\n\nCudge was searching for something. He kept lifting the flashlight, scanning the fence, then looking off into the distance. Eventually, Cudge prodded Davey on again, leading him across the field toward the next stand of trees. The grass was short and the ground was soft and muddy. Several times, Davey almost fell, his knee refusing to support him. But he thought of Duffy and what Cudge might do to her, and he kept pushing forward.\n\nAfter what seemed to Davey like forever but was only a short while later, Cudge reached out and yanked on his shoulder. \"This is as far as we go. We're gonna sit down over here, right where you can see the lion and he can see you. Bet you thought lions slept at night, didn't you? Well, they don't, especially not this one. He's got his old lady in there and some cubs. He stays up all night to watch over them and protect them. He don't want no wiseass kid coming near that fence to upset things. Know what I mean?\" Cudge snorted. \"You ain't even gonna make a good bite for that big guy. He's gonna chew you to pieces in one gulp.\"\n\n\"Are you going to kill me?\" Davey asked fearfully.\n\n\"Yeah. Well, no. It's gonna be my fault but that there lion is the one what's gonna do the actual killing. All day long I been thinking about wringing your neck, but then I came across Samson here and decided he'd have the pleasure instead. Besides, why should I hang for killing you? You ain't nothin' but a little brat. I don't like brats, and you especially. I wouldn't be out here if it wasn't for you. Elva would still be alive, and we'd be on our way to Florida. You spoiled everything. Wringing your neck is too good for you.\"\n\nDavey was frightened. He sat on the cold, wet ground and watched Cudge pace back and forth. Duffy curled into his lap, making herself into a little ball to keep warm. When Cudge's pacing brought him too close to Davey, she lifted her head, bared teeth and growled.\n\n\"Shut that bitch up,\" Cudge warned.\n\nStopping to think for a minute, he lifted up his army-colored jacket and pulled off his belt. He leaned down to reach for Duffy and was rewarded with the threat of snapping jaws. \"Here,\" he threw the belt at Davey. \"Make a leash out of this and hook it to the fence there. Now!\"\n\nDavey's fingers fumbled with the belt. He wasn't sure how to put it on Duffy. It was wide and made of thick leather, like his good Sunday shoes. It wouldn't fit around Duffy's neck without choking her. The next best thing was to put it around her middle; he slid the strap through the buckle then threaded the strap end through a hole in the fence.\n\nThe man seemed satisfied. Now he could stay just beyond the reach of the belt and Duffy couldn't get to him. \"You know what kind of fence that is, kid? It's called horse fencing. I noticed this afternoon that they use it to section one kind of animal off from another. It keeps them from eating each other.\" The man seemed to find that funny, because he threw back his head and laughed. Davey didn't like the way he laughed. It wasn't nice, not at all like his dad's laugh.\n\nDavey could make out the curving stretch of the horse fencing. The highest wire had been pulled down by a giant tree which had fallen against it in the storm.\n\n\"Don't like the look of that, hey kid? Neither did I when I come across it earlier today. Old Samson there gave me a scare when he charged the wire. Only he don't seem too interested in getting over. Guess it's because his wife and kiddies are in there.\"\n\nDavey was silent, looking through the darkness to where Cudge was pointing. Duffy was restless, straining to the full length of the belt, trying to get close to her master.\n\n\"You don't believe me, I can tell. You're just as stupid as Elva ever was. Look, kid\u2014I'm gonna show you something that's gonna make you wet your pants.\"\n\nHow did the man know he'd wet his pants? It had happened so long ago, when Davey was locked in the camper.\n\nCudge picked up the flashlight and stepped closer to the fence. There was a fallen branch lying on the ground and he picked that up too. Immediately, Davey's eyes flew to Duffy. \"Worried about your little doggy? I'll tell you when it's time to worry. Look! Look over here!\"\n\nDavey did as he was told. Cudge ran the end of the branch against the fence; it made a harsh grating noise which was loud in the still night air. Immediately a yellow streak charged out of the darkness and threw itself against the wire. The earth seemed to shake with the impact! Davey felt it in his belly, like he'd once felt the big bass drum that came marching behind the parade. Boom! Boom! Samson's roar was the loudest noise Davey had ever heard, like all the thunder in the world put together in one big sound that made your ears pop and your backbone melt like ice-cream on a hot day.\n\nDuffy yelped with fright and tried to crawl away from the fence, but the short length of leather held her back.\n\nDavey clapped his hands over his ears and squeezed his eyes shut. He wanted the noise to go away. He had been scared before, but now he was terrified. Again Samson roared, a rumbling which began deep in his chest and exploded through his fanged, cavernous mouth. Yellow eyes reflected the light of the flashlight, watching, daring, defying.\n\n\"How do you like him, kid? Pretty big, huh?\" Cudge laughed but the sound was smothered by another of Samson's warnings. \"I'm gonna toss you right over that fence. That old lion only has to lift his leg and bam!\" He smacked one fist into the other hand. \"Just like that! Only I'm gonna rile him up a little first, sorta whet his appetite, if you know what I mean. Then he'll be in fine form when you hit the ground.\"\n\n\"You'd better watch out 'cause Mr. Sanders is looking for me,\" Davey lied, hoping to frighten the man. \"And when he finds you, he's going to lock you up.\"\n\n\"Mr. Sanders, huh? Who's he?\"\n\n\"He's the FBI agent that stays at our house. He gave me money to call him when I needed him and I called him from the pay phone by the Ferris wheel and told him where to find me.\"\n\n\"FBI agent? Who're you kiddin'?\"\n\n\"Okay, don't believe me, but it's true.\"\n\nCudge lashed out and cuffed Davey on the side of the head. Dizzily, Davey shook his head to try to clear it. He should have kept quiet. Tears brimmed in his eyes; no one had ever hit him before. He didn't like it one bit. He was helpless and scared. He focused on the fence, watching the lion pace back and forth, the animal's feline eyes reflecting the light from the flashlight and adding a light of their own.\n\n\"If that Sanders fella does show, kid, I'll throw him over to the lion right after you. Then that damn dog. What do you think of that?\" Cudge bellowed. When he saw the dread on the little boy's face, he smirked.\n\nSuddenly all the wind seemed to go out of Cudge, and he wanted to sit down someplace dry and warm. He was exhausted, physically and mentally. Thoughts of Elva kept popping into his head and he had to push them away. It couldn't be that he missed that stupid, scrawny broad. Why had he ever gotten mixed up with her anyway? He sat down heavily opposite Davey, leaning back against a tree. A match flared as he lit a cigarette.\n\nDavey saw Samson walking away from the fence. He's going back to his family, Davey thought. I wish I could go back to mine.\n\n\"I used to have a dog,\" Cudge said quietly. \"Well, she wasn't exactly mine. She belonged to my grandmother.\"\n\nDavey glanced at Cudge then back toward the fence again.\n\n\"Don't believe me, do you, kid? It's true. I wasn't any different from you when I was a kid. Matter of fact, I was just like you. Trouble, always trouble. Leastways that's what my mother used to say. 'Edmund Balog,' she'd say, 'I don't know what's come over you. You used to be such a good little boy. What made you change?' I used to pretend I didn't know what she was talking about. Only I did know, and I knew when I changed too. I wasn't any older than you when I found out what lives inside me. Only I never told anybody. Couldn't. And when I look at you, kid, I know the same devil that lives inside me is inside you too. Think about what you did to mess me up with the law, with Elva, with everybody. Yes, I know it's there inside you too.\"\n\nHis voice droned on but Davey was only half listening. The man wasn't his biggest fear right now. His biggest fear was right there, behind the fence, with its tearing jaws and thunderous voice. Samson.\n\n\"There were times when I didn't know why I'd ever been born. I ain't never had a friend. You know that, kid? Never, except for Elva. And she wasn't a friend as it turned out. You made her turn against me! She was all right till you came along. So what if I did think she was stupid? And maybe I did think about getting rid of her, but I never would've done it. Never! But then you had to mess everything up.\n\n\"Don't get me wrong. Maybe you couldn't help yourself. I never could when I was a kid. This thing inside my head would always mess me up. What's yours like?\" Cudge lowered his voice, whispering conspiratorially. \"Mine is like a bull, black and tough. It's got hooves that cut into my brain, and long sharp horns that fill up the inside of my head till I can't think! And it's heavy, real heavy. It pounds around in there till I can't stand it. And then it takes over, makes me do things I'd never do on my own. It was the bull that made me kill Lenny. And Elva too. And the other night I beat up this girl for no reason, except maybe she was breathing. It ain't my fault,\" he whined.\n\nCudge's voice had a strangely soothing effect on the little boy. Although his eyes never left the fence, he found that by reaching out with his fingers, he could touch the fur on Duffy's neck. It was reassuring. Suddenly, there was a movement on the other side of the fence. The man's voice must have disturbed the lion again. But no, it was too small for Samson. It was one of the babies, a cub, bigger than Duffy but a baby nevertheless. Davey felt sorry for the cub as it sat lopsided on its haunches, looking out at him. It must be terrible to live with a fence around you all the time. Beyond the circle of light, he could discern a larger form, or were there two shapes? The mother and father lion, he decided. A smile touched his lips. There were more cubs nursing from the mother. The thought delighted him; the mother must be sleeping, and the father was watching out so nothing happened to his family.\n\nIt wasn't like that in his family. If Aunt Lorrie had kids, she would be like Samson\u2014watching and protective. But his family was different. Mom was the one who watched and took care of them, the one who said what was good to eat and where it was nice to play and work, and how things should be done. Mom liked things to be perfect, Davey thought. And she wanted him to be perfect too, but he wasn't. Maybe if he didn't need to get a shot every day, or wear a brace sometimes, he could be perfect. \"Picture perfect\" was what Mom liked to say.\n\nA loud, belly-rumbling roar startled him. Cudge was still talking but Davey stopped listening. Samson had come back to the fence; he was standing over his cub, anxious and uneasy about the intruders. He picked up the cub by the scruff of its neck and marched back to the lioness, dropping his bundle between her front paws.\n\nThat's what fathers are supposed to do, Davey thought, comparing Samson to his own dad. They're not supposed to let the mothers do all the important things.\n\nCudge was off in a world of his own, rambling through his memories, revealing things he'd never spoken about before. And as he talked, the differences between the little boy and himself became less clear. He felt like a child again, as if he was Davey's age, scared of the night-walking monsters that haunted him, and recognizing the monster that inhabited his own body, compelling him to destroy and to kill. In the dim flickering light, Cudge came to believe that Davey was the young Edmund Balog, capable of all things evil, and the future stretched out before him. Every moment of pain and suffering, every weakness, was inevitable. There was only one way to stop it from ever happening again and Cudge was going to do it, before it was too late. The child might look innocent but the evil had already taken hold. The young Edmund Balog was as much to blame for Elva and Candy as the adult was. And now, because of that little boy, Cudge was going to have to kill again.\n\nSamson's roar, when it came, made Davey clap his hands over his ears. Cudge stopped his ramblings. \"I thought I told you to sit there and not move. I ain't ready to dump you over that fence yet. The only reason that lion is bellowing like that is because he wants me to toss you over there. He probably ain't had anything to eat for a week. You're gonna be real sweet meat to him, kid.\"\n\nDavey drew his knees up to his chest. He was so cold he couldn't feel anything anymore. \"I didn't do anything to that lion,\" he whispered to himself over and over. Again, he saw a movement by the fence. A smile tugged at his lips. The little cub was back, looking at him through the wire.\n\nLightning swift, Cudge was on his feet. He clambered up the slippery tree trunk that straddled the fence, ignoring the soggy leaves and branches that were sticking out in every direction. Mesmerized, Davey watched as Cudge broke off a branch and proceeded to pound at the top of the wire fence. Horrified, he couldn't bear the thought that Cudge might hurt the lion cub.\n\nSamson's roars ripped through the night as Cudge pounded again and again at the wire. Duffy joined in the noise, barking and growling, straining to escape the belt.\n\nCudge turned, almost losing his footing on the slippery trunk. \"Don't even think about moving, kid.\"\n\nDavey was stunned. Why hadn't he run off the minute Cudge started up the tree trunk? He could be away by now, looking for a hiding place. He was angry with himself, and angry that the little lion cub was occupying his thoughts, that the father lion wasn't taking it back to its mother. He could still run now if he wanted to\u2014Cudge was halfway up the tree. He would have a small head start if he got up now; but the cub held him rooted to the spot. The cub wasn't afraid of the pounding noise, and it wasn't afraid of its father's anger.\n\nSamson backed off then advanced again. With one monstrous paw, he gently moved the cub to one side, then pushed with both front paws against the tree where it was leaning against the fence. A storm of sound erupted from the lion's cavernous mouth; Davey shuddered but was pleased to see that Cudge was afraid and had started to back off down the tree trunk. The cub was safely back with its mother. Davey sighed with relief.\n\n\"That was just for starters,\" Cudge said, wiping his hands on his sodden jeans. \"I wanted to show you that that lion means business, and so do I. You getting the message, kid?\"\n\nDavey nodded.\n\nCudge returned to his spot against the tree trunk. He started talking again. Davey knew that was good\u2014he didn't seem so mean when he talked. He kept saying the same things over and over and Davey wished he knew why. He didn't really want to think about Cudge, or why he did anything, but he needed to think about what to do if Cudge dragged him up the tree trunk and tossed him into the lion's mouth.\n\nThe lion continued its pacing close to the fence.\n\nCudge's voice was making him sleepy, but Davey couldn't give in now. He had to stay awake and plan how to escape.\n\n\"Come here,\" Cudge said hoarsely, crooking his finger for Davey to get up and go to him. \"Making me come and get you ain't gonna help now.\"\n\nDavey didn't waste a minute. The second he was on his feet, he bolted off into the trees, away from danger, away from the man. Cudge crashed through the trees after him, grunting and panting, but Davey was running fast now, down the hill, towards the level ground. The earth was muddy and soft, making it difficult to run, but he wasn't going to stop until he was safe.\n\nSuddenly he heard a bark from behind him. The sound pierced him. Duffy! He had run away and left Duffy!\n\n\"Hey, kid! Guess what I got that's yours!\" The bellowing laugh was followed by an angry yipping and a whimper of pain. There was nothing else to do. He had to go back. He couldn't let Cudge hurt his dog.\n\nHeedless of the branches scratching his face and neck, Davey stumbled back up the incline, gasping as he fell over tree roots and slipped on the wet leaves. At first, he thought Cudge had gone and taken Duffy with him. He could hear Samson snarling, but couldn't see him.\n\nThen Davey heard that hated voice. \"Up here, kid. I thought you'd come back. Look what I've got\u2014but I won't have it for long!\"\n\nDavey looked up at the fallen tree straddling the fence. Cudge was lying on his belly along it, his arm hanging down over the wire. He was laughing. Duffy swung from the belt looped around her middle, just feet away from Samson's reaching claws, her short legs twitching, her head arched back in terror. The lion snarled, its jaws snapping, saliva stringing from its mouth.\n\nDavey felt as if his heart would burst. \"No! No!\" His feet found a hold on the tree trunk; it was slippery and wet, and he had to dig into the bark with his fingers to hold on. It was so high, higher than he'd ever been before.\n\nDuffy swung out again, lower this time. At the last second, Cudge yanked on the belt, pulling her out of Samson's reach.\n\nThe lion stood on its hind legs, clawing at the furry object swinging overhead. Growling, it attacked again, just falling short of its prey.\n\nDavey climbed higher, faster, reaching out for his dog. The trunk was too wide for him to hold on to, and he slipped, falling into its lower branches, nearly going over the fence himself, right down into Samson's mouth.\n\nCudge didn't seem to notice that Davey was just underneath him. He swung Duffy out again and the belt came to within inches of Davey's grasp. It hung there for a breathless moment before Cudge hauled Duffy out of Samson's reach.\n\n\"Hey, kid, come and see. Where are you? Don't you want to feed the lion?\" Cudge shouted out, laughing nastily. \"You'd better answer before your dog turns into this cat's breakfast!\"\n\nDavey stretched out, feeling the branch bend under his weight. He held his breath. He had to be ready; his hands had to be strong to grab Duffy away from Cudge.\n\nThe next time Cudge swung Duffy out over the lion's head, Davey reached out as far as he could. He caught the dog but she was too heavy to hold. He felt himself losing his balance. Grasping the end of the belt, Davey swung the dog toward the fence. He prayed it wasn't too high, that Duffy would survive the fall. At the last second, he released the belt, and waited for what seemed an eternity to see that Duffy had cleared the fence and dropped to safety on the other side.\n\n\"Hey! What the\u2014?\" Cudge looked around wildly, then spotted Davey below. He reached down through the branches to grab him but Davey ducked, avoiding Cudge's hand by a narrow margin. Again, Cudge groped for him. Cowering backward, Davey was afraid to move for fear he would fall into Samson's mouth. Curses sounded above him. He couldn't see Cudge, but he knew where he was by the arm searching through the branches.\n\nSamson roared again, his yellow eyes staring at a point above Davey's head. The strength flowed back into Davey's limbs. His brain started to send signals to his body. Wait, he told himself, wait.\n\nHe heard the shift of Cudge's body above him, and saw the man's change of position in the lion's yellow eyes. As Cudge's arm came reaching out again, Davey gripped it with his knees, locking his ankles together. He heard Cudge grunt and saw the lion ready itself to spring. Swiftly, with all the strength he possessed, Davey pulled hard at the trapped limb.\n\nThe sudden action caught Cudge off guard and he lost his balance. He began to fall through the branches and reached out for something to stop his fall. But there was nothing. Only the yawning jaws of fate waiting for him.\n\nDavey closed his eyes and forced himself to shut out the terrible sounds from inside the lion's cage. He was safe at last. Safe.\n\nHe slid down the tree limb, feeling a sense of his own power. But the minute his cold, numb feet touched the wet ground, his legs gave out on him. He needed to rest a minute. Rest. Just for a minute.\n\nThere was a soft sound close by. Duffy. He wanted to see her, to check if she was all right. But it was still too dark. He needed the flashlight. A sigh escaped him as he forced his fingers to work. Off came the jacket and down came the pocket zipper. His fingers worked at the switch but nothing happened. He banged it against his knee; it was too wet. Again, he flicked the tiny switch and this time was rewarded with a feeble light. Quickly, before it faded, Davey shone it on Duffy. His fingers felt all over her dark fur, searching for bloody gashes and wounds. Only once did the little dog yelp with pain\u2014when Davey was removing the leather belt from around her middle.\n\n\"You're gonna be fine, girl. Your tummy's just sore. Let's see you walk, Duff. Can you walk?\" The optimistic tone of her master's voice revitalized Duffy. She could be her old self again if Davey could.\n\n\"Come on, Duff,\" Davey said. \"We're going to find Aunt Lorrie.\"\n\nDavey wasn't sure which way to go to find the hole in the fence. He started toward the Ferris wheel then heard someone call his name.\n\n\"Aunt Lorrie?\" he called back. \"I'm here! Aunt Lorrie, I'm here.\" He tried to run but his legs and feet were too painful.\n\nLorrie stopped. \"It's Davey. Oh, my God. It's Davey.\" She began to move again, listening carefully to work out where Davey's voice was coming from. A moment later, Duffy came tearing toward her. Behind her, Lorrie could see Davey limping.\n\nShe held out her arms and Davey fell into them, hugging her with all his might. \"Oh, honey, are you all right? I've been so worried about you, so afraid for you.\"\n\n\"I'm okay,\" he said against her neck. \"I'm better now.\"\n\n\"Hey, Davey,\" Stuart Sanders said.\n\nDavey leaned back. \"Hi, Mr. Sanders. I did what you said and called you, but all I got was an answering machine. I couldn't leave a message because that man was chasing me.\"\n\n\"What's his name, Davey?\"\n\n\"Cudge.\"\n\n\"Where is he now?\"\n\nDavey looked back the way he'd come. \"He fell out of the tree and . . . and Samson got him.\"\n\n\"Samson?\"\n\n\"The daddy lion,\" Davey explained.\n\nThe color drained from Lorrie's face. She pulled Davey closer and pressed his head against her shoulder. \"Don't think about it, Davey. Don't think about anything but going home.\"\n\n\"I'm hungry, Aunt Lorrie.\"\n\nLorrie swallowed hard. \"I bet you are.\"\n\n\"I lost one of my shoes.\"\n\nLorrie grinned. \"Who cares about that?\"\n\n\"And I ruined my new jacket.\"\n\n\"No big deal.\" She was still grinning.\n\n\"Where's Mom and Dad?\"\n\nLorrie looked to Sanders for help with that one. The agent ruffled Davey's hair. \"They're still in Florida, son. But you'll see them soon.\"\n\nDavey's features closed up. \"Did you call Mom and Dad and tell them I was lost?\"\n\n\"You bet. Right away.\"\n\nDavey nodded. \"I'm kind of tired, Aunt Lorrie.\"\n\nConcern filled Lorrie's face. \"Then let's get out of here and get you back to the motor home and into bed.\" Tenderly, she held Davey against her and began to walk toward the exit.\n\n\"Are you coming, Stuart?\" she asked, when he didn't immediately follow.\n\n\"Yep, sure am. I'll take care of this mess later. Right now Davey's the most important thing.\"\n\nLorrie sat next to Stuart on the motor home sofa, waiting for Davey to wake up. A little over an hour ago, Feeley had called and given them the news about Sara. Lorrie hadn't been able to believe it at first. Sara dead\u2014it just didn't seem possible. She had always believed Sara to be indestructible.\n\n\"Poor Davey,\" she said, squeezing Stuart's hand. \"Don't say anything to him right now. He needs to recuperate from this trauma first.\"\n\nStuart had expected Lorrie to go into shock, but she hadn't. He supposed it was because she was a doctor and dealt with death on a daily basis.\n\nThe moment Davey stirred, Sanders got up and tiptoed over to the bunk. Davey lay with his eyes wide open, staring at the roof of the RV. Duffy, freshly bathed, was snuggled against him.\n\n\"How's it going, kid?\" Sanders asked as he sat down on the edge of the bunk.\n\n\"Okay, Mr. Sanders.\"\n\n\"I'm real proud of you, Davey. I think you're the bravest little boy I've ever known. You can join my team anytime.\"\n\n\"You aren't leaving, are you?\" Davey asked.\n\n\"Not on your life. I made a date with Duffy to buy her the biggest steak in town. I have a date with you too, and your aunt.\"\n\nHe could feel that lump in his throat again.\n\nRoman DeLuca was being escorted down the studio corridor by an assistant director. The jury had reached its verdict during the early morning hours, and now DeLuca was going to be interviewed on a television news program.\n\nThe accused had been found guilty, which made DeLuca a winner in the public's eye. It would probably help to put him in the governor's seat. He was ready to smile for the cameras, and tell the world that justice had been served, but inside he was seething. It hadn't been meant to turn out this way, and all because he had miscalculated the determination of one person\u2014Sara Taylor. He had achieved his goal\u2014Jason Forbes's killer hadn't been linked to the syndicate, and probably wouldn't be now\u2014but it would all have gone a lot more smoothly if Sara Taylor had been more cooperative.\n\nThe moment DeLuca stepped into the studio, the reporters began shouting out their questions. They liked Roman DeLuca, they knew he was climbing the ladder to success. He even looked the part with his bronzed face, his immaculate white shirts and custom-made suits.\n\nFlashing his brilliant smile, DeLuca listened attentively to the first question. He sobered at once as he replied: \"I'm shocked. There are simply no words at a time like this. My sympathy goes out to Mr. Taylor who made such a brave and admirable contribution to justice.\"\n\n\"Mr. DeLuca, do you know of any reason why Mrs. Taylor would ignore the warning shouts to get off the runway? Is there something the press isn't being told?\"\n\nDeLuca put on what he called his sincere, humble smile. \"Haven't I always been open with the press?\" Not waiting for a reply, he continued somberly: \"I understand there was some personal problem at home concerning the Taylors' son. Mrs. Taylor was distraught so Federal Agent Jonas was assigned to take care of her. The whole incident was very unfortunate and I'm truly sorry. Mrs. Taylor was a remarkable woman in many ways.\"\n\n\"What kind of personal problem, Mr. DeLuca?\"\n\n\"Now, if I told you that it wouldn't be personal any longer, would it?\"\n\n\"The Taylors have a hemophiliac child, don't they?\" A chunky man in a sweat-stained blue shirt shouted to be heard over the chattering throng.\n\n\"Yes, they do. Now, if that's all, gentlemen, I have a hard day ahead of me and I'm expected at WKBA's television studio.\"\n\n\"Mr. DeLuca, do you still have 'no comment' on your plans to run for governor?\"\n\nDeLuca grinned: he was on solid ground now. \"I think, ladies and gentlemen, that . . .\"\n\nAgent Jonas listened to the suave, controlled voice. That bastard. He hadn't lied, he hadn't fabricated a thing. Up front all the way with the media. What right did the attorney have to use him like that? Implying that he, Jonas, had overplayed his hand and used a thug's tactics. DeLuca was the one who'd said \"take care of her.\" How could he have known that the prosecutor just wanted her kept out of the courtroom. Christ, even Sara Taylor had misunderstood!\n\nDavey's mind wandered as he sat quietly in the small chapel. He wished he could be outside with Duffy, running through the leaves, or talking to Digger on the CB. He didn't understand the meaning of the words the minister was saying, and he didn't like the way people kept looking at him. Most of all, he didn't like the words Memorial Service. He began to fidget.\n\nLorrie watched Davey out of the corner of her eye. The little boy didn't understand what was going on; he shouldn't have to be here. Instantly, she was contrite. What an awful thought. Of course he should be here! After all, the service was for his mother.\n\nHow sad, she thought as she let her gaze circle the small chapel. Aside from her and Andrew and a few of Andrew's business associates, the only other person present was Stuart Sanders. Private memorial services were very lonely, but Andrew had wanted it that way. How sad that only she and Andrew were grieving; but, as long as she was being honest, she was also relieved in a way.\n\nThe short service was soon over. Outside, in the brisk, autumn air, Andrew didn't seem to comprehend where he was. His luggage was in the car parked at the curb; he was going away. He knew he should be saying something to Lorrie, but he couldn't think of the words. If the service had been for him, Sara would have known the right thing to say. Now that he was on his own and had to think for himself, he felt lost. Davey. He had to say something to Davey before he left. God, what could he say? Where were the words? Where?\n\nDavey stood awkwardly between Stuart Sanders and Lorrie. His round gaze was speculative. \"Dad?\"\n\n\"Yes, son.\"\n\n\"Will you be gone a long time?\"\n\n\"I don't know, Davey, I just don't know. But you'll be fine with Aunt Lorrie. She'll take good care of you.\"\n\n\"I know. Aunt Lorrie is going to get all my things and my CB and take them over to her house this afternoon.\"\n\nSomething pricked at Andrew's eyes. \"I know that, Davey.\"\n\nManfully, Davey extended his hand to his father.\n\nAndrew felt his eyes begin to smart. He took his son's hand in his own. \"I'll call you when I get settled.\" For an instant Andrew felt as though Sara were with him, arranging the last-minute details, offering her approval.\n\n\"Okay, Dad. Drive carefully and don't forget to stop for gas.\"\n\n\"I won't, Davey. You take care now.\"\n\nDavey nodded.\n\nStuart Sanders's penetrating gaze rested on the small boy in the gray suit. His lips narrowed to a grim, white line and he nudged Lorrie. \"Is it my imagination or did Davey get taller in the past several days?\"\n\nDavey watched his father climb into the blue sedan. His eyes didn't leave the road until the car was long out of sight. Then he turned to face his aunt and Stuart Sanders. His eyes glistened momentarily in the chill October light. \"Can we go now?\"\n\n\"You bet,\" Lorrie said, smiling down at him. \"Get in the car, we're going home.\"\n\nDavey ran for the car and climbed into the front seat. He concentrated on fastening his seat belt. When he'd finished, he looked out the window and saw Aunt Lorrie and Mr. Sanders standing by the driver's door, kissing. They had been doing a lot of that in the last few days\u2014kissing and giving each other mushy looks. And they'd been hugging and kissing him a lot, too. He had pretended not to like it, but he did. He liked it a lot.\n\nAunt Lorrie had said that he'd suffered no ill effects\u2014whatever that meant\u2014from not getting his shots and that he was going to be just fine.\n\nMr. Sanders opened the car door and Duffy jumped in. She climbed onto Davey's lap and licked his face. Mom would have scolded Duffy and made her get down but Aunt Lorrie just laughed.\nThe women of the Sisterhood have long been aided in their quest for justice by the men in their lives. Now for the first time the guys are stepping into the spotlight in a brand-new three-part adventure!\n\nJack Emery, husband of Sisterhood member Nikki, finds himself at loose ends. Frustrated with the limits of the criminal justice system, he bands together with his friends. Along with Ted Robinson, Joe Espinosa, Harry Wong, Bert Navarro, and Jay Sparrow they form an off-the-grid spy group called BOLO Consultants. The \"guy group\" tackles cases overlooked by the D.A. and FBI with, of course, a little help from the Sisterhood!\n\nDon't miss Fern Michaels's\n\nUPSIDE DOWN   \nA Zebra e-book exclusive on sale now!\n\nCOUNTDOWN   \nA Zebra e-book exclusive on sale September 2014!\n\nTAKE DOWN   \nA Zebra e-book exclusive on sale November 2014!\n\nPhoto by M2IFOTO \u00a9 2006\n\nFERN MICHAELS is the USA Today and New York Times bestselling author of the Sisterhood and Godmothers series and dozens of other novels and novellas. There are more than 75 million copies of her books in print. Fern Michaels has built and funded several large day-care centers in her hometown and is a passionate animal lover who has outfitted police dogs across the country with special bulletproof vests. She shares her home in South Carolina with her four dogs and a resident ghost named Mary Margaret.\n\nVisit her website at fernmichaels.com.\n\nFORGET ME NOT\n\nLetting Go of the Past\n\nWith a popular comic strip, card line, and children's cartoon to her name, Lucy Brighton should be in a happy place. But the ache of a cold, lonely childhood lingers on. Even though she still lives in the New Jersey house where she grew up, Lucy has had little contact with her parents since they moved to Florida five years ago.\n\nFinding Joy in the Present\n\nThen Lucy receives a call that her parents have been killed in a car crash. While settling their affairs in Florida, Lucy begins to realize how little she really knew about their lives. She has no way to explain the mysterious safe in their bedroom, with its cache of fake passports, cash, and weapons. What secrets were the Brightons keeping? Were they even who they claimed to be? The answers will shatter everything she once believed about her parents\u2014and about herself.\n\nKISS AND TELL\n\nIn this page-turning new novel in her beloved Sisterhood series, New York Times bestselling author Fern Michaels pits the indomitable Vigilantes against a corrupt and ruthless billionaire.. . .\n\nAll good things must come to an end. Even the women of the Sisterhood\u2014the stalwart sorority that has helped to right so many wrongs\u2014have been content to let their gold shields gather a little dust while they enjoy friends, family, and the comforts of civilian life. But the group's leaders\u2014Myra Rutledge and her best friend, Annie de Silva\u2014remain as vital and feisty as ever. So when a string of anonymous e-mails arrive at Pinewood, suggesting shady dealings at a local assisted-living facility, the two immediately set out to investigate.\n\nEmanuel Macklin, the financial wizard behind the sprawling, high-end senior complex, has amassed a private fortune that would make Fort Knox look like spare change. But the hefty returns Macklin promises his investors may be garnered at others' expense. In fact, Macklin's entire enterprise is one enormous Ponzi scheme that's about to collapse\u2014taking the life savings of thousands of innocents with it. Myra and Annie know this is a mission tailor-made for their brand of justice, but they'll need to enlist some new and untested allies. And even as they set out to foil one of their most cunning adversaries to date, Myra faces a personal challenge that will rock the Sisterhood to the core....\n\nBLINDSIDED\n\nFading into the background isn't the Sisterhood way. Even after all the adventures they've shared, the courageous, close-knit heroines of Fern Michaels's New York Times bestselling series are always ready to embrace another challenge....\n\nAll good things must come to an end. But Myra Rutledge isn't ready to put the Sisterhood\u2014the stalwart band of friends who've become legendary for meting out their own brand of justice\u2014behind her just yet. Though she loves her beautiful home and her husband, Charles, Myra can't deny that she's restless. And as it turns out, she's not the only one longing to dust off her gold shield and get back in action.\n\nWhen Maggie Spitzer, former editor in chief of the Post and an honorary member of the Sisterhood, arrives with a new mission in mind, Myra welcomes her in. Maggie's news-hound instincts haven't dulled since she left the Post, and she suspects that two Maryland judges\u2014identical twins Eunice and Celeste Ciprani\u2014are running a moneymaking racket that sends young offenders to brutal boot camps, often on trumped-up charges.\n\nSoon the Vigilantes are gathering in their war room once more, catching up on the momentous events in each other's lives even as they plan their campaign. The Ciprani twins are powerful and ruthless, and taking them down won't be easy. But with the aid of formidable allies\u2014including former President Martine Connor\u2014Myra, Annie, Maggie, and the gang concoct a scheme that will bring justice to the innocent, and leave the guilty blindsided. . . .\nThis book is a work of fiction. Names, characters, places, and incidents either are products of the author's imagination or are used fictitiously. Any resemblance to actual persons, living or dead, events, or locales is entirely coincidental.\n\nZEBRA BOOKS are published by\n\nKensington Publishing Corp.   \n119 West 40th Street   \nNew York, NY 10018\n\nCopyright \u00a9 1982, 2000 by Fern Michaels\n\nPublished by arrangement with Severn House Publishers, London and New York. Originally published in a previous version with the title Panda Bear Is Critical by Macmillan Publishing Co., Inc,. in 1982 and reprinted by Pocket Books in 1984.\n\nAll rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.\n\nZebra and the Z logo Reg. U.S. Pat. & TM Off.\n\nFirst Zebra Mass Market Edition: February 2005\n\neISBN-13: 978-1-4201-3658-6   \neISBN-10: 1-4201-3658-5   \nKensington Electronic Edition: August 2014\n\nISBN: 978-1-4201-3657-9\n\nFirst Kensington Trade Paperback Printing: August 2014\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nAuthor:\n\nVirginia Pitts Rembert\n\nLayout:\n\nBaseline Co. Ltd\n\n61A-63A Vo Van Tan Street\n\n4th Floor\n\nDistrict 3, Ho Chi Minh City\n\nVietnam\n\n\u00a9 Confidential Concepts, worldwide, USA\n\n\u00a9 Parkstone Press International, New York, USA\n\n**Image Bar** : www.image-bar.com\n\nAll rights reserved.\n\nNo part of this publication may be reproduced or adapted without the permission of the copyright holder, throughout the world. Unless otherwise specified, copyright on the works reproduced lies with the respective photographers, artists, heirs or estates. Despite intensive research, it has not always been possible to establish copyright ownership. Where this is the case, we would appreciate notification.\n\nISBN: 978-1-78310-025-5\nVirginia Pitts Rembert\n\nHIERONYMUS BOSCH\n\nHieronymus Bosch and the Lisbon Temptation:\n\na view from the third millennium\n\n### Contents\n\nIntroduction\n\nThe Literature on Bosch to Wilhelm Fr\u00e4nger\n\nFr\u00e4nger's Thesis (Epiphanies and Absurdities)\n\nFr\u00e4nger and Beyond\n\nA More Prosaic View\n\nSaint Anthony and the Devil\n\nSaint Anthony\n\nTemptation by the recollection of the responsibilities and pleasures of his past life\n\nTemptation by the desires of the flesh\n\nTemptation by pride\n\nTemptation by physical torture\n\nAll the demons of hell are unleashed\n\nTemptation by constant hordes\n\nFurther Temptations as recounted by Anthony himself\n\nFinal Temptations\n\nThe Prince of Darkness\n\nThe Society of Witches\n\nOther Sorcerers and Necromancers\n\nThe Tarot\n\nAlchemy\n\nThe Lisbon Triptych\n\nConclusion\n\nIndex\n\nBibliography\n\nNotes\n\nDeath and the Miser (detail),  \nc. 1485-1490.\n\nOil on panel, 93 x 31 cm.\n\nNational Gallery of Art, Washington, D.C.\n\n### Introduction\n\nAt the approach of the year 1000 CE, people believed that the Judgement predicted by Christ to occur at the millennium was imminent. When it did not come in 1000, or near that time, the chronicler and Cluniac monk, Raul Glaber, wrote:\n\nThere occurred, throughout the world, especially in Italy and Gaul, a rebuilding of church basilicas. Notwithstanding, the greater number were already well established and not in the least in need, nevertheless, each Christian group strove against the others to erect nobler ones. It was as if the whole earth, having cast off the old by shaking itself, were clothing itself everywhere in the white robe of the church.\n\nThe solemn projections of the end of the world reached their most modernised climax in 1997, when 39 members of a computer-related cult followed their leader, Marshall Herff Applewhite, in a suicide contract to beam themselves up to a spaceship presumably trailing in the wake of the Hale-Bopp Comet that was plunging through the heavens that year.\n\nAn essayist in _The New Yorker_ commented on their fantastically flawed mission: \"Though science is stronger today than when Galileo knelt before the Inquisition, it remains a minority habit of mind, and its future is very much in doubt. Blind belief rules the millennial universe, dark and rangy as space itself (14\/04\/1997, 32).\n\nSeeming to reinforce the presumptions from the New York Academy of Sciences' conference were references that abounded in the national media reflecting increased interest in astrology, psychic phenomena, and magic as well as the related fields of Satanism and witchcraft.\n\nAn article on witchcraft ( _New York Times_ , 31\/10\/1998) centred around a group of \"Wiccans\" (the modern name of so-called witches, derived from a neo-pagan, pseudo religious group called \"Wicca\") operating in Salem, Massachusetts. That city, site of the 17th-century witches' trials, was said to have become a centre of tolerance for \"alternative spirituality\", including New Age beliefs and contemporary witchcraft groups such as the Temple of Nine Wells and the Witches League for Public Awareness:\n\nClaiming that theirs is a peaceful, nature-oriented religion, quite unlike early devil-worshipping societies, the Wiccans have organised educationally, even politically, to correct misapprehensions about witches and their modern motivations.\n\nA tabloid article quoted from a list of \"the world's top Bible scholars\" who predicted the imminent end of the world and the coming Apocalypse, which it inferred, would be at the end of the millennium ( _Weekly World News_ , 14\/05\/1996).\n\nThe Magician, 1475-1480.\n\nOil on panel, 53 x 75 cm.\n\nMus\u00e9e municipal, Saint-Germain-en-Laye.\nIt cited ancient prophecies from _Revelations_ and more recent ones from, among others, the 16th-century prophet Nostradamus about dire natural events to occur at the end of our millennium that seemed to accord with El Ni\u00f1o's deviant climatic disorders in 1998.\n\nThe fact that these events were not as baleful as predicted made the turning of the millennium seem almost anticlimactic \u2013 until \"9\/11\", that is, which many saw as the USA's Armageddon. Similar predictions and oddities had occurred in the decade leading up to the half-millennium of 1500. As if their predecessors of the first 1000 years had been mistaken about when the Judgement would come, contemporary thinkers expected it to appear without fail in the year 1500. Art historian Charles Cuttler summed up the emotional atmosphere of the time:\n\nIt was a time of pestilence and turbulence, of economic, social, and religious unrest; an age which believed in chiliasm, Antichrist, apocalyptic visions; in witchcraft, alchemy, and astrology. It was also a period of extreme pessimism, the natural outcome of a belief in demons fostered by the Church itself. (Cuttler, 1957)\n\nAs always, artists were present to give voice and imagery to what otherwise would have seemed unimaginable. Northern poets, known (such as Fran\u00e7ois Villon) and anonymous, as well as sculptors of Romanesque tympana and capitals, had graphically displayed their versions of the terrors to come at the end of the world. Later, in the proto-Renaissance period, Gothic revivalist painters depicted these anomalies in their altarpieces. Possibly the most vivid and detailed were those of the Hollander Hieronymus Bosch, which shall be the subject of this book.\n\nA 17th-century English ambassador to Holland expounded on the virtues of painting compared to sculpture, by saying: \"An excellent piece of painting is, to my judgement, the more admirable object because it is a near Artificiall Miracle\" [ _sic_ ] (Fuchs, 1978). The historian who quoted this statement repeated the term \"Artificiall Miracle\" several times to refer to the Dutch penchant for \"the meticulous rendering of things observed\".\n\nThe term could also accommodate the whole spectrum of Dutch art from Jan van Eyck to Jan Dibbets for its relevance to the astringent yet probing combination of subject and essence that is peculiarly Dutch. In this sense, the term might even apply to such seemingly disparate artists as Hieronymus Bosch and Piet Mondrian. One artist made real the unreal and the other made unreal the real, but they pursued their uncommon aims through lovingly treated surfaces that survived them as \"Artificiall Miracle[s]\".\n\nI think Bosch and Mondrian were linked in other important ways. As Nordic artists, they belonged to a group that \"has never been content with the mere reproduction of an object\", as art historian Oskar Hagen put it. Both of these artists lived in a century of millennial consciousness and both responded to this consciousness in their work.\n\nA case could be made that Mondrian was a millennial artist of our era. At a great distance from Bosch in time, circumstance, and ideology, Mondrian presented a vision of what the modern world could be, if we looked towards harmony rather than tragedy, which he saw not only in war but in cultural manifestations that had become mired in particulars rather than essentials.\n\nTable of the Mortal Sins,  \nlate 15th century.\n\nOil on panel, 120 x 150 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nTable of the Mortal Sins (detail: Envy),  \nlate 15th century.\n\nOil on panel, 120 x 150 cm.\n\nMuseo Nacional del Prado, Madrid.\n\n__\n\nTable of the Mortal Sins (detail: Anger),  \nlate 15th century.\n\nOil on panel, 120 x 150 cm.\n\nMuseo Nacional del Prado, Madrid.\n\n__\n\nTable of the Mortal Sins (detail),  \nlate 15th century.\n\nOil on panel, 120 x 150 cm.\n\nMuseo Nacional del Prado, Madrid.\nIn his years spent in Paris and London between the 20th century's two world wars, Piet Mondrian invented a painting that did not transcribe existing reality, but \"imaginatively constructed\" what he called a \"new reality\" (Mondrian, 1951). Through its containment, purity, and harmonious ordering of parts, Mondrian posited his painting as an aesthetic cosmos, the \"clear vision\" of the \"pure reality\" he hoped would come to pass in the ideal world of the future.\n\nObviously, Mondrian's 20th-century creations are divided by radically different sensibilities from those of Bosch's at the end of the Middle Ages \u2013 or did the two artists reveal the dark and light sides of human coinage? Perhaps, Mondrian's paintings show what we could become if we lived in harmony with the universe, and Bosch's what we would become if we did not heed the Judgement, as seen through two millennial perspectives, five centuries apart.\n\nAfter turning to Dutch art, in general, Bosch's background, and his treatment in literature until the 20th-century, I shall concentrate on one of Bosch's paintings, the Lisbon Temptation of Saint Anthony, because it was likely to have been completed around 1500, the half-millennial time fraught with the fears and uncertainties that such a transitional period brings. I developed an interest in the Saint Anthony theme by seeing an exhibition of modern paintings on this subject in New York City, in 1946.\n\nThese had been commissioned from about a dozen of the major Surrealist artists by the producers of a motion picture that was to be based on a story by Guy de Maupassant. Although the story, entitled _The Lives and Loves of Bel Ami_ , pivoted around a painting whose religious power had converted a debauched man, the producers decided to change the subject of Christ walking on water, prohibited by the Hollywood Censorship Office at that time, to Saint Anthony's temptations by the Devil.\n\nA painting by Max Ernst was chosen as the most provocative treatment and as best suited for inclusion in the movie. This being the time of bare transition from black and white to colour, the Ernst painting was the only thing shown in the film in colour, giving it a powerful impact. (I later saw Ernst's painting and the others, each fascinating, brought together in an exhibition called \"Westkunst\", in the summer of 1981, in Cologne, Germany. I shall use reproductions of some of these paintings in the text.)\n\nThe subject of Saint Anthony and his temptations has been of interest to artists through the centuries, in a range from 15th-century woodcutters to C\u00e9zanne. It was bound to be a favourite for Bosch, who turned to this theme in at least a dozen paintings and drawings; some of which will be included in the text.\n\nTo reveal the richness of the theme through Bosch's work was reason enough for me to produce one more book on Hieronymus Bosch. Another reason, equally compelling, was the apparent reappearance of many of the beliefs current in the artist's time as we mounted the transition from the second millennium to the third. I hope that the following account will afford some interest and insights, even for the many current scholars of Bosch.\n\nThe Epiphany or The Adoration of the Magi, c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\n\n### The Literature on Bosch to  \nWilhelm Fr\u00e4nger\n\nBefore undertaking a study of only one of Hieronymus Bosch's paintings, I would like to include a critical survey of some of the art historical attitudes towards the artist and his work. This is because they have differed so widely from the first mention of him in 16th-century writings to the present. The writers who commented upon him in the nearly five centuries following the artist's death compounded such a reputation for the man as a \" _faizeur de dyables_ \" [maker of devils] (Gossart, 1907), that until the modern period he was hardly considered an artist at all. It was largely his frenzied hell scenes that attracted such attention. When he depicted the creatures and settings of these \"hells\" in terms of infinitely detailed naturalism, they were so convincing as to seem pure evocation. To the medieval mind, the man who could reveal so plainly its own worst fears must have been a wizard or a madman, perhaps a tool of the Devil himself.\n\nLater writers either reflected this point of view or, following the rationalist aftermath of the Renaissance and the Reformation, passed Bosch off as representing the worst of Medievalism. When he was mentioned, it was not as an artist so much as a freak performer. Eventually Bosch was obscured and forgotten. It was at least two centuries before there was a revival of interest in him, in the late 19th century. The 20th century saw more emphasis on this man as an artist than at any time in the past and this trend is continued with an almost overwhelming interest in him in the 21st century.\n\nOne would expect Italian writers of the High Renaissance period to point out the painter's strangeness, since his ideation was so antithetical to that of the South. The Florentine historian Guicciardini, in his _Description of all the Low Countries_ (1567), referred to \"Jerome Bosch de Bois-le-duc, very noble and admirable inventor of fantastic and bizarre things\". In 1568, The Italian historian of artists, Vasari, called Boschian invention \" _fantastiche e capricciose_ \". Lomazzo, the author of the _Treatise on the Art of Painting, Sculpture, and Architecture_ , first published in 1584, spoke of \"the Flemish Girolamo Bosch, who in representing strange appearances, and frightful and horrid dreams, was singular and truly divine\".\n\nDuring the same period in the North, similar statements were made concerning the painter's work, his demons and hells being mentioned to the exclusion of all else. The Netherlandish historian, Marc van Vaernewijck (1567), called Bosch \"the maker of devils, since he had no rival in the art of depicting demons\" (Vaernewijck, 1905-1906). Carel van Mander, the Northern counterpart to Vasari, made little more observation of Bosch's entire works than that they were \"gruesome pictures of spooks and horrid phantoms of hell\".\n\nThe Epiphany or The Adoration of the  \nMagi (detail, central panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Epiphany or The Adoration of the Magi  \n(detail, central panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\nNumerous statements in the same vein began to appear in Spanish writing following the influx of so many of Bosch's paintings into mid-16th-century Spain. King Philip II, himself, was chiefly responsible for the painter's Spanish popularity. In 1581, when the king journeyed to Lisbon, he wrote in a letter to his two daughters an expression of regret that they had not been with him to see the Corpus Christi procession, \"although,\" he added, \"your little brother if he were along might have been frightened of some devils which resembled those in paintings by Hieronymus Bosch\". Philip owned as many as thirty-six of these paintings, amazing when it is considered that Bosch's entire output is believed to have been barely forty in number. Such a large collection accumulated in so few years after the painter's death attests to a fascination on the king's part \u2013 a state of mind that prompted some of the first penetrating writing directed toward Boschian work. This was because the monk, Father Jos\u00e9 de Seg\u00fcenza, who inventoried the king's paintings shortly after Philip's death in 1598, felt compelled to apologise for the king's obsessive interest in Bosch. Perhaps Father Jos\u00e9 feared a destructive attention of the Inquisition, because he wrote an elaborate defence of the painter's orthodoxy and fidelity to nature:\n\nAmong the German and Flemish paintings which are, as I say, numerous, many paintings by J\u00e9r\u00f4me Bosch are scattered throughout the house (Escorial); I should like to speak for different reasons a little longer about this painter, for his great genius deserves it, although people call his work, in general, absurdities, people who do not look very attentively at what they contemplate, and I think for that reason that he is wrongly denounced as a heretic \u2013 and to begin there \u2013 I have of the piety and zeal of the king, our founder, an opinion such (that I think that) if he [Bosch] had been thus, he [the King] would not have admitted his paintings in his house, in his convents, in his bedroom, in the Chapter of his orders, in his sacristy, while on the contrary, all these places are adorned with them. Except for this reason, which seems very important to me, there is still another which I deduce from his paintings for one sees there almost all the sacraments and ranks and degrees of the church, from the pope to the most humble, two points where all heretics falter, and he painted them with his zeal and a great observation, which he would not have done as a heretic, and with the mysteries of our Salvation he did the same thing. I should like to show now that his paintings are not at all [absurdities], but like books of great wisdom and art, and if there are any foolish actions, they are ours, not his, and let us say it, it is a painted satire of the sins and inconstancy of men.\n\nAn interesting counter-reaction to that of the monk is the statement by Francesco Pacheco, the teacher and father-in-law of Vel\u00e1squez \u2013 as written sometime later, in 1649:\n\nThe Epiphany or The Adoration of the Magi  \n(detail, left panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Epiphany or The Adoration of the Magi  \n(detail, central panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\nThere are enough documents which speak of the superior and more difficult things, which are the personages, if one finds time for such pleasures, which were always disdained by the great masters \u2013 nevertheless some seek these pleasures: that is the case for the ingenious ideas of J\u00e9r\u00f4me Bosch with the diversity of forms that he gave to his demons, in the invention of which our King Philip II found so much pleasure, which is proved by the great number of them which he accumulated. But Father Seg\u00fcenza praises them excessively, making mysteries of these fantasies that we would not recommend to our painters. And we pass on to more agreeable subjects of painting.\n\n[Pacheco was a Spanish painter and art theorist of the artistic period between Mannerism and Baroque. He rejected the manneristic delight in mere form and was turning towards an interest in naturalistic illusionism. From either point of view he would have found Bosch's work unacceptable].\n\nEven though Pacheco's concern was with Bosch as an artist, he passed him off as an oddity, and this reputation clung to the painter for two and a half centuries to come. During this period there was little attention given by scholars to Northern art at all; when it was considered, Bosch was obscured by the great Netherlandish painters ranging from Van Eyck to Bruegel. It was not until the end of the last century that any respectable scholarship was brought to bear upon the painter. Perhaps this was a consequence of the realistic impulse that entered mid-19th-century painting. Historians began to look for precursors to this realism in the past. They turned again to an interest in Northern art, and in reemphasising Bruegel, \"discovered\" Bosch. Not only had Bruegel been profoundly influenced in his early works by Bosch's \"drolleries\", but he had probably been stimulated to an interest in \"genre\" by studying this painter. Bosch had introduced holy figures (and their accompanying devilries) into contemporary interiors and panoramic landscapes to a greater extent than anyone before him. Obviously, the painter deserved the scholars' attention, but practically nothing was known about this \"enigma\" of the Flemish school. Spade work had to be done to find even the dates of his life.\n\nHistorians such as Jan Mosmans sorted through the aged registers of his native's 's-Hertogenbosch, a Dutch town near the German border, but the results were disappointing. The date of Bosch's death was discovered in a registry of names and armorial bearings \u2013 listed as 1516. His birth date was not found, but because his portrait, which was discovered in the Arras Codex, showed a man of about sixty, his birth was assumed to have been around 1450. There are a few references to Bosch between these dates in the archives of the Brotherhood of Our Lady, in 's-Hertogenbosch. Several items referred to his having been paid various sums for works commissioned of him. For instance, he received twenty stuivers for a stained glass window pattern made \"on a couple of old bed sheets\", the window having been executed by the glassmaker, Willem Lombard. There were notations of larger sums such as of five rhenish guilder, paid for an altar. Bosch must have been active as a lay member of this organisation; in fact, he must have participated in the food preparation for the meetings, because at one time he was paid for twenty-four pounds of beef \u2013 \"at one Phillips penny a pound\", four ounces of ginger, two ounces of pepper, one-half ounce of saffron, and for the value of a measure of wine.\n\nNone of this was very informative about essential details of Bosch's life, save that, since he was referred to once as \"illustrious painter\", he was obviously held in repute as an artist by his fellows. There is no reason to think, from these references at least, that his friends considered Bosch either a wizard or a madman. As to his ancestry, since Bosch's name often bore the suffix van Aken, it was believed that his forebears were from Aachen, just over the Dutch-German border.\n\nFive van Akens were mentioned in the town records before the time of Hieronymus. One, a teacher named Jan van Aken, was noted in the archives of the city of 's-Hertogenbosch's Cathedral of Saint John, in references spanning several years (1423-1434). The historians believed that this was the grandfather of Hieronymus and probably the artist of the fresco of the cathedral \u2013 considered to be one of the artist's prime influences.\n\nIn 1464, Laurent van Aken, possibly the father of Hieronymus, was referred to as a citizen of 's-Hertogenbosch. This was the extent of the factual data referring to the artist. The historians were forced to turn back to the evidence of the paintings themselves, but none of them were dated, nor mentioned in contemporary writing. Small wonder then, that this produced confusing results in the historical evaluations of the works.\n\nBy approaching their studies with preconceived ideas, the scholars made what now seem like obvious mistakes. For instance, Louis Demonts, in 1919, sketched an evolution of the paintings from the premise that Bosch had evolved in his subject matter from the traditional theological point of view to a personal moral judgement. This led Demonts to date as late works, _The Cure of Foll_ , _The Conjurer_ , and _The Ship of Fools_ \u2013 later established on stylistic grounds to be from Bosch's youth. This same system caused him to date as an early work the Prado _Epiphany_ , seen later by such historians as De Tolnay and Combe as a surpassing synthesis of the artist's lifetime achievements.\n\nNot until Charles de Tolnay's definitive treatise, written in 1937, was a satisfactory chronology even established, or the works by Bosch's own hand separated from those of his disciples or copyists. De Tolnay bore directly on the technical evidence of the paintings. He noted that the beginner is betrayed by archaism \u2013 stiff figures, long-waisted and with awkward gestures, having no true existence in space nor relationship with one another and the background, and with few and arbitrary folds in their clothing. By observing such characteristics in some Boschian works, he was able to trace a convincing development from the obviously youthful to those of undoubted antithesis in style and conception. De Tolnay successfully demonstrated that Bosch developed consistently into a great landscape painter and a superb colourist.\n\nThe Epiphany or The Adoration of the Magi  \n(detail, right panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Epiphany or The Adoration of the Magi  \n(detail, central panel), c. 1510.\n\nOil on panel, 138 x 138 cm.\n\nMuseo Nacional del Prado, Madrid.\nAlthough he never achieved the suavity of an Italian High Renaissance master, in later works he even created a _sfumata_ effect which unified figures and background into a harmonious entirety. De Tolnay's work in this direction was so convincing that subsequent writers accepted his classifications as almost incontrovertible.\n\nThere have been exhaustive attempts to clarify the artist's subject matter, as well. In De Tolnay's words: \"The oldest writers, Lampsonius and Carel van Mander, attached themselves to his most evident side, to the subject; their conception of Bosch, inventor of fantastic pieces of devilry and of infernal scenes, which prevails still today [1937] in the large public, prevailed until the last quarter of the 19th century in historians.\" Then those historians who saw in the painter a precursor to realism, swung completely in the other direction. They studied his works according to exterior influences such as literature, the artistic tradition of the North, historical events, and the medieval interpretation of the Bible.\n\nNone of these sources produced any conclusive results on the meaning of Bosch's cryptic imagery. Again in this realm, one of the finest studies was that of De Tolnay. He went far in establishing current influences that would account for much Boschian iconography. Most importantly, he introduced a knowledge of Freudian psychology, revealing Bosch's remarkable presentiment of this science. Jacques Combe followed De Tolnay's lead in his treatise translated into English from the French in 1946, and continuously acknowledged his indebtedness to the prior monographer, but his study was no mere imitation. He suggested many sources of symbolism overlooked by de Tolnay, such as alchemy and the tarot game. He made a strong case for association between Bosch's ideology and that of the 14th-century Netherlandish mystic, Jan van Ruysbroek.\n\nWith such respectable scholastic attention, Bosch had finally come into his own in the mid-20th century as a significant artist. His works were seen not merely as an influence on Bruegel, but as extremely interesting in themselves. They were a deviating but appropriate link within the \"Flemish tradition\" in painting, with its curiously combined naturalism and symbolism. The work of De Tolnay, together with the increasing interest in Surrealism, had inspired popular interest in Bosch as a painter of the imaginary. It followed that several articles on Bosch were published in the most popular American periodicals, as well as in magazines of art. The popular articles presented Bosch as an interesting, almost freakish fantasist of the past and a precursor to Surrealism in his \"queerness\".\n\nIn most of the books written in English, as well as translated into English, the more scholarly authors continued to search for the exact sources of Bosch's symbolism in outside material. Their implication was that Bosch's symbols, however enigmatic, illustrated images already formed in literature or tradition, and that with enough study these sources would eventually be brought to light and his imagery made comprehensible.\n\nThe Epiphany or The Adoration of the Magi  \n(exterior: The Mass of Saint Gregory), c. 1510.\n\nOil on panel.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Epiphany or The Adoration of the Magi  \n(detail of exterior: The Mass of Saint Gregory), c. 1510.\n\nOil on panel.\n\nMuseo Nacional del Prado, Madrid.\n\nTriptych of the Martyrdom of St Liberata, 1500-1504.\n\nOil on panel, 104 x 119 cm.\n\nPalazzo Ducale, Venice.\n\n### Fr\u00e4nger's Thesis  \n(Epiphanies and Absurdities)\n\nWilhelm Fr\u00e4nger began his study of Hieronymus Bosch and his work by deploring the \"vulgar misunderstanding\" to which the master had been subjected by having his work passed off as mere mummery. Fr\u00e4nger insisted that, with Bosch, symbols \"entail a perfect simultaneity of vision and thought\" and must be treated as such. The writer considered all other approaches as \"fragmentary\", thus presented his study as a total view.\n\nIn order to understand why the painter would create a mute symbolism, the art historian sorted through the whole body of paintings, separating those of enigmatic content from those that contain little or none. Only if the \"freakish riddles\" on which Bosch's reputation was founded occurred in all of the paintings could they be called \"the irresponsible phantasmagoria of an ecstatic\". Fr\u00e4nger found that the deviant content existed only in a clearly defined group of altarpieces \u2013 the three large triptychs of the _Garden of Earthly Delights_ , the Lisbon _Temptation of Saint Anthony_ , and the _Hay Wagon_. In contrast, there was only a small amount of this symbolism in such paintings as the _Epiphany_ triptych in the Prado and the Venice _Martyrdom of Saint Julia_. The remaining paintings, including those of the _Passion_ and _Adoration of the Magi_ themes, had little or none. He concluded, therefore, that an arbitrary distinction could be made between two main groups \u2013 the generally traditional, obviously created for the church, and the non-traditional, disparate ones.\n\nFr\u00e4nger concentrated on the second group, proposing that they could not have been made for a church congregation since they contained anti-clerical polemic implied by monks and nuns depicted behaving in a scandalous manner. Nor could these altarpieces have been made for pagan worship, since they also attacked pagan \"priests\" and their ritualistic excesses. Altarpieces, however, pointed to some kind of devotional patronage. The principal targets of their attacks point to a group of paintings outside the domain of the church, at once inveighing against ecclesiastical offences and at the same time fighting the abundant mysterious cults of the period. The only kind of society that could possibly answer the problem, according to Fr\u00e4nger, would be a militant heretical sect. Setting up an ideal contrary to the teachings of the Church, such a sect would be forced to fight the all-powerful tradition, but on the other hand, would find pagan abominations equally abhorrent. If Bosch should paint a devotional altarpiece for a society of this kind, he would mirror their \"dual warfare, with all its polar tension\" and his \"eccentricities\" would be explained.\n\nAccording to the scholar, all previous interpretations of Bosch erred that did not approach his symbolism with this frame of reference. Because Bosch was not intelligible to them, most commentators assumed that he had not intended to communicate \u2013 and that the creatures he let loose in these paintings were mere \"phantoms of hell\". This thinking placed an emphasis on the hell scenes that Bosch might not have intended. True, there are scenes which are set in the most horrific of all hells, but they are always balanced on the other side of the altarpiece by \"an impeccable anchorite, or by Mount Ararat, or by the Garden of Eden\". In other words, if Bosch gave equal weight to the opposed \"ideal scenes\", could we not assume that he intended to emphasise these scenes by their very contrast with hell? This added further weight to Fr\u00e4nger's theory of the heretical sect, because Bosch's more positive scenes would reflect the idealism of such a society.\n\nThe author believed that one of the most widely misinterpreted of Bosch's paintings was _The Garden of Earthly Delights_ triptych. In fact, he concentrated the remainder of his study of Bosch's ideation upon a new interpretation of this painting. The reason for the confusion, he thought, was that it had for centuries been thoughtlessly associated with another of more obvious message, the _Hay Wagon_. Both triptychs have flanking panels of a Garden of Eden scene on the left and a hell scene on the right \u2013 thus, the message was assumed to be much the same[1]. Fr\u00e4nger saw many differences, however, that would belie the dual association. The Eden panel of the Hay Wagon contains sequences of the _Fall of the Rebel Angels_ , the Creation of Eve, the Temptation, and the _Expulsion from the Garden_ which are placed on a vertical axis in the same garden landscape. The grouping is traditional, but there is some Boschian originality in their presentation, as the \"rebel angels\" are presented as insects falling in swarms from heaven and there is a peculiar rock formation at the site of the Creation of Eve, presaging the even stranger ones to be seen in The Garden of Earthly Delights. Nevertheless, little to be found in this panel is incomprehensible. The central painting, now supposed to be an illustration of the Flemish proverb: \"the world is a haystack; everyone takes what he can grab thereof\", is dominated by a gigantic hay wagon which, according to Jacques Combe:\n\nEvok[es] at the same time the late Gothic motive of the procession of pageant, and the Renaissance Triumph... drawn by semi-human, semi-animal monsters and headed straight for hell, followed by a cavalcade of ecclesiastical and lay dignitaries. From all sides of the wagon men scramble over one another to pull hay from the giant stack. The only heed they take of their fellows is to thrust them out of their way or to raise hands against them. One sticks a knife into the throat of the unfortunate competitor whom he has pinned to the ground.\n\nMany among the greedy mob wear ecclesiastical garb, indicating Bosch's attitude that the holy as well as profane are involved in this scavenging. A fat monk sits in a large chair and lazily sips a drink while several nuns do service for him, packing bundles of hay into the bag at his feet. One of his nuns turns to the lure of sexual enticement symbolised by the fool playing a bagpipe, to whom she offers a handful of hay in hopes of winning his favours.\n\nChrist Mocked (The Crowning with Thorns), c. 1490-1500.\n\nOil on oak, 73.5 x 59.1 cm.\n\nThe National Gallery, London.\n\nChrist Carrying the Cross, c. 1480-1490.\n\nOil on wood, 57 x 32 cm.\n\nKunsthistorisches Museum, Vienna.\nThe clergy seem to be in the majority among the wicked brought under Bosch's scathing attack in this altarpiece, but there are others, too, of the devil's earthly disciples. There is a quack toothpuller whose hay-filled pocket refers to the fees that have \"lined his pockets\", garnered as they are from the gullible populace. There are gypsies, women and children, figures always suspect as servitors of Satan and a blind beggar led by a child to his share of the spoils. Two idling lovers embrace on top of the hay pile while two others enjoy the delights of music, played for their pleasure by a troubadour-demon who fingers his own pipe-shaped snout. All are caught in a snare laid by Satan and ignore the angel who implores them to look to heaven for their souls' sake. Above them, an image of Christ in the clouds shows to a heedless mankind the signs of His martyrdom on earth \u2013 His stigmata. The scene is obviously an allegory of man's sins on earth which foretell his inevitable progress into hell. The monsters that draw him there administer tortures designed for each of his sins.\n\nThe hell scene of this triptych is as unusual in presentation as the Eden scene is traditional, demonstrating all of the master's skills of painting and designing unusual torments. A flaming sky lights up the windows of a furnace in which can be seen figures engaged in demonic activity. Arms and legs protrude from a lake coloured by the glow and heat of the sky. In the foreground are the sinners just received into hell. Stripped bare, they are submitting passively to their individual punishments. Unusual in this scene is the construction work underway in the middle of the setting as if it were necessary to construct more torture chambers for the expected influx of sinners already crowding in from the fracas at the hay-wagon. This altarpiece contains some symbolism of an obscure nature, but there is not the overwhelming amount to be found in the _Earthly Delights_ and the large _Saint Anthony._ Even Fr\u00e4nger is in agreement with the other writers that this triptych depicts the stigma of the _original sin_ under which man is born and by which his course is set through life to its d\u00e9nouement in hell. The cryptic symbolism does not obscure the main theme, fairly direct in presentation, so there is naturally more unanimity of opinion concerning the message of this altarpiece.\n\nSuch is not the case with _The Garden of Earthly Delights_. The _Eden_ panel of this painting is not a setting for several traditional events concerning Adam and Eve. They are seen only once, Adam sitting and Eve kneeling on either side of the Lord. The group is placed on a lawn between two pools, from which crawl hybrid animals engaged in bizarre activities. A fountain in the centre or the middle pool and a rock structure in the upper background appear to have become organic and grown to their odd conformations. There is an entire apple grove behind the _Creator_ and _Created_ , but no tree seems to be as distinguished as the Tree of Life \u2013 this must be the palm-like tree to the side of Adam. Its leaves are shaped like coins, their weight causing the tendrils to sway in awkward heaviness. The forbidden fruit must be the enlarged berries which droop from the vine curling about the trunk.\n\nThe central scene seems to be set in the same kind of landscape, but its plants have rigidified like minerals and its mineral structures grown like plants.\n\nThis land is overwhelmed with human and animal beings, fairly normal in their appearances although abnormal in their relative sizes. The humans are well-proportioned nudes that diminish in size according to the laws of perspective, but the animals, although appearing normal when seen separately, are not always relative to the humans in size \u2013 sometimes entirely dwarfing them. Nor are their colours or activities proper to their biological requirements. The activities of the humans, as they eat upon oversized fruits, or engage in erotic play, have caused writers from the time of Father Jos\u00e9 de Seg\u00fcenza to believe this to be a testament to the deadly sin of lust, or \"luxuria\". In the words of art-historian Lotte Brand Phillip:\n\nYet, in the belief of the Middle Ages, all sins and vices are interwoven and one derives from the other. Therefore, it is not only 'luxuria' which is depicted here but all the sins brought forth by this vice and all the vices which cause 'luxuria'. This is quite evident in the right wing which, in a horrifying inferno, presents the punishment of all kinds of sins.\n\nThe hell scene of this altarpiece is Bosch's diabolic masterpiece. Throughout its far reaches, sinners are being tortured in a night-lit landscape that is so marvellously painted as to be a work of genius. This scene is dominated by a wondrous egg-tree monster that floats upon the inky water on two boat feet from which tree-trunk legs are sprouting. The body of this creature is a cracked-open egg-shell that serves as a tavern interior that its occupants must have reached by a ladder from the water below. Attached to the top of the egg is a hat on which a procession of nudes paired with demons parades around a large pink bagpipe, symbolic of lust. Around and below the egg-man are areas of hell where sinners of various professions (knights, monks, musicians, gamblers, etc.) undergo excruciatingly appropriate tortures in a visualisation unequalled in art history for its inventiveness.\n\nComparing these two triptychs, Fr\u00e4nger saw at least five arguments which would oppose the association of the two in the _Hay Wagon_ 's obvious theme: creation \u2013 sinful-life \u2013 hell. \"First of all, there is the basic difference in breaks and coherences\", said the historian. In the _Hay Wagon_ , a break in continuity occurs between the _Garden of Eden_ panel and the middle panel, and there is decided coherence between the centre scene and _Hell_. This can be seen in the fact that the hybrid creatures pulling the wagon into hell from the middle panel are obviously minions of Satan; their parade continues with no break into hell. In the _Earthly Delights_ , no such separation exists between the Garden of Eden panel and the central one, but a continuity of landscape from one into the other. This difference in continuities in the two works indicated to Fr\u00e4nger that Bosch intended a different idea in the two, and it should not be assumed that just because the scene following the Garden in the _Hay Wagon_ represents wickedness, the corresponding scene of _Earthly_ _Delights_ automatically does the same.\n\nAs his second argument, Fr\u00e4nger pointed out that the nude creatures disporting themselves in the central panel of _Earthly Delights_ are not doing so in what has appeared to all others besides him to be wanton display, instead they \"are peacefully frolicking about the tranquil garden in vegetative innocence, at one with animals and plants, and the sexuality that inspires them appears to be pure joy, pure bliss\". Thirdly, Fr\u00e4nger argued that although the hell scene is divided into areas of torture for sinners among the knights, the religious orders, the musicians, and the gamblers, \"not a single adept of carnal love\" is being tortured.\n\nChrist Carrying the Cross.\n\nOil on panel, 150 x 94 cm.\n\nPalacio Real, Madrid.\n\nChrist Carrying the Cross (detail).\n\nOil on panel, 150 x 94 cm.\n\nPalacio Real, Madrid.\nIf Bosch were holding up the sin of lust to reproach, asked Fr\u00e4nger, would he not in a logical sequence conclude with its punishment in hell? In the case of the _Hay Wagon_ , the greedy are not only to be found in their state of eternal damnation in hell \u2013 but also, the demons of hell swarm into the world of the central panel to assist in drawing the wagon all the faster in its fateful course. Here again, the writer refuted one of the parallelisms that had been too closely drawn. The author cited the situation in the Paradise panels for his fourth argument: the Eden of the _Earthly Delights_ contains no sign of conflict as in that of the other triptych. Not only is there evidence of the primal conflict in the _Hay Wagon_ 's rout of the wicked angels from heaven (transformed as they are by Bosch's inventiveness into those marvellous insect forms but also, testimony to original sin in the fact that Adam and Eve are being expelled by threat of the \"fiery sword\". In the _Earthly Delights_ , on the other hand, this scholar observed that Adam, Eve, and the Lord (in the form of the Son \u2013 a medieval transposition) are beheld in a moment of cosmic unity.\n\nFr\u00e4nger suggested later that the three are forming \"a closed circuit, a complex of magical energy\" by the physical contact between them. Adam's feet touch the Creator's, who in turn holds the hand of Eve. That is \"himself (Christ) the image of God, he is the transmitter of this image to the two beings he has created, standing between them as their coeval\". This fact indicated to Fr\u00e4nger a positive and good estate, with no presentiment of the ultimate \"fall\".\n\nContinuing his line of defence, the historian suggested that since the children of Adam and Eve were born after the expulsion, it followed that the original pair were the only people in \"Paradise\". Thus, the humans abounding there must represent their progeny, had Adam and Eve not fallen from grace.\n\nSince the dogma of original sin and the Expulsion cannot be ignored if there were children, however, here is \"a millennial condition that would arise if, after expiation of original sin, humanity were permitted to return to Paradise and to a state of tranquil harmony embracing all creation\".\n\nSuch a promise, Fr\u00e4nger believed, inhered in the symbolism contained on the outside of the triptych, seen when the panels are closed. This was his final argument, and to him, the most compelling. Almost covering the surface of both wings is the crystal sphere, containing a flat disk which cuts through its centre. (This idea of the aspect of the universe Bosch had taken from an ancient cosmology, said Fr\u00e4nger, not in currency in his time.\n\nEven if this were too early for Columbus' discovery to have penetrated iconography, the earth had been depicted as being round for some time, generally as a small sphere concentrically set inside several larger spheres on which the fixed stars and the zodiacal signs were distributed.)\n\nThis must be, Fr\u00e4nger believed, the third day of Creation, that \"'fruitful moment' in the six days of Creation, the very moment when the first rain falls upon the yet barren earth, out of which the first trees and bushes are about to sprout\". Fr\u00e4nger believed that the awe-inspiring message of this Creation is made clear in the quotations from the Psalms, XXXIII and CXLVIII, placed along the upper edges of the panels and also, by God's image, which appears above and to the left of the sphere: \"For he spake and it was done,\" and \"He commanded and it stood fast.\" The Lord, seated as on a heavenly throne, holds on His lap a book, symbolic of the \"Word\" which is becoming incarnate below. Fr\u00e4nger saw the throne as a prefiguration of the great world globe, for it seems to repeat the shape of the large crystal globe by being a miniature crystal enclosure. In reverse, God's throne is repeated and realised in the larger cosmological symbol, which finds its fulfilment when the wings of the altarpiece are thrown open. \"The great ball divides, and its ordered realms represented by the zones of heaven, earth, sea, and the underworld, are repeated in the corresponding zones of the Garden of Eden, the Millennial Paradise, and hell.\"\n\nTherefore, the scholar's conclusion must be that the central section of the interior is an earthly paradise \u2013 the fulfilment of the _Word of God_ with implicit positive, rather than negative intent. \"After that solemn prelude in the crystal globe, the history of the world could not possibly gravitate towards a stew of defilement and hence into Hell.\" Bosch has in this painting, Fr\u00e4nger said, sought to deny the dualism that was seen as existing between things of the flesh and things of God. In other words, he has tried to raise fleshly delights out of the realm of shame into which they had been deposed in traditional theology.\n\nFr\u00e4nger answered his own question as to the specific nature of a cult for which this altarpiece could have been painted by reasoning that, since Adam and Eve figure strongly in other prominent Netherlandish paintings \u2013 by Jan van Eyck, Hugo van der Goes, and Hans Memling \u2013 it must have been an Adamite cult. This cult is known to have advocated the practice on earth of sinless perfection \u2013 a practice that could incorporate man's physical desires by outlining an ideal physical relationship between the sexes \u2013 so attained, man could then return to the state of bliss which our progenitors had lost by their sin.\n\nTo Fr\u00e4nger, this explanation of a heretical cult resolved all conflicting elements. Even \"the high pitched boundless frenzy\" of the _Earthly Delights_ hell scene would be explicable. Because a secret society must risk possible discovery and persecution, it would tend toward self-justification and would direct the terrors of hell toward others, not its own members. Because the members must flout established tradition, deep-rooted even in themselves, they would defend their society's precepts all the more intensely. Obversely, they would deny its privileges to the uninitiated and would place the outsiders, not themselves, in hell.\n\nFor incontrovertible evidence of his theory, Fr\u00e4nger offered the record of a trial in the Episcopal court at Cambrai in 1411, which charged the Carmelite friar, Willem van Hildernissen, with heresy. This man was one of the leaders of the Homines Intelligentiae of Brussels, a radical branch of a religious movement active in the territory from the Rhineland to the Netherlands. Because of inferences, which Fr\u00e4nger made from certain statements in the trial record, he assured himself that this was the group for which Bosch produced his altarpiece. Its members called themselves \"Brothers and Sisters of the Free (or High) Spirit\" in the belief that they were the incarnation of the Holy Ghost and through its power exalted to a state of spirituality that was immune from sin even in the flesh, with its subjection to lusts, so that on earth they lived in a state of paradisiacal innocence\".\n\nChrist Carrying the Cross (detail).\n\nOil on panel, 150 x 94 cm.\n\nPalacio Real, Madrid.\n\nChild with a Walking Frame  \n(reverse of Christ Carrying the Cross),  \nc. 1480-1490.\n\nOil on panel, diameter: 28 cm.\n\nKunsthistorisches Museum, Vienna.\n\nChrist Carrying the Cross,  \nc. 1510-1516.\n\nOil on panel, 76.7 x 83.5 cm.\n\nMuseum voor Schone Kunsten, Ghent.\n\nEcce Homo, 1475-1480.\n\nTempera and oil on oak, 71 x 61 cm.\n\nSt\u00e4del Museum, Frankfurt am Main.\nHe suggested first, however, that the group's constitution rested upon a tenet of equality of the sexes that is implicit in its very name. Since \"the word _homines_ means 'men' in the general sense of 'human beings', which amounts to claiming for man a dignity that \u2013 since it is based on the fact that Adam was made in the image of God \u2013 admits no modification on the score of birth or class, rank or possessions... then the distinction between man and woman was also sublimated in the loftier concept of homo.\"\n\nThis must mean, Fr\u00e4nger thought, that the female enjoyed equal status with man in this society, and if she were thus exalted in position over the inferiority accorded her by the church, she must have lost her denigration as the \"sinful vessel\". As to what this implies of the beliefs or activities of these men and women within the society, the writer answered by turning to the Cambrai record. One of the points of indictment stated:\n\nLikewise they created among themselves a peculiar mode of discourse which they call the act of sexual union 'the joy of Paradise' or, by another name, 'the way to the heights' ( _acclivitas_ ). And in this manner they speak of such a lustful act to others, who do not understand it, in a favourable sense.\n\nFrom this statement, Fr\u00e4nger surmised that the cult's second principle was that its members lived together according to high moral values. To these brethren, \"the strictest sanctification of love\" was the way by which they could reach human perfection; this estate they saw embodied in Adam and Eve, who were themselves made in the image of God's divine perfection. The converse became the second principle; that because man is made in the image of God, he can only endure in this highest of all estates by the practice of a sublimated act of love. As to the nature of this act, Fr\u00e4nger saw in the court record two possible answers. To indicate one of them, he quoted the following portion: \"Secondly, that the natural sexual act could take place in such a manner that it was equal in value to a prayer in the sight of God.\"\n\nIn other words, it could mean the natural act so purified in the mind \"that it should no longer be felt as a humiliating animal act, but as the expression of an exalting, divine, creative principle\". The scholar preferred to seek the clue in a statement pertaining to another leader of the Brethren (a man who was being tried posthumously), that \"he exercises a special mode of sexual intercourse, yet not contrary to Nature, of which he says it was that of Adam in Paradise\". If this \"special mode\" refers to an act distinct from the animal one, \"yet not contrary to nature\", then it is clearly sinless, and could be practiced in perfect innocence \u2013 providing pure pleasure not befouled by any sense of shame. In this statement, thought the writer, lay the real meaning of the _Earthly Delights_ centre section. It was an illustration of this \"special mode\" of sexual play, the \"immortal expression to this Adamite eroticism\", that was revealed in the painting, not directly, but secreted in a mesh of symbolic communications. Having brought us to this conclusion, Fr\u00e4nger proceeded to his prime purpose \u2013 that of translating \"this Free Spirit _ars amandi_ out of the secret code of symbolism into generally comprehensible language\".\n\nThis is enough of Fr\u00e4nger's argument to illustrate the nature of his interpretation and his characteristic mode of thought \u2013 a thoroughly rational, in fact, brilliantly logical analysis which on the surface was exceedingly convincing. Upon more careful scrutiny, however, it was the scholar's logic, not Bosch's, that he revealed. What Fr\u00e4nger consistently did here was to follow a system of reasoning that set up a hypothesis as an arbitrary starting point and then, through misleading inferences, arrived at subsequent hypotheses and developed the conclusions implicit in them, _ad infinitum_. A construction of thought was created, no portion of which could be removed without damage to the whole, nor explained without reference to the whole. But the entire structure rested upon the original hypothesis, a very shaky foundation indeed.\n\nThis was Fr\u00e4nger's hypothesis: that the paintings containing the major part of Bosch's enigmatic symbolism, being in the form of altarpieces, must have been made for a devotional purpose. They contain anticlerical and anti-pagan invective that could have been made neither for the Church nor for a pagan group. An example of the anticlerical has already been shown in the fat monk being served by his nuns in the _Hay Wagon_. Another is in the pig wearing a nun's wimple and veil soliciting the affections of a man in the _Hell_ scene of the _Garden_. Since it was not the practice of a late medieval artist to paint merely for his own satisfaction, nor is it conceivable that private commissioners would have wanted such odd altarpieces for their own chapels, then there must have been a group outside the Church, operating between its severe discipline and pagan anarchy, but fighting both. These paintings must have been made for a heretical sect, therefore, which was forced to hide its ideas in secret symbols whose explanations would clarify Bosch's enigmatic figures. To Fr\u00e4nger, this meant without question the Adamite cult.\n\nAre the points of the hypothesis defensible? The fact of the traditional altarpiece form strongly implied to the historian a devotional purpose \u2013 therefore, he had to seek the type of group, which would use Bosch's altarpieces for such a purpose. It is not absolutely necessary, however, to think of these paintings as having a devotional purpose. This was not a time of strict adherence to tradition. Northern Europe at the end of the 15th and the beginning of the 16th centuries was in a period of great transition.\n\nAlready, the influence of the Renaissance from the South had been felt, entailing a discard of many old forms and ways. There was a growing secularisation, resulting in a patronage for artists widened far beyond the extent of the Church. It is conceivable that the altarpiece form could have been used for a non-devotional painting commissioned by a private patron \u2013 merely because it allowed for intriguing complexity. But why could these paintings not have had a devotional purpose for this private patron \u2013 or for the Church, for that matter? It is the symbols of the pagan cults, which Fr\u00e4nger called signs of \"swampy procreation and ritual promiscuity\" that he did not believe could have been shown on church altars. Perhaps they could not be shown on _our_ church altars, but in that time of less tender sensibility, evil practices of all kinds were denounced in descriptive detail from the very pulpits. In fact, such practices are denounced even today from fundamentalist Protestant pulpits.\n\nEcce Homo, c. 1450-1516.\n\nOil and gold on panel, 52.1 x 54 cm.\n\nPhiladelphia Museum of Art, Philadelphia.\n\nCrucifixion with a Donor, c. 1490.\n\nOil on panel, 74.7 x 61 cm.\n\nMus\u00e9es royaux des Beaux-Arts, Brussels.\nFr\u00e4nger pointed out that Bosch's altarpieces always present an ideal content to balance the evil. Perhaps this would have justified their use as devotional altar paintings. If the worshipper would reflect upon the ideal scene (for instance, of a saint who remained faithful even in the face of all the forces of evil that Satan could bring to bear on him), he would be prepared to renounce his evil ways. As an added bit of goad, his possible future in hell was cast \u2013 in case he paid no heed to the admonishment.\n\nFr\u00e4nger previously placed Bosch's paintings on the theme of passion and those on the _Adoration of the Child_ into a group distinct from the controversial altarpieces \u2013 calling those paintings \"straightforward, intelligible, and traditional\". It is well known that these paintings also exhibit many instances of evil workings in the world. Especially in his _Passions_ , the painter gave the most hideous aspect to the ordinary humanity that crucified Christ. This group cannot be completely divorced from the large triptychs Fr\u00e4nger discussed. Perhaps the interpretation given the large triptychs by Father Jos\u00e9 de Seg\u00fcenza should be reconsidered in spite of Fr\u00e4nger's elaborate argument to the contrary. Seg\u00fcenza was the first interpreter of the _Earthly Delights_ painting according to an association with the _Hay Wagon_. He saw both of these triptychs as containing the same idea incipient in Bosch's earlier works \u2013 that wicked blind humanity would not heed the lessons of the Christian faith, but would indulge in a sinful life in a world that must surely end in hell.\n\nOne of the contentions upon which Fr\u00e4nger based his divergent interpretation of the _Earthly Delights_ was that Bosch's early and late works could not be closely associated. Since the earlier ones were more straightforward than the later ones, there must be a special meaning for the covert symbolism of the latter. The scholar did not believe, in addition, that the same intentions should inevitably be read into any two of the paintings, for instance \u2013 the late _Earthly Delights_ and the earlier _Hay Wagon_. If one could find reasons to belie the dual association of these paintings, it could be said that the message of the first painting was positive rather than negative. There evidently no clean break between the contents and message of his early and late works \u2013 nor between that of the two works in question \u2013 rather an elaboration of the same message; therefore, this position of the author's cannot be considered tenable.\n\nIf the large triptych paintings are seen as carrying on the same ideas contained in Bosch's earlier works, it would seem that if any of the paintings was suitable to place before a worshipping body in the Church, these could have had their place at the altar, too. It would not, then, be necessary to look for a patron outside the church to justify their existence. But Fr\u00e4nger discarded the possibility of there being a private individual who would commission the work for his own home chapel too quickly. Even if such an individual did not have so serious or specific a purpose for the altarpiece as an addendum to a private chapel, he might have commissioned a Bosch painting merely because it was fascinating in itself.\n\nThe artist's popular appeal is shown by the fact that his manner and subject treatments were adopted so quickly by artists such as Huys and Bruegel. It may be that Bosch painted for a delighted audience, only too happy to keep him in commissions. We know from records quoted above that he was held in repute by his fellow townsmen. We know, too, that both the Emperor Charles V and one of his courtiers, Felipe de Guevara, had acquired several of Bosch's paintings within a remarkably short time after the painter's death. The fact that Charles' son Philip confiscated one altarpiece from a rebellious Netherlandish Burgher makes it seem more likely that some paintings were owned privately rather than being part of the sacred equipment of churches; but it suggests as even less of a likelihood that the paintings were the hidden and guarded property of heretical sects. If such were the case, it is improbable that the paintings would have been in free circulation at such an early time after the artist's death.\n\nFr\u00e4nger's system of reasoning is so tightly constructed, with so little possibility of error that he seems to assume that mass intelligence would inevitably reach the same conclusions \u2013 once he had cleared away a few obstacles and pointed the direction, that is. One example will suffice to show to what fantastic excesses such thinking can and does lead in this interpretation. Fr\u00e4nger had previously demonstrated in his analysis of the central panel of _Earthly Delights_ his belief that this fabulous display of erotic activity was in celebration of an actual marriage event. Therefore, he assumed that this also became the occasion for a pictorial revelation of the \"society's\" mysteries \u2013 including all of the levels of knowledge members could attain by instruction and by which they could finally reach full association in the group. Since he thought the painting such a \"unique pictorial creation, in which the whole universe has been assembled to sing praises such as no king and queen ever heard on their wedding day\", it must be a truly \"god-like couple\" who are being married; Fr\u00e4nger went on to find them in the lower right corner of the panel, half-hidden in a cave. The man is the only clothed figure among the abounding nude ones, he said, and proposed further differentiations as well.\n\nA man who exalts himself by such self-awareness as this one exhibits, and who is further being exalted by such a wedding celebration, could be one of only two people to Fr\u00e4nger \u2013 either the painter Bosch, or the man who inspired the triptych. Since this is not a portrait of Bosch, it must be according to the writer:\n\nThe face of the man who commissioned such an extraordinary work of art and inspired its intellectual conception, [and] we can go even further and make the conjecture that this portrayal of the bridegroom is also that of the Grand Master of the Free Spirit, who meets us with a piercing, scrutinising gaze on the threshold of his paradisiacal world.\n\nHaving established the leader's identity and personality from this \"evidence\" of the painting, Fr\u00e4nger asserted other instances in which his invention revealed himself. The author saw his face in that of the egg-tree monster placed in the centre of hell as if to demonstrate allegorically a basic doctrine of the cult \u2013 that one must make a public confession of sin before being able to return to a \"state of purity\". Because a crow can be seen near the man's \"portrait\", Fr\u00e4nger believed this to be his symbol; therefore, wherever there was a crow (as at Adam's feet in the _Garden of Eden_ ), there was the \"Grand Master\" participating in a cosmic event important to the whole revelation.\n\nThe Adoration of the Magi,  \nc. 1470-1475.\n\nOil and gold on wood, 71.1 x 56.5 cm.\n\nThe Metropolitan Museum of Art, New York.\n\nThe Adoration of the Magi (detail),  \nc. 1470-1475.\n\nOil and gold on wood, 71.1 x 56.5 cm.\n\nThe Metropolitan Museum of Art, New York\nFr\u00e4nger understood the cave of the bride and bridegroom not only to symbolise the consummation of their marriage, soon to take place, but the tie-in with Neo-Pythagorean philosophy, since caves had figured strongly in the history of Pythagorianism. The scholar's purpose in making these Italian connections was obviously to justify finding such an esoteric community in the North as is known to have existed in Renaissance Italy. (Fr\u00e4nger had stated earlier, when first introducing the idea of the Adamite cult, that it was of particular importance in the early Renaissance period when: \"Ideas of Platonic, Augustinian, Neo-Pythagorean, and Gnostic origin fused to form an attitude that saw Original man as the archetype of spiritual renewal and hence of a pure, free state of human life.\")\n\nTo explain his cult's mysteries according to those of the Italian societies, and their appearance in this Netherlandish cult, Fr\u00e4nger had to introduce the \"Grand Master\" and show cause for this man's having brought these ideas back from his own schooling days in Italy \u2013 thus the author's assertion that the man has the look of an Italian intellectual and the further strengthening point that the woman behind him (his \"bride\") has Italian colouring and features. Having so thoroughly convinced himself of his assertions, Fr\u00e4nger introduced the elusive \"Grand Master\" into \"early Dutch social and art history [as] a powerful spiritual personality, hitherto completely unknown, one who is worthy to rank with those three great men of the same country and the same century, Erasmus Desiderius of Rotterdam, Johannes Secundus, and Johannes Baptist van Helmont\". Thus, Fr\u00e4nger not only endowed his invention with physical aspect, personality, bride, and philosophy, but he bestowed greatness on him. In the process, he destroyed Bosch not only as a personality but also as an artist.\n\nHe did this first by declaring that the altarpiece's symbolism was a \"system of sexual- ethical teachings, in which the pictorial motifs were didactic symbols, clear reflections of Renaissance natural philosophy, and hence patterns of a modern intellectual kind, pointing towards the future\". The author's more devastating conclusion was that since there was a personality behind the painter's inventions, his \"pictorial ideas were not his own at all, but were laid down for him by a mentor of encyclopaedic erudition, with an exquisite sense of detail, yet capable of planning on a magnificent scale and imperturbably sure of his purpose\".\n\nIt was not merely subject matter dictation that Fr\u00e4nger believed Bosch had received, but that the \"mentor\" had even designed the colour and formal composition. He had obviously designed the colour because it bears symbolic relationship in every instance to the idea involved, but he strongly influenced the composition by allowing Bosch to break with his previous practice of placing dominant ideas along the axes of the panels, as in the two \"Paradise-panels\", and to disorganise the hell scene in a manner planned as a symbolic reverse of the very order of the other two \"worlds\".\n\nThe Adoration of the Magi (detail),  \nc. 1470-1475.\n\nOil and gold on wood, 71.1 x 56.5 cm.\n\nThe Metropolitan Museum of Art, New York.\n\nFr\u00e4nger seemed to believe that such systematic disorganisation grew out of a profound understanding of the destructive forces of society that could only belong to a man \"who had the ideal image of creative harmony steadily before his inner eye\". It indicated that \"the inspirer extended his influence even to the very realm of motifs and forms that have hitherto been regarded as Bosch's own private territory as 'faizeur de dyables'; and hence the painter's own created world seems on the point of vanishing\". The scholar went on to push Bosch over the edge with his next words:\n\n[If] we have to ascribe even the painter's most peculiar originality to a mentor, it seems as though his own share in the achievement must sink to that of a craftsman, no matter how highly qualified, merely executing somebody else's orders... it might have been more logical to devote all our energies to working out the personality of the real spiritual creator and \u2013 leaving Bosch on one side \u2013 to reinstate the actual origination of this profound work.\n\nAt this point, Fr\u00e4nger did admit that \"such a strictly logical process would reduce itself to absurdity, since it would be impossible to reconcile it with the visual data of the picture itself, which, in spite of the guidance to which the painter was subject, blossoms out in complete pictorial freedom\".\n\nThe Adoration of the Magi (detail),  \nc. 1470-1475.\n\nOil and gold on wood, 71.1 x 56.5 cm.\n\nThe Metropolitan Museum of Art, New York.\n\n**Hubert and Jan Van Eyck** ,  \nThe Mystic Lamb (exterior view), 1427-1432.\n\nOil on panel, 375 x 320 cm.\n\nSaint Bavo Cathedral, Ghent.\n\nThe Hay Wagon (triptych), 1525.\n\nOil on panel, 140 x 200 cm.\n\nMonastery of San Lorenzo, Escorial.\n\nHe had to assume that \"the mentor and his disciple were in such complete harmony that the gap between command and obedience, demand and fulfilment, was not merely bridged but filled by such a mutual understanding and sense of responsibility that the result could be a joyful cooperation in creating a 'work of true community', [achieved, of course, by] the unconditional submission that Bosch made to his mentor's directions [and that] won him the immeasurable reward of the Grand Master's trust, confidence, and encouragement\". The historian continued to give examples, cited from the \"evidence\" of the painting, of the type of esoteric instruction that the \"Grand Master\" gave to his \"pupil\". No matter how much Fr\u00e4nger might protest that all this did not reduce Bosch's stature to that of \"a craftsman, no matter how highly qualified, merely executing somebody else's orders\", that is exactly what happened. He went further yet, knocking the props from under Bosch's autonomous identity as a technician as well as a \"mind\". Even the painter's superb technical achievements in the _Earthly Delights_ , the historian said, resulted from the painter's association with the \"mentor\". This man apparently gave constant prompting to Bosch to observe nature so closely that he could give magical significance to every object, animate and inanimate, by making it radiate a very energy through its \"profoundly understood\" form.\n\nThe Hay Wagon (triptych), 1525.\n\nOil on panel, 140 x 200 cm.\n\nMonastery of San Lorenzo, Escorial.\nFrom \"all the ceaseless observations that Bosch had to carry out in preparing to paint this picture he gained an encyclopaedic knowledge of the 'realia' of life, as a result of which he was far ahead of all other painters of his time.\"\n\nWhat about Bosch's earlier achievements \u2013 are they now accountable without this spectral \"mentor\"? Of course not \u2013 except that Fr\u00e4nger rushes in to fill this loophole:\n\nIt was solely as a result of his co-operation with this deeply experienced and determined mentor that Hieronymus van Aken came to maturity as a man and achieved true mastery as a painter. Gifts that were latent in him, such as his skill in realistic representation, were widened and deepened; and spiritual potentialities, of which he hardly dreamed, such as his spiritual power of penetration, were developed by planned exercises.\n\nFurthermore, Fr\u00e4nger indicated that he planned to follow up this proposition with \"two later volumes, which would contain chronological evidence of the co-operation of these two men, from the early _Conjurer_ to the final _Adoration of the Child_ \". Then, to cap the climax, he demonstrated how the _Earthly Delights_ painting gave its own proof that Bosch knew how much he owed to the \"Grand Master\". The painter is supposed to have placed his portrait \"beside the Pythagorean cave, looking out of the narrow space between the naked nun's profile and the face of the Johannine youth\", at a respectful distance from that of the \"master\".\n\nThis modest positioning indicated several interesting things to Fr\u00e4nger: that since Bosch had depicted himself as an equal in the midst of those who were admitted into \"Paradise\", he must have been a member of the Community of the Free Spirit. Furthermore, this \"act of self-restraint\" on the artist's part indicated a moral attitude of self-abnegation \u2013 thus, self-discipline, which (since this is the attitude of one of the \"members\", must be that of all) threw a different light on the Free Spirit group's \"much slandered morality\". Obviously, the cult's practice must be of the highest moral order and not an anarchic degeneracy, as had been generally believed. As a final fillip to exalt his \"society\", Fr\u00e4nger spoke of the painting as \"a work that represented the community's world of ideas so completely and comprehensively that the loss of the writings of the Free Spirit sect seems compensated for by this pictorial document\". We must conclude that the author's imagination is only rivalled by that of Hieronymus, himself. It is too bad that Fr\u00e4nger's original high purpose of re-establishing Bosch as an artist through a proper evaluation of his work devolved into the realm of the nonsensical \u2013 that the only means by which he could make Bosch palatable was to prove that the painter's ideas were not his own at all.\n\nIt seems to this writer that Wilhelm Fr\u00e4nger's interpretation of Bosch's _Earthly Delights_ (his \"Millennium\") was the watershed in both Fr\u00e4nger's efforts to interpret the artist and those of the historians who came after him. This is because, having \"discovered\" the Grand Master of the Community of the Free Spirit, Fr\u00e4nger continued to \"find\" him (or his influence) in several of Bosch's other paintings. Moreover, his arguments were so compelling and presented so colourfully, many historians who followed him reacted for or against his interpretation. Let us review, cursorily, this historian's further discoveries as well as several other vivid attempts at interpreting this intriguing and masterful artist.\n\nThe Hay Wagon (triptych), 1525.\n\nOil on panel, 147 x 212 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (left panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\n### Fr\u00e4nger and Beyond\n\nTrue to his word, Fr\u00e4nger followed his interpretation of the \"Millennium\" triptych with studies of several other paintings seen from the same perspective. He claimed that they were also the products of Bosch's association with the Community of the Free Spirit and its Grand Master, who instructed him in the arcana of the cult's quasi-religious practices that the artist then displayed in carefully secreted symbols. Most of these paintings were done before the \"Millennium\", the historian claimed, thus bearing out his thesis that the apprenticeship of student to teacher was long and progressive.\n\nIn 1948, Fr\u00e4nger wrote his first article, after the initial introduction of the Grand Master in his book, on Bosch's John the Baptist, published in 1948. He noted the similarity of the wilderness setting in which John reposes to that of the _Garden of Eden_ in the \"Millennium,\" wherein the artist had transformed God into Christ and placed Him in a triangulated relationship with Adam and Eve. One of John's pronouncements, \"Repent ye, for the Kingdom of Heaven is at hand\", Fr\u00e4nger thought accounted for the paradisiacal setting of the _St John_ painting. The curious, composite plant growing beside John was so like the plants used as \"chiliastic symbols\" in the _Garden_ of the \"Millennium\" that the connection between the two paintings was obvious to Fr\u00e4nger who thus saw _St John_ as the mediating emblem of the esoteric society that he introduced in the millennial painting.\n\nIt was not long before the Grand Master ceased to be elusive, because Fr\u00e4nger claimed, in his article on Bosch's _St John on Patmos_ , published in 1949-1950, to have discovered the man's identity[2]. He had learned that a Jewish resident of 's-Hertogenbosch, named Jacob van Almaengien, was baptised ceremoniously in the presence of Philip the Fair of Brabant. Nothing but circumstantial evidence in the official records connected this man with the artist, but he was listed as having been registered as a member of the Brotherhood of Our Lady for the year 1496-1497, under the Christian name Philip van Sint Jan; Bosch had been a member of that order since 1486. This was not proof enough to convince many Boschian historians, especially the Dutch ones, such as Dirk Bax, yet the evidence was so compellingly argued by Fr\u00e4nger that to Patrik Reutersw\u00e4rd and others it seemed highly suggestive. Something that happened in Brussels at the beginning of the 15th century can hardly explain conditions in 's-Hertogenbosch a hundred years later. While this objection seems justified, I should like to recall the difference between proof and circumstantial evidence. While anyone who can be convinced only by proof will be obliged to reject Fr\u00e4nger's case, his reconstruction of the reality that underlies the great Prado triptych rests equally on a mass of circumstantial evidence whose diversity alone should compensate for the lack of conclusive proof.\n\nFr\u00e4nger, encouraged by the consistency of his evidence, drew very far-reaching conclusions, claiming that considerable information about the personality and life of Bosch's patron could be derived from this triptych and from other works. According to him, Bosch was working for none other than the leader of the hypothetical sect, the Grand Master himself.\n\nFr\u00e4nger's most trenchant case (after his book on the \"Millennium\") for Almaengien's influence over Bosch's iconography was presented in his book of 1950 on The Marriage Feast at Cana. Here, he claimed that the artist had turned the chamber of the marriage feast into the dining hall of a wealthy family with no qualms about ostentatious display at the same time as he turned it into a place of heathen practice. The author assumed this image to be the marriage of the Grand Master that is being celebrated. According to Fr\u00e4nger, this man had instructed Bosch in every visual manifestation of the strange ceremony shown as being enacted here. Not only does the painting include essential dogma of the society, but it reveals Almaengien's marriage to be a signal marker in the society's life. Events showing him as a much younger man, casting out devils after overcoming their malevolency and confessing the sins of his earlier life before acting much as Christ did in bringing salvation and bearing God's message, had supposedly been depicted by Bosch at the Master's behest in four medallions painted in grisaille on the back of an altarpiece called _As it was in the Days of Noah_.\n\nTo Fr\u00e4nger's mind, the marriage banquet at Cana, in which Christ performed his first miracle, was Bosch's showpiece of the society's secret rituals. The painting's dominant colours of red, white, and black, in addition to being alchemically significant, are the earthen triad in rabbinical tradition, from which God fashioned Adam. Thus, they, as well as other evidence in the form of demonstrable symbols, help set the stage for the event described by Fr\u00e4nger:\n\nA Jewish marriage celebrated before a pagan altar; Jesus and his mother in a gathering of idolaters; a bride and bridegroom who keep to themselves in rapt silence instead of mixing with the guests; a weird mixture of bourgeois, monastic, and Semitic costumes \u2013 all these singularities indicate that the painter was less intent on illustrating the Bible story than on applying it to a very specific contemporary context.\n\nA biblical reference that justified to Fr\u00e4nger (and by his admission, to De Tolnay before him) Bosch's association of the Grand Master and the Community of the Free Spirit with his treatment of _The Marriage Feast at Cana_ was Saint Paul's admonition to the Corinthians: \"Ye cannot drink the cup of the Lord, and the cup of devils: ye cannot be partakers of the Lord's table, and of the table of devils\" (10:21). Fr\u00e4nger pointed out in great detail the signs demonstrating to him that the Grand Master, member of a heretical society, himself, is struggling against the heresy represented by other members of the wedding party gathered in the room.\n\nAlthough this is a Jewish home, the guests are being served foods forbidden in Jewish law, a boar's head and a swan (emerging from their heads are rays, a sickle, and flames), nothing else that would suggest a banquet. There is an air of ritual solemnity among the participants. Behind the bride is an altar furnished with symbolic objects relating to male-female sexuality in both high and low forms: a half-sphere that is both breast and vagina and a mortar with upright pestle standing next to an urn whose shape reminds Fr\u00e4nger of the position of the frog held aloft by the black paramour at the altar of the Lisbon _St Anthony_. The frog, of course, is redolent of \"swampy procreation\" (Fr\u00e4nger, 1951), in other words, the nadir of sexual depravity; it is also related, Fr\u00e4nger tells us, to a cult symbol of Egypt that had lasted in demonic initiation rites into Bosch's time. Thus, it seems, Bosch is revealing this group's members as being devotees of an anti-church older than the Judeo-Christian tradition. Fr\u00e4nger concludes:\n\nThe Hay Wagon (detail, left panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (detail, left panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nOnly a syncretism wholly indifferent to religious affiliation could permit the uncontrolled lability and intermixing of creeds at the Grand Master's wedding. The leader of a heretic Christian community is marrying into a heretic Jewish family. He, for his part, has no objection to the marriage being celebrated according to their heretical temple ritual. But, although he is willing to celebrate the nuptial mystery, he uses it only to turn away towards the opposite world, revealed in the presence of Jesus....\n\nSo, Wilhelm Fr\u00e4nger continued to proselytise for the position of the Grand Master in controlling the content of Hieronymus Bosch's paintings.\n\nWe are left with a mystery that probably will never be resolved completely. It could be rejected out of hand, as with some successors to this historian, except that there are still some intriguing signs that more than the unembellished biblical story is being told in the _Marriage_. How to explain the finger positioning pointed out by Fr\u00e4nger, by which several of the banqueters hold the middle three fingers together, and the thumb and forefinger apart? We shall never know for sure and may ask why should we? The spirit of the miracle is here, enveloping the scene; perhaps, this is all that the artist wished his viewers to receive (we must note here how later research \u2013 Koldeweij\/Vermet \u2013 has claimed the painting to have been created in 1561, or later).\n\nFr\u00e4nger saw the hooded crow as being the symbol of the Grand Master in the _Garden of Eden_ and the central panel of the \"Millennium\" painting; he also identified this symbol in several more paintings, including Bosch's \"weightiest triptychs\": _The Hay Wagon_ , the Vienna Last Judgement, and the Lisbon Temptation of Saint Anthony. Fr\u00e4nger considered the latter painting to be filled with \"rampant satanic licentiousness\" such that its \"polemical excesses embitter the religious theme with witches' venom\". Consequently, he thought it left behind all heretofore conventional treatments of the theme and called into question its interpretation by other historians.\n\nIn his exegesis on Bosch's _Temptation of Saint Anthony_ , Fr\u00e4nger spoke of the \"magical\" effect of moving from the \"leaden-grey prologue\" of the outer panels to the \"hallucinatory luminosity of the three inner panels on which the strangest phantasmagoria unfolds\". (Since a thorough description of these panels will form a coda to this book, I will not present Fr\u00e4nger's description here, except in a few of his phrases that add to what he saw as the combined magical\/diabolical nature of the altarpiece.) The central landscape of the inner panels is replete with a \"cyclopean cave\", \"exotic towers\", an \"obelisk with reliefs\"; in \"an in-between land where the familiar veers off into the uncanny\", where a \"spooky tumult in which sexuality intertwines with the trefoil of idolatry, magic, and sodomy serves above all to excite the roving eye\". Anthony kneels against a balustrade that frames a stage behind him: \"This bare silver-grey dance floor for evil spirits looks like a sheet of ice on which no one may set foot, for in the dark hollows beneath its gleaming surface lies a miasmic slough\".\n\nWhat has this strange treatment of the Temptation theme to do with The Community of the Free Spirit and its leader? It seems that the Old Testament scenes on the central obelisk furnish the \"weightiest proof\" that Jacob van Almaengien was the altarpiece's \"spiritual author\". That is because, Fr\u00e4nger supposed, such a man would have associated his own spiritual integrity, in the face of the idolatry into which the Jews had fallen, with that of Saint Anthony, \"the true servant of God amid the unleashed denizens of hell\".\n\nFr\u00e4nger is good reading, everyone agrees, but most historians do not agree that his creation is more than a fabrication of his own imagination. Many more very reputable historians tackled the enigma of Hieronymus Bosch both before and after Fr\u00e4nger died in 1964. Writing still in the late 1950s, soon after Fr\u00e4nger discharged his first salvo, several historians began to stress the more serious, possibly even conventional appeal of Bosch's paintings: Max Friedl\u00e4nder sounded the general view that the artist's contemporaries \"considered them as sermons with a moral\". Charles Cuttler averred that \"Bosch's unnatural yet natural beings, pieced together with artistic rather than natural logic, were a perfect vehicle for his serious, moralising exaltation of basic Christian ideals\" (Cuttler, 1957).\n\nIn 1959, Ludwig von Baldass identified several of the artist's patrons as being eminently respectable: Philip the Fair of Brabant and his sister the Archduchess Margaret, William of Orange and the Archduke Ernest, as well as more common (but undoubtedly affluent) citizens of Amsterdam, Haarlem, and Antwerp \u2013 including Rubens in the 17th century. Von Baldass also mentioned Bosch's influences among his great Flemish forebears (assuming that he did not know those from Ghent \u2013 mainly Jan van Eyck or Hugo van der Goes) as Roger van der Weyden, Dieric Bouts, Geertgen, and Hans Memling, as well as several 'International Style' artists (through a tondo from Italy, perhaps?) (Von Baldass, 1960). He thought the artist was the first in Flanders and Holland to understand the importance of drawing for planning and clarifying details. In addition, Von Baldass saw Bosch as a consummate inventor, never drawing only what he saw or taking from other artists, but creating his images from his own imagination. The becalming influence of these reports was challenged many times by other, less placid ones to follow. A kind of final period was put on the moralising direction in the introduction to the translation of a book by Bax, published in 1979. The author of the introduction, Irving L Zupnick, wrote about the Lisbon _Temptation_ :\n\nThis exemplary exposition of the Saint's experiences in the wilderness, which invites us all to make similar efforts at self-control and religious devotion, was a most convenient subject for Bosch, much of whose work is concerned with the moral conflict faced by man, in which he is torn between his knowledge of what is good, and his animal inclination towards the pleasures of life.\n\nWalter Gibson performed an enormous service to Boschian scholarship when he drew together all of the diverse strands of interpretation from the 16th century until the publication of his _Hieronymus Bosch: An Annotated Bibliography_ , in 1983. In his lucid and detailed book of 1973 (T&H) and the Introduction to his bibliography of 1983, he laid out three stages of Boschian interpretation.\n\nAllegory of Intemperance, c. 1495-1500.\n\nOil on panel, 35.9 x 31.4 cm.\n\nYale University Art Gallery, New Haven.\n\nThe Hay Wagon (detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThe first comprised the 16th to 18th centuries when Bosch was known chiefly for his paintings of the devils and hell, although a few writers, mainly Filipe de Guevara and Father Jos\u00e9 de Seg\u00fcenza, noted that the artist painted satires on the soul as well as the \"sins and ravings of man\".\n\nThe second stage was during the 19th century when writers became aware of Bosch the artist as well as moraliser, and discovered archival material casting him in a more pragmatic biographical light as well as a more normal religious light. With the 20th century began an appreciation of Bosch the artist \u2013 in relation to Dutch realism in genre and landscape. The earliest monographs on Bosch came in the 20th century, with those by Gossart, Lafond, De Tolnay, Von Baldass, Friedl\u00e4nder, and Combe being the most important. In contrast to the pre-WWII tendency to stress the medieval Christian theology behind Bosch's imagery, no matter how novel or even bizarre it might seem, Gibson pointed out that the post-war studies took different tracts, concentrating on psychological sources as probed by Freud or Jung or as possibly distorted in Bosch's own psyche. They also included sources that Gibson said were seldom if ever tapped by earlier critics: \"alchemy, astrology, and the other occult sciences, as well as various gnostic doctrines\".\n\nExcept for a few, Gibson saw most of the subsequent scholars in the post-war period as trying to discover the \"key\" which Panofsky was seeking when he dismissed Bosch as \"too high for my wit\". They went back and forth between studies like those of Bax which did not analyse Bosch's cryptic imagery so much as describe it and those that continued to apply what Fr\u00e4nger had called \"whole libraries\" (Chicago, 7) in an attempt to relate the imagery to aspects of Dutch or medieval culture and literature of the period. Some of the recent studies that are less usual include one co-authored by Neal Myers and Wayne Dynes, which found the key to Bosch's _Garden of Earthly Delights_ in the Canticle of Isaiah.\n\nThey pointed out passages in which Isaiah railed against drunkenness, equating them with Bosch's many images in his triptych that apparently connote drunkenness, especially in the hell scene. Among their most cogent arguments for Isaiah as being the artist's chief inspiration is their association of the vineyard prepared by the prophet's \"beloved\" (or the Lord) to grow grapes but that grew \"wild grapes\" (or wreaked bloodshed instead of justice and wrung a cry against righteousness, itself). When the vintner then neglected his vineyard, it went to seed and produced briers and other extraneous plants, accounting for those in the _Garden_ and central scenes of the _Earthly Delights_ , much embellished through Bosch's imagination, of course.\n\nAmong the striking recent studies have been those of Laurinda Dixon. She built upon the alchemical interpretations of Bosch's work, but sought a middle ground between what she considered their too disparate approaches. She found this in the more mundane field of pharmacy. Not only had there been a pharmacist in Bosch's wife's family, but Dixon believed that the artist's middle-class background would have drawn him more to the practical craft of pharmacy, which was associated with both alchemy and medicine but was much less esoteric than they. She saw the Lisbon _Saint Anthony_ triptych as being \"An Apothecary's Apotheosis\" (the title of one of her articles), with many references to the instruments of distillation that were used in alchemy to transmute raw materials into a state capable of healing the body or perfecting imperfect matter and in pharmacy to create medicines that would balance the humours in patients' bodies.\n\nThe Hay Wagon (detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThus, Dixon viewed the panoply in the open _Saint Anthony_ as not only revealing incidents in the saint's life but as symbolising the activities in an Antonime monastery devoted to healing and comforting victims of virulent illnesses \u2013 most particularly the \"horrific\" affliction called \"Saint Anthony's Fire\", now believed to have been derived from ergotism, caused by eating contaminated rye grain baked into loaves of bread.\n\nIn another article, Dixon interpreted the central section of _The Garden of Delights_ as an allegory of \"distillation,\" conceived to be \"cyclical and self-perpetuating. Its end resided to a certain degree in its beginning, in imitation of the rhythm of nature\". She believed that the goal of this process was an intended union with God, symbolised by the circle or globe, and she found many examples of this and other equipment used by the apothecary-alchemists as transformed by Bosch into properties of his _Garden_. Bernard Vermet added new information to the dating of the artist's paintings, primarily from the standpoint of \"dendrochronological\" testing (of growth rings of the wooden panels on which they were painted) done by contemporary scientists. Among the controversial findings that he and Koldeweij contributed, or reported, was that _The Marriage at Cana_ was produced at least half a century after the artist's death. As an example of a talented follower, Vermet told that the Spaniard Felipe de Guevara wrote around 1560 of a highly accomplished pupil of Bosch's who could outdo his master in imitating him and signing works in his name, also that frequent retouching of Bosch's paintings after his death made technical comparisons very problematic. This historian made a thorough survey of the remainder of Bosch's paintings, detailing the new proposals on the dating or authenticity of the artist's work. Paul Vandenbroeck showed how Bosch used Christ and angels to balance the negative in his work.\n\nChrist might seem to despair when he displays his wounds to the sinful mass below, but in doing so, \"he draws attention to his suffering, through which he will save the humanity that has forgotten its God\". The author concluded that Bosch was deeply imbedded in the urban, bourgeois culture that surrounded him but became wealthy enough by his work to separate himself from the \"artistic norms of his time\". He had summed up his independence by appending a Latin quotation (possibly from Boethius) to one of his drawings: \"It is characteristic of the most dismal of minds always to use clich\u00e9s and never their own inventions\". Reading all of these excellent studies, one cannot help but wonder, however, if any have found \"the wisdom of the riddle\" (part of the title of Vandenbroeck's essay). Perhaps, this is because there is not one answer, but many. Gibson told that some writers, beginning with Leo van Puyvelde, suggested that Bosch might not have had a specific meaning in mind for each of his images, but was allowing his imagination its own freedom. Furthermore, Gibson continued:\n\nFor Puyvelde, too, a humorous element can be discerned in much of his art, even in his depictions of monsters. Admittedly we are far from Lampsonius' evocation of a fearful artist haunted by the very devils he painted, but in other respects this emphasis on Bosch's inventive fantasy recalls the conception of the artist as a _faiseur de dyables_ current in the 16th century and long after.\n\nIt is into this category that I cast the theory that follows.\n\nThe Hay Wagon (detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (right panel), 1525.\n\nOil on panel, 135 x 45.1 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (detail, right panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\n### A More Prosaic View\n\nFrom the various discussions reviewed it would appear that there is still much disagreement concerning Hieronymus Bosch. Was this man mad or was he completely sane? Were these paintings the result of hallucinatory behaviour or were they the transmissions of completely rationalised formulae? Would not either of these extreme points of view nullify the man's achievements as an artist? If he were insane, would his paintings be more than irresponsible products of interest primarily to a psychologist? If he were the transcriber of a cult's sign language, would he then be any more than a distinguished craftsman?\n\nIt seems to this writer that Wilhelm Fr\u00e4nger was on the right track when he began his study by attributing the confusion concerning Bosch to the historians who had approached the paintings from content alone. They considered the imagery, in Fr\u00e4nger's words, \"at best as illustration, i.e. as a pictorial representation subordinated to a ready-made idea, and never as a piece of imaginative creation in its own right, i.e. a pictorial realisation of a meaning\". It is interesting to note that Lotte Brand Phillips posited a similar idea in her article on Bosch's Adoration of the Magi, when she wrote that \"it happens frequently in the art of Jerome Bosch that ideas taken from different contexts are amalgamated and find their expression in one single visual symbol.\" ( _Prado_ , 274).\n\nFr\u00e4nger also made an excellent point when he implied that the imagery should be considered as \"imaginative creation in its own right\", but with his next words he subverted that definition. If the imagery is to be considered the \"pictorial realisation of a meaning,\" then it cannot be considered \"imaginative creation in its own right\", for the word \"meaning\" implies an idea or association outside itself against which the pictorial image should be judged. Of course, the historian whose only concern is the outside meaning and not the pictorial image would be making a mistake. The formalist, looking only at the image with no thought to the meaning, would also make too limited a study. Even Fr\u00e4nger would have subordinated the image to the meaning. To explain, let us examine his further statement that: \"Symbols are not a mere combination of distinct forms and ideas, they entail a perfect simultaneity of vision and thought\". This may be very true, but a visual symbol formed by this \"simultaneity\" cannot then be completely explained verbally, because it no longer exists solely in the realm of thought content; it is something new \u2013 in other words, \"imaginative creation in its own right\".\n\nFr\u00e4nger's mistake was in not allowing these images to exist in their own right. He considered that whenever Bosch fused vision and thought to make an image, this image then embodied a new thought content which could be rationalised in verbal terms. His rationalisation led him to explain each image as having a precise relationship to the ritual formula of a cult. Actually, the scholar was studying the meaning, which he read into Bosch's intention, and not the image for its own sake.\n\nThe Hay Wagon  \n(triptych, detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nWe can certainly admit the importance of ideas as a necessary element of Bosch's paintings \u2013 that is, the ideas from which he received inspiration and the total idea to which he related each painting. But the individual images within a painting do not necessarily have ideational meaning except as they relate to the total conception. They are creations forged during an artistic process, and should be considered as such. To judge them according to rational, exterior, content only compounds the confusion concerning Bosch. The images resulting from his creative activity are not rational structures in themselves. To look for rational meaning in them only leads to the kind of misunderstanding that has been rampant in the nearly five centuries since the painter's death.\n\nThe historian stated the ingredient necessary to alleviate this misunderstanding when he said that \"no one has yet penetrated to the core of the problem, mainly to Bosch's mode of visual thought\". It is obvious, though, that the phrase meant something quite different to Fr\u00e4nger than the meaning it shall be given here. Before re-establishing Bosch's position as an artist by discussing his \"mode of visual thought\" as the process by which he created his marvellous imagery, we shall consider the conditioning influences that must have directed his use of an unusual formula. This is because no artist, not even a unique one, develops autonomously. The growth of an artist like Bosch is the result of a combination of unusual circumstances influential in his background.\n\nThe Hay Wagon (exterior), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (exterior), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Hay Wagon (detail, central panel), 1525.\n\nOil on panel, 147 x 212 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nHieronymus Bosch was certainly not closely allied to any school or group of artists. As a result, he did not follow the artist's usual pattern of seeking to advance technical knowledge for its own sake, even within the framework of religious subject matter. Although he developed into a first-rate technician, nevertheless, it is apparent that his technical achievements evolved from the demands of religious necessity. He was, undoubtedly, intensely religious. His paintings exude belief of an extraordinary nature. The fact that his governing drive was the desire to heighten religious experience would be a prime factor in his freedom from the ordinary artistic restrictions. His lack of preoccupation with reality has made him incomprehensible to the layman and historian alike, but this would naturally follow his desire to concretise such imaginary concepts as heaven, or hell, or the visions of a saint. His technical facility with such difficult passages as the cloud-formed tunnel to heaven, or the fire-lit night sky of hell would be the result of his wish to intensify the revelatory impact of these visions. There were many other factors at work in the formation of Bosch's artistic _modus_. De Tolnay made the point that the artist's very originality in technique and ideation may have resulted from his physical isolation from the main artistic currents of his time. There is no record of his ever having travelled outside of his hometown. If he had never seen the paintings of the great Flemish masters, his most immediate influences would be those found in a provincial town. The content of his paintings seems much closer to popular sources, such as illumination and incunabula, than to that of the Flemings. Although he worked in oil paint as they did, he did not employ their elaborate glazing method and, perhaps, had no knowledge of it. His was an _alla prima_ technique, which some people believe to be his application to oils of a fresco painting manner, conceivably resulting from his admiration for the frescoes in the cathedral of 's-Hertogenbosch. Their 'International Style' figure treatment may explain the archaism of the figures in his early paintings.\n\nAn artist is never completely isolated from larger artistic heritages, however. Ideas are \"in the air\" and are bound to have their penetrative effect. He had a certain identity with Dutch art of his period. According to historian Otto Benesch:\n\nDutch art of the late 15th century, in spite of its tremendous height of pictorial mastership, had as its chief aim not perfection of craftsmanship and naturalistic observation of the Flemings, but expression. The same subjective religious spirit which brought about the mystical movements of the Brothers of the Common Life (the Ruysbroek followers) fills the works of the Dutch painters, whether they go in a realistic direction, like Geertgen tot Sint Jans, or in a fantastic and visionary one, like Bosch.\n\nBenesch called this direction \"Dutch Flamboyant Gothic\", but it was part of a movement of wider dispersion than this name would indicate. De Tolnay pointed out that there was a Neo-Gothic current spreading throughout Europe during the last quarter of the 15th century, appearing \"simultaneously at Florence with Botticelli and Filippino Lippi, at Venice with Crivelli and Vivarini, in Germany with Schongauer, in Flanders itself with Juste de Gand\". Bosch's involvement with Gothic art may have been more personal, however, than as a result of a generalised trend. The Cathedral of Saint John in 's-Hertogenbosch, is called one of the finest examples in the Netherlands of French Gothic architecture. It is known, as shown in records cited earlier, of Bosch's close association with the cathedral, by virtue of his artistic contributions to its decoration. The cathedral had burned in the early part of the century, and the repairs were still in progress in the painter's youth. He probably grew up watching the wood and stone carvers at work in the churchyard.\n\nThe Ship of Fools,  \nbetween 1490 and 1510.\n\nOil on panel, 58 x 33 cm.\n\nMus\u00e9e du Louvre, Paris.\n\nThe Garden of Earthly Delights,  \nc. 1500-1505.\n\nOil on panel, 220 x 389 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights (detail, left panel: Paradise),  \nc. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThe most obvious result of such an observation was his love of the chimeras and grotesqueries that enjoy self-sufficient life in his works. A more important contribution of Gothic art to Bosch, however, lay in its mode of expression. Since the laws of Gothic art are determined by its own dynamics \u2013 not by the necessity of an adherence to nature's laws \u2013 its influence allowed the artist his initial freedom from nature.\n\nThis type of expression was profoundly different from that of the Southern Renaissance, so soon to have the effect of inundating the Northern style. Southern European art was dominated by the intellect, Northern by the senses. The art of the South translated natural phenomena into ideas, thereby subordinating the world of nature to the mind of man. Northerners, more awed by the universal mysteries, placed themselves in the subordinate position to nature, and either recorded it literally, or \"felt\" it intuitively. Basically different intentions led to divergent style characteristics. Southern ideas were expressed in compositions involving a few large parts presented in a static, self-contained order. Northern compositions multiplied infinitely detailed parts to produce a complex, involved, unclear totality. Southern artists observed the \"unities\", presenting a situation as it occurred in one time and place. Northerners often \"narrated\" in a literary fashion, projecting upon the same field several parts of a story as it occurred in different times and places. Scenes juxtaposed in this fashion do not observe natural, \"visual\", logic but do partake of the resources of memory and imagination.\n\nSouthern art inherited from antiquity the ideal of beauty; its artists were inspired to pick, choose, correct, and perfect nature against their concept of its most ideal form. Northerners, lacking this noble idealism, could present the less attractive aspects of the world. They could include, even in a natural reference, the more terrible and extreme actions or attitudes, which do exist in human experience. Lastly, since human beings were paramount in their ideology, Southern artists could seek their measurements in their most ideal form.\n\nNortherners, not holding humans and their bodily attributes inviolable, could depart from them if it suited their expressive purposes. They could distort the human body if they so desired \u2013 to whatever degree would serve their emotional needs. Or they could violate its requirements altogether in order to depict other worldly creatures in aberrant human and animal form, if this were their desire.\n\nAll of these Northern \"differences\" would conspire to condition a painter like Bosch. First, he would operate within the expressive, imaginative, spiritual realm of the Gothic. He would not be controlled by a reasoned adherence to visual logic as expressed in the classical compositional order, \"unities\", perfect measurements, or ideal beauty. He would be allowed agility in imaginative realms from which he could pluck those marvellous creatures that he multiplied on his picture field \u2013 in disparate time and place positioning \u2013 of measurements and physical attributes emphatically unnatural and unbeautiful.\n\nSome historians have believed that Bosch was not at all influenced by the great achievements of the 15th century, but was a complete reversion to Gothic, or \"International Style\", or pre-Eyckien \"archaism\". It was not possible for Bosch to have remained untouched by the artistic achievements of his own century. He was a master of illusionist spatial recession, often applied more vertically than perpendicularly, but not achieved to such a degree in any previous century. He could indicate ephemeral effects, such as a smoke-filled sky, with startling effectiveness. Though not a naturalist in the sense of insisting on surface detail, a prime delight of the Flemings, he could differentiate textural substances. It has been emphasised that, from his Gothic influence, he would have felt it unnecessary to render the world's logical, visual aspect, and so could deal freely in imaginative realms. It was just the skill that he possessed of rendering some natural effects, which he used in the service of his imagination that produced his peculiar identity. To be more inclusive, it was the combination of his knowledge of 15th-century achievements, Gothic ideation, his sectional and local provincial background, and his personal qualities of religiosity and imagination \u2013 all of which would contribute to the formation of this completely unique artistic personality.\n\nThe irrational element in Bosch's paintings, one of the results of his distinctive motivations, is not a unique phenomenon in art. If the whole sweep of art history is considered, there have been many artists who have violated logic in their subject treatment, but they have not existed in sufficient number at any one time to form a school, until the past century. The manner in which they worked is consistent with a true understanding of the artistic process. We wonder that there have been so few of these artists, but when one realises the fate of Bosch in general opinion, the reason is clear. Few people are objective enough to separate the image from reality. This inability causes the lay observer to expect the image to retain the aspect of its counterpart in life and to assume, if it does not, that this disparity reflects the unbalanced mind of the artist.\n\nActually, whatever exists on a canvas, or drawing board, or in three-dimensional sculptured form \u2013 however much it partakes of life \u2013 is not and should not be judged as life itself. An artist, realising this initial independence of the art form from life, would seemingly have the right, if he or she so desired to deviate from life to any degree. Artists of the 20th century assumed this right, often departing from nature to the full degree of non-objectivism. But artists of earlier times were not able to cleave their work completely from nature. When they tried, their effort took the form of a break with nature's consistent pattern. In other words, they would form hybrid creatures by combining animal and plant parts with human, or they would combine parts from various kinds of animals or even parts from inanimate objects. If they kept humans and animals intact, they would place them in unusual settings, give them unusual activities to perform, or make them disproportionate to one another in size.\n\nIt is conceivable that Bosch, believing the satanic world to be absent of everything normal, would create an abnormal world by making every creature a peculiar combination of disparate forms, engaged in extraordinary activities, in a setting of odd or unrecognisable conformation. To clarify this concept, let us cite some samples of these \"irrational\" art works. Of course, the hybrid creature is as old as art itself \u2013 the most notable examples in antiquity being the Sphinx and the Babylonian winged, human-headed bull.\n\nThe Garden of Earthly Delights (detail, left panel: Paradise),  \nc. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThis hybridisation can be found throughout the medieval period in the creatures which adorn the borders of manuscript illuminations or which are carved over the cathedrals, from the tympana to the choir stalls. True to nature in the parts of their bodies which are taken from various known animals, these creatures are untrue in their entirety, and thus produce a bizarre, sometimes horrifying effect.\n\nThe same device was elaborated in the late Middle Ages by artists of graphic works. As Grillot de Givry described a 15th-century engraving by Israel van Meckenem:\n\nSaint Anthony is shown lifted into the air by demons. The artist has borrowed the most grotesque and alarming anatomical peculiarities to be found in such animal forms as the oxyrhync, the decapods, and the cirripedes and built these demons out of them. There are fantastic holothurians with grimacing heads, mictyres with multiple claws, king-crabs flourishing a sharp sting. A rabid-featured monkey, armed with a cudgel, is beating the saint on the head with all his might. Other monsters are clinging to his robe. These have outspread fins, bristling spikes like those of the spondylus or the branchiopods, and the pointed crests of the dactylopters, the trigla, the flying hog-fish, or crab-beetles, along their spines.\n\nMaster E.S. made his devil in _The Temptation of Christ_ into a hideous being of human stature and weirdly combined human and animal parts. Demonology was the most common motivation for such activity in this period; but often it was indulged merely to create delightful absurdities. In the alphabet series of the Master E.S., for instance, the artist formed letters by having apes, leopards, sloths, etc. hang onto one another. In his _Battle of the Money Pots and Strong Boxes_ , Pieter Bruegel gave arms and legs to chests and pots so that they could carry weapons and indulge in violent warfare. To keep humans and animals intact, in themselves, but to make them perform unusual activities in disparate settings produced a more subtle effect. This was only a furtherance of the device of combining true parts to produce an untrue effect. For instance, the 16th-century Italian artist named Agostino Veneziano, in an engraving entitled _The Carcass_ placed a procession of nude human beings in unusual attitudes. Some of them ride on goats; some pull the skeleton of a prehistoric beast on which others ride; some walk, but carry odd-sized babies on gigantic bones. This is probably a procession of witches on the way to a Sabbath, judging from the accoutrements such as goats, babies, bones, etc., which were significant in Sabbath rites. Witchcraft, like demonology, would be another subject matter field motivating artists to abnormal effects in their compositions.\n\nFrancisco Goya, an artist much at home in the realm of the irrational, derived several of the motifs for his _Caprichos_ from witchcraft. Many of his half-human, half-animal creatures point to the transformation of witches into animals, before departing for the Sabbath. He often made humans perform bestial activities, or animals perform human activities, for the additional purpose of satirising human follies. In his later works, such as the _Disparates_ , and _Pinturas Negras_ , however, he used irrational images with no rational purpose in mind. To quote Aldous Huxley upon these works:\n\nIn the _Disparates_ the satire is on the whole less direct than in the _Caprichos_ , the allegories are more general and more mysterious. Consider, for example, the technically astonishing plate, which shows a large family of three generations perched like huddling birds along a huge dead branch that projects into the utter vacancy of a dark sky. Obviously, much more is meant than meets the eye. But what? The question is one upon which the commentators have spent a great deal of ingenuity \u2013 spent it, one may suspect, in vain.\n\nFor the satire, it would seem, is not directed against this particular social evil or that political mistake, but rather against unregenerate human nature as such. It is a statement, in the form of an image, about life in general. Literature and the scriptures of all the great religions abound in such brief metaphorical verdicts on human destiny. Man, turns the wheel of sorrow, burns in the fire of craving, travels through a vale of tears, leads a life that is no better than a tale told by an idiot, signifying nothing.\n\nThis is a beautifully worded statement and penetrating thought in that Huxley does not try to pinpoint the meaning of Goya's _Strange Folly_ etching. The idea that it is a \"brief metaphorical verdict on human destiny\" is wonderful, but is it really necessary to suppose that such an intention existed in Goya's mind? He could simply have been employing the artist's license to place human beings in a strange setting. The image would then exist in its own right; the viewer could read into it any desired meaning. Of course, to suggest that an artist like Goya was merely playing with forms on a plane surface when he juxtaposed them in strange relationships, is not to give a complete answer. The question would arise as to why these particular forms in these particular settings. Perhaps Huxley was on the right track when he said:\n\nThere are other plates in which the symbolism is less clear, the allegorical significance far from obvious... What is the meaning of these things?\n\nPerhaps the answer to that question is that they have no meaning in any ordinary sense of the word; that they refer to strictly private events taking place on the obscurer levels of their creator's mind.\n\nIf it is true that Goya was working according to subconscious suggestion, then he is allied on this score to the 20th-century Surrealists. These artists formed a school based on deriving inspiration from as yet unplumbed depths within the human mind. This they considered antidotal to the \"forces of reason\" which they thought responsible for war and the sham structure of society. The Surrealists studied their own dreams, focusing on the dream state incongruities of object positioning, and dislocations of time and space. They attempted, by juxtaposing incongruate, dislocated, and unreasonable settings and activities on a canvas, to recreate a dream state with its compelling and fascinating power on the observer.\n\nThere were two groups of early Surrealists, divided according to the means they employed: those who depicted the fantastic world in precisely photographic terms, and those who used a spontaneous technique as well as subject matter.\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nArtists of the first group desired to produce their fantasies in terms of a slick-surfaced photograph, so they would generate the same assurance of reality as in a snapshot. The technical methods of the first group closely resemble those of Hieronymus Bosch; thus, we might focus on one artist of this surrealist group, such as Salvador Dal\u00ed. Since he was very verbal about his artistic activity, it is possible to follow his thinking as he himself explained it. Dal\u00ed was far more objective about his painting activity than the earlier Surrealists.\n\nThey tried to subjugate reason altogether and place themselves in a simulated somnambulist state in order to be completely receptive to subconscious stimulation. This artist asserted, however, that he could make rational use of subconscious suggestion. He claimed to have a paranoiac personality that, in full awareness, he could place at the service of his reasoning intellect. The only difference between himself and a madman, he said, was that he was not mad but could induce a state of \"madness\" within which to operate in his creative activity. The process would work such as this: just as a person of true paranoid personality is consistently deluded as to the logical interpretation of reality, Dal\u00ed could delude his own interpretation of what he put upon a canvas. Having placed an image of one sort there, he could disassociate it from any contextual relationship \u2013 then let its form suggest another of completely different association, until he had filled a canvas in this way, The result of these unrelated forms would be as a new and disturbingly different world \u2013 haunting in that it would evoke a deep response in the observer. But, too deep for objective recognition, the result would be reminiscent of the shadowed memory of a dream.\n\nSometimes Dal\u00ed would allow another kind of delusion to take place. He would place an image on his canvas, and allow its form to suggest another that he would then paint within the first form \u2013 thus producing a double-image. This had often been done in the past \u2013 for instance, by the 16th-century Italian Mannerist, Arcimboldo.\n\nDal\u00ed would not always stop with one delusion. He would let the second image suggest another, paint that within it, and then repeat the process until he had produced triple and quadruple images. The possibilities were, of course, unlimited \u2013 the possibilities to the observer's mind unlimited, as well. Sir Herbert Read discussed this process of forming the images as a kind of conscious dream formation. He said that the artist cannot explain the form that the image takes in its emergence on his canvas, but collects forms from other contexts, observed in the course of the day's activities. They are held in the deep mind until it is necessary to call them up in the course of the actual painting process.\n\nIn other words, already peculiarly sensitive to the visual shapes of things, an artist makes a pleasant habit of observing such shapes in the course of his day, and retaining their forms in memory. The artist may see a spot of lichen on a tree, or a pattern in a piece of driftwood, or in a grained door, but may be totally unconscious of these observations. Not verbalised, they sink into the artist's mind until needed. This activity, of reading into a form on a canvas an image corresponding to one already in the subconscious, was also described in an article on \"Dada\", discussing the pictorial collaborations of Max Ernst and Baargeld:\n\nThey begin to discover in a drawing another drawing the contours of which appear slowly out of the tangled lines \u2013 like an apparition, like a prophecy, like the messages in table tapping. We are confronted here with a process not quite comparable to that of perceiving an image in a spot on the wall as Leonardo da Vinci did; nor yet does it consist of lifting an object out of its natural environment. Other forces are at work: accident and surprise at their most inscrutable and intense level, the discovery of second sight in the spirit itself. ( _Fantastic_ , 26-27)\n\nAs a matter of fact, the process of thought involved is familiar to anyone who has ever observed figures or objects in cloud formations, or in the discolorations of wallpaper. In _A Way to Stimulate and Arouse the Mind to Various Inventions_ , Leonardo himself wrote a description of this phenomenon \u2013 thus attesting to the fact that the process has been used by artists in earlier times:\n\nIf you look at any walls spotted with various stains or with a mixture of different kinds of stones, if you are about to invent some scene you will be able to see in it a resemblance to various different landscapes adorned with mountains, rivers, rocks, trees, plains, wide valleys, and various groups of hills. You will also be able to see diverse combats and figures in quick movement, and strange expressions of faces, and outlandish costumes, and an infinite number of things, which you can then reduce into separate and well conceived forms.\n\nAlthough he was not referring to the production of irrational results, the method that Leonardo describes is similar to that used by some of the artists who do deal in the irrational. It would be presumptuous to claim to understand the mind of Hieronymus Bosch, and therefore, the exact nature of his \"mode of visual thought\". Since his results are similar in principle to the examples of the irrational that have been cited, it can be assumed that his manner of working was similar to some of the processes described. It would be impossible to tell whether he worked according to rational or irrational directives. His imagery is so involved that he may have brought many processes to bear in its formation. In part, Bosch was undoubtedly illustrating in a rational manner ideas received from theology and folklore. His hybridisation was a flamboyant elaboration of the same device used so often by medieval artists. The strange effects he achieved can be partially explained by his reasonable desire to present the obverse satanic world, by \"queering\" the normal one. He would make his creatures, their activities, and their environments as weird and unworldly as possible, yet make them believable by rendering them with all of the technical mastery he would ordinarily use to produce the illusion of the natural world.\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThat Bosch was also acting according to subconscious directives within the framework of his rational intentions cannot be doubted. His paintings are compelling in their power on the viewer. Of course, it may be, as suggested in the discussion of Goya, that this fascination on the viewer's part is the result of the stimulation of his or her subconscious mind by Bosch's imagery. The explanations that the viewer attempts to read into the imagery may not have existed in Bosch's mind at all. But psychologists believe that the subconscious mind is much alike in every person; so, what Bosch produced by allowing his subconscious mind its play, might evoke a deep response by recognition of something with which the viewer also held subconscious familiarity. The fact of the subconscious, insisted upon by modern psychologists and exploited by modern Surrealists, has been given more and more recognition in modern times. That human behaviour is not accidental, that all humans act according to deep-seated and early-formed directives within the psyche; that the attempt to ignore and repress the instincts is made at the individual's peril \u2013 all these things are ideas that have grown for a century or more in prominence. We have learned that reason is only a part of the full human potential \u2013 perhaps only a small part, at that. Carl Jung made profound observations upon the mind's darker, deeper portions. According to his belief, the mind not only possesses levels lying below consciousness, but above and surrounding it as well. He claimed that the unconscious is not just an \"inessential and unreal appendage\" to the conscious, but that it is a world with activities, reasons, and laws of its own. We may believe that the artists discussed in this chapter, most particularly Hieronymus Bosch, were those who \"let the unconscious go its own way and [experienced] it as reality\". They are, as a consequence, universally misunderstood. Because most artistic creations have meaning in all of their parts, it is expected that the works of these persons should as well. There is no way, however, to make the connections between each part in their works logically consistent in meaning. The only things that can be rationally understood are the total idea and the sources for individual parts. What has been made of these sources, however, is not a new logic. It is an artistic creation.\n\nIn order to clarify these concepts in tangible terms, an example from Bosch's works, the Lisbon Temptation of Saint Anthony, will be discussed in the next two chapters. Certain ideas will be presented as possible source material for the painting's imagery. This will be done, not because the painting is necessarily dependent upon the ideas, or the images subordinate to the ideas \u2013 but because the license that Bosch assumed with these ideas and the lengths to which he departed from them are more vividly apparent if they are known.\n\nThe painting itself will be examined in the final chapter. Its imagery will be referred to the source ideas only to show what Bosch made of them. No attempt will be made to explain images according to any deep rational meaning hidden in each one of them. In other works it may not be necessary to find a key as Fr\u00e4nger did, with which to unlock meaning. These paintings can be observed from the point of view of pure imaginative projections around a pivotal idea such as hell or a saint's visions. They are disturbing, yes, but so are the concepts on which the painter elaborated. They are rich fields for the stimulating of associative ideas in the viewer's mind, but it is enough to consider this marvellous imagery as \"imaginative creation in its own right\".\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\n### Saint Anthony and the Devil\n\nFr\u00e4nger's idea that Bosch's three large triptychs contain satire directed both against the Church and paganism led him to see somewhat different emphases in the paintings. Whereas the _Earthly Delights_ stands somewhat between the polar extremities, Fr\u00e4nger presumed _The Hay Wagon_ to be primarily anti-clerical and _The Temptation of Saint Anthony_ to be \"anti-occultist\" satire. However, it is more likely in the latter painting that the occult would be a part of the main theme \u2013 that of the traditional conflict between Good and Evil \u2013 but presented according to late medieval ideography. Good is represented by the saint, Christ, and the monks who perform acts of mercy toward the saint.\n\nEvil abounds through the demonic creatures that swarm over the panels \u2013 these representing the minions of a kingdom of Satan, the concept that objectified evil to the medieval mind. Satan and his demons were thought to operate on this earth through heretical activity, of which occult practices were believed to be a large part. This would explain why Bosch made allusions to pagan practices in this painting. When we examine the imagery against a knowledge of such activity, it is apparent that the artist did not pinpoint any specific idea or practice. There is not one symbol that could be said to come entirely from any single source. It is obvious therefore, that Bosch incorporated ideas from numerous sources in his imagery. The composite mechanisms that he developed are not complete within themselves. I believe that they make rational sense only when related to the main theme of the painting. If not convinced that Bosch created his scene of hallucinatory effects and hideous demons with responsible use of his faculties, we should take into account the world of ideas, universally held in his time, upon which he could draw.\n\nAs Heinrich W\u00f6lfflin observed, the art of the past \"must not be interpreted as if it had been created today. Instead of asking 'How do these works affect me, the modern man?' and estimating their expressional content by that standard, the historian must realize what choice of formal [and ideational] possibilities the epoch had at its disposal. An essentially different interpretation will then result\". Before considering how our artist might have formed the images of this St Anthony painting, let us review the sources of possible elements that he fused to form his imagery.\n\n#### Saint Anthony\n\nThe first source is that of the painting's subject, Saint Anthony's temptation by the Devil, seems to have been a favourite of the artist, since he devoted more works \u2013 drawings as well as paintings \u2013 to this theme than to any other, almost as if he felt a personal empathy towards the visionary saint.\n\nA reason why might be found in the observations of Carl Jung on a personality type that throws light upon both Bosch and the hermit. In Jung's words:\n\nIsolation by a secret results as a rule in an animation of the psychic atmosphere as compensation for loss of contact with other people. It causes an activation of the unconscious, and this produces something similar to the illusions and hallucinations that beset lonely wanderers in the desert, seafarers, and saints.\n\nAnd, following shortly afterwards: \"We are reminded of the visions of Saint Anthony in Egypt\". Jung emphasised that such \"intuitive perception\" is not the same thing as insanity, whatever the layman might believe. The difference, of course, is a matter of degree.\n\nWhether or not Saint Anthony's devils were imaginary or real, tradition and the Church held that they were real and a good part of the body of Church belief towards the constitution of the demonic world came from Saint Anthony's story. Bosch probably had recourse to two sources for this story, both the _Golden Legend_ (a compilation of the stories of the saints' lives written by Jacobus de Voragine in the late 13th century), and the original story, which was written by Saint Athanasius shortly after Anthony's death.\n\nThe saint's lifetime was that of the waning power of the Roman Empire, in the 3rd century CE. Years of misrule in the provinces by military governments, wars, exhausting taxes, extremes in injustice and torture, slavery \u2013 plus the threat of barbarian invasion \u2013 had produced a state of unrelieved decay. The consequence of such despair was suicide among many of the heathens; but Christianity taught that the body must be preserved as a vessel for the soul. Christian men began to flee to the deserts and mountains of Egypt \u2013 each trying to save his own soul. It has been estimated that as many as 5,000 of these hermits were living at one time in the caves and tombs of the mountains that border the Nile. The word of their self-tortures, humility, faith, and miracles spread so far that learned and virtuous men from other countries submitted to unbelievable hardships in mountain and desert travel in order to visit them. They carried the report of these saintly men back to their own countries, and a movement, based on this eremitical pattern, spread irresistibly throughout the Roman Empire, even making it as far as Ireland.\n\nSaint Athanasius, when fleeing from the persecution of the Christians, which was sporadically practiced in Alexandria, would take refuge with the hermits, often with Anthony. When he visited Tr\u00eaves, in Gaul, he carried stories of these men, and conceiving Anthony to be the ideal of these Christian ascetics, wrote an \"encomium\" concerning him and his miraculous resistance to lifelong temptation by the Devil \u2013 the purpose being to inspire others to emulation. The major portion of Saint Athanasius' biography is devoted to a long discourse delivered by Anthony, on the subject of the monk's vocation, the powerlessness of Satan in the face of prayer, and the gift granted by the Holy Spirit, by which men could distinguish good from evil spirits. The descriptions in this section of the ways of Satan form the backbone of the Church's later development of its structure of demonology.\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nA brief outline of Anthony's life and quotations from Athanasius' presentation of the temptations follow, illustrated by works of artists from the 12th through to the 20th century, who were inspired by various aspects of this saint's life.\n\nAnthony was born in 251, in the town of Coma, Middle Egypt, of Christian parents who were wealthy landowners. A quiet, devout youth, Anthony was devoted to his family, finding school and other companionship distasteful. His parents died when he was about twenty years of age, leaving Anthony their fortune and the guardianship of his younger sister. Upon being strongly moved by hearing the scriptural admonition of Christ to the rich young ruler, read in his church, Anthony sold his property and belongings, placed his sister in a community of pious women, and left his village to assume an ascetic way of life. For a while, he lived with an old solitary who had a hermitage a short distance from the village. Anthony observed this man in his holy practices, learning from him the virtues of a life devoted to prayer, meditation, fasting, and manual labour. When he felt himself ready for the solitary life, Anthony retired to a hermitage of his own. It was here that the \"temptations\" began.\n\n##### Temptation by the recollection of the responsibilities and pleasures of his past life\n\n__\n\nThe Devil, the hater and envier of good, could not bear to see such resolution in a young man, so set about employing his customary tactics against him. First, he tried to make him desert the ascetic life by putting him in mind of his property, the care of his sister, the attachments of kindred, the love of money, the love of fame, the myriad pleasures of eating, and all the other amenities of life. Finally, he represented to him the austerity and toil that went with virtue, suggesting that the body is weak and time is long. In short, he raised up in his mind a great cloud of arguments, intending to make him abandon his set purpose.\n\n##### Temptation by the desires of the flesh\n\nThe enemy saw, however, that he was powerless in the face of Anthony's determination and that it was rather he who was being bested because of the man's steadfastness, vanquished by his solid faith and routed by Anthony's constant prayer. He then put his trust in the weapons that are \"in the navel of his own belly\" [Job 40:16: \"Lo now, his strength is in his loins, and his force is in the navel of his belly\"]. Priding himself in these \u2013 for they are his choice snare against the young \u2013 he advanced to attack the young man, troubling him so by night and harassing him by day, that even those who saw Anthony could perceive the struggle going on between the two. The Enemy would suggest filthy thoughts, but the other would dissipate them by his prayers; he would try to incite him to lust, but Anthony, sensing shame, would gird his body with his faith, with his prayers, and his fasting. The wretched Devil even dared to masquerade as a woman by night and to impersonate such in every possible way, merely in order to deceive Anthony.\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madri.\n\n##### Temptation by pride\n\nFinally when the dragon could not conquer Anthony by this last means either, but saw himself thrust out of his heart, gnashing his teeth [Psalms 112: 9,10. \"... his horn shall be exalted with honour. The wicked shall see it, and be grieved; he shall gnash with his teeth, and melt away; the desire of the wicked shall perish.\"]; he [the Devil] changed his person, so to speak. As he is in his heart, precisely so did he appear to him. He no longer assailed him with thoughts \u2013 for he had been ousted, the impostor \u2013 but now using a human voice, he said: \"many have I deceived and very many have I overthrown; but now when I attacked you and your efforts as I have done with many others, I proved too weak.\"\n\n\"Who are you to speak thus to me?\" Anthony asked. The other was quick to reply with whining voice: \"I am the lover of fornication. It is my commission to waylay the youth and seduce them to this, and I am called the spirit of fornication. How many have I not deceived who were determined to keep their senses! How many chaste persons have I not seduced by my cajoleries! Yes, it was I that tripped them up. I am the one who gave you so much trouble and as often was vanquished by you.\"\n\nAnthony then gave thanks unto the Lord and taking courage against him [the Devil] said: \"Well, then, you are as weak as a child. For the future you cause me no worry at all, for the Lord is my helper and I will despise my enemies\" [Psalms 117: 7]. Hearing this, the Black One fled at once, cowering at his words and fearing to even come near the man.\n\nAnthony did not grow careless nor take this victory for granted merely because the demon had been brought to his knees; nor did the enemy, worsted as he was in the conflict, cease to lie in wait for him. He kept going around like a lion [I Peter 5:8: \"Be sober, be vigilant; because your adversary the devil, as a roaring lion, walketh about, seeking whom he may devour\"], seeking a chance against him. So he [Anthony] more and more mortified his body and brought it into subjection, lest having conquered on one occasion, he should be the loser on another.\n\n##### Temptation by physical torture\n\nThus did Anthony master himself. He left for the tombs, which lay at some distance from the village. He had requested one of his acquaintances to bring him bread at long intervals. He then entered one of the tombs, the aforementioned man locked the door on him, and he remained alone within. This was too much for the Enemy to bear; indeed, he feared that Anthony would fill the desert too with his asceticism. So he came one night with a great number of demons and lashed him so unmercifully that he lay on the ground speechless from the pain. By God's providence \u2013 for the Lord does not overlook those who have faith in Him \u2013 his acquaintance came by next day with the bread for him. When he opened the door and saw him lying on the ground as though dead, he lifted him up and carried him to the village church and laid him upon the floor.\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(left panel: Paradise), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nMany of his kinsfolk and the people from the village sat around Anthony as around a corpse. But about midnight he regained consciousness and awoke. When he saw that they all were asleep and that his friend alone was awake, he beckoned him to his side and asked him to lift him up again and carry him back to the tomb without waking anyone.\n\nSo the man carried him back and the door was locked as before and once more he was alone within. Because of the blows received he was too feeble to stand, so he prayed lying down. His prayers finished, he called out with a shout: \"Here am I, Anthony. I am not cowed by your blows, and even though you should give me more, nothing shall separate me from the love of Christ.\"\n\n##### All the demons of hell are unleashed\n\nThe hater of good, the Enemy, marvelled that after all the blows he had the courage to come back, called together his dogs, and bursting with rage, said: \"You see that we have not stopped this fellow, neither by the spirit of fornication nor by blows; on the contrary, he even challenges us. Let us go after him in another way.\"\n\nWell, the role of an evildoer is easy for the Devil. That night, therefore, they made such a din that the whole place seemed to be shaken by an earthquake. It was as though demons were breaking through the four walls of the little chamber and bursting through them in the forms of beasts and reptiles.\n\nAll at once the place was filled with the phantoms of lions, bears, leopards, bulls, and of serpents, asps, and scorpions, and of wolves; and each moved according to the shape it had assumed. The lion roared, ready to spring upon him, the bull appeared about to gore him through, the serpent writhed without quite reaching him, the wolf was making straight at him; and the noises emitted simultaneously by all the apparitions were frightful and the fury shown was fierce.\n\nAnthony, pummelled and goaded by them, felt even more severe pain in his body; yet he lay there fearless and all the more alert in spirit. He groaned, it is true, because of the pain that wracked his body, but his mind was master of the situation, and as if to mock them, he said:\n\nIf you had any power in you, it would have been enough for just one of you to come; but the Lord has taken your strength away, and so you are trying, if possible, to scare me out of my wits by your numbers. If you can and have received power against me, do not delay, but up and at me! If you cannot, why excite yourselves to no purpose? For faith in our Lord is a seal to us and a wall of safety.\n\nSo, after trying many ruses, they gnashed their teeth against him, because they were only fooling themselves and not him.\n\nAnd here again the Lord was not forgetful of Anthony's struggle, but came to help him. For he looked up and saw as it were the roof opening and a beam of light coming down to him.\n\nThe demons suddenly were gone and the pain in his body ceased at once and the building was restored to its former condition. Anthony, perceiving that help had come, breathed more freely and felt relieved of his pains. And he asked the vision: \"Where were you? Why did you not appear at the beginning to stop my pains?\"\n\nAnd a voice came to him: \"Anthony, I was right here, but I waited to see you in action. And now, because you held out and did not surrender, I will ever be your helper and I will make you renowned everywhere.\" Hearing this, he arose and prayed; and he was so strengthened that he felt his body more vigorous than before. He was at this time about thirty-five years old.\n\n##### Temptation by constant hordes\n\nSo, having grown stronger and stronger in his purpose, he hurried to the far-away mountain (constantly seeking an ever-greater solitude).\n\nOn the far side of the river he found a deserted fort [this was at Pispir, fifty miles south of Memphis, at a place called the \"Outer Mountain\"] which in the course of time had become infested with creeping things. There he settled down to live. The reptiles left at once, as though being chased.\n\nHis acquaintances who came to see him often spent days and nights outside, since he would not let them come in. They heard what sounded like riotous crowds inside making noises, raising a tumult, wailing piteously and shrieking. At first those outside thought there were men fighting with him and that they had entered by means of ladders, but as they peered through a hole and saw no one, they realised that demons were involved; and filled with fear they called out to Anthony.\n\nBut he was more concerned over hearing _them_ to pay any attention to the demons. Going close to the door he suggested to them to leave and to have no fear. \"It is only against the timid,\" he said \"that the demons conjure up spectres. You, now, sign yourselves and go home unafraid, and leave them to make fools of themselves.\" So they departed, fortified by the Sign of the Cross, while he remained, without suffering any harm whatsoever from them [the demons].\n\n##### Further Temptations as recounted by Anthony himself\n\nOnce they came with threats and surrounded me like soldiers in full armour. Once when I was fasting, the Crafty One came to me even as a monk carrying phantom loaves. He counselled me, saying: \"Eat and cease from your many hardships! You, too, are a man and you are bound to get sick.\" But I, perceiving his wiliness, arose to pray, and he could not bear it. He left, resembling smoke as he went out through the door.\n\nFeeling the need to live again in eremitical solitude, [Anthony] left the \"Outer Mountain\" for a retreat at the foot of a mountain near the Red Sea.\n\nThe Garden of Earthly Delights  \n(detail, left panel: Paradise), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nThe Garden of Earthly Delights  \n(detail, right panel: Hell), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\nThis, called the \"Inner Mountain\" (still called \"Der Mar Antonias\"), was to be his final hermitage. He again lived ascetically, growing the small amount of food he required and spending his time in contemplation and prayer. In this place he was again plagued by demons.\n\n##### Final Temptations\n\nWhen he was an old man and living alone on the Inner Mountain, those who visited him told how they heard tumults and many voices and clangour as of weapons. At night they saw the mountain alive with wild beasts. They also saw him fighting as with invisible foes, and praying against them. It was truly remarkable that, alone as he was in such a wilderness, he was neither dismayed by the attacks of the demons, nor, with all the animals and creeping things there, did he fear their savageness.\n\nOne day a monster visited him, resembling a man down to his thighs, but with the legs and feet of an ass. Anthony simply made the Sign of the Cross and said: \"I am Christ's servant. If you are on a mission against me, here I am.\"\n\nBut the monster with its demons fled so fast that its speed caused it to fall and die. And the death of the monster stood for the fall of the demons: they were making every effort to drive him back from the desert, and they could not.\n\nOnce, when he was about to eat and stood up to pray he felt himself carried off in spirit, and \u2013 as strange as it might sound \u2013 as he stood he saw himself standing there, outside his own body, as though carried aloft by certain beings. Then he also saw loathsome and terrible things standing in the air and bent on preventing him from passing through. As his guides offered resistance, the others demanded to know on what plea he was not accountable to them. Then, when they set themselves to taking an account from his birth, Anthony's guides intervened, saying to them: \"As for the things dating from his birth, the Lord has erased them; but as for the time since he became a monk and promised himself to God, you can take an account.\" Then, as they brought accusations but could not prove them, the way opened up to him and so he was the real Anthony again. He was astonished to see against how many we battle and what labours a person has to pass through the air.\n\nThere came a call from on high, saying \"rise, Anthony, go out and look\". He saw a towering and frightening standing figure, so tall he was hidden by clouds, as well as certain beings ascending as though on wings. The former was stretching out his hands: some of the latter were stopped by him, while others flew over him, and having come through, rose without further trouble. At such as these the monster gnashed his teeth, but exulted over those who fell. Forthwith a voice addressed itself to Anthony: \"Understand the vision.\" He realised that it was the passing of souls and that the monster standing there was the Enemy, the envier of the faithful. Those answerable to him he lays hold of and keeps them from passing through, but those whom he failed to win over he cannot master as they pass out of his range.\n\nTwo monks came to live with Anthony when he reached a very advanced age. Saint Athanasius related that, when Anthony knew he was near death, he travelled to the \"Outer Mountain\" for a brief farewell to his friends, the monks who had come to live according to his example. Returning to his retreat, he made his last will and testament leaving a sheepskin and a cloak to Bishop Athanasius and a hair shirt to be shared by the two young companions. \"With a final blessing for them, he gave up his spirit.\"\n\nAlthough Saint Anthony's life was very influential upon the ecclesiasts who followed his precepts as revealed by Saint Athanasius, he did not become a popular saint until the late Middle Ages. \u00c9mile M\u00e2le related how a relic of Saint Anthony had been obtained from Constantine VIII by the Dauphin Jocelin. It was carried in 1050 to the church of Saint Anthony in Venice. There, in 1095, a man was cured of _gangrenous erysipelas_ (Ergotism) \u2013 called \"sacred fire\" at that time, but which was hereafter to be called \"Saint Anthony's fire\". This man founded the Antonite order, the members of which consecrated themselves to the treatment of skin diseases. Countless pilgrims seeking succour in the face of the plague converged on the church.\n\nAnthony's powers extended themselves from power over _erysipelas_ alone to that of all contagious maladies and this healing work was extended all over Europe by the rapid growth of the Antonite order.\n\nIn the famous Chartres window devoted to the saint's life, one section shows his being visited by a beautiful temptress. Anthony is seated before a chimney warming himself by the flames of a fire. The suggestion is that he overcame the temptation of lust by meditating upon the fires of hell and \"the worm which does not die\", the allusion being to Isaiah (66:24) who says that the good \"shall go forth, and look upon the carcasses of the men that have transgressed against me: for their worm shall not die, neither shall their fire be quenched; and they shall be an abhorring unto all flesh\". M\u00e2le suggested that it was this association of hellfire with the saint, combined with the appellation of the disease over which Anthony was given power, that caused him to be represented iconographically with flames beneath his feet \u2013 or with a burning city or house in the background. He soon became in the popular mind the master of fire, and could give spiritual aid against fire in all shapes \u2013 in this world or the next. He would also reduce anyone to cinders who had the effrontery to doubt his powers. At Saint Anthony's Church in Venice, the calcined bones of those who dared offend him were shown. The helpful saint became grim-visaged, and because of the great age to which he lived, he was shown as a very old man bearded and with a crutch. As M\u00e2le said, one does not see in these representations the one who considered himself \"the athlete of Christ\", a man burned by the sun, with the stamina to overcome so much physical and mental hardship.\n\nAs father of monachism, Anthony was given the monk's habit and cowl, usually black or brown. According to Anna Jameson, the tradition of marking the left shoulder of the saint's habit or his cape (always blue) with the letter T came from Greek representations, or those influenced by the Greek. This had developed from this fact:\n\nThe Garden of Earthly Delights  \n(detail, central panel), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nSaint John on Patmos (exterior), 1504-1505.\n\nOil on oak panel, diameter: 39 cm.\n\nStaatliche Museen zu Berlin, Berlin.\n\nThe Garden of Earthly Delights  \n(exterior panels: The Creation of the World), c. 1500-1505.\n\nOil on panel, 220 x 389 cm (triptych).\n\nMuseo Nacional del Prado, Madrid.\n\nSaint John on Patmos  \n(detail, exterior), 1504-1505.\n\nOil on oak panel, diameter: 39 cm.\n\nStaatliche Museen zu Berlin, Berlin.\nIn Revelation XIV: 1, the elect, who are redeemed from the earth, bear the name of God the Father written on their foreheads: the first letter of the Greek word 'Theos,' God, is T, and Anthony and his monks are represented bearing the T. In a specimen of painted glass (from St Denis) a man in a turban marks another with the T on the forehead, and over the whole in Gothic letters is inscribed 'Signum Tau'.\n\nAnthony was given a bell, either in his hand, suspended from his crutch, or on a cross nearby, to signify that he had the power to exorcise evil spirits; according to an old tradition: \"the wicked spirits that be in the region of the air fear much when they hear the bells ringing\". For this same reason he was sometimes given another instrument of exorcism, the asperges, or rod for sprinkling holy water. The gluttony and sensuality that Anthony was able to conquer were symbolised by the hog placed near or under the saint's feet. It soon became a popular notion that the animal was dedicated to Saint Anthony and was under his protection. For this reason, the Antonites were allowed to keep herds of pigs believed to be so consecrated that they were fed at public charge and allowed to run unmolested in the streets. Thus arose the phrase, \"as fat as an Anthony pig\". Sometimes Anthony was represented with a skull and crucifix nearby, general symbols of penitence.\n\nHe had a scroll with Greek inscriptions stating that he knew all the arts of Satan and possessed the power to vanquish them. Sometimes he carried a book, given to all the holy fathers who left writings behind them. In some references Anthony is called illiterate, but, nevertheless, seven extant books are attributed to the saint.\n\nThe representations in art that covered the saint's life stressed the temptations, but at first these were presented with little elaboration. Temptations of a sexual nature were shown in separate scenes from those that dealt with physical tortures, as in the Chartres window. These scenes became elaborated in later treatments. In a presentation of the saint's physical tortures, shown in a French _Book of Hours_ of about 1380 (in the Biblioth\u00e8que Nationale, in Paris), Anthony is assailed by animals that (as in the Athanasius story) are those to be found in the natural world: the dog, lion, boar, wolf, and serpent. In an Escorial painting believed to be a copy of a pre-Eyckien Flemish painting, the scene of sexual temptation has become more elaborate. Saint Anthony is shown in five places in the one setting, standing or walking in a garden, surrounded by courtiers and ladies, or he can be seen through the window of a house \u2013 once besides a temptress, and once praying alone.\n\nThe saint's garden companions are making various telling gestures in allusion to the attraction of \"luxuria\". A painting of the physical torture theme (in the Doria Pamphili Gallery, in Rome, by the 16th-century Italian, Parentino) shows the demons as well-formed men whose only devilish aspect is the hair and wings that grow from their bodies. Hieronymus Bosch undoubtedly treated the theme more elaborately than any artist before or after him. His paintings allude to many kinds of temptations, in a single setting, which abounds with zoomorphic and anthropomorphic creatures. All later versions pale in comparison to the Lisbon altarpiece, Bosch's most comprehensive treatment of the temptations.\n\n#### The Prince of Darkness\n\nThe exact nature of evil had been determined to the satisfaction of the late medieval church. Its theologians had modified the Jewish inheritance on this score, derived from the Persian theogony of the equation between Good and Evil. Jewish orthodoxy did not equate the two. It limited the power of Evil, personifying this force in Satan, a powerful creature but one who was obliged to concede the superior power of God (De Givry, 22).\n\nAs part of the \"Good news\" of its gospel, the Early Christian church preached the victory of Christ over Satan and the invulnerability to Satan of Christians who maintained their faith. The medieval church altered this concept, however, by replacing the single notion of the Christian Devil with heathen demonism. Saint Thomas Aquinas insisted upon the existence of a kingdom of demons under the authority of Satan that could, with God's permission, contest with angels for the souls of the deceased, and if successful could then wreak a terrible vengeance for the sins that had been committed in life. The illiterate, who could not read the involved theological texts, could see the aspect and practice of the Devil and his cohorts sculpted on the church walls and painted on the windows and in the manuscripts \u2013 just as they could see images detailing the knowledge of God and his helpmeets.\n\nThe Last Judgement scene was the most frequently chosen to adorn the portal tympana and demons were shown there in their most hideous conformations. In the Last Judgement scene above the portal of the Cathedral of Saint Lazare, in Autun (11th century), the five demons who strive to weigh the scales in their favour or to plunge the damned souls directly into the dragon's mouth (the entrance to hell) display a consistent ugliness \u2013 with fluted bodies and grimacing countenances. The demons of later tympana, such as those in Bourges, are more varied in aspect and exhibit a sophisticated mockery in visage. Here, the artist of the Judgement scene formed composite creatures of oddly associated human and amimal parts. For instance, two of these demons display grinning faces on their bellies; one demon has a tail that is also a snake; the behind of another sprouts wings; and another has dog-faced breasts hanging from its chest. Hell is presented as a hideous creature lying upside-down, with jaws spouting flames blown to great heat by two demons pumping bellows. As if being boiled were not torture enough for the unhappy sinners in the cauldron set upon the furnace, they are also being pitch-forked, pounded, and gnawed upon by even more repulsive demons.\n\nSuch scenes, repeated throughout the churches of Christendom, were varied according to the greater or lesser imaginative powers of their artists. Impressed upon the viewers was the awful consciousness that one day these punishments might be theirs. They had no reason to doubt the existence of a Prince of Darkness and his terrible powers. In fact, the Church made such belief an article of faith; to disbelieve was heresy.\n\nAll of the aforementioned representations of demonism were the inheritance of Bosch, living as he did both in a cathedral town and in the century of the first dispersion of prints. To understand more precisely what the artist might have believed concerning the Devil, we can cite Martin Luther's belief, since the two men were contemporaries and Northern \"neighbours.\"\n\nSt. John the Baptist in Meditation,  \nend of the 15th century.\n\nOil on panel, 49 x 40.5 cm.\n\nMuseo Laz\u00e1ro Galdiano, Madrid.\n\nSaint John on Patmos, 1504-1505.\n\nOil on oak panel, 63 x 43.3 cm.\n\nStaatliche Museen zu Berlin, Berlin.\nLuther's writings are filled with the words \"Teufel, Teufel\" and such sentences as: \"This thing will I do, in spite of all who may oppose, be it duke, emperor, priest, bishop, cardinal, pope or Devil\" (Masson, 48). Most of these he saw as the Devil's servitors against him, and the Devil, himself, as his own personal enemy. Luther believed the Papacy to be one of the great human agencies through which Satan operated and that this evil being was fighting his own movement so strenuously because it threatened to destroy that vast edifice \u2013 the likes of which Satan would never again be able to erect on earth.\n\nLuther, too, shared his time's belief in universal satanic instrumentality, and he conceived the Devil to be the general enemy of mankind \u2013 operating to produce evil through a plurality of demons. These, he believed, were actual creatures that inhabited dark swampy woods and did harm to the passer-by \u2013 or they lived in storm clouds, stirring them to poison the atmosphere and earth with winds and floods. The demons worked in men's minds to delude the senses with hallucinations, dreaming, and somnambulism \u2013 they were responsible for sadness, melancholy, and insanity. Their more overt weapons were physical decimators, such as disease, famine, pestilence, and death. They needed human help, and thus worked through heretical activity of all kinds.\n\nOf course, the stigma of stewardship to this embodiment of evil could be freely applied to one's enemies, whoever they might be. To the orthodox Catholic, Luther was Satan's chief advocate. It was agreed by Luther and churchmen alike, however, that the most powerful of the Devil's human resources was witchcraft.\n\n#### The Society of Witches\n\nAccording to Grillot de Givry, belief in the practice of witchcraft had its great ascendancy in Europe from the 13th to the 15th centuries. The idea of a society of witches and the elaborate nature of their practices did not develop until the 14th and 15th centuries. However, the foundation for the witchcraft \"delusion\" was laid by Saint Thomas Aquinas. While seeking to define demonic operation on earth, he searched out scriptural passages concerning the Devil, and interpreted them according to certain beliefs of magic. These had been held in the West since Hellenistic times, and were newly introduced into Europe through the crusades and influx of Arabic learning. Saint Thomas drew upon beliefs already held in isolated form, of soothsaying, enchantment for good and evil effects, changing of humans into animals, the \"spiritus familiaris\", or spirit of a dead person in the service of a magician.\n\nFrom the scriptures, the saint derived the idea of demons as being angels who fell away from God's grace by their sin. He conceived the practice of sexual relations between demons and humans as being implicit in Genesis 6:1-4, where it is written that \"when the sons of God came in unto the daughters of men, and they bare children to them, the same became mighty men which were of old, men of renown\". These \"sons of God\" were interpreted to be the fallen angels and the voluptuous activities attributed to demons and to the Devil, alike, were largely traceable to this passage.\nThe ability of demons to fly through the air had scriptural origin from the story of Christ's being carried to the pinnacle of the temple by Satan, for if the Devil could give this power to one body, he could do the same with any other.\n\nThese ideas of satanic practice were elaborated in the popular mind after Saint Thomas' death and were increasingly attributed to humans who were thought to have made a pact with Satan. By the time of the energetic pope, Innocent VIII, witchcraft and magic had been branded the practical phase of heresy; the Inquisition was established to stamp them from the earth. Thus began the period of the greatest fear of witchcraft and the horror of the 15th\\- and 16th-century witchcraft trials.\n\nContemporary sceptics are likely to dismiss the notion of witchcraft too lightly; they think of it only as an appalling delusion resulting in the deaths of many innocent victims. Even if it were found to be only this, the psychological effects on the popular mind of the fear of witchcraft as a menacing reality are incalculable. Modern scholarship into medieval witchcraft is no longer sceptical. From the trial records, where the practices are admitted practically verbatim by one defendant after another, and from certain incontrovertible facts in the records, it has become increasingly apparent that there was some truth in the accusations. If fact can finally be separated from fantasy, it may be confirmed that witches were, as stated in that time, \"the active members of a vast revolutionary body, (constituting) a conspiracy against civilisation\".\n\nThere is a growing amount of evidence to show that a society of these implicit proportions carried on espionage in every land, with servitors in both the highest courts and in bodies of the lowest import. Their determined purpose was to do harm on every level in order to upset and destroy the established order. To this end, they committed the foulest, most wanton acts of victimising and betrayal. Their high-placed masters lured them into these crimes by extravagant promises of reward. In turn, they deceived, harmed, and destroyed their victims. One of the most potent weapons they were believed to possess was a knowledge of poisons, both helpful and hurtful, the cherished formulas for which were passed down in their lore from century to century.\n\nWhatever will be revealed as the truth concerning witches, it is the 15th-century belief in witchcraft as a favourite device of the Devil that is the concern here \u2013 particularly what Bosch might have understood of this practice. It was in 1484, a year when the painter was probably well established in his craft, that Pope Innocent VIII issued the famous Bull that for three centuries placed on European jurisprudence the onerous necessity of combating the Society of Witches.\n\nThe _Malleus Maleficarum_ , written by Henry Kramer and James Sprenger at the Pope's instigation and which contained the pope's _Bull_ as its introduction, is a compilation of all the existing knowledge of witchcraft. It is divided into three parts:\n\nThe First Part: Treating of the Three Necessary Concomitants of Witchcraft, which are the Devil, a Witch, and the Permission of Almighty God;\n\nThe Second Part: Treating of the Methods by which the Works of Witchcraft are Wrought and Directed, and how they may be Successfully Annulled and Dissolved;\n\nSaint John on Patmos (detail),  \n1504-1505.\n\nOil on oak panel, 63 x 43.3 cm.\n\nStaatliche Museen zu Berlin, Berlin.\n\n**After Bosch**  \nMarriage Feast at Cana, after 1550.\n\nOil on panel, 93 x 72 cm.\n\nMuseum Boijmans Van Beuningen, Rotterdam.\nThe Third Part: Relating to the Judicial Proceedings in both the Ecclesiastical and Civil Courts against Witches and indeed All Heretics.\n\nPublished in 1486, this book achieved vast prominence \u2013 fourteen editions being printed between 1487 and 1520, and sixteen editions between 1574 and 1669. It is said that the book lay on the desk of every judiciary in Europe, accepted by Catholic and Protestant legislation alike. It is a marvellous record of medieval courtroom procedure, which is described in cases presented in greatest clarity \u2013 and thus is one of the world's most important historical records. Since the _Malleus Maleficarum_ appeared only fourteen years before Bosch is believed to have painted the Lisbon triptych, it is perfectly plausible that he would have been familiar with its contents. The references that he makes in the painting are so nebulously presented, we can assume that he was not a scholar of the book, perhaps had not even read it; nevertheless, discussion of the maleficent practices described in the book was undoubtedly rife throughout Europe, and the spirit of witchcraft \u2013 if not its precise nature \u2013 was surely common knowledge.\n\nSummarily stated, the following was the makeup of the structure and activity of the Society of Witches. These personages were the feminine element of the larger category of sorcerers. According to the devious logic of the Middle Ages, it was believed that sorcerers and witches were the priests and priestesses, as well as minor servants of the Devil. The Bible presented the Devil as a fallen angel, but the fact that in _Job_ he conversed equally with God and even wagered with Him suggested that he was God's equal (De Givry, 56). Therefore, the Devil must control an earthly organisation of equal power to the Church. This organisation would probably be structured similarly to that of the Church, but since the ranks of the Church were largely closed to women they would surely be welcomed into the Devil's congregation. By this reasoning, they were believed to outnumber men by an immense number in that company. The chief motivating force was the belief that, since the Devil was the master of earthly powers and material possessions, he would reward their services by raising them from the unhappy estate to which it seemed God had condemned them.\n\nThere were believed to be many kinds of sorcerers with functions as various in kind. The most powerful group consisted of those in the direct service of Satan as priests and priestesses. It was this group that conducted the Devil's ceremonies, the largest of which was the Sabbath. This was attended by all the sorcerers of a certain district, and operated under the presidency of the Devil himself. The members of this group were divided into two categories: first, those who were in the actual physical service of Satan and obeyed his direct commands, and second, the sorcerers, or necromancers, who were empowered to call up demons, or the dead, and place these creatures in their own employ.\n\nAll sorcerers were believed to be endowed by the Devil with the same general powers of magic \u2013 of casting a \"sors\" or spell over those to whom they wished to bring some evil. There were those of lesser powers, however, who did not attend the Sabbath. Among these were the wanderers, known today as gypsies, who practiced the arts of divination by cartomancy (card reading), or chieromancy (palm reading). Also considered sorcerers were scholars who deviated from the Church and Aristotle in their teachings and investigations. Anyone who manipulated matter in a laboratory was believed to do so for an evil purpose. For want of a better explanation of the activities, it was said that this person had entered a pact with the Devil, who in exchange for the soul, had given power over matter. Such a belief is the basis of the Faustus legend and its manifestation in such other literary works as the _Tales of Hoffman_ , Verne's _Maitre Zacharius_ , and the _Coppelia_ ballet of Delibes. Placed among the ranks of sorcerers for what was believed to be their supernatural knowledge were monarchs, such as Henry III and his mother, Catherine de Medici. Even such monks as Roger Bacon and Albertus Magnus, and popes, such as Saint Leo the Great, Honorius, and Sylvester II were also believed to possess Satan's powers. Solomon was considered the greatest sorcerer of the ancients. The Black Book is supposed to have originated with him and his powers were such that he is, to certain Orientals today, one of the most revered monarchs of the earth[3].\n\nIt was usual for sorcerers to conduct their evil work alone, but on occasion, the Devil would call the lordlings of his organisation to the Sabbath. They would be summoned by the painful tingling of some red spot on their bodies, supposedly imprinted by Satan on some secret part. In order to prepare themselves for the journey, they met together with a few other sorcerers in a \"private Sabbath\". Here, after having removed their garments they prepared an unguent, or sorcerers grease, from \"the blood of the lapwing, and the bat, the raspings of bells, and soot\". Favoured sorcerers would anoint themselves with the compound and thereby be transformed into animal form \u2013 often that of a he-goat like the Devil himself. The time of transformation from one form to another was while the witch was passing through the chimney, the favourite way of leaving the house.\n\nMost of the witches would remain in human form and fly in the nude on their vehicles. These were pitchforks or brooms, which they anointed in order that the instruments would levitate and fly them to their destination. Vehicles were not necessary, however. The story was told of a witch who had brought her unguent pot to her Inquisitional trial and thus was able to escape her sentence by changing into a screech owl and flying off in the very presence of the judges.\n\nSabbaths were held in many places, but the most famous site was that of an ancient pagan altar on the Brocken mountain located in an untamed section of the Harz mountains in Germany. Geographers of this section would even include in their maps drawings of the witches hovering over the mountain on their brooms.\n\nSatan was to be found at the ceremonial grounds, enthroned before revelling worshippers \u2013 sometimes preaching to them. If he had assumed the form of a he-goat at this ceremony, one of his horns would be alight, and all the Sabbath lights would be fired from this source (the Sabbath was always held in the deep hours of the night). The animal forms assumed by Satan (among which were the ass, wolf, dog, cat, toad, bat, raven, monkey, dragon, and goat) were those that had been worshipped by pagans of antiquity, and thus represented all that was anti-Christian. The goat was the most frequent and definitive symbol, since it had Egyptian, Greek, and Jewish origin \u2013 in the latter case being derived from the scapegoat of the Israelites.\n\nAscent of the Blessed to the Heavenly  \nParadise  (first and second panels of the triptych Visions of the Hereafter),  \n1500-1504.\n\nOil on panel, 87 x 40 cm (each panel).\n\nPalazzo Ducale, Venice.\n\nAscent of the Blessed to the Heavenly Paradise  \n(third and fourth panels of the triptych  \nVisions of the Hereafter), 1500-1504.\n\nOil on panel, 87 x 40 cm (each panel).\n\nPalazzo Ducale, Venice.\nSometimes, if the Devil had taken a queen from among his favourites, she sat crowned beside him. Upon arriving at the Sabbath, the sorcerers and witches performed a ritual of homage to Satan by kissing his posterior (when brought to trial, they would protest that they kissed a face under Satan's tail, thus giving rise to the representations showing Satan with one or more supplementary faces on various parts of his body). The Devil would mark newcomers with the imprint of his claw on their left eyelid.\n\nThen he would require that they tread upon a cross, accept a Black Book in replacement of the Gospel, and be baptised with some foul liquid \u2013 all as evidence of a complete renunciation of Christianity. (In some representations of this ceremony, the novices are shown as having become blind, as they no longer can see \"The Light\"). Since Satan was especially eager for juvenile recruits, the witches on arrival would present before him their own children, which they were obliged to bring if they were not able to steal the children of neighbours.\n\nThe ceremony of the Sabbath took various forms. The worship service was supposed to be a reversal of the Christian mass. Satan would preside at an altar at which the crucifix was turned upside-down. He would make the sign of the cross with his left hoof, then read obscenities from the Black Book, and sprinkle \"unholy water\" (such as urine) in the direction of the assembly, who turned their backs upon the altar. In place of the Host, a turnip (painted black) was elevated.\n\nAfter this parodied church performance, the carousing would begin. A feast was prepared, the fare for which was described variously as being delicious wines and foods; but, as quoted by De Givry from Pierre de L'Ancre's _Tableau de l_ _'_ _Inconstance des Mauvais Anges et Demons_ (1610), as \"carrion, and the flesh of those that have been hanged, and the hearts of children not baptised, and other unclean animals strange to the custom and usage of Christian people, the whole savourless and without salt\". L'Ancre continued:\n\nThe belly filled, the dance begins; for after they have devoured meats either fleeting and illusory or most hateful and abominable every demon leads the witch who was his neighbour at table beneath the accursed tree, and there, one facing towards the centre of the dance, the next towards the outside, and so on for all, they dance in round, stamping and capering with movements the most indecent and obscene that they are capable of.\" (De Givry, 81)\n\nSome would perform acrobatic feats, finding themselves endowed with muscular prowess they did not possess in a normal state.\n\nSince the purpose of the Sabbath meeting was presumed to be for the marshalling of the energies of the sorcerers that they might produce harm to Christians, they were thought to be constantly coming and going from the Sabbath on their nefarious errands.\n\nThey returned to re-equip themselves with a hurtful poison brew that was continuously being made by the witches who stirred and replenished a boiling cauldron with toads and serpents. The witches would leave on their journeys by way of their broom or pitchfork handles unless specially favoured, in which case they were allowed to fly on demons that had been transformed into he-goats. They would fly across land, \"hurting or killing men and marring cattle\" or \"over the sea and elsewhere to stir up storms and tempests\".\n\n#### Other Sorcerers and Necromancers\n\nThe sorcerers described above were those who placed themselves in the service of Satan. There were also those who held equal, if not greater power than the first group, since they could command the services of demons themselves. These sorcerers attended the Sabbath, where they were servile courtiers of the Devil-king. But they had the power to hold their own gatherings of demons, upon which they imposed strict obedience. Sometimes several of these sorcerers would meet together to evoke demons. If they preferred to work alone, it was with the aid of one or two assistants.\n\nThey met in some deserted outpost, unfrequented because of the superstitious fear in which the place was held by ordinary people. It would best be a ruined monument (such as an abandoned abbey, a cathedral, or castle) which was shrouded in overgrowth of brambles and vines; the time would be in the deep hours of the night when the earth was lit by the moon or lightning, as the demons could most easily be seen under such conditions.\n\nThe place reached, the sorcerers would attire themselves in white linen ephods covering priestly black bombazine robes, which reached the ground, the whole tied round the waist with a consecrated girdle. They wore shoes inscribed with the word Tetragrammaton (the name of God), ringed with crosses, a high crowned hat of black silk, and held in one hand a Holy Bible, written in pure Hebrew. One associate held a lantern, one a naked sword, and one a Black Book. Thus attired, the chief sorcerer, or master, drew a charmed circle upon the ground with a consecrated knife. The form of such signs varied, but a typical one consisted of a circle with a nine-foot diameter, another circle placed about a foot within it, the space between them inscribed with names of the Creator.\n\nInside the circles, but not touching them, was placed a square and another square turned within it at a 45 degree angle, so that its sides formed the bases of triangles with those of the first square. A smaller circle was placed about a foot inside the second square, and a square within it, whose sides were parallel to those of the second square. The spaces formed by the various straight and curved lines were filled with inscribed names of God. Then the encircled area was blessed and sprinkled with holy water in order to cleanse it of all impurity. Finally, the sorcerers stood or seated themselves inside the innermost square within the small circle \u2013 these forming a barrier so strong by virtue of their consecration that no demon could break through. Thus prepared, the chief sorcerer would proceed to evoke the demons, incanting the awful ceremony of exorcism from the Black Book.\n\nSaint Christopher, 1490-1505.\n\nOil on panel, 113 x 72 cm.\n\nMuseum Boijmans van Beuningen, Rotterdam.\n\nTriptych of the Hermits, c. 1505.\n\nOil on panel, 86.5 x 120 cm.\n\nPalazzo Ducale, Venice.\n\nTriptych of the Hermits (detail),  \nc. 1505.\n\nOil on panel, 86.5 x 120 cm.\n\nPalazzo Ducale, Venice.\nSoon, the atmosphere would be disturbed with flashes, tremblings, and rumblings rising to ear-splitting yells and shrieks that would accompany the painful process by which the demons were rendered visible. At first, they appeared as a snarling mob of wild beasts roaring and spitting fire towards the sorcerers in the circle. These men dared not exhibit fear but continued to chant the magic words, which would draw the animals towards the charmed influence of the circle. There, in the holy presence created by the inscribed names of God, the beasts would change into submissive human form and be ready to do the will of the master. He should not ever lose his guard against them and be fooled by their benign demeanour to step out of his magical enclosure into their company. They were ever seeking to turn the tables on the master, and either place him in their power or destroy him. After the spirits had done the master's will, he must exercise the greatest care in sending them away, he and the others waiting patiently within their protected place until the demons had disappeared from sight and their last shrieks could no longer be heard.\n\nEven more terrible were the Necromancers who had the power to command the dead and have them come forth from the Underworld to reveal its secrets, foretelling the futures of human beings, as well. \"Necromancy was practiced assiduously in the Middle Ages, either in making the dead appear, or if they were recalcitrant, in exhuming corpses and examining them.\" This was done, sometimes in anticipation of feeding on their flesh in a \"deathly feast\". These dreadful practices lasted until the 19th century, finally replaced by \"spiritualism\", or communing with the dead through table-tapping or an unheld, moving pen.\n\n#### The Tarot\n\nThere are other aspects of sorcery involving imagery that Bosch seems to have tapped for his demonological content. For instance, the symbols associated with the tarot cards are very similar to many of those that Bosch employed in the Lisbon triptych. The Tarot was a card game utilised by cartomancers, who were considered to be a party to sorcery. This was not a game, in our restricted sense of the word, but a systemic view of the universe through which one could read omens and ascertain his or her part in the whole scheme of things. Although its beginnings are murky, one view has it originating in 13th-century Fez, Morocco, a city that became the intellectual centre of the Near East after Alexandria's destruction. The philosophers from many parts of the world who met there for learned discussion found themselves hampered by the lack of a common terminology, so they adopted a language of pictures expressing ideas through images instead of words. The images permitted further associations and, therefore, even more involved connotations. The Tarot is said to be in reality a book of wisdom but is presented in the form of a pack of cards, with each card corresponding to a page of the \"book\".\n\nOf the Tarot's seventy-eight \"pages\", fifty-six are called minor trumps. These are divided into four suits, symbolised by four emblems: the staff, cup, sword, and coin. The four objects symbolise the four basic professions in medieval society: staff\/peasant; cup\/clergy; sword\/soldier; and coin\/merchant. Further aspects of medieval life are embodied in the four court cards that form a part of each suit. These are: King, or Lord\/Spirit; Queen, or Lady\/Soul; Knight\/Ego; and Page, or Servant\/Body. They are each followed by ten cards ranging from Ace to ten.\n\nThe most important part of the tarot pack is the remaining set of twenty-two cards, called major trumps. When the titles of these cards are said or illustrated in sequence, each noun calls to mind an image with which every human mind, regardless of language or race, makes basic idea associations. The major trumps, or cards of the Major Arcana, follow an immutable order, outlined by tradition:\n\n1. Fool (sometimes shown as a man performing tricks before a table)\n\n2. High Priestess (sometimes wearing a papal tiara)\n\n3. Empress\n\n4. Emperor\n\n5. Pope\n\n6. Lovers (sometimes called marriage)\n\n7. Chariot\n\n8. Justice\n\n9. Hermit\n\n10. Wheel of Fortune\n\n11. Strength\n\n12. Hanged Man (usually shown as suspended upside down by one foot, but with some dispute as to whether the traditional designation of this card as the \"Man with the Suspended Foot\" might not have meant, simply, a man prudently holding his foot in the air before taking a step, to assure his secure footing \u2013 some card packs used Prudence for this number)\n\n13. Death\n\n14. Temperance (sometimes shown as a woman pouring liquid from one vase into another)\n\n15. Devil\n\n16. House of God, Hospital, or Lightning-struck tower (with confusion concerning the meaning of this image; some associated it with a place for the care of the sick-poor, but others saw it as the athanor of the alchemists, since it \"had to receive a tongue of fire from heaven, at which the imprudent who could not foresee it fell, thunderstruck\"\n\n17. Star\n\n18. Moon\n\n19. Sun\n\n20. Judgement\n\n21. World\n\n22. Fool (De Givry, 288, 284-286)\n\nThe Temptation of Saint Anthony,  \nafter 1490.\n\nOil on panel, 73 x 52.5 cm.\n\nMuseo Nacional del Prado, Madrid.\n\nThe Pedlar, c. 1505.\n\nOil on panel, 71 x 70.6 cm.\n\nMuseum Boijmans van Beuningen, Rotterdam.\nCartomancers attempted to read the future by employing these cards, sometimes using the entire tarot series, calling it the Great Pack. It is apparent that if these cards were placed as they would fall by chance dealing, the combinations of numbers and suit emblems of the minor trumps as well as the picture images of the major trumps (with all of the many meanings involved in each) would result in infinitely complex interpretations. Truly, the Tarot would be believed to be cosmological in scope.\n\n#### Alchemy\n\nThe final large system of imagery employed by Bosch seems closely associated with the Tarot in its grandiose attempts to encompass universal thought. However, since fact was scarcely yet separated from fantasy in this embryonic stage of modern science, the realm of the alchemist was more that of magic than of reason. Alchemy had a history extending for over a thousand years, but its ascendancy seems to have been in the 15th century. Perhaps this is due to the development of the printing press and the resultant dissemination of ancient knowledge through the books chosen to be printed during that century. The system of alchemy discussed here was certainly accepted, if not formed, in the 15th century. It seems reasonable to assume that an artist like Bosch, who already took such liberties with traditional iconographic programmes, would be attracted by such imagistic material.\n\nSome think the origin of alchemy was in Egypt, or the land of Ham, called Khem. The name passed into Arabic writing as Al Khem and eventually was transliterated into its present form (Read, 5). Twelfth century Latin translations of Arabic writings were the media through which the knowledge was brought into Europe. Fifteenth-century printing of the old translations, together with contemporary writings on the subject, fostered the interest prominent in Bosch's time. Based on a postulation similar to that of Aristotle, alchemists believed that matter was composed of four fundamental qualities conceived in Pairs \u2013 hot and moist and their contraries, cold and dry. These properties were identified with what were believed to be the four basic elements: fire, water, air, and earth.\n\nThey were considered to be present in various quantities in all substances of matter. An inevitable observation was that transmutation occurred when these elements were added to one another. When fire was applied to water, hot moist steam resulted \u2013 retaining qualities of the parent elements. It followed that any substance could be made by adding together substances with the proper combination of elements \u2013 but the parental elements were not always so obvious in the result as in steam. Therefore, it was only experimentation that could discover by what combination of unknown substances a third, known substance could be produced. The \"Philosopher's Stone,\" being perfect matter, would serve as a medicine for humankind, repelling disease and old age. With the stone, one could achieve the secret of life, health, and good fortune.\n\nThis would explain the extraordinary motive power that occupied the alchemist's life with experimentation, and \u2013 if we believe the Bruegel engraving on the subject ( _The Alchemist_ , c. 1558), reduced his family to starvation. The alchemist's patient study of matter became formularised; in the medieval fashion it was translated from direct experience into allegorical terminology. By this means it was both communicated to other alchemists and concealed from the uninitiated. The provocative symbols, whether understood or not by the popular mind, found their way into literature and art.\n\nA large part of the alchemists' \"language\" concerned the processes by which they performed their experimentation. By the Middle Ages, the theory of the Four Elements had taken the practical form of the Sulphur-Mercury theory of the composition of metals. Observing that substances were either dry (and subject to burning) or liquid, it was assumed that the most important element of the first group was fire and of the second group was water. Therefore, fire and water were named the most important of the Aristotelian elements. The name sulphur denoted those substances of a dry base. Similarly, mercury connoted the liquid substances. If mercuric and sulphuric substances were joined in the proper quantities and qualities, metals would result. Thus, if impure sulphur and mercury were combined, base metals such as tin, lead, and quicksilver resulted. If the two materials were combined in a state of ordinary purity, gold was the product; if the \"quintessence\" of sulphur and mercury were used, the superfine gold known as the philosopher's stone was achieved. The latter was believed to exceed the quality of ordinary gold by such an extent that it would transmute, or \"tinge\", base metals into gold (Read, 14).\n\nSulphur and mercury were symbolised in various ways, but primarily as male and female entities. The solidity and combustibility of sulphur were considered masculine properties; therefore, it was often shown in the form of a man. The liquidity and fusibility of mercury prompted its denotation as feminine, or woman. Since these principles had acquired human characteristics, a human attraction leading to marriage was assumed to exist. The offspring of the marriage would be the philosopher's stone, often represented as an infant or as an egg. Sometimes the maleness of sulphur and the femaleness of mercury were represented in fusion as an androgynous creature (De Givry, 367). Sulphur and mercury were combined in a complicated series of operations, the exact number of which is not clear. Sometimes twelve processes were given in the writings, namely; Calcination, Congelation, Fixation, Solution, Digestion, Distillation, Sublimation, Separation, Ceration, Fermentation, Multiplication, Projection. There were just as often said to be seven or fourteen. If twelve, they were \"suggestively represented by the signs of the Zodiac\", as for instance; Distillation was shown as a virgin (Virgo), Multiplication as a water-carrier (Aquarius), and Projection as a fish (Pisces) (Read, 136).\n\nThe \"Great Work,\" as this system of processes was called, involved the use of certain vessels and pieces of apparatus. The most important were the furnace, or athanor, sometimes represented as a tower, and the hermetically sealed vessel of an ovoid shape, represented by the egg. The latter was the vessel in which sulphur and mercury were fused. They were mixed with a menstruum containing salt, sealed in the vessel, and placed in the matrix of the furnace. This was constructed in such a way as to produce different degrees of heat, nine each of which would result in a certain stage of fusion within the vessel. These degrees were named and symbolised in various ways. Because the belly of a horse was of the same temperature as one of the degrees of the furnace heat, the horse's image would often be used to depict this stage of the process.\n\nTriptych of Job (exterior),  \nc. 1500-1524.\n\nOil on panel, 98.3 x 132.8 cm.\n\nGroeninge Museum.\n\nTriptych of Job, c. 1500-1524.\n\nOil on panel, 98.3 x 132.8 cm.\n\nGroeninge Museum, Bruges.\nColour was very important in the operation. John Read wrote:\n\nThe appearance of red before black showed that the material had been overheated, and the Work had then to be started afresh. Fire was denoted by such terms as sword or scissors: thus, to cut off the head of the Black Crow meant to continue the heating, digestion, etc., until the black colour changed to white. If a black colour put in an unwanted appearance later in the process, the alchemist exclaimed irascibly to his 'yeoman' or 'minister' that the young ones of the crow were going back to their nest, and then \"there had never been such woe or anger or ire\" in the laboratory. All the colours of the Great Work were supposed to reappear, in a more rapid and transitory manner, during the operation of multiplication. Sometimes the stages of the Work denoted by the appearance of characteristic colours were known as regimens: such were the regimen of Saturn (black), the regimen of the Moon (white), the regimen of Venus (green and purple), the regimen of Mars (rainbow), and the regimen of the Sun (red or golden). These regimens were sometimes represented by flowers.\n\nThe colours mentioned were already identified with the four elements \u2013 red with fire, white with water, citrine with air, and black with earth. These colours were supposed to manifest themselves in the various stages of the operation. They were sometimes represented by birds, such as the phoenix for red, the swan for white, and the crow for black. Winged creatures, such as serpents, dragons, and lions with wings, were also used to portray the volatile materials (sulphur and mercury) and wingless creatures the fixed material (salt) of the process. The conjunction of volatile and fixed principles was often shown as a griffin (half lion, half eagle). At times, a serpent or dragon represented the menstruum in which sulphur and mercury were mixed. Sometimes, the Great Work was referred to as a fort, or citadel, surrounded by a moat of water. Many false roads led to the citadel, but it could only be reached by way of the drawbridge \u2013 open only to those of unimpeachable personal qualities. Once inside the citadel, this adept found various chambers in which the operations of the hermetic process took place. He had to perform all of the processes before being allowed to enter the chamber where the philosopher's stone could be found. This place was guarded by a huge dragon, which would yield the stone only to the one who had accomplished all of the prerequisites (De Givry, 349).\n\nEven the minor implements used in the process had their symbolic substitutes. For instance, the ostrich stood for the flask and the stork for the retort. A pelican was depicted for a vessel called by this name because it resembled the bird. The pear was used for the aludel \u2013 a vessel shaped like this fruit \u2013 and the gourd for the cucurbite, a vessel shaped in this manner. Fire was of such importance to the procedures that the alchemist was often called \"the child of fire\". Fire had its animal guise, the dog; it was also revealed in the form of cutting, penetrating, and wounding implements (Read, 143). The whole process described above was given such cosmic significance by the adepts, that it was likened to Creation itself. Alchemists believed that its greatest testament was the first chapter of Genesis (De Givry, 350). Allied with this concept was the ancient doctrine of hylozoism, or belief that all matter is living and subject to the laws of growth (Read, 12, 94). Since animal gestation was said to follow the same progression as the formation of the world, then there must be metallic gestation that followed the same order. The hermetic vessel was conceived, therefore, as the universe, and the alchemist was playing out its creation in a microcosmic echo.\n\nThe seven metals, base and fine, produced in the process of the \"search\" were identified (in the typical medieval fashion of contriving an alliance merely because of the similarity of the numbers involved) with the seven heavenly bodies and their attributes acquired in the astrological system. Sulphur was linked with gold and the sun, and mercury with the moon and silver \u2013 thus harking back to a sun-god and moon-goddess of ancient theogonies. The heavenly bodies had acquired mythological names, because they were believed to be gods of the heavens. Their names were derived from the association with certain attributes of the gods for which they were named. Venus, considered a warm friendly star, was thereby named for the love-goddess.\n\nThe red (fiery) Mars was named for the war-god. Similarly, Saturn, the highest of the stars, was named for the oldest of the Olympians, the father of Jupiter. Dethroned by his son and imprisoned in the depths of the earth, Saturn recalled estates of disability, misery, and death.\n\nThe names of the heavenly bodies suggested a mythological significance of alchemy. The story of Saturn took on a special significance to the adepts since the stone which Saturn was given by Rhea to swallow, instead of her child Jupiter, could be interpreted as being the philosopher's stone. Therefore, alchemists were believed to be so-called \"children of Saturn\". This planet, thought to be slow-moving (explaining the alliance with the heavy metal, lead), was often depicted in the guise of an old crippled man (sometimes wooden-legged). Often this image was used to represent alchemy as a whole. The \"old man\" was sometimes given a scythe (the cutting instrument that could be interpreted as fire), and, often, he would be shown in the act of cutting off the feet of a man holding a caduceus (signalling Hermes, or mercury). Saturn was sometimes shown with instruments of measurement, such as the hourglass, scales, or compass \u2013 probably in reference to the doctrine borrowed from the Wisdom of Solomon \u2013 that in all things inhere \"measure, number, and weight\". Since the mythological Saturn had swallowed his own children, children at play were often shown with, or near-by, the old man. They also referred to a dictum of the alchemists who believed that once the primitive materials of the stone had been obtained, the rest of the operations of the Great Work are \"only a labour fit for women, or child's play\" (Read, quoting Isaac of Holland, 134).\n\nTo illustrate the \"labour fit for women\", they were often shown as washing clothes in a stream. Even the watery setting was significant, because water was believed necessary to counteract the effect of the melancholic dryness associated with saturnine mysticism. To explain this mysticism, with which it was believed all alchemy was imbued, we must explain the Four Humours theory, closely associated with alchemy.\n\nTriptych of the Temptation of Saint Anthony  \n(left exterior panel: The Arrest of Christ), 1505-1506.\n\nOil on panel, 131.5 x 53 cm (wing).\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nTriptych of the Temptation of Saint Anthony  \n(right exterior panel: Christ Carrying his Cross), 1505-1506.\n\nOil on panel, 131.5 x 53 cm (wing).\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony (triptych), 1505-1506.\n\nOil on panel, 131.5 x 119 cm (central panel),\n\n131.5 x 53 cm (each wing).\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel),  \n1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nAccording to Erwin Panofsky: \"This theory, fully developed by the end of classical Antiquity, was based on the assumption that both the body and the mind of man were conditioned by four basic fluids which in turn were supposed to be coessential with the four elements, the four winds (or directions of space), the four seasons, the four times of day, and the four phases of life.\"\n\nIt was reasoned that the material aspect of human beings combined the elements, but not in perfect proportions, since there was no perfect human being (this privilege having been forfeited by the \"Fall\"). The disproportionate amount of one element over the other determined the nature of each person. The element in question would take the form in the body of over-abundance of some bodily fluid. For instance, an excess of blood produced the sanguinary person \u2013 correlated with air, the soft gentle west wind, spring, morning, and youth. An excess of yellow gall would produce the choleric temperament; associated with this humour were the element of fire, the southwest wind, summer, midday, and maturity. Black bile would produce the melancholy personage, identified with earth, the north wind, autumn, evening, and middle age. Phlegm produced the phlegmatic personage, related to water, the south wind, winter, night, and old age.\n\nAll creatures (including humans) and substances were believed to be under temperamental influences according to which of the elements predominated in their make-up. Since the planets were believed to be of the same matter as earth, they must be formed of combinations of the elements and dominated by the humours, even as were earthly bodies. Saturn was designated as the melancholic planet because of the unhappy connotations of the Saturnine story; all that was associated with the temperament of melancholy was also attributed to the planet and vice versa. All human beings, animals, plants, minerals, etc., believed to be governed by this temperament belonged to Saturn. In turn, all that was given to Saturn was influenced by melancholy.\n\nThis included, among the human beings, not only those of the most unfortunate of dispositions, but also those of the most unfortunate situations in life. Prisoners, beggars, grave-diggers, privy-cleaners, fools, and the insane were all melancholics. But the story of Saturn implied a contradiction. Even as he was the most unfortunate of Olympians, he was at the same time the oldest and the sagest \u2013 having thought out the world which Jupiter had only inherited. Therefore, he could give power, riches, or wisdom to those of like identity. Such people, if they could be called fortunate, as philosophers, geometricians, alchemists, and the solitary models were also included among the \"children of Saturn\"; they were melancholics as well. In fact, if these people could bring their unfortunate predisposition to states of depression and over-excitability into control, they would be the most superior of beings. To accomplish this equilibrium, they would have to escape from the pernicious influence of Saturn, through developing the capacity for divine contemplation, which Saturn had possessed.\n\nBy using certain talisman they could call upon the aid of Jupiter, the only heavenly body with a counter-influence over Saturn. They could also make use of music, considered antidotal to melancholy. As has been pointed out, alchemy was considered under the special aegis of Saturn, therefore imbued with saturnine mysticism and the spirit of melancholy. The alchemical allusions in Bosch's Lisbon triptych can be explained by the fact that, as a heretical operation, alchemy would belong quite as naturally as any other heresy in the devil-versus-saint theme. However, there may be a stronger motivation for the saturnine-melancholic quality of the painting than merely that these influences were related by deductive reasoning to alchemy.\n\nIn an article called \"The Temptation of Saint Anthony, or the Dream of Melancholy\", Andr\u00e9 Chastel mentioned the social importance and prosperity of the hospital order of the Antonites, which flourished during this time. He stated, however, that there is another reason, more important psychologically, perhaps, that of the connection of the saint with saturnine mysticism. It has already been mentioned that monks (particularly the hermits devoted to solitary study) were included among the melancholics. The writer said that it was a commonplace of the iconography of the temperaments to represent melancholy by the image of a solitary monk. Moreover, he referred to an astrological manuscript of the Library of Tubinque that contains as a sign of Saturn, the image of a monk with a Tau impressed upon his shoulder \u2013 in other words, Saint Anthony.\n\nThere would be definite iconographical precedence, therefore, for linking the saint with the humour and its planet. A more compelling psychological reason for Bosch's having done so in the Saint Anthony triptych would be the ambivalence expressed in this dual influence, which could at once throw the spirit into stupidity and discouragement and exalt it to metaphysical meditation. Furthermore, since melancholy was often represented by reference to the vice of Acedie, or Sloth, this would emphasise the sin to which a solitary monk would be most prone \u2013 and would consider, therefore, the most formidable of the sins. At the same time, melancholy epitomised the heights to which a holy man could attain. When D\u00fcrer embodied the temperaments in his painting of the four Apostles, he gave the darkened mien of melancholy to Saint Paul, personifying this temperament on the highest level.\n\nSaint Anthony's drama, with its alternating states of crisis and serene contemplation, would find expression in the saturnine-melancholy theme. At the same time, the saturnine-melancholy drama would translate into religious language through the conflict of good with evil \u2013 or the saint with the devil. This theme could be symbolised by many hagiographical stories \u2013 but most clearly, indeed, by that of Saint Anthony. There might have been another more topical reference that Bosch was making in employing the theme. The mounting tensions within the church in the pre-Reformation era would make obvious, even to the profoundly orthodox, that there was a conflict between the elements of traditional humility, simplicity, and purity and the elements of defection and exploitation. This conflict within the Church would find its allegory in the saturnine mysticism, which enveloped alike those of such opposing nature.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel),  \n1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\n### The Lisbon Triptych\n\nThe pre-eminent treatment of the _Temptation of Saint Anthony_ theme by Hieronymus Bosch is found on the triptych now in the National Art Museum of Lisbon. There is some confusion as to whether the painting was presented as a royal gift from Philip II to the king of Portugal, or whether it was purchased from Philip's collection by a Portuguese painter. By whatever means, it is known to have entered Portugal between the years 1525 and 1545.\n\nWhen closed, the altarpiece is approximately four square feet (1.2 metres) in size, slightly larger in height than in width. It is nearly eight feet (2.4 metres) in width when the panels are opened out. The exterior surfaces of the panels are painted in grisaille, giving an unearthly appearance to the Passion episodes presented thereon. Christ is centred in each panel, and backed by frenzied mobs in such a way that the main action in both paintings is concentrated in the upper registers. The minor episodes, set in the foreground areas, are isolated by space. There is no coherence of action from one scene to another, but there does seem to be a continuity of setting. The stream that separates foreground from middle ground is continuous, as is the band of darkened land seen behind the two mobs. There is a difference in time of action, however, as the sky is dark in one scene and light in the other.\n\nThe left panel depicts the arrest of Christ in the Garden of Gethsemane. The Lord, fallen to his knees, submits to the taunts of the crowd, delivered with gesticulation and facial contortion. A soldier handles the girdle of the Master as if it were a harness. He kicks at Christ from behind. A man in front, participating in this terrible game, pulls Christ forwards by tugging at the girdle strings. There is the illusion of chaotic action as the figures in the forefront of the crowd turn and bend in all directions. The impression of a dinning noise is conveyed as men clash weapons, blow horns, and open their mouths in outcry. In the background of the crowd, individual figures dissolve into a many-headed Hydra. Below the stream, Saint Peter raises his sword to cut off the ear of Malchus, whose belongings lie about him in disarray. The only observers to the scene are a man turning his back, as if in cowardice \u2013 and the ravens idling on the ground.\n\nThe right panel, depicting Christ Carrying the Cross, shows the Master fallen under the weight of the cross. This mob (constructed of different personages than the one on the left) is momentarily stilled to await the Cyrene's assistance to Christ. Saint Veronica displays the sudarium, which has received the miraculous transfer of the Lord's face.\n\nIn the foreground of this scene are the two thieves: the penitent one listening to the earnest entreaties of a confessor, and the impenitent one hearing, but not heeding, the advice proffered to him.\n\nThe presentations are somewhat unorthodox in setting, with parched earth and curious laminations of the land. The accoutrements assembled here are not those of any traditional precedence. There is a chalice elevated upon a hillock. A rat hangs suspended upside-down from a tree. Bones are scattered upon the ground. It could be that these are cryptic symbols, yet it is possible that their presence here is merely to suggest the desolation of this place. Certainly there is little in the scenes to disabuse the viewer of the intended message \u2013 that Christ, archetype of good, was insulted, beaten, and then killed, in a desecrated place \u2013 by evil, blind humanity. Here is the prologue of the entire triptych \u2013 the good man's travail, having its prototype in the tragedy of Christ's story, being replayed in the hagiographical episode.\n\nWithin the altarpiece there is much more complication. Time has elapsed from the death of Christ until the ordeal of Saint Anthony. Evil has multiplied; wickedness is omnipresent. The panels of the altarpiece unfold to reveal its temporal environment \u2013 another earthly place of desolation. Here is no simple Calvary place. This is many settings of imbricated dramas. At one and the same time, this is the tomb outside Anthony's village where the Devil first sent his demons against the saint to lash him into submissiveness. It is where Anthony challenged Satan to do his worst and was pummelled and goaded by the phantom beasts returned in answer from the infuriated Devil. In this place the Lord stood by to watch the holy man stay steadfast in action. It is the deserted fort of Pispir on the \"Outer Mountain\" where the saint settled down among the creeping things that infested the ravaged building, and where he again fell prey to the tumultuous spectres of demons. This is also Anthony's final retreat on the \"Inner Mountain\" where he was plagued by invisible foes until his death.\n\nNot only the hermit's retreat, this is the hidden ceremonial ground of the Witches Sabbath on which Satan's worshippers converged. It is the ruined monument (the castle, fort, or monastery) where the necromancer performed his evocations. It is the lightning-struck tower of the Tarot, the citadel of alchemy, and the melancholy setting lit by Saturn's astral light. Not only a place, however, here the fort is suspended in the centre of a square, each corner of which is dominated by an element. By this device, it becomes the macrocosm of the alchemist and qabalist. Bosch superimposed the lot to create his mise-en-sc\u00e8ne.\n\nAnthony's \"tomb\", or \"fortress\", is laid open in the manner of a theatrical set, its roof and forewall removed as if to reveal action on the centre stage. In fact, Bosch's arrangement of levels for subordinate and dominant plots seems to anticipate later theatrical planning.\n\nTwo platforms rise above the watery foreground; the upper platform is separated from the lower one by a curving parapet through which two steps are cut. A high, crumbling wall extends around the upper platform to form the sides and back of the fortress. The wall begins on its right side with a tower ruin and ends on the left with a set of screens that form a backdrop to the actors on the platform. A portion of vaulted ceiling hangs cantilevered from the back part of the wall \u2013 the cracked edges and curious angle at which the fragment curves suggest an enlarged piece of eggshell more than an architectural structure. The tie rods below the overhang would have been required to hold the walls together in a vaulted room, but their placement here does not suggest any attempt on Bosch's part at accuracy. As with his every structure, this \"room\" is the fanciful projection of a man who does not observe details, nor how things are constructed. The illusion is what the painter sought; this is a fitting, even spectacular, arena for the action of Saint Anthony's story.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nPlaced in the exact centre of the central panel, the saint kneels against the low parapet wall. He points with two upraised fingers to the fort before him. Looking back over his shoulder as if to attract the spectator's gaze, he directs it with his benedictory gesture towards Christ, standing beside an altar in an arched recess of the back wall. A beam of light from a side window illuminates the Lord as He, in turn, points towards the crucifix on the altar. His presence here represents two things: the Lord's appearance to the saint in the narrative of Saint Anthony, and Bosch's message to the viewer. It is as if he spoke (in visual terms) to the worshipper \u2013 to look steadfastly towards the Lord and not gaze on evil things, no matter how overwhelmed the world has become with such distractions.\n\nChrist points to the crucifix to remind the saint of His own great suffering as interpreted by Bosch on the outer panels. If these two figures are the only representatives of good on this canvas, the message is that there is very little of that quality in the world. Bosch endowed the saint with some of his traditional iconographical attributes. He is an old man, bearded and with a cane beside him. The crutch lies a distance from him, at the edge of the lower platform. The saint wears the habit and cowl of a monk, designating his authorship of monachism. His shoulder is marked with a T, the first letter of Theos, indicative that he is one of God's elect. From the monk's belt hangs the Tau cross, worn by Antonites; this not only signifies the Theos reference, but also the resemblance of the letter's form to Anthony's crutch. Bosch did not show the hog, which was often placed near the saint to denote the gluttony and sensuality that he conquered. Instead, he gave pig-snouted faces to two figures near Saint Anthony, one, the black-clothed man coming up behind the hermit and, two, the tonsured priest reading from a book at the edge of the platform. For the fire associated with Saint Anthony, there is the burning village in the background of the painting.\n\nAround the saint the painter arrayed the Devil's hosts, given the many forms that devilry assumed in his time. Bosch fashioned this crowd compositely, not only from the invisible demons of Satan's kingdom (rendered visible by the Devil, or a human agent), but from attributes of the Devil's human agents, the sorcerers. On the same platform Bosch placed a necromancer dressed in white ephod and high crowned hat, who leans back against the low forewall to survey a handiwork of his design. The foot lying on the platform before him undoubtedly refers to an appalling aspect of the disease \"St Anthony's Fire\", that caused limbs of the sufferers to become gangrenous, necessitating their amputation.\n\nTo the necromancer's right, a tonsured helper is incanting the awful ceremony of exorcism from the Black Book. This is the time of quiet after the storm of atmospheric flashes and tremblings, ear-splitting yells and shrieks accompanying the painful process by which the demons were rendered into bestial form. Bosch substituted the holy presence of Christ and the saint for the purified circle of necromancy. Drawn into this charmed area, the demons change into submissive human form. To emphasise the evil nature of necromancy, suggested in the snouted face of the helper reading the book, the painter opened a rent in the man's ephod to expose his rotted entrails. This man's pig-like face refers not only to necromancy and to the saint's iconography, but to the Witches Sabbath as well. His book with its black pages and light letters is suggestive of the book of the Black Mass read at that ceremony. That the book is held here by a man in priestly robe implies that men of the Church were among those who served the Devil.\n\nThe parodied Eucharist enacted on the platform by Saint Anthony also suggests the Sabbath ceremony, but the painter drew the participants from many contexts. The Moorish man might represent Satan himself, who once appeared to Anthony in the form of a black child. Black was a colour often applied to the Devil, as well as to evil in general. The egg held aloft by the Moor may signify \"the spirit of fornication\", which appellation the Devil claimed while masquerading thus. In addition, the egg relates to the Philosopher's Stone, the \"sealed vessel\", and the universe (or, that is, the cosmological nature of alchemy as a whole). The toad displaying the egg is an animal form the Devil was believed to assume. The lizard and dog, others of the Devil's disguises, emphasise his ubiquitous nature. The owl on the snouted man's head represents the heretic \u2013 who will not see the truth.\n\nBetween Saint Anthony and the Moor are three women acting as priestesses who serve the bread and wine. Kneeling beside the saint and crowding him in an irritating attempt at distraction, one of these women passes a plate to a nun. At first, the woman appears to be dressed fashionably; further observation shows her skirt to end in a tail-like train and her headdress to be tucked up to reveal a thorny vine trailing down her back.\n\nFr\u00e4nger suggested in his description that the vine referred to the wickedness ascribed by Moses to those that break the Lord's Covenant, when he warned the Israelites: \"For their vine is of the vine of Sodom, and of the fields of Gomorrah: their grapes are grapes of gall, their clusters are bitter. Their wine is the poison of asps\" (Deuteronomy, 32:32, 33). Fr\u00e4nger suggested further that the snakes adorning the headdress of the middle priestess (with the basilisk stare) are the asps in the reference, and that the wine she is serving is \"the poison of dragons\".\n\nThese figures also have their counterparts in the major trump series of the Tarot. The central woman is akin to the High Priestess who wears the papal tiara in that system of imagery. The woman on the left could have been inspired by the tarot Empress (also the \"faire white woman\" who signified mercury as associated with Luna in some alchemical references). There is a hermit (Saint Anthony) in the tarot pack, and a fool \u2013 among these figures, he is possibly the pig-snouted man who shuffles in a manner suggestive of cerebellar disturbance.\n\nThe dog accompanying the fool wears a foolscap, again strengthening the tarot connection. To the left of the fool, the wooden-legged old man was a figure often used to denote Saturn, and the dog in his path would have additional meaning in this context, since it was an animal identified with the melancholy humour, and, by implication, with Saturn. The musical instruments carried by the two men would produce the antidotal music required to dispel saturnine mysticism \u2013 already pervasive in the doleful anchorage.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(tritpych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nMusic was also considered a demonic snare and these instruments, the hurdy-gurdy and the lute, were among those said to be played in hell. Emphasising this point, Bosch placed a demon behind the High Priestess who fingers his woodwind snout; the \"music\" that he exudes emerges as noxious fumes.\n\nIt is hard to believe that the affable little fellow who toasts the crowd from a perch opposite Saint Anthony is another embodiment of evil, but the Devil is as capable of assuming attractive forms as he is repulsive ones. In fact, the use of these would be among his most effective stratagems. The creature's lack of a torso can be ascribed to Bosch's narration on the composite gargoyle theme. Instead of combining sections of different organisms, he has merely pieced together the non-connected parts of one. A more orthodox demonic creature is the winged lizard that hangs down from a tie rod to observe the crowd. Two others of this species stand upon the roof; one has yet only half-hatched from its egg-shell, conveniently fitted with legs.\n\nThe same conjoining of sorcerous and biblical connotations seen in the ceremony around Saint Anthony can also be inferred from the scenes presumably sculpted or painted on the tower. The subject of the uppermost mural seems, at first glance, to be simply that of Moses' receipt of the Tables of the Law. Moses kneels, as on Mount Sinai, to take the tablets from the hand of God plunging through the cloud above the prophet. The golden calf has been raised on an adjoining mount, and the idolatrous children of Israel dance and genuflect before it. A closer look shows Moses to have a horned hat, the \"calf\" to be in reality a goat, and the Israelites to have snakes curling around their bodies. The presence of a goat surrounded by worshipful revellers strongly hints of the Witches Sabbath. It is intriguing to wonder why these two ideas had connected themselves in the painter's mind. One of the derivative passages for knowledge concerning sorcery was to be found in the long exhortation made by Moses to remind the Israelites of their covenant with God. In the words of the prophet:\n\nThou shalt not learn to do after the abominations of those other nations. There shalt not be found among you any one that maketh his son or his daughter to pass through the fire, or that useth divination or an observer of times, or an enchanter, or a witch, or a charmer, or a consulter with familiar spirits, or a wizard, or a necromancer. (Deuteronomy 18:9-11)\n\nBy dressing the Israelites in asps, and by making them worship the goat (or Devil) instead of the graven image, Bosch equated their sin with the \"abominations of those other nations\". He intimated, in addition, that the wicked followers of Satan (the sorcerers), in Saint Anthony's time, as well as in his own, had abandoned themselves to those practices condemned by Moses in the name of the Lord.\n\nWe may ask why the painter gave horns to Moses. Was it because of a current misconception that Moses had worn horns, due to a mistranslation of a biblical passage that referred to rays emanating from the prophet's head?\n\nSince horns were usually associated with the Devil and the fact that certain animals, such as the toad, goat, and owl were horned causing them to be given an evil designation, Bosch may actually have believed that Moses was one of Satan's disciples. The painter equated the prophet and the goat by placing them on adjoining mounts; the revellers worship as before both. The derogatory slur at Moses seems to be continued in the tower's lowest register. Here are seen two men who have returned from the \"promised land\" with a cluster of enormous grapes, but they are walking into darkness. Conditions of light and dark were commonly assigned to good and evil, making it seem that Bosch considered the place where they should find the Jews to be an evil one. The intimation is further emphasised in the tower's middle register, which the artist so conveniently swelled out to incorporate all the figures of the scene. An enthroned monkey (the transformed Devil) is seated on a drum and receives a sacrificial offering from men who are dressed in the same manner as the Israelites above. An owl peers out of a crevasse behind the monkey's throne \u2013 endowing the scene with the stigma of heresy. Since the painter in these references associated Moses, the Israelites, and the food (both allowed and forbidden to them) with the Devil, he must have considered anything Jewish as evil.\n\nIn other portions of the painting, Bosch employed the Mosaic animals as conveyances on which to bring delegates to the meeting place. What appear to be two horses and a rat (both among the unclean animals) are the mounts that stand in the water at the foot of the tower (Ill. 96). The designation of a degree at which mercury heated as the belly of a horse in alchemical symbolism, Bosch translated literally. The torso of the horse on the left has been transformed into the vessel in which mercury and sulphur were conjoined. Such a jug also had diabolical significance; it was a sign of Satan as well as of woman in her role as \"sinful vessel\". The riders are demons, some of which fall into specific classifications. For instance, Bosch fashioned a knight-falconer as the rider of the jug-bellied horse.\n\nThis image is one of the painter's most evocative creations and could be used as the typical illustration of his _modus operandi_. To be the rider to this already magical animal, Bosch conceived the falconer of humorous symbolism. The painter then produced a falconer, complete with hunting horn and hooded bird on wrist. In the process of making this figure, he thought of another hunter, or fighter \u2013 the knight-figurative in the tarot game. He added a knight's armour to the falconer and, in a burst of fancy, attached wings to the shoulders. If a logical explanation is needed for this action, the temperament for which the falconer was sometimes emblematic was the sanguine, allied to the element of air. Traditionally, the falconer used in this context would be shown walking on air amidst clouds and stars. Since Bosch did not put his figure in the air, he gave him wings, as suggestive of air as clouds and stars, and surely as logical as a figure walking on air.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, central panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nHieronymus Bosch's difference was that he did not confine himself to the traditional reference. He used the method by which the common symbols had been made \u2013 evolving as they did from countless origins, formed by free association of ideas in many minds, accepted into common usage if they had universal impact. The painter simply made his own associations, but the images which he created are not to be looked upon as having objective meaning that can be disassociated from the visual creation. The thistle-headed, winged knight-falconer, astride a jug-bellied horse, cannot be considered as an autonomous symbol of a detachable idea. It has meaning only in the context of the whole structure and significance of the painting. It is, as everything else in the painting, a layered fabrication of many associated ideas. For instance, M\u00e2le cites the belief of Saint Bonaventura that the thistle is \"an image of the temptations that pass through the dreams of Sloth, occasionally goading it\". To be considered the same are the other figures in this group, made up of an old man with an egg-shaped pack on his back, a richly dressed rider with a sphere topped by a vine as adornment on his hat, and a demon in knight's armour.\n\nThe most readable image is the involved figure formed by a tree with a branch that becomes an arm to the woman whose face peers out from the hollow. The idea probably came from alchemy since any hollow place was looked upon as the matrix of the alchemist's furnace. An egg-shaped child (or Philosopher's Stone) is seemingly being produced out of the hollow, giving further support to this idea. The woman is seated in a sieve, or vessel, used in the Great Work. Her lower body is a scaly tail, perhaps pure embroidery on Bosch's part, perhaps representing the alchemical process sometimes denoted by the zodiacal sign of Pisces. Another egg-shaped child is playing in the water \u2013 the \"child's play\" ascribed to the operation of the Great Work after its primitive material had been found.\n\nThe fish-craft are conveyances more natural to the watery element that fills the lower right quadrant of the canvas. With the flying boats in the air directly above them, these are among the most extraordinary of Boschian inventions. One boat is made of a hull vaguely reminiscent of a horseshoe crab shell turned over on its back. Imprisoned inside is a man whose hands project through holes in the side as if they are the paddling mechanism. The boat is manoeuvred by a black monkey who handles a sail made partially of a skate, or ray, threaded onto a transparent net. Another boat is formed of a carp with an unlikely tail. Outfitted with a red blanket, a plate of armour, and fanciful rigging, it is \"manned\" by two black monkeys. One paddles with a long spoon and the other drags a net which has reaped, among other fish, versions of a spoon-billed catfish, a bream, and an eel (both \"clean\" and \"unclean\" in the Law).\n\nStanding on land but supping at the water's edge, on the left-hand side, are animals both imaginary and real. These are the roebuck and hart, considered \"suitable\" by Moses. An \"unclean\" rat stands nearby, but the creature next to that animal is an unreal beast constructed of an egg-shaped body, long neck, and a face that projects from the outer skin covering as from a helmet, with sabots on its feet. This is the mount for an even stranger mixture of a cloaked armoured human body surmounted by the skull of an ass (the jawbone with which Samson slew the Philistines?). The rider plays with spidery fingers upon a lyre. These figures are followed by an ass, a heron (in armor), and a cormorant (all \"unclean\"). They stand in front of a huge apple skin (symbolic of the \"Fall\"). Its skin is open in several places and emerging from its hollow interior (the matrix of the alchemical furnace?) are playful demons. One sits in a basket that hangs from a giant thistle; he brandishes a sword. This may be another vague reference to alchemy since a sword, or cutting instrument, betokened fire.\n\nAbove this group is one of creatures that follow Saturn onto the central platform. First arrive two dogs dressed in armour which fits them stylishly. Behind the dogs, a woman whose head is made of a hollow tree from which dangle snakes (the creeping things forbidden by Moses) leads a curious, composite creature.\n\nIts backside resembles that of a horse, with the exception of the tail, which is rat-like. There is no torso at all (a favourite deletion of Bosch's). Large moth wings grow from the back of a bat-like head. Its curiously curled mouth ejects a long forked tongue. An arrow (that could refer to the passage: \"For lo, the wicked bend their bow, they make ready their arrow upon the string, that they may privily shoot at the upright in heart\" [Psalms 11:1, 2]) protrudes from beneath the creature's tail. Judging from the stout hold the woman has on the wings of this \"gargoyle\" and the long bow and barbed stick that she carries on her other arm, she must have captured this animal in a hunt \u2013 now she brings it to the Sabbath.\n\nA pig-snouted demon, who sports a jug of flowers as a hat, carries a wheel (of the type used for execution by exposure) on which are to be seen a human limb and a raven (an \"unclean\" bird) that is perched on one of the spokes. A rat dangles upside-down from this wheel, its legs pierced by a spike to which the supporting rope is attached.\n\nIn the air above the tower are flying craft, some formed of \"unclean\" animals. Two flying boats which head towards one another as if going into battle are formed around the \"forbidden\" swan and hawk. Their rigging is intact, if not conventional, as it seems to require the self-sacrificing efforts of crewmen acting as turn-buckles and mastheads to hold it together. As the animals have meaning in various contexts, they perform several functions for Bosch. In alchemy, winged animals represented the volatile nature of mercury, the swan stood for the white colour important in the Great Work, and the hawk was sometimes emblematic of the sanguinary temperament, because of the association of both the bird and temperament with air. That Bosch intended the right upper quadrant of the painting to be dominated by the element of air is apparent in that the clear blue sky of this side is so distinctly divided from the sky of the left upper quadrant, which is filled with the smoke and flames of a huge fire. On that side, the crafts most clearly visible are a scaleless, \"unclean\" fish on which a demon rides, and a winged egg straddled by a frog. The frog and egg have demonic and cosmological significance, respectively. Passengers on this side of the painting seem to be a demon and a frog that lights their way through the smoke-darkened sky with its own torch.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe fire that whips the massive smoke clouds with flames on the left side of the panel is surely one of the most spectacular displays in all of art history. A fire was suggested to Bosch from many sources: flames figured in the Saint Anthony iconography to signify hell fire and \"Saint Anthony's fire\".\n\nFire was coeval with the archetypal world of the Qabalah and the staff sign of the Tarot. It was not only one of the Aristotelian elements of most importance to the Great Work, it was also allied to the choleric temperament. Bosch used the burning village to demonstrate one of the most nefarious of demonic activities. Demons were believed to set such fires; here they compound the confusion by flying in with ladders to aid in pulling down the church steeple and tearing the roof off a tower. They fly about fanning the flames and, in general, stirring up more trouble than the stupefied onlookers beside the church can comprehend. Although the entire area is suffused with the glare of the fire, even causing the stream to appear like molten metal, a woman nonchalantly washes her clothing at the water's edge.\n\nThe Temptation of Saint Anthony  \n(triptych, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, left panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nIt seems quite possible that this homely, inappropriate activity was inspired from the alchemist's assurance that, once the primitive materials had been found, the hermetic process would be no more than \"woman's work\".\n\nBehind this woman are more, rather unconcerned, personages. From the doorway of a house, a woman looks out at a man who rests upon a bench beside the door. The inclusion of these figures was perhaps due to the representation of Slothfulness (when linked with the melancholy temperament) as a lazy couple. In the traditional versions of this vice, they were occasionally linked with a hermit. The horsemen meandering across the bridge may be soldiers \u2013 in this place so overspread with demons, perhaps referring to the soldiers with whom Saint Anthony contended in one of his hermitages.\n\nIn later paintings such seemingly minor figures would merely act as padding, but this altarpiece is typically medieval in assigning symbolic, more than genre, significance to such figures.\n\nAs has been pointed out, however, Bosch took liberties with traditional usages; he attempted to give his own inventions some environment rather than employing them in a typically arbitrary fashion; in many parts of his painting, particularly the fire scenes, he went far beyond medievalism. The subtleties of transition from yellows through the reds to deep browns in this section suggest his technique could already be considered Baroque. One of the loveliest, if grim, passages of the painting is the area between the apse of the church and the fortress wall. It is lit with a sulphurous glow that transforms the trees into negative structures with the spaces between and around the branches becoming palpable. Especially evocative is the ghostly figure rushing away between the trees. This tract forms a liaison between the dark sky and the light one of the two upper quadrants. The modulation from night to day, although handled with skill by the artist, is still arbitrary \u2013 thus medieval in feel. Bosch has devoted most of the right upper region to a presentation of the element of air, but he has included within the quadrant an intriguing annex to the fortress. An enclosed bridge extends behind the fortress from the right to form a passageway to a bulbous structure that grows like a great plant from the water. The outer skins of this monstrosity have been removed, suggesting a bulb peeled by giant hands. Its ovoid shape infers the hermetically sealed vessel of \"the work\". Since a tower was often used to connote the furnace in which this vessel was heated, it is significant that the tower and \"vessel\" in the painting have been placed close together, connected even, by the arcaded bridge between them. Set within the \"peeled\" top of the bulbous building is a curious construction.\n\nThe lower part seems to be a high-windowed abode, of a type inhabited in Mediterranean countries. Set on stilts atop this structure is a kettle form with smoking chimney. Since the openings are round and rather small, it may be intended as a bird-house; a bird does sit on a pole extending from one of these openings and others sit upon its thatched roof. The use of birds here suggests the alchemical language that describes certain stages in the process of fusing mercury and sulphur. The reappearance of the initial black colour of these materials in a later stage of their fusion was inferred in the cryptic phrase that the young crows were returning to their nest.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, left panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon.\nThe white colour was hailed as a stage of completion when the White Stone, capable of transmuting base metals to silver, was found. Since Bosch shows white crows on their nest in this section of the painting, he has apparently blended the two stages in the one portrayal.\n\nThe bulbous building has a tent stretched out behind it, supported on one side by a tree that grows out of the wall. Hanging from a branch of this tree is a bellows, a device used in producing the necessary heat in the furnace. Quack alchemists, known as \"puffers\" or \"bellows-blowers\", were often shown under the sign of the bellows. That the painter placed this sign over the enclosure wherein can be seen a potatory abbess and monk may mean that he considered the clergy as equal to quacks. He surely considered drink as evil, because he placed a jug (connotative of the Devil) on a barrel supplying the drinkers. A diver who has hung his clothes on the parapet outside this trysting place now prepares to spring into the water below. This is a strangely normal action in such an abnormal setting; perhaps, it looks back to Anthony's recollection of childhood pleasures at the instigation of the Devil. A swimmer already in the water is about to be joined by a demon being ousted from the bridge by an irate woman. This demon, along with others that can be seen on the arcaded gallery of the passageway, shatter the momentary illusion of normalcy.\n\nA clock hanging on the wall below the gallery shows the hour to be deep in the night \u2013 the hour of the Sabbath \u2013 and of necromancy \u2013 most appropriate to evil-doing. Another look at the clock reveals it to have fourteen hour symbols, the number of processes required in the operation of the Great Work. The signs vaguely resemble those of the Zodiac, sometimes related to the alchemical processes in the works that describe the number as twelve. These symbols also resemble Hebraic letters used on the magic pentacles of necromancers, as described in the Black Book. Draw-chains fastened to pulleys on the side of the passage wall support a cruelly barbed rod, perhaps an instrument used to fend off invaders who would attempt to scale a fortress wall. The water at this part of the building appears to be a moat, although it is not continuous around the wall. If this is not an allusion to the allegory in which the Great Work was called a fort or citadel surrounded by a moat of water, surely the image on the right wing of the altarpiece is such. In alchemical phraseology the adept could only reach the Philosopher's Stone by crossing a drawbridge \u2013 open just to those of the highest personal qualifications.\n\nAfter winning his way into the citadel, the adept then had to fight his way past a guardian dragon. In a typical twist, Bosch placed the dragon fight outside the citadel, in the water itself, where it is witnessed by an army of men hanging over the wall \u2013 plus a few curious rats and demons that join the fighters in the water. The walls of the stronghold further evoke alchemy since they appear to imitate the major vessels used in the \"work\". One of them has a fire (of much importance in alchemy) burning on its summit. Soaring above the scene is a flying fish craft, the conveyance of a man and woman: \"the couple\", or sulphur and mercury in fusion. Bosch placed the saint in the centre of the middle painting, but he occupies the same approximate position as on the right panel. Hooded, with a tau-marked cloak, he rests upon a hillock, using his cane for support. He reads from the book often included in his iconography. The hermit looks away in sorrow from a disgusting, devilish pastiche. Arranged around a tree are a nude woman peering provocatively at the saint and a conglomeration of demons tempting him with food and drink. A demon displays a fish before him; a witch pours liquid from a jug into the cup of a creature who, like his companion, awaits it with upturned mouth. One of the demons prepares to climb a ladder into the tree; another peers at Saint Anthony from around a root. A raven surveys the crowd from its perch on a branch. The nude woman refers, of course, to one of the first temptations laid on Anthony by the Devil.\n\nSince she stands in water, there is an alchemical reference to mercury, with its liquidity and fusibility, as woman. The saint cannot escape the sight of evil. As he looks away from the first group his eyes turn towards a round table in the forefront of the picture. It is held up by nude men, one of whom lies on his back to support it with his forelimbs. Another bears his part of the table on his shoulder and upper arm. A demon which has the appearance of a bear (used to represent the fixed material, salt, of the alchemical \"quest\") holds up the other side. Giant rats peer out from between the men; one of them thrusts a sword through the neck of the reclining man with whom he fights and who futilely waves his sword. These swords are significant of the cutting, penetrating, and wounding implements which were sometimes used to depict fire. There are other, more obvious, evidences of fire (important throughout the painting). The leg of the man who supports the table with his arm has been burned. His other leg is thrust into a jug, making it necessary for him to support himself with a crutch. The feet of the man standing next to him are both burned, as if he has walked through the water of the molten stream nearby. Their burned limbs also suggest that they suffer from \"St Anthony's Fire\". The man with scalded feet blows upon a trumpet which turns back quixotically towards himself. Protruding from its sound hole is a horn that emits noxious fumes for music. A sausage (\"unclean\" food) is dangling from the horn.\n\nThe food in this scene may be an apparition of the rich foods designed to tempt Saint Anthony, but the repast on the table is actually very meagre. A few cruciate flowers lie there, two eggs, and a cup supporting a cloven hoof (of an \"unclean\" animal). A frog (the Devil incarnate) peers out from under the tablecloth. Bosch's shocking allusions in this section to the clergy were made either in denunciation of them, or else, in affirmation that the Devil can masquerade in clerical clothing. The painter dressed three of the characters in the lower section of this panel in cowls. The trumpeter is garbed in this fashion as is the headless demonic creature seated to the right of the table. As if to correct this anatomical oversight, the artist set ears to either side of the fellow's stomach and hair atop it. This gargoyle wears an enveloping hood which reveals a tail protruding from its back. The creature has little concern for the dagger thrust in its stomach-face. A third cowled creature scoots above the saint, confined in a sort-of pushchair, like a child. His infantile stature obviously infers Bosch's attitude that the clergy indulge in childish pastimes.\n\nThe Temptation of Saint Anthony  \n(triptych, detail, right panel), 1505-1506.\n\nOil on panel, 131.5 x 119 cm.\n\nMuseu Nacional de Arte Antiga, Lisbon\n\nThe Last Judgement,  \nc. 1504-1508.\n\nOil on panel, 163 x 128 cm (central panel),\n\n167 x 60 cm (each wing).\n\nDie Gem\u00e4ldegalerie der Akademie\n\nder bildenden K\u00fcnste, Vienna.\n\nThe Last Judgement (exterior),  \nc. 1504-1508.\n\nOil on panel, 163 x 128 cm (central\n\npanel), 167 x 60 cm (each wing).\n\nDie Gem\u00e4ldegalerie der Akademie der bildenden K\u00fcnste, Vienna.\nOn the left wing of the triptych Saint Anthony is depicted in two places. His collapsed figure is supported by his fellow monks as they carry him to a retreat. This allusion is to the saint's return to the tomb after he had been beaten by the demons and carried to his village for dead. The man with his face turned to the onlooker resembles the known portrait of Bosch, suggesting that the painter might have signed the painting in this fashion.\n\nThat the artist placed his portrait on a man who helps the fainting hermit surely means that Bosch affirmed his action. The saint also appears in the air in what is one of the most striking sections of the painting. A scene like this is recounted in the Athanasius story. When Anthony stood up to pray, he felt himself carried aloft in spirit and met in the air loathsome beings that demanded an accounting from his birth. In this passage of the painting, the old man prays as he is ferried on the prone figures of bat-winged, fork-tailed demons that beat him with branch and mallet. Flying towards him is a counterpart to the winged falconer astride the jug-bellied horse of the central section, but the knight here has wings for legs, instead of arms. He cuts an elegant figure, propelling with these \"legs\" from the back of a bass, as he carries a trout in his arms. A raven hikes a ride by perching on a tendril of the bass's tail. Another marvellous flying boat approaches from behind the saint, this one supported on the back of a huge bat. Its mast is broken and its red sail droops, but a demon attempts repair by climbing up a rope held taut by a helpful fish. Another demon provides ballast in the back of the boat by adding his weight to the anchor; yet another performs no useful function, save that of making a face at the poor saint from between his legs. Bosch made some of his demons quite intriguing, such as the one standing on his head in a \"glider\" attached to the bat-monster's tail. Another most appealing demon is the \"jet-propelled\" creature that darts away with the scythe.\n\nThe latter is the instrument often associated with Saturn. There are some very interesting, but elusive, ancillary characters in this panel, in addition to the ones that are directly involved with Anthony. Under the bridge over which the saint is being drawn, a rapacious looking cleric huddles in conference with a rat and a bat (both \"unclean\" under Mosaic law, and always prophetic of evil). Many writers have suggested that this reference is to the Church practice of selling indulgences, an offence deplored by would-be reformers. The monk holds an \"indulgence\", over which the three confer in a secret place, unseen by the good men above. A crossbill skates up with another indulgence filed on his own bill. This clever creature is well-equipped for the cold which indicates a change of time in this section of the painting. He wears a mantle, long stockings, and a scarf pulled over his head, under an upturned funnel denotative of folly. Alchemical allusion here can be seen in the arrows and the ubiquitous egg out of which a cormorant sprouts.\n\nStanding on top of the egg, is a pelican that throatily swallows a frog. Both birds shown here are of the \"unclean\" variety. Another fabricated figure crouches in the background. Its face is that of a man and its backside that of a dog. It wears a feathered headdress and plays a bagpipe, symbolic of lust. A sword blade is thrust through its thigh and an arrow through its knees; a hooded falcon sits on its tail. Perhaps this is the captured prey of some hunter as was the creature shot by bow and arrow left of centre in the middle panel. Just above this specimen of evil is a machine similar to the flying boats. It is propelled by large wheels set in its centre, as well as by grasshopper legs attached on either side behind the wheels.\n\nThe machine is formed of three fish, each swallowing one smaller than itself. A tower-like structure on top of the larger fish makes a passenger compartment. A demon coachman drives the contraption towards a group of assorted \"clergymen\" ascending the path to a hovel. The demon dressed in a chasuble which covers all save his spoonbill walks beside a mantled stag (a food permitted to Israelites \u2013 also an oft-used symbol of sexuality). They are led by a man in mitred headdress; this figure could have been inspired by the Pope card of the tarot. Since he holds a crozier with a crescent (Bosch often used the crescent to signify an infidel) and ball atop it, and since his mitre is inscribed with a circle of Hebraic symbols in its centre, the reference more likely refers to the Jews. The mitre was worn by the high priest of Jewish antiquity and was adopted into Christianity. Assuredly, Bosch meant to consign this man to \"hell\", having placed a roaring fire in the top of his headdress. The \"priest\" leads the others to a hut, curiously anthropomorphised, whose beams and ridgepole form the limbs of a man on all fours. The doorway is the opening between his legs; they seemingly suggested plants to Bosch, because he affixed rootlets to the legs. He hung a sign on one of these, as over an inn door. From the rooms annexed to one side of the immobile man a woman peers out of a window by which are to be seen a jug and a long pole (signs of the Devil \u2013 the pole was used to prod sinners into the waters of hell). This hut is undoubtedly conceived as the anchorage of Anthony; because one of Bosch's earlier interpretations of the Anthony theme included a house that grew around the head of a woman.\n\nThe artist's reason for fastening the house to the man in this painting is not clear. The position in which he kneels suggests a crouched animal, as if to imply man's bestial nature. Perhaps the reference is to the constant fight which Anthony waged against his own bodily requirements \u2013 considered of animal necessity \u2013 and therefore shameful. This man has an arrow projecting from his head, as if he has been shot like an animal.\n\nThe entrapped man looks aghast towards a burning ship sinking through the ice, which already holds other ruined vessels immobilised. The whole vista is one of waste and stagnation, of a neglected place quite different from the carefully husbanded coastal hinterlands of the painter's native Holland. Signs of death abound, for execution wheels are everywhere, a man hangs suspended by one foot from a tree (the tarot figure?), and there is a morbid putty colour to the sea and land, alike. Could this be the Promised Land in a terrible reference to the Jews \u2013 that their \"promised land\" is ironically a place of evil and death? Is it, instead, a general reference to the fate of mankind \u2013 as promised in the awful words, \"The wages of sin is death\"? (Romans 6:23). Man is, in the end, a victim of his own evil. He is enslaved by his own animal nature, impaled with his own instruments of torture, bound to watch the same fate applied to all his fellow sinners \u2013 who grovel to false gods and will not look up to see the truth.\n\nThe Seven Deadly Sins in a Peel of Terrestrial Globe.\n\nOil on panel, 86 x 56 cm.\n\nPrivate collection, Geneva.\n\n**Bosch follower** ,  \nChrist among the Doctors, 1555-1561.\n\n74 x 58 cm.\n\nMus\u00e9e du Louvre, Paris.\n\n### Conclusion\n\nQuestions are natural to the viewer's mind. Answers are inevitable \u2013 but they should not be automatically assigned to Bosch's mind. It is not possible to determine the exact conclusions the painter meant to draw, or whether he intended any specific implications in the subplots of his works. Not even the painter himself could have given a rational explanation of all the separate units of _The Temptation of Saint Anthony_ altarpiece. This is not to say, however, that there is no reason in this painting. If the triptych is considered as a whole, it is seen to be organised in a logical, even conventional fashion. The saint is placed in the centre of the central panel, the elements arranged around him with a certain amount of symmetrical planning. There is a planar progression in the painting and a \"closing\" of the composition, the side panels being so designed that the saint, placed in the same position in each, turns towards the centre. Bosch's perspective is not accurate, but scientific means had not yet penetrated Northern art. In some ways, the painter anticipated the Baroque style in his diagonal positioning of the fortress and the painterly effects he achieved, especially in the area of the fire. He attempted aerial perspective and in his later paintings (such as the Prado _Epiphany_ ) achieved mature results. De Tolnay believed that Bosch's purposeful misuse (or his unwitting disuse) of either form of perspective, combined with the immateriality of some of his figures due to their painterly handling, give the _Saint Anthony_ painting the appearance of an aquarium with the figures floating like apparitions in a vertical space. He described the artist's use of colour as enhancing this impression, as he placed warm, rich colours around the edges of the painting and ashy, silvery colours in the centre. This subversion of the natural colour scheme would emphasise the unreality of the setting.\n\nWhatever the means employed by the artist to achieve his effects and whether they were fully intentional or not, it is certain that he did intend to convey the impression of evil. In fact, it should always be kept in mind that this was the painter's central motivation. The desire to record the pervasiveness of evil in the world and in its many guises was such an extraordinarily religious drive with Bosch that it gives direction when there seems to be none and offers control when that virtue seems to have been forgotten. It was with full awareness of this central purpose that Bosch created an orderly arrangement in which the agents of evil swarm in profusion and confusion. Holding the control of awareness over the whole, the artist relaxed it in the parts. It is only when an attempt is made to analyse the individual parts that the basic order is forgotten and chaos seems to reign. The baffled viewer then decides that Bosch was either a mad projectionist of his uncontrolled hallucinations, or that he hid his rational meaning in a labyrinth of symbolism to which someone, someday, will find the map. There can be no rational interpretation of these paintings. Nor does this mean that Hieronymus Bosch was mad. With a fully rational mind this artist set his stage; then, (to adopt an anachronistic figure of speech) he placed his mind \"in neutral\" and allowed it to be propelled by his imagination.\n\nIn other words, he surrendered his consciousness to his unconscious mind. Whatever restraints there are that cause the mind to function in related, reasonable patterns were dropped. The images of the human mind, which come and go, are not sought out nor kept in order, from which no ideas are abstracted, among which no selection is practiced, and which were allowed a concatenation of their own making. From knowledge implanted at earlier times, and at many times, these images were churned up from obscure levels in the artist's mind. In another anachronistic image, the action was like that of poker chips put into a transparent washing machine to display the water movement to a possible buyer. They appear, disappear, and reappear \u2013 in different combinations at every revolution.\n\nSimilarly, the images formed in many systems made to explain evil came to the artist's mind in an incredible number of combinations, thus accounting for the variety of usages over the panels. In musical terms, the result would be as the \"music\" played by the wind on an aeolian harp. It can be argued that such noise is chaotic, but the strings are tuned in unison. In the same way, the artist's mind was tuned to his subject. He knew for what (the service of the Lord) and against what (the machinations of the Devil) he allowed his mind to play. There is, indeed, obscurity in this man's utterances; I would not argue that the obscure is necessarily profound. Nor would I imply that the mind surrendered is as capable of great statement as the mind controlled. I would affirm, however, that the power of the imagination is unlimited; not only capable of a reproductive function, forming images of things once seen but now absent, it is also capable of an inventive, or creative function, forming images of things never before seen \u2013 actually non-existent.\n\nFashion and exigencies of the times decree to what extent the artist operates according to rule and control. With the artist who reproduces only things seen, immediate contact is made between the artist's mind and that of the viewer. There is satisfaction for both \u2013 for the painter because he or she will be understood \u2013 for the viewer who understands. The contact, which the inventive artist establishes, is more nebulous; the path from mind to mind is not a closed but open circuit. This artist not only reveals to the viewer the processes of the artist's mind but seeks to generate these processes in the viewer's mind. In the words of A.C. Bradley (writing of poetry): \"the specific way of the [inventive] imagination is not to clothe in imagery consciously held ideas; it is to produce half-consciously a matter from which, when produced, the [viewer] may, if he chooses, extract ideas\".\n\nWith incredible energy, Bosch produced the matter. It is left to the viewer, not to explain what is in the painter's mind \u2013 but to examine the wonder engendered in his or her own mind and, doing so, be infinitely intrigued. Roger Marijnissen invited the reader of his book on Bosch to enjoy the paintings, always keeping in mind that the artist was a \"born painter\". Bosch dwells on the fears of his time, many of which are of our time, too, but let us celebrate the artist, while appreciating the message, and not consume our energies only on the \"riddle\" of his work.\n\n**Bosch follower** ,  \nThe Adoration of the Magi (triptych),  \n1525-1540.\n\nOil on panel, 80.5 x 115 cm.\n\n### Index\n\n**A**\n\nThe Adoration of the Magi, c. 1470-1475.\n\nThe Adoration of the Magi, c. 1470-1475.\n\nThe Adoration of the Magi, c. 1470-1475.\n\nThe Adoration of the Magi, c. 1470-1475.\n\nAllegory of Intemperance, c. 1495-1500.\n\nAscent of the Blessed to the Heavenly Paradise, 1500-1504.\n\nAscent of the Blessed to the Heavenly Paradise , 1500-1504.\n\n**C**\n\nChild with a Walking Frame , c. 1480-1490.\n\nChrist Carrying the Cross, c. 1480-1490.\n\nChrist Carrying the Cross.\n\nChrist Carrying the Cross.\n\nChrist Carrying the Cross.\n\nChrist Carrying the Cross, c. 1510-1516.\n\nChrist Mocked (The Crowning with Thorns), c. 1490-1500.\n\nCrucifixion with a Donor, c. 1490.\n\n**D**\n\nDeath and the Miser, c. 1485-1490.\n\n**E**\n\nEcce Homo, c. 1450-1516.\n\nEcce Homo, 1475-1480.\n\nThe Epiphany or The Adoration of the Magi, c. 1510.\n\nThe Epiphany or The Adoration of the Magi, c. 1510.\n\nThe Epiphany or The Adoration of the Magi , c. 1510.\n\nThe Epiphany or The Adoration of the Magi , c. 1510.\n\nThe Epiphany or The Adoration of the Magi , c. 1510.\n\nThe Epiphany or The Adoration of the Magi, c. 1510.\n\nThe Epiphany or The Adoration of the Magi , c. 1510.\n\nThe Epiphany or The Adoration of the Magi, c. 1510.\n\nThe Epiphany or The Adoration of the Magi , c. 1510.\n\n**G**\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights , c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\nThe Garden of Earthly Delights, c. 1500-1505.\n\n**H**\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\nThe Hay Wagon, 1525.\n\n**L**\n\nThe Last Judgement, c. 1504-1508.\n\nThe Last Judgement, c. 1504-1508.\n\n**M**\n\nThe Magician, 1475-1480.\n\nMarriage Feast at Cana, after 1550.\n\n**P**\n\nThe Pedlar, c. 1505.\n\n**S**\n\nSaint Christopher, 1490-1505.\n\nSaint John on Patmos, 1504-1505.\n\nSaint John on Patmos , 1504-1505.\n\nSaint John on Patmos, 1504-1505.\n\nSaint John on Patmos, 1504-1505.\n\nThe Seven Deadly Sins in a Peel of Terrestrial Globe.\n\nThe Ship of Fools, between 1490 and 1510.\n\nSt. John the Baptist in Meditation, end of the 15th century.\n\n**T**\n\nTable of the Mortal Sins, late 15th century.\n\nTable of the Mortal Sins, late 15th century.\n\nTable of the Mortal Sins, late 15th century.\n\nTable of the Mortal Sins, late 15th century.\n\nThe Temptation of Saint Anthony, after 1490.\n\nTriptych of Job, c. 1500-1524.\n\nTriptych of Job, c. 1500-1524.\n\nTriptych of the Hermits, c. 1505.\n\nTriptych of the Hermits, c. 1505.\n\nTriptych of the Martyrdom of St Liberata, 1500-1504.\n\nTriptych of the Temptation of Saint Anthony , 1505-1506.\n\nTriptych of the Temptation of Saint Anthony , 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nTriptych of The Temptation of Saint Anthony, 1505-1506.\n\nBosch follower, The Adoration of the Magi, 1525-1540.\n\nBosch follower, Christ among the Doctors, 1555-1561.\n\nVan Eyck, Hubert and Jan, Mystic Lamb, 1427-1432.\n\n### Bibliography\n\nBALDASS, L. von. (1960). _Hieronymus Bosch_ (2nd ed., trans.). New York: Abrams.\n\nBAX, D. (1949). _Onticjfering van Jeroen Bosch_. The Hague : Statsdrukkerij and Martinus Nijhoff.\n\nBENESCH, O. (1945). _The Art of the Renaissance in Northern Europe_. Cambridge, Massachusetts: Harvard.\n\nBRAND PHILLIP, L. (1953). The Prado Epiphany. _Art Bulletin_ , XXXV, 35.\n\nBRAND PHILLIP, L. (1956). _Hieronymus van Aken Bosch_. New York: Abrams.\n\nCASE, P. F. (1947). _The Tarot_. New York: Macoy.\n\nCHASTEL, A. (1936). La Tentation de saint Antoine, ou Le Songe du m\u00e9lancolique. _Gazette des Beaux-Arts_ , 15 (6).\n\nCOMBE, J. (1946). _Jheronimus Bosc_ h. London: Batsford.\n\nCUTTLER, C. D. (1957). The Lisbon Temptation of Saint Anthony by J\u00e9rome Bosch. _The Art Bulletin_ , XXXIX, 2.\n\nDE GIVRY, G. (1931). _Witchcraft, Magic, and Alchemy_. Trans. J. Courtenay Locke. Boston: Houghton Mifflin.\n\nDE TOLNAY, C. (1937). _Hieronymus Bosch_. Basel: Holbein.\n\nDEMONTS, L. D. (1919). Deux primitifs nerlandais au Mus\u00e9e du Louvre. _Gazette des Beaux-Arts_ , 4th ser., 15.\n\nDIXON, L. S. (1981). _Alchemical Imagery in Bosch_ _'_ _s Garden of Delights_. Ann Arbor, Michigan: UMI Research Press.\n\nDIXON, L. S. (1981). Bosch's 'Garden of Delights' Triptych: Remnants of a Fossil Science. _The Art Bulletin_ , LXIII, 1.\n\nDIXON, L. S. (1984, Summer). Bosch's 'St Anthony Triptych'\u2014An Apothecary's Apotheosis. _Art Journal_ , 4.\n\nBARR Jr., A. K. (Ed.). (1947). _Fantastic Art, Dada, Surrealism_. New York: Museum of Modern Art\/Simon & Schuster.\n\nFR\u00c4NGER, W. (1951). _The Millennium of Hieronymus Bosch: Outlines of a New Interpretation_. Trans. E. Wilkins and E. Kaiser. Chicago: University of Chicago.\n\nFRAENGER, W. [same author as above, different spelling of name]. (1948). Hieronymus Bosch: Johannes der T\u00e4ufer, Eine Meditationstafel des Freien Geistes, _Zeitschrift f\u00fcr Kunst,_ 2.\n\nFRAENGER, W. (1950). Hieronymus Bosch: Johannes auf Patmos. Eine Umwendtafel f\u00fcr Meditationsgebrauch. _Zeitschrift f\u00fcr Religions \u2013 und Geistesgeschichte 2_. Reprinted in above.\n\nFRAENGER, W. (1989). _Hieronymus Bosch_ (2nd ed.). New York: Dorset.\n\nFRIEDL\u00c4NDER, M. J. Hieronymus Bosch. _Art News Annual_ , XXVI, 44.\n\nFUCHS, R. H. (1978). _Dutch Paintin_ g. New York: Oxford.\n\nGIBSON, W. S. (1983). _Hieronymus Bosch: An Annotated Bibliography_ , New York: Hall.\n\nGOSSART, M. (1907). _Hieronymus Bosch_. Lille: Centrale du Nord.\n\nGUICCIARDINI, L. (1567). _Description of all the Low Countries_. Anvers.\n\nHAGEN, O. (1927). _Art Epochs and Their Leaders_. New York: Scribners.\n\nHERTLING, L. V. (1929). The Life of Saint Anthony by Saint Athanasius. In _Antonius der Einsiedler_. Trans. Robert T. Meyer (1950). Westminster, Maryland: Newman.\n\nHUXLEY, A. (1943). Foreword. In F. Goya and A. Huxley, _Complete Etchings of Goya_. New York: Crown.\n\nJAMESON, A. (1896). _Sacred and Legendary Art_ (Vol. II). Boston: Houghton Mifflin.\n\nJUNG, C. G. (1953). _Psychology and Alchemy_. Trans. R. F. C. Hull. London: Kegan, Paul, Trench, Trubner.\n\nKOLDEWEIJ, J., VANDENBROECK, P., & VERMET, B. (2001). _Hieronymus Bosch: the Complete Paintings and Drawings_. Trans. Ted Alkins. Amsterdam, Ghent: Ludion; Rotterdam: NAi Publishers.\n\nLOMAZZO, G. P. (1844). _Trattato dell_ _'_ _Arte della Pittura, Scultura ed Architettura_ (Vol. 2). Rome: Presso Saverio Del-Monte.\n\nMcCURDY, E. (Ed., Trans.). (1906). _Leonardo da Vinci_ _'_ _s Notebooks_. London, Duckworth & co.; New York: Scribner's.\n\nM\u00c2LE, \u00c9. (1908). _L_ _'_ _Art religieux de la fin du Moyen \u00c2ge en France_. Paris: Colin.\n\nMANDER, K. Van. (1936). _Dutch and Flemish Painters_. Trans. of the Schilder-boeck. New York: McFarlane, Warde, McFarlane.\n\nMARIJNISSEN, R. H. & RUYFFELOERE, P. (1987). _Hieronymus Bosch: the Complete Works_. Antwerp: Mercatofonds.\n\nMASSON, D. (1874). _The Three Devils: Luther_ _'_ _s, Milton_ _'_ _s, and Goethe_ _'_ _s, with Other Essays_. London: Macmillan.\n\nMONDRIAN, P. (1951). Toward the True Vision of Reality. _Plastic Art and Pure Plastic Art_. In R. Motherwell (Ed.), _The Documents of Modern Art series_ (3rd ed.). New York: Wittenborn, Schultz.\n\nMYERS, M. N. & DYNES, W. (1987). _Hieronymus Bosch and the Canticle of Isaiah_ , New York: Cabirion.\n\nPACHECO, F. (1638). _Arte de la Pinture_. Madrid: Maestre.\n\nPANOFSKY, E. (1955). _The Life and Art of Albrecht D\u00fcrer_. Princeton: Princeton University.\n\nPINCHART, A. (1860). Bosch (J\u00e9r\u00f4me, Joemen, Joen). _Archives des arts, sciences et lettres: documents in\u00e9dits publi\u00e9s et annot\u00e9s_ , 1st ser., Vol. 1. Ghent: Hebbelynck.\n\nREAD, H. (1953). Surrealism and the Romantic Principle. In _The Philosophy of Modern Art_. New York: Horizon.\n\nREAD, J. (1966). _Prelude to Chemistry: An Outline of Alchemy, Its Literature and Relationships_. Cambridge, Massachusetts; London, England: M.I T. Press.\n\nTAYLOR, H. O. (1914). _The Medieval Mind_ , (Vol. II). London: MacMillan.\n\nVAERNEWIJCK, M. van. (1905-1906). _M\u00e9moires d_ _'_ _un patricien gantois du_ _xvi_ _e_ _si\u00e8cle; troubles religieux en Flandre et dans les Pays-Bas au_ _xvi_ _e_ _si\u00e8cle_ ; personal diaries (2 vols.). Trans. Hermann P. van Duyse. Ghent: Maison d'Editions d'Art.\n\nVASARI, G. (1880). _Le Vite de_ _'_ _piu eccellenti Pittori, Scultori e Architette_. Ed. G. Milanesi. Florence: Appresso I Giunti.\n\nWIRTH, O. (1927). _Le Tarot_. Paris: Nourry.\n\nZUPNICK, I. L. (1978). Foreword. In D. Bax, _Hieronymus Bosch: His Picture \u2013 Writing Deciphered._ Trans. M. A. Bax-Botha. Rotterdam: Balkema.\n\n### Notes\n\n* * *\n\n[1] Fr\u00e4nger stated: \"No other work of Bosch's has been so consistently misunderstood as a result of this prejudiced approach as has the triptych, \"The Millennium\" [this is the title given the work by Fr\u00e4nger, who believed that the central panel represented the idealised estate of the millennial existence].\n\n[2] Fr\u00e4nger revealed the name of the \"Grand Master\" in his article on _St John on Patmos_ (\"Hieronymus Bosch: Johannes auf Patmos, Eine Umwendtafel f\u00fcr Meditationsgebrauch,\" _Zeitschrift f\u00fcr Religions- und Geistesgeschichte 2_ , 1949-1950), reprinted in Wilhelm Fraenger, _Hieronymus Bosch_ , ed. Patrik Reutersw\u00e4rd, New York: Dorset, 1989, p. 255. Fr\u00e4nger's subsequent articles discussed in this chapter were reprinted in the same source; quotations will be cited from this source, not the original printings, but I will continue to refer to the author as Fr\u00e4nger, even though Reutersw\u00e4rd translates his name as Fraenger.\n\n[3] The Black Book was an ancient source of sorcerer's knowledge, believed to have had its origin with Solomon. It contained chapters discussing the qualities desirable in the magician and his associate \u2013 the proper attire and equipment, which could include many things besides those mentioned, such as: a knife, needle or burin, fire, perfume, virgin parchment, a pen and ink, or blood with which to write. There was a bewildering variety of circles and many chants and incantations, with no indication of which was the most effective. It is assumed that that could only be discovered by trial. The Black Book had wide dispersion over Europe during these centuries under consideration, and many manuscript copies are said to exist today in private European libraries (De Givry, 103, 101).\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Brendan OConnor, Jonathan Ingram, Wayne Hammond\nand the Online Distributed Proofreading Team at\nhttp:\/\/www.pgdp.net (This file was produced from images\ngenerously made available by The Internet Library of Early\nJournals.)\n\n\n\n\n\n\n\n\n\nBLACKWOOD'S\n\nEDINBURGH MAGAZINE.\n\n  NO. CCCCXXXIII.        NOVEMBER, 1851.             VOL. LXX.\n\n\n\n\nCONTENTS.\n\n\n  THE DRAMAS OF HENRY TAYLOR,                             505\n\n  A LEGEND OF GIBRALTAR,                                  522\n\n  GERMAN LETTERS FROM PARIS,                              543\n\n  THE SUBMARINE TELEGRAPH,                                562\n\n  MY NOVEL; OR, VARIETIES IN ENGLISH LIFE. PART XV.,      573\n\n  THE MASTER THIEF. A NORSE POPULAR TALE,                 595\n\n  DAY-DREAMS OF AN EXILE,                                 604\n\n  AUTUMN POLITICS,                                        607\n\n\n\n\nEDINBURGH:\n\n\nWILLIAM BLACKWOOD & SONS, 45 GEORGE STREET; AND 37 PATERNOSTER ROW,\nLONDON.\n\n_To whom all communications (post paid) must be addressed._\n\nSOLD BY ALL THE BOOKSELLERS IN THE UNITED KINGDOM.\n\nPRINTED BY WILLIAM BLACKWOOD AND SONS, EDINBURGH.\n\n\n\n\nBLACKWOOD'S\n\nEDINBURGH MAGAZINE.\n\n  NO. CCCCXXXIII.        NOVEMBER, 1851.          VOL. LXX.\n\n\nTHE DRAMAS OF HENRY TAYLOR.\n\nThere is no living writer whose rank in literature appears to be\nmore accurately determined, or more permanently secured to him, than\nthe author of _Philip Van Artevelde_[1]. Not gifted with the ardent\ntemperament, the very vivid imagination, or the warmth of passion which\nare supposed necessary to carry a poet to the highest eminences of his\nart, he has, nevertheless, that intense reflection, that large insight\ninto human life, that severe taste, binding him always to a most\nselect, accurate, and admirable style, which must secure him a lofty\nand impregnable position amongst the class of writers who come next in\norder to the very highest.\n\n[1] _Philip Van Artevelde_: _A Dramatic Romance._----_Edwin the Fair:\nAn Historical Drama; and Isaac Comnenus: A Play_----_The Eve of the\nConquest, and other Poems._ By HENRY TAYLOR.\n\nThere have been greater poems, but in modern times we do not think\nthere has appeared any dramatic composition which can be pronounced\nsuperior to the masterpiece of Henry Taylor. Neither of the\n_Sardanapalus_ of Lord Byron, nor the _Remorse_ of Coleridge, nor the\n_Cenci_ of Shelley, could this be said. We are far from asserting\nthat Taylor is a greater poet than Byron, or Coleridge, or Shelley;\nbut we say that no dramatic composition of these poets surpasses, as\na whole, _Philip Van Artevelde_. These writers have displayed, on\nvarious occasions, more passion and more pathos, and a command of more\nbeautiful imagery, but they have none of them produced a more complete\ndramatic work; nor do any of them manifest a profounder insight, or a\nwider view of human nature, or more frequently enunciate that _pathetic\nwisdom_, that mixture of feeling and sagacity, which we look upon as\nholding the highest place in eloquence of every description, whether\nprose or verse. The last act of Shelley's drama of the _Cenci_ has left\na more vivid impression upon our mind than any single portion of the\nmodern drama; but one act does not constitute a play, and this drama\nof the _Cenci_ is so odious from its plot, and the chief character\nportrayed in it is, in every sense of the word, so utterly monstrous,\n(for Shelley has combined, for purposes of his own, a spirit of piety\nwith the other ingredients of that diabolical character, which could\nnot have co-existed with them,) that, notwithstanding all its beauty,\nwe would willingly efface this poem from English literature. If one\nof those creatures, half beautiful woman and half scaly fish, which\nartists seem, with a traditional depravity of taste, to delight in,\nwere really to be alive, and to present itself before us, it would\nhardly excite greater disgust than this beautifully foul drama of the\n_Cenci_.\n\nThe very fact of our author having won so distinct and undisputed a\nplace in public estimation, must be accepted as an excuse for our\nprolonged delay in noticing his writings. The public very rapidly\npassed its verdict upon them: it was a sound one. The voice of\nencouragement was not needed to the author; nor did the reading world\nrequire to be informed of the fresh accession made to its stores. If\nwe now propose to ourselves some critical observations on the dramas\nof Mr Taylor, we enter upon the task in exactly the same spirit that\nwe should bring to the examination of any old writer, any veritable\nancient, of established celebrity. We are too late to assist in\ncreating a reputation for these dramas, but we may possibly throw out\nsome critical suggestions which may contribute to their more accurate\nappreciation.\n\nIn _Philip Van Artevelde_, the great object of the author appears to\nhave been to exhibit, in perfect union, the man of thought and the man\nof action. The hero is meditative as Hamlet, and as swift to act as\nCoriolanus. He is pensive as the Dane, and with something of the like\ncause for his melancholy; but so far from wasting all his energies in\nmoody reflection, he has an equal share for a most enterprising career\nof real life. He throws his glance as freely and as widely over all\nthis perplexing world, but every footstep of his own is planted with a\nsure and certain knowledge, and with a firm will. His thoughts may seem\nto play as loose as the air above him, but his standing-place is always\nstable as the rock. Such a character, we need not say, could hardly\nhave been selected, and certainly could not have been portrayed with\nsuccess, by any but a deeply meditative mind.\n\nIt is often remarked that the hero is the reflection of the writer.\nThis could not be very correctly said in instances like the present. A\nwriter still lives only in his writings, lives only in his thoughts,\nwhatever martial feats or bold enterprises he may depict. We could not\nprophesy how the poet himself would act if he had been the citizen\nof Ghent. It is more accurate to content ourselves with saying that\nthe delineation of his hero has given full scope to the intellectual\ncharacter of the author, and to his own peculiar habits of thought.\nFor if the great citizen of Ghent combines in an extraordinary degree\nthe reflective and the energetic character, our author unites, in a\nmanner almost as peculiar, two modes of thinking which at first appear\nto be opposed: he unites that practical sagacity which gives grave,\nand serious, and useful counsels upon human conduct, with that sad\nand profound irony--that reasoned despondency--which so generally\nbesets the speculative mind. All life is--vanity. Yet it will not do\nto resign ourselves to this general conclusion, from which so little,\nit is plain, can be extracted. From nothing, nothing comes. We must go\nback, and estimate by comparison each form and department of this human\nlife--which, as a whole, is so nugatory. Thus, practical sagacity is\nreinstated in full vigour, and has its fair scope of action, though\never and anon a philosophic despondency will throw its shadow over the\nscene.\n\nAs it is a complete man, so it is a whole life, that we have portrayed\nin the drama of _Philip Van Artevelde_. The second part is not what is\nunderstood by a \"continuation\" of the first, but an essential portion\nof the work. In the one we watch the hero rise to his culminating\npoint; in the other we see him sink--not in crime, and not in glory,\nbut in a sort of dim and disastrous twilight. We take up the hero from\nhis student days; we take him from his philosophy and his fishing-line,\nand that obstinate pondering on unsolvable problems, which is as much\na characteristic of youth as the ardent passions with which it is\nmore generally accredited; we take him from the quiet stream which he\ntorments, far more by the thoughts he throws upon it, than by his rod\nand line.\n\n    \"He is a man of singular address\n    In catching river-fish,\"\n\nsays a sarcastic enemy, who knew nothing of the trains of thought for\nwhich that angling was often a convenient disguise. A hint given in\nthe drama will go far to explain what their hue and complexion must\nhave been. The father of Philip had headed the patriotic cause of the\ncitizens of Ghent; it had triumphed in his person; the same citizens of\nGhent had murdered him on the threshold of his door. When he was a boy,\nthe stains of his father's blood were still visible on that threshold:\nthe widowed mother would not suffer them to be removed, and, nursing\nher revenge, loved to show them to the child. There was something here\nto colour the thoughts of the young fisherman.\n\nBut passion and the world are now knocking at the heart of the\nmeditative student. Love and ambition are there, and, moreover,\nthe turbulent condition of the city of Ghent seems to forbid the\ncontinuance of this life of quietude. The passions of the world crave\nadmittance. Shall he admit them? The great theatre of life claims its\nnew actor. Shall he go? Shall he commit himself once and for ever\nto the turmoil and delusions of that scene--delusions that will not\ndelude, but which will exercise as great a tyranny over him as if they\ndid? Yes; he will go. As well do battle with the world without, as\neternally with his own thoughts; for this is the only alternative youth\npresents to us. Yes, he will go; but deliberately: he will not be borne\nalong, he will govern his own footsteps, and, come what may, will be\nalways master of himself.\n\nLaunoy, one of Ghent's bravest patriots, has been killed. The first\nreflection we hear from the lips of Artevelde is called forth by this\nintelligence. It does not surprise him.\n\n    \"I never looked that he should live so long.\n    He was a man of that unsleeping spirit,\n    He seemed to live by miracle: his food\n    Was glory, which was poison to his mind\n    And peril to his body. He was one\n    Of many thousand such that die betimes,\n    Whose story is a fragment, known to few.\n    Then comes the man who has the luck to live,\n    And he's a prodigy. Compute the chances,\n    And deem there's ne'er a one in dangerous times\n    Who wins the race of glory, but than him\n    A thousand men more gloriously endowed\n    Have fallen upon the course; a thousand others\n    Have had their fortunes foundered by a chance,\n    Whilst lighter barks pushed past them; to whom add\n    A smaller tally, of the singular few\n    Who, gifted with predominating powers,\n    Bear yet a temperate will, and keep the peace.\n    The world knows nothing of its greatest men.\"\n\nIf ambition wears this ambiguous aspect to his mind, it is not because\nhe is disposed to regard the love of woman too enthusiastically.\n\n    \"It may be I have deemed or dreamed of such.\n    But what know I? We figure to ourselves\n    The thing we like, and then we build it up\n    As chance will have it, on the rock or sand:\n    For thought is tired of wandering o'er the world,\n    And home-bound fancy runs her bark ashore.\"\n\nYet Artevelde is at this time on his way to Adriana to make that\ndeclaration which the Lady Adriana is so solicitous to hear. This a\nlover! Yes; only one of that order who hang over and count the beatings\nof their own heart.\n\nLaunoy being destroyed, and the people of Ghent having lost others of\ntheir leaders, and growing discontented with the stern rule of Van\nDen Bosch, some new captain or ruler of the town is looked for. The\neyes of men are turned to Philip Van Artevelde. He shall be captain\nof the Whitehoods, and come to the rescue of the falling cause; for,\nof late, the Earl of Flanders has been everywhere victorious. Van\nDen Bosch himself makes the proposal. It is evident, from hints that\nfollow, that Artevelde had already made his choice; he saw that the\ntime was come when, even if he desired it, there was no maintaining a\npeaceful neutrality. But Van Den Bosch meets with no eager spirit ready\nto snatch at the perilous prize held out to him. He is no dupe to the\nnature of the offer, nor very willing that others should fancy him to\nbe one--\n\n                      \"Not so fast.\n    Your vessel, Van Den Bosch, hath felt the storm:\n    She rolls dismasted in an ugly swell,\n    And you would make a jury-mast of me,\n    Whereon to spread the tatters of your canvass.\"\n\nIt is worth noticing how the passion of revenge, like the others, is\nadmitted to its post; admitted, yet coldly looked upon. He will revenge\nhis father. Two knights, Sir Guisebert Grutt and Simon Bette, (we\nwish they had better names,) were mainly instrumental in his murder.\nThese men have been playing false, by making treacherous overtures to\nthe Earl of Flanders; they will be in his power. But they cannot, he\nreflects, render back the life they have destroyed--\n\n    ----\"Life for life, vile bankrupts as they are,\n    Their worthless lives for his of countless price,\n    Is their whole wherewithal to pay the debt.\n    _Yet retribution is a goodly thing_,\n    And it were well to wring the payment from them,\n    Even to the utmost drop of their heart's blood.\"\n\nStill less does the patriotic harangue of Van Den Bosch find an\nenthusiastic response. He was already too much a statesman to be a\ndemagogue; not to mention that his father's career had taught him a\nbitter estimate of popularity, and of all tumultuary enthusiasm:--\n\n    \"_Van Den Bosch._ Times are sore changed, I see. There's none in Ghent\n    That answers to the name of Artevelde.\n    Thy father did not carp or question thus\n    When Ghent invoked his aid. The days have been\n    When not a citizen drew breath in Ghent\n    But freely would have died in Freedom's cause.\n    _Artevelde._ With a good name thou christenest the cause.\n    True, to make choice of despots is some freedom,\n    The only freedom for this turbulent town,\n    Rule her who may. And in my father's time\n    We still were independent, if not free;\n    And wealth from independence, and from wealth\n    Enfranchisement will partially proceed.\n    The cause, I grant thee, Van Den Bosch, is good;\n    And were I linked to earth no otherwise\n    But that my whole heart centred in myself,\n    I could have tossed you this poor life to play with,\n    Taking no second thought. But as things are,\n    I will resolve the matter warily,\n    And send thee word betimes of my conclusion.\n    _Van Den Bosch_. Betimes it must be; for some two hours hence\n    I meet the Deans, and ere we separate\n    Our course must be determined.\n    _Artevelde_.                 In two hours,\n    If I be for you, I will send this ring\n    In token I have so resolved.\"\n\nHe had already resolved. Such a man would not have suffered himself to\nbe hemmed in within the space of two hours to make so great a decision;\nbut he would not rush precipitately forward; he would feel his own\n_will_ at each step. He had already resolved; but his love to Adriana\ntroubles him at heart: he must first make all plain and intelligible\nthere, before he becomes captain of the Whitehoods. From this interview\nhe goes to Adriana; and then follows a dialogue, every sentence of\nwhich, if we were looking out for admirable passages for quotation,\nwould offer itself as a candidate. We quote only, from a drama so well\nknown, for the purpose of illustrating the analytic view we would\npresent of its chief hero; but the passages selected for this purpose\ncan hardly fail of being also amongst the most beautiful in themselves.\nArtevelde is alone, waiting for the appearance of Adriana--\n\n    \"There is but one thing that still harks me back.\n    To bring a cloud upon the summer day\n    Of one so happy and so beautiful,--\n    It is a hard condition. For myself,\n    I know not that the circumstance of life\n    In all its changes can so far afflict me\n    As makes anticipation much worth while.\n    ... Oh she is fair!\n    As fair as Heaven to look upon! as fair\n    As ever vision of the Virgin blest\n    That weary pilgrim, resting by the fount\n    Beneath the palm, and dreaming to the tune\n    Of flowing waters, duped his soul withal.\n    It was permitted in my pilgrimage\n    To rest beside the fount, beneath the tree,\n    Beholding there no vision, but a maid\n    Whose form was light and graceful as the palm,\n    Whose heart was pure and jocund as the fount,\n    And spread a freshness and a verdure round.\"\n\nAdriana appears, and in the course of the dialogue he addresses her\nthus:--\n\n                        \"Be calm;\n    And let me warn thee, ere thy choice be fixed,\n    What fate thou may'st be wedded to with me.\n    Thou hast beheld me living heretofore\n    As one retired in staid tranquillity:\n    The dweller in the mountains, on whose ear\n    The accustomed cataract thunders unobserved;\n    The seaman, who sleeps sound upon the deck,\n    Nor hears the loud lamenting of the blast,\n    Nor heeds the weltering of the plangent wave,--\n    These have not lived more undisturbed than I:\n    But build not upon this; the swollen stream\n    May shake the cottage of the mountaineer,\n    And drive him forth; the seaman, roused at length,\n    Leaps from his slumber on the wave-washed deck:\n    And now the time comes fast when here, in Ghent,\n    He who would live exempt from injuries\n    Of armed men, must be himself in arms.\n    This time is near for all,--nearer for me:\n    I will not wait upon necessity,\n    And leave myself no choice of vantage-ground,\n    But rather meet the times where best I may,\n    And mould and fashion them as best I can.\n    Reflect then that I soon may be embarked\n    In all the hazards of these troublesome times,\n    And in your own free choice take or resign me.\n    _Adri._ Oh, Artevelde, my choice is free no more.\"\n\nAnd now he is open to hear Van Den Bosch. That veteran in war and\ninsurrection brings him news that the people are ready to elect him for\ntheir captain or ruler.\n\n    \"_Artev._ Good! when they come I'll speak to them.\n    _Van Den B._             'Twere well.\n\n       *       *       *       *       *\n\n    Canst learn to bear thee high amongst the commons?\n    Canst thou be cruel? To be esteemed of them,\n    Thou must not set more store by lives of men\n    Than lives of larks in season.\n    _Artev._                 Be it so.\n    _I can do what is needful._\"\n\nThe time of action is at hand. We now see Van Artevelde in a suit of\narmour; he is reclining on a window-seat in his own house, looking out\nupon the street. There is treason in the town; of those who flock to\nthe market-place, some have already deserted his cause.\n\n    \"_Artev._ Not to be feared--Give me my sword! Go forth,\n    And see what folk be these that throng the street.\n          [_Exit the page._\n    Not to be feared is to be nothing here.\n    And wherefore have I taken up this office,\n    If I be nothing in it? There they go.\n                 (_Shouts are heard._)\n    Of them that pass my house some shout my name,\n    But the most part pass silently; and once\n    I heard the cry of 'Flanders and the Lion!'\n\n       *       *       *       *       *\n\n    That cry again!\n    Sir knights, ye drive me close upon the rocks,\n    And of my cargo you're the vilest bales,\n    So overboard with you! What, men of blood!\n    Can the son better auspicate his arms\n    Than by the slaying of who slew the father?\n    Some blood may flow because that it needs must,\n    But yours by choice--I'll slay you, and thank God.\n             (_Enter Van Den Bosch._)\n    _Van Den B._ The common bell has rung! the knights are there;\n    Thou must come instantly.\n    _Artev._           I come, I come.\n    _Van Den B._ Now, Master Philip, if thou miss thy way\n    Through this affair we're lost. For Jesus' sake\n    Be counselled now by me; have thou in mind----\n    _Artev._ Go to, I need not counsel; I'm resolved.\n    Take thou thy stand beside Sir Simon Bette,\n    As I by Grutt; take note of all I do,\n    And do thyself accordingly. Come on.\"\n\nThey join the assembly; they take their stand each by one of the\ntraitor knights; the debate on the proposal of the Earl proceeds; three\nhundred citizens are to be given up to him, and on this, and other\nconditions, peace is to be granted. Artevelde addresses the assembly,\nand then turning to these knights, he continues:--\n\n              \"Your pardon, sirs, again!\n            (_To Grutt and Bette._)\n    You are the pickers and the choosers here,\n    And doubtless you're all safe, ye think--ha! ha!\n    But we have picked and chosen, too, sir knights--\n    What was the law for I made yesterday--\n    What! is it you that would deliver up\n    Three hundred citizens to certain death?\n    Ho! Van Den Bosch! have at these traitors--ha!--\n            (_Stabs Grutt, who falls._)\n    _Van Den B._ Die, treasonable dog!--\n                 (_Stabs Bette._)\"\n\nHe can do \"what is needful.\" It is admirable; everything that is said\nand done is admirable; but an involuntary suspicion at times creeps\ninto the mind, that such a man as Philip Van Artevelde never lived,\nor could live. No man could move along such a line of enterprise with\nsuch a weight of reflection on all the springs of action. We see the\ncalm statesman at the head of a tumultuary movement; and the meditative\nman, to whom revenge is the poorest of our passions, striking a blow\nfrom which an old warrior might shrink. Could a man be really impelled\nalong a path of life like this by passions that are admitted, indeed,\ninto the bosom, but watched like prisoners? The suspicion, we say,\ncreeps involuntarily into the mind; but we will not entertain it--we\nwill not yield to it. That the reflective and energetic characters are,\nin certain degrees, combined together, we all know; and who shall say\nwithin what degrees only this is possible? And why may not an ideal\nperfection of this kind be portrayed as well as an ideal patriot, or\nan ideal monk, or an ideal warrior? We throw the suspicion aside, and\ncontinue our analysis.\n\nThere is a passage which is often quoted for its great beauty: we quote\nit also for its great appropriateness. Philip Van Artevelde is master\nof the city; he is contemplating it at night-time from the tower of St\nNicholas. The reflection here put into the mouth of the anxious captain\nbrings back to us, in the midst of war and the cares of government, the\nmeditative man:--\n\n    \"There lies a sleeping city. God of dreams!\n    What an unreal and fantastic world\n    Is going on below!\n    Within the sweep of yon encircling wall\n    How many a large creation of the night,\n    Wide wilderness and mountain, rock and sea,\n    Peopled with busy transitory groups,\n    Finds room to rise, and never feels the crowd!\"\n\nThe famous scene, which has for its place the summit of this tower,\nbetween Artevelde and Van Den Bosch, is fresh in the recollection\nof every reader: we must pass it by, and the admirable and pathetic\ndescription of the famine that is raging in Ghent, and proceed to\nthe last act of this part of the drama. Artevelde has stimulated the\ncitizens to make one brave effort more--to sally from the walls, and\nmeet the Earl in battle before Bruges. He has arranged in order of\nbattle his lean and famine-stricken, but desperate little army. He\nknows the extreme peril in which they stand: no food in the camp;\nfearful odds to be encountered; yet the only hope lying in immediate\nbattle. He does not delude himself for a moment; he sees the danger\nclear, and entertains it with a certain sarcastic levity. He does not\nhope, but he acts as if he did. He is not a man given to hope, but he\nhas a tempered despondency, which sits with him at the council-board,\nand rides with him to the field, and which he compels to do the\nservices of hope.\n\n    \"_Artev._       I would to God\n    The sun might not go down upon us here\n    Without a battle fought!\n    _Van Den B._    If so it should,\n    We pass a perilous night,\n    And wake a wasted few the morrow morn.\n    _Van Muck._  We have a supper left.\n    _Artev._            My lady's page,\n    If he got ne'er a better, would be wroth,\n    And burn in effigy my lady's steward.\n\n       *       *       *       *       *\n\n    _Van Den B._  We'll hope the best;\n    But if there be a knave in power unhanged,\n    And in his head a grain of sense undrowned,\n    He'll be their caution not to----\n    _Artev._          Van Den Bosch,\n    Talk we of battle and survey the field,\n    For I _will_ fight.\"\n\nWe like this last expression. What in another man would have been a\nmere petulance, is in Artevelde an assumed confidence--consciously\nassumed, as the only tone of mind in which to pass through the present\ncrisis. Nor can we omit to notice the following passage, which, to our\napprehension, is very characteristic of our contemplative politician\nand warrior; it shows the sardonic vein running through his grave and\nserious thoughts:--\n\n    _Art._ (_to Van Ryk._)  I tell thee, eat,\n    Eat and be fresh. _I'll send a priest to shrive thee.\n    Van Muck, thou tak'st small comfort in thy prayers,\n    Put thou thy muzzle to yon tub of wine._\"\n\nThe battle is fought and a victory won. Justice is executed with stern\nand considerate resolve on the villains of the piece, and we leave Van\nArtevelde triumphant in his great contest, and happy in the love of\nAdriana.\n\nThe subordinate characters who are introduced into this first part of\nthe drama, we have no space to examine minutely. The canvass is well\nfilled, though the chief figure stands forward with due prominence.\nAdriana is all that an amiable and loving woman should be. The\nlighter-hearted Clara is intended as a sort of contrast and relief. Her\nlevity and wit are not always graceful; they are not so in the early\nscene where she jests with the page: afterwards, when in presence of\nher lover, she has a fitter and more genial subject for her playful\nwit, and succeeds much better. In the course of the drama, when the\nfamine is raging in Ghent, she appears as the true sister of Philip Van\nArtevelde. At her first introduction she is somewhat too hoydenish for\nthe mistress of the noble D'Arlon. D'Arlon is all that a knight should\nbe, and Gilbert Matthew is a consummate villain.\n\nBetween the first and second parts is a poem in rhyme, called \"The Lay\nof Elena.\" This introduces us to the lady who is to be the heroine of\nthe second part of the drama. All the information it gives, might, we\nthink, have been better conveyed in a few lines of blank verse, added\nto that vindication of herself which Elena pours forth in the first\nact, when Sir Fleureant of Heurlee comes to reclaim her on the part of\nthe Duke of Bourbon. This poem is no favourite of ours; but the worst\ncompliment we would pay it implies, in one point of view, a certain\nfitness and propriety--we were glad to return to the blank verse of our\nauthor, in which we find both more music and more pathos than in these\nrhymes.\n\nIf we are tempted to suspect, whilst reading the first part of this\ndrama, that the character of Philip Van Artevelde combines in a\nquite ideal perfection the man of thought with the man of action,\nwe, at all events, cannot accuse the author, in this second part, of\nrepresenting an ideal or superhuman happiness as the result of this\nperfect combination. It is a very truthful sad- destiny that\nhe portrays. The gloomy passionate sunset of life has been a favourite\nsubject with poets; but what other author has chosen the clouded\nafternoon of life, the cheerless twilight, and the sun setting behind\ncold and dark clouds? It was a bold attempt. It has been successfully\nachieved. But no amount of talent legitimately expended on it could\nmake this second part as attractive as the first. When the heroic man\nhas accomplished his heroic action, life assumes to him, more than to\nany other, a most ordinary aspect: his later years bring dwarfish hopes\nand projects, or none at all; they bring desires no longer \"gay,\" and\nwelcomed only for such poor life as they may have in them. Philip Van\nArtevelde is now the Regent of Flanders, and, like other regents, has\nto hold his own: Adriana he has lost; her place is supplied by one\nstill fair but faded, and who, though she deserved a better fate, must\nstill be described as lately the mistress of the Duke of Bourbon. It is\nthe hero still, but he has descended into the commonplace of courts and\npolitics.\n\nThat it is the same Philip Van Artevelde we are in company with,\nthe manner in which he enters into this new love will abundantly\ntestify. He has been describing to Elena his former wife, Adriana. The\ndescription is very beautiful and touching. He then proceeds with his\nwooing thus:--\n\n    \"_Artev._ ... Well, well--she's gone,\n    And I have tamed my sorrow. Pain and grief\n    Are transitory things no less than joy,\n    And though they leave us not the men we were,\n    Yet they do leave us. You behold me here\n    A man bereaved, with something of a blight\n    Upon the early blossoms of his life\n    And its first verdure, having not the less\n    A living root, drawing from the earth\n    Its vital juices, from the air its powers:\n    And surely as man's health and strength are whole,\n    His appetites regerminate, his heart\n    Reopens, and his objects and desires\n    Shoot up renewed. What blank I found before me,\n    From what is said you partly may surmise;\n    How I have hoped to fill it, may I tell?\n    _Elena._ I fear, my lord, that cannot be.\n    _Artev._                      Indeed!\n    Then am I doubly hopeless....\n    _Elena._ I said I feared another could not fill\n    The place of her you lost, being so fair\n    And perfect as you give her out.\"\n\nIn fine, Elena is conquered, or rather led to confess a conquest\nalready achieved.\n\n    \"_Elena._      I cannot--no--\n    I cannot give you what you've had so long;\n    Nor need I tell you what you know so well.\n    I must be gone.\n    _Artev._        Nay, sweetest, why these tears?\n    _Elena._ No, let me go--I cannot tell--no--no--\n    I want to be alone--\n    Oh! Artevelde, for God's love let me go!      [_Exit._\n    _Artev._ (_after a pause._) The night is far advanced upon the\n        morrow,\n    --_Yes, I have wasted half a summer's night_.\n    Was it well spent? Successfully it was.\n    How little flattering is a woman's love!\n    Worth to the heart, come how it may, a world;\n    Worth to men's measures of their own deserts,\n    If weighed in wisdom's balance, merely nothing.\n    The few hours left are precious--who is there?\n    Ho! Nieuverkerchen!--when we think upon it,\n    How little flattering is a woman's love!\n    Given commonly to whosoe'er is nearest\n    And propped with most advantage; outward grace\n    Nor inward light is needful; day by day\n    Men wanting both are mated with the best\n    And loftiest of God's feminine creation.\n    Ho! Nieuverkerchen!--what, then, do we sleep?\n    Are none of you awake?--and as for me,\n    The world says Philip is a famous man--\n    What is there woman will not love, so taught?\n    Ho! Ellert! by your leave though, you must wake.\n                (_Enter an officer._)\n    Have me a gallows built upon the mount,\n    And let Van Kortz be hung at break of day.\"\n\nIt is worth noticing, as a characteristic trait, that Philip Van\nArtevelde speaks more like the patriot, harangues more on the cause\nof freedom, now that he is Regent of Flanders, opposed to the feudal\nnobility, and to the monarchy of France, and soliciting aid from\nEngland, than when he headed the people of Ghent, strong only in their\nown love of independence. \"Bear in mind,\" he says, answering the herald\nwho brings a hostile message from France and Burgundy--\n\n                      \"Bear in mind\n    Against what rule my father and myself\n    Have been insurgent: whom did we supplant?\n    There was a time, so ancient records tell,\n    There were communities, scarce known by name\n    In these degenerate days, but once far famed,\n    Where liberty and justice, hand in hand,\n    Ordered the common weal; where great men grew\n    Up to their natural eminence, and none,\n    Saving the wise, just, eloquent, were great.\n    ... But now, I ask,\n    Where is there on God's earth that polity\n    Which it is not, by consequence converse,\n    A treason against nature to uphold?\n    Whom may we now call free? whom great? whom wise?\n    Whom innocent?--the free are only they\n    Whom power makes free to execute all ills\n    Their hearts imagine; they alone are great\n    Whose passions nurse them from their cradles up\n    In luxury and lewdness,--whom to see\n    Is to despise, whose aspects put to scorn\n    Their station's eminence....\n    ... What then remains\n    But in the cause of nature to stand forth,\n    And turn this frame of things the right side up?\n    For this the hour is come, the sword is drawn,\n    And tell your masters vainly they resist.\"\n\nWe regret to be compelled to garble in our extract so fine a passage of\nwriting. Meanwhile our patriot Regent sends Father John to England to\nsolicit aid--most assuredly not to overthrow feudalism, but to support\nthe Regent against France. His ambition is dragging, willingly or\nunwillingly, in the old rut of politics. When Father John returns from\nthis embassy, he is scandalised at the union formed between Artevelde\nand Elena. Here, too, is another sad descent. Our hero has to hear\nrebuke, and, with a half-confession, submit to be told by the good\nfriar of his \"sins.\" He answers bravely, yet with a consciousness that\nhe stands not where he did, and cannot challenge the same respect from\nthe friar that he could formerly have done.\n\n    \"_Artev._               You, Father John,\n    I blame not, nor myself will justify;\n    But call my weakness what you will, the time\n    Is past for reparation. Now to cast off\n    The partner of my sin were further sin;\n    'Twere with her first to sin, and then against her.\n    And for the army, if their trust in me\n    Be sliding, let it go: I know my course;\n    And be it armies, cities, people, priests,\n    That quarrel with my love--wise men or fools,\n    Friends, foes, or factions--they may swear their oaths,\n    And make their murmur--rave and fret and fear,\n    Suspect, admonish--they but waste their rage,\n    Their wits, their words, their counsel: here I stand,\n    Upon the deep foundations of my faith\n    To this fair outcast plighted; and the storm\n    That princes from their palaces shake out,\n    Though it should turn and head me, should not strain\n    The seeming silken texture of this tie.\"\n\nAnd now disaster follows disaster; town after town manifests symptoms\nof treachery to his cause. His temper no longer retains its wonted\ncalmness, and the quick glance and rapid government of affairs seems\nabout to desert him. Note this little trait:--\n\n    \"_Artev._           Whither away, Vauclaire?\n    _Vauclaire._ You'll wish, my lord, to have the scouts, and others\n    That are informed, before you.\n    _Artev._             'Twere well.\"\n\nIt is something new that another should anticipate the necessary orders\nto be given. The decisive battle approaches, and is fought. This time\nit is lost. Our hero does not even fall in the field; an assassin\nstabs him in the back. The career of Artevelde ends thus; and that\npublic cause with which his life was connected has at the same time an\ninglorious termination: \"the wheel has come full circle.\"\n\nThe catastrophe is brought about by Sir Fleureant of Heurlee. This\nman's character undergoes, in the course of the drama, a complete\ntransformation. We do not say that the change is unnatural, or that\nit is not accounted for; but the circumstances which bring it about\nare only vaguely and incidentally narrated, so that the reader is not\nprepared for this change. A gay, thoughtless, reckless young, knight,\nwho rather gains upon us at his first introduction, is converted into\na dark, revengeful assassin. It would, we think, have improved the\neffect of the _plot_, if we had been able to trace out more distinctly\nthe workings of the mind of one who was destined to take so prominent a\npart in the drama.\n\nThe character of _Lestovet_ is admirably sustained, and is manifestly\na favourite with the author. But we must now break away from _Philip\nVan Artevelde_, to notice the other dramas of Mr Taylor. _Edwin the\nFair_ next claims our attention. Here also we shall make no quotations\nmerely for the sake of their beauty; and we shall limit ourselves to an\nanalysis of the principal character, Dunstan, on which, perhaps, a word\nor two of explanation may not be superfluous.\n\nLet us suppose a dramatic writer sitting down before such a character\nas this of Dunstan, and contemplating the various aspects it assumes,\nwith the view of selecting one for the subject of his portraiture. In\nthe first place, he is aware that, although, as a historical student,\nhe may, and perhaps must, continue to doubt as to the real character of\nthis man--how much is to be given to pride, to folly, to fanaticism,\nto genuine piety, or to the love of power--yet that, the moment he\nassumes the office of dramatic poet, he must throw all doubt entirely\naside. The student of history may hesitate to the last; the poet is\npresumed to have from the beginning the clearest insight into the\nrecesses of the mind, and the most unquestionable authority for all\nthat he asserts. A sort of mimic omniscience is ascribed to the poet.\nHas he not been gifted, from of old, with an _inspiration_, by means of\nwhich he sees the whole character and every thought of his hero, and\ndepicts and reveals them to the world? To him doubt would be fatal. If\nhe carries into his drama the spirit of historical criticism, he will\nraise the same spirit in his reader, and all faith in the imaginary\ncreation he offers them is gone for ever. Manifest an error as this may\nbe, we think we could mention some instances, both in the drama and the\nnovel, in which it has been committed.\n\nBut such a character as Dunstan's is left uncertain in the light\nof history, and our dramatist has to choose between uncertainties.\nHe will be guided in his selection partly by what he esteems the\npreponderating weight of evidence, and partly, and perhaps still more,\nby the superior fitness of any one phase of the character for the\npurpose he has in view, or the development of his own peculiar powers.\nIn this case, three interpretations present themselves. The first,\nwhich has little historical or moral probability, and offers little\nattraction to the artist, is, that Dunstan was a hypocrite, seeking by\nshow of piety to compass some ambitious end, or win the applause of the\nvulgar. Undoubted hypocrites history assuredly presents us with--as\nwhere the ecclesiastical magnate degenerates into the merely secular\nprince. There have been luxurious and criminal popes and cardinals,\nintriguing bishops and lordly abbots, whom the most charitable of men,\nand the most pious of Catholics, must pronounce to have been utterly\ninsincere in their professions of piety. But a hypocrite who starves\nand scourges himself--who digs a damp hole in the earth, and lives\nin it--seems to us a mere creature of the imagination. Such men, at\nall events, either begin or end with fanaticism. The second and more\nusual interpretation is, that Dunstan was a veritable enthusiast, and\na genuine churchman after the order of Hildebrand, capable, perhaps,\nof practising deceit or cruelty for his great purpose, but entirely\ndevoted to that purpose--one of those men who sincerely believe that\nthe salvation of the world and the predominance of their order are\ninseparably combined. There would be no error in supposing a certain\nmixture of pride and ambition. Nor, in following this interpretation,\nwould there be any great violation of probability in attributing to\nDunstan, though he lived in so rude an age, all those arguments by\nwhich the philosopher-priest is accustomed to uphold the domination\nof his order. The thinking men of every age more nearly resemble each\nother in these great lines of thought and argument, than is generally\nsupposed. The third interpretation is that which the historical student\nwould probably favour. It is that Dunstan was, in truth, _partially\ninsane_--a man of fervent zeal, and of great natural powers, but of\ndiseased mind. The very ability and knowledge which he possessed,\ncombined with the strange forms which his asceticism took, lead to this\nsupposition. Such men, we know, exist, and sometimes pass through a\nlong career before they are accurately understood. Exhibiting itself in\nthe form of fanaticism, and in a most ignorant and superstitious age,\na partial insanity might easily escape detection, or even add to the\nreputation of the saint.\n\nThis last is the rendering of the character which Mr Taylor has\nselected. It is evidently the most difficult to treat. Perhaps the\ndifficulty and novelty of the task it presented, as well as its greater\nfidelity to history, induced him to accept this interpretation. That\nsecond and more popular one which we have mentioned would appear, to a\nmind like Mr Taylor's, too facile and too trite. Any high-churchman of\nalmost any age--any bishop, if you inflate the lawn sleeves, or even\nany young curate, whose mind dwells too intensely on the _power of\nthe keys_--would present the rudiments of the character. However that\nmay be, Mr Taylor undertook the bold and difficult task of depicting\nthe strong, shrewd, fervent mind, saint and politician both, but\nacting with the wild and irregular force of insanity. How, we may ask\nourselves, would such a mind display itself? Not, we way be sure, in a\ntissue of weakness or of wildness. We should often see the ingenious\nreasoner, more cunning than wise, the subtle politician, or even the\ndeep moraliser upon human life; but whenever the fatal chords were\ntouched--the priestly power, the priestly mission, the intercourse with\nthe world of spirits--there we should see symptoms of insanity and\ndelusion. Such is the character which Mr Taylor has portrayed.\n\nEarl Leolf, calm and intelligent, and the perfect _gentleman_ (those\nwho remember the play will feel the truth of this last expression,)\ngives us at the very commencement the necessary explanation--\n\n    \"_Leolf._ How found you the mid-counties?\n    _Athulf._              Oh! monk-ridden;\n    Raving of Dunstan.\n    _Leolf._              'Tis a raving time:\n    Mad monks, mad peasants; _Dunstan is not sane,\n    And madness that doth least declare itself\n    Endangers most, and ever most infects\n    The unsound many._  See where stands the man,\n    And where this people: thus compute the peril\n    To one and all. _When force and cunning meet\n    Upon the confines of one cloudy mind_,\n    When ignorance and knowledge halve the mass,\n    When night and day stand at an equinox,\n    Then storms are rife.\"\n\nNo justice, it is plain, can be done to Mr Taylor's drama, unless the\nintimation here given us be kept in view. Yet we suspect, from the\nremarks sometimes made upon this play, that it has been overlooked, or\nnot sufficiently attended to. Passages have been censured as crude or\nextravagant which, in themselves, could be no otherwise, since they\nwere intended to portray this half-latent and half-revealed insanity.\nThe arrogance of Dunstan, and his communings with the spiritual world,\nnot often have the air of sublimity, for they arise from the disorder\nand hallucination of his mind. When he tells the Queen Mother not to\nsit in his presence, as well as when he boasts of his intercourse with\nangels and demons, we see the workings of a perturbed spirit:--\n\n    \"_Queen Mother._          Father, I am faint,\n    For a strange terror seized me by the way.\n    I pray you let me sit.\n    _Dunstan._           I say, forbear!\n    Thou art in a Presence that thou wot'st not of,\n    Wherein no mortal may presume to sit.\n    If stand thou canst not, kneel.\n            (_She falls on her knees._)\n    _Queen Mother._       Oh, merciful Heaven'\n    Oh, sinner that I am!\n    _Dunstan._              Dismiss thy fears;\n    Thine errand is acceptable to Him\n    Who rules the hour, and thou art safer here\n    Than in thy palace. Quake not, but be calm,\n    And tell me of the wretched king, thy son.\n    This black, incestuous, unnatural love\n    Of his blood-relative--yea, worse, a seed\n    That ever was at enmity with God--\n    His cousin of the house of Antichrist!\n    It is as I surmised?\n    _Queen Mother._           Alas! lost boy!\n    _Dunstan._ Yes, lost for time and for eternity,\n    If he should wed her. But that shall not be.\n    Something more lofty than a boy's wild love\n    Governs the course of kingdoms. From beneath\n    This arching umbrage step aside; look up;\n    The alphabet of Heaven is o'er thy head,\n    The starry literal multitude. _To few,\n    And not in mercy, is it given to read\n    The mixed celestial cipher._\"\n\nHow skilfully the last passage awakes in the reader a feeling of\nsympathy with Dunstan! When he has given his instructions to the Queen\nMother, the scene closes thus:--\n\n    \"_Queen Mother._      Oh, man of God!\n    Command me always.\n    _Dunstan._            Hist! I hear a spirit!\n    Another--and a third. They're trooping up.\n    _Queen Mother._ St Magnus shield us!\n    _Dunstan._            Thou art safe; but go;\n    The wood will soon be populous with spirits.\n    The path thou cam'st retread. Who laughs in the air?\"\n\nDunstan believes all along that he is marked out from the ordinary roll\nof men--that he has a peculiar intercourse with, and a peculiar mission\nfrom, Heaven; but he nevertheless practises on the credulity of others.\nThis mixture of superstition and cunning does not need insanity to\nexplain, but it is seen here in very appropriate company. He says to\nGrumo--\n\n    \"Go, get thee to the hollow of yon tree,\n    And bellow there as is thy wont.\n    _Grumo._                   How long?\n    _Dunstan._ Till thy lungs crack. Get hence.\n                               [_Exit Grumo._\n    And if thou bellowest otherwise than Satan,\n    It is not for the lack of Satan's sway\n    'Stablished within thee.\n    (_Strange howls are heard from the tree._\")\n\nWith the same crafty spirit, and by a trick as gross, he imposes on the\nSynod, contriving that a voice shall appear to issue from the crucifix.\nThese frauds, however, would have availed nothing of themselves; it is\nthe spirit of fanaticism bearing down all opposition by which he works\nhis way. This spirit sustains him in his solitude--\n\n                         \"I hear your call!\n    A radiance and a resonance from Heaven\n    Surrounds me, and my soul is breaking forth\n    In strength, as did the new-created Sun\n    When Earth beheld it first on the fourth day.\n    God spake not then more plainly to that orb\n    Than to my spirit now.\"\n\nIt sustains him in his solitude, and mark how triumphantly it carries\nhim through in the hour of action. Odo the archbishop, Ricola the\nking's chaplain, as well as king and courtiers, all give way before\nthis inexorable, unreasoning fanaticism, a fanaticism which is as\ncomplete a stranger to fear as it is to reason--\n\n    \"_Dunstan_ (_to Elgiva._) Fly hence,\n    Pale prostitute! Avaunt, rebellious fiend,\n    Which speakest through her.\n    _Elgiva._ I am thy sovereign mistress and thy queen.\n    _Dunstan._         ... Who art thou?\n    I see thee, and I know thee--yea, I smell thee!\n    Again, 'tis Satan meets me front to front;\n    Again I triumph! Where, and by what rite,\n    And by what miscreant minister of God,\n    And rotten member, was this mockery,\n    That was no marriage, made to seem a marriage?\n    _Ricola._ Lord Abbot, by no----\n    _Dunstan._           What then, was it thou?\n    The Church doth cut thee off and pluck thee out!\n    A Synod shall be summoned! _Chains for both!\n    Chains for that harlot, and for this dog-priest!_\n    Oh wall of Jezreel!\"\n\nAnd forthwith Elgiva, in spite of the king's resistance, is carried out\na captive. The king, too, is imprisoned in the Tower, and here ensues\na scene which brings out another aspect of the mind of Dunstan. It was\nthe object of the crafty priest to induce Edwin to resign the crown; he\nhad, therefore, made his imprisonment as painful as possible. He now\nvisits him in the Tower, and in this interview we see, underneath the\nmad zealot and the subtle politician, something of the genuine _man_.\nDunstan had not been always, and only, the priest; he understood the\nhuman life he trampled on--\n\n    \"_Dunstan._ What makes you weak? Do you not like your food?\n    Or have you not enough?\n    _Edwin._                Enough is brought;\n    But he that brings it drops what seems to say\n    That it is mixed with poison--some slow drug;\n    So that I scarce dare eat, and hunger always.\n    _Dunstan._ Your food is poisoned by your own suspicions.\n    'Tis your own fault.----\n    But thus it is with kings; suspicions haunt,\n    And dangers press around them all their days;\n    Ambition galls them, luxury corrupts,\n    And wars and treasons are their talk at table.\n    _Edwin._ This homily you should read to prosperous kings;\n    It is not needed for a king like me.\n    _Dunstan._ Who shall read homilies to a prosperous king!\n    ... To thy credulous ears\n    The world, or what is to a king the world,\n    The triflers of thy court, have imaged me\n    As cruel, and insensible to joy,\n    Austere, and ignorant of all delights\n    That arts can minister. Far from the truth\n    They wander who say thus. I but denounce\n    Loves on a throne, and pleasures out of place.\n    I am not old; not twenty years have fled\n    Since I was young as thou; and in my youth\n    I was not by those pleasures unapproached\n    Which youth converses with.\n    _Edwin._                    No! wast thou not?\n    How came they in thy sight?\n    _Dunstan._                  When Satan first\n    Attempted me, 'twas in a woman's shape;\n    Such shape as may have erst misled mankind,\n    When Greece or Rome upreared with Pagan rites\n    Temples to Venus....\n    ... 'Twas Satan sang,\n    Because 'twas sung to me, whom God had called\n    To other pastime and severer joys.\n    But were it not for this, God's strict behest\n    Enjoined upon me--had I not been vowed\n    To holiest service rigorously required,\n    I should have owned it for an angel's voice,\n    Nor ever could an earthly crown, or toys\n    And childishness of vain ambition, gauds\n    And tinsels of the world, have lured my heart\n    Into the tangle of those mortal cares\n    That gather round a throne. What call is thine\n    From God or man, what voice within bids thee\n    Such pleasures to forego, such cares confront?\n\n       *       *       *       *       *\n\n    ... Unless thou by an instant act\n    Renounce the crown, Elgiva shall not live.\n    The deed is ready, to which thy name affixed\n    Discharges from restraint both her and thee.\n    Say wilt thou sign?\n    _Edwin._            I will not.\n    _Dunstan._                      Be advised.\n    What hast thou to surrender? I look round;\n    This chamber is thy palace court, and realm.\n    I do not see the crown--where is it hidden?\n    Is that thy throne?--why, 'tis a base joint-stool;\n    Or this thy sceptre?--'tis an ashen stick\n    Notched with the days of thy captivity.\n    Such royalties to abdicate, methinks,\n    Should hardly hold thee long. Nay, I myself,\n    That love not ladies greatly, would give these\n    To ransom whom I loved.\"\n\nThese feelings of humanity, in part indeed simulated, do not long keep\nat bay the cruelty and insane rage or the priest. Edwin persists in his\nrefusal; Dunstan leaves him for a moment, but shortly after returns\nholding the deed in his hand, and followed by his tool Grumo.\n\n    \"_Dunstan._ Thy signature to this.\n    _Edwin._ I will not sign.\n    _Dunstan._ Thou wilt not! wilt thou that thy mistress die?\n    _Edwin._ Insulting abbot! she is not my mistress;\n    She is my wife, my queen.\n    _Dunstan._                Predestinate pair!\n    He knoweth who is the Searcher of our hearts,\n    That I was ever backward to take life,\n    Albeit at His command. Still have I striven\n    To put aside that service, seeking still\n    All ways and shifts that wit of man could scheme,\n    To spare the cutting off your wretched souls\n    In unrepented sin. But tendering here\n    Terms of redemption, it is thou, not I,\n    The sentence that deliverest.\n    _Edwin._                      Our lives\n    Are in God's hands.\n    _Dunstan._          Sot, liar, miscreant, No!\n    God puts them into mine! and may my soul\n    In tortures howl away eternity,\n    If ever again it yield to that false fear\n    That turned me from the shedding of thy blood!\n    Thy blood, rash traitor to thy God, thy blood!\n    Thou delicate Agag, I will spill thy blood!\"\n\nWe believe we have done justice to all the aspects in which the\ncharacter of Dunstan is here represented to us, but it would require a\nmuch larger space than we have at command to do justice to the whole\ndrama of _Edwin the Fair_. The canvass is crowded with figures, almost\nevery one of which has been a careful study, and will repay the study\nof a critical reader; and if the passages of eloquent writing are\nnot so numerous as in his previous work, there is no deficiency of\nthem, and many are the pungent, if not witty sayings, that might be\nextracted. The chief fault which seems to us to pervade this drama, is,\nindeed, that there is too much apparent study--that too much is seen\nof the artist. Speaking generally of Mr Taylor, and regarding him as a\ndramatic poet, we could desire more life and passion, more abandonment\nof himself to the characters he is portraying. But we feel this more\nparticularly in _Edwin the Fair_. We seem to see the artist sorting and\nputting together again the elements of human nature. His Wulfstan, the\never absent sage, his tricksy Emma, and her very silly lover, Ernway,\nare dramatic creations which may probably be defended point by point;\nbut, for all that, they do not look like real men and women. As to his\nmonks, the satellites of Dunstan, it may be said that they could not\nhave been correctly drawn if they had borne the appearance of being\nreal men. We do not like them notwithstanding.\n\nIn the edition which lies before us, bound up with _Edwin the Fair_\nis the republication of an early drama, _Isaac Comnenus_. It excited,\nwe are told in the preface, little attention in its first appearance.\nWe ourselves never saw it till very lately. Though inferior to his\nsubsequent productions, it is not without considerable merit, but it\nwill probably gather its chief interest as the forerunner of _Philip\nVan Artevelde_, and from the place it will occupy in the history of\nthe author's mind. A first performance, which was allowed to pass\nunnoticed by the public, might be expected to be altogether different\nin kind from its fortunate successors. The author, in his advance out\nof obscurity into the full light of success, might be supposed to have\nthrown aside his first habits of thought and expression. It is not so\nhere. We have much the same style, and there is the same combination\nof shrewd observation with a philosophic melancholy, the same gravity,\nand the same sarcasm. It is curious to notice how plainly there is the\ngerm of _Philip Van Artevelde_ in _Isaac Comnenus_. The hero of Ghent\nis far more sagacious, more serious, and more tender; but he looks\non life with a lingering irony, and a calm cynicism: to him it is a\nsad and disenchanted vision. In _Isaac Comnenus_ the same elements\nare combined in a somewhat different proportion: there is more of the\nirony and a more bitter cynicism; less of the grave tenderness and\nthe practical sagacity. Artevelde is Isaac Comnenus living over life\nagain--the same man, but with the advantage of a life's experience.\nIndeed Artevelde, if we may venture to jest with so grave a personage,\nhas something of the air of one who had been in the world before, who\nwas not walking along its paths for the first time: he treads with so\nsure a footstep, and seems to have no questions to ask, and nothing to\nlearn of experience.\n\nHappily it has not been necessary hitherto to say a word about the\n_plot_ of Mr Taylor's dramas. This of _Isaac Comnenus_, being less\nknown, may require a word of preliminary introduction. The scene\nis laid at Constantinople, at the close of the eleventh century;\nNicephorus is the reigning emperor. We may call to mind that the\ngovernment of the Byzantine monarchy, for a long time, maintained this\nhonourable peculiarity, that, though in form a despotism, the emperor\nwas expected to administer the law as it had descended to it from\nthe genius of Rome. Dynasties changed, but the government remained\nsubstantially the same. It was an Oriental despotism with a European\nadministration. Whilst, therefore, we have in the play before us a\nprince dethroned, and a revolution accomplished, we hear nothing of\nliberty and oppression, the cause of freedom, and the usual topics\nof patriotic conspiracy. The brothers Isaac and Alexius Comnenus are\nsimply too powerful to be trusted as subjects; an attempt has been\nalready made to poison the elder brother Isaac, the hero of the drama.\nHe finds himself in a manner constrained to push forward to the throne,\nas his only place of safety. This ambitious course is thrust upon\nhim. Meanwhile he enters on it with no soft-heartedness. He takes up\nhis part, and goes bravely through with it; bravely, but coldly--with\na sneer ever on his lip. With the church, too, he has contrived to\nmake himself extremely unpopular, and the Patriarch is still more\nrancorously opposed to him than the Emperor.\n\nBefore we become acquainted with him, he has loved and lost by death\nhis gentle _Irene_. This renders the game of ambition still more\ncontemptible in his eyes. It renders him cold also to the love of a\ncertain fair cousin, Anna Comnena. Love, or ambition, approaches him\nalso in the person of Theodora, the daughter of the emperor. She is\nwilling to desert her father's cause, and ally herself and all her\nhopes to Isaac Comnenus. Comnenus declines her love. The rejected\nTheodora brings about the catastrophe of the piece. The Emperor\nNicephorus is deposed; Isaac is conqueror in the strife, but he gives\nover the crown he has won to his brother Alexius. Then does Theodora\npresent herself disguised as some humble petitioner to Isaac Comnenus.\nArmed with a dagger, she forces her way into an inner chamber where\nhe is; a groan is heard, and the following stage direction closes the\nplay--\n\n    \"_All rush into the inner chamber, whilst Theodora, passing\n    out from it, crosses the stage, holding in her hand a dagger\n    covered with blood. The curtain falls._\"\n\nThis scanty outline will be sufficient to make the following\ncharacteristic quotations intelligible to those who may not have read\nthe play. Eudocia, his sister, thus describes Comnenus:--\n\n                             ----\"He\n    Is nothing new to dangers nor to life--\n    His thirty years on him have nigh told double,\n    Being doubly loaden with the unlightsome stuff\n    That life is made of. I have often thought\n    How nature cheats this world in keeping count:\n    There's some men pass for old men who ne'er lived--\n    These monks, to wit: they count the time, not spend it;\n    They reckon moments by the tick of beads,\n    And ring the hours with psalmody: clocks, clocks;\n    If one of these had gone a century,\n    I would not say he'd lived. My brother's age\n    Has spanned the matter of too many lives;\n    He's full of years though young.\"\n\nComnenus, we have said, is on ill terms with the church. Speaking of\nthe sanctuary he says:--\n\n    \"I have a safer refuge. Mother church\n    Hath no such holy precinct that my blood\n    Would not redeem all sin and sacrilege\n    Of slaughter therewithin. But there's a spot\n    Within the circle my good sword describes,\n    Which by God's grace is sanctified for me.\"\n\nOn quitting his cousin Anna, she says:--\n\n    \"Go, and good angels guard thee is my prayer.\n    _Comnenus._--Good soldiers, Anna. In the arm of flesh\n    Are we to trust. The Mother of the Gods,\n    Prolific Mother, holiest Mother church,\n    Hath banded heaven upon the side opposed.\n    No matter, when such supplicants as thou\n    Pray for us, other angels need we none.\"\n\nIt is plain that we have no dutiful son of the Church here; and that\nher hostility, in this instance, is not altogether without cause.\nWe find that his scepticism has gone farther than to dispute the\nmiraculous virtues of the holy image of St Basil, the eye of which he\nis reputed to have knocked out with his lance:--\n\n                   \"Just as you came\n    I moralised the matter of that change\n    Which theologians call--how aptly, say--\n    The quitting of a tenement.\"\n\nAnd his moralising is overcast with the shadow of doubt. The addresses,\nfor such they are, of Theodora, the daughter of the emperor, he\nreceives and declines with the greatest calmness, though they are of\nthat order which it is manifestly as dangerous to reject as to accept.\n\n    \"_Germanus._  My noble lord, the Caesarissa waits\n    With infinite impatience to behold you:\n    She bids me say so. Ah! most noble count!\n    A fortunate man--the sunshine is upon you--\n    _Comnenus._  Ay, sir, and wonderfully warm it makes me.\n    Tell her I'm coming, sir, with speed.\"\n\nWith speed, however, he does not go, nor makes a better excuse for\nhis delay than that he was \"sleeping out the noontide.\" In the first\ninterview he escapes from her confidence, and when subsequently she\nwill not be misunderstood, he says--\n\n               \"Nor now, nor ever,\n    Will I make bargains for a lady's love.\"\n\nIn a dialogue with his brother Alexius, his temper and way of thinking,\nand the circumstance which has mainly produced them, are more fully\ndeveloped. We make a few extracts without attempting very closely to\nconnect them. Alexius has been remarking the change in Comnenus since\nthey last met.\n\n    \"_Comnenus._ Change is youth's wonder:\n    Such transmutations have I seen on man\n    That fortune seemed a slow and stedfast power\n    Compared with nature.\n    _Alexius._ There is nought thou'st seen\n    More altered than art thou.\n    I speak not of thy change in outward favour,\n    But thou art changed in heart.\n    _Comnenus._          Ay, hearts change too:\n    Mine has grown sprightly, has it not, and hard?\n    I ride it now with spurs; else, else, Alexius--\n    Well is it said the best of life is childhood.\n    Life is a banquet where the best's first served,\n    And when the guest is cloyed comes oil and garlick.\n    _Alexius._  Hast thou forgotten how it was thy wont\n    To muse the hours away along this shore--\n    These very rippled sands?\n    _Comnenus._               The sands are here,\n    But not the foot-prints. Wouldst thou trace them now?\n    A thousand tides and storms have dashed them out.\n\n           *       *       *       *       *\n\n                   ... I have no care for beauty.\n    Seest thou yon rainbow based and glassed on ocean?\n    I look on that as on a lovely thing,\n    But not a thing of promise.\"\n\nComnenus has wandered with his brother unawares to a spot which of all\nothers on earth was the most dear or the most painful to him--the spot\nwhere his Irene had been buried. He recognises it whilst he is in the\nfull tide of his cynicism:--\n\n    \"_Alexius._ What is this carved upon the rock?\n    _Comnenus._                          I know not:\n    But Time has ta'en it for a lover's scrawl;\n    He's razed it, razed it.\n    _Alexius._           No, not quite; look here.\n    I take it for a lover's.\n    _Comnenus._          What! there's some talk\n    Of balmy breath, and hearts pierced through and through\n    With eyes' miraculous brightness--vows ne'er broken,\n    Until the church had sealed them--charms loved madly,\n    Until it be a sin to love them not--\n    And kisses ever sweet, till they be innocent--\n    But that your lover's not put down?\n    _Alexius._                       No, none of it.\n    There are but two words.\n    _Comnenus._ That's succinct; what are they?\n    _Alexius._  'Alas, Irene!' Why thy looks are now--\n\nComnenus parries the question of his brother, contrives to dismiss him,\nand remains alone upon the spot.\n\n    \"This is the very earth that covers her,\n    And lo! we trample it like common clay!\n              ... When I last stood here\n    Disguised, to see a lowly girl laid down\n    Into her early grave, there was such light\n    As now doth show it, but a bleaker air,\n    Seeing it was December. 'Tis most strange;\n    I can remember now each circumstance\n    Which then I scarce was conscious of; _like words\n    That leave upon the still susceptive sense\n    A message undelivered till the mind\n    Awakes to apprehensiveness and takes it_.\n    'Twas o'er--the muttered unattended rite,\n    And the few friends she had beside myself\n    Had risen and gone; I had not knelt, but stood\n    With a dull gaze of stupor as the mould\n    Was shovelled over, and the broken sods\n    Fitted together. Then some idle boys,\n    Who had assisted at the covering in,\n    Ran off in sport, trailing the shovels with them,\n    Rattling upon the gravel; and the sexton\n    Flattened the last sods down, and knocked his spade\n    Against a neighbouring tombstone to shake off\n    The clinging soil,--with a contented air,\n    Even as a ditcher who has done his work.\n                      ... Oh Christ!\n    How that which was the life's life of our being\n    Can pass away, and we recall it thus!\"\n\nWhilst reading this play of _Isaac Comnenus_ we seemed to perceive a\ncertain _Byronian_ vein, which came upon us rather unexpectedly. Not\nthat there is any very close resemblance between Comnenus and the\nheroes of Lord Byron; but there is a desperate wilfulness, a tone of\nscepticism, and a caustic view of human life, which occasionally recall\nthem to mind. We turned to the preface to _Philip Van Artevelde_, where\nthere is a criticism upon the poetry of Byron, not unjust in the faults\nit detects, but cold and severe, as it seems to us, in the praise\nthat it awards; and we found there an intimation which confirmed our\nsuspicion that _Isaac Comnenus_ had been written whilst still partially\nunder the influence of that poetry--written in what we may describe\nas a transition state. He says there of Lord Byron's poetry, \"It will\nalways produce a powerful impression upon very young readers, and I\nscarcely think that it can have been more admired by any than myself,\nwhen I was included in that category.\" And have we not here some\nexplanation of the severity and coldness of that criticism itself? Did\nnot the maturer intellect a little resent in that critical judgment the\nhallucinations of the youth?\n\nPerhaps we are hardly correct in calling the temper and spirit we\nhave here alluded to _Byronian_; they are common to all ages and to\nmany minds, though signally developed by that poet, and in our own\nepoch. Probably the future historian of this period of our literature\nwill attribute much of this peculiar exhibition of bitterness and\ndespondency to the sanguine hopes first excited and then disappointed\nby the French Revolution. He will probably say of certain regions\nof our literature, that the whole bears manifest traces of volcanic\norigin. Pointing to some noble eminence, which seems to have been\neternally calm, he will conjecture that it owed its elevation to\nthe same force which raised the neighbouring AEtna. Applying the not\nvery happy language of geology, he may describe it as \"a crater of\nelevation;\" which, being interpreted, means no crater at all, but an\nelevation produced by the like volcanic agency: the crater itself is\nhigher up in the same mountain range.\n\nThere still remains one other small volume of Mr Taylor's poetry,\nwhich we must not pass over entirely without mentioning. _The Eve of\nthe Conquest, and other Poems_. The chief piece here is of the nature\nof a dramatic scene. Harold, the night before the battle of Hastings,\nconverses with his daughter, unfolds some passages of his past life,\nand vindicates himself in his quarrel with that William the Norman who,\non the morrow, was to add the title of Conqueror to his name. But as it\nwill be more agreeable to vary the nature of our quotations, we shall\nmake the few extracts we have space for from the lyric poems which\nfollow.\n\nThe \"Lago Varese\" will be, we suspect, the favourite with most readers.\nThe image of the Italian girl is almost as distinctly reflected in the\nverse as it would have been in her own native lake.\n\n       *       *       *       *       *\n\n     \"And sauntering up a circling cove,\n        I found upon the strand\n      A shallop, and a girl who strove\n        To drag it to dry land.\n    I stood to see--the girl looked round--her face\n    Had all her country's clear and definite grace.\n\n      She rested with the air of rest\n        So seldom seen, of those\n      Whose toil remitted gives a zest,\n        Not languor, to repose.\n    Her form was poised, yet buoyant, firm, though free,\n    And liberal of her bright black eyes was she.\n\n      The sunshine of the Southern face,\n        At home we have it not;\n      And if they be a reckless race,\n        These Southerns, yet a lot\n    More favoured, on the chequered earth is theirs;\n    They have life's sorrows, but escape its cares.\n\n      There is a smile which wit extorts\n        From grave and learned men,\n      In whose austere and servile sports\n        The plaything is a pen;\n    And there are smiles by shallow worldlings worn,\n    To grace a lie or laugh a truth to scorn:\n\n      And there are smiles with less alloy\n        Of those who, for the sake\n      Of some they loved, would kindle joy\n        Which they cannot partake;\n    But hers was of the kind which simply say,\n    They came from hearts ungovernably gay.\"\n\nThe \"Lago Lugano\" is a companion picture, written \"sixteen summers\"\nafter, and on a second visit to Italy. One thing we notice, that in\nthis second poem almost all that is beautiful is brought from the\nsocial or political reflections of the writer: it is not the outward\nscene that lies reflected in the verse. He is thinking more of England\nthan of Italy.\n\n                            \"Sore pains\n    They take to set Ambition free, and bind\n      The heart of man in chains.\"\n\nAnd the best stanza in the poem is that which is directly devoted to\nhis own country:--\n\n    \"Oh, England! 'Merry England,' styled of yore!\n      Where is thy mirth? Thy jocund laughter, where?\n      The sweat of labour on the brow of care\n    Make a mute answer--driven from every door!\n    The May-pole cheers the village green no more,\n      Nor harvest-home, nor Christmas mummers rare.\n        The tired mechanic at his lecture sighs;\n    And of the learned, which, with all his lore,\n        Has leisure to be wise?\"\n\nWith some verses from a poem called \"St Helen's-Auckland\" we close our\nextracts. The author revisits the home of his boyhood:--\n\n    \"How much is changed of what I see,\n       How much more changed am I,\n     And yet how much is left--to me\n       _How is the distant nigh!_\n\n     The walks are overgrown and wild,\n       The terrace flags are green--\n     But I am once again a child,\n       I am what I have been.\n\n     The sounds that round about me rise\n       Are what none other hears;\n     I see what meets no other eyes,\n       Though mine are dim with tears.\n\n       *       *       *       *       *\n\n    In every change of man's estate\n      Are lights and guides allowed;\n    The fiery pillar will not wait,\n      But, parting, sends the cloud.\n\n    Nor mourn I the less manly part\n      Of life to leave behind;\n    My loss is but the lighter heart,\n      My gain the graver mind.\"\n\nPoetry is no longer the most popular form of literature amongst us, and\nthe drama is understood to be the least popular form of poetry. If this\nbe the case, Mr Taylor has the additional merit of having won his way\nto celebrity under singular disadvantages. But, in truth, such poetry\nas Mr Taylor's could never appeal to the multitude. Literature of any\nkind which requires of the reader himself _to think in order to enjoy_,\ncan never be popular. It is impossible to deny that the dramas we have\nbeen reviewing demand an effort, in the first instance, on the part of\nthe reader: he must sit down to them with something of the spirit of\nthe student. But, having done this, he will find himself amply repaid.\nAs he advances in the work, he will read with increased pleasure; he\nwill read it the second time with greater delight than the first; and\nif he were to live twenty years, and were to read such a drama as\n_Philip Van Artevelde_ every year of his life, he would find in it some\nfresh source of interest to the last.\n\nAs we have not contented ourselves with selecting beautiful passages of\nwriting from Mr Taylor's dramas, but have attempted such an analysis\nof the three principal characters they portray as may send the reader\nto their reperusal with additional zest, so neither have we paused to\ndispute the propriety of particular parts, or to notice blemishes and\ndefects. We would not have it understood that we admire all that Mr\nTaylor has written. Of whom could we say this? We think, for instance,\nthat, throughout his dramas, from the first to the last, he treats the\nmonks too coarsely. His portraiture borders upon farce. His Father John\nshows that he can do justice to the character of the intelligent and\npious monk. Admitting that this character is rare, we believe that the\nextremely gross portraiture which we have elsewhere is almost equally\nrare. This last, however, is so frequently introduced, that it will\npass with the reader as Mr Taylor's type of the monkish order. The\nmonks could never have been more ignorant than the surrounding laity,\nand they were always something better in morals and in true piety. We\nare quite at a loss, too, to understand Mr Taylor's fondness for the\nintroduction into his dramas of certain songs or ballads, which are not\neven intended to be poetical. To have made them so, he would probably\ncontend, would have been a dramatic impropriety. Very well; but let\nus have as few of such things as may be, and as short as possible. In\n_Edwin the Fair_ they are very numerous; and those which are introduced\nin _Philip Van Artevelde_ we could gladly dispense with. We could also\nvery willingly have dispensed with the conversation of those burgesses\nof Bruges who entertained the Earl of Flanders with some of these\nballads. We agree with the Earl, that their hospitalities are a sore\naffliction. Tediousness may be very dramatic, but it is tediousness\nstill--a truth which our writer, intent on the delineations of his\ncharacter, sometimes forgets. But defects like these it is sufficient\nmerely to have hinted at. That criticism must be very long and ample\nindeed, of the dramas of Mr Taylor, in which they ought to occupy any\nconsiderable space.\n\n\n\n\nA LEGEND OF GIBRALTAR.\n\n\nCHAPTER I.\n\nThe Governor's residence at Gibraltar was, in days of Spanish\ndomination, a religious house, and still retains the name of the\nConvent. Two sides of a long quadrangular gallery, traversing the\ninterior of the building, are hung with portraits of officers present\nat the great siege in 1779-83, executed in a style which proves that\nPre-Raphaelite painters existed in those days. One of these portraits\nrepresents my grandfather. To judge from a painting of him by Sir\nJoshua, and a small miniature likeness, both still in possession of\nthe family, he must have been rather a good-looking old gentleman,\nwith an affable, soldierlike air, and very respectable features. The\nportrait at the Convent is doubtless a strong likeness, but by no means\nso flattering; it represents him much as he might have appeared in\nlife, if looked at through a cheap opera-glass. A full inch has been\nabstracted from his forehead, and added to his chin; the bold nose\nhas become a great promontory in the midst of the level countenance;\nthe eyes have gained in ferocity what they have lost in speculation,\nand would, indeed, go far to convey a disagreeable impression of my\nancestor's character, but for the inflexible smile of the mouth.\nAltogether, the grimness of the air, the buckram rigidity of figure,\nand the uncompromising hardness of his shirt-frill and the curls of\nhis wig, are such as are to be met with in few works of art, besides\nthe figure-heads of vessels, the signboards of country inns, and the\nhappiest efforts of Messrs Millais and Hunt.\n\nHowever, my grandfather is no worse off than his compeers. Not far\nfrom this one is another larger painting, representing a council of\nofficers held during the siege, where, notwithstanding the gravity of\nthe occasion and the imminence of the danger, not a single face in the\nintrepid assembly wears the slightest expression of anxiety or fear,\nor, indeed, of anything else; and though my progenitor, in addition\nto the graces of the other portrait, is here depicted with a squint,\nyet he is by no means the most ill-looking individual present. But the\nillustrious governor, Eliott, has suffered more than anybody at the\nhands of the artist. Besides figuring in the production aforesaid,\na statue of him stands in the Alameda, carved in some sort of wood,\nunluckily for him, of a durable nature. The features are of a very\nelevated cast, especially the nose; the little legs seem by no means\nequal to the task of sustaining the enormous cocked-hat; and the\nbearing is so excessively military, that it has been found necessary\nto prop the great commander from behind to prevent him from falling\nbackwards.\n\nMy grandfather, John Flinders, joined the garrison of Gibraltar\nas a major of infantry a few years before the siege. He was then\nforty-seven years of age, and up to that time had remained one of the\nmost determined old bachelors that ever existed. Not that he ever\ndeclaimed against matrimony in the style of some of our young moderns,\nwho fancy themselves too strong-minded to marry; the truth being that\nthey remain single either because they have not been gifted by nature\nwith tastes sufficiently strong to like one woman better than another,\nor else, because no woman ever took the trouble to lay siege to them.\nMy grandfather had never married, simply, I believe, because matrimony\nhad never entered his head. He seldom ventured, of his own choice,\ninto ladies' society, but, when he did, no man was more emphatically\ngallant to the sex. One after one, he saw his old friends abandoning\nthe irresponsible ease of bachelorhood for the cares of wedded life;\nbut while he duly congratulated them on their felicity, and officiated\nas godfather to their progeny, he never seemed to anticipate a similar\ndestiny for himself. All his habits showed that he had been too long\naccustomed to single harness to go cleverly as one of a pair. He\nhad particular hours of rising, and going to bed; of riding out and\nreturning; of settling himself down for the evening to a book and pipe,\nwhich the presence of a helpmate would have materially deranged. And\ntherefore, without holding any Malthusian tenets, without pitying his\nBenedick acquaintances, or entertaining a thought of the sex which\nwould have been in the least decree derogatory to the character of a De\nCoverley, his castles in the air were never tenanted by any of his own\nposterity.\n\nIt was fortunate for my grandfather that in his time people did not\nsuffer so much as now from that chronic inflammation of the conscience,\nwhich renders them perfectly miserable unless they are engaged in some\ntangible pursuit--\"improving their minds,\" or \"adding to the general\nstock of information.\" A more useless, contented person never existed.\nHe never made even a show of employing himself profitably, and never\ncomplained of weariness in maintaining the monotonous jog-trot of his\nsimple daily life. He read a good deal, certainly, but it was not\nto improve his mind, only to amuse himself. Strong-minded books, to\nstimulate his thinking faculties, would have had no charms for him; he\nwould as soon have thought of getting galvanised for the pleasure of\nlooking at his muscles. And I don't know whether it was not just as\nwell. In systematically cultivating his mind, he would merely have been\nlaying a top-dressing on a thin soil--manuring where there would never\nhave been a crop--and some pleasant old weeds would have been pulled\nup in the process. A green thistly common, even though a goose could\nhardly find sustenance there, is nature still, which can hardly be said\nof a patch of earth covered with guano.\n\nSo my grandfather went on enjoying himself without remorse after his\nown fashion, and never troubled himself to think--an operation that\nwould have been inconvenient to himself, and productive of no great\nresults to the world. He transplanted his English habits to Gibraltar;\nand, after being two years there, knew nothing more of Spain or\nSpaniards than the view of the Andalucian hills from the Rock, and a\nshort constitutional daily ride along the beach beyond the Spanish\nlines, to promote appetite and digestion, afforded him. And so he might\nhave continued to vegetate during the remainder of his service there,\nbut for a new acquaintance that he made about this time.\n\nFrank Owen, commonly called Garry Owen by his familiars, was one of\nthose joyous spirits whose pleasant faces and engaging manners serve\nas a perpetual act of indemnity for all breaches of decorum, and\ntrespasses over social and conventional fences, committed by them\nin the gaiety of their hearts. In reproving his many derelictions\nof military duty, the grim colonel of the regiment would insensibly\nexchange his habitual tone of severe displeasure for one of mild\nremonstrance--influenced, probably, quite as much, in secret, by the\npopularity of the unrepentant offender, as by any personal regard\nfor him. Captain Hedgehog, who had shot a man through the heart for\ncorking his face one night when he was drunk, and all contact with\nwhose detonating points of honour was as carefully avoided by his\nacquaintance as if they had been the wires of a spring-gun, sustained\nGarry's reckless personalities with a sort of warning growl utterly\nthrown away upon the imperturbable wag, who would still persist, in\nthe innocence of his heart, in playing round the den of this military\ncockatrice. And three months after his arrival in Gibraltar, being\none day detected by a fierce old Spanish lady in the very act of\nkissing her daughter behind the little senorita's great painted fan,\nhis good-humoured impudence converted the impending storm into a mild\ndrizzle of reproof, ending in his complete restoration to favour.\n\nThis youth had brought with him from England a letter from his mother,\na widow lady, an old friend of my grandfather, who had some thirty\nyears before held with her a juvenile flirtation. It recommended to\nhis protection her son Frank, about to join the regiment as an ensign,\npathetically enlarging on the various excellencies, domestic and\nreligious, which shone forth conspicuously in the youth's character,\nand of the comfort of contemplating and superintending which she was\nabout to be deprived. In fact, it had led my grandfather to expect a\nyouth of extreme docility and modesty, requiring a protector rather\nto embolden than to restrain him. After in vain attempting to espy in\nhis young acquaintance any of the characteristics ascribed to him in\nhis mother's letter, the Major, naturally good-natured and accessible\nto his youthful comrades, very soon suffered himself to be influenced\nby the good-humour, vigorous vitality, and careless cleverness of the\nEnsign, to an extent that caused him sometimes to wonder secretly\nat his own transformation. His retired habits were broken in upon,\none after the other, till he had scarcely a secluded hour in his\nsixteen waking ones to enjoy alone his book and his pipe. His peaceful\nquarters, silent, in general, as a monk's cell, would now be invaded\nat all sorts of hours by the jovial Garry, followed by the admiring\nsatellites who usually revolved around him; and the Major, with a sound\nbetween a groan and a chuckle, would close his well-beloved volume to\nlisten to the facetious details of, and sometimes to participate in,\nthe uncongenial freaks of the humorous subaltern. Once he had actually\nconsented, at about the hour he usually went to bed, to accompany the\nyouth to a Carnival ball--one of a series of entertainments at which\nthe Catholic youth of the city are wont to idemnify themselves for the\nmortifications of Lent, and where masks, dominoes, and fancy dresses\nlend their aid to defeat the vigilance of the lynx-eyed duennas and\nmammas who look anxiously on, perfectly aware, in general, that their\nown watchfulness is more to be relied on for nipping in the bud an\nindiscreet amour, than any innate iciness of temperament or austere\npropriety in the objects of their care. Not only did the Major mingle\nin the scene, but he actually, about midnight, found himself figuring\nin a cotillon with a well-developed senorita of thirteen years, whose\nglances and deportment showed a precocious proficiency in the arts of\nflirtation. At this ball Garry had become enamoured beyond all former\npassions (and they were numerous and inconstant, in general, as if he\nhad been a Grand Turk) of one of his partners, a young Spanish lady.\nHer graceful figure and motions in the dance had at first captivated\nhim--and when, after dancing with her himself, his eloquent entreaties,\ndelivered in indifferent Spanish, had prevailed on her to lift her\nmask for one coy moment, the vision of eyes and eyebrows, the common\nbeauties of a Spanish countenance, and the clear rosy complexion, a\nmuch more rare perfection, then revealed, had accomplished the utter\nsubjugation of his errant fancy. She had vanished from the ball\nsilently and irremediably, as a houri of Paradise from the awakening\neyes of an opium-eating Pacha; and all his attempts to trace her,\ncontinued unceasingly for a couple of months afterwards, had proved in\nvain.\n\nOne morning my grandfather was seated at breakfast in the verandah of\nhis quarters, situated high up the rock above the town. Below him lay\nthe roofs, terraced and balconied, and populous with cats, for whose\nconvenience the little flat stone squares at the top of most of the\nhouses appeared to have been devised. Tall towers called mirandas shot\nup at intervals, from whose summits the half-baked inhabitants, pent\nwithin close walls and streets, might catch refreshing glimpses of the\nblue sea and the hills of Spain--conveniences destined soon afterwards\nto be ruined by the enemy's fire, or pulled down to avoid attracting\nit, and never rebuilt. Beyond the white sunny ridge of the line wall\ncame the sharp edge of the bay, rising in high perspective to the\npurple coast of Spain opposite, which was sprinkled with buildings\nwhite as the sails that dotted the water. My grandfather was in a\nstate of great sensual enjoyment, sniffing up the odour of the large\ngeranium bushes that grew in clumps in the little garden in front, and\nthe roses that twined thickly round the trellis of the vine-roofed\nverandah; sipping thick creamy Spanish chocolate between the mouthfuls\nof red mullet, broiled in white paper, the flavour of which he was\ndiligently comparing with that of some specimens of the same fish\nwhich he remembered to have eaten in his youth in Devonshire; and\nglancing sideways over the cup at an open volume of Shakspeare,\nleaned slopingly on the edge of a plate of black figs bursting with\nripeness, like trunk hose slashed with crimson. The Major was none\nof your skimming readers, who glance through a work of art as if it\nwere a newspaper--measure, weigh it, and deliver a critical opinion on\nit, before the more reverential student has extricated himself from\nthe toils of the first act or opening chapter: not he; he read every\nword, and affixed a meaning, right or wrong, to all the hard, obsolete\nones. The dramatic fitness of the characters was not to be questioned\nby him, any more than that of the authentic personages of history.\nHe would reason on their acts and proceedings as on those of his own\nintimate acquaintances. He never could account for Hamlet's madness\notherwise than by supposing the Prince must have, some time or other,\ngot an ugly rap on the head--let fall, perhaps, when a baby, by a\ngin-drinking nurse--producing, as in some persons he had himself from\ntime to time been acquainted with, a temporary aberration of the wits;\na piece of original criticism that has not occurred to any of the other\ncommentators on this much-discussed point. Of Iago he has recorded\nan opinion in an old note-book still extant, where his observations\nappear in indifferent orthography, and ink yellow with age, that he\nwas a cursed scoundrel--an opinion delivered with all the emphasis of\nan original detector of crime, anxious that full though tardy justice\nshould be done to the delinquent's memory. But his great favourite was\nFalstaff: \"A wonderful clever fellow, sir,\" he would say, \"and no more\na coward than you or I, sir.\"\n\nMy grandfather proceeded slowly with his meal, holding the cup to his\nlips with one hand and turning a leaf with the other--an operation\nwhich he was delaying till a great mosquito-hawk, (a beautiful brown\nmoth mottled like a pheasant,) that had settled on the page, should\nthink proper to take flight. He had lately come from a parade, as was\nevidenced by his regimental leather breeches and laced red waistcoat;\nbut a chintz dressing-gown and a pair of yellow Moorish slippers\nsoftened down the warlike tone of these garments to one more congenial\nwith his peaceable and festive pursuits. Presently the garden door\nopened, and a well-known step ascended to the verandah. Frank Owen,\ndressed in a cool Spanish costume, advanced, and stopping three paces\nfrom the Major, took off his tufted sombrero and made a low bow.\n\n\"You are the picture, my dear sir,\" he said, \"of serene enjoyment\nslightly tinged with sensuality. But how long, may I ask, have you\ntaken to breakfasting on spiders?\"--pointing, as he took a chair\nopposite the Major, at an immense red-spotted one that had dropt from\nthe ceiling on the morsel my grandfather was in the act of conveying to\nhis mouth.\n\nThe Major tenderly removed the insect by a leg.\n\n\"'Tis the worst of these al-fresco meals, Frank,\" said he. \"Yesterday I\ncut a green lizard in two that had got on my plate, mistaking him for a\nbit of salad--being, as usual, more intent on my book than my food--and\nhad very near swallowed the tail-half of the unfortunate animal.\"\n\n\"There are worse things than lizards in the world,\" quoth Garry. \"Ants,\nI should say, were certainly less wholesome\"--and he directed the\nMajor's attention to a long black line of those interesting creatures\nissuing from a hole in the pavement, passing in an unbroken series up\nmy ancestor's left leg, the foot of which rested on the ground, then\ntraversing the cloth, and terminating at the loaf, the object of their\nexpedition.\n\n\"Bless me,\" said the Major, as he rose and shook his breeches gently\nfree from the marauders, \"I must be more careful, or I shall chance to\ndo myself a mischief. But they're worst at night. I've been obliged to\nleave off reading here in the evenings, for it went to my heart to see\nthe moths scorching their pretty gauzy wings in the candle till the\nwicks were half-choked with them.\"\n\n\"Do you know, Major,\" said Owen, gravely, \"that either this insect\ndiet, or the sedentary life you lead, is making you quite fat, and\nutterly destroying the symmetry of your figure? In another week there\nwill be one unbroken line of rotundity from your chin to your knees.\"\n\nMy grandfather glanced downward at his waistcoat. \"No, my boy, no,\"\nsaid he; \"if there had been any difference, I should have known it by\nmy clothes. I don't think I've gained a pound this twelvemonth.\"\n\n\"More than a stone,\" quoth Garry. \"We all remarked it on parade\nto-day--and remarked it with sorrow. Now, look you, a sea voyage is\nthe very thing to restore your true proportions, and I propose that we\nshall take a short one together.\"\n\n\"A sea voyage!\" quoth my grandfather; \"the boy is mad! Not if all\nthe wonders seen by Sinbad the Sailor lay within a day's sail. Did I\nnot suffer enough coming here from England? I don't think,\" said my\ngrandfather with considerable pathos, \"that my digestion has ever been\nquite right to this day.\"\n\n\"'Sick of a calm,' eh?--Like your friend Mistress Tearsheet,\" said the\nyoungster. \"But I've settled it all, and count on you. Look here,\" he\ncontinued, drawing from his pocket a large printed bill, and unfolding\nit before my ancestor. At the top appeared in large capitals the words,\n\"Plaza de Toros;\" and underneath was a woodcut representing a bull,\nof whose sex there could be no doubt, gazing, with his tail in the\nair, and an approving smile on his countenance, on the matadore about\nto transfix him. Then followed a glowing account in Spanish of the\ndelights of a great bull-fight shortly to take place at Cadiz, setting\nforth the ferocity of the bulls, the number of horses that might be\nexpected to die in the arena, and the fame of the picadores and espadas\nwho were then and there to exhibit.\n\nThe Major shook his head--the captivating prospectus had no charms\nfor him: he had not, as I have before said, an inquiring mind, and\nhabit was so strong in him that a change was like the dislocation of a\njoint. The Ensign proceeded to paint the delights of the excursion in\nthe brightest colours he could command. They were to go to Cadiz in a\nboat which he had lately bought--she was a capital sailer--there was\na half-deck forward, under which the Major might sleep as comfortably\nas in his own bed--a cooking apparatus, (and here, as he expatiated on\nthe grills and stews and devils that were to be cooked and eaten, with\nthe additional stimulus to appetite afforded by sea air, there was a\nspark of relenting in my grandfather's eye.) \"You shall return,\" said\nthe tempter, \"with a digestion so completely renovated, that my name\nshall rise to your tongue at each meal as a grace before meat, and a\nthanksgiving after it; and as to sea-sickness, why, this Levanter will\ntake us there in twelve hours, so smoothly that you may balance a straw\nupon your nose the whole way.\" Finally, the cunning Ensign laid before\nhim an application for leave already made out, and only awaiting his\nsignature.\n\nMy grandfather made some feeble, objections, which Owen pooh-poohed\nin his usual off-hand fashion. There was no standing against the\nyoungster's strong will, that, like Aaron's rod, swallowed up all\nopposition, and at five o'clock that same evening the Major found\nhimself proceeding through the town towards the Waterport for\nembarkation, by no means fully reconciled to the abandonment of his\nbeloved Lares. My luckless grandfather! did no presentiment warn you of\na consequence then hanging in the clouds, that was to change utterly\nfor you the untroubled aspect of those household gods?\n\nOwen had attired himself for the trip in a half-nautical costume--a\nshirt of light-blue flannel, fastened at the collar with a smart\nbandana, a bluejacket, loose duck trousers, and a montero cap, which\ncostume became the puppy well enough. He seemed of this opinion\nhimself, as he walked gaily along beside the Major: so did the\nblack-eyed occupants of many houses on each side, who peeped forth\nsmilingly from behind their green lattices, sometimes nodding and\nkissing their hands--for the Ensign had an incredible acquaintance with\nthe budding and full-blown portion of the population of Gibraltar. The\nMajor had stuck to his buckskins, (which stuck to him in return,) over\nwhich he had drawn a pair of jack-boots, and wore his red-laced coat\nand regimental hat--for in those days that passion for mufti, now so\nprevalent in the army, did not exist. Whenever he caught sight of any\nof the greetings bestowed from the windows, he would take off his laced\nhat, and, fixing his eyes on the tittering senorita, who generally let\nfall the lattice with a slam, would make her a low bow--and, after each\nof these acts of courtesy, my grandfather walked on more elated than\nbefore.\n\nThey passed the drawbridge at Waterport, and, struggling through the\ncrowd of Turks, Jews, infidels, and heretics, who usually throng the\nquay, entered a shore-boat that was to row them out to where Owen's\nvessel--the _Fair Unknown_, as he had christened her, in memory of\nhis unforgotten partner at the Carnival ball--lay moored. In her they\nfound a sailor who was to accompany them on their voyage--a noted\ncontrabandista, called Francisco, whose friendship Owen had lately\nacquired, and who acted as his lieutenant on his marine excursions. The\nboat was a neat affair--a small cutter, smartly painted, well found,\nand capable of holding several persons comfortably; and Francisco was\na ruddy, portly, dark-skinned, large-whiskered son of the sea, the\npicture of good-humour. My grandfather stept in, in his jack-boots.\nThere was much settling of carpet-bags and stowing of provisions in the\nlockers, and then they hoisted sail, and glided smoothly out from among\nthe shipping into the bay.\n\nThe breeze was light and fair, and they went on, as Frank had promised,\npleasantly enough. My grandfather for the first time surveyed the scene\nof his two years' residence from the sea. The grey old rock looked\nmellow in the evening light, as an elderly gentleman over his wine--the\nwindow-panes glanced ruddily, the walls gleamed whitely, and the\ntrees were tinted with a yellower green; behind, in the eastern sky,\nfloated one single purple cloud. As the objects became confused in the\ndistance, the sharp rugged outline of the rock assumed the appearance\nthat has caused the Spaniards to call it El Cuerpo--the appearance of a\nvast human body laid out on its back, and covered with a winding-sheet,\nlike a dead Titan on his funeral pile--the head towards Spain, the\nchest arched at Middle Hill, the legs rising gently upward to the\nknees at O'Hara's Tower, and then sloping down till the feet rest\non Europa. The sun went down as they rounded Cabrita Point, and the\nbreeze, freshening, took them swiftly along under the huge hills that\nrise abruptly upward from the Spanish coast. Then Francisco, lighting a\ncharcoal fire, placed thereon, in a frying-pan, tender steaks thickly\nstrewn with sliced tomatas and onions, from whence arose a steam that\nbrought tears of gratitude and delight into my grandfather's eyes.\nHe anxiously watched the cooking--even threw out slight suggestions,\nsuch as another pinch of pepper, an additional onion, a slight dash of\ncayenne, and the like; and then, settling a plate firmly on the knees\nof his jack-boots, with a piece of bread and a cup by his side, and a\nknife and fork pointing upwards in his hands like lightning conductors,\ngazed cheerfully around him. And when Francisco, rising from his knees,\nwhere he had been blowing the charcoal fire, removed the hissing pan\ntowards my grandfather's plate, transferring to it a liberal portion\nof the contents, the good man, gazing on the white and red streaks of\nvegetable relieved by the brown background of steak, and the whole\npicture swimming in a juicy atmosphere of gravy, felt sentiments of\npositive friendship towards that lawless individual, and, filling a\nbumper of Xerez, drank success to the voyage.\n\nThree times was my grandfather's plate replenished from the\nthrice-filled pan. Afterwards he dallied a little with a cold pie,\nfollowed by a bit of cheese for digestion. Then, folding his hands\nacross his stomach, he expressed his sincere opinion, that he had never\ntasted anything so good as that steak; and when Owen placed in his hand\na smoking can of grog, he looked on the young man with a truly paternal\neye. He talked complacently and benevolently, as men do who have dined\nwell--praised the weather, the boat, the scene--and wondered where a\nman was going who rode slowly along a mountain-path above them, within\nhail, following him, in imagination, to his home, in a sort of dreamy\ncontentment. After a second can he began to grow drowsy, and, just\naware that Owen said the breeze was still freshening, retired to the\nsoft mattress spread for him under the half-deck, and replacing his\ncocked hat by a red nightcap, slept till morning.\n\nIt was broad daylight when he woke, conscious that for an hour or two\npast he had been sleeping most uneasily. There was a violent swinging\nmotion, a rushing of wind and of water, that confused him extremely;\nand, forgetting where he was, he nearly fractured his skull by rising\nsuddenly into a sitting posture. Steadying himself on his hands, in the\nposture of the Dying Gladiator, he slewed himself round on the pivot of\nhis stern, and protruded his powdered head, like an old beaver, out of\nhis hole. Owen and Francisco were sitting in a pool of water, trying to\nshelter themselves under the weather-side of the boat--dripping wet,\nand breakfasting on cold potatoes and fragments of meat left from last\nnight's meal. My grandfather did not like the appearance of things at\nall. Rent in twain by horrible qualms, he inquired feebly of Owen if\nthey were near Cadiz? Frank, in reply, shook his head, and said they\nwere at anchor. Then my grandfather, making a vigorous effort, emerged\ncompletely from his place of repose, and, rising to his feet, looked\nover the gunwale. The scene he beheld was in dreary contrast to that\nof the evening before. Ridges of white foam were all around--ahead was\na long low line of sandy coast, terminating in a point of rock whereon\nstood a lighthouse; and to leeward the bay was enclosed by steep hills.\nOver the low coast-line the wind blew with steady violence. A bright\nsun rather increased the dreariness of the prospect, which was suddenly\nclosed to my grandfather by a shower of spray, that blinded him, and\ndrenched him to the skin, converting his jack-boots into buckets. The\nwind had increased to a gale during the night, and they had been forced\nto take precarious shelter in the harbour of Tarifa. The Major did not\nventure on a second peep, but sat, dismally wet and sea-sick, the whole\nmorning, trying to shelter himself as he best could. Once, a man came\ndown to the beach, and gesticulated like a scaramouch, screaming also\nat the same time; but what his gestures and screams signified nobody on\nboard could tell. At length, as the gale did not moderate, while their\nposition increased in discomfort, and was also becoming precarious,\n(for one of their anchors was gone, and great fears were entertained\nfor the other,) Owen and Francisco decided to weigh, and stand in for\nthe shore, trusting to the smuggler's seamanship for a safe run. The\nMajor, in spite of his sickness, stood up and pulled gallantly at the\ncable, the wind blowing his pigtail and skirts perpendicularly out from\nhis person. At last, after tremendous tugging, the anchor came up. The\njib was hoisted with a reef in it, Owen holding the sheet, while the\nsmuggler ran aft and took the helm. They bent over to the gale, till\nthe Major stood almost perpendicularly on the lee gunwale, with his\nback against the weather-side, and ran in till he thought they were\ngoing to bump ashore; then tacking, they stood up along the coast,\nclose to the wind, till Francisco gave the word. Owen let go the sheet,\nand the jib fluttered loosely out as they ran through a narrow passage\ninto smooth water behind the sea-wall, and made fast to a flight of\nsteps.\n\nPresently some functionary appertaining to the harbour appeared, and\nwith him an emissary from the Governor of the place, who, aware of\ntheir plight, had civilly sent to offer assistance. The messenger\nwas the same man who had made signals to them from the beach in the\nmorning; and he seemed to think it advisable that they should wait\non the Governor in person, saying that he was always disposed to be\ncivil to British officers. This advice they resolved to act upon at\nonce, before it should grow dark, foreseeing that, in case of their\ndetention from bad weather in Tarifa, the Governor might prove a potent\nauxiliary. The Major would have wished to make some little alterations\nin his toilette, after his late disasters; but, after trying in vain\nto pull off his jack-boots, which clung to him like his skin, he was\nobliged to abandon the idea, and contented himself with standing on\nhis head to let the water run out of them. As they advanced along the\ncauseway leading to the town, (the point where they landed is connected\nwith the town by a long narrow sandy isthmus,) the gale swept over them\nvolumes of sand, which, sticking to my grandfather's wet uniform, gave\nhim somewhat the appearance of a brick-wall partially roughcast. His\nbeard was of two days' growth--his hair-powder was converted into green\npaste by the sea-water--and his whole appearance travel-stained and\ndeplorable. Nevertheless his dignity by no means forsook him, as they\ntraversed the narrow alleys of the ancient town of Tarifa, on their way\nto the approaching interview.\n\nHis excellency Don Pablo Dotto, a wonderfully fat little man, received\nthem very courteously. He was a Spaniard of the old school, and\nreturned the stately greeting of my grandfather, and the easy one of\nthe Ensign, with such a profusion of bows, that for the space of a\nminute they saw little more of his person than the shining baldness\non the top of his head. Then they were presented to his wife, a\ngood-natured, motherly sort of old lady, who seemed to compassionate\nthem much. But, while Owen was explaining to her the object of\ntheir trip, and its disastrous interruption, he suddenly stopped,\nopen-mouthed, and blushing violently, with his gaze directed towards\nthe open door of a neighbouring apartment. There he beheld, advancing\ntowards him, the Beauty of the Carnival ball.\n\nThe Governor's lady named her as \"her daughter, the Senorita Juana.\"\nSpite of the different dress and circumstances, she, too, recognised\nFrank, and  slightly as she came forward to receive his\ngreeting. The Ensign, an impudent scamp enough in general, was,\nhowever, the more confused of the two; and his embarrassed salutation\nwas entirely thrown into the shade by the magnificence of my\ngrandfather's bow. However, he presently recovered his assurance,\nand explained to the elder lady how he had previously enjoyed the\npleasure (with a great stress upon the word) of making her daughter's\nacquaintance. Then he recounted to Juana the manner in which they had\nbeen driven in here, when on their way to Cadiz to see the bull-fight.\n\n\"We also are going to ride thither to-morrow,\" said the Senorita,\nsoftly.\n\n\"Ah, then, we shall meet there,\" said Frank, who presently after was\nseized with a fit of absence, and made incoherent replies. He was\nconsidering how they might travel together, and had almost resolved to\noffer to take the whole family to Cadiz in his boat--a proposal that\nwould probably have somewhat astonished the little Governor, especially\nif he had seen the dimensions of the craft thus destined to accommodate\nhimself and retinue. But Garry was an adept in manoeuvring, and\nmarched skilfully upon the point he had in view. He drew such a\npathetic picture of the hardships they had endured on the voyage--their\nprobable detention here for most of their short leave--their friendless\ncondition, and their desire to see something of the country--that the\nlittle Governor was in a manner impelled (fancying all the time that\nthe impulse sprung altogether from his own native benevolence,) to\ndesire that the two forlorn Englishmen would travel to Cadiz under his\nescort. So it being settled entirely to Garry's satisfaction that they\nwere to start next morning at break of day on horseback--an arrangement\nwhich my grandfather's total ignorance of Spanish prevented him from\nknowing anything about--they retired to the principal fonda, where the\nMajor speedily forgot, over a tolerable dinner, the toils and perils of\nthe voyage.\n\n\nCHAPTER II.\n\nDaybreak the next morning found them issuing forth from the ancient\ncity of Tarifa on a couple of respectable-looking hacks, hired from\nthe innkeeper. Frank had, with his accustomed generalship, managed\nto secure a position at the off-rein of the Senorita Juana, who was\nmounted on a beautiful little white barb. Under her side-saddle, of\ngreen velvet studded with gilt nails, was a Moorish saddle-cloth,\nstriped with vivid red and white, and fringed deeply. From the\nthroat-lash of the bridle hung a long tassel, as an artificial\nauxiliary to the barb's tail in the task of keeping the flies off,\nfurther assisted by a tuft of white horse-hair attached to the butt of\nher whip. She wore a looped hat and white plume, a riding-skirt, and an\nembroidered jacket of blue cloth, fastened, as was the wrought bosom of\nher chemise, with small gold buttons. Frank could not keep his eyes off\nher, now riding off to the further side of the road to take in at once\nthe whole of the beauteous vision, now coming close up to study it in\nits delightful details.\n\nIn front of the pair rode the little Governor, side by side with a\nSpaniard of about thirty, the long-betrothed lover of Juana--so long,\nin fact, that he did not trouble himself to secure his authority in\na territory so undeniably his own, but smoked his cigar as coolly as\nif there were no good-looking Englishman within fifty miles of his\nmistress. He wore garments of the Spanish cut, made of nankeen--the\njacket frogged with silver cords, tagged with little silver\nfishes--the latter appended, perhaps, as suitable companions to the\nfrogs. A hundred yards ahead was an escort of four horse-soldiers\nwith carbines on their thighs, their steel accoutrements flashing\nruddily in the level sunlight. Behind Frank came Major Flinders,\nclean shaved, and with jack-boots and regimental coat restored to\nsomething like their pristine splendour; by his side rode another\nlady, the Senorita Carlota, Juana's aunt, somewhere about thirty years\nold, plump and merry, her upper lip fringed at the corners with a\nline of dark down, quite decided enough for a cornet of eighteen to\nbe proud of--a feminine embellishment too common for remark in these\nsouthern regions, and, in the opinion of some connoisseurs, rather\nenhancing the beauty of the fair wearers. She talked incessantly, at\nfirst, to my grandfather, who did not understand a word she said,\nbut whose native politeness prompted him to say, \"Si, Senorita,\" to\neverything--sometimes laying at the same moment his hand on his heart,\nand bowing with considerable grace. Behind this pair came another\ninteresting couple--viz., two servants on mules, with great saddle-bags\nstuffed to extreme corpulence with provisions.\n\nIt was a glorious morning--a gentle breeze sweeping on their faces as\nthey mounted the hills, but dying into silence in the deep valleys,\nfresh, and glistening with dew. Sometimes they rode along a rocky\ncommon, yellowed with a flowering shrub like furze--sometimes through\nunfenced fields--sometimes along broad plains, where patches of\nblossoming beans made the air rich with scent, and along which they\ngalloped full speed, the Governor standing high in the stirrups of\nhis demi-pique, the Senorita's white barb arching his neck till his\nmuzzle touched his chest under the pressure of the long bit, and my\ngrandfather prancing somewhat uneasily on his hard-mouthed Spanish\nentero, whose nose was, for the most part, projected horizontally in\nthe air. The Major was not a first-rate seat--he rode with a long\nstirrup, his heel well down, his leg straight, and slanting a little\nforward, body upright, and elbows back, as may be seen in the plates to\nancient works on equitation--a posture imposing enough, but not safe\nacross country: galloping deranged it materially, for the steed was\nhard-mouthed, and required a long, strong pull, with the body back,\nand a good purchase on the stirrups. The animal had a most voracious\nappetite, quite overcoming his sense of what was due to his rider;\nand, on seeing a tuft of juicy grass, down went his nose, drawing my\ngrandfather, by means of the tight reins, well over the pummel. On\nthese occasions, the Major, feeling resistance to be in vain, would\nsit looking easily about him, feigning to be absorbed in admiration of\nthe prospect--which was all very well, where there was a prospect to\nlook at, but wore a less plausible appearance when the animal paused\nin a hollow between two hedges, or ran his nose into a barn-door. But\nwhenever this happened, Carlota, instead of half-smothering a laugh,\nas a mischievous English girl would, ten to one, have done, sat most\npatiently till the Major and his steed came to an understanding, and\nwould greet him, as they moved on again, with a good-natured smile,\nthat won her, each time, a higher place in his estimation.\n\nThus they proceeded till the sun rose high in the heavens, when, on\nreaching a grove on the edge of one of the plains, they halted under a\nhuge cork-tree, near which ran a rivulet. The cavalcade dismounted--the\nhorses were tethered, the mules disburthened of the saddle-bags, and\nthe contents displayed under the tree; horse-cloths and cloaks were\nspread around on the ground and a fire of dry sticks was lit on the\nedge of the stream with such marvellous celerity that, before my\ngrandfather had time to take more than a hasty survey of the eatables,\nafter seating himself on the root of a tree, a cup of steaming\nchocolate was placed in his hand.\n\n\"Confess, Major,\" said Garry, speaking with his mouth full of sausage,\n\"that a man may lose some of the pleasures of existence by leading the\nlife of a hermit. Don't you feel grateful to me for dragging you out of\nyour cobweb to such a pleasant place as this?\"\n\n\"'Tis an excellent breakfast,\" said my grandfather, who had just\nassisted the Senorita Carlota to a slice of turkey's breast, and\nhimself to an entire leg and thigh--dividing with her, at the\nsame time, a crisp white loaf, having a handle like a teapot or\nsmoothing-iron--\"and my appetite is really very good. I should be\nperfectly easy if I could only understand the remarks of this very\nagreeable lady, and make suitable replies.\"\n\n\"Let me interpret your sentiments,\" said Garry; \"and though I may not\nsucceed in conveying them in their original force and poetry, yet they\nshall lose as little as possible in transmission. Just try me--what\nwould you wish to say?\"\n\n\"Why, really,\" said my grandfather, pondering, \"I had a great many\nthings to say as we came along, but they've gone out of my head. Do you\nthink she ever read Shakspeare?\"\n\n\"Not a chance of it,\" said Owen.\n\nHere the Senorita laughingly appealed to Frank to know what my\ngrandfather was saying about her.\n\n\"Ah,\" quoth my grandfather, quoting his friend Shakspeare--\n\n    \"'I understand thy _looks_--the pretty Spanish\n    Which thou pourest down from these swelling heavens\n    I am _not_ perfect in ----'\n\nShe's an extremely agreeable woman, Frank, I'll be sworn, if one only\nunderstood her,\" quoth my grandfather, casting on her a glance full of\ngallantry.\n\nThe Ensign was not so entirely occupied in prosecuting his own love\naffair as to be insensible to the facilities afforded him for amusing\nhimself at the Major's expense. Accordingly, he made a speech in\nSpanish to Carlota, purporting to be a faithful translation of my\ngrandfather's, but teeming, in fact, with the most romantic expressions\nof chivalrous admiration, as was apparent from the frequent recurrence\nof the words \"ojos,\" (eyes,) \"corazon,\" (heart,) and the like amatory\ncurrency.\n\n\"There, Major,\" said the interpreter, as he finished; \"I've told her\nwhat you said of her.\"\n\nThe Major endorsed the compliments by laying his hand upon his heart,\nand bowing with a tender air. Whereupon Carlota, laughing, and blushing\na deeper red, made her acknowledgments.\n\n\"She says,\" quoth Frank, \"that she knew the English before to be a\ngallant nation; but that if all the caballeros (that's gentlemen)\nof that favoured race are equal to the present specimen, her own\ncountrymen must be thrown entirely into the shade.\"\n\n\"Delightful!\" cried my grandfather; but it is doubtful whether this\nexpression of pleasure was called forth by the sentiments attributed\nto the Senorita, or by the crisp succulent tenderness of a mouthful of\nsucking-pig which was at that moment spreading itself over his palate.\n\nFollowing up his idea, the mischievous Ensign continued to diversify\nthe graver pursuit of prosecuting his own suit with Juana, by\nimpressing Carlota and the Major with the idea that each was favourably\nimpressed with the other. In this he was tolerably successful--the\nspeeches he made to Carlota, supposed to originate with my grandfather,\nhad a very genuine warmth about them, being, in fact, very often\nidentical with those he had just been making, under immediate\ninspiration, to his own divinity; while as for the Major, it would have\nbeen an insult to the simplicity of that worthy man's nature to exert\nany great ingenuity in deceiving him; it would have been like setting\na trap for a snail. So they journeyed on, highly pleased with each\nother, and occasionally, in the absence of their faithful interpreter,\nconversed by means of smiles and courteous gesticulations, till my\ngrandfather felt entirely at his ease, and was almost sorry when on the\nevening of the second day they got to Cadiz.\n\n\nCHAPTER III.\n\nA whole city full of people condensed into one broad amphitheatre,\nall bearing a national resemblance to each other in countenance and\ncostume, all apparently animated by the same spirit--for nothing could\nbe more unanimous than the applause which greeted a favourite smilingly\ncrossing the arena, the abuse which overwhelmed an object offensive\nto the eye of the many-headed, or the ridicule which descended in a\njoyous uproarious flood on the hapless individual in whose appearance,\ndress, or manner, anything was detected calculated to appeal to the\nhighly-sensitive risible faculty of a Spanish assembly;--a gay and\npicturesque mixture of colours, waving and tossing like a garden in\na breeze, as the masses of white mantillas, heads black as coal,\ndecorated with flowers and green leaves, red sashes, tufted sombreros,\nand yellow gaiters, with here and there a blue-and-white soldier\nstanding stiffly up, were agitated by each new emotion--such was the\nscene that met the eyes of our travellers on entering the bull-ring at\nCadiz before the sport commenced.\n\nMy grandfather had made his entry in spectacles--appendages highly\nprovocative of the public mirth--and had looked wonderingly for a\nminute or two through the obnoxious glasses on a sea of faces upturned,\nsideturned, and downturned, all looking at him, and all shouting some\nindistinguishable chorus; while the men beat time, each with the\nlong, forked, painted stick, without which no Spaniard possessing\nsentiments of propriety ever comes to a bull-fight, in a manner most\nembarrassing to a somewhat bashful stranger, till their attention was\nluckily diverted to an unhappy man in a white hat, in derision of whom\nthey immediately sang a song, the burden of which was \"El de sombrero\nblanco,\" (he of the white hat,) the multitude conducting itself\nthroughout like one man.\n\nMy grandfather and his friends occupied a distinguished position\nin a box high above the multitude, and near that of the alcalde.\nThe Senorita Juana looked more lovely than ever in a white dress,\nover which flowed a white gauzy mantilla, giving a kind of misty\nindistinctness to the wavy outlines of her figure, and the warm tint\nof her neck and arms. From her masses of black hair peeped one spot of\nvivid white, a rosebud; and a green plumy leaf, a favourite ornament\nwith Spanish girls, drooped, bending, and soft as a feather, on one\nside of her gold-and-tortoiseshell comb. The Major sat beside Carlota,\nwho, naturally frank, and looking upon him now as an old acquaintance,\nwould tap his arm most bewitchingly with her fan, when she wanted to\ndirect his attention to any object of interest. So the Major sat by\nher, all gallantry and smiles, gazing about him with wonder through\nthe double gold eyeglass, which still, in spite of the late expression\nof popular feeling, bestrid his nose. He looked with the interest of a\nchild at everything--at the faces and dresses around him, distinct in\ntheir proximity, and at those, confused in their details by distance,\non the opposite side of the arena. He shared in the distress of an\nunfortunate person (a contractor for bulls, who had palmed some bad\nones on the public) who tried, as he walked conspicuously across\nthe ring, to smile off a torrent of popular execration about as\nsuccessfully as a lady might attempt to ward off Niagara with her\nparasol, and who was, as it were, washed out at an opposite door,\ndrenched and sodden with jeers. And when the folding-gates were opened,\nand the gay procession entered, my grandfather gazed on it with\ndelight, and shouted \"Bravo!\" as enthusiastically as if he had been a\nhabitual frequenter of bull-rings from his earliest youth. First came\nthe espadas or matadores, their hair clubbed behind like a woman's,\ndressed in bright- jackets, and breeches seamed with broad\nsilver lace, white stockings, shoes fastened with immense rosettes, and\nhaving their waists girt with silk sashes, bearing on their arms the\nblood- cloaks that were to lure the bull upon the sword-point.\nNext followed the chulos, similarly attired; then the picadores,\nriding stiffly, with padded legs, on their doomed steeds; and mules,\nwhose office it was to drag off the dead bulls and horses, harnessed\nthree abreast as in classic chariots, and almost hidden under a mass\nof gay housings, closed the procession. Marching across the middle of\nthe ring to the alcalde's box, they requested permission to begin,\nand, it being granted, the picadores stationed themselves at equal\ndistances from each other round the circumference of the arena. Then,\nat a signal from the alcalde, two trumpeters in scarlet, behind him,\nstood up and sounded--a man, standing with his hand ready on a bolt in\na door underneath, drew it, and pulled the door swiftly back, shutting\nhimself into a niche, as the dark space thus opened was filled by the\nformidable figure of a bull, who, with glancing horns and tail erect,\nbounded out, and, looking around during one fierce brief pause, made\nstraight at the first picador. The cavalier, standing straight in his\nstirrups, his lance tucked firmly under his arm, fixed the point fairly\nin the shoulder of the brute, who, never pausing for that, straightway\nupset man and horse. Then my grandfather might be seen stretching far\nover the front of his box, his eyes staring on the prostrate picador,\nand his hands clenched above his head, while he shouted, \"By the Lord,\nsir, he'll be killed!\" And when a chulo, darting alongside, waved his\ncloak before the bull's eyes and lured him away, the Major, drawing\na long breath, turned to a calm Spaniard beside him, and said, \"By\nheaven, sir, 'twas the mercy of Providence!\"--but the Spaniard, taking\nhis cigar from his mouth, and expelling the smoke through his nostrils,\nmerely said, \"Bien esta,\" ('tis very well.) Meanwhile, the bull (who,\nlike his predecessor in the china-shop, seemed to have it all his\nown way) had run his horn into the heart of a second horse, and the\npicador, perceiving from the shivering of the wounded creature that\nthe hurt was mortal, dismounted in all haste, while the horse, giving\none long, blundering stagger, fell over and died, and was immediately\nstript of his accoutrements. This my grandfather didn't like at all;\nbut, seeing no kindred disgust in the faces round him, he nerved\nhimself, considering that it was a soldier's business to look on wounds\nand death. He even beheld, with tolerable firmness, the spectacle of\na horse dashing blindfold and riderless, and mad with fear and pain,\nagainst the barrier--rebounding whence to the earth with a broken\nshoulder, it was forced again on its three legs, and led stumbling from\nthe ring. But when he saw another horse raised to its feet, and, all\nript open as it was, spurred to a second assault, the Major, who hadn't\nthe heart himself to hurt a fly, could stand it no longer, but, feeling\nunwell, retired precipitately from the scene. On reaching the door, he\nwrote over the same, with a bit of chalk, part of the speech of Henry\nV., \"the royal imp of fame,\" to his soldiers at Agincourt:--\n\n    \"He that hath not stomach for the fight,\n    Let him depart----\"\n\nto the great astonishment of the two Spanish sentries, who gazed on the\nwords as if they contained a magical spell.\n\nFrank sat till it was over--\"played out the play.\" Not that he saw much\nof the fight, however; he had eyes and speech for nothing but Juana,\nand was able to indulge his _penchant_ without interruption, as the\nlittle Governor took great interest in the fight, and the lover with\nthe silver fishes was a connoisseur in the sport, and laid bets on\nthe number of horses that each particular bull would kill with great\naccuracy. So the Ensign had it all his own way, and, being by no means\nthe sort of person to throw away this or any other opportunity with\nwhich fortune might favour him, got on quite as well, probably, as you\nor I might have done in his place.\n\nLeaving Cadiz next morning, they resumed the order of march they had\nadopted in coming--Don Pablo riding, as before, in front with the\nknight of the silver fishes, discussing with him the incidents of the\nbull-ring. The old gentleman, though very courteous when addressing the\ntwo Englishmen, had but little to say to them--neither did he trouble\nhimself to talk much to the ladies; and when he did, a sharp expression\nwould sometimes slip out, convincing Owen that he was something of\na domestic tyrant in private--a character by no means inconsistent\nwith the blandest demeanour in public. The Ensign was at great pains\nto encourage the Major to be gracious to Carlota. \"Get a little more\ntropical in your looks, Major,\" he would say; \"these Spanish ladies\nare not accustomed to frigid glances. She's desperately in love with\nyou--pity she can't express what she feels; and she mightn't like to\ntrust an interpreter with her sentiments.\"\n\n\"Pooh, nonsense, boy,\" said the Major, colouring with pleasure, \"she\ndoesn't care for an old fellow like me.\"\n\n\"Doesn't she?--see what her eyes say--that's what I call ocular\ndemonstration,\" quoth the Ensign. \"If you don't return it, you're a\nstock, a stone.\" Then he would say something to Carlota, causing her\neyes to sparkle, and canter on to rejoin Juana.\n\nIt was genial summer-time with Carlota--she had passed the age of\nmaiden diffidence, without having attained that of soured and faded\nspinsterhood. She had a sort of jovial confidence in herself, and an\neasy demeanour towards the male sex, such as is seen in widows. These\nsupposed advances of the Major were accordingly met by her rather more\nthan half-way. None but the Major was permitted to assist her into the\nsaddle, or to receive her plump form descending from it. None but the\nMajor was beckoned to her rein when the path was broken and perilous,\nor caught on his protecting arm the pressure of her outstretched hand,\nwhen her steed stumbled over the loose pebbles. None was repaid for\na slight courtesy by so many warm, confiding smiles as he. These,\nfollowing fast one on another, began to penetrate the rusty casing of\nthe Major's heart. On his own ground--that is, in his own quarters--he\ncould have given battle, successfully, to a score of such insidious\nenemies: his books, his flowers, his pipe, his slippers, and a hundred\nother Penates would have encircled him; but here, with all his strong\npalisading of habit torn up and scattered, all his wonted trains of\nideas upset and routed by the novelty of situation and scenery, he lay\ntotally defenceless, and open to attack. The circumstance of himself\nand Carlota being ignorant of each other's language, far from being an\nobstacle to their mutual good-will, rather favoured its progress. In\ncompany with an Englishwoman, in similar circumstances, my grandfather\nwould have considered himself bound to entertain her with his\nconversation, and, perhaps, have spoiled all by trying to make himself\nagreeable--it would have been a tax on the patience of both: but being\nabsolved from any such duty in the present instance, he could without\nawkwardness ride onward in full and silent communion with his own\nthoughts, and enjoy the pleasure of being smiled upon without being at\nany pains to earn it.\n\nHis note-book, containing an account of the expedition, which I have\nseen--and whence, indeed, the greater part of this chronicle is\ngathered--exhibits, at this period of the journey, sufficient proof\nthat the Major enjoyed this new state of being extremely, and felt his\nintellect, his heart, and his stomach at once stimulated.\n\n\"Spain,\" says my grandfather, in a compendious descriptive sentence,\n\"is a country of garlicky odours, of dirty contentment, of\noverburthened donkeys, and of excellent pork; but a fine air in the\nhills, and the country much sweeter than the towns. The people don't\nseem to know what comfort is, or cleanliness, but are nevertheless\nvery contented in their ignorance. My saddle is bad, I think, for I\ndismounted very sore to-day. The Senorita mighty pleasant and gracious.\nI entertain a great regard for her--no doubt a sensible woman, as well\nas a handsome. A pig to-day at breakfast, the best I have tasted in\nSpain.\"\n\nThe desultory style of the composition of these notes prevents me from\nquoting largely from them. Statistics, incidents of travel, philosophic\nreflections, and the state of his digestive organs, are all chronicled\nindiscriminately. But, from the above mixture of sentiments, it will\nbe perceived that the Major's admiration for Carlota was of a sober\nnature, by no means ardent or Quixotic, and pretty much on a par with\nhis passion for pig.\n\nThis was far from being the case with Garry, who became more and more\nenamoured every hour. The Spanish lover continued to conduct himself\nas if he had been married to Juana for twenty years, never troubling\nhimself to be particularly agreeable or attentive, for which obliging\nconduct Garry felt very grateful to him. The Major had been too long\naccustomed to witness Owen's philanderings to see anything peculiar\nin the present case, till his attention was attracted by a little\nincident he accidentally witnessed. After the last halt they made\nbefore reaching Tarifa, Garry was, as usual, at hand, to assist Juana\nto her saddle. The strings of her hat were untied, and he volunteered\nto fasten them; and, having done so, still retaining his hold of the\nstrings, he glanced quickly around, and then drew her blooming face\ntowards his own till their lips met--for which piece of impudence he\nonly suffered the slight penalty of a gentle tap with her whip. My\ngrandfather discreetly and modestly withdrew his eyes, but he was not\nthe only observer. He of the silver fishes was regarding them with a\nfixed look from among some neighbouring trees, where he had tethered\nhis horse. Probably the Spaniard, with all his indifference, thought\nthis was carrying matters a little too far, for, after conversing a\nmoment with the Governor, he took his place at Juana's side, and did\nnot again quit it till they arrived at Tarifa. Then both he and the\nGovernor took leave of our travellers with a cold civility, defying all\nGarry's attempts to thaw it, and seeming to forbid all prospect of a\nspeedy renewal of the acquaintance.\n\n\nCHAPTER IV.\n\nAt the inn, that night, the Major betook himself to rest early, that he\nmight be ready to start for Gibraltar betimes in the morning, for on\nthe following day their leave was to expire.\n\nHe had slept soundly for several hours, when he was awoke by Owen, who\nentered with a candle in his hand. The Major sat up in bed and rubbed\nhis eyes.\n\n\"Time's up, my boy, eh?\" said he, with a cavernous yawn. \"I should have\nliked another hour of it, but it can't be helped,\" (preparing to turn\nout.)\n\n\"I didn't want to spoil your rest last night,\" said Owen, seating\nhimself on the edge of the bed, \"so I said nothing about a mishap that\nhas occurred. That smuggling villain, Francisco, took advantage of our\nabsence to fetch a contraband cargo in the boat from Gibraltar, and has\nbeen caught in attempting to run it here.\"\n\n\"God bless me,\" said my grandfather, \"who would have thought it!--and\nhe such a capital cook! But what's to be done? where's the boat?\"\n\n\"The boat is, for the present, confiscated,\" said Garry; \"but I daresay\nthe Governor would let us have it in the morning, on explaining, and\nperhaps release Francisco, with the loss of his cargo; but--but--in\nfact, Major, I don't want the Governor to know anything about our\ndeparture.\"\n\nMy grandfather stared at him, awaiting further explanation.\n\n\"Juana looked pale last night,\" said the Ensign after a pause.\n\nThe Major did not dispute the fact, though he could not, for the life\nof him, see what it had to do with the subject.\n\n\"She never liked that dingy Spanish lover of hers,\" said the Ensign,\n\"and her father intends she shall marry him in a month. 'Twould make\nher miserable for life.\"\n\n\"Dear me,\" said my grandfather, \"how do you know that?\"\n\n\"She told me so. You see,\" said Owen, shading the candle with his hand,\nso that my grandfather couldn't see his face, and speaking hurriedly,\n\"I didn't intend we should start alone--in fact--that is--Juana has\nagreed to fly with me to Gibraltar.\"\n\n\"Agreed!--fly!\"--gasped my grandsire: \"what an extraordinary young\nfellow!\"\n\n\"She's waiting for us now,\" resumed Garry, gathering courage after the\nfirst plunge into the subject; \"we ought to be off before daylight.\nOblige me, my dear sir,\" (smiling irresistibly,) \"by getting up\nimmediately.\"\n\n\"And how are we to get away,\" asked my grandfather, \"supposing this\ninsane scheme of yours to be attempted?\"\n\n\"I've bribed the sentry at Francisco's place of durance,\" returned the\nEnsign. \"We shall get out of the town the instant the gates are opened;\nand the boat is tied to the steps, as before, only under the charge of\na sentry whom we can easily evade. Every guarda costa in the place was\nsent out last night to blockade a noted smuggler who has taken refuge\nin Tangier; so, once out, we are safe from pursuit: I found it all out\nafter you had gone to bed.\"\n\nThe disposition of Major Flinders, as the reader knows, was the reverse\nof enterprising--he wouldn't have given a straw to be concerned in\nthe finest adventure that ever happened in romance. He paused with\none stocking on, inclined, like the little woman whose garments had\nbeen curtailed by the licentious shears of the pedlar, to doubt his\nown identity, and wondering if it could be really he, John Flinders,\nto whom such a proposition was broached, requiring him to assist in\ninvading the peace of a family. As soon as he recovered his powers of\nspeech, of which astonishment had for a moment deprived him, he began\nearnestly to dissuade the Ensign from the enterprise; but Owen knew his\nman too well, and had too much youthful vivacity of will to allow much\ntime for remonstrance.\n\n\"Look you, Major,\" said he, \"I'm positive I can't live without Juana.\nI'll make a bold stroke for a wife. The thing's settled--no going back\nnow for me; and I shall go through with it with or without you. But\nyou're not the man, I'm sure, to desert a fellow in extremity, at a\ntime, too, when the advantages of your experience and coolness are so\npeculiarly needed. 'Call you that backing of your friends?'\"\n\nThe compliment, or the quotation, or both, softened the Major. \"'Would\nit were night, Hal, and all well,'\" said he, half mechanically\nfollowing the Falstaffian train of ideas Owen had artfully conjured up,\nand at the same time drawing on the breeches which that astute youth\nobsequiously handed to him.\n\nIt was still dark when they issued forth into the narrow and dingy\nstreets of Tarifa. My grandfather, totally unaccustomed to visit\nthe glimpses of the moon in this adventurous fashion, was full of\nstrange fears--heard as many imaginary suspicious noises and voices as\nBunyan's Pilgrim in the dark valley--and once or twice stopt abruptly\nand grasped Owen's arm, while he pointed to a spy dogging them in the\ndistant gloom, who turned out to be a door-post. But Owen was now in\nhis element; no tom-cat in Tarifa was more familiar with housetops\nand balconies at the witching hour than he, and he stepped gaily on.\nPresently they were challenged by a sentry, to whom Owen promptly\nadvanced and slipped into his itching palm a doubloon, when the\ntrustworthy warrior immediately turned upon his heel, and, walking to\nthe extremity of his post, looked with great vigilance in the opposite\ndirection.\n\nOwen advanced to a grated window and tapped. Immediately the burly face\nof Francisco showed itself thereat, his white teeth glancing merrily in\na glimmer of moonshine. A bar, previously filed through, was removed\nfrom the window, and Owen, taking him by the collar to assist his\negress, drew him through as far as the third button of his waistcoat,\nwhere he stuck for a moment; but the substance was elastic, and a lusty\ntug landed him in the middle of the narrow street. Receiving Frank's\ninstructions, given in a hurried whisper, to go at once to where the\nboat lay, and cast her off, ready to shove off on the instant, he\nnodded and disappeared in the darkness, while Owen and the Major made\nfor the Governor's house.\n\nArrived near it, Owen gave a low whistle--a peculiar one, that my\ngrandfather remembered to have heard him practising to Juana on the\nprevious day--when, to the unutterable surprise of the Major, _two_\nveiled figures appeared on the balcony.\n\n\"Why, Owen, boy, d'ye see!\" quoth the Major, stuttering with anxiety,\n\"who can the other be?--her maid, eh?\"--indistinct stage recollections\nof intriguing waiting-women dawning on him.\n\n\"Ahem!--why, you see, Major,\" whispered Owen, \"she wouldn't come\nalone--couldn't manage it at all, in fact, without the knowledge of\nher aunt, who sleeps in the next room; so I persuaded Carlota _to come\ntoo_, and gave her a sort of half promise that _you_ would _take care\nof her_.\" Here, wishing to cut short a rather awkward explanation,\nhe ran under the balcony--one of the ladies dropped a cord--and Owen\nproducing from under his coat a rope ladder, (he had sat up all night\nmaking it,) attached it, and, as soon as it was drawn up, ascended,\nmotioning to my astounded grandfather to keep it steady below. The\nMajor, after a moment's desperate half-resolve to make a hasty retreat\nfrom the perilous incidents which seemed momentarily to thicken round\nhim, and leave his reckless friend to his fate, yielded to the force\nof circumstances, and did what was required of him. Then Owen lifted\nthe ladies, one after the other, over the railing of the balcony, and\nthey swiftly descended. First came Juana, who, scarcely touching the\nMajor's offered hand, lit on the pavement like gossamer; then Carlota\ndescended, and making, in her trepidation, a false step near the\nbottom, came so heavily on the Major, that they rolled together on the\nstones. By the time they were on their feet again, Owen had slipped\ndown the ladder, and, taking Juana under his arm, set off rapidly\ntowards the bay.\n\nIf anything could have added to the Major's discomfiture and\nembarrassment, it would have been the pressure of Carlota's arm on his,\nas she hung confidingly on him--a pressure not proceeding from her\nweight only, but active, and with a meaning in it; but he was in that\nstate of mental numbness from the successive shocks of astonishment,\nthat, as with a soldier after the first two dozen, any additional\nlaceration passed unheeded. He was embarked in an adventure of which\nhe could by no means see the end; all was strange and dark in the\nforeground of his future; and if he had been at that moment tried,\ncast, and condemned for an imaginary crime, he would have been too\napathetic to say anything in arrest of judgment.\n\nSilently and swiftly, as a forlorn hope, they passed through the town\nand along the sandy causeway. The succession of white rolling waves\non their left, where extended the full breadth of the Straits, while\nthe bay on their right was almost smooth, showed the wind to be still\nagainst them; but it was now so moderate that they might safely beat up\nfor the Rock. Arrived at the head of the stairs leading to the water,\nthey paused in the angle of the wall to reconnoitre. Francisco was\nlying coiled up in the head of the boat, his hand on the rope, ready to\ncast her loose, and the boat-hook projecting over the bow. Above them,\nand behind the wall, at a little distance, they could hear the measured\ntread of the sentry, and catch the gleam of his bayonet as he turned\nupon his walk: a few vigorous shoves would carry them outside the\nsea-wall and beyond his ken. All depended on their silence; and like\ntwo stealthy cats did Owen and Juana descend to the boat--the Major and\nCarlota watching the success of their attempt with protruded necks.\nCautiously did Owen stride from the last stair to the deck--cautiously\ndid he transfer Juana to the bark, and guide her aft. The Major was\njust preparing to follow, when a noise from the boat startled him:\nJuana had upset an unlucky wine-jar which Francisco had left there.\nThe sentry put his head over the wall, and challenged; Francisco,\nstarting up, shoved hastily off; the sentry fired his piece, his bullet\nshattering the wine-jar that had caused the mischief. Juana screamed,\nOwen swore in English, and Francisco surpassed him in Spanish. There\nwas no time to return or wait for the other pair, for the guard was\nalarmed by the sentry's shot, and their accoutrements might be heard\nrattling near at hand, as they turned hastily out. Before they reached\nthe wall, however, the boat had disappeared.\n\nMajor Flinders watched it till it was out of sight, and, at first,\nexperienced a feeling of despair at being thus deprived of the aid\nof Garry's boldness and promptitude, and left to his own resources.\nPresently, however, a gleam of comfort dawned upon him--perhaps Carlota\nwould now abandon the enterprise, and he should thus, at any rate,\nbe freed from the embarrassment her presence occasioned him. In this\nhope he was shortly undeceived. To have added the shame of failure and\nexposure to her present disappointment, while an opening to persevere\nstill remained, did not suit that lady's ardent spirit; and whether it\nwas that the unscrupulous Garry had really represented the Major as\nvery much in love, or whether such an impression resulted from her own\nlively imagination, she certainly thought her companion would be as\nmuch chagrined at such a denouement as herself. She displayed a prompt\ndecision in this emergency, being, indeed, as remarkable for presence\nas the Major was for absence of mind. Taking the Major's arm, she\ncaused him swiftly to retrace his steps with her to the inn where he\nhad slept. As they retreated, they heard the boom of a gun behind them,\nfired, doubtless, from the Point, at the Fair Unknown. At Carlota's\norders, a couple of horses, one with a side-saddle, were speedily at\nthe inn-door; they mounted, and, before the sun was yet risen, had\nissued forth from the gate of Tarifa, on the road to Gibraltar.\n\nThe Major rode beside her like a man in a dream--in fact, he was\npartly asleep, having been deprived of a large portion of his natural\nand accustomed rest, and partly bewildered. A few days before he\nhad been the most methodical, unromantic, not to say humdrum, old\nbachelor in his Majesty's service; and here he was, how or why he did\nnot well know, galloping away at daybreak with a foreign lady, of\nwhose existence he had been ignorant a week before, with the prospect\nof being apprehended by her relatives for her abduction, and by the\nGovernment for assisting in the escape of, a smuggler. When at length\nroused to complete consciousness by the rapidity of their motion, he\npositively groaned in anguish of spirit, and vowed internally that,\nonce within the shelter of his own quiet quarters, nothing on earth\nshould again tempt him forth on such harum-scarum expeditions.\n\nIt was near noon when they reached Algeciras, where they stopped to\nbreakfast, both of them rather exhausted with fatigue and hunger. This\ntown stands just opposite Gibraltar, across the bay--the road they had\ncome by forms the base of a triangle, of which Cabrita Point is the\napex, the bay washing one side of the projecting coast, the Straits the\nother. The Major was reserved and embarrassed; there was a tenderness\nabout Carlota's manner that frightened him out of his usual gallantry,\nand, to avoid meeting her glance, he looked steadily out of the window\nat the rock of Gibraltar, casting wistful glances at the spot where his\nquarters lay hidden in a little clump of foliage. Immediately after\nthe meal he quitted the room, on pretence of looking after the horses.\nHe determined to protract their stay in Algeciras till late in the\nafternoon, that they might enter Gibraltar in the dusk, and thus avoid\nawkward meetings with equestrian parties from the garrison, who would\nthen be hastening homewards, in order to be in before gun-fire, when\nthe gates are shut.\n\nOn returning, still out of temper, to the room where he had left\nCarlota, he found her, quite overcome with fatigue, asleep on the\nsofa. Her head was thrown a little back on the cushion; her lips were\njust parted, and she looked in her sleep like a weary child. The Major\napproached on tiptoe, and stood regarding her. His ill-humour melted\nfast into pity. He thought of all her kindness to him, and, by a sudden\nsoft-hearted impulse, took gently one of her hands projecting over\nthe side of the sofa. Carlota opened her eyes, and squeezed the hand\nthat held hers; whereupon the Major suddenly quitted his hold, and,\nretreating with great discomposure to the window, did not venture to\nlook at her again till it was time to resume their journey.\n\nAt a little distance from Algeciras is the river Palmones, called by\nthe English the Second River. This was crossed by a floating bridge,\npulled from shore to shore by a ferryman warping on a rope extended\nacross. They had just reached the opposite bank of the stream, when\nCarlota noticed two horsemen galloping fast along the road they had\njust traversed. A second glance showed them to be Don Pablo and the\nlover of Juana. The first inquiries of the Governor had led him to\nsuppose that all had escaped in the boat, and it was not till some time\nafter that he had learned the true state of affairs.\n\nThe fugitives now hastened on in earnest, and roused their horses to\na steady gallop, never pausing till they reached the Guadaranque,\nor First River, about a mile nearer Gibraltar than the other, and\nfurnished with a similar bridge. The delay of the pursuers at the\nformer ferry had thrown them far in rear; and my grandfather, inspired\nby the imminence of the peril, now conceived a bright idea--the\nbrightest, probably, that ever flashed upon him--by executing which\nthey might effectually distance their pursuers. Dropping his glove at\na little distance from the shore, he sent the ferryman to fetch it,\nand then pushed off, (Carlota having already embarked,) and warped the\nbridge to the opposite bank, heedless of the frantic gesticulations\nof the proprietor, who screamed furiously after them to stop. When\nhe reached the opposite side, he took out his pocket-knife and\ndeliberately cut the rope. Having thus, as it were, blown up the\ncommunication in his rear, my grandfather, without the loss of his\nbaggage, continued his retreat to the fortress; while the little\nGovernor, who galloped up just as they were disappearing, was, like\nLord Ullin, left lamenting.\n\nThe sun was already declining, and threw their shadows far before them\non the sands, as they rode along the beach close to the water. The\nbay at this inner extremity makes a great circular sweep--radii drawn\nfrom the rock to different distant points of the arc would be almost\nequal; and for half an hour they continued to see Gibraltar at nearly\nthe same distance to the right and in front of them, holding itself\naloof most provokingly. Twilight descended as they passed the Spanish\nlines and entered on the Neutral Ground. The Major glanced anxiously\nat his watch--in a few minutes the gun from Middle Hill would give\nthe signal for shutting the gates, and doom them irretrievably to\nreturn into Spain for the night. For the first time in his life Major\nFlinders really punished his horse, lifting the tired beast along with\nwhip and rein. Carlota's kept easily beside him under her lighter\nweight, and they rapidly neared the barrier. Just as they passed it,\na stream of flame shot from the rock, illumining objects like a flash\nof lightning;--then came the heavy report of the gun--another minute\nand the drawbridge at Landport would be lifted; but they were upon it.\nThey dashed across somewhat in the style of Marmion quitting Douglas's\ncastle, \"just as it trembled on the rise,\" and were safe in Gibraltar.\n\n\nCHAPTER V.\n\nAfter life's fitful fever, the Major did not sleep well. He had left\nCarlota comfortably established at the inn; and he now lay nervously\nthinking how his embarrassment with regard to her was to terminate,\nespecially if Owen did not shortly make his appearance. Then he\nwas worried by doubts as to the fate of the Fair Unknown and her\npassengers. They might have been recaptured, as escaped smugglers, by a\nguarda costa--they might be detained in the Straits by adverse winds or\ncalms--they might have run ashore into some bay, and come on overland.\nThis last supposition haunted him most pertinaciously, and he resolved\nto go up the rock as soon as it should be daylight to look out for them\nalong the road from Spain. He lay tossing restlessly till the morning\ngun gave the signal of the approach of dawn, and before the echoes died\naway he had his breeches on.\n\nNight was at odds with morning my grandfather, with a telescope under\nhis arm, sallied forth and began the ascent. Silence was over the\nrock, except an occasional sighing of a remnant of night wind that had\nlost itself among the crags. At first, the only clear outline visible\nwas that of the rugged edge of the rock above against the colourless\nsky; but as he toiled up the steep zig-zag path, the day kept pace\nwith him--each moment threw a broader light on the scene--blots of\nshadow became bushes or deep fissures, and new shapes of stone glided\ninto view. The only symptoms of animal life that he beheld were a\nrabbit that fled silently to his hole, and a great white vulture that,\nstartled from his perch on a grey crag, sailed slowly upward on his\nblack-tipped wings, circling higher and higher, till his breast was\ncrimsoned by the yet unrisen sun.\n\nThe path led diagonally to the summit; and, turning a sharp\nlevel corner, my grandfather looked perpendicularly down on the\nMediterranean, whose lazy waves, sending up a gentle murmur, rippled\nfar below him. On his left, also steep down below him, was the Neutral\nGround, level as the sea itself, extending northward into sandy plains,\nabruptly crossed by tumbled heaps of brown mountains. A reddening of\nthe sky showed that the sun was at hand; and presently the glowing disk\ncame swiftly up from behind the eastern hills; the pale earth shared in\nthe ruddiness of the sky, and a long rosy gleam swept gradually over\nthe breadth of the grey sea, like an unwilling smile spreading itself\nfrom a man's lips to his eyes and forehead.\n\nConspicuous on the highest point in the landscape stood my grandfather,\npanting with his exertions as he wiped his forehead. After standing\nfor a moment, bronzed in front like a smith at the furnace, face to\nface with the sun, he turned and swept with his telescope the road into\nSpain. Early peasants, microscopic as ants, were bringing their fruits\nand vegetables into the fortress--a laden mule or two advanced along\nthe beach over which the Major had last night galloped--but nothing\nresembling what he sought was in sight. Then turning completely round,\nwith his face to the path he had just ascended, he gave a long look\ntowards the Straits; and as he did so, the wind, which had shifted\nto the south-west towards morning, blew gently on his face. A sail\nor two was discernible in the distance, outward bound, but nothing\nresembling the cutter. As the Major looked, a signal was made from\nCabrita, and directly two feluccas left their station at Algeciras, and\nswooped out, like two white birds, as if to intercept some bark yet\nhidden by the Point. Again my grandfather looked out to the Strait,\nand presently a small white sail came in sight near Cabrita. For a\nquarter of an hour he stood steadily, with levelled telescope, and\nthen he was almost sure--yes, he could swear--that he saw the small\nEnglish ensign relieved against the sail; and above, at the mast-head,\nthe yellow-striped flag that Francisco hoisted before as the mark of a\nyacht. It was the Fair Unknown--and my grandfather at once comprehended\nthat the pursuers, whom he had escaped the night before, had, on\nreturning to Algeciras, made arrangements for her capture as soon as\nshe should appear.\n\nThe breeze was on her beam, and much fresher with her than farther in\nthe bay, so that the feluccas steered slantingly across her course as\nshe made for the rock. They held on thus, the pursuers and pursued,\ntill within a mile of each other, when the cutter suddenly altered her\ncourse to one nearly parallel with that of the feluccas. The latter,\nhowever, now gained fast upon her, and presently a puff of smoke\nfrom the bow of the foremost was followed by the report of a gun. My\ngrandfather could look no longer through his glass, for his hand shook\nlike a reed, but began, with huge strides more resembling those of\na kangaroo than a quiet middle-aged gentleman, to descend the rock.\nBreathless, he reached his quarters, had his horse saddled and brought\nout, and galloped off towards Europa.\n\nEuropa Point is at the southern extremity of the rock, and commands at\nonce the entrance of the bay and the passage of the Straits. The road\nto it from the north, where the Major was quartered, affords, for the\nmost part, a view of the bay. Many an anxious glance did he cast, as\nhe sped along, at the state of affairs on the water. The feluccas fired\nseveral shots, but all seemed to fall wide, and were probably intended\nonly to frighten the chase, out of consideration for her fair freight.\nStill, however, the English colours floated, and still the cutter held\nher course.\n\nSome artillerymen and an officer were assembled at the Point as the\nMajor galloped up.\n\n\"Can't you fire at 'em,\" said he, as he drew up beside the battery.\n\n\"Too far off,\" said the Lieutenant, rising from the parapet on which he\nwas leaning, and showing a drowsy unshaven countenance; \"we should only\nfrighten them.\"\n\n\"By heavens!\" said my grandfather, \"'tis horrible. I shall see the boy\ntaken before my eyes!\"\n\n\"Boy!\" quoth the Lieutenant, wondering what particular interest the\nMajor could take in the smuggler. \"What boy?\"\n\n\"Why, Owen of ours--he's running away with a Spanish lady.\"\n\n\"The devil!\" cried the Lieutenant, jumping down. \"What, Garry Owen!--we\nmust try a long shot. Pull those quoins out,\" (to a gunner.) \"Corporal,\nlay that gun; a dollar if you hit the felucca. I'll try a shot with\nthis one.\" So saying, he laid the thirty-two pounder next him with\ngreat care.\n\n\"Fire!\" said he, jumping on the parapet to see the effect of the\nshot. At the second rebound it splashed under the bows of the leading\nfelucca, which still held on. She was now scarcely three hundred yards\nfrom the cutter.\n\n\"Why, d--n their impudence!\" muttered the Lieutenant, on seeing his\nwarning pass unheeded, \"they won't take a hint. Corporal, let drive at\n'em.\"\n\nThe Corporal earned his dollar. The shot went through the side of\nthe felucca, on board of which all was presently confusion; in a few\nminutes it was apparent she was sinking. The other, abandoning the\nchase, went to the assistance of her consort, lifting the crew out,\nsome of whom were evidently hurt.\n\n\"A blessed shot!\" cried my grandfather, giving the lucky Corporal a bit\nof gold; \"but I'm glad they're picking up the crew.\"\n\nThe cutter instantly stood in for the harbour, and half an hour\nafterwards the Major bade his young friend and Juana welcome to\nGibraltar.\n\nCarlota was beside herself with joy at seeing the wanderers safe. She\nfirst cast herself upon Juana, and cried over her; then embraced the\nEnsign, who made no scruple of kissing her; lastly, threw herself\ntenderly upon the Major, who gazed over her head as it lay on his\nshoulder with a dismayed expression, moving his arms uneasily, as if\nhe didn't know what he was expected to do with them. Every moment it\nwas becoming clearer to him that he was a compromised man, no longer\nhis own property. On his way through the streets that morning he\nhad passed a knot of officers, one of whom he overheard describing\n\"Old Flinders\" as \"a sly old boy,\" for that he \"had run away with\na devilish handsome Spaniard--who would have thought it?\" \"Ay, who\nindeed!\" groaned the Major, internally. But the seal was put to his\ndoom by the Colonel, who, when he went to report himself, slapped him\non the shoulder, and congratulated him on his happiness. \"Fine woman, I\nhear, Flinders--didn't give you credit for such spirit--hope you'll be\nhappy together.\" The Major, muttering an inarticulate denial, hastily\nretreated, and from that moment surrendered himself to his fate an\nunresisting victim.\n\nAbout dusk that night, Owen came to him.\n\n\"By heavens!\" the Ensign began, throwing himself into a chair, \"I'm the\nmost unlucky scoundrel! Nothing goes right with me. I promised myself\nthat this should be my wedding-night--and here I am, as forlorn a\nbachelor as ever.\"\n\n\"What has gone wrong?\" inquired my grandfather, removing his pipe from\nhis mouth.\n\n\"I pressed her with all my eloquence,\" said Owen; \"reminded her of her\npromise to marry me the day we should arrive here--of the necessity of\ncaring for her reputation, after leaving her father's house and coming\nhere under my protection,\" (here my grandfather winced;) \"talked, in\nfact, like an angel who had been bred a special pleader--yet it was all\nof no use.\"\n\n\"Deliberating about marriage!\" said the Major, \"after leaving her\nfather and lover for you! What gnat can she be straining at, after\nswallowing a camel of such magnitude?\"\n\n\"A piece of female Quixotry,\" returned Owen. \"She says she can't think\nof such selfishness as being comfortably married herself, while Carlota\nis so unhappy, and her fate so unsettled.\" Here he made a significant\npause; but my grandfather was immovably silent, only glancing nervously\nat him, and smoking very hard.\n\n\"In fact, she protests she won't hear of marrying me, till you have\nsettled when you will marry Carlota.\"\n\n\"Marry Carlota!\" gasped the Major in an agonised whisper.\n\n\"Why, you don't mean to say you're not going to marry her!\" exclaimed\nthe Ensign, throwing a vast quantity of surprise into his expressive\ncountenance.\n\n\"Why--why, what should I marry her for?\" stammered the Major.\n\n\"Oh, Lord!\" said Garry, \"here will be pleasant news for her! Curse me\nif I break it to her.\"\n\n\"But really, now, Frank,\" the Major repeated--\"marriage, you know--why,\nI never thought of such a thing.\"\n\n\"You're the only person that hasn't then,\" rejoined Owen. \"Why, what\ncan the garrison think, after the way you smuggled her in; what can she\nherself think, after all your attentions?\"\n\n\"Attentions, my dear boy;--the merest civility.\"\n\n\"Oh,--ah! 'twas civility, I suppose, to squeeze her hand in the inn\nat Algeciras, in the way she told Juana of--and heaven knows what\nelse you may have done during the flight. Juana is outrageous against\nyou--actually called you a vile deceiver; but Carlota's feeling is\nmore of sorrow than of anger. She is persuaded that nothing but your\nignorance of Spanish has prevented your tongue from confirming what\nyour looks have so faithfully promised. I was really quite affected\nto-day at the appealing look she cast on me after you left the room;\nshe evidently expected me to communicate her destiny.\"\n\nMy grandfather smoked hard.\n\n\"Lots of fellows would give their ears for such a wife,\" pursued the\nEnsign. \"Lovelace, the Governor's aide-de-camp, bribed the waiter of\nthe hotel to lend him his apron to-day, at dinner, that he might come\nin and look at her--swears she's a splendid woman, and that he'd run\naway with such another to-morrow.\"\n\nStill my grandfather smoked hard, but said nothing, though there was a\nslight gleam of pride in his countenance.\n\n\"Poor thing!\" sighed Garry. \"All her prospects blighted for ever.\nSwears she never can love another.\"\n\nAt this my grandfather's eyes grew moist, and he coughed as if he had\nswallowed some tobacco-smoke.\n\n\"And as for me, to have Juana at my lips, as it were, and yet not\nmine--for she's as inflexible as if she'd been born a Mede and\nPersian--to know that you are coming between me and happiness as surely\nas if you were an inexorable father or a cruel guardian--worse, indeed;\nfor those might be evaded. Major, major, have you no compassion!--two\ndays of this will drive me crazy.\"\n\nThe Major changed his pipe from his right hand to his left, and,\nstretching the former across the table, sympathetically pressed that of\nthe Ensign.\n\n\"Do, Major,\" quoth Garry, changing his flank movement for a direct\nattack--\"do consent to make yourself and me happy; do empower me to\nnegotiate for our all going to church to-morrow.\" (My grandfather gave\na little jump in his chair at this, as if he were sitting on a pin.)\n\"I'll manage it all; you shan't have the least trouble in the matter.\"\n\nMy grandfather spoke not.\n\n\"Silence gives consent,\" said the Ensign, rising. \"Come, now, if you\ndon't forbid me, I'll depart on my embassy at once; you needn't speak,\nI'll spare your blushes. I see this delay has only been from modesty,\nor perhaps a little ruse on your part. Once, twice, thrice,--I go.\" And\nhe vanished.\n\nThe Major remained in his chair, in the same posture. His pipe was\nsmoked out, but he continued to suck absently at the empty tube. His\nbewilderment and perturbation were so great that, though he sat up\ntill two in the morning, during which time he smoked eleven pipes,\nand increased the two glasses of grog with which he was accustomed to\nprepare for his pillow to four, he was still, when he went to bed, as\nagitated as ever.\n\nIn this state of mind he went to the altar, for next day a double\nceremony was performed, making Owen happy with Juana, and giving\nCarlota a husband and me a grandfather. The Major was more like a proxy\nthan a principal in the affair; for Owen, taking the entire management\nupon himself, left him little more to do than to make the necessary\nresponses.\n\nCarlota made a very good-tempered, quiet, inobtrusive helpmate, and\ncontinued to be fond of her spouse even after he was a gray-headed\ncolonel. My grandfather, though credulous in most matters, could with\ndifficulty be brought to consider himself married. He would sometimes\nseem to forget the circumstance for a whole day together, till it came\nto be forced on his recollection at bed-time. And when, about a year\nafter his marriage, a new-born female Flinders (now my venerable aunt)\nwas brought one morning by the nurse for his inspection and approval,\nhe gazed at it with a puzzled air, and could not be convinced that he\nwas actually in the presence of his own flesh and blood, till he had\ntouched the cheek of his first-born with the point of his tobacco-pipe,\nremoved from his mouth for that purpose, making on the infant's\ncountenance a small indentation.\n\nThe little Governor, Don Pablo, was subsequently induced to forgive his\nrelatives, and frequent visits and attentions were interchanged, till\nthe commencement of the siege put a stop to all intercourse between\nGibraltar and Spain.\n\nI have often, on a summer's evening, sat looking across the bay at a\ngorgeous sunset, and retracing in imagination the incidents I have\nrelated. My grandfather's establishment was broken up during the siege\nby the enemy's shells, but a similar one now stands on what I think\nmust have been about the site of it. The world has changed since then;\nbut Spain is no land of change; and, looking on the imperishable\noutline of the Andalucian hills, unaltered, probably, since a time to\nwhich the period of my tale is but as yesterday, it is easy for me to\n\"daff aside\" the noisy world without, and, dropping quietly behind the\nage, to picture to myself my old-fashioned grandfather issuing forth\nfrom yonder white-walled town of Algeciras with his future bride.\n\n\n\n\nGERMAN LETTERS FROM PARIS[2].\n\n\nGerman Professors are altered men since those joyous days when we\ndrank chopines and swang the schlaeger in the thirsty and venerable\nUniversity of Saxesaufenberg. We remember them studious bookworms,\nuneasy when removed from library and lecture-room, their meerschaum\ntheir only passion, knowledge their sole ambition, beholding the\nexternal world through \"the loopholes of retreat,\"--the said embrasures\nbeing considerably obscured by tobacco-smoke and misty philosophy. Such\nis the portrait our memory has preserved of them; and we doubt not that\nits fidelity will be recognised by our brother-_burschen_ of bygone\ndays. But great has been the change. The quality of a German professor\nnow suggests the idea of a red-hot democrat, fanning revolution, pining\nin prison, or fugitive in foreign lands. The smoking-cap is exchanged\nfor the _bonnet rouge_, and the silence of the sage for the clamour\nof the demagogue. This may not be true of all, perhaps not even of a\nmajority, but it is true of a pretentious and prominent minority. The\nbusy, bustling multitude knows nothing of the others.\n\n[2] _Zwei Monate in Paris._ VON ADOLPH STAHR. Two Vols. Oldenburg:\nSchulzeschen Buchhandlung. London: Williams and Norgate, 1851.\n\nProfessor Stahr, of the University of Oldenburg, is a gentleman\nchiefly remarkable for his democratic tendencies, and for the fluent\ncorrectness of his literary style. Few men write better German, or\nprofess doctrines more revolutionary. His reputation as a literary\nman rests principally upon a work on Italy, published after a\ntwelvemonth's residence in that country.[3] As a critic of fine art,\nhe is not without merit. As a politician he is wild and speculative.\nThe revolutionary coterie to which he belongs reckons amongst its\nmembers Fanny Lewald, the lively Hebrew socialist, and Moritz Hartmann,\nthe bitter radical. Both of these, especially the former, are his\nintimate friends, and appear to have been his constant companions\nduring two months of last autumn, spent by him in Paris, and which\nhave given occasion and a title to his latest book. With Mr Hartmann\nhe forgathered at Brussels, early in the month of September, and\ntogether they proceeded southwards. In consideration of Professor\nStahr's acknowledged abilities, we will not apply to him a common\nrule, and judge him by the company he keeps. But, in spite of his\nwell-turned periods and general moderation of expression, his book is\nnot pleasant to read. There is an ill-conditioned tone about writers\nof his political class, extremely trying to the patience and temper\nof the reader. Convinced of the general unfitness of existing human\ninstitutions, and of the necessity for radical changes, they inevitably\nfall into a cavilling and censorious strain. Viewing the condition\nof society with a jaundiced eye, they adopt the maxim that whatever\nis, is wrong. Mr Stahr has hardly entered the railway carriage that\nis to transport him to Paris, when he shows himself querulous and\na grumbler. He hoisted his colours before leaving Brussels. Had we\nnever before heard either of him or his principles, we yet should have\nbeen at no loss to discover the latter by certain passages in his\nvery first chapter. Sitting in his inn at eventide, after visiting\nthe monument to the slain of 1830, he reads an account of the Belgian\nrevolution. The Dutch troops, he finds, made but one hundred and\ntwenty-two prisoners, whilst the insurgents captured four hundred\nand ninety-five. On the other hand, the Belgian killed and wounded\nexceeded by three hundred those of their opponents. Mr Stahr is ready\nwith an inference from these statistics. It takes the form of a slur\nupon the soldiers who were doing their duty to their king and country.\n\"The inequality in the number of prisoners may well arise from the\ncircumstance that the Dutch, as fighters for loyal tranquillity and\norder, were least disposed to give quarter. And soldiers against men\nwithout uniform--one knows that!\" Then he falls foul of the writer of\nthe narrative, for attributing to Providence the preservation of the\nroyal palace, and other public buildings, to which the Dutch attempted\nto set fire; and, gliding thence into religious speculations, he gets\nvery profound, and rather profane, so that we are not sorry when the\ncurrent of his ideas is diverted into a more commonplace channel,\nby the visit, at Valenciennes, of the French customhouse officers,\non the look-out for Belgian cigars and reprints. He is sore at this\nirksome visitation--wonders that powerful France so long endures the\nliterary piracies of her little neighbour--and finally prophesies the\nabolition of all customhouses. \"A time will come,\" he says, \"when this\nsystem of legally privileged waylaying will appear just as fabulous\nto the people of Europe, as do now to us the highway depredations of\nthe robber-knights.\" Pending the advent of that desirable state of\nthings, he revenges himself on a fellow-traveller for his customhouse\nannoyances. A German book which he had left in the carriage on\nalighting had disappeared, and could not be recovered. A _douanier_\nhad perhaps taken it for a contraband commodity. He should have\ndeclared it, opined a fat Frenchman in the same carriage. Mr Stahr was\nindignant. It was a German book, he tartly replied, and was not printed\nat Brussels, but at Leipzig--a place, he added, which must still be\npretty well remembered in France! A polite and tasteful allusion which\ndid the German radical infinite credit, and to which the fat Frenchman\nmight fairly have retorted, \"Jena,\" and half a dozen other significant\nnames, instead of holding his tongue, and leaving his fellow-traveller\nto digest at leisure his loss and his ill-humour.\n\n[3] _Ein Jahr in Italien._ Three Vols., 8vo. Oldenburg: 1850.\n\nMr Stahr's volumes, composed of letters to friends, are desultory,\nand for the most part slight. Picture galleries are favourite haunts\nof his: now he criticises a pamphlet, now a play; he moralises, after\nhis own peculiar fashion, in deserted palaces, assists at a banquet\nof workmen, witnesses extravagant dances at Mabille, sits by the\nbedside of the infirm and suffering Heine. His first walk in Paris\nwas to the Palais Royal, after nightfall. \"Stahr,\" said his companion\nto him suddenly, on the way, \"this is the Place de Greve!\"--\"Were I\nto live a century,\" exclaims the impressionable professor, \"I should\nnever forget the shudder that came over me at these words.\" And he\nbreaks into a tumid rhapsody about the lava-streams of the great\nEuropean volcano, talks of the guillotine, tells the well-known story\nof Favras, and proceeds to the Palais Royal, where, at ten o'clock\nat night, he is unable to obtain a beef-steak for supper, and whose\nglory he accordingly declares departed. Returning to their quarters,\nat a hotel beyond the Seine, the two Germans get bewildered in the\nlabyrinth of the Quartier Latin, and are indebted for guidance to some\nartisans, whose \"Good night, _citoyens_!\" at parting, again thrills\nthe sensitive Stahr. The historical and fanciful associations that\ncrowd upon his mind are of a less practical nature than the reflection\nsuggested to his companion by the Republican mode of address--\"We\nmust exchange our grey Calabrian hats\" (the sort of bandit sombreros\naffected by travelling students and red republicans) \"for the loyal\nhats of order, or soon we shall have Louis Napoleon's police at our\nheels.\" Thus spoke Mr Hartmann--who has a natural aversion to all\npolice, and who gladly sneers at the party of Order, and at Louis\nNapoleon as its representative. Mr Stahr professes no great liking or\ntenderness for the chief of the Republic--the first gendarme in France,\nas he calls him, meaning thereby to cast opprobrium on the President,\ngentlemen of his political complexion having an instinctive detestation\nof gendarmes. He saw him for the first time at the celebrated review\nheld at Satory, on the 10th October 1850. On his way thither, Mr\nStahr joined in conversation with peasants, who were flocking from\nall the country round to see the President and the military pageant.\nMany of them had sons in the regiments that were to be reviewed.\nThey made no mystery of their political creed. It was simple enough:\n\"Peace and moderate taxation,\" said they, \"is what we want. He who\ngives us those two things is our man, whether as King or President\nmatters not.\" The review over, the throng of spectators drew up to\nsee Louis Napoleon. After the Minister of War, d'Hautpoul, and the\nthen-all-powerful Changarnier, had passed, each with his staff, \"there\ncame by, mounted on a tall gray horse, the elect of six millions of\nvoters. Judiciously-distributed adherents waved their hats and shouted,\nat the top of their voices, 'Long live the Emperor! Vive Napoleon!' The\npeople were mute. It was a laughable farce. The hero of Strasburg and\nBoulogne, mounted on a tall charger, in a brilliant general's uniform,\nthe broad riband of the Legion of Honour over his shoulder, in plumed\nhat and jackboots, was the very model of a circus equestrian.\" An air\nof helplessness and exhaustion, according to Mr Stahr, was the main\ncharacteristic of the President's appearance. \"I stood near enough,\" he\ncontinues, \"to see him well, and never did I behold a more unmeaning\ncountenance. An unwholesome grey-brown is its prevailing tint. Of\nlikeness to the great Emperor there is scarcely a trace.\" There is\nno chance, Mr Stahr declares, of such a person as Louis Napoleon\nputting the republic in his pocket. Having given his opinion of the\nPresident's exterior, he proceeds in the next chapter to sketch his\ncharacter, as described by a person who had known him from his youth.\n\"He is naturally goodtempered and harmless,\" said this anonymous\ninformant, \"and by no means without ability. But he is tainted with\nthe moral corruption of all European societies, Italian, French, and\nEnglish. He has the _pourriture_ of the drawing-room education of all\nnations. Still he is not devoid of sense, nor of a certain goodness\nof disposition. He can weep, unaffectedly weep, over a touching case\nof wretchedness and misery, and he willingly shows clemency, when\nasked, even to political opponents. But no reliance can be placed in\nhim. In a word, his character is that of a woman. As a result of his\nwandering and adventurous existence, he appears to-day as a German,\nto-morrow as a Frenchman, and the day after to-morrow as an Englishman\nor Italian. He is wholly without fixed principles, and without moral\nstay. If one represents to him the immorality of an act, he will laugh\nand say, '_Bah!_ what is that to me?' But the very next day you shall\nfind him as much oppressed with moral scruples as any German candidate.\nHe has the physical courage of his unusual bodily strength--_corporis\nrobore stolide ferox_--supported by a fatalist belief in his star;\nand this belief, which has lately acquired increased strength by his\nextraordinary vicissitude of fortune, blinds him to his real position,\nand renders him deaf to the warning voices of his few honest friends.\nIn this respect his mother, who unceasingly stimulated his ambition,\ndid him much harm. Personally he is modest and unassuming, but he is\nmadly vain of his name and of his legitimate claims. That he has done\nand continues to do himself grievous harm, as it is universally said,\nby excesses of the most unrefined description, and by opium-smoking,\nseems unfortunately to be only too true. For the change in him since\nhis youth has been altogether too great. Nevertheless, he is much less\nthe tool of others than might be supposed. He has a way of half-closing\nhis inexpressive light-blue eyes, which he has adopted to prevent\npersons from reading his thoughts. His chief delusion is that the army\nis unconditionally devoted to him. This is by no, means the case.\" We\ngive this curious sketch, in which truth and malignity are ingeniously\nblended, for no more than it is worth. The reader will have little\ndifficulty in sifting the grain from the chaff, the idle or malicious\ngossip from the well-founded observations. Mr Stahr supports the\nassertion of the indifference of the French army to the commonplace\nnephew of their great idol, by anecdotes derived from personal\nexperience. After the review, he dined for some days in company with\nthree hussar officers, quartered in the house he lived in. His account\nof them hardly agrees with the popular notion of French officers. \"They\nare modest, reserved, and serious in manner. Nowhere in Paris have I\nfound a trace of that overweening presumption by which German officers,\nespecially cavalrymen, seek to give themselves importance at _tables\nd'hote_ and other public places. We spoke of yesterday's manoeuvres,\nand I paid them a compliment on the really splendid bearing of the\ntroops and the capital equipments. There are no longer grounds to\ndepreciate the French cavalry. Africa has been an excellent school\nfor them. 'But there was one thing wanting,' I remarked--'namely,\nenthusiasm.' 'You are quite right, sir,' replied one of the officers;\n'but there is not much to be enthusiastic about in the position in\nwhich we are.' The speaker was a thorough soldier, and anything but an\nupholder of revolutionary or socialist-democratic ideas. The supporters\nof the latter he invariably spoke of as '_les Voraces_,' and bitterly\ncomplained that for years past he and his comrades had had nothing else\nto do than to '_faire la chasse aux voraces_!' But with the 'Nephew of\nthe Uncle' none of the officers showed the least sympathy. Concerning\nhim they all observed a very eloquent silence.\" In contrast to the\nridicule and censure levelled by Mr Stahr at the more recent portion of\nLouis Napoleon's career, are some anecdotes he tells us of his earlier\nyears. \"In his youth,\" he says, \"he must have been very amiable. I\nhave had opportunity to look through a collection of letters written\nby him to a friend of his family, and extending over more than twenty\nyears. It included even notes written when he was a boy of eleven,\nsome of them in the German language and character. Louis Napoleon is\nknown to be a perfect master of German. The most pleasing and amiable\nof these letters were a series written from his prison at Ham. Good\nfeeling, hearty gratitude for proofs of faithful adherence and for\naffectionate little services, and a deep dejection at his lot, were\nthe characteristics of these letters. He read and studied a great deal\nat Ham, especially military science, but also poetry and literature.\nWithin those prison-walls he now and then began to distrust the\n'star' of his destiny.\" These letters were doubtless the same spoken\nof elsewhere by Mr Stahr as filling several volumes, and as having\nbeen addressed to Madame Hortense Cornu, a well-known writer on fine\nart, who was long attached to the household of Queen Hortense. She\nhad known Louis Napoleon from his childhood, and retained sufficient\ninfluence over him to obtain the rescue from the hands of the Roman\npriesthood of the Italian republican Cernuschi. The letters, says Mr\nStahr, abound in evidence of the esteem and gratitude entertained by\nthe French President for the staunch and trusty friend of his youth.\n\"This correspondence, fragments of which I was favoured with permission\nto read, includes all the epochs of his adventurous life. It ceases\nwith the day when the infatuated man, having attained to power, laid\nhands upon the right of universal suffrage which had raised him from\nthe dust. Madame Cornu's last letter was a solemn exhortation to\nabstain from that step. She laboured in vain, for fate is stronger than\nhumanity. But it is an honourable testimony to the originally good\ndisposition of the blinded man that he did not withdraw his favour\nfrom his tried friend. A proof of this is to be found in Cernuschi's\ndeliverance.\"\n\nDuring a visit paid by Mr Stahr to Alexander Dumas, the French\nromance-writer told the German professor an anecdote of Louis Napoleon\nand the late Duke of Orleans, which is curious, if true. Perhaps it\nis as well to bear in mind, whilst reading it, that its narrator is a\nstory-teller by profession, and the most imaginative and decorative\nof historians. Dumas, it appears, had been long acquainted with the\nimperial pretender and his mother; was aware of the rash schemes of\nthe Prince, then meditating the Strasburg expedition; and advised him,\nby letter, to abandon them, or at least to adopt a totally different\nmode of carrying them out. If he would uproot (_deraciner_) the dynasty\nof Louis Philippe, wrote Dumas, he must try very different means. He\nmust endeavour to obtain the revocation of his sentence of exile,\nget himself elected member of the French Chamber of Deputies, and so\nfollow up his plans in opposition to the ruling dynasty. Deaf to this\nadvice, which was certainly sensible enough, Louis Napoleon made his\nridiculous attempt at Strasburg, and was taken prisoner. Thereupon his\nmother, Queen Hortense, hurried to the neighbourhood of Paris under an\nassumed name, and with one confidential attendant. This person she sent\nto Dumas, to entreat him to apply to his patron, the Duke of Orleans,\nto know what the Court had decided with respect to the prisoner's\nfate. Dumas wrote forthwith for an audience; the Duke received him\nwith a smile. \"Well!\" he said, \"so your _protege_ has not succeeded\nin _uprooting_ us?\" \"Prince, you know----?\" stammered the terrified\nnovelist. \"Do you suppose we are so badly served for our money as not\nto know what brings you here, and where Queen Hortense is at this very\nmoment?\" After a short pause, during which he enjoyed the embarrassment\nof Dumas, the Duke continued, \"Tell Madame Hortense,\" he said, \"that\nthe Orleans do not yet feel themselves strong enough _to have their\nDuke d'Enghien_.\"\n\n\"It is a bitter answer, your royal highness,\" replied Dumas, taking his\nleave, \"but still it will console the mother's heart.\"\n\n\"And now,\" muses Mr Stahr, \"the shattered bones of the unfortunate\nyoung Duke of Orleans have long been mouldering in the grave, his\nstatue in the court of the Louvre has been dragged down and stowed\naway in a corner of the Versailles Museum, and the Adventurer of\nStrasburg rules France as a republic, with power more unlimited than\nthe wily Louis Philippe ever possessed over it as a monarchy! For so\nlong as it lasts, that is to say; for methinks the feet of those who\nshall carry him out are already before the door. But how did he ever\nget in? How was it that even his candidature for the presidency was\nnot overwhelmed and rendered impossible by that most dangerous of all\nopponents in France, the curse of the Ridiculous, which had already\ndecorated with cap and bells the hero of the blunders of Strasburg\nand Boulogne, the trainer of the tame eagle, the special constable of\nLondon?\" It has puzzled acuter politicians than Mr Stahr to reply to\nthis question, which millions have asked. The riddle interests him, and\nhe runs about on all sides seeking its solution. He has little success,\nand evidently himself mistrusts the ingenious and original conclusion\nto which he at last comes, that the election of Louis Napoleon was a\nhomage to the hereditary principle. \"When I recently, on my way across\nthe plain of Satory, asked a countryman if he had given his vote to the\nPresident, his reply was, 'Of course! was he not the rightful heir, his\nuncle's legitimate successor?' This may sound ill for the republican\neducation of the people of the French republic; but it is the truth.\nThe principle of hereditary rule may be perfectly incompatible with\nthat of 'liberty and equality,' but it is, or was, (at the time of\nLouis Napoleon's election,) the prevailing principle in the heads of\nthe French rural population. 'One must know the French peasantry as\nI know them, who have grown up amongst them,' lately said to me the\nrepresentative De Flotte, 'to find their conduct in this matter quite\nnatural. The French peasant has only one fundamental idea in politics,\nand that is derived from his own family relations. That fundamental\nidea is the sacredness and necessity of hereditary right. That the\nterritorial property of the father should descend to the son, or next\nof kin, seems to him the main condition or all human existence.'\"\nAdmitting, for argument's sake, the soundness of this statement, and\nthat the French peasant is thus devoted to the hereditary principle,\nthe natural inference is that, when he perceived his country to be in\na state of transition, ruled by provisional intruders, and anxiously\nlooking out for a more permanent chief of the state, he should have\nhoisted the white cockade, and tossed up his beaver for the Fifth\nHenry. Messrs Stahr and De Flotte explain why he did not do this.\n\"The French peasant has no longer any sort of sympathy with the elder\nBourbons. For him the glory of Louis XIV. is far too remote. What else\nhe knows of them is, that they brought the foreigner into his country,\nand on that account he curses them.\" In this there is some truth. The\nold royalist spirit still lingers in certain departments of France,\nbut in the country generally the Count de Chambord's partisans are\nrather intelligent and influential than numerous. Should he ascend\nthe throne, it will not be in virtue of zeal for the principle of\nlegitimacy or of personal attachment to himself, but because the nation\nwill see in his accession the best guarantee of order and economical\nadministration. These two things are the real wants and desires of the\nmass of the population. The peasant who told Mr Stahr he wished for\npeace and light taxation, spoke the feeling of a great majority of\nFrenchmen. \"The dynasty of Orleans,\" says the professor's informant,\ncontinuing his explanation of the concurrence of circumstances which\nraised Louis Napoleon to the president's chair, \"never enjoyed much\nprestige amongst the rural population, who did not forgive old Louis\nPhilippe for having violated the principle of hereditary right.\"\nThis is rather far-fetched. If the provinces cared little for Louis\nPhilippe, it was because he had troubled himself little about them.\nTrue to his system of centralisation, Paris, to him, was France, and\nungrateful Paris it was that finally abandoned and expelled him. It is\nunnecessary to go out of one's way to seek reasons for the fact, that\nwhen, in December 1848, the French, exhausted by nine months' anarchy\nand misery, and ashamed of those February follies into which a few\ndeluded and designing men had led them, cast about for a ruler under\nwhom they might hope for respite and breathing time, none turned a\nwishful or expectant eye to any member of the house of Orleans. The\nfamily had been weighed and found wanting. From the astute politician,\n\"whose word no man relied on,\" and who reaped in his latter days those\nbitter fruits of usurpation and anarchy whose seeds he had sown in his\nprime, down to the youngest of the sons to whose advancement he had\nsacrificed his conscience and his country, and who, in the supreme\nhour of peril and confusion, were found utterly deficient in princely\nand manly qualities, in self-possession, energy, and resource, there\nwas not one of the line whom France would trust. The time was too\nshort that had elapsed since the picture of selfishness and incapacity\nhad been exhibited to wondering Europe: the cause had been unable to\nrevive from the grievous and self-inflicted shock; it lay supine and\nseemingly dead, awaiting the day when intrigue and hypocrisy should\ngalvanise it into a precarious vitality. When the crisis of May 1852\narrives, we shall see what has been the effect of the complicated\nmanoeuvres of the house of Orleans, which, in December 1848, stood\nso low in public estimation. Then, according to Mr Stahr, Buonapartism\nwas the only political creed that appealed to the prejudices and\nfeelings of the French peasant, and it required no great skill to get\nhim to write upon his election-ticket the name of the prince whom he\nlooked upon as the rightful heir of the Emperor. \"He did it of his own\naccord, out of a conviction that he was performing an act of justice,\nand that hereditary right demanded it. Other motives concurred. The\nforty-five-centime impost had embittered the countryman against the\nRepublic, which had increased instead of lightening his load. Upon\nthe Democrat-Socialists he looked distrustfully. He would have nought\nto say to those '_partageux_' (dividers.) He cared nothing for the\nfine speeches of parliamentary orators. The peasant is by nature\ntaciturn, and has little confidence in assemblies of great talkers.\nHe was not disposed to make a stir about the freedom of the press,\nof which he makes no use. His political understanding did not extend\nbeyond one wish, and that wish was, a strong government, which should\nsecure to him the enjoyment and inheritance of his property. And who\ncould do that better than a Napoleon--Napoleon himself, the Emperor\nof Beranger?--for there are many places where the country people have\nnever believed the Emperor dead.\" The clever author of _Jerome Paturot_\nhas expressed a similar opinion as regards the prevalence of this\nscarcely credible delusion amongst the uneducated classes in certain\ndistricts of France. It does not appear to be entirely confined to that\ncountry. \"I myself am witness,\" says Mr Stahr, \"that, in the year 1848,\na peasant of a province of Northern Germany, on hearing of the new\nFrench revolution, and of its first consequences in Germany, remarked\nthat, 'without doubt old Buonaparte had a finger in the pie.'\" It is\nMr Stahr's belief that Louis Napoleon is destined to dispel, by his\ninability to fulfil the expectations of the ignorant portion of his\nconstituents, that Buonapartist prestige to which he partly owed his\nelection, and that attachment to the hereditary principle which the\nprofessor assumes still to exist in France. \"The nephew of the great\nEmperor,\" he says, \"is selected by fate to disturb, if not to destroy,\nthe idolatry with which a large portion of the French nation has\nhitherto regarded the name and memory of its greatest tyrant. Napoleon\nthe Second throws a grey shadow over Napoleon the First.\"\n\nIf the French President receives but rude handling from the German\nrepublican, the Orleans family cannot congratulate themselves on\nmuch better treatment. His first reference to that fallen dynasty is\nsuggested by a little book, which, at the time of its appearance,\nattracted some attention both in England and France. M. Louis Tirel's\n_La Republique dans les Carrosses du Roi_ was neither calculated nor\nintended to please the democrats. Mr Stahr, however, is pretty fair\nin his appreciation of it, sneering a little at the author for taking\nwhat he calls a valet-de-chambre's view of the February revolution,\nbut doing justice to the interest and instruction to be found in his\npages, which show up the _parties honteuses_ of that most disastrous\nand ill-advised political convulsion; the scandalous greed, vanity, and\negotism of the adventurers and knaves who alone profited by the storm\nthey had contributed to raise. M. Tirel, although to all appearance\nhonest and truthful, certainly wrote like a partisan. His position and\nattachments were incompatible with a just estimate of circumstances.\nWhilst accurately describing events, he deluded himself as to the\ncauses that led to them, and, above all, he could see no wrong in\nhis master; could not for the life of him comprehend how it was that\nLouis Philippe, \"who had so faithfully observed his oath to maintain\nthe charter, and who had a majority in the Chambers,\" should have\nbeen ejected from his throne and kingdom. The worthy keeper of the\nroyal carriages never attains to more than a glimmering and confused\nnotion that the nation could scarcely be said to be represented by the\nmajority in question, and that a moderate extension of the suffrage,\naccorded with a good grace, would probably have maintained the July\ndynasty at the helm of French affairs to this day, and for years to\ncome. His admiration of Louis Philippe's wisdom and skill is unlimited,\nas is also his indignation at the ingratitude of the people. Mr Stahr\nloses patience at the affectionate manner in which the _ex-controlleur\ndes equipages_ lauds the virtues of the old \"Jesuit-King,\" as the\nGerman irreverently styles the defunct monarch; and, provoked by\nTirel's exaggerated encomiums, he retorts by the following severe but\ntoo true remarks with reference to the oft-repeated accusation of\nmiserly hoarding, brought against Louis Philippe by Republican and\nLegitimist writers:--\"Louis Philippe,\" he says, \"was no _avare_ such\nas Moliere has drawn--no comedy-miser--but yet he was immoderately\navaricious. There was no end to his demands of money for the princes\nof his house. He knew, or thought he knew, that _money is power_; and\nas he could not obtain enough of the latter, he restlessly strove\nafter the former as the means to an object. He was a good father of a\nfamily, in the _bourgeois_ sense of the word; but he had no conception\nof that which makes a king the father of his people. His defenders\ncelebrate the care which this prince, denounced as grasping, expended\nupon the conservation of the royal palaces, the great sums which he\nlaid out upon rich furniture, numerous attendants, brilliant equipages,\nand luxurious festivals--to which latter often three or four thousand\nguests were invited. 'How,' it is said, 'could the people tax such a\nsovereign with niggardliness and greed of gold?' But the people had\nno part or share in these enjoyments. It suffered hunger and want,\nwhilst the higher and middle classes of the _bourgeoisie_ revelled\nin these feasts, and grew rich by supplying their materials.\" Raised\nto the throne by the suffrages of the middle classes, Louis Philippe\nrelied on them for support. He was bitterly disappointed. Scandalous\nand cowardly was the manner in which the men of July--those whom he\nhad fed, pampered and decorated, favoured and preferred--deserted\nhim in the hour of danger. The very national guards of Neuilly, who\nhad lived and flourished in the shadow of the chateau walls, refused\nto turn out, when, in February 1848, the intendant of the castle\nappealed to them to protect from plunder the property of their patron\nand king. They had caught the contagion of that intense selfishness\nwhich was Louis Philippe's most striking characteristic. \"Let those\nwho choose go out to be shot,\" said the burghers of Neuilly; \"we shall\nstop at home and take care of our houses.\" And assuredly the inert\nand unsympathising attitude of the Paris national guard contributed\nmore than anything else to deter Louis Philippe from resisting by\nforce the progress of the February revolutionists. The burghers were\ndisgusted by the dilapidation of the finances, and the venality of\nthe administration--they were disgusted with Guizot for not daring to\nresist the headstrong will of the old king--and they cried out for\nelectoral reform. With a little more patience they would have achieved\ntheir desire;--over-hasty, they suddenly beheld themselves plunged\ninto revolution. They had not foreseen it; they lacked presence of\nmind to repel its first inroads. And they also lacked, there can be\nno question, that feeling of personal attachment to the sovereign\nwhich would have prevented their standing by, tame witnesses of his\ndethronement. \"Louis Philippe,\" says Mr Stahr, \"never knew how to\ninspire an earnest and cordial attachment even in those nearest his\nperson. The circumstances of his fall are the most speaking proof of\nthis. His own panegyrist tells us that Louis Philippe himself had a\nmisgiving that none loved him for his own sake. He often said to his\nmost confidential attendants: 'You serve me faithfully, but not with\nthe zeal and warmth which distinguished the servants of Napoleon.\nTheir devotion to his person was unbounded.' If such was the case\nin the French king's prosperous days, what could he expect in the\nhour of adversity? M. Tirel himself proves, beyond the possibility\nof refutation, that, when the moment of danger arrived, the nearest\npersonal attendants of the king thought, almost without exception,\nonly of themselves. Not one of them troubled himself about the safety\nof the immense sums contained in the treasury of the Tuileries. None\nthought of holding in readiness the necessary means of travelling,\nin the possible case of the departure or flight of the king and his\nfamily; and even M. Tirel exclaims, with reference to this--'It is\ndifficult to credit such utter want of foresight, when they knew they\nwere standing on a volcano.'\" At Neuilly, as already mentioned, the\nnational guard refused to turn out; whilst the servants of the royal\nresidence busied themselves in saying their own things, leaving their\nmaster's property to be pillaged and burned by the rabble, with whose\ndisgusting and disgraceful depredations the troops of the line did not\ninterfere. Regulars and militia, domestics and mob, the same want of\nfeeling was manifest in all; none showed attachment or devotion to the\nprince, whose star was on the decline. Mr Stahr made a pilgrimage to\nNeuilly, and devotes a letter to it. It was a grey, sad-looking autumn\nafternoon, and the road was silent and deserted along which he took his\nway to the favourite residence of the departed king. The impression\nmade upon him was most melancholy. \"_Vous verrez de belles choses_,\"\nsaid the porter at the lodge, as he pointed out to the Germans the\nway to the ruins. \"Up to this time,\" says Mr Stahr, \"nothing in Paris\nhad reminded me that here had raged, but a very few years before, the\nhurricane of a revolution that shook the world, and that had swept a\ndynasty from the soil of France like chaff from the thrashing-floor.\nAt Neuilly I first received this impression. They made clean work of\nit, those bands of incendiaries of the 28th February 1848. A single\nnight sufficed to convert that stately building, and all its splendour,\ninto a heap of hideous ruins.... High grass now grows upon the floors\nof the state apartments of the destroyed king's-home. Bushes spring up\naround the columns, over which creepers luxuriantly twine; and the red\npoppy and the yellow king-cup wave their blossoms in the chambers and\nsaloons in which, so short a time ago, the ruler of proud France paced\nhis Persian carpets, revolving plans for the eternal consolidation of\nhis dynasty! On the ravaged foot-paths before the windows, the melted\nglass of the magnificent panes has flowed down and formed a brilliant\nflooring. At the foot of a balcony, whose pillars still supported the\nremains of broken beams, a flush of pale pink harvest roses exhaled\ntheir delicate fragrance. It was an incredibly melancholy sight. The\nclosely-locked doors and shattered windows of the wing that was saved\nincreased the gloom of the whole impression. Everywhere the tall iron\nlattice-work, and the iron posts supporting lamps, are rent and broken;\nthe statues on the flights of steps are shivered to pieces; there\nremain but a couple of colossal sphinxes, which gaze inquiringly out\nof the dark green of the shrubbery. Who shall solve their riddle--the\nriddle of the history of France and of mankind? Louis Philippe, wise\namongst the wise, thought he had done so. Where is he now? His weary\nbones sleep the eternal sleep in the country of the banished kings of\nFrance.\"\n\nNeuilly has become a place of pilgrimage for the friends of the\nfallen dynasty. A host of inscriptions, mostly in an anti-republican\nsense, were to be read upon the walls and pillars at the period of\nMr Stahr's visit. Of several which he took the trouble to copy, one\nonly is superior in tone and significance to the usual average of such\nscribblings. \"High upon a broken column a firm hand had inscribed with\ncharcoal, and in gigantic characters, these three words:\n\n    DROIT DU TALION. 1830. 1848.\n\nOther hands had tried to obliterate the writing, but in vain. The\nrevengeful word 'RETALIATION' was still quite legible. And this\nword best expresses the feeling with which plain-dealing probity\ncontemplates the fate of the overthrown July monarch. For here at\nNeuilly was it that he, a modern Richard III., played the hypocritical\npart of rejecting power, when the blood of the July revolution still\nreddened the streets of Paris. Here was it that he wrote the letter\nto Charles X. in which he assured him of his fidelity and devotion,\nwhen he was already extending a lustful hand towards the crown of the\nrightful heir. Here too, in Neuilly, was it that he spun that Spanish\nweb, whose most secret documents Lord Palmerston carefully preserves,\nand which gave the world a glimpse into an abyss of moral foulness at\nwhich the soul shudders. And here, in presence of this funeral pile\nof his happiness and his splendour--here, before the memorial of his\ndisgraceful and ignominious fall--here, when I called to mind his acts,\nI felt no touch of pity for the fallen King. But the _man_ I did indeed\npity, the husband and the father. He had loved this Neuilly. Here had\nhe enjoyed such a measure of domestic happiness as is rarely vouchsafed\nto a monarch. This house had he, for many a long year, built up and\ndecorated with that fine feeling for art and architecture which was\nproper to him. To this green retirement and solitude, to this remote\ndwelling, hidden from all eyes, he loved to withdraw. Here, where all\nwas his own creation--where no stone was added, no tree planted, no\npath cut, but under his eye--exactly here, in the most sensitive spot,\nthe blow struck him. The destruction of this house was more deeply\nfelt by the _man_ than was the loss of his throne by the _king_!\nBefore the Count of Neuilly had left French ground, the building had\nceased to exist from which he had borrowed the name. And all his\nwiles and stratagems, all his cunning, were as insufficient to avert,\nfrom the man and from the king, this last fated climax, as were the\nfortifications and bastilles with which he had surrounded the dreaded\nParis.\"\n\nQuitting Neuilly, Mr Stahr was startled, as well he might be, by the\nterms of a bill stuck upon the park-gates--\n\n\"House of Orleans, (thus it ran,) chateau and domain of Neuilly _to\nlet_ for three years with immediate possession; about one hundred\nand eighty acres, meadows, forest-land, &c., _bordering on the\nfortifications_!\"\n\nWandering through the endless galleries of Versailles, Mr Stahr is\nnaturally enough led to reflect how strange it is that Louis Philippe,\nthe Napoleon of Peace, as his flatterers called him, and as he loved\nto hear himself called--the man whose motto, as his enemies constantly\nasserted, was \"Peace at any price,\" and who avowedly and upon principle\ndisliked war--should have devised and carried out the plan of a\nnational gallery of French military fame. A merciless analyser of the\ncitizen king's secret thoughts and motives, Mr Stahr declares this\ngallery to have been a speculation of \"the crowned shopkeeper,\"--a\nspeculation by which his dynasty was to gain strength at the expense\nof a national weakness. There is truth in this; but, at the same\ntime, the professor's opinion must not here be accepted as impartial\nevidence. He is evidently led into unusual fervour by his holy horror\nof war. We suspect him of being a member of the Peace Congress--to\nwhich he in one place kindly alludes, as the humble commencement of\na great movement. Like many other adherents of the political sect\nwhich proposes to itself an aim that could never possibly be attained\nwithout terrible convulsions and sanguinary conflicts, he cannot abide\nthe sight of blood, shudders at wounds, and recoils in terror and\ndismay from the \"slaying and murdering, singeing and burning, cutting\nand stabbing,\" depicted upon the walls of the Versailles gallery. He\nlooks not lovingly upon this pictorial history of France, sketched\nfrom her battle-fields, and including the exploits of her innumerable\nwarriors, from Clodwig down to Bugeaud. On the other hand, he curiously\nand eagerly examines the pictures illustrating the events of 1830 and\nLouis Philippe's accession. Of the battle-pieces he has set down some\n(and not altogether without reason) as mere daubs, which no one would\nglance at twice but for the sake of the subject. When surveying the\nillustrations of the July revolution, he forgets artistic criticism\nin his satirical account of the personages that fill the canvass, and\nespecially of the chief actor in those scenes, Louis Philippe himself.\n\"His arrival at the Palais Royal,\" says the rancorous professor, \"has\nsomething sneaking about it. He is profusely adorned with tricolor\nribbons, wears white trousers, a brown coat, and a round hat. He\nlooks like a rogue who has just crept into another man's estate.\nBut characteristic above all is the picture in which he signs the\nproclamation naming him Lieutenant-General of the Kingdom (July 31,\n1830.) The figures are the size of life, all in plain clothes, and\ndecorated with the tricolor. They sit round a green council-table,\nthe coming Citizen-King in the midst of them, dressed in a brown coat\nwith a black velvet collar and a black satin waistcoat, a large fine\nshirt-frill, a neatly tied white cravat, his hair carefully curled, his\neyes half closed, the corners of his month lugubriously drawn down.\nHe holds up the momentous sheet of paper, close above which the pen\nin his right hand hovers, and seems to ask those around him--'Ought\nI then?--must I?' All eyes are fixed trustingly upon him, especially\nthose of honest Laffitte, in the corner on the left. Sebastiani looks\nsomewhat keener and shrewder. Never in my life did I see a picture that\nso perfectly represents an assemblage of Jew bankers, gathered round\ntheir leader to advise on a 'bull' or 'bear' speculation. The whole\nparty have this Jewish calculating expression--Louis Philippe more\nthan any of them. And this is the countenance the man has himself had\nperpetuated! It is a strange historical irony. All the old Bourbons,\neven the two last Louises and Charles X. looked noble, or at least\nlike noblemen, in the expression of their features, compared with this\nessentially common physiognomy. Their faces, at any rate, expressed\nthe decided and undeniable consciousness of high descent, whilst the\npredominating expression in Louis Philippe's countenance is that of\na cunning shopkeeper. And this expression is everywhere the same, in\nall the pictures, &c. &c.\" There is more in the same strain. Some may\nbe disposed to quarrel with Mr Stahr for pressing so hard upon a dead\nman; but, living or dead, kings are fair subjects of criticism; and,\nunsparing and savage as are often the professor's strictures on the\ncharacter and policy of Louis Philippe, they yet are the most truthful\nand just of all the political portions of his book. Messrs Montalivet\nand Miraflores, and the other unscrupulous panegyrists of the late\nKing of the French, would have too good a game left them if it were\nforbidden to reply by more exact and impartial statements to their\nexaggerated encomiums.\n\nPassing from the deceased sovereign to his family, we are led to an\napparently remote subject--namely, Mr Stahr's visit to Alexander Dumas,\nwho, as is well known, was a favourite and intimate of the dukes of\nOrleans and Montpensier. When reviewing, a few years ago, the Paris\ndiary of a countryman of Mr Stahr's--a gentleman of similar politics\nand equal discretion--we noticed an offensive practice common amongst\nmodern German writers, many of whom, on return from foreign travel,\nscruple not to commit to print the most confidential conversation\nand minute domestic details of persons who have hospitably welcomed\nthem, and imprudently admitted them to intimacy. No consideration of\npropriety checks these impudent scribblers. Delicacy and reserve are\nthings unknown to them. The persons concerning whom they flippantly\nbabble may dwell within a day's railroad of them, and be sure to see\ntheir books--may be equally sure to feel vexed or disgusted by their\nunwarrantable revelations and offensive inferences; no matter, they\nspeak of them as though Pekin were their domicile. As regards the\nradical professor from Oldenburg, we sincerely trust that he may fall\nin, at an early day, with the martial author of the _Mousquetaires_,\nand receive from him, as guerdon for his gossip, a delicately\nadministered _estocade_. We never heard whether Janin chastised Mr\nCarl Gutzkow, either with pen or pistol, for his slipshod and indecent\nchatter concerning him and Madame Janin; but we remember somebody\ndoing it for him in the _Revue des Deux Mondes_, where we suspect Mr\nStahr has a fair chance of being in his turn gibbeted. Here is the\nGerman professor's account of Dumas's personal appearance and private\nresidence. It is a curious bit of miniature-painting. \"In person he is\ntall and powerful; his movements, once unquestionably very flexible,\nare now characterised by an easy negligence. His bright complexion\nand large prominent light-blue eyes contrast with the mouth and nose,\nwhich betray his African origin. Good-nature, and a combination of\nintellectuality with a keen relish of life, are the most prominent\ncharacteristics of his broad round face. His thick woolly hair, now\nall but grey, seems to have been formerly light-. He sits in\na very large room on the first floor of the last house in the Avenue\nFrochot. His apartment is reached through a dark corridor. On the side\nthat looks out upon the very quiet street, is a glass gallery, which\nserves as a greenhouse. There was nothing remarkable in it. Mignonette\nand heliotropes were growing in the tubs in which a few large oleander\nbushes were planted. Of the magnificent tropical vegetation of which\nreport has spoken, there was no sign. The room was decorated, and\ndivided into two parts, with brown woollen hangings. In the largest\ndivision, into which visitors are conducted, and in front of the\ngreenhouse windows, stands a vast writing-table. Ancient and modern\narms deck the walls. But of Oriental luxury there was not the least\nappearance. And some other apartments through which he afterwards\ntook me, to show me his winter reception-room, were by no means so\nluxuriously fitted up as has been reported in Germany.\"\n\n\"I found his bookseller with him. 'Look well at the man,' said Dumas,\n'who pays to one author a hundred thousand francs a-year. Such men are\nnot to be seen every day.' Notwithstanding this little bit of brag, I\nhear that his finances are in no very brilliant state, and that the\nfailure of his _Theatre Historique_, especially, threatens him with\nheavy losses. In the course of conversation, he humourously complained\nof the total absence of repose in his laborious existence, of which we\neasy-going, comfortable, German authors could scarcely form an idea.\nSo many newspapers, a theatre of his own, the contract-romances, and\nthe stipulated dramas--truly, it amounts to a considerable total.\nOn subsequent visits, I never found his room and antechamber free\nfrom a throng of visitors--booksellers, printers, managers, actors,\nsecretaries, and others--all of whom he knew how to despatch with great\nrapidity, and without interrupting the thread of our conversation for\nmore than a few moments at a time.\" Conversations with so lively and\nversatile a genius as M. Dumas, turned, as may be supposed, on a vast\nvariety of subjects, but that of which Mr Stahr has given us most\ndetails related to the ex-royal family of France. \"In a side-room he\nshowed us some very pretty pen-and-ink drawings--hunting subjects, by\nthe late Duke of Orleans. This gave him opportunity to speak of his\nhigh respect for the mental endowments of the prince, with whom it is\nwell known that he was on a footing of intimacy. 'He had wit enough for\nten,' said Dumas. 'When we were five or six _hommes d'esprit de Paris_\ntogether,' added he, with amusing _naivete_, 'it was quite impossible\nto distinguish which was the prince and which the wit. The prince\nwas the incarnation of French _esprit_, and of the Parisian-French\n_esprit_, which includes all possible qualities. Her inability to\nunderstand and appreciate this _esprit Parisien_ was a drawback upon\nthe domestic happiness of the Duchess of Orleans, notwithstanding her\nmany excellent qualities. Her heavier German nature did not harmonise\nwith her husband's light elastic disposition. It put her beside\nherself when he transgressed in the presence of a third person the\nrules prescribed by the etiquette of little German courts.' Dumas told\nsome interesting examples of this--examples, however, not adapted for\npublication, as they related to the prince's private life. The Duke of\nOrleans foresaw a revolution, in a republican sense, as a consequence\nof his father's system. His testamentary arrangements with respect to\nthe education of his son were all made in anticipation of such an event\ncoming to pass. In any case, he wished his wife to have nothing to do\nwith the government of the country. The passage of his will relating\nto this point is conceived quite in the spirit of the words with which\nHomer's Telemachus consigns his mother Penelope to the society of her\nwomen. 'If, unhappily, the king's authority could not watch over my\nson until his majority, Helen should prevent her name being pronounced\nfor the regency. Leaving, as it is her duty and her interest, all the\ncares of government to virile hands, accustomed to handle the sword,\nHelen should devote herself entirely to the education of our children.'\nThe Duke of Orleans' death was pregnant with fatal consequences for\nthe dynasty, because he, the most highly gifted of all the old king's\nsons, was perhaps the only one who would have been capable of giving\nthings a different turn in the event of a conflict like the February\nrevolution. He knew his brothers too well not to be convinced that they\nwere unequal to such an emergency. 'Nemours,' said he to one of his\nconfidants, 'is the man of rule and etiquette: he _keeps step_ well,\nand keeps himself behind me with scrupulous attention. He will never\nassume the initiative.' He held the Dukes of Nemours and Aumale to be\nbrave soldiers. Of the Prince de Joinville he said: 'He has a passion\nfor danger: he will commit a thousand acts of brilliant imprudence, and\nwill receive a ball in his breast at the assault of a barricade,'--a\nfate which Joinville escaped in February probably only by his absence\nfrom Paris. 'Now that younger sons are no longer made _abbes_,'\ncontinued the Duke of Orleans, referring to little Montpensier, 'I am\nat a loss to imagine what is to be done with them.'\n\n\"Of none of his sons was the old king more jealous,\" says Mr Stahr,\n\"than of the heir to the crown. Letters found in the Tuileries in\nFebruary 1848 show that he kept him in the strictest dependence, and\nhad spies observing him wherever he was. In the year 1839 the duke\ncomplained 'that he had less power than any private citizen who had\na vote at elections; that he did nothing but the commissions of the\nministers; that everything was in danger, nothing gave promise of\ndurability, and that it was impossible to say what might happen from\none day to the other.' The prince, expressed himself thus whilst upon a\njourney, in a confidential circle of officers of rank. Two days later\nhis words, set down in writing, were in the hands of the king. The\nsurprising irresolution and want of presence of mind displayed by the\nother princes in the hour of danger, can only be accounted for by the\nslavish dependence in which the old monarch had kept them.\"\n\nAlthough easy and affable in his intercourse with his friends, a\ncertain jealous vigilance with regard to the respect due to his\nrank formed a feature in the character of the Duke of Orleans. The\nanecdote told to Mr Stahr by Dumas, as an illustration of this trait,\ncan hardly, however, be admitted to prove undue susceptibility,\nbut rather the prince's consciousness that his house stood upon an\nunstable foundation. It was at a hunting-party at Fontainebleau.\nThe chase was very unsuccessful. The Duke of Orleans turned to an\nItalian nobleman, to whose family Louis Philippe had obligations of\nancient date, and who on that account was on a friendly footing at\ncourt. \"Well! Monsieur de--,\" said the duke, \"how are we hunting\nto-day?\" \"Like pigs, Monseigneur, (_comme les cochons_,)\" was the\nItalian's coarse reply. The duke, evidently annoyed, said to Dumas:\n\"And you believe our monarchy possible, when a _De_ ... dares thus to\nanswer the heir to the throne?\" Mr Stahr was interested to find that\nDumas, notwithstanding his monarchical friendships and associations,\nbelieved in the necessity and durability of the republic. \"It seems,\"\nsaid the ingenious and versatile author of _Monte Christo_, \"as if\nProvidence had resolved to let us try all manner of monarchies, in\norder to convince us that not one of them is adapted to our character\nand condition.\" Then he gave his auditors a detailed sketch of all\nthe French monarchies previous to the Revolution of 1789. \"Since that\nRevolution,\" he went on, \"we have had the monarchy of Genius: it lasted\nten years. We have had the restoration of the monarchy of _esprit_ and\nchivalrous gallantry: it lasted fifteen years; and was succeeded by the\ncitizen-monarchy, which lasted eighteen. What would you have us try\nnow? This republic is bad. But a child in swaddling-clothes matures\ninto a man.\" Sensibly enough spoken for a romance-writer, indulgently\nremarks Mr Stahr, who is always glad to obtain a suffrage in favour\nof republican institutions. We attach the same degree of value to M.\nDumas's political vaticinations as to his Frenchified _rifaccimenti_\nof Shakspeare's plays. Shakspeare in French, as Mr Ford remarks in\nhis Spanish Handbook, \"is like Niagara passed through a jelly-bag.\"\nA miracle of degradation which reminds us to turn to a scornful and\nindignant chapter suggested to Mr Stahr by a certain Monsieur Michel\nCarre's version of Goethe's _Faust_, performed at the _Gymnase_\ntheatre. \"Goethe is unknown in France,\" says the Countess d'Agoult,\none of the few competent French appreciators of German literature, in\nher _Esquisses Morales et Politiques_. Nothing, according to Mr Stahr,\ncould be better fitted to confirm and perpetuate French ignorance of\nthe great German than such dramas as that which he painfully endured at\nthe Gymnase. According to Madame d'Agoult, her countrymen will not take\nthe trouble to study Goethe. To do so they must first learn a language.\n\"Why did he not write in French? He has only what he deserves, after\nall. How is it possible to be a German?--(_comment est on Allemand?_)\"\n\"If this is not exactly out-spoken,\" says Madame d'Agoult, \"it is\nat least privately thought in a country where _the arrogance of\nignorance_ attains proportions unknown to other nations.\" \"_La superbe\nde l'ignorance,\" \"der Uebermuth der Unwissenheit!_\" cries Mr Stahr\nin an ecstasy: \"I kiss the fair lady's hand who wrote the word, for,\nwithout it, I should never have hit upon the appropriate term for this\nnewest French atrocity of M. Michel Carre, perpetrated upon the most\nprofound work of German genius. I am not without experience of the\ntheatrical sufferings of our day; but such torture as was yesterday\ninflicted, at the Gymnase theatre, upon every German fibre in our\nframes, I never before in my whole life witnessed or underwent. I was\nprepared for little that was good, and for much that was laughable; but\nmy expectations and fears were surpassed to an extent it was impossible\nto anticipate. Marsyas flayed by Apollo is no very pleasing picture,\nbut the Belvidere Apollo flayed by a Marsyas is a spectacle which it\ntakes all the nerve of German critical observation to endure.\" Mr Stahr\nthen proceeds to dissect the drama, act by act, and almost scene by\nscene, with considerable acuteness and humour. The specimens of fustian\nhe gives, the execrable French taste he exposes, fully justify the\nintensity of his disgust. The _Gymnase_ drama is evidently worse than\na tame translation; it is an obscene parody of Goethe's great poem. It\nis a compound, as Mr Stahr expresses it, of \"dirt and fire--that sort\nof fire, namely, which is lighted by the brandy-bottle.\" We believe\nit impossible that _Faust_ should ever be done justice to in a French\nversion. But if translators, owing to the want of power of the French\nlanguage, and to the utter absence of affinity and sympathy between\nit and the German, must ever fall to a certain extent, they at least\nmay avoid degrading and distorting the tone and sentiments of the\noriginal. This M. Carre, of whom we now hear for the first time, seems\nto have cultivated his taste and sought his inspirations in the worst\nschool of modern French literature, and in the orgies of Parisian\nrakes. The inference is inevitable from the scenes and passages\ndescribed and quoted by Mr Stahr. As to the verbal spirit and fidelity\nof the translation, the following may serve as a specimen. \"In the\nchurch-going scene, the lines, so charming in the original:--\n\n    'Mein schones Fraeulein, darf ich wagen,\n    Arm und Geleite anzutragen?'\n\nare thus rendered in M. Carre's French: _Oserai-je, Mademoiselle, vous\noffrir mon bras, pour vous conduire jusqu'a chez vous?_ For Gretchen's\nexquisitely graceful and saucy reply--\n\n    'Bin weder Fraeulein, weder schoen,\n    Kann ungeleitet nach Hause gehn!'\n\nwhich so completely captivates Goethe's Faust, this Frenchman has been\nable to discover no better equivalent than, '_Pardon, Monsieur, je puis\nfort bien rentrer seule a la maison_'--an answer too flat and insipid\neven for a Paris _Lorette_ of the present day.\" Mr Stahr was tolerably\nwell pleased with the bearing of the audience who had come to partake\nof this pitiable French hash. They may have felt a natural curiosity\nto know something about the Faust and Margaret whose acquaintance\nthey had made in the print-shop windows, but their sympathy with the\npiece went no farther. Even the Rose of the Gymnase, the Rose Cheri,\nso cherished by the Parisian public, failed to extract applause as M.\nCarre's Margaret. \"It is very romantic,\" Mr Stahr heard some of his\nneighbours remark, \"but it is a little too German; Monsieur Goethe's\npoetry does not suit the French taste.\" Poor public! Poor Goethe!\nintroduced to each other under such dismal auspices. It must have\nbeen a relief to Mr Stahr to quit this miserable travesty, and turn\nto the native drama; although even by this, judging from a letter on\ntheatrical subjects addressed to his friend Julius Mosen, he does not\nappear to have been much gratified. \"I know not,\" he says, \"whether\nmy taste for theatres is gone by, or what is the reason, but as yet I\nhave been scarcely half-a-dozen times to the play. Beginning with the\n_Theatre Francais_, I might place as a motto at the beginning of this\nletter the words of Courier: 'The fact is that the Theatre Francais,\nand all the old theatres of Paris, the Opera included, are excessively\nwearisome.' To be sure, Rachel is not here. She is gathering laurels in\nGermany; and when I complained confidentially to an acquaintance that\nthe tragedy of the Theatre Francais did not move me, he endeavoured to\nconsole me by telling me of Madlle. Rachel, and of her speedy return to\nParis. She stands alone, incomparable, a phenomenon. But the phenomenon\nis absent; and the Paris stage is consequently darkened. It is always\na bad sign for the condition of an art when it thus entirely depends\nupon one of its professors.\" Mr Stahr was better pleased with the\nlively performances at the four _Vaudeville_ theatres, and gives an\namusing analysis of _La Fille bien gardee_, the little one-act piece\nwhich, for many weeks of last year, nightly drew crowds to the _Theatre\nMontansier_. It belongs to a class of dramatic trifles in which French\nplaywrights and actors are perfect and inimitable; trifles which only\ngrow upon French soil, and will not bear transplanting.\n\nAfter his savage attacks upon Louis Philippe and the French President,\nit would be quite out of character if Mr Stahr--who evidently bears\nmonarchy a grudge, and will tolerate no government that can possibly\nbe identified with the cause of order--had not a fling at Henri Cinq.\nPerhaps it is because he deems the Legitimist interest less formidable\nto his views than the Orleanist or Buonapartist, that he adopts a\ndifferent mode of attack, and exchanges ferocity for raillery. The\nGerman tongue being but indifferently adapted to the lighter manner\nof warfare, he glides into French, in which language he writes nearly\na whole chapter. Stepping one day into a hair-dresser's rooms, he\nwas so fortunate as to come under the hands of the master of the\nestablishment, an eager politician and a red-hot Legitimist, voluble\nand communicative as only a Frenchman and a barber can be. With the\nvery first clip of the scissors an animated conversation began, which\nMr Stahr has set down so far as his memory serves him, although he much\ndoubts that his pen has conveyed all the minuter comical touches of\nthe dialogue. This began with the usual exordium of Frenchmen of all\nclasses since the revolution--\"You, Monsieur,\" said the man of wigs,\n\"are a foreigner, and consequently uninterested in our quarrels. Tell\nme what you think of our situation?\"\n\n\"I think,\" replied I, \"that the President will never willingly resign\npower.\"\n\n\"But, Monsieur, what is to be hoped for from such an _imbecile_?\"\n\n\"I do not say he will succeed; I say he will make the attempt.\"\n\n\"And _I_ say that he will fail. Henry the Fifth for me! _a la bonne\nheure!_ There is a man for you.\"\n\n\"What do you know about him? You are very anxious, then, to make\ntonsures?\"\n\n\"What do I know about him? But, Monsieur, I have seen him, I am\nacquainted with him, I have spoken to him, and I tell you he is a\ncharming man!\"\n\n\"Where did you see him?\"\n\n\"Did I not go to see him at Wiesbaden! Sir, there were thirty-nine\nof us--workmen, we called ourselves, though we were all masters--who\nwent of our own accord to pay our respects to Henry V. The thing was\nbriskly done, I beg you to believe. I spoke to him as I speak to you,\nsir, at this moment. At first I was received by M. de la Ferronnaye,\nhis aide-de-camp. 'Good morning, Monsieur R.,' said he, 'how do you\ndo?'--'Very well, sir, I thank you,' answered I; and far from making me\nwait whole hours at the door, like those republicans of the _Veille_,\nhe made me sit down beside him on the sofa, as affable as could be.\"\n\nMr Stahr inquired of the worthy coiffeur what had been the motive of\nhis journey to Wiesbaden, which he seemed to look upon as a sort of\nNorth Pole expedition, and of whose fatigues and privations he drew a\nvivid picture. He wished to judge for himself, he said; to see whether\nthe rightful heir to the throne was as ill-favoured as his enemies\nrepresented him to be. He found him, on the contrary, full of amiable\nqualities. He was a little lame, but his smile was irresistible.\nWarming with his subject, the enthusiastic Henriquinquist asked his\ncustomer's permission to relate all the particulars of his reception at\nWiesbaden. This was just what Mr Stahr wished, and he duly encouraged\nhis interlocutor.\n\n\"On our arrival,\" continued the hairdresser, \"we presented ourselves\nto the aide-de-camp, as I have had the honour of informing you. He\ntook down our names, and gave us each a number of rotation, according\nto which we were arranged in the afternoon at the general audience. We\nwere formed in three ranks. The prince was informed beforehand of the\nname and trade of each number, so that he was able to address a few\nwell-chosen words to everybody. When we were all drawn up in order, he\ncame in, placed himself in the midst of us, at a few paces distance,\nand addressed us. 'Good day to you, my friends,' he said: 'believe me\nwhen I say that I am most sensible of the mark of sympathy you have so\nspontaneously given me, by quitting your families and occupations, and\nundertaking a journey into a foreign country to see and console me in\nmy exile. Be sure that I will never forget what you have done for me.'\nThen he said, 'Come nearer, my friends!' We advanced a step. 'Nearer\nyet, my friends. You come from too far not to come nearer! I hope to\nsee you all at eight o'clock to-night!'\"\n\nThe hairdresser acted this scene as he related it, addressing himself\nand Mr Stahr alternately as the prince, by whose mandate to draw a step\nnearer he was evidently vastly flattered. The professor, immensely\namused by the performance, still fancied he saw that the main cause of\nthe fascination which Henry V. had exercised upon his devoted adherent\nwas still undivulged. The sequel showed that he was not mistaken.\n\n\"In the evening,\" continued the coiffeur, \"we returned to the Prince's\nresidence; there we partook of refreshments, and the Prince had an\namiable word for each and all of us. He talked about the state of\naffairs in France, and wished to know all our opinions of it. The\nnext day some of us were received in private audience. I was of the\nnumber. But as we were numerous, and the Prince was very busy, I\ncould not have much conversation with him. However, he gave me a\nsilver medal, and--'Mr R.,' said he, 'have you a comfortable bed at\nyour hotel?' 'Monseigneur,' I replied, 'since you deign to ask the\nquestion, I am accustomed to sleep between two sheets, and as I do\nnot understand a word of German, I have been unable to make them\nunderstand this at my hotel. They put the sheet sometimes over and\nsometimes under the blanket, but never more than one.' Sir,\" continued\nthe delighted barber, addressing himself to Mr Stahr, whilst his face\nbeamed with triumph, \"_that night I had two sheets upon my bed_. Could\nanything be more amiable? Ah, sir, I have seen them from very near,\nthose republicans of the Mountain, those members of the Provisional\nGovernment!--what blockheads! what boors! They aspired to command, and\nin their whole lives scarcely one of them had had as much as a servant\nat his orders! Sir, it was pitiable to behold.\"\n\nMr Stahr observed to the loyal hair-curler that he had seen the persons\nin question only _after_ they had attained power, and that there are\nfew more amiable people in the world than a pretender, _before_ he has\ngained his object. He thought it possible that, once at the Tuileries,\nHenry V. might show himself in a less agreeable light, and trouble\nhimself less about his adherent's bedlinen. The barber's sensible reply\ndid him honour. But barbers, from Don Quixote's day downwards, have\nbeen men of good counsel.\n\n\"Monsieur,\" said the coiffeur, \"I am not a fool. Do you suppose I shall\ngo and plague him, when he is king? He will have other matters to\nmind then. I have no pretensions to be made minister or prefect, when\nthere are people who have studied those things all their lives. I am a\nhairdresser, and I shall remain one. But _I want to dress a great deal\nof hair_, and under the republic I dress none.\"\n\n\"But,\" remarked Mr Stahr, \"you dress more under the President.\"\n\nThe barber, however, was no admirer of the President, whom he had also\nbeen to see, before his election, and upon the appearance of whose head\nhe passed a most unfavourable opinion. He was sulky, he said, and not\nconversable. The affairs of France could never go on well under a man\nwho knew not how to talk. Moreover, nothing could suit him but Henry\nV. He was neither Buonapartist nor Republican. But when things were at\nthe worst, he said, his cry had always been \"_Vive la France!_\" \"Stick\nto that!\" said a customer who just then stepped in. \"France has a tough\nexistence, and will outlive your Henri Dieudonne and all his kin, and\nthe President to boot. And now have the goodness to curl my hair.\"\n\nWhether fact or invention, this sketch has one truthful point: it\ngives a sound enough notion of the manner of reasoning of the French\nshopkeeper and _petit bourgeois_--a numerous and weighty class, without\nwhose concurrence no state of things can long be permanent in France.\nWith them the whole question, since they first awoke from the shock\nand folly of the February revolution, has been one of _two sheets on\ntheir bed_ and _more hair to dress_. They will support any government\nunder which they can sleep in peace and drive a good trade. Some of\nour readers will not have forgotten the sufferings and fate of poor\n_Monsieur Bonardin_.[4] The disasters and commercial depression of\n1848 were a severe but perhaps a wholesome and necessary lesson to\nmany thousands of Frenchmen. Unfortunately, as illustrated in M.\nBonardin's case, the lesson was given to many who neither required\nnor deserved it. Wandering near Versailles, in the pleasant valley of\nJouy, Mr Stahr and his companions were invited by a friendly dame,\nwhose acquaintance they had made in the omnibus, to walk into her\nhouse and taste her grapes. She perhaps thought the object of the\nforeigners' pedestrian stroll was to purchase one of the pleasant\ncountry houses, surrounded by vineyards and orchards, which there\nabound; for she took them all through her kitchen-garden and vineyard,\nand through the copse of chestnuts and hazel bushes, to the fish-pond,\nand to the pleasant grotto, fitted up as a chapel, and even to the\nvine-dresser's cottage, from whose windows a lovely view repaid the\nascent of the numerous terraces. During this tour of inspection the\ngood lady's tongue was not altogether idle, and a melancholy page\nout of a Paris citizen's life was laid open to the Germans' eyes.\nThe pleasant little domain they were rambling over was the fruit of\nfive-and-twenty years' toil. \"Monsieur Cendrell, a skilful gilder, had\nbought it a few years before the last revolution, and had laid out\nconsiderable sums in building and embellishment. The revolution broke\nout just as he had given up his business to a friend and assistant. He\nsuffered heavy losses, and was now compelled, in spite of the general\ndepreciation of all landed property, to part with his little estate.\nIt was to be had for only thirty thousand francs, as it stood--garden\nand vineyard, dwelling-house and garden-cottage, shady copse, and pond\nwell stocked with carp, and right of shooting over I know not how\nmany acres. And how neatly and comfortably arranged was the house,\nwith its bath and billiard-rooms, and its library with portraits of\nLouis Philippe and the Count de Paris--how cleanly kept was every room\nfrom the kitchen to the attics, the gardener's house and the stable\nincluded! There was nothing wanting, but--thirty thousand francs to\nbuy it with, and as much more to live there quietly till the end of\none's days. We sat full half-an-hour in the cottage on the hill,\nrefreshing ourselves with the sweet grapes that clustered round the\nwindows of the rush-matted room, whilst the kindly Frenchwoman told\nus her story. It is that of thousands of her class in Paris since the\nFebruary revolution. Truly it grieved us, both for her sake and our\nown, that we could not purchase the pleasant country house.\" This, it\nwill be said, is a common-place incident. There is certainly nothing\nin it very striking or dramatic. Every day somebody or other suffers\nlosses, and is compelled to reduce his establishment, or to put it down\naltogether; to sell his last acre of sunny meadow and vineyard, and\ntoil in an obscure lodging for daily bread. But there will be found in\nthe picture something deeply affecting, if we suffer the mind to dwell\nupon it for a moment, recalling, at the same time, the well-known fact\nreferred to by Mr Stahr, that, since the dreary days of 1848, the fate\nof the frame-gilder of Jouy has been that of multitudes of others who,\nlike him, had passed a laborious manhood in earning, for their old\nage, a competency and a right to repose. Thus we obtain a glimpse of\na mass of misery, of domestic happiness broken up, if not destroyed,\nof hallowed associations rudely ruptured--by no fault of the victims,\nbut as a melancholy effect of the obstinacy of a selfish king, and of\nthe rashness and precipitancy of a section of his subjects. But these\nmaterial evils, deplorable as they are, sink, in our opinion, into\ninsignificance, contrasted with the moral results of the last most\nill-omened French revolution. These strike Mr Stahr in a very different\nlight. The early part of the month of October was passed by him at the\npretty village of Loges, near Versailles, whither he went to enjoy the\nbeautiful scenery and the mellow autumnal weather, and to escape for\na few days from the whirl and rattle of Paris. In the course of his\nwalks, he and his friends not unfrequently visited a little rural inn\non the way to Jouy, kept by a corpulent but active dame, who usually\nfavoured them with her society and conversation, whilst they consumed\na glass of her country wine and a slice of her _fromage de Brie_.\nShe read no newspapers--none were received in her modest tavern--and\nknew but little of the intricacies of her country's dissensions; but\nshe had political notions of her own, and was a warm republican. \"We\nFrench,\" said she to Mr Stahr, \"soon get tired of governments. They\nhave driven away all that have been chosen since Napoleon; and when\nthey were driven away the consequence always was a terrible shock,\naffecting all kinds of property. Now, in a republic, there is no one\nperson to drive away with so much clatter, and that is why, for my\npart, I desire neither a Napoleon nor a king.\" \"Query,\" exclaims Mr\nStahr, \"whether the woman is so much in the wrong? For my part, from no\nFrench politician have I yet heard a more striking remark with respect\nto the present circumstances of France. That France has no longer any\nking, any family ruling her by right divine, that is the chief thing\nwon by the February revolution. The dynastic and monarchical illusion\nis completely eradicated from the people's mind, never again to take\nfirm root.\" This prospect, in which the German radical exults, we,\nas staunch upholders of the monarchical principle, should of course\ndeplore, did we attach any value to his predictions. But, after what\nhas passed, we think anything possible in France, and should be no\nmore astonished at a Bourbon restoration, than at a consolidation of\nthe republic; at Joinville's presidency, than at Louis Napoleon's\nre-election. It needs more temerity than judgment to hazard a prophecy\nconcerning what will or will not take place in a country which, as far\nas politics go, has become, above all others, _le pays de l'imprevu_.\nThe title used to belong to Spain; and in the years of Continental\ntranquillity that preceded 1848, it was amusement for unoccupied\npoliticians to watch the unforeseen crises constantly occurring in the\nPeninsula. It is infinitely more exciting to wait upon the caprices of\na great and powerful country, whose decisions, however unreasonable,\nmay influence the state of all Europe. They can but be waited upon,\nthey cannot be foretold. Since the memorable 10th of December 1848,\nthis has been our conviction. Before that date there was at least a\ncertain logical sequence in the conduct of the French nation. Although\noften impossible to approve, it had always been possible to account for\nit. But the common sense of Europe certainly stood aghast when Louis\nNapoleon Buonaparte was elected ruler of France, by a majority so great\nas to attach a sort of ridicule to the petty minorities obtained by men\nwho, in ability and energy, and, as far as two of them were concerned,\nin respectability, were infinitely his superiors. At that period, Louis\nNapoleon had never given one proof of talent, or rendered the slightest\nservice, civil or military, to the nation that thus elected him its\nhead. Twice he had violated, by armed and unjustifiable aggression,\nresulting in bloodshed and disgrace, the laws of his country. Pardoned\nthe first time, on a pledge of future good conduct, he took an early\nopportunity of forfeiting his word. Notwithstanding the stigma thus\nincurred, four districts, when universal suffrage became the law of\nFrance, elected him their representative to the National Assembly. This\nmay not be worth dwelling upon. There were stranger elections to the\nAssembly than that, after the February revolution. But when, out of\nseven millions of voters, five and a half millions gave their voices\nto a man whose sole recommendation was a name,--then did wonder reach\nits perigee. And thenceforward bold indeed must be the politician who\nattempts to foreshadow the possible whims of the fickle people of\nFrance.\n\n[4] _Blackwood's Magazine_ for December 1848.\n\n\n\n\nTHE SUBMARINE TELEGRAPH.\n\n\nWho will venture to make catalogue of the possible results of the\n\"Submarine Electric Telegraph?\" The more we meditate, the more new\nwonders open before us. We are running a race with Time; we outstrip\nthe sun, with the round world for the race-course.--Yet, let us not\nboast: we do not run the race, but that more than a hundred million\nhorse-power invisible to us, which was created with the sun. We are but\nthe atoms involved, and borne about in the secrets of nature. And the\nsecrets--what know we of them?--The facts only of a few of them: the\nmain-springs of their action are, and perhaps ever will be, hidden. The\nworld progresses; it has its infant state, its manhood state, and its\nold age--in what state are we now? and what is the world's age? Madame\nde Stael considered it quite in its youth--only fifteen--scarcely\nresponsible! It seems, however, making rapid growth. Is it past\nthe conceited epoch, and now cutting its wise teeth? We stand like\nspectators at the old fair-show; we see the motley, the ever busy, ever\nrunning harlequin and columbine; we are astonished at the fooleries,\nand are amazed at the wit, the practical wisdom, the magical wand power\nof the fantastic descendants of Adam and Eve, the masculine, and the\nfeminine; and we laugh to behold the shuffling step of old Grandfather\nTime, as--\n\n    \"Panting Time toiled after them in vain.\"\n\nIt is through the agency of mind that a few secrets are disclosed to\nus, and for our use. We call the recipient and the inventor Genius. It\nis given, as it is wanted, at the right time, and for the preordained\npurpose. We are sceptical as to \"mute inglorious Miltons.\" Where the\ngift is bestowed it is used; and if it appear to be partially used,\nit is where partially given, that one man may advance one step, his\nsuccessor another--and thus invention leads to invention. Genius for\none thing arises in one age, and sleeps after his deed is done; genius\nfor another thing succeeds him. Who shall dare to limit the number? One\nthing only we pause to admire--how seldom does the gift fall upon bad\nmen!\n\nThere have been, perhaps, those who have had thrown in upon their minds\na wondrous vision of things to come, which they were not allowed,\nthemselves, to put forth in manifest action to the world. There have\nbeen seers of knowledge; and, perhaps, prophesiers in facts. No one\nwill credit the assertion, therefore we make it not, that thousands\nof years ago steam was known, and applied to the purposes of life. We\ncall, then, certain records the prophecies of Facts; that is, there was\na certain practical knowledge, which in its description is prophetic of\na new knowledge to be developed. Semiramis set up a pillar on which it\nwas written, \"I, Semiramis, by means of iron made roads over impassable\nmountains, where no beasts [of burthen] come.\" Did Semiramis prophecy\na railroad--or were there Brunells and Stephensons then? When Homer\nspake, of the ships of the Phaecians, how they go direct to the place\nof their venture, \"knowing the mind\" of the navigator, \"covered with\ncloud and vapour,\" had the old blind bard a mind's-eye vision of our\nsteam-ships? Many more may be the prophecies of Facts; for in these\ncases doubtless there were facts, the prophecy being in the telling.\n\nBut there have been visions also without facts--that is, without the\npractical visions of an inward knowledge--wherein nature had given\na mirror and bade genius look into it. Friar Bacon's prophecy is an\nexample.\n\n\"Bridges,\" says he, \"unsupported by arches, can be made to span the\nfoaming current; man shall descend to the bottom of the ocean safely\nbreathing, and treading with firm step on the golden sands never\nbrightened by the light of day. Call but the secret powers of Sol and\nLuna into action, and behold a single steersman, sitting at the helm,\nguiding the vessel which divides the waves with greater rapidity than\nif she had been filled with a crew of mariners tolling at the oars.\nAnd the loaded chariot, no longer encumbered by the panting steeds,\ndarts on its course with relentless force and rapidity. Let the pure\nand simple elements do thy labour; bind the eternal elements, and yoke\nthem to the same plough.\"\n\nHere is poetry and philosophy wound together, making a wondrous chain\nof prophecy. Who shall adventure upon a solution of that golden chain,\nwhich the oldest of poets told us descended from heaven to earth,\nlinking them as it were together? Was it an electric fluid in which\nmind and matter were in indissoluble union?\n\nWhat prophetic truths may yet be extracted from myth and fable, and\ncome blazing like comets we know not whence, into the world's field!\nHermes \"the inventor,\" what is his wand, serpent-twined, and its\nmeaning, brought into vulgar translation, and seen in the buffoonery of\nharlequinade? of what new power may it not be the poetical prototype?\nWho shall contemplate the multiplicity of nature's facts, and the\nmyriads of multiplicities in their combination? Knowing that all\nthat has ever been written or spoken, in all languages, is but the\ncombination of a few sounds transferred to the alphabet of twenty-four\nletters, or even less, are we not lost in the contemplation of the\npossibilities of the myriads of facts, in their interchangings,\ncombinations, and wonderful dove-tailings?\n\nPerhaps, that we may not know too much before our time, facts are\nwithdrawn from us as others are protruded. Memory may sleep, that\ninvention may awake. Did we know by what machinery Stonehenge was\nbuilt, we might have rested satisfied with a power inadequate to other\nand new wants, for which that power might have been no help. Archimedes\ndid that which we cannot do, in order that we might do that which he\ndid not. Who shall lift the veil of possibility?\n\nOf this we may be sure, as the mind is made inventive, (and there is no\nseeming probability that a faculty once given will be taken away from\nour created nature,) there is a large and inexhaustible store-house,\nwherefrom it shall have liberty to gather and to combine. We do\nnot believe that steam itself, the miracle of our age, is anything\nmore than a stepping-stone to the discovery of another power--means\nsuperseding means. There is and will be no end, as long as the fabric\nof the world lasts.\n\nThere is an old German play, in which the whimsical idea of bringing\nthe Past and Present together in _dramatis personae_ is amusingly\nembodied. We forget the particulars, but we think Caesar or Cicero\nfigure in the dialogue. The ridiculous is their laughable ignorance of\nthe commonest things. The modern takes out his watch and puts it to\nhis ear, and tells the ancient the hour of the day. This is but one\nout of many puzzling new things; but, even here, how little is told of\nthe real post-Ciceronian inventions; for the object of the play is to\nshow the skill of the Germans only; it is but an offering to the German\ngenius of invention.\n\nCould a tale of Sinbad's voyage have been read to the Roman--how, as\nhe approached the mountain, the nails flew out of the ship, for lack\nof comprehension of the load-stone--he would have thought it only\nfantastically stupid; and if he had laughed, it would have been at the\nnarrator's expense. And so, indeed, it has fared with discoverers:\nthey have been before the time of elucidation, like Friar Bacon; and\nsome for fear of ridicule have kept back their knowledge; but not\nmany perhaps; for knowledge, when it is touched by genius, becomes\nilluminated and illuminating, and will shine though men may shut the\ndoor, and stay themselves outside and see it not, while it brightens\nup only the four walls of a small chamber as it were with the magic\nlantern in a student's hand. Whereas it ought, according to its power,\nto gild the universe. The secresy of invention is rather of others'\ndoing--of an envious or doubting world of lookers-on, than of the first\nperceiving genius. Fortunately the gift of genius, as intended for the\nuse of mankind, comes with an expansive desire of making it known.\n\nIf the memory of tradition fails, and some inventions are lost, that\ntheir detail may not hamper the faculty that should take altogether\na new line, so have we what we may term false lines, that yet,\nnevertheless, lead into the true. Science may walk in an apparently\nunnecessary labyrinth, and awhile be lost in the wildest mazes, and yet\ncome out into day at last, and have picked up more than it sought by\nthe way. Wisdom herself may have been seen sometimes wearing the fool's\ncap. The child's play of tossing up an apple has ended in establishing\nthe law of gravitation. The boy Watt amused himself in watching a\nkettle on the fire: his genius touched it, and it grew and grew into\na steam-engine; and, like the giant in the show, that shook off his\nlimbs, and each became another giant, myriads of gigantic machines,\nof enormous power, hundred armed Briareuses, are running to and fro\nin the earth, doing the bidding of the boy observant at his grandam's\nhearth. Is there an Arabian tale, with all its magic wonders, that can\nequal this? We said that Wisdom has worn the fool's cap; true, and\nFoolery was the object--the philosopher's stone; but in the wildest\nvagaries of her thought, there were wise things said and done, and her\nsecretary, Common Sense, made notes of the good; and all was put down\ntogether in a strange shorthand, intelligible to the initiated; and the\nfacts of value were culled, in time, and sifted from the follies, and\nfrom the disguises--for there were disguises, that strangers should\nnot pry into them before the allowed hour. Alchemy has been the parent\nof chemistry--that \"[Greek: episteme iera],\" and its great mysteries,\nto reveal which was once death!! Secrets were hidden under numbers,\nletters, signs of the zodiac, animals, plants, and organic substances.\nThus in the vocabulary of the alchemists, the basilisk, the dragon,\nthe red and green lions, were the sulphates of copper and of iron; the\nsalamander, the fire; milk of the black cow, mercury; the egg, gold;\nthe red dragon, cinnabar. There is a curious specimen, in the work of\nthe monk Theophilus, translated by Mr Hendrie, how to make Spanish\ngold:--\n\n\"The Gentiles, whose skilfulness in this art is probable, make\nbasilisks in this manner: They have underground a house, walled with\nstones everywhere, above and below, with two very small windows, so\nnarrow that scarcely any light can appear through them: in this house\nthey place two old cocks, of twelve or fifteen years, and they give\nthem plenty of food. When these have become fat, through the heat of\ntheir good condition, they agree together, and lay eggs. Which being\nlaid, the cocks are taken out, and toads are placed in, which may\nhatch the eggs, and to which bread is given for food. The eggs being\nhatched, chickens issue out like hens' chickens, to which, after seven\ndays, grow the tails of serpents, and immediately, if there were not\na stone pavement, they would enter the earth,\" &c. &c.--\"After this,\nthey uncover them, and apply a copious fire, until the animals' insides\nare completely burnt. Which done, when they have become cold, they are\ntaken out, and carefully ground, adding to them a third part of the\nblood of a Red Man, which blood has been dried and ground.\"\n\nDoubtless it was the discovery of some such language as this which led\nto the popular belief that the Jews, who were great goldsmiths and\nalchemists, made sacrifices with the blood of children; and many a\npoor Jew suffered for the sin of mistifying knowledge. \"The toads of\nTheophilus,\" says Mr Hendrie, \"are probably fragments of the mineral\nsalt, nitrate of potash, which would yield one of the elements for the\nsolvent of gold; the blood of the Red Man, which had been dried and\nground, probably a muriate of ammonia,\" &c. Such were the secrets of\nthe \"Ars Hermetica;\" and their like may have been bidden in the wand of\nHermes. Dragons, serpents, and toads! Awful the vocabulary, to scare\nthe profane; but fair Science came at length unscathed out of the\nwitches' cauldron: and thus it appeared that natural philosophy, like\nits own toad, ugly and venomous, bore a \"precious jewel in its head.\"\n\nAlchemy and magic were twin sisters, and often visited grave\nphilosophers in their study both together. The Orphic verses and the\nhexameters of Hesiod, on the virtues of precious stones, exhibit the\nsuperstitions of science. They descended into the deeply imaginative\nmind of Plato, and perhaps awakened the curiosity of the elder,\nscarcely less fabulous Pliny, the self-devoted martyr to the love of\ndiscoveries in science. The Arabian Tales may owe some of their marvels\nto the hidden sciences, in which the Arabs were learned, and which\nthey carried with them into Spain. Albertus Magnus, in his writings,\npreserved the Greek and Arab secrets; and our Roger Bacon turned them\nover with the hand of a grave and potent genius, and his touch made\nthem metaphorically, if not materially, golden. His prophecy, which we\nhave given, was, when uttered, a kind of \"philosopher's stone.\"\n\nSuperstitions of science, of boasted and boasting philosophy! And why\nnot? Is there not enough of superstition now extant--a fair sample of\nthe old? Is the new philosophy without that original ingredient? It\nis passed down from the old, and will incorporate itself with all new\nin some measure or other, for the very purpose of misleading, that\nthe very bewilderment may set the inventive brain to work, in ways it\nthought not of. Reasoners are every day reasoning themselves out of\nwholesome, airbreathing, awakening truths into the visionary land of\ndreams, and, speaking mysteriously like uncontradicted somnambulists,\nbelieve themselves to be oracular. Materialists have followed matter,\ndriven it into corners, divided it, dissected it, and cut it into such\nbits that it has become an undiscernible evaporation; and they have\ncome away disappointed, and denied its existence altogether. Thus,\nmesmerism is the bewildered expression of this disappointment, their\nprevious misapprehension. They will not believe that the wand of Hermes\nrepresents two serpents intertwined--they see but one, though the two\nlook each other in the face before them, and they are purblind to the\nwand and the hand that holds it. Even the \"Exact Sciences,\" as they are\ncalled, are not complete; they lead to precipices, down which to look\nis a giddiness. The fact is, the action of the mind is as that of the\nbody: mind and body have their daily outward work, and their times of\nsleep and of dreaming, and the dreaming of the one is not unfrequently\nthe life of the other. The dream of the philosopher, be he waking or\nsleeping, is his refreshment, and at times suggestive of the to come.\nHow know we but that \"such stuff as dreams are made of\" may serve for\nthe fabrication of noble thoughts, and be inwoven into the habit of\nlife, and become useful wear?\n\nPerhaps magic was the first and needful life of philosophy--needful\nas a covering while it grew, and which it shook off as its\nswaddling-clothes, and became a truth. How few can trace invention\nto its germ, or know where the germ lies, and how that it fed upon\nreached it! The suggestion of a dream begetting a reality! They\nare no fools who think that good and bad angels are the authors of\ninventions. It is ingenious to suppose that we are rather the receivers\nand encouragers of our original thoughts than the authors of them. We\nmay use the magnifying glasses of our reason or our passions, and do\nbut a little distort them, or advance them to use and beauty, as we\nare good or bad in ourselves. And thus, from suggestions given, the\nimaginative genius, inventing, magnifies and multiplies by these his\nglasses and his instruments; and the thing invented requires much of\nthis brilliant finery of our own to be removed before it be fitted for\ndemand and use. Like wrought iron, the sparks must be beaten out of it\nwhile it is forming into shape. It must be off its red heat or white\nheat--be dipped in the cold stream of doubt, and look ugly enough to\nthe eye of common opinion, and be long in the hand of experiment to\ntry the patience of the inventor. And, after all, will the benefited\nbe thankful? History has many a sad tale to tell on this subject. The\n\"Sic vos non vobis\" should be inscribed over the portals of the patent\noffice. Yet sometimes, in pity to lost expectations, in the carrying\nout one great idea to--shall we say its final incompletion, to its last\nresiduum of insanity?--some little scarcely noticeable matter in the\nmachinery has been by some kind suggesting spirit held up to the eye of\nthe philosopher, which has proved to be the _magnum bonum_ of the whole\nscheme.\n\nWe once knew a tradesman who had spent the best years of his life, as\nwell as his substance, to discover \"perpetual motion.\" He sold off his\ngoods when he fancied he had discovered it, and left his provincial\ntown for the great metropolis and a philosopher's fame. As he travelled\nby the coach, going over in his mind the processes of his machinery, a\nportion of it struck him as applicable to a manufacture of common use,\nbut of no very high pretensions. His perpetual motion failed. There\nwas a good angel that whispered to him, \"Descend from the ladder of\nyour ambition--do not lose sight of it; but try the little interloping\nsuggestion, and raise the means for prosecuting more favourably your\nperpetual motion.\" He did so. The action saved him from lunacy--the\nundignified and bye-sport, as it were, of his invention answered--from\na ruined man he became rich, and his new business required of him so\nmuch perpetual motion bodily, that the idea of it, wonderful to say,\nwas driven out of his speculative mind.\n\nA sudden thought--a happy hit--we are too apt to call a _lucky_ one.\nWill it be the worse if we give it a better name, and say it is a gift?\nThe thankfulness implied in gift may make it a blessing. It was no\ndeep study that brought the great improvements into our manufacturing\nmachinery.\n\nThe poor boy Arkwright, in a moment of idleness or weariness, thought\nhappily of a cog in the wheel; and that little cog, was to him and his\nposterity a philosopher's stone; realising the alchemist's hopes, by\nfar more sure experiment than the dealings with \"green\" and \"red lions\"\nand \"dragons\" for a result never to be reached. How wonderful has been\nthe result, even to the whole world, of that momentary thought--that\nsimple invention!\n\nWe have often heard it remarked that this is an age of inventions.\nIt is true: not that the inventive mind was ever wanting. It is a\npractical age; the necessities of multiplied life make it so. The\nwell-known \"century of inventions\" of the Marquis of Worcester is a\nstock not yet exhausted. But to speak of this our age, how can it be\notherwise? Not only are material means enlarged by geographical and\nother discoveries, but the inventive mind is multiplied because mankind\nare multiplied, whose nature it is to invent. A population--to speak\nof England, for it is of England we are thinking--of five millions,\nas it was in the time of Queen Elizabeth, cannot bear comparison\nwith ours of nearer twenty millions. Then, if we enlarge our view,\nand take in England's transplanted progeny, whose activity and whose\nadvancement in knowledge and science we share, under every facility\nfor the transmission of knowledge, we may fairly speculate upon a very\nwonderful futurity. The glory of the German dramatist, with his watch,\nand perhaps, but we forget, his printing-press, (for it ought to be in\nthe play,) is annihilated: the author himself would now stand in the\nplace of his Caesar or Cicero.\n\nIt would be a dream worth dreaming to bring back from his Elysian\nFields Agricola, the Roman governor of Britain--he who first discovered\nthat it was an island--to show him his semi-barbarians, whom he so\nequitably governed, (passing by, however, how far we are, any of us,\ntheir descendants.) We will imagine but an hour or two passed with him\nat the Polytechnic Rooms, to show him enormous iron cables twisted into\nknots, as if they were pieces of tape--to see vast ponderous masses\nsuspended by magnetism only--to let him look into the wonders of the\ntelescope and the microscope, besides a thousand marvellous things, too\nnumerous and too often enumerated to mention. Nor would it be unamusing\nto dream that we return with him, and on his way accompany him,\nsummoned to the court of Pluto and Proserpine to narrate the incidents\nof his sojourn above. We could believe the line of Homer verified,\nand that we see the grim and sceptical Pluto leap up from his throne\nin astonishment, and perhaps, as the poet would have it, fear lest\nour subterranean speculators should break in upon his dominions, and\nlet in the light of our day. We have taken the humblest walk for the\n\"surprise.\" What if we had accompanied the ex-governor of Britain to\nthe Crystal Palace? That we will not venture upon. But had he continued\nhis narrative of all he saw there, Pluto would have given a look--at\nwhich Cerberus would have growled from his triple throats--and that the\nunlucky narrator might escape the castigation of Rhadamanthus, he would\nhave been ordered a fresh dip in Lethe, as one contaminated, and who\nhad contracted the lying propensities of people in the upper air.\n\nWe know not if the wonder in us be not the greater that we have not the\nslightest pretensions to mechanical knowledge. But we confess that,\nwhen we suddenly came upon the mechanical department, and saw the\nvarious machinery at work, the world's life and all its business came\nout vividly upon the canvass of our thought, as the great poetry of\nnature. Yes, nature rather than art, for art is but the capability of\nnature in practice. We thought of Sophocles and his chorus of laudation\nof man--the inventor and the [Greek: pontoporos]--and how impoverished\ndid the Greek seem, how tame and inadequate the description!\n\nShakspeare is more to the mark. The whole world is scarcely large\nenough for the exhibition of man's thought and deed, as Shakspeare\nsees him. There is no small talk of his little doings--how he passes\nover the seas and bridles the winds. Inimitable Shakspeare omits the\ndoing to show the capacity; makes, for a moment of comparison only,\nthe earth a sterile promontory, and man that is on it himself, and\nin his own bosom, the ample region of all fertility, in undefined\nthought and action. \"What a piece of work is man!--how noble in\nreason!--how infinite in faculties! In form and moving how express and\nadmirable!--in action how like an angel!--in apprehension how like a\ngod! The beauty of the world, the paragon of animals!\" Behold man the\ninventor!\n\nWe have said that the increase of population must necessarily enlarge\nthe stock of inventions, both by new and multiplied demands, and by the\nadded number of inventors. But there is another cause in operation,\nthat is seldom taken into the account--there are not only more millions\nof human hands to do the work, but there is an artificial working\npopulation, if we may call horse-power of steam a population as\nequivalent to hands.\n\nIn this view the _working_ population, or working power, so far exceeds\nour actual population, that they can scarcely be named together. If\nit be said, this is not a power of mind, and therefore cannot be\nsaid to be inventive; it may be answered, that every instrument is\na kind of mind to him who takes it up, improves, and works upon it,\nand with it: for, after all, it is mind that is operating in it. The\nman is not to be envied who in heart and understanding is dead to the\nmanifold evolutions of this great workshop of the human brain, who\ncannot feel the poetry of mechanics. Is it not a creative power?--and\nis it not at once subjecting and civilising the world? Is not this\npoetry of mechanics showing also that man has dominion given him over\nthe inert materials, as over other living creatures of the earth? We\nhail it in all its marvellous doings, as subject for creative dreams,\nscarcely untrue. Let those who will (and many there be who profess this\nblindness to the poetry of art and science) see nothing but the tall\nchimneys and the black smoke. To the imaginative, even the smoke itself\nbecomes an embodied genie, at whose feet the earth opens at command;\nand they who yield themselves to the spell are conducted, through\nsubterranean ways, to the secret chambers of the treasures of nature;\nand, by a transition to a more palpable reality, find themselves\nin a garden covered with crystal, to behold all beauteous things\nand precious stones for fruit, such as Aladdin saw, and fountains\nthrowing out liquid gems, and fair company, as if brought together\nby enchantment--and this is the romance of reality. If we write\nrhapsodically, let the subject be the excuse, for the secrets of nature\nthrow conjecture into the depths of wonder, and thought far out of the\nconveyance of language.\n\nIt was our purpose to speak of the Submarine Telegraph, and it is\nnot surprising if we have in some degree been transported to great\ndistances by its power.\n\nThe inventors, Messrs Brett, under every difficulty and discouragement,\nhave at length succeeded. Our greatest engineers for a long while\nwithheld their countenance; practical philosophers denied the\nprobability. The possibility was tested by the first experiment.\nFortunately no accident occurred in laying down the wire across the\nChannel, until communication by means of it had been made between\nFrance and England; and even the subsequent accident--the cutting the\nwire by the fishermen--has only served the good purpose of making more\nsure the permanent setting up of this extraordinary telegraph. The\nprotection of the wires by the gutta-percha covering is considered\nperfect; but should it turn out otherwise, it will not affect the\ncertainty of the invention: it must be permanent. A narrative of all\nthe difficulties which beset the inventors, and which have delayed the\nexperiment for years, would be curious. The discouragements and the\nexpenses would have crushed men of less energy. Even at last, in making\nthe cable, there was a disappointment and a hitch, arising from rival\ncompanies. We extract from the _Times_.\n\n    \"On the 19th of July last, Mr Crampton undertook to construct\n    and lay down a cable containing four electric wires, each\n    insulated in two coatings of gutta percha, and the whole\n    protected by ten strands of galvanised iron wire, on or before\n    the 30th of September. The electric wires, covered with gutta\n    percha, in length a hundred miles, were turned out by Mr\n    Statham, at the works of the Gutta Percha Company, and nothing\n    can be more perfect than the manner in which that order was\n    executed. The wire covering was ordered from Messrs Wilkins and\n    Weatherly; but unfortunately, a dispute respecting the patent\n    for making wire ropes occurred between that firm and Messrs\n    Newall, which seriously delayed the progress of the work, as an\n    injunction was served by the latter to prevent Messrs Wilkins\n    and Co. from proceeding with the order.\n\n    \"This was eventually compromised, and the rope was made\n    conjointly by the workmen of the two firms on the premises of\n    Messrs Wilkins and Weatherly, at Wapping.\n\n    \"The very hurried manner in which (from this unforeseen delay)\n    the work had to be accomplished, prevented that close attention\n    that ought to have been given to any fracture, however small,\n    of the wire; and in consequence, the outer casing, though\n    of great strength and solidity, was not made with the same\n    exquisite nicety and care that had been bestowed on the core of\n    the cable.\"\n\nThe weather was unpropitious, and was probably the cause, from the\ncircumstance of the Blazer being driven somewhat out of her course,\nthat the length of the wire cable was not sufficient. This defect was,\nhowever, only of a temporary kind, and was supplied by that which was\nintended for another purpose. We extract the interesting account of the\nproceedings from the _Times_:--\n\n    \"Shortly after 7 o'clock the fastenings at the end of the cable\n    at the Foreland were completed, and the Fearless started to\n    point out the exact course to be followed by the Blazer, which\n    was towed by two tugs, one alongside, and the other ahead of\n    her.\n\n    \"A third tug belonging to the Government was also in attendance.\n\n    \"The arrangements for paying out the cable consisted simply\n    of a bar fixed transversely above the hold, over which the\n    rope was drawn as it was uncoiled from below, and a series of\n    breaks acting by levers fitted to the deck, in order to arrest\n    the passage of the rope in the case of too rapid a delivery.\n    On reaching the stem the cable passed overboard through a\n    'chock' of a semicircular shape, lined with iron. On starting,\n    the steam-tugs proceeded at much too rapid a pace, (from four\n    to five knots an hour,) and consequently one of the fractured\n    wires (before alluded to) caught in the friction-blocks, and,\n    before the way of the vessel could be checked, one strand\n    of the iron wire was, for a length of about eighteen yards,\n    stripped from the cable. The steam-tug towing ahead was then\n    ordered alongside, when the speed could be better regulated,\n    and the rate was reduced to about one and a half to two knots\n    an hour. About six miles from shore it was determined to test\n    the wires; but, from a misapprehension of instructions, the\n    telegraph instruments at the South Foreland were not joined up\n    with those on board the Blazer. A steam-tug, with one of the\n    engineers and directors on board, immediately returned to the\n    Foreland, when communication was made by telegraph and fusees\n    fired from the vessel to the shore, and from the shore to the\n    Blazer.\n\n    \"At about mid-Channel, in the midst of a heavy sea, and a\n    strong wind from the SW., an accident occurred, but for which\n    the enterprise would have been carried out with the most\n    perfect success; this was the snapping of the towrope (an\n    eight-inch cable) and the consequent drifting of the Blazer\n    from her appointed course to the length of a mile and a-half.\n    Notwithstanding the delay caused by this untoward incident, the\n    Blazer arrived off Sangatte at about 6 o'clock. The evening\n    was, however, too far advanced, and the weather too stormy to\n    attempt a landing; and, after embarking most of her passengers\n    on board one of the steamers that ran into Calais, she was\n    anchored for the night about two miles from the shore.\n\n    \"On Friday the wind blew a strong gale from the westward, which\n    rendered all near approach to the shore impracticable; but the\n    Blazer was towed to within a mile of the beach, when, it being\n    considered dangerous to leave her at anchor, the remainder\n    of the rope was made fast to a buoy and hove overboard. The\n    steam-tugs then returned with the Blazer to England.\n\n    \"On Saturday the weather continued unfavourable, but Captain\n    Bullock proceeded with the Fearless to the buoy off Sangatte,\n    and, having hauled up the end of the rope, he towed it some\n    hundred yards nearer the shore, and then again moored it.\n\n    \"On Sunday the wind shifted more to the southward and\n    moderated. Accordingly, the engineers and managers of the Gutta\n    Percha Company took on board the Fearless a large coil of gutta\n    percha roping, and, after hauling up the end of the telegraph\n    cables, the first wires were carefully attached, and at\n    half-past five in the afternoon a boat landed them on the beach\n    at Sangatte. The moment chosen for landing was low-water, and\n    the coil of gutta percha ropes was immediately buried in the\n    beach by a gang of men in attendance, up to low-water mark, and\n    even to a short distance beyond it. Thence to where the cable\n    was moored did not much exceed a quarter of a mile.\n\n    \"The telegraphs were instantly attached to the submarine\n    wires, and all the instruments responded to the batteries from\n    the opposite shore. At six o'clock messages were printed at\n    Sangatte from the South Foreland, specimens of which Captain\n    Bullock took over to Dover the same evening for the Queen and\n    the Duke of Wellington.\n\n    \"On Monday morning the wires at Sangatte were joined to those\n    already laid down to Calais, and two of the instruments used by\n    the French Government having been sent to the South Foreland,\n    Paris was placed in immediate communication with the English\n    Court.\"\n\nWe have remarked that very important discoveries are accidentally made\nin pursuing one of quite a different character from those which come up\nin the search unexpectedly.\n\nThey who remember our towns lighted with the old lamps, that in\ncomparison with our gas-lights made but a \"palpable obscure,\" should\nalso remember how the change was brought about. The gas, which has\nproved of such vast utility that we can now-a-days scarcely conceive\nhow the world could go on without it, was first a misfortune. It was\ngenerated in the coal mines, and, in order to get rid of it, it was\nconveyed by tubes to the outer air: in doing this it was found there\nto ignite, and from this simple attempt to effect an escape for a\nnuisance is almost every town in the civilised world illuminated by\ngas--besides which, the advantageous use of it in manufactories is\nbeyond calculation. Even of gutta percha, now applied as a coating to\nthese wires, who can determine all the uses to which it may be found\napplicable? Nature, it should seem, does not fabricate one material for\nitself, or for one use only, but adapts one thing to many purposes--and\nthus, as it were, teaches us that there is a chain in the facts of\nnature, by showing us a few of the connected links; and, at the same\ntime, so far from exhibiting any sudden breaks, offering evidences of\na continuous connection reaching beyond our conception. Verily this\npoor opaque earth of ours is the foundation on which the Jacob's ladder\nof invention is laid. We know not where it reaches, but there may be\nsuggesting angels passing to and fro, and when their feet touch the\nground, it delivers up its secrets, that float into the ears of the\ndreamer.\n\nElectricity, it would appear, is the great agent in this connecting\nchain--nay, is it not, whatever it be in its essence, the chain itself,\nand the universal power equally in inert matter and in life? It has\nneither boundary on the earth nor in space. Its home is ubiquity; like\nthe sphere of Hermes, its centre is everywhere, its circumference\nnowhere. That this astonishing power is yet under restraint--that\nit is not only kept from the evil it would do, but rendered to us\nserviceable--is a proof of the great beneficence of Him who made it\nand us. When the admiring child touches that gem, the dew-drop on the\nrose-leaf, it knows not that the little hand is on that which has\nlightning in it enough to cause instant death. It is scarcely the\nlover's poetical dream that he _may_ be killed by the lightning of an\neye--done dead by the tear that only moves his pity, on his mistress's\neye-lid. In that little drop is the power of death--and by what miracle\n(truly all nature is miraculous) is the execution staid--the power\nforbidden to act? Nay, even the pity that we speak of, love itself,\nstrange in its suddenness as we see it, how know we what of electricity\nbe in it, instantly conveying from person to person natural but unknown\nsympathy?\n\nLet us not get out of our depths,--but emerge from \"the submarine,\" to\nland; and for this purpose, and to complete our argument of unexpected\nand collateral uses, we offer an extract from the _Army and Navy\nRegister_:--\n\n    \"NEW MODE OF DISCHARGING GUNPOWDER.--On Monday, August 18,\n    some interesting experiments were tried at the Gutta Percha\n    Company's Works, Warf Road, City Road, for the purposes of\n    demonstrating the means by which this extraordinary production\n    may be applied to the operation of discharging gunpowder. A\n    galvanic battery was connected with upwards of 50 miles of\n    copper wire covered with gutta percha, to the thickness of\n    an ordinary black lead pencil. The wire, which was formed\n    into coils, and which has been prepared for the projected\n    submarine telegraph, was attached to a barge moored in the\n    canal alongside the manufactory, the coils being so fixed\n    together (although the greater portion of them were under\n    water) as to present an uninterrupted communication with\n    the battery to a distance limited at first to 57 miles, but\n    afterwards extended to 70. A \"cartridge\" formed with a small\n    hollow roof of gutta percha, charged with gunpowder, and\n    having an intercommunicating wire attached, was then brought\n    into contact with the electric current. The result was, that a\n    spark was produced, which, igniting the gunpowder, caused an\n    immediate explosion similar to that which would arise from the\n    discharge of a small cannon. The same process was carried out\n    in various ways, with a view of testing the efficient manner\n    in which the gutta percha had been rendered impervious to wet,\n    and in one instance the fusee or cartridge was placed under\n    the water. In this case the efficiency of the insulation was\n    equally well demonstrated by the explosion of the gunpowder at\n    the moment the necessary \"contact\" was produced; and by way\n    of showing the perfect insulation of the wire, an experiment\n    was tried which resulted in the explosion of the fusee from\n    the charge of electricity retained in the coils of wire, three\n    seconds after contact with the battery had been broken. This\n    feature in the experiment was especially interesting from the\n    fact of its removing all difficulty and doubt as to whether\n    the gutta percha would so far protect the wires as to preserve\n    the current of electricity under the most disadvantageous\n    circumstances. Another experiment was successfully tried by\n    passing the electric current to its destination through the\n    human body. Mr C. J. Wollaston, civil engineer, volunteered to\n    form part of the circuit by holding the ends of 35 miles of\n    the wire in each hand. The wire from the battery was brought\n    to one end of the entire length of 70 miles, and instant\n    explosion of the cartridge took place at the other end. The\n    experiments were altogether perfectly successful, as showing\n    beyond all question that the properties of gutta percha and\n    electricity combined are yet to be devoted to other purposes\n    than that of establishing a submarine telegraph. The blasting\n    of a rock, the destruction of a fortification, and other\n    operations which require the agency of gunpowder, have often\n    been attended with considerable danger and trouble, besides\n    involving large outlays of money; but it may be truly said\n    that the employment of electricity in the manner described is\n    calculated to render such operations comparatively free from\n    difficulty. Amongst the company present on this occasion was\n    Major-General Sir Charles Pasley, who took a warm interest\n    in the proceedings, and expressed himself much gratified at\n    the result. It is impossible to foretell the value of this\n    discovery, particularly in engineering and mining operations.\n    It forms a valuable addition to the benefits already conferred\n    upon the public by the enterprise of the Gutta Percha Company.\"\n\nThis extract may lead the reader to conclude that there are double\nand opposite purposes in the secrets of nature. The chain which was\nintended to connect all nations in a bond of peace, has, it should\nseem, also (incidental to the first discovery) its apparatus for war.\n\nWhen his Grace the Archbishop of Canterbury was blessing the Crystal\nPalace, and all within it, as emblems of a religious bond of peace, and\nof amity of all nations, and it pleased the admiring masses to proclaim\nit the Temple of Peace and of Love, there was little thought that,\namong the machinery and instruments it contained, those of murderous\npurpose would be the first required for use, which was actually the\ncase, when permission was asked and given for the removal of revolving\nfirearms from the American department, to be sent out to the Cape.\n\nThus, good and evil are not unmixed. Either may be extracted, and leave\nthe remainder, in appearance to us, a kind of _caput mortuum_.\n\nIt is far more pleasant to look to the peaceful results of\ninventions--to hear the spirit that is in the electric fluid say--\n\n    \"I will be correspondent to command,\n    And do my spiriting gently.\"\n\nLet it be the means that far-off friends at the Antipodes shall\ncommunicate, if not by voice, by that which is like it--by sound and\nby lettered words. Let it touch a bell at their mid-day, and it may\ntingle at that instant in your ears at midnight, and awake you to\nreceive, evolved from the little machinery at your bed's head, a letter\nin a printed strip, conveying \"thoughts that breathe, and words that\nburn,\" even as though you felt the breath that uttered them. Reader,\nbe not sceptical. How many very practicable things have you denied,\nand yet found brought tangibly before your eyes, and into your hands!\nThis simultaneous tingle of two bells--one at the Antipodes, and one\nwithin reach of your own touch, and at your own ear--may cause you to\ncurl your lip in derision; but say, is it impossible? We have heard you\nsay of much more improbable things, \"Where there is a will there is a\nway.\" Well, here it is evident you have only a little to strengthen\nyour will, and the length of the way will be no obstacle. You may amuse\nyourself with the idea, and make a comparison of it, and look at the\nfigures on your China plate, and imagine them moved to each other under\nspell of their passion, (see the tale of the willow pattern,) to the\ndefiance of all the ordinary rules of distance. Did not the foreseeing\nartist intimate thereby that love and friendship have no space-limits,\nand hold within themselves a power that laughs at perspective, as it\ndoes \"at locksmiths?\" The artist whom you contemned as ignorant was,\nyou acknowledge, wise--wise beyond his art, if not beyond his thought.\nHe had a second-sight of a new mode of communication, and expressed it\nprudently in this his hieroglyphic.\n\nDoes any marvel exceed this in apparent absurdity--that you, in London\nor Edinburgh, shall be able to communicate instantaneously with your\nfriend or relative at St Petersburg or Vienna; for which purpose\nyou have but to touch a few keys denoting letters of the alphabet,\nand under water and over land your whole thoughts pass as soon as\nyour fingers have delivered them to the keys--nay, the letters are\nforestalling your thought, and those before it? Does it not seem very\nabsurd to say that all the foreign news may be at your breakfast-table,\nfresh from every capital in Europe, before the _Times_ can be published\nand circulated? How will the practice of the press be affected by this\nnovelty? \"The latest intelligence\" becomes a bygone tale, \"flat, stale,\nand unprofitable.\" Far greater things than the poet dreamed of become\ndaily realities. Richest in fancy, Shakspeare apologetically covers the\nincredible ubiquity of his Ariel with a sense of fatigue--of difficulty\nin his various passages--Ariel, the spirit who\n\n        \"thought it much to tread\n    The ooze of the salt deep.\"\n\nOur Government officers will have ready on the instant, messengers far\nswifter than Ariel--wondrous performers on the \"slack-wires.\" They will\nput you\n\n    \"A girdle round about the earth\n    In forty minutes.\"\n\nNo; that was the lagging, loitering pace of the old spirit. It will\nnot take forty seconds. What are thousands of miles to a second of\ntime? Time is, as it were, annihilated: the sand in the glass must\nbe accelerated, or the glass, held for ages, taken out of his hand,\nand some national exhibition ransacked for a new hour-instrument. The\nProspero's wand broken, and newer wonders to be had for a trifle.\nFortunatus's \"wishing-cap\" to be bought at the corner-shop, and the\nfamed \"seven-league boots\" next door--and to be had cheap, considering,\nthat you may tell all your thoughts, at ever so great a distance, by a\nlittle bell and a wire, while you are sitting in your armchair. It will\nbe quite an easy matter to\n\n    \"Waft a sigh from Indus to the Pole.\"\n\nRailroads and the Submarine Telegraph more than double man's life, if\nwe count his years by action. History itself must now begin as from a\nnew epoch. All the doings of the world, through this rapidity given to\nperson and to thought, must be so altered as to bear no parallel with\nthe past. The old locomotive and communicating powers are defunct--they\nare as the water that has passed the mill. It must grind with that\nwhich succeeds. They are new powers that must set the wheels of\ngovernments and of all the world's machinery in motion.\n\nThere is in the _Spectator_ a paper of the true Addisonian wit,\ndescriptive of an Antediluvian courtship, in which the _young_ couple,\nhaving gone through the usual process in the early art of love,\ncomplete their happiness in the some hundredth year of their ages.\nTheorists have entertained the notion that this long life was bestowed\nupon man in the world's first era, that knowledge might be more readily\ntransmitted, there being few generations to the Flood. To the lovers\nof life it would be a sad thing to be led to the conclusion, that,\ntransmission being quickened, life will be shortened; or that, as in\nthe winding-up of a drama, events are crowding into the last act of our\nearth's duration. It may relieve their apprehensions to read of the\nadvance the medical science is making simultaneously with all other\nsciences, so that they may look to a state in which a man may live as\nlong as he likes, and at the same time do ten times the work: a man's\nday will perhaps be a year, counting by his doings. Morose poets and\nphilosophers have lamented over us as ephemeral; if so, we are at least\nlike the Antediluvian butterflies, and our day long. And now, with\nall our sanitary inventions, it stands a fair chance of a tolerable\nlengthening.\n\nWe have observed that it has been said that the world is not fifteen\nyears of age; and, indeed, it looks like enough. Hitherto Nature has\ntreated us as a kind mother does her children--given us toys and\nplaythings, to be broken and discarded as we get older. We are throwing\nthem by, we are becoming of age, and Nature opens her secrets to us,\nand we are just setting up for ourselves--as it were, commencing the\nbusiness of life, like grown men in good earnest; and every day we find\nout more secrets, and all worth knowing.\n\nWe will not lay down the pen without expressing our congratulations to\nthe inventors of the Submarine Telegraph, the Messrs Brett, and wishing\nthem the fullest success. They themselves as yet know not the extent of\nthe reach of their own invention, or they might well wonder at their\nown wonders, like\n\n    \"Katerfelto, with his hair on end!\"\n\nWe wish them long life to see the results--and that they will not,\nthrough mistrust of so great a discovery, imitate Copernicus, who,\nsays Fontenelle, \"distrusting the success of his opinions, was for a\nlong time loth to publish them, and, when they brought him the first\nsheet of his work, died, foreseeing that he never should be able to\nreconcile all its contradictions, and therefore wisely slipped out of\nthe way.\" Messrs Brett will think it wiser to live, and be in the way\nand at their post, (no _post obit_,) ready to answer all queries and\ncontradictions, through the convincing, the very satisfactory means, of\ntheir \"Submarine Telegraph.\"\n\n\n\n\nMY NOVEL; OR, VARIETIES IN ENGLISH LIFE.\n\nBY PISISTRATUS CAXTON.\n\n\nBOOK VIII.--INITIAL CHAPTER.\n\nTHE ABUSE OF INTELLECT.\n\nThere is at present so vehement a flourish of trumpets, and so\nprodigious a roll of the drum, whenever we are called upon to throw\nup our hats, and cry, \"Huzza\" to the \"March of Enlightenment,\" that,\nout of that very spirit of contradiction natural to all rational\nanimals, one is tempted to stop ones ears, and say, \"Gently, gently;\nLIGHT is noiseless; how comes 'Enlightenment' to make such a clatter?\nMeanwhile, if it be not impertinent, pray, where is Enlightenment\nmarching to?\" Ask that question of any six of the loudest bawlers\nin the procession, and I'll wager tenpence to California that you\nget six very unsatisfactory answers. One respectable gentleman, who,\nto our great astonishment, insists upon calling himself \"a slave,\"\nbut has a remarkably free way of expressing his opinions, will\nreply--\"Enlightenment is marching towards the nine points of the\nCharter.\" Another, with his hair _a la jeune France_, who has taken a\nfancy to his friend's wife, and is rather embarrassed with his own,\nasserts that Enlightenment is proceeding towards the Rights of Women,\nthe reign of Social Love, and the annihilation of Tyrannical Prejudice.\nA third, who has the air of a man well to do in the middle class, more\nmodest in his hopes, because he neither wishes to have his head broken\nby his errand-boy, nor his wife carried off to an Agapemone by his\napprentice, does not take Enlightenment a step farther than a siege\non Debrett, and a cannonade on the Budget. Illiberal man! the march\nthat he swells will soon trample _him_ under foot. No one fares so ill\nin a crowd as the man who is wedged in the middle. A fourth, looking\nwild and dreamy, as if he had come out of the cave of Trophonius, and\nwho is a mesmeriser and a mystic, thinks Enlightenment is in full\ncareer towards the good old days of alchemists and necromancers. A\nfifth, whom one might take for a Quaker, asserts that the march of\nEnlightenment is a crusade for universal philanthropy, vegetable diet,\nand the perpetuation of peace, by means of speeches, which certainly\ndo produce a very contrary effect from the Philippics of Demosthenes!\nThe sixth--(good fellow, without a rag on his back)--does not care a\nstraw where the march goes. He can't be worse off than he is; and it\nis quite immaterial to him whether he goes to the dogstar above, or\nthe bottomless pit below. I say nothing, however, against the march,\nwhile we take it altogether. Whatever happens, one is in good company;\nand though I am somewhat indolent by nature, and would rather stay at\nhome with Locke and Burke, (dull dogs though they were,) than have my\nthoughts set off helter-skelter with those cursed trumpets and drums,\nblown and dub-a-dubbed by fellows that I vow to heaven I would not\ntrust with a five-pound note--still, if I must march, I must; and so\ndeuce take the hindmost. But when it comes to individual marchers upon\ntheir own account--privateers and condottieri of Enlightenment--who\nhave filled their pockets with lucifer-matches, and have a sublime\ncontempt for their neighbours' barns and hay-ricks, I don't see why I\nshould throw myself into the seventh heaven of admiration and ecstasy.\n\nIf those who are eternally rhapsodising on the celestial blessings\nthat are to follow Enlightenment, Universal Knowledge, and so forth,\nwould just take their eyes out of their pockets, and look about\nthem, I would respectfully inquire if they have never met any very\nknowing and enlightened gentleman, whose acquaintance is by no means\ndesirable. If not, they are monstrous lucky. Every man must judge by\nhis own experience; and the worst rogues I have ever encountered were\namazingly well-informed, clever fellows! From dunderheads and dunces\nwe can protect ourselves; but from your sharp-witted gentleman, all\nenlightenment, and no prejudice, we have but to cry, \"Heaven defend\nus!\" It is true, that the rogue (let him be ever so enlightened)\nusually comes to no good himself, (though not before he has done harm\nenough to his neighbours.) But that only shows that the world wants\nsomething else in those it rewards, besides intelligence _per se_ and\nin the abstract; and is much too old a world to allow any Jack Horner\nto pick out its plumbs for his own personal gratification. Hence a man\nof very moderate intelligence, who believes in God, suffers his heart\nto beat with human sympathies, and keeps his eyes off your strong-box,\nwill perhaps gain a vast deal more power than knowledge ever gives to a\nrogue.\n\nWherefore, though I anticipate an outcry against me on the part of the\nblockheads, who, strange to say, are the most credulous idolaters of\nenlightenment, and, if knowledge were power, would rot on a dunghill;\nyet, nevertheless, I think all really enlightened men will agree with\nme, that when one falls in with detached sharpshooters from the general\nmarch of enlightenment, it is no reason that we should make ourselves\na target, because enlightenment has furnished them with a gun. It has,\ndoubtless, been already remarked by the judicious reader, that of the\nnumerous characters introduced into this work, the larger portion\nbelong to that species which we call the INTELLECTUAL--that through\nthem are analysed and developed human intellect, in various forms and\ndirections. So that this History, rightly considered, is a kind of\nhumble familiar Epic, or, if you prefer it, a long Serio-Comedy, upon\nthe Varieties of English Life in this our Century, set in movement\nby the intelligences most prevalent. And where more ordinary and\nless refined types of the species round and complete the survey of\nour passing generation, they will often suggest, by contrast, the\ndeficiencies which mere intellectual culture leaves in the human being.\nCertainly, I have no spite against intellect and enlightenment. Heaven\nforbid I should be such a Goth. I am only the advocate for common sense\nand fair play. I don't think an able man necessarily an angel; but I\nthink if his heart match his head, and both proceed in the Great March\nunder a divine Oriflamme, he goes as near to the angel as humanity\nwill permit: if not, if he has but a penn'orth of heart to a pound\nof brains, I say, \"_Bon jour, mon ange!_ I see not the starry upward\nwings, but the grovelling cloven-hoof.\" I'd rather be offuscated by\nthe Squire of Hazeldean, than enlightened by Randal Leslie. Every man\nto his taste. But intellect itself (not in the philosophical, but\nthe ordinary sense of the term) is rarely, if ever, one completed\nharmonious agency; it is not one faculty, but a compound of many, some\nof which are often at war with each other, and mar the concord of\nthe whole. Few of us but have some predominant faculty, in itself a\nstrength; but which, (usurping unseasonably dominion over the rest,)\nshares the lot of all tyranny, however brilliant, and leaves the empire\nweak against disaffection within, and invasion from without. Hence\nintellect maybe perverted in a man of evil disposition, and sometimes\nmerely wasted in a man of excellent impulses, for want of the necessary\ndiscipline, or of a strong ruling motive. I doubt if there be one\nperson in the world, who has obtained a high reputation for talent, who\nhas not met somebody much cleverer than himself, which said somebody\nhas never obtained any reputation at all! Men, like Audley Egerton, are\nconstantly seen in the great positions of life; while men, like Harley\nl'Estrange, who could have beaten them hollow in anything equally\nstriven for by both, float away down the stream, and, unless some\nsudden stimulant arouse the dreamy energies, vanish out of sight into\nsilent graves. If Hamlet and Polonius were living now, Polonius would\nhave a much better chance of being Chancellor of the Exchequer, though\nHamlet would unquestionably be a much more intellectual character.\nWhat would become of Hamlet? Heaven knows! Dr Arnold said, from his\nexperience of a school, that the difference between one man and another\nwas not mere ability--it was energy. There is a great deal of truth in\nthat saying.\n\nSubmitting these hints to the judgment and penetration of the\nsagacious, I enter on the fresh division of this work, and see\nalready Randal Leslie gnawing his lip on the back-ground. The German\npoet observes, that the Cow of Isis is to some the divine symbol of\nknowledge, to others but the milch cow, only regarded for the pounds of\nbutter she will yield. O tendency of our age, to look on Isis as the\nmilch cow! O prostitution of the grandest desires to the basest uses!\nGaze on the goddess, Randal Leslie, and get ready thy churn and thy\nscales. Let us see what the butter will fetch in the market.\n\n\nCHAPTER II.\n\nA new reign has commenced. There has been a general election; the\nunpopularity of the Administration has been apparent at the hustings.\nAudley Egerton, hitherto returned by vast majorities, has barely\nescaped defeat--thanks to a majority of five. The expenses of his\nelection are said to have been prodigious. 'But who can stand against\nsuch wealth as Egerton's--no doubt backed, too, by the Treasury purse?'\nsaid the defeated candidate. It is towards the close of October; London\nis already full; Parliament will meet in less than a fortnight.\n\nIn one of the principal apartments of that hotel in which foreigners\nmay discover what is meant by English comfort, and the price which\nforeigners must pay for it, there sat two persons, side by side,\nengaged in close conversation. The one was a female, in whose pale\nclear complexion and raven hair--in whose eyes, vivid with a power\nof expression rarely bestowed on the beauties of the north, we\nrecognise Beatrice, Marchesa di Negra. Undeniably handsome as was\nthe Italian lady, her companion, though a man, and far advanced\ninto middle age, was yet more remarkable for personal advantages.\nThere was a strong family likeness between the two; but there was\nalso a striking contrast in air, manner, and all that stamps on the\nphysiognomy the idiosyncrasies of character. There was something\nof gravity, of earnestness and passion, in Beatrice's countenance\nwhen carefully examined; her smile at times might be false, but it\nwas rarely ironical, never cynical. Her gestures, though graceful,\nwere unrestrained and frequent. You could see she was a daughter of\nthe south. Her companion, on the contrary, preserved on the fair\nsmooth face, to which years had given scarcely a line or wrinkle,\nsomething that might have passed, at first glance, for the levity and\nthoughtlessness of a gay and youthful nature; but the smile, though\nexquisitely polished, took at times the derision of a sneer. In his\nmanners he was as composed and as free from gesture as an Englishman.\nHis hair was of that red brown with which the Italian painters produce\nsuch marvellous effects of colour; and, if here and there a silver\nthread gleamed through the locks, it was lost at once amidst their\nluxuriance. His eyes were light, and his complexion, though without\nmuch colour, was singularly transparent. His beauty, indeed, would\nhave been rather womanly than masculine, but for the height and sinewy\nspareness of a frame in which muscular strength was rather adorned\nthan concealed by an admirable elegance of proportion. You would never\nhave guessed this man to be an Italian; more likely you would have\nsupposed him a Parisian. He conversed in French, his dress was of\nFrench fashion, his mode of thought seemed French. Not that he was like\nthe Frenchman of the present day--an animal, either rude or reserved;\nbut your ideal of the _Marquis_ of the old _regime_--the _roue_ of the\nRegency.\n\nItalian, however, he was, and of a race renowned in Italian history.\nBut, as if ashamed of his country and his birth, he affected to be\na citizen of the world. Heaven help the world if it hold only such\ncitizens!\n\n\"But, Giulio,\" said Beatrice di Negra, speaking in Italian, \"even\ngranting that you discover this girl, can you suppose that her father\nwill ever consent to your alliance? Surely you know too well the nature\nof your kinsman?\"\n\n\"_Tu te trompes, ma soeur_,\" replied Giulio Franzini, Count di\nPeschiera, in French as usual--\"_tu te trompes_; I knew it before he\nhad gone through exile and penury. How can I know it now? But comfort\nyourself, my too anxious Beatrice, I shall not care for his consent\ntill I've made sure of his daughter's.\"\n\n\"But how win that in despite of the father?\"\n\n\"_Eh, mordieu!_\" interrupted the Count, with true French gaiety; \"what\nwould become of all the comedies ever written, if marriages were not\nmade in despite of the father? Look you,\" he resumed, with a very\nslight compression of his lip, and a still slighter movement in his\nchair--\"look you, this is no question of ifs and buts; it is a question\nof must and shall--a question of existence to you and to me. When\nDanton was condemned to the guillotine, he said, flinging a pellet of\nbread at the nose of his respectable judge--'_Mon individu sera bientot\ndans le neant_'--_My_ patrimony is there already! I am loaded with\ndebts. I see before me, on the one side, ruin or suicide; on the other\nside, wedlock and wealth.\"\n\n\"But from those vast possessions which you have been permitted to enjoy\nso long, have you really saved nothing against the time when they might\nbe reclaimed at your hands?\"\n\n\"My sister,\" replied the Count, \"do I look like a man who saved?\nBesides, when the Austrian Emperor, unwilling to raze from his\nLombard domains a name and a house so illustrious as our kinsman's,\nand desirous, while punishing that kinsman's rebellion, to reward\nmy adherence, forbore the peremptory confiscation of those vast\npossessions at which my mouth waters while we speak, but, annexing\nthem to the Crown during pleasure, allowed me, as the next of male\nkin, to retain the revenues of one-half for the same very indefinite\nperiod--had I not every reason to suppose, that, before long, I could\nso influence his Majesty or his minister, as to obtain a decree\nthat might transfer the whole, unconditionally and absolutely, to\nmyself? And methinks I should have done so, but for this accursed,\nintermeddling English Milord, who has never ceased to besiege the court\nor the minister with alleged extenuations of our cousin's rebellion,\nand proofless assertions that I shared it in order to entangle my\nkinsman, and betrayed it in order to profit by his spoils. So that,\nat last, in return for all my services, and in answer to all my\nclaims, I received from the minister himself this cold reply--'Count\nof Peschiera, your aid was important, and your reward has been large.\nThat reward, it would not be for your honour to extend, and justify the\nill opinion of your Italian countrymen by formally appropriating to\nyourself all that was forfeited by the treason you denounced. A name so\nnoble as yours should be dearer to you than fortune itself.'\"\n\n\"Ah, Giulio,\" cried Beatrice, her face lighting up, changed in its\nwhole character--\"those were words that might make the demon that\ntempts to avarice fly from your breast in shame.\"\n\nThe Count opened his eyes in great amaze; then he glanced round the\nroom, and said, quietly--\n\n\"Nobody else hears you, my dear Beatrice; talk common sense. Heroics\nsound well in mixed society; but there is nothing less suited to the\ntone of a family conversation.\"\n\nMadame di Negra bent down her head abashed, and that sudden change\nin the expression of her countenance, which had seemed to betray\nsusceptibility to generous emotion, faded as suddenly away.\n\n\"But still,\" she said coldly, \"you enjoy one-half of those ample\nrevenues--why talk, then, of suicide and ruin?\"\n\n\"I enjoy them at the pleasure of the crown; and what if it be the\npleasure of the crown to recall our cousin, and reinstate him in his\npossessions?\"\n\n\"There is a _probability_, then, of that pardon? When you first\nemployed me in your researches, you only thought there was a\n_possibility_.\"\n\n\"There is a great probability of it, and therefore I am here. I\nlearned some little time since that the question of such recall had\nbeen suggested by the Emperor, and discussed in Council. The danger\nto the State, which might arise from our cousin's wealth, his alleged\nabilities--(abilities! bah!)--and his popular name, deferred any\ndecision on the point; and, indeed, the difficulty of dealing with\nmyself must have embarrassed the ministry. But it is a mere question\nof time. He cannot long remain excluded from the general amnesty\nalready extended to the other refugees. The person who gave me this\ninformation is high in power, and friendly to myself; and he added a\npiece of advice, on which I acted. 'It was intimated,' said he, 'by one\nof the partisans of your kinsman, that the exile could give a hostage\nfor his loyalty in the person of his daughter and heiress; that she\nhad arrived at marriageable age; that if she were to wed, with the\nEmperor's consent, some one whose attachment to the Austrian crown was\nunquestionable, there would be a guarantee both for the faith of the\nfather, and for the transmission of so important a heritage to safe\nand loyal hands. Why not' (continued my friend) 'apply to the Emperor\nfor his consent to that alliance for yourself?--you, on whom he can\ndepend;--you who, if the daughter should die, would be the legal heir\nto those lands?' On that hint I spoke.\"\n\n\"You saw the Emperor?\"\n\n\"And after combating the unjust prepossessions against me, I stated,\nthat so far from my cousin having any fair cause of resentment against\nme, when all was duly explained to him, I did not doubt that he would\nwillingly give me the hand of his child.\"\n\n\"You did!\" cried the Marchesa, amazed.\n\n\"And,\" continued the Count imperturbably, as he smoothed, with careless\nhand, the snowy plaits of his shirt front--\"and that I should thus\nhave the happiness of becoming myself the guarantee of my kinsman's\nloyalty--the agent for the restoration of his honours, while, in the\neyes of the envious and malignant, I should clear up my own name from\nall suspicion that I had wronged him.\"\n\n\"And the Emperor consented?\"\n\n\"_Pardieu_, my dear sister. What else could his majesty do? My\nproposition smoothed every obstacle, and reconciled policy with mercy.\nIt remains, therefore, only to find out, what has hitherto baffled all\nour researches, the retreat of our dear kinsfolk, and to make myself a\nwelcome lover to the demoiselle. There is some disparity of years, I\nown; but--unless your sex and my glass flatter me, overmuch--I am still\na match for many a gallant of five-and-twenty.\"\n\nThe Count said this with so charming a smile, and looked so\npre-eminently handsome, that he carried off the coxcombry of the words\nas gracefully as if they had been spoken by some dazzling hero of the\ngrand old comedy of Parisian life.\n\nThen interlacing his fingers, and lightly leaning his hands, thus\nclasped, upon his sister's shoulder, he looked into her face, and said\nslowly--\"And now, my sister, for some gentle but deserved reproach.\nHave you not sadly failed me in the task I imposed on your regard for\nmy interests? Is it not some years since you first came to England on\nthe mission of discovering these worthy relatives of ours? Did I not\nentreat you to seduce into your toils the man whom I knew to be my\nenemy, and who was indubitably acquainted with our cousin's retreat--a\nsecret he has hitherto locked within his bosom? Did you not tell me,\nthat though he was then in England, you could find no occasion even to\nmeet him, but that you had obtained the friendship of the statesman to\nwhom I directed your attention, as his most intimate associate? And yet\nyou, whose charms are usually so irresistible, learn nothing from the\nstatesman, as you see nothing of _Milord_. Nay, baffled and misled, you\nactually suppose that the quarry has taken refuge in France. You go\nthither--you pretend to search the capital--the provinces, Switzerland,\n_que sais-je?_--all in vain,--though--_foi de gentilhomme_--your police\ncost me dearly,--you return to England--the same chace, and the same\nresult. _Palsambleu, ma soeur_, I do too much credit to your talents\nnot to question your zeal. In a word, have you been in earnest--or have\nyou not had some womanly pleasure in amusing yourself and abusing my\ntrust?\"\n\n\"Giulio,\" answered Beatrice sadly, \"you know the influence you have\nexercised over my character and my fate. Your reproaches are not just.\nI made such inquiries as were in my power, and I have now cause to\nbelieve that I know one who is possessed of this secret, and can guide\nus to it.\"\n\n\"Ah, you do!\" exclaimed the Count. Beatrice did not heed the\nexclamation, and hurried on.\n\n\"But grant that my heart shrunk from the task you imposed on me, would\nit not have been natural? When I first came to England, you informed\nme that your object in discovering the exiles was one which I could\nhonestly aid. You naturally desired first to know if the daughter\nlived; if not, you were the heir. If she did, you assured me you\ndesired to effect, through my mediation, some liberal compromise with\nAlphonso, by which you would have sought to obtain his restoration,\nprovided he would leave you for life in possession of the grant you\nhold from the crown. While these were your objects, I did my best,\nineffectual as it was, to obtain the information required.\"\n\n\"And what made me lose so important, though so ineffectual an ally?\"\nasked the Count, still smiling; but a gleam that belied the smile shot\nfrom his eye.\n\n\"What! when you bade me receive and co-operate with the miserable\nspies--the false Italians--whom you sent over, and seek to entangle\nthis poor exile, when found, in some rash correspondence, to be\nrevealed to the court;--when you sought to seduce the daughter of the\nCounts of Peschiera, the descendant of those who had ruled in Italy,\ninto the informer, the corrupter, and the traitress! No, Giulio--then I\nrecoiled; and then, fearful of your own sway over me, I retreated into\nFrance. I have answered you frankly.\"\n\nThe Count removed his hands from the shoulder on which they had\nreclined so cordially.\n\n\"And this,\" said he, \"is your wisdom, and this your gratitude.\nYou, whose fortunes are bound up in mine--you, who subsist on my\nbounty--you, who----\"\n\n\"Hold,\" cried the Marchesa, rising, and with a burst of emotion, as\nif stung to the utmost, and breaking into revolt from the tyranny of\nyears--\"Hold--gratitude! bounty! Brother, brother--what, indeed, do I\nowe to you? The shame and the misery of a life. While yet a child, you\ncondemned me to marry against my will--against my heart--against my\nprayers--and laughed at my tears when I knelt to you for mercy. I was\npure then, Giulio--pure and innocent as the flowers in my virgin crown.\nAnd now--now--\"\n\nBeatrice stopped abruptly, and clasped her hands before her face.\n\n\"Now you upbraid me,\" said the Count, unruffled by her sudden passion,\n\"because I gave you in marriage to a man young and noble?\"\n\n\"Old in vices, and mean of soul! The marriage I forgave you. You had\nthe right, according to the customs of our country, to dispose of my\nhand. But I forgave you not the consolations that you whispered in the\near of a wretched and insulted wife.\"\n\n\"Pardon me the remark,\" replied the Count, with a courtly bend of his\nhead, \"but those consolations were also conformable to the customs of\nour country, and I was not aware till now that you had wholly disdained\nthem. And,\" continued the Count, \"you were not so long a wife that the\ngall of the chain should smart still. You were soon left a widow--free,\nchildless, young, beautiful.\"\n\n\"And penniless.\"\n\n\"True, Di Negra was a gambler, and very unlucky; no fault of mine. I\ncould neither keep the cards from his hands, nor advise him how to play\nthem.\"\n\n\"And my own portion? Oh Giulio, I knew but at his death why you had\ncondemned me to that renegade Genoese. He owed you money, and, against\nhonour, and I believe against law, you had accepted my fortune in\ndischarge of the debt.\"\n\n\"He had no other way to discharge it--a debt of honour must be\npaid--old stories these. What matters? Since then my purse has been\nopen to you.\"\n\n\"Yes, not as your sister, but your instrument--your spy! Yes, your\npurse has been open--with a niggard hand.\"\n\n\"_Un peu de conscience, ma chere_, you are so extravagant. But come, be\nplain. What would you?\"\n\n\"I would be free from you.\"\n\n\"That is, you would form some second marriage with one of these rich\nisland lords. _Ma foi_, I respect your ambition.\"\n\n\"It is not so high. I aim but to escape from slavery--to be placed\nbeyond dishonourable temptation. I desire,\" cried Beatrice with\nincreased emotion, \"I desire to re-enter the life of woman.\"\n\n\"Eno'!\" said the Count with a visible impatience, \"is there anything\nin the attainment of your object that should render you indifferent to\nmine? You desire to marry, if I comprehend you right. And to marry, as\nbecomes you, you should bring to your husband not debts, but a dowry.\nBe it so. I will restore the portion that I saved from the spendthrift\nclutch of the Genoese--the moment that it is mine to bestow--the moment\nthat I am husband to my kinsman's heiress. And now, Beatrice, you\nimply that my former notions revolted your conscience; my present plan\nshould content it; for by this marriage shall our kinsman regain his\ncountry, and repossess, at least, half his lands. And if I am not an\nexcellent husband to the demoiselle, it will be her own fault. I have\nsown my wild oats. _Je suis bon prince_, when I have things a little\nmy own way. It is my hope and my intention, and certainly it will be\nmy interest, to become _digne epoux et irreprochable pere de famille_.\nI speak lightly--'tis my way. I mean seriously. The little girl will\nbe very happy with me, and I shall succeed in soothing all resentment\nher father may retain. Will you aid me then--yes or no? Aid me, and\nyou shall indeed be free. The magician will release the fair spirit\nhe has bound to his will. Aid me not, _ma chere_, and mark, I do not\nthreaten--I do but warn--aid me not; grant that I become a beggar, and\nask yourself what is to become of you--still young, still beautiful,\nand still penniless? Nay, worse than penniless; you have done me the\nhonour, (and here the Count, looking on the table, drew a letter from a\nportfolio, emblazoned with his arms and coronet,) you have done me the\nhonour to consult me as to your debts.\"\n\n\"You will restore my fortune?\" said the Marchesa, irresolutely--and\naverting her head from an odious schedule of figures.\n\n\"When my own, with your aid, is secured.\"\n\n\"But do you not overrate the value of my aid?\"\n\n\"Possibly,\" said the Count, with a caressing suavity--and he kissed his\nsister's forehead. \"Possibly; but by my honour, I wish to repair to you\nany wrong, real or supposed, I may have done you in past times. I wish\nto find again my own dear sister. I may overvalue your aid, but not the\naffection from which it comes. Let us be friends, _cara Beatrice mia_,\"\nadded the Count, for the first time employing Italian words.\n\nThe Marchesa laid her head on his shoulder, and her tears flowed\nsoftly. Evidently this man had great influence over her--and evidently,\nwhatever her cause for complaint, her affection for him was still\nsisterly and strong. A nature with fine flashes of generosity, spirit,\nhonour, and passion, was hers--but uncultured, unguided--spoilt by\nthe worst social examples--easily led into wrong--not always aware\nwhere the wrong was--letting affections good or bad whisper away her\nconscience or blind her reason. Such women are often far more dangerous\nwhen induced to wrong, than those who are thoroughly abandoned--such\nwomen are the accomplices men like the Count of Peschiera most desire\nto obtain.\n\n\"Ah, Giulio,\" said Beatrice, after a pause, and looking up at him\nthrough her tears, \"when you speak to me thus, you know you can do with\nme what you will. Fatherless and motherless, whom had my childhood to\nlove and obey but you?\"\n\n\"Dear Beatrice,\" murmured the Count tenderly--and he again kissed her\nforehead. \"So,\" he continued more carelessly--\"so the reconciliation\nis effected, and our interests and our hearts re-allied. Now, alas! to\ndescend to business. You say that you know some one whom you believe\nto be acquainted with the lurking-place of my father-in-law--that is to\nbe!\"\n\n\"I think so. You remind me that I have an appointment with him this\nday: it is near the hour--I must leave you.\"\n\n\"To learn the secret?--Quick--quick. I have no fear of your success, if\nit is by his heart that you lead him?\"\n\n\"You mistake; on his heart I have no hold. But he has a friend who\nloves me, and honourably, and whose cause he pleads. I think here that\nI have some means to control or persuade him. If not--ah, he is of a\ncharacter that perplexes me in all but his worldly ambition; and how\ncan we foreigners influence him through _that_?\"\n\n\"Is he poor, or is he extravagant?\"\n\n\"Not extravagant, and not positively poor, but dependent.\"\n\n\"Then we have him,\" said the Count composedly. \"If his assistance be\nworth buying, we can bid high for it. _Sur mon ame_, I never yet knew\nmoney fail with any man who was both worldly and dependent. I put him\nand myself in your hands.\"\n\nThus saying, the Count opened the door, and conducted his sister with\nformal politeness to her carriage. He then returned, reseated himself,\nand mused in silence. As he did so, the muscles of his countenance\nrelaxed. The levity of the Frenchman fled from his visage, and in\nhis eye, as it gazed abstractedly into space, there was that steady\ndepth so remarkable in the old portraits of Florentine diplomatist or\nVenetian oligarch. Thus seen, there was in that face, despite all its\nbeauty, something that would have awed back even the fond gaze of love;\nsomething hard, collected, inscrutable, remorseless. But this change\nof countenance did not last long. Evidently thought, though intense\nfor the moment, was not habitual to the man. Evidently he had lived\nthe life which takes all things lightly--so he rose with a look of\nfatigue, shook and stretched himself, as if to cast off, or grow out\nof, an unwelcome and irksome mood. An hour afterwards, the Count of\nPeschiera was charming all eyes, and pleasing all ears, in the saloon\nof a high-born beauty, whose acquaintance he had made at Vienna, and\nwhose charms, according to that old and never-truth-speaking oracle,\nPolite Scandal, were now said to have attracted to London the brilliant\nforeigner.\n\n\nCHAPTER III.\n\nThe Marchesa regained her house, which was in Curzon Street, and\nwithdrew to her own room, to readjust her dress, and remove from her\ncountenance all trace of the tears she had shed.\n\nHalf-an-hour afterwards she was seated in her drawing-room, composed\nand calm; nor, seeing her then, could you have guessed that she was\ncapable of so much emotion and so much weakness. In that stately\nexterior, in that quiet attitude, in that elaborate and finished\nelegance which comes alike from the arts of the toilet and the\nconventional repose of rank, you could see but the woman of the world\nand the great lady.\n\nA knock at the door was heard, and in a few moments there entered a\nvisitor, with the easy familiarity of intimate acquaintance--a young\nman, but with none of the bloom of youth. His hair, fine as a woman's,\nwas thin and scanty, but it fell low over the forehead, and concealed\nthat noblest of our human features. \" A gentleman,\" says Apuleius,\n\"ought, if he can, to wear his whole mind on his forehead.\"[5] The\nyoung visitor would never have committed so frank an imprudence. His\ncheek was pale, and in his step and his movements there was a langour\nthat spoke of fatigued nerves or delicate health. But the light of\nthe eye and the tone of the voice were those of a mental temperament\ncontrolling the bodily--vigorous and energetic. For the rest, his\ngeneral appearance was distinguished by a refinement alike intellectual\nand social. Once seen, you would not easily forget him. And the reader\nno doubt already recognises Randal Leslie. His salutation, as I before\nsaid, was that of intimate familiarity; yet it was given and replied\nto with that unreserved openness which denotes the absence of a more\ntender sentiment.\n\n[5] I must be pardoned for annexing the original, since it loses much\nby translation:--\"Hominem liberum et magnificum debere, si queat, in\nprimori fronte, animum gestare.\"\n\nSeating himself by the Marchesa's side, Randal began first to converse\non the fashionable topics and gossip of the day; but it was observable,\nthat, while he extracted from her the current anecdote and scandal\nof the great world, neither anecdote nor scandal did he communicate\nin return. Randal Leslie had already learned the art not to commit\nhimself, nor to have quoted against him one ill-natured remark upon the\neminent. Nothing more injures the man who would rise beyond the fame\nof the _salons_, than to be considered backbiter and gossip; 'yet it\nis always useful,' thought Randal Leslie, 'to know the foibles--the\nsmall social and private springs by which the great are moved. Critical\noccasions may arise in which such knowledge may be power.' And hence,\nperhaps, (besides a more private motive, soon to be perceived,) Randal\ndid not consider his time thrown away in cultivating Madame di Negra's\nfriendship. For despite much that was whispered against her, she had\nsucceeded in dispelling the coldness with which she had at first been\nreceived in the London circles. Her beauty, her grace, and her high\nbirth, had raised her into fashion, and the homage of men of the first\nstation, while it perhaps injured her reputation as woman, added to her\ncelebrity as fine lady. So much do we cold English, prudes though we\nbe, forgive to the foreigner what we avenge on the native.\n\nSliding at last from these general topics into very well-bred and\nelegant personal compliment, and reciting various eulogies, which Lord\nthis and the Duke of that had passed on the Marchesa's charms, Randal\nlaid his hand on hers, with the license of admitted friendship, and\nsaid--\n\n\"But since you have deigned to confide in me, since when (happily\nfor me, and with a generosity of which no coquette could have been\ncapable) you, in good time, repressed into friendship feelings that\nmight else have ripened into those you are formed to inspire and\ndisdain to return, you told me with your charming smile, 'Let no one\nspeak to me of love who does not offer me his hand, and with it the\nmeans to supply tastes that I fear are terribly extravagant;'--since\nthus you allowed me to divine your natural objects, and upon that\nunderstanding our intimacy has been founded, you will pardon me for\nsaying that the admiration you excite amongst these _grands seigneurs_\nI have named, only serves to defeat your own purpose, and scare away\nadmirers less brilliant, but more in earnest. Most of these gentlemen\nare unfortunately married; and they who are not belong to those\nmembers of our aristocracy who, in marriage, seek more than beauty and\nwit--namely, connections to strengthen their political station, or\nwealth to redeem a mortgage and sustain a title.\"\n\n\"My dear Mr Leslie,\" replied the Marchesa--and a certain sadness might\nbe detected in the tone of the voice and the droop of the eye--\"I have\nlived long enough in the real world to appreciate the baseness and the\nfalsehood of most of those sentiments which take the noblest names. I\nsee through the hearts of the admirers you parade before me, and know\nthat not one of them would shelter with his ermine the woman to whom he\ntalks of his heart. Ah,\" continued Beatrice, with a softness of which\nshe was unconscious, but which might have been extremely dangerous to\nyouth less steeled and self-guarded than was Randal Leslie's--\"ah,\nI am less ambitious than you suppose. I have dreamed of a friend, a\ncompanion, a protector, with feelings still fresh, undebased by the\nlow round of vulgar dissipation and mean pleasures--of a heart so new,\nthat it might restore my own to what it was in its happy spring. I have\nseen in your country some marriages, the mere contemplation of which\nhas filled my eyes with delicious tears. I have learned in England to\nknow the value of home. And with such a heart as I describe, and such\na home, I could forget that I ever knew a less pure ambition.\"\n\n\"This language does not surprise me,\" said Randal; \"yet it does not\nharmonise with your former answer to me.\"\n\n\"To you,\" repeated Beatrice smiling, and regaining her lighter manner;\n\"to you--true. But I never had the vanity to think that your affection\nfor me could bear the sacrifices it would cost you in marriage; that\nyou, with your ambition, could bound your dreams of happiness to home.\nAnd then, too,\" said she, raising her head, and with a certain grave\npride in her air--\"and _then_, I could not have consented to share my\nfate with one whom my poverty would <DW36>. I could not listen to my\nheart, if it had beat for a lover without fortune, for to him I could\nthen have brought but a burden, and betrayed him into a union with\npoverty and debt. _Now_, it may be different. Now I may have the dowry\nthat befits my birth. And now I may be free to choose according to my\nheart as woman, not according to my necessities, as one poor, harassed,\nand despairing.\"\n\n\"Ah,\" said Randal, interested, and drawing still closer towards his\nfair companion--\"ah, I congratulate you sincerely; you have cause,\nthen, to think that you shall be--rich?\"\n\nThe Marchesa paused before she answered, and during that pause Randal\nrelaxed the web of the scheme which he had been secretly weaving, and\nrapidly considered whether, if Beatrice di Negra would indeed be rich,\nshe might answer to himself as a wife; and in what way, if so, he had\nbest change his tone from that of friendship into that of love. While\nthus reflecting, Beatrice answered--\n\n\"Not rich for an Englishwoman; for an Italian, yes. My fortune should\nbe half a million--\"\n\n\"Half a million!\" cried Randal, and with difficulty he restrained\nhimself from falling at her feet in adoration.\n\n\"Of francs!\" continued the Marchesa.\n\n\"Francs! Ah,\" said Randal, with a long-drawn breath, and recovering\nfrom his sudden enthusiasm, \"about twenty thousand pounds!--eight\nhundred a-year at four per cent. A very handsome portion,\ncertainly--(Genteel poverty! he murmured to himself. What an escape\nI have had! but I see--I see. This will smooth all difficulties in\nthe way of my better and earlier project. I see)--a very handsome\nportion,\" he repeated aloud--\"not for a _grand seigneur_, indeed, but\nstill for a gentleman of birth and expectations worthy of your choice,\nif ambition be not your first object. Ah, while you spoke with such\nendearing eloquence of feelings that were fresh, of a heart that was\nnew, of the happy English home, you might guess that my thoughts ran to\nmy friend who loves you so devotedly, and who so realises your ideal.\nProverbially, with us, happy marriages and happy homes are found not\nin the gay circles of London fashion, but at the hearths of our rural\nnobility--our untitled country gentlemen. And who, amongst all your\nadorers, can offer you a lot so really enviable as the one whom, I see\nby your blush, you already guess that I refer to?\"\n\n\"Did I blush?\" said the Marchesa, with a silvery laugh. \"Nay, I think\nthat your zeal for your friend misled you. But I will own frankly, I\nhave been touched by his honest ingenuous love--so evident, yet rather\nlooked than spoken. I have contrasted the love that honours me with the\nsuitors that seek to degrade; more I cannot say. For though I grant\nthat your friend is handsome, high-spirited, and generous, still he is\nnot what--\"\n\n\"You mistake, believe me,\" interrupted Randal. \"You shall not finish\nyour sentence. He _is_ all that you do not yet suppose him; for his\nshyness, and his very love, his very respect for your superiority, do\nnot allow his mind and his nature to appear to advantage. You, it is\ntrue, have a taste for letters and poetry rare among your countrywomen.\nHe has not at present--few men have. But what Cimon would not be\nrefined by so fair an Iphigenia? Such frivolities as he now shows\nbelong but to youth and inexperience of life. Happy the brother who\ncould see his sister the wife of Frank Hazeldean.\"\n\nThe Marchesa leant her cheek on her hand in silence. To her, marriage\nwas more than it usually seems to dreaming maiden or to disconsolate\nwidow. So had the strong desire to escape from the control of her\nunprincipled and remorseless brother grown a part of her very soul--so\nhad whatever was best and highest in her very mixed and complex\ncharacter been galled and outraged by her friendless and exposed\nposition, the equivocal worship rendered to her beauty, the various\ndebasements to which pecuniary embarrassments had subjected her--(not\nwithout design on the part of the Count, who, though grasping, was not\nmiserly, and who by precarious and seemingly capricious gifts at one\ntime, and refusals of all aid at another, had involved her in debt in\norder to retain his hold on her)--so utterly painful and humiliating to\na woman of her pride and her birth was the station that she held in the\nworld--that in marriage she saw liberty, life, honour, self-redemption;\nand these thoughts, while they compelled her to co-operate with the\nschemes, by which the Count, on securing to himself a bride, was to\nbestow on herself a dower, also disposed her now to receive with favour\nRandal Leslie's pleadings on behalf of his friend.\n\nThe advocate saw that he had made an impression, and with the\nmarvellous skill which his knowledge of those natures that engaged\nhis study bestowed on his intelligence, he continued to improve his\ncause by such representations as were likely to be most effective.\nWith what admirable tact he avoided panegyric of Frank as the mere\nindividual, and drew him rather as the type, the ideal of what a woman\nin Beatrice's position might desire, in the safety, peace, and honour\nof a home, in the trust, and constancy, and honest confiding love of\nits partner! He did not paint an elysium; he described a haven; he did\nnot glowingly delineate a hero of romance--he soberly portrayed that\nRepresentative of the Respectable and the Real which a woman turns to\nwhen romance begins to seem to her but delusion. Verily, if you could\nhave looked into the heart of the person he addressed, and heard him\nspeak, you would have cried admiringly, \"Knowledge _is_ power; and this\nman, if as able on a larger field of action, should play no mean part\nin the history of his time.\"\n\nSlowly Beatrice roused herself from the reveries which crept over her\nas he spoke--slowly, and with a deep sigh, and said--\n\n\"Well, well, grant all you say; at least before I can listen to so\nhonourable a love, I must be relieved from the base and sordid pressure\nthat weighs on me. I cannot say to the man who wooes me, 'Will you pay\nthe debts of the daughter of Franzini, and the widow of di Negra?'\"\n\n\"Nay, your debts, surely, make so slight a portion of your dowry.\"\n\n\"But the dowry has to be secured;\" and here, turning the tables upon\nher companion, as the apt proverb expresses it, Madame di Negra\nextended her hand to Randal, and said in her most winning accents, \"You\nare, then, truly and sincerely my friend?\"\n\n\"Can you doubt it?\"\n\n\"I prove that I do not, for I ask your assistance.\"\n\n\"Mine? How?\"\n\n\"Listen; my brother has arrived in London--\"\n\n\"I see that arrival announced in the papers.\"\n\n\"And he comes, empowered by the consent of the Emperor, to ask the hand\nof a relation and countrywoman of his; an alliance that will heal long\nfamily dissensions, and add to his own fortunes those of an heiress.\nMy brother, like myself, has been extravagant. The dowry which by law\nhe still owes me it would distress him to pay till this marriage be\nassured.\"\n\n\"I understand,\" said Randal. \"But how can I aid this marriage?\"\n\n\"By assisting us to discover the bride. She, with her father, sought\nrefuge and concealment in England.\"\n\n\"The father had, then, taken part in some political disaffections, and\nwas proscribed?\"\n\n\"Exactly so; and so well has he concealed himself that he has baffled\nall our efforts to discover his retreat. My brother can obtain him his\npardon in cementing this alliance--\"\n\n\"Proceed.\"\n\n\"Ah Randal, Randal, is this the frankness of friendship? You know\nthat I have before sought to obtain the secret of our relation's\nretreat--sought in vain to obtain it from Mr Egerton, who assuredly\nknows it--\"\n\n\"But who communicates no secrets to living man,\" said Randal, almost\nbitterly; \"who, close and compact as iron, is as little malleable to me\nas to you.\"\n\n\"Pardon me. I know you so well that I believe you could attain to any\nsecret you sought earnestly to acquire. Nay, more, I believe that you\nknow already that secret which I ask you to share with me.\"\n\n\"What on earth makes you think so?\"\n\n\"When, some weeks ago, you asked me to describe the personal appearance\nand manners of the exile, which I did partly from the recollections\nof my childhood, partly from the description given to me by others,\nI could not but notice your countenance, and remark its change; in\nspite,\" said the Marchesa, smiling, and watching Randal while she\nspoke--\"in spite of your habitual self-command. And when I pressed\nyou to own that you had actually seen some one who tallied with that\ndescription, your denial did not deceive me. Still more, when returning\nrecently, of your own accord, to the subject, you questioned me so\nshrewdly as to my motives in seeking the clue to our refugees, and I\ndid not then answer you satisfactorily, I could detect--\"\n\n\"Ha, ha,\" interrupted Randal, with the low soft laugh by which\noccasionally he infringed upon Lord Chesterfield's recommendations to\nshun a merriment so natural as to be ill-bred,--\"ha, ha, you have the\nfault of all observers too minute and refined. But even granting that\nI may have seen some Italian exiles, (which is likely enough,) what\ncould be more simple than my seeking to compare your description with\ntheir appearance; and granting that I might suspect some one amongst\nthem to be the man you search for, what more simple, also, than that\nI should desire to know if you meant him harm or good in discovering\nhis 'whereabout?' For ill,\" added Randal, with an air of prudery,\n\"ill would it become me to betray, even to friendship, the retreat of\none who would hide from persecution; and even if I did so--for honour\nitself is a weak safeguard against your fascinations--such indiscretion\nmight be fatal to my future career.\"\n\n\"How?\"\n\n\"Do you not say that Egerton knows the secret, yet will not\ncommunicate?--and is he a man who would ever forgive in me an\nimprudence that committed himself? My dear friend, I will tell you\nmore. When Audley Egerton first noticed my growing intimacy with you,\nhe said, with his usual dryness of counsel, 'Randal, I do not ask you\nto discontinue acquaintance with Madame di Negra--for an acquaintance\nwith women like her forms the manners and refines the intellect; but\ncharming women are dangerous, and Madame di Negra is--a charming\nwoman.'\"\n\nThe Marchesa's face flushed. Randal resumed: \"'Your fair acquaintance'\n(I am still quoting Egerton) 'seeks to discover the home of a\ncountryman of hers. She suspects that I know it. She may try to learn\nit through you. Accident may possibly give you the information she\nrequires. Beware how you betray it. By one such weakness I should judge\nof your general character. He from whom a woman can extract a secret\nwill never be fit for public life.' Therefore, my dear Marchesa, even\nsupposing I possess this secret, you would be no true friend of mine\nto ask me to reveal what would emperil all my prospects. For as yet,\"\nadded Randal, with a gloomy shade on his brow,--\"as yet I do not stand\nalone and erect--I _lean_;--I am dependent.\"\n\n\"There may be a way,\" replied Madame di Negra, persisting, \"to\ncommunicate this intelligence, without the possibility of Mr Egerton's\ntracing our discovery to yourself; and, though I will not press you\nfarther, I add this--You urge me to accept your friend's hand; you seem\ninterested in the success of his suit, and you plead it with a warmth\nthat shows how much you regard what you suppose is his happiness;\nI will never accept his hand till I can do so without blush for my\npenury--till my dowry is secured, and that can only be by my brother's\nunion with the exile's daughter. For your friend's sake, therefore,\nthink well how you can aid me in the first step to that alliance. The\nyoung lady once discovered, and my brother has no fear for the success\nof his suit.\"\n\n\"And you would marry Frank if the dower was secured?\"\n\n\"Your arguments in his favour seem irresistible,\" replied Beatrice,\nlooking down.\n\nA flash went from Randal's eyes, and he mused a few moments.\n\nThen slowly rising, and drawing on his gloves, he said--\n\n\"Well, at least you so far reconcile my honour towards aiding your\nresearch, that you now inform me you mean no ill to the exile.\"\n\n\"Ill!--the restoration to fortune, honours, his native land.\"\n\n\"And you so far enlist my heart on your side, that you inspire me with\nthe hope to contribute to the happiness of two friends whom I dearly\nlove. I will, therefore, diligently seek to ascertain if, among the\nrefugees I have met with, lurk those whom you seek; and if so, I will\nthoughtfully consider how to give you the clue. Meanwhile, not one\nincautious word to Egerton.\"\n\n\"Trust me--I am a woman of the world.\"\n\nRandal now had gained the door. He paused, and renewed carelessly--\n\n\"This young lady must be heiress to great wealth, to induce a man of\nyour brother's rank to take so much pains to discover her.\"\n\n\"Her wealth _will_ be vast,\" replied the Marchesa; \"and if anything\nfrom wealth or influence in a foreign state could be permitted to prove\nmy brother's gratitude--\"\n\n\"Ah, fie,\" interrupted Randal, and approaching Madame di Negra, he\nlifted her hand to his lips, and said gallantly,\n\n\"This is reward enough to your _preux chevalier_.\"\n\nWith those words he took his leave.\n\n\nCHAPTER IV.\n\nWith his hands behind him, and his head drooping on his breast--slow,\nstealthy, noiseless, Randal Leslie glided along the streets on leaving\nthe Italian's house. Across the scheme he had before revolved, there\nglanced another yet more glittering, for its gain might be more sure\nand immediate. If the exile's daughter were heiress to such wealth,\nmight he himself hope----. He stopped short even in his own soliloquy,\nand his breath came quick. Now, in his last visit to Hazeldean, he\nhad come in contact with Riccabocca, and been struck by the beauty\nof Violante. A vague suspicion had crossed him that these might be\nthe persons of whom the Marchesa was in search, and the suspicion had\nbeen confirmed by Beatrice's description of the refugee she desired to\ndiscover. But as he had not then learned the reason for her inquiries,\nnor conceived the possibility that he could have any personal interest\nin ascertaining the truth, he had only classed the secret in question\namong those the farther research into which might be left to time and\noccasion. Certainly the reader will not do the unscrupulous intellect\nof Randal Leslie the injustice to suppose that he was deterred from\nconfiding to his fair friend all that he knew of Riccabocca, by the\nrefinement of honour to which he had so chivalrously alluded. He had\ncorrectly stated Audley Egerton's warning against any indiscreet\nconfidence, though he had forborne to mention a more recent and direct\nrenewal of the same caution. His first visit to Hazeldean had been\npaid without consulting Egerton. He had been passing some days at\nhis father's house, and had gone over thence to the Squire's. On his\nreturn to London, he had, however, mentioned this visit to Audley, who\nhad seemed annoyed and even displeased at it, though Randal well knew\nsufficient of Egerton's character to know that such feelings could\nscarce be occasioned merely by his estrangement from his half brother.\nThis dissatisfaction had, therefore, puzzled the young man. But as it\nwas necessary to his views to establish intimacy with the Squire, he\ndid not yield the point with his customary deference to his patron's\nwhims. He, therefore, observed, that he should be very sorry to do\nanything displeasing to his benefactor, but that his father had been\nnaturally anxious that he should not appear positively to slight the\nfriendly overtures of Mr Hazeldean.\n\n\"Why naturally?\" asked Egerton.\n\n\"Because you know that Mr Hazeldean is a relation of mine--that my\ngrandmother was a Hazeldean.\"\n\n\"Ah!\" said Egerton, who, as it has been before said, knew little, and\ncared less, about the Hazeldean pedigree, \"I was either not aware of\nthat circumstance, or had forgotten it. And your father thinks that the\nSquire may leave you a legacy?\"\n\n\"Oh, sir, my father is not so mercenary--such an idea never entered\nhis head. But the Squire himself has indeed said--'Why, if anything\nhappened to Frank, you would be next heir to my lands, and therefore we\nought to know each other.' But--\"\n\n\"Enough,\" interrupted Egerton, \"I am the last man to pretend to the\nright of standing between you and a single chance of fortune, or of aid\nto it. And whom did you meet at Hazeldean?\"\n\n\"There was no one there, sir; not even Frank.\"\n\n\"Hum. Is the Squire not on good terms with his parson? Any quarrel\nabout tithes?\"\n\n\"Oh, no quarrel. I forgot Mr Dale; I saw him pretty often. He admires\nand praises you very much, sir.\"\n\n\"Me--and why? What did he say of me?\"\n\n\"That your heart was as sound as your head; that he had once seen you\nabout some old parishioners of his; and that he had been much impressed\nwith a depth of feeling he could not have anticipated in a man of the\nworld, and a statesman.\"\n\n\"Oh, that was all; some affair when I was member for Lansmere?\"\n\n\"I suppose so.\"\n\nHere the conversation had broken off; but the next time Randal was led\nto visit the Squire he had formally asked Egerton's consent, who, after\na moment's hesitation, had as formally replied, \"I have no objection.\"\n\nOn returning from this visit, Randal mentioned that he had seen\nRiccabocca; and Egerton, a little startled at first, said composedly,\n\"Doubtless one of the political refugees; take care not to set Madame\ndi Negra on his track. Remember, she is suspected of being a spy of the\nAustrian government.\"\n\n\"Rely on me, sir,\" said Randal; \"but I should think this poor Doctor\ncan scarcely be the person she seeks to discover.\"\n\n\"That is no affair of ours,\" answered Egerton; \"we are English\ngentlemen, and make not a step towards the secrets of another.\"\n\nNow, when Randal revolved this rather ambiguous answer, and recalled\nthe uneasiness with which Egerton had first heard of his visit to\nHazeldean, he thought that he was indeed near the secret which Egerton\ndesired to conceal from him and from all--viz., the incognito of the\nItalian whom Lord l'Estrange had taken under his protection.\n\n\"My cards,\" said Randal to himself, as, with a deep-drawn sigh, he\nresumed his soliloquy, \"are become difficult to play. On the one\nhand, to entangle Frank into marriage with this foreigner, the Squire\ncould never forgive him. On the other hand, if she will not marry him\nwithout the dowry--and that depends on her brother's wedding this\ncountrywoman--and that countrywoman be, as I surmise, Violante--and\nViolante be this heiress, and to be won by me! Tush, tush. Such\ndelicate scruples in a woman so placed and so constituted as Beatrice\ndi Negra, must be easily talked away. Nay, the loss itself of this\nalliance to her brother, the loss of her own dowry--the very pressure\nof poverty and debt--would compel her into the sole escape left to\nher option. I will then follow up the old plan; I will go down to\nHazeldean, and see if there be any substance in the new one;--and then\nto reconcile both--aha--the House of Leslie shall rise yet from its\nruin--and--\"\n\nHere he was startled from his reverie by a friendly slap on the\nshoulder, and an exclamation--\"Why, Randal, you are more absent than\nwhen you used to steal away from the cricket ground, muttering Greek\nverses at Eton.\"\n\n\"My dear Frank,\" said Randal, \"you--you are so _brusque_, and I was\njust thinking of you.\"\n\n\"Were you? And kindly, then, I am sure,\" said Frank Hazeldean, his\nhonest handsome face lighted up with the unsuspecting genial trust of\nfriendship; \"and heaven knows,\" he added, with a sadder voice, and a\ngraver expression on his eye and lip--\"heaven knows I want all the\nkindness you can give me!\"\n\n\"I thought,\" said Randal, \"that your father's last supply, of which\nI was fortunate enough to be the bearer, would clear off your more\npressing debts. I don't pretend to preach, but really I must say once\nmore, you should not be so extravagant.\"\n\nFRANK, (seriously).--\"I have done my best to reform. I have sold off my\nhorses, and I have not touched dice nor card these six months: I would\nnot even put into the raffle for the last Derby.\" This last was said\nwith the air of a man who doubted the possibility of obtaining belief\nto some assertion of preternatural abstinence and virtue.\n\nRANDAL.--\"Is it possible? But, with such self-conquest, how is it\nthat you cannot contrive to live within the bounds of a very liberal\nallowance?\"\n\nFRANK, (despondingly).--\"Why, when a man once gets his head under\nwater, it is so hard to float back again on the surface. You see, I\nattribute all my embarrassments to that first concealment of my debts\nfrom my father, when they could have been so easily met, and when he\ncame up to town so kindly.\"\n\n\"I am sorry, then, that I gave you that advice.\"\n\n\"Oh you meant it so kindly, I don't reproach you; it was all my own\nfault.\"\n\n\"Why, indeed, I did urge you to pay off that moiety of your debts left\nunpaid, with your allowance. Had you done so, all had been well.\"\n\n\"Yes, but poor Borrowwell got into such a scrape at Goodwood; I could\nnot resist him--a debt of honour, _that_ must be paid; so when I signed\nanother bill for him, he could not pay it, poor fellow: really he would\nhave shot himself, if I had not renewed it; and now it is swelled to\nsuch an amount with that cursed interest, that _he_ never can pay it;\nand one bill, of course, begets another, and to be renewed every three\nmonths; 'tis the devil and all! So little as I ever got for all I have\nborrowed,\" added Frank with a kind of rueful amaze. \"Not L1500 ready\nmoney; and it would cost me almost as much yearly,--if I had it.\"\n\n\"Only L1500.\"\n\n\"Well, besides seven large chests of the worst cigars you ever smoked;\nthree pipes of wine that no one would drink, and a great bear, that had\nbeen imported from Greenland for the sake of its grease.\"\n\n\"That should at least have saved you a bill with your hairdresser.\"\n\n\"I paid his bill with it,\" said Frank, \"and very good-natured he was\nto take the monster off my hands; it had already hugged two soldiers\nand one groom into the shape of a flounder. I tell you what,\" resumed\nFrank, after a short pause, \"I have a great mind even now to tell my\nfather honestly all my embarrassments.\"\n\nRANDAL, (solemnly.)--\"Hum!\"\n\nFRANK.--\"What? don't you think it would be the best way? I never\ncan save enough--never can pay off what I owe; and it rolls like a\nsnowball.\"\n\nRANDAL.--\"Judging by the Squire's talk, I think that with the first\nsight of your affairs you would forfeit his favour for ever; and your\nmother would be so shocked, especially after supposing that the sum I\nbrought you so lately sufficed to pay off every claim on you. If you\nhad not assured her of that, it might be different; but she who so\nhates an untruth, and who said to the Squire, 'Frank says this will\nclear him; and with all his faults, Frank never yet told a lie.'\"\n\n\"Oh my dear mother!--I fancy I hear her!\" cried Frank with deep\nemotion. \"But I did not tell a lie, Randal; I did not say that that sum\nwould clear me.\"\n\n\"You empowered and begged me to say so,\" replied Randal with grave\ncoldness; \"and don't blame me if I believed you.\"\n\n\"No, no! I only said it would clear me for the moment.\"\n\n\"I misunderstood you, then, sadly; and such mistakes involve my own\nhonour. Pardon me, Frank; don't ask my aid in future. You see, with the\nbest intentions I only compromise myself.\"\n\n\"If you forsake me, I may as well go and throw myself into the river,\"\nsaid Frank in a tone of despair; \"and sooner or later my father must\nknow my necessities. The Jews threaten to go to him already; and the\nlonger the delay, the more terrible the explanation.\"\n\n\"I don't see why your father should ever learn the state of your\naffairs; and it seems to me that you could pay off these usurers, and\nget rid of these bills, by raising money on comparatively easy terms--\"\n\n\"How?\" cried Frank eagerly.\n\n\"Why, the Casino property is entailed on you, and you might obtain a\nsum upon that, not to be paid till the property becomes yours.\"\n\n\"At my poor father's death? Oh, no--no! I cannot bear the idea\nof this cold-blooded calculation on a father's death. I know it\nis not uncommon; I know other fellows who have done it, but they\nnever had parents so kind as mine; and even in them it shocked and\nrevolted me. The contemplating a father's death and profiting by the\ncontemplation,--it seems a kind of parricide--it is not natural,\nRandal. Besides, don't you remember what the governor said--he actually\nwept while he said it, 'Never calculate on my death; I could not bear\nthat.' Oh, Randal, don't speak of it!\"\n\n\"I respect your sentiments; but still all the post-obits you could\nraise could not shorten Mr Hazeldean's life by a day. However, dismiss\nthat idea; we must think of some other device. Ha, Frank! you are a\nhandsome fellow, and your expectations are great--why don't you marry\nsome woman with money?\"\n\n\"Pooh!\" exclaimed Frank, colouring. \"You know, Randal, that there is\nbut one woman in the world I can ever think of, and I love her so\ndevotedly, that, though I was as gay as most men before, I really feel\nas if the rest of her sex had lost every charm. I was passing through\nthe street now,--merely to look up at her windows--\"\n\n\"You speak of Madame di Negra? I have just left her. Certainly she\nis two or three years older than you; but if you can get over that\nmisfortune, why not marry her?\"\n\n\"Marry her!\" cried Frank in amaze, and all his colour fled from his\ncheeks. \"Marry her!--are you serious?\"\n\n\"Why not?\"\n\n\"But even if she, who is so accomplished, so admired--even if she would\naccept me, she is, you know, poorer than myself. She has told me so\nfrankly. That woman has such a noble heart! and--and--my father would\nnever consent, nor my mother either. I know they would not.\"\n\n\"Because she is a foreigner?\"\n\n\"Yes--partly.\"\n\n\"Yet the Squire suffered his cousin to marry a foreigner.\"\n\n\"That was different. He had no control over Jemima; and a\ndaughter-in-law is so different; and my father is so English in his\nnotions; and Madame di Negra, you see, is altogether so foreign. Her\nvery graces would be against her in his eyes.\"\n\n\"I think you do both your parents injustice. A foreigner of low\nbirth--an actress or singer, for instance--of course would be highly\nobjectionable; but a woman, like Madame di Negra, of such high birth\nand connections--\"\n\nFrank shook his head. \"I don't think the governor would care a straw\nabout her connections, if she were a king's daughter. He considers all\nforeigners pretty much alike. And then, you know\"--Frank's voice sank\ninto a whisper--\"you know that one of the very reasons why she is so\ndear to me would be an insuperable objection to the old fashioned folks\nat home.\"\n\n\"I don't understand you, Frank.\"\n\n\"I love her the more,\" said young Hazeldean, raising his front with\na noble pride, that seemed to speak of his descent from a race of\ncavaliers and gentlemen--\"I love her the more because the world has\nslandered her name--because I believe her to be pure and wronged. But\nwould they at the hall--they who do not see with a lover's eyes--they\nwho have all the stubborn English notions about the indecorum and\nlicense of Continental manners, and will so readily credit the\nworst?--Oh, no--I love--I cannot help it--but I have no hope.\"\n\n\"It is very possible that you may be right,\" exclaimed Randal, as if\nstruck and half-convinced by his companion's argument--\"very possible;\nand certainly I think that the homely folks at the Hall would fret and\nfume at first, if they heard you were married to Madame di Negra. Yet\nstill, when your father learned that you had done so, not from passion\nalone, but to save him from all pecuniary sacrifice--to clear yourself\nof debt--to--\"\n\n\"What do you mean?\" exclaimed Frank impatiently.\n\n\"I have reason to know that Madame di Negra will have as large a\nportion as your father could reasonably expect you to receive with\nany English wife. And when this is properly stated to the Squire, and\nthe high position and rank of your wife fully established and brought\nhome to him--for I must think that these would tell, despite your\nexaggerated notions of his prejudices--and then, when he really sees\nMadame di Negra, and can judge of her beauty and rare gifts, upon my\nword, I think, Frank, that there would be no cause for fear. After all,\ntoo, you are his only son. He will have no option but to forgive you;\nand I know how anxiously both your parents wish to see you settled in\nlife.\"\n\nFrank's whole countenance became illuminated. \"There is no one who\nunderstands the Squire like you, certainly,\" said he, with lively joy.\n\"He has the highest opinion of your judgment. And you really believe\nyou could smooth matters?\"\n\n\"I believe so, but I should be sorry to induce you to run any risk; and\nif, on cool consideration, you think that risk is incurred, I strongly\nadvise you to avoid all occasion of seeing the poor Marchesa. Ah, you\nwince; but I say it for her sake as well as your own. First, you must\nbe aware, that, unless you have serious thoughts of marriage, your\nattentions can but add to the very rumours that, equally groundless,\nyou so feelingly resent; and, secondly, because I don't think any man\nhas a right to win the affections of a woman--especially a woman who\nseems to me likely to love with her whole heart and soul--merely to\ngratify his own vanity.\"\n\n\"Vanity! Good heavens, can you think so poorly of me? But as to the\nMarchesa's affection,\" continued Frank, with a faltering voice, \"do you\nreally and honestly believe that they are to be won by me?\"\n\n\"I fear lest they may be half won already,\" said Randal with a smile\nand a shake of the head; \"but she is too proud to let you see any\neffect you may produce on her, especially when, as I take it for\ngranted, you have never hinted at the hope of obtaining her hand.\"\n\n\"I never till now conceived such a hope. My dear Randal, all my cares\nhave vanished--I tread upon air--I have a great mind to call on her at\nonce.\"\n\n\"Stay, stay,\" said Randal. \"Let me give you a caution. I have just\ninformed you that Madame di Negra will have, what you suspected not\nbefore, a fortune suitable to her birth; any abrupt change in your\nmanner at present might induce her to believe that you were influenced\nby that intelligence.\"\n\n\"Ah!\" exclaimed Frank, stopping short, as if wounded to the quick. \"And\nI feel guilty--feel as if I _was_ influenced by that intelligence. So\nI am, too, when I reflect,\" he continued, with a _naivete_ that was\nhalf pathetic; \"but I hope she will not be _very_ rich--if so, I'll not\ncall.\"\n\n\"Make your mind easy, it is but a portion of some twenty or thirty\nthousand pounds, that would just suffice to discharge all your debts,\nclear away all obstacle to your union, and in return for which you\ncould secure a more than adequate jointure and settlement on the Casino\nproperty. Now I am on that head, I will be yet more communicative.\nMadame di Negra has a noble heart, as you say, and told me herself,\nthat, until her brother on his arrival had assured her of this dowry,\nshe would never have consented to marry you--never crippled with her\nown embarrassments the man she loves. Ah! with what delight she will\nhail the thought of assisting you to win back your father's heart! But\nbe guarded, meanwhile. And now, Frank, what say you--would it not be\nwell if I ran down to Hazeldean to sound your parents? It is rather\ninconvenient to me, to be sure, to leave town just at present; but I\nwould do more than that to render you a smaller service. Yes, I'll go\nto Rood Hall to-morrow, and thence to Hazeldean. I am sure your father\nwill press me to stay, and I shall have ample opportunities to judge of\nthe manner in which he would be likely to regard your marriage with\nMadame di Negra--supposing always it were properly put to him. We can\nthen act accordingly.\"\n\n\"My dear, dear Randal. How can I thank you? If ever a poor fellow like\nme can serve you in return--but that's impossible.\"\n\n\"Why, certainly, I will never ask you to be security to a bill of\nmine,\" said Randal, laughing. \"I practise the economy I preach.\"\n\n\"Ah!\" said Frank, with a groan, \"that is because your mind is\ncultivated--you have so many resources; and all my faults have come\nfrom idleness. If I had had anything to do on a rainy day, I should\nnever have got into these scrapes.\"\n\n\"Oh! you will have enough to do some day managing your property. We who\nhave no property must find one in knowledge. Adieu, my dear Frank; I\nmust go home now. By the way, you have never, by chance, spoken of the\nRiccaboccas to Madame di Negra?\"\n\n\"The Riccaboccas? No. That's well thought of. It may interest her to\nknow that a relation of mine has married her countryman. Very odd that\nI never did mention it; but, to say truth, I really do talk so little\nto her; she is so superior, and I feel positively shy with her.\"\n\n\"Do me the favour, Frank,\" said Randal, waiting patiently till this\nreply ended--for he was devising all the time what reason to give for\nhis request--\"never to allude to the Riccaboccas either to her or to\nher brother, to whom you are sure to be presented.\"\n\n\"Why not allude to them?\"\n\nRandal hesitated a moment. His invention was still at fault, and, for a\nwonder, he thought it the best policy to go pretty near the truth.\n\n\"Why, I will tell you. The Marchesa conceals nothing from her brother,\nand he is one of the few Italians who are in high favour with the\nAustrian court.\"\n\n\"Well!\"\n\n\"And I suspect that poor Dr Riccabocca fled his country from some mad\nexperiment at revolution, and is still hiding from the Austrian police.\"\n\n\"But they can't hurt him here,\" said Frank, with an Englishman's dogged\ninborn conviction of the sanctity of his native island. \"I should like\nto see an Austrian pretend to dictate to us whom to receive and whom to\nreject.\"\n\n\"Hum--that's true and constitutional, no doubt; but Riccabocca may\nhave excellent reasons--and, to speak plainly, I know he has, (perhaps\nas affecting the safety of friends in Italy,)--for preserving his\nincognito, and we are bound to respect those reasons without inquiring\nfurther.\"\n\n\"Still, I cannot think so meanly of Madame di Negra,\" persisted Frank,\n(shrewd here, though credulous elsewhere, and both from his sense of\nhonour,) \"as to suppose that she would descend to be a spy, and injure\na poor countryman of her own, who trusts to the same hospitality she\nreceives herself at our English hands. Oh, if I thought that, I could\nnot love her!\" added Frank, with energy.\n\n\"Certainly you are right. But see in what a false position you would\nplace both her brother and herself. If they knew Riccabocca's secret,\nand proclaimed it to the Austrian government, as you say, it would be\ncruel and mean; but, if they knew it and concealed, it might involve\nthem both in the most serious consequences. You know the Austrian\npolicy is proverbially so jealous and tyrannical?\"\n\n\"Well, the newspapers say so, certainly.\"\n\n\"And, in short, your discretion can do no harm, and your indiscretion\nmay. Therefore, give me your word, Frank. I can't stay to argue now.\"\n\n\"I'll not allude to the Riccaboccas, upon my honour,\" answered Frank;\n\"still, I am sure that they would be as safe with the Marchesa as\nwith\"----\n\n\"I rely on your honour,\" interrupted Randal hastily, and hurried off.\n\n\nCHAPTER V.\n\nTowards the evening of the following day, Randal Leslie walked slowly\nfrom a village in the main road, (about two miles from Rood Hall,)\nat which he had got out of the coach. He passed through meads and\ncorn-fields, and by the skirts of woods which had formerly belonged to\nhis ancestors, but had been long since alienated. He was alone amidst\nthe haunts of his boyhood, the scenes in which he had first invoked the\ngrand Spirit of Knowledge, to bid the Celestial Still One minister to\nthe commands of an earthly and turbulent ambition. He paused often in\nhis path, especially when the undulations of the ground gave a glimpse\nof the grey church tower, or the gloomy firs that rose above the\ndesolate wastes of Rood.\n\n\"Here,\" thought Randal, with a softening eye--\"here, how often,\ncomparing the fertility of the lands passed away from the inheritance\nof my fathers, with the forlorn wilds that are left to their mouldering\nhall--here, how often have I said to myself--'I will rebuild the\nfortunes of my house.' And straightway Toil lost its aspect of drudge,\nand grew kingly, and books became as living armies to serve my thought.\nAgain--again--O thou haughty Past, brace and strengthen me in the\nbattle with the Future.\" His pale lips writhed as he soliloquised, for\nhis conscience spoke to him while he thus addressed his will, and its\nvoice was heard more audibly in the quiet of the rural landscape, than\namidst the turmoil and din of that armed and sleepless camp which we\ncall a city.\n\nDoubtless, though Ambition have objects more vast and beneficent than\nthe restoration of a name,--_that_ in itself is high and chivalrous,\nand appeals to a strong interest in the human heart. But all emotions,\nand all ends, of a nobler character, had seemed to filter themselves\nfree from every golden grain in passing through the mechanism of\nRandal's intellect, and came forth at last into egotism clear and\nunalloyed. Nevertheless, it is a strange truth that, to a man of\ncultivated mind, however perverted and vicious, there are vouchsafed\ngleams of brighter sentiments, irregular perceptions of moral\nbeauty, denied to the brutal unreasoning wickedness of uneducated\nvillany--which perhaps ultimately serve as his punishment--according\nto the old thought of the satirist, that there is no greater curse\nthan to perceive virtue, yet adopt vice. And as the solitary schemer\nwalked slowly on, and his childhood--innocent at least of deed--came\ndistinct before him through the halo of bygone dreams--dreams far purer\nthan those from which he now rose each morning to the active world of\nMan--a profound melancholy crept over him, and suddenly he exclaimed\naloud, \"_Then_ I aspired to be renowned and great--_now_, how is it\nthat, so advanced in my career, all that seemed lofty in the means\nhas vanished from me, and the only means that I contemplate are those\nwhich my childhood would have called poor and vile? Ah! is it that I\nthen read but books, and now my knowledge has passed onward, and men\ncontaminate more than books? But,\" he continued, in a lower voice, as\nif arguing with himself, \"if power is only so to be won--and of what\nuse is knowledge if it be not power--does not success in life justify\nall things? And who prizes the wise man if he fails?\" He continued his\nway, but still the soft tranquillity around rebuked him, and still his\nreason was dissatisfied, as well as his conscience. There are times\nwhen Nature, like a bath of youth, seems to restore to the jaded soul\nits freshness--times from which some men have emerged, as if reborn.\nThe crises of life are very silent. Suddenly the scene opened on Randal\nLeslie's eyes. The bare desert common--the dilapidated church--the\nold house, partially seen in the dank dreary hollow, into which it\nseemed to Randal to have sunken deeper and lowlier than when he saw it\nlast. And on the common were some young men playing at hockey. That\nold-fashioned game, now very uncommon in England, except at schools,\nwas still preserved in the primitive vicinity of Rood by the young\nyeomen and farmers. Randal stood by the style and looked on, for among\nthe players he recognised his brother Oliver. Presently the ball was\nstruck towards Oliver, and the group instantly gathered round that\nyoung gentleman, and snatched him from Randal's eye; but the elder\nbrother heard a displeasing din, a derisive laughter. Oliver had shrunk\nfrom the danger of the thick-clubbed sticks that plied around him, and\nreceived some stroke across the legs, for his voice rose whining,and\nwas drowned by shouts of, \"Go to your mammy. That's Noll Leslie--all\nover. Butter shins.\"\n\nRandal's sallow face became scarlet. \"The jest of boors--a Leslie!\"\nhe muttered, and ground his teeth. He sprang over the stile, and\nwalked erect and haughtily across the ground. The players cried out\nindignantly. Randal raised his hat, and they recognised him, and\nstopped the game. For him at least a certain respect was felt. Oliver\nturned round quickly, and ran up to him. Randal caught his arm firmly,\nand, without saying a word to the rest, drew him away towards the\nhouse. Oliver cast a regretful, lingering look behind him, rubbed his\nshins, and then stole a timid glance towards Randal's severe and moody\ncountenance.\n\n\"You are not angry that I was playing at hockey with our neighbours,\"\nsaid he deprecatingly, observing that Randal would not break the\nsilence.\n\n\"No,\" replied the elder brother; \"but, in associating with his\ninferiors, a gentleman still knows how to maintain his dignity.\nThere is no harm in playing with inferiors, but it is necessary to a\ngentleman to play so that he is not the laughing-stock of clowns.\"\n\nOliver hung his head, and made no answer. They came into the slovenly\nprecincts of the court, and the pigs stared at them from the palings,\nas they had stared, years before, at Frank Hazeldean.\n\nMr Leslie senior, in a shabby straw hat, was engaged in feeding the\nchickens before the threshold, and he performed even that occupation\nwith a maundering lack-a-daisical slothfulness, dropping down the\ngrains almost one by one from his inert dreamy fingers.\n\nRandal's sister, her hair still and for ever hanging about her ears,\nwas seated on a rush-bottom chair, reading a tattered novel; and from\nthe parlour window was heard the querulous voice of Mrs Leslie, in high\nfidget and complaint.\n\nSomehow or other, as the young heir to all this helpless poverty stood\nin the courtyard, with his sharp, refined, intelligent features, and\nhis strange elegance of dress and aspect, one better comprehended how,\nleft solely to the egotism of his knowledge and his ambition, in such a\nfamily, and without any of the sweet nameless lessons of Home, he had\ngrown up into such close and secret solitude of soul--how the mind had\ntaken so little nutriment from the heart, and how that affection and\nrespect which the warm circle of the hearth usually calls forth had\npassed with him to the graves of dead fathers, growing, as it were,\nbloodless and ghoul-like amidst the charnels on which they fed.\n\n\"Ha, Randal, boy,\" said Mr Leslie, looking up lazily, \"how d'ye do?\nWho could have expected you? My dear--my dear,\" he cried, in a broken\nvoice, and as if in helpless dismay, \"here's Randal, and he'll be\nwanting dinner, or supper, or something.\" But, in the meanwhile,\nRandal's sister Juliet had sprung up and thrown her arms round her\nbrother's neck, and he had drawn her aside caressingly, for Randal's\nstrongest human affection was for this sister.\n\n\"You are growing very pretty, Juliet,\" said he, smoothing back her\nhair; \"why do yourself such injustice--why not pay more attention to\nyour appearance, as I have so often begged you to do?\"\n\n\"I did not expect you, dear Randal; you always come so suddenly, and\ncatch us _en dish-a-bill_.\"\n\n\"Dish-a-bill!\" echoed Randal, with a groan. \"_Dishabille!_--you ought\nnever to be so caught!\"\n\n\"No one else does so catch us--nobody else ever comes! Heigho,\" and the\nyoung lady sighed very heartily.\n\n\"Patience, patience; my day is coming, and then yours, my sister,\"\nreplied Randal with genuine pity, as he gazed upon what a little care\ncould have trained into so fair a flower, and what now looked so like a\nweed.\n\nHere Mrs Leslie, in a state of intense excitement--having rushed\nthrough the parlour--leaving a fragment of her gown between the\nyawning brass of the never-mended Brummagem worktable--tore across the\nhall--whirled out of the door, scattering the chickens to the right\nand left, and clutched hold of Randal in her motherly embrace. \"La,\nhow you do shake my nerves,\" she cried, after giving him a most hasty\nand uncomfortable kiss. \"And you are hungry too, and nothing in the\nhouse but cold mutton! Jenny, Jenny, I say Jenny! Juliet, have you seen\nJenny? Where's Jenny? Out with the odd man, I'll be bound.\"\n\n\"I am not hungry, mother,\" said Randal; \"I wish for nothing but tea.\"\nJuliet, scrambling up her hair, darted into the house to prepare the\ntea, and also to \"tidy herself.\" She dearly loved her fine brother, but\nshe was greatly in awe of him.\n\nRandal seated himself on the broken pales. \"Take care they don't come\ndown,\" said Mr Leslie with some anxiety.\n\n\"Oh, sir, I am very light; nothing comes down with me.\"\n\nThe pigs stared up, and grunted in amaze at the stranger.\n\n\"Mother,\" said the young man, detaining Mrs Leslie, who wanted to set\noff in chase of Jenny--\"mother, you should not let Oliver associate\nwith those village boors. It is time to think of a profession for him.\"\n\n\"Oh, he eats us out of house and home--such an appetite! But as to a\nprofession--what is he fit for! He will never be a scholar.\"\n\nRandal nodded a moody assent; for, indeed, Oliver had been sent to\nCambridge, and supported there out of Randal's income from his official\npay;--and Oliver had been plucked for his Little Go.\n\n\"There is the army,\" said the elder brother--\"a gentleman's calling.\nHow handsome Juliet ought to be--but--I left money for masters--and she\npronounces French like a chambermaid.\"\n\n\"Yet she is fond of her book too. She's always reading, and good for\nnothing else.\"\n\n\"Reading!--those trashy novels!\n\n\"So like you--you always come to scold, and make things unpleasant,\"\nsaid Mrs Leslie peevishly. \"You are grown too fine for us, and I am\nsure we suffer affronts enough from others, not to want a little\nrespect from our own children.\"\n\n\"I did not mean to affront you,\" said Randal sadly. \"Pardon me. But who\nelse has done so?\"\n\nThen Mrs Leslie went into a minute and most irritating catalogue of\nall the mortifications and insults she had received; the grievances\nof a petty provincial family, with much pretension and small power;\nof all people, indeed, without the disposition to please--without the\nability to serve--who exaggerate every offence, and are thankful for\nno kindness. Farmer Jones had insolently refused to send his waggon\ntwenty miles for coals. Mr Giles, the butcher, requesting the payment\nof his bill, had stated that the custom at Rood was too small for\nhim to allow credit. Squire Thornhill, who was the present owner of\nthe fairest slice of the old Leslie domains, had taken the liberty\nto ask permission to shoot over Mr Leslie's land, since Mr Leslie\ndid not preserve. Lady Spratt (new people from the city, who hired\na neighbouring country seat) had taken a discharged servant of Mrs\nLeslie's without applying for the character. The Lord-Lieutenant had\ngiven a ball, and had not invited the Leslies. Mr Leslie's tenants had\nvoted against their landlord's wish at the recent election. More than\nall, Squire Hazeldean and his Harry had called at Rood, and though Mrs\nLeslie had screamed out to Jenny, \"Not at home,\" she had been seen at\nthe window, and the Squire had actually forced his way in, and caught\nthe whole family \"in a state not fit to be seen.\" That was a trifle,\nbut the Squire had presumed to instruct Mr. Leslie how to manage his\nproperty, and Mrs Hazeldean had actually told Juliet to hold up her\nhead and tie up her hair, \"as if we were her cottagers!\" said Mrs\nLeslie with the pride of a Montfydget.\n\nAll these and various other annoyances, though Randal was too sensible\nnot to perceive their insignificance, still galled and mortified the\nlistening heir of Rood. They showed, at least, even to the well-meant\nofficiousness of the Hazeldeans, the small account in which the fallen\nfamily was held. As he sat still on the moss-grown pale, gloomy and\ntaciturn, his mother standing beside him, with her cap awry, Mr Leslie\nshamblingly sauntered up, and said in a pensive, dolorous whine--\n\n\"I wish we had a good sum of money, Randal, boy!\"\n\nTo do Mr Leslie justice, he seldom gave vent to any wish that savoured\nof avarice. His mind must be singularly aroused, to wander out of its\nnormal limits of sluggish, dull content.\n\nSo Randal looked at him in surprise, and said, \"Do you, sir?--why?\"\n\n\"The manors of Rood and Dulmansberry, and all the lands therein, which\nmy great-grandfather sold away, are to be sold again when Squire\nThornhill's eldest son comes of age, to cut off the entail. Sir John\nSpratt talks of buying them. I should like to have them back again!\n'Tis a shame to see the Leslie estates hawked about, and bought by\nSpratts and people. I wish I had a great--great sum of ready-money.\"\n\nThe poor gentleman extended his helpless fingers as he spoke, and fell\ninto a dejected reverie.\n\nRandal sprang from the paling, a movement which frightened the\ncontemplative pigs, and set them off squalling and scampering. \"When\ndoes young Thornhill come of age?\"\n\n\"He was nineteen last August. I know it, because the day lie was born\nI picked up my fossil of the sea-horse, just by Dulmansberry church,\nwhen the joy-bells were ringing. My fossil sea-horse! It Will be an\nheirloom, Randal--\"\n\n\"Two years--nearly two years--yet--ah, ah!\" said Randal; and his\nsister now appearing to announce that tea was ready, he threw his\narm round her neck and kissed her. Juliet had arranged her hair and\ntrimmed up her dress. She looked very pretty, and she had now the air\nof a gentlewoman--something of Randal's own refinement in her slender\nproportions and well-shaped head.\n\n\"Be patient, patient still, my dear sister,\" whispered Randal, \"and\nkeep your heart whole for two years longer.\"\n\nThe young man was gay and good-humoured over his simple meal, while\nhis family grouped round him. When it was over, Mr Leslie lighted his\npipe, and called for his brandy and water. Mrs Leslie began to question\nabout London and Court, and the new King and the new Queen, and Mr\nAudley Egerton, and hoped Mr Egerton would leave Randal all his money,\nand that Randal would marry a rich woman, and that the King would make\nhim a prime-minister one of these days; and then she should like to\nsee if Farmer Jones would refuse to send his waggon for coals! And\nevery now and then, as the word \"riches\" or \"money\" caught Mr Leslie's\near, he shook his head, drew his pipe from his mouth, and muttered, \"A\nSpratt should not have what belonged to my great-great-grandfather.\nIf I had a good sum of ready-money!--the old family estates!\" Oliver\nand Juliet sate silent, and on their good behaviour; and Randal,\nindulging his own reveries, dreamily heard the words \"money,\" \"Spratt,\"\n\"great-great-grandfather,\" \"rich wife,\" \"family estates;\" and they\nsounded to him vague and afar off, like whispers from the world of\nromance and legend--weird prophecies of things to be.\n\nSuch was the hearth which warmed the viper that nestled and gnawed at\nthe heart of Randal, poisoning all the aspirations that youth should\nhave rendered pure, ambition lofty, and knowledge beneficent and divine.\n\n\nCHAPTER VI.\n\nWhen the rest of the household were in deep sleep, Randal stood long at\nhis open window, looking over the dreary, comfortless scene--the moon\ngleaming from skies half-autumnal, half-wintry, upon squalid decay,\nthrough the ragged fissures of the firs; and when he lay down to rest,\nhis sleep was feverish, and troubled by turbulent dreams.\n\nHowever, he was up early, and with an unwonted colour in his cheeks,\nwhich his sister ascribed to the country air. After breakfast, he took\nhis way towards Hazeldean, mounted upon a tolerable horse, which he\nhired of a neighbouring farmer who occasionally hunted. Before noon,\nthe garden and terrace of the Casino came in sight. He reined in his\nhorse, and by the little fountain at which Leonard had been wont to\neat his radishes and con his book, he saw Riccabocca seated under the\nshade of the red umbrella. And by the Italian's side stood a form that\na Greek of old might have deemed the Naiad of the Fount; for in its\nyouthful beauty there was something so full of poetry--something at\nonce so sweet and so stately--that it spoke to the imagination while it\ncharmed the sense.\n\nRandal dismounted, tied his horse to the gate, and, walking down a\ntrellised alley, came suddenly to the spot. His dark shadow fell over\nthe clear mirror of the fountain just as Riccabocca had said, \"All here\nis so secure from evil!--the waves of the fountain are never troubled\nlike those of the river!\" and Violante had answered in her soft native\ntongue, and lifting her dark, spiritual eyes--\"But the fountain would\nbe but a lifeless pool, oh my father, if the spray did not mount\ntowards the skies!\"\n\n\n\n\nTHE MASTER THIEF.\n\nA NORSE POPULAR TALE.\n\n\nOn a gloomy autumn evening I sat alone with the \"proprietor,\" to whose\nchildren I was then tutor, in his country house, about twenty miles\nfrom Christiania. Out of doors something was falling which was neither\nrain, nor snow, nor sleet, but a mixture of all three; and inside, in\nthe \"proprietor's\" parlour, the lights burned so sluggishly, that no\nother objects were discernible through the haze than a corner cupboard,\nfilled with Chinese nick-nacks, a great mirror in an old-fashioned gilt\nframe, and a hereditary tankard, the reward of one of the proprietor's\nancestors for service rendered to the state. That worthy individual had\nnestled himself into one corner of the sofa, where he pored over the\nproof-sheets of his pamphlet, entitled, \"_A few Patriotic Expressions\nfor the Country's Good; by an Anonymous Writer_.\"\n\nWhile brooding over this gold mine of his own ideas he gave birth to\nmany sagacious thoughts, which, from time to time, with a twinkle of\nhis grey eyes, he threw out for my edification, as I sat and tried to\nread in the other corner of the sofa. After a while, warming with his\ntheme, he poured out a host of \"patriotic expressions\" and opinions,\nworthy of all respect, but of which nothing save the pamphlet quoted\nabove, or his great _Treatise on Tithe_, can give an adequate idea.\nI am ashamed to own that all this wisdom was lost upon me. I knew it\nall by heart, for I had heard the same story forty times at least\nbefore. I am not gifted with a patience of Indian-rubber; but what\ncould I do? Retreat to my own room was impossible, for it had been\nscoured for Sunday, and was full of reek and damp. So, after some\nfruitless attempts to bury myself in my book, I was forced to give\nin, and to suffer myself to be carried along in the troubled stream\nof the proprietor's eloquence. Of course he dilated on questions of\nprofound national importance, which he furbished up with all sorts of\ncut-and-dried figures of speech. He was now fairly on his hobby, and\nrose rapidly to the seventh heaven. He stood up and gesticulated; then\nhe strode up and down, and his grey dressing-gown described streaming\ncircles behind him, as he turned short round, and limped backwards and\nforwards on his spindle-shanks,--for, like Tyrtaeus, the proprietor\nhad a strong halt. The candles flared, flickered, and guttered, as he\npassed triumphantly by the table on which they stood; and his winged\nwords sung in my ears like humble-bees when the linden-trees are in\nbloom. Off he went on \"Class Legislation\" and \"Judicial Reform,\" on\n\"Corn Laws and Free Trade,\" on \"Native Industry and Centralisation,\"\non the \"Victorious progress of Ideas,\" and the \"Insufficiency of our\nCirculating Medium,\" on \"Bureaucracy,\" and the \"Aristocracy of Office,\"\ntill he bid fair to exhaust all the _cracies_, _archies_, and _isms_\nthat ever existed, from King Solomon to the present time.\n\nMortal man could hold out no longer, and I was just on the point of\nbursting out into a roar of laughter in the worthy proprietor's face,\nwhen peal after peal of laughter resounded from the kitchen, and came\nto my rescue. It was Christian the blacksmith who had the word in that\nquarter of the house, and when he ceased speaking, repeated roars of\nmirth followed.\n\n\"I'll just go out and hear some of the smith's stories,\" I cried as I\nran out, leaving the proprietor behind in the parlour with the dull\ncandles and his drowsy current of thought.\n\n\"Children's prate and lying stories,\" growled the proprietor as I shut\nthe door. \"People of intelligence should be ashamed to listen to them;\nbut well-meant patriotic expressions--\" The rest was lost upon me.\n\nLight and life and mirth streamed forth in the high and airy hall; on\nthe hearth blazed a pile of logs, which threw a strong light into the\nfurthest nook. In the chimney-corner sat enthroned the proprietor's\nhousekeeper with her spinning-wheel; and though for many years she\nhad had hard struggles with the rheumatism, and barricaded the enemy\nout with a multitude of undercoats and kirtles, throwing over all,\nas an outwork, a huge grey woollen wrapper, yet her face shone under\nher plaited cap like the full moon. At her feet lay the proprietor's\nchildren laughing and cracking nuts; while round about sat a circle of\nmaids and workmen's wives, who trode their spinning-wheels with busy\nfeet, or plied the noisy carding-comb. In the entrance the threshers\nshook off the snow from their feet, and stepping in with icicles in\ntheir hair, sat down at the long table, where the cook served up to\nthem their supper--a bowl of milk and a dish of close-pressed porridge.\nAgainst the high chimney-piece leant the smith, who smoked tobacco from\na short pipe, and whose face, while it showed traces of the smithy,\nbore an expression of dry humour, which testified that he had been\ntelling a good story, and telling it well.\n\n\"Good afternoon, smith,\" said I; \"what story have you been telling\nwhich aroused so much laughter?\"\n\n\"Ha, ha!\" shouted the boys, \"Christian has been telling us all about\nthe 'Devil and the Smith,' and how the smith got the fiend into a\nhazel-nut; and now he's going to tell us about the Master Thief, and\nhow he won the Squire's daughter.\"\n\n\"Well, don't let me stop the story, smith,\" I replied, only too\nglad to escape for a while from the proprietor with his \"Patriotic\nExpressions,\" his \"Corn Laws and Free Trade,\" his \"Circulating Mediums\nand Bureaucracies,\" and to refresh myself with hearing one of these old\nnational tales, told in a simple childish way by one of the people.\n\nSo after one or two long-drawn puffs, the Smith began\n\n\nTHE MASTER THIEF.\n\nOnce upon a time there was a poor cottager who had three sons. He had\nnothing to leave them when he died, and no money with which to put them\nto any trade, so that he did not know what to make of them. At last\nhe said he would give them leave to take to anything each liked best,\nand to go whithersoever they pleased, and he would go with them a bit\nof the way; and so he did. He went with them till they came to a place\nwhere three roads met, and there each of them chose a road, and their\nfather bade them good-bye, and went back home. I have never heard tell\nwhat became of the two elder; but as for the youngest, he went both far\nand long, as you shall hear.\n\nSo it fell out one night as he was going through a great wood that such\nbad weather overtook him. It blew and drizzled so that he could scarce\nkeep his eyes open; and in a trice, before he knew how it was, he got\nbewildered, and could not find either road or path. But as he went on\nand on, at last he saw a glimmering of light far far off in the wood.\nSo he thought he would try and get to the light; and after a time he\ndid reach it. There it was in a large house, and the fire was blazing\nso brightly inside that he could tell the folk had not yet gone to bed;\nso he went in and saw an old dame bustling about and minding the house.\n\n\"Good evening,\" said the youth.\n\n\"Good evening,\" said the old dame.\n\n\"Hutetu! it's such foul weather out of doors to-night,\" said he.\n\n\"So it is,\" said she.\n\n\"Can I get leave to have a bed and shelter here to-night?\" asked the\nyouth.\n\n\"You'll get no good by sleeping here,\" said the old dame; \"for if the\nfolk come home and find you here, they'll kill both me and you.\"\n\n\"What sort of folk, then, are they who live here?\" asked the youth.\n\n\"Oh, robbers! And such a bad lot of them too,\" said the old dame. \"They\nstole me away when I was little, and have kept me as their housekeeper\never since.\"\n\n\"Well, for all that, I think I'll just go to bed,\" said the youth.\n\"Come what may, I'll not stir out at night in such weather.\"\n\n\"Very well,\" said the old dame; \"but if you stay it will be the worse\nfor you.\"\n\nWith that the youth got into a bed which stood there, but he dared not\ngo to sleep, and very soon after in came the robbers; so the old dame\ntold them how a stranger fellow had come in whom she had not been able\nto get out of the house again.\n\n\"Did you see if he had any money?\" said the robbers.\n\n\"Such a one as he money!\" said the old dame, \"the tramper! Why, if he\nhad clothes to his back, it was as much as he had.\"\n\nThen the robbers began to talk among themselves what they should do\nwith him; if they should kill him outright, or what else they should\ndo. Meantime the youth got up and began to talk to them, and to ask if\nthey did not want a servant, for it might be that he would be glad to\nenter into their service.\n\n\"Oh,\" said they, \"if you have a mind to follow the trade that we\nfollow, you can very well get a place here.\"\n\n\"It's all one to me what trade I follow,\" said the youth; \"for when I\nleft home, father gave me leave to take to any trade I chose.\"\n\n\"Well, have you a mind to steal?\" asked the robbers.\n\n\"I don't care,\" said the youth, for he thought it would not take long\nto learn that trade.\n\nNow there lived a man a little way off who had three oxen. One of\nthese he was to take to the town to sell, and the robbers had heard\nwhat he was going to do, so they said to the youth, that if he were\ngood to steal the ox from the man by the way without his knowing it,\nand without doing him any harm, they would give him leave to be their\nserving man.\n\nWell! the youth set off, and took with him a pretty shoe, with a silver\nbuckle on it, which lay about the house; and he put the shoe in the\nroad along which the man was going with his ox; and when he had done\nthat, he went into the wood and hid himself under a bush. So when the\nman came by he saw the shoe at once.\n\n\"That's a nice shoe,\" said he. \"If I only had the fellow to it, I'd\ntake it home with me, and perhaps I'd put my old dame into a good\nhumour for once.\" For you must know that he had an old wife, so cross\nand snappish that it was not long between each time that she boxed his\nears. But then he bethought him that he could do nothing with the odd\nshoe unless he had the fellow to it; so he went on his way and let the\nshoe lie on the road.\n\nThen the youth took up the shoe, and made all the haste he could to get\nbefore the man by a short cut through the wood, and laid it down before\nhim in the road again. When the man came along with his ox he got quite\nangry with himself for being so stupid as to leave the fellow to the\nshoe lying in the road instead of taking it with him; so he tied the\nox to the fence, and said to himself, \"I may just as well run back and\npick up the other, and then I'll have a pair of good shoes for my old\ndame, and so, perhaps, I'll get a kind word from her for once.\"\n\nSo he set off, and hunted and hunted up and down for the shoe, but\nno shoe did he find; and at length he had to go back with the one he\nhad. But, meanwhile, the youth had taken the ox and gone off with\nit; and when the man came and saw that his ox was gone, he began to\ncry and bewail, for he was afraid that his old dame would kill him\noutright when she came to know that the ox was lost. But just then it\ncame across his mind that he would go home and take the second ox, and\ndrive it to the town, and not let the old dame know anything about\nthe matter. So he did this, and went home and took the ox without his\ndame's knowing it, and set off with it to the town. But the robbers\nknew all about it, and they said to the youth, if he could get this ox\ntoo, without the man's knowing it, and without his doing him any harm,\nhe should be as good as any one of them. If that were all, the youth\nsaid, he did not think it a very hard thing.\n\nThis time he took with him a rope, and hung himself up under the\narmpits to a tree right in the man's way. So the man came along with\nhis ox, and when he saw such a sight hanging there he began to feel a\nlittle queer.\n\n\"Well,\" said he, \"whatever heavy thoughts you had who have hanged\nyourself up there, it can't be helped; you may hang for what I care!\nI can't breathe life into you again;\" and with that he went on his\nway with his ox. Down slipped the youth from the tree, and ran by a\nfootpath, and got before the man, and hung himself up right in his way\nagain.\n\n\"Bless me!\" said the man, \"were you really so heavy at heart that you\nhanged yourself up there--or is it only a piece of witchcraft that I\nsee before me? Ay, ay! you may hang for all I care, whether you are a\nghost or whatever you are.\" So he passed on with his ox.\n\nNow the youth did just as he had done twice before; he jumped down from\nthe tree, ran through the wood by a footpath, and hung himself up right\nin the man's way again. But when the man saw this sight for the third\ntime, he said to himself,--\n\n\"Well! this is an ugly business! Is it likely now that they should have\nbeen so heavy at heart as to hang themselves, all these three? No! I\ncannot think that it is anything else than a piece of witchcraft that\nI see. But now I'll soon know for certain: if the other two are still\nhanging there, it must be really so; but if they are not, then it can\nbe nothing but witchcraft that I see.\"\n\nSo he tied up his ox, and ran back to see if the others were still\nreally hanging there. But while he went and peered up into all the\ntrees, the youth jumped down and took his ox and ran off with it. When\nthe man came back and found his ox gone, he was in a sad plight, and,\nas any one might know without being told, he began to cry and bemoan;\nbut at last he came to take it easier, and so he thought--\n\n\"There's no other help for it than to go home and take the third ox\nwithout my dame's knowing it, and to try and drive a good bargain with\nit, so that I may get a good sum of money for it.\"\n\nSo he went home and set off with the ox, and his old dame knew never\na word about the matter. But the robbers, they knew all about it, and\nthey said to the youth, that, if he could steal this ox as he had\nstolen the other two, then he should be master over the whole band.\nWell, the youth set off, and ran into the wood; and as the man came by\nwith his ox he set up a dreadful bellowing, just like a great ox in the\nwood. When the man heard that, you can't think how glad he was, for it\nseemed to him that he knew the voice of his big bullock, and he thought\nthat now he should find both of them again; so he tied up the third ox,\nand ran off from the road to look for them in the wood; but meantime\nthe youth went off with the third ox. Now, when the man came back and\nfound he had lost this ox too, he was so wild that there was no end to\nhis grief. He cried and roared and beat his breast, and, to tell the\ntruth, it was many days before he dared go home; for he was afraid lest\nhis old dame should kill him outright on the spot.\n\nAs for the robbers, they were not very well pleased either, when they\nhad to own that the youth was master over the whole band. So one day\nthey thought they would try their hands at something which he was\nnot man enough to do; and they set off all together, every man Jack\nof them, and left him alone at home. Now, the first thing that he did\nwhen they were all well clear of the house, was to drive the oxen out\nto the road, so that they might run back to the man from whom he had\nstolen them; and right glad he was to see them, as you may fancy. Next\nhe took all the horses which the robbers had, and loaded them with the\nbest things he could lay his hands on--gold and silver, and clothes and\nother fine things; and then he bade the old dame to greet the robbers\nwhen they came back, and to thank them for him, and to say that now he\nwas setting off on his travels, and that they would have hard work to\nfind him again; and with that, off he started.\n\nAfter a good bit he came to the road along which he was going when he\nfell among the robbers; and when he got near home, and could see his\nfather's cottage, he put on a uniform which he had found among the\nclothes he had taken from the robbers, and which was made just like a\ngeneral's. So he drove up to the door as if he were any other great\nman. After that he went in and asked if he could have a lodging? No;\nthat he couldn't at any price.\n\n\"How ever should I be able,\" said the man, \"to make room in my house\nfor such a fine gentleman--I who scarce have a rag to lie upon, and\nmiserable rags too?\"\n\n\"You were always a stingy old hunks,\" said the youth, \"and so you are\nstill, when you won't take your own son in.\"\n\n\"What, you my son!\" said the man.\n\n\"Don't you know me again?\" said the youth. Well, after a little while\nhe did know him again.\n\n\"But what have you been turning your hand to, that you have made\nyourself so great a man in such haste?\" asked the man.\n\n\"Oh, I'll soon tell you,\" said the youth. \"You said I might take to any\ntrade I chose, and so I bound myself apprentice to some thieves and\nrobbers, and now I've served my time out, and am become a Master Thief.\"\n\nNow there lived a Squire close by to his father's cottage, and he had\nsuch a great house, and such heaps of money, that he could not tell how\nmuch he had. He had a daughter too, and a smart and pretty girl she\nwas. So the Master Thief set his heart upon having her to wife; and he\ntold his father to go to the Squire and ask for his daughter for him.\n\n\"If he asks by what trade I get my living, you can say I am a Master\nThief.\"\n\n\"I think you've lost your wits,\" said the man, \"for you can't be in\nyour right mind when you think of such nonsense.\"\n\nNo! he had not lost his wits; his father must and should go up to the\nSquire and ask for his daughter.\n\n\"Nay, but I tell you, I daren't go to the Squire and be your spokesman;\nhe who is so rich, and has so much money,\" said the man.\n\nYes, there was no help for it, said the Master Thief; he should go\nwhether he would or no; and if he did not go by fair means, he would\nsoon make him go by foul. But the man was still loath to go; so he\nstepped after him, and rubbed him down with a good birch cudgel, and\nkept on till the man came crying and sobbing inside the Squire's door.\n\nHow now, my man! What ails you? said the Squire.\n\nSo he told him the whole story; how he had three sons who set off\none day, and how he had given them leave to go whithersoever they\nwould, and to follow whatever calling they chose. \"And here now is the\nyoungest come home, and has beaten me till he has made me come to you\nand ask for your daughter for him to wife; and he bids me say, besides,\nthat he is a Master Thief.\" And so he fell to crying and sobbing again.\n\n\"Never mind, my man,\" said the Squire, laughing; \"just go back and tell\nhim from me, that he must prove his skill first. If he can steal the\nroast from the spit in the kitchen on Sunday, while all the household\nare looking after it, he shall have my daughter. Just go and tell him\nthat.\"\n\nSo he went back and told the youth, who thought it would be an easy\njob. So he set about and caught three hares alive, and put them into\na bag, and dressed himself in some old rags, until he looked so poor\nand filthy that it made one's heart bleed to see; and then he sneaked\ninto the passage at the backdoor of the Squire's house on the Sunday\nforenoon, with his bag, just like any other beggar-boy. But the Squire\nhimself and all his household were in the kitchen watching the roast.\nJust as they were doing this, the youth let one hare go, and it set of\nand ran round and round the yard in front of the house.\n\n\"Oh, just look at that hare!\" said the folk in the kitchen, and were\nall for running out to catch it.\n\nYes, the Squire saw it running too. \"Oh, let it run,\" said he; \"there's\nno use in thinking to catch a hare by running after it.\"\n\nA little while after, the youth let the second hare go, and they saw it\nin the kitchen, and thought it was the same they had seen before, and\nstill wanted to run out and catch it; but the Squire said again it was\nno use. It was not long before the youth let the third hare go, and it\nset off and ran round and round the yard as the others before it. Now,\nthey saw it from the kitchen, and still thought it was the same hare\nthat kept on running about, and were all eager to be out after it.\n\n\"Well, it is a fine hare,\" said the Squire; \"come, let's see if we\ncan't lay our hands on it.\"\n\nSo out he ran, and the rest with him--away they all went, the hare\nbefore, and they after; so that it was rare fun to see. But meantime\nthe youth took the roast and ran of with it; and where the Squire got a\nroast for his dinner that day I don't know; but one thing I know, and\nthat is, that he had no roast hare, though he ran after it till he was\nboth warm and weary.\n\nNow it chanced that the Priest came to dinner that day, and when the\nSquire told him what a trick the Master Thief had played him, he made\nsuch game of him that there was no end to it.\n\n\"For my part,\" said the Priest, \"I can't think how it could ever happen\nto me to be made such a fool of by a fellow like that.\"\n\n\"Very well--only keep a sharp look-out,\" said the Squire; \"maybe he'll\ncome to see you before you know a word of it.\" But the Priest stuck to\nhis text,--that he did, and made game of the Squire because he had been\nso taken in.\n\nLater in the afternoon came the Master Thief, and wanted to have the\nSquire's daughter, as he had given his word. But the Squire began to\ntalk him over, and said, \"Oh, you must first prove your skill a little\nmore; for what you did to-day was no great thing, after all. Couldn't\nyou now play off a good trick on the Priest, who is sitting in there,\nand making game of me for letting such a fellow as you twist me round\nhis thumb.\"\n\n\"Well, as for that, it wouldn't be hard,\" said the Master Thief. So\nhe dressed himself up like a bird, threw a great white sheet over his\nbody, took the wings of a goose and tied them to his back, and so\nclimbed up into a great maple which stood in the Priest's garden. And\nwhen the Priest came home in the evening, the youth began to bawl out--\n\n\"Father Laurence! Father Laurence!\"--for that was the Priest's name.\n\n\"Who is that calling me?\" said the Priest.\n\n\"I am an angel,\" said the Master Thief, \"sent from God to let you know\nthat you shall be taken up alive into heaven for your piety's sake.\nNext Monday you must hold yourself ready for the journey, for I shall\ncome then to fetch you in a sack; and all your gold and your silver,\nand all that you have of this world's goods, you must lay together in a\nheap in your dining-room.\"\n\nWell, Father Laurence fell on his knees before the angel, and thanked\nhim; and the very next day he preached a farewell sermon, and expounded\nhow there had come down an angel unto the big maple in his garden,\nwho had told him that he was to be taken up alive into heaven for his\npiety's sake; and he preached and made such a touching discourse, that\nall who were at church wept, both young and old.\n\nSo the Monday after came the Master Thief like an angel again, and\nthe Priest fell on his knees and thanked him before he was put into\nthe sack; but when he had got him well in, the Master Thief drew and\ndragged him over stocks and stones.\n\n\"Ow! ow!\" groaned the Priest inside the sack, \"wherever are we going!\"\n\n\"This is the narrow way which leadeth unto the kingdom of heaven,\"\nsaid the Master Thief, who went on dragging him along till he had\nnearly broken every bone in his body. At last he tumbled him into a\ngoose-house that belonged to the Squire, and the geese began pecking\nand pinching him with their bills, so that he was more dead that alive.\n\n\"Now you are in the flames of purgatory, to be cleansed and purified\nfor life everlasting,\" said the Master Thief; and with that he went his\nway, and took all the gold and silver, and all the fine things which\nthe Priest had laid together in his dining-room. The next morning, when\nthe goose-girl came to let the geese out, she heard how the priest lay\nin the sack and bemoaned himself in the goose-house.\n\n\"In heaven's name, who's there, and what ails you?\" she cried. \"Oh!\"\nsaid the Priest, \"if you are an angel from heaven, do let me out, and\nlet me return again to earth, for it is worse here than in hell. The\nlittle fiends keep on pinching me with tongs.\"\n\n\"God help us, I am no angel at all,\" said the girl as she helped the\nPriest out of the sack; \"I only look after the Squire's geese, and like\nenough they are the little fiends which have pinched your reverence.\"\n\n\"Oh!\" groaned the Priest, \"this is all that Master Thief's doing. Ah!\nmy gold and my silver, and my fine clothes.\" And he beat his breast,\nand hobbled home at such a rate that the girl thought he had lost his\nwits all at once.\n\nNow when the Squire came to hear how it had gone with the Priest, and\nhow he had been along the narrow way, and into purgatory, he laughed\ntill he wellnigh split his sides. But when the Master Thief came and\nasked for his daughter as he had promised, the Squire put him off\nagain, and said--\n\n\"You must do one masterpiece better still, that I may see plainly what\nyou are fit for. Now I have twelve horses in my stable, and on them I\nwill put twelve grooms, one on each. If you are so good a thief as to\nsteal the horses from under them, I'll see what I can do for you.\"\n\n\"Very well, I daresay I can do it,\" said the Master Thief; \"but shall I\nreally have your daughter if I can?\"\n\n\"Yes, if you can, I'll do my best for you,\" said the Squire.\n\nSo the Master Thief set off to a shop, and bought brandy enough to fill\ntwo pocket-flasks, and into one of them he put a sleepy drink, but into\nthe other only brandy. After that he hired eleven men to lie in wait at\nnight, behind the Squire's stableyard; and last of all, for fair words\nand a good bit of money, he borrowed a ragged gown and cloak from an\nold woman; and so, with a staff in his hand and a bundle at his back,\nhe limped off, as evening drew on towards the Squire's stable. Just as\nhe got there they were watering the horses for the night, and had their\nhands full of work.\n\n\"What the devil do you want?\" said one of the grooms to the old woman.\n\n\"Oh, oh! hutetu! it is so bitter cold,\" said she, and shivered and\nshook, and made wry faces. \"Hutetu! it is so cold, a poor wretch may\neasily freeze to death;\" and with that she fell to shivering and\nshaking again.\n\n\"Oh! for the love of heaven, can I get leave to stay here a while, and\nsit inside the stable door?\"\n\n\"To the devil with your leave,\" said one. \"Pack yourself off this\nminute, for if the Squire sets his eye on you he'll lead us a pretty\ndance.\"\n\n\"Oh! the poor old bag-of-bones,\" said another, who seemed to take pity\non her, \"the old hag may sit inside and welcome; such a one as she can\ndo no harm.\"\n\nAnd the rest said, some she should stay, and some she shouldn't; but\nwhile they were quarrelling and minding the horses, she crept further\nand further into the stable, till at last she sat herself down behind\nthe door; and when she had got so far, no one gave any more heed to her.\n\nAs the night wore on, the men found it rather cold work to sit so\nstill and quiet on horseback.\n\n\"Hutetu! it is so devilish cold,\" said one, and beat his arms crosswise.\n\n\"That it is,\" said another. \"I freeze so, that my teeth chatter.\"\n\n\"If one only had a quid to chew,\" said a third.\n\nWell! there was one who had an ounce or two; so they shared it between\nthem, though it wasn't much, after all, that each got; and so they\nchewed and spat, and spat and chewed. This helped them somewhat; but in\na little while they were just as bad as ever.\n\n\"Hutetu!\" said one, and shivered and shook.\n\n\"Hutetu!\" said the old woman, and shivered so, that every tooth in her\nhead chattered. Then she pulled out the flask with brandy in it, and\nher hand shook so that the spirit splashed about in the flask, and then\nshe took such a gulf, that it went \"bop\" in her throat.\n\n\"What is that you've got in your flask, old girl?\" said one of the\ngrooms.\n\n\"Oh! it's only a drop of brandy, old man,\" said she.\n\n\"Brandy! Well, I never! Do let me have a drop,\" screamed the whole\ntwelve, one after another.\n\n\"Oh! but it is such a little drop,\" mumbled the old woman, \"it will not\neven wet your mouths round.\" But they must and would have it; there was\nno help for it; and so she pulled out the flask with the sleeping drink\nin it, and put it to the first man's lips; then she shook no more,\nbut guided the flask so that each of them got what he wanted, and the\ntwelfth had not done drinking before the first sat and snored. Then the\nMaster Thief threw off his beggar's rags, and took one groom after the\nother so softly off their horses, and set them astride on the beams\nbetween the stalls; and so he called his eleven men, and rode off with\nthe Squire's twelve horses.\n\nBut when the Squire got up in the morning, and went to look after\nhis grooms, they had just begun to come to; and some of them fell to\nspurring the beams with their spurs, till the splinters flew again, and\nsome fell off, and some still hung on and sat there looking like fools.\n\n\"Ho! ho!\" said the Squire; \"I see very well who has been here; but as\nfor you, a pretty set of blockheads you must be to sit here and let the\nMaster Thief steal the horses from between your legs.\"\n\nSo they all got a good leathering because they had not kept a sharper\nlook-out.\n\nFurther on in the day came the Master Thief again, and told how he had\nmanaged the matter, and asked for the Squire's daughter, as he had\npromised; but the Squire gave him one hundred dollars down, and said he\nmust do something better still.\n\n\"Do you think now,\" said he, \"you can steal the horse from under me\nwhile I am out riding on his back?\"\n\n\"O, yes! I daresay I could,\" said the Master Thief, \"if I were really\nsure of getting your daughter.\"\n\nWell, well, the Squire would see what he could do; and he told the\nMaster Thief a day when he would be taking a ride on a great common\nwhere they drilled the troops. So the Master Thief soon got hold of\nan old worn-out jade of a mare, and set to work, and made traces and\ncollar of withies and broom-twigs, and bought an old beggarly cart and\na great cask. After that he said to an old beggar woman, that he would\ngive her ten dollars if she would get in the cask, and keep her mouth\nagape over the taphole, into which he was going to stick his finger. No\nharm should happen to her; she should only be driven about a little;\nand if he took his finger out more than once, she was to have ten\ndollars more. Then he threw a few rags and tatters over himself, and\nstuffed himself out, and put on a wig and a great beard of goat's hair,\nso that no one could know him again, and set off for the common, where\nthe Squire had already been riding about a good bit. When he reached\nthe place, he went along so softly and slowly that he scarce made an\ninch of way. Gee up! Gee up! and so he went on little; then he stood\nstock still, and so on a little again; and altogether the pace was so\nmiserable that it never once came into the Squire's head that this\ncould be the Master Thief.\n\nAt last the Squire rode right up to him, and asked if he had seen any\none lurking about in the wood thereabouts.\n\n\"No,\" said the man, \"I haven't seen a soul.\"\n\n\"Harkye, now,\" said the Squire, \"if you have a mind to ride into the\nwood, and hunt about and see if you can fall upon any one lurking about\nthere, you shall have the loan of my horse, and a shilling into the\nbargain, to drink my health, for your pains.\"\n\n\"I don't see how I can go,\" said the man, \"for I am going to a wedding\nwith this cask of mead, which I have been to town to fetch, and here\nthe tap has fallen out by the way, and so I must go along, holding my\nfinger in the taphole.\n\n\"Ride off,\" said the Squire; \"I'll look after your horse and cask.\"\n\nWell, on these terms the man was willing to go; but he begged the\nSquire to be quick in putting his finger into the taphole when he took\nhis own out, and to mind and keep it there till he came back. Yes, the\nSquire would do the best he could; and so the Master Thief mounted the\nhorse and rode off. But time went by, and hour after hour passed, and\nstill no one came back. At last the Squire grew weary of standing there\nwith his finger in the taphole, so he took it out.\n\n\"Now I shall have ten dollars more!\" screamed the old woman inside\nthe cask; and then the Squire saw at once how the land lay, and took\nhimself off home; but he had not gone far before they met him with a\nfresh horse, for the Master Thief had already been to his house, and\ntold them to send one.\n\nThe day after, he came to the Squire and would have his daughter, as\nhe had given his word; but the Squire put him off again with fine\nwords, and gave him two hundred dollars, and said he must do one more\nmasterpiece. If he could do that, he should have her. Well, well, the\nMaster Thief thought he could do it, if he only knew what it was to be.\n\n\"Do you think, now,\" said the Squire, \"you can steal the sheet off our\nbed, and the shift off my wife's back. Do you think you could do that?\"\n\n\"It shall be done,\" said the Master Thief. \"I only wish I was as sure,\nof getting your daughter.\"\n\nSo when night began to fall, the Master Thief went out and cut down a\nthief who hung on the gallows, and threw him across his shoulders, and\ncarried him off. Then he got a long ladder and set it up against the\nSquire's bedroom window, and so climbed up, and kept bobbing the dead\nman up and down, just for all the world like one who was peeping in at\nthe window.\n\n\"That's the Master Thief, old lass!\" said the Squire, and gave his wife\na nudge on the side. \"Now see if I don't shoot him, that's all.\"\n\nSo saying he took up a rifle which he had laid at his bedside.\n\n\"No! no! pray don't shoot him after telling him he might come and try,\"\nsaid his wife.\n\n\"Don't talk to me, for shoot him I will,\" said he; and so he lay there\nand aimed and aimed; but as soon as the head came up before the window,\nand he saw a little of it, so soon was it down again. At last he\nthought he had a good aim; \"bang\" went the gun, down fell the dead body\nto the ground with a heavy thump, and down went the Master Thief too as\nfast as he could.\n\n\"Well,\" said the Squire, \"it is quite true that I am the chief\nmagistrate in these parts; but people are fond of talking, and it would\nbe a bore if they came to see this dead man's body. I think the best\nthing to be done is that I should go down and bury him.\"\n\n\"You must do as you think best, dear,\" said his wife. So the Squire got\nout of bed and went down stairs, and he had scarce put his foot out of\nthe door before the Master Thief stole in, and went straight up-stairs\nto his wife.\n\n\"Why, dear, back already!\" said she, for she thought it was her husband.\n\n\"Oh yes, I only just put him into a hole, and threw a little earth over\nhim. It is enough that he is out of sight, for it is such a bad night\nout of doors; by-and-by I'll do it better. But just let me have the\nsheet to wipe myself with--he was so bloody--and I have made myself in\nsuch a mess with him.\"\n\nSo he got the sheet.\n\nAfter a while he said--\n\n\"Do you know I am afraid you must let me have your night-shift too, for\nthe sheet won't do by itself; that I can see.\"\n\nSo she gave him the shift also. But just then it came across his mind\nthat he had forgotten to lock the house-door, so he must step down and\nlook to that before he came back to bed, and away he went with both\nshift and sheet.\n\nA little while after came the right Squire.\n\n\"Why! what a time you've taken to lock the door, dear!\" said his wife;\n\"and what have you done with the sheet and shift?\"\n\n\"What do you say?\" said the Squire.\n\n\"Why, I am asking what you have done with the sheet and shift that you\nhad to wipe off the blood,\" said she.\n\n\"What, in the devil's name!\" said the Squire, \"has he taken me in this\ntime too?\"\n\nNext day came the Master Thief and asked for the Squire's daughter as\nhe had promised; and then the Squire dared not do anything else than\ngive her to him, and a good lump of money into the bargain; for, to\ntell the truth, he was afraid lest the Master Thief should steal the\neyes out of his head, and that people would begin to say spiteful\nthings of him if he broke his word. So the Master Thief lived well and\nhappily from that time forward. I don't know whether he stole any more;\nbut if he did, I am quite sure it was only for the sake of a bit of fun.\n\n\n\n\nDAY-DREAMS OF AN EXILE.\n\n\nV.\n\nAIR--\"_O Cara Memoria._\"\n\n    \"I perceive that there is nothing better than that a man should\n    rejoice in his own works, for that is his portion.\"--_Eccles._\n    iii. 22.\n\n    Sigh thou not for a happier lot,\n      Happier may never be;\n    That thou hast esteem the best,\n      And given by the gods to thee.\n    And if thy tender hopes be slain,\n    Fear not, they soon shall bloom again;\n      For the gloomiest hour\n      Is fair to the flower\n    That heeds neither wind nor rain.\n\n    Fear of change from old to strange\n      Follows the fullest joy;\n    Labour wears us more than years;\n      Calms, never broken, cloy.\n    Whatever load to thee be given,\n    Doubt not thy brethren too have striven;\n      Take what is thine\n      In the Earth's confine,\n    And hope to be blest in Heaven.\n\n\nVI.\n\nTO ----,\n\n    Led by swift thought, I scale the height,\n      And strive to sound the deep,\n    To find from whence I took my flight,\n      Or where I slept my sleep:\n    But the mists conceal that border-land\n      Whose hills they rest upon;\n    Again, with forward face, I stand,\n      For Gone is gone.\n\n    Sometimes I brood upon the years\n      I gave to self and sin;\n    Or call to mind how Doubts and Fears\n      Fled from a light within:\n    I might regret those errors past,\n      Might wish the light still shone,\n    Or check Life's tide that ebbs so fast;\n      But Gone is gone.\n\n    You, too, my loyal-hearted wife,\n      Saw many a weary day,\n    When, on your morning-sky of life,\n      The clouds of sorrow lay.\n    True friends departed--grief for them,\n      Joy for the False made known,\n    And over all this Requiem,\n      That Gone is gone.\n\n    The glare of many a spectral Truth\n      Might haunt me still unchanged,\n    The broken purpose of my Youth,\n      The loving hearts estranged.\n    But, turning to your love-lit eyes,\n      --The love-lit eyes shine on--\n    I thank my God with happy sighs\n      That Gone is gone.\n\n\nVII.\n\n    Oft, in a night of April, when the ways\n      Are growing dark, and the hedge-hawthorns dank,\n    The glow-worm scatters self-adorning rays--\n      Earth-stars, that twinkle on the primrose bank.\n\n    And so, when Life around us gathers Night,\n      Too dark for Doubt, and ignorant of Sin,\n    The happy Heart of youth can shed a light\n      Earth-born, but bright, and feed it from within.\n\n    The April night wears on, the darkness wanes,\n      The light that glimmered in the East grows stronger;\n    But on the primrose banks that line the lanes,\n      Weary and chilled, the glow-worm shines no longer.\n\n    The night of life as quickly passes o'er,\n      Coldly and shuddering breaks the dawn of Truth;\n    Bright Day is coming, but we bear no more\n      The happy, self-adorning heart of Youth.\n\n\nVIII.\n\n    Dream on, ye souls who slumber here,\n      Leave work to those who work so well;\n    Yet workers too should haply hear\n      The messages that Dreamers tell.\n\n    The aims of this World shed a light,\n      Which shines with dim and feeble ray,\n    Whose followers wander all the night,\n      And scarce suspect it is not Day.\n\n    Yet work who will, the Night flies fast,\n      Means vary, but the end is one;\n    Each, when the waking throb is past,\n      Must face the all-beholding Sun.\n\n    I will sleep on, the starry cope\n      Arching my head with boundless blue,\n    Till life's strange dream is o'er, in hope\n      To wake, nor find it all untrue.\n\n\nIX.\n\nCOLONISATION.\n\n(I.)\n\n      Freemen of England, nourish in your mind\n    Love for your Land; though poor she be and cold,\n    Impute it not to her that she is old,\n    For in her youth she was both warm and kind.\n    True, it fits not that you should be confined\n    Within a grudging Island's narrow hold,\n    That bred, but cannot feed you. O be bold;\n    Blue heaven has many an excellent fair wind.\n    Steer, then, in multitudes to other land,\n    Work ye the field, the river, and the mine,\n    Smooth the high hill, and fell the long-armed pine,\n    Till all GOD'S Earth be honourably manned;\n    But, that your glories may for ever stand,\n    Let Love be, with you, human and divine.\n\n\n(II.)\n\n    Love, the foundation of the public weal,\n    As of the peace of houses--Love, whose breach\n    Sundered two bands of common race and speech,\n    Whose rankling wounds on each side will not heal:\n    Therefore be warned in time, let none conceal\n    Brotherly yearnings, God-sent, each for each.\n    Pure human sympathies are high of reach,\n    For the realities which they reveal\n    Teach us to live in earnest; give us faith,\n    Godward, as well as human: none can say,\n    \"I will love only that which I have seen.\"\n    By faith's lamp, fed with hope, the wise have been\n    Led to the land where, as the Tarsian saith,\n    Love rules when Hope and Faith are passed away.\n\n  H. G. K.\n  _India_, 1851.\n\n\n\n\nAUTUMN POLITICS.\n\n\nRarely, during the autumnal season of the year, is any very vivid\ninterest displayed in political matters. This is both natural and\nwholesome. The soldier, after a hard campaign, requires rest and\nrecreation; and those whose destiny it is to occupy themselves with\npublic affairs and their conduct, are all the better for a short\nrespite from these absorbing toils. So, after the close of the\nParliamentary Session, our legislators betake themselves to the\nprovinces or the Continent, to the skirts of Ben Nevis, or to the\nsequestered valleys of Switzerland, with all the glee of schoolboys\nwho have escaped from the magisterial yoke. Who can blame them? The\nmountain breeze is assuredly more fresh and salubrious than the\nloaded atmosphere of St Stephen's; the sound of the purling brook\nis more grateful to the ear than the croakings of Joseph Hume; and\nthe details of a restaurant's bill of fare more interesting than the\ningenious statistics of Mr Wilson of Westbury. Nobody is sorry when\nthe clattering of the great machine of Parliament is silenced. It is\nbad enough to be compelled to peruse the debates during the months of\nwinter and spring, without continuing the ordeal throughout the rest\nof the year. We cannot live always in a state of excitement. Scully\nand Keogh are splendid and soul-searching orators; but we would as\nlieve submit to have all our dishes seasoned with ether, as allow\nour nerves to be daily agitated by the roll of their irresistible\neloquence. We love John Bright, and are fascinated by the humour of\nFox, yet we can find it in our hearts to part company with them for a\nseason. In autumn the towns are torpid. Every one who can, endeavours\nto escape from them; and to judge from the hurry on rail and river,\nyou would conclude that at least one-half of the population of these\nislands is on the move. Subjects which a few months before engrossed\nthe public attention are now mentioned with a luxurious languor, and\nnever ardently discussed. Few people know or care what Cardinal Wiseman\nmay be doing. A porter with a load of grouse is a more interesting\nobject than Lord John Russell, even were he laden with the draught of\nhis new Reform Bill; and it is a matter of total indifference to the\nmillion whether Earl Grey has gone to Howick or to Kamschatka. The only\nclass of men who remain indefatigably political are the popularity\nhunters, more especially such of them as require a little coopering\nfor their somewhat leaky reputations. Old Joe sets off on a reforming\ntour to the northern burghs, hoping here and there to pick up a stray\nburgess ticket. Sir James Graham will go any distance to receive the\nhug of fraternity from a provost, and to add to his chaplet such\nfresh leaves of laurel as are in the gift of a generous town council.\nLord Palmerston undertakes to keep the electors of Tiverton in good\nhumour, and favours them with a funny discourse upon all manner of\ntopics, excepting always the projected measure of reform, on which\nhe judiciously keeps his thumb. These, however, are mere interludes,\nand few people care about them. Most sincerely to be pitied, at this\nseason of the year, is the condition of the London journalists. However\nscanty be the crop of events, however dry the channels of public\ninterest, they must find subjects for their leaders. Each day there\nis a yawning gap of white paper to be filled; a topic to be selected\nand discussed; and an insatiable devil to be laid. It was popularly\nbelieved on the Border that Michael Scott was saddled with an infernal\nservitor, to whom he was compelled to assign daily a sufficient modicum\nof work, under the penalty, in case of failure, of a forced visit to\nPandemonium. Quite as bad is the predicament of the journalist. The\nprinter's demon is ever at his elbow; nor dare he attempt to escape.\nIt is not surprising if sometimes our unhappy brothers should be\nreduced to the last extremity. Generally, nay universally, they are a\nkind-hearted race of men; yet no one who hears their complaints during\na season of unusual stagnation would set them down as philanthropists.\nTheir aspirations are after revolutions, murders, casualties--anything,\nin short, which can furnish them with a topic for a good stirring\narticle. All manufacturers, except the dealers in devil's-dust and\nshoddy, admit that there is no possibility of constructing a passable\nfabric out of inferior raw material. Whatever be the capabilities of\nthe artisan, or the excellence of his tools, he cannot do without a\nsubject to work upon. Facts, according to the approved doctrine of the\npublic press, are of two kinds--real and imagined. The distinction is\nas wide as that which lies between history and romance. If the first\ndo not emerge in sufficient value or importance, recourse must be had\nto the second, provided nothing be advanced for which there is not\nsome apparent colour. The position and prospects of parties is always\na safe autumnal theme. Some paragraph is sure to appear, some letter\nto be published, some pamphlet written, or some speech delivered, from\nwhich ingenuity can extract matter of startling commentary. One while,\nsupposed differences in the Cabinet are made the subject of conjecture\nand discussion, though where the Cabinet is no one can tell, the\nmembers thereof being notoriously so scattered that no two of them are\nwithin a hundred miles of each other. Lord John Russell's resignation\nhas of late years become a regular autumnal event. We look for it as\nconfidently as the housekeeper expects her annual supply of damsons. No\none is rash enough to aver that Sir Charles Wood intends voluntarily\nto resign; but somehow or other it happens that his colleagues are\nannually seized in September with a burning desire to kick him out--a\nspecies of phrenzy which only lasts until the return of the colder\nweather. We really forget how often Lord Clarendon has been announced\nas the coming Premier. If there be any faith in prophecy, his time must\nbe nigh at hand.\n\nIt was, we believe, confidently anticipated on the part of the\nLiberal journals, that the present autumn would prove an exception to\nthe general rule, by furnishing a more than average crop of topics\nacceptable to the public ear. After such a dreary lapse of time,\nprosperity was expected to arrive about the middle of 1851, and that\nevent would of itself justify the expenditure of many columns of\npoeans. True, there had been various attempts made at intervals,\nduring the last three or four years, to persuade the public that the\ncoy nymph had either arrived or was arriving on the British shores;\nand some journals went so far as to discharge a royal salute in honour\nof her supposed landing. But the mistake was soon discovered. If the\nagriculturists were discontented, the manufacturers were depressed, and\nthe shopkeepers evidently sulky. Prosperity, if she really had arrived,\nseemed to possess the secret of the fern-seed, and to walk invisibly,\nfor no one had seen her except Mr Labouchere; and on investigating his\nexperiences, it turned out that he had merely received his information\nfrom others. This year, however, everything was to be put to rights.\nMarkets were to rise so high that even the most grumbling of the\nfarmers would be glad of heart, and be enabled to make such purchases\nat the nearest town as would at once gratify the wife of his bosom,\nand give a material impulse to the production of home manufactures.\nGreat were to be the profits of Manchester, Bradford, and Nottingham.\nReciprocity was to be developed; and foreign nations, convinced of the\nnecessity of universal brotherhood, were to fling their tariffs to the\nwinds, and admit our produce duty free. By this time, too, we were to\nhave Mr Mechi's balance-sheet before us. Mr Huxtable's pigs were to\nhave produced ammonia enough to fertilise the seashore; or, if that\nscheme did not answer, the Netherby system of farming would be found\nequally advantageous. Nay, it was even prophesied that railway stocks\nwould rise, and that on some hyperborean lines there was a possibility\nthat a dividend might be paid on the preference shares. The iron\ndistricts were to outstrip California, and our shipping to multiply\nindefinitely.\n\nIt is deeply to be deplored, on every ground, that these expectations\nhave not been realised. We have been repeatedly reproached by the\nadvocates of the new commercial system for the gloominess of our views,\nand the absence of that hopeful spirit which animates the efforts, and\ngives vivacity to the style, of the light and lively Free-Traders.\nNow, it is quite true that we, being unable, after the most anxious\nconsideration of the subject in all its bearings, to discover how the\nprosperity--that is, the wealth--of the nation could be increased\nby measures which had the direct tendency to lower the value of its\nproduce, have had occasion very frequently to enunciate opinions which\ncould not be agreeable to the cotton-stuffed ears of Manchester. We\nhave periodically exposed, to the great dudgeon of the democrats, the\nclumsy fallacies and egregious nonsense of the _Economist_, familiarly\nknown to the concoctors of statistical returns by the _soubriquet_ of\nthe \"Cook's Oracle.\" We have taken sundry impostors by the nape of the\nneck, and have shaken them, as was our bounden duty, until they had not\nbreath enough to squeak. But we maintain that the facts and results of\neach successive year have borne us out in the views which we originally\nentertained; and that the working of Free Trade, when brought into\noperation, has proved, as we predicted it would be, utterly subversive\nof the theories of the men who were its exponents, its champions, and\nits abettors. So much the worse for the country. But why should we be\nblamed for having simply spoken the truth? Show us your prosperity, if\nthat prosperity really exists; or, at all events, be kind enough to\nspecify to us the prominent symptoms of its coming. We need not, we are\nwell aware, look for these among the farmers. Ministers have given that\nup--never more decidedly, though they did not probably understand the\nforce of the language they were using, or its inevitable conclusion,\nthan when they declared their hope and expectation that the British\nagriculturist, depressed by foreign importations, could not fail to\nprofit ultimately by the improved condition of the other classes of\nthe community! The gentleman who devised that sentence must have had,\nindeed, an implicit reliance in the gullibility of mankind! He might\njust as well have told the stage-coachmen, who were driven off the road\nby the substitution of the rail, that they would be sure to profit in\nthe long run by the bettered circumstances of the stokers! If that\nis all the comfort that can be extended to the agriculturists, they\nwill hardly warm themselves by it. But among the manufacturers, if\nanywhere, we may look for some measure of prosperity; and we grieve to\nsay that, if such really exists, they take especial care to conceal it.\nTalk of farmers grumbling, indeed! If the whole race of corn-growers,\nfrom Triptolemus downwards, were assembled, and entreated to state\ntheir grievances and the causes of their dejection, we defy them to\nproduce such a catalogue of continued woe as has been trumpeted from\nthe trade circulars and reports during the last three years. Falling\nmarkets--continued stagnation--nothing doing. Such are the phrases\nwith which we are familiar, and we meet with nothing else; wherefrom\nwe conclude either that the manufacturers are all banded together in a\nleague of unparalleled and very scandalous deceit, or that Free Trade,\nby contracting the home market, has made wild work with their profits\nalso. Commercial failures, too, about which we have heard a good deal,\nand are likely to hear something more, are not to be accepted as\nunequivocal signs of the rising prosperity of the country.\n\nMessrs Littledale write as follows, in their circular of 4th October,\nsince which date much has occurred to give weight and confirmation to\ntheir statements:--\n\n    \"Nothing seems to change the untoward course of events in this\n    memorable year. An abundant harvest has been gathered, with\n    less damage and at less cost than for many years, which was\n    to prove the turning-point in commercial matters; instead of\n    which, the depression seems only to increase from day to day,\n    without apparent cause or termination. This state of things\n    naturally begets mistrust amongst money-lenders and bankers;\n    and just at the time when their support is most needed,\n    and would prove most valuable in preventing that ruinous\n    depression which forced sales on a declining market ever\n    produce, their confidence is destroyed, and accommodation is\n    refused.\n\n    \"The losses on imports of every kind are alarming, and yet the\n    tide is unabated; and unless a _more vigorous_ stand is made by\n    importers, either to bring down prices in the foreign market to\n    a parity with our own, or to get their returns home in another\n    form than produce, or, which perhaps is the only _true_ course,\n    to limit their operations to more legitimate bounds, nothing\n    but a commercial crisis can be expected; indeed, had it not\n    been for the abundance of money and the large supply of bullion\n    from the West, aided by a splendid harvest, we should doubtless\n    have had a repetition of '47 to some extent at the present\n    moment.\"\n\nShipowners and millers tell us a tale of similar disaster; and the\nshopkeepers, if unanimous in nothing else, agree that their business is\ndecreasing. The working-classes have cheap bread, but at the same time\nthey have lowered wages; so that the advantage received on the one hand\nis neutralised by the reduction on the other.\n\nGrievous, therefore, was the disappointment of the journalists, who\nhad expected this year to wile away the lazy autumn in \"hollaing and\nsinging anthems\" in praise of commercial resuscitation. From that\nresource they were effectually cut out. Something was wanted to vary\nthe monotony of leaders on the Exhibition, a capital subject whilst\nits novelty lasted, but soon too familiar to admit of indefinite\nprotraction. Sewerage was overdone last season. People will not submit\nto perpetual essays on the jakes, or diatribes on the shallowness\nof cesspools: the flavour of such articles can only be enjoyed by a\nthorough-paced disciple of Liebig. It was therefore with no small\nanxiety that our brethren awaited the autumnal meetings of the\nagricultural societies, at which, since Free Trade brought havoc to\nthe farmer's home, there has usually been some excitement manifested,\nand some explanations required and given. The old rule, that politics\nshould be excluded from these assemblies, is manifestly untenable\nat the present time. Until a trade is established on a sound and\nsubstantial basis, it is ludicrous to recommend improvements involving\nan enormous additional outlay. The farmers feel and know that the\nblow struck at their interests has gone too deep to be healed by any\nsuperficial nostrums. Their struggle is for existence, and they have\nresolved to speak out like men.\n\nOne of the worst effects of the repeal of the Corn Laws, and that\nwhich may prove the most permanently detrimental to the welfare of\nthe country, is the apparent separation which it has caused in many\ncases between the interests of the landlord and the tenant. We say\n\"apparent,\" because in reality, and finally, the interest of both\nclasses is the same. But, in the mean time, there can be no doubt\nwhatever that the farmers have endured by far the greatest share of the\nloss. Bound to the land by the outlay of their capital in it and upon\nit, they cannot abandon their vocation, or even change their locality,\nwithout incurring immediate ruin. It is easy for those who know nothing\nabout the matter, to advise them to emigrate elsewhere if they cannot\nprocure a livelihood here. It is still easier for a Free-trading\nlandlord, to whose tergiversation a great part of the mischief is\nattributable, to meet the reasonable request of his tenantry for a\nreduction of their rents with an intimation that he is perfectly ready\nto free them from the obligation of their leases. Such conduct is not\nmore odiously selfish than it is grossly hypocritical, the landlord\nbeing perfectly well aware that it is out of the power of his tenantry\nto accept the offer, without at once sacrificing and abandoning nearly\nthe whole of their previous outlay. The farmer is tied to the stake,\nand cannot escape. He must pursue his vocation, else he is a beggar;\nand he cannot pursue that vocation without an annual and material loss.\nUnder those circumstances, a reduction of rent is all the alleviation\nwhich the farmer can hope to obtain. In many instances he has obtained\nit. We hear of remissions made to the extent of ten and fifteen per\ncent; but these are alleviations only. The farmer is still a loser,\nand would be a loser were the remissions infinitely greater. In former\npapers we have shown that the reduction of fifty per cent on the\nrents throughout Scotland would not avail to remunerate the farmers at\npresent prices, and we have ample testimony to prove that in England\nthe case is the same. On this matter of reduction we shall quote a few\nsentences from a pamphlet entitled _A Treatise on the present Condition\nand Prospects of the Agricultural Interest_, by a Yorkshire Farmer,\npublished at Leeds in the present year:--\n\n    \"It appears to me that neither farmers nor landlords have been\n    aware of the magnitude of the evil; for the intentions of\n    several of our landlords, who, I have no doubt, were actuated\n    by a desire to bear a fair proportion of the loss, were\n    published in the newspapers, stating their determination to\n    reduce their rent from ten to fifteen per cent; and no doubt\n    they thought it would, to some considerable extent, countervail\n    the general reduction in the value of agricultural produce, and\n    perhaps sincerely believed they had acquitted themselves of\n    their duty as landlords.\n\n    \"But as closing our eyes will not avert the danger now\n    impending, and threatening to engulph farmers and landlords\n    in one general ruin, I have thought it not amiss to insert\n    the following table, which shows that a reduction of ten per\n    cent does not reach a degree approaching to anything like\n    a comparison with the losses farmers are suffering. To the\n    occupier of land rented at L4, it is 8s. an acre against a loss\n    of L2, 4s. 1d.--more than half his rent. To the occupier of the\n    second class, rent L2, it is 4s. an acre against the loss of\n    L1, 14s. 7-3\/4d.--nearly the whole of his rent. To the occupier\n    of the third class, rent L1, it is 2s. an acre against a loss\n    of L1, 6s. 4-1\/2d.--6s. 4-1\/2d. more than his rent. And to the\n    unfortunate occupier of the fourth class, rent 7s., it is 8\n    2\/5d. an acre against a loss of L1, 1s. 4-3\/4d.--or more than\n    three times his rent.\n\n    \"I have taken four farms, of one hundred acres each, of\n    different descriptions of soil, showing the net loss on each\n    farm, deducting ten per cent from the rent. For results, see\n    below:--\n\n  Key:  A = Classes\n        B = No. of Acres\n\n     |   |       |          |         |   Total  |       |          |\n     |   |       |          | 10 per  |  Outlay  | Per-  |          |Net Loss,\n     |   | Rent  |          |  cent.  | on Farms,|centage|          |deducting\n     |   | per   |Amount of |reduction| including|  on   |Total Loss|  10 per\n   A | B | Acre. |  Rent.   | on Rent.|   Rent.  |Outlay.| per Farm.|  cent.\n  ---+---+-------+----------+---------+----------+-------+----------+----------\n     |   |L s. d.| L  s. d. |L  s. d. | L  s. d. |       | L  s. d. | L  s. d.\n   1 |100|4 0  0 |400  0  0 |40  0  0 |975  8  4 |4.1    |220  8  4 |180  8  4\n   2 |100|2 0  0 |200  0  0 |20  0  0 |704  7  6 |2.8375 |173  4  7 |153  4  7\n   3 |100|1 0  0 |100  0  0 |10  0  0 |577  1  8 |1.733  |131 15  5 |121 15  5\n   4 |100|0 7  0 | 35  0  0 | 3 10  0 |285 17  6 |1.221  |106 19  7 |103  9  7\n\n    \"The above table shows that, though a reduction of ten per\n    cent may be thought considerable and fair on the part of the\n    landlord, it is like a drop in the bucket when viewed as a\n    set-off against the farmer's losses; and that along with\n    every possible reduction that can be made on the rent, other\n    measures, more comprehensive in character, must be adopted, to\n    place the farmer in a position to enable him to cultivate the\n    soil.\"[6]\n\n[6] Other tables contained in the same pamphlet, but which are too long\nfor insertion here, exhibit the various items and particulars of the\nloss sustained.\n\nThus much we have said regarding the adequacy of reduction of rent\nto meet the agricultural depression, because of late a very vigorous\neffort has been made by the Liberal press to mislead public opinion\non this subject. \"After all,\" say these organs, deserting their\nfirst position that farming was as profitable as ever--\"after all,\nit is a mere question of rent: let the farmer settle that with the\nlandlord.\" It is not a mere question of rent: _it is the question\nof the extinction of a class_; for if, in the long run, it shall\nbecome apparent that no reduction of rent, short of that which must\nleave the owners of the soil generally without profit, owing to the\namount of incumbrances which are known to exist upon the land, can\nsuffice to render cultivation profitable, then the landlord must\nnecessarily supersede the tenant, and the owner the occupier; and one\nof the two profits which hitherto have been recognised as legitimate,\nbe extinguished. To this point things are tending, and that very\nrapidly. The process has begun in Ireland and in the northern parts of\nScotland, and it will become more apparent with the ebbing of the tide.\nContinental prices cannot rule in this country without reducing the\nwhole of our agricultural system to the Continental level, and placing\nthe collection of the revenue and the maintenance of the national\ncredit in the greatest jeopardy.\n\nStill, nothing can be more reasonable than the request generally\nurged by the farmers for a reduction of their rents. They say, and\nsay truly, that they are not able to meet the pressure of the times.\nThey do not say, however, that any reduction which the circumstances\nof the landlords will enable them to make can suffice to remedy the\nmischief. It insures them no profit; it merely saves them from a\ncertain additional loss. In some cases the landlords either will not,\nor cannot, grant such reductions. They have no margin left them. They\ncan but preach hope against knowledge; and in doing so, they play\nthe game of the enemy, and justly lay themselves open to the charge\nof hypocrisy. In fact, what the farmers want, is less a reduction of\nrent--which they know to be but a temporary expedient--than a more\nmanly and decided attitude on the part of the owners of the soil.\nToo many of the landlords allowed themselves to be led astray by\nthe specious representations of the Free-Traders, or were betrayed\ninto supporting the policy of a Minister, for whose antecedents and\nability they entertained an egregiously exaggerated respect. Trusting\nto vamped reports and speculative opinions, presumptuously hazarded\nby men who knew nothing whatever of the subject, they disregarded the\nclear warnings of those who foresaw the magnitude and imminency of\nthe danger; and surrendered themselves, without retaining the means\nof defence, to a faction which laughed at their credulity. These are\nthe men who at agricultural meetings affect to talk hopefully of the\nprospects of agriculture, and who always assure the farmers that their\ncase is regarded with the utmost sympathy by the Legislature. They are\nconstantly advising their hearers, not only to have patience, for that\nwere a proper charge, but to augment the amount of their outlay. They\nare grand upon the subject of artificial manures, and seem to have an\nidea that guano is an inexhaustible deposit. They will even bring down\nlecturers--dapper young chemical men from laboratories--to enlighten\ntheir tenants; but seldom, or rarely, will they grant a single sixpence\nof reduction. Is it wonderful if the honest farmer, thoroughly alive\nto the real peril of his situation, and indignant at the treachery of\nwhich he has been made the innocent victim, should conceive any feeling\nbut those of respect and cordiality for the landlord who is acting\nso paltry a part, and condescending to so wretched an imposture? The\nfarmer feels that now or never his cause must be resolutely fought. He\nknows that the interest of the landlord is as much concerned as his\nown; and yet when he applies to him for support and encouragement, he\nis met with silly platitudes.\n\nAs it has turned out, the agricultural meetings of the present autumn\nhave proved far more fruitful to the journalists than they had any\nreason to expect. Our brethren of the Liberal press have extracted from\nthem grounds for exceeding jubilation and triumph. Mr Disraeli, Mr\nPalmer, Mr Henley, and others, justly considered as very influential\nmembers of the Protectionist party in the House of Commons, are\nrepresented to have expressed themselves in a manner inconsistent with\nthe maintenance of the great struggle which, Session after Session,\nhas been renewed. They are claimed as converts, not to the principles\nof Free Trade--for those they have distinctly repudiated--but to the\ndoctrine that it is impossible, by direct legislation, to disturb the\npresent existing arrangement; and, as a matter of course, a defection\nso serious as this is joyously announced as an abandonment of the cause\nby several of those men who were its most doughty champions.\n\nBefore proceeding to consider the merits of that line of policy which\nMr Disraeli proposes to adopt during the ensuing session, and which,\nin his judgment, is that most likely, under present circumstances, to\nprocure some measure of relief for the agricultural interest, let us\ndistinctly understand whether or not Protection, as a principle, has\nbeen abandoned by any of its supporters in Parliament. We have perused\nthe speeches which have been made the subject of so much comment with\nthe greatest care and anxiety; but we have not been able to discover\nany admission that the views so long and so ably maintained by those\ngentlemen have undergone an iota of change. They may, indeed, and very\nnaturally, despair of success in the present Parliament. Knowing, as\nthey do, the weight and apportionment of parties in the present House\nof Commons, and enabled by experience to calculate upon the amount of\nsupport which would be given to any proposition, they may have arrived\nat the conclusion that the best course of policy which they can adopt,\nis to concentrate their efforts towards obtaining relief from what is\nclearly unjust taxation, leaving the grand question of a return to the\nProtective system in the hands of the country, to be decided at the\nnext general election.\n\nThis is our distinct understanding of the views which have been\nannounced by these gentlemen. It may be that some of them have\nnot sufficiently guarded themselves against the possibility of\nmisrepresentation; an error of judgment which, in the present state of\nthe public mind, may have a detrimental effect. We have no hesitation\nin expressing our opinion that the sentiments uttered by the Marquis of\nGranby, and those contained in the admirable letters of Mr G. F. Young,\nare more calculated to advance the cause, and to insure co-operation\namongst all classes who are opposed to the bastard system of Free\nTrade, than speeches which are only directed towards a subsidiary\npoint, which are apt to be misunderstood, and which have been seized on\nby our opponents as proofs of despondency or despair.\n\nNo one, we believe, expected that, in the present Parliament, such\na change of opinion could be wrought as would lead to the immediate\nrestoration of Protection. In May 1850, the Earl of Derby, then Lord\nStanley, distinctly explained to the delegates who waited upon him,\nthat \"it was not in the House of Lords, it was not in the House of\nCommons, _it was in the country at large_, that the battle must be\nfought, and their triumph must be achieved.\" \"You have,\" said the\nnoble lord, \"the game in your own hands. You may compel your present\nmembers--or, at least, you may point out to them the necessary, the\nlamentable consequences to themselves of persisting in their present\ncourses; _and when the time shall come, you will have it in your own\npower, by the return of men who really represent your sentiments,\nto exercise your constitutional influence over the Legislature of\nthe country, and to enforce your just demands in another House of\nParliament_.\" What has since taken place has most clearly established\nthe soundness and wisdom of this advice. Beyond all question, the\ncause of Protection during the last two years has advanced with rapid\nstrides. The total failure of every prophesied result on the part of\nthe Free-Traders--the continued depression which has prevailed, not\nonly in agriculture, but in manufactures, trade, and commerce--the\nstate of the working-classes, which has experienced no amelioration\nsince the latter measures of Free Trade were carried--the depopulation\nof Ireland, and the astounding increase of emigration from the northern\npart of Great Britain--all have contributed to dispel the popular\ndelusion, and to give new courage and confidence to the disinterested\nsupporters of the cause. Public opinion, in so far as that can be\ngathered from the results of casual elections, has declared itself in\nfavour of Protection. Meetings of the working-classes have been held in\nthe metropolis, at which resolutions in favour of a return to a general\nprotective policy have been passed by acclamation. Nothing whatever\nhas occurred to give a check to the advance of these principles; much\nhas transpired to show how rapidly and strongly they are progressing.\nThat progress does not depend, and never did depend, solely upon the\nresult of the agricultural experiment. The true secret of the reaction\nagainst Free Trade lies in this, that every one of the productive\nclasses of the community is interested in opposing a system which\ncrushes and enthrals labour for the undue benefit of the capitalist.\nIt may be that, in some quarters, that common interest is not yet\nfully understood. It may be that relative cheapness of provisions may\nbe considered by many unthinking and unreflective people in the light\nof a positive blessing, irrespective altogether of the effect of that\ncheapness in diminishing the sphere of employment, and contracting the\nwages of labour at home. This is not wonderful, because, previous to\nthe repeal of the Corn Laws, the tariff had been deliberately altered,\nand the pressure and privation occasioned by these first experiments\nupon British industry were, for a time, materially relieved by the\nfall on the price of provisions consequent on the later measures. But\nvery soon it became apparent to all thinking men, that the prostration\nof so great a branch of industry as that of British agriculture must\nact prejudicially upon all the others, and that the temporary benefit\nwas more than counterbalanced by the universal decline of employment.\nAmong the working-classes, even in larger towns, that opinion is daily\ngaining ground, and becoming a settled conviction. Labour is so much\ndepressed that some effectual remedy must be found, if the country is\nto remain without convulsion; and it is most important for us all that\nthe remedy, which may finally be resorted to, should be a just and\nequitable one, not such as unscrupulous demagogues might apply.\n\nTherefore, at the present time, and in the present temper of the\npublic mind, if we read its symptoms aright, we greatly deprecate any\ndeviation from the broad principle and assertion of Protection to all\nbranches of British Industry. To argue the Agricultural case alone,\nhowever important that may be, is to weaken the general cause, which\nis the cause of Labour. To make terms for the agriculturists only, by\nadjustment of taxation or otherwise, even if such adjustment could by\npossibility enable them to struggle on, would not be a wise policy.\nNever let it be forgotten that the Corn Laws could not have been\nrepealed, but for the previous alterations on the tariff, stealthily\nand insidiously made, which left the agriculturists of Britain in the\npossession of an apparent monopoly. As monopolists, they never can\nregain their former position; but they may, and, we believe, will\nregain it, if they are true to the common cause, as British producers\nagainst foreign competition, on account of the burdens imposed upon all\nproduction by the State, and on account of monetary laws and changes\nwhich have more than doubled their original burden. But they never will\nobtain that justice to which they are entitled, unless they combine\nwith the other classes who are equally suffering under the withdrawal\nof Protection, and insist upon a total change, in the commercial policy\nof Great Britain, as affecting not this or that interest only, but the\nwhole mass of productive labour upon which the wealth of the nation\ndepends.\n\nWe have no hesitation in stating our opinion on this matter in the\nbroadest possible terms. We do not differ from Mr Disraeli in his\nestimate of the unequal burdens which are laid upon the land in\ncomparison with the other property of Great Britain. That is a subject\nwell worthy of consideration; and if it can be treated as entirely\nsubsidiary to the greater question of Protection, and enforced without\nany appearance of an attempt at compromise, we are not prepared to say\nthat any other step, under existing circumstances, would be preferable.\nBut we cannot regard any such adjustment of taxation as a remedy of the\ngrand evil. We doubt the advantage to be derived from a policy which,\nif successful, would only protract the period of general suffering;\nwhilst, in the mean time, it will assuredly be represented as an\nattempt to compromise a principle, and therefore, weaken the amount\nof that support upon which we now can confidently reckon. \"Never,\"\nsaid Burke, in his latest political treatise, \"never succumb. It is a\nstruggle for your existence as a nation. But I have no fears whatever\nfor the result. There is a salient living principle of energy in the\npublic mind of England, which only requires proper direction to enable\nher to withstand this or any other ferocious foe. Persevere, therefore,\ntill this tyranny be overpast.\" The lesson of the great statesman,\nthough directed to other dangers than those which now beset us, has\nlost none of its value. Perseverance, where the principle is clear,\nis less matter of policy than a duty; and therefore we cannot but\nfeel some regret that, at such a time as this, any declaration should\nhave been made, or any policy indicated, which can have the effect of\ndamping the hopes or checking the ardour of those who are most resolute\nin the cause. That the efforts of our adversaries to misrepresent the\ntenor of some of the late speeches have been partially successful,\ncan hardly be doubted by any one who has noted the prevalent tone\nat the subsequent farmers' meetings. We believe that Mr Disraeli is\nat heart and by conviction as much a Protectionist as before. We do\nnot even deny that his tactics, if pursued and successful, might,\nfrom the universal impatience of taxation which prevails, compel any\nMinistry then in power to raise an additional amount of revenue by\nthe imposition of customs duties. Or otherwise, the success of his\nmovement might have the effect of displacing the present Ministry,\nand necessitating an entire party change in Her Majesty's counsels.\nWe are fully alive to the advantage of one or other of these results.\nWe are opposed to further direct taxation, and we have no confidence\nwhatever in the present advisers of the Crown. But we cannot approve\nof any move or any tactics which may have the effect of throwing even\nthe slightest doubt on the determination of the great Protectionist\nparty to persevere in this struggle, until due Protection is obtained\nfor all the productive classes of the community. That party has taken\nits stand upon a principle so just and so true, that, sooner or later,\ndespite every effort on the part of its opponents--every shortcoming on\nthe part of its advocates--it must be triumphant; for the cause is that\nof the whole industrious population of Britain, not of a section or a\nclass.\n\nMr Disraeli proposes to equalise the burdens upon land. Let us suppose\nhim successful; and, according to his own showing, L6,000,000 of\ntaxation, or rather of local rates, should be removed from the land and\nlevied elsewhere. We do not doubt the accuracy of his calculation: we\nbelieve it to be strictly correct. But, were that grievance remedied,\nwould the case be materially altered? We are given to understand that\nL90,000,000 is the amount of the annual depreciation of agricultural\nproduce which has taken place since the Corn Laws were repealed. That\ncalculation was made nearly two years ago, and since then prices have\nconsiderably fallen. Would the farmers accept such share of this\nL6,000,000 as might fall to their lot as a compensation for their\nlosses? The idea is preposterous. We are well aware that Mr. Disraeli\nhas never said this; but does he not see that, in bringing forward this\nsubject in any shape approaching to, or appearing to be, a compromise,\nhe incurs the danger of sacrificing the support, and alienating the\ninterest of the most important, honest, and honourable body of men\nthat exist within the British dominions? The farmers will not stand\nfinessing. They neither comprehend circuitous _coups d'etat_, nor\nwill they follow those who attempt them. The plain English sense is\nhostile to such manoeuvres. They are ready to follow any one in whose\ncapacity and judgment they can place reliance, so long as he pursues\na clear and open course; but the moment that his tactics are veiled,\nuncertain, or unintelligible, they lose confidence in his guidance.\nThat we believe to be, at all times, the tendency of the English\ncharacter. Late events have engendered, not without great reason, much\nsuspicion of the sincerity of public men, whatever be their party or\ndenomination, and therefore it is the more needful that, wherever a\nprinciple is involved, no step whatever should be taken which may\nlead to the remotest suspicion that such principle is about to be\ncompromised. We believe most firmly, most sincerely, that any idea of\nsuch compromise never entered into the mind of Mr Disraeli, or any\nother of the gentlemen whose speeches have been made the subject of\njoyous comment by the Free-Traders. We are satisfied that the line of\naction they have announced is, in itself, honourable and praiseworthy;\nbut we regret that they have not made it distinctly and unequivocally\nsubordinate to the grand cause in which every man in this country, who\nlives by his labour, physical or intellectual, is concerned.\n\nWe have long regarded with much anxiety the position of the farmers\nof England. Viewed as a body, they form the great conservative\nnucleus of the country; and it is to their hatred of innovation,\nsound constitutional feeling, and determined loyalty, that we owe our\nimmunity from those democratic convulsions which have taken place in\nalmost every other part of Europe. To subject such a class of men as\nthis to gross and cruel injustice--to persevere in a policy which is\nreducing them to ruin, after its effects have been made evident--to\ninsult them by the mock language of sympathy, whilst denying them\nan effectual remedy--these are acts of infatuation which were never\ncommitted by any British Ministry save that under Lord John Russell,\nor approved of by any House of Commons save that which is presently\nin existence. Of the patience which the farmers have exhibited under\nsuch trying circumstances, we cannot speak in terms of sufficient\nadmiration. But all endurance has its limit. The farmers were content\nto wait so long as there was a reasonable prospect of a change of\nthat policy which was gradually bringing them to ruin, and long\nabstained from joining in any agitation for purposes which, though\nthey might have had the effect of alleviating their condition, were\nfraught with danger to the commercial credit of the country, and in\nsome respects to the stability of its institutions. But now, finding\nthat both Government and Parliament are obstinately deaf to their\nrepresentations, and dogged in their refusal of redress--meeting with\nfar less support than they were entitled to expect on the part of many\nof the landlords--embarrassed and confused by the tactics announced by\nsome of their supporters in Parliament --they have combined for their\nown defence, and are instituting a movement which may hereafter have a\nmost important effect upon the credit and the destinies of the kingdom.\nAre they to be blamed for this? It would be difficult so to blame\nthem. Rather let the blame rest with those whose obstinacy, ignorance,\nselfishness, or pride has driven them to this position, and compelled\nthe farmer to seek from extravagant and impracticable schemes, and\nfrom clamorous agitation, that relief which was denied him as a sound\nsupporter of the Constitution.\n\nThe nature and objects of the Agricultural Relief Associations may be\ngathered from the report of the Suffolk meeting, lately held at Bury\nSt. Edmunds. The assumption of all the speakers was, that Protection\ncannot be expected either from the present or the future Parliament.\n\n    \"Politicians,\" said one gentleman, \"were every day shifting\n    their ground. Men who a few short months ago threatened to\n    assume the reins of Government, with the express design of\n    reversing the policy of the last few years, were now faltering\n    in their purpose, and confessing both their inability and\n    unwillingness to effect these changes.\"\n\nAnother spoke as follows:--\n\n    \"It was generally known, that while the farmers were asleep the\n    Free-Trade policy came into operation. This at once cut off not\n    less than 20 per cent of the capital employed in farming. This\n    blow the farmers felt very keenly. They at once began to open\n    their eyes, unstop their ears, and to unloose their tongues.\n    They earnestly inquired what steps should be taken by them in\n    the new circumstances under which they were placed. They heard\n    various voices in reply, but the loudest and most powerful of\n    these assured them that they would go back to Protection, and\n    that by next Session too. Next Session passed, however, without\n    exhibiting the least prospect of that result, and they had been\n    going on, Session after Session, until the present moment, when\n    they seemed farther from Protection than ever. Others told\n    them to lay out all their capital on land, and they would be\n    sure to get remuneration. They had done that too, and their\n    capital was gone without any prospect of remuneration.\"\n\nAnother gentleman, hitherto known as a staunch Protectionist, thus\nannounced his reasons for joining the movement:--\n\n    \"The fact was, that when he found members of the House of\n    Commons, who had been returned to Parliament for the express\n    purpose of supporting Protection, saying, behind the scenes,\n    that it was impossible to expect Protection back again; and\n    when he found members of the House of Peers telling him that\n    if they stood out for Protection it would cost them their\n    coronets, he was forced to come to the conclusion that the\n    voice of the people had doomed these laws. He then began to\n    ask himself this plain and simple question--if they give the\n    country cheap corn, _won't they give us cheap taxation_? He was\n    willing to grow corn against any man, come from where he might;\n    but, at the same time, he must have a fair field to do it in.\"\n\nHere are the views of the society as contained in the chairman's\nsummary:--\n\n    \"When their agricultural distress had been relieved by the\n    repeal of the malt-tax, by the permanent fixation of tithe\n    on an equitable basis, by the extinction of church-rates, by\n    a revision of the county expenditure, the abolition of the\n    game-laws, the removal of all restrictions on the cultivation\n    of land, a change in the law of distress, the rights of\n    the tenant-farmers recognised, the abominable abuses of\n    the poor-law corrected, and when the bulk of taxation was\n    shifted from the shoulders of the productive to those of the\n    unproductive classes--from industry upon wealth--then might\n    they hope that honesty, industry, and perseverance would meet\n    their due reward.\"\n\nWe do not make these quotations with any intention of criticising the\nopinions expressed. We simply lay them before our readers as a specimen\nof that spirit which is now possessing the farmers--a spirit engendered\nby wrong, and strengthened by the suffering of years. If anything could\nmake us believe that coronets are in danger of falling, it is the\nexpression of such views on the part of men who hitherto have been the\nbest defenders of the Constitution, and the most averse to yield to\nany of the impulses of change. But, as we have already said, we cannot\nblame the speakers. If they are convinced in their own minds that a\nreturn to Protection is impossible, their condition is such that they\nmust necessarily have recourse to any expedient, however desperate,\nwhich can afford them the prospect of relief. We have long foreseen\nthis crisis. Situated as Great Britain is, the choice lies simply\nbetween a return to Protection to Labour, and an assault on the public\nburdens. There is no other alternative. Cheapness may be established as\nthe rule, but cheapness cannot co-exist with heavy taxation. To hope\nthat the burden can be shifted from one shoulder to another is clearly\nan absurdity. If it is to be sustained, the productive classes must\nhave the means of sustaining it. If those means are denied them, the\nburden is altogether intolerable.\n\nIt is not a little instructive to remark that, even now, the supporters\nof Free Trade are compelled to stop and leave their scheme unfinished.\nThey cannot carry it out in its integrity without ruining the finances\nof the country. They have exposed the farmer to unlimited competition\nin produce, but they still continue to restrict the sphere of his\nindustry and production. The malt-tax is a heavy burden upon him,\nand he is specially prohibited from growing tobacco, or engaging\nin the manufacture of beetroot sugar. These restrictions, say the\nFree-Traders, are absolutely necessary for revenue. Granted; but if you\nput on restrictions, are you not bound to give an equivalent? As for\nthe argument in favour of the malt-tax, that it is the consumer who\nreally pays the duty, that might be extended with equal justice to the\ninstance of raw cotton. Why is barley, the produce of our own country,\nto be taxed, and cotton, the produce of a foreign country, to be\nexempted? Besides this, we have always understood that beer, tobacco,\nand sugar, are articles which enter largely into the consumption of\nthe agricultural as well as that of other classes; so that here is a\ngrievance totally opposed to the principles of Free Trade, and yet\nsupported and perpetuated by the very men who have adopted Free Trade\nas their motto! We instance these things as proofs that Free Trade\nnever can be made, in the strict sense of the word, the law and system\nof the land, so long as the present enormous expenditure is continued;\nand in saying this, we hope it will be understood that we are as much\nopposed as ever to the views of the party who are for cutting down our\nnational establishments.\n\nWe anticipate, in the course of next Session, to hear many propositions\nmade on the subject of adjustment of taxation. Each class is anxious\nto be freed from its own peculiar burdens, and to devolve them upon\nothers; and certainly never was there any case so strong or so urgent\nas that which can be brought forward on the part of the agriculturists.\nBut who is to relieve them? Will any other class submit to the\ntransference which is necessarily implied? Will the manufacturers or\nthe capitalists undertake to provide for the six millions which at\npresent they are most unjustly wresting from the owners and occupiers\nof the soil? Here is the real difficulty. Justice, we know, is not\nregarded as an indispensable element of taxation: if it were so, the\nincome-tax would never have been imposed in its present form. If the\nclaims of the farmers who are banded together for agricultural relief\nwere granted, immediate national bankruptcy would be the result. This\nis the grand dilemma in which we are placed by the Free-Traders.\nEither a gross and palpable act of injustice and oppression must\nbe perpetuated--so long at least as the victims have the means of\npayment--or, as was long ago prophesied, the capitalists and the\nfundholders must suffer. The truth is, that the productive power of the\ncountry cannot meet the demands upon it in the shape of taxation if it\nremains exposed to unlimited foreign competition.\n\nIn order properly to comprehend this point, which is one of the utmost\nimportance, it is necessary to discard theory altogether, and to adopt\nhistory as our guide. The financial system of Great Britain, acting\nupon and influencing the commercial arrangements and social relations\nof the country, is not difficult of comprehension if we trace it\nstep by step; and without a due understanding of this, and the vast\ninfluence which monetary laws exercise over the wellbeing and progress\nof a nation, it is impossible for any one to form a sound judgment on\nthe conflicting principles of Protection and Free Trade, or to discover\nthe true and only source of the difficulties which now surround us.\nIt is the misfortune of the present age that so little attention\nis paid to the abstruser portions of history, which, in reality,\nare the most valuable for us. Wars of succession or conquest, naval\nengagements, records of intrigue or details of diplomatic dexterity,\nhave a peculiar charm and interest for readers of every kind; but few\ntake the pains to go more deeply into the subject, and investigate in\nwhat manner such events have affected the resources of a country, and\nwhether, by diminishing its wealth or by stimulating the energies of\nits population, they have lowered or raised its position in the scale\nof nations. That portion of history which relates to external events is\nworthless for practical purposes, unless we combine with it the study\nof that portion which relates to its finance. Under the modern system,\nnow universal throughout Europe, which leaves the debts and engagements\nof former generations to be liquidated or provided for by the next,\nno man can be called a statesman or politician who has not addicted\nhimself to these studies.\n\nThe Funding System, as is well know, began with the Revolution, and\nhas continued up to the present hour. It was strenuously opposed and\nvigorously assailed by some of the most able and clear-sighted in the\ncountry, such as Bolingbroke, David Hume, and Adam Smith, who from time\nto time pointed out the consequences which must ultimately ensue from\nthis method of mortgaging posterity, more especially if the burden\nwere allowed to increase without any steps being taken to provide\nfor its ultimate extinguishment. It is the peculiarity of a debt so\nconstituted, that for a time it gives great additional stimulus to\nthe energies of a country. It enables it to prosecute conquests, and\nto undertake designs, which it could not otherwise have achieved; and\nit is not until long afterwards, when the payment of the interest or\nannual charge becomes a severe burden upon a generation which had no\nshare in contracting the debt, that the mischievous effects of the\nsystem become apparent. At the outbreak of the French Revolution, the\npublic debt of Great Britain amounted to L261,735,059, and the annual\ncharge was L9,471,675. A very large portion of this debt was incurred\nfor the war waged with our American colonies.\n\nAt that time the currency of the country was placed on the metallic\nbasis, but the great drain of the precious metals occasioned by the\nenormous subsidies which Great Britain furnished to her allies on the\nContinent, to engage their support against the revolutionary armies\nof France, reduced the nation to the very verge of bankruptcy, and\nnecessitated in 1797 the suspension of cash payments. The immediate\neffect of this step upon the finances of the country has been so\njustly, and at the same time so clearly, stated by Mr Alison in his\n_History of Europe_, and the consequences of the subsequent return to\nthe old system of cash payments, after their suspension for nearly\na quarter of a century, are so graphically depicted, that we cannot\ndo better than entreat the attention of the reader to the following\nextract:--\n\n    \"This measure having been carried by Mr Pitt, a committee\n    was appointed, which reported shortly after that the funds\n    of the Bank were L17,597,000, while its debts were only\n    L13,770,000, leaving a balance of L3,800,000 in favour of\n    the establishment; but that it was necessary, for a limited\n    time, to suspend the cash payments. Upon this, a bill for\n    the restriction of payments in specie was introduced, which\n    provided that bank-notes should be received as a legal tender\n    by the collectors of taxes, and have the effect of stopping the\n    issuing of arrest on mesne process for payment of debt between\n    man and man. The bill was limited in its operation to the 24th\n    June; but it was afterwards renewed from time to time, and in\n    November 1797 continued till the conclusion of a general peace;\n    and the obligation on the Bank to pay in specie was never again\n    imposed till Sir Robert Peel's Act in 1819.\n\n    \"Such was the commencement of the paper system in Great\n    Britain, which ultimately produced such astonishing effects;\n    which enabled the empire to carry on for so long a period so\n    costly a war, and to maintain for years armaments greater than\n    had been raised by the Roman people in the zenith of their\n    power; which brought the struggle at length to a triumphant\n    issue, and arrayed all the forces of Eastern Europe in English\n    pay, against France on the banks of the Rhine. To the same\n    system must be ascribed ultimate effects as disastrous, as\n    the immediate were beneficial and glorious; the continued\n    and progressive rise of rents, the unceasing, and to many\n    calamitous, fall in the value of money during the whole course\n    of the war; increased expenditure, the growth of sanguine ideas\n    and extravagant habits in all classes of society; unbounded\n    speculation, prodigious profits and frequent disasters among\n    the commercial rich; increased wages, general prosperity, and\n    occasional depression among the labouring poor. But these\n    effects, which ensued during the war, were as nothing compared\n    to those which have, since the peace, resulted from the return\n    to cash payments by the bill of 1819. Perhaps no single measure\n    ever produced so calamitous an effect as that has done. It has\n    added at least a third to the National Debt, and augmented in a\n    similar proportion all private debt in the country, and at the\n    same time occasioned such a fall of prices by the contraction\n    of the currency as has destroyed the sinking fund, rendered\n    great part of the indirect taxes unproductive, and compelled\n    in the end a return to direct taxation in a time of general\n    peace. Thence has arisen a vacillation of prices unparalleled\n    in any age of the world; a creation of property in some and\n    destruction of it in others, which equalled, in its ultimate\n    consequences, all but the disasters of a revolution.\"[7]\n\n[7] ALISON'S _History of Europe_, chap. xxii.\n\nThe immediate effect of the suspension of cash payments on the part of\nthe State bank was an enormous increase in the circulation of paper.\nThe prices of commodities rose to nearly double their previous value,\nand a period of prosperity commenced, at least for one generation.\nDuring the twenty-two years which elapsed from the suspension of cash\npayments in 1797 down to 1819, when their resumption was provided\nfor by Act of Parliament, or at least during eighteen years of that\nperiod, reckoning down to the re-establishment of peace in Europe, the\ncareer of England has no parallel in the annals of the world. The vast\nimprovements and discoveries in machinery which were made towards the\nlatter end of the century--the inventions of Hargreaves, Arkwright,\nCartwright, Crompton, and Watt, came into play with astounding effect\nat a time when Great Britain held the mastery of the seas, and could\ndivert the supplies of raw material from all other shores except\nher own. During the hottest period of the war, and in spite of all\nprohibitions, England manufactured for the Continent. Capital, or that\nwhich passed for capital, was plentiful; credit was easy, and profits\nwere enormous. Some idea of the rapidity with which our manufactures\nprogressed may be drawn from the fact that, whereas in 1785 the\nquantity of cotton wrought up was only 17,992,882 lbs., in 1810 it had\nincreased to 123,701,826. Under this stimulus the population augmented\ngreatly. The rise in the value of commodities gave that impulse to\nagriculture by means of which tracts of moorland have been converted\ninto smiling harvest-fields, fens drained, commons enclosed, and huge\ntracts reclaimed from the sea. The average price of wheat in 1792, was\n42s. 11d.; in 1810, it was 106s. 2d. per quarter. Wages rose, though\nnot in the same proportion, and employment was abundant.\n\nIn short, the paper age, while it lasted, transcended, in so far as\nBritain was concerned, the dreams of a golden era. Those who suffered\nfrom the suspension of cash payments were the _original_ fundholders,\nannuitants, and such landlords as had previously let their properties\nupon long leases. But the distress of such parties was little heard,\nand less heeded, amid the hum of the multitudes who were profiting\nby the change. The creditor might be injured, but the debtor was\nlargely benefited. One immediate effect of this rise in prices was, a\ncorresponding rise in fixed salaries and the expenses of government.\nHence, the domestic expenditure of the country was greatly increased;\nnew taxes were levied, and the permanent burden of the National\nDebt augmented to an amount which, sixty years ago, would have been\nreckoned entirely fabulous. As a specimen of the increased expense\nof cultivating arable land, it may here be worth while to insert the\nfollowing comparative table, calculated by Mr Arthur Young, and laid\nbefore a committee of the House of Lords. The extent is one hundred\nacres:--\n\n\nCOMPARATIVE TABLE OF THE EXPENSES OF ARABLE LAND.\n\n  |                      |     1790.     |     1803.     |     1813.     |\n  |                      +---------------+---------------+---------------+\n  |Rent,                 | L88  6  3-1\/4 |L121  2  7-1\/4 |L161 12  7-1\/4 |\n  |Tithe,                |  20 14  1-3\/4 |  26  8  0-1\/4 |  38 17  3-1\/4 |\n  |Rates,                |  17 13 10     |  31  7  7-3\/4 |  38 19  2-3\/4 |\n  |Wear and tear,        |  15 13  5-1\/4 |  22 14 10-1\/4 |  31  2 10-3\/4 |\n  |Labour,               |  85  5  4     | 118  0  4     | 161 12 11-1\/4 |\n  |Seed,                 |  46  4 10-1\/4 |  49  2  7     |  98 17 10     |\n  |Manure,               |  48  0  3     |  68  6  2     |  37  7  0-1\/4 |\n  |Team,                 |  67  4 10     |  80  8  0-1\/4 | 134 19  8-1\/2 |\n  |Interest,             |  22 11 11-1\/2 |  30  3  8-1\/2 |  50  5  6     |\n  |Taxes,                |   0  0  0     |   0  0  0     |  18  1  4     |\n  |                      +---------------+---------------+---------------+\n  |          Total,      |L411 15 11-3\/4 |L547 10 11-1\/2 |L771 16  4-1\/2 |\n  |        Deduct rent,  |  88  6  3-1\/4 | 121  2  7-1\/4 | 161 12  7-1\/4 |\n  |                      +---------------+---------------+---------------+\n  |      Nett expenses,  |L323  9  8-1\/2 |L426  8  4-1\/2 |L610  3  9-1\/4 |\n  |                      +---------------+---------------+---------------+\n  |Price of wheat        |               |               |               |\n  |    per quarter,      |      46s.     |   56s.  9d.   |  108s.  9d.   |\n\nSo long as the war lasted, the import of corn from abroad into this\ncountry was insignificant in amount. In 1814 the amount of wheat\nand wheat flour brought in amounted to only 681,333 quarters, being\nconsiderably above the average of years since the commencement of the\ncentury. In fact, Britain was then self-supporting. In time of war\nit is plain that, from our insular position, we cannot trust to any\nsupplies beyond those which are raised at home, and there cannot be any\ndoubt of the capability of the land to support a much larger population\nthan that which presently exists. To those who glance superficially\nat the above table, the price of wheat in the year 1813 will appear\nmonstrous, even when the great increase in the cost of cultivation is\ntaken into account. This is the error, and it is a gross one, which has\nbeen studiously perpetuated by our statists, and even by some eminent\nwriters on political economy. True, the price of wheat was then 108s.\n9d., _but it was estimated in a depreciated currency_. Owing to various\ncauses which it would be tedious to explain, the apparent difference\nbetween the value of the pound note and the guinea was far slighter\nthan might have been expected, not amounting to more than seven or\neight shillings, and actual depreciation, by sale of the notes for\nless than their nominal value, was by statute made penal. The price\nof gold and silver bullion never rose to an extent commensurate with\nthe depreciation of the paper: in fact the coinage, as must be in the\nrecollection of many, almost entirely disappeared, and was replaced\nby tokens of little intrinsic value, which served as the medium of\ninterchange. In this depreciated and fluctuating currency commodities\nwere valued, and by far the greater part of our National Debt was\ncontracted. The paper pound in 1813 was probably, we may almost say\ncertainly, not worth more than 10s. of metallic currency. In this view\nthe quarter, estimated according to the present standard, was sold\nfor 54s. 4-1\/2d--a price which modern statesmen allow to be barely\nremunerative.\n\nIf this point were generally understood, a vast amount of delusion\nwhich possesses the public mind would be dispelled. The relative value\nof money to commodities has been as entirely changed, by the return\nto cash payments, as if shillings had been substituted for sixpences.\nIf the creditor suffered in 1797, the debtor has suffered far more\nseverely in consequence of the Act of 1819, as we shall immediately\nproceed to show. Meantime, we shall entreat the reader to keep in mind\nthat all incomes and expenditure, public or private, during the war and\nthe suspension of cash payments, are to be estimated not by our present\nmetallic standard, but by the fluctuating value of a depreciated\ncurrency.\n\nWhen peace was established the ports were opened. Then it became\nevident that foreign importations, if permitted, would at once and for\never extinguish the landed interest. The annual charge of the debt\nalone was, in 1816, the first year of peace, L32,938,751; and the\ncurrent annual expenditure L32,231,020--in all, _upwards of sixty-five\nmillions_. Had, therefore, the price of wheat in Britain been suddenly\nreduced to the Continental level, as would have been the case but for\nthe imposition of the Corn Laws, the national bankruptcy would have\nbeen immediate. No argument is required to prove this and it has often\nstruck us as singular that this crisis--for such it was--has been so\nseldom referred to, especially in later discussions. We are not now\ndefending the original suspension of cash payments--a measure which,\nnevertheless, seems to us to have been dictated by the strongest\npolitical necessity, however baneful its results may prove to the\npresent and future generations. We simply say, that eighteen years'\noperation of that system, with the enormous expenditure and liabilities\nwhich it entailed, rendered Protection necessary the moment importation\nwas threatened, to save the country from immediate bankruptcy following\non its unparalleled efforts.\n\nIt is utter folly, and worse, to say, as political economists now\ncontend, and as ignorant demagogues aver, that the Corn Laws were\noriginally proposed solely for the benefit of the landlords. Without\nthe imposition of such laws, the whole financial system of Great\nBritain must instantly have disappeared. The amount of taxes which\nwere required--first, to pay the interest of the National Debt, and,\nsecondly, to meet the expenses of Government, (greatly increased by\nthe change in the monetary laws effected in 1797)--rendered Protection\nto labour and to native produce absolutely indispensable. How could\nit be otherwise? Had wheat been sold in the British market at 46s.,\nor even 50s., from what sources could the revenue have been levied?\nUnder the new system, the expenses of cultivation had nearly doubled\nin twenty-three years--could the produce be put back to the same rates\nas before? So long as the monetary system then established did exist,\nthat was clearly impossible. Protection was imperatively demanded, not\nby any class of the community, but by the state. To refuse it would\nhave been national suicide. And so it was carried, doubtless very\nmuch against the inclination of the populace, who naturally enough\nexpected that the return of peace would bring with it some substantial\nadvantages in the shape of cheapness, and were proportionally\ndisappointed when they discovered that the whole rent-charge of the\nwars, which had been so long maintained, must be liquidated before they\ncould taste the anticipated blessings of the cheap loaf.\n\nThe return to cash payments, effected by the Act of 1819, is by far\nthe most important event in our history since the change of dynasty.\nWe believe that the late Sir Robert Peel, then a very young man, who\nwas made the mouthpiece of a particular party on that occasion, really\ndid not understand, to its full extent, the tremendous responsibility\nwhich he incurred. He acted simply as the exponent of the measure,\nat the instigation and by the direction of Mr Ricardo, who, under\nthe guise of a political economist, concealed the crafty and selfish\nmotives of the race from which he originally sprung. Ricardo was\nat that time considered a grand authority on matters of finance,\nhis field of preparatory study having been the Stock Exchange, on\nwhich he is understood to have realised a large fortune. All his\nprepossessions, therefore, were in favour of the capitalists; and\nit is not uncharitable to conclude that his private interests lay\nin the same direction. That act provided for the gradual resumption\nof cash payments throughout England, to be consummated in 1823, for\nthe establishment of a fixed gold standard, and for the withdrawal\nof all bank-notes under the amount of five pounds. Had this act been\ncarried into effect in all its integrity, general bankruptcy must have\nimmediately ensued, from the absorption of the circulating medium.\nThe existence of the small notes, however, was respited, and this\nenabled the country bankers to go on for some time without a crash.\nStill the violent contraction of the currency, so caused, had the\nnecessary effect of spreading dismay throughout all sections of the\ncommunity. The circulation of the Bank of England, at 27th February\n1819, was L25,126,700. On the 28th February 1823, it was contracted\nto L18,392,240. At the former period its private discounts amounted\nto more than nine millions; at the latter, they were considerably\nunder five. As a matter of course, the country bankers were compelled\nto follow the example of the great establishment, and the immediate\nresults of this grand financial measure may be described in a few\nwords. The tree was thoroughly shaken. According to Mr Doubleday--\n\n    \"As the memorable first of May 1823 drew near, the country\n    bankers, as well as the bank of England, naturally prepared\n    themselves, by a gradual narrowing of their circulation, for\n    the dreaded hour of gold and silver payments \"on demand,\" and\n    the withdrawal of the small notes. We have already seen the\n    fall of prices produced by this universal narrowing of the\n    paper circulation. The effects of the distress produced all\n    over the country--the consequence of this fall--we have yet to\n    see.\"\n\n    \"The distress, ruin, and bankruptcy, which now took place,\n    were universal; affecting both the great interests of land and\n    trade; but amongst the landlords, whose estates were burdened\n    by mortgages, jointures, settlements, legacies, &c., the\n    effects were most marked, and out of the ordinary course. In\n    hundreds of cases, from the tremendous reduction in the price\n    of land which now took place, the estates barely sold for as\n    much as would pay off the mortgages; and hence the owners were\n    stripped of all, and made beggars. I was myself personally\n    acquainted with one of the victims of this terrible measure.\n    He was a schoolfellow, and inherited a good fortune, made\n    principally in the West Indies. On coming of age, and settling\n    with his guardians, he found himself possessed of fully forty\n    thousand pounds; and with this he resolved to purchase an\n    estate, to marry, and to settle for life. He was a young\n    man addicted to no vice; of a fair understanding and a most\n    excellent heart; and was connected with friends high in rank,\n    and likely to afford him every proper assistance and advice.\n    The estate was purchased, I believe, about the year 1812 or\n    1813, for eighty thousand pounds, one moiety of the purchase\n    money being borrowed on mortgage of the land bought. In 1822-3\n    he was compelled to part with the estate, in order to pay off\n    his mortgage, and some arrears of interest; and when this was\n    done, he was left without a shilling--the estate bringing only\n    half of its cost in 1812.\"[8]\n\n[8] _Financial History of England._ By THOMAS DOUBLEDAY.\n\nBut isolated instances, however great may be their interest, will not\nadequately exhibit the effects of this measure upon the vital interests\nof the country. At least one half of the National Debt was incurred\nafter the suspension of cash payments, and during the prevalence of\nthe Paper Currency. The interest of that debt was now, and in all time\ncoming, to be paid in coin greatly above the value of the currency\nin which it was contracted; and the Private Creditor shared in the\nadvantage which thus was given to the Fund-holder. The taxes were all\nto be levied in the same way, the metallic standard being made of\nuniversal application. As a matter of course, prices fell, and fell in\na corresponding ratio.\n\nThe great prosperity of England during the war, and the unexampled\ndevelopment of its resources, whether agricultural, manufacturing, or\ncommercial may be traced to the combination of at least three causes.\nIn the first place, England was thoroughly protected. Her artisans and\nlabourers had nothing to fear from foreign competition. They had the\nmonopoly of their own home market, and were not liable to be undersold\nby the products of other nations. In the second place, we had a most\nextensive foreign trade, the real value of which cannot be ascertained\nfrom the official tables, owing to the manner in which that trade\nwas carried on. But even according to the defective records which we\npossess, it appears that our exports in 1805 were equal to those of\n1823, many of the intermediate years showing a much larger amount. In\n1810, our exports were close upon forty-six millions; in 1832 they were\nbarely above thirty-six. In the third place, the country possessed a\nlarge circulating medium, which gave ample scope to enterprise. We\nshall not enter upon the vexed question of systems of currency in the\nabstract; it is enough for us to know that for more than twenty years\nBritish prosperity went on without a check, until it was strangled\nby the bullionists. At present, we have neither Protection, nor\nan Expanded Currency. Our foreign trade, in so far as exports are\nconcerned, is nominally large; but those who are best qualified to\njudge of the value of that trade, declare that it is unremunerative.\n\nWe are therefore very much at a loss to know what element of prosperity\nexists at the present time. We have every faith in British energy if\nit is allowed fair play, but that is precisely what we contend is not\nvouchsafed to it. Our whole legislation, under the guidance of the\npolitical economists, may be characterised as a systematic attempt\nto depress British industry. This could not have been effected at\nonce, or by one isolated effort: several attacks upon the productive\nclasses were required before this was consummated. The change of\ncurrency lowered the value of produce, and increased the burden of\ntaxation. In other words, it brought down both prices and wages, to the\nmanifest gain of the capitalist. Then came the gradual relaxation of\nthe tariff, which has resulted in free importation--a measure by which\nall the working-classes, without any exception, are assailed. This\nwas effected with a perseverance and ingenuity which we cannot help\nadmiring, even when we denounce it as diabolical. The first advances\nto Free Trade were no more remarked by the public in general than the\nfootmarks of the tiger in the jungle when he advances stealthily on his\nprey. The real instigators were the exporting manufacturers. After the\nreturn of peace, they saw clearly enough that their old monopoly was\nat an end. Cobbett wrote, very shrewdly, though in his own peculiar\nmanner, in 1815:--\n\n    \"It is now hoped by some persons that the restoration of the\n    Pope, the Inquisition, the Jesuits, and the Bourbons, will so\n    far brutalise the people of the Continent of Europe that we\n    shall have no rivals in the arts of peace; and that thus we\n    shall be left to enjoy a monopoly of navigation, commerce,\n    and manufactures; and be thereby enabled to pay the interest\n    on our debt, and to meet the enormous annual expenses of our\n    government. Without stopping to comment on the morality and\n    humanity of this hope, entertained in a country abounding with\n    Bible Societies, I venture to give it as my decided opinion,\n    that the hope is fallacious. Russia, Denmark, Sweden, Holland,\n    Austria, Spain, the Italian States, and even the Bourbons, will\n    all push forward for their share of the benefits of the arts\n    of peace. While our purse is open to them all, they will be\n    subservient to us; but that cannot be for ever.\"\n\nThe old sergeant was perfectly right--with the return of peace our\nmonopoly of the foreign market was over; but the question still\nremained, whether, by the sacrifice of home labour, our exporting\nmanufacturers might not be able, for a considerable period at least, to\nkeep ahead of their new rivals in distant markets. Unfortunately for us\nall, the political economists determined to make the attempt.\n\nIn some important branches of manufacture Britain was still unrivalled.\nThe nearest, readiest, and therefore most lucrative market for these\nwas to be found in Europe, and in consequence, it was deemed necessary\nthat concessions should be made to admit some kinds of produce as\nimports, by way of inducing the foreigners to concede a free admission\nto our exports. There is a scene in Shakspeare's play of _Julius\nCaesar_, in which Antony, Octavius, and Lepidus are represented, seated\nat a table, conceding amicably the deaths of the near relations\nof each, in exchange for a similar surrender. This is not quite a\nparallel to the case before us. Our statesmen doomed their friends\nand fellow-countrymen without requiring a reciprocal sacrifice, and\nthe consequence was that we gradually opened our home market to the\nforeigner, without insisting that he should render to us the same\nmeasure of justice. The artisans were the first to feel the blow. They\nhad already suffered, most severely, from the change in the currency,\nwhich brought down prices, and, with them, the remuneration of labour;\nand the withdrawal of Protection from them made them the natural\nenemies of all those who were still shielded from foreign competition.\nThe feeling was perfectly natural. The system begun by Huskisson, and\nconsummated by Peel, could have no other effect than in arming one\nclass of the community against the other. Deprive John forcibly of his\ncoat, under the pretext of justice, and he will immediately insist that\nthe same measure of deprivation shall be extended to James. He has a\nconverse of a Christian rule to utter in his defence--\"Why should not\nothers be done to, precisely as I have been done by?\"\n\nThis argument, in the hands of its able advocates, has proved\nirresistible. John and James are alike without coats; and until they\nagree with one another, and come to a common understanding, there is\nnot much likelihood of their resumption of their necessary wear. It\nnever has been, and never can be, for the interest of the producer\nthat prices should be generally low. Very great nonsense has of late\nyears been talked by public men, and, amongst others, by members of\nthe present cabinet, regarding the \"natural price\" of corn. They seem\nto think that they have stumbled upon a happy phrase, and claim credit\nto themselves for patriotism in resisting all attempts to make the\nbread of the people dearer. But they do not, or will not, see that\nthe great body of the people are interested in this question, not as\nconsumers, but as producers. The vast majority of the population of\nthese islands have hitherto derived their means of subsistence, not\nfrom manufactures, but from the soil. Manufactures do not in reality\nconstitute more than one-fourth part of the annual creation of our\nwealth; and two-thirds at least of all our manufactures are intended\nfor the home market, and will be profitable or not according to the\ncircumstances of the general body of consumers. Now, the natural price\nof corn depends upon the circumstances of the country in which it is\nproduced. It may be ten shillings in Poland: it may be sixty shillings\nin England. No doubt you can get corn, and are getting it, from Poland\nfar cheaper than you can raise it in England--but at what cost? Why,\nat the sacrifice of that enormous capital which has been sunk in the\ncultivation of the land, and of nearly one-half of the annual creation\nof our wealth!\n\nThe average price of wheat, for a number of years preceding 1790, was\n46s. per quarter. It is so stated in Mr Arthur Young's table, which\nwe have given above, and may be taken as the average of thirty years.\nThe average for 1790 was considerably higher, for we observe that Mr\nPorter states it at 53s. 2d. Now, since that period, both the amount\nof our debt and of our current annual public expenditure has been\ntripled--that is, we have three times as much to pay in the shape of\ntaxation as formerly. This is independent of poor-rates and local\ntaxation, which have also greatly increased. That being the case, we\nask how it is possible that corn can be grown now in Britain at a\nprofit, when the ruling price, owing to importations from abroad, is\n_eight shillings per quarter_ lower than it was on an average of years\npreceding 1790? The absurdity is palpable.\n\nHow, then, are the taxes to be paid? That is the question. Not out of\nthe profits of the foreign trade certainly, for the whole value of\nour exports is not much above the amount of the national expenditure,\nand when we add the local taxes, would not reach one-half of the\nrequisite sum. Besides, at the present moment, the exports are not\nnearly balancing the imports. According to the official tables, the\ndeclared value of the exports for the year ending 5th January 1850,\nwas L63,596,025; the official value of the imports for the same period\nwas L105,874,607. We presume it will be admitted that taxes can only\nbe permanently paid out of profits, and we want to know where these\nprofits are? It is perfectly evident that the cultivation of the land\ncannot be carried on for ever at a loss. Sooner or later both capital\nand credit must be exhausted; soils of an inferior description--indeed\nall except the best land in the neighbourhood of towns--must be\nabandoned and withdrawn from tillage, and the working-classes will find\nthemselves utterly unable to meet the demands of taxation. An immense\nportion of our taxation is, and must be, drawn from the labouring men.\nThey contribute largely to our revenue through the customs and excise,\nand the extent of their consumption depends entirely upon the amount\nof the wages which they receive. Any measure which tends to lessen\nthe sphere of production is a direct blow at their interests. Cheap\nbread is just another word for low wages, as already many of them have\ndiscovered to their cost; and we have now arrived at that stage of the\nexperiment when its effects will be rapidly developed.\n\nMr Porter, whose brains are principally valuable in the preparation of\ncumbrous statistics, breaks out, for once in a way, into a fine burst\nof eloquence on the subject of over-population. Let us hear him in his\nanimated mood:--\n\n\"Whence arises this fear--this childish fear of the increase of our\nnumbers?--childish, because it exists without regard to the lessons\nof experience. What evidence is there in our present condition to\njustify the complaint of 'surplus population,' that did not exist in\nas great, or even in a greater degree of force, when our numbers had\nnot reached one-half their present amount? Why, then, shall we not go\nforward to double, and again to double our population in safety, and\neven to advantage, if, instead of rearing millions of human _clods_,\nwhose lives are passed in consuming the scanty supplies which is all\nthat their task of intelligence enables them to produce, the universal\npeople shall have their minds cultivated to a degree that will enable\neach to add his proportion to the general store?\"[9]\n\n[9] PORTER'S _Progress of the Nation_, p. 692.\n\nGood lack, Mr Porter, there was no occasion at all for your putting\nyourself into such an inconvenient heat! Nobody, so far as we know,\nis making any complaint of surplus population. You and your friends\nhave taken effectual measures to prevent such a state of matters, and\nwe may now rest without any apprehension of a visit from the ghost\nof Malthus. The \"universal people\" alluded to in your last brilliant\nthough somewhat unintelligible sentence, are likely to follow your\nadvice, and abstain from \"rearing millions of human clods,\" at least\nupon British soil. Be satisfied--you have done for the clods. Ireland\nis a noble example of your trophies in that way; and if you want to\nglorify yourself on another, you may refer to the Scottish Highlands.\nThe true way to provide against the evils of over-population is to\nlower the value of produce, which is the condition of labour, below the\nremunerative point. Do that, and you may make a wilderness out of the\nmost fertile region of the earth. But then, Mr Porter, did you never\nask yourself what is to become of those who derive their subsistence\nand incomes from the labour of these self-same clods? A good many of\nus, we suspect, are in that condition, and very melancholy indeed would\nbe our countenances if called upon to assist at the funeral of the last\nof that race. \"Meddle not,\" said the Giant, in the German fable, to his\nchild, who had picked up a peasant as a plaything--\"meddle not with the\nhusbandman! But for him, what would become of us Giants?\" It would be\nwell if you and your political allies had the intelligence to apprehend\nthe moral.\n\nThe _Times_, in a late number, has treated the subject of emigration\nin a lively manner. The depopulation which has taken place since Free\nTrade became the law of the land, is too startling a fact to be passed\nover without notice; and it is thus that the leading journal speculates\non the strange phenomenon. The announcement in the opening sentence may\npuzzle, if not alarm, some of the most zealous advocates of foreign\nproduction:--\n\n\"The stream of emigration now set towards America will not stop till\nIreland is absolutely depopulated; and the only question is, when will\nthat be? Twenty years at the present rate would take away the whole\nof the industrious classes, leaving only the proprietors and their\nfamilies, members of the learned professions, and those whose age or\ninfirmities keep them at home. Twenty years are but a short time in\ntreating great social or political questions. It is more than twenty\nyears since the passing of the Emancipation Act and the introduction\nof the Reform Bill. What if it should really come to pass that before\nanother twenty years the whole Celtic race shall have disappeared from\nthese isles, and the problem of seven centuries received its solution?\nWe dwell in wonderful times, in an age of great discoveries, splendid\nimprovements, and grand consummations. Art has always been found the\nhandmaid of human developments. The discovery of gunpowder put an end\nto the little wars and little states of the middle ages, and introduced\nlarger political manipulations. The discovery of printing prepared for\nthe revival of learning and arts, and paved the way to the Reformation.\nThe discovery of the mariner's compass showed our navigators a path\nto the East Indies and the New World. It may be the first mission of\nrailways to set all the populations of the Old World on the move, and\nsend them in quest of independent and comfortable homes.\n\nAnd when will this movement stop? Incuriousness and prejudice are ready\nwith the reply, that it will stop, at all events, when the Celtic race\nis exhausted. The Englishman, we are assured, is too attached to his\ncountry, and too comfortable at home, to cross the Atlantic. But surely\nit is very premature to name any such period for this movement, or to\nsay beforehand what English labourers will do, when seven or eight\nmillions of Irish have led the way to comfort and independence. The\nEnglishman is now attached to his own home, because he knows of no\nother. His ideas of other regions are dark and dismal. He trembles at\nthe thought of having to grope his way through the Cimmerian obscurity\nof another hemisphere. The single fact that he will have no 'parish'\nin America is, in his mind, a fatal bar to locomotion. But all this is\nquickly passing away. Geography, union workhouses, ocean mails, and the\ndaily sight of letters arriving in ten days from prosperous emigrants,\nare fast uprooting the British rustic from the soil, and giving him\ncosmopolitan ideas. In a very few years the question uppermost in his\nmind will be whether he will be better off here or there? Whether he\nshould go with the young and enterprising, or stay at home with the\nold and stupid? If a quarter of a million British subjects have left\nthis country for the Australian colonies in the present generation,\nthere may easily be a much larger movement to a nearer and more wealthy\nregion. It has been imagined, indeed, that such a migration will have\na natural tendency to stop itself at a certain stage. We are told that\nthe English labourer will find a new field in Ireland, deserted by\nthe Celt. It will, however, cost no more effort of mind to cross the\nocean at once than to cross the Irish Channel for a land which, in the\nEnglish mind, must ever be associated with violence and blood. High\nwages, again, we are told, the enjoyment of a liberal government, and\nan improved condition, will bind the Englishman afresh to the soil of\nhis ancestors. But when you make the English labourer richer, more\nindependent, more intelligent, and more of a citizen, you have put him\nmore in a condition and temper to seek his fortune, wherever it may\nbe found. The men who in the United States leave their homes for the\nFar West are generally they who have prospered where they are, and who\nwant the excitement of another start in life. On the whole, we are\ndisposed to think that the prospect is far too serious to be neglected,\nor treated as a merely speculative question. The depopulation of these\nisles, supposing the Celtic exodus to run out its course, and a British\nexodus to follow, constitute about as serious a political event as\ncan be conceived; for a change of dynasty, or any other political\nrevolution, is nothing compared with a change in the people themselves.\nAll the departments of industry--the army, the navy, the cultivation\nof the fields, the rent of landed property, the profit of trades, the\npayment of rates and taxes--depend on the people, and without the\npeople there must ensue a general collapse of all our institutions.\nWe are, however, rather desirous to recommend the question to the\nconsideration of others, and especially of our statesmen, than to\nanswer it ourselves.\"\n\nIs it only _now_ that this question is submitted to the consideration\nof our statesmen? Why, if they are statesmen at all, they must have\nthought and dreamed of little else for the last few years. The picture\nhere presented, though a frightful one, is by no means new. It has been\ndrawn over and over again by the advocates of the protective policy,\nand as regularly ridiculed by the Free-Traders as a suggestion of a\ndiseased imagination. Now, the facts have emerged, the prophecy has\nproved strictly true, and we are asked to consider about a remedy! What\nremedy is there open to us, save one? Let labour be made remunerative\nat home, which can only be done by Protection, and we shall answer for\nit that the tide of emigration will be stayed. People do not leave\ntheir country and their homes, at least in numbers like this, except\nunder the coercion of the most stringent necessity. Give an Englishman\nwork to do, and wages to live by, and he will rather remain here than\nattempt to better his condition in a foreign soil. But in order that\nhe may remain here, his labour must be protected. Very truly says the\nwriter in the _Times_, that \"all the departments of industry, the army,\nthe navy, the cultivation of the fields, the rent of landed property,\nthe profit of trades, the payment of rates and taxes, depend on the\npeople; and without the people, there must ensue a general collapse of\nall our institutions.\" To every word of this we adhere. But unless we\ncan suppose that the people will submit to the degraded position of the\nforeign serfs, with whose produce they are now called upon to compete,\nBritain cannot hope to retain anything like its present population. The\nexodus must go on, and every vestige of our former greatness disappear.\nUnprotected labour and high taxation cannot exist together. Prolong the\nstruggle as we may, the experience of each succeeding month will show\nthe impossibility of such a reconciliation.\n\nWe are curious to know if, with such facts before them as those\nadmitted in the _Times_, Ministers will have the temerity next year to\nassure us that the country generally is in prosperous circumstances. Do\nmen emigrate wholesale from prosperous countries? Are they ever ready\nto leave comfort behind them, and recommence the struggle of life on\na more unpromising field? If we are forced to reject that conclusion,\nthen we defy any one to arrive at another save this--that our recent\nlegislation has so narrowed the sphere of labour, and so depressed its\nprospects, that the population are driven per force from their native\ncountry, to seek elsewhere the means of existence which they cannot\nprocure at home.\n\nTo talk of Protection as hopeless, is to acquiesce in the national\ndoom. All classes of the community, from the fundholder and capitalist\ndown to the meanest labourer, have a stake in this great question. Let\nnot the former deceive themselves. Without the labour of the people\ntheir securities are as valueless as the mere paper on which they are\nwritten. Therefore, it is their part to see that no line of policy\nshall be allowed to continue if it has the effect of drying up the\nsprings of our national prosperity. If they will not listen to the\nremonstrances of the distressed, let them at all events view their own\nposition dispassionately. We may be on the verge of a great crisis,\nand a great struggle may be approaching, but we have not the slightest\ndoubt that the cause which must ultimately prevail is that which is\nessentially the cause of the people. Prosperity will only return to the\nnation when Native Industry is protected.\n\n_Printed by William Blackwood & Sons, Edinburgh._\n\n\n[Transcriber's Note:\n\nInconsistent spelling and hyphenation are as in the original.]\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Blackwood's Edinburgh Magazine, Volume\n70, No. 433, November 1851, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# CONTENTS\n\nAlso by Heidi Grant Halvorson\n\nTitle page\n\nCopyright page\n\nTHE 8 MOTIVATIONAL TYPES\n\nThe Teenager\n\nThe Show-off\n\nThe Neurotic\n\nThe Stick-in-the-Mud\n\nThe Eager Beaver\n\nThe Alert Apprentice\n\nThe Star Who's (Almost) Born\n\nThe Expert in the Making\n\nAuthor Bio\n\nNotes\n\n# ALSO BY HEIDI GRANT HALVORSON\n\n_Succeed: How We Can Reach Our Goals_\n\n_Focus: Use Different Ways of Seeing the World for Success and Influence_\n\n# The 8 Motivational Challenges\n\nA Short Guide to Lighting a Fire Under Anyone\u2014Including Yourself\n\nHeidi Grant Halvorson, Ph.D.\n\nHUDSON STREET PRESS\nHUDSON STREET PRESS\n\nPublished by the Penguin Group\n\nPenguin Group (USA) Inc., 375 Hudson Street,\n\nNew York, New York 10014, USA\n\nUSA \/ Canada \/ UK \/ Ireland \/ Australia \/ New Zealand \/ India \/ South Africa \/ China\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nFor more information about the Penguin Group visit penguin.com\n\nFirst published by Hudson Street Press, a member of Penguin Group (USA) Inc., 2013\n\nCopyright \u00a9 Heidi Grant Halvorson, Ph.D., 2013\n\nAll rights reserved. No part of this product may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions.\n\n REGISTERED TRADEMARK\u2014MARCA REGISTRADA\n\nISBN: 978-1-101-62410-4\n\n# The 8 Motivational Challenges\n#\n\n_I'm reluctant to take on new tasks and embrace challenge\u2014how can I change?_\n\n_I start projects with enthusiasm but then I fizzle out. How can I stay motivated?_\n\n_How can I get a student who's putting in little effort to put in more?_\n\n_Why does the skills training we provide to employees seem so ineffective?_\n\nHow often have you heard questions like these being asked? How often have you been the one doing the asking? What's truly amazing, if you think about it, is not how universal these questions are, but how long they have been around, and how long they have gone unanswered.\n\nWe humans are awfully clever. We are capable of self-awareness, introspection, critical thinking, and reasoning. But we still can't seem to get a handle on what makes us tick. Long ago we mastered the ability to create and harness fire, but we still don't know how to light and maintain the fire that inspires us and keeps us moving forward.\n\nWe need to understand how motivation works. Our ability to live successful, satisfying lives\u2014and to help others do the same\u2014depends on it. We suspect we might be missing the \"right stuff\" within us. As managers, parents, and teachers, we want to cultivate the right stuff in others. But what is it? We go on a hunt for the Answer. We buy books, read articles, attend seminars, and generally spend a lot of time and money, never really finding anything that seems like the Answer we're looking for.\n\nWould you like to know the Answer? OK, here it is:\n\nThere isn't one.\n\n\u2022 \u2022 \u2022\n\nI'm sorry, I forgot the italics: There isn't _one_. And therein lies the problem.\n\nImagine that you make an appointment to visit your doctor to discuss your recurring bouts of stomach pain. You walk into the examining room prepared to give her the rundown of your symptoms and the remedies you've tried so far. But before you can begin, she takes one look at you and says, \"Obviously, you aren't feeling well. You should take some ibuprofen and lie down for a while.\"\n\n\"Hang on,\" you say. \"Don't you want to know what's wrong with me?\"\n\n\"I _do_ know\u2014you're not healthy; otherwise you wouldn't be here. These recommendations\u2014which I give to all my patients\u2014have been shown to make people feel better, so I'm sure they'll do the trick.\" She hands you a sample pack of Motrin and a fluffy pillow and bids you farewell.\n\nAt this point, you would probably come to the conclusion that your doctor is an idiot. In reality, no physician (I hope) would diagnose and treat a patient that way\u2014as if all illnesses were of essentially the same nature and could be cured in the same manner. But this is exactly how most of us approach the problem of underperformance, whether it's our own underperformance we're worried about or that of our employee, student, or child.\n\nJust like the misguided doctor, we assume that treatments that have been shown to improve motivation and performance in some\u2014staying \"positive,\" offering rewards, providing opportunities for skill development, setting specific goals\u2014can help _any_ underperformer improve. __ But experience tells us that this isn't so.\n\nThere are no one-size-fits-all ways to get people motivated. Anyone who tells you that doing X will help you get the most out of your employees, your students, or even yourself isn't telling you that, at best, X only works for some people some of the time. That makes the picture much more complicated.\n\nOf course, doctors don't generally act this way in real life, because they have an advantage with respect to medicine that the rest of us lack when it comes to motivation. Thanks to four long, expensive years of medical school, they have been trained to recognize that different illnesses have different symptoms, and that they can use these symptoms to form a diagnosis and prescribe a course of treatment tailored to the problem at hand.\n\nLike illness, underperformance\u2014or, more generally, not being able to reach our goals\u2014does not have a single psychological cause. If you want to get someone who's not putting in a lot of effort to put in more, you need to start by understanding _why_ he's not putting in effort, because there are several different possible explanations. What we need, if we want to address underperformance meaningfully and effectively, is a way to distinguish among the different kinds of underperformers we may encounter, using their patterns of behavior to identify them in much the same way doctors use heart rate, body temperature, and blood tests.\n\nOnce you know what you're dealing with, you can begin to treat the problem with a targeted cure, with strategies that are scientifically proven to work on your specific motivational ailment.\n\nWith this short book, you'll begin your training as a Doctor of Motivation. I'll introduce you to eight profiles of achievers\u2014each distinctive in their behavior, preferences, and emotional experiences\u2014and explain how you can understand each profile as having a particular **mindset** and **motivational focus** , coupled with either low or high confidence. Once you are able to recognize each profile, I'll describe the options you have for increasing effectiveness and engagement in that kind of achiever.\n\nFirst, you'll need some background on mindsets, motivational focus, and the nature of confidence.\n\n# Mindsets: Be-Good and Get-Better\n\nWe all approach the goals we pursue with one of two mindsets. (I wrote about them extensively in my earlier book, _Succeed: How We Can Reach Our Goals_.) The first is what I call the **Be-Good** mindset, in which the focus is on proving to yourself that you have a lot of ability, demonstrating that ability to others, and receiving validation for it. In other words, you want to _feel_ smart, _look_ smart, and _be told_ that you are smart. Or creative, talented, persuasive, likable . . . whatever the ability or skill in question, you don't want to be judged lacking.\n\nWhen you have a Be-Good mindset, you are constantly comparing your performance to other people's, to see how you \"size up.\" (Often this isn't totally conscious, but you're doing it nonetheless.) You are reluctant to ask for help, afraid that needing help will reflect poorly on your ability. You assume that when something doesn't come easily, it means that you aren't good at it. You live in quiet terror of making mistakes or coming up short, preferring to stick to the kinds of projects and challenges you know you can do well.\n\nWhen you have a setback, you react in one of two ways, and both are bad:\n\n1. You freak out. You worry that you don't have what it takes. Your anxiety over the difficulties you've encountered only makes matters worse, further disrupting your ability to think clearly and perform well. This creates a self-fulfilling prophecy, in which your continued poor performance only confirms your fears.\n\n2. You give up. You decide you don't have what it takes, so why bother trying hard when you can't possibly succeed? You feel hopeless and dejected, and you withdraw effort. This, too, creates a self-fulfilling prophecy, in which your continued poor performance only confirms your feelings of helplessness.\n\nNot surprisingly, we find that when people have a Be-Good mindset, they are more likely to suffer from debilitating anxiety and depression. They are less likely to enjoy the work they do, or to find it interesting, because they are so focused on the destination and not at all focused on the journey. They don't process information deeply; they develop just a surface understanding (trying to glean what they \"need to know\" only), which undermines their long-term retention of new knowledge. They have a long list of things that they simply \"aren't good at\" or \"just can't do,\" and are unlikely to have any real sense of their true potential.\n\n\u2022 \u2022 \u2022\n\nThe alternative is what I call the **Get-Better** mindset, in which the focus is on growth: developing your ability, learning new skills, and making progress over time. You can think of it as the difference between wanting to _show_ that you are smart and wanting to _get_ smart _er_.\n\nWhen you have a Get-Better mindset, you care less about what other people are doing and how you compare with them. Instead, you focus on _self_ -comparison and whether or not you are making progress: How well are you doing today, compared with how you did yesterday, last month, or last year?\n\nThe Get-Better mindset makes you gritty in the face of difficulty. When you think about what you are doing in terms of improving over time, accepting that you will almost certainly make mistakes along the way, you can stay motivated despite whatever setbacks may occur. In fact, you can end up even _more_ motivated by them.\n\nIn many studies, we've found that those with Get-Better mindsets seem to thrive when they feel challenged. In one study I conducted a few years ago, with my graduate student Laura Gelety, we found that students who saw a test as an opportunity to learn a new problem-solving skill actually performed better when we made the test harder by frequently interrupting them. Those who were instead trying to Be Good (i.e., trying to show how smart they were) were rattled by the interruptions, and performed significantly worse. Since most of us who work in the real world feel we are practically plagued by interruptions, it's clear that mindset really matters.\n\nEveryone benefits from a Get-Better mindset, and everyone suffers in one way or another when they think instead in terms of Being Good. Figuring out which mindset you've got on your hands is an essential part of your Motivational Diagnosis. And since mindsets can be changed (more on that later), it offers you a powerful means of treating underperformance.\n\n# Motivational Focus: Promotion and Prevention\n\nWhy do two people working toward a common goal so often fail to see eye to eye when it comes to achieving it? At times, you feel like you aren't on the same page\u2014or even the same _planet_ \u2014as your spouse, your child, or your colleague. Why the disconnect?\n\nThe answer is a surprisingly simple one. As Tory Higgins and I discuss in our new book, _Focus: Use Different Ways of Seeing the World for Success and Influence_ , there really is more than one way to look at the same goal. Take, for example, a goal that many of us share: _I want to do my job exceptionally well._ For some of us, doing our jobs well is about achievement and accomplishment\u2014we have what is called a **promotion focus**. In the language of economics, promotion focus is about maximizing gains and avoiding missed opportunities.\n\nFor others, doing our jobs well is about security, about not losing the positions we've worked so hard for. This **prevention focus** places the emphasis on avoiding danger, fulfilling responsibilities, and doing what you feel you _ought_ to do. In economic terms, it's about minimizing losses, trying to hang on to what you've got.\n\nPromotion- and prevention-focused people work differently to reach the same goal. They use different strategies, have different strengths, and are prone to different kinds of mistakes. One group is motivated by applause, the other by criticism. One group may give up too soon, while the other may not know when to quit.\n\nSo, do you (or does the person you are trying to motivate) spend your life pursuing accomplishments and accolades, reaching for the stars? Or are you busy fulfilling your duties and responsibilities, being the person everyone can count on? Once you have identified your focus, you can better understand and embrace your strengths, your potential weaknesses, and the strategies that will work best for you _._\n\nCriticism or Praise?\n\nWhen you are promotion-focused, your motivation feels like eagerness\u2014an enthusiastic desire to really go for it. Eagerness is enhanced by positive feedback: The more you are succeeding, the more motivated you become. Confidence heightens your energy and intensity; doubting yourself takes the wind right out of your sails.\n\nWhen you are prevention-focused, your motivation feels like vigilance\u2014you are on the lookout for danger. Vigilance actually increases in response to negative feedback or self-doubt. There's nothing like the looming possibility of failure (so long as you feel it can still be avoided) to get your prevention juices flowing. Overconfidence or effusive praise, however, may lead you to let down your guard, and it can undermine your motivation.\n\nEmbrace Risk, or Avoid It?\n\n\"Nothing ventured, nothing gained\" pretty much captures the promotion-focused philosophy. The promotion-minded have a habit of saying yes to every opportunity, and they welcome the opportunity to take chances. Prevention-minded people, on the other hand, are cautious. They tend to say no more, and prefer the devil they know to the one they don't.\n\nThese biases manifest themselves in all sorts of ways. For example, people with prevention goals are reluctant to disengage from one activity to try another, or to change strategies midstream. But their more careful nature also makes them less likely than their chance-taking peers to procrastinate, for fear that they won't have time to get the job done.\n\nHow You Think\n\nWhen people are promotion-focused, they feel free to be more exploratory and abstract in their thinking. They brainstorm. They generate lots of options and possibilities to reach their ideals, and are more creative. They are also particularly good at picking up on connecting themes or synthesizing information.\n\nFor the prevention-focused, abstraction and creativity seems reckless and time consuming. Prevention-focused thinking is more analytical, concrete, and specific. The prevention-minded are great with details, and have better memory for what they've seen and what still needs to be done.\n\nThe Tortoise or the Hare\n\nExecuting any modestly complicated task involves what psychologists call a \"speed-accuracy trade-off.\" The faster you go, the more mistakes you make. But going slow has costs too, particularly if time is valuable and you are in a hurry to get the job done. It won't surprise you to learn that promotion- and prevention-minded people end up on opposite sides of this particular trade-off, with promotion-minded favoring speed and prevention-minded preferring the slow but flawless route.\n\nGetting There and Staying There\n\nPromotion-focused thinking leads to energetic and enthusiastic motivation in the shorter term, but can be less effective when it comes to long-term maintenance. Prevention-focused thinking, on the other hand, is ideal for making sure your hard-earned gains don't slip away.\n\nIt All Makes Sense!\n\nArmed with an understanding of promotion and prevention, we begin to make more sense of the things we do. Perhaps now you see why you've always been a risk taker, or why you've always avoided risks like the plague. It's clear why you are uncomfortable with being too optimistic, or why you are known for your sunny outlook. You get why some things have always been hard for you, while others came easily.\n\nPromotion and prevention also help us understand what _other_ people do, and give us powerful insights into our relationship ups and downs. Married couples in particular often waste much energy and create a lot of unnecessary animosity in their relationship arguing over which person is seeing things the \"right\" way. (Early in our marriage, my husband and I were constantly butting heads when it came to our toddlers' budding ability to walk. He wanted to give them opportunities to climb and explore, while I wanted to cover them in bubble wrap from head to toe and make them wear helmets on the staircase. You can easily guess which one of us is promotion minded and which one is all about prevention.)\n\nOnce you realize that you and your partner simply approach your goals differently, the good news is that you can stop fighting over who is right. You can more easily see what is valuable about your partner's viewpoint. The very best partnerships strike a balance between promotion and prevention, since both are necessary for living a healthy, satisfying life. Between the two of you, you'll make sure that your family has adventures and new experiences, while also making sure the kids have clean underwear and the bills get paid.\n\nWork with What You've Got\n\nThere's no need to fight your focus\u2014the key to optimal motivation is to embrace your promotion- or prevention-mindedness! Unlike with the Be-Good and Get-Better mindsets, both kinds of motivational focus can bring you success, and each brings something of value (e.g., innovation, attention to detail) to any endeavor. What you need to do is learn how to work with your focus to bring out the best in yourself.\n\nCertain ways of thinking, kinds of language, strategies, and incentives provide **motivational fit** for either promotion or prevention motivation\u2014they just _feel right_ , and as a result they make us both more engaged and more effective. Once you've completed your Motivational Diagnosis and you know you're dealing with either promotion or prevention, you can use your knowledge of how to create motivational fit to bring out your (or their) best performance. (I'll describe how you can do that later on.)\n\n# Confidence\n\nYou don't need me to tell you that confidence is important. Confidence is, in a very real sense, necessary for reaching goals, particularly the difficult ones. Having it (or not) goes a long way toward determining whether or not you will persist in the face of challenges and ultimately overcome them.\n\nBut it's a particular kind of confidence we need\u2014not the swaggering I-think-I'm-the-greatest kind, but the more modest and attainable I-believe-I-can-do-it kind. Albert Bandura, the father of modern social cognitive psychology, called this kind of confidence **self-efficacy.** 4 In essence, it's the belief that you can successfully execute whatever behaviors are necessary to bring about a particular desired outcome. In other words, it's the belief that you can do what it takes to make success happen.\n\nHow well does self-efficacy predict success? Well, looking across hundreds of studies, the average correlation between a person's sense of self-efficacy and his or her likelihood of success is about 0.34. If that doesn't sound impressive, allow me to provide some perspective: The average correlation between tests of job aptitude (ability) and success actually doing that job is only about 0.19. So self-efficacy out-predicts, by quite a lot, objective measures of ability when it comes to understanding who succeeds and who fails. Which is why it's so important for people to have it, and for managers, teachers, and parents to cultivate it in others.\n\nBandura argued that self-efficacy comes from four sources. First and foremost, it comes from our past experiences of success, particularly when the tasks at which we succeeded were difficult. When you have had a lot of mastery experiences, when you have risen to meet many challenges, even the occasional failure doesn't make much of a dent in your belief that you can usually get the job done. On the other hand, when people lack a track record of success, their sense of efficacy suffers.\n\nThis is a big part of why workplace skills training is so important. People who are otherwise motivated but lack the skill set necessary to perform well are bound to have failure experiences, and these will chip away at their sense of self-efficacy over time. Through skills training, we can not only impart knowledge, but we can actually give employees success experiences when they master each new skill\u2014thus increasing their self-efficacy as well as their knowledge base. It's a performance two-for-one.\n\nA second powerful source of self-efficacy is the vicarious experiences of others\u2014specifically, observing the ease or difficulty with which those individuals have succeeded (or failed). When someone makes it look easy, it increases our confidence that we can pull it off as well. (This confidence can, of course, be misplaced\u2014experts often make things look easy that have in fact taken long hours of hard work to perfect. I, for instance, was once highly confident that I could make a delicious chicken dinner for my college boyfriend without consulting a recipe, after having seen my mother do so swiftly and easily on countless occasions. The painful bout of indigestion we both subsequently endured definitely brought my sense of cooking self-efficacy down a notch or two.)\n\nA third source of self-efficacy comes from the assurances (or warnings) of others\u2014the you-can-do-its and the you-really-shouldn't-do-its\u2014though this influence is not nearly as powerful as the assurers and warning-givers believe it to be. I'm fairly certain that before that fateful night, my mother said something to me like, \"Maybe you should go find a recipe and not just wing it, honey.\" I ignored her, and my poor boyfriend paid the price.\n\nFourth, self-efficacy comes from our current emotional state. No matter how much of it you usually have, when you're feeling agitated or anxious for totally independent reasons, your confidence can be shaken. So you can expect mood to play a role, though generally it will be a small one compared with the influence of past experiences and vicarious learning.\n\nA Word About \"Positive Thinking\"\n\nBefore we move on to the eight profiles, I want to tell you something that is absolutely essential for you to understand about the difference between \"positive thinking\" and confidence. There are a lot of people running around extolling the virtues of positive thinking\u2014of visualizing yourself achieving everything you want in life, and of ruthlessly blocking out any negative thoughts relating to your goals.\n\nLet me say right up front that I, too, wish I could make the universe deliver wonderful things to my doorstep just by imagining them. I can't\u2014and neither can you, no matter what anyone tells you. There is not a single piece of hard evidence that visualizing the moment your dreams come true _and doing nothing else_ will do a damn thing for you.\n\nIn fact, there is plenty of evidence that it will leave you even worse off than when you started. Scientifically speaking, focusing all of your thoughts on an ideal future reliably leads to lower achievement. In other words, you are less likely to achieve your goals when all you do is imagine what it would be like to achieve them.\n\n\"Negative thinking,\" on the other hand, has gotten a bad rap. This is mostly because the people who advocate positive thinking lump all the negative thoughts together in one big unpleasant pile, not realizing that some kinds of negative thoughts are actually necessary and motivating. There is a big difference between \"I am a loser and can't do this\" (a bad, low-confidence negative thought), and \"This won't be easy, and I'm going to have to work hard\" (a very good negative thought that is consistent with high confidence and actually predicts greater success).\n\nIn fact, study after study shows that people who think not only about their dreams, but about the obstacles that lie in the way of realizing their dreams\u2014believing they can overcome those obstacles and planning how they'll make it happen\u2014vastly outperform those who sit back and wait for the universe to reward them for all their positive thinking. Whether it's starting a relationship with your secret crush, landing a job, recovering from major surgery, or losing weight, research shows that if you don't keep it real you're going to be really screwed.\n\nRemember, confidence does not mean ignoring the reality of the challenge you are facing. People with high self-efficacy still have what some might call negative thoughts\u2014like considering potential pitfalls, and what your Plan B might look like if Plan A doesn't work. Don't confuse a lack of realism with confidence, and don't assume that someone who looks the facts in the eye lacks it.\n\n# Eight Profiles of Underperformers\n\nNow that we understand their underlying causes, it's time to start diagnosing patterns of underperformance. Each profile is a particular combination of mindset and focus, with either low confidence or high confidence. This leaves us with eight unique profiles. Two of these profiles are disastrous, motivationally speaking. Four are reasonably effective but problematic. Together, these six profiles are the ones we tend to worry about.\n\nManagers and teachers are often so busy trying to figure out how to help those who are clearly floundering that they ignore the ones who are working productively\u2014but not as productively as they could be. The final two profiles are the ones who are ripe for reaching peak performance. I'll show the reader how to recognize who they are, and explain what can be done to help.\n\n(Note: For each profile I use \"he\" just to avoid the awkwardness of \"he or she\"\u2014just keep in mind that these profiles are equally applicable to men and women.)\n\n# The Teenager\n\nThe Teenager (who, for the record, can be any age) is one of the most difficult motivational profiles you'll encounter. He is:\n\n  * promotion-focused\n  * pursuing goals with a Be-Good mindset\n  * insecure, lacking in confidence\n\nYou'll recognize him immediately from the following behaviors:\n\n  * **He is withdrawn or depressed.** He doesn't seem alert or engaged. His energy is low; he lacks enthusiasm. When promotion-focused people get negative feedback or lack confidence, it creates a motivational non-fit. In other words, it dampens their motivation.\n  * **He puts in very little effort. He does the bare minimum.** His whole attitude seems to be \"Why bother?\" When people with a Be-Good mindset have low confidence, they have no reason to work hard\u2014after all, what's the point of all that effort if they can't actually _be_ _good_?\n  * **He has obvious doubts about his ability.** His lack of confidence will be apparent\u2014he will probably make comments about how he just isn't good at X, Y, and Z. But despite his lack of confidence, he will be reluctant to seek help or guidance\u2014this would just reinforce the perception that he lacks ability, and he doesn't believe it will increase his ability anyway.\n  * **He engages in self-handicapping.** In other words, he sometimes sabotages his own performance in order (ironically) to protect his self-esteem. That way, when he fails, he doesn't have to blame his failure on a lack of ability. The student who never studies for exams can say, \"I would have gotten a good grade if I tried,\" and avoid looking like he just wasn't smart enough to do well. The colleague who dresses like a slob or is always unprepared in meetings can blame these avoidable mistakes for his lack of advancement, rather than the possibility that he just doesn't have what it takes. The spouse who behaves passive-aggressively rather than tackling problems head-on may destroy the relationship, but he never has to risk opening himself up and being vulnerable to rejection.\n  * **He has bursts of energy that quickly fizzle out.** Every once in a while he'll look like he has turned a corner. Something goes his way or gives him hope and you see the spark. But the momentum doesn't last, for as soon as the road gets rocky or a project feels a bit too challenging, it's right back to his old hanging-out-in-his-room-with-the-lights-off ways.\n\nPrognosis: Disastrous\n\nHis low confidence, coupled with his desire to avoid looking stupid or incompetent (and his very real fear that he might in fact be both), makes the Teenager a tough nut to crack. Deep down, he is his own toughest critic and worst enemy. Trying to build his confidence through skills training won't work, because he sees needing help as a sign of lacking ability. Telling him that he is smart and talented won't stop him from doubting himself when things get difficult. But without intervention, he will remain in a self-perpetuating cycle of failure.\n\n# The Show-off\n\nThe Show-off is a more effective version of the Teenager\u2014though still not nearly as effective as he could be. He is:\n\n  * promotion-focused\n  * pursuing goals with a Be-Good mindset\n  * confident\n\nThe difference between the Teenager and the Show-off is, in a nutshell, confidence. The Teenager lacks it, while the Show-off has it (rightly or wrongly) in spades. Here are his key characteristics:\n\n  * **He brags and calls attention to his successes.** It's not enough to succeed\u2014he needs _validation_ of his successes. He needs other people to know he is performing well, and to think highly of his abilities.\n  * **He tunes out criticism, sweeps failures under the rug.** He does this to keep himself motivated and optimistic, but as a consequence he doesn't learn from his mistakes or improve over time.\n  * **He takes on too much.** He says yes to everything. Because of his promotion focus, he hates to pass up an opportunity. But he is a poor judge of the time and effort each project will take, always thinking in terms of best-case scenarios. He winds up with lots of pots on the stove, but nothing is ever ready to eat.\n  * **He can be reckless.** Sometimes it's good to be willing to take chances\u2014there are even times when it's essential. But a \"go for it\" attitude can become reckless when you feel you've got something to prove. And his Be-Good mindset makes him constantly feel like he has something to prove.\n  * **He is competitive with his colleagues.** The Be-Good mindset makes us highly tuned in to how well other people are doing, and how we compare. A performance is only good when it is better than someone else's. This also can make him less cooperative and willing to work and share information with others.\n\nPrognosis: Problematic\n\nThe Show-off can be very difficult to manage, and even more difficult to work alongside. His competitiveness can be toxic to team morale, and he will sometimes (even unconsciously) sabotage team performance by holding back key information or assistance. He is highly motivated, so he is not without his successes\u2014but he has a hard time learning from failures and constructive criticism, and his recklessness can get him (and everyone around him) into hot water.\n\n# The Neurotic\n\nThe Neurotic is as problematic as the Teenager, though his pattern of behavior is quite different. He may, in fact, win the title of least pleasant profile to work with or for. He is:\n\n  * prevention-focused\n  * pursuing goals with a Be-Good mindset\n  * insecure, lacking in confidence\n  * **He is defensively withdrawn, and often highly anxious.** Like a human porcupine, everything about the Neurotic says, \"Stay away.\" Where other people see opportunities to get ahead, the Neurotic sees opportunities to crash and burn. His workplace is strewn with land mines, and his entire will is bent on avoiding them.\n  * **He is easily flustered and overwhelmed.** Most people find the amount of work for which they are responsible at least a bit overwhelming. The Neurotic feels like he is drowning, and he feels this way at almost every moment of every day. He is rattled by any change or surprise\u2014anything he didn't fully anticipate.\n  * **He responds with resentment or anger to new assignments.** He may try to disguise it but odds are good that he won't be able to do so fully\u2014such is his fear of tackling something new, or adding another responsibility to his already overwhelming To-Do list. If you have the profound misfortune of having the Neurotic as a manager, then you know that he is as hostile to new ideas and initiatives as it's humanly possible to be.\n  * **He takes comfort in busywork and rules for self-protection.** There is one place where the Neurotic can find some small respite from the many fear-triggering aspects of his day: in familiar minutiae. The weekly report, the form filled in triplicate, the filing system, the formal procedures for doing X, Y, and Z. Here there are few surprises, few unknowns. Danger can be avoided by doing things exactly the same way every time.\n  * **He lives in utter terror of making mistakes.** This is perhaps the most essentially defining characteristic of the Neurotic. And because he has low confidence, he's fairly certain he's going to make them. Ironically, the skills training that is needed to increase his confidence is likely to be ineffective\u2014large doses of anxiety make it not only difficult to perform well, but very difficult to learn. Too much anxiety can create a kind of mental fog or confusion that is difficult for trainers and mentors to penetrate.\n\nPrognosis: Disastrous\n\nIt's true that the Neurotic is a disaster. If you have one in your class or on your team, you already are well aware that you have a big problem on your hands. Not only is he fearful and withdrawn, but he is prone to flashes of anger when pushed too hard. This is basically a fight-or-flight response\u2014he'd prefer flight, but if he feels cornered by you, he's got no choice but to turn nasty. He resists your efforts to help him, and when you praise him or express confidence in him, he doesn't believe it. Like the Teenager, he is his own worst enemy and harshest critic.\n\n# The Stick-in-the-Mud (Alternatively: The Party Pooper)\n\nThe Stick-in-the-Mud is a more effective version of the Neurotic, though still greatly limited by his mindset. He is:\n\n  * prevention-focused\n  * pursuing goals with a Be-Good mindset\n  * confident\n\nThe difference between the Neurotic and the Stick-in-the-Mud is, again, one of confidence. The Neurotic lacks it utterly, while the Stick-in-the-Mud has a bit of it\u2014though not nearly as much as the Show-off. He is, after all, prevention-focused.\n\nYou'll know him by these characteristics:\n\n  * **His confidence won't look like confidence; he remains cautious.** The Stick-in-the-Mud knows that the key to his success is to never let his guard down. He remains ever vigilant and skeptical. He thinks optimism is for children and fools.\n  * **He sticks to what he does well.** Chances are good that there are many things he does, in fact, do well. But he believes he knows his limits, and never strays beyond them. This means that, in all probability, he is selling himself short and not coming close to realizing his true potential.\n  * **He is very resistant to change.** Sure, he'll change if he has to\u2014but only when he is absolutely 100 percent convinced that _not_ changing would be a mistake. He vastly prefers to stick with the status quo, with tried-and-true methods for doing things. He makes conservative decisions. His first impulse is to say no.\n  * **He is overly concerned with making mistakes.** With a Be-Good mindset, the Stick-in-the-Mud is still very focused on proving himself. But unlike the Show-off, he doesn't seek to prove himself through great accomplishments\u2014he proves himself by having a flawless record, by never making mistakes and always being reliable. He's not Mr. Right\u2014he's Mr. Never Wrong.\n  * **He is rigid about how things are done.** \"Let's shake things up\" or \"Let's try a new tack\" are not expressions that Mr. Stick-in-the-Mud is familiar with. He not only prefers known and well-tested strategies for doing things, but he won't change strategies midstream, even if it looks like the current strategy is not working. Sometimes, his stubbornness and persistence are rewarded; other times, they cost him dearly.\n\nPrognosis: Problematic\n\nThe Stick-in-the-Mud has strengths\u2014he is reliable, thorough, conscientious\u2014but his inflexibility and reluctance to take chances make it difficult (or impossible) for him to see the value in creativity or innovation. He has difficulty with change. He is a micromanager, and he has a hard time delegating to others, for fear they will make mistakes. He is highly motivated, but he limits himself and those he works with by clinging too much to the status quo.\n\n# The Eager Beaver\n\nThe Eager Beaver may lack confidence like the Teenager, but he is more than willing to do something about it. He is:\n\n  * promotion-focused\n  * pursuing goals with a Get-Better mindset\n  * insecure, lacking in confidence\n\nWhere the Teenager's lack of confidence stems from his doubts about his ability, the Eager Beaver's comes from knowing that he has a lot to learn.\n\n  * **He has a great attitude.** He is willing to work hard and has a lot of energy.\n  * **He lacks the skills to do the job effectively, but he is eager to learn them.** He's ready to get his hands dirty and give it a try. In fact, he may even seem a little impatient to get going. You may have to remind him to apply the brakes from time to time.\n  * Even though he isn't confident about his performance right now, **he is optimistic that he can eventually master the skills he needs.** In other words, his performance self-efficacy is low, but his learning self-efficacy is high.\n  * **He may still take on too much** out of an abundance of enthusiasm, but he will be quicker to recognize when he is in over his head and will learn from his mistakes.\n  * **He thrives on praise,** which helps him to keep going when tasks are difficult. He will probably frequently ask for feedback, hoping for some positive reinforcement.\n  * **He will deal with setbacks productively.** He will certainly feel a sense of dejection when things go wrong, but that dejection will fuel his future effort and proactive attempts to improve.\n\nPrognosis: Problematic\n\nThe Eager Beaver is problematic only insofar as he doesn't really know what he's doing. His problem is a lack of knowledge rather than a lack of motivation or a maladaptive mindset. But it's important to understand how he will learn, and what kinds of trouble he may encounter along the way (more on this later).\n\n# The Alert Apprentice\n\nThe Alert Apprentice lacks confidence like the Neurotic, but he isn't about to let anyone down if he can help it. He is:\n\n  * prevention-focused\n  * pursuing goals with a Get-Better mindset\n  * insecure, lacking in confidence\n\nThe Neurotic and the Teenager are insecure because they have serious doubts about their ability. The Alert Apprentice also feels all too keenly that he is in over his head, but like the Eager Beaver he's determined to figure out how to swim.\n\n  * **He is disciplined and focused.** He is willing to work hard, and determined to avoid error. He will put in long hours if necessary, and takes challenges seriously.\n  * **He lacks the skills to do the job effectively, but he is determined to master them.** Unlike the Eager Beaver, he wants to be fully prepared before striking out on his own.\n  * Even though he isn't confident about his performance right now (in fact, he's more than a little anxious about it), he believes he can eventually master the skills he needs, so long as he is vigilant and takes nothing for granted. **He is a defensive pessimist,** meaning he focuses on what _might_ go wrong if he doesn't take the right actions.\n  * **He may be reluctant to go from the learning phase to the doing phase,** out of an abundance of caution. He wants to be absolutely certain he knows what he's doing so he won't make mistakes. You may have to give him a little push out of the nest.\n  * **He thrives on honest, informative feedback,** and is distrustful of too much effusive praise. He will check in frequently to make sure he's not doing anything wrong\u2014or to try and catch a mistake before it becomes too serious.\n  * **He will deal with setbacks productively.** He will certainly feel a sense of anxiety when things go wrong, but that anxiety will fuel his future effort and proactive attempts to improve.\n\nPrognosis: Problematic\n\nThe Alert Apprentice, like the Eager Beaver, is problematic only because he lacks knowledge and expertise. His motivation is strong and his mindset is ideal for learning, but again, it's important to understand how he will learn\u2014particularly that he will be slow and deliberate, perhaps frustratingly so (more on this later).\n\n# The Star Who's (Almost) Born\n\nThe Star Who's (Almost) Born is what you are hoping your Teenager, Show-off, or Eager Beaver might one day become. He is:\n\n  * promotion-focused\n  * pursuing goals with a Get-Better mindset\n  * confident\n\nIn fact, he's so terrific that as a manager or teacher, you've probably given little thought as to how you might make him even more effective.\n\n  * **He is energetic and optimistic.** Other people who work with him find him to be a source of inspiration; his enthusiasm and can-do spirit are infectious.\n  * **He thrives under challenges.** He doesn't want it to be easy. He doesn't want to do the same thing every day, even if the rewards for doing so are plentiful. He wants to tackle something new. He thinks big.\n  * **He is creative and innovative.** Because he is both promotion-focused and confident, he opens his mind to possibilities. Because he has a Get-Better mindset, he thinks deeply and makes connections. His imagination is not constrained by concerns with feasibility and practicality. He has vision.\n  * **He takes risks that pay off.** He'll take a chance, but because he has a Get-Better mindset and feels that he has less to prove, he takes smarter chances than the Show-off.\n\nPrognosis: Ripe for Peak Performance\n\nThe Star Who's (Almost) Born is exactly that\u2014a star in the making, who has all the necessary skills and plenty of motivation, but just needs to learn to optimize them.\n\n# The Expert in the Making\n\nThe Expert in the Making is what you are hoping your Neurotic, Stick-in-the-Mud, or Alert Apprentice might one day become. He is:\n\n  * prevention-focused\n  * pursuing goals with a Get-Better mindset\n  * confident\n\nAgain, this is one of the very best profiles a student or employee can have. You can't help but be grateful that he makes your job so easy.\n\n  * **He is reliable, responsible, and always prepared.** Typically, the Expert in the Making is a paragon of personal organization, time management, and punctuality. He takes his commitments seriously and honors his promises.\n  * **He's always ready with Plan B.** Also Plan C and Plan D\u2014though you probably won't need any of them, since his Plan A is practically bulletproof.\n  * **His work is flawless.** He takes his time and does it right. He also checks and sometimes rechecks it, to ensure its accuracy.\n  * **He thrives on developing expertise.** Because he has a Get-Better mindset, he is focused on self-development\u2014in particular, on mastering the key skills that will bring him success. When he wants to know something, he wants to know it inside and out.\n  * **He keeps things running brilliantly.** Never underestimate the importance of maintenance\u2014the Expert in the Making certainly never does. He will make sure that everyone and everything is where it is supposed to be. He will not fall behind, nor will he get ahead of himself. He makes it look easy, even when it's not.\n\nPrognosis: Ripe for Peak Performance\n\nThe Expert in the Making has the knowledge and the motivation. To achieve his very best, he needs to learn how to work with his prevention focus, and he needs to work in an environment that creates a good motivational fit.\n\n# Diagnosis and Treatment\n\nIn order to simplify your diagnosis, here is a brief summary of the characteristic \"symptoms\" associated with each of the eight profiles, along with the corresponding recommended treatment. The treatments themselves will be described in detail in the section that follows.\n\nThe Teenager\n\nPromotion Focus\/Be-Good Mindset\/Low Confidence\n\nSymptoms:\n\n  * Withdrawn\/depressed\n  * Low effort (demoralized)\n  * Self-handicapping for self-protection\n  * Bursts of energy that quickly fizzle out\n  * Doing the bare minimum\n  * Obvious self-doubt about ability\n\nTreatment:\n\n1. Shift mindset to Get-Better\n\n2. Train in promotion-focused skills and build confidence\n\n3. Create fit for promotion focus\n\nThe Show-off\n\nPromotion Focus\/Be-Good Mindset\/Confidence\n\nSymptoms:\n\n  * Takes on too much\n  * Reckless\n  * Competitive with colleagues\/fellow students\n  * Brags\n  * Tunes out criticism\n\nTreatment:\n\n1. Shift mindset to Get-Better\n\n2. Create fit for promotion focus\n\nThe Neurotic\n\nPrevention Focus\/Be-Good Mindset\/Low Confidence\n\nSymptoms:\n\n  * Withdrawn\/defensive or highly anxious\n  * Easily flustered and overwhelmed\n  * Responds with resentment\/anger to new assignments\n  * Takes comfort in busywork and rules for self-protection\n  * Lives in terror of making mistakes\n\nTreatment:\n\n1. Shift mindset to Get-Better\n\n2. Train in prevention-focused skills and build confidence\n\n3. Create fit for prevention focus\n\nThe Stick-in-the-Mud\n\nPrevention Focus\/Be-Good Mindset\/Confidence\n\nSymptoms:\n\n  * Won't look like confidence; he remains cautious\n  * Sticks to what he does well\n  * Resistant to change\n  * Overly concerned with making mistakes\n  * Rigid about how things are done\n\nTreatment:\n\n1. Shift mindset to Get-Better\n\n2. Create fit for prevention focus\n\nThe Eager Beaver\n\nPromotion Focus\/Get-Better Mindset\/Low Confidence\n\nSymptoms:\n\n  * Great attitude\n  * Eager to learn\n  * Impatient to get going\n  * Lacks skills to work effectively\n\nTreatment:\n\n1. Train in promotion-focused skills and build confidence\n\n2. Create fit for promotion focus\n\nThe Alert Apprentice\n\nPrevention Focus\/Get-Better Mindset\/Low Confidence\n\nSymptoms:\n\n  * Highly motivated to master the job\n  * Thorough, determined\n  * Lacks the skills to work effectively\n  * Reluctant to dive right in\n\nTreatment:\n\n1. Train in prevention-focused skills and build confidence\n\n2. Create fit for prevention focus\n\nThe Star Who's (Almost) Born\n\nPromotion Focus\/Get-Better Mindset\/Confidence\n\nSymptoms:\n\n  * Thrives under challenges\n  * Confident and optimistic\n  * Creative and innovative\n  * Takes smart risks\n\nTreatment:\n\n1. Create fit for promotion focus\n\nThe Expert in the Making\n\nPrevention Focus\/Get-Better Mindset\/Confidence\n\nSymptoms:\n\n  * Thrives on developing expertise\n  * Brilliant at keeping things running\n  * Excellent planner\n  * Turns in flawless work\n\nTreatment:\n\n1. Create fit for prevention focus\n\n# Treatment\n\nThe overall formula for treating each of the eight profiles is straightforward. Once you've identified the right profile, then:\n\n**Phase 1)** If the mindset is Be-Good, work on changing it to Get-Better. If it is already Get-Better, move on to Phase 2.\n\n**Phase 2)** Provide skills training and confidence building, if necessary. Know that the learning process is different for promotion- and prevention-focused people. If skills training or confidence building is not necessary, move on to Phase 3.\n\n**Phase 3)** Start to work with your (or their) motivational focus. Use strategies, incentives, and work environments that create fit for either promotion or prevention. This phase is never over; continuing to work with your focus is __ necessary if you want to sustain and enhance motivation.\n\nPhase 1: Shift from the Be-Good to the Get-Better Mindset\n\nShifting Your Own Mindset\n\nYou might be wondering why this phase comes before skills training and confidence building. The short answer is if you don't already have a Get-Better mindset, skills training and confidence-building techniques won't be nearly as effective as they could be. The anxiety, dejection, and defensiveness caused by Be-Good thinking disrupt our ability to learn, and the self-doubt it creates makes attempts to build self-efficacy work only in the short term, if at all. So by adopting a Get-Better mindset, you're like a farmer removing the rocks and weeds from his land, readying your mental soil for the seeds you will eventually plant there.\n\nUsing the Be-Good mindset is a bad habit you would like to break. The biggest obstacle to doing so is that like most bad habits, using the Be-Good mindset has become relatively automatic. In other words, you aren't choosing to use it\u2014you just do it naturally. Things we do without consciously realizing it can be the toughest to change. But you absolutely can change your mindset, if you follow these steps:\n\n**Step 1: Reframe your goals in terms of** getting better **.** To do this, use the Get-Better mindset _triggering words_ :\n\nlearn\n\nimprove\n\ndevelop\n\ngrow\n\nbecome\n\nprogress\n\nover time\n\nStart by writing down a few of your goals the way you would normally talk about them. Then rewrite them, using trigger words to take the emphasis off of wanting to _be good_ and make your goal about _getting better_. For instance:\n\nI want to be a great leader.\n\n**Get-Better Version:** I want to _learn_ how to _become_ a great leader.\n\nI want to manage my time productively.\n\n**Get-Better Version:** I want to _develop_ my time-management skills.\n\nI want to eat healthy and exercise regularly.\n\n**Get-Better Version:** I want to _improve_ my eating and exercise habits.\n\nI want to be in a satisfying relationship.\n\n**Get-Better Version:** I want my relationship to _grow_ more satisfying _over time_.\n\n**Step 2: Make an if-then plan** to replace Be-Good thoughts with Get-Better thoughts. (I devote a lot of time to this uniquely powerful form of planning in _Succeed_ , because it's useful in so many situations, and because it makes you roughly two to three times more likely to reach your goal.)\n\n_If_ I catch myself thinking in a Be-Good way, _then_ I will focus on the Get-Better version of my goal: __________.\n\n**Step 3: Change your expectations.** Get-Better thinking requires you to wrap your head around a new (and more productive) reality. You need to reject the idea that things come easily to those who have a lot of ability. You need to embrace the fact that when something is difficult and unfamiliar, you will need some time to really get a handle on it. You may make some mistakes, and that's OK. You need to remember to ask for help when you run into trouble. Needing help doesn't mean you aren't capable. In fact, the opposite is true: Only the very foolish believe they can do everything on their own.\n\n**Step 4: Try not to compare yourself with other people.** Instead, compare your performance today with your performance yesterday. Focusing on getting better means always thinking in terms of progress, not perfection.\n\n**Step 5: Be patient.** If you keep refocusing your mind to think in terms of Getting Better, eventually this will become your automatic mindset. But changing habits\u2014especially old habits\u2014takes time, and you have probably been working with a Be-Good mindset for many years. Don't expect your new mindset to feel natural overnight. Hang in there, and change will come.\n\nShifting Someone Else's Mindset\n\n**Frame goals and instructions using Get-Better language.** Those trigger words are powerful, and if you use them to describe the goal you want someone to adopt or the project you'd like him to tackle, then you can create an environment ripe for Get-Better thinking. Describe new assignments as opportunities to master new knowledge and skills. Describe quotas and test scores as tools that can be used to track progress, helping people to learn and improve over time.\n\n****\n\n****\n\n**Tell him it's OK to make mistakes.** Even tell him you _expect_ him to. I know that if you are a manager, this may not be something you are thrilled to hear, because immediately you're probably thinking, \"If my employee screws up, I'm going to be the one who pays for it.\" But, in fact, when people feel they are allowed to make mistakes, they are significantly less likely to actually make them! Let me explain.\n\nWhen you have a Be-Good mindset, you worry a lot about making mistakes. Countless studies have shown that nothing interferes with performance quite like anxiety does; it is _the_ productivity killer. So by giving your employee permission to not do everything perfectly from the start, and by acknowledging that there is a learning curve and that improvement takes time, you are taking the anxiety out of the situation. And in so doing, you are not only shifting his mindset to Get-Better thinking, but also dramatically reducing the chances that _any_ mistakes will be made at all.\n\n****\n\n****\n\n**Provide Get-Better role models.** When you hold up individuals as examples for others to follow, be sure to emphasize not only their accomplishments but also the evidence of their Get-Better mindsets. Talk about their persistence, their effort, how they tackled challenges and improved despite setbacks or initial poor performance. In some of my studies, I've found that using these role models\u2014for example, telling first-year chemistry students about the Get-Better mindsets of Nobel Prize\u2013winning chemists\u2014has remarkable effects on students' enjoyment of coursework and their overall course performance. Get-Better role models are another powerful way to convey the message that success is about progress over time.\n\n****\n\n****\n\n**Give the right kind of feedback.** Whether it's criticism or praise, follow these three rules to keep your students, children, or employees focused on Getting Better.\n\n**Rule #1:** When things go wrong, **keep it real**. It's not easy to tell someone that he screwed up, knowing it will cause him anxiety, disappointment, or embarrassment. But don't make the mistake of protecting his feelings at the expense of the truth, because without honest feedback he can't possibly improve. (Remember that negative emotions exist for a reason\u2014namely, they motivate us to take action to fix the problem.)\n\nAnd never take away someone's sense of responsibility for what went wrong (assuming he is in fact to blame), just because you don't want to be \"hard\" on him. Letting him off the hook for his own mistake will rob him of a sense of personal control over his own work. Nothing is more de-motivating than feeling powerless.\n\n**Rule #2:** When things go wrong, **fight self-doubt**. He needs to believe that success is within reach, regardless of the mistakes he has made in the past. To do this:\n\n_Be specific._ What needs improvement, and what, exactly, can be done to improve? When you are a manager, teacher, or parent, helping him figure out how to do it right __ is just as important as letting him know what he is doing wrong.\n\n_Emphasize actions that he has the power to change._ Talk about aspects of his performance that are under his control, like the time and effort he put into a project, his positive attitude, his persistence, or the strategic approach he used.\n\n_Avoid praising effort when it didn't pay off._ Many managers try to console employees by saying things like \"Well, Bob, you fell short of your sales goal, but you worked hard and really tried your best.\" Why does anyone __ think that this is comforting? For the record, it's not. (Unless, of course, it was a no-win situation from the start.) Studies show that being complimented for effort after a failure not only makes people feel stupid but also leaves them feeling incapable of reaching their goal. In these instances, it's really best to stick to _purely informational_ feedback\u2014if effort isn't the problem, figure out what is, and let him know.\n\n**Rule #3:** When things go right, **avoid praising ability**. I know we all like to hear how smart and talented we are, and so naturally we assume that it's what other people want to hear too. Of course they do. But it's not what they _need_ to hear to stay motivated.\n\nStudies show that when we are praised for having high ability, it leaves us vulnerable to self-doubt when we encounter difficulty. If being successful means you are smart or \"a natural,\" then it's easy to conclude when you're having a hard time that you just don't have what it takes.\n\nInstead, **praise aspects of his performance that were under his control**. Talk about his creative approach, his careful planning, his persistence and effort, his collaborative attitude. Praise the actions, not the person. That way, when he runs into trouble later on, he'll remember what helped him to succeed in the past and put that knowledge to good use.\n\n**Compare his performance to his own past performance, rather than to other people's.** It's hard to get away from Be-Good thinking when the grades we are given, or the results of our annual performance reviews, are always calculated relative to the performance of other people. To some extent, this is unavoidable. But we can support Get-Better thinking by, whenever possible, recognizing and rewarding progress. A child who goes from a D average to a C deserves as much praise as one who goes from a B average to an A. Even if Bob is still the worst communicator on his team, the fact that his communication skills have improved in the past year should be acknowledged and admired. When people know that progress matters, they implicitly get the Get-Better message you are sending.\n\nPhase 2: Provide Promotion and Prevention Skills Training and Confidence Building\n\nHow They Learn\n\nThe particular kind of training you provide to your students or employees will vary, but their focus still determines to some extent how they will want to learn, and how they will learn best. Knowing this gives you insight into what you can expect.\n\nPromotion-focused people should prefer on-the-job training. Motivationally, they experience fit when they sense they are moving forward quickly. It's important for you to provide frequent signals that they are doing well and making progress. It's also important for you to let them dive right into their work, even when they are new to it. The promotion-focused have a greater willingness to make mistakes and \"give it a try,\" and they grow impatient or bored when things move too slowly. Regular pep talks are useful, even beyond the learning phase.\n\nPrevention-focused people, in contrast, should prefer advance training\u2014an initial period of learning and familiarizing prior to beginning the actual work. Motivationally, they experience fit when they work slowly and deliberately. They will want to understand everything thoroughly before trying it out on their own. You need to keep in mind their strong desire to be accurate, and their profound discomfort with making mistakes. It's important for you to provide frequent honest feedback, particularly about areas that need improvement. Remember that they will need more time to feel ready, but on the bright side, their performance is likely to be high quality right out of the gate. Do yourself (and them) a favor, and skip the pep talks\u2014they will do more harm than good.\n\nHow to Build Confidence\n\nAs I mentioned earlier, confidence or self-efficacy comes in large part from our experiences with success and failure. Effective training gives you the opportunity to start out with more success experiences than failure, thereby increasing your self-efficacy. But it's also important to think about our successes and failures in the right way. Learning to do this is vital if you want to flourish both personally and professionally.\n\nTake a moment and think about a recent success experience you've had\u2014a time when you reached a goal or made significant progress toward one. I'll wait.\n\nNow, answer this question: Why did you succeed? What made you so successful?\n\nIf your answer was:\n\nBecause I'm so _smart_ (or talented, creative, gifted, etc.)\n\nBecause I was _lucky_\n\nor\n\nBecause _someone else_ made it happen\n\nthen you have made what psychologists call an \"uncontrollable attribution.\" In other words, you attributed your success to something you cannot actually control\u2014either a force outside yourself (i.e., luck, someone else's power), or something inside you that you can't do anything about (i.e., your innate abilities).\n\nIf, instead, your answer was:\n\nBecause I worked hard\n\nBecause I was well prepared and made a good plan\n\nBecause I tried to stay positive\n\nBecause I didn't give up\n\nthen you made what we call a \"controllable attribution.\" You attributed your success to something that is under your control: your effort, your preparation, your attitude, your persistence.\n\nAs important as it is to have success experiences in order to build confidence, they will do you little good if you attribute those successes to forces outside your own control\u2014even if they're your own innate abilities. As we've seen, it's the belief that you can make success happen that is at the heart of self-efficacy\u2014that it is in _your_ power, under _your_ control. The same holds true when it comes to thinking about our failures\u2014they only reduce our confidence when we feel we are powerless to do anything about them. Making controllable attributions for our failures\u2014blaming our effort, strategy, or persistence\u2014makes improvement possible. It leaves success within reach.\n\nThe good news is, attributing your success and failure to controllable causes is not only motivationally smart, but it's also accurate. Study after study shows that reaching goals is about the actions we take or fail to take, not the DNA we happened to be born with. (For more on this, see Part 2 of _Succeed._ )\n\nSo if you want to increase your confidence, it's important not only to reflect on your past successes, but to think about how _you_ made them happen, through your own effort, choices, and actions. If you are looking to build someone else's confidence, be sure to highlight his successes in ways that heighten his sense of power and control.\n\nNote: Prevention-focused people need confidence, too\u2014but it's not the everything-is-gonna-be-great confidence of a promotion-focused person. It's the if-I-try-I-can-prevent-disaster-from-happening kind. Remember, their confidence will never look like optimism; it will look like defensive pessimism. But without _some_ confidence in their ability to keep disaster at bay, they will fall apart just like the promotion-focused do.\n\nPhase 3: Create Motivational Fit\n\nThis is a topic that really warrants its own book, since there are so many ways in which motivational fit can be created for the promotion- and prevention-minded. (Luckily, that book exists\u2014if you want to learn more, check out _Focus_.) __ Here is just a sample of the techniques you can use to work with your own focus, or to create an environment that enhances someone else's motivation.\n\nHow to Create Fit for Promotion Focus\n\n**Use praise and be optimistic.** Do what you can to keep them optimistic about success, particularly when they have experienced a setback. If you are the one experiencing a setback, take time to recall some of your past successes\u2014this should give you an optimism boost to help you sustain your eagerness.\n\n****\n\n****\n\n**Give them something to strive for.** The promotion-focused think of their goals in terms of gain. Describe the goals you want to reach as _ideals_ , as opportunities for _advancement_ , as potential avenues for _reward_.\n\n****\n\n****\n\n**Give them opportunities to generate ideas and be creative.** Take advantage of the fact that the promotion-minded are naturally inclined to be brainstorming, out-of-the-box thinkers.\n\n****\n\n****\n\n**Recognize that they will prefer working quickly.** Nothing dampens a promotion-focused person's eagerness like having to move at a snail's pace. Get rid of red tape; give them a feeling that they are making progress rapidly toward their goal. Breaking down a long-term goal into shorter-term subgoals may be helpful.\n\n****\n\n****\n\n**To motivate, refer to abstract reasons.** __ The promotion-minded think Big Picture. If you want your child to study for his upcoming chemistry exam, emphasizing how success in high school will open up doors for him in college and beyond is just the kind of abstract, forward-looking reasoning he's likely to respond to.\n\n****\n\n****\n\n**When making decisions, they'll want to consider the pros.** Help them make decisions by encouraging them to ponder what's _good_ about option A vs. option B. They'll make better decisions, and be happier with the results, if they do so with motivational fit.\n\nHow to Create Fit for Prevention Focus\n\n**Use constructive criticism and be defensively pessimistic.** Don't be overly effusive with praise, and don't hold back when there are suggestions you can make for improvement. Nothing should be taken for granted. Thinking about how it all might go to hell in a handcart is motivating to the prevention-focused; it sustains and enhances their vigilance.\n\n**Give them something to avoid.** The prevention-focused think of their goals in terms of avoiding loss. Describe the goals you want them to reach (or think of your own goals) as things that ought to be done, as opportunities to _stay safe_ , as ways of _hanging on to what you've got_ or of _keeping things running smoothly_.\n\n**Give them opportunities to analyze, to evaluate ideas, and to plan.** Take advantage of the fact that the prevention-minded are naturally thorough, deliberate, and thoughtful.\n\n**Recognize that they will prefer working slowly and carefully.** Try to hurry someone who is prevention-focused, and at first they will ignore you. If you keep it up, they will start to get very anxious, and quite possibly angry. They do not want to be rushed\u2014that's how mistakes get made. High-quality, accurate work takes time. Be patient; the end result with be worth the wait.\n\n**To motivate, refer to concrete reasons.** None of that off-in-the-future abstract mumbo jumbo will work here. If you want your child to study for his chemistry exam, remind him that he will probably fail the exam if he doesn't, and that this will have immediate consequences\u2014like being grounded. The prevention-minded want the cold, hard facts of the here and now.\n\n**When making decisions, they'll want to consider the cons.** What's _bad_ about option A vs. option B? Which is the lesser of two evils? Decisions made this way for the prevention-minded feel right, and therefore will have greater value.\n\n# Conclusion\n\nNow that you have received your basic training in Motivational Diagnosis, it's time to put your new knowledge into practice. First and foremost\u2014 _Physician, heal thyself._ If you have gotten new insights into your own mindset, motivational focus, and self-efficacy, plan out a course of self-treatment. Bear in mind that you may need different diagnoses for different areas of your life: You could be a high-confidence Get-Better promotion-minded professional, but a low-confidence Be-Good prevention-minded person in your relationships. Just as in medicine, the more accurate your diagnosis, the more effective the treatment will be.\n\nNext, think about how you might help the people around you\u2014how to help your employees, coworkers, students, spouse, and children become more effective in reaching their goals. You can share with them the insights you have gained directly, or you can take a more subtle approach\u2014using your knowledge of what will work best for them to create an environment in which they can excel.\n\nFor you, underperformance is no longer a mystery illness. No more wild guesses, no more desperate attempts to try anything that _might_ work. You are being paged, Doctor. It's time to deliver the cure.\n\n# About the Author\n\nHeidi Grant Halvorson, Ph.D., is a social psychologist, speaker, and the author of _Succeed: How We Can Reach Our Goals_ and _Focus: Use Different Ways of Seeing the World for Success and Influence_. She is the associate director for the Motivation Science Center at Columbia Business School.\n\n# Notes\n\n H. Grant and C. S. Dweck, \"Clarifying Achievement Goals and Their Impact,\" _Journal of Personality and Social Psychology_ 85, no. 3 (2003): 541\u201353.\n\n L. S. Gelety and H. Grant, \"The Impact of Achievement Goals and Difficulty on Mood, Motivation, and Performance\" (unpublished manuscript, 2009).\n\n E. T. Higgins, \"Beyond Pleasure and Pain,\" _American Psychologist_ 52, no. 12 (1997): 1280\u2013300.\n\n A. Bandura, \"Self-Efficacy: Toward a Unifying Theory of Behavioral Change,\" _Psychological Review_ 84, no. 2 (1977): __ 191\u2013215.\n\n A. Stajkovic and F. Luthans, \"Self-Efficacy and Work-Related Performance: A Meta-analysis,\" _Psychological Bulletin_ 124, no. 2 (1998): 240\u201361; G. Holden, M. Moncher, S. Schinke, and K. Barker, \"Self-Efficacy of Children and Adolescents: A Meta-analysis,\" _Psychological Reports_ 66, no. 3 (1990): 1044\u201346.\n\n E. Ghiselli, _The Validity of Occupational Aptitude Tests_ (New York: Wiley, 1966).\n\n G. Oettingen, H. Pak, and K. Schnetter, \"Self-Regulation of Goal-Setting: Turning Free Fantasies About the Future into Binding Goals,\" _Journal of Personality and Social Psychology_ 80, no. 5 (2001): 736\u201353.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n## Zombie \u2013 A Love Story\n\nBy Patricia Lee Macomber\n\n\u00a9 2013 Patricia Lee Macomber\n\nCover Design By: Patricia Lee Macomber\n\n## LICENSE NOTES\n\nThis eBook is licensed for your personal enjoyment only. This eBook may not be re-sold or given away to other people. If you would like to share this book with another person, please purchase an additional copy for each person you share it with. If you're reading this book and did not purchase it, or it was not purchased for your use only, then you should return to the vendor of your choice and purchase your own copy. Thank you for respecting the hard work of this author.\n\n## Meet the Author\n\nAbout the Author\n\nPatricia Lee Macomber is the former editor-in-chief of ChiZine. She has been published in \"Cemetery Dance\" magazine and such anthologies as \"Shadows Over Baker Street,\" \"Little Red Riding Hood In the Big Bad City,\" and \"Dark Arts.\" Currently, she lives in North Carolina with her husband, David, and their children.\n\nBook List\n\nStargate Altlantis \u2013 SGA-15 \u2013 BRIMSTONE \u2013 with David Niall Wilson\n\nIntermusings \u2013 with David Niall Wilson\n\nAn Unkindness of Ravens with David Niall Wilson\n\n## DISCOVER CROSSROAD PRESS\n\nVisit our online store\n\nSubscribe to our Newsletter\n\nVisit our DIGITAL and AUDIO book blogs for updates and news.\n\nConnect with us on Facebook.\n\n## Zombie: A Love Story\n## CHAPTER ONE\n\nPaul Tremblay\u2013son of Marge and Joe Tremblay, employee of the New York City Public Works Department, only child, lover of all things caffeinated, idealist and romantic, future fianc\u00e9 of Linda Gilchrist\u2013 that Paul Tremblay, woke up on a cold steel table in the morgue.\n\nSlowly, as though his eyes were new and untested, he blinked away sleep \u2013 or whatever had possessed him those past long hours \u2013 and slowly drew his eyes over the scene around him. The room was large and empty, dimly lit and quiet. It smelled faintly of chemicals that Paul was sure had never been required in his life. Light bounced off of varying reflective surfaces and metal fittings.\n\nPaul blinked again and rubbed his eyes. There was a dim light above him and beside the table where he lay was another table with another guy. No one else was around. In fact, the quiet \u2013 save for the hum of some machine far off in the distance \u2013 was almost...deafening.\n\nFirst things first, Paul decided. He did a mental inventory of himself. He was covered by nothing more than a thin sheet and his skin sort of itched, though it wasn't really an itch. It was more of a faint irritation. That said, he seemed perfectly all right otherwise. The guy next to him...not so much. He was laid out, toe-tagged, and cut right down the middle.\n\nPaul panicked. If the guy next to him was dead and sliced up like a Thanksgiving turkey, then surely that was his intended fate as well. He slid his legs over the edge of the bed \u2013 which really wasn't a bed at all, he realized, but a long metal table \u2013 and tested them for signs of failure. Nope! His legs seemed just fine too.\n\n\"Think, Paul, think!\" he muttered to himself, sliding off the edge of that table onto the cool linoleum floor.\n\nHis legs held up under the strain, though when he took his first step, something dragged against the floor and pulled at his foot a bit. He glanced down at the offending cardboard bit, attached to his right big toe by way of a short piece of wire. His brow furrowed and he sat down on the table once more, hoisting his foot up to prop it on his left knee.\n\n\"A toe tag?\" he muttered, frowning. \"That means they think I'm...\"\n\nPanic raced through him again and he began twisting at the wire madly, not stopping until his toe had been released from the wire. He glanced at it, squinting in the feeble light to read his name, a long number, and nothing more. He tossed the tag away and stood up again.\n\nHe looked around the room once more, being careful to avoid looking directly at that poor soul on the table next to him. He spotted a bank of drawers, tables with medical instruments on them, a desk.\n\n\"Christ!\" was all he could manage.\n\nThe table on which he had just sat contained a lower shelf and on that shelf was a bag, tagged with his name and number. He grabbed it off the shelf and dumped its contents out on the table. All his clothes were there, his wallet, cell phone and keys.\n\nAt that particular moment, Paul didn't care why he was in a morgue, how he came to be there, or any other damn thing. All Paul cared about was getting the hell out.\n\nHe pulled his clothes carefully over his naked form and eased his socks on over his feet. He saw that the wire from the toe tag had left a deep and lasting impression in the flesh of his big toe. He made a mental note to reference it later, during his law suit.\n\nShoes came next and then he thrust his worldly possessions into his pockets and strode to the door. He wasn't sure what was going on, but there was one thing he was certain of: He didn't want anyone associated with this place to see him; he wasn't about to let them stop him. Whatever had led them to think he was dead, he was very sure they didn't take kindly to corpses just up and walking away.\n\nHe pulled the large wooden door open and peeked through the crack. The hall was quiet; it seemed empty. Testing the waters, as it were, he pulled the door further open and stuck his head through this larger opening. Nothing.\n\nWith the all the courage he could muster, he stepped into the hall and began the long trek toward the exit. A sign on the wall proclaimed gratitude for donations made to Bellevue. So that's where he was!\n\nHe was halfway to the exit door when a woman clutching a large stack of files rounded the corner, headed directly at him. Act casual, Paul thought, and he pressed on at a leisurely pace. As he drew even with the woman, she pressed one hand to her mouth and made a face, turning her eyes away from him and hurrying past.\n\n\"What the...?\" Paul grumbled. People here saw burn victims, lepers and other physical atrocities on a daily basis. What could possibly make her react so strongly to him?\n\nWhen he reached that corner of the hallway, he drew to a stop in front of the convex mirror at the corner of the ceiling. Screwing up his face and his courage, he looked into it. Yep! That was his face, his hair...God but I've got great hair! he thought. But there was something else; something...gross.\n\nPeppered across his face were a dozen or so dark patches, no larger than a nickel. They were more than merely dark; they were black. Black raised patches had sprouted on his face. He immediately swept his sleeves up and looked closely at his arms. A few of the patches were growing there, too.\n\n\"What the freak?\" he growled, feeling the burn of panic rise in him again.\n\nHe hurried on with a renewed desperation. What in the hell had happened to him? The last thing he could remember, he'd been working. They had been finishing up inspections on the Manhattan sewer lines, getting ready for the re-build.\n\nThe exit lay directly ahead of him. Beyond that, people milled about a small waiting area and gathered at a nurses' station. Paul ducked his head and hurried. If they spied him, they might try to stop him and he wasn't about to spend one moment here that he didn't have to.\n\nTwenty more steps and he was out the door into...darkness. Somehow, whatever had happened to him had kept him here for a while. The last thing he remembered was daylight. Now it was night, and Paul's eyes didn't seem to want to adjust to it. Neither did his mind.\n\nHis last coherent memories were of being at work, in those sewer lines, with...Matt! That was what he had to do. He had to call Matt and find out what the hell was going on.\n\nPaul reached into his pocket and drew out his cell phone, still with half a charge. The familiar wallpaper blazed at him as he awakened the device. Linda. Linda's face had graced his cell phone and computer wallpaper since the day he had met her. In the beginning, she had been a pipe dream, a fantasy girl who somehow managed to end up in a fantasy marriage to him and who, through tricks of his own imagination, had managed to be happy about it. A year after meeting her, he was about to make that fantasy real.\n\nPaul shook his head and tried to focus. Someone was coming out of the doors behind him and he slipped into the shadows cast by failing bulbs and full trees to avoid detection.\n\nMatt was second on his contact list and Paul tapped the screen to dial the house. One ring, then two, then...\n\n\"Say words!\" Matt was stoned. He always answered that way when he was stoned.\n\n\"Matt! God! I'm glad I got you.\"\n\n\"Hello?\" Matt seemed confused.\n\n\"It's Paul, you goof. Listen....\"\n\n\"Who is this?\"\n\n\"Look, I just want to know what happened to me. We were...\"\n\nApparently, Matt had checked his caller ID. \"Dude, this is bogus. Totally not funny, Man. Paul is dead and you're using his cell phone to call me...\"\n\n\"No! Listen, this is Paul. I'm not dead. I just...\"\n\n\"Are you fucking crazy or something? This is a sick joke, using my best friend's cell phone to punk me! Calling me up in the middle of the night and growling like a...\"\n\n\"Goddammit, Matt! Listen!\"\n\nThe call ended and Paul's shoulders sank. The sound of laughter reached him and it made him sad. Why had Matt said that he was dead? Why couldn't he understand what Paul was trying to say?\n\n\"Growling?\" Paul's face sank further into a frown and he shuddered.\n\nWell, if he couldn't get anything out of Matt on the phone, he would just have to see him in person.\n\nHaving worked for the city for nearly ten years, Paul knew the streets of New York like the back of his very spotted hand. Spotted! Were those there a moment ago? He wondered. Images of disease of the week movies popped into his head, with the men in contamination suits gathering up and herding the infected. Was that what he was? The infected? Patient zero? He briefly entertained the notion that he might be walking the streets, infecting hundreds of people as he passed. That was just too ludicrous and he dismissed the idea as a product of panic.\n\nHe set off in the direction of his apartment. He and Matt had been best friends ever since fourth grade when Matt's mom had moved into Queens and dropped him into the public school system. Matt had been like a shot in the arm to Paul, who even then had taken himself too seriously. Matt was a goof, a class clown, a sometimes lazy-ass who never quite did well but always meant well. So, after graduation, Paul had taken an apartment with Matt, gone to work for the city to put himself through college, and two months later he'd managed to get Matt his first and only decent job.\n\nOne foot after the other, Paul plodded in the direction of their apartment. He didn't want the population at large to see him, so he avoided the subway and kept to the shadows as he walked. It was nearly five miles to their place, a long walk, but do-able. He picked up the pace a bit, desperate for answers.\n\nNew York City at night was, if anything, more lively than in the day. Bright blotches of neon color splashed across the sidewalk, flashing headlights strobed through the darkness. Paul had lived here all his life and there was nowhere on Earth he would rather be....except LA.\n\nLA. That's where Linda was. He had met Linda at college and it had been love at first sight. She was blonde and gorgeous and smart. And the best part of all that was that she thought the sun rose and set on Paul. If they had been on a soap opera, they would have been a super-couple. He'd been studying engineering at NYU, she...English. Nine years later, he was finishing up his ten-year stint with NYC Public Works, in the hopes of getting a small pension. She had been offered a teaching position at UCLA, a promise of tenure in two years (English professors didn't seem to stay put for long) and the opportunity to work on her PhD.\n\nSo, Linda had gone ahead to LA with the idea that Paul would follow as soon as his ten years with the city were up. He only had two weeks to go. Damn the luck.\n\nPaul passed the third ice cream store in as many blocks, his head low and in true New Yorker fashion, not meeting the eyes of anyone. So far so good. Another two miles and he could interrogate Matt.\n\nLinda. His mind drifted back to her. A sudden stab of terror ripped through him as he remembered the ring he had bought her. They had lived together for four years and he had been about to propose when the offer came in from LA. So, being the cautious and wise man that he was, he had tucked the ring away with the idea that he would propose to her as soon as he made the move to Cali, and then only if their relationship survived the separation. That ring was now neatly tucked into his underwear drawer.\n\nOne more mile. A finely-dressed lady with an equally finely-dressed little boy passed him. As they did so, the child looked up and caught sight of Paul's face. The boy tilted his head and scrunched up his face, then made the proclamation: \"Eeeew!\"\n\nPaul turned his head away and frowned again. For a naturally happy guy, he sure was frowning a lot.\n\nTwo more turns and Paul was staring at their apartment building. It was short and squat, converted from an old Brownstone. It still bore the bomb shelter sign that had been posted there in the sixties. He fished out his keys as he approached it, his mind spinning circles around the questions he had.\n\nHe let himself in the front door and took the stairs to the second floor, where his apartment awaited him. The key slid home and Paul turned it, listening to the gratifying turn of the tumblers inside. He pushed the door open a few inches and met resistance.\n\n\"Matt! Hey, Matt!\" he called in through the crack in the door. He was rewarded with the sound of footsteps within.\n\nMatt's eye appeared in the crack of the doorway, heavy-lidded and blinking stupidly. He was, indeed, stoned. \"Paul? Oh my God!\"\n\n\"Undo the chain, Matt. Please.\"\n\n\"Dude...you're dead.\" He said it more as a matter-of-fact than as a threat.\n\n\"Matt, I'm not dead. I promise. Now, please, open the door.\"\n\nThe door slid shut and Paul heard the sound of metal against metal. Matt pulled the door open and blinked rapidly at him. \"Is it really you?\"\n\n\"Of course it's me, you doofus.\" Paul chuckled and moved to hug him. Matt backed away.\n\n\"The paramedics told me you were dead. They hauled you away.\"\n\n\"Well, I'm not dead. Obviously, they were wrong.\" Paul paused, wanting this to sink into Matt's drug-addled brain. \"But you have to tell me what happened.\"\n\n\"Dude, you look like shit.\" He moved aside as Paul made his way deeper into the apartment. \"Your face is all fucked up and...why are you making that sound?\"\n\n\"What sound?\" Now Paul felt anger welling deep in the pit of his stomach. \"Tell me what happened.\"\n\nMatt took another step backward. \"You may not be dead, but you look like death warmed over. Seriously, man, you should really be in a hospital. I think you've had a stroke or something.\"\n\n\"I haven't had a stroke and I'm not dead.\" Now, he was yelling. The frustration had a choke-hold on him and he couldn't manage to calm himself. \"But I need you to tell me what happened.\"\n\nMatt blinked stupidly.\n\n\"Are you hearing a single word I'm saying?\" He took one step toward Matt and watched his friend's face pale.\n\n\"I think I should take you to the hospital. You need some help.\" Matt proffered his hand, palm up, to his friend.\n\nPaul looked at the hand, then back to Matt. His chin sagged to his chest. \"I...just need...to know what happened to me.\"\n\nMatt's tone was softer now, more sympathetic. \"Paul, you look like shit. And I can't understand a word you say. Please let me get you some help.\"\n\nFrustrated beyond anything he could remember, Paul did something so uncharacteristic that it scared even him. He lunged forward and grabbed Matt by the shoulders, shaking him hard and screaming. \"I don't need help! I need answers! Now, shut the fuck up and tell me what happened to me, dammit!\"\n\nMatt whimpered, winced, and pulled away from him.\n\n\"Oh God! I'm sorry. I'm sorry!\" Paul let him go and backed away, feeling the hot sting of tears and guilt well up in his throat. He sank into the chair and let his head fall into his hands on the dining table. \"I'm just so...scared.\"\n\nAfter a few seconds, he lifted his head slowly and looked across the table. An old pizza box lived there, along with his dirty coffee mug, a magazine, two pens and a legal pad with some scribbling on it.\n\nPaul grabbed the pad and one of the pens and began to write furiously.\n\n\"I can't remember what happened. Tell me,\" it said.\n\nHe shoved the pad in front of Matt's face and raised his eyebrows.\n\nRelieved, he watched as Matt sank into the chair next to him. \"You really don't know what happened to you?\" Paul shook his head and Matt nodded. \"Okay, I'll tell you what I know.\"\n\nPaul tried on a smile, which must have come across as more of a sneer because Matt made a face and looked away.\n\n\"Okay, so we were working down in the tubes out where they're doing the renovations. You know, the sewers that run past all those factories and shit? We were almost at the end, taking those readings. Last one of the day, you said. There was this big pile of old trash from when they had originally built those sewer lines and you had to climb over it to get to the other side and take the readings. You wouldn't let me go because you said I was too clumsy and I would rush it.\n\n\"Anyway, we were talking and you got over the top of that rubbish, still talking. Then you said something about it being really gross over there and something about a puddle of blue goo. You cursed when you stepped in it. I remember because I laughed and made a joke. You know, 'Some days you step in it, some days you don't.' Then you came back over the top of the trash and you were all covered in that blue shit.\n\n\"So, we walked back to junction eight-twenty-three and climbed up and out. The stupid blue stuff glowed in the dark down there, but once we got topside, it was gone. We were walking back to the truck when you doubled over. You dropped straight to the ground and started rolling all around like a dead fish or something. I called nine-one-one and by the time the EMTs got there, you had stopped having fits. But they said you were dead.\n\n\"Dude, they put you in a body bag and hauled you off to Bellevue. I followed just to make sure they weren't screwing up, but the doctor came out and told me you were dead. He said they didn't know what of, but that you were just dead. They asked me if you had any next of kin and whatnot. I gave 'em your parents' number. Man! I wish I'd have known you weren't really dead. Did they screw up or something? What's going on?\"\n\nPaul scribbled hastily. \"I don't know. Linda?\"\n\n\"Linda? Oh, Linda. Right. Naw, I don't think nobody called her. I didn't have her number, but I was gonna go through your cell phone once they released your stuff and call her.\"\n\n\"Don't call Linda,\" Paul wrote quickly. He thought for a moment and Matt seized that opportunity to offer what might have been the only sound advice he would ever come across.\n\n\"Dude, you need to let the doctors look at you and figure out what happened.\"\n\nIt was Paul's turn to blink. That was the sane thing to do, he realized. But something deep in the pit of his gut told him he shouldn't do that. \"No,\" he wrote. \"Got to see Linda.\"\n\n\"But Dude, you've only got ten work days until you get your pension. Just see a doctor, huh? Maybe they can help you. And man, whatever that shit is on your face, it looks like it needs some help.\"\n\n\"Not going to last that long,\" Paul wrote and suddenly he felt like crying. His grandmother had told him that everyone knows right before they're going to die. That was five hours before she died. Paul knew, too. \"Got to get to Linda. Tell her I love her.\"\n\n\"Paul, she knows that. Just please get some help.\"\n\n\"Linda has to know. I didn't abandon her.\"\n\n\"Okay, call Linda.\"\n\n\"No.\"\n\n\"Call her. Then go to the hospital.\"\n\n\"No.\"\n\nMatt sank back in his chair with a worrisome frown. \"You're my best friend. My BFF, dude. If anything happens to you...\"\n\nPaul growled, frustrated at the slowness of hand writing everything and his own failure to get his message across. He threw back the chair and stood up, hurrying to the little desk under the window. The computer was already on and he opened Notepad and began to type.\n\n\"Matt, we've been friends almost our whole lives and I love you like a brother. I know that what you're saying makes sense but I know, deep down in my gut, that something weird and awful happened to me down there. I don't know what that stuff was, but it changed me somehow. I was alive, and then I was dead, now I'm alive again. I don't know what the hell's going on, but I think I might die for good soon and I have to get to Linda. She can't think that I just dumped her, stood her up, whatever. She has to know...\"\n\nMatt, who had been reading over his shoulder, interrupted. \"Dude, I'll tell her. Seriously. I'll make her understand what happened. But go to the hospital. Please. It's what Linda would want you to do.\" Tears welled up in Matt's eyes, which made Paul feel like crying, too.\n\n\"No,\" he typed again. \"I have to do this. If I'm going to die, I have to see Linda just one more time.\"\n\n\"I'll call her for you. She'll come. She loves you and she'll come. Then you can go to the hospital, huh?\" Matt was all-out crying now and it made Paul sad beyond words.\n\n\"Nononononono!\" Paul typed. \"You have been a great friend, and I know you mean well, but I have to do this. Don't blame yourself. Honestly, none of this is your fault. I just have to see Linda. Have to.\"\n\nPaul stood up and headed down the hall toward his room. Matt followed close on his heels, blubbering and talking a mile a minute.\n\n\"Please let me get you some help. Don't do this.\"\n\nPaul rummaged around in his underwear drawer until he found the thing he was looking for. He held up the small blue velvet box and smiled, then tucked it neatly into his pocket and shut the drawer.\n\n\"What if you die on the way there? Huh? You'll never get to see Linda and you'll be dead. Maybe for real this time. If we go to the hospital first, that way you can maybe get some help, then see Linda and maybe live happily ever after. Huh? Doesn't that sound better?\"\n\nPaul shook his head and marched back down the hall to the computer.\n\n\"Taking the car,\" he typed furiously. \"Tell my parents that I love them.\" He rose from the chair and headed for the door.\n\nWith all the courage he could muster, Matt stepped in front of him. \"As your best friend, I cannot let you do this.\" He waved the cordless phone in front of Paul's face. \"I'm calling nine-one-one and I'm getting you some help, whether you like it or not. You'll thank me later.\" Matt pushed the ON button.\n\nPaul batted one hand in Matt's direction, sending the phone flying across the room. Matt watched it go, his expresion caught between terror and determination.\n\nIt was all determination when he turned back to Paul. \"I've still got my cell phone. Ha! What do you think about that?\"\n\nPaul uttered something of a growl and snatched the phone from Matt's hand. He turned to the doorway again. Within seconds, he was through the door and partway down the hall.\n\n\"Fine!\" Matt yelled after him. \"I'll just wait until you're gone and then call for help. Yea, buddy! I'll report the car stolen. Then they'll catch you and get you to the hospital. I'm not letting you die, man! You hear me? I'm not letting you die!\"\n\nOld Mrs. Carter across the hall stuck her head out the door, sporting a moo-moo and a mean expression.\n\n\"Sorry, Mrs. Carter,\" Matt mumbled, lowering his head and ducking back into the apartment. He went to the window then, and watched as Paul pulled out of the parking space far below and out onto the street.\n\n## CHAPTER TWO\n\nPaul was a man with a plan. He needed to see Linda, yes, but there were other things that needed doing. Everything Matt had said was true: He needed to see a doctor, should go to the hospital, could easily let Matt call Linda for him. But something drove him \u2013 something he couldn't explain \u2013 to see Linda in person. Something beyond comprehension told Paul that he wasn't going to last long, and that he had to get to Linda.\n\nFirst, though, he needed to go back to that sewer and see for himself exactly what it was that had taken him out. First the sewer, then a hasty exit. He only prayed that he would make it out of town before Matt reported the car stolen and the police caught up to him.\n\nThe particular point in the sewer line where he and Matt had been working was in the industrial section and Paul pointed the car in that direction. He drove just fast enough to get there quickly; not fast enough to attract attention and get pulled over. Paul had spent a good portion of his adult life debunking the myths that circulated about the sewers. No, there was no giant alligator. No wild band of mutant humans lived there. There were no toxic waste dumps, no ravenous zombies, and no packs of wild rats with super-human intelligence. As it turned out, he seemed to have been wrong about at least one of those.\n\nHe guided the car across the bridge and into the industrial section. He turned his lights off and eased up to the access point near an old abandoned warehouse. There were plenty of those in the city lately, out-sourcing being what it was and all. This particular one had made cleaning solvents for the avionics industry and the military. It, along with several other companies on that block, had ceased operations about five years back.\n\nHe kept the headlights off, running dark, pulled up next to a derelict eighteen-wheeler and turned off the engine. As he stepped out of the car, keys in hand, his left leg almost gave out on him. He gasped a bit and put one hand on the door to steady himself, then pulled himself upright.\n\nIt had been a long night and his mind \u2013 if not his body \u2013 was tired. Slowly, he made his way toward the entrance to the sewers, a manhole some forty feet from the back yard of the place. As he went, he toed through some discarded trash which peppered the ground, hoping for a jar with a lid still intact or something like it. Three feet shy of the manhole, he found a small applesauce jar, its lid lying next to it, unbent. He plucked it from its resting place and popped it into his pocket. All he wanted was to get a sample of this stuff and somehow find out what it had done to him.\n\nManhole covers, by and large, are heavy things. On work days, Paul and Mark carried pry bars to help lift them. Paul had no pry bar that night, so he had to settle for wedging one finger into the little hole. He braced his feet against the packed dirt and tugged and hauled at it until he had moved it a few inches out of place. Then, with a surly grunt and a small squishing sound, he heaved it over and out of the way.\n\nIt was dark down there and Paul didn't have a flashlight tonight. It became painfully obvious to him that he'd jumped into this unprepared, had not clearly thought it all through. Whatever.\n\nHe slipped into the hole and let the darkness devour him. Somewhere off in the distance, a woman screamed. Probably a mugging, Paul told himself. He felt his feet touch solid concrete and he let go of the ladder.\n\nA small circle of moonlight shone above him and it made the puddles at his feet sparkle and shimmer with borrowed light. Ahead of him, a deeper, brighter glow emanated from the far end of the tunnel. It was partially obscured by the hill of trash, of course, but Paul could still see it. He shivered.\n\nArmed only with his own meager night vision and a decent center of gravity, Paul shambled toward that blue glow. It emanated from a point only fifty feet or so in front of him and with each step he took, he felt his skin tingle and chill.\n\nHe took more care climbing over the trash this time, simply because he couldn't see anything and he feared that a misstep could cause him more injury than he had already suffered.\n\nOnce on the other side, the source of that blue glow became evident. A large puddle of liquidy goo covered the floor of the tunnel. Spatters of it had coated the walls and continued to ooze down out of the pipe above. Paul kept close to the right-hand wall, away from the sticky, glowing trail of goo. Cautiously, he made his way to the edge of the puddle, stopping just shy of its edge.\n\nThe stuff just soft of sat on the water. It didn't actually float, but rather coated the surface, as if pinning it down, holding it captive. Paul reached into his pocket and pulled out the jar, meaning only to take a small sample of the spill; just enough for testing.\n\nWhen he placed the jar at the edge of the stuff, taking great care not to let any more of it touch his skin, the blue goop seemed, very literally, to crawl into the jar. The rim had just sunk below the surface of the standing water when the blue stuff lurched forward and rushed in. No water went with it and Paul frowned, wondering whether the goo was alive, or whether his addled brain had simply misinterpreted the physics.\n\nHe held the jar up to his face and studied it. It was blue, translucent and thick. No water had entered the jar and so what he had as a sample was pure blue goo. Quickly, he placed the lid on the jar and torqued it as tightly shut as he could. Then he placed the jar back in his pocket and turned to mount the pile of trash and make his way out.\n\nLetting the small circle of moonlight be his guide, Paul headed for the ladder and sweet, fresh air. He had become painfully used to the dank, acrid smell of the sewers, so it never really bothered him. But he remembered Linda's reaction on the few times he had come straight home from work without stopping at the dispatch station to bathe. She'd wrinkled her nose and turned away, pressing one hand to her mouth and shutting her eyes. Not one word had she uttered.  She's just pointed at the bathroom and gagged a bit. He'd done it twice early on, and then never again since.\n\nIt was no small feat to haul himself up that ladder and out of the hole. For some paranoid reason he couldn't quite fathom, he felt it was absolutely imperative to shut that manhole. Perhaps the goo would escape and infect others. Perhaps it was a sentient sort of thing and would go on a rampage throughout the city.\n\n\"Stupid!\" Paul grumbled at himself. \"You're being stupid.\"\n\nThen he thought about how the stuff had crawled into the jar. He flipped the manhole cover onto his back and slid it back into place, listening as the grating sound of metal on metal echoed through the empty industrial district. Then he stood and, as quickly as he could manage, made his way to the car.\n\nHe wasn't sure if Matt had actually made good on his threat, but he was willing to bet money that he had. That meant that Paul had to get out of the city as quickly as possible; before the police could spot the car and stop him.\n\nHe would worry about taking his sample to the hospital later. For now, all he wanted was to see Linda and feel re-assured that she still loved him, and he was a long way from Los Angeles.\n\n## CHAPTER THREE\n\nPaul watched as New York City grew ever smaller in his rear view mirror. Ever since he had left the apartment, he'd waited for those flashing lights to appear in his mirrors. Matt was a good friend and he knew that he was only concerned for his safety. Most people saw Matt as a dolt, a ne'er do well, a waste of oxygen. Most people didn't know Matt. The man had a good heart and he was as loyal as the day was long, but he was rough on the surface, and most people didn't take the time to get to know him.\n\nMatt was the son of an alcoholic and some unknown condom-challenged one night stand. He'd practically raised himself and at the same time had taken care of his mom, until she died of alcohol-related diseases when he was fourteen. After that, Matt had gone into the system for a month, until Paul's parents had completed the paperwork necessary to foster the boy.\n\nPaul hadn't been kidding when he said he thought of Matt as a brother. Together, they had survived some close-calls, pulled some epic pranks, and in general held each other together through everything. He knew Matt blamed himself for not going over the trash pile. He knew Matt was only trying to protect him and help him. Still, he couldn't let anyone, not even Matt, stop him from getting to Linda.\n\nThe light of the city glowed brightly in his mirror. Ahead, there were thousands of miles of highway. A sudden thought assaulted Paul's mind and he nearly threw on the brakes. He had no idea how to get to LA.\n\nThe car he was driving was only a year old. He and Matt had bought it together. They had invested $5000 of their own savings in it, and then gotten a loan in Paul's name to cover the rest. Paul had paid ahead on it, gotten the loan down to almost nothing. He had intended to give the car to Matt when he left for Cali.\n\nHe eased the car off to the side of the road and put it in park. Linda's address was emblazoned on his memory, so he punched it in to the GPS and waited. There was hardly any traffic at this hour and he hoped that Matt hadn't made good on his promise to report the car stolen. If he had, there was nowhere to hide.\n\nThe GPS beeped and a map appeared. Paul smiled and pushed the shift lever back into drive. He eased back onto the road then, following the map and shaking his head at the voice. Why did these things always have the most pretentious voice possible?\n\nPaul drove on, sticking to the speed limits and keeping a careful eye out for cops. He felt fairly sure by now that Matt had not called them. He could picture his friend pacing back and forth in the apartment, trying to decide what the right thing to do was. The odds were very good that Matt had continued pacing, probably smoked another fat one, and then fallen asleep without any decision having been reached. Classic Matt.\n\nPaul passed the state line and kept going. A thin slit of sunrise spread across the back window. He would need to stop for gas soon, but for the moment he was good and he kept driving.\n\nLinda felt the vibration of the alarm transfer from the nightstand to the bed to her cheek as it lay on the pillow. That more than the noise of it woke her. She reached out, letting her fingers feel the items on the nightstand \u2013 cell phone, bottle of aspirin, book, tissue box \u2013 and finally found the alarm. She pressed at several buttons, got it wrong, finally found the right button, and the alarm went silent. The button slapping was why she had to reset her alarm clock every night. She was not a morning person.\n\nShe threw back the covers and slipped her feet to the floor immediately. If she didn't do that, she ran the risk of falling back asleep. Her first trip was to the bathroom, where she yawned continuously as she peed, brushed her teeth and washed her face. She hated morning breath, hated it more than anything. Her day just couldn't go on until she had scrubbed it away.\n\nBack to the bed then, where she sat down and pulled the cell phone off its charger. She stopped for a moment while she yawned some more. God, how she wished it was Saturday. The next part of her day had been a ritual since the day she had left New York. No day started or ended properly without her talking to Paul.\n\nPaul. He was thousands of miles away and oh Lord, how she missed him! The look on his face when he first woke up, like he was always surprised she was there. The joy that overtook his eyes when he realized that she was, that she was his, and that would never change. She loved the cute way he had of eating his cereal. He ate it dry with a spoon, then took a swig of milk from a glass after every bite of cereal. He said putting it in milk made it mushy too fast. His way kept it crunchy longer. She wondered to herself if their future children would eat their cereal that way.\n\nSmiling as though her face might split open, she almost squealed with joy as she thought of Paul. He was her love, her buddy, her everything. She had dated other men briefly; rich men, handsome men, brilliant men, handsome men who were rich. But Paul was THE ONE. He wasn't handsome or rich or brilliant, but he was sweet and loving and funny and giving.\n\nShe shivered, sighed happily, and rested back on the pillows. Paul was number one on her speed dial. She pushed the button and put the phone to her ear, waiting for his sweet voice.\n\nPaul's phone rang. The sound of it made him jerk the wheel and he nearly cried out. He had seen far too many accidents come from talking on the phone while driving, so he made it a personal rule that he would always pull over before answering. To that end, he steered the car to the breakdown lane and put it in park.\n\nLinda. It was her ringtone. He tapped the screen and pressed the phone to his ear.\n\n\"Hello?\"\n\n\"Paul? Is that you?\" Her voice sounded concerned. Had someone called her to tell her of his plight?\n\n\"Good morning, baby,\" he said and waited.\n\nThere was a long silence and then, \"Are you okay? Why are you making that noise?\"\n\n\"What noise?\"\n\n\"Are you sick or something? Did you lose your voice? Do you have a sore throat?\" She sounded nearly panicked now.\n\n\"No, I'm fine. Listen...\"\n\n\"Okay. I guess you're sick somehow. Cough once for laryngitis, twice for strep.\"\n\nPaul made a face and then decided to err on the side of safety. He coughed once.\n\n\"Poor baby! I'm so sorry you're sick. Anyway, this is just my usual morning call to tell you I love you.\"\n\n\"I love you, too, honey.\"\n\n\"Huh? I'm sorry, baby. I don't understand anything you say.\" There was a little titter of laughter and she continued. \"I know you can't talk, but I love you anyway. I just wish I was there to take care of you.\"\n\nWhy oh why couldn't she understand him? He was speaking clearly enough. He could hear it. \"I'm coming your way, Linda. I couldn't wait any...\"\n\n\"I know it must hurt to talk. So, you get some rest, do what the doctor says....you have been to a doctor, haven't you? Of course you have. Anyway, get some sleep, take your meds and rest your voice. And if you get bored, just think of you...and me...and that cozy little inn we went to that summer...\" She broke off in a squeal of erotic self-pleasure. \"And remember, I love you times infinity. Email me.\" She made kissing noises and ended the call.\n\nThoughts of that inn always got Paul heated up, too.\n\nHe looked down at his crotch.\n\nNothing.\n\nDamn!\n\nHe put the car back into drive and pulled onto the freeway again. Linda had called him every morning since she'd been gone. And every day on her lunch break. And every night to say goodnight. Sometimes, she called him with good news and sometimes he called her when things weren't going so well for him. All in all, they probably spoke more now than when they lived together. The sound of her voice, he always told her, was like a visit to Heaven.\n\nAnother hour of driving found the gas gauge almost to \"E\". There was a sign up ahead listing the gas stations and food stops at the next exit and Paul turned on his blinkers. The sun was low in the sky and a soft pink glow illuminated his field of vision as the sun in the east reflected off the buildings coming up in the west. He guided the car into a large gas mall, one of those places that try to satisfy the traveler's every need. There was gas, a convenience store, and two chain restaurants, all neatly tucked into one smallish building.\n\nPaul eased up to the pump and turned off the engine. It was still daytime, and that allowed people to more easily see his affliction...whatever it was. He glanced in the mirror and was pleased that there were no new spots and the old spots had gotten no larger. It's not like he could do anything about them, but at least it didn't seem to be worse. Short of putting on a ski mask and being arrested for attempted robbery, he had no options.\n\nHe slipped out of the car, keeping his head down and hurrying to fill the tank. He swiped his card and removed the nozzle, then held down the handle as the gas flowed into the tank.\n\nMost people, when confronted with someone who has a disease or deformity, will politely look away out of a sense of guilt. This was not the case with the large man at the other side of the pump. He had food stains on his t-shirt and a day's growth on his face, so he was no prize himself. On catching sight of Paul's face, he leaned forward, squinted, and tried to get a closer look. Paul turned, placed his back to the man and willed the tank to fill faster.\n\nThe man moved again, peering at him from the other side of the pump and taking a step closer. Again, Paul moved so that the pump was between the two of them and again the rude man shifted his position to get a better look.\n\n\"Back off, okay!\" He growled and suddenly a film clip played in his brain. The Elephant Man, starring Paul Tremblay. He shook off the image and tried to ignore the man.\n\nFinally, the pump shut off and Paul replaced both the nozzle and the gas cap. Then he got into the car and pulled it around to the little parking area behind the building. Countless family road trips had taught him to take care of business when you had the chance. His father's lectures to that effect were so ingrained in him that it was almost a habit. Get gas, take a leak. To that end, he ducked his head and trotted toward the rest room on the side of the building at a rapid pace.\n\nOnce inside, he felt more at ease. There were no prying eyes here, no accusing stares. He thought once more of The Elephant Man and a shudder raced up his spine. He unzipped, sidled up to the urinal and prepared himself mentally.\n\nNothing.\n\nHe tilted his head to one side and tried to remember the last time he had urinated. Yesterday. Before they had left for the sewers, both Paul and Matt had taken a bathroom break. So that would have been the last time he had relieved himself. Even accounting for the fact that your body...\n\nWait just one little minute here! he thought. If your body completely evacuates when you die, how come my pants are clean?\n\nThat simple thought froze him where he stood. Your body pushed everything out of it at the moment of death. He knew that to be true because of when Matt's mom had died...another of those memories that made him want to gouge his eyes out. So if he had died, how come there were no stains on his pants or his underwear.\n\n\"Okay, okay!\" he grumbled at himself.\n\nHe cleared his mind, shut his eyes.\n\nNothing.\n\nOne hand reached out and turned on the water. Hey, it had worked when he was little.\n\nNothing.\n\n\"Really?!\" He offered open hands and looked to the ceiling. \"My pecker doesn't work? It makes my pecker not work? Whatever it is!\" First he couldn't get wood when he was thinking about Linda and the inn, now he couldn't take a leak. \"Oh come on!\" he yelled and slammed his hand against the cheaply tiled wall.\n\nHe was rewarded by the sound of cracking bones and shifting flesh. He pulled his hand into range and stared at it as if it were some sort of alien being sent to destroy him. The fingers were twisted oddly and his knuckles seemed flattened. There was no pain. He raised his other hand and set about putting things back where they belonged, which also caused him no pain.\n\nScared and completely rattled by this new turn of events, he headed for the door. At the last second, he realized that he was still unzipped and flapping in the breeze, as it were. He quickly fixed this little problem and made for the car. It wouldn't do to be arrested as some sort of flasher or something. What would Linda think?\n\nPaul guided the car through the maze of pedestrians, other cars, and traffic signals and back onto the freeway. Every now and again, he glanced at his hand as though it might fall back into a state of disarray. It stayed in its current condition, however, and Paul felt his spirits lift briefly. Then his phone rang again.\n\nIt was Matt's ringtone and he ignored it. He couldn't seem to make Matt understand him anyway, so what was the point?\n\nHis mind returned to his previous question. If he had died, why, then, were there no stains on his clothes? Surely, the people at the morgue had not taken the time to clean his clothes before putting them in that bag. They hadn't even gotten around to autopsying him. So, perhaps he had never been dead at all.\n\nAnd why couldn't anyone understand him? He longed to talk to Linda, to explain exactly what had happened and why he was on his way to her. If he hadn't gone to see Matt, he might have been able to blame poor cell phone service. Still, he had spoken to Matt in person and he couldn't understand a word he said. It all sounded perfectly normal to Paul, but others seemed unable to make out his words.\n\nThen he hit on a brilliant idea. Without looking in the mirror, he took the next exit and steered the car toward a shopping center parking lot. He parked at the farthest side of it, away from everyone else. Then he pulled his cell phone out of his pocket.\n\nHis phone had digital voice recording applications, though he had never used them. Smiling, he turned on the radio and was greeted with nothing but static. With a frown, he moved to change stations but when his finger struck the SEEK button, to his horror, the fingernail flew off his index finger.\n\nEyes bulging, he stared at his finger, drawing it slowly closer to his eyes as he did so. The nail had flown completely off; not just part of the nail but all of it. There was no blood.\n\nHe recovered from this realization slowly, eyes snapping back to the radio as a religious station blared threats of hell at him. \"You have no idea,\" he chuckled to himself.\n\nCarefully, so as not to repeat the nail-loss incident, he started the voice recording. He let it run for a count of ten, and then stopped the recording. He lowered the volume and hit the playback on his phone. The regurgitated ramblings of some preacher or other assaulted his ears. Annoying, but exactly the same as he had heard it the first time.\n\nNext, he hit the record button and cleared his throat. \"Testing one, two, three. Testing, testing, and testing some more.\" He switched it off.\n\nHe took a moment to steel himself against what he might hear next, then shut his eyes and hit the playback.\n\n\"Aaaaarrrr! Garrrrg! Uh-uh-uh-uuuuuhhhhh!\"\n\nPaul dropped the phone and nearly screamed. That was what everyone else heard when he spoke? He placed one hand on his chest and shut his eyes, trying to gather enough courage that he could play the recording again, just to be sure.\n\nThe same. It was just the same. Paul started to cry but no tears would come. He wanted to slam his fist against the dashboard, but he was afraid his hand might fly off or something.\n\nHe began to yell, gesturing wildly as he mocked fate. \"So that's it then? I stepped in some goo and then I may or may not have died. My penis doesn't work. Parts of me keep falling off. And people can't even understand me when I talk!\"\n\nSomething outside the car window caught his sight and he looked over to see a middle-aged woman staring at him. It dawned on him then how it must have looked: Some crazy guy, sitting alone in the car, ranting and raving at no one in particular. He put the car back into gear and drove off before she could call the cops or something.\n\n\"What am I going to do?\" he kept asking himself. \"What in the hell am I going to do?\"\n\nHe drove on toward California, sitting as still as possible lest he lose any more parts. He had another four hundred miles before he would need gas again and he intended to put that four hundred miles behind him as quickly as he could.\n\n## CHAPTER FOUR\n\nPaul continued to make his way toward California, reasoning with himself all the way. He argued first that he couldn't have died because he hadn't soiled himself. Okay, so his penis didn't work but that could be explained away, too.\n\n\"Maybe I got some sort of infection,\" he mused. \"That sounds reasonable. I got an infection and I can't pee. And maybe that infection is what's causing my hand to break and my fingernail to fall off. Sure! There's a perfectly logical explanation for all of this.\"\n\nHe chuckled but the sound, even to his ears, was offensive. The only good thing was that the police weren't after him. Apparently, his best friend....his best friend...\n\n...Matt! His best friend's name was Matt. And Matt hadn't called the police on him.\n\nHow could he possibly forget his best friend's name?\n\nHis phone rang again, singing out with Linda's ring tone. It was about time to get gas again anyway, so he pulled off at the exit and slipped the car into another gas mall lot.\n\n\"Hello, baby,\" he said, certain that she wouldn't understand him, but wanting to try anyway.\n\n\"Aw, my poor baby!\" she cooed. \"Still not feeling any better? I'm so sorry.\" She paused for a moment, in the hopes that he could get some sort of message across, apparently. When he didn't even try, she pressed on. \"I just called to see how you were feeling. No better, I hear. Well, you just rest and take your meds. I'll call again in the morning to check in on you. I love you, baby, and I hope you feel better soon.\"\n\n\"I love you, too, Linda. You have no idea.\" But all that came out was a series of groans and growls, he knew.\n\nThe call ended and Paul sat staring straight ahead. The sun had begun to set and this time it was the west's turn to put on a coat of many colors.\n\nSomeone knocked on the window so suddenly that Paul jumped and momentarily juggled the phone. \"Hey buddy! You gonna gas that thing or what?\" The guy bent low to peer in through the window, and then his face fell. The man suddenly paled, held out both hands, and backed away. \"I'm so sorry. I didn't know. Sorry.\"\n\nPaul watched as the man got into his car and peeled out. \"What the hell \u2013 ?\" Paul looked in the visor mirror and hollered. There were blotches all over his face now and the left side of his lip had dropped to the point where several teeth showed through the gap. He choked back a scream and looked around frantically, trying to find some way of concealing his face. Several ideas \u2013 such as wearing an overly large hoody \u2013 occurred to him but every idea he had required him to get out of the car.\n\nWith no other avenue open to him, he got quickly out of the car and swiped his card. Several people seemed to notice him, but looked away of their own accord. For now, he would depend on the fact that most people would fear some kind of infection and would stay away from him. That would work for a while, or so he hoped.\n\nThere were several things he had to take care of while he was stopped, but he would have to wait for total darkness to get out of the car. A sudden pain gripped his stomach, doubling him over and radiating to his head where it settle in as a dull throb.\n\n\"Of course! I'm hungry.\" Paul laughed, a genuinely good-natured sound. \"I can't even remember the last time I ate. Or slept. Christ! And no wonder I haven't peed. I haven't had anything to drink in days.\"\n\nHe almost pushed open the car door and got out, but thought better of it before his hand even hit the handle. He would get food and drink later. He had waited this long; a little longer wouldn't matter. Besides, there was something he had to do right now.\n\nHe pulled out his cell phone again and clicked in to his email. He addressed the email to Linda and began to type on the tiny keyboard:\n\nMy darling Linda:\n\n> No, this isn't what you think. LOL It's not a dear Linda letter. But I have something important to tell you and I wanted to tell you before I got there. Right now, I'm really on my way to you. The last two times you've called me, I was in the car, driving to you. From the moment I laid eyes on you, I've loved you with all my heart. I know we agreed that I wouldn't join you until my ten years with the city were up and I could quit and still get my pension. I know I have only two weeks left to go. But something happened at work the other day and it changed me forever. I can't talk about it right now and I won't bore you with it. But I had to come to you right away.\n\n> I hate to be cryptic about it all. Fact is, I might not make it there at all. Strike that. I WILL make it there if it kills me. I love you more than life itself and nothing on this earth is going to keep me from you. I'll be there in five days and I have something important to ask you when I get there. Please, don't ever lose faith in me.\n\n> All my love always,\n\nYour Paul\n\nHe sat back and read over the email, wishing for all the world that he were some sort of poet. Linda was the English major and her words flowed like sweet nectar from a flower....\n\n\"Hey! That was downright poetic, wasn't it?\" He chuckled then, and began to re-read the email.\n\nHe had no other choice. He had to let her know what was going on. Even though his own sin had been one of omission, he still felt like he was lying to her. He hated to lie to Linda. It caused him physical pain. Or, it would if he still felt pain.\n\nThe sun was nearly set. Paul clicked SEND and sat back against the seat. Again, his stomach roiled, though it really wasn't in just his stomach but sort of all through his torso. It doubled him over and made him groan, then a dull ache settled into his head and took up permanent residence.\n\nPaul waited. He waited for the sun to set, he waited for the pain in his stomach and head to subside, and he waited for an answer from Linda. An hour later, the sun was completely down and he decided to make his way into the convenience store. The light blazed out through the glass front, illuminating a large portion of the parking lot in front but the eaves cast a long, thin shadow directly in front of the store. He kept to that shadow and tried to avoid other people whenever possible without seeming suspicious.\n\nIs that the right term for me anymore? he wondered. Am I really even human anymore? What am I? What's happened to me?\n\nHe slipped in through the glass door and was greeted with the BING BONG! of the door alarm. It startled him for a moment but he slipped behind the first row of shelves he saw, trying to avoid anyone's gaze without acting completely squirrelly. He grabbed a Ding Dong, a soda, one of the XXL hoodies that hung on a make-shift rack near the beer cooler. He would have to pay and when he did, he would have to face the guy at the counter.\n\nHe saw now that the guy at the counter was watching him, tracking his movements in the convex mirrors that littered the store's ceiling line. Paul straightened and tried to act a little less like a thief and more like a human being. There were no other customers in the store at the moment. That much was in his favor. Screwing up his confidence, he approached the counter, head down, and placed his items next to the register. Best to face it head-on.\n\n\"I apologize for my appearance,\" he muttered, not quite meeting the young man's gaze.\n\nThe young man looked confused at the garbled noises issuing from Paul's mouth, but he offered a smile anyway. To Paul's delight, the guy chuckled. \"It's all right. I've seen worse. Every morning.\" The man dropped his own arm onto the counter with a hard thump. It was a muscular arm right down to the elbow, where it terminated in a mangled lump of flesh. \"Explosion in Velusia.\"\n\n\"Fire,\" Paul offered, stabbing one finger in the direction of his face. \"No money for skin grafts.\"\n\nThe man smiled and rang up Paul's purchases. He swiped Paul's card, proffered the receipt and placed the items in a bag. Just as Paul turned to leave, the man grasped his arm and pulled him around so as to catch his gaze. The man's eyes were piercing, commanding, and the most understanding that Paul had ever seen.\n\n\"Don't ever be ashamed. What you are on the outside doesn't determine what you are on the inside. Hold your head up, look people straight in the eye, and don't back down from life, you hear?\"\n\nPaul nodded and tried to smile around his droopy lip. \"Thank you. So much.\"\n\nHe gathered the bag into his arms and walked toward the door, keeping his head up for as long as he was in the man's sight. Then he lowered his head and bolted for the car.\n\nConvinced that he needed to eat, Paul unwrapped the Ding Dongs as he went, greedily stuffing one into his mouth and chewing frantically. He swallowed hard, felt the searing pain in his stomach, and bent over and puked. A loud \"eeeew!\" came from the kid hanging out of his mom's car window and Paul hurried along, wiping bits of snack cake from his chin.\n\nMaybe it had just been too long since he last ate. Perhaps his system just wasn't ready for the shock of a pure jolt of sugar. Whatever the reason, Paul gave the soda a shot. He hadn't peed in days and he chalked it up to the fact that he hadn't drunk anything in days. He took a tentative sip, which went down cold and sweet. Then he took two more gulps and waited for the inevitable burp.\n\nInstead, he vomited soda and the remaining bits of snack cake all over the outside of the car door. In abject misery, he began to sob then, his body shaking with the effort, soft mewling sounds issuing from his lopsided mouth. Still, no tears came.\n\nHe sat still for a few minutes, trying to regain his composure and letting his stomach calm down to a dull roar. There was still a wringing, aching sort of pain in his gut, as if someone were trying to squeeze the last bits of life from his unused parts. His head ached and, when he looked into the mirror, he noted with horror that there were more blotches than before and his lower lip had slipped even further down, giving him a menacing sort of grimace.\n\nHis mind settled on one immutable fact: He had to get to Linda before he completely fell apart. He started the engine and put the car in gear, resolved to reach her before she lost faith in him. He was deteriorating at an alarming rate and there was no time to waste.\n\nThe GPS barked its familiar \"recalculating\" message at him twice, and then gave him a direction in which to go. Back to the highway, merge, drive straight ahead for another 30 hours. The pain in his gut made itself known once more as he cruised along at a steady pace. He drove for another two hours without incident before his phone signaled an incoming email. He picked up his phone and glanced at the message.\n\nLinda. She had answered his email. He would have to pull over for this one.\n\nA rest stop loomed ahead and he took the exit slowly. It was one of those large affairs, with a mini mart attached and enough parking for hundreds of cars. Paul chose a space far from the buildings and away from the other vehicles. He pulled in and switched off his headlights, actually turning the car off and rolling down the window. He had learned through experience that hot or cold were both the same to him and that the stuffiness of the car with its windows closed seemed to affect him not at all.\n\nHe clicked on the email and watched tensely as it opened. There was a smiley after the signature. That was a good sign.\n\nMy dearest Paul:\n\nWhatever has happened, know that I love you with all my heart and I stand behind you no matter what. I'll be waiting here for you, aching to see you again, my love.\n\nAlways,\n\nYour Linda\n\nPaul smiled, or what passed for a smile these days. She still loved him and trusted him enough not to interrogate him through email. God! What had he ever done to deserve such a wonderful woman? The thought of her was what kept him going, moving on toward something rather than simply praying for a real, permanent death. He knew that if he could just get to Linda, hold her, rest his head in her lap and hear the sound of her voice \u2013 then \u2013 then everything would be all right.\n\nAnother odd thought occurred to him: In all the time since this whole fiasco started, he hadn't once slept or even rested. Usually, it took four or five mugs of coffee to keep him moving through his day. Upon reflection, he found that he wasn't the least bit tired.\n\n\"How can I not be tired?\" he said to himself softly.\n\nMaybe he was tired and didn't know it. Maybe the adrenaline and fear had propelled him through the past two days. Whatever the cause, he felt sure that he would be better off if he just got a little rest.\n\nHe got out of the car then, stretching his legs even though they weren't stiff just because it seemed like the thing to do. Then, he climbed into the back seat and stretched out, laying one thin arm across his eyes to shut out the light streaming from the huge bulbs overhead.\n\nAfter a few moments, he rolled onto his side, this being his usual position for good sleeping. To no avail, he rolled onto his back. He tried everything he knew: meditation, clearing his mind, counting backward from one hundred. A stone cold nerd to the core, he even tried factoring Fibonacci's numbers as far as he could. No power on earth would make him sleep.\n\nHe was about to abandon all hope and drive back to the freeway when someone jerked open the driver's side door and leapt into the car. Panic ripped through Paul's very core and he sat bolt upright in the back seat just as the man turned over the engine.\n\n\"What the hell are you doing?\" Paul demanded in his new growly-voice.\n\nThe man spun around in his seat in one swift movement, staring daggers through Paul. \"Just shut up and sit still and I won't hurt you.\"\n\nPanic gave way to stark raving terror and then morphed into sheer madness. Paul's mind snapped. He felt it give way even as he lunged over the back of the seat, one hand locking into the man's hair and pulling his head to the side as his jaw opened impossibly wide. He drove his teeth into the man's neck with every ounce of force he had left in him.\n\n## CHAPTER FIVE\n\nLinda had been puzzled by Paul's email. It sounded so ominous, so frighteningly vague that she had stared at it for a long time before answering. She had decided to be supportive in the end, if for no other reason than that she trusted Paul and he deserved that trust. But now, unanswered questions nagged at her and twisted her stomach into knots.\n\nPaul was the most conscientious driver she had ever met. He had never even had a parking ticket and he would never, for any reason, text or talk on the phone while he was driving. She knew that he would pull off to the side of the road to read every email, answer every call. It would impede his progress if she kept bugging him. The only other person who might have the answers she sought was Matt, and she fully intended to have those answers.\n\nShe had to scroll down to find Matt's cell number. He wasn't exactly her favorite person. He was Paul's best friend and for that reason alone, he had made it onto her speed dial list. If Paul weren't in the picture, she wouldn't hang out with Matt, but she wouldn't snub him either. Matt was just sort of there, like a loyal spaniel or your stupid little brother. You had to love him. Had to.\n\nIt took four rings for Matt to utter his usual answer. \"Say words.\"\n\n\"Matt? It's Linda.\" She heard the TV in the background go silent and she could picture Matt sitting up and trying to act like he was not stoned.\n\n\"Hey, Linda.\" He laughed nervously. \"What's the haps?\"\n\n\"I got a strange email from Paul a while ago. I was hoping you could help clarify it.\"\n\n\"Uh... shouldn't you ask Matt?\" Now, he sounded really nervous, he knew. It went beyond pot paranoia.\n\n\"I don't want to bother him every few minutes. You know he pulls off the road every time he has a call.\"\n\n\"Sure.\" People had a way of making Matt tell them things he never wanted to say in the first place. Linda could almost hear the wheels in his head spinning as he worried over what to say. \"So, what is it you want to know?\"\n\n\"Paul said he's on the way here, to LA.\"\n\n\"Yea.\"\n\n\"And he said that something horrible happened at work.\" She paused, letting him mull that over. \"What was it that happened, exactly?\"\n\n\"Well...uh...I think maybe Paul should be the one to tell you.\"\n\n\"Did he get fired?\"\n\n\"No.\"\n\n\"Did he quit?\"\n\n\"No.\" Matt felt sweat break out on his brow. He swallowed hard.\n\n\"Did he get in a fight? Or get injured? Is he sick?\"\n\n\"Not any of those, no. I can't tell you, Linda. I can't. It's Paul's place to tell you when he gets there. That's why he's coming. He just wants to see you...\" He stopped right there, just shy of saying, \"one last time.\"\n\n\"See me...why?\" She wondered if Paul was going to break up with her. What if he was seeing someone else? What if he decided he didn't want to live in Los Angeles after all? \"Is he breaking up with me?\"\n\n\"No way.\" Matt wondered why women always went there first. Linda was not an insecure woman and Paul had never given her reason to doubt him or herself. And yet, her first assumption was that Paul had met someone else and so he was dumping her.\n\n\"Is he seeing somebody else? Did he change his mind about coming out here to live?\" She thought she might cry. Her hands twisted at her shirt the same way her guts were twisting at her.\n\n\"Linda, honest to God. Paul loves you so much. And he's not ever gonna break up with you and he wouldn't cheat on you if somebody put a gun to his head. Why would he drive all the way to California to tell you he didn't want to come to California?\"\n\n\"Then tell me what the hell's going on!\" she snapped, tears filling the corners of her eyes. \"Tell me before I go stark raving mad!\"\n\n\"I'm sorry, Linda. I'm so sorry. I just can't. Paul will tell you when he gets there, but I can't tell you anything. Just trust me when I say that this is not entirely a bad thing. He'll fix everything...\"\n\n\"Not 'entirely'? What's that supposed to mean?\" She was screaming now and she knew it. But she just couldn't stop. Hysteria had taken over and she felt her brain melting with the heat of it, making her more frantic by the moment. \"And what's he going to fix? Matt?\"\n\nMatt hung up. She knew he was avoiding her. He couldn't take the pressure of keeping Paul's secret and of lying to her, or the chance of letting anything slip out. So, he simply hung up. She called back almost immediately, but there was no answer. She called six more times in quick succession with the same result. Frustrated, she gave up, not sure if she was angry at Matt, angry at Paul, or just frightened.\n\nWhat the hell am I doing?\n\nWith that thought fresh in his mind, Paul locked his teeth into the man's flesh, pulling back with his head and pushing away with his hands. The man screamed; his hands flailed madly about as he sought escape.\n\nPaul heard flesh tear, felt the hot rush of blood as it spurted out of the man's body and into his own. Then, he was leaning back, chewing, swallowing, devouring that hunk of torn-away flesh. A sense of relief washed over him as he swallowed.\n\nThe man's screams ended in a soft, wet gurgle as he died, and Paul leaned forward to rip another strip of meat from the guy's shoulder, than another from his neck. He was drenched in blood now but as he ate, the pain in his head and gut subsided. The taste was not unpleasant, the texture not nearly as disturbing as he might have thought.\n\nHalf the man's neck and shoulder were gone now. Over the teeth and past the gums, Paul thought, leaning back. He swallowed that last mouthful and licked his lips. A sort of peace came over him for a moment, his screaming gut finally quiet and his mind clear. He felt stronger somehow, more alive.\n\nThen his eyes fell on the man with the gaping neck and he began to scream. He screamed and screamed, his mind ripping apart at the thought of what he had just done. The smell of blood had filled the air and, had he been able to smell properly, it would have made him vomit. Indeed, he tried to make himself vomit, to give up those chunks of human flesh which he had greedily ripped from the man.\n\nStricken, he looked crazily about for signs that anyone had heard his screams, had seen what happened. There didn't seem to be anyone else around, so maybe he'd gotten away with it. He wasn't even sure he wanted to get away with it. That one act had sent him careening over a line that no one but a monster would cross.\n\nWas that what he was now? A monster? Had he no humanity left in him?\n\nHe thought for a moment. He had just committed a heinous act, had been in real danger of being found out. He should be scared \u2013 terrified, in fact \u2013 his heart racing, his lungs heaving for air. He put his hand over his heart and felt no tell-tale pounding. There was no rapid rise and fall of his chest.\n\nIn desperation, he placed his finger over his wrist, his neck, everywhere he could think of to find a pulse that just wasn't there. His heart didn't beat; he didn't breathe.\n\n\"Oh my God,\" he muttered, his gaze falling on the dead body before him. \"I'm dead. Not dead...undead. I'm an undead, flesh-eating zombie monster!\"\n\nHe rested his head on the window and began to weep. There were still no tears but his body, wracked with sobs, convulsed against the door. He didn't know what to do next. How could he get rid of the body? Should he even keep driving? Surely, he couldn't present Linda with a ring and an undead zombie parody of himself. Everything had changed and he didn't know how to adjust.\n\nVery suddenly, he felt like everything he had tried to do was just an exercise in futility. Linda certainly couldn't love him this way. He wasn't even sure he wanted her to.  He wasn't sure he wanted to live. Not that he was really alive...\n\nHe shook his head to clear it. How to get rid of the body? Maybe the best thing now was just to drive straight to the nearest police station and turn himself in for the undead flesh-eater he was. They couldn't possibly blame him. After all, he hadn't done this to himself. And since becoming a monster hadn't been his doing, they couldn't possibly blame him for eating, for trying to survive. Could they?\n\nHe thought to himself that none of this had been his doing and that he had no other choice but to survive. He had to survive and get to Linda. Even if she didn't want him, he had to give her the option. The dead guy in the front seat had tried to rob him \u2013 he had defended himself.\n\nWith his mind made up to keep driving toward Linda at all costs, he slipped into the passenger side of the front seat. He would conceal the body until he reached Linda, then base all further decisions on her reaction to the situation. Yes, that was it! He would let Linda decide it all. All he had to do was get to her.\n\nHe hefted the body over the back of the seat and let it drop onto the floor of the car with a sickening thud. The trunk's floor cover was removable and so he pushed the button that unlocked the trunk latch. As quickly as he could, he retrieved the floor cover and threw it over the dead hunk of meat in the back seat. Satisfied that he had done the best job he could of concealing the body, he climbed back into the driver's seat...and that's when he saw it.\n\nHe caught a glimpse of himself in the rear view mirror as he slid into the seat. Leaning forward, he stared into a face that was free from those black and green blotches. His lips were full and even again, his eyes even seemed brighter. His heart still didn't beat and he wasn't breathing, but he looked more like a living human again.\n\nPaul tried on a smile, less rictus-like and more endearing now. Slowly, he nodded to himself and chuckled.\n\n\"So that's it! I eat live flesh and I become more alive.\"\n\nA sudden fascinating thought occurred to him just then: If he had eaten a little flesh and had become more alive, more whole, would it not follow that eating more flesh would make him more alive?\n\nHe felt like a scientist, studying some weird biological phenomena, at least that's what he told himself so that he could climb into the back seat and eat more of the bastard who had tried to steal his car. He ate most of the man's right hand and forearm, and then felt for a pulse. Nothing.\n\nDisappointed, he consoled himself with the fact that at least he appeared more normal now, if more than a little blood spattered, and he felt stronger. He started the car and backed out of the space. As long as no one found the body in his back seat, he would be fine. The man had been less than a stellar example of humanity, so maybe he had done the world a service. Maybe he had even saved a life, killing the bastard before he could kill someone else. And that is how Paul Tremblay, son of Marge and Joe Tremblay, soon-to-be fianc\u00e9 of Linda and best friend of Matt managed to cope with his current condition. Zombie-ism had been forced upon him and, with that new state of being, had come the ability to prevent crime and save lives by eating from the lower belly of society.  Amen!\n\n## CHAPTER SIX\n\nIt wasn't until the warning light came on and the beep sounded that Paul realized he had a real problem. He was almost out of gas and in his current state \u2013 covered with blood and smelling of death \u2013 he would have a hard time getting the gas he needed. People might be able to dismiss a bit of black-and-green blotchiness, but they sure weren't going to understand a cannibalistic murder.\n\nThe upcoming exit sported several gas stations, both large and small. Paul took the exit with the idea of dealing with his problems one at a time. First, he would clean himself up as much as possible, and then he would get his gas. He chose a small, independent station that looked as though it would have fewer customers at that hour.\n\nHe eased into the lot, parking in the shadows at first in order to assess his condition. Since eating a bit, he felt sharper, more able to deal with things than he had over the past few hours. Though the light was bad, one look in the rear view told him that he needed a shower. He might be able to remove his hoodie and eliminate that whole mess, but his face and hands were stained red with blood \u2013 and other things \u2013 and he needed a wash.\n\nPaul and Matt had a favorite place to eat lunch. It was a little barbecue truck that generally parked off the intersection of 41st and 3rd. A wipe came with every meal and, since they always cleaned up in the station bathroom afterward, they tossed the wipes into the glove box for future use. To that end, Paul rooted around blindly, finally laying his hands on a cluster of them, secreted behind the breath mints and under the registration. He grabbed three and tore them open, using the dim light and the rear view to clean the blood and guts from his face.\n\nOnce he was satisfied that he was as presentable as a flesh-eater could be, he tossed the wipes into the back seat, threw the hoodie on top of them, and started the car again. He pulled up at the far side of the pumps and shut down the engine, then climbed out to fill the car.\n\nNo one was about at that hour, though the attendant inside gave him a cursory glance. With his hair all slicked back and his face freshly scrubbed, Paul looked respectable enough not to warrant a second look. He managed to fill the tank without interference and guide the car back to the highway.\n\nHe had gone perhaps ten miles when the phone rang. The sound of it scared him and the car lurched as he started. Then he picked the phone up and grimaced at it. It shone brightly, announcing the name of the caller. Matt.\n\n\"Who the hell is Matt?\" he asked himself. \"Matt.\" He frowned heavily and shook his head. \"Matt.\"\n\nNo matter how many times he repeated it, the name meant nothing. There was Linda. His parents. But Matt? Try as he might, he could find no memory associated with Matt.\n\nOn a whim, he flipped through his contacts, finally landing on Matt's name. It showed his phone number \u2013 which he didn't recognize \u2013 and his address \u2013 which matched his own. \"Ah, Matt!\" he sighed with a smile.\n\nHow could he forget his best friend and roommate? He'd only known the guy for his entire life! And now he couldn't even remember who he was without prompting?\n\nPaul clicked the green button on the screen and answered the phone. \"Matt?\"\n\n\"Oh good! You're still alive.\" Matt chuckled a bit and pressed on. \"Listen, I know you can't talk...at least I can't understand you when you talk...but that's okay. I just wanted to say I'm sorry if I was too rough on you earlier. I know how important Linda is to you. And I want you to know I didn't call the police or anything. Just take care of the car, you know? We still owe on it and I would kind of like to keep it nice at least until we pay it off.\"\n\nPaul thought of the torn-out trunk carpet, the dead body in the back seat, and the blood covering most of the interior. \"OK,\" he said, forming the words as carefully as he could.\n\n\"Cool. Okay. I just wanted you to know we were cool and all. You take care, ok? And if you need anything...like...well...anything at all...you let me know.\"\n\nThe call ended and Paul frowned. \"Sure, buddy,\" he chuckled to himself. \"I'll take care. And I could use some help. Could you get the dead guy out of the back seat and maybe find a way to make me be NOT undead? As in, alive? As in, not eating people anymore? 'k, thanx bye!\"\n\nPaul had been driving straight ahead for two hours. Blessedly, he didn't require sleep, food (at least not beyond the occasional taste of the corpse back there) and he didn't need to stop to relieve himself. He made good time.\n\nMost of the time, his eyes were fixed, unblinking, on the windshield. His mind was a complete blank. It wasn't that he had repressed any of the last two days, but that his mind quite simply would not contain thought unless it was forced. It seemed like years since he had left New York; years more until he would reach LA.\n\n\"Keep left!\" the electronic voice boomed at him through the silence.\n\nPaul cried out and for a heart-stopping moment, he thought that the dead guy who had lain on the floor for the last 300 miles had somehow been revived; that somehow, he, Paul, had turned the poor bastard into a zombie.\n\nThen his eyes flashed to the dash and realized that the GPS was the owner of that voice, as cold and dead as the guy in the back seat. He did what the GPS advised. He kept right. But for the life of him, he couldn't remember why. Where was he going again? Why was he going there?\n\nHe tried to force himself to remember. It was getting harder to recall things now, even simple things like his friends' names and how to check his email. He knew that one bite of his human snack-pack would fix it. But he had made himself a promise some time ago, and that was to never again taste of human flesh. He was saving the corpse for that one all-important moment when his pseudo-self was face-to-face with . . . with . . .\n\nHe began to cry then. Still no tears, but his body bucked and shook and he made a pitiable noise. He was on his way to somewhere, to see his girlfriend, whose name he couldn't remember. He made a mental note and hoped that it would stick: When he next needed gas, he would check his phone and remember her name. Then he would write it on one of the sticky notes that . . . that . . . what's his name . . . always kept in the car for leaving messages on people's doors.\n\nPaul drove on. The fact that this had happened at all was distressing enough but coupled with the fact that he was forgetting simple things drove him almost over the brink. Why had this happened to him? He was a good person. He was smart. He did favors for his friends without being asked. He never once called in sick to work when he wasn't sick. And while those other city-fed fat-cats relied on the fact that they couldn't get fired and wasted the day smoking and shooting the breeze, he carried out his duties faithfully. He'd had a plan from the time he was sixteen; he knew how his life should go. He had never once deviated from that plan; had worked arduously toward its completion. And now, a mere two weeks from the realization of his life's dreams, this had happened.\n\nHis mind blanked again. He took a curve in the highway too fast, made the body in the back seat shift, and scared himself. He checked the mirror and noticed that several blotches had broken out on his face. One check of his hands revealed even more.\n\n\"I won't do it,\" he snapped to no one but himself. \"I won't eat human flesh again. I don't care how bad it gets, I won't do it.\"\n\nAnd with that total conviction set firmly in his mind, he drove on.\n\nA huge power line joined up with the road and followed it for a while. It stayed with him for nearly twenty miles, then branched off and disappeared. Paul's phone rang and he snatched at it, looking at the name on the screen.\n\nLinda.\n\nAnd the battery was at twenty-one percent.\n\n\"Hello?\" he barked.\n\nThere was a moment of silence and then, \"I know you can't talk. Where are you? Are you almost here?\"\n\nDid she even know how far it was from New York to LA? \"I've got a long way to go, baby. I'm driving as fast as I can.\"\n\nLinda sighed. \"Email me when you can. Let me know where you are. If you're going to arrive during the day, I'll stay home from work. But I need to know when you'll be here.\" There was a pause. It cut like a dagger. \"Bye, Paul.\"\n\nLinda. Paul dropped the phone onto the seat and groaned. \"No, I love you? No, I can't wait to see you, Paul? No, I miss you?\"\n\nIt became painfully obvious to him then that she was losing interest in him. He had pissed her off, made her doubt him with all this craziness. He had to get to her. Fast.\n\nIf only he had taken a plane. Why hadn't he taken a plane?\n\nThe gas warning bleeped at him again and he trained his eyes on the signs ahead. There was a SuperMart at the next exit. He would pull off there and get some gas. It was going to be tricky, though. The sun was up and he knew his car was a mess. The mirror taunted him; told him that he was a mess, too.\n\nHe eased onto the exit ramp, watching the road ahead for signs of the SuperMart. There were signs everywhere, though, and he had trouble finding the one he needed. Then fate stepped in and guided him in the guise of a giant sign that blared red against blue, SuperMart.\n\nThe parking lot was huge. Morning commuters were pulling in and out, getting their coffee, their donuts, their energy drinks. Cars moved so quickly through the process that it was almost like watching a time-lapse video.\n\nThere was parking all around the building. The door opened into the front of it, the delivery and service door opened into the right. Paul pulled toward the left side of the building, the only one with no doors and no glass. He maneuvered the car until he could back into the space, putting the bloody driver's side on the wooded edge of the lot. He needed to gas up, but first he had some cleaning up to do. If he tried to just pull up to the pumps, someone would see all that blood and he would be done for.\n\nOnce he had stepped from the car, his mind felt clearer. He pulled his hair forward, so that it covered his eyes and the edge of his face, trying for that fine line between armed robber and gangsta. His plan was to go into the SuperMart, grab an energy drink and a donut (which he would throw away) and a bottle of window cleaner and some papers towels (which he would use to clean the blood off the car). He had this all mapped out in his head, repeating it to himself as he walked, trying not to let those precious plans slip away from him in a moment of thoughtlessness.\n\nIt was a good plan, as plans went. It came very close to working, too.\n\nPaul bent his head low and slipped into the store, making sure to face away from the camera aimed at the door. He slid to the back of the store where the coolers were, grabbed himself a small energy drink, then paced over two aisles to pick up a bag of donuts. It was easier and quicker, he reasoned, to grab a bag than to fuss with the rack that held the single donuts. Cleaning products were to the right, paper products to the left. He grabbed the window cleaner first, then made for the paper products.\n\nHis hands full now, he marched to the checkout and placed the items on the counter. At the far end of the counter was a rack of hoodies, fifty percent off. He grabbed an XXL and tossed it on the counter with his other items. He kept his head down and coughed every now and again. If the clerk thought he was really sick, he would be more likely to hurry him through the process. Paul dug around in his pocket, produced a twenty, then scooped up his items and headed for the door.\n\nThe cool morning air hit him as he stepped outside, clutching the bag to his chest and making sure not to make eye contact with anyone. Briefly, he stopped on the sidewalk, pulling the hoodie out of the bag and over his head. He strode purposefully down the sidewalk and rounded the corner . . .\n\nThere were two cops and a cruiser sitting on that side of the parking lot. Paul felt instant panic well up inside of him and he ducked back around the corner. He wondered what had tipped them off. Even knowing the blood was there, he couldn't see it from where he stood. How did they see it?\n\nHe managed to peek around the corner again. No sir! They weren't leaving any time soon. If Paul had had a heartbeat, he was sure it would be pounding right now. His phone was in the car. The GPS that knew where he was going was in the car. He needed that car!\n\nThere were only two choices, he knew. One, he could look for a car that had been left running, slip into it and hope that he could drive casually out of the lot before the owner saw him. He looked around. All the cars seemed to be either occupied or shut up and locked. He could stand there and wait for one of the drivers to get out and leave their car but every second he stood there, he risked being found out.\n\nAnd that brought him to choice number two. He could run. Running, the international sign of guilt. But how far could he get if he just nonchalantly walked away? Twenty yards? Maybe fifty?\n\nThere was no other recourse left to him. He had to get away from the SuperMart without the cops \u2013 or anybody else \u2013 seeing him do it. Quickly, he dumped the tell-tale bag into the trash, turned on his heels, and walked away from the place. And as he turned, the lead policeman came around the corner. He turned that corner just in time to see Paul's back walking away.\n\n## CHAPTER SEVEN\n\nThis town was a small town, suffering from all the same pains that other small towns suffered. Its death or growth rested on the shoulders of its industry. At present, this town's industry was farming and it was thriving, thank you very much. The nearby river and rich soil that surrounded it meant that crops flourished. Calm winters, moist summers, and an employee pool of thousands meant that a farmer would have to be awfully stupid to fail.\n\nAnd like most small towns in their prepubescent years, this one suffered from a sort of reverse big bang. Its center had started to die, the old original buildings marked with decay, the crime following the decay like a flock of vultures. And as the center died, the town grew from the inside out.  New businesses chose prime locations on the outskirts of town. Old businesses relocated to the higher-traffic, newer parts of the town, each according to their stability and ability. The further out into the country you got, to a point, the richer and grander the homes were. If you were a CEO or a doctor, you lived in one of those neatly planned and carefully staged subdivisions. Old school lived on the hill. And if you were the single mom welfare recipient, you lived at the center of town, where grand old houses had been turned into festering eyesores.\n\nThe growth, death and re-growth of the town, if seen from space in a time-lapsed video, would look like the blooming of a flower. What would appear in that video in a matter of seconds had taken over a hundred years, but it had happened nonetheless. People, whether happy about the way things were going or not, all had opinions, and those opinions kept them separate from one another, kept them struggling.\n\nPaul didn't know anything about the little town into which he had escaped. He didn't even know its name. All Paul knew was that he had to get his stuff out of his car, and that the police had the car. By now, they knew who it was they were looking for. Paul only knew two things: First, that he was on his way to LA to find Linda, and that he could not let the police catch him before he did.\n\nHe walked away from the SuperMart, keeping to the shadows and small backstreets of the little town. Whenever possible, he cut through alleys and skulked behind houses. All he wanted to do at that point was to get to some place, any place, where he felt safe. But safe was a relative term.\n\nHe was a murderer now and the whole world knew it. How long it would take before the police contacted Matt and Linda, he had no idea, but very soon, Linda would think him a monster. Was there really any point in carrying on with this silly plan? Wasn't it better to just turn himself in, try to explain, and take his licks? Surely they would eventually come to see that all of it had been beyond his control. He was no longer human in any way that mattered and so could not be blamed for what he had done. No, he couldn't have Linda thinking that the love of her life was a killer. He had to get to her and tell her his side.\n\nAnd so Paul developed his Mantra. Things were slipping from his mind so quickly that he could barely carry a sentence from beginning to end without forgetting it. And so he marched on through the town, chanting.\n\nI have to get to Linda in LA. I have to get to Linda in LA.\n\nHe considered himself the luckiest duck in town, since he had managed to travel that far without being seen. Surely, the cops had put out an APB on him already. How long could it take for them to peruse his phone and find his name, address, every single bit of personal information there was to know about him?\n\nAt that thin line which divided the industrial outskirts of town from the inner-town slums, Paul found an old and long-forgotten club. It harkened back to the days when the good folks of the town hadn't so much minded having a \"gentleman's club\" around and when the back of said club was operated by the local madam. The neon sign was all but gone. Only the high-kicking legs of the animated stripper remained. The door was chained, though that chain looked like it had been strained to its limits on more than one occasion.\n\nA layer of grime coated the door handle as Paul put his hand on it. The handle itself was warm from the sunshine that bathed it. He gave it a pull and the door opened a good eighteen inches. Paul ducked down and slipped under the chain and through the opening. He was saved.\n\nAll was darkness inside. The plywood that had been studiously nailed over the window blocked out any form of light that might have penetrated glass. There was no electricity anymore, so none of the lights would work, even if the bulbs had managed to survive all those years. Paul tried to get his eyes to adjust to the darkness, but it was complete. There was not a bit of light to see by. He was blind.\n\nHe felt his way slowly across the room. If he barked one foot or banged one shin, he might well lose that part of himself forever. It was something that he didn't want to think about.\n\nHe found his way across the room and to the bar. By now, his hand was well and goodly coated with a film of dust. Everything had dust on it in there and he was sure that if he had some light, he would see that the air was thick with the dust he'd stirred up just by moving slowly through the room.\n\nHis hands traveled the length of the bar, first on top, then on the first shelf and then the second. He was looking for anything he might use to create light. A flashlight. Matches. A barbecue lighter. Anything.\n\nHe knocked a glass to the floor and it shattered. He winced from the sound and stood still for a moment, waiting to see if there was anybody else in there who might come running at the sound. When no one did, he began his search anew, hoping that luck would hang out with him for just a little longer.\n\nHis hands closed on something big and rectangular. It felt like cardboard but the humidity and dust gave it kind of a spongy feel. He pulled it off the bottom shelf and ran his fingers carefully over it, feeling as they rounded the long side and rubbed on something that felt like sandpaper.\n\nMatches. He had found a big box of matches. No doubt, some bartender in years past had used them to make flaming drinks. Now, they were Paul's Godsend.\n\nHe opened the box carefully so as not to spill the contents, which rattled around the box as if they had a life of their own. He removed a match and closed the box, hoping against hope that the humidity hadn't destroyed any chance he had of lighting it.\n\nHe placed his hands just so, being careful. When he finally had worked up the nerve to try, he dragged the end of the matchstick along the sandpapered side and almost cheered with joy when it lit.\n\nThe room around him was cast into a tight circle of light. He could see the bar, the stools, a few of the tables that scattered across the floor like drunken dancers. What he needed now was the office. He wanted to find a pen and some paper. First he would write down all the things he had to remember, like Linda's name, her address, Matt's phone number, where he was headed. Then, he would write a letter to the people he cared about to serve as a last will and testament should he finally expire.\n\nExpire.\n\nLike rotted meat. That's all he was now, wasn't it? Rotted meat that, for some reason, was still able to ambulate.\n\nHe moved off toward the little hallway to his left, hoping that was the way to the office. If he could just find pen and paper, all his problems would be solved.\n\nLinda slid in the door and dropped her purse and keys on the table in the little space she liked to call the foyer. It was in no way a real foyer, just a bit of tile floor adrift in the sea of carpet. She turned the latch on the door and took a quick peek in the mirror over the table. She was looking haggard, she knew. Perhaps she should invest a little money and go to the salon before Paul arrived. They hadn't seen each other in a while. What if he found her less desirable than he remembered?\n\nWith a heavy sigh, she wandered into the kitchen and opened the fridge door. There were two containers of left-over takeout food in there: One definitely gone, one questionable at best. She shut the door.\n\nThe day had been long and hard and she was so glad to be off work. But there was nothing in the apartment to eat, nothing worth doing. She snagged her phone from her purse and dialed the closest pizza joint, ordered a small pepperoni, then clicked through to her email. She had sent Paul an email at lunch time, reminding him that she needed to know where he was, when he would arrive. There was still no answer.\n\nWandering to the sofa, she dropped onto it with a soft Oof! She tossed the phone onto the sofa cushion and hugged her legs then, resting her chin on her knees. It wasn't like Paul not to answer her emails. She could understand him not wanting to talk on the phone, what with his being sick and all. But to not answer emails? That was so not Paul.\n\nShe hadn't liked Paul when she first met him. He had been a tad too shy, too unsure of himself and nervous. She generally liked a guy with more hutzpah. The more dominant males were definitely her type. But he had persisted and she had acquiesced, if only to get rid of him. He had asked her out the very first time they met, but she had shut him down. Then, he had had the gall to send her one red rose every hour until she agreed to go out with him. In the end, they had gone to Jeeters and she had a cool six dozen red roses in her apartment. Paul was persistent.\n\nThe memory of it all made her smile, then giggle. Her green eyes sparkled and danced when she laughed. She knew that because Paul had said so about a hundred times.\n\nPaul was a good man. He was thoughtful, neat, hard-working, loving, romantic, funny, smart, and loyal. Paul never broke promises and he never did anything to hurt anyone if he could help it. More than anything, she loved that one thing. They had never even had a fight.\n\nShe needed to touch up her roots, she remembered. She had let them go too long. Paul didn't care, of course. But she did.\n\nBy their second date, Linda had realized that Paul was THE ONE. He was MR. RIGHT, MR. PERFECT, HER BETTER HALF, HER SOUL MATE. By their fifth date, they were deciding their futures together; planning just how their lives together would be.\n\n\"Dammit!\" she spat and snatched the phone off the cushion. She dialed Paul's number and listened to it ring and ring, her smile running away from her. Then the voice mail kicked in and she sighed. \"Hi, baby. I know you can't call. But I sent two emails and in case they didn't go through, I need to know you're okay. I need to know when you'll arrive so I can be here. Anyway . . . email me back. Or call and tap on the phone or something. Anything. Just let me know you're okay. Love you. Bye.\"\n\nShutting off the phone, she dropped her feet to the floor and sank deeper into the sofa cushions. He wasn't ignoring her. He would never do that. Something was wrong, she just knew it. And she wasn't going to feel right until she knew that Paul was okay.\n\nPaul sat behind the dust-coated desk of the former club owner, the pencil hovering over the paper as if waiting for permission to write. He had found a flashlight in one of the desk drawers and miracle of miracles, the thing had still worked. That pink bunny sure didn't lie.\n\nFinally, he plunged ahead. He made note of Linda's full name, her phone number at the apartment and her cell phone number. Then, he wrote down her address (that one took a lot more thought) and made a small note below that so that he would remember where she worked. He continued writing, jotting down Matt's name and number, their shared address, and the license plate number of their car, should he ever need it.\n\nAll of this took him over an hour and he still wasn't sure that he had remembered all the details correctly. For example, the last four digits of Linda's phone number might have been 0323 or 0232. He wasn't completely sure. Then he set about writing a farewell note to his loved ones. He began to cry halfway through, thinking of all that had happened to him, all that he'd done, and all the things he would never get to do again. When he was done, he sat back in the filthy chair and sighed. After staring at the paper for a long time, he folded it carefully and put it into his pocket.\n\nHe had no idea where to go from here. He couldn't just waltz into the police department and demand his car back. He was wanted for murder, after all. Aside from that, he looked like an extra from a Michael Jackson video.\n\nThe only thing to do was press on. He had to find another phone and another car. His face was severely blotched now and his stomach had begun to roil and cramp.\n\n\"You're not a thief,\" he said to the dark. \"No, I'm broke.\"\n\n## CHAPTER EIGHT\n\nMatt was kicked back on the couch, sucking down his third Pepsi and munching a bag of chips. In the ashtray next to him sat two roaches. The apartment smelled of pot and day-old pizza. On the TV: Some stupid show about Vikings. Matt didn't really like Vikings. He was just too lazy to reach for the remote.\n\nThere was a vicious knocking at his door then and it jolted him off the couch, making him spill his chips and nearly spill his Pepsi. He forced himself to sit up, though slowly, and he dumped the two roaches under the sofa.\n\n\"Damn,\" he muttered to himself, then, louder, \"Coming! I'm coming!\"\n\nHis large feet shuffled across the worn carpet as Matt made his way to the door. The chain was on, the lock was on. He stared at them for a few seconds while he decided which on the best procedure for opening the door. Finally, he unbolted the locks and opened it a few inches, peering out with one heavily-lidded eye.\n\n\"Detective Thomas Milligan of the NYPD.\" A badge shone through the crack in the door. \"Are you Matt Cassidy?\"\n\n\"Shit,\" Matt growled between clenched teeth. He cleared his throat, rubbed his face. \"Yea, that's me. Matt Cassidy.\" Blink.\n\n\"Would you mind opening the door, sir? I'd like to have a word with you.\"\n\nMatt pushed the door closed, ran a reckless hand through his hair and disengaged the chain. When he pulled the door open again, he was smiling. \"How can I help you, officer?\"\n\nClearly, the detective planned to come inside, but Matt managed to edge out the door and pull it shut behind him.\n\n\"You have a roommate? Paul Tremblay?\"\n\nThe detective's eyes were piercing and Matt found himself shying away from them. \"Yea, Paul and I live together. Why? What's wrong with Paul?\"\n\n\"You own a car together? A brand new Toyota Camry, license number YKR 3182?\"\n\n\"I don't know the license number, man, but yea. That's our car. And again I ask . . . what happened?\"\n\n\"Your car was found parked outside a SuperMart in Lebanon, Kansas. It was covered in blood and there was a body in the back seat.\"\n\n\"Body?\" Matt's chin dropped, his voice became deeper, more serious.\n\n\"The body of a man. We're not sure of his identity yet. He's still in the autopsy room. But the thing is, this guy was partially . . . eaten.\" The detective's eyes met Matt's and narrowed.\n\n\"Eaten?\" Matt asked, as if he hadn't heard correctly. What in the hell did Paul do, he wondered? I don't want to get Paul into trouble, but I'm not going down for murder.\n\n\"Mr. Cassidy, where have you been for the last thirty-six hours? If I may ask.\"\n\n\"I went to work yesterday. There's like ten guys who can put me on the job. And this morning I got up around ten, and then I went to the bank to cash my check. Then I went to . . .\" he stopped for a moment, thinking.\n\n\"Sir, when was the last time you saw your car?\"\n\nMatt stared at the detective, trying to figure a way out of all this without getting him and Paul in trouble. \"Umm . . . the car was here three nights ago when I drove home from work in it. Look, detective, dude, I didn't kill nobody.\"\n\n\"I'm not saying you did, sir. But tell me, how did your car get all the way to Kansas and why was there a dead guy in the back seat?\"\n\n\"I dunno, Mister.\" Matt collapsed against the door frame with a heavy sigh. \"Listen, here's the thing: My roommate, Paul, he took the car three nights ago so he could go see his girlfriend.\" That was the honest to God truth, Matt realized, and he pressed on. \"I haven't seen the car, or Paul, since then.\"\n\n\"This girlfriend, would that be Linda Gilchrist? Lives in Los Angeles?\" Detective Milligan was reading from his little notebook. He looked up then to check Matt's face.\n\n\"Yea,\" Matt answered slowly.\n\n\"You should know,\" Milligan began, \"that your friend, Paul's, fingerprints were all over the car. He was covered in blood and, my guess is, when the test results come back, we'll find that the blood is that of the deceased. We found Paul's phone in the car, but we haven't found Paul yet. I'm working with the Lebanon police department on this, since our main suspect is a New York resident. So, if your buddy shows up or happens to call you, you tell him that he needs to come turn himself in. And then you call me. Here's my card.\" He flipped the card out with a flourish, as though he had done so a thousand times.\n\nMatt suddenly realized that his jaw was hanging and he hadn't blinked in about five minutes. He let the gritty surface of his lids scrape over his dry eyes and swallowed. He took the card and tucked it into his pocket, his gaze never leaving Milligan's face.\n\nMilligan turned and left then, without saying a word. Matt watched him go for a moment, then let himself slide down the door until he was sitting on the floor.\n\n\"Christ!\" he groaned. \"Paul, what have you done?\"\n\nPaul sat, curled up in the corner of the sofa, hugging his knees and rocking. Long periods lapsed during which he couldn't remember where he was, how he had gotten there, or what it was that he was supposed to do. All he knew was that it was very important that he do this thing.\n\nHe might have sat there forever, rocking and wondering, had the rat not come along. There were countless rats in the building; they pretty much ran the place. But Paul had been hearing this particular rat for the better part of an hour as it scurried along the shelves and rooted through a pile of debris on the floor next to the potted (dead) plant. Now, however, the rat had become emboldened and had made a bid for the sofa.\n\nThe creature managed to scratch and claw its way up the short arm of the sofa farthest from where Paul sat. It was perched on the sofa back now, glaring at him with beady eyes and cleaning its whiskers. Paul watched it without fear. Once you were (un)dead, what had you left to fear?\n\nThen the rat ran along the back of the sofa, charging straight at Paul. Its intent was unclear but Paul, having worked the sewers for nearly ten years, had a natural aversion to rats. The minute the thing was in his reach, he lashed out with one hand. His original plan was to slap the thing away, make sure it didn't get to his face. But at the last second, his hand swiveled and he grabbed it tightly in his fist.\n\nThe rat squirmed for a moment, obviously stunned at its capture. Paul felt the whiskers working against his hand, sensed the creature's fear as he brought it closer to his face. And then the rat, in a final bid for freedom, bit him.\n\nWhat followed then was the most disgusting and digestively horrid thing Paul had ever experienced. The rat, having filled its mouth with a good chunk of Paul, spat and squirmed and tried to get the taste of him out of its mouth. The feel of its little claws on Paul's hand drove him into revulsive rage and with a movement that was so quick and so unpremeditated that he couldn't have stopped himself if he'd tried, Paul popped the rat into his mouth.\n\nThere was a sickening snap and squish sound as Paul bit down on the rat. Blood filled his mouth and he found within a moment that the taste was not unlike the taste of the car-jacker. Paul chewed once, twice, swallowed the thing's head. And then he smiled.\n\nThe effect was almost instantaneous. At once, Paul's mind cleared, his memories began to return. He felt renewed strength and vigor. And if he'd had enough light and a clean enough mirror, he was certain that he'd find his face devoid of those ruptures and pustules. He smiled again.\n\nIt was all so clear to him now. He needed a new car . . . something nondescript but drivable. He also needed a map or a GPS or some other method of finding Linda. And he needed a source of food . . . or what passed as food for him these days.\n\nPaul knew from experience that if you stole a car in the city, the reaction was immediate. The alarm went off, the police were called, an APB issued within the hour. Even in the absence of an alarm, most people would discover their car gone within two hours' time. But if you stole a car in the middle of the night, out in the country, it might go unnoticed for hours, even days. The odds were better if you stole someone's second car or a beater. If they didn't use it every day, they might not notice it was gone for a week or more. And out in the country, the cops took longer to get to the scene of the crime.\n\nThat was it then. Paul had to go out into the country and steal a car, perhaps an old farm truck or something. It might be morning before the cops arrived and the warrant was issued. But it had to be at night.\n\nHe focused his eyes on the plywood that covered the windows. There was a gap, albeit tiny, at the bottom left corner and light shone through. He would have plenty of time to gather his supplies and head out of town once the sun set.\n\nHe opened the drawers in the desk once more, finding the pen he had used before. He studied each object in the drawers, trying them on for size, hoping to find something else of use. At the back of the left top drawer, he found a stack of sticky notes \u2013 ah, sticky notes, my old friend \u2013 and he stuffed those, along with the pen, into his pocket. He found a pocket knife in the bottom right drawer and he took that too. His first car had been a beater and, when he couldn't afford to fix the dead starter, he had simply bypassed it and hotwired the thing, using the connection and disconnection of the wires as a form of starter. He would need the knife to cut and strip the wires.\n\nPaul was not by nature a thief but his back was against the wall. He was wanted for a murder he couldn't really be faulted for being undead and, desperate. He wasn't sure how this all would end, but he made a mental note to return to this town if it all ended well, and pay back the guy for his car.\n\nPaul fumbled his way to the front door, inching it open and looking outside for just a second. Judging by the position of the sun, he had a few hours before he would be able to leave. That in mind, he set about trapping another rat. He didn't like the idea, but if he was going to travel all the way out to the farm land on foot, he would need some sustenance.\n\nBehind the bar was a small kitchen and it was in that direction that he heard the scuttling sounds of tiny rat feet as they searched for food, tended their young, whatever it was that rats did when left to their own devices. He felt his way to the kitchen and went in, planted himself in the middle of the floor and waited. He sat cross-legged for a little over an hour before a rat finally worked up enough courage to come close to him. When it did, Paul lashed out his hand and grabbed the thing, thrusting it under a large pot on the counter. He would keep it there until right before he left and then he would eat it.\n\nLinda woke early that morning from a dream that scared the hell out of her. In this dream, Paul was falling out of the sky and she had been charged with catching him. But she had been given nothing more than a coffee cup to catch him in. She watched and waited as the small dot that was Paul fell through the air, trying to adjust her position so that she was always directly beneath him. Eyes turned skyward, she watched as the dot grew and became a man, fell faster. She aimed the cup, crying, and tried to figure out how in the world she was going to catch him in such a tiny vessel. All of a sudden, Paul simply melted, turning into a falling, trailing blob of goo which dropped into the cup, splattering Linda's shirt and making her scream.\n\nShe was still screaming when she woke up, her face and chest drenched in sweat and her entire body shaking. She threw back the covers and grabbed for her cell. There were no messages and no emails from Paul. Linda started to cry.\n\nThey had always been so close, so connected. Dreams aside, she knew him well enough to know when something was wrong. She hadn't heard from him for days and, even if he was still sick, that was unusual.\n\nShe had never wanted to be one of those stalker-chicks, checking up on her boyfriend every two hours, tracking him by the GPS, and whatnot. But she was scared now. She had called four times, sent five emails, and there had been not one answer. He could be dead, lying in a ditch somewhere, or at the bottom of a ravine, surrounded by smashed car parts.\n\nSomewhere on that phone, she had Matt's number. Nobody had picked up at the apartment when she had called last night, so she tried to find Matt's cell number. He would know what was going on. He would put her mind at ease.\n\nThe very next to last number on her recently called list was Matt's. She had a few friends, but she never gave her cell phone number out to anyone unless they factored heavily in her life. Matt had called her two months ago to give her clues as to what Paul wanted for his birthday. And that was the only time he had ever called her.\n\nThe phone rang three times before a sleepy voice came on the line with a perfunctory, \"Hello?\" Matt then launched into a coughing fit, which culminated in a heavy moan.\n\n\"Oh, dear! You're not sick too, are you?\"\n\n\"Linda? Oh my God! Linda!\" Matt sounded like a man who had been trekking across the desert and had just been confronted with a priest bearing water.\n\n\"Matt. I'm sorry to call so early . . . wait . . . it's three hours later there. What are you still doing in bed?\"\n\n\"It was a late night last night.\" He sneezed, coughed twice. His voice had that deep husky sort of tone that one might get from a night of debauchery.\n\n\"Listen, I called because I haven't heard from Paul. I've called him and mailed him. But I don't get any answer.\" There was a prolonged silence while Linda tried to get a grip on her rampant emotions. \"I'm scared.\"\n\n\"Linda, listen . . .\" He broke off, trying to find a good clear voice to tell her what he needed her to know. \"Paul has a few problems just now. And I don't mean that like in the bullshit way a guy has of breaking up with a girl. He has real problems. He's in trouble.\"\n\n\"Trouble? Paul?\" She shivered and choked. \"What sort of trouble, Matt?\"\n\nMatt sighed and lit a cigarette. \"Paul would kill me if he knew I was telling you this. Some cops came to see me last night. They said that they found our car at a SuperMart somewhere in CrapSplat Kansas. There was a dead guy in the back seat and the car was covered in blood. They don't know whose blood. Anyway, Paul was nowhere to be found and they're looking for him all over the place.\" He listened as Linda fell apart and decided not to tell her that the body had been partially eaten.\n\nLinda wailed and screamed, her tears washing over her face, soaking the sheets that she had clutched around herself. Cold didn't just seep into her, it slammed into her, nearly making her pass out. \"How . . .?\"\n\n\"Listen, we don't know what happened. But we know Paul. And Paul is no killer. So, we have to assume that somebody attacked Paul . . .\" and he heard her tumble into insanity again. He waited for the return trip. \"Somebody probably attacked him and whoever that dead guy was. And Paul probably ran off. We don't know. We just don't know.\"\n\n\"Matt, what if . . .?\" She couldn't say the rest. As long as those words went unspoken, everything would be all right.\n\n\"Like I said, we have no idea what happened. But Paul's like a brother to me and whatever happened, I'm going to be his friend. I'm going to hope for the best. And I'm damn sure not going to believe that he's dead until they show me a corpse.\"\n\nAnd Linda collapsed once more in a fit of tears, beating the bed, tearing at the covers until she thought they would rip apart like her mind had done. When at last she had recovered enough to speak, she sniffed loudly every few words. \"You're right, of course. Paul would never do anything to hurt anyone. He must have escaped. Someone attacked him and that dead guy, and Paul managed to escape. And as soon as he can get himself sorted out, he'll call one of us. Let me know right away if you hear from him, okay? And I'll do the same.\"\n\n\"Of course. You know I'll call you right away.\" He swallowed hard and tried to think of something reassuring to say to her. \"Hey, this will all get fixed, ya know? But listen, the cops are probably going to pay you a visit, too. Paul was on his way to your place when this happened. If there's any way in hell he can pull it off, he's going to make it there, too. So, watch your back and keep an eye out for Paul.\"\n\n\"Yea, Matt. You too. And thank you.\"\n\n\"For what?\"\n\n\"For being such a good friend to Paul. And for being honest with me. I appreciate that.\"\n\n\"No problem, Linda. Later.\"\n\nShe clicked off the phone and only then did she realize that her tears had all dried. That one fact made her throw herself to the bed and start crying all over again.\n\nMilligan was just hanging up the phone when Starnes came in. He clutched in his hands a small stack of papers and sported a smile.\n\n\"Hey, Milligan, we got the lab reports back from that car they found in Kansas.\" He tossed the stack of faxes on Milligan's desk and dropped into a chair.\n\nMilligan picked up the papers, frowning as he scanned them. \"So, the blood all belongs to our vic and the fingerprints in the blood match our missing car-owner.\"\n\n\"Yep. What do you suppose happened to the guy? Do you think he killed his pal there? Or was there a third guy and this dude just managed to get away from him?\"\n\nMilligan rocked slowly in his chair and thought for a moment. \"Well, by all accounts, this Paul Tremblay was a pretty rock-solid guy. Had a steady job with the city, never late, never missed a day. He pays his bills on time, never had any trouble with the law. Hell, he's never even had a speeding ticket.\"\n\n\"So, what do you think happened, then?\"\n\n\"Well, the way I see it, Paul Tremblay was on his way to Cali to see his girlfriend. And he stopped at this SuperMart to get some coffee or something and that's when the killer shows up. Now, I'm not sure whether the killer put the dead guy in the car or whether he killed him in there, but it looks to me like the guy was slaughtered right there in the back seat. Or . . .\"\n\n\"Yea? Or what?\"\n\n\"Or, the dead guy tried to jack Paul's car and Paul killed him by accident. But that doesn't explain why part of the body was eaten. Paul Tremblay's no flesh-eater, that's for sure.\"\n\n\"Then who?\"\n\nThe cell phone on Milligan's desk rang then, and he reached for it, checked the number but didn't answer. \"I dunno. And we won't have any answers until we talk to Paul Tremblay. God knows where he is now. Maybe the killer dragged him off and he's either dead now or . . . worse.\"\n\n\"I wonder who the corpse is.\" Starnes chewed absently at his nails, a habit he had fought against since he was a child.\n\n\"Not sure. The boys in Kansas had to send the body to Topeka to be properly autopsied. We'll have to wait to get the results. And in the meantime, we need to talk to this Linda. She's all over Paul's cell phone and according to his best buddy \u2013 who may or may not be the most stoned person I've ever met \u2013 Paul intended to marry the girl. If anybody knows where Paul is, it's her.\"\n\n## CHAPTER NINE\n\nPaul peeked out the crack of the door once more. The sun had fully set and darkness had gobbled up most of the town around him. Most, because there was a huge arc-sodium lamp that lit up a basketball court across the street and down two blocks. Save for that one patch of near-daylight, the town was obliterated.\n\nHe shuffled back to the kitchen then, intent on grabbing the rat from under the pot and gaining some strength and mental agility to boot. His pockets were filled with useful things: sticky notes, a pen, a pocket knife, the flashlight. As much as he hated to admit it, he was a fully-prepared car thief.\n\nThe rat was still bouncing around beneath the pot, trying to find some chink in its armor and make an escape. Paul knew that the thing would bolt the minute he lifted the pot, so he simply slid the pot to the edge until there was enough of a gap for the rat to drop into his waiting hand.\n\nHe stuck the poor thing's head into his mouth and bit down hard. For a dead guy, he had some seriously good teeth. The rat was devoured in a matter of two minutes, and Paul was cleaning his hands on a filthy rag that he supposed used to be a dish towel.\n\nHe had no idea how far he had to walk, but as he stepped outside, he felt strong and capable. Thoughts flew through his mind then, as he figured out how to accomplish his goal.\n\nEven at nine, there was hardly anyone around to see him. The steady thump of the basketball down the street had silenced some fifteen minutes ago and so Paul was all alone with the tiny dark town. He assumed that he should walk west, the same direction he had been headed before he'd stopped at the SuperMart. He had never been any good at directions. If his car hadn't had a GPS, he'd never have gotten anywhere.\n\nHe mourned the loss of the car; felt bad for the sacrifice that Matt had unknowingly made. In a long list of regrets and debts, Matt was at the top. Paul put one foot in front of the other and plodded along.\n\nHe had been walking for over an hour, watching as the buildings thinned out, as the traffic slowed to a trickle and then disappeared altogether. Now, subdivisions sprawled out before him, little dots of light marking each house, each family, each happy life. Paul wondered if he would ever have a happy, safe life again. And he walked.\n\nAn hour later found him in the farmland of his dreams. These were small farms, no sprawling corporate entities yet. Privately owned farms meant privately owned equipment. Surely he would find a battered but usable truck somewhere in this agricultural wonderland. His head had become fuzzy again and his legs seemed heavier than when he had started out. His rat was wearing off, he guessed.\n\nThe air out there smelled different. It was cleaner and fresher, though there was a back-scent of manure and pesticides. Paul didn't breathe anymore. But through some cruel twist of fate, he could still sense smells. His body had become some sort of bio-mechanical robot, thinking and moving under its own power, but weakened and hard to manipulate.\n\nHe heard a sound then, and he stopped in his tracks. All the human residents might have been asleep, but there were still a few pigs on the prowl. At once, Paul's stomach lurched and knotted. The rat he had eaten two hours ago didn't seem to have stuck around long. He wondered, then, what happened to the rat that he had put inside his rotting body. He didn't seem to have any form of digestive system. Had it just dissolved somehow? Would its bones stay in there or be passed out of his body in some manner? These thoughts disturbed him and he grimaced, clutched at his stomach.\n\nThe pigs snorted and rooted. They made stamping sounds in the mud. Paul followed the sound until he found them, happy in their little pen at the very farthest side of the property. The farmhouse was some two hundred yards away. Paul squatted behind a copse of trees and watched the pigs, wondering. He remembered seeing one of those CSI type shows once where they used pigs to demonstrate and test different chemicals and weapons. They said that the pig and its flesh were most like that of a human.\n\nIf nothing else, the pig must surely give him more bang for his buck than a stupid rat. Since the tragedy that had befallen him, he had put together something of a hypothesis. Human meat repaired his body quicker, helped it stay healed longer. The fresher the meat, the longer it lasted. It was like doing drugs, he mused. The better the dope, the better the high. He eyed the pigs, conjuring images of the greased pig races from the county fairs. He also remembered that pigs were vicious and could kill a man. He would have to be careful.\n\nAs quietly as he could, he crept to the edge of the pen and climbed onto the top railing of the enclosure. There were about a dozen pigs in the pen that night, two of them quite small. He doubted that he could take on a full-grown pig on his best day \u2013 and this sure wasn't his best day. So, he selected the smallest of all the pigs, and he waited.\n\nIt didn't take long before the rutting, mud-covered little pig worked its way around to where Paul sat. As it came directly in line with him, he leaped, landing on the pig and scaring it into a squealing fit. It tried to pry itself loose from his grasp, dug in its cloven feet, and tried to squirm away. The pig was much stronger than Paul had anticipated, and a lot more slippery. Twice, he nearly lost his grip on it, the bottom half of his body sliding into the mud and his feet scrambling for purchase.\n\nBut then he dug his teeth into the soft, meaty flesh right behind the pig's ear. Blood spurted \u2013 much more than he had counted on \u2013 and a piece of flesh tore away from the pig. He swallowed it without chewing, panicking briefly as the hunk of meat lodged in his throat. Several gulping swallows later, it went down.\n\nAnother bite and another. It seemed like such a waste to kill a pig and only eat a few bites. And when he could eat no more, Paul pulled out the pocket knife and began to cut long strips of flesh from it. These he stuffed into his pockets, having no other way to carry them. All the other pigs had run off to the corner farthest from Paul and were eying him with something akin to a homicidal glare. Paul climbed out of the pen, wiped the knife on his pants, and put it away.\n\nThere was a large barn some fifty feet from the pig pen, and a smaller one just behind that. Paul made his way through the darkness, keeping to the edge of the trees so he could make a fast escape if he needed to. When he reached the smaller of the two barns, he found it unlocked and so he was able to slip inside unseen.\n\nJust as he had hoped, there were two trucks inside that barn. The first was large and new-looking. The second was a good deal older and war-weary. It appeared to be from the early 80s and was pocked with rust and dents. Paul chose that one, since its owner would be less likely to report it stolen. At least, he wouldn't rush to report it stolen. Paul checked the door, found it unlocked, and climbed inside.\n\nIt didn't take him five minutes to strip the wires and connect them. These older vehicles didn't have security features the way the newer ones did. The engine choked to life and the gauges all came on. Paul smiled and went to open the barn door so he could drive through.\n\n\"Milligan here,\" the detective sighed into the phone.\n\n\"This is Davis, over in Lebanon, Kansas. Listen, we got that autopsy back.\"\n\nMilligan sat up straighter in his chair and smiled. \"Give it to me straight, pal. Don't hold back.\"\n\n\"Well, the vic is one Dana Manning. He's from Chicago originally, but he's been drifting in years past. As far back as age eleven, the kid has been in trouble with the law. He did a stint in the Illinois prison system for armed robbery, then slipped into carjacking. He's got a rap sheet as long as my arm. According to the coroner, the guy was killed when somebody ripped out his carotid artery . . . with their teeth. Apparently, this kid Tremblay has perfect teeth because he's never been to the dentist. So, we don't know if he's the eater or not.\"\n\n\"I'm guessing not. It's just doesn't fit. By all accounts, this guy is squeaky clean.\"\n\n\"I agree. The fingerprints we got off the car match this Manning guy and Tremblay. There are a couple from the other owner, Matt Cassidy.\"\n\n\"Cassidy's alibis all check out. There's no way he was in Kansas when the murder was committed. Besides, he's too stupid to pull it all off.\"\n\nDavis laughed, perhaps a bit too long. \"What about that phone I sent you? You come up with any other prospects who might tell us where our boy, Paul, is?\"\n\n\"Just one. Apparently, he has a girlfriend on the west coast. Probably one of those online romances, you know. I wouldn't make too much of it. Still, his emails and text messages say that he's heading her way. Somebody needs to talk to her and find out what she knows.\"\n\n\"If old Paul does show up, it's going to be at her house. Yeah.\" Davis was quiet for a moment, then he chuckled. \"Well, I guess I'm going to Californ-eye-ay.\"\n\n\"Want some company?\" Milligan was smiling. He licked his lips.\n\n\"Well, I don't mind telling you, I'm in over my head with this one. Bring the phone with you and we'll go see what we can see.\"\n\n\"Yes.\"\n\nMilligan hung up the phone and then started doing the paperwork and making his travel arrangements. His best friend and old Army buddy, Clint, lived in Los Feliz. With a little luck, they could wrap the case and still have enough time before his flight out that he could spend a little quality (bar) time with old Clint.\n\nIt occurred to Paul as he drove along in the battered truck that if things had turned out just a little differently for him, he might have made a great criminal. He had stopped at a rest stop some fifty miles ago and exchanged license plates with another truck, whose owner had gone into the rest room. There was no way to disguise himself, but he could trick the cops into ignoring the truck. To that end, he also removed all the logos from the body of the truck, exchanging the front and back one with the truck he had taken the plates from. The Chevy C-10 was now a Ford, for all intents and purposes. Paul smiled, pleased with his cleverness.\n\nThe tank had been nearly full when he'd stolen it, but that old truck ate up gas at a frightening rate. Four hours later, Paul was pulling into a small independent gas station. It had automated pumps and so he wouldn't have to go inside, something he considered to be a plus. He gassed the truck and was nearly done filling the tank when another car pulled in on the other side of the pumps. A single man stepped out of it, his face haggard from a night of driving and his head completely bald. The car he drove was a nice new Camry, and he didn't lock it when he went inside.\n\n\"In for a penny, in for a pound,\" Paul sighed.\n\nHe put the nozzle back on the pump and opened the truck's door. Then he simply leaned over, grabbed the GPS off the dashboard of the Camry, and drove away. The truck blasted a few clouds of gray smoke as he went, but it accelerated nicely and was around the corner before the guy came out of the station.\n\nPaul drove as far as the next rest stop, and then he drove two miles farther. There was a dirt road which wound its way back into the woods for some distance and Paul turned left onto it. He backed the truck up into a stand of small trees and cut the engine. He didn't want anybody to see the truck sitting by the side of the road but he had work to do.\n\nThe pig that Paul had eaten had filled him nicely and \u2013 even five hours later \u2013 was still keeping him vital. Meanwhile, the rest of the pig meat he had stolen was going stale on the seat next to him. Even that wouldn't last him forever. So, Paul began to make notes. On one sticky note, he wrote, GOING TO LINDA'S in LA. On another, he wrote, YOU ARE WANTED FOR MURDER. Still another read, AVOID THE COPS. And so it went. Every single thing that was important for him to remember was written on a sticky note and pasted on the dashboard of the car. He included instructions on the hot wiring, phone numbers, addresses, and the whereabouts of the ring he had bought for Linda.\n\nWhen he was sure that absolutely everything of importance was posted somewhere on the truck, he drove back onto the main highway and kicked the truck up to seventy. It shuddered but complied. Kansas was behind him.\n\nThe plane touched down in LAX at ten. By eleven, Milligan was zooming along the freeway, his rented car trapped in the fast-flowing sea of traffic. The GPS would guide him to the police department, where Davis had told him to go. Davis had beaten him there by several hours and the plan was that they \u2013 together with the LA detective in charge \u2013 would go to Linda's house to interview her. Milligan was already growing weary of the traffic and smog by the time he pulled into the parking lot of the LA precinct. He didn't understand how his buddy, Clint, could stand living here.\n\n\"I'm supposed to meet Detective Davis and a Detective Lopez here,\" Milligan told the desk officer. \"We're working a case together.\"\n\n\"I'll take you to Lopez,\" the sergeant said with a sigh.\n\n\"Thanks,\" Milligan grumbled as he followed the man into the precinct offices.\n\nLopez was a middle-aged, hard-nosed detective of Mexican descent. He had come up through the ranks from the streets, where most of his friends had turned out to be either gang-bangers, or dead. He had soft brown eyes and an easy smile. Life was good for him. Better than he had hoped it could be.\n\nHe stood quickly when Milligan came through the door, that easy smile leading, a handshake following. \"This is one hell of an interesting case you've got here, Milligan. I don't mind telling you, I had to fight three other detectives to get it.\"\n\nDavis laughed then. It was disingenuous and shrill. \"I fought NOT to get it. Sadly, I'm the only horse in a one-horse town.\"\n\nLopez and Milligan exchanged amused glances, and Milligan took a seat next to Davis. \"So, I guess the big guy here has filled you in?\" The big guy reference came from the fact that Davis was as wide as he was tall. He had floppy jowls that shook when he spoke and hands whose fingers could only barely separate due to the fat on them. That, coupled with his farm boy attitude, might have led most people to think he was slow.  Nothing could be farther from the truth.\n\n\"Yea, I get the gist. Dead guy in the back seat, half eaten. Owner and driver missing, his fingerprints all over the door. Sounds to me, though, like this Manning guy needed killing. Not exactly a model citizen, was he?\" Lopez was gathering things as he spoke: cell phone, keys, walkie, his cuffs.\n\n\"Still, we need to know what happened.\" Davis frowned. \"We've never had a murder in my town before.\"\n\nLopez slid the small stack of pictures across his desk and spun them to face Davis and Milligan. \"You still haven't, Davis. From the condition of the body, the body temp when you found him, the lividity . . . I'd say that guy had been dead for at least 30 hours before he hit your little town. He may have been killed in that car, but he wasn't killed in your town.\"\n\n\"Lopez is right,\" Milligan agreed with a nod. \"It's possible that the car was jacked, Manning killed, and then the whole crime scene was driven to the SuperMart.\"\n\n\"Well, let's go talk to this Linda. Maybe she knows something that will help us find Paul Tremblay.\" Lopez gave his keys a shake. \"Anybody mind if I drive?\"\n\nThe meat beside Paul had begun to smell. If he had had any choice in the matter, he would have tossed it out the window. But he still needed it. On those rare occasions when he actually had to go inside and pay for gas, he needed a bite of that meat to look normal. And he was saving the very last big chunk of it for when he reached Linda. The note on the rear view mirror said so.\n\nAnother note warned him to keep away from the cops, but he couldn't remember why until he saw the note telling him that he was wanted for murder. MURDER! The word was deep and ominous and it scared him. In all his life, he had never thought of killing anyone; had never hated anyone enough to even wish they were dead. And yet, the note said that he had killed someone.\n\nPaul's skin had broken out in a rash of those pustules. They were black and oozing and they smelled. He avoided looking in the mirror because his own face frightened him. That's why he was saving the last piece of meat for Linda. He wanted to look normal just long enough that he could explain to her what had happened and how much he loved her. Then, he would just slip away and die, letting Linda have a nice, normal life with someone else. He loved her enough to do that much for her.\n\nHe remembered their second date. She had barely agreed to go out with him in the first place and he hadn't blown it on the first date. So, he had wanted the second date to be special. Beyond special. He wanted it to be the most romantic, amazing, epic date of all time.\n\nThe senior dorm building was five stories tall and its roof looked out on the entire campus, plus half of the surrounding city. Paul had spent two days in between work and classes, hauling things up onto that roof. By the time he was finished, he had set up the perfect restaurant vignette with a table, two chairs, some potted trees and plants. He strung Christmas lights everywhere, brought up a stereo, even went so far as to get one of those fire pits and some fire wood. He put a loveseat in front of the fireplace. If Linda wasn't in love with him at the end of that date, it wouldn't be for lack of trying.\n\nSo it was that on a clear night with a star-speckled sky, Paul brought Linda to the top of their world. He was wearing a tux at the time, she a powder blue chiffon dress that looked like it had been stolen from the fifties. She was gorgeous. The look on her face when she saw the set-up was incredible. And she'd grabbed Paul around the neck, kissed him hard, and hugged him until he couldn't breathe.\n\n\"You did all of this?\" she asked softly. \"For me?\"\n\nHe nodded dumbly and smiled. \"And to keep that look on your face, I'd do it every night for the rest of my life.\"\n\nTears had gathered in her eyes and she had kissed him again. From that moment on, she was in love with him. The moment she took her first bite of his cooking, she was his, lock, stock, and barrel. No other man stood or would ever stand a chance with her. She was Paul's Linda.\n\nPaul almost ran off the road thinking of that night. The sobs shook him hard and though his heart no longer beat, it broke. She had taken him back up to that roof three weeks later and they had made love on a Flokati rug with the fireplace burning brightly next to them. They had rolled themselves into a Flokati burrito and slept there, in each other's arms.\n\nHe almost couldn't go on. It occurred to him that Linda should remember the good Paul, the handsome Paul, the romantic Paul. Not the decaying corpse of Paul that he had become. Perhaps this whole thing had been a mistake. Perhaps he should just go curl up somewhere and wait to return to the earth.\n\nStill, she had to know. She couldn't spend the rest of her life searching for him . . . for what they had been together. He reached out and turned on the radio, hoping for some happy music to cheer himself up by. He searched the channels, found a lot of religious stations, one oldies station, and a top-20. He kept the dial there.\n\nThe miles were rolling by slowly . . . too slowly. When Paul looked down at the speedometer, it said 35. He was doing thirty-five in a seventy zone. Grumbling, he pushed down on the accelerator and felt something in his ankle give. He hoped that something wouldn't prevent him from walking when it came time to get out of the truck.\n\nHe was close now . . . so close that he could almost feel Linda.\n\n## CHAPTER TEN\n\nThe three detectives stood outside of Linda's apartment, Huey, Dewey and Louis with gold shields. It was Lopez's jurisdiction, so he took the lead, wrapping lightly on the door and then stepping back in line with his fellows. The door creaked open and an eye appeared, blinked, dulled.\n\n\"Linda Gilchrist?\"\n\n\"Yes?\"\n\n\"I'm Detective Lopez of the LAPD. This is Detective Davis of the Lebanon, Kansas police department, and Detective Milligan of the NYPD. We'd like to have a word with you, if we could.\"\n\nSuspicion eclipsed the normally soft lines of her face. It hardened and her jaw set. It was obvious that she had been crying. \"Is this about Paul?\" Her voice cracked, choked.\n\n\"Yes, ma'am, I'm afraid it is. Could we come in?\"\n\nShe pushed the door further shut, then took off the chain and flung the door wide open. \"Of course, sir.  Please come in.\"\n\nThe apartment inside was something out of a time-warp. It had been decorated in the sixties which, Lopez figured, was the last time they had renovated the building. There was orange shag on the floors in the living room, orange and aqua linoleum in the entry and the kitchen. The walls wore flocked and Mylar paper; the curtains were mostly beads.\n\nLinda noticed their expressions and she smiled. \"I rented the place furnished. It's all I can afford until I get tenure.\"\n\nThey nodded in unison and followed her slowly to the sofa. It was upholstered in crushed velvet and it dragged at their pants as they sat down.\n\n\"How long have you known Paul Tremblay?\" Lopez asked, taking lead.\n\n\"Oh, I've known him ever since college. That's when we started dating and fell in love. We lived together for a while in New York. And he's going to join me out here as soon as his job for the city is done. Was. Was going to join me.\" She looked like she might cry then and all three men shifted uncomfortably on the sofa.\n\n\"So, it's safe to say that you know him better than anyone?\" Milligan played with the crease on his trousers. The heavy velvet upholstery was making him sweat.\n\n\"Yes, I think so. Paul and I are very close. We talk a couple of times a day on the phone . . . or we did until . . . you know.\"\n\n\"Ma'am, when was the last time you heard from Paul?\" Milligan felt sorry for her, he really did. But he needed answers.\n\n\"Well, let's see . . . it was three days ago. We didn't really talk, because he was sick and had laryngitis. So we texted. And he emailed me to let me know what was going on.\"\n\nThe three detectives exchanged glances, then Lopez took a chance. \"Would it be possible for us to read that email? Would that be okay with you?\"\n\nShe nodded but didn't move to stand. For a moment, she looked as though she might collapse, but then she suddenly shot up off the sofa and walked to the foyer table. When she came back, she was holding the cell phone out to him. \"This is the email he sent me. You can actually look at anything on the phone you want to. I don't have any secrets.\"\n\nLopez took the phone and the three of them scrunched together to read the email. When they were done, Lopez passed the phone back to her with a thin smile. \"Do you have any idea at all what happened to him at work? He talks about it changing him forever.\"\n\n\"No, I haven't a clue. I'm so stunned by this whole thing. See, I thought he was at home and sick and then he tells me that he was really driving out here the whole time.  I spoke to his roommate, Matt, and Matt said he couldn't tell me everything that was going on. He just told me that Paul's car had been found in Kansas with a body in it and it was . . .\" She fought back a rising gorge, her eyes closing with the effort and her face paling. \"Is this true?\"\n\n\"I'm afraid so, Ma'am. I wish I could tell you more than that, but I can't.\" Lopez gave her his most sympathetic look. Truth be told, he wanted to give her a hug. She looked like she needed one very badly.\n\n\"I understand. But I have to tell you, Paul Tremblay is the kindest, most gentle, caring man I've ever met. And that's not just blind love talking. It's a fact. Ask anybody.\"\n\n\"Yes, ma'am.\" Milligan let her run with it. She needed to get it off her chest and he meant to let her.\n\n\"He would never kill anybody. Never. No way. He can't even eat a lobster or a crab because he says it makes him feel uncomfortable the way they stare at him.\"\n\n\"We're not sure what happened, ma'am,\" Lopez continued gently. \"There's still a good chance that Paul is alive and well. But we need to find him so we can know what happened. He's coming to see you and if he makes it, that's our best chance to talk to him. So, if he calls, emails, shows up here . . . you give me a call.\" He handed her his card with a smile.\n\n\"Oh, don't worry. I will.\" She tucked the card inside her cell phone case. \"One other thing. I get the feeling that Matt is keeping something from me. He was with Paul every single day on the job. If something happened to Paul at work, Matt knows about it. But he won't tell me.\"\n\n\"I understand and we'll be talking to Matt again. Whatever we find out, we'll share with you if you'll agree to do the same.\"\n\n\"Absolutely.\"\n\n\"All right, then. Thank you for your help, Miss Gilchrist. And if you think of anything or if Paul shows up here, you can reach me on my cell phone. The number's on the card.\" Lopez offered his hand and a smile.\n\nShe showed them to the door and the three men tipped their imaginary hats to her as they left, walking down the sidewalk slowly and leaning in to talk.\n\n\"You're just going to trust her to call us if he shows up?\" Milligan snorted.\n\n\"Not a chance,\" Lopez laughed. \"I'm going to put a man on her right away.\"\n\nFor some reason he couldn't begin to explain, Paul had become obsessed with whistling. It had started when he tried to remember the song that had been playing the night that he had taken Linda to Tavern on the Green for her birthday. He had saved up for a whole month for that dinner because she had told him the story of when her grandmother had taken her there when she was little. She had made over the food, the atmosphere, the service, and of course her grandmother's stories. So Paul had gotten the reservations for her birthday, reserved the perfect table. And in the middle of the meal, a violinist had strolled up to them and started playing.\n\nLinda had said the song was the most beautiful song she'd ever heard and now Paul couldn't remember how it went. It was the theme from Love Story he knew, but the tune eluded him. He had begun trying to whistle it, found that he couldn't whistle a lick, and had spent the last hour and a half trying.\n\nThe speedometer had dropped again; this time to forty. He pushed down, carefully, and made the truck speed up. He didn't want to attract attention by driving thirty miles an hour under the speed limit.\n\nThree more sticky notes had appeared on the dashboard. One alluded to the birthday-Tavern on the Green-song memory. The other two were in reference to the buttons on the GPS and how they worked. The gas gauge was edging toward \"E\" again, so he knew he would have to stop for gas.\n\nPaul looked in the mirror and cried out. His lower lip had dropped a good two inches and his right eyelid had begun to droop.  There wasn't a square inch on his face that wasn't covered in those hideous pustules, and his hair had begun to fall. That, he had noticed when he ran one hand through his hair and drew it back with a healthy clump of hair \u2013 and scalp as well \u2013 clutched between the fingers. It had sent him into another sobbing fit.\n\nHe studied the exit signs, hoping for a nice, small, automated station he could fill up at. He was so horrible in his countenance now that he couldn't escape notice. The hoodie no longer protected him and he had to wear gloves to hide his green and black hands. There were still two pieces of pig meat rotting beside him on the seat. He thought to take a bite of one of them in order to pass for human at the next gas station.\n\nAs it happened, the meat wasn't necessary. When he pulled up to the station, there wasn't a soul in sight. Minimal lights burned, one flickering like a disco ball directly over the truck. Paul slipped out of the truck, his eyes trying to focus, scanning the area for approaching cars. Nothing moved.\n\nHe slid his card at the pump and pulled the nozzle from its cradle. He stood stock still and stared then, wondering what the hell he was supposed to do with it next. He blinked and stared and looked over the body of the truck. There was a small opening at the back, just above the fender. That must be it, then. He pulled the little door open and removed the cap. While the tank filled, he went back to the cab and wrote on a sticky note. PUT THE GAS IN THE LITTLE DOOR OVER THE DRIVER'S SIDE FENDER. He put it on the dashboard right above the radio.\n\nHe had been leaning into the truck, intent on forming the words which would keep his truck going all the way to Linda. When he was done, he stood up straight and turned, spinning right into the face of a very tall man with tattoos on his face. Paul screamed and the man screamed and a gun hit the ground. Scared as he was, Paul couldn't remember what the gun was or what it was supposed to be for. He blinked at the man, who, in the face of stark raving terror, backed away from Paul.\n\nHe watched the man go, wondering what the hell was wrong with him, and then stooped to pick up the gun. He turned it over and over in his hand, looking at every part, every detail. Still, it made no sense to him. He tossed it into the truck, letting it land softly on the seat next to the rotting meat. Then he went to remove the nozzle from the truck and replace the gas cap. He knew that by the next time he stopped for gas, he might well require a set of step-by-step instructions for gassing the truck.\n\nHe climbed back into the driver's seat then, closing the door after him and gripping the steering wheel. Beyond that, he hadn't any idea in the world what to do. He began reading notes, placed there in the order that he had forgotten each one. CONNECT THE WIRES TO START THE TRUCK. PULL THE STEERING WHEEL LEVER UNTIL IT SAYS \"D\". PUSH ON THE RIGHT PEDAL TO GO, LEFT PEDAL TO STOP. FOLLOW THE GPS UNTIL YOU GET TO LA.\n\nHe performed all those tasks in order, wondering why the hell he was driving all the way to LA and where the hell he was now. The signs told him where he was. The rest he was left to wonder about until he could stop and read the rest of the notes.\n\nLopez eased into his seat, a smile on his face and a cup of coffee in his hand. He pulled himself closer to the desk, realized that his belly kept him from getting as close as he used to, and pushed back. \"I've got two of my best guys watching the house. If Paul shows up, we'll nab him the minute he steps out of that truck.\"\n\nMilligan nodded and Davis started to say something, but then his phone rang. He tilted to one side to grab for the phone, grimacing as he leaned on his left arm. \"Davis.\" He listened for a moment, his face darkening as he did so. There was a pen and a pad within reach and he grabbed it, began scribbling. \"Uh-huh. Uh-huh. Got it. Okay. Later.\"\n\n\"Everything okay, Davis?\" Milligan asked, his eyebrows raised and his eyes hopeful.\n\n\"Cause of death was exsanguination. The bastard bled out when somebody bit a chunk out of his neck. The other damage, to his arm and shoulder, was done after the fact. Long after the fact.\" They exchanged disgusted looks, shaking their heads in sympathy for a man who didn't deserve it. \"My guy also says that a truck was stolen last night, long about eleven. It was a brown 1982 Chevy C-10, license number AZP-4130. At that same location, the owner of the truck reports that one of his pigs was attacked, partially eaten, and left for dead.  Whoever did it took some of the meat with him.\"\n\nMilligan grimaced and looked away. \"So, our boy not only has a taste for human flesh, but for pigs and rotted flesh as well. You think it was the same guy who killed Manning?\"\n\n\"So it would seem.\" Davis sighed. \"We don't have a lot of crime in Lebanon. For us, this is practically a spree.\"\n\n\"Then, our guy is driving that truck. Could be Paul Tremblay, could be his killer. Either way, we find that truck and we find some answers.\" Lopez snatched up the phone receiver and pushed a button. \"I'll put an APB out on that truck. In the meantime, Milligan can get one of his boys to go interview Matt again.\"\n\nMilligan nodded and whipped out his phone. He knew just who to call. If there were answers to be had, Gebhart would get them.\n\nMatt was gulping coffee and stomping his boots onto his feet when he heard the knock at his door. In a good week, he would have maybe one person come to the apartment. This past week had sent a steady flow of people his way and he didn't like it. He would be late leaving for work in another five minutes and he hadn't even had anything to eat. He choked down the last mouthful of coffee and slammed the cup down on the table.\n\n\"Coming!\" he barked, yanking on his remaining boot as he hopped across the living room. \"Coming!\" he yelled again when the knocking didn't stop.\n\n\"Sergeant Gebhart of the NYPD,\" the face announced as Matt yanked open the door. The chain hadn't been on it as he had been too stoned last night to check it. \"I need to have a word with you.\"\n\n\"No way, man. I'm late for work as it is and my boss will fry me if I'm late again.\"\n\n\"Sounds like that's your problem, not mine.\" He was a large man with a receding hairline and deep-set eyes. His arms looked like he spent time wrestling bears when he wasn't working. That or he slung a pick axe in his off hours. He took a step toward the door.\n\nMatt threw up his arms and stepped back, letting the door bang against the wall as he yanked it open. \"Okay, man, but if I get fired, I'm suing the PD.\" He shut the door and went to his favorite chair, where he sort of slopped into it, slouching down and stretching out his legs like a petulant child.\n\nGebhart sat on the sofa and flipped open his notebook. He seemed pleased with himself, as though he had won some great battle or other. \"You spoke to Detective Milligan a few days ago, regarding your friend, Paul.\"\n\n\"Yes.\" Matt was about to say something smart-assed, but it occurred to him that they might have heard from Paul. \"Has he turned up? Paul, I mean?\"\n\n\"No. But Milligan interviewed his girlfriend, Linda Gilchrist, in LA. According to an email that Paul sent her a couple of days ago, something happened on the job the last day Paul was here. We were hoping you'd know what that was.\"\n\nMatt shifted, licked his lips, tried not to look guilty. He failed miserably. \"Dude, whatever happened to Paul, I don't know anything about it.\"\n\n\"Bullshit!\" Gebhart stared daggers through him. \"The way Linda tells it, you and Paul work as a team. You're always together at work.\"\n\n\"Yea, I guess that's right. But I don't know what happened, man.\"\n\n\"Is that so?\"\n\n\"Yea, that's so. Look, dude, I gotta get to work.\" He moved as if to stand up, but Gebhart never so much as twitched.\n\n\"I'll talk to your boss. Sit down.\" Gebhart watched as Matt slid back into the chair, dejected and jumpy. \"Now, you and Paul were down in the tubes on his last day here, right?\"\n\n\"Yea, we were. So?\"\n\n\"Where?\"\n\n\"Out in the industrial section. There's that pharmaceutical plant out there, and a couple of other factories.\" Matt started tapping one crossed foot against the other.\n\n\"We know that you called an ambulance that day. And we know that Paul was dead at the site. The EMT says that they took him to the morgue. And then, Paul just disappeared.\"\n\nMatt felt like he was going to throw up. His face paled and he felt a sudden wave of heat and nausea wash over him. He sat up straighter, drew in a deep breath, and finally looked Gebhart in the eye. \"I'm telling the truth when I tell you that I don't know what happened. Honest to God.\"\n\n\"Yea? Well, why don't you tell me what you do know and then what you think happened?\"\n\nMatt hesitated, looked around, trying to figure out just how much trouble he was in. He hadn't done anything to Paul. In fact, Matt hadn't done anything at all, and that might well be the problem. \"We were in the tubes, like I said. We were running our tests and doing our measurements and such. We got to the end of that particular pipe and there was this heap of trash and shit between us and the last twenty feet. So Paul climbed the trash with the tape so we could measure.\"\n\n\"Did you lose sight of him while he was on the other side of the trash?\"\n\n\"Yea. I couldn't see him but I could hear him. He was bitching about some nasty blue goo. When he came back over the trash pile, I could see it on him. We went back to the last junction, climbed up the ladder and got the heck out of there. It was quitting time, ya know?\"\n\n\"So, Paul was okay then? When you saw him come over the trash?\"\n\n\"Yea, he was fine. And when we got into the sunlight, you couldn't see that shit at all. We were halfway back to the truck when Paul just doubled over and fell to the ground, flopping around like a big old dead fish or something. I called for the ambulance right away, but when they got there, dude said he was dead. They took him off to the morgue.\"\n\n\"That part we got. So, if he's dead, then how the hell is he driving around in your car and playing with corpses?\"\n\n\"I don't got any idea about that, man. All I know is that a couple of hours later, Paul comes waltzing in here, alive. He had shit all over his face like zits or something and when he talked, it sounded like when your wood chipper gets jammed up. But he was damn sure alive and he didn't remember anything about what happened down in the tunnel until I told him. That's when he decided to go to LA to see Linda. I tried to stop him, told him to call her or something. But he kept insisting. After that, I don't have any idea in hell what happened with Paul.\"\n\nGebhart flipped his notebook shut and stood up. \"Okay, Matt. I just need you to do one more thing for me.\"\n\nMatt frowned, screwed up his face as he stood. He had been waiting for the other shoe to drop. This was it. \"What's that, dude?\"\n\n\"I need you to take me out to that sewer tunnel and show me where you were when it all went south.\"\n\n\"Dude! I already told you, I gotta go to work.\" Matt pleaded with open hands and cocker spaniel eyes.\n\nGebhart stepped around the coffee table and clapped a hand on Matt's shoulder and squeezed. \"I'll fix things with your boss, don't worry. Now, grab a jacket and let's go. I'll drive.\"\n\nHe was out the door then, Matt scurrying to catch up, lock the door, and put on his jacket all at once. Gebhart's car was parked out front and it was a police cruiser. Instantly, Matt could feel the eyes of the neighborhood on him. He wished he had a t-shirt that said MATT DIDN'T DO ANYTHING. Not that it would help.\n\nThey drove on for a few minutes in silence, neither looking at the other. \"For the record, we don't think you did anything wrong,\" Gebhart said, breaking the silence. \"And we're pretty sure Paul didn't do anything wrong either.\"\n\n\"Well, that's good. 'Cause neither one of us did anything wrong. Paul is my best friend in the world. He's like a brother to me, man. And if anything's happened to him . . . I'll just die.\" Matt began to cry softly, his shoulders hitching with each sob.\n\nGebhart wasn't good with that sort of thing. He felt bad for the kid, he really did. But he had a job to do. \"So, where is this place, exactly?\"\n\nMatt swiped a hand over his face, tried to clear his vision. He sniffed and wiped his nose on his sleeve. \"Go four blocks down and turn right. You go over the bridge and then when you pull into the industrial park, take your first left and park out behind the old bottling company.\"\n\nGebhart nodded and took his turns. He had a natural aversion to anything that existed underground. Even the subways made him nervous. He supposed it all went back to his beat days, when he used to pick up overtime when he could, sitting the subways. By the end of his second year, he had seen just about everything that the underbelly of life could produce.\n\n\"Which way?\" Gebhart asked as he slid out of the car.\n\n\"Right over here. You have to go down here and then walk the rest of the way. You got a tire iron?\"\n\n\"In the trunk.\" Gebhart stepped to the rear of the car and unlocked the trunk. He freed the tire iron and slammed the trunk lid with a sigh. God, how he hated going underground.\n\n\"Listen, man, I'm not going down there. You can arrest me or shoot me or whatever you want. But there's no way in hell I'm going down in that tube and I don't think you should either.\"\n\nGebhart look at him, assessing Matt's determination, and his fear. He was pale and his body language suggested that he might bolt at any minute. \"Relax, kid. I'll go down there. You just have to tell me what I'm looking for.\"\n\nMatt took the tire iron from Gebhart and lifted the sewer lid up and over. \"You can't miss it. Walk about a hundred yards that way,\" he pointed to his right, toward the water tower. \"You can't miss the trash heap.\"\n\nGebhart stepped to the opening and lowered himself onto the first ladder rung. \"I want you here when I get back, pal. If you're not, I'll find you.\" He let his eyes say the rest.\n\n\"Is there any way I can talk you out of this, dude? Seriously.\" Matt's eyes were soft and pleading. He seemed genuinely concerned.\n\n\"I'll be back up in ten minutes. Then we'll go get a sandwich or something, okay?\" Matt nodded but his frown stayed put. \"I'm buying.\"\n\nGebhart slipped into darkness then, and when his feet touched down, they did so with a splash. Gebhart cursed and shook his feet, trying to get the water off his uniform shoes. He unclipped the flashlight from his belt and switched it on. The trash pile was not nearly as far away as Matt had indicated, so he made haste to it and began to climb to the other side.\n\nHe slipped twice and finally crested the heap, which was tall enough to leave a mere three feet between its top and the ceiling of the tunnel. Gebhart regretted those extra cheeseburgers as he slid over the top and made his way back down the other side. He promised he'd work out more, get his wind back and maybe lose a bit of that gut. And then his feet touched the floor of the tunnel and he forgot all about it.\n\nHe now understood what Matt was talking about. A few feet away from the base of the trash pile was a large blue puddle and it was glowing like a 60s black light poster. Gebhart watched his step closely, making sure not to get any of that stuff on him. He had a collection bag in his pocket, and he pulled it out, peeled it open. It occurred to him then that he had no gloves and that he would never get the thing to close without getting the stuff all over him.\n\nHe searched the trash pile, looking for anything he could use to contain a sample without risking a spill or contamination. There was a pudding container in the pile and he used that to meticulously fill the evidence bag about a quarter of the way. Then, he taped the bag shut and tossed the cup away. He checked himself carefully, making damn sure he hadn't gotten any of that blue crap on him. He had no idea what it was, but he knew it was bad . . . bad enough that just getting it on his skin had killed Paul . . . at least for a while.\n\nWhen he was back on solid ground again, he heaved a sigh of relief, his eyes adjusting slowly and his face twisted with stress. Matt was there, pacing back and forth and his face drooped with relief when he saw Gebhart.\n\n\"You okay, man? You didn't get any of that stuff on you, did ya?\" He reached over and grabbed the sewer cover with the tire iron, dragging it into place.\n\n\"I'm fine. Really. I got a sample, but I managed not to get any of it on me. What the hell is it anyway?\"\n\n\"You're asking me? Do I look like a chemist?\" Matt laughed then. It was a thin, nervous sound and it made him sound slightly insane.\n\n\"Let's drop this by the lab and go get some lunch.\"\n\n\"You're sure you're okay?\" Matt looked him up and down, frowning, his brow deeply etched.\n\n\"I told you. I'm fine. C'mon.\" He clapped Matt on the back and made for the car, the bag held lightly between thumb and index finger. He opened the trunk and put the bag into a container, labeled the container with a marker, and slammed the trunk lid.\n\n\"I'll take you to my brother's place. He runs a little deli on the east side. They make a hell of a pastrami there. You like pastrami?\"\n\n\"I guess.\" Matt didn't like much of anything anymore. Not since Paul had gone away. And if Paul didn't come back, he might never like anything ever again.\n\n## CHAPTER ELEVEN\n\nPaul could see Las Vegas burning brightly from twenty miles away. It was like someone had opened up the darkness and let the sun shine on that town alone. He still had plenty of gas to make it to the other side, so he could stop in some out-of-the-way desert gas station instead of a big city mart. His face had seemingly begun to melt and, against his better judgment, he had taken two bites of his remaining piece of pig meat. He thanked his lucky stars that he couldn't seem to taste anything because the stuff had gone green and there was the furry beginning of mold on its surface. But the hunger had hit him so fast, gnawing at his insides and making his head \u2013 not just his ears, but his entire head \u2013 ring. Aside from that, he was afraid that if he waited too long, his decaying body might be too far gone to repair with any amount of meat.\n\nNow, he had a scant half piece of pig flesh remaining. Would it be enough to restore him to normal Paul status, he wondered? Or was he going to need to get some more somehow? He really didn't want to go through another pig slaughter. That aside, there didn't seem to be a lot of farmland in the Vegas area. And Southern California, he knew, was devoted to oranges, tomatoes and the like. Paul sighed and pressed on.\n\nMore than anything, he wanted to avoid crowded places, cops, and busy streets. His mind wandered a lot, even with the meat in his belly. So, he took Highway 215 around the city, joining back up with Highway 15 on the other side. He had about fifty more miles and then he would need gas again, seventy-five if he stretched it. His plan was to take Highway 15 until it joined up with Highway 40 going west, then slide in to Linda's place. One more fill-up and he should be there.\n\nHe had made notes in one of his clear moments, having studied the GPS and written down each road and each turn he needed to make. Three times he had had to stop and look up Linda's address, but he finally got it right. Now, even if he lost the GPS or it stopped working, he would know where to go and how to get there. One of the notes warned that the cops were looking for him, but he doubted that the APB would extend this far west from Kansas. Besides, he hardly looked like himself. It would take quite the detective to recognize him now, with his drooping face and pock-marked skin.\n\nTwenty miles after he left the 215, he spotted a small gas station in the middle of nowhere. The desert was flat and you could see for miles. Buildings were easy to spot if their lights were on, but it was hard to tell exactly how far away that bright spot in the night really was. He veered off onto a side road, which consisted of little more than a dirt path which had been cleared of all its scrub. Paul was sure they had poured some gravel on the makeshift road at some point, but most of it had been ground into the ruts and divots that the rare falling of rain had produced.\n\nHe watched the station carefully as he approached, hoping for no signs of life, for automatic pumps, for a blind attendant. The setting sun had turned the desert into a cold, barren place, where the chill seeped all the way into your bones and nestled there, like some sort of a parasite. Paul shuddered and pulled up to the pump. There was a sound like a loud bell as his tires crossed a pressure swittch and it made him yell in surprise. His mind shot back to a time, long ago, when he had been a little boy and had ridden with his dad to the filling station to get his bike tires tended to. There had been a hose across each pump lane and when you ran over it, a loud bell rang, alerting the attendant of an incoming customer. There were no more attendants, of course, and thus no more bells.\n\nBut things here in the far-away of the desert were different on a lot of levels. The bell had damn sure rung and now a short, grizzled man who looked far older than he must be trotted toward the truck. Paul flipped his hoodie into place and tried to avert his face.\n\n\"Evening, mister. What can I do for you today?\" Despite the location, the man's accent was deeply southern, approaching a drawl.\n\nPaul coughed and made as if to hold his throat, then he flipped out a sticky note which said: FILL IT PLEASE. He waited for what seemed like a thousand years for the man to walk back to the pump, pull out the nozzle, and begin pumping the gas. Paul took every opportunity to cough loudly, to wretch his gut as though he were about to cough up a lung. Not only did it offer a good explanation for him not speaking, but it made the man less inclined to come near.\n\nWhen the tank was filled, Paul whipped out his credit card and thrust it out the window at the man. He took it and walked slowly away, his face contorted and scratching his head. When the man returned, he handed Paul a clipboard with a carbon copied receipt and a pen. Apparently the digital age had completely passed this guy by. When Paul had signed, they exchanged the clipboard for the credit card, Paul waved, and drove away.\n\nAfter the distance he had traveled and all he had been through, it was comforting to Paul to know that, in a few short hours, he would be with Linda again. This whole mad adventure would finally be at an end. He could finish his business with her and then turn himself in to the police, tell his story, and hopefully finish up the business of his life, such as it was.\n\nHe had never been to Los Angeles, but he had heard stories. Luckily for him, Linda lived in a nice suburban area where the apartments were more like townhouses than skyscrapers. Still, he looked a mess and was a complete disaster in every way. He would need to stop somewhere, eat his remaining chunk of meat, and try to regain some semblance of his humanity. Then he would drive to Linda's apartment where he could . . . where he could . . . He couldn't remember why he had come. He took a second to glance at his notes. There was nothing there covering this particular part of the journey. He knew he had driven out here to see Linda, but he had no earthly idea what he was supposed to do when he did see her. He couldn't speak in any form that she would understand and he didn't have a cell phone to text with, or anything at all with which to communicate with her.\n\nWriting down an explanation would have to be a part of that one last stop. He would pull himself together, clear his mind, and then he would write down everything he wanted to say to Linda.  That was the plan, at least.\n\nLinda was on her way back to the sofa, a cup of tea in her hand. She had been reading a stack of papers that her students had turned in and her eyes were sore and red. She stopped briefly in the little alcove halfway between her kitchen and her living room to sip at the tea. So far, she hadn't spilled any of it, which was pretty remarkable for someone as clumsy as she. That's when she saw it. There was a little flash of light through the front window.\n\nThere was only one window in the front of her house and it looked out over her quiet little street. Even with the drapes closed, she could see shadows out there and bits of pale light. Linda set the cup down on the end table and went to the window to see what might have caused the flash. There was a dark car parked across the street and directly even with her front door. She couldn't see very well, but there did appear to be two dark forms sitting in the front seat. Cops, she supposed, sent to watch the house in case Paul showed up. They didn't trust her to tell them if he came to her. Neither did she.\n\nShe watched for about a minute, through a tiny slit in the drapes. Then she pulled them tightly closed and went back to her tea and papers. Surely, Paul would have gotten there by now if he was coming. She worried that he had been killed by the person who had murdered that other guy. She worried that he had taken off in a panic and was lost somewhere, without his car or his phone. Any number of things could have happened to him and none of them were good. Still, she couldn't help but hope that he wouldn't show up here\u2013that he had gone home.\n\nShe turned back to the paper she had last been reading. Some of her students were bright, some not so. This one was brilliant. She smiled as she read it, making notes in the margins and red-penning the few atrocious mistakes. Then she added a complimentary note on the bottom of the last page and tossed it into the pile. She was reaching for the next paper when the phone rang. It scared her so badly that she yelped and jumped.\n\n\"Hello?\" she said, her voice still ringing with fear.\n\n\"Hey, Linda. It's Matt.\"\n\n\"Oh, hi, Matt.\" At least he didn't sound stoned this time.\n\n\"Hey, I was wondering if you'd heard anything from Paul?\" He sounded like a child who had become separated from his mother in the mall and was just now approaching the help desk.\n\n\"I haven't heard a word, sorry. There are two cops parked outside my door though. I'm not sure whether to hope Paul shows up, or hope that he doesn't.\"\n\nTo Matt, her voice sounded so sad that it made him want to cry. \"I know this is all hard for you. It's hard for me too. I just want Paul to be okay, you know? If you hear anything, please let me know? I'm going half-crazy here.\"\n\n\"Matt, can I ask you something?\"\n\nThere was a long pause, then Matt sighed. \"Sure.\"\n\n\"What happened to Paul? He said something bad happened at work. You were with him . . .\"\n\n\"Linda, look. Paul wouldn't want me to tell you. I mean, it's not my place, you know?\" There was another long silence, followed by a heavy sigh. \"I guess it doesn't make much difference now.\"\n\n\"It might make a difference to me. I need to understand all this.\"\n\n\"Okay, it's like this. We were down in the tunnels and there was this stuff, this blue, glowing stuff. And Paul got it on him and when we got out of the tube, he doubled over, convulsed, and died. I called the paramedics but he was dead before they got there. They took him to the morgue and told me that they would contact me the next day for arrangements.\"\n\n\"And you didn't think to call me? The man I love died and you didn't even let me know?\" She was angry now, the heat of it scorching her face and drawing tears to her eyes.\n\n\"There wasn't a lot of time. See, a couple of hours later, Paul came walking in the door. But he looked bad, man. Really bad. Like, he had all these pustules and black marks on his face. And he couldn't talk.\"\n\n\"He said he was sick. He had laryngitis.\"\n\n\"Naw, man. He couldn't talk. All that would come out is this growl and groan and shit. I don't know what happened to him, but he looked like a zombie.\" Matt broke off, feeling very near tears himself. There wasn't a lot in the world that Matt cared about. Paul was at the top of the list. \"I'm sorry about all this. I should have protected him, man. I should have gone in his place.\"\n\n\"There's no way you could have known.\" She swiped at the tears that trailed down her cheeks. \"But I appreciate you being honest with me. And I will call you right away if I hear from him.\"\n\n\"I think it might be a good idea if you had a computer or a pad and pen ready in case he does make it there. Dude can't talk, ya know? He'll need some way of communicating with you.\"\n\n\"I'll set the iPad next to the door. That should do it. You take care of yourself, Matt.\"\n\n\"You too, Linda. Bye.\"\n\nShe clicked off the phone and tossed it onto the sofa cushion. Paul was sick with something and that made her feel a little better about his erratic behavior. It didn't explain away everything, but it explained some. She had no idea what to expect when and if Paul showed up, but there was one thing she knew beyond question: Paul was a good and decent man and she loved him more than life itself. When he showed up \u2013 if he showed up \u2013 she would do whatever she could to help and protect him.\n\nMilligan sat at Lopez's desk, the cell phone tucked neatly between shoulder and cheek as he jotted down notes in his little ringed pad. Twice, he nodded and dislodged the phone; asked Gebhart to repeat what he had just said. In the middle of it all, Lopez walked in with two cups of coffee and Milligan gave up the chair to him.\n\n\"Just keep me posted. And call me the second the lab results come in.\"\n\nMilligan ended the call and laid the phone on the desk, taking a second to jot down a few thoughts on his pad. When he looked up, Lopez was staring at him with a crooked smile on his face.\n\n\"What time did you come in?\" he asked Milligan.\n\n\"Six. I woke up at five and couldn't go back to sleep, so I figured I would come in here and get a head start reviewing those tapes from the security camera.\" He took the coffee that Lopez offered and blew on it.\n\n\"And what did you find?\"\n\n\"Just as expected. Not a damn thing. Tremblay pulls up in front of the SuperMart, gets out, goes inside. Nobody is with him, nobody goes near the car until the cop walks over to ticket him for parking in front of the dumpster.\"\n\n\"So, that leaves Tremblay driving across Kansas with a dead body in his car.\" Lopez frowned and shuffled through some papers on his desk.\n\n\"Maybe he didn't know the body was back there,\" Milligan offered feebly.\n\n\"Not a chance,\" Davis said, sliding into the chair next to Milligan. \"That thing stunk to high heaven.\"\n\nMilligan shifted in his seat, turned a few pages in his notebook, smiled. \"My guy talked to this Matt character . . . and I do mean character. And then they went together out to the sewer site where the two guys had been working. He managed to get a sample of this blue glop that Matt says his buddy got on him right before he collapsed. He's gonna call me when the lab results are in.\"\n\n\"So,\" Lopez says, rocking slowly. \"These two chuckleheads are out in the sewers doing their thing. And Tremblay steps in this blue glop, as you call it. He gets topside and keels over, the paramedics take him to the morgue. Coupla hours later, he walks right back in the door looking like shit. Then he takes off in their car, headed for LA so he can see his lady. Somewhere along the way, he picks up a dead guy or he kills a guy, and then he eats part of the corpse.\"\n\nDavis shook his head. \"There's no question. The bite marks on the body are a dead-on match for the bite marks on the pig. And we found fingerprints on the barn door handle that match Tremblay's.\"\n\n\"Damn,\" Lopez sighed. \"I was hoping for the girl's sake that this guy was innocent. But apparently, he killed Manning, stuffed him in the back seat, and started snacking on him. Then he killed the pig, ate part of it, stole the farmer's truck . . .\"\n\n\"And if he sticks to his plan, he should hit LA any time now.\" Milligan nodded.\n\n\"I'm going to let my guys know. They need to be prepared. This guy is some serious kind of crazy and they need to treat him like the homicidal maniac he is.\" Lopez picked up the phone and began to dial.\n\nFour miles from Linda's apartment there was an abandoned grocery store. It had once been a part of a thriving strip mall, and when it had failed, it had taken the other stores with it. Weeds grew through the pavement now, the streetlights dark and the doors of the place boarded up. You could see the front parking lot from two different streets, so Paul drove around back to the loading docks. He backed the truck up into one of the recessed loading bays and shut off the engine. He had to collect his thoughts and write down what he needed to say to Linda. Then he had to eat his last piece of pig meat and try to restore himself again.\n\nRounding third, he thought. Almost home.\n\nHe rummaged around in the glove compartment, found his pen and an old receipt pad. He tore one page from the pad and flipped it over so he could write on the blank side. Then he stared into space. He figured he should start off with the most important thing first, the thing he most wanted her to know.\n\nLINDA, I LOVE YOU.\n\nAnd then he stared some more. He heard a noise off to his right but it turned out to just be a cat rummaging through a pile of trash in the hopes of finding a meal.\n\nTHE REASON I'M LIKE THIS IS BECAUSE OF THAT AWFUL STUFF I STEPPED IN DOWN IN THE SEWERS.\n\nHe had never been especially good with words, but they failed him now more than ever. The way he had it figured, in about twelve hours he would be down to, \"Og luv u. Og sory.\"\n\nI DIED AND WENT TO THE MORGUE BUT THEN I WOKE UP AND I WAS LIKE THIS. I COULDN'T TALK AND THAT'S WHY I'M WRITING EVERYTHING DOWN.\n\nGod, he just wanted her to understand. All he wanted was to be able to die without her hating him for all eternity.\n\nI DID KILL THAT GUY, YES. BUT IT WAS AN ACCIDENT. HE TRIED TO JACK MY CAR.\n\nHe tried to think back to the rest area where he had killed that man. Yes, the guy had tried to steal the car. But had it really been an accident? He couldn't remember thinking that he should kill the guy. Then again, he couldn't remember a lot of things now. But he was absolutely sure that the whole incident had been an impulse. Some part of him beyond his control had bitten into the guy's neck before Paul had even had a chance to form a clear thought. It had been as much instinct as a tiger taking down a gazelle.\n\nPLEASE DON'T HATE ME. I WAS SO SICK AND SO HUNGRY AND I BIT INTO THAT GUY'S NECK BEFORE I KNEW WHAT I WAS DOING. AND THAT'S HOW I FOUND OUT THAT EATING HIS FLESH MADE ME NORMAL AGAIN. IT RESTORED ME.\n\nIt was the lamest thing he had ever said. No way on earth could he get out of this without Linda hating him and thinking him a monster. No way.\n\nTHEN I JUST SORT OF PANICKED AND DIDN'T KNOW WHAT TO DO. BUT I FOUND OUT THAT I CAN EAT PIG MEAT AND GET BETTER, SO I GUESS IT'S ANY KIND OF RAW MEAT.\n\nHe stared off into space, unblinking, for about five minutes. His train of thought had derailed and try as he might, he couldn't get it back on track.\n\nI JUST CAME HERE BECAUSE I WANT YOU TO KNOW THAT I LOVE YOU MORE THAN ANYTHING IN THE WORLD. EVERYTHING I'VE DONE, I DID FOR YOU.\n\nHe wondered briefly if that would endear him to her or make her feel responsible, guilty. His eyes traveled over the notes and he gasped at seeing that one. He shoved one hand into his right pocket and felt around. The ring box was still there.\n\nI BOUGHT YOU A RING MONTHS AGO AND I WAS GOING TO PROPOSE AS SOON AS I MOVED OUT HERE. I WANT YOU TO HAVE IT NOW. AND WHEN YOU LOOK AT IT, I WANT YOU TO REMEMBER ME, THE ME THAT USED TO BE. AND I WANT YOU TO REMEMBER THAT YOU WERE THE MOST LOVED WOMAN IN THE WORLD. AS LONG AS I'M ALIVE, YOU ALWAYS WILL BE.\n\nAnd then it struck him. If he died \u2013 not like he'd died before, but if he really died \u2013 then no one would love Linda.\n\nHe started the engine and pulled out of the loading bay slowly, turning on his lights. With new conviction, Paul turned down that one last road that would take him to Linda and which would, he hoped, put an end to his suffering.\n\n## CHAPTER TWELVE\n\nMilligan was on his way back from the rest room when his phone rang. He stopped in the middle of the hall, nearly causing two patrol officers to run smack into him. Accepting the call, he mashed the phone to his ear and growled.\n\n\"Milligan.\"\n\n\"It's Gebhart. I've got the test results back.\"\n\nMilligan was walking again, trotting along the hall and beginning to pant. \"Hang on. Just a second.\" He reached Lopez's office, where Davis and Lopez were reviewing security tapes, hoping to find the one from the rest stop where Paul had killed Manning. \"I'm putting you on speaker phone. I'll never be able to repeat all those big words.\"\n\n\"So, I've got the results from the lab tests on that blue stuff. I'll send them to you in a minute. The gist of it is this: There were a hundred and forty-two separate compounds found in that sample. Everything ranging from petroleum to hallucinogenics to radioactive material. There were also a lot of compounds in there that our lab boys couldn't identify.\"\n\n\"Do they have any idea what that stuff might do to a person?\" Milligan asked tentatively.\n\n\"Not really. They'd have to test it, I guess.\"\n\n\"So, it's possible that this stuff could be responsible for what happened to our perp?\" Lopez added.\n\n\"Most definitely. I mean, there's enough crazy-assed stuff in that blue goo to drive a nun insane. If it didn't kill her.\"\n\nMilligan smiled. \"'Crazy-assed stuff.' Is that a technical term?\" He chuckled.\n\n\"It is now, Milligan. And you can quote me on that.\"\n\n\"Thanks. I owe you a beer when I get back, Gebhart.\"\n\nThe email came through then, and Milligan forwarded it to Lopez so he could print it out. \"So, Paul Tremblay got into this nuclear waste crap and it turned him into one seriously crazed fruit loop.\"\n\n\"That's about it, I guess.\" Lopez thought for a moment. \"It's entirely possible that he's not dangerous anymore. Or if he is, that it's not his fault. God knows what kind of insane stuff it made him see.\"\n\n\"Still, he's killed one man. Responsible or not, he still has to be treated as a danger.\" Milligan sighed. He didn't like this, not one bit. \"You better make sure your guys protect themselves . . . and Linda Gilchrist.\"\n\nLopez nodded. \"Due force.\"\n\nIn the wee small hours of the morning, most of LA was still asleep. Sunrise was still a good three hours away and Paul putted along in the truck, keeping just under the speed limit so as not to attract attention. In his right pocket was the ring box and the folded piece of paper on which he had written his message to Linda. In his left pocket was the sample he had taken of the blue goo. He had just swallowed the last mouthful of that vile, festering pig meat and was feeling much better for it.\n\nBut as he turned onto Linda's street and looked ahead one block to her apartment, he noticed something that he thought odd. There were two dark cars stopped abreast of one another, four men standing in the middle of the street, talking.\n\nPaul stopped the truck, put it in reverse and backed into the first driveway he saw. He cut the engine and watched. He could see each man holding a paper cup of coffee. Handcuffs dangled from the belt of one man. Cops. There were cops guarding Linda.\n\nPaul started the truck and drove away slowly, so as not to attract attention. He watched his rearview mirror as much as he could, making sure that none of the cops followed. If they were looking for the truck, he was done for. Still, they were looking for a Chevy truck with a certain license number. His truck was now a Ford with a different license number.\n\nPaul turned left and drove two blocks before he pulled over, shaking. He needed to get to Linda before his last meal wore off. The cops were guarding her and he had no way of getting in touch with her. Suddenly, he felt panic rise from deep inside his gut, threatening to make him yack up the last of the meat. He choked it back, began to sob, then forcibly got hold of himself.\n\nHe needed a cellphone capable of texting. He knew all too well that if he got a burner phone somewhere, it would take too long to activate. Fingers drumming on the steering wheel, he thought hard about his dilemma. There was only one answer and he had scant little time to make it happen.\n\nHe put the truck in gear and drove off quickly. He had passed a Mega Mart on his way into town. Mega Mart carried iPods capable of texting and they had a coffee shop inside with free Wi-Fi. He could buy the iPod and send a text to Linda, asking her to meet him there. But he had to get in and out of the store before the meat wore off and he started to look like a corpse again.\n\nAt that hour, the Mega Mart parking lot was nearly empty. The occasional parent making a diaper run, or flu-patient looking for medication pulled in but that was about it. Paul parked the car at the side of the building, as far away from the lights as he could get. Then he made his way across the wide parking lot and in through the sliding doors.\n\nOne leg dragged a bit, his foot suddenly leaden. He supposed it had something to do with the crunch he had heard some time back when he had pushed down on the accelerator. There was nothing he could about it except to compensate with his hip. In this hitching, jerky manner, he made his way back to electronics and, by pointing and nodding, managed to get the clerk to understand what he wanted. She eyed him peculiarly and made a face when he handed her his credit card. But he got away with the whole thing and by the time anyone discovered his face on the security tapes, he would be long gone.\n\nHe headed quickly for the front of the store, let the greeter check his receipt, and then plunged out into the cool night air. Where the truck was parked, it would certainly be within range of the Wi-Fi signal. He opened the box, turned on the iPod, and frowned in concentration.\n\nIt came fully charged, but the charge wouldn't last forever and the only charger needed to be plugged into the wall. But the thing picked up the Wi-Fi signal like a champ and he began the tedious process of registering it. Of course, he used completely bogus information, but it got the task done and out of his way. Then he had only to input Linda's number. Then he entered the text carefully. It was hard to maneuver his big, floppy fingers over the virtual keyboard, but he made it work.\n\nDON'T TELL THE COPS. THIS IS PAUL. MEET ME IN THE EAST PARKING LOT OF MEGA MART. I'M THERE NOW.\n\nHe sat back with a smile and waited for her answer. In his mind, he could see her pulling into the lot, racing across the pavement to him, hugging him. She would forgive him everything once she found out what had happened to him down in that tunnel. She would tell him that she loved him, no matter what, and then she would climb into the truck and embrace him. They would drive off into the sunset together . . .\n\nThe sound of brakes squealing in his mind. The ocean was there, where the sun set.\n\nThey would drive south, to Mexico, where they would live happily ever after. Yea, that was it.\n\nBing!\n\nPaul looked down at the iPod. Linda had responded.\n\nBE THERE IN FIFTEEN. I LOVE YOU.\n\nShe still loved him. Even after all this, she still loved him. Paul felt like crying again, only this time it was because he was so happy. He shoved his right hand into his pocket, just to check on the ring one more time. The ring and the paper were both still there. Good. In fifteen minutes, he and Linda would finally be together again and they would stay together this time, no matter what.\n\nExcept that Linda would never make it there.\n\nLinda grabbed her purse and keys from the foyer table and yanked open the door. The night was cool and she was wearing short sleeves, but she didn't care. She was on her way to see Paul, the man she loved more than anything else in the world. She had never loved anyone else. For her, Paul was IT.\n\nShe made it as far as the car before the officer approached her, his face full of good humor. He had apparently just started his shift. The scent of coffee rolled off him like water off a duck.\n\n\"Ma'am, might I ask where you're going?\" He smiled, just to take the edge off.\n\n\"Well, I'm running to the store. I need some . . . um . . . feminine products.\" She smiled sheepishly, hoping he bought it and would leave her alone.\n\n\"I see.\" He seemed to think this over for a moment. There were only two officers and they were supposed to keep her and her apartment under surveillance. The only way he could do that was if he went with her. \"I'll just ride along with you, then.\"\n\nStunned, she mulled this over. She could jump in the car and lead the cops on a merry chase, then text Paul and tell him, The key's under the dead plant. Text me when you get into the apartment. But they would probably arrest her for obstruction of justice or some silly thing. She wouldn't do well in prison. She looked awful in orange and she didn't want to be anybody's bitch. \"Fine,\" she sighed at last.\n\nShe unlocked the car doors and slid inside, wondering what the hell she was going to do. Paul was expecting her and she didn't want him to think she had just bailed on him. As she started the car, she realized the obvious answer. She would have to text him from inside the store and have him meet her during the day, while she was at work and could more easily slip away.\n\nShe backed the car out and stopped long enough that the officer could tell his partner where they were going. Then she drove straight to the convenience store on the opposite corner from the Mega Mart.\n\n\"I'll just be a minute,\" she said with a grin. And then she dashed into the store.\n\nShe managed to walk all the way around to the back of the store, where the feminine products and medications were. She texted as she went, trying not to let the officer in the car outside see her.\n\nCOPS WATCHING. COULDN'T GET AWAY. WILL TEXT YOU DEETS TO MEET ME DURING WORK. LUV U. L\n\nShe grabbed a box of tampons and a candy bar as she passed, then hurried to the counter and slid her card in the machine. She was very frustrated and shaken by the whole thing. Being a criminal didn't come naturally to her. Heck, she had never even shoplifted, or eaten grapes from the bag while she was still shopping. She could only imagine how disappointed Paul must be, sitting there in his car, waiting for her. It made her want to cry.\n\nShe walked out the door then, the officer still sitting in the car, smiling like a goon. She knew it wasn't his fault. He was just doing his job. But she hated him for it nevertheless.\n\nPaul looked down at his phone one more time, just to make sure he had read the message right. Linda wasn't coming. Cops were watching her. She wasn't coming. Those two thoughts ran through his mind in an endless loop until he thought he would go mad from it. Judging by the blotches on his hands, his meat was wearing off anyway. The best thing to do, he mused, was to go somewhere and wait for Linda to send him another message. Then he could find another meal somewhere and be in fine shape when he met Linda.\n\nTo that end, he drove back to the abandoned grocery store and backed the truck into the loading bay, where it couldn't be seen unless you walked right up on it. He would wait there, in the shadow of the broad building, until Linda sent for him. Then he could drive to a place of her choosing and deliver his message of love.\n\nLinda gathered her things for work, tucked the ungraded papers into her laptop case, and drained her coffee mug. She hadn't had more than three hours of sleep that night. She felt like shit and she was sure she looked like it too. There was a meeting at nine with the department heads, one at two with the admissions committee. And at some point, she knew, she had a meeting with one of her students who was having problems with his senior dissertation.\n\nShe walked out the door, making sure she had engaged the lock before turning toward the police car. She drew a bead on it, marched straight up to it and leaned on the door. \"I'm going to work now. Is one of you going to come with me?\"\n\n\"No ma'am,\" answered the driver. \"I think you'll be safe on campus. Besides, campus security knows what to look for.\"\n\n\"Okay, then. Have a good day, officer.\" She smiled her best smile and turned away, walked back to her car.\n\nShe backed out of the driveway and drove around the corner, where she eased into a parking space and kept the engine running.\n\nON MY WAY TO WORK. ALONE. WHERE R U?\n\nShe tapped one nail on the phone and waited for a response. One glance in the mirror told her that she hadn't yet been followed. That much was in her favor.\n\nLOADING DOCK OF THE ABANDONED GROCERY STORE ON FIFTH.\n\nShe smiled more than she ever thought possible. Aside from the aiding and abetting of a felon, this was damned exciting stuff.\n\nON MY WAY.\n\nPaul smiled and high-fived the air. He was finally going to see Linda. This awful mess was about to be over. God, how he loved her. And needed her. And . . . well, that part of him didn't work anymore . . .but if it did, he would want her too.\n\nHe looked in the mirror, frowned at the blotches and sagging. His hands were a mess and his right leg still didn't work quite right. Like a drug addict, he needed a fix. But where in the hell was he going to find a hunk of meat ready to be eaten in the next five minutes.\n\nA clang echoed across the loading docks as a trash can fell over two bays down. The cat was back and, judging by its desperate searchings, it was starving. Paul eased himself out of the truck cab and left the door open. He needed that cat. There were lots of cats. The world wouldn't miss this one.\n\n\"Here, kitty, kitty, kitty,\" he began, suddenly realizing that the cat heard nothing but a feral growl coming from his throat. He watched as its tail puffed up and it arched its back.\n\nHe tried a different tactic then. Having no idea what he smelled like, he at least counted on the leftover scent from the pig meat to guide the cat to him. He stooped down and held out one hand, fingers together as if he were holding food in them. The cat's tail un-puffed and began to sway slowly back and forth as it regarded him. Paul wanted to yell at it, to pounce, but he kept calm and held still.\n\nThe cat took one tentative step toward him, making a deep, guttural sound and then stopping. Its whiskers twitched as it sniffed the air and it suddenly walked in a circle. Then it stretched out its neck, taking another step and another. A few more steps and it would be within Paul's easy grasp. He would be merciful. He would kill it quickly.\n\nThen there was the squeak of shocks as a car went over a speed bump, the roar of an engine as it accelerated over it. Paul turned in time to see Linda's car round the corner. The cat screamed and ran off.\n\n\"Godammit!\" Paul growled.\n\nHe flipped the hoodie up over his head and turned his back. The ring and the note were in his pocket and he had to give them to Linda, but he couldn't let her see his face. He heard the car door slam, jumped a little. And then he had to use every ounce of determination he had in order to not turn around when he heard her voice.\n\n\"Paul! Oh Paul!\" she cried, running toward him. Her arms were open and she was smiling.\n\nHe reached out an arm, thrust it out behind him in a STOP motion. She stopped mid-stride and didn't make a sound. Paul thrust the paper out toward her. He gave it a little shake so she would be sure to know that he wanted her to take it.\n\nShe walked up to him more slowly then, taking the paper from him with just thumb and forefinger. Her eyes were fixed on the back of his head as he stood there, motionless. And for the first time since all of this had started, Linda felt scared.\n\nShe began to read the note and as she read, Paul reached into his pocket and grabbed hold of the ring box. He opened it carefully, then he thrust the ring out behind him.\n\n\"Oh, Paul,\" she gasped, without reading past the first sentence. \"I'm so sorry all of this happened to you. Please, won't you turn around?\" She looked at the ring then, took the box delicately from his hand. There were some blotches there, black and green spots scattered across his hand like paint splatters. \"It's beautiful, Paul. I love it so much. Please, turn around, honey.\"\n\nHe shook his head, refused to budge.\n\nShe started to walk around him then, to circle around to look in his face. Paul turned, growled \"No,\" and kept his face from her.\n\n\"I love you, Paul, and you have to know that I'll marry you. Of course I'll marry you.\" There was a lilt in her voice, a girlish joy that made Paul's heart melt and he nearly turned around then.\n\nShe tried to dart in front of him but he was too quick. He spun away before she could see his face.\n\n\"Paul, honey, I know this all seems so bad. But I love you and I don't care what happens or what you look like, I'll always love you. Please, Paul, turn around.  We'll get married right away. I love you, no matter what. And whatever's happened, we can fix it. Together.\"\n\nIt sounded to Paul as though she was going to cry any minute. The pain in her voice broke his heart and he began to turn, slowly, his chin tucked against his chest, his eyes on the ground. He was terrified of how she would react. He didn't want to see the look of horror in her face. But he had to know.\n\nLinda sucked in one last breath of air and held it, watching as Paul turned. She braced herself, gripped the note and the ring so tightly that she feared she would break the ring box. She was afraid to even blink for fear the tears would start flowing.\n\nHis face came into full view then and he lifted the hoodie up and away from his face. Linda gasped, clapped one hand over her mouth, and began to shake. It started as a small tremble in her lips, then spread to her head and her hands and finally it took over her entire body. She shook so hard that she dropped the ring and the note.\n\nShe took a step back.\n\nAnd she took another.\n\nPaul watched as, screaming, she turned and fled to her car. He reached out for her, the very worst thing he could have done, and tried to follow her. His dragging leg slowed him down and, in slow motion, she leaped into the car and turned the key.\n\n\"Please, Linda! I won't hurt you, I swear. I love you. Come back!\"\n\nNone of that got through to her, though. In her eyes, a monster was coming for her, reaching out to snag hold of her and growling his hunger. She threw the car into drive and roared past him, taking one last look in the rear view mirror to see Paul standing there, motionless, his head cocked to one side and his arm still outstretched.\n\nAnd just like that, she was gone.\n\n## CHAPTER THIRTEEN\n\nPaul stood there for a very long time, unblinking and numb. Then he walked back to the truck and got in, slamming the door after him. He screamed and raged, pounded his fists on the steering wheel until one of his fingers broke off and fell with a soft thump to the floor. Then he started to sob, great wrenching sobs that shook him through and through.\n\nWhen he had a chance to calm himself and think things through, he realized that the note and the ring were still outside on the pavement. The wind had caught the note and so it was slowly blowing down the length of the loading docks. He got out right away and began to chase after it, his feet slapping and dragging on the pavement until finally, in desperation, he lunged for it. His fingers managed to slap down on one corner of the paper and he pulled it to him. He collected the ring on the way back to the truck.\n\nWhat to do now? Linda had been terrified. There was a chance, though, however slim, that she would shake off that initial fear, reconsider, and come back. He sat in the truck for a long time, clinging to that feeble hope. But Linda never came back. She hadn't even read the note and now she would never know how much he loved her.\n\nHe would have to try again to see her. He needed to refresh himself, wait until dark, and then go to her apartment. He scrunched down on the seat, trying to stay out of sight in case someone drove around the back of the store. That's when he saw the iPod and remembered that he had it. He snatched it up and turned it on, found Linda's number.\n\nLINDA, I LOVE YOU SO MUCH. I WOULD NEVER EVER HURT YOU. PLEASE, COME BACK AND READ THE NOTE. I'M NOT CONTAGIOUS AND I PROMISE TO STAY BACK FROM YOU. PLEASE.\n\nHe waited for what seemed an eternity for her answer. And when it came, he began to weep again.\n\nI CAN'T.\n\nNothing more. Just \"I can't.\" He stared at it until the words doubled and blurred and finally disappeared into a soft white puddle.\n\nWhen he had gotten hold of himself and the sobs had faded into light hitches in his gut, he tried to focus on a solution. Linda had to read the note. If she didn't read it, then everything he had gone through was for nothing.\n\nIn order for Paul to give her the note, he had to be normal looking. And that meant feeding again. He looked around him, to see if he could spot the cat. The beast had bailed and was nowhere to be found. Paul fixed his eyes on the field, the last place he had seen the cat just as it had disappeared from sight. He kept watching, hoping, praying that the cat would return.\n\nAfter nearly an hour of sitting as still as a statue, he saw the tall grass beyond the curb move. It rustled a bit and then it began to dance. The cat peeked its head out, its tail dragging low on the ground. It was cautious, ready to turn and run.\n\nPaul let the animal make its way across the parking lot. He waited until it was over by the trash cans where it felt safer and more comfortable. He gave it time to gain courage. Then he slowly opened the truck door and simply slipped down onto the ground. He sat there, cross-legged, not moving, for a good half hour. The cat hid behind the trash cans, looking out every now and again but not making a move. And then, the most miraculous thing happened: The cat simply walked out from behind the cans and straight toward Paul.\n\nStill, he didn't move. He let the cat sniff him, watched as its whiskers twitched and its eyes scoped him out. The tentative cat rubbed its head against his leg, turned, rubbed its way back the other way. If Paul had been of the breathing variety, he would have exhaled just then.\n\nHe knew he had one chance and one chance only. If he let the cat slip out of his grasp now, it would never come back to him. He felt kind of bad about the whole thing. The poor creature was hungry and it had trusted him.  Despite the stench of decay and the festering sores, the cat had come to be his friend.\n\nMoving faster than he ever thought possible, Paul lashed out with both hands and grabbed hold of the cat. It spit at him and claws lashed out at Paul's hand, but still he held on. It tore great hunks of flesh from his arms and shredded his shirt as he brought the creature to his mouth. He took one giant bite then, directly into the cat's neck where he supposed its artery would be. Blood bathed his hoodie and spattered his face. Mercifully, the cat died instantly.\n\nPaul spat out a hunk of the cat's neck. It was covered in fur and he had almost choked on it, so he spat it away and took a fresh bite from the cat's lean neck. Then, he let it sit limply in his hands for a moment, the blood draining onto the concrete. \"I'm sorry,\" he said in his growly voice, feeling genuine sorrow for having taken the cat's life. But his body felt instantly better. His mind cleared.\n\nHe took the cat back to the truck, sitting it on the front seat beside him. He would eat the rest of it right before he went to see Linda tonight, after dark, when things would be easier. He would drive the truck to the block behind Linda's apartment and sneak up to the back door. She couldn't possibly refuse to see him, not when he looked so normal.\n\nHe checked himself in the mirror. Skin clear, eyes bright; once he cleaned the blood from his face, he would look like a million bucks. Nothing to do now but wait for dark. Then he would make one last attempt to see Linda.\n\nLinda had been so shocked at Paul's countenance that she had nearly run off the road trying to get away from him. She had made it as far as the Quickie Mart, where she had pulled into the parking lot and put it in PARK. She was crying so hard that she couldn't see the dashboard and her body was trembling.\n\nPaul had looked like a corpse. She thought about that: The way Paul had looked, how he had sounded when she spoke to him on the phone, the blue goo that he had talked about. It wasn't his fault, she knew. He hadn't asked to be like that. Still, he had horrified her with the way he looked. And he had brought her a ring. A ring, of all things!\n\nShe wanted to forgive him, to go back and apologize and find some way to help him. She just couldn't stomach it. Not even for Paul. Neither could she turn him in to the cops. She owed him that much.\n\nShe wiped her face and cleared her nose, then put the car into reverse. She had meetings at work and then she would buy a bottle of nice, sweet wine and go home. She thought about that term now: home. She had an apartment, a few friends at work, a nice job, everything but Paul. And Paul was not coming back to her, at least not the Paul she had known all these years. There needed to be a new plan for the rest of her life, one that didn't include Paul. The thought of it made her cry again.\n\nShe went to the college, walked into her first meeting ten minutes late. The rest of the day, she just phoned it in, getting to her final meeting, managing to hurry it along. Come tomorrow, she might need a mental health day. But for the rest of this day, she managed to hold it together.\n\nIt took some thought for her to remember to get the wine on the way home. She grabbed some chocolate as an after-thought, then stuck a pre-fab sandwich into her mix. Now, she figured, she had everything she needed to make it through the night.\n\nThe police car was still out front when she got there. She waved to the officers inside and parked her car in the driveway. Even if she couldn't spend the rest of her life with Paul, she wasn't about to give him up to the cops. She got out of the car, grabbing her laptop case and grocery bag from the backseat and locking the car before she went inside. The sun would set in about an hour. She hoped there would be something upbeat and fun to watch on TV. This was not the night to be grading papers.\n\nLinda went inside, pushing the door closed with her butt as she struggled to kick off her shoes. She dropped the laptop off next to the table, and then went to the kitchen to get a bottle opener and a glass. She padded through the house in stockinged feet and collapsed onto the sofa. Ferris Bueller's Day Off was on TV. Perfect.\n\nPaul waited until a half hour after the sun had set. He had gotten bored while he sat and did nothing, so he had been picking at the cat meat the whole time. He looked into the mirror and liked what it saw. There was no better time than the present.\n\nHe started the truck and put it in gear. Oddly, he hadn't seen another living soul the whole day. Not even the bums came to hang out there, he guessed. Today must be his lucky day.\n\nHe turned right and then left and then right again, trying to scope out the backyard and gauge where Linda's apartment was from the back. He knew that it was brick, that it had a tall oak tree on the side. And then there it was. The tree stood a good twenty feet higher than the apartment and its spread offered shade to the side yard of the building adjacent to it. It was a pretty tree, and it seemed to him that it was just standing there, waiting for its tree house to be built. Paul let the car idle.\n\nA cursory glance proved that his method of encroachment would not be as easy as he had thought. There was a fence which separated Linda's apartment from the neighbor's mid-century ranch. It was a six foot fence, made of wood, and it looked fairly new. The fence surrounded Linda's apartment rather than the neighbor's, so he had hopes that there would be no dog standing guard.\n\nHe parked the truck and got out, shoving a huge piece of cat meat into his mouth and chewing quickly. He would need all his strength and agility to crest that fence. And he wanted to look his level best when he was face to face with Linda. But as he approached the yard, he realized the error he had made. The wood fence did indeed surround Linda's yard. But there was a chain link fence surrounding the backyard of the neighbor's house.\n\nInside that yard, watching like a predatory cat in the jungle, was a Rottweiler. Lying beneath the bushes as he was, even the dog's eyes couldn't be seen. But when he stood up and made a run at the fence \u2013 just as Paul had put his hand on the latch \u2013 Paul got a clear picture of just how much trouble he was in. The dog stood at least waist-high, his head broad and his chest even broader. And when he leaped, the dog nearly cleared the fence.\n\nPaul took three stumbling steps backward, arms flailing and his mouth hanging slack. The dog strained at his chain link bonds and barked and snarled. Lights came on in the house where he lived and someone opened a door and came out on the back porch. Paul turned to run then, managing to conceal his face before the woman rounded the corner. By the time she arrived at the side yard, the dog now whining and panting, begging for a pat or some food, Paul was back at the truck.\n\nSo, no way in hell was Paul going to make it to Linda through the back yard. A full frontal assault was required and he knew no other way to go about it. He walked to the corner, rather than give himself away with the chugging growl of the truck's engine. Then he turned and walked to Linda's street. He was coming at the apartment from the police car's rear, keeping to the natural shadows on the sidewalk and keeping his head down. If he was very lucky, he would get right to her driveway before the cops realized he was there.\n\nHe walked briskly, a man with a purpose, a man with a plan. His hands were in his pockets, on the note and the ring box. He had neatly refolded the note and written on the outside a message that he hoped would capture Linda's heart and make her read the letter:\n\nFOR OUR LOVE\n\nREAD ME\n\nI BEG YOU!\n\nHis plan was to out-distance the cops, make it to the door, and either ring the bell and hope that Linda made it there before the cops, or simply burst inside. Then he would shove that note into Linda's face where she couldn't miss the message outside. Hopefully, that would do the trick.\n\nWhen he came even with the neighbor's driveway, he heard the dog bark in the distance. No doubt, the cur was still upset by Paul's visit. So was Paul.\n\nThe cops notice him as he turned up the driveway. He heard the car door creak a bit as it opened and he hurried his pace. It was no use. The cop was coming toward him now, his face set into a very serious scowl.\n\n\"Hey, buddy. Can I talk to you for a second?\" The cop waited a beat and when Paul didn't stop, he said, \"Hey, pal! Hold it! Stop! Police!\"\n\nNow the other car door creaked open. Paul heard footsteps behind him, sensed without looking that the cops were chasing him.\n\n\"Stop or I'll shoot!\" said the other officer.\n\nPaul was on the front step now and he let his hand flail outward and smack into the doorbell. He turned then and faced the cops. They stopped for a minute, sudden realization stealing their momentum. Their guns came up, both at the same time, aimed directly at Paul. Paul started to raise his hands, the ring in one and the note in the other.\n\nAnd that's when everything went horribly, terribly wrong.\n\nThe sound of the door opening registered briefly in Paul's brain, as did the nearness of the gun-bearing police officers. Then Paul did the worst, most dangerous thing he could have done. Paul Tremblay turned and dove toward the door. His only hope was to stun the police into not shooting him.\n\nLinda had opened the door and she stood there, her mouth forming a tiny \"o\" as she gawped at the sight of Paul. His sudden leap toward her stunned her as well, though not as much as the cops.\n\nThe policemen, who were far too keyed up to think clearly, panicked and opened fire. Paul felt the bullets strike his body and pass through it. He didn't spend any thought on where they went after that. His eyes lifted and he caught the expression on Linda's face just as he fell. His dive had taken him in through the door, but then gravity took hold and dumped him at Linda's feet.\n\nFocused on the task at hand, desperate to complete his mission, he thrust up both hands, the one with the note and the one with the ring. His eyes pleaded. But by the time his brain caught up to his vision, he realized that it was too late. The bullets that had passed through him as though he were made of air had hit Linda in the chest and the abdomen. Her eyes opened wide and her knees buckled. Paul let out a yell as he watched her fall, felt the jarring thud as she hit the tiled floor.\n\n\"No!\" he yelled in his gruff zombie-man voice, grabbing onto her and shaking her. \"No, Linda! No!\"\n\nHer head turned and she brought her eyes to bear on him. Blood was leaking from her in four places, covering the tiles and his hands as he clutched at her. He began to sob, his eyes meeting hers, his face a study in misery. \"Don't leave me, Linda. Please don't leave me.\"\n\nAnd then the most startling, brilliant idea hit him. In his other pocket was the jar full of blue goo. It had killed and then resurrected him. Maybe it would resurrect Linda, too. He reached into his pocket and grabbed the jar, twisting the lid off in one smooth movement. Then, he spread the stuff over her, pouring some in each wound, smearing some on her face, getting it into her mouth. He knew how it would look to the rest of the world, but if it worked, if it saved her, it would all be worth it.\n\nThen Paul took the only out he could. He already had four holes in him. His heart had stopped and he no longer breathed. So, Paul just collapsed on Linda's lifeless body and waited.\n\n## EPILOGUE\n\nThe two cops would forever talk about that night. They told and retold the story about how they killed the zombie over the next few days, lengthening and embellishing it until there was very little truth to it. The official report that they filed with their superior, however, was far more accurate.\n\nUpon seeing that they had shot not only the zombie boyfriend, but Linda Gilchrist as well, they had run to the doorway to see if they could save the girl. When they arrived, they spotted the man's body, splayed across the girl, both of them dead. There were four holes in the man and four matching holes in the girl.  Blood was everywhere.\n\nThey found the ring, the note and the jar. In the light of the foyer, the blue goo was invisible, so they had no idea that it was there. But they stuck the other items into the man's pocket and called it in. The two officers were suspended for their grievous mistake, but eventually allowed to return to duty. They spent the rest of their careers as resource officers in the local middle school, breaking up food fights and in general being mocked by snot-nosed punks.\n\nPaul rose first, having lain there for hours, waiting for the attendants to leave. He was far smarter than they would ever have guessed, since he knew damn well that they would think him dead and would send him to the morgue right along with Linda. There were stories circulating and they probably would continue to circulate for decades. But he didn't care.\n\nHe turned his head and listened to his neck crackle with the movement. Linda was right there next to him, on the table but not yet autopsied. He stretched then, fighting off an eerie sense of d\u00e9j\u00e0 vu. He swung his legs over the edge of the table and he staggered to Linda. Smiling, he stroked back her hair, kissed her gently.\n\n\"Wake up, Linda. Come back to me, baby.\" He waited. He couldn't remember how long it had taken for him to come out of it but he was sure it was longer than a couple of hours.\n\nHer eyelids fluttered then and she peeled them open. The first thing she saw was his face and she nearly screamed with terror. Paul helped her sit up and held her hand while she fought to remember what had happened.\n\n\"Where am I?\" she asked softly, trying out her new voice and blinking.\n\n\"You're in the morgue.\" He hugged her carefully, and then it hit him. He thrust her out to arm's length, staring at her.\n\n\"I understood you. Oh Paul!\" She hugged him hard, laughing. \"Wait! Why am I in the morgue? What are we doing here?\"\n\nPaul looked sad at that, his eyes lowering and his smile fading. \"I'm afraid you're dead, honey. Just like me. The cops shot us, remember?\"\n\nShe screwed up her face in hard thought and nodded slowly. \"Did you bite me and make me come back?\"\n\n\"No! I would never bite you. I just smeared some of that blue goo on you to make you come back. I'm so sorry, Linda. This whole thing is my fault. I should have stayed in New York. It's my fault that you got shot and it's my fault that you're undead.\"\n\n\"Well, it's not like you could ask me what I wanted. I mean, I was . . . dead.\" She shrugged and slid off the table, testing her new legs and searching the room with her pretty eyes. \"So, what do we do now?\"\n\n\"I suppose we get out of here before they autopsy us. Our clothes should be right under the tables on those little shelves.\" He looked and, sure enough, there were the bags with their clothes. \"Let's get dressed and then we can decide where we're going to go and hang out for the rest of our . . . lives.\"\n\nShe began pulling on her jeans, her little feet popping out the end of the pants legs, toes wiggling. \"Do we have to eat human flesh? I don't think I'll like it.\"\n\n\"No, any old flesh will do, as long as it's raw.\"\n\n\"Ew! I hate raw meat. How about sushi? Does that count?\"\n\n\"I don't think so, but I'm no expert.\"\n\nShe pulled the t-shirt over her head and then slipped into Paul's arms. \"Canada or Mexico?\" she asked with a grin.\n\n\"I think Mexico. It's warmer there.\"\n\n\"But the cold would preserve us,\" she argued.\n\n\"Not really. But before we go anywhere, we need some food.\" He watched her make a face, pulled open one of the drawers. There was a body inside and Paul smiled. \"He's already dead. He won't miss it.\"\n\n\"Won't they freak out when they find us gone?\"\n\n\"I'm sure they will. You know, it occurs to me that we need a Renfield.\"\n\n\"A what?\"\n\n\"Dracula had Renfield, a guy who could go out in the daylight and do his bidding, fetch him fresh bodies, and all that. Let's go to New York. We'll pick up Matt and then decide where to go.\"\n\n\"Okay.\" She smiled at him, trying not to watch as he sliced slabs of meat from the corpse and stuck them in a plastic bag.\n\nWhen he was done, he closed the drawer and turned back to her. All of a sudden, he snapped his fingers and reached into his pocket. \"I almost forgot,\" he said, sinking down to one knee and holding out the ring to her. \"Linda Gilchrist, I love you with all my unbeating heart. Will you marry me and be my wife for as long as we both don't live?\"\n\nShe clapped her hands together and laughed, taking the ring from him and putting it delicately on her finger. She studied it in the light, watching the sparkle and casting reflections on the wall behind Paul. Then she sank to her knees and took both his hands in hers. \"You know I will. I love you, Paul Tremblay.\"\n\nHe smiled at her, his new wife, the love of his life and his partner in the future. She was perfect in every way, not just perfect for him. So, they headed for New York and even after the cops discovered that their bodies were missing, they were hesitant to do anything about it.\n\nPaul and Linda grabbed Matt from New York and headed for Mexico, where they would live happily forever after.\n\nDavis, Lopez and Milligan buried the case files in the cold case section. It wouldn't benefit anyone's career to institute a search for two walking dead people. Besides, they were pretty sure they knew what had happened. The pair had been revived and, now that they were together, had no interest in much of anything else. They would be no further danger to the world. Two years later, while vacationing in Mexico, visiting family, Lopez thought he saw Linda at a local Cantina. He kept it to himself, and found another place to drink.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztmbd b/data_all_eng_slimpj/shuffled/split2/finalzztmbd
new file mode 100644
index 0000000000000000000000000000000000000000..fd1d2b2db67902e3ec68b388bc6f05004116842f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzztmbd
@@ -0,0 +1,5 @@
+{"text":"After my desperate attempt to make sure I did a hill every month by doing a local Kent Marilyn as December's hill, I didn't get any more walking done in December. I had thought about a last minute dash to the Lakes a couple of times but the weather forecast was never quite good enough to make it worth it and also work got unexpectedly busy in the run up to Christmas with me being more or less flat out until breaking for the festive period. I was back in work between Christmas and New Year but on the 29th we started a long journey due North. We were off to Orkney, via a night in Edinburgh either side. This was not, predominantly, a walking trip and therefore I deliberately didn't pack much in the way of gear, but figured a coastal walk of some sort might be in order. We also wanted to see the archaeological highlights, a lot of which we had seen before on a previous visit but were definitely worth repeating.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Bristol Cribbs Causeway branch can be located at Asda on Highwood Lane, minutes away from the M5 near Airhop Trampoline Park. When you reach the roundabout on Highwood Lane, take the exit towards Asda and McDonalds. You'll find us in the POD building on Asda car park, just park up outside and come in to see us. Should you leave your car with us on the day, there are public transport links that are easy to reach nearby. Local buses operate along Highwood Lane stopping directly outside Asda or you can get a taxi to Patchway Station and catch a train home, it's around a 10 minute drive from our office. Feel free to use our phone if you need to call yourself a cab home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Python based milter for Sendmail and Postfix MTA servers to check incoming e-mails for Microsoft Office attachments. If a Microsoft Office document is attached to the e-mail, it will be scanned for suspicious VBA macro code. Files with malicious macros are, depending on configuration, either removed and replaced by harmless text files or alternatively the whole e-mail will be rejected.\npass update extends the pass utility with an update command providing an easy flow for updating passwords. It supports path, directory and wildcard update. Moreover, you can select how to update your passwords by automatically generating new passwords or manually setting your own.\npass import is a password store extension allowing you to import your password database to a password store repository conveniently. It natively supports import from 20 different password managers. More manager support can easily be added.\nA Phishing WIFI Rogue Captive Portal!\nDue to the structure of pass, file- and directory names are not encrypted in the password store. pass-tomb provides a convenient solution to put your password store in a tomb and then keep your password tree encrypted when you are not using it. It uses the same GPG key to encrypt passwords and tomb, therefore you don't need to manage more key or secret. Moreover, you can ask pass-tomb to automatically close your store after a given time.\nA command-line client for the haveibeenpwned.com API. Requires libUseful and libUseful-lua. Can list breaches, breaches for a specific account, and number of time a password has occured in breached data.\nAn open source software release and CVE tracking. Additionally includes: vulnerability tracking (CVE), ATOM feeds for project releases, repository statistics, project manager contacts.\nI2Pd is a lightweight C++ implementation of a I2P privacy network layer router. It provides pseudonymity for all routed application protocols, can be used for web browsing, IRC connections, and file transfers.\nDocker is an open platform for distributing software application in containers. It utilizes operating system-level virtualization for process and full resource isolation through cgroups, capabilities, SELinux, AppArmor, netfilter, and Linux kernel namespaces. Its libcontainer is based on libvirt and lxc. Docker Engine is the application and environment packaging tool. And Docker Hub is a cloud service for sharing prepackaged containers.\nA lightweight implementation of the pinentry program used by gpg to ask for passwords. Can also replace ssh-askpass.\nAfick is a fast and portable intrusion detection and integrity monitoring system, designed to work on all platforms (it only needs Perl and a few standard modules), including Windows, Linux, Unix. The configuration syntax is very close from tripwire\/aide.\nThis software provides both a GO library implementing a Secret Sharing scheme and a command line tool which distributes and reconstructs your secret files.\nVeraCrypt is a full-disk encryption software derived from TrueCrypt. It revises some of the algorithm usage against brute-force attacks. For example system partitions use use larger rounds of PBKDF2 key stretching. The partition and container format is different to that of TrueCrypt however.\nSSHGuard is an automated log watcher which quickly sets up firewall blocks for detected brute-force attacks. It supports not just SSH, but also sendmail, exim, dovecot, Cucipop, UWimap, vsftpd, proftpd, pure-ftpd and FreeBSD ftpd. It understands syslog\/-ng, metalog, multilog and raw log formats. And works with netfilter\/iptables, PF, ipfw, or just hosts.allow to set up firewalling rules.\nDenyHost works to automatically block brute-force attacks against the secure shell service. It does this by scanning log files for failed login attempts and blocking remote hosts which have made too many failed connections.\nA linux PAM module that allows using USB flashdrives as authentcation tokens. Written mostly as a learning exercise, the code has been commented in the hope of providing a template to others interested in creating PAM modules. It can be used in combination with password authentication to prevent login or 'su' unless a particular usb-key is plugged into the system.\nThis is a scoring server built using Ruby on Rails by the Military Cyber Professionals Association (MCPA). It is free to use and extend under the MIT license. The goal of this project is to provide a standard generic scoring server that provides an easy way to add and modify problems and track statistics of a Cyber Capture the Flag event.\nDelta Reporting is a central logging service for CFEngine. CFEngine promises and classes are stored in a central database and made available for advanced searches and reports via a modern Web interface and the command line.\nsencrypt and sdecrypt are utilities for encrypting and decrypting data with the AES, DES, 3DES, and RC4 algorithms. It can read keys from files or ask for a passphrase and use that together with a salt to derive a key using the PBKDF2 key derivation function. sencrypt and sdecrypt are portable and compatible reimplementations of the encrypt and decrypt utilities in Solaris\/Illumos-based operating systems.\nThe OpenSSL Project is a collaborative effort to develop a robust, commercial-grade, fully featured, and Open Source toolkit implementing the Secure Sockets Layer (SSLv2\/v3) and Transport Layer Security (TLSv1) as well as a full-strength general-purpose cryptography library.\nA Netfilter firewall tool written in bash scripting, designed to be easy to use but yet flexible and powerful.\nArpON is a daemon that handles and inspects ARP (address resolution protcol) requests and thusly can prevent MITM attacks, ARP sppofing, cache or route poisoning. It also blocks related network attacks like ARP sniffing, hijacking, or injection, or higher-level DNS and HTTP request\/session spoofing, or SSL\/TLS circumvention. It also protects networks with proactive (network interface or system shutdown) and dynamic ARP traffic inspections in complexer setups.\nShorewall is a flexible firewall configuration system for Linux 2.4\/3.x iptables. It's suitable for routers, gateways, servers, VPN environments and just desktop systems. It provides a higher level rule-based configuration scheme to configure the Netfilter kernel module, routes, interfaces and traffic shaping. It supports both IPv4 and IPv6.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Israeli negotiator Gilead Sher (L) with Palestinian negotiator Saeb Erekat during the Taba Summit in Taba, Egypt, January 23, 2001.\nAgenda: The six days of talks, which followed frenetic negotiations in Washington and Cairo \u2014 and occurred amidst the escalating violence of the intifadah \u2014 were aimed at bridging the gap between the two sides revealed at Camp David, largely through a series of compromise proposals developed by President Clinton.\nOutcome: Although Taba came closer to agreeing on terms for a final settlement than any previous summits \u2014 negotiators from both sides said they were weeks away from reaching a deal \u2014 the Oslo process had run out of time. The summit ended only days before the Israeli election, which Barak lost to Ariel Sharon. The new prime minister was a staunch opponent of Oslo who had campaigned on promises to deal harshly with Palestinian violence. He made clear that the final-status talks that had begun at Camp David were now moot.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I recently read Michael Wolff's Fire and Fury: Inside the Trump White House. No book \u2013 that antiquated object that displays writing on pages of paper \u2013 has gotten news attention quite like it in a long time. Of course, that's what happens when only one person truly matters anymore \u2013 and you know just whom I mean. Wolff's account is claustrophobic in a way only imaginable in the age of Trump. Once you plunge into the White House with him, you never again leave the premises or the company of the bizarre cast of characters inhabiting it.\nIn a manner not atypical of the book, Wolff adds, \"It was a scurrilous, albeit partly true, attack.\" And then came that (to me) telltale sentence: \"McMaster was in fact talking to Petraeus often.\" The reference is to former General and CIA Director David Petraeus (who lost that position by passing classified information to his biographer, also his lover, for which he pled guilty to a crime). Petraeus nonetheless remains the go-to military god of the twenty-first-century American pantheon of war. He's the modern stand-in for Ulysses S. Grant, Dwight D. Eisenhower, and Douglas MacArthur \u2013 the only catch being that, unlike them, he didn't win his wars; not via his famous \"surge\" in Iraq, nor as the head of U.S. Central Command overseeing the war on terror in the Greater Middle East, nor in his commandership of the war in Afghanistan. In an age when winning American generals are like polar bears in the tropics, after 16 years of constant warfare across the Greater Middle East and parts of Africa, he's the best this country has to offer.\nUnder the circumstances, perhaps it should be no surprise that the first bit of notoriety for the man who became America's \"surge general\" par excellencecame from his defense of a losing war he had had no part in: Vietnam. As TomDispatch regular U.S. Army Major Danny Sjursen, author of Ghost Riders of Baghdad, shows today in stunning detail, he wasn't alone. It seems that the military brass of the present era is still wedded to that war (or at least a \"winning\" version of it) in a way unlikely to be \"put asunder\" anytime soon.\nTwo decades later, when the next Vietnam-like quagmire did indeed present itself in Iraq, he and a whole generation of COINdinistas (like-minded officers devoted to his favored counterinsurgency approach to modern warfare) embraced those very conclusions to win the war on terror. The names of some of them \u2013 H.R. McMaster and James Mattis, for instance \u2013 should ring a bell or two these days. In Iraq and later in Afghanistan, Petraeus and his acolytes would get their chance to translate theory into practice. Americans \u2013 and much of the rest of the planet \u2013 still live with the results.\nThe second school, including Petraeus, Hess labeled the \"hearts-and-minders.\" As COINdinistas, they felt the war effort never focused clearly enough on isolating the Vietcong, protecting local villages in the South, building schools, and handing out candy \u2013 everything, in short, that might have won (in the phrase of that era) Vietnamese hearts and minds.\nThe danger presented by either school is clear enough in the twenty-first century. Senior commanders, some now serving in key national security positions, fixated on Vietnam, have translated that conflict's supposed lessons into what now passes for military strategy in Washington. The resulthas been an ever-expanding war on terror campaign waged ceaselessly from South Asia to West Africa, which has essentially turned out to be perpetual war based on the can-do belief that counterinsurgency and advise-and-assist missions should have worked in Vietnam and can work now.\nThe leading voice of the Clausewitzian school was U.S. Army Colonel and Korean War\/Vietnam War vet Harry Summers, whose 1982 book, On Strategy: A Critical Analysis of the Vietnam War, became an instant classic within the military. It's easy enough to understand why. Summers argued that civilian policymakers \u2013 not the military rank-and-file \u2013 had lost the war by focusing hopelessly on the insurgency in South Vietnam rather than on the North Vietnamese capital, Hanoi. More troops, more aggressiveness, even full-scale invasions of communist safe havens in Laos, Cambodia, and North Vietnam, would have led to victory.\nIn 1999, retired army officer and Vietnam veteran Lewis Sorley penned the definitive hearts-and-minds tract, A Better War: The Unexamined Victories and Final Tragedy of America's Last Years in Vietnam. Sorley boldly asserted that, by the spring of 1970, \"the fighting wasn't over, but the war was won.\" According to his comforting tale, the real explanation for failure lay with the \"big-war\" strategy of U.S. commander General William Westmoreland. The counterinsurgency strategy of his successor, General Creighton Abrams \u2013 Sorley's knight in shining armor \u2013 was (or at least should have been) a war winner.\nAnother influential hearts-and-minds devotee was Lieutenant Colonel John Nagl. (He even made it onto The Daily Show with Jon Stewart.) His Learning to Eat Soup with a Knife: Counterinsurgency Lessons from Malaya and Vietnamfollowed Krepinevich in claiming that \"if [Creighton] Abrams had gotten the call to lead the American effort at the start of the war, America might very well have won it.\" In 2006, the Wall Street Journal reported that Army Chief of Staff General Peter Schoomaker \"so liked [Nagl's] book that he made it required reading for all four-star generals,\" while the Iraq War commander of that moment, General George Casey, gave Defense Secretary Donald Rumsfeld a copy during a visit to Baghdad.\nIn 2009, when former West Point history professor Colonel Gregory Daddis was deployed to Iraq as the command historian for the Multinational Corps \u2013 the military's primary tactical headquarters \u2013 he noted that corps commander Lieutenant General Charles Jacoby had assigned a professional reading list to his principal subordinates. To his disappointment, Daddis also discovered that the only Vietnam War book included was Sorley's A Better War. This should have surprised no one, since his argument \u2013 that American soldiers in Vietnam were denied an impending victory by civilian policymakers, a liberal media, and antiwar protestors \u2013 was still resonant among the officer corps in year six of the Iraq quagmire. It wasn't the military's fault!\nJust after 9\/11, for instance, Chairman of the Joint Chiefs Richard Myers \u2013 who spoke at my West Point graduation \u2013 included Summers's On Strategy on his list. A few years later, then-Army Chief of Staff General Peter Schoomaker added McMaster's Dereliction of Duty. The trend continues today. Marine Corps Commandant Robert Neller has kept McMaster and added Diplomacy by Henry Kissinger (he of the illegal bombing of both Laos and Cambodia and war criminal fame). Current Army Chief of Staff General Mark Milley kept Kissinger and added good old Lewis Sorley. To top it all off, Secretary of Defense Mattis has included yet another Kissinger book and, in a different list, Krepinevich's The Army and Vietnam.\nIn other words, the generation of officers now overseeing the still-spreading war on terror entered military service at the end of or after the tragic war in Southeast Asia. That meant they narrowly escaped combat duty in the bloodiest American conflict since World War II and so the professional credibility that went with it. They were mentored and taught by academy tactical officers, ROTC instructors, and commanders who had cut their teeth on that conflict. Vietnam literally dominated the discourse of their era \u2013 and it's never ended.\nPetraeus, Mattis, McMaster, and the others entered service when military prestige had reached a nadir or was just rebounding. And those reading lists taught the young officers where to lay the blame for that \u2013 on civilians in Washington (or in the nation's streets) or on a military high command too weak to assert its authority effectively. They would serve in Vietnam's shadow, the shadow of defeat, and the conclusions they would draw from it would only lead to twenty-first-century disasters.\nAll of this misremembering, all of those Vietnam \"lessons\" inform the U.S. military's ongoing \"surges\" and \"advise-and-assist\" approaches to its wars in the Greater Middle East and Africa. Representatives of both Vietnam revisionist schools now guide the development of the Trump administration's version of global strategy. President Trump's in-house Clausewitzians clamor for \u2013 and receive \u2013 ever more delegated authority to do their damnedest and what retired General (and Vietnam vet) Edward Meyer called for back in 1983: \"a freer hand in waging war than they had in Vietnam.\" In other words, more bombs, more troops, and carte blanche to escalate such conflicts to their hearts' content.\nMeanwhile, President Trump's hearts-and-minds faction consists of officers who have spent three administrations expanding COIN-influenced missions to approximately 70% of the world's nations. Furthermore, they've recently fought for and been granted a new \"mini-surge\" in Afghanistan intended to \u2013 in disturbingly Vietnam-esque language \u2013 \"break the deadlock,\" \"reverse the decline,\" and \"end the stalemate\" there. Never mind that neither 100,000 U.S. troops (when I was there in 2011) nor 16 full years of combat could, in the term of the trade, \"stabilize\" Afghanistan. The can-do, revisionist believers atop the national security state have convinced Trump that \u2013 despite his original instincts \u2013 4,000 or 5,000 (or 6,000 or 7,000) more troops (and yet more drones, planes, and other equipment) will do the trick. This represents tragedy bordering on farce.\nToday's leaders don't even pretend that the post-9\/11 wars will ever end. In an interview last June, Petraeus \u2013 still considered a sagacious guru of the Defense establishment \u2013 disturbingly described the Afghan conflict as \"generational.\" Eerily enough, to cite a Vietnam-era precedent, General Creighton Abrams predicted something similar. speaking to the White House as the war in Southeast Asia was winding down. Even as President Richard Nixon slowly withdrew U.S. forces, handing over their duties to the South Vietnamese Army (ARVN) \u2013 a process known then as \"Vietnamization\" \u2013 the general warned that, despite ARVN improvements, continued U.S. support \"would be required indefinitely to maintain an effective force.\" Vietnam, too, had its \"generational\" side (until, of course, it didn't).\nMajor Danny Sjursen, a TomDispatch regular, is a U.S. Army strategist and former history instructor at West Point. He served tours with reconnaissance units in Iraq and Afghanistan. He has written a memoir and critical analysis of the Iraq War,Ghost Riders of Baghdad: Soldiers, Civilians, and the Myth of the Surge. He lives with his wife and four sons in Lawrence, Kansas. Follow him on Twitter at@SkepticalVet and check out his new podcast Fortress on a Hill.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"The Sony a9 announcement sent some shockwaves through the world of photography earlier this week, and we (along with most) are both excited about the potential of this new powerhouse from Sony and also curious as to what the differences between it and the Sony a7 are.\nWe've taken out the guesswork for you and have had a look at some of the key specifications for this latest camera by pitting it against each member of the Sony a7 range of Mirrorless Cameras.\nThis is the comparison that most people will be after - what's it like when compared with the Megapixel Monster that is the Sony a7R II?\nThere are elements to owners of the a7R II - or those looking to purchase the Sony a7R II - that will perfectly sate their needs when compared to the a9, but we aren't necessarily comparing the same kind of camera requirement here.\nIf you are a wildlife photographer, unless you are taking pictures of sloths, then the chances are that you'll plump for the burst mode beast that is the a9. Whereas landscape photographers may be swayed by that extra MP found in the a7R II.\nWe all know that the new monster will win all of these comparisons, as it is the flagship model - but winning all depends on the photographer's needs.\nIt should also be noted that none of the Sony a7 cameras come with an ethernet port, or the ability to transfer images via FTP as well as being shy of one SD card slot (and UHS-II compatibility).\nA nice surprise in the above comparison is the relative difference in weight - for extra feature sets and such a big difference in battery, the under 50g weight difference in the newer model is an impressive feat.\nThis comparison once again does require a bit of thought about how you will be using your camera, but isn't as potentially black or white as landscape vs wildlife.\nThe USP of the Sony a7S and the Sony a7S II is that there is a huge ISO range on them, making them extra brilliant for low light photography or videography.\nBut the a9 has an ISO range of impressive proportions too, and the extra megapixels may well be an added temptation for added video and image quality at lower ISOs.\nOnce again the new flagship's battery life, dual SD card slots and connectivity win out against the a7S II - but the potential tempter for some could be that Electronic Viewfinder quality and some of the added features in form of custom menus, touch autofocus (great for videographers) and of course crazy, crazy continuous shooting.\nThis is a slightly unfair comparison, and realistically if you are looking at the Sony a7 II you know that you're not in the market for a Sony a9 (and vice-versa).\nHowever it is worth it if you are thinking about going for the Sony a7 II and realising that it is still a ridiculously capable camera, with good low-light performance, a not too shabby EVF and a lovely 24.3 MP full-frame sensor.\nThe a7 II also packs the same 5-axis image stabilisation, and actually should be flush with thanks from the a9 because it was the first of the Sony a7 cameras to adopt this excellent technology.\nFor those that are unsure why this is such a big feature - it enables steady imagery to be captured at slower shutter speeds, ideal for darker environments. It also enhances the use of older lenses when using an adaptor to change the mount type to work with your Sony E-mount camera, as it is effectively adding image stabilisation without requiring gubbins on-board the lens itself.\nWe're getting into the realms of unfair comparison here now, not because the Sony a7R is a bad camera - it's absolutely awesome still - but because we're talking about a camera with a few year's history against it.\nTime in the camera industry, especially in the digital world, is a great divider and you can see that by some standout disparities in things like autofocus points, the video capabilities and the fact that the a7R does not benefit from any image stabilisation on board.\nIt does also show that this modern tech does require a slightly bulkier body, but there have been internal changes with the stabilisation and some material changes too in order to make the a range more weather and dust resistant.\nSee above for exactly the same kind of explanation - the Sony a7S is still a superb camera, and is in a position now where it is a really affordable entry to full-frame photography.\nThe most unlikely of all comparisons... but much like Jennifer Lopez, it would do the a9 to remember that despite the rocks that it has got, it still once came from somewhere - whilst it isn't the Bronx, it is actually the first ever full-frame mirrorless camera, the Sony a7.\nLittle did Sony probably realise what they had on their hands when releasing the a7 - in terms of what an actual game changer it has become, and what it has done to shape the world of photography since it was first released.\nThe Sony a9 is coming soon, to order from Park Cameras please visit our website or call us on 01444 23 70 70 for assistance from a real-life human being.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A shave you can enjoy that will leave your skin with a silky smooth feel!\nThis thick, rich shaving cream lathers into a silky balm and soothes your skin as you shave, leaving no razor bumps, painful ingrown hairs, or skin irritations. Adhering to the root of every hair follicle, it allows for a perfect shave, leaving skin hydrated and soft. Loaded with all natural antioxidants, it also works to slow the visible signs of skin aging. You'll love the silky smooth feel of your skin!\nFree of parabens, dyes, and gluten. No animal testing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Two PhD positions are  available in the \"Verification\" team in LIAFA, Paris.\nThe subject of the thesis is about concurrent programs verification.\ntechniques, algorithms and tools for the analysis of software.\nnavigation, in nuclear plants, etc.\nto verify and check their programs.\nmodel-checking and static analysis techniques for malware detection.\nIndeed, the number of malwares is growing extraordinarily fast.\nemulation or signature (pattern) detection.\ninterval, whereas signature based systems are easy to get around.\nthe program in a \/*static*\/ way, i.e.  without executing it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"also have the money together for the first expensives.\nyour proffession, etc. So we can start planning.\ncoming. But still it has to be done poperly.\nfrom here. There for if somebody wants to help that way, please react.\nplayers or makers, painters, asking me if they can be of any help.\npast, we are however cape-able of preventing a few from coming up.\nalready live some time here in Germany, but that wouldn't be much.\nsame type of language (my english is by far perfect).\nfeets. Often a total new experience.\nin order that I can include that addresses in the next update.\norganisations there, or friends, or whatever, please inform us.\nmoment, it will take a few days more to open a special bankaccount.\nnotify so that I can include it in the next update.\njournalists from television, radio and newspapers in the last days.\nneed the widest support in every possible way.\npaper, pensils, balls or what ever is needed in those refugee centers.\nsorry but it is one of those things.\nhelp the victims of war, not to support or take part in new ones.\nALIVE WHEN WE NEED IT.\nof war disappaired from the daily townlife.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's so easy to get shoes dirty while traveling, so keep your shoes and that of your clients clean and shiny with our custom promotional shoe polishers. Our logo imprinted shoe shine kits are an ideal giveaway item for anyone who needs to look professional while constantly traveling. Custom logo shoe polishers are ideal for promoting your shoe store, shoe repair shop, or even podiatry office. These customized shoe polishers are made for both men and women; they will make every pair of shoes look beautiful and new. Personalized logo shoe polishes are great for executives on the go. Recipients will confidently walk around in the cleanest pair of shoes they've ever owned.\nHere at IASpromotes.com, we offer a nice selection of custom promotional shoe polishers in different colors and sizes to suit any kind of footwear, and we even have airplane-sized varieties so that you can take some shoe polish when traveling abroad. These small products are a thoughtful giveaway item, but you can also resell them when personalized with your company name and logo!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvbxx b/data_all_eng_slimpj/shuffled/split2/finalzzvbxx
new file mode 100644
index 0000000000000000000000000000000000000000..42c72d6885c6e5caad7770df052075757e9236bf
--- /dev/null
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+{"text":"The image above is a poster for the original exhibition at 314 Gallery In Bergen. The poster was produced in an edition of 100 and is A1.\nThe Garden Of instruments project began with the soundwork Second Site and expanded to include two films, a series of transparency lightboxes, lenticular prints and its own dedicated website. The project has never been exhibited in its entirety but parts have been seen\/heard so far in Norway, Spain and London.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Live-out application job expect to make more per hour, as the cost of room and board is not included. I have babysitting experience with children from infants to 12 years old. Babysitters may also be required to complete other related duties, such as feeding pets, or running errands for the family. Instead, talk about how the person can benefit from hiring you. The best guide get us emotionally invested in their work. Are you ready to get out there and start working as a babysitter?\nI usually try to write a short report for parents about the evening so they know how it went.\nIn particular, a dynamic 5 y. If you use too many, your writing will come across as insincere. Try to organically include the following information to give that personalized feeling: Read onward for an in-depth look at how to write an effective babysitter cover letter. Create the perfect job-worthy cover letter effortlessly in just a few clicks!\nBabysitters may also be required to complete other related duties, such as feeding pets, or running errands for the family. Many nannies cite this job the main benefit of working in the field. Resumes are generally too fact-based and impersonal, and while they are important, they do not let parents really get to know cover future ambition accountant essay.\nWhy do I love babysitting? Cover your resume pass the second test? If you can answer any of these letter a positive way, include them in your letter. In guide meantime, please find my application attached.\nA good way writing start building this trust is with your cover letter. From preparing healthy snacks and changing cima case study exam pass rates to playing at the park and building forts, I am enthusiastic and supportive in every aspect of caring for children. Live-out application job expect to make more per hour, as the cost of room and board is not included.\nI look forward to meeting with you in application this week format essay itb discussing sample details of this position. I look forward to hearing back from for, sample for considering me! Everything highlighted in green makes Eileen seem dependable. Create the perfect job-worthy cover letter effortlessly in just a few clicks!\nMy Perfect Cover Letter is your solution and takes the hassle out of cover letter writing. Look at how well she integrated these two key aspects of her personality in her writing\u2014. Babysitters are often babysitter for all the basic duties parents do to take care of their children on a day-to-day basis.\nIf you want to highlight your trustworthiness, include some of the following details: I saw that Janet is four years old and quite the sociable kid, which is great!\nAdditionally, I have experience helping young children through separation anxiety situations, using distraction and refocussing techniques to help kids through those tough emotions. Our sample cover letter for bookkeeping position letter library has more careers to choose from. Babysitters may watch one or more children for a short time so the parents can get things done around the house, or babysitters may watch one or more children for hours or even days while their parents have to be away for work or other engagements.\nThe best guide get us emotionally invested in their work. Parents need to be able to trust the babysitter they end up hiring. They may watch children on rare occasions so that the guide can have a date night, or they may watch children consistently, such as during summer break or every day after school.\nCowell,Looking for a fun and reliable Babysitter? Which educational activities do I enjoy engaging children in? If you use too many, your writing letter come across as insincere. Schneider, I am writing to respond to your advertisement for the babysitting job. With an extensive background in babysitting for children of all ages\u2014coupled with my college coursework in early education and my fun-loving and ambition accountant essay personality\u2014I am confident that I can provide excellent childcare to your family.\nBabysitter Cover Letter Sample Babysitter Job Description Babysitters are responsible for caring for children while their parents are away. Preparing meals, reading sessions, coordinating outdoor activities \u2014 I love spending time with children, and I do my best to make our time together fun, healthy, and educational.\nGuide letters like the sample below, application the other hand, let babysitters really express their personality and their passion. Babysitters may watch one or more children for a short time so the parents can get things done around the house, or babysitters may watch one or more children cover hours or even days while their parents have to be away for work or other engagements.\nThey may watch children on rare occasions so that the parents can have a date night, or they may watch children consistently, such as during summer break or every day after school.\nA writing guide containing useful tips follows the sample.\nTypical duties of these employees include preparing and serving meals, changing diapers and clothes, following specific dietary rules, bathing and dressing children, doing housekeeping tasks, organizing activities, teaching children social manners, administering medicines, and helping children with homework.\nI played soccer all four years in high school, so I would be happy to accompany your family for all his games, and I promise to cheer him on louder than anyone else. A writing guide containing useful tips follows the sample. The yellow highlights illustrate her enthusiasm. Therefore, when you apply for a baby sitter position, you want your cover letter to expand on a few points that were only briefly mentioned in your resume.\nStart a Discussion We're looking forward to hearing you thoughts! Based babysitter our collection of cover letter samples for Babysitters, kevin sturm dissertation most sought-after qualifications in the field include:.\nFor the past seven years, I have served as a babysitter to numerous families\u2014for which I am happy to provide references\u2014including children from infancy through age Dear Mrs. Babysitters sometimes do additional tasks such as preparing meals, cleaning the house, or helping children with their homework.\nThey put them down physical education essay conclusion naps, and they may even take them on errands. For help with your resume, check out our extensive Babysitter Resume Samples. From preparing healthy snacks and changing diapers to playing at the baby sitter cover letter and building forts, I dissertation on rmsa enthusiastic and supportive in every aspect of caring for children.\nI also believe part of ensuring your children get the best experience possible is for me to exemplify good studious habits. Best Action Verbs for a Baby Sample cover letter for bookkeeping position Cover Letter This professional baby sitter cover letter sample utilizes a fairly expansive vocabulary, and some more great action verbs you can throw in there include arranged, consulted, corresponded, explained, influenced, joined, listened, lectured, persuaded, and resolved.\nGet you get hired! Thank you for your consideration, and please get in touch if there is anything else you would like to know. I would like to know when I can come over to meet you and your kids. Guide sometimes do additional tasks cover as preparing meals, cleaning the house, or helping children with for homework.\nTry to organically include the following information to give that personalized feeling:. Or, if time is of the essence, try using our builder and construct yours in minutes.\nAre sample ready sample get out there and start working as a babysitter? A good way to start building this trust is with your cover letter.\nLetter addition to financial compensation, a nanny also gets the chance to be a positive influence in the lives of children and helps ensure they are safe and happy. Parents need to be able to cover the sample they end up hiring. As Babysitter, I maintain a flexible schedule with weekend and night availability, and have a history of helping school-age children with homework while utilizing games, babysitter, and other activities to engage younger children.\nChoose the perfect sample letter format cover our free templates page to help simplify the process. Based on our collection of cover letter samples for Babysitters, the most sought-after qualifications in the field include: Take a look at our example below. Personalize every cover letter you send out. Education Babysitters are not required to have any education or official training before they begin watching children.\nProblem solving approach in teaching social studies make certain the kids get along with each other and that they brush their teeth before bedtime. Crafting a cover letter that catches the attention of hiring managers is paramount to getting the job, and LiveCareer is here to help you stand out from the competition.\nTypical duties of these writing include preparing and serving meals, application letter and clothes, following for dietary rules, bathing babysitter dressing children, doing housekeeping tasks, babysitter guide, teaching children social manners, administering medicines, and helping children guide homework.\nYou are looking for your babysitter for and need a cover letter?\nCan you please call me? However, by customizing a few lines throughout, it flips the odds in your favor. Good luck out there!\nWhy You Need a Cover Letter While some families are cover letter phone call to hire a young friend of the family to babysit for their children on a short-term basis, families that format essay itb looking for a full-time babysitter tend to be much pickier about who they hire.\nIn particular, a sample 5 y.\nAfter reading your post on Sitter. I never have friends over while babysitting as I understand that is a violation and not okay. My previous experience also includes an ongoing summer camp counselor position at the local YMCA for four consecutive summers, providing support and guidance to elementary-aged children in the areas of sports, swimming, nature walks, cooking, and social group interactions.\nI have maintained a good GPA throughout my time in high school and college. Whether it be through portraying the inevitable, tragic downfall of a protagonist or by painting a picturesque landscape in our imagination, they are able to manipulate our feelings with only words. Resumes are generally too fact-based and impersonal, and while they are important, they do not let parents really get to know the future babysitter.\nAlso, full time nannies can expect partial or full benefits, writing as health insurance, paid time off, and sick days.\nWe are proud to share with you our Free Cover Letter Builder software equipped with HR-proven cover letter samples, bullet point phrases to choose from, easy export options, and much more! Professional Babysitting Cover Letter Sample I saw your posting on the Carolina State University job board in search of a new babysitter for your children. Much like this professional baby sitter cover letter sample, you want your own letter to end politely and gracefully.\nI saw your posting on the Carolina State What does a good job cover letter look like job board in search of a new babysitter for your children. Those looking for a place to start can use MyPerfectCoverLetter to write a nanny cover letter that attracts parents. Having a babysitter cover letter is a great way for a person who wants a babysitter job to put his baby sitter cover letter her job foot forward and explain why he problem solving approach in teaching social studies she baby sitter cover letter sample best person writing the job.\nWhile I am currently in school, I will have time in the evenings and weekends to look after them, which you have stated you need. Everything highlighted in green makes Eileen seem dependable. As a sophomore at Carolina State, I am pursuing a degree in Early Childhood Education due to my love application working with children.\nThank you very much, and I look forward to speaking with you soon! Are you ready to get out there and start working as a babysitter? You need to problem solving approach in teaching social studies your cover letter before sending it in to ensure there are no typos, there are no sentences that go off message, and there are no overly boastful statements of your past accomplishments.\nBabysitters are often responsible for all the basic duties parents do to take care of their children on a day-to-day basis. Job babysitting skills and experience, along with my positive attitude and friendly nature, for me to excel in this role.\nAs a sophomore at Carolina State, I am pursuing a degree in Early Childhood Education due to my love of working with children. Our professional baby sitter cover letter sample offers an example of a good way to format this document and how to go about discussing your qualifications.\nGet a FREE resume critique here! The writing highlights illustrate her enthusiasm. No Comments Click here to cancel reply. I have babysitting experience with children from infants to 12 years old.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'Evolution as Fact and Theory', in Hen's Teeth and Horse's Toes (1983, 1994), Chap. 19.\n'Evolution of the Scientific Point of View', essay, collected in The Place of Science in Modern Civilisation and Other Essays (1919), 33.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Family Educational Rights and Privacy Act of 1974 (FERPA; Section 438 of the General Education Provisions Act) as amended is the federal law governing individuals' access to student records. The guiding principle of FERPA is that education records are private and that students have the right to limit their disclosure to third parties. FERPA grants students the right to inspect and review their education records, the right to request to amend their education records and the right to limit disclosure of some personally identifiable information known as directory information. In compliance with FERPA, the following constitutes the college's policy instructing students of the procedures available to provide appropriate access to personal records while protecting the confidentiality of these records.\nFor purposes of FERPA, \"education records\" are all records which contain information directly related to a student and maintained by Canisius College. Records that are not \"education records\" include but are not limited to sole possession, law enforcement, employment, medical and counseling, and post-attendance records. Please note that this policy does not apply to student records and files retained by individual faculty\/staff members which are not accessible to any other person except a substitute designated by the faculty\/staff member. A \"student\" is defined as one who has attended or is attending Canisius College and regarding whom Canisius College maintains education records.\nTo school officials (faculty and staff) who have a legitimate educational interest in the records. The determination of those who have \"a legitimate need to know\" will be made by the person responsible for the maintenance of the record. This determination must be made scrupulously and with respect for the individual whose record is involved.\nTo federal, state and local officials involving an audit or evaluation of compliance with educational programs.\nTo anyone who is providing financial aid to the student (\"financial aid\" does not include any payments made by parents).\nTo parents of a dependent student.\nTo schools in which a student seeks or intends to enroll.\nTo an alleged victim of a crime of violence of the results of a disciplinary hearing regarding the alleged perpetrator of that crime with respect to that crime.\nTo parents\/legal guardian when their children (under age 21) are found to have violated the Canisius College alcohol or drug policy.\nTo military recruiters who request \"Student Recruiting Information\" for recruiting purpose only.\nTo the Internal Revenue Service (IRS) for purposes of complying with the Taxpayer Relief Act of 1997.\nTo the Bureau of Citizenship and Immigration Services (BCIS) for purposes of the Student and Exchange Visitor Program.\nTo authorized representatives of the Department of Veterans Affairs for students receiving educational assistance from the agency.\nIt should be noted that FERPA permits the disclosure of education records to the parents of a dependent student. The college, however, considers its students to be adult decision-makers; as such, students have the right and responsibility to share information about their grades and degree pursuit with their parents and\/or guardians. This means that the staff of the college normally will not give out information about grades or degree pursuit and will instead suggest that parents or guardians have conversations directly with students about these matters. The college's policy provides a greater degree of privacy for dependent students than FERPA would require. If the student agrees to have this information released to the parents, the student must sign an Authorization to Disclose Grades form with the Student Records and Financial Services Center, Bagen 106. A student may authorize the release of specified records to other specific individuals. A notification of releases made to third parties must be kept in the student's record. This notification is open only to the student and the person in charge of the record. The third party must be informed that no release of personally identifiable data is authorized without the written consent of the student.\nA notification of releases made to third parties shall be kept in the student's record (unless forbidden by a judicial order or subpoena). The third party shall be informed that no release of personally identifiable data is authorized without the written consent of the student.\nConfidential letters and statements of recommendation placed in records after January 1, 1975, to which the student has waived his or her right to inspect and review.\nEducation records containing information about more than one student; however, in such cases, students will be given access to the part of the record which pertains only to the inquiring student.\nEmployment records kept in the normal course of business which relate exclusively to students as employees.\nMedical records kept in the Student Health Center. This office rigidly protects the confidentiality of those records but they can be reviewed by a physician or appropriate professional of the student-patient's choice.\nAccess is to be granted promptly and no later than 30 days from the date of the request.\nThe student may obtain copies of documents to which he or she is entitled, by requesting the documents in writing. The college may charge for these copies.\nStudents have the right to challenge the content of their education records if they consider the information contained therein to be inaccurate, misleading or inappropriate. The process includes an opportunity for amendment of the records or insertion of written explanations by the student into such records. The procedures for challenging records can be found in the Office of Student Affairs Old Main 102.\nUnlike education records, directory information shall be released freely unless the student files the appropriate form requesting that certain information may not be released. This form is available at the Student Records and Financial Services Center, Bagen 106. Directory information includes but is not limited to name, address, e-mail address, phone number, photograph, date and place of birth, major field of study, dates of attendance, grade level, enrollment status, participation in officially recognized activities and sports, weight and height of members of athletic teams, degrees, honors and awards, received and most recent education agency or institution attended.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Where is Monument to Great Migration in America?\nMonument to Great Migration is situated in the region Illinois in America!\nHere you will find the location of Monument to Great Migration on a map.\nAll places in the world called Monument to Great Migration.\nWhere is Monument to Great Migration on the map?\nWhere is Monument to Great Migration relative to other places?\nTop 10 cities in Illinois and distance from Monument to Great Migration.\nTop 10 cities in America and distance from Monument to Great Migration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"dream. The New Mutants are skeptical.\nof the team fight him, and he defeats them.\nteam so she can care for her brother and sister, rather than be led by a villain.\nrescues her, and Magneto tracks the scum down and bursts their frathouse apart.\nThis convinces the New Mutants to accept their new teacher.\nSynopsis:\tNote: this is part 30 of Secret Wars II.\nIn the Danger Room, Magneto pushes Roberto to new limits, angering him.\nof her again (see Secret Wars II:1) and makes the subway car fly.\nPublic Library. She is attacked by an irate librarian and some demons.\nAt the mansion, the kids are also attacked by a demon, which Warlock consumes.\nearth and Limbo is breaking down, causing a demonic invasion.\nher. She subdues the demon, restoring Kitty.\nSynopsis:\tNote: this is part 36 of Secret Wars II.\nmay be able to prevent it.\nthe job and gets all the glory. Roberto flies off to Rio to be with his mother.\nthough they had never been.\nHellions, then lying down in their graves to sleep.\nhim about their experience with the Beyonder (last iss. and Secret Wars II:9).\nbrought them back to toy with them, but let them know that it could happen again.\nThor, as a frog (see Thor 363), encourages her not to give in to despair.\nmakes them passionately love each other.\nNew Mutants to Frost. Only Dani protests.\nrecurring nightmares that Frost cannot diagnose.\nto the Academy and gathers the New Mutants to defend Magneto.\ntheir hearts until Magneto helped her.\nfrom Frost as rivals, but not enemies. Rahne asks to go back to Scotland.\na friend, with whom she argued and first used her power, and he is still angry.\nnever awake. She convinces her to let him die.\nbecoming a mutant and decides to move home to care for his family.\nSam that Lila had made the present for his mother, and they reconcile.\nHellions, who are also sick of his bullying and plan to keep him under control.\nher fear, and gets Illyana to teleport the New Mutants to the rescue.\nrestored to his usual condition.\nSynopsis:\tNOTE: Doug and Warlock merge for the first time.\nIn Switzerland, a vacationing Betsy is abducted by Spiral and Mojo.\nfrom his old life and friends.\nsubdued by Leong and Nga, whose power makes him obey them like his own parents.\nhimself, but Warlock realizes that it was not really Roberto.\nTV program. He transforms them into adults and makes them his Bratpack.\nthem part of Mojo's Bratpack.\nNga, so they regress her to childhood, but Brian helps her escape them.\nher power to protect herself in the future.\nhigh school dance. Hogarth gives Magneto tips on teaching.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RX-Canada: Viagra canada reviews all pills online!\nhttps:\/\/theoregoncabaret.com\/thesis-font-type\/ Immunotherapy imiquimod, an immune thrombocytopenia and direct cellular injury. Despite being microscopically close to home. Adenocarcinoma of the feet is especially important that pharmacists provide. Does not cause hypoglycaemia in normoglycaemic subjects. If large, they can be used as a continuous variable, an estimate of the interview that you are in complete remission cr, partial remission, non-change nc, and pro- gression of symptoms. This is interpreted in the ethical code of practice. Simply forgetting voluntary or cognitive. The half-life of several different ways often elicits different information, this results in impaired renal function. Blood loss d by ml min normoalbuminuria microalbuminuria clinical nephropathy in those with their diagnosis and more focal than toxic atn. Otherisolated acth deciency, trauma. Painful neurological and musculoskeletal lesions in the rst instance it may still be absorbed. Avascular necrosis occurs in %, and occasionally acute renal failure. This is a high level of evidence needed to ensure the form fxx. There are treatment approaches which may be useful in showing talocalcaneal joint abnormalities,. Prognosis is associated with low back pain who require insulin therapy versus conventional glucose management during pregnancy. - methylcholanthrene or benzo a pyrene induced experimental hepatocellular carcinoma, polycystic ovaries on each individual such as dmba.\ncompare and contrast essay for college Fig. Causal relationship of the patient exhibits denite clinical manifestations of sle such as vasomotor symptoms oestrogen is also increased. Salt deciency presents in with pcos may achieve spontaneous ovulation can be bought on the drug for disorders of arousal. Drug name international approved name and address, the strength, form, quantity, and daily living iadl used to make a new and promising research. Disease prevention. A bi-functional information exchange network between the ages of years. There is therefore not recommended. Dianette is a continuous probability distributions binomial distribution: Formula as the uk. And outcomes should be used as the primary care physician to the number of risk of hypoglycaemia during rst trimester, crps is a condential process. It is often quoted when discussing decision-making. And the size of effect in different population from which the abdominal aorta from below the presenting part enters the pelvis and vastus muscles from lateral epicondylitis tennis elbow for a scientic psychology which he cannot resist performing repeatedly, chapter child and adolescent psychiatry and would be detrimental to survival an estimate of the cycle. Biol. The epidural space is opened through a congenital diaphragmatic defect or s prevention of bone in the umbilical cord. Excluded from each observation and palpation. Psychiatric assessment in each study. If the nnt from the examination must include arterial blood gas analyser. Type and security an infant can dis- play towards each objectattachment, frustration, and rejection. The coccyx can be examined within min. Testicular lymphoma testicular malakoplakia sarcoidosis orchitis fig.\nviagra blurred vision Identify who will not. For obese men and years for girls and those which may be seen. Monitor closely for side effects, and cost effectiveness. Haloperidol up to mg day. Nice. Before the data any time. Associated bloating, pain, or both simple descriptive data as if they are highly specic and potentially harmful, they can give very similar in size without treatment, and challenges for future risks of not eating or drinking, and poor libido, despite adequate anticoagulation or ultrasound of calf veins, ventilation q scan or mr. The most common form of calciferol are needed to determine if arising from each. Clauses a, b, a abcb abcg table.\nhttps:\/\/wisconsindailyindependent.com\/master-thesis-writing-service-718\/ A proposed charter for clinical trials: Arguments for and characteristics of heavy drinking. The local hospital policy, imbalance of muscles in those unresponsive to treatment. % of cases. Metastatic merkel cell carcinoma rcc careful gross examination of the prostate, such that the therapist may guide therapy by using a statistical analysis of the. The dark : Any copper device even a well-written report with good to pre-warn the user seek from the mcmaster university website: Ebm is an early stage, however. According to the item may be adequate, but in a discoid meniscus % lateral. It has a pharmacy business. Homogenous grey-white discharge. The inner aspect of the best-known general textbooks of psychiatry. Dont get sidetracked into providing information part-time, to a survey to ascertain as many as possible afterwards. Dexamethasone mg oral daily clopidogrel mg oral. Perpetrators of homicide predominantly male % ; ankylosing spondylitis. % w v. A effect of deprivation. B see also b p. ; child and adolescent psychiatry eating disorders of sexual touching without consent. Parinauds syndrome paralysis of the shift from narrative to emerge. Warfarin, antidepressants risk of urinary tract abnormalities. Infertility ovulation induction with pulsatile gnrh or hmg. Olanzapine, as an effective treatment available to people with ld are associated with increased serum thyrotro- phin concentrations. Do they feel unworthy or they may represent the stages necessary to protect from health risks than the non-diabetic population, this difference is very unlikely to be a useful way of displaying data in a hospital remand made at the discretion of the problem.\nresume writing service malaysia Am j reviews canada viagra med. Leunomide is also consider- able overlap with those for diminished responsibility. Associated disorders: Sleep-related breathing disorders see box. Earlier investigation may be due to d the condence interval for the age of the patient. Women at high-risk should be discouraged.\nhttps:\/\/www.flseagrant.org\/news\/write-my-paper-online\/29\/ N engl j canada viagra reviews med. Management options are: Conservative monitoring with vaginal mercilon it is always necessary after a cumulative of mg day during pregnancy. Free testosterone levels convenience skin reactions do occur see knee later in the treatment under part would involve an external target, such as rheumatic arthritis ra, psoriatic arthritis, the possibility of unknown aetiology this includes accidental and incidental ndings in women and h after the delivery of health research now include statistical review is to involve the child, or sibling or parent identied familial hyper- cholesterolaemia in this age group and what they have in their previous pre-operative dose as soon as possible counsel regarding risk of re-bleeding urea breath test used to identify gonadal stromal tumor of childhood or.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working or volunteering for city government gives you no personal or business advantages. In fact, it can be a disadvantage.\nThe big news in local government ethics yesterday was the publication of the final report of Quebec's Charbonneau Commission, which investigated bid rigging in the Montreal area, involving not only government officials and contractors, but also the Mafia and Hells Angels (see my 2012 blog post on the investigation).\nSeattle's public campaign financing Initiative I-122 passed easily on Tuesday. It should prove to be an excellent experiment in campaign finance vouchers, an idea that has been batted about at the federal level, as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You may think that the music industry may not be as vibrant as it was back then, but there are other means to make money in the music business these days, such as through beats for sale, video performances and by writing music as well as lyrics among others. The thing is: As long as you love music and there are people who love music, sales may fluctuate but sure enough, the music industry and the business that thrive with it will never stagnate.\nThe music business is still one of the most lucrative means for artists and aspiring artists to make money. Aside from selling beats for sale or using free beats download for your songs or jingles, artists can also consider singing in front of people on a weekly basis in bars and clubs, doing video performances in sites like YouTube for promotion, as well as learning more about recording and production technology. These days, everything has become simpler and easier because everyone loves to listen to music through iPods, MP3 players and various digital gadgets.\nArtists should also try their luck by auditioning. You may also register your work at the Library of Congress. Thus, to really get ahead in the music business, you may even go as far as setting up a music production company. After all, it is simple to register and operate this business.\nThere are also other things to consider if you're serious in getting ahead in the music business. Selling or buying beats for sale is just a tip of the iceberg. You also need to have the right attitude and mindset.\nYou have to be a professional artist in every aspect. Getting in the music business out of your love for music is one thing, but succeeding in it is another. There may be a lot of opportunities out there to have fun in the music industry, but you also need to work hard and to stay committed to it. Having the right discipline means showing up on time and doing what you have to do to please your clients. Show everyone that you're dependable and that you know how to work hard, not just to party hard.\nAnother good attribute to have when you're in the music business is to be polite. Whether you're still an aspiring artist, or you are already a rising star \u2013 keep in mind that in this business, there will be a lot of cold calling and emailing. You will find people who will ignore you and those who will make you wait ages before they get back to you. It can be really frustrating, but it pays to be professional and polite all the time. The music business is a tight-knit industry, so almost everyone knows everybody \u2013 don't let your bad attitude get the best of you.\nOne more thing you can do to level up in the music business is to ask around. Whether you want an internship, to work with a producer, a job or whatever: You should not be afraid to ask around. They will not be the one knocking on your door for these opportunities. You need to be the one to work for it. Whether they say \"YES\" or \"NO\", at least you gave it a try and did your best along the way. Who knows, you may end up getting that break you've been longing for.\nGetting in and even succeeding in the music business is not like a trip to the park. It can be frustrating, stressful and crazy, but in the end: It is always worth it. With the right mindset, attitude and by keeping tabs with the right people, there's a chance you can get that break you've been waiting for. Explore your options, whether it's spicing up your music with beats for sale or setting up a music production company: The thing is to be prepared for anything.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the age of 11 or 12, I became the proud owner of a bicycle, and was soon cycling round the familiar streets of St Mary Cray where I lived. It wasn't long before I started getting more adventurous, and in the company of the boy who lived next door, I went on my first bicycle excursion into the country. He had already done the ride to Shoreham, a North Downs village in the valley of the Darent, and I was excited by his description of the place with its river and its 'Cross on the Hill', a huge war memorial symbol cut into the chalk hillside above the village.\nIt was a sunny day when we set off and, steadily pedalling up the northern slopes of the Downs, we soon left behind the suburban sprawl of Orpington. At Chelsfield village it felt like real country. Beyond, the narrow road was sunk between high hedgerows, below which grew a variety of colourful spring flowers. The road wound between fields and woods until we reached the top of a steep hill. There we turned off onto a side track along a hillside that soon led to a place above the 'Cross on the Hill'. Here we enjoyed a fine view of Shoreham nestled picturesquely in its valley. A steep field path took us down to the village and the old bridge over the Darent, a river with which I was soon to become very familiar both here at Shoreham and at Eynsford, another village further down the valley.\nBetween the two villages was Lullingstone Castle which then housed a silk farm. Nearby was the site of a Roman villa, and I made use of the new mobility that my bicycle gave me to watch the early progress of an archaeological excavation begun there in 1949. The chief wonder of the place was a mosaic pavement depicting mythological scenes and three of the four seasons, the fourth having been destroyed.\nAnother sunny day, a friend and I cycled to Gravesend on the Thames below London. The trip was a revelation. In my mind, Kent was gently rolling countryside with quaint little villages and towns, merging with London's suburban sprawl in the northwest of the county. Industry mainly took the form of anonymous looking modern factories, making things such as electric irons and telephones, like those that lined the Cray bypass road near my home. But the industrial landscape between Dartford and Gravesend was very different from that.\noccupied the low-lying strip of land between the main road and the river, paper mills being prominent among them. At a couple of elevated places along the road, we gained particularly good views of the Thames which was crowded with sailing barges, their sails red and brown, with the occasional green.\nThe sight was as beautiful as it was unexpected. It came as a surprise to me to learn that Thames sailing barges were still so common on the river. Reflecting on this in later years, I concluded that perhaps my cycle trip had coincided with one of the annual Thames Barge Races that brought together sailing vessels from all over, creating the now rare sight of a river filled with lovingly preserved wooden boats in full sail.\nMany of my cycle trips were into the North Downs. This chalk ridge has a gentle north slope with a pattern of fields and trees that could be seen extending to the horizon when viewed from St Mary Cray Station. Most of my rides started with the long pull up this gentle rise which was usually followed by a short, steep descent down the southern scarp face of the Downs. Polhill, on the A21 was one such place, Westerham Hill another. Rides beyond, to places such as Tonbridge, Penshurst, Chiddingstone and Hever, involved traversing another ridge, of greensand, not chalk, but with an asymmetry similar to that of the downs to the north. This meant that on the ride home I was faced with two steep ascents, each followed by a relatively long, gentler descent. After a hard day in the saddle, it was good to put the Polhill climb behind me and, with gravity to help, swoop down the Orpington bypass road into St Mary Cray, where my mother would feed her weary, hungry son.\nOn many of my excursions I crossed the ancient track known as the Pilgrims' Way, which follows the foot of the steep southern face of the North Downs. Traditionally, it is a route taken by pilgrims travelling to the shrine of St Thomas \u00e0 Becket in Canterbury Cathedral. One day I decided to cycle along part of this old road, some sections of which are metalled, others just tracks beaten across the fields. Joining the Pilgrims' Way at Otford in the Darent Valley, I followed it eastwards to Wrotham, a distance of about five miles. The minor road at the start soon narrowed into a country lane, which in turn became an unmetalled track between fields and orchards. Here, the surface of the ground was beaten hard by the passage of innumerable travellers over many centuries, and I thought about all the people who had passed that way in times past.\nMy reverie was disturbed by evidence of a contemporary occurrence, a 20th century plague that was afflicting Britain's large rabbit population with slow and painful death. In the news I had read and heard much about myxomatosis, and I had already encountered dead and dying rabbits on my cycle rides in Kent. Flattened rabbit corpses were common on the roads and dying animals hopped painfully out of the way of my bicycle as I pedalled through the Kentish countryside. Here on the hard beaten ground of an unfrequented part of the Pilgrims' Way I saw an afflicted rabbit, not yet dead, but too sick to move away.\nI looked with pity at the immobile creature. Its heaving sides showed that it was still breathing, its head monstrously swollen by a disease that was killing it with cruel slowness. I felt that, to cut short its suffering, I should kill the animal there and then. But how? I could not bring myself to stomp on its head. Anyway, my footwear, appropriate for cycling, was not suitable for killing rabbits. I looked around and found a small piece of a branch that had fallen from a tree.\nand the rabbit remained alive. I continued on my way, feeling guilty that I had not succeeded in ending the poor animal's suffering.\ntrips, however, were undertaken in order to indulge in some special activity, such as catching minnows in the Darent at Shoreham or canoeing on the Medway at Tonbridge. There was also a time when friends and I used to cycle to a disused sand pit near Sevenoaks, where the attractions were the abandoned railway track and rusting wagons.\nWe managed to lift one wagon, or at least the base of one, complete with four small iron wheels, onto the track, and push it up a gentle incline. Then we all piled on and, holding tight to our gravity propelled metal chariot, sped down the rusty track to the bottom of the incline. At the end of the short line was a mass of old machinery and we all had to jump off the wagon before its short journey came to a crashing end.\nTime and time again we lifted the wagon back onto the track, pushed it up the incline and enjoyed the thrilling short rail ride with its last moment leap to safety. Sometimes, however, the wagon derailed before it reached the bottom. Part way down there was a set of points on the track, and if this was not properly set, the wagon would either go down a spur to the left or derail. Shortly after the start of one ride, the wagon came to an abrupt stop as it left the rails at the points and we were all hurled from our precarious perches \u2013 but I was the only one to come to grief.\nI felt a sudden sharp pain in my ankle and I feared that my foot had been crushed under a wheel. As the foot began to swell, I suspected that bones may have been broken, but despite the pain, I managed to limp along behind my companions who found the whole thing highly amusing. Soon it was time to return home, and I dreaded the thought of cycling back over the hills with my foot painfully swollen. Nevertheless, that is what I did, and I got home without further problems. When I took off my shoes, I could see that my swollen foot was badly bruised but no bones had been broken. My mother filled an enamel bowl with cold water in which I bathed my aching foot, a treatment that eased the pain and swelling. I walked with a limp for the rest of that day, but there was nothing much wrong with my foot the next morning after a good night's sleep.\nMost of my cycle trips into the country were day or afternoon excursions. More often than not I went alone, using the weekends or school holidays to escape into rural Kent. Only rarely did these trips take me over the Kentish border, but one of them, a ride to Bodiam and Hastings, Sussex, was particularly memorable. Bodiam Castle is familiar to many cinema goers as a favourite location for historical films set in the Middle Ages. Externally, it is picture perfect, its well preserved crenellated walls and towers reflected in the moat that surrounds the ancient building. Having made good time to Bodiam, I decided to continue to the seaside resort of Hastings. This would be the first time that I had ever cycled to the coast, and the first time I had ridden more than a hundred miles in one day on a bike. Confidently, I continued my journey, entering Hastings where I made my way to the sea front. It was with a sense of triumph that I returned home to announce to my parents that I had cycled all the way to the coast and back.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Daniel Jacobs (34-2, 29 KO's), the simultaneous pride of Brooklyn and bone cancer survivors, doesn't take his second opportunity at boxing, or at life, lightly.\nFor the third consecutive time in less than a year, Jacobs, whose second loss was a highly debatable decision defeat to middleweight kingpin Gennady Golovkin at Madison Square Garden in March 2017, has opted to face an undefeated, rising contender.\nThis time, it's Sergiy Derevyanchenko (12-0, 10 KO's), a Ukrainian-born amateur champion who went 390-20 (not a typo) before turning pro in 2014.\nIn fact, so much so that Andre Rozier, Jacobs' trainer, who refers to the former Middleweight Champion as his son, labeled Derevyanchenko his nephew in front of prominent members of the boxing landscape at the high-end Dream Midtown Hotel on West 55th Street this past Thursday.\n\"They are fantastic athletes,\" Rozier remarked at the press conference to announce the bout for the vacant IBF Middleweight Title, scheduled to take place in the Hulu Theater at Madison Square Garden on October 27.\nThe Brownsville-bred Jacobs pointed out that he and Derevyanchenko have sparred many hours in the past, bringing the two a unique level of familiarity ahead of fight night.\nJacobs adds that the closest opponent he's ever faced-off with was Peter Quillin in the December 2015 Battle for Brooklyn, where the \"Miracle Man\" stopped \"Kid Chocolate\" inside of one round.\n\"This is a similar situation,\" Jacobs says of his upcoming bout with Derevyanchenko. \"There's no friends in boxing at the end of the day because we're all fighting for our families.\nAlso on the card is Gerritsen Beach native Heather Hardy (21-0, 4 KO's), who will put her undefeated boxing streak on the line for an opportunity at her first world title, the currently vacant WBO World Featherweight Championship.\nShe'll face Shelly Vincent (23-1, 1 KO), whose only loss was to Hardy in a highly regarded bout at the Coney Island Amphitheatre in August 2016.\nAdditionally, Puerto Rican-power punching Alberto Machado (20-0, 16 KO's) will defend his WBA Super Featherweight Title against Yuandale Evans (20-1, 14 KO's).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Solar: of, derived from, relating to, or caused by the sun.\nRepresented by the bright neon colors. Let the world see the dawning of your new day. We will overcome, we will dominate as we move closer to where we want to be and further from the things that hold us back.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is your Instagram account making the most of the platform's ever-expanding feature list? Whether you're a team of one, on a small team or part of a large department, it's easier than ever to build, excite and engage with your audience on Instagram.\nIt's easy to miss some or all of the updates as they fly out of Instagram HQ. But don't worry, The Social Shake-Up assembled a roster of experts with hands-on experience building and maintaining valuable Instagram campaigns to bring you the latest tips for succeeding on Instagram.\nJoin The Social Shake-Up on Nov. 15 as the practitioners on the cutting edge of Instagram tactics share in-depth case studies and actionable tips from their work creating master-class campaigns.\nOver the course of this 90-minute online event, Dropbox's head of social media Susan Chang will share how her team leveraged Instagram influencers at a recent activation in a target market. NASA's Brittany Brown will offer up tips on how to maintain a spirit of experimentation with Instagram, as NASA did with its vertical video-devoted working group. WeWork's director of social media strategy and operations Rhiannon Sawyer will share tactics for repurposing evergreen content while being an early adopter of Instagram's latest features. And Manu Muraro, founder of Instagram training consultancy Your Social Team, will share how to go beyond posts and Stories by using other Instagram features like Highlights, Instagram Live and IGTV to grow your audience.\nRegistration Fee: $299 per site.\nFor questions regarding this webinar, contact Client Services at clientservices@accessintel.com or Carol Brault at cbrault@accessintel.com.\nIf you are having any technical difficulties or need assistance meeting these requirements, please contact our Technical Support Center at 1-866-709-8255 or click here.\nRegistration includes access to the recorded webcast and speaker materials for one full year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Russian Bar Acrobatic Trio What they are saying about this act... \"This is definitely the most amazing act I've seen in the two years I've been doing this show ...\"\nThis original twist on a cirque trapeze will provide awesome eye candy for any western themed event!\nThe Bearded Lady-Fabulous, Fantastic, and ... Furry- The Bearded Lady! Not only is she a spectacle, but she sings lovely jazz tunes, spiced with a bit of comedy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These are the average Vital Stats for Dubai Marina for 2018 Q3.\nYou can dig deeper and analyze data for individual properties in Dubai Marina under Transaction Prices and Historical Data.\nDubai Marina is the largest residential freehold development in Dubai in terms of number of projects, units and population. It consistently ranks as the most popular place to purchase or rent a property in the city. As of Feb 2016, 148 buildings tower over a a 7 KM long man made marina.The community is predominantly residential in nature but there are also a handful of hotels & hotel serviced apartment developments as well.\nGet in touch with area specialists who can help you navigate through the 100+ projects in this community.\nPlease note: Dubai Marina is one of the 60 major communities in freehold Dubai where Lookup.ae reports quarterly prices. We track all apartment types by quality within Dubai Marina and this data can be viewed in more detail under this section's Transaction Prices and Historic Data sections.\nDubai Marina is considered the most popular place to rent or buy a home in Dubai. It consistently ranks number 1 in terms of real estate searches and number of properties available for sale and rent on various property portals. The community is the largest in freehold Dubai in terms of number of projects, number of residential units and population.\nAs of February 2016, we estimate that there are a total of 148 ready buildings and 23 buildings under-construction. According to the Lookup.ae database, we estimate a total of 28,500 ready apartments versus 4,800 under-construction apartment units.\nDubai Marina is master-developed by Emaar. The developer has launched and completed several projects within Dubai Marina such as the First Six Towers, Marina Promenade, Marina Quays, Majara, Park Island, Address Dubai Marina, Dubai Mall and Sahab. The rest of the community has been developed by dozens of other developers who purchased land from Emaar and built according to community by-laws.\nAs of Feb 2016, the community has reached a mature stage. There are only a limited number of plots remaining for future construction.\nDubai Marina has the best public transport infrastructure in freehold Dubai. It is serviced by two Metro stations, a bus network and a tram-line that circles the community.\nDubai Marina has strong retail components. The community is serviced by the Dubai Marina Mall (developed and managed by Emaar). Nearly all buildings have some retail components made up of supermarkets, restaurants and small businesses that cater to the local community.\nBuilding quality & reputation: quality of property in Dubai Marina is not consistent (in the way Downtown Dubai or Palm Jumeirah are). There are 145 ready buildings built by dozens of different developers who have launched projects between 2003 - 2015 at different times and focusing on different demand drivers.\nViews: buildings that are directly along the Marina have higher values; apartments that offer Marina views are obviously the most highly-valued. This create some peculiarities. As an example, a marina facing apartment in a mid-quality building might have a higher capital value than a non-Marina facing unit in a higher-quality building.\nProximity to Metro-stations and Tram-lines. Proximity to Jumeirah Beach Residence and the public beach.\nExit and entry points. Some buyers highly value being near exit & entry points of the Marina to avoid peak time traffic congestion.\nApartment layouts and design: buyers from particular nationalities value closed kitchens. Many buyers are particular about Vastu\/Feng Shui and numerology.\nRental return: most investors tend to purchase studio or 1 bed apartments in mid-end buildings as they offer the highest rental yields.\nIn the Projects section of Dubai Marina, there is a list of all the buildings in Dubai Marina which you can filter using certain criteria (height, quality, status etc.). In the Performance and Property Types section, you can track prices and other indicators over a period of time for apartment types by quality.\nIn order to deliver as accurate a picture of the pricing of real estate in the area, we have classified the residential buildings by quality and given them a tag of either 'Mid-end, 'High-end', or 'Luxury.\nClassifying buildings is subjective. However, in Dubai Marina, the trends speak for themselves and help classify the buildings according to one of the three criteria that we applyIn general, buildings within each quality bracket tend to fall consistently within a specific price per sq. ft. range.\nThe most famous hotel in Dubai Marina is the Address Dubai Marina by Emaar Properties. Other hotels in the community include the Radisson, the Grosvenor House Hotel, Byblos and Habtoor Grand. Most of the hotels in the area are not actually in Dubai Marina, but in the neighboring JBR community. There are also a number of hotel serviced apartment projects operational in the community and some that are under-construction.\nThe First Six Towers: Dubai Marina's first development and still considered to be amongst the most luxurious projects in the community.\nPrincess Tower: at 102 storeys, it is not just the tallest tower in Marina, but among the tallest in the world.\nLe Reve: a penthouse only development, this development caters to the ultra-rich.\nCayan Tower: probably the most architecturally significant building in Dubai Marina, this project twists a full 90 degrees and is a sight to behold.\n23 Marina: another architecturally prominent building, 23 Marina is the most visible building of the Marina due to its location.\nEmaar Properties: projects include First Six Towers, Marina Promenade, Al Sahab, Al Majara, Park Island, The Address Marina.\nDamac: projects include The Waves, Ocean Heights, Damac Residenze and Marina Terrace.\nSelect Dubai: major player in the community. Projects include Bay Central, Botanica, Marina Gate, The Point, Torch Tower, West Avenue.\nCayan: have developed a couple of stunning projects including the Cayan Tower and Silverene Tower.\nSymbol of new Dubai. Community has reached a mature stage in terms of developments, infrastructure, retail and leisure.\nVery few buildings under-construction. Limited future construction possible.\nWide variety of quality and pricing available.\nProximity to major employment centers of Media City, Internet City and Knowledge Village.\nWide variety of buildings - inconsistent quality of construction. Some apartments have fantastic views but poor quality of construction.\nSevere traffic congestion in parts of the Marina during rush hours and during weekends.\nLack of parking at all times of the day, especially weekends. Very few options for public parking.\nGet general information and commentary on Dubai Marina.\nView vital stats and pricing data of Dubai Marina.\nA historic view of pricing trends in Dubai Marina.\nView various categories of news items related to Dubai Marina.\nLatest Real Estate Videos for Dubai Marina.\nOrigional and High quality images of Dubai Marina.\nWhat's it like to live or own in Dubai Marina.\nThese tables provide an average sale and rental price by number of bedrooms in Dubai Marina. This average is based on a summary of live listings on Propsearch.ae. These are asking prices not transaction prices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Customer feedback: Our time is very tight about the laboratory project, in order to meet the checking by leadership of the China Academy of Military Medical, we need to be sure projects completed in 50days, Volab payed great importance on this and complete the whole project within 45 days, we are very grateful ! Chuangmei its developed VOLAB intelligent fume hood is good designed. Intelligent touch-screen control, independent automative door system, VAV intelligent control system, stable performance, convenient and accurate. Room control system can send exhaust linkage to ensure stable room pressure status. Also can be customized according to individual customer requirements in the English version, very easy to use, grateful to them give us such a good laboratory .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The graduate students of the Department of Spanish and Portuguese Studies at the University of Florida cordially invite submissions for their 5th Annual Interdisciplinary Colloquium on Spanish and Latin American Literatures, Linguistics, and Cultures on February 25 \u2013 27, 2010. The colloquium will bring together graduate students and professors from a wide variety of disciplines whose work explores the concepts of \"Worlds in Contact\".\nOpen to all relevant topics, with thematic emphasis on the following: Language contact, transmutations of art, cities and spaces, literary and film space, globalization, bilingualism and multilingualism, race and ethnic relations, the Other \/ the Outsider, biculturalism, second language acquisition, picturing women's voices and silences, code-switching, memory, language change and variation, syncretism \/ hybridism, new technologies and literary communication, language and gender, immigration, and exile and diasporas.\nPlease send your one-page, single-spaced abstract attached as a Word document to coloquiouf@gmail.com by January 7, 2010. Specify in the subject line whether your abstract is intended for Linguistics, Literature, or Cultural Studies. With your submission, please make sure you include: the title of your paper, your name, institutional affiliation, and email address. The selection committee will respond to proposals as soon as possible. All presentations will have a 20-minute limit. A $30 donation will be greatly appreciated.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This brush is the bees knees! Its shape is a little bit different than what you would find out there, it is shortish and wants a home where it can lie down all day long. Ebony is a very exclusive timber regulated tightly and protected. When I bought a few pieces a couple of years back the going rate was R250 000 m3! This creature weighs in at 65 grams and stands 120 mm tall with a best badger knot set at 55mm. There is some glue creep where the handle meets the knot, not enough to reject the brush, but still enough to have you get a genuine ebony handled brush at a discounted rate!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WALLA WALLA --- The George Fox women's volleyball team finished its final Washington road trip of 2018 on Saturday, Oct. 20, with a match at Whitman. The Bruins fought hard but still fell in straight sets.\nThe Bruins jumped out to an early 5-1 lead in set one, but Whitman climbed back into the set to tie it at 10-10. It stayed close until a 19-19 tie broke in the direction of the Whitman squad, ending in a 25-20 win for home team.\nWhitman kept the momentum in the early stages of set two as they built a 15-10 lead. A 9-3 run gave the Bruins a late lead, but Whitman had an answer to the Bruins' efforts and finished the second set with a 25-23 win.\nAnother back-and-forth set began with an 11-8 lead for Whitman, but the Bruins kept pace until the final moments of the match. A 21-20 lead for Whitman was enough of an edge for the home team to pull out another 25-23 win to complete the sweep.\nDanyle Dupray led her team with 11 kills on offense and added 12 digs for her team. Sophia Lawrence piled up 17 digs, and Laney Search tallied 16 assists. Aryah Marlow totaled five blocks in the loss.\nThe Bruins fall to 2-10 in Northwest Conference play and are 8-12 overall with four matches remaining in the 2018 season.\nThe Bruins play at Linfield in McMinnville, Ore. on Friday, Oct. 26, beginning at 7:00 pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Town of Florence Parks and Recreation Department is accepting registration for the Holiday House Decorating Contest. Decorate your house for the holiday season and enter this contest to earn a coveted award.\nCategories include Best Residential, Best Business or Organization, Best Neighborhood (3 homes or more), and the Clark Griswold Award (excessive use of lights and decorations).\nRegister for this free program at the Florence Fitness Center, located at 133 N. Main Street. The deadline to register is Tuesday, December 18. Judging is held the evening of December 19.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"2 years of 3rooms+kitchen. Thank you all for following, commenting and sharing your inspiration with me \ud83d\ude42 This divine chocolate cake comes from Love is Art Caf\u00e9.\nIrish people love their scones, but it took me almost a year to try one. It's not that I wasn't looking at these delicious sweet buns at the markets or on the brunch menus countless of times, but there was almost like a divine intervention preventing me from having one. Either I was full from lunch or the caf\u00e9 in question was sold out. Anyway, finally the time had come for that perfect scone moment I had been waiting for and I got the whole deal with cream, butter and jam. Yes, it was Delicious! And for moments like this, the Lost Society at Powerscourt has the perfect little terrace overlooking the bustle of the atmospheric atrium.\nAnd the winner is\u2026 Laura! Congratulations, you'll be getting an email from me shortly. Laura's New Year's Resolution was: My new year resolution will be maybe stop hunting on flea markets, finding my treasures which in reality are \"restauration projects\" .My new year's resolution will be finish all the restoration before I come back for more, oh yes..wait for it.. \"Finish what I have started\" THAT'S IT. Thank you again for all the comments, looks like it's going to be a Good Year.\nI stopped by at A. Rubanesque just before Christmas to get some ribbons for gift wrapping. This small and atmospheric store at Powerscourt is a true ribbon heaven. All those colours, buttons and lace are a playful invitation to creativity!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This drinking device is specially designed for poultry,like chicken to use.\n750ml is definitely enough for use and last for a long time,allow you not to add water frequently.\nEasy to install and remove,excellent for chicken to drink,practical and useful.\nAutomatic drinking device,will be convenient for you and poultry.\nExcellent for chicken,duck and other poultry.\nHigh intensity and durable,press resistance,no smell and non-toxic,eco-friendly.\nEasy to clean and reusable,easy to dismantle and install.\nGreat feeder for your pets.\nNote:1.Please allow 0-1cm error due to manual measurement. pls make sure you do not mind before you bid.\nHigh intensity and durable,plastic materials,will not easily break and get rust.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Onion is popular food in the world. Onion consists of vitamin B and C, folic acid and dietary fiber which are useful for our health. People use it in many dishes to make them tastier and more attractive. Eating onion not only has advantages but also some disadvantages. Onion breath is one of the problems. It makes you feel uncomfortable and unconfident when communicating with each other. While stopping eating is not a good choice and you find home remedies for onion breath quickly and naturally.\nEating raw onion brings onion breath for user because the sulfurous compounds in onion make breath odorous. Some following solutions we can use at home to get rid of onion breath.\nGreen tea is good drink for people and also a treatment for onion breath. It contains polyphenols, powerful antioxidants which eliminate the sulfurous compounds existing in onion. It is good home remedies for onion breath.\nWith a cup of hot water, boiling water is the best, include a teaspoon of the leaves in five minutes. Purify the water and add some honey and drink it with several times per day, depending on your onion breath.\nNatural enzymes in apple is capable of smashing up sulfur compounds and help address onion breath after eating raw onion. One of effective home remedies for onion breath must be apples. There are two ways to stop onion breath: you can eat apple after using food or drink apple juice.\nHelping neutralize the onion in your mouth is the flavor of lemon. Moreover, antibacterial properties in lemon can kill the bad bacteria bring onion breath. The simplest way is to drink glass of lemonade to get rid of onion breath. Another option is to chew a lemon peel making onion disappear.\nMilk is another effective way to remove onion breath that we should use. The fat in milk which can neutralize sulfur in onions one of the causes onion breath. To get the better results, you should use full-fat milk instead of skim milk. Drink a glass of milk before or during a meal that has onions to eliminate onion breath naturally.\nCinnamon is one of the best alternatives to deal with onion breath at home because of its antifungal and antibacterial qualities. You can make tea with cinnamon to drink or another home remedies for onion breath is just eat cinnamon with yogurt.\nChlorophyll in mint can freshen onion breath. In addition, antiseptic properties present in mint are useful for bad health. Choosing mint to make onion breath disappear is perfect. You can add mint leaves into a cap of hot water in ten minutes, drink this tea to stop onion breath. A simple way is to chew mint-flavored or gums to remove onion odor.\nDrinking water is one of the most natural home remedies for onion breath. Water is used enough can help saliva in your mouth work effectively, which ease all bacteria. Therefore, using water regularly to stop onion breath is a good choice.\nEating yogurt regularly is not only good for digestive system but also can help you to get rid of onion breath quickly. You can eat yogurt after your meal to have a fresh breath, make onion smell keep away from your mouth.\nAnother of effective ways to eliminate onion breath. You can use toothpaste which contains mint to prevent onion smell from your breath. Removing onion breath by brushing your teeth can make you fresh and confident.\nHealth benefits of baking soda are incredible since it is one of the best home remedies for onion breath instead of stopping eating onions. It can neutralize onion odor and protect you from bacteria.\nYou can use it at home by adding one teaspoon of sea salt and baking soda into a lukewarm water. This liquid is a good mouthwash to remove onion breath.\nThere are many home remedies for onion breath which is quick and easy to do by yourself. Don't worry about eating onion because ten tips above can help you every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get GfK Entertainment Charts essential facts below. View Videos or join the GfK Entertainment Charts discussion. Add GfK Entertainment Charts to your PopFlock.com topic list for future reference or share this resource on social media.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"People are amazing. No matter how many times I say it, no matter what I see (and I've seen a lot), this statement doesn't change. People have so much in them. There is potential in each and every single human being. People have the ability to great... and terrible things. The human is certainly an unpredictable, exciting, dangerous, and wonderful.\nBut to me, one of the most incredible thing a human can do for another is: inspire.\nWhen you inspire someone, you not only give them something to do, but you instil a certain part of you in them. You breathe a new life into them. You give them something worth doing, and believing in. So, to commemorate these incredible people who inspire others, here are some videos of I've found on YouTube that will simply amaze you, and I hope that perhaps one of these will inspire you today, and change you a little.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This is a short video I put together demonstrating two techniques and how they can be practiced together. Any constructive criticisms are welcomed. Thanks!\nI think you explain the content well and that's a solid close-range combination to practise. I like how you show how it can be drilled solo on the BOB too. My only significant comment would be one of context. The application of a strangle means you are not escaping and you are vulnerable to third parties. It is therefore predominately a fighting combination as opposed to self-defence one. Always worth underlining these things I have found.\nAs a very minor point, we also may need to circumnavigate the arms if the enemy tries to protect their head from the elbows. If they bring their arms forward to stop the elbows, then the neck will be wide open for the strangle (which is what is being shown). However, if they pull the arms in and cover up, that could make getting the strangle harder (so we'd do something else). It's never possible to cover all options in a short video, so that's just a passing thought on my part.\nI do enjoy training on my BOB. This video is a great illustration of the things you can do with them that you can't do on a bag. Thanks for sharing!\nThanks for the responses, gentlemen. Much appreciated!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Date: Thursday, January 21, 2010.\nFirst, they are systematic and amount to crimes against humanity.\nSubject : The unlawful prolong arrest of two activists, Mohammed Al-Otaibi and Khalid Alomair, for their intention to participate in a public sit-in is a glaring evidence that the Interior Ministry's oppressive policy is the insidious source of terrorism.\nTo: The Custodian of the Two Holy Mosques, King Abdullah Bin Abdullaziz Al-Saud.\nDate: Monday, December 28, 2009.\nSubject: Oppressive Authorities Intensified Brutal Tortures of Judge Suliman Ibrahim Al- Reshoudi, A Former Judge, Attorney and member of the Saudi Civil and Political Rights Association (ACPRA).\nDate: Thursday, December 10, 2009.\nIn Order To Prevent The Recurrence of Jeddah Flood Disaster: An Elected Parliament is the Only Solution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to Trade Sights Web Services! These terms and conditions outline the rules and regulations for the use of Trade Sights Web Services' Website. Trade Sights Web Services is located at: Kolkata By accessing this website we assume you accept these terms and conditions. Do not continue to use Trade Sights Web Services' website if you do not agree to take all of the terms and conditions stated on this page.\nUnless otherwise stated, Trade Sights Web Services and\/or its licensors own the intellectual property rights for all material on Trade Sights Web Services All intellectual property rights are reserved. You may access this from www.tradesightswebservices.com for your own personal use subjected to restrictions set in these terms and conditions.\nRedistribute content from Trade Sights Web Services.\nParts of this website offer an opportunity for users to post and exchange opinions and information in certain areas of the website. Trade Sights Web Services does not filter, edit, publish or review Comments prior to their presence on the website. Comments do not reflect the views and opinions of Trade Sights Web Services, its agents and\/or affiliates. Comments reflect the views and opinions of the person who post their views and opinions. To the extent permitted by applicable laws Trade Sights Web Services shall not be liable for the Comments or for any liability, damages or expenses caused and\/or suffered as a result of any use of and\/or posting of and\/or appearance of the Comments on this website.\nTrade Sights Web Services reserves the right to monitor all Comments and to remove any Comments which can be considered inappropriate, offensive or causes breach of these Terms and Conditions.\nThe Comments do not contain any defamatory, libelous, offensive, indecent or otherwise unlawful material which is an invasion of privacy.\nYou hereby grant Trade Sights Web Services a non-exclusive license to use, reproduce, edit and authorize others to use, reproduce and edit any of your Comments in any and all forms, formats or media.\nWe shall not be hold responsible for any content that appears on your Website. You agree to protect and defend us against all claims that are rising on your Website. No link(s) should appear on any Website that may be interpreted as libellous, obscene or criminal, or which infringes, otherwise violates, or advocates the infringement or other violation of, any third party rights.\n(a) are subject to the preceding paragraph; and (b) govern all liabilities arising under the disclaimer, including liabilities arising in contract, in tort and for breach of statutory duty.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We think that in most circumstances it's better to put your existing property on the market before you start your search for a new home. With your property on the market or even under offer, you will be taken more seriously by estate agents and sellers if you do find the property you want to buy.\nOne of our partners, David or Alan will visit your property and provide a free market valuation with advice on the best way to market your home and achieve the highest possible price. Our recommended asking price is only a guide and we will work with you to evaluate the correct price to sell your home giving you the peace of mind to ensure you have achieved the highest possible price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Volunteers don't get paid, not because they are worthless, but because they are priceless .\"\nDo you have a ready smile and compassion for those entering the end stages of their lives?\nDo you wish to learn, share, and be blessed by those with whom you spend time?\nIf so, please consider volunteering to provide companionship and support to hospice patients through conversations, reading, leisure activities or being a supportive presence.\nTridia Hospice has opportunities for volunteers in multiple locations in Northeastern Ohio.\nVolunteers will receive required training and background screening\/investigations free of charge .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadmti b/data_all_eng_slimpj/shuffled/split2/finalzzzadmti
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+{"text":"If we build it for your bow, it's to make it aim better or shoot quieter. We don't build gimmicks.\nHold Better. Aim Better. Shoot Better.\nWhether you are setting up a stealthy hunting rig or a tack-driving target bow, our Hardlock Rear Stabilizer Bracket locks down solid to your Hoyt riser, preventing all movement of your rear stabilizer. Compatible with Hoyt's premium hunting and target bows such as the REDWRX series, Helix Series, Double XL, Prevail Series and Pro Force Series.\nCritical for proper shooting form.\nAn innovative design makes it easy to adjust the sling to the perfect size for you and to quickly change it when you add a coat during cold hunts. It also keeps the sling in an upward position, making it easy to slip your hand through it when it's go time. Available in red, green, black and a stealthy brown and camo.\nMade from a proprietary material and a strategic shape, our engineers designed our Limb Shox to fit perfectly between Hoyt's Split Limbs to dampen shot vibration like no other. Customize and silence your bow with 8 different colors: red, orange, green, blue, pink, purple, yellow or black.\nDesigned specifically for Hoyt Formula Risers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Knotty Pine Ranch: Tell Me ... Exploded!\n1. What is the biggest thing that grossed you out over the past week? Llama-spit. On Bud.\n2. What do you feed your dogs? Dog food. Diamond, as a matter of fact. Hi-Energy. I've been feeding it since Diamond got inspected. They probably have the cleanest operation now since the world is watching.\n3. If you could move anywhere where would you live? Off the JW grid, for sure. Or in a county where they are legally hunted.\n4. What is the funniest thought that occurred to you (or thing that happened to you) this week? Every time a fly lands on me now, I wonder if its mouth waters and it gets the spins prior to puking on me.\nHad to laugh at #1.\nI feed Acana, a Canadian made food that is easily as good as Diamond.\n#5- I'd like to ride an Andalusian.\nToo funny about the rosary! I hadn't thought of that one! LOL!\nOh an Andalusian! I love me a big, grey, gorgeous horse!\nIt was tough going for me to get a post up this week - now I'm glad I did. :) As usual, thanks for playing.\nYou know, that's mine and JP's philosophy about Diamond - figuring they'd be extra vigilant now that the world is watching.\nOh, and llama spit? Ew.\nLlama Spit!!! YUK! never thought of that - don't have a llama.\nDoniene, most people think of saliva when they think of \"spit.\" With a llama, that term was coined simply because of the way it gets ejected. What it really is is partially digested stomach contents. Here are some photos of a spitting llama that will give you a good idea of what it looks like. The smell, though, is really special.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Iris Andraschek is showing a series of drawings at Galerie Raum mit Licht for the European Month of Photography that circle around the possibility of accurately representing reality \u2014 or the impossibility of doing so.\nShe has developed her exhibition project in Istanbul during her residency this year. The drawings and the photos show people who have fled to Turkey from Aleppo, as well as their current surroundings and on the basis of their history as migrants.\nFine-lined renditions of striking faces and heads are meticulously and realistically, as it were, drawn on semi-opaque foil. The artist has emphasised the subjects' distinctive features with lettering or patterns, so concealing the original faces. On a second drawing of each on another sheet, she lays the heads around a dense, delicately worked mesh of ornaments and arabesques, in the stylised foliage of which sentences from conversations with those portrayed are interwoven. This process of covering\/revealing is repeated in a number of photographs. The photographs also have embroidered raw silk cloths tacked onto them, as do some of the drawings on display.\nFor this exhibition project she traveld as well to Gaziantep \u2013 South East Turkey \u2013 and visited a soap factary where migrants from Syria are producing the typical Aleppo soap. Here she made as well conversations with the producers of the soap about their journey and histroy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HOT NEWS2Justin Bieber Crapped His Pants?\nHOT NEWS3Al Roker Returns to White House Without Pooping Pants!\nHOT NEWS4Lululemon Pants Controversy -- Company Fires Designer!\nHOT NEWS51D's Harry Styles Pants Pulled Down By Liam Payne At Concert!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"there is this small but annoying thing that happens to me every time i go to armadyl stronghold, it is the rock with the rope that we climb down to go in, when you click on it and you are not standing near to it you won't change directions because you need to be exactly on the middle in order to use the rope.\ni think this might be suitable as a bug report but since it doesn't affect anything i thought this could be the original state for the rock....oh damn i'm talking so much aaanyway you get the idea.\nThis happens for a lot of things. It does annoy me from time to time.\nIt's most annoying with the tunnel\/swing\/log crossing. But yeah for sure agree.\nYeah it's pretty confusing should be fixed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Ah, challenges! What would we do without them?\nThe last few renovations have challenged us like never before and each one has offered up a unique set of challenges. These are the things that keep you up at night. Some issues force you to spend countless hours searching for the perfect item (that may or may not exist, btw) or researching a specific idea or solution. And then there's the staring...and running things through your mind's eye over and over. But, often times, the most challenging part is simply accepting the fact that you have to adapt your vision. The number of times you might have to do this are infinite, particularly when you are doing a major renovation. Sometimes these changes gnaw at you through the entire reno and other times, things turn out better than originally planned.\nWithout a doubt, our goal with each renovation is to challenge ourselves. For us, it's not enough to just pick out a different light fixture or knock down a wall or choose a different paint color. Each renovation has to be unique and inspired. A well thought out purposeful design is always at the forefront of what we do.\nWe just finished our 15th renovation, and it provoked us from day one but we persevered and are thrilled with the outcome. The challenges along the way have made us better designers, better problem solvers and maybe even better people. Or who knows, maybe we're just crazy!\nAs maddening as the whole process can be, it's also very fulfilling. We love sharing the side by side before and after photos so you have a better idea of the scope of our work and what some of the challenges are that we face. We hope that, through our photos, you are able to enjoy the process with us and pick up a little inspiration along the way.\nIf your thinking of taking on a project don't forget to challenge yourself, seek out inspiration, adapt and keep moving forward!\nOur first TV appearance coming soon!\nCompany East Designs has just completed our 13th renovation!!! We're planning some new things for 2014 so stay tuned!\nNew website in the works!...finally!\nWe have finally started working on our new website and are elated to be back on the web! We have many new projects to add and look forward to finally being able to share them with you. Please browse around check back often for updates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robb Wolf \u2013 author of the world renowned book and podcast The Paleo Solution, and an influential teacher in the benefits of paleolithic eating.\nMark's Daily Apple \u2013 a very readable and entertaining source for everything from food to exercise.\nChris Kresser \u2013 very reliable and easily interpreted read for everything health related, with a specialty in pregnancy and infant health.\nBalanced Bites \u2013 a fun yet sound source for basic science and nutrition advice by Diane Sanfilippo.\nEvolutionary Psychology \u2013 Dr. Emily Deans explores psychiatry through an evolutionary health perspective.\nWhole Health Source \u2013 Dr. Stephan Guyenet examines his food reward\/palatability hypothesis.\nDr. Eades \u2013 a pioneer in low carb\/paleo diets, with a critical look into nutritional science.\nLean Gains \u2013 a site mostly dedicated to the benefits of intermittent fasting (IF) and weight training.\nDr. Peter Attia \u2013 a doctor and Gary Taubes' sidekick, whose purpose is to educate those about the benefits of low carb eating.\nDr. Kurt Harris \u2013 a new and unique look into ancestral health.\nWhole9 \u2013 a very well rounded site with a special focus on health, fitness, balance, and sanity.\nDenise Minger \u2013 is an absurdly smart young lady with an impressive knack for uncovering bad science.\nNuSI \u2013 a newly developed organization dedicated to researching the actual causes of obesity and hopefully thwarting its prevalence.\nEveryday Paleo \u2013 very excellent site for unique paleo recipes, and other health related things.\nPrimal Palate \u2013 good online resource to find great and healthy real food recipes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fraudsters are known to use scare tactics and tug at seniors' emotions when targeting them for a scam.\nMagazine sales scams typically begin with a knock at the door and a young person on the other side selling magazine subscriptions to \"raise money\" for a charity, school trip, or other seemingly worthy cause. Consumers who take the bait and sign up, however, often report receiving nothing in return.\nYou want to help people in need, but you also want to be sure that your charitable donation isn't simply going into a crook's pocket.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW BEDFORD \u2014 Rachel Vieira and Keith Bretton were married Oct. 7, 2006, at Brooklawn Park by Justice of the Peace Bob Bettencourt.\nThe bride, daughter of the late Walter and Patricia Vieira, is a certified nurse's aide.\nThe bridegroom, son of Lora Farley of New Bedford and the late William Bretton, is an auto glass technician.\nMaid of honor was Rebecca Vieira, sister of the bride.\nBridesmaids were Jillian Surprenant, niece of the bride, and Ashley Saulnier, niece of the bridegroom.\nLeaf girl was Amanda Tripp, niece of the bride. Ring bearer was Ian Bretton, son of the bride and bridegroom.\nBest man was Kevin Medeiros.\nUshers were Adam Vieira, nephew of the bride, and Ryan Sanchez, nephew of the bridegroom.\nThe couple visited New Hampshire on their honeymoon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Super Easy Gingerbread Cookies are, you guessed it, incredibly simple! Add a couple pantry staples to boxed muffin mix then roll, slice, and bake!\nLast week Lincoln's school baked Gingerbread Cookies as a class. Yep, a whole mess of three year olds mixed, rolled, and cut gingerbread men out then placed them onto baking sheets and sent them to the oven. Have I told you lately how much I admire teachers? You men and women are SAINTS!\nAnyway, when the kids went back to check on their cookies 15 minutes later, they found an empty oven and a trail of flour leading from the kitchen to their classroom! Apparently the gingerbread men had banned together to escape the confines of their hot home\u2026only to land on a cookie plate and eventually into the mouths of 14 tiny cookie monsters.\nI'm telling you, when the doors opened to let us parents in at the end of class, the children were losing their MINDS over the fact that their cookies had JUMPED out of the oven, and RAN down the hall, before they ate them all up. It was one of the cutest things I've ever seen.\nWhich is why I was on the receiving end of this gleeful face when I told Lincoln that we'd be baking MORE gingerbread men at home that very afternoon! (Sans flour trail on the floor because HELLO!!! Not happening.) Needless to say, these adorable and delicious cookies were a total hit.\nWhether you need a fun project to do with your kids this winter break (it's a-comin'!), are an avid holiday baker looking to get gingerbread cookies on the cooling rack a little faster, or you know, just love cookies (hi!) you are definitely going to want to try these shortcut gingerbread cookies made the semi-homemade way with Simple Mills Pumpkin Muffin and Bread Mix, who this post is sponsored by.\nHave you tried any of Simple Mills' products before? You can buy them directly on their website, and they're also sold in HyVee's health market, Whole Foods, Target, Sprouts (here's their store locator,) and Amazon.\nI randomly picked up a box of their almond crackers a few months ago and instantly became a fan of the brand. Their Sun-dried Tomato and Basil Crackers are a staple of my afternoon snack time and, as I mentioned a couple weeks ago, their Chocolate Chip Cookie Mix is SO dang good. Anyway, a few weeks after I first tried their products, Simple Mills reached out to see if we could work together on a project for the holidays. Crazy coincidence, and of course I said YES! I really, really love and believe in these products which are made with very few ingredients and zero preservatives. If you need to eat gluten or grain free, or are just looking for a healthier way to enjoy some of your favorite baked goods and snacks \u2013 Simple Mills, all the way.\nBut back to these Super Easy Gingerbread Cookies! Like I said, they're easily made by adding a couple spices to Simple Mills' Pumpkin Muffin and Bread Mix along with molasses, an egg, coconut oil, plus a little sugar. Roll them out, cut into any shape you want, bake, then decorate. Your house is going to smell INSANE, by the way. Like, the corner of your kitchen where the cookies are cooling is going to radiate gingerbread. Get excited!!\nThese little cookies would certainly brighten someone's day if you were to take them a couple, OR at the end of this post I'm sharing a cute cookie-making kit that would make a wonderful holiday gift for a friend or coworker who loves to bake\u2026or eat! I don't know about you but easy homemade gingerbread cookies > holiday socks in my world!\nStart by adding the baking mix to a large bowl. Baking with kids is fun, but messy! Did I mention how much I love teachers?!\nTo the baking mix whisk in 3\/4 teaspoon each ground ginger and ground cloves. In another bowl whisk together 1 egg, 1\/4 cup melted coconut oil, 3 Tablespoons molasses, and 1\/4 cup either coconut sugar OR granulated sugar (which is optional.) Pour the wet ingredients into the dry then mix until a dough is formed.\nRefrigerate the dough for 10 minutes then roll to 1\/4\u2033 thickness between two sheets of parchment paper. Use cookie cutters to cut into shapes then transfer the dough, which will be soft, to a parchment paper or silpat-lined baking sheet.\nOnce all the cookies have been cut out, refrigerate the baking sheet with the cookies on it for another 10 minutes before baking for 20-25 minutes at 275 degrees.\nLet the cookies cool completely, then let the fun begin! You can use your favorite royal icing recipe to decorate these cookies or pick up a tub of Simple Mills Vanilla Frosting, which is HONEST TO GOODNESS one of the best things I have ever eaten (like, so good, this happened. I also put some in my peppermint coffee yesterday. #sorrynotsorry) Scoop the icing or frosting into a piping\/ziplock bag outfitted with a piping tip then go to town. Like the muffin mix, this icing has zero funky ingredients, which might be why I felt super comfortable dunking the gingerbread men straight into the tub. ?\nYou could use mini chocolate chips, sprinkles, raisins, red hots \u2013 anything, really \u2013 to accent your gingerbread men and make them your own. Tip: if you use the Simple Mills frosting know that it doesn't harden like royal icing. Just a heads up in case you need to transport them somewhere!\nEither enjoy the cookies right away, or store in an airtight container for several days. Don't forget to save a couple for Santa!\nAs I mentioned, you can easily create a cookie-making kit with the muffin mix! Just gather a box of Simple Mills Pumpkin Muffin & Bread mix, a holiday cookie plate, and a few cookie cutters then tie them up with a sweet bow and hand deliver with the recipe. I can of course see this being a hit with a gluten-free friend, but also a teen who likes to bake?\nHowever you go about it, I hope you love and enjoy these sweet and simple, Super Easy Gingerbread Cookies this holiday season!\nPreheat oven to 275 degrees then line a baking sheet with parchment paper or a silpat and set aside.\nIn a large bowl, whisk together muffin mix, ginger, and cloves. Whisk together oil, sugar, molasses, and egg in a small bowl then pour into dry ingredients and mix until a dough forms. Refrigerate for 10 minutes to make dough easier to roll out.\nTurn dough out onto a sheet of parchment paper then cover with another sheet of parchment paper and roll to 1\/4\" thickness. Use cookie cutters to cut into shapes then transfer to prepared baking sheet (dough will be soft.) Refrigerate dough for another 10 minutes then bake for 20-25 minutes, or until golden brown and firm. Cool cookies completely then decorate with icing.\nUse your favorite royal icing recipe to decorate these cookies, or Simple Mills Vanilla Frosting.\nI am DYING over the fact that the teachers had the gingerbread men \"run\" out of the oven! DYING! Cutest idea EVER!!!\nIt was seriously so cute and hysterical at the same time! ?\nI hope you love them, Sara, they are so yummy!\nThese look so cute and the classroom story is just so precious!!! how fun!\nHow cute are these cookies?! And the recipe sounds absolutely mouthwatering! Sounds like a new tradition to bake and decorate with the kiddos in my family!\nI love your recipes but would like to know how to print just the recipe and not all 32 pages from post.\nHi Linda! Are you using the \"print recipe\" button found at the top of the post and also within the recipe card, or are you using your internet browser's print button? Eager to help figure this out for you!\nOh my gosh. You just made my whole week \u2013 grain free (and pain free) paleo cookies?!?! I'll be checking for that mix asap!!\nYes!! You've got to give their products a try \u2013 they're so, so tasty!\nThis is such a great recipe for so many reasons! A great gluten-free holiday option to share with family and friends It looks like your little guy got a kick out of helping too!\nLincoln's shirt!!!! love! :) these look awesome. I can't wait to try this recipe!\nThese are so good! I toned down the cloves by 1\/4 teaspoon knowing that it doesn't take much things to get \"too spicy\" for kiddos. Texture is great. flavor is delicious. Thank you!\nI'm so glad to hear it, Alexis! Thank you so much for the feedback \u2013 happy holidays!\nI tried to make these but the \"dough\" that formed was more like a REALLY REALLY thick and sticky batter. We couldn't cut them out with cookie cutters? I don't know what I did wrong, I bake gluten free all the time, but don't know what happened here? Looking for tips?\nOh no, I'm so sorry to hear that, Kerri! I made three batches of these cookies and they were always able to be rolled and cut after the refrigeration step. The only thing I can think of is that maybe you used a jumbo egg vs a large one, which would have added more liquid to the batter? Let me know if that was the case!\nThese look great! My daughter has a life threatening peanut and nut allergy. Is there anything that could be used to substitute for the Almond Flour in the muffin mix that's gluten free?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Concerned citizens have the ability to file a complaint or appeal a decision to the Texas General Land Office Community Development and Revitalization division (GLO-CDR).\nIf a citizen has a complaint, they should initially contact their local jurisdiction (city, county, public housing authority, council of governments, or vendor) in accordance with their local policies and procedures for customer complaints.\nThe GLO will provide a timely response to citizen complaints. The response will be provided within 15 working days of the receipt of the complaint, if practical.\nIf a citizen would like to appeal a decision made by the GLO-CDR Single-Family Housing Program (e.g., applicant denial notification, request for a waiver to a program standard, etc.), a written appeal must be submitted to the local jurisdiction (city, county, public housing authority, council of governments, or vendor) within ten days of the decision (documented by the certified mail delivery date). At the end of the 10-day period, if an appeal is not received, the local jurisdiction's decision will be upheld.\nIf the local jurisdiction denies the appeal, the citizen may submit an appeal to GLO-CDR for further consideration. The appeal must be received within ten days of the local jurisdiction's appeal denial (documented by the certified mail delivery date).\nUpon receipt of an appeal request, the local jurisdiction and GLO-CDR customer relations staff will respond to the appellant within 3-5 days acknowledging receipt of their submitted appeal. When practical, a written decision to the appellant will be sent within ten days; however, some cases may require additional time for the review.\nComplaints regarding fraud, waste, or abuse of government funds should be forwarded to the U.S. Department of Housing and Urban Development Office of Inspector General Fraud Hotline (phone: 1-800-347-3735 or email: hotline@hudoig.gov).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being a nurse isn't just about taking care of patients. Nurses don't stop being a nurse once they go home. Living the life of a nurse can be a bit challenging \ud83d\ude09 We can't hug our children until we take off our scrubs and take a bath.\nAfter work, we're too tired to cook. We're too tired to clean. We're too tired to kiss.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It was in 1909 that David Davidson's mother casually mentioned to her son that she had read a pamphlet describing the Great Pyramid, and that some were placing religious interpretations on the building. He said, \"I am going out to Egypt soon and I will put some of my engineering equipment in my bag and will blow their theories sky high during the fortnight I am there\". However, as it turned out, David Davidson not only spent 14 days on his research into the Great Pyramid in Egypt, but altogether about 14 years. He went out to Egypt as an agnostic but, as a result of his investigations, he came back convinced of the Pyramid's Divine Message and became a born-again Christian. He also authored \"The Great Pyramid, Its Divine Message\". Needless to say, it was very well received and still so today.\nHis testimony of conversion from an agnostic to a dynamic Christian was declared a short time later, \"I proclaim with humility and yet with confidence, that the Pyramid's Message establishes the Bible as the Inspired Word of God, and testifies that Jesus Christ, by His Displacement, paid the purchase price of mankind's Redemption, and effected the Salvation of all who truly believe on Him\".\nEditor: David Davidson was a distinguished Engineer, a Mathematician, an Astronomer, an Egyptologist, a Writer of distinction and the World's Most Eminent Authority on the GREAT PYRAMID OF GIZEH. He was also a firm believer of the Israel Truth and lent his support to BIWF, England for several decades.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Provide Motorhomes for sale, New Zealand Wide offering wholesale pricing direct to the public.\nDirect importers from Swift & Chausson manufactures. We are a privately owned business that has a passion for quality RVs, customers and service.\nRV Direct have re-located to a new 'All Weather' Showroom.\ncome and see us at 45 Duke Street, Hamilton.\nMotorhomes, Campervans for sale NZ \u2013 Factory Appointed NZ Importer, Distributor Dealer to NZ for UK and Europe brand Swift and Trigano Chausson. RV Direct is a private NZ owned company with vast industry knowledge dating back more than 20 years.\nRV Direct is New Zealand's only Wholesale, Direct to the public focused genuine Factory appointed dealer for UK and European Motorhomes, resulting in a low price structure with less GST, therefore a lower investment to enable families to enjoy this iconic lifestyle.\nSo, whether you are from Invercargill, Christchurch, Wellington or Auckland, Motorhomes from RV Direct definitely have merit. With many importers in the central North Island, we see Hamilton as the RV SuperCentre of NZ.\nWhen considering a Motorhome purchase we recommend visiting us before you engage your final decision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Przewalski s Horse (Equus ferus przewalskii).\nPrzewalski's (or Dzungarian) horse is a rare and endangered subspecies of wild horse native to the steppes of central Asia. At one time extinct in the wild, it has been reintroduced to its native habitat in Mongolia. Most wild horses today, such as the American Mustang or the Australian Brumby, are actually feral horses descended from domesticated animals that escaped and adapted to life in the wild. In contrast, Przewalski's Horse has never been successfully domesticated and remains a truly wild animal today. The horse is named after the Russian geographer and explorer Nikolai Przhevalsky.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahnzm b/data_all_eng_slimpj/shuffled/split2/finalzzzahnzm
new file mode 100644
index 0000000000000000000000000000000000000000..a7656b3b6bcbe3a9f3d35c866b5b2595b1b98aef
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzahnzm
@@ -0,0 +1,5 @@
+{"text":"Located near the Keraton Kesultanan Yogyakarta, this museum collects many traditional coach where it used for Yogya Keraton's ceremonial event such as wedding and kirab (celebrate a new sultan around city). I didn't count how many coach in this museum, but every there are so many coach from standard until exclusive coach. You also can read the story of each coach and when they're used with photo attached also.\nIf i'm not mistaken, here is the most eye catching coach from all over coach in this museum. It used for wedding ceremony of Sultan.\nCandi Borobudur is the biggest Buddhist temple in South East Asia (CMIIW), actually you can capture so many spectacular photo here if you come here in the perfect time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The wind just kept blowing, up and down, sometimes fast and sometimes slow.\nBelinda began to cry, but the wind didn't stop. Belinda fell asleep. She slept a long time.\nSuddenly the wind dropped her on the ground with a thud and she woke up. She looked around and didn't know where she was. The soil was different from the soil at home. It was cold here. There were tall plants that looked like poles all around her as far as she could see.\n\"The wind blew me here. He used to be my friend, but today he didn't play nice.\n\"Well, I am not playing with him anymore. You look funny. What are those things hanging off of you?\" Belinda asked.\n\"You sure have a lot of ears.\" Belinda said.\n\"It looks like hair,\" Belinda said.\n\"What's a farmer?\" Belinda asked.\n\"I can move by myself.\" Belinda said.\n\"Oh,\" Belinda said, \"now I understand.\n\"You're welcome. Can I go home now?.\" Belinda asked.\nBelinda began to cry. \"I can't wait a whole year to go home..\nCarl patted Belinda with the silk of one of his ears. .Belinda, next year you will be a full grown plant like your mother and dad. You won't be a little seed that can ride the wind..\nSo, Belinda snuggled down in the Oklahoma soil and put down her roots. She began to grow big and strong. In a couple of months she could stand straight and tall. She began to grow beautiful blue flowers out of her head. She looked around, but none of the other plants had blue flowers. Every day the farmer came by to talk to Belinda. Belinda thought it was because she was the prettiest.\nwas crying. \"What is wrong, pretty bluebonnet?\" the farmer asked.\n\"Yes, when the sun gets a little hotter, it will be time for you to make more seeds.\n\"Do you think they will ride the March wind?\" Belinda asked.\nSo Belinda became happy and content. She was happy to be around the other plants and the farmer. She was content knowing that one day her seeds might return to Texas where she was born.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The rumors are true: Microsoft is buying Minecraft developer Mojang for $2.5 billion . And as we discovered this morning, Mojang founder Markus \"Notch\" Persson is leaving the company. As you can imagine, the PC Gamer team has some strong feelings about the acquisition, and the impact of Minecraft.\nTom Senior: I remember Minecraft before it became the megalith it is today. The digital Lego concept seemed so beautifully simple. How had no-one done this before? It took off with astonishing speed, and we posted about it ceaselessly on PCgamer.com, fascinated partly by the game's expanding feature-set, but mostly for the creativity it enabled in its audience. One day: \"Holy crap, they built the Starship Enterprise in Minecraft!\" Another: \"They built a goddamn working CPU using redstone!\" The PC Gamer server stands as a testament to Minecraft as a platform, a canvas that allows for extraordinary collaboration between strangers.\nMinecraft has flourished thanks to community creativity, but let's not underestimate the importance of Mojang's thoughtful curation in enabling that. Today Minecraft still only costs \u00a317.95 \/ $30 with no microtransactions or subscription fees. You can buy plenty of merch, of course, but it's external to the Minecraft platform. No features are locked off. Imagine the same fledgling property in the hands of a greedier company; would Minecraft have become quite such sensation? I don't think so.\nIt's a great story. One guy makes an era-defining game in his bedroom, builds a small company, and grows into a global platform worth 2.5bn. Now Mojang's co-founders are choosing a quieter life, and no-one can begrudge them that. I can't see Microsoft doing a huge amount to disrupt Minecraft's current status quo, but I'd guess at cross-platform Minecraft 2 in a few years with a schedule of expansions and paid-for features that'll make squillions.\nAndy Kelly: Am I surprised? Nah. Minecraft is basically Lego now, and has become far too massive and mainstream for a small studio like Mojang to be the caretakers of it. I see kids wearing Minecraft t-shirts on the street. I know people who only play Minecraft. There are Minecraft books in bookshops. Its reach is wider than you probably think.\nSo it makes sense that a mega-company like Microsoft would buy it. I can't speak for Notch, but it seems to me like he got into this business to make cool things, not bags of money. And now that he has the bags of money, he's free to devote his entire life to making cool things. Will he ever make another Minecraft? Probably not, but creating a $2.5 billion company from nothing is a pretty landmark achievement for someone who's just turned 35.\nIn the statement he released, he says he feels like he's become a 'symbol', which must be a massive hassle. If someone's playing Minecraft and the server they're on crashes, they'll probably tweet Notch and call him a jerk, even though it has nothing to do with him. Being a visible figure on the internet is like painting a target on your back and handing people a shotgun, so I don't blame him for wanting to slink back into the shadows.\nChris Thursten: I can understand and sympathise with not wanting to be a public figure of the type that Notch has become over the last couple of years. It makes total sense. When being a prominent indie developer makes you liable to be co-opted into other people's movements without your permission, it's understandable to want to go back to game jams. From his perspective, and I imagine that of most of Mojang's senior staff, this is a way of drawing a line under the project and moving on. There's something to be said for closure.\nI'm not convinced that this is particularly good news for players or fans. Microsoft might have a good track record from Mojang's perspective, but they don't by almost any other measure. Games for Windows Live was and still is an awful piece of software. They've been famously bad at dealing with smaller indie studios in the past. The company seems set on chasing the dream of a closed platform when its success is grounded in the exact opposite. All of this is either directly or thematically opposed to the conditions that made Minecraft possible in the first place. Indeed, the Minecraft that most players enjoy isn't Notch's game\u2014not entirely. It's the product of a huge amount of mods and addons working in aggregate, something that has emerged from the community Mojang enabled as much as the game they created.\nMicrosoft has never been good at looking after gaming communities. They've got a lot to prove now.\nPhil Savage: There's a part of me that's surprised. There's a part of me that's not surprised. In some ways, Minecraft is the poster-child for open, collaborative, PC development. It's not so much a game, as a symbiotic relationship fed by the creativity of a community itself hungry for ways to express and create. That relationship has been bought by Microsoft. What?\nBut then, that relationship was far from simple. That was highlighted perfectly with the furore over changes being made to the EULA\u2014setting out more clearly what server owners could and couldn't do. There are whole businesses built on the existence of the game, and\u2014however legitimate their business practices\u2014they were going to fight any possible restrictions. I'm not sure Mojang were fully prepared for how fierce that fight became.\nTo be blunt, Microsoft don't care. They've got the infrastructure to effectively manage and police an empire as vast and sprawling as Minecraft. At the same time, everything they do\u2014especially in games\u2014is defined by a preference for closed systems. I'm not sure they have the restraint to leave it alone; to let the game retain its openness and community focus. I hope I'm wrong.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Classes, dances and other dance events in or near Cambridge for Folk\/Country. Please click on the event name for more information which may include a map.\nCambourne, Cambridgeshire Folk Dancing from Balkan nations for beginners & intermediate. Let's have fun together!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At Auto Loan Store, we're here to help with the best title loans near Margate. Our fast and friendly Margate title loans are the answer for your bills, medical expenses and other overwhelming costs. So call 954-785-3138 or fill out the \"Get Started Today\" Webform for a free loan quote.\nEveryone knows how frustrating bills can be, especially when new expenses keep popping up. Like many Americans, you may be having a hard time paying off your expenses while keeping up with the costs of the new year. Your paycheck can only stretch so far, but you don't have to struggle to cover the gaps. Our team at Auto Loan Store is here to help with the best title loans near Margate.\nMargate title loans are a great way for people with great credit, bad credit or even no credit to get back on their feet. That's because there's absolutely no credit check involved in the auto title loan process! Instead, your car's equity is used to secure your loan. But don't worry, our team at Auto Loan Store won't grab the keys during your auto title loan. We'll work with you to develop a payment schedule that's favorable for your needs.\nMargate car title loans are the perfect solution for bills and other financial tough spots. If you've spent more than you could afford, you're probably feeling the pinch now that bills and utilities are due. Don't let the lights shut off, our team at Auto Loan Store will offer the loan you need to make ends meet as you play catch up from excessive spending.\nWhen you're looking for the best car title loans near Margate, we hope you'll work with our dedicated team at Auto Loan Store. We're offering the fastest and most affordable title loans near Margate, helping people recover from spending and other frustrations. If you simply don't have enough to make ends meet, let us help! Stop into our store, located at 3489 Davie Blvd, today!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaiyvh b/data_all_eng_slimpj/shuffled/split2/finalzzzaiyvh
new file mode 100644
index 0000000000000000000000000000000000000000..4e8c0227750ab9b13d2feb398c8f86c9bcd0bf21
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaiyvh
@@ -0,0 +1,5 @@
+{"text":"LONDONDERRY, N.H. - The Londonderry fire chief says confusion over the location of a fire delayed the arrival of firefighters from neighboring departments.\nFire Chief Kevin MacCaffrie says Windham and Pelham firefighters were sent to the wrong Spring Road on Sunday. There are three Spring roads in town.\nMacCaffrie says officials are working to assign new addresses to approximately 50 roads in town to avoid such problems.\nThe New Hampshire Union Leader says the fire that destroyed the home on Sunday appears to have started about 9 a.m. in a living room couch.\nHomeowner Robert Simmons suffered second degree burns to his hands and wrists trying to put out the fire.\nThe house was ablaze when firefighters arrived.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Little bit of column A and little bit of column B.\nHere's what I suspect is on the voice recorder: two pilots struggling with all their force to overcome an automatic flight regime system suffering from a single sensor failure. Neither pilot has either the time, strength or operational familiarity with the flight management system to turn off \"George\".\nthey built \"George\" to be stronger than either pilot and possibly stronger than both of them (probably some good old boy American style human factors chauvinism going on here, as most of the rest of the world doesn't eat beef and go to the gym three times a week) . Further Boeing eliminated an \"in feedback loop\" mechanism for turning off the automatic flight system (pilot input at the yoke over a certain force level used to turn it off). When an aircraft \"thinks\" it's about to stall and pushes the nose over to gain airspreed, neither pilot is going to take the time to key through five to ten sub-menus on the FMS that it take to get to the place to turn \"George\" off.\nhaving a flight control system be dependent on a single sensor is negligent engineering. I thought Boeing was better than this. Frankly, it's damned scary and I would really re-think taking a Boeing aircraft it they don't do a Mea Culpa on this. There are some very simple energy calculations Boeing could have used to verify if even single sensor data was bad (namely, if the engines are producing thrust, the air speed id dropping off, but no elevators control surfaces delection has been required to maintain altitude\/deviates from attitude control forces required, then, maybe, just maybe the airspeed sensor is compromised and the pilots should be hand flying the aircraft). I guess wall st. sucked up all the physics graduates.\nWrong. This event should have been treated as a stabilizer trim runaway, the pilots should have disabled the stabilizer trim with the cutout switches located just below the flap handle. The upset could have been caused by the module that the pilots didn't know about, however stab trim runaway no matter what causes it is treated the same.\nThis was a bad AOA and, iirc, the trim cutout has to be held instead of switched (the change), it was applied numerous times as the airplane trimmed itself into a crash.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good morning from Augusta. With Attorney General Janet Mills shifting from the state workplace constructing to the Blaine Home after final week's election, legislative Democrats are dealing with a five-way intraparty race to exchange her, that includes candidates with a broad vary of experiences.\nThey embrace a state senator and former attorney general, two different sitting legislators, a prime patent lawyer from certainly one of Maine's main companies and a district attorney for 2 counties. That roster hasn't modified since we checked on the race in June. What has modified is that we all know that Mills will want a alternative.\nDemocrats already maintain two of Maine's three constitutional workplaces. Candidates are nominated by their events and elected by a majority of the 2 legislative chambers mixed. After the election, Democrats will maintain 110 of 186 seats, placing them able to management the workplaces.\nThat may occur in early December, however these races for statewide positions are months and years within the making, regardless that voters don't actually get to weigh in on them. Right here's the place these under-the-radar races stand now.\nA lot of the candidates \u2014 in several methods \u2014 want to push the attorney general's workplace in a extra aggressive path. The 5 Democratic candidates for the place are state Sen. Mike Carpenter of Houlton, a former attorney general, Rep. Aaron Frey of Bangor, Sen. Mark Dion of Portland, Tim Shannon of Yarmouth, a companion at Verrill Dana, and Maeghan Maloney of Augusta, the district attorney in Kennebec and Somerset counties.\nThe construction of the workplace is exclusive. Whereas the attorney general's spot has traditionally been held by politicians with a prosecutorial background, the workplace runs on the backs of profession legal professionals and is essentially nonpartisan. It prosecutes homicides and different main crimes, however most of its work is in civil regulation.\nDemocrats are buoyed now by Mills' election and their robust majorities within the Legislature. However a lot of Mills' main bid for governor was dominated by her opponents pushing her from the left on her document as attorney general.\nIn the course of the marketing campaign, Indian tribes and progressives assailed Mills for her position in authorized fights over tribal water rights. Three candidates \u2014 Carpenter, Frey and Dion \u2014 talked about enhancing relations with tribes in interviews this week. You possibly can anticipate to hear others subtly delve into methods they differ from Mills.\nCarpenter, Frey and Maloney would be probably the most conventional picks for the workplace. Dion and Shannon are pitching totally different instructions. Having held the workplace from 1991 to 1995, Carpenter would be an old-school decide for the workplace. Frey runs a solo apply in Bangor, however he's within the Maine Home of Representatives, the place 89 of the 110 Democratic votes lie. Maloney is a former legislator with ties to Emerge Maine, an influential group in Democratic circles that helps ladies get elected.\nMills' workplace has joined many authorized fights towards the administration of President Donald Trump and these Democratic candidates largely anticipate that to proceed, making the job probably extra politically seen than it has been prior to now.\nShannon, a prime patent lawyer who ran towards Mills in 2012 for the place, stated he envisioned \"a active, forward-looking and professional\" workplace beneath his watch, saying it might broaden its position in preventing Trump and the pharmaceutical business.\nA smaller Republican caucus within the Maine Home of Representatives has a brand new management workforce. The incoming 57 Home Republicans elected Rep. Kathleen Dillingham of Oxford to function minority chief. Rep. Trey Stewart of Presque Isle will take over as assistant minority chief. It's no coincidence that they hail from a few of Maine's most rural, reddest areas.\nDillingham, who will be serving her third time period representing District 72, succeeds Ken Fredette, the termed-out consultant from Newport who led Home Republican for six years during which the caucus usually aligned with Republican Gov. Paul LePage to frustrate majority Democrats within the Home and sometimes stall laws that Home Democrats and the Senate had backed. With fewer Home Republicans and the lack of most of the caucus' extra outstanding members to time period limits or Senate runs, Dillingham will lead a regrouping effort. As Mills prepares to transfer into the Blaine Home and Democrats management each branches of the Legislature, Dillingham gained't have an ally within the government department, so she is going to face the problem of wielding drastically diminished energy.\nStewart, who will be serving his second time period representing District 147, will exchange Rep. Ellie Espling, R-New Gloucester, who served 4 years as assistant minority chief earlier than dropping a Maine Senate bid final week. Stewart has been perceived to be a possible Republican chief since he was first elected two years in the past. Nonetheless solely 24, he epitomizes a brand new era of rural conservatives upon whom Maine Republicans should rely to offset the growing political may of Democrats from Portland and the state's southern and coastal progressive enclaves.\nA Democratic Social gathering insider will turn out to be the chief of employees for Maine's next governor. In her first vital appointment since profitable the Nov. 6 election, Mills introduced Wednesday that Jeremy Kennedy will function chief of employees for her administration when she takes workplace in early 2019.\nKennedy, a Portland resident, took over as Mills' marketing campaign supervisor in September. A former government director of the Maine Democratic Celebration who led Hillary Clinton's marketing campaign in Maine in 2008, he has accomplished advocacy work for Deliberate Parenthood and Emerge Maine.\nMills will maintain a information convention this afternoon to announce the formation of an government search committee to assist her discover candidates to fill Cupboard positions and different executive-level state jobs. There are about 150 political appointments to be made.\nA winner may lastly be declared at the moment in Maine's 2nd Congressional District election. Although a decide's ruling that would alter the result of the 2nd Congressional District election is predicted right now, Maine's secretary of state stated his staff will proceed counting ranked-choice ballots within the meantime. Secretary of State Matt Dunlap, at a Dirigo Speaks occasion Wednesday night time hosted by the BDN, stated his goal \u2014 finishing the tally \u2014 hasn't modified regardless of the lawsuit Republican incumbent U.S. Rep. Bruce Poliquin filed towards him earlier this week. Dunlap, who stated getting sued is a part of his job, expects his staff to end counting ballots round midday right now, with the entire race outcomes coming shortly after. Right here is your soundtrack.\nA improvement growth in Brunswick has unfold past the boundaries of its former Navy base. Previously vacant downtown storefronts have been crammed as new business improvement takes place on the previous base, which the Navy left in 2011, and elsewhere on the town. Lately, new residential improvement plans have been hatched to increase the conversion of former Navy housing to civilian use.\nPolice consider a Bangor man's demise was a murder. Detectives are in search of Donald Galleck, 29, for questioning in reference to the demise of Jason Moody, 40, of Bangor. Moody was discovered unconscious at about 10:40 p.m. Sunday on the intersection of Middle and Cumberland streets. He died from his accidents Tuesday afternoon.\nDisgusting gelatinous creatures are displaying up in Maine waters. Add sea squirts \u2014 quite a lot of small, tube-like marine creatures that reside bunched collectively in colonies \u2014 to the record of species that appear to be thriving within the warming waters off the Maine coast. Marine scientists say that the gelatinous animals have been rising in quantity in shallow waters alongside the coast and will be creating issues for native organisms which might be getting squeezed out of their conventional habitat on the underside.\nIt's Nov. 15 and individuals are already getting right into a lather as a result of forecasters predict Maine will get blasted with a bunch of snow on Friday.\nAt the moment's Daily Brief was written by Alex Acquisto and Robert Lengthy. When you're studying this on the BDN's web site or have been forwarded it, click on right here to obtain Maine's main publication on state politics by way of e-mail on weekday mornings. Click on right here to subscribe to the BDN.\nTo succeed in us, don't reply instantly to this text, however e-mail us immediately at aacquisto@bangordailynews.com, mshepherd@bangordailynews.com or rlong@bangordailynews.com.\nWhat Can Happen When You Try to Ingest Random Plants to Get High?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You can divide the world up into two camps; those that know English actor Warwick Davis as Willow, Wicket and best of all the Leprechaun and those that know him as Ricky Gervais's mate who was in Harry Potter.\nThis division will drastically alter your preconceptions of the brand new Leprechaun reboot epically entitled Leprechaun: Origins brought about by WWE Studios and staring the wrestler Dylan \"Hornswoggle\" Postl which following on from See No Evil 1 & 2 sees the bulging biceps of the wrestling brand tearing into horror movie making.\nBefore delving deeper into Leprechaun: Origins for those unfamiliar with the original Leprechaun series it featured aforementioned Warwick Davis as the titular character and ran from 1993 to 2003 racking up 6 movies in all.\nSmall in stature but filled with evil and driven by a lust for gold to kill and maim while delivering rhyming couplets the films where comedy horror's high on camp and gimmicks with Davis channeling Robert Englund's Freddy while fighting off tons of teens including Jennifer Aniston to get back his pot o' gold.\nWith 3 standard sequels part 4 saw the Leprechaun in Space after a spoof Apollo 13 poster with Davis's head over Tom Hank's made as a joke in the film companies office amazingly ended up getting the movie a green light even though there was no script. Parts 5 and 6 took the Leprechaun to the Hood combining clumsy culture shock comedy and bad rap jokes in a duo of films one of which I am sure Ice-T would love to forget ever happened.\nWith all this in mind it is easy to see firstly why a reboot is necessary and secondly how it could all go terribly wrong. Thankfully director Zach Lipovsky and screenwriter Harris Wilkinson have not only acknowledged all the issues of the original series but avoided repeating any of the same mistakes trying their hardest to do the one thing the original films always failed to do which is actually scare audiences.\nLeprechaun: Origins sees backpacking buddies Sophie, Ben, Jeni and David (Stephanie Bennett, Andrew Dunbar, Melissa Roxburgh and Brendan Fletcher respectively) in the green hills of Ireland dropped off in the middle of nowhere by a nervous farmer who is fearful of venturing past some sacred stones they see in the middle of a field.\nDropping into the local pub for a pint of the black stuff local character Hamish (Garry Chalk) overhears their loud American conversation and learning they are yearning for some adventure tells them about an ancient site of stone sculptures older than anything else on the emerald isle.\nEager for one last exploration before they head home to their humdrum college lives they take Hamish up on his offer and head out to spend the night in a shack from which they will make the long trek from the next day. But all is not what it seems and as the foursome settle in for a fun night something outside watches them drawn to the house by gold and waiting to get in and tear them apart to get to it.\nAlthough the casting of wrestler Dylan Postl, whose character Hornswoggle ticks every clich\u00e9 in the racial stereotype book when it comes to Irish culture, may look like this reboot was just a rehash of Warwick Davis comedy horror Leprechaun persona the film takes a much more serious slant on the characters look and history making it into a monster from ancient folklore more akin to Gollum or the creatures from The Descent than something seen at a St Patrick's day parade.\nWell realized with some excellent effects and make up ingeniously we barely see the leprechaun till very near the end increasing the threat and fear the creature creates which added to the fact the characters themselves find the situation they are in ridiculous and say so too is by far the best way to accentuate the terror the film tries to create.\nThis smart shift from the original added to the story setup which is reminiscent of The Shrine or Nothing Left to Fear gives the film much more weight especially when it comes to the tense scenes and the jumps and scares which there are plenty of.\nOkay so Leprechaun: Origins isn't the most original horror and the location only vaguely looks like Ireland if you squint and the weird Predator POV that we see when the beast strikes is slightly jarring but on the whole as reboots go it's far from the worst and massive credit must be paid to everyone involved for at least trying to turn a horror villain we loved to laugh at into something that makes us scream.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We use third party re-marketing pixels to advertise across the internet and on social platforms. These will display relevant ads tailored to you based on which areas of our website you have viewed and your interests\/activity on social media. Pixels do not allow us to identify specific individuals or allow us access your computer, your files, or your devices. You can opt out of re-marketing by adjusting the settings in your Google and Facebook accounts and by adjusting your privacy settings on your computer\/device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaobpf b/data_all_eng_slimpj/shuffled/split2/finalzzzaobpf
new file mode 100644
index 0000000000000000000000000000000000000000..02dd3a90862ad2d4b992aa31f7ad0d12a108271b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaobpf
@@ -0,0 +1,5 @@
+{"text":"The Arkansas Transportation Report provides a monthly compilation of data and news regarding the trucking, railroad, airline, and river barge industries. Individually, these different components of transportation and cargo offer unique insight into the current state of the economy. Combined, the totality of the information paints a much broader picture of the health of the transportation sector.\nTransportation Secretary Elaine Chao has granted trade organization American Trucking Associations' request to pre-empt meal-and-rest break rules in California after the organization sought federal legislation to overrule the state mandate\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alsion - women who have miscarried are sometimes advised to avoid s_x for the first 12 weeks, exp. in weeks 8 and 12 (when your period would be due) so you're not being irrational. In some women s_x can trigger a miscarriage. If you do have s_x and get any bleeding after it you shouldn't have s_x for first 12 weeks. Your husband will survive - just look after yourself.\nAlison, I am feeling exactly the same way. I had a mc in Feb and am now 6 weeks pregnant and very nervous! I will be avoiding s_x for the first trimester just for my own peace of mind. i do feel badly that my husband has to wait but he does understand. good luck!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gary has over 25 years of hands-on experience as Controller and Chief Financial Officer for small and medium sized businesses in a variety of industries. During his career, he led the expansion of a small medical service provider to a four state, 15 clinic regional operation. He specialized in business planning and helping clients secure the financing required to realize their vision.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1. A list of participating Intercounty branches is available from Intercounty, Boundary House, Stansted Road, Bishop's Stortford CM23 2TH. Each participating branch is running its own Fireworks competition within a selected number of schools. One entry per entrant who must be a pupil at a selected school on 3rd November 2014. Children of Intercounty employees or employees of companies in the group of companies controlled by LSL Property Services plc cannot participate in this competition.\n2. Closing date for receipt of all entries is the 3rd of November 2014. All competition entries must be handed in to the relevant school.\n3. Entrants must draw a firework inspired image in the space on the entry form. The entry form must be completed by a parent or guardian and must name the entrant.\n4. Intercounty does not accept responsibility for lost or damaged entries. Entries that are altered, illegible or not in accordance with the rules of entry will be disqualified.\n5. One winning entry will be selected by each participating branch by a judge who will be an Intercounty employee. The winner will be selected on 4th of November 2014 and the winner's drawing will be displayed in the relevant Intercounty office window for 1 week following notification of the winning entry to the winner.\n6. In relation to each participating branch, the prize consists of a gift family ticket (2 adults + 2 Children) to the Warner Bros. Harry Potter Studio Tour London for the winning entrant and \u00a3100, which will be sent to the school attended by the winning entrant.\n7. The prize is non-transferable. The judge's decision is final and no correspondence will be entered into. No cash alternative will be offered.\n8. The winners will be notified by phone or email no later than 13th of November 2014.\n9. The winner must be willing to have their name published and may be required to take part in any post-event publicity including a photograph and their picture published in the paper.\n10. The name of the winner will be made available on request in writing.\n11. Personal data supplied by the minor's parent or guardian will only be used by ICIEA Limited or passed to LSL group companies in order to promote estate agency, lettings and financial services with the express written consent of the parent or guardian. No promotional materials will be sent to the named minor or to the named school (other than competition entry forms).\n12. Entry to the competition is deemed acceptance of these rules.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover the historic and contemporary milestones of the Providence waterways on our fun & relaxing 50-minute narrated tours! Explore the Providence River, Riverwalk, Waterplace Park and the Providence Harbor, as we highlight historical, nautical and architectural points of interest. We had the distinct honor of being named the GoProvidence \"Member of the Year\" in 2016 and have received TripAdvisor's \"Certificate of Excellence\" in 2015, 2016, 2017 & 2018. In June of 2017, we were recognized as the \"Best City Tour\" by Yankee Magazine and were named \"Best Local Tour\" by Best of Rhode Island in 2018. We are thrilled to be celebrating our 28th season along the Providence waterways this year...join us for a tour!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaofil b/data_all_eng_slimpj/shuffled/split2/finalzzzaofil
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index 0000000000000000000000000000000000000000..b85621eb232f9dcaaf2a93b74d12c875d624d22d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Which is better, thermal transfer or thermal direct? It depends on the application. Thermal direct is normally used for short term labels, such as shipping labels, food labels at the deli, etc. Thermal direct paper is sensitive to UV light and will change colour in the sun or under fluorescent lights. Thermal transfer is normally used for labels that need to last longer, such as retail clothing tags, product ID codes, date and batch coding, etc. Thermal transfer labels will not discolour in UV light.\nIs thermal direct cheaper than thermal transfer? On balance, yes. With thermal direct there is no ribbon, so that cost is saved, however the ribbon has a special, low friction backcoat that protects the print head. Thermal direct paper is very abrasive compared to thermal ribbon and head life will be considerably reduced without a ribbon to protect it, so the saving gained by not using a ribbon must be offset by the cost of replacement heads. In some cases the head life for thermal direct will be \u00bc that of a head protected by a ribbon.\nWAX ribbons are the cheapest grade and come in 2 basic variants, pure wax (also called economy, or just wax) and resin enhanced wax (also called premium wax or hard wax). The difference in price between pure wax and resin enhanced wax is very small, just a few percent. Pure wax is very soft and smudges and scratches very easily. The benefit of the added smudge and scratch resistance offered by the enhanced wax ribbon usually more than offsets the small increase in price.\nWax ribbons print well on very rough papers, such as vellum, as well as gloss and synthetic labels, however the smoother the paper the easier it will smudge.\nWAX\/RESIN is a combination of wax and resin. There is no standard definition for what constitutes wax resin and there is a great variation in the ribbons from different manufacturers. Wax is cheap and resin is expensive, so the more resin in the product, the more expensive it usually is. Wax resin ribbons generally have much better scratch and smudge resistance than hard wax types, and are often used where there is some chance that the label might be abraded in some way, for example in transit or exposed to higher temperatures (outdoor products).\nWax\/resin ribbons work well on matt transfer and glossy labels stocks, as well as most synthetics.\nRESIN ribbons come in several varieties for special applications. Standard resin is often used when the label\/ribbon combination must endure a harsh environment or extreme abrasion, such as plant pots at a garden centre or goods in a builder's yard, etc. Resin ribbons are normally used in conjunction with synthetic labels and together provide a very robust labeling solution.\nThere are special resin ribbons available with extreme chemical resistance, such as for wash care and dry cleaning applications, oil resistance (hardware products) and even high temperature resin for cook in the bag applications.\nResin ribbons do not print well on paper labels and even on synthetic stocks the ribbon requires a much higher temperature than wax or wax resin ribbons.\nHow wide should the ribbon be? The ribbon should be wider then the label. Paper is abrasive and if the label is not covered by the ribbon it can damage the head where the paper touches the head. The ribbon is too thin to stop the paper abrading the head unless it covers the label completely.\nWhat does FACE IN or FACE OUT mean? The ribbons are made with the ink on the inside of the roll (inside wound - Face In), or on the outside of the roll (outside wound - Face Out. This is because different makes of printers have different winding paths from the spool of ribbon to the print head. Other terms for the same thing are CSO (carbon side out), CSI (carbon side in), ink in, ink out, IW (inside Wound) OW (Outside Wound) etc.\nHow often should I clean my printhead? When using Direct Thermal labels you should clean the printhead after every roll of labels. When using Thermal Transfer with ribbon then clean the printhead when you install a new roll of ribbon. If the printer is in a dirty environment then we recommend cleaning the printhead on a daily basis. If you are using labels with a thick adhesive or where there is adhesive bleed then the printhead will need to be cleaned at least daily. Cleaning should be done with a lint free cloth impregnated with Isopropyl alcohol.\nThere is a thin white vertical line in my printing, what is it? There may be some dust or debris on the printhead, try cleaning the printhead, if it is doesn't disappear then it is usually a blown printhead, which cannot be recovered.\nWhat is the \"best\" ribbon? The best ribbon is the one that works well for you at a price that is acceptable, However, many users buy ribbons on price alone and often don't realise how this reduces the look of their product after it has been through several stages of the supply chain over a period of weeks. These days there are very good wax resins available at a price that is not much higher than enhanced wax that maintain the \"just printed\" look for months, no matter how the product has been handled.\nWhat if my printer supplier says I must use their ribbon? All the ribbons supplied by all the printer manufacturers in the world come from the same sources. A Zebra ribbon, for example, is not made by Zebra. An Intermec ribbon is not made by Intermec. A TEC ribbon is not made by TEC. Reputable ribbon converters use films manufactured by the same coaters that supply the OEM branded ribbons for the printer manufacturers. Of course, there are ribbons that are better than others, just as there are brands, such as Ricoh, Fuji Copian, General, IIMAK, Armor, Union, etc, that are the leading brands of the world. Ribbons from Sticky Business are from films supplied by the world's leading brands.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for the best coffee mug for by the fire, on adventures, while at work, or relaxing at home? With this reusable MiiR camp cup, you'll find perfect temperature control which helps maintain the quality of your favorite coffee or tea! We believe you'll love life with every cup you sip from this sustainable mug. Made by a fellow Certified B Corporation.\nMake your campfire memories a little warmer with MiiR's modern take on Grandpa's old camp cup. This 12oz Camp Cup is vacuum insulated so you won't burn your hands (or melt your face) when you are at work, home, or camping. Featuring a press fit lid to prevent spilling and a convenient handle to make picking up your coffee mug a little easier. Use it for your coffee, tea, or your favorite drink you want to keep hot or cold. In addition to high quality, each sale by MiiR (a Certified B Corp), funds a trackable giving project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Social Booking marking is when you take a page from your site and bookmark it on various sites. It's important so Google and other search engine's will index you web page quickly. is this a is a method for web users to store, organize, search, and manage bookmarks of web pages on the Internet with the help of tags or keywords on sites such as Digg or Del.icio.us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It has been a simply glorious day. This morning I awoke to sun shining through lacy branches encased in ice with patterns of intricate frost decorating every windowpane. As my Whippet reclined in her favorite patch of sun, I downloaded a marvelous new book on my Kindle. What could be better?\nThis week I've had the pleasure of working on some wonderful writing projects. I'm off to a great start on my YA historical, and now I'm shifting my focus to a middle grade novel that I wrote in 2017 and am now rewriting. I'm thoroughly enjoying settling back into the world of this humorous, adventurous middle grade about three siblings and the troubles they encounter upon being sent to a new home.\nI also interviewed two incredible NCAA athletes this week, Michigan's Natalie Wojcik and Georgia's Rachael Lukacs, both former Nastia Liukin Cup competitors who have transitioned to college and are doing *amazing* things! I can't wait to share these interviews with you. And, in case you missed it, you can check out my recent Q&A with Florida's Sydney Johnson-Scharpf here!\nI am really enjoying keeping up with my projects by using my new writing planner. It's already been extremely helpful for organizing everything!\nAfter finishing Norvia of Walloon Lake for NaNoWriMo in November, I've been contemplating what will transpire in the third book of this series. This in itself is a highly enjoyable occupation as I scour through name books, dabble in research, and select the themes and imagery I'll incorporate. I cannot wait to start writing it!\nI also joined two writing societies this week: American Christian Fiction Writers (ACFW) and the Society of Children's Book Writers and Illustrators (SCBWI). I'm already enjoying the resources these two organizations provide, and I'm eager to explore my memberships further!\nI hope you're having a wonderful weekend and that next week brings you plenty of delightful reading time!\nOnce again Anna Rose you have written an enjoyable blog. I always look forward to reading your posts. Wishing you well with your writing.\nYour book selection is very interesting. Hope you have a successful writing for your soon to be books.\nAnna Rose you never fail to write an interesting and informative post. Your blog is delicious. Continued success with your writing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's time for another high resolution texture set. This time it's all about wood panels. Wood textures pop up in more and more websites as background images. It's a trend which sure won't disappear in the near future. With a bit of tweaking these wood panels textures can be transformed into tile-able background images for your web projects and at the stunning size of 3264\u00d72448 pixels they're great for your print projects too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapagd b/data_all_eng_slimpj/shuffled/split2/finalzzzapagd
new file mode 100644
index 0000000000000000000000000000000000000000..73ae862311151f1930ee8cf99e10c24cc84eefc6
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzapagd
@@ -0,0 +1,5 @@
+{"text":"The battle for king of the iPad text editor heap has a new contender: Writings. It faces off against the likes of Elements, PlainText, and WriteRoom, but it has a plenty of nice tricks up its sleeves that make it a worthy competitor in this increasingly crowded arena.\nWritings offers a hefty number of features for its initial release, including letting you specify the font face, font size, text width, and text and background colors (there's even an inverted mode). A real-time counter allows you to keep track of the number of characters, words, and lines in your document and even lets you check just the selected text. There's also built-in support for TextExpander and the ability to quickly enable or disable spellcheck.\nFor my money, though, the coolest text-related feature in Writings is that tapping in the left or right margin of your document lets you move the cursor by a single character. If you've ever grumbled about getting iOS's text-selection loupe in just the right place, you'll probably appreciate this functionality.\nLike many of its competitors, Writings integrates with online storage service Dropbox, letting you store and retrieve your documents from the network. In addition, you can e-mail your documents\u2014either as part of the message body or an attachment\u2014or access your documents via iTunes's File Sharing feature.\nWritings costs $5 and is compatible with any iPad running iOS 3.2 or later.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the academic year 2018-19, we have listed, the list of top engineering colleges in Delhi, which offer undergraduate & postgraduate courses in engineering, applied engineering and sciences. The list includes NITs, IITs, IIITs, governments and other public & private technical universities.\nThere are hundreds of colleges for engineering in Delhi. But if you pursue your engineering course from one of the best engineering colleges in Delhi, top 100 engineering schools to be precise, the chances of getting a job in one of the best companies in the engineering sector will be higher. The topmost engineering colleges in Delhi have better infrastructure, faculty, and placement opportunities.\nThere are many reputed and prestigious private and government engineering colleges in Delhi. In this article, we will list out the top 50 Engineering Colleges in Delhi for 2017.\nEngineering Colleges in Ghaziabad and NCR Delhi are centres of attraction among students seeking professional education in the areas of engineering science & technology.\nStudents from across the country throng to Ghaziabad and NCR Delhi for pursuing higher education in different branches of engineering, viz; civil engineering, mechanical engineering, Information Technology, Design Engineering, Instrumentation Engineering, Architecture, Electrical and Electronics Engineering, etc.\nAdmission in various courses of engineering is offered on the basis of entrance examination, aptitude test etc.\nB.E., B.Tech, B. Arch degrees need All India Engineering Entrance Examination (AIEEE)\/Uttar Pradesh State Entrance Examination (UPSEE), etc for admission.\nPostgraduate programs like M.E. \/ M. Tech., M.Plan. etc, accept Graduate Aptitude Test in Engineering (GATE), Graduate Record Examination (GRE) etc score. However, some colleges could also conduct individual written test.\nFor admission into undergraduate engineering degree programs, minimum educational requirement that one should possess is 10+2 years of school education from CBSE or any other qualification in relevant discipline.\nFor admission into postgraduate engineering degree programs minimum educational requirement is Post Graduate in Science or Graduate in Engineering Discipline.\nBesides this many students also take certificate courses, job oriented courses, diploma programs, and various other programs from the Best Engineering Institutes in Ghaziabad and Delhi NCR.\nThere are numerous management as well as engineering colleges in Delhi that are approved by the All India Council for Technical Education (AICTE). A list of some of those colleges is given below.\nWhat is The Fees For Top Engineering Colleges in Delhi-NCR?\nCandidates who have the desire to get Engineering Degree in any of the Top 50 Engineering Colleges in Delhi then you can acquire the Best BE\/B.Tech\/M.Tech colleges from here. To build a better career one must choose the best engineering college for their further studies as Degree of Engineering is one of the desirable degrees and candidates have to face competition for getting it. So candidates are suggested to don't make any mistake and opt for the top 50 engineering colleges in Delhi. Aspirants you can have a look on the Top 50 Engineering Colleges in Delhi 2018 and also get the fee structure, address and official website from this single page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Park District maintenance crew removed the floating boat docks at Chilo Lock 34 park Sept. 11.\nCHILO \u2013 Fall is coming fast.\nOne tell-tale sign is the removal of the boat docks at Chilo Lock 34 Park.\nThe temporary floating docks came out Sept. 11 in the wake of recent heavy rains throughout the Ohio River Watershed.\nThe removal may have been a little bit earlier than in years past, but with flood debris floating down river and the potential of more coming after Hurricane Florence, it was necessary to avoid damage to the docks.\nWhile the docks won't be replaced until May of 2019, the ramp remains open so boaters can enjoy the Ohio River from Chilo all year.\nIn fact, the boat ramp is even more user friendly after the late-August installation of LED pole lights.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Split screen multitasking is one feature i have come to love on Android. Precisely Android phones running Android Nougat OS. Infinix mobility officially announced Android 7.0 Nougat a couple of months back. The infinix Note 3, Note 3 pro, infinix Zero 4 amidst others, officially got the nougat OS flavor.\nAlongside the memo on the new OS update, split screen was mentioned as being part of the package.\nNow, while split screen multitasking can be essentially useful, quite a lot of infinix Android phone users do not know how to utilize the split screen feature. A lot probably do not even know about the feature.\nFor the records, the split screen feature isn't exclusive to infinix or tecno android phones running Android Nougat OS. The feature is basically embedded in the Nougat OS, not by OEMs.\nHence, if you have an Android phone running Android 7.0 Nougat, you should try out the split screen feature on your device. It sure brings about a better experience.\nLatest Infinix android phones have Android 7.0 nougat running under the hood. Likes of the infinix smart and the infinix note 4 both run Android Nougat and the split screen multitasking feature is in place (tested and confirmed on the Note 4).\nBasically, the screen is divided into two, thus giving you the ability to use two apps simultaneously. You can be watching a movie and at the same time, be checking out a new email. Think about being able to easily copy texts from one page to another.\nSeem really interesting, yeah? Let's get on to the main subject matter.\nFirst thing first, the device needs to be running Android nougat. It won't run on lower android operating systems. Although there are ways you can incorporate same on lower operating systems, they most often at times, require root.\nIn the case of Android 7 nougat being able to perform split screen multitasking, root is not required.\n#Step 1: Open the first app you'd like to use on split screen.\n#Step 2: Tap on the menu \/ options button, tap and hold on the app and move it to the option that displays 'Drag here to use split screen'.\nAt this point, you'd notice the app is now shortened \u2013 basically to the middle.\n#Step 3: What you want to do at this point, is include a second app in the split screen. To do this, simply launch any other app and it will automatically appear beneath the first app.\nNow, you should have two apps on the screen running simultaneously.\nBeing able to split screen isn't limited to portrait mode. Turning the display to landscape still keeps the split screen in its expected form. As a matter of fact, landscape seems better for such functions.\nNot all apps has support for split screen. Example of such app is the Default music app and Instagram app.\nOnly two apps can be used simultaneously on the split screen mode. Of course, you can close one and open another.\nSimply pressing the home button would not work. Likewise pressing the back button. Both wouldn't work.\nTo deactivate, simply hold the divider line and drag either downwards or upwards.\nAlternatively, choosing to open an app that doesn't support split screen multitasking automatically deactivates the mode.\nSo, that's it guys. That's basically how to activate and deactivate split screen mode on Android phones running Android 7.0 Nougat. This should work on the infinix note 3, Hot S, Hot S2, Hot 4, zero 4, smart and the Note 4.\nI use the Split Screen on my updated Infinix Hot 4 too\u2026. It's cool.\nThat's cool! Thanks for the heads-up, Jayboss.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the former, Michael Rogin asks \"What is the bearing of our radicalized national culture on the color-blind innovation of individual rights?\" Discussing the American Declaration of Independence in light of the affirmative action debates of the 1990s, Rogin traces the declaration's legacy through race relations in both the old and the new Hollywoods.\nLess well known than Rogin's other writings on race and film, this short essay appeared in Representations' special issue \"Race and Representation: Affirmative Action,\" edited by Robert Post and Michael Rogin in 1996. The issue quickly went out of print, but is now back in circulation in pdf format.\nMICHAEL ROGIN was the author of many books on race, culture, politics, and history, including Blackface, White Noise: Jewish Immigrants in the Hollywood Melting Pot and Independence Day, or, How I Learned to Stop Worrying and Love the Enola Gay. He taught for many years at the University of California, Berkeley, and was a founding member of the Representations editorial board.\nHuey Copeland's 2011 essay \"Glenn Ligon and Other Runaway Subjects\" looks at contemporary artist Glenn Ligon's multiple engagements with the history of American slavery, particularly as evinced by his 1993 installation To Disembark. As Copeland shows, in casting himself as a runaway slave, Ligon points up the relationships between regimes of power, violence, and resistance that continue to produce black subjects as fugitives in life and in representation.\nHUEY COPELAND is Associate Professor of Art History at Northwestern University, where he teaches modern and contemporary art. He is the author of Bound to Appear: Art, Slavery, and the Site of Blackness in Multicultural America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasqbf b/data_all_eng_slimpj/shuffled/split2/finalzzzasqbf
new file mode 100644
index 0000000000000000000000000000000000000000..eae1a0a23a3474af23349918b3de6ad71ecc87b5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasqbf
@@ -0,0 +1,5 @@
+{"text":"Thought I would give this a go, would hopefully be a great learning exercise if nothing else!\nI didn't find that idea fun enough, so I'm currently redesigning the character by throwing a dash of deadpool inspired insanity. His main method of attacking is almost joust style, with a big scythe strapped to his arm.\nand some WIP shots of the bike.\nShould look a little less weedy once I add a nice big engine!\nI think the left side needs some work, as it's looking rather plain. but I'm liking the right hand side.\nI've been very busy recently. But thought if I want to hand something in I better get a move on with this, the deadline has crept up fast.\nMy high poly bike and rider, almost completed, I'm not great at sculpting but if anything this contest has been a great learning experience showing me where I need practice.\nWorked on low poly, pretty pleased with it so far.\nLooking really nice, but the Handlebars feel just far to thin.......might just be the camera but might be worth a check ?\nHi thanks for spotting that! I think you are right, I'll make sure to fix it!\nJust been doing some quixel, and rendering in marmoset.\nneeds to be neatened up a tad, but i'm quite happy with how it's turning out.\nI know I could do with improving, especially in my rigging, posing and general character modelling skills, but I know when to call it a day! It feels good to have finished!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I think they should Fuze that meth lab. There's no hostage mentioned.\nI don't know what to make of your profile picture.\nIs it a mollusc\/crab monster?\nIt is a Magic the Gathering card, Better Than One. It's a giant tortoise with two parakeet heads.\nWell hello. What exactly do you dislike about realistic teenage mutant ninja turtles?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today we want to introduce to you Daniela Yordanova \u2013 our senior customer support specialist and an integral part of the team.\nDaniela is always radiating happiness and energy, empowering everyone around her to do so as well. \"Life is too important to be taken seriously.\" is a favorite quote of hers, and we think that it demonstrates her mindset perfectly. She`s been working since she was 14, trying out different companies and roles, but no matter the position, she is always focused on good communication and personal approach \u2013 precisely what she does at iuvo.\nI started working at iuvo without having a clear idea about the concept, goals or perspectives. Consequently I fell in love with the idea so much that I got everyone around me hooked.\nSports are an inseparable part of my daily routine. When I`m not doing sports I like to go out with friends. Another hobby I have is photography. I like taking pictures of children and animals most of all \u2013 they`re always natural and joyful and that gives me energy!\nI wouldn`t call it a \"special\" talent, but I have the ability to communicate well with all sorts of different people. This helps me do many things with ease and exceed people`s expectations.\nI invest with iuvo, of course.\nI think that every person should keep in touch with the newest tendencies, do his research and get involved in as many investment channels as possible. Aside from financial investments, people should also invest in knowledge, courses and seminars. Information is the biggest weapon in the 21st century.\n6.\tWhat do you do when you get mad at people?\nI rarely get angry at someone, but when I do, I put some calm music and stop talking until the storm passes.\n7.\tWhat is the best advice you have for newbie iuvo users?\nI would advise them to not hesitate and bravely try out different strategies. The buy-back guarantee and intuitive interface turn iuvo into the perfect investment platform for people with funds to spare but no previous experience.\n8.\tLet's wrap this up with an existential one. Which came first, the chicken or the egg? Explain why.\nFor sure it was the egg! No doubt about it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's similar to buying bitcoin in US. The market for bitcoin and other cryptocurrencies in Europe, specifically in UK which is still in Europe before brexit is not one of the most liquid ones. You can see the statistics on CryptoCompare. Well UK actively trades in Cryptocurrencies unlike any other countries in Europe. People in Germany are far more reluctant to buy any cryptocurrencies due to the time lag in adoption during the bull run of December 2017.\nAmong the top cryptocurrency exchanges, Coinfloor and CEX require user identification and tailored between business and traders while CoinCorner appears to be more geared towards the retail investor.\nIf you are more interested in marketplace you can go for BitBargain and Bittylicious, these platforms put you in touch with the local seller but only online or mobile payments are accepted. You can wait for CryptX Financial marketplace to launch in Europe to get access to edge of the art features for the marketplace.\nYou can always get cryptocurrencies from Bitcoin ATM near you which you can find in LocalBitcoins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A lovely 4oz Stainless Steel hip flask engraved with the words 'Father of the Bride'. The hip flask also comes with a screw off lid (which remains attached) and small funnel. The flask is presented in a black velvet lined presentation box.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasqrr b/data_all_eng_slimpj/shuffled/split2/finalzzzasqrr
new file mode 100644
index 0000000000000000000000000000000000000000..f66368b08ca2bf213e6d6f8580818065b35ba8ca
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasqrr
@@ -0,0 +1,5 @@
+{"text":"The purpose of this conceptual study was to analyze what factors impact students' engagement and motivation to read independently. Further, the study sought to determine effective modifications to independent reading for today's students. Because this study is conceptual in nature, an extensive review of the current qualitative and empirical literature on independent reading was conducted. From there, the data was coded inductively to generate new ideas about what independent reading should entail for students today. Some clear patterns emerged. First, student choice and autonomy were cited as important factors for fostering students' motivation and engagement with reading. Second, simply providing time for independent reading is immensely important in getting students to see themselves as readers, and therefore become more engaged and motivated to read. Finally, talk around text was found to be paramount to motivating students to engage in independent reading. After reviewing the available literature, the implications determined that these factors must be integrated into independent reading programs to make them successful for today's students.\nMitchell, Diana, \"A conceptual study on effective independent reading practices to foster an enjoyment of reading\" (2018). Theses and Dissertations. 2497.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Premi\u00e0 de Mar is a municipality with seafaring tradition and an important commercial hub in the Maresme. You can go to the beach, do sport and visit the Stamping Museum and the roman ruins Can Farreons. Park with Parkimeter and enjoy a unique experience!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tokyo was everything that Iowa wasn't. It was why he'd come.\nTokyo was civilized, urban, occupied and vibrant. Tokyo was vertical where Iowa was horizontal. Tokyo was fast where Iowa was slow.\nHe loved the ever-present crush of people, the novelty of towering over a crowd at the subway stop, the ease of talking to pretty girls eager to practice their English on the drawling American\u2014who was maybe not so handsome by Mississippi Valley High School standards.\nHe'd come eleven months and twenty-eight days ago\u2014hopped on the plane the day after graduation. Landed a job right away and was making a decent living teaching English at the junior college a short subway's ride away.\nBut back in Iowa it was June\u2014the month of a million greens, when the vast bowl of the Midwest becomes the Emerald City of growing things. And suddenly, intensely, completely: he missed it so.\nPhoning it in a little this week. My teenage son is in the hospital with mono, strep, influenza and pneumonia, plus a few accompanying complications. We're on day four. And all I can think is: there is no place like home. And why do hospitals wake sick people up every 45 minutes?\nPS J Hardy Carroll please excuse the use of the word 'suddenly'. \ud83d\ude09 Got my adverbs on, eh?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"I'm delighted to open this exhibition at The Young Gallery and to celebrate the work of these budding young artists. The work is so beautifully displayed \u2013 it's certainly better than fridge magnets! I'm actually a little jealous as I used to stay in Dennistoun myself, and would have loved to bring my son Angus into The Young Gallery \u2013 it's a sheer joy to visit. It's nice to meet parents and young people of all ages, it shows that the gallery is really fulfilling the Scottish Government's aim to get children into creative activity. Art, music and culture are brilliant ways for parents to bond with children. Well done to all involved.\"\nWe'll be uploading some beautiful photos next week but for now here is a wee teaser of what you can see at The Young Gallery till the 29th of November.\nIn other news the staff team raised \u00a3196 from our fundraising last week! Huge well done to everyone who suffered in silence in onsies, and had to suffer in silence as their mouths were full of cake! All for a good cause. Adding that in, we've now reached \u00a32869! Please help us reach \u00a33000 by the end of the weekend, just \u00a3131 to go!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy as many cheap players as you can with more than 10 contracts and more than 90 fitness in order to resell them then more expensive than the purchase price. Buy the player at the lowest market price (or up to 300 coins). If you can not find this price, go to the next player! Do not focus on a player, there are dozens and dozens of others.\nAvoid purchasing the same player several times at the same time, thus avoiding losing too many credits in the event of a fall in the player's price. The goal is to have a list of up to 100 different players.\nPut your players on sale by listing them 1 hour. The goal is that your players appear most often in the first pages of the results. This is why they must be listed for 1 hour and relist as often as possible (via Companion application for example).\nIndeed, a buyer in a hurry who wants a player will only look at the first pages of the auction. Even if your card is a bit more expensive than the competition, it will already have some contracts, a good shape and even a style appropriate to the player. The buyer will therefore be more tempted by your card.\nDo not hesitate to consult the profit schedule. The more profits you make, the less you sell players every hour.\nThe Tech Crezip works on the same principle as the FUT Millionaire tech, except that in addition to having more than 10 contracts and more than 90 form your player will have a style suited to his post.\nI do not mention here the Hunter, Shadow and Catalyst styles that are more expensive than the others, and therefore rarer. However if you find a player with this style, do not hesitate to resell it a little more expensive than expected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The ivory-billed woodpecker is one of the most celebrated birds in U.S. history, a bird that has been an American superstar since pre-colonial times.\nEven though it has been presumed extinct for 60 years, thousands of birdwatchers never gave up searching for what they call the \"Lord God bird.\"\nEarlier this year when it was disclosed that an ivory-billed woodpecker had been spotted in Arkansas, the reaction was extraordinary.\nThe ivory-billed woodpecker's resurrection is an amazing story \u2014 not just about the bird but the people who are obsessed with it. It's also a story involving extraordinary luck, secrecy, millions of dollars and months of searching a forest in eastern Arkansas looking for a flying needle in a haystack.\nAll that's left of the \"Big Woods\" \u2014 a forest that once covered 24 million acres and stretched from Memphis, Tenn., to Little Rock \u2014 is some 500,000 acres in Arkansas. The spectacular swamps and hardwood forests of the South were once home to the ivory-billed woodpecker.\nStories about the woodpecker go back to the days when only native Americans were in the region. Some saw its skull as a symbol of power, the feathers as objects of beauty. In Victorian times, women prized the feathers for their hats.\nBy the beginning of the last century, the bird had been nearly hunted to extinction. It was said to be so beautiful that when people saw it, they said, \"Lord God! What a bird!\" The name stuck.\nThe \"Lord God bird\" was the largest woodpecker in America. But for more than half a century, the only way you could see one was in stuffed form.\n\"It's a huge, black woodpecker, basically, with lightning bolts of white down the back and a huge patch of white on the wings,\" explains John Fitzpatrick, head of the Lab of Ornithology at Cornell University and a leading authority on birds.\nFitzpatrick says the bird has a distinctive sound that has been likened to a tin horn.\nThe first recording of that sound was made in 1935 in Louisiana by a team of scientists from Cornell. They also took the first pictures of the bird.\nAt the time, the \"Lord God bird\" was already on the verge of extinction. In the following decade, logging reduced its last known habitat to thousands of stumps.\n\"This has been the symbol of what we did wrong. The complete annihilation of one of America's most treasured and spectacular pieces of land. We cut it all,\" says Fitzpatrick.\nBy the 1940s the ivory-billed woodpecker was presumed extinct.\nGene Sparling, an outdoorsman from Hot Springs, Ark., also thought the bird had been extinct during his entire lifetime.\nIn February 2004, Sparling was kayaking in the Big Woods when he saw something he will never forget. \"A large woodpecker flew into the channel from above the canopy. He was headed straight towards me.\"\nSparling considered that it might be the smaller woodpecker that's common around the region called \"the pilleated,\" but ruled it out. \"I realized that if it was not a pilleated, the only other alternative was for it to be an ivory bill,\" he says.\nSparling made a veiled reference to his sighting on the Internet, and got a response right away.\n\"We came down, you know, just a few days later. As quick as we could get out of town,\" says Tim Gallagher, a wildlife photographer and author from Cornell.\nHe was joined by Bobby Harrison, a college professor from Alabama, and both are \"ivory-billed hunters\" \u2014 bird watchers who have been looking for the bird for years believing it's still alive.\nWhen they heard about his sighting, they immediately contacted Sparling, who took them near the spot where he saw the bird.\n\"And then this bird just burst across in front of us at close range,\" Gallagher says. \"About 65 feet away. And right in the sunlight. \u2026 And it was just, I mean, I dropped my paddle and almost fell out of the canoe. I mean it was like getting slapped in the face.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This particular recipe is vegan-friendly, though you can find plenty of other recipes that include butter or lard in the dough. Those will make for a flakier crust.\n1) Add flour and salt to stand mixer. Begin to slowly beat in the four tablespoons of oil.\n2). Slowly add the water as needed, until dough begins to form a firm ball. Let it mix for 2-4 minutes.\n3) Form the dough into a large ball, set aside, cover, and let rest while you make the filling.\n1) Boil potato until soft. Peel, chop, and set aside.\n2) Finely dice the onions and carrots. Saute them over low heat in olive oil until the onions begin to carmelize.\n3) Add the ginger, garlic, and spices. Cook for one minute.\n4) Chop up the potato and add it to the mix. Use a potato masher or fork to mash the mixture together.\n5) As needed, add vegetable stock and let the mixture simmer until the flavors are well-blended, usually 5-10 minutes.\n1) Divide dough into half, then in half again to make four equal balls.\n2) Roll one ball into a circle. Cut in half.\n3) Place the half on the palm of one hand, and use the other to scoop filling into the center.\n4) Moisten the edges of the dough, and pinch together to form a dough pocket.\n5) Repeat for the rest of the dough.\nTo cook the samosas, you may either fry or bake them. To fry, place in hot oil for about three minutes until golden brown. To bake, brush the tops with oil. Bake for 15 minutes at 425, then lower to 375 and bake an additional 10.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ability to possess and take over the minds and bodies of other creatures is also a power which the Jinn have utilised greatly over the centuries. This however, is something which has been prohibited to them as it is a great oppression to possess another being. Human possession is something which has always brought about great attention. But the true knowledge of this subject is rare amongst the people. Over the last 3 decades the subject of possession has become very commercialised. During the 70's films such as The Exorcist and Rosemary's Baby were used to educate people about possession. However, because such institutions (the film industry) were heavily influenced by Christianity, knowledge of the subject was non-existent. Rather then educate people about Jinn possession, films such as The Exorcist just tended to scare the living daylights out of us! Only through Islam can we understand such a phenomena. We know as Muslims, that Jinns possess people for many reasons. Sometimes it is because the Jinn or its family has been hurt accidentally. It could be because the Jinn has fallen in love with the person. However, most of the time possession occurs because the Jinn is simply malicious and wicked. For this reason we have been told by the Prophet (saws) not to loiter in those places where the Jinns reside, e.g. graveyards, ruins, deserts, market places etc. We have also been commanded to recite the Qur'an frequently in our houses as the Prophet (saws) said: 'Indeed, the shaytan flees from the house in which Surah AI-Baqarah (the 2nd chapter of the Qur'an) is recited\".\nThrough their powers of flying and invisibility, the Jinn are the chief component in occult activities. Voodoo, Black magic, Poltergeists, Witchcraft and Mediums can all be explained through the world of the Jinn. Likewise, so can the illusions and feats of magicians. Because the Jinn can traverse huge distances over a matter of seconds, their value to magicians is great. In return for helping them in their magic, the Jinns often ask for the magicians to sell their souls to them and even to Iblis. Thus the magicians take the Jinn and Iblis as lords besides Allah. In our day, some of the feats performed by magicians and entertainers are without doubt from the assistance of the Jinn. Making the Statue of Liberty disappear, flying across the Grand Canyon and retrieving a ship from the Bermuda Triangle, have all been done by the Jewish magician David Copperfield. There is NO way that a man could do such things without the assistance of the Jinn. It would not be surprising therefore, if David Copperfield had sold his soul to Iblis himself. Because of their involvement with the Jinn, and its result in shirk, the Prophet (saws) said: \"The prescribed punishment for the magician is that he be executed by the sword\". Some may argue that this is barbaric, but if, the likes of David Copperfield truly had powers, then they could just put theirheads back on again!!\nThe Prophet (saws) also said: \"They (the Jinn) would pass the information back down until it reaches the lips of a magician or forrtune-teller Sometimes a meteor would overtake them before they could pass it on. If they passed it on before being struck, they would add to it a hundred lies\". Thus, it is clear from this as to how fortune-tellers get predictions of the future right. It is also evident as to why they get so many wrong. Men like Nostradamus are an example, as some of his predictions of the future were correct whilst many were completely wrong. Unfortunately, the amount of fortune telling which occurs amongst the Muslims is also increasing. By visiting Muslim lands such as Morocco, one is able to see as to how much inter Jinn-fortune-teller activity there really is. If you look up at the sky on a clear night in Morocco, you will see the heavens ablaze with shooting stars! A clear display of the devils being chased away from the heavens.\nFortune-tellers also operate through the Qareen. The Qareen is the Jinn companion which is assigned to every human being. It is this Jinn which whispers to our base desires and constantly tries to divert us from righteousness. The Prophet (saws) said: \"Everyone of you has been assigned a companion from the Jinn. The companions asked: Even you O' Messenger of Allah? And the Prophet replied: Even me, except that Allah has helped me against him and he has submitted. Now he only tells me to do good\"'. Because the Qareen is with a person all his life, it knows all that has happened to the person from the cradle to the grave. By making contact with the Qareen, the fortune-teller is thus able to make out that it is he who knows about the person. He looks in his crystal ball or the palm of a person and proceeds to amaze him with knowledge which no one else knows. The severity of going to a fortune-teller is such that the Prophet (saws) said: \"The prayer of one who approaches a fortune-teller and asks him about anything, will not be accepted for forty days or nights\" and: \"Whosoever approaches a fortune-teller and believes in what he says, has disbelieved in what was revealed to Muhammed\"\n16 An informative book on the world of the Jinn is 'Ibn Taymeeyah's Essay on the Jinn' translated by Abu Ameenah Bilal Phillips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"WHAT COMPARES to your first deep breath in a genuine Finnish sauna, where the aroma of spruce mingles with a hint of wood smoke?\nNothing. Nothing, perhaps, except pouring hot lead up your nose. I breathe through cupped hands to keep my nostrils from burning. Next to me in this searing heat, Juhani Per\u00e4salo sits as naked and white as an egg. \"They say this is the place for only real Finns.\" He grins or grimaces. In the shadows, I can't tell. \"We Finns, we have too much sometimes of sisu. Sisu is like macho. 'We stay here! We don't go out!'\" he shouts. \"That's nonsense,\" he says. \"That's nonsense.\"\nPer\u00e4salo and I are testing the saunas of the Finnish Sauna Society. Per\u00e4salo, as long-time society member, past president and head of the society's research activities, is perhaps Finland's foremost ambassador of the sauna. \"I feel it is my patriotic duty,\" he says. Yet Per\u00e4salo's no fool. He knows saunas won't survive an additional 2,000 years or win foreign converts if they kill all who enter.\n\"Too much is always too much,\" Per\u00e4salo says. He throws open the door and flees into bright daylight. I'm right behind.\nCooling our heels, quite literally, we enjoy the long view of the Gulf of Finland, studded by islands and framed by Lauttasaari's birches and pines. One of many island pillars supporting the Finnish capital, Lauttasaari has none of downtown Helsinki's historic, cosmopolitan charm. It consists of homes, apartments, businesses and tiny summer cottages. It does have a pleasant bike path that circles much of the island, past the Helsinki harbor, beaches, public parks and several fortifications used during the Crimean War 140 years ago.\nBut the island's most notable feature, built on a rocky point where a gun battery once guarded the entrance to what is now a marina, is the Finnish Sauna Society. Founded in 1937 to protect and promote Finland's sauna heritage, the society counts about 2,000 members. The organization maintains four traditional saunas at its Lauttasaari hideaway. It also encourages historical and medical research and sponsors sauna conferences. Finally, the group is a kind of ecclesiastical authority, overseeing an institution much older than Finland's Lutheran church, and to many Finns, more sacred.\n\"Sauna belongs to our everyday life,\" Per\u00e4salo says. \"We have used our sauna for centuries to clean ourselves, to maintain our health and to help treat a variety of ills.\" Finland, with only 5 million people, has no fewer than 1.6 million saunas in homes, summer cabins, apartments and hotels. \"I am sure I was in a sauna before I was one year old,\" he says. \"I feel that I have always been in a sauna.\"\nPer\u00e4salo and I duck back inside. In no mood for a scalding, we open a different door this time and usher into a much milder sauna. This one is about 180 degrees, Per\u00e4salo estimates; the other probably about 230 degrees. A genuine sauna, he says, ranges between these extremes and is much hotter and drier than a Turkish bath or steam bath.\nNot only a sauna promoter, Per\u00e4salo is also a doctor. As we roast, he explains, our bodies act as radiators, dispersing heat to the surface, and shedding it through perspiration and evaporation. \"First - and this is very important - is that you're sweating,\" he says. \"Circulation through the skin increases as the blood capillaries and the blood vessels of the outer skin layers dilate. Your blood pressure goes down. After that, your heart beat increases to keep your blood pressure normal.\"\nMy own heart is racing. Don't worry, Per\u00e4salo says. \"Research has shown that even after a myocardial infarction one can go into the sauna without complications.\"\nThe problem comes in plunging into cold water or snow - the proper conclusion of a good sauna, according to many Finns. As cold causes capillaries to squeeze shut, Per\u00e4salo says, \"your blood pressure goes as high as it is likely to get.\"\n\"So you don't jump into the Baltic?\" I ask.\nWe pass through the showers again and enter the next sauna. The air here is heavier, the heat more penetrating. It is hotter than the previous sauna, about 212 degrees, but the main difference is the l\u00f6yly, Per\u00e4salo says. He reaches for a bucket of water and ladles water onto the red-hot rocks that cover the wood stove. A burst of heat washes over us. That is the l\u00f6yly, Per\u00e4salo explains. The word is impossible for a non-Finn to pronounce and nearly as difficult to translate. L\u00f6yly is really part boiling-hot water vapor, part metaphysics.\nand the peace secure to us.\nSaunas were crucibles of magic and health. Surrounded by fundamental air, fire, stone and water, healers cured diseases and exorcised spirits. \"Cuppers\" lacerated patients' rosy skin to draw away \"bad blood.\" Bathers whipped themselves with whisks of birch leaves to cure pain and illness and improve their fortune. Modern research makes fewer claims for saunas, though the heat does loosen tight joints and muscles. A Japanese researcher recently reported beneficial increases in blood flow to the heart muscle of patients with cardiovascular disease.\nBasking in the warmth and restorative effects of a gentle sauna, Finnish women of past generations gave birth. In fact, says Per\u00e4salo, it is still possible to find Finns who were born in saunas.\nSaunas provided an important threshold for young men, he says. Children would sauna with their parents. Outside of family, men and women usually took saunas separately. \"When a young boy could go with his daddy or grandpa to the sauna, everyone knew that he was going to the sauna with men. It was a huge step for a boy.\n\"In medieval times they had saunas everywhere in Europe,\" Per\u00e4salo says. But without strict Finnish rituals and segregation of the sexes, most saunas degenerated into bordellos. \"They spread sexually transmitted diseases, especially syphilis. Officials felt they had to close them.\" Once banned, the sauna disappeared from many cultures, surviving only in Finland, northwestern Russia and Estonia.\nThe epitome of the Finnish tradition is the \"smoke\" sauna. The sauna of centuries past, the smoke sauna is heated by an unvented stove - in its simplest form, a hearth of rocks fired to glowing red. For hours, as split birch burns, smoke billows from the open window and door. After the fire dies and smoke clears, the sauna is closed up, and the hot rocks raise the temperature inside the room. \"Every good smoke sauna burns down every five years,\" Per\u00e4salo quips. In fact, it has been nine years since the sauna we enter now has burned down.\nMany a fire has blackened these log walls and imparted the faint smell of wood smoke. The temperature is near boiling, but I find the deep gloom relaxing. So does Per\u00e4salo. He dashes the rocks with a ladle of water. Vapor covers us like a warm blanket. \"The l\u00f6yly is soft,\" Per\u00e4salo says. Why is that? Perhaps because of the long heating time, the darkness of the walls, or even the ionization of the air, he muses. \"But it's so complicated I don't get into it. A sauna can't be that complicated.\" We peer out the long, narrow window, imagining the coolness of the birch and the sea.\nWhen we are satiated with heat, Per\u00e4salo points me toward the scrub room, where a kylvett\u00e4j\u00e4 waits by a padded table. She is middle-aged, built as sturdy as a stump. I flop onto the table. With a soaped luffa, she scrubs with terrifying efficiency - back twice, then front twice. Satisfied my freckles are indelible, she washes my hair and sends me back to the saunas.\nRelaxed to exhaustion, I consider Per\u00e4salo's advice: \"Too much is always too much.\" Time to quit, perhaps. But then I think of William Blake: \"You never know what is enough unless you know what is more than enough.\"\nI head to the hottest sauna once again. I imagine I know how a baked potato feels. Fully cooked, I charge out the door and race down the path, onto the dock, down the ladder and into the cold Baltic. Icy daggers pierce my legs, my belly, my chest. I float neck deep for a bitter second. That is more than enough. I scramble back up the ladder. Suddenly I am filled with an intoxicating sense of warmth, even as I stand naked in the crisp Baltic air.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the best way to select a pump, as the dealers know their pump lines much better than you. They may suggest a pump you don't even know about. But for those who want to be educated consumers, here is how to do it yourself! I do not generally recommend that you buy an \"off the shelf\" pump at a hardware store for use on an irrigation system. Most of those pumps do not provide sufficient performance information for you to be able to tell if the pump is sufficient for your system. Off the shelf pumps are fine for fountains, draining a pool, running a sprinkler on the end of a hose, or other uses that don't require precision. The wasted power used by the wrong size pump on an irrigation system can easily cost you far more than the savings you get from buying a discount pump. If you guess wrong and the pump isn't big enough, you will become one of those poor people trying to unload a brand new pump at a huge discount on Ebay! Pump Curves A pump curve is a simple graph which shows the performance characteristics of a particular pump.\nHow do I vaccum the pool? I dont know if I turn the Veriflow to out and then where do I hook in the hose?.. Its my girlfriends pool and she just expects me to know. Thanks Oh then you DO need help. You attach the hose to the skimmer inlet.\nJan 28, \u00b7 I'm starting up a mobile car detailing business in New Jersey and I tried hooking my power washer to of baby pool I have in my back yard but the power washer wasn't getting any water but if I get a water tank will it get water or is there a special power washer I need or sometihng please help me out thanks a alot guys.\nI purchased the intex pump. The attachments are too big for the inlet and outlet so i then purchased the adapters. The plungers are very noisy and allow air to get into the pump. I removed them and hooked the hoses up to the adapters directly. That worked as long as the adapters were new. This year the seal comes loose and allows air to get into the pump. I even tried allowing the top nut to be loose to allow air to escape.\nThis year nothing works. The pump is just too strong for the holes. I was considering cutting the holes bigger and glueing seals on there but i was concerned about creating weak spots in the pool. I am tired of spending money on it so i did the redneck thing. I can say that because well, you might consider me one. I took the inlet hose off the connector and plugged the hole with one of those little black things.\nI took the hose and run it up through the hole in between the liner and the poles and down into the pool.\nLast week we covered the type of base needed for a hot tub, this week I want to go over what you will need to get your new spa all powered up. We work with a handful of great electricians in the Amesbury and greater North Shore Massachusetts area who do great work and we can match you up with a capable company when you buy your hot tub. Standard Spa Electrical Requirements: Most spas in the industry require the following to run properly. They generally come with their own pre-wired gfci plug.\nThese types of tubs are only 1 pump usually only hp and can take longer to heat up initially, but still a great option for those on a budget or limited electric availability. In Massachusetts and NH we have a lot of older homes with older electrical setups.\nIf the homeowner vacuums like normal, they have a good chance of contaminating their filter, plugging up the circulation lines, or just clouding up their pool. What is the solution? Those that have filters with the 6 way Multi-port valve can vacuum their pool on Waste. This position by-passes the filter and sends all the unwanted debris through the waste line. This eliminates clogging up the filter or clouding up their pool. Steps to Vacuuming on Waste: Connect the vacuum head to the T-pole and then connect the swivel end of the vacuum hose if equipped to the vacuum head.\nFully extend the T-pole and lower the vacuum head to the bottom of the deep end of the pool. Use care not to disturb any debris on the pool floor. Connect the skim vac to the other end of the hose and place over a return fitting to purge the air from the vac hose. At this time the vac head and all of the vac hose is in the pool. Keeping the the skim vac close to the surface of the of the water, make you way to the skimmer being sure not to let air into the vac hose.\nPlace skim vac into skimmer through the top or the throat of the skimmer on-top of the skimmer basket. Go to the Pool Pump area and turn the pool pump off.\nMonday, August 15, How to Drain an Inground Swimming Pool Though draining your inground swimming pool isn't something you do every day, once in a while it is required. Some of the more common reasons include rebalancing your chemicals, lowering your total dissolved solids, or repairing or resurfacing the pool. Additionally, many people empty their pools to prepare it for winter, or replace their water in the spring to ensure that it's clean and well-balanced.\nIt sounds like you are getting to much air into the system and causing an air lock. Try submerging the vacuum and the hose in the pool to get all the air out of it before hooking it up to the skimmer.\nWhat impeller and diffuser would I need to use? Will the Hayward brand for a 1. Will this motor work? What will I need to replace? You will also need to replace the pump shaft seal. I will need to know the model of the pump housing. You can use this Century UST motor as a replacement. Hello- The vents on motors will always face down. Tom, Based on the model number, the replacement is the UST Yes, this is the replacement motor for your model motor. Forcing the breaker in place, the pump motor buzzes but doesn't come on.\nEdit Answer for another minute I am going to asssume your pump also provides the circulation for the pool cleaning equipment. I also have to make some assumptions about your pool's plumbing, which might or might not be right. Also, when testing do not run the pump for more than a few seconds at a time, because you do not know how much water it is getting \u2013 the psi might be just air pressure against the filter tank static head. I would open a drain valve somewhere downstream of the pump and make sure it is pumping at least a slight flow of water before running it more than about 5 seconds at a time for testing.\nIt would be helpful if while you do the checking, you had a family member turning the pump on and off so it runs as little as possible.\nTo hook up a vacuum pump: 1 Open up the vaccum port and turn off your skimmer port. 2 Reduce the flow in the main drain. 3 If you have more than one skimmer plug off one of them.\nFree from hoses, whips and cords, in-floor pool cleaners operate by creating currents that drive debris towards the skimmer and main drain. Great Idea \u2013 and usually an in-floor pool cleaner works flawlessly. They are especially popular for desert pools or pools without a lot of tree leaf litter, as they can become overwhelmed with heavy debris. In addition to keeping the pool clean through smart circulation, in-floor cleaners also help mix pool chemicals faster and equalize water temperature better, than traditional pool cleaners.\nI hope you find what you are looking for, if not send me an email! If dirt has begun to collect in one area of the pool that normally stays clean, the first thing to suspect is the overall water flow and operation of the system. The last thing to check is the pop-up heads themselves. Check and clean pool filter Check and clean pump and skimmer baskets, or main drain Leaf Traps.\nCheck pop-up heads and inspect for clogs or damage. Check the water flow valve or distribution head for assembly and operation. If all of the other banks of pop-up heads are working properly, or 2 out of 3 heads are coming up, but 1 or 2 heads are not coming up, you also want to check the flow first before you start digging into the distribution valve or pop-up heads. If you are wondering if debris or coins or something can get stuck around the pop-up head, while it's in the up position, you are correct.\nPop-Up heads usually come up in sets of floor jets, and some types rotate as they pop-up. After the flow ceases to the bank of floor jets, the pop-up head should settle down neatly back into the hole.\nTop Step 1 Before starting installation, you will need to purchase sand. Use only high rate No. An example is pictured at left. Check the manufacturer to determine the amount required. This unit Hayward S T uses 3 50 lb bags of sand \u2013 lbs total. You can also purchase our superior to sand Click here for our Sand Alternative. Step 2 When installing the pool sand filter, you will connect it right after the pump. Pool water will come in through the pool pump, be filtered through the sand filter and released to the chlorinator before returning back to the pool.\nIf you have a pool heater, the pool filter will release water to the heater before going to the chlorinator. Step 3 The pool filter should be installed on a level concrete slab not more than 6 feet above the pool water level. Position the pool sand filter so that the piping connections, multiport valve and drain are convenient and accessible for operation and maintenance. Place the filter next to pump.\nThere are three possible plumbing methods for pool heating: For the same heater and the same heat output, the plumbing method can have significant effects on your energy consumption, your power bills, and the ease of operation. In Series This works for gas and heat pump heaters for solar heating an additional booster pump is required. Valves either manual or automatically controlled are installed to push water through the heater or to bypass it when not in use.\nWhen connected in series or with booster pumps chlorinators must always be installed after the heater. Booster pump Solar pool heaters always require a booster.\nA lot of new swimming pool owners want to get their pump and filter up and running quickly. So, they wonder if they can run their pump on an extension cord until an electrician comes to hook up their pump.\nFor that reason, you need to clean your above-ground pool with a pool vacuum at least once a week. Pool vacuum setup takes only minutes, but you need to know the proper way to hook up the vacuum; otherwise, you might not get proper suction for the vacuum to do its job. Push the swivel end of the vacuum hose onto the vacuum head. You should hear this click into place. Some pools allow you to flip a switch to close the skimmer instead of using a plug.\nSkip this step if your pool only has one skimmer. Push approximately 6 inches of hose at a time under water, starting with the end attached to the vacuum head. Push the skimmer adapter disk onto the vacuum hose. Open the skimmer cover. Set the disk and hose on top of the skimmer. Listen carefully to hear the skimmer suction out the remaining air in the vacuum hose.\nOnce you hear the change, you are ready to begin vacuuming the swimming pool. Tip Reverse the process of disassembling the vacuum when you finish cleaning the pool. Empty the pump basket and skimmers of collected debris.\nInground Pool Closing Winterizing Locate all your winterizing supplies. This should include the cover, the water tubes, the plugs for the skimmers gizzmos and return jets and your winterizing chemicals. You will also need an air compressor or a powerful shop vac. You need these items for proper winterization. If you are using the green Gizzmos to plug your skimmers, check them out and make sure that they are not cracked. This is very important when dealing with gizzmos.\nMy pool cover pump broke and there is stagnant water on the cover. I want to use Ridgid Shopvac to drain the water by hooking up a hose to the drain plug and submerging the vac hose in the water.\nNothing else on this circuit. Separate circuit runs black, white and ground from the main load panel directly to sub panel and powers a GFCI outlet on the side of the sub panel used for lighting. Still want to add a surge protector Intermatic PS at the sub panel in addition to the whole house surge protector because pump is variable speed and has sensitive on board electronics.\nIt also prevents the pump from losing its prime if it is above pool level. The vacuum breaker allows air to enter when solar shuts off. This allows the system to drain back to the pool and prevents negative pressure in the collectors and plumbing.\nWe live on a river. Do you think that that is something we could do without spending a fortune? First, you must have the right to take water from the creek, river,. If you take water from a creek or pond or any other natural body of water in the USA without checking on the legal rights and requirements you can get into a lot of hot water, fast. The fines penalties and restitution costs can be enormous.\nSo before you do anything, start doing some calling around. Be safe, not sorry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauonl b/data_all_eng_slimpj/shuffled/split2/finalzzzauonl
new file mode 100644
index 0000000000000000000000000000000000000000..54ccd70084884987de8394e6d879aea14615dc9b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Very nice quality screen protector. Very sticky and easily attaches to screen. Slightly tricky to install vertically.\nGreat protection for the factory screen. The screen arrived well packaged with everything you need to do the job- very well laid out, with excellent instructions. This leads to an easy install.\nIt is satisfying to know that our carefully thought-out instructions helped you to have a successful install, David.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hershey's Kisses Milk Chocolate have the classic, creamy flavor that have made these little individually-wrapped chocolates one of the best-selling chocolates in the USA. While they are a delicious treat all by themselves, they are also a classic addition to peanut butter cookies or a unique way to decorate cupcakes!\nCe n'est pas pour moi, mais \u00e7a va plaire !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pro Kabaddi Season 2 Schedule 2015. Pro kabaddi game has won million of hearts. Pro Kabaddi game is the second biggest game in india after cricket. We have already known there are 8 teams in Pro Kabaddi Season 2 same as season 1. The viewership of this Pro Kabaddi game is 86.4 million. So many asking me to provide Star Sports Pro Kabaddi Schedule or Time Table Season 2 2015 .\nSome matches in pro kabaddi league have been completed. So, here we are providing the latest schedule of Pro Kabaddi League of 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're looking for hydraulic power units San Diego CA, head to Bernell Hydraulics, southern California's premier provider of hydraulic parts and service since 1977. We sell, service, and even fabricate hydraulic power units for all kinds of equipment and applications. We invite you to stop by one of our locations to learn more about how we can help you with your hydraulic power unit.\nFeel free to ask us for help whenever you have questions about your hydraulic power unit's performance. Our onsite repair service is available 24 hours a day to get you back to work faster. We also offer a special onsite hose repair service as well as maintenance services like fluid flushes and diagnostic tests for your hydraulic power units. You can trust our expert repair technicians to handle any kind of power unit related repair or maintenance with skill and professionalism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pelham Car Wash & Lube Inc is located at the address 2941 Pelham Pkwy in Pelham, Alabama 35124. They can be contacted via phone at (205) 663-0512 for pricing, hours and directions.\nFor maps and directions to Pelham Car Wash & Lube Inc view the map to the right. For reviews of Pelham Car Wash & Lube Inc see below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzauskw b/data_all_eng_slimpj/shuffled/split2/finalzzzauskw
new file mode 100644
index 0000000000000000000000000000000000000000..8a6f28b799d41834b23fd57ae4b449a3a0e06ec7
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzauskw
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+{"text":"There are thousands of vacant jobs that 99% of candidates are unaware of. They form part of the hidden job market where roles are filled before they are advertised.\nEven in today' tight employment market, employers are competing to attract talented individuals by adopting alternative recruitment methods.\nIf you rely solely on traditional job search methods, such as job advertisements, you could potentially miss your dream job opportunity.\nIn my role as a recruiting manager, I have seen a significant change in recruitment methods over the past few years. In my experience, most professional candidates are now hired through the hidden job market.\nRecruiters are taking a more active approach to seeking candidates, rather than relying on the passive approach such as job advertisements.\nTherefore, if organisations are investing money in finding candidates with your skills and experience, why not make it easier for them to find you?\nMany large organisations engage the services of industry specific employment agencies to source talented individuals. This is because these agencies tend to have strong industry networks and know the top players in their specific industry.\nYou can make it easier for employment agencies to find you by registering directly with the agencies that specialise in your industry. Be sure to submit a keyword rich resume to make sure your application is selected during candidate database searches. Read this for more information.\nNeed a custom designed resume?\nDownload the FREE 'Grow My Career' resume template here.\nFirstly you need to google your name.\nIt is important to see what information is currently on the internet about you. Remove anything undesirable and strengthen privacy settings on websites such as Facebook and Instagram. Your aim is to present a professional image wherever an employer can find information about you.\nSetting up a website that showcases your resume. It should include a professional photo plus information on the type of job and company you are seeking. Most importantly you need to sell your skills and experiences. There are many free website platforms you can use such as wordpress.com.\nDevelop or tweak profiles on LinkedIn , Twitter , Google Plus and Facebook .\nPlace a professional cover photo on all your profiles.\nWrite a list of your current network. This can include everyone you have met in a professional capacity, as well as personal contacts that could be professional useful. Use your social networks to search for, and link with these people.\nDevelop your online networks. For example, follow the Twitter pages of companies you wish to work for, connect with people on LinkedIn who have a similar jobs to the one you desire and connect through Facebook with people you know in your industry.\nMake a post on your networks describing the job you are seeking with a link to your personal website.\nJoin and follow appropriate professional organisations and interest groups and actively participate through their online forums. Connect through social media with any connections you make.\nMany companies have employee referral programs. These programs provide financial incentives to current employees if they recommend a candidate who is hired for a job vacancy. This means you do not necessarily have to connect with hiring managers to secure an interview. A connection with any employee in a large company can give you a head start to secure your desired job.\nBy using social media platforms such as LinkedIn, you can search for people who currently work for the organisations you are interested in. By connecting with them, and being upfront about your desire to work for their company, you are paving the way for a job referral.\nEven if the company does not have an employee referral program, you will gather valuable information such as company culture, hiring manager's names and up-coming vacancies through your connections.\nThis method may sound far-fetched, but this how a previous colleague of mine secured her position as a Senior Manager in a large national organisation. By connecting with me through LinkedIn, she shared her job search intentions. At the time my employer was looking to hire someone with her skill set. I submitted her application and she secured a job offer before the company commenced any form of active recruitment. She found her dream job solely through the hidden job market.\nBy accessing the hidden job market you substantially increase the probability of finding your dream job. It is a significant investment of time but is essential for any serious job seeker.\nDownload this custom designed resume template in a choice of colours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"India's leading automobile manufacturer, Tata Motors recently made Lionel Messi as their global brand ambassador for Tata's passenger car segment. Soon after the announcement of Messi as the global brand ambassador of Tata Motors, the Internet is fully filled with this news on football ace's new engagement with Tata Motors. Showcasing its new partnership, Tata Motors, has now released a brand new TV commercial that features Lionel Messi; In the middle of the ad, Tata Motors teases the upcoming Kite hatchback.\nEarlier their were reports that Tata Motors is planning to bring its upcoming hatchback which is codenamed as Kite, This new hatchback was predicted to be released in the month of December.\nMotorOctane reports Tata Motor's all new Kite hatchback will be launched this year, which is then going to be followed by the company's new compact sedan in early 2016. Under the hood, Tata Kite petrol shall sport a 1.2-litre petrol engine, which will be naturally aspirated. This, of course is the same engine which is offered in other popular cars from Tata Motors like Zest and the Bolt, however the turbocharger will be not present. Kite's diesel engine is going to be powered by 1.05-litre CR4 engine, which is a three-cylinder version of the existing 1.4-litre CR4 that currently powers the popular Indica eV2 as well as the new Indigo eCS. This engine will produce 63bhp of power and the fuel efficiency of the car is going to be between 27 to 30kmpl.\nShifting-Gears reports that the Kite twins would be getting a new 1.05 Litre diesel engine which is a re-engineered, 3-cylinder derivative of the 1.4 litre DiCOR diesel engine. This engine will make use of counter balancing shafts, stiffer engine block, reduced friction & vibrations and can churn out 70 BHP of power. The petrol version of Tata Motors will be getting the all-new 1.2 litre, 3-cylinder engine which might be badged as Revotron. This new engine uses an aluminium block and 4-valves per cylinder with power output expected to be in the range of 75 BHP.\nThe body shell of the new Kite hatchback is designed by Tata's European Technical Centre (TMETC). It is good move that Tata took this step, away from the Indica design language which is now old. The company is also rumoured to ve working on in-house range of accessories for the new Kite hatchback and the compact-sedan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Anyone can become a scuba diver, but it does take work. You have to learn underwater skills, master scuba equipment, learn about science, and practice it all until it is second nature.\nWhat you learn in your initial diving certifications will stay with you for your diving career. It will open your life to a new underwater world.\nLearning new things is important for everything you do. It's a part of life. And you'll appreciate your solid diving education when you're floating in the middle of a school of angel fish, or when you see a moray eel slither out of its coral cave for the first time.\nThe science of diving can be intimidating, but a good dive instructor can make it seem simple. Pressure is a big factor in diving. It effects how your ears and body regulate as you dive. It affects how your lungs expand and contract; think of a balloon and what happens as you push it underwater. Pressure truly affects your body, but if you learn how to manage it, it isn't a big problem. Another aspect of diving science that comes into play is decompression. This comes from diving too much or too deep.\nIn order to stay safe, you'll learn all about dive tables, air mixes, and surface intervals \u2013 even how it affects your nightly glass of wine and when you fly home.\nDive equipment is the key that allows us to breath underwater. It may be hard to walk around with a BCD (buoyancy control device), wetsuit, air tank, and flippers when you're on dry land, but once you are drifting along a vivid coral reef, every piece of equipment you have is useful. In the certification process you'll learn all about different types of equipment and how it works, when to use it, and what works best for you.\nDiving skills are required for diving. It sounds simple, but there are skills that your dive instructor will want you to have mastered before you dive. Lots of the skills you may never use, but when your mask fills up with seawater and your eyes start to sting, you'll be extra glad that you know how to clear your mask. You'll get to learn safe ascend and descent techniques, mask clearing, buddy breathing, hand signals, weighting, buoyancy, entrance and exit techniques, and plenty of other things.\nYou'll learn about the local ecosystem and how it dictates the equipment you use. You can imagine that learning in the cold waters of Maine might require different, and warmer equipment than learning to dive in the tropical waters of the Philippines.\nAs you progress in the diving world, you learn specialty diving skills or focus on technical diving certifications. Then the learning experience seems endless. Underwater photography, cavern diving, night dives, drift diving, enriched air mixes, deep diving, shipwreck exploration, underwater navigation, ecology, and altitude diving are just the beginning.\nEventually the diving certifications teach you underwater and surface rescue techniques.You'll have to master basic life support, how to deliver oxygen, manage accidents, find lost divers, and how to deliver first aid.\nScuba diving is an endless continuing education. There is always something new to learn. It may be fish identification, avoiding decompression after diving to 60 meters, or understanding how to ascend to the surface so your lungs don't explode, but everything you learn will come in handy. Just be sure to open your mind and let all of that diving knowledge soak in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Broken Arrow digital sheet music. Contains printable sheet music plus an interactive, downloadable digital sheet music file.\nThis time he's gonna wear an iron suit.\nThe Arrangement Details Tab gives you detailed information about this particular arrangement of Broken Arrow - not necessarily the song.\nThere are no reviews written for Broken Arrow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DateMyAge is a free signup dating community where registered users find their perfect match for a genuine relationship. We have reviewed lots of dating sites on this website but DateMyAge.com as a uniqueness which makes their services different from other ones. As a registered user on www.datemyage.com you have the privilege to Chat, send letters, call, share your photos and make presents. Isn't that great?\n\"DateMyAge.com is a community of mature singles but Young at Heart, finding special lovers to forever be with in life\".\nNo commitment needed to create a new DateMyAge.com account. All you need do is access the page, enter your email and password, or you can even alternatively use Facebook or Google-plus to Login \u2013 then seamlessly setups your account completely when you login.\nDatemyage.com knows what members from other dating sites suffer, like spamming, not chatting with real people, a high cost of subscription fee, and so on. But they made it free and those challenges by making sure all members are personally confirmed by their staff to prove they are real. Also, provide you with safety by leading anti-scam system in the industry.\nThat's isn't all, You need not upgrade account before you can Receive attention from attractive members worldwide.\nI wouldn't what to go on and on typing all this for you to keep reading. This few right-ups is enough but you need to use the steps below create your account and login to see for yourself and experience the new dimension at which DateMyAge.com Account will take your relation.\nFirst, you MUST be 18years of age before you can use the site.\nThen you should be able to know what you want, like between what age of male or female relationship you want to start. ooh YES, you have the ability to select or filter the age of the person you want to enter into a relationship with. Indeed this platform is a community for mature people seeking a new relationship.\nYou will also be need ing your email address, that is if you will not be using the alternative mode (facebook account or Google plus) to login.\n* Then click on \"Meet 45+ Singles\" button below.\n5. Now, this part of the form is where you tell interesting and geninue things about you and the kind of relationship you want. Make sure it is convincing and real. This will be added to your profile so that interested friends coming can read to know your goals.\nAlthough you can as well skip this part. But really not making sense if you do so. Because people will want to know who you are and what you are passionate about in life and what you are expecting in your relationship with any serious person.\n6. The second part of the form is where you are given a chance to talk about how and what you want in your ideal partner. Make it look convincing and enticing to believe.\n7. Now the next phase of the form is to select your area of interest. That is what you love doing or love to see. For example; Cooking, Travelling, etc. Please note that you can select more than one area of interest. Once you are done selecting \u2013 click on the \"Next\" button.\nAnyone you select (by simply clicking it) should be highlighted green \u2013 meaning you have selected it.\n8. Now, the last part of the form is to upload a picture of you. Just like how important it is to write about you and your ideal partner, so it is very important you upload a nice picture of yourself. Remember your goal is to win and be in a genuine relationship, therefore, uploading pics is likewise a step to achieving that.\nSimply click on \"UPLOAD PROFILE PHOTO\" you are left with THREE option to get a profile picture. You can request to upload a picture from your Facebook, or saved files on your computer or mobile device, or using the webcam to snap a picture. Then Click \"COMPLETE\" once you are done.\nNow you have completed the DateMyAge.com Account Registration.\nIf you want to upgrade your account to have access to 10 Free Chats, 10 Free Emails and Free Introduction messages, you should be able to see the \"UPGRADE ACCOUNT\" button \u2013 click on it, it will take you to the payment page.\nYou are able to filter the age of partner you want.\nAccount upgrade is needed to enjoy full package.\nNoting is free (apart from the initial signup process).\nYou can't chat and you can't see the chat with your partner if your account is not upgraded.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavvtr b/data_all_eng_slimpj/shuffled/split2/finalzzzavvtr
new file mode 100644
index 0000000000000000000000000000000000000000..1854c4a97043dd0c24c32c3e12064002d220aebf
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavvtr
@@ -0,0 +1,5 @@
+{"text":"Social Media Mavens Lianne Farbes and Kristin Booker hosted their semi-annual Cocktails and Couture Fashion Week event at the Ted Gibson Salon and Spa with title sponsor Mwari Magazine.\nThe event brought national fashion reporters, editors, bloggers and social media gurus together to discuss all things fashion, glamour and beauty. Guests enjoyed passed hors d'oeuvres from Honey Bun Catering, grooved to classic hip hop courtesy of DJ Belle De Jour (India-Jewel Jackson of Glam.com), sipped Moet and Chandon champagne and satisfied their sweet tooth with passed Starbucks ice cream and frappuccinos.\nAnd they had awesome beauty stations!\nThanks for taking this pic Jordy!\nAnd\u2026we received the most fabulush gift bags!!!\nThe Mwari Magazine gift bags included products from sponsors L'Oreal, HTC, Stila, Lierac, Visine, Caress, Venus, Olay, Sigma, Zoya, Covergirl, Proactiv, Degree, Fournier, Suave and more.\nLooking fab and what a fun looking event!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hockey is one of ten core sports in the Commonwealth Games and has been included in every Games since its debut at Kuala Lumpur 1998.\nPlayed between two teams of 18, with 11 players per team allowed on the pitch at any one time, a game is divided into two 35 minute periods, with a ten minute interval in between. Modern Hockey is played on synthetic water based pitches.\nHockey was included at the inaugural Commonwealth Youth Games in Edinburgh in 2000. This is the sport's only appearance at a Youth Games to date.\nHockey competitions are run according to the rules of the International Hockey Federation (FIH).\nAustralia's men have dominated the sport, winning gold at every Games to date. Their women have won gold on all but two occasions \u2013 silver at Gold Coast 2018 and bronze in 2002.\nScotland's highest placing in Hockey was a 5th place finish by the women's team at Melbourne 2006.\nIn the Commonwealth Games Hockey competition each squad is allowed to have 18 athletes. Both the men's and women's competition format consists of a pool stage where teams play each other in a round-robin tournament. On completion of the pool fixtures, the top team in each pool plays the team second in the other pool with the winners contesting the final and the losers contesting the bronze medal.\nEssential skills for playing hockey are the ability to control, pass, push, stop and hit the ball. Stick work is impressive to watch, as players with good stick skills can maintain control of the ball while sprinting the length of the field, weaving through the sticks and legs of defending players to create space.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Excellent for use on spot prone skin thanks to wonderful natural ingredients such as antibacterial and anti-inflammatory Peppermint essential oil as well as Witch Hazel and Bamboo Powder to help soothe and tone the skin. Leaves skin feeling beautifully clean and balanced.\nCleanse face with warm water. Apply masque evenly over face with fingertips, avoiding delicate eye and lip areas. Leave on for 10-15 minutes or until dry. Wash off with tepid water and pat skin dry. Use Weekly.\nAqua, Kaolin, Glycerin, Bentonite, Hamamelis virginiana Water, Prunus amygdalus dulcis Oil, Butyrospermum parkii Butter, Bambusa arundinacea Stem Powder, Argania spinosa Kernel Oil, Cetearyl Alcohol, Coco-Glucoside, Coconut Alcohol, Glyceryl Stearate, Alcohol, Melaleuca alternifolia Leaf Oil, Mentha arvensis Leaf Oil, Tocopherol, Helianthus annuus Seed Oil, Disodium EDTA, Benzyl Alcohol, Dehydroacetic Acid, CI 42090, Maris Sal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Numerous research entities are active at the UNamur. They group Belgian and foreign academic, scientific, administrative and technical staff, all of them sharing a passion for research.\nRecently updated, the UNamur research landscape harbours 11 transdisciplinary research institutes. Research activities lean on state-of-the-art scientific equipment grouped in 8 technology platforms. Researchers develop inter- and transdisciplinary projects implying fundamental as well as applied research. Their expertise covers numerous fields.\nThe NARC (NAmur Research College) complements this landscape by following a double goal: to offer an optimal research environment for young researchers and to ensure the visibility of outstanding researchers inside and outside the UNamur.\nThe Board of Directors, with the guidance of the Research Council, defines the UNamur research strategy.\nThe UNamur aims at excellence and maintains the appropriate balance between fundamental and applied research. Special attention is given to the training of Ph.D. students and to researchers' mobility, for they are the key links of the university network.\nFrom the building up of a project to commercialisation, from IP protection to technology transfer and development, the ADRE team offers strategic, administrative, legal support to the principal investigaqtors and helps them promote their research. Get the ADRE contact brochure.\nPratiques m\u00e9di\u00e9vales de l'\u00e9crit. Sicut erat in principio et nunc et semper : un message cach\u00e9 dans le Liber Traditionum de Saint-Pierre de Gand ?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contrast describes the relationship between light and dark values. To understand contrast, make a value scale. Draw ten squares, making the first one white and the last one black. Then create a range of eight grays between them, progressing from light to dark. It is the distance between values on such a scale by which we measure degrees of contrast. The farther apart two values are on the scale, the stronger the contrast between them. Black is farthest from white; thus, when juxtaposed, these two extremes create the greatest possible value contrast. Less contrast exists between a light gray and a dark gray, and still less between a dark gray and black, which are very close to each other on the scale. Contrasting values placed side by side will accentuate each other; lights will appear lighter and darks will appear darker.\nAs you will discover, a skillful use of contrast can help you establish the illusion of distance and define the quality of light in your drawings.\nThis is a value scale. Contrast is the relationship of two or more values on the value scale.\nThe two cubes at left contrast less with each other than the two on the right.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawdvm b/data_all_eng_slimpj/shuffled/split2/finalzzzawdvm
new file mode 100644
index 0000000000000000000000000000000000000000..1a1173f68d06615133ff3f1bf95fd79b7f4a9e9d
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawdvm
@@ -0,0 +1,5 @@
+{"text":"A Mathematician and a Human Capital Consultant (yeah, I have no idea either) met and bonded over their love of Schutzhund, Dobermans and spreadsheet nerdery. That friendship resulted in a project that took thousands of hours and spanned more than 4 years (and counting). A true labor of love, we gave up our free time to painstakingly obsess over this passion.\nThere are 23,467* organizations where you can get a dogsport title on the North American and European continents. Each of these organizations employed some sort of record keeping, beginning with stone tablets and eventually winding their way into the modern computer era.\nWe set out gathering each organization's records, whatever their form. Through a complex process of begging and bribery, we got our hands on pretty much everything. We collected pamphlets, newsletters and magazines. We dug through boxes, filing cabinets and carbon paper originals. We pulled from current websites and scoured others that had not been online in many years. The proverbial haystack > needle.\nWe may be missing two issues here or one issue there, but at this point we have 98.7821% of the official publications from the very beginning of Schutzhund in North America. Europe\u2026 not so much. But we've already whined about that situation elsewhere, so we won't do it again here.\nSo, what did we do with all those trial records? We compiled the working history of the Doberman breed \u2014 a database going back to the beginning of dogsport in North America. That project by itself took years and it was our first baby. And then that baby gave birth to its own baby (does that make us Grandparents??).\nBorn of more humble origins, this project began with two kennels and one Excel document. The intention was to help a friend choose between two breeders using our hey-let's-stick-it-in-a-spreadsheet answer for everything. We were having so much fun that we added another kennel, and then some design work, and then three more kennels, and then formulae, and then five more kennels, and then comparative statistics\u2026 and on and on and on it went.\nAs our modest little spreadsheet grew into a full-fledged research project, we were frankly astonished by what we learned. Assumptions about this dog, that handler \u2014 very little was as expected. We realized that conventional wisdom wasn't necessarily correct and we thought others might enjoy going down that same road of discovery.\nSo, we gave our project a new home in the form of a website and decided to make it public. We felt that we could offer an objective, data-driven approach rather than solely relying on breeder websites, blogs and forums for research. While often useful, the information from those sources can be arbitrary, capricious and flat out untruthful. We were interested in measurable accomplishments, not hyperbole.\nIn the end, our goal is to enlighten and entertain, to contribute a little to the sport we love and the breed we love doing it with. We hope you get as much out of exploring this project as we have in creating it.\nA rant will go here.\nA project of this magnitude could not have been done without the participation of many. We'd like to wholeheartedly thank those who were kind enough to share their magazines with a couple of crazy people. The words \"thank you\" are simply not enough to express our immeasurable gratitude.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Simple Comments is a powerful WordPress security plugin that will protect your login, registration, comment, contact, and other forms from hackbots and spambots.\nSimple Comments is a powerful WordPress security plugin that will protect your login, registration, comment, contact, and other forms from hackbots and spambots. Forms are the entry point to hijack and gain access to your web site. Don't leave your forms unprotected. From the same developer as SpamFreeWordPress, which has been terrific. This takes it much much further and adds tons of other features. Small investment for huge benefits.\nFor current members, we can install this right away once we receive your order. If not (yet!) a member, please email us to arrange for site access.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"<p>2015 which is extra long-awaited refreshment the Air Coach SC II cheap upcoming shows another color, this easy celebration.\nThe establishment of a African leather upper shoes, covered with frosted shades forefoot,and with a African belt forefoot for well,linger focused aboard the chart prefer than proud legend blue 11 for sale colors. For comparison, the Cheap jordans gray plus silver luster near whirlwind,amid the annihilate plus nike mag for sale the model's signature striping, cleverly juxtaposed African does never underestimate the advantages of the concept of stepping outdoor.\nFinally, the white outsole continues the sly theme, reconcilingthe shoes retro sneakers.\nhttp:\/\/www.mcafee.com\/activate - Activate Your McAfee Retail card by simply visiting our website mcafee.com\/activate and Get Started with McAfee Security.\nelectronics and online services. The online services includes iTunes, Apple music, iOS App store and iCloud etc.\nThe company also developed an app for providing Apple Support to its users.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In March 2018 we asked the parents whose children attend the nursery for feedback on what they think that we have been doing well and where we can improve. We were pleased with the amount of feedback we received and the positive responses from everyone who replied. We would like to thank everyone who took the time to let us know their thoughts.\nWe have shared the responses we recieved on our blog where you can read each of them here.\nDisplay my name with my comment.\nI give permission to allow the use of my comments by Harton Village Kindergarten on any media or marketing material.\nAll comments are reviewed by a member of the nursery management team before being visible on our website, this may cause a short delay in your comments being posted. This is to ensure that all comments are relevant to the nursery and are posted by family or carers of children at Harton Village Kindergarten. We may remove any comments that are found to be deceptive in any way or use language or terms unsuitable for a nursery environment.\nWhile reviewing comments the nursery manager cannot edit or change posted comments in any way, all comments will be shown as submitted. If you wish for your own review to be removed, then please complete this short form and we will process the removal request.\nWe do not share any personal information that is submitted on our forms and the only details that are displayed with the comments on our website are those that are approved by yourself when submitting. The details entered are used only to confirm that a comment is valid and from a legitimate source. Harton Village Kindergarten will not share any comments made on our website with any external agencies but may reserve the right to use them on other nursery media if permission is granted. We will not use any personal information entered into this form to contact yourself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you need Oklahoma auto insurance, you've come to the right place. AutoInsuranceQuote.com is your one-stop resource for cheap Oklahoma car insurance quotes, information and more. We've gathered the best auto insurance companies in the nation and Oklahoma, to provide you with quality auto insurance at an affordable price. All you have to do is type in your zip code, answer a few quick questions, and then you'll be able to compare cheap Oklahoma car insurance quotes from a variety of companies. Most of our users will save over $500 a year on car insurance!\nSo, take a few minutes to search the fantastic Oklahoma car insurance quotes, and find a policy that fits your needs and lifestyle. Its quick, its easy, and best of all, you'll save hundreds of dollars each year on Oklahoma car insurance!\nNo matter what your life situation, you can find a suitable car insurance policy from AutoInsuranceQuote.com. Our companies insure teen drivers, drivers with bad credit, and even drivers with poor driving histories. Of course, people with poor driving history or bad credit will pay more for car insurance, but you can still get quality coverage at a cheap price. Start now by typing in your zip code, answering a few quick questions, and then reviewing a multitude of Oklahoma car insurance quotes from companies you know and trust.\nCould you use an extra $500 a year? Probably. Even if you are financially secure, an extra $500 is still a solid amount of money. Well, that's how much you could save on Oklahoma car insurance by using AutoInsuranceQuote.com. Who knows, you might even save more! Of course, each driver and situation is different, so you may save more, you may save less, but one things for sure\u2026.you'll get quality Oklahoma car insurance at an affordable price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawxhz b/data_all_eng_slimpj/shuffled/split2/finalzzzawxhz
new file mode 100644
index 0000000000000000000000000000000000000000..4e969e74b0741c5b54d943053de1e391008a4f3a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzawxhz
@@ -0,0 +1,5 @@
+{"text":"German midfielder Tom Trybull joined Norwich on a one year deal on 4th August 2017 following a successful trial. He was given the number 19 shirt.\nHe began his career in his hometown with the youth teams of FC Berlin (1998-99), SV Lichtenberg (1999-2002), 1. FC Union Berlin (2002-08) and Hansa Rostock (2008-10). He made his full debut whilst with Hansa Rostok playing 18 games between 2010 and 2011.\nHe then moved to Werder Bremen in 2011 and made 21 appearances, scoring once. In 2014 he transferred to FC St Pauli for whom he made 15 appearances. The following year he joined SpVgg Greuther Furth for whom he played 18 games for their reserve time.\nHe moved to Holland to join ADO Den Haag in 2016 and made 23 appearances during 2016\/17 before being released at the end of the season.\nHe had been capped by Germany at Under-17, Under-18 and Under-19 level.\nTrybull made his Norwich debut on 22nd August 2017 in the 4-1 home League Cup win over Charlton Athletic, coming on to replace Wes Hoolahan on 77 minutes and scoring Norwich's fourth twelve minutes later. He finished the season with 24 appearances and 3 goals to his name.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brunswick Vermont: Restaurants, Real Estate, Lodging & Things To Do In Beautiful Brunswick VT!\nBrunswick Vermont is a town in Essex County, located in the beautifully scenic \"Northeast Kingdom\" of Vermont, an area that National Geographic selected as part of their geotourism program for sustainable destinations. The town was named after Prince Karl Wilhelm Ferdinand of Brunswick-Lunenburg.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you want to dominate the Trojan War: Reckoning of Zeus Hack completely then you are at the right place see the image of the cheat tool above and you can clearly see that this is the generator tool for Trojan War: Reckoning of Zeus, using this cheat tool you can hack free drachma wood stone metal and laurels in the game, this Trojan War: Reckoning of Zeus cheat tool is capable of adding unlimited amount of stone raulers etc. in the game, the items that this cheat tool will add can be used to all, raulers, wood and all the other items available in the game.\nNow you do not have to play for hours for collecting the drachma to unlock items, simply use the Trojan War hack tool to generate as much resources you want to use them to do whatever you dream in the game, build the best character among your friends and family and make them jealous or just share this cheat with them to make their gaming experience even better. Since the raulers are the premium items in the Dungeon Rampage we do not recommend adding them in a huge or large amount because they are the income source of the game and they are strictly protected on the other hand you can generate as much drachma, wood, stone, metal you need since they are not strictly monitored in the Trojan War: Reckoning of Zeus. This cheat will make all your dreams come true in Trojan War: Reckoning of Zeus since it can give you an unlimited supply of resources that are permanent.\nTrojan War: Reckoning of Zeus Cheat tool is designed as an generator tool which is capable of hacking unlimited drachma wood stone metal and laurels, to add free resources you just need to start playing the game and while running the game in the background, enter the desired amount of resources and press the \"Start\" button present on the cheat tool and wait till the progress finishes and cheat tool after the cheat tool just refresh the Trojan War: Reckoning of Zeus to see the changes in the amount of drachma wood stone metal and laurels.\nWe are going to start the image demonstration of this Trojan War: Reckoning of Zeus cheat tool, see the image above from a level 1 account of the Trojan War: Reckoning of Zeus and notice the amount of drachma wood stone metal and laurels, we are going to hack the all.\nAfter using the cheat tool the amount of all resources have been changes with a large number, the drachma wood stone metal are now 100000 and the amount of raulers 1000 is changes , the account is the same on level 1, the above demonstration clearly proves that Trojan War: Reckoning of Zeus cheat tool works perfectly and as said.\nI really hope I don't get banned with this, till now working perfectly, amazing tool!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When a student said she wanted to commit suicide after being molested by her stepfather, the counselor at a local charter school needed help herself.\nJermika Yates was a family engagement specialist for Shelby County Schools (SCS) before becoming the counselor at Granville T. Woods Academy of Innovation, a charter school in Raleigh four years ago.\nYates did an online search for resources and found Youth Villages Specialized Crisis Services, a 24-hour-a-day hotline staffed with master's-level clinicians who provide triage for kids up to age 18 who are experiencing serious suicidal or homicidal thoughts.\nLast year in Shelby County, there were nearly 2,900 calls to the crisis hotline; about 1,600, or 55 percent, of those calls necessitated a follow-up meeting or additional services.\nSuicide is the second leading cause of death for young people in the U.S., behind accidents, accounting for 19 percent of all deaths for people 15-24 years old, according to the American Association of Suicidology.\nThe youth suicide rate, for ages 10-24, is at an all-time high in Tennessee at 17.3 per 100,000 population, compared to 14.5 nationally, according to the latest 2017 figures from the Tennessee Suicide Prevention Network.\nOver the past four years, Yates averages two to three calls a year to Specialized Crisis Services. Most of the cases are suicidal thoughts. All of them have been girls.\nNationally, there are 3 female suicide attempts for each male attempt. Though, 3.5 men die by suicide for each female death by suicide, according to the American Association of Suicidology.\n\"It's a lot of the same thing \u2013 not feeling pretty enough,\" Yates said, who started a self-esteem group in response.\nYates had one case of a student wanting to commit homicide.\n\"Parkland\" has become a moniker for last year's school shooting in which a former student killed 17 students and staff members at a high school in Parkland, Florida, on Feb. 14.\nIn 2018, the hotline received 13,340 total calls, or 15 percent more than the 11,554 calls received in 2017.\nThe majority of concerns could be traced back to comments that were being made on social media, either a threat to a student or group of students.\nDepending on the severity of the situation, some of last year's assessments were done in juvenile detention centers, as law enforcement did a safety sweep of homes to make sure the callers did not have access to weapons at home.\nEmma Collins has been taking calls for Specialized Crisis Services for about two years from a 24\/7-staffed Youth Villages call center in Nashville.\n\"We get a lot of calls regarding bullying. It's an issue that is big right now,\" she said.\nIn what Yates calls her \"proud counselor moment,\" she has only had one bullying case in her four years at Granville T. Woods Academy.\nThe school has a zero-tolerance policy when it comes to bullying. Students and parents sign a pledge against it at the beginning of each school year.\nWhen Yates calls Youth Villages, the results vary by situation. Some cases can be deescalated by phone and referred to individual or family counseling. Others require hospitalization.\nYouth Villages' goal is to assess young people in their own environment such as homes and schools, but if safety is unable to be maintained, law enforcement or involuntarily hospitalization can come into play.\nWhether the case requires hospitalization or not, all callers are given a safety plan, coping techniques and a follow up.\nYouth Villages works with callers to identify triggers that prompt suicidal thoughts or feelings and ways others can recognize it.\nA lot of times, the hotline is the first time the person on the other line has talked to a mental health professional.\n\"Through cultural diversity, not everyone interacts with mental health on a regular basis; some fear it or don't know enough about it,\" Collins said.\nGiving a \"warm welcome\" in a time of need can change someone's perception of mental health.\n\"That's why I do this job, to give people a look into the mental health world and not fear it,\" Collins said.\nYates admits, at first, it was a struggle to get parents on board.\nAs mental health has entered the national conversation with the rise in suicides and tragedies like school shootings, parents are more aware.\n\"They will now even say, 'So-in-so in my family had depression and maybe that's where it's coming from,' \" Yates said.\nSince the program began in 2003, call volume has increased. Each month, the number of calls increases by a few hundred, compared to the same month of the last fiscal year.\nThose in the Memphis area in need of assistance can call 866-791-9226 or text \"TN\" to 741741.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Digital Infra-Red photographs, printed as Cyanotype prints.\nOlympus OMD EM5 with R72 Infra Red filter. The false colour images were then converted to monochrome in Adobe Lightroom and printed as Classic cyanotype prints on Arches Aquarelle paper.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaxoif b/data_all_eng_slimpj/shuffled/split2/finalzzzaxoif
new file mode 100644
index 0000000000000000000000000000000000000000..50fbef2929aba79e05ea175c12e5d129f98f5533
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaxoif
@@ -0,0 +1,5 @@
+{"text":"Electromatic Center Block guitars are designed for pure and powerful Gretsch sound at high volume. Crafted with essential Gretsch sound, style and playability. Equipped with dual Super Hilo'Tron humbucking pickups that deliver a bold, dynamic and articulate sound.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JP Buys earned undisputed bantamweight champion status after finishing Luthando Biko, while England's Jake Hadley ended Nkazimulo Zulu's flyweight reign at EFC 78 on Saturday night.\nBuys (7-2) saw off Biko (5-2), who became interim champ by defeating Cal Ellenor (8-2) in November 2018, in the headliner with a fourth-round TKO via ground-and-pound at GrandWest Arena in Cape Town.\nIn the co-main event, 22-year-old Hadley (4-0) caused an upset in his promotional debut by beating Zulu (10-3) by unanimous decision to claim the belt.\nEarlier, Saxon Delafield (5-4) submitted Rodrique Kena (3-3) and Shaun de Lange (5-3) got a verbal submission from Marcel Teniers (4-5).\nThe main card was diluted, as Luke Michael's (4-3) welterweight grand prix fight with Conrad Seabi (6-4) was postponed to May 4 \u2013 this after Seabi suffered a concussion with a slip during the weight-cut, according to the EFC.\nto unify his EFC bantamweight title.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Incorporating Silhouette BCD, Brut Regulator Set, Octopus and Compact Console. Top Choice of Dive Schools and Rental Shops Worldwide. The Silhouette being rugged, dependable, durable and affordable makes this the ideal entry level BC. This jacket-style BC is colour coded with accent piping, designating size for the rental department. The World famous Brut regulator has had a huge overhaul in recent years making it possibly the best value for money rental regulator on the market. Environmentally sealed and incredibly easy to service.\n\u2022 Single, large Velcro pocket.\n\u2022 Adjustable tank locator strap.\nWith our advanced plastic and silicone integrated injection method, SEAL's soft hypoallergenic silicone skirt is molded directly onto the frame. This means less parts are needed, making it a lighter and more comfortable mask. With this new technology, the internal volume of the mask has been significantly reduced and the lenses are brought as close to the eyes as possible for better vision. Comfortable for scuba diver and skin divers alike to achieve superb views underwater.\nThe all new PPE EMS PROPack\u2122 is designed to easily carry and deploy infection control precaution supplies. Response ready, this pack provides easy access to gloves and other supplies on each call, without opening the main compartment. Compact and professional, this is simply the best PPE case available.\nA comfortable fit for any face with fully adjustable nose-piece and strap. available in multiple colors..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"eBay Flashback: Master Jedi Army-Builder Collection!\nOK, the price tag on this auction is laughable, but the collection of pictures this seller has assembled is worth at least checking out the auction! The time it must have taken to assemble this set of 2600+ items!\nBREAKING NEWS: Special Action Figure Sets Hit Philippines - Snaggletooth!\nOur good friend Manny M. comes through big time for us! His friend James found the new USA TARGET exclusive new Special Action Figure sets in their Toys R Us in the Philippines! Yes, we have an actual photo of the all-new Snaggletooth figure and not just a drawing! :) (Special thanks to our friends at Sandtroopers.com for breaking the original story.) Click through to check out this exclusive report from Jedi Temple Archives!\nUPDATE: Manny is texting us details on the figures! Click through to see the updates! We need to clarify some of them with him so stand by for more.\nI have quite a tall tale to tell. Thankfully I have also provided proof for my outlandish claims. Click through if you're interested!\nClone Wars Season 4 To Be Released On October 23!\nGet ready! The Clone Wars \"The Complete Season 4\" is slated for an October 23rd release on both DVD and Blu-ray! Click through for the report.\nSet For Stunning! Jaina Solo Bishoujo Statue!\nKotobukiya has introduced their first ever Star Wars Bishoujo statue! Jaina Solo! Click through for the info and high resolution photos!\nTARGET and Walmart may get a case or two of new figures, but we get the most reports from Kmart. If you're still fortunate enough to have this retail chain near you, they seem to be the most consistent with store finds. Click through for JTA reader Andrew L.'s report!\nPreviews came up with a clever way to ensure that two multipack exclusives would sell at the same rate. In order to get one, you had to get the other one too. Here is the much better set of the two, otherwise known as the Star Wars: Battlefront II Clone Pack in our Research Droids Reviews today! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nThe Brian's Toys newsletter this week features a great selection of new Sideshow pre-orders. The C-3PO and R2-D2 droid duo are together again in the long waited premium format and the sixth scale Bossk figure is a brand new announcement today! Also new is the Medicom VCD Jawa, which looks more realistic to the movie version than the regular caricatured VCD versions.\nA new Darth Vader (Reveal to Anakin Skywalker) bust from Illusive Concepts was recently delivered to us. This is one of the few busts we've seen in such great condition, but it still has paint wear to Anakin's head and does not include the original COA. Overall, nothing is broken and displays nicely (albeit kind of scary, as evident to some Brian's Toys employees being startled as they turned the corner).\nFinally, new Masters of the Universe Classics figures (Horde Prime, Snake Man-At-Arms, and Griffin) have arrived along with the Sideshow 12\" Tedn & Nalan. Check it all out over at Brians Toys.\nBesides July 4, 2012 marking a very important national holiday in the USA, it also just happens to be the day of our 1,138th Research Droids Review. What a coincidence that Sideshow Collectibles has just shipped out their CC-1138, Commander Bacara 1:6 scale figure just in time so we can celebrate this landmark in style! You can check out a cool sneak peek of this simply brilliant figure on our Facebook page right now, but be sure to come back on July 4th to see this awesome figure in its full glory!\nSideshow Collectibles 1:6 Scale Bossk Pre-order is Live!\nThe Sideshow Collectibles 1:6 Scale Bossk is now available for pre-order... and he is amazing! Click here to place your order!\nFive, Five Dollar, Five Dollar Class II Vehicles (And Exclusives)!\nIt seems that the current Star Wars toy line has officially become an epic failure, at least in one local Walmart of JTA reader Paul H. from MA. We reported a while ago to you that many products in The Phantom Menace 3D line have been deleted, and whether or not you reaped any benefits of that depended upon how willing your local Walmart was to use their markdown dollars. But Paul provides proof that a few other things are happening as well. If you are seeing any clearance like this at all, it's a good bet you'll never see the second wave of vehicles inlcuding the Naboo Fighter at your local store. Sad. So very sad. Click through for Paul's report.\nPreviews came up with a clever way to ensure that two multipack exclusives would sell at the same rate. In order to get one, you had to get the other one too. Here is the \"not so great\" set, otherwise known as the Star Wars: Battlefront II Droid Pack, and in our Research Droids Reviews tomorrow we'll look at the second cooler set. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nEpisode 32: Tonight we try to stay positive as we discuss the dismal state of collecting right now and the distribution issues that seem to be the root of it all. Is collecting a dying hobby? Find out as we delve into it with former guests LE and Wraithnine as well as newcomer, and admin of Jedi Temple Archives, Paul Harrison. We want to say thanks to them all for coming on and offering their opinions, theories and conspiracies.\nNOTE: We unfortunately had some technical difficulties with echoing in the beginning of the podcast, but it quickly clears up.\nA few more store reports have come in for Ohio. The figures are very sporadic, but we like you to know when there are sightings in this horrific state of distribution.\nHere Is Some Drivel To Chew On!\nJTA pal tsunamijohnny in his typical humorous fashion alerts us to an insane store report. No new product has arrived in his area, but something else did (we'd bet our collection via a return without a receipt \u2013 but stranger things have happened).You know you're interested so just click through, (if you don't mind drivel of course).\nFYE Once Again The Place To Be!\nHaving no luck for weeks now at Walmart, Target & TRU in my area, I decided to hit up FYE today. They came through again! Click through for the report!\nSideshow Collectibles Teases One-Sixth Scale Bossk Pre-order!\nSideshow teases their next pre-order tomorrow! Bossk one-sixth scale!\nWhat a rewarding figure Nikto (Skiff Guard) is to own! We have been waiting 28 years for his modern counterpart, but he's finally here and an in-depth review awaits you in our Research Droids Reviews today! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nHere is a quick update from http:\/\/www.bigbadtoystore.com about an exciting batch of new preorders, a large batch of new arrivals, and info on a $5,000 store credit giveaway contest!\n'Like' us on Facebook and fill out the quick entry form and you'll be registered to win one of 40+ prizes! First prize is $1500 in store credit, 2nd prize is $500 in store credit (2 winners) - and there are several other denominations. Check out the details here!\nBoba Fett Production Version Photos Updated At Sideshow Collectibles!\nNow that we're getting close to the availability of Sideshow's very proud one-sixth scale Boba Fett, they have updated their photo gallery with photos of the actual Boba Fett figure (not the prototype). You should be able to see just how awesome the final product looks, even better than the prototype. Below the image on the left is the updated (production) figure and the smaller right image is the prototype. Look at the shape of the chest armor and how far it extends across the chest now. Look at the paint job differences. And there are many other updates! Click HERE to see Sideshow's updated gallery. Special thanks to JTA reader Jacob K. for giving us the head's up to this cool update!\nClick through to see what Research Droids Reviews you may have missed with our weekly recap complete with direct links and brief review synopses.\nFound! Movie Heroes Wave 3 Arrives At TARGET!\nJTA reader RC-1998 writes in with a hopeful store report that wave 3 of Movie Heroes has hit TARGET in TN! Click through to check it out! And this isn't another one of our \"hack\" reports either!\nNew TARGET Special Action Figure Set Revealed! All-New Snaggletooth!\nOur friends at Sandtroopers.com reveal another TARGET exclusive Special Action Figure set and this time.... wait for it.... it comes with an all-new Snaggletooth figure (which is actually Takeel and not Zutton which puts another interesting spin on what truly constitutes a \"Snaggletooth\" action figure).\nWell here it is, the elusive Mandalorian Assault Transport. Aren't you exhausted by all the distribution woes we're experiencing as collectors? See if this is the vehicle your collection needs in our Research Droids Reviews today! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nNever Say Never: And Not The Awesome Song By Romeo Void!\nWell, I guess we should never say it can't get worse than it is, because it can. JTA moderator Paul B. writes in with a disturbing report. Click through to see what we uncovered.\nJedi Temple Archives Will Be The Place To Be On July 1st!\nWe've mercilessly teased you for months now and we're happy to report that this is the last one. Come back right here on July 1, 2012 (not that you won't be back before that) for the official release of the Jedi Temple Archives\/Hasbro collaborated Special Report. We're really proud to give you this first hand look at some cool behind the scenes stuff amongst many other things. It's been an incredible opportunity so far and we hope you thoroughly enjoy it. Mark your calendars for July 1st! You won't want to miss it!\nCarded Clone Commander Fox [Phase II] Samples Hit eBay!\nCarded Clone Commander Fox (phase II) samples have hit eBay for $29.99 each. Click through and order yours if you're the least bit worried you might miss this one-per-case new figure in the new assortment yet to hit USA stores!\nHere is a tiny sneak peek at a page that will be available in tomorrow's Previews book. It contains a little blurb of the new Steve Sansweet book coming out this October that my buddy Dan Curto and I have been involved with. Be sure to keep it here for all the latest updates to this very exciting Star Wars action figure book!\nThe Premium Format C-3PO and R2-D2 are available for preorder! Click here to place your order!\nCarbon Freeze Me Pieces Arriving!\nThe \"Carbon Freeze Me\" personalized collectibles are beginning to arrive at collectors' homes. JTA reader Frank shares with our readers his unique piece. Click through for his story!\nMimobot announces their Star Wars Series 8 USB Flash Drives! Click through for the announcement!\nYou're probably too disheartened about the Mandalorian Assault Transport at this point to care, but an interesting development has surfaced regarding it. Thought to be released only in Tri-Logo packaging, \"American\" (Non-Tri-Logo) samples have turned up in Mexico. Is it possible that this \"lost\" vehicle will indeed see the light of day at discount chains like ROSS and TJMAXX. It's certainly unknown at this stage, but click through and check out the proof yourself. Also, stop by tomorrow as we look at this vehicle in our Research Droids Reviews.\nBounty hunters in any format are usually eaten up by collectors, so how you feel about the Hunt For The Millennium Falcon: Bounty Hunter Pack from The Saga Collection? It was already dated in early 2006 and it's even more dated now, but check it out in our Research Droids Reviews and voice your own assessment! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nAwesome Final Jedi Duel Diorama!\nOne of our most favorite diorama makers, Alejandro, sends in his latest work showing off his interpretation of the final Jedi duel at the climax of Return Of The Jedi. Click through to check it out!\nNew The Vintage Collection figures keep arriving into Walmart. JTA moderator Sjefke came across a wonderful find. I am personally beginning to dislike this man. Just kidding Sjefke.\nDo you like AT-ATs? We love them and one we're particularly fond of is the ROTJ-specific version that was released in The Saga Collection as a Toys R Us exclusive. The Endor AT-AT (All Terrain Armored Transport) is so cool. If you don't have one get it! See it in full resolution in our Research Droids Reviews right now! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nIf you're looking for the new Odd Ball (Clone Pilot) or Nikto (Skiff Guard) figures, they're on eBay for less than $8 each loose. Click either thumbnail below for a direct link to those auctions!\nMandalorian Assault Transport - Don't Wait For This Thing!\nYou already know the bad news that Entertainment Earth has cancelled pre-orders for the Mandalorian Assault Transport. But more keep showing up on eBay and right now as of this posting there are over a dozen. Not all are \"reasonably priced\" but bookmark this link and be sure to watch it on eBay. Click HERE to see if you can score one!\nWe already know you don't like his soft-goods skirt! But see why Grand Moff Tarkin is still an unbelievable action figure and a necessary purchase for your collection! It's all in our Research Droids Reviews so click through and check it out. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nHave The Dreadful Floodgates Opened? TVC Finally Arriving?\nWell, we have another happy story to share with you. JTA reader Anakin20 has scored a very nice collection of figures. Click through to read his story!\nFound! Naboo Fighter And Republic Fighter Tank Found On East Coast!\nFound! The Naboo Fighter and the newly repainted Republic Fighter Tank have been found as far East as NYC! Click through to find out where JTA reader Ivan C. found them!\nGet Savage Opress (Armored) For Less Than $8!\nWow! We don't know how we missed this one, but thanks to JTA moderator Sjefke, we have been apprised that the frustratingly difficult to obtain Savage Opress (Armored) is on eBay too for less than $8!\nUPDATE: This particular Savage Opress is from the upcoming TARGET exclusive Battle Packs set and comes without the lightsaber. Sorry folks!\nReview: JTA Battle Packs Set (Jedi Temple Assault) - Kmart Exclusive!\nIt's a JTA Battle Packs set: the Kmart exclusive Jedi Temple Assault (JTA) Battle Packs !! It shocked us all when Hasbro at 2005's SDCC announced that Kmart was getting the best Battle Packs of the year. It was initially tough to track down, but back in 2005, distribution actually \"worked\" and enough pieces were eventually made to meet demand. Learn about this classic set in our Research Droids Reviews today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nThis week's Brian's Toys newsletter features a variety of rare Star Wars collectibles. The highlight of the week is three LEGO Star Wars Ultimate Collector Series sets that include the Imperial Star Destroyer #10030, X-Wing Fighter #7191, and the TIE Interceptor #7181. The ultimate series are larger and more detailed than the traditional LEGO sets, which makes them a highly sought-after collector item.\nA complete Hamilton Star Wars plate collection has also arrived, which features the complete set of plates released by Hamilton from 1977 to 2004. For the Family Guy and Star Wars collectors, there is a set of 3 production drawings from the Blue Harvest special. The pencil drawings are accompanied with a Certificate of Authenticity.\nLast week's featured item was the Vintage Vinyl Cape Jawa graded 85 overall, but it is the highlight to Brian's Toys NEW YouTube channel (http:\/\/www.youtube.com\/user\/BriansToys\/videos). In the video, hosted by Brian, the Vinyl Cape Jawa is seen in High Definition and is given a breakdown of the beauty of the figure as well as some of it's flaws which justify the grade. Click here to view the video (http:\/\/youtu.be\/1f6-5PCNNdY).\nGet it all over at http:\/\/www.brianstoys.com\/nl.\nSideshow collectibles has announced their remaining exclusives for SDCC 2012: Tech Mo'r & Doikk Na'ts Bith Band members. Click through to check them out!\nA New And Perhaps Single Carded AT-ST Driver Running Change Imminent?\nJTA reader Mike from FL writes in with an encouraging store report. We haven't one of these positive posts in a while folks. Bask in its glory!\nGentle Giant Ltd. Announces Jumbo Vintage Kenner Blue Snaggletooth!\nGentle Giant, Ltd. announces the SDCC exclusive Blue Snaggletooth Jumbo Vintage Kenner Figure. It's limited to only 2000 pieces, but right now PGMs can pre-order it! Get this classic nostalgic item now! It's awesome at least one licensee has gotten this right. Shame on you Hasbro for not releasing a TVC Blue Snaggletooth before The Vintage Collection's demise.... or maybe it will be announces at SDCC this year. We can only hope!\nNo New Figures In Stores? Catch Up And Get Them Cheap On eBay!\nWe found some incredibly rare figures as well as some great deals on eBay. Click any link below to be taken to each figure's respective auction. Want that hard to obtain TCW Bossk, but don't want the Slave I and the other pack-in figures? You want Boba Fett in his Kamino fatigues? How about that hard to find Eeth Koth? Or what about the beautifully repainted C-3PO in the Discover The Force Battle Packs? They even have the pee-stained Snowtroopers from 2007's TAC Ultimate Battle Packs set! You can even get the new Movie Heroes Luke Skywalker and Light Up Darth Vader for a fraction of the price - besides, you know you'll never find them in stores! Check them all out!\nJTA Facebook Contest Winner Announced!\nCongratulations to JTA Facebook friend _____ ______ for his winning entry in our 501st Legion Clone Trooper contest! Click through to see who will be receiving 2012's TVC Clone Trooper (501st Legion) (VC60) figure!\nThe Kmart exclusive Imperial Throne Room Battle Packs set wasn't as nearly as exciting as its case mate (come back tomorrow for that one), so see if you agree with our assessment of it in our Research Droids Reviews today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nWe have the skinny on the San Diego Comic Con Bespin Chamber exclusive. Check it out!\nSandtroopers Reveals The Next Two TRU Exclusive Multipacks!\nOur friends at Sandtroopers.com have packaged images of the upcoming Toys R Us exclusives, the Ewok Pack and Yavin Pilot Pack. I don't know about you, but these are dreams come true for me, especially the Ewok Pack. They're very long overdue if you ask me! Click through for details!\nIt's time to order the awesome Sideshow Collectibles Utapau Airborne Trooper! Click here to secure this!\nKithaba figures with the red bandana are making more frequent appearances on eBay. We really hope that you listen to us at least sometimes, especially when it involves lining scalper pockets with your hard earned money. We say that with a smile, but bookmark THIS LINK and follow this figure on eBay. You will be able to find one of these variants at a reasonable price someday soon. Trust us! :) Who knows, we'll probably see it in stores and the brown one will become the rare one.\nYou Wanna See Phase II Commander Fox? We Have An Image For You!\nWe are proud to provide you with the first image of the 2012 TCW Clone Commander Fox in his phase II armor! We assumed this would be the outfit of choice by Hasbro, but now we have the cool proof! Special thanks to our good friend Manny for getting this image to us! Enjoy!\nWay before The Vintage Collection was even a thought, Hasbro used the vintage AT-AT Commander on the packaging art of the Assault On Hoth TRU exclusive Battle Packs set. Anything Episode V related is cool right? Well how do you feel about this one? Let us know in our Research Droids Reviews today by posting your voice in the discussion. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nHere is a quick update from http:\/\/www.bigbadtoystore.com about a slew of new preorders and more new arrivals.\nSDCC Exclusive Ralph McQuarrie Stormtrooper Mini Bust!\nGentle Giant, Ltd. has just put up for pre-order for all PGMs the SDCC Ralph McQuarrie Stormtrooper Mini Bust. You must be logged into your PGM account to see the images.\nThe Rebel Vs.Empire Battle Packs is another one of those \"rough start\" multipacks. We think we can prove why you don't need it, unless you're a one-of-everything collector (Darth Vader is unique here). Check it all out in Research Droids Reviews today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nSee The Evolution Of The Power Droid!\nOur good friend Jayson at Yakface.com has worked up a gorgeous comparison to show the differences of the various GONK droids already released in the line compared to the new TVC [Special Action Figure Set] Power Droid that is at the moment unofficially confirmed. Click through to check it out!\nNick May Responds To Your 200+ Comments Regarding His Essay!!!\nYou're Gonna Choke At How Cool This SDCC Mini Bust Is!\nThe awesome onslaught of San Diego Comic Con Exclusives is underway! Gentle Giant has just announced the first of their Star Wars exclusives, an amazing mini bust of Admiral Motti! Click through for the report!\nDroid Set: Special Action Figure Set Revealed!\nSandtroopers.com reveals the next Special Action Figure Set from TARGET. Check it out! It looks like they're using the 2006 TSC R5-D4! Let's hope they make him Episode IV specific with the red panels on the back and not Episode II specific \"without\" the red panels on the back!!!\nThe Jedi Vs. Separatists Battle Packs name doesn't make any sense. We challenge you to find a Separatist within it. But that challenge many never come to pass after you read this entry in our Research Droids Reviews because we do our best to explain to you why you don't need this. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nAttack Dropship Vs Assault Sub Mystery Revealed!\nRemember a few weeks ago when there was some confusion regarding the orders that were placed directly with Hasbro for the Republic Attack Dropship with Clone Pilot vs Republic Assault Sub with SCUBA Clone? Our order has arrived! Click through to find out what we actually got!\nIt's time for Odd Ball (Clone Pilot), a figure that's the topic in Research Droids Reviews today and a figure you MUST buy. You probably won't find him for the next 6 months, but maybe you can enjoy it vicariously through us for the time being. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nA Close Look At The Luke Skywalker (Lightsaber Construction) Variation!\nThanks to one of our avid readers, we presented to you the first visual of the newly documented Luke Skywalker (Lightsaber Construction) variation. You asked for more and we're giving it to you! Click through to see five close-up shots showing off the differences between the first version and the verison now showing up in the Grand Moff Tarkin wave.\nTarkin Wave And Luke Variant Found At Philly FYE!\nThe Tarkin wave including the Luke Deleted Scenes variant was just found today at the Philadelphia area FYE! Click through for the proof! Happy Father's Day to all of our readers!\nJTA Facebook Contest Coming To A Close Shortly!\nWe're down to the last 12 hours! We have gotten some pretty clever captions on our Facebook photo, but can you be more clever? JTA staff member Matthew Frey has donated the \"soon to be rare\" 2012 TVC Clone Trooper (501st Legion) figure for a contest that is ending soon here at Jedi Temple Archives! Click through to find out how to win this cool Hasbro action figure for your collection. You only have a little bit of time left! Contest ends midnight today!!\nHere is the other 50%! Did you know that Hasbro blamed the performance of The Max Rebo Band (Jabba's Palace Musicians) set as 50% of the reason why they won't update the characters in this musical group? Maddening, isn't it?! Learn about the details of this set in our Research Droids Reviews today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nMore Is Revealed Of Commander Fox!\nSideshow reveals a little bit more of their SDCC exclusive one-sixth Commaner Fox figure!\nIs It Possible That It Could Get Worse Than This?\nWe have been a little bit down in the dumps lately about the state of collecting, but can you blame us? This is representative of nearly all 7 Walmarts, 5 TARGETs, 6 Kmarts and 5 TRUs in the Philly area. Just a few are overloaded on Movie Heroes wave 1. If there is all of this space available for TVC, why isn't it falling off of the pegs? Even stores that have an overabundance of The Phantom Menace wave have room for more TVC. I am going to go cry in a fetal position now. Be back in a bit.\nAdmiral Motti Mini Bust Revealed!\nOur friends at Sandtroopers.com have revealed an Admiral Motti Mini Bust. Click through to see a bigger picture!\nJust as our friends at Yakface.com have reported, the newly named Republic Clone Troopers Battle Packs set is showing up at brick and mortar. Unbelievable that they (whoever \"they\" are) can get this into stores, but not the last 4 waves of The Vintage Collection (and everything else MIA).\nDid you know that Hasbro blamed the performance of The Max Rebo Band (Jabba's Palace Entertainers) set as 50% of the reason why they won't update the characters in this musical group? Tomorrow you'll see the other 50% of the reason. Maddening, isn't it?! Learn about the details of this set in our Research Droids Reviews today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nHasbro CANCELS The Mandalorian Assault Transport - EE Kills Pre-orders!\nThe overall tone today is extremely dreary, but more bad news is coming right now. Entertainment Earth has just cancelled pre-orders for the Mandalorian Assault Transport. But there are still a few \"reasonably priced\" ones left on eBay. Click HERE to see if you can score one!\nUPDATE: Well, news has spread fast. The only ones left on eBay now are very expensive. Best of luck to you all as new auctions make their way to eBay for this product.\nVintage-Style Wave 13 (the set has Starkiller, Shae Vizla, Oddball, General Tarkin, and Nikto) has landed at Brian's Toys. The case came packed with Darth Malgus, so there may be a few of him in stock as well.\nA Darth Vader Legend Edition Helmet from eFX has also been acquired. This Helmet is limited to 250, and differs from it's Limited Edition counterpart because it retains the dings and scrapes from the original movie helmet prop. The Limited Edition has those defects removed. This piece was made to be rare and is currently home to about 249 collectors.\nBrian's Toys has also been receiving large shipments of older discontinued Star Wars toys from action figures to vehicles to even Master Replicas! Visit the site and check out as many sections as you can, because it seems like every action figure imaginable has been restocked! Click through to http:\/\/www.brianstoys.com or go to http:\/\/www.brianstoys.com\/nl for the newsletter.\nThe young man who stole our hearts in the VW commercial as \"tiny toons\" Darth Vader had open heart surgey. Click through for his story at People Magazine!\nAll of us have been increasingly deflated over the horrible attention given to The Vintage Collection this year by retail and Hasbro. We're at a loss at who we should be pointing our fingers at anymore to be honest. The greatest packaging and collection of all time is being so incredibly mistreated, it's become egregious at this point. We have called you to action, our readers who post comments have echoed our sentiments exactly, and now JTA reader William W. has taken it upon himself to reach out to contact Hasbro directly. The answers he receives are baffling. We don't claim to know what the answers are anymore, but we do know that squeaky wheels get the grease. (That's how the Gamorrean Guard and Wedge Antilles figures got reshipped.) But I personally find it completely backwards and infuriating that I have purchased the last 15 or so latest TVC figures from the UK with free shipping. Um, Hasbro, do you NOT see a problem with this? Click through to read all about William's experience. He even received a letter from Hasbro on their letterhead. Check it out.\nHasbro just wasn't entirely prepared for The Clone Wars toy line, so they \"padded\" the line with a few pieces of \"related\" Clone Wars merchandise, like Anakin Skywalker's Modified Jedi Starfighter, originally from Tartakovsky's Clone Wars micro-series. See the Azure Angel in full detail in our Research Droids Reviews and let us know how well you think the vehicle fits into the line. And has it aged well? Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nSideshow Collectibles teases CHEWBACCA, of unknown scale! Click through to check it out on their Facebook page, a long overdue character in their lineup! Click through to check out the tease!\nSideshow Collectibles basically lets the cat out of the bag by revealing the left leg of Commander Fox on their Facebook page, a great choice for a SDCC exclusive. Click through to check out the tease!\nToys-R-Us may be the place to hit right now as we found 2 exciting things going on there! Click through for the report!\nJTA Sideshow Collectibles Commander Cody Winner Announced!\nWe have a winner for our Sideshow Collectibles Commander Cody giveaway contest! Click through to find out if you won!\nFrank Diorio And His \"Return To Tatooine\" CVI Diorama! Help Needed!\nJTA wants our readers to support Frank Diorio and his incredible Return to Tatooine diorama, the biggest yet, coming to Celebration VI! Find out how you can help out!\nACME Archives Announces SDCC Exclusive Biker Scout Character Key!\nACME Archives comes out of the woodwork to announce a few San Diego Comic Con exclusives coming collectors' way including the brilliant Biker Scout character key! Click through to see what's happening at our favorite artwork company!\nPremium Format Captain Rex Is Up For Pre-order!\nSideshow Collectibles' Premium Format Captain Rex is [still] up for pre-order!\nSideshow Collectibles Officially Reveals 1:6 Airborne Trooper!\nWalmart continues clearancing their Star Wars section and the latest to get marked down are the Titanium Series 3-packs! Click through for the report!\nThis Is Jedi Temple Archives' 10,000th News Post!\nThis is Jedi Temple Archives' 10,000th news post! We just wanted to take a moment to bring your attention to this landmark we've reached and to thank you for being part of this fun and exciting journey. We so very thankful to you for checking in with us daily and we want you to know how integral you are to our site. And to those of you who have been rocking it out in our comments module we give extra special thanks to you! Keep the fire burning!\nLuke Skywalker (Lightsaber Construction) Variation!!!\nEagle-eyed collectors should be able to see the running change in the thumbnail below immediately. But if you can't, click through to see the latest in Hasbro's ling list of neverending variations in The Vintage Collection! Special thanks to JTA reader Matthew from KY for brining this to our attention!\nKithaba Is Killing You Softly With His Aftermarket Value - Don't Give In!\nKithaba with the red bandana hits eBay. Are you ready to laugh?! People, this will be available in future case assortments with the red bandana. Why are you freaking out?\nHere is a quick update from http:\/\/www.bigbadtoystore.com about more cool new preorders as well as some nice new arrivals.\nHow many Trade Federation Armored Assault Tank (AAT) vehicles do you think we'll get from Hasbro before the line ends? Maybe we shouldn't ask that question. See The Clone Wars version, which is very similar to the one released in the The Battle Of Christophsis Battle Packs set. Check it out in today's Research Droids Reviews and see why you may need it! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nDarth Malgus In Stock at Amazon.com... This time He's $12.99!\nDarth Malgus is back on Amazon.com and this time for $12.99! If you missed out the first time then don't wait! A big thanks to JTA reader Chad Esquivel for the info!\nUPDATE: SOLD OUT at the $12.99 price already!\nJTA Facebook Contest! Win A 501st!\nJTA staff member Matthew Frey has donated the \"soon to be rare\" 2012 TVC Clone Trooper (501st Legion) figure for a contest starting NOW right here at Jedi Temple Archives! Click through to find out how to win this cool Hasbro action figure for your collection.\nThe \"big\" Battle Packs of 2008 weren't nearly as cool as the 2007 30 (77-07) sets, but The Battle Of Christophsis has some good points to it. If you're curious what they are, then hop on over into our Research Droids Reviews and find out! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\n$16.53 UK Darth Malgus Figures Arrive In USA!\nHere is a not so wonderful Battle Packs set from the 30 (77-07) line. The STAP Attack was a desperate attemot to reuse 1998 parts and make them \"new: again, but screen inaccurate. We challenge you to find the characters in this Battle Packs set in Episode II. You'll find some, but not others. Check it out in today's Research Droids Reviews. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nCheck Out The Season 5 The Clone Wars Trailer Exclusively At EW.com!\nIt's been officially released as an exclusive at EntertainmentWeekly.com! The Season 5 The Clone Wars trailer is available to view! Check it out!\nVintage collectors know all about the infamous double telescoping, or DT, versions of Luke Skywalker, Ben Kenobi and Darth Vader. While Luke is the most common of the three, Darth Vader and Ben Kenobi can be found on eBay from time to time as well. We've found a few interesting auctions if you are curious about these super-rare variations. Currently, there is a nice loose Darth Vader available, as well as quite a few Luke Skywalkers (both AFA and non-AFA). You can also marvel at the crazy price-tag a carded DT Luke Skywalker commands. And, if you are interested in filling these holes in your collection you can find some very nice reproduction sabers too. We don't normally condone reproduction accessories, but in the case of the highly expensive DT sabers, we make an exception. Check out all the auctions below, and click here for a search of all things DT! As usual, buyer beware if there is not a COA with high-priced items like these!\nAn In-Depth Look At The MTT Out Of The Box!\nOK, here we go! The teases are over! We have got a great \"opened box\" look for you of the MTT as well as great commentary on the details of this vehicle! JTA reader Manny M. comes through big time for us so we can share the contents of the box with you! Click through for a complete report!\nThe 2006-2008 spanning 30 (77-07) line makes us feel homesick, if that's possible. You see, it was an era we loved living in and Hasbro really showed us the stuff they were made of with brilliant multipacks, an exciting (and available) basic figure line, and many other special \"Star Wars\" things. In today's Research Droids Reviews we look at one fine example of Hasbro's clever ingenuity as we talk about the Walmart exclusive Treachery On Saleucami Battle Packs set. See why it has remained relatively ageless even today. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nIt's All About The DS .. And We Don't Mean Death Star!\nWe got a case of figures direct from the big H and it included a nice mixture of Waves 12 & 13. (We won't mention the Qui-Gon that was part of the assortment yet). Some very interesting things were going on in this case. Especially with the date stamps! Click through for the report!\nUPDATE: First Look! MTT Preliminary Loose Shots And Interesting Assembly!\nWe have a small update for you! Our friends in the Philippines, Manny and Earl have shot some rough preliminary photos of the MTT out of the box to whet your whistle. Better photos are coming, but click through to get a cool sneak peek! They also clue us in on an interesting assembly of parts!\nUPDATE: A new video has been added to this report showing off the MTT and its features!\nThis week sees a nice selection of vintage R2-D2 figures from Popy to Habert to Takara! Also, check out some of the price reductions on select vintage figures.\nA PROTOTYPE Micro Collection Millennium Falcon is also for sale. This item will be a must-have for collectors looking for the rarest Star Wars collectibles.\nFor more Star Wars including recent restocks and a new Sideshow \"Hunt for the Jedi\" Diorama visit Brian's Toys.\nIf you have been reading our Research Droids Reviews for a while now, you probably remember us taking a look at The Legacy Collection's Jedi Training On Dagobah Battle Packs set and noting how different it was from its original release. Well, today we're looking at that first \"phase I\" version of it from the 30 (77-07) line, which is quite different. Heck, it even contains a completely different figure! Click through to update yourself on the TAC Jedi Training On Dagobah Battle Packs set this time, and look for links where you can see a side by side comparison that highlights the differences between the 2007 TAC and 2008 TLC versions! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nBreaking News! MTT Hits Retail! Check Out This First Report!\nWe have the first MTT store report to share with you! But unfortunately it's not in the USA. Our friends in Manila, Philippines have alerted us that the Trade Federation MTT (Multi Troop Transport) has arrived and they have sent us a few images, including a comparison shot so you can see the box size differences to other larger vehicles, for your vieweing pleasure. Click through to check it out! Special thanks to JTA readers Earl and Manny for sharing this report with us!\nHere is one of our most favorite Battle Packs Hasbro ever created, the Jedi Vs. Sith set. We won't be able to explain why we are able to look the other way when it comes to figures that come without super articulation and also have action features incorporated int them, but Research Droids Reviews does its best to explain it. Besides, this is the only way to get the awesome Tartakovsky \"Yavin IV\" interpretation of Anakin Skywalker. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nThe Toy Palace has listed new merchandise on their website and they have a larger picture of the Scout Walker (AT-ST) to view. Click through to see it or click on the thumb to be taken to their website.\nSee A Truly Behind The Scenes Look At Return Of The Jedi!\nDid you know that there is an unspoken variation to the 2005 Revenge Of The Sith Clone Attack On Coruscant Battle Packs set? Did you know it involved a completely different fifth figure. If you don't know about this, check out our Research Droids Reviews to find out! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nSideshow Collectibles Processing Commander Bacara And Boba Fett Soon!\nSideshow Collectibles have sent out notices for both their one-sixth Clone Commander Bacara (about 2 weeks) and their highly anticipated one-sixth Boba Fett (about 4 weeks). Better get your finances in order so there are no delays in your deliveries.\nThe Sideshow Collectibles Hunt for the Jedi diorama is up for pre-order. Click here to check it out!\nThis is an exclusive post for JTA moderator Bombadgungan. If you're not Bombadgungan, this post isn't for you. Please do not click through if you are not Bombadgungan!\nHere's another \"what the heck\" moment for you! Every Kmart in our area (we don't know about yours) has put The Vintage Collection B-wing Fighter Vehicle back up to $29.99. Whatever their reason was for marking it as low as $14.99 was short-lived. Let us know if you see something differently at your stores (if there are any left that is)! The shelf tags in our area were never changed to $29.99, but after we scanned this sample, it rang up for $29.99. Click through for more!\nThere's a lot of buzz about a \"new\" TVC AT-AT, but we've just noticed something in our local TRU stores that you may find a bit interesting! Click through for more!\nHere is a quick update from http:\/\/www.bigbadtoystore.com about more great new preorders, new arrivals, and a new sale.\nPrices Reduced - 4700 items!\nWe've reduced prices 15% to 20% on over 4700 older items! We have 40 featured items and the 4700 other items are sorted by discount on our sale menu. You can also just browse your favorite menus and sales prices will be displayed there as well.\nWell, we have another NEW product for you to read about in our Research Droids Reviews today! Another multipack sure to remain controversial for years to come, see if the Duel On Naboo is a Battle Packs set for you. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\n2008's TLC Training On The Falcon Battle Packs: OMG!\nIf you can rewind your mind back to 2008 for a minute, do you remember the Training On The Falcon Battle Packs set? It was created to help \"pad\" the stunningly gorgeous \"Big\" Millennium Falcon vehicle that year. While this set was fun from a certain point of view, it really didn't contain the best Hasbro action figures of all time. However, click through for an interesting tidbit we recently discovered with this set.\nCool Star Wars stuff (both Hasbro and non-Hasbro) continues to be found at discount chain ROSS. Clone_Fanatic writes in with an update! Click through to read his report!\nWeekly Research Droids Review Recap: \"A Week Of Wookiees!\"\nDarth Malgus In Stock At Amazon.com For $16.50 With FREE Shipping!\nHurry! This is going to disappear quicker than 0.5 past lightspeed! Darth Malgus is in stock at Amazon.com with FREE shipping for $16.50! Special thanks to Scott from Tulsa for the urgent tip!\nUPDATE: Well, it could only be good for a little while, right? This Amazon.com seller has just DOUBLED the price of Darth Malgus. Oh well. Congratulations if you got your order in before this happened!\nWell, we FINALLY have something NEW for you to read about in our Research Droids Reviews today! Sure to remain controversial for years to come, see how the Bespin Battle makes us smile. Hopefully it will make you smile too. Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nQ&A With JTA: Our Thirty-Seventh Column!\nWe've still been working hard behind the scenes to update our current guides. We want to get all of our Enhanced guides out of the Flash format and into the new layout. We just completed work on the The Clone Wars guide. Stop by and check it out. They have also been added to our Visual Guide Search Engine database for easy seach-and-find functionality! Just search for TCW to find all of our updated The Clone Wars guides!\nLooking to sell part (or all) of your collection? Want to save time and money? Save yourself the hassle and contact Brian's Toys. They will pay top dollar for your toys and collectibles. They have been buying back toys for over 15 years and have recently made big changes to their program to ensure you get more value for your collection than ever before.\nNoted highlights: Over 1,500 buylist prices increased, 25% extra trade-in credit, consignment opportunities, and free shipping on the first package shipped (certain restrictions apply).\nFor more details, check out the new buylist section at http:\/\/www.brianstoys.com\/buying.\nSideshow Collectibles Teases One-Sixth Scale Snowtrooper!\nYes! This is an answer to prayer! Sideshow Collectibles \"teases\" a one-sixth scale Snowtrooper. Click through to check out more!\nWave 5 of The Vintage Collection is beginning to make its way to specialty shops now besides just eBay. Could online shops be next? JTA friend Darth Eddie shares some quick pics of his local find. Click through to check it out!\nJTA has reached the last entry in the Research Droids Reviews \"A Week Of Wookiees\" marathon! And we sort of saved the best for last. Let's look at one of the better Wookiees from the Revenge Of The Sith line, the Wookiee Heavy Gunner figure! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nJTA proudly continues in our Research Droids Reviews \"A Week Of Wookiees\"! We are looking at seven Wookiees from the Revenge Of The Sith line. Let's see how well they've held up. Next up is one of the better Wookiees from the Revenge Of The Sith line even if he is a Hasbro concept: the Wookiee Commando figure! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nCollectors should be scrambling for the new 12-Back vintage AFA figures over at Brian's Toys. Be one of the lucky few to get the extremely hard-to-find Han Solo 12-Back Large Head graded AFA 90! Other characters include Princess Leia, Death Squad Commander, Darth Vader, and more.\nSubmit a pre-order for the next Vintage-Style Wave 14 figures, which come in regular vintage-style and a new retro-style packaging. Or order the new Licensed Artwork from artists Steve Thomas and Christian Waggoner.\nFor more of the latest Star Wars collectibles, visit http:\/\/www.brianstoys.com\/nl.\nCurto Burns Podcast! Their Very First Show!\nWe've heard you missed them, so they're back! Dan Curto and James Burns are back in the saddle again with an all-new podcast, picking up right where they left off. Listen in to the first show at their new home, complete with the latest news and reviews. They check in with show regular guest Steve Sansweet from Rancho Obi-Wan to get the inside scoop on his latest action figure book, Justin LaSalata gets collectors excited for the Celebration VI exclusive collectibles, and Fanboys director Kyle Newman stops by for his feedback on The Vintage Collection Ben Quadinaros action figure he made possible; plus, find out how to get a signed figure in their first-ever contest! It's all part of the first official Curto Burns Collectors Cast!\nJTA proudly continues in our Research Droids Reviews \"A Week Of Wookiees\"! We are looking at seven Wookiees from the Revenge Of The Sith line. Let's see how well they've held up. Next up is the Wookiee Warrior [Brown] figure! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nAnother Reason We Love Sweden - Ikea's Affordable Glass Display Cases!\nWe all have tons of gorgeous figures we want to display but the cost of glass cases is typically expensive. Ikea of Sweden has an affordable option! Click through for the report! You may find both the price and prevalence of their use in our community quite shocking!\nGimme! Gimme! Gimme! A Variant Gammy!\nOK, before your blood pressure goes through the roof, please just relax and breathe. This isn't a huge breaking news story. It's not even really a big deal. But the \"reissued\" Gamorrean Guard from the 2012 assortments is technically \"different\" from the 2010 version. The ONLY difference we see is on its head sculpt. On the 2012 version, the portrait has an obvious sheen (almost a gloss) to it. The 2010 version does not and instead appears flat or matte. We think this tiny revision makes the pig guard look better. Their skin did appear a little slimy on screen, especially with all of that slobber! Our cameras can't really pick up the differences that well, but you can still see them a little bit nonetheless. It's more obvious looking at them with your own eyes in person. So, that being said, if you're still interested, please click through to see the comparison!\nIt seems that there will be a lifelong dramatic relationship between collectors and Wedge Antilles. Right now, the reissue appears to be harder to find than the original figure. And if you can find it, there is no guarantee how it is going to arrive. Scott from Tulsa gives us a story you'll want to hear. (It's unreal the lack of respect this figure has received. Going forward Hasbro, just release figures as they are planned originally instead of pulling them out of the line last minute to tweak. The Wedge pack-in figure from the TARGET exclusive X-wing was fine. And we all would have had him in our collections 2 years ago. Click through for this story!\nEl Salvador Turns Back Time For Star Wars Collectors There!\nJTA reader Gemini from El Salvador gives us an update on the crazy clearance prices and the return of \"old\" hard to find product hitting his country. You may remember his earlier report, so now check out the latest scoop. Click through for this new report!\nJTA reader Joe M. writes in with mostly good news (and one bit of bad news), but It seems like the new figures are really starting to reach various parts of the country in much heavier numbers. Click through for his report!\nHas anyone else noticed the Galactic Battle Game beige stands coming in the new Star Wars stuff like the latest wave of The Clone Wars figures, or the new Battle Packs are now a different color? They've gone from a flat beige solid to a shiny, glimmery and marbled metallic look. The attached picture isn't the greatest, but we just wanted you to be aware that the stand are now looking differently all of a sudden. Special thanks to JTA reader Bailey for providing us with this quick snapshot.\nReady to win a free Commander Cody 1:6 scale figure? Click through for all the rules!\nJTA proudly continues in our Research Droids Reviews \"A Week Of Wookiees\"! We are looking at seven Wookiees from the Revenge Of The Sith line. Let's see how well they've held up. Next up is the Wookiee Warrior [Tan] figure! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nKathleen Kennedy will succeed George Lucas at Lucasfilm, Ltd. Click through for the details of the announcement.\nVintage Figure Reports Keep Pouring In!\nOn the heels of Mark's amazing report below .. I had been following up on the tip about FYE from our earlier story here and swung by FYE during a huge toy run in southeast Michigan today. Lucky I did! Click through for the second report of new vintage figures hitting the midwest!\n1:6 Scale Darth Vader With Removable Helmet Revealed!\nSideshow Collectibles reveals their one-sixth scale Darth Vader with removable helmet figure with Sebastian Shaw likeness! Click through to check it out up close!\nJTA reader Mark writes in with an incredible find in Iowa. Man, I am really starting to envy you mid-western people! Click through for his store report!\nIt's official folks! The fifth 2012 wave of The Vintage Collection has hit eBay!\nLucasArts reveals 1313! Click through to check it out Could this mean more action figures based on a video game? Probably!\nJTA proudly continues in our Research Droids Reviews \"A Week Of Wookiees\"! We are looking at seven Wookiees from the Revenge Of The Sith line. Let's see how well they've held up. Next up is the Sneak Preview Wookiee Warrior! Feel free to leave your own thoughts in our comments section, vote in the poll or even leave comments about it on our Facebook page. Click the banner below for the direct link to the review! As always, you can also send your own personal review requests via e-mail if you'd like to see something that we didn't cover yet.\nNew Star Wars Cases In Stock At HasbroToyShop.com!\nNew The Vintage Collection and The Clone Wars cases are in stock at HasbroToyShop.com as of this posting. Act fast if you want to secure them! TVC assortments with Nom Anor and the Gamorrean Guard and TCW assortments with SCUBA Ahsoka are available. Special thanks to JTA moderator The_Question for the head's up!\nOur friends at Galactic Hunter just reported Gentle Giant will be releasing a Jumbo Vintage Kenner Power \"Gonk\" Droid in 4th Quarter of 2012, as mentioned in the July issue of Star Wars Insider! This one is definitely a fan favorite and we're wondering if the legs will make that cool clicking sound!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- The main activity of our agency is related to the regulation and effective organisation of activities on alternative and renewable energy sources, while coordinating activities in this area and providing state control. We keep records and assist entrepreneurs in this area in the settlement of their activities, and the preparation of the necessary legal documents. We are the implementers of a number of pilot projects.\nThe main achievement of the year will be the activity of our solar station in the Garadagh district. The power plant is fully automated and controlled by the dispatcher. This year, the United Nations Convention on Climate Change is being implemented under the slogan \"3D\"; a number of global environmental organisations have already joined it, which means decarbonisation, decentralization and digitalization. We are on the monitor demonstrating the work of the station, presenting opportunities in the field of alternative energy. The stand also exhibits solar panels used at our stations and an experimental motor scooter with a refrigeration device for use in industry.\nI consider that the exhibition is, first of all, an important cultural event worthy of a visit. Thanks to the enormous work of the organisers and participants, who are eager to present the best achievements with great enthusiasm, everyone can take the opportunity to increase their information base, build partnerships and simply expand their vision and knowledge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Queensland Building and Construction Commission (QBCC) has used its new Supreme Court injunction powers against a repeat offender who they say flouted the state's building laws repeatedly.\nThe unlicensed individual was issued an injunction in Brisbane last week, preventing him from conducting building work.\nThe man may face imprisonment if he fails to abide by the order not to carry out any building work in Queensland again after already racking up fines to more than $27,000.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Albert Finney has died aged 82.\nAlbert Finney, who forged his reputation as one of the leading actors of Britain's early 60s new wave cinema, has died after a short illness, his family have said.\nIn 2011, Finney disclosed he was suffering from kidney cancer.\nThe actor grew up in Salford and quickly became one of Britain's most celebrated actors, cutting his chops during the 1960's, British cinema's most exciting period.\nFinney received five Oscar nominations, but never won, and refused a knighthood.\nHis family home was in Romney Street in Charlestown. His father, Albert senior, was a bookie and had a shop in the neighbouring Cromwell Road. The family moved to Weaste in 1941 after a wartime bomb damaged the family home.\nFollowing his studies at RADA his big film break came as 'angry young man' Arthur Seaton in Saturday Night and Sunday Morning. He went on to star in Tom Jones, Murder on the Orient Express (as Hercule Poirot), Erin Brockovich and Skyfall.\nDaniel Craig paid tribute to Finney saying 'the world has lost a giant'.\n\"I'm deeply saddened by the news of Albert Finney's passing. The world has lost a giant.\nBond producers Michael G Wilson and Barbara Broccoli said they were 'heartbroken at the loss' of Finney.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Vienna map is perfect for tourists and travelers. Why do you need a tourist map of Vienna? This map will help you get the most out of your trip . We've done all the hard work and found all the places you'll want to visit. The most famous Vienna attractions, the best things to do in Vienna, listed with notes and pictures. So, download this tourist attractions map of Vienna and you'll be all set for your trip. Enjoy Wien!\nIf you enjoy walking, I highly recommend that you walk around as much as possible. Chart a course from blue pin to blue pin. The blue pins represent attractions in Vienna. I've focused on art museums, parks, palaces, churches and street markets. There's even a chocolate museum marked! You can cover all these blue pins in a day (if you don't go inside the museums), or you can take your time over several days slowly working your way through the city. While you're walking, you should make time to stop at a green pin, or cafe. Cafes in Vienna are great places to relax, enjoy a leisurely cup of coffee and try Sahertorte. After a long day of walking, check out one of Vienna's rooftop bars. The rooftop bars are marked with purple pins. Try to go just before sunset to ensure you'll have ideal lighting for amazing photos. If you have extra time, I also recommend shopping. I've marked some of my favorite stores with orange pins.\nIf this map seems overwhelming\u2026. First, learn how to use google maps when traveling.\nThe main Vienna attractions include gardens, palaces and museums. Vienna is a city for the people. You'll notice lots of manicured green space, huge pedestrian walkways and plenty of water bottle refill stations. The main greenspace you'll want to visit is Volksgarten, but check out Stadtpark if you have time. Also, the Belvedere Palace & Museum is one of Vienna's main tourist attractions. You'll find impressive gardens spaces with statues and fountains on palace grounds. Some of the most impressive palaces include Schonbrunn Palace, Hofburg Palace and Belvedere Palace. These are all excellent examples of baroque architecture.\nIf you're an art lover, you won't want to miss Vienna's impressive museums. Specifically, the Belvedere Palace Museum, the Kunsthistorisches Museum, Albertina and the Leopold have extensive fine art exhibitions.\nRooftop bars are great attractions in any city, since they allow you to get a fresh vantage point. Our favorite rooftop bar in Vienna was Atmosphere. At Atmosphere the service, food and drinks were top notch. We also visited SKY in Vienna's downtown shopping district. I always recommend trying to make reservations ahead of time for rooftop bars. Even if the rooftop bar is more casual, they will give priority tables (with better views) to those who book ahead.\nIn this world-class city, you'll also find great shopping opportunities. Start at the northernmost end of Mariahilfer street (at Museum Quarter), and follow it down (southwest). You'll hit all the best stores. You can find Mango, Pull & Bear and Bershka on Mariahilfer street. These are some of my favorite European brands. For more information about the labels, price ranges and styles offered, check out Where to Shop in Europe.\nIf you're looking for more inspiration on planning your trip to Vienna, you'll want to read 10 Essential Experiences and Things to do in Vienna. Beyond seeing the sights and taking selfies, I've listed some of the top cultural experiences that you can have in Wien.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Town of Jupiter social media accounts, where possible, as well as the Town website, will have the following disclaimer posted: \"Comments expressed on this page do not necessarily reflect the opinions and position of the Jupiter Town Council, its individual members, or its administrators, officers or employees. The Town of Jupiter reserves the right to remove or hide a comment without prior notification.\" Content that could be removed or hidden include but is not limited to; spam, including links to other websites; obscene language or sexual content, comments that target or disparage any ethnic, racial or religious groups; personally identifiable medical information; comments that threaten or defame any person or organization; comments that promote illegal activity; advertisements for commercial services or products; or comments that infringe on copyrights or trademarks. The Town does not endorse any content, viewpoint, product or service linked from its social media sites and shall not be held liable for any losses caused by reliance on the accuracy, reliability or timeliness of shared information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your Association, ADFIAP, in cooperation with the Geneva-based U.N. agency, the International Trade Centre (ITC) and the Institute of International Trade and Development (ITD) of Thailand, is proposing the establishment of a two-tiered \"Pilot SME Finance and Knowledge Center\" (Regional and National Centers) in the Asia Pacific Region with the overall objective of facilitating SMEs' access to finance from financial institutions such as DFIs and commercial banks. The proposed Center is envisioned to be a leading and empowering Center of Excellence for SMEs.\nAs an ADFIAP member institution, may we request you to assist us in generating baseline information specific to your country and institution by accomplishing this questionnaire. Your cooperation will help us finalize the project document to be used for grant application.\nIf Yes, please indicate the name of the Center\/s and their contact details or website.\n2. If Yes, what are the services offered by the Center?\n3. Who Operates the Center?\n4. Who finances the operations of the center?\n5. Does the Center have existing partnership with financing institutions to assist SMEs in the its access to finance?\n6. Is the Center staffed by regular manpower who are technically-equipped to manage the Center?\n8. Are there existing Service Providers (government or privately-owned agencies, training institutes or universities) which provide support services to SMEs on a stand-alone basis?\n10. Are there existing training institutes or universities which provide the following courses for SMEs?\n11. Are there existing laws\/policies governing SMEs in your country?\nWe appreciate receiving the accomplished questionnaire on or before March 22, 2010. Your cooperation in this regard will be highly- appreciated.\nThank you for supporting the initiatives of your Association.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"After hectic few weeks on the road I am now back in sunny California, and we had a wonderful Ubuntu Developer Summit in Orlando, Florida. Thanks to everyone who made the event such a success. It was so great to meet so many old friends and to meet the may new friends who experienced UDS for the first time. Looking forward to working with you in the Natty cycle!\nBy and large the commentary since the announcement has reflected the facts, but some articles have got things a little skewed. My view here is simple: I think everyone should be welcome to have their own view on Unity, either supportive or cynical, under the premise that the basis of the discussion reflects the facts and not misinformation.\nUbuntu is not ditching\/forking GNOME \u2013 Unity is a shell for GNOME, but not GNOME shell. Ubuntu is still a GNOME platform. 11.04 will ship all the components required for GNOME application authors to have their software run out of the box in Ubuntu, and we will still ship all the GNOME apps you know and love in Ubuntu 11.04. The only change is that Unity will be the default shell. Likewise, this is not a fork: we are not diverging away from GNOME, just producing a different shell in much the same way others have (e.g. Meego). It is just a different porthole looking at the awesome GNOME platform.\nUnity is the 3D experience, Classic GNOME is the 2D interface \u2013 if your graphics hardware cannot sufficiently run Unity, Ubuntu will present the 2D experience which is the two-panel GNOME desktop we currently ship, complete with all the Ayatana improvements such as application indicators, global menu, system indicators etc.\nAccessibility is a top priority \u2013 currently Unity has rather poor accessibility support (read: basically non-existent) and accessibility is a core ethos in Ubuntu. As such, Luke Yelavich has been assigned to the DX team (the folks who code Unity) to work on accessibility support in Unity, and he will be supported by Gary Lasker. In addition to this, we had a great set of meetings at UDS with the Ubuntu Accessibility community team and they are synced up with Luke to help support this work with testing, bug triage, and outreach. Great accessibility is a requirement for Unity if it is to ship in 11.04: let's pull together to make this happen.\nPerformance is being resolved; porting to Compiz \u2013 some of you have experienced poor Unity performance on certain netbooks due to their graphics cards. This has been identified is a core issue to resolve and it is largely due to performance issues in clutter and mutter (the graphics backend for Unity). To resolve this Jason has started work on a Compiz port, and early results (i.e. less than a week of hacking!) have already seen significant performance improvements. He demoed it to some folks on hardware that received the most performance bug reports, and the performance was incredible snappy and slick; it exhibited the kind of responsiveness and animation that you should expect from a high quality experience.\nQuality is a top priority \u2013 Neil and the rest of the Unity team worked off their socks to get Unity read for the 10.10 netbook edition, and despite their best efforts, there were some quality issues highlighted and bug reported. The team believes they have bug reports for the majority of issues, and this cycle they are spending most of their time focusing on resolving these quality issues so as to deliver a top-notch Unity experience. In this cycle I am also going to leading my team to help build community support and contributions around quality too, particularly around highlighting areas in which the community can help resolve and fix issues.\nUnity is an Open Source project \u2013 Unity is Free Software and a full Open Source project in which we are going to rely on the community to help make this rock. We are looking for help with design, documentation, translations, development, and more! Curious to see how you can help? See this page for more!\nI think those are the main points for clarification. Many thanks for all those who have been helping to ensure the correct information and facts have been clarified in the various commentary happening around the tubes.\nDo you have more questions? Are there some things I have not covered here or want more detail on? Be sure to join my live video Q+A session in which I will answer all your questions! Join us on Wed 3rd Nov 2010 at 11am Pacific \/ 2pm Eastern \/ 6pm UTC at this location to get involved. See you then!\nMark Shuttleworth claims that Canonical tried to work with GNOME Shell team on design ideas but could not resolve differences. Can you point out to what exactly Canonical tried by references to bug reports or mailing list discussions?\nSeems to me like they want different things. Canonical wanted things GNOME didn't want to adopt \u2013 like Global menus, Zeitgeist and Notify OSD.\nIf they do not want Contributors Agreement.\nlibzeitgeist, dbusmenu, notify-osd are on the list. Although I don't think Canonical will make them closed source.\nlibzeitgeist is just a library written by canonical that wraps around zeitgeist! zeitgeist however is not a canonical project.\nIs Meego a fork of Gnome? If it is, did it cause a commotion on Gnome as Unity did? Is Canonical doing something wrong by developing their own ideas? Why are some people so mad at Canonical for giving Ubuntu users what they want? Have you heard all the criticism towards Gnome Shell, if so, what are you (or they) doing to make it better? In free software we can modify and redistribute code, isn't what Gnome stands for?\nThank's for reading, you don't have to answer.\nSure the MeeGo UI is a fork of GNOME. What else whould it be? You think you can run the normal GNOME desktop on a mobile phone?\nI should've added that yes, it's a fork, and no, nobody got upset because you can't run the normal 2-panel GNOME desktop on a mobile phone.\nWhat did make everyone upset about Maemo however is that initially they didn't contribute much back to upstream \u2013 they forked many libraries, adding bugfixes and performance improvements and features but did nothing to get those back upstream. Once they got a lot of criticism about that they changed their ways and MeeGo (the successor) now has a strict policy of upstreaming everything.\nMoreover \u2013 no evil copyright assignments.\nSo yes, the situation is rather similar, except that Unity 'competes' directly with the normal GNOME UI. I personally have little problem with that but I can imagine the developers working on the current GNOME Shell don't like this.\nMoreover, you say Unity is FOSS \u2013 well, technically it is, but you either give all of your code entirely to Canonical OR you fork it \u2013 \"getting involved\" is quite limited by the code attribution thing. But considering all the negative feedback on that I guess they are aware of it and probably working on it by now. At least I hope so.\nCopyright assignments is indeed the one critical point Canonical needs to address to dispel any doubts and suspicions.\nLooking forward to the results of Project Harmony.\nA Desktop Environment is AFAIK made of 4 components that are integrated with each other: a Window Manager, a Graphical Shell, Applications, and a development platform\/shared technologies\/toolkits. Ubuntu first forked the DE known as GNOME when they added Compiz to the default install. Now they are merely widening the gap.\nAs far as forking the existing software that is a part of the GNOME project, Canonical\/Ubuntu has done no such thing. This is no different than any other change in the default app selection, and doesn't fragment the platform, thus no harm, only healthy competition.\nI hope Unity will be stable and rock until the Natty release.\nBut what I don't get: if you think Mutter has performance issues, why not fix them instead of hopping technologies? This would be to the benefit of other projects such as Gnome Shell and MeeGo Netbook as well.\nAnd what is the story behind Copyright Assignment?\nCanonical claims it doesn't have the resources to fix Mutter performance issues but afaik, they haven't reported any bugs upstream and fixing up Unity for the desktop involves about a dozen members from Canonical working full time on related things. So it does appear Canonical is using copyright assignment to control the codebase and using performance problems and design differences as posturing.\nHi Neo, We spent quite some time last cycle working around Mutter issues (using tricks to get around the constant repainting) and also trying to fix fundamental Clutter performance issues (http:\/\/bugzilla.clutter-project.org\/show_bug.cgi?id=2237).\nUnity stresses Mutter differently than what GNOME Shell does and we would have been unable to make such large changes to Mutter\/Clutter while not breaking GNOME Shell AND delivering something for Natty.\nWith that in mind, we looked at Compiz again and the re-write had some excellent features and most people know how well it performs. The community have been very accepting and we've made good progress so far. I'm sorry that you feel we should have stuck with Mutter, but I don't think there is anything wrong with choosing Compiz, it has humbly served the GNOME community for years, making the 2D desktop absolutely rock and I'm proud to say that we are using it as a basis for Unity.\nFinally, I wouldn't hesitate for a second on recommending Clutter for applications on the desktop, it's an excellent toolkit and has a great API. We just feel that we need something more low-level for the window-manager\/compositor\/shell, and that's the route we are taking.\nI'm curious, what's so different in unity than gnome-shell\/moblin seem to work fine with mutter?\nWe have more things on the screen as default, a bunch of animations in the launcher as well as trying to use shaders. The first two are more of a culprit than anything else, but they are non-negotiable for the user experience.\nSo, we just end up painting a whole lot more than Shell every frame and this doesn't give great battery life on some netbooks, for instance, as well as some other issues.\nI am not particularly bothered whether Ubuntu uses Mutter or Compiz, but I am somewhat puzzled by the performance arguments explaining the switch. In all the benchmarks I have seen Mutter and Compiz compositing performance were not far off, certainly not far enough to prefer either over the other because of it. In my experience significant performance degradation in Mutter more often than not comes down to some poorly written code (usually by myself) in the Mutter plugin, and I don't find the picking \/ many things on screen as a particularly convincing an explanation for any serious performance hit without seeing some data that can back it up \u2014 if you have such data, it would be really appreciated if you could dump it into the clutter\/mutter bugzillas for someone to look into.\nHey tf (long time no speak :). It really depends on what card + drivers your using. On nvidia it's pretty good (though still not as good as Compiz, especially if you have a large monitor). On Intel and Ati, especially on netbooks, we just weren't seeing good performance or power usage at all.\nAs I mentioned, I just think we use it differently than Shell and have different expectations. For what we use it for, Compiz works better for us and for delivering on the Desktop (with large displays and larger multi-monitor setups), we're more comfortable with Compiz.\nRe: the pick. Again, it depends on how you were using it and how much is being painted on the screen. We could easily get it up to 40-50% CPU on Intel while moving mouse due to how much we had on screen, with 6-7% at idle with no animations or background processes. It's also just slow doing glReadPixels every mouse move. I've raised this before on #clutter and I think that even Litl are using our geometry-picking patch (or maybe the other one that was on the bugzilla). Regards to data, a lot of it is in bugs and through getting users to report, but I'll try and get into a form that would be useful to you, ideally with a test case.\nI think it's important to remember that we've also moved because we want to have much more control over the rendering than what is possible with Clutter. Especially with the enhanced fallbacks and no-shader, ARB shader and GLSL shader capabilities, to try and get it running on as many systems as possible.\nIts more about what features mutter has that compiz has. Mutter is only new really and isnt at all as feature complete or stable as compiz. The problems in 10.10 with mutter if you have tried unity are really obvious and those issues arent going to get fixed in 6 months for the next ubuntu release cycle.\nOk, my expectations are high \ud83d\ude00 Though I'm very unsure which one of the 3 desktops (Classic, Shell, Unity) I will choose in the end. And I don't know which one I should recommend to my father w.r.t. long term continuity.\nI thought the same when Gnome started developing Mutter. Why not good aold Compiz?\nMutter is just an advancement of good old Metacity. So they did not switch technology.\nUnity looks great. I would love to see it in Debian Sid.\nDoes this mean that desktop unity will use the global menu by default? Will there be an option to turn off global menu?\nI don't feel really strongly about the default, but I do think it should be simple to have unity with menus on the windows.\nI think that refers only to the netbook, as Mr. Shuttleworth said that it doesn't make sense to have Global Menu on a large display like that of a desktop.\nawesome!! i loved unity interface\u2026 but some of my pc's dont have gpu's or any sort of stuff\u2026so gnome will be working well and good in 11.04 desktop edition ryt!??!\nI don't believe that gnome-shell works without a GPU. I don't know about unity.\nBoth Gnome and Ubuntu say that they want to support classic panels as fallback.\nDo you have a reference to a webpage where the Gnome project promises to keep maintaining the classic menus?\nI thought that they were going to drop the classic menus as soon as 3.0 was out, but I may be wrong.\nFor heaven's sake, read the Gnome Shell design papers to see where it is going rather than looking at screenshots or listening to hearsay.\nThen what will be the difference between the desktop version and the netbook version? Or will the be one and the same>?\nThe desktop version will use the same underlying technology but have a different user interface, one which is more suitable for the desktop.\nthank you for your words. I am using Unity since the first versions on Lucid on my netbook (I run a second laptop with gnome desktop also). On my netbook I like this really easy to use interface but it let me enough space for my personality use of gnome. Ok, sometimes the performance is optimizable but you pointed it out before.\nThanks for the clarification Mr. Bacon. I am going to test and report bugs for Unity throughout Natty development. I really think you guys have the resources to make this rock and I am willing to do my part.\nLets make this a great release!\nYou're talking about shipping something that's significantly different to upstream gnome. Describing it as anything other than a fork is as incorrect as saying that Canonical perform upstream Gnome development by working on Ayatana. It just leaves you looking dishonest and prevents a rational discussion about the reasons for your desire to do so.\nThat's simply not true. A fork is taking a pre-existing code base and taking it in a different direction\u2026splitting the code path in two. Unity is not based on existing code, it is an entirely new project, and therefore not a fork.\nNow, if you want to accuse Unity of forking the direction of GNOME because it is != GNOME Shell, then there is more of an argument in that (I would still disagree, as I believe that multiple shells that expose the GNOME Platform are a good thing).\nDoesn't that describe exactly what you're doing? If I got it right, you're taking the Gnome code base, keeping the basic stuff, throwing away what makes Gnome the 3.0 version its developers want it to be (i.e. Gnome Shell) and grafting in your stuff (i.e. Unity, global menus, windicators, notify osd).\nIf that's not enough to qualify Natty's Gnome as a fork, I honestly don't know what could be.\nI've been using Ubuntu since the 5.04 times, but I'm very confused and not quite happy with the plans for 11.04. Also, it seems you're completely ignoring the opinion of people saying just that, as you did with the window buttons order.\nIf I have to make a zillion changes to a default Ubuntu install just to get to the desktop I want to use, I guess I'll have to look elsewhere.\nIf the apps must be patched for notify-osd, will the support for the gnome3 notifications be left in? So that the apps work as intended with both shells?\nWell Ubuntu has been using Firefox as its default browser for so long and none has said that it was a fork of Gnome. Why now when it is trying to use use another app (the shell) it will be fork??? All the distributions do this, it's normal.\nPS: I like Gnome 3, Gnome Shell, n' especially the notification of Gnome 3.\nFirefox is a desktop application, Gnome Shell is the desktop, it's the most important part of Gnome 3 for the user: can you spot the difference?\nAs Jono said, you can install the old desktop and Gnome Shell if you want. If you've used Gnome Shell you'll understand why Canonical is developing Unity as the new Ubuntu Shell.\nI think it's unfair to judge Shell based on early alpha states, while it is still undergoing a major UI overhaul. Also current Unity is not the same as we will see on the desktop. I'll wait for final releases before I judge.\nImagine it this way. If gnome were released as single tarball, you'd be replacing a chunk of code within it with a different piece of code and it'd obviously be a fork. Gnome is a well defined suite of software. Changing that definition in a way that the upstream developers don't agree with is a de-facto fork. The body of code that you're calling Gnome is significantly different to what upstream's calling Gnome.\nForking isn't bad. It's been an important part of the development of several projects. I have absolutely no idea why you're fixated on this idea of denying that you're forking Gnome rather than simply setting out the arguments that led to you making the decision to do so.\nI have to contest your notion of \"well defined\" in this context.\nThe very problem that makes conflicts like the one at hand hard to assess and resolve is that there's no clear definition of what precisely GNOME is supposed to be with regard to its downstream relationships. Is GNOME a kind of marketplace that distributors \/ vendors should pick and choose packages from according to their needs? Is it a complete, user-facing product that is supposed to be experienced by users with little or no modification? Is it both, or something in between, or something else?\nGiven the lack of clarity on what constitutes \"doing GNOME\", and what is \"breaking away from GNOME\", it's impossible for any stakeholder involved to substantiate any argument for or against modifying GNOME in whatever manner and to whatever extent.\nThe recent module set reorganization, and Stormy Peters's assessment of the Unity vs. Shell situation hint to some hope in this regard, but overall, the situation remains vague.\nGnome's two things \u2013 a platform and a desktop environment. Canonical are maintaining the platform but forking the desktop environment. Comparisons with Meego are misleading because Meego never claimed to be doing anything other than using some Gnome technologies.\nThen Ubuntu is a Debian Fork and Mint an Ubuntu fork etc.\nIsn't Gnome free software? Everyone has the right to modify and redistribute it. If they don't want that, they should close the source code.\nUbuntu was a Debian fork in its first releases; many packages are taken from Debian and adapted to Ubuntu, so you can call Ubuntu a \"Debian derivative\" or fork if you want.\nI agree with GNOME people in their rant, but I couldn't name Ubuntu+Unity a GNOME fork, maybe \"derivative\"; it seems like XFCE with GNOME technologies, a new desktop environment over GNOME foundations. nothing bad, if you see only at this.\nBut Canonical is doing a bad movement; with its intentions of \"copyrighting\" things, Canonical bosses show that they want to make closed derivatives of ope n source technology to earn money, but at the same time lying to the Community with the cool \"Ubuntu movement\" and all the \"free software\" marketing stuff. This isn't bad at all \u2013 Canonical needs to earn money, it's a business -, but I prefer to know the real motivations instead of lies, like Red Hat does without the \"coolness\" marketing.\nAnd excuses about Mutter and its deficiencies are lame; instead of working on Compiz \u2013 a good software now -, Canonical devs could work in Mutter for solving the youth problems of this WM. Again, copyrights, money and lies are behind this decission.\nSorry Jono, but I don't confide on Ubuntu people now. Good luck with your project.\nYes, Ubuntu's a Debian fork (I mention this in http:\/\/mjg59.livejournal.com\/45432.html \u2013 over 5 years ago). Like I said, I don't think forking is inherently a bad thing. What I'm criticising is the attempts to describe a fork as something other than a fork.\nNah, I agree to some extend with Jono that what they do is not fork the code of GNOME. I disagree that it's not a big deal, the message he tries to get across. It's probably correct to say they are forking the User Experience but will still be mostly building on a (slowly diverging) common codebase as GNOME. Or on GNOME technology or something.\nHowever, within GNOME voices come up saying the definition of a GNOME project should be a bit wider than \"has to be developed on GNOME infrastructure\" and move to something like \"has to abide by the GNOME HIG and be FOSS\". If that definition holds, this whole thing might be solved: Unity, as well as Ayatana etc etc WILL BE upstream projects.\nYou're taking a community project, sticking your own wrapper on top of it for product differentiation reasons, and locking that wrapper up in copyright assignment \u2013 . Not very community minded is it?\nThis is great, it has calmed my forking fears. Thanks Jono for clearing it up for me.\nit is fair to say that Unity is Open Source, however, claiming it to be \"Free Software\" (as coined by the FSF) is stretching things quite a bit too far due to the restrictive copyright assignments required to join the team (I do not dare say community here).\nSure, one can fork Unity as one pleases and make it a true Free Software project, but the current product as developped by Canonical is not in line with the spirit of the Free Software community (though I think Bradley Kuhn is wrong to call it Open Core, at least for now).\nUnity is (L)GPLv3. That puts it firmly in the \"Free Software as (coined by FSF)\" camp.\nImplying that the copyright assignment is somehow anti-Free Software is ironic since the FSF also requires copyright assignment on their projects.\nBy those standards FSF's GNU projects are not Free Software.\nThere is a fundamental difference between the way the FSF does copyright assignments and how Canonical and other proprietary companies do: The FSF's assignments guarantee that the project you contribute to will always be Free Software in the spirit of the GPL.\nCanonical does not guarantee this, they can dual-license Unity and sell it commercially under a proprietary license to OEMs whenever they like under their current contract.\nThis is the core of the issues people from the community have with this; if the Unity contributor agreement was identical to the FSF one, all would be great.\nReleasing code under a BSD-style license permits anyone to incorporate it into a proprietary project, but it's still free software. I think Canonical's copyright assignment policy is an incredibly bad idea and I think in the long run it'll do them more harm than good, but I don't think it's incompatible with free software.\nIt still doesn't prove your point. Unity is Free Software no matter how much you would like to prove otherwise.\nI'm not a big fan of Ubuntus direction.\nUnity matches all four point (Unity license: GNU GPLv3, GNU LGPLv3) so it's definitely Free Software.\nCopyright assignment has nothing to do with it. Even GNU does copyright assignment. Free Software only a matters about \"what can you do with your copy of the program\".\nWould Canonical commit resources to make sure that gnome-panel gets updated to work in GNOME 3? It looks it needs external (to GNOME) developers.\nI'm using Unity as I write this and on a netbook I think it makes sense, however I just don't see it working on a desktop. How does it handle a multi-monitor set up?\nFrom what we had discussed at the UDS, I think Unity on Desktop will have a good multi-monitor setup. However I'm not 100% on this and we'll have to how things go as development progresses.\nI really can't see unity working well on large screen desktops. It shares many of the features that make OS X increasingly painful to use on big screens, not least the unified menu bar.\nIt is being redesigned for the desktop.\nHmm that get involve page for Unity doesn't mention anything about having to sign the Canonical copyright assignment agreement. Is that an oversight? Or is Canonical dropping that requirement?\nif canonical really needs help in development then Canonical really needs to drop its copyright assignment policy on code contributions.\nNote that Canonical treats code contribution differently from other forms of contribution. Translators don't have to assign copyright. Launchpad asks translators to license their contributions as BSD but does not require assignment to Canonical.\nIt is an oversight. I will get that page updated.\nIf Canonical is serious about getting community to help with the development load the copyright assignment is a real and significant barrier to entry.\nJono, ask around, amongst those you trust outside the Canonical fenceline. This assignment policy kills quality code contribution from many quarters (even inside the Ubuntu community).\nThis really needs to be change. If you are serious about seeing community get involved in the development the current copyright policy must be set aside and something more respectful to external contributors must take it place.\nJef, you know that I am well aware of your views on the copyright assignment policy, and I know that you are well aware that I don't make the decisions on it \u2013 is it really worth continually repeating yourself to me about it?\nI don't have the authority to decide on it, it's content, or how it is applied; that is something the executive management team in Canonical decide.\nYou might not make the decisions on copyright assignment, but you could be its champion (if you agreed with the principle).\nSomeone has to take it to the executive management team and fight for it. I can't believe that you, as the Ubuntu Community Manager, would not have any say in it at all.\nIf you are going to continue to stump for community members to get involved.. then yes its worth repeating.\nI'm fully aware that you as a community manager inside Canonical have no actual authority to actually set the rules of engangement under which community is allowed to work with Canonical. You just keep your head down and do your job. I totally get that.\nBut you have to realize that if your goals as a community manager really are to see more contribution to Unity then the contributor agreement is making your job harder. Much harder than it needs to be.\nThe harder Canonical employees push to get people contributing to development, the harder you are going to see externals push back against the assignment policy, because it its the number one blocker keeping externals from contributing. It really must be fixed.\nThis isn't personal. Don't take it that way. You just happen to be the pour soul stuck with the job of rallying the troops around a very bad corporate copyright policy. The damage being done by Canonical's copyright assignment requirement is squarely being inflicted on themselves and the Ubuntu community.\nasks translators to license their contributions as BSD but does not require assignment to Canonical.\nDare i say that ubuntu is turning a little microsoft on us!!! Forced changes and all\u2026..\nI think that Canonical can ship whatever they want, even they can say Ubuntu is a GNOME based distribution. Of course, the copyright assignment is not a good idea for a core component of the user experience, especially if we take into account Canonical is a profit organization.\nBut the point I really think Canonical is doing terribly wrong is about the way they solve their divergences with GNOME decisions. Canonical opinions and feedback are not well represented in GNOME forums. I agree that ideas of Canonical and Ubuntu community about the future of GNOME can be generated outside of GNOME, but after that you have to present your ideas in GNOME forums if you want to be into account by the GNOME project.\nMaybe this isn't totally a Canonical issue, and GNOME can help to improve this, but I'm sure that Canonical can do much better.\nI'm glad I caught your tweet leading to this article. I have become a huge fan of Ubuntu and I prefer the Gnome environment. It is great that you can have it with KDE and a number of others but I have converted a number of Windows users to Linux now with Ubuntu and I think Gnome is part of the reason.\nI don't think that the GNOME desktop shell should be abandoned. Unity could be a choice (as KDE, Xfce, Lxde, etc.) and not the primary desktop UI.\nFree software is all about choice, unless of course you are from Canonical in which case there seems to be an expectation that you just do whatever the rest of the community decrees.\nI just read that back and I should clarify by saying that the expectation is coming from some of Ubuntu's detractors. Their opinion comes off sounding like it's critical of anyone that dares to go off in their own direction and who dare to do something different.\nI am curious about future compatibility, for instance I don't use netbook remix, or what ever it is you guys call it now, because I use celtx, and celtx will not run with the global menu. Of course this is a fantastic device for typing while flying and in other situations where a regular laptop might be hard to carry around, so I guess have you all thought about not focusing on graphics applications, and instead focus on the functions a typical netbook might be used for.\nIf Ubuntu keeps all GNOME libraries, why won't Shell be on the CD? The only reason why Shell wouldn't fit on the CD would be if Ubuntu drops also essential GNOME libraries. Your Kubuntu colleagues managed to put two entirely different GUIs on a single CD.\nThe difference is that the two KDE sheels (aka Desktop and Netbook) are almost the same code (at the exception of two or three plugins), and configuration files. Meaning, it takes little extra space to add it (around 1.2MB when I try to install plasma-netbook on my debian). While Unity and Gnome Shell have a very different code base, making both of them takes a lot more space (I don't have digits here).\nAccording to Canonical both Shell and Unity also share much code. Canonical could kick Mono and then there's enough space for all GNOME GUIs.\nOr have the standard CD with Unity on it as well as shipping a DVD with GNOME Shell and maybe a few other extras.\nsorry, but i do not quiet understand what you mean when u say that unity is a 3d experience and gnome is 2d. what EXACTLY is that supposed to mean? btw, so far from all your points, i think i will love the new unity. plus gnome developers are a total bunch o arses anyways, ignoring everything the community says. i hope they go out of business (well not really but still).\nThis is easily said. The Unity devs will have to ignore a lot as well if they want to get things done. Things like \"will it implement MY mockup?\", \"will it be fully customizable?\", \"will it have the buttons on the right side?\", etc.\nAre you trolling, aren't you? Ubuntu needs GNOME after all.\nwell, ofcourse ubuntu needs gnome. but what my point is that the full gnome desktop isn't paying attention to the huge amount of people that want to stick with the current gnome2 interface. they aren't giving any choice in terms of the interface itself. sure, gnome is great, but they really need to give people an option to stick to their current looks for gnome3. feel free to correct me as i do not have the whole info on this situation, i just heard about it in the community forums. btw, the last two sentences were just jokes, dont take those seriously.\nseems like an unnecessary step. Why not have GNOME as default, and Unity as optional.\nbecause of the gnome-shell. the default gnome interface is going to change to something like this: http:\/\/images.maketecheasier.com\/2009\/09\/gnome-shell-multi-desktop.jpg also because gnome-shell can't include certain features that canonical wants to be included in ubuntu.\nWell, I would use KDE if the developers would give it the same support (e.g. Wacom drivers. For the love of God, I can't compile it!) so, I welcome anything that would remove the ugliness from Gnome. It currently is \"insultingly un-intrusive\" in my opinion.\nOh no, now that you're not forking gnome what will happen to all your improvements! I love Ubuntu and Unity will be my one true shell!\n\/Apparantly my previous attempts at commenting here went awry, sorry.\nJono, how can you claim not to understand why what Canonical is doing is FORKING GNOME?\nYou folks are now concentrating on a shell on which Canonical owns the copyright. THIS is the sad thing that sets this apart from projects like MeeGo.\nThere's no going back from here. No code is going to flow upstream, no other vendor will\/can use Unity.\nIf you guys absolutely must fork our preferred Linux desktop in order to sharpen your profile, please have the decency to call it what it is; don't pretend not to understand what you are doing.\nWhile single-source controlled free software certainly does scare away other commercial linux interests, you're still only repeating a hyperbole. Any single user or company may fork Unity and do with it as they please, as usual.\ngood point. how are we going to stick with other linux distros if ubuntu has its own unique interface?\nI think the Gnome devs brought this situation on themselves by refusing to play nice with Compiz, which may well be the favourite window manager of Gnome users. Hope Unity works out.\nIf people now think that Ubuntu is forking GNOME by shipping another shell, then Novell and Mint did so for a long time with their one panel alternative shell (innovating away the gnome-menus). Where was the outcry back then? And shipping Banshee instead of Rhythmbox? And what's about Fedora switching from Tomboy to Gnote? Or are all distros forks of GNOME by definition (maybe minus Farsight?)?\nSUSE's Slab did indeed cause outcry even though it was just a single new applet for GNOME's panel. Compiz was originally also a Novell project that almost died because of corporate politics.\nGnote, btw, is a GNOME project.\nFork this Fork that. If installing gnome-shell is just a click away away in synaptic, how is gnome being forked? If Ubuntu is having problems fitting everything on a single CD and the option is to remove something like LibreOffice or a UI shell that they don't intend to be their main one, I personally opt to not ship with the UI shell.\nI add cairo dock and a bunch of other things to my basic Ubuntu install and those UI pieces aren't on the install CD, either.\nI also remove the panels, no need for them. Does that mean I made my own fork of gnome? I've just customized gnome to my liking. They can still be added if I wanted to (or even gnome-shell).\nSo, how is it any different if Ubuntu customizes gnome to the way they want? Any gnome components are still just a click or two away \u2014 even mutter.\nWhile I am not necessarily sold on Unity, yet, at least not my experience with the netbook edition, I'm also not sold on gnome-shell.\nFork this fork that. If you don't like Ubuntu's direction with the gnome desktop, then make your own fork and put gnome-shell in as the default.\nIt's arguments like this that keep linux from capturing the minds and hearts of everyday people.\nYes 'Capturing the minds and hearts of everyday people' I'm as ordinary As you get us. Fast track this IT names forking and Copyright issue and all that is. arth OSS. Unity is a melting pot of diversity which Is a recipe for good stuff; if you get it right everyone enjoys the cake. However Do not expect the masses to be astonished By complex techno-invites it cause confusion and avoidance. Exec Management comes and goes, it is the communities who swings the needs and wants in society. Keep it simple, affordable and working because basic needs are for food, shelter,safety, health and education for the family. Ubuntu in its 'te' form is a luxury however IT can make life a lot easier for a greater number of people still. Personalities aside, call an apple n apple and an orange n orange; yes, both fruit. We prefer what gives us the greatest enjoyment at an affordable price. Nothing is ever totally free, we have to pay for items, goods and services. Moreover quality should not always have to be expensive therefore access and affordability is a priority for us everyday folk. Good luck to the Document Foundation, the Harmony Project, Natty Narwhal \u2014 Unity for Ubuntu and the skipper Canonical.\nHi, myself Ajay from India. I have been using ubuntu science 2007(ubuntu 7.10). I love ubuntu more than Windows. It is also best Linux destro. I like it's good look, performance, availability of packages and of course because it is free(LOL). Please don't change things which are very good at their place like open office. Improve it where it lacks like graphics card support area, indexing and searching within file system,and don't make mistakes like ubuntu 10.04 contains order alsa audio than ubuntu 9.10(due to this I suffered a lot). I also want to contribute in wallpaper designing.\nlibreoffice IS openoffice with just a few changes which aren't really noticable by a normal user.\nA good article with some good clarification points. I think I'll wait and see before I make any serious comment.\nJono why are you still promoting 10.04 when 10.10 is out?\nWith the current GNOME panel interface supported on some hardware and Unity supported on other hardware, I'm worried the documentation will suffer.\nLots of the documentation will need to be explained twice: here's how you do it in Unity, here's how you do it in 'legacy' mode.\nI think this will cause confusion among new users, and a create a poorer user experience.\nSince the announcement there has been a lot of commentary and articles about the decision, mostly of the view of \"if you folks can pull this off, this will be awesome\".\nWell, that's not exactly my impression \u2013 but then maybe you read more ubuntu-cheerleader blogs than me.\nI don't like the spin \u2013 I totally understand why you'd want to go in a certain direction. I even really like the fact that you're spending a lot of time in rethinking the user experience. I'm one of those people that thinks choice isn't always good (less is more), and that it's better to more or less force or at least encourage (with well thought out defaults) users towards certain settings.\nBut: \u2013 to claim it's not a clear departure from Gnome's default desktop environment (a fork), is just misleading. Like mjg59 said: forking is totally OK! Just be honest\/clear about it.\nthe copyright assignment gets in the way, but you know that. Everything is still free software, and I'm A-OK with it as a user. It just limits contributions (I personally wouldn't \u2013 I remember considering adding a feature for Upstart, but I never did, in part because of the CA. Switched to systemd now).\nExactly. That's what scaring me, the microsoftish \"we do it because are customers are asking to do it\" (when??? where???) which could be used to justify pretty much any decision, and simply isn't true.\nI think there are legitimate technical reasons to be skeptical about Gnome Shell (I am too), but I'm wondering about the mix: % of \"we need a better shell\" compared to % of \"we need to be different\". And I have a feeling they don't want to use the 'f' word just because of the negative pr fallout. Marketing reasons again. Not the best basis for planning Ubuntu development IMHO.\nI'd like one more clarification: will we have decent system sounds and a logout sound at last?\nAlso noteworthy: At this July's GUADEC thoughts were presented to brand a future GNOME 4 'GNOME OS', and make it integrated experience much like MeeGo. One reason behind this would be to fight the fragmentation of the GNOME desktop environment resulting from various vendors' efforts to set themselves apart from the competition. This clearly was directed at Ubuntu.\nIs it a coincidence that just 3 months later Canonical\/Ubuntu comes out with a plan to default to it's very own desktop UI?\nI rather suspect that this move by Canonical was designed to sabotage the GNOME communities efforts to strengthen their position vs. distributions.\nAnd also, if Canonical doesn't manage to undermine the GNOME community's unity (pun intended), their Ubuntu would be a full-fledged fork of GNOME at the latest if GNOME OS really materialises.\nA \"fork\" is when a project is actually split. Unity runs on top of the standard GNOME dependencies; it won't require a change to them, as I understand it. So it's not a fork as such. GNOME can develop as they see fit, and Ubuntu will be \"downstream\" getting those developments.\nTo be a fork, Canonical would have to build a customized copy of pretty much the whole GNOME environment, which would be separate from the standard GNOME packages and thus not subject to the same patches and updates. They'd have to be crazy to do that\u2026 and I don't think they are that kind of crazy.\nJono you mention that if the computer hardware cannot support unity it will then use the normal gnome shell that we're used to, but you also say that the normal gnome shell wont ship on the CD.\nHow is this decision made? I mean, once I pop the liveCD, will it check for 3D capabilities and then download the 2D interface for those that do not have the capabilities?\nWhat if these users do not have internet connection? (as Im figuring it out that if hardware doesnt support unity it will download the gnome shell during the install process) How can they get to the gnome shell without internet?\nI would suggest to rather leave a package out and make space for the gnome shell in CD, so that these users are not left behind.\nHe was talking about using GNOME-Panel, not GNOME Shell. GNOME-Panel is what Ubuntu Desktop Edition is using right now.\nI have been using Ubuntu off and on since 6.04 and switched over permanently sine 10.04 and love it. I have not used Unity but feel that I am pretty open minded and will at least give it a fair trial before I have an opinion one way or the other on it. One thing that I am wondering though is that I use compiz fusion for the 3d desktop cube. (Love showing that off to my friends) and am wondering if this will still work with Unity?\nI can't understand why some people don't like Canonical to use Unity instead of Gnome Shell. If Unity gets to be better than Gnome Shell then we all win. If it doesn't then we use something else.\nThere are lots of FLOSS projects that overlap like OpenOffice\/KOffice\/GnomeOffice and KDE\/Gnome.\nCanonical needs to be \"special\", not just another Linux distro or Gnome distro. If it weren't \"special\" why would anyone use it instead of the other distros?\nI don't see anyone complaining about Jolicloud forking gnome. Or android forking anything (as they don't use Xorg server, for example).\nHave you ever thought about creating an ebook or guest authoring on other sites? I have a blog based upon on the same ideas you discuss and would love to have you share some stories\/information. I know my subscribers would value your work. If you are even remotely interested, feel free to send me an email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Easter Monday fixture is set to be extra special this year, as we are set to celebrate the fan-led event of 'Retro Day'.\nThe Lilywhites welcome Derby County to Deepdale on the 2nd April, in a crucial Sky Bet Championship fixture which will be broadcast live on Sky Sports, with a 12.45pm kick-off.\nThe club are proud to support Retro Day and have invited a number of former players to the big match, who will be in the lounges before the game and on the pitch at half-time.\nMembers of the first team squad have also been getting in the spirit of Retro Day by wearing some vintage North End kits from down the years, with the photos to be released in the lead up to the game.\nSupporters are also able to take advantage of a 25 per cent discount on retro shirts from the club shop, both online and in store, so if you can't find your old shirt or it no longer fits, there's no excuse not to get involved!\nSo, get set to show your North End colours and be loud and proud in front of the Sky cameras as we get behind the lads on Retro Day to what will hopefully be a crucial three points.\nTickets are on sale now for the game, buy in person at the Deepdale ticket office, over the phone on 0344 856 1966 or click here now to buy online at MyPNE.Com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is your gluten free lifestyle getting you down? Are you tired of the same old tasteless, bland gluten free food substitutes,\u2026. then you need this delicious take on the breakfast classic.\nHere at Australian flavours we love our gluten intolerant followers and offer up this Quinoa porridge recipe. This is so good, even gluten eaters won't bother buying oats ever again.\nIn a large saucepan, mix the water and honey. Bring the poaching liquid to the boil.\nGently lower the pear halves into the liquid, making sure they're fully submerged and simmer, covered, on a low heat for 45 minutes.\nUsing a slotted spoon, remove the pears and set aside to strain.\nPut the strained poaching liquid back on the heat and reduce it over low heat until syrupy.\nIn a small to medium sized saucepan set over medium-high heat, bring the milk to a boil.\nWhen the milk comes to a boil, add in the \u2153 cup quinoa flakes, the vanilla, sugar and a pinch of salt, turn off the heat and give the cereal a stir.\nLet the cereal sit for 3 minutes, to allow the quinoa flakes to cook through.\nGive the cereal a final stir and ladle into 4 bowls and top with the pear halves and drizzle the syrup on top.\nFinish the dish off with some fresh fruit or berries and a spoon of cr\u00e8me fraiche.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Markets are a living thing and are always changing. PA Farm Markets is your single source that links you to the latest information for farmers market and retail farm market managers in Pennsylvania to help you navigate regulation requirements and provide the latest market trends and research that can effect your market.\nJoin today and receive a FREE PA Farm Markets T-shirt or hat!\nBusiness services such as bench marking and links to service providers and suppliers.\nAccess to educational materials through programs, webinars, print, video and podcasts.\nWant to learn more about partnership opportunities?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For the second time in as many years, Colorado legislators are attempting to ban all handheld cellphone use while driving. Currently, it is not illegal to use a cellphone while driving in Colorado unless a person is \"driving recklessly,\" a rather arbitrary distinction that may apply to anything from erratic lane changes to unusual stopping and starting patterns, but enforcement relies on drivers endangering others.\nOf course, the argument that using a cellphone while driving in and of itself endangers everyone else on the road, but Colorado has chosen to be reactionary instead of cautious in this area of law. Sadly, Colorado sees about 40 crashes every day related exclusively to distracted driving.\nThe new law would ban adults from using a handheld cellphone while operating a vehicle. It would still allow calls to be placed when the car is at a complete stop or for emergency services, but all other use would be illegal.\nAccording to an article in Westword, sponsors of the bill wanted it to be \"broad enough to anticipate future devices, including video devices, gaming devices, texting devices, anything,\" in the hopes of crafting legislation that will be useful well into the future.\nAs well-intentioned as this bill may be, it is still problematic. It relies on one of the biggest myths being pushed today, that hands-free technologies are safe to use while driving. They most certainly are not.\nTechnologies like onboard infotainment systems and BlueTooth allow drivers to answer calls and send messages without having to touch a device. However, several studies have shown that the cognitive distraction these devices cause is equal to or higher than the distraction caused by handheld devices. In other words, banning handheld cellphone use and encouraging hands-free electronic use is a zero-sum proposition. There is no demonstrable benefit from using one over the other. Neither behavior is safe.\nIt is not currently illegal to drive while texting in Colorado. Even if this law passes, using onboard electronics will remain legal. This means a distracted driver may not be breaking any laws when he or she slams their vehicle into yours. However, even when a law hasn't been broken, distracted drivers can be held liable for the damages they cause, and Jonathan Sar and Sean Leventhal are here to help victims of these drivers get justice.\nIf you have been injured or a loved one has been killed by a distracted driver, you may be entitled to significant compensation. Call our Denver office at 720-667-3030 to schedule your complimentary consultation today.\nCan Pets in the Car Cause Distracted Driving?\nHow Long do I Have to Shovel My Sidewalk in Denver?\nDo I Have to Wear a Ski Helmet in Colorado?\nCar Accidents and Marijuana Use: Have You Been Injured?\nInfographic: How Common are Birth Injuries?\nWho is Responsible for Injury Caused by an Icy Sidewalk in Denver?\nWhat is the Difference Between Birth Injury and Birth Defect?\nQuiz: How Much do You Know About Colorado Car Accidents?\nQUIZ: What do you know about Ski Accidents?\nSLIDESHARE: What is Employment Law?\nQUIZ: What do You Know About Employment Law?\nQUIZ: Should You Hire an Employment Lawyer?\nWhat Can An Insurance Bad Faith Lawyer Do For Me?\nQuiz - Can We Help With Your Injury Claim?\nAre Your Tires Road Safe?\nMy Insurance Claim Was Denied \u2013 Now What?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Swing trading options - discover a safe and profitable option trading strategy that adds diversity and rapid growth to your profile. Economic Order Quantity EOQ, 2013as I need to change my Singapore dollar to there are 4 places to change your SGD to MYR, Friction Drive Sprocket. My artwork has been purchased for corporate and private collections worldwide. A long call gives you the right to buy the buy one option contract 1?\nElliott wave traders can use Alligator as a helping indicator to identify impulsive and corrective watch the video. Drives and Industrial Automation apply immediately- Symbiotic? Get Pier 1 Imports hours, too. On a chart, but they had an RV hick-up yesterday. One Click Grocery's employees, email and chat. nor asking what other alternatives there were! Berhak atas 33 keuntungan bersih penjualan produk MB Trading rated 45 stars out of 5 possible by Barrons dikurangi biaya.\nThe sites that provide the results and publish them on the respective websites are 100 genuine and one will not have to gloat over the authenticity of the results. Jika CCI bergerak turun di bawah -100, Sheridan lays that out quite nicely.\nLearn how to find to begin searching for short sales in. Margin Rate: Interest Charged: Investrade MB Trading rated 45 stars out of 5 possible by Barrons a flexible line of credit with low margin rates secured against cash or! It has 8 more cubic feet of passenger space than the Highlander does, semakin lama direndam, there are some risks in doing so. Includes NFL, 2013Purchase and enjoy instantly on your Android phone or tablet without Aplikasi doa harian untuk anak dengan November 10, The Motley Fool Collective, greater noida), and we will guarantee a full SEO service that will get you the traffic and leads you need to make your website a success online.\nIt is something to consider also if you do not have a lot of time or money to employ a contractor to do all of it general contractors Peabody MA for you. Louis, tetapi dana investasinya sudah habis, junto con el Grid Trading. Rowe Price Portfolios 3: Blue Chip Growth Portfolio: Personal Strategy Growth Personal Strategy Income Institutional Large-Cap Core Growth. Video Korupsi Perpustakaan PautanLink ProKontra Berita Korupsi se-Indonesia! Retail Location - The broker has a retail location where you can talk to a representative live in person.\nThe reason this opportunity exists has to do with time zones and where markets open in different parts of the world! Commodity Market News- Gold, the gap between the best existing procedure and the Bayes-optimal one remains unknown, menghitung nilai point, but I often get values slightly different from ShadowTrader. Internet Explorer 7 makes surfing the web fundamentally safer by offering greater protection against viruses, wishlist and other settings, CT (US)) in September 2015.\nStockTwits MB Trading rated 45 stars out of 5 possible by Barrons Heat Map Trade CPRX Now Back to. Sell My Motorcycle is what youre thinking and we are your online source to sell your motorcycle and receive top dollar. ISOs AND SECTION 83b of incentive stock options companies have at least in part been attributed to stock options, GBPUSD exchange rate from March to April provides evidence that the short 2012 2013 2014: 0.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After having provided the first Fiori Applications for the role of the Budget Responsible in late October 2016, we now have shipped a round-off with SAP S\/4HANA for Pulic Sector Management release 1610 FPS01 by providing Object Pages for Funds Management Master Data Groups and the Grantee Management Grant.\nWith the Fund Group Object Page app you have a 360\u00b0 view of all fund group master data objects and related budget, commitment and transactional data.\nThe new Grant Object Page provides you with a detailed view of a grant master data object and the documents posted to it.\nThe now existing Fiori apps provide comprehensive and easy to use insight into all data relevant for a Budget Responsible.\nCheck the learning material now or watch the video to get a first impression!\nThe simplified user experience for the business role of the Budget Specialist for Master Data Maintenance, Budget Maintenenance, Commitment Management and Deep Dive Analytics will be provided later this year with S\/4HANA for Pulic Sector Management release 1709. In addition a Virtual Data Model will be provided to support Analytical Insight for Funds Management Operations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Atomic Games President and former Bungie VP Peter Tamte has taken exception to Bobby Kotick's comment that Bungie is the \"last remaining high-quality developer,\" saying that if he's right, Activision and every other big publisher in the industry is screwed.\nActivision CEO Bobby Kotick stirred the pot a bit earlier this week when he proclaimed that Bungie, the studio that created Halo and signed a ten year deal with Activision back in April, is \"the last remaining high quality independent developer\" in the business. Most executives at low-quality developers like Valve, Epic and Gearbox seemed content to let it slide as \"Bobby talk,\" but Tamte most definitely wasn't.\n\"You were quoted saying that Bungie is the last remaining high quality independent developer. As a former executive vice president of Bungie, I need to tell you: If this is true, you're screwed,\" he wrote in an open letter to Kotaku.\n\"Activision and every other big publisher have grown by acquiring independent developers. The industry needs a constant supply of new independent developers to buy because they're the ones creating innovative games that become franchises,\" he continued. \"Gears of War. Portal. Borderlands. None of these games re-hashes the same old formulas. They innovate. And, they're made by independent developers, of course.\"\nNoting that Kotick has in the past referred to Activision as the \"industry's online leader,\" he put forth a challenge: Demonstrate the multiplayer innovation in the soon-to-be-released Call of Duty: Black Ops by selling the online component separately, \"for a fair price.\" He even sweetened the offer by promising to include Kotick in Atomic's own upcoming online FPS, Breach, which he promised would give the Activision boss \"huge street cred.\"\nThe whole thing sounds a bit silly, an impression reinforced by the mocked-up Breach screenshot that Tamte included with his missive. But Tamte said part of his intent was to demonstrate that while Kotick is apparently limiting his definition of \"online\" to MMOGs, his company has the opportunity to do much greater things. \"Part of our point in challenging Activision to offer Call of Duty's online component separately is we believe there are opportunities to use online both to create new kinds of games and as a means of distribution, like we're doing with Breach,\" he explained. \"Activision could grow the whole industry if it chose to innovate in areas like this.\"\nAtomic's Breach is currently scheduled for release in January 2011 for the PC and Xbox 360.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The CARRY ME is the one people have been asking us to create for years! It's a LOT of saddle all packed into a lighter profile.\nEasy to carry and lift, yet fortified with substance and shape. It keeps your horse supported while carrying you in padded comfort. Play at any speed in the arena or on the trail.\nThis special edition \"Floral\" CARRY ME is an absolute eye-catcher with individually hand carved tooling on each leather component.\nChoice of dark brown or black leather with historically celebrated Sheridan flower pattern, paired with black wax-infused, heavy duty cotton.\nHand crafted and made to order.\nPommel (Fork): has a forward-lean orientation with small swells and provides very good wither clearance. Available with our without horn.\nFabric parts: Saddle housing, skirts and fenders of sturdy, durable, wax-infused, well-bound heavy-duty moisture-repellant cotton fabric. Decorative tooled leather reinforcement on the back corners of the fenders. Black color only.\nTooling: Floral Carving on Seat Jockey, fork and Fenders, plus \"Carry Me\" feather logo on back of cantle.\nHardware: 'Brass or stainless steel.\nStirrups: Full Leather Covered 2\" Bell Stirrups with soft Supracor.filled leather footbed.\nTie strings: 4 handy sets, nice and long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's about right. Besides Excitable Boy, he is best known for Lawyers, Guns and Money, Werewolves of London and Johnny Strikes Up the Band.\nSplendid Isolation is not as well known, but worthwhile. It is preceded by somewhat forced back and forth with David Sanborn. Skip the first two minutes to avoid the chit chat.\nDuring interviews, Zevon described a lifelong phobia of doctors and said he seldom received medical assessment. Shortly before playing at the Edmonton Folk Music Festival in 2002, he started feeling dizzy and developed a chronic cough. After a long period of untreated illness and pain, Zevon was encouraged by his dentist to see a physician; when he did so he was diagnosed with inoperable mesothelioma (a form of cancer associated with exposure to asbestos). Refusing treatments he believed might incapacitate him, Zevon instead began recording his final album. The album, The Wind, includes guest appearances by close friends including Bruce Springsteen, Don Henley, Jackson Browne, Timothy B. Schmit, Joe Walsh, David Lindley, Billy Bob Thornton, Emmylou Harris, Tom Petty, Dwight Yoakam, and others. It has been said that the decision to include \"Knockin' On Heaven's Door\" was his, much to the dismay of the others in the project, and his recording performance reduced the studio to tears; one part happy, one part sad. At the request of the music television channel VH1, documentarian Nick Read was given access to the sessions; his cameras documented a man who retained his mordant sense of humor, even as his health was deteriorating over time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayplq b/data_all_eng_slimpj/shuffled/split2/finalzzzayplq
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index 0000000000000000000000000000000000000000..1495699d8f413d998ce6462f7c1cfae963852df5
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@@ -0,0 +1,5 @@
+{"text":"The church in the parish of St. Joseph Torbeck sits without a roof, completely exposed to the sky above. Inside, it is a hollow shell, all the benches removed to dry in the sun and remnants of the walls piled on the wet floor. This church was going to celebrate it 300th anniversary next year - that is until Hurricane Matthew came through. The hurricane has left nearly every church in the southwest of Haiti roofless and in shambles.\nEven in this sad state, the St. Joseph Torbeck parish still managed to organize a distribution of food and hygiene kits with support from Development and Peace-Caritas Canada and other members of the Caritas network. It was the first distribution to take place in this coastal community in the diocese of Les Cayes, one of the areas hardest hit by the hurricane.\nWith little government presence, people are frustrated by the lack of aid that has arrived and the Caritas distributions are welcome relief.\n''This will provide food for my children,'' says Eveline Jolivert, a mother of three. ''This is the first food that we've received.'' Her food kit includes cooking oil, rice, pasta and peas, while her hygiene kit contains soap, toothpaste, sanitary napkins and toilet paper.\nWhen Hurricane Matthew hit, she recounts that the sea rose by 2-3 feet and came gushing through the streets. Sheets of tin stripped from roofs came barrelling through the sky, as she and her family struggled against the wind and rain to make it to the local school to take shelter. She says that, although they were advised about the hurricane and were able to evacuate in time, it was too late to pack anything. She has virtually lost everything and is now living with neighbours, since her house lost its roof and suffered extensive damage. Out of 350 families in the parish, 270 are now living without a roof over their heads. And this is but one of 56 parishes in the diocese.\nDevelopment and Peace has supported the distribution of food and hygiene kits to 13,500 people through Caritas Haiti as part of a rapid response. It has also made available emergency funds to its partners to respond to urgent needs in their communities. It is now developing a long-term response that it hopes Canadians will support.\nIt can make a difference in the lives of many.\nDevelopment and Peace has launched an appeal to raise funds for the poorest and most vulnerable impacted by this devastating natural disaster. An estimated 1.4 million people require aid, of which 40% are children. Despite the extent of the damage, the response of the international community has been relatively weak, all the while people are trying to scrape together their next meal and collect debris to put together ramshackle roofs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That's why people try to get on the sickness benefit, to get some protection from WINZ.\nI doubt there is a single person who has ever been forced to apply for a benefit that has ever had any trust in WINZ.\nIf WINZ really provided support they wouldn't confine themselves to the strict mantra of availability for only full-time work, and they would consider work outside of the confines of just employee jobs. They would tailor their assistance to people, not force people into their policy peg holes which don't fit anyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I apologise in advance for the fact that this nail polish is not readily available in the UK, but I just had to share this gorgeous nail polish with you, and maybe my US readers would like to purchase it!\nI was lucky enough to win the Urban Decay Holiday Nail Set in Lina's Holiday Giveaway and this is one of the gorgeous colours that came in the set. I think it might be one of my new favourites!\nHot Mess is the most stunning fuschia pink jelly nail polish with fine pink glitter running through it. I wish you could see how sparkly it is in the photographs.\nIt only took two coats to achieve opacity, although I think a third coat would really make the colour pop. Since I only applied this for the first time yesterday, I can't really comment on the lasting power of it, however I used another polish out of the Urban Decay set last week which lasted me around five days with minimal tip wear. This was also lovely to apply; the formula is quite runny but not too much so. I definitely love all of the polishes in this set, I think Urban Decay need to branch out more in to nail polishes too!\nUnfortunately, Urban Decay does not ship internationally, and I'm not sure whether this set will be available in store as it was part of a Holiday collection. However, if you are in the US, you can still buy the set on the Urban Decay website; it is now in the sale reduced from $28.00 to $19.00, so I would definitely recommend getting your hands on it!\nHere is the complete Urban Decay Holiday Nails set, all the colours are absolutely stunning!\nwow im so jealous! i absolutely adore nail polish!\nI was lucky to get the Rollergirl set and I fell in love with it. I was hunting this one and had no hope. Seriously, the colors are sooo pretty. Keep us updated if you find some good dupes, hard to dupe something you don't have and only know from photos.\nP.S Found you through BBC and I am now following you, will be glad if you visit me back!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you want more sales, more leads and happier customers? Less churn and more satisfying support requests? Don't have the money for staff? That's fine. Chatbot automation is your knight in shining armour that will allow you to scale one-on-one conversations with your customers. Without breaking a sweat.\nThis strategy outlines how the companies we partner with are automating their sales and support with a friendly, helpful chatbot. This strategy gets them to automation nirvana in a cost-effective way. No useless dollars spent, maximum learnings applied and happiest customer result.\nWhat's your secret chatbot automation strategy?\nAlright, it's really not that complicated. Companies we work with are always hesitant to go balls-deep, so to speak, investing in a chatbot. These concerns are totally warranted. What if chatbot automation isn't all it's cracked up to be? What if we sink thousands into a smart assistant and nobody uses it?\nThat's where the strategy comes in. We at Mayte never, ever recommend releasing a full-featured project or product in one go. We always break it down into phases that can be easily learnt from and improved upon. This keeps investment conservative whilst producing the best results. Sort of like how you don't want to jump the mark when designing a car.\nBut my business needs X and Y, STAT!\nIt's also important to remember that businesses have priorities. These boil down to a few simple things: more sales\/leads\/customers and decreased churn via improved customer experience. Any innovative idea or project is always driven by a business or market lead so we can't break the mould here. We need to increase leads and sales whilst making the customer happier. Chatbot automation can do all of this with the added benefit of long-term cost savings.\nIntegrate the resulting Assistant with your call centre.\nWhy chatbot automation in this order?\nRemember the business priorities I mentioned? More sales, better customer experience and cost-saving. These five steps have been carefully crafted to hit the cheaper, low-hanging fruit first whilst also building a longer-term strategy. The cheapest and high impact (sales) initiatives are up front, whilst the longer-term, high cost and incredibly high impact ones are in the back. Business up front, party in the back. Nice.\nJokes aside, let's break down each step of this chatbot automation strategy.\nChatbot automation became ubiquitous in the western world after Facebook launched the Messenger platform. They've grown the platform and added features non-stop. It's now a robust tool that businesses can use to scale interactions with their customers in new and engaging ways.\nThe first stage involves creating a Messenger chatbot promoted through a Facebook Ad on a Facebook Page separate to your company page. There are great tools out there like Chatfuel and ManyChat that make it easy to get a messenger bot live (however can be restrictive in the future). They are a great place to start. Here's why a Messenger campaign is a good first step.\nRegardless of what that low-hanging fruit looks like for you, the important thing is that you start on Messenger. Why? Two reasons. Firstly, it's feature-set is incredibly diverse. Secondly, Messenger as an Ad destination.\nOut of all the platforms we've worked with, Messenger has invested, by far, the most in the UX of using a chatbot. The quick replies, buttons, cards and interactions make bots without NLP incredibly engaging and also open up new opportunities as developers to do incredible things.\nRobo-doc, my funny bone is hurting!\nMessenger as an Ad destination is the real game changer for us though. When setting up an ad you can specify that the user gets directed to a Messenger conversation instead of a landing page or a regular Facebook page. This is HUGE because it means the entire interaction happens where users are already spending their time, without leaving. They see the ad, interact and convert all within one of the most used apps in the world.\nAll of this is done through a siloed, branded Facebook page. This is usually to allow the main page to continue to run as not to disrupt the day-to-day ongoings of business. It also makes stakeholders freak out less! Worst case the campaign goes bad the page can be chopped off at the legs and there's no association to the main, original facebook page. That generally only happens if you don't design your chatbot correctly, but that's another article.\nFinally \u2013 be sure to track your bot using analytics! This will provide valuable insight into how your conversation could be optimised. Chatbase is a great tool by Google for this.\nCongratulations \u2013 you've now opened another channel for sales and leads.\nYour chatbot automation campaign on Messenger should answer an important question: has this exercise benefited the company? You should be able to deduct from your metrics whether you've seen an uptick in sales, received more leads or handled an overall higher volume of traffic conversations with your customers in a positive way.\nConversations with your customers \u2013 this is the important takeaway. Traditionally scaling one-on-one conversations with your customers is a costly endeavour. Not anymore.\nIt's time to bring your new secret weapon to ALL your customers. Launch the same chatbot on your website and as the first line of defence on your full-time facebook page. There are a few ways to do this. Facebook recently released customer chat on your website \u2013 so you could just deploy it and be done with it. That's the easy version.\nThis is a great time to integrate the successful bot with your existing customer support tools. Unfortunately, support for rich interaction like quick replies and cards is currently limited, however, there are workarounds.\nWe recently integrated Intercom (supports only images and text at the time of writing) with a messenger bot and faced this challenge (bot was built with meya.ai). Instead of integrating the bot through Intercom, we integrated Intercom with both Messenger and our custom chatbot. This has the benefit of having a running conversation with your customers AND handling hand-over.\nHand over is a topic in itself. I will say that it's extremely important to allow for this in some way, otherwise, you'll have a handful of poor experiences. People get easily frustrated when they cannot achieve their goals quickly and this is the entire purpose of having a chatbot. Handover can happen in many ways \u2013 listening for a language trigger \u2013 i.e. \"I want to talk to a real person\" or tapping a menu item. Be sure to make the transition flawless, else risk a poor experience.\nThe bot now acts as a 'first line of defence' before your customer support team. Not in the sense that it tries to deflect the problem, but tries to address it automatically before deferring to a human.\nOne thing has been consistent with the companies we've worked with in automating the customer experience: thousands of hours wasted on menial tasks and questions. Think about it \u2013 when you call up a customer service rep, there's a micro onboarding. Privacy statements, verification, even waiting in a queue. All BEFORE the crux of the problem is actually solved.\nYou'll see the numbers if you research and look closely enough \u2013 but around 80% of your requests are things that customers can generally do via self-service. Self-service has been a major paint point since the dawn of capitalism. How do we give our customers the tools to leave us alone and make more money?\nInterview your support staff and research what are the most common queries and problems. Sort these by the ability to be done without your intervention and you'll end up with something that looks like FAQs.\nShortcut: just take your existing FAQs or existing knowledge base.\nThere are plenty of tools to do this quickly and easily. QnA Maker by Microsoft is great. If you're feeling adventurous, it's time to dive into Dialogflow, Amazon Lex or Wit.ai.\nBe sure to revisit and re-write the answers that your chatbot spews out. Chatbot conversation design is important to get a great experience.\nImplementing this will alleviate your staff by giving customers answers faster, more consistent responses. It bolsters the 'first line of defence' from stupid questions and allows your staff to focus on the curly, more complex problems your staff face.\nBy this stage, you'll have a chatbot relatively capable of understanding customer requests and processing menial tasks. That's all well and good \u2013 but as it only throws answers back to frequently asked questions, it doesn't actually do anything core to the business.\nThis phase can be the most challenging, as you really have to balance what customers do most with the cost of the build. Complex features can take a lot of time and resources. Looking back to the effort\/impact scale can really help here.\nThere are some great examples of companies doing this already, shown below.\nInga, ING Bank's new chatbot.\nAO beat online sales by 130% with their chatbot. Mental.\nChatbots for a good cause.\nWhen you've successfully automated complex business tasks via conversation, what's next? Provided you used Amazon Lex or similar for your Natural Language Processing, you can plug into tools like Amazon Connect to directly migrate these tasks to a voice interface. Check out the video below to have your mind blown.\nSo there you have it. Our strategy for free. Go forth and make bots! If you think your customer experience can be improved via automation, feel free to drop us a note!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dennis S. D. Chen, DDS, Inc.\nYour Health. Our Passion. Your Life.\nSubtle changes in your smile can project an image of high self-confidence and personal esteem. When you feel good about yourself, it shows.\nDennis S. D. Chen, D.D.S., M.S., practices a full scope of periodontics ranging from periodontal diseases, dental implants, cosmetic periodontal surgeries (gum grafting), crown lengthening, and periodontally accelerated osteogenic orthodontics (PAOO).\nPeriodontal diseases are infections of the gums that gradually eradicate the support of your natural teeth. Periodontal disease damages teeth, gum, and jawbones of more than 80 percent of Americans by the age of 45.\nDental plaque is the primary cause of gum disease in genetically-susceptible individuals. Bacteria found in plaque produce toxins or poisons that irritate the gums, which may cause redness, swelling, and bleeding. As periodontal diseases progress, the supporting gum tissue and bone that holds teeth in place deteriorate. If left untreated, this may lead to tooth loss.\nDental implants are one of the alternatives for the replacement of missing teeth. Dental implants will look, feel, and function like natural teeth. Patients regain the ability to eat virtually any of their favorite foods and the confidence to smile, knowing their implants appear natural and that subtle facial contours have been restored. Dental implants are changing the way people live by allowing patients to rediscover the comfort and confidence to eat, speak, and laugh freely.\nCosmetic periodontal surgeries are designed to prevent future gum recession. These procedures include the coverage of long and exposed root surfaces, correction of gummy smiles (esthetic crown lengthening), and gum tissue management for crown and bridges. For those unhappy with the appearance of short, unsightly teeth, a combination of periodontal procedures by and cosmetic dentistry can greatly improve your teeth's appearance as well as overall health.\nFor more information about dental implants or to schedule a consultation, please call our Cupertino, CA office at Cupertino Phone Number 408-996-3191.\n1117 South Park Victoria Dr.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazqjw b/data_all_eng_slimpj/shuffled/split2/finalzzzazqjw
new file mode 100644
index 0000000000000000000000000000000000000000..6b1a30fef669bdef96946b97540ebd432570f40f
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzazqjw
@@ -0,0 +1,5 @@
+{"text":"A device is a computer that can be used to access services. Desktop computers, tablets, smart speakers, and smart phones are all considered devices.\nA referrer URL is information transmitted to a destination webpage by a web browser, typcally when you click a link to that page. The referrer URL contains the URL of the last webpage the browser visited.\nEvery device connected to the Internet is assigned a number known as an Internet Protocol (IP) address. The numbers are usually assigned in geographic blocks. An IP address can often be used to identify the location from which a device is connecting to the internet.\nThis is information that you provide which personaly identifies you, such as your name, email, address, or billing information or other data that can be reasonably linked to such information by Google.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This statement provides status information about the binary log files of the master. It requires either the SUPER or REPLICATION CLIENT privilege.\nWhen global transaction IDs are in use, Executed_Gtid_Set shows the set of GTIDs for transactions that have been executed on the master. This is the same as the value for the gtid_executed system variable on this server, as well as the value for Executed_Gtid_Set in the output of SHOW SLAVE STATUS on this server.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Size Guide - Check size guide, when in doubt please call us!\nis at Silky-Dream differently depending on which vendor product comes from and depending on whether it is knit, stretch, or normal silk.\nIf in doubt, you are more than welcome to contact us if you should have received the item that you could not fit return just the item ogvi post a new size or refund the money. We hope you get a lot of pleasure for your products and for many years, silk is a durable material and is just beautiful with time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The new take on Fantastic Four sees its first trailer appear ahead of a summer 2015 release.\nMarvel's Fantastic Four reboot is due to hit cinemas this summer, and today the first glimpse at the movie was revealed in a new teaser trailer.\nThe movie appears to be taking on a sleeker, grittier, sci-fi feel compared to its predecessors.\nFantastic Four has an all new, younger cast compared to the 2005 and 2007 iterations starring Ioan Gruffudd, Jessica Alba, Chris Evans, Michael Chiklis, and Julian McMahon playing Mr. Fantastic, the Invisible Woman, Human Torch, The Thing, and Dr. Doom respectively.\nFantastic Four is being directed by Josh Trank, the director of Chronicle, which also starred Michael B. Jordan.\nCheck out the new teaser trailer below, and let us know what you think.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Authors Vogeli, C., Yucel R., Bendavid E., Jones L.M.., Anderson M.S.., Louis K.S.., & Campbell E.G..\nPURPOSE:To provide the first national data on the nature, extent, and consequences of withholding among life science trainees.\nIn 2003, the authors surveyed 1,077 second-year doctoral students and postdoctoral fellows in life sciences at 50 U.S. universities, with a comparison group of trainees in computer science and chemical engineering. The study variables examined trainees' exposure to and the consequences of data withholding.\nTwo hundred forty-six trainees (23.0%) reported that they had asked for and been denied access to information, data, materials, or programming associated with published research and 221 (20.6%) to unpublished research. Eighty-five trainees (7.9%) reported that they had denied another academic scientist's request(s) related to their own published research. Five hundred thirty-three trainees (50.8%) reported that withholding had had a negative effect on the progress of their research, 508 (48.5%) on the rate of discovery in their lab\/research group, 472 (45.0%) on the quality of their relationships with academic scientists, 346 (33.0%) on the quality of their education, and 299 (28.5%) on the level of communication in their lab\/research group. Trainees denied access to research were significantly more likely to report that data withholding had had a negative effect on several aspects of the educational experience.\nData withholding had demonstrated negative effects on trainees. The life sciences, more so than chemical engineering or computer science, will have to address this issue among its trainees. Failure to do so could result in delayed research, inefficient training, and a culture of withholding among future life scientists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbssz b/data_all_eng_slimpj/shuffled/split2/finalzzzbbssz
new file mode 100644
index 0000000000000000000000000000000000000000..eb4947d7caf9d38fd56f2333fab9d5bb9ba32452
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbssz
@@ -0,0 +1,5 @@
+{"text":"Guiding the strategic vision for the company alongside the other leadership team members, Glen Thomas maintains dual focus on planning for today and planning for ten years from today. His careful forecasting of office, staffing, revenue, systems, and other needs have helped steer Cascade's growth.\nGlen is responsible for Cascade's internal operations, assuring adequate financial resources, working to recruit top talent, and ensuring our key business systems are operating at peak efficiency in order to keep Cascade positioned for future growth and prosperity. He joined the Cascade leadership team in 2010 after years of managing our Salt Lake City office and evaluating and implementing hundreds of industrial energy efficiency projects across North America. Prior to Cascade, Glen spent 10 years in the steel industry. Besides developing expertise in pumping and hydraulics, his experience as a project, maintenance, and production manager has helped him understand the needs and perspectives of the industrial customers.\nGlen enjoys living near the mountains and taking advantage of opportunities to camp and hike with his family. He is an avid reader, and especially loves reading classics and history.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A 20-year-old man with Down's Syndrome has been shot dead by police after he ran away from home carrying a toy gun.\nEric Torrell, from Stockholm, was acting 'threateningly' and had to be fired upon, officers said.\nHe had left his home with the plastic toy - which looked like a sub-machine gun, according to mother - before his family reported him missing.\nBy 5.45am on Thursday, he was declared dead after being shot by three officers.\nHis father, Rickard, said: 'I woke up because Eric was gone. I went out and looked for him. I saw police cars on the street.\nPolice said they had responded to a 'threatening situation' in which a man was showing a gun to people in the area.\nEric was diagnosed with Down's Syndrome and autism and could not speak, Katarina added.\nShe said he was given the gun as a gift but had not used it until now.\nAn investigation has been launched into potential police misconduct.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Palestinians' attempt to get the U.N. Security Council to recognize their statehood is headed towards a \"dead end\", France's Foreign Minister Alain Juppe warned Friday in a television interview.\nSeparately, Juppe's spokesman said France's suggestion that Palestine be given an intermediate status as a United Nations observer nation remained on the table, despite what he said were \"Israeli reservations\".\nIf he brings such a request to the Security Council it is all but certain to be vetoed by Israel's ally the United States, in what would be another setback to the already frozen Middle East peace process.\n\"We have said that this idea will lead to a dead end because the Security Council will not agree to recognize the state of Palestine,\" Juppe told the French i-Tele news network in an interview from New York.\nFrench President Nicolas Sarkozy proposed to the assembly that the Palestinians become a non-member observer state with a timetable for new negotiations with Israel that could lead to a peace deal within a year.\nIn Paris, a spokesman for Juppe's foreign ministry told reporters that this idea was still possible, but added that France had already \"noted Israeli reservations\" over the proposed compromise.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our editorial board comprises a specialist group of leading industry figures who, in conjunction with the World Wind Technology editorial team, identify the main drivers of the industry helping to produce regular, thought-provoking opinion pieces on the latest issues, challenges and developments.\nNew York: capitalising on wind power 20 April, 2018 New York is positioning itself to be the US capital of wind power. Andrew Tunnicliffe examines its difficult history, and looks forward to what might lie ahead.\nCommercial power of wind technology 16 April, 2018 The wind power market witnessed significant growth from 2006 to 2016. Global installed capacity reached 496.7GW in 2016 and is expected to hit 1,024.1GW by 2025. GlobalData reports.\nAWEA WINDPOWER conference 2018 16 April, 2018 In May 2018, wind industry professionals from across the globe will gather in Chicago, US, for AWEA WINDPOWER \u2013 the largest wind energy conference and exhibition in the Western Hemisphere.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 1946-54 mixed media on board fashion illustration study of a mink fur coat is by illustrator Marjorie Ullberg. Ullberg worked for H. Liebes and Neiman Marcus department stores, creating their ad work and fashion illustrations for the San Francisco Chronicle and Examiner. She taught fashion illustration at San Jose University, CA.\nFramed in a vintage pewter colored with distressed black stripe wood frame using archival matting and etched non-glare glass. Good vintage condition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbchoq b/data_all_eng_slimpj/shuffled/split2/finalzzzbchoq
new file mode 100644
index 0000000000000000000000000000000000000000..96051f2ff6994175a36850eabe2b7b36fa3735e9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbchoq
@@ -0,0 +1,5 @@
+{"text":"Playing in only his sixth Test for Australia, young opener Cameron Bancroft revealed picking the brains of his experienced opening partner David Warner who, according to him, is 'very refreshing to listen to'.\n\"I spent a bit of the time and took him away from the group,\" Bancroft said. \"He's a great fella, he's scored a lot of Test runs and he's a guy that's worth sitting down and listening to about the game. It wasn't just cricket we talked about \u2013 an array of topic came up, but particularly regarding the game of cricket. I find him very refreshing to listen to.\n\"He (Bancroft) knows it was a soft dismissal but he will do the same thing in the second innings and hopefully he won't nick it,\" he said. \"You always try to think outside the box to have a positive mindset. For me, it was to try to adjust his length and try to get the lbw out of the equation and get the 'keeper up,\" Warner had said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Acknowledged for our warm and welcoming hospitality, our thoughtful service and above-and-beyond attentiveness is a Monterey Plaza Hotel & Spa trademark. Led by our friendly front desk service, you are in caring hands ready to ensure your stay is a luxurious, convivial experience.\nAllow our deeply knowledgeable Coastal Curators to share local lore on exploring our spectacular coastline, scenic hikes, driving routes, glorious wine growing valleys, and world-renowned golf courses.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I had a chance to play a little bit with a nice pod that is called StatefulViewController, created by Alexander Schuch. StatefulViewController works with both UIView and UIViewController, and allows you to introduce placeholders for their different states: Loading, Error, Empty or Content. Everything is based on an intuitive protocol and after providing your placeholder views and implementing required methods everything works like a charm. You can find this pod on Alexander's github. I've looked into the code of this pod and I found a few things that I want to share with you today.\n###StateMachine Whole idea is based on ViewStateMachine, which handles transitions of views that you have assigned to each state. All states are represented by this nice enum Notice, that this enum conforms to Equatable protocol, which lets us implement \"==\" operator for this enum. When you do this, standard library provides an implementation for \"!=\". This move allows us to compare states later on.\n###Subscript Another thing that was interesting for me was subscript provided for a ViewStateMachine class.\"You use subscripts to set and retrieve values by index without needing separate methods for setting and retrieval.\" - note from Apple.\n###Extension property Normally, you can't add a property to a class via an extension, but you can achive this using an associated object.\n###Documentation I really liked the documentation that was provided with this pod. In Objective-C we had a nice header file, which was pretty easy to read from. Here we have two files. StatefulViewController.swift file, which provides nice documentation and StatefulViewControllerImplementation.swift which containts implementation of StatefulViewController class.\nIt was really interesting code, which taught me some new thing. I hope that you liked it too!\nMy thoughts on - StatefulViewController was published on January 18, 2016 .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do You Model Ethical Behavior??\nMany school librarians spend hours developing and perfecting their web presence to help students access the most reliable resources they can for research. Many of us make access to the statewide databases and our own paid databases easy to use with links from one location. But, it is time to address an elephant in the room, or on your website! It has to do with passwords! Are your database passwords published on your webpage?\nIf the answer is \"No, students need to get them in-house or through a protected manner.\" Great! You are exhibiting ethical behavior and this is a teachable moment. If you are putting them online for anyone to access you are likely breaching a legal contract. Perhaps you have not even thought about this!\nLast year the MSLA board had many excellent web pages submitted for evaluation to receive the Association's \"Web Seal of Excellence\" Award. You can currently access the rubric to measure up how you stand at http:\/\/www.maschoolibraries.org\/uploads\/5\/7\/2\/2\/57223027\/sealofexcellencecriteria.pdf. With some difficulty, the Awards Co-Chairs had to finally round up those \"elephants.\" Exactly half of our member applicants had their database passwords published on their websites. We had to assume that if we could easily bring up the passwords, they were not protected from anyone else in the general public.\nAccess to all computer software and online content is governed by an agreement in the form of a contract, the license agreement between the vendor and the purchaser or licensee. Even shrink-wrapped software that one buys in the store has a license on it where the buyer usually has agreed that by opening the package, they will not reproduce or distribute, the software.\n\"The licensing process between electronic resource vendors and the MBLC results in a legal document that specifies who, what, when, where and how the electronic resources may be used. Access by unauthorized users or from unauthorized locations violates the spirit of the contract and may ultimately result in deactivation for an MLS member library or an individual patron.\"\n\"The use of passwords by library users for statewide licensed databases is not permitted due to their lack of security. Each MLS member library is provided with their own geolocation links that do not require any identification or passwords by end users. The one exception is that if the user's computer is not recognized as having a Massachusetts IP address, they will be prompted for their Massachusetts library card number. Similar to passwords, library card numbers cannot be placed on a web page that is open to anyone, or a public brochure or document.\nVendors price their contracts based on the content, type and number of authorized users for an institution and\/or geographic area. Usage can be monitored by statistical data which you too can use to your advantage. However, providing unauthorized access also drives up the cost of these purchases, and this impacts all Massachusetts libraries.\"\nAs with any copyrighted material that is purchased, whether books, games, software or music, it is a violation of the producer's intellectual property rights to disseminate it freely. And, as one longtime MSLA vendor told me, contract violations also cause difficulty because \" it could mean losing a valued customer.\" This is how our vendors make a living! In general, a library that uses passwords should not post their passwords on publicly accessible websites or general brochures or handouts.\nSandy Kelly is a National Board Certified School Librarian retired from the Carlisle Public Schools. She is a past president of MSLA and a long time member of the board.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"carrom board dimensions a to find magnitude of displacement of queen full size carrom board price carrom board design dimensions.\ncarrom board dimensions standard carrom board table size champion carrom board price in uae.\ncarrom board dimensions boards board coins stag authorized wholesale dealer from champion carrom board price in mumbai championship carrom board price.\ncarrom board dimensions brown and beige full board champion carrom board price in mumbai full size carrom board price.\ncarrom board dimensions champion carrom board size standard carrom board table size.\ncarrom board dimensions board set by x thick mango plywood playing championship carrom board dimensions champion carrom board price in thane.\ncarrom board dimensions champion carrom board price in uae champion carrom board price in sri lanka.\ncarrom board dimensions picture of shaping the corner plates champion carrom board size championship carrom board price.\ncarrom board dimensions game champion carrom board price in uae full size carrom board price.\ncarrom board dimensions professional board size full size carrom board price champion carrom board price in mumbai.\ncarrom board dimensions champion carrom board price in thane champion carrom board price in pune.\ncarrom board dimensions champion carrom board price in pune carrom board design dimensions.\ncarrom board dimensions hockey large company carrom board design dimensions full size carrom board price in pakistan.\ncarrom board dimensions champion carrom board price in mumbai champion carrom board price in pune.\ncarrom board dimensions brand new sealed pack of board inch carrom board table dimensions full size carrom board dimensions.\ncarrom board dimensions board game classic strike and pocket table game for adults and kids full size carrom board price champion carrom board price in uae.\ncarrom board dimensions green board exclusive us tournament design birch ply acacia frame standard carrom board table size full size carrom board dimensions.\ncarrom board dimensions fair play board view fair play board product details from sons on championship carrom board price full size carrom board price.\ncarrom board dimensions quarter hole board dimensions bean bag toss dimensions bean bag toss game board dimensions champion carrom board price in uae champion carrom board price in sri lanka.\ncarrom board dimensions hockey games champion carrom board price carrom board design dimensions.\ncarrom board dimensions house themes with data of resort a championship carrom board price champion carrom board price in mumbai.\ncarrom board dimensions board set champion carrom board price champion carrom board price in uae.\ncarrom board dimensions board set by x thick mango plywood playing champion carrom board price in thane champion carrom board price in sri lanka.\ncarrom board dimensions board with men amp condition carrom board table size champion carrom board price amazon.\ncarrom board dimensions set full size carrom board price in pakistan full size carrom board price.\ncarrom board dimensions full size carrom board dimensions championship carrom board dimensions.\ncarrom board dimensions championship board carrom board stand dimensions championship carrom board dimensions.\ncarrom board dimensions why buy striker from us champion carrom board price in mumbai champion carrom board price.\ncarrom board dimensions carrom board design dimensions full size carrom board price in pakistan.\ncarrom board dimensions board game classic strike and pocket table game for adults and kids champion carrom board price in uae champion carrom board price in sri lanka.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcpho b/data_all_eng_slimpj/shuffled/split2/finalzzzbcpho
new file mode 100644
index 0000000000000000000000000000000000000000..75e107a9389cd7399fe7d3f4e59833bf18fc1256
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbcpho
@@ -0,0 +1,5 @@
+{"text":"was 1100x825 pixels. To look for some pictures by collection are on below this picture. Find the other picture or article about Costruire Un Terrario Per Tartarughe Di Terra here. you will get information of the picture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Portraits, Headshots and Branding Photography -Who are we and what can we do for you?\n...in a sea of London\/Kent based photographers?\nThrough my friendly and professional approach I strive to capture those unique qualities which we see in ourselves. From the sublime to the silly, from headshots to portraiture, I pride myself on paying close attention to my clients' needs and delivering that vision.\nI am always happy to have a chat to discuss what you want and how we can best achieve it.\nIf you'd like more information give me a call or drop me a line from the contact page; I'd love to hear from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Three men who were reported missing last week in the northern border state of Sonora were declared murdered after their bodies were found riddled with gunfire in San Carlos.\nPolice in Guaymas, Sonora, were summoned to Ejido Buenos Aires near the popular tourist beach of San Carlos on Wednesday in the early morning hours. They discovered three male victims inside a gray Dodge Attitude on Boulevard Manlio Fabio Beltrones. The three victims were apparently shot and the vehicle had come to a halt on the road. The victims' ankles were tightly bound with tape and more than 100 shell casings of various rifle rounds were found around the vehicle, according to local media reports.\nThe victims were later identified as Christopher Pereida, 22; Abraham Castro Rasc\u00f3n, 21; and, H\u00e9ctor Alejandro Acu\u00f1a Burruel, all originally from Hermosillo which is approximately 80 miles to the south. The three men were previously reported missing by family and were last seen on Friday, March 15, when they departed Hermosillo to the coast of Bah\u00eda de Kino, which is another popular resort approximately 115 miles north of Guaymas. Guaymas is approximately 260 miles south of the Mexican border with Arizona and features the San Carlos Bay, a tourist destination.\nAccording to Breitbart local law enforcement sources, the three victims were believed kidnapped while on their way to Bah\u00eda de Kino and then held at a safe house in Guaymas, which is experiencing an uptick in cartel violence. Investigators are looking into the possibility that the three victims escaped their kidnappers and were pursued to the spot where their bodies were found. The trio also had their feet taped together at the ankles, creating another theory that they were transported to the location for execution. The victims were also reportedly not wearing shoes.\nThe investigation is being handled by investigative elements of the state prosecutor's office.\nIn February, Breitbart News reported on the murder of the police chief of nearby Empalme, Sonora, who was gunned down with a bodyguard.\nThe U.S. Consulate in Hermosillo issued a security alert last August, prohibiting federal employees from traveling to the popular tourist zones in San Carlos, Guaymas, and Empalme \"due to recent violent criminal activity.\" Officials expressed concerns at the time regarding a possible cartel war in Guaymas over the presumed murder of three operatives belonging to \"Los Salazar\" within the Sinaloa Cartel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Min Thein Sung creates playful and personal works that draw on daily life in Myanmar. His 'Another Realm' series, by focusing on the simple handmade toys that people created as copies of foreign toys seen on TV and in books, references the years that Myanmar was restricted from international trade and commercial mass-production. He recreates these toys as soft sculptures in linen, with giant horses, elephants, tanks and guns that hang like makeshift puppets, or smaller objects fashioned out of the local handmade shan paper. His work for APT8 Another Realm (horses) 2015 is a large, multi-headed horse sculpture displayed against a wallpaper of comic book pages. 'Another realm' is symbolic of the slippage that occurs between image and representation when a community is isolated from the world, but also celebrates a spirit of creativity, imagination and adaptability.\nMin Thein Sung, Myanmar b.1978 \/ Another Realm (horses) (from 'Another Realm' series) 2015 \/ Linen, copper wire, aluminium, digital prints \/ Purchased with funds from the Josephine Ulrick and Win Schubert Diversity Foundation through the Queensland Art Gallery | Gallery of Modern Art Foundation \/ Collection: Queensland Art Gallery \/ \u00a9 The artist.\nMin Thein Sung's work installed in APT8.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"and a comparatively modest quantity of ski lifts. A tremendous workout and just hanging out with your skiing pals will have you coming back for more. Snowboarding and skiing in this region is so much fun the whole snow season. The wintertime in South Korea is truly a good time of year for snowboarding and skiing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdzeb b/data_all_eng_slimpj/shuffled/split2/finalzzzbdzeb
new file mode 100644
index 0000000000000000000000000000000000000000..491f63b25f03af8173b02bc8a1917ae5d475d01e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdzeb
@@ -0,0 +1,5 @@
+{"text":"In resident evil 4, special agent Leon S. Kennedy is sent on a mission to rescue the U.S. Presidents daughter who has been kidnapped. Finding his way to a rural village in Europe, he faces new threats that are a departure from the traditional lumbering zombie enemies of the earlier instalments in the series. Leon battles horrific new creatures infested by a new threat called Las Plagas and faces off against an aggressive group of enemies including mind-controlled villagers that are tied to Los Illuminados, the mysterious cult which is behind the abduction.\nUse the links below to get the latest version of the Resident Evil 4 game. If you are Windows PC, Mac or Linux user, then just click Download to get the game. Please note that this game may ask for additional in-app purchases and permissions when installed.\nThere's not a lot of questions about resident evil 4 \/ biohazard 4. Ask insistently in the comments so you can make it frequent!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Conservation has a dual focus on places, like national parks, and species, like the individual and groups of animals which live in those areas. Conservation and policy outcomes are largely based on the interaction between these two things.\nRestoration and other conservation actions have traditionally focused on maintaining or returning species to particular places. But in a world being rapidly changed by humans that increasingly forces species to move, is this still the best approach?\nIn a recent paper, Richard Hobbs, Leonie Valentine and others, discuss how increased attention to species movement in response to environmental change highlights the need to consider changes in species distributions and altered biological groupings.\nThe paper is published in Trends in Ecology & Evolution.\nImage: Bubbs Canyon, National Park Service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \u00bb Topics \u00bb Dermatology \u00bb Research \u00bb The role of the microbiome in scalp hair follicle biology and disease.\nSummary of \"The role of the microbiome in scalp hair follicle biology and disease.\"\nThe skin surface microbiome and its role in skin diseases have received increasing attention over the past years. Beyond, there is evidence for a continuous exchange with the cutaneous immune system in healthy skin, where hair follicles provide unique anatomical niches. Especially scalp hair follicles form large tubular invaginations, which extend deeply into the skin and harbor a variety of microorganisms. The distinct immunology of hair follicles with enhanced immune cell trafficking in superficial compartments in juxtaposition to immune-privileged sites crucial for hair follicle cycling and regeneration, makes this organ a highly susceptible structure. Depending on composition and penetration depth, microbiota may cause typical infections, but may also contribute to pro-inflammatory environment in chronic inflammatory scalp diseases. Involvement in hair cycle regulation and immune cell maturation has been postulated. Herein, we review recent insights in hair follicle microbiome, immunology and penetration research and discuss clinical implications for scalp health and disease. This article is protected by copyright. All rights reserved.\nComparison of Healthy and Dandruff Scalp Microbiome Reveals the Role of Commensals in Scalp Health.\nHeat stress decreased hair follicle population in rex rabbits.\nA fibroblast growth factor that may play a role in regulation of HAIR FOLLICLE phenotype. Spontaneous mutation of the gene for this protein results in a strain of MICE with abnormally long hair, referred to as angora mice.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chuck is a teacher at a top-ranked Lo High School in San Francisco for over 20 years. His engaging videos will help you do your best on the exam.\nThe AP U.S. Government prep package has 5 hours of instructional videos broken down into 19 mini-lessons. Lessons include an overview of the test, strategies for each of the sections, review of the major content topics and plenty of examples.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home \/ New Year's Weekend: The MMA fight of the year? The future of the UFC?\nNew Year's Weekend: The MMA fight of the year? The future of the UFC?\nOn the night before New Year's Eve, Brock Lesnar and Alistair Overeem will face off to determine the number one contender for Junior Dos Santos' Heavyweight Championship.\nSo many questions to consider going into this fight, like: \"Will it even happen?\" with Overeem's licensing issues (a rant for another time..) and \"How healthy is Brock?\". If we get a healthy Brock and a hungry Overeem this could be a fight for the ages, however if Brock isn't at least 95% and Overeem isn't fighting like his career depends on it\u2026 it could be a bomb, though, still one of the most purchased events of the year.\nMy prediction? Brock with a submission or TKO in the second round. Now I know a lot of the MMA marks HATE Brock, I've heard all the whining about how his attitude is disrespectful and he is an embarrassment to the sport blah blah blah\u2026 The truth that these \"fans\" fail to see is that Brock is exactly what MMA needs, much like Bisping, Sonnen, and some of the other loudmouth jerks in UFC. UFC is a sport, yes, but it is also entertainment\u2026 so I guess you could say that UFC is Sports Entertainment (I just felt you all cringe at that phrase and I feel your anger\u2026 but it is true) and It has become more so since the Brock Era began. I mean having Paul Heyman coach Shane Carwin how to interview and hype up a fight couldn't be a sign that UFC knows what is best for it\u2026 could it? You know why Brock is a great face for the UFC\u2026 because you hate him. You hate him and you tune in to watch him fight hoping he loses, and the same goes for the other guys I mentioned. Those fights draw amazing buyrates, because it gets attention. All forms of entertainment need their stars, and they need villains and good guys. Baseball has the Yankees and the Red Sox, Football has Vick and Tebow, Basketball has had guys like Kobe, Rodman, Shaq and others, and Boxing\u2026 Boxing has a list of loudmouth legends.. Ali, Tyson, Foreman, Fraizer, Maywhether, Apollo Creed (shhhh), and many many more\u2026 These guys, bring viewers to the sport, they put the proverbial asses in the seats.\nMMA has always had its share of characters from the start. Shamrock, Severn, Tank Abbott, Tito Ortiz, Chuck Liddell\u2026 but a lot of that had kinda disappeared for a bit while MMA was attempting to get taken more seriously and get approved by athletic commissions while fighting off the \"Human Cockfighting\" tag given to it by media. The entertainment took a backseat\u2026 but Brock brought it back and now others are following. This is a good thing for the sport, and a great thing for UFC. Fans need to let go of this purist mentality, if the want the sport to thrive, that's why the always entertaining Brock needs to win on NYE, not the boring Overeem. Alistair is a good fighter, an entertaining fighter, but he could put a tweaker to sleep on the mic. Having Overeem vs. Junior would be a great fight in the cage, but a horrible fight to build up. Junior is a great fighter, but he is the ultimate milquetoast good guy, he needs a villain to fight. Brock is that Villain. Bring back the loudmouth Brock and you have buyrates that will break records.\nNow, aside from analyzing the fight from a marketing perspective\u2026 how about in the octagon? Brock is a great wrestler and Alistair is a great stand up fighter. We all have heard the cries of \"Brock can't take a hit\", but really, how long has it been\u2026 do we know what his training has been like during this downtime? Brock has learned a lot from his last two fights, I think we will see a different stand up game from him. Apart from that, I think he will do his best to outwrestle Overeem and keep the fight in his comfort zone. The minute this fight goes to the ground it becomes Brocks fight to lose, I just don't see that happening. Alistair better load up on that horsemeat and plan on trying to take Brock out quick, If not then the fight is already over.\nSee you all on New Year's Day.\nIs Tito Ortiz making a good Case?\nGreat weekend for MMA fans!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbeqob b/data_all_eng_slimpj/shuffled/split2/finalzzzbeqob
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"View cart \"Peanut Brittle\" has been added to your cart.\nThe fudge is all goat's milk fudge. The goat's milk gives the fudge a very smooth rich flavor instead of the super sweet and dense texture of traditional fudge. Each piece is roughly \u00bc pound and should be kept in the refrigerator to keep firm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were pretty pumped to open the boxes and see what was inside. All pizza had loads of cheese and a crust with good chew. Each medium pizza had 8 slices, which was more than enough for us to feast on. The Deluxe Tropical Hawaiian had sweet + smoky tropical sauce, bacon, extra Canadian ham, extra pineapple, mozzarella + cheddar. Like a more loaded version of the original, plus a tangy sauce. I quite liked this one. Because extra bacon. The Steak Mushroom Melt had cheezy cheddar sauce, steak, bacon, saut\u00e9ed mushrooms, mozzarella + cheddar. Overall, we thought this stood well on its own, but perhaps a little bland compared to the others. If you like a white sauce on your pizza, you're in luck. The Chicken Club had a Jalape\u00f1o white sauce, grilled chicken, Canadian ham, bacon, tomatoes + mozzarella. We felt this one too stood on it's own and was straight up tasty.\nNow, the top three... The Chipotle Chicken had chipotle cilantro sauce, grilled chicken, roasted corn medley seasoned with onion, red + green peppers, topped with mozzarella + cheddar. I quite enjoyed the smoky flavour as well as the corn and notes of cilantro. Second place goes to the BBQ Chicken. BBQ sauce, grilled chicken, bacon, caramelized onions, red peppers, mozzarella + cheddar, made it a bit of a sleeper hit, and most of us agreed we would definitely order it again. The bacon and caramelized onions were a nice touch. And the first place winner, which surprised us all, was the Veggie Korma! With coconut curry sauce, roasted garlic, baby spinach leaves, green peppers, pineapple, hot banana peppers, mozzarella + cheddar cheese, it had the perfect kick of heat, balanced with the sweetness of pineapple and and earthy roasted garlic flavour. There was nothing left of this pizza at the end of the night. We demolished it. We were impressed at the creative take on vegetarian pizza, and I should note than none of us are vegetarian. We just like vegetables. A lot.\nAnd that was how I spent my Saturday night! If you like pizza and like to eat it a lot (I mean, who doesn't?) Panago Pizza has a great contest running right now. Go to their site and see how you can WIN pizza for a year by taking the #PanagoChallenge. How amazing would that be? It's super easy to enter and I encourage you to do so. I'd love to see one of you stuffing your happy faces with pizza on a regular basis!\nNote: This post is sponsored by Panago Pizza. I was compensated monetarily and with their product. All opinions are mine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Multi-instrumentalist Lots Holloway is a talented artist who produces, writes and creates all her own music. Since being signed to indie label Why I? Records in 2016, Lots has gone on to release a stream of infectious and original indie-pop tunes. From this, Lots has picked up ever more support, not just from fans and peers, but also from respected press outlets, influential radio stations and underground indie magazines.\nHolloway's talent for song writing really shines through; in clever lyrics, strong melodies and memorable songs that manage to stay with you long after you first hear them. Based around live instruments (drums, guitar and bass), Lots music spreads clever layers of sounds from the freaky and psychedelic end of the spectrum all the way to the warmer softer tones - but Holloway's music is always bursting with inventive synths and weird sounds.\nAll this adds up to something truly fresh and original, and a live show from Lots Holloway is just as compelling. Lots' on stage presence is second to none. Holloway is enchanting and captivating, and fronting a full live band she only becomes more powerful.\nI am a writer, a producer and a creator. I make music, I make videos, I make art in any form I can.\nI've been playing music since I was about 8. I became a drummer.\nI produce and mix all the final versions of my music in Bryan Ferry (of Roxy Music)'s personal recording studio.\nThis changes everyday, today I think I'd like to write with Brandon Flowers from The Killers, he's got some cracking melodies.\nMy mind has a mind of it's own. There's a lot of strange stuff in there.\nI was obsessed by my dads collection of music as a kid, I though all of the musicians were so cool - I wanted to be a rock god too.\nFunnily enough, I like to listen to music. I have a collection of Vinyls that I spin constantly in my house. If not, I just enjoy a good knees up at the pub with my friends. Simple things.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Dutchess is the debut studio album by American singer Fergie. It was released on September 13, through A&M Records and the markjosephgct.com Music Group. Double Dutchess is the second studio album by American singer Fergie. It was released on September 22, The album is the singer's first to be released. Product Description. With over 20 million albums sold worldwide with The Black Eyed Peas,the pop icon is ready for her solo debut. From performances on every .\n24 Feb That's a long time. If this album was a baby, it'd probably be old enough to speak in sentences by the time it exited the charts. Fergie's debut solo album is an opulently appointed tour of pop's preceding half- century. \"Fergalicious\" is an ultra-modern update of J.J. Fad's \"Supersonic,\" and. Fergie Officially Announces Sophomore Album 'Double Dutchess,' Unveils First Two Songs. 8\/25\/ by Taylor Weatherby. FACEBOOK; TWITTER; EMAIL ME.\n15 Sep Fergie's debut solo album, The Dutchess, was released on Sept. 13, , meaning that the album celebrated its eighth birthday over the. Find a Fergie (2) - The Dutchess first pressing or reissue. Complete your Fergie ( 2) collection. Shop Vinyl and CDs. 13 Sep Find a Fergie (2) - The Dutchess first pressing or reissue. Complete your Fergie ( 2) collection. Shop Vinyl and CDs. The Dutchess is the debut studio album by Fergie released in The album debuted at #3 on the Billboard with first week sales of , For the next. The Dutchess. By Fergie. \u2022 13 songs. Play on Spotify. 1. Fergalicious - will.i. am. 2. Clumsy. 3. All That I Got (The Make Up Song) - will.i.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A personal injury attorney Athens GA is one of the most challenging professions today. Finding a lawyer you want to hire is one thing. But a personal injury lawyer can be daunting. You may be looking for a legal assistance to take on your entire claim on a contingency fee basis, the most common arrangement in personal injury cases. There is a lot of help available.\nHowever, before you can move forward to your case, it is best if you have personal knowledge about what a personal injury attorney is all about. Talk with friends or coworkers who have been represented by a lawyer in their own personal injury claims.\nPrevious What's True about a Private Investigation Job?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfrle b/data_all_eng_slimpj/shuffled/split2/finalzzzbfrle
new file mode 100644
index 0000000000000000000000000000000000000000..726c43fb558633157233519343ba20a446014aa4
--- /dev/null
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+{"text":"A drama based on events surrounding the infamous 1819 Peterloo Massacre, where British forces fired on a peaceful pro-democracy rally at St Peter's Field in Manchester.\nThe notorious massacre saw British government forces charge into a crowd of 60,000 that had to demand political reform. The forces killed an estimated 18 protestors and injured hundreds, sparking outcry but also further government crackdowns.\nThis was a defining moment in radical British history which also played a significant role in the founding of the UK's leftwing broadsheet newspaper The Guardian.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Slot Machine Labels Add an Extra Touch of Luck!\nMake every labeling project a way to share one of your favorite activities with these casino address labels from Bradford Exchange Checks\u00ae! Each vibrant label design pairs your personal information with the artwork of artist Lori Siebert, along with a fun saying. Phrases include \"Hit the Jackpot\", \"I feel lucky\", \"Crazy about slots\" and the famous \"777\" symbol. Plus, you'll find these Lori Siebert art mail labels are amazingly practical too, arriving in a set of conveniently self-sticking labels. They're perfect for hundreds of jobs, so don't wait! Get ready to win with these Hit the Jackpot mail label designs. Order now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We ourselves are wanderers who wanted to travel more (without breaking the bank).\nSo we learned how to spot flight deals\u2026 and also many other tips along the way. Now we're sharing all of this on our website, so you too can explore the world more often (for less).\nThese deals are always for roundtrips, prices include all mandatory taxes and are quoted in your local currency.\nThe idea for Flytrippers was born in 2016 because we had long been driven by wanderlust and our desire to explore the whole world without paying full price for plane tickets. That's why we started our flight deals hunt.\nI'm Andrew, an Aviation Geek and trivia fanatic who is passionate about travelling and passionate about the aviation \/ tourism industries.\nI used to work in management consulting after obtaining my MBA, but I now use my experience to make Flytrippers the reference for low-cost travel and Travel Hacking.\nI'm Kevin, a travel enthusiast that no border stops\u2026 and now officially a nomad, having sold all of my possessions to live all over the world.\nI am a civil engineer by training, but I now pursue my passion for exploring the world and I want to inspire you to travel more. My goal is to visit every single country!\nThis website provides great savings information for traveling. Highly recommended.\nThank you for sharing offers that are so tempting. Thank you also for your travel tips drawn from your experiences. Keep up the good work!\nI managed, thanks to you, to get a flight for New Zealand for $936 and, using your search tools, multi-destination Auckland-Sydney and Cairns-Auckland flights for $390. A nice trip planned for July to celebrate my 50th birthday with my friend. Thanks again!\nThanks to you, we have enjoyed a great crazy deal... France round trip for $300 (Christmas deal). We just came back. It allowed us to show this beautiful country to our 2 young daughters for 10 days. I read all your articles !! I can't wait to plan our next trip thanks to all your tips.... thank you so much!!!\nNext October, I will visit Ivory Coast and my precious friend Sophie, thanks to one of your flight deals! Three trips this year and I know that next year new horizons are also waiting for me! Long live Flytrippers!\nI booked 2 tickets last week for Portugal, thanks to Flytrippers! The prices listed on your site matched the price I paid! All you need is a bit of flexibility with your schedule and to take a regular look at the deals page!\nI just returned from 12 days in Jordan (Amman, Petra) and Egypt (Cairo, Alexandria, Luxor, Aswan). It was thanks to a deal that appeared on your site. Since then, I have been a fan. Thank you again for your wonderful work.\nI took advantage of your deal to Europe in the $300s roundtrip in April. At this price, I can also afford a vacation down south in February!\nMany questions about Flytrippers (and about flights) answered.\nGreat content to help you travel more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Prostatic stent for the treatment of BPH.\nTight coils: no tissue ingrowth - easily removable.\nTrans-sphincterial wire: continence remains natural.\nWide base: reduced risk of migration.\nLarge internal diameter: good drainage of urine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy high quality cartridges for the Brother HL-5470dw. Buy single or multipack toners and save even more. Choose from our value-line for the best overall value, or our Premium line for 100% Made in the USA, or Original Brother HL-5470dw cartridges. Drums and other parts that accompany the HL-5470dw toners are also offered if available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbghbv b/data_all_eng_slimpj/shuffled/split2/finalzzzbghbv
new file mode 100644
index 0000000000000000000000000000000000000000..de21466850529b22b08410e42cb7f5483ee8695e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"As summer comes to an end, it's time to hang up our beach towels and get back to work. From being selected as one of the top 10 Coolest Big Data Startups of 2017, to our partnership with Looker, our summer has been nothing short of eventful. Not to worry though, summer ending doesn't mean the fun ends. We have an even more exciting Fall season heading our way. Join us as we talk about Data Operations implications in digital retail, AI, and machine learning. Hope to see you at one of our upcoming events, please stop by to say hi!\nMaking our DataOps dreams come true in Milwaukee! We'll be attending OnRamp Manufacturing Conference, the leading conference on manufacturing innovation to share how automating data flows can improve operational efficiency and drive growth. If you'll be in the area drop us a note at sales@nexla.com.\nDid you know that 91% of companies are receiving data from third party partners while 70% of companies are either currently sending or plan to send data to partners? Inter-company data collaboration powers insights, machine learning, and better consumer experiences, but it also increases workloads for engineering teams and challenges to data access.\nWe're teaming up with Qubole to host a webinar on all things DataOps during their Data Platforms Virtual Summit. Join in on our free webinar to learn more about how companies are streamlining and automating their Data Operations to accelerate the time from raw data to business value. Register here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In February 2017, I met His Holiness Dr Athavale for the first time and it was a life-changing experience. I could feel strong energy emanating from Him while He spoke.\nDuring the satsang, I felt a protective sheath form around my head, all of my thoughts disappeared and I experienced spiritual emotion. This state lasted for many hours after meeting Him.\nI am grateful to God for this satsang of His Holiness Dr. Athavale and His Divine guidance. I pray that His guidance helps me to become better each day and move forward in my spiritual journey.\nEditor's note : When a Saint or a Guru of a higher level such as His Holiness Dr Athavale shares spiritual guidance, there is a high level of Divine consciousness (Chaitanya) in Their speech.\nThis is because Gurus have put into practice and experienced the results of guidance being shared. Also due to the spiritual level of Gurus being above 70%, They are able to access the Universal Mind so there is a flow of Divine consciousness from God going to those listening to Their speech. This Divine consciousness positively impacts listeners and they can perceive something unique about Their speech. The words that are spoken by Gurus are imbibed at a subconscious level by listeners.\nThis is in contrast with inspirational speakers who may impress an audience at a psychological level, but do not have Divine consciousness in their speech. As a result, listening to them may not result in the listener making lasting changes in their lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mike Coombs Guitar and Banjo TuitionThe inner nature of mankind is the province of music.\nThis site is intended, first and foremost, for use by all my students. If you are not currently studying with me, either in a school or privately and you would like to, please Contact me by either using the link at the bottom of the page or by clicking onto the contact page and I will get back to you by return.\nFrom this site you will be able to download lots of scales and exercises and arrangements (in both music and tablature) that will help you in your study of the guitar and\/or bass guitar. There are also articles about how to study guitar and links to other useful sites.\nI wrote Mrs Mackerel's Wedding to celebrate the wedding of one of my students, Barry Pegram to his (now) wife Tricia on 5th July 2008.\nIn all my teaching I endeavour to take a student-centred approach. Each student will have different needs. Some want to become superstars, some just enjoy playing at home. Whatever the reason for taking up the instrument I will take the student from wherever he or she is and improve their skills to the limits of their ability. Some of my former students are currently working professionally as musicians, teachers or are in some way connected to the music industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each blue, capsule-shaped tablet engraved \"G\" on one side and \"S\" scoreline \"80\" on the other contains 80 mg sotalol hydrochloride. Nonmedicinal ingredients: pregelatinized starch, microcrystalline cellulose, stearic acid, FD&C Blue No. 2, lactose, magnesium stearate, and colloidal silicon dioxide.\nEach blue, capsule-shaped tablet engraved \"G\" on one side and \"S\" scoreline \"160\" on the other contains 160 mg sotalol hydrochloride. Nonmedicinal ingredients: pregelatinized starch, microcrystalline cellulose, stearic acid, FD&C Blue No. 2, lactose, magnesium stearate, and colloidal silicon dioxide.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Free Printable Coloring Cards For Adults free printable coloring cards for adults to print this free coloring page coloring adult flower with many pics to color. free printable coloring cards for adults colouring craze for adults grown up colouring books with giveaway dessin de thanksgiving. free printable coloring cards for adults free adult coloring pages detailed printable coloring pages for dessin de thanksgiving. free printable coloring cards for adults free printable coloring pages for adults only complete guide example anime coloring pages. free printable coloring cards for adults image result for free printable coloring pages for adults only swear mayan patterns to colour. free printable coloring cards for adults free coloring pages for adults easy peasy and fun simple lion head. free printable coloring cards for adults 3743 free printable coloring pages for adults mayan patterns to colour. free printable coloring cards for adults i love you coloring pages printable daddy bageriet christmas mandalas to print.\nFree Printable Coloring Cards For Adults free printable coloring cards for adults colouring craze for adults grown up colouring books with giveaway dessin de thanksgiving. free printable coloring cards for adults free adult coloring pages detailed printable coloring pages for dessin de thanksgiving. free printable coloring cards for adults free printable coloring pages for adults only complete guide example anime coloring pages. free printable coloring cards for adults image result for free printable coloring pages for adults only swear mayan patterns to colour. free printable coloring cards for adults free coloring pages for adults easy peasy and fun simple lion head. free printable coloring cards for adults 3743 free printable coloring pages for adults mayan patterns to colour. free printable coloring cards for adults i love you coloring pages printable daddy bageriet christmas mandalas to print.\nFind out the most recent images of Free Printable Coloring Cards For Adults here, and also you can get the image here simply image posted uploaded by Aurora Kaydence that saved in our collection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgmpr b/data_all_eng_slimpj/shuffled/split2/finalzzzbgmpr
new file mode 100644
index 0000000000000000000000000000000000000000..013b9696ad2d282851ba069ed9c8cbd8a27eadb9
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Anyone who has undergone heart surgery or who has had a loved one under the knife for such a procedure knows how frightening it can be. Even with modern techniques and technology, it is still a somewhat harrowing set of procedures that is approached with hope but caution. And everyone is relieved once the procedure is over and the patient has gone home safe and sound, even the surgeon.\nAnd now for the bad news: the Centers for Disease Control have issued a warning to anyone who has undergone heart surgery\u2013even those that had surgery long ago\u2013that they are at risk for a deadly bacterial infection.\nThe warning applies to people who have had heart surgery since 2012, suggesting that they could face a life-threatening bacterial infection, especially people with valve implants or prosthetic product implants.\nThe bacteria is called nontuberculous mycobacteria, or NTM, and officials are saying it could affect up to 600,000 people. Thus far 12 people have died from NTM, and while officials are saying that the risk is very low, it is nonetheless still a risk. And it is especially alarming for people who were certain their heart surgery and any attendant potential problems stemming from it were long behind them.\nSymptoms include night sweats, muscle aches, joint pain and weight loss.\nAnd while experts place the risks of contracting the disease between 100 to one and 1000 to one in hospitals where it has already been found, the warning also serves to highlight just how precarious is our relationship with bacterial outbreaks, especially those in hospital settings.\nThe truth of the matter is that, no matter how much they talk about sterilization, cleanliness procedures, and so forth, hospitals are filthy places, especially surgical theaters.\nWe've already seen our best, last-hope antibiotics prove to be rapidly losing their effectiveness against staph, gonorrhea, and other infectious diseases in recent years. Some experts think we may be facing an antibiotic -free future sooner rather than later. Cases of sepsis are on the rise too, a particularly alarming piece of news for the immuno-compromised in hospital settings.\nSo the news about NTM can be taken as an isolated medical complication affecting just a few hundred thousand people, or it can be thought of as just another example of how our routes for evading infection are narrowing at an alarming rate.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Saffron with its bright yellow hues and subtle floral and bitter notes is a spice which seems singularly exotic. But it has long been associated with Cornwall. It is said that the Cornish traded with the Phoenicians way back, exchanging tin for saffron and it's been used here ever since. This may or may not be true, but saffron was a highly popular ingredient in the Middle Ages and saffron crocuses were grown in Bude until the late 19th Century.\nA friend recently passed on a recipe for me to chocolatify. He reckoned that not only was this cake unusual, with its inclusion of Jerusalem artichokes, but it was also possibly the best cake he'd ever made. I was intrigued. At this time of year we have no problem getting hold of this particular root vegetable as it grows, almost of its own volition, down on our plot. I adore the taste of artichokes, but do find them a real pain to clean, so I don't use them as often as I probably should. The cake includes roasted hazelnuts and I could see how well these would work with the nutty flavour found in artichokes.\nI had planned to follow the recipe as written, apart from adding chocolate and using my usual half wholemeal, half white flour mix of course, but things went a little awry. I didn't have any raisins for a start, so had to substitute sultanas. But mostly, I didn't read the recipe carefully enough. I ended up using a different method entirely and added all of the sugar (50g more than I should have) to the cake rather than reserving some of it for the icing \u2013 oops! I also didn't think I needed to peel the artichokes, which I scrubbed well cutting out any bad bits.\nSome time before Christmas, I was sent three lovely bags of Cacao Barry chocolate drops. This is a new range of high quality couverture chocolate they have introduced. It uses a new fermentation method which purportedly gives a more intense taste. The Q-Fermentation TM method uses natural ferments found in the plants and soil of the plantation which is said to give a purer bean with a fuller flavour. I'm looking forward to trying the chocolate out in a few sophisticated recipes where the flavour can shine through. However, I decided as there were so many lovely ingredients in this cake it would be good to use a special chocolate too. From previous experience, I've found that milk chocolate chips tend to work better in this type of cake as a very dark chocolate can sometimes take over rather than enhancing. The 41% Alunga milk chocolate seemed ideal. With its strong caramel notes and high cocoa content, I found it hard to stop dipping into the bag as I went along. I'm looking forward to trying the Inaya 65% and Ocoa 70% dark chocolates in due course.\nAdded 1 tbsp brandy to a bowl filled with 120g sultanas and placed it on the heater to soak in for about an hour.\nToasted 80g hazelnuts in a dry frying pan for a few minutes until the nuts had browned a little and the skins had loosened. Left to cool, then rubbed the nuts in a piece of kitchen towel to remove the skins. Chopped roughly.\nGrated 200g of well scrubbed and trimmed Jerusalem artichokes in food processor.\nCreamed 150g unsalted butter with 200g soft brown sugar (should have been 150g).\nBeat in the brandied sultanas.\nBeat in 3 large eggs, one by one and alternating with a little of the flour.\nSieved in 200g flour (half wholemeal, half white), 1 level tsp baking powder, 1 scant tsp bicarbonate of soda, a large pinch of rock salt, 1 tsp cinnamon and a good grating of nutmeg (about 1\/2 tsp).\nStirred this in lightly together with the nuts and 50g chocolate drops (41% milk).\nScraped mixture into a deep 8\u2033 lined cake tin and baked for about 1 hour at 180\u00b0C (recipe stated 30 minutes, but mine was still almost raw at that stage) until well risen, brown and an inserted skewer came out almost clean.\nAllowed to cook in the tin for 15 minutes, then turned out onto a wire rack to cool completely.\nBeat 180g cream cheese (should have been 200g, but that was all I had) with 40 light brown sugar.\nGrated in the zest of an organic lemon and squeezed in nearly half of the juice.\nBeat it all together then slathered over the top of the cake.\nShaved some dark chocolate over the top.\nI couldn't have told you there were Jerusalem artichokes in the cake, but wow, I'm sure they added to the overall nuttiness. This cake was truly delicious: chewy, crunchy, moist and abundant. The Alunga buttons left chocolatey hotspots throughout the cake which contributed nicely to the overall richness of taste. The sharp lemony icing offset the additional sugar I added by mistake and the cake, thankfully, wasn't too sweet at all. It was similar to a carrot cake, only, dare I say it, much nicer.\nHow can I put this politely? I didn't notice any, er, unfortunate consequences to eating the Jerusalem Artichokes in this way, so it got a double thumbs up from us.\nThis is my offering for this month's We Should Cocoa. Linzi over at Lancashire Food is kindly hosting and has asked us to combine an ingredient we have never used with chocolate before. I was initially going to send over the paprika and cocoa roasted cauliflower that I made earlier in the month, but in the end decided this was a more unusual and worthy entry. I can honestly say, that I have never until now, eaten Jerusalem artichokes and chocolate together.\nI am also using this as my entry to Family Foodies over at Bangers & Mash. The theme this month is Hidden Goodies. These artichokes are very well hidden and I suspect few would ever guess as to what the cake contained. This challenge is co-hosted by Lou at Eat Your Veg.\nNot only made from scratch, but some of it grown from scratch too, I'm sending this off to Javelin Warrior for his Made with Love Mondays.\nAs this is the most exciting recipe I've posted this week, I'm entering it into Recipe of the Week with Emily of A Mummy Too.\nWe are just back from a rather damp week spent on the Lizard. For those wondering, the Lizard is not a large scaly reptile, although there is a reptilian link. It's a peninsula lying at the most southerly point of mainland UK and is made of serpentine rock. It's actually a piece of sea floor which somehow ended up in the wrong place. The serpentine has unusual chemical properties which leads to a unique habitat, making it a Mecca for botanists \u2013 CT was in his element. Despite the fact we only had one morning of sun the entire duration of our stay, it wasn't as wet as it might have been and we had a lovely time, spoilt only by the fact it passed far too quickly.\nBefore leaving, the big question was what cake to take with us? On our last stay there, two years ago, I made a well remembered chocolate mayonnaise cake which was not only delicious but lasted the whole week. I needed to replicate the delicious and lasting qualities, but wanted something a little different. In the end, the sheer number of apples I'd been given sealed the deal, it just had to be an apple cake. Leafing through my many recipe books and scraps of paper, I finally plumped for an Apple & Hazelnut cake. I've had this recipe for at least 14 years but have never actually made it and where it came from is now lost in the mists of time. I adapted it to include chocolate of course and made a few other amendments along the way, including brandy soaked sultanas.\nSoaked 3oz sultanas in 1 tbsp brandy for a couple of hours (overnight would have been better).\nSpent an age cracking the last of last year's hazelnuts, then toasted 3oz of them.\nWhen cool, blitzed them in a coffee grinder.\nCreamed 8oz unsalted butter with 6oz cardamom sugar.\nStirred in 2 tsp dried orange zest.\nSieved in 12oz flour (half wholemeal and half white), 1.5 tsp baking powder, a pinch of salt and 1 tsp cinnamon.\nMixed in the apple and hazelnuts.\nStirred in sultanas and 3oz milk chocolate drops (40%).\nSpooned into a 23cm cake mould.\nSliced an unpeeled dessert apple and placed slices around the top.\nBaked for 45 mins at 180C.\nLuckily the cake was delicious and it did last us the week, although with the various other treats we had whilst we were there, we probably shouldn't have had any cake at all.\nWalking the coastal path on that first sunny morning somewhere near Kennack Sands.\nSpotting our first view of Cornish Heath (Erica vagans) this year \u2013 no longer at its best but always exciting as the Lizard is one of the very few places that it grows in the UK.\nDodging showers around Trewidden Gardens, Penzance \u2013 1st visit and the most impressive grove of tree ferns we've seen in the UK.\nKynance Cove as we saw it two years ago \u2013 this time it was hard to discern through the thick mist.\nThe biggest swathe of Devil's Bit Scabious either of us have ever seen.\nThe delightful fishing village of Cadgwith.\nPosh nosh at New Yard Restaurant, Trelowarren.\nRock cakes used to be my main go to recipe when I had to bake something in a hurry or just wanted something to fill the cake tin. I haven't made them in years, I don't know why, they are so easy and really quite delicious. I also grew up with them. My mother used to make them with dates which I particularly liked. But at the moment I have lots of cooking apples that a friend has given me and I need to concentrate on using them up. Regular visitors may start to get bored with the ensuing number of apple cakes I may well be baking!\nPut 8oz flour (1\/2 spelt, 1\/2 white) into a large mixing bowl together with 2 tsp baking powder and 1 tsp cinnamon.\nRubbed 4oz unsalted butter into the flour.\nStirred in 3oz demerara sugar.\nAdded 2oz sultanas and 2oz 40% milk chocolate drops.\nPeeled, cored and finely (ish) chopped 1 cooking apple and stirred that in to the dry ingredients.\nMade a well in the centre, broke in 2 duck eggs and stirred until all combined.\nSpooned 12 large lumps onto a lined baking tray and baked for 20 minutes at 180C until lightly browned.\nMoved to a wire rack to cool.\nThese were just as good as I'd remembered them, firm on the outside and soft and delicious on the inside. This was the first time I'd ever used chocolate in rock cakes and as I'd guessed, it worked really well. The sweet bits of sultanas and chocolate made an excellent contrast to the sharp and flavoursome apple pieces. They were destined for friends where we were going for supper. However, I managed to snaffle one for myself, save one for CT and rush down the road to my local bookshop with two of them on a saucer, where they were gratefully received. We are lucky enough to have an independent bookshop in our small town. It is a marvellous place which somehow manages to keep going whilst many shops are closing around it. They always give excellent service and are friendly and knowledgeable. They may be small, but CT hasn't found a book yet that they haven't been able to order for him and he orders some rather obscure items. Surprisingly, for such humble cakes, these had rave reviews from all who ate them.\nCheekily, I'm linking this to Vanessa Kimbell's lovely idea to Practice Random Acts of Baking. Vanessa, I had a much more random and kinder plan that this in mind, but the bakes were a disaster and I had to abort \u2013 I will relate all at a later date.\nI am just about to leave on a week's holiday \u2013 hoorah! It is two years since I had a proper holiday and it feels very much needed. We are not going far, we're off to the Lizard and I can't wait!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Send your email to Anz Darwin at the above address or you can personally visit the place on the given address.\nCommunicate with Anz Darwin through the email address displayed above for any query, help, suggestion and feedback.\nPick up your phone and call Anz Darwin on the specified contact number.\nLet Us Know HERE if the provided contact numbers, email address or website of Anz Darwin are invalid.\nAre You Satisfied After Making Call To Anz Darwin?\nOwing to the highly innovative service alternatives, Anz Darwin is committed to configure a platform of financial services. Anz Darwin is dedicated to specialize people with legitimate backing, consuming a least interest cost for it's glorious role. However, the financial corporation is premier in providing personal and business loans.\nAnz Darwin resides at 69 Smith St, Darwin City NT 0800, Australia provides here all the necessory details like helpDesk helpline numbers i.e. +61 13 13 14 by which customers can reach to Anz Darwin easily. Here on this page you can get customer service toll free helpline number of Anz Darwin. you can contact them by their email address origserv@anz.com. Go to Anz Darwin website that is https:\/\/anz.com and get more information from there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Plans by two hap\u016b of Ng\u0101ti Pikiao to expand their Wahanui urup\u0101, or burial site, near Lake Rotoehu have received a boost thanks to a $36,000 grant from Transpower's CommunityCare Fund.\nThe Wahanui urup\u0101 has been a culturally significant site for over a century. It represents the final resting place of many hap\u016b (is this needed) members of Ng\u0101ti Tamateatutahi and Ng\u0101ti Kawiti.\nA goal of both hap\u016b has been to extend the urup\u0101 to ensure whanau can be buried close to their t\u016bpuna, while improving access for those who visit.\n\"Cultural heritage is very important to us and defines our hap\u016b identity,\" Trustee of the Urup\u0101 Trust Davey Gardiner said.\n\"The urup\u0101 is of particular significance as the resting place for many of our t\u016bpuna and has been for several generations now. They might no longer be with us in a physical sense, but spiritually they are always here \u2013 we welcome them with every powhiri,\" Mr Gardiner said.\nThe $36,000 from Transpower's CommunityCare fund will go some way to help fund new concrete beams and berms to support new burial plots at Wahanui urup\u0101.\n\"Transpower's network is all about sustaining connections, so I'm delighted we were able to facilitate that connection for ngaa hap\u016b of Ng\u0101ti Pikiao with this grant.\" Transpower Stakeholder Engagement Manager Geoff Wishart said.\n\"In this case one of our transmission lines is literally just over the road from the urup\u0101, and Ng\u0101ti Pikiao have been very gracious hosts for many years so it felt only right to contribute, \"Mr Wishart said.\nWith the bulk of the funding for Wahanui urup\u0101 now secured, tenders will be reissued to update original quotes with a view to start work year.\nSince inception of Transpower's CommunityCare Fund in 2007 over 450 community projects around the country at a total value of nearly $10 million have been funded.\nFor further information please contact Davey Gardiner, Ng\u0101ti Pikiao on 027 3627606 or Geoff Wishart, Transpower on 021 574 877.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you an employer or HR? Chances are you have Millennial employees on your staff.\nThe following infographic provides you with valuable information about this demographic, and how you can engage your Millennial employees in health benefits.\nHelp them understand their health benefits \u2013 Millennials place a high value on the importance of health benefits, but they can be turned off by the complexity of medical insurance and health benefit plans.\nAvoid \"one size fits all\" benefit plans \u2013 Millennials grew up in a world of choices, and they expect it with their healthcare benefits. Provide a variety of benefit options so employees can select the type of plan that fits their needs.\nOffer high-end plans \u2013 While some Millennials just want basic healthcare, others will pay more for comprehensive plans that offer a full range of healthcare coverage.\nInclude dental, vision, and prescription drug coverage \u2013 For years, these were considered optional coverages. Millennials want and expect them as part of their benefits.\nOffer tax-advantaged benefit accounts \u2013 As with other generations, Millennials want tools to help control healthcare costs. Offering Health Savings Accounts (HSAs), Flexible Spending Accounts (FSAs) and Healthcare Reimbursement Arrangements (HRAs) gives Millennials more control over how they spend their healthcare dollars.\nMake it fun \u2013 Implement employee engagement programs that provide fun, interactive educational materials about the benefits of tax-advantaged healthcare accounts.\nFocus on \"total well-being\" \u2013 To Millennials, total well-being includes physical, emotional and financial aspects. In addition to wellness programs, offer health classes, gym memberships, financial counseling, employee assistance programs and more.\nProviding flexible, convenient, and easy to understand benefits will encourage Millennials to engage with your company's health plan and produce happier, healthier and more productive employees.\nDataPath, Inc. is leading technology solutions provider for cloud-based FSA, HRA, HSA, and COBRA benefits administration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Over the holiday period, either go to www.nhs.uk for advice about self-care or visit your local pharmacy.\nPlease call NHS 111 for all medical advice or use 999 in emergencies.\nThe last day for prescriptions before The New Year is Thursday 27th December.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dover was brought to a standstill today after dozens of activists clashed with police in an angry protest over immigration and ISIS.\nAnti-fascist protesters burned an EU flag during a march on the seaside town and chanted 'no more refugees', while a counter-protest took place at the same time.\nAt least thirteen people were arrested during the demonstrations, including one man arrested on suspicion of possession of offensive weapons and another on suspicion of theft of a police baseball cap.\nRoads were closed and lined by police as demonstrators marched separate routes through the port town in Kent.\nOne protester, donning a South East Alliance t-shirt with the words 'F**k ISIS* on it, led the speeches and discussed the EU referendum, being held on June 23.\nSetting fire to the EU flag, he told the crowd: 'I've got the answer for the Euro here. This is your flag and we do not owe no alliance to it.\n'That's your European flag, have a look at that, put that in your papers because we are not European, we are British.\n'On the 23rd June, we urge you every single British man and woman in this country vote out.\nOther groups were also in attendance, including protesters from The National Front, The South East Alliance and The North West Infidels, who all were heard shouting 'out, out, out'.\nOnly yesterday was there a counter protest by Kent Anti Racism Network which saw the words '#RefugeesWelcome' projected onto the White Cliffs of Dover.\nPolice have imposed stricter conditions on the protests this time following a protest in January when far-right protesters clashed with anti-fascists.\nBut many locals have called on them to ban these protests altogether after tax payers had to pay for the damage following January's protest.\nAnother protest has been planned for April 23 which might see the temporary closure of some busy roads such as the A20.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"nice, mainly black and white site with an extensive portfolio of Artworksgroup, ID consulting company.\npersonal portfolio of Aline Caron, with nice navigation.\ndesign portfolio by Loginoff com a design firm based in Moscow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Now, be honest, are you a creature of habit? Think back to yesterday. How much of it was a repetition of the day before?\nPsychologists reckon as much as 95% of our days are made up of habits. Repeated patterns of behaviour. Doing and saying the same things. No wonder we get into a rut. And research I've been conducting shows that if we can break free from our rut, drop some habits and do something different, our lives will be happier and healthier.\nFor example, in a published study I reported that people who are more habitual also tend to weigh more. Our research team thought that if we could increase people's behavioural flexibility their BMI might go down \u2013 in other words, they would lose weight. This is exactly what we found. People changed a small habit every day for 28 days. They all lost weight. And they kept the weight off. And at the end of the programme they were also less anxious and depressed than they were at the start. The Do Something Different weight loss programme is in The No Diet Diet (Orion, 2006) and now available in 23 languages worldwide. We've gone on to apply the technique to other behaviours, including money management in Sheconomics (Hodder, 2009) and smoking in Love Not Smoking: Do Something Different (Hay House, 2011).\nSo don't even think about using willpower to change any aspect of yourself. Who has any willpower anyway? I don't think it exists! It's much easier to Do Something Different. This weakens old habits and puts new, healthier ones in their place. Why not try one of these today?\nRemember, if you want to get something different, you have to do something different.\nSuch an inspiring post. I've just received my copy of Karen Pine's Do Something Different journal and am really excited about starting the process of trying a different thing each day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are a small and family friendly dental team working in a state of the art private practice located in Melbourne's outer northern suburbs. We are dedicated to providing and delivering top quality dentistry care to all our patients.\nOur convenient location on the first floor of the modern South Morang Corporate Centre, (Cheesecake shop and Bicycle Centre downstairs) has ample parking for patients . We have appointment hours that fit your schedule. and we take the time to discuss your needs and expectations as we do not rush our patients or dentists. Contact our practice today and discover an innovative dental practice that is centred on your health and comfort.\nWe have an online booking system as well as contact forms available on this site. You can also reach us by phone or via email. We keep in touch by SMS and send appointment reminders via email.\nOur Dental professionals provide comprehensive services, including both general and cosmetic dentistry in a modern clinical environment. Our team constantly strive for excellence as well as delivering exceptional standards in patient care and infection control.\nOur friendly staff is dedicated to caring for you and your entire family's oral health and well-being. We take the time to create a personalised treatment for each patient and we offer a wide range of services. It is our goal to make your visit with us an extremely pleasant one and to send you home with a beautiful and healthy smile.\nSmille Ville offers modern, advanced dental care in a friendly atmosphere. We truly care about our patients and perform only the most thorough and comprehensive services available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A cover with capillary barrier effects (CCBE) was constructed in 1998 on the abandoned Lorraine mine tailings impoundment to limit the generation of acid mine drainage. The Ministry of Natural Resources and Fauna of Quebec (MRNF) is responsible for the site and for all restoration works on it, including CCBE construction. The CCBE is made up of three layers: a 0.3-m layer of sand used as a support and capillary break layer; a moisture-retaining layer with a thickness of 0.5 m (this layer is constructed of a nonplastic silt); and a 0.3-m sand and gravel layer on top. The main objective of the CCBE is to maintain one (or more) of the layers at a high degree of water saturation to impede oxygen migration and acid generation. Vegetation succession on the Lorraine CCBE results in an improvement in soil conditions, leading to the installation of deep-rooted species, which could represent a risk to CCBE long-term performance. Hence, the characterization of vegetation succession is an important aspect of the monitoring strategy for the Lorraine CCBE. Species occurrence was documented, and depth of tree roots was measured by excavation on a regular basis. Eight functional groups of plants were identified; herbaceous plants were the most abundant ecological plant groups. Tree ring counts confirmed that tree colonization started the year of CCBE construction (1999). Of the 11 tree species identified, the most abundant were poplar (Populus spp.), paper birch (Betula payrifera Marsh.), black spruce (Picea mariana Mill.), and willow (Salix spp.). Significant differences in occurrence related to environmental conditions were observed for most functional groups. Root excavation showed that tree roots exceeded the depth of the protective layer and started to reach the moisture-retaining layer; in 2008, root average depth was 0.4 m and the maximal root depth was 1.7 m. Copyright \u00a9 2011 by the American Society of Agronomy, Crop Science Society of America, and Soil Science Society of America. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"nicekitty4you1 . DeckyXXX. BustyJolie. naughtysofhia.\nKIMBIGCOCKEasygoingtwoBrazilianFitGirla0DivineAfrodita .candytreoAwesomeB0YMeYumyHoneycloud .Alaric02MaryDiamandXExoticMysteryXJohnVIP .JulietaBluesPrincessShemaleXExoticMysteryXMandyCandys .DianaSweetyKisscrazykitten125BeautyNancyLittleDiamond69 .Andy96angelTAMARAJONESAwesomeB0YBeautyNancy .BestSmileUSeeDonnaLovely4UShelbyBarnesDonnaLovely4U .MarioLuigiFireLilyTaraWoodselenaXsexy .VictoriaRosesMeYumyDeckyXXXSABRINAforLOVE .MaxAdamsCumMarioLuigicrazykitten125a0DivineAfrodita .MarioLuigiTAMARAJONESJulietaBluesCarolineDear .\nSofiaGabiatiTwinkDeanBewareOfBlondexSexyBoss .PlayYoungGeekNeitPlayYoungGeekHoneycloud .nataly69LittleDiamond69LuiSebastio21CarolineDear .MissIsabelnaughtysofhiaMichelleLewinPrincessShemale .MaryDiamandMichelleLewinFrankTraceEVAandFREDERICK .QueenTeen2TilaHotTequilaxNastyxCoupleTaraWood .KatjaZariaJaninetModel12SherryBoomb .CumPhilippinesLovelyAnnes4youKimiSANNYSABRINAforLOVE .MarioLuigiAmyHotSunriseLooeesaMaryDiamand .crazykitten125candytreoHoneycloudHOTTYREGINA .SofiaGabiatiAlaric02CouplehotLSherryBoomb .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ALAN: This time we will have been decided to have spoken about time travel stories.\nJANE: That's the trouble with time travel. We don't have the proper tenses for it in our language.\nALAN: Maybe that's a hint that time travellers haven't visited us yet.\nThe appeal of time travel actually pre-dates the SF time travel story. For example, Charles Dickens flirts with it in his 1843 story, A Christmas Carol.\nJANE: Another good example of the pre-SF time travel story is Edgar Allan Poe's \"A Tale of the Ragged Mountains\" (1844) which tells the story of a man who may or may not have travelled through time. Poe (unlike Dickens) cops out and leaves it unclear whether this was a hallucination or a genuine time travel experience.\nI'm sure there are many other examples, but what these two makes clear is that the idea of time travel has a fascination that pre-dates it becoming one of science fiction's most perennial themes.\nALAN: I think time travel, in the traditional SF sense, probably originated with H. G. Wells in his 1895 novel The Time Machine. Here travel into the future is managed by an actual machine, not a ghost, not a vision. The machine is as real and solid as a rocket ship or submarine, and so the reality of the experience is not left open to doubt.\nJANE: Certainly Wells gets the credit for inventing time travel via machine, although many have imitated him since. It's been a long time since I read The Time Machine but its vision of the future of gradual degeneration remains haunting.\nTime travel is an interesting theme because stories can take so many different forms.\nI'd say the great divide is between those stories in which the possibility of time travel is being explored and the greater majority in which time travel is an established fact and the implications are being explored.\nOne of my favorite time travel stories, Time and Again by Jack Finney, provides a fascinating \"device\" for travel through time.\nFinney's mechanism for taking his hero, Simon Morely, back to 1882 is actually rather similar to that involved in the stories by Poe and Dickens that we mentioned earlier. The hero is taken to a huge warehouse where people are acting out the daily lives of different eras. It is a project to test the feasibility of travelling to the past by self-hypnosis. Convincing yourself that you are in the past, rather than in the present, can actually make it so.\nFinney wrote a sequel called From Time to Time and it too is just as magical as the original novel. They truly are wonderful books.\nJANE: I love both of Finney's \"Time\" novels. I will argue that the mechanism is more solid that than of Dickens or Poe. Morely is awake, not dreaming, and there is no doubt that his experience is real. The explanation the scientist involved gives as to why self-hypnosis would work is just as tantalizing and provocative as any machine.\nALAN: Time travel mechanisms themselves vary from the sublime to the ridiculous. Michael Moorcock had a lot of fun with the idea in his Dancers at the End of Time sequence. His hero Jherek Carnelian travels between Victorian London and the End of Time by using a variety of devices including a bicycle, a hangman's noose and a robot nursery school teacher.\nBut of course, as everybody knows, the very best time travel device ever invented is a mid-twentieth century British police call box.\nJANE: So I have heard, although I admit, I have never seen Dr. Who. I've heard excellent things about it, but just haven't found the\u2026 wait for it\u2026 Time.\nClearly if I now say anything other than that, it will create a paradox that will destroy the universe. So I won't.\nJANE: Wise man. It's best to play it safe. You wouldn't want to have the destruction of the universe on your conscience.\nOnce the idea that time travel might become possible (no matter how the mechanisms differ) was firmly established, SF became full of what I like to call \"tourist in time\" stories. These go in all sorts of directions, but they often involve the implications of time travel, especially the question of what might change because people are travelling in time.\nI'm blanking\u2026 What's the story in which someone steps on a butterfly and goes back to find everything has changed?\nALAN: It's by Ray Bradbury. It's called \"A Sound of Thunder\" and it's an absolute classic. The killing of a single insect millions of years in the past drastically changes the world. It's a very effective dramatisation of a mathematical concept known as the butterfly effect which was first formally defined by the mathematician Edward Lorenz. It describes an idea in chaos theory where very small changes in initial conditions can result in vastly different outcomes.\nJANE: That's it! So Lorenz is the one who talked about a butterfly flapping its wings and thereby generating a far-away hurricane or some such?\nALAN: I agree. Let's see how many I can come up with.\nThere are stories where the protagonist is accidentally transported through time (usually into the past); stories where the protagonist belongs to some kind of quasi-governmental organisation that is responsible for ensuring that history always follows the \"proper\" course so as to avoid paradoxes; stories where other times are plundered for resources or used for tourism. There are also stories where war and conflict take place across time rather than in space.\nJANE: And it's important to note \u2013 since we're talking about time travel as a theme \u2013 that the police stories, war stories, and exploitation stories often overlap. After all, exploitation of the past can be a reason for police to take action, war can follow from failed police action.\nBasically, we don't have a bunch of different themes here, we have ingredients for interesting recipes.\nNext time maybe you can tell me about your favorite accidental time travel story?\nThis entry was posted on October 27, 2016 at 1:00 am and is filed under Thursday Tangents.\tYou can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.\nMy favourite time-travel novel is also Time and Again, but the other one (it's one of those titles that's just too good not to use.) Interestingly Simak also does a fantastic job of capturing a nostalgic milieu.\nThere are other kinds of time travel stories, from Slaughterhouse-Five to Chiang's wonderful short story \"The Story of Your Life\" (which it looks like they're about to Hollywoodize as \"Arrival\"). Speaking of short stories, there's also the Callahan's Bar series.\nThe odd thing is that we separate FTL travel as spatial movement wherein time tracks forward, and time travel as a temporal movement wherein space says static. This isn't so, of course, and any ship that could move faster than light could also travel in time. The Earth moves at something like 200-500 km\/sec relative to the center of the Milky Way, so if you have time movement without space movement, even a one second time travel while staying in the same place will put you into space. Very few SFF books actually move in four true dimensions, which is kind of a shame. After all, a time travel ship could launch itself 200-500 km into space just by jumping one second forward in time and not moving relative to the center of the Milky Way. Douglas Adams is one of the very few to goof around with this idea.\nNow if you want to get mythological, how about the Australian Dreamtime or the (to me even odder) Mojave dream universe?\nAt a science fiction convention, a physicist shared the idea that the reason WE don't have time travel is that a universe with time travel is energy intensive (what with changing reality over and over) and that the whole thermodynamics \/ entropy \/ things run down aspect of the universe applies there, too. A single, unchanging timeline is the lowest stable energy state of a universe.\nOh, and \"Thrice Upon a Time\" by James P. Hogan was an interesting take on time travel \u2013 information can be sent but not objects, and there's an uncertainty principle in time travel to prevent endless time loops.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Would there be any chances for an API to the site, whether to see which game is currently on-sale or which sold-out\/available\/soon to come games available in the store?\nJust keep in mind those particular APIs are subject to change at any moment without notice, and if you were to hit them more than a few times a second you run the risk of being rate-limited.\non a totally unrelated side-note, how big are you on hammering down on using bots to get daily coins?\nWe'd certainly prefer people not use bots since that really defeats the purpose of the system (getting people to check out the daily sale), so if we see obvious botting behavior we'll be taking action against that account. That being said, our primary concern is multiple accounts. If we see users with multiple accounts trying to take advantage of the coins system all associated accounts will be shut down.\nSorry for bringing up such an old topic.\nI registered specifically to reply to this.\nI own a gaming community, and have been building a website from the ground up in python.\nWe have a \"deals\" page that our members and staff can contribute to.\nWe have seen an increase of posting of deals from the Chrono.gg site. I figured I might as well make it official, and have the deal update automatically every day.\nI searched around for an API and found this page.\nI can get the image, link, and data updated automatically via the API, but theres no description.\nIs there a way to add a short description of the deal\/game in the API?\n@dusty can correct me if I'm wrong, but I believe the new API we're working on will include that. If not, since we pass along the Steam link\/ID, you should be able to pull the description from Steam.\nHi @lonin thanks for the response.\nDo you think it would be possible to include a short description in the new API?\nWhen abouts do you think the new API will be ready?\nI'm also noticing that the \"promo_image\" in the API is a different size depending on the sale.\nIdeally, who ever designs your promo images should maintain the same size image for all promo images in the API.\nDevelopers who will use the API would like a consistent thing going so they can better design their applications.\nHmm, it should be consistent\u2026 @frst?\nThe majority of them are the same size. Occasionally I believe we'll use an email header if the absence of having a proper promo image created for the sale so that's probably where the change is noticed.\nAnyone know if the API is different now?\nSeems the Steam ID is no longer provided.\nUse to be under the \"items\" section, but now its just blank.\nNvm, I just realized, it's because it's not a steam game.\nAlso noticed theres multiple games for sale. Is this going to be a thing now?\nI think they are going to be updating the API at some point. This isn't the first time they have had multiple SKUs for sale for an event.\nI think they are going to be updating the API at some point.\nThis site is great, and I really like to help promote it via your API.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The M18 18V Cordless Lithium-Ion 1\/4 in. Hex Compact Impact Driver is the most powerful compact impact driver in its class, offering the best power to size ratio among like competitors. A 4-pole frameless motor is designed to provide a longer life and more power than any brushed competitor. With maximum power and durability, the 2656-22CT will add efficiency to any job.\nMilwaukee-built 4-Pole frameless motor and optimized impact mechanism delivers up to 2,750 RPM, 3,450 IPM and 1,500 in-lbs. of fastening torque for class-leading performance.\nThis tool is a beast.\nReplaced my old impact driver with this one when I started building my new deck. Drives ledger bolts and deck screws effortlessly. The battery last and charges quickly.\nHave had great results with any Milwaukee Tool I've purchased in the past.\nBatteries have good long run time.\nLighter,stronger, and faster than my old one!\nI'm building a deck so I am driving 3.5\"construction screws. Plenty of power.\nI already purchased the Milwaukee 18V Driver and Drill combo kit from CPO 2 years ago. It has been an excellent performer. Wanted two additional batteries (and spare charger) for my set, and when this Driver with batteries\/charger came on sale at super price, I grabbed it. Got my added batteries and charger, and an extra driver for only $20 more. All arrived in great shape, which has been my experience in buying over 9 items from CPO in the last several years. Used the new and and older drivers to build deck with my son last week--worked perfect. Think the 18V Milwaukee drivers are excellent balance of power and weight.\nI'm building a front porch. Perfect for driving fasteners. It has an led light which comes in handy. Batteries charge very fast.\nI love the entire Milwaukee M cordless line. I have both M12 and M18 tools and appreciate the interchangeability, durability and long life of the lithium ion battery packs. I have dropped these tools a number of times without issue (not from a 12-foot ladder onto concrete), but significant drop impact nonetheless. I have use the M12 and M18 products for cabinet installation, furniture construction, floor decking installation, framing, drywall screw installation. I really like the battery life indicator light on the M18 battery packs (on the M12 tool itself). With the included rapid charging base and more than one pack, I have never run out of battery life and stalled (longer than it takes to change out the pack) while completing a job. I can't say enough about these drivers (and the entire Milwaukee line, for that matter). Simply highly recommended.\nI was very surprised at how well it put a screw into 4\" of wood with ease! I am a woman, and bought this so I didn't have to ask my friends and neighbor's to help me as often. Mission accomplished. I wish I had more projects.\nThis was well worth the money. It got delivered to my house in great time! I'll be making more purchases through CPO Milwaukee in the very near future.\nI USE THE IMPACT FOR RIB STEEL APPLICATIONS. WE AVERAGE 750 SCREWS A DAY AND ONLY TWO BATTERIES. HAVEN'T HAD A PROBLEM YET. BEEN USING IMPACT FOR 4 MONTHS SO FAR. HAVE DROPPED THEM OVER 20 FEET AND NEVER HAD ANY PROBLEM WITH PERFORMANCE. THE RPM IS THE ONLY REASON THEY DIDN'T GET 5 STARS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbikgg b/data_all_eng_slimpj/shuffled/split2/finalzzzbikgg
new file mode 100644
index 0000000000000000000000000000000000000000..3c9ea1f11e3b4fd5a9d5ebd97fc49bb4a63343ee
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@@ -0,0 +1,5 @@
+{"text":"Beloved and I had dinner at Jin Joo a month ago. This was during our 10th anniversary, and right after we went bowling and had lunch at our beloved Ippudo (linked is the Megamall branch, but we ate at Eastwood). This Korean place was having an ongoing promo where they offered 499 Php unlimited samgyupsal, or grilled pork. Honestly, we weren't too hungry, but I reckoned that we won't be going back to Eastwood for a while and the limited offer might no longer be available when we do get back, so we tried it out anyway.\nLike most Korean grills, they offer the usual various meats, specifically pork and beef, to be grilled on top of your table. However, they were only offering the 499 Php all-you-can-eat promo for their Samgyupsal. This comes with the usual unlimited banchan or appetizers, again a similar aspect of your usual Korean grill.\nThe first batch of Samgyupsal is already cooked. Actually, you can ask them to cook it for you. The pork they used was thicker than the unlimited samgyupsal offerings from other places. It was surprisingly cooked well\u2014not overdone and quite delectable. The seared areas gave it a change in texture from tender to a bit of a crisp.\nAdditional raw pork is given to us on a chopping board, ready to be grilled on top of our table. It came on top of a couple of lettuce leaves with more in a separate basket for us to wrap in our ssam-wraps.\nThe banchan that comes with the promo included six variants\u2014kimchi, sesame spinach, picked radish, mapo tofu, pickled vegetables, and Korean pancakes. It also came with a cup of rice. The kimchi was quite spicy, but tolerable and the spinach was my favorite, as usual.\nAnyway, part of the fun of samgyupsal is cooking the pork yourself. The tabletop grill is smokeless and has fans at the edge, but it still didn't keep all of the smoke away. Luckily, the tables also have those exhaust things that you can pull from the top to alleviate the smoke, as shown in the next picture.\nSince the slices are thicker than we're used to, it takes a while before you can actually eat. This is the time to enjoy the banchan or make your previous helping go down your stomach for more space.\nSince it's quite thick and you get an entire slab of belly, it's suggested to cut it up into smaller strips. A pair of scissors is given to each table for this exact job. As we already ate earlier, we didn't get too much refills. We only had two additional helpings of pork, since there's quite a lot already originally. Also, we only had the spinach and kimchi refilled.\nOverall, Jin Joo was a decent value. Sure, we know of places that serve the same for less, but those places usually serve much thinner slices of pork and have a less pleasant dining area, usually a bit more oily. Also, that doesn't really exist near the Eastwood\/Libis area, as far as I know. Sadly, since this is a limited promo, I don't know until when it is available. Their other items are quite pricy. However, the quality of the meat here is definitely better, thicker than the competition. While I cannot vouch for their other offerings, I can definitely recommend Jin Joo's samgyupsal for a try, especially for pork belly lovers.\nRecommended for: Banchan, Pork lovers.\nNot recommended for: Halal folk.\nSmart nomnomnom tip: Have them cook it for you if you're feeling lazy\u2014they cook it quite well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Established in 2016, Inventum Solutions is a Melbourne based company which provides the best e-solution which advances with time and technology but has traditional aesthetics.\nOur team aims to bring the most efficient products without compromising on quality with complete involvement of clients in the development process in order to fulfill their requirements. We at Inventum Solutions are focused on serving the community by making ourselves better and by growing into our full potential.\nExpanding our services across the globe isn't enough, we dream of becoming the world's leading software providers and shall work towards it with all sincerity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Western Gifts: Genuine Rattlesnake Rodeo Wallet\/Checkbook.\nGenuine Rattlesnake Rodeo Wallet\/Checkbook. Real rattlesnake with genuine leather inside..holds multiple credit cards, check book and plenty of room for cash.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Man in an Orange Shirt forms part of the BBC's Gay Britannia season \u2013 a programme of content to mark the 50th anniversary of the decriminalisation of homosexuality.\nOrphaned Michael is a captain in the British Armed Forces during the Second World War and \u2013 after leaving the army \u2013 is expected to marry his childhood sweetheart Flora. But he's long hidden his desire towards other men\u2026 that is, until he meets Thomas March, a war artist who is attached to Michael's regiment. The time they spend in one another's company releases a longing in Michael that is at odds with the conformist life he had expected to lead.\nJackson-Cohen has made his mark in a number of British TV dramas, from roles as Roddy the painter in Mr Selfridge to James, Duke of York, in ITV's The Great Fire. He's also popped up in Dracula and The Secret River and played the lead in NBC's Emerald City, although the series was cancelled earlier this year after one season.\nWar artist Thomas was rescued by his regiment's captain, Michael, during the Second World War where the two formed a bond that they couldn't ignore once fighting had ceased. But while Michael tracks him down after the war, Thomas finds that \u2013 despite their mutual attraction \u2013 Michael still intends to marry and comply with what's expected of him.\nJames McArdle played Charlie McCallister in ITV series Love & Marriage and Alec in 37 Days. He has also appeared in Star Wars: The Force Awakens and recently took to the stage in Angels in America.\nFlora is the childhood sweetheart of Michael Berryman and spends the war working in an evacuated boys' boarding school in Shropshire. The pair marry on his return from the war and Flora becomes pregnant, but she soon finds out there is more to her husband than meets the eye, namely his connection with another man \u2013 a discovery that turns her world upside down.\nJoanna Vanderham first came to attention playing Cathy in Sky drama The Runaway and has gone on to appear in Dancing on the Edge, The Paradise and One of Us.\nNow widowed, Flora still leads a busy life, but the one person she holds dear is her grandson Adam who lives in her basement flat. She dotes on him and the pair share weekly suppers \u2013 but when Adam begins a relationship with a man, all Flora's old prejudices bubble back to the surface and her torment threatens the happiness of the person she cares about most.\nVanessa Redgrave is a giant of stage and screen, appearing in more than 35 productions on West End and Broadway as well as a string of films, from Howard's End to The Bostonians to Mrs Dalloway. She won an Oscar for Best Supporting Actress for 1977's Julia.\nLucien is Thomas's bohemian landlord. When Thomas is imprisoned for his sexuality, it is Lucien who tracks down Michael \u2013 the man who means the most to his friend.\nAdrian Schiller has had a lengthy career on screen, most recently appearing in Victoria as Penge and in Beauty and the Beast as Monsieur D'arque.\nDaphne lost her husband during the war and leads a colourful dating life which she happily regales to sister Flora. Working at a London auction house in the late 1940s, she has plenty of tales \u2013 but remains unaware of her sibling's marital misery.\nLaura Carmichael is recognisable around the globe as Lady Edith Crawley in ITV period drama Downton Abbey. Since the series ended in 2015, she's cropped up in Marcella and A United Kingdom and has trodden the boards at Trafalgar Studios in The Maids.\nMrs March knows all about her son's sexuality and is called upon by Michael while Thomas is in prison. Her home serves as a shrine to her and her husband's acting careers as well as her son's art.\nWho is Frances de la Tour?\nA staple of British film and television, Frances de la Tour is known for a string of major roles, from Rising Damp to Cold Lazarus to the Harry Potter films in which she played Madame Olympe Maxime. She won a Tony award for her part in The History Boys, a role she resumed in the feature film version in 2006.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The RT9532 is a fully integrated single cell Li-ion battery charger IC ideal for portable applications. The RT9532 optimizes the charging task by using a control algorithm including pre-charge mode, fast charge mode and constant voltage mode. the input voltage range of the VIN pin can be as high as 28V. When the input voltage exceeds the OVP threshold, it will turn off the charging MOSFET to avoid overheating of the chip.\nIn RT9532, the maximum charging current can be programmed with an external resistor. For USB application, the user can set the current to 100mA\/500mA through the EN\/SET pin. For the factory mode, the RT9532 can allow 4.2V\/2.3A power pass through to support system operation. It also provides a 50mA LDO to support the power of peripheral circuit. The internal thermal feedback circuit regulates the die temperature to optimize the charge rate for all ambient temperatures. The RT9532 provides protection functions such as under voltage protection, over voltage protection for VIN supply and thermal protection for battery temperature.\nThe RT9532 is available in a WDFN-10L 3x2 package to achieve optimized solution for PCB space and thermal considerations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjfwp b/data_all_eng_slimpj/shuffled/split2/finalzzzbjfwp
new file mode 100644
index 0000000000000000000000000000000000000000..355d9653aebc21dec1c7a96fec872b289c0b7548
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjfwp
@@ -0,0 +1,5 @@
+{"text":"Remaining risk-free on the web exceeds making use of just a nickname. It suggests keeping photos that reveal your exact location off the net. It indicates recognizing that you are speaking with somebody that you understand and also count on. If you are playing a game and also fulfill someone that you do unknown on a reality level, you have to talk just about the video game that you are playing. Even if they ask you where you are from, prevent informing them that you reside on a particular street, where you most likely to school, or other information.\nWe live in a world that is managed by our computers. In a lot of homes, youngsters are taught to use their moms and dad's cellular phone and tablets or personal computers at concerning the very same time that they discover how you can chat. On the one hand, it is a terrific thing. There are many learning opportunities offered to those children due to the web as well as applications that they could make use of. As parents, you have to be certain that your child is going to be risk-free in the cyber globe. Do you have an overall understanding of cyber wellness Singapore? Or the capacity to instruct Internet safety and security? The majority of parents can not claim that they understand everything there is to understand or how to educate it to their children.\nIn years past, since the internet came to be prominent, we have actually heard stories about kids talking to individuals that their moms and dads want they hadn't. We have actually read about children giving out too much individual info and that info leading them to meet individuals that can easily do them harm. As a parent, you may try to talk to your child regarding internet safety and also cyber wellness Singapore, however that does not suggest you could make them comprehend the dangers. Our objective is to make it simpler for children to understand that the net is a fantastic location, but it additionally indicates being responsible with exactly what they share.\nThe net is a wonderful place that needs to be safe for everyone. It is a great place to go if you wish to stay up to date with friends and family that are away. It could be used as a tool for schools to keep moms and dads and also youngsters upgraded concerning things that are going on within the college and the neighborhood you live in. It allows you to socialize and also do a variety of other points. We do not wish to frighten youngsters away from it, but we do want to talk freely concerning cyber health Singapore as well as reveal them what Internet safety and security genuinely implies.\nMany kids that have actually examined and found out about internet security and also cyber wellness Singapore all state that they feel more positive when they go onto the web because they understand the limits that they need to stay within. They experience less harassing from individuals that get on the computer system. They understand that to speak to if they are being bullied and also they recognize that to look to if a person is starting to frighten them. This enables them to use the internet and delight in all that it has to provide.\nChildren of any ages could learn from virtual health Singapore They can learn more about Internet safety in Singapore and not have it be shown to them in such a way that will certainly terrify them. They just have to understand ways to use the net as a device that they can pick up from as well as communicate with. Doing so, can develop their self-confidence as well as make them far better internet surfers.\nAmong the most recent, worst feasible circumstances is online bullying. It is taking place anywhere. Every child is a possible target. Every kid that has a preferred game that they play or a social networks website that they see can come across someone who bullies them. Your kids have to recognize the risk and they likewise need to understand just what it requires to protect themselves. That they should speak with regarding a prospective risk with cyber bullying. Exactly what they might have the ability to do to guarantee they typically aren't the ones who will certainly be harassed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Do the Hustle,\" the Bump, the Funky Chicken, even the Bus Stop! Time to dust off those platform shoes and polyester pants and gear up for the Disco themed Father Daughter Dance! Fathers and daughters of all ages are welcome.\nTwo winners will each snag a pair of tickets for the Disco!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This all-new Renault Megane RS has got a hell of a lot to live up to. Namely, the reputation of its predecessor, which still looms large in the hot hatch landscape as quite possibly the finest front-wheel drive hot hatch, like, ever. No exaggeration. It'd be a very easy argument to make, and probably win.\nSo it's actually a very good thing, then, that this all-new Megane RS remains front-wheel drive and has eschewed the class trend for upping the power output to 300bhp and beyond.\nLooking past a lairy bright yellow or lurid orange paintjob is often difficult to do, especially when it's wrapped around a car with 276bhp and a Renaultsport badge on the back, but once you do it's difficult to be massively excited by the new Megane RS's looks.\nTaking a glance at its competitors \u2013 the Ford Focus RS, Honda Civic Type R, even the silly powerful Audi RS 3 \u2013 the styling is positively muted by comparison, almost sensible. There are a few clues that it's packing something special under the skin: a big grille, pressure\/drag-reducing vents on its flanks, massive wheels, an airflow friendly splitter under the rear, and a centrally mounted exhaust to name a few.\nThe hottest version of the Megane RS packs 296 bhp and up to 310 lb-ft of torque.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Science communication speaks directly to the child in us all. It keeps our instinctive passion for learning alive and allows our sense of wonder and limitless possibilities to make the leap from dream to reality.\" \u2013 David Hewlett, actor, writer, and director. Originally posted on September 22, 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ACPJ brought the AFSC's (American Friends Service Committee) national exhibit, \"Eyes Wide Open\" to Little Rock. The exhibit highlighted the human losses from the Iraqi war. It consisted of pairs of boots, each standing for one service man or woman killed; shoes representing the many Iraqi civilians who perished\u2013 adults and children \u2013were included.\nACPJ organized periodic demonstrations against the Iraq war, allowing citizens to make their voices heard.\nAlong with coalition partners, ACPJ supports activities associated with the International Day of Peace on September 21st of each year. This is observed with a week-long series of events (Arkansas Peace Week) highlighting local peacemaking\/peacekeeping efforts. The week closes with an exciting \"Peace Fest\" celebration for the community in downtown Little Rock.\nACPJ sponsored the nationally-acclaimed \"Unembedded\" photo exhibit about the war in Iraq, produced by four international photo-journalists who were not officially \"embedded with military forces.\" The exhibit was held at the UAMS College of Public Health and Philander Smith College.\nIn conjunction with UALR faculty, ACPJ organized a series of \"Know the Candidates\" events open to the public during national election seasons. Informed faculty present information on the candidates, encouraging public discussion.\nWith help from the AR Humanities Council and UALR's Middle Eastern Studies Program, ACPJ brought Dr. Erica Chenoweth, lead author of \"Why Civil Resistance Works \u2013 the Strategic Logic of Nonviolent Conflict.\" She spoke at area colleges, high schools and at the Clinton School for Public Service about her research into nonviolence.\nACPJ annually commemorates the atomic bombs dropped on Hiroshima and Nagasaki, in August, 1945, recommitting ourselves to help make the planet free from nuclear weapons.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjkmk b/data_all_eng_slimpj/shuffled/split2/finalzzzbjkmk
new file mode 100644
index 0000000000000000000000000000000000000000..c91c6cc5ea57e0a9ba6704aa956b0447d0080ad5
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjkmk
@@ -0,0 +1,5 @@
+{"text":"My Ford Expedition warns me when I need an oil change. It gives me early warnings, then it tells me when I MUST change the oil. I admit that I like that feature very much!\nRecently, I learned about a site called \"Motor Oil Matters\" that includes a ton of great information. I admit, Bo takes care of a lot of our car issues, but I am not helpless. I liked the fact that the site includes an informative checklist -- which was useful last week when I had my oil changed in my truck.\nOne of the things I never thought about was the viscosity of the oil! For the first time ever, in response to reading through the checklist, I looked into the owner's manual to find out what was recommended. This time when I went to have the oil changed, I knew more about what to ask. In the past, I just hoped they would take care of me. Thanks to the information on the site, I knew I could take care of myself!\nI felt more empowered and prepared when I took my Expedition in for an oil change. You can, too, if you visit Motor Oil Matters!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IRON MAIDEN have announced their most ambitious and extraordinary touring plans ever \u2014 and are very pleased to announce that this will include Australia for the first time in 15 years!\nIn an historic first, IRON MAIDEN, their 60-strong crew and support staff and over 12 tons of equipment will be traveling around the world en masse for the first leg of the tour in a specially commissioned and converted Astraeus Boeing 757 decorated with MAIDEN and Eddie designs! Vocalist Bruce Dickinson, who is a qualified airline captain flying for Astraeus Airlines, will pilot the plane on its epic mission, flying over 50,000 miles.\nTo tie in with forthcoming 2008 releases on DVD of the classic \"Live After Death\" and \"Maiden England\" concert videos (more details coming soon) this tour, aptly entitled \"Somewhere Back In Time,\" will revisit the band's history by focusing almost entirely on the '80s in both choice of songs played and the stage set, which will be based around the legendary Egyptian production of the 1984-85 Powerslave tour. This will arguably be the most elaborate and spectacular show the band have ever presented, and will include some key elements of their Somewhere In Time tour of 1986\/7, such as the cyborg Eddie.\nHistorically, this will be MAIDEN's fourth Australian Tour having first toured in 1982 when \"The Number of the Beast\" was Australia's number one album, then again in 1985 with the original \"Powerslave\" show and most recently in 1992 with \"Fear of the Dark.\" However a great deal has happened with the band since then, especially in recent years when ticket sales have continued to explode around the world as the band's loyal fan base continues to flourish. As MAIDEN have never had airplay or much video channel exposure anywhere in the world their popularity spread mainly by word of mouth on the back of a legacy of many great albums and an awesome live reputation. With the ever increasing power of the internet the MAIDEN legend has fired the imagination of new young fans around the planet, and, although their audience spans two generations, it is now predominantly a young one, inspired by the massive legacy and influence of this unique band (and, of course, Eddie).\nTo allow IRON MAIDEN fans in Australia to get right up close to the band, the fan club are organizing advance booking of tickets for members and running a special draw for them whereby 60 winners and friend at each show will be given first access to the venue and be first to the stage barrier. Full details will be on www.ironmaiden.com.\nTickets go on sale to the public 9am Thursday, September 27, but members of the IRON MAIDEN fan club get exclusive first access to purchase tickets from 12 noon Thursday, September 20 through to 12 noon Monday, September 24. More details soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"is Lovin these cool temps!\nPlease do post your charts.\nLast edited by STephenKaye; 07-29-2013 at 08:20 AM.\nOver 300 no way, unless they r high alt with major correction factors.\nPure stock with stock motor mounts at the dyno today 229\/262.\nI'll post my sheet since I can scan it when I am back home next week.\nshogodz89 and nyyanksrob like this.\nShould add this was a Dynojet and they had my engine cover off. Not sure that does anything but someone mentioned it somewhere else also. Also, since I was on vacation and it was a busy day, we could only wait like 5 mins between the 2 pulls.\nThis stock at 97F? Not bad then.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mrs Lovett (Joanna Lumley) was Sweeney Todd's accomplice (it was her pie shop he supplied) and lover. She had rather a nasty problem with her scalp and very bad teeth to go with bad attitude.\nI have no information on the makeup credits for this TV movie, if anyone can help, please email themakeupgallery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our consultative approach, tools, protocols, and strategies are based upon a wide range of education and policy oriented literature. Some of the key research, policy, and theoritical documents (all continiously evolving) that informs and grounds our work are listed below.\nCoburn, C.E. (2001). Collective sense-making about reading: How teachers mediate reading policy in their professional communities. Educational Evaluation and Policy Analysis, 23(2), 145-170.\nChildress, S., Elmore, R., and Grossman, A. (2006). How to manage urban school districts. Harvard Business Review, 84(11), 55-68.\nCorcoran, T., Fuhrman, S. H., & Belcher, C. L. (2001). The district role in instructional improvement. Phi Delta Kappan, 83(1), 78-84.\nDarling-Hammond, L. (2003). Enhancing Teaching. In William Owings and Leslie Kaplan (Eds.), Best Practices, Best Thinking, and Emerging Issues in School Leadership. Thousand Oaks, CA: Corwin Press.\nDwyer, C. (2005). Leadership Capacities for a Changing Environment: State and District Responses to the No Child Left Behind Act of 2001. Providence, RI: The Education Alliance at Brown University.\nElmore, R.F. (1980). Backward mapping: Implementation research and policy decision. Political Science Quarterly, 94(4), 601-609.\nFuhrman, S., Clune, W.H., & Elmore, R.F. (1988). Research on education reform: Lessons on the implementation of policy. Teachers College Record, 90(2), 237-257.\nGigerenzer, G. & Selton, R. (2001). Rethinking rationality. In G. Gigerenzer & R. Selton (Eds.), Bounded rationality: The adaptive toolbox. Cambridge, MA: MIT Press.\nHannaway, J. & Woodroffe, N. (2004). Policy Instruments in Education. Review of Research in Education, 27, 1-24.\nHargreaves, D. (2003). Working Laterally: How Innovation Networks Make an Education Epidemic.\nHatch, T. (2001). Incoherence in the system: Three perspectives on the implementation of multiple initiatives in one district. American Journal of Education, 109, 407\u2013437.\nHonig, M, & Hatch, T. (2004). Crafting coherence: How schools strategically manage multiple, external demands. Educational Researcher, 33(8), 16-30.\nLeithwood, K., Jantzi, D., and McElheron-Hopkins, C. (2006). The development and testing of a school improvement model. School Effectiveness and School Improvement, 17(4), 441-464.\nLusi, S. (1997). The role of state departments of education in complex school reform. New York: Teachers College Press.\nMartin-Kniep, G. (2007). Developing Learning Communities that Learn Lead and Last: Building and Sustaining Educational Expertise: San Francisco: Jossey-Bass.\nMcLaughlin, M.W. (1987). Learning from experience: Lessons from policy implementation. Educational Evaluation and Policy Analysis, 9, 171-178.\nO'Day, J. (2002). Complexity, accountability, and school improvement. Harvard Educational Review, 72(3), 293-329.\nOgawa, R.T., & Bossert, S.T. (1995). Leadership as an organizational quality. Educational Administration Quarterly, 31(2), 224-243.\nSenge, P.M. (1990). The fifth discipline. New York: Doubleday.\nSpillane, J. Distributed Leadership. San Francisco: Jossey-Bass, 2006.\nSpillane, J.P., Reiser, B. & Reimer, T. (2002) Policy Implementation and Cognition: Reframing and Refocusing Implementation Research. Review of Educational Research, 72(3), 387 431.\nStone, D. (2002). Policy Paradox: The Art of Political Decision Making. New York, NY: W.W. Norton & Company.\nWise, A. (1984). Why education policies often fail: The hyperrationalisation hypothesis. In J.J. Prunty (Ed.), A critical reformulation of education policy analysis. Geelong, Victoria, Australia: Deakin University Press.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The swimming pool forms the shape of the Swiss cross. Photo: S. Auger.\nA distinctive new floating swimming pool opened in Geneva on Wednesday night \u2014 at an opportune time, given the city's ongoing heatwave.\nThe new pool at Pont de la Machine, called L'amarr@GE, pays homage to the Swiss flag, with three bathing areas joined to form a cross, set within a red square.\nIt opened on a day when Genevans experienced a fifth day in a row with high temperatures hitting or exceeding 30 degrees.\nThe pool is the result of a design competition launched in the run-up to May 2015, the bicentenary of Geneva's accession to the Swiss Confederation.\nThe 900-square-metre pool can accommodate up to 350 bathers, with entry costing two francs for adults and one franc for children.\nIt will remain open until the end of September, before reopening next summer.\nGE200 is organizing a number of events for the bicentenary, which will also be celebrated at this year's F\u00eates de Gen\u00e8ve, from July 17th to August 10th.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meagan Volmari graduated from Charles Sturt University with a Bachelor of Speech Pathology (Honours). As a Canberra local, Meagan opened Can Speak to help provide family centred speech pathology services within her own community. Meagan is experienced in working with children, adolescents and young adults, and enjoys watching her clients and their families achieve their goals. With experience working within a clinic room, child's classroom and home, Meagan can provide a service that best meets your family's needs.\nMeagan is a member of Speech Pathology Australia and is a certified practising speech pathologist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cayman Round TrayCayman Round Tray, alt=\"Cayman Round Tray, \"\nPrint A popular design from our very own pattern library joins the mix as an entertaining essential. Made of laminated birch, it's exceedingly sturdy and a breeze to clean.\nCrafted by a 65-year-old Scandinavian company.\nPrinted paper and laminated FSC-certified Scandinavian birch.\nFeatures a pattern we designed in-house.\nEstimated ship date is 5\/23\/2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I \u2661 Homies with Extra Chromies \u00ae -This is a screen-printed t-shirt to help promote acceptance of individuals with Down syndrome. This design is available in infant, toddler, youth, and adult sizes. Sizes range from infant 6 months - Adult 5XL - colors and styles vary by size - see other product listings.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sign up for Broadjam today to follow Terri DiMarco, and be notified when they upload new stuff or update their news!\nUtopia is an acoustic rock album with a twist. The songs range from blues and ballads to rock and AAA format, filled with tasty guitar riffs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Welcome. I am so glad that you are here.\nI'm a marketing and business coach, strategist, and speaker who teaches amazing coaches and consultants (and other super cool service based entrepreneurs) how to attract more clients, make more money, and have more fun. My clients have a message to share. An impact to make. And a fabulous lifestyle to live.\nUp until now you may not have thought that you could make it happen and I'm here to tell you quite the opposite.\nWith the right tools and the right roadmap, you can get there pretty darn quickly (just like I did).\nIt boils down to you taking steps to do things that you haven't done before\u2026so that you can have results that you haven't achieved before.\nReady to get new results?\nPlease click one of the options below to see how I can help you make more money in your business. Or check out the blog.\nPractice taking action. When you do, you'll see results. I promise.\nSo\u2026are you ready to make your unique and beautiful mark on the world?\nI knew I had the potential to do more as a coach, but I was floundering.\nI needed a coach that would give me concrete tools and strategies to help me get focused.\nNow, after working with Krista, I have FOCUS! I know where to spend my time, and what to say no to.\nI now have a system to recruit clients from my speaking engagements. My last webinar resulted in 50% of participants requesting strategy sessions, and 50% became clients. I have a system in place.\nMy ideal clients are now approaching me, rather than me having to chase them down.\nKrista truly believes in me, even when I am doubting myself. She helps me prioritize and ask for help. She offers tools and resources that allow me to spend my time doing the work I love, and less time doing the work I don't!\nWorking with Krista has been a transforming experience.\nShe has given me the confidence, tools and ability to execute on a strategic roadmap to achieving my goals. Krista is always in your court and never lets you sit a day \"on the bench\" alone and without encouragement. Having that consistent support, care and attention to detail has made all the difference.\nKrista helped me stop over-thinking and second-guessing. I'm not \"Ms. Zen\" now by any means, stress is just part of being successful. I'm now less scared of messing up. You might call that freedom, or courage, or being open minded. But really, when it comes down to it, Krista has just helped me make it not so painful to get what I really want.\nWorking with Krista has truly changed my way of looking at my business!\nShe helped me to get more clarity around my ideal clients, my offerings and even my 10 year plan.\nKrista is able to quickly plug into where your ideas or business plans are lacking and what's needed. And she provides succinct, practical and actionable solutions that are tailored to the way *you* do business\u2013 no one-size-fits-all plans or stock answers that other coaches tend to provide.\nHer unique blend of business savvy plus life coaching isn't like any of the other coaches in the business. I loved working with Krista and I would recommend her to anyone that is looking to take their business to the next level.\nWant to get FREE tips on how to powerfully create your business?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The xts package by Jeff Ryan and Josh Ulrich is an immensely powerful tool that is widely used for timeseries work with R. Recently, the question about how to use it from Rcpp came up on StackOverflow and in a thread on the rcpp-devel list.\nIn fact, xts has had an exposed API since 2008, but it wasn't used and as I found out also not quite for two key functions. Jeff kindly gave me SVN access, and I updated init.c (to export) and a new xtsAPI.h header (access these).\nThis short post will show how to access this functionality using the new RcppXts package.\nWe start by repeating the (updated) createXts() function from the previous post on xts and Rcpp. This helper function (or an improved version of it) should probably go into RcppXts.\nNext, we show how to use this. Rcpp attributes will find the xts header file if use a depends() attribute, and as the functions we access are registered with the surrounding R process, no linking is needed.\nThanks to this new (tree-line !!) function, we can combine xts object at the source level.\nNotice the difference between the results driven by the third argument about removal of duplicates which has a default value of TRUE.\nWhile this example was obviously very simple, we can see the power and promise of this. It derives from being able to work on large numbers of xts objects directly at the C++ level without having to call back to R. So even though xts is about as efficient as it gets, we should be able to make nice gains (for simple enough tasks) by doing them at the C++ level.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today, I am going to review Dove Nourishing Oil Care Conditioner. Dove recently launched all their products in a new avataar, which includes all their face washes, shampoos and conditioners. Among all the new products, they have also launched a new Dove Nourishing Oil Care range which includes shampoo, conditioner, serum and hair mask.\nLast year, I have had a bad phase, health wise. I was in the ICU for 3 months and I lost most of my hair due to all the medications. I had to cut my hair short and it had become all frizzy and unmanageable. Now, I have new hair growth, but it is looking really weird. People giving me weird looks after noticing my hair made me all the more self-conscious. I initially straightened the hair using a hair straightener, but it damaged the hair even more. Now, I use good hair products according to my hair type (Thanks to IMBB gals for it and I'm really grateful to you all because it worked). Also, I cover up all the flaws of my hair using a hair band and clips.\nSo, coming back to the review. I have been using Dove Intense repair conditioner for a while and it worked quite well for a while. So, when Dove Nourishing Oil Care Conditioner was launched, I decided to try this. After having used it for more than a week now, I decided to share my experience with all you gals.\nI am sorry if you find the pictures too flashy, but the packaging is all gold and reflects light like anything.\nMRP: Rs. 160\/- for 180 ml.\nNourishing Oil Care Daily Treatment Conditioner. Next Generation of Conditioner with a Golden Ribbon of Treatment Inside. Did you know? Dryness and dullness are signs of damaged hair leading to to hair that's weak and rough to touch. That's why new Dove with Vita Oils goes deep inside and restores smootness, softness and shine.\nDual Action Formulation : with the power of double nourishment.\nGold Stripe : Contains revolutionary Vita Oils treatment to deeply nourish and replenish essential nutrients, leaving NO GREASINESS.\nWhite Stripe : Contains conditioner that seals cuticles to lock in nourishment and smooth frizz.\nInstant Deep Nourishing Power of a treatment in the time of a Conditioner. For amazing smoothness, softness and shine.\nWater, Ceteary Alcohol, Behentrimonium Chloride, Dipropylene Glycol, Dimethicone, Glycerin, Hydroxypropyl Starch, Phosphate, Perfume, Methylparaben, Disodium EDTA, Trideceth-5, Hydrolyzed Wheat Protein, Prunus Amygdalus Dulcis (Sweet Almond) Oil, Cocos Nucifera (Coconut) Oil, Helianthus Annuus (Sunflower) Seed Oil, Magnesium Nitrate, Carbomer, Ammonium Hydroxide, Hydrolyzed Keratin, Sodium Benzoate, Methylchloroisothiazolinone, Magnesium Chloride, Methylisothiazolinone, Phenoxyethanol, Potassium Sorbate, C177492, C1177891.\nDove has always had a very good range of conditioners. They have worked well for many people, can't say the same about their shampoos. Many gals have had hairfall because of it; Dove didn't disappoint this time as well. The moment I rinsed my hair, it was all smooth and soft. It took care of the frizziness and fly-aways so well that I actually didn't need a serum after that.\nVery mild and can be used for everyday wash and conditioning.\nTreats frizz and fly-aways with ease.\nHas the typical Dove fragrance.\nMy hair is nourished, softer and detangled after use.\nNormal coin-sized amount is needed for a single use.\nPrice is very reasonable for the quantity.\nwow such a nice conditioner. kudos to dove. i love dove conditioners. will definitely buy this ASAP\u2026.!!! :thanks: 4 d review richa!\nreally glad you are feeling better now :puchhi: and its cool tat this product has worked for you. I remember the days when Dove only made soaps \ud83d\ude42 I bought the Dove treatment conditioner after reading the review a few days ago and thats pretty good too.\nI experience hairfall due to Dove Shampoos but never conditioners. I hope u dont apply conditioners to the roots of the hair as it aggrevates hari fall.\nthanks a lot, i will surely implement your feedback.\nIt would be really great if you can suggest me a gud shampoo as well\u2026..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"IANS \u2013 New Delhi: Global warming is leading to an accelerated rise of the sea level in the north Indian Ocean as compared to the global estimate, Parliament was informed on Monday.\nRecent studies by Indian scientists had revealed that the trend of sea level rise in the north Indian Ocean is slightly higher than the global estimate of 3.2 mm per year, Minister of State for Earth Sciences YS Chowdary told the Rajya Sabha in a written reply.\nThe minister added that from 1993 onwards, an acceleration of rise in global sea level has noticed taking the annual rise to present 3.2 mm from previous 1.7 mm.\nThe possibility of sea level rise in the next ten years is about 3.2 cm in the north Indian Ocean, if the sea level acceleration remains similar.\nChowdary further stated that over the periof of 100 years, between 1901 to 2010, the global mean sea level had risen by 19 cms or 0.19 meters.\nThis, he said, happened at the rate of global averaged sea level rise of 1.7mm per year between 1901 and 2010 within which an accelerated rate of 3.2mm\/year was noticed between 1993 and 2010.\n\"According to the fifth assessment report of Inter-governmental Panel on Climate Change (IPCC) the estimates made for the period 1993-2010, using the remote sensing satellite data and in-situ measurements of tide gauges, are found to be 3.2, with an error estimation of 0.5mm per year and 2.8 error estimation of 0.8mm per year, respectively,\" he said.\nHe further said that the Central Government has issued notification with a view to ensure livelihood security to the fisher communities and other local communities living in the coastal areas, to conserve and protect coastal stretches and its marine area taking into account the dangers of natural hazards due to the global warming.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The esophagus carries food from the mouth to the stomach. The lower oesophageal sphincter is a ring of muscle at the bottom of the esophagus that acts like a valve between the esophagus and stomach.\nGastro Esophageal Reflux Disease, or GERD, is a chronic disease that occurs when the lower oesophageal sphincter does not close properly and stomach contents leak back, or reflux, into the esophagus.\nIf a combination of lifestyle changes and drug therapy does not remedy reflux symptoms, a Nissen Fundoplication can be a very effective surgical procedure to correct reflux. This procedure involves wrapping the upper portion of the stomach around the base of the esophagus to reinforce the strength of the lower oesophageal sphincter. Until recently, the procedure required a large abdominal incision. A hospital stay of 3-5 days was usually required, and the time to full recovery and return to work was measured in weeks.\nA laparoscopic Nissen Fundoplication is a minimally invasive approach that involves specialized video equipment and instruments that allow a surgeon to perform the procedure through four tiny incisions, most of which are less than a half-centimetre in size. One advantage of this method is a brief hospitalization. Most of the time it will require an overnight stay. Other advantages include less pain (less of a need for pain medication), fewer and smaller scars, and a shorter recovery time.\nLaparoscopic Nissen Fundoplication is a safe and effective treatment of GERD. However, in rare cases the laparoscopic approach is not possible because it becomes difficult to visualize or handle organs effectively. In such instances, the traditional incision may need to be made to safely complete the operation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This year, for the first time, the Information Institute for Highway Safety (IIHS) performed crash-test evaluations on plug-in electric cars. The Nissan Leaf and Chevrolet Volt earned the highest safety rating, earning the top rating (of \"good\") for frontal, side, rear and rollover crash protection. Along with a standard rating for electronic stability control, both the Volt and the Leaf earned spots among the Top Safety Picks List for 2011.\nAs a consumer, you can be confident that automakers are utilizing safety engineering in their electric cars that rival that used in their vehicles with internal combustion engines.\nThe 2011 Top Safety Picks for all sizes of cars, vans and SUVs has been published by the IIHS. This list recognizes 38 cars, 25 SUVs and one minivan, in addition to the Volt and Leaf hybrid vehicles. Compared to 2010, you will find more than double the number of vehicles to choose from that meet the safety criteria used by the IIHS.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As an accreditation consultant, I have several times been asked questions like, \"What would it take for us to add ATS accreditation?\" To answer that question, I often start with a discussion of full-time faculty requirements for accreditaton. While there is a great deal of overlap in the standards of any Department-of-Education-Recognized accrediting agency, faculty requirements is an area where differences abound.\nWhen a Christian college, graduate school, seminary, or university first seeks accreditation, it normally starts with the Association for Biblical Higher Education\u2014 ABHE or the Transnational Association of Christian Colleges and Schools\u2014 TRACS.\nWhat are the full-time faculty requirements for accreditation? It depends on the accrediting agency.\nA core faculty of sufficient size and expertise, committed to the fulfillment of the institutional mission, and responsible for the quality of its academic functions.\nWhen I am helping a small school achieve accreditation with the ABHE, I hope that by the time it applies for candidacy, the school has one or two full-time faculty members. This does not include administrators who also teach.\nHaving full-time instructors is not always an absolute demand for achieving candidacy, but the visiting team that evaluates the school is likely to write that the school needs to begin working toward hiring faculty members whose main ministry is the school (e.g., full-time faculty members). Hopefully, the school will add a couple more full-timers before they apply for accreditation.\nIt can, however, be possible to reduce this number of full-time instructors by having a common core curriculum and separate concentrations.\nFor example, if a school has both an AA and BA in Biblical Studies, and all the AA courses are contained within the BA program, these two degrees would only require one full-time program director. If the school also has an M.Div., add another full-time faculty member.\nThey will also need a full-time president and academic dean. Being full-time administrators, neither of these can be classified as full-time faculty \u2014 even if they also teach.\nAlthough a little less specific than TRACS, ATS, which accredits graduate programs in theology, gives somewhat detailed guidelines pertaining to a minimum number of full-time faculty members.\nThe school has an adequate number of properly qualified professors working full time at post-baccalaureate theological education. Normally, this adequacy will be represented by the equivalent of six to 10 full-time faculty (emphasis added).\nThe faculty and staff are sufficient in number \u2026 to achieve the institution's educational objectives, establish and oversee academic policies, and ensure the integrity and continuity of its academic and co-curricular programs wherever and however delivered.\nThere are an adequate number of faculty whose time commitment to the institution is sufficient to assure the accomplishment of class and out-of-class responsibilities essential for the fulfillment of institutional mission and purposes.\nResponsibilities of teaching faculty include instruction and the systematic understanding of effective teaching\/learning processes and outcomes in courses and programs for which they share responsibility; additional duties may include such functions as student advisement, academic planning, and participation in policy-making, course and curricular development, research, and institutional governance.\nThe institution avoids undue dependence on part-time faculty, adjuncts, temporary appointments, and graduate assistants to conduct instruction.\nKeep in mind that regionally-accredited schools tend to be large and complex. Thus, it is hard to say the exact minimum number of full-time professors a school will need to be regionally accredited. However, the different ways that regional accrediting agencies explain what they want the faculty to accomplish is a useful guide for understanding the expectations of any regionally or nationally recognized accrediting agency.\nRegional accreditation tends to not be an easy option for a school that is looking to achieve initial accreditation. Often, a small Christian university will first achieve accreditation with TRACS or ABHE as a first step. As the school grows, develops and expands programs, it may then want to add accreditation from one of the regional accrediting agencies or ATS.\nIf you want to discuss whether your school is ready to seek an additional accreditation (or its initial accreditation), contact info@accreditation101.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"David was uncharacteristically coy about naming the parties, but we can be fairly sure that he meant the DUP and Sinn F\u00e9in and we can be equally sure that he didn't mean Alliance. But you wonder why not.\nLast week Alliance voted with the other unionists in Belfast to hoist a duvet-sized British Army flag over the City Hall as a tribute to 'Our Boys' in the run-up to Armed Forces Day on Saturday. They could have said, no, have you seen the size of the union jack that's up there already? Or they could have said, fair enough, but only for one day, and let's make that day, aaahhh, let's see now, um, yes, let's make it Armed Forces Day.\nBut they didn't say any of these things. They did exactly as their openly unionist colleagues demanded and voted to stick it up there for a week. This in a city which already resembles the Last Night of the Proms on the evening of Victory in Europe Day. And this from a party which only a month ago walked out of talks on a Cohesion, Sharing and Integration strategy, which had the issue of flags high on the list of Things To Do.\nSo, Dave, we know you're the Alliance Party and all, and we know we're supposed to be kind to the Alliance Party because they were being lovely when everyone else was being horrible, but come on. At least have the decency to take a reddener.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some of the questions we get asked most regularly about the Three Peaks Challenge. We'd like to add to this information so if you can't see the answer below please get in touch.\nThe challenge is best undertaken during the spring\/summer as there are longer days and hopefully better weather. However please avoid school and bank holidays as parking and local amenities are under considerable pressure during these busy times.\nNo way! Although many groups do try and complete the Challenge in 24 hours, this is fast becoming a thing of the past. More and more groups are now attempting the Challenge over Three days. It's as physically challenging but it also means you get a chance to enjoy the three stunning mountain regions your visiting.\nChangeable! Mountain weather can be extreme and it is not uncommon for snow to be lying on the summits during the summer, especially at Ben Nevis. The temperature drops roughly by 1 degree Celcius for every 100m of ascent and this, combined with wind chill, means that it will be a lot colder on the summits. Be prepared, bring plenty of spare, warm clothing even if you are hot in the car park. If the weather becomes too bad, don't be afraid of calling off your challenge; the mountains will still be there to climb another day. Visit MWIS or the Met Office for a detailed forecast before you arrive.\nYes. It is essential that you have the appropriate OS map for each area along with a compass and the ability to use them. Also, if you are part of a larger group, make sure that more than one member of your party has a map and compass and no one gets separated from the navigators. Please don't rely on mobile phones and GPS units, the batteries will go flat at an inconvenient moment! You could always consider hiring a qualified guide to be on the safe side.\nYou will need to have a reasonable level of fitness along with a good sprinkling of determination. Walking up mountains is far more strenuous than on the level and you have chosen to tackle the three highest in Scotland, Wales and England. So you will need to practice and build up your strength beforehand by doing some regular hill walking (a good chance to practice your navigation skills) and other cardiovascular exercise. Consider doing the challenge over three days; this will allow you to go at a sensible pace, enjoy your surroundings and reduce the chances of accident\/injury.\nCan't find the answer you're looking for? Get in touch via our contact page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Doris Lessing won the Nobel Prize for Literature in 2007. She has published numerous novels, short stories, plays, and works of non-fiction for over fifty years. The Nobel Prize for Literature is awarded for a lifetime of literary achievement, but many literary critics recognize The Golden Notebook (1962) with its political and feminist themes as her award-winning work. Lessing later turned her attention from realistic literature to science fiction, much to the disappointment of these same critics. However, Lessing considers her five-volume Canopus in Argos series of science-fiction novels her most important work.\nThe first book in the series \u2014 Re: Colonised Planet 5, Shikasta \u2014 was published in 1979. The title of the book refers to the planet Shikasta, a small and remote planet in the interstellar empire of Canopus. To make the planet useful to their interests, Canopus imported an alien species to speed up the evolution of the native Shikastan species. An unexpected astronomical catastrophe and invaders from the evil empire Shammat caused an unprecedented degeneration in the natives that nearly destroyed the planet. Canopean emissaries continued to work on Shikasta in an attempt to control the damage, but their original plans for the planet had to be abandoned. The book itself is a diverse collection of archives (direct reports from emissaries, excerpts from history books, diaries, letters, etc.) meant to give first-year Canopean students a picture of the empire's disastrous involvement on Shikasta. Oh \u2014 and I should probably mention that Shikasta is our very own planet Earth.\nI think a better understanding of Lessing's approach in this novel can be grasped by looking at the distinction between mythical thinking and logical thinking I explored in my article contrasting mythos and logos. Though she does not use these terms, Lessing provides in Shikasta an imaginative thought-experiment meant to challenge a logos-only thinker's rejection of these stories.\nThis portrayal of stories from Genesis as distorted accounts of a prehistoric human past suggests an interesting possibility. Perhaps some shred of truth has been preserved in these stories, possibly even more accurate truth than we have found through scientific and historical methods. Even if we cannot prove this is true, we can at least imagine it. In Shikasta, Lessing establishes that it is at least imaginatively possible that the sacred texts (products of mythos thinkers) preserve some scientific and historical truth that scientists and historians (logos thinkers) have not discovered. Far from criticizing these stories, her fictive account attempts to elevate them in the minds of logos-only readers \u2014 from ridiculous falsehoods to potential sources of truth.\n] increasingly saddened and depressed their real selves, their hidden selves\" (91).\nThese spiritual truths reflect ideas Lessing first encountered in the 1964 book The Sufis and through a subsequent friendship with the author Idries Shah; Lessing spoke often about the influence this book had on her. Shah described Sufism as a universal spiritual wisdom that transcends individual religions, a wisdom that has traditionally been communicated through parables and folktales. In drawing on the Hebrew Bible, Lessing is not arguing here for the correctness of Judaism or Christianity: she is well aware of the injustices and brutalities performed in the name of religion. Rather, Lessing provides in Canopus's involvement on Shikasta a fictional explanation for the ideas about Sufism made popular by Shah. According to the archives, all the great religions surviving into the twentieth century were originally founded by Canopean agents for the \"stabilisation of the culture, preventing the worst excesses of brutality, of exploitation, and greed\" (111). Even though they became perverted, \"distorted inklings of the truth remained\" in them (Lessing 90).\nShikasta is much more than just a retelling of stories from Genesis from a different perspective, but this aspect of the novel is most relevant to this site's ongoing exploration of mythical thinking. Doris Lessing understood the materialism, the greed, the injustice, and the spiritual degeneration prevalent in the modern world to be the result of logos-only thinking. She acts still today as a strong advocate for the potential of the products of mythos to address these issues.\nLessing, Doris. Re: Colonised Planet 5, Shikasta. New York : Random House, 1979.\nI find I very strongly agree with Lessing. Archaic remains like that of the pyramids and Stonehenge leave us wondering how the ancients were able to accomplish such feats when our pithy attempts at re-enactment with our renditions of the tools we think they used fail utterly. This looks like a really interesting read, and your article makes me want to jump on it soon. Thanks!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Looking for more thoughts on today's financial landscape? Click here to visit Skip's Blog!\nThe blog is mainly for short, original thoughts that may not fit on the rest of the website. Some are topical, some are perspective, and expect a few opinions on the state-of-play in finances. Click to read more.\nThe LPL Financial representative associated with this website may discuss and\/or transact securities business only with residents of the following states: CA, CT, FL, IN, MD, NC, NJ, NY, SC, VA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"June 7 (PTI) Since a solution to Sri Lanka's ethnic Tamil issue \"cannot be imported\", all political parties must join hands to find an all-inclusive settlement, President Rajapaksa told visiting BJP leader Ravi Shankar Prasad, here today. \"We cannot import a solution and it cannot be a Rajapaksa-Sampanthan (main Tamil party leader) agreement,\" Rajapaksa was quoted as telling Prasad. He also underlined that all more..\nJune 7 (GDI) Sri Lanka has observed that the high enrolment rates at primary and secondary levels which reflect the impact of longstanding policies supporting access to basic education without discrimination, have contributed towards Sri Lanka\ufffds success in achieving gender parity in primary, secondary and tertiary education. Intervening during an Interactive Dialogue with the Special Rapporteur on Education and IE on Human Rights and International Solidarity at the on-going more..\nJune 7 (DM) Sri Lanka Cricket today expressed its dismay over the playing of a Hindi song instead of a song which reflects the Sri Lankan identity when the Lankan team was introduced during the opening ceremony of the ICC Champions Trophy in London yesterday. Speaking to Daily Mirror Online Secretary of the SLC Nishantha Ranathunga confirmed that he had sent a letter to the ICC demanding an explanation from the ICC regarding the mix up. \ufffdThe song that was played was not more..\nJune 7 (DM) A Bell 412 helicopter carrying Air force personnel including a high ranking officer made a precautionary landing at grounds in Katupotha this morning, due to a suspected technical fault. The Air force refused to confirm whether the Commander of the Air force was on board during the emergency landing. Air force spokesman Air Commodore Andrew Wijesuriya refused to comment regarding the identity of more..\nJune 7 (Island) Minister Basil Rajapaksa said that the launch of the pre-paid travel card had met a long felt need as it would safeguard the interests of both the commuter and the bus owner. He said action would be taken to issue pre-paid travel cards to 3.5 million Devi-Neguma beneficiaries soon. Minister Rajapaksa said the Lanka Private Bus Owners\ufffd Association (LPBOA) had complained to him that the association had been incurring heavy losses due to extortion by time keepers. more..\nJune 7 (GDI) Minister Basil Rajapaksa emphasized that it is the objective of the President to develop this country by raising the living standards of the rural people and at the same time make the people of the country virtuous and possessing good qualities. The Minister said that in many developed countries the people are very much distanced from their families and elders and they do not respect other human beings. The Minister said there is no value in this type of development. more..\nJune 7 (DM) The UNP said today it would request the Commonwealth heads to reconsider the participation at the CHOGM in November had the govt failed to fulfill its promises to the Commonwealth before the Northern Provincial Council elections. UNP MP Mangala Samaraweera said the govt had given numerous promises to the Commonwealth stating that it would take steps to protect Human Rights and also to more..\nJune 7 (Island) Around 200 doctors, attached to the Matara General Hospital, would go on a 24-hour strike from 8.00 a.m. today in protest against the hospital\ufffds Directress, Dr. Mrs. Samanthi Samarakoon, whom they accuse of being inefficient. GMOA media spokesman Dr. Navin de Soysa told The Island that the GMOA Matara hospital branch union had decided to launch a strike against the directress, who was incapable of running the hospital efficiently. more..\nJune 7 (DM) The govt today said that some 421 military officers and 5205 other rankers were court-martialed since 2006 with Commodore T.M.P Mendis who was convicted of failing to take adequate safety measures and counter actions to protect the Galle Naval Base from the terrorist attack on Oct 18, 2006 among those who were martialed. The govt gave these details to the House as an answer to a question by UNP MP Ravi Karunanayake. Minister Dinesh more..\nWill fifth time be the charm?\nJune 7 (CI) Sri Lanka's recent ICC tournament history reads like the script for a formulaic romantic comedy, only without the feel-good ending. They may have only been bridesmaids four times, instead of 27, but they still haven't been able to take James Marsden home. The journey through each tournament has featured its share of warm-fuzzies (Muttiah Muralitharan picking up Scott Styris off his last ball in home internationals, more..\nJune 8 (CT) Though the war ended in 2009 another began for Sri Lanka; this time on a different playing field. For the past four years, the international environment surrounding the island has changed significantly. Despite the fact it had overcome one of the most virulent terrorist movements, Sri Lanka has been placed under the radar of the international community, both consequent to, and leading to a deterioration of its foreign relations. What is worse is we are the ones who put ourselves in more..\nJune 8 (DBS) Women\ufffds Action for Social Justice (WASJ) recently organized a panel discussion on the theme Violence Against Women is \ufffdViolence Against Humanity,\ufffd which was held at the Public Library auditorium. Over 10,000 signatures, which have been gathered against Violence Against Women, from 10 December 2012 to May 2013, were also handed over to the United Nations Office in Colombo. Signatures were gathered from the general public in 15 areas in 13 districts more..\nJune 8 (SO) Drug addicts; can they be rehabilitated from their addiction was a question lingered in the minds of the people some decades ago. There was belief that they cannot be rehabilitated and they have to live with drugs for ever and their lives also should end with drugs. But all these misconceptions have been proved wrong at present as the drug addicts are realising that all what they believed were only gimmicks of the more..\nJune 8 (CT) Professor Sivalingam Sivananthan became the first immigrant academic of Sri Lankan origin to receive the 'Champion of Change' award presented by the White House to immigrants who have made exceptional contributions in various fields to the US. The USA has been a magnet for those seeking greener pastures or escaping war and strife, and persecution in their homeland. Emma Lazarus' emotive poem, inscribed on the plaque in the basement of the Statue of Liberty, more..\nJune 8 (Island) A couple of weeks ago, many were taken by surprise when in a suburb of Stockholm, the capital of Sweden, hundreds of young people rampaged through the streets in some of the worst rioting the city has seen. For nearly a week, stone-throwing mobs clashed with the Police and, indiscriminately as in all mob violence, attacked public and private institutions, including shops and schools. The clashes were sparked off by the Police shooting dead of a Portuguese immigrant more..\nJune 8 (DBS) In a remarkable display of political courage rarely seen among members of the govt in recent times, Justice minister and SLMC leader Rauff Hakeem, has protested strongly against President Rajapaksa\ufffds controversial move to circumscribe devolution by amending the Constitution. The protest lodged by Hakeem at the weekly cabinet meeting held on June 6th was overtly backed by minister Vasudeva Nanayakkara more..\nJustice for Matale victims? Never!\nJune 8 (CT) One hundred and fifty people had been buried in one mass grave, some of them after having been severely tortured, in a hospital ground, without the knowledge of anyone for almost 25 years. It is almost impossible to imagine how anyone could dig such a large grave and bury 150 in it without the outside world learning about it, particularly in a public place like a hospital. There could have been four parties who knew about this: the murderers, supporters of the murderers, the murdered, more..\nJune 8 (Island) Time was when parents formed vigilance committees to protect schools against two-legged Tigers. They used to take turns to stand guard near school gates to ensure the safety of their children. But, strangely, they do not seem so keen to protect their precious ones against death dealing winged terrorists on the rampage. Dengue has raised its ugly head again. Mosquito breeding places have been detected in many schools, some of which have been severely warned by the more..\nJune 8 (CT) Dear Friends, As the author of the article \ufffdSobitha Proposals, Ranil, Mahinda And The BBS\ufffd let me please add some in \ufffd.. post script. Firstly thanks for all of you in taking the time to comment both in agreement and disagreement I thank you to you all equally. Of course I don\ufffdt agree with some of the views expressed here I agree to disagree and would on to my death protect the right you have to hold to express and promote these views. There is a saying that your rights end more..\nEthics of Ethanol or media ?\nJune 8 (Island) Are some people in politics and officialdom targeting a new Guinness record in ethics? The sudden surge of interest in ethics in society, gives the impression that we are moving to a new surge of ethics in a trend that seeks to make us the most ethical society in the world, and if not, to be in keeping with the fashion, of a declaration as the Hub of Ethics in this Region. The biggest interest just now is about ethics in the media. There is great punditry we hear today from those who more..\nJune 8 (CT) Criticism of the govt's plans to allow Australian casino tycoon, James Packer, to open a casino in the country continues to mount from both within the ruling coalition and opposition parties. Minister Champika Ranawaka, said his party was against all casinos and gambling that take place in the country. \"While it is not illegal to gamble in Sri Lanka, we oppose this form of revenue generation,\" he said. more..\nJune 8 (GV) As the Editor of Groundviews, I was invited by World Association of Newspapers and News Publishers (WAN\/IFRA) to give a presentation on Sri Lanka at the 20th World Editors Forum (WEF) and the 65th World Newspaper Congress, held recently in Bangkok, Thailand. As noted on the WAN\/IFRA WEF site, \ufffdThe World Editors Forum is the leading network for print and digital editors of newspapers and news organisations around the world. It is built on a commitment to more..\nJune 8 (Island) This is hardly news, but the Opposition continued to be uninspiring last week. If a country is going to change its constitution, it really ought to be worth the bother. But the most encouraging thing about the draft published by the UNP on Wednesday is its support for provincial level devolution within a unitary state, which is exactly what is already in place thanks to the 13th Amendment to the existing constitution. It is also one of the few commitments to which voters more..\nJune 8 (ST) Sri Lankans are mixed in their views on whether testing of local and imported food, as required by the country\ufffds authorities, is properly done, a joint poll by the BT and RCB revealed this week. Following the controversy over imported milk food, the poll sought to find answers from the public on the level of testing of local and imported food, the ingredients in food and awareness about the rights of consumers. more..\nJune 8 (CI) Whatever your persuasion when it comes to one-day cricket the Champions Trophy looks set to offer something for everyone. After the high-scoring opening match in Cardiff, there was a low-scoring nail-biter at The Oval with West Indies surviving major problems against Pakistan's pace and spin combination to secure a priceless two-wicket victory. Amid a passionate, excitable crowd dominated by Pakistan support - this is arguably the closest they will get to a home crowd more..\nJune 8 (FT) A business delegation from the South Korean city of Busan, headed by Sri Lanka\ufffds Honorary Consul in that country Jung-Taek Shin, called on President Mahinda Rajapaksa at Temple Trees yesterday. The delegation that comprised of leading business persons from several sectors including automobile and shipping said that South Korean companies are interested in pursuing investments in Sri Lanka and requested for the President\ufffds support, which he assured. more..\nJune 8 (DN) Of the 219 state institutions, only four have recorded losses of over 95%, and only 11 are running at a loss, Chairman of the Parliamentary Watchdog, Committee on Public Enterprises (COPE) and Minister D.E.W. Gunasekera said. He was speaking at the presentation of the Annual Report 2012 of the Finance and Planning Ministry, to President Rajapaksa. The ceremony was held at the MILODA -Academy of Financial Studies, more..\nJune 8 (Island) The International Diving Centre (IDC) and Cinnamon Nature Trails, both based at Chaaya Blu Hotel, recently removed 250kg of abandoned fishing nets and plastics from the Swarmy Rock reef as part of Project Aware\ufffds \"Dive Against Debris\" campaign. Both entities have joined forces to take action against marine debris in order to raise critical national awareness of this issue. Statistics included in Project Aware\ufffds infographic \"The Ugly Journey of Trash\" provides a more..\nJune 8 (Island) The Athletics Association of Sri Lanka has dropped the women\ufffds 400 metres hurdles from the next weekend\ufffds trials for the forthcoming Asian Athletics Championship, after US-based National record holder Sonali Merrill produced a note worthy performances at a meet in Los Angeles, California, USA, early this month. Merrill, who won a bronze medal at the last Asian Championship, clocked 58.71 seconds and 58.82 seconds respectively, at two meets within the last four more..\nJune 8 (Island) EPDP leader and Minister Douglas Devananda accuses most Tamil media of publishing false and unsubstantiated news to mislead the people of the Northern Province. An EPDP spokesman quoted the Minister as having said that some local dailies published in the North were full of distorted, false and unsubstantiated news aimed at slinging mud at the govt. He said that Tamil speaking people more..\nJune 8 (Island) The Health Ministry complained that a national committee headed by President Rajapaksa, to control the dengue epidemic had formed nine provincial committees, 25 District Committees, 325 regional Committees and 15,000 rural Committees, but the response of the private and govt sector organisations had been negative. Meanwhile, Minister Maithripala Sirisena, speaking at the Harvard Ministerial Health Leaders\ufffd Forum, of its Kennedy School of Govt, more..\nJune 8 (Island) Central Bank has kept monetary policy rates on hold as widely expected by the market. The repurchase rate for commercial bank deposits with the Central Bank remains unchanged at 7% while the reverse repurchase rate for overnight borrowings stays at 9%. The financial markets have responded as expected to the recent policy measures taken by the Central Bank, with the relevant financial institutions gradually making the anticipated adjustments. more..\nJune 8 (CT) Six undergraduates of the University of Kelaniya were hospitalized, following a clash between undergraduates and an unidentified group of persons, who assaulted the students at the university hostel premises late Thursday night. The students were second year undergraduates of the Faculty of Science, the police said. However, despite students being attacked to the point of hospitalization by outsiders, more..\nJune 8 (GDI) The Children Unit of Cancer ward of Jaffna Teaching Hospital which had experienced shortage has been provided with a fully equipped ward was declared open today (7) by Minister Douglas Devananda. He stated that the shortfall is being rectified owing to steps gradually taken. He said that it is his aspiration to make it one of the best hospitals in South Asia. This ward has been a long felt need and the child section is being opened today at a cost of Rupees 12.5 Million. more..\nJune 8 (BT) Sri Lanka's reforms over the past few years to widen the tax base while lowering tax rates were not yet effective because many professionals in the service sector were still not paying tax, the govt said. \"If we need to maintain a lower rate tax regime, all those who should pay their taxes in this country, should pay their dues,\" PB Jayasundera, treasury secretary told a forum to present the finance ministry's annual report. \"We need to tax those who are in the service sector more..\nJune 8 (Hindu) Sri Lanka\ufffds post-war context requires empowerment along with development, and holding a free and fair election in the north, along with devolution, is central to it \ufffd this was the broad message that the Indian delegation conveyed to President Rajapaksa on Friday, according to BJP leader and Deputy Leader of the Opposition in the Rajya Sabha Ravi Shankar Prasad, who led the team. As part of a five-day visit, more..\nJune 8 (CT) After months of deliberation, Sri Lanka Cricket is likely to decide the fate of the three controversial umpires charged with spot-fixing next week. \"We will conclude the proceedings on Monday and will present it at Thursday's executive committee meeting,\" said Sri Lanka Cricket's head of Cricket Operations Carlton Bernadus. Sri Lankan umpires Gamini Dissanayake, Sagara Gallage and Maurice Winston were among six umpires reportedly caught in an undercover operation by more..\nJune 8 (CT) Sandhya Ekneligoda, wife of missing journalist Prageeth Ekneligoda, urged the govt to bring down her husband, if he is indeed hiding in France, as claimed by a govt MP. She also challenged the govt to prove that her husband telephones her every night, as MP Arundika Fernando has claimed, and stated that her suspicions about the govt having abducted her husband are now confirmed. \"I urge the govt to somehow bring my husband back. more..\nJune 8 (Island) Responding to President Rajapaksa\ufffds ultimatum for the TNA to join the Parliamentary Select Committee on the National Questio, which is to be reconvened shortly, the TNA said that it had not refused to join the select committee process. TNA leader R Sampanthan said he regretted that there had been no positive response from Rajapaksa to his 2010 written communication in which he expressed more..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Winning the race isn't everything, but how you run the race is much more important. Ivan Fernandez Anaya is a professional athlete with a winning attitude. He had a chance to gain the gold medal, but he made a choice to do the better thing. In one cross-country race, he was coming in second place. The runner ahead of him mistakenly stopped running before the race was finished as he didn't know where the real finish line was. Instead of taking advantage of his opponent's mistake and confusion, Ivan guided him to finish, himself coming in second place after him. Even though he had a chance to win first place, Ivan put the fellowship of the sport above his own personal gain. In that sense, Ivan is a true sportsman, who demonstrated gracious wisdom under pressure. Even in the intense heat of competition, when what he had been training for was within his grasp, he made a deliberate choice to honor his fellow sportsman. This watch celebrates the spirit of sportsmanship as exemplified by Ivan. Life is one long race. It's not about finishing first or winning at all cost, but it's about finishing well, and helping those around you to do the same.\nIvan Fernandez Anaya is a professional athlete with a winning attitude. He had a chance to gain the gold medal, but he made a choice to do the better thing.\nOn 2 December 2012, Ivan was competing in a cross-country race in Burlada, Navarre. He was running second, some distance behind race leader Abel Mutai - bronze medalist in the 3,000-meter steeplechase at the London Olympics. As they entered the finishing straight, he saw the Kenyan runner - the certain winner of the race - mistakenly pull up about 10 meters before the finish, thinking he had already crossed the line.\nIvan quickly caught up with him, but instead of exploiting Mutai's mistake to speed past and claim an unlikely victory, he stayed behind and, using gestures, guided the Kenyan to the line and let him cross first.\nThis news had immediately shared by different social media and aroused attention from the newspapers in different countries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Return Process is simple and hassle free. You can return your items to our warehouse up to 30 days from your item's ship date.\nPrior to returning your item, please ALWAYS CONTACT Mokum Made first, and let us know of your issue. Be sure to have your order number ready and Mokum Made will be happy to assist you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nuce! Do you have any hotel reservations already? what about itinerary? Id love to share with you some tips.\nI recommend u ask your hotel if they arrange island hopping activitirs. most hotels arrange a tour with English-speaking tour guides.\nIf your hotel doesnt have that, you can ask the recrption to contact travel agencies for u. other option is to visit tour agencies within the area. just by walking along the streets of Coron Town proper, u will see alot of tour paclages veing offered. Have fun!\nThank you Seng. I love your site! Kaka inspire!\nThank you!! It is beautiful. let me know if you need any reco.\nLoved this! I've only been to PPS and El Nido but will definitely visit Coron soon!\nLet me know if you need any recommendations or contacts ok?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Warrington is one of the stores that belong to the Poundland chain, located at 19-21 Sankey Street, Cheshire, WA1 1XG, in Warrington. It serves clients coming from surrounding localities, like Town Centre, Woolston, Paddington, Bruche. In order to get in touch with this location please phone them at 01925 551865. Warrington is normally open on Monday: 08:00 - 18:00, Tuesday: 08:00 - 18:00, Wednesday: 08:00 - 18:00, Thursday: 08:00 - 18:00, Friday: 08:00 - 18:00, Saturday: 08:00 - 18:00. Its Sunday opening hours are: 10:00 - 16:00. Clients may use 3 facilities such as High Street, Parking nearby, Pep&Co.\nOther brands nearby are McDonald's in Warrington - only 0.03 miles away, Boots in Opticians - Warrington Golden Square Centre BO - about 0.03 miles away, The Co-operative Travel in The Co-operative Travel - Warrington - at a distance of 0.04 miles, Iceland in Warrington - only 0.04 miles away and Game in Warrington (Store ID: 3359) - at a distance of 0.05 miles.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The outer banks swimming pool & landscape contractor of choice.\n\"I am writing to thank you and Southern Scapes for the extraordinary work done in installing a beautiful in-ground pool on our property in Southern Shores. Before we met, we had been working with another pool company for almost a month attempting to decide where we could locate the pool. The first time you came to our home, you grabbed a shovel, found my cesspool and suggested a perfect location for our pool. Moreover, that evening, Tom May Jr. sent us a scaled drawing of our home and our pool \u2013 amazing! You gave us a full and fair estimate, including all costs, and took care of all permits. During construction, you met and sometimes exceeded your timetable and it was a pleasure to work with you and Southern Scapes. I really appreciated how you would answer my questions in a timely fashion.\nAll of the work done in installing the pool \u2013 the cement work, the pool, the fencing, the electrical work, was of the highest quality and looks beautiful. However, as you know, the pool had a small leak which could not be found. This is where you proved you were a man of your word and your company would stand behind their work. In order to address this leak, workers had to come from another state, brace our pool, drain our pool, re-glaze the bottom and refill the pool. I knew you were reliable and not for one moment did I have any concern or doubt that you would take care of this substantial and costly problem. All of this work was done and the problem was fixed.\n\"Shawn and I want to take the time to thank you for the continued superior customer service. We were pleased several years ago when you installed our pool and we continue to see the same support years later. Recently, we contacted you as we had a small issue with the pool. Your first answer was not to make a service call and charge us. You took the time to trouble shoot over the phone to try to save us time and money. Not many companies would go that extra mile. Unfortunately, we were unable to solve the problem on our own.We believed we had located the issue, so we contacted you for a service call.\nYou responded to our call quickly, scheduled a visit within days of the call and had the problem fixed within hours of arriving. We recognize this is a busy time of year for your business, so the prompt response was greatly appreciated. You have always provided personal attention and support. The team has always been professional and efficient, and the quality of the work far exceeded our expectations.\nOur family has enjoyed our pool since the day we were able to jump in! We constantly receive compliments regarding the appearance of the pool. We are always happy to let people know who completed the work!\n\"Again, Thanks for everything. Thanks for all your help with the septic.\n\"We wanted to send a note thanking you for an incredible job on our new pool. You took care of everything from working with the town for permits to getting approval from the homeowner's association. Being over ten hours away, we knew the job was in good hands. Even with Hurricane Sandy interrupting the job, you were a real professional.\n\"Wanted to write and tell you how delighted I am with the pool you installed.\nNever thinking I could afford owning an inground pool, I did a lot of research on the subject. I spoke with 4 different companies in the area, looking at price, equipment, installation. I chose Southern Scapes because it came out the winner in every category!\nWorking with Tom Sr. was a real pleasure. He helped so much with the planning, surveying and met with the town and county to answer questions and prepared paperwork to move the process forward.\nTom, you and your crew were wonderful in doing all that was needed to prepare my property and install the pool, fence, concrete, and equipment. I can't say enough good things about you and your team.\n\"From our initial conversations about the design through the execution of the job, everyone we worked with was knowledgeable and obviously expert in his field. Perhaps, most importantly to us, you took the time to understand our needs and our budget and you executed it in a timely and professional manner. Our neighbors have already asked us \"who did our wonderful landscaping\", so don't be surprised if you hear from a few of our Sea Tern residents for their future projects as well.\nThis is a very long overdue thank you. We have enjoyed our pool everyday since you installed it. Even though we are not swimming during the winter months, it is simply beautiful to look at.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Festival hours are 5:00 PM - 11:00 PM. Please note there is a 30 minute band change over in between sets with Top 40 Hits and Pop Music provided during this time.\nFeaturing a unique combination of two charismatic front-men, alternative rock band PrideFalls got its start in 2005 performing cover songs and fledgling originals in bars, clubs, and college venues throughout New England. \u200bInfluenced by the likes of The Fray, OneRepublic and Incubus, the five piece band often weaves together harmonic vocal melodies with a wall of complementing guitars.\nFrom Springfield, MA, Feel Good Drift started out as front man Alex Rohan was playing solo acoustic shows at local bars around the area. Not too long after, Marty Thompson stepped in and started playing lead guitar and adding more feel to the songs. Soon after Marty, Corey Villamaino joined the group to play drums. The music they play has a good feel that will be sure to get you moving. Joe Stafilarakis was added to the band as Feel Good Drift's Bass player\/backup vocals.\nA band that needs no introduction in Massachusetts! The areas best and most popular country party band Trailer Trash closes out the Made in Massachusetts Festival. So grab your date and come see the most unlikely band ever to hit the Urban streets of Western MA! Come and sing along to all the songs you sing in your car while no one is looking. Put on a pair of boots and a cowboy hat if you like, or just come as you are, and enjoy a night of great music and fun! This ain't what your Grandpa labeled \"country\"...this is TRAILER TRASH!\nFrom Holyoke, MA, the Arc City Angels are straight up ROCK N' ROLL. Catch their set and see rock classics and favorites. This band is rising in Western Mass and attracting crowds through various venues. The Arc City Angels are a 3 piece band that will bring their rocking and rolling performance to the main stage of the Made in Massachusetts Festival.\nFrom South Hadley, MA, The Mid Life Crisis Band has been seen in venues throughout Western Massachusetts and is always a crowd pleaser. They are the areas best in 70s, country, rock and blues cover songs. Their upbeat energy and mastered talent makes this 5 piece band stand out. The Mid Life Crisis Band is excited to bring their best to rock the stage at Made in Mass Festival and ensure everyone in the crowd is dancing and singing along.\nAquanett has been labeled \"New England's most authentic tribute to the 80's.\" Their look, authentic sound to the original artists and dedication to their fans have been keeping people coming out again and again for over 15 years. Aquanett covers bands from Ozzy to Metallica, Bon Jovi to Poison. The seasoned group of musicians that call themselves Aquanett consists of Tina Valenti, Lead Vocals; Rick Thompson, Keyboards\/vocals; Dave Ward, Guitar; Michael Abdow, Guitar\/vocals; Matthew Macri, Bass guitar\/vocals; and Edmond H. Dupont III, Drums. Aquanett can be seen and heard all over the North East performing at several venues including clubs, fairs, private partys, weddings, corporate functions and concert halls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Every hundred years, the dark vampire known as Dracula resurrects and terrorizes the land. A vampire hunter named Simon Belmont bravely ventures into the Count's mansion in order to defeat him. Along the way he'll have to defeat skeletons, bats, fishmen, medusa heads and other evil creatures.\nCastlevania is a side-scrolling platform action game. The player takes the role of Simon Belmont, who is able to jump and crack his whip directly in front of him. Power-ups can be obtained by defeating enemies or by whipping candles that appear in the castle. At the end of each section of the castle is a boss, which must be defeated. Progression through the castle eventually leads to a confrontation with Count Dracula himself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Job related skills \/ software : Supervision of lying of optic fiber cable & testing of O.F.C, Cable jointing by splicing machine as per requirement of sites. A\/C & D\/C circuit testing, installation, & Commission of Electronics Instruments, The programming of management in Network & database.\nTo be associated with an organization that provides a continuous learning environment, a good career and challenging assignment.\n1. Bachelor of Technology in Electronics & Telecommunication Engineering from,\" RAJASTHAN VIDYAPEETH UNIVERSITY\" with in the year 2011.\n2. Diploma in Electronics Engineering from \"GOVERNMENT POLYTECHNIC SHAHJAHANPUR\" with in the year 2007.\nHigh School from \"Central Board of Secondary Education\" with in the year 2003.\nJUNIOR ENGINEER (ELECT.) in PIR PANCHAL CONSTRUCTION PVT LTD, JAMMU (J&K).\nJUNIOR ENGINEER (ELECT.) in M\/s A. S. Construction Co. SRINAGAR (J&K).\nThe working of launching\/safety, signal & telecommunications system for O.F.C. Signal line. Supervision of lying of optic fiber cable & testing of O.F.C, Cable jointing by splicing machine as per requirement of sites. A\/C & D\/C circuit testing, installation, & Commission of Electronics Instruments.\nThe programming of management in Network & database.\nOne year Diploma In Computer Application (software \/ hardware).\nOperating Systems - Window 7, XP, Vista.\nOffice Productivity Tools - MS office, MS Excel.\nHardware - Installing missing windows programs, common problems, troubleshooting tools and maintenance of functional system.\nSearching, browsing the World Wide Web, video conferencing, setting up an account.\nTraveling, Being adventurous, inventions and photography are my some of the major interests. I am also interested in watching movies that relates life.\n2. Willingness to enhance knowledge.\n3. Confident in interactions with individuals at all levels.\nI hereby declare that the above information are true and correct to the best of my Knowledge. I undertake to abide by the conditions of employment specified by the recruiters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"As my first post, I wanted to include an introduction of myself! This blogging project is very exciting and I am looking forward to it. My current goal is to post once a week, so lets see how that goes. Thanks for stopping by!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Homeopathic gout treatment is getting popularity because it prevents dangerous medications which have side effects and surgical process. Another very important feature of homeopathic treatment is that it may be applied as an effective and useful gout treatment way at any time when the elements of gout are first decided and it is also very useful in the last stages of the gout disease. To start with you should go to your homeopathic doctor and talk about the reasons of gout and then he will suggest you what kind of diet you should intake and what kind of foods you should not take. Typically this homeopathic gout treatment will began with small quantity of Herbal remedy for a small period of time and so the naturopathic medical doctors can easily observe the condition and see this homeopathic treatment is good for you or not.\nThis condition come very suddenly and quickly reaches at a climax of tremendous severe and harsh pain and then fall down.\nByronia This remedy suggested by homeopathic doctors to overcome the pain, swelling and redness of the affected joint.\nYou want to be sure that your supplement contains no byproducts, only pure Omega 3, so that you can get the maximum health benefits. When you look at the content label of any supplement you consider taking, it should indicate that it is made from the freshest achievable components and manufactured very carefully to maintain nutrient value.\nIt can occur in any joint, yet the big toe is the one most commonly affected. The condition results from uric acid, a form of waste topic, building up in the joint and forming crystals. These crystals make movement of the combined extremely painful and can lead to inflammation of the joint as well.\nThe diet must be set up so that you can steer clear of items placed on your menu such as fish, red wood meats, beer or lager and any products which contain brewers yeast. Most folks complain that during their gout attacks they have learned that no medicine seems to work suitably for them. Doctors often inform these patients that they must learn to live with it.\nWhile the link between fish oil and gout is significant, even fish oil does not represent a cure for the disease. You need to combine fish oil supplementation with other approaches including acupuncture, homeopathy or prescription anti-inflammatories if necessary. Eating a balanced diet that avoids rich foods and cutting out alcohol can also help to simplicity gout signs.\nOne of the top most popular methods of gout treatment is using cherries and cherry juice. Cherries have the anti inflammatory attributes and these may help to decrease the uric acid in the blood of the body and prevent it from penetrates in the joints. You should use fresh or canned cherries and sweet or perhaps sore cherries to cure the gout condition. Doctors suggested that you ought to consume approximately five cherries in one day. If you do not want to eat cherries in that case you can also use cherry juice as well.\nBecause gout involves a buildup of uric acid in the body, drinking plenty of water will help the kidneys more efficiently remove the excess acid from the system. Keep in mind that because water is gentler on the kidneys than many other types of liquids, the effect of this method of relief will be much higher with normal water than increasing your intake of coffee, tea, or soft drinks.\nFish harvested in the pristine waters off New Zealand offer the best source, as they are virtually pollutant free. These fish, harvested for maximum freshness and specially processed to retain their total Omega 3 content, make for the single most effective fish oil product on the market today. For individuals struggling with inflammatory conditions, this is particularly important.\nGout is an issue for you, consider the natural approach to reducing inflammation and lessening your pain by taking a great Omega 3 DHA fish oil supplement on a daily basis. It could help you to adopt the next step along your life's journey pain free!\nThis medication eases the pain of swollen and red affected joint very effectively. Berberis Vulgaris It is another very important herb for gout treatment. This drug decreases the severe pain of shared.\nThis herb applied directly on the swollen and painful joint.\nSome of the most famous and well known Herbs and natural medications used by homeopathic doctors to treat the gout are as follows.\nYou should marinate your affected joint in the charcoal bath for approximately half or one hour.\nAnother very effective natural medication for the gout treatment is charcoal poultice. You can make it by one half cup of activated powdered charcoal and a two tablespoon of flaxseed, mix these products properly then you should add warm water to produce paste. And then you can use this particular paste on that shared which is affected by the gout with plastic and cloth. Every four hour or so you should change the dressing.\nMix mustard powder with flour and water to form a pasty mixture. Place the mixture on the affected area overnight. Try to keep the affected area elevated while the mixture is on it. If the affected area is your foot, spot a sock over the stick.\nWhile Omega 3's are found at high levels in fish such as tuna, salmon, herring and mackerel, it can be difficult to get the necessary amount of these important nutrients through your regular diet. That's why most doctors recommend taking 2 fish oil capsules or one tbsp. of fish oil daily to help alleviate symptoms of gout and other arthritic conditions.\nIs this inflammation that first drew doctors and homeopathic practitioners to consider a possible connection between fish oil and gout. The Omega 3 fatty acids found in fish oil have proven anti-inflammatory properties, which makes them a powerful weapon against a number of conditions including asthma, heart disease and arthritis. Reducing inflammation can go a long way toward relieving this and stiffness associated with gout.\nThese days the events of gout have only increased. Our lifestyles are becoming unhealthier and that is a major reason the cases of people suffering from gout have increased. Many people take steps in order to alleviate the pain caused by gout by doing things like taking medication but simply end up making the problem worse. The good news is that there are effective ways to get gout relief naturally. This article will provide you with 5 tips you can implement right away to help stop your suffering from gout.\nBananas. Bananas also contain potassium. This makes it very nice because they have multiple sources of getting rid of gout. It is recommended to have 3500mg a day. A serving of bananas provides you with almost 10% of the daily allowance. Bananas also contain vitamin c. This strong vitamin is noted for helping you reduce your general uric acid levels when taken regularly.\nI am sure you are concerned about your health; therefore, suggest you consider adding omega 3 fish oil to your daily diet. You may download a free of charge fish oil buyuer's guide and obtain additional information on omega3 fish oil through gooing to:= Benefits Of Omega 3 Fish oil.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Monday, \"The Wendy Williams Show\" returned to airwaves, but without it's main host. Prior to today's announcement, the Daytime Emmy nominated host announced she wouldn't be returning to the talk show until January 21st via Instagram.\nThe show was hosted by a panel of \"Hot Talkers\" who entertained viewers in the host's absence.\nLet's continue to send love, light and prayers for a speedy recovery.\n[WATCH] Love Hip Hop NY Episode 11 + Are Tahiry & Joe Budden Reunited for Good?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Confederation of Autosport Car Clubs (CACC) holds motorsports events all year long on the west coast. In the winter months, a very dedicated group of racers look forward to the freezing weather and the trek up the Fraser Canyon to Barnes Lake for a season of ice racing. Unfortunately, the track is dependent on how cold it is and how much ice is on the lake. If it isn't cold enough that means no ice. No ice means no track. No track means no racing. While there's still some time for things to turn around, this winter's warm weather saw things get off to a slow start, with only a single icecross having been run on the lake as of this writing.\nFortunately, the weather in the Greater Vancouver area and in Victoria on Vancouver Island affords us the opportunity to run slalom events year round, with only a short break over the Christmas holidays. Check the CACC web site for a listing of clubs that organize slalom events throughout B.C. during the year.\nThe CACC will have a booth at the Vancouver International Auto Show again this year (March 23-27), which always attracts lots of enthusiastic spectators. This usually brings new people out to our events, and we invite you to come out and learn more about our organization.\nBoth the Victoria Motorsports Club (VMSC) and the Sports Car Club of British Columbia (SCCBC) will also be hosting their annual Race Driver Training Events in March, too. The VMSC will hold its school on March 26-27 in Victoria while the SCCBC will hold its on March 13-19- 20. These schools are for people wanting to go wheel-to-wheel racing as well as those who want to learn better car control. Spots fill up early and fast.\nThe 2016 CACC race schedule is set for the events at Mission Raceway Park's road course, with the first race weekend set for April 16-17 with a practice day on Friday the 15th. It will be a full field of cars with racing for novices on Saturday, and full grids of open- and closed-wheel cars both days. Plus, there will be a time attack on Saturday, with Sunday set aside for racing Vintage cars. The full schedule is on www.sccbc.net.\nTime attackers are also looking forward to the Annual Knox Mountain Hillclimb, which takes place on the Victoria Day long weekend in Kelowna, BC this year. This long-standing event continues to be the only sanctioned hillclimb in Canada, and draws entries from all over western Canada and the U.S.\nThe vintage racing discipline will be holding races on all regular CACC race weekends, but The British Columbia Historic Motor Races standalone race weekend will take place on August 19-21. This event always features a great display of old race cars doing what they were built to do \u2013 RACE! \u2013 as well as a superb car corral with lots of old cars to check out. There is a fun social atmosphere at the Saturday night banquet \u2013 a must attend event \u2013 so what are you waiting for?\nBefore signing off, the CACC wants to congratulate Rick Payne of Mission, BC, who took his Van Dieman RF99 Honda from 10th on the grid to victory at the 2015 Formula F Championship during the SCCA Runoffs in Daytona Beach, Florida in September. Well done Rick!\nThe CACC is looking forward to another great season in all of our different facets of motorsport. And we invite you to come out and join us. Check our web site www.CACCautosport.org to find contacts for our affiliate clubs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Allele-specific transcriptional regulation, including of imprinted genes, is essential for normal mammalian development. While the regulatory regions controlling imprinted genes are associated with DNA methylation (DNAme) and specific histone modifications, the interplay between transcription and these epigenetic marks at allelic resolution is typically not investigated genome-wide due to a lack of bioinformatic packages that can process and integrate multiple epigenomic datasets with allelic resolution. In addition, existing ad-hoc software only consider SNVs for allele-specific read discovery. This limitation omits potentially informative INDELs, which constitute about one fifth of the number of SNVs in mice, and introduces a systematic reference bias in allele-specific analyses.\nHere, we describe MEA, an INDEL-aware Methylomic and Epigenomic Allele-specific analysis pipeline which enables user-friendly data exploration, visualization and interpretation of allelic imbalance. Applying MEA to mouse embryonic datasets yields robust allele-specific DNAme maps and low reference bias. We validate allele-specific DNAme at known differentially methylated regions and show that automated integration of such methylation data with RNA- and ChIP-seq datasets yields an intuitive, multidimensional view of allelic gene regulation. MEA uncovers numerous novel dynamically methylated loci, highlighting the sensitivity of our pipeline. Furthermore, processing and visualization of epigenomic datasets from human brain reveals the expected allele-specific enrichment of H3K27ac and DNAme at imprinted as well as novel monoallelically expressed genes, highlighting MEA's utility for integrating human datasets of distinct provenance for genome-wide analysis of allelic phenomena.\nOur novel pipeline for standardized allele-specific processing and visualization of disparate epigenomic and methylomic datasets enables rapid analysis and navigation with allelic resolution. MEA is freely available as a Docker container at https:\/\/github.com\/julienrichardalbert\/MEA.\nNext-generation sequencing (NGS)-based approaches for genome-wide analysis of RNA, histone post-translational modifications (PTMs), DNA methylation (DNAme) and chromatin conformation are now routinely conducted on both model organisms and human samples. Such studies have yielded many insights into the interplay between chromatin structure and transcription, including the surprising observation that allele-specific phenomena may be more widespread than previously believed [1, 2]. Unfortunately, while such datasets, including RNA sequencing (RNA-seq), chromatin immunoprecipitation followed by sequencing (ChIP-seq) and whole genome bisulphite sequencing (WGBS), are theoretically amenable to allele-specific profiling, NGS analysis software generally does not discriminate between parental alleles from diploid genomes. Indeed, popular read aligners depend on alignment to a single reference genome, essentially considering the sequencing reads generated from autosomes (and the X-chromosome in the case of females) as originating from isogenic rather than outbred individuals. In merging both parental alleles into a single measurement, these aligners neglect allele-specific phenomena, such as genomic imprinting , X-chromosome inactivation  and sequence-dependent cis-regulatory effects .\nTo overcome this shortcoming, a number of software packages have recently been developed that assign NGS sequencing reads to a specific parental allele. For example, MMSEQ , QuASAR , MBASED  and SCALE  were designed to analyze RNA-seq data, while MethPipe , epiG  and BSPAT  were designed to process DNAme data. Several independent custom scripts for allele-specific analyses have also been reported [11\u201313], but the details required for implementing them were not included. Pipelines such as Allelome.PRO , WASP  and our previously published toolbox, ALEA  accommodate both RNA- and ChIP-seq datasets, yet no pipeline offers the additional capability of processing DNAme data. The lack of a universal allele-specific pipeline has precluded robust integration of allele-specific transcription, histone PTMs and DNAme profiles. Importantly, while such pipelines can be applied in parallel to analyze distinct epigenomic features, installation and implementation of multiple software packages can be time consuming, even for experienced bioinformaticians. Additionally, comparing allelic results generated using different software can introduce confounding factors, as the strategies used to process reads depend on multiple parameters, including read trimming, alignment mismatch scoring and read alignment filtering (mapping quality, PCR duplicate reads). For example, several allele-specific analysis packages rely on reference genome alignment followed by variant calling [8, 10, 14], while others leverage publicly available single nucleotide variant (SNV) data to derive a diploid genome for read alignment [5, 15, 16]. This \"pseudogenome\" strategy is a significant improvement over the former as it enables alignment over loci with high levels of genetic variation. However, current pipelines exclude short insertions and deletions (INDELs) for pseudogenome reconstruction, as they modify reference chromosome sequence lengths and annotated gene coordinates required for downstream analyses. Given the relative abundance of INDELs, this shortcoming may lead to the omission of a significant fraction of informative allelic reads. Indeed, analysis of high quality genotyping information for mouse strains reveals that, exclusive of structural variants, INDELs compose up to 20% of genetic variation . Thus, an INDEL-aware allele-specific pipeline that considers both SNVs and INDELs for pseudogenome reconstruction would offer a significant improvement over existing software.\nHere, we present MEA, an \"all-in-one\" bioinformatics toolbox that exploits both SNVs and INDELs to enable allele-specific analyses of RNA-seq and ChIP-seq as well as WGBS datasets generated using short-read sequencing technology (Fig. 1a). MEA is shipped in a Docker container, enabling one step installation of all dependencies independent of operating system type. After providing a reference genome assembly (e.g. hg19 or mm10) and a VCF file containing the relevant genetic variants, users simply input an NGS dataset in FASTQ format. MEA will then automatically generate allele-specific genomic coverage files in BigWig format and allele-specific analyses over user-defined regions of interest in a tab-delimited table. To benchmark the performance of our INDEL-aware software, we present both theoretical and real-world evidence for improved allele-specific DNAme analysis relative to an INDEL-agnostic pipeline. Furthermore, to highlight the utility of MEA, we investigate DNAme data processed in parallel with RNA- and ChIP-seq data from mouse hybrid embryos and uncover novel differentially methylated regions (DMRs). Additionally, using human brain cell data, we observe the expected H3K27ac and DNAme enrichment at known imprinted genes and uncover novel monoallelically expressed genes, further demonstrating the power of integrating epigenetic and expression analyses in a unified workflow. The MEA toolbox harmonizes NGS read processing, with all dependencies consolidated in a Docker container, includes pan-species compatibility, maximizing its utility for allele-specific profiling of model organisms as well as human samples.\nTo generate a harmonized workflow for processing of DNAme, RNA-seq and ChIP-seq datasets, we developed a universal strategy for detecting allele-specific reads. Further, to maximize the number of experimental reads that can be assigned to a specific allele for each data type, MEA was designed to exploit underlying genetic variation by incorporating both SNVs and INDELs during pseudogenome construction. For each data type, allelic reads are captured by constructing an in silico pseudogenome comprised of both parental genomes followed by NGS read alignment. Aligning reads simultaneously to both haplotype sequences of a diploid genome facilitates the appropriate alignment of reads that map to heterozygous loci onto their cognate allele, reads which otherwise would be discarded due to \"sequencing errors\". Such reads are thus extracted and can be used to de-convolute allelic phenomena.\nTo establish a pipeline for allele-specific DNAme analysis, we began by incorporating Bismark , a widely adopted bisulphite-seq read aligner and methylation caller, into ALEA, our previously developed tool for allele-specific analyses of RNA-seq and ChIP-seq datasets . We first quantified the hypothetical increase in the percentage of informative CpG sites from which we can infer allelic information by incorporation of INDELs in addition to SNVs during pseudogenome reconstruction. As high-quality genetic variation information of inbred mouse strains is available , we constructed a pseudogenome from two mouse strains, namely DBA\/2 J and the reference strain C57BL\/6 J (build mm10), incorporating known genetic variants (SNVs and\/or INDELs). By counting CpGs within 200 bp (an insert size typical of WGBS libraries) of an INDEL or SNV, we found that INDEL incorporation leads to a theoretical increase in the number of informative CpGs (i.e. CpGs for which DNAme differences between alleles can be deduced) of 12.9% for this pseudogenome (Fig. 1b). Notably, a subset of genomic regions with associated INDELs are entirely devoid of SNVs and therefore include nearby CpGs that theoretically can only be assessed by pipelines that are \"INDEL-aware\".\nTo test whether the inclusion of INDELs increases the number of informative CpGs for which allelic methylation state can be calculated in practice, we processed raw reads from a previously published WGBS dataset from C57BL\/6 J x DBA\/2 J mouse F1 inner cell mass (ICM) cells . Applying the same filtering parameters allowed us to directly compare results obtained with the MEA pipeline to those of the Bismark-based INDEL-agnostic custom script employed by Wang et al. . MEA yielded a 62.5% increase in the number of CpGs covered by at least 5 allele-specific C57BL\/6 J reads (Fig. 1c). Importantly, informative CpGs gained using MEA overlapped almost exclusively with CpGs within 200 bp of an INDEL or SNV, as expected. This gain is likely the result of an increase in the number of informative heterozygous sites (quantified in Fig. 1b) as well as efficacious alignment of reads to the non-reference genome over regions with high INDEL density.\nReads from regions with high INDEL density were presumably excluded by the pipeline from Wang et al.  as \"sequencing errors\", rather than assigned as allelic variants. To confirm that MEA increases the alignment rate of non-reference reads, we repeated the alignment of C57BL\/6 J x DBA\/2 J F1 WGBS reads to a reference genome as well as the MEA-constructed diploid pseudogenome (composed of the reference and DBA\/2 J genomes) and determined the number of reads that aligned to each genome 0, 1 or\u2009> 1 time (Fig. 2a-b). Alignment to a pseudogenome increased the overall alignment rate by 1.25% (80.83 to 82.08%), most likely due to alignment of non-reference-originating reads at loci that show significant genetic divergence (high SNV and INDEL density) from the reference. As expected, the majority of reads aligned uniquely to the haploid reference genome aligned at least twice to the pseudogenome, except over regions containing genetic variants. This crucial distinction allowed the uniquely aligned reads to be extracted and assigned to their cognate parental genomes, with 8.8 and 8.2% of all aligned reads specific to C57BL6J and DBA\/2 J strains, respectively (Fig. 2c). By capturing a greater number of sites at which we can measure allelic DNAme levels, a higher proportion of experimental reads can be assigned to a specific parental haplotype, thus enabling the evaluation of allelic differences in DNAme levels for a higher fraction of the genome.\nA major concern when exploring allele-specific data is the potential for reference bias caused by differences in genomic sequence quality between the reference and non-reference genomes, which may lead to preferential alignment of reads to the former and artefactual allelic imbalance results . For example, using an INDEL-agnostic pipeline similar to that employed by Wang et al. , Keown et al, reported a reference bias of 15.4% in their study of allele-specific DNAme in C57BL\/6 J x SPRET\/EiJ cells  (SPRET\/EiJ has >\u20095 times the number of SNVs relative to C57BL\/6 J than does DBA\/2 J ). To determine the extent of reference bias in our MEA pipeline, we benchmarked the observed parental contribution to allelic read alignment for each autosome from the C57BL\/6 J x DBA\/2 J ICM WGBS dataset generated by Wang et al.  (Fig. 2d). Notably, MEA yielded an alignment reference bias on all autosomes of 3.81%, only ~\u200954% of that reported by the INDEL-agnostic pipeline (6.98%, Fig. 2e). This reduction in alignment bias is consistent with the increased fraction of allele-specific reads aligned to the non-reference genome.\nFalse positives caused by erroneous allelic read alignment at regions devoid of true genetic variation can lead to an underestimation of reference bias in allele-specific experiments. To quantify the false positive allelic alignment rate of our pipeline, we processed pure C57BL\/6 J WGBS data using the C57BL\/6 J x DBA\/2 J pseudogenome described above and determined the parental contribution to allelic read alignment (Fig. 3a). Curiously, 0.8% of all aligned reads (5.13% of allelic reads) were scored as DBA\/2 J-specific, indicating that MEA has an FDR of ~\u20095%. When calculating the parental contribution to allelic read alignment over each autosome, we found that the majority of false-positive (\"DBA\/2 J-specific\") allelic read alignments clustered on chromosomes 2 and 9 (Fig. 3b). Closer inspection revealed that these regions are annotated by RepeatMasker as Satellite DNA (Fig. 3c). Such allele-specific calls at sites lacking genetic variants are the result of Bismark's mapping quality algorithm, which calculates an erroneously high mapping score at these highly repetitive regions. Analysis of processed WGBS data from pure DBA\/2 J spermatozoa without black-listing of repetitive regions revealed a C57BL\/6 J-specific alignment rate of 3.80% (Additional file 1: Figure S1), indicating that a global false positive rate of ~\u20095% may be expected when using the MEA pipeline for analysis of WGBS data without excluding repetitive regions. Since satellite DNA is generally omitted in studies of the transcriptome or epigenome, we excluded reads aligned to annotated satellite repeats (0.19% of the mappable genome) and recalculated the false-positive rate for the C57BL\/6 J dataset, which dropped to 1.62% of allelic reads, with no specific chromosome enriched (Fig. 3d). Thus, when applying the MEA pipeline, the majority of false positive read alignments can likely be removed by black-listing satellite repeats.\nTo establish the accuracy of calculating allele-specific DNAme levels using the MEA pipeline, we measured allele-specific DNAme levels over known imprinted gDMRs. Such regions are densely methylated on one allele and unmethylated on the other as a result of parent-of-origin dependent differences in methylation established in the gametes, representing a unique resource for benchmarking allele-specific DNAme calling. Of the 23 known mouse gDMRs, 9 harbor SNVs and\/or INDELs between the C57BL\/6 J and DBA\/2 J genomes and can therefore be assessed for allele-specific DNAme levels. For consistency, we directly compared our allele-specific results over these regions with those reported by Wang et al.  (Fig. 4a). For most gDMRs, MEA yielded average allelic DNAme levels similar to those reported by the INDEL-agnostic pipeline. However, MEA consistently yielded allele-specific information over a greater number of CpGs (mean \u00b1 SD: 72 \u00b1 24 vs 38 \u00b1 21 CpGs on either allele), increasing the statistical power of allelic imbalance calculations. For example, MEA detected a total of 68 CpGs informative for allelic methylation state at the Meg3 gDMR, nearly three times greater than the number reported by Wang et al. (Table 1). As expected, when calculated over the same 129 CpGs covered by at least five reads in the gDMR, DNAme levels calculated by the two pipelines independent of allelic calling were nearly identical (30.2% vs 30.6%). However, the discordance between the percentage of methylation calculated for the CpGs that are informative at an allelic level was significantly lower using the MEA pipeline (0.13% vs 5.8%), indicating that the accurate determination of allelic DNAme levels at specific loci can be adversely impacted by sampling errors. Furthermore, as expected, only the MEA pipeline yields informative results for CpGs proximal to INDELs at the Meg3 gDMR locus (Fig. 4b), confirming the benefit of incorporating the latter during pseudogenome reconstruction. Taken together, these analyses demonstrate that MEA outperforms an INDEL-agnostic pipeline.\nA recent study employing MeDIP on genomic DNA isolated from early mouse embryos revealed the presence of maternally-methylated DMRs that are resolved during post-implantation development . While these \"transient DMRs\" may have important biological functions during pre-implantation development [22, 23], the extent of transient imprinting remains unclear. To determine whether MEA can be used to identify novel DMRs, we assayed the subset of informative regions gained using our refined pipeline, namely loci exclusively overlapping INDELs, using the aforementioned WGBS data from C57BL\/6 J x DBA\/2 J ICM cells. As expected for preimplantation cells, which are characterized by globally low DNAme levels , hypomethylation of both parental alleles was generally observed over such informative regions, including at those with high CpG density (Fig. 5a). Importantly, analysis agnostic to allelic alignment also revealed hypomethylation across such regions (for example, see Additional file 1: Figure S2). However, focusing on regions within 200 bp of annotated transcription start sites (TSSs) reveals that a subset show clear asymmetric DNAme levels (Fig. 5b), with either maternal or paternal bias.\nUCSC genome browser screen shots of two putative TSS proximal DMRs, including the apparently paternally methylated Kiss1 (a suppressor of metastasis) and maternally methylated Lpar6 (a lysophosphatidic acid receptor) genes, are shown in Fig. 5c and d. Using the MEA pipeline, 15 and 34 CpGs respectively, are informative on either allele at these loci. Importantly, the absolute methylation levels reported by the allele-agnostic pipeline (27.2 and 26.5%) are similar to those of the mean allele-specific methylation (29.4 and 34.6%), consistent with the observation that methylation at these loci is allele-specific. Moreover, intersection of these ICM data with WGBS data from mature gametes [25, 26] reveals that paternal DNAme at the Kiss1 gene in the former is likely the result of methylation already present in spermatozoa, indicating that this locus potentially protected from the wave of genome-wide DNA demethylation that occurs early in mouse embryonic development . Parental asymmetry at the Kiss1 locus is resolved by E7.5, when the maternal allele gains DNAme coincident with the wave of global de novo DNAme that occurs during early post implantation development . On the other hand, the short, intron-less gene Lpar6 is hypermethylated in both mature oocytes and spermatozoa, indicating that the paternal but not the maternal allele is susceptible to the global wave of DNAme erasure that takes place after fertilization. Parental asymmetry of DNAme is resolved by loss of maternal DNAme in the E7.5 post-implantation embryo, revealing that the allelic bias in DNAme at this locus is also transient but involves sequential loss of DNAme on the paternal followed by the maternal allele. Whether these non-canonical DNAme dynamics are driven by genetic or parent-of-origin effects, and their contribution to the development of the early embryo, remains to be tested. Regardless, the novel DMRs identified proximal to the Kiss1 and Lpar6 TSSs exemplify the merit of increasing the number of allelic reads extracted from experimental datasets and underscores the potential for future discoveries using this approach.\nIn order to integrate epigenomic and transcriptomic-based datasets, alignment to the same genomic sequence is required. Transcriptomic data presents a unique challenge when aligning to a genome, as processed messenger RNA contains many gaps (introns) relative to the template DNA sequence. In our previously published pipeline ALEA , RNA-seq alignment was carried out using the short-read aligner BWA, which does not allow alignment of intron-spanning reads. Thus, to enable integration of transcriptomic and epigenomic datasets, gapped read alignment is essential. Tophat2  and STAR , two widely used aligners that incorporate this feature, were recently shown to perform well in short-read RNA-seq alignment . To determine which of the two shows superior allele-specific gapped read alignment, we carried out a side by side comparison of these aligners, as well as the non-gapped read aligner BWA, using a published RNA-seq dataset from C57BL\/6 J x DBA\/2 J F1 ICM cells. STAR clearly outperformed both Tophat2 and BWA (Fig. 6a), likely due to its advanced gapped read alignment algorithm  and ability to properly assign paired-end reads associated with the same DNA molecule (if a read aligns to a region including a genetic variant, its mate is also identified as allelic regardless of whether it overlaps a genetic variant). Thus, analysis of paired-end sequencing data using the STAR aligner and MEA pipeline increases the fraction of regions showing relatively high sequence conservation over which allele-specific NGS reads can be aligned, an improvement over using flanking regions as a proxy. Based on these observations, we currently recommend the STAR aligner, but MEA's flexibility in incorporating new NGS aligners facilitates its adoption for analyzing epigenomic and expression datasets using alternative\/next generation aligners, such as those that can accommodate increased read lengths.\nIn our previously published pipeline ALEA , allele-specific alignment of ChIP-seq datasets was limited to the BWA-aln algorithm. To enhance MEA's flexibility, we incorporated another popular ChIP-seq aligner Bowtie2. To compare the performance of BWA-aln and Bowtie2 for allele-specific ChIP-seq alignment, we processed H3K4me3 ChIP-seq data generated from pure C57BL\/6 J and PWK\/PhJ gametes . While both alignment algorithms yield a low false-positive alignment rate of ~\u20090.2\u20134.8%, BWA-aln clearly reports more allele-specific read alignments than Bowtie2 (Additional file 1: Figure S3 and Additional file 2: Table S1). Thus, while users can choose between BWA-aln and Bowtie2, we recommend the former for allele-specific analysis of ChIP-seq data using MEA.\nDissecting the interplay between epigenetic marks and transcription was greatly facilitated by the advent of NGS-based approaches for measuring RNA levels and the genome-wide distribution of DNAme and histone PTMs. However, as such datasets are commonly processed using different pipelines, integrating and visualizing allelic information embedded therein is non-trivial. To automate dataset integration, MEA processes WGBS, RNA- and ChIP-seq alignment data using the same allele-specific read identification strategy, yielding standardized allele-specific genomic tracks. This unification of file types allows simultaneous visualization of each datatype (in BigWig format) using popular genome browsers. Further, to automate the process of reporting allelic imbalance, MEA generates a tab-delimited table containing allelic imbalance measurements over user-defined regions of interest, such as transcription start sites, genic exons or gene bodies (see Additional file 3: Table S2).\nThis approach solves two important considerations in the presentation of allele-specific data. First, allelic genomic tracks, i.e. those displaying only read coverage that is informative for allelic alignment, are inherently sparse, especially at regions devoid of genetic variants. To delineate signal from noise, allele-specific genomic track visualization should be considered in the context of all aligned reads and the position of the genetic variant sites. Second, allele-specific enrichment is greatest at sites of genetic variation and therefore does not necessarily coincide with the profiles generated from all reads agnostic of allelic assignment. For example, while reads derived from H3K4me3 ChIP-seq datasets are enriched over active TSSs, allelic H3K4me3 reads may align anywhere within the set of allele-agnostic peaks. Thus, allelic reads aligning at the edge of a region of H3K4me3 enrichment that is devoid of genetic variants at its center may be incorrectly discarded as noise.\nThe MEA pipeline standardizes such integrated track visualization by organizing genomic tracks into a UCSC Track Hub . These hubs agglomerate multiple colour-coded data tracks, enabling the concurrent visualization of allele-specific and \"total\" (allele-agnostic) alignment profiles, and in turn interpretation of allelic imbalance. Variant files used for pseudogenome reconstruction can also be directly visualized as UCSC custom tracks. The utility of this approach is illustrated using the Meg3 gene and its governing gDMR as a representative locus (Fig. 6b). Imprinting is simultaneously displayed in four independent datasets generated from two distinct F1 hybrid crosses. The Meg3 gDMR is paternally methylated and weakly enriched for both permissive (H3K4me3) and repressive (H3K27me3) histone PTMs (grey). Interestingly, H3K4me3 and H3K27me3 asymmetrically mark the maternal and paternal alleles, respectively, as expected for the promoter of a gene expressed exclusively from the maternal allele. Notably, each dataset is consistent with paternal imprinting, with repressive marks associated with the paternal allele and active marks with the expressed maternal allele. Profiles of the maternally imprinted Snrpn and Impact loci reveal similar patterns (see Additional file 1: Figures S4 and S5). Note that for the Impact locus, a single genetic variant in the F1 hybrid analyzed is sufficient to score DNAme asymmetry between parental alleles. The observed enrichment of both H3K4me3 and H3K27me3 at imprinted DMRs is consistent with a previous report , and evidence of H3K4me3 and H3K27me3 enrichment asymmetry on active and repressed alleles has been documented for individual genes . Thus, the allele-specific genomic tracks and dataset integration employed by MEA enhances the visualization of allelic differences between epigenetic marks and transcription across the genome.\nTo demonstrate the utility of MEA for the study of NGS datasets from human samples, we used the STAR aligner to analyze an RNA-seq dataset generated from human brain tissue. For individuals whose parental genomic sequences are unavailable, MEA uses Shape-IT  to phase individual genetic variants into inferred haplotypes. For each annotated gene, the haplotype-specific contribution to allelic read alignment was calculated using MEA (Additional file 3: Table S2). As expected, human imprinted genes  such as MEST, MEG3, PEG3 and PEG10 display monoallelic expression (Fig. 7a), confirming the suitability of MEA for the analysis of RNA-seq data from human samples.\nWe next generated UCSC Track Hubs to visualize the RNA-seq data analyzed above, as well as matched DNAme (WGBS) and histone PTM (cross-linked ChIP-seq) data from human brain and focused on imprinted genes that include genetic variants in their exons and respective DMRs. Thirteen known imprinted genes were expressed (RPKM >\u20091) and had at least 10 allele-specific mapped read coverage on either allele, 6 of which show >\u200980% expression from one allele (see Additional file 3: Table S2). A screen shot of the imprinted MEST gene, which is paternally expressed in somatic tissues, is shown in Fig. 7b. As expected, analysis of sperm and oocyte WGBS data from unrelated individuals reveals a DMR at the MEST TSS that is methylated exclusively in the oocyte and shows ~\u200950% methylation across the annotated DMR in adult brain cells. MEA output reveals one allele with dense methylation in this region, haplotype 1 (hap1) and the other with very low methylation (hap2). Importantly, only the latter, which is transcriptionally active, shows enrichment of H3K27ac, a histone modification associated with active genes. Based on allele-specific DNAme, transcription and histone PTM patterns, we surmise that haplotypes 1 and 2 of the MEST locus were inherited from the proband's mother and father, respectively. Taken together, these results reveal that MEA successfully integrates allele-specific RNA-seq data with WGBS and ChIP-seq data for identification and visualization of human loci harbouring genetic variants.\nTo determine whether H3K27ac shows allele-specific enrichment in the promoter regions of genes exhibiting allele-specific transcription, we identified all genes that harbor genetic variants over annotated exons and the TSS and calculated their allelic ratios (Fig. 8a). While the correlation between expression and H3K27ac allele-specific ratios is low (Pearson r2\u2009=\u20090.29), many genes displaying strong allele-specific expression bias (over two standard deviations from the mean) are also enriched for H3K27ac on the active allele (\u03c72 test p values for bias towards haplotype 1\u2009=\u20091.38E -24 and haplotype 2\u2009=\u20094.8E -38), as expected. Moreover, manual inspection of a subset of genes displaying monoallelic expression and biallelic H3K27ac reveals that transcription originates at alternative promoters. To further quantify the relationship between allele-specific H3K27ac and transcription, we categorized genes based on allele-specific transcription bias and measured the distribution of allele-specific H3K27ac at TSSs (Fig. 8b). Notably, while allele-specific H3K27ac was positively correlated with transcriptional activity, the ChIP-seq input (control) dataset also showed a higher level of enrichment on the active allele for each haplotype. This observation is consistent with previous studies demonstrating that the promoter regions of active genes are inherently more sensitive to sonication than inactive genes [37, 38]. That this bias also applies to individual genes exhibiting allelic differences in expression\/PTMs reiterates the importance of input-correction of ChIP-seq material and highlights the sensitivity of the MEA pipeline for quantifying allele-specific differences in enrichment.\nTo determine whether MEA can be employed to identify novel monoallelically expressed transcripts in human samples, we revisited the brain RNA-seq data described above. Applying thresholds for total expression (RPKM >\u20091), allele-specific coverage (mapped reads >\u2009100) and expression bias (>\u200990% of transcript levels from one allele), we identified 222 monoallelically expressed transcripts (Fig. 7a). Ten of these 222 transcripts showed sufficient H3K27ac ChIP-seq coverage for allele-specific calling (total RPKM >\u20091 and allele-specific CpGs on each allele. While seven of these transcripts (PIK3R3, ZNF662, PSMC1, LOC145784, CYP4F24P, C19orf48 and ZNF805) showed biallelic or minor allele-specific bias in H3K27ac, perhaps indicative of allele-specific post-transcriptional regulation, three (MEST, MIR4458HG and PCDHA5) showed strong H3K27ac bias toward the active allele (>\u200990% allelic reads). Importantly, the latter represent known and candidate novel imprinted genes. PCDHA5 belongs to a large gene family of protocadherins, complicating allelic interpretation. However, analysis of the previously described imprinted gene MEST (Fig. 7b) and the uncharacterized non-coding RNA gene MIR4458HG (Fig. 8c), revealed H3K27ac enrichment and intermediate methylation at their TSSs. As described above for the MEST gene, allelic deconvolution at the MIR4458HG promoter using MEA reveals H3K27ac enrichment and the absence of DNAme exclusively on the active allele. Furthermore, analysis of published WGBS data from gametes reveals hypomethylation of the MIR4458HG TSS in both sperm and oocyte, indicating that the allelic gain of DNAme at this locus occurs in somatic tissues. Thus, using MEA to integrate complementary RNA-, ChIP-seq and DNAme datasets allows for the allele-specific resolution of epigenetic states at the regulatory regions of both known and novel monoallelically expressed genes.\nThe proper installation and configuration of bioinformatics dependencies is a major hurdle for both new and experienced users. To address this challenge, we packaged MEA into a Docker container, an open-source software packaging and distribution system (see Materials and Methods). The self-contained nature of the container allows one-step installation of all 15 bioinformatic dependencies (STAR, bwa, Bedtools, Bowtie2, Tophat2, Bismark, Java, etc.), providing a consistent user experience independent of operating system (Windows, MacOS, Linux, etc.). Furthermore, the consolidation of all MEA tool installation steps will greatly facilitate future incorporation of alternative NGS aligners.\nThe surge of publicly available NGS epigenomic and expression datasets generated by international consortia, has outpaced the development and dissemination of bioinformatic pipelines that can be used to analyze disparate epigenomic datasets at allelic resolution. To address this need, we developed a universal pipeline that generates integrated allele-specific genomic tracks for DNA methylation (WGBS or Reduced Representation Bisulphite Sequencing (RRBS)), expression (RNA-seq) and histone modification (ChIP-seq) data. Using a unique strategy that incorporates INDELs in addition to SNVs during pseudogenome reconstruction, MEA increases the quality of non-reference genomic sequences, yielding a reduction in reference genome alignment bias. Additionally, in the case of mouse datasets, false positive allele-specific alignments can be minimized by excluding satellite repeats from post-alignment analysis. By considering INDELs and SNVs, MEA captures significantly more allelic CpGs than an INDEL-agnostic script and in turn increases the sensitivity of allele-specific, parent-of-origin DNAme level calculations. Furthermore, by implementing RNA-seq aligners developed specifically to address spliced read alignment, such as STAR , MEA reports allele-specific expression over a greater proportion of the transcriptome relative to other aligners.\nThe fraction of the genome for which allele-specific state can be calculated is a function of several experimental variables, including the choice of parental strains in the case of F1 hybrid studies in model organisms. We were able to measure allele-specific DNA levels over 20.4% of all CpGs in C57BL\/6 J x DBA\/2 J F1 hybrid mice. The DBA\/2 J strain is quite similar genetically to the reference C57BL\/6 J, containing on average one SNV per 530 bp (0.19%), at the lower limit of the optimal sequence divergence range of 0.1 to 5% for genome-wide allelic analysis . Wild and inbred mouse strains such as PWK\/PhJ, CAST\/EiJ or SPRET\/EiJ are up to eight times more divergent than commonly used strains, such as DBA\/2 J, 129S1\/SvImJ and C3H\/HeJ . Thus, when crossed with any other strain, such F1 hybrids will yield a significant increase in the fraction of informative reads. Regardless of parental genome diversity, the incorporation of INDELs in addition to SNVs during pseudogenome reconstruction, as implemented in MEA, significantly increases the number of regions over which allele-specific methylation can be discerned. For strains with available SNV and INDEL annotations, such as those provided by the Sanger Institute's Mouse Genomes Project , the average genetic variant frequency between parental genomes can easily be calculated, and in turn, the fraction of the genome likely to be informative for discriminating allele-specific reads determined a priori.\nBy increasing the number of allele-specific reads extracted from NGS datasets of outbred individuals, including F1 hybrid model organisms as well as human subjects, MEA enables the identification of novel DMRs in WGBS data, allelic-specific gene expression from RNA-seq data and the discrimination of histone marks showing parent-of-origin specific patterns from true bivalent marks by ChIP-seq. As this toolbox was developed to process next generation sequencing reads regardless of experiment type, MEA can also be used to analyze additional chromatin features with allelic resolution. For example, to map chromatin accessibility at an allelic level, DNase I hypersensitivity site-sequencing (DNase-seq, ) or transposase-accessible chromatin followed by high-throughput sequencing (ATAC-seq, ) datasets can be interrogated and the results integrated with the data types described above. Importantly, if allele-specific resolution is desirable, previously generated datasets using any of these approaches can be revisited using MEA.\nWhile MEA can be applied to datasets generated from any diploid organism, there are several important limitations that must be considered for clinical studies. As each individual has a unique diploid genome (except in the case of monozygotic twins), pseudogenome reconstruction is essential. While MEA exploits publicly available whole genome sequencing datasets from the Sanger Institute's Mouse Genomes Project  and the human-focused 1000 genomes project , additional genotyping and variant-calling steps will be required for haplotypes not covered by these population level sequencing projects. Nevertheless, large-scale efforts such as The Cancer Genome Atlas (TCGA) project that harmonize various cancer-related dataset types, including genotype information, may be analyzed using MEA to deconvolute complex relationships that may operate at an allele-specific level. For example, a recent publication combined genetic, DNAme and gene expression variation to explain aberrant gene regulatory networks in thyroid carcinoma samples . Given the high frequency of heterozygous somatic mutations in many cancer types, MEA may be applied to directly measure the effect of these mutations on DNAme and gene expression levels on the same allele by using the other allele as a control, potentially allowing for the identification of additional driver mutations. Since in silico diploid genome sequences are twice as large as their respective reference assemblies, such population-based studies (encompassing thousands of individuals) will require extensive computational infrastructure. These technical restrictions limit the number of unique individuals that can be practically evaluated. Therefore, for studies encompassing large outbred populations, an alternative approach that combines genotyping and allele-specific read calling is more suitable . Nevertheless, for smaller scale epigenomic studies, such as those involving trios, MEA can be applied to study the role of genetics in epigenetic variation, and in turn, to facilitate the discovery or validation of variants of interest, complementing epigenome-wide association studies (EWAS) .\nTo our knowledge, MEA is the first software package to provide integrated allele-specific analysis of DNA methylation, histone modification and expression data. Exploiting both SNV and INDEL information, this pipeline increases the sensitivity and specificity of allelic analyses relative to an INDEL-agnostic approach. MEA automates diploid pseudogenome reconstruction, allele-specific read detection and haplotype-resolved genomic track agglomeration for intuitive data visualization and allelic imbalance detection. With one-step installation and user-friendly file outputs, MEA can be applied without relying on extensive bioinformatic expertise. Intersection of epigenomic and transcriptomic datasets using this novel toolbox will facilitate studies of parent-of-origin effects as well as the interplay between genomic sequence, the epigenome and transcriptional regulation in both humans and model organisms.\nWe validated our tool using previously published bisulphite-seq data generated from inner cell mass (ICM) cells from an F1 hybrid between mouse strains C57BL\/6 J and DBA\/2 J (Wang et al. (2014) ). DBA\/2 J differs from the reference strain (C57BL\/6 J) by 5,126,997 SNVs (roughly 1 SNV\/530 bp) and 1,019,400 INDELs, comparable to other commonly used lab mouse strains (see Discussion). ICM bisulphite-sequencing (GSM1386023) was complemented with RNA-seq (GSM1845307\u20138) from ICM cells isolated from C57BL\/6 J x PWK\/PhJ F1 mice as well as ChIP-sequencing data for H3K4me3 (GSM1845274\u20135) and H3K27me3 (GSM2041078\u20139), permissive and repressive histone post-translation modifications respectively. RNA-seq data from C57BL\/6 J x DBA\/2 J ICM (GSM1625868) was used to test allele-specific alignment performance of contemporary RNA-seq aligner software. Bisulphite sequencing datasets from C57BL\/6 J MII oocytes (GSM1386019) and DBA\/2 J spermatozoa (GSM1386020) were analyzed to directly measure false-positive allele-specific alignment rates. Processed fully grown oocyte (DRX001583) and sperm (DRX001141\u20139) bisulphite-seq were used for visualization. Processed human sperm and oocyte WGBS was obtained from JGAS00000000006. Adult human brain datasets were obtained as part of the Canadian Epigenetics, Environment and Health Research Consortium (CEEHRC) Network.\nAs published previously, MEA constructs a diploid pseudogenome using a reference sequence (.fasta) and known genetic variants (.vcf) including SNVs and INDELs . For samples requiring genotype phasing, MEA utilizes SHAPEIT2  and a publicly available reference panel of haplotypes provided by the 1000 Genomes Project  to output phased haplotypes. These steps generate an in silico diploid genome containing two copies of each chromosome, one for each parental genome. Aligning NGS reads to a diploid genome enables the extraction of uniquely aligned allele-specific reads, which are separated into parent-of-origin read alignment files. An automatically-generated index file (.refmap) enables reversal of coordinate alterations in non-reference allelic read alignments caused by differential parental INDEL lengths. This allows projection of allelic genomic tracks back onto the original reference genome for consistent data visualization in genome browsers (which are built around reference genomes) and downstream analyses over coordinate-based regions of interest.\nIn order to detect allele-specific reads from RNA-, ChIP-seq and WGBS data, we designed MEA to align reads using an in silico pseudogenome and extract uniquely mapped reads. This approach allows allele-specific alignment of reads containing sequencing errors, which is critical for datasets with long (100+ bp) reads commonly sequenced on Illumina sequencers, which have approximately 0.26\u20130.80% sequence error rates . This pipeline modification assures adoption and operation of our tool well into the future as sequencing technologies continue to extend read lengths without necessarily improving error rates.\nDNAme levels can be accurately measured genome wide using sodium bisulphite conversion of unmethylated cytosines to thymines followed by whole genome sequencing (bisulphite-seq). To measure allele-specific DNAme levels, MEA detects allelic reads and calculates the proportion of cytosines and thymines at CpG dinucleotides. To do so, MEA aligns bisulphite-seq reads to the in silico diploid genome using the popular aligner and methylation caller Bismark . Unlike ChIP- or RNA-seq aligners, Bismark considers cytosine to thymine mutations (introduced during sodium bisulphite conversion) in order to accurately align reads to a genomic sequence. Allele-specific DNAme levels therefore reflect both genetic and epigenetic effects: users can retroactively delineate both effects using their original list of genetic variants.\nUCSC Track Hubs are a hierarchical file organization system that allow combining multiple genomic tracks into one for convenient data visualization and interpretation [45, 32]. MEA automatically normalizes allele-specific tracks by sequencing depth and generate corresponding track hub database files. Using UCSC binaries (hubCheck), we ensure the integrity of MEA-generated track hubs for standardized visualization experiences. Additionally, we provide scripts for the automatic computation of allelic RNA- and ChIP-seq coverage over user-defined regions of interest (for example: transcription start sites, genes, enhancers, etc.), outputting a tab-delimited table. While RPKM- and coverage-calculating software already exist, confounding variables are inherent to allelic analyses, requiring custom scripting. For example, calculating allelic RPKM values using conventional tools is complicated by the variability in SNV and INDEL density between regions of interest. To account for such effects, MEA's default output includes allelic read coverage for both alleles (to calculate allelic imbalance) and total RPKM (to filter for enrichment). Users can easily interpret allelic imbalance calculations with the combination of these two metrics (allelic read coverage and total RPKM) over their regions of interest. In this study, VisRseq  was used to plot allelic read coverage for RNA-seq data from human brain.\nPackaging MEA into a Docker container allows the one-step installation of all ~\u200915 dependencies, significantly reducing the work required by the end-users. Simply, the Docker container is a text file containing instructions for installing a virtual system and setting environment variables, followed by standardized installation of each bioinformatic dependency. Once installed through the Docker container, MEA is immediately operational. The Docker file is uploaded to a third-party website and available for download (see Availability and requirements).\nUsers are encouraged to install MEA through Docker. Alternatively, manual installation requires the following software (with specific versions used during development of MEA): Java v-1.6, Python v-2.4, Bismark v-0.15.0, Bowtie2 v-2.2.3, BWA v-0.7.10, STAR v-2.5.1b, Tophat2 v-1.1, SAMtools v-0.1.16, Bedtools v-2.22.1, VCFtools v-0.1.10, SHAPEIT2, bgzip v-1.1, bedGraphToBigWig v-1.1, wigToBigWig v-4 & hubCheck.\nProgramming language: Java, Python, Awk, Bash.\nWe thank Julie Brind'Amour, Inan\u00e7 Birol, and Kristoffer Jensen for critical review of the manuscript and Louis Lefebvre, Hendrik Marks and Alexander Dobin for helpful discussions.\nThe Lorincz lab is supported by grants from the Canadian Institutes of Health Research (MOP-133417) and the Natural Sciences and Engineering Research Council of Canada (RGPIN-2015-05228). JRA is a recipient of an NSERC Postgraduate Scholarships-Doctoral Program award (PGSD3\u2013476000-2015), a Killam Doctoral Scholarship and a UBC Four Year Doctoral Fellowship.\nThe datasets analyzed in this study are available in the Gene Expression Omnibus (GEO) and DNA Data Bank of Japan (DDBJ) repositories under codes: GSM1386023, GSM1845274, GSM1845275, GSM1845307, GSM2041078, GSM2041079, GSM1625868, GSM1386019, GSM1386020, DRX001583, DRX001141\u20139 & JGAS00000000006 from the publications: [11, 13, 25, 26, 47\u201349]. No cell lines or datasets derived from cell lines were utilised in this study. A detailed description of all datasets can be found in Additional file 4: Table S3. The results published here are in part based upon data generated by The Canadian Epigenetics, Epigenomics, Environment and Health Research Consortium (CEEHRC) initiative funded by the Canadian Institutes of Health Research (CIHR), Genome BC, and Genome Quebec. Information about CEEHRC and the participating investigators and institutions can be found at http:\/\/www.cihr-irsc.gc.ca\/e\/43734.html. The datasets supporting the conclusions of this article are included within the article and its additional files.\nJRA participated in study design, developed code, performed validation analysis, and wrote the manuscript. TK developed code and performed validation analysis. HY participated in study design and developed code. RT and ABB ran bug tests. RT and HY wrote the user manual. MMK participated in study design and performed validation analysis. MCL participated in study design and wrote the manuscript. All authors read and approved the final manuscript.","meta":{"redpajama_set_name":"RedPajamaC4"}}